[    {        "title": "1910.14489v1.Gyrokinetic_investigation_of_the_damping_channels_of_Alfvén_modes_in_ASDEX_Upgrade.pdf",        "content": "Gyrokinetic investigation of the damping channels of Alfv\u0013 en modes in\nASDEX Upgrade\nF. Vannini1, A. Biancalani1, A. Bottino1,\nT. Hayward-Schneider1, Ph. Lauber1, A. Mishchenko2, I. Novikau1, E. Poli1and the ASDEX\nUpgrade team1.\n1Max-Planck-Institut f ur Plasmaphysik, 85748 Garching, Germany\n2Max-Planck-Institut f ur Plasmaphysik, 17491 Greifswald, Germany\nfrancesco.vannini@ ipp.mpg.de\nAbstract\nThe linear destabilization and nonlinear saturation of energetic-particle driven Alfv\u0013 enic\ninstabilities in tokamaks strongly depend on the damping channels. In this work, the collision-\nless damping mechanisms of Alfv\u0013 enic modes are investigated within a gyrokinetic framework,\nby means of global simulations with the particle-in-cell code ORB5, and compared with the\neigenvalue code LIGKA and reduced models. In particular, the continuum damping and the\nLandau damping (of ions and electrons) are considered. The electron Landau damping is\nfound to be dominant on the ion Landau damping for experimentally relevant cases. As an\napplication, the linear and nonlinear dynamics of toroidicity induced Alfv\u0013 en eigenmodes and\nenergetic-particle driven modes in ASDEX Upgrade is investigated theoretically and compared\nwith experimental measurements.\n1arXiv:1910.14489v1  [physics.plasm-ph]  31 Oct 20191 Introduction\nIn burning plasmas relevant for magnetic fusion energy (MFE) research, an important role is\nplayed by energetic particles (EPs). With the term EPs, we refer to fusion reaction products\n(like alpha particles) or super-thermal ions or electrons resulting from plasma heating. Such\nparticles possess higher velocities compared to those typical of the background plasma. In typ-\nical tokamaks the frequency associated with the gyrocenter motion of the EPs falls inside the\nmagnetohydrodynamic (MHD) domain ( O(10\u00002\nci)), being \n cithe ion cyclotron frequency. Be-\ncause of that, through resonant wave-particle interactions, one of the three solution satisfying the\nMHD dispersion relation can be excited. Among them, the most detrimental are the shear Alfv\u0013 en\nwaves (SAWs), having a group velocity parallel to the equilibrium magnetic \feld and satisfying\nthe dispersion relation: !=kjjvA, where the Alfv\u0013 en speed has been de\fned vA=p\nB=(4\u0019\u001am0)\n(being\u001am;0the background plasma mass density and B the background magnetic \feld strength).\nThe excitation of these modes creates a transport channel for the EPs, which can lead to loss of\nEPs before their thermalization, causing a less e\u000bective heating and also possibly damaging the\nvessel of the machine. they are also believed to be responsible of large abrupt events (ALE) ob-\nserved in the Japanese tokamak (JT-60U) (see Ref.[1]). We can simply outline the whole zoology\nof SAWs basically in discrete Alfv\u0013 en eigenmodes (AEs) and energetic particle continuum modes\n(EPMs), (non-normal modes of the SAW continuum spectra, merging as discrete \ructuations at\nthe frequency that maximizes the wave-EP power exchange, above the threshold condition of the\ncontinuum damping [2]).\nIn this paper the attention will be mainly focused on toroidal Alfv\u0013 en eigenmodes (TAE, Alfv\u0013 en\neignemodes lying in the frequency gap caused by the tokamak toroidicity) and EPM. The main\ngoal of this paper will be to understand what are the damping mechanisms of the modes of\ninterest comparing (when possible) the simulations results with the prediction of MHD theory.\nThe proper domain to take into account all the nonlinear e\u000bects, like wave-wave and wave-\nparticle interaction, as well as \fnite-Larmor-radius and \fnite-orbit-width e\u000bects, is represented\nby the gyrokinetic theory. Because of that, the simulations have been principally carried with the\nglobal, nonlinear, electromagnetic, gyrokinetic, PIC code ORB5 [3, 4] whose model, if properly\nset, contains the MHD equations as a subset. Some comparisons with the linear gyrokinetic code\nLIGKA [5] have also been performed.\nThe paper is structured as follows. In Section 2 a description of the model implemented in the\ncode ORB5 is given. Section 3 and Section 4 will be dedicated to the description of two di\u000berent\ndamping mechanisms a\u000becting the SAWs: the continuum damping and the Landau damping.\nThey will be brie\ry explained analytically, starting from the MHD equations. The theory in\nuse will be compared with the results from the simulations carried with ORB5. In Section3\nthe numerical simulations will be performed in the cylinder limit using simpli\fed pro\fles. In\nthe simulations in Section 4, a small but \fnite inverse aspect ratio will be considered, using the\nequilibrium pro\fles of the International Tokamak Physics Activity (ITPA, see Ref.[6]). In Section\n5 the studies on the linear and nonlinear growth rate and frequency spectra conducted considering\nexperimental pro\fles from the NLED-AUG case [7] will be presented. Finally, summary and future\noutlook will be shown in Section 6.\n22 Model\nSince the Alfv\u0013 en waves have a frequency much smaller then the typical ion cyclotron frequency\n(\nci) and their amplitude in the core is small compared to the background quantities, a good\ndescription of their propagation and interaction with the bulk plasma can be given through the\ngyrokinetic theory. This allows to retain a kinetic description of the events under consideration,\nreducing the 6D problem to a 5D one, by averaging the fast gyromotion. In this way the numerical\ncosts are sensibly reduced.\nORB5 is a global, nonliner, gyrokinetic, electromagnetic, PIC code, which can take into\naccount collisions and sources [3, 8]. The gyrokinetic model of ORB5 [9] contains the reduced\nMHD equations as a subset [10]. In this section we give a brief description of the gyrokinetic model\nimplemented in ORB5 and brie\ry show how the implemented equations are solved. We refer for\nmore exhaustive explanations to Ref.[8], which also give a more complete description of the\nrecent updates in ORB5. Concerning the magnetic equilibrium in use, ORB5 can either upload\nan ideal-MHD equilibrium (solution of the Grad-Shafranov equation) from the CHEASE code\n[11], or consider ad-hoc equilibria constituted by circular, concentric magnetic surfaces. It deals\nwith a straight-\feld line set of coordinates. The magnetic surfaces are labeled by s=p\n = edge,\nwhich plays the role of radial coordinate. Here  is the poloidal magnetic \rux function. The\nangular dependence is given by the toroidal coordinate 'and by the poloidal magnetic angle:\n\u001f=1\nq(s)Z\u0012\n0d\u00120B\u0001r'\nB\u0001r\u00120 (1)\nbeing\u00120the geometrical poloidal angle and q(s) the safety factor, de\fned as:\nq(s) =1\n2\u0019Z2\u0019\n0d\u00120B\u0001r'\nB\u0001r\u00120: (2)\nAll the quantities in the code are normalized through four reference parameters: the ion mass\n(mi), the ion charge ( qi=eZi, beingethe electric charge and Zithe atomic number), the\nvalue of the magnetic \feld strength on axis ( B0=jB(s= 0)j) and the value of the electron\ntemperature at a speci\fed reference position s0,Te(s0). All other normalized quantities are\nobtained through these: the time units are provided in the inverse of the ion-cyclotron fre-\nquency, \n ci=qiB0=(mic), the velocity units are normalized through the ions sound velocity\n(cs=p\nqsTe(s0)=mi, being the temperature measured in keV), the length units through the ion\nsound Larmor radius ( \u001as=cs=\nci) and the densities are normalized by means of their average\nin space. The Vlasov-Maxwell gyrokinetic equations are derived through variational principles\nfrom a discrete gyrokinetic Lagrangian. This choice allows to take into account all the simpli-\n\fcation needed in the model under consideration, directly in the Lagrangian and then derive\nthe gyrokinetic equation. An immediate consequence is that it is possible to consistently derive\nconserved quantities (like the energy, see Ref.[12]), that are also used in ORB5 to test the quality\nof the simulations performed. This choice also allows to derive naturally the weak gyrokinetic\nform of the \feld equations. In the Lagrangian an ordering is present, separating the e\u000bects given\nfrom the geometry of the non-uniform magnetic \feld, from those related to the \ructuations of\nthe electromagnetic perturbation. This means that (as can be derived, see Ref.[9, 13]) the small\nparameter related to the variation of the background magnetic \feld \u000fB=\u001athLB, (being\u001aththe\nthermal Larmor radius and LBthe typical variation length of the magnetic \feld) and the small\n3parameter related to the \ructuating electromagnetic \feld ( \u000f\u000e) are related through:\n\u000fB=O(\u000f2\n\u000e): (3)\nIn this way the action functional, written in \\ pz-formulation\" appears to be the following:\nA=Zt1\nt0Ldt=X\nsZ\ndtd\n\u0012qs\ncA\u0003\u0001_X+msc\nqs\u0016_\u0012\u0000H0\u0013\nfs+\n\u0000\u000f\u000eX\ns6=eZ\ndtd\nH1fs\u0000\u000f\u000eZ\ndtd\nHdk\n1fe+\n\u0000\u000f2\n\u000eX\ns6=eZ\ndtd\nH2feq;s\u0000\u000b\u000f2\n\u000eZ\ndtd\nHdk\n2feq;e\u0000\u000b\u000f2\n\u000eZ\ndtdV\f\fr?A1;jj\f\f2\n8\u0019(4)\nwhere\u000b= 0 gives the electrostatic model, while \u000b= 1 the electromagnetic one. In Eq.4 d\n =\ndV dW , beingdW =B\u0003\njjd\u0016dpz. A sum over the species \\ s\" also appears. The symplectic\nmagnetic \feld is de\fned through the symplectic magnetic potential A\u0003=A+ (c=qs)pz^bbeing ^b\nthe unitary vector parallel to the background magnetic \feld. The canonical gyrocenter momentum\nispz=msvk+\u000b\u000f\u000e(qs=c)A1k. In the action functional, some approximations have been done.\nThe quasi-neutrality allows to consider in Eq.4 only the contribution given from the magnetic\npotential, neglecting the one given from the perturbed electric \feld. Also the incompressibility\nof the parallel perturbed magnetic \feld is assumed B1;jj=o(B1;?) and only the perpendicular\ncomponent of the perturbed magnetic potential is retained: B1;jj=r\u0002(A1;jjb)\u0018rA1;jj\u0002b.\nIn Eq.4 it must be noted that while H0,H1multiply the total distribution functions fs;fe,H2\nis related only to the equilibrium distribution function. Thanks to this choice nonlinear second\norder terms do not appear in the gyrocenter dynamics and the \feld equations are linear. The\ngyrocenter hamiltonians appearing are:\nH0=p2\nz\n2ms+\u0016B H 1=qs\u001c\n\u001e1\u0000\u000bA 1;kpz\nmsc\u001d\n(5)\nH2=\u0000msc2\n2B2jr?\u001e1j2+\u000bq2\ns\n2msc2\nA1;k\u000b2\nwhere the gyroaveraging operator has been introduced hfi=1\n2\u0019R2\u0019\n0d\u0012f. The gyroaveraging is\nremoved for the electrons that are treated as drift-kinetic:\nHdk\n1=\u0000e\u0012\n\u001e1\u0000\u000bA 1;kpz\nmsc\u0013\nHdk\n2=\u000be2\n2mec2A2\n1;k (6)\nFor the distribution of the species sthe linear gyrokinetic Vlasov equation is:\ndfs\ndt=@fs\n@t+_X\u0001rfs+ _pz@fs\n@pz= 0 (7)\nwhere the gyrokinetic characteristics can be derived from Eq.4 and are:\n8\n>>>>><\n>>>>>:_X=c^b\nqsB?\nk\u0002rH+@H\n@pzB?\nB?\nk\n_pz=\u0000B?\nB?\nk\u0001rH(8)\n4The \feld equations, quasineutrality and Amp\u0012 ere, are both derived from Eq.4 via functional\nderivatives on the perturbed \feld. ORB5 splits the total distribution function in a background\ndistribution function f0and in a time dependent one \u000efand discretize this latter through numer-\nical particles (markers) used to sample the phase space. Through an operator splitting approach\nthe code solves \frst the conlisionless dynamics (using a 4th-order Runge-Kutta method) and\nthen treats the collisions with a Lanngevin approach. The quasineutrality and Amp\u0012 ere equations\nare solved using the Galerkin methods and the perturbed \felds are discretized through cubic\nB-splines \fnite elements de\fned on a grid ( Ns;N\u001f;N\u001e). Finally it is important to mention that\nfrom the numerical side, recently the mixed-representation (\\pullback\" scheme [14]) has solved\nthe so-called \\cancellation problem\" for electromagnetic simulations.\n53 Continuum damping\nIn the present section, the continuum damping will be studied. The tokamak con\fguration will be\nselected in order to have the continuum damping as main damping mechanism a\u000becting the Alfv\u0013 en\nwaves. In order to do so, it is \frst important to understand what are the equations governing\nthe Alfv\u0013 en waves. These will be obtained under the validity of the ideal magnetohydrodynamic\n(MHD) theory by treating MHD equations with a perturbative approach. The Alfv\u0013 en wave's\ndynamics can be expressed starting from the quasi-neutrality condition r\u0001\u000eJ= 0, (being\u000eJ\nthe perturbed current) that rewritten in terms of its components parallel and perpendicular to\nthe background magnetic \feld ( ^b=B=B) reads:\nr\u0001\u000eJ?+B\u0001r\u000eJk\nB= 0: (9)\nFollowing Ref.[15], in order to obtain a simpli\fed but relevant set of equations, modes with\nk?\u001dkkare considered, so that the time scale between incompressible shear Alfv\u0013 en waves and\ncompressional waves can be separated. To further simplify the problem, we consider a pressureless\nplasma (P= 0) obtaining the following vorticity equation:\nB\u0001r\u00141\nBr2\n?\u00121\nBB\u0001r\u000e\u001e\u0013\u0015\n\u0000r\u0001\u00121\nv2\nA@2\n@t2r?\u000e\u001e\u0013\n= 0: (10)\nA di\u000berential equation for the perturbed scalar potential \u000e\u001eis thus obtained. It is linked to the\nperturbed magnetic potential ( \u000eA\u0019\u000eA^b) through the condition \u000eEk= 0, derived from the\nideal Ohm's law. In this section a non-uniform plasma equilibrium with cylindrical limit, will be\nconsidered. awill denote the typical length scale perpendicular to the equilibrium magnetic \feld\nwhileR0will represent the typical length scale parallel to it. The equilibrium magnetic \feld,\nin a coordinate system ( r;\u0012;z ) will be assumed to be B= (0;B0;\u0012(r);B0;z(r)). The geometrical\nradiusrcan be used in the cylindrical limit instead of the \rux coordinate s. By assuming a shear\nAlfv\u0013 en oscillation of the scalar potential \u000e\u001e(r;\u0012;\u001e;t ) of the form:\n\u000e\u001e=X\nm;n\u000e\u001em;n(r)ei(m\u0012\u0000n z\nR0\u0000!t); (11)\nwheremis the poloidal mode number, we can now write Eq.10 in cylindrical coordinates:\n1\nr@\n@rr\"\u0012m\nq(r)\u0000n\u00132\n+R2\n0\nv2\nA@2\n@t2#\n@\n@r\u0012\u000e\u001e\nr\u0013\n=m2\nr2\"\u0012m\nq(r)\u0000n\u00132\n+R2\n0\nv2\nA@2\n@t2#\n\u000e\u001e ; (12)\nwhere the local safety-factor pro\fle has been de\fned:\nq(r) =rB 0;z\nR0B0;\u0012: (13)\nThe shear Alfv\u0013 en wave dispersion relation is then found to be:\n!2\nA=v2\nAk2\nm;n=v2\nA\nR2\n0\u0012m\nq(r)\u0000n\u00132\n(14)\nEquation 14 proves that the shear Alfv\u0013 en waves are local plasma oscillations, having a frequency\nspectrum that varies continuously throughout the plasma radial direction. Due to the hypothesis\n6k?\u001dkkthe local nature of the continuum plasma oscillation can be exploited by reducing Eq.12\nto:\n1\nr@\n@rr\"\u0012m\nq(r)\u0000n\u00132\n\u0000!2R2\n0\nv2\nA#\n@\u000e\u001e\n@r= 0; (15)\nwhich integrated in the radial domain, becomes a di\u000berential equation for the radial electric \feld\nEr:\u0012\n!2\nA+@2\n@t2\u0013\nEr= 0)Er=E0e\u0000i!A(r)t: (16)\nAssuming now a dispersion relation of the form !A(r) =!A0+!0\nA(r\u0000r0) and by Fourier\ntransforming the radial electric \feld in the radial coordinate the following relation is obtained:\n(FEr)(kr) =p\n2\u0019E 0e\u0000i(!A0\u0000!0r0)t\u000e(kr+!0\nAt)kr/\u0000!0\nAt (17)\nThe obtained linear dependence in time of the radial wave number is a proof of the phase mixing\ntogether with the fact that, being Er(r;t) =\u0000ikr(t)\u001e(r;t), the scalar potential exhibits the\ncharacteristic decay called continuum damping:\n\u000e\u001e/\f\f!0\nAt\f\f\u00001(18)\nas it was proved in Ref.[15] (see also Ref.[16, 17, 18] for the application to Geodesic Acoustic\nmodes, GAMs). To see evidence of this mechanism, the simulations presented in this section\nhave been run with simpli\fed geometries and pro\fles, without the presence of EPs. In order to\nbe in the cylinder limit the inverse aspect ratio has been chosen to be \u000f= 0:01 and \rat density\n(ne=ni= 2:22\u00011020m\u00003) and temperature ( Te=Ti= 0:01keV) pro\fles have been taken\ninto account. This leaves all the radial dependence of the dispersion relation in the safety factor\npro\fle. Moreover, in this temperature regime the Landau damping can be neglected [19]. In\nthe simulations under examination only one axysimmetric perturbation n= 0; m= 1 peaked\nat the radial position r= 0:6, has been considered, together with a linear safety-factor pro\fle\nq=q0+q1\u0001rso that:!A=vA\nR01\nq0+q1r. The other important parameters in the simulations (minor\nand major radius, value on axis of the equilibrium magnetic \feld, ion cyclotron frequency, Alfv\u0013 en\nfrequency on-axis and the ratio of the last two), are reported in Tab.1.\nTable 1: Main simulation's parameters of the reference case for the study of the continuum\ndamping.\na0[m]R0[m]B0[T]\nci[rad=s ]!A0[rad=s ]\nci=!A0\n0:1 10 3 2:87\u00011084:38\u0001105655\nIn Fig.1 on the left, the measured values for the wave numbers krare shown for a simulation\nhavingq0= 1:75 andq1= 0:5. They have been measured interpolating the mode structure with\na sinusoidal function at times where a maximum has been reached at the radial position r= 0:6.\nBy linearly inetrpolating the measured wave numbers it is possible to calculate the coe\u000ecient\nkr;1, which is found to be in reasonable agreement with the theoretical expectations Eq.17. In\nFig.1 the dynamic of the scalar potential at some radial positions is shown, together with the\npredicted decay, Eq.18.\n7Figure 1: Left: Radial wave number dependence on time. The results of ORB5 are given by dots.\nThe theoretical prediction for this simulation ( q0= 1:75 andq1= 0:5) is thatkr;1= 0:123!A0,\nwhile the measured value is kr;1= 0:107!A0. Right: Perturbation amplitude dependence on time.\nAnalytical estimation are given by the dashed lines (curves decaying in time as \u001e\u0018j!0\nAtj\u00001).\nThe scalar potential measured at di\u000berent radial positions is given by continuous lines. No EPs\nare present here.\nIn Fig.2 \fnally the obtained values of the coe\u000ecients kr;1have been plotted against di\u000berent\nvalues of the slope of the safety factor pro\fles ( q1) in use in the di\u000berent simulations and com-\npared with Eq.17. Given the reasonable agreement found between the results of the numerical\nsimulations and the theory, we can say to have veri\fed the relevance of the continuum damping\nas main damping mechanism for this speci\fc case and to have observed the presence of phase\nmixing.\nFigure 2: Dependence of kr;1on the slope of the safety factor pro\fle. Analytical estimation are\ngiven by the dashed lines and the results of ORB5 are given by dots. No EP are present here.\n84 Landau damping\nIn the present section bulk species temperatures high enough to make the continuum damping\nnegligible with respect to the Landau damping will be considered, so that the latter becomes the\nmain damping mechanisms.\nThe attention will be focused also on a particular Alfv\u0013 en eigenmode, the toroidal Alfv\u0013 en\neignemode (TAE). Its characteristic frequency lies in the gap created in the continuum spectra\nby two close poloidal modes ( m;m +1) which are coupled because of the \fnite tokamak toroidicity,\n[20]. A TAE is located at a radial position r0satisfying:q(r0) =2m+1\n2n. The theoretical derivation\nexposed in Ref.[20, 21] will be now followed. Here, a kinetic transverse part of the wave-induced\ncurrent\u000eJk\n?is added to the ideal MHD current, so that Eq.9 becomes:\nr\u0001(\u000eJMHD+\u000eJk\n?) = 0: (19)\nEquation 19 is then multiplied by \u000e\u001eand integrated in the overall plasma volume obtaining:\nZ\ndx\u000eJMHD\u0001r\u000e\u001e+Z\ndx\u000eJk\n?\u0001r?\u000e\u001e= 0 (20)\nwhere it was assumed as boundary conditionsR\ndx\u0001\u000eJ\u000e\u001e= 0. Calling !0the frequency of the\nwave solution of the ideal MHD vorticity equation, we can consider !=!0+\u000e!the solution\nof the new vorticity equation Eq.19, being \u000e!\u001c!0. Following a perturbative approach, an\nexpression for \r= Imf!gis obtained from Eq.20:\n\r=2\u0019\nc2P\nm;nR\nVd3x\u000eJk\n?m;n\u0001r\u000e\u001e\u0003\nm;n\nP\nm;nR\nVd3x1\nv2\nAh\f\f\u000e\u001e0m;n\f\f2+\u0000m\nr\u00012j\u000e\u001em;nj2i(21)\nwhere all the appearing perturbed quantities have been decomposed in Fourier components in\nthe poloidal plane. In order to obtain a simpli\fed equation for \r, some further calculations have\nbeen done and will be now described. Eq.21 is then written in cylindrical coordinates, after\nwriting the perturbed current in terms of the perturbed distribution function and this in terms of\nthe unperturbed distribution function F0. Assuming a Maxwellian distribution function F0and\nfocusing our attention on TAE (that is assuming to have a perturbation \u000e\u001estrongly peaked at\nthe radial position where we expect to have a TAE), we obtain:\n\r=X\nj\rj\rj=\u0000\fjq2\n0vA\n2q0R0\u0014\nGmj+nq0rL\u0012;j1\nn0;j@n0;j\n@r(Hmj+\u0011Jmj)\u0015\n: (22)\nBeing:\n\n\u0012;j=eBp\nmjcrL\u0012=vth;j\n\n\u0012;j\fj= 8\u0019n0;jTj\nB2\n0\u0011j=@log(Tj)\n@log(n0;j)\u0015j=vA=vth;j: (23)\n9And:\n8\n>>>>>>>>><\n>>>>>>>>>:gm;j(\u0015j) =\u0019\n2\u0015j(1 + 2\u00152\nj+ 2\u00154\nj)e\u0000\u00152\njGmj=gm;j(\u0015j) +gm;j(\u0015j=3)\nhm;j(\u0015j) =\u0019\n2(1 + 2\u00152\nj+ 2\u00154\nj)e\u0000\u00152\njHmj=hm;j(\u0015j) +1\n3hm;j(\u0015j=3)\njm;j(\u0015j) =\u0019\n2\u00123\n2+ 2\u00152\nj+\u00154\nj+ 2\u00156\nj\u0013\ne\u0000\u00152\njJmj=jm;j(\u0015j) +1\n3jm;j(\u0015j=3):(24)\nIn Eq.22,\rhas been decomposed in the species contributions (the sum over j). It is formally\nidentical to the one derived in Ref.[22]. The di\u000berence lies in the fact that in Ref.[22] the authors\nhave obtained the estimation for \rstarting from energy principles, while here everything has\nbeen done by adding a correction to the MHD quasi-neutrality equation and thus to the Alfv\u0013 en\ndynamics [23]. Since we are interested in the study of the Landau damping, we will not consider\nthe EPs contribution, which actually drives the mode unstable. Eq.22 depends on the ratio\nbetween the Alfv\u0013 en speed and the thermal velocity of the considered species. This means that\n\u0015j\u0018m1=2\njand, since the bulk ion mass is bigger than the electron mass (for Hydrogens mH\u0018\n2000me) one can understand that the ion contribution is negligible with respect to the electron\ncontributions (because of the presence of the exponential terms in the polinomia). Because of\nthat we will focus our attention on the electron Landau damping in this analytical derivation.\nIn this section an equilibrium with small, but \fnite value of inverse aspect ratio will be\nconsidered, \u000f= 0:1. The temperature pro\fles are \rat. When considered, the fast particles have a\ndensity pro\fle peaked on axis (see Fig.3 on the left). The magnetic equilibrium and pro\fles are\nthose of the ITPA-TAE international benchmark case [6] and the safety factor pro\fle is shown\nin Fig.3 on the right. The main results that will be displayed in this section, have been obtained\nconsidering heavier electrons: me=mH=200. This has been checked to be at convergence. In\nTab.2 other important details of the simulations are presented.\nTable 2: Main simulation's parameters of the ITPA-TAE case.\na0[m]R0[m]B0[T]\nci[rad=s ]!A0[rad=s ]\nci=!A0\n1 10 3 2:87\u00011081:46\u0001106196\nFigure 3: Density pro\fles and safety factor of the ITPA-TAE case ( q(r)'1:71 + 0:15r2).\n10Figure 4: Left:Initial mode structure.Right: Frequency spectra without energetic particles.\nThe chosen initial potential perturbation is peaked around r= 0:5 and is constituted by one\nsingle toroidal mode number n= 6 while the poloidal mode numbers 9 \u0014m\u001412 are considered\n(see Fig.4 on the left). A TAE is located at r= 0:5 in the gap of the continuum spectra created\nby the coupling of the poloidal modes m= 10 andm= 11, as it is shown in Fig.4 on the right.\nIn Fig.5 the dependence of the damping rate against the value of the (\rat) electron temperature\nis shown. The damping rate is found to increase with the increasing electron temperature. This\nis an evidence that the dominant damping is the electron Landau damping. The errorbar of\nthe measured points correspond to 20% of their value. This because, as it is shown in Fig.6\non the left, the damping rate value has a dependence on the chosen width of the perturbation.\nFor completeness in Fig.5 the approximated analytical electron Landau damping formula is also\nshown (dashed line). A reasonable qualitative agreement is found between the predicted decay and\nthe simulation results. Finally in Fig.6 on the right, the dependence of the measured damping\nrate of ORB5 simulations against the electron mass has been shown. For decreasing electron\nmasses, the absolute value of the damping rate is shown to decrease, consistently with theory of\nthe electron Landau damping. In summary, it has been proved that the bulk electrons provide\nthe main damping mechanism of the observed Alfv\u0013 en modes in this particular regime. Several\napproximations have been done in the analytical theory, inter alia only passing particles are\nconsidered thus neglecting the contributions of barely trapped electrons, which are thought to be\nimportant, and which are included in our numerical simulations.\nFigure 5: Landau damping dependence versus the electron temperature.\n11Figure 6: Left:Damping rate dependence on the width of the initial Gaussian beam. Right:\nDamping rate dependence on the electron mass for ORB5 simulations.\n125 NLED-AUG case\nIn the present section the results of numerical simulations involving a realistic scenario will be\npresented.\nThe shot number #31213 of ASDEX-Upgrade (AUG) has been selected within the Non-Linear\nEnergetic-particle Dynamics (NLED) Eurofusion enabling research project [7]. Here an early o\u000b-\naxis NBI (with TEP\u001893keV) occurs with an injection angle (angle between the horizontal\naxis and the beam-line) of 7 :13\u000e. The magnetic equilibrium measured at the time t= 0:84sis\nconsidered in the present simulations (see Fig.7 on the bottom left). This case is referred to as\n\\NLED-AUG case\". Further description of this case can be found in Ref.[7]. The NLED-AUG\ncase is found to be of great interest because of its rich linear and nonlinear dynamics arising from\nthe interaction of the modes with the EPs.\nTab.3 contains the details of the main parameters considered in the simulations. Tab.4 shows\nthe values of the bulk species pro\fles on axis in the absence of EPs. The bulk ions, as well as\nthe EPs (when considered), are constituted by deuterium. The EP temperature will be always\nconsidered to be radially \rat and equal to TEP= 113keV. For the EPs density pro\fles, an o\u000b-\naxis density pro\fle \ftting the experimental pro\fles is considered with Maxwellian distribution\nfunction. For comparison we also run simulations with an on-axis EPs density pro\fles. Note that\nwhen EPs are included the electron density pro\fles is changed in order to match quasi-neutrality\nne=Zi\u0001ni+ZEP\u0001nEP.\nIn Fig.7 the safety factor pro\fle is shown, together with the temperature pro\fles of the bulk\nspecies. The safety factor pro\fle has a reversed shear, with qmin(r= 0:5)'2:28. The density\npro\fles in use will be shown in the following subsection. For numerical reasons, the electron mass\nis chosen to be me=mD=500, beingmDthe deuterium mass.\nThis section is divided in two subsections. In the \frst, the results of numerical simulations\nonly involving the linear dynamics will be discussed. In particular, the dependence of \ragainst\nthe electron temperature will be shown, with and without EPs contribution. Also the results\nof a benchmark with LIGKA are presented. In the second subsection instead, the results of\nsimulations involving also the nonlinear dynamics will be presented. Finally, it is important to\nremind that, unless speci\fcally written, the bulk and energetic ions will be treated as drift-kinetic.\nThe initial perturbation considered will take into account just one toroidal mode number ( n= 1)\nand the poloidal mode number 0 \u0014m\u00147.\nTable 3: Main simulation's parameters of the NLED-AUG case.\na0[m]R0[m]B0[T]\nci[rad=s ]!A0[rad=s ]\nci=!A0\n0:482 1:666 2:202 1:0539\u00011084:98\u000110621:15\nTable 4: Pro\fle's parameters of the NLED-AUG case.\nTe(s= 0) [keV]Ti(s= 0) [keV]ne(s= 0) [m\u00003]ni(s= 0) [m\u00003]nf(s= 0) [m\u00003]\n0:709 2:48 1:672\u000110191:6018\u000110196:98\u00011017\n131.5 2.0\nR [m]0.8\n0.6\n0.4\n0.2\n0.00.20.40.60.8Z [m]\n1.82.02.22.42.62.83.03.2B [T]Figure 7: Top Left: Safety factor pro\fle. Top Right: Bulk species temperature pro\fles. Bottom\nleft: Poloidal view of the magnetic equilibrium in use.\n145.1 Linear simulations\nIn the present subsection the results of numerical simulations involving only the wave linear\ndynamics will be presented. Fig.8 and Fig.10 show the frequency spectra, mode structure and\npoloidal view of the scalar potential \u001e, obtained considering respectively o\u000b-axis and on-axis\ndensity pro\fle for the EPs and a concentration equal to 3%. The frequency spectra have been\nanalyzed in the same temporal domain, when a clearly growing mode is observed. In both Fig.8\nand Fig.10 the continuum spectra obtained with the linear gyrokinetic code LIGKA [5] is shown\n(red crosses), together with the analytical curve for the continuum spectra calculated in cylindrical\ncoordinates and including the toroidicity e\u000bects, [20] (green dotted line).\nWhen the EPs possess an o\u000b-axis pro\fle, a mode sitting at the radial position r'0:22 is\nobserved. The dominant poloidal component of the scalar potential appears to be that having\nm= 2. Due to the measured frequency lying on the branch of the continuum, this mode is\nidenti\fed as an EPM. The frequency is measured as:\nf= 129kHz (25)\nWhen \fnite Larmor radius are take into account, a slightly change in the frequency (from 129 to\n131kHz) is observed. The frequency measured in numerical simulations can be compared with\nthe experimental measurements. In Fig.9, the spectrogram obtained with Mirnov coils is shown.\nA big variety of EPs driven modes can be found. At t= 0:84s, the modes with frequencies around\n50kHz have been identi\fed as EGAMs (see Ref.[24, 25, 26]). We focus here on the Alfv\u0013 en modes\nwith frequency lying in the domain between 100 and 150 kHz. The numerical result shows that\nthese modes are indeed EPMs (see the white cross in Fig.9). Despite the approximation of the\nEPs distribution function we notice that the results of the numerical simulations appear to be in\ngood agreement with the modes observed in the spectrogram.\n15Figure 8: Numerical results for o\u000b-axis EPs pro\fle. EPs concentration of 3%, TEP= 113KeV .\nFigure 9: Experimental spectrogram obtained with Mirnov Coil compared with theoretical pre-\ndiction at one selected time. The theoretical prediction is obtained treating the fast ions as\ngyrokinetic.\n16Figure 10: Numerical results for on-axis EPs pro\fle with a concentration of 3%, TEP= 113KeV\nAn on-axis density pro\fle is also considered for the EPs. The radial dependence of the EPs\ndensity pro\fles is expressed by the formula: nEP'(1\u0000r\u000b)\f. The coe\u000ecients \u000b;\f have been\nchosen in order to have the second derivative of nEPequal to zero at the position where an Alfv\u0013 en\nmode is expected. The numerical analysis shows that a mode lying in the gap of the continuum\nspectra, created by the poloidal modes m= 2 andm= 3 is observed. It appears to be peaked at\nthe radial position r'0:738. Due to the radial localization and frequency this is identi\fed as a\nTAE.\nIn Fig.11 the dependence of \ragainst the value of the bulk species temperature is shown.\nIn the plot on the top left, the dependence of the growth rate against the electron temperature,\nkeeping the bulk ions temperature constant ( Ti(r= 0) = 3:5keV), is shown. In the plot on\nthe top right, the dependence of the growth rate against the bulk ion temperature, keeping the\nelectron temperature constant ( Te(r= 0) = 0:707keV), is shown. The EPs temperature is \rat\nTEP= 113KeV and the EPs have a concentration of 3%. In the plot on the bottom instead,\nthe dependence of the damping rate (simulations without fast particles) is shown, against the\nvalue of the electron temperature. This study of the dependence of the growth or damping rate\nagainst the electron temperature shows that the electrons are the main responsible of the damping\nof the Alfv\u0013 en modes even in this realistic scenario, which is identi\fed here as electron Landau\n17damping. Since a realistic scenario is considered here, the approximate theoretical predictions for\nthe Landau damping described in the previous sections is outside its validity regime and therefore\nis not shown.\nFigure 11: Top: respectively on the left and on the right, two scans in the electron and bulk ions\ntemperature for a growing mode are shown. EPs concentration 3%, TEP= 113KeV . Bottom:\nScan in the electron temperature for a damped mode.\nIn Fig.12 a theoretical estimation of the regions in the radial domain where the Landau damp-\ning is supposed to dominate over the continuum damping is presented for simulations without\nEPs. In order to perform this calculation, the characteristic radial structure has been measured\nin a simulation with an EPM, giving the value of kr;0. This allows us to calculate the analytical\nprediction for the half decay time of the continuum damping. The half decay time due to Landau\ndamping, on the other hands, can be measured in a simulation without EPs:\nt1=2;continuumdamping =j2\u0000kr;0j\f\f@!\n@s\f\ft1=2;Landaudamping = log 2=\r (26)\nThe radial regions where the Landau damping dominates over the continuum damping are those\nwhere the half decay time of the Landau damping is smaller than the half decay time of the\ncontinuum damping: t1=2;continuumdamping >t1=2;Landaudamping . Equivalently it can be said that\nthe Landau damping dominates over the continumm damping in those regions where:\n\r>\f\f\f\flog(2)\n2\u0000kr;0\f\f\f\f\u0001\f\f\f\f@!\n@s\f\f\f\f(27)\nIn Fig.12 on the left the continuum spectra calculated with the code LIGKA are shown. They\nare used to calculate the derivative of the frequency to obtain the estimation of the regions\n18where the Landau damping dominates over the continuum damping. To do so, the frequency has\nbeen divided in two branches, denoted as upper and lower branch. In Fig.12 on the right the\nregions where t1=2;continuumdamping >t1=2;Landaudamping are shown. Therefore the green regions\ncorrespond to the radial domain where the Landau damping is dominant over the continuum\ndamping if the frequency of the mode is sitting on the lower branch. The cyan regions instead\nrepresent the domains where the Landau damping is dominating on the continuum damping if\nthe frequency of the mode is sitting on the upper branch.\nFigure 12: Left: continuum spectra calculated with the code LIGKA. It has been\ndivided into two branches (denoted as upper and down). Right: regions where\nt1=2;continuumdamping >t1=2;Landaudamping .\nFinally in Fig.13, the results of a \frst benchmark between ORB5 and the code LIGKA are\nshown. Here a scan in the EPs temperature is depicted. The EPs have an on-axis pro\fles and\ntheir concentration is kept constant and equal to 3%. A reasonable agreement has been found\nbetween the two codes for the measured growth rate and frequency.\nFigure 13: Scan in TEP. TAE growth rate and frequency, calculated with LIGKA and ORB5 for\nan EPs concentration equal to 3 % (same density, temperature pro\fles in use).\n195.2 Nonlinear simulations\nIn this subsection results involving the nonlinear dynamics of the Alfv\u0013 en waves are presented,\nwhen both on-axis (Fig.14) and o\u000b-axis (Fig.15) density pro\fles for the EPs are considered. With\nan on-axis density pro\fles of the EPs, a mode sitting in the frequency gap is observed (TAE),\nFig.14. Its mode structure and frequency spectra are not observed to change passing from the\nlinear to the nonlinear phase, con\frming its nature of an eignemode of this system, which is only\nweakly perturbed by EPs.\nFigure 14: Mode structure and frequency spectra in the linear phase (left) and in the saturation\nphase (right). EPs have ah on-axis pro\fle.\nWhen an o\u000b-axis density pro\fle for the EPs is considered, a mode with dominant poloidal\nmode number m= 2 and peaked around r'0:22 is observed (see Fig.15). This is consistent with\nwhat was observed in the previous sections, when just the linear e\u000bects in the simulations were\ninvolved. Passing to the nonlinear phase a secondary mode with m= 2 andm= 3 is observed\nto grow around the radial position r'0:738. This second mode is identi\fed as the previously\ndescribed TAE. This happens, because in the \frst linear phase the EPs drive the EPM unstable,\nwhich appears in fact to be dominant. In the nonlinear phase, the coexistence of the EPM and\nTAE is observed, due to an earlier saturation of the EPM (see Fig.16) .\n20Figure 15: Mode structure and frequency spectra in the linear phase (left) and in the saturation\nphase (right). EPs have an o\u000b-axis pro\fles.\nFigure 16: Time evolution of the dominant poloidal modes of the scalar potential ( m= 2;3) at\nthe radial positions where the TAE and EPM are located. EPs have an o\u000b-axis pro\fles.\n216 Conclusion\nThe presence of Alfv\u0013 en modes in burning plasma can a\u000bect negatively the energy con\fnement\nand can also cause a damage in the con\fning machine. Because of their importance, the present\npaper has dealt with the main damping mechanisms a\u000becting the Alfv\u0013 en modes, trying to outline\nthem and to understand in which regime they are acting. Among the great zoology of existing\nAlfv\u0013 en modes, the attention has been focused on toroidal Alfv\u0013 en eigenmodes and energetic particle\nmodes. These studies have been mainly carried by means of numerical simulations conducted with\nthe code ORB5. The obtained results have been compared, when possible, with the presented\nanalytic theory developed in a simpli\fed geometry, and with the results of the linear code Ligka.\nIn Section 3 simulations with very small inverse aspect ratio ( \u000f= 0:01) have been considered,\nin order to lead the analysis in the cylinder limit. Simpli\fed pro\fles have been taken into account\nand very low electron temperature has been considered in order to have the continuum damping\ndominant over the Landau damping. The developed theory has been used to analyze the results\nof simulations without energetic particles. The dependence of the radial wave number of the\nmode against the time have been observed (phase mixing). Also the scalar potential has been\nfound to decay as \u000e\u001e/j!0\nAtj\u00001(continuum damping).\nIn Section 4 higher bulk ion and electron temperatures have been considered, in order to\nobserve the Landau damping to be dominant over the continuum damping. The numerical simu-\nlations have been conducted using plasma equilibrium and pro\fles from the ITPA-TAE interna-\ntional benchmark case. In order to separate the ions and electrons contributions to the damping,\na kinetic term to the ideal MHD vorticity equation has been added. Following a perturbative\napproach, a simpli\fed estimation for the Landau damping has been obtained using cylindrical\ncoordinates. The developed analytical theory has been compared with the dependence found in\nnumerical simulations of the damping rate against the bulk electron temperature. A reasonable\nagreement has been found and and this has also proved that the electron are the main responsible\nfor the damping.\nIn Section 5 a realistic plasma equilibrium taken from a shot in ASDEX Upgrade has been\nconsidered. The results of the linear numerical simulations have shown the dependence of the\ndamping rate against the bulk electron temperature describing, also in this case, the action of\nthe Landau damping. A theoretical estimation of the radial regions where the Landau damping\nis expected to be dominant over the continuum damping has been presented, for simulations\nnear marginal stability. A benchmark with the code LIGKA has shown good agreement for the\nfrequency and growth rate dependence on the EPs temperature. Finally, the nonlinear simulations\nhave shown the interaction of an EPM and a TAE in the scenario with o\u000b-axis EPs density pro\fle.\nThe future works will extend the developed theory in order to \fnd a better agreement be-\ntween the predicted estimation of the decaying mode and the numerical simulations. In Ref.[2]\nit was suggested that all the Alfv\u0013 en \ructuations can be explained within the framework of a\nsingle general \fshbone-like dispersion relation (GFLDR). This could represent the starting point\nto improve the anaytical prediction of the damping rate and it would be a very interesting ana-\nlytical and numerical task. Future and deeper benchmark with the code Ligka and the Hybrid\nMagnetohydrodynamics Gyrokinetic code HYMAGYC [27] would be of great interest in order to\nbetter understand the linear and nonlinear dynamics contained in the NLED-AUG case.\n227 Acknowledgments\nSimulations presented in this work were performed on the CINECA Marconi supercomputer\nwithin the ORBFAST and OrbZONE projects.\nOne of the authors, F. Vannini, would like to thank Xin Wang, Zhixin Lu and Gregorio Vlad\nfor useful, interesting discussions and for great help provided in understanding Alfv\u0013 en dynamics,\nGyrokinetic and MHD theory. The authors wish to acknowledge stimulating discussions with\nF. Zonca, G. Fogaccia, A. K onies, J. Gonzalez-Martin and A. Di Siena. This work was partly\nperformed in the frame of the \\Multi-scale Energetic particle Transport in fusion devices\" ER\nproject.\nThis work has been carried out within the framework of the EUROfusion Consortium and has\nreceived funding from the Euratom research and training program 2014-2018 and 2019-2020 under\ngrant agreement number 633053. The views and opinions expressed herein do not necessarily\nre\rect those of the European Commission.\n23References\n[1] A. Bierwage, Kouji Shinhoara, Yasushi Todo, Nobuyuki Aiba, Masao Ishikawa, Go Mat-\nsunaga, Manabu Takechi and Masatoshi Yagi, \\Simulation tackle abrupt massive migrations\nof energetic beam ions in tokamak plasma\", Nature Communications 9, (2018)\n[2] Liu Chen and Fulvio Zonca, \\Physics of Alfv\u0013 en waves and energetic particles in burning\nplasmas\", Rev. Mod. Phys. 88, (2016)\n[3] S. Jolliet, A. Bottino, P. Angelino, R. Hatzky, T.M. Tran, B. McMillan, O. Sauter, K. Appert,\nY. Idomura and L. 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We then confront the\ntrans-Planckian question of inflationary cosmology. When d issipation increases with the\nenergy, the quantum field describing adiabatic perturbatio ns is completely damped at the\nonset of inflation. However it still exists as a composite ope rator made with the additional\nfields. And when these are in their ground state, the standard power spectrum obtains\nif the threshold energy is much larger that the Hubble parame ter. In fact, as the energy\nredshifts below the threshold, the composite operator beha ves as if it were a free field\nendowed with standard vacuum fluctuations. The relationshi p between our models and\nthe Brane World scenarios studied by Libanov and Rubakov dis playing similar effects is\ndiscussed. The signatures of dissipation will be studied in a forthcoming paper.\n11 Introduction and presentation of the settings\nEven though relativistic QFT provides an excellent description of par ticle physics, being non-\ncompact, Lorentz symmetry can only be tested up to a certain ene rgy scale. Thus one cannot\nexclude that some unknown high energy processes break the invar iance under boosts, thereby\nintroducing a threshold energy Λ LV, and a preferred frame. It is therefore of interest to deter-\nmine what would be the signatures when this possibility is realized [1].\nThis question is particularly relevant for inflationary cosmology beca use primordial density\nfluctuations arise from vacuum fluctuations which had very short w ave lengths (very large\nproper frequencies) at the onset of inflation[2]. Indeed, the initial frequencies of the modes we\nobserve today in the CMB anisotropies were all larger than\nΩin=HeNextra, (1)\nwhereHis the value of the Hubble parameter during inflation, and where Nextrais the number\nof e-folds from the onset of inflation till t0, the moment when the (comoving) scale of our\nvisible universe exited the Hubble radius. (The total number of e-fo lds is thus N0+Nextra\nwhereN0∼60 is the number of e-folds from t0till the end of inflation.) Irrespectively of the\ninflationary scenario, Ω inis thus larger than the Planck mass MPlwhenNextra>lnMPl/H.\nSinceHshould be of the order of 10−5MPl, the initial frequencies of observable modes were all\ntrans-Planckian when Nextra>5ln10∼12. Moreover, since in most scenarios Nextra≫12,\nΩinwas generically much larger than the Planck mass.\nIn the absence of Quantum Gravity, there is no understanding of t he nature of thedegrees of\nfreedom at these scales: we neither know what is their Hilbert space , nor how they propagate,\nand this, because the notions of differential geometry used when d escribing (quantum) fields\nin expanding universes might not apply at very short distances. In p articular eq. (1) which is\nbased on a relativistic mode equation in an accelerating universe might loose its validity above\na certain Ultra-Violet scale Λ LV. Since this scale might significantly differ from the Planck\nmass, we shall keep it distinct from the Planck mass. We shall also use the abbreviations LI\nandLVfor Lorentz Invariance and Lorentz Violation respectively.\nIn this context, it is instructive to phenomenologically parameterize the deviations from LI\nand determine their signatures on inflationary spectra. This line of t hought has been proposed\nin the context of Hawking radiation from black holes, wherein the asy mptotic quanta also arise\nfromvacuum fluctuations withexponentially growing frequencies [3 , 4]. The deviations from LI\nhave been characterized by non-linear dispersion relations imposed in a particular rest frame,\nΩ2=Fn(p2) =p2(1±p2n/Λ2n\nLV+O(p2n+2)), (2)\nwhere Ω and pare respectively the frequency and the norm of the spatial momen tum measured\nin that frame. The first deviation is characterized by a power of p/ΛLVand the sign defining\nsuper-luminous (+) and sub-luminous ( −) cases. When Λ LVis much larger than Hawking\ntemperature, and when the frame is freely falling, it was shown that the asymptotic properties\nof Hawking radiation are unmodified, even though the near horizon p ropagation is radically\nmodified when Ω is larger than Λ LV. The robustness of asymptotic properties relies on the\n2adiabatic character of the evolution of the vacuum (ground) stat e [5]. The same program was\nthen applied to the inflationary spectra [6, 7], and similar results were obtained because, in this\ncase as well, the ground state adiabatically evolves when σ=H/ΛLV≪1 [8]. However, so far,\nonly dispersive models have been systematically studied.\nThe aim of the present work is to extend this analysis to dissipative mo dels. To this end,\nwe shall first construct QFT displaying dissipative effects in the UV. I ndeed, unlike dispersion,\ndissipation requires enlarged dynamical settings because if one trie s to introduce dissipation\nfrom the outset in eq. (2), one looses both unitarity and predictab ility. To preserve them, we\nshall work with Hamiltoniantheories in which dissipative effects arecau sed by interactions with\nadditional fields. Doing so, we shall establish that dissipative effects aregeneric. That is, when\nstarting with a Lagrangian in which LIis broken in the UV, the effective theory unavoidably\ndevelops dissipation above a certain energy scale, simply because no thing can prevent this.\n(With relativistic QFT instead, LIdid prevent it).\nSince we want to construct generalized QFT, we should decide what t o keep. First, we want\nto preserve the unitarity of the description because the calculatio n of the power spectrum of\n(adiabatic) density fluctuations [2] rests on the identification of a s calar field, hereafter called\nφ, which obeys Equal Time Commutators (ETC). This identification is ne cessary to fix the rms\namplitude of density fluctuations in the vacuum. Explicitely the power spectrum is given\nPp(t)≡4πp3/integraldisplay/parenleftBigdx\n2π/parenrightBig3\neipxGa(t,x;t,0), (3)\nwhereGais the anti-commutator of φevaluated in the asymptotic (Bunch-Davies) vacuum\nmuch after horizon exit. In the absence of dissipation, Ppcan be deduced from the norm of\nφin\np(t), the Fourier mode of φwith asymptotic positive frequency. However in the presence\nof dissipation, this notion of (free) mode disappears. Hence the kn owledge of Gabecomes\nnecessary since eq. (3) gives the only way to obtain the power spec trum.\nTo get dissipation we shall thus introduce additional degrees of fre edom, hereafter called Ψ.\nThen the whole system ( φ+ Ψ) will evolve unitarily, by construction. This guarantees that the\nETC ofφis exactly preserved (in a non trivial way since φundergoes dissipation). Moreover,\nthe (dressed) 2pt function of φwill be given by the usual QM trace\nGW(x,y) = AR/bracketleftBig\nˆρTˆφ(x)ˆφ(y)/bracketrightbig\n, (4)\nwhere ˆρTis the initial matrix density of the total system, where ˆφ(x) is the Heisenberg field\noperator evolved with the time ordered exponential of the total H amiltonian, and where the\ntrace is taken over both Ψ and φ. The anti-commutator Gadetermining the power spectrum\nin eq. (3) is simply the symmetric part of GW.\nOur second requirement concerns the properties of dissipative eff ects. When considering\nthe theory in vacuum and in Minkowski space-time, we impose that th ese effects preserve\nthe stationary, the homogeneity, and isotropy of flat space-time . These requirements define\na preferred frame which is inertial and globally defined. Then irrespe ctively of the choice of\nΨ and Ψ- φinteractions, the Fourier transform of the retarded Green func tion,Gr(x,y) =\n3θ(tx−ty)2ImGW(x,y), is of the form\nGr(ω,/vector p) =1/parenleftBig\n−ω2+p2+Σr(ω,p)/parenrightBig. (5)\nIn the true vacuum, at the level of the 2 pt functions, the dissipat ive (dispersive) effects are\nthus completely characterized by the imaginary and odd part in ω(real and even part) of the\nself energy Σ r(ω,p).\nIn these expressions, the energy ωand the momentum square p2are defined in the preferred\nframe. The novelty is that Σ ris a function of ωandpseparately, and not only a function of\nthe relativistic invariant ω2−p2as it is the case in LIQFT. Therefore dissipation can become\nsignificant above a critical energy on the mass shell , i.e. along the minima of the denominator\nof eq. (5); a possibility forbidden in LItheories. For instance, one verifies that the following\nself-energies induce significant dissipation only above Λ LV\nImΣ(n)\nr(ω,p) =−ω\nΛLVp2/parenleftbiggp2\nΛ2\nLV/parenrightbiggn\n. (6)\nWiththisequation we have identified therelevant (lowest order) qua ntity governing dissipation,\nthe equivalent of the first order deviation in eq. (2). In the body of the paper, we shall provide\nLagrangians of Ψ and φwhich give this class of self-energies labeled by n. Rather than focusing\non a particular case, we shall describe the whole class of dissipative b ehaviors (at the level\nof 2-point functions). We shall thus follow a phenomenological appr oach based on minimal\nassumptions (i.e. unitarity), rather than some ”inspired” approac h (e.g. by string theory, or\nbranes scenarios [9, 10]) which would lead to a particular sub-class of models.\nOur thirdrequirement concerns theextension ofourmodels fromM inkowski space tocurved\nspace-times. TodefineourQFTincurvedspace, wesimplyimplementt heEquivalencePrinciple\n(EP). It fixes the action density our models to be the covariantized version of that we had in\nMinkowski space (up to some non-minimal coupling). To perform the covariantization, it is\nuseful to characterize the preferred frame in a coordinate invar iant way by lµ, a unit time-like\nvector field [11]. In terms of this vector, ωandp2are given by\nω≡lµpµ, p2≡⊥µνpµpν, (7)\nwhere⊥µν≡gµν+lµlνis the (positive definite) metric in the spatial sections orthogonal t olµ.\nThe covariant action will be a sum of scalar functions of the four loca l fieldsφ, Ψ,gµνandlν\nwhich reduces to the Minkowski model in the zero curvature limit. Ve ry importantly for us, we\nshall see that the EP guarantees the adiabaticity of the evolution o f the (interacting) ground\nstate as long the gradients of the metric are much smaller than the U V scale Λ LV.\nFor the interested reader, we mention that additional comments w hich place the present\nwork in broader contexts can be found in the introduction of [12]. T hese comments concern\nQuantum Gravity and the description of ”mode creation” in expandin g universes.\n42 Dissipation in Minkowski space from LVeffects\nIn this Section, we provide a class of models defined in Minkowski spac etime which exhibit\ndissipative effects above a certain energy scale Λ LV. Stationarity, homogeneity and isotropy\nwill be exactly preserved. Therefore, the only invariance of relativ ist QFT which is broken is\nthat under boosts. These theories define a preferred rest fram e which is globally defined, as it\nis the case in FLRW space-times. For the simplicity of the presentatio n, we shall first work in\nthe preferred frame. At the end of this Section we shall covariant ize our action and generalize\nit to curved spacetime.\n2.1 Free field settings\nWe start with a brief presentation of the free field quantization to in troduce notations and to\npoint out what are the properties which will be lost in the presence of dissipation.\nThe action of our free massless field φis the usual one:\nSφ=1\n2/integraldisplay\ndtd3x(∂tφ2−∂xφ·∂xφ), (8)\nwheretandxare Cartesian coordinates. Due to the homogeneity of space, the equation of\nmotion can be analyzed mode by mode:\nφ(t,x) =/integraldisplayd3p\n(2π)3/2eip·xφp(t). (9)\nThe Fourier mode φp(t) obeys\n(∂2\nt+ω2\np)φp= 0, (10)\nwhereω2\np=p2=p·pis the standard relativistic dispersion relation. Notice that eq. (10) is\nsecond order, homogeneous (no source term), and time reversib le (no odd power of ∂t), three\nproperties we shall loose when introducing interactions breaking LI.\nIn homogeneous space-times, the Equal Time Commutator betwee n the Heisenberg field\noperator and its momentum implies that the mode operator (q-numb er)φp(t) obeys\n[φp(t),∂tφ†\np′(t)] =iδ3(p−p′). (11)\nWhen decomposing this operator as\nφp(t) =apφp(t)+a†\n−pφ∗\np(t), (12)\nwhere the destruction and creation operators satisfy the usual commutators\n[ap,a†\np′] =δ3(p−p′),[ap,ap′] = 0, (13)\neq. (11) is verified because the Wronskian of the positive frequenc y (c-number) mode\nφp(t) =e−iωpt/(2ωp)1/2, (14)\n5is constant (and conventionally taken to be unity).\nHad an odd term like γ∂tbe present in eq. (10) the constancy of the Wronskian would have\nbeen lost. Hence the possibility of realizing the ETC (11) with the help o f eq. (13) would have\nbeen lost as well. This already indicates that, unlike dispersive (real) e ffects, dissipative effects\nrequire more general settings than the above.\n2.2 Interacting models breaking LI, general properties\nWe now introduce additional degrees of freedom, here after collec tively named Ψ, which induce\ndissipationabovetheenergyΛ LV. Weshallworkwithaparticularclassofmodelsinordertoget\nanexact (non-perturbative) expression for the two-point func tion of eq. (4). Before introducing\nthese models, we derive general results valid for all unitary QFT’s po ssessing dissipative effects\nabove Λ LVin the ground state (the interacting vacuum).\nWe assume that the total action decomposes as\nST=Sφ+SΨ+Sφ,Ψ, (15)\nwhere the first action is that of eq. (8), the second one governs t he evolution of the Ψfields,\nand the last one the coupling between φand these new fields. We also impose that the last two\nactions preserve the homogeneity and isotropy of Minkowski spac e but break the invariance\nunder boosts. From now on, the coordinates t,xare at rest with respect to the preferred frame\ndefined by SΨ+Sφ,Ψ, i.e.,∂t≡lµ∂µin the covariant notation of eq. (7).\nWhen the state of such system is homogeneous, the Fourier trans form of the 2pt function\nof eq. (4) is of the form\nGp,p′(t,t′) = Tr[ˆ ρTˆφp(t)ˆφ†\np′(t′)],\n=GW(t,t′;p)δ3(p−p′). (16)\nAt this point, an important remark should be made. In the presence of interactions, the notion\n(and the usefulness) of the time-dependent modes of eq. (14) ha s disappeared whereas the\nfunction GW(t,t′;p) of eq. (16) is always well-defined, for all choices of ΨandSΨ+Sφ,Ψ.\nWhen the situation is stationary, GW(p;t,t′) further simplifies in the frequency representa-\ntion:\nGW(t,t′;p) =/integraldisplaydω\n2πe−iω(t−t′)GW(ω,p). (17)\nWhen working in the ground state, one finds the simplest case, beca use whatever the Ψfields\nmay be, the Fourier transform of the time ordered (Feynman) pro pagator,\n2GF= ReGW+iImGWsign(t−t′), (18)\nis always of the same form as Grin eq. (5), and therefore characterized by a single (complex)\nfunction Σ F(ω,p). When restricting attentionto Gaussian models, Σ F(ω,p)is given by a 1-loop\ncalculation, whereas in general, it contains a series of 1PI graphs.\n6In non-vacuum states and in non-stationary situations, the 2pt f unctions and self-energies\nhave a more complicated structure [20]. To understand this struc ture it is useful to study\nseparately the commutator Gc=iImGWand the anti-commutator Ga= ReGW.\nLet us conclude with two remarks. First, the effective dispersion re lation of φisa posteriori\ndefined by the poles of eq. (5) which are governed by Σ( ω,p) [13, 14]. In this way, non-\ntrivial dispersion relations arise from dynamical processes rather than from being introduced\nfrom the outset. The present work therefore provides physical foundations (and restrictions,\nas later discussed) to the kinematical approach which is usually adop ted [1]. Second, from\nanalyzing dynamical models, we shall see that, even in the vacuum, on-shelldissipative effects\n(i.e. dissipation arising along the minima of the denominator of eq. (5)) are unavoidable when\nLIis broken in the UV by the action SΨ+Sφ,Ψ, in complete opposition with the fact that\non-shell dissipation is forbidden when working (in the vacuum) with LIactions.\n2.3 Gaussian models\nTo simplify the calculation of Σ( ω,p) and to get non perturbative expressions, we assume that\nthe action STis quadratic in all field variables. At first sight, this could be considere d as an\nartificial hypothesis. However, it should be recalled that we are not after computing Σ from\nfirst principles. Rather we aim to compute the signatures of the pow er spectrum (3) giventhe\nproperties of Σ, following the approach adopted in [3, 5, 6, 7].\nGiven that wearepreserving thehomogeneity of Minkowski space, theGaussianassumption\nimplies that the total action splits as\nST=/integraldisplay\nd3pST(p), (19)\nwhere each action ST(p) depends only on φpand the p-th Fourier component of Ψ. The\nstructure of these actions is\nST(p) =1\n2/integraldisplay\ndt φ∗\np(−∂2\nt−ω2\np)φp+1\n2Σi/integraldisplay\ndtΨ∗\ni(p)(−∂2\nt−Ω2\ni(p))Ψi(p)\n+Σi/integraldisplay\ndt gi(p)φpΨ∗\ni(p), (20)\nwhereiis a discrete (or continuous) index, where Ω i(p) is the energy of the quanta of the\noscillators Ψ i(p), and where gi(p) is the coupling constant at fixed p,i. Since φ†\np=φ−p,\nΨ†\ni(p) = Ψ i(−p),ST(p) +ST(−p) is real. Instead of choosing a priori one model, it is more\ninstructive to solve the equations of motion without specifying the s et of Ψ i(p), their energy\nΩi(p) and their coupling gi(p). We shall choose them in due course. To preserve stationarity\nin Minkowski space, the Ω’s and the g’s must be time independent. When Ω2\ni(p)∝ne}ationslash=M2\ni+p2,\nthe kinetic action of Ψ i(p) breaks LIand defines the preferred frame. On the contrary when\nΩ2\ni(p) =M2\ni+p2the preferred frame is only defined by SφΨthrough the p-dependence of the\ncoupling functions gi(p).\nModels of this type have been used for different purposes. They ha ve been introduced\n(in their continuous version) to study non-pertubatively atomic tr ansitions see [15] and refs.\n7therein, see also [16, 17] for an application to the Unruh effect. The y have been used in Quan-\ntum Optics [18], to model quantum Brownian motion, and to study dec oherence effects [19].\nDepending on the aims, they can be solved and analyzed by means of d ifferent methods. In\nwhat follows we shall use the simplest approach based on Heisenberg picture.1Since we shall\nwork at zero temperature, this approach offers a simple characte rization of the state of the\nsystem in terms of the ground states of the free modes before φ−Ψ interactions are turned on.\nTo prepare the application to inflation, we shall treat ω2\np, Ω2\niandgias arbitrary functions\nof time (in cosmology these quantities become time dependent throu gh their dependence in the\nscale factor a(t)). The equations of motion are\n(∂2\nt+ω2\np)φp= Σigi(p)Ψi(p), (21)\n(∂2\nt+Ω2\ni)Ψi(p) =gi(p)φp. (22)\nThe general solution of the second equation reads\nΨi(p,t) = Ψo\ni(p,t)+/integraldisplay\ndt′Ro\ni(t,t′;p)gi(t′;p)φp(t′), (23)\nwhere Ψo\ni(p,t) is a free solution which depends on initial conditions imposed on Ψ i(p). The\nsecond term contains Ro\ni(t,t′;p), the (free) retarded Green function of Ψ i(p). It obeys\n(∂2\nt+Ω2\ni(p))Ro\ni(t,t′;p) =δ(t−t′), (24)\nand vanishes for t < t′. Injecting eq. (23) in eq. (21) one gets\n(∂2\nt+ω2\np)φp= Σigi(t;p)Ψo\ni(p,t)+Σigi(t;p)/integraldisplay\ndt′Ro\ni(t,t′;p)gi(t′;p)φp(t′).(25)\nThe solution of this equation has always the following structure\nφp(t) =φd\np(t)+/integraldisplay\ndt′Gr(t,t′;p)[Σigi(t′;p)Ψo\ni(p,t′)]. (26)\nThe first term is the ”decaying” solution. It contains all the informa tion about the initial\nconditionof φp. Thesecondtermisthe”driven”solution. Itisgovernedbytheinitia lconditions\nof Ψi(p) and by the (dressed) retarded Green function, the solution of\n/integraldisplay\ndt′[δ(t−t′)(∂2\nt′+ω2\np)−Σigi(t;p)Ro\ni(t,t′;p)gi(t′;p)]Gr(t′,t1;p) =δ(t−t1).(27)\nNotice that φd\np(t′) is an homogeneous solution of this equation. Therefore the evolut ion of\nbothφdandGrfully takes into account, through the non-local term in the above b racket, the\n1Even though legitimate, we shall not use the general methods (Infl uence Functional, Master Equation,\nClosed Time Path Integral) [18] which have been developed to study ” open quantum systems” because they\nsomehow hide the simplicity of the present models. Moreover we are p lanning (in a subsequent work) to study\nthe correlations between φand Ψ i. Therefore we shall treat φand Ψ ion equal footing as in [15, 16, 17].\n8back-reaction due to the coupling to the Ψ i. In Gaussian models, it is quadratic in gi. Hence\nφdandGrare series containing all powers of gi. Moreover since gi(t;p) are arbitrary functions\nofpandt, at this point, there is no reason to consider non-Gaussian models.\nTo conclude this subsection, we notice that eq. (26) also furnishes the exact solution for the\n(Heisenberg) mode operator φp(t) because the equations we solved were all linear. Since the\npower spectrum in inflation is obtained from vacuum fluctuations, ins tead of further analyzing\nthe time dependence of the mode (as one would do in classical terms) , it is more relevant to\nstudy the correlation functions of φp(t).\n2.4 Structure of two-point correlation functions\nThis sub-section mainly contains well-known results which follow from t he linearity of eq. (26).\nThe key result we shall later use is given in eq. (31).\nSince our models are Gaussian, the 2pt function of eq. (16) govern sallobservables built\nwith the Heisenberg field operator φ. To analyse it, as already mentioned, it is appropriate to\nstudy separately the commutator and the anti-commutator.\nWe start with the simple part, the commutator\nGc(t,t′;p)δ3(p−p′)≡Tr[ρT[φp(t),φ†\np′(t′)]−]. (28)\nFrom eq. (26) one sees that it decomposes into two terms, one due to the non-commuting\ncharacter of φd, the other due to that of Ψ0\ni. In addition, since both commutators are c-\nnumbers, it is independent of ρT, the state of the system. Hence, for all Gaussian models, one\nhas\nGc(t,t′;p) = [φd(t),φd(t′)]−+/integraldisplay/integraldisplay\ndt1dt2Gr(t,t1)Gr(t′,t2)D(t1,t2), (29)\nwhere the ”dissipative” kernel D(t1,t2) is given by\nD(t1,t2) = ΣiΣjgi(t1)gj(t2)[Ψo\ni(t1),Ψo\nj(t2)]−= Σigi(t1)Go\nc,i(t1,t2)gi(t2).(30)\nNotice how this kernel combines the coupling giand the non-commuting properties of Ψ i.\nThe next property of Gcis more relevant. To all orders in giand for all sets of gi,Ωi(even\nwith arbitrary time dependence), one obtains\ni∂tGc(t,t′;p)|t=t′= 1. (31)\nThis identity corresponds to the ETC of eq. (11). The 1 on the rhsis guaranteed by the\nHamiltonian character of the evolution of the entire system φ+Ψ. It is therefore this equation\nwhich replaces the constancy of the Wronskian that was relevant in the case of free evolution.\nEq. (31) is crucial for us for two reasons. First, since the operat orφd(t) in eq. (26)\ndecays in t−tinwheretinis the moment when the interactions are turn on –because it is an\nhomogeneous solution of eq. (27)– the first term in eq. (29) decay s as exp−γ(t+t′−2tin).\nTherefore when γ(t+t′−2tin)≫1, the non-commuting properties φpareentirely due to those\nof the environment degrees of freedom, Ψo\ni. Secondly, these sum up exactly to 1, as if the driven\nterm of eq. (26) were a canonical degree of freedom.\n9We now analyze the anti-commutator,\nGa(t,t′;p)δ3(p−p′)≡Tr[ρT{φp(t),φ†\np′(t′)}+]. (32)\nWhen the (initial) density matrix factorizes, ρT=ρφρΨ, as it is the case in the ”free” vacuum\nbefore the interactions are turned on, Gaalso splits into two terms,\nGa(t,t′;p) = Tr[ρφ{φd(t),φd(t′)}+]+/integraldisplay/integraldisplay\ndt1dt2Gr(t,t1)Gr(t′,t2)N(t1,t2).(33)\nThe first term only depends on the initial state of φ. Similarly, the driven term only depends\non the state of the environment through the ”noise” kernel\nN(t1,t2) = ΣiΣjTr[ρΨ{gi(t1)Ψo\ni(t1),gj(t2)Ψo\nj(t2)}+]. (34)\nAs for the commutator, in the presence of dissipation, the first te rm exponentially decays,\nexpressing the progressive erasing on the information contained in the initial state of φ. At\nlate times therefore it is the state of Ψ which fixes the anti-commuta tor ofφ. This allows to\nremove the restriction that initially the density matrices factorizes . If one is interested by the\nlate time behavior, only Nmatters.\nIn brief, we have recalled two important results. First, at late time, the Heisenberg field\nφreduces to its driven term, the second term of eq. (26), since bot h its commutator and\nanti-commutator are determined by those of Ψo\ni. Second, only two (real) kernels determined\nby the environment govern the two-point functions of φ, namely DandNof eqs. (30, 34).\nTherefore the set of environments (Gaussian or notGaussian) possessing the same kernels will\ngive rise to the same 2pt functions. Hence they should be viewed as f orming an equivalent\nclass. The degeneracy can be lifted by considering correlations with observables containing the\noperators Ψ i, or higher order correlations functions of φ(for non-Gaussian environments), two\npossibilities we shall not discuss in this paper.\nTo compute GcandGa, two different routes can be adopted. When gi(p) and Ω( p) are\nconstant, one should work in Fourier transform because the equa tions can be algebraically\nsolved, in full generality. Instead when gi(p) and/or Ω( p) are time-dependent, as it will be the\ncase in expanding universes and in curved space-times, it is appropr iate (but not necessary) to\nexploit the above mentioned degeneracy by choosing the Ψ iand their frequencies Ω2\ni(p) so as\nto simplify the time dependence of the equations. In the text, we pr oceed with time dependent\napproach. In Appendix A, we present the Fourier analysis which is st raightforward. We invite\nthe reader unfamiliar with the Quantum Mechanical treatment of dis sipation to read it.\n2.5 Time dependent settings\nIn Appendix B, we provided a class of models characterized by the po wer ofp/ΛLVwhich\nspecifies how dissipative effects grow, see eq. (85). This class cove rs the general case and can\nbe used as a template to study the consequences of dissipative effe cts. In addition, as noticed\nafter eq. (76), the dissipative effects are governed by D= Σig2\niRo\ni. Therefore all environments\ndelivering the same kernel give rise to the same (stationary) pheno menology.\n10In this Section we exploit this freedom having in mind the transposition of our model from\nMinkowski space to cosmological, and thus time-dependent, metric s. Therefore the selected\nmodels should possess two-point functions with simple properties wh en expressed in the terms\nof time (as opposed to Fourier components). The core of the prob lem is that Gadepends, see\neq. (33), on the retarded Green function of φwhich is not known. Indeed Gris only implicitly\ndefined as a solution of eq. (27) which is, in general, a non-local differential equation. We\nare thus led to choose Ψ in order for this equation to be local. This implie s that the retarded\nGreen function of Ψ appearing in eq. (27) be proportional to δ(t−t′). (Since this requirement\nconcerns the Green function of the environment it does not restr ict the phenomenology of φ.)\nGiven this aim, we select the models defined by the action\nS(n)\nT(p) =1\n2/integraldisplay\ndt φ∗\np(−∂2\nt−ω2\np)φp+1\n2/integraldisplay\ndt/integraldisplay∞\n−∞dkΨ∗(p,k)(−∂2\nt−(πΛLVk)2)Ψ(p,k)\n+gΛLV/integraldisplay\ndt/integraldisplay∞\n−∞dk/parenleftbiggp\nΛLV/parenrightbiggn+1\nφp∂tΨ∗(p,k). (35)\nInSφΨhave factorized out a factor of Λ LVso that the coupling constant gis dimensionless.\nWhen compared with the action of eq. (20), we have replaced the dis crete index iby the\nintegral over the dimensionless variable k. As recalled in Appendix A, the spectrum of the\nenvironment must be continuous to have proper dissipation, see dis cussion after eq. (76). The\nvariablekcan be viewed as a momentum, in the units of Λ LV, in a flat extra spatial dimension.\nThe relationship with some Brane World Scenarios in then clear [9, 10]. I n the ’atomic’ version\nof this model [16, 17] which has inspired us, the radiation field Ψ is a ma ssless 2 dimensional\nfield propagating in the dimension associated with k.\nWe have also introduced an additional time derivative acting Ψ in Sφ,Ψ. This choice leads\nto the above mentioned δ(t−t′). Indeed, on the one hand, taking account this extra derivative,\nthe continuous character of k, and the fact that gis independent of k, eq. (25) becomes\n(∂2\nt+ω2\np)φp=gn∂t/integraldisplay\ndkΨo(p,k,t)−gn∂t/integraldisplay\ndt′/integraldisplay\ndkRo(t,t′;k,p)∂t′/parenleftbig\ngnφp(t′)/parenrightbig\n,(36)\nwheregn≡gΛLV(p/ΛLV)n+1. On the other hand, for each 3-momentum p,Ψ=/integraltext\ndkΨ(k) is a\nmassless 2-dimensional free field. In Fourier components, its reta rded Green function is given\nbyRo(ω,k) = 1/(−(ω+iǫ)2+(πΛLVk)2). Hence Ro(t,t′) obeys\n∂tRo(t,t′)≡∂t/integraldisplay∞\n−∞dω\n2π/integraldisplay∞\n−∞dkRo(ω,k)e−iω(t−t′)=δ(t−t′)\nΛLV, (37)\nwhich is the required property to simplify eq. (36).\nWhengis constant, the retarded Green function of φassociated to eq. (36) obeys the\nfollowing localequation\n[∂2\nt+g2\nn\nΛLV∂t+ω2\np]Gr(t,t′,p) =δ(t−t′), (38)\n11To make contact with Appendix A and B, let us rewrite this equation in F ourier transform:\n[−ω2−ig2ω\nΛLVp2/parenleftbigp\nΛLV/parenrightbig2n+ω2\np]Gr(ω,p) = 1. (39)\nWe thus see that ReΣ r= 0 and that ImΣ ris (exactly) given by g2times that of eq. (85).\nThus, even though we have chosen a simple form for Ro(t,t′), the above action delivers the n\ndissipative behaviors of Appendix B by choosing the appropriate pow er ofp/ΛLVinSφΨ.\nWhengandω2\nparearbitrarytime-dependent functions, theFourieranalysis loos esitspower.\nHowever, in time dependent settings, our Grstill obeys a local equation:\n/bracketleftbig\n∂2\nt+2˜γn∂t+[ω2\np+∂t˜γn]/bracketrightbig\nGr(t,t′) =δ(t−t′). (40)\nwhere the n-th decay rate ˜ γn(t) =g2(t)γnis now a definite time dependent function.\n2.6 Covariant description\nWe now provide the covariantized expression of the action ST=/integraltext\nd3pST(p), where ST(p) is\ngivenineq. (35). This expression will thenbeusedtodefineourtheo ryincurved backgrounds.2\nTwo steps should be done. We need to go from pconsiderations to a local description, and\nexpress the various actions in terms of the unit vector field lµand the spatial metric ⊥µν. Both\nare straightforward and, in arbitrary coordinates, the action re ads\nST=−1\n2/integraldisplay\nd4x√−ggµν∂µφ∂νφ\n+1\n2/integraldisplay\nd4x√−g/integraldisplay\ndk/bracketleftBig/parenleftbig\nlµlν−c2\nΨ⊥µν/parenrightbig\n∂µΨ(k)∂νΨ(k)−(πΛLVk)2Ψ2(k)/parenrightbig/bracketrightBig\n+gΛLV/integraldisplay\nd4x√−g/parenleftBig/parenleftbig∆\nΛ2\nLV/parenrightbig(n+1)/2φ/parenrightBig\nlµ∂µ/integraldisplay\ndkΨ(k), (41)\nwhere the symbol ∆ is the Laplacian on the three surfaces orthogo nal tolµ.\nWe have slightly generalized the action of eq. (35) by subtracting c2\nΨ⊥µνto the kinetic term\nof Ψ, where c2\nΨ≪1. With this new term, the Ψ( k) are now massive fields which propagate\nwith a velocity whose square is bounded by c2\nΨ. In addition they now possess a well defined\nenergy-momentum tensor which can be obtained by varying their ac tionwith respect to gµν. To\nobtain the simplified expressions we have used (and shall still use), t he (regular) limit c2\nΨ→0\nshould be taken.\n2This procedure perhaps requires further explanation since Ψ is not a fundamental field but ”nothing more\nthanaconvenientparameterizationofsomeenvironmentdegrees offreedom.”First,Ψhasnotbeenintroducedto\nparameterizethedissipativeeffectsarisingfromanytheory, buto nlythosefromtheoriesobeyingtheEquivalence\nPrinciple, see [9] for a prototype. Second, from the point of view of cond-mat physics, it could a priori seem\ninappropriate to proceed to a covariantization, since in most situat ions (heat bath) there is a preferred frame\nwhich is globally defined. However, when the system is non-homogene ous (e.g. a fluid characterized by a non-\nhomogeneous flow), low energy fluctuations effectively live in a curve d geometry[28]. Moreover, in this case,\nshort distance effects, i.e. dispersive (or dissipative) effects, are covariantly described when using this metric\nbecause they arise locally[3, 11]. Hence the covariantization is both a necessary step to implem ent the EP in\nour settings and a property that emerges in cond-mat physics.\n12In this limit, the (free) retarded Green function of the Ψfield obeys a particularly simple\nequation when expressed in space-time coordinates:\nlµ∂\n∂xµ/integraldisplay\ndkRo(x,y;k) =lµ∂\n∂xµRo(x,y) =1\nΛLVδ4(xµ−yµ)√−g. (42)\nOn the r.h.s, one finds the delta function with respect to the invarian t measure d4x√−g. This\nequation is nothing by the covariantized and ”localized” version of eq . (37). Its physical\nmeaning is clear. In our model, the back-reaction of φ(x) onto itself through Ψis local.\nIn spite of this, the equations of motions do not have a particularly s imple form. Using the\ncondensed notion Ψ=/integraltext\ndkΨ, one gets\n1√−g∂µ√−ggµν∂νφ(x) =gΛLV1√−g/parenleftbig∆\nΛ2\nLV/parenrightbign+1\n2√−glµ∂µΨ(x), (43)\nwhere the interacting Ψ(x) field is\nΨ(x) =Ψo(x)−gΛLV/integraldisplay\nd4y√−gRo(x,y)/bracketleftBig1√−g∂µ/parenleftBig\nlµ√−g/parenleftbig∆\nΛ2\nLV/parenrightbign+1\n2φ(y)/parenrightBig/bracketrightBig\n.(44)\nWhen inserting eq. (44) in eq. (43), using eq. (42), one verifies tha t the dissipative term is local\nand first order in lµ∂µ. Therefore, as expected, dissipation occurs along the preferre d direction\nspecified by the vector field lµ.\n2.7 Dissipative effects in curved background geometries\nTo define a dissipative QFT in an arbitrary curved geometry, one nee ds some principles. From\na physical point of view, we adopt the Equivalence Principle, or bette r what can be considered\nas its generalization in the presence of the unit time-like vector field lµ. We are in fact dealing\nwith two (set of) dynamical fields, the φfield we probe, and the Ψfield we do not; but also\nwith two background fields gµνandlµ. The Generalized Equivalence Principle means that the\naction densities of the dynamical fields be given by scalar functions ( under general coordinate\ntransformations) which coincide to those one had in Minkowski spac e time for a homogeneous\nand static lµfield, i.e. those of eq. (41).\nHowever, the densities are not completely fixed by the GEP. In this w e recover what was\nobtained withthe EP: Fora scalar field ina curved geometry, there w as always thepossibility of\nconsidering a non-minimal coupling to gravity by adding to the Lagran giana term proportional\ntoRφ2. In the present case, the ambiguity is larger because lµdefines new scalars, the first of\nwhich is the expansion Θ = ∇µlµ, where∇µis the covariant derivative with respect to gµν. The\nambiguity can only be resolved by adopting some additional principle, s uch as the principle of\nminimal couplings which forbids adding densities containing these scala rs.\nRather than adopting it, we shall choose the non-minimal coupling so as to keep eq. (42),\ni.e. so that the localityof the back-reaction effects of φthroughΨbe preserved. This choice\nmaintains the simplicity of the equations of motion in curved backgrou nds, but is by no means\nnecessary. Starting from eq. (41), the locality is preserved by re placing in SΨandSΨφ\nlµ∂µΨk→ DlΨk≡lµ∂µΨk+Ψk\n2Θ =1\n2(lµ∇µΨk+∇µ[lµΨk]). (45)\n13To simplify the forthcoming equations, we use the fact that one can always work in ”pre-\nferred” coordinate systems in which the shift livanishes and in which the preferred time is such\nthatl0= 1. (We assume that the set of orbits of lµis complete and without caustic. In this\ncase, every point of the manifold is reached by one orbit.) In these c oordinate systems, it is\nuseful to work with rescaled fields Ψr≡(−g)1/4Ψ because the above equation simplifies\nDlΨk= (−g)−1/4lµ∂µ((−g)1/4Ψk) = (−g)−1/4lµ∂µΨr\nk, (46)\nsince Θ = ( −g)−1/2∂µ[(−g)1/2lµ]. Notice also that there exists a subclass of background fields\n(g,l), for which one can find coordinate systems such that boththe shift liandgoivanish. In\nthesecomoving coordinate systems, the above equations further simplify since on ly the spatial\npart of the metric matters because −g=hcwhereh≡det(⊥ij).3\nHaving chosen this non-minimal coupling, one verifies that the kinetic term of the rescaled\nfields Ψ ris insensitive to the ”curvature” of both gµνandlµ(when the limit c2\nΨ→0 is taken).\nMoreover the differential operator which acts on the retarded Gr een function of Ψ rin the\nequation of motion of φ, see eqs. (43, 44), is also ”flat” thereby guaranteeing that the m odified\nversion eq. (42) still applies, that is\nDlRo(x,y) = (−g(x))−1/4lµ∂\n∂xµ/parenleftBig\nRo\nr(x,y)/parenrightBig\n(−g(y))−1/4\n=1\nΛLVδ4(xµ−yµ)√−g, (47)\nwhereRo\nr(x,y) is the retarded Green function of the rescaled field Ψ r. It obeys (in pre-\nferred coordinate systems) ∂tRo\nr=δ4/ΛLV, and ”defines” the retarded Green function Ro=\n(−g)1/4Ro\nr(−g)1/4which is a bi-scalar. Hence the equations of motion in an arbitrary bac k-\nground ”tensor-vector metric” specified by the couple ( gµν,lµ), are given by eqs. (43, 44) with\nthe substitution of eq. (45).\nSeveral remarks should be made. First, from the simplified equation ∂tRo\nr=δ4/ΛLVit\nmight seem that the background tensor metric gµνplays no role. This is not true, since it is\nused to normalize the field lµat every point.\nSecond, in the limit c2\nΨ→0, the (rescaled) Ψ kfields define a new kind of field. They\npropagate in an effective space-time given by the time development o f the 3-dimensional set\nof orbits of the lµfield. Indeed, at fixed k, Ψk(x) can be decomposed in non-interacting local\nfield-oscillators, each of them evolving separately along its orbit. Th is situation is similar to\nthe long wave length (gradient-free) expression of [32]. In the ab sence oflµ, the geometry must\nbe (nearly) homogenous for the action to posses this decompositio n. However, when lµis given,\n3It is clear that this is the case when lµcoincides with the cosmological frame and when one uses comoving\ncoordinates ds2=−dt2+a2dx2sincelµ= (1,0). However, in certaincases one should searchfor the ”comoving ”\ncoordinate system. To illustrate this point, consider the former sit uation in Lemaˆ ıtre coordinates X=ax. In\nthis case one has ds2=−dt2+ (dX−Vdt)2, where the velocity is V=HX. The spatial sections are now\nthe Euclidean space with h= 1, and the (contravariant) components of the unit vector field a relµ= 1,V.\nTo compute hcone should solve the equation of motion of comoving (free falling) obs erversdX−Vdt= 0,\nand use the initial position as new coordinates. This procedure is exp licitely done in [12] when starting with\nPainlev´ e-Gullstrand coordinates to describe the black hole metric a nd using a freely falling frame.\n14one can identify, even in non-homogeneous metrics, each space-t ime point in an invariant way\nby the spatial positionof the corresponding ”preferred” orbit at some time, andthe proper time\nalongtheorbit(aslongas lµhasnocaustic). Wecanthus buildcovariant actionsexploiting this\npossibility and consider fields composed of a dense set of local oscillat ors at rest with respect\ntolµ. The fields Ψ kwe use belong to this class of fields.\n3 Dissipative effects in cosmology\n3.1 The action\nOur aim is to describe dissipative effects in an expanding homogeneous universe when the\nvector field lµis aligned along the cosmological frame, when the dissipative effects a re known\nin Minkowski space, and when implementing the Equivalence Principle.\nIn this case, to get the action we simply consider eq. (41) (with the c urved metric modifi-\ncations discussed in the former subsection) in a FLRW metric. Using c omoving coordinates,\nds2=−dt2+a2(t)dx2, (48)\nthe components of lµare (1,0). To simplify the notations, we use the conformal time dη=dt/a\nand work with the rescaled fields φr=aφand Ψ r=h1/2\ncΨ =a3/2Ψ. Dropping these rindices,\nworking in Fourier transform with respect to x, the non-minimally coupled action associated\nwith the replacement of eq. (45) is\nS(n)\nT(p) =1\n2/integraldisplay\ndη φ∗\np(−∂2\nη−ω2\np(η))φp\n+1\n2/integraldisplay\ndt/integraldisplay\ndkΨ∗(p,k)(−∂2\nt−(πΛLVk)2)Ψ(p,k)\n+/integraldisplay\ndηgn(η)φp∂η/integraldisplay\ndkΨ∗(p,k). (49)\nThe conformal frequency ω2\np(η) =p2−∂2\nηa/ais that of a rescaled minimal coupled massless\nfield. In this expression, as everywhere in this Section, pis now the conformal (dimensionless\nand constant) wave vector. The time dependent coupling coefficien t is\ngn≡g a1/2ΛLV(p/aΛLV)n+1. (50)\nIts dependence in a(given that of p) follows from having implemented the Generalized Equiv-\nalence Principle (GEP) which determines the powers of afor each term in the action. As we\nshall see, this relative power guarantees that all φpwill be damped at a fixed and common\nproper scale (in the adiabatic approximation). Had we started with a n action of the type (49)\nwithout relying on the GEP, the dependence in awould have been arbitrary, and the proper\nscale at which modes would have been damped would have run as well.\nIn the same vein, one sees that the proper frequency of the Ψ field s stays constant. This\nfollowsfromtheGEPbut also fromourchoice ofnon-minimal couplings . Inthispaper, we want\n15indeed to analyse the phenomenology of dissipative effects when the new degrees of freedom\nare and stay in their ground state. One could have chosen more com plicated models in which\nΨ is parametrically excited. This would be the case when introducing a n on zero velocity cΨ,\nand with minimally coupling (by dropping Θ on the rhs of ≡in eq. (45)). However, when Ψ is\ntaken massive (i.e. by restricting the krange to |k|>1), or discretizing kas it would be the\ncase for Kaluza-Klein modes, and if Λ LV≫H, the phenomenology of all these models coincide\nsince the amplitude for parametric excitations will be exponentially da mped. In this regime\nthere is thus no gain in studying more complicated actions than that g iven in eq. (49).\n3.2 Equation of motion\nUsing the results of the subsection 2.5, the equation of motion of He isenberg operator φpis\n/parenleftbig\n∂2\nη+2γn∂η+(ω2\np(η)+∂ηγn)/parenrightbig\nφp=gn∂ηΨo(p), (51)\nwhere the decay rate in conformal time is,\nγn(η) =g2\nn\n2ΛLV=1\n2(aΛLV)/parenleftbigp\naΛLV/parenrightbig2n+2. (52)\nIt is dimensionless, as it should be. It should be compared with the com oving frequency pto\nget the relative strength of dissipation. One obtains\nγn(η)\np=1\n2/parenleftbigp\naΛLV/parenrightbig2n+1=1\n2/parenleftbigpphys\nΛLV/parenrightbig2n+1. (53)\nIn the last equality we have re-introduced the proper momentum pphys=p/a. With this\nequation we verify that, at any time in an expanding universe and for every mode φp, the\nrelativestrengthofdissipationissimplythatofMinkowski spaceeva luatedatthecorresponding\nenergy scale, see eq. (86). This directly results from having impleme nted the GEP.\nNotice however that the equation of motion in an expanding universe contains a frequency\nshift\ng2\nn\nΛLV∂η(lngn) =p2/parenleftbigaH\np/parenrightbig/parenleftbigp\naΛLV/parenrightbig2n+1. (54)\nWe have factorized out the unperturbed frequency square to ge t the relative value of the shift.\nIt vanishes both when dissipation is negligible and when aH/p≪1, i.e. when the expansion\nrateHis negligible with respect to the proper momentum. Fromthis express ion we can already\nconcludethatitcannotplayanyrolewhenthetworelevantscales HandΛLVarewell separated,\ni.e. when\nσ≡H\nΛLV≪1. (55)\nIndeed when the physical momentum is high and of the order of Λ LV, the relative frequency\nshift proportional to σ, and when the physical momentum is of the order of H(at horizon exit,\nsee Figure 1), it is proportional to σ2n+1.\nIn quantum settings however, it is not sufficient that the equation o f motion possesses its\nMinkowskian form because the state of the system, and therefor e the observables, might be\naffected by the combined effect of dissipative effects and the expan sion rate.\n163.3 Power Spectrum\nLet us consider the power spectrum of a scalar massless test field ( which is the relevant case\nfor gravitational waves and density perturbations) both in the st andard free field settings and\nin the presence of dissipative effects.\n3.3.1 Free settings\nThe equation of motion is the same as in eq. (10) with\nω2\np=p2−2\nη2. (56)\nFor simplicity we consider the inflationary background to be de Sitter wherea(η) =−1/Hη\nandη <0. At the onset of inflation, when pphys/H=p/ainH=p(−ηin)→ ∞, the positive\nfrequency modes are\nφin\np=1\n(2p)1/2/parenleftbigg\n1−i\npη/parenrightbigg\ne−ipη. (57)\nWhenp|η|= 1, the physical wave length λ=a/pbecomes larger than the Hubble radius. Near\nthat ”horizon-exit”, ω2\npflips sign, the mode stops oscillating and starts to grow like a. At late\ntime with respect to horizon exit, when |pη| ≪1, this implies that the power spectrum becomes\na constant, as we now recall.\nWhen inflation lasts long enough, i.e. when the number of extra e-fold s obeysNextra≫1,\nsee eq. (1), all observable modes φpare in their ground state at the onset of inflation (simply\nbecause this is the only state compatible with inflation [27]). In this cas e, the anti-commutator\nof the free field, see eq. (32), when evaluated at equal time ηis simply given by\nGfree\na(η,p) =|φin\np|2=1\n(2p)/parenleftbigg\n1+1\n(pη)2/parenrightbigg\n. (58)\nThen the power spectrum of the physical (un-rescaled) field given by\nPfree\np(η) =p3\n2π2Gfree\na(η,p)\na2(η)=/parenleftbigH2\np\n2π/parenrightbig2(1+(pη)2), (59)\nbecomes constant after horizon exit. We have added a psubscript to Hbecause in slow\nroll inflation, the relevant value of Hfor thep-mode is that evaluated at horizon exit, i.e.\nHp=H(tp), where tpis given by p/a=H. The above equation shows that Ppacquires\nsome scale dependence only through Hp. Similarly the deviations from this standard behavior\nstemming from some UV modification of the theory will also depend on pthroughHp(and its\nderivatives). For explicit examples, we refer to [30] where the mod ifications of the spectrum\nstem from the fact that pηinis taken large but finite, and also to [31] wherein the UV cutoff is\nendowed with a finite width. This last case bears many similarities with th e dissipative settings\nwe now study.\n173.3.2 Dissipative settings\nIn the presence of dissipation, the expression for Garadically differs from the above.\nAt the level of the Heisenberg operator, when dissipative effects g row with the energy (as\nwe suppose it is the case), the decaying solution of eq. (51) is comple tely erased (unless one\nfine-tunes Nextra, see eq. (1), so as to keep a residual amplitude). That is, in inflation the mode\noperator is entirely given by its driven term, the second term in eq. ( 26). Then the power\nspectrum is also purely driven and given by the second term of eq. (3 3):\nGdriven\na(η,p) =/integraldisplay\ndη1/integraldisplay\ndη2Gr(η,η1,p)Gr(η,η2,p)N(η1,η2,p), (60)\nwhereGris the retarded Green function, solution of eq. (51) with δ(η−η1) as a source, and\nwhere the kernel Nis the anti-commutator of gnlµ∂µΨp, the source of eq. (51).\nEq. (60) tells us that only the state of the environment matters. I n other words, because of\nthe strong dissipation at early times, the power spectrum is indepen dent of the initial state of\nφ, what ever it was. In spite of this, when eq. (55) is satisfied, and wh en the environment is in\nits ground state, the predictions are unchanged, i.e. the power sp ectrum obtained from Gdriven\na\ncoincides with that obtained with Gfree\na. To show this let us study GrandN.\n3.3.3 Dissipative Green functions and noise kernel\nThe retarded Green function is of the form\nGr(η,η0,p) =θ(η−η0)e−Rη\nη0dη′γ(η′)×2Im/parenleftBig\n˜φ∗\np(η)˜φp(η0)/parenrightBig\n, (61)\nwhere (in the under-damped regime) the modes ˜φpare unit Wronskian positive frequency\nsolutions of eq. (56) with a frequency square given by\n(ωeff\np)2=ω2\np−γ2\nn=p2(1−2\np2η2−γ2\nn\np2). (62)\nIn this, we obtain a time-dependent version of the stationary case , see eq. (87). With the\nsecond expression, we verify that when eq. (55) is satisfied, the t wo corrections terms are never\nsimultaneously relevant, which guarantees, as we shall discuss belo w, that dissipative effects\ncan be studied in the quasi-stationary approximation. Notice also th at the frequency shift ∂ηγ\npresent is eq. (54) drops out from ωeff.\nEq. (61) implies that\nGr(η,η0,p)→0 when η0→ηΛ\np, (63)\nwhereηΛ\npis the ”Λ LV-exit” time of the p-mode defined by p/a(ηΛ\np) = Λ LV, orγ/p= 1/2, see\neq. (53) and the Figure.\n18Ln a\nLn d\nFigure caption. We have represented in a log-log plot and by a dashed line the e volution of\ndH=RH/a, the comoving Hubble radius, as a function of a, both during inflation (dH∝1/a)\nand during the radiation era (dH∝a). We have represented by a thick line the trajectory\nof the cutoff length dΛ= 1/aΛLVof a fixed proper scale which obeys 1/ΛLV≪RHduring\ninflation. The dotted line corresponds to an intermediate fix ed proper length λwhich obeys\n1/ΛLV≪λ≪Rinfl.\nH. The vertical line represents a fixed comoving scale dp= 1/p. Below\nthe cutoff length, all modes are over-damped. When a mode exit s the cutoff length, it becomes\nunder-damped and starts propagating. When it crosses the in termediate length λ, it behaves\nas a free mode, and it gets amplified only when exiting the Hubb le radius. Adiabaticity, which\nis guaranteed by 1/ΛLV≪RH, guarantees in turn that, near λ, modes are all born in the\nBunch-Davies vacuum when the environment Ψis in its ground state.\nEq. (63) guarantees that no quantum coherence is left between la te timeη, e.g. horizon-\nexit, and what happened for times earlier than Λ LV-exit. The physical implication of this is\nthat, what ever transitions happened, what ever was the quantu m state, no record is kept at\nlate time.4Having understood the structure of the retarded Green functio n, let us now briefly\nconsider the commutator Gc(η,η0). It is given by\nGc(η,η0,p) =−i/parenleftBig\nGr(η,η0,p)−Gr(η0,η,p)/parenrightBig\n,\n=e−|Rη\nη0dη′γ(η′)|×2iIm/parenleftBig\n˜φp(η)˜φ∗\np(η0)/parenrightBig\n. (64)\nIt possesses two noteworthy properties. First, because of the absolute value in the damping\nexponential term, it obeys two different local differential equation s depending of the sign of\nη−η0. Whenη > η0,Gcis an homogeneous solution of eq. (51), whereas when η < η0, it obeys\nthe equation wherein the sign of the dissipative term has been chang ed. These two differential\n4This behaviorhasbeen described, drawn, andsometimes named anti-dissipation , see[5,11,7,33, 34, 35, 14].\nIn fact when tracing backwards in time vacuum configurations from the freely propagating regime down to the\ndissipative one, dissipation arises towards the past. When viewed as a function of η0,Gr(η,η0) indeed increases\nto the future, thereby showing anti-dissipation. From eq. (61) we understand the origin of this quantum\nproperty (without classical counterpart). As a function of η0, first,Grobeys eq. (51) with the ”wrong” sign for\nthe dissipative term, and second, the boundary condition is a final o ne and not an initial one, thereby explaining\nwhy when η0→η, the value of the current is ”adjusted” to become exactly one.\n19equations can be groupedinto a single non-local equation. Second, because of this ”quasi”-local\nproperties, Gcis independent of ηin, the moment when φwas put in contact with Ψ.\nThese properties of Gcfollow from the fact that its source term, the Dkernel of eq. (29),\nis ”ultra” local for the Ψfield of eq. (49):\nD(η1,η2)≡[gn∂η1Ψ(η1), gn∂η2Ψ(η2)] =gn(η1)i∂η2δ(η1−η2)\nΛLVgn(η2). (65)\nThis should be contrasted with the kernel N, the anti-commutator, which is non local. Indeed,\nin the vacuum, it is given\nN(η1,η2) =gn(η1)−a(η1)a(η2)\nπΛLV(t1−t2)2gn(η2), (66)\nwhere 1/(t1−t2)2should be understood as the derivative of the principal part of 1 /(t1−t2).\nOne gets a simple expression in terms of the proper time tbecause the proper frequency of Ψ k\nis constant in our model (49). We also remind the reader that it is only in the high temperature\nlimit that Nwould become proportional to δ(t1−t2).\nInthispaper weshallcomputethepower spectrumwhentheΨ kareallintheirgroundstate.\nThe reason for this choice is as follows: As pointed out in the footnot e 2, the Ψ kfields should\nbe conceived as parametrizing some degrees of freedom in a fundam ental theory obeying the\nEquivalence Principle. In these theories, when inflation lasts long eno ugh, all relevant degrees\nwill be in their ground state, as is the case when dealing with free mode s, see eq. (58). We see\nno reason why dissipation could possibly invalidate this conclusion.\nWhen the Ψ kare all in their ground state, the kernel Nis non-local. Therefore the anti-\ncommutator Gaobeys a non-local equation: eq. (51) with Nas the source. The physical\nmeaning is that the environment is driving φpin an non-instantaneous way (this is unavoidable\ninthevacuumsince onlyonesignofthefrequency ispresent). Thec onsequence isthatthevalue\nof power spectrum depends (to a certain extend) on the history o f the combined evolution of\nφ+Ψ. The mathematical implication is that anexact calculationof Gaiseffectively impossible.\nNevertheless when H/ΛLV≪1, these non local effects give only but sub-leading (non-\nadiabatic) corrections, because the evolution of φ+Ψconsists in a parametric (adiabatic)\nsuccession of stationary states ordered by the scale factor a.\n3.3.4 Scale separation and adiabatic evolution\nGiven the importance of this result, we shall spend some time to expla in its root and its\nvalidity. We proceed in two steps. We first show that adiabaticity is su fficient for obtaining\nthe standard power spectrum, and then show that scale separat ion, eq. (55), is sufficient to\nguarantee adiabaticity, thereby generalysing the results of [8].\nAdiabaticity means that the evolution proceeds slow enough for indu cing no (non-adiabatic)\ntransition (in molecular physics, they are called Landau-Zener tran sitions, in the present cos-\nmological context they correspond to pair creation of φquanta). When this is the case, the\nobservables take, at any time, the value they have in the correspo nding stationary situation.\n20As of retarded Green function and the commutator, this principle d oes not bring any new\ninformation because in the WKB approximation, they are local funct ions since these 2pt func-\ntions obey local equations. For the anti-commutator, adiabaticity guarantees that when ηand\nη′are close (in the sense that 1 −a(η)/a(η′)≪1), its value is well approximated by\nGdriven\na(η,η′,p)≃Gstatio\na(η−η′;ωp(a),gp(a)) =/integraldisplaydω\n2πeiω(η−η′)Ga(ω;ωp(a),gp(a)),(67)\nwhereGa(ω;ωp(a),gp(a)) is the Fourier component calculated with the values of the freque ncy\nωp(a) and the coupling gp(a) both evaluated with a=a(η). In Appendix A, these Fourier\ncomponents have been algebraically solved for all frequencies and a ll couplings, see eq. (81).\nMoreover, since by hypothesis, we are in the vacuum, eq. (79) also applies. In other words, the\nvalue ofGafollows from that of the commutator Gc. This is sufficient to guarantee that when\nthe mode φpbecomes free, i.e. much after Λ LV-exit but before horizon-exit ( H≪p/a≪ΛLV),\neq. (84) applies. Thus, irrespectively of what was the coupling with t he environment,\nGdriven\na(η,η,p)→Gfree\na(η,η,p) =1\n2ωp(a). (68)\nWith this equation we reach our first conclusion: in the adiabatic appr oximation and when\nthe environment is in its ground state, the modes φpare born in the usual in-vacuum (Bunch-\nDavies) as they become, one after the other, freely propagating after Λ LV-exit, see the Figure.5\nIt now behooves to usto show that scale separation guarantees t hat eq. (67) provides a valid\napproximationfor Gdriven\nabeforehorizon-exit, i.e. whenthesecondtermintheparenthesis in eq.\n(62)canstillbeneglected. Tothisend, weprovideanupperboundf ortheprobabilityamplitude\nof getting a non-adiabatic transition in the under-damped regime, i.e .H≪p/a≤ΛLV. This\namplitude is governed by the relative frequency change\n∂ηωeff\np\n(ωeff\np)2=−γ2\nn∂ηlnγn\n(p2−γ2n)3/2= (2n+1)Hγ2\nphys\n(p2\nphys−γ2\nphys)3/2, (69)\nwherepphys=p/aandγphys=γ/aare the physical (proper) momentum and decay rate.\nTherefore, going backwards in time from the free regime up to p2\nphys≥4γ2\nphys(i.e.pphys≤ΛLV),\nthe non-adiabatic parameter steadily grows but stays bounded by\n∂ηωeff\np\n(ωeff\np)2<3(2n+1)σ/parenleftBigpphys\nΛLV/parenrightBig4n+1\n<3(2n+1)σ≪1. (70)\nThis guarantees that the amplitude for the system to jump out of t he ground state is bounded\nbyσ(up to an overall factor which plays no role). There is no need to stu dy the stability\nof the ground state in the transitory regime from under-damped t o the over-damped modes\nbecause what ever transitions happened their residual impact at la te time would be suppressed\nby a factor e−Rγdt≃exp(−1/σ(2n+1))≪1. This completes the proof that scale separation\nguarantees adiabaticity, in agreement with the conclusion reached in [9, 10].\n5This behavior is in accord with the procedure of [29, 30, 31] wherein t he initial state is imposed on modes\nwhen their proper momentum reaches a given value. This radically diffe rs with the “initial slice” approach [36]\nwherein the state is imposed at the same time on all modes, irrespect ively of the value of their momentum.\nFrom what we see, the Equivalence Principle allows only the first to be d ynamically realized.\n213.3.5 Perspectives: Beyond the adiabatic approximation an d scale crossing\nIt is first interesting to point out that dissipative models are more ”r obust” than dispersive\nmodels in that the ”driven” power spectrum determined by eq. (60) is well defined even\nwhenH >ΛLV, contrary to the fact that most dispersive models make sense only when scale\nseparation ( H≪ΛLV) is realized. Therefore, with dissipative models, one can study scale\ncrossing, i.e. the crossing of Hpthrough Λ LVfrom above to below during slow roll inflation.\nThe power spectrum has been studied in this regime when using a part icular Brane World\nscenario in [10]. We agree with most of the conclusions but not with tha t reached after Eq.\n(35) according to which “towards the end of inflation, perturbatio ns on the brane are the sum\nof two independent fields”. We do not agree because in those settin gs as well the field operator\nis purely driven. Hence we do not see how to decompose it as a superp osition of two commuting\noperators. Moreover, with the approximation used in [10], it seems that the ETC, eq. (11), is\nviolated by a amount proportional to the reported change of the p ower spectrum.\nIn any case, the calculation of the modifications of the power spect rum introduced by\ndissipation is difficult because the non-local properties of eq. (60) c annot be neglected. We are\nnot aware of any analytical treatment, and we are currently stud ying the modifications using\nnumerical techniques [38].\nAdded note. Since the present paper was submitted, we have obtained several results [38].\nIn particular, the leading modification of the power spectrum with re spect to the standard\nresult in the regime σ=H/Λ≪1, i.e. the signature of UV dissipation, scales with a power of\nσwhich is equal to that of P/Λ inγ/p, see eq. (53). In this respect, dissipative models behave\nlike dispersive models where the leading modification scales with a power ofσwhich is that\nof the first non-linear term in the dispersion relation, i.e. 2 nusing the parameterization of eq.\n(2), see [39]. In the opposite regime, when H >Λ, no universal behavior is obtained.\n4 Conclusions\nIn this paper we have obtained the following results.\nFirst, we have provided a class of unitary models defined in Minkowski space which are\ncharacterized by the power of p/ΛLVwhich weighs the growth of dissipation in the preferred\nframe, see eq. (85). Unitarity is achieved by introducing a dense se t of additional fields\nΨwhich induce dissipation through interactions with φ. In Appendices A and B we have\ngiven a thorough analysis of the Green functions of these Gaussian models which cover the\nphenomenology of UV dissipative effects in Minkowski space at the lev el of 2pt functions.\nSecond, amongst the variousactions delivering the same stationar yphenomenology, we have\nselected one which gives rise to a local differential equation for the r etarded Green function\nofφ, see eqs. (36, 37). Using the Equivalence Principle, we have extend ed this class of\nmodels to arbitrary curved backgrounds in subsection 2.7, thereb y allowing to confront the\ntrans-Planckian question of inflationary cosmology and black hole ph ysics.\nThird, we have applied our dissipative models to inflationary cosmology . At early times,\nhence at very high energy, the dissipative effects are so strong th at all information about the\ninitial state of the φis erased, see eq. (63). Nevertheless, when the UV scale Λ LVis much\n22larger than the Hubble parameter, we have demonstrated that th e standard expression of the\npower spectrum is found when the environment is in its ground state . The reason for this is\nthe following: even though the field oscillator φpis purely driven by Ψp, i.e. it is given by the\nsecond term eq. (26), as its proper momentum p/aredshifts under Λ LV, the composite operator\nbehaves as if it were a free operator, see eqs. (31, 84), thereby guaranteeing eq. (68).\nLet us also note that our models can be used for phenomenological p urposes in the sense\nof [1], and that we are planning to apply them to study Hawking radiatio n in the presence of\ndissipation. We are presently completing the calculation of the power spectrum beyond the\nadiabatic approximation so as to determine the signatures of dissipa tion [38] and to compare\nthem to those of dispersion [39]. Finally it would be interesting to compu te the VEV of the\nstress-tensor of the Ψfields in non-trivial backgrounds. This would allow to take into accoun t\nthe back-reaction on the cosmological metric engendered by the fi eldsΨandlµ[40, 41, 42].\nAcknowledgments.\nI am grateful to Dani Arteaga and Enric Verdaguer for common wo rk allowing me to deepen\nmy understanding of dissipative effects. I am also grateful to Julian Adamek, David Campo,\nTed Jacobson, Jean Macher, Jens Niemeyer, and Valeri Rubakov f or interesting discussions. I\nwish to thank the organizers of the workshops on ”Micro and Macro structure of spacetime”\nheld inPeyresq in June 2005, 2006, and 2007where this work has bee n presented and discussed.\nThis work has been supported by the Agence Nationale de la Recherc he (projet 05-1-41810).\n5 Appendix A :\nStationary states and vacuum 2pt functions\nIn this Appendix, we recall the (well-known) relationships between Gc,Gawhich always hold\nin stationary states. In these states, Gc, Gaand the kernels D, Nare related to each other in\na universal way, generally referred as a Fluctuation-Dissipation re lation. We explain its origin\nand its physical implications in the present context. We start with th e most basic object: the\nretarded Green function Gr.\n5.1 The retarded Green function\nThe Fourier transform of eq. (25) gives\n(−ω2+ω2\np)φp(ω) = Σigi(p)Ψo\ni(p,ω)+Σig2\ni(p)Ro\ni(ω;p)φp(ω), (71)\nwhere\nRo\ni(ω;p) =/parenleftbig\n−(ω+iǫ)2+Ω2\ni(p))/parenrightbig−1, (72)\nis the Fourier transform (defined as in eq. (17)) of the retarded G reen function of Ψ i. As usual,\nits retarded character is enforced by the imaginary prescription o f the two poles to lay in the\nlower half plane ( ǫ >0). The solution of the above equation is\nφp(ω) =φd\np(ω)+Gr(ω,p)Σigi(p)Ψo\ni(p,ω), (73)\n23where the Fouriertransform ofthe retardedGreen functionof φ, thesolution ofeq. (27), always\ntakes the form\nGr(ω,p) =/parenleftbig\n−(ω+iǫ)2+p2+Σr(ω,p)/parenrightbig−1. (74)\nAll effects of the coupling to the Ψ i’s are thus encoded in the (retarded) self-energy Σ r(ω,p).\nFor Gaussian theories, it is algebraically given by\nΣr(ω,p) =−Σig2\ni(p)Ro\ni(ω,p). (75)\nThe dissipative effects are governed by the imaginary part of Σ r(ω,p). In the present case, one\nhas\n2ImΣr(ω,p) =−Σig2\ni(p)Go\nc,i(ω,p) =−D(ω,p). (76)\nTo get the first equality we have used the fact that in stationary st ates the retarded Green\nfunction and the commutator are related by 2Im Gr(ω) =Gc(ω) for all degrees of freedom, free\nor interacting. In the second equality, we have introduced D(ω), the Fourier transform of the\nkernel of eq. (30).\nSeveral observations should be made here. First, from eq. (72), we obtain that D(ω) is\nproportional to Σ ig2\niδ(ω−Ωi). Therefore there is no dissipation for lower frequencies than the\nlowest value of Ω i. This simply follows from energy conservation. Second, to obtain ”t rue”\ndissipation, D(ω,p) should be a continuous function and not a sum of delta. This can only\nhappen when the Ψ iform a dense ensemble. In Section 2.5, we shall thus replace the disc rete\nsum oniby an integral over a continuous variable, k. We shall not consider the discrete cases\neven though these could display interesting properties. Third, fro m a phenomenological point\nof view, only D(ω,p) matters. Hence we cannot disentangle the spectrum of the envir onment,\nwhich is given by Ro\ni(ω,p), from the coupling strength g2\ni(p). This is a good thing, because\nwhen working intime-dependent settings, we shall exploit this equiva lence to chose the simplest\nmodel of Ψ i’s which gives the kernel D(ω,p) we want.\nIt is also worth noticing that the dispersive (real) effects are not dir ectly related to D(or\nN). These are governed by the even part of Σ r(ω,p) which is given by\nReΣr(ω,p) =/integraldisplaydω′\n2πD(ω′,p)\nω−ω′, (77)\nwhere the integral should beunderstood asa principal value. This in tegral relationship explains\nwhy one often founds that dispersive effects appear before dissip ative effects (for increasing ω).\nWe also learn that the dispersive models studied in the literature violat e the above relation\nsince they assume ReΣ r∝ne}ationslash= 0 and ImΣ r≡0. Therefore these models are incoherent and cannot\nresult from dynamical processes.\n5.2 Fluctuation-Dissipation relations and vacuum self-en ergy\nIn this subsection, we derive the relationships between Gc,Gaand Σ Fwhich exist in the true\n(interacting) ground state.\n24In interacting theories, the only stationary states are thermal s tates, see e.g. [37]. In these\nstates, the Fourier transform of DandNare related by\nN(ω) =D(ω) coth(βω/2),\n=D(ω)sign(ω)[2n(|ω|)+1]. (78)\nIn the second line, n(ω) is the Planck distribution. It gives the mean occupation number of\nΨo\niquanta as a function of the frequency (measured in the rest fram e of the bath). The above\nrelation directly follows from the fact that the individual commutato rs and anti-commutators\nof the free fields Ψo\niobey this relation, as any free oscillator would do. It implies that the\nFourier transform of GcandGaare also related by\nGa(ω) =Gc(ω)sign(ω)[2n(ω)+1]. (79)\nIt should be stressed that this equation is exact, i.e. non-perturb ative, and valid for all theories,\nGaussian or not. (It indeed directly follows from the cyclic propertie s of the trace defining\nGβ(t,t′) =Tr[e−βHTφ(t)φ(t′)]).\nFor Gaussian models, there exists an alternative proof of eq. (79) . It suffices to note that\nin steady states the decaying terms of eqs. (29) and (33) play no r ole, and that the Fourier\ntransform of the driven terms are respectively given by\nGc(ω) =|Gr(ω)|2D(ω), (80)\nGa(ω) =|Gr(ω)|2N(ω), (81)\nsince the Fourier transform of the retarded Green function obey sGr(ω) =G∗\nr(−ω), see eq.\n(73). Irrespectively of the complexity of Gr, i.e. irrespectively of the functions gi(p), Ωi(p) and\nthe set of the Ψ ifields,GcandGaare thus related to each other by the FD relation (79).\nThese universal relations will be relevant for inflationary models whe rein only the ground\nstate contributes. In particular, they imply that in the true vacuum , i.e. when n(ω) = 0,Gc\nandGaare exactly related by Ga(ω) =Gc(ω)sign(ω). Hence the Wightman function\nGW=1\n2(Gc+Ga) =Gcθ(ω), (82)\nis determined by the commutator and contains only positive frequen cy, as in the free vacuum.\nEquations(80,81)alsoallowtocomputethevacuumself-energyof theFeynmanGreenfunction.\nFor Gaussian models it is given by\n2ImΣF(ω,p) =−D(ω,p)sign(ω). (83)\nWiththelastequalitywerecoverthefactthatinthevacuum, itissuffi cienttoconsiderFeynman\nGreen functions. In non-vacuum states, and in non-stationary s ituations, this is no longer true,\nthereby justifying the use of the in−inmachinery [20] (the Schwinger-Keldish formalism).\nBefore specializing to a specific class of models giving rise to dissipation at high frequency,\nwe make a pause by asking the following important question: What sho uld be known about\n25the Ψifields to get eqs. (80, 81, 82) ? We have proven that it is sufficient fo r the Ψ i’s to be\ncanonical fields, but is it necessary ?\nThe answer is two fold. On one hand, the Ψ icannot be stochastically fluctuating quantities\n(i.e. commuting variables) because this would lead to a violation of eq. ( 78) that would imply\nthe violation of eq. (79) and the ETC eq. (31).6They cannot be either a combination of\nquantum and stochastic quantities because this would still lead to a v iolation of the ETC.\nHence they must be built only from quantum (canonical) degrees of f reedom.\nOntheotherhand, theΨ i’scanbecompositeoperators, i.e. polynomialsofsome(unknown)\ncanonical fields. Indeed, their commutators would still be all relate d to their anti-commutators\nby the FD relation eq. (78), and this even though they depend non- linearly on n(ω) in non-\nvacuum states. The difference with Gaussian models is that these no n-linear operators posses\nnon-vanishing higher order correlation functions. Hence, the self -energies Σ r,ΣFwill be series\nin powers of g2\ni, and not just a single quadratic term as in eq. (75, 83). Neverthele ss these\nhigher loops corrections preserve the validity of eq. (82) in the gro und state, as well as that\nof eqs. (80, 81) in any thermal state, when properly understood , i.e. with Dnow defined by\n-2ImΣ r(as the effective dissipation kernel), and Nrelated to it by the FD relation.\nIn brief, we have reached/recalled the following results. Firstly, th e q-number combination\nΨ= ΣigiΨi, the fluctuating source term of φ, must obey the FD relation (78). This can\neither be postulated, or better, be viewed as resulting from the fa ct thatΨis entirely made\nout of quantum degrees of freedom. Secondly, to lowest order te rm ing, the self-energy can\nbe obtained by treating Ψas a Gaussian variable, what ever its composition may be. Thirdly,\nwhen dealing with non-Gaussian theories, once having computed Σ r(ω), the resulting equations\nfor the 2pt functions have the same structure and the same mean ing as in Gaussian theories,\nwithDreplaced by -2ImΣ r. Therefore, the phenomenology of two-point functions is entirely\ncovered by Gaussian settings .\n5.3 The double limit: g2T→ ∞followed by g2→0.\nTo perform a phenomenological analysis of dissipation, we need to un derstand how the theory\nbehaves intransitoryregime fromdissipative tofreepropagation. Similarly, to studyprimordial\nspectra in inflation or Hawking radiation, we also need to understand how free motion emerges\nas the proper frequency get red-shifted. It is therefore releva nt to study the behavior of the\ntwo-point function in the following double limit.\nOne first takes g2T→ ∞, whereT=t−tin,tinbeing the moment when the interactions\nare turn on, and tthe moment when one studies the field properties. This limit implies that\nthe decaying term in eq. (26) plays no role. Therefore, near time t, the Heisenberg field φ(t) is\na composite operator which only acts in the Hilbert space of Ψ.\nSecondly, one considers the ”free” limit g2→0 of that composite operator. One could\nnaively conclude that GcandGaof eqs. (80, 81) would vanish since both DandNare\n6This constitutes the simplest proof that it is inconsistent to couple q uantum variables to stochastic (or\nclassical) ones. If one does so, the ETC of the dressed quantum va riables will always be dissipated after a time\nof the order of γ−1. One can therefore view the experimental evidences for the ETC o f some degrees of freedom\nas a very strong indication that alldynamical variables in our world are quantum mechanical in nature.\n26proportional to g2. However, this is not the case, because the common prefactor, |Gr|2, is\nsingular in this second limit. In fact, one verifies that it scales in 1 /g2in such a way that, in\nthe (interacting) vacuum, one always recovers\nGW(ω)g2→0=1\n2ωp2πδ(ω−ωp). (84)\nThis is the standard vacuum fluctuations of a free massless mode of momentum p.\nTwo important lessons have been reached. First we learned is that e ven though φactsonly\non theΨ-Hilbert space, when g2→0, it behaves as if it were a free mode possessing its own\nHilbert space, with no reference to Ψ-dynamics. Secondly, the quantum state in this would be\nHilbert space is still exactly that of Ψ. Therefore, in stationary situations, the only ”souvenir”\nkept by the composite operator is the equilibrium distribution n(ω) inherited from its parents.\nLet us now emphasize that the above limit is relevant for non-Gaussia n models as well.\nIndeed, in the limit g2→0, there will always be a value of g2sufficiently small that the\nmodel can be well approximated by a Gaussian model. Therefore the behavior of the 2-point\nfunctions in the transitory regime from dissipation to free propaga tion can be analyzed by\nstudying Gaussian models (at least in the quasi-static limit).\n6 Appendix B:\nDissipative effects above ΛLV. The Phenomenology\nWe now have all the tools to construct models giving rise to dissipation in the vacuum above a\ncritical energy scale Λ LV. In this Appendix we work from a purely phenomenological point of\nview, and provide the class of dissipative models wherein the imaginary part of the self-energy\nis governed by a single term, in analogy with the dispersion relations of eq. (2).\nFromaphenomenologicalpointofview, ifoneconsidersonlystationa rysituations(i.e. static\nmetrics and stationary states), one can simply choose the functio nD(ω,p) entering eq. (76)\nand eq. (80) as one wishes . There is indeed no restriction on D(ω,p) besides its constitutive\nproperties, namely being odd in ωand giving rise to poles in Grall localized in the lower half ω\nplane. In this we have reached our first aim, namely identify how to ge neralize the free settings\nso as to incorporate some arbitrary dissipative effects.\nWe can thus consider the dispersive models which correspond to tho se defined by eq. (2).\nThey are characterized by a single term giving rise to dissipation abov e ΛLV. In the vacuum,\nthey are fully specified by the imaginary part of the (retarded) self -energy\n−ImΣ(n)\nr(ω,p) =ω\nΛLVp2/parenleftbiggp\nΛLV/parenrightbigg2n\n= 2ωγn. (85)\nIn these models, the decay rate (inverse life time) on the mass shell is\nγn=p\n2/parenleftbiggp\nΛLV/parenrightbigg2n+1\n. (86)\n27To verify it, assuming that ReΣ r= 0, the two poles of Gr(ω) in eq. (74) are located in\nω±(p) =±/radicalBig\nω2p−γ2−iγ. (87)\nFrom this, by inverse Fourier transform Gr(ω), one obtains that the decay rate is indeed γin\nthe under-damped regime, for ω2\np> γ2. In the overdamped regime, for γ2> ω2\np, the decay rates\nof the two independent solutions of G−1\nrφd= 0 are Γ ±=γ±/radicalbigγ2−ω2\np.\nOne thus have the following behavior as pgrows. For p≪ΛLV,ω±≃p, and one has\na free propagation which is slightly damped with a life time in the units of t he frequency\ngiven by (Λ LV/ω)n+1≫1. In the opposite regime of high momenta p≫ΛLV, deep in the\noverdamped regime, the two roots ω±are real and the notion of propagation (in space-time)\nis absent. In anticipation to what will occur in inflation (or in black hole p hysics), we invite\nthe reader to study the migration of the poles of Grwhen extrapolating backwards in time a\nmode, i.e. as pincreases. (Remember that the physical momentum of a mode in cos mology is\npphys(t)∝p/a(t) wherepis the norm of the conserved comoving wave vector (near a black ho le\none finds p(r)∝ω/xwherex=r−rSis the proper distance from the horizon measured in a\nfreely falling frame, and ωthe conserved Killing frequency.))\nOne could of course generalize the above class by considering in eq. ( 85) polynomials in\npdimensionalized by different UV scales. However, unless fine tuning, t he phenomenology of\nthe transition from the IR dissipation-free sector to the dissipativ e sector will be dominated a\nsingle term. One should also consider the possibility that ImΣ strictly v anishes below a certain\nfrequency Ω 1, as this would be the case when ever the spectrum of the Ψ fields pos sesses this\ngap, see the remarks after eq. (76).\nHaving the phenomenology of dissipative and unitary models under co ntrol (with dispersive\nand dissipative effects related by Kramers relation, eq. 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Anglin, “Thermal equilibrium from the Hu-Paz-Zhang maste r equation,”\narXiv:hep-th/9210034.\n[38] J. Adamek, D. Campo, J. Niemeyer, and R. Parentani, “Inflatio nary Spectra from Lorentz\nViolating Dissipative Models,” to appear .\n[39] J. Macher and R. Parentani, “Signatures of trans-Planckian d ispersion in inflationary\nspectra,” arXiv:0804.1920 [hep-th].\n[40] T. Jacobson and D. Mattingly, Phys. Rev. D 63(2001) 041502.\n[41] M. Lemoine, M. Lubo, J. Martin and J. P. Uzan, Phys. Rev. D 65(2002) 023510.\n[42] S. Shankaranarayanan and M. Lubo, Phys. Rev. D 72(2005) 123513.\n30"    },    {        "title": "2203.01632v1.Stability_results_of_locally_coupled_wave_equations_with_local_Kelvin_Voigt_damping__Cases_when_the_supports_of_damping_and_coupling_coefficients_are_disjoint.pdf",        "content": "arXiv:2203.01632v1  [math.AP]  3 Mar 2022STABILITY RESULTS OF LOCALLY COUPLED WAVE EQUATIONS WITH LO CAL\nKELVIN-VOIGT DAMPING: CASES WHEN THE SUPPORTS OF DAMPING AN D\nCOUPLING COEFFICIENTS ARE DISJOINT\nMOHAMMAD AKIL1, HAIDAR BADAWI1, AND SERGE NICAISE1\nAbstract. In this paper, we study the direct/indirect stability of loc ally coupled wave equations with local\nKelvin-Voigt dampings/damping and by assuming that the sup ports of the dampings and the coupling coeffi-\ncients are disjoint. First, we prove the well-posedness, st rong stability, and polynomial stability for some one\ndimensional coupled systems. Moreover, under some geometr ic control condition, we prove the well-posedness\nand strong stability in the multi-dimensional case.\nContents\n1. Introduction 1\n2. Direct and Indirect Stability in the one dimensional case 4\n2.1. Well-Posedness 4\n2.2. Strong Stability 5\n2.3. Polynomial Stability 9\n2.3.1. Proof of Theorem 2.6 9\n2.3.2. Proof of Theorem 2.7 13\n3. Indirect Stability in the multi-dimensional case 16\n3.1. Well-posedness 16\n3.2. Strong Stability 17\nAppendix A. Some notions and stability theorems 20\nReferences 21\n1.Introduction\nThe direct and indirect stability of locally coupled wave equations with lo cal damping arouses many interests in\nrecent years. The study of coupled systems is also motivated by se veralphysicalconsiderationslike Timoshenko\nand Bresse systems (see for instance [ 10,6,3,2,1,15,14]). The exponential or polynomial stability of the wave\nequation with a local Kelvin-Voigt damping is considered in [ 20,23,13], for instance. On the other hand, the\ndirect and indirect stability of locally and coupled wave equations with lo cal viscous dampings are analyzed in\n[8,18,16]. In this paper, we are interested in locally coupled wave equations wit h local Kelvin-Voigt dampings.\nBefore stating our main contributions, let us mention similar results f or such systems. In 2019, Hayek et al.in\n[17], studied the stabilization of a multi-dimensional system of weakly cou pled wave equations with one or two\nlocally Kelvin-Voigt damping and non-smooth coefficient at the interfa ce. They established different stability\n1Universit ´e Polytechnique Hauts-de-France, CERAMATHS/DEMAV, Valencien nes, France\nE-mail address :Mohammad.Akil@uphf.fr, Haidar.Badawi@uphf.fr, Serge.N icaise@uphf.fr .\nKey words and phrases. Coupled wave equations, Kelvin-Voigt damping, strong stab ility, polynomial stability .\n1results. In 2021, Akil et al.in [24], studied the stability of an elastic/viscoelastic transmission problem of\nlocally coupled waves with non-smooth coefficients, by considering:\n\n\nutt−/parenleftbig\naux+b0χ(α1,α3)utx/parenrightbig\nx+c0χ(α2,α4)yt= 0,in (0,L)×(0,∞),\nytt−yxx−c0χ(α2,α4)ut= 0, in (0,L)×(0,∞),\nu(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞),\nwherea,b0,L >0,c0/\\e}atio\\slash= 0, and 0 < α1< α2< α3< α4< L. They established a polynomial energy decay\nrate of type t−1. In the same year, Akil et al.in [5], studied the stability of a singular local interaction\nelastic/viscoelastic coupled wave equations with time delay, by consid ering:\n\n\nutt−/bracketleftbig\naux+χ(0,β)(κ1utx+κ2utx(t−τ))/bracketrightbig\nx+c0χ(α,γ)yt= 0,in (0,L)×(0,∞),\nytt−yxx−c0χ(α,γ)ut= 0, in (0,L)×(0,∞),\nu(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞),\nwherea,κ1,L >0,κ2,c0/\\e}atio\\slash= 0, and 0 < α < β < γ < L . They proved that the energy of their system decays\npolynomially in t−1. In 2021, Akil et al.in [4], studied the stability of coupled wave models with locally\nmemory in a past history framework via non-smooth coefficients on t he interface, by considering:\n\n\nutt−/parenleftbigg\naux+b0χ(0,β)/integraldisplay∞\n0g(s)ux(t−s)ds/parenrightbigg\nx+c0χ(α,γ)yt= 0,in (0,L)×(0,∞),\nytt−yxx−c0χ(α,γ)ut= 0, in (0,L)×(0,∞),\nu(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞),\nwherea,b0,L >0,c0/\\e}atio\\slash= 0, 0< α < β < γ < L , andg: [0,∞)/ma√sto−→(0,∞) is the convolution kernel function.\nThey established an exponential energy decay rate if the two wave s have the same speed of propagation. In\ncase of different speed of propagation, they proved that the ene rgy of their system decays polynomially with\nratet−1. In the same year, Akil et al.in [7], studied the stability of a multi-dimensional elastic/viscoelastic\ntransmission problem with Kelvin-Voigt damping and non-smooth coeffi cient at the interface, they established\nsome polynomial stability results under some geometric control con dition. In those previous literature, the\nauthors deal with the locally coupled wave equations with local dampin g and by assuming that there is an\nintersection between the damping and coupling regions. The aim of th is paper is to study the direct/indirect\nstability of locally coupled wave equations with Kelvin-Voigt dampings/d amping localized via non-smooth\ncoefficients/coefficient and by assuming that the supports of the d ampings and coupling coefficients aredisjoint.\nIn the first part of this paper, we consider the following one dimensio nal coupled system:\nutt−(aux+butx)x+cyt= 0,(x,t)∈(0,L)×(0,∞), (1.1)\nytt−(yx+dytx)x−cut= 0,(x,t)∈(0,L)×(0,∞), (1.2)\nwith fully Dirichlet boundary conditions,\n(1.3) u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, t∈(0,∞),\nand the following initial conditions\n(1.4) u(·,0) =u0(·), ut(·,0) =u1(·), y(·,0) =y0(·) and yt(·,0) =y1(·), x∈(0,L).\nIn this part, for all b0,d0>0 andc0/\\e}atio\\slash= 0, we treat the following three cases:\nCase 1 (See Figure 1):\n(C1)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) =d0χ(d1,d2)(x),\nwhere 0< b1< b2< c1< c2< d1< d2< L.\nCase 2 (See Figure 2):\n(C2)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) =d0χ(d1,d2)(x),\nwhere 0< b1< b2< d1< d2< c1< c2< L.\n2Case 3 (See Figure 3):\n(C3)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) = 0,\nwhere 0< b1< b2< c1< c2< L.\nWhile in the second part, we consider the following multi-dimensional co upled system:\nb1b2c1c2d1d2Lb0c0\n0d0\nFigure 1. Geometric description of the functions b,canddin Case 1.\nb1b2 0 d1d2c1c2Lb0d0c0\nFigure 2. Geometric description of the functions b,canddin Case 2.\n0b1b2c1c2Lb0c0\nFigure 3. Geometric description of the functions bandcin Case 3.\nutt−div(∇u+but)+cyt= 0 in Ω ×(0,∞), (1.5)\nytt−∆y−cyt= 0 in Ω ×(0,∞), (1.6)\nwith full Dirichlet boundary condition\n(1.7) u=y= 0 on Γ ×(0,∞),\nand the following initial condition\n(1.8) u(·,0) =u0(·), ut(·,0) =u1(·), y(·,0) =y0(·) andyt(·,0) =y1(·) in Ω,\n3where Ω ⊂Rd,d≥2 is an open and bounded set with boundary Γ of class C2. Here,b,c∈L∞(Ω) are such\nthatb: Ω→R+is the viscoelastic damping coefficient, c: Ω→Ris the coupling function and\n(1.9) b(x)≥b0>0 inωb⊂Ω, c(x)≥c0/\\e}atio\\slash= 0 inωc⊂Ω and c(x) = 0 on Ω \\ωc\nand\n(1.10) meas( ωc∩Γ)>0 and ωb∩ωc=∅.\nIn the first part of this paper, we study the direct and indirect sta bility of system ( 1.1)-(1.4) by consider-\ning the three cases ( C1), (C2), and (C3). In Subsection 2.1, we prove the well-posedness of our system by using\na semigroup approach. In Subsection 2.2, by using a general criteria of Arendt-Batty, we prove the stron g\nstability of our system in the absence of the compactness of the re solvent. Finally, in Subsection 2.3, by using\na frequency domain approach combined with a specific multiplier metho d, we prove that our system decay\npolynomially in t−4or int−1.\nIn the second part of this paper, we study the indirect stability of s ystem (1.5)-(1.8). In Subsection 3.1,\nwe prove the well-posedness of our system by using a semigroup app roach. Finally, in Subsection 3.2, under\nsome geometric control condition, we prove the strong stability of this system.\n2.Direct and Indirect Stability in the one dimensional case\nIn this section, we study the well-posedness, strong stability, and polynomial stability of system ( 1.1)-(1.4).\nThe main result of this section are the following three subsections.\n2.1.Well-Posedness. In this subsection, we will establish the well-posedness of system ( 1.1)-(1.4) by using\nsemigroup approach. The energy of system ( 1.1)-(1.4) is given by\nE(t) =1\n2/integraldisplayL\n0/parenleftbig\n|ut|2+a|ux|2+|yt|2+|yx|2/parenrightbig\ndx.\nLet (u,ut,y,yt) be a regular solution of ( 1.1)-(1.4). Multiplying ( 1.1) and (1.2) byutandytrespectively, then\nusing the boundary conditions ( 1.3), we get\nE′(t) =−/integraldisplayL\n0/parenleftbig\nb|utx|2+d|ytx|2/parenrightbig\ndx.\nThus, if ( C1) or (C2) or (C3) holds, we get E′(t)≤0. Therefore, system ( 1.1)-(1.4) is dissipative in the sense\nthat its energy is non-increasing with respect to time t. Let us define the energy space Hby\nH= (H1\n0(0,L)×L2(0,L))2.\nThe energy space His equipped with the following inner product\n(U,U1)H=/integraldisplayL\n0vv1dx+a/integraldisplayL\n0ux(u1)xdx+/integraldisplayL\n0zz1dx+/integraldisplayL\n0yx(y1)xdx,\nfor allU= (u,v,y,z)⊤andU1= (u1,v1,y1,z1)⊤inH. We define the unbounded linear operator A:D(A)⊂\nH −→ H by\nD(A) =/braceleftbig\nU= (u,v,y,z)⊤∈ H;v,z∈H1\n0(0,L),(aux+bvx)x∈L2(0,L),(yx+dzx)x∈L2(0,L)/bracerightbig\nand\nA(u,v,y,z)⊤= (v,(aux+bvx)x−cz,z,(yx+dzx)x+cv)⊤,∀U= (u,v,y,z)⊤∈D(A).\nNow, ifU= (u,ut,y,yt)⊤is the state of system ( 1.1)-(1.4), then it is transformed into the following first order\nevolution equation\n(2.1) Ut=AU, U(0) =U0,\nwhereU0= (u0,u1,y0,y1)⊤∈ H.\n4Proposition 2.1. If (C1) or (C2) or (C3) holds. Then, the unbounded linear operator Ais m-dissipative in\nthe Hilbert space H.\nProof.For allU= (u,v,y,z)⊤∈D(A), we have\nℜ/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−/integraldisplayL\n0b|vx|2dx−/integraldisplayL\n0d|zx|2dx≤0,\nwhich implies that Ais dissipative. Now, similiar to Proposition 2.1 in [ 24] (see also [ 5] and [4]), we can prove\nthat there exists a unique solution U= (u,v,y,z)⊤∈D(A) of\n−AU=F,∀F= (f1,f2,f3,f4)⊤∈ H.\nThen 0∈ρ(A) andAis an isomorphism and since ρ(A) is open in C(see Theorem 6.7 (Chapter III) in [ 19]),\nwe easily get R(λI−A) =Hfor a sufficiently small λ >0. This, together with the dissipativeness of A, imply\nthatD(A) is dense in Hand that Ais m-dissipative in H(see Theorems 4.5, 4.6 in [ 22]). /square\nAccording to Lumer-Phillips theorem (see [ 22]), then the operator Agenerates a C0-semigroup of contrac-\ntionsetAinHwhich gives the well-posedness of ( 2.1). Then, we have the following result:\nTheorem 2.2. For allU0∈ H, system ( 2.1) admits a unique weak solution\nU(t) =etAU0∈C0(R+,H).\nMoreover, if U0∈D(A), then the system ( 2.1) admits a unique strong solution\nU(t) =etAU0∈C0(R+,D(A))∩C1(R+,H).\n2.2.Strong Stability. In this subsection, we will prove the strong stability of system ( 1.1)-(1.4). We define\nthe following conditions:\n(SSC1) ( C1) holds and |c0|<min/parenleftbigg√a\nc2−c1,1\nc2−c1/parenrightbigg\n,\n(SSC3) ( C3) holds, a= 1 and |c0|<1\nc2−c1.\nThe main result of this section is the following theorem.\nTheorem 2.3. Assume that ( SSC1) or(C2) or(SSC3) holds. Then, the C0-semigroupofcontractions/parenleftbig\netA/parenrightbig\nt≥0\nis strongly stable in H; i.e. for all U0∈ H, the solution of ( 2.1) satisfies\nlim\nt→+∞/ba∇dbletAU0/ba∇dblH= 0.\nAccording to Theorem A.2, to prove Theorem 2.3, we need to prove that the operator Ahas no pure imaginary\neigenvalues and σ(A)∩iRis countable. Its proof has been divided into the following Lemmas.\nLemma 2.4. Assume that ( SSC1) or (C2) or (SSC3) holds. Then, for all λ∈R,iλI−Ais injective, i.e.\nker(iλI−A) ={0}.\nProof.From Proposition 2.1, we have 0 ∈ρ(A). We still need to show the result for λ∈R∗. For this aim,\nsuppose that there exists a real number λ/\\e}atio\\slash= 0 and U= (u,v,y,z)⊤∈D(A) such that\nAU=iλU.\nEquivalently, we have\nv=iλu, (2.2)\n(aux+bvx)x−cz=iλv, (2.3)\nz=iλy, (2.4)\n(yx+dzx)+cv=iλz. (2.5)\nNext, a straightforward computation gives\n(2.6) 0 = ℜ/a\\}b∇acketle{tiλU,U/a\\}b∇acket∇i}htH=ℜ/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−/integraldisplayL\n0b|vx|2dx−/integraldisplayL\n0d|zx|2dx.\n5Inserting ( 2.2) and (2.4) in (2.3) and (2.5), we get\nλ2u+(aux+iλbux)x−iλcy= 0 in (0 ,L), (2.7)\nλ2y+(yx+iλdyx)x+iλcu= 0 in (0 ,L), (2.8)\nwith the boundary conditions\n(2.9) u(0) =u(L) =y(0) =y(L) = 0.\n•Case 1: Assume that ( SSC1) holds. From ( 2.2), (2.4) and (2.6), we deduce that\n(2.10) ux=vx= 0 in ( b1,b2) andyx=zx= 0 in ( d1,d2).\nUsing (2.7), (2.8) and (2.10), we obtain\n(2.11) λ2u+auxx= 0 in (0 ,c1) and λ2y+yxx= 0 in ( c2,L).\nDeriving the above equations with respect to xand using ( 2.10), we get\n(2.12)/braceleftiggλ2ux+auxxx= 0 in (0 ,c1),\nux= 0 in ( b1,b2)⊂(0,c1),and/braceleftiggλ2yx+yxxx= 0 in ( c2,L),\nyx= 0 in ( d1,d2)⊂(c2,L).\nUsing the unique continuation theorem, we get\n(2.13) ux= 0 in (0 ,c1) and yx= 0 in ( c2,L)\nUsing (2.13) and the fact that u(0) =y(L) = 0, we get\n(2.14) u= 0 in (0 ,c1) and y= 0 in ( c2,L).\nNow, ouraim is to provethat u=y= 0 in (c1,c2). For this aim, using ( 2.14) and the fact that u,y∈C1([0,L]),\nwe obtain the following boundary conditions\n(2.15) u(c1) =ux(c1) =y(c2) =yx(c2) = 0.\nMultiplying ( 2.7) by−2(x−c2)ux, integrating over ( c1,c2) and taking the real part, we get\n(2.16) −/integraldisplayc2\nc1λ2(x−c2)(|u|2)xdx−a/integraldisplayc2\nc1(x−c2)/parenleftbig\n|ux|2/parenrightbig\nxdx+2ℜ/parenleftbigg\niλc0/integraldisplayc2\nc1(x−c2)yuxdx/parenrightbigg\n= 0,\nusing integration by parts and ( 2.15), we get\n(2.17)/integraldisplayc2\nc1|λu|2dx+a/integraldisplayc2\nc1|ux|2dx+2ℜ/parenleftbigg\niλc0/integraldisplayc2\nc1(x−c2)yuxdx/parenrightbigg\n= 0.\nMultiplying ( 2.8) by−2(x−c1)yx, integrating over ( c1,c2), taking the real part, and using the same argument\nas above, we get\n(2.18)/integraldisplayc2\nc1|λy|2dx+/integraldisplayc2\nc1|yx|2dx+2ℜ/parenleftbigg\niλc0/integraldisplayc2\nc1(x−c1)uyxdx/parenrightbigg\n= 0.\nAdding ( 2.17) and (2.18), we get\n(2.19)/integraldisplayc2\nc1|λu|2dx+a/integraldisplayc2\nc1|ux|2dx+/integraldisplayc2\nc1|λy|2dx+/integraldisplayc2\nc1|yx|2dx≤2|λ||c0|(c2−c1)/integraldisplayc2\nc1(|y||ux|+|u||yx|)dx.\nUsing Young’s inequality in ( 2.19), we get\n(2.20)/integraldisplayc2\nc1|λu|2dx+a/integraldisplayc2\nc1|ux|2dx+/integraldisplayc2\nc1|λy|2dx+/integraldisplayc2\nc1|yx|2dx≤c2\n0(c2−c1)2\na/integraldisplayc2\nc1|λy|2dx\n+a/integraldisplayc2\nc1|ux|2dx+c2\n0(c2−c1)2/integraldisplayc2\nc1|λu|2dx+/integraldisplayc2\nc1|yx|2dx,\nconsequently, we get\n(2.21)/parenleftbigg\n1−c2\n0(c2−c1)2\na/parenrightbigg/integraldisplayc2\nc1|λy|2dx+/parenleftbig\n1−c2\n0(c2−c1)2/parenrightbig/integraldisplayc2\nc1|λu|2dx≤0.\nThus, from the above inequality and ( SSC1), we get\n(2.22) u=y= 0 in ( c1,c2).\n6Next, we need to prove that u= 0 in (c2,L) andy= 0 in (0 ,c1). For this aim, from ( 2.22) and the fact that\nu,y∈C1([0,L]), we obtain\n(2.23) u(c2) =ux(c2) = 0 and y(c1) =yx(c1) = 0.\nIt follows from ( 2.7), (2.8) and (2.23) that\n(2.24)/braceleftiggλ2u+auxx= 0 in ( c2,L),\nu(c2) =ux(c2) =u(L) = 0,and/braceleftiggλ2y+yxx= 0 in (0 ,c1),\ny(0) =y(c1) =yx(c1) = 0.\nHolmgren uniqueness theorem yields\n(2.25) u= 0 in ( c2,L) andy= 0 in (0 ,c1).\nTherefore, from ( 2.2), (2.4), (2.14), (2.22) and (2.25), we deduce that\nU= 0.\n•Case 2: Assume that ( C2) holds. From ( 2.2), (2.4) and (2.6), we deduce that\n(2.26) ux=vx= 0 in ( b1,b2) andyx=zx= 0 in ( d1,d2).\nUsing (2.7), (2.8) and (2.26), we obtain\n(2.27) λ2u+auxx= 0 in (0 ,c1) and λ2y+yxx= 0 in (0 ,c1).\nDeriving the above equations with respect to xand using ( 2.26), we get\n(2.28)/braceleftiggλ2ux+auxxx= 0 in (0 ,c1),\nux= 0 in ( b1,b2)⊂(0,c1),and/braceleftiggλ2yx+yxxx= 0 in (0 ,c1),\nyx= 0 in ( d1,d2)⊂(0,c1).\nUsing the unique continuation theorem, we get\n(2.29) ux= 0 in (0 ,c1) and yx= 0 in (0 ,c1).\nFrom (2.29) and the fact that u(0) =y(0) = 0, we get\n(2.30) u= 0 in (0 ,c1) and y= 0 in (0 ,c1).\nUsing the fact that u,y∈C1([0,L]) and (2.30), we get\n(2.31) u(c1) =ux(c1) =y(c1) =yx(c1) = 0.\nNow, using the definition of c(x) in (2.7)-(2.8), (2.26) and (2.31) and Holmgren theorem, we get\nu=y= 0 in ( c1,c2).\nAgain, using the fact that u,y∈C1([0,L]), we get\n(2.32) u(c2) =ux(c2) =y(c2) =yx(c2) = 0.\nNow, using the same argument as in Case 1, we obtain\nu=y= 0 in (c2,L),\nconsequently, we deduce that\nU= 0.\n•Case 3: Assume that ( SSC3) holds. Using the same argument as in Cases 1 and 2, we obtain\n(2.33) u= 0 in (0 ,c1) and u(c1) =ux(c1) = 0.\nStep 1. The aim of this step is to prove that\n(2.34)/integraldisplayc2\nc1|u|2dx=/integraldisplayc2\nc1|y|2dx.\n7For this aim, multiplying ( 2.7) byyand (2.8) byuand using integration by parts, we get\n/integraldisplayL\n0λ2uydx−/integraldisplayL\n0uxyxdx−iλc0/integraldisplayc2\nc1|y|2dx= 0, (2.35)\n/integraldisplayL\n0λ2yudx−/integraldisplayL\n0yxuxdx+iλc0/integraldisplayc2\nc1|u|2dx= 0. (2.36)\nAdding ( 2.35) and (2.36), taking the imaginary part, we get ( 2.34).\nStep 2. Multiplying ( 2.7) by−2(x−c2)ux, integrating over ( c1,c2) and taking the real part, we get\n(2.37)−ℜ/parenleftbigg/integraldisplayc2\nc1λ2(x−c2)(|u|2)xdx/parenrightbigg\n−ℜ/parenleftbigg/integraldisplayc2\nc1(x−c2)/parenleftbig\n|ux|2/parenrightbig\nxdx/parenrightbigg\n+2ℜ/parenleftbigg\niλc0/integraldisplayc2\nc1(x−c2)yuxdx/parenrightbigg\n= 0,\nusing integration by parts in ( 2.37) and (2.33), we get\n(2.38)/integraldisplayc2\nc1|λu|2dx+a/integraldisplayc2\nc1|ux|2dx+2ℜ/parenleftbigg\niλc0/integraldisplayc2\nc1(x−c2)yuxdx/parenrightbigg\n= 0.\nUsing Young’s inequality in ( 2.38), we obtain\n(2.39)/integraldisplayc2\nc1|λu|2dx+/integraldisplayc2\nc1|ux|2dx≤ |c0|(c2−c1)/integraldisplayc2\nc1|λy|2dx+|c0|(c2−c1)/integraldisplayc2\nc1|ux|2dx.\nInserting ( 2.34) in (2.39), we get\n(2.40) (1 −|c0|(c2−c1))/integraldisplayc2\nc1/parenleftbig\n|λu|2+|ux|2/parenrightbig\ndx≤0.\nAccording to ( SSC3) and (2.34), we get\n(2.41) u=y= 0 in ( c1,c2).\nStep 3. Using the fact that u∈H2(c1,c2)⊂C1([c1,c2]), we get\n(2.42) u(c1) =ux(c1) =y(c1) =yx(c1) =y(c2) =yx(c2) = 0.\nNow, from ( 2.7), (2.8) and the definition of c, we get\n/braceleftbiggλ2u+uxx= 0 in ( c2,L),\nu(c2) =ux(c2) = 0,and/braceleftbiggλ2y+yxx= 0 in (0 ,c1)∪(c2,L),\ny(c1) =yx(c1) =y(c2) =yx(c2) = 0.\nFrom the above systems and Holmgren uniqueness Theorem, we get\n(2.43) u= 0 in ( c2,L) and y= 0 in (0 ,c1)∪(c2,L).\nConsequently, using ( 2.33), (2.41) and (2.43), we get U= 0. The proof is thus completed. /square\nLemma 2.5. Assume that ( SSC1) or (C2) or (SSC3) holds. Then, for all λ∈R, we have\nR(iλI−A) =H.\nProof.See Lemma 2.5 in [ 24] (see also [ 4]). /square\nProof of Theorems 2.3. From Lemma 2.4, we obtain that the operator Ahas no pure imaginary eigenvalues\n(i.e.σp(A)∩iR=∅). Moreover, from Lemma 2.5and with the help of the closed graph theorem of Banach,\nwe deduce that σ(A)∩iR=∅. Therefore, according to Theorem A.2, we get that the C 0-semigroup ( etA)t≥0\nis strongly stable. The proof is thus complete. /square\n82.3.Polynomial Stability. In this subsection, we study the polynomial stability of system ( 1.1)-(1.4). Our\nmain result in this section are the following theorems.\nTheorem 2.6. Assume that ( SSC1) holds. Then, for all U0∈D(A), there exists aconstant C >0 independent\nofU0such that\n(2.44) E(t)≤C\nt4/ba∇dblU0/ba∇dbl2\nD(A), t >0.\nTheorem 2.7. Assume that ( SSC3) holds . Then, for all U0∈D(A) there exists a constant C >0 independent\nofU0such that\n(2.45) E(t)≤C\nt/ba∇dblU0/ba∇dbl2\nD(A), t >0.\nAccording to Theorem A.3, the polynomial energy decays ( 2.44) and (2.45) hold if the following conditions\n(H1) iR⊂ρ(A)\nand\n(H2) limsup\nλ∈R,|λ|→∞1\n|λ|ℓ/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble\nL(H)<∞withℓ=/braceleftigg1\n2for Theorem 2.6,\n2 for Theorem 2.7,\naresatisfied. Sincecondition( H1)isalreadyprovedin Subsection 2.2. We stillneedtoprove( H2), let usproveit\nbyacontradictionargument. Tothisaim,supposethat( H2)isfalse,thenthereexists/braceleftbig/parenleftbig\nλn,Un:= (un,vn,yn,zn)⊤/parenrightbig/bracerightbig\nn≥1⊂\nR∗\n+×D(A) with\n(2.46) λn→ ∞asn→ ∞and/ba∇dblUn/ba∇dblH= 1,∀n≥1,\nsuch that\n(2.47) ( λn)ℓ(iλnI−A)Un=Fn:= (f1,n,f2,n,f3,n,f4,n)⊤→0 inH,asn→ ∞.\nFor simplicity, we drop the index n. Equivalently, from ( 2.47), we have\niλu−v=f1\nλℓ, f1→0 inH1\n0(0,L), (2.48)\niλv−(aux+bvx)x+cz=f2\nλℓ, f2→0 inL2(0,L), (2.49)\niλy−z=f3\nλℓ, f3→0 inH1\n0(0,L), (2.50)\niλz−(yx+dzx)x−cv=f4\nλℓ, f4→0 inL2(0,L). (2.51)\n2.3.1.Proof of Theorem 2.6.In this subsection, we will prove Theorem 2.6by checking the condition ( H2),\nby finding a contradiction with ( 2.46) by showing /ba∇dblU/ba∇dblH=o(1). For clarity, we divide the proof into several\nLemmas. By taking the inner product of ( 2.47) withUinH, we remark that\n/integraldisplayL\n0b|vx|2dx+/integraldisplayL\n0d|zx|2dx=−ℜ(/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH) =λ−1\n2ℜ(/a\\}b∇acketle{tF,U/a\\}b∇acket∇i}htH) =o/parenleftig\nλ−1\n2/parenrightig\n.\nThus, from the definitions of bandd, we get\n(2.52)/integraldisplayb2\nb1|vx|2dx=o/parenleftig\nλ−1\n2/parenrightig\nand/integraldisplayd2\nd1|zx|2dx=o/parenleftig\nλ−1\n2/parenrightig\n.\nUsing (2.48), (2.50), (2.52), and the fact that f1,f3→0 inH1\n0(0,L), we get\n(2.53)/integraldisplayb2\nb1|ux|2dx=o(1)\nλ5\n2and/integraldisplayd2\nd1|yx|2dx=o(1)\nλ5\n2.\nLemma 2.8. The solution U∈D(A) of system ( 2.48)-(2.51) satisfies the following estimations\n(2.54)/integraldisplayb2\nb1|v|2dx=o(1)\nλ3\n2and/integraldisplayd2\nd1|z|2dx=o(1)\nλ3\n2.\n9Proof.We give the proof of the first estimation in ( 2.54), the second one can be done in a similar way. For\nthis aim, we fix g∈C1([b1,b2]) such that\ng(b2) =−g(b1) = 1,max\nx∈[b1,b2]|g(x)|=mgand max\nx∈[b1,b2]|g′(x)|=mg′.\nThe proof is divided into several steps:\nStep 1. The goal of this step is to prove that\n(2.55) |v(b1)|2+|v(b2)|2≤/parenleftigg\nλ1\n2\n2+2mg′/parenrightigg/integraldisplayb2\nb1|v|2dx+o(1)\nλ.\nFrom (2.48), we deduce that\n(2.56) vx=iλux−λ−1\n2(f1)x.\nMultiplying ( 2.56) by 2gvand integrating over ( b1,b2), then taking the real part, we get\n/integraldisplayb2\nb1g/parenleftbig\n|v|2/parenrightbig\nxdx=ℜ/parenleftigg\n2iλ/integraldisplayb2\nb1guxvdx/parenrightigg\n−ℜ/parenleftigg\n2λ−1\n2/integraldisplayb2\nb1g(f1)xvdx/parenrightigg\n.\nUsing integration by parts in the left hand side of the above equation , we get\n(2.57) |v(b1)|2+|v(b2)|2=/integraldisplayb2\nb1g′|v|2dx+ℜ/parenleftigg\n2iλ/integraldisplayb2\nb1guxvdx/parenrightigg\n−ℜ/parenleftigg\n2λ−1\n2/integraldisplayb2\nb1g(f1)xvdx/parenrightigg\n.\nUsing Young’s inequality, we obtain\n2λmg|ux||v| ≤λ1\n2|v|2\n2+2λ3\n2m2\ng|ux|2and 2λ−1\n2mg|(f1)x||v| ≤mg′|v|2+m2\ngm−1\ng′λ−1|(f1)x|2.\nFrom the above inequalities, ( 2.57) becomes\n(2.58) |v(b1)|2+|v(b2)|2≤/parenleftigg\nλ1\n2\n2+2mg′/parenrightigg/integraldisplayb2\nb1|v|2dx+2λ3\n2m2\ng/integraldisplayb2\nb1|ux|2dx+m2\ng\nmg′λ−1/integraldisplayb2\nb1|(f1)x|2dx.\nInserting ( 2.53) in (2.58) and the fact that f1→0 inH1\n0(0,L), we get ( 2.55).\nStep 2. The aim of this step is to prove that\n(2.59) |(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2≤λ3\n2\n2/integraldisplayb2\nb1|v|2dx+o(1).\nMultiplying ( 2.49) by−2g/parenleftbig\naux+bvx/parenrightbig\n, using integration by parts over ( b1,b2) and taking the real part, we get\n|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2=/integraldisplayb2\nb1g′|aux+bvx|2dx+\nℜ/parenleftigg\n2iλ/integraldisplayb2\nb1gv(aux+bvx)dx/parenrightigg\n−ℜ/parenleftigg\n2λ−1\n2/integraldisplayb2\nb1gf2(aux+bvx)dx/parenrightigg\n,\nconsequently, we get\n(2.60)|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2≤mg′/integraldisplayb2\nb1|aux+bvx|2dx\n+2λmg/integraldisplayb2\nb1|v||aux+bvx|dx+2mgλ−1\n2/integraldisplayb2\nb1|f2||aux+bvx|dx.\nBy Young’s inequality, ( 2.52), and (2.53), we have\n(2.61) 2 λmg/integraldisplayb2\nb1|v||aux+bvx|dx≤λ3\n2\n2/integraldisplayb2\nb1|v|2dx+2m2\ngλ1\n2/integraldisplayb2\nb1|aux+bvx|2dx≤λ3\n2\n2/integraldisplayb2\nb1|v|2dx+o(1).\nInserting ( 2.61) in (2.60), then using ( 2.52), (2.53) and the fact that f2→0 inL2(0,L), we get ( 2.59).\n10Step 3. The aim of this step is to prove the first estimation in ( 2.54). For this aim, multiplying ( 2.49) by\n−iλ−1v, integrating over ( b1,b2) and taking the real part , we get\n(2.62)/integraldisplayb2\nb1|v|2dx=ℜ/parenleftigg\niλ−1/integraldisplayb2\nb1(aux+bvx)vxdx−/bracketleftbig\niλ−1(aux+bvx)v/bracketrightbigb2\nb1+iλ−3\n2/integraldisplayb2\nb1f2vdx/parenrightigg\n.\nUsing (2.52), (2.53), the fact that vis uniformly bounded in L2(0,L) andf2→0 inL2(0,1), and Young’s\ninequalities, we get\n(2.63)/integraldisplayb2\nb1|v|2dx≤λ−1\n2\n2[|v(b1)|2+|v(b2)|2]+λ−3\n2\n2[|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2]+o(1)\nλ3\n2.\nInserting ( 2.55) and (2.59) in (2.63), we get\n/integraldisplayb2\nb1|v|2dx≤/parenleftbigg1\n2+mg′λ−1\n2/parenrightbigg/integraldisplayb2\nb1|v|2dx+o(1)\nλ3\n2,\nwhich implies that\n(2.64)/parenleftbigg1\n2−mg′λ−1\n2/parenrightbigg/integraldisplayb2\nb1|v|2dx≤o(1)\nλ3\n2.\nUsing the fact that λ→ ∞, we can take λ >4m2\ng′. Then, we obtain the first estimation in ( 2.54). Similarly,\nwe can obtain the second estimation in ( 2.54). The proof has been completed. /square\nLemma 2.9. The solution U∈D(A) of system ( 2.48)-(2.51) satisfies the following estimations\n(2.65)/integraldisplayc1\n0/parenleftbig\n|v|2+a|ux|2/parenrightbig\ndx=o(1) and/integraldisplayL\nc2/parenleftbig\n|z|2+|yx|2/parenrightbig\ndx=o(1).\nProof. First, let h∈C1([0,c1]) such that h(0) =h(c1) = 0. Multiplying ( 2.49) by 2a−1h(aux+bvx),\nintegrating over (0 ,c1), using integration by parts and taking the real part, then using ( 2.52) and the fact that\nuxis uniformly bounded in L2(0,L) andf2→0 inL2(0,L), we get\n(2.66) ℜ/parenleftbigg\n2iλa−1/integraldisplayc1\n0vh(aux+bvx)dx/parenrightbigg\n+a−1/integraldisplayc1\n0h′|aux+bvx|2dx=o(1)\nλ1\n2.\nFrom (2.48), we have\n(2.67) iλux=−vx−λ−1\n2(f1)x.\nInserting ( 2.67) in (2.66), using integration by parts, then using ( 2.52), (2.54), and the fact that f1→0 in\nH1\n0(0,L) andvis uniformly bounded in L2(0,L), we get\n(2.68)/integraldisplayc1\n0h′|v|2dx+a−1/integraldisplayc1\n0h′|aux+bvx|2dx= 2ℜ/parenleftbigg\nλ−1\n2/integraldisplayc1\n0vh(f1)xdx/parenrightbigg\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=o(λ−1\n2)\n+ℜ/parenleftigg\n2iλa−1b0/integraldisplayb2\nb1hvvxdx/parenrightigg\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=o(1)+o(1)\nλ1\n2.\nNow, we fix the following cut-off functions\np1(x) :=\n\n1 in (0 ,b1),\n0 in ( b2,c1),\n0≤p1≤1 in (b1,b2),andp2(x) :=\n\n1 in ( b2,c1),\n0 in (0 ,b1),\n0≤p2≤1 in (b1,b2).\nFinally, take h(x) =xp1(x)+(x−c1)p2(x) in (2.68) and using ( 2.52), (2.53), (2.54), we get the first estimation\nin (2.65). By using the same argument, we can obtain the second estimation in (2.65). The proof is thus\ncompleted. /square\nLemma 2.10. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations\n(2.69) |λu(c1)|=o(1),|ux(c1)|=o(1),|λy(c2)|=o(1)and|yx(c2)|=o(1).\n11Proof.First, from ( 2.48) and (2.49), we deduce that\n(2.70) λ2u+auxx=−f2\nλ1\n2−iλ1\n2f1in (b2,c1).\nMultiplying ( 2.70) by 2(x−b2)¯ux, integrating over ( b2,c1) and taking the real part, then using the fact that\nuxis uniformly bounded in L2(0,L) andf2→0 inL2(0,L), we get\n(2.71)/integraldisplayc1\nb2λ2(x−b2)/parenleftbig\n|u|2/parenrightbig\nxdx+a/integraldisplayc1\nb2(x−b2)/parenleftbig\n|ux|2/parenrightbig\nxdx=−ℜ/parenleftbigg\n2iλ1\n2/integraldisplayc1\nb2(x−b2)f1uxdx/parenrightbigg\n+o(1)\nλ1\n2.\nUsing integration by parts in ( 2.71), then using ( 2.65), and the fact that f1→0 inH1\n0(0,L) andλuis uniformly\nbounded in L2(0,L), we get\n(2.72) 0 ≤(c1−b2)/parenleftbig\n|λu(c1)|2+a|ux(c1)|2/parenrightbig\n=ℜ/parenleftig\n2iλ1\n2(c1−b2)f1(c1)u(c1)/parenrightig\n+o(1),\nconsequently, by using Young’s inequality, we get\n|λu(c1)|2+|ux(c1)|2≤2λ1\n2|f1(c1)||u(c1)|+o(1)\n≤1\n2|λu(c1)|2+2\nλ|f1(c1)|2+o(1).\nThen, we get\n(2.73)1\n2|λu(c1)|2+|ux(c1)|2≤2\nλ|f1(c1)|2+o(1).\nFinally, from the above estimation and the fact that f1→0 inH1\n0(0,L), we get the first two estimations in\n(2.69). By using the same argument, we can obtain the last two estimation s in (2.69). The proof has been\ncompleted. /square\nLemma 2.11. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimation\n(2.74)/integraldisplayc2\nc1|λu|2+a|ux|2+|λy|2+|yx|2dx=o(1).\nProof.Inserting ( 2.48) and (2.50) in (2.49) and (2.51), we get\n−λ2u−auxx+iλc0y=f2\nλ1\n2+iλ1\n2f1+c0f3\nλ1\n2in (c1,c2), (2.75)\n−λ2y−yxx−iλc0u=f4\nλ1\n2+iλ1\n2f3−c0f1\nλ1\n2in (c1,c2). (2.76)\nMultiplying ( 2.75) by 2(x−c2)uxand (2.76) by 2(x−c1)yx, integrating over ( c1,c2) and taking the real part,\nthen using the fact that /ba∇dblF/ba∇dblH=o(1) and/ba∇dblU/ba∇dblH= 1, we obtain\n(2.77)−λ2/integraldisplayc2\nc1(x−c2)/parenleftbig\n|u|2/parenrightbig\nxdx−a/integraldisplayc2\nc1(x−c2)/parenleftbig\n|ux|2/parenrightbig\nxdx+ℜ/parenleftbigg\n2iλc0/integraldisplayc2\nc1(x−c2)yuxdx/parenrightbigg\n=\nℜ/parenleftbigg\n2iλ1\n2/integraldisplayc2\nc1(x−c2)f1uxdx/parenrightbigg\n+o(1)\nλ1\n2\nand\n(2.78)−λ2/integraldisplayc2\nc1(x−c1)/parenleftbig\n|y|2/parenrightbig\nxdx−/integraldisplayc2\nc1(x−c1)/parenleftbig\n|yx|2/parenrightbig\nxdx−ℜ/parenleftbigg\n2iλc0/integraldisplayc2\nc1(x−c1)uyxdx/parenrightbigg\n=\nℜ/parenleftbigg\n2iλ1\n2/integraldisplayc2\nc1(x−c1)f3yxdx/parenrightbigg\n+o(1)\nλ1\n2.\nUsing integration by parts, ( 2.69), and the fact that f1,f3→0 inH1\n0(0,L),/ba∇dblu/ba∇dblL2(0,L)=O(λ−1),/ba∇dbly/ba∇dblL2(0,L)=\nO(λ−1), we deduce that\n(2.79) ℜ/parenleftbigg\niλ1\n2/integraldisplayc2\nc1(x−c2)f1uxdx/parenrightbigg\n=o(1)\nλ1\n2andℜ/parenleftbigg\niλ1\n2/integraldisplayc2\nc1(x−c1)f3yxdx/parenrightbigg\n=o(1)\nλ1\n2.\n12Inserting ( 2.79) in (2.77) and (2.78), then using integration by parts and ( 2.69), we get\n/integraldisplayc2\nc1/parenleftbig\n|λu|2+a|ux|2/parenrightbig\ndx+ℜ/parenleftbigg\niλc0/integraldisplayc2\nc1(x−c2)yuxdx/parenrightbigg\n=o(1), (2.80)\n/integraldisplayc2\nc1/parenleftbig\n|λy|2+|yx|2/parenrightbig\ndx−ℜ/parenleftbigg\niλc0/integraldisplayc2\nc1(x−c1)uyxdx/parenrightbigg\n=o(1). (2.81)\nAdding ( 2.80) and (2.81), we get\n/integraldisplayc2\nc1/parenleftbig\n|λu|2+a|ux|2+|λy|2+|yx|2/parenrightbig\ndx=ℜ/parenleftbigg\n2iλc0/integraldisplayc2\nc1(x−c1)uyxdx/parenrightbigg\n−ℜ/parenleftbigg\n2iλc0/integraldisplayc2\nc1(x−c2)yuxdx/parenrightbigg\n+o(1)\n≤2λ|c0|(c2−c1)/integraldisplayc2\nc1|u||yx|dx+2λ|c0|\na1\n4(c2−c1)a1\n4/integraldisplayc2\nc1|y||ux|dx+o(1).\nApplying Young’s inequalities, we get\n(2.82) (1 −|c0|(c2−c1))/integraldisplayc2\nc1(|λu|2+|yx|2)dx+/parenleftbigg\n1−1√a|c0|(c2−c1)/parenrightbigg/integraldisplayc2\nc1(a|ux|2+|λy|2)dx≤o(1).\nFinally, using ( SSC1), we get the desired result. The proof has been completed. /square\nLemma 2.12. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations\n(2.83)/integraldisplayc1\n0/parenleftbig\n|z|2+|yx|2/parenrightbig\ndx=o(1)and/integraldisplayL\nc2/parenleftbig\n|v|2+a|ux|2/parenrightbig\ndx=o(1).\nProof.Using the same argument of Lemma 2.9, we obtain ( 2.83). /square\nProof of Theorem 2.6.Using (2.53), Lemmas 2.8,2.9,2.11,2.12, we get /ba∇dblU/ba∇dblH=o(1), which contradicts\n(2.46). Consequently, condition (H2) holds. This implies the energy decay estimation ( 2.44).\n2.3.2.Proof of Theorem 2.7.In this subsection, we will prove Theorem 2.7by checking the condition ( H2),\nthat is by finding a contradiction with ( 2.46) by showing /ba∇dblU/ba∇dblH=o(1). For clarity, we divide the proof into\nseveral Lemmas. By taking the inner product of ( 2.47) withUinH, we remark that\n/integraldisplayL\n0b|vx|2dx=−ℜ(/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH) =λ−2ℜ(/a\\}b∇acketle{tF,U/a\\}b∇acket∇i}htH) =o(λ−2).\nThen,\n(2.84)/integraldisplayb2\nb1|vx|2dx=o(λ−2).\nUsing (2.48) and (2.84), and the fact that f1→0 inH1\n0(0,L), we get\n(2.85)/integraldisplayb2\nb1|ux|2dx=o(λ−4).\nLemma 2.13. Let0< ε <b2−b1\n2, the solution U∈D(A)of the system (2.48)-(2.51)satisfies the following\nestimation\n(2.86)/integraldisplayb2−ε\nb1+ε|v|2dx=o(λ−2).\nProof.First, we fix a cut-off function θ1∈C1([0,c1]) such that\n(2.87) θ1(x) =\n\n1 if x∈(b1+ε,b2−ε),\n0 if x∈(0,b1)∪(b2,L),\n0≤θ1≤1 elsewhere .\nMultiplying ( 2.49) byλ−1θ1v, integrating over (0 ,c1), using integration by parts, and the fact that f2→0 in\nL2(0,L) andvis uniformly bounded in L2(0,L), we get\n(2.88) i/integraldisplayc1\n0θ1|v|2dx+1\nλ/integraldisplayc1\n0(ux+bvx)(θ′\n1v+θvx)dx=o(λ−3).\n13Using (2.84) and the fact that /ba∇dblU/ba∇dblH= 1, we get\n1\nλ/integraldisplayc1\n0(ux+bvx)(θ′\n1v+θvx)dx=o(λ−2).\nInserting the above estimation in ( 2.88), we get the desired result ( 2.86). The proof has been completed. /square\nLemma 2.14. The solution U∈D(A)of the system (2.48)-(2.51)satisfies the following estimation\n(2.89)/integraldisplayc1\n0(|v|2+|ux|2)dx=o(1).\nProof.Leth∈C1([0,c1]) such that h(0) =h(c1) = 0. Multiplying ( 2.49) by 2h(ux+bvx), integrating over\n(0,c1) and taking the real part, then using integration by parts and the fact that f2→0 inL2(0,L), we get\n(2.90) ℜ/parenleftbigg\n2/integraldisplayc1\n0iλvh(ux+bvx)dx/parenrightbigg\n+/integraldisplayc1\n0h′|ux+bvx|2dx=o(λ−2).\nUsing (2.84) and the fact that vis uniformly bounded in L2(0,L), we get\n(2.91) ℜ/parenleftbigg\n2/integraldisplayc1\n0iλvh(ux+bvx)dx/parenrightbigg\n= 2/integraldisplayc1\n0iλvhuxdx+o(1).\nFrom (2.48), we have\n(2.92) iλux=−vx−/parenleftbig\nf1/parenrightbig\nx\nλ2.\nInserting ( 2.92) in (2.91), using integration by parts and the fact that f1→0 inH1\n0(0,L), we get\n(2.93) ℜ/parenleftbigg\n2/integraldisplayc1\n0iλvh(ux+bvx)dx/parenrightbigg\n=/integraldisplayc1\n0h′|v|2dx+o(1).\nInserting ( 2.93) in (2.90), we obtain\n(2.94)/integraldisplayc1\n0h′/parenleftbig\n|v|2+|ux+bvx|2/parenrightbig\ndx=o(1).\nNow, we fix the following cut-off functions\nθ2(x) :=\n\n1 in (0 ,b1+ε),\n0 in ( b2−ε,c1),\n0≤θ2≤1 in (b1+ε,b2−ε),andθ3(x) :=\n\n1 in ( b2−ε,c1),\n0 in (0 ,b1+ε),\n0≤θ3≤1 in (b1+ε,b2−ε).\nTakingh(x) =xθ2(x)+(x−c1)θ3(x) in (2.94), then using ( 2.84) and (2.85), we get\n(2.95)/integraldisplay\n(0,b1+ε)∪(b2−ε,c1)|v|2dx+/integraldisplay\n(0,b1)∪(b2,c1)|ux|2dx=o(1).\nFinally, from ( 2.85), (2.86) and (2.95), we get the desired result ( 2.89). The proof has been completed. /square\nLemma 2.15. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations\n(2.96) |λu(c1)|=o(1)and|ux(c1)|=o(1),\n(2.97)/integraldisplayc2\nc1|λu|2dx=/integraldisplayc2\nc1|λy|2dx+o(1).\nProof.First, using the same argument of Lemma 2.10, we claim ( 2.96). Inserting ( 2.48), (2.50) in (2.49) and\n(2.51), we get\nλ2u+(ux+bvx)x−iλcy=−f2\nλ2−if1\nλ−cf3\nλ2, (2.98)\nλ2y+yxx+iλcu=−f4\nλ2−if3\nλ+cf1\nλ2. (2.99)\n14Multiplying ( 2.98) and (2.99) byλyandλurespectively, integrating over(0 ,L), then using integration by parts,\n(2.84), and the fact that /ba∇dblU/ba∇dblH= 1 and /ba∇dblF/ba∇dblH=o(1), we get\nλ3/integraldisplayL\n0u¯ydx−λ/integraldisplayL\n0ux¯yxdx−ic0/integraldisplayc2\nc1|λy|2dx=o(1), (2.100)\nλ3/integraldisplayL\n0y¯udx−λ/integraldisplayL\n0yx¯uxdx+ic0/integraldisplayc2\nc1|λu|2dx=o(1)\nλ. (2.101)\nAdding ( 2.100) and (2.101) and taking the imaginary parts, we get the desired result ( 2.97). The proof is thus\ncompleted. /square\nLemma 2.16. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following asymptotic behavior\n(2.102)/integraldisplayc2\nc1|λu|2dx=o(1),/integraldisplayc2\nc1|λy|2dx=o(1)and/integraldisplayc2\nc1|ux|2dx=o(1).\nProof.First, Multiplying ( 2.98) by 2(x−c2)¯ux, integrating over ( c1,c2) and taking the real part, using the\nfact that /ba∇dblU/ba∇dblH= 1 and /ba∇dblF/ba∇dblH=o(1), we get\n(2.103) λ2/integraldisplayc2\nc1(x−c2)/parenleftbig\n|u|2/parenrightbig\nxdx+/integraldisplayc2\nc1(x−c2)/parenleftbig\n|ux|2/parenrightbig\nxdx=ℜ/parenleftbigg\n2iλc0/integraldisplayc2\nc1(x−c2)y¯uxdx/parenrightbigg\n+o(1).\nUsing integration by parts in ( 2.103) with the help of ( 2.96), we get\n(2.104)/integraldisplayc2\nc1|λu|2dx+/integraldisplayc2\nc1|ux|2dx≤2λ|c0|(c2−c1)/integraldisplayc2\nc1|y||ux|+o(1).\nApplying Young’s inequality in ( 2.104), we get\n(2.105)/integraldisplayc2\nc1|λu|2dx+/integraldisplayc2\nc1|ux|2dx≤ |c0|(c2−c1)/integraldisplayc2\nc1|ux|2dx+|c0|(c2−c1)/integraldisplayc2\nc1|λy|2dx+o(1).\nUsing (2.97) in (2.105), we get\n(2.106) (1 −|c0|(c2−c1))/integraldisplayc2\nc1/parenleftbig\n|λu|2+|ux|2/parenrightbig\ndx≤o(1).\nFinally, from the above estimation, ( SSC3) and (2.97), we get the desired result ( 2.102). The proof has been\ncompleted. /square\nLemma 2.17. Let0< δ <c2−c1\n2. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following\nestimations\n(2.107)/integraldisplayc2−δ\nc1+δ|yx|2dx=o(1).\nProof.First, we fix a cut-off function θ4∈C1([0,L]) such that\n(2.108) θ4(x) :=\n\n1 if x∈(c1+δ,c2−δ),\n0 if x∈(0,c1)∪(c2,L),\n0≤θ4≤1 elsewhere .\nMultiplying ( 2.99) byθ4¯y, integrating over (0 ,L) and using integration by parts, we get\n(2.109)/integraldisplayc2\nc1θ4|λy|2dx−/integraldisplayL\n0θ4|yx|2dx−/integraldisplayL\n0θ′\n4yx¯ydx+iλc0/integraldisplayc2\nc1θ4u¯ydx=o(1)\nλ2.\nUsing (2.102) and the definition of θ4, we get\n(2.110)/integraldisplayc2\nc1θ4|λy|2dx=o(1),/integraldisplayL\n0θ′\n4yx¯ydx=o(λ−1), iλc 0/integraldisplayc2\nc1θ4u¯ydx=o(λ−1).\nFinally, Inserting ( 2.110) in (2.109), we get the desired result ( 2.111). The proof has been completed. /square\nLemma 2.18. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations\n(2.111)/integraldisplayc1+ε\n0|λy|2dx,/integraldisplayc1+ε\n0|yx|2dx,/integraldisplayL\nc2−ε|λy|2dx,/integraldisplayL\nc2−ε|yx|2dx,/integraldisplayL\nc2|λu|2dx,/integraldisplayL\nc2|ux|2dx=o(1).\n15Proof.Letq∈C1([0,L]) such that q(0) =q(L) = 0. Multiplying ( 2.98) by 2q¯yxintegrating over (0 ,L), using\n(2.102), and the fact that yxis uniformly bounded in L2(0,L) and/ba∇dblF/ba∇dblH=o(1), we get\n(2.112)/integraldisplayL\n0q′/parenleftbig\n|λy|2+|yx|2/parenrightbig\ndx=o(1).\nNow, take q(x) =xθ5(x)+(x−L)θ6(x) in (2.112), such that\nθ5(x) :=\n\n1 in (0 ,c1+ε),\n0 in ( c2−ε,L),\n0≤θ1≤1 in (c1+ε,c2−ε),andθ2(x)\n\n1 in ( c2−ε,L),\n0 in (0 ,c1+ε),\n0≤θ2≤1 in (c1+ε,c2−ε).\nThen, we obtain the first four estimations in ( 2.111). Now, multiplying ( 2.98) by 2q/parenleftbig\nux+bvx/parenrightbig\nintegrating over\n(0,L) and using the fact that uxis uniformly bounded in L2(0,L), we get\n(2.113)/integraldisplayL\n0q′/parenleftbig\n|λu|2+|ux|2/parenrightbig\ndx=o(1).\nBy taking q(x) = (x−L)θ7(x), such that\nθ7(x) =\n\n1 in ( c2,L),\n0 in (0 ,c1),\n0≤θ7≤1 in (c1,c2),\nwe get the the last two estimations in ( 2.111). The proof has been completed. /square\nProof of Theorem 2.7.Using (2.85), Lemmas 2.14,2.16,2.17and2.18, we get /ba∇dblU/ba∇dblH=o(1), which\ncontradicts ( 2.46). Consequently, condition (H2) holds. This implies the energy decay estimation ( 2.45)\n3.Indirect Stability in the multi-dimensional case\nIn this section, we study the well-posedness and the strong stabilit y of system ( 1.5)-(1.8).\n3.1.Well-posedness. In this subsection, we will establish the well-posednessof ( 1.5)-(1.8) by usinf semigroup\napproach. The energy of system ( 1.5)-(1.8) is given by\n(3.1) E(t) =1\n2/integraldisplayL\n0/parenleftbig\n|ut|2+|∇u|2+|yt|2+|∇y|2/parenrightbig\ndx.\nLet (u,ut,y,yt) be a regular solution of ( 1.5)-(1.8). Multiplying ( 1.5) and (1.7) byutandytrespectively, then\nusing the boundary conditions ( 1.9), we get\n(3.2) E′(t) =−/integraldisplay\nΩb|∇ut|2dx,\nusing the definition of b, we get E′(t)≤0. Thus, system ( 1.5)-(1.8) is dissipative in the sense that its energy\nis non-increasing with respect to time t. Let us define the energy space Hby\nH=/parenleftbig\nH1\n0(Ω)×L2(Ω)/parenrightbig2.\nThe energy space His equipped with the inner product defined by\n/a\\}b∇acketle{tU,U1/a\\}b∇acket∇i}htH=/integraldisplay\nΩvv1dx+/integraldisplay\nΩ∇u∇u1dx+/integraldisplay\nΩzz1dx+/integraldisplay\nΩ∇y·∇y1dx,\nfor allU= (u,v,y,z)⊤andU1= (u1,v1,y1,z1)⊤inH. We define the unbounded linear operator Ad:D(Ad)⊂\nH −→ H by\nD(Ad) =/braceleftbig\nU= (u,v,y,z)⊤∈ H;v,z∈H1\n0(Ω),div(ux+bvx)∈L2(Ω),∆y∈L2(Ω)/bracerightbig\nand\nAdU=\nv\ndiv(∇u+b∇v)−cz\nz\n∆y+cv\n,∀U= (u,v,y,z)⊤∈D(Ad).\n16/tildewideΩ\nΩωc ωb •x0\nΓ0Γ1 •xν\nFigure 4. Geometric description of the sets ωbandωc\nIfU= (u,ut,y,yt) is a regular solution of system ( 1.5)-(1.8), then we rewrite this system as the following first\norder evolution equation\n(3.3) Ut=AdU, U(0) =U0,\nwhereU0= (u0,u1,y0,y1)⊤∈ H. For all U= (u,v,y,z)⊤∈D(Ad), we have\nℜ/a\\}b∇acketle{tAdU,U/a\\}b∇acket∇i}htH=−/integraldisplay\nΩb|∇v|2dx≤0,\nwhich implies that Adis dissipative. Now, similar to Proposition 2.1 in [ 7], we can prove that there exists a\nunique solution U= (u,v,y,z)⊤∈D(Ad) of\n−AdU=F,∀F= (f1,f2,f3,f4)⊤∈ H.\nThen 0∈ρ(Ad) andAdis an isomorphism and since ρ(Ad) is open in C(see Theorem 6.7 (Chapter III) in\n[19]), we easily get R(λI−Ad) =Hfor a sufficiently small λ >0. This, together with the dissipativeness of Ad,\nimply that D(Ad) is dense in Hand that Adis m-dissipative in H(see Theorems 4.5, 4.6 in [ 22]). According\nto Lumer-Phillips theorem (see [ 22]), then the operator Adgenerates a C0-semigroup of contractions etAdin\nHwhich gives the well-posedness of ( 3.3). Then, we have the following result:\nTheorem 3.1. For allU0∈ H, system ( 2.1) admits a unique weak solution\nU(t) =etAdU0∈C0(R+,H).\nMoreover, if U0∈D(A), then the system ( 2.1) admits a unique strong solution\nU(t) =etAdU0∈C0(R+,D(Ad))∩C1(R+,H).\n3.2.Strong Stability. In this subsection, we will prove the strong stability of system ( 1.5)-(1.8). First, we\nfix the following notations\n/tildewideΩ = Ω−ωc,Γ1=∂ωc−∂Ω and Γ 0=∂ωc−Γ1.\nLetx0∈Rdandm(x) =x−x0and suppose that (see Figure 4)\n(GC) m·ν≤0 on Γ 0= (∂ωc)−Γ1.\n17The main result of this section is the following theorem\nTheorem 3.2. Assume that (GC)holds and\n(SSC) /ba∇dblc/ba∇dbl∞≤min/braceleftigg\n1\n/ba∇dblm/ba∇dbl∞+d−1\n2,1\n/ba∇dblm/ba∇dbl∞+(d−1)Cp,ωc\n2/bracerightigg\n,\nwhereCp,ωcis the Poincarr´ e constant on ωc. Then, the C0−semigroup of contractions/parenleftbig\netAd/parenrightbig\nis strongly stable\ninH; i.e. for all U0∈ H, the solution of (3.3)satisfies\nlim\nt→+∞/ba∇dbletAdU0/ba∇dblH= 0.\nProof.First, let us prove that\n(3.4) ker( iλI−Ad) ={0},∀λ∈R.\nSince 0∈ρ(Ad), then we still need to show the result for λ∈R∗. Suppose that there exists a real number\nλ/\\e}atio\\slash= 0 and U= (u,v,y,z)⊤∈D(Ad), such that\nAdU=iλU.\nEquivalently, we have\nv=iλu, (3.5)\ndiv(∇u+b∇v)−cz=iλv, (3.6)\nz=iλy, (3.7)\n∆y+cv=iλz. (3.8)\nNext, a straightforward computation gives\n0 =ℜ/a\\}b∇acketle{tiλU,U/a\\}b∇acket∇i}htH=ℜ/a\\}b∇acketle{tAdU,U/a\\}b∇acket∇i}htH=−/integraldisplay\nΩb|∇v|2dx,\nconsequently, we deduce that\n(3.9) b∇v= 0 in Ω and ∇v=∇u= 0 in ωb.\nInserting ( 3.5) in (3.6), then using the definition of c, we get\n(3.10) ∆u=−λ2uinωb.\nFrom (3.9) we get ∆ u= 0 inωband from ( 3.10) and the fact that λ/\\e}atio\\slash= 0, we get\n(3.11) u= 0 in ωb.\nNow, inserting ( 3.5) in (3.6), then using ( 3.9), (3.11) and the definition of c, we get\n(3.12)λ2u+∆u= 0 in/tildewideΩ,\nu= 0 in ωb⊂/tildewideΩ.\nUsing Holmgren uniqueness theorem, we get\n(3.13) u= 0 in/tildewideΩ.\nIt follows that\n(3.14) u=∂u\n∂ν= 0 on Γ 1.\nNow, our aim is to show that u=y= 0 inωc. For this aim, inserting ( 3.5) and (3.7) in (3.6) and (3.8), then\nusing (3.9), we get the following system\nλ2u+∆u−iλcy= 0 in Ω , (3.15)\nλ2y+∆y+iλcu= 0 in Ω , (3.16)\nu= 0 on ∂ωc, (3.17)\ny= 0 on Γ 0, (3.18)\n∂u\n∂ν= 0 on Γ 1. (3.19)\n18Let us prove ( 3.4) by the following three steps:\nStep 1. The aim of this step is to show that\n(3.20)/integraldisplay\nΩc|u|2dx=/integraldisplay\nΩc|y|2dx.\nFor this aim, multiplying ( 3.15) and (3.16) by ¯yand ¯urespectively, integrating over Ω and using Green’s\nformula, we get\nλ2/integraldisplay\nΩu¯ydx−/integraldisplay\nΩ∇u·∇¯ydx−iλ/integraldisplay\nΩc|y|2dx= 0, (3.21)\nλ2/integraldisplay\nΩy¯udx−/integraldisplay\nΩ∇y·∇¯udx+iλ/integraldisplay\nΩc|u|2dx= 0. (3.22)\nAdding ( 3.21) and (3.22), then taking the imaginary part, we get ( 3.20).\nStep 2. The aim of this step is to prove the following identity\n(3.23) −d/integraldisplay\nωc|λu|2dx+(d−2)/integraldisplay\nωc|∇u|2dx+/integraldisplay\nΓ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u\n∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndΓ−2ℜ/parenleftbigg\niλ/integraldisplay\nωccy(m·∇¯u)dx/parenrightbigg\n= 0.\nFor this aim, multiplying ( 3.15) by 2(m·∇¯u), integrating over ωcand taking the real part, we get\n(3.24) 2 ℜ/parenleftbigg\nλ2/integraldisplay\nωcu(m·∇¯u)dx/parenrightbigg\n+2ℜ/parenleftbigg/integraldisplay\nωc∆u(m·∇¯u)dx/parenrightbigg\n−2ℜ/parenleftbigg\niλ/integraldisplay\nωccy(m·∇¯u)dx/parenrightbigg\n= 0.\nNow, using the fact that u= 0 in∂ωc, we get\n(3.25) ℜ/parenleftbigg\n2λ2/integraldisplay\nωcu(m·∇¯u)dx/parenrightbigg\n=−d/integraldisplay\nωc|λu|2dx.\nUsing Green’s formula, we obtain\n(3.26)2ℜ/parenleftbigg/integraldisplay\nωc∆u(m·∇¯u)dx/parenrightbigg\n=−2ℜ/parenleftbigg/integraldisplay\nωc∇u·∇(m·∇¯u)dx/parenrightbigg\n+2ℜ/parenleftbigg/integraldisplay\nΓ0∂u\n∂ν(m·∇¯u)dΓ/parenrightbigg\n= (d−2)/integraldisplay\nωc|∇u|2dx−/integraldisplay\n∂ωc(m·ν)|∇u|2dx+2ℜ/parenleftbigg/integraldisplay\nΓ0∂u\n∂ν(m·∇¯u)dΓ/parenrightbigg\n.\nUsing (3.17) and (3.19), we get\n(3.27)/integraldisplay\n∂ωc(m·ν)|∇u|2dx=/integraldisplay\nΓ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u\n∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndΓ andℜ/parenleftbigg/integraldisplay\nΓ0∂u\n∂ν(m·∇¯u)dΓ/parenrightbigg\n=/integraldisplay\nΓ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u\n∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndΓ.\nInserting ( 3.27) in (3.26), we get\n(3.28) 2 ℜ/parenleftbigg/integraldisplay\nωc∆u(m·∇¯u)dx/parenrightbigg\n= (d−2)/integraldisplay\nωc|∇u|2dx+/integraldisplay\nΓ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u\n∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndΓ.\nInserting ( 3.25) and (3.28) in (3.24), we get ( 3.23).\nStep 3. In this step, we prove ( 3.4). Multiplying ( 3.15) by (d−1)u, integrating over ωcand using ( 3.17), we\nget\n(3.29) ( d−1)/integraldisplay\nωc|λu|2dx+(1−d)/integraldisplay\nωc|∇u|2dx−ℜ/parenleftbigg\niλ(d−1)/integraldisplay\nωccy¯udx/parenrightbigg\n= 0.\nAdding ( 3.23) and (3.29), we get\n/integraldisplay\nωc|λu|2dx+/integraldisplay\nωc|∇u|2dx=/integraldisplay\nΓ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u\n∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndΓ−2ℜ/parenleftbigg\niλ/integraldisplay\nωccy(m·∇¯u)dx/parenrightbigg\n−ℜ/parenleftbigg\niλ(d−1)/integraldisplay\nωccy¯udx/parenrightbigg\n= 0.\nUsing (GC), we get\n(3.30)/integraldisplay\nωc|λu|2dx+/integraldisplay\nωc|∇u|2dx≤2|λ|/integraldisplay\nωc|c||y||m·∇u|dx+|λ|(d−1)/integraldisplay\nωc|c||y||u|dx.\n19Using Young’s inequality and ( 3.20), we get\n(3.31) 2 |λ|/integraldisplay\nωc|c||y||m·∇u|dx≤ /ba∇dblm/ba∇dbl∞/ba∇dblc/ba∇dbl∞/integraldisplay\nωc/parenleftbig\n|λu|2+|∇u|2/parenrightbig\ndx\nand\n(3.32) |λ|(d−1)/integraldisplay\nωc|c(x)||y||u|dx≤(d−1)/ba∇dblc/ba∇dbl∞\n2/integraldisplay\nωc|λu|2dx+(d−1)/ba∇dblc/ba∇dbl∞Cp,ωc\n2/integraldisplay\nωc|∇u|2dx.\nInserting ( 3.32) in (3.30), we get\n/parenleftbigg\n1−/ba∇dblc/ba∇dbl∞/parenleftbigg\n/ba∇dblm/ba∇dbl∞+d−1\n2/parenrightbigg/parenrightbigg/integraldisplay\nωc|λu|2dx+/parenleftbigg\n1−/ba∇dblc/ba∇dbl∞/parenleftbigg\n/ba∇dblm/ba∇dbl∞+(d−1)Cp,ωc\n2/parenrightbigg/parenrightbigg/integraldisplay\nωc|∇u|2dx≤0.\nUsing (SSC) and (3.20) in the above estimation, we get\n(3.33) u= 0 and y= 0 in ωc.\nIn order to complete this proof, we need to show that y= 0 in/tildewideΩ. For this aim, using the definition of the\nfunction cin/tildewideΩ and using the fact that y= 0 inωc, we get\n(3.34)λ2y+∆y= 0 in/tildewideΩ,\ny= 0 on ∂/tildewideΩ,\n∂y\n∂ν= 0 on Γ 1.\nNow, using Holmgren uniqueness theorem, we obtain y= 0 in/tildewideΩ and consequently ( 3.4) holds true. Moreover,\nsimilar to Lemma 2.5 in [ 7], we can prove R(iλI− Ad) =H,∀λ∈R. Finally, by using the closed graph\ntheorem of Banach and Theorem A.2, we conclude the proof of this Theorem. /square\nLet us notice that, under the sole assumptions ( GC) and (SSC), the polynomial stability of system ( 1.5)-(1.8)\nis an open problem.\nAppendix A.Some notions and stability theorems\nIn order to make this paper more self-contained, we recall in this sh ort appendix some notions and stability\nresults used in this work.\nDefinition A.1. Assume that Ais the generator of C0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space\nH. TheC0−semigroup/parenleftbig\netA/parenrightbig\nt≥0is said to be\n(1) Strongly stable if\nlim\nt→+∞/ba∇dbletAx0/ba∇dblH= 0,∀x0∈H.\n(2) Exponentially (or uniformly) stable if there exists two positive co nstantsMandεsuch that\n/ba∇dbletAx0/ba∇dblH≤Me−εt/ba∇dblx0/ba∇dblH,∀t >0,∀x0∈H.\n(3) Polynomially stable if there exists two positive constants Candαsuch that\n/ba∇dbletAx0/ba∇dblH≤Ct−α/ba∇dblAx0/ba∇dblH,∀t >0,∀x0∈D(A).\n/square\nTo show the strong stability of the C0-semigroup/parenleftbig\netA/parenrightbig\nt≥0we rely on the following result due to Arendt-Batty\n[9].\nTheorem A.2. Assume that Ais the generatorof a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space\nH. IfAhas no pure imaginary eigenvalues and σ(A)∩iRis countable, where σ(A) denotes the spectrum of\nA, then the C0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is strongly stable. /square\nConcerning the characterization of polynomial stability stability of a C0−semigroup of contraction/parenleftbig\netA/parenrightbig\nt≥0we\nrely on the following result due to Borichev and Tomilov [ 12] (see also [ 11] and [21])\n20Theorem A.3. Assume that Ais the generator of a strongly continuous semigroup of contractio ns/parenleftbig\netA/parenrightbig\nt≥0\nonH. IfiR⊂ρ(A), then for a fixed ℓ >0 the following conditions are equivalent\n(A.1) limsup\nλ∈R,|λ|→∞1\n|λ|ℓ/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble\nL(H)<∞,\n(A.2) /ba∇dbletAU0/ba∇dbl2\nH≤C\nt2\nℓ/ba∇dblU0/ba∇dbl2\nD(A),∀t >0, U0∈D(A),for some C >0.\n/square\nReferences\n[1] F. Abdallah, M. Ghader, and A. Wehbe. 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Tasson\nPhysics Department, St. Olaf College, Northfield, MN 55057, USA\nThis contribution to the CPT’16 proceedings summarizes rec ent tests of\nLorentz violation in the pure-gravity sector with cosmic ra ys and reviews recent\nprogress in matter-gravity couplings.\n1. Introduction\nLorentz violation1as a signalof new physics at the Planckscale2is actively\nsought in a wide variety of tests3within the general test framework of the\ngravitational Standard-Model Extension (SME).4A subset of these tests\ninvolvegravitationalphysics,whichcanprobeLorentzviolationinth epure-\ngravitysectorassociatedwithbothminimal5andnonminimal6–8operators,\nas well as Lorentz violation in matter-gravity couplings.9,10While Lorentz\nviolation in gravity has been sought in a number of systems,3we focus\nhere on recent tests exploring gravitational ˇCerenkov radiation in Sec. 2,\nand review the status of searches for the α(aeff)µcoefficient in the matter\nsector in Sec. 3, including work with superconducting gravimeters.11\n2. Gravitational ˇCerenkov radiation\nTheˇCerenkov radiation of photons by charged particles moving faster than\nthe phase speed of light in ponderable media is a well-known phenomeno n\nin Nature. If the analogous situation of particles exceeding the pha se speed\nof gravity were to occur, as might be the case in General Relativity ( GR)\nin the presence of certain media,12the gravitational ˇCerenkov radiation of\ngravitons would be expected.13In the presence of suitable coefficients for\nLorentz violation in the SME, both electrodynamic14and gravitational6\nˇCerenkov radiation become possible in vacuum. In this section, we re view\nthe tight constraints that have been achieved by considering cosm ic rays\nand mention other possible implications of vacuum gravitational ˇCerenkov\nradiation.6Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\nAs with all SME searches for Lorentz violation, our analysis begins wit h\nthe expansion about GR and the Standard Model provided by the SM E ac-\ntion. Here we consider the linearized pure-gravity sector, which ha s now\nbeen written explicitlyforoperatorsofarbitrarymassdimension dthat pre-\nserve the usual gauge invariance of GR.8Except where noted, we assume\nthat theothersectorsofthetheoryareconventional. Explorat ionofthe dis-\npersion relation generated by this action reveals that a class of coe fficients\nfor Lorentz violation manifest as a momentum-dependent metric pe rturba-\ntion that generates a momentum-dependent effective index of ref raction for\ngravity. With an appropriate sign for the coefficients for Lorentz v iolation,\nthis index is greater than 1 and particles may exceed the speed of gr avity\nand radiate gravitons. Hence each observation of a high-energy p article can\nplace a one-sided constraint on a combination of these coefficients.\nTo obtain constraints, we perform a calculation of the rate of grav iton\nemission that parallels standard methods assuming for simplicity and d ef-\niniteness that only coefficients at one arbitrary dimension dare nonzero.\nThe calculation is provided for photons, massive scalars, and fermio ns, the\nonly differences in the three cases being the details of the matrix elem ent\nfor the decay and the form of a dimensionless function of din the results.\nThe rate ofpowerloss canthen be integratedto generatearelatio nbetween\nthe time of flight tfor a candidate graviton-radiating particle, the energy\nof the particle at the beginning of its trip Ei, and the observed energy at\nthe end of its trip Ef. This relation takes the form\nt=Fw(d)\nGN(s(d))2/parenleftBigg\n1\nE2d−5\nf−1\nE2d−5\ni/parenrightBigg\n, (1)\nwhereGNis Newton’s constant, Fw(d) is a species wdependent function of\nd, ands(d)isacombinationofcoefficientsforLorentzviolationatdimension\ndthat depends in general on the direction of travel for the particle . This\nresult is distinguished from earlier work on the subject of gravitatio nal\nˇCerenkov radiation15by the connection to the field-theoretic framework\nof the SME, the consideration of anisotropic effects, the explorat ion of\narbitrary dimension d, and the treatment of photons and fermions.\nConservative constraints can be placed using Eq. (1) by setting\n1/E2d−5\ni= 0, solving for s(d),\ns(d)(ˆp)≡(s(d))µνα1...αd−4ˆpµˆpνˆpα1...ˆpαd−4</radicalBigg\nFw(d)\nGNE2d−5\nfL,(2)\nand using suitable data on cosmic ray observations.16HereLis the travelProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\ndistance. Given the dependence on Efin Eq. (2), the highest energy events\nyield the tightest bounds. The highest energy cosmic rays are believ ed to\nbe nuclei, and continuing toward conservative constraints, we ass ume that\nthe gravitons are radiated by a partonic fermion in an iron nucleus ca rrying\n10% of observed energy E⊕, which leads to Ef=E⊕/560. A consideration\nof the likely origin of these particles leads to a conservative estimate of\nL= 10 Mpc. We then use the available data on cosmic ray energies and\ndirection oforigin16to placeconstraintson sixmodels. Threeofthe models\nare constructed as the isotropic limit at d= 4,6,8 respectively. In each\nof these models we place a one-sided limit on the one isotropic coefficien t\ninvolved at the level of 10−14, 10−31GeV−2, and 10−48GeV−4respectively.\nThe other three models involve two-sided constraints on the anisot ropic\ncoefficients at each d. Atd= 4 we place eight constraints at the 10−13\nlevel, while at d= 6 we constrain 24 coefficients at the level of 10−29\nGeV−2, and atd= 8 we constrain 48 coefficients at the 10−45GeV−4level.\nThe paper concludes by discussing some ways in which our work might\nbeextended. Topicsconsideredincludetheroleofthemattersect or,theim-\npact of gravitational ˇCerenkov radiation by photons on cosmological mod-\nels, and gravitons emitting electromagnetic ˇCerenkov radiation.\n3. Matter-gravity couplings\nIn Ref. 9 the phenomenology of matter-gravity couplings was deve loped\nwith a focus on spin-independent coefficients, particularly the coun ter-\nshaded10α(aeff)µcoefficients. As of the CPT’13 meeting,17constraints on\nα(aeff)µhad been placed using the following systems:3,17precession of the\nperihelion of Mercury9and Earth,9torsion pendula,10a torsion strip bal-\nance,18atom interferometry,19and co-magetometry.20This work resulted\nin a number of measurements of the time component reaching the lev el of\n10−11GeV on both the neutron and the proton plus electron coefficients.\nFor the spatial components, two combinations of the nine coefficien ts were\nconstrained at the level of 10−6GeV, and four combinations were weakly\nconstrained at the 10−1GeV level. Note that coverage is sufficient to span\nthe space that is accessible without charged-matter experiments .\nSince CPT’13, α(aeff)µ(as well as sµν) has been considered in an anal-\nysis of planetary ephemerides.21This work considerably extends the level\nof the independent constraints on α(aeff)Jcoefficients to 10−5GeV to 10−3\nGeV, and the analysis of gravimeter experiments extends the maxim um\nreach for these coefficients even further.11The consideration of boundProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n4\nkinetic energy in equivalence-principle tests22has also been used to fur-\nther separate the α(aeff)Tcoefficients and other matter-sector coefficients.\nThough the α(aeff)µcoefficient space accessible with ordinary matter has\nnow been covered more uniformly with initial constraints, opportun ities for\nfurther improvements with currently available methods remain.9,23\nReferences\n1. For a review, see, J.D. Tasson, Rep. Prog. Phys. 77, 062901 (2014).\n2. V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 39, 683 (1989).\n3.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2016 edition, arXiv:0801.0287v9.\n4. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998); V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n5. Q.G. Bailey and V.A. Kosteleck´ y, Phys. Rev. D 74, 045001 (2006).\n6. V.A. Kosteleck´ y and J.D. Tasson, Phys. Lett. B 749, 551 (2015).\n7. Q.G. Bailey, V.A. Kosteleck´ y, and R. Xu, Phys. Rev. D 91, 022006 (2015).\n8. V.A. Kosteleck´ y and M. Mewes, Phys. Lett. B 757, 510 (2016).\n9. V.A. Kosteleck´ y and J.D. Tasson, Phys. Rev. D 83, 016013 (2011).\n10. V.A. Kosteleck´ y and J.D. Tasson, Phys. Rev. Lett. 102, 010402 (2009).\n11. N. Flowers, C. Goodge, and J.D. Tasson, arXiv:1612.0849 5.\n12. P.C. Peters, Phys. Rev. D 9, 2207 (1974).\n13. P. Szekeres, Ann. Phys. 64, 599 (1971); A.G. Polnarev, Sov. Phys. JETP 35,\n834 (1972); D. Chesters, Phys. Rev. D 7, 2863 (1973); M. Pardy, Phys. Lett.\nB336, 362 (1994).\n14. R. Lehnert and R. Potting, Phys. Rev. Lett. 93, 110402 (2004); Phys. Rev.\nD70, 125010 (2004); B. Altschul, Phys. Rev. Lett. 98, 041603 (2007); Phys.\nRev. D75, 105003 (2007); Astropart. Phys. 28, 380 (2007); Nucl. Phys. B\n796, 262 (2008); M. Schreck, arXiv:1310.5159; M.A. Hohensee et al., Phys.\nRev. Lett. 102, 170402 (2009); Phys. Rev. D 80, 036010 (2009).\n15. G.D. Moore and A.E. Nelson, JHEP 09, 023 (2001).\n16.Catalog of Cosmic Rays, http://eas.ysn.ru/catalog; D.J. Bird et al., Astro-\nphys.J.441, 144 (1995); R.U. Abbasi et al., Astropart. Phys. 30, 175 (2008);\nA. Aabet al., Astrophys. J. 804, 15 (2015); U.R. Abbasi et al., Astrophys.\nJ.790, L21 (2014).\n17. J.D. Tasson, in V.A. Kosteleck´ y, ed., CPT and Lorentz Symmetry VI , World\nScientific, Singapore, 2014; arXiv:1308.1171.\n18. H. Panjwani, L. Carbone, and C.C. Speake, in V.A. Kostele ck´ y, ed., CPT\nand Lorentz Symmetry V , World Scientific, Singapore, 2011.\n19. M.A. Hohensee et al., Phys. Rev. Lett. 106, 151102 (2011).\n20. J.D. Tasson, Phys. Rev. D 86, 124021 (2012).\n21. A. Hees et al., Phys. Rev. D 92, 064049 (2015).\n22. M.A. Hohensee, H. M¨ uller, and R.B. Wiringa, Phys. Rev. L ett.111, 151102\n(2013).\n23. R.J. Jennings, J.D. Tasson, and S. Yang, Phys. Rev. D 92, 125028 (2015)."    },    {        "title": "2205.02594v3.Lorentz_symmetry_breaking____classical_and_quantum_aspects.pdf",        "content": "arXiv:2205.02594v3  [hep-th]  12 Jan 2023Lorentz symmetry breaking – classical and\nquantum aspects\nT. Mariz, J. R. Nascimento, A. Yu. Petrov\nThis is a preprint of the following work: Tiago Mariz, Jose Roberto Nas cimento,\nAlbert Petrov, Lorentz Symmetry Breaking – Classical and Quantu m Aspects, 2022,\nSpringer, reproduced with permission of Springer Nature Switzerla nd AG 2022. The final\nauthenticated version is available online at: https://doi.org/10.1007 /978-3-031-20120-2.\n1Contents\n1 Introduction 7\n2 Classical aspects 11\n2.1 Review of LV extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n2.2 Wave propagation in LV theories . . . . . . . . . . . . . . . . . . . . . . . 15\n2.3 Duality issues in LV theories . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n2.4 Spontaneous breaking of Lorentz symmetry . . . . . . . . . . . . . . . . . . 20\n2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n3 Perturbative generation 27\n3.1 Problem of ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n3.2 LV contributions in the scalar sector . . . . . . . . . . . . . . . . . . . . . 30\n3.3 Mixed LV scalar-vector contribution . . . . . . . . . . . . . . . . . . . . . . 32\n3.4 The Carroll-Field-Jackiw term . . . . . . . . . . . . . . . . . . . . . . . . . 36\n3.4.1 General situation and zero temperature case . . . . . . . . . . . . . 36\n3.4.2 Finite temperature case . . . . . . . . . . . . . . . . . . . . . . . . 41\n3.4.3 Non-Abelian generalization . . . . . . . . . . . . . . . . . . . . . . . 43\n3.5 CPT-even term for the gauge field . . . . . . . . . . . . . . . . . . . . . . . 44\n3.6 Higher-derivative LV terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0\n3.7 Divergent contributions in gauge and spinor sectors of LV QED . . . . . . 53\n3.8 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55\n4 Lorentz symmetry breaking and noncommutativity 57\n4.1 Formulations for noncommutativity . . . . . . . . . . . . . . . . . . . . . . 58\n4.2 LV impacts of the Moyal product . . . . . . . . . . . . . . . . . . . . . . . 5 9\n4.3 Seiberg-Witten map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62\n4.4 Noncommutative fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65\n4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68\n35 Lorentz symmetry breaking and supersymmetry 71\n5.1 Deformation of the supersymmetry algebra . . . . . . . . . . . . . . . . . . 72\n5.2 Lorentz symmetry breaking through introducing extra superfi elds . . . . . 77\n5.3 Straightforward breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79\n5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82\n6 Lorentz and CPT symmetry breaking in gravity 85\n6.1 Motivations for 4D gravitational CS term . . . . . . . . . . . . . . . . . . . 86\n6.2 Perturbative generation of the gravitational CS term . . . . . . . . . . . . 89\n6.3 Problem of spontaneous Lorentz symmetry breaking in gravity . . . . . . . 93\n6.4 Possible LV terms in gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 95\n6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98\n7 Experimental studies 101\n7.1 Non-gravitational studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102\n7.2 Gravitational effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5\n4Abstract\nIn this book, we review various aspects of the Lorentz symmetry b reaking, both classical\nandquantumones, withthespecialinteresttoperturbativegene rationofLorentz-breaking\nterms. We present impacts of Lorentz symmetry breaking in nonco mmutative and super-\nsymmetric theories. Also, we discuss the problem of Lorentz symme try breaking in a\ncurved space-time. The book is closed with a review of experimental studies of Lorentz\nsymmetry breaking.\n56Chapter 1\nIntroduction\nThe Lorentz symmetry, being certainly one of the most important s ymmetries of the\nquantum field theory, naturally requires detailed studies. One of th e most important\nrelated issues is the determination of its limits of applicability. While validit y of this\nsymmetry continues to be verified with a high degree of precision [1], s ome profound\ntheoretical motivations for its breaking must be discussed. Alread y in 1989, in [2], it\nhas been supposed that, in the low-energy limit of the bosonic string , some vector, or,\nin general, tensor fields can acquire non-zero expectation values, which, in their turn,\nintroduce some privileged directions in the space-time, and, theref ore, break the Lorentz\nsymmetry. As a next step, the first Lorentz-violating (LV) gener alization of a known field\ntheory model was proposed, that is, the modification of the usual four-dimensional QED\nby the following additive term called the Carroll-Field-Jackiw (CFJ) ter m [3]:\nLCFJ=ǫαβγδkαAβ∂γAδ. (1.1)\nHere, thekαis a constant vector, whose components are assumed to be small, in troducing\nthe anisotropy in the space-time and thus breaking the Lorentz sy mmetry1. This term\ncan be treated as a natural four-dimensional extension of the we ll-known Chern-Simons\nterm, being gauge invariant up to a total derivative. Evidently, the introduction of the\nCFJ term called an interest to investigation of its implications both at t he classical level,\nespecially dispersion relations and issues related with unitarity and ca usality, and at the\nquantum level, that is, possibilities for its perturbative generation, studies of renormal-\nization and renormalizability, relation with possible anomalies, etc., and some important\nresults obtained within these studies are reviewed in this book. Furt her, other LV exten-\nsions have been proposed, not only for the electromagnetic field, b ut for spinor and scalar\n1Within this book we discuss only small LV modifications of known field the ory models and do not\naddress the theories characterized by strong space-time asymm etry like the Horava-Lifshitz-like ones [4].\n7onesaswell, whichallowedtoformulatetheLVstandardmodelexten sion (LVSME) [5,6],\nmaking thus a Lorentz symmetry breaking to be an experimentally te stable phenomenon.\nIt is important to say here that the CFJ term breaks the CPT symme try as well, due to\nthe presence of the constant vector. In general, the LV terms in volving constant tensors\nof odd ranks break also the CPT symmetry, while the presence of co nstant tensors of even\nranks maintains the CPT invariance, see e.g. [7].\nClearly, it is necessary here to discuss the reasons allowing for the a pplicability of\nthe usual relativistic formalism within LV theories. Really, in a LV theor y there can\nbe two types of Lorentz transformations [5, 6, 8, 9]: the observe r ones and the particle\nones. The background fields, that is, just those ones represent ed by constant vectors or\ntensors, transform under the observer Lorentz transformat ions but not under the particle\nLorentz transformations, while the dynamical fields transform bo th under observer and\nparticle Lorentz transformations. Therefore, the Lagrangians of LV theories continue\nto be invariant under observer Lorentz transformations but not invariant under particle\nLorentz transformations (the last ones are sometimes also denom inated as active Lorentz\ntransformations) [5, 6]. This allows to apply the relativistic methodolo gy within the\nframework of the observer Lorentz transformations.\nBesides of the string theory, there are other reasons for the Lo rentz symmetry break-\ning. The most important of them are, first, the possible variability of the fundamental\nconstants, especially, the speed of light, which began to be discuss ed in order to explain\nthe cosmic acceleration (see e.g. [10]), second, the space-time non commutativity, which in\nthe most used, Moyal-type version involves a constant second-ra nk antisymmetric tensor\nΘµν, which clearly is not Lorentz-invariant [11], third, the loop quantum g ravity proposed\nas an attempt to quantize the geometry (see for a review [12]), fou rth, the concept of the\nspace-time foam originally proposed in [13] and further discussed, e.g., in [14, 15] (three\nlast scenarios evidently require existence of some minimal distance s cale), and, probably,\nother reasons.\nThe main implications of the LV extensions of field theory models, espe cially, of the\nelectrodynamics, at the tree level consist in nontrivial behavior of the possible classical\nsolutions, which can display in a vacuum the phenomena characterist ic for propagation of\nelectromagnetic waves in nontrivial media, such as birefringence [16 , 17] and rotation of\nthe polarization plane [16], which can allow to use LV models for an effect ive description\nfor certain condensed matter phenomena, some examples of such studies are given in\n[18, 19, 20]. Besides of this, one of the important issues at the tree level is also related\nwith the problems of unitarity and causality, which, in general, are no t guaranteed for LV\ntheories, so that, typically, to achieve unitarity and causality, the LV parameters should\nsatisfy special conditions which vary for different theories, see e.g . [21, 22, 23]. However,\n8it turnsouttobethatformany examples oftheLVtheories neither unitarity norcausality\nare broken for some characteristic, usually small, values of LV para meters.\nAt the perturbative level, the main idea consists in a possibility to gene rate certain LV\nadditive terms which can be employed within extensions of known field t heory models, as\nquantum corrections, through possible couplings of these fields to some matter, typically\nspinor one. This method is a reminiscence of an old concept of emerge nt dynamics [24],\nand its first application in the context of Lorentz symmetry breakin g has been carried out\nto generate the CFJ term [25]. This term turned out to be finite, alth ough it is formally\nsuperficially divergent, and ambiguous, which called a discussion abou t possible origins of\nthese ambiguities and their relation with the chiral anomaly [26]. Furt her, the ambiguity\nwas shown to occur not only for the CFJ term but also for some othe r LV terms, being\nalso related with anomalies – straightforward generalizations of the chiral anomaly and\ngravitational anomalies [27]. The possibility to generate different LV t erms, especially\nthe modifications for the electrodynamics, has been studied in differ ent dimensions of the\nspace-time, from two to five, and will be one of the main issues consid ered in this book.\nWe already noted that one of the reasons for interest to LV theor ies is motivated\nby development of the concept of space-time noncommutativity [28 ] (for a discussion\nof a relation between Lorentz symmetry breaking and noncommuta tivity, see also [29];\ngeneralization of this correspondence to a curved space-time cas e is considered in [30]).\nIn this book we review the LV impacts of most popular formulations of noncommutative\nfield theories, that is, the Moyal product formulation and the Seibe rg-Witten map. We\nnote that unlike other LV theories, the noncommutative field theor ies are non-polynomial\nwith respect to the LV parameters even at the classical level. We als o describe the\nnoncommutative field approach which is more similar to the usual LV me thodology.\nOne more important point in studying the LV theories consists in poss ibility of de-\nvelopment of their supersymmetric extensions. There was a commo n belief that breaking\nof the Lorentz symmetry would necessarily imply breaking of the sup ersymmetry as well.\nHowever, the prescription allowing to construct the LV deformatio n of the supersymme-\ntry algebra was proposed already in 2001 [31]. Further, two other m anners to construct\nthe supersymmetric LV theories were proposed: first, introducin g of the extra super-\nfield(s) whose component(s) are related with some constant vect ors (tensors), second,\ndirect adding of LV terms where derivatives acting on superfields ar e contracted with\nsome LV vectors and tensors. We note that in all these cases, all p owerful machinery of\nsupergraphs can be applied to obtain quantum corrections.\nAn important issue of studying the Lorentz symmetry breaking is de voted to the pos-\nsibility of constructing LV extensions of gravity. The main difficulty in t his study consists\nin the fact that a straightforward LV modification of gravity models would break the\n9general coordinate invariance as well, while it would be probably prefe rable to maintain\nthe general coordinate invariance, playing the role of the gauge sy mmetry for the gravi-\ntational field. The only known Lorentz-CPT breaking additive term in gravity possessing\nthis property is the four-dimensional gravitational Chern-Simons term [32]. Many issues\nrelated to this term have been studied in great details, including poss ibilities of its pertur-\nbative generation, arising of ambiguities, which, in this case, are linke d with gravitational\nanomalies, and verification of consistency of known exact solutions of the usual general\nrelativity within the modified gravity model whose action is given by a su m of this term\nand the usual Einstein-Hilbert action, that is, the Chern-Simons mo dified gravity (for a\nreview of various results obtained in this theory, see also [33]). In th e book, we discuss\nthis term and some other possible Lorentz and/or CPT breaking ter ms in gravity.\nOur book has the following structure. In the Chapter 2, we review t he tree-level as-\npects of LV theories; in the Chapter 3, we discuss perturbative ge neration of LV quantum\ncorrections depending on scalar and vector fields, in space-times o f different dimensions\nfrom 2 to 5; the Chapter 4 will be devoted to discussing of the space -time noncommuta-\ntivity within the context of the Lorentz symmetry breaking; in the C hapter 5, we discuss\nsupersymmetric LV theories; in the Chapter 6, we discuss LV exten sions of gravity, espe-\ncially the four-dimensional gravitational Chern-Simons term; and, in the Chapter 7, we\npresent a review of experimental studies of the Lorentz symmetr y breaking.\n10Chapter 2\nClassical aspects of the Lorentz\nsymmetry breaking\nThe aim of this chapter consists in discussing of typical phenomena c haracteristic for LV\ntheories at the classical level. There is a variety of manners to cons truct LV extensions\nfor different theories describing scalar, spinor, electromagnetic a nd gravitational fields (a\nvery extensive review of corresponding additive terms can be foun d in [34]), however,\nthe possible impacts of all these extensions are very similar. One of t he most important\nconsequences of the Lorentz symmetry breaking consists in a def ormation of dispersion\nrelations, which leads to the concept of variable speed of light (see e .g. [10]). Another\nits important implications consists in unusual behavior of plane wave s olutions which\ndisplay the phenomena typical for propagation of electromagnetic waves in nontrivial\nmedia, such as birefringence of waves and rotation of their plane of polarization. In this\nchapter, we, first, review the most important LV modifications of d ifferent field theory\nmodels and review their tree-level properties, second, describe t he characteristic features\nof LV theories, third, consider their duality aspects, where the dis persion relations play\nan important role, fourth, discuss the spontaneous Lorentz sym metry breaking.\n2.1 Review of most important LV extensions of field\ntheory models\nHere, we present the most studied terms among those ones propo sed in [34], and review\nsome other interesting terms. We start our consideration with the more generic renormal-\nizable LV extension of the four-dimensional QED without higher deriv atives, discussed in\n11[35]:\nL=¯ψ(iΓνDν−M)ψ−1\n4FµνFµν−1\n4κµνλρFµνFλρ+ǫµνλρkµAν∂λAρ,(2.1)\nwhere\nΓν=γν+cµνγµ+dµνγµγ5+eν+ifνγ5+1\n2gλµνσλµ,\nM=m+aµγµ+bµγµγ5+1\n2Hµνσµν, (2.2)\nDµ=∂µ−ieAµis the simplest covariant derivative, and κµνλρ,kµ,aµ,bµcµν,dµν,eµ,\nfµ,gλµν,Hµνare the constant (pseudo)tensors of corresponding ranks. In what follows,\nwe discuss different terms contributing to this Lagrangian, and fur ther, we consider as\nwell non-minimal operators whose dimension is higher than four, so t hat they yield non-\nrenormalizable couplings or higher-derivative extensions of quadra tic actions. Also, we\nwill introduce the LV modifications for scalar field theories and some o ther models.\nThe first consistent example of a LV theory is the electrodynamics w ith the additive\nCFJ term (1.1). It should be noted that, besides of the Lorentz sy mmetry, the CPT\nsymmetry is also broken in this case, as it occurs for any term involvin g a constant\ntensor of an odd rank [7], while for an even-rank constant tensor t he CPT invariance is\nmaintained. The important aspect of this theory consists also in its g auge invariance. So,\nthe Lagrangian of the simplest LV extension of the spinor QED, being the particular case\nof the generic theory (2.1), can be written as (see e.g. [25]):\nLQED,LV=−1\n4FµνFµν+ǫαβγδkαAβ∂γAδ+¯ψ(i∂ /−eA /−m+b /γ5)ψ. (2.3)\nThe dispersion relation in the purely gauge sector of this theory wer e found already in [3]\nand looks like\np4+p2k2= (k·p)2, (2.4)\nwhere thepαis a 4-momentum. Anticipating the further discussions, we can say t hat in\nprinciple, there are two solutions for it. However, there is no actua l birefringence in this\ncase since one of solutions is non-causal.\nThe CFJ term can be naturally generalized to a non-Abelian case. Ind eed, it is well\nknown that the non-Abelian gauge invariant generalization of the Ch ern-Simons (CS)\nterm, for the Lie-algebra valued vector field Aµ=AµaTa, whereTaare the generators of\nthe algebra, has the form [36]:\nLCS=ǫαβγtr(Aα∂βAγ+2\n3AαAβAγ). (2.5)\n12Promoting this expression to four dimensions, in the same manner in w hich one can treat\nthe CFJ term as an extension of the usual CS term, one arrives at\nLna=ǫαβγδkδtr(Aα∂βAγ+2\n3AαAβAγ). (2.6)\nThis term has been originally proposed in [37], in order to obtain the mos t generic renor-\nmalizable LV extension of the spinor QED. And, in [38], this term was sho wn to arise as\na one-loop quantum correction.\nOne more important LV extension of electrodynamics is CPT-even. I ts standard form\nlooks like (cf. [34]):\nLeven=−1\n4καβγδFαβFγδ. (2.7)\nTheκαβγδisaconstant fourth-ranktensor whose symmetry isthesame aso fthecurvature\ntensor. In one of the most typical cases, that is, the aether one [39], this tensor is chosen\nto be expressed in terms of a single constant vector uα:\nκαβγδ=−(uαuγηβδ−uαuδηβγ+uβuδηαγ−uβuγηαδ), (2.8)\nwhich allows to write the Levenas [40]:\nLeven=uαuγFαβFγβ. (2.9)\nWe note that the form (2.8) is actiually a particular case of the ansat z proposed in [41],\nwhere, instead of uµuν, the generic second-rank tensor kµνis used. In the scalar sector,\none could define an aether-like term for the scalar field [39], given by\nLsc,aether=1\n2φ(u·∂)2φ. (2.10)\nIn the spinor sector one can introduce, first, the CPT-Lorentz b reaking term ¯ψb /γ5ψwhich\nalready has been considered within the model (2.3), second, the ae ther-like contribution\nto the Lagrangiian of the spinor field [39], that is,\nLsp,aether=iuαuβ¯ψγα∂βψ. (2.11)\nIn the gravity sector, the most important term is certainly the fou r-dimensional gravita-\ntional Chern-Simons term [32]:\nICS=ǫαβγδkδ(Γν\nαµ∂βΓµ\nγν+2\n3Γν\nαµΓµ\nβρΓρ\nγν). (2.12)\n13Besidesoftheseterms, itisworthtomentionthehigher-derivative LVterms, characterized\nby dimensions 5 and more, with the most important among them is the M yers-Pospelov\n(MP) term [42], the simplest dimension-5 one, looking like\nLMP=ξ\nMnαFαβ(n·∂)nλǫµνλβFµν. (2.13)\nHereξis a dimensionless constant, Mis some mass scale, usually the Planck mass, and\nnµis a dimensionless LV vector. It is important here that this vector ca n be chosen\nin a manner allowing to avoid higher time derivatives in (2.13) (e.g. with n0= 0) and\nthus eliminating the ghost states, and this approach can be genera lized to other higher-\nderivative LV terms, both in QED and other field theory models, which further will be\nreferred as Myers-Pospelov-like ones (a detailed discussion of unit arity in Myers-Pospelov-\nlike theories can be found, e.g. [43, 44, 45, 46, 47], for many issues r elated to dispersion\nrelations in these theories see also [48] and references therein). T his possibility opens the\nway to construct consistent higher-derivative field theories witho ut use of the Horava-\nLifshitz methodology.\nThe terms listed above are well defined either in four dimensions (tho se ones involving\nthe Levi-Civita symbol), or in any dimensions (aether-like ones). At t he same time, there\nis one more LV term which is well defined only in three dimensions [49], aris ing in the\ncontext of the Julia-Toulouse mechanism (see [50] and references therein), and, therefore,\nsometimes called the Julia-Toulouse (JT) term, namely,\nLJT=ǫαβγvαFβγφ, (2.14)\nintroducing the “mixing” between dynamics of scalar and gauge fields , withvαbeing a\nconstant vector, and even the two-dimensional LV term [51], which is written as\nL2d=ǫαβkαφ∂β¯φ. (2.15)\nIn the next chapter we discuss the manners to generate these qu adratic terms. Typically,\nit can be done with use of appropriate LV couplings.\nWe already noted that the most popular of these couplings, used in t he paradigmatic\nLagrangian (2.3), is given by the term ¯ψb /γ5ψ. Its importance is related with the fact\nthat its presence allows to generate the Levi-Civita symbol, and hen ce, the CFJ term\nand its generalizations. However, recently the non-minimal (i.e. non -renormalizable)\ninteractions, allowing for simpler ways to generate many new terms, also began to attract\nthe interest. The first one is the magnetic coupling involving the coup ling constant g1:\nLmagn=g1ǫµνλρbµjνFλρ, (2.16)\n14wherejρ=¯ψγρψis the usual Dirac current. Namely, this interaction has been used in\n[40] to generate the aether term. The non-renormalizability of this vertex, as well as of\nother non-minimal vertices, in principle can be controlled through re striction of the study\nto the fermionic determinant, that is, to the one-loop order, with t he external lines are\npurely gauge (or scalar, in an extension of the Yukawa model) ones.\nOne can introduce as well the analogue of this coupling involving the us ual vectoraµ\nrather than the axial one bµ, and a coupling constant g2, given by the Lagrangian\nLv=g2aµFµνjν, (2.17)\nwhich was used to generate the axion term in [52].\nAnother vertex, playing an essential role in the scalar sector, is th e Yukawa-like LV\ncoupling with the coupling constant h[40]:\nLYukawa=h¯ψa /ψφ. (2.18)\nThis vertex is renormalizable in four dimensions and allows us to genera te the aether term\nfor the scalar field.\nThere can be other LV couplings as well. A list of all possible LV terms, c ontributing\neither to quadratic action or to interaction vertices, with dimension s up to 6, and, in the\npurely gauge sector, with dimensions up to 8, is given in [53]. The typica l examples of\nsuch terms are, first, L(4+n)\nA=−1\n2k(4+n)κλα1...αnµνFκλD(α1...Dαn)Fµνin the gauge sector,\nbeing the natural higher-derivative extensions of the CPT-even t erm−1\n4κµναβFµνFαβ, and\nsecond,L(3+n)\nψD=1\n2c(3+n)µα1...αn¯ψγµiD(α1...iDαn)ψin the spinor sector, that is, higher-\nderivative extensions for the term ¯ψcµνiγµDνψcontributing to the theory (2.1). The\ngeneralizations of other terms from the spinor sector of (2.1) can be constructed in a\nsimilar manner. In the Chapter 3, we use the simplest examples of LV c ouplings to\ngenerate the quadratic LV terms in scalar, spinor and vector sect ors.\n2.2 Wave propagation in LV theories\nAlready in [10], the variability of the speed of light has been proposed a s one of possible\nexplanationofacosmicacceleration. Itiseasytoseethatitcanocc uronlyifthedispersion\nrelation for the electromagnetic field differs from the usual one E=p). The deformation\nof dispersion relations can be implemented also in massive theories [54], implying the\nformE2=/vector p2+m2+f(E,p,m,Λ), wheref(E,p,m,Λ) is some function suggested to be\nsuppressed in the limit Λ → ∞, with Λ is a characteristic energy scale. This function\ncharacterizes the deviation from the usual relativistic scenario. S ometimes it is supposed\n15that Λ is the Planck mass, For example, in [54], the relation\nE2=m2+p2−LPp2E (2.19)\nwas considered, with LPbeing a Planck length. In this case the usual phase speed is\nfound to be v≃1+LP|p|/2, so, even in the massless case, it is not a constant more but\na parameter depending on the reference frame. Other interestin g examples of dispersion\nrelations, discussed in [54], are E2=p2−L2\nPp2E2andE2=p2−L2\nPE4. Also, it is worth\nto mention the following generalized dispersion relations used in [55] f or cosmological\nstudies:\nE2−p2f2(E) = 0. (2.20)\nDifferent forms of f(E) were discussed in [55], e.g., f1(E) = (1+λE)γ, with 2/3≤γ≤1,\nandf2(E) =2λE\n1−e−2λE. Withinthesecontexts, thekeymotivationfortheLorentzsymme try\nbreaking stems from the idea of the minimal length (and presence of the corresponding\nenergy scale) whose existence clearly imposes natural limits of validit y on the Lorentz\nsymmetry. The most important concepts based on this idea are the loop quantum gravity\nand the space-time foam. Many examples of modified dispersion relat ions constructed in\nsimilar manners were discussed also in[56], forming a base of so-calledg eneralized Lorentz\ninvariance with an invariant energy scale, further referred as dou ble special relativity.\nWhile inthepapersmentioned above, thedeformeddispersion relatio nswere discussed\non their own base, the natural question consists in possibility for th eir arising in some\nconsistent field theory models. As an example, we study the relation (2.4), which de-\nscribes the LV electrodynamics with the CFJ term. There are the fo llowing characteristic\nsituations.\n1. The vector kµ= (µ,0,0,0) is time-like. In this case we have E2=/vector p2±|/vector p||µ|[3],\nwhere the solution with the minus sign is unstable at |/vector p|<|µ|, making the wave vector\nto be space-like and implying instability of the solution, and should be dis regarded by\nimposing appropriate boundary conditions.\n2. The vector kµ= (0,µ′,0,0) is space-like. In this case, if we, for the sake of the\nsimplicity, suggest /vector pto be directed along xaxis, we have E2=/vector p2+µ′2\n2±/radicalBig\n/vector p2µ′2+µ′4\n4.\nHowever, in this case again the solution with the minus sign implies E2< /vector p2, so the\ncorresponding 4-momentum is space-like, and this solution is unstab le.\nThe possibility for the unstable solutions allowing a 4-momentum to be s pace-like,\nwhich can be eliminated through boundary conditions, frequently ta kes place in LV theo-\nries. Moreover, althoughinthesecasesthebirefringenceisruledo utwiththerequirements\nof stability, in certain situations two solutions consistent with stabilit y conditions can ex-\nist. For example, it was proved in [6], that in an extended LV QED with th e quadratic\n16Lagrangian involving both CFJ and CPT-even terms:\nL=−1\n4FαβFαβ+ǫαβγδkαAβ∂γAδ−1\n4καβγδFαβFγδ, (2.21)\nand with/vector p= (0,0,p), one finds the dispersion relations\nE=p(1+ρ)±/radicalBig\nσ2p2+τ2, (2.22)\nwhereρandσare constants constructed from components of καβγδ, andτis a constant\nconstructed of components of kα, their explicit form is given in [6]. If ρ,σ, andτare small\nenough, the stability of both solutions is guaranteed (at the same t ime, we note that it\nwas shown in [57] that, for kα= 0 with the special form of καβγδ, the birefringence does\nnot occur).\nAnother important effect relatedtothewave propagationinLVthe ories isrelatedwith\nthe rotation of a plane of light polarization. Its possibility was noted in [16], and here\nwe demonstrate this situation by the example of the electrodynamic s with an additive\nMyers-Pospelov (MP) term [42]. Indeed, if we consider the sum of th e usual Maxwell\nterm and the MP term (2.13), with again /vector p= (0,0,p) andnα= (1,0,0,0), one will have\nthe following dispersion relations, with /vector ǫx,ybeing the polarization vectors along xandy\naxes [42]:\n(E2−p2±2ξ\nMp3)(/vector ǫx±i/vector ǫy) = 0. (2.23)\nWe see that the polarizations along the axes xandysatisfy different dispersion relations\nand thus have different velocities, so, the polarization plane will rota te.\nIn principle, these effects justify the possibility to apply the Lorent z symmetry break-\ningasaneffectiveanalogicaldescriptionofcondensed matterphen omena. Somesuccessful\nexamples of such applications are presented in [18].\n2.3 Duality issues in LV theories\nAn important issue in the context of the Lorentz symmetry breakin g is related with\ndiscussions of the possible LV generalizations of the known duality be tween self-dual and\nMaxwell-Chern-Simons theories. Originally, this duality was establishe d in [58]. The\nessence of the idea of the duality is as follows: if one consider the Lag rangians of self-dual\nand Maxwell-Chern-Simons theories, given respectively by the expr essions:\nLSD=m2\n2fαfα−m\n2ǫαβγfα∂βfγ;\nLMCS=−1\n4FαβFαβ+m\n2ǫαβγAα∂βAγ, (2.24)\n17the dynamics of these theories is related through the mapping\nfα↔1\nmFα≡1\nmǫαβγ∂βAγ. (2.25)\nMoreover, if one introduces to both these Lagrangians the intera ction terms, fαjαand\nAαGαrespectively, one will arrive at the equations of motion:\n−mǫαβγ∂βfγ+m2fα=jα;\n−mǫαβγ∂βFγ+m2Fα=Gα, (2.26)\nwhich, besides of the duality between fαand1\nmFα, establishes also the mapping between\nthe currents jαandGα. This duality has been discussed in great details in [59], where it\nwas confirmed to occur not only for the case of the usual spinor co upling characterized\nby the current jα=¯ψγαψ, but also for coupling to scalar matter whose contribution to\na Lagrangian cannot be represented as fαjα. The natural question is the possibility to\nconstruct the LV extensions of these results, so that new couplin gs for fields and currents\nturn out to be implied by the duality. The simplest example where such a coupling arises,\nwithout Lorentz-breaking effects yet, is given in [59], where it was sh own that the dual\nterm to the usual coupling fαjαis the magnetic coupling1\nmFµjµ.\nWe start with the following simple generalization of the self-dual theo ry [60]:\nLSD=−m2\n2fαfα+m\n2ǫαβγfα∂βfγ+1\n2∂µφ∂µφ+fµ(2mφvµ+jµ), (2.27)\nwhere we have introduced the LV quadratic term, 2 mφvµfµ. Using the gauge embedding\nmethod (which is one of most efficient methods to construct a dual t o a given theory, see\ne.g. [59]), we arrive at the following dual Lagrangian:\nLMCS=1\n2FαFα−m\n2ǫαβγAα∂βAγ+1\n2∂µφ∂µφ+φǫµνλvµFνλ+2φ2vµvµ+\n+1\n2m2jµjµ+1\nmjµFµ+2\nmφvµjµ. (2.28)\nFirst, the duality of these two theories is confirmed with use of the m apping\nfα↔1\nmFα+2φ\nmvµ+1\nm2jµ, (2.29)\nwhich means that two theories are dynamically equivalent. Second, o ur important result\nconsists in generation of new terms, not only the Thirring current- current interaction\nand the magnetic coupling, but also the remarkable mixed quadratic t erm (2.14), which\nlinks the dynamics of gauge and scalar fields and has a key role within th e Julia-Toulouse\nmechanism [49].\n18Third, one can discuss as well the dispersion relations in these theor ies. For the self-\ndual theory, one has three types of dispersion relations, namely,\n(i) :E2=/vector p2; (ii) :E2=/vector p2+m2;\n(iii) : (E2−/vector p2)2−(E2−/vector p2)m2+4m2v2= 0; (2.30)\nwhile for the dual MCS-like theory, one has\n(i) :E2=/vector p2; (ii) :E2=/vector p2+m2; (iii) :E2−/vector p2= 4v2; (2.31)\n(iv) : (E2−/vector p2−m2)(E2−/vector p2−4v2)+v2(E2−/vector p2)−\n−(/vector v·/vector p−v0E)2= 0.\nFormally, some of these dispersion relations are different, however , only those ones cor-\nresponding to physical degrees of freedom should coincide for bot h theories [59]. In this\ncase, only the relations (i) and (ii) are common for these theories, a nd they correspond\nto the scalar field and the only physical degree of freedom of the th ree-dimensional vector\nfield respectively.\nIn the paper [61], the similar analysis has been carried out also for CPT -even LV\nextensions of the self-dual and Maxwell-Chern-Simons theories, w hose Lagrangians look\nlike\nLSD=m2\n2fαhαβfβ−m\n2ǫαβγfα∂βfγ+fµjµ;\nLMCS=−1\n4FαβFαβ+m\n2ǫαβγAα∂βAγ−α\n8(ǫµνλbµFνλ)2\n−1\n2m2jµ(h−1)µνjν+1\nmjµ(h−1)µνFµ. (2.32)\nHere,hαβ=ηαβ−βbαbβand (h−1)αβ=ηαβ+αbαbβ, withα=β\n1−βb2, where we assumed\nsmallness of Lorentz-breaking effects requiring |β| ≪1, andbβis a constant vector imple-\nmenting theLorentzsymmetry breaking. Inthiscase weobtain, am ongall, theaether-like\nterm (2.9), arising from the ( ǫµνλbµFνλ)2term. The equivalence of the equations of mo-\ntion in these theories can be shown as well [61]. In this case there is on ly one physical\ndegree of freedom, and only one physical dispersion relation, which is common in both\ntheories. It looks like\nE2−/vector p2−m2+α[b2(E2−/vector p2)−(/vectorb·/vector p−b0E)2] = 0. (2.33)\nThe detailed analysis of stability and causality issues can be done in this case, it is\npresented in [61]. We note again that if the Lorentz-breaking param eters are enough\nsmall we have no problems with unitarity and causality.\n19Moreover, the Lorentz symmetry breaking opens the way to gene ralize the duality for\nthe four-dimensional case. In this case we start with the self-dua l model extension\nLSD=m2\n2fαfα−1\n2ǫαβγδkαfβ∂γfδ+fαjα, (2.34)\nwhere we have replaced the Chern-Simons term by its four-dimensio nal analogue, that\nis, the CFJ term. Under the gauge embedding procedure [59], we arr ive at the following\ndual action:\nLnew=1\n2ǫαβγδkαfβ∂γfδ+1\n2m2jαjα+1\n2m2ǫαβγδkαjβFγδ+k2\n4m2FαβFαβ−\n−1\n2m2kαkβFβγFαγ. (2.35)\nIn this case, the dual action turns out to involve the new terms, th at is, the aether term\n(2.9) and the LV magnetic coupling (2.16). Thus, we see that, first, the aether term arises\ntogether with the coupling used for its generation [40], second, the dual of the CPT-odd\ntheory is CPT-even. The dispersion relations in these theories [62], w ith the first of them\nvalid only in the Maxwell-like theory (2.35), are given by\n(i)k2p2−(k·p)2= 0,\n(ii)k2p2−(k·p)2+m4= 0. (2.36)\nThe second of these relations is common for both models and thus co rresponds to the\nphysical degree of freedom. This duality can be verified in different m anners [62]. To\nclose this section, we emphasize that the dual immersion of Lorentz -breaking theories\nindeed allows to generate new LV terms.\n2.4 Spontaneous breaking of Lorentz symmetry\nIn principle, there are two main manners to break the Lorentz symm etry, that is, explicit\none, where a new term proportional to a constant vector or tens or is added to a Lorentz\ninvariant action, and the spontaneous one, when the original actio n is Lorentz invariant,\nwhile its minimum introduces a privileged space-time direction. While in mos t part of\nthis book we suggest that the Lorentz symmetry breaking is explicit , it is important to\nconsider the spontaneous one as well, because, first, it provides a natural mechanism\nfor originating the LV theories, second, it is deeply related with strin g theory, and thus\npossesses profound motivations [2], third, as we will see in the Chapt er 6, it plays a\nfundamental role within the curved space-time context. Following t his paper, namely\nthe spontaneous breaking of Lorentz symmetry in a low-energy limit of a string theory\n20is responsible for arising the LV theories which are thus treated as lo w-energy effective\ntheories.\nThe paradigmatic mechanism for breaking the Lorentz symmetry in a spontaneous\nmanner is based on the bumblebee model originally introduced within th e gravity context\n[34]. However, this model can be treated as well in the plane space-t ime. By definition,\nthis model describes the vector field Bµcalled the bumblebee field, with a self-coupling\npotential:\nS=/integraldisplay\nd4x/parenleftBig\n−1\n4BµνBµν−V(BµBµ∓b2)/parenrightBig\n. (2.37)\nHereBµν=∂µBν−∂νBµis a stress tensor, and b2>0. The potential V(BµBµ∓b2) is\nsuggested, by renormalizability reasons, to be either of the form V=f\n4!(BµBµ∓b2)2with\nfbeing a coupling constant, or of the form V=1\n2σ(BµBµ∓b2), withσbeing a Lagrange\nmultiplier field; for the (+ −−−) signature, the sign in the expression BµBµ∓b2is chosen\nto be (−) for the time-like Bµ, and (+) for the space-like one.\nIt is clear that for both these forms of the potential the minimum is a chieved for\nBµBµ=±b2. Therefore one concludes that the value of Bµequal to the constant vector\nbµ(satisfying the relation bµbµ=±b2), which can be chosen to be, e.g., parallel to one\nof coordinate axes, say x0orxi, introduces a privileged direction. Afterward, one can\nproceed in a manner similar to the Higgs mechanism, that is, we can exp and theBµnear\nthe minimum as Bµ→˜Bµ+bµ, and as a result, we get a mass term for the new field ˜Bµ\nlooking likef\n4!(4˜Bµ˜Bνbµbν+˜B2b2). Thus, we arrive at the essentially Lorentz-breaking\ncontribution to the quadratic action, looking like ˜Bµ˜Bνbµbν. Namely this mechanism is\nresponsible for arising of LV terms in a low-energy limit of string theor y [2], so that it\nis natural to treat LV theories as effective theories for a low-ener gy limit of some more\nfundamental theory. If the bumblebee action, including, what is es pecially important,\nits potential, is induced through radiative corrections, the Lorent z symmetry is said to\nbe broken dynamically. This is the simplest example of a situation where the Lorentz\nsymmetry breaking can be treated as an emergent phenomenon. A lso, in [63], it was\nshown that if the action (2.37) is extended to a curved space-time, one can introduce the\nbumblebee perturbation ˜Bµthrough the replacement Bµ→bµ+˜Bµ, so thatbµis the\nv.e.v. ofBµ, the induced metric is gµν=ηµν+βbµbν, withβis some constant, ˜Bµis\nsmall, and both the potential Vand its derivative vanish at Bµ=bµ(i.e. at˜Bµ= 0). As\na result, one gets the LV action for the ˜Bµ, composed by the sum of the usual Maxwell\nterm, the aether-like term (2.7), where κµνλρis constructed on the base of the vector bµ,\nand the corresponding potential. Therefore, this mechanism allows to generate a CPT-\neven LV terms for the vector field. In [64], this approach was gene ralized for the case of\nthe antisymmetric tensor field.\n21Let us consider now examples of dynamical Lorentz symmetry brea king, for theories\ndescribing vectors and antisymmetric tensors. Some first discuss ions of this approach can\nbe found in [65], and its further development, for the vector field, w as presented in [66],\nand for the antisymmetric tensor one, in [67]. Although these studie s involve calculations\nof quantum corrections, it is very natural to discuss their results namely here, within a\ngeneral context of spontaneous Lorentz symmetry breaking.\nWe start with the following four-fermion model, chosen to be massles s for the sake of\nthe simplicity [66]:\nL0=i¯ψ∂ /ψ−G\n2(¯ψγµγ5ψ)2. (2.38)\nAsusual, weintroducethevectorfield Bµinordertoavoidarisingofthenon-renormalizable\nfour-fermion vertex:\nL=g2\n2BµBµ+¯ψ(i∂ /−eB /γ5)ψ. (2.39)\nIt is immediate to see that, eliminating the Bµfrom this action through purely algebraic\nequations of motion, we recover (2.38) with G=e2\ng2. We can write down the one-loop\ncorrected effective potential for this theory as\nVeff=−g2\n2BµBµ+iTrln(p /−eB /γ5). (2.40)\nTo break theLorentz symmetry, we must first findthe minimum of th eVeff. It isobtained\nfrom the conditiondVeff\ndBµ|eBµ=bµ= 0. The resulting bµvector will introduce the privileged\nspace-time direction. As a result, we obtain\ndV\ndBµ|eB=b=−g2\nebµ−iΠµ= 0, (2.41)\nwhere Π µ= tr/integraltextd4k\n(2π)4i\nk /−b /γ5(−ieγµγ5). Calculating the trace in this expression with use of\nthe exact propagator of the fermionic field ( k /−b /γ5)−1instead of its expansion in power\nseries inbµ(we note that in the massless case this exact propagator has a ver y simple\nform; different expressions for the spinor propagator will be discu ssed in the next chapter,\ndetails of the calculation presented here can be found in [66]), we find that\nΠµ=ieb2\n3π2bµ. (2.42)\nWe note that the Πµis finite: during its calculation, instead of an expected UV divergence\none finds a removable singularity, poleterms cancel out, andno ren ormalization is needed.\nIn next chapters we will see that removable singularities sometimes a rise within one-loop\ncalculations in LV theories, providing finiteness of corresponding co ntributions.\n22So, the gap equation (2.41) yields\ndVeff\ndBµ/vextendsingle/vextendsingle/vextendsingle\neBµ=bµ=/parenleftBigg\n−1\nG+b2\n3π2/parenrightBigg\nebµ= 0, (2.43)\nand its integration gives the effective potential:\nVeff=e4\n12π2B4−e2b2\n6π2B2+α, (2.44)\nwhereαis an arbitrary constant, and for α=b4\n12π2, withb2=3π2\nG, one hasVeff=\n1\n12π2(e2B2−b2)2, that is, the positively defined potential perfectly reproducing th e bum-\nblebee form. If the finite temperature is introduced, one can find t hat this potential\ngenerates phase transitions [66].\nIt is interesting now to consider the dynamics of Bµexpanded about the non-zero\nvacuum< Bµ>=bµ\ne. In this case, we make a shift Bµ=bµ\ne+Aµ, withAµis the new\ndynamical field. In this case, the effective action of Aµis\nSeff=g2\n2(bµ\ne+Aµ)(bµ\ne+Aµ)−itr/integraldisplayd4k\n(2π)4ln(k /−b /γ5−eA /γ5). (2.45)\nAfter one-loop calculations (see [66] for details), we find\nL=−1\n4FµνFµν+e2\n24π2bµǫµνλρAνFλρ−\n−e2\n48π2(∂µAµ)2−e4\n12π2/parenleftbigg\nAµAµ+2\neA·b/parenrightbigg2\n. (2.46)\ni.e. we obtained an appropriate structure of the bumblebee Lagran gian, involving a\nMaxwell-like kinetic term (we note that such a form of the kinetic term is necessary\nto avoid ghosts, see [68]), a positively defined potential, and also, P roca-like and CFJ\nterms.\nNow, let us consider the antisymmetric tensor field case. We start w ith another four-\nfermion model:\nL=¯ψ(i∂ /−m)ψ−G\n2JµνJµν, (2.47)\nwhereJµν=i¯ψγ[µ∂ν]γq\n5ψ, withq= 1,2, is a (pseudo)tensor current which can be coupled\nto an antisymmetric tensor field in the minimal manner, i.e. through th e termBµνJµν.\nWe note that, unlike the above case of the vector bumblebee model, in this theory the\nchoice of zero mass will not simplify calculations essentially. Again, we in troduce the\nbumblebee field, in this case, tensorial one, Bµν, in order to eliminate the four-fermion\n23coupling:\nL=L0+g2\n2/parenleftBigg\nBµν−e\ng2Jµν/parenrightBigg2\n=g2\n2BµνBµν+¯ψ(i∂ /−ieBµνγ[µ∂ν]γq\n5−m)ψ. (2.48)\nSimilarly to (2.40), we can write the one-loop corrected effective pot ential of our theory\nas\nVeff=−g2\n2BµνBµν+itr/integraldisplayd4p\n(2π)4ln(p /−eBµνγ[µpν]γq\n5−m). (2.49)\nIn this case, we find that at q= 2, the one-loop effective potential can be obtained exactly\n[67]. However, for study of spontaneous Lorentz symmetry brea king, assuming that fields\nare small, it is sufficient to find only the terms up to the fourth order in dynamical fields.\nIn [67] our effective potential was shown to look like, after renorma lization:\nV(q=2)\neff=m4\n4(e2BµνBµν−bµνbµν)2+..., (2.50)\nwhere thebµνis the non-zero v.e.v. of Bµν.\nForq= 1, we can find the one-loop effective potential only order by order . The\nrelevant contribution looks like\nV(q=1)\neff=7m4\n12(e2BµνBµν−bµνbµν)2+7e4m4\n48(Bµν˜Bµν)2+..., (2.51)\nwhere˜Bµνis the dual of Bµν:˜Bµν=1\n2ǫµνρσBρσ. So, in both cases we have continuous\nsets of minima allowing for spontaneous symmetry breaking. The der ivative dependent\ncontributions to the effective action, both for q= 1 andq= 2, can be obtained as well.\nExplicitly, the low-energy effective Lagrangians for these theories , after wave function\nrenormalizations and adding some finite constants, look like\nLB,1=−1\n12HµνλHµνλ−7m4\n12(e2BµνBµν−bµνbµν)2−2\n5(∂αBµα)2−\n−7e4m4\n48(Bµν˜Bµν)2+O(B3), (2.52)\nand\nLB,2=−1\n12HµνλHµνλ−m4\n4(e2BµνBµν−bµνbµν)2−e4m4\n16(Bµν˜Bµν)2+\n+O(B3). (2.53)\nThus, we foundthateach of these Lagrangiansdisplays thesame s tructure asinthevector\nfield case (2.46), i.e. it includes a gauge invariant kinetic term, a poten tial allowing to\nbreak the Lorentz symmetry spontaneously, and some other ter ms.\n24To close the section, we note, first, that the Hamiltonian formulatio n of theories with\nspontaneous Lorentz symmetry breaking displays some peculiaritie s [69, 70] (for the dis-\ncussion of degrees of freedom in such theories, see also [71]), sec ond, that the spontaneous\nLorentz symmetry breaking plays a special role in a curved space-t ime where it appears\nto be the only consistent mechanism to break the Lorentz symmetr y, third, that in some\ncases, dynamical Lorentz symmetry breaking can generate a non trivial topology [72].\nThe detailed discussion of Lorentz symmetry breaking in gravity will b e presented in the\nChapter 6.\n2.5 Conclusions\nInthischapter, wediscussed themainclassical impactsoftheLore ntzsymmetry breaking,\nrelated with general structure of classical actions of various field theory models, corre-\nsponding propagators and plane wave solutions. It is clear that the list of results obtained\nat the classical level is much wider. For example, one can mention man y studies of exact\nsolutions and geometric phases. One of the first important results in this direction is the\nexplicit expression for static solutions of the modified Maxwell equat ions in the three-\ndimensional LV extended QED, whose action involves the Julia-Toulou se term (2.14). It\nwas shown in [73] that in this case the static equations are the fourt h order ones, and\nwhile at small distance the scalar potential displays the usual logarit hmic behavior, at\nlarge distance, instead of the usual exponential-like decay occurr ing in the usual Chern-\nSimons modified QED, the potential is dominated by the LV term and log arithmically\ngrows. Further, solutions of classical equations of motion in differe nt LV extensions of\nQED were studied also, e.g. in [74] (see also references therein). Th e paper [75], where a\npossibility of arising the Aharonov-Casher phase in the extended QE D, with the LV term\nwas introduced through the already mentioned magnetic coupling (2 .16), was studied,\nstarted a line of investigation of geometrical phases in LV theories. Further, the impact\nof Lorentz symmetry breaking in solutions of the Dirac equation has been also considered\nin [76, 77, 78] and many other papers; for the review of these stud ies, we recommend also\n[79].\nAll these results clearly show that the LV effects should have nontr ivial implications\nalready at the tree level. At the same time, one of the most importan t problems is –\nwhat possible mechanisms could be responsible for arising the LV modifi cations of known\nfield theory models? One of these mechanisms uses the spontaneou s Lorentz symmetry\nbreaking [2] discussed briefly in this chapter, with its key feature co nsists in arising of\nLV terms at the tree level as a consequence of choosing some minimu m of the potential\n25introducingtheprivilegedvectorortensor, sothattheLagrangia nofthetheoryconsidered\nnear this minimum displays dependence on this vector (tensor), and the another one,\nbased on the old idea of emergent dynamics [24], is a perturbative ge neration approach,\nintensively usedtojustifythepossibilityforarisingtheLVmodificatio nsofthefieldtheory\nmodels, and applied for the first time already in [25]. Following this idea, t here is some\nfundamental spinor matterfieldcoupledinaLVmanner withotherfie lds(scalar, vector or\ngravityones), sothatdifferent LVadditive termsforthesefieldsa riseasone-loopquantum\ncorrections. We note that, in a certain sense, dynamical symmetr y breaking, involving\nboth generating of a quantum correction and choosing of one minimu m, combines these\napproaches. At the same time we note that the coupling of a gauge fi eld to a scalar field\ninstead of a spinor one is less convenient for perturbative generat ion of LV terms since the\ncorresponding action will not include the Dirac matrices necessary t o generate the Levi-\nCivita symbol which is present in CPT-Lorentz breaking terms discus sed in this book.\nThe perturbative generation methodology allows to obtain many of t he terms proposed\nin [34], and we will discuss it in the next chapter.\n26Chapter 3\nPerturbative generation of LV terms\nIn this chapter, we demonstrate the methodology allowing for gene rating various LV\nterms, involving scalar and gauge fields. Conceptually, we follow the a lready mentioned\nidea of the emergent dynamics [24], suggesting that the new contrib utions to Lagrangians\nof different field models can arise as quantum corrections in some fun damental theories.\nNamely this method has been used in[25], where anappropriate LV cou pling of the spinor\nfield to a constant axial vector bµwithin a LV extension of QED was applied to generate\nthe CFJ term. Just this approach became further the main tool allo wing to obtain new\nLV terms. Besides of this, it is worth to mention also other methodolo gies allowing to do\nthis, such as: dimensional reduction [80], spontaneous Lorentz sy mmetry breaking, that\nwas briefly discussed in the previous chapter, and known, in some ca ses, like in [81], to\noccur in the one-loop corrected effective action of a correspondin g theory, so that one can\nspeak about the dynamical Lorentz symmetry breaking, and the n oncommutative fields\nmethod [82]. However, the perturbative generation seems to be th e most natural manner\nfor obtaining LV terms. We note nevertheless that this way is espec ially appropriate just\nin the case when the corresponding LV quantum corrections are fin ite, since otherwise one\nshould, inordertoensurethemultiplicativerenormalizability, introdu cethecorresponding\nLV term already at the tree level, which clearly contradicts to the int erpretation of the\ncorresponding new LV term as an emergent phenomenon.\n3.1 Problem of ambiguities\nFirst of all, we should discuss the reasons for the most controvers ial phenomenon charac-\nteristicforfour-dimensionalLVgaugecontributions, thatis, the arisingoftheambiguities.\nThe characteristic situation [26] is the following one: the one-loop int egral over the inter-\nnal momenta is an expression looking like a sum of two divergent terms , with the pole\n27parts of these terms mutually cancel. However, any divergent ter m represents itself as a\nsum of a pole part and a finite part. In principle, this finite part can be fixed with use of\nan appropriate normalization condition [83]. Nevertheless, in genera l, there is no reasons\nto prefer any of different normalization conditions in LV theories, es pecially taking into\naccount that the corresponding terms arising after the calculatio ns are actually finite and\nthus do not depend on any renormalization scale. Therefore, within different regulariza-\ntions, or, as is the same, different normalization conditions, the finit e parts turn out to\ndepend on the regularization scheme, and hence the result for the corresponding integral\nover internal momenta is ambiguous. From the formal viewpoint, th is ambiguity is a per-\nfect example of the indeterminate form ∞−∞, whose fixing requires special conditions.\nTypically, two most known ambiguous integrals are considered. The fi rst of them arises\nwithin study of the CFJ term, see e.g. [38], looking like\nJµν=/integraldisplayd4p\n(2π)4ηµνp2−4pµpν+3m2ηµν\n(p2−m2)3. (3.1)\nOne can easily verify that by carrying out the replacement pµpν→1\n4ηµνp2, with a sub-\nsequent integration in four dimensions, and the replacement pµpν→1\n4+ǫηµνp2, with a\nsubsequent integration in 4+ ǫdimensions, and, afterward, taking the limit ǫ→0, one\narrives at different results for Jµν[38]. This difference stays at the finite temperature\nas well, and, moreover, these two possibilities do not exhaust a list of other manners of\ncalculating this integral, with more results can be obtained.\nAnother ambiguous integral was found in [40]. It reads as\nIµν=/integraldisplayd4p\n(2π)4m2ηµν+2pµpν−ηµνp2\n(p2−m2)2, (3.2)\nand also yields different results under the same replacements as abo ve. Thus, we can\nconclude that in LV theories, in certain cases, formally divergent co ntributions can be\nactually finite, but display ambiguities as a consequence of arising the indeterminate form\n∞−∞.\nIn principle, more ambiguous integrals can exist. For example, in a man ner similar to\nthe above discussions, one can show that logarithmically divergent in tegrals of the form\nJ(n)\nµν=/integraldisplayd4p\n(2π)4p2n(ηµνp2−4pµpν+3m2ηµν)\n(p2−m2)n+3, (3.3)\nfor all non-negative n, will be ambiguous by the same reasons, and one of the possi-\nbilities for an analogous generalization for Iµνis described by the following superficially\nquadratically divergent integrals:\nI(n)\nµν=/integraldisplayd4p\n(2π)4p2n(m2ηµν+2pµpν−ηµνp2)\n(p2−m2)n+2. (3.4)\n28However, up to now, there is no known constructive examples of ph ysically motivated\nscenarios where both these ambiguous integrals really arise, excep t of the usual case n= 0\ncorresponding to eqs. (3.1,3.2) and occurring in the above-mention ed LV extensions of\nQED. In principle, other, more involved, ambiguous loop integrals can exist as well, see\nf.e. [84].\nAt the same time, it should be noted that the chiral QED, whose Lagr angian is given\nby [85]\nL=¯ψ(i∂ /+eA /(1−γ5)+b /(1+γ5)−m)ψ, (3.5)\ndisplays a highly nontrivial behavior. Indeed, since the spinor propa gator in this theory\nis the common one, that is,i\nk /−m, the CFJ-like contribution to the effective Lagrangian\nwill be given by the expression:\nS2(p) =−e2\n2tr/integraldisplayd4k\n(2π)4/bracketleftBig1\nk /+p /−mA /(−p)(1−γ5)1\nk /−mb /(1+γ5)×\n×1\nk /−mA /(p)(1−γ5)+ (3.6)\n+1\nk /+p /−mb /(1+γ5)1\nk /+p /−mA /(−p)(1−γ5)×\n×1\nk /−mA /(p)(1−γ5)/bracketrightBig\n.\nLet us now take into account only the first order in the external mo mentump, use the\nexpression1\nk /−m=k /+m\nk2−m2, and consider e.g. only the first contribution of S2. Then, we get\nS2,1(p) =e2\n2tr/integraldisplayd4k\n(2π)4k /+m\n(k2−m2)4p /(k /+m)A /(−p)×\n×(1−γ5)(k /+m)b /(1+γ5)(k /+m)A /(p)(1−γ5). (3.7)\nThis expression is superficially logarithmically divergent and involves th e factor (1 −\nγ5)(k /+m)b /(1 +γ5). However, γ5commutes with even degrees of γµand anticommutes\nwithoddones, hence, theUVleadingmomentumdependentfactor, whichonlycontributes\nto the divergence, turns out to be proportional to (1 −γ5)(1 +γ5) = 0. Thus, the\npotential divergence disappears. The similar situation occurs in ano ther contribution to\n(3.6), implying also vanishing of the leading (divergent) contribution. As a result, unlike\nthe usual calculation of the CFJ term, after the calculation of the t race, the resulting\nexpression is superficially finite and hence contains no ambiguity. The refore, in the chiral\nQED (3.5) the CFJ term is ambiguity-free. The explicit result for it is [85 ]:\nLCFJ=e2\n3π2ǫµνλρbµAν∂λAρ. (3.8)\n29However, one can show that in this case, the absence of an ambiguit y is paid by a price\nof existence of the gauge anomaly. Indeed, it is easy to see that in t his case the gauge-\nbreaking Proca-like divergent term is generated. Hence, the cons istent elimination of\nambiguities requires a strong extension of the model which has been performed in [86].\n3.2 LV contributions in the scalar sector\nIn this section, we discuss the simplest possible LV modifications for t he scalar field mod-\nels. There are two typical terms, namely, the aether-like term (2.1 0) and the essentially\ntwo-dimensional Chern-Simons-like term (2.15).\nThe simplest manner to generate the aether-like term is based on th e use of the LV\nYukawa-like action\nS=/integraldisplay\ndDx/bracketleftBig¯ψ(i∂ /−m)ψ+1\n2(∂µφ∂µφ+m2φ2)−g¯ψa /ψφ/bracketrightBig\n, (3.9)\nwhere the LV vector aµis dimensionless, and the new vertex, given by (2.18), is one of\nthe ingredients of the LV SME [6].\nThe lower contribution to the aether-like term is given by the Feynma n diagram de-\npicted by the Fig. 3.1 [40]:\n••\nFigure 3.1: Two-point function of the scalar field.\nAfter calculating thetraceanddisregarding thetermsproportion altoa2(denoted here\nby dots), which do not break the Lorentz symmetry, we arrive at t he following result:\nS2(p) =−d\n2g2φ(p)φ(−p)[ηµνηρσ+ηµσηρν]aµaρ×\n×/integraldisplaydDk\n(2π)Dkν(kσ+pσ)\n[k2−m2][(k+p)2−m2]+..., (3.10)\nwheredisadimensionofDiracmatricesinthecorrespondingspace-time, and Disaspace-\ntime dimension. It is clear that the aether-like contribution from this integral is finite\nin two- and three-dimensional space-times, and, within the framew ork of the dimensional\nregularization, also in all space-times with higher odd dimensions. Exp licitly, one gets,\nafter returning to the coordinate space:\nS2=−dg2Γ(2−D\n2)\n6(4π)D/2(m2)2−D/2φ(a·∂)2φ. (3.11)\n30This term evidently replays the structure (2.10). We see that at D= 4, the aether-\nlike term should be present already at the tree level, in order to intro duce the one-loop\ncounterterm of the form1\n2Zaetherφ(a·∂)2φ, withZaether= 1−g2\n12π2ǫ. We note that in this\ncase the renormalizability of the theory is not spoiled since no coupling s with a negative\nmass dimension are present. In principle, the aether term for the s calar field can be\ngenerated also through other couplings to a spinor matter, e.g. th e CPT-even one\nS=/integraldisplay\ndDx/bracketleftBig¯ψ(i∂ /+ikµνγµ∂ν−hφ−m)ψ+1\n2(∂µφ∂µφ+m2φ2)/bracketrightBig\n, (3.12)\nwithkµνis a constant tensor, and again, the aether-like term will be finite in t wo, three\nand all higher odd dimensions, and divergent in four dimensions.\nThereisonemorepossibleLVadditivetermforthescalartheory, th atis, theessentially\ntwo-dimensional term(2.15)(see[51]foritsapplicationsforstudie s oftopologicaldefects).\nTo generate it, we start with the following two-dimensional action, s ee [87]:\nS=/integraldisplay\nd2x¯ψ/bracketleftBig\ni∂ /−m−ga /φ−mgγ5χ/bracketrightBig\nψ, (3.13)\nwhereγ5=γ0γ1now is a two-dimensional analogue of the chirality matrix, φis a scalar\nfield (denoted by a solid line), and χis a pseudoscalar one (denoted by a dashed-and-\ndotted line).\n...◦•\nFigure 3.2: Two-point function of φandχfields.\nHere the dark circle is for a /insertion, and the light one is for γ5one. The contribution\nof the Fig. 3.2 looks like\nΓ =/integraldisplayd2p\n(2π)2φ(−p)Π(p)χ(p), (3.14)\nwith Π(p) being a self-energy tensor given by\nΠ(p) =−img2tr/integraldisplayd2k\n(2π)2/bracketleftBig\na /i\nk /−mγ5i\nk /+p /−m/bracketrightBig\n. (3.15)\nAfter straightforward calculations, we find that, in the first orde r in the external pµ,\nΠ(p) =−ig2\n2πǫµνaµpν. (3.16)\n31So, our final result for this contribution to the effective action is\nΓ =g2\n2π/integraldisplay\nd2xaµǫµνφ∂νχ. (3.17)\nThis is just the term (2.15). We note that it is ambiguity-free as is sho uld be, since the\nresult is superficially finite. In principle, such a term, for the special case of a space-like\naµ(namely, the case a0= 0), can be also generated through the noncommutative field\nmethod proposed in [82].\nIn principle, other LV additive terms, involving scalar fields only, can b e introduced,\ne.g., the Myers-Pospelov-like ones (cf. [42]):\nLMP,scal=1\nΛN¯φ(n·∂)Nφ, (3.18)\nwhere Λ Nis some scale factor of an appropriate mass dimension, nµis a dimensionless\nvector, and N≥3. The importance of these terms consists in the fact that for a sp ace-like\nnµ, there will be no higher time derivatives and hence no ghosts, so, on e has a consistent\nscheme to incorporate higher derivatives to a field theory model wit hout use of a Horava-\nLifshitz approach. These terms can be generated through schem es we presented above,\nat higher orders of a derivative expansion of the corresponding eff ective action.\n3.3 Mixed LV scalar-vector contribution\nIn three space-time dimensions, there is an unique possibility to cons truct the quadratic\nLVadditiveterm, involvingfieldsofdifferentnatures, scalaroneand gaugeone,thatis, the\nJulia-Toulouse term (2.14). In the section 2.3, we already noted tha t this term naturally\nemerges within the gauge embedding of the special Lorentz-CPT br eaking extension of\nthe self-dual model. However, this is not an unique way to generate the term (2.14).\nFollowing [88], we start with the following three-dimensional action:\nS=/integraldisplay\nd3x¯ψ/bracketleftBig\ni∂ /−m−eA /−ga /φ/bracketrightBig\nψ. (3.19)\nHereaµis a LV constant vector, and φis a scalar. The one-loop effective action is clearly\ngiven by the functional trace:\nΓ(1)=−iTrln/bracketleftBig\ni∂ /−m−eA /−ga /φ/bracketrightBig\n. (3.20)\nThe desired two-point “mixed” function is described by the Feynman diagram given by\nthe Fig. 3.3.\nHere the black dot is for the a /insertion, the dashed line is for a spinor propagator,\nand solid and wavy lines denote external scalar and gauge fields, res pectively.\n32•\nFigure 3.3: Mixed two-point vector-scalar function.\nThe contribution of this diagram yields\nI=−egtr/integraldisplayd3p\n(2π)3/integraldisplayd3k\n(2π)3A /(−p)(k /+m)φ(p)a /(k /+p /+m)×\n×1\n(k2−m2)[(k+p)2−m2]. (3.21)\nForthediag(+−−) signature, andDiracmatrices ( γ0)α\nβ=σ2,(γ1)α\nβ=iσ1,(γ0)α\nβ=iσ3,\nobeying relations: {γµ,γν}= 2ηµν, tr(γµγνγλ) = 2iǫµνλ, we arrive at\nI=−2iǫαµνegm/integraldisplayd3p\n(2π)3pαAµ(−p)φ(p)aν×\n×/integraldisplayd3k\n(2π)31\n(k2−m2)[(k+p)2−m2]. (3.22)\nThen, after integrating over internal momenta, in the first order inpµwe obtain\nI=ǫλµνegm\n4π|m|/integraldisplayd3p\n(2π)3pλAµ(−p)φ(p)aν. (3.23)\nThis expression can be represented as\nI=−eg\n8πsgn(m)/integraldisplay\nd3xǫλµνFλµaνφ. (3.24)\nThis is exactly the Julia-Toulouse term (2.14). We note that here the sign of the mass\narises. It is rather typical for the three-dimensional theories, s ee e.g. [89], where the\nsimilar situation was shown to take place for the Chern-Simons term.\nThis calculation can be performed through evaluating (3.20) not only with the Feyn-\nman diagram approach as we did now, but also through the proper-t ime method. To\nproceed with it, we add to the (3.20) the field independent term −iTrln(i∂ /+m), in order\nto have a trace of an operator of second order in derivatives, as t he proper-time method\nrequires. So, our one-loop effective action becomes:\nΓ(1)=−iTrln(−✷−m2−eA /(i∂ /+m)−gφa /(i∂ /+m)). (3.25)\nSince we need only the first order in aµ, we expand this expression up to the first order\ninaµand then employ the Schwinger proper-time representation A−1=1\ni/integraltext∞\n0dseisA. We\narrive at\nΓ(1)\n1=−gTr/bracketleftBig/integraldisplay∞\n0dseis(✷+m2+eA /(i∂ /+m))φa /(i∂ /+m)/bracketrightBig\n. (3.26)\n33Then we expand this expression up to first derivatives and to the fir st order in Aµ. We\nobtain\nΓ(1)\n1=−egTr/bracketleftBig/integraldisplay∞\n0dseism2/parenleftBig\nisA /(i∂ /+m)−s2(∂µA /)(i∂ /+m)∂µ/parenrightBig\n×\n×φa /(i∂ /+m)eis✷/bracketrightBig\n. (3.27)\nAfter keeping only the terms with three Dirac matrices, with the sub sequent Fourier\ntransform and integrating over momenta, we arrive at the result ( 3.24).\nWe note, that, in principle, our result for the Julia-Toulouse term co uld be obtained\nwithin usual calculations of the Chern-Simons term, where one could do the replacement\nAµ→Aµ+aµφ. Also, if one would like to consider the free scalar-vector theory wh ose\naction involves the usual kinetic term for the scalar field, the Maxwe ll term, and the\nJulia-Toulouse term, it is possible to show that the dispersion relation s are just those\nones given by (2.31).\nThere aresomeother mannerstogeneratethemixed term(2.14). Oneofthemconsists\nin a dual immersion of the LV extension of the self-dual theory, nam ely this way has been\nemployed in the section 2.3. Another one consists in a dimensional red uction of the\nelectrodynamics with the CFJ term carried out in [80] (see also [90]). W e start with its\nfour-dimensional Lagrangian looking like\nL=−1\n4FµνFµν+ǫµνλρkµAν∂λAρ, (3.28)\nand “freeze” the dynamics along the third spatial dimension (perha ps, this method can\nbe an useful tool to describe essentially two-dimensional quantum systems, for example,\ngraphene), so, one can split the indices like ν= (a,3), with the index atakes values 0 ,1,2,\nandA3=φ, and the dependence on x3will be suppressed (thus, ∂3φ=∂3Aa= 0), so,\none arrives at the following dimensionally reduced Lagrangian (with ǫabc≡ǫ3abc):\nLred=−1\n4FabFab−1\n2∂aφ∂aφ+ǫabc(µAa∂bAc−kaFbcφ). (3.29)\nWe see that within this reduction, the Chern-Simons term with a mass µ=k3and the\nJT term are generated.\nOne more way to generate this term is based on the dynamical Loren tz symmetry\nbreaking approach, presented in [81]. We proceed in a manner similar t o that one de-\nscribed in the section 2.4 (see also [65]). The key idea is the following one : in the\nthree-dimensional space-time, we consider several four-fermio n interactions which, as it\nis usually done within the Gross-Neveu approach, afterwards are p resented in terms of\nvertices involving corresponding different scalar or vector auxiliary fieldsAI, whereIis a\n34generalized summation index. As a result, we have the equivalent the ory represented by\nthe following Lagrangian:\nL=g2\nI\n2AIAI+¯ψ(i∂ /−m−eIAIΓI)ψ, (3.30)\nwhereAI={Vµ,Θ,Aµ,Φ,Tµ}is a set of the vector and scalar fields coupled to spinors\nwith use of matrices Γ I={γµ,γ3,γµγ5,γ3γ5,γµγ3γ5}playing here the role of generalized\nDirac matrices, and ψis the four-component spinor. Here we use 4 ×4 matrices, whose\nexplicitformisgivenin[81], so, ourspinorrepresentationoftheLore ntzgroupisreducible.\nThe one-loop corrected effective potential of the theory defined in a manner analogous\nto that one used in the section 2.4, i.e. as\nVeff=−g2\nI\n2AIAI+itr/integraldisplayd3k\n(2π)3ln(k /−m−eIAIΓI), (3.31)\nwhere only derivative independent terms are taken into account, e vidently possesses non-\ntrivial minima at some ∝an}b∇acketle{tAI∝an}b∇acket∇i}ht=bI/eI, withbI={0,0,bµ,φ0,0}. Now, let us make the\nshift of the vacuum by the rule AI→ AI+bI/eI, so that now ∝an}b∇acketle{tAI∝an}b∇acket∇i}ht= 0. As a result, the\none-loop contribution to the effective action (we note that here th e overall sign is changed\nas it must be!) takes the form\nSeff[A,b] =−iTrln(i∂ /−m−b·Γ−eIAIΓI). (3.32)\nIn this effective action we take into account only the lower, quadrat ic order in fields AI,\ngiven by\nS(2)\neff[A,b] =i\n2/integraldisplay\nd3xAIΠIJAJ, (3.33)\nwith the corresponding self-energy tensor ΠIJwritten within the derivative expansion\nmethodology[91](thedetaileddiscussionofthisapproachwillbegive ninthenextsection)\nlooks like\nΠIJ=−eIeJtr/integraldisplayd3p\n(2π)3i\ni∂ /−m−b·ΓΓIi\np /−i∂ /−m−b·ΓΓJ. (3.34)\nExpanding this expression up to the first order in derivatives and ca lculating correspond-\ning traces and integrals, we obtain the following mixed and convention al Chern-Simons\nterms:\nLCS=−eVeV3\n2πmΘǫµλνbµ∂λVν−eVeT\nπǫµλνVµ∂λTν+eV3eT\nπΘgµνbµTν\n+eAeA3\n6πmΦǫµλνbµ∂λAν+e2\nA\n6πmφ0ǫµλνAµ∂λAν. (3.35)\n35We note here also the presence of Julia-Toulouse terms, i.e., the firs t and fourth terms.\nByanalyzing theabove Chern-Simons-like Lagrangian, we candiscus s two models that\nexhibit charge fractionalization without the breaking of Tsymmetry. First, if we restrict\nourselves to the case Tµ=Aµ= Φ = 0, i.e. we have only the usual vector Vµand the\nscalar Θ, we arrive at\nLCS=eV3eV\n2πmϑǫµλν∂µΘ∂λVν, (3.36)\nwhere we have introduced ϑ(x) =xµbµ+C, withCbeing a constant. At the same way,\nif we choose Vµ= Θ =Tµ= 0, we have\nLCS=−eA3eA\n6πmϑǫµλν∂µΦ∂λAν, (3.37)\nwhich, upto anadjustment ofnumerical constants, isthe low-ene rgy mixed Chern-Simons\nterm forgraphene discussed in[92, 93]. Byconsidering the propert ies of the bilinear terms\nassociated to the fields AI, with respect to C,P, andTtransformations, we note that\nboth terms (3.36) and (3.37) violate the Csymmetry, whereas the PandTsymmetries\nare preserved. This, obviously, leads to a CPT symmetry violation, w ith theCsymme-\ntry violation is a direct consequence of the charge fractionalization [81], whereas the T\nsymmetry is preserved. This is another manner to generate the mix ed scalar-vector term.\nIn principle, it can be obtained also in an essentially non-perturbative way, that is, the\nJulia-Toulouse technique used for study of condensation of topolo gical defects [88]. To\nconclude this section, we note that this term, first, can arise by diff erent reasons, second,\napparently can be useful to study condensed matter systems.\n3.4 The Carroll-Field-Jackiw term\n3.4.1 General situation and zero temperature case\nThe CFJ term (1.1) is, without doubts, a paradigmatic example of the Lorentz-breaking\nterm. As we noted in the beginning of this chapter, it can be generat ed as a quantum cor-\nrection from the following in the appropriate LV extension of the spin or QED. Explicitly,\nthis theory is described by the Lagrangian:\nLQED=−1\n4FµνFµν+¯ψ(i∂ /−eA /−m+b /γ5)ψ. (3.38)\nThis possibility was for the first time described in [25] and, afterwar d, discussed in numer-\nous papers. The one-loop effective action is clearly given by the ferm ionic determinant:\nΓ(1)=−iTrln(i∂ /−eA /−m+b /γ5). (3.39)\n36The CFJ term is evidently generated by the expansion of this determ inant up to the\nsecond order in Aµ, and to the first order in derivatives. Its final form is\nLCFJ=Ce2ǫµνλρbµAν∂λAρ, (3.40)\nwithCbeing some numerical constant. To obtain the CFJ term, and conse quently the\nconstantC, one considers the contributions to the two-point function descr ibed by the\nfollowing Feynman diagrams:\n•\n•\nFigure 3.4: Contributions to the CFJ term.\nThe external lines are the gauge fields, and the •symbol here is for the b /γ5insertion.\nSo, let us describe some ways to calculate the contributions of thes e two graphs.\nWithin the first approach we use the usual spinor propagators. In this case, the sum\nof these graphs yields\nΓ2[A] =−e2\n2/integraldisplayd4p\n(2π)4Aµ(−p)Πµν(p)Aν(p), (3.41)\nwhere\nΠµν(p) = tr/integraldisplayd4k\n(2π)4/bracketleftBig\nγµ1\nk /−mb /γ51\nk /−mγν1\nk /+p /−m+\n+γµ1\nk /−mγν1\nk /+p /−mb /γ51\nk /+p /−m/bracketrightBig\n(3.42)\nis the self-energy tensor for this theory up to the first order in bµ. Following one manner,\nto obtain the CFJ term, one should expand the integrand up to the fi rst order in the\nexternalmomentum p. Namelythiswaywasemployedintheoriginalpaper[25]. However,\nalready inthat paper, it was arguedthat the CFJterm, being super ficially logarithmically\ndivergent, is actually finite but ambiguous, Indeed, this procedure yields the result\nΠµν=1\n2π2ǫµναβbαpβ/parenleftBiggθ\nsinθ−1\n4/parenrightBigg\n, (3.43)\nwhereθ= 2arcsin(√p2/2m). Therefore, in the low-energy limit p→0, one has Πµν=\n3\n8π2ǫµναβbαpβ, and, then, C=3\n16π2. One can develop as well another procedure, within\nwhich thebµbecomes an external non-constant field depending on one more ex ternal\n37momentum, so, the only gauge invariant result for the CFJ contribu tion to the effective\nLagrangian is zero. This suggestion was originally made in [94] and discu ssed in [25].\nFurther, in [95, 96] it has been argued that, in a certain sense, the zero result for the CFJ\nterm ispreferable since, if onewould treat bµasa dynamical field, even the contribution of\nthe CFJ term to the effective action will break the gauge invariance, so, the zero value for\nthe coefficient Cis necessary to maintain the gauge symmetry. The same argumenta tion\ncan be applied to the four-dimensional gravitational CS term as well [95].\nInstead of expanding the effective action in series in bµ, one can perform the sum over\nall possible b /γ5insertions and obtain the exact propagator of the spinor field [97]:\nS(p) =i(p /−m+b /γ5)−1=ip2+m2−b2−2(b·p+mb /)γ5\n(p2+b2−m2)2−4[(b·p)2−m2b2]×\n×(p /+m−b /γ5), (3.44)\ntogether with the dimensional regularization, one arrives at the re sult [97]:\nΠµν=−1\n2π2ǫµναβbαpβ\n1−θ(−b2−m2)/radicalBigg\n1+m2\nb2\n. (3.45)\nThus, forb2>0 (that is, time-like bµ), one has Πµν=−1\n2π2ǫµναβbαpβ, andC=1\n4π2,\nwhile for the zero mass and the space-like bµone finds a zero result, so, C= 0 (see\ndetails in [97]). We note that the exact propagator (3.44) is especially convenient for\nthe calculations in the massless fermions case since it displays no infra red singularities at\nm= 0. As we already mentioned in the section 3.1, the ambiguity is a price w e must pay\nfor a finiteness of the superficially (logarithmically) divergent result . It was argued in [26]\nthat this ambiguity is strongly related with the Adler-Bell-Jackiw anom aly.\nLet us also discuss some other ways to calculate the CFJ term. One o f the main\nmanners to do it is based on the derivative expansion method [91]. The essence of this\nmethod is as follows. We carry out a formal Fourier transformation of the determinant\n(3.39) which we then rewrite as\nΓ(1)=−iTrln(p /−eA /−m+b /γ5) =\n=−iTr(1−eA /1\np /−m+b /γ5)+···, (3.46)\nwhere dots are for the field-independent contribution. However, now, to extract possi-\nble contributions depending on derivatives of Aµ, we use the following identity which is\nstraightforward in the coordinate representation, it is the key ide ntity of this approach\nreflecting the fact that the fields and derivatives do not commute, withpµbeing an inte-\ngration momentum in the corresponding loop:\nAαpµ=pµAα−i(∂µAα). (3.47)\n38This identity can be used to put all internal momenta in positions allowin g to integrate\nover them (with the ”correct” position of internal momenta is at th e left-hand side from\nall fields). It can be straightforwardly generalized for higher orde rs in derivatives. Equiv-\nalently, we can start with the expression (3.46) with the subsequen t replacements: i∂µ→\npµ,Aµ(x)→Aµ(x−i∂\n∂p), andfurtherweexpand Aµ(x−i∂\n∂p) =Aµ(x)−i(∂νAµ)(x)∂\n∂pν+...,\nand the derivatives with respect to pµmust act on propagators1\np /−m+b /γ5. Keeping only\nthe terms of the second order in fields Aαand of the first order in their derivatives and\nin the constant bµvector, we arrive at the following expression for the one-loop effec tive\nLagrangian [98]:\nL(1)= 2ie2/integraldisplayd4p\n(2π)41\n(p2−m2)3[ǫνλµρ(3m2+p2)−4ǫνλµαpαpρ]bρAν∂λAµ\n=e2ǫνλµαkαAν∂λAµ, (3.48)\nwherekµ=Cbµ, andCis defined by the Eq. (3.40). The constant Cis easily shown to be\nfinite and ambiguous, just as we discussed in the Section 3.1: if we rep lacepµpν→p2\n4ηµν\nand integrate in 4 dimensions, we get C=3\n16π2, and if we replace pµpν→1\n4+ǫηµνp2and\nintegrate in 4+ ǫdimensions, we arrive at C=1\n4π2(the same values of Care presented\nin [38], arising not only within calculating the usual CFJ term but also with in computing\nits non-Abelian extension). Besides of these results, the coefficien tCwas calculated with\nuse of the exact propagator (3.44) in the massless case [99] and fo und to be equal to1\n8π2.\nWithin the Pauli-Villars regularization, the coefficient Cvanishes [6].\nOne more methodology to treat the CFJ term in the massless case wa s proposed in\n[100]. To do it, we express the exact propagator (3.44) for m= 0 as\nSexact(p) =i\nk /+b /γ5=i\np /+b /PL+i\np /−b /PR, (3.49)\nwherePR=1+γ5\n2andPL=1−γ5\n2are the left and right chiral projectors Actually, namely\nthese projectors were used in the section 3.1. The self-energy te nsor is expressed in terms\nof exact propagators in the simplest form,\nΠµν(p) =1\n2tr/integraldisplayd4q\n(2π)4γµSexact(q)γνSexact(q+p), (3.50)\nso, it remains to substitute here directly the exact massless propa gator (3.49) and extract\nonly the CFJ-like terms, that is, those ones involving only one γ5, which clearly can yield\nthe Levi-Civita symbol. The corresponding result is [100]:\nΠµν\nCFJ(p) = 4ibβǫµναβ/integraldisplayd4q\n(2π)4(pα+qα)\n(q+b)2(q+p+b)2, (3.51)\n39with the integral over kis supposed to be regularized in some way. Afterwards, we apply\nthe methodology of the implicit regularization (see e.g. [101]), which a llows us to find\nthat\nΠµν\nCFJ(p) =−4iα2bβpαǫµναβ, (3.52)\nwhereα2is some completely undetermined constant, finite but strongly depe nding on\nthe regularization prescription. It is clear that this is the only possib le one-derivative\ncontribution to the self-energy tensor proportional to ǫµναβ. In [100] the α2is treated as\na surface term arising within the integration, explicitly defined throu gh the relation\nα2gµν=/integraldisplayd4q\n(2π)4∂\n∂qµ/bracketleftBiggqν\n(q2−λ2)2/bracketrightBigg\n,\nwithλis a mass scale. From a formal viewpoint the arising of α2is related with the\nfact that integrals I0α=pα/integraltextd4q\n(2π)41\n(q+b)2(q+p+b)2andIα=/integraltextd4q\n(2π)4qα\n(q+b)2(q+p+b)2diverge, and\nthe Πµν\nCFJwill be proportional to their linear combination looking like a momentum pα\nmultiplied by some undetermined expression behaving as ∞−∞, withα2will be just the\nvalue of this expression, which is not fixed by any physical reasons.\nOnce more manner to argue that the CFJ term is strongly ambiguous being propor-\ntional to a completely undetermined constant isbased onthe funct ional integral argumen-\ntation [102]. To do it, we start with the functional integral corresp onding to the one-loop\neffective action (3.39):\nZ[A] =/integraldisplay\nDψD¯ψexp(iS[A]) =\n=/integraldisplay\nDψD¯ψexp(i/integraldisplay\nd4x¯ψ(iD /−m−b /γ5)ψ), (3.53)\nwhereDµ=∂µ−ieAµis a usual gauge covariant derivative. Then, after the changes of\nvariablesψ(x)→eiα(x)γ5ψ(x),¯ψ(x)→¯ψ(x)eiα(x)γ5, the integral measure is modified as\nDψD¯ψ→DψD¯ψexp/parenleftBig\n−i\n8π2/integraltextd4xα(x)˜FαβFαβ/parenrightBig\n, see details in [102, 103]. The action S[A]\nunder this changes of variables transforms to S′[A] =/integraltextd4x[¯ψ(iD /−me2iα(x)γ5−b /γ5)ψ−\n∂βα(x)Jβ\n5], withJβ\n5being a conserved axial current which, however, can be determine d\nin an ambiguous manner like Jβ\n5=¯ψγβγ5ψ+c˜FβγAγ, with an arbitrary constant c, and\nthe˜Fβγis a tensor dual to Fµν, since the only restriction for this axial current is that it\nshould satisfy the conservation law ∂βJβ\n5= 0. Choosing α(x) =−xαbα, one rests with\nZ[A] =ei(c−1\n4π2)/integraltext\nd4xbα˜FαβAβ×\n×/integraldisplay\nDψD¯ψexp(i/integraldisplay\nd4x¯ψ(iD /−me2iγ5xµbµ)ψ). (3.54)\nWe calculate the functional determinant of iD /−me2iγ5xµbµup to the first order in bµand\nthesecondorder inthefield Aµ. Inthemomentum space, multiplicationby thecoordinate\n40xµis equivalent to differentiation with respect to the corresponding mo menta, so we can\nperform the loop integration as above in this section, with the result for the exponential\nof the fermionic determinant will be [102]:\n/integraldisplay\nDψD¯ψexp(i/integraldisplay\nd4x¯ψ(iD /−me2iγ5xµbµ)ψ) =\n= exp(i\n4π2/integraldisplay\nd4xbα˜FαβAβ). (3.55)\nSubstituting this result to the generating functional Z[A] (3.54), we find that the only\nsurviving term is that one proportional to the constant c. Thus, the corresponding contri-\nbution to the effective action Γ[ A] =−ilnZ[A] is completely undetermined being equal\nto\nΓ[A] =c/integraldisplay\nd4xbα˜FαβAβ, (3.56)\nthat is, the CFJ coefficient Cfrom (3.40) within this prescription is equal to the factor c.\nWe note that this fact completely matches the conclusion achieved in [100] on the base\nof the implicit regularization. Regarding the argumentation present ed in [83], we note\nthat it is based on choosing appropriate normalization conditions, wh ereas there is no\nunique prescription to define these conditions once and forever, s o, the ambiguity can be\neliminated only within a separate calculation, but there is no profound reasons to say that\none value of the constant Cis correct but others are not. Actually, the papers [102, 100]\nargued that the ambiguity of the CFJ term is intrinsic. We note that t he resultC=1\n4π2\nhas been unambiguously obtained also within the Schwinger proper tim e approach [104].\nHowever, using of the proper time formulation itself involves some sp ecific regularization\nprescription.\n3.4.2 Finite temperature case\nThe ambiguity we found for the CFJ term can be naturally generalized for the finite\ntemperature case. To illustrate the situation, we study the contr ibution (3.48) at the\nfinite temperature within two different regularization schemes analo gous to those ones\nused in a previous subsection, that is, we suggest the purely spatia l dimension dto be\neither 3 or 3 + ǫwithǫ→0, so, actually, we again face the result which is finite but\nundetermined at ǫ= 0, i.e. a removable singularity.\nAs usual within the finite temperature approach, we follow the Mats ubara formalism,\nthat is, we assume that the system stays in a thermal equilibrium at t he temperature\nT=1\nβ. Then, we carry out the Wick rotation and suggest that the time-lik e component\nof the moment is discrete, p4= 2πT(n+1\n2), withnis integer. In this case we can replace\n41the integral over p0by the sum over n. It is instructive to consider (3.48) separately for\nthe time-like and space-like bµ.\nWithin one manner we start with the expression (3.48), and, after a pplying the Mat-\nsubara formalism, promote spatial integral to ddimensions, and choose bµto be time-like.\nWe arrive at\nk4=b4T∞/summationdisplay\nn=−∞/integraldisplaydd/vector p\n(2π)d3ω2\nn−/vector p2\n(/vector p2+ω2n)3=b4\n4π2, (3.57)\nthat is, in this case we have C=1\n4π2which matches one of zero-temperature results\nobtained in [97]. If within (3.48) we suggest the vector bµto be space-like, we replace\npipj→δij/vector p2\nd, withd= 3+ǫis a spatial dimension, and arrive at\nki=biT∞/summationdisplay\nn=−∞/integraldisplaydd/vector p\n(2π)d4m2−ω2\nn+(4\nd−1)/vector p2\n(/vector p2+ω2\nn)3, (3.58)\nwhich is not ambiguous within this calculation, and the final result is\nki=bi\n4π2(1+F(ξ)), (3.59)\nwhereξ=m\n2πT, and\nF(ξ) = 2π2/integraldisplay∞\n|ξ|dz(z2−ξ2)1/2tanh(πz)\ncosh2(πz)(3.60)\nis a temperature depending function vanishing at T→0, i.e.ξ→ ∞, and tending to 1\natT→ ∞.\nAlternatively, we could first replace pµpν→1\n4ηµνp2in (3.48), and only afterward apply\nthe Matsubara formalism. In this case we would arrive at\nkµ=3bµ\n16π2(1+F(ξ)), (3.61)\nwhereF(ξ) is given by (3.60). We conclude that the CFJ term continues to be am biguous\nat non-zero temperature as well, so, the finite temperature is not sufficient to eliminate\nthe ambiguity of the result.\nIt is clear that this list of possible finite-temperature values for kµwe gave here is\nnot complete since other regularizations exist as well. Moreover, it is interesting to note\nthat only within some specific schemes one finds that all components of thekµvector\ncharacterizing the CFJ term are proportional to the original LV ve ctorbµwith the same\ntemperature dependent coefficient α, that is,kµ=α(T)bµ, as occurs, for example, in\nthe second scheme we used here, whereas in general case it is not s o. The fact that\nin generalkµis not proportional to bµ, clearly reflects the intrinsic Lorentz symmetry\n42breaking occurring in a finite temperature case due to the existenc e of the privileged\nreference frame of the thermal bath, which could break the prop ortionality between kµ\nandbµ. In [105] the strong asymmetry between spatial and temporal co mponents of the\nkµstemming from this breaking of the proportionality has been obtaine d with use of\nnon-covariant splitting the integration momentum, applied by the ru lepµ→ˆpµ+δµ0p0\nso that ˆp0= 0, with subsequent arising the ”covariant” and ”non-covariant” CFJ-like\ncontributions (this asymmetry has been further confirmed in [106 ]). Actually, a lot of\nmanners to calculate the CFJ term, starting from the Lagrangian ( 3.38), on the base of\ndifferent regularization schemes, including the finite temperature c ase, is known (see e.g.\n[106] and references therein).\n3.4.3 Non-Abelian generalization\nThe natural generalization of the CFJ term consists in its non-Abelia n generalization\nwhichcanbeintroducedbyanalogywiththewell-knownnon-AbelianCS term. Originally,\nthe non-Abelian extension of the CFJ term (2.6) has been proposed in [37]. To show that\nit could emerge as a perturbative correction, we start with the exp ression for the one-loop\neffective action of the gauge field (3.39), where, however, we sugg est that the gauge field\nis Lie-algebra valued, Aµ=AµaTa, whereTaare generators of some Lie group satisfying\nthe relations [ Ta,Tb] =ifabcTcand tr(TaTb) =δab. To find the non-Abelian CFJ term we\nshould, besides of the contribution involving two external gauge fie lds and one derivative\nacting on one of these fields, as in the subsection 3.4.1, consider also the contribution with\nthree external fields and no derivatives.\nThis scheme of calculations has been pursued in [38]. In this case, the corresponding\ngeneralization of the fermionic determinant (3.39) is\nΓ(1)=−iTrln/bracketleftBig\n(i∂ /−m+b /γ5)δij−gγµAa\nµ(Ta)ij/bracketrightBig\n. (3.62)\nAfter expansion of this determinant up to relevant orders and calc ulating traces and\nintegrals, the non-Abelian CFJ term was found to be\nSCFJ,non −Abelian=/integraldisplay\nd4xǫνλρµkµ(Aa\nν∂λAa\nρ−2\n3igfabcAa\nνAb\nλAc\nρ), (3.63)\nwith the constant vector kµis just the same one that can be read off from the expression\n(3.48). Repeating the discussion of the previous subsections, we c onclude that this vector\nis finite and ambiguous. Moreover, in the finite temperature case it b ehaves just in the\nsame way as in the Abelian theory (see the subsection 3.4.2).\nAnother way to obtain this term is based on the Schwinger proper tim e approach\n[104]: we consider the same expression for the one-loop effective ac tion (3.39), but with\n43the Lie-algebra valued vector field Aµ=AµaTa. After adding the field-independent factor\n−iTrln(i∂ /−m−b /γ5) in order to deal with the second-order operator of the standar d\nform✷+...which is the most convenient one within the proper time method, and\nexpanding the effective action up to the first order in the LV vector bµ, we obtain the\ndesired contribution to the effective action in the form\nΓ =−iTr(✷+igA /∂ /+mgA /+m2)−1[(gA /+2m)b /γ5−2i(b·∂)γ5].(3.64)\nNow, we employ the Schwinger representation by the rule\n(✷+igA /∂ /+mgA /+m2)−1=/integraldisplay∞\n0dse−s(✷+igA /∂ /+mgA /+m2), (3.65)\nand, after manipulations described in [104], we see that the divergen t part of this effective\naction completely vanishes, and the result identically reproduces th e expression (3.63),\nwithkµ=g2\n4π2bµ, this is one of the results obtained in [38]. Formally this result is\nambiguity-free, however, we note that the Schwinger represent ation itself plays the role\nof the specific regularization eliminating the ambiguity. We note never theless that the\nnumber of methods to generate the CFJ term, both in Abelian and no n-Abelian versions,\nis not exhausted by the approaches described here, since it can be generated from much\nmore sophisticated theories, including non-minimal ones, see f.e. [84 ].\nAnother interesting question is the study of topological propertie s of the non-Abelian\nCFJ term. In principle, one could consider the simplest topological st ructure, that is,\nto suggest the vector kµto be directed along the zaxis, therefore, for any value of z\ncoordinate one has a topological structure of the three-dimensio nal non-Abelian Chern-\nSimons theory, and the topological structure of the whole four-d imensional theory is\nsome foliation obtained as a direct sum of the three-dimensional top ological structures.\nHowever, this issue certainly requires further studies.\n3.5 CPT-even term for the gauge field\nThe CPT-even extension for the free gauge field has been originally s uggested [6] to look\nlike\nLeven=−1\n4κµνρσFµνFρσ, (3.66)\nwhereκµνρσis a constant tensor possessing the symmetries of the Riemann cur vature\ntensor. One of the most interesting particular cases is that one wh en this tensor is\nexpressed intermsofonlyoneconstant vector(orpseudovecto r)uµthroughtheexpression\n(2.8). In this case the term (3.66) takes the form:\nLeven=uµuρFµνFρν. (3.67)\n44Namely this expression was discussed in [39] within the context of high er dimensions.\nThe natural question evidently consists in the possibility for its pert urbative generation.\nCertainly it could be done in different ways.\nThe most straightforward way to obtain this term is based on use of the magnetic\n(non-minimal) coupling, namely this way was followed in [40]. The magnetic coupling has\nbeen introduced in [75] in the four-dimensional space in the form (2.1 6). We have showed\ninSection2.4that just thiscoupling ariseswithin thedual embedding p rocedureoftheLV\nself-dual model (see (2.35)), where the current is the usual spin or one, that is, jµ=¯ψγµψ.\nHowever, it is clear that, first, it would be interesting to consider th e theory involving\nnot only the non-minimal coupling only but also the usual interaction v ertexe¯ψγµψAµ,\nsecond, it is possible to obtain not only a purely non-minimal contribut ion to the aether\nterm but other ones as well. Although the non-minimal coupling is non- renormalizable,\nwe note that if we restrict our study by the one-loop order, or, as is the same, by the\nfermionic determinant, the non-renormalizability does not give any p roblems.\nTo demonstrate how the aether term can arise, let us consider the extended QED with\nthe magnetic coupling whose action, in the spinor sector, is [107]:\nS=/integraldisplay\nd4x¯ψ(i∂ /−m−eA /−gǫµνρσγµbνFρσ−b /γ5)ψ, (3.68)\ni.e., the role of uµused in (3.67) is played by the axial vector bµ. The corresponding\nfermionic determinant would be\nΓ(1)=−iTrln(i∂ /−eA /−gǫµνρσγµbνFρσ−m−b /γ5). (3.69)\nThere will be three different one-loop aether-like contributions in th is theory: the first\nof them is a purely non-minimal one, the second one is mixed, involving b oth minimal\nand non-minimal vertices, and the third one involves two minimal vert ices. Let us study\nall these cases.\nFirst, the purely non-minimal contribution is given by the Feynman dia gram depicted\nat Fig. 3.5.\n••\nFigure 3.5: Contributions to the aether term for the electromagne tic field with non-\nminimal vertices.\nThe corresponding analytic expression for this diagram looks like [40 ]:\nS2(p) =−g2\n2ǫµνρσǫµ′ν′ρ′σ′bµFνρbµ′Fν′ρ′/integraldisplayd4k\n(2π)41\n[k2−m2]2×\n45×tr[m2γσγσ′+kαkβγαγσγβγσ′]. (3.70)\nThere are several manners to calculate this expression. Here we d iscuss two of them,\nthose ones developed in [107]. To start, we evaluate the trace in the four-dimensional\nspace-time and obtain\nS2(p) =−2g2ǫµνρσǫµ′ν′ρ′σ′bµFνρbµ′Fν′ρ′/integraldisplayd4k\n(2π)41\n[k2−m2]2[m2ησσ′+\n+kαkβ(ηασηβσ′−ηαβησσ′+ηασ′ηβσ)]. (3.71)\nActually, the integral in this expression has just the structure giv en by (3.2), up to the\noverall factor. Now, let us perform its calculation.\nWithin the first manner, we replace kαkβ→1\n4ηαβk2, as it should be done in the four-\ndimensional space-time, and only after that, we introduce the dime nsional regularization,\npromoting the integral to d= 4−ǫdimensions. Carrying out Wick rotation, we arrive at\nthe Euclidean result\nS2(p) =−ig2ǫµνρσǫµ′ν′ρ′σ′bµFνρbµ′Fν′ρ′ησσ′/integraldisplayd4−ǫkE\n(2π)4−ǫk2\nE+2m2\n[k2\nE+m2]2. (3.72)\nIn this case we have the mutual cancellation of divergences, repro ducing the scenario\ndiscussed in the Section 3.1, so, our integral turns out to be finite:\n/integraldisplayd4−ǫkE\n(2π)4−ǫk2\nE+2m2\n[k2\nE+m2]2=m2(1−ǫ/2)\n(4π)2−ǫ/2/bracketleftbigg\nΓ(ǫ\n2)+Γ(−1+ǫ\n2)/bracketrightbigg\n=\n=−m2\n16π2+O(ǫ). (3.73)\nCollecting all together, multiplying two Levi-Civita symbols, disregard ing the Lorentz-\ninvariant term proportional to b2, in order to keep track of the aether term only, and\nreturning to the Minkowski space, we arrive at\nS2(p) =g2m2\n4π2(bµFµν)2. (3.74)\nWithin another manner, we first promote our integral to d= 4−ǫdimensions, then we\nreplacekαkβ→1\ndηαβk2. In this case, in the Euclidean space we have\nS2(p) =−ig2ǫµνρσǫµ′ν′ρ′σ′bµFνρbµ′Fν′ρ′ησσ′/integraldisplayddkE\n(2π)d2(d−2\nd)k2\nE+2m2\n[k2\nE+m2]2.(3.75)\nIf one substitutes d= 4−ǫ, withǫ∝ne}ationslash= 0 in this expression, the integral identically vanishes.\nSo, the purely non-minimal contribution displays a removable singular ity atǫ= 0. In\nprinciple, otherresultscanbeobtainedfortheexpression(3.70), seethediscussionin[108],\n46where also the finite temperature behavior of the generated aeth er term is considered. So,\nfor these two examples one can write this result as\nSmixed=C0g2m2(bµFµν)2, (3.76)\nwithC0is a finite ambiguous constant.\nWenotethatinotherspace-timedimensions onecanintroducethen on-minimal vertex\nas well, it will look like Smagn=g¯ψǫνρσbνFρσψin 3DandSmagn=g¯ψǫλµνρσσλµbνFρσψin\n5D. In both cases, the aether term will be finite within the dimensional r egularization\nframework [40].\nSecond, we study the contributions involving onenon-minimal verte x and one minimal\none, and one insertion b /γ5in one of the propagators [107]. In this case one can consider\njust the same Feynman diagrams given by Fig. 3.4, used in the Section 3.4.1 for the\ncalculation of the CFJ term, with the only difference that one of the e xternalAµfields in\nthis case is replaced by ǫµνρσbνFρσ, and the coupling eaccompanying just this field – by\ng. The structure of the integral over the internal momenta contin ues to be the same as\nin the case of the CFJ term, that is, (3.48). As a result, we repeat a ll calculations from\n[98] and arrive at\nSmixed=−2Ceg(bµFµν)2, (3.77)\nwith the finite constant Cis just that one characterizing the value of the CFJ term, it is\ndefined in (3.40), and its value is discussed in details in the Sec. 3.4.1. We note that this\ncontribution arises only in the four-dimensional space-time, where the vectorǫµνρσbνFρσ\ncan be defined, and has no analogues in other space-time dimensions .\nThird, there is a purely minimal aether-like contribution. In this case we have two\nb /γ5insertions (some preliminary studies of this contribution were also ca rried out in [83]\nwithin the renormalization group context), and the explicit form of t his contribution is\nSAA(p) =ie2\n2/integraldisplayd4l\n(2π)4(γµ1\nl /−mγν1\nl /+p /−mγ5b /1\nl /+p /−mγ5b /1\nl /+p /−m+\n+γµ1\nl /−mγ5b /1\nl /−mγν1\nl /+p /−mγ5b /1\nl /+p /−m+\n+γµ1\nl /−mγ5b /1\nl /−mγ5b /1\nl /−mγν1\nl /+p /−m)Aµ(−p)Aν(p), (3.78)\nwherepis an external momentum.\nThis contribution is a sum of Feynman diagrams given by Fig. 3.6.\nTaking into account only the terms of the second order in p, we get for the sum of\nthese graphs:\nSAA=−e2\n6m2π2bµFµνbλFλν. (3.79)\n47••• •\n• •\nFigure 3.6: Contributions to the aether term for the electromagne tic field with minimal\nvertices.\nWe note that this expression is superficially finite and hence ambiguity -free.\nSo, the complete aether-like contribution is a sum of (3.76,3.77,3.79), and it is finite\nand involves two ambiguous constants C0andC. While the constant Cis related with\nthe CFJ anomaly, the question of relation of C0with any possible anomaly is still open.\nThis calculation can be straightforwardly generalized for a non-Abe lian case, see [109].\nFor the presence of a minimal coupling only, the result for the aethe r term is obtained\nthrough expansion of the functional determinant (3.62) up to rele vant orders, and the\nresult is a straightforward generalization of (3.79):\nSAA=−e2\n6m2π2bµFµνabλFa\nλν, (3.80)\nwhereFa\nλν=∂µAa\nν−∂νAa\nµ−igfabcAb\nµAc\nνis the non-Abelian field strength. It is clear\nthat including of the non-Abelian analogue of magnetic coupling will yield non-Abelian\nanalogues of (3.76) and (3.77).\nBesides of the study performed in [40, 107], a very interesting disc ussion of ambiguity\nof the aether term for the gauge field is presented in [110]. In this p aper, it was argued\nthat the same ambiguous integral over momenta as in (3.71), that is ,\nIσσ′=/integraldisplayd4k\n(2π)41\n[k2−m2]2×\n×[m2ησσ′+kαkβ(ηασηβσ′−ηαβησσ′+ηασ′ηβσ)]. (3.81)\nwill accompany not only the aether term, but also the CFJ term and t he Proca term, so\nthat the whole contribution will look like\nΓ2=−2(eAσ+gǫσλµνbλFµν)Iσσ′(eAσ′+gǫσ′λ′µ′ν′bλ′Fµ′ν′), (3.82)\nand it is natural to choose the value of the Iσσ′to be zero in order to rule out the AµAµ\nterm, thus avoiding the breaking of the gauge symmetry. Neverth eless, since this integral\nisambiguous, nobodyforbidstochooseitsvaluestobedifferent for allthreecontributions,\nthat is, to bezero when it accompanies theProca term, andnon-ze ro when it accompanies\nthe gauge invariant aether and CFJ terms.\n48To close the discussion of generating the aether term from the ext ended QED La-\ngrangian given by (3.68), we note its advantage consists in the fact that only in this\ntheory one can arrive at finite CPT-even contributions. All anothe r couplings will yield\ndivergent aether-like results in the four-dimensional space-time. There are two most im-\nportant examples of obtaining these results. In the paper [111], th e starting point is the\nfollowing extension for the spinor sector of QED:\nS=/integraldisplay\nd4x¯ψ(i∂ /−eA /+λ\n2κµνρσσµνFρσ−m)ψ. (3.83)\nWe see that the LV coupling is CPT-even. The corresponding one-loo p effective action of\nthe gauge field, that is, the fermionic determinant, is given by the ex pression:\nΓ(1)[A] =−iTrln(i∂ /−eA /+λ\n2κµνρσσµνFρσ−m). (3.84)\nIn this case, it is easy to calculate the contributions of first and sec ond orders in κµνρσto\nthe two-point function of the gauge field. They diverge, explicitly loo king like [111]:\nI1=meλ\n8π2ǫκµναβFµνFαβ;\nI2=−λ2\n16π2ǫκµνρσκρσ\nαβFµνFαβ. (3.85)\nIt is clear that if the constant LV tensor κµναβis characterized by only one vector, looking\nlike (2.8), both these results reproduce the aether term (2.9). We note that, first, to\nsubtract the arising divergence in a consistent manner, one should introduce the aether\nterm in the tree-level action from the very beginning, second, for the dimensionless κ, the\ncouplingλhas a negative mass dimension, so, this theory is actually non-renor malizable\nand can be treated only as a kind of some low-energy effective model.\nAnother interesting LV extension of the QED is characterized by th e following action\nof fermions:\nS=/integraldisplay\nd4x¯ψ[i(∂ /+ieA /)+icµνγµ(∂ν+ieAν)−m]ψ. (3.86)\nHere, thecµνis a constant tensor which is usually assumed to be symmetric (actua lly,\nifcµνis antisymmetric, it can be ruled out through an appropriate field red efinition\n[112]). Since it is dimensionless, the corresponding theory is all-loop re normalizable. The\ncorresponding fermionic determinant is\nΓ(1)[A] =−iTrln[i(∂ /+ieA /)+icµνγµ(∂ν+ieAν)−m]. (3.87)\nExpanding it up to the second order in Aµ, for the simplest choice cµν=uµuνwe arrive\nat the following result [113]:\nS(2)\neff=−e2\n6π2ǫ/integraldisplay\nd4x1\n4/parenleftBig\n1+u2/parenrightBig−1˜Fµν˜Fµν, (3.88)\n49where\n˜Fµν= (gµα+uµuα)(gνβ+uνuβ)Fαβ, (3.89)\nso, we arrive at the divergent aether-like contribution, and again t he aether term must be\npresent in the tree-level action from the very beginning for the sa ke of the multiplicative\nrenormalizability of the theory. The analogues of this result for oth er choices of cµνcan\nbe obtained as well [113]. In the five-dimensional case, the corresp onding result is finite\nwithin the dimensional regularization.\n3.6 Higher-derivative LV terms\nThe next step of our study consists in generating the higher-deriv ative terms, attracting\ngreat attention due to highly nontrivial dispersion relations allowing f or birefringence of\nelectromagnetic waves, rotation of polarization plane in a vacuum (s ee section 2.2) and\nmany other interesting effects, a general review on wave propaga tion in higher-derivative\nLV extensions of QED can be found in [114] for the gauge sector, an d in [115] for the\nspinor sector, more results on higher-derivative LV extensions of QED and their relation\nto massive photons are discussed in [116]. As we already mentioned, t he first known\nexample of such terms is the Myers-Pospelov (MP) term (2.13) which is very convenient\nsince it allows to conciliate higher derivatives with unitarity, for certa in choices of the LV\nvector. Another important example is the higher-derivative CFJ-lik e term\nLhd\nCFJ=α\nM2ǫµνρσbµAν✷∂νAρ. (3.90)\nHereαis some dimensionless constant, and Mis a mass scale. Both the MP term and the\nhigher-derivative CFJ-like term can be generated as perturbative corrections within the\nsameLVextended QEDwhose actionisgiven by(3.68). Westartagain withthefermionic\ndeterminant (3.69). Expanding it in power series in bµwe find that the three-derivative\ntermisdescribedbythreecontributions, thatis, thoseoneswitht wonon-minimalvertices,\nwith one minimal vertex and one non-minimal one, and, finally, with two minimal ones\n[117].\nFor the first contribution, we have only one insertion in one of propa gators and there-\nfore consider again the Feynman diagrams given by the Fig. 3.4, so, w e reproduce iden-\ntically the situation described in the Section 3.4.1, with the only differen ce is that in our\ncase we should replace both external eAµlegs by external gǫµνρσbνFρσlegs, while the\nintegral over momenta is given again by the (3.42). After extractin g only the first order\nin external momenta which is the only relevant one for this fragment , we arrive at the\nsame integral over momenta as in (3.48), whose result, in a pure ana logy with the Section\n503.4.1, is\nΓ(1)\n1= 2g2Cǫνλµαbα˜Aν∂λ˜Aµ, (3.91)\nwhere˜Aµ=ǫµνρσbνFρσ, andCis the same finite ambiguous constant arising within calcu-\nlations of the CFJ term (see Section 3.4) and defined by (3.40). Carr ying out contractions\nof the Levi-Civita symbols, we arrive at\nΓ(1)\n1= 2g2C/bracketleftBig\nbαFαµ(b·∂)bβǫβµνλFνλ+b2bβǫβµνλAµ✷Fνλ/bracketrightBig\n. (3.92)\nWe see that this contribution is, first, finite and ambiguous (which re flects the fact that\nthe Adler-Bell-Jackiw (ABJ) anomaly can be extended to involve highe r-derivative terms\n[118]), second, involves boththe MP term (the first one in this expre ssion) and the higher-\nderivative CFJ-like term (the second one in this expression). We not e that here the last\nterm is of third order in the vector bµwhile in principle one can obtain the analogous term\nwithout the constant b2multiplier, that is, the one given by (3.90), using the one-loop\neffective action of the minimal LV QED (3.39) as a starting point [119]. I n that case the\nhigher-derivative CFJ-like contribution is finite and ambiguity-free.\nFor the second contribution, we consider Feynman diagrams with tw o insertions in\npropagators. The corresponding contributions are again depicte d at Fig. 3.6, and in this\ncase we repeat the calculation from the previous section and replac e only oneeAµleg by\nthe external gǫµνρσbνFρσleg in the expressions (3.78) and (3.79). As a consequence, we\narrive at the expression analogous to (3.79), where this replaceme nt of one external leg is\nperformed. Hence, the corresponding result can be written as\nSb2=eg\n6π2m2/integraldisplay\nd4x/bracketleftBig\nbαFαµ(b·∂)bβǫβµνλFνλ+b2bβǫβµνλAµ✷Fνλ/bracketrightBig\n. (3.93)\nAgain, we obtain both the MP and the higher-derivative CFJ-like term s. We note that\nthis result is non-ambiguous.\nFor the third contribution, we have three insertions in the propaga tors. The corre-\nsponding Feynman diagrams are depicted at Fig. 3.7.\n• ••• •\n•• ••\n• ••\nFigure 3.7: Higher-derivative contributions with three insertions.\nThe corresponding result looks like\nSb3=4e2\n45π2m4/bracketleftbigg\nbαFαµ(b·∂)bβǫβµνλFνλ+5\n4b2bβǫβµνλAµ✷Fνλ/bracketrightbigg\n. (3.94)\n51The sum of these three contributions (3.92,3.93,3.94) gives our final result for the higher-\nderivative contribution to the effective Lagrangian. It reads as\nShd=/parenleftBigg\n2g2C+eg\n6π2m2+4e2\n45π2m4/parenrightBigg\nbαFαµ(b·∂)bβǫβµνλFνλ\n+/parenleftBigg\n2g2C+eg\n6π2m2+e2\n9π2m4/parenrightBigg\nb2bβǫβµνλAµ✷Fνλ. (3.95)\nWe see that this result is finite and gauge invariant. Also, one can obs erve that for the\nlight-likebµ, the CFJ-like term vanishes, and only the MP term survives. The finit e\ntemperature behavior of this term is studied in [120].\nThere are other possibilities to generate the higher-derivative LV t erms. First of\nall, it is worth to mention the study based on using the third-rank ten sorgµνρ(it is\nantisymmetric with respect to two first indices), so that the followin g extension of the\nQED is introduced [121]:\nS=/integraldisplay\nd4x¯ψ[i(γµ+1\n2gκλµσκλ)(∂µ+ieAµ)−m]ψ. (3.96)\nWe note that this theory is renormalizable since gκλµis dimensionless. In this case, the\nfirst-order LV contribution is given by the sum of Feynman diagrams given by Fig. 3.8,\nwhere the dark ball is for gµνλinsertion.\n•\n•• •\nFigure 3.8: Higher-derivative contributions with gµνλinsertion.\nThe propagator in this theory is usual, so, one can straightforwar dly sum the con-\ntribution of these diagrams. As a result, we arrive at the following hig her-derivative\ncontribution [121]:\nΓg=e2\n12mπ2Aµ(gµνα✷∂α−gµρα∂ρ∂α∂ν−gρνα∂ρ∂α∂µ)Aν. (3.97)\nWe note that a priorithe only restriction for the gµνλis its antisymmetry with respect\nto the first two indices. Therefore we can present the tensor gµναas a sum of irreducible\n(totally antisymmetric and partially symmetric) parts as gµνα=ǫµναβgβ+ ¯gµνα, where\ngγ=−1\n6gµναǫµναγ. In this case the first term in (3.97) will yield the higher-derivative\nCFJ-like result (3.90); evidently, if gµνλis totally antisymmetric, only the first term of\n52(3.97) is non-trivial. It was claimed in [121] that other components of (3.97) can describe\nsome MP-like corrections.\nAlso, we have noted above that the higher-derivative CFJ-like term (3.90) can arise\nalready in the first order in bµ. The corresponding prescription has been discussed in\n[119]. Our starting point is the usual fermionic determinant (3.39), w hich, as we already\nnoted, correspondstothesimplest LVextension oftheQED.Then weapplythederivative\nexpansion approach just as in the section 3.4, but in this case we exp and our fermionic\ndeterminant up to the third derivative of the background field Aµkeeping at the same\ntime only the first order in bµ. Effectively it means that we should expand the self-energy\ntensor given by(3.42) upto thethirdorder inthe external pµ. The result will befiniteand\nambiguity-free. Its explicit form is given by (3.90), withα\nM2=e2\n24m2π2. In principle, since\nthe constant vector bµis small, this term will dominate in comparison with the CFJ-like\npart of (3.95) which we calculated above. The finite temperature be havior of this term is\nalso described in [119].\n3.7 Divergent contributions in gauge and spinor sec-\ntors of LV QED\nUp to now, we, being motivated by the concept of emergent dynamic s, paid our main\nattention to finite contributions to the one-loop effective action. T he key idea of our\napproach consists in suggesting that various LV terms in the gauge sector are generated\nas a consequence of integrating over a spinor field in some fundamen tal LV extension of\nQED, which can be done consistently only if the results are finite, oth erwise one must\nintroduce these terms from the very beginning, jeopardizing the c oncept. At the same\ntime, possible divergent contributions to the effective action for diff erent LV extensions of\nQEDalso must beconsidered, inorder to obtaina morecomplete pert urbative description\nof a corresponding theory. We note that study of divergent corr ections is especially\nimportant for non-minimal LV theories because of their non-renor malizability.\nThe first step in study of divergent LV contributions has been perf ormed already in\n[35], where the one-loop renormalization of the minimal LV extension o f QED (2.1) in\nthe first order in LV parameters was carried out. Further discuss ions of these first-order\nresults, including the methodology allowing to adapt the known Lehma nn-Symanzik-\nZimmermann (LSZ) formalism to LV theories, are presented in [122]. C learly, the natural\ncontinuationsofthisstudyare, first, calculatingdivergent corre ctionsofsecondandhigher\norders in LV constant vectors (tensors), second, obtaining qua ntum corrections in the\nspinor sector of corresponding theories, third, investigation of h igher loop LV corrections.\n53Let us briefly review the main results achieved along these directions .\nThe divergent corrections in effective action, involving second and h igher orders in LV\nparameters, were studied in a number of papers. In [40], the aeth er-like contribution to\nthe effective action in the spinor sector of the non-minimal LV QED wit h the action\nS=/integraldisplay\nd4x¯ψ(i∂ /−m−eA /−gǫµνρσγµbνFρσ)ψ,\nsimilar to (3.68), but without ¯ψb /γ5ψterm, was found to look like\nΓ(1)\nsp∝ig2m2\nπ2ǫ¯ψb /(b·∂)ψ, (3.98)\nwhich matches the form of the CPT-even aether term for the spino r fieldiuµuν¯ψγµ∂νψ\nproposed in [39]. It worth mentioning that in three- and five-dimens ional space-times\nthe analogous aether-like corrections arise as well, being finite in the se cases within the\nframework of the dimensional regularization. It is clear that the ae ther term for the\nspinor field will arise as well in the case of presence of the ¯ψb /γ5ψadditive term. The\ndivergent aether-like results were also shown to arise in the spinor s ector in [123] where\nthe contributions of the second order in LV parameters of the minim al LV extension of\nQED[35], givenby(2.1),wereobtained. Also, divergentcontribution sinthespinorsector,\nup to the first order in derivatives, and proportional to the first o rder of non-minimal LV\nparameters listed in [53], were obtained in the paper [124] and in some o ther papers.\nIn the gauge sector, besides of divergent results for the CPT-ev en LV term discussed\nin the section 3.5, the most interesting divergent contributions of t he second order in LV\nparameters were obtained in the paper [125], where the minimal LV ex tension of QED\n[35], given by (2.1), was studied, and the one-loop divergent correc tions were shown to\nhavetheaether-likeform(3.66). Itisworthtomentionalsohigher- derivative divergent LV\ncontributions obtained in [111, 126] for the CPT-even case (of fou rth order in derivatives),\nand in [127], for the CPT-odd case (of third order in derivatives).\nAs for higher-loop calculations, it should be noted that, up to now, t here are a few\nhigher-loop results for LV theories. The examples are, first, two- loop renormalization in\nthe LV scalar field theory involving the aether-like LV term (2.10), pe rformed in [128],\nsecond, discussion ofhigher-loopcontributions inthesimplest LVex tension ofQED(3.38)\npresented in[129], whereit wasarguedthatnoCFJtermcanbegene ratedinhigherloops.\nThere are other interesting discussions of perturbative aspects of LV theories and\nrelated papers, we discussed only part of results obtained up to no w. Nevertheless, still\nthere are many perturbative calculations to be performed, espec ially, those ones treating\nimpacts of various non-minimal vertices listed in [53].\n543.8 Discussion and conclusions\nWe discussed the possibilities of perturbative generation of differen t LV terms. Explicitly,\nwe saw that these terms arise within the context of the emergent d ynamics in a theory\ninvolving coupling of scalar, gauge and, as it will be discussed in the Cha pter 6, gravi-\ntational fields to some primordial spinor field which afterwards is bein g integrated out.\nHere we were mostly interested in the calculation schemes allowing to o btain essentially\nfinite results, to avoid problems with the renormalization. Indeed, in an opposite case\nthe corresponding terms in gauge/scalar sector should be introdu ced already at the tree\nlevel, to get the multiplicatively renormalizable theory, hence these t erms cannot be more\nconsidered as an emergent phenomenon.\nWe showed that it is possible to obtain both the new LV terms defined in many space-\ntime dimensions, as e.g. the aether terms for gauge and scalar fields , and the new terms\nwell defined only in specific dimensions, such as the CFJ-like term for s calar fields defined\nonly in two dimensions, the mixed scalar-vector term defined only in th ree dimensions,\nand the CFJ term defined only in four dimensions. We demonstrated t hat there are\nmanners to generate various terms suggested in [53], which appar ently shows that the\nperturbative mechanism is a consistent way to get new LV terms. It is interesting to note\nthat in certain cases the LV couplings nevertheless can yield non-ze ro Lorentz-invariant\nquantum corrections. Besides of simple arising the terms proportio nal tobµbµ, as occurs\ne.g. within studies of the aether terms for scalar and gauge fields [40 ], a very interesting\nexample is presented in [52], where the Lorentz-invariant axion term has been generated\nin the LV Abelian gauge theory from the triangle Feynman diagram simila r to that one\nused in this chapter to obtain the CFJ term.\nAn interesting feature of some of our results is their ambiguity. Act ually, we showed\nthat at the one-loop order there are two typical ambiguities, the fi rst of them, being a\nparadigmatic example, arises within the calculation of the CFJ term, a nd of the terms\nwhichcanbeobtainedfromtheCFJtermthroughreplacingexterna llegs,whilethesecond\none arises in the purely non-minimal sector of the LV extended QED. We indicated that,\nin principle, other ambiguous integrals can exist, and the ambiguity pe rsists in the finite\ntemperature case as well. It is interesting to note that while the firs t ambiguity is closely\nrelated with the Adler-Bell-Jackiw (ABJ) anomaly (for details, see [26 ]), the possible\nanomaly related to the second ambiguity is not known yet, and, more over, the problem\nof its existence is still open, see also the discussion in [118].\nBesides of finite terms, which are of special interest, providing a po ssible mechanism\nfor generating LV terms in the gauge sector, there are many diver gent LV corrections\nwhich arise in various LV extensions of QED. We reviewed briefly some r esults related to\n55obtaining these terms presented in a number of papers. Still, many s tudies of divergent\nLV terms, as well as of finite ones, are to be done. It is natural to e xpect that one of\nmain lines in these studies will consist in explicit calculations of perturba tive impacts of\nvarious LV terms listed in the review [53]. In this context, we can ment ion the paper [84]\nwhere perturbative generation of the CFJ term in the extended sp inor QED involving\nall dimension-5 LV operators was studied, with the result was shown to be finite and\nambiguous.\nWe close the discussion claiming that the perturbative generation sh ows itself to be\nan appropriate mechanism for obtaining at least some of the LV exte nsions of known\ntheories, especially, of the spinor electrodynamics. In the Chapte r 6, we demonstrate that\nthis methodology can be applied in the case of gravity as well.\n56Chapter 4\nLorentz symmetry breaking and\nspace-time noncommutativity\nAs it was claimed in [28], one of important motivations for the Lorentz s ymmetry break-\ning stems from the space-time noncommutativity. Indeed, the initia l statement for the\nnoncommutative field theory is the suggestion that the commutato r of two space-time\ncoordinates ˆ xµ,ˆxνis the constant antisymmetric matrix Θµν(see [11]):\n[ˆxµ,ˆxν] =iΘµν. (4.1)\nThis matrix is clearly not Lorentz-invariant, so, this form of the spa ce-time noncommu-\ntativity breaks the Lorentz symmetry. At the same time, some of v arious additive LV\nterms proposed in [34, 53] are based on second-rank constant te nsors, which can be, in\nmany cases, antisymmetric as well. Hence, an action of a classical no ncommutative field\ntheory expanded up to a first order in Θµν, can be naturally treated as a particular case of\ncertain models discussed in [34]. Following [11], this relation emerges fro m the low-energy\nlimit of the superstring action in the presence of the constant antis ymmetric tensor field\nBµν, so that the Θµνis related with Bµνthrough the algebraic relation:\nΘµν= 2πα′/parenleftBigg1\ng+2πα′B/parenrightBiggµν\nA, (4.2)\nwhere the subscript Ais for the antisymmetric part, and α′is a constant describing a\ncoupling of the string to the Bµν. Here, the space-time metric gµνis assumed to be\nthe Minkowski one. We note that this manner of introducing the Lor entz symmetry\nbreaking within the string context essentially differs from that one u sed in [2], based\non the spontaneous Lorentz symmetry breaking. Therefore, th e problem of systematic\ndescription of noncommutative field theories is based on the method ology distinct from\n57that one described in previous chapters, but, nevertheless, the re are some strong analogies\nbetween noncommutative field theories and ”usual” LV theories, wh ich will be discussed\nin this chapter.\n4.1 Formulations for noncommutativity\nAs it was argued in [11], a systematic study of field theories defined in a noncommutative\nspace-time can be done through an appropriate mapping of these t heories to the usual\nspace-time. There are two known manners to map theories formula ted in a noncommuta-\ntive space-time to theories formulated in the usual space-time, th at is, the Seiberg-Witten\n(SW) map [11] and the Moyal product [11, 130].\nThe Moyal product is a universal formulation for noncommutative fi eld theories al-\nlowing to define noncommutative extensions for any field theory exc ept of gravity, since\nin that case one should generalize this product to the curved space -time as well, and the\nconsistent manner of such a generalization is still unknown. To intro duce the Moyal prod-\nuct, one assumes [11], that any field φ(ˆx) in a noncommutative D-dimensional space-time\ncan be expressed in the form of the Fourier integral:\nφ(ˆx) =/integraldisplaydDk\n(2π)D˜φ(k)eikˆx, (4.3)\nwithkare usual momenta, and ˜φ(k) is a Fourier transform of this field, so, using the\nHausdorff formula, for the product of two noncommutative fields o ne has\nφ1(ˆx)φ2(ˆx) =/integraldisplaydDk1dDk2\n(2π)2Dφ1(k1)φ2(k2)eik1ˆxeik2ˆx=\n=/integraldisplaydDk1dDk2\n(2π)2Dφ1(k1)φ2(k2)ei(k1+k2)ˆxe−i\n2Θµνk1µk2ν. (4.4)\nIt is clear that this product is associative and can be straightforwa rdly generalized to an\narbitrary number of fields, with the only difference with the usual co mmutative theories\nconsistsinthepresence ofanadditionalfactorlike e−iΘµνk1µk2ν. Thisallowsustointroduce\na following rule: the product of fields on the noncommutative space- time must be mapped\ninto their Moyal product on the usual space-time defined as\nφ1(x)∗φ2(x)∗...∗φn(x) =/integraldisplaydDk1dDk2...dDkn\n(2π)nDφ1(k1)×\n×φ2(k2)...φn(kn)ei(k1+k2+...+kn)xexp[−i\n2Θµν/summationdisplay\ni<j≤nkiµkjν]. (4.5)\n58We see that the impact of the space-time noncommutativity is conce ntrated now in the\nΘ-dependent phase factor. This formula evidently can be applied to a product of any\nfields.\nAnother manner to treat the noncommutativity is based on the SW m ap which is an\nespecially powerful tool for the gauge theories. In this case we su ggest that the noncom-\nmutative gauge theory is described in terms of the noncommutative gauge field ˆAµwhich\ncan be represented as a power series in Θµν. Let our starting point be a noncommutative\n(NC) Yang-Mills theory whose action is\nS=−1\n4tr/integraldisplay\ndDxFµν∗Fµν, (4.6)\nwithFµν=∂µˆAν−∂νˆAµ−ig[ˆAµ,ˆAν]∗, so, the commutator is both algebraic and Moyal\none. We see that it is nontrivial even for the U(1) case. The ˆAµis a noncommutative\ngauge field.\nThen we, following [11], suggest that the NC gaugefieldis projected in to a usual gauge\nfield in such a way that if the noncommutative field ˆAµis mapped in some manner to\nthe usual commutative field Aµ, the infinitesimal gauge transformation of ˆAµis related\nthrough this mapping to the infinitesimal gauge transformation of Aµ, so that\nˆA(A)+δˆλˆA(A) =ˆA(A+δλA), (4.7)\nwith the parameter ˆλof the noncommutative gauge transformation is linked with the\nusual parameter of the gauge transformation λin some way. It was shown in [11] that the\nexplicit expressions for ˆA,ˆλandˆFcan be obtained as series in Θ order by order, with\nthe lower terms of these series look like:\nˆAµ=Aµ−1\n4Θνλ{Aν,∂λAµ+Fλµ}+O(Θ2);\nˆλ=λ+1\n4Θνλ{∂νλ,Aλ}+O(Θ2); (4.8)\nˆFµν=Fµν+1\n4Θκρ(2{Fµκ,Fνρ}−{Aκ,DρFµν+∂ρFµν})+O(Θ2).\nThis mapping can be used to construct the SW analogue of the nonco mmutative Yang-\nMills theory which is known to display many interesting effects at the tr ee level.\nIn the next sections we discuss some impacts of the noncommutativ ity within the LV\ncontext, both within Moyal product and SW map approaches.\n4.2 LV impacts of the Moyal product\nThe Moyal product approach is the most used formulation of the no ncommutative space-\ntime being especially convenient for calculating of the loop correction s. It is easy to show\n59that the quadratic part of the action for any noncommutative mod el is not affected by\nthe noncommutativity, which nontrivially contributes to interaction vertices only. Then,\nin many cases, due to presence of (anti)symmetrized products of fields in vertices, the\nMoyal phase factors are reduced to sine or cosine forms. As a res ult, we can find new\nforms of contributions from different Feynman diagrams.\nAs the simplest example, we consider the noncommutative φ4theory [131]:\nS=/integraldisplay\ndDx/bracketleftBigg\n−1\n2φ(✷+m2)φ+λ\n4!φ∗φ∗φ∗φ/bracketrightBigg\n. (4.9)\nIt is instructive to give here the explicit form of the noncommutative quartic vertex in\nmomentum space reflecting the symmetry between all four φlegs:\nV4=λ\n72/integraldisplaydDp1dDp2dDp3ddp4\n(2π)4Dφ(p1)φ(p2)φ(p3)φ(p4)× (4.10)\n×[cos(p1∧p2)cos(p3∧p4)+cos(p1∧p3)cos(p2∧p4)+\n+ cos(p1∧p4)cos(p2∧p3)].\nHere we define p∧q=1\n2Θµνpµqν.\nThe lower quantum correction is presented by an usual seagull gra ph given by Fig.\n4.1.\nFigure 4.1: Higher-derivative contributions with three insertions.\nAfter carrying out all contractions, the contribution of this grap h to the self-energy\ntensor, in the Euclidean space, can be cast as [131]\nΣ(p) =−λ\n6/integraldisplaydDk\n(2π)D2+cos(2k∧p)\nk2+m2. (4.11)\nWe see that this is a sum of two terms called planar and nonplanar cont ributions. The\nreasons for the names ”planar” and ”nonplanar” can be found in [13 1] and in papers\ncited there. The planar contribution is the one not affected by the n oncommutativity. It\ncoincides with the contribution of the whole graph for the commutat ive analogue of this\ntheory, up to the constant factor – indeed, for Θµν= 0, the numerator reduces to 3 – and,\nin this case, the planar contribution displays the same divergence, u p to a constant factor,\nas in the commutative counterpart of this theory. In principle, in so me noncommutative\n60field theories, contributions from certain Feynman diagrams can be completely planar\ndue to cancellation of the exponential factors stemming from vert ices, for example, such\na situation occurs for a two-point function of the gauge superfield in the supersymmetric\nCPN−1model in the fundamental representation [132].\nIt is clear that since the planar part does not depend on Θµν, it is perfectly Lorentz\ninvariant, for D= 4+ǫbeing equal to\nΣpl(p) =λm2\n48π2ǫ+fin. (4.12)\nThe second, nonplanar term essentially depends on the noncommut ativity matrix Θµν.\nTo proceed with it, we can use the formula [133], with p◦p≡1\n4pµΘµνΘνλpλ:\n/integraldisplaydDk\n(2π)Deik∧p\n(k2+M2)N= (−1\n2)N1\n(2π)D/21\n(N−1)!(M2)D/2−N×\n×KD/2−N(√M2p◦p)\n[√M2p◦p]D/2−N. (4.13)\nIt is well known that for small arguments x→0, the asymptotics of the modified Bessel\nfunction is Kn(x)|x→0∼x−n. Therefore we see that if D/2≥N, the r.h.s. of the\nexpression above is singular at p→0 being proportional to1\n(p◦p)D/2−N. In particular, for\nN= 1 andD= 4, as in our case, the nonplanar contribution is proportional to1\np◦p, thus\ndisplaying a quadratic infrared singularity and being at the same time o f the order Θ−2,\nthat is, the singularity of the same order as the UV divergence in a pla nar counterpart\nof the contribution. This mechanism for arising infrared singularities of such a form in\nnoncommutative theories with ultraviolet divergences is called the ult raviolet/infrared\n(UV/IR) mixing [134], and these singularities are called the UV/IR infra red ones.\nBesides of this, we note that in general, ΘµνΘνλis not required to be proportional to\nδµ\nλ, hence we find that the nonplanar contributions, first, essentially break the Lorentz\nsymmetry, second, are large if the UV/IR infrared singularity is pre sent, since the Θµνis\nnaturally treated to be small. Hence we face the problem of large LV q uantum correc-\ntions, similarly to the situation occurring in higher-derivative LV theo ries [135]. From a\nformal viewpoint, the large quantum corrections reflect the fact that the noncommutative\ndeformation of the theory plays the role of some regularization use d to eliminate the ul-\ntraviolet divergences, similarly to higher-derivative additive terms, therefore, at Θµν→0,\nas well as if higher derivative additive terms in [135] are switched off, the regularization is\nremoved, and the singularities are recovered. Therefore we see t hat together with break-\ning of the perturbative expansion, we violate the restriction for LV terms to be small.\nMoreover, using the mechanism of the UV/IR mixing [134], one can obt ain terms with\nan arbitrary large negative degree of ΘµνΘνλ, so, the resulting effective theory cannot\n61be well-defined if we suggest LV parameters to be small. This is the main problem of\nthe noncommutative field theories, which, as is well known, is typically solved through an\nappropriate supersymmetric extension of a noncommutative theo ry which typically allows\nto eliminate dangerous quadratic and linear divergences, both ultra violet and the related\nUV/IR infrared ones. The paradigmatic example is the noncommutat ive Wess-Zumino\nmodel [136]. However, to close this section, we emphasize again that the form of non-\ncommutative theories, either supersymmetric or non-supersymm etric ones, is essentially\ndistinct from conventional LV theories considered throughout th is book, whose action is\na sum of a usual Lorentz-invariant action and a small additive LV ter m involving only\nsome lower degrees of LV constant parameters.\n4.3 Seiberg-Witten map\nAnother description of noncommutative gauge field theories is base d on the SW map. As\nwe already noted in Section 4.1, the SW map is based on the gauge-pre serving mapping\nof the noncommutative gauge field to the commutative one. It is eas y to see [11] that\nwhile the action of the usual NC gauge theory is polynomial with respe ct to the fields\nˆAµ, see (4.6), its commutative image will look like an infinite series in Aµ, and there is\nno closed form for it. Nevertheless, the SW map is known to obtain ma ny interesting\ntree-level results.\nThe SW mapping for the noncommutative Maxwell action up to the lowe r nontrivial\norder in Θαβyields the following Lagrangian:\nLSW=−1\n4FµνFµν+1\n8ΘαβFαβFµνFµν−1\n2ΘαβFµαFνβFµν+.... (4.14)\nFirst of all, one can show that the wave propagation in the theory (4 .14) displays some\ninteresting features. Indeed, the corresponding Euler-Lagran ge equations will be highly\nnonlinear, therefore, whereas the first pair of the Maxwell equat ions, that one stemming\nfrom the antisymmetry of Fµν, will be the same as in the usual electrodynamics, the\nsecond one, corresponding to Euler-Lagrange equations, will be s trongly modified [137].\nSo, let us briefly discuss the Maxwell equations in the new theory.\nIf we define as usual that F0i=Ei, andFij=−ǫijkBk, and choose the noncommu-\ntativity to be purely spatial, that is, Θ0i= 0 and Θij=ǫijkΘk, while the first pair of\nMaxwell equations has the standard form\n/vector∇·/vectorB= 0;\n∇×/vectorE=−1\nc∂/vectorB\n∂t, (4.15)\n62the second pair can be formally written as\n/vector∇·/vectorD= 0;\n∇×/vectorH=1\nc∂/vectorD\n∂t, (4.16)\nbut in this case the vectors /vectorHand/vectorDare strongly nonlinear in /vectorEand/vectorB, so that up to\nthe first order in /vectorΘ, one finds [138]:\n/vectorD= (1−/vectorΘ·/vectorB)/vectorE+(/vectorΘ·/vectorE)/vectorB+(/vectorE·/vectorB)/vectorΘ;\n/vectorH= (1−/vectorΘ·/vectorB)/vectorB+1\n2(E2−B2)/vectorΘ−(/vectorΘ·/vectorE)/vectorE. (4.17)\nThe solutions of these equations display essentially new effects beca use of their nonlinear\nnature. For example, even for the simplest electric field described b y the plane wave\n/vectorE=/vectorE(ωt−/vectork·/vector r), one has possibilities to obtain very nontrivial effects. First, the k nown\ntransversality relations for electric and magnetic fields, and similarly for/vectorHand/vectorD, look\nlike [138]:\n/vectorB=/vector κ×/vectorE+/vectorb;\n/vectorD=−/vector κ×/vectorH+/vectord. (4.18)\nHere/vector κ=c/vectork\nωis a dimensionless wave vector, and the vectors /vectorb,/vectordare time-independent\nones, they serve to describe some primordial constant electric an d magnetic fields.\nOne of the conclusions is that now the electric field is no more a transv ersal one, with\nits longitudinal component ELis related with transversal ones Ei\nTas\nEL=−ˆκiβijEj\nT, (4.19)\nwith ˆκis a normalized wave vector, and βij= Θibj+Θjbi.\nThe dispersion relation in this theory isω\nck= 1−/vectorΘT·/vectorbT, where/vectorbTis a transversal part\nof the background magnetic field /vectorb(4.18), and /vectorΘT– of the vector /vectorΘ (we note that, fol-\nlowing [82], the space-time noncommutativity itself can give an origin fo r the background\nmagnetic field). So, superluminal propagation and causality violation are ruled out for\ncertain choices of /vectorΘTand/vectorbT.\nA more detailed discussion of dispersion relations in the SW formulation of the non-\ncommutativetheorieswasdevelopedin[139]. Followingthispaper, wes tartwithcovariant\nequations of motion which can be obtained from the Lagrangian (4.14 ):\n∂µFµν+ΘαβFµ\nα(∂βFν\nµ+∂µFν\nβ) = 0. (4.20)\n63Then, it is easy to show that for the plane wave solution Fµν(x) =˜Fµν(kx), withkµbeing\na constant vector, Eq. (4.20) trivially becomes\nkµ˜F′µν+Θαβ˜Fαµ(kβ˜F′ν\nµ+kµ˜F′ν\nβ) = 0, (4.21)\nwhere the prime in ˜F′\nµνis for differentiation with respect to kx. In this context, remaining\nMaxwell equations are given by the standard Bianchi identity,\n∂ρFµν+∂νFρµ+∂µFνρ= 0, (4.22)\nand contracting it with ∂ρ, we obtain\nk2˜F′\nµν+kρkν˜F′\nρµ+kρkµ˜F′\nνρ= 0, (4.23)\nwhere we have used also the fact that our field is the plane wave solut ion. Now, by using\nEq. (4.21) in the above expression, we finally get\nk2˜F′\nµν+2Θαβ˜Fρ\nαkβkρ˜F′\nµν= 0. (4.24)\nWe note that since kρ˜Fρ\nαis already of the first order in Θ because of (4.21), the second\nterminaboveequationcanbedisregarded. Then, weconcludethat thedispersionrelation\nin this theory has the usual form k2= 0, and, by considering again Eq. (4.22), the plane\nwave continues to be transversal as in the commutative case, i.e., kµ˜F′µν= 0.\nHowever, if we make the simplest modification by considering the case of a superpo-\nsition of a constant background Bµνand a plane wave ˜Fµν(kx), the field equation (4.20)\nbecomes\n˜kµ˜F′µν= 0, (4.25)\nwhere˜kµ=kµ+ ΘαβBαµkβ−ΘµαBαβkβ. Thus, by substituting the above expression\ninto Eq. (4.23), we obtain the modified dispersion relation k2=−2ΘαβBρ\nαkβkρ, which in\ncertain cases reduces to the results of [138]. Moreover, one can s how as well that there\nexists a mapping between the solution of the equations of motion in th e Moyal picture,\nwhere the Moyal product is expanded up to the first order, and th e superposition of a\nconstant background and a plane wave in the SW picture. This justifi es that at the tree\nlevel, these two approaches are equivalent for a certain class of so lutions, that is, the\nsolutions described by a superposition of a constant field and a plane wave.\nNevertheless, it should be noted that at the quantum level these t wo approaches are\nessentially distinct, since, unlike the Moyal product approach, firs t, the SW map yields\nan essentially non-renormalizable theory, second, it can be made to be consistent with the\nsupersymmetry only with use of a nonlinear representation of the S USY algebra [140].\n64We note that the behavior rather similar to that one described abov e is displayed\nby the plane wave solutions in another LV deformation of the electro dynamics based on\nthe Heisenberg-Euler action [141], so, it is quite typical for nonlinear extensions of the\nelectrodynamics. As a by-product, we note that the noncommuta tivity and/or Lorentz\nsymmetry breaking can be treated as possible ingredients for analo gue models aimed to\ndescribe the propagation of electromagnetic field in a condensed ma tter as it was also\nsuggested in [139].\n4.4 Noncommutative fields\nBesides of the Moyal product and Seiberg-Witten map, there is one more manner to im-\nplement anoncommutativity withinthefield theorymodels, thatis, th eso-callednoncom-\nmutative field approach proposed initially in [142]. Its starting point is the deformation\nof canonical commutation relations between two fields or two momen ta. Following [82], if\none considers a theory of two free real scalar fields φ1andφ2with equal masses described\nby the usual action corresponding to the Hamiltonian density\nH=1\n22/summationdisplay\ni=1/parenleftBig\nπiπi+/vector∇φi·/vector∇φi+m2φiφi/parenrightBig\n. (4.26)\nThen we deform the canonical commutation relation between two mo menta as\n[πi(/vector x),πj(/vector y)] =iΘijδ(/vector x−/vector y), (4.27)\nwhereas other canonical commutation relations stay untouched ( in [87], the similar anal-\nysis was performed for an equivalent theory involving a complex scala r fieldφ, and the\nconjugated one φ∗). In the case of free fields a deformation of the relation between t wo\nfields implies in analogous results, however, in a general theory such a deformation could\nyield much more complicated, in general nonpolynomial, expressions. In principle, such\na deformation can be treated as a kind of Bopp shift. Also, it is clear t hat non-trivial\nnoncommutative relations between space-time coordinates xµthemselves will imply as\nwell nontrivial noncommutative relations between their functions, in particular, fields\nand corresponding canonical momenta. Therefore, this approac h can be naturally in-\nterpreted as another, alternative description for noncommutat ivity, although there is no\nstraightforward mapping between Moyal product and noncommut ative fields frameworks.\nAfter the above mentioned deformation of commutation relations, we should find new\ncanonical variables Π iwhose commutator is the usual one, [Π i(/vector x),Πj(/vector y)] = 0. They are\ngiven by the definition\nΠi=πi−1\n2Θijφj, (4.28)\n65which allows us to write immediately the deformed Lagrangian, which, a fter expressing\nof momenta in terms of velocities, looks like\nLdef= Πi˙Ai−H=−2/summationdisplay\ni=1φi(✷+m2)φi−1\n22/summationdisplay\ni,j=1Θij˙φiφj. (4.29)\nIt is clear that the additive term proportional to Θ ijbreaks the Lorentz symmetry involv-\ning only the time derivative, which clearly will modify the dispersion relat ions.\nIf we apply this methodology to the free electrodynamics [82], the sit uation becomes\nmore interesting. We follow the usual Dirac prescription for quantiz ation of constrained\ntheories, so, we get the primary constraint Φ(1)=π0≃0, whereπµ=F0µis a momentum\ncanonically conjugated to Aµ. The Hamiltonian density is\nH=1\n2πiπi+1\n4FijFij+A0∂iπi. (4.30)\nTherequirement ofconservationoftheprimaryconstraintyieldst hesecondaryone, Φ(2)=\n∂iπi≃0, after which the Dirac algorithm ends, there is no new constraints more.\nNow, we again deform the canonical commutation relations to the fo rm (4.27). The\nkey point of the noncommutative approach now that we want the ne w model, obtained\nthrough the deformation of the gauge theory, to be gauge invaria nt as well. To see the\nway to do it, we note that the gauge transformation of Aifield with a parameter α(/vector x)\ncan be presented as\nδAi(/vector x) = [Ai(/vector x),/integraldisplay\nd3/vector yα(/vector y)Φ(2)(/vector y)] =−∂iα(x). (4.31)\nIn the usual, non-deformed case, one finds the following gauge tra nsformation of the\ncanonical momentum:\nδπi(/vector x) = [πi(/vector x),/integraldisplay\nd3/vector yα(/vector y)Φ(2)(/vector y)] = 0,\nas it should be. We want to have the same gauge transformations of the field and the\nconjugated momentum in the deformed case as well which is very nat ural, while it is clear\nthat in this case [ πi(/vector x),/integraltextd3/vector yα(/vector y)Φ(2)(/vector y)]∝ne}ationslash= 0 since the πi’s do not commute more. To\nachieve, for fields and momenta, the same gauge transformations as in the undeformed\ncase, we extend the constraint Φ(2)by an additive term so that the new constraint ˜Φ(2)\nreads as\n˜Φ(2)=∂iπi−Θjk∂jAk. (4.32)\nIt is easy to check that in this case one has [ Ai(/vector x),/integraltextd3/vector yα(/vector y)˜Φ(2)(/vector y)] =−∂iα(x) and\nπi(/vector x),/integraltextd3/vector yα(/vector y)˜Φ(2)(/vector y)] = 0 as it should be. For the sake of the convenience, we introduce\nthe vector Θ isatisfying the relation Θ i=1\n2ǫijkΘjk, so, Θjk=ǫijkΘi.\n66Again, one needs to get a canonical momentum Π isatisfying the relation [Π i,Πj] = 0,\nwhile other commutation relations stay unchanged, as above. The Π ievidently looks like\nΠi=πi−1\n2ΘijAj, (4.33)\nand the resulting Lagrangian is\nL=πi˙Ai−H, (4.34)\nwhere the Hamiltonian His given by (4.30), and the momenta are afterward expressed\nin terms of velocities. As a result, we arrive at the new Lagrangian\nLnew=−1\n4FµνFµν+1\n4ǫµνλρΘµAνFλρ, (4.35)\nwhere the constant 4-vector Θ µis defined as Θ µ= (0,Θi). Therefore, we see that the\nelectrodynamics with the noncommutative deformation of the cano nical algebra is equiv-\nalent to the electrodynamics with usual commutation relations, but with the CFJ term.\nIt is interesting to note that performing the analogous noncommut ative deformation of\nthe canonical algebra in the electrodynamics with the CFJ term will imp ly in modifica-\ntion of the CFJ coefficient only, but without arising other new terms. Also, if we apply\nthis methodology in the three-dimensional case, instead of a LV ter m, the usual Lorentz-\ninvariant Chern-Simons term will arise, with the mass is proportional to Θ =1\n2ǫijΘij\n[143].\nThis methodology can be applied as well to non-Abelian theories [144] where the non-\nAbelian CFJ term (2.6) was generated. Therefore, the noncommut ative deformation of\nthe canonical algebra can be treated as one more manner to gener ate the LV terms, which\ndiffers from the perturbative generation approach based on using additional LV couplings.\nMoreover, this methodology can be applied as well even in the linearize d gravity case\n[145], where, however, the study is much more sophisticated – first , the components of a\nmetric fluctuation tensor have two indices, which makes calculations to be more involved,\nsecond, to achieve the transversality, one should implement the tr ansverse projectors into\ndeformed commutation relations playing the role of analogues of (4.2 7). The extended\nlinearized gravity theory with a resulting additive term, under imposin g some gauge fixing\nconditions, yields the same dispersion relations as the extended linea rized gravity theory\nwiththeadditive termgivenby(6.34), which justifies theequivalence ofthetermobtained\nby noncommutative fields approach and the term (6.34).\n674.5 Conclusions\nWe considered the LV impacts of noncommutative extensions for fie ld theory models. It\nshould be noted that we treated the simplest form of the space-tim e noncommutativity\nbased on the Moyal product together with its mapping to a nonpolyn omial commuta-\ntive theory through the SW map. In principle, one can introduce as w ell other forms of\nnoncommutativity which are not based on constant noncommutativ ity parameters, e.g.,\nthe dynamical noncommutativity approach [146]. However, this app roach is Lorentz in-\nvariant since all Θµν, being new coordinates in this approach, are integrated out at fina l\nsteps of all calculations. As a result of using the Moyal product or t he SW map, one\narrives at a theory essentially depending on the LV constant antisy mmetric tensor Θµν.\nWe see, however, that while the SW map description is rather similar to the approach\ndeveloped for constructing the usual LV SME [5, 6], so that the act ion is a sum of a\nknown Lorentz-invariant classical action and small LV corrections , the approach based\non the Moyal product is essentially different, because, first, inste ad of calculating order\nby order, it allows to obtain the exact dependence of quantum corr ections on the non-\ncommutativity parameters Θµν, second, in many cases, as a consequence of the UV/IR\nmixing, the results display singularities in a commutative limit. Neverthe less, both these\nsituations have certain analogies in ”usual” LV theories: the exact d ependence on the LV\nparameters is treated, e.g., within the exact propagator of the sp inor field (3.44) allowing\nto take into account all orders of its expansion in the LV vector bµ[97], and the singulari-\nties in the Lorentz-invariant limit, which in NC theories corresponds t o the commutative\nlimit, can occur in some higher-derivative LV extensions of the field th eory models [135].\nThis similarity is rather natural since the noncommutative field theor ies by their essence\nare nothing more as infinite-derivative LV extensions of usual field t heories. We argued\nthat, at the classical level, in lower orders in expansion in the Θµνparameter, for some\nsolutions of equations of motion, there is some mapping between SW m ap and Moyal\nproduct which justifies their classical equivalence in certain cases. However, the situation\nis more involved when quantum corrections are studied, since, unlike the Moyal product\napproach, within the SW map framework, the QED is clearly non-reno rmalizable.\nAlso, weintroducedamethodologyofnoncommutative fieldsallowing f ormapping ofa\ntheory formulated in terms of noncommuting momenta into a theory formulated in terms\nof canonical variables, whose action differs from the original action by additive terms of\ndifferent orders in noncommutativity parameters Θ ij, and their presence explicitly breaks\nthe Lorentz symmetry in four- and higher-dimensional space-time s. The relevance of the\nnoncommutative fields approach within the LV context is confirmed b y the fact that, as\nwe have noted in previous chapters, it allows to generate some of kn own LV additive\n68terms, such as the CFJ term, its non-Abelian generalization, and th e two-dimensional\nCS-like term for scalar fields.\nWe noted already that to solve the main problem of noncommutative t heories, that\nis, the problem of nonintegrable infrared singularities caused by the UV/IR mixing mech-\nanism, one should carry out the supersymmetric extension of thes e theories. This is,\nclearly, one of motivations to the interest in LV extensions of super symmetric, especially\nsuperfield, theories. We discuss this problem in the next chapter.\n6970Chapter 5\nLorentz symmetry breaking and\nsupersymmetry\nThe supersymmetry began to be treated as a fundamental symme try of Nature already\nin 1970s by many reasons: first, it essentially improves the renorma lization behavior of\nfield theory models, second, it allows to unite Poincare symmetry with internal symme-\ntries in a nontrivial manner, third, it found broad applications within t he string theory.\nHowever, for a long time there was a common belief among field theoris ts that since the\nsupersymmetry represents itself as a nontrivial extension of the Poincare symmetry, in\nLV theories the supersymmetry should be broken as well. Neverthe less, a first manner\nto construct the LV extension of the supersymmetry algebra was proposed already in\n2001 [31]. In 2003, another manner to conciliate the Lorentz symme try breaking with an\nunbroken supersymmetry, based on introduction of some extra s uperfield(s) with some of\nwhose components depending on the LV parameters [147], was prop osed, we note that\nnamely within this approach the supersymmetric extension of the CF J term was success-\nfully constructed. In [148], an idea of a straightforward LV exten sion of supersymmetric\nfield theories where additive LV terms are introduced already at the level of a superfield\naction, was proposed. In this chapter we review all these approac hes, including their ad-\nvantages and disadvantages, and present corresponding main re sults obtained with their\nuse, especially, examples of quantum calculations. We essentially use the superfield for-\nmalism for the description of LV supersymmetric field theories, since it, first, allows for\nvery compact formulation of superfield theories, second, is highly c onvenient for quan-\ntum calculations (for a review of superfield methodology in supersym metric field theories\nsee e.g. [149]). As we claimed in the previous chapter, the noncommut ative (super)field\ntheories also display the Lorentz symmetry breaking, the methodo logy of their studies is\nwell developed, however, in this chapter we will restrict ourselves b y theories where the\n71Lorentz symmetry breaking is implemented in a conventional manner , that is, via small\nadditive terms.\n5.1 Deformation of the supersymmetry algebra\nThe Kostelecky-Berger method of LV extension for superfield the ories based on the de-\nformation of the supersymmetry algebra is perhaps the most elega nt one. Its main idea\nlooks like follows [31]. Let us start with the usual supersymmetry alge bra whose anticom-\nmutation relations, in the four-dimensional case, are (see e.g. [14 9]):\n{Qα,¯Q˙β}= 2iσµ\nα˙α∂µ, (5.1)\nso, the supersymmetry generators can be chosen, e.g., in the for m\nQα=−∂α+i¯θ˙β¯σµ\n˙βα∂µ;\n¯Q˙α=∂˙α−iθβσµ\n˙αβ∂µ (5.2)\nOther representations of the supersymmetry algebra in a four-d imensional space-time\nare possible as well, the only requirement is that in any case, generat ors must satisfy\nthe relation (5.1). The θα,¯θ˙αare the Grassmannian (anticommuting) coordinates of the\nsuperspace, and ∂α=∂\n∂θα, with an analogous definition for ∂˙α. All other anticommutators\nare:{Qα,Qβ}= 0,{¯Q˙α,¯Q˙β}= 0, and both Qαand¯Q˙αcommute with usual space-time\nderivatives ∂µ. The supersymmetry transformation of an arbitrary superfield Σ can be\nrepresented as superspace translations: δΣ = (ǫαQα+ ¯ǫ˙α¯Q˙α)Σ, where ǫα, ¯ǫ˙αare the\ninfinitesimal constant Grassmannian parameters.\nThen, let usdeformthese generatorsthroughasimple replacemen t:∂µ→∂µ+kµν∂ν=\n∇µ, with∇µcan be called a “twisted” derivative, i.e. instead of generators (5.2) , one\nmust use new, ”twisted” generators:\nQα=−∂α+i¯θ˙β¯σµ\n˙βα∇µ;\n¯Q˙α=∂˙α−iθβσµ\n˙αβ∇µ. (5.3)\nThekµνis a constant tensor breaking the Lorentz symmetry. Without loss of generality,\nwe can require all its components to be much less than 1, enforcing t he Lorentz symmetry\nbreaking to be small. Such replacement is linear with respect to deriva tives, in order to\nbe consistent with the Leibnitz rule. In principle, in some simple situatio ns one can also\nconstruct a consistent deformation of the supersymmetric field t heory abandoning the\nLeibnitz rule [150], however, in many cases the linearity of supersymm etry generators in\nderivatives is essential.\n72To construct thesuperfield theory, oneneedstointroducesupe rcovariant derivatives in\nsuch a manner that, for any superfield Σ, the supercovariant der ivatives of this superfield\nwould be superfields, i.e. these derivatives should anticommute with s upersymmetry\ngenerators: {Qα,Dβ}= 0,{¯Q˙α,Dβ}={Qα,¯D˙β}={¯Q˙α,¯D˙β}= 0. Explicitly, the\n”twisted” supercovariant derivatives consistent with the deform ed supersymmetry look\nlike\nDα=∂α+i¯θ˙β¯σµ\n˙βα∇µ;\n¯D˙α=−(∂˙α+iθβσµ\n˙αβ∇µ), (5.4)\nand satisfy the following anticommutation relations:\n{Dα,¯D˙β}=−2iσµ\nα˙α∇µ. (5.5)\nIt is clear that the Leibnitz rule is maintained for these derivatives. F rom this relation,\none can derive the following useful identities:\nD2¯D2D2= 16✷DD2;DαDβDγ= 0, (5.6)\nwith the analogous identities can be obtained after replacements of Dαby¯D˙α. Here\n✷D=∇µ∇µ=✷+2kµν∂µ∂ν+kµνkρν∂µ∂ρis the deformed d’Alembertian operator.\nAs a next step, one can introduce a chiral superfield Φ satisfying th e relation ¯D˙αΦ = 0,\nand similarly the antichiral one ¯Φ so thatDα¯Φ = 0. Their components can be defined\nthrough projections:\nφ= Φ|, ψα=1\n2DαΦ|, F=D2\n4Φ|, (5.7)\nwhere the |symbol means that after the differentiation one puts θ=¯θ= 0, and the\nexpressions for components of ¯Φ are the analogous ones. This allows us to define the LV\nWess-Zuminomodelwhoseactionbeingwrittenintermsofsuperfield sformallyreproduces\nthe usual expression\nS=/integraldisplay\nd8zΦ¯Φ+/bracketleftBigg/integraldisplay\nd6z/parenleftBiggm\n2Φ2+λ\n3!Φ3/parenrightBigg\n+h.c./bracketrightBigg\n, (5.8)\nbut the component form of this action involves extra, LV terms and looks like\nS=/integraldisplay\nd4x/parenleftBig\nφ✷D¯φ+F¯F+i¯ψ˙ασµα\n˙α∇µψα+\n+/parenleftBig\nm(ψ2+φF)+λ(φψ2+1\n2Fφ2)+h.c./parenrightBig/parenrightbigg\n. (5.9)\nOne can immediately see that this action involves the aether-like term for the spinor field\ni¯ψ˙ασµα\n˙αkµν∂νψαwhich for the simplest, aether-like choice kµν=αuµuν, withαis some\n73number, and uµis a constant vector (it is natural to require |α| ≪1, withuµuµis either\n−1, 0 or 1), exactly reproduced the aether term introduced in [39], and, under the same\nchoice, the aether-like term αφ(u·∂)2¯φand the new term of fourth order in the vector\nuµfor the scalar field. Actually, the last term also displays the aether f orm looking like\nα2u2φ(u·∂)2¯φ. So, we see that this theory involves aether terms for scalar and s pinor\nfields as ingredients.\nThe superfield propagators in the theory (5.8) are exact analogue s of those ones in the\nusual Wess-Zumino model, with only difference in replacement of a simp le derivative ∂µ\nby a ”twisted” one ∇µ:\n<Φ(z1)¯Φ(z2)>=−i\n✷D−m2δ8(z1−z2); (5.10)\n<Φ(z1)Φ(z2)>=<¯Φ(z1)¯Φ(z2)>∗=−im\n✷D−m2(D2\n4✷D)δ8(z1−z2),\nwithD2,¯D2factors are associated with the vertices by the same rules as in the usual\nWess-Zumino model [149], that is, a vertex with an external chiral fi eld carries the factor\n−¯D2\n4, and with an external antichiral one – the factor −D2\n4.\nHere, some words about dispersion relations are in order. Disregar ding the factor1\n✷D\nwhich is really needed only to rewrite integrals over a chiral (antichira l) subspace as those\nones over a whole superspace, and does not correspond to new de grees of freedom, one\nfinds the physical spectra are completely described by the denomin ator✷D−m2, or, after\nthe Fourier transform, ˜ p2+m2, where ˜p2= (pµ+αuµuαpα)(pµ+αuµuβpβ) is a twisted\nscalar square of the momentum. We find that for different choices o fuµ, we have different\ndispersion relations:\n(i) For the time-like uµ= (1,0,0,0), one has E2(1−α)2=/vector p2+m2.\n(ii) For the space-like uµ= (0,1,0,0), one has E2=/vector p2+m2+(2α+α2)(/vector u·/vector p)2.\n(iii)Forthelight-like uµ= (1,1,0,0),and/vector palongthexaxis, onehas E=1\n1−2α(−2αp±/radicalBig\np2(1+2α+4α2)+m2.\nIt is clear that the dynamics in all these cases is consistent if αis enough small.\nAs an example of quantum calculation in this theory we present here t he computation\nof the one-loop low-energy effective action which is completely descr ibed by the K¨ ahlerian\neffective potential, by the definition depending only on the superfield s themselves but not\non their derivatives. We employ the methodology of summation over a n infinite number\nof Feynman supergraphs discussed in details in [151].\nInsupergraphs given by theFig. 5.1, theFeynman rulesaremodified : weincorporated\nthe mass into background fields Ψ = m+λΦ,¯Ψ =m+λ¯Φ denoted here by double lines,\n74✧✦★✥\n✧✦★✥\n✧✦★✥\n\u0000\u0000\u0000\u0000 ❅❅❅❅❅❅❅❅ \u0000\u0000\u0000\u0000\n...\nFigure 5.1: Contributions to the K¨ ahlerian effective potential.\nso, we have\n<Φ(z1)¯Φ(z2)>=−i\n✷Dδ8(z1−z2);\n<Φ(z1)Φ(z2)>=<¯Φ(z1)¯Φ(z2)>∗= 0. (5.11)\nThese Feynman supergraphs can be summed, just as it is done for u sual superfield\ntheories, yielding the following expression for the one-loop K¨ ahleria n effective potential\nK(1)=−i\n2∞/summationdisplay\nn=11\nn/integraldisplay\nd4θ/bracketleftBigg\nΨ¯ΨD2¯D2\n16✷2\nD/bracketrightBiggn\nδ(z−z′)|z=z′, (5.12)\nWe simplify this expression taking into account that theD2¯D2\n16✷Dis a projecting operator\neven in our LV case, so, [D2¯D2\n16✷D]n=D2¯D2\n16✷D. Then, we shrink the loop into a point through\ntheknown identityD2¯D2\n16δ(θ−θ′)|θ=θ′= 1 [149], and, after Fourier transform, remembering\nthe structure of the deformed d’Alembertian operator, we arrive at\nK(1)=i\n2/integraldisplay\nd4θ/integraldisplayd4q\n(2π)41\n(qµ+kµνqν)2ln/parenleftBigg\n1−¯ΨΨ\n(qρ+kρσqσ)2/parenrightBigg\n. (5.13)\nTo integrate, we make a change of variables by the rule qµ+kµνqν→˜qµ, which implies\narising the Jacobian of this replacement through the rule d4q= Ξd4˜q, with Ξ = det∂qν\n∂˜qµ=\ndet−1(δµ\nν+kµ\nν). This Jacobian is a constant, it does not depend on momenta. We no te\nthat if one suggests kµνto beantisymmetric andsmall, which does not imply any inconsis-\ntencies, this Jacobian reduces to 1. In the case of the effective po tential, the only impact\nof the Lorentz symmetry breaking consists just in the presence o f this Ξ multiplier. After\nthis change of variables and the Wick rotation, we have\nK(1)=−1\n2Ξ/integraldisplay\nd4θ/integraldisplayd4˜q\n(2π)41\n˜q2ln(1+¯ΨΨ\n˜q2). (5.14)\nNamely this expression (of course, except of the Ξ factor) arises within the analogous\ncalculations in Lorentz-invariant theories (see e.g. [151]), so, we imm ediately can write\ndown the result:\nK(1)=−1\n32π2Ξ/integraldisplay\nd4θΨ¯ΨlnΨ¯Ψ\nµ2. (5.15)\n75We see that the result differs from that one in the Lorentz-invarian t theory only by the\nJacobian factor. This allows us to formulate the following rule: for an y supersymmetric\ntheory whose superfield action does not involve explicit space-time d erivatives, within\nthe Kostelecky-Berger deformation of the supersymmetry algeb ra, the results of quantum\ncalculations can be obtained from the usual ones through replacem ent of all derivatives\n(bothspinor supercovariant andusual ones) bytheir ”twisted” a nalogues, andmultiplying\nof theL-loop expression by ΞL, with Ξ is the above-mentioned Jacobian. This rule has\nbeen also explicitly verified for the coupling of the chiral matter with a gauge superfield\n[152]. Moreover, the same rule is valid in the three-dimensional case a s well, where there\nis only one type of Grassmannian variables θα, and the supercovariant derivatives are\ngiven by\nDα=∂α+iθβγµ\nβα∇µ, (5.16)\nand the Dirac matrices for the three-dimensional space are the fo llowing 2 ×2 matrices:\n(γ0)α\nβ=−iσ2,(γ1)α\nβ=σ1,(γ2)α\nβ=σ3. One can see that since in the three-dimensional\nspace the same deformed d’Alembertian ✷Darises, the dispersion relations in three- and\nfour-dimensional superfield theories within this methodology are th e same.\nIn this context, some interesting observations for supersymmet ric Lorentz-breaking\nfield theories within this approach can be done. First of all, this formu lation possesses a\nnontrivial geometrical interpretation. Indeed, if we consider the three-dimensional theory\nof the scalar superfield Φ, with the action\nS=/integraldisplay\nd5z/parenleftBigg1\n2Φ(D2+m)Φ+λ\n3!Φ3/parenrightBigg\n(5.17)\nand calculate the lower ”fish” contribution to the two-point functio n, in the low-energy\nlimit we arrive at [152]\nΓ2=λ2Ξ\n48π|m|/integraldisplay\nd5zΦ(D2−2m)Φ. (5.18)\nProjecting this expression to components, we have:\nΓ2=−λ2\n48π|m|/integraldisplay\nd3xΞ(ηµν∇µφ∇νφ+...), (5.19)\nwhereφis a lower component of the scalar superfield, and dots are for othe r terms.\nIf we remind that ∇µ=∂µ+kµν∂ν, we can introduce a new ”inverse” metric gρσ=\nηµν(δρ\nµ+kρ\nµ)(δσ\nν+kσ\nν), and it is clear that Ξ = |det(gρσ)|1/2≡/radicalBig\n|g|. So, this quantum\ncorrection can be represented as\nΓ2=−λ2\n48π|m|/integraldisplay\nd3x/radicalBig\n|g|gµν∂µφ∂νφ+..., (5.20)\n76so, this result formally replays the action of the scalar field in a curve d space-time, thus,\nactually the Lorentz symmetry breaking generates a new geometr y, which, however, also\ncorresponds to a flat space since the new metric gµνis composed by constant components,\nso, our new geometry is affine. The question about possibility of a fur ther generalization\nof this concept in order to get a space with a nontrivial curvature is still open since if\nthe tensorkµνwill not be constant, the possible extension of the supersymmetry algebra\nevidently would be much more involved. For a four-dimensional theor y of a chiral scalar\nsuperfield, the same situation occurs as well with the only difference that the lower scalar\ncomponent of this superfield is complex. However, a Lorentz-brea king gauge superfield\ntheory apparently does not admit a geometrical interpretation, s ince there is no way to\nobtain the modified metric gµνcontracted with vector fields.\nAnotherobservationisthatifweconsiderthegeneralizationofthe freesupersymmetric\nQED with an action of a gauge superfield V:\nS=−1\n16/integraldisplay\nd8zVDα¯D2DαV, (5.21)\nits component form will be\nS=−1\n4d4x¯Fµν¯Fµν, (5.22)\nwhere¯Fµν=∇µAν−∇νAµis a deformed field strength of the electromagnetic field. It\nis invariant under the Abelian transformations δAµ=∇µξ, so, the gauge transformation\nitself in this case depends on the LV parameter. We note that, unlike the cases of scalar\nand spinor fields, this action strongly differs from the QED with an add itive aether term\n(2.9) which is invariant under usual gauge transformations δAµ=∂µξ. The one-loop\nK¨ ahlerian effective potential for a super-QED deformed in this way was calculated also\nin [152]. The generalization of these results for a non-Abelian superg auge theory can be\neasily developed.\n5.2 Lorentz symmetry breaking through introducing\nextra superfields\nOne more approach to implement Lorentz symmetry breaking in supe rfield theories is\nbased on introducing Lorentz breaking parameters through some (extra) superfields. Ini-\ntially this idea was proposed in [147], and it got further development in [ 153, 154].\nThe key idea of this approach is the following one. Let us consider the gauge superfield\nwhose component form, in the gauge sector, is (see e.g. [149]):\nV(x,θ,¯θ) =¯θ˙ασµ\n˙ααθαAµ(x)+..., (5.23)\n77whereAµistheusualvectorgaugefield, θα,¯θ˙αaretheGrassmanniancoordinates, anddots\nare for terms irrelevant for our purposes. The corresponding Ab elian superfield strength\nWα=−1\n4¯D2DαV, in a purely vector sector, has the following component structure :\nWα(x,θ,¯θ) = (σµν)β\nαθβFµν−i\n2(σµν)β\nα(σλ)β˙αθ2¯θ˙α∂λFµν+.... (5.24)\nThen, let us consider the chiral scalar superfield S(i.e.¯D˙αS= 0):\nS(x,θ,¯θ) =s(x)+..., (5.25)\nwiths(x) is a scalar, and the lower component of the corresponding antichir al field is\ns∗(x). We suggest the superfield Sto be purely external, displaying no dynamics.\nLet us extend the super-QED described by the usual supersymme tric Maxwell action\n(cf. [149])\nSMaxw=1\n4/integraldisplay\nd6zWαWα, (5.26)\nthrough adding the following term:\nSodd=/integraldisplay\nd8z(SWαDαV+¯S¯W˙α¯DαV). (5.27)\nIn components, one has (cf. [147]):\nSodd=/integraldisplay\nd4x/parenleftBig\n−1\n2(s+s∗)FµνFµν+i\n2∂µ(s−s∗)ǫµνλρFνλAρ/parenrightBig\n+.... (5.28)\nHere the dots are for other terms which do not contribute to the p urely gauge sector of\nthis theory. We can choose s(x) =−ikµxµ, and, consequently, s∗(x) =ikµxµ, withkµis\na constant vector. Therefore, s+s∗= 0, and we rest with\nSodd=/integraldisplay\nd4xǫµνλρkµFνλAρ+.... (5.29)\nAs a result, we see that the CFJ term is the only purely gauge contrib ution to the Sodd\n(actually, it turns out to be that the gauge invariance of Soddin the component form is\nmuch more transparent than in the superfield form). Further, in [ 153], the dispersion\nrelations for the theory with such an additive term were studied in diff erent sectors. It\nis interesting to notice that, up to now, namely this approach is the o nly one allowing\nto generate the supersymmetric extension of the CFJ term, since the Kostelecky-Berger\napproachdiscussedintheprevioussectionisessentiallyCPTeven, a ndtheapproachbased\non applying extra derivatives to superfields, which we discuss in the n ext section, can\nonly generate the higher-derivative terms for the “main” compone nt of the corresponding\n78superfield, that is, for the scalar swithin the chiral superfield and the vector Aµwithin\nthe real one.\nFurther, the CPT-even term can also be introduced within this meth od. It was shown\nin [154] that, if we for the same superfields WαandSas above (and the conjugated ones)\nconsider the additive term\nSeven=/integraldisplay\nd8z(DαS)Wα(¯D˙α¯S)¯W˙α, (5.30)\nafter projecting this action to components we arrive at [154]\nSeven=/integraldisplay\nd4x/bracketleftBig\n−4FµνFµν∂λs∂λs∗−8FµλFλ\nν(∂µs∂νs∗+∂µs∗∂νs)/bracketrightBig\n.(5.31)\nFors(x) =ikµxµands∗(x) =−ikµxµ, we arrive at\nSeven=/integraldisplay\nd4x/bracketleftBig\n−4k2FµνFµν−16kµkνFµλFλ\nν/bracketrightBig\n, (5.32)\nwhere the first term replays the Maxwell term with the extra multiplie r depending on\nthe square of the LV vector, and the second one – the CPT-even c ontribution introduced\nin [40]. We see that the aether-like structure arises naturally within t his method. In\n[154], some applications of this term were studied. It is natural to ex pect that within\nthis approach some other LV terms can arise as well. Also, a natural problem consists in\nsearching for the possibility of emerging of such a term as a quantum correction from an\nappropriate coupling. Up to now this scenario never was demonstra ted.\n5.3 Straightforward Lorentz symmetry breaking in a\nsuperfield action\nThe third manner to break the Lorentz symmetry in superfield theo ries is based on\nstraightforward adding terms proportional to constant LV vect ors (tensors) to the su-\nperfield action, it was firstly proposed in [148]. The simplest way to do it consists in\nconsideration of additive terms like\nδS=/integraldisplay\nd8zkµν∂µΦ∂ν¯Φ, (5.33)\nwith the analogous extensions are possible as well for chiral Lagran gians (some examples\nwill be given further) and for terms depending on gauge superfields . Nevertheless, the\nexamples for LV extensions of superfield theories we consider here differ from those ones\nproposed in [148], moreover, unlike that paper, here we concentra te on calculating the\none-loop effective potential. It is clear that within this approach the supersymmetric LV\n79terms essentially involve higher derivatives, e.g. for the expression above, one gets terms\nlike/integraltextd4xkµν∂µφ✷∂νφ. However, for an appropriate choice of the LV parameters, in this\ncase it isk00=k0i=ki0= 0, one can avoid the presence of the higher time derivatives,\nand, hence, eliminate the ghosts, so, in a certain sense in this case w e can speak about an\nattempt of supersymmetric extension of Horava-Lifshitz-like the ories introduced in [4].\nOur first example will be the theory proposed in [155]:\nS=/integraldisplay\nd8zΦ(1+ρ∆z−1)¯Φ+(/integraldisplay\nd6zW(Φ)+h.c.), (5.34)\nwithρissome constant characterizing theenergy scale at which higher de rivatives become\nrelevant, andthenumber z≥2(typically, integer), inanalogywith[4], iscalledthecritical\nexponent. It is clear that for z= 2, the additive LVtermreproduces theexpression (5.33),\nwithk00=k0i=ki0= 0 andkij=−ρδij. We note that for any z, this LV term is CPT-\neven. One can consider W(Φ) =m\n2Φ2+λ\n3!Φ3as in the Wess-Zumino model, but it is not\nmandatory.\nOne can obtain the component structure of this action:\nS=/integraldisplay\nd4x/bracketleftBig¯φ✷(1+ρ∆z−1)φ−i¯ψ˙ασm\n˙αα∂m(1+ρ∆z−1)ψα+ (5.35)\n+¯F(1+ρ∆z−1)F−/parenleftBiggm\n2(φF+1\n2ψαψα)+λ\n2(φ2F+1\n2φψαψα+h.c./parenrightBigg/bracketrightBigg\n.\nWe see that in the scalar sector, one has two time derivatives and, m aximally, 2zspatial\nones, so, the critical exponent for the scalar component is equal toz. At the same time,\nthe critical exponent for the spinor field is 2 z−1, i.e. the critical exponents for different\ncomponents of the superfield do not coincide.\nThe propagators in this theory look like\n<Φ(z1)¯Φ(z2)>=i1+ρ∆z−1\n✷(1+ρ∆z−1)2−m2δ8(z1−z2); (5.36)\n<Φ(z1)Φ(z2)>=−im\n✷(1+ρ∆z−1)2−m2(−D2\n4✷)δ8(z1−z2).\nAs usual, Φ( z1)Φ(z2)>=<¯Φ(z1)¯Φ(z2)>∗. It is not difficult to show that for z≥2 the\ntheory is finite.\nTo find the K¨ ahlerian effective potential, we proceed just as in the S ec. 5.1: consider\nthe same sequence of the supergraphs given by Fig. 5.1, again we inc orporate the mass\ninto the background field Ψ = −W′′, and modify the propagators to be\n<Φ(z1)¯Φ(z2)>=i\n✷(1+ρ∆z−1)δ8(z1−z2);\n<Φ(z1)Φ(z2)>= 0. (5.37)\n80Doing the D-algebra transformations, Wick rotation and summation as in the Sec. 5.1\ntogether with the replacement ρ→ρ(−1)z−1, we arrive at\nK(1)=−1\n2/integraldisplay\nd4θ/integraldisplaydk0Ed3/vectork\n(2π)41\nk2ln/bracketleftBigg\n1+Ψ¯Ψ\nk2(1+ρ(/vectork2)z−1)2/bracketrightBigg\n. (5.38)\nUnfortunately, this integral cannot be evaluated exactly. Using t he scheme of approxi-\nmated computations based on disregarding of subleading orders in /vectork2, we find [155]:\nK(1)=1\n12πcsc(π\nz)(4ρ\n3)−1/z/integraldisplay\nd4θ(Ψ¯Ψ)1/z. (5.39)\nThis expression displays singularity at z→1 as it should be, since the Lorentz-invariant\ncase, where the theory displays a divergence, corresponds name ly to this value of the\ncritical exponent.\nIf we introduce the similar deformation into the chiral effective pote ntial, we have the\naction\nS=/integraldisplay\nd8zΦ¯Φ+/bracketleftBigg/integraldisplay\nd6z(1\n2Φ(m+a(−∆)z)Φ+λ\n3!Φ3)+h.c./bracketrightBigg\n. (5.40)\nProceeding in a similar way as above, after the Wick rotation we have t he one-loop\nK¨ ahlerian effective potential in the form\nK(1)=−1\n2/integraldisplay\nd4θ/integraldisplaydk0Ed3/vectork\n(2π)41\nk2ln\n1+(Ψ+a/vectork2z)(¯Ψ+a/vectork2z)\nk2\n0E+/vectork2\n, (5.41)\nand, under the same approximate manner, we get\nK(1)=−1\n8πa1/zcsc(π\n2z)/integraldisplay\nd4θ(Ψ¯Ψ)1/2z. (5.42)\nIn this case the singularity occurs at z= 1/2. However, this is not unusual since namely\nforz= 1/2 in this theory spatial and time derivatives in UV leading terms enter t he\ntheory in the same order making the behaviour of loop integrals to be similar to the\nLorentz-invariant case. Coupling of the chiral matter to gauge fie lds yields the analogous\nresults.\nA slightly different approach within this line is based on use of Myers-Po spelov-like\nterms (see Section 2.1) where some derivatives are contracted to constant LV vectors\n(tensors) whose specific choice allows to avoid arising the higher time derivatives. In this\ncase we start with the following action [156]:\nS=/integraldisplay\nd8z[Φ(1−1\nΛ2(n·∂)2)¯Φ+φ¯φ]+\n+/bracketleftbigg/integraldisplay\nd6z(M\n2Φ2+1\n2λΦφ2+1\n2fφΦ2)+h.c./bracketrightbigg\n. (5.43)\n81Here theφis a light (massless) chiral superfield which we treat as a purely backg round\none, and Φ is a heavy superfield which we assume to be a purely quantu m one. Here\nMis a large mass, and Λ is an energy scale at which the higher derivatives become\nimportant (from phenomenological estimations [156], it is reasonable to choose Λ to be\nthe Planck mass, and Mto be a characteristic string mass, so, M≃10−2Λ). Thenµis a\ndimensionless LV vector, in the Euclidean space nµnµ= 1. After summation of the same\ngraphs depicted at Fig. 5.1, we find the one-loop K¨ ahlerian effective potential to be\nK(1)=−1\n2/integraldisplay\nd4θ/integraldisplayd4k\n(2π)41\nk2ln\nk2/parenleftBigg\n1+(n·k)2\nΛ2/parenrightBigg2\n+|Ψ|2\n. (5.44)\nThis integral can be calculated approximately. Since, as we already s aid,M≃10−2Λ, we\ncan expand this expression in series in M/Λ and 1/M. As a result, we find\nK(1)=λ2\n32π2/integraldisplay\nd4θφ¯φ[3+lnM2\n4Λ2]+..., (5.45)\nwhere dots are for suppressed terms. We see that in this case the contributions arising\ndue to the Lorentz symmetry breaking are significant, so, we obse rve the effect of large\nquantum corrections. However, this is rather natural since the L V term here effectively\nplays the role of the higher-derivative regularization. The same situ ation occurs within\nthe first approach described in this section. Indeed, if we compare this result with (5.39)\nwith rescaling ρ=α\nΛ2z−2, in order to have a dimensionless αand a scale Λ, we see that\nthe result (5.39) is proportional to (M\nΛ)2\nz−2, thus, for z≥2 the quantum correction in\nthis case is also not suppressed, being large instead of this. To close this section, we\nconclude that within this “straightforward” approach, both the H orava-Lifshitz-like and,\nfor a space-like nµ, the Myers-Pospelov-like theories display no ghosts being therefo re\nperturbatively consistent.\n5.4 Conclusions\nNow, let us compare the results obtained within different approache s aimed to introduce\nLorentz symmetry breaking in superfield theories. First of all, it is int eresting to note\nthat all these ways allow for introducing CPT-even terms into classic al actions, and,\nfurther, into quantum corrections, whereas only the way based o n introducing of the\nextra superfield [147] can yield CPT-odd expressions, that is, first of all, the CFJ term.\nUp to now, no other manner to construct a supersymmetric exten sion of the CFJ term is\nknown.\nSecond, a nontrivial fact consists in a possibility to generate a nont rivial geometry\nfrom quantum corrections. A discussion on this fact is presented a lso in [157] where it\n82is noted that some LV modifications of Dirac matrices can be defined a s well. However,\nan open question is whether one can, starting from some LV superfi eld gauge theory,\nobtain the action of the gauge field coupled to the new geometry con sistently, so that not\nonly derivatives, but also vector component fields would be contrac ted to a new metric.\nAnother possible line of studies could consist in a generalization of this approach for the\ncase of a space-time with a non-zero curvature through suggest ing the tensor kµνto be\na fixed function of space-time coordinates rather than a constan t. However, in this case\nthe deformed supersymmetry algebra will clearly be much more comp licated, and other\ndifficulties, intrinsic for studies of Lorentz symmetry breaking in a cu rved space-time,\nwhich will be discussed in the next chapter of the book, will arise as we ll.\nThird, the problem of consistent supersymmetric extension of Hor ava-Lifshitz-like the-\nories continues to be open. The reason is that usual superfield mod els always involve\nlower (second) spatial derivatives in a kinetic term, therefore, on e cannot arrive at a the-\nory involving only terms with two time derivatives and just 2 zspatial derivatives, while\nimplementing of higher spatial derivatives into a spinor supercovaria nt derivative clearly\nbreaks the Leibnitz rule. Some attempts to proceed in this case are presented in [150],\nhowever, this methodology is still under development.\nIt must be mentioned that there is one more manner to implement Lor entz symmetry\nbreaking specific for supersymmetric field theories, that is, the fe rmionic noncommuta-\ntivity [158]. Following this approach, the anticommutation relations be tween fermionic\ncoordinates are deformed as {θα,θβ}= Σαβwhere Σαβis a constant symmetric matrix\nclearly breaking the Lorentz symmetry (e.g., in a three-dimensional space-time this ma-\ntrix is equivalent to the constant vector Σµ=1\n2(γµ)αβΣαβ). However, this methodology\nis known to meet its own difficulties. We close this section with a conclusio n that the\nproblem of construction of a consistent supersymmetric extensio n for LV theories is still\nopen.\n8384Chapter 6\nLorentz and CPT symmetry\nbreaking in gravity\nIn this chapter we describe some first steps in study of CPT and/or Lorentz breaking\nextensions of gravity. Implementation of the Lorentz symmetry b reaking within the grav-\nity context is certainly one of the most complicated issues within gene ral studies of LV\ntheories, facing many difficulties. The most important reason for th ese difficulties is the\nfollowing one. In the curved space-time, the symmetry group is tha t one of general co-\nordinate transformations like xµ=xµ(x′), which, at the same time, represents itself as\nthe extension both of the Lorentz group and of the gauge group. As we noted in previ-\nous chapters, in many cases the Lorentz symmetry breaking implies the CPT symmetry\nbreaking as well, whereas the gauge symmetry is not broken, the pa radigmatic example\nis the CFJ term. Therefore, it is natural to require for the most int eresting CPT, and\nin certain cases Lorentz breaking extensions of gravity to be cons istent with the general\ncoordinate (that is, gauge) invariance, and there is only a few know n examples where this\nconsistency is possible. The most important of such examples, is the four-dimensional\nChern-Simons modified gravity which we discuss in this chapter. Anot her possible ap-\nproach could consist in consideration of the weak (linearized) gravit y limit, where we\nconsider only the dynamics of the symmetric metric fluctuation tens orhµν, and we can\napply the methods similar to those ones used for studies of LV exten sions of QED. In this\nchapter we consider both these approaches within the Riemannian f ramework.\nBesides of this, we note that introducing Lorentz symmetry break ing in a curved\nspace-time faces one difficulty more. As we noted in previous chapte rs, in a flat space-\ntime the Lorentz symmetry can be violated explicitly through introdu cing new terms\nproportional to constant vectors (tensors) which cannot be int roduced consistently in\na non-zero curvature case. Indeed, let us consider for example a constant vector kµ.\n85In the flat space-time it satisfies the condition ∂νkµ= 0, but this condition evidently\nbreaks general covariance. A straightforward covariant gener alization of this condition,\nlooking like ∇νkµ= 0, imposes additional restrictions on space-time geometry (so-c alled\nno-go constraints), which are hard to satisfy in general (see the discussion in [159]). And\nif no such conditions are imposed, one faces an infinite tower of term s proportional to\n∇ν1...∇νnkµwhich makes all calculations to beextremely complicated (some lower t erms\nof this form are discussed in [160], where renormalization of LV QED in a curved space-\ntime is performed). Therefore, the most promising way of breaking Lorentz symmetry in\nthe presence of gravity is based on spontaneous Lorentz symmet ry breaking, instead of\nthe explicit one. The Einstein-aether and bumblebee models allowing to break Lorentz\nsymmetry spontaneously on a curved background, will also be discu ssed in this chapter.\n6.1 Motivations for 4Dgravitational Chern-Simons\nterm\nWe start our study of LV modifications in gravity with the paradigmat ic example of\nthe additive CPT-, and in certain cases Lorentz-breaking term, co nsistent at the same\ntime with the gauge symmetry, that is, the four-dimensional gravit ational CS term. It\nrepresents itself as a natural generalization of the three-dimens ional Lorentz-invariant\ngravitational CS term introduced in [36] and looking like\nSCS=1\n2µκ2/integraldisplay\nd3xǫµνα/bracketleftBig\n(∂µωνab)ωab\nα+2\n3ωc\nµaωνcbωab\nα/bracketrightBig\n, (6.1)\nwhereµis a mass,κ2is a gravitational constant (of mass dimension −1, in 3D), andωab\nµ\nis a connection. Here and further in this section, we follow notations adopted in [104], so,\nthe Greek indices are for the curved space-time, while the Latinindic es arefor the tangent\none. As our geometry is assumed to be Riemannian, the connection c an be expressed in\nterms of vielbeins or metrics through well-known expressions. One c an straightforwardly\nverify that the action (6.1) is gauge invariant. We note that there is no/radicalBig\n|g|factor in this\nintegral since the invariant Levi-Civita tensor in a curved space isǫµνα√\n|g|rather than ǫµνα.\nIntroducing a constant (pseudo)vector bµ, it is easy to generalize the CS action to\nthe four-dimensional case. From the formal viewpoint, this gener alization is similar to\npromoting the usual CS term to the CFJ term through the replacem entǫµνλ→bρǫρµνλ\ncorrespondingtoaddingofonemoredimension. Asaresult, treatin gωab\nµasaRiemannian\nconnection, wearriveatthe4 DCSaction(wenotethatinfourdimensions, no µmultiplier\n86is needed):\nSCS=/integraldisplay\nd4xǫρµναbρ/bracketleftBig\n(∂µωνab)ωab\nα+2\n3ωc\nµaωb\nνcωa\nαb/bracketrightBig\n. (6.2)\nOriginally, this term has been introduced in [32], and in this section we r eview some\nresults obtained in [32]. To prove gauge invariance of this term, that is, its invariance\nunder general coordinatetransformations, onecan writedown it s equivalent form: indeed,\nwe can define the vector\nKρ=ǫρµνα/bracketleftBig\n(∂µωνab)ωab\nα+2\n3ωc\nµaωb\nνcωa\nαb/bracketrightBig\n, (6.3)\nso that∂µKµ=1\n2∗RR, where\n∗RR=1\n2ǫαβγδRµν\nαβRγδµν (6.4)\nis a pseudoscalar Pontryagin density. So, suggesting that the vec torbµis expressed as\nbµ=∂µϑ, whereϑis a pseudoscalar called the CS coefficient, which, for the first step, is\ntreated as an external function, one can rewrite the 4 Dgravitational CS term as\nSCS=1\n4/integraldisplay\nd4xϑ∗RR. (6.5)\nWe see immediately that this term is both Lorentz and gauge invariant , but parity-\nbreaking. If we require the ϑto be linear in coordinates, just as we did in the section 3.3\nduring the study of three-dimensional CS terms, that is, ϑ=bµxµ, withbµbe constant,\nafter a simple integration by parts we arrive at the particular Loren tz-breaking CS term\n(6.2). We note that this is the lower gauge-invariant CPT-Lorentz b reaking term in\ngravity consistent withthegaugeinvariance requirement, andoth er possible CPT-Lorentz\nbreaking gauge-invariant terms for the gravity would necessarily in volve higher orders in\nderivatives.\nThe complete action of the CS modified gravity involving both the usua l Einstein-\nHilbert term and the CS term, looks like [32]:\nSCSMG=1\n16πG/integraldisplay\nd4x(/radicalBig\n|g|R−1\n2bµKµ). (6.6)\nComparing the theory (6.6) and the LV electrodynamics with the CFJ term (2.3), we can\nconclude that the 4 Dgravitational CS term is related with the gravitational anomalies\noriginallyintroduced in[27], without anyconcerning oftheLorentzsy mmetry breaking, in\nthe manner similar to that one relating the CFJ term with the Adler-Be ll-Jackiw anomaly\n(see the discussion of this correspondence in [26]).\n87The equations of motion for this theory also were obtained originally in [32], they have\nthe form\nGµν+Cµν=−8πGTµν, (6.7)\nwhereTµνis the energy-momentum tensor of a matter, Gµν=Rµν−1\n2Rgµνis the usual\nEinstein tensor, and Cµνis a Cotton tensor defined as\nCµν=−1\n2/radicalBig\n|g|ǫσµαβ/bracketleftBig\nbσ∇αRν\nβ+bστRτν\nαβ/bracketrightBig\n+(µ↔ν). (6.8)\nHere∇αis a covariant derivative, and bµ=∇µϑ,bµν=∇µ∇νϑare constructed on the\nbase of the CS coefficient.\nWe can find the divergence of the modified Einstein equations (6.7). I n the r.h.s. we\nobtain zero due to the conservation of the energy-momentum ten sor of the matter, then,\nthe identity ∇µGµν= 0 continues to be valid because of the Bianchi identities. Finally\nwe arrive at\n∇µCµν=1\n8/radicalBig\n|g|bν∗RR. (6.9)\nThis is a constraint for the possible solutions of Eq. (6.7). In many re levant cases, e.g.\nspherically symmetric metrics, one has∗RR= 0, so, the Cotton tensor will be conserved\n(see e.g. [161] for a general discussion).\nOne can suggest the CS coefficient to be not an external object bu t a dynamical field.\nIn this case one starts with the action of the dynamical CS modified g ravity [161]:\nSDCSMG=1\n16πG/integraldisplay\nd4x(/radicalBig\n|g|R+1\n4ϑ∗RR+1\n2/radicalBig\n|g|∇µϑ∇µϑ). (6.10)\nIn this case, one finds an additional contribution to the energy-mo mentum tensor of the\nmatter, that is, the energy-momentum tensor of the scalar field ϑ.\nIt remains to write down the linearized form of the gravitational CS t erm. Suggesting\nthat the metric looks like gµν=ηµν+hµν, wherehµνis a small metric fluctuation, and\nrelabelingbµbyvµ, we have [32, 162]:\nSCS=1\n4/integraldisplay\nd4xhµνvλǫαµλρ∂ρ(✷hα\nν−∂ν∂γhγα). (6.11)\nWe see that the gravitational CS term involves higher derivatives. H owever, it was proved\nin [32] that for physical degrees of freedom there is no higher time d erivatives in the\ncorresponding equation of motion, hence, neither unitarity nor ca usality are violated.\nSo,letusdiscusssomeaspectsrelatingtothisterm,namely, itsper turbativegeneration\nand the ambiguity of the result.\n886.2 Perturbative generation of the gravitational CS\nterm\nNow, letuspresentsomeschemestogeneratethegravitationalC Sterminfourdimensions.\nThefirst ofthese schemes hasbeenproposedin[162]. Thestarting pointofthecalculation\nis the following classical action of the spinor field ψon a curved background:\nS=/integraldisplay\nd4xeeµ\na¯ψ/parenleftBig1\n2iγa↔\n∂µ+1\n4iωµcdΓacd−bµγaγ5−m/parenrightBig\nψ. (6.12)\nHere Γacd=1\n6γaγcγd+...is the antisymmetrized product of three Dirac matrices. The\ncorresponding one-loop effective action of the gravitational field c an be cast as\nΓ(1)=−iTrln(i∂ /−m−b /γ5+ω /). (6.13)\nTo consider the weak field approximation, we write gµν=ηµν+hµν, so, the vielbein is\neµ\na=δµ\na+1\n2hµ\na, ande= 1 +1\n2ha\na(again, we note that Greek indices are for the curved\nspace-time, and Latin ones – for the tangent one). So, omitting th e irrelevant terms, in\nparticular, those ones proportional to hµ\nµ, we get the action of ψcoupled to hµν:\nS=/integraldisplay\nd4x¯ψ/parenleftBig1\n2iΓµ↔\n∂µ+hµνΓµν−bµγµγ5−m/parenrightBig\nψ, (6.14)\nwhere Γµ=γµ−1\n2hµνγν, and Γµν=1\n2bµγνγ5−i\n16(∂ρhαβ)ηβνΓρµα.\nThe nontrivial contributions to the two-point function of the metr ic fluctuation hµν\nare given by the Feynman diagrams depicted at Fig. 3.4, the only differ ence is that now\nthe external lines are for the metric fluctuation. It is clear that th e quartic vertex and\nthat one involving the bµhµνcontraction evidently will not yield CS-like results.\nThese Feynman diagrams are superficially divergent. However, aft er long and very\ninvolved calculations described in [162], adopting a special calculation s cheme based on\n’t Hooft-Veltman prescription [163], we find that all divergences can cel out, and, in the\nzero mass limit, where possible divergent contributions of the above mentioned Feynman\ndiagrams vanish, arrive at the finite result\nSCS=1\n192π2/integraldisplay\nd4xhµνbλǫαµλρ∂ρ(✷hα\nν−∂ν∂γhγα), (6.15)\nthat is, the vector vµ(6.11) is expressed as vµ=1\n48π2bµ. So, this result appears to be\nunique, at least within this approach. However, this is not sufficient t o conclude whether\nthe gravitational CS term we obtained is indeed unambiguous in gener al. First of all, we\nshould emphasize that the ’t Hooft-Veltman prescription adopted w ithin [162] is nothing\nmore that a fixing of the regularization scheme so that in this case on e obtains a definite\n89result. Besides, as we observed in the Chapter 3, the finiteness of a superficially divergent\nresult typically signalizes about its ambiguity.\nTo demonstrate that our result is indeed ambiguous, we use anothe r manner of calcu-\nlating the gravitational CS term based on the proper-time method w hich was for the first\ntime carried out in [104]. In this case, we can consider the complete ac tion of the spinor\non a curved background (6.12), thus avoiding use of the weak field a pproximation, so, we\ncan obtain the full-fledged CS term.\nTo compute the CS term, we can put e≃1. This approximation is sufficient for ob-\ntaining the gravitational CS term since all higher terms in expansion o fewill be irrelevant\nfor our calculations. Then, in order to complete the operator whos e trace we study, up\nto the quadratic one, as it is required by the proper-time technique (cf. Section 3.3), we\nadd to the one-loop effective action (6.13) the constant of the for m\nC0=−iTrln(i∂ /+m+b /γ5). (6.16)\nAs a result, after multiplication of determinants with use of the fact that detAB=\ndetAdetB, we arrive at the following expression for the one-loop effective act ion [104]:\nΓ(1)=−iTrln/bracketleftBig\n−✷+iω /∂ /+mω−m2+(ω /−2m)b /γ5+\n+ 2i(b·∂)γ5−b2/bracketrightBig\n. (6.17)\nNow, we expand this expression up to the first order in the LV vecto rbµand employ the\nSchwinger proper-time representation T−1=/integraltext∞\n0dse−sT.\nAfterward, the divergent parts easily can be found to cancel. The n, the finite contri-\nbution of the second order in connections ωlooks like\nS(2)\nfin= tr/integraldisplay\nd4x/integraldisplay∞\n0dse−sm2/bracketleftBig\ns2m2ω /(∂ /ω /)b /γ5+2m2s3ω /∂ /(∂αω /)∂αb /γ5\n+2m2s3ω /(∂αω /)∂α∂ /b /γ5/bracketrightBig\ne−s✷δ(x−x′)|x′=x. (6.18)\nTo perform calculations, we employ the following representation of t he geodesic bi-scalar\nσ(x,x′) [164]:\nδ(x−x′) =/integraldisplayd4k\n(2π)4eikα∇ασ(x,x′). (6.19)\nCalculating the trace (see details in [104]), we arrive at\nS(2)\nCS=i\n4/integraldisplay\nd4x/integraldisplay∞\n0dse−sm2bµωνab∂λωρcd/integraldisplayd4k\n(2π)4esk2s2m2ǫµνλρ(gacgbd−gadgbc)\n=1\n32π2/integraldisplay\nd4xǫµνλρbµ∂νωλabωρba, (6.20)\n90withǫµνλρ=eeµaeνbeλceρdǫabcd.\nThe relevant term of third order in connections ωis\nS(3)\nfin=itr/integraldisplay\nd4x/integraldisplay∞\n0dse−sm2/bracketleftBiggs2\n2m2ω /ω /ω /b /γ5+\n+s3\n3m2(ω /∇/ω /∇/ω /+ω /∇/ω /ω /∇/+ω /ω /∇/ω /∇/)b /γ5 (6.21)\n−s3\n3m2(ω /∇/ω /ω /+ω /ω /∇/ω /+ω /ω /ω /∇/)(b·∇)γ5−s3\n3m4ω /ω /ω /b /γ5/bracketrightBigg\n×\n×e−s✷δ(x−x′)|x′=x.\nAfter calculating the trace we arrive at\nS(3)\nCS=−i\n16/integraldisplay\nd4x/integraldisplay∞\n0dse−sm2bµωνabωλcdωρef/integraldisplayd4k\n(2π)4esk2/parenleftBiggs2\n2m2ǫµνλρ+\n+s3\n3m2ǫµνλρk2+s3\n3m2ǫανλρkαkµ−s3\n3m4ǫµνλρ/parenrightBigg\n×\n×/bracketleftBig\ngaf(gbcgde−gbdgce)+gae(gbdgcf−gbcgdf)+\n+gad(gbfgce−gbegcf)+gac(gbegdf−gbfgde)/bracketrightBig\n.\nBy integrating over the momenta kand the Schwinger parameter s, we find\nS(3)\nCS=−1\n48π2/integraldisplay\nd4xǫµνλρbµωνabωλbcωρca, (6.22)\nso that, summing this expression with (6.20), we find that our comple te result yields the\ngravitational CS term [32]:\nSCS=1\n32π2/integraldisplay\nd4xǫµνλρbµ/parenleftbigg\n∂νωλabωρba−2\n3ωνabωλbcωρca/parenrightbigg\n. (6.23)\nIt is easy to verify that in the weak field approximation this term does not reproduce\nthe value of the numerical coefficient presented in (6.15). We conclu de that the four-\ndimensional gravitational CS term is ambiguous. Unlike the approach developed in [162],\nin this case there is no need to use the zero mass limit to obtain our res ult.\nAt the same time, using the methodology based on transformations of the functional\nintegral, developed in [102], we can show that the gravitational CS te rm (6.23) is accom-\npanied by a completely undetermined constant C:\nSCS=C/integraldisplay\nd4xǫµνλρbµ/parenleftbigg\n∂νωλabωρba−2\n3ωνabωλbcωρca/parenrightbigg\n. (6.24)\nThis constant arises from the modification of a CS conserved curre nt which is totally\narbitrary [102]. Some more details of this calculation can be found in [1 65]. At the same\n91time, it has been argued in [95, 96], that the result C= 0, i.e. vanishing of the one-loop\ncontribution to the gravitational CS term, is preferable in a certain sense. Following\nthese arguments, if one assumes bµto be a dynamical field, instead of a given vector or a\ngradient of a given scalar, the gravitational CS term loses its gauge invariance, therefore,\nthe consistent manner of calculations should imply a zero result for it . As we noted in\nthe section 3.4.1, the CFJ term displays the analogous behaviour.\nAnother interesting discussion of ambiguity of the four-dimensiona l gravitational CS\nterm has been presented in [166]. In this paper, the methodology o f the implicit regu-\nlarization [100] is applied to generation of the gravitational CS term, where the starting\npoint is the action of the spinors on a weak gravity background (6.14 ) taken in the zero\nmass case, m= 0. Within this scheme of calculations, the self-energy tensor Πµναβis\nfound to be (see [166] for details):\nΠµναβ=−i\n8ǫλρβµbλpρ/bracketleftBig\n(i\n48π2−64σ0−4v0+4ξ0)pαpν− (6.25)\n−(i\n48π2+32σ0)ηανp2/bracketrightBig\n+(α↔β)+(µ↔ν)+(α↔β,µ↔ν).\nHereσ0,v0andξ0are finite but yet undetermined parameters of implicit regularization .\nThe next step consists in imposing the requirement of transversalit y for this self-energy\ntensor,pµΠµναβ= 0, which implies a requirement for these parameters to satisfy the\nrelationξ0−v0= 24σ0, so, one has\nΠµναβ=−i\n24ǫλρβµbλpρ(i\n16π2+96σ0)[pαpν−ηανp2]+ (6.26)\n+ (α↔β)+(µ↔ν)+(α↔β,µ↔ν).\nTherefore, we recover the explicit formof thegravitational CSte rm given by (6.11), where\nvµ= (1\n24π2−64iσ0)bµ. We see that the result unavoidably depends on an arbitrary regu-\nlarization parameter σ0, thus, this calculation is consistent with the above-mentioned fact\nthat the constant accompanying the gravitational CS term is comp letely undetermined,\ncf. [165].\nTo close the discussion of the four-dimensional gravitational CS te rm, we note again\nthat this term is related with the gravitational anomaly [27] just in t he same manner as\nthe CFJ term is related with the Adler-Bell-Jackiw anomaly (see [26] an d the Section 3.4\nof this book).\n926.3 ProblemofspontaneousLorentzsymmetry break-\ning in gravity\nAs we noted in the beginning of this chapter, constant vectors (te nsors) used to construct\nLV terms in a flat space-time, in general cannot be defined for a cur ved one – in [159],\npossible restrictions for geometry arising from requirement for su ch vectors and tensors\nto be constant, called no-go constraints, have been discussed in g reat details, and it was\nargued that in general, these restrictions cannot be satisfied, pe rhaps except of some fine-\ntuned situations. Moreover, these restrictions typically contrad ict to Bianchi identities\nand to conservation of the energy-momentum tensor. Hence, th e explicit Lorentz sym-\nmetry breaking, in general, does not appear to be an appropriate m echanism in gravity\n(nevertheless, some interesting results have been obtained within this approach as well,\nsee f.e. [167] where conservation laws for nondynamical backgrou nds were studied, and\n[168] where some schemes allowing to maintain Bianchi identities in the e xplicit Lorentz\nsymmetry breaking case were proposed), and one must employ the spontaneous Lorentz\nsymmetry breaking mechanism, within which, the vector (tensor) in troducing a privileged\nspace-time direction, is not required to be constant.\nSo, let us discuss models used to break the Lorentz symmetry spon taneously in a\ncurved space-time. There are two known theories considered in th is context, the Einstein-\naether gravity and the bumblebee gravity.\nLet us discuss first the Einstein-aether gravity. Originally, it was int roduced in [169].\nThe action of this theory is [170]:\nS=−1\n16πG/integraldisplay\nd4x/radicalBig\n|g|/bracketleftBig\nR+λ(uµuµ−1)+Kαβ\nµν∇αuµ∇βuν/bracketrightBig\n, (6.27)\nwhereuµis a dynamical vector satisfying the constraint uµuµ= 1, so, choosing of a\npossible value of this vector breaks the Lorentz symmetry sponta neously, the λis the\nLagrange multiplier field, the tensor Kαβ\nµνlooks like\nKαβ\nµν=c1gαβgµν+c2δα\nµδβ\nν+c3δα\nνδβ\nµ+c4uαuβgµν, (6.28)\nandc1,c2,c3,c4are constants.\nThe corresponding equations of motion are:\ngαβuαuβ= 1;∇αJα\nµ−c4˙uα∇µuα=λuµ;\nTαβ=−1\n2gαβLu+∇µ/parenleftBig\nJα\n(µuβ)−Jµ\n(αuβ)−J(αβ)uµ/parenrightBig\n+ (6.29)\n+c1[(∇µuα)(∇µuν)−(∇αumu)(∇βuµ)]+c4˙uα˙uβ+[uν∇µJµν−c4˙u2]uαuβ.\nHere ˙uµ=uα∇αuµ,Jα\nµ=Kαβ\nµν∇βuν, andLuisu-dependent part of the Lagrangian.\n93Consistency of some known metrics, in particular, cosmological and black hole ones, in\nthis theory has been verified in a number of papers (see e.g. [171] an d references therein).\nIt is clear that this theory can be extended through adding differen t terms where the\nvarious degrees of the uµvector are coupled to different fields, for example, scalar, spinor\nand gauge ones, for example, it is possible to implement aether terms discussed in the\nChapter 3, like that one for the gauge field, uαFαµuβFβµ. In the flat space limit, the uµ\nvectors can (but are not restricted to) be constant ones.\nIn the Einstein-aether theory, the spontaneous Lorentz symme try breaking is intro-\nducedwithuseoftheconstraint. Atthesametime, itisknownthatp resenceofconstraints\nmakes perturbative studies of such theories more complicated req uiring special method-\nologies like 1 /Nexpansion, which, moreover, cannot be applied in our case as the uµfield\npossesses only four components. The bumblebee gravity, where t he spontaneous Lorentz\nsymmetry breaking isimplemented in other manner, that is, through choosing a minimum\nfor some potential, proposed in [34], is free of this problem. The actio n of the bumblebee\ngravity looks like\nS=−/integraldisplay\nd4x/radicalBig\n|g|/parenleftbigg1\n16πG(R+ξBµBνRµν)−1\n4BµνBµν−V(BµBµ±b2)/parenrightbigg\n.(6.30)\nHereVisthepotential, typically onecanuse V=λ\n4(BµBµ±b2)2, andb2>0 isaconstant.\nTheBµν=∂µBν−∂νBµis the stress tensor, and ξis a coupling constant characterizing\nthe magnitude of the non-minimal coupling between the bumblebee fie ld and the Ricci\ntensor.\nLet usbriefly review themost importantresults obtainedwithinstud ies ofthebumble-\nbee gravity. As we have already noted, the main direction of resear ch in various modified\ngravity theories, including the bumblebee gravity, is study of consis tency of known results\nfound within general relativity, especially, cosmological solutions an d black holes, within\nthe new modified gravity. First we discuss the solutions originally trea ted within the\nbumblebee context in [172]. In this case, the action of the model is giv en by (6.30). In\nthis theory, the static spherically symmetric metric was considered and shown to imply\na modification of the Schwarzschild solution so that its 00 and 11 comp onents behave as\n−g00=g−1\n11= 1−2GLm\nr1−L, withGLis a modified gravitational constant, L≃b2\n0/2, and\nthe dimensionless b0characterizes the radial component of the vector implementing th e\nspontaneous Lorentz symmetry breaking. Another important so lution is the G¨ odel one,\nwhich, as it is known, closed timelike curves (CTCs) can arise. In the b umblebee gravity\nthis metric was shown to be consistent for certain vacua [173], so, t his Lorentz-breaking\nscenario does not exclude CTCs. The third solution usually tested fo r various modified\ngravity models is the Friedmann-Robertson-Walker cosmological me tric, and it has been\n94proved in[174] thatwithin thebumblebee gravity, cosmic accelerat ion, including late-time\nde Sitter expansion, can occur.\n6.4 Possible LV terms in gravity\nThe four-dimensional gravitational CS term we have considered ab ove is certainly a\nparadigmatic example of the CPT-Lorentz breaking term in gravity. Nevertheless, other\nLV additive terms in gravity, although perhaps less advantageous, can be considered as\nwell. The list of possible terms with dimensions up to 8 can be found in [15 9]. Let us\nbriefly characterize their general features.\nAs an example of a Lorentz-breaking extension for the Einstein gra vity, we can discuss\nthe aether-like term for the gravity [39] in a space-time of an arbit rary dimension D,\nlooking like\nSgrav\naether=α/integraldisplay\ndDx/radicalBig\n|g|uµuνRµν, (6.31)\nwithαis a small constant. This term is CPT-even. Actually, this is a particula r\nform of the CPT-even term Seven=/integraltextd4x/radicalBig\n|g|sµνRµν, whose different aspects, includ-\ning Hamiltonian formulation and cosmological issues, were studied in va rious papers, see\nf.e. [175, 176, 177]. As we noted in the previous section, to avoid the explicit breaking\nof the diffeomorphism invariance, one should suggest that the vect oruµis not a constant\nbut, instead of this, arises as a result of the spontaneous Lorent z symmetry breaking\noccurring for some potential. As an example, we can choose the bum blebee-like one, see\ne.g. [34]:\nV(uµ) =λn(uµuµ±v2)n(6.32)\nwherev2is some positive constant, so, we should add this potential to the ac tion (6.31),\ntherefore, the complete action would be\nS=MD−4\n∗/integraldisplay\ndDx/radicalBig\n|g|(1\nκR+αuµuνRµν+Lkin\nbumb[u]+λn(uµuµ±v2)n).(6.33)\nHere,Lkin\nbumb[u] =−1\n4Fµν[u]Fµν[u] is the Maxwell kinetic term for the bumblebee field,\ndenoted here as uµinstead ofBµused in the previous section, and the M∗is a parameter\nwith the mass dimension 1 introduced to ensure a correct dimension o f the action. We\nnote that this action essentially differs from the Enstein-aether th eory (6.27) discussed\nin the previous section. The term (6.31), proposed originally in [39] to couple the aether\nterm with gravity is, first, more convenient for studies within the we ak gravity framework\nsince it allows, in particular, to treat the vector uµasconstant one, second, is more similar\n95with other aether-like terms defined in [39, 40] for scalar, spinor an d gauge fields. One can\ncheck that the gauge symmetry in this theory can be maintained for special restrictions\nongµν(fot the weak gravity case, hµν) anduµ. For example, if we consider the five-\ndimensional space-time choosing a vector uµdirected along the extra (fifth) dimension,\nandremindthat,fortheweakgravity, thegaugetransformation sforthemetricfluctuation\nand the LV vector are δhµν=∂µξν+∂νξµandδuµ=∂5ξν, we can impose restrictions\nha5= 0 (witha= 0,1,2,3) and∂5ξa= 0 [39]. We note that for this term, as well as for\nmany other Lorentz and/or CPT-breaking terms in gravity propos ed in [34, 160], with\nthe only known exception in a full-fledged gravity is the gravitational CS term, the gauge\ninvariance requires imposing of special conditions (no-go constrain ts), e.g., the terms like\ntµνλρRµνλρare in general not gauge invariant by the same reasons as (6.31). M oreover,\nthe term of the first order in derivatives, being generated as a qua ntum correction in\nthe linearized theory described by the action (6.14), is essentially div ergent [145]. The\nsame is valid for terms linear in Riemann and Ricci tensors in a full-fledge d gravity with\nLV terms (i.e. the terms of second order in derivatives of hµνin the weak gravity case,\ncf. [160]. This emphasizes the importance of the gravitational CS te rm which, as we\nnoted above, is one-loop finite. In principle, one can abandon as well the restriction for\nthe geometry to be Riemannian, through introduction of the torsio n, but up to now,\nneither perturbative generation of any LV term involving torsion no r studies of impact of\nCPT-Lorentz breaking terms involving torsion were carried out.\nBesides of the aether-like term (6.31), other terms involving couplin gs of the LV vector\nuµwith various geometrical objects can be introduced, such as, e.g., sµνRµν,tµνλρCµνλρ,\nwithsµνandtµνλρare constructed on the base of the uµ, andCµνλρis the Weyl tensor,\nsee f.e. [178]. In that paper, the lower contributions from these te rms to the parametrized\npost-newtonian (PPN) expansion are found, and classical dynamic s of the vector field in\nthis theory is studied in [179].\nThese terms are the examples of dimension-4 LV terms in the gravita tional sector. A\nlist of possible LV terms in the gravitational sector with dimensions up to 8 is presented\nin [159]. Among most important terms, we can emphasize the following o nes: first, the\ngravitational Chern-Simons term, second, the terms proportion al to various orders of\ncovariant derivatives of the Riemann tensor, like e.g. (⌣\nk(n)\nD)αβγδµ 1...µnD(µ1...Dµn)Rαβγδ,\nthird, those ones proportional to direct products of various deg rees of Riemann tensor,\nlike e.g. (⌣\nk(6)\nR)α1β1γ1δ1α2β2γ2δ2Rα1β1γ1δ1Rα2β2γ2δ2, fourth, those ones involving various non-\nzero degrees both of the Riemann tensor and of its covariant deriv atives. Except of the\ngravitational Chern-Simons term, all these terms, being consider ed in the weak field limit,\nwith the LV vectores (tensors) assumed to be constant, break t he gauge symmetry (the\n96importance of gauge symmetry breaking within the gravity context has been discussed in\n[180]). The dispersion relations for linearized gravity models whose ac tion is given by the\nsum of the Einstein-Hilbert action and each of these terms, for a sim plified case when the\nconstant tensors, e.g. (⌣\nk(n)\nD)αβγδµ 1...µn, are completely characterized by only one constant\nvector, have been studied in [181], and were shown to display a usual formE2=/vector p2, being\nunaffected by the Lorentz-breaking vectors. Studies of dispers ion relations for a more\ngeneric form of Lorentz-breaking tensors are presented in [182].\nTo illustrate the problem with the gauge (non)invariance we turn aga in to the weak\ngravity limit, where the dynamical field is the metric fluctuation, and c onsider the term\ninitially introduced in [145]:\nLlingrav=−2ǫλµνρbρhνσ∂λhσ\nµ. (6.34)\nThe main motivation to define this term in [145] was the generalization o f the noncommu-\ntative field method discussed in the section 4.4 and based on deforma tion of the canonical\ncommutation algebra between fields and their canonically conjugate d momenta applied in\n[82] for the gauge fields, to the case of the (linearized) gravity, an d namely this structure\nrests when we impose a special gauge allowing to rule out the non-loca l contributions,\nwhose presence is a price we should pay for the gauge symmetry. If we consider the usual\ngauge transformations for this term, we will see that it is not invaria nt. Among other\nproperties of this term we should emphasize as well the birefringenc e of the gravitational\nwaves. Also, if one would try to generate this term from the usual t heory of spinors on\na curved background (6.14), the result will be divergent, which, ho wever, is natural since\nthe theory of spinors on a curved background (6.12) is non-renor malizable.\nNevertheless, many gauge invariant Lorentz-breaking terms in th e linearized gravity\ncase are possible as well. Besides of the gravitational Chern-Simons term discussed in the\nsection 6.2, there are other gauge invariant CPT-odd LV terms and CPT-even LV terms\npresented in [181]. However, all these terms are higher-derivative ones. Some examples\nare\nLeven=1\n2KαΠαβKβ, (6.35)\nwithKα=bµΠµνhνα, where Πµν=ηµν✷−∂µ∂νis the projection-like operator, and\nLodd=ǫαβγδbαKβ∂γKδ. (6.36)\nThe dispersion relations in both these cases are E2=/vector p2, being unaffected by the Lorentz-\nbreaking vectors. Other higher-derivative gauge invariant LV ter ms for the linearized\ngravity can be constructed as well, they involve more derivatives (e .g. one can consider\n97linear combinations of terms ǫαβγδbαKβ∂γ✷nKδ,1\n2KαΠαβ✷nKβfor various n≥1, being\nstraightforward generalizations of above terms). However, it is c lear that more studies of\nhigher-derivatives LV additive terms in gravity are still to be done.\n6.5 Conclusions\nIn this chapter, we described some first steps carried out to cons ider the CPT-Lorentz\nbreaking extensions in the gravity. We demonstrated that the fou r-dimensional gravita-\ntional CS term is apparently the best possible CPT-Lorentz breakin g extension of the\nEinstein gravity, since it, first, can be perturbatively generated in a consistent manner\nbeing finite, second, does not display any problems with unitarity and /or causality, third,\nis gauge invariant. Clearly, it calls the interest to studying of possible classical solutions\nof the CS modified gravity, that is, first of all, verification of consist ency of known general\nrelativity solutions within the CSMG, either in dynamical or non-dynam ical cases.\nForthefirst time, thisstudy was carriedout alreadyintheseminal w ork [32] where the\nconsistency of the Schwarzschild metric within the CSMG was proved . As a continuation,\nin [161] it was proved that the spherically symmetric solutions of gene ral relativity, whose\nmost important example are clearly the cosmological solutions, and t he static solutions\nwith axial symmetry, as well as their generalizations, so-called stat ionary solutions where\nthe metric depends on time only through the factor t−αφ, withαis some constant,\ncontinue to be consistent within the CSMG, where the Kerr metric is n o more consistent.\nFurther, it was shown in [183] that to achieve the consistency within the DCSMG, the\nKerr metric should be deformed by additive terms proportional to t he CS coefficient ϑ,\nwith the desired consistency can be shown order by order. A great review on different\nexact solutions in CSMG is presented in [33]. Other solutions, which do not match\nthese examples, are the G¨ odel-type solutions characterized by t he possibility of the closed\ntime-like curves [184, 185]. Their consistency within the CSMG was ch ecked both in\nnon-dynamical and dynamical cases in [186, 187], where it was shown that, in the CSMG,\nboth in non-dynamical and dynamical cases, besides of the known G ¨ odel-type solutions,\nthe new solutions, e.g., completely causal ones, can arise.\nHowever, it is clear that the problem of possible manners to implement the Lorentz\nsymmetry breaking within the gravity, and, further, to study the resulting theories at\nthe quantum level, is still open. As we already noted, one important is sue is the gauge\nsymmetry, that is, the general covariance, whose maintaining in an expected LV extension\nof a full-fledged gravity requires special efforts. For example, as w e already noted above,\none of the main difficulties is related to the development of an appropr iate manner for\n98introduction of privileged directions in a curved space-time without b reaking the general\ncovariance; it was claimed in [172] that a consistent manner to do it is b ased on a spon-\ntaneous Lorentz symmetry breaking introduced within the bumbleb ee model in a curved\nspace-time, however, up to now, only some first studies in this direc tion were carried\nout. Another problem is the search for the renormalizable model of the gravity itself.\nCertainly, constructing of Lorentz-breaking non-Riemannian gra vity models, which could\ninvolve torsion and nonmetricity, as well as development of LV massiv e gravity theories,\nis also an important direction of studies (some first steps along this w ay are discussed in\n[188, 189, 176]). Therefore we conclude that constructing of con sistent LV extension of\ngravity certainly requires more active studies.\n99100Chapter 7\nExperimental studies of the Lorentz\nsymmetry breaking\nIt isclear thatall new theories shouldbeverified throughsomeexpe riments. So, this book\nwould be clearly incomplete without a review of different experimental studies of possible\nLV phenomena and measurements of LV parameters characterizin g various LV extensions\nof known field theory models. Therefore, in this last chapter we pre sent this review, in\nwhich we discuss both macroscopic studies of LV effects, especially g ravitational ones,\nand the microscopic studies, especially quantum ones. Also, one of t he main reasons for\nexperimental studies of the Lorentz symmetry breaking is the det ermination of possible\nlimits of applicability for special relativity, which, as well as those ones for any physical\ntheory, require a careful study. Besides of this, there are stro ng cosmological motivations\nfor these studies, such as, e.g., the hypothetic possibility for cosm ic radiation anisotropy\ndiscussed in [190] where it was called the ”axis of evil”. While in [191] it was argued that\nit is more reasonable to attribute this anisotropy to inappropriate m ethods of statistical\nanalysis rather than to fundamental physical effects, the possib ility of this anisotropy\ncertainly requires further studies.\nOne more (and historically, the first) effect treated as a possible ma nifestation of the\nLorentz symmetry breaking was the Greisen-Zatsepin-Kuzmin (GZ K) effect [192, 193]\nwhich showed that the flux of the cosmic rays strongly decreases w ith the energy more\nthan about 1020eV, which naturally called the interest to the idea of deformed disper sion\nrelations which essentially involve a certain energy scale as it is sugges ted within the\ndouble special relativity [194] (see also the discussion in the section 2 .2 of this book\nand [56]). Other important effects within this context are the possib le birefringence of\na light in a vacuum and rotation of a plane of polarization of light in a vacu um which\ncan occur in certain LV extensions of electrodynamics (see discuss ions in Section 2.2)\n101and the Cherenkov radiation which in the LV case is possible even in a va cuum, see\nf.e. [195, 196] (various aspects of Cherenkov radiation in different LV extensions of QED\nare discussed also, e.g. in [41, 197, 198, 199], see also references t herein, in [200], the\nCherenkov radiation for massive photons is treated, and in [201], it is studied for LV\nfermions).\nThe most appropriate theory to treat all these issues related with Lorentz symmetry\nbreaking, is, clearly, the minimal LV Standard Model extension (LV S ME) [5, 6] which\nhas been discussed throughout this book and includes as ingredient s almost all terms\nconsidered here – the CFJ term (both in Abelian and non-Abelian form s), the aether\nterm for scalar, gauge and spinor fields, and different LV spinor-sc alar and spinor-vector\ncouplings. In [34], this model was generalized to include the gravity. T his model clearly\nallows various manners for experimental measurements of differen t LV effects. Treating\nnumerical estimations of the parameters of LV SME, we, first of all, should refer to the\nData Tables [1] which collect the results for estimated values of all LV parameters known\nup to now. Certainly, non-minimal extensions of this model discusse d in [53, 159] also\nmust be studied experimentally.\nWe note that, within verifying the Lorentz symmetry breaking and/ or measuring its\nparameters, two directions of studies can be emphasized: (i) non- gravitational studies,\nsuch as, first, the determinations of unusual behavior of electro magnetic waves which\ncan originate from the Lorentz symmetry breaking, such as birefr ingence and rotation\nof the plane of polarization, second, the possible modifications of dis persion relations,\nthird, measurements of the parameters of the Lorentz-violating extension of the Standard\nModel; (ii) gravitational and cosmological studies.\n7.1 Non-gravitational studies\nOne of the most known experiments aimed to study possible unusual behavior of elec-\ntromagnetic waves in a vacuum is the PVLAS experiment (see e.g. [202 ]) in which the\nrotation of plane of polarization of light in an external magnetic field w as measured. In\nthe paper [203] it was claimed that this rotation can be attributed to space-time noncom-\nmutativity, which, as we noted in the Chapter 4, represents itself a s one of the known\nforms of Lorentz symmetry breaking, with the noncommutativity p arameter is estimated\nas Θ≃(30GeV)−2. Nevertheless, further [204] it was argued that this rotation sho uld\nbe attributed to the axion-photon coupling, whereas the impact of the noncommutativity\nmust be unobservable.\nAnother important line of experimental studies for the possible Lor entz symmetry\n102breaking is based on studies of cosmic rays. Indeed, as we already n oted, one of inter-\npretations of the GZK effect was based on the idea that it can be exp lained by a strong\nLorentz symmetry breaking at some energy scale, as is suggested by the double special\nrelativity, therefore the dispersion relations should be modified by e xtra terms becoming\nimportant at very large energies. In the paper [205], one of the firs t reviews of studies\nof cosmic rays within this context was presented. Following [205], the ultra-high energy\ncosmic rays (UHECRs) with the energy E≥1018.5eV, mostly have the extragalactic\norigin. Actually they are frequently related with the γ-ray bursts, see e.g. [206]. The\nmain experimental studies of UHECRs presented in [205] are perfor med by the Pierre\nAuger Collaboration. While the experiments discussed in [205] showed that the Lorentz\ninvariance isvalid uptothe valueofthe usual Lorentz γfactorup to1011, we canconclude\nthat these experiments impose strong restrictions on the parame ters of Lorentz symmetry\nbreaking but do not rule it out, and, as it is claimed in [205], new studies in this direction\nclearly will establish new questions, and, among the suggestions give n in that paper, one\nshould emphasize the need to study, first, γ-ray bursts, second, high-energy neutrinos.\nWithin this context, the discussion of the γ-ray bursts, first time observed in 1967\nand representing themselves as processes characterized by ext remely high energy scale, is\nespecially important. In [207], it was suggested that namely by this r eason, it is natural\nto expect the LV effects within the γ-ray bursts observations, with the main line of study\ncould consist indetermination of polarizationof the electromagnetic waves emitted within\nthe bursts. Also, in [207], some spectroscopic experiments in order to determine the LV\nparameters kµandκµνλρaresuggested; theresultsofsometheseexperimentsareprese nted\nin [1] where the typical bound for kµobtained from different experiments is about 10−43\nGeV, and for the dimensionless κµνλρ– about 10−15÷10−17. More recent results for study\nof theγ-ray bursts in this context are presented in [208], where the resu lts are obtained\nwith use of the Fermisatellite, where it is shown on the base of these observations that\nthe characteristic energy of Lorentz symmetry breaking should b e no less than about\n7,6MPl, that is, larger than the Planck energy. Besides of this, one should mention as\nwell the paper [209], where the constraints for the LV coefficients, including the higher-\ndimensional coefficients, obtained on the base of the γ-ray bursts studies are given. Also,\nin [210], on the base of studies of γ-ray bursts some estimations for upper limits for\nthe vacuum birefringence effect were obtained, however, these lim its are too large, e.g.,\nforξcharacterizing the Myers-Pospelov term (2.13) they give ξ <2.6×108. Besides\nof this, in [206] estimations for modification of dispersion relations an d for rotation of\nthe polarization plane of the light in a vacuum based on study of the γ-ray burst GRB\n041219A were carried out, and the possibility of Horava-Lifshitz-lik e deformations of the\ndispersion relations, with additive αz/vectork2zterms, was estimated, and it was shown that\n103time delay expected from suggestions of existence of additive term s characterized by the\ncritical exponent z= 2, calculated on the base of results obtained for this γ-ray burst, is\nbounded by 10−21s.\nThe LV parameters can be estimated as well on the base of astroph ysical observations.\nIn this context, it is important to mention results obtained from obs ervations of astro-\nphysicalγ-ray sources which allowed to estimate the bounds for cµνcoefficients from the\nspinor sector of LV QED to be of the order of 10−16÷10−17[211].\nIt should benotedthat the LVmodifications of dispersion relations f or theelectromag-\nnetic field can imply as well in the Cherenkov radiation, see e.g. [41, 21 2]. For example,\nin [212] it was argued that the Cherenkov radiation can occur for th e Myers-Pospelov-like\ndispersion relationfor thephoton E2=p2±ξp3\nM, andfor theelectron E2=p2+m2+ηR,Lp3\nM,\nwithMis a Planck-order mass scale, ηR,Lcorrespond to different electron helicities, and\nthe sign±is for two photon helicities. It was argued in [212] that it follows from o bser-\nvations of the Crab nebula that |ξ| ≤10−3, and|ηL−ηR|<4.\nVarious experiments with elementary particles are used for tests a nd measurements of\nparameters of Lorentz symmetry breaking. In this context, the experiments with Penning\ntraps, allowing to detect a possible particle-antiparticle asymmetry , play the special role.\nIt was proved in these studies [213], that the bounds for componen ts of the LV axial\nvectorbµfrom the minimal LV extended QED (2.1) are about 10−24GeV. In the same\npaper, some estimations for non-minimal LV parameters are also pr esented. More results\nfor non-minimal LV parameters can be found in [214]. In [208], some o ther experiments\nwithin the particle physics allowing to measure the LV parameters wer e proposed.\nManyotherstudiesofelementaryparticlesaimedtodetecttheimpa ctsofLorentz-CPT\nsymmetry breaking were performed as well. Among them, experimen ts with neutrinos\nhave a special role. First of all, it should be noted that the neutrino o scillations also can\nbe explained with use of the Lorentz symmetry breaking, as it was ar gued in [215, 216].\nIt is interesting to notice that despite the famous “discovery of su perluminal neutrinos”\nwithin the OPERA experiment was proved to be an experimental erro r, it called much\nattention namely to studying of possible LV effects in neutrino physic s. Further, studies\nof neutrino oscillations were used to obtain estimations for LV param eters in [217], where\nit was shown that the bound for the parameter aµdefined in (2.1), estimated on the base\nof these oscillations, is about 10−23GeV. Various estimations for the Lorentz-breaking\nparameters were obtained also within the framework of the DUNE ex periment, see f.e.\n[218], and the IceCube experiments, see f.e. [219]. As another app lication of neutrinos, in\n[220] it was argued that the neutrinos are the very convenient obj ects for measurements\nallowing for imposing very strong bounds on LV parameters from gra vity experiments\nwhich we consider in the next section.\n1047.2 Gravitational effects\nPurely gravitational studies presented by an important line of expe riments have been\ndescribed in [221]. Within these studies, the LV extensions for theor y of fermions and\nbumblebee model coupled to the gravity were considered. Different tests, representing\nthemselves either as free-fall gravimeter tests, where falling cor ner cubes or matter in-\nterferometers were used as free-fall gravimeters, or as the te sts of the weak equivalence\nprinciple (WEP) were carried out. Other gravimeters used within the study performed\nin [221] were the force-comparison gravimeters based on comparin g of the gravitational\nforce with anappropriate electromagnetic force. As a result, e.g., for the force-comparison\ngravimeter tests, the experimental bounds were showed to be: a bout 10−8GeV for the\ncoefficientaµcharacterizing the term aµeeµ\na¯ψγaψ, and about 10−8for the coefficients cµν\nentering the term −1\n2icλνeeµ\naeνaeλ\nb¯ψ↔\nDµψ. In [221], also other tests for the WEP were\npresented, e.g., the satellite-based ones, the Solar system ones b ased on the precession\nof the perihelion, which yield the results: |aµ| ≤10−6GeV, and |cµν| ≤10−7, tests with\nantimatter, and a great number of photon tests. Some possible sa tellite tests have been\ndiscussed as well in [222].\nOne of the most important experimental discoveries of recent yea rs is, certainly, the\ndetection of gravitational waves [223]. Therefore, it is natural to suggest that the LV im-\npacts can be studied within the context of gravitational waves as w ell. In particular, as we\nalready mentioned in the Chapter 6, some LV extensions of gravity c an allow for birefrin-\ngence of gravitational waves (see e.g. [145]). The results of first s tudies in this direction\nbased on study of the gravitational-wave event GW150914 are pre sented in [224], for a\nmore detailed discussion see [182]. Later experimental studies of bir efringence of gravi-\ntational waves, for a certain higher-derivative extension of the E instein gravity, can be\nfound in [225]. It is interesting to note that some studies in that pape r suggest to consider\nhigher-derivative extensions of gravity, while up to now, except of the Chern-Simons mod-\nified gravity, no other examples of higher-derivative LV gravity mod els were sufficiently\nstudied. It worth mentioning that within this context, the gravitat ional Cherenkov ra-\ndiation can exist as well and will produce the energy losses, see the d iscussion in [226],\nwhere a typical extension of gravity implying in the modified Einstein eq uations\nGµν= 8πGTµν+ˆ¯sαβ˜Rαµβν,\nwith˜Rαµβνis a double dual of the Riemann tensor obtained through its contrac tion with\ntwo Levi-Civita symbols, and ˆ¯sαβis a differential operator involving LV parameters of\ndimensions d= 4,6,8,...likeˆ¯sµν=/summationtext\nd(¯s(d))α1...αd−4µν∂α1...∂αd−4, was considered, and\nobservations of cosmic rays were used to estimate the LV paramet ers (¯s(d))α1...αd−4µν , e.g.,\n105it was found that the bound for the dimensionless (¯ s(4))µνis of the order 10−14. In [227],\nit was estimated with use of the lunar laser range, to be about 10−9(the lunar laser\nrange was used to estimate LV parameters also in [228]), and in [229] this parameter was\ndetermined by gravity tests of the Solar system objects, where t he typical bound was\nfound to be about 10−9...10−11. Some aspects of gravitational Cherenkov radiation are\ndiscussed also in [230]. And in [231], estimations for the coefficients up t o (¯s(10)) are done\non the base of different cosmic ray observations, so that the typic al bound of (¯ s(10))ijturns\nout to be about 10−61GeV−6. As an aside observation, we once more see that the studies\nof cosmic rays and γ-ray bursts allow to obtain the most important information about th e\nscale of LV parameters. Further, in [232], a methodology allowing for estimation of LV\nparameters from observations of gravitational waves was prese nted, and it was claimed in\nthis paper that this analysis is currently on the way. Among other ex perimental studies\nof LV effects in gravity, the short-range gravity tests [233] are worth mentioning. The\nmost recent results of various experiments on Lorentz and CPT br eaking, both in flat and\ncurved spacetime, can be found in [234].\nTo close this book, we note that there are numerous experiments p erformed in order\nto study different LV extensions of various field theories, either of gravity, QED or other\nmodels, and apparently, more new experiments are to be performe d. And in principle, we\nnote that the complete picture of a possible LV extension of the Sta ndard Model is still\nvery far from its conclusion, and many problems in this context cont inue to be open and\nwill be open during many years, and the main attention in nearest yea rs certainly should\nbe paid to studying LV extensions of gravity.\n106Acknowledgements. Authors are grateful to J. F. Assun¸ c˜ ao, A. P. Baeta Scarpelli,\nH. Belich, L. H. C. Borges, F. A. Brito, L. C. T. Brito, M. 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Soc. 000, 1–??(2011) Printed 7 November 2018 (MN L ATEX style file v2.2)\nOn the spectrum of the pulsed gamma-ray emission from\n10MeV to 400GeV of the Crab pulsar.\nN. Chkheidze1,⋆G. Machabeli1and Z. Osmanov2\n1Centre for Theoretical Astrophysics, ITP, Ilia State Unive rsity, Tbilisi, 0162, Georgia\n2Free University of Tbilisi, Tbilisi, 0183, Georgia\n7 November 2018\nABSTRACT\nIn the present paper a self-consistent theory, interpreting the VERITAS and the\nMAGIC observations of the very high energy pulsed emission from th e Crab pulsar is\nconsidered. The photon spectrum between 10MeV and 400GeV can be described by\ntwo power-law functions with the spectral indexes equal to 2 and 3 .8. The source of\nthe pulsed emission above 10MeV is assumed to be the synchrotronr adiation, which is\ngenerated near the light cylinder during the quasi-linear stage of th e cyclotron insta-\nbility. The emitting particles are the primary beam electrons with the L orentz factors\nup to 109. Such high energies by beam particles is supposed to be reached due to\nLandau damping of the Langmuir waves in the light cylinder region. This mechanism\nprovides simultaneous generation of low (radio) and high energy (10 MeV-400GeV)\nemission on the light cylinder scales, in one location of the pulsar magne tosphere.\nKey words: instabilities - plasmas - pulsars: individual (PSR B0531+21) - radiation\nmechanisms: non-thermal\n1 INTRODUCTION\nThe recent observations of the Crab pulsar in the very\nhigh energy (VHE) domain with the VERITAS array of\natmospheric Cherenkov telescopes revealed pulsed γ-rays\nabove 100GeV (Aliu et al. 2011), which was later con-\nfirmed by measurments of the MAGIC Cherenkov telescope\n(Aleksi´ c et al. 2011, 2012). Prior to the work Aliu et al.\n(2011) the highest energy of the pulsed emission from the\nCrab pulsar was 25GeV (Aliu et al. 2008). The detection\nof such a high energy pulsed γ-rays cannot be explained\non the basis of current pulsar emission models. It is gener-\nally assumed that the VHE emission is produced either by\nthe Inverse Compton scattering or by the curvature radi-\nation. By analyzing the aforementioned emission processes\n(Machabeli & Osmanov 2009, 2010), we have found that for\nCrab pulsar’s magnetospheric parameters even very ener-\ngetic electrons are unable to produce the photon energies\nof the order of 25GeV. Studying the curvature radiation, it\nwas shown that the curvature drift instability efficiently re c-\ntifies the magnetic field lines, leading to a negligible role o f\nthe curvature emission process in the observed VHE domain\n(Osmanov et al. 2009). In previous work Chkheidze et al.\n(2011) we have explained the origin and the measured spec-\ntrum of the Crab pulsar in the high energy (HE) domain\n(0.01−25)GeV, relying on the pulsar emission model first\n⋆E-mail: nino.chkheidze@iliauni.edu.gedeveloped by Machabeli & Usov (1979). According to these\nworks, in the electron-positron plasma of a pulsar magne-\ntosphere the low frequency cyclotron modes, on the quasi-\nlinear evolution stage create conditions for generation of the\nHE synchrotron radiation.\nIt is well known that the distribution function of rela-\ntivistic particles is one dimensional at the pulsar surface , be-\ncause any transverse momenta ( p⊥) of relativistic electrons\nare lost in a very short time ( /lessorequalslant10−20s) via synchrotron\nemission in very strong magnetic fields. But plasma with an\nanisotropic one-dimensional distribution function is uns ta-\nble which inevitably leads to the wave excitation process.\nThe main mechanism of the wave generation in plasmas of\nthe pulsar magnetosphere is the cyclotron instability, whi ch\ndevelops at the light cylinder length-scales (a hypotheti-\ncal zone, where the linear velocity of rigid rotation exactl y\nequals the speed of light). During the quasi-linear stage of\nthe instability a diffusion of particles arises as along, als o\nacross the magnetic field lines. Therefore, plasma particle s\nacquire transverse momenta and, as a result, start to radiat e\nin the synchrotron regime.\nIn Chkheidze et al. (2011), it was shown that near the\nlight cylinder the radio waves are generated, provoking the\nre-creation of the pitch angles and the subsequent syn-\nchrotron radiation in the HE domain. Thus, in the frame-\nwork of this model generation of low and high frequency\nwaves is a simultaneous process and it takes place in one\nlocation of the pulsar magnetosphere. This explains the ob-\nc/circlecopyrt2011 RAS2\nserved coincidence of the HE emission pulses with the ra-\ndio signals. The recent VERITAS and MAGIC observations\nhaveshownthattheaforementioned coincidencetakesplace s\nin the VHE domain ( >100GeV) as well (Aliu et al. 2008,\n2011; Aleksi´ c et al. 2011). According to Aliu et al. (2011)\ndetection of pulsed γ-ray emission of the order of 100GeV\nrequires that the emission should be produced far out in the\nmagnetosphere. Thus, we suppose that the pulsed high and\nthe very high energy radiation of the Crab pulsar is gener-\nated through the synchrotron mechanism at the light cylin-\nder length-scales, switched on due to the quasi-linear diffu -\nsion. The resonant particles are the primary beam electrons\nwith the Lorentz-factor γb∼108−9, giving the synchrotron\nemission in the (0 .01−400)GeV energy domain.\nAccordingtoAliu et al. (2008) ajoint fittothe EGRET\n(10MeV to 10GeV) and MAGIC ( >25GeV) data predicted\na power-law spectrum with a generalized exponential shape\nfor the cutoff, described as Fǫ∝ǫ−αexp(−(ǫ/ǫ0)β), where\nα= 2.022±0.014. We provided a theoretical confirmation\nof the measured spectrum, which yielded β= 1.6 and the\ncutoff energy ǫ0= 23GeV (Chkheidze et al. 2011). Recent\nVERITAS observations (100 −400)GeV combined with the\nFermi-LAT data (0 .1−10)GeV favor a broken power law as\na parametrization of the spectral shape. The good fit results\nare also obtained if one uses a log-parabola function, but it\nfails to describe the spectrum below 500MeV. Although, the\nFermi-LAT and Magic data below 60 GeV can be equally\nwell parameterized by broken power law and exponential\ncutoff. In the energy range between 100GeV and 400GeV\nmeasured by VERITAS and MAGIC, the spectrum is well\ndescribed by a simple power law with the spectral index\nequal to 3.8 (Aliu et al. 2011; Aleksi´ c et al. 2012).\nThe paper is organized as follows. In Sect. 2 we describe\nthe emission model, in Sect. 3 we derive the theoretical syn-\nchrotron spectrum for the high and the very high energy\nγ-ray emission of the Crab pulsar and in Sect. 4 we discuss\nour results.\n2 EMISSION MODEL\nAny well known theory of pulsar emission suggests that,\nthe observed radiation is generated due to processes tak-\ning place in the electron-positron plasma. It is generally a s-\nsumed that the pulsar magnetosphere is filled by dense rela-\ntivistic electron-positron plasma with an anisotropic one -\ndimensional distribution function (see Fig. 1 from Arons\n(1981)) and consists of the following components: a bulk of\nplasma with an average Lorentz-factor γ∼γp, a tail on the\ndistribution function with γ∼γt, and the primary beam\nwithγ∼γb. The distribution function is one-dimensional\nand anisotropic and plasma becomes unstable, which might\ncause a wave excitation in the pulsar magnetosphere. The\nmain mechanism of wave generation in plasmas of the pul-\nsar magnetosphere is thecyclotron instability. The cyclot ron\nresonance condition can be written as (Kazbegi et al. 1992):\nω−k/bardblV/bardbl−kxux+ωB\nγr= 0, (1)\nwhereux=cV/bardblγr/ρωBis the drift velocity of the particles\ndue to curvature of the field lines with the curvature radius\nρ,ωB≡eB/mcis the cyclotron frequency, eandmare theelectron’s charge and the rest mass, cis the speed of light,\nkxis the wave vector’s component along the drift and B\nis the magnetic field induction. During the wave generation\nprocess, one also has a simultaneous feedback of these waves\non the resonant electrons (Vedenov et al. 1961). This mech-\nanism is described by the quasi-linear diffusion (QLD), lead -\ning to a diffusion of particles as along as across the magnetic\nfield lines. Therefore, resonant particles acquire transve rse\nmomenta (pitch angles) and, as a result, start to radiate\nthrough the synchrotron mechanism.\nThe wave excitation leads to redistribution process of\nthe resonant particles via the QLD. The kinetic equation\nfor the distribution function of the resonant electrons can\nbe written as (Chkheidze et al. 2011):\n∂f0\n∂t+∂\n∂p/bardbl/braceleftbig\nF/bardblf0/bracerightbig\n+1\np⊥∂\n∂p⊥/braceleftbig\np⊥F⊥f0/bracerightbig\n=\n=1\np⊥∂\n∂p⊥/braceleftbigg\np⊥D⊥,⊥∂f0\n∂p⊥/bracerightbigg\n. (2)\nwhere\nF⊥=−αsp⊥\np/bardbl/parenleftbigg\n1+p2\n⊥\nm2c2/parenrightbigg\n, F /bardbl=−αs\nm2c2p2\n⊥,(3)\nare the transversal and longitudinal components of the syn-\nchrotron radiation reaction force, where αs= 2e2ω2\nB/3c2\nandD⊥,⊥is the transverse diffusion coefficient which is de-\nfined as follows (Chkheidze et al. 2011)\nD⊥,⊥=πe4np\n8mcω2\nBγ3p|Ek|2. (4)\nHere|Ek|2is the density of electric energy in the waves\nand its value can be estimated from the expression |Ek|2≈\nmc2nbγbc/2ωc, whereωcis the frequency of the cyclotron\nwaves. From Eq. (1) it follows that\nωc≈ωB\nδγr, (5)\nwhereδ=ω2\np/(4ω2\nBγ3\np),ωp≡/radicalbig\n4πnpe2/mis the plasma\nfrequency and npis the plasma density.\nThe transversal QLD increases the pitch-angle, whereas\nforceF⊥resists this process, leading to a stationary state\n(∂f/∂t= 0). The pitch-angles acquired by resonant electrons\nduring the process of the QLD satisfies ψ=p⊥/p/bardbl≪1.\nThus, one can assume that ∂/∂p⊥>> ∂/∂p /bardbl. In this\ncase the solution of Eq.(2) gives the distribution function\nof the resonant particles by their perpendicular momenta\n(Chkheidze et al. 2011)\nf(p⊥) =Cexp/parenleftbigg/integraldisplayF⊥\nD⊥,⊥dp⊥/parenrightbigg\n=Ce−/parenleftbigg\np⊥\np⊥0/parenrightbigg4\n, (6)\nwhere\np⊥0≈π1/2\nBγ2p/parenleftbigg3m9c11γ5\nb\n32e6P3/parenrightbigg1/4\n. (7)\nAnd for the mean value of the pitch angle we find ψ0≈\np⊥0/p/bardbl≃10−6. Synchrotron emission is generated as the\nresult of appearance of pitch angles.\nThe synchrotron emission flux of the set of electrons in\nthe framework of the present emission scenario is written as\n(see Chkheidze et al. (2011))\nc/circlecopyrt2011 RAS, MNRAS 000, 1–??3\nFǫ∝/integraldisplayp/bardblmax\np/bardblminf/bardbl(p/bardbl)Bψ0ǫ\nǫm/bracketleftBigg/integraldisplay∞\nǫ/ǫmK5/3(z)dz/bracketrightBigg\ndp/bardbl.(8)\nHeref/bardbl(p/bardbl) is the longitudinal distribution function of elec-\ntrons,ǫm≈5·10−18Bψ0γ2GeV is the photon energy of the\nmaximum of synchrotron spectrum of a single electron and\nK5/3(z) is the Macdonald function. After substituting the\nmean value of the pitch-angle in the above expression for\nǫm, we get\nǫm≃5·10−18π1/2\nγ2p/parenleftbigg3m5c7γ9\nb\n4e6P3/parenrightbigg1/4\n. (9)\nAccordingly, the beam electrons should have γb≃(6·\n108−109) to radiate the photons in the energy domain\n∼(10−100)GeV energy. The gap models provide the\nLorentz factors up to 107, which is not enough to explain\nthe detected pulsed emission. Consequently, the additiona l\nparticle acceleration mechanism should be invoked to accel -\nerate the fastest electrons to even higher energies.\nAccording to Aliu et al. (2011) the observations of the\nCrab pulsar indicates that the HE pulsed emission should\nbe produced far out in the magnetosphere. Therefore, we\nassume that the Langmuir waves (L) generated via the\ntwo-stream instability at the light cylinder length-scale s\nundergo Landau damping on the fastest beam electrons,\nwhich results in their effective acceleration. Excitation o f\nL waves through the two-stream instability in the rela-\ntivistic electron-positron plasma is considered in a serie s\nof works (see e.g., Lominadze & Mikhailovskii (1979); Usov\n(1987); Asseo & Melikidze (1998); Ursov & Usov (1988);\nGogoberidze et al. (2008)), where it is assumed that the in-\nstability develops due to overlapping of the fast and slow\npair plasma particles at distances r∼108cm from the star\nsurface. For typical parameters of pair plasma in the pulsar\nmagnetospehers the growth rate of the instability is quite\nsufficient (see Gogoberidze et al. (2008)), that inevitably\nprovides existence of L waves in the vicinity of the light\ncylinder zone. The phase velocity of the excited L waves is\nasymptotically close tothespeedoflight (Gogoberidze et a l.\n2008). Therefore, these waves can be only damped on the\nfastest electrons of the primary beam, which velocity equal s\nthe phase-velocity of the waves and the distribution functi on\nsatisfies∂fb0/∂p/bardbl<0. Taking into account the equipartition\nof energy among the plasma components npγp=nbγb/2,\none can estimate the energy density of L waves as the\nhalf of the energy density of the primary beam particles.\nThrough the Landau damping process the energy of the\nwaves is transferred to the small fraction of the beam elec-\ntrons with the highest Lorentz factors. If we assign the\ndensity of these electrons as n∗and the Lorentz factors as\nγ∗, and equate the energy densities of particles and the L\nwaves, we will get 2 n∗/nb≈γb/γ∗. The total density of\nthe beam electrons is equal to the Goldreich-Julian density\nnb=B/Pce≈2·107cm−3(Goldreich & Julian 1969) and\nalso if we take into account that γb∼107andγ∗∼109,\nwe findn∗∼105cm−3. As we see, if the wave energy is\ntransferred to the fastest beam electrons, which number is\ntwo orders of magnitude smaller than the total number of\nthe primary beam particles, they will gain the Lorentz fac-\ntors up to 109. The observational fact, that the emission fluxabove25GeV decreases shouldbe caused byreducednumber\nof emitting particles with the highest Lorentz factors.\nIt should be mentioned that during the Landau damp-\ning process the beam distribution function will be alongate d\nandwill form ahighenergy’tail’on thedistributionfuncti on.\nThe final shape of the distribution function after the quasi-\nlinear relaxation is the plateau, and the stationary state i s\nreached. But in our case this might not be achieved as in the\nsame region where the L waves are damped the cyclotron in-\nstabilityis developed,which involvesthebeam electrons i nto\nthe cyclotron resonance process. This complicated process\ncausing the redistribution of the resonant particles needs a\nmore detailed investigation, which we plan to perform in\nour future work. In the present paper, we estimate the final\nshape of the distribution function which provides explana-\ntion of the measured spectrum.\nThe beam particles lose their energy through the syn-\nchrotronradiation, whichsets theupperlimit ontheLorent z\nfactors that can be achieved during their acceleration pro-\ncess (de Jager et al. 1996). The maximum achievable value\nofγcan be estimated by equating the synchrotron radiative\nlooses to the power of the emitting particles gained through\nthe acceleration process, which in our case can be written\nas:\n2\n3e4B2ψ2\n0γ2\nm2c3=mc2γΓLD. (10)\nHere\nΓLD=nbγbωb\nnpγ5/2\np, (11)\nis the Landau damping rate, where ωb=/radicalbig\n4πe2nb/m\n(Volokitin et al. 1987). The estimations show, that the\nLorentz factor of the beam electrons that can be reached\nthrough the acceleration process γ/lessorsimilar1019. Consequently,\nthe limit on the energy of synchrotron photons in our case\nshould beǫmax∼1054eV (see Eq. (9)), which inevitably al-\nlows generation of the observed ∼100GeV photons from the\nCrab pulsar through the synchrotron emission mechanism.\n3 SPECTRUM OF THE SYNCHROTRON\nRADIATION\nTo obtain the synchrotron emission spectrum in our case,\nwe need to solve the integral (8). For this reason, first let us\nfind the parallel distribution function of the beam electron s\nf/bardblduring the QLD process of the cyclotron instability. By\nmultiplying both sides of Eq. (2) on p⊥, integrating it over\np⊥and taking into account that the distribution function\nvanishes at the boundaries of integration, Eq. (2) reduces t o\n∂f/bardbl\n∂t=∂\n∂p/bardbl/parenleftBigαs\nm2c2π1/2p2\n⊥0f/bardbl/parenrightBig\n. (12)\nForγψ≪1010, a magnetic field inhomogeneity does\nnot affect the process of wave excitation. The equation that\ndescribes the cyclotron noise level, in this case, has the fo rm\n(Lominadze et al. 1983)\n∂|Ek|2\n∂t= 2Γc|Ek|2, (13)\nwhere\nc/circlecopyrt2011 RAS, MNRAS 000, 1–??4\nΓc=π2e2\nk/bardblf/bardbl(pres), (14)\nis the growth rate of the instability. Here k/bardblcan be found\nfrom the resonance condition (1)\nk/bardblres≈ωB\ncδγres. (15)\nCombining Eqs. (10) and (11) one finds\n∂\n∂t/braceleftBigg\nf/bardbl−α∂\n∂p/bardbl/parenleftBigg\n|Ek|\np1/2\n/bardbl/parenrightBigg/bracerightBigg\n= 0, (16)\nα=/parenleftbigg4\n3e2\nπ5c5ω6\nBγ3\np\nω2p/parenrightbigg1/4\n. (17)\nConsequently, one can write\n/braceleftBigg\nf/bardbl−α∂\n∂p/bardbl/parenleftBigg\n|Ek|\np1/2\n/bardbl/parenrightBigg/bracerightBigg\n=const. (18)\nTakingintoaccountthatfor theinitial moment(themoment\nwhen the cyclotron instability arises) the energy density o f\ncyclotron waves equals zero, the corresponding expression\nwrites as\nf/bardbl−α∂\n∂p/bardbl/parenleftBigg\n|Ek|\np1/2\n/bardbl/parenrightBigg\n=f/bardbl0. (19)\nLet us assume that |Ek| ∝γ−m(as there is no direct way\nto calculate the dependence |Ek(γ)|, we can only make an\nassumption and check its plausibility by fitting the theoret -\nical emission spectrum with the observed one), in this case\nfor the parallel distribution function we will have\nf/bardbl∝f/bardbl0+γ−m−3\n2. (20)\nThe initial distribution f/bardbl0of the beam particles in this case\nis the redistributed one after the Landau damping process.\nThe final shape of the distribution function of resonant par-\nticles via the Landau damping when the stationary state is\nreached is the plateau. But in our case this might not be\nreached as in the same region develops the cyclotron insta-\nbility. Thus we consider f/bardbl0∝γ−n, wherenis not close\nto zero. Consequently, the parallel distribution of the bea m\nelectrons is proportional to two power-law function with th e\nindexesm+3/2 andn.\nThe effective value of the pitch angle depends on |Ek|2\nas follows (Chkheidze et al. 2011)\nψ0=1\n2ωB/parenleftBigg\n3m2c3\np3\n/bardblω2\np\nγ3p|Ek|2/parenrightBigg1/4\n. (21)\nUsing expressions (8), (18) and (19), and replacing the inte -\ngration variable p/bardblbyx=ǫ/ǫm, we will get the synchrotron\nemission spectrum\nFǫ∝ǫ−2m+4n−1\n5−2m/braceleftbigg\nG1/parenleftbiggǫ\nǫm/parenrightbigg\nmax−G1/parenleftbiggǫ\nǫm/parenrightbigg\nmin/bracerightbigg\n+\n+ǫ−6m+5\n5−2m/braceleftbigg\nG2/parenleftbiggǫ\nǫm/parenrightbigg\nmax−G2/parenleftbiggǫ\nǫm/parenrightbigg\nmin/bracerightbigg\n,(22)\nwhere\nG1(y) =/integraldisplay∞\nyx2m+4n−1\n5−2m/bracketleftbigg/integraldisplay∞\nxK5/3(z)dz/bracketrightbigg\ndxG2(y) =/integraldisplay∞\nyx6m+5\n5−2m/bracketleftbigg/integraldisplay∞\nxK5/3(z)dz/bracketrightbigg\ndx. (23)\nThe energy of the beam particles vary in a broad range\nγb∼106−109in which case, we have ( ǫ/ǫm)max≪1\nand (ǫ/ǫm)min≫1. Under such conditions the functions\nG1(y)≈G2(y)≈G(0). Consequently, we can assume that\nthe synchrotron emission spectrum (Eq. (20)) is propor-\ntional to two power-law functions ǫ−2m+4n−1\n5−2mandǫ−6m+5\n5−2m.\nAccording to VERITAS observations the spectrum mea-\nsured in the energy domain (100 −400)GeV is well described\nbypower-lawwiththespectral indexequalto3 .8(Aliu et al.\n2011). When m≈1, we have −(6m+5)/(5−2m) =−3.8.\nAt the same time, when m= 1 andn= 1.2, we find\n−(2m+4n−1)/(5−2m) =−2 that is in a good agreement\nwith the observations in the 10MeV - 25GeV energydomain,\nwhich shows the power-law spectrum F(ǫ)∝ǫ−2.022±0.014\n(Aliu et al. 2008).\n4 DISCUSSION\nThe interesting observational feature of the Crab pul-\nsar of the coincidence of pulse-phases from different fre-\nquency bands, ranging from radio to VHE gamma-rays\n(Manchester & Taylor 1980; Aliu et al. 2008, 2011) implies\nthat generation of these waves occur in one location of\nthe pulsar magnetosphere. This consideration automatical ly\nexcludes the generally accepted HE emission mechanisms,\nthe Inverse Compton (IC) scattering and the curvature\nradiation, which are not localized (Machabeli & Osmanov\n2009, 2010). These particular issues have been considered\nby Machabeli & Osmanov (2010). Studying the curvature\nradiation it was found that the curvature drift instability is\nefficient enough to rectify the magnetic field lines (curvatur e\ntends to zero) in the region of the generation of high and the\nVHE emission, making the curvature emission process neg-\nligible. At the same time by analyzing the IC scattering, it\nwas found that for Crab pulsar’s magnetospheric parame-\nters even very energetic electrons are unable to produce the\nobserved HE photons.\nIn Lyutikov et al. (2011) it is argued that the main gen-\nerationmechanism oftheVHEemission oftheCrabpulsar is\nthe IC scattering of the soft UV photons by the secondary\nplasma particles. Particularly, it is assumed that the sec-\nondary plasma, which is produced due to cascades in the\nouter gaps of the magnetosphere is responsible for the soft\nUV emission via the synchrotron mechanism. This radiation\nplays the target field role for the IC scattering process. As\na result, the VHE γ-ray emission is produced extending to\nhundreds of GeV. Let us consider the equation that gives\nthe frequency of the photon after the IC scattering (e.g.\nRybicki & Lightman (1979)):\nω′=ω(1−βcosθ)\n1−βcosθ′+(1−cosθ′′)/planckover2pi1ω/γmc2, (24)\nwhereωis the frequency before scattering, β≡υ/c,θ=\n(/hatwidestPK),θ′= (/hatwidestPK′) andθ′′= (/hatwidestKK′) (byPwe denoted the\nmomentum of relativistic electrons before scattering, Kand\nK′denotes the three momentum of photon before and af-\nter scattering, respectively). The emission maximum comes\nalong the magnetic field lines. The pitch angles of the rela-\nc/circlecopyrt2011 RAS, MNRAS 000, 1–??5\ntivistic electrons moving along the magnetic field lines are\nvery small ( ψ∼10−6see Eq.(7)), accordingly as the UV\nemission is generated through the synchrotron regime, the\nangleθshould also be very small. Since we observe the well-\nlocalized pulses of the VHE emission, the angle θ′≪1. Tak-\ning into account the observational fact of the phase coinci-\ndence of signals from different frequency bands, one should\nassume that θ≈θ′. At the same time the coincidence of\npulse signals of UV and the VHE emission gives θ′′= 0.\nConsequently, the IC scattering of the soft UV photons by\nthe secondary plasma electrons in case of the Crab pulsar\nshould only cause the redistribution of the electrons, butc an\nnot providethe increase in theemission frequency,especia lly\nup to the VHE band. For significantly increasing the photon\nenergy by the IC scattering processes without violating the\ncondition of the pulse-phase coincidence, the angle θmust\nbe large enough, which can not be provided by our model.\nApparently, the IC scattering should play the main role in\nthe generation of the high energy emission for pulsars with\nthe large pulse profiles in soft energy domains.\nThe emission model proposed in the present paper im-\nplicitly explains the observed pulse-phase coincidence of low\n(radio) and high frequency (10MeV-400GeV) waves, as their\ngeneration is a simultaneous process and it takes place in\nthe same place of the pulsar magnetosphere. In previous\nworks (Machabeli & Osmanov 2010; Chkheidze et al. 2011)\nwe applied this model to explain the HE (0.01-25GeV)\npulsed emission of the Crab pulsar observed by the MAGIC\nCherenkov Telescope (Aliu et al. 2008). It was found that\non the light cylinder length-scales the cyclotron instabil ity\nis arisen, which on the quasi-linear stage of the evolution\ncauses re-creation of the pitch angles and as the result the\nsynchrotron radiation mechanism is switched on. We assume\nthat the source of the high and the VHE pulsed emission of\nthe Crab pulsar is the synchrotron radiation of the ultrarel -\nativistic primary beam electrons. To explain the observed\nhigh frequency gamma-rays by synchrotron mechanism the\nLorentz factors of the emitting particles should be of the or -\nder of 108−109(see Eq. (9)). The highest Lorentz factor for\nthe typical pulsar is ∼107. Thus, we assume that such an\neffective particle acceleration is caused by existence of th e\nLangmuir waves with the phase velocities υϕ/lessorsimilarcclose to the\nvelocities of the fastest beam electrons in the region close to\nthe light cylinder. Consequently, the L (electrostatic) wa ves\nare efficiently damped on the most energetic primary beam\nelectrons. The Landau damping causes the growth of a HE\ntail on the distribution function of the resonant electrons\nand inevitably throws the most energetic particles to highe r\nLorentz factors (up to γ∼109). At the same time the beam\nelectrons acquire the pitch angles due to the cyclotron in-\nteraction with the transverse waves, which causes the syn-\nchrotron radiation processes giving the observed high and\nthe VHE emission up to 400GeV. The distribution func-\ntion tends to form the plateau (due to Landau damping),\nthough the number of processes impede this. The reaction\nforce of the synchrotron emission, scattering of L waves and\nthe Compton scattering of photons on the beam particles\nalso take place in process of formation of the distribution\nfunction of beam electrons. As the result, it is unlikely for\nthe distribution to reach the shape of plateau and thus, we\nrepresent it as f/bardbl0∝γ−n.\nThe calculation of the synchrotron emission spectrumby taking into account the processes described above (see\nEq. (20)) and matching it with the observations shows that\nthe emission spectrum in the 10MeV-25GeV energy domain\ndepends on the power-law index nof distribution function of\nthe beam particles. At the same time the emission spectrum\nin (100−400)GeV energy domain does not depend on nbut\nonly depends on m(see Eq. (18)). When m= 1 andn= 1.2\nthe emission spectrum well matches the measured one in\nboth high (0 .01−25GeV) and the very high (100 −400GeV)\nenergy domains. In previous paper (Chkheidze et al. 2011)\nexplaining the HE (10MeV-25GeV) spectrum we obtained\nthe power-law function with the exponential cutoff Fǫ∝\nǫ−2exp[−(ǫ/23)1.6], asγb∼108were the highest considered\nLorentz factors of the emitting electrons. We assume that\ndetection of the exponential cutoff at higher ( >400GeV)\nphoton energies can not be excluded, the exact location of\nthe cutoff energy is defined by the highest energy of the\nbeam electrons that can be reached through the Landau\ndamping before the particles reach the light cylinder. This\nparticular problem needs more detailed investigation, whi ch\nis the topic of our future work.\nREFERENCES\nAleksi´ c J. et al., 2011, ApJ, 744, 43\nAleksi´ c J. et al., 2012, A&A, 540, 69\nAliu E. et al., 2008, Sci, 322, 1221A\nAliu E. et al., 2011, Sci., 334, 69\nArons J., 1981, in Proc. Varenna Summer School and\nWorkhop on Plasma Astrophysics, ESA, 273\nAsseo E., Melikidze G. I., 1998, MNRAS, 301, 59\nChkheidze, N., Machabeli, G., Osmanov, Z., 2011, ApJ,\n730, 62\nde Jager, O. C., Harding, A. K., Michelson, P. F., Nel, H.\nI., Nolan, P. L., Sreekumar, P., & Thompson, D. J. 1996,\nApJ, 457, 253\nGogoberidze G., Machabeli G. Z., Usov V. 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V., 1987, ApJ, 320, 333\nc/circlecopyrt2011 RAS, MNRAS 000, 1–??6\nVedenov A.A., Velikhov E.P. & Sagdeev R.Z., 1961, Soviet\nPhysics Uspekhi, Volume 4, Issue 2, 332\nVolokitin, A.S., Krasnoselskikh, V.V. & Machabeli, G.Z.,\n1987, Soviet Journal of Plasma Physics, 11, 310\nc/circlecopyrt2011 RAS, MNRAS 000, 1–??"    },    {        "title": "2203.06360v1.Asymptotic_expansion_of_solutions_to_the_wave_equation_with_space_dependent_damping.pdf",        "content": "arXiv:2203.06360v1  [math.AP]  12 Mar 2022ASYMPTOTIC EXPANSION OF SOLUTIONS TO THE WAVE\nEQUATION WITH SPACE-DEPENDENT DAMPING\nMOTOHIRO SOBAJIMA AND YUTA WAKASUGI\nAbstract. We study the large time behavior of solutions to the wave equa tion\nwith space-dependent damping in an exterior domain. We show that if the\ndamping is effective, then the solution is asymptotically ex panded in terms\nof solutions of corresponding parabolic equations. The mai n idea to obtain\nthe asymptotic expansion is the decomposition of the soluti on of the damped\nwave equation into the solution of the corresponding parabo lic problem and\nthe time derivative of the solution of the damped wave equati on with certain\ninhomogeneous term and initial data. The estimate of the rem ainder term is\nan application of weighted energy methods with suitable sup ersolutions of the\ncorresponding parabolic problem.\nContents\n1. Introduction 2\n1.1. Problem and backgrounds 2\n1.2. Main result 5\n1.3. A rough descripsion of strategy 6\n1.4. Construction of the paper 6\n1.5. Notations 7\n2. Preliminaries 7\n2.1. Weight functions 7\n3. Justification of the decomposition 10\n3.1. Well-posedness and regularity of solutions for the damped wave\nequation 11\n3.2. Regularity of solutions for the corresponding heat equation 12\n3.3. A decomposition lemma 14\n3.4. Derivation of the asymptotic expansion 14\n4. Energy estimates for the heat equation 15\n5. Energy estimates for the damped wave equation 19\n5.1. First order energy estimates 19\n5.2. Higher order energy estimates 24\n6. Proof of the asymptotic expansion 28\nAcknowledgements 31\nReferences 31\n2020Mathematics Subject Classification. 35L20; 35C20; 35B40.\nKey words and phrases. wave equation, space-dependent damping, asymptotic expan sion.\n12 M. SOBAJIMA AND Y. WAKASUGI\n1.Introduction\n1.1.Problem and backgrounds. Let Ω be an exterior domain with a smooth\nboundary∂Ω inRNwithN≥2, or Ω = RNwithN≥1. We consider the initial-\nboundary value problem of the wave equation with space-dependen t damping\n\n\n∂2\ntu−∆u+a(x)∂tu= 0, x ∈Ω,t>0,\nu(x,t) = 0, x ∈∂Ω,t>0,\nu(x,0) =u0(x), ∂tu(x,0) =u1(x), x∈Ω.(1.1)\nHere,u=u(x,t) is a real-valuedunknownfunction, and a(x) denotes the coefficient\nof the damping term.\nWeassumethat a(x) isasmoothpositivefunctionon RNhavingboundedderiva-\ntives and satisfying\nlim\n|x|→∞|x|αa(x) =a0 (1.2)\nwith some constants α∈[0,1) anda0>0. Here, the precise meaning of (1.2)\nis limr→∞sup|x|>r||x|αa(x)−a0|= 0, that is, the convergence is uniform in the\ndirection. In this case, the damping is called effective, and, as we will s ee later,\nthe asymptotic behavior of the solution is closely related to a certain corresponding\nparabolic problem. Here, we remark that it is sufficient that a(x) is defined on ¯Ω,\nbecause we can extend it to RNso that it has the same property as above.\nThe initial data ( u0,u1) are assumed to belong to ( H2(Ω)∩H1\n0(Ω))×H1\n0(Ω).\nThen, it is known that (1.1) admits a unique solution\nu∈C([0,∞);H2(Ω))∩C1([0,∞);H1\n0(Ω))∩C2([0,∞);L2(Ω))\n(see [7, Theorem 2]). In our main result, we shall put stronger assu mptions on the\ndata.\nThe aim of this paper is to prove the asymptotic expansion of the solu tion as\ntime tends to infinity. In particular, we show that the solution uis asymptotically\nexpanded in terms of a sequence of solutions to corresponding par abolic equations\nwith certain inhomogeneous terms.\nThe asymptotic behavior of solutions to the damped wave equation h as long\nhistory after a pioneering work by Matsumura[15]. He studied the Ca uchy problem\nof the wave equation with constant damping\n∂2\ntu−∆u+∂tu= 0,(x,t)∈RN×(0,∞), (1.3)\nand applied the Fourier transform to obtain the L∞andL2estimates of solutions.\nIn particular, he showed that the decay rates are the same as tho se of the corre-\nsponding heat equation\n∂tv−∆v= 0,(x,t)∈RN×(0,∞). (1.4)\nAfter that, the precise asymptotic profile ofsolutions werestudie d by Hsiao and Liu\n[6] for the hyperbolic conservation laws with damping, and by Karch [1 3] and by\nYang and Milani [47] for (1.3), and the so-called diffusion phenomena was proved,\nthat is, the solution uof (1.3) is asymptotically approximated by a solution of the\nheatequation(1.4) astime tendstoinfinity. Moredetailedasymptot icbehaviorwas\nstudied by many mathematicians, and we refer the reader to [5, 14, 20, 21, 29] for\nthe asymptotic behavior involving the decomposition of solution into t he heat-part\nand wave-part.WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 3\nFor the higher order asymptotic expansions of the Cauchy problem of (1.3),\nGallay and Raugel [4] determined the second order expansion when N= 1 by the\nmethod of scaling-variables. Moreover, by Fourier transform, Ta keda [37] studied\nthe caseN≤3 and obtained the expansion of any order in terms of the Gaussian.\nMichihisa [17] also gave another expression of expansion for any N≥1 by the\nFourier transform method.\nOn the other hand, the asymptotic behavior of solutions to the initia l-boundary\nvalue problem of the wave equation with constant damping in an exter ior domain\nis also well-studied. This problem firstly studied by Ikehata [8], and he p roved\nthe diffusion phenomena, that is, the asymptotic profile of solution t o the extorior\nproblem is given by the exterior heat semigroup with Dirichlet boundar y condition.\nAfterthat, thisresultwasextendedbyIkehataandNishihara[10], ChillandHaraux\n[3], and Radu, Todorova, and Yordanov [27] to the abstract proble m\nu′′(t)+Au(t)+u′(t) = 0, t>0, (1.5)\nwhereAis a nonnegative self-adjoint operator in a Hilbert space. Recently, the\nfirst author [31] proved the higher order asymptotic expansion of the solution to\n(1.5) in terms of the solutions of the corresponding first order equ ation. Radu,\nTodorova, and Yordanov [28] studied the diffusion phenomena for m ore general\nabstract equation\nCu′′(t)+Bu(t)+u′(t) = 0, t>0\nwith a nonnegative self-adjoint operator Band a positive bounded operator C\nby the method of diffusion approximation. Nishiyama [23] also studied a similar\nproblem by the method of resolvent estimates.\nWe also refer the reader to Wirth [41, 42, 43, 44, 45], Yamazaki [46], and [40],\nfor the diffusion phenomena of the wave equation with time dependen t damping.\nFortheinitial-boundaryvalueproblemofthewaveequationwithspac e-dependent\ndamping (1.1), under the assumption of (1.2), it is expected that th e damping is\nclassified in the following way:\n•(scattering)When α>1,thesolutionbehaveslikethatofthewaveequation\nwithout damping.\n•(effective) When α<1, the solution behaves like that of the corresponding\nheat equation.\n•(critical) When α= 1, the equation is formally invariant under hyperbolic\nscaling and the behavior of the solution may also depend on the const ant\na0.\nThe scattering case α >1 when Ω = RN(N/ne}ationslash= 2) or Ω ⊂RNis an exterior\ndomain (N≥3) was studied by Mochizuki [18], Mochizuki and Nakazawa [19], and\nMatsuyama [16], and they proved that there exist initial data such t hat the energy\nof the corresponding solution does not decay to zero, and it appro ached a solution\nof the wave equation without damping in the energy norm. The case N= 2 seems\nstill open.\nThe critical case α= 1 with the assumption on a(x) replaced by b0/an}b∇acketle{tx/an}b∇acket∇i}ht−1≤\na(x)≤b1/an}b∇acketle{tx/an}b∇acket∇i}ht−1(b0,b1>0) was studied by Ikehata, Todorova and Yordanov [11].\nThey proved that, when Ω = RNwithN≥3 and the initial data are in C∞\n0(RN),\nthe energy of the solution decays as O(t−b0) if 1< b0< N, andO(t−N+δ) with\narbitrarysmall δ>0 ifb0≥N. Moreover,the decayrate O(t−b0) when 1<b0<N\nis optimal under some additional assumptions on a(x). A similar result was also4 M. SOBAJIMA AND Y. WAKASUGI\nobtained for the case N≤2. This indicates that the behavior of the solution\ndepends on the coefficient b0.\nFor the effective case α<1, Todorova and Yordanov [38] developed a weighted\nenergy method with an exponential-type weight function\nexp/parenleftbigg\nm(a)A(x)\nt/parenrightbigg\n,\nwhich is a refinement of the method by Ikehata [9]. Here, A(x) is a solution\nof the Poission equation ∆ A(x) =a(x) satisfying A(x)∼ /an}b∇acketle{tx/an}b∇acket∇i}ht2−α, andm(a) =\nliminf |x|→∞a(x)A(x)\n|∇A(x)|2. They showed that if Ω = RN(N≥1),a(x) is radially sym-\nmetric and satifies (1.2) with some α∈[0,1) anda0>0, and the initial data belong\ntoC∞\n0(RN), then we have m(a) =N−α\n2−α, and the following estimates hold:\n/ba∇dblu(t)/ba∇dblL2≤Cδ(1+t)−N−α\n2(2−α)+α\n2(2−α)+δ,\n/ba∇dbl(∂tu(t),∇u(t))/ba∇dblL2≤Cδ(1+t)−N−α\n2(2−α)−1\n2+δ,\nwhereδ >0 is an arbitrary small loss of decay. Radu, Todorova, and Yordano v\n[25, 26] studied the energy decay of higher order derivatives and e xtended the result\nto more general second-order hyperbolic equations. The assump tion of the radial\nsymmetry on a(x) was removed by the authors [33] by modifying the function A(x)\nabove. Moreover, the authors [39, 32, 34] proved the diffusion ph enomena in the\ncase ofα∈(−∞,1) and exterior domains. The asymptotic profile of solution uis\ngiven by the corresponding parabolic problem\n\na(x)∂tV0−∆V0= 0, x ∈Ω,t>0,\nV0(x,t) = 0, x ∈∂Ω,t>0,\nV0(x,0) =u0(x)+a(x)−1u1(x), x∈Ω.(1.6)\nRecently, the authors [35, 36] developed a different kind of weight ed energy\nmethod applicable to a wider class of initial data including polynomially dec aying\nfunctions. Roughly speaking, the suggested weight functions for m the inverse of\nthe self-similar solutions Φ βof the equation |x|−α∂tΦ−∆Φ = 0 given by\nΦβ(x,t) =t−βϕβ(ξ(x,t)),\nwhereβ∈[0,N−α\n2−α) is a parameter,\nϕβ(z) =e−zM/parenleftbiggN−α\n2−α−β,N−α\n2−α;z/parenrightbigg\n, ξ(x,t) =|x|2−α\n(2−α)2t\nwith the Kummer confluent hypergeometric function M(b,c;z) (see Section 2 for\nthe precise definition). Moreover, the relation between the order of the weight of\ninitial data and the decay rates of the solution was revealed. It is wo rthly noticing\nthat these weight functions have a polynomial growth which enables us to take\ninitial data having a polynomial decay, and the endpoint β=N−α\n2−αprovides the\nexponential type solution\nΦN−α\n2−α(x,t) =t−N−α\n2−αexp/parenleftbigg\n−|x|2−α\n(2−α)t/parenrightbigg\nwhich corresponds to the exponential type weight function introd uced in [38].\nAs mentioned above, the sharp decay estimates of solutions and th e diffusion\nphenomena for the effective case α <1 is now known very well. In contrast, the\nhigher order asymptotic expansion of the solution remains open.WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 5\nHere, we mention a result by Orive, Zuazua, and Pazoto [24] and Joly and Royer\n[12] for periodic and asymptotically periodic coefficient cases from diff erent aspects.\nFor the exterior problem with decaying damping such as (1.1), it seem s difficult to\napply the Fourier analysis which is the strong tool for the whole spac e case, and\nto apply the spectral analysis because of the appearance of unbo unded diffusion\noperators and non-comutativity.\nIn the present paper we introduce a new method (inspired by [31]) t o reach the\nasymptotic expansion in terms of solutions of the corresponding pa rabolic equation\nwith certain inhomogeneous terms (see a description of the idea in Su bsection 1.3).\n1.2.Main result. The followingis our main result which desribes the higher order\nasymptotic expansion of the solution uof (1.1).\nTheorem 1.1. Letnbe a nonnegative integer. Assume (1.2)for someα∈[0,1)\nanda0>0. Ifn+1<N−α\n2αandλ∈[2α\n2−α(n+1),N−α\n2−α), then there exist a positive\nintegers=s(n)and a constant m=m(n,α,λ)>0such that the following holds:\nSuppose the initial data u0andu1satisfy\nu0∈Hs+1,m(Ω)∩Hs,m\n0(Ω), u1∈Hs,m\n0(Ω). (1.7)\nThen there exist profiles /tildewideV1...,/tildewideVn∈C([0,∞);L2(Ω))and a positive constant C\nsuch that the solution uof(1.1)satisfies\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu(t)−V0(t)−n/summationdisplay\nj=1/tildewideVj(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Ω)≤C(1+t)−λ\n2−(2n+1)(1−α)\n2−α+α\n2(2−α)(1.8)\nfort>0, whereV0is given by (1.6). Moreover, the profiles are successively deter-\nmined as /tildewideVj=∂j\ntVjwith the unique solutions Vjof\n\n\na(x)∂tVj−∆Vj=−∂tVj−1, x∈Ω,t>0,\nVj(x,t) = 0, x ∈∂Ω,t>0,\nVj(x,0) =−(−a(x))−j−1u1(x), x∈Ω.(1.9)\nforj= 1,...,n.\nRemark 1.2. For eachVj(j= 0,1,...,n), we also have\n/ba∇dbl∂j\ntVj(t)/ba∇dblL2(Ω)≤C(1+t)−λ\n2−2j(1−α)\n2−α+α\n2(2−α)\n(see Section 6). We remark that when a(x)≡1, the expansion in Theorem 1.1\ncoincides with the known result [31].\nRemark 1.3. One can also represent the profiles /tildewideVjforj= 1,...nin terms of the\nsemigroupetLgenerated by L=a−1∆. For instance, the second profile /tildewideV1can be\nwritten as\n/tildewideV1(t) =−LetL[a−2u1]−a−1LetL[u0+a−1u1]−/integraldisplayt\n0Le(t−s)L/bracketleftBig\na−1LesL[u0+a−1u1]/bracketrightBig\nds.\nIfa≡1, thenet∆anda−1commutes, and therefore, the above description can be\nsimplified to /tildewideV1(t) =−∆et∆u1−∆(1+t∆)et∆[u0+u1]as in[31], but the semigroup\netLanda−1do not commute in general. To avoid such a complicated situat ion, we\nhave chosen the parabolic equations (1.8)for the determinination of the profiles /tildewideVj.6 M. SOBAJIMA AND Y. WAKASUGI\nRemark 1.4. (i)About the explicit values of s=s(n)andm=m(n,α,λ)in\nTheorem 1.1, a rough computation shows that we can take s= 5(n+1)andm=\n(λ+2n+1)2−α\n2+(6n2+14n+8)α. However, we omit the detailed computation,\nand do not discuss the optimality of them here.\n(ii)Ifu0,u1∈C∞\n0(Ω), then the assumptions on the initial data of Theorem 1.1 are\nautomatically fulfilled.\n1.3.A rough descripsion of strategy. By the previous studies [39, 32, 36], the\nsolution of (1.6) is known to be the first asymptotic profile of the solu tion of (1.1).\nTo investigate the asymptotic behavior of solution to (1.1), we follow the idea of\n[31]. First, the fact that V0is the first asymptotic profile implies that u−V0is a\nremainderterm. In [31], it is found that the remainderterm u−V0can be expressed\nas the time derivative of the solution of the damped wave equation wit h a certain\ninhomogeneous term. More precisely, let U1be the solution of\n\n∂2\ntU1−∆U1+a(x)∂tU1=−∂tV0, x ∈Ω,t>0,\nU1(x,t) = 0, x ∈∂Ω,t>0,\nU1(x,0) = 0, ∂tU1(x,0) =−a(x)−1u1(x), x∈Ω.\nNext, we have the decomposition u=V0+∂tU1(see Lemma 3.9). Then we further\nconsider the asymptotic profile of U1. By experience, it is natural to choose V1\nvia (1.8) with n= 1 (V1andU1has the same inhomogeneous term in respective\nequations). Then, inasimilarway U1canalsobealsodecomposedas U1=V1+∂tU2\nwith the second auxiliary function U2via\n\n∂2\ntU2−∆U2+a(x)∂tU2=−∂tV1, x ∈Ω,t>0,\nU2(x,t) = 0, x ∈∂Ω,t>0,\nU2(x,0) = 0, ∂tU2(x,0) = (−a(x))−2u1(x), x∈Ω.\nThe relation\nu=V0+∂tU1=V0+∂tV1+∂2\ntU2\ncanbe expectedtodeterminethesecondexpansion. Continuously , usingthe ( n+1)-\nth auxiliary function Un+1given by\n\n∂2\ntUn+1−∆Un+1+a(x)∂tUn+1=−∂tVn, x ∈Ω,t>0,\nUn+1(x,t) = 0, x∈∂Ω,t>0,\nUn+1(x,0) = 0, ∂tUn+1(x,0) = (−a(x))−n−1u1(x), x∈Ω,(1.10)\none can obtain the relation\nu=V0+∂tV1+∂2\ntV2+···+∂n\ntVn+∂n+1\ntUn+1. (1.11)\nMore precise discussion will be given in Section 3. Note that even if the initial\ndata (u0,u1) are compactly supported, V0,...,V nandUn+1do not have compact\nsupports in general. Thereforethe finite propagatoinpropertyd oes not workin this\nsituation. Applying a weighted energy method developed by the auth ors’ previous\npapers [30, 36], we prove that ∂n+1\ntUn+1decays faster than the other terms in\n(1.11), and this implies that the solution uis asymptotically expanded by the sum\nofV0,∂tV1,...,∂n\ntVn.\n1.4.Construction of the paper. This paper is constructed as follows. In the\nnext section, we prepare the weight functions used in the energy m ethod in subse-\nquent sections. In section 3, we state the well-posedness and reg ularity of solutions\nof the problem (1.1) and discuss the validity of the decomposition (1.1 1) (formally\nexplained in Subsection 1.3) in a suitable weighted Sobolev space. In Se ction 4, weWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 7\ndiscuss the weighted energy estimates for the corresponding par abolic equations.\nIn Section 5, we prove the weighted energy estimates for the damp ed wave equation\n(1.1) with an inhomogeneous term. Finally, in Section 6, we complete th e proof of\nTheorem 1.1 by adapting the energy estimates prepared in Sections 4 and 5 to the\noriginal problem (1.1).\n1.5.Notations. We finish this section with some notations used throughout this\npaper. The letter Cindicates a generic positive constant, which may change from\nline to line. We also express constants by C(∗,...,∗), which means this constant\ndepends on the parametersin the parenthesis. The symbol f/lessorsimilargstands forf≤Cg\nholds with some constant C >0, andf∼gmeans both f/lessorsimilargandg/lessorsimilarfhold.\nWe denote /an}b∇acketle{tx/an}b∇acket∇i}ht=/radicalbig\n1+|x|2forx∈Rn. LetL2(Ω) be the usual Lebesgue space\nwith the norm\n/ba∇dblf/ba∇dblL2(Ω)=/parenleftbigg/integraldisplay\nΩ|f(x)|2dx/parenrightbigg1/2\n,\nandC∞\n0(Ω) stands for the space of infinitely differentiable functions with co mpact\nsupport in Ω. For a nonnegative integer kandm∈R, we introduce the weighted\nSobolev spaces by\nHk,m(Ω) ={f: Ω→R;/an}b∇acketle{tx/an}b∇acket∇i}htm∂α\nxf∈L2(Ω)for any α∈ZN\n≥0with|α| ≤k},\n/ba∇dblf/ba∇dblHk,m(Ω)=/summationdisplay\n|α|≤k/ba∇dbl/an}b∇acketle{tx/an}b∇acket∇i}htm∂α\nxf/ba∇dblL2(Ω),\nwhere we used the notion of multi-index and the derivatives are in the sense of\ndistribution. When m= 0, we denote Hk(Ω) =Hk,0(Ω) for short. Also, Hk,m\n0(Ω)\nis the completion of C∞\n0(Ω) with respect to the norm /ba∇dblf/ba∇dblHk,m(Ω).\n2.Preliminaries\n2.1.Weight functions. Throughout this section, we slightly generalizethe condi-\ntions ona(x) and assume that a(x) is a smooth positive function on RNsatisfying\nlim\n|x|→∞|x|αa(x) =a0 (2.1)\nwith some constants α∈(−∞,min{2,N}) anda0>0.\nWe prepare weight functions constructed in [36]. First, we introduc e a suitable\napproximate solution of the Poisson equation ∆ A(x) =a(x).\nLemma2.1 ([33, Lemma2.1],[36, Lemma3.2]) .Leta(x)be asmooth positive func-\ntion onRNsatisfying the condition (2.1)with some constants α∈(−∞,min{2,N})\nanda0>0. Then for every ε∈(0,1), there exist a function Aε∈C2(RN)and\npositive constants cεandCεsuch that\n(1−ε)a(x)≤∆Aε(x)≤(1+ε)a(x), (2.2)\ncε/an}b∇acketle{tx/an}b∇acket∇i}ht2−α≤Aε(x)≤Cε/an}b∇acketle{tx/an}b∇acket∇i}ht2−α,\n|∇Aε(x)|2\na(x)Aε(x)≤2−α\nN−α+ε (2.3)\nhold forx∈RN.8 M. SOBAJIMA AND Y. WAKASUGI\nRemark 2.2. The above type function Aε(x)was firstly introduced by Ikehata [9],\nTodorova and Yordanov [38], and Nishihara [22]. In particular, in [38], a solution\nof∆A(x) =a(x), that is, the equation obtained by taking ε= 0in(2.2), was ap-\nplied for weighted energy estimates for the damped wave equa tion(1.1)with radially\nsymmetric a(x). Lemma 2.1 is a refinement of the method of [38]to remove the\nassumption of radial symmetry on a(x).\nThe following definitions are connected to the supersolution of a(x)vt−∆v= 0\nconstructed in [36], which plays a crucial role to obtain several estim ates verifying\nasymptotic expansion.\nDefinition 2.3 (Kummer’s confluent hypergeometric functions) .Forb,c∈Rwith\n−c /∈N∪{0}, Kummer’s confluent hypergeometric function of first kind is defined\nby\nM(b,c;s) =∞/summationdisplay\nn=0(b)n\n(c)nsn\nn!, s∈[0,∞),\nwhere(d)nis the Pochhammer symbol defined by (d)0= 1and(d)n=/producttextn\nk=1(d+\nk−1)forn∈N; note that when b=c,M(b,b;s)coincides with es.\nDefinition 2.4. (i)Forε∈(0,1/2), we define\n/tildewideγε=/parenleftbigg2−α\nN−α+2ε/parenrightbigg−1\n, γε= (1−2ε)/tildewideγε. (2.4)\n(ii)Forβ≥0andε∈(0,1/2), define\nϕβ,ε(s) =e−sM(γε−β,γε;s), s≥0.\nRemark 2.5. (i)We slightly modify the definition of ˜γεandγεfrom those of [36]\nin order to gain a positive term in the right-hand side of Prop osition 2.8 (iv). This\nmodification enables us to unify the proof of energy estimate s for the case N= 1\nandN≥2(see Sections 4 and 5).\n(ii)We note that ϕβ,ε(s)is a unique (modulo constant multiple) solution of\nsϕ′′(s)+(γε+s)ϕ′(s)+βϕ(s) = 0 (2.5)\nwith bounded derivative near s= 0.\nLemma 2.6. The function ϕβ,εdefined in Definition 2.4 satisfies the following\nproperties.\n(i)If0≤β <γε, thenϕβ,ε(s)satisfies the estimates\nkβ,ε(1+s)−β≤ϕβ,ε(s)≤Kβ,ε(1+s)−β\nwith some constants kβ,ε,Kβ,ε>0.\n(ii)For everyβ≥0, the estimate\n|ϕβ,ε(s)| ≤Kβ,ε(1+s)−β\nholds with some constant Kβ,ε>0.\n(iii)For everyβ≥0,ϕβ,ε(s)andϕβ+1,ε(s)satisfy the recurrence relation\nβϕβ,ε(s)+sϕ′\nβ,ε(s) =βϕβ+1,ε(s).WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 9\n(iv)If0≤β <γε, then we have\nϕ′\nβ,ε(s) =−β\nγεe−sM(γε−β,γε+1;s)≤0,\nϕ′′\nβ,ε(s) =β(β+1)\nγε(γε+1)e−sM(γε−β,γε+2;s)≥0.\n(v)If0≤β <γε, thenϕ′\nβ,εsatisfies\n−ϕ′\nβ,ε(s)≥kβ,ε(1+s)−β−1\nholds with some constant kβ,ε>0.\nProof.The proof of the assertions (i)–(iv) are completely the same as tha t of [36,\nLemma 3.5], and we omit the detail. The property (v) follows from the e xpression\nin (vi) and the fact M(γε−β,γε+ 1;s)∼Γ(γε+1)\nΓ(γε−β)s−β−1esass→ ∞(see, for\nexample, [36, Lemma 2.2 (ii)] or [1, p.192, (6.1.8)]). /square\nHere, we give a family of supersolutions of a(x)vt−∆v= 0, which we use later.\nDefinition 2.7. Forβ≥0and(x,t)∈RN×[0,∞), we define\nΦβ,ε(x,t;t0) = (t0+t)−βϕβ,ε(z), z=/tildewideγεAε(x)\nt0+t,\nwhereε∈(0,1/2),/tildewideγεis the constant given in (2.4),t0≥1,ϕβ,εis the function\ndefined by Definition 2.4, and Aε(x)is the function constructed in Lemma 2.1.\nFort0≥1 and (x,t)∈RN×[0,∞), we also define\nΨ(x,t;t0) :=t0+t+Aε(x). (2.6)\nProposition 2.8. The function Φβ,ε(x,t;t0)defined in Definition 2.7 satisfies the\nfollowing properties:\n(i)For everyβ≥0, we have\n∂tΦβ,ε(x,t;t0) =−βΦβ+1,ε(x,t;t0)\nfor any(x,t)∈RN×[0,∞).\n(ii)Ifβ≥0, then there exists a constant Cα,β,ε>0such that\n|Φβ,ε(x,t;t0)| ≤Cα,β,εΨ(x,t;t0)−β\nfor any(x,t)∈RN×[0,∞).\n(iii)Ifβ∈[0,γε), then there exists a constant cα,β,ε>0such that\nΦβ,ε(x,t;t0)≥cα,β,εΨ(x,t;t0)−β\nfor any(x,t)∈RN×[0,∞).\n(iv)For everyβ≥0, there exists a constant cα,β,ε>0such that\na(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0)≥cα,β,εa(x)Ψ(x,t;t0)−β−1.\nfor any(x,t)∈RN×[0,∞).10 M. SOBAJIMA AND Y. WAKASUGI\nProof.The properties (i)–(iii) are the same as [36, Lemma 3.8] and [36, Lemma\n5.1]. Thus, we omit the detail. For (iv), we put z= ˜γεAε(x)/(t0+t) and compute\n(t0+t)β+1(a(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0))\n=−a(x)/parenleftbigg\nβϕβ,ε(z)+zϕ′\nβ,ε(z)+ ˜γε∆Aε(x)\na(x)ϕ′\nβ,ε(z)+ ˜γε|∇Aε(x)|2\na(x)Aε(x)zϕ′′\nβ,ε(z)/parenrightbigg\n.\nUsing the equation (2.5) with (2.4), we rewrite the right-hand side as\n˜γεa(x)/parenleftbigg\n1−2ε−∆Aε(x)\na(x)/parenrightbigg\nϕ′\nβ,ε(z)+a(x)/parenleftbigg\n1−˜γε|∇Aε(x)|2\na(x)Aε(x)/parenrightbigg\nzϕ′′\nβ,ε(z).\nBy (2.2) and (2.3) in Lemma 2.1, we have\n1−2ε−∆Aε(x)\na(x)≤ −ε,\n1−˜γε|∇Aε(x)|2\na(x)Aε(x)≥ε/parenleftbigg2−α\nN−α+2ε/parenrightbigg−1\n>0.\nCombining them to the properties (iv) and (v) in Lemma 2.6, we conclud e\na(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0)≥ −ε˜γεa(x)(t0+t)−β−1ϕ′\nβ,ε/parenleftbigg˜γεAε(x)\nt0+t/parenrightbigg\n≥Ca(x)(t0+t)−β−1/parenleftbigg\n1+˜γεAε(x)\nt0+t/parenrightbigg−β−1\n≥Ca(x)(t0+t+Aε(x))−β−1\n=Ca(x)Ψ(x,t;t0)−β−1,\nwhich completes the proof. /square\nFinally, we prepare a useful lemma for our weighted energy method.\nLemma 2.9 ([30, Lemma 2.5]) .LetΦ∈C2(Ω)be a positive function and let\nδ∈(0,1/2). Then, for any u∈H2(Ω)∩H1\n0(Ω), we have\n/integraldisplay\nΩu∆uΦ−1+2δdx≤ −δ\n1−δ/integraldisplay\nΩ|∇u|2Φ−1+2δdx+1−2δ\n2/integraldisplay\nΩu2(∆Φ)Φ−2+2δdx,\nprovided that the right-hand side is finite.\n3.Justification of the decomposition\nIn this section, we justify the decomposition\nu=n/summationdisplay\nj=0∂j\ntVj+∂n+1\ntUn+1\nwhich is explained in Subsection 1.3. Here we need to clarify existence, uniqueness\nand also an expected regularity of respective components V0, ...VnandUn+1.\nTherefore we discuss it in the following way: we first prepare the well- posedness of\nthe initial-boundary value problem of the damped wave equation. Nex t, we show a\nkeydecompositionlemma which statesthat a solutionofthe damped w aveequation\ncan be decomposed into a solution of the corresponding parabolic eq uation and the\nderivative of a solution of the damped wave equation with another inh omogeneousWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 11\nterm. Finally, using the decomposition lemma repeatedly, we explain ho w the\nhigher order asymptotic profiles are determined.\n3.1.Well-posedness and regularity of solutions for the damped w ave\nequation. We consider the initial-boundary value problem of the damped wave\nequation with a general inhomogeneous term\n\n\n∂2\ntw−∆w+a(x)∂tw=F, x ∈Ω,t>0,\nw(x,t) = 0, x ∈∂Ω,t>0,\nw(x,0) =w0(x), ∂tw(x,0) =w1(x), x∈Ω.(3.1)\nWe first prepare the well-posedness and the regularity of solutions for (3.1).\nWe recall the following well-posedness result by Ikawa [7].\nTheorem 3.1 ([7, Theorem 1]) .For any (w0,w1)∈(H2(Ω)∩H1\n0(Ω))×H1\n0(Ω)\nandF∈C1([0,∞);L2(Ω)), there exists a unique solution\nw∈C([0,∞);H2(Ω))∩C1([0,∞);H1\n0(Ω))∩C2([0,∞);L2(Ω))\nof(3.1).\nBy applying the above theorem to /an}b∇acketle{tx/an}b∇acket∇i}htmw, (/an}b∇acketle{tx/an}b∇acket∇i}htmw0,/an}b∇acketle{tx/an}b∇acket∇i}htmw1), and/an}b∇acketle{tx/an}b∇acket∇i}htmF, we\nhave the well-posedness of the problem (3.1) in weighted Sobolev spa ces.\nTheorem 3.2. For any (w0,w1)∈(H2,m(Ω)∩H1,m\n0(Ω))×H1,m\n0(Ω)andF∈\nC1([0,∞);H0,m(Ω)), there exists a unique solution\nw∈C([0,∞);H2,m(Ω))∩C1([0,∞);H1,m\n0(Ω))∩C2([0,∞);H0,m(Ω))\nof(3.1).\nNext, we discuss the regularity of the solution. We first recall the f ollowing\nregularity theorem by Ikawa [7]:\nTheorem 3.3 ([7, Theorem 2]) .Letk≥1be an integer and let m≥0,w0∈\nHk+2(Ω),w1∈Hk+1(Ω), andF∈/intersectiontextk\nj=0Cj+1([0,∞);Hk−j(Ω)). We successively\ndefine\nwp= ∆wp−2−a(x)wp−1+∂p−2\ntF(x,0)\nforp= 2,...,k+1, and assume the k-th order compatibility condition\n(wp,wp+1)∈(H2(Ω)∩H1\n0(Ω))×H1\n0(Ω)\nforp= 0,1,...,k. Then, the solution wto(3.1)obtained by Theorem 3.1 belongs\nto\nC([0,∞);Hk+2(Ω))∩\nk+1/intersectiondisplay\nj=1Ck+2−j([0,∞);Hj\n0(Ω))\n∩Ck+2([0,∞);L2(Ω)).\nFrom the above theorem and the same argument as Theorem 3.2, we have the\nfollowing regularity theorem in weighted Sobolev spaces.\nTheorem 3.4. Letk≥1be an integer and let m≥0,w0∈Hk+2,m(Ω),w1∈\nHk+1,m(Ω), andF∈/intersectiontextk\nj=0Cj+1([0,∞);Hk−j,m(Ω)). We successively define\nwp= ∆wp−2−a(x)wp−1+∂p−2\ntF(x,0)12 M. SOBAJIMA AND Y. WAKASUGI\nforp= 2,...,k+1, and assume the k-th order compatibility condition\n(wp,wp+1)∈(H2,m(Ω)∩H1,m\n0(Ω))×H1,m\n0(Ω)\nforp= 0,1,...,k. Then, the solution wto(3.1)obtained by Theorem 3.2 belongs\nto\nC([0,∞);Hk+2,m(Ω))∩\nk+1/intersectiondisplay\nj=1Ck+2−j([0,∞);Hj,m\n0(Ω))\n∩Ck+2([0,∞);H0,m(Ω)).\n3.2.Regularity of solutions for the corresponding heat equatio n.Follow-\ning our previous study [32, section 2], we prepare the well-posednes s and regularity\nof solutions for the initial-boundary problem of the corresponding h eat equation\nwith a general inhomogeneous term\n\n\na(x)∂tv−∆v=G, x∈Ω,t>0,\nv(x,t) = 0, x ∈∂Ω,t>0,\nv(x,0) =v0(x), x∈Ω.(3.2)\nLetdµ=a(x)dxand we define\nL2\ndµ(Ω) =/braceleftBigg\nf∈L2\nloc(Ω);/ba∇dblf/ba∇dblL2\ndµ=/parenleftbigg/integraldisplay\nΩ|f(x)|2dµ/parenrightbigg1/2\n<∞/bracerightBigg\n,\n(f,g)L2\ndµ:=/integraldisplay\nΩf(x)g(x)dµ.\nThe operator −a(x)−1∆ is formally symmetric in L2\ndµ(Ω), and its bilinear closed\nform is defined by\na(u,v) =/integraldisplay\nΩ∇u(x)·∇v(x)dx,\nD(a) =/braceleftbigg\nu∈L2\ndµ(Ω)∩˙H1(Ω);/integraldisplay\nΩ∂u\n∂xjϕdx=−/integraldisplay\nΩu∂ϕ\n∂xjdxfor allϕ∈C∞\n0(RN)/bracerightbigg\n.\nFrom [32], we have the Friedrichs extension −Lof the operator −a(x)−1∆ in\nL2\ndµ(Ω).\nLemma 3.5 ([32, Lemma 2.2]) .The operator −LinL2\ndµ(Ω)defined by\nD(L) =/braceleftBig\nu∈D(a);∃f∈L2\ndµ(Ω)s.t.a(u,v) = (f,v)L2\ndµfor anyv∈D(a)/bracerightBig\n,\n−Lu=f\nis nonnegative and selfadjoint in L2\ndµ(Ω). Therefore, Lgenerates an analytic semi-\ngroupT(t)onL2\ndµ(Ω)satisfying\n/ba∇dblT(t)f/ba∇dblL2\ndµ≤ /ba∇dblf/ba∇dblL2\ndµ,/ba∇dblLT(t)f/ba∇dblL2\ndµ≤1\nt/ba∇dblf/ba∇dblL2\ndµ\nfor anyf∈L2\ndµ(Ω).\nWe alsorecall the followingproperty proved in [32], which will be used in S ection\n6:WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 13\nLemma 3.6 ([32, Lemma 2.3]) .We have\n/braceleftBig\nu∈H2(Ω)∩H1\n0(Ω);a(x)−1/2∆u∈L2(Ω)/bracerightBig\n⊂D(L).\nFor the inhomogeneous problem (3.2), applying [2, Lemma 4.1.1, Propo sition\n4.1.6], we have the following well-posedness result.\nTheorem 3.7. Assume that v0∈D(L)anda(x)−1G∈C1([0,∞);L2\ndµ(Ω)). Then,\nthe function vdefined by\nv(t) =T(t)v0+/integraldisplayt\n0T(t−s)(a(x)−1G(s))ds\nis the unique solution to the problem (3.2)satisfying\nv∈C([0,∞);D(L))∩C1([0,∞);L2\ndµ(Ω)).\nNext, we discuss the higher order regularity in time for the solution o f (3.2). We\nnote that, by a formal straightforward computation, the initial v alues of∂j\ntvfor\nj≥1 are given by\n∂j\ntv(x,0) = [a(x)−1∆]jv0(x)+j−1/summationdisplay\nl=0[a(x)−1∆]la(x)−1∂j−1−l\ntG(x,0).(3.3)\nTheorem3.8. Letk≥1be an integer, v0∈D(L), anda(x)−1G∈Ck+1([0,∞);L2\ndµ(Ω)).\nAssume that ∂j\ntv(x,0)defined by the right-hand side of (3.3)satisfies∂j\ntv(x,0)∈\nD(L)forj= 1,...,k. Then, the solution vto(3.2)obtained by Theorem 3.7\nbelongs to\nCk([0,∞);D(L))∩Ck+1([0,∞);L2\ndµ(Ω)). (3.4)\nProof.Whenk= 1, letψ=ψ(x,t) be the solution of (3.2) with the inhomogeneous\nterm∂tGand the initial data ψ(x,0) =a(x)−1∆v0(x) +a(x)−1G(x,0). Then, by\nTheorem 3.7, ψis given by\nψ(t) =T(t)[a(x)−1∆v0(x)+a(x)−1G(x,0)]+/integraldisplayt\n0T(t−s)(a(x)−1∂sG(s))ds\n=∂tT(t)v0+T(t)[a(x)−1G(x,0)]\n+/bracketleftbig\nT(t−s)(a(x)−1G(s))/bracketrightbigt\n0+/integraldisplayt\n0∂tT(t−s)(a(x)−1G(s))ds\n=∂tT(t)v0+a(x)−1G(t)\n+∂t/integraldisplayt\n0T(t−s)(a(x)−1G(s))ds−a(x)−1G(t)\n=∂tv(t).\nSinceψ∈C([0,∞);D(L))∩C1([0,∞);L2\ndµ(Ω)), we have\nv∈C1([0,∞);D(L))∩C2([0,∞);L2\ndµ(Ω)),\nthat is, the assertion when k= 1 is proved. The general case k≥1 can be proved\nin the same way with induction, and we omit the detail. /square14 M. SOBAJIMA AND Y. WAKASUGI\n3.3.A decomposition lemma. In the following two subsections, we give the idea\nof the asymptotic expansion of the solution of the damped wave equ ation (1.1). To\nsimplify the discussion, we only give formal computation here. The ju stification\nand the complete proof of the asymptotic expansion will be given in Se ction 6.\nRelated to the initial-boundary value problem of the damped wave equ ation\n(3.1), we consider the parabolic problem with the same inhomogeneou s termFand\nthe initial data w0(x)+a(x)−1w1(x):\n\n\na(x)∂tV−∆V=F, x ∈Ω,t>0,\nV(x,t) = 0, x ∈∂Ω,t>0,\nV(x,0) =w0(x)+a(x)−1w1(x), x∈Ω.(3.5)\nFor the solution Vof the above problem, we further consider the following initial-\nboundary value problem of the damped wave equation with the inhomo geneous\nterm−∂tVand the initial data ( U,∂tU)(x,0) = (0,−a(x)−1w1(x)):\n\n\n∂2\ntU−∆U+a(x)∂tU=−∂tV, x ∈Ω,t>0,\nU(x,t) = 0, x ∈∂Ω,t>0,\nU(x,0) = 0, ∂tU(x,0) =−a(x)−1w1(x), x∈Ω.(3.6)\nThen, we have the following decomposition of the solution wto (3.1).\nLemma 3.9. Letwbe the solution of the damped wave equation (3.1)with the\ninhomogeneous term Fand the initial data (w0,w1). LetVbe the solution of the\nparabolic problem (3.5), and letUbe the solution of the problem (3.6). Then, we\nhave\nw=V+∂tU.\nProof.Let ˜w=V+∂tU. Then, we have\n˜w(x,0) =V(x,0)+∂tU(x,0) =w0(x)+a(x)−1w1(x)−a(x)−1w1(x) =w0(x).\nAlso, by (3.6), we obtain\n∂t˜w=∂tV+∂2\ntU= ∆U−a(x)∂tU. (3.7)\nThis implies ∂t˜w(x,0) = ∆U(x,0)−a(x)∂tU(x,0) =w1(x). Finally, differentiating\n(3.7) again, and using the relation ∂2\ntU=∂t˜w−∂tV, we deduce\n∂2\nt˜w= ∆∂tU−a(x)∂2\ntU\n= ∆(˜w−V)−a(x)(∂t˜w−∂tV)\n= ∆˜w−a(x)∂t˜w+F.\nConsequently, ˜ wis the solution of (3.1), and hence, the uniqueness shows ˜ w=w.\nThis completes the proof. /square\n3.4.Derivation of the asymptotic expansion. Letube the solution of (1.1).\nTo expand uin terms of solutions of the corresponding parabolic problem, we con -\nsiderfunctions V0,V1,...,V nand the remainderterms U1,U2,...,U n+1successively\ndefined in the following way: first, we define V0by\n\n\na(x)∂tV0−∆V0= 0, x ∈Ω,t>0,\nV0(x,t) = 0, x ∈∂Ω,t>0,\nV0(x,0) =u0(x)+a(x)−1u1(x), x∈Ω.WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 15\nandU1by\n\n\n∂2\ntU1−∆U1+a(x)∂tU1=−∂tV0, x ∈Ω,t>0,\nU1(x,t) = 0, x ∈∂Ω,t>0,\nU1(x,0) = 0, ∂tU1(x,0) =−a(x)−1u1(x), x∈Ω.\nThen, by Lemma 3.9 we have the first decomposition u=V0+∂tU1. According to\nthe experiences, we expect that this is the first-orderasymptot ic expansionof uand\ntherefore∂tU1can be regarded as a perturbation. Next, to obtain the second-o rder\nexpansion, we further consider the decomposition of U1in terms of corresponding\nparabolic problem. Namely, we define V1by\n\n\na(x)∂tV1−∆V1=−∂tV0, x∈Ω,t>0,\nV1(x,t) = 0, x ∈∂Ω,t>0,\nV1(x,0) =−a(x)−2u1(x), x∈Ω.\nandU2by\n\n\n∂2\ntU2−∆U2+a(x)∂tU2=−∂tV1, x ∈Ω,t>0,\nU2(x,t) = 0, x ∈∂Ω,t>0,\nU2(x,0) = 0, ∂tU2(x,0) =a(x)−2u1(x), x∈Ω.\nThen, by Lemma 3.9 again, we have the decomposition U1=V1+∂tU2, which\nimplies\nu=V0+∂tV1+∂2\ntU2.\nWe expect that this gives the second-order asymptotic expansion ofu. Repeating\nthis procedure for j= 2,3,...,nwe successively define Vjby\n\n\na(x)∂tVj−∆Vj=−∂tVj−1, x∈Ω,t>0,\nVj(x,t) = 0, x ∈∂Ω,t>0,\nVj(x,0) =−(−a(x))−j−1u1(x), x∈Ω\nandUn+1by\n\n\n∂2\ntUn+1−∆Un+1+a(x)∂tUn+1=−∂tVn, x ∈Ω,t>0,\nUn+1(x,t) = 0, x∈∂Ω,t>0,\nUn+1(x,0) = 0, ∂tUn+1(x,0) = (−a(x))−n−1u1(x), x∈Ω.(3.8)\nThen, we can have the expected higher order decomposition\nu=V0+∂tV1+∂2\ntV2+···+∂n\ntVn+∂n+1\ntUn+1. (3.9)\nBy Theorems 3.2 and 3.4, the existence, uniqueness, and regularity of the solution\nUn+1to (3.8) can be obtained from the assumptions on the initial data of T heorem\n1.1. The detail will be discussed in Section 6.\nIn the following sections, we give energy estimates for V0,V1,...,V nandUn+1\nto justify that (3.9) actually gives the n-th order asymptotic expansion of u.\n4.Energy estimates for the heat equation\nWe apply the weighted energy method to obtain the decay estimate o f the par-\nabolic problem\n\n\na(x)∂tv−∆v=G, x∈Ω,t>0,\nv(x,t) = 0, x ∈∂Ω,t>0,\nv(x,0) =v0(x), x∈Ω.(4.1)16 M. SOBAJIMA AND Y. WAKASUGI\nThe goal of this section is the following weighted energy estimates fo r higher\norder derivatives of solutions to (4.1).\nTheorem 4.1. Letk≥0be an integer, δ∈(0,1/2),ε∈(0,1/2),λ∈[0,(1−2δ)γε)\n(see(2.4)for the definition of γε),t0≥1andβ=λ/(1−2δ). Letv0∈D(L),\na(x)−1G∈Ck+1([0,∞);L2\ndµ(Ω)), and letvbe the corresponding solution of (4.1).\nMoreover, we assume that ∂j\ntv(x,0)given by the right-hand side of (3.3)satisfies\n∂j\ntv(x,0)∈D(L)∩H0,(λ+2j)2−α\n2−α\n2(Ω),\n∇∂j\ntv(x,0)∈H0,(λ+1+2j)2−α\n2(Ω),\nand/integraldisplay\nΩa(x)−1|∂j\ntG(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞)\nforj= 0,1,...,k. Then, we have/integraldisplay\nΩa(x)|∂j\ntv(x,t)|2Ψ(x,t;t0)λ+2jdx∈L∞(0,∞), (4.2)\n/integraldisplay\nΩ|∇∂j\ntv(x,t)|2Ψ(x,t;t0)λ+2jdx∈L1(0,∞), (4.3)\n/integraldisplay\nΩ|∇∂j\ntv(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L∞(0,∞), (4.4)\n/integraldisplay\nΩa(x)|∂j+1\ntv(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞) (4.5)\nforj= 0,1,...,k.\nWe note that Theorem 3.8 ensures the regularity property (3.4) fo r the solution\nv. Thus, it suffices to show the estimates (4.2)–(4.5).\nThe proof of Theorem 4.1 is based on an induction argument. The follo wing\nlemma is the first step.\nLemma 4.2. Under the assumptions on Theorem 4.1 with k= 0, we have (4.2)–\n(4.5)fork= 0.\nProof.ByLemma2.9, Proposition2.8(iii), andtheSchwarzinequality,wecalcu late\nd\ndt/integraldisplay\nΩa(x)v2\nΦ1−2δ\nβ,εdx= 2/integraldisplay\nΩv∆v\nΦ1−2δ\nβ,εdx−(1−2δ)/integraldisplay\nΩa(x)v2∂tΦβ,ε\nΦ2−2δ\nβ,εdx\n+2/integraldisplay\nΩvG\nΦ1−2δ\nβ,εdx\n≤ −2δ\n1−δ/integraldisplay\nΩ|∇v|2\nΦ1−2δ\nβ,εdx−(1−2δ)/integraldisplay\nΩ(a(x)∂tΦβ,ε−∆Φβ,ε)v2\nΦ2−2δ\nβ,εdx\n+C/parenleftbigg/integraldisplay\nΩa(x)v2Ψλ−1dx/parenrightbigg1/2/parenleftbigg/integraldisplay\nΩa(x)−1G2Ψλ+1dx/parenrightbigg1/2\n.\nFrom the Young inequality and Proposition 2.8 (iv), we obtain\nd\ndt/integraldisplay\nΩa(x)v2\nΦ1−2δ\nβ,εdx=−2δ\n1−δ/integraldisplay\nΩ|∇v|2\nΦ1−2δ\nβ,εdx+C/integraldisplay\nΩa(x)−1G2Ψλ+1dx.WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 17\nIntegrating it over [0 ,t] and using Proposition 2.8 (iii), we deduce\n/integraldisplay\nΩa(x)v(x,t)2Ψ(x,t;t0)λdx∈L∞(0,∞),\n/integraldisplay\nΩ|∇v(x,t)|2Ψ(x,t;t0)λdx∈L1(0,∞), (4.6)\nThus, we have (4.2) and (4.3) in the case j= 0. We next compute\nd\ndt/integraldisplay\nΩ|∇v|2Ψλ+1dx= (λ+1)/integraldisplay\nΩ|∇v|2Ψλdx+2/integraldisplay\nΩ(∇∂tv·∇v)Ψλ+1dx.(4.7)\nBy (4.6), the first term of the right-hand side belongs to L1(0,∞), and by integra-\ntion by parts, the second term is estimated as\n2/integraldisplay\nΩ(∇∂tv·∇v)Ψλ+1dx\n=−2/integraldisplay\nΩ(a(x)∂tv−G)∂tvΨλ+1dx−(λ+1)/integraldisplay\nΩ∂tv(∇v·∇Aε(x))Ψλdx\n≤ −2/integraldisplay\nΩa(x)|∂tv|2Ψλ+1dx\n+1\n2/integraldisplay\nΩa(x)|∂tv|2Ψλ+1dx+2/integraldisplay\nΩa(x)−1G2Ψλ+1dx\n+1\n2/integraldisplay\nΩa(x)|∂tv|2Aε(x)Ψλdx+2(λ+1)2/integraldisplay\nΩ|∇Aε(x)|2\na(x)Aε(x)|∇v|2Ψλdx\n≤ −/integraldisplay\nΩa(x)|∂tv|2Ψλ+1dx\n+C/integraldisplay\nΩ|∇v|2Ψλdx+2/integraldisplay\nΩa(x)−1G2Ψλ+1dx. (4.8)\nHere, we have used the property (2.3) in Lemma 2.1 and the relation Aε(x)≤\nΨ(x,t;t0), which follows from the definition Ψ (see (2.6)). By (4.6) and the as-\nsumption on G, the last two terms of the right-hand side of above are in L1(0,∞).\nThus, integrating (4.7) over [0 ,t], we conclude\n/integraldisplay\nΩ|∇v(x,t)|2Ψ(x,t;t0)λ+1dx∈L∞(0,∞),\n/integraldisplay\nΩa(x)|∂tv(x,t)|2Ψ(x,t;t0)λ+1dx∈L1(0,∞),\nthat is, (4.4) and (4.5) in the case j= 0, and the proof is now complete. /square\nRemark 4.3. It should be noted that the integration by parts in the above p roof can\nbe justified completely by the approximation argument in the same as [36, Section\n4].\nNext, we prove the following lemma, which is the main part of the induct ion\nargument of the proof of Theorem 4.1.\nLemma 4.4. Assumea(x)satisfies (1.2). Letσ≥0,t0≥1,v0∈D(L)and\na(x)−1G∈C1([0,∞);L2\ndµ(Ω)). Letvbe the corresponding solution of (4.1). We18 M. SOBAJIMA AND Y. WAKASUGI\nfurther assume\nv0∈H0,2−α\n2σ−α\n2(Ω),∇v0∈H0,2−α\n2(σ+1)(Ω),\n/integraldisplay\nΩa(x)−1|G(x,t)|2Ψ(x,t;t0)σ+1dx∈L1(0,∞), (4.9)\nand also/integraldisplay\nΩa(x)|v(x,t)|2Ψ(x,t;t0)σ−1dx∈L1(0,∞). (4.10)\nThen, we have\n/integraldisplay\nΩa(x)|v(x,t)|2Ψ(x,t;t0)σdx∈L∞(0,∞), (4.11)\n/integraldisplay\nΩ|∇v(x,t)|2Ψ(x,t;t0)σdx∈L1(0,∞), (4.12)\n/integraldisplay\nΩ|∇v(x,t)|2Ψ(x,t;t0)σ+1dx∈L∞(0,∞), (4.13)\n/integraldisplay\nΩa(x)|∂tv(x,t)|2Ψ(x,t;t0)σ+1dx∈L1(0,∞). (4.14)\nRemark 4.5. In the proof of Theorem 4.1, we will choose σ=λ+2jforj∈N.\nProof of Lemma 4.4. Suppose (4.9) and (4.10). Similarly as the proof of Lemma\n4.2, we compute\nd\ndt/integraldisplay\nΩa(x)|v(x,t)|2Ψ(x,t;t0)σdx\n= 2/integraldisplay\nΩv(∆v)Ψσdx+σ/integraldisplay\nΩa(x)|v|2Ψσ−1dx\n+2/integraldisplay\nΩvGΨσdx (4.15)\nThe second term of the right-hand side is in L1(0,∞) due to the assumption (4.10).\nNoting the relation 2 v∆v=−2|∇v|2+∆(|v|2) and using the integration by parts,\nwe calculate the first term of the right-hand side as\n2/integraldisplay\nΩv(∆v)Ψσdx=−2/integraldisplay\nΩ|∇v|2Ψσdx+/integraldisplay\nΩ|v|2∆(Ψσ)dx.\nThe last term of the above is further estimated by\n/integraldisplay\nΩ|v|2|∆(Ψσ)|dx=σ/integraldisplay\nΩ|v|2/vextendsingle/vextendsingle∆AεΨ+(σ−1)|∇Aε|2/vextendsingle/vextendsingleΨσ−2dx\n≤C/integraldisplay\nΩa(x)|v|2Ψσ−1dx,\nwhere we have used (2.2), (2.3), and Aε≤Ψ. Therefore, this term also belongs to\nL1(0,∞) by the assumption (4.10). Finally, we apply the Schwarz inequality to the\nlast term of (4.15) and obtain\n2/integraldisplay\nΩvGΨσdx≤/integraldisplay\nΩa(x)|v|2Ψσ−1dx+/integraldisplay\nΩa(x)−1|G|2Ψσ+1dxWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 19\nand these are in L1(0,∞) due to the assumptions (4.9) and (4.10). Consequently,\nintegrating (4.15) over [0 ,t], we have\n/integraldisplay\nΩa(x)|v(x,t)|2Ψ(x,t;t0)σdx∈L∞(0,∞),\n/integraldisplay\nΩ|∇v(x,t)|2Ψ(x,t;t0)σdx∈L1(0,∞).\nThus, we have (4.11) and (4.12).\nNext, we compute\nd\ndt/integraldisplay\nΩ|∇v(x,t)|2Ψ(x,t;t0)σ+1dx= (σ+1)/integraldisplay\nΩ|∇v|2Ψσdx\n+2/integraldisplay\nΩ(∇∂tv·∇v)Ψσ+1dx.(4.16)\nThe first term of the right-hand side belongs to L1(0,∞) by (4.12). The second\nterm can be estimated in completely the same way as (4.8), and we hav e\n2/integraldisplay\nΩ(∇∂tv·∇v)Ψσ+1dx≤ −/integraldisplay\nΩa(x)|∂tv|2Ψσ+1dx\n+C/integraldisplay\nΩ|∇v|2Ψσdx\n+C/integraldisplay\nΩa(x)−1|G|2Ψσ+1dx.\nBy using (4.9), and (4.12), the last two terms of above belong to L1(0,∞). Finally,\nintegrating (4.16) over [0 ,t], we conclude\n/integraldisplay\nΩ|∇v(x,t)|2Ψ(x,t;t0)σ+1dx∈L∞(0,∞),\n/integraldisplay\nΩa(x)|∂tv(x,t)|2Ψ(x,t;t0)σ+1dx∈L1(0,∞).\nThis completes the proof of (4.14) and (4.13). /square\nProof of Theorem 4.1. We note that the case k= 0 has been already proved by\nLemma 4.2. Let k≥1 be an integer. Then, by Lemma 4.2, we have (4.2)–(4.5) in\nthe casej= 0.\nNext, forj= 1, we apply Lemma 4.4 with σ=λ+2jand with the replacement\nofvandGby∂tvand∂tG, respectively. We remark that the condition (4.10) with\nσ=λ+2jis fulfilled by virtue of (4.5) with j= 0. Then, we obtain (4.11)–(4.14)\nforσ=λ+2jwith the replacement of vby∂tv, namely, we reach the conclusions\n(4.2)–(4.5) for j= 1.\nThe properties (4.2)–(4.5) for j= 1 allow us to apply again Lemma 4.4 with\nσ=λ+2j,j= 2andwith the replacementof vandGby∂2\ntvand∂2\ntG, respectively.\nThen, we can see that (4.2)–(4.5) for j= 2 hold. Repeating this argument until\nj=k, we complete the proof of Theorem 4.1. /square\n5.Energy estimates for the damped wave equation\n5.1.First order energy estimates. In this section, we discuss the energy esti-\nmate for the general damped wave equation (3.1).20 M. SOBAJIMA AND Y. WAKASUGI\nThe results of this section will be used in the next section by putting w=Un+1,\nF=−∂tVn,w0= 0, andw1= (−a(x))−n−1u1(x) (see (3.8)) to derive the energy\nestimate of ∂n+1\ntUn+1.\nWe start with the definition of the weighted energy of w.\nDefinition 5.1. Forδ∈(0,1/2),ε∈(0,1/2),λ∈[0,(1−2δ)γε)(see(2.4)for the\ndefinition of γε),β=λ/(1−2δ),t0≥1, andν >0, we define\nE1[w](t;t0,λ) :=/integraldisplay\nΩ/parenleftbig\n|∇w(x,t)|2+|∂tw(x,t)|2/parenrightbig\nΨ(x,t;t0)λ+1dx,\nE0[w](t;t0,λ) :=/integraldisplay\nΩ/parenleftbig\n2w(x,t)∂tw(x,t)+a(x)|w(x,t)|2/parenrightbig\nΦβ,ε(x,t;t0)−1+2δdx,\nE[w](t;t0,λ,ν) :=νE1[w](t;t0,λ)+E0[w](t;t0,λ)\nfort≥0.\nWe note that, for any ν >0, there exists t1>0 such that\nE[w](t;t0,λ,ν)∼E1[w](t;t0,λ)+/integraldisplay\nΩa(x)|w(x,t)|2Ψ(x,t;t0)λdx\nholds for any t0≥t1. Indeed, the Schwarz inequality implies\n|2w∂tw| ≤a(x)\n2|w|2+2a(x)−1|∂tw|2,\nand Proposition 2.8 (iii) and (iv) lead to\n2a(x)−1|∂tw|2Φ−1+2δ\nβ,ε≤CΨα\n2−α|∂tw|2Ψλ≤Ct−1+α\n2−α\n0|∂tw|2Ψλ+1≤ν\n2|∂tw|2Ψλ+1\nfor sufficiently large t0.\nThe main theorem of this subsection is the following:\nTheorem 5.2. Assume that a(x)satisfies(1.2). Letδ∈(0,1/2),ε∈(0,1/2),λ∈\n[0,(1−2δ)γε), andβ=λ/(1−2δ). Then, there exist constants ν=ν(N,α,δ,ε,λ )>\n0andt∗=t∗(N,α,δ,ε,λ,ν )≥1such that for any t0≥t∗, the following holds: Let\nm= (λ+1)2−α\n2and assume F∈C1([0,∞);H0,m(Ω))satisfies\n/integraldisplay\nΩa(x)−1|F(x,t)|2Ψ(x,t;t0)λ+1dx∈L1(0,∞).\nLetwbe the solution of (3.1)with the initial data (w0,w1)∈(H2,m(Ω)∩H1,m\n0(Ω))×\nH1,m\n0(Ω)given in Theorem 3.2. Then, we have\nE[w](t;t0,λ,ν)∈L∞(0,∞),\n/integraldisplay\nΩ|∇w(x,t)|2Ψ(x,t;t0)λdx∈L1(0,∞),\n/integraldisplay\nΩa(x)|∂tw(x,t)|2Ψ(x,t;t0)λ+1dx∈L1(0,∞).\nThe proof of Theorem 5.2 is a bit lengthy, but the outline is as follows. W e\nshall derive good terms −/integraldisplay\nΩ|∇w|2Ψλdxand−/integraldisplay\nΩa(x)|∂tw|2Ψλ+1dxfrom the\ncomputations ofd\ndtE0[w](t;t0,λ) andd\ndtE1[w](t;t0,λ), respectively (see the right-\nhand sides of Lemmas 5.3 and 5.4). Then, we sum up them with sufficient ly smallWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 21\nνand sufficiently large t0so that the other bad terms are absorbed by these good\nterms.\nWe first give estimates of E0[w](t;t0,λ).\nLemma 5.3. Under the assumptions on Theorem 5.2, for any t0≥1andt >0,\nwe have\nd\ndtE0[w](t;t0,λ)≤ −η0/integraldisplay\nΩ|∇w(x,t)|2Ψ(x,t;t0)λdx\n−η0/integraldisplay\nΩa(x)|w(x,t)|2Ψ(x,t;t0)λ−1dx\n+C/integraldisplay\nΩ|∂tw(x,t)|2Ψ(x,t;t0)λdx\n+C/integraldisplay\nΩa(x)−1|F(x,t)|2Ψ(x,t;t0)λ+1dx\nwith some constants η0=η0(ε,δ)>0andC=C(N,α,δ,ε,λ )>0.\nProof.By the definition of E0[w](t;t0,λ) and using the equation (3.1), we calculate\nd\ndtE0[w](t;t0,λ) = 2/integraldisplay\nΩ|∂tw|2Φ−1+2δ\nβ,εdx+2/integraldisplay\nΩw/parenleftbig\n∂2\ntw+a(x)∂tw/parenrightbig\nΦ−1+2δ\nβ,εdx\n−(1−2δ)/integraldisplay\nΩ/parenleftbig\n2w∂tw+a(x)|w|2/parenrightbig\n(∂tΦβ,ε)Φ−2+2δ\nβ,εdx\n= 2/integraldisplay\nΩ|∂tw|2Φ−1+2δ\nβ,εdx+2/integraldisplay\nΩw(∆w+F)Φ−1+2δ\nβ,εdx\n−(1−2δ)/integraldisplay\nΩ/parenleftbig\n2w∂tw+a(x)|w|2/parenrightbig\n(∂tΦβ,ε)Φ−2+2δ\nβ,εdx.\nApplying Lemma 2.9, we have\nd\ndtE0[w](t;t0,λ)≤2/integraldisplay\nΩ|∂tw|2Φ−1+2δ\nβ,εdx−2δ\n1−δ/integraldisplay\nΩ|∇w|2Φ−1+2δ\nβ,εdx\n−(1−2δ)/integraldisplay\nΩ|w|2(a(x)∂tΦβ,ε−∆Φβ,ε)Φ−2+2δ\nβ,εdx\n−2(1−2δ)/integraldisplay\nΩw∂tw(∂tΦβ,ε)Φ−2+2δ\nβ,εdx\n+2/integraldisplay\nΩwFΦ−1+2δ\nβ,εdx. (5.1)\nBy Proposition 2.8 (iii) and (iv), the third term of the right-hand side o f (5.1) is\nestimated as\n−(1−2δ)/integraldisplay\nΩ|w|2(a(x)∂tΦβ,ε−∆Φβ,ε)Φ−2+2δ\nβ,εdx≤ −η/integraldisplay\nΩa(x)|w|2Ψλ−1dx\nwith someη >0. Moreover, by Proposition 2.8 (iii), the second term of the right-\nhand side of (5.1) is estimated as\n−2δ\n1−δ/integraldisplay\nΩ|∇w|2Φ−1+2δ\nβ,εdx≤ −η′/integraldisplay\nΩ|∇w|2Ψλdx (5.2)22 M. SOBAJIMA AND Y. WAKASUGI\nwith some η′>0. Moreover, we can drop the third term of the right-hand side,\nand we also have |∂tΦβ,ε|=|βΦβ+1,ε| ≤CΨ−β−1, which implies\n/vextendsingle/vextendsingle/vextendsinglew∂tw(∂tΦβ,ε)Φ−2+2δ\nβ,ε/vextendsingle/vextendsingle/vextendsingle≤C|w||∂tw|Ψλ−1.\nTherefore, from the above inequality with the Schwarz inequality, w e estimate the\nfourth term of the right-hand side of (5.1) as\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩw∂tw(∂tΦβ,ε)Φ−2+2δ\nβ,εdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/integraldisplay\nΩ|w||∂tw|Ψλ−1dx\n≤C/parenleftbigg/integraldisplay\nΩa(x)|w|2Ψλ−1dx/parenrightbigg1/2/parenleftbigg/integraldisplay\nΩa(x)−1|∂tw|2Ψλ−1dx/parenrightbigg1/2\n≤η1/integraldisplay\nΩa(x)|w|2Ψλ−1dx+C(η1)/integraldisplay\nΩ|∂tw|2Ψλdx\nfor anyη1>0. Similarly, the last term of the right-hand side of (5.1) is estimated\nas\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay\nΩwFΦ−1+2δ\nβ,εdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/integraldisplay\nΩ|w||F|Ψλdx\n≤/parenleftbigg/integraldisplay\nΩa(x)|w|2Ψλ−1dx/parenrightbigg1/2/parenleftbigg/integraldisplay\nΩa(x)−1|F|2Ψλ+1dx/parenrightbigg1/2\n≤η2/integraldisplay\nΩa(x)|w|2Ψλ−1dx+C/integraldisplay\nΩa(x)−1|F|2Ψλ+1dx(5.3)\nfor anyη2>0. Therefore, by applying (5.2)–(5.3) to (5.1), we have the desired\nestimate. /square\nLemma 5.4. Under the assumptions on Theorem 5.2, there exists t2≥1such that\nfor anyt0≥t2andt>0, we have\nd\ndtE1[w](t;t0,λ)≤ −/integraldisplay\nΩa(x)|∂tw(x,t)|2Ψ(x,t;t0)λ+1dx+C/integraldisplay\nΩ|∇w(x,t)|2Ψ(x,t;t0)λdx\n+C/integraldisplay\nΩa(x)−1|F(x,t)|2Ψ(x,t;t0)λ+1dx\nwith some constant C=C(N,α,δ,ε,λ,t 2)>0.WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 23\nProof.By the definition of E1[w](t;t0,λ) and the equation (3.1), we calculate\nd\ndtE1[w](t;t0,λ) = 2/integraldisplay\nΩ/parenleftbig\n∇∂tw·∇w+∂tw∂2\ntw/parenrightbig\nΨλ+1dx\n+(λ+1)/integraldisplay\nΩ/parenleftbig\n|∇w|2+|∂tw|2/parenrightbig\nΨλdx\n= 2/integraldisplay\nΩ∂tw/parenleftbig\n−∆w+∂2\ntw/parenrightbig\nΨλ+1dx\n−2(λ+1)/integraldisplay\nΩ∂tw(∇w·∇Ψ)Ψλdx\n+(λ+1)/integraldisplay\nΩ/parenleftbig\n|∇w|2+|∂tw|2/parenrightbig\nΨλdx\n=−2/integraldisplay\nΩa(x)|∂tw|2Ψλ+1dx+2/integraldisplay\nΩ∂twFΨλ+1dx\n−2(λ+1)/integraldisplay\nΩ∂tw(∇w·∇Ψ)Ψλdx\n+(λ+1)/integraldisplay\nΩ/parenleftbig\n|∇w|2+|∂tw|2/parenrightbig\nΨλdx. (5.4)\nHere, we note that the integration by parts in the second identity is justified, since\n∂tw∇wΨλ+1∈L1(Ω) for each t≥0. For the second term of the right-hand side of\n(5.4), we apply the Schwarz inequality to obtain\n|2∂twF| ≤a(x)\n4|∂tw|2+4a(x)−1|F|2. (5.5)\nNext, by the Schwarz inequality, the third term of the right-hand s ide of (5.4) is\nestimated as\n|−2(λ+1)∂tw(∇w·∇Ψ)| ≤a(x)\n4|∂tw|2Ψ+C|∇w|2|∇Ψ|2\na(x)Ψ\n≤a(x)\n4|∂tw|2Ψ+C|∇w|2,\nwhere we have also used\n|∇Ψ|2\na(x)Ψ≤|∇Aε(x)|2\na(x)Aε(x)≤2−α\nN−α+ε,\nwhich follows from (2.3). Moreover, for the last term of the right-h and side of\n(5.4), we note that Ψ−1≤t−1+α\n2−α\n0Aε(x)−α\n2−α≤Ct−1+α\n2−α\n0a(x)≤a(x)\n2(λ+1)holds for\nt0≥t2, provided that t2is sufficiently large. Thus, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(λ+1)/integraldisplay\nΩ|∂tw|2Ψλdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n2/integraldisplay\nΩa(x)|∂tw|2Ψλ+1dx. (5.6)\nFinally, applying (5.5)–(5.6) to (5.4), we have the desired estimate. /square\nNow we are in the position to prove Theorem 5.2.24 M. SOBAJIMA AND Y. WAKASUGI\nProof of Theorem 5.2. Lett2be the constant given in Lemma 5.4. For t0≥t2, by\nLemmas 5.3 and 5.4, we calculate\nd\ndtE[w](t;t0,λ,ν)≤/integraldisplay\nΩ/parenleftbig\n−νa(x)+CΨ−1/parenrightbig\n|∂tw|2Ψλ+1dx\n+(νC−η0)/integraldisplay\nΩ|∇w|2Ψλdx\n+C/integraldisplay\nΩa(x)−1|F|2Ψλ+1dx. (5.7)\nBy takingν >0 sufficiently small so that νC−η0<0. After that, taking t∗≥t2\nsufficiently large so that\n−νa(x)+CΨ−1≤ −νa(x)+Ct−1+α\n2−α\n0a(x)≤ −ν\n2a(x)\nholds fort0≥t∗. Therefore, integrating (5.7) over [0 ,t], we have\nE[w](t;t0,λ,ν)+η∗/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tw|2Ψλ+1dxdτ\n+η∗/integraldisplayt\n0/integraldisplay\nΩ|∇w|2Ψλdxdτ\n≤E[w](0;t0,λ,ν)+C/integraldisplayt\n0/integraldisplay\nΩa(x)−1|F|2Ψλ+1dxdτ\nwith some constant η∗>0. By the assumptions of the theorem, the right-hand\nside is bounded with respect to t>0. Thus, we complete the proof. /square\n5.2.Higher order energy estimates.\nDefinition 5.5. Letδ∈(0,1/2),ε∈(0,1),λ∈[0,(1−2δ)γε), whereγεis defined\nin(2.4), and letν >0. For an integer j≥1, we define\nE(j)\n1[w](t;t0,λ) :=/integraldisplay\nΩ/parenleftbig\n|∇w(x,t)|2+|∂tw(x,t)|2/parenrightbig\nΨ(x,t;t0)λ+1+2jdx,\nE(j)\n0[w](t;t0,λ) :=/integraldisplay\nΩ/parenleftbig\n2w(x,t)∂tw(x,t)+a(x)|w(x,t)|2/parenrightbig\nΨ(x,t;t0)λ+2jdx,\nE(j)[w](t;t0,λ,ν) :=νE(j)\n1[w](t;t0,λ)+E(j)\n0[w](t;t0,λ)\nfort≥0.\nWe remark that there exists t1>0 such that for any t0≥t1andt≥0, we have\nE(j)[w](t;t0,λ,ν)∼/integraldisplay\nΩ/bracketleftbig/parenleftbig\n|∇w|2+|∂tw|2/parenrightbig\nΨλ+1+2j+a(x)|w|2Ψλ+2j/bracketrightbig\ndx.\nThe main result of this subsection is the following energy estimates fo r higher\norder derivatives of the solution of the damped wave equation (3.1) .\nTheorem 5.6. Letk≥1be an integer. Assume that a(x)satisfies (1.2). Let\nδ∈(0,1/2),ε∈(0,1/2), andλ∈[0,(1−2δ)γε). Then, there exist constants\nν(j)=ν(j)(N,α,λ,j )>0andt(j)\n∗=t(j)\n∗(N,α,δ,ε,λ,ν(j),j)≥1forj= 1,...,kWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 25\nsuch that for any t0≥max1≤j≤kt(j)\n∗, the following holds: let m= (λ+1+2k)2−α\n2\nand assume F∈/intersectiontextk\nj=0Cj+1([0,∞);Hk−j,m(Ω))satisfies\n/integraldisplay\nΩa(x)−1|∂j\ntF(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞)\nforj= 0,1,...,k. Letwbe the solution of (3.1)in Theorem 3.4 with the initial\ndata(w0,w1)∈Hk+2,m(Ω)×Hk+1,m(Ω)satisfying the k-th order compatibility\ncondition in the sense of Theorem 3.4. Then, we have\nE(j)[∂j\ntw](t;t0,λ,ν(j))∈L∞(0,∞),\n/integraldisplay\nΩa(x)|∂j+1\ntw(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞)\nforj= 1,...,k.\nThe proof of Theorem 5.6 is based on an induction argument, which is s imilar\nto that of Lemma 4.4. The main part of the induction argument is the f ollowing\nlemma.\nLemma 5.7. Letj∈N. Assumea(x)satisfies (1.2). Letλ≥0andm= (λ+\n1+2j)(2−α)/2. Then, there exist constants ν(j)=ν(j)(N,α,λ,j )>0andt(j)\n∗=\nt(j)\n∗(N,α,λ,ν(j),j)≥1such that for any t0≥t(j)\n∗, the following holds: Assume F∈\nC1([0,∞);H0,m(Ω))and letwbe the solution of (3.1)with initial data (w0,w1)∈\n(H2,m(Ω)∩H1,m\n0(Ω))×H1,m\n0(Ω). If\n/integraldisplay\nΩa(x)−1|F(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞),\n/integraldisplay\nΩa(x)|w(x,t)|2Ψ(x,t;t0)λ−1+2jdx∈L1(0,∞)\nare satisfied, then\nE(j)[w](t;t0,λ)∈L∞(0,∞),\n/integraldisplay\nΩa(x)|∂tw(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞)\nhold.\nFor the proof of Lemma 5.7, we further prepare the following two lem mas.\nLemma 5.8. Under the assumptions of Lemma 5.7, we have\nd\ndtE(j)\n0[w](t;t0,λ)≤C/integraldisplay\nΩ|∂tw|2Ψλ+2jdx−/integraldisplay\nΩ|∇w(x,t)|2Ψ(x,t;t0)λ+2jdx\n+C/integraldisplay\nΩa(x)|w(x,t)|2Ψ(x,t;t0)λ−1+2jdx\n+C/integraldisplay\nΩa(x)−1|F(x,t)|2Ψ(x,t;t0)λ+1+2jdx\nwith some constant C >0.26 M. SOBAJIMA AND Y. WAKASUGI\nProof.We compute\nd\ndtE(j)\n0[w](t;t0,λ) = 2/integraldisplay\nΩ|∂tw|2Ψλ+2jdx+2/integraldisplay\nΩw/parenleftbig\n∂2\ntw+a(x)∂tw/parenrightbig\nΨλ+2jdx\n+2(λ+2j)/integraldisplay\nΩ/parenleftbig\n2w∂tw+a(x)|w|2/parenrightbig\nΨλ−1+2jdx\n= 2/integraldisplay\nΩ|∂tw|2Ψλ+2jdx+2/integraldisplay\nΩw(∆w+F)Ψλ+2jdx\n+2(λ+2j)/integraldisplay\nΩ/parenleftbig\n2w∂tw+a(x)|w|2/parenrightbig\nΨλ−1+2jdx\n= 2/integraldisplay\nΩ|∂tw|2Ψλ+2jdx−2/integraldisplay\nΩ|∇w|2Ψλ+2jdx\n−2(λ+2j)/integraldisplay\nΩw(∇w·∇Ψ)Ψλ−1+2jdx\n+2(λ+2j)/integraldisplay\nΩ/parenleftbig\n2w∂tw+a(x)|w|2/parenrightbig\nΨλ−1+2jdx\n+2/integraldisplay\nΩwFΨλ+2jdx.\nThe Schwarz inequality implies\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2(λ+2j)/integraldisplay\nΩw(∇w·∇Ψ)Ψλ−1+2jdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay\nΩ|∇w|2Ψλ+2jdx+C/integraldisplay\nΩa(x)|w|2|∇Ψ|2\na(x)ΨΨλ−1+2jdx\n≤/integraldisplay\nΩ|∇w|2Ψλ+2jdx+C/integraldisplay\nΩa(x)|w|2Ψλ−1+2jdx,\nwhere we have used ∇Ψ =∇Aε(x), Ψ(x,t;t0)≥Aε(x), and (2.3) in Lemma 2.1.\nSimilarly, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2(λ+2j)/integraldisplay\nΩ2w∂twΨλ−1+2jdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/integraldisplay\nΩ|∂tw|2Ψλ+2jdx+C/integraldisplay\nΩa(x)|w|2a(x)−1Ψλ−2+2jdx\n≤C/integraldisplay\nΩ|∂tw|2Ψλ+2jdx+C/integraldisplay\nΩa(x)|w|2Ψλ−1+2jdx,\nand\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay\nΩwFΨλ+2jdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/integraldisplay\nΩa(x)|w|2a(x)λ−1+2jdx+C/integraldisplay\nΩa(x)−1|F|2Ψλ+1+2jdx.\nThis completes the proof. /squareWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 27\nLemma 5.9. Under the assumptions of Lemma 5.7, there exists t2≥1such that\nfor anyt0≥t2andt>0, we have\nE(j)\n1[w](t;t0,λ)≤ −/integraldisplay\nΩa(x)|∂tw(x,t)|2Ψ(x,t;t0)λ+1+2jdx\n+C/integraldisplay\nΩ|∇w(x,t)|2Ψ(x,t;t0)λ+2jdx\n+C/integraldisplay\nΩa(x)−1|F(x,t)|2Ψ(x,t;t0)λ+1+2jdx\nwith some constant C=C(N,α,δ,ε,λ,j,t 2)>0.\nThe proof is completely the same as that of Lemma 5.4 by replacing λbyλ+2j.\nThus, we omit the detail.\nProof of Lemma 5.7. By Lemmas 5.8 and 5.9, taking ν(j)>0 sufficiently small,\nand then, taking t(j)\n∗≥t2sufficiently large depending on ν(j), we have\nd\ndtE(j)[w](t;t0,λ,ν(j)) =ν(j)d\ndtE(j)\n1[w](t;t0,λ)+d\ndtE(j)\n0[w](t;t0,λ)\n≤ −η1/integraldisplay\nΩa(x)|∂tw|2Ψλ+1+2jdx−η2/integraldisplay\nΩ|∇w|2Ψλ+2jdx\n+C/integraldisplay\nΩa(x)|w|2Ψλ−1+2jdx+C/integraldisplay\nΩa(x)−1|F|2Ψλ+1+2jdx\nfort0≥t(j)\n∗andt >0 with some constants η1,η2>0. By integrating the above\ninequality on [0 ,t] and using the assumptions, we conclude\nE(j)[w](t;t0,λ,ν(j))+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tw|2Ψλ+1+2jdxdτ+/integraldisplayt\n0/integraldisplay\nΩ|∇w|2Ψλ+2jdxdτ\n≤E(j)[w](0;t0,λ,ν(j))+C/integraldisplayt\n0/integraldisplay\nΩa(x)|w|2Ψλ−1+2jdxdτ\n+C/integraldisplayt\n0/integraldisplay\nΩa(x)−1|F|2Ψλ+1+2jdxdτ,\nand the proof is complete. /square\nProof of Theorem 5.6. By Theorem 5.2, there exist ν >0 andt∗≥1 such that\nE[w](t;t0,λ,ν)∈L∞(0,∞),\n/integraldisplay\nΩa(x)|∂tw(x,t)|2Ψ(x,t;t0)λ+1dx∈L1(0,∞) (5.8)\nhold fort0≥t∗. Now, thanks to the property (5.8), we apply Lemma 5.7 with\nj= 1 and the replacement of wandFby∂twand∂tF, respectively. Then, there\nexistν(1)>0 andt(1)\n∗≥1 such that\nE(1)[∂tw](t;t0,λ,ν(1))∈L∞(0,∞),\n/integraldisplay\nΩa(x)|∂2\ntw(x,t)|2Ψ(x,t;t0)λ+1+2dx∈L1(0,∞)28 M. SOBAJIMA AND Y. WAKASUGI\nhold fort0≥t(1)\n∗. The latter property allows us to apply again Lemma 5.7 with\nj= 2 and the replacement of wandFby∂2\ntwand∂2\ntF, respectively. Repeating\nthis argument until j=k, we reach the conclusion of Theorem 5.6. /square\n6.Proof of the asymptotic expansion\nIn this section, we give the estimates of the right-hand side of (3.9) and complete\nthe proof of Theorem 1.1.\nLetn∈Nbe fixed, and let δ∈(0,1/2),ε∈(0,1/2),λ∈[0,(1−2δ)γε), and\nt0≥1. Moreover, we define λj=λ−2α\n2−αjforj= 1,...,n+ 1, and we assume\nthatλn+1∈[0,(1−2δ)γε), that is,λ∈[2α\n2−α(n+1),(1−2δ)γε). Here, we note that\nthe assumption n+1<N−α\n2αensures that this interval is not empty, provided that\nδandεare sufficiently small. We also put ˜ m:= (λn+1+1+2n)2−α\n2.\nAs in (1.7), we assume that the initial data u0andu1satisfy\nu0∈Hs+1,m(Ω)∩Hs,m\n0(Ω), u1∈Hs,m\n0(Ω)\nwith sufficiently large sandm.\nNote that, in what follows, we retake the parameter t0≥1 suitably larger from\nline to line.\nStep 0: Estimates of V0:Wefirstgivetheestimatesof V0, whichisthesolution\nof (1.6). We apply Theorem 4.1 with k=n,v=V0,G= 0,v0=u0+a(x)−1u1.\nNoting Lemma 3.6 and taking sandmsufficiently large, we have\n∂j\ntV0(x,0)∈D(L)∩H0,(λ+2j)2−α\n2−α\n2(Ω),\n∇∂j\ntV0(x,0)∈H0,(λ+1+2j)2−α\n2(Ω)\nforj= 0,1,...,n, where∂j\ntV0(x,0) is defined by the right-hand side of (3.3) with\nv0=u0+a(x)−1u1andG= 0. Therefore, the assumptions of Theorem 4.1 are\nfulfilled, and we have\n/integraldisplay\nΩa(x)|∂j\ntV0(x,t)|2Ψ(x,t;t0)λ+2jdx∈L∞(0,∞), (6.1)\n/integraldisplay\nΩa(x)|∂j+1\ntV0(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞) (6.2)\nforj= 0,1,...,n. In particular, (6.1) with j= 0 implies the L2-estimate of V0:\n/ba∇dblV0(t)/ba∇dblL2(Ω)≤C(1+t)−λ\n2+α\n2(2−α).\nStep 1: Estimates of V1:Next, we consider the estimate for V1, which is\nthe solution of (1.9) with j= 1. We apply Theorem 4.1 with k=n,v=V1,\nv0=−a(x)−2u1(x),G=−∂tV0, and the replacement of λbyλ1=λ−2α\n2−α.\nSimilarly as before, If we take sandmsufficiently large, then we have\n∂j\ntV1(x,0)∈D(L)∩H0,(λ1+2j)2−α\n2−α\n2(Ω),\n∇∂j\ntV1(x,0)∈H0,(λ1+1+2j)2−α\n2(Ω)WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 29\nforj= 0,1,...,n, where∂j\ntV1(x,0) is defined by the right-hand side of (3.3) with\nv0=−(−a(x))−2andG=−∂tV0. Moreover, from (6.2), one obtains\n/integraldisplay\nΩa(x)−1|∂j\nt(−∂tV0)(x,t)|2Ψ(x,t;t0)λ1+1+2jdx\n≤/integraldisplay\nΩa(x)|∂j+1\ntV0(x,t)|2Ψ(x,t;t0)λ+1+2jdx∈L1(0,∞)\nforj= 0,1,...,n(this is the reason why we define λ1=λ−2α\n2−α). Therefore, the\nassumptions of Theorem 4.1 are fulfilled, and we deduce/integraldisplay\nΩa(x)|∂j\ntV1(x,t)|2Ψ(x,t;t0)λ1+2jdx∈L∞(0,∞), (6.3)\n/integraldisplay\nΩa(x)|∂j+1\ntV1(x,t)|2Ψ(x,t;t0)λ1+1+2jdx∈L1(0,∞)\nforj= 0,1,...,n. In particular, (6.3) with j= 1 implies the L2-estimate of ∂tV1:\n/ba∇dbl∂tV1/ba∇dblL2≤C(1+t)−λ1\n2−1+α\n2(2−α)≤C(1+t)−λ\n2−2(1−α)\n2−α+α\n2(2−α).\nStepn: Estimates of Vn:Continuing this argument until j=n, we can esti-\nmateVn. Indeed, weapplyTheorem4.1with k=n,v=Vn,v0=−(−a(x))−n−1u1(x),\nG=−∂tVn−1, and the replacement of λbyλn=λ−2α\n2−αn. Similarly as before, if\nwe takesandmsufficiently large, then we have\n∂j\ntVn(x,0)∈D(L)∩H0,(λn+2j)2−α\n2−α\n2(Ω),\n∇∂j\ntVn(x,0)∈H0,(λn+1+2j)2−α\n2(Ω)\nforj= 0,1,...,n, where∂j\ntVn(x,0) is defined by the right-hand side of (3.3) with\nv0=−(−a(x))−n−1andG=−∂tVn−1. Furthermore, by the ( n−1)-th step, we\nhave the estimate of inhomogeneous term −∂tVn−1:\n/integraldisplay\nΩa(x)−1|∂j\nt(−∂tVn−1)(x,t)|2Ψ(x,t;t0)λn+1+2jdx\n≤/integraldisplay\nΩa(x)|∂j+1\ntVn−1(x,t)|2Ψ(x,t;t0)λn−1+1+2jdx∈L1(0,∞).\nThus, the assumptions of Theorem 4.1 are fulfilled, and we have/integraldisplay\nΩa(x)|∂j\ntVn(x,t)|2Ψ(x,t;t0)λn+2jdx∈L∞(0,∞), (6.4)\n/integraldisplay\nΩa(x)|∂j+1\ntVn(x,t)|2Ψ(x,t;t0)λn+1+2jdx∈L1(0,∞) (6.5)\nforj= 0,1,...,n. In particular, (6.4) with j=nimplies the L2-estimate of ∂n\ntVn:\n/ba∇dbl∂n\ntVn(t)/ba∇dblL2(Ω)≤C(1+t)−λn\n2−n+α\n2(2−α)=C(1+t)−λ\n2−2n(1−α)\n2−α+α\n2(2−α).\nMoreover, we shall check that Vnhas the regularity\n∂tVn∈n/intersectiondisplay\nj=0Cj+1([0,∞);Hn−j,˜m(Ω)) (6.6)\nwhich will be required in the next step. This follows if both the initial valu e\n∂tVn(x,0)andtheinhomogeneousterm −∂tVn−1haveenoughregularityandbelong\nto a suitable weighted Sobolev space. Note that ∂tVn(x,0) can be computed from30 M. SOBAJIMA AND Y. WAKASUGI\n(3.3), and the regularity of −∂tVn−1is obtained by a similar iteration argument as\nSteps 0,1,...,n. Thus, we can obtain (6.6) if sandmare sufficiently large.\nFinal step: Estimates of Un+1:Finally, we estimate ∂n+1\ntUn+1defined by\n(1.10) with j=n. First, to check the compatibility condition on Un+1, we prepare\nthe following lemma:\nLemma 6.1. LetU(0)\nn+1= 0,U(1)\nn+1=−(−a(x))−n−1u1(x), and letU(p)\nn+1forp=\n2,...,n+1be successively defined by\nU(p)\nn+1(x) = ∆U(p−2)\nn+1−a(x)U(p−1)\nn+1−∂p−1\ntVn(x,0).\nThen, we have, for p= 2,...,n+1,\nU(p)\nn+1(x) = (−a(x))−(n−p+2)u1(x)−∂tVn−p+2(x,0)−···−∂p−1\ntVn(x,0).\nProof.Whenp= 2, the conclusion is obvious. Suppose that the conclusion is true\nup top−1, and consider the case p. Using the assumption of the induction, we\nhave\nU(p)\nn+1(x) = ∆U(p−2)\nn+1−a(x)U(p−1)\nn+1−∂p−1\ntVn(x,0)\n= ∆/parenleftBig\n(−a(x))−(n−p+4)u1(x)−∂tVn−p+4(x,0)−···−∂p−3\ntVn(x,0)/parenrightBig\n−a(x)/parenleftBig\n(−a(x))−(n−p+3)u1(x)−∂tVn−p+3(x,0)−···−∂p−2\ntVn(x,0)/parenrightBig\n−∂p−1\ntVn(x,0).\nRecallinga(x)∂tVj−∆Vj=−∂tVj−1andVj(x,0) =−(−a(x))−j−1u1(x), we see\nthat the right-hand side becomes\n(−a(x))−(n−p+2)u1(x)−∂tVn−p+2(x,0)−···−∂p−1\ntVn(x,0),\nwhich completes the proof. /square\nSinceu1∈Hs,m\n0(Ω) with sufficiently large sandm, we easily see U(1)\nn+1=\n−(−a(x))−n−1u1∈Hn+1,˜m(Ω) and\n(−a(x))−(n−p+2)u1∈H2,˜m(Ω)∩H1,˜m\n0(Ω) (p= 0,...,n),\n(−a(x))−1u1∈H1,˜m\n0(Ω).\nMoreover, from (3.3) and a similar induction argument, we can see th at the func-\ntions∂p\ntVj(x,0) satisfy the following conditions, provided that sandmare suffi-\nciently large: for Vj(j= 2,...,n),\n∂p\ntVj(x,0)∈H2,˜m(Ω)∩H1,˜m\n0(Ω) (p= 1,...,j−1),\n∂j\ntVj(x,0)∈H1,˜m\n0(Ω);\nforV1,\n∂tV1(x,0)∈H1,˜m\n0(Ω).\nTherefore, by applying Lemma 6.1, we see that the data U(p)\nn+1forp= 0,1,...,n+1\nsatisfy then-th order compatibility condition\n(U(p)\nn+1,U(p+1)\nn+1)∈(H2,˜m(Ω)∩H1,˜m\n0(Ω))×H1,˜m\n0(Ω)WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 31\nforp= 0,...,n. Combining this with the fact (6.6), we can apply Theorem 3.4\nwithk=n,m= ˜mto obtain the regularity of Un+1and the weighted energy of\n∂n\ntUn+1is well-defined. Moreover, it follows from (6.5) that\n/integraldisplay\nΩa(x)−1|∂j\nt(−∂tVn(x,t))|2Ψ(x,t;t0)λn+1+1+2jdx∈L1(0,∞)\nforj= 0,1,...,n. Hence, we can apply Theorems 5.2 and 5.6, and there exist\nν(n)>0 andt(n)\n∗≥1 such that for any t0≥t(n)\n∗we have\nE(n)[∂n\ntUn+1](t;t0,λn+1,ν(n))∈L∞(0,∞).\nIn particular, the above bound yields/integraldisplay\nΩ|∂n+1\ntUn+1(x,t)|2dx≤C(1+t)−λn+1−1−2n.\nNamely, we conclude/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu(t)−n/summationdisplay\nj=0∂j\ntVj(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Ω)=/ba∇dbl∂n+1\ntUn+1(t)/ba∇dblL2(Ω)\n≤C(1+t)−λ\n2−(2n+1)(1−α)\n2−α+α\n2(2−α),\nwhich completes the proof of Theorem 1.1.\nAcknowledgements\nThis work was supported by JSPS KAKENHI Grant Numbers JP16K17 625,\nJP18H01132, 18K13445, JP20K14346.\nReferences\n[1]R. Beals, R. Wong , Special functions, A graduate text. Cambridge Studies in A dvanced\nMathematics 126, Cambridge University Press, Cambridge, 2 010.\n[2]T. Cazenave, A. 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Math. 124(2000), 415–433.\nDepartment of Mathematics, Faculty of Science and Technolo gy, Tokyo University\nof Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan\nEmail address :msobajima1984@gmail.com\nLaboratory of Mathematics, Graduate School of Engineering , Hiroshima University,\nHigashi-Hiroshima, 739-8527, Japan\nEmail address :wakasugi@hiroshima-u.ac.jp"    },    {        "title": "1207.6484v1.The_effect_of_non_uniform_damping_on_flutter_in_axial_flow_and_energy_harvesting_strategies.pdf",        "content": "The e\u000bect of non-uniform damping on \rutter in axial \row and energy harvesting\nstrategies\nKiran Singh,1,\u0003S\u0013 ebastien Michelin,1,yand Emmanuel de Langre1,z\n1Department of Mechanics, LadHyX,\n\u0013Ecole Polytechnique, 91128, Palaiseau, France\nThe problem of energy harvesting from \rutter instabilities in \rexible slender structures in axial\n\rows is considered. In a recent study, we used a reduced order theoretical model of such a system to\ndemonstrate the feasibility for harvesting energy from these structures. Following this preliminary\nstudy, we now consider a continuous \ruid-structure system. Energy harvesting is modelled as\nstrain-based damping and the slender structure under investigation lies in a moderate \ruid loading\nrange, for which the \rexible structure may be destabilised by damping. The key goal of this work\nis to analyse the e\u000bect of damping distribution and intensity on the amount of energy harvested\nby the system. The numerical results indeed suggest that non-uniform damping distributions may\nsigni\fcantly improve the power harvesting capacity of the system. For low damping levels, clustered\ndampers at the position of peak curvature are shown to be optimal. Conversely for higher damping,\nharvesters distributed over the whole structure are more e\u000bective.\nI. INTRODUCTION\nIncreasing energy demands motivate the interest in energy harvesting concepts, where the idea is to harness the\nenergy of naturally occurring phenomena. At the scale of kilowatts, concepts include energy harvesting from tidal\ncurrents [1] and ocean waves [2]. At the lower end of the power spectrum, concepts based on photo/thermovoltaics\nand magneto/piezoelectrics show the scope for powering sensors and mobile electronic devices [3]; these include energy\nscavenging from ambient vibrations in structures such as buildings and bridges and oscillatory motion of wheels in\nautomobiles or turbines in engines [4, 5]. Energy harvesting from \ruid-structure interactions (FSI) includes concepts\nsuch as vortex induced vibrations (VIV) of blu\u000b bodies in cross-\row [6, 7], resonant vibrations induced in aerofoils\nmounted on elastic supports [8, 9], \rutter of \rexible plates [10{12], and combinations thereof, as for the coupled\nVIV-\rutter energy harvester examined by [13]. In this work we focus on harvesting energy from \rutter instabilities\nof slender structures in axial \row.\nThe classical description of \rutter instabilities is self-sustained oscillations that arise due to the unstable coupling of\n\ruid dynamic pressure and structural bending modes, where for undamped structures the critical speed at \rutter onset\ndepends on \ruid as well as structural properties [14]. Flutter instabilities are observed in a diversity of con\fgurations,\nwhich may be broadly classi\fed as internal or external \rows [14, 15]. Internal \row instabilities are observed in \rexible\npipes and channels and are invariably motivated by biological phenomena such as \row induced oscillations in airways\nand veins [16, 17]. The pipe-conveying \ruid is a canonical problem that yields deep insights into FSI; in particular\n[18] examined the relationship between local and global instabilities and showed that locally stable con\fgurations may\nbecome unstable due to wave re\rections at \fnite boundaries; this destabilising mechanism was recently predicted for\ncompliant channels as well [19]. External \row based instabilities include \rapping \rags [20{22] and panels [23] in a\nsteady \row. For all aspect ratios, the plate can become unstable due to \ruid loading, de\fned as the ratio of \ruid and\nstructure inertia, and may be represented as a nondimensional length [24] or mass ratio [25]. [26] and more recently\n[27] examined the occurrence of instabilities in slender structures. They used experimental and theoretical techniques\nto examine the role of inviscid and viscous drag on static and dynamic instabilities that arise in \rexible cylinders in\naxial \row. For a recent review on the \rutter dynamics of \rexible bodies in external \row, the reader is referred to [28]\nand references therein.\nIn this work, we analyse the scope for harvesting energy in slender elastic cantilevered structures that \rutter in\na steady \row. From the point of view of the \ruid-solid system, energy harvesters are essentially an energy sink,\ntherefore for the theoretical approach adopted here they are modelled as internal damping in the structure. As a \frst\nstep, we examined the feasibility of this concept using a nonlinear model of a reduced-order system, consisting of a\nslender cylinder pair connected by discrete springs and dampers [29]. It was shown that the optimal con\fguration for\n\u0003Electronic address: kiran.singh@cantab.net; Present address:OCCAM, The Mathematical Institute, 24-29 St Giles, OX13LB, Oxford\nyElectronic address: sebastien.michelin@ladhyx.polytechnique.fr\nzElectronic address: delangre@ladhyx.polytechnique.frarXiv:1207.6484v1  [physics.flu-dyn]  27 Jul 20122\nFIG. 1: Slender cantilevered structure placed in a steady axial \row of velocity U1exand \fxed in O. The instantaneous\ndeformation of the inextensible beam is measured by the tangent angle \u0012(s;t) with respect to exof the local tangent \u001c.\nthis two-degree-of-freedom system is one with energy harvesters positioned away from the instability source; such a\ncon\fguration maintains self-sustained \rapping in the presence of structural damping.\nIn this work we generalise this approach for a continuous system, and seek to maximise energy harvesting through\ncarefully-tailored distributions of structural damping. Conventionally, structural instabilities are stabilised by damping\n[16, 30] . However, damping may be destabilising under moderate to heavy \ruid loading conditions, as observed in\nin\fnite plates [31], \rags [11] and \ruid-conveying pipes [32]. [29] show that energy harvesting requires the presence\nof two traditionally competitive elements { \rutter oscillations and damping. A con\fguration for which \rutter is\ndestabilised by damping is especially interesting from an energy harvesting perspective. This idea has been exploited\nfor piezoelectric based energy harvesting from \rapping \rags: [11] used linear theory to show the gain in conversion\ne\u000eciency for \rags with high \ruid loading destabilised by piezoelectric based damping. In this work, we explore this\nidea using a general model for structural damping (equivalently energy harvesting) and a nonlinear model of the \ruid\nstructure interaction of a slender structure in a mean \row.\nIt is worth noting that existing insights on damping are generally based on the assumption of a constant distribution\nof damping in the structure [see for example 10], whilst little is known about how nonuniform damping distributions\na\u000bect the dynamics. In this work we seek clarity on the role of damping distribution on the \rutter response of slender\nstructures. The speci\fc motivation is to identify physical mechanisms that maximise this dissipated (i.e. harvested)\npower.\nThis paper is organised as follows: in Section II, the model used for the dynamics of slender structures with\nnon-uniform damping in an axial \row is presented. In Section III, the case of uniform damping is considered as a\nreference con\fguration and the role of \ruid loading on destabilisation by damping is discussed. Computations are\nthen performed for a neutrally buoyant slender cylinder with moderate \ruid loading, and the role of damping on the\nharvested power and \rutter response is examined. Section IV investigates non-uniform damping distributions and\nseeks optimals on two di\u000berent families of damping functions, either distributed over the whole structure or focused\non a particular region. In Section V, the impact of damping distribution on the \rutter dynamics is investigated further\nto understand the fundamental di\u000berence between optimal con\fgurations at low and intermediate damping.\nII. FLUID-SOLID MODEL\nWe consider a cantilevered (clamped-free, \fxed at O) slender structure of length Lwith crosswise dimension D,\ndensity\u001as, sti\u000bnessB, and nonuniform structural damping B\u0003(s). The slender solid is immersed in a stream of \ruid of\ndensity\u001amoving at mean speed U1, and the solid motion is con\fned to the ( ex;ey) plane (Figure 1). The equations\nof motion are nondimensionalised by the characteristic system scales: \u001a; L;U1.\nA. Nonlinear beam model\nThe \rexible structure is modelled as an inextensible Euler-Bernoulli beam, where r(s) is the position vector in the\n\fxed coordinate system ( ex;ey) andsis the curvilinear coordinate. At each point along the beam, the orientation\n\u0012(s;t) is de\fned as the angle of the tangent vector \u001c(s) with the horizontal; n(s) is the local normal. The nonlinear3\nequation of motion for the beam subjected to a \ruid force, f, is:\n1\nM\u0003@2r\n@t2=@\n@s\u001a\n\u0017\u001c\u00001\nM\u0003U\u00032\u0014@2\u0012\n@s2+@\n@s\u0012\n\u0018(s)@2\u0012\n@s@t\u0013\u0015\nn\u001b\n+f; (1)\nwhere the internal tension, \u0017(s;t), is essentially a Lagrange multiplier to satisfy the inextensibility condition @r=@s=\u001c\nand\u0018(s) =U1B\u0003(s)=(BL) is the non-dimensional damping distribution. A Kelvin-Voigt damping model is considered\nhere, generalising previous contributions [e.g. 10, 14, 32, 33] to nonuniform damping distributions.\nThe clamped-free boundary conditions must also be satis\fed, namely at the \fxed end ( s= 0):\n\u0012= 0; r= 0; (2)\nand at the free end ( s= 1):\n@\u0012\n@s+\u0018@2\u0012\n@s@t= 0;@2\u0012\n@s2+@\n@s\u0012\n\u0018@2\u0012\n@s@t\u0013\n= 0; \u0017 = 0: (3)\nConsistently with prior work [21, 25], the nondimensional mass ratio ( M\u0003) and \row speed ( U\u0003) are de\fned as:\nM\u0003=\u001aDL\n\u001asA; U\u0003=U1L\u0012\u001asA\nB\u00131=2\n; (4)\nwithAthe cross-sectional area of the structure.\nB. Fluid dynamic model\nIn the limit of slender structures ( D\u001cL), and for purely potential \row upstream of the structure's trailing edge,\nLighthill's large amplitude elongated-body theory [34] provides a leading order expression for the `reactive' force fi\napplied by the \row on the \rapping body, associated with the local transverse motion of each cross-section\nfi=\u0000ma\nM\u0003\u0012@(unn)\n@t\u0000@(unu\u001cn)\n@s+1\n2@(u2\nn\u001c)\n@s\u0013\n; (5)\nwhereu\u001c\u001c+unn=@r=@t\u0000exis the solid's local velocity relative to the incoming \row, and ma=Ma=\u001asA, where\nMais the dimensional added mass per unit length of the cross-section. [35] showed that Lighthill's theory compares\nwell with Reynolds-averaged Navier{Stokes (RANS) simulations to compute the forces on a swimming \fsh during\ntransient manoeuvres. This reactive force does not include any \row separation associated with the transverse motion\nof each cross-section, and as emphasised in [35], for freely \rapping bodies, an additional contribution for the \ruid\nforce must be included to account for such dissipative e\u000bects. Here, an empirical `resistive' force model is therefore\nadded following [36] and [33]:\nfv=\u00001=2Cdunjunjn; (6)\nwhereCDis the empirical drag coe\u000ecient, and CD= 1 is used in the following for circular cross-sections [see 29, for\na discussion of the impact of this coe\u000ecient on the \rapping dynamics]. [37] tuned the drag coe\u000ecients to show good\nagreement with direct numerical simulations.\nThus the \ruid force, f, in Eq. (1) is modelled as the sum of the reactive ( fi) and resistive ( fv) components. In the\nremainder of the paper we assume a neutrally buoyant circular cylinder, therefore \u001a=\u001asandA=\u0019D2=4. Unless\notherwise stated, we assume D=L = 0:1.\nNote that the \ruid force description is purely local here, and does not explicitly account for wake e\u000bects. When the\nslender body assumption is not veri\fed, and in particular, in the case of two-dimensional plates, an explicit description\nbecomes necessary [see for example 20, 21, 38].\nC. Energy harvester model\nAs noted earlier, energy harvesting is represented as a strain-based damping \u0018(s). We focus here on the mean\nnondimensional harvested power:\nP=P\n\u001aDLU31=1\nM\u0003U\u00032Z1\n0\u0018(s)\n_\u00142\u000b\nds; (7)4\n(a)\n (b)\nFIG. 2: (a) Maximum de\rection ymax(solid) and orientation \u0012max(dashed) at the free end as a function of the non-dimensional\n\row velocity U\u0003forM\u0003= 12:7. (b) Snapshots of the beam response for U\u0003= 10, 13 and 19 (from top to bottom).\nwherePis the mean dimensional harvested power, _ \u0014is the time derivative of the local curvature \u0014andh\u0001iis the time\naverage taken over a period Tof the limit-cycle oscillation. Pcan also be understood as the e\u000eciency of the system.\nAs discussed in Section I, the presence of structural damping can a\u000bect the \rutter response. The intensity and\ndistribution of damping are characterised by\n\u00180=Z1\n0\u0018(s)ds;and ~\u0018(s) =\u0018(s)=\u00180; (8)\nEquation (8) allows us to independently evaluate the impact on the system response of (i) the amount of damping \u00180\nand (ii) its spatial distribution.\nD. Numerical solution\nEquation (1) is solved numerically together with boundary conditions (2){(3) using an iterative second-order implicit\ntime-stepping scheme [39], and spatial derivatives are computed using Chebyshev collocation [40]. Conservation of\nenergy is ensured by verifying that _E=Wf\u0000Q;where\nE=1\n2Z1\n0 \nj_rj2\nM\u0003+\u00142\nM\u0003U\u00032!\nds; (9)\nQ=1\nM\u0003U\u00032Z1\n0\u0018(s) _\u00142ds; Wf=Z1\n0f\u0001_rds; (10)\nare respectively the mechanical energy of the system, the dissipated power and the rate of work of the \ruid forces,\nand is classically obtained by projecting the equation of motion (1) on the solid velocity _rand by integrating over\nthe entire beam. Also note from (7), P=hQi. In the numerical implementation the beam and the \row are initially\nat rest; the \row speed is ramped up to its steady state value and a small perturbation is applied to the vertical \row.\nE. Nonlinear response of an undamped beam\nPrior to analysing the energy harvesting properties of the system, we \frst examine the undamped \rutter response\nof the structure. In Figure 2, we plot the system response for increasing nondimensional \row speed, for a circular5\nFIG. 3: Variation of the critical \rutter speed, Ucfwith damping \u00180forM\u0003= 2:5 (dotted), 5 :1 (dash-dotted), 8 :5 (dashed), 12 :7\n(solid) and 20 (thick solid) obtained from linear stability analysis for uniform damping. Results of nonlinear computations for\nM\u0003= 12:7 are also presented (squares).\ncylinder (M\u0003\u001912:7). The critical \row speed at which \rutter ensues is veri\fed from linear stability analysis and\nis con\frmed with [26]. Consistent with \rutter in plates [33], we note that this instability is a supercritical Hopf\nbifurcation with \row speed. The jumps in the bifurcation curve correspond to the mode switching reported in [41],\nthis may also be discerned from the snapshots of the beam at three di\u000berent \row speeds. For the energy harvesting\ncomputations performed in the rest of the paper we set the \row speed at U\u0003= 13, corresponding to well-developed\noscillations and moderate de\rections for the undamped con\fguration.\nIII. UNIFORM DAMPING\nHere, a uniform damping distribution \u0018(s) =\u00180is considered. The dependence of critical \rutter speed on the\ndamping intensity \u00180is \frst analysed for varying values of \ruid loading M\u0003. Based on these results, a moderate \ruid\nloading is selected to study the power harvesting capacity of the con\fguration.\nA. Destabilisation by damping: impact on critical \rutter speed\nAs noted in Section I, damping can destabilise \rexible structures at su\u000eciently high \ruid loading. [32] shows that\ndestabilisation of long pipes with a high mass ratio is associated with a drop in the critical \rutter speed. Based on\nthis work, we analyse the linear stability of the system and in Figure 3 compare the variation of the critical \rutter\nspeed,Ucf, with damping, \u00180, at di\u000berent values of \ruid loading, M\u0003. For lightly loaded structures ( M\u0003<5:5)Ucf\nmonotonically increases with damping; at higher values ( M\u0003\u00158:5),Ucfdecreases with damping until a speci\fc value\n(\u0018m) above which Ucfincreases rapidly; note that \u0018mincreases with M\u0003.\nFor the remainder of the energy harvesting analysis, we settle on a value of M\u0003= 12:7 (corresponding to D=L =\n0:1): this choice allows us to analyse the scope for harvesting energy from a system at moderate \ruid loading with\ndestabilisation by damping.\nB. Harvesting power with constant damping\nFigure 4(a) presents the evolution of harvested power, P, with damping for 10\u00003<\u00180<10. Equation (7) becomes\nP=1\nM\u0003U\u00032\u00180kKk 1; (11)6\nFIG. 4: (a) Evolution of the harvested power P(solid) and curvature norm k\u0016Kk1(dashed) with the uniform damping intensity\n\u00180. (b) Distribution of curvature-change \u0016K(s) along the beam for \u00180= 0, 0:022, 0:22 and 0:47 (from left to right and top to\nbottom, respectively). ( M\u0003= 12:7; U\u0003= 13).\nwhereK(s) =h_\u00142iandkKk 1=R1\n0Kdsis theL1-norm ofK(s). The evolution of the \rutter response with damping\nis examined by plotting the rescaled curvature term, \u0016K=K=kK0k1(whereK0(s) is the value ofKfor\u00180= 0) for\nincreasing\u00180(Figure 4b). At small \u00180, the response is virtually no di\u000berent from the undamped case, but for \u00180>0:1\na perceptible change is observed. First, the curvature is redistributed along the entire beam, and as the damping is\nincreased further, the response of the beam is damped out globally. This can also be seen from the variation of kKk 1\nwith\u00180in Figure 4(a). Notably from (11) it is kKk 1that directly impacts the harvested power.\nAs a result, for small damping, the change in system response is virtually imperceptible from the undamped case,\nand power simply scales linearly with \u00180. However, damping has a strong e\u000bect on the \rutter response at larger \u00180: it\nreduces the amplitude of curvature-change signi\fcantly and causes a sharp reduction in P. The strategy to optimally\nharvest power is to \fnd the upper bound on \u00180below whichkKk 1can be maintained close to (or ideally enhanced\nabove) its undamped value.\nIV. NONUNIFORM DAMPING\nThe results for constant damping suggest that as long as kKk 1is maintained at undamped levels, the harvested\npower increases linearly with \u00180. Figure 4(b) clearly shows that K(s) varies signi\fcantly along the length of the beam.\nConcentrating harvesters around the zone of peak curvature could therefore enhance the harvested power. This idea\nis tested in this section on a reduced functional space for the damping distribution \u0018(s). Two families of damping\nfunctions are considered, namely (a) dispersed and (b) focused damping distributions. Optimisation of the harvested\npower is performed within each family, in anticipation of insights into the global optimal.\nA. Dispersed harvester distribution\nIn this section, we are interested in simple non-homogeneous distributions of damping of the form:\n\u0018(s) =\u00180\u0000\n1 +\u00181(s\u00001=2)\u0001\n(12)\ncharacterised by the total damping, 10\u00003< \u00180<10, and slope,\u00002\u0014\u00181\u00142. This function family corresponds to\na dispersed distribution, where the damping is signi\fcant over the entire length of the structure. Figure 5(a) shows\nthe variation of rescaled power, \u0016P=P=Pmax\nhwith (\u00180;\u00181), wherePmax\nhis the maximum harvested power for constant\ndamping (Section III). Figure 5 shows that for all \u00180, the optimal distribution corresponds to \u00181= 2, when damping\nis distributed increasingly from \fxed to free end, and that this linear distribution of damping leads to an increase7\nFIG. 5: Linear distribution (12): (a) rescaled power contours P=Pmax\nhfor varying \u00180and\u00181. (b) Rescaled harvested power\nP=Pmax\nhand (c)k\u0016Kk1as a function of the damping intensity \u00180for linearly decreasing ( \u00181=\u00002, dashed), constant ( \u00181= 0,\ndotted) and linearly increasing ( \u00181= 2, solid) damping distributions. ( M\u0003= 12:7; U\u0003= 13).\nof the maximum harvested power by 50% compared to the uniform distribution ( \u00181= 0). Through non-uniform but\nsimple distributions of damping on the structure, it is indeed possible to enhance the \rutter response above that of\nthe undamped con\fguration (Figure 5c).\nB. Focused harvester distribution\nThe results from the reduced order analysis [29] suggest that the optimal distribution ought to be localised at\nspeci\fc points on the beam. The chief drawback of the two parameter functions examined in the previous section is\nthe inability to consider localised distributions of damping in speci\fc regions. To consider such peaked distributions,\nwe now turn to the following three-parameter family of Gaussian damping distribution:\n\u0018(s) =\u00180\u0018g\nk\u0018gk1; \u0018g= e\u0000\u000b(s\u0000so)2; (13)\nwhere\u00180is the total damping in the system, sois the centre of the distribution and 1 =\u000bis a measure of its spread in\ns. For increasing \u000benergy harvesters are increasingly concentrated around so.\nFigure 6 presents the rescaled power P=Pmax\nhin the (\u000b;so)-space for small ( \u00180= 0:01) and moderate damping\n(\u00180= 0:47). For small damping, one sees that power is optimally harvested for dampers focused at so= 0:8, the\nposition of the maximum of K0along the beam (Figure 4b). This is quite distinct from the high damping optimal,\nwhich corresponds to a dispersed distribution ( \u000b= 2:7; so= 1). Of particular note is that for all \u00180, a uniform\ndistribution ( \u000b= 0) is preferred over a concentration of harvesters at so<0:5.\nWe next examine a wider range of damping, 10\u00003<\u00180<50, and plot the optimal values of ( \u000b;so), in Figure 7(a).\nFor moderate to large damping, \u00180>0:3, a dispersed distribution with peak at so= 1 is optimal. Conversely for\n\u00180<0:02, harvesters concentrated at so\u00190:8 (position of peak K) is the optimal con\fguration. Note that for these\ncomputations we set 0 <\u000b< 500, and at these values of \u00180the upper bound is reached; more detailed insights and\ncomputations on the small damping regime are presented in Section V. Figure 7(b) shows the evolution with \u00180of\nthe optimal normalised distribution and illustrates the transition from focused to dispersed pro\fles when the total\ndamping is increased.8\n(a)\n (b)\nFIG. 6: Gaussian distribution (13): Maps of rescaled power P=Pmax\nhfor varying \u000bandsoat (a)\u00180= 0:01 and (b) \u00180= 0:47.\n(M\u0003= 12:7; U\u0003= 13).\n(a)\n (b)\nFIG. 7: Gaussian function optimal con\fguration: (a) Spread parameter, \u000b, (open circles) and centre location, so, (\flled\nboxes) corresponding to the optimal Gaussian distribution for a given \u00180. (b) Rescaled damping distribution at discrete\n\u00180= 0:004;0:025;0:25;0:47 (solid, dashed, dashed-dot and dotted curves, respectively, indicated on (a) with vertical lines of\ncorresponding description); superimposed is the linear optimal distribution ( \u00181= 2; thick solid curve). ( M\u0003= 12:7; U\u0003= 13).\nIt is possible to determine the optimal harvested power for each value of \u00180; in Figure 8 these optimals are compared\nwith the results for uniform and linear distributions. Strikingly, the peak power values are coincident for linear\nand Gaussian distributions, which is consistent with the similarity in distributions (Figure 7(b)). Because of the\nfundamentally di\u000berent structures of the distribution used, this result suggests that the optimal obtained with such\nsimple functions represents a good approximation of the absolute optimal. Departing from this maximum of harvested\npower, in particular at small damping we see that concentrated harvesting is superior to the optimal linear distribution.\nThese results con\frm that nonuniform damping distributions can be advantageously employed to enhance the power\nharvesting capacity. Focused damping distributions are optimal for small damping, whereas dispersed distributions\nare preferential at higher damping. The following section investigates in more details this transition by considering\nthe impact of damping on the nonlinear dynamical response of the system.9\nFIG. 8: Optimal power values corresponding to Gaussian optimal con\fguration in Figure 7 (squares) compared to the optimal\npower obtained for linear distribution ( \u00181= 2; thick curve) and constant distribution ( \u00181= 0; dotted curve).\nV. DISCUSSION: IMPACT OF LOCALISED DAMPING\nThe above computations suggest a very di\u000berent impact of damping on the dynamics of the structure depending\non the magnitude of \u00180, thereby leading to quite di\u000berent optimal strategies to maximise the harvested power. In this\nsection, we \frst consider the limit of asymptotically small damping before studying the e\u000bect of focused dampers on\nthe body's dynamics.\nA. Optimal distribution for asymptotically small damping\nFor asymptotically small damping ( k\u0018k1\u001c1, withk\u0018k1the maximum value of \u0018(s) fors2[0 1]), the \rapping\ndynamics is not modi\fed at leading order so that K=K0+O(k\u0018k1). Thus\nP\u00181\nM\u0003U\u00032Z1\n0\u0018(s)K0ds\u0014\u00180\nM\u0003U\u00032kK0k1 (14)\nand this upper bound can be approached asymptotically using, for example, the Gaussian distribution (13) centred\non the maximum of K0and increasing \u000b. More precisely, any peaked distribution of damping of L1-norm\u00180centred\non this maximum and with a typical width (here \u000b\u00001=2) much smaller than the length scale \u0015associated with the K0\npeak width, should approach the theoretical upper bound Pth=\u00180kK0k1=(M\u0003U\u00032), provided that the maximum\ndamping remains small enough that the approximation for Pin Eq. (14) still holds (which in the present case is\nequivalent to keeping \u000b1=2\u00180bounded).\nThis conjecture is tested numerically using a Gaussian distribution, Eq. (13), with so= 0:8, and by computing the\nharvested power for small damping: 2 :2\u000210\u00005< \u00180<2:2\u000210\u00002. Figure 9 shows P=Pthand con\frms that this\nconjecture indeed holds since for increasing \u000b, the power asymptotically approaches its optimal for small enough \u00180.\nWhen\u00180\u00152:2\u000210\u00003, the asymptotic value Pthcan not be reached any more: as \u000bis increased, the e\u000bect of the\nmaximum damping \u000b\u00180on the \rutter dynamics is important even before the dampers are focused enough for Pto\napproachPth.\nTwo essential results are illustrated here. For su\u000eciently small damping, the nonlinear response of the structure\nremains unchanged so the best strategy is to focus all the harvesters in the region of maximum curvature-change.\nHowever \fnite levels of damping signi\fcantly modify the nonlinear dynamics of the beam, so at these levels a narrow\nfocusing of damping is sub-optimal; next we investigate further the behaviour at \fnite damping.10\nFIG. 9: Focused harvesters at so= 0:81 for small damping: Curves show the normalised power for increasingly focused\nharvesters for small damping values: \u00180= 2:2\u000210\u00005: 10\u00002(solid, dashed, dash-dot, dotted curves, respectively). ( M\u0003=\n12:7; U\u0003= 13).\nB. Impact of \fnite and localised damping\nThe computations from Section IV show that for Gaussian distributions and \u00180>0:02 a defocusing of harvesters is\npreferable, as the system response becomes increasingly in\ruenced by damping. In order to understand the impact on\nthe system's response, the Gaussian distribution (13) is used in the high damping range and \u000bis varied over a range\nthat transitions the distribution from dispersed to focused (0 <\u000b< 102) with\u00180= 0:47 andso= 0:45 (corresponding\nto the position of kKk1for\u000b= 0). Consistent with Figure 6, the harvested power in Figure 10(a) decreases with\n\u000b, and Figure 10(b) shows the evolution of K(s).\nFor dispersed distributions ( \u000b<10),K(so) decreases with \u000band the zone of maximum curvature-change shifts to\nregions of reduced damping. Nonetheless nonzero damping in this zone is adequate to retrieve some of the energy.\nFor\u000b>10 damping becomes increasingly focused and the \rexible body does not deform anymore near the damper\n(K(s0)\u00190): no energy is harvested anymore since no deformation occurs near the harvester's position, and this is\nre\rected in the sharp drop in power in Figure 10(b). This illustrates the main e\u000bect of a focused distribution for\n\fnite damping: we see a redistribution of the solid's deformation to regions with little or no damping. This is also\nthe reason why a focused damping distribution is not appropriate for optimal energy harvesting in the \fnite damping\nrange.\nVI. CONCLUSIONS\nIn this work, we considered the possibility of harvesting energy from a slender body \ruttering in an axial \row, and\nin particular the impact of harvester-distribution on the performance of the system as well as potential optimisation\nstrategies. To this end, a simpli\fed \ruid-solid model was proposed with energy harvesting represented as nonuniform\nstructural damping. Depending on the \ruid-to-solid inertia ratio, damping can actually enhance the \rutter response\nof the structure, as well as reduce the critical \row velocity above which the system can operate.\nFor uniform damping distributions, maximising the harvested power appears as a trade-o\u000b on the damping intensity:\nfor small damping, the \rutter response is only weakly modi\fed and the harvested power increases as more damping\nis added to the system. For higher damping however, the dynamical response can be strongly modi\fed and the self-\nsustained oscillations are eventually mitigated. The e\u000bect of a nonuniform distribution of harvesters along the structure\nwas considered next. We show that even simple nonuniform distributions such as linear and Gaussian functions, can\nlead to an increase in harvested peak power on the order of 50%. The similarity in the optimal distribution and\nperformance obtained through an optimisation on two fundamentally di\u000berent families of distributions suggests that\nsimple strategies can capture rather well the characteristics of the global optimal distribution.\nInvestigating further into the relationship between damping and \rutter response, we showed that for small damping,\nlocalised harvesting is optimal as it takes full advantage of the system's maximum curvature without impacting its11\n(a)\n (b)\nFIG. 10: Focused harvesters at so= 0:45 for large damping ( \u00180= 0:47): (a) power dependence on \u000band (b) \u0016K(s) for\n\u000b= 0;1;10;102(solid, dashed, dashed-dot, dotted curves, respectively). ( M\u0003= 12:7; U\u0003= 13).\ndynamics signi\fcantly. On the other hand, for \fnite damping, focused distributions perform rather poorly as the\nbeam response adapts to rigidify the damped region, leading to negligible harvested power. Instead, nonuniform\ndistributed damping over the entire length of the system becomes optimal.\nReturning to the result from the simple bi-articulated model [29], we note a clear di\u000berence in the optimal con-\n\fgurations. Whilst the reduced order model optimal has dampers focused at the \fxed end, the continuous optimal\nrequires a dispersed distribution with minimal damping at the \fxed end and increasing to a maximum at the free\nend. However, we \fnd a crucial di\u000berence between the two system con\fgurations: whilst the bi-articulated system\nhas a single moving region of curvature (the second articulation) that is responsible for driving the instability, defor-\nmations may occur all along the length of the beam in the continuous system. Therefore in the latter, curvature can\nbe displaced away from regions with focused damping while still maintaining the \rutter dynamics. 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Fluids Struct. , 16:739{759, 2002."    },    {        "title": "2312.09140v1.Nonlocal_damping_of_spin_waves_in_a_magnetic_insulator_induced_by_normal__heavy__or_altermagnetic_metallic_overlayer__a_Schwinger_Keldysh_field_theory_approach.pdf",        "content": "Nonlocal damping of spin waves in a magnetic insulator induced by normal, heavy, or\naltermagnetic metallic overlayer: A Schwinger-Keldysh field theory approach\nFelipe Reyes-Osorio and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n(Dated: December 15, 2023)\nUnderstanding spin wave (SW) damping, and how to control it to the point of being able to\namplify SW-mediated signals, is one of the key requirements to bring the envisaged magnonic tech-\nnologies to fruition. Even widely used magnetic insulators with low magnetization damping in their\nbulk, such as yttrium iron garnet, exhibit 100-fold increase in SW damping due to inevitable con-\ntact with metallic layers in magnonic circuits, as observed in very recent experiments [I. Bertelli\net al. , Adv. Quantum Technol. 4, 2100094 (2021)] mapping SW damping in spatially-resolved\nfashion. Here, we provide microscopic and rigorous understanding of wavevector-dependent SW\ndamping using extended Landau-Lifshitz-Gilbert equation with nonlocal damping tensor , instead\nof conventional local scalar Gilbert damping, as derived from Schwinger-Keldysh nonequilibrium\nquantum field theory. In this picture, the origin of nonlocal magnetization damping and thereby in-\nduced wavevector-dependent SW damping is interaction of localized magnetic moments of magnetic\ninsulator with conduction electrons from the examined three different types of metallic overlayers—\nnormal, heavy, and altermagnetic. Due to spin-split energy-momentum dispersion of conduction\nelectrons in the latter two cases, the nonlocal damping is anisotropic in spin and space, and it can\nbe dramatically reduced by changing the relative orientation of the two layers when compared to\nthe usage of normal metal overlayer.\nIntroduction. —Spin wave (SW) or magnon damping\nis a problem of great interest to both basic and ap-\nplied research. For basic research, its measurements [1–4]\ncan reveal microscopic details of boson-boson or boson-\nfermion quasiparticle interactions in solids, such as:\nmagnon-magnon interactions (as described by second-\nquantized Hamiltonians containing products of three or\nmore bosonic operators [5, 6]), which are frequently en-\ncountered in antiferromagnets [4, 5] and quantum spin\nliquids [7], wherein they play a much more important\nrole [8] than boson-boson interactions in other condensed\nphases, like anharmonic crystalline lattices or superflu-\nids [5]; magnon-phonon interactions [3], especially rel-\nevant for recently discovered two-dimensional magnetic\nmaterials [2]; and magnon-electron interactions in mag-\nnetic metals [1, 9–12]. For the envisaged magnon-\nbased digital and analog computing technologies [13–\n17], understanding magnon damping makes it possible\nto develop schemes to suppress [18] it, and, further-\nmore, achieve amplification of nonequilibrium fluxes of\nmagnons [19–22]. In fact, overcoming damping and\nachieving amplification is the keyto enable complex\nmagnon circuits where, e.g., a logic gate output must\nbe able to drive the input of multiple follow-up gates.\nLet us recall that the concept of SW was introduced by\nBloch [23] as a wave-like disturbance in the local mag-\nnetic ordering of a magnetic material. The quanta [6] of\nenergy of SWs of frequency ωbehave as quasiparticles\ntermed magnons, each of which carries energy ℏωand\nspin ℏ. As regards terminology, we note that in magnon-\nics [13] SW is often used for excitations driven by an-\ntennas [24–27] and/or described by the classical Landau-\nLifshitz-Gilbert (LLG) equation [9, 10, 28, 29], whereas\nmagnon is used for the quantized version of the same ex-\ne\nee\nee\neFIG. 1. (a) Schematic view of bilayers where a metallic over-\nlayer covers the top surface of magnetic insulator, as often\nencountered in spintronics and magnonics [13, 30]. Three\ndifferent energy-momentum dispersion of conduction elec-\ntrons at the interface are considered, with their Fermi sur-\nfaces shown in panel (b)—normal metal (NM); heavy metal\n(HM) with the Rashba SOC [31, 32], and altermagnetic metal\n(AM) [33, 34]—with the latter two being spin-split. The rel-\native alignment of the layers is labeled by an angle θ[33, 34],\nmeaning that the wavevector qof SWs within FI is at an an-\ngleθaway from the kx-axis.\ncitation [5], or these two terms are used interchangingly.\nIn particular, experiments focused on SW damp-\ning in metallic ferromagnets have observed [1] its de-\npendence on the wavevector qwhich cannot be ex-\nplained by using the standard LLG equation [28,\n29],∂tMn=−Mn×Beff\nn+αGMn×∂tMn(where ∂t≡\n∂/∂t), describing dynamics of localized magnetic mo-\nments (LMMs) Mnat site nof crystalline lattice (also\nused in atomistic spin dynamics [28]) viewed as classi-\ncal vectors of unit length. This is because αG, as the\nGilbert damping parameter [35, 36], is a local scalar (i.e.,\nposition-independent constant). Instead, various forms\nof spatially nonuniform (i.e., coordinate-dependent) and\nnonlocal (i.e., magnetization-texture-dependent) damp-\ning due to conduction electrons have been proposed [9,arXiv:2312.09140v1  [cond-mat.mes-hall]  14 Dec 20232\n10, 37–39], or extracted from first-principles calcula-\ntions [40], to account for observed wavevector-dependent\ndamping of SWs, such as ∝q2(q=|q|) measured in\nRef. [1]. The nonlocal damping terms require neither\nspin-orbit coupling (SOC) nor magnetic disorder scatter-\ning, in contrast to αGwhich is considered to vanish [41]\nin their absence.\nThus, in magnonics, it has been considered [30] that\nusage of magnetic insulators, such as yttrium iron gar-\nnet (YIG) exhibiting ultralow αG≃10−4(achieved on\na proper substrate [42]), is critical to evade much larger\nand/or nonlocal damping of SWs found in ferromagnetic\nmetals. However, very recent experiments [24–27] have\nobserved 100-fold increase of SW damping in the segment\nof YIG thin film that was covered by a metallic overlayer.\nSuch spatially-resolved measurement [24] of SW damp-\ning was made possible by the advent of quantum sensing\nbased on nitrogen vacancy (NV) centers in diamond [43],\nand it was also subsequently confirmed by other meth-\nods [25–27]. Since excitation, control, and detection of\nSWs requires to couple YIG to metallic electrodes [13],\nunderstanding the origin and means to control/suppress\nlarge increase in SW damping underneath metallic over-\nlayer is crucial for realizing magnonic technologies. To\nexplain their experiments, Refs. [24–27] have employed\nthe LLG equation with ad hoc intuitively-justified terms\n(such as, effective magnetic field due to SW induced eddy\ncurrents within metallic overlayer [24]) that can fit the\nexperimental data, which is nonuniversal and unsatisfac-\ntory (many other examples of similar phenomenological\nstrategy exist [1, 44]).\nIn contrast, in this Letter we employ recently derived\n∂tMn=−Mn×Beff\nn+Mn×X\nn′(αGδnn′+λR)·∂tMn′,\n(1)\nextended LLG equation with all terms obtained [45]\nmicroscopically from Schwinger-Keldysh nonequilibrium\nquantum field theory [46] and confirmed [45] via exact\nquantum-classical numerics [47–50]. It includes nonlo-\ncal damping as the third term on the right-hand side\n(RHS), where its nonlocality is signified by dependence\nonR=rn−rn′, where rnis the position vector of lat-\ntice site n. Equation (1) is applied to a setup depicted in\nFig. 1 where conduction electron spins from three differ-\nent choices for metallic overlayer are assumed to interact\nwith LMMs of ferromagnetic insulator (FI) at the inter-\nface via sdexchange interaction of strength Jsd, as well\nas possibly underneath the top surface of FI because of\nelectronic evanescent wavefunction penetrating into it.\nNote that FI/normal metal (NM) bilayer directly mod-\nels recent experiments [24] where FI was a thin film of\nYIG and NM was Au, and SW damping within FI was\nquantified using quantum magnetometry via NV centers\nin diamond. Next, the FI/heavy metal (HM) bilayer,\nsuch as YIG/Pt [18, 27], is frequently encountered in\n0 2 4\nK/J0.51.0q (1/a)\n kF= 0\n.92kF= 0\n.99kF= 1\n.08kF= 1\n.15kF= 1\n.22kF= 1\n.30kF= 1\n.38(a)\n1.0 1.2 1.4\nkF0.81.01.21.4qmax(1/a)\n∝kF(b)FIG. 2. (a) Wavevector qof SW generated by injecting spin-\npolarized current in TDNEGF+LLG simulations of NM over-\nlayer on the top of 1D FI [Fig. 1(a)] as a function of anisotropy\nK[Eq. (3)] for different electronic Fermi wavevectors kF. (b)\nMaximum wavevector qmaxof SWs that can be generated by\ncurrent injection [21, 57] before wavevector-dependent SW\ndamping becomes operative, as signified by the drop around\nkFin curves plotted in panel (a).\nvarious spintronics and magnonics phenomena [13, 30].\nFinally, due to recent explosion of interest in altermag-\nnets [33, 34], the FI/altermagnetic metal (AM) bilay-\ners, such as YIG/RuO 2, have been explored experimen-\ntally to characterize RuO 2as a spin-to-charge conver-\nsion medium [51]. The Schwinger-Keldysh field theory\n(SKFT), commonly used in high energy physics and cos-\nmology [52–54], allows one to “integrate out” unobserved\ndegrees of freedom, such as the conduction electrons in\nthe setup of Fig. 1, leaving behind a time-retarded dis-\nsipation kernel [48, 55, 56] that encompasses electronic\neffects on the remaining degrees of freedom. This ap-\nproach then rigorously yields the effective equation for\nLMMs only , such as Eq. (1) [45, 56] which bypasses the\nneed for adding [1, 24, 44] phenomenological wavevector-\ndependent terms into the standard LLG equation. In\nour approach, the nonlocal damping is extracted from\nthe time-retarded dissipation kernel [45].\nSKFT-based theory of SW damping in FI/metal bilay-\ners.—The nonlocal damping [45] λRin the third term\non the RHS of extended LLG Eq. (1) stems from back-\naction of conduction electrons responding nonadiabati-\ncally [48, 59]—i.e., with electronic spin expectation value\n⟨ˆsn⟩being always somewhat behind LMM which gener-\nates spin torque [60] ∝ ⟨ˆsn⟩×Mn—to dynamics of LMMs.\nIt is, in general, a nearly symmetric 3 ×3 tensor whose\ncomponents are given by [45]\nλαβ\nR=−J2\nsd\n2πZ\ndε∂f\n∂εTr\u0002\nσαAnn′σβAn′n\u0003\n. (2)\nHere, f(ε) is the Fermi function; α, β =x, y, z ;σα\nis the Pauli matrix; and A(ε) = i\u0002\nGR(ε)−GA(ε)\u0003\nis\nthe spectral function in the position representation ob-\ntained from the retarded/advanced Green’s functions\n(GFs) GR/A(ε) =\u0000\nε−H±iη\u0001−1. Thus, the calcula-\ntion of λRrequires only an electronic Hamiltonian H\nas input, which makes theory fully microscopic (i.e.,3\n−5 0 5\nX−505Y\nλNM\nR(a)NM\n−1 0 1\nλα\nR\n−5 0 5\nX−505Y\nλxx\nR(b)HM t SOC= 0.3t0\n−5 0 5\nX−505Y\nλzz\nR(c)\n−5 0 5\nX−505Y\nλ⊥\nR(d)AM t AM= 0.5t0\n0.0 0.5 1.0 1.5\nq (1/a)0123Γq≡Im(ωq) (J/¯ h)×10−1\nEq.(6)(e)\nη= 0.1\nη= 0\n−5 0 5\nX−505Y\nλyy\nR(f)\n−5 0 5\nX−505Y\nλxy\nR(g)\n−5 0 5\nX−505Y\nλ/bardbl\nR(h)\nFIG. 3. (a)–(d) and (f)–(h) Elements of SKFT-derived nonlocal damping tensor in 2D FI, λRwhere R= (X, Y, Z ) is the\nrelative vector between two sites within FI, covered by NM [Eq. (5)], HM [Eqs. (8)] or AM [Eqs. (9)] metallic overlayer. (e)\nWavevector-dependent damping Γ qof SWs due to NM overlayer, where the gray line is based on Eq. (6) in the continuous\nlimit [58] and the other two lines are numerical solutions of extended LLG Eq. (1) for discrete lattices of LMMs within FI. The\ndotted line in (e) is obtained in the absence of nonlocal damping ( η= 0), which is flat at small q.\nHamiltonian-based). Although the SKFT-based deriva-\ntion [45] yields an additional antisymmetric term, not\ndisplayed in Eq. (2), such term vanishes if the system\nhas inversion symmetry. Even when this symmetry is\nbroken, like in the presence of SOC, the antisymmet-\nric component is often orders of magnitude smaller [56],\ntherefore, we neglect it. The first term on the RHS of ex-\ntended LLG Eq. (1) is the usual one [28, 29], describing\nprecession of LMMs in the effective magnetic field, Beff\nn,\nwhich is the sum of both internal and external ( Bextez)\nfields. It is obtained as Beff\nn=−∂H/∂Mnwhere His\nthe classical Hamiltonian of LMMs\nH=−JX\n⟨nn′⟩Mn·Mn′+K\n2X\nn(Mz\nn)2−BextX\nnMz\nn.(3)\nHere we use g= 1 for gyromagnetic ratio, which sim-\nplifies Eq. (1); Jis the Heisenberg exchange coupling\nbetween the nearest-neighbors (NN) sites; and Kis the\nmagnetic anisotropy.\nWhen nonlocal damping tensor, λRis proportional\nto 3×3 identity matrix, I3, a closed formula for the\nSW dispersion can be obtained via hydrodynamic the-\nory [58]. In this theory, the localized spins in Eq. (1),\nMn= (Re ϕn,Imϕn,1−m)T, are expressed using com-\nplex field ϕnand uniform spin density m≪1. Then,\nusing the SW ansatz ϕn(t) =P\nqUqei(q·rn−ωqt), we ob-\ntain the dispersion relation for the SWs\nωq= (Jq2+K−B)\u0002\n1 +i(αG+˜λq)\u0003\n, (4)\nwhere qis the wavevector and ωis their frequency. Thedamping of the SW is then given by the imaginary part\nof the dispersion in Eq. (4), Γ q≡Imωq. It is comprised\nby contributions from the local scalar Gilbert damping\nαGand the Fourier transform of the nonlocal damping\ntensor, ˜λq=R\ndrnλrneiq·rn.\nResults for FI/NM bilayer. —We warm up by extract-\ning Γ qfor the simplest of the three cases in Fig. 1, a\none-dimensional (1D) FI chain under a 1D NM over-\nlayer with spin-degenerate quadratic electronic energy-\nmomentum dispersion, ϵkσ=t0k2\nx, where t0=ℏ2/2m.\nThe GFs and spectral functions in Eq. (2), can be\ncalculated in the momentum representation, yielding\nλ1D\nR=2J2\nsd\nπv2\nFcos2(kFR)I3, where vFis the Fermi velocity,\nR≡ |R|, and kFis the Fermi wavevector. Moreover, its\nFourier transform, ˜λq=2J2\nsd\nv2\nF[δ(q) +δ(q−2kF)/2], dic-\ntates additional damping to SWs of wavevector q=\n0,±2kF. Although the Dirac delta function in this ex-\npression is unbounded, this unphysical feature is an ar-\ntifact of the small amplitude, m≪1, approximation\nwithin the hydrodynamic approach [58]. The features of\nsuch wavevector-dependent damping in 1D can be cor-\nroborated via TDNEGF+LLG numerically exact simu-\nlations [47–50] of a finite-size nanowire, similar to the\nsetup depicted in Fig. 1(a) but sandwiched between two\nNM semi-infinite leads. For example, by exciting SWs\nvia injection of spin-polarized current into the metallic\noverlayer of such a system, as pursued experimentally in\nspintronics and magnonics [21, 57], we find in Fig. 2(a)\nthat wavevector qof thereby excited coherent SW in-4\ncreases with increasing anisotropy K. However, the max-\nimum wavevector qmaxis limited by kF[Fig. 2(b)]. This\nmeans that SWs with q≳kFare subjected to additional\ndamping, inhibiting their generation. Although our an-\nalytical results predict extra damping at q= 2kF, finite\nsize effects and the inclusion of semi-infinite leads in TD-\nNEGF+LLG simulations lower this cutoff to kF.\nSince SW experiments are conducted on higher-\ndimensional systems, we also investigate damping\non SWs in a two-dimensional (2D) FI/NM bilayer.\nThe electronic energy-momentum dispersion is then\nϵkσ=t0(k2\nx+k2\ny), and the nonlocal damping and its\nFourier transform are given by\nλNM\nR=k2\nFJ2\nsd\n2πv2\nFJ2\n0(kFR)I3, (5)\n˜λNM\nq=kFJ2\nsdΘ(2kF−q)\n2πv2\nFqp\n1−(q/2kF)2, (6)\nwhere Jn(x) is the n-th Bessel function of the first kind,\nand Θ( x) is the Heaviside step function. The nonlo-\ncal damping in Eqs. (5) and (6) is plotted in Fig. 3(a),\nshowing realistic decay with increasing R, in contrast to\nunphysical infinite range found in 1D case. Addition-ally, SW damping in Eq. (6) is operative for wavectors\n0≤q≤2kF, again diverging for q= 0,2kFdue to arti-\nfacts of hydrodynamic theory [58]. Therefore, unphysical\ndivergence can be removed by going back to discrete lat-\ntice, such as solid curves in Fig. 3(e) obtained for n=1–\n100 LMMs by solving numerically a system of coupled\nLLG Eq. (1) where λRin 2D is used [45]. In this numer-\nical treatment, we use kF= 0.5a−1where ais the lattice\nspacing; k2\nFJ2\nsd/2πv2\nF=η= 0.1;K= 0; Bext= 0.1J;\nandαG= 0.1.\nResults for FI/HM bilayer. —Heavy metals (such as of-\nten employed Pt, W, Ta) exhibit strong SOC effects due\nto their large atomic number. We mimic their presence at\nthe FI/HM interface [31] by using 2D energy-momentum\ndispersion ϵk=t0(k2\nx+k2\ny) +tSOC(σxky−σykx), which\nincludes spin-splitting due to the Rashba SOC [31, 32].\nUsing this dispersion, Eq. (2) yields\nλHM\nR=\nλxx\nRλxy\nR0\nλxy\nRλyy\nR0\n0 0 λzz\nR\n, (7)\nfor the nonlocal damping tensor. Its components are, in\ngeneral, different from each other\nλxx\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n+ cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8a)\nλyy\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8b)\nλzz\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8c)\nλxy\nR=−J2\nsdsin(2θ)\n4π\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\n, (8d)\nwhere kF↑andkF↓are the spin-split Fermi wavevec-\ntors [Fig. 1(b)], and θis the relative orientation angle\n[Fig. 1(b)] between the SW wavevector qand the kxdi-\nrection. Thus, the nonlocal damping tensor in Eq. (7)\ngenerated by HM overlayer is anisotropic in spin due to\nits different diagonal elements, as well as nonzero off-\ndiagonal elements. It is also anisotropic in space due to\nits dependence on the angle θ. Its elements [Eqs. (8)]\nare plotted in Figs. 3(b), 3(c), 3(f), and 3(g) using\ntSOC= 0.3t0. They may become negative, signifying the\npossibility of antidamping torque [21] exerted by conduc-\ntion electrons. However, the dominant effect of nearby\nLMMs and the presence of local scalar αGensures that\nLMM dynamics is damped overall. Although there is no\nclosed expression for the SW dispersion in the presence of\nanisotropic λHM\nR, we can still extract SW damping Γ qin-duced by an HM overlayer from the exponential decay of\nthe SW amplitude in numerical integration of extended\nLLG Eq. (1) using SW initial conditions with varying q.\nFor an HM overlayer with realistic [31, 32] tSOC= 0.1t0\nthe results in Fig. (4)(a) are very similar to those ob-\ntained for NM overlayer with the same Fermi energy.\nAlso, the spatial anisotropy of λHM\nRdid not translate into\nθ-dependence of the SW damping.\nResults for FI/AM bilayer. —Altermagnets [33, 34] are\na novel class of antiferromagnets with spin-split elec-\ntronic energy-momentum dispersion despite zero net\nmagnetization or lack of SOC. They are currently in-\ntensely explored as a new resource for spintronics [51,\n61, 62] and magnonics [63, 64]. A simple model for\nan AM overlayer employs energy-momentum dispersion\nϵkσ=t0(k2\nx+k2\ny)−tAMσ(k2\nx−k2\ny) [33, 34], where tAMis5\n0.0 0.5 1.0 1.5\nq (1/a)1.01.52.02.5Γq≡Im(ωq) (J/¯ h)×10−1\n(a)NM\nHM\n0.0 0.5 1.0 1.5\nq (1/a)1234Γq≡Im(ωq) (J/¯ h)×10−1\n(b)NM\nAM :θ= 45◦\nAM :θ= 0◦\nFIG. 4. (a) Wavevector-dependent damping Γ qof SWs under\nNM or HM overlayer with the Rashba SOC of strength tSOC=\n0.1t0. (b) Γ qof SWs under AM overlayer with tAM= 0.8t0\nand for different relative orientations of FI and AM layers\nmeasured by angle θ[Fig. 1]. All calculations employ η= 0.1\nand Fermi energy εF= 0.25t0.\nthe parameter characterizing anisotropy in the AM. The\ncorresponding λAM\nR= diag( λ⊥\nR, λ⊥\nR, λ∥\nR) tensor has three\ncomponents, which we derive from Eq. (2) as\nλ⊥\nR=J2\nsd\n4πA+A−\u0014\nJ2\n0\u0012rϵF\nt0R+\u0013\n+J2\n0\u0012rϵF\nt0R−\u0013\u0015\n,\n(9a)\nλ∥\nR=J2\nsd\n2πA+A−J0\u0012rϵF\nt0R+\u0013\nJ0\u0012rϵF\nt0R−\u0013\n, (9b)\nwhere A±=t0±tAMandR2\n±=X2/A±+Y2/A∓is\nthe anisotropically rescaled norm of R. They are plot-\nted in Figs. 3(d) and 3(h), demonstrating that λAM\nRis\nhighly anisotropic in space and spin due to the impor-\ntance of angle θ[61, 65, 66]. Its components can also\ntake negative values, akin to the case of λHM\nR. It is inter-\nesting to note that along the direction of θ= 45◦[gray\ndashed line in Figs. 3(d) and 3(h)], λ⊥\nR=λ∥\nRso that\nnonlocal damping tensor is isotropic in spin. The SW\ndamping Γ qinduced by an AM overlayer is extracted\nfrom numerical integration of extended LLG Eq. (1) and\nplotted in Fig. (4)(b). Using a relatively large, but real-\nistic [33], AM parameter tAM= 0.8t0, the SW damping\nfor experimentally relevant small wavevectors is reduced\nwhen compared to the one due to NM overlayer by up to\n65% for θ= 0◦[Fig. 4(b)]. Additional nontrivial features\nare observed at higher |q|, such as being operative for a\ngreater range of wavevectors and with maxima around\n|q|= 2p\nϵF/t0and|q|= 3p\nϵF/t0. Remarkably, these\npeaks vanish for wavevectors along the isotropic direction\nθ= 45◦[Fig. 4(b)].\nConclusions. —In conclusion, using SKFT-derived non-\nlocal damping tensor [45], we demonstrated a rigorous\npath to obtain wavevector damping of SWs in magnetic\ninsulator due to interaction with conduction electrons\nof metallic overlayer, as a setup often encountered in\nmagnonics [13–17, 30] where such SW damping was di-\nrectly measured in very recent experiments [24–27]. Ouranalytical expressions [Eqs. (5), (7), and (9)] for nonlo-\ncal damping tensor—using simple models of NM, HM,\nand AM overlayers as an input—can be directly plugged\ninto atomistic spin dynamics simulations [28]. For more\ncomplicated band structures of metallic overlayers, one\ncan compute λRnumerically via Eq. (2), including com-\nbination with first-principles calculations [40]. Since we\nfind that λHM,AM\nR is highly anisotropic in spin and space,\nthe corresponding SW damping Γ qthus understood mi-\ncroscopically from SKFT allows us to propose how to\nmanipulate it [Fig. 4]. For example, by using HM or AM\noverlayer and by changing their relative orientation with\nrespect to FI layer, Γ qcan be reduced by up to 65%\nat small wavevectors q[Fig. 4], which can be of great\ninterest to magnonics experiments and applications.\n∗bnikolic@udel.edu\n[1] Y. Li and W. Bailey, Wave-number-dependent Gilbert\ndamping in metallic ferromagnets., Phys. Rev. Lett. 116,\n117602 (2016).\n[2] L. Chen, C. Mao, J.-H. Chung, M. B. Stone, A. I.\nKolesnikov, X. Wang, N. Murai, B. Gao, O. De-\nlaire, and P. Dai, Anisotropic magnon damping by\nzero-temperature quantum fluctuations in ferromagnetic\nCrGeTe 3, Nat. Commun. 13, 4037 (2022).\n[3] P. Dai, H. Y. Hwang, J. Zhang, J. A. Fernandez-Baca,\nS.-W. Cheong, C. Kloc, Y. Tomioka, and Y. 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B 108, 054511 (2023)."    },    {        "title": "2211.00257v1.Modeling_of_three_dimensional_betatron_oscillation_and_radiation_reaction_in_plasma_accelerators.pdf",        "content": "Modeling of three-dimensional betatron oscillation and radiation reaction in plasma\naccelerators\nYulong Liu and Ming Zeng\u0003\nInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China and\nUniversity of Chinese Academy of Sciences, Beijing 100049, China\n(Dated: November 2, 2022)\nBetatron oscillation is a commonly known phenomenon in laser or beam driven plasma wake\feld\naccelerators. In the conventional model, the plasma wake provides a linear focusing force to a rela-\ntivistic electron, and the electron oscillates in one transverse direction with the betatron frequency\nproportional to 1 =p\r, where\ris the Lorentz factor of the electron. In this work, we extend this\nmodel to three-dimensional by considering the oscillation in two transverse and one longitudinal\ndirections. The long-term equations, with motion in the betatron time scale averaged out, are\nobtained and compared with the original equations by numerical methods. In addition to the lon-\ngitudinal and transverse damping due to radiation reaction which has been found before, we show\nphenomena including the longitudinal phase drift, betatron phase shift and betatron polarization\nchange based on our long-term equations. This work can be highly valuable for future plasma based\nhigh-energy accelerators and colliders.\nI. INTRODUCTION\nThe new generation of accelerators, using plasma as\nthe acceleration media, o\u000ber high acceleration gradient\nin the order of 10{100 GV =m and strong transverse\nfocusing \feld [1{3]. Depends on the driver type, the\nplasma accelerators are named laser wake\feld acceler-\nators (LWFAs), which are driven by laser pulses, and\nplasma wake\feld accelerators (PWFAs), which are driven\nby charged particle beams. When a high intensity laser\npulse (>\u00181018W=cm2) or a high current particle beam\n(>\u00181 kA) propagates through an underdense plasma, the\nradiation pressure of the laser or the space charge of the\nbeam expels all plasma electrons away from axis radi-\nally, leaves behind a nearly uniform ion channel. This\nhigh-intensity three-dimensional (3D) regime has been\nreferred to as the blowout regime [4, 5]. In this regime,\nthe expelled electrons are pulled back by the ion chan-\nnel and thereby bubble-like plasma wake wave is created.\nThe wake consists of a longitudinal electric \feld that is\na function of distance behind the driver, and transversal\nelectromagnetic \felds that are proportional to the o\u000b-\naxis distance. Consequently, in addition to the longitu-\ndinal acceleration / deceleration, the electrons reside in\nthe wake also perform radial oscillation, called betatron\noscillation (BO), under the action of transverse focusing\n\feld, with the frequency !\f=!p\u0014=p\r, where!pis the\nplasma frequency, \ris the relativistic factor of the elec-\ntron, and\u0014is the focusing constant which takes 1 =p\n2\nfor the blow-out regime [6, 7].\nElectrons emit synchrotron radiation when performing\nBO [8{10], which a\u000bects the electrons in return. Such ef-\nfect is called the radiation reaction (RR) and its classical\nexpression is the Lorentz{Abraham{Dirac (LAD) equa-\ntion or the Landau-Lifshitz equation [11, 12]. Because\n\u0003Corresponding author: zengming@ihep.ac.cnthe RR force is proportional to the classical electron ra-\ndiusre\u00192:81\u000210\u000015m, it is generally negligible unless\nunder extreme conditions [13, 14] or for su\u000eciently long\ninteraction time [15]. The BO in a plasma accelerator is\nanother good case for such long interaction time. The ra-\ndiation leads to the energy loss of electrons and in return\na\u000bects the energy-dependent betatron frequency, as well\nas the other beam properties, such as the energy spread\nand emittance [16{23].\nAlthough there are many established theories on the\nlong-term RR damping e\u000bect of BO, their models assert\nthe electron only moves in one plane, spanned by the\nlongitudinal direction and one transverse direction, thus\nonly linear polarization is considered. Moreover, these\nmodels usually neglect the longitudinal and energy os-\ncillations during one betatron period. In this paper, we\nestablish a 3D BO model with RR e\u000bect, which gener-\nalizes the betatron polarization from linear to elliptical,\nand considers both the longitudinal and energy oscilla-\ntions. Long-term equations (LTEs), without resolving\nthe betatron period, are derived and veri\fed by numer-\nical methods. The LTEs reproduce the previous results,\nsuch as longitudinal and transverse damping due to RR,\nas the special cases, and meanwhile reveal new phenom-\nena such as betatron phase shift and polarization change.\nThe rest of this paper is organized as follows. Sec. II\ngives the original form of the force, and shows the equa-\ntions of motion expressed by the transverse motion only.\nSec. III derives the LTEs by averaging the equations of\nmotion through one betatron period. Sec. IV discusses\nthe di\u000berent phenomena in the RR dominant regime and\nthe betatron phase shift dominant regime. Sec. V nu-\nmerically veri\fes the LTEs by comparing with the code\nPTracker which solves the equations with the original\nform of force. Before start, it is worth noting that we use\nplasma normalization units described in Appx. A, and\nsome symbols and calculation rules used often during the\nderivation are described in Appx. B.arXiv:2211.00257v1  [physics.acc-ph]  1 Nov 20222\nII. THE ELECTROMAGNETIC FIELD AND\nTHE EQUATIONS OF MOTION\nConsider an electron with \r\u001d1 is trapped in a plasma\nwake\feld with the longitudinal co-moving coordinate \u0010=\nz\u0000\fwt, where the wake is moving in the z+ direction with\nthe phase velocity \fw. Neglect the interaction between\nthe beam particles, the electromagnetic \feld provided by\nthe wake can be modeled as [7]\nEz=Ez0+\u0015\u00101; (1)\n~E?=\u00142(1\u0000\u0015)~ r; (2)\nB\u0012=\u0000\u00142\u0015r; (3)whereEz0=Ezj\u0010=h\u0010i,\u0015=dEz=d\u0010j\u0010=h\u0010i, and~ r= (x;y)\nis the transverse o\u000bset. The force can be expressed as\nfz=\u0000Ez0\u0000\u0015\u00101+\u00142\u0015(x\fx+y\fy) +frad\nz;(4)\nfx=\u0000\u00142(1\u0000\u0015+\u0015\fz)x+frad\nx; (5)\nfy=\u0000\u00142(1\u0000\u0015+\u0015\fz)y+frad\ny; (6)\nwhere\fx= _x,\fy= _y,\fz= _z=\fz0+_\u00101,\fz0=\fw+\n_h\u0010i=h\fzi, and~fradis the RR force. The formulas of\n3D BO with RR can be written in the form of transverse\nterms only (see Appx. C)\n_\r=\u0000Ez0\fz0+\u0012\u0015\fz0\n4+\u00142\u0015\u0000\u00142\u0013\n(x\fx+y\fy)\u00002\n3re\r2\u00144\u0000\nx2+y2\u0001\n; (7)\n_pz=\u0000Ez0+\u0015\u00121\n4+\u00142\u0013\n(x\fx+y\fy)\u00002\n3re\r2\u00144\u0000\nx2+y2\u0001\n; (8)\n_px=\u0000\u00142x+\u00142\u0015\n2\u0010\nh\ri\u00002+\f2\nx+\f2\ny\u0011\nx\u00002\n3re\r2\u00144\u0000\nx2+y2\u0001\n\fx; (9)\n_py=\u0000\u00142y+\u00142\u0015\n2\u0010\nh\ri\u00002+\f2\nx+\f2\ny\u0011\ny\u00002\n3re\r2\u00144\u0000\nx2+y2\u0001\n\fy; (10)\nwhere\n~ p=\r~\f; (11)\n\fz0= 1\u00001\n2\u0010\nh\ri\u00002+\n\f2\nx\u000b\n+\n\f2\ny\u000b\u0011\n; (12)\nandreis also normalized to k\u00001\np. One may note the\nsecond terms in Eqs. (7) - (10), which come from the\noscillation of \fzand the modulation of \rdue to trans-\nverse oscillation, were neglected in previous works. In\nthe following sections we show these terms lead to new\nphenomena.\nIII. THE LONG-TERM EQUATIONS OF 3D\nBETATRON OSCILLATION\nIn this section we use the same averaging method as\nRef. [20]. We \frstly introduce two complex variables\nU=\u0010\nx\u0000i\u0014\u00001\r1\n2\fx\u0011\ne\u0000i'; (13)\nV=\u0010\ny\u0000i\u0014\u00001\r1\n2\fy\u0011\ne\u0000i'; (14)where\n'=Z\n!\fdt=\u0014Z\n\r\u00001\n2dt (15)\nis the betatron phase. Obviously jU1j\u001cjhUijandjV1j\u001c\njhVijare satis\fed, and we apply the rules in Appx. B\noften in the following. Because the equations for xandy\ndirections are symmetric, we may derive for xdirection\nonly, then exchange xandy,UandVfor theydirection.\nWith the help of Eqs. (7), (9) and (11), we may write the\ntime derivative of Eq. (13) as\n_U=\u0000i1\n2\u0014\u00001\r\u00001\n2Ez0\fz0\fxe\u0000i'+i1\n3re\u00143\r3\n2\u0000\nx2+y2\u0001\n\fxe\u0000i'\n+i1\n2\u0014\u00001\r\u00001\n2\u0014\u0015\fz0\n4+\u00142(\u0015\u00001)\u0015\n(x\fx+y\fy)\fxe\u0000i'\u0000i1\n2\u0014\u0015\r\u00001\n2\u0010\nh\ri\u00002+\f2\nx+\f2\ny\u0011\nxe\u0000i':(16)\nIn the following, we omit hionUandVfor conve- nience, so that all UandVactually meanhUiandhVi.3\nWe perform average on Eq. (16) to obtain (note only the terms with ei0'survive after averaging)\n_U=1\n4Ez0\fz0h\ri\u00001U\u00001\n24re\u00144h\ri\u0010\njUj2U+ 2jVj2U\u0000V2U\u0003\u0011\n+i1\n64\u0014\u0015\fz0h\ri\u00003\n2\u0010\njUj2U+V2U\u0003\u0011\n\u0000i1\n16\u00143h\ri\u00003\n2h\u0010\njUj2+ 2\u0015jVj2\u0011\nU\u0000(2\u0015\u00001)V2U\u0003i\n\u0000i1\n4\u0014\u0015h\ri\u00005\n2U:(17)\nBy asserting V= 0 and omitting the last three terms in\nEq. (17), which comes from the second terms in Eqs. (7)\n- (10), we can reproduce Eq. (19) in Ref. [20].\nThe average of Eq. (7) leads to\nh_\ri=\u0000Ez0\fz0\u00001\n3re\u00144h\ri2\u0010\njUj2+jVj2\u0011\n; (18)\nwith the second term reproduces Eq. (B2) in Ref. [18].\nEz0is a function ofh\u0010i, which obeys\n_h\u0010i=1\n2\r\u00002\nw\u00001\n2h\ri\u00002\u00001\n4\u00142h\ri\u00001\u0010\njUj2+jVj2\u0011\n;(19)\nwhere\rw=\u0000\n1\u0000\f2\nw\u0001\u00001=2and we have used Eq. (C2).The above averaged equations Eqs. (17), (18) and (19)\nare already enough to predict the long-term behavior of\nBO. However, the equations for the complex variables are\nnot explicit. To make them more physically meaningful,\nwe introduce\nU=jUjei\bx; (20)\nV=jVjei\by; (21)\n\u0001\b = \by\u0000\bx: (22)\njUjhas the meaning of the BO amplitude in the xdirec-\ntion, and \b xthe phase shift. For the ydirection they are\nsimilar. Thus \u0001\b is the phase di\u000berence of the two di-\nrections. By Applying djUj=dt=\u0010\n_UU\u0003+U_U\u0003\u0011\n=2jUj\nand _\bx=\u0010\n_UU\u0003\u0000_U\u0003U\u0011\n=2ijUj2we get\ndjUj\ndt=1\n4Ez0\fz0h\ri\u00001jUj\u00001\n24re\u00144h\rih\njUj3+jVj2jUj(2\u0000cos 2\u0001\b)i\n\u00001\n16\u0014\u00141\n4\u0015\fz0\u0000\u00142(1\u00002\u0015)\u0015\nh\ri\u00003\n2jVj2jUjsin 2\u0001\b;(23)\n_\bx=1\n24re\u00144h\rijVj2sin 2\u0001\b +1\n64\u0014\u0015\fz0h\ri\u00003\n2h\njUj2+jVj2cos 2\u0001\bi\n\u00001\n16\u00143h\ri\u00003\n2h\njUj2+ 2\u0015jVj2+ (1\u00002\u0015)jVj2cos 2\u0001\bi\n\u00001\n4\u0014\u0015h\ri\u00005\n2;(24)\nd\u0001\b\ndt=\u00001\n24re\u00144h\ri\u0010\njVj2+jUj2\u0011\nsin 2\u0001\b +1\n8\u0014\u00141\n4\u0015\fz0\u0000\u00142(1\u00002\u0015)\u0015\nh\ri\u00003\n2\u0010\njVj2\u0000jUj2\u0011\nsin2\u0001\b: (25)\nNote when doing exchange of UandVfor theydirection,\none also has to change the \u0006sign of \u0001\b.To further simplify we notice Eq. (23) can be rewritten\nwith the help of Eq. (18)\ndjUj\ndt=\u00001\n4h_\ri\nh\rijUj\u00001\n8re\u00144h\ri\u0014\njUj3+4\u0000cos 2\u0001\b\n3jVj2jUj\u0015\n\u00001\n16\u0014\u00141\n4\u0015\fz0\u0000\u00142(1\u00002\u0015)\u0015\nh\ri\u00003\n2jVj2jUjsin 2\u0001\b;\n(26)\nwhich reproduces Eq. (66) in Ref. [22] if V= 0. Introduce\nSx=\u0014h\ri1\n2jUj2; (27)\nSy=\u0014h\ri1\n2jVj2; (28)which have the physical meaning of the areas (divided by\n2\u0019) of the ellipses encircled by the particle trajectory in\nx-pxandy-pyphase spaces. Then Eqs. (18), (19), (23),4\n(24) and (25) can be rewritten as\nh_\ri=\u0000Ez0\fz0\u00001\n3re\u00143h\ri3\n2(Sx+Sy); (29)\n_h\u0010i=1\n2\r\u00002\nw\u00001\n2h\ri\u00002\u00001\n4\u0014h\ri\u00003\n2(Sx+Sy); (30)\n_Sx=\u00001\n4re\u00143h\ri1\n2\u0012\nS2\nx+4\u0000cos 2\u0001\b\n3SxSy\u0013\n\u00001\n8\u00141\n4\u0015\fz0\u0000\u00142(1\u00002\u0015)\u0015\nh\ri\u00002SxSysin 2\u0001\b; (31)\n_\bx=1\n24re\u00143h\ri1\n2Sysin 2\u0001\b +1\n64\u0015\fz0h\ri\u00002(Sx+Sycos 2\u0001\b)\n\u00001\n16\u00142h\ri\u00002[Sx+ 2\u0015Sy+ (1\u00002\u0015)Sycos 2\u0001\b]\u00001\n4\u0014\u0015h\ri\u00005\n2;(32)\nd\u0001\b\ndt=\u00001\n24re\u00143h\ri1\n2(Sy+Sx) sin 2\u0001\b +1\n8\u00141\n4\u0015\fz0\u0000\u00142(1\u00002\u0015)\u0015\nh\ri\u00002(Sy\u0000Sx) sin2\u0001\b: (33)\nIt is generally safe to take \fz0= 1 here. But Eq. (12), or\n\fz0= 1\u00001\n2h\nh\ri\u00002+1\n2\u0014h\ri\u00003=2(Sx+Sy)i\n, gives a bet-\nter accuracy. The above long-term equations, Eqs. (29)\n- (33), show that the BO experiences acceleration (for\nEz0<0) or deceleration (for Ez0>0), radiation damp-\ning, longitudinal phase drift, and betatron phase shift.\nThese equations may be used for the long-term behavior\nof BO without resolving the betatron period.\nIV. DISCUSSION ON TWO REGIMES\nFrom Eqs. (31) - (33) we note two regimes. One is the\nRR dominant regime, where reh\ri5=2\u001d1, so that the\n\frst terms in Eqs. (31) - (33) dominate. This regime has\nbeen discussed before [20], although only for the linearly\npolarized case \u0001\b = 0 (so that the ratio between Sx\nandSyis a constant). The other is the betatron phase\nshift dominant regime, where reh\ri5=2\u001c1, so that the\nremaining terms in Eqs. (31) - (33) dominate. These\nterms were previously proposed [22], but the betatron\nphase shift is found for the \frst time in the present work.\nIn the RR dominant regime, an interesting phe-\nnomenon is that an elliptical polarization (in the x-y\nplane) always approaches linear polarization, because\n\u0001\b always approaches the nearest integer multiple of\n\u0019according to Eq. (33). This phenomenon can also be\nviewed by rotating the xaxis to the major axis of the el-\nlipse, so that Sx>Syand \u0001\b =\u0019=2. De\fneR=Sy=Sx\nand perform time derivative with the help of Eq. (31)\n_R=\u00001\n6re\u00143h\ri1\n2R(Sx\u0000Sy)<0; (34)which suggests that the ellipse monotonically becomes\nthinner.\nThe betatron phase shift dominant regime requires a\nmoderate\r, or straightforwardly re!0, which corre-\nsponds to very dilute plasma case, leads to a constant\nS\u0011Sx+Sy. It can be proved that the time integral\nof Eq. (30) reproduces Eq. (6) in Ref. [21], which is the\nh\u0010idrift, in the case that h\rilinearly depends on t. In\nanother case that h\u0010idrifts around the zero point of Ez0,\nthe drift frequency can be obtained by using Eqs. (29)\nand (30), and asserting Ez0=\u0015h\u0010i\n!h\u0010i=s\n\u0015\fz0\u0012\n1 +3\n8\u0014h\ri1\n2S\u0013\nh\ri\u00003: (35)\nWe also note the angular momentum Lz=\rx\fy\u0000\ry\fx\nand its changing rate\nhLzi=\u0000S1\n2xS1\n2ysin \u0001\b; (36)\n_hLzi=\u00001\n3re\u00143h\ri1\n2(Sx+Sy)hLzi; (37)\nwhich obeys the law of conservation of angular momen-\ntum ifhLzi= 0 initially, or re!0. Especially for re!0,\nthe particle trajectory in the x-yplane generally encir-\ncles an ellipse with constant area and shape, which also\nhas precession leading to the rotation of the major and\nminor axes of the ellipse.5\nV. NUMERICAL COMPARISON OF\nLONG-TERM EQUATIONS AND THE\nORIGINAL ONES\nTo verify the LTEs, we solve them numerically us-\ning the backward-di\u000berentiation formulas (BDF) in the\nSciPy integration package [24]. Meanwhile, the original\nequations of motion with the force expressions Eqs. (4)\n- (6) are solved by Runge-Kutta 4th order method using\nthe code PTracker (PT) [25]. We choose four cases with\ntheir parameters and initial values listed in Tab. I, and\nthe comparison results are plotted from Fig. 1 to 4. Note\nthat \bxcannot be obtained directly from PT. Thus we\nperform the following treatment to the PT results\nx\u0001cos'=jUj\n2[cos \bx+ cos (2'+ \bx)]; (38)\nbecausex=jUjcos ('+ \bx), where'is obtained by\nnumerical integral based on Eq. (15). Then cos \b xcan\nbe obtained by a low-pass \flter. Similar treatment is\nperformed to obtain cos \u0001\b, according to\nx\u0001y=jUjjVj\n2[cos \u0001\b + cos (2 '+ \bx+ \by)]:(39)\nA case in the betatron phase shift dominant regime\nis shown in Fig. 1. We see Sx+Syis a constant, al-\nthoughSx,Syand \u0001\b change gradually. The approxi-\nmate \\phase-locking\" is chosen, i.e. \rw\u0019\rz0, thush\u0010i\noscillates near the zero point of Ezwith the drift fre-\nquency!h\u0010iaccording to Eq. (35). h\rioscillates with\nthe same drift frequency as shown in Fig. 1 (b).\nA second case during the transition of the two regimes\nis shown in Fig. 2. Sx+Syis approximately a constant\ninitially, and starts to decrease near the regime transition\n\r=r\u00002=5\ne.\nA third case in the RR dominant regime is shown in\nFig. 3. The initial values are chosen so that the particle\ntrajectory in the x-yplane is a ellipse with its major axis\nlaying on the xaxis. As shown in Fig. 3 (a), R=Sy=Sx\ndecreases monotonically, as predicted by Eq. (34).\nThe last case shown in Fig. 4 is also in the RR domi-\nnant regime, but the particle trajectory in the x-yplane\nis a oblique ellipse. As shown in Fig. 4 (c), \u0001\b gradually\napproaches \u0019, which is in accordance with the discussion\nin Sec. IV.\nIn all these plots, the results from PT and LTEs show\nagreement with high accuracy, demonstrating the cor-\nrectness of LTEs. Because the BO frequency is the high-\nest frequency in our physical process, the LTEs largely\nreduce the numerical complexity and meanwhile keep the\nlong-term accuracy.\nVI. CONCLUSIONS\nWe have established a three-dimensional betatron os-\ncillation model including radiation reaction to study the\n0 1 2 3 4\nt 1e52.55.07.510.012.5Sx, SySx, PT\nSx, LTE\nSy, PT\nSy, LTE\n0 1 2 3 4\nt 1e39698100102104\nPT\nLTE\n1 2 3 4\nt 1e51.0\n0.5\n0.00.51.0xy\nPT\nPT filtered\n|U||V|\n2cos, LTE\n1 2 3 4\nt 1e51.5\n1.0\n0.5\n0.00.51.0x cos\nPT\nPT filtered\n|U|\n2cosx, LTE\n(a) (b)\n(c)(d)\nFIG. 1. The numerical comparison of the LTEs and the orig-\ninal equations solved by PTracker in the betatron phase shift\ndominant regime. (a) SxandSychange with time, but Sx+Sy\nis a constant. (b) \rhas oscillation frequencies of 2 !\f\u00190:14\ndue to the BO and !h\u0010i\u00193:11\u000210\u00003due to the drift oscilla-\ntion ofh\u0010i. (c) The gray curve shows x\u0001yobtained from PT,\nand the black curve shows its low-pass \fltered result, which\nis compared with the LTE solution according to Eq. (39). (d)\nThe gray curve shows x\u0001cos'obtained from PT, and the\nblack curve shows its low-pass \fltered result, which is com-\npared with the LTE solution according to Eq. (38).\n0.0 0.5 1.0 1.5 2.0\nt 1e7010203040Sx, SySx, PT\nSx, LTE\nSy, PT\nSy, LTE\n0.0 0.5 1.0 1.5 2.0\nt 1e70.00.51.01.52.0\n1e4\nPT\nLTE\n0.5 1.0 1.5 2.0\nt 1e70.00.20.40.60.8xy\nPT\nPT filtered\n|U||V|\n2cos, LTE\n0.5 1.0 1.5 2.0\nt 1e70.25\n0.000.250.500.751.00xcos\nPT\nPT filtered\n|U|\n2cosx, LTE\n(a)(b)\n(c) (d)\nFIG. 2. The numerical comparison of the LTEs and the origi-\nnal equations solved by PTracker in the transition between the\nbetatron phase shift dominant and the RR dominant regimes.\n(a)Sx+Syis initially approximately a constant, but decreases\nlater. (b)\rincreases due to the acceleration \feld, and passes\nthe regime transition at \r=r\u00002=5\ne = 104. (c) and (d) show\nthe same treatments as in Fig. 1 (c) and (d).6\n0 2 4 6 8\nt 1e70.760.770.780.790.800.81RPT\nLTE\n0 1 2 3\nt 1e60.00.20.40.60.81.0\n1e6\nPT\nLTE\n1 2 3\nt 1e60.005\n0.0000.0050.0100.015xy\nPT\nPT filtered\n|U||V|\n2cos, LTE\n1 2 3\nt 1e60.000.050.100.150.20xcos\nPT\nPT filtered\n|U|\n2cosx, LTE\n(a)(b)\n(c) (d)\nFIG. 3. The numerical comparison of the LTEs and the origi-\nnal equations solved by PTracker in the RR dominant regime.\n\u0001\b =\u0019=2 andSx> Sy, thus the major axis of the particle\ntrajectory ellipse lays on the xaxis. (a)R=Sy=Sxdecreases\nwith time due to Eq. (34), thus the ellipse is getting thinner.\n(b)\rincreases due to the acceleration \feld. (c) and (d) show\nthe same treatments as in Fig. 1 (c) and (d).\n0 1 2 3 4\nt 1e70510152025Sx, SySx, PT\nSx, LTE\nSy, PT\nSy, LTE\n0 1 2 3 4\nt 1e70.500.751.001.251.501.75\n1e6\nPT\nLTE\n1 2 3 4\nt 1e72.0\n1.5\n1.0\n0.5\n0.02xy/|U||V|\nPT\nPT filtered\ncos, LTE\n1 2 3 4\nt 1e70.00.51.01.52xcos/|U|\nPT\nPT filtered\ncosx, LTE\n(a) (b)\n(c) (d)\nFIG. 4. The numerical comparison of the LTEs and the origi-\nnal equations solved by PTracker in the RR dominant regime.\nSx=Sy, thus the particle trajectory in the x-yplane is an\noblique ellipse. (a) SxandSydecrease with the same rate.\n(b)\rinitially decreases with time because the longitudinal\nRR damping is stronger than the acceleration. Later the RR\ndamping becomes weaker due to the decrease of SxandSy,\nand\rincreases. (c) and (d) show the same treatments as in\nFig. 1 (c) and (d), but the oscillation amplitudes are divided\nso that the changes of \u0001\b and \b xare clearer. \u0001\b is between\n\u0019=2 and\u0019, thus \u0001\b gradually approaches \u0019.TABLE I. The cases for comparing PT and LTEs\nCaseParameters Initial Values\nEz\u0015\u0014re\rwjUjjVj\bx\byh\rih\u0010i\nFig. 1\u0015\u00101\n41p\n20 141.12 0.87 0\u0019\n6102-0.05\nFig. 2 -0.001 01p\n210\u0000101041.12 0.87 0\u0019\n61030\nFig. 3\u0015\u00101\n21p\n210\u0000101040.20.18 0\u0019\n2103-0.1\nFig. 4 -0.1 01p\n210\u0000101040.2 0.2\u0019\n4\u00191060\nlong-term behavior of an electron in laser or beam driven\nplasma wake\feld. The original equations of motion have\nbeen expressed by the transverse oscillation terms as\nEqs. (7) - (10), and then averaged in one betatron pe-\nriod to obtain the long-term equations Eqs. (29) - (33).\nThe conditions of our model are r2\u001c\r,r2\r\u001d1 and\nr\rre=2\u000b\u001c1, as discussed in Appx. C. Our model, on\none hand, reproduces previous results such as longitu-\ndinal deceleration and transverse damping, and on the\nother hand reveals new phenomena such as longitudinal\nphase drift oscillation, betatron phase shift and betatron\npolarization change. Two regimes with distinct behav-\niors, determined by re\r5=2, are discussed in Sec. IV, and\nare demonstrated by numerical methods in Sec. V. The\nnumerical comparisons of the long-term equations and\nthe original equations of motion show the high accuracy\nof our model. This model can be fundamental for future\nplasma based high-energy accelerators and colliders [26].\nACKNOWLEDGMENTS\nMZ greatly appreciates the fruitful discussion on the\naveraging method with Igor Kostyukov and Anton Golo-\nvanov from Institute of Applied Physics RAS, Russia.\nThis work is supported by Research Foundation of Insti-\ntute of High Energy Physics, Chinese Academy of Sci-\nences (Grant Nos. E05153U1, E15453U2).\nAppendix A: Plasma normalization units\nThe plasma normalization units are used throughout\nthe paper, as listed in Tab. II, where cis the speed of\nlight in vacuum, !pis the plasma frequency, eis the\nelementary charge, and meis the electron mass. For\nexample, the time is normalized to !\u00001\np, means any time\nrelated quantity such as tin this paper actually means\n!ptin the unnormalized form.7\nTABLE II. The plasma normalization units\nPhysical quantities Variables Normalization units\ntime t !\u00001\np\nfrequency ! !p\nlength x;y;z;re c=!p\nvelocity v c\nmomentum p mec\nangular momentum L mec2=!p\nelectric \feld E mec!p=e\nmagnetic \feld Bme!p=e(in SI)\nforce f mec!p\nAppendix B: Symbols and rules\nIf any variable X, either real or complex, can be ex-\npressed asX=hXi+X1, wherehimeans taking average\nin the betatron period time scale, and X1is the BO term,\ntaking average and derivative can permute\n\u001cd\ndtX\u001d\n=d\ndthXi: (B1)\nWe use a dot on the top to express the time derivative if\nthere is no ambiguity. We have the order-of-magnitude\nestimation\n_X1\u0018!\fX1\u0018\r\u00001\n2X1: (B2)\nIfjX1j\u001cjhXij, for any power \u000bwe have\nhX\u000bi=hXi\u000b\"\n1 +O \nX2\n1\nhXi2!#\n: (B3)\nAnd if another variable Y=hYi+Y1also hasjY1j\u001c\njhYij,\nhXYi=hXihYi\u0014\n1 +O\u0012X1Y1\nhXihYi\u0013\u0015\n: (B4)\nIfXis a complex, taking average and modulus can per-\nmute\nhjXji=jhXij\"\n1 +O \njX1j2\njhXij2!#\n: (B5)\nHowever, taking modulus and derivative cannot permute.\nAppendix C: Equations of motion expressed by\ntransverse oscillations\nIn Eqs. (4) - (6), the longitudinal and transverse os-\ncillations are coupled. As shown in the following, thelongitudinal variables \u00101and\fzare dependent variables\nwhich can be expressed by the transverse ones.\nWe treat~fradas a perturbation and omit it \frst. On\none hand we have\n\r\u00002= 1\u0000\f2\nz\u0000\f2\nx\u0000\f2\ny\n=\r\u00002\nz0\u00002\fz0_\u00101\u0000\f2\nx\u0000\f2\ny+O\u0010\n_\u001012\u0011\n;(C1)\nwhere\rz0=\u0000\n1\u0000\f2\nz0\u0001\u00001=2. By taking average we get\n\r\u00002\nz0\u0019h\ri\u00002+\n\f2\nx\u000b\n+\n\f2\ny\u000b\n: (C2)\nWrite\r=h\ri+\r1in the form\n\r\u00002=h\ri\u00002\"\n1\u00002\r1\nh\ri+O \n\r2\n1\nh\ri2!#\n; (C3)\nwe have\n\r1\u0019\"\n\f2\nx\u0000\n\f2\nx\u000b\n2+\f2\ny\u0000\n\f2\ny\u000b\n2+\fz0_\u00101#\nh\ri3:(C4)\nOn the other hand,\n_\r=\u0000Ez0\fz0\u0000\u0015\fz0\u00101\u0000\u00142(1\u0000\u0015) (x\fx+y\fy) (C5)\nby applying _ \r=\u0000~\f\u0001~Eand Eqs. (1) and (2), or\n_\r1=\u0000\u0015\fz0\u00101\u0000\u00142(1\u0000\u0015) (x\fx+y\fy): (C6)\nNote Eq. (B2), Eq. (C4) seams incompatible with\nEq. (C6), unless\n_\u00101=\u0000\f2\nx\u0000\n\f2\nx\u000b\n2\u0000\f2\ny\u0000\n\f2\ny\u000b\n2; (C7)\nwhich leads to\n\u00101=\u0000x\fx+y\fy\n4; (C8)\nwhich is a general form of Eq. (18) in Ref. [22]. Then\n1\u0000\fz= 1\u0000\fz0\u0000_\u00101=1\n2\u0010\nh\ri\u00002+\f2\nx+\f2\ny\u0011\n;(C9)\nand the formulas of 3D BO with negligible RR are\n_\r=\u0000Ez0\fz0+\u0012\u0015\fz0\n4+\u00142\u0015\u0000\u00142\u0013\n(x\fx+y\fy);\n(C10)\n_pz=\u0000Ez0+\u0015\u00121\n4+\u00142\u0013\n(x\fx+y\fy); (C11)\n_px=\u0000\u00142x+\u00142\u0015\n2\u0010\nh\ri\u00002+\f2\nx+\f2\ny\u0011\nx; (C12)\n_py=\u0000\u00142y+\u00142\u0015\n2\u0010\nh\ri\u00002+\f2\nx+\f2\ny\u0011\ny: (C13)8\nFrom Eq. (C10) we may write\n\r=h\ri+\u0012\u0015\fz0\n4+\u00142\u0015\u0000\u00142\u0013x2\u0000\nx2\u000b\n+y2\u0000\ny2\u000b\n2;\n(C14)\nindicating the prerequisite of the above derivation, which\nhas used Eq. (B3), is r2\u001ch\ri.\nNow we consider RR as a perturbation. The LAD\nequation for the RR four-force is [11]\nFrad\n\u0016=2\n3re\u0014d2P\u0016\nd\u001c2+\u0012dP\u0017\nd\u001cdP\u0017\nd\u001c\u0013\nP\u0016\u0015\n: (C15)with the metric (1 ;\u00001;\u00001;\u00001), wherereis the classical\nelectron radius (also normalized to k\u00001\np),P\u0016is the four-\nmomentum, and \u001cis the proper time ( d\u001c=dt=\r). Use\nEqs. (C10) - (C13) we can verify\ndP\u0017\nd\u001cdP\u0017\nd\u001c=\r2\u0012\n_\r2\u0000\f\f\f_~ p\f\f\f2\u0013\n\u0019\u0000\r2\u0000\n_px2+ _py2\u0001\n(C16)\nas long as r2\r2\u001d1. We can also prove that the \frst\nterm in Eq. (C15) is negligible compared with the second\nterm as long as r2\r\u001d1. Finally the equations of motion\nexpressed by the transverse oscillations are obtained as\nEqs. (7) - (10). As it has been discussed in Ref. [22] and\n[23], this classical RR model is valid as long as r\rre=2\u000b\u001c\n1, where\u000bis the \fne structure constant.\n[1] T. Tajima and J. M. 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Troncoso, and Arne Brataas\nCenter for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: July 3, 2019)\nGilbert damping is a key property governing magnetization dynamics in ordered magnets. We present a\ntheoretical study of intrinsic Gilbert damping induced by magnon decay in antiferromagnetic metals through\ns-dexchange interaction. Our theory delineates the qualitative features of damping in metallic antiferromagnets\nowing to their bipartite nature, in addition to providing analytic expressions for the damping parameters. Magnon-\ninduced intraband electron scattering is found to predominantly cause magnetization damping, whereas the Néel\nfield is found to be damped via disorder. Depending on the conduction electron band structure, we predict that\nmagnon-induced interband electron scattering around band crossings may be exploited to engineer a strong Néel\nfield damping.\nIntroduction.— The dynamical properties of a harmonic\nmode are captured by its frequency and lifetime [ 1,2]. While\nthe eigenfrequency is typically determined by the linearized\nequations of motion, or equivalently by a non-interacting de-\nscription of the corresponding quantum excitation, the lifetime\nembodies rich physics stemming from its interaction with one\nor more dissipative baths [ 1,3]. Dissipation plays a central\nrole in the system response time. In the context of magnetic\nsystems employed as memories, the switching times decrease\nwith increasing damping thereby requiring a stronger dissi-\npation for fast operation [ 4–6]. The dissipative properties of\nthe system also result in rich phenomena such as quantum\nphase transitions [ 7–10]. Furthermore, the formation of hybrid\nexcitations, such as magnon-polarons [ 11–18] and magnon-\npolaritons [ 19–24], requires the dissipation to be weak with\nrespect to the coupling strengths between the two participating\nexcitations [ 25]. Therefore, in several physical phenomena that\nhave emerged into focus in the recent years [ 12,16,26–30],\ndamping not only determines the system response but also the\nvery nature of the eigenmodes themselves. Understanding,\nexploiting and controlling the damping in magnets is thus a\nfoundational pillar of the field.\nThe success of Landau-Lifshitz-Gilbert (LLG) phenomenol-\nogy [ 31,32] in describing ferromagnetic dynamics has inspired\nvigorous e \u000borts towards obtaining the Gilbert damping param-\neter using a wide range of microscopic theories. The quantum\nparticles corresponding to magnetization dynamics - magnons\n- provide one such avenue for microscopic theories and form\nthe central theme in the field of magnonics [ 33,34]. While\na vast amount of fruitful research has provided a good under-\nstanding of ferromagnets (FMs) [ 35–54], analogous studies on\nantiferromagnets (AFMs) are relatively scarce and have just\nstarted appearing [ 55,56] due to the recently invigorated field\nof antiferromagnetic spintronics [ 57–62]. Among the ongoing\ndiscoveries of niches borne by AFMs, from electrically and\nrapidly switchable memories [ 63], topological spintronics [ 60],\nlong range magnonic transport [ 64] to quantum fluctuations\n[65], an unexpected surprise has been encountered in the first\nprinciples evaluation of damping in metallic AFMs. Liu and\ncoworkers [ 56] and another more recent first-principles study\n[66] both found the magnetization dissipation parameter to bemuch larger than the corresponding Néel damping constant,\nin stark contrast with previous assumptions, exhibiting richer\nfeatures than in FMs. An understanding of this qualitative\ndi\u000berence as well as the general AFM dissipation is crucial\nfor the rapidly growing applications and fundamental novel\nphenomena based on AFMs.\nHere, we accomplish an intuitive and general understanding\nof the Gilbert damping in metallic AFMs based on the magnon\npicture of AFM dynamics. Employing the s-d, two-sublattice\nmodel for a metallic AFM, in which the dandselectrons\nconstitute the magnetic and conduction subsystems, we derive\nanalytic expressions for the Gilbert damping parameters as\na function of the conduction electron density of states at the\nFermi energy and s-dexchange strength. The presence of spin-\ndegenerate conduction bands in AFMs is found to be the key\nin their qualitatively di \u000berent damping properties as compared\nto FMs. This allows for absorption of AFM magnons via s-\ndexchange-mediated intraband conduction electron spin-flip\nprocesses leading to strong damping of the magnetization as\ncompared to the Néel field [ 67]. We also show that interband\nspin-flip processes, which are forbidden in our simple AFM\nmodel but possible in AFMs with band crossings in the conduc-\ntion electron dispersion, result in a strong Néel field damping.\nThus, the general qualitative features of damping in metallic\nAFMs demonstrated herein allow us to understand the Gilbert\ndamping given the conduction electron band structure. These\ninsights provide guidance for engineering AFMs with desired\ndamping properties, which depend on the exact role of the\nAFM in a device.\nModel.— We consider two-sublattice metallic AFMs within\nthes-dmodel [ 35,36,44]. The delectrons localized at lat-\ntices sites constitute the magnetic subsystem responsible for\nantiferromagnetism, while the itinerant selectrons form the\nconduction subsystem that accounts for the metallic traits. The\ntwo subsystems interact via s-dexchange [Eq. (3)]. For ease of\ndepiction and enabling an understanding of qualitative trends,\nwe here consider a one-dimensional AFM (Fig. 1). The re-\nsults within this simple model are generalized to AFMs with\nany dimensionality in a straightforward manner. Furthermore,\nwe primarily focus on the uniform magnetization dynamics\nmodes.arXiv:1907.01045v1  [cond-mat.mes-hall]  1 Jul 20192\nFIG. 1: Schematic depiction of our model for a metallic AFM.\nThe red and blue arrows represent the localized delectrons\nwith spin up and down, respectively, thereby constituting the\nNéel ordered magnetic subsystem. The green cloud illustrates\nthe delocalized, itinerant selectrons that forms the conduction\nsubsystem.\nAt each lattice site i, there is a localized delectron with spin\nSi. The ensuing magnetic subsystem is antiferromagnetically\nordered (Fig. 1), and the quantized excitations are magnons\n[68,69]. Disregarding applied fields for simplicity and as-\nsuming an easy-axis anisotropy along the z-axis, the magnetic\nHamiltonian, Hm=˜JP\nhi;jiSi\u0001Sj\u0000KP\ni(Sz\ni)2, wherehi;ji\ndenotes summation over nearest neighbor lattice sites, is quan-\ntized and mapped to the sublattice-magnon basis [69]\nHm=X\nqh\nAq\u0010\nay\nqaq+by\nqbq\u0011\n+By\nqay\nqby\nq+Bqaqbqi\n; (1)\nwhere we substitute ~=1,Aq=(2˜J+2K)SandBq=\n˜JS e\u0000iq\u0001aP\nh\u000eieiq\u0001\u000e, where S=jSij,ais the displacement\nbetween the two atoms in the basis, and h\u000eidenotes sum-\nming over nearest neighbor displacement vectors. aqandbq\nare bosonic annihilation operators for plane wave magnons\non the A and B sublattices, respectively. We diagonalize\nthe Hamiltonian [Eq. 1] through a Bogoliubov transforma-\ntion [ 69] toHm=P\nq!q\u0010\n\u000by\nq\u000bq+\fy\nq\fq\u0011\n;with eigenenergies\n!q=q\nA2q\u0000jBqj2. In the absence of an applied field, the\nmagnon modes are degenerate.\nTheselectron conduction subsystem is described by a tight-\nbinding Hamiltonian that includes the “static” contribution\nfrom the s-dexchange interaction [Eq. (3)] discussed below:\nHe=\u0000tX\nhi;jiX\n\u001bcy\ni\u001bcj\u001b\u0000JX\ni(\u00001)i\u0010\ncy\ni\"ci\"\u0000cy\ni#ci#\u0011\n:(2)\nHere ci\u001bis the annihilation operator for an selectron at site\niwith spin\u001b.t(>0)is the hopping parameter, and J(>0)\naccounts for s-dexchange interaction [Eq. (3)]. The (\u00001)i\nfactor in the exchange term reflects the two-sublattice nature of\nthe AFM. The conduction subsystem unit cell consists of two\nbasis atoms, similar to the magnetic subsystem. As a result,\nthere are four distinct electron bands: two due to there being\ntwo basis atoms per unit cell, and twice this due to the two\npossible spin polarizations per electron. Disregarding applied\nfields, these bands constitute two spin-degenerate bands. We\nlabel these bands 1 and 2, where the latter is higher in energy.The itinerant electron Hamiltonian [Eq. (2)] is diagonalized\ninto an eigenbasis (c1k\u001b;c2k\u001b)with eigenenergies \u000f1k=\u0000\u000fk\nand\u000f2k= +\u000fk, where\u000fk=p\nJ2S2+t2j\rkj2, where\rk=P\nh\u000eie\u0000ik\u0001\u000e. The itinerant electron dispersion is depicted in Fig.\n2.\nThe magnetic and conduction subsystems interact through\ns-dexchange interaction, parametrized by J:\nHI=\u0000JX\niSi\u0001si; (3)\nwhere si=P\n\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0is the spin of the itinerant elec-\ntrons at site i, where \u001bis the vector of Pauli matrices. The term\nwhich is zeroth order in the magnon operators, and thus ac-\ncounts for the static magnetic texture, is already included in He\n[Eq. (2)]. To first order in magnon operators, the interaction\nHamiltonian can be compactly written as\nHe\u0000m=X\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n+h.c.;(4)\nwhere\u0015and\u001aare summed over the electron band indices. As\ndetailed in the Supplemental material, WA;\u0015\u001a\nkk0qandWB;\u0015\u001a\nkk0q, both\nlinear in J, are coe \u000ecients determining the amplitudes for\nscattering between the itinerant electrons and the aqandbq\nmagnons, respectively. Specifically, when considering plane\nwave states, WA=B;\u0015\u001a\nkk0qbecomes a delta function, thereby enforc-\ning the conservation of crystal momentum in a translationally\ninvariant lattice. Inclusion of disorder or other many-body\ne\u000bects results in deviation of the eigenstates from ideal plane\nwaves causing a wave vector spread around its mean value [ 2].\nThe delta function, associated with an exact crystal momentum\nconservation, is thus transformed to a peaked function with\nfinite width (\u0001k). The\u0015\u001acombinations 11and22describe\nintraband electron scattering, while 12and21describe in-\nterband scattering. Intraband scattering is illustrated in Fig.\n2. Interband scattering is prohibited within our model due to\nenergy conservation, since the uniform q=0magnon energy\nis much smaller than the band gap.\nThe scattering described by He\u0000m[Eq. (4)] transfers spin\nangular momentum between the magnetic and conduction sub-\nsystems. The itinerant electrons are assumed to maintain a\nthermal distribution thereby acting as a perfect spin sink. This\nis consistent with a strong conduction electron spin relaxation\nobserved in metallic AFMs [ 70,71]. As a result, the magnetic\nsubsystem spin is e \u000bectively damped through the s-dexchange\ninteraction.\nGilbert damping.— In the Landau-Lifshitz-Gilbert (LLG)\nphenomenology for two-sublattice AFMs, dissipation is ac-\ncounted via a 2\u00022 Gilbert damping matrix [ 72]. Our goal here\nis to determine the elements of this matrix in terms of the\nparameters and physical observables within our microscopic\nmodel. To this end, we evaluate the spin current “pumped”\nby the magnetic subsystem into the sconduction electrons,\nwhich dissipate it immediately within our model. The angu-\nlar momentum thus lost by the magnetic subsystem appears\nas Gilbert damping in its dynamical equations [ 72,73]. The3\n-\n/2 -\n /4 0\n /2\n /4\ne\ne\nkF,1a/epsilon1=µ1\nm\ne\ne\nm\n/epsilon1=µ2\nkF,2a\nFIG. 2: The selectron dispersion in metallic AFM model,\nwhere the red and blue dispersions depict electron bands 1 and\n2, respectively. Illustrations of intraband electron-magnon\nscattering at two di \u000berent Fermi levels, \u00161and\u00162, are added.\nThe depicted momentum transfer is exaggerated for clarity.\nsecond essential ingredient in identifying the Gilbert damping\nmatrix from our microscopic theory is the idea of coherent\nstates [ 74,75]. The classical LLG description of the magne-\ntization is necessarily equivalent to our quantum formalism,\nwhen the magnetic eigenmode is in a coherent state [ 74–76].\nDriving the magnetization dynamics via a microwave field,\nsuch as in the case of ferromagnetic resonance experiments,\nachieves such a coherent magnetization dynamics [73, 77].\nThe spin current pumped by a two-sublattice magnetic sys-\ntem into an electronic bath may be expressed as [78]\nIz=Gmm(m\u0002˙m)z+Gnn(n\u0002˙n)z\n+Gmn\u0002(m\u0002˙n)z+(n\u0002˙m)z\u0003;(5)\nwhere mand nare the magnetization and Néel field nor-\nmalized by the sublattice magnetization, respectively. Here,\nGi j=\u000bi j\u0002(M=j\rj), where\u000bi jare the Gilbert damping co-\ne\u000ecients,\ris the gyromagnetic ratio of the delectrons\nandMis the sublattice magnetization. Considering the uni-\nform magnetization mode, Izis the spin current operator\nIz=i[He\u0000m;Sz][79], where Sz=P\niSz\ni. We get\nIz=iX\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n\u0000h.c.:(6)\nThe expectation value of this operator assuming the uniform\nmagnetization mode to be in a coherent state corresponds to\nthe spin pumping current [Eq. (5)].\nIn order to evaluate the spin pumping current from Eq. (6),\nwe follow the method employed to calculate interfacial spin\npumping current into normal metals in Refs. [ 73,77,78], and\nthe procedure is described in detail therein. Briefly, this method\nentails assuming the magnetic and conduction subsystems to\nbe independent and in equilibrium at t=\u00001, when the mu-\ntual interaction [Eq. (4)] is turned on. The subsequent timeevolution of the coupled system allows evaluating its physical\nobservables in steady state. The resulting coherent spin-current\ncorresponds to the classical spin current Izthat can be related\nto the motion of the magnetization and the Néel field [Eq. (5)].\nAs a last step, we identify expressions for (m\u0002˙m)z,(m\u0002˙n)z\nand(n\u0002˙n)zin terms of coherent magnon states, which enables\nus to identify the Gilbert damping coe \u000ecients\u000bmm,\u000bnnand\n\u000bmn.\nResults.— Relegating the detailed evaluation to Supplemen-\ntal Material, we now present the analytic expression obtained\nfor the various coe \u000ecients [Eq. (5)]. A key assumption that\nallows these simple expressions is that the electronic density of\nstates in the conduction subsystem does not vary significantly\nover the magnon energy scale. Furthermore, we account for a\nweak disorder phenomenologically via a finite scattering length\nlassociated with the conduction electrons. This results in an\ne\u000bective broadening of the electron wavevectors determined by\nthe inverse electron scattering length, (\u0001k)=2\u0019=l. As a result,\nthe crystal momentum conservation in the system is enforced\nonly within the wavevector broadening. By weak disorder we\nmean that the electron scattering length is much larger than\nthe lattice parameter a. Ifkandk0are the wave vectors of the\nincoming and outgoing electrons, respectively, we then have\n(k\u0000k0)a=(\u0001k)a\u001c1. This justifies an expansion in the wave\nvector broadening (\u0001k)a. The Gilbert damping coe \u000ecients\nstemming from intraband electron scattering are found to be\n\u000bmm=\u000b0(\u0018J)\u0000\u000b0(\u0018J)\n40BBBBBBBB@1+\u00182\nJ\u0010\n\u00182\nJ+8\u00004 cos2(kFa)\u0011\n\u0010\n\u00182\nJ+4 cos2(kFa)\u001121CCCCCCCCA[(\u0001k)a]2;\n\u000bnn=\u000b0(\u0018J)\n40BBBBBB@1+sin2(kFa)\ncos2(kFa)\u00182\nJ\u0010\n\u00182\nJ+4 cos2(kFa)\u00111CCCCCCA[(\u0001k)a]2:\n(7)\nwhere\u0018J=JS=t,kFis the Fermi momentum and ais the lattice\nparameter, and where\n\u000b0(\u0018J)=\u0019v2J2\n8g2(\u0016)j˜Vj24 cos2(kFa)\n\u00182\nJ+4 cos2(kFa): (8)\nHere, vis the unit cell volume, g(\u000f)is the density of states\nper unit volume, \u0016is the Fermi level, and !0is the energy of\ntheq=0magnon mode. ˜Vis a dimensionless and generally\ncomplex function introduced to account for the momentum\nbroadening dependency of the scattering amplitudes. It satisfies\n˜V(0)=1and0\u0014j˜V(\u0001k)j\u00141within our model. These analytic\nexpressions for the Gilbert damping parameters constitute one\nof the main results of this letter.\nDiscussion.– We straightaway note that \u000bnn=\u000bmm\u0018\n[(\u0001k)a]2\u001c1.\u000bnnis strictly dependent upon (\u0001k)a, and is non-\nzero only if there is disorder and a finite electron momentum\nbroadening. \u000bmmis large even when considering a perfectly\nordered crystal. This latter result is in good accordance with\nrecent first-principles calculations in metallic AFMs [ 56,66].\nWe moreover observe that both \u000bmmand\u000bnnare quadratic\ninJandg(\u0016). This result is shared by Gilbert damping ow-\ning to spin-pumping in insulating ferrimagnet |normal metal4\ne\n e\nm\nkFa/epsilon1=µ\nFIG. 3: A schematic depiction of magnon-induced interband\nscattering in a band crossing at the Fermi level.\n(NM) and AFM |NM bilayers with interfacial exchange cou-\npling [ 78]. Metallic AFMs bear a close resemblance to these\nbilayer structures. There are however two main di \u000berences:\nThes-dexchange coupling exists in the bulk of metallic AFMs,\nwhereas it is localized at the interface in the bilayer structures.\nAdditionally, the itinerant electron wave functions are qual-\nitatively di \u000berent in metallic AFMs and NMs, owing to the\nmagnetic unit cell of the AFM. Indeed, these di \u000berences turn\nout to leave prominent signatures in the Gilbert damping in\nmetallic AFMs.\nThe uniform mode magnon energy is much smaller than the\nelectron band gap within our simple model. Interband scat-\ntering is thus prohibited by energy conservation. However,\nin real AFM metals, the electron band structure is more com-\nplex. There may for instance exist band crossings [ 80–82].\nIn such materials, magnon-induced interband electron scatter-\ning should also contribute to Gilbert damping, depending on\nthe position of the Fermi surface. Motivated by this, we now\nconsider Gilbert damping stemming from interband scattering,\nwhile disregarding the energy conservation for the moment,\nlabeling the coe \u000ecients\u000bI\nmmand\u000bI\nnn. We then find the same\nexpressions as in Eq. (7) with the roles of \u000bI\nmm;nninterchanged\nwith respect to \u000bmm;nn. This implies that \u000bI\nnnis large and inde-\npendent of electron momentum broadening, whereas \u000bI\nmmis\nproportional to the electron momentum broadening squared.\nAlthough arriving at this result required disregarding the en-\nergy conservation constraint, the qualitative e \u000bect in itself is\nnot an artifact of this assumption. In materials with a band\ncrossing, as depicted in Fig. 3, \u000bI\nnn=\u000bI\nmm> \u000b nn=\u000bmmis a gen-\neral result. This generic principle derived within our simple\nmodel provides valuable guidance for designing materials with\nan engineered Gilbert damping matrix.\nWe now provide a rough intuitive picture for the damping\ndependencies obtained above followed by a more mathemati-\ncal discussion. Consider a conventional di \u000braction experiment\nwhere an incident probing wave is able to resolve the two\nslits only when the wavelength is comparable to the physical\nseparation between the two slits. In the case at hand, the wave-\nfunctions of electrons and magnon participating in a scatteringprocess combine in a way that the wavenumber by which the\nconservation of crystal momentum is violated becomes the\nprobing wavenumber within a di \u000braction picture. Therefore,\nthe processes conserving crystal momentum have vanishing\nprobing wavenumber and are not able to resolve the opposite\nspins localized at adjacent lattice sites. Therefore, only the aver-\nage magnetization is damped leaving the Néel field una \u000bected.\nWith disorder, the probing wavenumber becomes non-zero and\nthus also couples to the Néel field. The interband scattering,\non the other hand, is reminiscent of Umklapp scattering in a\nsingle-sublattice model and the probing wavenumber matches\nwith the inverse lattice spacing. Therefore, the coupling with\nthe Néel field is strong.\nThe Gilbert damping in metallic AFMs here considered is\ncaused by spin pumping from the magnetic subsystem into\nthesband. This spin pumping induces electron transitions\nbetween spin\"/#states among the selectrons. The Gilbert\ndamping coe \u000ecients depend thus on transition amplitudes pro-\nportional to products of itinerant electron wave functions such\nas y\n\u0015k\"(x) \u001ak0#(x). The damping e \u000bect on sublattice A depends\non this transition amplitude evaluated on the A sublattice, and\nequivalently for the B sublattice. Assuming without loss of gen-\nerality that site i=0belongs to sublattice A, we find in the one-\ndimensional model that the damping on sublattice A is a func-\ntion ofP\njcos2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj), whereas the damping\non sublattice B is a function ofP\njsin2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj).\nEquivalently, by straightforwardly using that m=(mA+mB)=2\nandn=(mA\u0000mB)=2, this analysis predicts that \u000bmmis a\nfunction ofP\nj y\n\u0015k\"(xj) \u001ak0#(x), whereas\u000bnnis a function of\nP\njcos\u0010\u0019xj\na\u0011\n y\n\u0015k\"(xj) \u001ak0#(x). Assuming plane wave solutions\nof the electron wave functions, and if we consider intraband\nscattering only, we more concretely find that \u000bmmis a function\nof(1\u0000i(\u0001k)a), where iis the imaginary unit, whereas \u000bnnis a\nfunction of ( \u0001k)a. This coincides well with Eq. (7).\nAbove, we presented a discussion of interband scattering in\nthe minimal model where the band gap artificially was set to\nzero. In this limit, the upper electron band is a continuation\nof the lower band with a \u0006\u0019=amomentum shift. We may then\nwrite 2k\u001b= 1;k+\u0019=a;\u001b. Under the assumption of a disappear-\ning band gap, momentum-conserving interband scattering at\nmomentum kis therefore equivalent to intraband scattering be-\ntween kandk\u0006\u0019=a. This is the exact phase shift which results\nin a small\u000bmmand a large \u000bnnconsistent with the discussion\nabove. In real metallic AFMs with complex band structures,\nthe exact wave function relations unveiled above do not apply.\nHowever, interband transition amplitudes will undoubtedly\ncarry a position dependent phase. This position dependence\nresults in a dephasing of transition amplitudes at neighboring\nlattice sites, which gives rise to a non-negligible \u000bnn. The pre-\ncise damping coe \u000ecients in real metallic AFMs depend on the\ndetailed electron wave functions. We may however generally\nconclude that \u000bI\nnn=\u000bI\nmm>\u000b nn=\u000bmm.\nConclusion.— We have provided a microscopic derivation\nof Gilbert damping resulting from magnon decay through s-d\nexchange interaction in metallic antiferromagnets. Analytic5\nexpressions for Gilbert damping coe \u000ecients resulting from in-\ntraband electron scattering are presented, while Gilbert damp-\ning resulting from interband electron scattering is discussed on\na conceptual level. We find that intraband electron scattering\ngives rise to a large magnetization damping and a negligible\nNéel field damping. The intraband Néel field damping is pro-\nportional to the inverse electron scattering length squared, and\ndisappears exactly if there is no crystal disorder. 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Gal’tsov(2,3)\n(1)Institute of Physics and Research Centre of Theoretical Phy sics and Astrophysics,\nSilesian University in Opava, Bezruˇ covo n´ am.13, CZ-7460 1 Opava, Czech Republic\n(2)Faculty of Physics, Moscow State University, 119899, Mosco w, Russia and\n(3)Kazan Federal University, 420008 Kazan, Russia\nDraft version March 28, 2018\nAbstract\nIn many astrophysically relevant situations radiation reaction forc e acting upon a charge can not\nbe neglected and the question arises about the location and stability of circular orbits in such regime.\nMotion of point charge with radiation reaction in flat spacetime is desc ribed by Lorenz-Dirac (LD)\nequation, while in curved spacetime – by DeWitt-Brehme (DWB) equat ion containing the Ricci term\nand the tail term. We show that for the motion of elementary partic les in vacuum metrics the DWB\nequation can be reduced to the covariant form of the LD equation w hich we use here. Generically,\nthe LD equation is plagued by runaway solutions, so we discuss compu tational ways to avoid this\nproblem in constructing numerical solutions. We also use the first ite ration of the covariant LD\nequation which is the covariant Landau-Lifshitz equation, comparin g results of these two approaches\nand showing smallness of the third-order Schott term in the ultrare lativistic case. We calculate the\ncorresponding energy and angular momentum loss of a particle and s tudy the damping of charged\nparticle oscillations around an equilibrium radius. We find that dependin g on the orientation of the\nLorentz force, the oscillating charged particle either spirals down t o the black hole, or stabilizes the\ncircular orbit by decaying its oscillations. The later case leads to an int eresting new result of shifting\nof the particle orbit outwards from the black hole. We also discuss th e astrophysical relevance of the\npresented approach and provide estimations of the main paramete rs of the model.\nSubject headings: black hole physics, magnetic fields, radiation reaction\n1.INTRODUCTION\nSynchrotron radiation emitted by a charged particle\nleads to appearance of the back-reaction force which can\nsignificantly affect its motion. The purpose of this pa-\nper is to study the motion of charged particles in the\ncombined magnetic and gravitational fields, taking into\naccount the radiation reaction force. There is convincing\nevidence that magnetic fields are indeed present in the\nvicinity of black holes. The observations of the Galac-\nticCentre(Eatough et al.(2013))havedemonstratedthe\nexistenceofstrongmagneticfieldofhundredGaussinthe\nvicinity of the supermassive black hole at the Galactic\ncentre. Recent studies of the quasi-periodic oscillations\n(QPOs) observed in the black hole microquasars have\nshown possible signatures of Galactic magnetic fields in\nthe vicinity of three microquasars (Koloˇ s et al. (2017)),\nif ”magnetic” generalizations of the geodesic models of\ntwin high frequency (HF) QPOs (Stuchl´ ık et al. (2013))\nare applied to data. Observed HF QPOs and relativistic\njets in microquasars indicate the presence of an external\nmagnetic field, influencing both oscillations in the accre-\ntion disk and creation of jets. The presence of radiation-\nreaction force and its relevance in the vicinity of black\nholes can make sufficient contribution for the shifts in\nthe QPO frequencies, which are usually observed. More-\nover, the radiation reaction can support the accretion of\ncharged particles from accretion disk towards the black\nhole. Depending on the magnitude of the external mag-\nnetic field, the radiation-reaction force can considerably\nshift the stable orbits of a particle which can sufficientlyinfluence the predictions of black hole parameters.\nThe weakness of magnetic field in our study is under-\nstood in the sense that it does not perturb the spacetime\nmetric; the corresponding condition on the field magni-\ntudeBin the vicinityofSchwarzschildblackholeofmass\nMreads (Gal’tsov & Petukhov (1978)):\nB << B G=c4\nG3/2M⊙/parenleftbiggM⊙\nM/parenrightbigg\n∼1019M⊙\nMGauss.(1)\nThis weakness is compensated by large ratio e/mfor\nelectrons and protons, whose motion will be essentially\naffected by magnetic fields already of the order of few\ngauss.\nStudy of the particle motion and electrodynam-\nics in magnetized black holes began long ago, see\ne.g. Gal’tsov & Petukhov (1978); Blandford & Znajek\n(1977), for a review of early results see Aliev & Gal’tsov\n(1989). Equatorial orbits are of primary interest for\ntheory of accretion disks. It was shown that the in-\nnermost stable circular orbits (ISCO) in the field of\nmagnetized black hole are shifted towards the hori-\nzon (Gal’tsov & Petukhov (1978)) for suitable direction\nof rotation. Recently, detailed studies of the equa-\ntorial motion of charged particles in weakly magne-\ntized Schwarzschild and Kerr black holes were per-\nformed in Frolov & Shoom (2010); Koloˇ s et al. (2015);\nTursunov et al. (2016), revealing a number of new types\nof motion.\nEstimates show that in many physically interesting\ncases the radiation reaction force is non-negligible, and2\nthis has prompted us to consider what happens once the\nreaction force is taken into account. The general prob-\nlem of the synchrotron radiation and treatment of the\nradiation-reaction force in curved spacetime was widely\nstudied in literature. Below, we remind only few of\nthem. The electromagnetic radiation emitted by rela-\ntivistic particles in the presence of strong gravity has\nbeen studied, e.g., in Poisson (2004); Johnston et al.\n(1973); Zerilli (2000); Price et al. (2013). The problem\nof synchrotron radiation in curved spacetime has been\nstudied, in Sokolov et al. (1983, 1978) using covarianti-\nzation of the flat space results. Actually, we will use here\nthe same approach to describe the reaction force. For\nthe recent treatment of radiation reaction in flat space\nsee Gal’tsov & Spirin (2006) where the derivation of the\nShott term was given explicitly along the lines of the\nTeitelboim idea to associate it with the Coulomb part of\nthe electromagnetic field of a point charge. It is worth\nnoting that the problem of the Shott contribution to re-\naction force (the third derivative term) still remains the\nsource of discussion (see e.g. Poisson (2004)), so we will\ntouch it in this paper too.\nGeneralizationofradiationreactionequationtocurved\nspace was given by DeWitt & Brehme (1960). Their re-\nsult (corrected by Hobbs (1968)) shows that radiation\nreaction in curved spacetime is essentially non-local and\nappeals to solve the integral equation. But contrary\nto gravitational radiation reaction in collisions of black\nholes, electromagnetic radiation from point particles can\nstill be simplified to a local theory.\nRecently, generalizationof the radiation reactionequa-\ntion to higher dimensions was discussed in Gal’tsov\n(2002); Gal’tsov & Spirin (2007) which may be rele-\nvant to extra-dimensional models. Radiation from hy-\npothetical massless charges was considered in Gal’tsov\n(2015) bringing new light on the relationship be-\ntween the ultrarelativistic and the massless limits in\nsynchrotron radiation and radiation reaction equa-\ntion. The motion of point particles with scalar,\nelectric and mass charges, has been reviewed in\nPoisson (2004). The problem of radiation-reaction\nfor extended charged particles has been studied in\nCremaschini & Tessarotto (2011), its quantum aspects\nare discussed in Cremaschini & Tessarotto (2015). Sta-\ntistical treatment of the radiation-reaction problem is\ngiven in Cremaschini & Tessarotto (2013). The problem\nof gravitational self-force of test particles is summarized\nin Barack (2014). Nevertheless, despite the active in-\nterest in the topic, and comprehensive literature review,\nthe successful attempts to integrate the equations of mo-\ntion in the curved background combined with external\nelectromagnetic fields is quite rare.\nIn spite of weakness of radiation reaction force in or-\ndinary stellar astrophysical situations, it was found re-\ncently that in high energy plasma processes in pulsar\nmagnetospheres, black-hole accretion disks, hot accre-\ntion flows in X-ray Binaries and Active Galactic Nu-\nclei, relativistic jets, and so on, which are dominated\nby magnetic reconnection in plasma, radiation reaction\nmayplayacrucialrole. The newtheory, named radiative\nmagnetic reconnection (for a recent review see Uzdensky\n(2016)), predicts notable radiative effects in astrophysi-\ncal reconnection such as radiation-reactionlimits on par-\nticle acceleration, radiative cooling, radiative resistivity,braking of reconnection outflows by radiation drag, radi-\nation pressure, viscosity, and even pair creation at high-\nest energy densities. The radiative reconnection theory\nis based on the flat-space description of radiation reac-\ntion adapted to phenomena in strong gravitational fields\nwhich is well justified in dense plasmas.\nHowever, in the case of diluted media, when individ-\nual particle motion is relevant, it seems more adequate\nto use description of radiation reaction in curved space-\ntime from the very beginning. This is the main goal of\nthe present paper. Here, we would like both to discuss\nconceptualproblemsofsuchadescription,andtoanalyse\nnumerically the modification of particle motion within a\nsimple model of weakly magnetized Schwarzschild black\nhole. Classifying the motion of charged particles by the\ngiven set of parameters, we find the explicit trajecto-\nries for each class of orbits and study the evolutions of\nthe particle energy, angular momentum, etc. Motion of\ncharged particles in the combined magnetic and gravi-\ntatational field without radiation reaction reveals inter-\nesting new types of trajectories. We intend to apply\nthe full machineryofcurvedspacetime radiationreaction\ntheory to these problems. We will also discuss possible\napplications of our results in some astrophysical scenar-\nios.\nThe paper is organized as follows. In Sec. 2, we test\nthe dynamical equations with radiation reaction force\nin flat spacetime using two main approaches and com-\npare the results. In Sec. 3, we present the properties of\nthe circular motion of non-radiating charged particles in\nweakly magnetized Schwarzschildblack hole, concentrat-\ning attention on the bounded orbits and charged parti-\ncle oscillations. In Sec. 4, we give the general relativistic\ntreatmentofthe radiation-reactionforcein curvedspace-\ntime, and under reasonable assumptions we give explicit\nform of the equations ofmotion ofchargedparticle in the\nvicinity of Schwarzschild black hole immersed in exter-\nnal asymptotically uniform magnetic field. In Sec. 5, we\nanalyse the trajectories of charged particles for different\nclasses of orbits and find the evolutions of the relevant\nparameters of the system during and after the decay of\nparticle oscillations. Switching to the Gaussian units in\nSec.6, we estimate the characteristic decay times of the\ncharged particle oscillations and discuss the relevance of\nthe model in astrophysicalsituations. We summarize our\nresults in Sec. 7.\nThroughout the paper, we use the spacetime signature\n(−,+,+,+), and the system of geometric units in which\nG= 1 =c. However, for expressions having an astro-\nphysicalrelevance, we use the constants explicitly. Greek\nindices are taken to run from 0 to 3.\n2.RADIATION REACTION IN FLAT SPACETIME\nFor completeness, we first summarize the results of the\nmotion of charged particle in the flat spacetime. The\nequation which describes the motion of charged particle\nin magnetic field in general contains two forces\nduµ\ndτ=fµ\nL+fµ\nR, (2)\nwherefµ\nL= (q/m)Fµνuνis the Lorentz force, Fµν=\n∂µAν−∂νAµis the tensor of electromagnetic field and\nuµ(τ) =dxµ/dτis a four-velocity of the particle. The3\nlast term fµ\nRis the radiation reaction force which in\nthe non-relativistic case has to lead to the expression\n3q2\n2md2uα\ndτ2. Moreover any force vector has to satisfy the\ncondition fµ\nRuµ= 0. This implies that the correct co-\nvariant form of the expression for the radiation reaction\nforce is\nfµ\nR=2q2\n3m/parenleftbiggd2uµ\ndτ2+uµuνd2uν\ndτ2/parenrightbigg\n. (3)\nThis expression was found by Dirac and motion equa-\ntion is sometimes called as Lorentz-Abraham-Dirac\nor Lorentz-Dirac (LD) equation. The first term\nin the parentheses, also known as the Schott term,\narises from the particle electromagnetic momentum\n(Gal’tsov & Spirin (2006)). The second term in paren-\ntheses is the radiation recoil term which corresponds to\nthe relativistic correction of the radiation reaction force.\nThe four-velocity of the charged particle satisfies the fol-\nlowing equations\nuαuα=−1, uα˙uα= 0, uα¨uα=−˙uα˙uα.(4)\nThus, the motion of radiating charge is described by a\nthird order differential equation in coordinates, rather\nthan habitual second order. This may lead to unphys-\nical solutions - the existence of pre-acceleration in the\nabsence of external forces. Even though the unphysi-\ncal solutions can be removed by properly chosen initial\nconditions, the integration of exact form of equations of\nmotion (2) and (3) is inconvenientdue to the exponential\nincrease of the computational error in practical calcula-\ntions.\nHowever, one can reduce the order of LD equation by\nthe method proposed in Landau & Lifshitz (1975), i.e.,\nby rewriting the self-force in terms of the external force\nand the four-velocity of a particle. Substituting higher\norder terms in ( 3) by the derivatives of the Lorentz force\nwe get the equation in the following form\nduµ\ndτ=fµ\nL+2q2\n3m(δµ\nα+uµuα)dfα\nL\ndτ. (5)\nThis equation, usually referred as Landau-Lifshitz (LL)\nequation, has important consequences: it is of the sec-\nond order, does not violate the principle of inertia, and\nthe self-force vanishes in the absence of the external\n(Lorentz) force, Rohrlich (2001); Poisson (1999). The\nself-contained derivation of the equation ( 5) in terms of\nretarded potentials is given in Poisson et al. (2011). The\nequation ( 5) can be applied for the cases with any ex-\nternal force acting on a charged particle instead of the\nLorentzforce. In case when fµ\nL=q\nmFµνuν, the radiation\nreaction force can be rewritten in the form\nfµ\nR=k˜q/bracketleftbig\nFµν\n,αuαuν+ ˜q(FµνFνρ−FβαFβρuαuµ)uρ/bracketrightbig\n,\n(6)\nwhere ˜q=q/mis the specific charge of the particle,\nk= (2/3)˜qq, and the comma in the first term denotes\nthe partial derivative with respect to the coordinate xα.\nItwasconcludedinSpohn(2000)thatusingtheLLequa-\ntionisidenticaltoimposingDirac’sasymptoticcondition\nlimτ→∞˙uµ= 0 to the LD equation. It was later con-\nfirmed by Rohrlich (2001) that the reduced form of the\nequation of motion is exact, rather than approximative,\nthough LL equation was proposed in Landau & Lifshitz(1975) as an approximative solution to the third order\nLD equation. More details on the treatment of radia-\ntion reaction of charged particles in flat spacetime can\nbe found in a book of Spohn (2004). In our numerical\nstudy we found that the LL approximation is perfectly\napplicable if the Schott term is small with respect to the\nradiation recoil term, which is the case we consider here.\nBelowweshowthe representativeexampleofthe charged\nparticle motion in external uniform magnetic field, inte-\ngrating both LD and LL equations. Results of numerical\nstudiesofLDand LLequationsforthe motion ofcharged\nparticle in uniform magnetic field in flat spacetime are\nin accord with the analytical treatment of the radiation\nreaction force performed in Spohn (2000).\n2.1.Charged particle in uniform magnetic field\nLetusconsiderthemotionofachargedparticleinaho-\nmogeneous magnetic field aligned along z-axis, such that\nthe independent nonvanishing component of the electro-\nmagnetic tensor is Fxy=B. We introduce the new pa-\nrameter in the form B=qB/(2m), where the factor 1 /2\nis added in order to correspond to the similar parameter\nintroduced in the curved spacetime case.\nBoth LD and LL equations, lead to equivalent results,\ndiffering in the number of initial conditions. In case of\nLD equation, one needs to set the values of 9 constants -\narbitrary independent components of initial position, ve-\nlocity and the acceleration of the charged particle. The\nthreeother constantsaregivenby the normalizationcon-\ndition (4). However, the direct integration of higher or-\nder equations leads to the exponential increase of the\ncomputational error in very short times. It is interest-\ning to note that the problem of the time dispersion er-\nror can be greatly reduced by integrating equations of\nmotion backwards in time. Similar method of solving\nLorentz-Dirac equations has been proposed in the past\nby Huschilt & Baylis (1976).\nOn the other hand, the reduced-orderequations of mo-\ntion (5) can be written explicitly in the form\ndux\ndτ=2Buy−4kB2/parenleftbig\n1+u2\n⊥/parenrightbig\nux, (7)\nduy\ndτ=−2Bux−4kB2/parenleftbig\n1+u2\n⊥/parenrightbig\nuy, (8)\nduz\ndτ=−4kB2u2\n⊥uz, (9)\ndut\ndτ=−4kB2u2\n⊥ut. (10)\nHereu2\n⊥= (ux)2+(uy)2is square of the particle veloc-\nity in the plane orthogonal to the magnetic field and z\naxis. The representative trajectory of radiating charged\nparticle corresponding to both LD and LL cases is shown\nin Fig.1. Initial conditions are chosen as follows: initial\nplane velocity is u⊥0= 0.8c, velocity in vertical direction\nisuz\n0= 0.5c, magnetic field is aligned along zaxis and\nmagnetic parameter is chosen to be B= 1, radiation pa-\nrameterk= 0.01. The estimations of the parameters in\nrealistic situations are given in Sec 6. Energy and angu-\nlar momentum loss leads to decay of the plane velocity of\nthe particle, while the vertical component remains con-\nstant. Apparent deceleration along zaxis shown in Fig. 1\n(solid thick curve in the middle plot), while no forces are4\nFig. 1.— Motion of radiating charged particle in flat spacetime. Left figure represents an example of 3D trajectory of charged part icle.\nMiddle plot represents the evolution of different component s of 3-velocity in proper time τ: the damping harmonic oscillations correspond\ntoux(solid thin) and uy(dashed) components of velocity, tangential to them (dotte d) is a plane velocity of a particle v⊥orthogonal to z\naxis, vertical uzcomponent of velocity is shown by solid thick line and dot-da shed curve of the middle plot shows the evolution of 3-veloci ty\n(/radicalBig\nu2x+u2y+u2z). Right figure shows the change of the energy, angular moment um and gyroradius of charged particle in time twith respect\nto the static observer. Energy decreases up to the certain va lue (12), while angular momentum and radius of gyration asym ptotically tend\nto zero.\nappliedinthe zdirection, appearsonlyin theframemov-\ning with the particle. The velocity with respect to the\nstatic observer, vz=dz/dt≡uz/ut=const, remains\nconstant.\n2.2.Energy and momentum loss\nThe rate of the energy loss of the particle can be eval-\nuated from Eq.( 10). Modifying it for a static observer,\nwe get\ndE\ndt=−kB2˜q2u⊥(t)2. (11)\nIntegrating this equation, one can obtain the energy of\nthe particle in a given moment of time. The evolution\nof the energy in time is shown in Fig. 1for the given tra-\njectory. Thus the energy loss will be given only by the\nchange of the plane velocity u⊥in time, while the kinetic\nenergy associated with the motion in the z-direction will\nbe conserved. The plane velocity u⊥of the particle de-\ncreases in time and asymptotically tends to zero as rep-\nresented by the dotted curve of the middle plot of Fig. 1.\nThis implies that there exists an irreducible (specific) en-\nergy of radiating charged particle which corresponds to\nthe final state of the particle, having the following simple\nform\nE0=/parenleftbig\n1−(vz\n0)2/parenrightbig−1\n2, (12)\nwherevz\n0is the vertical velocity of the particle along the\nz-axis, measured by the static observers, which is con-\nstant during the radiation process.\nIn orderto find the rate ofthe angularmomentum loss,\none can fix the motion in a plane by taking uz= 0. In\ngeneral, the specific angular momentum for the motion\nin a plane is defined by the formula L=ρ2dφ/dτ+˜qAφ,\nwhereρis gyroradius of the particle trajectory. Re-\nminding that dφ/dτ=u⊥γ/ρandρ=u⊥/ωL, where\nωL=qB/m≡2Bis the Larmorfrequency, one can write\nthe specific angular momentum of the radiating chargedparticle in given moment of time in the form\nL=u⊥(τ)2\n4B[2γ(τ)+1], γ= (1−u2\n⊥)−1\n2.(13)\nSolving first two equations of motion ( 7) and (8), and\nsubstituting into Eq.( 13), we get the evolution of the an-\ngular momentum in time, which is represented in Fig. 1.\nUnlike the energyof the particle, the angular momentum\nasymptotically tends to zero for large τ. This occurs due\nto the reason that the gyroradius ρof the charged parti-\ncle tends to zero as well, while Lis proportional to ρ.\nOne can find the ratio between the angular momentum\nloss˙L=dL/dtandthe energyloss ˙E=dE/dtin theform\n˙L\n˙E=r2Ωu2\n⊥+1\nu2\n⊥. (14)\nwhere Ω is the angular frequency of the charged particle\nmeasured by the observers at rest. In Cartesian coordi-\nnates Ω takes the form\nΩ =x˙y−y˙x\nx2+y2, (15)\nwhere dots denote the derivative with respect to the co-\nordinate time t.\n2.3.Decay time\nOne can find the characteristic time required to decay\nthe energyofa radiatingchargedparticle in the following\nway. Since the velocity in the magnetic field direction is\nconstant, one can consider only the planar motion of the\nparticle by taking uz= 0. This implies that according\nto condition uαuα=−1, we get u2\n⊥= (ut)2−1. Thus,\nequation ( 10) can be rewritten in the form\ndE\ndτ=−K/parenleftbig\nE3−E/parenrightbig\n,K= 4kB2.(16)5\nIntegrating this equation, we get the particle energy in a\ngiven moment of time\nE(τ) =EieKτ\n/radicalbig\n1+E2\ni(e2Kτ−1), (17)\nwhere the integration constant Eiis the initial specific\nenergy of the particle. Asymptotically in time the spe-\ncific energy tends to the particle rest energy, being equal\nto 1. Thus, the decay time during which the specific en-\nergy will be lowered from EitoEfdue to radiation takes\nthe following form\nT=1\n2KlnE2\nf(E2\ni−1)\nE2\ni(E2\nf−1). (18)\nwhereK= 4kB2. Since the energy in ( 17) is the expo-\nnentially decreasing function, the particle energy cannot\nbe reduced to 1 in practical calculations.\n3.CHARGED PARTICLE ORBITING BLACK HOLE\nWITHOUT RADIATION-REACTION\nIn this section we shortly summarize previous re-\nsults related to the particle motion in the field of mag-\nnetized black holes presented in Koloˇ s et al. (2015);\nStuchl´ ık & Koloˇ s(2016); Tursunov et al. (2016). The in-\nterval in the Schwarzschild black hole spacetime in sper-\nical coordinates ( t,r,θ,φ) reads\nds2=−f(r)dt2+f−1(r)dr2+r2(dθ2+sin2θdφ2),(19)\nwhereMis the black hole mass and the function f(r) is\nthe lapse function given by\nf(r) = 1−2M\nr. (20)\nHereafter, without loss of generality we put the mass of\nthe black hole to be equal to unity, M= 1. Let us\nconsider the black hole immersed into external asymp-\ntotically uniform magnetic field. In the case when this\nfield is weak implying that the metric of Schwarzschild\nblack hole is not violated (see Eq.( 1)), one can write the\nsolution of the Maxwell equation for the four-vector po-\ntential of electromagnetic field Aµin the following form\n(Wald (1984))\nAφ=B\n2gφφ=B\n2r2sin2θ, (21)\nwhich is the only nonzero component of four-vector po-\ntentialAµandBis the strength of the magnetic field\nat spatial infinity, which is taken to be constant. The\nantisymmetric tensor of the electromagnetic field Fµν=\nAν,µ−Aµ,νinthis casehasonlytwoindependent nonzero\ncomponents\nFrφ=Brsin2θ, Fθφ=Br2sinθcosθ.(22)\n3.1.Bounded motion around black hole\nThe motion of charged particles is described by the\nLorentz equation in curved spacetime. In case when the\nradiative processes can be neglected, one can write for\nthe particle of mass mand charge qthe Lorentz equation\nof motion\nDuµ\ndτ≡duµ\ndτ+Γµ\nαβuαuβ=q\nmFµ\nνuν,(23)whereuµis the four-velocity of the particle, normalized\nby the condition uµuµ=−1,τis the proper time of\nthe particle and components of Γµ\nαβare the Christoffel\nsymbols.\nSymmetry of the Schwarzschildblack hole ( 19) and ex-\nternal magnetic field give us a right to find the conserved\nquantities associated with the time and space compo-\nnents of the generalized four-momentum πα=pα+Aα.\nThus, the energy and angular momentum of the charged\nparticleinthepresenceofexternaluniformmagneticfield\ntake the following form\nE=−πt=mf(r)dt\ndτ, (24)\nL=πφ=mr2sin2θ/parenleftbiggdφ\ndτ+qB\n2m/parenrightbigg\n.(25)\nIn this case, the circular motion of charged particles is\nalways bounded in the plane orthogonal to the magnetic\nfield lines, which corresponds to the equatorial plane\nθ=π/2. The boundary of the motion is governed by\nthe shape of the effective potential, determined by the\nequation\nE2=Veff(r,θ;L,B). (26)\nUsing the notation\nE=E\nm,L=L\nm,B=qB\n2m. (27)\nwe can write the effective potential in the form\nVeff(r,θ)≡f(r)/bracketleftBigg\n1+/parenleftbiggL\nrsinθ−Brsinθ/parenrightbigg2/bracketrightBigg\n.(28)\nThe effective potential ( 28) shows clear symmetry\n(L,B)↔(−L,−B) that allows to distinguish the fol-\nlowing two situations\n-minus configuration , withL>0,B<0 (equiva-\nlent toL<0,B>0) - magnetic field and angular\nmomentum parametershaveoppositesigns and the\nLorentz force is attracting the charge towards the\nblack hole.\n+plus configuration , withL>0,B>0 (equivalent\ntoL<0,B<0) - magnetic field and angular mo-\nmentum parameters have the same signs and the\nLorentz force is repulsive, acting outward the black\nhole.\nThe last case appears only in the combination of gravity\nwith electromagnetic field and cannot exist in flat space-\ntime (for details, see Koloˇ s et al. (2015); Tursunov et al.\n(2016); Stuchl´ ık & Koloˇ s(2016)). Particular trajectories\nof charged particles in magnetized Schwarzschild black\nhole are compared with radiating particle motion in Sec-\ntion5.\n3.2.Charged particle oscillations\nCharged particles can undergo stable quasi-harmonic\noscillationsinradialandverticaldirectionsinthevicinity\nof magnetized black hole. In the case when the oscilla-\ntions are small enough in comparison with the radius of\nthe corresponding stable circular orbit, one can find the6\nFig. 2.— Dependence of angular momentum L, energy Eand azimuthal velocity uφof a charged particle on the location of the circular\norbitrwithout radiation for different values of magnetic paramete rB.\nlocally measured frequencies of vertical ωθand radial ωr\noscillations equal to\nω2\nθ=L2\nc\nr4−B2, (29)\nω2\nr=1\n(r−2)r5/bracketleftbig\n(r−2)2/parenleftbig\nB2r4+3L2\nc/parenrightbig\n−2r/parenleftbig\nBr2−Lc/parenrightbig2−2r3/bracketrightbig\n(30)\nwhereLcis the specific angular momentum at the circu-\nlar orbit, for details see Koloˇ s et al. (2015). In addition\nto the frequencies givenabove, one can find alsothe Kep-\nlerian axial frequency ωφand Larmor angular frequency\nωLwhich are given by\nωφ=dφ\ndτ=Uφ=Lc\ngφφ−B, ωL=qB\nm= 2B.(31)\nAs one can see, the frequency ωLdoes not depend on r\ncoordinate and plays crucial role in the regions detached\nfrom the black hole. The characteristic oscillations of\nchargedparticlesandrepresentativeplotsofchargedpar-\nticle trajectories around Schwarzschild black hole can be\nfound in Fig.7 of paper Koloˇ s et al. (2015). For the par-\nticle to oscillate in the vicinity of a black hole, its energy\nhas to be larger than the minimum of the effective po-\ntential, however, keeping the finite type of the motion.\nThus,theminimalenergyofachargedparticleintrapped\nstates corresponds to the stable circular orbits given by\nthe minimum of the effective potential ( 28). The maxi-\nmal energy at the trapped states is given by the unstable\ncircular orbit with the corresponding angular momen-\ntum. Energy of the charged particle at the circular orbit\nis given by\nE2\n±=rf2\n(r−3)2/parenleftBig\nr−3+2B2r3f±2Br/radicalbig\nr−3+B2r4f2/parenrightBig\n,\n(32)\nwhere the signs correspond to the maximal and minimal\nenergy of a particle at the circular orbit, and f= 1−2/r\nis the lapse function. The difference between E+and\nE−can be very large for large values of magnetic pa-\nrameter B, representing the charged particles acceler-atedup to ultrarelativisticvelocities(Koloˇ s et al.(2015);\nStuchl´ ık & Koloˇ s (2016)).\nSome types of the quasi-circular epicyclic motion are\npossible only in the presence of magnetic field. One of\nthe interesting and illustrative examples of the effect of\nmagnetic field on the charge particle motion in the black\nhole background is the appearance of curled trajectories,\nas demonstrated in Fig. 4which corresponds to the plus\nconfiguration with repulsive Lorentz force. The conser-\nvation of the angular momentum ( 25) in the absence of\nradiation leads to the equation\n˙φ=L\nr2−B. (33)\nInthe non-magneticcase, therighthandsideoftheequa-\ntion above is positive for positive L. The presence of\nmagnetic field with B>0 decreases the velocity in φ-\ndirection that can become negative, if\nL>L∗(r;B)≡ Br2. (34)\nFor the energy, this condition means\nE>E∗(L;B)≡/radicalBig\n1−2/radicalbig\nB/L. (35)\nThus, decreasingthe azimuthalvelocityofaparticlewith\nthe influence of magnetic field, and keeping the above\ngiven conditions, we get so called curled trajectories,\nwhich are not possible in the absence of magnetic field.\nThis type of orbits does not appear in the case of attrac-\ntive Lorentz force corresponding to the minus configura-\ntions.\nIn the next sections we show that the oscillations of\ncharged particles in magnetic field near a Schwarzschild\nblack hole will be damped due to the synchrotron radi-\nation and, in particular cases the charged particle orbits\nwill not be able to stay stable, bringing the particle to\nthe black hole due to the effect of the radiation reaction\nforce.\n4.RADIATION REACTION IN THE FIELD OF\nMAGNETIZED BLACK HOLES\n4.1.Radiation-reaction force7\nThe motion of a relativistic charged particle is gov-\nerned by the Lorentz-Dirac equation which includes the\ninfluence of the external electromagnetic fields and cor-\nresponding radiation-reaction force. The last force arises\nfrom the radiative field of the charged particle and the\nequations ofmotion in generalcan be written in the form\nDuµ\ndτ= ˜qFµ\nνuν+ ˜qFµ\nνuν, (36)\nwhere the first term on the right hand side of Eq.( 36)\ncorresponds to the Lorentz force with electromagnetic\ntensorFµνgiven by ( 22), while the second term is the\nself-force of charged particle with the radiative field\nFµν=Aν,µ− Aµ,ν. The vector potential of the self-\nelectromagnetic field of the charged particle satisfies the\nwave equation\n✷Aµ−Rµ\nνAν=−4πjµ, (37)\nwhere✷=gµνDµDν, andDµis the covariant differenti-\nation and Rµ\nνis the Ricci tensor. The retarded solution\nto Eq.(37) for the vector potential takes the form\nAµ(x) =q/integraldisplay\nGµ\n+λ(x,z(τ))uλdτ, (38)\nwhereGµ\n+λis the retarded Green function and the in-\ntegration is taken along the worldline of the particle z,\ni.e.,uµ(τ) =dzµ(τ)/dτ. For details, see, e.g. Poisson\n(2004). The covariant generalization of the dynamics of\nradiating charged particle in curved spacetime has been\nderived in DeWitt & Brehme (1960) and completed in\nHobbs (1968), using the tetrad formalism. The explicit\nform of Eq.( 36) for the motion of charged particle under-\ngoing radiation-reaction force in curved spacetime reads\nDuµ\ndτ= ˜qFµ\nνuν+2q2\n3m/parenleftbiggD2uµ\ndτ2+uµuνD2uν\ndτ2/parenrightbigg\n+q2\n3m/parenleftbig\nRµ\nλuλ+Rν\nλuνuλuµ/parenrightbig\n+2q2\nmfµν\ntailuν,(39)\nwhere the last term of Eq.( 39) is the tail integral\nfµν\ntail=/integraldisplayτ−0+\n−∞D[µGν]\n+λ′/parenleftbig\nz(τ),z(τ′)/parenrightbig\nuλ′dτ′.(40)\nDetailed derivation of the equations of motion of radi-\nating charged particles can be found in Hobbs (1968);\nPoisson (2004). The integral in the tail term is evalu-\nated over the past history of the charged particle with\nprimes indicating its prior positions. All other quanti-\nties are evaluated at the current position of the particle\nz(τ). The term containing the Ricci tensor vanishes in\nthe vacuum metrics, so this term is irrelevantin our case.\nThe existence of the ”tail” integral in ( 39) implies that\nthe radiation reaction in curved spacetime has non-local\nnature, because the motion of the charged particle de-\npends on its whole history and not only on its current\nstate. The radiation field Fµνin Eq.(36), emitted by the\ncharged particle, interacts with the curvature of back-\nground spacetime and comes back to the particle with\na delay corresponding to the tail integral in ( 39). In\nsuch sense, the radiated electromagnetic field of charged\nparticle carries the information about the history of the\nparticle. Even in the absence of external forces, such asthe Lorentz force, the free trajectory of the charged par-\nticle does not follow the geodesics, which is one of the\nmost important consequences of equation ( 39).\nHowever, for purposes of the present paper, the\ntail term can be neglected as we show below. The\ntail terms can be estimated based on the results by\nDewitt & Dewitt (1964), Smith & Will (1980), as well\nas multiple subsequent papers. For a particle with the\nchargeqand mass m, the ratio of the tail force Ftail∼\nGMq2/(r3c2) to the Newton-force FN∼GMm/r2in the\nvicinity of a black hole ( r∼rH= 2GM/c2) of the stellar\nmassM∼10M⊙is\nFtail\nFN∼q2\nmMG∼10−19/parenleftBigq\ne/parenrightBig2/parenleftBigme\nm/parenrightBig/parenleftbigg10M⊙\nM/parenrightbigg\n,(41)\nwhereeandmeare the charge and the mass of an elec-\ntron. For supermassive black holes (SMBH) with the\nmassM∼109M⊙this ratio is 8 orders lower. On the\nother hand, the radiation reaction force (second term\non the right hand side of ( 39)) depends on the presence\nof external force, arising in our case from the external\nmagnetic field. According to Piotrovich et al. (2011);\nBaczko et al.(2016), thecharacteristicvaluesofthemag-\nnetic fields near the stellar mass black holes and SMBH\nareB∼108G, forM= 10M⊙andB∼104G, for\nM= 109M⊙. Thus, for the particle with velocity com-\nparable to the speed of light, v∼c, the ratio of the radi-\nation reaction force FRR∼q4B2/(m2c4) to the Newton-\nforce gives an order\nFRR\nFN∼q4B2MG\nm3c8∼103/parenleftBigq\ne/parenrightBig4/parenleftBigme\nm/parenrightBig3/parenleftbiggB\n108G/parenrightbigg2/parenleftbiggM\n10M⊙/parenrightbigg\n.\n(42)\nThe ratio of FRRtoFNfor SMBH with M= 109M⊙\nand magnetic field B∼104G gives the same order of\nmagnitude.\nThe above estimations of the tail term apply in the\nnon-relativistic, or moderately relativistic case when the\nLorentzfactorisoftheorderofunity. Moregeneralargu-\nment is based on the treatment of the gravitational self-\nforce in the Schwarzschild and Kerr spacetimes. As was\nshown by one of the present authors in Gal’tsov (1982),\nthe radiative part of the self-force (based on the half-\ndifference of the retarded and advanced Green’s func-\ntion) in the Kerr metric satisfies the (averaged on time)\nbalance equations for the energy and angular momen-\ntum of radiation of all spins s= 0,1,2 in the sense that\nlocal work of the self-force is equal to radiated fluxes\nat infinity and the black hole horizon. This is valid\nindependently of the particle velocity and extends to\nthe ultrarelativistic case. Later on it was shown that\nsimilar balance is valid for the Carter’s constant (Mino\n(2003)), which completes the set of quantities determin-\ning geodesic motion in the Kerr field. The conservative\npart of the self-force (half-sum of the retarded and ad-\nvanced potentials) is more difficult to compute since this\ndemands proper elimination of divergences. This caused\na vivid discussion in the literature from the mid-90-ies\nto early 2000-ies, nicely reviewed in Tanaka (2006) (for\nmore recent work on the same subject containing fur-\nther references see Fujita et al. (2017)), resulting in a\nconsensus opinion that this contribution is small with\nrespect to to radiative self-force especially with growing8\nLorentz factor (Pound et al. (2005); Sago et al. (2005)).\nTherefore, if one is interested to estimate the gravita-\ntional and electromagnetic tail terms in the ultrarela-\ntivistic limit, using the above results one can revoke\nthe gravitational synchrotron radiation (GSR), which\nwas computed earlier for spins s= 0,1,2 most no-\ntably in Chrzanowski & Misner (1974) and in Breuer\n(1975) with comparison to flat-space synchrotron radi-\nation (SR). From these results one can see that GSR in\nthe ultrarelativistinc limit is suppressed by a square of\nthe Lorentz factor with respect to SR. These arguments\nlead us to conclude that for elementary particles mov-\ning with any velocity in both gravitational and electro-\nmagnetic field the purely gravitational tail term can be\nneglected in comparison with the electromagnetic (prop-\nerly covariantized) radiation reaction force. Thus, for\npurposes of the present paper, the equation of motion\n(39) can be simplified to the following covariant form of\nLD equation:\nDuµ\ndτ= ˜qFµ\nνuν+fµ\nR, (43)\nwith the radiation reaction force given by\nfµ\nR=2q2\n3m/parenleftbiggD2uµ\ndτ2+uµuνD2uν\ndτ2/parenrightbigg\n.(44)\nIntroducing the four-acceleration as a covariant deriva-\ntive of four-velocity, aµ=Duµ/dτ, one can rewrite the\ntermD2uµ/dτ2as follows\nD2uµ\ndτ2≡Daµ\ndτ=daµ\ndτ+Γµ\nαβuαaβ\n=d\ndτ/parenleftbiggduµ\ndτ+Γµ\nαβuαuβ/parenrightbigg\n+Γµ\nαβuα/parenleftbiggduβ\ndτ+Γβ\nρσuρuσ/parenrightbigg\n=d2uµ\ndτ2+/parenleftBigg\n∂Γµ\nαβ\n∂xγuγuβ+3Γµ\nαβduβ\ndτ+Γµ\nαβΓβ\nρσuρuσ/parenrightBigg\nuα.\n(45)\nThus, the general relativistic equations of radiating\ncharged particle motion are given by Eqs ( 44) and (45).\nHowever, the full form of the equations is plagued by\nrunaway solutions. One can avoid this problem in a sim-\nilar way, as in the flat spacetime case, namely reducingthe order of differential equations. In the absence of the\nradiation-reactionforce, the motion of the charged parti-\ncle in external electromagnetic field is governed by equa-\ntion (23). Taking the covariant derivative with respect\nto the proper time from both sides of Eq.( 23), we get\nD2uα\ndτ2= ˜qDFα\nβ\ndxµuβuµ+ ˜q2Fα\nβFβ\nµuµ,(46)\nSubstituting ( 46) into Eq.( 44), we get the radiation re-\naction force in the form\nfα\nR=k˜q/parenleftbiggDFα\nβ\ndxµuβuµ+ ˜q/parenleftbig\nFα\nβFβ\nµ+FµνFν\nσuσuα/parenrightbig\nuµ/parenrightbigg\n,\n(47)\nwhere the covariant derivative from the second rank ten-\nsor reads\nDFα\nβ\ndxµ=∂Fα\nβ\n∂xµ+Γα\nµνFν\nβ−Γν\nβµFα\nν.(48)\nEquations( 43) and (47) givea covariantform ofLandau-\nLifshitz equations. Below we test these equations in par-\nticular case of the motion of charged particles around\nSchwarzschild black hole immersed into external asymp-\ntotically uniform magnetic field.\nThe motion of charged particles in the vicinity of\nmagnetized Schwarzschild black hole is generally chaotic\nKop´ aˇ cek & Karas (2014); Kop´ aˇ cek et al. (2010). How-\never, close to the minimum of the effective potential,\nwhich corresponds to the stable circular orbit, the mo-\ntion of the charged particle is regular, being of harmonic\ncharacter. The motion is also regular, if the particle is\nmoving entirely in the equatorial plane and the chaotic\nbehaviour appears with increasing the inclination angle.\nHere, we focus our attention on the regular motion only.\nIn order to represent the equations of motion explicitly,\nwe fix the plane of the motion at the equatorial plane,\nθ=π/2, of a magnetized black hole. However, for ex-\nploring the trajectories of particles, we use the full set of\nequations of motion and solve them numerically. With-\nout loss of generality, one can again equalize the mass of\na black hole to unity, M= 1. The non-vanishing com-\nponents of equations of motion of radiating charged par-\nticles moving around Schwarzschild black hole immersed\ninto external asymptotically uniform magnetic field take\nthe following form\ndut\ndτ=2utur\nr(2−r)−2kBut\nr/braceleftbig\n2Brf/bracketleftbig\nf(ut)2−1/bracketrightbig\n−uφ/bracerightbig\n, (49)\ndur\ndτ=uφ(2Brf+r−1)−1\nr2−2kBur\nr/braceleftbig\n2Brf2(ut)2−uφ/bracerightbig\n, (50)\nduφ\ndτ=−2ur/parenleftbig\nuφ+B/parenrightbig\nr+2kB\nr3/bracketleftbig\nr2(uφ)2+2Br3f2(ut)2uφ+1/bracketrightbig\n, (51)\nwherefisthelapsefunctiongivenby( 20),B=qB/(2m)\nis the magnetic parameter and k= 2q2/(3m) is the ra-\ndiation parameter.\n4.2.Energy and angular momentum lossThe total energy-momentum radiated by the charged\nparticle is equal to the integral of the radiation reac-\ntion force taken along the worldline of the particle. In\nflat spacetime, the radiated four momentum of a particle\nwith charge qis given by dPµ/dτ=2\n3q2aαaαuµ. Syn-\nchrotron radiation in curved spacetime has been studied,9\nin Sokolov et al. (1983), Sokolov et al. (1978) using co-\nvariantization of the flat space results. The problem has\nbeen revisited more recently in Shoom (2015), where,\nhowever, the radiation reaction force is not taken into\naccount. The evolutions of the particular components of\nthe four-momentum of the particle with radiation reac-\ntion force can be found from equations ( 43) and (47).\nFor the motion at the equatorial plane, the energy loss\nis given by\ndE\ndτ=−2kB/bracketleftbigg\n2BE3−E/parenleftbigg\n2Bf+uφ\nr/parenrightbigg/bracketrightbigg\n.(52)\nFor ultrarelativisticparticle with E ≫1, the leading con-\ntribution to the energy loss is given by the first term in\nsquare brackets of ( 52). However, for small velocities\nclose to the stable circular orbits, the last two terms can\nplay significant role. Both situations will be studied in\ndetails in the following section. Similarly, one can find\nthe rate of angular momentum loss as\ndL\ndτ= 4B2kuφ/parenleftbig\nf2(ut)2−f/parenrightbig\n−2uruφ/parenleftbig\nr−4B2k2/parenrightbig\n+2rBur.\n(53)\nIn the next section we will test the equations of mo-\ntion numerically for general bounded motion of radiat-\ning charged particle without restriction to the equatorial\nplane.\n5.DAMPING OF OSCILLATIONS\nIn this section we study the damping of charged par-\nticle oscillations due to radiation reaction force for both\n”+” and ” −” configurations. In particular, we demon-\nstrate that the particles at ” −” configurations spiral\ndown to the black hole, while at ”+” configurations the\nmotion remains stable. In the subsection 5.3, we study\nthe evolution of circular orbits under the influence of self\nforce. We show that in such a case the circular orbits\ncan be shifted outwards from the black hole.\nAs pointed out in Section 3, depending on the direc-\ntion of the Lorentz force one can distinguish two qualita-\ntively different types of the motion. In a radiating case\none can see such differences as well. The motion along\na general worldline highly depends on the initial energy\nand the position of the particle. Radiation effect on the\ncharged particle motion also has different timescales de-\npending on whether the particle is initially oscillating or\nnot. The representative comparison of trajectories of os-\ncillating charged particle in the presence and absence of\nradiation reaction force is illustrated in Fig. 3for attrac-\ntive Lorentz force and in Fig. 4, for repulsive one. Note\nthat the trajectories represented in Figs. 3and4are\nplotted by integration of the full form of equations of\nmotion ( 43) and (47) without restriction to the equato-\nrial plane. In both cases, the motion of the charge is\ninitially bounded. The charged particle starts its motion\nslightly abovethe equatorialplane (keeping regularchar-\nacter of the motion) in the vicinity of weakly magnetized\nSchwarzschild black hole. The initial energy and angular\nmomentum of the particle correspond to the state close,\nbut above the minimum of the effective potential which\ngenerates barrier where the particle bounces. Due to the\nactionoftheradiationreactionforce,thechargedparticle\nchanges its oscillatory characterof the motion because ofthe loss of the energy and angular momentum. The final\nstate of the charged particle depends on the direction of\nthe Lorentz force. For the Lorentz force directed towards\nthe horizon of the black hole, the synchrotron radiation\nleads to the collapse of the initially stable particle into\nthe black hole (see Fig. 3). In the inverse case, when the\nLorentz force is directed outwards the horizon (compen-\nsating thus the ”gravitational attraction”), the particle\nremainsin the bounded regionin the vicinity ofthe black\nhole. On the other hand, loss of the energy of the par-\nticle leads to the disappearance of curled trajectories as\nwell as any oscillations, as shown in Fig. 4.\nAnother representative example of the dependence of\nthetrajectoriesonthealignmentoftheexternalmagnetic\nfield is shown in Figs. 5and6, where the initial condi-\ntions differ only in the sign of the magnetic parameter B,\nwhilethe otherparametersarechosentobe the same. As\nbefore, for the attractive Lorentz force, the charged par-\nticle spirals down to the black hole (Fig. 5), while in the\nrepulsive case, the motion is stable (Fig. 6). Stability of\ntheplus configurations against collapse can be explained\ndue to the reason that a radiating particle is assumed to\nbe accelerated by the external Lorentz force only. This\nimplies that the radiation reaction in fact decreases the\ngyroradiusof the radiatingparticle. In the casewhen the\nLorentz force is directed outwards the black hole, the gy-\nrocenter of oscillating charged particle is located near its\norbit (center of ”curls”), while in the case of minus con-\nfiguration , the gyrocenter of the orbit is located inside\nthe horizon (it coincides with a black hole singularity for\nthe circular motion). Indeed, from the analogy with the\nflat spacetime formula ( 5), the sign of the radiation reac-\ntion force depends on the alignment of the Lorentz force\ndue to the reduction of order procedure performed in the\nbeginning. Thus, the ”Lorentz-repulsive”case with radi-\nationreactionrepresentsdamping ofoscillations, turning\nthe oscillatory epicyclic type of motion to nearly circu-\nlar orbit. A strong non-linearity of equations of motion\nshifts the location of stable circular orbits as well, how-\never, as we will see below, this effect is relatively slow in\ncomparisontotheprocessofradiativedampingofoscilla-\ntions. One can conclude that for relatively short periods\nof time an oscillating charged particle, undergoing repul-\nsive Lorentz force and radiation reaction force will damp\nits oscillations settling down to the circular orbit. This\nresult is represented in Fig. 7, being in accord with the\nqualitative predictions given in Shoom (2015).\n5.1.Evolution of energy and angular momentum\nIn fact, the synchrotron radiation carries out the ki-\nnetic energy of the particle. One can see in E−τplots\nof Fig.5and Fig.6, that the energy slightly decreases in\nboth cases. However, the azimuthal angular momentum\nof the particle changes differently in dependence on the\ndirection of the Lorentz force. While in the minus con-\nfigurations the charged particle decreases its angular mo-\nmentum due to radiation, in the plus configurations the\nangular momentum quasi-periodically increases. Quasi-\nperiodicity is connected with the quasi-harmonic oscilla-\ntions of the particle in the motion with ”curls”. Increase\nof the angular momentum of the particles in the plus\nconfigurations becomes clear if one compares the angular\nmomentum of charged particles in the motion with curls,\nL∗given by ( 34), with those of the circular orbits given10\nFig. 3.— Representative comparison of trajectories of non-radiati ng (first line) and radiating (second line) charged particle s withminus\nconfiguration corresponding to the attractive Lorentz force from differen t view angles around black hole in magnetic field. Initial con ditions\nin both cases are chosen to be same and shown in the second row p lots. Starting point is indicated by black dot. A radiating c harged\nparticle escapes the initial boundary of the motion (dashed contours in the third row) governed by an effective potential (28) and collapses\nto the black hole due to the loss of angular-momentum. Magnet ic field is aligned with z- axis. Trajectory of radiating charged particle\ncorresponds to the integration of full set of equations of mo tion (43) and (47) without fixing the plane of the motion.\nFig. 4.— Similar to Fig.3 comparison of trajectories of non-radiati ng and radiating charged particles in case of repulsive Lore ntz force,\ni.e.plus configuration . A radiating particle stays in the region of initial boundar y of the motion and radiates its oscillatory part of\nenergy-momentum tending to the stable circular orbit.11\nFig. 5.— Spirallingdown to the black holeof charged particle due to r adiationreaction ofa minus configuration particle, and corresponding\nevolution in local time τof the radius of orbit r, angular momentum L, energy E, the radial velocity ur, the angular azimuthal velocity uφ\nand gamma factor ut≡dt/dτ≡γ. The angular momentum and energy are measured with respect t o observer at rest at infinity. Starting\npoint of the particle is indicated as black dot.\nFig. 6.— Decay of oscillations due to radiation reaction of a plus configuration particle, and corresponding evolution in time τof the radius\nof orbit r, angular momentum L, energy E, the radial velocity ur, the angular azimuthal velocity uφand gamma factor ut≡dt/dτ≡γ.\nThe angular momentum and energy are measured with respect to observer at rest at infinity. Starting point of the particle i s indicated as\nblack dot.\nFig. 7.— Damping of oscillations due to radiation reaction of a plus configuration particle, and transition from curled motion to the\nnearly circular orbit. Corresponding evolution of the radi us of orbit r, angular momentum L, energy E, the radial velocity ur, the angular\nazimuthal velocity uφand gamma factor ut≡dt/dτ≡γare represented as in previous plots. The angular momentum a nd energy are\nmeasured with respect to observer at rest at infinity. Starti ng point of the particle is indicated as black dot.12\nFig. 8.— Number of revolutions of the oscillatingcharged particle\naround black hole given as a ratio of the maximal decay time to\nthe orbital time in dependence on the magnetic parameter Band\ndifferent values of radiation parameter k.\nby Koloˇ s et al. (2015)\nLc=−Br2+r/radicalbig\nB2r2(r−2)2+r−3\nr−3.(54)\nLet us now assume that the initial angular momentum of\nthe particle is ≈ L∗, while the angular momentum at the\nfinal state is nearly equal to those of the stable circular\norbit,≈ Lc. From the fact that that Lc>L∗(see details\nin Koloˇ s et al. (2015)) one can conclude that the parti-\ncle is actually gaining an angular momentum by reduc-\ning radial oscillations due to radiation reaction force. In\ntheminus configurations , the trajectories with curls, i.e.\nthe regions with negative angular velocity cannot be ob-\nserved. Therefore, a charged particle initially located at\na bounded orbit around a black hole continuously loses\nits angular momentum due to radiation reaction force.\nWhen angular momentum of the particle becomes lower\nthan the one corresponding to the innermost stable cir-\ncular orbit (ISCO), L<LISCO, the particle collapses to\nthe black hole. The value of angularmomentum at ISCO\nis given by Koloˇ s et al. (2015)\nLISCO=−2BrI+/radicalBig\nB2r2\nI(5r2\nI−4rI+4)+2rI,(55)\nwhererIis the ISCO radius. Detailed analysis of the\nboundaries of angular momentum separating different\ntypes of orbits in the vicinity of magnetized black holes\ncanbe found in Koloˇ s et al.(2015) fornon-rotatingblack\nholes and in Tursunov et al. (2016) for magnetized Kerr\nblack hole.\n5.2.Lifetime of oscillations\nOne can calculate the relaxation time τof a charged\nparticle required for decay of the radial oscillations due\nto radiation reaction force. Calculation of the relaxation\ntime makes sense in the plus configurations only, when\ntheradiationreactiondoesnotcausetheparticlecollapse\nto the black hole. The rate of the energy loss can be\nwritten as\n˙E=Ef−Ei\nτ, (56)whereEiandEfare the initial and final energies of the\nparticle. For the particle with velocity close to the speed\nof light, v∼c, the leading contribution for the energy\nlossisgivenbythefirsttermoftheexpression( 52), which\ncan be solved analytically giving\ndE\ndτ=−4B2kE3,E(τ) =Ei/radicalbig\n1+8B2kE2\niτ.(57)\nExtracting τfrom (57) at the final energy, we get\nτ=1\n4kB2E2\ni−E2\nf\nE2\niE2\nf. (58)\nOne can find the upper limit of the relaxation time re-\nquired to lower the maximal energy at the bounded orbit\nto the lowest energy corresponding to the circular orbit,\nwhere no oscillations exist. Thus, identifying Ei=E+\nandEi=E−, givenby( 32), andassuminglocationsofex-\ntremarE\nmax≈rE\nmin=r(indeed, according to Koloˇ s et al.\n(2015), for large values of Bthe locations of extrema are\nvery close), we get\nτmax=/radicalbig\nr−3+B2f2r4\nkB(1+4B2r2)f2. (59)\nNote that this equation corresponds to the particles with\nultrarelativistic velocities. This implies that for large\nvalues of magnetic parameter B, one can write Eq.( 59)\nin the following simple form\nτmax≈1\nkB2f(r),B ≫1, (60)\nwheref(r) = 1−2M/ris the lapse function. In partic-\nular, this equation shows that closer to black hole, the\ndecay of oscillations of charged particle is slower.\nIt is useful to compare the maximal relaxation time\nτmaxof the charged particle oscillations with the orbital\ntime of a charged particle around black hole. The or-\nbital frequency ωφis defined in Eq.( 31). Considering the\nstable circular orbits at the equatorial plane, we get the\ntime of one revolution around the black hole in the form\nτc=2πr2\nLc−Br2, (61)\nwhereLcis given by ( 54). Dividing ( 59) by (61), we get\nthe maximal number of revolutions around black hole\nduring which the radial oscillations decay\nNmax≡τmax\nτc=BrA\nπk(r−2)2(4B2r2+1)(Br(r−2)+A),\n(62)\nwhereA=/radicalbig\nr−3+B2r2(r−2)2. The dependence of\nNmaxon the magnetic parameter Bis given in Fig. 8, for\ndifferent values of the radiation parameter k. Since both\nthe parameters Bandkdepend on the specific charge\nof the charged particle, the parameters are not really in-\ndependent. This implies that the parameter kcan be\nexpressed as fraction of the parameter B. As one can see\nfrom Fig. 8, the radiation reaction of the charged particle\ncannot be neglected even for small values of the param-\neterk. For large values of B, initially oscillating charged\nparticle may decay its radial oscillations ending up at13\nFig. 9.— Damping of vertical oscillations of a charged particle revo lving near the circular orbit of a black hole in magnetic field and\ncorresponding evolution in time τof the angle θ, angular momentum L, energy E, the vertical velocity uθ, the angular azimuthal velocity\nuφand gamma factor ut≡dt/dτ≡γ. The angular momentum and energy are measured with respect t o observer at rest at infinity.\nStarting point of a particle is indicated as black dot.\nFig. 10.— Radiative widening of the stable circular orbit of a\ncharged particle due to radiation reaction force. Starting point of\na particle is indicated as black dot.\nthe circular orbit pathing only fraction of the total or-\nbit around the black hole. The estimations of both the\nparameters Bandk, and the lifetime of the charged par-\nticle oscillations in realistic scenarios for electrons and\nions will be given in Sec. 6.\nEquations of motion ( 43) and (47) allow us to repeat\nthe calculations of the damping of vertical oscillations\n(inθ) of charged particles as well. In the case when\nthe initial velocity of a particle is ultrarelativistic, the\nleading term responsible for the radiation reaction for all\ncomponents of velocities is the last term of Eq.( 47). This\nimplies that the decay rate of the vertical oscillations\nof charged particle is similar to the decay rate of the\nradial oscillations. The representative plots of relevant\nparameters of charged particles during decay of vertical\noscillations is presented in Fig. 9.\n5.3.Circular orbit widening\nAnother new consequence of the radiation reaction\nforce in the vicinity of a black hole and in presence of\nexternal magnetic field is the evolution of the circularFig. 11.— Evolution of radial position of charged particle in time\nfor two configurations corresponding to trajectories illus trated in\nFigs 5 and 6.\norbits of charged particles during the radiation process\nand shifting of the radius of circular orbit outwards from\nthe black hole. Obviously, the last case can be realized\nfor repulsive Lorentz force only, while in attractive case,\nas we have already seen, the particle spirals down to the\nblack hole and no closed circular orbits can be formed.\nLet usconsiderapurely circularmotion ofachargedpar-\nticle revolving around black hole at the equatorial plane,\nin presence of uniform magnetic field (see Fig. 10). While\nsmall oscillations can appear, they will be dumped rel-\natively fast and the energy of the particle can be con-\nsidered to be always located near the minimum of the\neffective potential corresponding to the circular orbits.\nThe radiation reaction force in the frame moving with\nthe particle vanishes, however, in the locally geodesic\nframe of reference, the self force is directed anti-parallel\nto the particle velocity and concentrated in the narrow\ncone along the motion. This implies that, in fact, the ra-\ndiation reaction reduces the angular velocity of the par-\nticle. Reducing the angular velocity while keeping the14\ncircular character of the motion causes the increasing of\nthe radius of the circular orbit. This, in turn, increases\nthe particle energy and angular momentum with respect\nto the observer at rest at infinity. Indeed, the energy\nand angular momentum of the charged particle for an\nobserver at infinity are given as\nE=/parenleftbigg\n1−2\nr/parenrightbigg\nut, (63)\nL=r2/parenleftbig\nuφ+B/parenrightbig\n, (64)\nwhich are not conserved for radiating particles. When\nthe radius of the orbit rincreases faster than the de-\nceleration of velocities utanduφ, then the energy and\nangular momentum of the charged particle will increase\naccordingly. Example of the widening of the circular or-\nbit of radiating charged particle is illustrated in Fig. 10.\nNote that the parameter responsible for the radiation re-\nactionkis taken to be unrealistically large in order to\nobtain representative plot, since for small values of the\nparameter kdue to slowness of the process the shift of\nthe orbit would be barely seen.\nThe process of the expansion of circular orbits is much\nslower in comparison to the process of decay of the os-\ncillations. Representative changes of the positions of\ncharged particles in time for attractive and repulsive\nLorentz forces are demonstrated in Fig. 11. In the case of\nthe attractive Lorentz force ( B<0), the charged particle\nplunges to the black hole, as described in the previous\nsection. The wiggles in B<0 case appear due to the\nprecession of the particle around circular orbit, as can\nbe also seen in r−τplot of Fig. 5. In the repulsive case,\nthe particle starts to oscillate inside a barrier given by\nthe effective potential at a given moment of time. Due\nto the radiation reaction force, it decays its oscillations\nending up at a circular orbit with larger radius than the\ninitial position. At the circular orbit, particle continues\nto slow down its orbital velocity, keeping the stable mo-\ntion which affects on the location of the circular orbit,\nshifting it towards infinity. Note that the timescale of\nthe plot is logarithmic, which implies that the process of\nshifting of the particle circular orbits is relatively slow.\nThe energy loss of this process is given by ( 52), where\nall terms are important since the motion is not neces-\nsarily ultrarelativistic. Since all quantities in ( 52) are\nimplicit functions of time, one can rewrite this equation\nin the form\n˙E=−A1E3+A2Ex(τ), (65)\nwhere dot denotes the derivative with respect to the\nproper time τ,A1= 4kB2andA2= 2kBare constants\nandx(τ) = 2kBf(τ)+uφ(τ)/r(τ). The analytical form\nof the solution is found to be\nE(τ) =EieA2X(τ)\n/parenleftbig\n1+2A1E2\ni/integraltextτ\n0e2A2X(τ′)dτ′/parenrightbig1\n2,(66)\nwhereX(τ) =/integraltextτ\n0x(τ)dτ. The equation ( 65) can be\nsolved numerically for the given set of equations of mo-\ntion, and it is performed for all E−τdependence plots of\nthe present paper. The detailed analysis of the effect of\nradiative widening of orbits is left for our future studies.\n6.RELEVANCE TO ASTROPHYSICSB (Gauss) τe(s) τp(s)\n101210−1610−6\n10810−8102\n1041 1010\n1 108101\n10−410161026\nTABLE 1\nTypical decay times of electrons τeand protons τp\norbiting a black hole for different values of magnetic\nfieldB.\nIn order to relate our results to realistic astrophysical\nscenarios, we perform the estimations of relevant param-\neters of the discussed model. Even thought the magnetic\nfield does not violate the geometry of the background\nspacetime, satisfying the condition ( 1), one cannot ne-\nglect its effect on the motion of charged particles due to\nthe large values of the specific charge (charge per mass\nratio) for elementary particles. Restoring the world con-\nstants, the dimensionless parameter B, widely used in\nthe present paper, takes the form\nB=|q|BGM\n2mec4, (67)\nreflecting the relative influence of the gravitational and\nmagnetic fields on the charged particle motion. Accord-\ning to Piotrovich et al. (2011); Baczko et al. (2016), the\ncharacteristic values of the magnetic fields near the stel-\nlar mass and supermassive black holes are B∼108G,\nforM= 10M⊙andB∼104G, forM= 109M⊙. For\nelectrons, one can estimate the parameter Bas\nBBH≈4.32×1010forM= 10M⊙,(68)\nBSMBH≈4.32×1014forM= 109M⊙.(69)\nAs a representative example one can estimate the pa-\nrameter Bfor an electron in the vicinity of the super-\nmassive black hole (SMBH) at the center of the Milky\nWay, which is currently the best studied SMBH candi-\ndate. The equipartition strength of the magnetic field\nnear the Galactic Center is usually considered to be of\ntens of Gauss (Johnson et al. (2015)). Moreover, multi-\nfrequencymeasurementsofpulsarnearthe GalacticCen-\nter by Eatough et al. (2013) demonstrate the existence\nof strong magnetic field of few hundred Gauss near the\nevent horizon of SMBH. The mass of the black hole can-\ndidate is measured to be 4 .3×106M⊙(for more de-\ntails about Sgr A*, see recent review by Eckart et al.\n(2017)). Thus, the parameter Bfor the electron sur-\nrounding SMBH at the center of our Galaxy can be es-\ntimated as\nBSgrA∗≈|e|BGM\n2mec4≈1.86×1010,(70)\nFor protons, the values of Bin (68) - (70) are lower by\nthe factor mp/me≈1836. Large values of the parameter\nBin realistic scenarios imply that the effects of magnetic\nfield on the dynamics of charged particles play one of the\nessential roles.\nRadiation reaction force acting on a charged particle is\nrepresented by the dimensionless parameter kwhich has15\nthe form\nk=2\n3q2\nmGM. (71)\nThe value of parameter kis much lower than those of\nB. For electrons orbiting stellar mass and supermassive\nblack holes we have respectively\nkBH∼10−19forM= 10M⊙,(72)\nkSMBH∼10−27forM= 109M⊙.(73)\nFor electron around Sgr A*, we have kSgrA∗∼10−25.\nFor protons, the values of kparameter is lower by the\nfactormp/me≈1836 as in case with B. Despite the\nweakness of parameter kas compared to B, it enters\ninto the equations for ultrarelativistic particles as kB2,\nwhich can make the effect of the radiation reaction force\nconsiderably large.\nOne can estimate the timescale of the decay of charged\nparticle oscillations. Typical orders of magnitude of the\noscillation decay time of an electron and proton orbiting\nblackholearegiveninTab. 1. Onecancomparetheseval-\nues with the timescale of one orbit of particle τcaround\nstellar mass and supermassive black holes. For the orbit\nat ISCO we have\nτc∼10−3s,forM= 10M⊙, (74)\nτc∼104s,forM= 109M⊙, (75)\nFor Sgr A* the electron decay time is about 104s, while\norbiting time at ISCO is ∼103s. It is interesting\nto note that for ions the decay time of oscillations is\nmuch less than for electrons, namely for the factor of\n(mp/me)3∼1010. Thus, one can conclude, that the ra-\ndiation reaction of electrons can be quite relevant for the\nastrophysically plausible magnetic fields providing rea-\nsonable decay times, while radiation reaction of ions can\nbe relevant only in the presence of magnetic fields much\nstronger than those corresponding to black holes, e.g., in\nthe vicinity of neutron stars.\nEstimations of the timescale of the radiative widening\nof charged particle orbit show that this process is slower\nthanthedampingofoscillationsduetoradiationreaction\nfor about 1010times in average, for the values of param-\netersBandkgiven by equations ( 72) and (68). However\ninsomeastrophysicalscenarioswith largemagneticfields\nthis process can be potentially measured.\nThe radiation reaction and related decay time of par-\nticle oscillations can have a significant effect in plasma\nsurrounding black hole, when the decay time of charged\nparticle becomes smaller than the average time between\ncollisions of particles in plasma.\n7.SUMMARY\nWe have studied the radiation reaction of charged par-\nticles moving around Schwarzschild black hole immersed\ninto external asymptotically uniform magnetic field. We\nstarted analysis from the study of the motion in flat\nspacetime, described by the Lorentz-Dirac equation and\nby its reduced form – the Landau-Lifshitz equation. We\nhavetestedboth equationsnumericallyandfoundthat in\nthe simplified framework of the asymptotically uniform\nmagnetic field both approaches lead to the same result.\nThe Landau-Lifshitz equation is more convenient as it is\na second order differential equation which can be solvedin usual manner, while for the Lorentz-Diracequationwe\nhave to perform the integration of dynamical equations\nbackwards in time, in order to avoid the exponential in-\ncrease of computational error.\nWehaveshownthat the tailterm appearingin the gen-\neral formalism for the motion in curved spacetime can\nbe neglected in the presence of magnetic field. Vanish-\ning the Ricci term for Schwarzschild black hole case en-\nablestousethecovariantformoftheLorentz-Diracequa-\ntion or its reduced form – the covariant Landau-Lifshitz\nequation. The motion of charged particles around non-\nrotating black hole in the presence of external uniform\nmagnetic field can be classified into two different types,\ndiffering by the orientation of the Lorentz force directed\ntowards and outwards from the black hole, which we\ncalled as minusandplus configurations , respectively. We\nconcentratedattentiononthe quasi-boundedorbitschar-\nacterized by the presence of maxima and minima of the\neffective potential. In such orbits, the charged particles\ncan undergo the stable quasi-harmonicoscillations in the\nradial and vertical directions. We have shown that the\npresence of the radiation reaction force leads to the de-\ncay of the particle oscillations, while the decay time in-\ncreases as the orbit approaches to the black hole. The\nfinal state of the particle depends on the orientation of\nthe Lorentz force with respect to the black hole. In case\nwhentheLorentzforceisdirectedtowardstheblackhole,\nthe radiationreactionleadsto the fall ofthe chargedpar-\nticle from initially stable orbit into the black hole. In-\nversely, when the Lorentz force is repulsive, the orbit of\nthe charged particle remains bounded while oscillations\ndecay.\nWe have shown that in the absence of oscillations, the\nradiation reaction force can shift the circular orbits out-\nwards from the black hole. This can occur only in case\nwhen the Lorentz force is repulsive. While the stability\nof the circular orbit is conserved, the radiation reaction\nacting against the particle motion leads to decreasing of\nthe linear velocity of the particle which in turn leads\nto the widening of the orbit. Even though the kinetic\nenergy of a particle decreases due to the synchrotron ra-\ndiation, its potential energy increases faster due to the\nwidening of the orbit. One can find efficiency of this\nprocess defined as the ratio between the gain energy re-\nquired for the shifting the particle from ISCO to infinity\nto its final energy. The value of the efficiency depends\non the initial position of the particle and the parame-\ntersBandk, which are governing the magnetic field and\nradiation reaction force, respectively. The efficiency can\nreach the values, close to the maximal 100%. The max-\nimal efficiency, however can never be achieved, since it\nwould require B → ∞, and the ISCO would coincide\nwith the black hole horizon. In the absence of magnetic\nfield, the formal efficiency of mechanical process of shift-\ning the particle orbit from infinity to the ISCO is 5 .7%.\nOne needs to note that the process of widening of orbits\nis many orders of magnitude slower that the process of\ncooling of the particle oscillations. The physical mecha-\nnism causing the gain in energy requires further investi-\ngations.\nWe have estimated the relevant parameters of the\nmodel as well as the decay times of the particle oscil-\nlations for typical values of the magnetic field for stellar\nmass and supermassive black holes. Assuming the test16\nparticle as an electron, we have found that the values of\nthe decay times are reasonable and can be measured in\nmost of the astrophysical scenarios. Despite to the slow-\nness of the process of the radiative widening of orbits the\nestimations show that for stellar mass black holes in the\npresence of magnetic fields of order 108G and more, the\nwidening of orbits can be potentially measured.\nWe expect that our results can be relevant in treating\nvariety of astrophysical phenomena observed in micro-\nquasars or active galactic nuclei, containing stellar mass\nor supermassive black holes. These results can put rele-\nvant limits on validity of recent models of high-frequency\nquasi-periodic oscillations (HF QPOs), or motion of jets\nobserved in microquasars. The limits implied by the ra-\ndiation reaction forces can be clearly relevant for boththe magnetized geodesic models of HF QPOs discussed\nin Koloˇ s et al. (2017); see also Stuchl´ ık et al. (2013), or\nthe string loop models of jets or HF QPOs discussed\nin Jacobson & Sotiriou (2009); Stuchl´ ık & Koloˇ s (2012);\nKoloˇ s & Stuchl´ ık (2013); Tursunov et al. (2014).\nACKNOWLEDGMENTS\nA.T.andM.K.acknowledgetheCzechScienceFounda-\ntion Grant No. 16-03564Yand the Silesian University in\nOpava Grant No. SGS/14/2016. Z.S. acknowledges the\nAlbert Einstein Centre for Gravitation and Astrophysics\nsupported by the Czech Science Foundation Grant No.\n14-37086G. 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J. 2000, Nuovo Cimento B Serie, 115, 687"    },    {        "title": "1505.04087v2.Reliable_Damping_of_Free_Surface_Waves_in_Numerical_Simulations.pdf",        "content": "Reliable Damping of Free Surface Waves in Numerical\nSimulations\nRobinson Peri\u0013 c1, Moustafa Abdel-Maksoud\nHamburg University of Technology (TUHH), Institute for Fluid Dynamics and Ship\nTheory (M8), Hamburg, Germany\nAbstract\nThis paper generalizes existing approaches for free-surface wave damping via\nmomentum sinks for \row simulations based on the Navier-Stokes equations.\nIt is shown in 2D \row simulations that, to obtain reliable wave damping,\nthe coe\u000ecients in the damping functions must be adjusted to the wave pa-\nrameters. A scaling law for selecting these damping coe\u000ecients is presented,\nwhich enables similarity of the damping in model- and full-scale. The in-\n\ruence of the thickness of the damping layer, the wave steepness, the mesh\n\fneness and the choice of the damping coe\u000ecients are examined. An e\u000ecient\napproach for estimating the optimal damping setup is presented. Results of\n3D ship resistance computations show that the scaling laws apply to such\nsimulations as well, so the damping coe\u000ecients should be adjusted for every\nsimulation to ensure convergence of the solution in both model and full scale.\nFinally, practical recommendations for the setup of reliable damping in \row\nsimulations with regular and irregular free surface waves are given.\nKeywords: Damping of free surface waves, absorbing layer, volume of \ruid\n(VOF) method, damping coe\u000ecient, scaling law\n1. Introduction\nIn free surface \row simulations based on the Navier-Stokes equations, it\nis often required to model an in\fnite domain, which basically means min-\nimizing undesired wave re\rections at wave-maker and domain boundaries,\n1Corresponding author. Tel. +49 40 42878 6031.\nE-mail adress: robinson.peric@tuhh.de\nPreprint submitted to Ship Technology Research May 22, 2017arXiv:1505.04087v2  [physics.flu-dyn]  19 May 2017while choosing the solution domain as small as possible in order to lower\nthe computational e\u000bort. Since such simulations are among the computa-\ntionally most e\u000bortful techniques to solve numerical \row problems, they are\nemployed when simpler approaches (e.g. potential \row methods) cannot be\nused or when higher accuracy is required. As the schemes contain numer-\nical errors (discretization, iteration, modelling (in some cases)), to achieve\nthe required accuracy it is necessary to minimize avoidable uncertainties; of\nthese, the performance of the wave damping approach is often among the\nmost critical.\nThe elimination of wave re\rections at the domain boundaries is commonly\nachieved by\n(i)increasing the domain size\n(ii)beaches (e.g. Lal and Elangovan (2008)): a slope in the domain bottom\nleads to wave breaking and energy dissipation as in experiments\n(iii) grid damping (e.g. Kraskowski (2010), Peri\u0013 c and Abdel-Maksoud (2015)):\ncontinuously increasing the cell size towards the corresponding domain\nboundary increases numerical discretization and iteration errors, thus\ndamping the wave (this approach is also called numerical beach or grid\nextrusion)\n(iv) active wave absorption techniques (e.g. Cruz (2008); Higuera et al.\n(2013); Sch a\u000ber and Klopman (2000)): a boundary-based wave-maker\ngenerates waves which eliminate the incoming waves via destructive\ninterference\n(v)solution-forcing orsolver-coupling (e.g. Ferrant et al. (2008), Guignard\net al. (1999), Kim et al. (2012), Kim et al. (2013), W ockner-Kluwe\n(2013)): the \row is forced to a known solution in the vicinity of the\nboundary or the Navier-Stokes-based \row solver is coupled to another\n(e.g. potential \row based) solver\n(vi) damping layer approaches (e.g. Cao et al. (1993); Choi and Yoon\n(2009); Ha et al. (2011); Israeli and Orszag (1981); Park et al. (1999)):\nthe damping layer (also called sponge layer, absorbing layer, damping\nzone, porous media layer) is a zone set up next to the corresponding\nboundaries, in which momentum sinks are included in the governing\nequations to damp the waves propagating through the zone\n2Apart from the above methods, further approaches have been developed for\nother governing equations, like potential \row with boundary element method\nor Boussinesq-type equations (see e.g. Grilli and Horrillo (1997) and ref-\nerences therein). However, many of these approaches have not yet been\ntransferred to Navier-Stokes-type equations. From the above mentioned ap-\nproaches, (i) is the least feasible due to its (in many cases enormous) inherent\nincrease in computational e\u000bort. With (ii) it is di\u000ecult to minimize re\rec-\ntions to less then 10% of the incoming wave, which is a problem in experi-\nments as well. Approaches (iv) and (v) have recently attracted much interest\nand produced good results; however, since they are often used to simultane-\nously generate and damp waves, it is likely that their damping performance\nvaries depending on the kind of waves they are currently generating. At the\ntime of writing, (iii) and (vi) are the most widely-used wave damping ap-\nproaches in Navier-Stokes-type equation \row solvers, implemented in most\ncommercial and research codes. Although both approaches have been used\nwith success, the disadvantage of (iii) is that its performance depends on\ngrid, time step, temporal/spatial discretization schemes, etc., which for suf-\n\fciently \fne discretization is not the case for (vi), as the results in this work\nsuggest. Thus (iii) is less predictable than (vi) regarding the damping qual-\nity. This work focuses exclusively on damping layer approaches (vi) based\non momentum source terms, since a detailed investigation and comparison\nof all previously mentioned approaches was not possible in the scope of this\nstudy.\nThis paper discusses momentum source terms based on linear or quadratic\ndamping functions as de\fned in Sect. 3. The amount and character of\nthe damping is controlled by coe\u000ecients in the damping functions and the\nthickness of the damping layer. Such approaches are widely used and already\nimplemented in several open source as well as commercial computational \ruid\ndynamics (CFD) solvers. Section 3 shows how the existing implementations\ncan be generalized. Two of the most widely used implementations, the ones\nfrom CD-adapco STAR-CCM+, based on Choi and Yoon (2009), and ANSYS\nFluent, based on Park et al. (1999), are discussed in Sect. 4. Various other\nimplementations of linear or quadratic damping exist, see e.g. Cao et al.\n(1993) and Ha et al. (2011).\nThe nomenclature from Sect. 4 is used throughout the work. For all\nsimulations in this study, the damping function from Choi and Yoon (2009)\nis used, since it is rather generic and can be set up to act similar to the\nother approaches mentioned. Thus the results are easily applicable to all\n3damping approaches used in other CFD codes which can be generelized as\ndescribed in Sect. 3. Although widely used, much about the damping func-\ntions remains unknown at the time of writing. It has been observed that the\ncoe\u000ecients in the damping functions, and thus the damping performance, are\ncase-dependent. In the above mentioned codes, these damping coe\u000ecients\ncan be modi\fed by the user. However, no guidelines seem to exist for this.\nThus in practice, either the default settings or values from experience are\nused as coe\u000ecients in the damping functions. If during or after the simula-\ntion it is observed that the damping does not work satisfactorily, then the\ndamping coe\u000ecients are modi\fed by trial and error and the simulation is\nrestarted. This procedure is repeated until acceptable damping is obtained,\na process which requires human interaction and can cost considerable addi-\ntional time and computational e\u000bort. Especially in the light of the increased\nautomation of CFD computations, it would be preferable to be sure about the\ndamping quality before the simulation. Similar to wave damping in experi-\nments, where according to Lloyd (1989) seldom data on beach performance is\npublished, also with CFD simulations, the performance of the wave damping\nis often not su\u000eciently accounted for. Thus with most CFD publications,\nit is di\u000ecult to judge on the damping quality even if the damping setup is\ngiven.\nThe aim of the present work is to clarify how the damping functions\nwork, on which factors the damping quality depends and how reliable wave\ndamping can be set up case-independently, so that no more \fne-tuning of\nthe damping coe\u000ecients is necessary.\nIn Sect. 3, the generalized forms of linear and quadratic damping func-\ntions are given. In the following Sect. 4, the damping approach used in\nthis work is described. Moreover, it is shown exemplarily for two widely\nused damping functions how these can be generalized to the equations given\nin Sect. 3 and how results from one implementation can be transferred to\nanother.\nStarting from the analogy of the damped harmonic oscillator, Sect. 5\nattempts to show similarities to damping phenomena outside the hydrody-\nnamics \feld and constructs scaling laws for wave damping based on this\ninformation. These scaling laws are fully formulated and veri\fed in Sects.\n13 and 14. Furthermore, recommendations for selecting the damping coe\u000e-\ncients are given.\nSections 8 to 12 investigate via 2D \row simulations how the damping qual-\nity and behavior is in\ruenced by the choice of the coe\u000ecients in the damping\n4functions, the thickness of the damping layer, the computational grid and\nthe wave steepness. As the investigations so far have been mainly concerned\nwith regular, monochromatic waves, recommendations for the damping for\nirregular waves are given in Sect. 15 and illustrated with simulation results.\nSince wave damping is also widely used to speed up convergence for ex-\nample in ship resistance computations, it is veri\fed in Sect. 16 via 3D \row\nsimulations that the presented scaling laws hold for such cases as well.\n2. Governing Equations and Solution Method\nThe governing equations for the simulations are the Navier-Stokes equa-\ntions, which consist of the equation for mass conservation and the three\nequations for momentum conservation:\nd\ndtZ\nV\u001adV+Z\nS\u001av\u0001ndS= 0; (1)\nd\ndtZ\nV\u001auidV+Z\nS\u001auiv\u0001ndS=\nZ\nS(\u001cijij\u0000pii)\u0001ndS+Z\nV\u001agiidV+Z\nVqidV : (2)\nHereVis the control volume (CV) bounded by the closed surface S, vis the\nvelocity vector of the \ruid with the Cartesian components ui,nis the unit\nvector normal to Sand pointing outwards, tis time,pis the pressure, \u001aand\u0016\nare \ruid density and dynamic viscosity, \u001cijare the components of the viscous\nstress tensor, ijis the unit vector in direction xj,gcomprises the body forces\nandqiis an optional momentum source term. For the present simulations,\nthe only body force considered was the gravitational acceleration, i.e. g=\n(0;0;\u00009:81m\ns2)T. Only incompressible Newtonian \ruids are considered in this\nstudy. Thus \u001cijtakes the form\n\u001cij=\u0016\u0012@ui\n@xj+@uj\n@xi\u0013\n:\nNo turbulence modeling was applied to the above equations since, unless\nwaves break, the \row inside them can be considered practically laminar and\nall structures of interest can be resolved with acceptable computational ef-\nfort. For all simulations, the software STAR-CCM+ 8 :02:008 was used. The\n5volume of \ruid (VOF) method implemented in the STAR-CCM+ software\nis used to account for the two phases (air and water). Further details on the\nmethod can be found in Muzaferija and Peri\u0013 c (1999). The governing equa-\ntions are applied to each cell and discretized according to the Finite Volume\nMethod (FVM). All integrals are approximated by the midpoint rule. The\ninterpolation of variables from cell center to face center and the numerical\ndi\u000berentiation are performed using linear shape functions, leading to ap-\nproximations of second order. The integration in time is based on assumed\nquadratic variation of variables in time, which is also a second-order approx-\nimation. Each algebraic equation contains the unknown value from the cell\ncenter and the centers of all neighboring cells with which it shares common\nfaces. The resulting coupled equation system is then linearized and solved\nby the iterative STAR-CCM+ implicit unsteady segregated solver, using an\nalgebraic multigrid method with Gauss-Seidl relaxation scheme, V-cycles for\npressure and volume fraction of water, and \rexible cycles for velocity cal-\nculation. For each time step, one iteration consists of solving the governing\nequations for the velocity components, the pressure-correction equation (us-\ning the SIMPLE method for collocated grids to obtain the pressure values\nand to correct the velocities) and the transport equation for the volume frac-\ntion of water. For further information on the discretization of and solvers for\nthe governing equations, the reader is referred to Ferziger and Peri\u0013 c (2002)\nor the STAR-CCM+ software manual.\n3. General Formulation for Linear and Quadratic Damping\nLinear wave damping is obtained by inserting the momentum source term\nqd;lin\niforqiin Eq. (2):\nqd;lin\ni=\u001aCi;linui; (3)\nwith coe\u000ecient Ci;lin, which usually depends on the spatial location. Ci;linreg-\nulates the strength of the damping and is used to provide a smooth blending-\nin of the damping, by increasing the amount of damping from weak damping\nwhere the waves enter the damping layer to strong damping at the end of\nthe damping layer. This is to prevent undesired re\rections at the entrance\nto the damping layer as shown in Sects. 8 and 9. Commonly, the blending-in\nis realized in an exponential or quadratic fashion.\nQuadratic wave damping takes the form\nqd;quad\ni =\u001aCi;quadjuijui; (4)\n6with coe\u000ecient Ci;quad, which regulates strength and blending-in of the damp-\ning. In contrast to Eq. (3), \ruid particles with higher velocities experience\ndisproportionately high damping.\nIn most CFD codes, Ci;linandCi;quadcan be adjusted by the user. Usually,\nbut not necessarily, the momentum source terms are only applied to the\nequation for the vertical velocity component.\n4. Common Implementations of Linear and Quadratic Wave Damp-\ning\nA widely used approach is the one in Choi and Yoon (2009), which is e.g.\nimplemented in the commercial software code STAR-CCM+ by CD-adapco.\nIt features a combination of linear and quadratic damping, which allows the\nuse of either one or a combination of both approaches. Taking zas the\nvertical direction (i.e. perpenticular to the free surface) and was the vertical\nvelocity component, then the following source term appears for qiin Eq. (2):\nqd\nz=\u001a(f1+f2jwj)e\u0014\u00001\ne1\u00001w ; (5)\n\u0014=\u0012x\u0000xsd\nxed\u0000xsd\u0013n\n: (6)\nHerexstands for the wave propagation direction with xsdbeing the start\nandxedthe endx-coordinate of the damping layer, thus the thickness of the\ndamping layer xd=jxed\u0000xsdj.f1is the damping constant for the linear\npart andf2is the damping constant for the quadratic part of the damping\nterm. The default values are f1= 10:0 s\u00001andf2= 10:0 m\u00001according to\nthe STAR-CCM+ manual (release 8 :02:008). The terme\u0014\u00001\ne1\u00001blends in the\ndamping term, i.e. it is zero at xsd, and it equals one at xed.\u0014describes\nvianthe character of the blending functions, i.e. for n= 1 the blending\nis nearly linear. When increasing n, the blending becomes smoother at the\nentrance to the damping layer and at the same time more abrupt at xed, see\nFig. 1.\n7Figure 1: The functione\u0014(x)\u00001\ne1\u00001evaluated for n= 1,n= 2 andn= 4 over the dimensionless\ndistancex\u0000xsd\nxed\u0000xsd; a wave enters the damping layer atx\u0000xsd\nxed\u0000xsd= 0 and is damped during\npropagation towardsx\u0000xsd\nxed\u0000xsd= 1, where the damping layer ends\nAnother widely used approach was presented by Park et al. (1999), which\nis implemented in ANSYS Fluent.:\nqd\nz=\u001a(0:5f3jwj)\u0014\u0012\n1\u0000z\u0000zfs\nzb\u0000zfs\u0013\nw ; (7)\nwith damping constant f3,\u0014as given in Eq. (6) with n= 2, vertical coordi-\nnatezandz-coordinates of the domain bottom zband the free water surface\nzfs. The default value for f3is 10:0 m\u00001.\nThis approach can be generalized to a quadratic damping function accord-\ning to Eq. (4). It corresponds to the quadratic part of Eq. (5), except for a\nquadratic instead of exponential blending in x-direction and an added verti-\ncal fade-in. The latter term is a linear fade between domain bottom, where\nno damping is applied, to free surface level, where full damping is applied;\nfor practical deep water conditions of several wavelengths water depth, the\nin\ruence of the z-fading term on the applied damping becomes small, since\nmost wave energy is concentrated in the vicinity of the free surface. Thus for\nsuch cases, this damping approach can be modeled using Eqs. (5) and (6)\n8with accordingly adjusted coe\u000ecients. In a similar manner, also the other\ndamping layer approaches from Sect. 1 (vi) can be modeled. Therefore in the\nfollowing, only the approach by Choi and Yoon (2009) will be considered to\nstudy wave damping. Thus the \fndings in this work can easily be transferred\nto other damping approaches.\n5. Dependence of Damping Coe\u000ecients\nFor gravity waves, all scaling laws should be consistent with Froude scal-\ning. Thus to obtain similar wave damping for waves of di\u000berent scale, the\nblending part of Ci;linandCi;quadfrom Eqs. (3) and (4) must be geomet-\nrically similar. Therefore, the thickness xdof the damping layer must be\ndirectly proportional to the wavelength \u0015. For the remaining part of Ci;lin\nandCi;quad, which regulates the magnitude of the damping, the scaling laws\ncan be obtained by dimensional analysis. Due to its dimension of [ s\u00001],Ci;lin\nis directly proportional to the wave frequency !, whereasCi;quadhas the di-\nmension [m\u00001] and is thus directly proportional to \u0015\u00001(or!2in deep water).\nThus to achieve similar damping when changing scale and/or wave, it is nec-\nessary to:\n9Scaling Laws\n1. Set the damping thickness xdso that it is geometrically similar to the\ndamping layer thickness xd;refof the reference case (i.e. scale xdwith\nthe percentual change in wavelength)\nxd=xd;ref\u0001\u0015\n\u0015ref; (8)\n2. For linear damping , scalef1with the change of wave frequency !\nf1=f1;ref\u0001!\n!ref; (9)\nwith wave frequency !refof the reference case. This holds for all damp-\ning formulations that can be generalized by Eq. (3).\n3. For quadratic damping , scalef2with the change of wavelength \u0015to the\npower of minus one\nf2=f2;ref\u0001\u0015ref\n\u0015; (10)\nwith wavelength \u0015refof the reference case. This holds for all damping\nformulations that can be generalized by Eq. (4).\n6. Numerical Simulation Setup\nFigure 2 shows the solution domain, a 2D deep water wave tank with\nlengthLx= 6\u0015and height Lz= 4:5\u0015, given in relation to wavelength\n\u0015. It is \flled with water to a depth of d= 4\u0015, the rest of the tank is\n\flled with air. The origin of the coordinate system is set in the bottom left\nfront corner of the tank in Fig. 2. The top boundary (i.e. z=Lz) is a\npressure outlet boundary, i.e. the pressure there is set constant and equal\nto atmospheric pressure. This corresponds to an open water tank, where air\ncan \row in and out through the tank top. The x=Lxboundary is a pressure\noutlet, where volume fraction and hydrostatic pressure are prescribed for an\nundisturbed free surface. The waves are generated by prescribing velocities\nand volume fraction of a 5th-order Stokes waves according to Fenton (1985)\nat thex= 0 boundary. The wave damping layer extends over a distance of\nxd= 2\u0015in boundary-normal direction from the x=Lxboundary. Thus the\n10waves are created at x= 0, propagate in x-direction, enter the damping layer\natx=Lx\u0000xd, are subjected to damping until x=Lxis reached, where\nthe remaining waves are re\rected and, while further subjected to damping,\npropagate back to x=Lx\u0000xd. If the damping was set up correctly, then\nthe re\rected waves are either completely damped or their height is decreased\nso much, that their in\ruence on the simulation results can be neglected.\nOtherwise they will be evident in the simulation results as additional error,\nwhich can be substantial as will be shown in the following sections.\nFigure 2: 2D wave tank \flled with water (light gray) and air (white). The damping layer\nis shown in dark grey\nLocal mesh re\fnement is used, so that the grid is \fnest in the vicinity\nof the free surface, where the waves are discretized by roughly 100 cells per\nwavelength and 16 cells per wave height, as seen in Fig. 3. The temporal\ndiscretization involved more than 500 time steps \u0001 tper wave period T, thus\nin all investigated cases the Courant number C= (ui\u0001t)=\u0001xiremains well\nbelow 0:5 for every cell with size \u0001 xiand corresponding \ruid velocity ui.\nWaves were generated for over 12 wave periods with 8 iterations per time\nstep and under-relaxation of 0 :4 for pressure and 0 :9 for all other variables.\nThe discretization is based on the grid convergence study conducted by Peri\u0013 c\n(2013).\n11Figure 3: Computational grid for the whole domain (right), and a close up of the dis-\ncretization around free surface level (left)\nThe numerical setup is the same for all simulations in this study, only\nthat the scale, wave and damping parameters are varied. Variations of the\nsetup are mentioned where they occur.\n7. Assessing the Damping Performance for Regular, Monochro-\nmatic Waves\nWhen wave re\rections occur within a damping zone, the waves will prop-\nagate back into the solution domain with a diminished height. Due to the\nresulting superposition of the partly re\rected wave and the original wave,\nthe original wave will seem to grow and shrink in a time-periodic fashion.\nFor regular, monochromatic waves, this occurs uniformly in all parts of the\ndomain where the two wave systems are superposed as seen later e.g. in\nSect. 11 from Fig. 9. In the present work, this phenomenon is called a\npartial standing wave , since the more the damping deviates from the optimal\nvalue, the more does the phenomenon resemble a standing wave, until for the\nbounding cases (i.e. zero damping or in\fnitely large damping) there is 100%\nre\rection, and a standing wave occurs.\nAssessing the amount of re\rections for regular monochromatic waves is\nan intricate issue. Since only a single wave period occurs, a wave spectrum\n12cannot be constructed. As seen from Fig. 9, the wave height envelope changes\nover space. However, this cannot be predicted since it is not known in advance\nat which point in the damping layer the re\rection mainly occurs. Thus\nfrom the recordings of a single wave probe, the height of the re\rected wave\ncannot be determined. However, the surface elevation in close vicinity to\nthe boundary to which the damping zone is attached shows how much of\nthe incident wave reaches the domain boundary despite the damping, since\na maximum ampli\fcation occurs at the domain boundary when the wave is\nre\rected.\nThus in this study, \frst the free surface elevation is recorded at x= 5:75\u0015,\ni.e. next to the domain boundary. The average wave height at this location is\nobtained and plottet in relation to the average wave height of the undamped\nwave, to show how much the wave height is reduced after propagating from\nthe entrance to the outer boundary of the damping layer. As shown in Sects.\n8 and 9, this is a good criterion for insu\u000ecient to optimal wave damping,\nwhere wave re\rections occur mainly at the outer boundary of the damping\nlayer, atx=Lx; however, this criterion will not detect re\rections if the\ndamping is too strong, since then the waves will be re\rected at the entrance\nto the damping layer.\nTherefore, secondly a re\rection coe\u000ecient CRis computed as proposed\nby Ursell et al. (1960). CRcorresponds to the ratio of the heights of the\nincident wave to the re\rected wave. It can be written as\nCR= (Hmax\u0000Hmin)=(Hmax+Hmin); (11)\nwhereHmaxis the maximum and Hminthe minimum value of the wave height\nenvelope. It holds 0 \u0014CR\u00141, so thatCR= 1 for perfect wave re\rection\nandCR= 0 for no wave re\rection.\nAs explained above, the wave height for a partial standing wave is not\nconstant, but oscillates around a mean value. This occurs uniformly in all\ndomain parts where the partial standing wave has fully developed. Thus\nHmaxandHminare obtained in the following fashion in this work: The free\nsurface elevation in the whole tank is recorded at 40 evenly timed instances\nper wave period starting at 10 periods simulation time for 2 wave periods.\nFor each recording j, the average wave height \u0016Hjis calculated for the inter-\nval 2\u0015\u0014x\u00144\u0015, which is adjacent to the damping zone but not subject to\nwave damping and su\u000eciently away from the inlet; furthermore, the partial\nstanding wave at this location is fully developed during this time interval,\n13while wave re-re\rections at the wave-maker have not yet developed signi\f-\ncantly. Therefore, the maximum \u0016Hmaxand minimum \u0016Hminof all \u0016Hjvalues\ncan be taken as HmaxandHmin, respectively. This approach detects all wave\nre\rections that propagate back into the solution domain. The accuracy of\nthe scheme can be increased if the surface elevation in the tank is recorded in\nsmaller time intervals, however this also increases the computational e\u000bort\nsigni\fcantly. Furthermore, the above procedure requires undisturbed regular\nmonochromatic waves. Yet in many applications, \ruid-structure interactions\ncreate \row disturbances, so the above scheme cannot be applied directly in\nthe simulations, but an additional 2D simulation of undisturbed wave prop-\nagation and damping for otherwise the same setup is required to obtain CR.\nAs long as this process is not fully automized, running and post-processing\nthe 2D simulation requires additional human e\u000bort. Overall, this procedure\nis rather e\u000bortful, which explains why CRis rarely computed in practice.\nWith the two presented approaches, it is possible to distinguish between\nthose wave re\rections, which occur mainly at the entrance to the damping\nzone, and those re\rections which occur mostly at the boundary to which the\ndamping zone is attached. Furthermore, the in\ruence of wave re\rections on\nthe solution can be quanti\fed.\n8. Variation of Damping Coe\u000ecient for Linear Damping\nTo investigate which range for the linear damping coe\u000ecient according\nto Eq. (5) produces satisfactory wave damping, waves with wavelength \u0015=\n4 m and height 0 :16 m are investigated. Only linear damping is considered,\nsof2= 0. Simulations are performed for damping coe\u000ecient f1between\nf12[0:625 s\u00001;1000 s\u00001]. The recorded surface elevations near the end of the\ndamping layer in Fig. 4 show that the damping is strongly in\ruenced by the\nchoice off1, while the wave phase and period remain nearly unchanged.\n14Figure 4: Surface elevation scaled by height Hof the undamped wave over time scaled by\nthe wave period T; recorded at x= 5:75\u0015for simulations with f12[0:625 s\u00001;80 s\u00001] and\nf2= 0\nFigure 5 shows how a variation of f1a\u000bects the re\rection coe\u000ecient CR,\nwhich describes the amount of re\rected waves that are present in the solution\ndomain outside of the damping zone, and the mean measured wave height\nHmeanrecorded in close vicinity to the boundary to which the damping layer\nis attached.\n\u000fFor smaller damping coe\u000ecients ( f1.20 s\u00001), wave re\rections occur\nmainly at the domain boundary to which the damping zone is attached.\nHere, the both show the same trend, with CR<H mean=(2H) since the\nre\rected waves loose further height until they leave the damping zone\natx= 4\u0015. Forf1<1:25 s\u00001, the in\ruence of the damping is so small\nthat a partial standing wave (as described in Sect. 7) higher than the\ninitial wave appears even inside the damping layer.\n\u000fFor stronger damping ( f1>20 s\u00001), wave re\rection occurs mainly at\nthe entrance to the damping zone. Thus Hmean continually decreases\nwhen increasing f1, since less of the incoming wave can pass through\nthe damping zone, so the water surface will be virtually \rat near the\ndomain boundary. This is also visible later in Figs. 13 from Sect.\n13. The aforementioned increase in the amount of wave re\rections\ndetectable in the solution domain can be seen in the curve for CR. This\nshows that, for a given fade-in function and damping layer thickness,\n15there is an optimum for f1so that the wave re\rections propagating\nback into the solution domain are minimized. The best damping was\nachieved for f1= 10 s\u00001, in which case the e\u000bects of wave re\rections\non the wave height within the solution domain are less than 0 :7%.\nFigure 5: Mean wave height Hmean recorded at x= 5:75\u0015scaled by twice the height Hof\nthe undamped wave and re\rection coe\u000ecient CRover damping coe\u000ecient f1, whilef2= 0\nThis provides the following conclusions: Hmean is not a suitable indicator\nfor damping quality if the damping is stronger than optimal, however it is\nuseful to characterize where the re\rections originate from: if CRshows that\nnoticeable re\rections are present within the solution domain, while at the\nsame timeHmeanis negligibly small, then the re\rections cannot occur at the\ndomain boundary, but must occur closer to the entrance to the damping layer.\nSigni\fcant re\rections ( CR>2%) in form of partial standing waves appear for\nroughlyf1<5 s\u00001andf1>80 s\u00001. The height of the partial standing wave\nincreases the more f1deviates from the regime where satisfactory damping\nis observed; however, the increase is slower if the damping is stronger than\noptimal instead of weaker.\n9. Variation of Damping Coe\u000ecient for Quadratic Damping\nTo investigate which range for the quadratic damping coe\u000ecient accord-\ning to Eq. (5) produces satisfactory wave damping, waves with wavelength\n16\u0015= 4 m and height 0 :16 m are investigated. Only quadratic damping is con-\nsidered, so f1= 0. Simulations are performed for damping coe\u000ecient f2\nbetweenf22[0:625 m\u00001;10240 m\u00001]. The recorded surface elevations near\nthe end of the damping layer in Fig. 6 show that the damping is strongly\nin\ruenced by the choice of f2, while the wave phase and period remain nearly\nunchanged.\nFigure 6: Surface elevation scaled by height Hof the undamped wave over time scaled by\nthe wave period T; recorded at x= 5:75\u0015for simulations with f22[0:625 m\u00001;10240 m\u00001]\nandf1= 0\nFigure 7 shows how a variation of f2a\u000bects the re\rection coe\u000ecient CR\nand the mean wave height Hmeanrecorded in close vicinity to the boundary\nto which the damping layer is attached.\nThe results show the same trends as the ones in Sect. 8. For the given\nfade-in function and damping layer thickness, there is an optimum for f2\nso that the wave re\rections propagating back into the solution domain are\nminimized. The best damping was achieved for f2= 160 m\u00001, in which case\nthe e\u000bects of wave re\rections on the wave height within the solution domain\nare less than 0 :7%. The e\u000bects of wave re\rections increase the further f2\ndeviates from the optimum. For smaller f2values, the waves are re\rected\nmainly at the domain boundary, for larger f2values the re\rection occurs\nmainly at the entrance to the damping zone. Signi\fcant re\rections ( CR>\n2%) in form of partial standing waves appear for roughly f2<80 m\u00001and\nf2>640 m\u00001.\n17Figure 7: Mean wave height Hmean recorded at x= 5:75\u0015scaled by twice the height Hof\nthe undamped wave and re\rection coe\u000ecient CRover damping coe\u000ecient f2, whilef1= 0\nCompared to the linear damping functions from the previous section, the\nuse of quadratic damping functions does not o\u000ber a signi\fcant improvement\nin damping quality. With an optimal setup, both approaches provide roughly\nthe same damping quality if the setup is optimized. However, the range of\nwave frequencies that are damped satisfactorily is narrower for quadratic\ndamping.\n10. In\ruence of Computational Mesh on Achieved Damping\nThe simulations from Sect. 8 are rerun with same setup except for a grid\ncoarsened by factor 2. Thus whereas the \fne mesh simulations discretize the\nwave with 100 cells per wavelength and 16 cells per wave height, the coarse\nmesh simulations have 50 cells per wavelength and 8 cells per wave height.\nAll coarse grid re\rection coe\u000ecients di\u000ber from their corresponding \fne grid\nre\rection coe\u000ecient by CR;coarse =CR;\fne\u00060:8%. This is also visible in Fig.\n8. Thus for su\u000ecient resolution, the damping e\u000bectiveness can be considered\ngrid-independent. This is expected since the damping-related terms in Eqs.\n(2) to (4) do not depend on cell volume. The used grids in this study are\nthus adequate.\n18Figure 8: Re\rection coe\u000ecient CRover damping coe\u000ecient f1for coarse and \fne mesh\nsimulations, with f2= 0\n11. In\ruence of the Thickness of the Damping Layer\nThe simulation for \u0015= 4:0 m,H= 0:16 m andf1= 10 s\u00001from Sect.\n8 is repeated for xd= 0\u0015;0:25\u0015;0:5\u0015;0:75\u0015;1:0\u0015;1:25\u0015;1:5\u0015;2:0\u0015;2:5\u0015with\notherwise the same setup. The evolution of the free surface elevation in the\ntank over time in Fig. 9 shows that xdhas a strong in\ruence on the achieved\ndamping. Setting xd= 0\u0015deactivates the damping and produces at \frst\na nearly perfect standing wave, which then degenerates due to the in\ru-\nence of the pressure outlet boundary, since prescribing hydrostatic pressure\nestablishes an oscillatory in-/out\row of water through this boundary which\ndisturbs the standing wave. For xd= 0:5\u0015, a strong partial standing wave oc-\ncurs, and for xd= 1:0\u0015only slight re\rections are still observable CR\u00191:6%.\nFor largerxd, the in\ruence of wave re\rections continues to decrease. This is\nevident from the plot of CRfor the simulations shown in Fig. 10.\n19Figure 9: Free surface elevation over x-location in tank shown for 40 equally spaced\ntime instances over one period starting at t= 16 s; from top to bottom xd=\n0\u0015;0:5\u0015;1:0\u0015;1:5\u0015;2:0\u0015;2:5\u0015; the damping zone is depicted as shaded gray\n20Figure 10: Re\rection coe\u000ecient CRover damping zone thickness xd\nSubsequently, the simulations from Sect. 8 have been rerun with the\ndamping thickness set to xd= 1\u0015. Comparing the resulting curves for CR\noverf1shows that increasing xdnot only improves the damping quality; it\nalso widens the range of damping coe\u000ecients for which satisfactory damping\nis obtained; thus the wave damping then becomes less sensitive to !the more\nxdincreases. However, this also increases the computational e\u000bort.\nFigure 11: Re\rection coe\u000ecient CRover damping coe\u000ecient f1forxd= 1\u0015andxd= 2\u0015,\nwhilef2= 0\n21The results show that, if the damping coe\u000ecients are set up close to the\noptimum and it is desired that CR<2%, thenxd= 1\u0015su\u000eces. This knowl-\nedge is useful, since by reducing xdthe computational domain can be kept\nsmaller and thus the computational e\u000bort can be reduced. However, if better\ndamping is desired or when complex \row phenomena are considered, espe-\ncially when irregular waves or wave re\rections from bodies are present, then\nthe damping layer thickness should be increased to damp all wave compo-\nnents successfully. The present study suggests that a damping layer thickness\nof 1:5\u0015\u0014xd\u00142\u0015can be recommended.\n12. In\ruence of Wave Steepness on Achieved Damping\nThe simulations in this section are based on those from Sect. 8, i.e.\n\u0015= 4:0 m,H= 0:16 m and varying f1in the range [0 :625 s\u00001;1000 s\u00001]. The\nsimulations were rerun with the same setup except for two modi\fcations: The\nwave height was changed to H= 0:4 m, resulting in a steepness of H=\u0015 = 0:1\ninstead of the previous H=\u0015 = 0:04. Furthermore, the grid was adjusted to\nmaintain the same number of cells per wave height as well as per wavelength,\nso that both results are comparable.\nAs can be seen from Figure 12, the in\ruence of the increased wave steep-\nness is comparatively small, except for the cases with signi\fcantly smaller\nthan optimum damping ( f1\u00142:5 s\u00001). For the rest of the range, i.e. 5 \u0014f1\u0014\n1000 s\u00001, the di\u000berence in re\rection coe\u000ecients is only CR(H=\u0015 = 0:1) =\nCR(H=\u0015 = 0:04)\u00061:7%. Therefore, although the damping performs slightly\nbetter for waves of smaller steepness, the in\ruence of wave steepness can be\nassumed negligible for most practical cases. If stronger wave steepness is con-\nsidered and less uncertainty is required, then the thickness of the damping\nlayer can be further increased to decrease CR.\n22Figure 12: Re\rection coe\u000ecient CRover damping coe\u000ecient f1for waves of same period\nbut di\u000bering steepness H=\u0015 = 0:04 andH=\u0015 = 0:1, whilef2= 0\n13. The Scaling Law for Linear Damping\nIn order to verify the assumed scaling law for linear damping from Sect.\n5, the simulation setup with the best damping performance ( f1= 10:0 s\u00001)\nfrom Sect. 8 was scaled geometrically and kinematically so that the generated\nwaves are completely similar, for wavelengths \u0015= 0:04 m;4 m;400 m and thus\ncorresponding heights of H= 0:0016 m;0:16 m;16 m. This corresponds to a\nrealistic scaling, since geometrically scaling by 1 : 100 is common in both\nexperimental and computational model- and full scale investigations. As\nnecessary requirement to obtain similar damping (i.e. similarity of CRand\nsurface elevation), the damping length xdis scaled directly proportional to\nthe wavelength, according to Eq. (8).\nAt \frst, the simulations are run with no scaling of f1, so thatf1= 10 s\u00001in\nall cases. Figure 13 shows the free surface in the tank after the simulations. In\nthe vicinity of the inlet (0 <x=\u0015< 1) the e\u000bects of wave re\rections have not\nyet fully established, thus the di\u000berences between the curves are small. This\nshows that the wave-maker generated similar waves in the three simulations.\nAs the plot shows an arbitrarily selected time instant, the e\u000bects of wave\nre\rections cannot be judged from it, since the partial standing waves were\nnot captured at their maximum ampli\fcation. Therefore, for \u0015= 400 m\nno noticeable di\u000berence to the reference case \u0015= 4 m is seen outside the\ndamping zone (0 < x=\u0015 < 4), whereas obvious re\rections are visible for\n23\u0015= 0:04 m. However regarding the surface elevation within the damping\nzone (4<x=\u0015< 6), the three curves di\u000ber considerably, and thus the wave\ndamping also does not act in a similar fashion. A close look at the surface\nelevation in the damping zones shows that wave re\rections will mostly occur\nat domain boundary x=\u0015= 6 for\u0015= 0:04 m, in contrast to \u0015= 400 m, where\nthey will mostly occur close to the entrance to the damping zone at x=\u0015= 4.\nFigure 13: Surface elevation in the whole domain after \u001912:6 periods; for similar waves\nwith wavelengths \u0015= 0:04 m;4 m;400 m; no scaling of f1, thusf1= 10 s\u00001andf2= 0 for\nall cases\nSubsequently, based on the \fndings in Sect. 5, the simulations are rerun\nwithf1scaled with wave frequency !according to Eq. (9), based on reference\ncase\u0015ref= 4 m. It is apparent from Fig. 14, that the surface elevations are\nsimilar everywhere in the tank and that the proposed scaling law is correct.\n24Figure 14: Surface elevation in the whole domain after \u001912:6 periods; for similar waves\nwith wavelengths \u0015= 0:04 m;4 m;400 m; scaling off1according to Eq. (9) , thusf1=\n10 s\u00001;3:16 s\u00001;1 s\u00001andf2= 0\nTable 1 compares the re\rection coe\u000ecients CRfor both unscaled and\ncorrectly scaled f1. Whenf1is held constant, an up- or down-scaling of\nthe wave with factor 100 will lead to a signi\fcant increase in re\rections.\nTherefore without adjusting f1, it is not possible to obtain similar damping\nin model- and full scale.\nIf insteadf1is scaled according to Eqs. (8) and (9), CRstays nearly\nthe same; slight \ructuations of CRare due to the scheme for obtaining CR.\nTherefore, the damping quality can be reliably reproduced when the scaling\nlaw is applied.\nThis is even evident in the CRvalues for \fxed f1: For the scaled waves\nconsidered, f1is either roughly 10 times larger ( \u0015= 400 m) or smaller ( \u0015=\n0:04 m) than the optimum value; compared with the 10 times larger or smaller\nthan optimal f1values from Fig. 5, the CRvalues coincide.\n25Table 1: Re\rection coe\u000ecient CRfor unscaled and correctly scaled simulations\n\u0015CR(\fxedf1)CR(scaledf1)\n0:04 m 39:8% 1 :1%\n4 m 0:7% 0 :7%\n400 m 2:4% 0 :6%\nPractical Recommendation\nWhen using linear damping according to Eqs. (5) and (6), the following\ndamping setup can be recommended based on the results from Sects. 8 to\n15:\nf1= \t 1! ; (12)\nwith \t 1=\u0019, wave frequency !,f2= 0,xd= 2\u0015andn= 2.\n14. The Scaling Law for Quadratic Damping\nIn order to verify the assumed scaling law for quadratic damping from\nSect. 5, the simulation setup with the best damping performance ( f2=\n160:0 m\u00001) from Sect. 9 was scaled geometrically and kinematically so that\nthe generated waves are completely similar, for wavelengths \u0015= 0:04 m;4 m;400 m\nand thus corresponding heights of H= 0:0016 m;0:16 m;16 m. As in the pre-\nvious section, the damping length xdis scaled directly proportional to the\nwavelength, according to Eq. (8).\nAt \frst the simulations are run with no scaling of f2, so thatf2= 160 m\u00001\nin all cases. Subsequently, based on the \fndings in Sect. 5, the simulations\nare rerun with f2scaled with wave frequency !squared according to Eq.\n(10), based on reference case \u0015= 4 m. It is apparent from Fig. 15, that the\nsurface elevations are similar everywhere in the tank and that the proposed\nscaling law is correct.\n26Figure 15: Surface elevation in the whole domain after \u001912:6 periods; for similar waves\nwith wavelengths \u0015= 0:04 m;4 m;400 m; scaling off2according to Eq. (10) , thusf2=\n160 m\u00001;16 m\u00001;1:6 m\u00001andf1= 0\nTable 2 compares the re\rection coe\u000ecients CRfor both unscaled and\ncorrectly scaled f2. Whenf2is held constant, an up- or down-scaling of\nthe wave with factor 100 will lead to a signi\fcant increase in re\rections. If\ninsteadf2is scaled according to Eqs. (8) and (9), CRstays nearly the same;\nslight \ructuations of CRare due to the scheme for obtaining CR. This shows\nthat the damping quality can be reliably reproduced when the scaling law is\napplied.\nTable 2: Re\rection coe\u000ecient CRfor unscaled and correctly scaled simulations\n\u0015CR(\fxedf2)CR(scaledf2)\n0:04 m 65:9% 1 :1%\n4 m 0:6% 0 :6%\n400 m 17:7% 0 :5%\nThe present results show that if the damping parameters are not ad-\njusted when performing model- and full-scale simulations, then unsatisfac-\ntory damping can be expected.\n27Compared to linear wave damping, quadratic damping has a narrower\nrange of wave frequencies for which satisfactory damping is obtained. There-\nfore, since the damping performance is more sensitive to changes in wave\nfrequency for quadratic (or higher order) damping functions, and since with\noptimum set up the damping performance is the same as with linear damp-\ning, it is recommended to use linear damping functions. For this reasons,\nonly linear damping was examined in the following Sects. 15 and 16.\nPractical Recommendation\nWhen using quadratic damping according to Eqs. (5) and (6), the following\ndamping setup can be recommended based on the results from Sect. 9:\nf2= \t 2\u0015\u00001; (13)\nwith \t 2= 2\u0019\u0001102, wavelength \u0015,f1= 0,xd= 2\u0015andn= 2.\n15. Damping of Irregular Free-Surface Waves\nThe previous results indicate that, to simulate irregular waves, the damp-\ning setup should be based on the wave component with the longest wave-\nlength. If the damping setup is optimal for the longest wave component, then\non the one hand the damping coe\u000ecient for all shorter wave components will\nbe smaller then optimal, which has a negative e\u000bect on the damping perfor-\nmance for the corresponding wave component as seen in Sect. 8. On the\nother hand, the damping layer thickness relative to the wavelength will be\nlarger for these wave components, which has a positive e\u000bect on the damping\nperformance as shown in Sect. 11. From the previous results, the latter e\u000bect\nis expected be the prevailing in\ruence on the damping quality. Furthermore,\nthe shorter wave components are discretized with less cells per wavelength,\nso the discretization errors are stronger for these wave components, which\nprovides a faster wave energy dissipation for these components which is also\nbene\fcial in this respect.\nExemplarily, linear damping of three superposed wave components with\nsame steepnessHi\n\u0015i= 0:03 and wavelengths di\u000bering by 400% is demonstrated\nin the following. At the wave-maker, the following surface elevation is pre-\nscribed\n\u0011(t) =3X\ni=1Hi\n2cos(\u0000!i(t+ 1:19 s) +\u001ei); (14)\n28with wave height Hi, wave frequencies !i, phase shifts \u001eiand timet. The\nactual parameters are shown in Table 3.\nTable 3: Parameters of the wave components\ni\u0015i(m)Hi(m)!i(rad\ns)\u001ei(rad)\n1 4:0 0:12 3:926 0:0\n2 2:0 0:06 5:551\u00000:9176\n3 1:0 0:03 7:851 3:543\nVelocity and volume fraction corresponding to Eq. (14) are taken from\nlinear wave theory and are applied at the inlet boundary x= 0 as linear\nsuperposition. The domain length is Lx= 3\u00151. The grid dimensions in free\nsurface zone are \u0001 x= 0:0195 m and \u0001 z= 0:0024 m with a time step of\n\u0001t= 0:001 s.\nThe damping setup is based on the wave component with the longest\nwavelength according to Eq. (12): The damping layer thickness is xd= 2\u00151\nand the damping coe\u000ecients are f1= 20 s\u00001andf2= 0. With this setup, all\nwave components were damped satisfactorily without noticeable re\rections.\nAs can be seen from Fig. 16, the waves enter the damping layer at x= 4 m\nand after propagating a distance of 2 \u00151into the damping layer, the wave\nheight is reduced by two orders of magnitude. Thus linear damping seems\nwell suited for the damping of irregular waves. Although the approach from\nSect. 5 to determine CRcannot be applied in this case, the results from\nSect. 8 show that for all wave components re\rections will mostly occur at the\ndomain boundary, and not at the entrance to the damping layer. Thus in this\ncase, measuring the surface elevation in close vicinity to the corresponding\ndomain boundary is enough to evaluate the damping performance.\n29Figure 16: Free surface elevation in the solution domain after 20 s; top: whole domain;\nbottom: close up of the last part of the damping layer\nHowever, for realistic wave spectra the above approach is not feasible, as\nthese consist of wave components over a broad range of wavelengths. Thus\nselecting the component with the longest wavelength for the damping zone\nsetup would not be practical, since this would require a very large damping\nzone and increase the total amount of grid cells and computational e\u000bort\nsubstantially (possibly one or more orders of magnitude if the mesh size is\nnot changed).\nFor practical purposes, an e\u000ecient approach is outlined in the following.\nAccording to Lloyd (1989), most realistic sea spectra are narrow banded.\nThis means that most wave energy is concentrated in a narrow band of wave\nfrequencies around a peak frequency. Although larger wavelengths occur as\nwell, these carry a comparatively small amount of energy. Thus we propose\nto set up the wave damping based on the peak wave frequency. As seen in\nSect. 11 the range of frequencies that are damped can be modi\fed with the\n30damping zone thickness, so the broader-banded the spectrum is, the larger\nhasxdto be.\nExemplarily, this approach is investigated for the widely used JONSWAP\nspectrum, which was originally developed by Hasselmann et al. (1973). Ir-\nregular waves are prescribed at the inlet boundary, with parameters peak\nwave period Tp= 1:6 s, signi\fcant wave height Hs= 0:12 m, peak shape pa-\nrameter\r= 3:3, spectral width parameter \u001b= 0:07 for!\u0014!pand\u001b= 0:09\nfor!>! p. The wave spectrum was discretized into 100 components.\nThe choice of xd= 2\u0015peak, with wavelength \u0015peakcorresponding to the\npeak wave frequency, seems adequate for such cases. No visible disturbance\ne\u000bects were noted in the simulation. The wave heights over time are shown\nrecorded close to the wave-maker in Fig. 17 and recorded close to the bound-\nary to which the damping zone is attached in Fig. 18. This shows that the\naverage wave height is reduced by roughly two orders of magnitude.\nFigure 17: Surface elevation over time recorded directly before the wave-maker\n31Figure 18: Surface elevation over time recorded in close vicinity to the boundary, to which\nthe damping layer is attached\nFrom the surface elevation in the tank in Fig. 19 no undesired wave\nre\rections can be noticed. Although a more detailed study of the error of\nthis approximation regarding di\u000berent parameters of the spectrum was not\npossible in the scope of this study, the proposed approach seems to work\nreasonably well for practical purposes. Further research in this respect is\nrecommended to reduce uncertainties regarding the re\rections.\nFigure 19: Surface elevation in the whole domain at t\u001915:6 s\n16. Application of Scaling Law to Ship Resistance Prediction\nThis section compares results from model and full scale resistance compu-\ntations in 3D of the Kriso Container Ship (KCS) at Froude number 0 :26. The\n32simulations in this section are based on the simulations reported in detail in\nEnger et al. (2010). For detailed information on the setup and discretiza-\ntion, the reader referred to Enger et al. (2010). In the following, only a brief\noverview of the setup and di\u000berences to the original simulations are given for\nthe sake of brevity. The present model scale simulation di\u000bers only in the\ndamping setup (and slight modi\fcations of the used grid) from the \fne grid\nsimulation in Enger et al. (2010). The KCS is \fxed in its \roating position\nat zero speed. To simulate the hull being towed at speed U, this velocity\nis applied at the inlet domain boundaries and the hydrostatic pressure of\nthe undisturbed water surface is applied at the outlet boundary behind the\nship. The domain is initialized with a \rat water surface and \row velocity U\nfor all cells. As time accuracy is not in the focus here, \frst-order implicit\nEuler scheme is used for time integration. Apart from the use of the k-\u000f\nturbulence model by Launder and Spalding (1974), the computational setup\nis similar to the one used in the rest of this work. The computational grid\nconsists of roughly 3 million cells. For comparability, we compare only the\npressure components of the drag and vertical forces, which are obtained by\nintegrating the x-component for the drag and z-component for the vertical\ncomponent of the pressure forces over the ship hull. The simulation starts\nat 0 s and is stopped at tmax= 90 s simulation time. The Kelvin wake of the\nship can be decomposed into a transversal and a divergent wave component.\nThe wave damping setup is based on the ship-evoked transversal wave (wave-\nlength\u0015t\u00193:1 m), the phase velocity of which equals the service speed U.\nWave damping according to Eqs. (5) and (6) has been applied to inlet, side\nand outlet boundaries with parameters xd= 2:3\u0015t,f1= 22:5 s\u00001,f2= 0,\nandn= 2. This setup provided satisfactory convergence of drag and vertical\nforces in model scale. The wake pattern for the \fnished simulation is shown\nin Fig. 20 and the results are in agreement with the \fndings from Enger et\nal. (2010). The obtained resistance coe\u000ecient CT;sim= 3:533\u000110\u00003compares\nwell with the experimental data ( CT;exp= 3:557\u000110\u00003, 0:68% di\u000berence to\nCT;sim) by Kim et al. (2001) and to the simulation results by Enger et al.\n(2001) (CT;Enger = 3:561\u000110\u00003, 0:11% di\u000berence to CT;exp).\n33Figure 20: Wave pro\fle for model scale ship with f1= 22:5 s\u00001att= 90 s\nAdditionally, full scale simulations are performed with Froude similarity.\nThe scaled velocity and ship dimensions are shown in Table 4. The grid is\nsimilar to the one for the model scale simulation except scaled with factor\n31:6. Assuming similar damping can be obtained with the presented scaling\nlaws,xdwas scaled according to Eq. 8 by factor 31 :6 as well. Otherwise the\nsetup corresponds to the one from the model scale simulation.\n34Table 4: KCS parameters\nscale waterline length L(m) service speed U(m=s)\nmodel 7:357 2 :196\nfull 232:5 12 :347\nFigure 21 shows drag and vertical forces over time when both model and\nfull scale simulations are run with the same value for damping coe\u000ecient\nf1. In contrast to the model scale forces, the full scale forces oscillate in a\ncomplicated fashion. Therefore without a proper scaling of f1, no converged\nsolution can be obtained for the full scale case.\n35Figure 21: Drag (top image) and vertical (bottom image) pressure forces on ship over time\nfor model (red) and full scale (grey); no scaling of f1, thusf1= 22:5 s\u00001is the same in\nboth simulations\nFinally, the full scale simulation is rerun with f1scaled according to Eq. 9\nto obtain similar damping as in the model scale simulation with f1= 22:5 s\u00001.\nThe correctly scaled value for the full scale simulation is thus f1;full=f1;model\u0001\n!full=!model = 22:5 s\u00001\u00011=p\n31:6\u00194 s\u00001. The resulting drag and vertical\nforces in Fig. 22 show that indeed similar damping is obtained, since in\nboth cases the forces converge in a qualitatively similar fashion. Note that a\nperfect match of the curves in Fig. 22 is not expected, since Froude-similarity\nis given, but not Reynolds-similarity. Thus although the pressure components\nof the forces on the ship will converge to the same values (if scaled by L3), the\nway they converge (amplitude and frequency of the oscillation) may not be\n36exactly similar, since this depends on the solution of all equations. However,\nthe tendency of the convergence will be qualitatively similar as shown in the\nplots.\nFigure 22: Drag (top image) and vertical (bottom image) pressure forces on ship over time\nfor model and full scale; the damping setup for the full scale simulation is obtained by\nscaling the model scale setup according to Eqs. 8 and 9\n17. Discussion and Conclusion\nIn order to obtain reliable wave damping with damping layer approaches,\nthe damping coe\u000ecients must be adjusted according to the wave parameters,\nas shown in Sects. 8, 9, 13 and 14. It is described in Sect. 7 that the\nprocedure to quantify the damping quality is quite e\u000bortful, and thus it\n37is seldom carried out in practice. This underlines the importance of the\npresent \fndings for practical applications, since unless the damping quality\ncan be reliably set to ensure that the in\ruence of undesired wave re\rections\nare small enough to be neglected, a large uncertainty will remain in the\nsimulation results. The optimum values for damping coe\u000ecients f1andf2\ncan be assumed not to depend on computational grid, wave steepness and\nthicknessxdof the damping layer as shown in Sects. 10, 11 and 12. As shown\nin Sect. 11, the damping layer thickness has the strongest in\ruence on the\ndamping quality. If it increases, the range of waves that will be damped\nsatisfactorily broadens and the re\rection coe\u000ecient for the optimum setup\nshrinks; unfortunately, the computational e\u000bort increases at the same time as\nwell, thus optimizing the damping setup is important. In contrast, Sect. 12\nshows that the wave steepness has a smaller e\u000bect on the damping, with the\ntendency towards better damping for smaller wave steepness. For su\u000eciently\n\fne discretizations, Sect. 10 shows that the damping can be considered not\na\u000bected by the grid. A practical approach for e\u000ecient damping of irregular\nwaves has been presented in Sect. 15. The scaling laws in Sect. 5 and\nrecommendations given in Sects. 13 and 14 provide a reliable way to set\nup optimum wave damping for any regular wave. Moreover, similarity of\nthe wave damping can be guaranteed in model- and full-scale simulations as\nshown in Sects. 13, 14 and 16. The \fndings can easily be applied to any\nimplementation of wave damping which accords to Sect. 3.\nReferences\n[1] Cao, Y., Beck, R. F. and Schultz, W. W. 1993. An absorbing beach for\nnumerical simulations of nonlinear waves in a wave tank, Proc. 8th Intl.\nWorkshop Water Waves and Floating Bodies , 17-20.\n[2] Choi, J. and Yoon, S. B. 2009. Numerical simulations using momentum\nsource wave-maker applied to RANS equation model, Coastal Engineer-\ning, 56, (10), 1043-1060.\n[3] Cruz, J. 2008. Ocean wave energy, Springer Series in Green Energy and\nTechnology , UK, 147-159.\n[4] Enger, S., Peri\u0013 c, M. and Peri\u0013 c, R. 2010. Simulation of Flow Around KCS-\nHull, Gothenburg 2010: A Workshop on CFD in Ship Hydrodynamics,\nGothenburg.\n38[5] Fenton, J. D. 1985. A \ffth-order Stokes theory for steady waves, J.\nWaterway, Port, Coastal and Ocean Eng. , 111, (2), 216-234.\n[6] Ferrant, P., Gentaz, L., Monroy, C., Luquet, R., Ducrozet, G., Alessan-\ndrini, B., Jacquin, E., Drouet, A. 2008. Recent advances towards the\nviscous \row simulation of ships manoeuvring in waves, Proc. 23rd Int.\nWorkshop on Water Waves and Floating Bodies , Jeju, Korea.\n[7] Ferziger, J. and Peri\u0013 c, M. 2002. Computational Methods for Fluid Dy-\nnamics, Springer.\n[8] Grilli, S. T., Horrillo, J. 1997. Numerical generation and absorption of\nfully nonlinear periodic waves. J. Eng. Mech. , 123, (10), 1060-1069.\n[9] Guignard, S., Grilli, S. T., Marcer, R., Rey, V. 1999. Computation of\nshoaling and breaking waves in nearshore areas by the coupling of BEM\nand VOF methods, Proc. ISOPE1999 , Brest, France.\n[10] Ha, T., Lee, J.W. and Cho, Y.S. 2011. Internal wave-maker for Navier-\nStokes equations in a three-dimensional numerical model, J. Coastal\nResearch , SI 64, 511-515.\n[11] Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright,\nD. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruse-\nman, P., Meerburg, A., M uller, P., Olbers, D. J., Richter, K., Sell, W.\nand Walden, H. 1973. Measurements of wind-wave growth and swell de-\ncay during the Joint North Sea Wave Project (JONSWAP), Deutsche\nHydrographische Zeitschrift , (8), Reihe A.\n[12] Higuera, P., Lara, J. L., Losada, I. J. 2013. Realistic wave generation\nand active wave absorption for Navier-Stokes models Application to\nOpenFOAM r,Coastal Eng. , 71, 102-118.\n[13] Jha, D. K. 2005. Simple Harmonic Motion and Wave Theory, Discovery\nPublishing House.\n[14] Kim, J., O'Sullivan, J., Read, A. 2012. Ringing analysis of a vertical\ncylinder by Euler overlay method, Proc. OMAE2012 , Rio de Janeiro,\nBrazil.\n39[15] Kim, J., Tan, J. H. C., Magee, A., Wu, G., Paulson, S., Davies, B.\n2013. Analysis of ringing ringing response of a gravity based structure\nin extreme sea states, Proc. OMAE2013 , Nantes, France.\n[16] Kim, W. J., Van, S. H., Kim, D. H. 2001. Measurement of \rows around\nmodern commercial ship models, Experiments in Fluids , 31, Springer-\nVerlag, 567-578.\n[17] Kraskowski, M. 2010. Simulating hull dynamics in waves using a RANSE\ncode, Ship technology research , 57, 2, 120-127.\n[18] Lal, A., Elangovan, M. 2008. CFD simulation and validation of \rap type\nwave-maker, WASET , 46, 76-82.\n[19] Launder, B.E., and Spalding, D.B. 1974. The numerical computation of\nturbulent \rows, Comput. Meth. Appl. Mech. Eng. , 3, 269-289.\n[20] Lloyd, A. R. J. M. 1989. Seakeeping: Ship Behaviour in Rough Weather,\nEllis Horwood Limited.\n[21] Muzaferija, S. and Peri\u0013 c, M. 1999. Computation of free surface \rows us-\ning interface-tracking and interface-capturing methods, Nonlinear Water\nWave Interaction, Chap. 2, 59-100, WIT Press, Southampton.\n[22] Park, J. C., Kim, M. H. and Miyata, H. 1999. Fully non-linear free-\nsurface simulations by a 3D viscous numerical wave tank, International\nJournal for Numerical Methods in Fluids , 29, 685-703.\n[23] Peri\u0013 c, R. 2013. Internal generation of free surface waves and application\nto bodies in cross sea, MSc Thesis, Schriftenreihe Schi\u000bbau, Hamburg\nUniversity of Technology, Hamburg, Germany.\n[24] Peri\u0013 c, R., Abdel-Maksoud, M., 2015. Assessment of uncertainty due to\nwave re\rections in experiments via numerical \row simulations, Proc.\nISOPE2015 , Hawaii, USA.\n[25] Sch a\u000ber, H.A., Klopman, G. 2000. Review of multidirectional active\nwave absorption methods, Journal of Waterway, Port, Coastal, and\nOcean Engineering , 88-97.\n40[26] Ursell, F., Dean, R. G. and Yu, Y. S. 1960. Forced small-amplitude\nwater waves: a comparison of theory and experiment, Journal of Fluid\nMechanics , 7, (01), 33-52.\n[27] W ockner-Kluwe, K. 2013. Evaluation of the unsteady propeller perfor-\nmance behind ships in waves, PhD thesis at Hamburg University of\nTechnology, Schriftenreihe Schi\u000bbau , 667, Hamburg.\n41"    },    {        "title": "2010.14148v2.Legolas__a_modern_tool_for_magnetohydrodynamic_spectroscopy.pdf",        "content": "Draft version June 2, 2021\nTypeset using L ATEX default style in AASTeX63\nLegolas: a modern tool for magnetohydrodynamic spectroscopy\nNiels Claes,1Jordi De Jonghe,1and Rony Keppens1\n1Centre for mathematical Plasma-Astrophysics, Celestijnenlaan 200B, 3001 Leuven, KU Leuven, Belgium\nABSTRACT\nMagnetohydrodynamic (MHD) spectroscopy is central to many astrophysical disciplines, ranging\nfrom helio- to asteroseismology, over solar coronal (loop) seismology, to the study of waves and insta-\nbilities in jets, accretion disks, or solar/stellar atmospheres. MHD spectroscopy quanti\fes all linear\n(standing or travelling) wave modes, including overstable (i.e. growing) or damped modes, for a\ngiven con\fguration that achieves force and thermodynamic balance. Here, we present Legolasa), a\nnovel, open-source numerical code to calculate the full MHD spectrum of one-dimensional equilibria\nwith \row, that balance pressure gradients, Lorentz forces, centrifugal e\u000bects and gravity, enriched\nwith non-adiabatic aspects like radiative losses, thermal conduction and resistivity. The governing\nequations use Fourier representations in the ignorable coordinates, and the set of linearised equations\nare discretised using Finite Elements in the important height or radial variation, handling Cartesian\nand cylindrical geometries using the same implementation. A weak Galerkin formulation results in a\ngeneralised (non-Hermitian) matrix eigenvalue problem, and linear algebraic algorithms calculate all\neigenvalues and corresponding eigenvectors. We showcase a plethora of well-established results, rang-\ning from p- and g-modes in magnetised, strati\fed atmospheres, over modes relevant for coronal loop\nseismology, thermal instabilities and discrete overstable Alfv\u0013 en modes related to solar prominences, to\nstability studies for astrophysical jet \rows. We encounter (quasi-)Parker, (quasi-)interchange, current-\ndriven and Kelvin-Helmholtz instabilities, as well as non-ideal quasi-modes, resistive tearing modes, up\nto magneto-thermal instabilities. The use of high resolution sheds new light on previously calculated\nspectra, revealing interesting spectral regions that have yet to be investigated.\n1.INTRODUCTION\nThe study of stability for plasmas and \ruids alike has been a major topic of research over the last century. Under-\nstanding how and why a given medium reacts to a linear perturbation is of central importance to many astrophysical\nphenomena. In incompressible or compressible \ruids, governed by hydrodynamic equations, notable instabilities are\nthe Kelvin-Helmholtz instability (KHI), which arises due to a velocity shear at the interface of two \ruids, and the\nRayleigh-Taylor instability (RTI) where gravitational strati\fcation can lead to an unstable con\fguration of layered\n\ruids of di\u000berent density (Chandrasekhar 2013; Choudhuri 1998). In plasmas, governed by the magnetohydrodynamic\n(MHD) equations, the study of waves and instabilities becomes much richer due to the inclusion of magnetic \felds,\nwith a modern overview provided in Goedbloed et al. (2019). Magnetic \felds modify the two aforementioned insta-\nbilities in various ways, and the combination of \row, magnetic \felds and pressure gradients introduces many new\nmodes, e.g. the magnetorotational instability (Balbus & Hawley 1991) relevant for (weakly magnetised) accretion\ndisks or the Trans-Slow-Alfv\u0013 en Continuum modes in disks of arbitrary magnetisation (Goedbloed et al. 2004). In the\nhighly magnetised solar corona, observed coronal loop oscillations (periods and damping times) are routinely used\nto infer loop parameters like their \feld strength (Nakariakov & Ofman 2001). Embedded in the hot solar corona,\nwe \fnd stable and long-lived quiescent prominences, with internal dynamics due to KHI (Hillier & Polito 2018) and\nRTI (Hillier 2018). The formation of prominences is due to the thermal instability (TI), as demonstrated in direct\nobservations by e.g. Berger et al. (2012) or in simulations by Xia & Keppens (2016); Claes et al. (2020). Together with\ncategorising all instabilities, knowing the stable eigenoscillations such as p-modes or g-modes in strati\fed atmospheres\nor stellar interiors, is of prime importance to link theoretical understanding with observed periodic phenomena. In\nall of these cases, we need to compute eigenoscillations and corresponding eigenfunctions from the linearised set of\ngoverning equations. Linear MHD spectroscopy, which encompasses the entirety of helio- and asteroseismology, but\na)The Legolas code is available on GitHub: https://github.com/n-claes/legolasarXiv:2010.14148v2  [astro-ph.SR]  1 Jun 20212\nincorporates laboratory fusion plasma MHD spectroscopy (Goedbloed et al. 1993), MHD spectroscopy of accretion\ndisks (Keppens et al. 2002) and jets, as well as solar coronal seismology (Roberts 2019), is thus a powerful tool for\nstudying many astrophysical processes.\nSince the advent of more powerful computational resources, the main focus of computational astrophysical research\nhas gradually shifted towards solving the fully non-linear MHD equations, where many non-adiabatic/non-ideal e\u000bects\nare incorporated, depending on the application at hand. While this approach successfully reproduced many physical\nphenomena, especially for realistic solar setups (e.g. sunspots (Rempel 2012), \rares (Ruan et al. 2019), or promi-\nnences (Xia & Keppens 2016)), it usually fails to answer which speci\fc perturbation produces the complex evolution\nas witnessed. At the same time, theoretical insight showed that MHD spectral theory actually governs the stabil-\nity of \rowing, (self-)gravitating single \ruid evolutions of nonlinear, time-dependent plasmas, and this at any time\nduring their nonlinear evolution (Demaerel & Keppens 2016). Hence, in order to predict the reaction of a certain\nphysical state to perturbations, we should really quantify all its waves and instabilities using linear theory. This has\nbeen recognised fully in laboratory fusion plasmas, where MHD spectroscopy is very successful for identifying waves\nand stability aspects of a given toroidal Grad-Shafranov equilibrium. That this can meaningfully be done for states\nthat include important non-adiabatic e\u000bects, like optically thin radiative losses, is important for investigations into\nprominences and their intriguing \fne structure, as revealed by means of direct observations (Engvold 1998; Mackay\net al. 2010; Ballester 2006) or through numerical simulations (Xia & Keppens 2016; Xia et al. 2017; Claes et al. 2020).\nIn that context, early analytical work by Van der Linden & Goossens (1991a) based on linear MHD suggests the\nhypothesis that \fnite perpendicular thermal conduction induces \fne structure in unstable linear eigenmodes. Since\nthis pioneering work of Van der Linden & Goossens (1991a), not much research has been done regarding the full MHD\nspectrum when non-adiabatic e\u000bects are at play, for the simple reason that to date, there existed no numerical tool to\nsolve the full system of linearised MHD equations with all the physical e\u000bects included. This is why we developed the\nnew and open-source Legolas solver.\nLegolas builds on the heritage of early numerical codes, most notably LEDA (Kerner et al. 1985), which allowed\nstudies of the ideal or resistive MHD spectrum for laboratory plasmas, approximated by a di\u000buse cylindrical plasma\ncolumn (or \rux tube) and CASTOR (Kerner et al. 1998), which applied to resistive spectra of general tokamak\ncon\fgurations. The latter has follow-up codes such as FINESSE (Beli en et al. 2002) and PHOENIX (Blokland\net al. 2007b), extending it to stationary and axisymmetric truly 2D con\fgurations. LEDA was later extended in\nVan der Linden et al. (1992), where non-adiabatic e\u000bects like anisotropic thermal conduction and optically thin\nradiative losses were added to the equations, using a simple analytic function to treat radiative cooling e\u000bects. A\ndi\u000berent branch of LEDA, called LEDAFLOW (Nijboer et al. 1997), was developed to investigate the resistive MHD\nspectrum, augmented with gravitational and \row e\u000bects, but omitting those non-adiabatic terms. Since these codes\nwere developed decades ago and focus shifted away from linear MHD, their further development was stalled, although\nin laboratory fusion context, tools to compute multidimensional equilibria and their linear modes are very important\nfor diagnosing experiments. The original codes, like LEDA(FLOW), were not \rexible, in the sense that adding di\u000berent\nequilibria or accounting for additional terms in the equations would be a major undertaking, as parts were hard-coded\nto (limited) computational resources of that time. Furthermore, programming languages and numerical tools like\nLAPACK (Anderson et al. 1999) to solve eigenvalue problems have come a long way. This prompted us to develop\na brand new, modern MHD spectral code which we named Legolas , short for \\Large Eigensystem Generator for\nOne-dimensional pLASmas\". The Legolas code is able to handle both Cartesian and cylindrical geometries, and\nintroduces many new features, e.g. selecting between modern cooling curves that treat optically thin radiative cooling\ne\u000bects. Furthermore, every aspect of the code is modularised, making it ready to be extended with additional physics\nor modern algorithmic requirements (such as mesh re\fnement). The main goal of this paper is to present the new\ncode in terms of its implementation details and to validate it against a plethora of test cases that ensure a correct\ntreatment of the governing equations.\nThese tests include eigenmode quanti\fcations of ideal, static MHD con\fgurations under adiabatic conditions, where\nthe static (that is, no equilibrium \row) and adiabatic linear MHD equations make the problem self-adjoint. When\nperforming a standard Fourier analysis in the ignorable directions, the resulting eigenvalue problem is then Hermitian,\nmeaning that all eigenfrequencies will be either fully real (stable waves) or fully complex (pure damped or unstable\nmodes), hence they are found on the real or imaginary axis of the complex eigenfrequency plane, and the full MHD\nspectrum will be both left-right and up-down symmetric. However, in nature, physical conditions may be far from\nideal. The inclusion of non-ideal e\u000bects like resistivity or thermal conduction lifts the self-adjointness of the eigenvalue3\nproblem, allowing the eigenmodes to move away from the axes into the complex plane, and the up-down symmetry\ngets broken. As long as the equilibrium con\fguration is static, all (adiabatic or non-adiabatic) modes will still\nhave a complementary mode that lies mirrored around the imaginary axis, making the entire spectrum left-right\nsymmetric. This is related to the forward and backward propagating mode symmetry, or the equivalent statement\non the parity-time (PT) symmetry. However, for typical astrophysical plasmas, the conditions are far from static:\ntokamak plasmas, astrophysical jets, solar coronal loops, accretion discs. . . all have equilibrium \rows. The inclusion\nof a background \row breaks the left-right symmetry of the MHD spectrum, resulting in an even more complicated\nstructure. However, the study of the ideal, linear MHD spectrum of \rowing plasmas is still governed by a pair of\nself-adjoint operators (Goedbloed 2011; Goedbloed et al. 2019), and it leaves the up-down symmetry of the spectrum\nintact (where every overstable mode has an equivalent damped counterpart at the same frequency). The combination\nof \row and non-adiabatic e\u000bects, where both left-right and up-down eigenfrequency symmetries are broken, has never\nbeen explored in earnest. All of the above makes it clear that a numerical approach becomes essential, especially\nwhen the equilibrium is no longer homogeneous. Since in reality, virtually no astrophysical con\fguration is spatially\nhomogeneous, we need a \rexible numerical tool to explore the spectrum systematically.\nLegolas solves the linearised MHD equations including various non-adiabatic e\u000bects, resistivity and gravity, assuming\na one-dimensional (1D) equilibrium pro\fle with the possibility of background \row. A standard Fourier analysis for\nthe perturbations is combined with a Finite Element representation of the eigenfunctions in the important coordinate.\nWe transform the original system into an eigensystem for the complex eigenfrequencies !. We use a general formalism\nto include two kinds of geometries, a plane Cartesian strati\fed slab or a (possibly also strati\fed) cylinder, through\nthe inclusion of a scale factor originating from the divergence, gradient and curl operators. Legolas can handle the\nhydrodynamic limit (where all equilibrium magnetic \feld components are set to zero), enabling us to investigate\nstability of hydrodynamic static and stationary equilibria. The resulting system of equations is solved in weak form,\ntransforming the original system into a non-Hermitian complex eigenvalue problem, which is solved using the QR\nalgorithm. This results in a calculation of all eigenfrequencies and corresponding eigenfunctions of the system, such\nthat a detailed analysis of mode stability, but also the entire overview on all supported linear wave modes, becomes\npossible.\nSection 2 introduces the system of equations, along with the linearisation procedure and Fourier mode representation.\nThe treatment of the \fnal system using the \fnite element method is given in Appendix A where we explain the basic\nmathematical formalism behind the FEM, with a complete treatment of the \fnite element matrix assembly process\nand the boundary conditions. Legolas is tested against a large amount of spectra found in the literature in Section\n3, which is subdivided into multiple categories. First we treat ideal MHD with only a gravitational term included,\nwhich has as main advantage that solutions can be obtained analytically. The \frst case handles gravito-MHD waves\nin a Cartesian slab, after which we quantify quasi-Parker instabilities in strati\fed atmospheres. Results obtained with\nLegolas are compared with results found in various modern textbooks, including results for cylindrical geometries,\ndiscussing modes for ideal \rux tubes and tokamak current equilibria where we show liability to interchanges. For\nthe inclusion of \row into the equations, we consider a non-trivial case related to astrophysical jet stability, looking\nat Kelvin-Helmholtz and current-driven instabilities. We demonstrate that we can compute Suydam cluster modes,\noriginating from a surface where the wave vector is locally perpendicular to the magnetic \feld. We then transit to the\ninclusion of non-ideal e\u000bects such as resistivity, looking \frst at the resistive spectrum for a homogeneous plasma. We\nthen extend to recover the resistive quasi-mode in a non-homogeneous case. This is further complemented by adding\nan inhomogeneous medium and background \row, in such a way as to give rise to resistive tearing modes, all of which\nin a Cartesian geometry. Additionally, by allowing for resistivity and current variation, we show that we can also\ncompute resistive rippling modes. Lastly, we treat non-adiabatic e\u000bects, including optically thin radiative losses and\nthermal conduction into the equations, revisiting some pioneering results of discrete Alfv\u0013 en waves and magnetothermal\ninstabilities. Along the way, we \fnd interesting extensions to the original published works, due to the much higher\nresolutions employed here. Since Legolas is the \frst, modern linear MHD code to investigate realistic astrophysical\nplasmas, this opens the door to further in-depth studies of non-ideal equilibrium con\fgurations at high resolutions,\nranging from loops, jets or accretion disks.4\n2.PROBLEM DESCRIPTION AND MODEL EQUATIONS\nThe MHD equations with the inclusion of non-adiabatic e\u000bects, resistivity and gravity can be written in a (nor-\nmalised) Eulerian representation as\n@\u001a\n@t=\u0000r\u0001 (\u001av); (1)\n\u001a@v\n@t=\u0000rp\u0000\u001av\u0001rv+ (r\u0002B)\u0002B+\u001ag; (2)\n\u001a@T\n@t=\u0000\u001av\u0001rT\u0000(\r\u00001)pr\u0001v\u0000(\r\u00001)\u001aL+ (\r\u00001)r\u0001(\u0014\u0001rT) + (\r\u00001)\u0011(r\u0002B)2; (3)\n@B\n@t=r\u0002(v\u0002B)\u0000r\u0002 (\u0011r\u0002B); (4)\nwhere\u001ais the plasma density, vis the velocity, Tis the temperature, pis the pressure, Bdenotes the magnetic \feld\n(satisfyingr\u0001B= 0),\u0011is the resistivity and gthe gravitational acceleration. To close the system the (normalised)\nideal gas law p=\u001aTis used, while \rdenotes the ratio of speci\fc heats, taken to be 5 =3. The symbol Lin Eq. (3)\nrepresents the heat-loss function, de\fned as energy losses minus energy gains due to optically thin radiative cooling\ne\u000bects (Parker 1953) and is given by\nL=\u001a\u0003(T)\u0000H; (5)\nwhereHrepresents the total energy gains. In general, Hcan be anything (for example heating through dissipative\nAlfv\u0013 en waves (van der Holst et al. 2014)), but since there is still no well-de\fned parametrisation for coronal heating\nto date this term is assumed to be as convenient as possible, that is, constant in time but possibly varying in space\nto ensure thermodynamic balance. Note that Hshould always be consistent with a given equilibrium pro\fle as to\nexactly balance out the radiative losses (and possibly the thermal conduction e\u000bects and/or ohmic heating e\u000bects),\nreaching a thermal equilibrium state. This indirectly implies that the heating term is not necessarily independent of\nlocation, but dependent on the connection between the radiative losses and equilibrium temperature pro\fle, which,\nin general, are both spatially dependent. The \frst term in (5) denotes the radiative losses, dependent on the cooling\ncurve \u0003(T). These curves are tabulated sets resulting from detailed calculations, and can hence be interpolated to high\ntemperature resolutions. Legolas has multiple cooling curves implemented, most notably those by Colgan et al. (2008)\nand Schure et al. (2009), where the latter one is extended to the low-temperature limit using Dalgarno & McCray\n(1972). In addition we also implemented a piecewise power law as described by Rosner et al. (1978), which is an explicit\n(piecewise) function over the entire temperature domain. In the solar and astrophysical literature, these cooling curves\ncollect detailed knowledge on radiative processes, which are all assumed to be in the optically thin regime. It is\nworth noting that the inclusion of time-dependent background heating or \rows can in\ruence the spectrum in itself,\nas shown in for example Barbulescu et al. (2019); Hillier et al. (2019), but due to the time-dependence of those e\u000bects\nthe assumption of a stationary background state is no longer applicable. A possibility however may be varying the\nbackground heating or \row in subsequent runs, provided the background state is known at every snapshot.\nThermal conduction in magnetised plasmas is highly anisotropic, as the e\u000bect is a few orders of magnitude stronger\nalong the \feld lines than across. We hence use a tensor representation to model this anisotropy, denoting the thermal\nconductivity tensor \u0014by\n\u0014=\u0014keBeB+\u0014?(I\u0000eBeB); (6)\nwhereIdenotes the unit tensor and eB=B=Bis a unit vector along the magnetic \feld. The coe\u000ecients \u0014kand\u0014?\ndenote the conductivity coe\u000ecients parallel and perpendicular to the local direction of the magnetic \feld. For typical\nastrophysical applications the Spitzer conductivity is used, given by\n\u0014k\u00198\u000210\u00007T5=2erg cm\u00001s\u00001K\u00001; \u0014?\u00194\u000210\u000010n2B\u00002T\u00003\u0014k;\n\u0014k\u00198\u000210\u000012T5=2W m\u00001K\u00001; \u0014 ?\u00194\u000210\u000030n2B\u00002T\u00003\u0014k;(7)\nwherendenotes the number density, given by \u001a=nmpwithmpthe proton mass. The \frst and second row in Eq.\n(7) give the thermal conduction coe\u000ecients in cgs and mks units, respectively. For the solar corona we typically \fnd\nthat\u0014kis about 12 orders of magnitude larger than \u0014?(Priest 2014), so perpendicular thermal conduction is usually\nignored. Nevertheless, in Legolas both parallel and perpendicular thermal conduction are implemented. For the5\nFigure 1. Unit vectors and examples of B0andv0for the Cartesian (left) and cylindrical case (right).\nresistivity\u0011one can in principle take any pro\fle. We implemented the Spitzer resistivity,\n\u0011=4p\n2\u0019\n3Zione2pmeln(\u0015)\n(4\u0019\u000f0)2(kBT)3=2; (8)\nwhereZiondenotes the ionisation taken to be unity, eandmedenote the electron charge and mass, respectively,\nand\u000f0andkBare the electrical permittivity and Boltzmann constant. The Coulomb logarithm is given by ln( \u0015)\nand is approximately equal to 22 for solar coronal conditions (Goedbloed et al. 2019). It is important to emphasise\nthat, since we will further linearise the governing non-linear equations, we can adopt fully realistic values for all the\nnon-ideal coe\u000ecients, such as the resistivity or thermal conduction coe\u000ecients. This is in contrast to fully nonlinear\ncomputations, which are severely restrained in reaching magnetic Reynolds numbers beyond 104\u0000105.\n2.1. Equilibrium conditions\nWe consider a general coordinate system denoted by ( u1;u2;u3), corresponding to three orthogonal basis vectors.\nThe main advantage of this approach is that it allows us to include two di\u000berent geometries with only one basic\nformalism (and implementation). First we consider a standard plane slab geometry in Cartesian coordinates, that is,\na plasma which is con\fned in height, and considered to be bounded by two horizontal, perfectly conducting walls at a\n\fxed distance apart, extending outwards to in\fnity in the other two ignorable coordinates. This case also approximates\nthe limit of a fully in\fnite free space when the walls are moved o\u000b to in\fnity. In Cartesian geometry the coordinate\nsystem can be written as ( x;y;z ) and the vectors fu1;u2;u3gare the standard Cartesian triad f^ex;^ey;^ezgalong the\naxes. This makes it quite convenient to include for example gravitational e\u000bects which will induce an equilibrium\nstrati\fcation in the u1coordinate. The second geometry is that of an in\fnitely long plasma cylinder encased by\na solid wall at a certain distance away from the cylinder axis, for which the coordinate system can be de\fned as\n(r;\u0012;z ). At each point the vectors fu1;u2;u3gare de\fned as the triad of tangent vectors, f^er;^e\u0012;^ezg, with ^eralong\nthe radial direction, ^ezin the direction of the cylinder axis and ^e\u0012=^ez\u0002^ertangent to the cylinder. A detailed view\nof both geometries and their corresponding coordinate systems is shown in Figure 1. The basic operators present in\nEquations (1)-(4), that is, the divergence, gradient and curl, introduce a scale factor \"=rfor cylindrical geometries,\nwhich is reduced to \"= 1 for a Cartesian coordinate system. Hence exploiting this scale factor in the mathematical\nformalism allows for one implementation, where one can conveniently switch between both cases. We note that the\ncylindrical setup is also applicable to the so-called cylindrical accretion disk limit, as for example exploited to study\nMHD instabilities in disks by Blokland et al. (2007a).\nLinearisation involves splitting variables into two parts: a time-independent part, usually denoted with subscript 0,\nand a perturbed part, denoted by a subscript 1. Legolas handles one-dimensional equilibria which depend only on\nu1, or, more speci\fcally, time-independent equilibria of the form\n\u001a0=\u001a0(u1);\np0=p0(u1);\nT0=T0(u1);v0=v02(u1)u2+v03(u1)u3;\nB0=B02(u1)u2+B03(u1)u3: (9)6\nIn general we have g=\u0000g(u1)u1, where the Cartesian case is for a strati\fed atmosphere or layer, and the cylindrical\ncase can also allow for gravitational strati\fcation of an accretion disk situated for u1=r2[1;R]. In the case of\na cylinder where u1=r2[0;R], this gravitational term is absent. Using these equations in combination with Eqs.\n(1)-(4) yields two conditions for the time-independent parts, given by\n\u0012\np0+1\n2B2\n0\u00130\n+\u001a0g\u0000\"0\n\"\u0000\n\u001a0v2\n02\u0000B2\n02\u0001\n= 0; (10)\n1\n\"(\"\u0014?T0\n0)0\u0000\u001a0L0= 0; (11)\nwhere the \frst condition originates from the momentum equation (2) and should always be satis\fed as it expresses a\nforce-balanced state. The second condition originates from the non-adiabatic terms in the energy equation and should\nbe accounted for if these terms are included, the prime denotes the derivative with respect to u1. It should be noted\nthat resistive terms are not considered here, which is justi\fed by considering that the time scales on which the magnetic\n\felds decay due to resistivity is much, much larger than the time scales of resistive modes. This is a consequence of\nlarge magnetic Reynolds numbers Rmin typical astrophysical cases, yielding magnetic decay time scales of \u001c\u0018Rm\u001cA\n(with\u001cAthe typical Alfv\u0013 en time in ideal MHD) compared to much faster resistive mode time scales of \u001c\u0018R\u000b\nm(where\ntypically 0<\u000b< 1). We can hence consider the equilibrium itself to be independent of resistivity, which removes some\nstringent extra conditions on the energy and induction equations. Also note that the third term in Eq. (10) is only\nincluded for a cylinder, since \"0= 0 in a Cartesian geometry. This translates to the well-known fact that the centrifugal\nand tensional part of the Lorentz force are absent for a Cartesian slab. Furthermore a cylindrical equilibrium pro\fle\nshould satisfy on-axis regularity conditions, meaning that v02;v0\n03;B02andB0\n03all have to be equal to zero at r= 0.\nWhen considering an accretion disk in the cylindrical limit, the inner edge of the disk is at r= 1, so lengths are then\nexpressed in this inner disk radius and no regularity conditions apply then.\n2.2. Linearised equations\nNow we linearise Eqs. (1)-(4) around the equilibrium speci\fed in (9), where the unperturbed time-independent parts\nare denoted with a subscript 0 and the perturbed time-dependent parts are denoted with a subscript 1. It follows from\nthe adopted equilibrium con\fguration that r\u0001v0= 0 andr\u0001B0= 0, such that the divergence-free condition on the\nmagnetic \feld is ful\flled and that the equilibrium \row \feld is incompressible. However, the perturbed quantities can\nrepresent both incompressible or compressible eigenoscillations. The no-monopole condition should also be taken into\naccount for the perturbed magnetic \feld. Therefore, we adopt a vector potential to write B1=r\u0002A1such that\nr\u0001B1= 0 is automatically satis\fed. The system of linearised equations is thus given by\n@\u001a1\n@t=\u0000\u001a0r\u0001v1\u0000v0\u0001r\u001a1; (12)\n\u001a0@v1\n@t=\u0000\u001a0v0\u0001rv1\u0000\u001a0v1\u0001rv0\u0000\u001a1v0\u0001rv0\u0000r(\u001a1T0+\u001a0T1)\n+ (r\u0002B0)\u0002(r\u0002A1) + [r\u0002(r\u0002A1)]\u0002B0+\u001a1g;(13)\n\u001a0@T1\n@t=\u0000\u001a0v1\u0001rT0\u0000\u001a0v0\u0001rT1\u0000(\r\u00001)\u001a0T0r\u0001v1\u0000(\r\u00001)\u001a1L0\u0000(\r\u00001)\u001a0(LTT1+L\u001a\u001a1)\n+ (\r\u00001)r\u0001(\u00140\u0001rT1) + (\r\u00001)r\u0001(\u00141\u0001rT0)\n+ 2(\r\u00001)\u0011(r\u0002B0)\u0001[r\u0002(r\u0002A1)] + (\r\u00001)d\u0011\ndTT1(r\u0002B0)2;(14)\n@A1\n@t=v1\u0002B0+v0\u0002(r\u0002A1)\u0000\u0011r\u0002(r\u0002A1)\u0000d\u0011\ndTT1(r\u0002B0); (15)\nwherep1is replaced by \u001a1T0+\u001a0T1resulting from the linearised ideal gas law. The perturbation \u00141of the thermal\nconduction tensor is obtained by linearising the expressions (6)-(7), while the derivatives of the heat-loss function with\nrespect to density and temperature are given by\nL\u001a=@L\n@\u001a\f\f\f\f\nT= \u0003(T0); LT=@L\n@T\f\f\f\f\n\u001a=\u001a0d\u0003(T0)\ndT; (16)\nwhich should be evaluated using the equilibrium quantities. The terms containing @\u0011=@T follow from a linearisation of\nthe resistivity parameter, which can be written in terms of the variable T1using the temperature dependence originating7\nfrom the assumed Spitzer resistivity in Eq. (8), that is, \u00111=T1@\u0011=@T . In addition, Legolas allows for an anomalous\nresistivity prescription in which we typically have \u0011(u1;j(u1;t)), that is, a resistivity pro\fle that is spatio-temporal\nin general, but for a \fxed time, depends on position and current pro\fle. This in turn implies that a total derivative\nshould be used for the resistivity, given by \u00110=@\u0011=@u 1+T0\n0@\u0011=@T . In most use cases a resistivity pro\fle \u0011(T(u1))\nis su\u000ecient, which is only temperature dependent (and hence indirectly spatially varying as well for inhomogeneous\ntemperature pro\fles). Also note that these linear equations (as also the nonlinear set above) assumed an external\ngravitational \feld, so we did not need to linearise the gravity term g(this is the so-called Cowling approximation). In\nthe future, we can extend the set of equations with the Poisson equation and also allow for self-gravity driven Jeans\ninstabilities.\nNext, we perform a Fourier analysis with an exponential time dependence, imposing standard Fourier modes on the\nu2andu3coordinates of a perturbed quantity f1, given by\nf1=bf1(u1) exp [i(k2u2+k3u3\u0000!t)]: (17)\nIn this form the wave numbers k2andk3correspond to kyandkzin Cartesian geometry, and to mandkin a cylindrical\ngeometry, respectively. Note that mis quanti\fed to integer values, since the \u0012-direction is periodic. Additionally, we\napply the following transformation to the perturbed quantities\n\"b\u001a1=e\u001a1; i\"bv1=ev1;bv2=ev2; \"bv3=ev3;\n\"bT1=eT1; ibA1=ea1; \"bA2=ea2;bA3=ea3;(18)\nThis particular transformation simpli\fes the resulting set of equations and has as additional e\u000bect that all terms are\nreal except for the non-adiabatic and resistive contributions, such that we are only dealing with imaginary terms\nwhen these physical e\u000bects are included. This is in analogy to the fact that the purely adiabatic case is governed by\nself-adjoint operators: one for the case without \row and two for the case with \row included (Goedbloed 2018a,b). The\n\fnal set of Fourier-analysed linearised equations is given below, where the tilde notation in Eqs. (18) is dropped for\nthe sake of simplicity. From now on, tildes will no longer be written explicitly since there is no confusion possible.\n!1\n\"\u001a1=\u00001\n\"\u001a0\n0v1\u00001\n\"\u001a0(v0\n1\u0000k2v2\u0000k3v3) +1\n\"\u00121\n\"k2v02+k3v03\u0013\n\u001a1; (19)\n!\u001a01\n\"v1=\u0012\u001a1T0+\u001a0T1\n\"\u00130\n+1\n\"B0\n03(a0\n2\u0000k2a1)\u0000(\"B02)0\n\"(a0\n3\u0000k3a1)\n\u0000B03(\nk2\n3\n\"a2\u0000k2k3\n\"a3\u0000\u00141\n\"(a0\n2\u0000k2a1)\u00150)\n+1\n\"\u0012k2v02\n\"+k3v03\u0013\n\u001a0v1\n+B02\u001ak2\n2\n\"2a3\u0000k2k3\n\"2a2\u00001\n\"h\n\"(a0\n3\u0000k3a1)i0\u001b\n\u00002\"0\n\"\u001a0v02v2\u0000\"0\n\"2v2\n02\u001a1+1\n\"g\u001a1;(20)\n!\u001a0v2=1\n\"2(\u001a1T0+\u001a0T1)k2+\u001a0\u0012k2v02\n\"+k3v03\u0013\nv2+1\n\"2(\"B02)0(k3a2\u0000k2a3)\n+B03\u0014\n\u0000\u0012\nk2\n3+k2\n2\n\"2\u0013\na1+k2\n\"2a0\n2+k3a0\n3\u0015\n\u00001\n\"2\u001a0(\"v02)0v1;(21)\n!\u001a01\n\"v3=1\n\"(\u001a1T0+\u001a0T1)k3+1\n\"\u001a0\u0012k2v02\n\"+k3v03\u0013\nv3+1\n\"B0\n03(k3a2\u0000k2a3)\n\u0000B02\u0014\n\u0000\u0012\nk2\n3+k2\n2\n\"2\u0013\na1+k2\n\"2a0\n2+k3a0\n3\u0015\n\u00001\n\"\u001a0v0\n03v1;(22)8\n!\u001a01\n\"T1=\u00001\n\"\u001a0T0\n0v1+1\n\"\u001a0\u0012k2v02\n\"+k3v03\u0013\nT1\u0000(\r\u00001)1\n\"\u001a0T0(v0\n1\u0000k2v2\u0000k3v3)\n\u0000i(\r\u00001)\u0000\n\u0014k;0\u0000\u0014?;0\u0001\n\"1\nB2\u0012k2B02\n\"+k3B03\u00132\nT1+i(\r\u00001)1\n\"\"\n\"\u0014?;0\u0012T1\n\"\u00130#0\n\u0000i(\r\u00001)\u0014?;0\n\"\u0012k2\n2\n\"2+k2\n3\u0013\nT1+i(\r\u00001)1\n\"(\"\u0014?;1T0\n0)0\n+i(\r\u00001)\u0000\n\u0014k;0\u0000\u0014?;0\u0001\n\"1\nB2T0\n0\u0014\u0012k2k3\n\"B02+k2\n3B03\u0013\na2\u0000\u0012k2\n2\n\"B02+k2k3B03\u0013\na3\u0015\n\u0000i(\r\u00001)L01\n\"\u001a1\u0000i(\r\u00001)1\n\"\u001a0(LTT1+L\u001a\u001a1)\n+ 2i(\r\u00001)\u0011(\nB0\n03\u0014\u00101\n\"(a0\n2\u0000k2a1)\u00110\n+k2k3\n\"a3\u0000k2\n3\n\"a2\u0015\n\u0000(\"B02)0\n\"2\u0014\u0010\n\"(a0\n3\u0000k3a1)\u00110\n+k2k3\n\"a2\u0000k2\n2\n\"a3\u0015)\n+i(\r\u00001)d\u0011\ndT1\n\"\"\n\u0000\nB0\n02\u00012+\u0000\nB0\n03\u00012+ 2\"0\n\"B02B0\n02+\u0012\"0\n\"B02\u00132#\nT1;(23)\n!a1=\u0000B03v2+1\n\"B02v3\u00001\n\"v02a0\n2\u0000v03a0\n3+\u0012k2v02\n\"+k3v03\u0013\na1+i\u0011\u0014\n\u0000\u0012k2\n2\n\"2+k2\n3\u0013\na1+k2\n\"2a0\n2+k3a0\n3\u0015\n; (24)\n!1\n\"a2=\u00001\n\"B03v1+k3v03\n\"a2\u0000k2v03\n\"a3+i\u0011\"\u00121\n\"(a0\n2\u0000k2a1)\u00130\n+k2k3\n\"a3\u0000k2\n3\n\"a2#\n+id\u0011\ndT1\n\"B0\n03T1; (25)\n!a3=1\n\"B02v1\u0000k3v02\n\"a2+k2v02\n\"a3+i1\n\"\u0011\u0014\u0010\n\"(a0\n3\u0000k3a1)\u00110\n+k2k3\n\"a2\u0000k2\n2\n\"a3\u0015\n\u0000id\u0011\ndT1\n\"2(\"B02)0T1: (26)\nThe perturbed thermal conductivity tensor \u0014?;1in Eq. (23) written in terms of the perturbed variables is given by\n\"\u0014?;1=@\u0014?;0\n@TT1+@\u0014?;0\n@\u001a\u001a1\u00002\"B02\u0000\na0\n3\u0000k3a1\u0001@\u0014?;0\n@(B2)+ 2B03\u0000\na0\n2\u0000k2a1\u0001@\u0014?;0\n@(B2): (27)\nAn interesting side note is that \u0014k;1does not appear in the equations, which is due to the fact that this term is\naccompanied by a B0\u0001rT0contribution, and this is zero due to the equilibrium pro\fle in Eq. (9) (Van der Linden &\nGoossens 1991a,b). We now have a system of eight ordinary di\u000berential equations in u1for the perturbed quantities\n\u001a1;v1;v2;v3;T1;a1;a2anda3.\n2.3. Boundary conditions\nThe above system of di\u000berential equations (19)-(26) has to be complemented by a set of boundary conditions on\nboth sides of the domain. For a Cartesian geometry we look at a domain enclosed by two conducting walls. Clearly,\nthe velocity component perpendicular to the walls has to be zero since there can not be any propagation into a solid\nboundary. Mathematically, this translates into v\u0001^en= 0, where ^enrepresents the normal vector to the wall. Following\nthe same reasoning we also require that B\u0001^en= 0, so in terms of a vector potential this implies ^en\u0002A= 0. Hence,\napplying this to the set of linearised equations this means that for the Cartesian case v1,a2anda3all have to be\nzero on the boundaries. Furthermore, since we are dealing with a perfectly conducting wall one has to take care when\nthermal conduction is included. In that case the rigid wall directly in\ruences the temperature, since it acts as an\nenergy reservoir essentially eliminating the temperature perturbation. Hence, if and only if perpendicular thermal\nconduction is taken into account we have to supplement the boundary conditions by the additional condition T1= 0\nat the boundary. In theory there is a second possibility, which is treating the wall as a perfect insulator instead of a\nperfect conductor. In that case there is no heat \rux, which translates to the boundary condition T0\n1= 0 instead of\nT1= 0. For now we only consider the latter condition, that is, the one corresponding to a perfectly conducting wall.9\nIn a cylindrical geometry we have the exact same boundary conditions as for the Cartesian case at the outer wall\nr=R, or at the outer edge of the accretion disk at R. The same is true at the inner disk edge, but for a \rux tube\nextending to r= 0 we have to take the regularity conditions into account when treating the cylinder axis r= 0, which\ncomes down to the fact that rvrshould go to zero when approaching r= 0. Looking back at the transformations (18)\nwe applied, it follows that this condition is equivalent to v1= 0. Analogously, the same holds true for a2anda3such\nthat these conditions are identical to the ones we applied for the Cartesian case, which is convenient implementation-\nwise. We again have to consider an additional condition if perpendicular thermal conduction is taken into account,\nsince thenrbT1= 0 should also hold on the cylinder axis, which, similarly as for v1, translates into T1= 0 atr= 0.\nIn the case of con\fnement by a perfectly conducting wall we thus have straightforward boundary conditions, that\nis,v1= 0;a2= 0;a3= 0 andT1= 0 for both the Cartesian and cylindrical geometries on both sides. This latter\nboundary condition should only be taken into account if and only if perpendicular thermal conduction is included.\n2.4. Solving the equations\nThe system of equations (19)-(26) is solved through usage of a Finite Element discretisation. Applying a weak\nGalerkin formalism turns this system of equations into a generalised matrix-eigenvalue problem. A detailed explanation\non how this is done can be found in Appendix A, where we describe the structure of the \fnite element approach and\nthe matrix assembly process, along with a detailed treatment of how the boundary conditions are handled.\n3.RESULTS\nAs is common practice when developing a new numerical code we tested Legolas against various results previously\nobtained in the literature. We divided this section into four subsections, each of which handles di\u000berent physical\ne\u000bects. To begin with, we discuss results for adiabatic equilibria where only gravity is included in Subsection 3.1. In\nthis case we can compare numerical spectra obtained through Legolas with analytical solutions acquired by solving\ndispersion relations, here we focus on strati\fed atmospheres containing p- andg-modes. We then move on to cylindrical\ngeometries in 3.2 where we \frst look at adiabatic \rux tubes, followed by the inclusion of \row e\u000bects by considering\nequilibria with Kelvin-Helmholtz instabilities and Suydam cluster modes. Next the focus shifts to non-adiabatic e\u000bects\nin 3.3 by looking at a resistive MHD computation for a case without gravity, where a quasi-mode is known analytically.\nResistive tearing modes are also discussed, combining the e\u000bects of \row and resistivity. The \fnal subsection 3.4 treats\nthe inclusion of thermal conduction and optically thin radiative cooling e\u000bects, where we look at non-adiabatic discrete\nAlfv\u0013 en waves and magnetothermal modes.\n3.1. Cartesian cases: waves in strati\fed atmospheres\nFirst of all we discuss multiple theoretical results for adiabatic equilibria in a Cartesian geometry, where only gravity\nis included. We consider p- andg-modes in strati\fed layers, and pay special attention to speci\fc unstable branches.\n3.1.1. Gravito-MHD waves\nThe \frst test case covers gravito-MHD waves as discussed in Goedbloed et al. (2019, \fg. 7.9), which handles an\nexponentially strati\fed atmosphere with constant sound and Alfv\u0013 en speeds. This magnetised atmosphere contains\nthe generalisation of the p- andg-modes of an unmagnetised layer, and the constancy of the sound and Alfv\u0013 en speed\nrenders it analytically tractable, since the slow and Alfv\u0013 en continua collapse to points. The geometry is Cartesian,\nwithx2[0;1] and an equilibrium con\fguration given by\n\u001a0=\u001ace\u0000\u000bx; p 0=pce\u0000\u000bx;B0=Bce\u00001\n2\u000bx^ez; \u000b =\u001acg\npc+1\n2Bc; (28)\nwherepcandBcare taken to be 0.5 and 1, respectively, as to yield a plasma beta equal to unity. The parameter\n\u000bis taken to be 20, which, together with g= 20, is used to constrain the value for the constant \u001ac. These four\nequations completely determine the equilibrium con\fguration, since the temperature is T0=p0=\u001a0, following the\nideal gas law. The spectrum discussed in Goedbloed et al. (2019) is actually the solution to the analytic dispersion\nrelation for gravito-MHD waves, which shows the squared eigenvalue as a function of wave number for a \fxed angle\n\u0012=\u0019=4 between the wave vector k0and the magnetic \feld B0. However, the spectrum as calculated by Legolas\ncorresponds to one single equilibrium con\fguration, meaning one value for kyandkz. In order to reproduce \fgure\n7.9 from Goedbloed et al. (2019) and compare the results, we performed 100 di\u000berent runs where the equilibrium10\nFigure 2. Spectrum of gravito-MHD modes, obtained through 100 Legolas runs of 351 gridpoints each. The fast (top), Alfv\u0013 en\n(middle) and slow (bottom) branches of the MHD spectrum are clearly visible. The inset shows unstable slow modes at low\nfrequencies.\nparameters in Eq. (28) remained unchanged, but kyandkztook on 100 di\u000berent values between 0 andp\n250 as to\nyield a wave number range for k2\n0between 0 and 500. Since the magnetic \feld is purely aligned with the z-axis we can\nwritekk=kz=k0cos(\u0012) andk?=ky=k0sin(\u0012). All runs were performed using 351 gridpoints, yielding a matrix\nsize of 5616\u00025616.\nOur results are shown in Figure 2, where every vertical collection of points at the same k2\n0value represents one single\nLegolas run. Since we are in an MHD regime with \f= 1, the three MHD subspectra can be clearly distinguished,\nshowing the fast p-modes (top-left branches), Alfv\u0013 en g-modes (middle branches) and slow g-modes (bottom branches).\nThe inset shows a zoom-in near the marginal frequency of the spectrum, showing unstable ( !2<0) slow MHD modes.\nThese long-wavelength unstable modes are related to the Parker instabilities, due to magnetic buoyancy, as we will\nshow in Section 3.1.2. Note that since this case is adiabatic and fully self-adjoint, every individual MHD spectrum\nis left-right and up-down symmetric in the complex eigenfrequency plane, but this aspect is hidden from the !2\u0000k2\n0\nview shown here.\n3.1.2. Quasi-Parker instabilities\nNext we discuss a modi\fed case of the gravito-MHD waves, namely a spectrum showing quasi-Parker instabilities as\ndone in Goedbloed et al. (2019, \fg. 12.2). The di\u000berence with the previous case is that a fully analytic description is\nno longer possible, since the introduction of magnetic shear leads to continuous ranges in the MHD spectrum. Instead\nof showing the spectrum for one single value for \u0012, we now vary the direction of the wave vector k0between 0 and \u0019.\nThe equilibrium con\fguration is similar to the one in Section 3.1.1, given in Cartesian geometry by\n\u001a0=\u001ace\u0000\u000bx; p 0=pce\u0000\u000bx; \u000b =\u001acg\npc+1\n2Bc;\nB02=Bce\u00001\n2\u000bxsin(\u0015x); B 03=Bce\u00001\n2\u000bxcos(\u0015x);(29)\nwhere magnetic shear was introduced through the parameter \u0015. The quantities \u000bandBcare assigned the same values\nas in Eq. (28), except that g= 0:5 andpc= 0:25 which yields a plasma beta \f= 0:5. The wave vectors are given by\nky=\u0019sin(\u0012) andkz=\u0019cos(\u0012), such that k2\n0\u001910. The angle \u0012was varied between 0 and \u0019for a total of 100 runs at\n351 gridpoints each, shown in Figure 3.11\nFigure 3. Spectrum showing Parker and quasi-Parker modes without (left) and with (right) magnetic shear. The slow and\nAlfv\u0013 en continua are shown in red and cyan, respectively, where the insets zoom into the region of quasi-interchange modes.\nThe bottom row of panels show the eigenfrequency view for the single case \u0012= 0:3\u0019. The continua are again annotated on the\n\fgures, visualising the collapsed single point values (left) as well as the genuine continuum ranges (right). The grey dashed line\nin the top two panels denotes != 0.\nThe left panels handle the case without magnetic shear, that is, \u0015= 0, which basically reduces to the one from\nthe previous subsection. In this case the slow and Alfv\u0013 en continua collapse into single point values, denoted in red\nand cyan, respectively. The right panels show the same con\fguration where \u0015= 0:3 was taken, introducing magnetic\nshear, which introduces genuine continua seen as bands. These continua a\u000bect the overall stability, and organise the\nentire MHD spectrum: all discrete modes are fully aware of the essential spectrum formed by these (slow and Alfv\u0013 en)\ncontinua and the (fast) accumulation points at in\fnite frequency. All features of the original \fgure in Goedbloed et al.\n(2019) are reproduced. The inset zooms into the region where both continua overlap, showing quasi-interchange and\ninterchange instabilities. Once more, each run shown here collectively in Figure 3 actually has a spectrum that is\nleft-right and up-down symmetric in the eigenfrequency plane. This is depicted on the bottom two panels, which show\nthe eigenfrequency view for one single case ( \u0012= 0:3\u0019). The continuum ranges separate nicely: the collapsed single\npoint values are denoted by cyan (Alfv\u0013 en) and red (slow) points on the left panel, the genuine continua are shown with\ncyan and red bands on the right panel. The instabilities themselves are situated on the (positive) imaginary axis, due\nto the self-adjointness of the eigenvalue problem mentioned earlier.\nAs explained in Goedbloed et al. (2019), we see from this eigenmode computation that the Parker instability, which is\nthere for k0parallel toB0, becomes a quasi-Parker instability away from perfect alignment, and connects smoothly to\nwell-known quasi-interchange instabilities that occur here (marginally) away from perpendicular orientation. Quantify-\ning how the equilibrium parameters in\ruence the growth rates of these unstable branches can only be done numerically,\ne.g. with Legolas .12\n3.2. Adiabatic, cylindrical cases\nNext we move on to cylindrical con\fgurations, which provide tests for the scale factor \"in the equations. Analytical\nresults from the literature are again well reproduced. Furthermore we look at di\u000berent spectra previously obtained by\nthe LEDA code, discussed in various papers, and compare those with the new spectra from Legolas .\n3.2.1. Magnetic \rux tubes\nThe \frst case that we describe in this subsection is a magnetic \rux tube embedded in a uniform magnetic envi-\nronment, discussed in Roberts (2019). The equilibrium con\fguration is simple, in the sense that we have a uniform\nmagnetic \feld aligned with the z-axis both inside and outside of the \rux tube, with a similar structure for the other\nequilibrium parameters:\nB0(r);\u001a0(r);p0(r);T0(r) =8\n<\n:B0;\u001a0;p0;T0; r<a\nBe;\u001ae;pe;Te; r>a(30)\nwhere the subscripts 0 and erefer to values inside the tube and for the environment, respectively. The outer radius of\nthe tube is denoted by aand hence represents a discontinuous interface between the tube itself and the environment.\nSince total pressure balance should be preserved across the boundary, which is something that follows from Eq. (10),\nthis yields a relation between pressures and magnetic \feld components inside and outside of the tube, which in turn\nimplies a connection between the plasma densities, sound speeds and Alfv\u0013 en speeds across the boundary:\np0+1\n2B2\n0=pe+1\n2B2\ne;\u001ae\n\u001a0=c2\ns+1\n2\rc2\nA\nc2se+1\n2\rc2\nAe; (31)\nwherec2\ns=\rp0=\u001a0andc2\nA=B2\n0=\u001a0denote the sound speed and Alfv\u0013 en speed, respectively, in which the values outside\nof the \rux tube are used if there is a subscript epresent.\nIt should be noted that this extremely simple equilibrium con\fguration is the standard case used in many solar\ncoronal loop seismology e\u000borts. Since it simply has two uniform media (one inside the tube and one in its exterior),\nit has no continuous spectra (they reduce to point values), but the interface makes it possible to again have surface\nmodes that would be a\u000bected by true radial variation. Also note that these \rux tubes have only stable waves, but\nwe can distinguish between body and surface waves, depending on the variation of the eigenfunctions within the \rux\ntube. In the exterior of the \rux tube all eigenfunctions are exponentially varying.\nWe should also clarify here that the original dispersion relation as given in Roberts (2019) assumes a \rux tube\nembedded in an environment extending towards in\fnity, while Legolas on the other hand assumes a \fxed wall\nboundary at the outer edge of the domain. Hence, we assume here that the domain is situated in r2[0;10] with the\ninner \rux tube wall at r= 1, in order to minimise the outer wall in\ruence. However, this introduces an additional\ncomputational challenge, in the sense that we are (mainly) interested in the behaviour of the inner modes, since we\nknow that the outer modes all have exponentially varying eigenfunctions which decay to in\fnity (or towards our far-\naway outer wall). Hence, in order to resolve those inner waves huge resolutions are needed due to the 1 : 10 ratio. In\norder to circumvent this issue we used a simple prescription for mesh re\fnement, that is, a 60 \u000030\u000010 division of\nthe initial nodes. This means that 60% of the gridpoints are used for the inner tube region r2[0;0:95], 30% of the\ngridpoints are located near the transition region r2[0:95;1:05], and the remaining 10% are used for the environment\nr2[1:05;10].\na) Photospheric \rux tube. First we look at a \rux tube under photospheric conditions, that is, an equilibrium for\nwhichcAe<cs<cse<cA. More speci\fcally, we take cAe=cs=2,cse= 3cs=2 andcA= 2csfollowing Roberts (2019,\n\fg. 6.5). The relations between the inner and outer regions of the \rux tube follow straightforward from Eq. (31), and\nhence we only have two degrees of freedom, namely \u001a0andp0which are both taken to be unity, with \r= 5=3. This\nresults in\u001a0\u00190:567\u001ae,ct\u00190:89cs,ck\u00190:63cA\u00191:27cs. Here we also introduced the tube and kink speeds, given\nby\nc2\nt=c2\nsc2\nA\nc2s+c2\nA; c2\nk=\u001a0c2\nA+\u001aec2\nAe\n\u001a0+\u001ae; (32)\nusing the same notation as in Eqs. (31). As described before we take r2[0;10] and place the \rux tube boundary at\na= 1. Next we perform 40 runs at 300 gridpoints each for four azimuthal wave numbers m= 0 tom= 3. For the\nwave number k3=kzwe take 40 values in such a way that the dimensionless wave number kzahas values in [0 ;6:2].13\nFigure 4. Panela: spectrum showing the dispersion relation for a \rux tube under photospheric conditions. The dimensionless\nphase speed !=k zcsis displayed as a function of the dimensionless wave number kza, for four values of the azimuthal wave\nnumberm= 0 (blue dots), m= 1 (red crosses), m= 2 (green squares) and m= 3 (yellow triangles). Both sound speeds, the\nkink speed and the tube speed are denoted using dashed grey horizontal lines, all normalised to cs. Panelb: zoom-in of panel\nabetween the tube speed ctand internal sound speed cs. Panelscandd: eigenfunctions of the modes annotated with circles\n(c) and squares ( d) on the top-left panel a. Panele: \frst three eigenfunctions of the m= 0 andm= 1 body wave sequence\non the bottom-left panel b. Panelf: \frst three eigenfunctions of the m= 2 andm= 3 body wave sequence. For both panels\neandfthen= 1,n= 2 andn= 3 modes are shown in solid, dashed and dotted lines, respectively, only the \frst mode is\nannotated on the bottom-left panel. Every eigenfunction on panels cthroughfis normalised to its maximum absolute value in\nthat particular grid interval.\nThe spectrum showing the dispersion relation, where the dimensionless phase speed !=kzcsis plotted as a function of\nkza, is depicted in Figure 4. All speeds indicated on the \fgure are normalised to the internal sound speed cs. Panela\nclearly shows the fast surface waves, including the sausage ( m= 0), kink ( m= 1) and \frst two \ruting modes ( m= 2\nandm= 3). The eigenfunctions in the top-right panel ccorrespond to the modes annotated with a transparent circle\non panela, with colours indicating the mode number m. The left and right side of panels cthroughfshow the\neigenfunctions for the inner and outer regions of the \rux tube, respectively. All eigenfunctions are normalised to their\nmaximum value, and all eigenvalues having a normalised phase speed larger than 1.5 are not shown. It should be\nnoted that the \frst three runs, that is, the \frst three dots according to the kzaaxis, were done using 1001 gridpoints.\nThe reason for this is that when we divide the eigenvalues !bykzsmall errors are increased selectively for small kz,\nexplaining why those modes seem slightly scattered, and hence why we have to employ such high resolutions in order\nto minimise said error.\nThe panel bof Figure 4 zooms in between the tube speed ( ct) and internal sound speed ( cs), showing a clear\nrepresentation of the various body waves that accumulate to the tube speed at long wavelengths. Panel eshows the\n\frst three modes of the m= 0 andm= 1 sequences in solid, dashed and dotted lines, respectively; that is, the \frst\nmode in the sequence (solid line) corresponds to the mode annotated on panel b. The next two modes in that sequence\nare the next two blue dots moving vertically downwards (same kzavalue), these are not annotated to avoid cluttering\nthe \fgure. Analogously, panel fshows the \frst three modes of the m= 2 andm= 3 sequences, with everything\ncolour-coded according to the legend. As indicated before all eigenfunctions in the outer region are exponentially\nvarying. The panels aandbin Figure 4 reproduce the analytical results in Roberts (2019, \fg. 6.5).14\nFigure 5. Panela: spectrum showing the dispersion relation for a \rux tube under coronal conditions. The dimensionless phase\nspeed!=k zcsis displayed as a function of the dimensionless wave number kza, for four values of the azimuthal wave number\nm= 0 (blue dots), m= 1 (red crosses), m= 2 (green squares) and m= 3 (yellow triangles). Both Alfv\u0013 en speeds, together\nwith the kink, tube and sound speeds are denoted by grey horizontal lines and are all normalised to cs. Panelbzooms in on the\ntube speed-related body mode sequences. Panel b: zoom-in of panel abetween the tube speed ctand internal sound speed cs.\nPanelscandd: eigenfunctions of the modes annotated with circles ( c) and squares ( d) on the top-left panel a. Panele: \frst\nthree eigenfunctions of the m= 0 andm= 1 body wave sequence on the bottom-left panel b. Panelf: \frst three eigenfunctions\nof them= 2 andm= 3 body wave sequence. For both panels eandfthen= 1,n= 2 andn= 3 modes are shown in solid,\ndashed and dotted lines, respectively, only the \frst mode is annotated on the bottom-left panel b. Every eigenfunction on panels\ncthroughfis normalised to its maximum absolute value in that particular grid interval.\nOne mode has not yet been discussed, and that is the horizontal line of modes between the internal sound speed and\nkink speed. The eigenfunctions are shown in panel dand correspond to the annotated squares on the top-left panel a.\nThese modes are not present in the original work, and it is not a priori clear how to interpret these. However, what\nwe do know is that their phase speed is approximately 0 :5(ck+cs) and that these modes are degenerate, meaning that\ntheir position does not change when the mode number mor wave number kzchanges. The position of the outer wall\nalso does not seem to have any in\ruence on their value. This, together with the fact that the eigenfunctions seem to\nindicate that these are surface waves, strengthens the belief that these solutions could be actual waves and not some\nnumerical remnant.\nb) Coronal \rux tube. The second application of the magnetic \rux tube is one under coronal conditions, that is, an\nequilibrium for which cse<cs<cA<cAe. More speci\fcally, we take cAe= 5cs,cse=cs=2 andcA= 2cs. Analogous\nto the previous case, the relations between the equilibrium values inside and outside of the \rux tube follow from Eq.\n(31), where we again take p0and\u001a0equal to one. This results in \u001a0\u00194:86\u001ae,ct\u00190:89csandck\u00191:38cA\u00190:55cAe,\nand we use the same values as for the photospheric case for kzand the \rux tube and outer wall boundaries. Similar to\ncasea)we perform 40 runs at 300 gridpoints each for four values of k2=m(where again the \frst three runs have 1000\ngridpoints) and plot the spectrum showing the dispersion relation in Figure 5. Again all speeds indicated on the \fgure\nare normalised to the sound speed cs. The top-left panel afocuses on the fast body waves, the bottom-left panel bon\nthe slow body waves. Panel cshows the eigenfunctions corresponding to the four body waves indicated with circles\non panela. Panelddepicts eigenfunctions of the modes annotated with squares between the internal Alfv\u0013 en ( cA) and\nkink (ck) speeds, representing the kink m= 1 and \frst two \ruting ( m= 2 andm= 3) modes.15\nSimilar to Figure 4 panels eandfrepresent the \frst three body modes in the m= 0 andm= 1 sequences (panel\ne), of which the \frst mode is indicated on panel b. Panelfshows the \frst three modes of the m= 2 andm= 3\nsequences, with the \frst, second and third mode indicated with a solid, dashed and dotted line, respectively. The\ncolours of panels cthroughfare consistent with the legend. Figure 5 reproduces the analytical results in Roberts\n(2019, \fg. 6.7), which are based on the analytic dispersion relation containing Bessel functions.\n3.2.2. Tokamak constant current\nNext we discuss an example initially given in Kerner et al. (1985), which shows the ideal MHD spectrum in the\npresence of an unstable m=\u00002 interchange mode in a cylindrical geometry. We start from a so-called tokamak current\npro\fle, in which an axial current density of the form j0= (0;0;j0\u0000\n1\u0000r2\u0001\u0017) is assumed, with j0a given constant.\nThis yields a twisted magnetic pro\fle in which the longitudinal component B0zis uniform and equals one, while the\npoloidal component B0\u0012is given by\nB0\u0012=j0\n2r(\u0017+ 1)h\n1\u0000\u0000\n1\u0000r2\u0001\u0017+1i\n; (33)\nfor a given value of \u0017. For the equilibrium considered here we take \u0017= 0, making the current pro\fle constant over\nthe \rux tube. This means that B0\u0012has a linear pro\fle in rsuch that the magnetic \feld lines have a constant pitch\nand the current is distributed equally in the plasma. An expression for the pressure (and hence temperature) can be\nfound by integrating (10) and assuming that, for example, the pressure vanishes at the outer boundary, resulting in\na parabolic pressure pro\fle. Hence, for a cylindrical geometry in which r2[0;1] this yields the following equilibrium\ncon\fguration:\n\u001a0= 1; p 0=1\n4j2\n0\u0000\n1\u0000r2\u0001\n; B 02=1\n2j0r; B 03= 1; (34)\nwhere we assumed a uniform density. We introduce an additional parameter q, called the safety factor, given by\nq(r) =rkzB0z\nB0\u0012=2kz\nj0: (35)\nWe performed 39 runs, varying the q-factor between 1.9 and 2.1 in order to probe the regime containing the unstable\nm=\u00002 interchange mode, which is associated with the vanishing of the factor m+kq, that is, the k\u0001Bproduct.\nThis implicitly constrains the value for j0, and we assigned k2=m=\u00002 andk3=k= 0:2 for all runs. It should\nbe noted that this particular equilibrium con\fguration requires a high resolution near q= 2 to correctly resolve the\nunstable modes. We thus used 501 gridpoints for all runs. The complete spectrum is shown in Figure 6, where the\nsquared eigenvalues are plotted as a function of the safety factor. The 3 main branches, that is, fast, Alfv\u0013 en and slow,\nare denoted on the right side of the \fgure, as well as the region where the modes become unstable ( !2<0). We\nsee that in the region near the m=\u00002 interchange instabilities the slow and Alfv\u0013 en modes collapse to zero, which\nis due to the vanishing of the combination F=mB\u0012=r+kBzin the ideal MHD equations (Goedbloed et al. 2019).\nThe slow and Alfv\u0013 en continua are annotated on the \fgure in red and cyan, respectively. The Alfv\u0013 en continuum is\ncollapsed to a single point for this equilibrium con\fguration, while the slow continuum covers a range in frequency.\nNote in particular how the full spectrum ranges over many orders of magnitude in the !2view shown here: an intrinsic\nproperty and challenge posed by MHD spectral theory.\n3.2.3. KH and CD instabilities\nAs a \frst test for the inclusion of \row into the equations, we look at the interaction between Kelvin-Helmholtz (KH)\nand current-driven (CD) instabilities in a magnetised astrophysical jet, following Baty & Keppens (2002). This model\nuses a cylindrical jet with a supersonic background \row aligned with the axis and sheared in the radial direction. The\nequilibrium con\fguration is taken such that KH surface modes can develop, and is generally given by\n\u001a0= 1; v 02= 0; v 03=1\n2Vtanh\u0012rj\u0000r\na\u0013\n;\nB02=B\u00120rrc\nr2c+r2; B 03=Bz0; T 0=p0\n\u001a0\u0000B2\n\u00120\n2\u001a0 \n1\u0000r4\nc\n(r2c+r2)2!\n:(36)\nHererjdenotes the jet radius and rcquanti\fes the radial variation, and they are taken to be rj= 1 andrc= 0:5,\nwithr2[0;2]. The parameter Vrepresents the amplitude of the velocity, given by V= 1:63, while the chosen velocity16\nFigure 6. Complete MHD spectrum for a tokamak current pro\fle in the presence of m=\u00002 interchange modes. The squared\neigenvalues are plotted against the safety factor q, instabilities correspond to !2<0. The various branches are indicated on the\nright side of the \fgure.\npro\fle ensures that the shear layer is situated at the jet radius with a radial width given by a= 0:1rj. Both the density\n\u001a0and the pressure on-axis p0are chosen to be equal to unity. The parameters B\u00120andBz0control the amplitude and\ntwist of the magnetic \feld, respectively, as explained in Baty & Keppens (2002). The original work discusses three\ndi\u000berent magnetic \feld con\fgurations, we choose the pro\fle with\nB\u00120= 0:4 \nr2\nc+r2\nj\nrjrc!\n; B z0= 0:25; (37)\nsuch thatB02= 0:4 at the jet radius. Furthermore, the azimuthal and longitudinal wave numbers are taken to be\nk2=m=\u00001 andk3=k=\u0019.\nThe entire spectrum is calculated at high resolution using 501 gridpoints and shown in Figure 7. Panel adepicts\nthe full slow and Alfv\u0013 en spectra with the KH and \frst three CD unstable modes ( !I>0) denoted on the \fgure itself.\nThe real and imaginary parts of the \u001a,rvrandvzeigenfunctions for each of these modes are shown on the subsequent\npanels. Those for the KH mode (panels c\u0000d) are localised around the jet radius r= 1, which is also the point\nwhere the sonic and Alfv\u0013 enic Mach numbers drop to zero (panel b). Panelsethroughjdepict the eigenfunctions of\nthe \frst three CD modes, with the \frst, second and third shown on the \frst, second and third row of the bottom\npanels, respectively. The left column shows the real part, the right column the imaginary part of the eigenfunction.\nThe CD modes have an increasing number of nodes on r2[0;1], which is most clearly visible by looking at the rvr\neigenfunction (green): no nodes for the \frst CD mode (panels e\u0000f), one node for the second CD mode (panels g\u0000h)\nand two nodes for the third CD mode (panels i\u0000j).\nNote that here, we are still adiabatic such that the up-down symmetry (relating to time reversal) is still present\nin the eigenfrequency plane, but the introduction of equilibrium \row caused left-right symmetry breaking between\nforwards and backwards propagating modes. As pointed out in Goedbloed (2018a), the study of MHD spectra of\nstationary (with \row) equilibria is still governed by two self-adjoint operators, but as seen in Figure 7, modes can17\nFigure 7. Panela: MHD spectrum for the equilibrium con\fguration given in (36). The KH and CD instabilities are indicated\non the panel. Panel b: sonic and Alfv\u0013 enic Mach numbers. The other panels show the real and imaginary parts of the \u001a(blue),\nrvr(green) and vz(red) eigenfunctions, for the KH mode ( c\u0000d), \frst CD mode ( e\u0000f), second CD mode ( g\u0000h) and third\nCD mode ( i\u0000j), respectively.\nenter the complex plane at various locations (identi\fed by the spectral web (Goedbloed 2018b)). The correspondence\nwith the original \fgure in Baty & Keppens (2002) is one-to-one for the KH and CD modes, but here we have a lot\nmore detail near the axes due to the higher resolution. We will discuss this resolution aspect in Section 4.\n3.2.4. Suydam cluster modes\nNext we look at Suydam cluster modes in a cylindrical geometry, which arise from the presence of a Suydam surface in\nthe equilibrium con\fguration, that is, a location where k\u0001B0= 0. Shear \row e\u000bects are included, and the equilibrium\nis given by\n\u001a0= 1; v 02= 0; v 03=vz0\u0000\n1\u0000r2\u0001\n;\nB02=J1(\u000br); B 03=p\n1\u0000P1J0(\u000br); p 0=P0+1\n2P1J2\n0(\u000br);(38)\nwhere\u000b= 2,P0= 0:05,P1= 0:1 andvz0= 0:14, the functions J0andJ1denote the Bessel functions of the \frst kind.\nThe wave numbers were chosen to be k2=m= 1 andk3=k=\u00001:2, ensuring a Suydam surface at r\u00190:483.\nThe spectrum is calculated using 501 gridpoints for r2[0;1] and is shown in Figure 8. The resulting locations\nof the various o\u000b-axis outer modes are in agreement with results given in Nijboer et al. (1997). However, since this\nspectrum is calculated using a \fve times higher resolution, we have much more intricate detail near the Suydam surface,\nrevealing even more o\u000b-axis modes (inset on panel a). The top two panels on the right side of Figure 8 show the sonic\nand Alfv\u0013 enic Mach numbers (panel b), together with the k\u0001B0=B02k2=r+B03k3andk\u0001v0=B02v02=r+B03v03\npro\fles as a function of radius (panel c), respectively. The location of the Suydam surface at r\u00190:483 is denoted18\nFigure 8. Panela: Suydam cluster spectrum for the equilibrium given in (38). Inset: zoom-in near the Suydam surface,\nrevealing more detail. Panel bdepicts the sonic and Alfv\u0013 enic Mach numbers as a function of radius, panel cshows the k\u0001B0\nandk\u0001v0curves where the red cross denotes the Suydam surface. The bottom row of panels shows the real part of the \u001aand\nrvreigenfunctions, for the four modes in the Suydam sequence annotated on panel a. The dotted red line denotes the location\nof the Suydam surface.\nwith a red cross. The bottom row of panels ( d\u0000g) show the real part of the \u001aandrvreigenfunctions, for the four\nmodes annotated on panel awith the location of the Suydam surface annotated with a vertical red line. !1and!3\ncorrespond to modes on the left side of the Suydam surface, the other two correspond to modes on the right side. All\neigenfunctions shown here show the speci\fc variation associated with their location relative to the Suydam surface:\n!1and!3have their localised behaviour on the left side of the red dashed line, while it is vice-versa for the other two\nmodes. For visual purposes the horizontal axis in panels eandfis di\u000berent from the one in panels candd, since\nthe former correspond to the next modes in the Suydam sequence, showing stronger radial variation. Note that the\noriginal Suydam criterion (Goedbloed et al. 2019) is related to static equilibria, the generalisation of these Suydam\ncluster criteria is presented in Wang et al. (2004).\n3.3. Resistive, Cartesian cases\nAll cases discussed up to now handled an adiabatic equilibrium con\fguration with or without the inclusion of \row.\nThe up-down symmetry of all the MHD spectra shown so far is perfectly maintained, related to the fact that these\ncases are in essence time-reversible. Every instability (or overstability in the case with \row) has a damped counterpart.\nWe now move on to include additional e\u000bects. Hence, we now compute spectra for time-irreversible cases, where either\nresistivity or other non-adiabatic e\u000bects enter, which will break the up-down symmetry. Here we \frst focus on the\ninclusion of resistivity.\n3.3.1. Resistive homogeneous plasma19\nFigure 9. Panela: spectrum for a homogeneous medium with constant resistivity. Panel bzooms in near the origin, showing\nthe start of the fast mode sequence. Panel czooms in further, revealing the semi-circles traced out by the Alfv\u0013 en and slow\nmodes (outer and inner semi-circles, respectively).\nFirst we look at the most simple con\fguration, that is, a homogeneous plasma in a Cartesian geometry with resistivity\nincluded. The uniform equilibrium is given by\n\u001a0= 1; B 02= 0; B 03= 1; T 0=1\n2\fB2\n0; (39)\nwhere we take a plasma beta of \f= 0:25,k2=ky= 0,k3=kz= 1 andx2[0;1]. The value for the resistivity is\nassumed to be constant and given by \u0011= 10\u00003, as described in Goedbloed et al. (2019). This spectrum is calculated\nusing 1001 gridpoints, the result is shown in Figure 9. In ideal MHD the fast modes form a Sturmian sequence (that is,\nthe oscillation of the eigenfrequencies increases when the number of modes increases, which in turn implies a larger real\npart of!) of stable fast magneto-acoustic waves with frequencies accumulating to in\fnite frequency (related to the p-\nmodes in our strati\fed example). The slow modes have an anti-Sturmian sequence towards their accumulation point (in\nessence the collapsed slow continuum) and the Alfv\u0013 en modes are degenerate (Goedbloed et al. 2019). When resistivity\nis included the fast modes become damped and the Alfv\u0013 en and slow modes trace out semi-circles in the bottom-half\nof the complex plane. The semi-circles and initial fast mode sequence shown in Figure 9 are in perfect agreement with\nthe spectrum depicted in Goedbloed et al. (2019, \fg. 14.6). These semi-circles can be quanti\fed analytically, their\nradius does not depend on the resistivity. The magnetic Reynolds number, calculated as Rm= (x1\u0000x0)cA=\u0011, is equal\nto 1000.\nDue to the rather extreme resolution employed here we trace much further into the fast mode sequence, where we\nsee something interesting: the fast modes appear on curves that loop around, breaking the purely Sturmian behaviour\nfor a moment, after which they continue again towards in\fnity. This implies that initially the fast modes become more\ndamped at higher mode frequencies up to a certain turning point at which they achieve maximal damping (the bottom\nof the loop). After passing this turning point the oscillation frequency of the modes increases again and the damping\nfrequency seems to converge towards one single value.\nOf course, the strong damping for the fast modes as we go further into the fast mode sequence must have consequences\nfor the original uniform (ideal) equilibrium state. In what follows, we will adopt the common practice to compute\nresistive MHD spectra about an ideal MHD state, which itself will evolve when resistivity is acting.20\n3.3.2. Quasi-modes in resistive MHD\nWe now turn to a non-adiabatic case, where resistivity is important. We will compute the resistive MHD spectrum for\na case where the equilibrium varies across an interface which gives rise to so-called quasi-modes. These are essentially\nsurface waves undergoing damping, due to the fact that the global quasi-mode overlaps in frequency with the continuum\nrange, causing resonant absorption. Quasi-modes are quite important in solar physics, since they can be indirectly\nrelated to the coronal heating problem as discussed in for example Poedts et al. (1989); Poedts & Kerner (1991). A\ndetailed analytical treatment of quasi-modes including theoretical growth rates is given in Priest (2014), where they\nstart from an inhomogeneous layer of width lconnecting two regions of uniform plasma. This can be reproduced by\nintroducing a linear density pro\fle between two homogeneous regions in a Cartesian geometry, however, this would\nmean that the density derivative shows rather strong discontinuities near the edges of the transition layer. We therefore\nopt for a smooth pro\fle by introducing a sine dependence such that\n\u001a0(x) =8\n>>>><\n>>>>:\u001a1 ifx0\u0014x<s\u00001\n2l;\n1\n2\u001a1\u0014\n1 +\u001a2\n\u001a1\u0000\u0012\n1\u0000\u001a2\n\u001a1\u0013\nsin\u0012\u0019(x\u0000s)\nl\u0013\u0015\nifs\u00001\n2l\u0014x\u0014s+1\n2l;\n\u001a2 ifs+1\n2l<x<x 1;(40)\nwheresdenotes the midpoint between x0andx1, representing the left and right edges of the Cartesian grid, respectively,\nand equal to 0 and 1 such that s= 0:5. The width of the transition region l= 0:1 is taken to be small as to reduce the\nin\ruence of the walls. If l!0 the inhomogeneous region disappears and we simply have a discontinuous jump in the\nplasma. Furthermore we take \u001a1= 0:9,\u001a2= 0:1,B02= 0,B03= 1 andT0= 0. The magnetic \feld is unidirectional\nalong thez-axis, and setting the pressure (and thus temperature) to zero provides an additional test on the handling\nof zero rows in the matrix. This zero-temperature case is frequently encountered in fully nonlinear MHD simulations\nwhich arti\fcially adopt a zero plasma beta. It has as an important consequence that the slow continuum collapses\nto marginal frequency, eliminating many interesting modes from the spectrum. Additionally we adopt wavenumbers\nk3= 0:05 andk2= 1:0, such that k3\u001ck2. As discussed in Priest (2014) the quasi-modes are damped, such that they\nmove away from the real eigenfrequency axis with complex eigenvalues !=!R+i!Igiven by\n!R=\u0006s\n\u001a1!2\nA1+\u001a2!2\nA2\n\u001a1+\u001a2; ! I=\u0000\u0019kzl\u0000\n!2\nA2\u0000!2\nA1\u0001\n8!R; (41)\nFigure 10. Panela: part of the MHD spectrum containing the quasi-mode, the red dot denotes the theoretical prediction of\nEq. (41). Panels bandcshow plots of the sinusoidal density pro\fle and Alfv\u0013 en frequency as a function of the grid, respectively.\nNote that these two panels only show part of the grid, near the transition region. The inset shows the real part of the vx\n(orange) and \u001a(blue) normalised eigenfunctions associated with the quasi-mode. The blue dot in panel cdenotes the squared\nquasi-mode frequency, matching the squared Alfv\u0013 en frequency at x= 0:5. The vertical dotted line denotes the middle of the\ngrid.21\nwith!2\nA=k2\nzv2\nA. This analytic result originates from a complex analysis of the linearised initial value problem.\nHowever, there is one caveat: it is impossible to obtain complex eigenvalues away from the real or imaginary axes in\nan ideal plasma, due to the matrix operator being hermitian in ideal MHD. Since the (ideal) quasi-mode is damped,\nwe therefore include a (small) value for the resistivity, \u0011= 10\u00004, which is su\u000ecient to make the matrix operator\nnon-Hermitian and allows for complex eigenvalues away from the horizontal axis.\nThe magnetic Reynolds number in this case varies between Rm\u0019104andRm\u00193\u0002104. Figure 10 shows the\nMHD spectrum on panel a, for 501 gridpoints, with the theoretical prediction for the global quasi-mode location\nannotated with a red dot. The actual quasi-mode is also denoted on the \fgure, and is slightly more damped than its\ntheoretical counterpart. This is to be expected, since the inclusion of resistivity imposes additional damping on the\neigenvalues, shifting them downwards. It should be noted that increasing the width lof the transition layer moves the\nquasi-mode further down and to the right on the \fgure, such that it eventually merges with the damped modes found\non the branch immediately to its right. This phenomenon, where the quasi-modes merge with the resistive branches\nis discussed in more detail in Van Doorsselaere & Poedts (2007). The inset on panel adepicts the \u001a(blue) and\nvx(orange) eigenfunctions associated with the quasi-mode, showing localised variation near the transition region as\nexpected. Note that the spectrum as shown in Figure 10, left panel a, is still left-right symmetric, but that resistivity\nhas broken the up-down symmetry, with all modes here found in the stable (damped) half plane.\n3.3.3. Resistive tearing modes\nNext we move on to an inhomogeneous medium with the inclusion of resistivity and an optional linear \row pro\fle,\ndiscussed in Goedbloed et al. (2019). The geometry is Cartesian, with x2[\u00000:5;0:5] and an equilibrium con\fguration\ngiven by\n\u001a0= 1; B 02= sin(\u000bx); B 03= cos(\u000bx);\nT0=1\n2\fB2\n0; v 02=Vx; v 03= 0;(42)\nwherek3=kz= 0,\f= 0:15 and\u000b= 4:73884, with a constant resistivity value of \u0011= 10\u00004. The magnetic\ncon\fguration chosen here is a linear force-free \feld with shear, with \u000bthe proportionality constant between the\ncurrent and magnetic \feld, that is, j=\u000bB0. This speci\fc choice of parameters violates the tearing mode stability\ncriterion (Goedbloed et al. 2019), which results in an isolated, unstable tearing mode. Spectra are calculated both for\nV= 0 (no \row) and V= 0:15, the inclusion of the linear velocity pro\fle in the latter introduces a Doppler shift in\nthe slow and Alfv\u0013 en continuum, the results are shown for 501 gridpoints in Figure 11. Panels aandbshow spectra for\nV= 0,ky= 0:49 (no \row) and V= 0:15,ky= 1:5 (linear \row pro\fle), respectively, revealing the intricate behaviour\nof the damped slow and Alfv\u0013 en sequences. The purely imaginary unstable tearing mode is annotated with an arrow.\nThe resistivity \u0011is equal to 10\u00004for both cases, yielding a magnetic Reynolds number of \u0019104. These results are in\nperfect agreement with the original spectra depicted in Goedbloed et al. (2019, \fg. 14.7-14.9). Note that these cases\nstill have the left-right symmetry maintained, despite the presence of equilibrium \row. However, this is purely because\nthe \row pro\fle and domain happens to be chosen in a symmetric way, such that forward and backward modes behave\nsymmetrically.\nFor an excellent discussion on tearing modes we refer to Goedbloed et al. (2019), where they perform a detailed\nanalysis on the resistive MHD equations to derive an analytical expression for the growth rate of the tearing mode,\ngiven by\nIm(!) =R\u00003=5\nm(KH)2=5\u0012\u00010\nC\u00134=5\nvA=a; in which \u00010=\u00002p\nH2\u0000K2cot\u00121\n2p\nH2\u0000K2\u0013\n; (43)\nwithRm=avA=\u0011the magnetic Reynolds number, athe width of the slab and vAthe Alfv\u0013 en velocity. The other\nparameters in this equation are variables introduced during the analysis, given by H=\u000ba,K=k0aandC=\n2\u0019\u0000(3=4)=\u0000(1=4)\u00192:1236. Panels canddof Figure 11 show a comparison of the tearing mode growth rate between\nLegolas results (using the no-\row case) and Eq. (43). Panel choldsky= 1:5 constant but varies the resistivity,\nreaching Reynolds numbers of \u0019108, a feat that is nearly impossible to achieve if fully nonlinear codes would be\nused instead. The correspondence between the theoretical and numerical growth rates is nearly one-to-one, except\nfor large resistivity values since the theoretical approximation starts to break down in those regimes. In panel dwe\nkeep\u0011= 10\u00006(magnetic Reynolds number of \u0019106) constant and vary kybetween 0.1 and 3.5, which again yields22\nFigure 11. Resistive spectra of the equilibrium given in (42), unstable to the tearing mode (annotated with an arrow). Panels\naandbshow the spectrum without and with a linear \row pro\fle, respectively, zoomed in to reveal the complex structure of the\nslow and Alfv\u0013 en modes (fast modes are not shown). Panels candd: growth rate of the tearing mode versus \fxed ky, varying\u0011\nand \fxed\u0011, varyingky, respectively. Both bottom panels are cases without \row, with 64 runs of 351 gridpoints each, per panel.\nThe red line in panels canddannotates the theoretical growth rate prediction given in Eq. (43).\nan excellent agreement between theory and numerical results. For both panels canddwe performed 64 runs of 351\ngridpoints each.\n3.3.4. Resistive rippling modes\nThis subsection is accompanied by an Erratum, see Appendix B. The original \fgure and text in this subsection are\nkept to match the original publication in ApJS 2020, 251, 25.\nAs an extension to the tearing modes described above we take a look at resistive rippling modes, which originate\nwhenever there is a spatially varying resistivity pro\fle as discussed in for example Priest (2014). To that extent we\ntake the same equilibrium and parameters as Eq. (42) (with V= 0, so without \row and ky= 0:49), but impose\na hyperbolic tangent pro\fle on the resistivity to let it smoothly drop down to zero near the edges of the domain.\nThis spatial variation in resistivity will excite (unstable) rippling modes, in a current-carrying equilibrium that is\nalso unstable to the tearing mode. The resulting spectrum is shown in Figure 12 for 1001 gridpoints, which is again\nleft-right symmetric. The adopted \u0011(x) pro\fle is given explicitly in Eq. (44) and depicted in Fig. 12, panel c. Here23\nFigure 12. Resistive spectrum using the same parameters as panel ain Figure 11, but with a spatially varying \u0011(x)-pro\fle\n(panelc). Panelb: full spectrum with modi\fed fast mode sequences. Panel a: zoom in on the slow and Alfv\u0013 en modes, the inset\nzooms in on the tearing and unstable rippling modes. Panel d:vx(solid) and \u001a(dotted) eigenfunctions of the tearing mode\n(inset, blue). Panels eandg:vxeigenfunctions of the \frst four modes associated with the left part of the \u0011(x) pro\fle (inset,\nred circles). Panels fandh:vxeigenfunctions of the \frst four modes associated with the right part of the \u0011(x) pro\fle (inset,\ngreen squares).\n\u00110= 10\u00004denotes the constant resistivity value, sL=\u00000:4 andsR= 0:4 represent the centre of the left and right\ntransition region, respectively, having a width of l= 0:1.\n\u0011(x) =8\n>>>>>>><\n>>>>>>>:\u00110\n2+\u00110\n2 tanh\u0019tanh\u00122\u0019(x\u0000sL)\nl\u0013\nifsL\u0000l\n2\u0014x\u0014sL+l\n2;\n\u00110 ifsL+l\n2<x<sR\u0000l\n2;\n\u00110\n2+\u00110\n2 tanh\u0019tanh\u00122\u0019(sR\u0000x)\nl\u0013\nifsR\u0000l\n2\u0014x\u0014sR+l\n2;\n0 elsewhere(44)\nThe inclusion of this relatively simple \u0011(x) pro\fle has a major in\ruence on the resulting spectrum: the fast mode\nsequences trace out intricate patterns in the complex eigenvalue plane. The semi-circles traced out by the slow and\nAlfv\u0013 en modes are still vaguely present, but are much more scattered than their constant- \u0011counterparts. The inset\non panelaof Figure 12 zooms in near the origin, revealing that the tearing mode is still present, annotated with an\narrow and a large blue dot. Panel dshows thevxand\u001aeigenfunctions of the tearing mode, with relatively sharp\ntransitions near the x= 0 point and a (minor) in\ruence from the wings of the \u0011(x) pro\fle. The rippling modes visible\non the inset come in two branches, annotated with red circles and green squares, which intersect near the top of the\nbranches. The red circles correspond to the in\ruence of the left wing of the \u0011(x) pro\fle, visible in the eigenfunctions\nwhich are shown in panels e(\frst and second dot) and g(third and fourth dot), and are very localised near the \u0011\ntransition region as can be seen by the grid indication on the horizontal axis. Analogously, the branch annotated by\ngreen squares corresponds to the right wing of the \u0011(x) pro\fle, with vxeigenfunctions shown in panels fandh.\nThe fact that the rippling modes appear more unstable than the tearing mode indicates that their importance should\nnot be underestimated, and that they will most likely be prominently present when fully realistic \u0011pro\fles are used.\nIndeed, in actual plasmas the temperature variation in the equilibrium can itself already cause a spatially varying24\nresistivity pro\fle, and this is accounted for in Legolas . This allows for future systematic studies of rippling versus\ntearing mode dominance in slabs or loop-like settings.\n3.4. Non-adiabatic, cylindrical cases\nSince Legolas is the \frst 1D code to simultaneously include non-adiabatic e\u000bects, resistivity and \row, there are no\nknown previously calculated spectra where all these physical e\u000bects are included. Nevertheless, there exist some spectra\nin the literature where solely non-adiabatic e\u000bects are included (and where the equilibrium is static and no resistivity\nis incorporated), so we will use these as a base comparison for testing the non-adiabatic terms in the implementation.\n3.4.1. Non-adiabatic discrete Alfv\u0013 en waves\nThe \frst spectrum we will look at is that of non-adiabatic discrete Alfv\u0013 en waves, described in Keppens et al. (1993).\nThe basic cylindrical equilibrium represents a solar coronal loop, and non-adiabatic e\u000bects included were optically thin\nradiative losses and parallel thermal conduction. It was pointed out how a cluster sequence of discrete Alfv\u0013 en waves\nmay become unstable, and could lead to disruptions or oscillatory behaviour in loops of prominences.\nThis particular equilibrium uses an axial current pro\fle in a cylindrical geometry, yielding the same magnetic\ncon\fguration as given in Eq. (33) although this time with \u0017= 2 such that a current distribution is present throughout\nthe loop. The pressure pro\fle can be obtained through integration of the equation for magnetostatic equilibrium (10)\nwithout \row. As a boundary condition we impose that the pressure vanishes at the plasma boundary r=R, which can\nbe used to constrain the integration constant. Furthermore a parabolic density pro\fle is used, yielding an equilibrium\ncon\fguration given by\n\u001a0= 1\u0000(1\u0000d)\u0010r\nR\u00112\n; B 02=1\n6j0r\u0000\nr4\u00003r2+ 3\u0001\n; B 03= 1;\np0=j2\n0\n720h\n12\u0000\nR10\u0000r10\u0001\n\u000075\u0000\nR8\u0000r8\u0001\n+ 200\u0000\nR6\u0000r6\u0001\n\u0000270\u0000\nR4\u0000r4\u0001\n+ 180\u0000\nR2\u0000r2\u0001i\n;(45)\nin whichd= 0:2 andj0= 0:125. The cylinder wall is taken at R= 1. The equilibrium temperature pro\fle can be\nderived using T0=p0=\u001a0andT0\n0= (p0\n0\u001a0\u0000\u001a0\n0p0)=\u001a2\n0, where the prime denotes the derivative with respect to r. Only\nconduction parallel to the magnetic \feld lines is taken into account, since cross-\feld thermal conduction acts on too\nlong a timescale in this case (Keppens et al. 1993).\nThe wave numbers k2=mandk3=kare taken to be 1 and 0 :05, respectively. Optically thin radiative losses\nare included as described in Eq. (5) in which we assume that the heating is such that it exactly balances out the\ncooling terms in the equilibrium state. We use the cooling curve introduced by Rosner et al. (1978), which represents a\npiecewise cooling law with predetermined coe\u000ecients. Since the inclusion of non-adiabatic e\u000bects requires us to specify\nunit normalisations in order to look up the dimensional values in the cooling tables, we take reference values of 50\nGauss for the magnetic \feld, 1 :5\u000210\u000015g cm\u00003for the density and 1010cm as a length scale. This automatically\nconstrains all other normalisations through the ideal gas law, assuming a fully ionised plasma.\nThe spectrum is calculated using 501 gridpoints, Figure 13 shows the discrete Alfv\u0013 en spectrum (panel a) along with\na plot of the Alfv\u0013 en continuum (panel b). In total six discrete modes (that is, modes that fall outside of the Alfv\u0013 en\ncontinuum) were found in contrast with the \fve discrete modes in the original paper (Keppens et al. 1993). The\ndiscrete modes are annotated according to their overtones, where !1represents the fundamental mode (FM) and !6\nthe \ffth overtone (OT). This last overtone does not show up for resolutions below \u0018200 gridpoints, which explains\nwhy it is not present in the original work. The imaginary part of the rvreigenfunction corresponding to each of\nthese six modes is shown in the bottom-right panels of Figure 13, where the location of the minimum in the Alfv\u0013 en\ncontinuum is indicated with a red dotted line. Note that the number of eigenfunction nodes increases when considering\nmodes further in the Alfv\u0013 en sequence: the eigenfunction corresponding to !1has no nodes, !2through!6have one,\ntwo, three, four and \fve nodes, respectively. Table 1 shows a detailed comparison between the discrete modes found\nbyLegolas and the ones from Keppens et al. (1993). We multiplied these latter by i, since they use the convention\nexp(st) rather than exp( \u0000i!t) in the Fourier analysis (thus !=is).\nIn reality, the mode sequence is expected to be an in\fnite sequence accumulating to the local minimum in the Alfv\u0013 en\ncontinuum, as indicated in panel bon Figure 13. Both the real and imaginary parts of the eigenvalues are in excellent\nagreement. It should be noted that when the non-adiabatic e\u000bects are omitted the imaginary parts of these discrete\nmodes become zero, such that they lie on the real axis, representing stable waves. The inclusion of non-adiabatic25\nFigure 13. Panela: discrete Alfv\u0013 en spectrum, the Alfv\u0013 en continuum is shaded in cyan. Six discrete modes were found,\nannotated!1through!6. The insets depict zoom-ins, the regions are annotated on the \fgure. Panel bshows a plot of the\nAlfv\u0013 en continuum versus radius, the minimum is denoted by a red cross. The six panels in the bottom-right corner show the\nimaginary part of the rvreigenfunctions for modes !1through!6, the red dotted line represents the location of the minimum\nof the Alfv\u0013 en continuum.\nTable 1. Comparison of discrete Alfv\u0013 en eigenvalues between Legolas and the original work.\nEigenvalue Legolas Keppens et al. (1993)\nFM!1\u00000:1044165837 + 1 :7260866\u000210\u000007i\u00000:10436875 + 1 :7\u000210\u000007i\n1stOT!2\u00000:1090793809 + 6 :5883040\u000210\u000008i\u00000:10907061 + 6 :7\u000210\u000008i\n2ndOT!3\u00000:1096033464 + 8 :3813211\u000210\u000009i\u00000:10960184 + 8 :4\u000210\u000009i\n3rdOT!4\u00000:1096779374 + 2 :5325093\u000210\u000009i\u00000:10967774 + 2 :5\u000210\u000009i\n4thOT!5\u00000:1096871498 + 3 :7770023\u000210\u000010i\u00000:10968708 + 4 :0\u000210\u000010i\n5thOT!6\u00000:1096883410 + 7 :4335660\u000210\u000011i |\ne\u000bects hence has almost no in\ruence on their oscillation frequency, and solely pushes these modes into the unstable\npart of the imaginary plane. In Keppens et al. (1993), it was pointed out how these discrete Alfv\u0013 en mode sequences\ncan be studied by means of a WKB analysis. However, this only correctly predicted the damped or overstable nature\nof the higher order modes: to determine whether the most global modes of the sequence are overstable or damped\nrequires a full numerical computation, possible with general tools like Legolas .\n3.4.2. Magnetothermal instabilities\nAs a \fnal test we look at magnetothermal instabilities, originally depicted in Van der Linden et al. (1992). The\ngeometry is cylindrical with r2[0;1] and an isothermal equilibrium pro\fle given by\nT0= 1; B 02=r\n1 +r2; B 03= 0; p 0=1\n2 (1 +r2)2; \u001a 0=p0\nT0; (46)\nwithk2=m= 0 andk3=k= 1. There is no Bz, such that this con\fguration actually represents a z-pinch (which is\nvery unstable), and we are looking at axisymmetric (sausage) m= 0 modes here. Field-aligned thermal conduction is\nincluded, cross-\feld thermal conduction is omitted. Optically thin radiative losses are accounted for, and we use the\nsame cooling curve and heating assumptions as for the non-adiabatic discrete Alfv\u0013 en waves in (45). Reference values\nare taken to be 2 :6 MK for the temperature, 10 Gauss for the magnetic \feld and 108cm for the length scale. As26\nFigure 14. Magnetothermal instabilities for m= 0 andk= 1. The fundamental modes and 14 overtones discussed in the\noriginal work are encircled by red and denoted on the left panel ( a). Panelbshows the thermal continuum as a function of\nradius. Panel czooms into the region denoted with dashed lines on panel a, revealing various other modes forming intricate\nstructures.\nbefore, this automatically constrains all other normalisations as well. These parameters are representative for solar\ncoronal loops and arcades.\nBoth the Alfv\u0013 en and the slow continuum collapse into marginal (zero) frequency. However, including \fnite parallel\nthermal conduction and radiative losses introduces the thermal continuum. Together with the marginal slow-Alfv\u0013 en\nfrequencies, this organises the modes such that thermal instabilities merge with magnetic modes, introducing mag-\nnetothermal branches. Figure 14 shows the MHD spectrum for 1001 gridpoints in the region of the magnetothermal\nbranches, where the bottom-right panel ( c) zooms in further near the origin. The modes denoted by I+1andT1\nrepresent fundamental magnetic and thermal modes, respectively, meaning they are not coalesced. The overtones of\nthese modes on the other hand are coalesced, these are denoted by ( I+2;T2)\u0000and (I+15;T15)\u0000for the \frst and 14th\novertones, respectively. The minus sign in superscript means that these are located on the negative part of the real\naxis. There are corresponding modes on the positive part of the real axis, since without \row, we maintain the left-right\nsymmetry of the spectrum. The \frst 14 overtones are in excellent agreement with those described in Van der Linden\net al. (1992), and are encircled in red on the left panel ( a). However, the original work only displays solutions up\nto the 14thovertone. The resolution used here is much higher than the originally published results, allowing us to\nprobe the region near the origin as well. Panel czooms in near the origin, denoted by the dotted rectangle on panel\na, revealing a complex mixture of di\u000berent branches and scattered modes. It seems that the original magnetothermal\nbranches split and a dense region covered with many magnetothermal modes appears. The analytically known thermal\ncontinuum is shaded in green on the \fgure, and is represented by a dense, continuous range of discrete eigenvalues\non the imaginary axis. Since this is the \frst time that high resolution is possible for computing the magnetothermal\nmodes, it is left to future work to clarify how the MHD spectrum allows for such complex mode interactions, revealing\nthe possible presence of areas covered by modes in the complex eigenfrequency plane.\n4.CONVERGENCE\nAs discussed in this paper, increasing the resolution can have a major in\ruence on the spectrum, depending on\nwhether or not all modes are su\u000eciently resolved. Generally speaking, the further one goes in a speci\fc sequence, that\nis, looking at larger mode numbers which represents overtones having more and more nodes in their eigenfunctions,\nthe higher the resolution that is required in order to resolve the mode completely. For eigenfunctions it is usually27\nFigure 15. Spectrum of the Kelvin-Helmholtz and current-driven equilibrium as shown earlier in Figure 7, using the same\nparameters as in Section 3.2.3, at various resolutions (indicated on the top-left of each panel). The inset on the top two rows\nof panels zooms near the horizontal axis, for vertical axis values between \u00000:1 and 0:1. The insets on the bottom row of panels\nhave slightly di\u000berent vertical bounds, from \u00000:03 to 0:03.\nimmediately clear if a mode is resolved or not, since higher mode numbers translate into more oscillations in the\neigenfunction. Hence, once the number of oscillations approaches the amount of gridpoints, the eigenfunction is no\nlonger resolved.\nFor the eigenfrequencies it is not a priori clear when a speci\fc !is resolved. One way to constrain an eigenvalue is\nto do multiple runs, each time increasing the resolution. Once an eigenvalue no longer shifts in the complex plane it\ncan be considered resolved, and increasing the resolution even further will not have (much) e\u000bect on its value. Now,\nthe question naturally arises how many gridpoints are typically needed to be able to speak of a \\resolved\" spectrum.\nThis will strongly depend on the type of equilibrium considered: for smooth equilibria without sharp transitions one\ncan get away with a few dozen gridpoints and already reach an acceptable accuracy for most modes. However, in the\ncase of equilibria with large gradients, or even with localised discontinuities (interfaces), one has to make sure that a\nsu\u000ecient number of gridpoints are taken in order to su\u000eciently resolve that jump.\nFurthermore, the amount of eigenvalues in the spectrum increases with resolution, since there are as many eigenfre-\nquencies as the dimension of the matrices (which is 16 times the number of gridpoints). It is therefore entirely possible\nthat one starts probing parts of the spectrum that were initially not visible at lower resolutions, simply because there\nwere no eigenvalues in that region for that amount of gridpoints. An example is given here, where we look back at\nthe Kelvin-Helmholtz equilibrium discussed in Section 3.2.3. Figure 15 shows this equilibrium for the same values as\nemployed earlier, but every panel shows the spectrum for a di\u000berent resolution, with the amount of gridpoints given in\nthe top-left corner. The \frst three panels start out at low resolution, increasing from 26 to 76 gridpoints. Three modes\nare annotated on the \fgure, corresponding to the Kelvin-Helmholtz (KH) and current-driven (CD) modes discussed\nearlier. These modes are already decently resolved even at the lowest resolutions, and can be considered completely\nresolved at around 100 gridpoints. However, the region near the horizontal axis is another matter entirely. At low\nresolutions (\frst three panels) we see that most of the modes here are almost randomly scattered in the complex\nplane. At around 100 gridpoints they start lining up and form a more intricate \\elliptic\" pattern. This is the point\nwhere the original paper by Baty & Keppens (2002) stopped, since it was not really feasible to run these codes at\nhigher resolutions, and they were only interested in the most unstable KH and CD modes that were resolved and that\ndetermined the early evolution of a full non-linear MHD run that they performed.28\nWhen the resolution is increased even further, we see that the near-axis modes shift even closer to the real eigenfre-\nquency axis, and as visible on the insets on each panel these modes start to form intriguing patterns at high resolutions.\nThe amount of \\scattered\" modes decreases considerably, and a helix-like structure is formed at very high resolutions\n(1001 gridpoints, lower right panel). These complex and almost geometric patterns may arise in various di\u000berent equi-\nlibria as well, and call for extensive research in this topic. A complementary means to identify which modes are actually\nresolved is to exploit the spectral web, see Goedbloed (2018a,b) and Goedbloed et al. (2019, chapter 12-13), which\nlocates eigenmodes on speci\fc curves and their intersections. These curves relate to the two self-adjoint operators\nat play in stationary, adiabatic MHD. Only a combined approach using high resolution Legolas runs, modern linear\nalgebra solvers and physical insight in MHD spectroscopy, will in time reveal the true importance of these modes.\nBased on the conclusions drawn here, it is clear that the resolution required depends on the part of the spectrum\nthat is to be investigated. For isolated modes about 100 gridpoints is more than su\u000ecient to resolve most of them,\nalthough this depends on the speci\fed equilibrium. For large-scale surveys of the spectrum large resolutions should\nbe employed, in order to reveal possible regions of interest.\n5.CONCLUSION AND OUTLOOK\nIn this paper we introduced the novel 1D \fnite element code Legolas , meant to tackle the complete MHD spectrum\nincluding all kinds of physical e\u000bects in order to be able to look at various realistic equilibrium con\fgurations. We\nperformed a Fourier mode analysis on the linearised MHD equations, after which the resulting matrix eigenvalue\nproblem is solved using a \fnite element approach. A general formalism was introduced to handle both Cartesian and\ncylindrical geometries. Legolas is the \frst spectral code to combine the e\u000bects of \row, resistivity, gravity, radiative\ncooling and anisotropic thermal conduction. This opens the door to novel, in-depth studies of the complete MHD\nspectrum, which was up to now impossible with existing numerical tools.\nWe tested Legolas against various previously established results from the literature, looking at the comparison to\nanalytical results as well as to spectra previously obtained by similar numerical codes. Correspondence with existing\nresults is in most cases one-to-one, greatly increasing the con\fdence in this new tool. Some cases were run in higher\nresolutions than their original counterparts, revealing interesting features and additional structure in the spectra. The\nresistive homogeneous case in Figure 9 is in perfect agreement with results from the literature near the origin, revealing\nthe semi-circles traced out by Alfv\u0013 en and slow modes. However, far along the fast mode sequence an intriguing loop\nappears, locally breaking the Sturmian behaviour of the fast modes. Even this case calls for further investigation, since\nthis phenomenon has not been described before as no extremely high resolution studies of that part of the spectrum\nhave been done to date.\nSomething similar can be seen when looking at the instabilities of a cylindrical magnetised jet \row in Section 3.2.3,\nwhere the outer KH and CD modes are in perfect correspondence with the original results. However, the high resolution\nrevealed complex structures which were not probed before. This becomes even more clear in the convergence study\nof Section 4, where we used the same setup but increased the resolution even further. It is clear that the spectrum\nevolves drastically at higher resolutions, when more and more modes are becoming properly resolved. Since a complete\nknowledge of the MHD spectrum of \rowing equilibria is lacking, tools like Legolas and careful converged studies will\nbe essential to unravel the role of as yet unresolved spectral structure. We speculate that these spectral structures have\na physical meaning, which can be corroborated in adiabatic \rowing cases with a complementary spectral web approach.\nOur speculation extends to cases that address MHD modes in cylindrical accretion disks (Keppens et al. 2002), where\nit is becoming clear that many more modes than the celebrated MRI instability enrich the spectral structure.\nAlso in the non-adiabatic cases we found something interesting: the outer modes of the magnetothermal sequence\nare again in perfect correspondence with modes found in the original work. However, near the origin - a region that has\nnot been investigated until now - there are indications of a splitting of the magnetothermal sequence in its thermal and\nmagnetic counterparts with various modes scattered in between. An in-depth study of magnetothermal instabilities is\ncalled for, a topic of research that essentially stopped progressing for the last two decades. Now that high resolution\nMHD simulations start to reveal the complexity of thermal instability driven evolutions (Xia & Keppens 2016; Claes\net al. 2020), a revival of this topic is urgently needed.\nAs a side note it can be argued that a standard Fourier decomposition as done in all cases discussed in this paper\nmisses out on non-modal growth, that is, so-called transient modes, which have been shown to be of signi\fcant impor-\ntance in resistive systems (MacTaggart 2018). However, it is the authors' opinion that a calculation of the complete\nspectrum is all that is needed. Transient growth may well be a consequence of solving an actual initial value problem29\nusing a Laplace transformation, while taking all discrete and continuous modes into account. A detailed proof is far\nfrom evident, however, in adiabatic cases we have self-adjoint operators (one for a static case, two if \row is included),\nsuch that it seems plausible that a full decomposition in terms of a complete basis of eigenfunctions is possible. The\ninclusion of \row makes these eigenfunctions non-orthogonal, while in ideal, static plasmas the eigenfunctions are\northogonal (Goedbloed et al. 2019, chapter 12). For non-ideal, stationary plasmas non-orthogonality can again be\nexpected. How this decomposition would be a\u000bected by the inclusion of non-adiabatic e\u000bects such as resistivity or\nviscosity is not a priori clear. The discussion of transient growth is usually done in the context of non-modal analysis\nand uses the concept of pseudo-spectra, that is, also allowing for modes that are nearly eigenvalues.\nAll of the above is a clear indication that a thorough investigation of the entire spectrum is in order to yield new\ninsights in (M)HD instabilities. For solar applications alone the e\u000bect of thermal conduction on the linear modes as\nindicated by Van der Linden & Goossens (1991a) has never been investigated in fully realistic setups, possibly allowing\nfor a deeper understanding of \fne structure in solar prominences. However, since Legolas is such a versatile code the\npossibilities are endless, from various hydrodynamic con\fgurations to more magnetically-oriented astronomical cases\nlike accretion disks or astrophysical jets. The discussion of the more \\advanced\" cases in this paper alone reveals how\nmuch of the MHD spectrum is still not thoroughly investigated. We plan to tackle various new cases in future work,\ncombining linear results from this code with state of the art nonlinear numerical simulations. Including additional\nphysical e\u000bects is also planned, with an extension to viscous (M)HD (Navier-Stokes), Hall MHD and ambipolar terms.\nAdditionally, we plan to include \\real\" atmospheric models, that is, numerically generated hydrostatic equilibria based\non a given temperature pro\fle. The possibilities are endless, and modern MHD spectroscopy will no doubt shed new\nlight on various physical phenomena.\nAcknowledgements. The authors would like to thank Hans Goedbloed for useful discussions and suggestions. This work is supported by\nfunding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme, Grant\nagreement No. 833251 PROMINENT ERC-ADG 2018; by the VSC (Flemish Supercomputer Center), funded by the Research Foundation\n{ Flanders (FWO) and the Flemish Government { department EWI; and by internal funds KU Leuven, project C14/19/089 TRACESpace.30\nAPPENDIX\nA.NUMERICAL APPROACH\nA.1. Finite element analysis\nThe system of equations (19)-(26) actually de\fnes a generalised eigenvalue problem in !of the formAX=!BX\nin which the state vector Xis given by\nX= (\u001a1;v1;v2;v3;T1;a1;a2;a3) = (x1;x2;x3;x4;x5;x6;x7;x8); (A1)\nwhere each superscript on the right hand side denotes the index of the unknown variable in the state vector. The\nmatricesAandBcontain the various equilibrium quantities and di\u000berential operators with respect to u1. The\neigenvalue problem is solved by using the Finite Element Method (FEM). The basic idea is to discretise the domain in\n(not necessarily equally spaced) subdomains that are separated by nodes, that is, prechosen \fxed points xjwithjthe\nnode number (0 to N) on the interval x(in the Cartesian case), after which a linear combination of basis functions is\nused to approximate the unknown variable xi. These basis functions are represented by local piecewise polynomials\non every subdomain, and vanish outside this subdomain. As shown in for example Rappaz (1977), using the same\nbasis functions for all variables leads to spectral pollution, while using a combination of linear and constant \fnite\nelements can yield inaccurate eigenvalues (Kerner et al. 1985). Hence, Legolas uses a combination of higher-order\n\fnite elements, namely quadratic basis functions for the variables \u001a1;v2;v3;T1;a1and cubic hermite basis functions for\nthe variables v1;a2;a3. This mixture of high-order \fnite elements ensures that the eigenfunctions can represent certain\nphysical properties of speci\fc eigenfunctions exactly, and can for example allow for truly incompressible modes that\nhaver\u0001v1= 0 everywhere on the domain. The trade-o\u000b in using higher-order basis functions is that this increases\nthe accuracy of the spectrum, but also increases the size of the \fnal matrices by a factor two compared to linear \fnite\nelements. Hence, the approximation of a state variable xifor a domain divided into Nsubdomains (nodes) is given\nby\nxi(u1)\u0019NX\nj=0\u0012\nH1\nj(u1)xi\nj1+H2\nj(u1)xi\nj2\u0013\n; (A2)\nwhereH1\nj(u1) andH2\nj(u1) denote the quadratic or cubic elements at node j. The hermite ( Cj) and quadratic ( Qj)\nbasis functions themselves are given by\nQ1\nj(x) =8\n>>>><\n>>>>:4 (x\u0000xj\u00001) (xj\u0000x)\n(xj\u0000xj\u00001)2;\n0;\n0;C1\nj(x) =8\n>>>>><\n>>>>>:\u0012x\u0000xj\u00001\nxj\u0000xj\u00001\u00132\u0012\n3\u00002x\u0000xj\u00001\nxj\u0000xj\u00001\u0013\nforxj\u00001\u0014x\u0014xj;\n\u0012xj+1\u0000x\nxj+1\u0000xj\u00132\u0012\n3\u00002xj+1\u0000x\nxj+1\u0000xj\u0013\nforxj\u0014x\u0014xj+1;\n0 elsewhere ;\nQ2\nj(x) =8\n>>>>><\n>>>>>:(2x\u0000xj\u0000xj\u00001) (x\u0000xj\u00001)\n(xj\u0000xj\u00001)2;\n(2x\u0000xj+1\u0000xj) (x\u0000xj+1)\n(xj+1\u0000xj)2;\n0;C2\nj(x) =8\n>>>>><\n>>>>>:(x\u0000xj)\u0012x\u0000xj\u00001\nxj\u0000xj\u00001\u00132\nforxj\u00001\u0014x\u0014xj;\n(x\u0000xj)\u0012xj+1\u0000x\nxj+1\u0000xj\u00132\nforxj\u0014x\u0014xj+1;\n0 elsewhere ;(A3)\nas in for example Kerner et al. (1998); Goedbloed et al. (2019), and are shown graphically along with their derivatives\nin Figure 16. Since we have two basis functions per subdomain, this allows for the approximation of both the original\nvariable and its derivative by di\u000berentiating Eq. (A2). To turn the problem algebraic in nature, we use the Galerkin\nmethod such that the eigenvalue problem is written as a set of integral equations by multiplying each of the eight\nequations by an appropriate element of the chosen basis, denoted by hj, and integrate over the relevant domain.\nMathematically, this can be written asZ\n\n(AX\u0000!BX)hjdu1= 0: (A4)31\nFigure 16. The quadratic (a) and cubic (b) basis functions for the interval [ xj\u00001;xj+1] along with their derivatives (c) and\n(d), respectively. The solid lines denote H1, the dashed lines H2.\nHowever, the matrix Acontains second-order derivatives with respect to u1. In order to reduce these to derivatives\nof \frst order, hence simplifying the integrals, we make use of the Galerkin weak formulation. This is achieved by\nperforming integration by parts, which introduces additional surface terms that have to be evaluated at the boundaries.\nThese surface terms can in turn be exploited to enforce boundary conditions, which will be discussed in Subsection\nA.3. Since there are eight unknowns in our eigenvalue system, together with two basis functions and Nsubdomains,\nthis implies that the \fnal matrix eigenvalue problem will have 16 Nequations, resulting in a 16 N\u000216Nsize matrix.\nBy extension, it becomes clear that using basis functions of even higher order will increase the size of this eigensystem\nconsiderably.\nA.2. Matrix assembly\nThe actual expression for the matrix elements can be obtained by applying Eq. (A4) to the system of di\u000berential\nequations (19)-(26). As an example we will look at the v1component of the linearised continuity equation (19),\ncorresponding to element (1 ;2) in the matrices. This \\number\" links to the indices of the state vector X, since the\ncontinuity equation is associated with \u001a1, or index 1 in the state vector (A1), and the v1component is associated with\nindex 2. How these indices translate to the actual position in each matrix will be discussed further in this section.\nIn the \fnite element representation adopted here, the ( \u001a1;v1) contribution can be expanded as\nNX\nj=0!Z1\n\"h1\njh1\nkdu1=\u0000NX\nj=0Z1\n\"\u001a0\n0h1\njh2\nkdu1\u0000NX\nj=0Z1\n\"\u001a0h1\njdh2\nk\ndu1du1; (A5)\nwhere theu1-dependence of the equilibrium density \u001a0and the basis functions h1andh2is implied. In this case h1\nis quadratic, since \u001a1is associated with a quadratic basis function; analogously h2is cubic. The matrix elements for\nthis particular contribution are hence given by\nBjk(1;1) =Z1\n\"h1\njh1\nkdu1;Ajk(1;2) =\u0000Z\u00121\n\"\u001a0\n0\u0013\nh1\njh2\nkdu1\u0000Z\u00121\n\"\u001a0\u0013\nh1\njdh2\nk\ndu1du1: (A6)\nIf this reasoning is applied to all equations in the linear system, it follows that Bwill only have equal-number elements,\nimplying thatBis fully symmetric and real. Aon the other hand will have cross-term elements such that it is, in\ngeneral, not symmetric. Furthermore, it might be complex, depending on the included physical e\u000bects. Terms that\ncontain derivatives of the state vector components, as for example ( v1;\u001a1) which corresponds to element (2, 1), will\nbe integrated by parts. It is exactly this integration that gives rise to the surface terms, that is, terms that do not\ncontain an integral which translate into the natural boundary conditions. These are discussed in the next subsection.\nThe actual assembly of both matrices AandBinLegolas is done by sequential iteration over the gridpoints. From\n(A3) we see that \fnite elements have a localised nature around every gridpoint j, meaning that only the elements\nassociated with a certain region (that is, the gridpoint jitself and its neighbours j\u00001 andj+ 1) yield a non-zero\ncontribution. However, it is actually easier implementation-wise to loop over the elements in the interval [ xj\u00001;xj]\ninstead of over those in [ xj\u00001;xj+1], since then the actual integration of the matrix elements can be done in the same\nway, independent of whether the basis functions are cubic or quadratic. If only the interval [ xj\u00001;xj] is considered we32\nend up with 16 possible combinations of the basis functions for every gridpoint. This translates into a 4 \u00024 sub-matrix\nfor every variable, where every one of the 16 elements corresponds to one speci\fc combination of the shape functions.\nAs there are eight variables in the state vector this implies a 32 \u000232 matrix block for every gridpoint, hereafter dubbed\na \\quadblock\" since it consists of four 16 \u000216 blocks (hereafter called \\subblocks\"). Every one of those subblocks\ninside a quadblock corresponds to a quarter section of the aforementioned sub-matrix, which is a 2 \u00022 block in every\nsubblock. Hence, to recap, a 2 \u00022 block times eight variables represents a 16 \u000216 subblock, of which four combined\nform a 32\u000232 quadblock for every gridpoint.\nOf course, every matrix element contains one or more integrals that still have to be calculated. Since the coe\u000ecient\nfunctions are in general complicated expressions depending on u1, this is done numerically using a 4-point Gaussian\nquadrature for which an integral in the interval [ xj\u00001;xj] can be expressed as\nZxj\nxj\u00001f(u1)du1\u00191\n2(xj\u0000xj\u00001)4X\ni=1wif\u00121\n2(xj\u0000xj\u00001)\u0018i+1\n2(xj\u00001+xj)\u0013\n; (A7)\nwhere\u0018iandwiare the evaluation points and weights of the Gaussian integration. These values can be found in various\ntextbooks, as for example given in Goedbloed et al. (2019). The function fdenotes the integral coe\u000ecients, which\nare essentially the equilibrium quantities and basis functions evaluated in the various evaluation points. This actually\nimplies that every grid interval [ xj\u00001;xj] is subdivided into four points, meaning that the equilibrium expressions are\nprobed using 4( N\u00001) points rather than Npoints. The way the matrices are then assembled is thus done on a\ndouble-loop basis, where the outer loop iterates over the intervals [ xj\u00001;xj] and the inner loop iterates over the four\nGaussian points. This inner loop will calculate the basis functions and matrix elements at every point, then multiply\nthe coe\u000ecients with the Gaussian weights and \fnally add them all together in a consistent manner.\nSince every quadblock corresponds to the interval [ xj\u00001;xj] we still have to account for the contribution of the xj+1\ngridpoint. This is done by partially overlapping the quadblock in the next gridpoint with the one from the previous\ngridpoint. Figure 17 shows a visual representation of the structure and assembly process for the Amatrix, using\nthe Kelvin-Helmholtz and current-driven equilibrium discussed in Section 3.2.3 with only six gridpoints here for the\npurpose of illustration. The left panel shows the general matrix, highlighting the block-tridiagonal structure. The\ndashed grey lines denote the 32 \u000232 quadblocks of the matrix, and every dot represents a non-zero value. In total\n\fve quadblocks can be distinguished for six gridpoints, one for every grid interval. The middle panel shows a zoom-in\nof the quadblock corresponding to the second grid interval as annotated on the left panel, where it can be seen that\nthe top-left corner of this quadblock overlaps with the bottom-right corner of the previous quadblock corresponding\nto the \frst grid interval.\nA single quadblock is further divided into four subblocks, with the location of the di\u000berent state variables annotated\non the middle panel of Figure 17. The matrix element A(2;5) is highlighted in every subblock which corresponds to\nthe (v1;T1) contribution, which are cubic ( v1) and quadratic ( T1) variables. The 2 \u00022 block in the top-left corner of\nthe quadblock corresponds to the top-left 2 \u00022 corner of the right panel, as indicated by the background colours. The\nright panel shows the various combinations of the regular basis functions for the A(2;5) contribution, where the blue\ncurve corresponds to the cubic basis functions and the orange curves to the quadratic ones. It should be noted that\nthe right panel shows one speci\fc case, that is, the regular h2\njh5\nkterms of the matrix element. If for example the h2\njh7\nk\nelement is calculated, corresponding to the a2term in (20), the quadratic basis functions have to be replaced by their\ncubic counterparts, since a2is also a cubic variable. A similar reasoning can be made for the other matrix elements.\nThe termpH\u000b\f\njH\u000b\f\njat the top of every sub-panel on the right of Figure 17 denotes which combination of the basis\nfunctions should be used, where pstands for the integral coe\u000ecients. The exponent \u000b\frefers to the expressions for\nthe basis functions in (A3), where \u000b= 1 stands for Q1\njorC1\njand\u000b= 2 stands for Q2\njorC2\nj, depending on the variable\nunder consideration. \fstands for which part of the basis function that should be taken, \f= 1 means the \frst equation\nin cases, while \f= 2 means the second one. As an example we can look at H12\njH21\nj: sinceA(2;5) represents a cubic\nand quadratic variable, this translates into C12\njQ21\nj. For the cubic part we therefore take the second equation of C1\nj,\ncorresponding to xj\u0014x\u0014xj+1. The quadratic part on the other hand is given by Q21\nj, which means that we take\nthe \frst equation of Q2\nj, corresponding to xj\u00001\u0014x\u0014xj. Boundary conditions are imposed after matrix assembly is\ncompleted.33\nFigure 17. General assembly and structure of the \fnite element (complex) Amatrix. Left: Example of a full matrix for six\ngridpoints, showing the block-tridiagonal structure where dots represent non-zero values. The middle panel zooms in on one\nquadblock, showing the dependence of di\u000berent subblock positions with respect to the variables. The 2 \u00022 blocks corresponding\ntoA(2;5) (cubic, quadratic) are highlighted. Right panel: 2 \u00022 block assembly for a general A(cubic, quadratic) matrix\nelement. Cubic elements are shown in blue, quadratic ones in orange. The dotted grey line denotes zero.\nA.3. Implementation of boundary conditions\nIntegration by parts on (A4) gives rise to additional surface terms, which should be evaluated at the boundaries.\nThese kind of conditions are called natural boundary conditions, since they emerge in a natural way by rewriting the\neigenvalue problem. The regularity and \fxed wall conditions considered earlier on the other hand are called essential\nboundary conditions, and have to be handled explicitly. Since the additional surface terms originate from reducing\nsecond-order derivatives to \frst order derivatives, we only have these terms for the variables v1;T1;a2anda3, as these\nare the only equations that contain derivatives of higher order. Hence, for the momentum equation, the additional\nsurface terms can be written as\nSv1=\"\nT0\n\"h2\njh1\nk+\u001a0\n\"h2\njh5\nk+\u0012\nB02k3\u00001\n\"B03k2\u0013\nh2\njh6\nk+B03\n\"h2\njdh7\nj\ndu1\u0000B02h2\njdh8\nk\ndu1#\n@\n=v1\u0014T0\n\"\u001a1+\u001a0\n\"T1+\u0012\nB02k3\u00001\n\"B03k2\u0013\na1+B03\n\"a0\n2\u0000B02a0\n3\u0015\n@\n;(A8)\nwhere the number in superscript on hj=kdenotes the index of the variable in the state vector X. The subscript @\nmeans that these terms should be evaluated at the left- or inner edge, as well as at the right- or outer edge. In a\nsimilar manner are the surface terms for the energy and induction equations given by\nST1=i(\r\u00001)\n\"T1\"\nT0\n0@\u0014?\n@\u001a\u001a1+\u0012\nT0\n0@\u0014?\n@T\u0000\"0\n\"\u0014?\u0013\nT1+\u0014?T0\n1\n+ 2\u0012\nT0\n0\u0000\n\"B02k3\u0000B03k2\u0001@\u0014?\n@(B2)\u0000\u0011B0\n03k2+\u0011k3(\"B02)0\u0013\na1\n+ 2\u0012\nT0\n0B03@\u0014?\n@(B2)+\u0011B0\n03\u0013\na0\n2\u00002\u0012\n\"T0\n0B02@\u0014?\n@(B2)+\u0011(\"B02)0\u0013\na0\n3#\n;(A9)\nSa2=i\u0011\n\"a2\u0010\n\u0000k2a1+a0\n2\u0011\n; (A10)\nSa3=i\u0011a3\u0010\n\u0000k3a1+a0\n3\u0011\n; (A11)\nwhich should all be evaluated at the boundaries. For the case of a solid wall we see that the natural boundaries\nsimplify considerably, since if v1;a2;a3are all zero at the wall, Sv1;Sa2andSa3are also zero and have to be omitted.\nThe natural boundary condition on T1is only relevant if resistivity or perpendicular thermal conduction is included.\nHowever, in the case of the latter, the additional essential boundary condition requires that T1= 0, in which case34\nST1drops out as well. The only combination in which the surface terms for the energy equation are non-zero is when\nresistivity is included, but perpendicular thermal conduction is omitted. In that case the resistive heating terms should\nbe included in the calculation, which is done by adding the appropriate terms to the matrix elements A(5;6);A(5;7)\nandA(5;8).\nCurrently only \fxed wall boundary conditions are implemented in Legolas . However, the surface terms described\nhere can be used to impose other types of boundary conditions as well. In the case of a plasma-vacuum-wall transition\nwe have for example Bessel functions at the outer boundary of a cylindrical geometry (Roberts 2019), which encode\nthe analytic vacuum solution for the electromagnetic \feld in the outer vacuum region. These expressions can then be\nused to rewrite the surface terms (A8)-(A11) in an appropriate way such that they can be added to their respective\nsubblock positions in the matrix. This is a planned extension to be included in future versions of Legolas . This\nfunctionality was previously available in some LEDA versions (Van der Linden et al. 1992).\nThe essential boundary conditions as described in Section 2.3 have to be implemented explicitly. This is done by\nomitting the relevant basis functions that do not satisfy the boundary conditions on the edges. Consider as an example\nthe variable v1, which is associated with a cubic basis function. From Figure 16 we see that the only cubic element\nwhich is non-zero at the left boundary is C12\nj, which implies that the matrix elements where it appears should be\nzeroed out. Looking back at how the quadblock is composed in Figure 17, the boundary condition v1= 0 corresponds\nto forcing the odd rows and columns of the v1contribution to zero, for subblocks 1, 2 and 3 on the left boundary (that\nis, the \frst node j) since these correspond to the 2 \u00022 blocks in blue, green and red on the right panel. Similarly, on\nthe right side (viz. the last node j) onlyC11\njis non-zero, which implies that the odd rows and columns of subblocks\n2, 3 and 4 have to be zeroed out (corresponding to green, red and yellow). Extending this reasoning to the essential\nboundary condition on T1, we see that in this case the even rows and columns should be handled since T1is associated\nwith a quadratic element. This is done for both matrices.\nOf course, \\just\" zeroing out rows and columns in a matrix has the unpleasant side-e\u000bect that the matrices become\nsingular. For the Amatrix this is not necessarily a problem, however, the Bmatrix can never be singular since it\nis inverted when solving the general eigenvalue problem using the QR algorithm. Therefore, we introduce a one on\nB's diagonal at the location that was zeroed out, and an element \u000eonA's diagonal. The e\u000bect of this is that one\nessentially \\forces\" the boundary condition, since this implies that \u000exi\nj=!xi\nj. By extension, if \u000eis taken to be a large\nnumber (we take \u000e= 1020), this means that xi\nj= 0, which corresponds to the essential boundary condition we wanted\nto impose. The only side-e\u000bect of this approach is that it introduces eigenvalues equal to \u000e. However, since \u000eis taken\nto be large, these will not in\ruence the spectrum in any way and they can be easily \fltered out during post-processing.\nThis method thus provides a relatively easy and straightforward way to impose Dirichlet boundary conditions at the\nedges. The imposed boundary conditions are noticeable on the left panel of Figure 17, especially for the \frst node.\nThe odd rows and columns that were zeroed out can clearly be seen, together with the large numbers introduced on\nthe main diagonal.\nB.ERRATUM: \\LEGOLAS: A MODERN TOOL FOR MAGNETOHYDRODYNAMIC SPECTROSCOPY\"\n(2020, APJS, 251, 25)\nIn this work, published in ApJS 251, 25 (2020), we reported on the Legolas code, where we gave a detailed overview\nof the code itself and discussed its application on various equilibria. The case in Section 3.3.4 treated so-called rippling\nmodes, which may arise whenever there is a spatially varying resistivity pro\fle present. Herein we imposed a hyperbolic\ntangent pro\fle for the resistivity \u0011(x), and showed a spectrum that has multiple unstable branches on the left and\nright side of the imaginary axis as depicted here in Fig. 18, middle left panel. Based on the very localized nature of\nthe eigenfunctions, we concluded that these were rippling modes alongside an already present tearing mode. However,\nduring an extension of the code we discovered a bug in one of the terms of the matrix elements A(7;5) andA(8;5),\nwhich both correspond to the T1resistive components in the linearized a2anda3equations. More speci\fcally, the\nresistivity derivative with respect to temperature d\u0011=dT was erroneously treated as a spatial derivative. This implies\nthat these two elements were nonzero, while they should vanish for the imposed temperature-independent resistivity\npro\fle. These nonzero values in turn led to a modi\fcation of the spectrum and gave rise to the two unstable branches.\nAfter these two terms were corrected, the rippling modes are no longer present as can be seen in the right-hand side\nof Figure 18, while the tearing mode and its associated eigenfunctions remain una\u000bected. The large-scale spectrum is\nbarely modi\fed, however on smaller scales it can be seen that there is a major e\u000bect on the damped slow and Alfv\u0013 en35\nFigure 18. Spectra before (left) and after (right) correction of the erroneous terms. The large-scale spectrum remains mostly\nuna\u000bected (top panels), as are the tearing mode and its eigenfunctions (bottom panels). The damped slow and Alfv\u0013 en sequences\n(middle panels) are in fact modi\fed, and there are no longer rippling modes present. The tearing mode is annotated with a\ncyan cross.\nsequences. After correction of the terms these sequences better trace out the semicircles in the complex plane, in much\ncloser resemblance to the original tearing mode spectra in Section 3.3.3 of the original work.\nIt should be noted that the general conclusions regarding rippling modes remain valid. For the spatially varying\nresistivity pro\fle that we imposed there are no rippling modes, but for a more general \u0011(x;T(x)) pro\fle the two matrix\nelements discussed here will indeed be nonzero. As such it is entirely possible that for some pro\fles rippling modes\nmay arise, and the question of rippling- versus tearing mode dominance in more realistic resistivity pro\fles remains\nrelevant.\nREFERENCES\nAnderson, E., Bai, Z., Bischof, C., et al. 1999, LAPACK\nUsers' Guide, 3rd edn. (Philadelphia, PA: Society for\nIndustrial and Applied Mathematics)\nBalbus, S. A., & Hawley, J. F. 1991, The Astrophysical\nJournal, 376, 214\nBallester, J. 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V., Meng, X., et al. 2014, The\nAstrophysical Journal, 782, 81\nVan der Linden, R., & Goossens, M. 1991a, Solar physics,\n134, 247\n|. 1991b, Solar physics, 131, 79\nVan der Linden, R., Goossens, M., & Hood, A. 1992, Solar\nphysics, 140, 317\nVan Doorsselaere, T., & Poedts, S. 2007, Plasma Physics\nand Controlled Fusion, 49, 261\nWang, C., Blokland, J., Keppens, R., & Goedbloed, J.\n2004, Journal of plasma physics, 70, 651\nXia, C., & Keppens, R. 2016, The Astrophysical Journal,\n823, 22\nXia, C., Keppens, R., & Fang, X. 2017, Astronomy &\nAstrophysics, 603, A42"    },    {        "title": "0707.0776v1.Damping_of_bulk_excitations_over_an_elongated_BEC___the_role_of_radial_modes.pdf",        "content": "arXiv:0707.0776v1  [cond-mat.other]  5 Jul 2007Damping of bulk excitations over an elongated BEC - the role o f radial modes\nE. E. Rowen,∗N. Bar-Gill, R. Pugatch, and N. Davidson\nWeizmann Institute of Science, Rehovot, Israel, 76100\n(Dated: October 31, 2018)\nWe report the measurement of Beliaev damping of bulk excitat ions in cigar shaped Bose Einstein\ncondensates of atomic vapor. By using post selection, excit ation line shapes of the total population\nare compared with those of the undamped excitations. We find t hat the damping depends on the\ninitial excitation energy of the decaying quasi particle, a s well as on the excitation momentum. We\nmodel the condensate as an infinite cylinder and calculate th e damping rates of the different radial\nmodes. The derived damping rates are in good agreement with t he experimentally measured ones.\nThe damping rates strongly depend on the destructive interf erence between pathways for damping,\ndue to the quantum many-body nature of both excitation and da mping products.\nEver since Wigner and Weisskopf first calculated the\ndamping rate of an atom coupled to the vacuum [1], it\nis known that coupling to a bath affects the decay of\nan otherwise stable quantum system. The structure of\nthe bath plays a key role in determining the rate of the\ndamping, and may even cause the damping to be sub-\nexponential or super-exponential [2]. Gaseous Bose Ein-\nstein condensates (BEC) are usually well isolated from\ntheir surroundings, leading to coherent evolution. Three\nbody loss, which is the main cause for decoherence in the\ngroundstate,canbeontheorderofseconds[3]. However,\ndamping of the elementary excitations over the conden-\nsate is much more rapid, but still accessible experimen-\ntally. The bath, which is coupled to the excitation, is\nin this case a quasi-continuum of unoccupied excitations.\nThese excitations, known as Bogoliubov quasi particles,\nare approximate eigen states of the Bosonic many-body\nHamiltonian. Coupling of an excited quasi particle (QP)\nto initially unoccupied QP modes gives rise to decoher-\nence via the Beliaev damping mechanism, in which a\nQP decays into two new QPs while fulfilling energy and\nmomentum conservation [4]. Another damping process,\nthat takes place at a finite temperature, is called Lan-\ndau damping. This process involves the annihilation of\na thermally excited QP together with the damped one,\nand the creation of a more energetic QP, also conserving\nmomentum and energy [5].\nDiscrete Beliaev coupling of excitations was measured\nin [6, 7]. Since the spectrum is discrete, such Beliaev\ncouplingoflowenergyexcitationsis rare,and evenatlow\ntemperatures, damping is usually governed by Landau\nprocesses [5, 8, 9, 10, 11]. Damping of bulk excitations\nis due mainly to Beliaev coupling at low temperatures\n[11]. Such damping was measured [12], and found to\nhave strong momentum dependence, in agreement with\n[4]. However, the dependence of the Beliaev damping on\nthe excitation energy was not measured.\nIn this letter we study the effects of the damping on\nthe excitation line shape in an elongated condensate.\nBy comparing the overall response to Bragg excitations,\nwith the response of the undamped fraction, the damp-\ning rates are quantified as a function of both momentumand energy of the decaying QP. In addition to the mo-\nmentum dependence of the damping rate, we observe a\ndependence on the excitation energy. By modelling the\ncondensate as an infinite cylinder [13],[14], we calculate\nthe damping ratesofthe different elementarymodes, and\nfind good agreement with the measured line shapes. Ac-\ncording to the model, the spatial dependence of the QP\nwave functions plays a key role in the difference between\nthe damping rates. The different energy dependence at\ndifferent momentum excitations is a result of quantum\ninterference due to many-body effects.\nWe create a BEC of N= 4×105 87Rb atoms in\nthe internal state |F= 2,mF= 2/angb∇acket∇ightin a cylindrically sym-\nmetric Ioffe-Pritchard trap with trapping frequencies of\nωρ= 2π×350Hz and ωz= 2π×30Hz. The Thomas-\nFermi radii of the condensate are RTF≈3µm and\nZTF≈36µm, andthechemicalpotentialis µ/h≈4kHz.\nWe excite the condensate by shining it with two off res-\nonant laser beams detuned by 0 .2nm from the D2 tran-\nsition for a period of 2ms (Bragg excitation). Then we\nrapidly shut the magnetic trap off, and image the atomic\ncloud after 38ms time of flight. By varying both the\nfrequency detuning ωand the angle αbetween the two\nbeams, we can measure the response of the condensate\nto an excitation with momentum /planckover2pi1k= 2/planckover2pi1kLsin(α/2)\nin the axial direction ˆ z, and with energy /planckover2pi1ω. Here\nkL= 2π/780nm−1is the wave number of the lasers.\nSincekLZTF≫1, the wave vector of the excitation k\nis a good quantum number.\nA unique feature of experiments with BEC is the abil-\nity to observe the damping products in a single image.\nIn the insets of Fig. 1 we present time of flight images of\nan excited condensate. In both images roughly one third\nof the condensate atoms were excited to a QP mode with\nmomentum 2 /planckover2pi1kL, but in Fig. 1b, the damping is much\nlarger. This is evident in the plot of the normalized mo-\nmentum distribution in Fig. 1. For the lower energy ex-\ncitation (a), the peak in the momentum distribution at\nk= 2kLis more pronounced. For the larger energy ex-\ncitation (b), the damping products are dominant. The\nstrong reduction of the condensate fraction along with\nthe tendency of the damping products toward lower ax-2\n−1012300.020.040.060.080.1\n(a)\nmomentum [kL]momentum distribution [a.u.]\n−1012300.020.040.060.080.1\n(b)\nmomentum [kL]momentum distribution [a.u.]\nexcitation excitationBEC BEC\nFIG. 1: Beliaev damping of condensates excited with momen-\ntum 2/planckover2pi1kL, and different frequencies: 14 kHz (a) and 18 kHz\n(b). Insets - absorption images of excited condensates afte r\n38ms time of flight. The graphs show the normalized momen-\ntum distribution p(k) along the axial direction ˆ z, which is a\nvertical integral of the absorption images in the insets. Th e\nfree-particle resonance is at 15kHz and the local density re s-\nonance for our condensate is 17 .5kHz. In (a) the undamped\nexcitation is larger than that of (b), while the cloud of Beli aev\ndamping products is larger in (b). As the Beliaev damping\nextracts atoms from the condensate, the condensate peak is\ngreatly suppressed in (b).\nial momentum are evidence of multiple collisions.\nIn order to quantify the damping of the excitations\nat different frequencies we employ a post-selection tech-\nnique [15]. We measure the response in two ways. First\nwe measure the average momentum along the ˆ zaxis, of\nall atoms in the expanded atomic cloud: /angb∇acketleftk/angb∇acket∇ight/2kL=/integraltext\ndk k·p(k)/2kL, wherep(k) is the momentum distri-\nbution. Since momentum along ˆ zis conserved during\ndamping, this is a measure of the excitation fraction in-\ncluding damped excitations and will be referred to as\noverall response. The second way to measure response is\nto count the fraction of atoms that remain in the excita-\ntion mode alone: N2kL/Ntot, where the occupation N2kL\nand the total number of atoms Ntotare determined by\na fit to the momentum distribution. This is referred to\nas the undamped population. In the absence of collisions\nthe two measuring methods are equivalent. The differ-\nence between the two measurements is a measure of the\nnumber of excitations that were damped.\nWe repeat this measurement for different values of ω.\nIn Fig. 2a we compare the obtained line shape for ex-\ncitations with momentum 2 /planckover2pi1kL. Empty circles are the\nmeasurements of the overall response, while the filled cir-\ncles are the undamped results of the same images. The\noverall response displays a peak at 16 .6kHz, less than\npredicted by the local density approximation [16]. This\nis due to the relatively large excitation fraction, which\nshifts the resonance towards the free particle value [15].\nThe width of the resonance is due mainly to the inhomo-\ngeneous density of the condensate. The response of the/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s114/s101/s115/s112/s111/s110/s115/s101\n/s32/s91/s107/s72/s122/s93/s40/s98/s41/s40/s97/s41\nFIG. 2: (a) Line shape at k= 2kLmeasured in two ways: ( ◦)\nThe overall response /angbracketleftk/angbracketright/2kL, and (•) the undamped pop-\nulationN2kL/Ntot. The dashed and solid lines are gaussian\nfits centered at 16 .6 kHz and 15 .3 kHz, with widths of 4 kHz\nand 2.6 kHz respectively. The resonance in the line shape of\nthe undamped population is shifted down in energy and is\nnarrower than that of the overall response. The dotted line\nmarks the center of the resonance of the undamped popula-\ntion. (b) The corresponding theoretical curves obtained fr om\nour model, for the line shape of the overall response (dashed\nline), and that of the undamped population (solid line). The\nradial modes leading to the line shape are marked by bars,\npositioned at their corresponding energy, and with a height\nproportional to their overlap with the condensate.\nundamped population (filled circles) peaks at 15 .3kHz,\nand is 5 times smaller than that of the total population,\ndue to Beliaev damping. There is a clear shift down\nin the resonance of the undamped population, implying\nthat the damping rate is larger for the more energetic\nexcitations. This can be understood intuitively in a local\ndensity approximation: more energetic excitations are\nin spatial regions of larger density, leading to a higher\ndamping rate.\nIn Fig. 3a we present the response of the condensate to\nexcitations with smaller momentum - 1 .1/planckover2pi1kL. A reduc-\ntion in the damping as compared to the case of 2 /planckover2pi1kLis\nevident. This is a result of the quantum interference due\nto the collective nature of the low momentum excitations\nand therefore is present in a homogeneous BEC as well\n[12]. A gaussian fit to the overall response is centered\nat 6.48±0.04 kHz. The fit to the undamped line shape\nhas a peak at 6 .63±0.07 kHz. Contrary to the 2 kLcase,\nthe line shape of the undamped population is not shifted\ndown in energy, and the naive linear dependence of the\ndamping rate on the local density fails.\nTo theoretically account for these effects we include3\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s40/s98/s41/s40/s97/s41/s114/s101/s115/s112/s111/s110/s115/s101\n/s91/s107/s72/s122/s93\nFIG. 3: (a) Line shape at k= 1.1kLmeasured both ways: ( ◦)\nThe overall response /angbracketleftk/angbracketright/1.1kL, and (•) the undamped popu-\nlationN1.1kL/Ntot. All circles are averages over 4 data points.\nThe dashed and solid lines are gaussian fits giving a central\nfrequency of 6 .5kHz for the overall response and 6 .6kHz for\nthe undamped population. The dotted line marks the cen-\nter of the line shape of the undamped population. There is\nno downward shift of the resonance, and no narrowing as are\npresent in Fig. 2. (b) The corresponding theoretical curves\nobtained from our model, for the line shape of the overall\nresponse (dashed line), and the undamped population (solid\nline). The bars mark the frequency and overlap with the con-\ndensate of the relevant radial modes.\nspatial dependence. We exploit the cigar shape of the\ncondensate to neglect the ˆ zdependence of the ground\nstate, thus reducing the problem to 1D in the radial di-\nrection [13]. A similar treatment is performed in [10]\nto describe Landau damping of quadrupole oscillations\nof an elongated BEC. The ground state ψ0(ρ) of the\nGross-PitaevskiiHamiltonian HGP=−∇2/2M+V(ρ)+\ngN/L|ψ0|2is calculated by imaginarytime evolution, for\na radially dependent potential V(ρ) =1\n2Mω2\nρρ2and our\nexperimental parameters. Excitations over the conden-\nsate are obtained by solving the Bogoliubov equations:\n/planckover2pi1ωνuν(r) =/bracketleftbigg\nHGP+gN\nL|ψ0|2/bracketrightbigg\nuν+gN\nLψ2\n0vν(1)\n−/planckover2pi1ωνvν(r) =/bracketleftbigg\nHGP+gN\nL|ψ0|2/bracketrightbigg\nvν+gN\nLψ∗\n02uν.(2)\nThe wave functions uν(r) of Eqs. 1,2 are decoupled to\nuν(r) =un,m,k(ρ)eimφeikz,and so arevν(r). The set of\nquantum numbers νare in this case: k- the momentum\nalong ˆz,m- the vorticity around ˆ z, andn- the number\nof radial nodes of the wave functions u(ρ) andv(ρ). The\nenergy of the radial modes /planckover2pi1ωk,m,nincreases with nas\nwell as|k|and|m|. There is a conservation law for k56700.511.522.53\nω/2π [kHz]Gamma [ms−1]\n14161800.511.522.53\nω/2π [kHz]Gamma [ms−1](a) (b)\nFIG.4: Dampingratesversustheenergyofthedifferentradia l\nmodes for the first few radial modes with k= 1.1kL(a), and\nk= 2kL(b). (•) - the calculated damping rate. ( ◦) - the\ndamping rate obtained by taking only the terms involving\nuνuν1uν2in Eq. 3.\nandm(kcan only decay into q,k−q, andmintom1,m2\nfulfillingm1+m2=m), butthequantumnumber nisnot\nconserved. We consider the damping of excitations with\nm= 0 as the Bragg pulse carries no angular momentum\nalong ˆz. Still, all mvalues need to be considered as\ndamping products.\nThe Beliaev coupling is part of the next order\nexpansion of the many-body Hamiltonian: HB=/summationtext\nν,ν1,ν2Aν\nν1;ν2ˆb†\nν1ˆb†\nν2ˆbν+H.C. The three QP overlap\nAν\nν1;ν2= 2g/radicalbig\nN0/integraldisplay\ndr(uνuν1uν2+uνuν1vν2\n+uνvν1uν2+vνuν1vν2+vνvν1uν2+vνvν1vν2) (3)\ninvolves interference of six different quantum pathways.\nUsing first order time dependent perturbation theory\nwe calculate the damping as a function oftime. Since the\ndamping is nearly linear, we can extract damping rates\nΓν=1\nt4\n/planckover2pi12/summationdisplay\nm,n1,n2/integraldisplay\ndq/vextendsingle/vextendsingleAν\nν1;ν2/vextendsingle/vextendsingle2sin2(∆ωt/2)\n∆ω2,(4)\nfrom the different initially excited radial modes ν=\n(n,0,k), to all possible pairs of modes ν1= (n1,m,q)\nandν2= (n2,−m,k−q), with an energy mismatch of\n∆ω=ων−ων1−ων2These rates are plotted in Fig. 4\nversus the corresponding eigen-frequencies ωn,0,k.\nThe radial mode dependence of the damping rate is\ndifferent for the different initial momenta. The damp-\ning rate from radial modes with momentum k= 2kL\nincreases with n. In this regime, /planckover2pi1ω≫µ, excitations\nare nearly single-particle in nature and v(r)→0, for\nboth the decaying excitation, and most of the damping\nproducts. Therefore the damping is mostly governed by4\nthe term involving only uνuν1uν2. This can be seen by\nthe fact that the damping rates in Fig. 4b (filled circles),\nfollow the term with only u’s (empty circles) [17]. The\nincrease of both rates with radial mode energy is a re-\nsult of the number of available energy-conservingpairs of\ndamping products. As the energy of the decaying QP is\nincreased, more pairs are available, and the damping in-\ncreases. Our calculations show that the n= 4 mode has\ntwice as many possible damping products as the mode\nn= 0, in agreement with the damping rate which is ap-\nproximately doubled.\nDamping of modes with momentum 1 .1/planckover2pi1kLis differ-\nent. First, the energy spacing between modes is com-\nparable to the 2 kLcase, but the energies are smaller.\nThis is manifested in a largerrelative increase of avail-\nable modes with radial mode. In fact, the n= 3 mode\nhas 5 times more possible damping target modes than\nthen= 0 mode. Alone, this would increase the damping\nfrom the higher radial modes as happens with the empty\ncircles in Fig. 4a. This effect is compensated by interfer-\nence between the different terms in Eq. 3. Nearly all the\ndamping products have /planckover2pi1ω/lessorsimilarµ, and therefore, as the ra-\ndial number increases, the wave functions overlap denser\nregions of the condensate and v(r) becomes more signif-\nicant. Together, the increase due to phase space, and\nthe reduction due to quantum interference between the\nterms in Eq. 3 cancel, and the damping is similar from\nall radial modes for k= 1.1kLas seen in filled circles of\nFig. 4.\nFor even smaller momentum excitations, our model\npredicts non-exponential damping. However, the damp-\ning is so slow, that the infinite cylindrical approximation\nis no longer valid for our trap parameters.\nThe line shapes of the overall response, and the un-\ndamped population obtained by the model are presented\nbelow the data in Figs. 2b, 3b. The bars are positioned\nat the eigen-frequencies of the radial modes with the cor-\nresponding k(form= 0). The height of the bars is pro-\nportional to the overlap with the condensate, that deter-\nmines the response to the Bragg pulse [13]. The dashed\nlinesaretheexpectedoverallresponsetotheBraggpulse,\nobtained by summation over the responses of the differ-\nent radial modes including Fourier broadening of each\nmode, as in [13]. The solid line is obtained in a similar\nmanner after multiplying each response function by the\ncorresponding damping e−Γn,0,kteff. We further convolve\nthe line shape with a gaussianof width 300Hz to account\nfor residual sloshing in the trap, that changes the effec-\ntive detuning [18]. The suitable effective damping time,\nteff= 5ms, is 3ms longer than the Bragg pulse. This\nmay be due to collisions during the expansion of the con-\ndensate and enhancement of the damping due to thermal\npopulation of the damping products.\nGaussian fits to the calculated line shapes give res-\nonances at 6 .3kHz for both overall response and un-\ndamped population for the 1 .1kLexcitations. For the2kLexcitations, the model gives a line shape centered at\n16.5kHz for the overall response and at 15 .6kHz for the\nundamped population in good agreement with the mea-\nsuredvalues. Themodeldisplaysthesamenarrowingand\ndownward shift in the line shape of the undamped popu-\nlation at high momentum, indicating the larger damping\nratefrommoreenergeticexcitations. Bothmodelandex-\nperiment also show a uniform, much smaller damping of\nthesmallmomentumexcitations. Thelowestradialmode\nis more separated from the others, and distinguishable\nfrom the rest both in the model and in the experimental\ndata.\nInconclusion,weuseapostselectiontechniquetomea-\nsure the Beliaev damping of QPs in an elongated conden-\nsate. We find a dependence of the damping rate on both\nthe excitation energy and the momentum of the decayed\nQP. We measure a shift downward in the excitation en-\nergy of the undamped excitation for high momentum,\nwhile for small momentum we measure no such shift.\nModelling our system as infinite along the axial direc-\ntion, we obtain quantitative agreement between theory\nand experiment. The overall damping is suppressed for\nlowmomentumexcitationsasaresultofinterferingquan-\ntum pathways within each damping channel.\nThe inhomogeneous treatment includes a quantization\nof the bath in the radial direction. Since our excitation\nenergiesare above /planckover2pi1ωρ, we are only indirectly sensitive to\nthis quantization. Beliaev damping becomes even more\nintriguingwhentheradialtrappingfrequencyisincreased\nand the system becomes one dimensional. In this regime\nit could be possible to observe effects such as quantum\nZeno[2], nonexponentialdampingandeventhecomplete\ninhibition of damping due to the convex excitation line\nshape.\nStudying damping in our system has the advantage\nof imaging which modes of the continuum are excited\nupon damping. This can serve as a spectroscopic tool\nto probe the QP spectrum of the damping products [19].\nOne can also differentiate between Beliaev damping and\nLandau damping according to the resulting momentum\ndistribution of damping products.\nThis work was supported by the DIP and Minerva\nfoundations.\n∗Electronic address: eitan.rowen@weizmann.ac.il\n[1] V. F. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930).\n[2] A. G. Kofman and G. Kurizki, Nature (London) 405,\n546 (2000).\n[3] E. A. Burt et al., Phys. Rev. Lett. 79, 337 (1997).\n[4] S. T. Beliaev, Sov. Phys. JETP 34, 299 (1958).\n[5] L. Pitaevskii and S. Stringari, Phys. Lett. A. 235, 398\n(1997).\n[6] E. Hodby et al., Phys. Rev. Lett. 86, 2196 (2001).\n[7] V. Bretin et al., Phys. Rev. Lett. 90, 100403 (2003).5\n[8] P. O. Fedichev et al., Phys. Rev. Lett. 80, 2269 (1998).\n[9] B. Jackson and E. Zaremba, New J. Phys. 5, 88 (2003).\n[10] M. Guilleumas and L. P. Pitaevskii, Phys. Rev. A 67,\n053607 (2003).\n[11] S. Giorgini, Phys. Rev. A 57, 2949 (1998).\n[12] N. Katz et al., Phys. Rev. Lett. 89, 220401 (2002).\n[13] C. Tozzo and F. Dalfovo, New J. Phys. 5, 54 (2003).\n[14] S. Stringari, Phys. Rev. A 58, 2385 (1998).[15] N. Katz et al., Phys. Rev. A 70, 033615 (2004).\n[16] F. Zambelli et al., Phys. Rev. A 61, 063608 (2000).\n[17] The discrepancy between the filled and empty circles is\ndue to destructive interference in damping events involv-\ning damping products with non-negligible v(r).\n[18] J. Steinhauer et al., Phys. Rev. Lett. 90, 060404 (2003).\n[19] N. Katz et al., Phys. Rev. Lett. 95, 220403 (2005)."    },    {        "title": "1004.4410v1.Asymptotic_behavior_of_Lorentz_violation_on_orbifolds.pdf",        "content": "arXiv:1004.4410v1  [hep-ph]  26 Apr 2010OU-HET 665/2010\nAsymptotic behavior of Lorentz violation on orbifolds\nNobuhiro Uekusa\nDepartment of Physics, Osaka University\nToyonaka, Osaka 560-0043 Japan\nE-mail: uekusa@het.phys.sci.osaka-u.ac.jp\nAbstract\nMomentum dependenceof quantumcorrections with higher-di mensional Lorentz\nviolation is examined in electrodynamics on orbifolds. It i s shown that effects of the\nLorentz violation are not decoupled at high energy scales. D espite the loss of the\nhigher-dimensional Lorentz invariance, a higher-dimensi onal Ward identity is found\nto be fulfilled for one-loop vacuum polarization. This impli es that gauge invariance\nmay be prior to Lorentz invariance as a guiding principle in h igher-dimensional field\ntheory. As a universal application of electrodynamics, an e xtra-dimensional aspect\nfor Furry’s theorem is emphasized.1 Introduction\nField theory with extra dimensions provides an interesting framewor k for physics be-\nyond the standard model. As in the four-dimensional case, one of t he fundamental keys\nthat characterize theory is symmetry which is preserved or broke n. In models with ex-\ntra dimensions, a variety of symmetry breaking have been provided [1]-[11]. It has also\nbeen shown that combinations of sources for extra-dimensional s ymmetry breaking are\nrelatively accommodating and yield various possibilities [12]-[14]. Associa ted with non-\nrenormalizable properties, it is still controversial whether quantu m corrections are validly\nextracted in the field-theoretical context. Higher-dimensional fi eld theory can be regarded\nasahigh-energyeffectivetheorywithadistinctultravioletcompletio n. Whileattemptsfor\nrealistic models have been developed, most of models such as orbifold models with a min-\nimal setup require higher-dimensional Lorentz invariance as a basic symmetry. However,\nextra dimensions are clearly different than our four dimensions. It in cludes potentially an\nextra-dimensional Lorentz violation.\nIf the Lorentz invariance in extra dimensions is violated, it is importan t to be taken\ninto account whether the symmetry breaking is spontaneous or no t. When a symmetry\nbreaking is described to be spontaneous in a certain theory, the co rresponding symmetry\nis expected to be recovered at high energies in its framework. The H iggs mechanism is\nthis type of symmetry breaking. Symmetry breaking in orbifolding inv olves the extra-\ndimensional origin. It is nontrivial whether symmetry is recovered a t high energy scales.\nEven if the starting action is Lorentz invariant, loop effects can give rise to a Lorentz vio-\nlation. If a model in the standpoint of effective field theory beyond t he standard model al-\nlows that the Lorentz invariance is lost at high energy scales, the st arting action should be\ndescribed in a Lorentz-non-invariant manner or only approximately in a Lorentz-invariant\nmanner with respect to extra dimensions. The extra-dimensional L orentz violation has\nbeen found to affect spectra, Kaluza-Klein parity and parity violatio n [15]. Therefore in\nthe field-theoretical context it should be clarified if the extra-dime nsional Lorentz invari-\nance on orbifold models is asymptotically preserved.\nIn this paper, we study momentum dependence of Lorentz violating terms in elec-\ntrodynamics on an orbifold S1/Z2. With an explicit analysis for loop diagrams and\nrenormalization, it is shown that effects of the Lorentz violation are not decoupled at\nhigh energy scales. As another notable feature, despite the loss o f the higher-dimensional\nLorentz invariance, a higher-dimensional Ward identity is found to b e fulfilled for one-\nloop vacuum polarization. This implies that higher-dimensional gauge in variance may\nbe prior to higher-dimensional Lorentz invariance as a guiding princip le in a high-energy\nfield theory. We also discuss an extra-dimensional aspect for Furr y’s theorem.\nThe paper is organized as follows. In Sec. 2, our Lorentz violent act ion is given. In\nSec. 3, a formalism of a renormalization is shown in the orbifold model. I n Sec. 4, the\nasymptotic energy dependence of Lorentz violating terms is given. It is also shown that\nhigher-dimensional Ward identity is fulfilled for one-loop vacuum polar ization. In Sec. 5,\na discussion about Furry’s theorem is given. In Sec. 6, we conclude w ith some remark.\nThe detail of loop corrections is summarized in Appendix A.\n12 Five-dimensional electrodynamics and Lorentz vi-\nolation\nWe start with the action for five-dimensional quantum electrodyna mics,\nS=SLI+SLV+SGF, (2.1)\nwith the Lorentz invariant action,\nSLI=/integraldisplay\nd4x·1\n2/integraldisplayL\n−Ldy/parenleftbigg\n−1\n4FMNFMN+¯ψiγMDMψ/parenrightbigg\n, (2.2)\nand the Lorentz violating action\nSLV=/integraldisplay\nd4x·1\n2/integraldisplayL\n−Ldy/parenleftbigg\n−λ\n2FµyFµy+k¯ψiγ5Dyψ/parenrightbigg\n, (2.3)\nwhereλandkare dimensionless coupling constants and their nonzero values indica te\nthe violation of the five-dimensional Lorentz invariance. After a re normalization, both of\nλandkare momentum-dependent. The Lorentz violating terms such as ¯ψγ5ψcan be\nabsorbed by the terms in Eq. (2.3) via a field redefinition [15]. The actio ns (2.2) and (2.3)\nhave gauge invariance although its form is not in a Lorentz-invariant way. The gauge\nfixing action is denoted as SGF, whose explicit form will be given after a field redefinition\nwith respect to renormalization factors. The fifth-dimensional Lo rentz violation is only\ntaken into account while the four-dimensional Lorentz invariance is preserved. The five-\ndimensional indices are denoted as M. Greek indices µrun over 0,1,2,3 and the fifth index\nis denoted as y. The gamma matrices are given by\nγµ=/parenleftbigg\nσµ\n¯σµ/parenrightbigg\n, γ5=/parenleftbigg\n−i12\ni12/parenrightbigg\n, (2.4)\nwhere the Pauli sigma matrices are used as σµ= (12,σi) and ¯σµ= (−12,σi). The five-\ndimensional covariant derivative is defined as DM=∂M−igAM. The extra-dimensional\nspace is compactified on S1/Z2, where the fundamental region is 0 ≤y≤L. The five-\ndimensional spacetime is flat with the metric (1 ,−1,−1,−1,−1). The orbifold boundary\nconditions for gauge fields and fermions are\nAµ(x,−y) =Aµ(x,y), Aµ(x,L−y) =Aµ(x,L+y), (2.5)\nAy(x,−y) =−Ay(x,y), Ay(x,L−y) =−Ay(x,L+y), (2.6)\nψ(x,−y) =iγ5ψ(x,y), ψ(x,L−y) =iγ5ψ(x,L+y), (2.7)\nsuch that the photon and left-handed Weyl fermion have zero mod e.\nIn order to perform renormalized perturbation, we define renorm alized fields as\nAµ=Z1/2\nAAµr, Ay=Z1/2\n5Ayr, ψ=Z1/2\nψψr. (2.8)\nThe Lagrangian terms for the gauge field are rewritten as\n−1\n4FMNFMN−λ\n2FµyFµy\n=−1\n4FMNrFMN\nr−λr\n2FµyrFµy\nr\n−1\n4δ1FµνrFµν\nr+1\n2δ2∂µAyr∂µAyr−δ4∂µAyr∂yAµ\nr+1\n2δ3∂yAµr∂yAµ\nr,(2.9)\n2whereλris the renormalized coupling for λ. Among the counterterms in the equation\n(2.9), the cross term ∂µAyr∂yAµ\nralso appears. The renormalization factors are given by\nδ1=ZA−1, δ 2= (1+λ)Z5−(1+λr), (2.10)\nδ3= (1+λ)ZA−(1+λr), δ 4= (1+λ)Z1/2\n5Z1/2\nA−(1+λr).(2.11)\nThe part of the gauge field has the original three coefficients λ,ZAandZ5. One of the\nfour renormalization factors δ1,···,δ4can be written in terms of the other factors. For\nexample,δ4is\nδ4= (1+λ)(1+δ1)/braceleftBigg/bracketleftbiggδ2−δ3\n(1+λ)(1+δ1)+1/bracketrightbigg1/2\n−1/bracerightBigg\n+δ3. (2.12)\nThe equation (2.9) has gauge invariance although it is not the five-dim ensional Lorentz\ninvariant form. It is convenient to choose the gauge fixing action as\nSGF=/integraldisplay\nd4x·1\n2/integraldisplayL\n−Ldy/parenleftbigg\n−1\n2ξ(∂µAµ\nr−ξ(1+λr)∂yAyr)2/parenrightbigg\n. (2.13)\nFor the gauge ξ= 1, the kinetic term and λrterm in Eq. (2.9) and the gauge fixing yield\n−1\n2[∂µAνr∂µAν\nr−(1+λr)∂yAµr∂yAµ\nr]+1\n2/bracketleftBig\n∂µ˜Ayr∂µ˜Ayr−(1+λr)∂y˜Ayr∂y˜Ayr/bracketrightBig\n,(2.14)\nwhere the rescaling has been employed as ˜Ayr≡√1+λrAyrfor the canonical normaliza-\ntion. Unless 1+ λ>0, tachyonic degrees arise. At the moment its positivity is assumed.\nThe cross terms of AµandAyare gathered into a total derivative −(1+λr)∂y(Aµ\nr∂µAyr),\nwhich is vanishing due to periodicity. From Eqs. (2.1) and (2.8), the La grangian terms\nfor the fermion are rewritten as\n¯ψriγM∂Mψr+kr¯ψriγ5∂yψr+δ5¯ψriγµ∂µψr+δ6¯ψriγ5∂yψr, (2.15)\nwherekris the renormalized coupling for k. Correspondingly to the two coefficients kand\nZψ, the renormalization factors are given by δ5=Zψ−1 andδ6= (1+k)Zψ−(1+kr).\nThe Lagrangian terms of interactions are rewritten as\ngrAµr¯ψrγµψr+iNrgr˜Ayr¯ψrγ5ψr+δ7Aµr¯ψrγµψr+δ8˜Ayr¯ψrγ5ψr, (2.16)\nwith the rescaled field ˜AyrforAyr. HereNr≡ −i(1+kr)/√1+λr. The renormalization\nfactorsareδ7=gZ1/2\nAZψ−grandδ8=δ8(δ1,···,δ7,kr,λr,gr). Forcouplingsandfields, the\nsubscriptrand tilde to indicate renormalized and rescaled quantities will be suppr essed\nhereafter.\nIn order to calculate quantum loop corrections, we write down the f our-dimensional\nLagrangian based on a mode expansion. From the equations of motio n, the mode expan-\nsion of fields is given by\nAµ(x,y) =1√\nLAµ0(x)+∞/summationdisplay\nn=1/radicalbigg\n2\nLAµn(x)cos/parenleftBignπ\nLy/parenrightBig\n, (2.17)\n3Ay(x,y) =∞/summationdisplay\nn=1/radicalbigg\n2\nLAyn(x)sin/parenleftBignπ\nLy/parenrightBig\n, (2.18)\nψL(x,y) =1√\nLψL0(x)+∞/summationdisplay\nn=1/radicalbigg\n2\nLψLn(x)cos/parenleftBignπ\nLy/parenrightBig\n, (2.19)\nψR(x,y) =∞/summationdisplay\nn=1/radicalbigg\n2\nLψRn(x)sin/parenleftBignπ\nLy/parenrightBig\n. (2.20)\nAfter the integration of the fifth space, the four-dimensional La grangian is obtained as\nL4D=Lquad\nAµ+Lquad\nAy+Lquad\ncross+Lquad\nψ+Lint. (2.21)\nHere the quadratic Lagrangians are given by\nLquad\nAµ=−1\n2∂µAν0∂µAν\n0−1\n2∞/summationdisplay\nn=1/parenleftbig\n∂µAνn∂µAν\nn−m2\nAnAµnAµ\nn/parenrightbig\n−1\n4δ1Fµν0Fµν\n0−1\n2∞/summationdisplay\nn=1/parenleftbigg1\n2δ1FµνnFµν\nn−δ3\n(1+λ)m2\nAnAµnAµ\nn/parenrightbigg\n,(2.22)\nLquad\nAy=1\n2∞/summationdisplay\nn=1/parenleftbig\n∂µAyn∂µAyn−m2\nAnAynAyn/parenrightbig\n+δ2\n2(1+λ)∞/summationdisplay\nn=1∂µAyn∂µAyn,(2.23)\nLquad\ncross=δ4\n(1+λ)∞/summationdisplay\nn=1mAn(∂µAyn)Aµ\nn, (2.24)\nLquad\nψ=¯ψ0iγµPL∂µψ0+∞/summationdisplay\nn=1¯ψn(iγµ∂µ−mψn)ψn\n+δ5¯ψ0iγµPL∂µψ0+∞/summationdisplay\nn=1¯ψn/parenleftbigg\nδ5iγµ∂µ−δ6\n(1+k)mψn/parenrightbigg\nψn. (2.25)\nThe Lagrangian Lquad\nAµforAµhas counterterms for δ1andδ3. The Lagrangian Lquad\nAy\nforAyhas a counterterm for δ2. For the Lagrangian Lquad\ncross, there is a cross term only\nfor the counterterm. The renormalization factor δ4is not independent of δ1,δ2,δ3. The\nLagrangian Lquad\nψforψhas counterterms for δ5andδ6. Then-th masses of bosons and\nfermion are\nmAµn=√\n1+λnπ\nL=mAyn≡mAn, mψn= (1+k)nπ\nL. (2.26)\nWe have defined Dirac fermions as\nψ0≡/parenleftbiggψL0\n0/parenrightbigg\n, ψn≡/parenleftbiggψLn\nψRn/parenrightbigg\n, (2.27)\nand introduced the left-chiral projection operator PL≡(12+iγ5)/2. The interaction\nterms of the Lagrangian are\nLint=g√\nL¯ψ0γµPLAµ0ψ0+∞/summationdisplay\nn=1g√\nL¯ψnγµAµ0ψn\n4+∞/summationdisplay\nn=1g√\nL/parenleftbig¯ψnγµPLAµnψ0+¯ψ0γµPLAµnψn/parenrightbig\n+∞/summationdisplay\nn,m,ℓ=1g√\n2L/braceleftbig¯ψnγµAµmψℓ(δn+m,ℓ+δn,m+ℓ)+¯ψnγµiγ5Aµmψℓδn+ℓ,m/bracerightbig\n+∞/summationdisplay\nn=1Ng√\nL/parenleftbig¯ψnPLAynψ0−¯ψ0PRAynψn/parenrightbig\n+∞/summationdisplay\nn,m,ℓ=1Ng√\n2L/braceleftbig¯ψnAymψℓ(δn,m+ℓ−δn+m,ℓ)+¯ψniγ5Aymψℓδn+ℓ,m/bracerightbig\n,(2.28)\nwhere counterterms for interactions have been omitted. The sum of modes for three\nindices is denoted as/summationtext∞\nn,m,ℓ=1=/summationtext∞\nn=1/summationtext∞\nm=1/summationtext∞\nℓ=1. At tree level, λandkaffect the\nKaluza-Klein spectrum given in Eq. (2.26). The equation (2.23) means thatAynhas no\ncounterterm for the mass. As an explicit consistency check, it will b e shown that the one-\nloop two-point function for Aynhas the bulk divergence only for a four-momentum term.\nIn Eq. (2.28), the terms ¯ψnPLAynψ0and¯ψ0PRAynψnhave relative sign and ¯ψnAymψℓhas\nthe factor ( δn,m+ℓ−δn+m,ℓ). The importance of their signs will be emphasized in Sec. 5.\n3 Renormalization on orbifolds\nIn this section, we give a formalism of the renormalization for two-po int functions for Aµ\nandAy. The one-loop vacuum polarizations for AµandAyare diagonal with respect to\nKaluza-Klein modes. The detail of a calculation is summarized in Append ix A.\nThe tree level propagators for the s-th fieldsAµsandAysare\nDµν(p2) =−iηµν\np2−m2\nAs+iǫ, D 55(p2) =i\np2−m2\nAs+iǫ, (3.1)\nwherep2=pµpµ. For simplicity, iǫwill be omitted hereafter. Exact propagators can\nbe decomposed with one-particle irreducible amplitudes. At one-loop level, diagrams of\nthe decomposition are shown in Figure 1, where an unshaded circle de notes a one-loop\ndiagram. The corresponding equations are written as\nGµν=Dµν+DµρΠρσGσν+DµρΠρ5G5ν, (3.2)\nG55=D55+D55Π55G55+D55Π5σGσ5, (3.3)\nG5ν=D55Π5σGσν+D55Π55G5ν, (3.4)\nGµ5=DµρΠρσGσ5+DµρΠρ5G55. (3.5)\nThe one-loop vacuum polarizations have the tensor structure give n by\nΠµν= Π1ηµν+Π2pµpν,Πµ5=−Π5\nµ= Π3pµ= Π5µ, (3.6)\nwhere the explicit forms of Π 1, Π2and Π 3will be given later. With these quantities, the\none-loop exact propagators are solved as\nGµν=−D55\n1+D55Π1ηµν\n5Figure 1: One-loop decomposition of exact propagators.\n+D55[(D55Π2)(1−D55Π55)+(D55Π3)2]pµpν\n(1+D55Π1)[(1−D55Π55)(1+D55(Π1+Π2p2))+(D55Π3)2p2],(3.7)\nG55=D55(1+D55(Π1+Π2p2))\n(1−D55Π55)(1+D55(Π1+Π2p2))+(D55Π3)2p2, (3.8)\nG5ν=D55(D55Π3)\n(1−D55Π55)(1+D55(Π1+Π2p2))+(D55Π3)2p2pν, (3.9)\nwhereGν5=G5ν.\nNow we perform the renormalization. From the Lagrangians (2.22), (2.23) and (2.24),\nthe contributions of counterterms are led to\nΠct\nµν(p) =−i(p2ηµν−pµpν)δ1+iδ3\n1+λm2\nAsηµν, (3.10)\nΠct\nµ5(p) =δ4\n1+λmAspµ,Πct\n55(p) =iδ2\n1+λp2. (3.11)\nOnly three renormalization factors among δ1,···,δ4are independent. All the divergence\nassociated with Π 1,Π2,Π3,Π55must be removed with three renormalization factors. As\nthe first step, it is convenient to fix the renormalization condition fo r the off-diagonal\ncomponent, G5ν(m2\nAs) = 0. This condition yield\nΠ3(m2\nAs) = 0, (3.12)\nwhich corresponds to the fixing of δ4. For Π 3= 0, the other propagators are simplified as\nGµν=−D55\n1+D55(Π1+Π2p2)/bracketleftbigg\nηµν+D55Π2\n1+D55Π1(p2ηµν−pµpν)/bracketrightbigg\n,(3.13)\nG55=D55\n1−D55Π55. (3.14)\nFor Eq. (3.13), Gµν, the term of ( p2ηµν−pµpν) is renormalized with the counterterm for\nδ1. The corresponding renormalization condition can be imposed as\nΠ2(m2\nAs) = 0. (3.15)\n6Aswewillshowexplicitly, thedivergentpartforΠ 1andΠ55satisfy((Π 1+p2Π2)/m2\nAs)div=\n(Π55/p2)divat one-loop level. This reduces to δ2=δ3. Thus the renormalization can be\nchosen as\nΠ1(m2\nAs) = 0. (3.16)\nOn the other hand, the finite part is ((Π 1+p2Π2)/m2\nAs)/ne}ationslash= (Π55/p2). This means that the\npropagator for Ayreceives finite mass corrections with Π55(m2\nAs)/ne}ationslash= 0. For the divergent\npart, it will be found in the following sections that at one-loop level, δ2=δ3=δ4=\n(1+k)2δ1. Thus the momentum-dependent vacuum polarizations Π 1(p2), Π2(p2), Π3(p2)\nand Π55(p2) can be achieved after the divergent part is fixed with the renorma lization\nconditions (3.12), (3.15) and (3.16). Fromthese equations, we can identify the asymptotic\nbehavior of the Π 1(p2), Π2(p2), Π3(p2) and Π55(p2). It needs to be checked if Lorentz\ninvariance is preserved at high energy scales.\nRenormalization for fermion self-energies would be given in a similar pro cedure. It\nmay be technically complicated since one-loop self-energies are not d iagonal with respect\nto Kaluza-Klein modes. This can be found from explicit one-loop amplitu des summarized\nin Appendix A. A feasible way to treat off-diagonal components has b een developed in\nRef. [16]. At the first step to address asymptotic behavior of the L orentz violation, we are\ninterest innotonlyLorentzinvariancebutalsogaugeinvariance. Bo thoftheseinvariances\ncanbesimultaneously examined whenthevacuumpolarizationrather thantheself-energy\nis analyzed. Therefore we focus on the effects on the vacuum polar ization forAµandAy\nand the issue for determining momentum-dependent amplitudes with external fermions\nwill be left for future work.\n4 Energy dependence of Lorentz violating terms and\nhigher-dimensional Ward identity\nFollowing the formalism of the previous section, we analyze explicit one -loop results for\nthe Lorentz violation. The one-loop contributions for the vacuum p olarization, Π(1)\n1, Π(1)\n2,\nΠ(1)\n3and Π55(1)are given via the dimensional regularization by\nΠ(1)\n1(p2) =8ig2\n(4π)2(1+k)/integraldisplay1\n0dx\n\n\nz4−∞/summationdisplay\nnp=1z3e−2z4\nz3·cos(2πnpxs)\n\n×x(1−x)(p2−m2\nψs)\n−1\n4∞/summationdisplay\nnp=1z3(z3+2z4)e−2z4\nz3(1−2x)mψssin(2πnpxs)\n\n, (4.1)\nΠ(1)\n2(p2) =−8ig2\n(4π)2(1+k)/integraldisplay1\n0dx\n\n\nz4−∞/summationdisplay\nnp=1z3e−2z4\nz3·cos(2πnpxs)\n\n×x(1−x)}, (4.2)\nΠ(1)\n3(p2) =−8ig2N\n(4π)2(1+k)/integraldisplay1\n0dx\n\n\nz4−∞/summationdisplay\nnp=1z3e−2z4\nz3cos(2πnpxs)\n\n7×x(1−x)mψs\n+1\n4∞/summationdisplay\nnp=1z3(z3+2z4)e−2z4\nz3(1−2x)sin(2πnpxs)\n\n, (4.3)\nΠ55(1)(p2) =8ig2N2\n(4π)2(1+k)/integraldisplay1\n0dx/braceleftbig\nz4x(1−x)p2\n−1\n4∞/summationdisplay\nnp=1/bracketleftbig\n3z2\n3(z3+2z4)+2z3(2x(1−x)m2\nψs)/bracketrightbig\ne−2z4\nz3cos(2πnpxs)\n+1\n4∞/summationdisplay\nnp=1z3(z3+2z4)e−2z4\nz3(1−2x)mψssin(2πnpxs)\n\n, (4.4)\nwherez3≡(1 +k)/(npL) andz4≡/radicalBig\nx(1−x)(m2\nψs−p2). In the above equations, the\nnp-independent part is finite due to the dimensional regularization in sp acetime with\nodd dimensions but it is potentially divergent. For the np-independent part, (Π(1)\n1+\np2Π(1)\n2)/m2\nAs= Π55(1)/p2is satisfied.\nFrom the renormalization conditions (3.12), (3.15) and (3.16), the r enormalization\nfactorsδ1,···,δ4are fixed. Then the renormalized vacuum polarizations are given by\nΠj(p2) = Π(1)\nj(p2)−Π(1)\nj(m2\nAs), (4.5)\nΠ55(p2) = Π55(1)(p2)\n−ip2\n1−iΠ(1)\n1(m2\nAs)\nm2\nAs−1\n1+iΠ(1)\n2(m2\nAs)/parenleftBigg\n1−Π(1)\n3(m2\nAs)−iΠ(1)\n1(m2\nAs)\nm2\nAs/parenrightBigg2\n,(4.6)\nwherej= 1,2,3. At high energies, the vacuum polarizations behave as\nΠµν(p2)→Πµν\nas(p2) =/bracketleftbig\n−(p2ηµν−pµpν)−N2m2\nAsηµν/bracketrightbig\nΠas\n2(p2), (4.7)\nΠµ5(p2)→Πµ5\nas(p2) =−iN2(pµmAs)Πas\n2(p2), (4.8)\nΠ55(p2)→Π55\nas(p2) =−N2p2Πas\n2(p2), (4.9)\nwhere Πas\n2(p2)≡ −8ig2(4π)−2(1 +k)−1/integraltext1\n0dxz4x(1−x). In obtaining the asymptotic\nvalues (4.7), (4.8) and (4.9), we have employed the renormalization f actorsZAandZ5\nand the renormalized coupling constant λso as to satisfy the renormalization conditions.\nExplicitly these constants are given by\nZA= 1+iΠ(1)\n2(m2\nAs), (4.10)\nZ5=1\n1+iΠ(1)\n2(m2\nAs)/parenleftBigg\n1−Π(1)\n3(m2\nAs)−iΠ(1)\n1(m2\nAs)\nm2\nAs/parenrightBigg2\n, (4.11)\nλr=λ−(1+λ)iΠ(1)\n1(m2\nAs)\nm2\nAs. (4.12)\nwhere the subscript rindicates a renormalized quantity again to avoid confusion. The\nequations (4.8) and (4.9) include Nrin whichλrobeys Eq. (4.12) and is generally nonva-\nnishing. Thus the extra-dimensional Lorentz invariance is violated in a generic region in\nthe parameter space at high energy scales. Especially λ= 0 does not mean λr= 0.\n8To identify the effect of the violation of translation invariance due to the brane, we\nconsider the limit L→ ∞. For this limit, the factor z3approaches zero, z3→0 asL−1\nso that the vacuum polarization become Eqs. (4.7), (4.8) and (4.9). Thenλris given in\nEq. (4.12). In the representation (4.12), the limit yields Π(1)\n1(m2\nAs)→0 asL−3. Thus the\ncouping constant for L→ ∞isλr→λ. Therefore the infinite compactification radius\nand zero original λcan recover the higher-dimensional Lorentz invariance.\nNow we move on to the issue of Ward identity. We compare the Lorent z violating case\nwith a simple extension of the four-dimensional quantum electrodyn amics. In a simple\nextension, the vacuum polarization has the form ΠMN\n5D= (pMpN−pLpLηMN)Π. This is\ndecomposed as\nΠµν\n5D=/bracketleftbig\n−/parenleftbig\np2ηµν−pµpν/parenrightbig\n+p2\n5ηµν/bracketrightbig\nΠ,Πµ5\n5D= (pµp5)Π,Π55\n5D=p2Π.(4.13)\nwhich satisfy the identities,\npµΠµν\n5D+p5Π5ν\n5D= 0, pµΠµ5\n5D+p5Π55\n5D= 0. (4.14)\nOn the other hand, the asymptotic vacuum polarization given in Eq. ( 4.7), (4.8) and (4.9)\nhave the relation\npµΠµν\nas+imAsΠ5ν\nas= 0, pµΠµ5\nas−imAsΠ55\nas= 0. (4.15)\nFrom the correspondence Πµν\nas↔Πµν\n5D, Πµ5\nas↔iΠµ5\n5Dand Π55\nas↔Π55\n5D, we find that the\none-loop vacuum polarizations satisfy the five-dimensional Ward ide ntity even without\npreserving the five-dimensional Lorentz invariance.\n5 Furry’s theorem on orbifolds\nSo far we have examined the properties of the vacuum polarizations with an explicit\ndiagrammatic calculation. In this section, we give a formal aspect in h igher-dimensional\ngauge theory.\nIn the four-dimensional electrodynamics, the charge conjugatio n is a symmetry of the\ntheory,C|Ω/an}bracketri}ht=|Ω/an}bracketri}ht, whereCdenotes the charge conjugation and |Ω/an}bracketri}htis the vacuum state.\nThe electromagnetic current, jµ=¯ψγµψchanges sign under the charge conjugation,\nCjµ(x)C†=−jµ(x) so that its vacuum expectation value is vanishing, /an}bracketle{tΩ|Tjµ(x)|Ω/an}bracketri}ht= 0.\nFurry’s theorem states that any vacuum vacuum expectation valu e of an odd number of\nelectromagnetic currents is vanishing.\nNow we consider a two-current function Mµ\nY≡ /an}bracketle{tTjµ(x1)jY(x2)/an}bracketri}htby introducing an-\nother operator jY=¯ψψand by imaging gauge and Yukawa interactions for external lines.\nHere the ground state of the free theory with the symmetry of th e charge conjugation is\ndenoted as /an}bracketri}ht. Because of the charge conjugation CjY(x)C†= +jY(x), the two-current\nfunction Mµ\nYis vanishing. At the first sight, the function Mµ\nYwith gauge and Yukawa\ninteractions seems to look like the vacuum polarizations Πµ5and Πµ5\n5D. On the other hand,\nthe vacuum polarization Πµ5\n5Dis not vanishing for nonzero Πµν\n5Das seen from Eq. (4.13).\nWe have also explicitly derived a nonzero Πµ5. Thus the structure of Πµ5needs to be\nclarified from the viewpoint of Furry’s theorem.\n9The one-loop two-point function for Aµj(x) andAys(w) is given by\nNg2\n2Lδjs/integraldisplay\nd4x1d4x2Dj,µρ(x−x1)Ds(w−x2)/an}bracketle{tTOρ\nY(x1,x2)/an}bracketri}ht, (5.1)\nwith the two-current operator\nOρ\nY(x1,x2)≡jρ\ns,0(x1)j0,s(x2)−jρ\n0,s(x1)js,0(x2)\n−jρ\nn,ℓ(x1)jℓ,n(x2)(δn+s,ℓ−δn,s+ℓ)−jρ5\nn,ℓ(x1)j5\nℓ,n(x2)δn+ℓ,s.(5.2)\nHere the currents with zero mode are given by jρ\nI,J=¯ψIγρψJandjI,J=¯ψIψJ, where\nI,J= 0,1,···,∞and the currents with γ5are given by jρ5\nn,ℓ=¯ψnγρiγ5ψℓandj5\nℓ,n=\n¯ψℓiγ5ψn. The charge conjugation yields a change of the overall factor and the interchange\nof indices,\nCjρ\nI,J(x)C†=−jρ\nJ,I(x), Cj I,J(x)C†= +jJ,I(x), (5.3)\nCjρ5\nn,ℓ(x)C†= +jρ5\nℓ,n(x), Cj5\nn,ℓ(x)C†= +jρ5\nℓ,n(x). (5.4)\nFrom these equations, we obtain COρ\nY(x1,x2)C†= +Oρ\nY(x1,x2). Therefore that Πµ5is\nnot necessarily zero is consistent with Furry’s theorem. In Eq. (2.2 8),¯ψnPLAynψ0and\n¯ψ0PRAynψnhave relative sign. If they have the same sign, the contribution fro m the\nfirst line in Eq. (5.2) would vanish. The role of the relative sign in the ter m¯ψnAymψℓin\nEq. (2.28) is similar.\nApplication of Furry’s theorem in orbifold models may be given not only f or two-\npoint functions but also for other functions. For example, the vac uum expectation\nvalue of one current is vanishing, /an}bracketle{tTjµ\nn,n(x)/an}bracketri}ht= 0, where the indices are the identical\nn. Because a nonzero Πµ5is expected from a nonzero ΠMN\n5D, Furry’s theorem in effec-\ntive four-dimensional theory may be related to the discrete symme try in the original\nhigher-dimensional theory. Due to the dependence of Lorentz tr ansformation on the di-\nmensionality of spacetime, it is nontrivial to introduce discrete symm etry such as P,C,T\nin higher-dimensional theory [17, 18]. We leave further exploration o f this issue for future\nwork.\n6 Conclusion\nWe have studied the momentum dependence of Lorentz violating ter ms in the field-\ntheoreticalcontext inelectrodynamicsonorbifolds. Hereanexplic it analysishasbeenper-\nformed for loop diagrams and renormalization. We have found that t he extra-dimensional\nLorentz invariance is violated in a generic region in the parameter spa ce at high energy\nscales. In particular, even if the original action is higher-dimensiona l Lorentz invariant,\nit is violated by loop effects. While the higher-dimensional Lorentz inva riance is lost,\na higher-dimensional Ward identity has been found to be fulfilled for t he one-loop vac-\nuum polarization. Therefore higher-dimensional gauge invariance m ay be prior to higher-\ndimensional Lorentz invariance as a guiding principle in a high-energy fi eld theory. We\nhave also discussed Furry’s theorem in orbifold models to confirm the consistency about\nthe vacuum polarizations.\n10The four-dimensional Lorentz violation has also been studied as a dis tinct topic of\nLorentz violation. In the four-dimensional electrodynamics with Lo rentz violation, it\nhas been discussed that Pauli-Villars regularization is a useful choice associated with\ngauge invariance [19, 20, 21]. On the other hand, it has been shown t hat propagators\ncorresponding to Pauli-Villars are radiatively generated in an orbifold model [22]. In\nthis light, the Pauli-Villars regulator may be the necessity of an extra -dimensional model\nrather than a choice. These relations should be examined further.\nAcknowledgments\nThis work is supported by Scientific Grants from the Ministry of Educ ation and Science,\nGrant No. 20244028.\n11A Loop corrections\nIn this appendix, the details of loop corrections are given.\nA.1 Diagrams and four-momentum integrals\nWe evaluate loop corrections by calculating the sum of diagrams for e ach Kaluza-Klein\nmode. Propagators are defined for four-dimensional fields. The t ree-level propagators are\ndiagonal with respect to Kaluza-Klein modes and are given by\nDµν(x−w) =/an}bracketle{tTAµ\n0(x)Aν\n0(w)/an}bracketri}ht=/integraldisplayd4p\n(2π)4−iηµν\np2+iǫe−ip·(x−w)(A.1)\nDµν\nn(x−w) =/an}bracketle{tTAµ\nn(x)Aν\nn(w)/an}bracketri}ht=/integraldisplayd4p\n(2π)4−iηµν\np2−m2\nAn+iǫe−ip·(x−w)(A.2)\nDn(x−w) =/an}bracketle{tTAyn(x)Ayn(w)/an}bracketri}ht/integraldisplayd4p\n(2π)4i\np2−m2\nAn+iǫe−ip·(x−w),(A.3)\nfor bosons and\nS(x−w) =/an}bracketle{tTψ0(x)¯ψ0(w)/an}bracketri}ht=/integraldisplayd4p\n(2π)4PLip /\np2+iǫe−ip·(x−w)(A.4)\nSn(x−w) =/an}bracketle{tTψn(x)¯ψn(w)/an}bracketri}ht=/integraldisplayd4p\n(2π)4i(p /+mψn)\np2−m2\nψn+iǫe−ip·(x−w),(A.5)\nfor fermions.\nThe vacuum polarizations for AµandAyinvolve the following momentum integrals:\nI1(µ,ν;m1,m2)≡ −g2\n2L/integraldisplayd4p1\n(2π)4tr/parenleftbiggp /1\np2\n1−m2\n1γµp /1+p /2\n(p1+p2)2−m2\n2γν/parenrightbigg\n,(A.6)\nI2(µ,ν;m1,m2)≡ −g2\n2L/integraldisplayd4p1\n(2π)4tr/parenleftbiggm1\np2\n1−m2\n1γµm2\n(p1+p2)2−m2\n2γν/parenrightbigg\n,(A.7)\nfor two four-indices,\nI1(µ;m1,m2)≡ −g2\n2L/integraldisplayd4p1\n(2π)4tr/parenleftbiggm1\np2\n1−m2\n1γµp /1+p /2\n(p1+p2)2−m2\n2/parenrightbigg\n,(A.8)\nI2(µ;m1,m2)≡ −g2\n2L/integraldisplayd4p1\n(2π)4tr/parenleftbiggp /1\np2\n1−m2\n1γµm2\n(p1+p2)2−m2\n2/parenrightbigg\n,(A.9)\nfor one four-index and\nI1(m1,m2)≡g2\n2L/integraldisplayd4p1\n(2π)4tr/parenleftbiggp /1\np2\n1−m2\n1p /1+p /2\n(p1+p2)2−m2\n2/parenrightbigg\n, (A.10)\nI2(m1,m2)≡g2\n2L/integraldisplayd4p1\n(2π)4tr/parenleftbiggm1\np2\n1−m2\n1m2\n(p1+p2)2−m2\n2/parenrightbigg\n, (A.11)\nfor no four-indices. These satisfy a property I1(µ;m2,m1) =−I2(µ;m1,m2).\n12The self-energies for ψinvolve the following momentum integrals:\nB1(m1,m2)≡ −g2\n2L/integraldisplayd4p2\n(2π)4γµ1\np2\n2−m2\n1p /1−p /2\n(p1−p2)2−m2\n2γµ, (A.12)\nB2(m1,m2)≡ −g2\n2L/integraldisplayd4p2\n(2π)4γµ1\np2\n2−m2\n1m2\n(p1−p2)2−m2\n2γµ, (A.13)\nE1(m1,m2)≡ −g2\n2L/integraldisplayd4p2\n(2π)41\np2\n2−m2\n1p /1−p /2\n(p1−p2)2−m2\n2, (A.14)\nE2(m1,m2)≡ −g2\n2L/integraldisplayd4p2\n(2π)41\np2\n2−m2\n1m2\n(p1−p2)2−m2\n2. (A.15)\nWith these integral expressions, the vacuum polarizations are sum marized as follows:\n0 0\n=∞/summationdisplay\nn=−∞{I1(µ,ν;mψn,mψn)+I2(µ,ν;mψn,mψn)}, (A.16)\nj s\n=∞/summationdisplay\nn=−∞{I1(µ,ν;mψn,mψ,n+s)+I2(µ,ν;mψn,mψ,n+s)}δjs,(A.17)\nj s\n=N∞/summationdisplay\nn=−∞{I1(µ;mψn,mψ,n+j)+I2(µ;mψn,mψ,n+j)}δjs,(A.18)\nj s\n=N2∞/summationdisplay\nn=−∞{I1(mψn,mψ,n+j)+I2(mψn,mψ,n+j)}δjs.(A.19)\nThe vacuum polarizations for AµandAydo not give rise to one-loop corrections for brane\nterms. The Kaluza-Klein modes for external lines are diagonal.\nThe fermion self-energies are summarized as follows:\n0 0=∞/summationdisplay\nn=−∞B1(mAn,mψn)PL+B1(0,0)PL, (A.20)\n0s=√\n2∞/summationdisplay\nn=1{B1(mAn,mψn)PL+B2(mAn,mψn)PR}δ2n,s,(A.21)\ns 0=√\n2∞/summationdisplay\nn=1{B1(mAn,mψn)+B2(mAn,mψn)}PLδ2n,s, (A.22)\nj s={B1(0,mψs)+B2(0,mψs)}δjs\n13+B1(mAj,0)iγ5δjs\n+∞/summationdisplay\nn=1{B1(mAn,mψ,j+n)+B2(mAn,mψ,j+n)}δj+2n,s\n+∞/summationdisplay\nn=1{B1(mAn,mψ,n+s)+B2(mAn,mψ,n+s)}δj,s+2n\n+∞/summationdisplay\nℓ=1{B1(mA,ℓ+s,mψℓ)+B2(mA,ℓ+s,mψℓ)}iγ5δj,s+2ℓ\n+∞/summationdisplay\nℓ=1{B1(mA,j+ℓ,mψℓ)−B2(mA,j+ℓ,mψℓ)}iγ5δj+2ℓ,s\n+∞/summationdisplay\nn=−∞{B1(mAn,mψ,j+n)+B2(mAn,mψ,j+n)}δjs,(A.23)\n0 0=N2/bracketleftBigg∞/summationdisplay\nn=−∞E1(mAn,mψn)PL−E1(0,0)PL/bracketrightBigg\n, (A.24)\n0s=−√\n2N2∞/summationdisplay\nn=1{E1(mAn,mψn)PL+E2(mAn,mψn)PR}δ2n,s,(A.25)\ns 0=−√\n2N2∞/summationdisplay\nn=1{E1(mAn,mψn)+E2(mAn,mψn)}PLδ2n,s,(A.26)\nj s=N2/bracketleftBig\niγ5E1(mAj,0)δjs\n−∞/summationdisplay\nn=1{E1(mAn,mψ,n+s)+E2(mAn,mψ,n+s)}δj,s+2n\n−∞/summationdisplay\nn=1{E1(mAn,mψ,j+n)+E2(mAn,mψ,j+n)}δj+2n,s\n+∞/summationdisplay\nℓ=1iγ5{E1(mA,j+ℓ,mψℓ)+E2(mA,j+ℓ,mψℓ)}δj+2ℓ,s\n+∞/summationdisplay\nℓ=1iγ5{E1(mA,ℓ+s,mψℓ)−E2(mA,ℓ+s,mψℓ)}δj,s+2ℓ\n+∞/summationdisplay\nn=−∞{E1(mAn,mψ,j+n)+E2(mAn,mψ,j+n)}δjs\n−{E1(0,mψj)+E2(0,mψj)}δjs/bracketrightBig\n. (A.27)\nFor (E1+E2), the mode sum with −∞ ≤n≤ ∞is regarded as a formal equation because\n14Aynhas no zero mode.\nA.2 Evaluation of momentum integrals\nWe calculate themomentum integrals by introducing Feynman parame ters andemploying\nthe dimensional regularization and the Poisson resummation.\nThe momentum integrals for Π µνare given by\n∞/summationdisplay\nn=−∞(I1(µ,ν;mψn,mψ,n+s)+I2(µ,ν;mψn,mψ,n+s))\n=8ig2\n(4π)2(1+k)/integraldisplay1\n0dx\n\n\nz4−∞/summationdisplay\nnp=1z3e−2z4\nz3·cos(2πnpxs)\n\n×x(1−x)/parenleftbig\n(p2\n2−m2\nψs)ηµν−p2µp2ν/parenrightbig\n−1\n4∞/summationdisplay\nnp=1z3(z3+2z4)e−2z4\nz3(1−2x)mψssin(2πnpxs)ηµν\n\n.(A.28)\nFors= 0, thefour-dimensionalWardidentity issatisfied. Itisalsoseenfr omthefollowing\nequation,\nI1(µ,ν;m1,m2)+I2(µ,ν;m1,m2) =−8g2\nL/integraldisplay1\n0dx/integraldisplayddℓ\n(2π)d1\n[ℓ2−∆]2\n×/bracketleftbigg\nx(1−x)(p2\n2ηµν−p2µp2ν)+1\n2(m1−m2)(xm2−(1−x)m1)ηµν/bracketrightbigg\n,(A.29)\nwhereℓ=p1+xp2and ∆ =xm2\n2+(1−x)m2\n2−x(1−x)p2\n2. In the main text, the letter of\nthe external momentum is denoted as pinstead ofp2. The momentum integrals for Π µ5\nare given by\n∞/summationdisplay\nn=−∞(I1(µ;mψn,mψ,n+s)+I2(µ;mψn,mψ,n+s))\n=−8ig2\n(4π)2(1+k)p2µ/integraldisplay1\n0dx\n\n\nz4−∞/summationdisplay\nnp=1z3e−2z4\nz3cos(2πnpxs)\nx(1−x)mψs\n+1\n4∞/summationdisplay\nnp=1z3(z3+2z4)e−2z4\nz3(1−2x)sin(2πnpxs)\n\n. (A.30)\nThe momentum integrals for Π 55are given by\n∞/summationdisplay\nn=−∞(I1(mψn,mψ,n+s)+I2(mψn,mψ,n+s))\n=8ig2\n(4π)2(1+k)/integraldisplay1\n0dx/braceleftbig\nz4x(1−x)p2\n2\n15−1\n4∞/summationdisplay\nnp=1/bracketleftbig\n3z2\n3(z3+2z4)+2z3(2x(1−x)m2\nψs)/bracketrightbig\ne−2z4\nz3cos(2πnpxs)\n+1\n4∞/summationdisplay\nnp=1z3(z3+2z4)e−2z4\nz3(1−2x)mψssin(2πnpxs)\n\n. (A.31)\nFor fermion self-energies, the momentum integrals with −∞ ≤n≤ ∞are given by\n∞/summationdisplay\nn=−∞(B1(mAn,mψ,n+s)+B2(mAn,mψ,n+s)) =−4ig2\n(4π)2/integraldisplay1\n0dx1√w\n×\n\n\nw2−∞/summationdisplay\nnp=1w1e−2w2\nw1cos/parenleftbigg\n2πnp(1+k)2xs\nw/parenrightbigg\n(1−x)/parenleftbigg\np /1−2(1+λ)mψs\nw/parenrightbigg\n+∞/summationdisplay\nnp=1(1+k)√ww1(w1+2w2)e−2w2\nw1sin/parenleftbigg\n2πnp(1+k)2xs\nw/parenrightbigg\n\n. (A.32)\nHere\nw1≡√w\nnpL, w 2≡/radicalBigg\nx(1−x)/parenleftbigg(1+λ)\nwm2\nψs−p2\n1/parenrightbigg\n, (A.33)\nw≡x(1+k)2+(1−x)(1+λ). (A.34)\nThe momentum integrals for E1andE2are obtained as with the relations\nE1(m1,m2) =−1\n2B1(m1,m2), E 2(m1,m2) =1\n4B2(m1,m2). (A.35)\n16References\n[1] J. Scherk and J. H. Schwarz, Phys. Lett. B 82, 60 (1979).\n[2] J. Scherk and J. H. Schwarz, Nucl. Phys. B 153, 61 (1979).\n[3] Y. Hosotani, Phys. Lett. B 126, 309 (1983).\n[4] Y. Hosotani, Annals Phys. 190, 233 (1989).\n[5] E. A. Mirabelli and M. E. Peskin, Phys. Rev. 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D 80, 055025 (2009) [arXiv:0905.1022\n[hep-ph]].\n[18] C. S. Lim, N. Maru and K. Nishiwaki, arXiv:0910.2314 [hep-ph].\n[19] R. Jackiw and V. A. Kostelecky, Phys. Rev. Lett. 82, 3572 (1999)\n[arXiv:hep-ph/9901358].\n[20] M. Perez-Victoria, JHEP 0104, 032 (2001) [arXiv:hep-th/0102021].\n[21] B. Altschul, Phys. Rev. D 70, 101701 (2004) [arXiv:hep-th/0407172].\n[22] N. Uekusa, Nucl. Phys. B 827, 311 (2010) [arXiv:0909.0825 [hep-ph]].\n17"    },    {        "title": "1501.04551v2.Relativistic_Lagrangians_for_the_Lorentz_Dirac_equation.pdf",        "content": "arXiv:1501.04551v2  [physics.class-ph]  13 Jun 2015Relativistic Lagrangians for the Lorentz-Dirac equation\nShinichi Deguchia,∗, Kunihiko Nakanoa, Takafumi Suzukib\naInstitute of Quantum Science, College of Science and Techno logy, Nihon University,\nChiyoda-ku, Tokyo 101-8308, Japan\nbJunior College Funabashi Campus, Nihon University, Narash inodai, Funabashi, Chiba\n274-8501, Japan\nAbstract\nWe present two types of relativistic Lagrangians for the Lorentz- Dirac equa-\ntion written in terms of an arbitrary world-line parameter. One of th e La-\ngrangians contains an exponential damping function of the proper time and\nexplicitly depends ontheworld-line parameter. Another Lagrangian includes\nadditional cross-terms consisting of auxiliary dynamical variables a nd does\nnotdependexplicitlyontheworld-lineparameter. Wedemonstratet hatboth\nthe Lagrangians actually yield the Lorentz-Dirac equation with a sou rce-like\nterm.\n1. Introduction\nA charged particle emitting electromagnetic radiation is subjected t o the\nreaction force caused by the particle’s own electromagnetic radiat ion. This\nphenomenon is well-known as the radiation reaction [1, 2, 3, 4, 5, 6]. It was\nfirst evaluated by Lorentz at the end of the 19th century [7] and s ubsequently\nargued by Abraham and Lorentz on the basis of the charged rigid sp here\nmodel of a charged particle [8, 9]. In the zero radius limit that this mod el\ntends to become a point charge, the classical non-relativistic equa tion of\nmotion for the charged particle located at the position x=x(t) is found to\nbe\nmd2x\ndt2=F+2\n3e2d3x\ndt3, (1.1)\n∗Corresponding author.\nEmail address: deguchi@phys.cst.nihon-u.ac.jp (Shinichi Deguchi)\nPreprint submitted to Elsevier April 4, 2018wheremis the physical mass of the particle, eits electric charge, and F\ndenotes the external Lorentz force. (In this paper, we employ u nits such\nthatc= 1.) Equation (1.1) is called the Lorentz-Abraham equation (or\nthe Abraham-Lorentz equation). A relativistic extension of the Lo rentz-\nAbraham equation was derived by Dirac in a manifestly covariant mann er\nby considering energy-momentum conservation [10], and is now oft en called\nthe Lorentz-Dirac equation [3, 6, 11, 12, 13]. With the spacetime co ordinates\nxµ=xµ(l) (µ= 0,1,2,3) of a charged particle propagating in 4-dimensional\nMinkowski space, the Lorentz-Dirac equation reads\nmduµ\ndl=eFµν(x)uν+2\n3e2/parenleftbig\nδµ\nν−uµuν/parenrightbigd2uν\ndl2. (1.2)\nHere,uµ:=dxµ/dl,Fµνisthefieldstrengthtensor ofanexternal electromag-\nnetic field, and ldenotes the proper time of the particle or, in other words,\nthe arc length of the world-line traced out by the particle. The metr ic ten-\nsor of Minkowski space is assumed to be ηµν= diag(1 ,−1,−1,−1), so that\nuµuµ=ηµνuµuν= 1 holds. Equations (1.1) and (1.2) are unusual ones in-\ncluding third-order time derivatives of the particle’s position coordin ates. In\nconnectionwiththisfact, theseequationsadmitphysicallyunaccep tablesolu-\ntions such as runaway and pre-acceleration solutions [1, 2, 4, 5, 6, 12, 13, 14].\nTo overcome this problem, various ideas have been proposed until r ecently\n[1, 6, 13, 14, 15, 16, 17, 18, 19, 20, 21]; however, it seems that an ultimate\nsolution to the problem has not been found yet.\nOnce the equations of motion (1.1) and (1.2) have been obtained, it is\nquite natural to seek Lagrangians corresponding to these equat ions in or-\nder to develop the Lagrangian and Hamiltonian formulations of a char ged\nparticle subjected to the radiation reaction force. If these form ulations are\nestablished, they might lead to a novel quantum-mechanical descr iption of a\ncharged particle undergoing radiation reaction and might give us new room\nto deal with the above-mentioned problem. As far as the present a uthors\nknow, there have been a few attempts to construct Lagrangians correspond-\ning to Eqs. (1.1) and (1.2) until now [22, 23, 24]. Carati constructe d an\nexplicitly time-dependent Lagrangian for Eq (1.1) with the use of aux iliary\ndynamical variables [22]. (Carati also considered a relativistic exten sion of\nthis Lagrangian in an extremely limited case.) Barone and Mendes deriv ed\nan explicitly time-independent Lagrangian for Eq. (1.1) by incorpora ting\nthe time-reversed copy of Eq. (1.1) into the original setting [23]. Ca rati’s\nand Barone-Mendes’s approaches are, respectively, based on lea rning from\n2the direct and indirect Lagrangian formulations of the damped harm onic os-\ncillator [25, 26, 27, 28, 29, 30].1It should be pointed out here that in these\napproaches, the external Lorentz force Fis assumed to be independent of\nthe velocity v:=dx/dt. Hence, it follows that in actuality, Carati’s and\nBarone-Mendes’s Lagrangians can describe only a charged particle being in\nthe particular situation in which the magnetic field vanishes or is paralle l to\nv.2\nIn this paper, we present two types of Lagrangians for the Loren tz-Dirac\nequation (1.2) that are constructed in such a fashion that the cor responding\nactions remain invariant under reparametrization of a world-line par ameter\nalong the particle’s world-line. These Lagrangians are completely rela tivis-\ntic and admit the general form of the external Lorentz force. Als o, the\nLagrangians are outside the scope of Kupriyanov’s proof [24], beca use they\ncontain auxiliary dynamical variables in addition to xµ. One of the La-\ngrangians contains an exponential damping function of the proper timel,\nwhile another Lagrangian includes additional cross-terms consistin g of two\nauxiliary dynamical variables. Both the Lagrangians include terms sim ilar\nto what can be seen in the Lagrangian that governs a certain model of a\nrelativistic point particle with rigidity [31, 32, 33]. We would like to em-\nphasize that our Lagrangians are not immediate extensions of Cara ti’s and\nBarone-Mendes’s Lagrangians.\nThis paper is organized as follows. In section 2, we introduce necess ary\ndynamicalvariablesanddefinetheirtransformationrulesunderre parametriza-\ntion of a world-line parameter. In section 3, we present a Lagrangia n that\ncontains an exponential damping function and show that the Lagra ngian ac-\ntually yields the Lorentz-Dirac equation with a source-like term. In s ection\n4, we consider a Lagrangian including additional cross-terms, inste ad of the\n1Thedirectformulationadoptsanexplicitlytime-dependent Lagrang ianofthedamped\nharmonic oscillator [25, 26, 27], while the indirect formulation adopts a n explicitly time-\nindependent Lagrangian for a system consisting of the damped har monic oscillator and its\ntime-reversed counterpart [25, 28, 29, 30].\n2In Ref. [24], Kupriyanov investigated the possibility of constructing Lagrangians\ncorresponding to Eqs. (1.1) and (1.2) and reached the conclusion t hat there exist no\ncorresponding Lagrangians. However, Kupriyanov’s proof of this conclusion considers\nLagrangians consisting only of the coordinate variables, such as xandxµ, and their\nfirst- and second-order time derivatives. Since Carati’s and Baron e-Mendes’s Lagrangians\ncontainextradynamicalvariables,theseLagrangiansareoutside thescopeofKupriyanov’s\nproof.\n3exponential damping function, and show that this Lagrangian also y ields the\nLorentz-Dirac equation with a source-like term. Section 5 is devote d to a\nsummary and discussion. Appendix A provides the Lorentz-Dirac eq uation\nwritten in terms of an arbitrary world-line parameter instead of the proper\ntimel.\n2. Preliminaries: dynamical variables andtheir transform ation rules\nLetτ(τ0≤τ≤τ1) be an arbitrary world-line parameter along the\nparticle’s world-line, being chosen in such a manner that dx0/dτ >0. The\nspacetime coordinates of a charged particle are now denoted as xµ=xµ(τ).\nUnder the reparametrization τ→τ′=τ′(τ) (dτ′/dτ >0), the coordinate\nvariables xµbehave as scalar fields on the 1-dimensional parameter space\nT:={τ|τ0≤τ≤τ1}:\nxµ(τ)→x′µ(τ′) =xµ(τ). (2.1)\nIn addition to xµ, we introduce auxiliary dynamical variables qµ\ni=qµ\ni(τ),\nλiµ=λiµ(τ) (i= 1,2), andξµ=ξµ(τ). They are assumed to transform\nunder the reparametrization as scalar-density fields of weight 1 on T:\nqµ\ni(τ)→q′µ\ni(τ′) =dτ\ndτ′qµ\ni(τ), (2.2)\nλiµ(τ)→λ′\niµ(τ′) =dτ\ndτ′λiµ(τ), (2.3)\nξµ(τ)→ξ′\nµ(τ′) =dτ\ndτ′ξµ(τ). (2.4)\nThe components of the vector resolute of the 4-vector (˙ qµ\ni) perpendicular to\n(qµ\ni) are given by\n˙qµ\ni⊥:= ˙qµ\ni−qi˙qi\nq2\niqµ\ni, (2.5)\nwhere ˙qµ\ni:=dqµ\ni/dτ,q2\ni:=qiµqµ\ni, andqi˙qi:=qiµ˙qµ\ni(no sum with respect to\ni). It can be shown by using Eq. (2.2) that unlike ˙ qµ\ni, the components ˙ qµ\ni⊥\ntransform homogeneously as\n˙qµ\ni⊥(τ)→˙q′µ\ni⊥(τ′) =/parenleftbiggdτ\ndτ′/parenrightbigg2\n˙qµ\ni⊥(τ). (2.6)\nWe thus see that under the reparametrization, ˙ qµ\ni⊥behave as scalar-density\nfields of weight 2 on T.\n43. A Lagrangian with an exponential damping function\nNow, from the dynamical variables xµ,qµ\ni,λiµ, andξµ, we construct the\nfollowing Lagrangian:\nLD=exp(−kl)\n(q2\n1q2\n2)1/4/bracketleftbigg1\n2/parenleftbigg˙q2\n1⊥\nq2\n1−˙q2\n2⊥\nq2\n2/parenrightbigg\n−λ1µ(qµ\n1−˙xµ)+λ2µ(qµ\n2−˙xµ)+ξµ(qµ\n1−qµ\n2)−3\n2eFµν(x)qµ\n1qν\n2/bracketrightbigg\n,\n(3.1)\nwherek:= 3m/2e2, ˙xµ:=dxµ/dτ, ˙q2\ni⊥:= ˙qi⊥µ˙qµ\ni⊥, andFµν(=−Fνµ) is again\nthe field strength tensor of an external electromagnetic field. In Eq. (3.1),\nthe proper time lis a function of τrepresented as\nl(τ) =/integraldisplayτ\nτ0d˜τ/radicalBig\n˙xµ(˜τ)˙xµ(˜τ). (3.2)\nHere,xµ(˜τ) is understood as a solution of the equation of motion for xµ\nobtained later, not as a dynamical variable whose variation is taken in to\naccount in varying the action\nSD=/integraldisplayτ1\nτ0dτLD. (3.3)\nThe Lagrangian LDexplicitly depends on τvia the exponential damping\nfunction exp( −kl). Sincel(τ) is geometrically the arc length of the particle’s\nworld-line, it is certainly reparametrization invariant.3Considering this fact\nand using the transformation rules in Eqs. (2.1), (2.2), (2.3), (2.4) , and (2.6),\nwe can show that the action SDremains invariant under the reparametriza-\ntionτ→τ′. We also see that LDremains invariant under the gauge trans-\nformation\nλ1µ→λ′\n1µ=λ1µ+θµ, λ2µ→λ′\n2µ=λ2µ+θµ, ξµ→ξ′\nµ=ξµ+θµ,\n(3.4)\n3Strictly speaking, l(τ) is a functional of xµas well as a function of τandτ0. In this\nsense,l(τ) should be read as l(τ,τ0;xµ). The reparametrization invariance of l(τ) can be\nexpressed as l(τ′,τ′\n0;x′µ) =l(τ,τ0;xµ).\n5with real gauge functions θµ=θµ(τ). The Lagrangian LDhas the antisym-\nmetric property\nLD(qµ\n1,˙qµ\n1,λ1µ;qµ\n2,˙qµ\n2,λ2µ) =−LD(qµ\n2,˙qµ\n2,λ2µ;qµ\n1,˙qµ\n1,λ1µ).(3.5)\nLet us derive the Euler-Lagrange equations for the dynamical var iables\nfromLD. Noting that xµ(˜τ) contained in l(τ) and hence l(τ) itself are not\nobjectsfortakingvariation,wecaneasilyobtaintheEuler-Lagran geequation\nforxµ:\nd\ndτ/bracketleftBigg\nexp(−kl)\n(q2\n1q2\n2)1/4(λ1µ−λ2µ)/bracketrightBigg\n+3exp(−kl)\n2e(q2\n1q2\n2)1/4∂µFνρ(x)qν\n1qρ\n2= 0.(3.6)\nThisequationincludesthegauge-invariantquantity λ1µ−λ2µasareflectionof\nthe gauge invariance of LD. Hence, λ1µandλ2µthemselves are not uniquely\ndetermined. The Euler-Lagrange equation for qµ\n1can be written as\nexp(−kl)\n(q2\n1q2\n2)1/4/bracketleftbigg1\n2/parenleftbiggd\ndτ∂K1\n∂˙qµ\n1−∂K1\n∂qµ\n1/parenrightbigg\n+λ1µ−ξµ+3\n2eFµν(x)qν\n2/bracketrightbigg\n+/parenleftBigg\nd\ndτexp(−kl)\n(q2\n1q2\n2)1/4/parenrightBigg\n1\n2∂K1\n∂˙qµ\n1+q1µ\n2q2\n1LD= 0, (3.7)\nwith\nK1:=˙q2\n1⊥\nq2\n1=q2\n1˙q2\n1−(q1˙q1)2\n(q2\n1)2. (3.8)\nApplying the formulas\n1\n2∂K1\n∂˙qµ\n1=˙q1⊥µ\nq2\n1=1/radicalbig\nq2\n1d\ndτq1µ/radicalbig\nq2\n1, (3.9)\nd\ndτ∂K1\n∂˙qµ\n1−∂K1\n∂qµ\n1=2\nq2\n1/parenleftbigg\n¨q1⊥µ−2q1˙q1\nq2\n1˙q1⊥µ/parenrightbigg\n, (3.10)\nd\ndτ1\n(q2\n1q2\n2)1/4=−1\n2(q2\n1q2\n2)1/4/parenleftbiggq1˙q1\nq2\n1+q2˙q2\nq2\n2/parenrightbigg\n(3.11)\nto Eq. (3.7) appropriately, we obtain\nkdl\ndτ1/radicalbig\nq2\n1d\ndτq1µ/radicalbig\nq2\n1=3\n2eFµν(x)qν\n2+(q2\n1q2\n2)1/4q1µ\n2q2\n1exp(−kl)LD\n+¨q1⊥µ\nq2\n1−/parenleftbigg5q1˙q1\nq2\n1+q2˙q2\nq2\n2/parenrightbigg˙q1⊥µ\n2q2\n1+λ1µ−ξµ.(3.12)\n6Here, ¨q1⊥µ, together with ¨ q2⊥µ, is defined by\n¨qi⊥µ:= ¨qiµ−qi¨qi\nq2\niqiµ, (3.13)\nwhere ¨qiµ:=d2qiµ/dτ2andqi¨qi:=qiµ¨qµ\ni(nosumwithrespectto i). Following\nthe same procedure as that used for deriving Eq. (3.12), we can de rive the\nEuler-Lagrange equation for qµ\n2as\nkdl\ndτ1/radicalbig\nq2\n2d\ndτq2µ/radicalbig\nq2\n2=3\n2eFµν(x)qν\n1−(q2\n1q2\n2)1/4q2µ\n2q2\n2exp(−kl)LD\n+¨q2⊥µ\nq2\n2−/parenleftbiggq1˙q1\nq2\n1+5q2˙q2\nq2\n2/parenrightbigg˙q2⊥µ\n2q2\n2+λ2µ−ξµ.(3.14)\nThe Euler-Lagrange equations for λ1µ,λ2µ, andξµare respectively found to\nbe\nqµ\n1= ˙xµ, (3.15)\nqµ\n2= ˙xµ, (3.16)\nqµ\n1=qµ\n2. (3.17)\nEquation (3.17) can also be found from Eqs. (3.15) and (3.16).\nSubstituting Eqs. (3.15) and (3.16) into Eq. (3.12) and noting\nLD(qµ\n1,˙qµ\n1,λ1µ;qµ\n2,˙qµ\n2,λ2µ) =LD(˙xµ,¨xµ,λ1µ; ˙xµ,¨xµ,λ2µ) = 0,(3.18)\nwe have\nkdl\ndτ1√\n˙x2d\ndτ˙xµ\n√\n˙x2=3\n2eFµν(x)˙xν+...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2+λµ\n1−ξµ,(3.19)\nwhere ¨xµ:=d2xµ/dτ2and...xµ:=d3xµ/dτ3. Similarly, substituting Eqs.\n(3.15) and (3.16) into Eq. (3.14) and using (3.18), we have\nkdl\ndτ1√\n˙x2d\ndτ˙xµ\n√\n˙x2=3\n2eFµν(x)˙xν+...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2+λµ\n2−ξµ.(3.20)\nComparing Eq. (3.19) with Eq. (3.20) leads to\nλµ\n1=λµ\n2. (3.21)\n7This equality is covariant under the gauge transformation (3.4). It follows\nfromEq. (3.21)thatEq. (3.6)isidenticallysatisfied, because ∂µFνρ(x)qν\n1qρ\n2=\n∂µFνρ(x)˙xν˙xρ= 0 holds by using Eqs. (3.15) and (3.16). Hereafter, taking\ninto account Eq. (3.21), we simply write λµ\n1andλµ\n2asλµ. Thereby, Eqs\n(3.19) and (3.20) can be written together as a single equation\nkdl\ndτ1√\n˙x2d\ndτ˙xµ\n√\n˙x2=3\n2eFµν(x)˙xν+...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2+Λµ,(3.22)\nwhereΛµ:=λµ−ξµ. Obviously, Λµis invariant under the gauge trans-\nformation (3.4). Equation (3.22) is precisely the equation of motion f orxµ\nmentioned under Eq. (3.2). Since xµcontained in lhas been assumed to\nbe a solution of Eq. (3.22), it can be identified with xµin Eq. (3.22).\nUpon considering this fact, substituting the τ-derivative of Eq. (3.2), i.e.,\ndl(τ)/dτ=/radicalbig\n˙xµ(τ)˙xµ(τ), into Eq. (3.22) and recalling k:= 3m/2e2, we\nobtain\nmd\ndτ˙xµ\n√\n˙x2=eFµν(x)˙xν+2\n3e2/parenleftbigg...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2/parenrightbigg\n+2\n3e2Λµ.(3.23)\nIfΛµ= 0, Eq. (3.23) is identical with the Lorentz-Dirac equation written\nin terms of the arbitrary world-line parameter τ; see Eq. (A.8) in Appendix\nA. For this reason, Eq. (3.23) can be said to be the Lorentz-Dirac e quation\nwith a source- liketerm 2e2Λµ/3. We thus see that the Lagrangian LDyields\nthe Lorentz-Dirac equation with a source-like term.\nNow we adopt the proper-time gauge τ=l, choosing τto be the proper\ntimel. Accordingly, ˙ xµ=uµ, ˙x2= 1, and ˙ x¨x= 0 are valid, so that Eq.\n(3.23) becomes\nmduµ\ndl=eFµν(x)uν+2\n3e2/parenleftbig\nδµ\nν−uµuν/parenrightbigd2uν\ndl2+2\n3e2Λµ. (3.24)\nThis is exactly what is defined by adding the source-like term 2 e2Λµ/3 to the\n(original) Lorentz-Dirac equation (1.2).\n84. A Lagrangian with additional cross-terms\nNext we consider an alternative Lagrangian defined by\nLA=1\n(q2\n1q2\n2)1/4/bracketleftBigg\n1\n2/parenleftbigg˙q2\n1⊥\nq2\n1−˙q2\n2⊥\nq2\n2/parenrightbigg\n−k\n2/parenleftBigg\n˙q1⊥µqµ\n2/radicalbig\nq2\n1−˙q2⊥µqµ\n1/radicalbig\nq2\n2/parenrightBigg\n−λ1µ(qµ\n1−˙xµ)+λ2µ(qµ\n2−˙xµ)+ξµ(qµ\n1−qµ\n2)−3\n2eFµν(x)qµ\n1qν\n2/bracketrightbigg\n.\n(4.1)\nHere it should be emphasized that LAincludes the additional cross-terms\nproportional to kinstead of the exponential damping function exp( −kl).\nAlso, it is worth noting that unlike LD, the Lagrangian LAdoes not depend\nexplicitly on τ. Using the transformation rules in Eqs. (2.1), (2.2), (2.3),\n(2.4), and (2.6), we can show that the action\nSA=/integraldisplayτ1\nτ0dτLA (4.2)\nremains invariant under the reparametrization τ→τ′. Just like LD, the\nLagrangian LAremains invariant under the gauge transformation (3.4) and\npossesses the antisymmetric property\nLA(qµ\n1,˙qµ\n1,λ1µ;qµ\n2,˙qµ\n2,λ2µ) =−LA(qµ\n2,˙qµ\n2,λ2µ;qµ\n1,˙qµ\n1,λ1µ).(4.3)\nWe now derive the Euler-Lagrange equations for the dynamical var iables\nfromLA. The Euler-Lagrange equation for xµis found to be\nd\ndτ/bracketleftBigg\n1\n(q2\n1q2\n2)1/4(λ1µ−λ2µ)/bracketrightBigg\n+3\n2e(q2\n1q2\n2)1/4∂µFνρ(x)qν\n1qρ\n2= 0.(4.4)\nThis equation includes the gauge-invariant combination λ1µ−λ2µowing to\nthegaugeinvarianceof LA, andhencecannotdetermine λ1µandλ2µuniquely.\nThe Euler-Lagrange equation for qµ\n1can be written as\n1\n(q2\n1q2\n2)1/4/bracketleftbigg1\n2/parenleftbiggd\ndτ∂K1\n∂˙qµ\n1−∂K1\n∂qµ\n1/parenrightbigg\n−k\n2/parenleftbiggd\ndτ∂J\n∂˙qµ\n1−∂J\n∂qµ\n1/parenrightbigg\n+λ1µ−ξµ+3\n2eFµν(x)qν\n2/bracketrightbigg\n+/parenleftBigg\nd\ndτ1\n(q2\n1q2\n2)1/4/parenrightBigg/parenleftbigg1\n2∂K1\n∂˙qµ\n1−k\n2∂J\n∂˙qµ\n1/parenrightbigg\n+q1µ\n2q2\n1LA= 0, (4.5)\n9whereK1andJare given by Eq. (3.8) and\nJ:=˙q1⊥µqµ\n2/radicalbig\nq2\n1−˙q2⊥µqµ\n1/radicalbig\nq2\n2. (4.6)\nApplying the formulas (3.9)–(3.11) and\n∂J\n∂˙qµ\n1=1/radicalbig\nq2\n1/parenleftbigg\nq2µ−q1q2\nq2\n1q1µ/parenrightbigg\n, (4.7)\nd\ndτ∂J\n∂˙qµ\n1−∂J\n∂qµ\n1=d\ndτq2µ/radicalbig\nq2\n2+1/radicalbig\nq2\n1/parenleftbigg\n˙q2µ−q1˙q2\nq2\n1q1µ/parenrightbigg\n(4.8)\nto Eq. (4.5) appropriately, we obtain\nk\n2/bracketleftBigg\nd\ndτq2µ/radicalbig\nq2\n2+1/radicalbig\nq2\n1/parenleftbigg\n˙q2µ−q1˙q2\nq2\n1q1µ/parenrightbigg/bracketrightBigg\n=3\n2eFµν(x)qν\n2+(q2\n1q2\n2)1/4q1µ\n2q2\n1LA+¨q1⊥µ\nq2\n1−/parenleftbigg5q1˙q1\nq2\n1+q2˙q2\nq2\n2/parenrightbigg˙q1⊥µ\n2q2\n1+λ1µ−ξµ\n+k\n4/radicalbig\nq2\n1/parenleftbiggq1˙q1\nq2\n1+q2˙q2\nq2\n2/parenrightbigg/parenleftbigg\nq2µ−q1q2\nq2\n1q1µ/parenrightbigg\n, (4.9)\nwhereq1q2:=q1µqµ\n2andq1˙q2:=q1µ˙qµ\n2. Similarly, the Euler-Lagrange equa-\ntion forqµ\n2is derived as\nk\n2/bracketleftBigg\nd\ndτq1µ/radicalbig\nq2\n1+1/radicalbig\nq2\n2/parenleftbigg\n˙q1µ−˙q1q2\nq2\n2q2µ/parenrightbigg/bracketrightBigg\n=3\n2eFµν(x)qν\n1−(q2\n1q2\n2)1/4q2µ\n2q2\n2LA+¨q2⊥µ\nq2\n2−/parenleftbiggq1˙q1\nq2\n1+5q2˙q2\nq2\n2/parenrightbigg˙q2⊥µ\n2q2\n2+λ2µ−ξµ\n+k\n4/radicalbig\nq2\n2/parenleftbiggq1˙q1\nq2\n1+q2˙q2\nq2\n2/parenrightbigg/parenleftbigg\nq1µ−q1q2\nq2\n2q2µ/parenrightbigg\n. (4.10)\nThe Euler-Lagrange equations for λ1µ,λ2µ, andξµare respectively found to\nbe\nqµ\n1= ˙xµ, (4.11)\nqµ\n2= ˙xµ, (4.12)\nqµ\n1=qµ\n2, (4.13)\n10which are compatible with one another.\nSubstituting Eqs. (4.11) and (4.12) into Eq. (4.9) and noting\nLA(qµ\n1,˙qµ\n1,λ1µ;qµ\n2,˙qµ\n2,λ2µ) =LA(˙xµ,¨xµ,λ1µ; ˙xµ,¨xµ,λ2µ) = 0,(4.14)\nwe have\nkd\ndτ˙xµ\n√\n˙x2=3\n2eFµν(x)˙xν+...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2+λµ\n1−ξµ. (4.15)\nSimilarly, substituting Eqs. (4.11) and (4.12) into Eq. (4.10) and using\n(4.14), we have\nkd\ndτ˙xµ\n√\n˙x2=3\n2eFµν(x)˙xν+...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2+λµ\n2−ξµ. (4.16)\nComparing Eq. (4.15) and Eq. (4.16) leads to\nλµ\n1=λµ\n2. (4.17)\nThen we see that Eq. (4.4) is identically satisfied owing to ∂µFνρ(x)qν\n1qρ\n2=\n∂µFνρ(x)˙xν˙xρ= 0. With Λµ:=λµ−ξµ(λµ:=λµ\n1=λµ\n2), Eqs. (4.15) and\n(4.16) can be written together as a single equation\nmd\ndτ˙xµ\n√\n˙x2=eFµν(x)˙xν+2\n3e2/parenleftbigg...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2/parenrightbigg\n+2\n3e2Λµ,(4.18)\nafter the substitution of k= 3m/2e2. This equation is completely the same\nas Eq. (3.23). In this way, it is established that the Lagrangian LAalso\nyields the Lorentz-Dirac equation with a source-like term.\n5. Summary and discussion\nWe have presented two relativistic Lagrangians LDandLAand have\ndemonstrated that the Euler-Lagrange equations derived from LD, or those\nderived from LA, together lead to the Lorentz-Dirac equation with a source-\nlike term. This equation is a differential equation for xµhaving the inhomo-\ngeneous term 2 e2Λµ/3. Hence it follows that its solutions naturally depend\nonΛµ. The Lorentz-Dirac equation itself can be obtained in a particular\nsituation such that Λµ= 0. For this reason, LDandLAcan simply be said\nto be the Lagrangians for the Lorentz-Dirac equation.\n11Contracting both sides of Eq. (4.18) with ˙ xµ, we have the orthogonality\ncondition\n˙xµΛµ= 0. (5.1)\nThis condition can be written as Λ0=v·Λ, withΛ:= (Λr) (r= 1,2,3) and\nthe velocity vector v:= (dxr/dx0). Accordingly, the 4-vector ( Λµ) can be\nexpressed as ( v·Λ,Λ). As can be seen from Eq. (4.18), the source-like term\n2e2Λµ/3 is regarded as a component of the force 4-vector ( fµ) = (v·f,f),\nprovided that f:= (2e2/3)Λis identified with an external (non-Lorentzian)\nforce acting on the charged particle. In this way, the source-like t erm can be\ntreated as a component of the force 4-vector of an external fo rce.\nIn the indirect formulation of the damped harmonic oscillator [28], a pa ir\nof two coordinate variables is introduced to describe the motion for ward in\ntime and that backward in time. A pair of qµ\n1andqµ\n2does not correspond to\nsuchapairofcoordinatevariables. Infact, qµ\n1andqµ\n2areincludedeveninthe\nLagrangian with an exponential damping function LD. Also,qµ\n1=qµ\n2= ˙xµ\nis eventually found from the Lagrangians LDandLA. For these reasons, qµ\n1\nandqµ\n2should simply be regarded as auxiliary variables useful for deriving\nthe Lorentz-Dirac equation.\nAs has been emphasized above, LDexplicitly depends on the parameter\nτ, whereas LAdoes not depend explicitly on τ. In a consistent quantiza-\ntion of the damped harmonic oscillator [25, 29, 30], the indirect formu lation\nbased on an explicitly time-independent Lagrangian is adopted rathe r than\nthe direct formulation based on an explicitly time-dependent Lagran gian.\nReferring to this fact, we should choose LAas a desirable Lagrangian when\nwe consider quantum theory of a charged particle described by the Lorentz-\nDirac equation. The Lagrangian and Hamiltonian formulations based o nLA\nand the subsequent quantization procedure are interesting issue s that should\nbe addressed in the future.\nAcknowledgments\nWe are grateful to Shigefumi Naka, Takeshi Nihei and Akitsugu Miw a for\ntheir useful comments. We thank Keita Seto for valuable informatio n on the\nLorentz-Dirac equation. One of us (T.S.) thanks Kenji Yamada, Ka tsuhito\nYamaguchi and Haruki Toyoda for their encouragement. The wor k of S.D. is\nsupported in part by Grant-in-Aid for Fundamental Scientific Rese arch from\nCollege of Science and Technology, Nihon University.\n12Appendix A. Lorentz-Dirac equation written in terms of an ar bi-\ntrary world-line parameter\nLet us recall the Lorentz-Dirac equation Eq. (1.2), i.e.,\nmduµ\ndl=eFµν(x)uν+2\n3e2/parenleftbig\nδµ\nν−uµuν/parenrightbigd2uν\ndl2, (A.1)\nwhich is written in terms of the proper time l. Noting that the infinitesimal\nproper time can be expressed as dl=√\n˙x2dτ, we can show that\nuµ:=dxµ\ndl=˙xµ\n√\n˙x2, (A.2)\nduµ\ndl=¨xµ\n⊥\n˙x2, (A.3)\nd2uµ\ndl2=1√\n˙x2/bracketleftbigg...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2−/parenleftbigg\n¨x2−(˙x¨x)2\n˙x2/parenrightbigg˙xµ\n(˙x2)2/bracketrightbigg\n,(A.4)\nwith\n¨xµ\n⊥:= ¨xµ−˙x¨x\n˙x2˙xµ,...xµ\n⊥:=...xµ−˙x...x\n˙x2˙xµ. (A.5)\nHere,\n˙xµ:=dxµ\ndτ,¨xµ:=d2xµ\ndτ2,...xµ:=d3xµ\ndτ3, (A.6)\n˙x2:= ˙xµ˙xµ,¨x2:= ¨xµ¨xµ,˙x¨x:= ˙xµ¨xµ,˙x...x:= ˙xµ...xµ.(A.7)\nSubstituting Eqs. (A.2) and (A.4) into Eq. (A.1), we obtain\nmd\ndτ˙xµ\n√\n˙x2=eFµν(x)˙xν+2\n3e2/parenleftbigg...xµ\n⊥\n˙x2−3(˙x¨x)¨xµ\n⊥\n(˙x2)2/parenrightbigg\n.(A.8)\nThis is the Lorentz-Dirac equation written in terms of anarbitrary w orld-line\nparameter τ. We can directly derive Eq. (A.8) by evaluating the reaction\nforce due to the particle’s own electromagnetic radiation without ad opting\nthe proper-time gauge τ=l.\n13References\n[1] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields , 4th\ned., Butterworth-Heinemann, Oxford, 1975.\n[2] J. D. Jackson, Classical Electrodynamics , 3rd ed., John Wiley & Sons,\nNew York, 1998.\n[3] A. O. Barut, Electrodynamics and Classical Theory of Fields and Par-\nticles, Dover, New York, 1980.\n[4] F. Rohrlich, Classical Charged Particles , 3rd ed., World Scientific, Sin-\ngapore, 2007.\n[5] F. Rohrlich, “The dynamics of a charged sphere and the electron ,” Am.\nJ. Phys. 65(1997) 1051-1056.\n[6] H. 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Caldirola, “Forze non conservative nella meccanica quantistic a,”\nNuovo Cimento 18(1941) 393-400.\n[27] E. Kanai, “Onthequantizationofthedissipative systems,” Prog .Theor.\nPhys.3(1948) 440-442.\n[28] H.Bateman, “Ondissipative systems andrelatedvariationalpr inciples,”\nPhys. Rev. 38(1931) 815-819.\n[29] H.FeshbachandY.Tikochinsky, “Quantizationofthedampedh armonic\noscillator,” Trans. N.Y. Acad. Sci., Ser. II 38(1977) 44-53.\n[30] R. Banerjee and P. Mukherjee, “A canonical approach to the quanti-\nzation of the damped harmonic oscillator,” J. Phys. A: Math. Gen. 35\n(2002) 5591-5598, arXiv:quant-ph/0108055.\n[31] M. Pavˇ siˇ c, “Classical motion of membranes, strings and poin t particles\nwith extrinsic curvature,” Phys. Lett. B 205(1988) 231-236.\n[32] M. Pavˇ siˇ c, “Thequantization ofa point particle with extrinsic curvature\nleads to the Dirac equation,” Phys. Lett. B 221(1989) 264-268.\n[33] M. S. Plyushchay, “Does the quantization of a particle with curv ature\nlead to Dirac equation?,” Phys. Lett. B 253(1991) 50-55.\n16"    },    {        "title": "2008.10381v3.Is_the_local_Lorentz_invariance_of_general_relativity_implemented_by_gauge_bosons_that_have_their_own_Yang_Mills_like_action_.pdf",        "content": "Is the local Lorentz invariance of general relativity implemented by gauge bosons that\nhave their own Yang-Mills-like action?\nKevin Cahill\u0003\nDepartment of Physics and Astronomy\nUniversity of New Mexico\nAlbuquerque, New Mexico 87131\n(Dated: October 18, 2022)\nGeneral relativity with fermions has two independent symmetries: general coordinate invariance\nand local Lorentz invariance. General coordinate invariance is implemented by the Levi-Civita con-\nnection and by Cartan's tetrads both of which have as their action the Einstein-Hilbert action. It is\nsuggested here that local Lorentz invariance is implemented not by a combination of the Levi-Civita\nconnection and Cartan's tetrads known as the spin connection, but by independent Lorentz bosons\nLab\nithat gauge the Lorentz group, that couple to fermions like Yang-Mills \felds, and that have\ntheir own Yang-Mills-like action. A nonsingular 4 \u00024 hermitian scalar \feld his needed to make the\naction of the Lorentz bosons invariant under local Lorentz transformations. Lorentz bosons couple\nto fermion number and generate a spin-dependent static potential that violates the weak equiva-\nlence principle. If a Higgs mechanism makes them massive, then the static potential also violates the\ninverse-square law. Experiments put upper bounds on the strength of such a potential for masses\nmL<20 eV. These upper limits imply that Lorentz bosons, if they exist, are nearly stable and\ncontribute to dark matter.\nI. INTRODUCTION\nGeneral relativity with fermions has two indepen-\ndent symmetries: general coordinate invariance and local\nLorentz invariance. General coordinate invariance is the\nwell-known, de\fning symmetry of general relativity. It\nacts on coordinates and on the world indexes of tensors\nbut leaves Dirac and Lorentz indexes unchanged. It is im-\nplemented by the Levi-Civita connection and by Cartan's\ntetrads.\nLocal Lorentz invariance is a quite di\u000berent symmetry.\nIt acts on Dirac and Lorentz indexes but leaves coordi-\nnates and world indexes unchanged. In standard formu-\nlations [1{5], the derivative of a Dirac \feld is made co-\nvariant (@i+!i) by a combination of the Levi-Civita\nconnection \u0000j\nkiand Cartan's tetrads ca\njknown as the\nspin connection\n!i=1\n8!ab\ni[\ra;\rb] (1)\nin which\n!ab\ni=ca\njcbk\u0000j\nki+ca\nk@icbk; (2)\naandbare Lorentz indexes, and i;j;k are world indexes.\nBecause it acts on Lorentz and Dirac indexes but\nleaves world indexes and coordinates unchanged, local\nLorentz invariance is more like an internal symmetry than\nlike general coordinate invariance. In theories with local\nLorentz invariance and internal symmetry, the covariant\nderivativeDiof a vector of Dirac \felds  has the spin\nconnection !iand a matrix Aiof Yang-Mills \felds side\nby side\nDi = (@i+!i+Ai) : (3)\n\u0003cahill@unm.eduJust as the Yang-Mills connection Aiis a linear combina-\ntionAi=\u0000it\u000bA\u000b\niof the matrices t\u000bthat generate the\ninternal symmetry group, so too the spin connection !i\nis a linear combination !i=1\n8!ab\ni[\ra;\rb]of the matrices\n\u0000i1\n4[\ra;\rb]that generate the Lorentz group.\nSo I ask: Does the independent symmetry of local\nLorentz invariance have its own, independent gauge \feld\nLi=1\n8Lab\ni[\ra;\rb] (4)\nwith its own \feld strength Fik=[@i+Li;@k+Lk]and\nYang-Mills-like action\nSL=\u00001\n4f2Z\nTr\u0010\nFy\nikhFikh\u00001\u0011pgd4x? (5)\nThis action is made suitably positive and invariant under\nthe local Lorentz transformations\nF0=D\u00001(\u0003)FD(\u0003) and h0=Dy(\u0003)hD(\u0003) (6)\nby a nonsingular 4 \u00024 hermitian scalar \feld h[6] whose\naction density is\nSh=\u0000M2Tr\u0002\n(Dih)h\u00001(Dih)h\u00001\u0003\n(7)\nin which its covariant derivative is [6]\nDih=@ih\u0000hLi\u0000Ly\nih: (8)\nThe covariant derivative of a Dirac \feld would then be\nDi =\u0000\n@i+1\n8Lab\ni[\ra;\rb]\u0001\n (9)\ninstead of the standard form (1{3)\nDi =\u0010\n@i+1\n8\u0000\nca\njcbk\u0000j\nki+ca\nk@icbk\u0001\n[\ra;\rb]\u0011\n :(10)arXiv:2008.10381v3  [gr-qc]  16 Oct 20222\nIf so, then the Lorentz bosons Lab\nicouple to fermion\nnumber and not to mass and lead to Coulomb and\nYukawa potentials that violate the inverse-square law\nand the weak equivalence principle. The hermitian scalar\n\feldhmust assume a nonzero value in the vacuum\nh0=h0jhj0ibecause it is a nonsingular matrix. It may\nplay the role of a new Higgs \feld. Its action (7) introduces\na new mass into the theory.\nExperiments [7{30] have set upper limits on the\nstrength of such Yukawa potentials for Lorentz bosons\nof mass less than 20 eV. These upper limits imply that\nLorentz bosons, if they exist, are nearly stable and con-\ntribute to dark matter. Whether fermions couple to\nLorentz bosons Liwith their own action SLor to the\nspin connection !iis an open experimental question.\nThis paper outlines a version of general relativity with\nfermions in which the six vector bosons of the spin con-\nnection!ab\niare replaced by six vector bosons Lab\nithat\ngauge the Lorentz group and have their own Yang-Mills-\nlike action. The theory is invariant under general co-\nordinate transformations and independently under local\nLorentz transformations.\nSection II sketches the traditional way of including\nfermions in a theory of general relativity. Section III\ndescribes the local Lorentz invariance of a theory with\nLorentz bosons. Section IV says why general-coordinate\ninvariance and local-Lorentz invariance are independent\nsymmetries. Section V describes the Yang-Mills-like ac-\ntion (5) of the gauge \felds Lab\niof the local Lorentz group.\nSections VI and VII suggest ways to make gauge the-\nory and general relativity more similar to each other.\nSection VIII discusses Higgs mechanisms that may give\nmasses to the gauge bosons Lab\niof the Lorentz group.\nSection IX describes some of the constraints that exper-\nimental tests [7{30] of the inverse-square law and of the\nweak equivalence principle place upon the proposed the-\nory. Section X discusses the stability and masses of L\nbosons and suggests that they may be part or all of dark\nmatter. Section XI summarizes the paper.\nII. GENERAL RELATIVITY WITH FERMIONS\nA century ago, Einstein described gravity by the action\nSE=1\n16\u0019GZ\nRpg d4x=1\n16\u0019GZ\ngikRikpg d4x\n(11)\nin whichG= 1=m2\nPis Newton's constant, the metric\nis (\u0000;+;+;+), letters from the middle of the alphabet\nare world indexes, g=jdetgikjis the absolute value of\nthe determinant of the space-time metric, and the Ricci\ntensorRik=R`\ni`kis the trace of the Riemann tensor\nRj\ni`k=@`\u0000j\nki\u0000@k\u0000j\n`i+ \u0000j\n`m\u0000m\nki\u0000\u0000j\nkm\u0000m\n`i(12)\nin which\n\u0000k\ni`=1\n2gkj(@`gji+@igj`\u0000@jgi`) = \u0000k\n`i (13)is the Levi-Civita connection which makes the covariant\nderivative of the metric vanish [31].\nThe standard action of general relativity with fermions\nis the sum of the Einstein-Hilbert action (11) and the\naction of matter \felds including the Dirac action\nZ\n\u0000\u0016 \raci\na(@i+!i+Ai) pgd4x: (14)\nIn what follows, it is proposed to replace the spin con-\nnection!iin the standard Dirac action (14) with an in-\ndependent gauge \feld Li=\u00001\n8Lab\ni[\ra;\rb]that has its\nown action (5) and to use\nSD=Z\n\u0000\u0016 \raci\na(@i+Li+Ai) pgd4x (15)\nas the action of a Dirac \feld. This change re\rects the\nindependence of general coordinate invariance and lo-\ncal Lorentz invariance and makes general relativity and\nquantum \feld theory somewhat more similar.\nIII. LOCAL LORENTZ INVARIANCE\nThe Einstein action (11) has a trivial symmetry under\nlocal Lorentz transformations that act on Lorentz indexes\nbut leave world indexes and coordinates unchanged. This\nsymmetry becomes apparent when Cartan's tetrads ca\ni\nandcb\nkare used to write the metric gikin a form\ngik(x) =ca\ni(x)\u0011abcb\nk(x) (16)\nthat is unchanged by local Lorentz transformations\nc0a\ni(x) = \u0003a\nb(x)cb\ni(x): (17)\nThe Levi-Civita connection (13) and the action (11) are\nde\fned in terms of the metric and so are also invariant\nunder local Lorentz transformations.\nMore importantly, the two Dirac actions (14) and (15)\nhave a nontrivial symmetry under local Lorentz trans-\nformations. Under such a local Lorentz transformation, a\nDirac \feld transforms under the (1\n2;0)\b(0;1\n2) represen-\ntationD(\u0003) of the Lorentz group with no change in its\ncoordinates x\n 0\n\u000b(x) =D\u00001\n\u000b\f(\u0003(x)) \f(x): (18)\nThe Lorentz-boson matrix Li=\u00001\n8Lab\ni[\ra;\rb]makes\n@i+Lia covariant derivative\n@i+L0\ni=D\u00001(\u0003)(@i+Li)D(\u0003): (19)\nIn more detail with \u0003 = \u0003( x), the matrix Litransforms\nas\nL0\ni=D\u00001(\u0003)@iD(\u0003) +D\u00001(\u0003)LiD(\u0003)\n=D\u00001(\u0003)@iD(\u0003)\u00001\n8D\u00001(\u0003)Lab\ni[\ra;\rb]D(\u0003)\n=D\u00001(\u0003)@iD(\u0003)\u00001\n8Lab\ni\u0003c\na\u0003d\nb[\rc;\rd] (20)3\nin which \u0003c\na= \u0003\u00001c\na. Since\nTr\u0000\n[\ra;\rb][\rc;\rd]\u0001\n= 16\u0000\n\u000ea\nd\u000eb\nc\u0000\u000ea\nc\u000eb\nd\u0001\n; (21)\nits components transform as\nL0ab\ni=\u00001\n2Tr(L0\ni[\ra;\rb]) (22)\n= \u0003a\nc\u0003b\ndLcd\ni\u00001\n2Tr\u0000\nD\u00001(\u0003)@iD(\u0003)[\ra;\rb]\u0001\n:\nUnder an in\fnitesimal transformation\n\u0003 =I+\u0015andD(\u0015) =I\u00001\n8\u0015ab[\ra;\rb]; (23)\nthe Lorentz bosons transform as\nL0ab\ni=Lab\ni+Lcb\ni\u0015a\nc+Lad\ni\u0015b\nd+@i\u0015ab: (24)\nThe components of the spin connection obey similar\nequations, and the conventional Dirac action (14) also\nis invariant under local Lorentz transformations.\nLocal Lorentz transformations operate on the Lorentz\nindexesa;b;c;::: of the tetrads, of the spin connection\n!ab\ni, and of the gamma matrices \ra[\rb;\rc], and also on\nthe Dirac indexes \u000b;\f;\r of the gamma matrices and of\nthe Dirac \felds \u0016 ; , but not upon the world index ior\nthe spacetime coordinates x. In this sense, the invariance\nof Dirac's action SDunder local Lorentz transformations\nis like an internal symmetry.\nIV. LOCAL LORENTZ INVARIANCE AND\nINVARIANCE UNDER GENERAL\nCOORDINATE TRANSFORMATIONS ARE\nINDEPENDENT SYMMETRIES\nThe Dirac action SDis invariant both under a local\nLorentz transformation \u0003( x) and under a general coordi-\nnate transformation x!x0. Under a local Lorentz trans-\nformation \u0003( x), the coordinates are unchanged, x0=x,\nand the \felds transform as\n 0\n\u000b(x) =D\u00001\n\u000b\f(\u0003(x)) \f(x)\nc0a\ni(x) = \u0003a\nb(x)cb\ni(x)\nL0ab\ni(x) = \u0003a\nc(x)\u0003b\nd(x)Lcd\ni(x)\n\u00001\n2Tr\u0000\nD\u00001(\u0003(x))@iD(\u0003(x))[\ra;\rb]\u0001\nh0(x) =Dy(\u0003(x))h(x)D(\u0003(x)) (25)\nin which \u0003a\nc(x) = \u0003\u00001a\nc. Under a general coordinate\ntransformation, the \felds transform as\n 0\n\u000b(x0) = \u000b(x)\nc0a\ni(x0) =@xk\n@x0ica\nk(x)\nL0ab\ni(x0) =@xk\n@x0iLab\nk(x)\nh(x0) =h(x): (26)The two transformations,  \u000b(x)!D\u00001\n\u000b\f(\u0003(x)) \f(x) and\nx!x0, are di\u000berent and independent; the coordinates x0\nand \u0003(x)xare unrelated.\nEvery conventional, local Lorentz transformation is\na general coordinate transformation, so one might be\ntempted to imagine that every general coordinate trans-\nformation is a conventional, local Lorentz transformation.\nBut one can see that this is not the case by comparing the\nin\fnitesimal form of a general coordinate transformation\ndx0i=@x0i\n@xkdxk(27)\nwhich has 16 generators with that of a conventional, local\nLorentz transformation\ndx0a= \u0003a\nbdxb=dxa+ (\u000fr\u0001Ra\nb+\u000fb\u0001Ba\nb)dxb(28)\nwhich has only 6 [32].\nSpecial relativity o\u000bers another temptation. In special\nrelativity, global Lorentz transformations \u0003 act on the\nspacetime coordinates and on the indexes of a Dirac \feld\nx0a= \u0003a\nbxb\n 0\n\u000b(x0) =D\u00001\n\u000b\f(\u0003) \f(\u0003x):(29)\nThis global Lorentz transformation leaves the specially\nrelativistic Dirac action density unchanged\n\u0002\n\u0000i y\r0\ra@a \u00030=\u0000i yD\u00001y\r0\raD\u00001@0\na \n=\u0000i y\r0D\raD\u00001\u0003c\na@c \n=\u0000i y\r0\rb\u0003a\nb\u0003c\na@c (30)\n=\u0000i y\r0\rb\u000ec\nb@c =\u0000i y\r0\rb@b :\nBut in general relativity with fermions, Cartan's\ntetradsci\naallow the action to be invariant under a local\nLorentz transformation without a corresponding general\ncoordinate transformation. The matrix D\u00001\n\u000b\f(\u0003(x)) repre-\nsents a local Lorentz transformation and acts (18) on the\nspinor indexes of the Dirac \feld but not on its spacetime\ncoordinates. Since\nD\raD\u00001= \u0003a\na0\ra0andD\raD\u00001= \u0003a0\na\ra0;(31)\na local Lorentz transformation (20) does not change\nD\raD\u00001c0i\na=\rb\u0003a\nb\u0003c\naci\nc=\rb\u000ec\nbci\nc=\rbci\nb: (32)\nBut the e\u000bect of a local Lorentz transformation (20) on\nthe Lorentz matrix Liis\nD(@i+L0\ni)D\u00001\u0000@i=\u00001\n8L0ab\niD[\ra;\rb]D\u00001\n=\u00001\n8\u0003a\nc\u0003b\ndLcd\niD[\ra;\rb]D\u00001\n=\u00001\n8\u0003\u00001a\nc\u0003\u00001b\ndLcd\ni\u0003e\na\u0003f\nb[\re;\rf]\n=\u00001\n8\u000ee\nc\u000ef\ndLcd\ni[\re;\rf]\n=\u00001\n8Lcd\ni[\rc;\rd]=Li (33)4\nso that\nD(@i+L0\ni)D\u00001=@i+Li: (34)\nA local Lorentz transformation therefore leaves the Dirac\naction density invariant\n\u0002\n\u0000i y\r0\raci\na(@i+Li) \u00030\n=\u0000i yD\u00001y\r0\rac0i\na(@i+L0\ni)D\u00001 \n=\u0000i yD\u00001y\r0\rac0i\naD\u00001D(@i+L0\ni)D\u00001 \n=\u0000i y\r0D\rac0i\naD\u00001(@i+Li) (35)\n=\u0000i y\r0\raci\na(@i+Li) :\nThe symmetry under local Lorentz transformations is\nindependent of the symmetry under general coordinate\ntransformations. They are independent symmetries.\nV. ACTION OF THE GAUGE FIELDS Li\nSince local Lorentz symmetry is like an internal sym-\nmetry, its gauge \felds Li=1\n8Lab\ni[\ra;\rb]should have an\naction like that of a Yang-Mills \feld\nSL=\u00001\n4f2Z\nTr\u0010\nFy\nikhFikh\u00001\u0011pgd4x (36)\nin which\nFik=[@i+Li;@k+Lk]; (37)\nandhis a 4\u00024 nonsingular hermitian matrix [6] .\nUnder local Lorentz transformations D(\u0003(x)), the \feld\nstrengthsFikand the matrix htransform as\nF0ik(x) =D\u00001(\u0003(x))Fik(x)D(\u0003(x))\nh0(x) =Dy(\u0003(x))h(x)D(\u0003(x))(38)\nand so the action SLis invariant under local Lorentz\ntransformations as well as under general coordinate\ntransformations [6] .\nOne might think that it would be su\u000ecient to set h=\n\f=i\r0, sinceDy(\u0003)\fD(\u0003) =\fandD(\u0003)\fDy(\u0003) =\f,\nbut the resulting action SLis not bounded below.\nIn terms of the gamma matrices\n\r0=\u0000i\u0012\n0 1\n1 0\u0013\n; \ri=\u0000i\u0012\n0\u001bi\n\u0000\u001bi0\u0013\n;\nand\r5=\u0012\n1 0\n0\u00001\u0013\n;(39)\nthe commutators in Li=\u00001\n8Lab\ni[\ra;\rb]are for spatial\na;b;c = 1;2;3,\n[\ra;\rb]= 2i\u000fabc\u001bcIand [\r0;\ra]=\u00002\u001ba\r5:(40)\nSo setting\nra\ni=1\n2\u000fabcLbc\niandba\ni=La0\ni; (41)the matrix of gauge \felds Liis\nLi=\u00001\n8Lab\ni[\ra;\rb]=\u0000i1\n2ri\u0001\u001bI\u00001\n2bi\u0001\u001b\r5;(42)\nand its \feld strength (37) is\nFik=[@i+Li;@k+Lk] (43)\n=\u0000i1\n2(@irk\u0000@kri+ri\u0002rk\u0000bi\u0002bk)\u0001\u001bI\n\u00001\n2(@ibk\u0000@kbi+ri\u0002bk+bi\u0002rk)\u0001\u001b\r5:\nWith the abbreviations\nRik= (@irk\u0000@kri+ri\u0002rk\u0000bi\u0002bk)\nBik=(@ibk\u0000@kbi+ri\u0002bk+bi\u0002rk);(44)\nthe actionSLis the trace\nSL=\u00001\n16f2Z\nTrh\u0000\nRik\u0001\u001bI+iBik\u0001\u001b\r5\u0001\nh\n\u0000\nRik\u0001\u001bI\u0000iBik\u0001\u001b\r5\u0001\nh\u00001ipgd4x:(45)\nThe 4\u00024 nonsingular hermitian matrix hmay play a\nrole in the action (15) of a spin-one-half \feld  because\nwe may take the quantity \u0016 either to be \u0016 = y\fas\nusual or to be \u0016 = yh. The covariant derivative (8) of\nhtransforms as\u0000\nDih\u00010=Dy(\u0003)DihD(\u0003) .\nVI. MAKING GENERAL RELATIVITY MORE\nSIMILAR TO GAUGE THEORY\nThere are three reasons to de\fne the covariant deriva-\ntive of a Dirac \feld in terms of Lorentz bosons\nLi=1\n8Lab\ni[\ra;\rb] (46)\nwith their own action (36) as\nDi = (@i+Li+Ai) (47)\nrather than in terms of the spin connection (1)\n!i=1\n8!ab\ni[\ra;\rb]\n=1\n8\u0010\nca\njcbk\u0000j\nki+ca\nk@icbk\u0011\n[\ra;\rb](48)\nas (3)\n(@i+!i+Ai) : (49)\nOne reason is that the symmetry of local Lorentz trans-\nformations is independent of the symmetry of general\ncoordinate transformations. So local Lorentz invariance\nshould have its own gauge \feld Liand action SLinde-\npendent of the tetrads and the Levi-Civita connection of\ngeneral coordinate transformations.\nA second reason to prefer the Lorentz connection Li\nto the spin connection !iis that the L-boson covariant\nderivative\n\u0000\n@i+1\n8Lab\ni[\ra;\rb]\u0001\n (50)5\nis simpler than the spin-connection covariant derivative\n\u0010\n@i+1\n8(ca\njcbk\u0000j\nki+ca\nk@icbk)[\ra;\rb]\u0011\n : (51)\nA third reason is that using the Lorentz connection\n(46), the Dirac covariant derivative (47), and the action\n(36) for the Lorentz connection, makes general relativity\nwith fermions more similar to the gauge theories of the\nstandard model.\nVII. MAKING GAUGE THEORY MORE\nSIMILAR TO GENERAL RELATIVITY\nUnder a local Lorentz transformation, the spin connec-\ntion!ichanges more naturally, more automatically than\ndoes the Lorentz connection Li. The automatic feature of\nthe spin connection is that its de\fnition (2) implies that\nunder in\fnitesimal (23) and \fnite local Lorentz transfor-\nmations it transforms as\n!0ab\ni=!ab\ni+!cb\ni\u0015a\nc+!ad\ni\u0015b\nd+@i\u0015ab(52)\nand as\n!0ab\ni= \u0003a\nc\u0003b\nd!cd\ni+ \u0003a\nc@i\u0003cb: (53)\nThe terms @i\u0015aband \u0003a\nc@i\u0003cboccur automatically\nwithout the need to put in by hand a term like\nD\u00001(\u0003)@iD(\u0003).\nTerms like D\u00001(\u0003)@iD(\u0003) are a common feature of\ngauge theories whether abelian or nonabelian. We can\nmake them occur automatically in local Lorentz trans-\nformations if we add to the Lorentz connection Lab\ni\nthe termua\n\u000b@iub\u000bin which the four Lorentz 4-vectors\nua\u000b(x) obey the condition\nua\u000b\u0011\u000b\fub\f=\u0011ab; (54)\nand\u000b= 0;1;2;3 is a label, not an index. It follows then\nfrom this condition (54) on the quartet of vectors ua\u000b\nthat the augmented Lorentz connection Lab\nnewi\nLab\nnewi=Lab\ni+ua\n\u000b@iub\u000b(55)\nautomatically changes under a local Lorentz transforma-\ntion \u0003a\nc= \u0003\u00001a\ncto\nL0ab\nnewi= \u0003a\nc\u0003b\ndLcd\ni+ \u0003a\ncuc\n\u000b@i(\u0003b\ndud\u000b)\n= \u0003a\nc\u0003b\nd(Lcd\ni+uc\n\u000b@iud\u000b) +uc\n\u000bud\u000b\u0003a\nc@i\u0003b\nd\n= \u0003a\nc\u0003b\ndLcd\nnewi+\u0011cd\u0003a\nc@i\u0003b\nd\n= \u0003a\nc\u0003b\ndLcd\nnewi+ \u0003a\nc@i\u0003cb\n= \u0003a\nc\u0003b\ndLcd\nnewi+ \u0003\u00001a\nc@i\u0003cb(56)\nwithout the need to explicitly add the last term\n\u0003\u00001a\nc@i\u0003cbby hand.\nIn matrix form, the condition (54) is the requirement\nua\u000b\u0011\u000b\fub\f=\u0011ab(57)that the matrix formed by the quartet of vectors ua\u000bbe\na Lorentz transformation\nxaua\u000b\u0011\u000b\fub\fyb=xa\u0011abyb: (58)\nThe augmentation of the Lorentz connection Lab\ni!\nLab\nnewiby the addition of the term ua\n\u000b@iub\u000b, which is\nsimilar to the tetrad term ca\nk@icbkof the spin connection\n(2), makes its change (56) under local Lorentz transfor-\nmations as automatic as that (53) of the spin connection.\nThe use of a more automatic connection makes gauge\ntheory more similar to general relativity with fermions.\nWe can extend the use of such terms to internal sym-\nmetries and so make the inhomogeneous terms appear\nautomatically rather than by hand or by \fat. For in-\nstance, we can augment the abelian connection Aito\nAnewi(x) =Ai(x) +e\u0000i\u001e(x)@iei\u001e(x)(59)\nin which\u001e(x) is an arbitrary phase. A U(1) transforma-\ntion\ne\u0000i\u001e(x)!e\u0000i(\u0012(x)+\u001e(x))and (x)!e\u0000i\u0012(x) (x) (60)\nwould then change the covariant derivative ( @i+Anewi) \nto\n[(@i+Anewi) ]0= (@i+Ai+e\u0000i(\u0012+\u001e)@iei(\u0012+\u001e))e\u0000i\u0012 \n=e\u0000i\u0012(@i\u0000i@i\u0012+Ai+i@i\u0012+e\u0000i\u001e@iei\u001e) (61)\n=e\u0000i\u0012(@i+Ai+e\u0000i\u001e@iei\u001e) =e\u0000i\u0012(@i+Anewi) :\nSimilarly, we can augment the nonabelian connection\nAi=\u0000it\u000bA\u000b\niforSU(n) to\nAnewi(x) =Ai(x) +u\u000b\f(x)@iu\u0003\n\u000b\r(x) (62)\nin which the n n-vectorsu\u000b\rare orthonormal\nu\f\u000bu\u0003\n\r\u000b=\u000e\f\r: (63)\nAnSU(n) transformation\nAi!gAigy; u!gu;and !g (64)\nwould then change the covariant derivative ( @i+Anewi) \nto\n[(@i+Anewi) ]0= (@i+gAigy+gu@i(uygy))g \n=g(@i+gy@ig+Ai+ (@igy)uuyg+u@iuy) \n=g(@i+gy@ig+Ai+ (@igy)g+u@iuy) \n=g(@i+Ai+u@iuy) =g(@i+Anewi) : (65)\nVIII. POSSIBLE HIGGS MECHANISMS\nThe 4\u00024 nonsingular hermitian matrix his needed to\nmake the action (36) gauge invariant. It also may replace6\n\fin the fermionic action (15). Since it is nonsingular, it\nmust assume a nonzero average value in the vacuum\nh0=h0jhj0i: (66)\nSo we have a new kind of Higgs mechanism that can give\nmasses to gauge \felds and fermions. This will be taken\nup in later papers.\nOther kinds of Higgs mechanisms are also possible with\ndi\u000berent Higgs \felds. The actions SLandSD(5 & 15)\nleave the gauge bosons Lmassless, but a Higgs mecha-\nnism is possible. An interaction with a \feld vathat is\na scalar under general coordinate transformations but a\nvector under local Lorentz transformations has as its co-\nvariant derivative\nDiva=@iva+La\nbivb: (67)\nIf the time component v0has a nonzero mean value in\nthe vacuumh0jv0j0i 6= 0, then the scalar \u0000DivaDiva\ncontains a mass term\n\u0000La\n0iv0L0i\nav0=\u0000La0\niv0La0iv0(68)\nthat makes the boost vector bosons bs\ni=Ls0\nimassive\nbut leaves the rotational vector bosons rs\ni=1\n2\u000fstuLtu\ni\nmassless. On the other hand, if the spatial components\nhave a nonzero mean value, h0jvj0i6= 0, then the mass\nterm is\n\u0000La\nsivsLsi\navs=\u0000Ls0s\nivsLs0sivs+L0s\nivsL0sivs:(69)\nAdding the two mass terms (68 and 69), we \fnd\n\u0000La\n0ivbL0i\navb=L0s\nivaL0siva\u0000Ls0s\nivsLs0sivs(70)\nwhich makes all six gauge bosons massive as long as the\nmean value is timelike\nh0jvavaj0i<0 (71)\nand at least two spatial components h0jvsj0i6= 0 have\nnonzero mean values in the vacuum. This condition holds\nin all Lorentz frames if three vectors va1;va2;andva3\nhave di\u000berent timelike mean values in the vacuum.\nIn the vacuum of \rat space, tetrads have mean values\nthat are Lorentz transforms of ca\ni=\u000ea\niand that produce\nthe Minkowski metric (16)\ngik=\u000ea\ni\u0011ab\u000eb\nk=\u0011ik: (72)\nSo it is tempting to look for a Higgs mechanism that\nuses the covariant derivatives D`ca\nkof the tetrads. For\n\u0000j\nk`= 0 andca\nk=\u000ea\nk, the term\n\u00001\n2m2\nL(Dick\na)Dica\nk=\u00001\n2m2\nLLbi\nack\nbLa\ncicc\nk\n=\u00001\n2m2\nLLi\nabLab\ni(73)\nmakes the rotational bosons rs\ni=1\n2\u000fstuLtu\nimassive but\nmakes the boost bosons bs\ni=Ls0\nitachyons. If weakly\ncoupled tachyons are unacceptable, then the Higgs mech-\nanism (67{70) that uses three world-scalar Lorentz vec-\ntors with di\u000berent time-like mean values in the vacuum\nva\n1,va\n2,va\n3is a more plausible way to make the gauge\nbosonsLab\nimassive.IX. TESTS OF THE INVERSE-SQUARE LAW\nIn the static limit, the exchange of six Lorentz bosons\nLab\niof massmLwould imply that two macroscopic bod-\nies ofFandF0fermions separated by a distance rwould\ncontribute to the energy a static Yukawa potential\nVL(r) =3FF0f2\n2\u0019re\u0000mLr: (74)\nThis potential is positive and repulsive (between fermions\nand between antifermions) because the L's are vector\nbosons. It violates the weak equivalence principle because\nit depends upon the number Fof fermions (minus the\nnumber of antifermions) as F= 3B+Land not upon\ntheir masses. The potential VL(r) changes Newton's po-\ntential to\nVNL(r) =\u0000Gmm0\nr\u0010\n1 +\u000be\u0000r=\u0015\u0011\n(75)\nin which the coupling strength \u000bis\n\u000b=\u00003FF0f2\n2\u0019Gmm0=\u00003FF0m2\nPf2\n2\u0019mm0; (76)\nand the length \u0015is\u0015=~=cmL. Couplings \u000b\u00181 are\nof gravitational strength. Experiments [7{30] that test\nthe inverse-square law and the weak equivalence principle\nhave put upper limits on the strength j\u000bjof the coupling\nfor a wide range of lengths 10\u00008< \u0015 < 1013m and\nmasses 2\u000210\u000020<mL<20 eV.\nExperiments that tested the inverse-square law at very\nshort distances, between 10 nm and 3 mm, were done\nwith masses of gold [7], of gold and silicon [8], of plat-\ninum [9], and of tungsten [10]. For a mass mofNatoms\nof gold which has FAu= 670 fermions (quarks and elec-\ntrons) in each atom of mass mAu= 196:966 u, the ratio\nFmP=mthat appears in the coupling \u000b(76) is\nNFAumP\nNm Au=FAumP\nmAu=670mP\n196:966u= 4:458\u00021019:(77)\nSo the coupling strength is \u000bAu=\u00009:490\u00021038f2\nfor gold. An atom of silicon has FSi= 98 fermions and\na mass ofmSi= 28:085 u, soFSimP=mSi= 4:573 and\n\u000bSi=\u00009:988\u00021038f2. Platinum has FPt= 663 and\nmPt= 195:084u, so\u000bPt=\u00009:474\u00021038f2. Tungsten\nhasFW= 626 and mW= 183:84u, so\u000bW=\u00009:510\u0002\n1038f2. For such test masses, f2\u0019j\u000bj\u000210\u000039.\nThe Riverside group [7] placed on the strength j\u000bAuj\nan upper limit (95% con\fdence) that drops from j\u000bAuj.\n1019toj\u000bAuj.1016as the length \u0015rises from 10\u00008m\nto 4\u000210\u00008m. The IUPUI group [8] put an upper limit\n(95% con\fdence) on the strength j\u000bAu-Sijthat drops from\nj\u000bAu-Sij.1016toj\u000bAu-Sij.105as the length \u0015rises from\n4\u000210\u00008m to 8\u000210\u00006m. These results of the Riverside\nand IUIPUI groups are plotted in Fig. 1 from Chen et\nal. [8].\nOther short-distance experiments [9, 10, 12{21, 27{29]\nhave tested the inverse-square law at the slightly longer7\ndistances of 2\u000210\u00006<\u0015< 3\u000210\u00003m. The Washington\ngroup [9] used test masses of platinum. The [10] used test\nmasses of tungsten. The upper limits (95% con\fdence) on\nthe strengthj\u000bjare shown for platinum in Fig. 2 from Lee\net al. [9] and for tungsten in Fig. 3 from Tan et al. [10].\nThe upper limit on the strength j\u000bjfalls fromj\u000bj.106\nat\u0015\u00182\u000210\u00006m toj\u000bj.104at\u0015\u00188\u000210\u00006m\nand then fromj\u000bj.103at\u0015\u001810\u00005m toj\u000bj.1 at\n\u0015\u00184\u000210\u00005m and toj\u000bj.10\u00003at\u0015\u00182\u000210\u00003m.\n10-710-610-510-410-1103107101110151019\nIUPUI\n    (2014)\nRadion\nStrange modulus\nStanfordYaleIUPUI (2005-2007)\nWashington\nGluon modulus\n  ||\n (meters)DilatonRiverside\nRegion\nexcluded by experiments\nGauged\nbaryons\nFIG. 1: Upper limits (95% con\fdence) on the strength\nj\u000bAujof Yukawa potentials that violate the inverse-square\nlaw at distances 10\u00008<\u0015< 2\u000210\u00004m. (Fig. 4 of Chen\net al. [8])\nOther groups [14, 21{26, 29, 30] have tested the\ninverse-square law over a huge range of longer distances,\n10\u00003<\u0015< 3\u00021015m. In 2012 the HUST group [29] put\nan upper limit of j\u000bj.10\u00003for 7\u000210\u00004<\u0015< 5\u000210\u00003\nm, while in 1985 the Irvine group [21] put an upper limit\nofj\u000bj.10\u00003for lengths 7\u000210\u00003<\u0015< 10\u00001m.\nFischbach and Talmadge [30] and Adelberger et al. [14]\nhave reported tests of the inverse-square law for distances\nin the range 10\u00002< \u0015 < 1015m [14, 21, 24, 25, 30]. As\nshown in Fig. 4 from Adelberger et al. [14], the upper\nlimit lies between j\u000bj<3\u000210\u00004andj\u000bj<2\u000210\u00003\nfor 10\u00002< \u0015 < 104m but drops from j\u000bj.10\u00004to\nj\u000bj.10\u000010as the length increases from 104to 108m.\nThe upper limit is about j\u000bj.5\u000210\u00009on planetary\nscales 1010<\u0015< 5\u00021011m.\nThe Washington group have used torsion-balance ex-\nperiments to look for Yukawa potentials that violate the\nweak equivalence principle in the range of distances 0 :3<\n\u0015 <109m [14]. They have put upper limits (95% con\f-\ndence) on the strength j\u000bjof the coupling to B,Z, and\nN\u0011B\u0000Lbut not explicitly on the coupling to fermion\nnumberF= 3B+L. ForB, their upper limit runs from\nj\u000bj.10\u00005at 10\u00001m toj\u000bj.6\u000210\u00008at 7\u0002105m and\nFIG. 2: Upper limits (95% con\fdence) on the strength\nj\u000bPtjof Yukawa potentials that violate the inverse-square\nlaw at sub-mm distances [9{21]. (Fig. 5b of Lee et al. [9])\nFIG. 3: Upper limits (95% con\fdence) on the strength\nj\u000bWjof Yukawa potentials that violate the inverse-square\nlaw at mm and sub-mm distances [8, 10, 12, 15, 16, 18{\n21, 27{29]. Light lines are theory [14, 23] (Fig. 6 of Tan\net al. [10]).\nthen falls toj\u000bj.10\u000010for 107<\u0015< 1013m as shown\nby the dashed lines in Fig. 5 from Berg\u0013 e et al. [11]. For Z\nandN, their upper limit runs from j\u000bj.10\u00006at 10\u00001m\ntoj\u000bj.6\u000210\u00009at 106m and then falls to j\u000bj.2\u000210\u000011\nfor 107<\u0015< 109m [14].\nMore recent satellite measurements by the MICRO-\nSCOPE mission have lowered the upper limit on the\nstrengthj\u000bjof Yukawa potentials that violate the weak\nequivalence principle by about an order of magnitude for8\nFIG. 4: Upper limits (95% con\fdence) on the strength\nj\u000bjof Yukawa violations of the inverse-square law at large\ndistances 10\u00002<\u0015< 1015m [14, 21, 24, 25, 30] (Fig. 10\nof Adelberger et al. [14]).\n107<\u0015< 109m [11]. The upper limit for coupling to B\nisj\u000bj.10\u000011for 107< \u0015 < 109m as shown in Fig. 5\nfrom Berg\u0013 e et al. [11]. Their limit for coupling to Nis\neven lower:j\u000bj.4\u000210\u000013for 107<\u0015< 109m [11].\nFIG. 5: Upper limits (95% con\fdence) on the strength j\u000bj\nof Yukawa potentials that violate the weak equivalence\nprinciple at long distances [11, 14, 21, 22, 24, 25] (Fig. 1\nof Berg\u0013 e et al. [11]).\nSome of these important results [7{30] are summarized\nin broad-brush fashion in Fig. 6. The upper bound (95%\ncon\fdence) onj\u000bjis the solid dark-blue curve which falls\n10-910-710-510-310-110110310510710910-1510-1010-510010510101015FIG. 6: The upper bound (95% con\fdence) on the\nstrengthj\u000bjof Yukawa potentials that violate the inverse-\nsquare law or the weak equivalence principle at various\ndistances\u0015is the solid dark-blue curve [7{30]. The re-\ngion under the dotted green line denotes Lbosons with\nlifetimes greater than the age of the universe for the case\nin which the fudge factor b= 1. The region below that\nline and between the vertical dashed blue lines denotes\nLbosons that are between 1 and 100% of dark matter.\nThe thin vertical gray solid line marks the wavelength of\nanLboson whose mass is 2 me\u000b\n\u0017e= 2:2 eV [33].\nfrom 1019for\u0015= 7\u000210\u00008m to 10\u000011at\u0015= 109m.\nThe (\u0015;j\u000bj) region above this curve is excluded. Points in\nthe allowed region that are below the blue-green dotted\nline correspond to Lbosons with lifetimes longer than\nthe age of the universe. Those that also are between the\nvertical dashed lines denote e\u000bectively stable Lbosons\nwhose masses could account for between 1 and 100% of\ndark matter.\nX. LORENTZ BOSONS AS DARK MATTER\nAnalysis of the double galaxy cluster 1E0657-558 (the\n\\bullet cluster\" at z= 0:296) suggests [34, 35] that\ndark matter interacts weakly, perhaps with gravitational\nstrengthj\u000bj\u00181. As of now, there has been no accepted\ndetection of dark matter in a laboratory.\nThe experiments [7{30] sketched in Sec. IX put no up-\nper limits on the mass mL=h=c\u0015 ofLbosons and no\nlower limits on their coupling j\u000bj. The proposed Lbosons\nare electrically neutral. If their mass is heavy enough and\nif their coupling is su\u000eciently weak, then they would be\nan e\u000bectively stable part of dark matter.\nBecause they couple to fermion number and not to\nmass, their coupling f2is much weaker than j\u000bjby a\nfactor related to Avogadro's number. For the metals (Au,\nSi, Pt, and W) used in many of the experiments [7{29],9\nthe relation is\nf2\u0018j\u000bj\u000210\u000039: (78)\nEven for the highest upper limit j\u000bj<1019shown in\nFig. 6, the coupling of the Lbosons is only f2.10\u000020.\nBecause they interact so weakly, Lbosons decay\nslowly. The decay width of the Zboson is \u0000 Z=\n3:7e2mZc2=4\u0019= 2:5 GeV, and its lifetime is \u001cZ=\n~=\u0000Z= 2:6\u000210\u000025s. The analog of the electromagnetic\ncouplinge2=4\u0019\u00181=137 forLbosons isf2=4\u0019. In terms\noff2andj\u000bj(78), the decay width of an Lboson of mass\nmLis roughly\n\u0000L\u0018bf2mLc2\n4\u0019=bj\u000bjmLc2\n4\u0019\u000210\u000039(79)\nin whichbis a fudge factor, 0 :1.b.10, that depends\non the decay channels. The Lboson lifetime then is\n\u001c=~\n\u0000L=8:3\u00021024\nbj\u000bjmLc2[eV]s =1:9\nbj\u000bj107\nmLc2[eV]t0(80)\nin whicht0= 4:356\u00021017s is 13.8 billion years, the age\nof the universe. An Lboson of mass mL<(19=bj\u000bj) MeV\nis e\u000bectively stable in that its lifetime exceeds the age of\nthe universe. If the fudge factor bis taken to be unity,\nthenLbosons of wavelength \u0015are e\u000bectively stable for\ncouplings\nj\u000bj.\u0000\n1:5\u00021013\u0001\n\u0015[m] (81)\nwhich is the dotted green line in Fig. 6. Points below it\ndenote e\u000bectively stable Lbosons.\nIf the lightest fermion has mass mlightest , thenLbosons\nof mass less than 2 mlightest would be absolutely stable.\nThe dash-dotted gray vertical line in Fig. 6 is the wave-\nlength\u0015= 5:6\u000210\u00007m of twice the upper limit on the\ne\u000bective mass of the electron neutrino, 2 m(e\u000b)\n\u0017e[33].\nThe mass density of cold dark matter is \u001acdm =\n(2:2414\u00060:017)\u000210\u000027kg m\u00003[36, col. 7, p. 15]. So\nif all of cold dark matter were made of Lbosons of mass\nmL, then their number density would be\nnL=\u001acdm\nmL=1:26\u0002109\nmLc2[eV]m\u00003: (82)\nTo estimate the present number density of each kind\nofLboson, I'll assume that the Lbosons are e\u000bectively\nstable and have not interacted since they dropped out of\nequilibrium in the very early universe.\nAt temperaures kT\u001dmLc2so high that the weakly\ninteracting Lbosons were in thermal equilibrium, the\nnumber density of each of the six Lbosons is given by\nthe Planck distribution as\nn(T) =3\u0010(3)(kT)3\n\u00192(~c)3=9:609\u0002107T3\n(m K)3: (83)\nIn the limit of vanishing coupling j\u000bj!0, the number\nn(t)a3(t) ofLbosons within a \fxed comoving box doesnot change with time. So the number now n(t0)a3(t0) =\nn(t0) is the number at any earlier time multiplied by a3(t)\nn(t0) =n(t)a3(t): (84)\nAtvery early times, we may approximate the integral for\nthe time as a function of the scale factor aas [37]\nt(a) =1\nH0Za\n0dxp\n\n\u0003x2+ \nk+ \nmx\u00001+ \nrx\u00002\n\u00191\nH0Za\n0xdxp\nr=a2\n2H0p\nr: (85)\nThe Hubble constant and the fraction \n r= 9:0824\u000210\u00005\nthen give us the scale-factor as\na(t) =q\n2H0p\n\nrt= 2:04\u000210\u000010p\nt[s]: (86)\nIfNtypes of particles made up the radiation at very\nearly times, then the time and the temperature were re-\nlated by [38]\np\nNtT2=s\n3c2\n16\u0019Ga r= 3:25924\u00021020s K2(87)\nin whicharis the radiation constant\nar=\u00192k4\n15~3c3= 7:56577(5)\u000210\u000016J m\u00003K\u00004:(88)\nSo in terms of the number density (83), the scale-factor\n(86), and the time-temperature relation (87), the number\ndensity is roughly\nn(t0) =n(t)a3(t) =4:8\u0002109\nN3=4m\u00003: (89)\nIn the standard model, N= 126, but the actual number\nrelevant at high temperatures may be much higher. If\nwe assume thatN= 44, thenN\u00003=4= 1=64, and the\npresent number density of each kind of Lboson would\nbe\nn(t0) = 7:5\u0002107m\u00003: (90)\nLet us further assume that all six Lbosons get the\nsame mass mL. In this case, if their mass density is not\nto exceed the density of dark matter, then the inequality\n6mLn(t0)<\u001acdm= 2:24\u000210\u000027kg m\u00003(91)\nimplies that the mass mLmust be less than\nmL<4:9\u000210\u000036kg = 2:8 eV=c2: (92)\nThe lifetime of an Lboson (80) would then be\n\u001cL>3:4\nbj\u000bj\u0002106t0 (93)10\nwhich forbj\u000bj<1:7\u0002106exceeds the age t0of the\nuniverse. The range \u0015L=h=mLcof the corresponding\nYukawa potential is\n\u0015L>4:5\u000210\u00007m: (94)\nPoints (\u0015;j\u000bj) below the dotted green line in Fig. 6 and\nbetween its vertical dashed blue-green lines denote L\nbosons constituting between 1 and 100% of the dark mat-\nter. The upper limit on the e\u000bective mass of the electron\nneutrino is m(e\u000b)\n\u0017e<1:1 eV [33]. The thin gray vertical\nline labelsLbosons of mass mL= 2me\u000b\n\u0017e.\nXI. CONCLUSIONS\nGeneral relativity with fermions has two indepen-\ndent symmetries: general coordinate invariance and local\nLorentz invariance. General general coordinate invariance\nacts on coordinates and on the world indexes of tensors\nbut leaves Dirac and Lorentz indexes unchanged. Local\nLorentz invariance acts on Dirac and Lorentz indexes but\nleaves world indexes and coordinates unchanged. It acts\nlike an internal symmetry.\nGeneral coordinate invariance is implemented by the\nLevi-Civita connection \u0000j\nkiand by Cartan's tetrads ca\ni.\nIn the standard formulation of general relativity with\nfermions, local Lorentz invariance is implemented by the\nsame \felds in a combination called the spin connection\n!ab\ni=ca\njcbk\u0000j\nki+ca\nk@icbk. These \felds all have the\nsame action, the Einstein-Hilbert action R.\nBecause local Lorentz invariance is di\u000berent from and\nindependent of general coordinate invariance, it is sug-\ngested in this paper that local Lorentz invariance is im-\nplemented by di\u000berent and independent \felds Lab\nithat\ngauge the Lorentz group and that have their own Yang-\nMills-like action.\nThe replacement of the spin connection with Lorentz\nbosons moves general relativity closer to gauge theory\nand simpli\fes the standard covariant derivative\n\u0010\n@i+1\n8\u0000\nca\njcbk\u0000j\nki+ca\nk@icbk\u0001\n[\ra;\rb]\u0011\n (95)\nto\n\u0000\n@i+1\n8Lab\ni[\ra;\rb]\u0001\n : (96)Whether the Dirac action has the spin-connection form\n(95) or the Lorentz-boson form (96) is an experimental\nquestion.\nBecause the proposed action (15) couples the gauge\n\feldsLab\nito fermion number and not to mass, it violates\nthe weak equivalence principle. It also leads to a Yukawa\npotential (74) that violates Newton's inverse-square law.\nExperiments [7{30] have put upper limits on the\nstrengthj\u000bjof the Yukawa potentials (75) that violate the\ninverse-square law and the weak equivalence principle for\ndistances 10\u00008<\u0015< 109m. The upper limit ranges from\nj\u000bj<1019at\u0015= 10\u00008m toj\u000bj<103at\u0015= 10\u00005m and\ntoj\u000bj<10\u000011at\u0015= 109m. There are no experimental\nlower limits on the coupling at any distance, so Lbosons\ncould have lifetimes that exceed the age of the universe.\nThere are no experimental upper limits on the masses of\nLbosons. Long lived, massive, weakly interacting, neu-\ntralLbosons would contribute to dark matter. From the\nobvious requirement that they could make up all of dark\nmatter but not more, we can infer a crude theoretical\nupper limit on their mass of mL.2:8 eV=c2if all 6 are\nstable and have the same mass.\nThe discovery of a violation of the inverse-square law\nby future experiments would not be enough to establish\nthe existence of Lbosons because the violation could be\ndue to the physics of a quite di\u000berent theory.\nIfLbosons are discovered, physicists will decide how\nto think about the force they mediate. The force might\nbe considered to be gravitational because it arises in a\ntheory that is a modest and natural extension of general\nrelativity. But the force is not carried by gravitons. It is\ncarried by Lbosons, and they implement a symmetry,\nlocal Lorentz invariance, that is independent of general\ncoordinate invariance. So the force is new and might be\ncalled a Lorentz force.\nACKNOWLEDGMENTS\nI am grateful to E. Adelberger, R. Allahverdi, D.\nKrause, E. 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Still, only a few implementations of physical damping exist in compliant robotic\nlegged locomotion. It remains unclear how physical damping can be exploited for locomotion\ntasks, while its advantages as sensor-free, adaptive force- and negative work-producing actuators\nare promising. In a simplified numerical leg model, we studied the energy dissipation from\nviscous and Coulomb damping during vertical drops with ground-level perturbations. A parallel\nspring- damper is engaged between touch-down and mid-stance, and its damper auto-decouples\nfrom mid-stance to takeoff. Our simulations indicate that an adjustable and viscous damper is\ndesired. In hardware we explored effective viscous damping and adjustability, and quantified the\ndissipated energy. We tested two mechanical, leg-mounted damping mechanisms: a commercial\nhydraulic damper, and a custom-made pneumatic damper. The pneumatic damper exploits a\nrolling diaphragm with an adjustable orifice, minimizing Coulomb damping effects while permitting\nadjustable resistance. Experimental results show that the leg-mounted, hydraulic damper exhibits\nthe most effective viscous damping. Adjusting the orifice setting did not result in substantial\nchanges of dissipated energy per drop, unlike adjusting the damping parameters in the numerical\nmodel. Consequently, we also emphasize the importance of characterizing physical dampers\nduring real legged impacts to evaluate their effectiveness for compliant legged locomotion.\nKeywords: damping, energy dissipation, legged locomotion, ground disturbance, drop test, rolling diaphragm\n1 INTRODUCTION\nWhile less understood, damping likely plays an essential role in animal legged locomotion. Intrinsic\ndamping forces can potentially increase the effective force output during unexpected impacts (M ¨uller et al.,\n2014), reduce control effort (Haeufle et al., 2014), stabilize movements (Shen and Seipel, 2012; Secer and\nSaranli, 2013; Abraham et al., 2015), and reject unexpected perturbations (Haeufle et al., 2010; Kalveram\net al., 2012), e.g., sudden variations in the ground level (Figure 1). Stiffness, in comparison, has been\nstudied extensively in legged locomotion. Its benefits have been shown both in numerical simulations, e.g.,\nthrough spring-loaded inverted pendulum (SLIP) models (Mochon and McMahon, 1980; Blickhan et al.,\n1arXiv:2005.05725v2  [cs.RO]  6 Jun 2020Mo et al. Effective damping in legged locomotion\nFigure 1. (A-C) Problem identification, and related research question. The limited nerve conduction\nvelocity in organic tissue (More et al., 2010) 2presents a significant hazard in legged locomotion.\nLocal neuromuscular strategies 6provide an alternative means of timely and tunable force and power\nproduction . Actuators like the indicated knee extensor muscle keep the leg extended during stance phase\n(muscle length L muscle ) by producing the appropriate amount of muscle force (F muscle ), correctly timed .\nNeuromuscular control 1plays a major role in initiating and producing these active muscle forces, but\nworks best only during unperturbed locomotion. Sensor information from foot contact travels via nerves\nbundles 2to the spinal cord, but with significant time delays in the range of t=40ms (More and Donelan,\n2018, for 1 mleg length) and more. Hence, the locomotion control system can become ‘sensor blind’ due\nto conduction delays, for half a stance phase , and can miss unexpected perturbations like the depicted\nstep-down. During step-down perturbations 3additional energy 4is inserted into the system. Viscous\ndamper-like mechanisms produce velocity dependent counter-forces, and can dissipate kinetic energy .\nLocal neuromuscular strategies 6producing tunable, viscous damping forces would act instantaneously\nand adaptively . Such strategies 6could also be robust to uncontrolled and harsh impacts of the foot after\nperturbations 5, better than sensor-based strategies. In this work (D), we are testing and characterizing\nspring-damper configurations mounted to a two-segment leg structure, during rapid- and slow-drop\nexperiments, for their feasibility to physically and instantaneously produce tunable, speed-dependent forces\nextending the leg. Work loops (E)will indicate how much effective negative work is dissipated, between\ntouch-down and mid-stance. Prior to impact 7and during the leg loading 8the spring-damper’s tendons\nact equally. Starting at mid-stance, the main spring extends the knee, leading to leg extension and leaving\nthe damper’s tendon slack 9.\n2007), and physical springy leg implementations (Hutter et al., 2016; Spr ¨owitz et al., 2013; Ruppert and\nBadri-Spr ¨owitz, 2019).\nWhat combines both mechanical stiffness and intrinsic, mechanical damping is their sensor- and\ncomputational-free action. A spring-loaded leg joint starts building up forces exactly at the moment\nof impact. Mechanical stiffness, or damping, acts instantaneously, and are not subject to delays from post-\nprocessing sensor data (Grimminger et al., 2020), delays from limited nerve conductive velocities (More\nand Donelan, 2018), or uncertainties in the estimation of the exact timing of swing-to-stance switching\n(Bledt et al., 2018).\nLegged robots commonly exploit virtual damping : actively produced and sensory-controlled negative\nwork in the actuators (Seok et al., 2015; Hutter et al., 2012; Havoutis et al., 2013; Kalouche, 2017;\nGrimminger et al., 2020). Virtual damping requires high-frequency force control, and actuators\nmechanically and electrically capable of absorbing peaks in negative work. In comparison, mechanical\ndamping based systems (Hu et al., 2019; Garcia et al., 2011) act instantaneously, share impact loads with\nthe actuator when in parallel configuration, and require no sensors or control feedback. The instantaneous\nmechanical response of a damper is especially relevant in biological systems, where the neuronal delay\nmay be as large as 5 %to40 % of the duration of a stance phase (More et al., 2010). In such a short\nThis is a provisional file, not the final typeset article 2Mo et al. Effective damping in legged locomotion\ntime-window, physical damping could help to reject the perturbation (Haeufle et al., 2010; Kalveram et al.,\n2012) by morphological computation, as it mechanically contributes to the rejection of the perturbation,\na contribution that otherwise would need to be achieved by a (fast) controller (Zahedi and Ay, 2013;\nGhazi-Zahedi et al., 2016). Hence, physical damping has the potential to contribute to the morphological\ncomputation (Zahedi and Ay, 2013; Ghazi-Zahedi et al., 2016) of a legged system.\nCompared to virtual damping with proprioceptive sensing strategies (Grimminger et al., 2020), a legged\nrobot with physical damping requires additional mechanical components, e.g., a fluidic cylinder, and the\nmechanics to convert linear motion to rotary output. In a cyclic locomotion task, the energy removed by\nany damper must also be replenished. Hence, from a naive energetic perspective, both virtual and physical\ndamping systems are costly.\nEnergy dissipation in the form of negative work has been quantified in running birds, and identified as a\npotential strategy to ‘... reduce the likelihood of a catastrophic fall.’ (Daley and Biewener, 2006, p. 185).\nIn virtual point-based control strategies for bipedal running, positive work is inserted into hip joints, and\nnegative work is then dissipated in equal amounts in the spring-damper leg (Drama and Spr ¨owitz, 2020).\nIn sum, either physical damping or virtual damping allows removing energy from a legged locomotion\nsystem. In this work, we focus on physical damping produced by a viscous damper. We aim towards an\nunderstanding of how physical damping can be exploited in legged locomotion and which requirements a\ndamper must fulfill.\nWe consider two damping principles: viscous damping and Coulomb damping. Viscous damping reacts\nto a system motion with a force that is linearly (or non-linearly) proportional to its relative acting speed.\nCoulomb damping generates a constant force, largely independent from its speed (Serafin, 2004). From a\ncontrol perspective, viscous damping can be beneficial for the negotiation of perturbations in locomotion\nas it approximates the characteristics of a differential, velocity-dependent term. Yet it is unknown how this\nintuition transfers into reality, where impact dynamics and non-linearities of the leg geometry alter the\nstance-phase dynamics of locomotion.\nDamping in legged locomotion can have other purposes, besides dissipating energy. The authors of\n(Werner et al., 2017, p. 7) introduced a damping matrix in the control scheme, which reduced unwanted\noscillations in the presence of modeling errors. Tsagarakis et al. (2013) mount compliant elements with\nsome damping characteristics, which also could reduce oscillations of the system’s springy components.\nIn this project, we focus our investigation on the effect of damping during the touch-down (impact)\nand mid-stance. We chose this simpler drop-down scenario as it captures the characteristics of roughly\nhalf a locomotion cycle. A complete cycle would require an active push off phase, and the leg’s swing\ndynamics. Hence, we study the effectiveness of physical damping on the leg’s energy dissipation within\none drop (touch-down to lift-off), by quantifying its effective dissipated energy Eeffective . We combine\ninsights from numerical simulations and hardware experiments (Figure 2). By studying the response\nof two damping strategies (viscous and Coulomb damping) in numerical drop-down simulations, we\ninvestigate how physical damping can influence the dynamics of the impact phase. We then examine\nhow these theoretical predictions relate to hardware experiments with two functionally different, physical\ndampers. Hence we explore and characterize the physical damper implementations in a robot leg for their\neffectiveness in drop-impacts.\nFrontiers 3Mo et al. Effective damping in legged locomotion\nHardware experiment: leg dropviscous    vs.   Coulomb friction\nvariation:                   ground level heightNumerical simulation: leg drop\nground level variation: valve weigh\nhydraulic diaphragm\n spring\nvariation:\nHardware experiment: hydraulic damper drop\nheight setting\ndamper only damper\nvalve\nsettingvalve\nsetting\n1\n 2\n3\n 4\n 5\n6\n 7\n 8\nFigure 2. Overview: We study the effective dissipated energy Eeffective in drop experiments, i.e., the energy\ndissipation within one drop cycle between touch-down and lift-off (Figure 6). We focus on a system\ndesign with a damper and a spring, both acting in parallel on the knee joint (Figure 1 Eand Figure 3).\nNo active motor is considered as it is not relevant for the drop scenario, but required for continuous\nhopping. In numerical simulations , we quantify the difference in energy dissipation between viscous 1\nand Coulomb damping for varying ground level heights 2(Section 2 and Figure 4). The first set of\nhardware experiments characterizes the industrial hydraulic damper. For this, we drop the isolated\ndamper (damper only, not mounted in the leg) on a force sensor and calculate the energy dissipation.\nWe vary the ground level height 3, the valve setting 4and the drop mass 5, to investigate its dynamic\ncharacteristics (Section 4.1 and Figure 7). For the second set of hardware experiments, we drop a\n2-segment leg with dampers mounted in parallel to knee springs . We investigate the energy dissipation\ndynamics of the hydraulic 6and diaphragm damper 7by comparing it to a spring-only condition 8, where\nthe damper cable is simply detached (Section 4.2 and Figure 8). We also vary the valve setting on the\ndampers to test the dynamic adjustability of damping (Section 4.3 and Figure 9).\n2 NUMERICAL SIMULATION\nWe use numerical simulations to theoretically investigate the energy dissipation in a leg drop scenario\n(Figure 2). In analogy to our hardware experiment (Section 3.3), a 2-segment leg with a damper and a\nspring in parallel on the knee joint is dropped vertically (Figure 3a). Once in contact with the ground, the\nknee flexes and energy is dissipated. We compare viscous vs. Coulomb damping to investigate which of\nthese two theoretical damping strategies may be more suited for the rejection of ground-level perturbations.\nAlso, we investigate how the adjustment of the damping characteristics influences the dissipated energy.\nIn all the damping scenarios investigated, the system is not energy conservative. As we investigate the\npotential benefit of damping in the initial phase of the ground contact, i.e., from touch-down to mid-stance,\nwe do not consider any actuation. Without actuation or control, the model’s dissipated energy is not refilled,\nunlike in, for example, periodic hopping (Kalveram et al., 2012).\nThis is a provisional file, not the final typeset article 4Mo et al. Effective damping in legged locomotion\nl0 = \nd\nk\nmβ\nλ1\nλ\n2\nFlegg\ny(t)α\nrkrd\n24.5cm\n(a)Leg model\n⑥\n⑦\n⑧\n⑨\n⑩\n⑪\n⑫⑬⑭⑮\n⑯ (b)Leg design\n (c)Drop test bench\nFigure 3. (a) 2-segment spring-damper-loaded leg model used for simulation. (b)Mechanical design of\nthe 2-segment leg. The knee pulley 11\ris fixed with the lower segment 12\r, coupled with the spring 8\rand the\ndiaphragm damper 15\ror hydraulic damper 16\rvia cables 9\r10\r.(c)Drop test bench with the 2-segment leg.\n2.1 Model\nThe numerical model is a modified version of the 2-segment leg proposed in Rummel and Seyfarth\n(2008) with an additional damper mounted in parallel to the knee-spring. The equation describing our leg\ndynamics is:\n¨y(t) =Fleg(t)\nm\u0000g (1)\nwhere gis the gravitational acceleration, mis the leg mass (lumped at the hip), and y(t)is the time-\ndependent vertical position from the ground. Fleg(t)is the force transmitted to the hip mass - and the\nground - through the leg structure. As such, the force depends on the current phase of the hopping cycle:\nFleg(t) =8\n<\n:0 , flight phase: y(t)>l0\ny(t)\nl1l2t(t)\nsin(b(t)), ground contact: y(t)\u0014l0(2)\nwith segment length liand knee angle b(t)(Figure 3a), l0is the leg length at impact. t(t)is the knee\ntorque which is produced by the parallel spring-damper element, as in\nt(t) =k r2\nk(b(t)\u0000b0)+td(t) (3)\nwith kandrkbeing the spring stiffness coefficient and lever arm, respectively. td(t)is the damping torque,\nwhich is set to zero during leg extension, i.e., the damper is only active from impact to mid-stance:\ntd(t) =0 if ˙b(t)>0 (4)\nThe modeled damper becomes inactive during leg extension, in accordance to our hardware: the tested\nphysical dampers apply forces to the knee’s cam via a tendon (Figure 1d, 9), and this tendon auto-decouples\nduring leg extension. By choosing different definitions of the damper torque td(t), we can analyse different\ndamper concepts. The model parameters are listed in Table 1.\nFrontiers 5Mo et al. Effective damping in legged locomotion\nSimulations were performed using MATLAB (the MathWorks, Natick, MA) with ODE45 solver (absolute\nand relative tolerance of 10−5, max step size of 10−5s). When searching for appropriate settings of the\nnumerical solver, we progressively reduced error tolerances and the maximum step size until convergence\nof the simulation results in Table 2 to the first non-significant digit.\nTable 1. Simulation and hardware parameters\nParameters Symbol Value Unit\nMass m 0.408 kg\nReference drop height h0 14 cm\nSpring stiffness k 5900 N/m\nLeg segment length l1;l2 15 cm\nLeg resting length l0 24.6 cm\nKnee resting angle b0 110 deg\nSpring lever arm rk 2.5 cm\nDamper lever arm rd 2 cm\n2.2 Damping characteristics\nWe compared two damping concepts in our numerical simulation: (1) pure Coulomb damping, i.e., a\nconstant resistance only dependent on motion direction, and pure viscous damping, i.e., a damper torque\nlinearly dependent on the knee angular velocity. Accordingly, we tested two different definitions of td:\ntd(t) =8\n<\n:\u0000dcrdsign(˙b(t)) , pure Coulomb damping\n\u0000dvr2\nd˙b(t) , pure viscous damping(5)\nwhere rdis the damper level arm, dc(inN) and dv(inNs/m ) the Coulomb damping and viscous damping\ncoefficients, respectively.\n2.3 Energy dissipation in numerical drop simulations\nWith this model, we investigate the difference in energy dissipation in response to step-up/down\nperturbations (cases 1and 2in Figure 2). For each drop test, the numerically modeled leg starts at\nrest ( ˙ y(t) =0) with a drop height\nh=y(t=0)\u0000l0 (6)\ncorresponding to the foot clearance at release. The total energy at release is ET(h) =mgh . Given that\nall model parameters in Table 1 are fixed, the energy dissipated in a drop becomes a function of the drop\nheight and the damping coefficients: ED=fED(h;dc;v).\nA simulated drop height hcan be seen as a variation Dhfrom a reference value h0:\nh=h0\u0006Dh (7)\nEqual to the hardware experiments, we use h0=14cm as reference drop height. In the reference drop\ncondition, i.e., h=h0, the energy dissipated by damping is ED0=ED(h0) = fED(h0;dc;v).ED0only\ndepends on the damping level, namely the chosen damping strategy (viscous or Coulomb damping) and\nassociated damping coefficient. We chose five different desired damping levels (set 1-5) as a means of\nThis is a provisional file, not the final typeset article 6Mo et al. Effective damping in legged locomotion\nscanning a range in which the damping could be adjusted: for each set, the amount of energy that is\ndissipated at the reference drop height ED0differs. The chosen ED0values (Table 2, column “Reference\nheight”) correspond to proportional levels ( [0:1;0:2;:::; 0:5]) of the systems potential energy in terms of\nthe leg resting length l0, as in\nED0=mg[0:1;0:2;:::; 0:5]l0 (8)\nThis corresponds to damping configurations that dissipate between \u001917% and\u001988% of the system’s\ninitial potential energy at the reference height ( ET0=ET(h0) =mgh 0=560mJ ), as shown in Table 2,\ncolumn “Reference height”. To achieve these desired damping levels, we adjusted the damper parameters\ndcanddvaccordingly (Table 2, column “Damping coeff.”). As an example: for set 3, both damping values\nwere adjusted such that at the reference height h0both dampers dissipate ED0=mg0:3l0=295mJ , which\ncorresponds to 53 % of the total energy ET0.\nIn the numerical simulations, we focus on the relation between a ground level perturbation Dhand the\nchange in energy dissipation – and their dependency on the damper characteristics. A drop from a height\nlarger than h0corresponds to a step-down ( Dh>0), and a drop from a height smaller than h0to a step-up\n(Dh<0). Each condition introduces a change of the total energy of DET=mgDh. The change in energy\ndissipation due to the perturbation is defined as\nDED(Dh) =ED(h0+Dh)\u0000ED0 (9)\nwhich is the difference between the dissipated energy when released from a perturbed height and the\ndissipated energy when released from the reference height. As a reference, we further define the full\nrejection case where\nDED(Dh) =DET=mgDh (10)\nIn human hopping a full recovery within a single hopping cycle is not seen during experimental drop down\nperturbations. Instead, a perturbation of Dh=0:1l0is rejected in two to three hopping cycles (Kalveram\net al., 2012, Figure 7a). In our results, this corresponds to the partial rejections observed with sets 2and3\nforDh=\u00062:5cm.\n2.4 Simulation results\nFigure 4a shows the relation between the change in drop height and the corresponding change in dissipated\nenergy by the simulated dampers for set 1,3and5(continuous line for pure viscous, dashed for pure\nCoulomb damping). For the range of simulated drop heights, pure viscous and Coulomb dampers change\nthe amount of dissipated energy with an almost linear dependence on the drop height. However, pure\nviscous damping has a slope closer to the full rejection scenario (blue line in Figure 4a), regardless of the\nset considered. In a step-down perturbation ( Dh>0in Figure 4a), pure viscous damping dissipates more\nof the additional energy DET, while in a step-up perturbation ( Dh<0) it dissipates less energy than pure\nCoulomb damping. As such, the results show that a viscous damper can reject a step-down perturbation\nfaster, e.g., within less hopping cycles, and it requires smaller correction by the active energy supply during\na step-up perturbation.\nAdjusting the damping parameters allows to change the reaction to a perturbation (Figure 4). Increasing\nthe damping intensity, i.e., dvanddcfrom set 1to5, allows to better match the full recovery behaviour (blue\nline in Figure 4a). However, this comes at the cost of a higher energy dissipation at the reference height,\nFrontiers 7Mo et al. Effective damping in legged locomotion\nTable 2. Numerical simulation Total dissipated energy ( ED) in one drop cycle for different drop heights\n(h).Reference height is the reference drop height h=h0=14cm . During step up(down) condition, the\ndrop height is reduced(increased) by Dh=2:5cm. Percentage values indicate the change in dissipated\nenergy ( DED) relative to the change in system total energy ( DET) due to the height perturbations. Each\nset simulates two separate mechanical dampers (pure viscous or pure Coulomb damping), with damping\ncoefficients chosen to dissipate the same energy at the reference condition, i.e., ED0. Results of set 1,3\nand5are further described in Figure 4. For all tested conditions, viscous damping outperforms Coulomb\ndamping, as indicated by the always higher percentage values.\nDamping coeff.Step up Reference height Step down\nh=h0\u0000Dh=11:5cm h=h0=14cm h=h0+Dh=16:5cm\ndv dc ED(DED=DET) ED0(ED0=ET0) ED(DED=DET)\nSet 1Viscous 29.5 Ns/m 0 N 82 mJ (15%) 97 mJ (17%) 112 mJ (15%)\nCoulomb 0 Ns/m 7.7 N 88 mJ (9%) 97 mJ (17%) 104 mJ (7%)\nSet 2Viscous 68 Ns/m 0 N 167 mJ (30%) 197 mJ (35%) 227 mJ (30%)\nCoulomb 0 Ns/m 17.3 N 178 mJ (19%) 197 mJ (35%) 214 mJ (17%)\nSet 3Viscous 119.4 Ns/m 0 N 249 mJ (46%) 295 mJ (53%) 341 mJ (46%)\nCoulomb 0 Ns/m 29.3 N 264 mJ (31%) 295 mJ (53%) 323 mJ (28%)\nSet 4Viscous 197.1 Ns/m 0 N 330 mJ (63%) 393 mJ (70%) 455 mJ (62%)\nCoulomb 0 Ns/m 46.1 N 346 mJ (47%) 393 mJ (70%) 436 mJ (43%)\nSet 5Viscous 349.4 Ns/m 0 N 411 mJ (81%) 492 mJ (88%) 572 mJ (80%)\nCoulomb 0 Ns/m 76.3N 423 mJ (69%) 492 mJ (88%) 556 mJ (64%)\ni.e., in absence of a ground perturbation (Table 2, column ‘reference height’). Increasing the damping rate\nalso affects the energetic advantage of viscous damping over Coulomb damping. Figure 4b shows this\nin detail for a specific step down perturbation (Dh=2:5cm): from set 1to set 3, the spread between the\nDEDvalues of the viscous damper and the Coulomb damper increases (from 8 mJ to18 mJ ). However, the\ndifference in dissipated energy DEDslightly reduces from set 3 to set 5 (from 18 mJ to 16 mJ).\nTable 2 quantifies the previous findings by indicating the percentage of energy perturbation DETthat\neach damping approach dissipates for Dh=\u00062:5cm and for all the tested sets of damping coefficients dv\nanddc. The data further confirms the observations from Figure 4, showing that:\n1.within each set, viscous damping outperforms Coulomb damping for all the simulated conditions - its\ndissipated energy is always the closest to 100 % of DET, which means the closest to full rejection ;\n2.the energetic benefit of viscous damping over Coulomb damping, i.e., the spread in percentage values\nofDED=DET,does not monotonically increase with higher damping rates, i.e., moving from set 1 to 5.\nFurthermore, Table 2 shows that for small damping rates, i.e., set 1, viscous damping introduces only\nmarginal benefits in energy management compared to Coulomb damping: <10% spread between the\ncorresponding DED=DETvalues.\n3 HARDWARE DESCRIPTION\nWith the previous results from our numerical simulation in mind, we tested two technical implementations\n(Figure 5) to produce adjustable and viscous physical damping. We implemented a 2-segment leg hardware\n(Figure 3b) and mounted it to a vertical drop test bench to investigate the role of physical damping. The\ndrop test bench produces velocity profiles during impact and stance phase similar to continuous hopping\nand allows us testing effective damping efficiently and repeatable.\nThis is a provisional file, not the final typeset article 8Mo et al. Effective damping in legged locomotion\nFigure 4. Numerical simulation Cases 1and 2from Figure 2, (a): Change of total energy vs. change\nof drop height for set 1,3and5, with damping coefficients as in Table 2. Continuous lines are viscous\ndamping results, dashed Coulomb damping. Positive perturbations, i.e., Dh>0, correspond to step-down\nperturbations; step-up perturbations, otherwise. The steepest line indicates the slope needed for a full\nrejection of aDhdeviation. For each set (1, 3, 5), the damping parameters are matched such that viscous\nand Coulomb damping dissipate the same energy at the reference height h0(see Table 2). Within each\nset, the viscous damping line is closer to the desired full rejection line than the corresponding Coulomb\ndamping line. This means that for the same cost (in the sense of dissipated energy at the reference height)\nviscous damping always rejects more of ground level perturbation than Coulomb damping. (b):DED\nforDh=2:5cm. The horizontal line indicates the amount of energy to dissipate for full rejection of\nDh. Energetic advantage of viscous damping over Coulomb damping, as indicated by the spread in the\ncorresponding DEDvalues, increases from set 1 to 3, and reduces from set 3 to 5.\n3.1 Rolling Diaphragm Damper\nThe most common designs of viscous dampers are based on hydraulic or pneumatic cylinders (viscous\ndamping) and can offer the possibility of regulating fluid flow by altering the orifice opening (adjustability).\nThese physical dampers can display high Coulomb friction, caused by the mechanical design of the sliding\nseal mechanisms. Typically, the higher the cylinder pressure is, the higher the Coulomb friction exists.\nIdeally, we wanted to test one physical damper concept with the least possible amount of Coulomb friction.\nInspired by the low-friction hydrostatic actuators (Whitney et al., 2014, 2016), we designed a low-Coulomb\ndamper based on a rolling diaphragm cylinder. Its cylinder is 3D printed from Onyx material. Figure 5a\nillustrates the folding movement of this rolling diaphragm mounted on a piston. The rolling diaphragm\nis made of an elastomer shaped like a top hat that can fold at its rim. When the piston moves out, the\ndiaphragm envelopes the piston. In the ideal implementation, only rolling contact between the diaphragm\nand the cylinder occurs, and no sliding contact. Hence, Coulomb friction between piston and cylinder is\nminimized. We measured FC\u00190:3Nof Coulomb friction for our rolling diaphragm cylinder, at low speed.\nFrontiers 9Mo et al. Effective damping in legged locomotion\n①\n②\n(a)Diaphragm damper\n (b)Hydraulic damper\nFigure 5. (a) -left-top: schematic of a diaphragm damper, illustrating the motion of rolling diaphragm,\nwhich includes an adjustable orifice 1\r, a cylinder 2\r, a piston 3\r, and a rolling diaphragm 4\r.(b)-left:\nschematic of a hydraulic damper, fluid is sealed inside the cylinder 2\rwith an recovery spring 5\rto reset\nthe piston 3\r.\nOur numerical simulation results promoted viscous and adjustable damping for use in vertical leg-drop.\nBy concept, both properties are satisfied by the diaphragm damper with an adjustable valve. When an\nexternal load Fextpulls the damper piston (Figure 5a), the fluid inside the cylinder chamber flows through\na small orifice, adjustable by diameter. This flow introduces a pressure drop DP(t), whose magnitude\ndepends on the orifice cross-section area Aoand piston speed v(t). As such, for a given cylinder cross\nsection area Ap, the diaphragm damper reacts to an external load Fextby a viscous force Fp(t)due to the\npressure drop DP(t):\nFp(t) =ApDP(t) =Apf(v(t);Ao) (11)\nWe mounted a manually adjustable valve (SPSNN4, MISUMI) to set the orifice size Ao. For practical\nreasons (weight, leakage, complexity of a closed circuit with two cylinders) we used air in the diaphragm\ncylinder as the operating fluid, instead of liquid (Whitney et al., 2014, 2016). Air is compressible, and\nwith a fully closed valve the diaphragm cylinder also acts as an air spring. This additional functionality\ncan potentially simplify the overall leg design. With the pneumatic, rolling diaphragm-based damper\nimplementation, we focused on creating a light-weight, adjustable damper with minimal Coulomb friction,\nand air as operating fluid.\n3.2 Hydraulic Damper\nIn the second technical implementation we applied an off-the-shelf hydraulic damper (1214H or 1210M,\nMISUMI, Figure 5b), i.e., a commercially available solution for adjustable and viscous damping. Tested\nagainst other hydraulic commercial dampers, we found these specific models to have the most extensive\nrange of adjustable viscous damping and the smallest Coulomb friction ( FC\u00190:7N). Similarly to the\ndiaphragm damper, these hydraulic dampers produce viscous damping by the pressure drop at the adjustable\norifice. The operating fluid is oil, which is in-compressible. Hence, the hydraulic damper should not exhibit\ncompliant behavior. Other than the diaphragm damper, the hydraulic damper produces damping force when\nThis is a provisional file, not the final typeset article 10Mo et al. Effective damping in legged locomotion\nits piston is pushed, not pulled. This design also includes an internal spring to recover the piston position\nwhen unloaded. In sum, the hydraulic damper features high viscous damping, no air-spring effect, and a\nhigher Coulomb friction compared the custom-designed pneumatic diaphragm damper.\n3.3 Articulated Leg Design\nThe characteristics of a viscous damper strongly depend on the speed- and force-loading profile imposed\nat its piston, because of the complex interaction of fluid pressure and compression, viscous friction, and\ncavitation (Dixon, 2008). We implemented a hardware leg to test our two physical dampers at loading\nprofiles (speed, force) similar to legged hopping and running.\nThe 2-segment hardware leg (Figure 3b) is designed with a constant spring and damper lever arm,\nparameters are provided in Table 3. In all experiments with the 2-segmented leg, the leg spring provides\nelastic joint reaction forces. Dampers are swapped in and out in a modular fashion, depending on the\nexperimental settings. The 2-segment leg design parameters are identical to those in our simulation model\n(Table 1). A compression spring 8\ris mounted on the upper leg segment 13\r. When the leg flexes, the spring\nis charged by a spring cap 7\rcoupled to a cable 10\rattached to the lower leg. Either damper 15\r16\ris fixed on\na support 6\ron the upper segment 13\r. The support 6\rcan be moved within the upper segment 13\r, to adjust\nthe cable 9\rpretension. Cables 9\r10\rlink the damper piston 3\rand the spring 8\rto the knee pulley 11\r, which\nis part of the lower segment 12\r.\nDuring the leg flexion, the cable under tension transmits forces instantly to the spring and damper. Spring\nand damper forces counteract the knee flexion. During leg extension, the spring releases energy, while\nthe damper is decoupled due to slackness of the cable. We included a hard stop into the knee joint to\nlimit the maximum leg extension, and achieve a fixed leg length at impact. At maximum leg flexion at\nhigh leg loading, segments can potentially collide. We ensured not to hit either hard stops during the\ndrop experiments. The hydraulic damper 16\rrequires a reverse mechanism 14\r, since its piston requires\ncompression to work. The piston of the diaphragm damper 15\rwas directly connected to the knee pulley.\nThe diaphragm damper 15\rincluded no recovery spring 5\r, hence we reset the piston position manually after\neach drop test. In sum, different spring-damper combinations can be tested with the 2-segment leg setup.\nNote that the here shown hardware leg has no actuation. If a motor would actuate the knee joint, in parallel\nmounted to the spring and the damper, the damper would share the external impact load, and consequently\nreduce an impact at the motor.\n3.4 Experimental set-up, data sampling and processing\nWe implemented an experimental setup for repetitive measurements (Figure 3c). A drop bench was\nused to constrain the leg motion to a single vertical degree of freedom, and linear motion. This allowed\nus to fully instrument the setup (slider position, and vertical ground reaction forces, GRF), and ensured\nrepeatable conditions over trials. Adjusting the drop height allowed us setting the touch-down speed. A\nlinear rail (SVR-28, MISUMI) was fixed vertically on a frame. The upper leg segment was hinged to a rail\nslider. The rail slider was loaded with additional, external weights, simulating different robot masses. We\nset the initial hip angle a0to align the hip and foot vertically. A hard stop ensured that the upper leg kept a\nminimum angle a>a0.\nFrontiers 11Mo et al. Effective damping in legged locomotion\nTwo sensors measured the leg dynamics: the body position yand the vertical ground reaction force are\nrecorded by a linear encoder (AS5311, AMS) and a force sensor (K3D60a, ME, amplified with 9326 ,\nBurster), respectively (Figure 3c). The duration from touch-down to mid-stance is very short, typically\nt\u0014100ms , and high-frequency data sampling was required. The encoder data was sampled by Raspberry\nPi 3B+ with f=8kHz sampling rate. Force data were recorded by an Arduino Uno, with a 10-bit internal\nADC at 1 kHz sampling rate. A high-speed camera (Miro Lab 110, Phantom) recorded the drop sequence\natf=1kHz sampling rate. We performed ten trials for each test condition. Sensor data was processed with\nMATLAB (the MathWorks, Natick, MA). Data was smoothed with a moving average filter, with a filter\nspan of 35samples for encoder data, and 200samples for force data. Repeated experiments of the same\ntest condition are summarized as an envelop defined by the average \u0006the standard deviation of the filtered\nsignals.\n4 HARDWARE EXPERIMENTS AND RESULTS\nIn the drop experiments, we characterize both the hydraulic and diaphragm dampers, and the 2-segment\nspringy leg (Figure 6). We chose three orifice settings (labeled as a, b, and c) for each damper, and focus on\nthe effects of viscous damping and adjustable dissipation of energy in the hardware setup. Table 3 lists an\noverview of the drop tests, and its settings (drop height, weight, orifice setting, damper type). To emphasize\nthe fundamental differences between the damper designs, we compare only one model of the hydraulic\ndamper (1214H) to the diaphragm damper (Section 4.1 - Section 4.3), and show the potential of the second\nhydraulic damper (1210M) in Section 4.4. Videos of the experiments can be found in the supplementary\nmaterial, and online1.\nTable 3. Drop test settings for experiments\nDrop test setup Drop height Drop weight Orifice\nFig. [cm] [g] [ \u0018]\nDamper (1214H)7a 3, 5, 7 280 b\n7b 5 280 a, b, c\n7c 3 280, 620 b\nDamper (1214H, diaphragm) & leg8a, 8b 14 408 c\n8c 14 408 damper detached\nDamper & leg (simulation)9a, 9b 14 408 a, c\n9c 14 408 viscous, Coulomb\nDamper (1210M) & leg 10 14 408 a, b\n4.1 Isolated damper drops, evaluation\nIn this experiment we characterized the hydraulic damper by dropping it under changing conditions of\nthe instrumented drop setup, without mounting it to the 2-segment leg. The experimental setup allows\ndifferentiating effects, compared to the 2-segment leg setup, and to emphasize the viscous damper behavior\nof the off-the-shelf component. We also applied the results to estimate the range of damping rates available\nwith changing orifice settings. The hydraulic damper was directly fixed to the rail slider into the drop bench\n(Section 3.4). The piston pointed downwards. We measure the vertical ground reaction force to determine\nthe piston force, and we recorded the vertical position of the slider over time, to estimate the piston speed\nafter it touches the force sensor.\n1https://youtu.be/F00Sma2BQ4c\nThis is a provisional file, not the final typeset article 12Mo et al. Effective damping in legged locomotion\n80 85 90 95 100\nLeg length [%]0123GRF vertical [BW]80 85 90 95 100\nLeg length [%]0123GRF vertical [BW]\nHydraulic damper Diaphragm damperTouch-down 215 msMid-stance 271 ms\nLift-off 364 ms\nTouch-down 216 msMid-stance 280 ms\nLift-off 371 ms0 ms 215 ms 271 ms 364 ms 400 ms 441 ms\n0 ms 216 ms 280 ms 371 ms 417 ms 468 ms\nFigure 6. High-speed snapshots of drop experiments starting from release to second touchdown. Leg with\nhydraulic damper is shown on the top row, leg with diaphragm damper the bottom row. Depicted are from\nleft to right: release, touchdown, mid-stance, lift-off, apex, second touchdown. The right plots illustrate the\ntiming of the events corresponding to the snapshots.\nFigure 7 shows the force-speed profiles for drop tests with different drop heights (Figure 7a), orifice\nsettings (Figure 7b), and drop loads (Figure 7c). Data lines in Figure 7 should be interpreted from high\nspeed (impact, right side of each plot) to low speed (end of settling phase, 0 m/s , left). The time from\nimpact to peak force (right slope of each plot) is ( \u001924 ms ), while the negative work (shown in legends)\nwas mainly dissipated along the falling slope in the much longer-lasting settling phase after the peak (left\nslope of each plot, \u0019200 ms).\n0 0.2 0.4 0.6 0.8 1\nPiston speed [m/s]0204060GRF vertical [N]3 cm: 56 mJ\n5 cm: 84 mJ\n7 cm: 116 mJ\n(a)Three drop heights\n0 0.2 0.4 0.6 0.8\nPiston speed [m/s]0204060GRF vertical [N]Orifice a: 89 mJ\nOrifice b: 82 mJ\nOrifice c: 81 mJ\n (b)Three valve settings\n0 0.2 0.4 0.6 0.8\nPiston speed [m/s]0204060GRF vertical [N] ~620 g: 241 mJ\n~280 g: 57 mJ\n (c)Two system weights\nFigure 7. Characterizing the hydraulic damper A single damper (not leg-mounted) drops onto the\nforce sensor. 10repeated experiments are plotted as an envelop, defined by the average \u000695 % of the\nstandard deviation data. The curves are read from right to left, i.e. from touch-down at maximum speed\nto zero speed at rest, also corresponding to the maximum damper compression. (a)280 g drop mass with\nmedium orifice in 3drop heights. (b)280 g drop mass with 5 cm drop height in 3orifice settings. (c)3 cm\ndrop height with medium orifice in 2 drop weights.\nThe results from tests with drop heights from 3 cm to7 cm show viscous damping behavior in the\nsettling phase after peak force (left slope), with higher reaction forces at higher piston speeds with higher\ndissipation, ranging from 45 N for maximum speeds of 0.6 m/s with 56 mJ to65 N at0.9 m/s with 116 mJ .\nThe piston force almost linearly depends on the piston speed (Figure 7a).\nFrontiers 13Mo et al. Effective damping in legged locomotion\nChanging the orifice setting at a constant drop height resulted in different settling slopes (Figure 7b).\nApplying a least-squares fit on the left-falling settling slope, we estimate an adjustable damping rate\nbetween 91 Ns/m and192 Ns/m . The dissipated energy changes from 89 mJ to81 mJ , respectively. Hence\nadjusting the orifice setting has an effect on the damping rate and the dissipated energy in the isolated\nhydraulic damper, but not as we intuitively expected.\nWe interpret the rising slope in the impact phase (right part of each curve, Figures 7a and 7b) as a build-up\nphase; the hydraulic damper takes time ( \u001924 ms ) to build up its internal viscous flow and the related piston\nmovement, after the piston impact. With heavier weights ( 620 g = heavy, 280 g = light, Figure 7c), the\nimpact phase equally lasts \u001924 ms . After the impact phase with heavy weight, the damper shows the same\ndamping rate in the settling phase, in form of an equal left slope.\nSimilar drop tests for the evaluation of the isolated diaphragm damper were not possible since the\norientation of the internal diaphragm only permits to pull the piston. In the following section, we test the\ndiaphragm (connected by a piston reverse mechanism) and the hydraulic damper directly on the 2-segment\nleg structure.\n4.2 Composition of dissipated energy\nWe performed drop tests of two damper configurations: one off-the-shelf hydraulic damper, and custom-\nmade pneumatic damper, each mounted in parallel to a spring at the 2-segment leg (Section 3.3, Figure 3b),\nto quantify the effect of viscous damping for drop dynamics similar to legged hopping.\nFor each drop, the effective dissipated energy Eeffective was computed by calculating the area enclosed\nby the vertical GRF-leg length curve from touch-down to lift-off (Josephson, 1985), i.e., the work-loop\narea. These work-loops are to be read counter-clockwise, with the rising part being the loading during leg\nflexion, and the falling part being the unloading, due to spring recoil. Eeffective does not only consist of the\nviscous loss Eviscous due to the damper, but also Coulomb friction loss in the leg ( Ecfriction ) and the impact\nlossEimpact due to unsprung masses:\nEeffective =Ecfriction +Eimpact +Eviscous : (12)\nWe propose a method to indirectly calculate the contribution of viscous damping, by measuring and\neliminating effects from Coulomb friction, and unsprung masses.\nTo quantify the Coulomb friction loss Ecfriction , we conducted ‘slow drop’ tests. The mechanical setup is\nidentical to ‘free drops’ test, where the leg is freely dropped from a fixed height. However, in the ‘slow\ndrop’ experiment the 2-segment leg is lowered manually onto the force plat, contacting and pressing the\nleg-damper-spring system onto the force plate. At slow speed only Coulomb friction in joints and damper\nact, but no viscous damping or impact losses occur. Consequently the dissipated energy calculated from the\nsize of the work loop is due to Coulomb friction losses Ecfriction .\nTo identify the impact loss Eimpact , we remove the viscous component first by detaching the damper\ncable on the setup. A ‘free drop’ test in this spring only condition measures the contribution of friction\nlossEcfriction and impact loss Eimpact combined. A ‘slow drop’ test of the same setup is able to quantify\nthe friction loss Ecfriction . The impact loss Eimpact is therefore estimated as the energy difference between\n‘free drop’ and ‘slow drop’ in the spring-only condition (Figure 8c). Since the effective dissipated energy\nThis is a provisional file, not the final typeset article 14Mo et al. Effective damping in legged locomotion\nEeffective is directly measured, and the friction loss Ecfriction and impact loss Eimpact are obtained separately,\nthe viscous loss Eviscous can be computed according to Equation (12).\nFigures 8a and 8b show the ‘free drop’ and ‘slow drop’ results of the hydraulic damper and diaphragm\ndamper, respectively. Both drop heights are 14 cm , at identical orifice setting. We calculated the negative\nwork of each work-loop (range indicated by the two vertical dash lines), as shown in Figure 8. To provide\nan objective analysis, the work-loop area of each ‘slow drop’ (manual movement) was cut to the maximum\nleg compression of the corresponding ‘free drop’ condition. The dissipated energy of the leg-mounted\nhydraulic damper is 150 mJ and60 mJ for ‘free drop’ and ‘slow drop’, respectively, and 100 mJ and67 mJ\nfor the diaphragm damper, respectively. According to Figure 8c, the impact loss Eimpact due to unsprung\nmasses play a large role, accounting for 31 mJ . The viscous loss Eviscous of the hydraulic and the diaphragm\ndamper are 59 mJ and 2 mJ, respectively.\n80 85 90 95 100\nLeg length [%]0123GRF vertical [BW]Free drop: 150 mJ\nSlow drop: 60 mJ\nleg flexion\n(a)Hydraulic damper and spring\n80 85 90 95 100\nLeg length [%]0123GRF vertical [BW]Free drop: 100 mJ\nSlow drop: 67 mJ\n (b)Diaphragm damper and spring\n80 85 90 95 100\nLeg length [%]0123GRF vertical [BW]Free drop: 91 mJ\nSlow drop: 60 mJ\n (c)Spring only\nFigure 8. Characterizing the contribution of velocity-dependent damping: Vertical GRF versus leg\nlength change, a 2-DOF leg with damper/spring drops onto the force sensor: Three different hardware\nconfigurations were tested, for slow and free drop speeds on the vertical slider. Yellow data lines indicate\nslow-motion experiments. Experiments ‘start’ bottom right, at normalized leg length 100 % . Reading goes\ncounter-clockwise, i.e. from touch-down to mid-stance is indicated by the upper part of the hysteresis curve,\nwhile the lower part indicates elastic spring-rebound, without damper contribution.\n4.3 Adjustability of dissipated energy\nWe tested the adjustability of energy dissipation during leg drops by the altering orifice setting for\neach leg-mounted damper, and quantified by calculating the size of the resulting work-loops. The drop\nheight was fixed to 14 cm and we used 2orifice settings. The identical same set-up but in spring-only\nconfiguration (damper cables detached) was tested for reference. Work-loop and corresponding effective\ndissipated energies are illustrated in Figures 9a and 9b. The hydraulic damper-mounted leg dissipated\n156 mJ and150 mJ energy on its two orifice settings, the pneumatic diaphragm damper dissipated 102 mJ\nand100 mJ . In Figure 9c, we display results from the numerical model introduced in Section 2 to estimate\nthe work-loop shape that either a pure viscous or pure Coulomb damper would produce, if dissipating the\nsame amount of energy as the hydraulic damper with orifice-a (Figure 9a). We set the damping coefficients\nof our numerical model to ED0\u0019156mJ , so that: (dv;dc) = ( 51Ns =m;0N)for pure viscous damping; and\n(dv;dc) = ( 0Ns=m;13:2N)for pure Coulomb damping. Work-loops from the numerical simulation differ\nnotably from the experimental data, suggesting that neither the hydraulic or diaphragm damper can easily\nbe approximated as pure viscous or pure Coulomb dampers. Both work loops in Figure 9c present about\nequal amount of dissipated energy. Yet, both differ greatly due to their underlying damping dynamics,\nvisible in their unique work-loop shapes. Their individual characteristics are different enough to uniquely\nidentify pure viscous or pure Coulomb dampers, from numerical simulation.\nFrontiers 15Mo et al. Effective damping in legged locomotion\n80 85 90 95 100\nLeg length [%]0123GRF vertical [BW]Spring only: 101 mJ\nOrifice a: 156 mJ\nOrifice c: 150 mJ\nleg flexion\n(a)Hydraulic damper and spring\n80 85 90 95 100\nLeg length [%]0123GRF vertical [BW]Spring only: 87 mJ\nOrifice a: 102 mJ\nOrifice c: 100 mJ\n (b)Diaphragm damper and spring\n75 80 85 90 95 100\nLeg length [%]0246GRF vertical [BW]Pure viscous\nPure Coulomb (c)Simulation\nFigure 9. Adjustability and tunability of damping: Vertical GRF vs leg length change, a 2-segment\nleg with damper and spring drops onto the force sensor. Two different hardware configurations were\ntested, for different orifice settings. (a) and (b) show the result from hydraulic damper and diaphragm\ndamper respectively, where the green data lines indicate the leg drop without damper for comparison. (c):\nSimulated approximation of hydraulic damper orifice a by a pure viscous and a Coulomb damper. Damping\ncoefficients are chosen to allows same dissipated energy, i.e., ED0=156mJ: respectively — pure viscous\ndamper: dc=0 Nanddv=51 Ns/m ; pure Coulomb damper: dc=13.2 N anddv=0 Ns/m . None of the two\ncurves can fully capture the work-loop of hydraulic damper.\n4.4 Damper selection choices\nIn accordance with the simulation results, we aim to use a viscous damper to dissipate energy introduced\nby a ground disturbance. How much energy could be dissipated by the damper, depended mainly on the\nselected viscous damper, and only to a limited degree on the orifice setting. Results from the hydraulic\ndamper 1214H showed significant energy dissipation capabilities: \u001911 % of the system’s total energy\n(59 mJ of560 mJ ) were dissipated (Figure 9a at orifice setting ’c’ and Table 4 ). At the drop, in sum\n150 mJ (27 % ) of the leg’s system energy were lost, due to Coulomb friction in the joints, impact dynamics,\nand viscous damping losses. Other dissipation dynamics are feasible, by selecting appropriate dampers.\nWe tested a second hydraulic damper (1210M, MISUMI) under equal conditions and compared it to\ndamper-1214H. The two applied orifice settings changed the observed work loop largely by shape, and\nlittle by area (Figure 10). The damper-1210M dissipated \u001960 % system energy, and the leg lost in sum\n(viscous+Coulomb+impact) 72 % of its system’s energy during that single drop. At other orifice settings,\nwe observed over-damping; the 1210M-spring leg came to an early and complete stop, and without rebound\n(data not shown here due to incomplete work loop).\nFor comparison, time plots of the vertical GRF and the impulse at stance phrase are shown in Figure 11.\nThe energy composition (Equation 12) is provided in Table 4. The ’spring only’ data correspond the\ncurves in Figure 8c. The diaphragm+spring data correspond to ‘orifice c’ in Figure 9b. The hydraulic\n(1214H)+spring data correspond to ‘orifice c’ in Figure 9a. The hydraulic (1210M)+spring data correspond\nto ‘orifice b’ in Figure 10. Among the tested dampers, the hydraulic 1210M damper showed the largest\nvertical GRF; peak vertical GRF of 6.3 BW are observed, almost twice as much as the ‘spring only’ case.\nThe viscous dampers 1214H and 1210M shifted the peak of their legs’ vertical GRF to an earlier point in\ntime, compared to the spring-leg and the spring+diaphragm-leg (Figure 11).\n5 DISCUSSION\nA primary objective of this study was to test how physical dampers could be exploited for locomotion\ntasks by characterizing multiple available technical solutions. Our numerical model predicted three crucial\naspects: (1) a pure viscous damper generally performs better than a pure Coulomb damper (Figure 4); (2)\nThis is a provisional file, not the final typeset article 16Mo et al. Effective damping in legged locomotion\n80 85 90 95 100\nLeg length [%]01234567GRF vertical [BW]Spring only: 101 mJ\nOrifice a: 395 mJ\nOrifice b: 401 mJ\nFigure 10. Higher energy dissipation with a different model of the hydraulic damper (1210M):\nVertical GRF vs. leg length change, a 2-DOF leg with a parallel damper and spring drops onto the\nforce sensor. Two damper orifice settings were tested (blue, red curves). The two resulting curves are\ncompared with the spring-only configuration, provided as reference.\n0 0.05 0.1 0.15 0.205 spring only\n0 0.05 0.1 0.15 0.205 diaphragm + spring\n0 0.05 0.1 0.15 0.205 Hydraulic (1214H) + spring\n0 0.05 0.1 0.15 0.2\nTime [s]05GRF vertical [BW]\nHydraulic (1210M) + spring\n(a)Vertical GRF over time\n0 0.05 0.1 0.15 0.201\nspring only\n0 0.05 0.1 0.15 0.201\ndiaphragm + spring\n0 0.05 0.1 0.15 0.201\nHydraulic (1214H) + spring\n0 0.05 0.1 0.15 0.2\nTime [s]01Impulse [Ns]\nHydraulic (1210M) + spring (b)Vertical Impulse over time\nFigure 11. Ground reaction forces and the corresponding, instantaneous impulse for leg drop\nexperiments . The corresponding work curves are provided in Figures 8 to 10.\nhigher damping rates result in better rejection of ground disturbances (Figure 4a), however at the cost\nof higher dissipation at reference height (Table 2); (3) characteristic work loop shapes for pure viscous\nand Coulomb damper during leg-drop (Figure 9c). Our hardware findings show that neither of the tested\nphysical dampers approximates as pure viscous or pure Coulomb dampers. The experiments also suggest\nthat the mapping between dissipated energy and damping rates is concealed by the dynamics of the impact\nand the non-linearity of the force-velocity characteristics of the leg in the stance phase. Therefore, it is vital\nto test damping in a real leg at impact because the behavior is not merely as expected from the data sheets\nand the simple model.\nFigure 7 characterizes how the hydraulic damper dissipates energy during a free drop. The experimental\nresults show that the dissipated energy of the hydraulic damper scales with drop height (Figure 7a) and\nweight (Figure 7c), but less intuitively, it reduces with increasing damping rates (Figure 7b). This can be\npartially interpreted in the context of an ideal viscous damper (as in Equation (5)), but linear) for which the\nFrontiers 17Mo et al. Effective damping in legged locomotion\nTable 4. Leg drop experiments and their individual energetic losses per drop. The system’s initial\npotential energy is 560 mJ .Eeffective : sum of all energetic losses visible as the area of the hysteresis curve,\ni.e. in Figure 8, Ecfriction : negative work dissipated by Coulomb friction, Eimpact : energetic losses from\nimpact (unsprung mass). The negative work dissipated by viscous damping in the physical damper is\nEviscous . The corresponding work curves are provided in Figures 8 to 10.\nDrop test setup Eeffective Ecfriction Eimpact Eviscous\n[mJ] [mJ] [mJ] [mJ]\nSpring only 91 60 31 0\nDiaphragm + spring 100 67 31 2\nHydraulic 1214H + spring 150 60 31 59\nHydraulic 1210M + spring 401 38 31 332\neffective dissipated energy Eeffective would be calculated as in,\nEeffective =Z\nFp(t)dyp=Z\n(dv\u0001vp(t))dyp (13)\nwhere Fp(t)is the damper piston force and ypis the piston displacement, vp(t)the corresponding velocity.\nWhen increasing the drop height, the velocity at impact is increased, so is vp(t). With the assumption of\nEquation (13), this results in higher damping forces Fp(t), and thus, dissipated energy Eeffective , as seen\nin Figure 7a. The heavier drop weight leads to slower deceleration. Therefore the velocity profile vp(t)is\nincreased, which also leads to higher dissipation Eeffective (Figure 7b). An orifice setting of high damping\nrate will increase the damping coefficient dv. However, the velocity profile vp(t)is expected to reduce due\nto higher resistance. This simple analogy shows that the coupling between damping coefficient dvand\nvelocity profile vp(t)makes it difficult to predict the energy dissipation by setting the orifice and serves as\nan interpretation of why adjusting the orifice generates a relatively small adjustment of 10 % (81 mJ -89 mJ )\nof the dissipated energy. Also, the impact phase (time for the damper to output its designed damping force\nunder sudden load) introduces additional non-linearity to the output force profile. Overall, the results in\nFigure 7 indicate that we can approximate the damping force produced by the hydraulic damper to be\nviscous and adjustable— as such dampers are typically designed (Dixon, 2008)—, but the mapping of\nenergy dissipation to orifice setting is difficult to predict in a dynamic scenario.\nThe approximation as a linear, velocity dependent damper allows us to rapidly estimate energy dissipation\nin simulation, over a range of parameters. However, the exact mapping of the hardware leg/spring/damper\nenergy dissipation to orifice setting is difficult to predict, when basing the estimation only on the isolated-\ndamper drop experiments from Figure 7. Instead, the leg/spring/damper experiments show that the energetic\nlosses from the impact remove 31 mJ energy, compared to 59 mJ damper losses. The high amount of force\noscillations at impact (up to 1 BW , Fig. 8a) during the first 3 %leg length change leads us to believe that\nthese impact oscillations move the damper’s dynamic working range, i.e., its resulting instantaneous force\nand velocity. The oscillations are likely caused by unsprung mass effects of the leg/spring/damper structure,\nand could not be captured in an isolated-damper setup, or—at least not easily—in a simulation.\nThe work loops of leg drop experiments (Figure 8) show the effects of our tested dampers on a legged\nsystem. From touch-down to mid-stance ( leg flexion ), the ‘free drop’ curves show a larger negative work\ncompared to the ‘slow drop’ curves, illustrating that the damper absorbs extra energy. The returning curves\n(mid-stance to lift-off) of the hydraulic damper aligns well with the ‘slow drop’ curve, indicating the\ndamper is successfully detached due to slackness cable while the spring recoil. Figure 8b shows that the\n‘free drop’ force of the diaphragm damper is slightly higher than ‘slow drop’ force in the first half of the leg\nThis is a provisional file, not the final typeset article 18Mo et al. Effective damping in legged locomotion\nextension phase. This discrepancy is likely caused by the elastic force component of the diaphragm damper\ndue to sudden expansion of the air chamber volume. The elastic component seems to dominate the damper\nbehavior, which thus acts mostly as an air spring. By separating its energetic components (Equation (12)),\nwe found that the hydraulic damper produces a viscous-like resistance higher than the diaphragm damper\n(59 mJ versus 2 mJ), indicating the hydraulic damper is more effective in dissipating energy under drop\nimpact. Hence, the hydraulic damper shows more viscous behavior, while the diaphragm damper is more\nelastic.\nPhysical damping in the system comes at the cost of energy loss, and to maintain periodic hopping, it\nbecomes necessary to replenish energy that is dissipated by damping ( ED0). Therefore, there is a trade-off\nto consider: simulation results show that higher damping results in faster rejection of ground perturbation\nat the price of more energy consumption at reference drop height (Table 2, Figure 4). An adjustable damper\nwould partly address this problem: on level ground, the damping rate could be minimal, and on rough\nterrain increased. The adjustability of the two dampers is illustrated in Figures 9a and 9b. We discuss the\nadjustability from both energy dissipation and dynamic behavior perspectives.\nCompared with the spring-only results, both the hydraulic and the diaphragm damper reduced the\nmaximum leg flexion and dissipated more energy. The orifice setting changes the shape of the work loop\ndifferently for the two setups. For the hydraulic damper (Figure 9a), orifice setting-c shrinks the work\nloop from left edge, indicating more resistance is introduced by the damper to reduce leg flexion. For\nthe diaphragm damper (Figure 9b), orifice setting-c not only shrinks the work loop, but also increases its\nslope. We interpret this as the elastic contribution of air compression: relatively fewer air leaves through\nthe smaller orifice, but instead acts as an in-parallel spring.\nConcerning energy dissipation, changes of orifice settings led to relatively small changes in effective\ndissipated energy Eeffective :150 mJ to156 mJ for the hydraulic damper, and 100 mJ to102 mJ for the\ndiaphragm damper. Even for the other damper model (1210M), which dissipates high amounts of energy,\nchanges in orifice setting change the work loop shape drastically, but not the dissipated energy ( 395 mJ\nversus 401 mJ).\nSimilar to the isolated damper drop, the data (Figures 9a and 9b) shows that specific orifice settings\nintroduce more resistance, but not necessarily lead to higher energy dissipation, for both hydraulic and\ndiaphragm damper. However, in our simplified numerical leg model, an increase in viscous damping\ncoefficients leads to a systematic increase of dissipated energy (Table 2), and a sharper tip at the left side of\nthe work loop (Figure 9c). The discrepancy is likely due to the non-linear coupling between the damper\nmechanics and the leg dynamics in the hardware setup: (1) The damping force generated by the fluid\ndynamics in the orifice only approximates a linear viscosity (Dixon, 2008). (2) The impact loading on\nboth the nonlinear leg structure and the damper. This makes the prediction of the energy dissipation not\nstraight-forward based on our simplified numerical leg model, and points towards the need of a combined\napproach between simulation and hardware testing to fully understand physical damping in a legged system.\nViscous, velocity dependent damping alters the leg’s loading characteristics, and leads to a peak force at\nthe instance of touch-down. As a result, the vertical GRF is increased in the early stance phase, shifting\nand increasing the peak vertical GRF before mid-stance (Figure 11a). When designing a legged system\nwith a viscous damper, its increasing load on the mechanical structure should be considered.\nThe selection of viscous dampers depends on the task. High damping can fully reject disturbances in a\nsingle cycle, but lower damping could have energetic benefits. Here we looked for a damper that would\ndissipate significant negative work (Eviscous\nET0\u001910%\u000015% ) in form of viscous damping. The air-filled\nFrontiers 19Mo et al. Effective damping in legged locomotion\ndiaphragm damper lead to insufficient energy losses ( 2 %), but the hydraulic dampers dissipated 10 % and\n60% of the system’s total energy (Table 4).\nDrawing conclusions about animal locomotion based on the here presented leg-drop experiments is\nsomewhat early. However, observations from (M ¨uller et al., 2014, Table 1, p. 2288) indicate that leg forces\ncan increase at unexpected step-downs during locomotion experiments. Further, Kalveram et al. (2012) Rev1,\nGC 1 suggests in a comparison of experimental human hopping and numerical simulations that damping may\nbethedriving ingredient in passive stabilization against ground-level perturbations. We are consequently\nexcited about the here presented results of viscous dampers mounted in parallel to a leg’s spring, producing\nadaptive forces without the need for sensing.\n6 CONCLUSION\nWe investigated the possibility to exploit physical damping in a simplified leg drop scenario as a template\nfor the early stance phase of legged locomotion. Our results from a) numerical simulation promote the\nuse of adjustable and viscous damping over Coulomb damping to deal with a ground perturbation by\nphysical damping. As such, we b) tested two technical solutions in hardware: a commercial, off-the-shelf\nhydraulic damper, and a custom-made, rolling diaphragm damper. We dissected the observed dissipated\nenergy from the hardware damper-spring leg drops, into its components, by experimental design. The\nresulting data allowed us to characterize dissipation from the early impact (unsprung-mass effects), viscous\ndamping, Coulomb damping, and orifice adjustments individually, and qualitatively . The rolling diaphragm\ndamper features low-Coulomb friction, but dissipates only low amounts of energy through viscous damping.\nThe off-the-shelf, leg-mounted hydraulic damper did exhibit high viscous damping, and qualitatively\nshowed the expected relationship between impact speed, output force and negative work. Changes in orifice\nsetting showed only minor changes in overall energy dissipation, but can lead to large changes in leg\nlength dynamics, depending on the chosen technical damper. Hence, switching between different viscous,\nhydraulic dampers is an interesting future option. Our results show how viscous, hydraulic dampers react\nvelocity-dependent, and create an instantaneous, physically adaptive response to ground-level perturbations\nwithout sensory-input.\nCONFLICT OF INTEREST STATEMENT\nThe authors declare that the research was conducted in the absence of any commercial or financial\nrelationships that could be construed as a potential conflict of interest.\nAUTHOR CONTRIBUTIONS\nAM contributed to concept, hardware design, experimental setup, experimentation, data discussion and\nwriting. FI contributed to concept, simulation framework, experimental setup, data discussion and writing.\nDH and ABS contributed to concept, data discussion and writing.\nACKNOWLEDGMENTS\nThe authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for\nsupporting AM, FI, the China Scholarship Council (CSC) for supporting AM, and the Ministry of Science,\nResearch and the Arts Baden-W ¨urttemberg (Az: 33-7533.-30-20/7/2) for supporting DH, and the Max\nPlanck Society for supporting ABS.\nThis is a provisional file, not the final typeset article 20Mo et al. Effective damping in legged locomotion\nREFERENCES\nAbraham, I., Shen, Z., and Seipel, J. (2015). A Nonlinear Leg Damping Model for the Prediction of Running\nForces and Stability. Journal of Computational and Nonlinear Dynamics 10. doi:10.1115/1.4028751\nBledt, G., Wensing, P. M., Ingersoll, S., and Kim, S. (2018). Contact model fusion for event-based\nlocomotion in unstructured terrains. 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Quantifying morphological computation. Entropy 15, 1887–1915. doi:10.\n3390/e15051887\nThis is a provisional file, not the final typeset article 22"    },    {        "title": "2210.09824v1.Concepts_in_Lorentz_and_CPT_Violation.pdf",        "content": "arXiv:2210.09824v1  [hep-ph]  18 Oct 2022Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n1\nConcepts in Lorentz and CPT Violation\nV. Alan Kosteleck´ y\nPhysics Department, Indiana University\nBloomington, IN 47405, USA\nThis contribution to the CPT’22 meeting provides a brief rev iew of some con-\ncepts in Lorentz and CPT violation.\n1. Introduction\nIn recent years, substantial advances have been made in the the ory and\nphenomenologyofLorentz and CPT violation, driven by the intriguing pos-\nsibility that experimental searches for the associated effects cou ld uncover\ntiny observable signals from an underlying unified theory such as str ings.1\nThis contribution to the CPT’22 proceedings summarizes a few conce pts\nin the subject, focusing on the approach that uses effective field t heory2to\nconstruct a general realistic framework describing physical effec ts.3–5\n2. Foundations\nA key property of Minkowski spacetime is its invariance under Loren tz\ntransformations. The Lorentztransformationsinclude spatialr otationsand\nvelocity boosts, which together can be viewed as generalized rotat ions in\nspacetime. Theories manifesting isotropy under these spacetime r otations\nhave Lorentz invariance (LI), so a theory with Lorentz violation (L V) incor-\nporates one or more spacetime anisotropies. Experiments testing LI seek\nto identify possiblespacetime anisotropiesby comparingphysicalqu antities\nat different spacetime orientations. For example, symmetry under spatial\nrotations can be explored by comparing at different spatial orienta tions the\nticking rates of two clocks or the lengths of two standard rulers.\nOur most successful fundamental theories describing nongravit ational\naspects of Nature are constructed on Minkowski spacetime. How ever, the\nUniverse contains gravitational interactions, which cannot be scr eened and\nhence are ubiquitous. Minkowski spacetime is therefore believed to be un-\nphysical in detail, with our Universe instead involving Riemann spacetim eProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n2\nor perhaps a generalization. A generic Riemann spacetime lacks globa l\nLI. Instead, the relevant spacetime symmetries are local Lorent z invariance\n(LLI) and diffeomorphism invariance (DI). A theory with LLI is isotro pic\nunder local spacetime rotations about every point, while one with DI is\nunchanged by local translations. Experiments can test these sym metries\nby comparing properties of objects at different orientations and lo cations\nin the neighborhood of a spacetime point.\nMost experiments and observations involve only weak gravitational\nfields and so take place in asymptotically flat spacetime, a limit of Rie-\nmann spacetime that reduces to Minkowski spacetime for zero gra vity. In\nasymptotically flat spacetime, the standard notion of LI turns out to be a\ncombination of LLI and DI.5Experiments searching for LV are therefore\nin reality sensitive to a combination of local Lorentz violation (LLV) an d\ndiffeomorphism violation (DV). The corresponding observables manif est a\nmixture of local spacetime anisotropy and spacetime-position depe ndence.\n3. Theory\nSince no compelling evidence for LV exists to date, a broad-based an d\nmodel-independent methodology is desirable in the search for possib le ef-\nfects. Any violations are expected to be small corrections to the k nown\nphysics of General Relativity (GR) and the Standard Model (SM), s o it\nis natural to study LV using the approach of effective field theory.2Typi-\ncally, integrating over high-energy degrees of freedom in a theory generates\na specific effective field theory applicable at low energies. In the cont ext\nof searches for LV, however, the approach can instead be used t o study\nsimultaneously a large class of underlying theories and determine pos sible\nobservable effects in a model-independent way.\nThe realistic coordinate-independent theory incorporating gener al LV\nthat is basedon GR coupled tothe SM is knownasthe Standard-Mode lEx-\ntension (SME).3,4In a realistic effective field theory, CPT violation comes\nwith LV both in Minkowski spacetime3,6and in asymptotically flat space-\ntime,4so the SME also describes CPT violation in a model-independent\nway. The Lagrange density Lof the theory includes LV operators of any\nmass dimension d, with the minimal theory defined as the subset of oper-\nators of renormalizable dimension d≤4. Each term in Lis constructed as\nan observer-scalarcontraction of a LV operator O(x) with a coupling coeffi-\ncientk(x) or its derivatives. The coefficients are expected to be suppresse d\neither by powers of a high-energy scale such as the Planck energy o r viaProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n3\na mechanism such as countershading.7The explicit forms of all minimal\nterms3,4and many nonminimal terms5,8–10are known.\n4. Backgrounds\nAny given coefficient k(x) inLcan be viewed as a prescribed background\nin spacetime,4which may arise as a vacuum expectation value of a field.\nAll indices carried by k(x) are contracted with those of the corresponding\noperator O(x), sok(x)iscovariantunderobserverlocalLorentztransforma-\ntions and general coordinate transformations. However, k(x) is unaffected\nby particle local Lorentz transformations and diffeomorphisms, wh ich act\nonly on dynamical fields. Covariant and contravariant local indices o nk(x)\nare physically equivalent because the local metric ηis Minkowski and non-\ndynamical. In contrast, covariant and contravariant spacetime in dices can\ngenerate physically distinct effects because the spacetime metric gis dy-\nnamical. Disregarding derivatives of k(x), a generic term in Lthus takes\nthe form L ⊃kµ...ν...a...(x)Oµ...ν...a...(x), wherespacetimeindicesareGreek\nand local indices areLatin. The term is LLVwhen k(x) carriesa localindex\nand is DV when k(x) carries a spacetime index or varies with x.\nIt is physically useful to distinguish two types of backgrounds k(x), de-\nnoted as /angbracketleftk/angbracketrightandk. Spontaneous backgrounds /angbracketleftk/angbracketrightarise dynamically from\nsolving equations of motion, so they satisfy the Euler-Lagrange eq uations\nand are thus on shell. Their dynamical origin means that small fluctua tions\naround/angbracketleftk/angbracketrightcan occur, so additional modes appear in the effective theory11\nincluding Nambu-Goldstone12and massive modes. Explicit backgrounds k\nare externallyprescribed and hence nondynamical. They areuncon strained\nby Euler-Lagrange equations and thus can be off shell, and no dynam ical\nfluctuations occur. The spontaneous or explicit nature of the LLV and DV\ncan be used to classify terms in Land determine their physical implica-\ntions.5The geometry of gravity is affected differently in the two scenarios.\nFor spontaneous violation, it can remain Riemann or Riemann-Cartan .4\nFor explicit violation, in contrast, the geometry typically cannot be R ie-\nmann and is conjectured instead to be Finsler,4,5,13,14leading to unique\ngravitational effects.15\n5. Applications\nBy virtue of its generality and model independence, the SME can be e x-\npected to contain the large-distance limit of any realistic theory with LV.\nThe background coefficients affect the behavior of experimentally k nownProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n4\nforce and matter fields and hence can be used to predict possible ob serv-\nable signals of Lorentz and CPT violation. The physical definition of a\ngiven particle species can be affected,16and both its free propagation and\ninteractions can be modified in a flavor-dependent way. The effects can\ndepend on the magnitude and orientation of momenta and spins and c an\ndiffer between particles and antiparticles. For spontaneous LV, th e dynam-\nical fluctuations around /angbracketleftk/angbracketrightcan also modify the physics and can even play\nthe role of the photon or graviton.11In any scenario, the coefficients are\nthe targets for experimental searches. Since a coefficient chang es under\nobserver frame transformations, the inertial frame used to pre sent results\nmust be specified. In practice no laboratory is inertial, and the Eart h’s\nrotation and revolution imply that LV measurements typically exhibit t ime\nvariations.17Instead, the canonical choice to report experimental results\nis the Sun-centered frame.18Many experiments have achieved impressive\nsensitivities to coefficients expressed in this frame.19\nThe SME framework also has applications in broader contexts. One\nconcernssearchesfornewLIphysicsbeyondGRandtheSM.Byvir tueofits\ninclusion of all effective background couplings, the framework can d escribe\nphysicaleffects fromanynewfieldproducingabackgroundoveras pacetime\nregionofexperimentalrelevance, andhenceitcanbeusedtodedu cebounds\non new LI physics from existing constraints on LV. This technique ha s\nled, for example, to tight constraints on torsion20and nonmetricity,21and\nsimilar ideas have been adopted for spacetime-varying couplings and ghost-\nfree massive gravity.22Another application in a different context involves\nthe description of emergent LI in certain condensed-matter syst ems. For\nexample, properties of Dirac and Weyl semimetals are calculable in the\nSME framework.23Future explorations in these and other contexts offer\nexcellent prospects for conceptual and practical advances.\nAcknowledgments\nThis work was supported in part by US DoE grant DE-SC0010120 and by\nthe Indiana University Center for Spacetime Symmetries.\nReferences\n1. V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 39, 683 (1989); V.A. Kost-\neleck´ y and R. Potting, Nucl. Phys. B 359, 545 (1991); Phys. Rev. D 51, 3923\n(1995).\n2. See, e.g., S. Weinberg, Proc. Sci. CD 09, 001 (2009).Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n5\n3. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998); V.A. Kosteleck´ y and R. Lehnert, Phys. Rev. D63,\n065008 (2001).\n4. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n5. V.A. Kosteleck´ y and Z. Li, Phys. Rev. D 103, 024059 (2021).\n6. 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Cim. 19,\n154 (1961); J. Goldstone, A. Salam, and S. Weinberg, Phys. Re v.127, 965\n(1962).\n13. R. Bluhm, Phys. Rev. D 91, 065034 (2015); Phys. Rev. D 92, 085015 (2015);\nR. Bluhm and A. Sehic, Phys. Rev. D 94, 104034 (2016); R. Bluhm, H. Bossi,\nand Y. Wen, Phys. Rev. D 100, 084022 (2019).\n14. M. Schreck, Phys. Lett. B 793, 70 (2019); B.R. Edwards and V.A. Kost-\neleck´ y, Phys. Lett. B 786, 319 (2018); D. Colladay and P. McDonald, Phys.\nRev. D85, 044042 (2012); V.A. Kosteleck´ y, Phys. Lett. B 701, 137 (2011);\nV.A. Kosteleck´ y and N. Russell, Phys. Lett. B 693, 2010 (2010).\n15. V.A. Kosteleck´ y and Z. Li, Phys. Rev. D 104, 044054 (2021).\n16. V.A. Kosteleck´ y, E. Passemar, and N. Sherrill, arXiv:2 207.04545.\n17. V.A. Kosteleck´ y, Phys. Rev. Lett. 80, 1818 (1998).\n18. R.Bluhm et al., Phys.Rev.D 68, 125008 (2003); Phys.Rev.Lett. 88, 090801\n(2002); V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 66, 056005 (2002).\n19. V.A. Kosteleck´ y and N. Russell, arXiv:0801.0287v15.\n20. R. Lehnert, W.M. Snow, and H. Yan, Phys. Lett. B 730, 353 (2014); V.A.\nKosteleck´ y, N. Russell, and J.D. Tasson, Phys. Rev. Lett. 100, 111102\n(2008).\n21. R. Lehnert, W.M. Snow, Z. Xiao, and R. Xu, Phys. Lett. B 772, 865 (2017);\nJ. Foster et al., Phys. Rev. D 95, 084033 (2017).\n22. V.A. Kosteleck´ y, R. Lehnert, and M.J. Perry, Phys. Rev. D68, 123511\n(2003); V.A. Kosteleck´ y and R. Potting, Phys. Rev. D 104, 104046 (2021).\n23. V.A. Kosteleck´ y, R. Lehnert, N. McGinnis, M. Schreck, a nd B. Seradjeh,\nPhys.Rev.Res. 4, 023106 (2022); A.G´ omez, A. Mart´ ın-Ruiz, andL. Urrutia,\nPhys. Lett. B 829, 137043 (2022)."    },    {        "title": "2308.01575v1.Quasinormal_modes_of_the_spherical_bumblebee_black_holes_with_a_global_monopole.pdf",        "content": "Quasinormal modes of the spherical bumblebee black holes with a\nglobal monopole\nRui-Hui Lin,1,∗Rui Jiang,1and Xiang-Hua Zhai1,†\n1Division of Mathematics and Theoretical Physics,\nShanghai Normal University, 100 Guilin Road, Shanghai 200234, China\nAbstract\nThe bumblebee model is an extension of the Einstein-Maxwell theory that allows for the sponta-\nneous breaking of the Lorentz symmetry of the spacetime. In this paper, we study the quasinormal\nmodes of the spherical black holes in this model that are characterized by a global monopole. We\nanalyze the two cases with a vanishing cosmological constant or a negative one (the anti-de Sitter\ncase). We find that the black holes are stable under the perturbation of a massless scalar field.\nHowever, both the Lorentz symmetry breaking and the global monopole have notable impacts on\nthe evolution of the perturbation. The Lorentz symmetry breaking may prolong or shorten the\ndecay of the perturbation according to the sign of the breaking parameter. The global monopole,\non the other hand, has different effects depending on whether a nonzero cosmological constant\npresences: it reduces the damping of the perturbations for the case with a vanishing cosmological\nconstant, but has little influence for the anti-de Sitter case.\n∗linrh@shnu.edu.cn\n†(corresponding author)zhaixh@shnu.edu.cn\n1arXiv:2308.01575v1  [gr-qc]  3 Aug 2023I. INTRODUCTION\nOne of the major challenges in theoretical physics is to reconcile Einstein’s general rel-\nativity (GR), the most successful theory of gravitation so far, with the standard model of\nparticle physics (SM), which unifies all other interactions. A possible clue to this problem is\nspontaneous symmetry breaking, which plays a key role in the elementary particle physics.\nIn the early universe, the temperature may have been high enough to trigger symmetry\nbreaking. Depending on the topology of the manifold Mof degenerate vacua, this process\nmay result in different types of topological defects[1, 2]. In particular, when second homo-\ntopy group π2MofMis nontrivial, point defects, i.e., monopoles are formed and can be the\nsource of inflation and seeds of the cosmic structures[3, 4]. A simplest model to describe a\nglobal monopole is to consider a triplet of scalar fields with a global O(3) symmetry sponta-\nneously broken to U(1). The gravitational field of a global monopole describes a spacetime\nwith a solid angle deficit[3, 5]. If such a global monopole is captured by an ordinary black\nhole, a new type of black hole with a global monopole charge will be formed. This charge\nmakes the black hole topologically different from the usual one, and may have interesting\nphysical implications (see, e.g., Refs. [6–14]).\nAnother possible symmetry breaking that may emerge in the quest of quantization of GR\nis the breaking of Lorentz symmetry. It is shown that this symmetry may be strongly vio-\nlated at the Planck scale ( ∼1019GeV) in some approaches to quantum gravity (QG)[15, 16],\nsuggesting that it may not be a fundamental symmetry of nature. Moreover, Lorentz symme-\ntry breaking (LSB) may also provide candidates of observable signals of the underlying QG\nframework at low energy scale [17–19]. Therefore, the occurrence of LSB has been explored\nin various scenarios, such as string theory (see, e.g., [17]), loop quantum gravity[20, 21],\nmassive gravity[22], Einstein-aether theory[23], and others[24–27]. The effective field theory\nto study violations of fundamental symmetries in SM and gravity is called the Standard-\nModel Extension (SME)[28, 29]. The gravitational couplings of SME is explored in a general\nRiemann-Cartan spacetime in Ref. [19], where it is shown that gravitational coupling spon-\ntaneous LSB can occur without geometrical incompatibilities.\nThe bumblebee model is a simple example to incorporate spontaneous LSB in the gravi-\ntational sector. It is a vector-tensor theory that generalizes the Einstein-Maxwell theory by\nintroducing a vector field Bµthat acquires a non-zero vacuum expectation value (VEV) [17–\n219, 30]. This vector field Bµcouples nonminimally to the spacetime and thus induces a spon-\ntaneous breaking of the local Lorentz symmetry by selecting a preferred spacetime direction\nin the local frames through its frozen VEV. When the bumblebee field is set to be a constant\nbackground, it can be viewed as the coefficients related to LSB in SME[31]. The bumblebee\nmodel has attracted recent interests since an exact Schwarzschild-like black hole solution\nwas found[32]. Other static[33–35] and rotating[36–38] solutions in the bumblebee model\nhave been obtained. The various physical aspects such as the gravitational lensing[39, 40],\nshadow[32, 36, 41], accretion[42], perturbations[43–49], and superradiance[50–52], have also\nbeen studied.\nIn particular, by considering the bumblebee model with a global monopole, G¨ ull¨ u and\n¨Ovg¨ un constructed a family of Schwarzschild-like black hole solutions and analyzed the in-\nfluence of the LSB parameter on the black hole shadow and the weak deflection angle[33].\nFurthermore, in the study of the generalized uncertainty principle correction of the bum-\nblebee black holes, Gogoi and Goswami proposed that a similar family of black hole so-\nlutions with both a global monopole and a non-vanishing cosmological constant can also\nbe constructed[49]. These two families of black holes, as strong field solutions, provide\noptimal environments for investigating spontaneous LSB and global monopoles within the\ngravitational context. When black holes are perturbed, the resulting oscillations and their\nevolution may reveal unique characteristics of the black hole, or equivalently, the spacetime\nand gravity. These oscillating modes are referred to as quasi-normal modes (QNMs) and\nhave become a widely utilized tool for analyzing the stability of black hole spacetime since\ntheir use in demonstrating the stability of the Schwarzschild black hole[53, 54]. In the cur-\nrent paper, we focus on the QNMs of aforementioned two families of black hole solutions,\nnamely the spherical bumblebee black holes with a global monopole in the presence or ab-\nsence of a cosmological constant. Our aim is to study how the spontaneous LSB and the\nglobal monopole affect the stability of the black hole spacetime.\nThis paper is structured as follows. In Section II, we provide a brief overview of spherical\nblack holes in the bumblebee model with a global monopole and establish the configuration\nof perturbation by considering an impinging massless scalar field. In Section III, we examine\nthe QNMs of these spherical bumblebee black holes with a global monopole for vanishing\nand non-vanishing cosmological constant. Our study is concluded in Section IV. Throughout\nthe paper, we follow the metric convention (+ ,−,−,−) and use the units G=ℏ=c= 1.\n3II. SPHERICAL BUMBLEBEE BLACK HOLES WITH A GLOBAL MONOPOLE\nAND THEIR SCALAR PERTURBATION\nA. Spherical bumblebee black holes with a global monopole\nThe Lagrangian of the bumblebee model is generally written as[32, 34]\nLB=√−g\u00141\n16π(R−2Λ + ξBµBνRµν)−1\n4BµνBµν−V\u0000\nBµBµ±b2\u0001\u0015\n+LM,(1)\nwhere Bµνdenotes the strength of the bumblebee field Bµand is defined as\nBµν=∂µBν−∂νBµ. (2)\nThe term with the parameter ξrepresents the quadratic coupling between the bumblebee\nfield and the Ricci tensor Rµν. Λ is the cosmological constant and LMis the Lagrangian of\nmatter. The term V(BµBµ±b2) refers to the potential of the bumblebee field that takes\na minimum at BµBµ±b2= 0 with b2>0. The ±signs here associate to a spacelike or\ntimelike bumblebee field, respectively. For a stable vacuum spacetime, one would expect\nthat the potential Vwill be minimized and the bumblebee field Bµwill be frozen at the\nVEV⟨Bµ⟩=bµwith bµbµ=∓b2. Due to the coupling featured by the parameter ξ, such a\nnonzero VEV of Bµthen will break Lorentz symmetry by selecting a preferred direction of\nspacetime. Variation of the Lagrangian (1) with respect to the metric gµνgives the equation\nof motion\n0 =Rµν+ Λgµν−κ\u0012\nT(M)\nµν−1\n2gµνT(M)\u0013\n+ξbµbαRαν\n+ξbνbαRαµ−ξ\n2gµνbαbβRαβ−ξ\n2∇α∇µ(bαbν)\n−ξ\n2∇α∇ν(bαbµ) +ξ\n2∇2(bµbν) +κ(gµνb2−2bµbν)V′,(3)\nwhere κ= 8πandV′denotes d V(x)/dx. For a global monopole field, the energy-momentum\ntensor T(M)\nµνhas the form[3]\nT(M)ν\nµ = diag\u0012η2\nr2,η2\nr2,0,0\u0013\n, (4)\nwhere ηis a constant related to the global monopole charge. For the usual spherical ansatz\nof metric\nds2=f(r)dt2−dr2\nh(r)−r2dθ2−r2sin2θdϕ2(5)\n4where f(r) and h(r) are the metric functions of the radial coordinate r, it is assumed that\nthe bumblebee field has a purely radial form bµ= (0, br(r),0,0), and hence, BµBµ=−b2\nr.\nIn the scenario with a vanishing Λ, a spherical solution can be constructed when the\npotential Vsatisfies V′= 0 at its minimum where BµBµ+b2= 0[32, 33]. This solution is\nshown to be Schwarzschild-like with a global monopole and can be given by[33]\nf(r) =1−κη2−2M\nr,\nh(r) =L+ 1\nf(r),\nbr(r) =|b|p\nh(r),(6)\nwhere L≡ξb2is the LSB parameter, and Mis an integration constant corresponding to\nthe Arnowitt-Deser-Misner mass. The event horizon can be easily found to be at rh=\n2M/(1−κη2), independent of the bumblebee field.\nFor the case where the cosmological constant Λ is nonzero, it is suggested that solutions\ncan be constructed by setting[34]\nV\u0000\nBµBµ+b2\u0001\n=λ\n2\u0000\nBµBµ+b2\u0001\n, (7)\nand viewing this term as a Lagrange multiplier term that ensures BµBµ+b2= 0 in the\nvariation of the Lagrangian (1). Following this scheme, we check that the Schwarzschild-\n(a)dS-like bumblebee black hole solution with a global monopole proposed in Ref. [49] can\nbe constructed, which is given by\nf(r) =1−κη2−2M\nr−Λ\n3r2,\nh(r) =L+ 1\nf(r),\nbr(r) =|b|p\nh(r),\nλ=ξΛ\nκ(L+ 1).(8)\nB. Scalar perturbation\nWe study the perturbation of the spherical bumblebee black holes (6) and (8) by con-\nsidering an impinging massless scalar field Φ. We restrict our attention to the cases with\n5Λ≤0, as the cases with Λ >0 have a cosmological horizon that limits the causal structure\nof the spacetime.\nWe assume that Φ is minimally coupled to gravity and satisfies the covariant Klein-Gordon\nequation,\n1√−g∂µ\u0000\ngµν√−g∂νΦ\u0001\n= 0. (9)\nUsing the separation of Φ in the spherical coordinates ( t, r, θ, ϕ ),\nΦ =ψl(r)\nrYlm(θ, ϕ)e−iωt, (10)\nwhere Ylm(θ, ϕ) is the spherical harmonic function, we can obtain the radial equation of Eq.\n(9) as\n1\n1 +Lfd\ndr\u0012\nfdψl\ndr\u0013\n+\u0002\nω2−U(r)\u0003\nψl= 0. (11)\nThe effective potential U(r) is defined as\nU(r) =f(r)\u0014l(l+ 1)\nr2+f′(r)\n(1 +L)r\u0015\n. (12)\nIntroducing the tortoise coordinate xdefined by\ndx\ndr=√\n1 +L\nf(r), (13)\none can rewrite the radial equation (11) as\nd2ψl\ndx2+\u0002\nω2−U(x)\u0003\nψl= 0. (14)\nTo keep track of the temporal evolution of Φ that is encoded in ωin the radial equation\n(11), one can, alternatively, adopt a different separation of Φ as\nΦ =Ψ(t, r)\nrYlm(θ, ϕ), (15)\nwhere its dependence on randtare kept in a single function Ψ( t, r). Then, the equation\nfor Ψ from Eq. (9) is\u0014∂2\n∂x2−∂2\n∂t2−U(x)\u0015\nΨ(x, t) = 0 . (16)\nWith the light cone coordinates\ndu=dt−dx, dv =dt+dx, (17)\n6L=-0.5L=0L=0.5L=1\n-20 0 20 40 60 800.000.020.040.060.080.100.120.14\nxUκη2=0.2\nκη2=0κη2=0.2κη2=0.4κη2=0.6\n-20 -10 0 10 20 30 40 500.000.050.100.150.200.25\nxUL=0.5FIG. 1. The effective potential for scalar perturbations with different Landκη2when l= 2, where\nwe have set M= 1 and Λ = 0.\nL=-0.5L=0L=0.5L=1\n0 2 4 6 8010203040\nxUκη2=0.2\nκη2=0κη2=0.2κη2=0.4κη2=0.6\n0 2 4 6 8010203040\nxUL=0.5\nFIG. 2. The effective potential for scalar perturbations with different Landκη2when l= 2, where\nwe have set M= 1 and Λ = −3.\none can rewrite Eq.(16) as\n\u0014\n4∂2\n∂u∂v+U(u, v)\u0015\nΨ(u, v) = 0 . (18)\nGenerally, based on different separations, either Eq. (14) or (18) can be solved numerically\nonce the boundary conditions are set, which we will do for the bumblebee black holes in\nthe next section. Here we concentrate on the effective potential Udefined in Eq. (12). The\nbehaviors of Ufor vanishing and negative Λ are shown in Figs. 1 and 2, respectively. The\ntortoise coordinate xranges from −∞ at the event horizon to + ∞at spatial infinity. For\nΛ = 0, the effective potential tends to zero both near the horizon and at large distances\nfrom the black hole. For Λ <0, however, the effective potential diverges at spatial infinity.\n7The effects of the LSB parameter Land the global monopole parameter κη2on the effective\npotential are evident.\nIII. QNMS OF THE BUMBLEBEE BLACK HOLES WITH A GLOBAL MONOPOLE\nA. The black hole with a vanishing cosmological constant\nAs seen in the previous section, the effective potential Ufor the scalar field approaches\nzero at x→ ±∞ in this case. Therefore, together with the physical consideration that the\nfield should be purely outgoing towards spatial infinity and purely ingoing at the horizon,\nthe asymptotic forms of the field can be written as\nψl∼\n\ne−iωx, x→ −∞ ,\neiωx, x→+∞.(19)\nThe radial equation (14) and the boundary condition (19) determine a discrete spectrum\nof frequencies {ωn}for a fixed pair of symmetry breaking parameters Landκη2. We use\nthe continued-fraction method[55] to solve this problem numerically. The frequency ωis\ngenerally complex, with a real part ωRcorresponding to the oscillation frequency of the field\nand an imaginary part ωIreflecting the evolution of the wave amplitude. A positive ωI\nimplies a growing scalar field with a growth rate of ωI. A negative ωIimplies a decaying\nscalar field with a damping rate of |ωI|.\nThe QNM frequencies as functions of κη2for different Lare shown in Fig. 3. The left\npanel shows that the real part of the frequencies, ωR, decreases with increasing Lorκη2. The\nright panel shows that the imaginary part of the frequencies, ωI, is negative for all values\nofLandκη2, indicating that the black hole is stable under the perturbation of a massless\nscalar field. However, we also observe that |ωI|becomes smaller as Lorκη2increases. This\nimplies that for the black hole with larger Lorκη2, the perturbation takes longer time to\ndecay away. The effect of Lon the stability in this case agrees with the results of other\nbumblebee black holes reported in Refs. [43, 44]. We note that κη2≥0, so the perturbation\nto the black hole without a global monopole ( κη2= 0) fades out more quickly than the case\nwith one ( κη2>0).\nWe also investigate the temporal evolution of the perturbation by solving Eq. (18). We\n8L=-0.5L=0L=0.5L=1\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.00.10.20.30.40.5\nκη2ωR\nL=-0.5L=0L=0.5L=1\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.000.020.040.060.080.100.120.14\nκη2-ωIFIG. 3. The real part (left) and imaginary part (right) of QNM frequencies of the spherical\nbumblebee black holes with a global monopole and a vanishing Λ, where we have set M= 1, l= 2\nandn= 0.\nset the initial conditions as\nΨ(u,0) = 0 , Ψ(0, v) = 1 . (20)\nThe numerical results are displayed in Fig. 4. They confirm that the black hole is stable\nunder the massless scalar field perturbation. The left panel shows the decay of the pertur-\nbation for different Lwith κη2= 0.2 and l= 2, in which it is obvious that the scalar field\ndecays slower for larger L. The right panel shows the decay of the perturbation for different\nκη2with L= 0.5 and l= 2, where one can see that the scalar field decays slower for larger\nκη2. This agrees with the results in the frequency domain. Another consistent observation\nfrom Figs. 3 and 4 is that the oscillation frequencies are more sensitive to the monopole\ncharges η2than to the LSB parameters L.\nWe use the Prony method[56] to extract the QNM frequencies from the time domain\nprofile and compare them with the results from the continued-fraction method. For instance,\nwe find the frequency ω= 0.342031 −0.0504228 iforκη2= 0.2, L= 0.5, l= 2 by using the\nProny method. The continued-fraction method gives ω= 0.342003 −0.0504359 ifor the\nsame case, which has an error less than one thousandth. Fig. 5 shows the comparison of the\nQNMs obtained by the two methods for different κη2, which demonstrates a good agreement\nbetween the two methods.\n9L=-0.5L=0L=0.5L=1150 200 250 300 35010-810-50.01\nt|Ψ|\nκη2=0κη2=0.2κη2=0.4κη2=0.6\n150 200 250 300 35010-810-50.01\nt|Ψ|FIG. 4. Temporal evolution of the scalar perturbations, with κη2= 0.2, l= 2 in the left panel and\nL= 0.5, l= 2 in the right panel.\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.00.10.20.30.40.5\nκη2ωR\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.000.020.040.060.08\nκη2-ωI\nFIG. 5. Comparison of the QNM frequencies obtained by the continued-fraction method (blue line)\nand those extracted from time domain analysis (red dots), with M= 1, L= 0.5, l= 2.\nB. The black hole with a negative cosmological constant\nFor Λ <0, the effective potential Uwill diverge at spatial infinity, as shown in Fig. 2.\nThis requires that Φ = 0 at spatial infinity. Therefore, the asymptotic forms of Ψ in this\ncase should be\nΨ∼\n\ne−iωx, x→ −∞ ,\n0 , x→+∞.(21)\nWe apply the Horowitz-Hubeny method[7] to solve Eq. (14) for the spectrum of frequencies\nwith these Dirichlet boundary conditions. The resulting QNM frequencies as functions of κη2\nfor various Lare depicted in Fig. 6. The left panel reveals that ωRdecreases with increasing\nLorκη2, implying that the scalar field oscillates at a lower frequency for larger Lorκη2.\n10L=-0.5\nL=0\nL=1\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.02.53.03.54.04.55.0\nκη2ωR\nL=-0.5\nL=0\nL=1\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.00.51.01.52.02.53.03.5\nκη2-ωIFIG. 6. The real part (left) and imaginary part (right) of QNM frequencies of the spherical\nbumblebee black holes with a global monopole and non-zero Λ, where we have set Λ = −3, l=\n2, n= 0 and a suitable Msuch that the horizon radius r+= 1.\nThe right panel demonstrates that ωIis negative for all values of Landκη2, indicating the\nstability of the bumblebee black hole under massless scalar field perturbations. Furthermore,\nwe observe that |ωI|decreases with increasing L, suggesting that the perturbation to the\nblack hole takes longer time to decay away for larger L. On the other hand, |ωI|is insensitive\ntoκη2, which differs significantly from the zero-Λ scenario.\nTo compute the temporal evolution of the QNMs in this setting, we need to limit the\ncalculation at a finite x=xmax, since the potential diverges at spatial infinity. This implies\nthat the calculation domain on the u-vplane is bounded by the line v−u= 2xmax[57], as\nillustrated in Fig. 7. Hence, we can impose the boundary condition\nΨ(v−u= 2xmax) = 0 . (22)\nFig. 8 illustrates the temporal evolution of scalar perturbations around the bumblebee\nblack hole with a global monopole and nonzero Λ for various values of Landκη2. The black\nhole is stable under massless scalar field perturbations, as evident from the decay of the\nscalar field. The left panel displays the evolution of the scalar perturbation for various L\nwith κη2= 0.2 and l= 2. The scalar field obviously decays slower for larger L, agreeing with\nthe results shown in Fig. 6. The right panel exhibits the evolution of scalar perturbation\nfor various κη2with L= 0.5 and l= 2. The decay speed of the scalar field is very similar\nfor different values of κη2.\n11FIG. 7. Schematic of the numerical grid and the calculation domain. The black dots denote the\ngrid points where the field values are given by the initial condition and the boundary condition.\nThe blue dots denote the grid points to be computed.[57]\nL=-0.5L=0L=1\n7.5 10.0 12.5 15.0 17.5 20.010-1010-810-610-4\nt|Ψ|\nκη2=0κη2=0.2κη2=0.4κη2=0.6\n6 7 8 9 10 11 1210-910-710-50.001\nt|Ψ|\nFIG. 8. Temporal evolution of the scalar perturbations in the scenario with nonzero Λ, where\nκη2= 0.2, l= 2 in the left panel and L= 0.5, l= 2 in the right panel.\nWe again employ the Prony method to extract the QNM frequencies from the time\ndomain profile. To verify the robustness of our calculations, in addition to the previously\nshown cases of l= 2, we use the cases of l= 0 as the examples to cross check the results\nfrom frequency and time domains, which are presented in Fig. 9. The lines correspond to\nthe Horowitz–Hubeny method, and the red dots represent the frequencies extracted by the\nProny method. The results exhibit good agreement between the two methods, confirming\nthe reliability of our results.\n12L=-0.5\nL=0\nL=1\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.701234\nκη2ωR\nL=-0.5\nL=0\nL=1\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70123\nκη2-ωIFIG. 9. Comparison of the QNM frequencies obtained by the Horowitz–Hubeny method (solid\nlines) and those extracted from time domain analysis (red dots), with Λ = −3, l= 0 and Mchosen\nsuch that the horizon radius r+= 1.\nIV. CONCLUSION AND DISCUSSION\nWe have investigated the stability of the spherical bumblebee black holes with a global\nmonopole in the two cases with a zero or a negative cosmological constant by examining the\nQNMs of the black holes under the perturbations of a massless scalar field. The analyses are\nperformed in both frequency domain and time domain by using different numerical methods.\nWe find that both families of black holes are stable under the perturbations of a massless\nscalar field. In particular, the scalar field around the black holes decays slower for larger\nLSB parameter L. On the other hand, the influence of the global monopole on the stability\nof two types of black holes is different. For the black hole with zero Λ, the global monopole\nfield prolongs the decay of the perturbation. Whereas for the black hole with negative Λ,\nthe effect of the global monopole on the stability is negligible.\nSuch a significant difference may be understood as follows. QNM frequencies are eigen-\nvalues ωthat satisfy the boundary conditions corresponding to the outgoing wave at spatial\ninfinity and the ingoing wave at the horizon. Thus, computing QNM frequencies is an eigen-\nvalue problem that depends on the boundary condition at spatial infinity. When Λ = 0, the\nboundary condition (19) at spatial infinity depends on both Landκη2through the tortoise\ncoordinate (13). Hence, both Landκη2affect QNM frequencies in this case. However,\nwhen Λ <0, the boundary condition is a Dirichlet type that is independent of Landκη2\ndue to the divergent potential. Therefore, κη2has little effect on the QNM frequencies.\n13Nevertheless, Lstill affects the QNM frequencies because it breaks the symmetry between\nspace and time. Moreover, the QNMs are derived by analyzing the outgoing wave signals of\nthe perturbation, which propagates differently depending on Lbecause the temporal-radial\nsymmetry of the metric is broken. The global monopole, on the other hand, only manifests\nitself in a solid deficit angle in the spacetime, which does not contribute to the breaking of\nthe temporal-radial symmetry. This can also be seen by rewritting the effective potential\nU(r) in Eq. (12) for large ras\nU(r) = (1 −κη2−2M\nr+r2)\u0014l(l+ 1)\nr2+2M\n(1 +L)r3+2\n1 +L\u0015\n=2r2\n1 +L+\u00142(1−κη2)\n1 +L+l(l+ 1)\u0015\n−2M\n(1 +L)r+O(1\nr2).(23)\nOne can see that Lcontributes to the coefficient of the r2term, while κη2only affects the\nconstant term. Therefore, the global monopole does not alter the shape of U(r) significantly\nat large rand hence, the stability of the black hole is almost independent of the global\nmonopole.\nIn the high energy regime, the wavelength of the massless scalar field is negligible com-\npared to the horizon scale of the black hole. Thus, the field follows the null geodesics, which\nare influenced by the global monopole and the LSB parameter[33]. 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With the feedback from a detec-\ntion system, five degrees of freedom of the coil, i.e. the\nhorizontal displacement x,yand the rotation angles \u0012x,\n\u0012y,\u0012z, can be controlled by the active damping device.\nHence, two functions, i.e. suppressing the undesired coil\nmotions and reducing the misalignment error, can be\nrealized with this active damping device. The principle,\nconstruction and performance of the proposed active\ndamping device are presented.\nIndex Terms —watt balance, joule balance, the\nPlanck constant, electromagnetic damping, misalign-\nment error.\nI. Introduction\nSeveral national metrology institutes (NMIs) across the\nworld are working hard on redefining one of the seven\nSI base units, the kilogram (kg), in terms of the Planck\nconstanth, by means of either the watt/joule balance\n[1-9] or the X-ray crystal density (XRCD) method [10].\nThe watt balance, which was proposed by Kibble in 1975\n[11], has been adopted by the majority of NMIs. The\nNational Institute of Metrology (NIM, China) is focusing\non the joule balance, which can be seen as an alternative\nrealization of the watt balance [12].\nThe unwanted coil motions in the watt/joule balance,\ne.g., horizontal and rotational movements, will lower the\nsignal to noise ratio in the measurement. With these\nmotions, the measurement would take a much longer time\nin order to achieve the type A relative uncertainty at\nan order of 10\u00008. Moreover, the noticeable undesired coil\nmotions will introduce a systematic bias [13]. Therefore,\ndamping devices are necessary to be employed to suppress\nthese undesired coil motions in the watt/joule balance. In\nwatt balances, e.g., NIST-4 at the National Institute of\nStandards and Technology (NIST, USA) [14], an active\ndamping system has been used during the measurement.\nThis work is supported by the China National Natural Science\nFoundation (Grant Nos.91536224, 51507088) and the China National\nKey Research and Development Plan (Grant No. 2016YFF0200102).\nJ. Xu, Q. You are with Tsinghua University, Beijing 100084, China.\nZ. Zhang and Z. Li are with the National Institute of Metrology\n(NIM), Beijing 100029, China and the Key Laboratory for the\nElectrical Quantum Standard of AQSIQ, Beijing 100029, China.\nS. Li is currently with the International Bureau of Weights and\nMeasures (BIPM), Pavillon de Breteuil, F-92312 S` evres Cedex,\nFrance. E-mail:leeshisong@sina.com.The damping coils in such systems are conventionally\nlocated on the framework of the suspended coil. To avoid\nadditional force in the weighing mode and unwanted in-\nduced voltage in the velocity mode, the damping device\nis turned off during both measurement modes. During the\ndynamic measurement mode, the undesired coil motions,\nhowever, is not taken care of. In the joule balance at NIM\nto suppress unwanted motions during the measurement\nwithout introducing any flux into the main magnet, a novel\nactive electromagnetic damping device, located outside\nthe main magnet, is designed and used. The merit of\nthe system is that the field of the damping device has\nno effect on the main field, and hence can be used in\nboth the dynamic and static measurement modes. With\nthe feedback from a detection system, five degrees of\nfreedom of the coil, i.e. the horizontal displacement x,y\nand the rotation angles \u0012x,\u0012y,\u0012z, can be controlled by\nthis device. Two functions, i.e. suppressing the undesired\ncoil motions and reducing the misalignment error, then\ncan be realized in the measurements. The original idea was\npresented in [15]. In this paper, the principle, construction\nand performance of the active electromagnetic damping\ndevice are presented.\nThe rest of this article is organized as follows: the\nprinciple of the joule balance is introduced in section\n2; the principle and construction of the active damping\ndevice are presented in section 3; the performance of the\nactive damping device in the joule balance is presented\nin section 4; some potential systematic effects using the\nactive damping device, the vertical force and the flux\nleakage at the mass weighing position, are discussed in\nsection 5.\nII. Principles of Joule Balance\nThe joule balance is the integral of the watt balance. The\nmathematical details are presented in [12]. The equation\nis expressed as\nZ\nLF\u0001dl+Z\nL\u001c\u0001d\u0012=I[ (B)\u0000 (A)]; (1)\nwhereFdenotes the magnetic force, \u001cthe torque relative\nto the mass center of the coil, Lthe vector trajectory\nwhen the coil is moved from position A to B, Ithe current\nthrough the coil,  (B)\u0000 (A)the flux linkage difference\nof the position B and A.\nOn the right side of equation (1), the product of the flux\nlinkage difference  (B)\u0000 (A)and the current Iis the\nmagnetic energy change. On the left side of equation (1),arXiv:1610.02799v1  [physics.ins-det]  10 Oct 2016IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 2\nFig. 1. The schematic diagram of the misalignment errors.\nthe two integral terms are the work done by the magnetic\nforce and torque. They can be expressed as\nwAB=Z\nLFzdz+Z\nLFxdx+Z\nLFydy\n+Z\nL\u001czd\u0012z+Z\nL\u001cxd\u0012x+Z\nL\u001cyd\u0012y:(2)\nNote that in equation (2) only the vertical force, i.e.\nFz, can be measured precisely. The integral terms of the\nhorizontal forces Fx,Fyand torques \u001cx,\u001cy,\u001czare parts\nof the misalignment errors in the joule balance. Other\nparts of the misalignment errors come from the position\nchange of the coil between the two measurement modes\nin the joule balance. In equation (1), it is assumed that\nthe positions A and B in the measurement of the flux\nlinkage difference are the same as the starting point and\nending point of the integral trajectory L. However, it is\nnot the case in the actual measurement. As shown in Fig.\n1, when current pass through the coil, the positions A and\nB will actually shift to A’ and B’ . Then, equation (1) can\nbe rewritten as\nwAA0+wA0B0+wB0B=I[ (B)\u0000 (A)]; (3)\nwherewAA0,wA0B0,wB0Bare the work done by the mag-\nnetic force and torque when the coil is respectively moved\nfrom position A to A’, A’ to B’ and B’ to B. It can be seen\nfrom equation (3) that the misalignment error includes\nthree parts: wAA0,wB0Band the integral terms of the\nhorizontal forces and torques in wA0B0.\nThe active damping device used in the joule balance\nis able to produce auxiliary horizontal forces and torques\non the coil to change the position of the coil. Therefore,\nwith the feedback from the detection system of the five\ndirections of displacement, the position of the coil can\nbe easily controlled. As a result, the positions A and B\ncan be adjusted to be the same as A’ and B’ by the\nfeedback of the damping device. In the meanwhile, the\nhorizontal displacement x,yand the rotation angles \u0012x,\n\u0012y,\u0012z, can be kept unvaried when the coil is moved along\nthe vector trajectory Lin the weighting mode of the\nFig. 2. Construction of the damping device. (a) The overall structure\nof the damping device with the top cover 1 open. (b) The permanent\nmagnets (5, 6, 7, 11,12, 13) and the inner yokes (8, 9, 10). (c) Com-\nbination of auxiliary coils 14 and 15. (d) Combination of auxiliary\ncoils 14 and 16.\njoule balance. These two improvements will significantly\nreduce the misalignment error. For an ideal case, i.e. the\nalignment is adjusted to be good enough, only the work\ndone by the vertical force in wA0B0is left on the left side\nof equation (3) and other terms can be ignored. Then\nequation (3) can be rewritten as\nZB0\nA0Fzdz=I[ (B)\u0000 (A)]: (4)\nIII. Principle and Construction of the Damping\nDevice\nA. The practical structure\nFig. 2 shows the practical structure of the active damp-\ning device used in the joule balance.The outer yokes 1, 2,\n3 and the inner yokes 8, 9,10 are made of soft iron with\nhigh permeability. The component 4 shown in Fig. 2(a)\nis made of aluminum, for mechanically supporting three\ninner yokes. The component 17 in Fig. 2(c) and Fig. 2(d)\nis the framework of the auxiliary coils which is made up\nof polysulfone.\nComponents 5, 6, 7, 11, 12 and 13 are permanent\nmagnets magnetized along the zaxis. It should be noted\nthat the magnetization direction of the three permanent\nmagnets 5, 12, 13 is opposite to the magnetization direc-\ntion of the other three permanent magnets 6, 7, 11. As a\nresult, in this magnetic circuit, the magnetic flux in the\nair gap originates from the yoke 8 and enters the yokes 9,\n10.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 3\nComponents 14, 15 and 16 are three orthogonal aux-\niliary coils. Fig. 2(c) and Fig. 2(d) show two different\ncombinations of the auxiliary coils. Coils 14 and 15 are\nglued together in Fig. 2(c) and coils 14 and 16 are glued\ntogether in Fig. 2(d). Since there are considerable mag-\nnetic gradients in x,yand the azimuth \u0012, torques and\nforces will be produced on the auxiliary coils when current\npasses through the auxiliary coils.\nThe cross sectional view of the inner yokes and the\nauxiliary coils in the active damping device is shown in\nFig. 3. The dotted lines denote the magnetic flux in the\nair gap. The coil 14 in the magnetic field shown in Fig.\n3(a) will produce a torque \u001cxaroundxaxis, relative to\nthe mass center. Coil 15 in Fig. 3(b) will produce a force\nFyalong theyaxis and coil 16 in Fig. 3(c) will produce\na forceFxalong thexaxis. The forces and torques of the\nthree coils along other directions are theoretically small.\nThe schematic diagram of the damping device is shown\nin Fig. 3(d), containing 3 damping segments. The coil is\nsuspended from the spider by three rods. Each damping\nsegment is fixed on a support with the auxiliary coils\nmechanically connected to the coil spider rod. In two\nof them, the combination of the auxiliary coils 14 and\n15 shown in Fig. 2(c) is used. In the other one, the\ncombination of the auxiliary coils 14 and 16 shown in Fig.\n2(d) is used. Hence, the torque produced by coil 14 in all\nthe three damping segments is used to adjust the rotation\nangle\u0012x,\u0012yof the suspended coil. The force of coil 15 in\ntwo segments points to the center of the suspended coil\nand can be applied to adjust the horizontal movement x,\ny. The force of coil 16 in the third segment is along the\ntangential of the suspended coil and can be employed to\nadjust the rotation along zaxis, i.e.\u0012z.\nThe auxiliary coils are connected to the current sources\nthrough the suspension system by fine wires called hair-\nsprings. To reduce the number of the hairsprings and keep\nthe symmetry of the suspended coil, six auxiliary coils are\ninstalled in the practical structure instead of nine in the\ninitial design [15]. Fig. 4(a) shows the six auxiliary coils\ninstalled on the suspended system. In theory, to suppress\nthe motions of the five degrees of freedom, the minimum\nnumber of auxiliary coils is five. Hence, only two of the\nthree coils 14 are connected to the current sources in\npractice. The three damping devices are placed at angles\nof 120 degrees around the magnet in the joule balance as\nshown in Fig. 4(b). The pallets of the damping devices are\nfixed with respect to the marble shelf in the joule balance\nwhich will be kept static.\nIn the joule balance, the suspended coil is kept static\nwhile the magnet is moved by a linear translation stage.\nFor the active damping device used in the joule balance,\nthe auxiliary coils are always kept static and do not require\ntoo much moving space. If such a damping device is used\nin the velocity mode of watt balances, the length of the\ninner yokes must be long enough for the movement of the\nauxiliary coils.\nFig. 3. The cross sectional view of the inner yokes and the auxiliary\ncoils. (a) The torque of coil 14. (b) The force of coil 15. (c) The force\nof coil 16. (d) The schematic diagram of three damping devices.\nFig. 4. (a) The auxiliary coils installed on the suspended system. (b)\nMechanical assembling of the damping device.\nB. The feedback control\nThe entire control circuit is shown in Fig. 5. The PID\ncontroller is realized with a LabVIEW program. Five\nchannels of a high-speed analog output (NI 6733) are used\nas the voltage control of the current sources. With an\nexternal voltage reference, the range of the analog output\nvoltage is \u00061 V. The circuit of a current source is shown in\nFig. 6, performing the U=Iconverter. A 10 \nfour-terminal\nresistor with a low temperature coefficient ( <1 ppm/\u000eC)\nis used as the sense resistor in the current source. Hence,\nthe range of the output current is \u0006100 mA.\nThe detection system of the relative position between\nthe coil and the main magnet is composed of a laser\ninterferometer and position sensitive devices (PSD). The\nrotation angles \u0012x,\u0012yare obtained with the laser interfer-\nometer. The resolution of \u0012xand\u0012yis less than 1 \u0016rad.\nThe horizontal displacements x,yand rotation angle \u0012zIEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 4\nFig. 5. The block diagram of the active control circuit.\nFig. 6. The circuit of the current source.\nFig. 7. The setup of auxiliary coils and the coordinate axis of the\ndetection system.\nare obtained from four PSDs. The resolution in xandyis\nabout 2\u0016m and the resolution of \u0012zis about 5\u0016rad.\nThe five auxiliary coils and the coordinate axis of the\ndetection system are shown in Fig. 7. The large circle\ndenotes the suspended coil in joule balance. Five auxiliarycoils are set concentric to the suspended coil with every 120\ndegrees. Since the horizontal displacement and rotation\nangle of the suspended coil are in a small range, the\nrelationship between the detection and the auxiliary coil\ncurrent is close to be linear. In order to check the depen-\ndence between the coil motion and the current through\nthe auxiliary coils, the transfer matrix Mis measured by\napplying a 50 mA current individually to each of the five\nauxiliary coils. The measurement is written as\n2\n66664dx\ndy\nd\u0012x\nd\u0012y\nd\u0012z3\n77775=M2\n66664I16\nI15\u00002\nI15\u00001\nI14\u00002\nI14\u000013\n77775\n=2\n666640:12 0:32\u00001:04 \u00000:20 0:16\n0:16\u00001:44 0:88 \u00000:24\u00000:24\n0:10 0:12 0:16 1:78 1:30\n0:04 0:10\u00000:08\u00001:16 1:78\n6:30 0:20\u00000:30 0:00 0:003\n777752\n66664I16\nI15\u00002\nI15\u00001\nI14\u00002\nI14\u000013\n77775;\n(5)\nwhere dx,dy,d\u0012x,d\u0012yandd\u0012zare the variations of the\ndetection system with units of \u0016m,\u0016m,\u0016rad,\u0016rad,\u0016rad\nrespectively, I16,I15\u00002,I15\u00001,I14\u00002andI14\u00001the currents\npassing through the five auxiliary coils with unit of mA.\nThe determined matrix Min equation (5) matches the\nanalysis of the above section: The horizontal displacement\nx,yis mainly controlled by coils 15-1 and 15-2. The\nrotation angle \u0012x,\u0012yis mainly adjusted by coils 14-1 and\n14-2. The rotation angles \u0012zis mainly controlled by coil\n16. It can be seen that due to some imperfection of the\nmagnetic field and the misalignment of five auxiliary coils,\nthere are some weak couplings between different channels.\nIn the control, we should use the inverse of the transfer\nmatrix to decouple the correlation of different loops, i.e.\n2\n66664I16\nI15\u00002\nI15\u00001\nI14\u00002\nI14\u000013\n77775=M\u000012\n66664dx\ndy\nd\u0012x\nd\u0012y\nd\u0012z3\n77775\n=2\n66664\u00000:03 0:02 0:00 0:00 0:16\n\u00000:78\u00000:87\u00000:16 0:07 0:04\n\u00001:22\u00000:26\u00000:08 0:14 0:03\n0:12 0:04 0:39\u00000:29\u00000:01\n0:07 0:06 0:26 0:37\u00000:013\n777752\n66664dx\ndy\nd\u0012x\nd\u0012y\nd\u0012z3\n77775:\n(6)\nWith a known difference of the detection and the target,\nthe output voltage of auxiliary coils will be determined\nby both the PID controller and the inverse matrix M\u00001.\nNote that since the response of auxiliary coils can easily\nexcite the suspension mechanism, fast and large feedback\ncurrents should be avoided.\nIV. Experimental Results\nA. Damping performance\nWhen the suspended coil receives external shocks such\nas loading of the test mass, ground vibration, etc., theIEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 5\nenergy blue transfer out through the suspension system is\nvery slow due to a high Qfactor. The suspended coil will\nswing for a long time without damping. Fig. 8 shows the\nmotions of the suspended coil when the mass is put on the\nmass pan with and without the proposed damper. Without\nany damping device, the maximal motion amplitudes of\nthe five degrees of freedom x,y,\u0012x,\u0012y,\u0012zare 250\u0016m,\n150\u0016m, 500\u0016rad, 550\u0016rad, 300\u0016rad respectively. After\nabout 5 minutes, the motion amplitudes will decrease\nto 10\u0016m, 10\u0016m, 15\u0016rad, 15\u0016rad, 50\u0016rad.The damping\nfactor is about 0.11 kg \u0001s\u00001. The steady state deviations\nbetween the mass on state and the mass off state of the\nfive degrees of freedom are 10 \u0016m, 35\u0016m, 50\u0016rad, 20\u0016rad,\n30\u0016rad respectively.\nWhen the active damping device is used, the motions of\nthe suspended coil will decay rapidly. As shown in Fig.\n8,x,y,\u0012x,\u0012y,\u0012zwill reduce to 8 \u0016m, 8\u0016m, 10\u0016rad,\n10\u0016rad, 20\u0016rad in 30 seconds. The damping factor is\nabout 1.1 kg \u0001s\u00001, which is about 10 times larger than that\nwithout the damper. The steady state deviations between\nthe mass on state and the mass off state of the five degrees\nof freedom are less than 2 \u0016m, 2\u0016m, 1\u0016rad, 1\u0016rad, 5\u0016rad.\nB. Reduction of the current dependence of the coil position\nWhen current passes through the suspended coil, the\npositions A and B in the flux linkage difference mea-\nsurement will change to A’ and B’, yielding the errors\nwAA0andwB0B. To generate a 5 N magnetic force, the\nsuspended coil current is about 7.5 mA. Fig. 9 shows the\nchanges of the five degrees of freedom x,y,\u0012x,\u0012y,\u0012z\nwhen the suspended coil current is reduced from 7.5 mA\nto 0 mA slowly. Without the active damping device, the\nmagnitude changes are 25 \u0016m, 5\u0016m, 15\u0016rad, 15\u0016rad,\n70\u0016rad respectively. Due to the restriction of the narrow\nair gap and the laser alignment in the present construction\nof the joule balance, it is very difficult to further adjust\nthe concentricity and horizontality of the suspended coil\nto reduce the magnitudes of the coil position change.\nTo reduce the errors wAA0andwB0B, the positions of A\nand B can be adjusted to be the same as A’ and B’ by the\nactive damping device. As shown in Fig. 9, when the active\ndamping device is added, the magnitude changes of x,y,\n\u0012x,\u0012y,\u0012zare less than 2 \u0016m, 2\u0016m, 1\u0016rad, 1\u0016rad, 5\u0016rad\nrespectively, which on average is one magnitude improved\nthan that without the electromagnetic damper.\nC. Improvement of the vertical movement\nIn both the weighing and dynamic modes of the joule\nbalance, the magnet is moved in the vertical direction zby\na translation stage [17]. However, due to the distortion of\nthe guide rail in the translation stage and other mechanical\nexcitations, the other five degrees of freedom x,y,\u0012x,\u0012y,\n\u0012zwill also change with the movement of the magnet.\nTherefore, the horizontal forces Fx,Fyand torques \u001cx,\n\u001cy,\u001czwill contribute to in the misalignment errors.\nFig. 10 shows the value of the five degrees of freedom\nx,y,\u0012x,\u0012y,\u0012zwith different zduring the movement of\nFig. 8. Coil motions when test mass is put on the mass pan with\nand without the proposed damper. The red lines (–) are without the\ndamper while the blue dash lines (- -) are with the damper.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 6\nFig. 9. Coil motions when the coil current is reduced from 7.5 mA\nto 0 mA slowly. The red lines (–) are without the damper while the\nblue dash lines (- -) are with the damper.\nFig. 10. Coil motions with different zduring the movement of the\nmagnet. The red lines (–) are without the damper while the blue dash\nlines (- -) are with the damper.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 7\nTABLE I\nThe unwanted vertical force of the five auxiliary coils\nwith 100 mA excitation.\nCoil No. Vertical force /mg uncertainty /mg\n16 0.08 0.01\n15-1 0.06 0.01\n15-2 0.06 0.01\n14-1 1.04 0.01\n14-2 0.85 0.01\nthe magnet. Without the active damping device, the peak-\nto-peak values of the five degrees of freedom are 15 \u0016m,\n15\u0016m, 10\u0016rad, 8\u0016rad, 40\u0016rad respectively. When the\nactive damping device is added during the movement, the\nunwanted motions of the magnet can be compensated by\nthe motions of the suspended coil. Hence, the changes of\nthe relative value of x,y,\u0012x,\u0012y,\u0012zcan be reduced. As\nshown in Fig. 10, the peak-to-peak values of the five de-\ngrees of freedom are reduced to 6 \u0016m, 6\u0016m, 4\u0016rad, 4\u0016rad,\n25\u0016rad with the active damping device. The measurement\nshows that all curves of the five degrees of freedom are\nnearly flat. Therefore, the misalignment errors, resulting\nfrom the work done by the horizontal forces and torques\ninwA0B0, are greatly reduced.\nV. Consideration of Potential Systematic\nEffects\nA. The vertical force\nWhen the active electromagnetic damping device is used\nin the weighing mode of the joule balance, any additional\nvertical force caused by the five auxiliary coils should be\nconsidered. Here the mass comparator has been used to\nmeasure the vertical force. With 100 mA current in the\nauxiliary coils, the measurement results are shown in Table\nI. As discussed in [15], the vertical forces of coil 16 and 15\nare very small when compared with that of coil 14. Since\nthe working current in the auxiliary coils is much less than\n100 mA in the weighing mode, the vertical forces will be\nmuch smaller than the results in Table I.\nWhen the force of the suspended coil is measured in the\nweighing mode, the suspended coil is static and the current\nin the auxiliary coils of the active damping device is nearly\nconstant. As the magnetic force is proportional to the cur-\nrent, the additional vertical force of the auxiliary coil can\nbe determined by measuring the current in the auxiliary\ncoil. Note that in this measurement, the uncertainty of the\nmass comparator is about 0.01 mg. A correction may then\nbe made and the effect of the unwanted vertical forces can\nbe reduced to several parts in 108.\nB. Magnetic flux leakage\nThe distance between the active damping devices and\nthe test mass is about 160 mm. Hence, the leakage mag-\nnetic field of the active damping devices should be consid-\nered. The interaction between the magnetic field and the\nmass is discussed in [16]. The equation of the vertical force\nFig. 11. The absolute magnitude of the magnetic field at the location\nof the mass. The measurement is fitted by a quadratic function.\non the mass with a volume susceptibility \u001fand permanent\nmagnetization Mis given by [18]\nFz=\u0000\u00160\n2@\n@zZ\n\u001fH\u0001HdV\u0000\u00160@\n@zZ\nM\u0001HdV: (7)\nThe magnetic field at the location of the mass is\nmeasured by a gauss meter. Fig. 11 shows the absolute\nmagnitude of the magnetic field as a function of the\ndistance from the top surface of the mass pan. Note that\nalthough the measured field is comparable to the earth’s\nmagnetic field, it produces a much larger field gradient,\nabout 1.3\u0016T/mm. The employed 500 g test mass is about\n50 mm in height and 40 mm in diameter. The magnetic\nsusceptibility \u001fof the mass material is less than 6\u000210\u00004\nand the permanent magnetization Mis about 0.1 A/m.\nWith the method used in [16], the calculated magnetic\nforce of the mass is less than 10 \u0016g. Hence, the relative\nsystematic error caused by the leakage magnetic field is\nless than 2\u000210\u00008.\nVI. Conclusions\nThe active electromagnetic damping device presented\nin this paper can be used in both the dynamic and\nstatic measurement modes of the joule balance. Five de-\ngrees of freedom of the suspended coil, i.e. the horizontal\ndisplacement x,yand the rotation angles \u0012x,\u0012y,\u0012z,\ncan be actively controlled by the damping device. The\nexperimental results show that the active damping device\nwell meets the design targets. The damping factor of the\nsystem is increased by a factor of 10 and the alignment\nis greatly improved by compensating for the non-vertical\nmovement of the coil. Two potential systematic effects, i.e.\nthe unwanted vertical force and the leakage magnetic field\nof the active damping device, are analyzed. It is shown that\nthese effects are small and compensable within few parts\nin108. At present, the misalignment error is about several\nparts in 107due to the restriction of the narrow air gap\nand the laser alignment in the joule balance project. The\npresented active damping device in this case is powerful\nto reduce misalignment uncertainties.IEEE TRANS. INSTRUM. MEAS., CPEM2016, JUNE 10, 2021 8\nReferences\n[1] Z. Zhang, Q. He and Z. Li, ”An approach for improving the\nwatt balance” in Digest of Conf. on Precision Electromagnetic\nMeasurement , Torino, Italy, Jul. 9-14, 2006, pp. 126-127.\n[2] A. Robinson, B. P. Kibble, ”An initial measurement of Planck’s\nconstant using the NPL Mark II watt balance. ” Metrologia ,\nvol.44, no. 6, pp.427-440, 2007.\n[3] A. G. Steele, J. Meija, C. A. Sanchez, et al, ”Reconciling Planck\nconstant determinations via watt balance and enriched-silicon\nmeasurements at NRC Canada.” , Metrologia , vol.48, pp.L8-L10,\n2012.\n[4] A. Picard, H. Fang, A. Kiss, et al, ”Progress on the BIPM watt\nbalance.” IEEE Transactions on Instrumentation and Measure-\nment , vol.58, pp.924-929, 2008.\n[5] S. Schlamminger et al, ”Determination of the Planck constant\nusing a watt balance with a superconducting magnet system at\nthe National Institute of Standards and Technology”, Metrologia ,\nvol.51, pp15-24, 2014.\n[6] A. Eichenberger, H. Baumann, B. Jeanneret, B. Jeckelmann, P.\nRichard,and W. Beer, ”Determination of the Planck constant\nwith the METAS wattbalance,” Metrologia , vol. 48, no. 3, pp.\n133ÂĺC141,2011.\n[7] M. Thomaset al, ”First determination of the Planck constantus-\ning the LNE watt balance”, Metrologia , vol.52, pp433-443, 2015.\n[8] M.C.Sutton, T.M.Clarkson. ”A magnet system for the MSL watt\nbalance.” Metrologia , vol.51, pp. 101-106,2014.\n[9] D.Kim, B.C.Woo, K.C.Lee, et al, ”Design of the KRISS watt\nbalance.” Metrologia , vol.51, pp. 96-100, 2014.\n[10] Y. Azumaet al, ”Improved measurement results forthe Avo-\ngadro constant using a28Si-enriched crystal”, Metrologia , vol.52,\npp360-375, 2015.\n[11] B. P. Kibble, ”A measurement of the gyromagnetic ratio of\nthe proton by the strong field method”, Atomic Masses and\nFundamental Constants5 , Springer US, pp545-551, 1976.\n[12] J. Xu et al, ”A determination of the Planck constant by the gen-\neralized joule balance method with a permanent-magnet system\nat NIM”, Metrologia , vol.53, pp86-97, 2016.\n[13] S. Li et al, ”Coil motion effects in watt balances: a theoretical\ncheck”, Metrologia , vol.53, pp817-828, 2016.\n[14] D. Haddadet al, ”A precise instrument to determine the Planck\nconstant, and the future kilogram”, Review of Scientific Instru-\nments ,vol.87, pp1-14, 2016.\n[15] J. Xu et al, ”A Magnetic Damping Device for Watt and Joule\nBalances”, To be published inProc. CPEM Dig., Jul. 2016 .\n[16] Seifert F, Panna A, Li S, et al,”Construction, Measurement,\nShimming, and Performance of the NIST-4 Magnet System”,\nIEEE Transactions on Instrumentation and Measurement ,vol.63,\npp.3027-3038, 2014.\n[17] Zhengkun Li, et al, ”The Improvement of Joule Balance NIM-\n1 and the Design of New Joule Balance NIM-2”, IEEE Trans-\nactions on Instrumentation and Measurement , vol.64, pp.1676-\n1684, 2015.\n[18] R. S. Davis, ”Determining the magnetic properties of 1 kg mass\nstandards”, J. Res. Nat. Inst. Standards Technol. , vol.100, no.3,\npp.209ÂĺC226, May. 1995.\nJinxin Xu was born in Yancheng, Jiangshu\nprovince, China, in 1990. He received the B.S.\ndegree from Harbin Institute of Technology,\nHarbin, China, in 2012. He is currently pursu-\ning the Ph.D. degree at Tsinghua University,\nBeijing, China. His dissertation will be a part\nof the joule Balance at the National Institute\nof Metrology, China.\nQiang You was born in Shandong province,\nChina. He is currently pursuing the Ph.D. de-\ngree with Tsinghua University, Beijing, China.\nHis dissertation will be a part of the joule bal-\nance with the National Institute of Metrology,\nBeijing, China.\nZhonghua Zhang was born in Suzhou,\nJiangshu, China in July 1940. He received his\nB.S. degree and M.S. degree in Electrical En-\ngineering from Tsinghua University, Beijing,\nChina separately in 1965 and 1967. In 1995, he\nbecame a member of the Chinese Academy of\nEngineering. In 1967, Zhonghua Zhang joined\nthe Electromagnetic Division of National In-\nstitute of Metrology (NIM), Beijing, China.\nSince then he has been involved in the re-\nsearch work on the Cross Capacitor standard,\nsuperconducting high magnetic field standard and Quantum Hall\nResistance standard. In 2006, he proposed the joule balance and led\nthe research work since then.\nZhengkun Li was born in Henan Province,\nChina in 1977. He received the B.S. degree in\ninstrument science and technology from China\nInstitute of Metrology, Hangzhou, China, in\n1999. He received the M.S. degree in quantum\ndivision of National Institute of Metrology\n(NIM), China in 2002. He received his Ph.D.\ndegree from Xi’an Jiaotong University, Xi’an,\nChina in 2006. In 2002, he became a perma-\nnent staff of NIM and worked on the establish-\nment of Quantum Hall Resistance standard.\nSince 2006, he has been involved in the research work on joule balance\nproject at NIM and focuses on the electromagnetic measurement\nincluding mutual inductance measurement for the joule balance.\nShisong Li (M’15) got the Ph.D. degree in\nTsinghua University, Beijing, China in July,\n2014. He then joined the Department of Elec-\ntrical engineering, Tsinghua University during\nJuly, 2014-September, 2016. He has been a\nguest researcher at the National Institute of\nMetrology in China since 2009, and he worked\nat the National Institute of Standards and\nTechnology, United States for 14 months dur-\ning 2013-2016. He is currently with the In-\nternational Bureau of Weights and Measures\n(BIPM), France. His research interests include modern precision\nelectromagnetic measurement and instrument technology."    },    {        "title": "2104.05789v1.Thermal_Radiation_Equilibrium___Nonrelativistic__Classical_Mechanics_versus__Relativistic__Classical_Electrodynamics.pdf",        "content": "arXiv:2104.05789v1  [physics.class-ph]  12 Apr 2021Thermal Radiation Equilibrium: (Nonrelativistic) Classi cal\nMechanics versus (Relativistic) Classical Electrodynami cs\nTimothy H. Boyer\nDepartment of Physics, City College of the City\nUniversity of New York, New York, New York 10031\nAbstract\nPhysics students continue to be taught the erroneous idea th at classical physics leads inevitably\nto energy equipartition, and hence to the Rayleigh-Jeans la w for thermal radiation equilibrium.\nActually, energy equipartition is appropriate only for nonrelativistic classical mechanics, but has\nonly limited relevance for a relativistic theory such as classical electrodynamics. In this article,\nwe discuss harmonic-oscillator thermal equilibrium from t hree different perspectives. First, we\ncontrast the thermal equilibrium of nonrelativistic mecha nical oscillators (where point collisions\nare allowed and frequency is irrelevant) with the equilibri um of relativistic radiation modes (where\nfrequency is crucial). The Rayleigh-Jeans law appears from applying a dipole-radiation approx-\nimation to impose the nonrelativistic mechanical equilibr ium on the radiation spectrum. In this\ndiscussion, we note the possibility of zero-point energy fo r relativistic radiation, which possibility\ndoes not arise for nonrelativistic classical-mechanical s ystems. Second, we turn to a simple electro-\nmagneticmodelofaharmonicoscillator andshowthattheosc illator isfullyinradiationequilibrium\n(which involves allradiation multipoles, dipole, quadrupole, etc.) with clas sical electromagnetic\nzero-point radiation, but is notin equilibrium with the Rayleigh-Jeans spectrum. Finally, we\ndiscuss the contrast between the flexibility of nonrelativi stic mechanics with its arbitrary poten-\ntial functions allowing separate scalings for length, time , and energy, with the sharply-controlled\nbehavior of relativistic classical electrodynamics with i ts single scaling connecting together the\nscales for length, time, and energy. It is emphasized that wi thin classical physics, energy-sharing,\nvelocity-dependent damping is associated with the low-fre quency, nonrelativistic part of the Planck\nthermal radiation spectrum, whereas acceleration-depend ent radiation damping is associated with\nthe high-frequency adiabatically-invariant and Lorentz- invariant part of the spectrum correspond-\ning to zero-point radiation.\n1I. INTRODUCTION\nA. Erroneous Textbook Claim\nThe current textbooks of modern physics still make the erroneou s claim that classical\nphysics leads inevitably tothe Rayleigh-Jeansspectrum fortherma l radiationequilibrium.[1]\nIndeed the treatment of classical physics in these texts seems to have progressed only so far\nas James Jeans’ Report on Radiation and the Quantum-Theory in 1914, but no further.[2]\nIn this article, we make another[3] attempt to counteract the err oneous textbook claim.\nSpecifically, we point out that a simple electromagnetic-model harmo nic oscillator is notin\nequilibrium with the Rayleigh-Jeans spectrum. In the discussion, we e mphasize the con-\ntrasting points of view between nonrelativistic classical mechanics a nd relativistic classical\nelectrodynamics in connection with equilibrium for harmonic oscillators .\nOur earlier attempt[3] to correct the misinformation regarding the rmal radiation equilib-\nrium in classical physics involved an extended historical survey. It w as pointed out that\nthe introduction of classical electromagnetic zero-point radiation led to modifications of the\nhistorical arguments. The modified arguments provide natural cla ssical explanations for the\nPlanck spectrum within relativistic classical physics. The Planck spec trum with zero-point\nradiation corresponds to a radiation energy per normal mode\nUrad(ω,T) =1\n2/planckover2pi1ωcoth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\n=/planckover2pi1ω\nexp[/planckover2pi1ω/(kBT)]−1+1\n2/planckover2pi1ω. (1)\nThe spectrum involves a transition from the relativistic high-freque ncy region,\n/planckover2pi1ω/(kBT)>>1, associated with adiabatically-invariant and Lorentz-invariant ze ro-point\nradiation where Urad(ω,T)→(1/2)/planckover2pi1ω, over to the low-frequency Rayleigh-Jeans re-\ngion,/planckover2pi1ω/(kBT)<<1, associated with nonrelativistic energy-sharing behavior where\nUrad(ω,T)→kBT.\nB. Three Analyses for Radiation Equilibrium\nThe present discussion is quite different from the previous historica l survey. Here we\npresent three different analyses of radiation equilibrium connected to harmonic oscillator\nsystems; these analyses include aspects of point-particle collisions , adiabatic invariance,\nscattering, and scaling which do not appear in earlier work.\n21. Point-Particle Collisions and Adiabatic Invariance\nAfter some basic preliminaries, we start our analysis by contrasting the equilibrium for\ntwo different situations; one is a box containing a finite number of mec hanical oscillators\nconnected through point collisions with a free particle, and the othe r involves a charged\nmechanical oscillator coupled to a divergent number of electromagn etic radiation modes.\nWhentheequipartitionideasofnonrelativisticmechanicsareimposed uponradiationtreated\nin the dipole approximation, the Rayleigh-Jeans spectrum appears. Our analysis also\nsuggests the possibility of classical zero-point energy for radiatio n associated with oscillator\nadiabatic invariance , something which is not possible for classical mec hanical systems which\nallow point-particle collisions.\n2. Equilibrium Under Scattering\nIn the second equilibrium analysis, we emphasize that a charged harm onic oscillator con-\nnected to radiation through the dipole approximation can come to eq uilibrium with any\nradiation spectrum. In order to obtain a specific radiation spectru m, some additional as-\nsumptionsareneeded. TheRayleigh-Jeansspectrumandzero-po intspectrumcorrespondto\nthe extremes allowed by Wien’s displacement theorem. If we turn to in variance under scat-\nteringasourcriterionforapreferredspectrum, thenagainther earetwo naturalpossibilities.\nIn one possibility, the mechanical behavior is modified to an anharmonic nonrelativistic os-\ncillator while maintaining the dipoleapproximation to radiation. The other possibility\ninvolves maintaining the harmonic mechanical behavior but going to electromagnetic in-\nteractions beyond the dipole approximation, giving an approximately relativistic oscillator\nsystem. We point out that the first nonrelativistic procedure leads to the Rayleigh-Jeans\nspectrum, while the second relativistic analysis leads to the classical zero-point radiation\nspectrum. For a harmonic oscillator, the Rayleigh-Jeans spectrum arising from the energy\nequipartition ideas of nonrelativistic classical mechanics is not tolera ted beyond the dipole\ncoupling to radiation.\n33. Scaling in Nonrelativistic Mechanics and in Relativisti c Electrodynamics\nFinally, in the third equilibrium analysis, we point out the sharp differenc es in scaling\nbehavior between nonrelativistic mechanics and relativistic electrod ynamics, and connect\nthese differences to the requirements for thermal equilibrium. The enormous flexibility of\nnonrelativistic classical mechanics contrasts with the strictly-con trolled radiation connec-\ntions of relativistic classical electrodynamics. And this contrast is c rucial in questions of\nthermal radiation equilibrium. We emphasize that, within classical phy sics, the full Planck\nspectrum requires both velocity-dependent damping and accelera tion-dependent radiation\ndamping.\nC. Missing Aspects in the Physics of 1900: Zero-Point Radiation and Relativity\nThe foundation for the erroneous claims regarding classical therm al radiation which per-\nvade the textbooks and the internet lies in the fact that the physic ists of the early 20th\ncenturywere(just as the physicists of today are) unaware of two crucial aspects of classical\nphysics existing within classical electrodynamics: 1)theexistence o fclassical electromagnetic\nzero point radiation, and 2)the importance of special relativity. In this article, we illustrate\nthe importance of these two aspects of classical physics by compa ring the ideas of thermal\nequilibrium which arise in classical mechanics and in classical electrodyn amics. The models\nfor our discussion are harmonic oscillators.\nII. BASIC PRELIMINARIES\nA. Some Contrasts Between (Nonrelativistic) Classical Mechanics and ( Relativ is-\ntic) Classical Electrodynamics.\nIn order to prepare the reader for our analysis, we first mention s ome of the important\ncontrasts between classical mechanics andclassical electrodyna mics. Classical mechanics al-\nlows point collisions between massive particles which provide velocity-d ependent damping,\nwhereas classical electrodynamics involves only long-range forces associated with electro-\nmagnetic fields. Classical mechanics allows particles to exist without a ny connection to\na spacetime-dependent field, whereas in classical electrodynamics , every charged particle is\n4connected to electromagnetic fields which introduce acceleration- dependent radiation forces.\nThereisnospecial roleforoscillation frequency when considering thermal equilibrium within\nnonrelativistic classical mechanics, whereas classical electrodyna mics determines thermal\nequilibrium through the frequency-dependent connection of char ges to radiation fields. Fi-\nnally, the relativistic theory of classical mechanical particles is an insignificant part of me -\nchanics; the relativistic theory is restricted to point collisions between particles and allows\nno potential functions.[4] Classical mechanics which allows the intera ction of particles be-\nyond point collisions satisfies Galilean symmetry, whereas classical ele ctrodynamics satisfies\nLorentz symmetry.\nB. Harmonic Oscillator in Action-Angle Variables\nA one-dimensional harmonic oscillator of natural (angular) freque ncyω0and mass mcan\nbe described by the Hamiltonian\nH(x,p) =p2/(2m)+(1/2)mω2\n0x2=Uosc. (2)\nWithin nonrelativistic mechanics, this Hamiltonian will give the equations of motion for the\nposition xand momentum pwhile the initial conditions will determine the actual motion,\nincluding theenergy Uoscandphase of the oscillation. When discussing adiabaticchanges in\nthe oscillator’s natural frequency ω0, it is convenient to reexpress the Hamiltonian in terms\nof action-angle variables wandJusing the canonical transformation with the connection\nx=/radicalbigg\n2J\nmω0sinwandp=/radicalbig\n2mω0Jcosw. (3)\nIn terms of the new variables wandJ, the Hamiltonian takes the form\nH=p2\n2m+1\n2mω2\n0x2=Jω0=Uosc. (4)\nIn terms of action-angle variables, the oscillator Hamiltonian no longe r refers to the particle\nmassmnor the amplitude of oscillation x. The initial conditions again give the oscillator\nenergyUosc=Jω0and the initial phase φinw=ω0t+φ.\n5C. Thermodynamics of the Harmonic Oscillator and Wien’s Theorem\nA harmonic oscillator (whether a mechanical oscillator or a radiation m ode) is such a\nsimple system that its thermodynamic behavior depends upononly tw o variables, its natural\nfrequency ω0and its temperature T. By considering the adiabatic change of the oscillator\nfrequency ω0, it is easy to derive Wien’s displacement theorem indicating that in ther mal\nequilibrium, the energy of the oscillator must take the form[5]\nUosc(ω0,T) =ω0f(ω0/T), (5)\nwheref(ω0/T)isanunknown functionoftheratio ω0/T. Theextremes ofenergy equiparti-\ntionandzero-point energy correspond to thelimiting situations whe re the energy Uosc(ω0,T)\nbecomes independent of one of the two variables. Thus if f(ω0/T)→const×(T/ω0), then\nwe have energy equipartition Uosc(ω0,T)→const×T; on the other hand, if f(ω0/T)→\nconst′, thenonefindstemperature-independent zero-pointenergy Uosci(ω0,T)→const′×ω0.\nIf we consider only the first two laws of thermodynamics, then the f unctionf(ω0/T) giving\nthe transition between these extremes is unknown. Only if we add fu rther criteria such\nas ”smoothness”[5] or inclusion of the third law,[6] do we obtain the P lanck spectrum with\nzero-point energy given inEq. (1). However, boththese additiona l criteria arenot currently\nin fashion in physics, and below we will consider other possibilities.\nD. Thermal Radiation Modes\nThe solutions of Maxwell’s equations involve electromagnetic fields aris ing both from\ncharge and current sources, and also from boundary conditions. The boundary-condition\nelectromagnetic fields satisfy the source-free Maxwell equations . Thermal radiation can be\ntreated as boundary-condition radiation, and can be expanded in t erms of plane waves with\nrandom phases satisfying periodic boundary conditions in a large cub ic box with sides of\nlengtha,\nE(r,t) =/summationdisplay\nk,λ/hatwideǫ(k,λ)/parenleftbigg8πUrad(ω)\na3/parenrightbigg1/21\n2{exp[ik·r−iωt+iθ(k,λ)]+cc},(6)\nB(r,t) =/summationdisplay\nk,λ/hatwidek×/hatwideǫ(k,λ)/parenleftbigg8πUrad(ω)\na3/parenrightbigg1/21\n2{exp[ik·r−iωt+iθ(k,λ)]+cc},(7)\n6where the sum over the wave vectors k=/hatwidex2πl/a+/hatwidey2πm/a+/hatwidez2πn/ainvolves integers\nl,m,n= 0,±1,±2,...running over all positive and negative values, there are two polar-\nizations/hatwideǫ(k,λ), λ= 1,2,and the random phases θ(k,λ) are distributed uniformly over the\ninterval (0 ,2π],independently for each wave vector kand polarization λ. The notation “ cc”\nrefers to the complex conjugate. We have assumed that the radia tion spectrum is isotropic,\nand that the energy per normal mode at radiation frequency ω=ckis given by Urad(ω).\nThere are a divergent number of radiation normal modes with no limit o n the frequency ω.\nE. Equilibrium Between a Charged Oscillator and Random Radiation in the\nDipole Approximation\nWhen treated in the dipole approximation, the interaction of a charg ed harmonic oscil-\nlator with the random radiation in Eqs. (6) and (7) can be written as N ewton’s second\nlaw\nm¨x=−mω2\n0x+mτ...x+eEx(0,t), (8)\ninvolving theharmonicrestoringforce −mω2\n0x, thedipoleradiationdampingforce mτ...xwith\nτ= 2e2/(3mc3), and the radiation driving force treated in dipole approximation eEx(0,t).\nThis equation of motion has been solved many times, beginning with Plan ck’s work at\nthe end of the 19th century.[7][8][9][10] It is found that in this dipole app roximation, the\noscillator acts essentially like a radiation mode at the oscillation freque ncyω0: the average\nenergy of the oscillator matches the average energy of the radiat ion modes at frequency ω0,\nUosc(ω0) =Urad(ω0), and the phase space distribution Posc(x,p) of the oscillator takes the\nform\nPosc(x,p) =const×exp[−H(x,p)/Urad(ω0)] (9)\nwhereH(x,p) is the Hamiltonian of the oscillator given in Eqs. (2) or (4).\nIn the dipole approximation, the harmonic oscillator is in equilibrium with anyspectrum\nUrad(ω) of isotropic random radiation. Thus, other than the requirement of isotropy in\ndirection, the spectrum Urad(ω) is arbitrary. The oscillator will enforce isotropic behav-\nior for the radiation spectrum, but the harmonic oscillator will not ch ange the frequency\ndistribution of radiation among the various frequencies.[9]\n7III. EXPLORING EQUILIBRIUM FOR OSCILLATORS IN MECHANICS AND\nIN ELECTROMAGNETIC RADIATION\nWe now turn from the preliminary aspects of our analysis over to spe cific treatments of\nradiation equilibrium for harmonic oscillators.\nA. Equilibrium for Systems of Oscillators\nSuppose that we have an uncharged free particle of mass Mand a special harmonic oscil-\nlator of (angular) frequency ω0at our disposal. We wish to contrast the equilibrium behav-\nior for two different systems. The first system involves a collection o fNnon-interacting,\nclassical-mechanical, one-dimensional oscillators of various freque nciesωin an elastic-walled\nbox; the second involves the electromagnetic radiation in a reflectin g-walled box. Both\nboxes can be described as containing harmonic oscillators inside, the one in terms of me-\nchanical oscillators and the other in terms of electromagnetic radia tion modes. We assume\nthat the oscillators in both boxes are in “thermal equilibrium,” and we w ish to explore this\nthermal equilibrium.\nB. Equilibrium for Mechanical Oscillators\n1. Equilibrium Through Point-Particle Collisions\nIfweintroduceourspecialoscillatorintotheboxof Nmechanicaloscillators, theoscillator\nwillnotcometoequilibriumintheboxunlessthereissomeinteractionbe tweenthisoscillator\nand the original oscillators in the box. However, if we now introduce o ur free particle M\ninto the box of mechanical oscillators, then the particle will collide with all the masses of the\nmechanical oscillators and will bring to equilibrium the entire collection ( including both our\nspecial oscillator, the Noscillators in the box, and also the mass M) by sharing the total\nsystem energy Utotalamong the finite number N+1 of mechanical oscillators and the mass\nM. Each oscillator of mechanical frequency ωwill come to equilibrium at the Boltzmann\ndistribution Posc(J,ω) =const×exp[−Jω/Uav] where here Jis the action variable of the\noscillator with Hamiltonian H=Jω, andUav=Utotal/(N+1+3/2) is the average energy,\n(from energy equipartition) with the average energy of each one- dimensional oscillator 2 /3\n8that of the mass Mwhich has three-dimensional motion. We notice that the natural\nfrequency ωof an oscillator is irrelevant for the average oscillator energy at equ ilibrium;\nour special oscillator at frequency ω0has the same average energy as any oscillator at any\nfrequency ω.However, we notice that the phase space distribution Posc(J,ω)for anoscillator\ndoesdependuponitsfrequency ωandalsoupontheaverageenergy Uavfortheentiresystem.\n2. Change of Equilibrium Due to Adiabatic Change of Oscillator Frequency\nIf we now isolate our special oscillator and carry out an adiabaticcha nge in the frequency,\nthe quantity of energy divided by frequency, Uav/ω0=J, is an adiabatic invariant and does\nnot change. When the frequency is increased from ω0toω′\n0, the energy of our special\noscillator is increased from Uavover toU′= (ω′\n0/ω0)Uav. This energy U′is now above the\naverage energy Uavof the other oscillators in the box. On reconnecting our special osc illator\nto the collection of the other mechanical oscillators and the mass M, the average energy of\neach oscillator will increase, and the phase space distribution of eac h oscillator will change\nto give a larger average value of its action variable J. Thus an adiabatic change in the\nfrequency of our special oscillator will indeed disturb the equilibrium o f the entire system.\nC. Equilibrium for Random Radiation\n1. Charged-Oscillator Equilibrium Depends on Oscillator Fr equency\nThisthermal-equilibrium situationfornonrelativistic mechanical oscilla tors(where point-\nparticlecollisionsareallowed, andthereisnoparticularroleregarding energyequilibrium for\ntheoscillatorfrequency) istobecontrastedwiththeelectromagn eticsituationwhichinvolves\na divergent number of massless time-harmonic radiation modes char acterized by frequency.\nIn analogy with the mechanical case, the introduction of a special h armonic oscillator of\nfrequency ω0intoaconducting-walled enclosure givesusnoinformationunless wes pecify the\ninteraction of the oscillator with the radiation modes of the enclosur e. A free but uncharged\nparticleMhas no interaction with the electromagnetic oscillators. However, w e canconnect\nour special mechanical oscillator to radiation by taking the oscillating particle as charged.\nAspointedoutbyPlanckattheendofthe19thcenturyandnoteda boveinourpreliminaries,\ntreated in the dipole approximation, this electromagnetic harmonic o scillator will interact\n9withtheradiationmodesatthesame frequency asthenaturalfre quency oftheoscillatorand\nwill come to equilibrium with these modes, but not with modes of differing frequency. Thus\nthe frequency-dependent connection between an oscillator and e lectromagnetic radiation is\na crucial aspect of electromagnetism.\n2. Possibility of Adiabatic Invariance in Zero-Point Radiat ion\nWe now consider an adiabatic change in the frequency ω0of our oscillator over to a new\nfrequency ω′\n0. During this adiabatic change, the ratio of energy to frequency, U0/ω0=J,\nof our special oscillator is unchanged. We can now have our oscillator (treated in the\ndipole approximation) interact with radiation at the new frequency ω′\n0. If the radiation\nmodes at ω′\n0have greater energy than our oscillator, then, on interaction, th ere will be\nenergy transferred from the radiation modes at ω′\n0over to our special oscillator. On the\nother hand, if the radiation modes at ω′\n0have less energy, then the transfer is reversed.\nHowever, we also have the possibility that our special oscillator has t he same average energy\nas the radiation modes at ω′\n0and so is already in equilibrium with these modes. This\nsituation of undisturbed equilibrium despite an adiabatic change of an oscillator’s frequency\nand energy is an entirely new possibility which does not arise for mecha nical oscillators\nwhere energy equipartition is involved. Here in the electromagnetic s ituation, we have\nthe possibility of making adiabatic changes in the frequency and ener gy of our oscillator\nand yet not transferring any average energy between the electr omagnetic radiation modes.\nIf this situation of thermal equilibrium holds for all radiation modes, t hen the classical\nelectromagnetic spectrum is Lorentz invariant and is termed “class ical electromagnetic zero-\npoint radiation.”[11]\n3. Introduction of Planck’s Constant /planckover2pi1\nThe scale of this zero-point radiation involves the ratio U/ωwhich is the same for all\nradiation frequencies. The scale which fits with nature correspond s to the introduction of\nPlanck’s constant /planckover2pi1as the scale invariant so that the average energy Urad(ω) of a radia-\ntion mode of frequency ωsatisfiesUrad(ω) = (1/2)/planckover2pi1ω. There is no question that, when\ninterpreted in terms of classical physics, nature contains classica l electromagnetic zero-point\n10radiation. Thus the introduction of Planck’s constant /planckover2pi1as the scale of classical zero-point\nradiation gives fully classical explanations for Casimir forces, van de r Waals forces, oscillator\nspecific heats, diamagnetism, and the Planck spectrum of thermal radiation.[12] Zero-point\nradiation involves the same phase space distribution for every radia tion mode no matter\nwhat its frequency,\nPradzp(J) =const×exp/bracketleftbigg\n−Jω\n(1/2)/planckover2pi1ω/bracketrightbigg\n=const×exp/bracketleftbigg\n−J\n/planckover2pi1/2/bracketrightbigg\n. (10)\n4. Planck’s Constant /planckover2pi1Can Appear in Classical or Quantum Theories\nThe appearance of Planck’s constant /planckover2pi1within classical physics provokes the indignation\nof some physicists. Some uninformed physicists continue to insist th at Planck’s constant /planckover2pi1\nis a “quantum constant,” and that any theory in which /planckover2pi1appears has a “quantum” element.\nThat this claim is untrue has been pointed out in work published in the Am erican Journal of\nPhysics.[13] Actually, Planck’s constant /planckover2pi1is a physical constant, like Cavendish’s constant\nG, and can appear in any theory which involves an appropriate scale, j ust as the constant G\ncan appear in both Newtonian gravity and in general relativity since Gsets the scale of the\ngravitational interaction. In modern theory, Planck’s constant a ppears both as the scale\nof action in quantum theory and as the scale of classical zero-point radiation in classical\nelectrodynamics.\nBecause Planck’s constant /planckover2pi1does not appear in Maxwell’s equations but only in the\nsource-free boundary condition[13] on the equations, there are two natural versions of clas-\nsical electrodynamics, one including /planckover2pi1and one not. The version which includes Planck’s\nconstant provides a natural classical explanation of the Planck sp ectrum[3] and of some\nother phenomena[12] with a scale set by /planckover2pi1.The version which appears in current textbooks\nof classical electromagnetism assumes that the source-free elec tromagnetic field vanishes and\nso offers no explanation of phenomena at the scale of /planckover2pi1.\nIt remains true that Planck’s constant /planckover2pi1has no place in classical mechanics because\nclassical mechanics (with point-particle collisions which share all ener gy) allows no zero-\npoint energy. Of course, classical mechanics also does not allow mas sless waves which carry\nenergy andmomentum. However, classical electromagnetism is quit e different fromclassical\n11mechanics. Classical electromagnetism includes massless waves, an d, indeed, has a natural\nplace for Planck’s constant as the scale of classical electromagnet ic zero-point radiation.\nIV. RADIATION EQUILIBRIUM FOR AN OSCILLATING CHARGED SYSTEM\nA. A Point Harmonic Oscillator Does Not Determine Radiation Equilibrium\nIn the discussion above, we have seen the reappearance of the sa me two oscillator ener-\ngies (equipartition and zero-point energy) which appeared from th e limits of Wien’s law (5),\narising from the thermodynamics of the harmonic oscillator. Again it m ust be emphasized\nthat since a charged harmonic oscillator treated in the dipole approx imation comes to equi-\nlibrium with anyradiation spectrum, the determination of the actual spectrum of thermal\nradiation equilibrium must be based upon other factors. Here we list s everal criteria.\nB. Specific Criteria for Radiation Equilibrium\n1. Adoption of the Nonrelativistic Oscillator Equilibrium\nIn the early years of the 20th century, just as in the modern phys ics texts of today, it\nwas assumed[1][2] that nonrelativistic statistical mechanics correc tly determines the phase\nspace of the harmonic oscillator according to the Boltzmann distribu tion\nPosc(x,p) =const×exp[−H(x,p)/(kBT)] (11)\nThen comparing equations (9) and (11), it was concluded that the s pectrum of random\nradiation in equilibrium with the oscillator must be Urad(ω) =kBT. In other words,\nnonrelativistic classical mechanics, which uses Boltzmann statistica l mechanics, must lead\nto the Rayleigh-Jeans spectrum for relativistic classical radiation.\nIn 1910, Einstein and Hopf[14][15] tried to avoid full Boltzmann statis tical mechanics\nand to use instead only the well-established average thermal kineticenergy of a massive\nnonrelativistic particle in one dimension as (1 /2)Mv2= (1/2)kBT. However, they again\narrived at the Rayleigh-Jeans spectrum for thermal radiation.\n122. Adoption of the Spectrum Invariant Under Adiabatic Change\nAn entirely different criterion involves adiabatic invariance. We saw ab ove that, under\nan adiabatic change of frequency, a charged harmonic oscillator of frequency ω0and energy\nU(ω0) (when treated in the dipole approximation) will not transfer energ y among radiation\nmodes if the radiation spectrum is Lorentz invariant. Thus the assu mption of adiabatic\ninvariance for the oscillator picks out a special radiation spectrum, that of classical electro-\nmagnetic zero-point radiation. This is also the spectrum allowed by th e limit of Wien’s law\n(5) which makes the equilibrium spectrum independent of the temper atureT.\n3. Adoption of the Spectrum Invariant Under Scattering\nA third criterion involves invariance under scattering. The charged harmonic oscillator\ntreated in the dipole approximation scatters radiation so as to make the radiation pattern\nisotropic; however, the scattering by the oscillator does not chan ge the frequency spectrum\nof the random radiation. In order to obtain a preferred spectrum based upon invariance\nunderscattering , we must go beyond use of the harmonic oscillator treated in the dipo le\napproximation.\na. Two Oscillator Extensions Leading to Equilibrium Radiat ion Spectra There are two\nnatural extensions of the harmonic oscillator model. The first poss ible extension retains the\nelectromagnetic aspect (continuing the use of the dipole approxima tion), but changes the\nmechanics (going beyond the harmonic oscillator over to an anharmo nic oscillator). The\nsecond possible extension retains the mechanics (continuing the us e of a purely harmonic\nmechanical oscillator), but changes the electromagnetic connect ion (going beyond the dipole\nradiation approximation over to full electromagnetic interaction at all harmonics). The\ntwo possible extensions give different preferred equilibrium spectra . The nonrelativistic\nmechanical extension while retaining the dipole radiation approximatio n gives the Rayleigh-\nJeans spectrum. The relativistic electromagnetic extension while re taining the harmonic\nmechanical behavior gives classical zero-point radiation. Both ext ensions will be discussed\nbelow.\n13C. Radiation Scattering by an Anharmonic Nonrelativistic Mechanical System in\nthe Dipole Approximation\nIn order to avoid the use of the ideas of classical statistical mecha nics, various physicists\nhave considered the transition beyond harmonic oscillators to nonr elativistic anharmonic\noscillators which will scatter radiation and change the spectrum of t he random radiation.\nIn 1924, Van Vleck[16] considered (and partially published) a genera l analysis of nonlin-\near nonrelativistic mechanical systems in scattering equilibrium wher e the radiation was\ntreated in the dipole approximation. Work by other authors has con sidered scattering by\na nonrelativistic nonlinear oscillator,[17] and scattering by nonrelativ istic potentials where\nthe charged particle momentum took relativistic form.[18] All of thes e classical scattering\ncalculations arrived at the Rayleigh-Jeans spectrum. None of thes e calculations treats a\nrelativistic scattering system.\nA nonrelativistic anharmonic oscillator with a nonlinear term in energy αx3and Hamil-\ntonian\nH(x,p) =p2/(2m)+(1/2)mω2\n0x2+αx3(12)\nprovides a simple example of a scattering system which will enforce th e Rayleigh-Jeans\nspectrum for radiation equilibrium. The equation of motion takes the form\nm¨x=−mω2\n0x+3αx2+mτ...x+eEx(0,t), (13)\nand continues the use of the dipole approximation in the radiation dam pingmτ...xand in\nthe electromagnetic force eEx(0,t). The mechanical motion of the oscillator now involves\na mechanical oscillation at frequency ω1which is different from ω0and which depends upon\nthe constant αgiving the scale of the nonlinear term.[19] The mechanical motion now\nincludes the harmonics[19] of the frequency ω1\nx(t) =D0+D1cos[ω1t+φ1]+D2cos[2ω1+φ2]+... (14)\nSince the mechanical motion has oscillations at all the harmonics nω1of the fundamental\nmechanical frequency ω1, the nonlinear oscillator will interact with radiation in the dipole\napproximation at all the harmonics of the oscillation frequency ω1. This interaction with\nradiation will transfer energy among the radiation modes, and so will lead to a unique\nequilibrium spectrum. We note that the radiation equilibrium is determin ed by the non-\nrelativistic mechanical system and not by any properties of the radiation with which the\n14system has minimal (dipole) interaction. In all treatments with nonr elativistic nonlinear\nmechanical scatterers connected to radiation through the dipole approximation, one finds\nthe Rayleigh-Jeans spectrum as the unique spectrum of radiation e quilibrium. When van\nVleck[16] partially published his analysis in 1924, the work was regarde d as confirming that\nthe Rayleigh-Jeans spectrum was the appropriate equilibrium radiat ion spectrum expected\nwithin classical physics.\nD. Radiation Scattering by a Charged Harmonic Oscillator with Full Electro-\nmagnetic Interactions\n1. One-Dimensional Electromagnetic Harmonic Oscillator\nIn order to emphasize electromagnetic interactions in contrast to arbitrary mechanical\npotentials, we mention here the possibility of a one-dimensional oscilla tor where the oscillat-\ning particle of mass mand charge emoves under purely electromagnetic forces. We imagine\na charge ewhich is constrained to move along the straight line between two char gesq(of\nthe same sign as e) separated by a distance l. The charge ewill then oscillate in one di-\nmension along the line between the charges q,but is in unstable equilibrium against motion\nto the side if the constraint were removed. For small oscillations, th e mechanical motion of\neis harmonic with (angular) frequency ω0= [32eq/(ml3)]1/2. For any finite amplitude of\noscillation, there will be non-harmonic contributions from the electr omagnetic forces on e\ndue to the charges q. However, as the amplitude of oscillation becomes ever smaller, the\noscillation becomes ever-more-nearly harmonic. Furthermore, on transformation to a new\ninertial frame S′, the electromagnetic system would have relativistic behavior in the lim it of\nzero-oscillation amplitude at a new Lorentz-transformed oscillation frequency ω′\n0.\n2. Relativistic Harmonic Oscillator\nIf we adopt this one-dimensional electromagnetic model for the os cillating charge ewhere\nthe restoring force arises from the electrostatic field in the rest f rame of the charges q, then\nthe relativistic equation for the constrained motion for the charge ein electromagnetic fields\nisdp/dt=eE(r,t).In the approximation that the velocity of the charge erelative to the\n15chargesqis small compared to c,one can approximate the relativistic change in momentum\nas mass times acceleration, accurate through first order in the ra tiov/c. Thus we have\ndp\ndt=d\ndt/bracketleftBigg\nmv\n(1−v2/c2)1/2/bracketrightBigg\n=d\ndt/bracketleftbigg\nmv/parenleftbigg\n1+v2\n2c2+.../parenrightbigg/bracketrightbigg\n≈d(mv)\ndt, (15)\nprovided that we can neglect terms of order ( v/c)2. Our electromagnetic oscillator can be\ntreated as a relativistic system provided that its oscillation velocity is small. The oscillator\nsystem is fully relativistic only in the limit v→0.\n3. Radiation Equilibrium Beyond the Dipole Approximation\nIn his textbook of classical electrodynamics, Jackson points out t hat[20] “Appreciable\nradiation in multiples of the fundamental [oscillatory frequency] can occur because of rela-\ntivistic effects ... even though thecomponents of velocity aretruly sinusoidal, or it can occur\nif the components of the velocity are not sinusoidal, even though pe riodic.” All of the previ-\nousscattering calculations have involved mechanical motionswhich are“not sinusoidal, even\nthough periodic,” as in our Eq. (14). Thus the harmonics appear in th e purely mechanical\nmotion of the charged particle. However, recent work by Huang an d Batelaan[21] regarding\nabsorption of radiation at harmonics due to relativistic effects sugg ested the possibility of\ndetermining radiation equilibrium not from nonrelativistic mechanical m otions but rather\nfrom purely relativistic effects in classical electromagnetism.\nInproblem14inChapter 14, Jackson[20]asks astudent tocalculat e theradiationemitted\nat the harmonics nω0due to a purely sinusoidal motion at ω0\nx=Dcos[ω0t+φ]. (16)\nThus the treatment of the multipole moments of the oscillating charg e can be extended\nbeyond the dipole term to include the quadrupole moment of the harm onic oscillator which\nwill emit radiation at frequency 2 ω0, and indeed can be extended to all higher multipoles.\nHowever, if the harmonic oscillator can emit quadrupole radiation at t he second harmonic\n2ω0of the oscillator’s natural frequency ω0, then it can also absorb energy at the sec-\nond harmonic. The force on the oscillator must be extended beyond the dipole inter-\nactioneEx(0,t) to include the next term in the expansion of the true force eEx(x,t) as\neEx(0,t) +ex/bracketleftBig\n∂Ex(/hatwideix′,t)/∂x′/bracketrightBig\nx′=0. Thus a purely sinusoidal mechanical motion of the\n16oscillator combined with the relativistic radiation analysis can lead to a specific equilibrium\nspectrum of random classical radiation. Here for the first time in th e classical physics\nscattering literature, radiation equilibrium is determined not by nonrelativistic mechanical\nconsiderations but by relativistic electromagnetic aspects.\n4. Zero-Point Radiation as the Oscillator-Scattering Equi librium Spectrum\nThe calculation for the equilibrium radiation spectrum of a electromag netic charged har-\nmonic oscillator of small amplitude has been carried out.[22] The equilibr ium spectrum is a\nLorentz-invariant radiation spectrum. The Rayleigh-Jeans spect rum is not Lorentz invari-\nant, and so is notthe spectrum of radiation equilibrium for a charged classical harmon ic\noscillator when treated beyond the dipole approximation for the rad iation interaction. Only\nLorentz-invariant zero-point radiation will serve as an equilibrium sp ectrum.\n5. Use of Nonrelativistic versus Relativistic Theory\nA harmonic oscillator is like a clock with frequency ω0. Thepointharmonic oscillator\nallows only a dipole connection to radiation, and is in equilibrium with anyspectrum of\nisotropic random radiation. In the dipole approximation, the radiatio n energy Urad(ω0)\nmerely determines the one number giving the oscillator energy Uosc(ω0). The point os-\ncillator clock itself can assume either Galilean or Lorentz transforma tion properties when\nviewed in a new inertial frame. However, if the oscillator motion has fin ite amplitude, then\nthe analysis of the oscillator must choose between the nonrelativist ic and the relativistic\ntransformations. If nonrelativistic mechanical motion beyond harmonic motion is involved,\nthen there is enormous flexibility in the choice of the mechanical inter action potential V(x);\nin equilibrium, the nonrelativistic nonlinear oscillator, with a dipole conne ction to radiation,\nenforces a frequency-independent constant energy on the rad iation modes, corresponding to\nthe frequency-independent energy kBTallowed by the Wien law (5). On the other hand, if\nrelativistic electromagnetic interactions are assigned to the harmonic oscillato r, then the ex-\ntension from the dipole moment to the quadrupole moment and beyon d allows no flexibility\nwhatsoever; the harmonic oscillator motion is completely determined , and the equilibrium\nradiationspectrumassumesaLorentz-invariantform, correspo ndingtothezero-pointenergy\n17limit (1/2)/planckover2pi1ω0allowed by the Wien law (5).\n6. The Relativistic Limit and Classical Zero-Point Radiation\nThe electromagnetic oscillator is fully relativistic only in the limit as its velo city goes to\nzero. However, it turns out that the radiation balance at each har monic does not depend\nupon the actual oscillation amplitude for small oscillations, provided t hat the amplitude is\nnon-vanishing.[22] Thus the radiation equilibrium continues to hold with out any change\nas the amplitude of oscillation goes toward zero, and the oscillator mo tion becomes ever\ncloser to the relativistic limit. We expect that the radiation equilibrium o btained from\napproximately relativistic oscillator motion will hold even in the relativist ic limit.\nIf one uses the action-angle variables wandJ, then the Hamiltonian in Eqs. (2) and (4)\nmakes no reference to the mass of the oscillator or to the amplitude of oscillation, and the\nphase space distribution for the oscillator given in Eq. (9) becomes\nPosc(w,J) =const×exp/bracketleftbigg\n−Jω0\nUrad(ω0)/bracketrightbigg\n. (17)\nThis phase space distribution gives equilibrium for the oscillator at all t he harmonics nω0\nin a Lorentz-invariant radiation spectrum Urad(ω). If we take the Lorentz-invariant radia-\ntion spectrum (required for full radiation equilibrium by the oscillator ) as that of classical\nelectromagnetic zero-point radiation Uzprzp(ω) = (1/2)/planckover2pi1ω, then the oscillator phase space\n(17) takes the form\nPosczp(w,J) =const×exp/bracketleftbigg\n−J\n/planckover2pi1/2/bracketrightbigg\n, (18)\nwhere the oscillator frequency ω0has cancelled out leaving a phase space which independent\nof frequency. This phase space distribution for the oscillator is the same as the phase space\nin Eq. (10) for each mode of electromagnetic radiationin the zero-p oint radiation field. The\nequilibrium phase space distribution in Eq. (18) is invariant under an ad iabatic change of\nfrequency ω0.\nV. SCALING IN NONRELATIVISTIC MECHANICS AND IN RELATIVISTIC\nELECTRODYNAMICS\nIn order to broaden our understanding of the contrasts arising in treatments of classical\nradiation equilibrium, we now turn to the matter of scaling. Scaling can be regarded as a\n18change in the fundamental unit used to evaluate some physical qua ntity, or as a multiplica-\ntive change in the physical quantity itself. The contrasting scaling a spects of nonrelativistic\nclassical mechanics and of relativistic classical electrodynamics are reflected in their deter-\nminations of thermal equilibrium.\nA. Scaling Aspects of Nonrelativistic Mechanics\n1. Separate Scalings in Length, Time, and Energy\nBecause nonrelativistic mechanics involves no fundamental consta nts, it allows separate\nscalings in length as σl, in time as σt, and in energy as σU. For example, any nonrelativistic\nclassical mechanical system can be reimagined as a system where all the lengths are twice\nas large ( σl= 2), the times are three times as long ( σt= 3), and the energies are four times\nas great ( σU= 4). This flexibility arises since the only forms of energy are kinetic en ergy\n(which can be scaled through the mass m), and potential energy V(x,y,z) (which can be\nrescaled through the constants connecting distance to energy) . For a nonlinear mechanical\noscillator of mass m, harmonic frequency ω0, and nonlinear parameter α, as in Eq. (12), the\nquantities m,ω0, andαare all freely adjustable, corresponding to allowing separate scalin gs\nσl,σt,σU.\n2. Thermal Equilibrium Reflects the Three Separate Scalings\nEquilibrium involving the interaction of classical mechanical systems m ust reflect the\nthree separate scalings allowed for nonrelativistic mechanical syst ems. In an equilibrium\nsituation for nonrelativistic physics, the energy kBTmust scale separately from the length\nand time parameters of the mechanical systems. If we rescale the energy (by a factor σU)of\na mechanical system which is in equilibrium, then the equili brium of the system should not\nbe disturbed by the energy rescaling. Furthermore, if we replace a member of a mechanical\nsystem by a new mechanical member involving new lengths and times bu t with the same\nenergy, then the mechanical equilibrium will be unchanged because t he total system energy\nremains unchanged. If these scaling ideas are applied to the Wien-law expression in Eq.\n(5), they pick out only the equipartition limit involving the energy-dep endent temperature\nTbut having no dependence upon the frequency-dependent ω. Indeed the Boltzmann\n19distribution reflects this idea of a separate energy scaling. The Bolt zmann probability on\nphase space involves the mechanical energies of the constituent s ystems with no other aspect\ninvolved. Interestingly enough, the Coulomb potential (which is the only potential which\ncan be extended to a fully relativistic classical electromagnetic theory) does not fit into the\nnonrelativistic Boltzmann analysis.\nTheRayleigh-Jeansspectrumreflectsthisnonrelativisticscaling pa tternwheretheenergy\nUrad(ω,T) =kBTof each radiation mode is the same, and is entirely independent of the\nfrequency ωof the radiation mode. It is interesting that in 1924, van Vleck[23] re marked\nwith surprise regarding his nonrelativistic calculations that “...in a field o f radiation whose\nspecific energy [ ρrad(ω,T) = [ω2/(π2c3)]Urad(ω,T)] does not vary with the frequency, we\nhave the rather surprising result that the mean absorption is indep endent of the form of\nthe force function V(x,y,z) which holds the electron in the atom.” Apparently, the idea\nofkineticenergy equipartition, completely independent of the force functio nV(x,y,z),\nwas a familiar and accepted idea of nonrelativistic mechanics, wherea s the directly-related\nresult associated with rates of energy absorption and loss from th e radiation field treated in\nthe dipole approximation was regarded as “surprising.” Van Vleck’s ex pressions for energy\nemission and absorption in the Rayleigh-Jeans spectrum can be show n to give kineticenergy\nequipartition.[24]\n3. No-Interaction Theorem in Relativistic Mechanics\nMany physicists are so accustomed to using nonrelativistic classical mechanics with its\nfreedom to choose interaction potentials V(x,y,z) at will, that they are surprised that rel-\nativity places very strong restrictions on systems. Indeed, most physicists have probably\nnever heard of the “no-interaction theorem” of Currie, Jordan, and Sudarshan[4] which\n“says that only in the absence of direct particle interaction can Lor entz invariant sys-\ntems be described in terms of the usual position coordinates and co rresponding canonical\nmomenta.”[26] Thestandardgraduate-level mechanics text then simply dismisses relativistic\nideas, continuing, “The scope of the relativistic Hamiltonian framewo rk is therefore quite\nlimited and so for the most part we shall confine ourselves to nonrela tivistic mechanics.”\n20B. Scaling Aspects of Relativistic Electrodynamics\n1. Single σltU−1-Scaling of Relativistic Electrodynamics\nIn contrast to the three separate scalings in length, time, and ene rgy appearing in non-\nrelativistic mechanics, electrodynamics allows only one single scaling co nnecting together\nlength, time, and energy. This situation arises because classical ele ctrodynamics involves\nseveral fundamental constants. Length and time are coupled th rough the speed of light c,\nwhile energy and length are coupled through the electronic charge e,or through Stefan’s\nconstant aS=U/(VT4) related to the total thermal-part Uof the radiation energy in a\nbox of volume V. Therefore (relativistic) classical electrodynamics allows only one s caling\nσltU−1which preserves the values of these fundamental constants.[25] We note that Planck’s\nconstant /planckover2pi1has the same dimensions as e2/cand so is also unchanged under σltU−1-scaling.\nSuchσltU−1-scaling should not disturb electromagnetic thermal equil ibrium.\nIn addition to preserving the fundamental constants c, e,andaS, theσltU−1-scaling also\npreserves the form of Maxwell’s equations. Thus invariance under σltU−1-scaling holds\nfor Gauss’s law ∇ ·E= 4πρprovided that the electric field Escales as charge divided by\nlength squared. Faraday’s law ∇×E=−(1/c)∂B/∂tsatisfies invariance under the scaling\nprovided that the magnetic field Balso scales as charge divided by length squared. Finally,\nthe scaling for the total energy Uin a volume V,U=uV, follows from the scaling for the\nenergy density u= [1/(8π)](E2+B2).\n2.σltU−1-Scaling Allows a Function of ω/Tfor Classical Thermal Radiation\nLorentz-invariant classical zero-point radiation is σltU−1-invariant, and has no preferred\nlength, time, or energy.[27] Thus under a σltU−1-scale transformation, zero-point radiation\nis mapped onto itself. For any radiation mode of frequency ω, and energy Uradzp(ω) =\n(1/2)/planckover2pi1ω, theσltU−1-scaling will carry the radiation mode into a new radiation mode where\nthe frequency is ω′=ω/σltU−1and the energy is U′=U′/σltU−1. But then we have\nU′\nradzp(ω′) =Uradzp(ω)/σltU−1= (1/2)/planckover2pi1ω/σltU−1= (1/2)/planckover2pi1ω′=Uradzp(ω′), so that the\nfunction connecting frequency and energy is completely unchange d. Note that under this\nσltU−1-scale transformation, the phase space distribution of each radia tion mode remains\nunchanged at Pradzp(Jrad) =const×exp[−Jrad/(/planckover2pi1/2)]. On the other hand, thermal\n21radiation at temperature T >0 is not invariant under σltU−1-scaling. Since temperature T\ntransforms as an energy, the information of Wien’s law in Eq. (5) und ergoes a σltU−1-scale\ntransformation from temperature Tto a new temperature T′=T/σltU−1, while making\nthe ratio ω/Tinvariant, ω/T=ω′/T′. Thus the σltU−1-scaling of relativistic classical\nelectrodynamics is consistent with Wien’s law.\n3.σltU−1-Scaling Allows Only Zero-Point Radiation for the Relativi stic Harmonic Oscillator\nBecause of the existence of the fundamental constants candewhich require the σltU−1-\nscaling of electromagnetism, a charged (relativistic) electrodynam ic system in thermal radi-\nation can have at most one freely-adjustable parameter for fixed temperature T. After the\none parameter of the charged system has been chosen, the cond itions of thermal equilibrium\nwill determine theotherphysical quantities. Forthe(relativistic) o ne-dimensional harmonic\noscillator with its one scaling parameter ω0, the oscillator energy Uosc(ω0) is determined by\nthe initial conditions in connection with the radiation spectrum at fre quencyω0.\nIn the calculation[22] for the full electromagnetic radiation equilibriu m of an oscillator\nof small amplitude, purely electromagnetic interactions are invoked to arrive at the same\nzero-point radiation spectrum which is associated with adiabatic tra nsformation of the os-\ncillator. However, the full Planck spectrum including zero-point rad iation does notappear.\nSome physicists have taken this limitation as a sign that there is somet hing wrong with the\nanalysis, and they feel justified in their contentment with the erro neous claim that classi-\ncal physics leads to the Rayleigh-Jeans spectrum. However, the r eason for the limitation\nin the calculation is that the small-amplitude oscillator becomes a relativ istic system only\nat zero oscillation velocity, and therefore does not show the variet y of behavior of a fully\nrelativistic electromagnetic system. Indeed, the restriction of th e (relativistic) oscillator to\nthe zero-oscillation-velocity limit necessarily excludes velocity-depe ndent damping which is\ncrucial for the low-frequency part of the Planck thermal spectr um.\n224.σltU−1-Scaling Allows a Function of mc2/(kBT)for the Relativistic Classical Hydrogen\nAtom\na. Scaling for the Classical Hydrogen Atom The limiting considerations for the (ap-\nproximately relativistic) harmonic oscillator do not appear for the re lativistic classical hy-\ndrogen atom. The classical hydrogen atom consists of a point char geeof massmin a\nCoulomb potential V(r) =−e2/rin the presence of random classical radiation. This\nsystem is fully relativistic when considered as part of classical electr odynamics. The one\nfreely-adjustable scaling parameter is the mass m.Using this mass and the fundamental\nconstants eandc, one can form the energy mc2, the length e2/(mc2), and time e2/(mc3).\nUnder a σltU−1-scale transformation, the mass mis carried into mass m′=m/σltU−1, but\nvelocities are unchanged. Since classical zero-point radiation is σltU−1-scale invariant, then\nthe classical hydrogen atom in zero-point radiation will have its equilib rium phase space\nunchanged by the transformation. Since velocities are unchanged under the transforma-\ntion, this fits with the absence of velocity-dependent damping in zer o-point radiation; only\nacceleration-dependent radiation damping is involved in equilibrium. It is familiar that for\ncircular orbits, the speed of a particle in a circular orbit is v=e2/JwhereJis the angular\nmomentum. Under a σltU−1-scale transformation, the quantities v, e,andJall remain\nunchanged.\nb. Scaling for Hydrogen in Zero-Point Radiation In zero-point radiation, the scale\nof the random radiation is given by /planckover2pi1which has the same dimensions as e2/c. Thus\nin the limit which removes all factors of c, we find that the classical hydrogen atom in\nzero-point radiation has a typical length [ e2/(mc2)][/planckover2pi1c/e2]2=/planckover2pi12/(me2), a typical time\n[e2/(mc3)][/planckover2pi1c/e2]3=/planckover2pi13/(me4), and a typical energy mc2[e2/(/planckover2pi1c)]2=me4//planckover2pi12. These quan-\ntities, which involve no factors of c, are familiar from the Bohr model of hydrogen. They\nalso appear in work involving both numerical simulations and the Fokke r-Planck equation\nfor the classical hydrogen atom in classical zero-point radiation.[28 ]\nc. Scaling for Hydrogen in Thermal Radiation with T >0 If the classical hydrogen\natom is in equilibrium with thermal radiation at temperature T >0, then there will be\nvelocity-dependent dampinganda σltU−1-scaletransformationwillindeedchangethesystem.\nThe relativistic particle in a Coulomb potential allows a phase space dist ribution dependent\nupon both the mass m, and also the fundamental constants eandc. Thus the phase space\n23for the Coulomb situation can involve the characteristic energy rat iomc2/(kBT) which is\nfreely adjustable in both mand inT.This freely-adjustable ratio allows the possibility of a\ntransition between a high-frequency region dominated by accelera tion-dependent radiation\ndamping and a low-frequency region dominated by velocity-depende nt damping.\nFor the Coulomb potential, when the orbital radius is small, the velocit y is high and\nthe frequency is high, corresponding to the zero-point radiation p art of the spectrum where\nacceleration-dependent radiation damping dominates and velocity- dependent damping is\nminor. On the other hand, when the orbital radius is large, the veloc ity and frequency\nare low, and acceleration-based radiation damping is also low. In this lo w-frequency re-\ngion, velocity-dependent damping due to thermal radiation for T >0 might well dom-\ninate. This situation can be illustrated[29] by considering relativistic circular orbits\ncharacterized by angular momentum Jwhere the velocity is v=e2/J,the energy is\nU(J) =mc2/radicalBig\n1−[e2/(Jc)]2, the radius is r= [e2/(mc2)](Jc/e2)2/radicalBig\n1−[e2/(Jc)]2, and\nthe frequency is ω=∂U/∂J= (mc3/e2)[e2/(Jc)]3//radicalBig\n1−[e2/(Jc)]2. The ratio of orbital\nfrequency ωto temperature Tcan be written as\n/planckover2pi1ω\nkBT=mc2\nkBT/parenleftbigg/planckover2pi1c\ne2/parenrightbigg/parenleftBigg\n1−/parenleftbigge2\nJc/parenrightbigg2/parenrightBigg−1/2/parenleftbigge2\nJc/parenrightbigg3\n. (19)\nWe see from Eq. (19), that for changing temperature Tand for fixed mass mand fixed\nangular momentum J(which corresponds to fixed frequency ω), the system changes from\nthe high-frequency region of the thermal spectrum /planckover2pi1ω/kBT >>1,to the low-frequency\nregion/planckover2pi1ω/kBT <<1 exactly in unison with the ratio mc2/(kBT).The transition between\nthe two regions of behavior is mediated by the ratio mc2/(kBT). An analogous ratio does\nnot exist for the (approximately relativistic) harmonic oscillator.\nVI. MISCELLANEOUS ASPECTS OF THERMAL EQUILIBRIUM\nA. Full Thermal Equilibrium Requires Both Velocity-Dependent Damping and\nAcceleration-Dependent Damping\nIndeed it is interesting to see that within classical physics, thermal radiation equilibrium\nfor non-zero temperature T >0 requires both acceleration-dependent radiation damping\nand velocity-dependent damping. The Einstein-Hopf analysis[14][15] of 1910 considered a\n24particle of large mass Mcontaining an electric dipole oscillator which experienced velocity-\ndependentdampingwhenmovingthroughrandomradiation,butthe rewerenonon-radiation\nforces and so the particle motion experienced no acceleration-dep endent damping. Assum-\ning that the average kinetic energy of the one-dimensional motion o f the large mass M\nwas (1/2)kBT, the Einstein-Hopf analysis led to the low-frequency, velocity-dep endent,\nRayleigh-Jeans part of the Planck law for thermal radiation equilibriu m. Indeed, the\nvelocity-dependent damping without acceleration-dependent rad iation damping is analo-\ngous to the situation which arises in nonrelativistic mechanics where thermal equilibrium\ndepends entirely upon velocity-dependent damping, and no acceler ation-dependent damp-\ning can appear. It was only later when the Einstein-Hopf analysis was extended[30] to\nalso include acceleration-dependent radiation damping of the partic le motion (arising from\nnon-radiation forces on the particle) that the classical Einstein-H opf analysis led to the full\nPlanckspectrumincluding zero-pointradiation. Furthermore, the inclusionofclassicalzero-\npoint radiation in the Planck spectrum suggests the possibility of sup erfluid-like behavior\nfor an Einstein-Hopf particle at low temperatures.[31]\nB. Information Contained within the Rayleigh-Jeans and Zero-Point Radiation\nSpectra\n1. Zero-Point Spectrum Has Least Information\nZero-point radiation involves the spectrum of random radiation with least possible infor-\nmation. The spectrum is Lorentz invariant and has no preferred ine rtial frame, and hence\nno velocity-dependent damping. The spectrum is also scale invariant , and has no preferred\nlength or time or energy. Even in curved spacetime, zero-point rad iation has correlation\nfunctions which involve only the geodesic separations between the s pacetime points where\nthe correlation is evaluated.\n2. Information in the Rayleigh-Jeans Spectrum\nThe Rayleigh-Jeans spectrum contains more information than the c lassical zero-point\nradiation spectrum. Both spectra are characterized by a single pa rameter: /planckover2pi1for zero-point\n25radiation and Tfor the Rayleigh-Jeans spectrum. For the Rayleigh-Jeans spectr um, the\ntemperature Tis freely-adjustable, and the energy per normal mode is UradRJ(ω,T) =kBT\nat anyfrequency ω.However, theRayleigh-Jeans spectrum determines exactly onepr eferred\ninertial frame in which it is isotropic. Furthermore, at fixed tempera tureT, one can\ndetermine whether one frequency is larger than another frequen cy by comparing their phase\nspace distributions. Thus the phase space distribution for electro magnetic radiation in the\nRayleigh-Jeans spectrum is PradRJ(J,ω,T) =const×exp[−Jω/(kBT)] so that a larger\nvalue of frequency ωcorresponds to a phase space distribution more concentrated ne ar\nJ= 0. Ontheother hand, theclassical zero-point radiationspectru mUradzp(ω) = (1/2)/planckover2pi1ω\nhas a phase space distribution which is the same for radiation modes o f any frequency ω,\nPradzp(J,ω) =const×exp[−J/(/planckover2pi1/2)].Zero-point radiation has no preferred inertial frame\nand is isotropic in every inertial frame. If a harmonic oscillator of sma ll amplitude were\nto scatter zero-point radiation so as to enforce any radiation spe ctrum other than the zero-\npoint radiation spectrum, the oscillator would impose a preferred ine rtial frame upon the\nscattered radiation.\n3. Thermal Radiation Spectrum as Giving Least Information in t he Presence of Zero-Point\nRadiation\nThermal radiation at T >0 involves a finite amount of energy spread over the modes\nof the zero-point radiation spectrum and so introduces a preferr ed inertial frame. The\nspectrum of zero-point radiation acts as a noise spectrum into whic h the finite amount of\nthermal energy is introduced, and the spectrum of thermal radia tion represents the mini-\nmal information consistent with the finite total available thermal en ergy and the divergent\nLorentz-invariant spectrum of zero-point radiation. The entrop y of thermal radiation at\nfrequency ωis determined by comparing the amount of thermal radiation to the a mount of\nzero-point radiation at that frequency.[3]\nC. Inconsistent Mixtures of Nonrelativistic and Relativistic Physics\nThe derivations of the Rayleigh-Jeans law for thermal radiation give n in the textbook\nliterature arise from attempts to combine nonrelativistic mechanics with relativistic elec-\n26trodynamics. Such inconsistent mixtures of nonrelativistic and rela tivistic theories satisfy\nneither Galilean invariance nor Lorentz invariance. Indeed, the res ults of the analysis de-\npend upon the inertial frame in which the analysis is done. For particle s undergoing point\ncollisions, this situation has already been exhibited explicitly in the litera ture.[32] Thus\nif one considers the point collision between two particles, using relativ istic physics for one\nand nonrelativistic physics for the other, one can solve for the mot ion of the particles after\nthe collision by using the energy and momentum conservation laws in an y inertial frame;\nhowever, the results satisfy neither the nonrelativistic conservation law for the uniform mo-\ntion of the center of mass nor the relativistic conservation law for the uniform motion of\nthe center of energy, and the results indeed depend upon the iner tial frame chosen for the\nanalysis.\nVII. CLOSING SUMMARY\nA. Two Natural Spectra Associated with Harmonic Oscillators\nAssociated with harmonic oscillators, there are two spectra of ran dom radiation which\narise in a natural manner. One of these is the Rayleigh-Jeans spect rum which is associated\nwith the equipartition ideas of nonrelativistic classical mechanics and is based uponvelocity-\ndependent damping. The other is the classical zero-point radiation spectrum which arises\nfrom an adiabatic transformation of oscillator frequency and is ass ociated with relativistic\nclassical electrodynamics which includes acceleration-dependent d amping.\nB. Oscillators in Translational Motion and Two Forms of Damping\nIf one considers not a stationary oscillator but rather an oscillator which is free to move in\nover-all translation, then one can obtain either the Rayleigh-Jean s spectrum (obtained for a\nsituation with velocity-dependent damping on the oscillator but no ac celeration-dependent\ndamping by Einstein and Hopf[14]) or the full Planck spectrum with ze ro-point radiation (if\none allows both types of damping on the oscillator[30]).\n27C. Thermal Radiation Equilibrium from Scattering\nA charged harmonic oscillator at rest with its electromagnetic intera ctions limited to the\ndipole approximation is in equilibrium with anyspectrum of radiation; the energy of the\noscillator merely matches the energy of the radiation modes at the n atural frequency of the\noscillator. If we wish to go beyond this ambiguous radiation situation a nd to determine\na preferred equilibrium radiation spectrum under scattering , then there are two natural\nscattering extensions. If we continue the limitation on the electromagnetic inte ractions\nto the dipole approximation but go to a more complex nonrelativistic mechanical motion,\nthen we arrive at the Rayleigh-Jeans spectrum. On the other hand , if we treat the charged\nharmonicoscillatorasafullyelectromagneticsystem but withonly approximately-relativistic\nharmonic oscillator mechanical behavior, then we arrive at classical zero-point radiation as\nthe preferred radiation spectrum. Scattering by a fully relativistic electromagnetic system,\nsuch as the classical hydrogen atom, has the possibility of giving the full Planck radiation\nspectrum with zero-point radiation as the equilibrium classical radiat ion spectrum.\nD. Relativity and Thermal Radiation\nIf one reads James Jeans’ Report on Radiation and the Quantum-Theory published in\n1914 or if one reads the modern physics textbooks published in the la st few years, one would\nfind no inkling that special relativity might have some relevance to the blackbody radia-\ntion problem. However, the equilibrium radiation spectrum is determin ed by assumptions\non both the mechanical motion (whether nonrelativistic or relativist ic) and the relativistic\nelectromagnetic interactions. In any case, the radiation equilibrium of the fully electromag-\nnetic charged harmonic oscillator of small amplitude does notlead to the Rayleigh-Jeans\nspectrum. It is an outright error to claim that classical physics lead s inevitably to the\nRayleigh-Jeans spectrum.\n[1] See for example, R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids,\nNuclei, and Particles 2nd ed. (Wiley, New York 1985); K. S. Krane, Modern Physics , 2nd\ned. (Wiley, New York 1996); R. Taylor, C. D. Zafiratos, and M. A . Dubson, Modern Physics\n28for Scientists and Engineers , 2nd ed. (Pearson, New York, 2003); S. T. Thornton and A.\nRex,Modern Physics for Scientists and Engineers , 4th ed. (Brooks/Cole, Cengage Learning,\nBoston, MA, 2013).\n[2] J. H. Jeans, Report on Radiation and the Quantum Theory (www.ForgottenBooks.org, 2013).\nThis is a reproduction from the report originally published in 1914.\n[3] T. H. Boyer, “Blackbody radiation in classical physics: A historical perspective,” Am. J. Phys.\n86, 495-509 (2018).\n[4] D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, “Relati vistic Invariance and Hamiltonian\ntheories of interacting particles,” Rev. Mod. Phys. 35, 350-375 (1963).\n[5] T. H. Boyer, “Thermodynamics of the harmonic oscillator : Wien’s displacement law and the\nPlanck spectrum,” Am. J. Phys. 71, 866-870 (2003).\n[6] T. H. Boyer, “Thermodynamicsof the harmonic oscillator : derivation of the Planck blackbody\nspectrum from pure thermodynamics,” Eur. J. Phys. 40,025101(16pp) (2019).\n[7] See for example, M. Planck, The Theory of Heat Radiation (Dover, New York 1959).\n[8] T. W. Marshall, “Random electrodynamics,” Proc. R. Soc. A276, 475-491 (1963).\n[9] T. H. Boyer, “Random electrodynamics: Thetheory of clas sical electrodynamics with classical\nelectromagnetic zero-point radiation,” Phys. Rev. D 11, 790-808 (1975).\n[10] B. H. Lavenda, Statistical Physics: A Probabilistic Approach (Wiley, New York 1991), pp.\n73-74.\n[11] T. H. Boyer, “Understanding zero-point energy in the co ntext of classical electromagnetism,”\nEur. J. Phys. 37, 055206(14) (2016).\n[12] A review of the work on classical electromagnetic zero- point radiation up to 1996 is provided\nby L. de la Pena and A. M. Cetto, The Quantum Dice - An Introduction to Stochastic Elec-\ntrodynamics (Kluwer Academic, Dordrecht 1996). For a more recent short r eview, see T.\nH. Boyer, “Stochastic Electrodynamics: The Closest Classi cal Approximation to Quantum\nTheory,” Atoms 7(1), 29-39 (2019).\n[13] T. H. Boyer, “The contrasting roles of Planck’s constan t in classical and quantum theories,”\nAm. J. Phys. 86, 280-283 (2018).\n[14] A. Einstein and L. Hopf, “Statistische Untersuchung de r Bewegung eines Resonators in einem\nStrahlungsfeld,” Annalen der Physik (Leipzig) 33, 1105-1115 (1910).\n[15] For modernnotation, see P. W. Milonni, The Quantum Vacuum: An Introduction to Quantum\n29Electrodynamics (Academic Press, Boston1994), pp. 11-14.\n[16] J. H. van Vleck, “The absorption of radiation by multipl y periodic orbits, and its relation to\nthe correspondence principle and the Rayleigh-Jeans law: P art II. Calculation of absorption\nby multiply periodic orbits,” Phys. Rev. 24, 347-365 (1924); “A correspondence principle for\nabsorption,” Jour. Opt. Soc. Amer. 9, 27-30 (1924).\n[17] T. H. Boyer, “Equilibrium of random classical electrom agnetic radiation in the presence of a\nnonrelativistic nonlinear electric dipole oscillator,” P hys. Rev. D 13, 2832-2845 (1976).\n[18] R. Blanco, L. Pesquera, and E. Santos, “Equilibrium bet ween radiation and matter for clas-\nsical relativistic multiperiodic systems. Derivation of M axwell-Boltzmann distribution from\nRayleigh-Jeans spectrum,”Phys.Rev.D 27, 1254-1287 (1983); “Equilibriumbetweenradiation\nand matter for classical relativistic multiperiodic syste ms. II. Study of radiative equilibrium\nwith Rayleigh-Jeans radiation,” Phys. Rev. D 29, 2240-2254 (1984).\n[19] M. Born, The Mechanics of the Atom (Ungar, New York 1970), pp. 66-71.\n[20] J. D. Jackson, Classical Electrodynamics 3rd ed. (John Wiley & Sons, New York, 1999), p.\n704, problem 14.22.\n[21] W. C-W. Huang and H. Batelaan, “Discrete Excitation Spe ctrum of a Classical Harmonic\nOscillator in Zero-Point Radiation,” Found. Phys. 45, 333-353 (2015).\n[22] T. H. Boyer, “Equilibrium for classical zero-point rad iation: detailed balance under scattering\nby a classical charged harmonic oscillator,” J. Phys. Commu n.2, 105014(17) (2018).\n[23] See ref. 16, p. 359. Van Vleck’s comment involves radiat ion absorption but is related to\nnonrelativistic kinetic energy and to the nonrelativistic Larmor formula for radiation emission.\nThus for a general oscillation such as suggested by our Eq. (1 4), the average kinetic energy\nin thenthharmonic involves ( nω1)2D2\nnwhereas the average radiation emission by the nth\nharmonic is ( nω1)2times as large, involving ( nω1)4D2\nn= (nω1)2/bracketleftBig\n(nω1)2D2\nn/bracketrightBig\n.\n[24] T. H. Boyer, “Statistical equilibrium of nonrelativis tic multiply periodic classical systems and\nrandom classical electromagnetic radiation,” Phys. Rev. A 18, 1228-1237 (1978).\n[25] T. H. Boyer, “Scaling symmetries of scatterers of class ical zero-point radiation,” J. Phys. A:\nMath. Theor. 40, 9635-9642 (2007); “Scaling symmetry and thermodynamic eq uilibrium for\nclassical electromagnetic radiation,” Found. Phys. 19, 1371-1383 (1989).\n[26] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics 3rd ed. (Addison-Wesley, New York,\n2002), p. 353.\n30[27] T. H. Boyer, “Conformal Symmetry of Classical Electrom agnetic Zero-Point Radiation,”\nFound. Phys. 19, 349-365 (1989).\n[28] For work on the classical hydrogen atom in classical zer o-point radiation, see D. C. Cole and\nY. Zou, “Quantum Mechanical Ground State of Hydrogen Obtain ed from Classical Electrody-\nnamics,” Phys. Lett. A 317, 14-20 (2003), and T. H. Boyer, “Relativity andRadiation Ba lance\nfor the Classical Hydrogen Atom in Classical Electromagnet ic Zero-Point Radiation,” Eur. J.\nPhys.42, 025205 (24pp) (2021).\n[29] T. H. Boyer, “Unfamiliar trajectories for a relativist ic particle in a Kepler or Coulomb poten-\ntial,” Am. J. Phys. 75, 992-997 (2004).\n[30] T. H. Boyer, “Derivation of the Blackbody Radiation Spe ctrum without Quantum Assump-\ntions,” Phys. Rev. 182, 1374-1383 (1969).\n[31] T. H. Boyer, “Particle Brownian motion due to random cla ssical radiation: Superfluid-like\nbehavior in classical zero-point radiation,” Eur. J. Phys. 41, 055103 (17pp) (2020).\n[32] T.H.Boyer, “Illustratingsomeimplications ofthecon servation lawsinrelativistic mechanics,”\nAm. J. Phys. 77, 562-569 (2009).\n31"    },    {        "title": "1709.03347v1.Comparison_of_damping_mechanisms_for_transverse_waves_in_solar_coronal_loops.pdf",        "content": "Draft version November 8, 2018\nTypeset using L ATEX default style in AASTeX61\nCOMPARISON OF DAMPING MECHANISMS FOR TRANSVERSE WAVES IN SOLAR CORONAL LOOPS\nMar\u0013\u0010a Montes-Sol \u0013\u0010s1, 2and I ~nigo Arregui1, 2\n1Instituto de Astrof\u0013 \u0010sica de Canarias, E-38205 La Laguna, Tenerife, Spain\n2Departamento de Astrof\u0013 \u0010sica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain\n(Received; Revised; Accepted)\nSubmitted to ApJ\nABSTRACT\nWe present a method to assess the plausibility of alternative mechanisms to explain the damping of magnetohydro-\ndynamic (MHD) transverse waves in solar coronal loops. The considered mechanisms are resonant absorption of kink\nwaves in the Alfv\u0013 en continuum, phase-mixing of Alfv\u0013 en waves, and wave leakage. Our methods make use of Bayesian\ninference and model comparison techniques. We \frst infer the values for the physical parameters that control the wave\ndamping, under the assumption of a particular mechanism, for typically observed damping time-scales. Then, the\ncomputation of marginal likelihoods and Bayes factors enable us to quantify the relative plausibility between the alter-\nnative mechanisms. We \fnd that, in general, the evidence is not large enough to support a single particular damping\nmechanism as the most plausible one. Resonant absorption and wave leakage o\u000ber the most probable explanations in\nstrong damping regimes, while phase mixing is the best candidate for weak/moderate damping. When applied to a\nselection of 89 observed transverse loop oscillations, with their corresponding measurements of damping times scales\nand taking into account data uncertainties, we \fnd that only in a few cases positive evidence for a given damping\nmechanism is available.\nKeywords: magnetohydrodynamics (MHD) | methods: statistical | Sun: corona | Sun: oscillations\n| waves\nCorresponding author: Mar\u0013 \u0010a Montes-Sol\u0013 \u0010s\nmmsolis@iac.esarXiv:1709.03347v1  [astro-ph.SR]  11 Sep 20172 Montes-Sol \u0013\u0010s & Arregui\n1.INTRODUCTION\nThe damping of magnetohydrodynamic (MHD) waves in solar coronal structures is a commonly observed phe-\nnomenon and a source of information about their physical conditions, dynamics, and energetics. The study of the\ndamping of transverse waves has attracted particular attention, since the \frst imaging observations of decaying trans-\nverse coronal loop oscillations by Aschwanden et al. (1999) and Nakariakov et al. (1999). High resolution imaging and\nspectroscopic observations with ground- and space-based observatories such as Hinode, CoMP, SDO/AIA, STEREO,\nHi-C, and IRIS have enabled us to measure transverse wave dynamics with increasing precision in almost all layers of\nthe solar atmosphere (see e.g., Okamoto & De Pontieu 2011; White & Verwichte 2012; Verwichte et al. 2013; Morton\n& McLaughlin 2013; Thurgood et al. 2014; Morton et al. 2015; Okamoto et al. 2015). Recent observational analyses\nhave permitted the creation of databases containing the oscillation properties for a large number events and with the\ninclusion of measurement errors (Verwichte et al. 2013; Goddard et al. 2016). The increase in the number of measured\nevents and their properties including their uncertainty has led to advances in statistical seismology of coronal loops.\nVerwichte et al. (2013) considered a statistical approach to obtain information on the loop cross-sectional structuring\nparameters by forward modelling of scaling laws between periods and damping times, showing that restrictions can\nbe found to the loop's density contrast and inhomogeneity layer. Following Bayesian methods, a number of studies by\nArregui & Asensio Ramos (2011); Arregui et al. (2013a,b); Asensio Ramos & Arregui (2013); Arregui & Asensio Ramos\n(2014); Arregui et al. (2015) have shown how information on the uncertainty of the inferred coronal loop parameters\nand the plausibility between alternative models can be assessed.\nAlthough damping is not an omnipresent phenomenon, see e.g., An\fnogentov et al. (2013, 2015) for examples of\ndecay-less oscillations or Wang et al. (2012) for an example of growing oscillations, theoretical explanations for the\nphysical origin of the damping of transverse waves are abundant. Because direct viscous and resistive difussion time\nscales are too long in uniform plasmas, mechanisms based on the cross-\feld or \feld-aligned inhomogeneity of the wave\nguides become relevant. In the context of coronal loop oscillations, the discussion rapidly focused on mechanisms\nsuch as resonant absorption (Goossens et al. 2002; Ruderman & Roberts 2002; Goossens et al. 2006), phase mixing\nof Alfv\u0013 en waves (Heyvaerts & Priest 1983), lateral wave leakage (Spruit 1982; Cally 1986; Roberts 2000; Cally 2003)\nor foot-point leakage at the chromospheric density gradient (Berghmans & de Bruyne 1995; De Pontieu et al. 2001;\nOfman 2002). Methods to assess the plausibility between alternative damping mechanisms are discussed in Roberts\n(2000); Ruderman (2005); Nakariakov & Verwichte (2005); Aschwanden (2005).\nRough estimates of damping time scales predicted by the mechanisms under consideration, for typical coronal loop\nphysical properties, and their comparison to observed damping time scales point to resonant damping as the most\nplausible mechanism, with phase mixing and wave leakage producing damping time scales that seem to be too long in\ncomparison to those observed. The drawback to this method is the di\u000eculty in de\fning what a typical coronal loop\nis, since the physical parameters of observed loops cannot be directly measured.\nThe modeling and analysis of wave properties for equilibrium states that enable the simultaneous occurrence of more\nthan one mechanism is another alternative. The studies by e.g., Terradas et al. (2006), Rial et al. (2013) and Soler\net al. (2009) indicate that resonant absorption in the Alfv\u0013 en continuum is a far more e\u000ecient mechanism than lateral\nwave leakage in curved loops or than damping in the slow continuum in prominence threads. Resonant damping is\nfrequency selective, with low-frequency waves being favored in front of high frequency waves (Terradas et al. 2010).\nA comparison between the outward to inward power ratio measurements in CoMP observations of coronal waves by\nTomczyk et al. (2007) and the theoretical modeling by Verth et al. (2010) gives additional support to the resonant\ndamping model.\nOfman & Aschwanden (2002) proposed a method based on the use of scaling laws between the observed periods\nand damping times. Their suggestion is based on the assumption that each damping mechanism is characterized by\na particular power law, with a distinct power index between the damping time and the oscillation period. By \ftting\nthe observed time-scales to those predicted by each damping mechanism, one can compare the theoretically predicted\nand \ftted power indexes to discriminate between damping mechanisms. The application of this method has given\nresults that support both phase mixing (Ofman & Aschwanden 2002) and resonant absorption (White & Verwichte\n2012; Verwichte et al. 2013). It was pointed out by Arregui et al. (2008) that the use of scaling laws to discriminate\nbetween damping mechanisms is questionable. For example, the resonant damping model is able to produce data\nrealizations for which di\u000berent scaling laws with di\u000berent indexes may be obtained. The major drawback to this\nmethod is the faultiness of the assignment of a single power index to a particular damping mechanism, even more soDamping mechanisms for transverse coronal waves 3\nwhen the damping time depends on a number of loop parameters that might have an intrinsic variability and values\nthat are highly uncertain.\nWe adopt a di\u000berent approach and consider Bayesian methods to compute the level of plausibility among alternative\ndamping models, conditional on observed data and considering their uncertainty. Inference and model comparison\nmethods are applied to three particular damping mechanisms: resonant damping in the Alfv\u0013 en continuum; phase\nmixing of Alfv\u0013 en waves; and wave leakage of the principal kink mode. These damping models are selected so as to\ngive continuity to the discussion initiated by Ofman & Aschwanden (2002), but the methods here presented can be\nextended in the future to include additional damping models in a straightforward manner. The methods are \frst\napplied to hypothetical data and then to a sample of real observations from the databases compiled by Verwichte et al.\n(2013) and Goddard et al. (2016).\nThe layout of the paper is as follows. Section 2 gives a description of the damping models being compared. In\nSection 3, we present our methodology, based on the use of Bayes' rule for parameter inference and model comparison.\nOur results are presented in Section 4. We \frst apply Bayes' rule to the problem of inferring the relevant physical\nparameters, under the assumption that each of the considered damping models is true. Then, we assess the plausibility\nof each damping model conditional on both hypothetical and real observations of transverse loop oscillations. Our\nsummary and conclusions are presented in Section 5.\n2.DAMPING MODELS\nIn this work, we have studied the degree of plausibility of three damping mechanisms in explaining the observations of\ntransverse loop oscillations. The \frst considered damping model is resonant absorption in the Alfv\u0013 en continuum. This\nmechanism consists of an energy transfer between the global kink mode of a magnetic \rux tube to Alfv\u0013 en oscillations\nat the boundary of the tube, due to the transverse variation of Alfv\u0013 en velocity within a layer that separates the tube\nand the background corona. Resonant absorption has been studied extensively and in great detail and is known to be\nable to produce time and spatial damping scales comparable to those observed.\nFollowing Goossens et al. (2002) and Ruderman & Roberts (2002), we focus on the fundamental kink mode of a\ncylindrical density tube of length L and radius R with a uniform magnetic \feld along the axis of the tube and a\nnon-uniform variation of the cross-\feld density on a length-scale l. Under the thin tube ( L>>R ) and thin boundary\n(l=R<< 1) assumptions the expression for the damping ratio is\n\u001cd\nP=2\n\u0019R\nl\u0010+ 1\n\u0010\u00001; (1)\nwith\u001cdthe damping time, P the oscillation period, lthe thickness of the non-uniform layer, and \u0010=\u001ai=\u001aethe density\ncontrast between the internal ( \u001ai) and external ( \u001ae) densities. The factor 2 =\u0019is due to the assumption of a sinusoidal\npro\fle for the density at the non-uniform boundary layer.\nAccording to Equation (1), the damping ratio due to resonant absorption is a function of two parameters, \u0010andl=R.\nAs shown by Goossens et al. (2002), considering a typical contrast of \u0010= 10, the mechanism is able to explain observed\ndamping time scales for values of l=Rin between 0.1 and 0.5. Considering values of \u0010andl=Rwithin plausible ranges,\n\u00102(1;10] andl=R2(0;2], the values for the damping ratio predicted by theory are in a wide range \u001cd=Ps(0:5\u0000104).\nThe mechanism is therefore able to predict damping properties compatible with observations.\nThe next considered mechanism is associated with the phase mixing of Alfv\u0013 en waves, \frst discussed by Heyvaerts &\nPriest (1983) in the context of coronal heating. The basic idea is that Alfv\u0013 en waves propagating along the magnetic\n\feld in a medium with a transverse gradient of Alfv\u0013 en velocity become rapidly out of phase. As time progresses,\nincreasingly shorter spatial scales are created. The mechanism has been discussed as a possible reason to explain\nthe observed damping in Roberts (2000); Nakariakov & Verwichte (2005) and Ofman & Aschwanden (2002). In his\nanalysis, Roberts (2000) obtained an analytical expression for the damping ratio of the form\n\u001cd\nP=\u00123\n\u00192\u0017\u00131=3\nw2=3P\u00001=3; (2)\nwhere\u0017= 4\u0002103km2s\u00001is the coronal kinematic shear viscosity coe\u000ecient and wthe transverse inhomogeneity\nlength scale. Note that in Equation (2), the damping ratio is a function of the period. Considering a period in the range\nP2[150;1250]s and w2[0:5;20]Mm, damping ratios in the range \u001cd=Ps[0:3\u00006] are obtained, so this mechanism\ncould also explain the observed damping time scales, although it has serious limitations of applicability to transverse4 Montes-Sol \u0013\u0010s & Arregui\nloop oscillations. Ofman & Aschwanden (2002) considered phase mixing as a mechanism involving the friction between\nadjacent unresolved thin loop strands, a hypothesis that remains to be demonstrated. The mechanism is here included\nin the interest of continuity in the discussion by considering the same damping mechanisms discussed by Ofman &\nAschwanden (2002), but presenting an alternative method to the scaling law approach for their comparison.\nThe third considered mechanism is wave leakage of the principal leaky mode, as discussed by Cally (2003) and\nTerradas et al. (2007). This theoretical solution consists of a radiating wave which oscillates with the kink mode\nfrequency and looses part of its energy to the background corona. Following Cally (2003), we consider the analytic\napproximate solution for the damping ratio in the thin tube limit and valid for \u001ai>>\u001a eof the form\n\u001cd\nP=4\n\u00194\u0012R\nL\u0013\u00002\n: (3)\nConsidering values of R/L in the plausible range R/L 2[10\u00004;0:3], theory predicts damping ratios in the range\n\u001cd=Ps(0:5\u0000105).\nFor simplicity, strong assumptions are made in the adopted damping mechanisms and formulas. The resonant\ndamping formula assumes the thin tube and thin boundary approximation and the adoption of a particular radial\ndensity pro\fle. The in\ruence of considering di\u000berent density pro\fles in the inferences using resonant absorption was\nanalyzed by Arregui et al. (2015). This in\ruence was found to be important in strong damping regimes. The phase-\nmixing damping formula is valid in the strongly developed regime and the value of kinematic viscosity is kept constant.\nFor wave-leakage, the possible in\ruence of the density contrast was ignored. These assumptions lead to the existence\nof hidden variables that may vary from loop to loop. All loops do not necessarily have the same radial density pro\fle,\ntemperature, viscosity and density contrast.\nOnce the three damping mechanisms to be considered in this study are presented, it is worth clarifying that they\nare going to be compared on the basis of their ability to reproduce (in a statistical way) the observed periods and\ndamping time scales, taking into account those observed time scales and their associated uncertainties.\n3.BAYESIAN METHODOLOGY\nIn this paper, we adopt the methods of Bayesian analysis to perform parameter inference and model comparison.\nBayesian reasoning goes back to the essay by Bayes & Price (1763) and was given its current mathematical basis\nby Laplace (1774). It o\u000bers a principled way to assess the plausibility of statements conditional on the available\ninformation. In recent years, it has become widely used to perform scienti\fc inference in areas such as the physical\nsciences (von Toussaint 2011), cosmology (Trotta 2008) or exoplanet research (Gregory 2005).\nWe have used Bayesian analysis tools to perform parameter inference, assuming a particular damping model is true,\nand to compare the relative plausibilities of the three damping models described above.\n3.1. Parameter Inference\nIn the Bayesian framework the inference of a parameter set \u0012that characterizes a model M conditional on observed\ndatadis performed by making use of the Bayes' theorem\np(\u0012jM;d) =p(\u0012jM)p(djM;\u0012)\np(djM): (4)\nIn this expression, p(\u0012jM;d) is the posterior probability of the parameters conditional on the assumed model and the\nobserved data; p(\u0012jM) is their prior probability, before considering the data; and p(djM;\u0012) is the likelihood function.\nThe denominator in Equation (4) is a normalization factor called marginal likelihood given the model M, integrated\nlikelihood or model evidence. It represents the probability of the data given the model, and is computed by integrating\nthe likelihood times the prior over the parameter space\np(djM) =Z\n\u0012p(\u0012jM)p(djM;\u0012)d\u0012: (5)\nWhen interested in obtaining information on one particular parameter \u0012iof model M, we compute its marginal\nposterior as\np(\u0012ijM;d) =Z\np(\u0012jM;d)d\u00121:::d\u0012i\u00001d\u0012i+1:::d\u0012n; (6)\nwhich is an integral of the full posterior over the rest of the n model parameters.Damping mechanisms for transverse coronal waves 5\nTable 1. Kass & Raftery (1995) evidence classi\fcation.\n2lnBF kj Evidence\n0-2 Not Worth more than a bare Mention (NWM)\n2-6 Positive Evidence (PE)\n6-10 Strong Evidence (SE)\n>10 Very Strong Evidence (VSE)\n3.2. Model Comparison\nThe Bayesian framework also o\u000bers a straightforward method to compare the relative goodness of alternative models\nin explaining the observed data. Application of Bayes' theorem to N alternative models leads to\np(Mkjd) =p(Mk)p(djMk)\np(d)fork= 1;2;:::;N: (7)\nThe posteriors ratio gives us a measure of the relative plausibility between two models MkandMjas\np(Mkjd)\np(Mjjd)=p(djMk)\np(djMj)p(Mk)\np(Mj);k6=j: (8)\nIf we do not have any a priori reason to prefer one model over another before considering the data, p(Mk) =p(Mj),\nso that we can de\fne\nBFkj=p(djMk)\np(djMj);k6=j: (9)\nThis expression is called the Bayes factor (see Kass & Raftery 1995) which equals the ratio of marginal likelihoods in\nEquation (6) in the case of equal priors.\nOnce the Bayes factors are computed, the level of evidence for one model against the alternative is assessed by\nfollowing the criteria for evidence classi\fcation developed by Kass & Raftery (1995), see Table 1.\nThese general concepts and methods are applicable as long as we \frst specify priors and likelihood functions. In\nthis work, we have adopted independent priors for model parameters, so that the global prior is given by the product\nof individual priors\np(\u0012jM) =nY\ni=1p(\u0012ijM): (10)\nFurther, we consider that each parameter lies on a given plausible range, with all values being equally probable a\npriori. This de\fnes uniform priors of the form\np(\u0012ijM) =1\n\u0012max\ni\u0000\u0012min\ni;\u0012i2(\u0012min\ni;\u0012max\ni) (11)\nand zero otherwise.\nAs for the likelihood functions, we assume Gaussian pro\fles, so that the data and the theoretically predicted values\nare related as\np(djM;\u0012) =1p\n2\u0019\u001b\u0001e\u0000(\u0001obs\u0000\u0001theor)2\n2\u001b2\n\u0001; (12)\nwith \u0001 representing the observable and \u001b\u0001its associated uncertainty.\n3.3. Numerical Integration\nThe computation of marginal likelihoods and marginal posteriors using Equations (5) and (6) requires the compu-\ntation of integrals in the parameter space. In low-dimensional parameter spaces, such as the ones considered in this\nwork, the use of direct numerical integration techniques is still feasible from the point of view of computational cost.\nNevertheless, we have additionally used Markov Chain Monte Carlo sampling and Monte Carlo integration (Robert &6 Montes-Sol \u0013\u0010s & Arregui\nTable 2. Inferred parameters from Figure 2.\nMethod \u0010 l=R w R=L\n(Mm)\nNumeric 4 :1+3:9\n\u00002:40:4+0:5\n\u00000:27:5+1:1\n\u00001:10:12+0:01\n\u00000:01\nMCMC 4 :1+3:8\n\u00002:50:4+0:5\n\u00000:17:4+1:1\n\u00001:00:12+0:01\n\u00000:01\nCasella 2004) to compute marginal posteriors and marginal likelihoods, respectively. The comparison between both\ntypes of approaches gives robustness to the obtained results.\nThe marginal posteriors in Equation (6) are computed from the Markov Chain Monte Carlo (MCMC) sampling of\nthe global posterior over the space of parameters for each considered model. The marginal likelihoods in Equation\n(5) result from Monte Carlo (MC) integration. In this case, MC integration is carried out by the average of all\nvalues resulting from the evaluation of the likelihood function, at the points of the parameter space from the MCMC\nsampling of the normalized prior. These two integration methods are applied employing the emcee package of Python\n(Foreman-Mackey et al. 2013).\n4.RESULTS\nBefore considering the problem of assessing the relative plausibility of the three considered damping mechanisms in\nexplaining the observed time scales, we perform the inference of physical parameters under the assumption that each\nmechanism is true. Then, our model comparison tools are applied for the computation of the relative plausibility of\nthe three damping models, using possible values for the damping rate and its uncertainty. Finally, these techniques\nare applied to a large sample of transverse loop oscillations events for which the Bayes factors are computed.\n4.1. Parameter Inference\nFor each mechanism, the damping is determined by di\u000berent coronal loop physical parameters that cannot be\ndirectly measured. The theoretical predictions for the observable damping ratio are given by Equations (1), (2), and\n(3). The inference using observed damping ratios can in principle provide information about the cross-\feld density\ninhomogeneity, in the case of resonant absorption and phase mixing, and the coronal loop radius to length ratio in\nthe case of wave leakage. We consider di\u000berent possible values for the damping ratio and its uncertainty and perform\nBayesian inference using Equation (4), de\fning for each model Gaussian likelihoods according to Equation (12) and\nuniform priors, de\fned in Equation (11). Once the full posteriors are known, the marginalization in the corresponding\nparameter space (Equation 6) leads to the probability density function for each unknown parameter. The resulting\nmarginal posteriors are shown in Figure 1. The left hand side panels show results for di\u000berent values of the observable\ndamping rate with a \fxed uncertainty of 10%. The right hand side panels show results for a \fxed observable damping\nrate and three di\u000berent values for the uncertainty on the observable damping ratio.\nConsider \frst resonant absorption. For this mechanism, such inversion problem was \frst solved by Arregui &\nAsensio Ramos (2011) using a MCMC sampling of the posterior by using observed values for periods and damping\ntimes. Later on, Arregui & Asensio Ramos (2014) presented solutions akin to those presented here, by making use\nof the properties of the joint probability of data and parameters. Figures 1a-d show the results for this mechanism.\nFor this inference problem, \u0012=f\u0010;l=Rgandd=\u001cd=P. In principle, both the density contrast, \u0010, and the transverse\ninhomogeneity length scale, l=R, can be properly inferred, although as already noted by Arregui & Asensio Ramos\n(2014) the posteriors for the density contrast show long tails. The posterior densities peak at lower values for both\nunknowns the larger the damping ratio is (see Figures 1a and 1c) and their distributions have broader shapes the\nlarger the uncertainty in the observed damping ratio (see Figures 1b and 1d). These results indicate that, in principle,\nthe assumption that resonant absorption is the true mechanism that produces the damping of coronal loop oscillations\ntogether with measurements of the damping ratio enables us to infer the two parameters that control the cross-\feld\ndensity inhomogeneity.\nFor phase mixing, the theoretical prediction in Equation (2) does not enable to factorize the damping rate as a sole\nfunction of model parameters since the oscillation period is present in the right-hand side of this equation. For this\nreason, we perform the inference by assuming particular values for the period in the range P2[150;1250]s. This\nleaves us with an inversion problem with one observable, d=\u001cd=P, and one unknown, \u0012=fwg. Figures 1e and 1fDamping mechanisms for transverse coronal waves 7\n2 4 6 8 10\n0.00.20.40.60.81.01.2p(|r)\n(a) r=1.5\nr=3.0\nr=10.0\n2 4 6 8 10\n0.00.10.20.30.4p(|r=3)\n(b)=0.30\n=0.75\n=1.50\n0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nl/R02468101214p(l/R|r)(c) r=1.5\nr=3.0\nr=10.0\n0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nl/R012345p(l/R|r=3)(d) =0.30\n=0.75\n=1.50\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nw (Mm)0.00.20.40.60.81.0p(w|r)(e) r=1.5\nr=3.0\nr=10.0\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nw (Mm)0.000.050.100.150.200.250.300.35p(w|r=3)(f) =0.30\n=0.75\n=1.50\n0.00 0.05 0.10 0.15 0.20 0.25 0.30\nR/L020406080100120p(R/L|r)(g) r=1.5\nr=3.0\nr=10.0\n0.00 0.05 0.10 0.15 0.20 0.25 0.30\nR/L010203040506070p(R/L|r=3)(h)=0.30\n=0.75\n=1.50\nFigure 1. Marginal posterior distributions for coronal loop physical parameters under the assumption of resonant damping,\n(a)-(d); phase-mixing, (e) and (f); and wave leakage, (g) and (h). The left-hand side panels show the results for di\u000berent values\nof the damping ratio, r, with an uncertainty of 10%. The right-hand side panels show the results for a \fxed damping ratio,\nr=3, and three values for its uncertainty.8 Montes-Sol \u0013\u0010s & Arregui\n0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nl/R012345p(l/R|r=3.0)(a) numeric\nMCMC\n2 4 6 8 10\n0.00.10.20.30.4p(|r=3.0)\n(b) numeric\nMCMC\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nw0.000.050.100.150.200.250.300.35p(w|r=3.0)(c) numeric\nMCMC\n0.00 0.05 0.10 0.15 0.20 0.25 0.30\nR/L010203040506070p(R/L|r=3.0)(d)numeric\nMCMC\nFigure 2. A comparison between the marginal posterior distributions obtained by direct numerical integration (in red) and\nMCMC sampling (blue histograms) for a case with r=3, and an uncertainty of 10%.\nshow the results for this mechanism. The posteriors computed for the case P=150 s are only shown to not deface the\nplot. The posteriors shift toward larger values of wfor longer periods. The inference results for this mechanism show\nthat the length scale wcan be properly inferred although a larger upper limit in the parameter range is needed for\nthe case r=10. The posteriors shift towards larger values of wthe longer the damping ratio is (see Figure 1e) and\nshow broader distributions the larger the uncertainty in the observed damping ratio (see Figure 1f). These results\nindicate that the assumption that phase mixing is the true damping mechanism of coronal loop oscillations together\nwith measurements of the damping ratio enables us to infer w.\nRegarding wave leakage, the theoretical damping rate in Equation (3) is only a function of the coronal loop radius\nto length ratio, R=L. This enables us to solve the inference with one observable, d=\u001cd=P, and one parameter,\n\u0012=fR=Lg. Figures 1g and 1h show the results for this mechanism. They show that the ratio between the radius and\nthe length of coronal loops can be properly inferred. The posterior distributions shift toward smaller values of R=L\nthe larger the damping ratio is (see Figure 1g) and have broader shapes the larger the uncertainty in the observed\ndamping ratio (see Figure 1h). These results indicate that assuming that wave leakage is the true mechanism that\ncauses the damping of coronal loop oscillations together with measurements of the damping ratio, a relation among\ntwo structural features of coronal loops can be inferred.\nComparing these results obtained from direct numeric integration with the results from MCMC sampling, we can see\nthe degree of applicability of each method to the inference problem. An example of this comparison is show in Figure\n2, and the corresponding median values of damping model parameters within a credible interval of 68% are presented\nin Table 2. The agreement between the posterior distributions and the minimal di\u000berences between the mean values,Damping mechanisms for transverse coronal waves 9\nindicate the robustness of results and the applicability of both direct numerical integration and MCMC to solve this\ntype of inference problems.\n4.2. Model Comparison\nIn this section, we show the results from the application of Bayesian model comparison to the three considered\ndamping mechanisms. The comparison is performed in two principal ways. First, we consider plausible values for the\nobserved damping rate and compute the marginal likelihoods according to Equation (5), and the Bayes factors given\nby Equation (9) for each model as a function of this observable. Then, the Bayes factors between the damping models\nare computed as a function of periods and damping times on a wide range of values and for a given observational\nuncertainty. Finally, the method is applied to a large sample of transverse loop oscillation data to determine in how\nmany of the real events evidence supporting either of the considered mechanisms is found.\nFigure 3a shows the results from the computation of marginal likelihoods for the three considered damping mecha-\nnisms, as a function of the damping ratio. Resonant absorption and wave leakage o\u000ber more plausible explanations for\nlow damping rate values, while the evidence for phase mixing attains larger values from low to intermediate values of\nthe damping rate. We note that for phase mixing, as the analytic expression for the damping ratio is a function of the\nwave period, its marginal likelihood are computed for particular values of the period. These results are determined for\na \fxed value of the uncertainty on the observed damping ratio. Figures 3b-d show the marginal likelihoods, separately\nfor each mechanism, for three di\u000berent values for the error on the measured damping ratio. In all three cases, increasing\nthe uncertainty produces a decrease on the maximum value of the marginal likelihood, as well as some spread out of\nthe distributions. The discussed marginal likelihoods are computed by direct numerical integration, using Equation\n(5). Just as with the inference in the previous section, we check that they can equally well be computed by means\nof Monte Carlo integration. For each damping ratio, we carry out the MCMC sampling of the global prior given by\nEquation (10), just as we sample the posterior for the inference. Then, we evaluate the likelihood function in Equation\n(12) over the sampled parameters, and \fnally, we average the results to obtain the marginal likelihood. This is shown\nin Figure 4, where a comparison between both methods to evaluate marginal likelihoods is presented. On the one hand,\nthis result con\frms the goodness of direct numerical integration. On the other, it demonstrates that the developed\nMonte Carlo integration method can be con\fdently used when the \frst approach will not be feasible, for example, in\ncases with a larger dimension of the parameter space.\nThe quantitative assessment of the relative performance between damping mechanisms is given by the computation\nof the Bayes factors in the one-to-one comparison between the three damping mechanisms. In the following, we use\nthe subscripts 0, 1, and 2, to identify resonant absorption, phase mixing, and wave leakage, respectively. Figure 5\nshows the distribution of Bayes factors as a function of the damping ratio corresponding to one-to-one comparisons\nbetween the three mechanisms. An error measurement on the damping ratio of 10% is assumed. In Figure 5a, we see\nthat very strong evidence for resonant absorption in front of phase mixing is obtained for the lowest and the largest\nconsidered values of the damping ratio, in between 0 and 0.5 and for r >9. The \frst interval corresponds to extremely\nstrong damping regimes in which the damping time is shorter than the oscillation period. The second to rather weak\ndamping regimes. For intermediate damping ratio values, the evidence favors phase mixing, with values for 2 lnBF 10\nin between 2 and 5. These Bayes factor values correspond to positive evidence for phase mixing in front of resonant\nabsorption, according to the levels of evidence in Table 1. For the comparison between resonant absorption and wave\nleakage, Figure 5b shows very strong evidence in favor of the resonant absorption again for the lowest values of the\ndamping ratio, followed by a decrease of BF02until both Bayes factors intersect. For intermediate and large values of\nthe damping ratio, the evidence is inconclusive since the magnitude of both Bayes factors is below 2. Finally, Figure 5c\nshows the confrontation between phase mixing and wave leakage, with results that are akin to those between resonant\nabsorption and phase mixing in Figure 5a. Wave leakage is the favored model for strong and weak damping regimes,\nwhile positive evidence for phase mixing is obtained in the intermediate damping regime, with r2(2\u00007).\nIt is customary to address the comparison between damping mechanism using data on periods and damping times\nin a scatter plot where these data are then \ftted to a straight line, according to the scaling law hypothesis (Ofman\n& Aschwanden 2002). We proceed di\u000berently and next evaluate the plausibility between damping mechanisms by\ncalculating the Bayes factor distributions in such a two-dimensional plane with the two observables of interest, period\nand damping time. Hence d=fP;\u001cdg. To do so we consider the damping ratios resulting for di\u000berent combinations\nof\u001cdandPon a wide range of values and a \fxed uncertainty. Figure 6 shows the obtained Bayes factor distributions.10 Montes-Sol \u0013\u0010s & Arregui\n0 2 4 6 8 10\nr0.00.20.40.60.81.0p(r|M)(a)resonant\nwave leakage\nphase mixing\n0 2 4 6 8 10\nr0.00.20.40.60.81.0p(r|M)(b) =10%\n=25%\n=50%\n0 2 4 6 8 10\nr0.000.050.100.150.20p(r|M)(c)=10%\n=25%\n=50%\n0 2 4 6 8 10\nr0.00.10.20.30.40.50.60.70.8p(r|M)(d) =10%\n=25%\n=50%\nFigure 3. (a) Marginal likelihoods computed using Equation (5) for the three considered models as a function of the observable\ndamping ratio for a \fxed value of \u001b= 10%. For phase mixing, three \fxed values of period (150 ;500;and 1250sfrom right to left)\nhave been taken. (b)-(d): Marginal likelihoods for each mechanism separately for three di\u000berent values of \u001b= 10%;25%;50%.\nFor each mechanism the same color as in panel (a) is applied.\n0 2 4 6 8 10\nr0.00.20.40.60.81.0p(r|M)\nresonant\nphase mixing\nleakage\nFigure 4. A comparison between the marginal likelihoods obtained by direct numerical integration (continuous lines) and the\nMonte Carlo integration (\flled circles) for the three damping mechanisms as a function of the observable damping ratio with\nuncertainty of 10%. For phase mixing, P= 150 s is \fxed.\nThe general conclusion is that the evidence for any of the considered models against an alternative depends on the\nparticular combination of observed periods and damping times.\nIn particular, for resonant absorption against phase mixing, Figure 6a shows that at the upper-left corner of this\npanel, corresponding to large values of the damping ratio, we obtain strong (purple) and very strong (gray) evidence inDamping mechanisms for transverse coronal waves 11\n0 2 4 6 8 10\nr20\n10\n010202ln(BFkj)\nBF01BF10(a)\n0 2 4 6 8 10\nr15\n10\n5\n0510152ln(BFkj)BF20\nBF02(b)\n0 2 4 6 8 10\nr20\n10\n010202ln(BFkj) BF12\nBF21(c)\nFigure 5. Bayes factors for the one-to-one comparison between resonant absorption, phase mixing, and wave leakage mech-\nanisms, here represented with the subscripts 0, 1, and 2 respectively, in the range r2[0:01;10] with uncertainty of 10%. For\nphase mixing, P=150 s is \fxed.\n200 400 600 800 1000 1200\nP (s)5001000150020002500300035004000d(s)\n(a) 2lnB01\n10\n6\n2\n2610\n200 400 600 800 1000 1200\nP (s)5001000150020002500300035004000d(s)\n(b) 2lnB02\n10\n6\n2\n2610\n200 400 600 800 1000 1200\nP (s)5001000150020002500300035004000d(s)\n(c) 2lnB12\n10\n6\n2\n2610\nFigure 6. Bayes factors in the one-to-one comparison between resonant absorption, phase mixing, and wave leakage mechanisms\nas a function of the observables period and damping time with uncertainties of 10% for each. The dashed lines indicate \u001cd=P.\nThe di\u000berent levels of evidence of Table 1 are indicated in the color bars. NW (yellow), PE (green/red), SE (blue/purple), VSE\n(white/gray).\nfavor of resonant absorption. The lower-right corner corresponding to low damping ratios shows very strong evidence\n(white) in favor of the phase mixing model. In the remaining regions di\u000berent colored bands denote positive (pink for\nresonant and green for phase mixing) or insigni\fcant evidence (yellow), depending on the particular combination of\nthe two observables.\nIn the comparison between resonant absorption and wave leakage, Figure 6b, most of the ( \u001cd;P)- plane is colored in\nyellow resulting in a lack of evidence for a particular model. Grey and purple regions (very strong and strong evidence)\nindicate the dominance of the resonant mechanism for the lowest values of the damping ratio and positive evidence\n(green for wave leakage and pink for resonant) is found in very narrow regions of the plane of observables.\nFinally, Figure 6c shows the results from the comparison between phase mixing and wave leakage. The white\n(very strong evidence) and the blue (strong evidence) regions are located at period and damping time combinations\ncorresponding to large damping ratios and point out to the dominance of the wave leakage model over phase mixing.\nFor low damping ratios, we \fnd very strong evidence (gray) in favor of phase mixing. The remaining combinations\nlead to positive (pink for phase mixing and green for wave leakage) or inconclusive evidence (yellow) depending on the\nobservables.\nIn the three panels of Figure 6, focusing in observations with values of the damping ratio larger than 1, which are\nlocated in the upper-left side of the dashed line, resonant absorption and wave leakage are more plausible than phase\nmixing but no distinction can be made to favor any of them. Phase mixing seems to be the most plausible mechanism\nin the intermediate region of the observable ( \u001cd;P)-plane (green color in Figure 6a and pink color in Figure 6c).\nUntil now, model comparison is pursued by considering hypothetical values for observed periods, damping times,\nand their measurement errors. The full potential of the method here presented is discernible when applied to real\nevents of damped transverse loop oscillations. For this reason, we consider a selection of 89 loop oscillation events for12 Montes-Sol \u0013\u0010s & Arregui\n1 2 3 4 5\nr4\n3\n2\n1\n01232lnB01\n(a)NWM PE>0 PE<0\n1 2 3 4 5\nr0.6\n0.4\n0.2\n0.00.20.40.62lnB02\n(b)NWM\n1 2 3 4 5\nr2\n1\n01232 lnB12\n(c)NWM PE>0 PE<0\nFigure 7. Representation of the Bayes factors computed for the 89 events selected from Verwichte et al. (2013) and Goddard\net al. (2016). The di\u000berent panels correspond to the three one-to-one comparisons between resonant absorption, phase mixing,\nand wave leakage, here represented with the subscripts 0, 1, and 2 respectively.\nwhich periods and damping times are listed in the databases presented by Verwichte et al. (2013) and Goddard et al.\n(2016), discarding events for which errors were not reported. The considered events and their oscillation properties\nare presented in Table 3 of Appendix A. They are sorted in increasing value for their corresponding damping ratio\nin order to locate the events easily according to results. We further include for each event the inferred parameters,\ncomputed following the methods described in Section 4.1.\nFigure 7 shows scatter plots that display Bayes factor and damping ratio values for the selected 89 events for each\none-to-one model comparison between our three damping mechanisms. The colors indicate the level of evidence, based\non the magnitude of the corresponding Bayes factor. The Bayes factor values are also listed in Table 3. The conspicuous\nresult is that the blue color, which corresponds to absence of evidence for any damping mechanism, dominates in all\nthree panels.\nFor the comparison between resonant absorption and phase mixing (left panel), in approximately 78% of the events\nwe have not su\u000ecient evidence to favor one model or the other (NWM). Some events (colored in red and yellow) show\npositive evidence for either resonant absorption or for phase mixing. The evidence is positive for resonant absorption\nin 8% of the events, with strong damping (low values of r) and positive evidence for phase mixing in about 14% of\nthe events, with damping ratio values in between 3 and 6. In the middle panel the analysis for resonant absorption\nvs. wave leakage is shown. In all events the Bayes factor and therefore the evidence is not strong enough to support\nany of the two mechanisms. Finally, the right panel shows the comparison between phase mixing and wave leakage.\nFor 79% of the events the evidence is inconclusive, with the events distributed among the full range of damping ratios.\nOnly for 3% of the events we obtain positive evidence in favor of wave leakage, for loop oscillations with very strong\ndamping, and for another 18% positive evidence in favor of phase mixing.\n5.SUMMARY AND CONCLUSIONS\nIn the last years, high resolution observations have enabled us to better characterize the damping of transverse\noscillations in coronal loops. More accurate estimates of periods and damping times are now obtained and the size\nof the databases has increased. However, the discussion about the mechanisms involved in the quick damping of the\nobserved oscillations remains latent. The possible mechanisms were proposed a few decades ago but no self-consistent\nmethod exists yet to assess which one better accounts for the observed damping phenomenon. The widespread approach\nhas been to resort to the use of scaling laws that would apparently be an intrinsic property of each considered theoretical\nmechanism. This has proven to be of little use since, physical properties of coronal loops being unknown and probably\ndi\u000berent, mechanisms such as resonant absorption are known to predict not only one but many scalings with di\u000berent\npower indexes.\nIn this paper, we have presented a method to compute relative probabilities between alternative damping mechanisms\nfor transverse coronal loop oscillations. We considered three mechanisms among those that have been proposed:\nresonant absorption, phase mixing of Alfv\u0013 en waves, and wave leakage. They all are in principle likely to occur because\nof the highly inhomogeneous nature of the corona across the oscillating waveguides. Their direct applicability to\nthe damping of transverse loop oscillations is di\u000berent, with phase-mixing presenting serious limitations. Together\nwith model comparison, the coronal loop physical parameters that characterize each mechanism have been inferred.Damping mechanisms for transverse coronal waves 13\nThe analysis was carried out using Bayesian analysis tools. Bayesian inference enabled us to obtain probability\ndensity functions for the parameters of interest, with correct propagation of uncertainty from observables to physical\nparameters. The computation of marginal likelihoods informs us on the likelihood of each mechanism to generate the\nobserved data. Finally, Bayes factors quantify the level of evidence for one mechanism against another alternative.\nOur inference results indicate that physical parameters such as the density contrast, the transverse density inhomo-\ngeneity length-scales, and the aspect ratio of coronal loops can be properly inferred. Considering typically observed\ndamping ratio values, the obtained distributions peak at reasonable values of the unknown parameters. Our model\ncomparison results indicate that, as a general rule, a single damping mechanism cannot explain the observed damping\nof coronal loop oscillations. However, the method enables us to assign a level of evidence to each considered damping\nmodel. Considering hypothetical observed damping ratios over a range of plausible values, we found that resonant\nabsorption and wave leakage o\u000ber the most probable explanation in strong damping regimes, while phase mixing is\nthe best candidate for moderate/weak damping. The method was then applied to a large selection of loop oscillation\nevents compiled in databases that provide accurate measurements of period and damping time uncertainties. A fre-\nquentist analysis of the obtained Bayes factors indicates that only in very few cases the evidence is large enough to\nsupport a particular damping mechanism. For the rest of the cases, nothing can be stated. Since the uncertainty on\nthe measured times-scales is essential when translated into levels of evidence, the future increase in the precision of\ndata becomes relevant in order to determine the associated damping processes.\nThe results here presented have not given a de\fnitive answer to the question of what mechanism is responsible for\nthe quick damping of coronal loop oscillations, but the method makes use of all the available information - models,\nobserved data with their uncertainty, and prior information - in a consistent manner. The method is applicable to\nadditional damping models and formulas. Approximate answers have been obtained by considering the pertinent\nquestion of quantifying the evidence for each model in view of data, rather than obtaining exact answers to the more\nmisleading question of how well the data \ft to questionable theoretical power laws.\nWe acknowledge \fnancial support from the Spanish Ministry of Economy and Competitiveness (MINECO) through\nprojects AYA2014-55456-P (Bayesian Analysis of the Solar Corona), AYA2014-60476-P (Solar Magnetometry in the\nEra of Large Telescopes), and from FEDER funds. M.M-S. acknowledges \fnancial support through a Severo Ochoa\nFPI Fellowship under the project SEV-2011-0187-03. I.A. acknowledges \fnancial support through a Ram\u0013 on y Cajal\nFellowship.\nSoftware: emcee (Foreman-Mackey et al. 2013)\nAPPENDIX\nA.ANALYSED CORONAL LOOP DATA14 Montes-Sol \u0013\u0010s & ArreguiTable 3 . Transverse loop oscillations data, inference results and Bayes factors.\nRef Ev.ID Lp.ID P \u001cd r l/R \u0010 wR=L 2lnBF 01 2lnBF 02 2lnBF 12\n(s) (s) (Mm)\n1 24 \u0001 \u0001 \u0001 895\u00062 521 \u00068 0:6\u00060:0 1:5+0:3\n\u00000:16:5+2:4\n\u00002:21:6+0:1\n\u00000:00:27+0:00\n\u00000:003:45\u00030:48 \u00002:97\u0003\n2 10 1 687 :6\u000610:2 481:2\u000665:4 0:7\u00060:1 1:4+0:3\n\u00000:26:1+2:6\n\u00002:31:9+0:4\n\u00000:30:25+0:02\n\u00000:022:80\u00030:56 \u00002:24\u0003\n2 56 7 849 :6\u000633:0 818:4\u0006235:8 1:0\u00060:3 1:2+0:5\n\u00000:35:8+2:8\n\u00002:63:4+1:5\n\u00001:30:22+0:04\n\u00000:031:43 0:56 \u00000:87\n2 48 3 964 :8\u000612:6 945:6\u0006185:4 1:0\u00060:2 1:1+0:4\n\u00000:35:5+3:0\n\u00002:53:6+1:1\n\u00001:00:21+0:03\n\u00000:021:21 0:56 \u00000:65\n2 9 1 308 :4\u000610:2 305:4\u000658:8 1:0\u00060:2 1:1+0:4\n\u00000:35:5+3:0\n\u00002:52:2+0:6\n\u00000:60:21+0:03\n\u00000:022:31\u00030:56 \u00001:75\n1 40 \u0001 \u0001 \u0001 302\u000614 306 \u000643 1:0\u00060:1 1:0+0:4\n\u00000:25:3+3:1\n\u00002:52:2+0:5\n\u00000:40:21+0:02\n\u00000:022:18\u00030:56 \u00001:62\n1 44 \u0001 \u0001 \u0001 1170\u00066 1218 \u000648 1:0\u00060:0 0:9+0:5\n\u00000:15:2+3:2\n\u00002:54:2+0:3\n\u00000:20:20+0:00\n\u00000:000:62 0:55 \u00000:07\n1 25 \u0001 \u0001 \u0001 452\u00061 473 \u00066 1:0\u00060:0 0:9+0:5\n\u00000:15:2+3:2\n\u00002:52:8+0:0\n\u00000:10:20+0:00\n\u00000:001:57 0:55 \u00001:02\n2 43 1 428 :4\u00064:2 451:8\u000687:0 1:1\u00060:2 1:0+0:5\n\u00000:25:4+3:1\n\u00002:52:8+0:8\n\u00000:70:21+0:03\n\u00000:021:72 0:55 \u00001:18\n1 32 \u0001 \u0001 \u0001 216\u000627 230 \u000623 1:1\u00060:2 1:0+0:4\n\u00000:25:3+3:2\n\u00002:52:0+0:5\n\u00000:40:20+0:02\n\u00000:022:32\u00030:55 \u00001:78\n1 29 \u0001 \u0001 \u0001 225\u000640 240 \u000645 1:1\u00060:3 1:1+0:5\n\u00000:35:6+3:0\n\u00002:62:1+0:8\n\u00000:70:21+0:04\n\u00000:022:40\u00030:54 \u00001:85\n2 3 2 217 :2\u00064:8 247:2\u000628:2 1:1\u00060:1 0:9+0:4\n\u00000:25:1+3:3\n\u00002:52:2+0:4\n\u00000:30:19+0:01\n\u00000:011:97 0:52 \u00001:45\n2 16 2 141 :0\u00064:2 161:4\u000638:4 1:1\u00060:3 1:0+0:5\n\u00000:35:4+3:1\n\u00002:61:9+0:6\n\u00000:60:20+0:03\n\u00000:022:57\u00030:53 \u00002:04\u0003\n2 49 5 481 :8\u000610:8 562:2\u000673:2 1:2\u00060:2 0:9+0:5\n\u00000:25:1+3:3\n\u00002:53:3+0:6\n\u00000:60:19+0:02\n\u00000:011:09 0:51 \u00000:57\n1 31 \u0001 \u0001 \u0001 213\u00069 251 \u000636 1:2\u00060:2 0:9+0:5\n\u00000:25:1+3:3\n\u00002:52:3+0:5\n\u00000:50:19+0:02\n\u00000:011:88 0:51 \u00001:37\n1 41 \u0001 \u0001 \u0001 565\u00064 666 \u000642 1:2\u00060:1 0:8+0:5\n\u00000:15:1+3:3\n\u00002:53:6+0:3\n\u00000:40:19+0:01\n\u00000:010:82 0:50 \u00000:31\n2 23 1 921 :6\u000624:0 1151:4\u000693:0 1:2\u00060:1 0:8+0:5\n\u00000:15:0+3:3\n\u00002:55:0+0:6\n\u00000:60:18+0:01\n\u00000:010:09 0:48 0:39\n2 18 2 571 :2\u00066:6 732:0\u0006208:2 1:3\u00060:4 1:0+0:5\n\u00000:35:4+3:1\n\u00002:64:3+1:9\n\u00001:60:19+0:04\n\u00000:030:76 0:50 \u00000:26\n1 34 \u0001 \u0001 \u0001 596\u000650 771 \u0006336 1:3\u00060:6 1:1+0:6\n\u00000:45:7+2:9\n\u00002:84:9+2:9\n\u00002:40:20+0:06\n\u00000:040:71 0:51 \u00000:20\n2 44 3 417 :0\u00068:4 540:0\u0006180:0 1:3\u00060:4 1:0+0:6\n\u00000:45:5+3:1\n\u00002:73:8+1:9\n\u00001:60:20+0:05\n\u00000:031:08 0:50 \u00000:57\n2 9 2 537 :0\u00068:4 709:8\u0006285:6 1:3\u00060:5 1:1+0:6\n\u00000:45:6+3:0\n\u00002:84:7+2:6\n\u00002:20:20+0:05\n\u00000:040:76 0:51 \u00000:25\n2 40 3 331 :8\u00062:4 439:2\u000664:8 1:3\u00060:2 0:8+0:5\n\u00000:25:0+3:3\n\u00002:53:3+0:7\n\u00000:70:18+0:02\n\u00000:010:92 0:46 \u00000:46\n1 30 \u0001 \u0001 \u0001 215\u00065 293 \u000618 1:4\u00060:1 0:7+0:5\n\u00000:14:9+3:4\n\u00002:52:8+0:3\n\u00000:20:18+0:01\n\u00000:001:14 0:44 \u00000:70\n1 35 \u0001 \u0001 \u0001 212\u000620 298 \u000630 1:4\u00060:2 0:8+0:5\n\u00000:24:9+3:4\n\u00002:52:9+0:6\n\u00000:60:18+0:01\n\u00000:011:09 0:43 \u00000:66\n1 33 \u0001 \u0001 \u0001 520\u00065 735 \u000653 1:4\u00060:1 0:7+0:5\n\u00000:14:8+3:4\n\u00002:54:5+0:5\n\u00000:50:17+0:01\n\u00000:010:09 0:42 0:32\n2 48 1 916 :8\u00069:6 1318:8\u0006936:0 1:4\u00061:0 1:1+0:6\n\u00000:55:8+2:9\n\u00002:88:2+6:0\n\u00004:80:20+0:06\n\u00000:05\u00000:17 0:50 0:67\n1 46 \u0001 \u0001 \u0001 150\u00065 216 \u000660 1:4\u00060:4 0:9+0:6\n\u00000:35:2+3:2\n\u00002:62:7+1:1\n\u00000:90:18+0:04\n\u00000:021:62 0:46 \u00001:16\n2 19 2 676 :2\u00067:2 993:0\u000686:4 1:5\u00060:1 0:7+0:5\n\u00000:14:8+3:5\n\u00002:55:4+0:8\n\u00000:70:17+0:01\n\u00000:01\u00000:33 0:40 0:73\n2 49 4 627 :0\u000610:2 922:8\u0006154:8 1:5\u00060:2 0:8+0:5\n\u00000:24:9+3:4\n\u00002:55:3+1:4\n\u00001:20:17+0:02\n\u00000:01\u00000:15 0:42 0:57\nTable 3 continued on next pageDamping mechanisms for transverse coronal waves 15Table 3 (continued)\nRef Ev.ID Lp.ID P \u001cd r l/R \u0010 wR=L 2lnBF 01 2lnBF 02 2lnBF 12\n(s) (s) (Mm)\n2 44 2 586 :8\u000611:4 877:2\u0006297:6 1:5\u00060:5 0:9+0:6\n\u00000:45:3+3:2\n\u00002:75:6+2:8\n\u00002:30:19+0:05\n\u00000:030:19 0:46 0:28\n1 27 \u0001 \u0001 \u0001 2418\u00065 3660 \u000680 1:5\u00060:0 0:7+0:5\n\u00000:14:8+3:5\n\u00002:510:6+0:4\n\u00000:30:17+0:00\n\u00000:00\u00001:78 0:38 2:15\u0003\n1 38 \u0001 \u0001 \u0001 115\u00062 1750 \u000630 1:5\u00060:3 0:7+0:5\n\u00000:24:8+3:4\n\u00002:52:5+0:6\n\u00000:60:17+0:02\n\u00000:021:40 0:40 \u00001:00\n2 24 1 1071 :6\u000618:0 1645:8\u0006255:6 1:5\u00060:2 0:7+0:5\n\u00000:24:8+3:5\n\u00002:57:4+1:8\n\u00001:60:17+0:02\n\u00000:01\u00000:90 0:39 1:29\n1 45 \u0001 \u0001 \u0001 623\u00064 960 \u000660 1:5\u00060:1 0:7+0:5\n\u00000:14:8+3:5\n\u00002:55:6+0:5\n\u00000:50:16+0:01\n\u00000:00\u00000:48 0:37 0:86\n2 25 1 307 :8\u00066:6 480:0\u0006300:0 1:6\u00061:0 1:1+0:6\n\u00000:55:7+2:9\n\u00002:85:4+4:1\n\u00002:90:20+0:06\n\u00000:050:64 0:48 \u00000:16\n2 1 1 205 :2\u00063:6 320:4\u000667:2 1:6\u00060:3 0:8+0:5\n\u00000:24:9+3:4\n\u00002:53:4+1:1\n\u00001:00:17+0:02\n\u00000:020:79 0:40 \u00000:39\n2 26 1 717 :0\u00067:8 1122:6\u0006270:0 1:6\u00060:4 0:8+0:5\n\u00000:25:0+3:3\n\u00002:66:4+2:3\n\u00002:10:17+0:03\n\u00000:02\u00000:39 0:41 0:81\n1 26 \u0001 \u0001 \u0001 630\u000630 1000 \u0006300 1:6\u00060:5 0:9+0:6\n\u00000:35:2+3:3\n\u00002:66:3+2:8\n\u00002:40:18+0:05\n\u00000:03\u00000:18 0:43 0:61\n2 56 2 712 :8\u00067:8 1177:2\u0006177:6 1:7\u00060:2 0:7+0:5\n\u00000:24:8+3:5\n\u00002:56:7+1:6\n\u00001:40:16+0:02\n\u00000:01\u00000:83 0:35 1:18\n2 34 1 597 :0\u000616:2 1002:0\u000661:8 1:7\u00060:1 0:6+0:5\n\u00000:14:7+3:5\n\u00002:56:2+0:7\n\u00000:60:16+0:01\n\u00000:00\u00000:83 0:32 1:15\n2 48 2 945 :6\u00067:2 1598:4\u0006130:2 1:7\u00060:1 0:6+0:5\n\u00000:14:7+3:5\n\u00002:57:8+1:0\n\u00000:90:16+0:01\n\u00000:01\u00001:31 0:32 1:63\n1 50 \u0001 \u0001 \u0001 491\u000618 834 \u00066 1:7\u00060:1 0:6+0:5\n\u00000:14:6+3:6\n\u00002:55:7+0:4\n\u00000:30:16+0:00\n\u00000:00\u00000:70 0:31 1:02\n2 24 3 1227 :6\u000634:8 2100:6\u0006386:4 1:7\u00060:3 0:7+0:5\n\u00000:24:8+3:5\n\u00002:59:3+2:6\n\u00002:40:16+0:02\n\u00000:02\u00001:46 0:35 1:81\n1 48 \u0001 \u0001 \u0001 273\u000654 468 \u000636 1:7\u00060:4 0:7+0:5\n\u00000:24:8+3:5\n\u00002:54:5+1:4\n\u00001:30:16+0:02\n\u00000:020:09 0:36 0:26\n1 36 \u0001 \u0001 \u0001 256\u000622 444 \u0006105 1:7\u00060:4 0:7+0:6\n\u00000:24:9+3:4\n\u00002:64:5+1:7\n\u00001:50:16+0:03\n\u00000:020:22 0:37 0:16\n2 56 5 810 :0\u00069:6 1450:2\u0006307:8 1:8\u00060:4 0:7+0:5\n\u00000:24:8+3:5\n\u00002:58:1+2:6\n\u00002:30:16+0:02\n\u00000:02\u00001:19 0:34 1:52\n2 43 3 501 :0\u00064:8 902:4\u0006108:6 1:8\u00060:2 0:6+0:5\n\u00000:14:6+3:6\n\u00002:56:4+1:2\n\u00001:10:15+0:01\n\u00000:01\u00000:92 0:29 1:22\n2 31 2 575 :4\u00065:4 1054:2\u0006141:0 1:8\u00060:2 0:6+0:5\n\u00000:14:6+3:6\n\u00002:57:0+1:5\n\u00001:30:15+0:01\n\u00000:01\u00001:12 0:29 1:41\n1 42 \u0001 \u0001 \u0001 222\u000618 420 \u0006360 1:9\u00061:6 1:1+0:6\n\u00000:65:7+2:9\n\u00002:97:2+5:9\n\u00004:30:19+0:07\n\u00000:060:21 0:44 0:23\n1 43 \u0001 \u0001 \u0001 474\u000612 900 \u0006120 1:9\u00060:3 0:6+0:5\n\u00000:14:6+3:6\n\u00002:56:8+1:4\n\u00001:40:15+0:01\n\u00000:01\u00001:09 0:27 1:36\n2 40 8 259 :8\u00064:8 540:6\u0006129:6 2:1\u00060:5 0:6+0:5\n\u00000:24:7+3:6\n\u00002:55:9+2:2\n\u00001:90:15+0:03\n\u00000:02\u00000:65 0:27 0:92\n2 29 1 222 :6\u00063:0 469:8\u000637:2 2:1\u00060:2 0:5+0:5\n\u00000:14:4+3:7\n\u00002:55:4+0:6\n\u00000:70:14+0:01\n\u00000:01\u00000:89 0:19 1:07\n2 40 9 370 :8\u00063:0 789:0\u0006159:6 2:1\u00060:4 0:6+0:5\n\u00000:24:6+3:6\n\u00002:57:2+2:2\n\u00002:00:14+0:02\n\u00000:01\u00001:22 0:23 1:45\n2 4 2 208 :2\u00061:8 446:4\u000660:0 2:1\u00060:3 0:5+0:5\n\u00000:14:5+3:7\n\u00002:55:4+1:1\n\u00001:10:14+0:01\n\u00000:01\u00000:83 0:19 1:02\n1 49 \u0001 \u0001 \u0001 282\u00066 606 \u0006186 2:1\u00060:7 0:7+0:6\n\u00000:34:9+3:4\n\u00002:76:6+3:0\n\u00002:50:15+0:04\n\u00000:02\u00000:65 0:30 0:95\n2 44 1 433 :8\u00063:6 945:0\u0006185:4 2:2\u00060:4 0:6+0:5\n\u00000:24:5+3:6\n\u00002:58:0+2:4\n\u00002:20:14+0:02\n\u00000:01\u00001:50 0:22 1:72\n2 56 1 544 :2\u00068:4 1242:6\u0006282:6 2:3\u00060:5 0:6+0:5\n\u00000:24:6+3:6\n\u00002:59:7+3:3\n\u00002:90:14+0:02\n\u00000:02\u00001:85 0:21 2:06\u0003\n1 37 \u0001 \u0001 \u0001 135\u00069 311 \u000685 2:3\u00060:6 0:6+0:6\n\u00000:24:7+3:5\n\u00002:65:0+2:2\n\u00001:80:15+0:04\n\u00000:02\u00000:30 0:25 0:55\n1 39 \u0001 \u0001 \u0001 103\u00068 242 \u0006114 2:3\u00061:1 0:9+0:7\n\u00000:45:3+3:2\n\u00002:85:1+3:3\n\u00002:50:17+0:07\n\u00000:040:37 0:35 \u00000:02\n2 40 7 343 :2\u00063:6 850:2\u0006163:8 2:5\u00060:5 0:5+0:5\n\u00000:14:4+3:7\n\u00002:58:6+2:4\n\u00002:30:13+0:02\n\u00000:01\u00001:87 0:13 2:00\u0003\nTable 3 continued on next page16 Montes-Sol \u0013\u0010s & ArreguiTable 3 (continued)\nRef Ev.ID Lp.ID P \u001cd r l/R \u0010 wR=L 2lnBF 01 2lnBF 02 2lnBF 12\n(s) (s) (Mm)\n1 51 \u0001 \u0001 \u0001 348\u00067 906 \u0006288 2:6\u00060:8 0:6+0:7\n\u00000:34:8+3:5\n\u00002:79:7+4:4\n\u00003:80:14+0:05\n\u00000:02\u00001:61 0:22 1:83\n2 1 2 246 :6\u00063:0 645:6\u0006167:4 2:6\u00060:7 0:5+0:6\n\u00000:24:5+3:6\n\u00002:58:1+3:1\n\u00002:70:13+0:03\n\u00000:02\u00001:57 0:16 1:72\n2 43 2 216 :0\u00061:8 566:4\u000655:2 2:6\u00060:3 0:4+0:5\n\u00000:14:3+3:8\n\u00002:47:3+1:1\n\u00001:00:13+0:01\n\u00000:00\u00001:85 0:05 1:90\n2 8 1 224 :4\u00064:2 600:0\u000660:0 2:7\u00060:3 0:4+0:5\n\u00000:14:3+3:8\n\u00002:47:7+1:2\n\u00001:10:13+0:01\n\u00000:01\u00001:97 0:04 2:01\u0003\n1 52 \u0001 \u0001 \u0001 340\u00063 930 \u0006144 2:7\u00060:4 0:4+0:5\n\u00000:14:3+3:8\n\u00002:49:8+2:4\n\u00002:10:13+0:01\n\u00000:01\u00002:41\u00030:05 2:46\u0003\n1 13 \u0001 \u0001 \u0001 448\u000618 1260 \u0006500 2:8\u00061:1 0:7+0:7\n\u00000:45:0+3:3\n\u00002:811:6+5:2\n\u00005:20:15+0:07\n\u00000:03\u00001:58 0:25 1:83\n1 47 \u0001 \u0001 \u0001 122\u00066 348 \u0006360 2:9\u00063:0 1:0+0:7\n\u00000:65:6+3:0\n\u00002:99:1+6:7\n\u00005:60:18+0:08\n\u00000:07\u00000:21 0:36 0:57\n1 16 \u0001 \u0001 \u0001 358\u000630 1030 \u0006570 2:9\u00061:6 0:9+0:7\n\u00000:55:3+3:2\n\u00002:910:9+5:8\n\u00005:80:16+0:08\n\u00000:05\u00000:99 0:32 1:31\n2 38 2 312 :0\u00064:8 913:8\u0006330:0 2:9\u00061:1 0:7+0:7\n\u00000:34:8+3:5\n\u00002:710:8+5:0\n\u00004:60:14+0:06\n\u00000:03\u00001:72 0:20 1:92\n1 17 \u0001 \u0001 \u0001 326\u000645 980 \u0006400 3:0\u00061:3 0:7+0:8\n\u00000:45:1+3:3\n\u00002:811:2+5:3\n\u00005:30:15+0:08\n\u00000:03\u00001:46 0:25 1:71\n2 20 1 321 :6\u000613:8 971:4\u0006460:2 3:0\u00061:4 0:8+0:7\n\u00000:45:2+3:3\n\u00002:811:1+5:5\n\u00005:50:15+0:08\n\u00000:04\u00001:28 0:27 1:55\n2 43 5 270 :0\u00061:2 840:0\u0006120:0 3:1\u00060:4 0:4+0:5\n\u00000:14:2+3:8\n\u00002:410:6+2:3\n\u00002:20:12+0:01\n\u00000:01\u00002:80\u0003\u00000:05 2:76\u0003\n2 4 1 137 :4\u00061:8 430:8\u000690:0 3:1\u00060:7 0:4+0:5\n\u00000:14:3+3:8\n\u00002:57:8+2:4\n\u00002:30:12+0:02\n\u00000:01\u00002:00 \u00000:01 2:00\n2 45 1 148 :8\u00062:4 469:2\u000699:6 3:2\u00060:7 0:4+0:5\n\u00000:14:3+3:8\n\u00002:58:2+2:5\n\u00002:40:12+0:02\n\u00000:01\u00002:10\u0003\u00000:01 2:09\u0003\n2 31 1 460 :2\u00062:4 1453:2\u0006121:2 3:2\u00060:3 0:4+0:5\n\u00000:14:1+3:9\n\u00002:414:0+1:8\n\u00001:80:12+0:00\n\u00000:00\u00003:49\u0003\u00000:08 3:41\u0003\n2 11 3 156 :0\u00063:0 530:4\u000690:0 3:4\u00060:6 0:4+0:5\n\u00000:14:1+3:9\n\u00002:49:3+2:4\n\u00002:30:11+0:01\n\u00000:01\u00002:61\u0003\u00000:09 2:52\u0003\n1 15 \u0001 \u0001 \u0001 382\u000612 1330 \u0006528 3:5\u00061:4 0:7+0:8\n\u00000:34:9+3:4\n\u00002:812:8+4:8\n\u00005:70:13+0:07\n\u00000:03\u00001:81 0:16 1:97\n2 3 1 147 :6\u00061:8 528:0\u0006108:0 3:6\u00060:7 0:4+0:5\n\u00000:14:1+3:9\n\u00002:49:8+3:0\n\u00002:70:11+0:02\n\u00000:01\u00002:70\u0003\u00000:10 2:60\u0003\n2 32 1 256 :8\u00061:2 933:0\u000673:2 3:6\u00060:3 0:3+0:5\n\u00000:14:0+3:9\n\u00002:412:8+1:6\n\u00001:50:11+0:00\n\u00000:00\u00003:57\u0003\u00000:18 3:39\u0003\n1 12 \u0001 \u0001 \u0001 249\u000633 920 \u0006360 3:7\u00061:5 0:7+0:8\n\u00000:44:9+3:5\n\u00002:812:2+5:1\n\u00005:60:13+0:08\n\u00000:03\u00001:81 0:15 1:96\n1 18 \u0001 \u0001 \u0001 357\u000689 1320 \u0006720 3:7\u00062:2 0:9+0:7\n\u00000:55:3+3:2\n\u00002:911:9+5:6\n\u00006:40:16+0:09\n\u00000:05\u00001:01 0:26 1:27\n2 54 5 288 :0\u00066:0 1183:2\u0006193:8 4:1\u00060:7 0:3+0:5\n\u00000:14:0+3:9\n\u00002:415:6+2:8\n\u00003:30:10+0:01\n\u00000:01\u00003:68\u0003\u00000:23 3:45\u0003\n2 7 1 101 :4\u00061:2 433:8\u000678:0 4:3\u00060:8 0:3+0:5\n\u00000:14:0+3:9\n\u00002:410:6+2:8\n\u00002:60:10+0:01\n\u00000:01\u00003:24\u0003\u00000:25 2:99\u0003\n2 40 2 336 :6\u00061:8 1489:8\u0006204:6 4:4\u00060:6 0:3+0:5\n\u00000:13:9+4:0\n\u00002:317:4+2:0\n\u00002:70:10+0:01\n\u00000:01\u00003:35\u0003\u00000:30 3:06\u0003\n1 14 \u0001 \u0001 \u0001 392\u000631 1830 \u0006790 4:7\u00062:0 0:6+0:8\n\u00000:44:9+3:5\n\u00002:813:5+4:6\n\u00006:30:13+0:09\n\u00000:03\u00001:45 0:09 1:53\n2 17 1 124 :2\u00062:4 599:4\u0006275:4 4:8\u00062:2 0:7+0:8\n\u00000:45:0+3:4\n\u00002:912:2+5:2\n\u00005:90:13+0:09\n\u00000:04\u00001:78 0:10 1:88\n1 22 \u0001 \u0001 \u0001 436\u00064:5 2129 \u0006280 4:9\u00060:6 0:2+0:5\n\u00000:13:9+4:0\n\u00002:318:4+1:2\n\u00002:30:09+0:01\n\u00000:00\u00000:88 \u00000:37 0:51\n1 23 \u0001 \u0001 \u0001 243\u00066:4 1200 4 :9\u00060:1 0:2+0:5\n\u00000:03:8+4:1\n\u00002:319:4+0:5\n\u00000:60:09+0:00\n\u00000:00\u00004:20\u0003\u00000:38 3:82\u0003\n2 32 2 202 :8\u00061:2 1146:6\u0006291:0 5:7\u00061:4 0:3+0:5\n\u00000:13:9+4:0\n\u00002:415:9+2:9\n\u00004:50:09+0:02\n\u00000:01\u00002:89\u0003\u00000:35 2:53\u0003\nTable 3 continued on next pageDamping mechanisms for transverse coronal waves 17Table 3 (continued)\nRef Ev.ID Lp.ID P \u001cd r l/R \u0010 wR=L 2lnBF 01 2lnBF 02 2lnBF 12\n(s) (s) (Mm)\nNote |Columns present the reference (Ref), the event ID (Ev.ID) and loop ID (Lp.ID) from references, the observed period (P), the\ndamping time ( \u001cd), damping rate (r), the parameter inferred median values of the three selected theoretical models ( \u0010, l/R, w, R/L) with\ntheir uncertainties, and Bayes factor. 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S., & Verwichte, E. 2012, A&A, 537, A49"    },    {        "title": "0805.1224v2.The_Lorentz_group_and_its_finite_field_analogues__local_isomorphism_and_approximation.pdf",        "content": "arXiv:0805.1224v2  [math-ph]  4 Aug 2008The Lorentz group and its finite field analogues:\nlocal isomorphism and approximation\nStephan Foldes\nTampere University of Technology\nPL 553, 33101 Tampere, Finland\nsf@tut.fi\n4 August 2008\nAbstract\nFinite Lorentz groups acting on 4-dimensional vector space s coordinatized\nby finite fields with a prime number of elements are represente d as homomor-\nphic images of countable, rational subgroups of the Lorentz group acting on\nreal 4-dimensional space-time. Bounded subsets of the real Lorentz group are\nretractable with arbitraryprecisionto finitesubsets ofsu ch rational subgroups.\nThese finite retracts correspond, via local isomorphisms, t o well-behaved sub-\nsets of Lorentz groups over finite fields. This establishes a r elationship of\napproximation between the real Lorentz group and Lorentz gr oups over very\nlarge finite fields.\n1 Finite Minkowski spaces and statement of the theorem\nThe purpose of this paper is to establish a relationship of approximat ion\nbetween Lorentz transformations of Minkowski space R4and Lorentz trans-\nformations of 4-dimensional spaces over very large finite fields, in t he precise\nsense of Theorem 1 below. In relation to theoretical alternatives in particle\nphysics, Lorentztransformationsover finite fieldsseem to haveb een first con-\nsidered by Coish in [C], then by Shapiro [S], Ahmavaara [A1, A2], Yahia [Y],\nJoos [Jo], Beltrametti and Blasi [BB1], and Nambu [N]. Much of the algeb ra\nofLorentzgroupsoverfinitefieldshadalreadybeendevelopedbyD ickson[D],\nandinthe context ofphysical applications thistheory wasbrought uptodate\nby Beltrametti and Blasi [BB2]. A finite analogue of the Alexandrov-Z eeman\ncharacterization of the Lorentz group via optical causality was es tablished\n1by Blasi, Gallone, Zecca and Gorini [BGZG]. The approximation describe d\nin Theorem 1 relies on the one hand on modifying a relationship of appro xi-\nmation between positive real numbers and quadratic residues modu lo a large\nprime established by Kustaanheimo [JK, K1, K2], and modifying J¨ arne felt’s\napproximation of collinear ranges of points in Euclidean space by colline ar\nranges in finite geometries [J¨ a]. The proof of Theorem 1 also makes e ssen-\ntial use of rotation-boost decomposition, and of representing Lo rentz groups\nover finite fields as quotients of appropriate subgroups of the rea l Lorentz\ngroup. The algebraic background and terminology used in this theor em are\nas follows.\nWe consider fields Fin which −1 is not a square, and we shall assume\nthroughout that Fhas this property. (The main cases of interest are the\nfieldRof real numbers and the p-element finite fields Fpfor prime numbers\np≡7mod8, although many of the basic facts are also true for p-element\nfields with primes p≡3mod8.) For any such field Fthe 4-dimensional\nvector space F4is endowed with the ( F-valued)Minkowski norm\nµ(t,x,y,z) =t2−x2−y2−z2\nAproper Lorentz transformation over Fisalineartransformation Tfromthe\nvectorspace F4toitselfwhichpreserves theMinkowski norm( i.e.,µ(T(v)) =\nµ(v) for all v∈F4) and which has determinant 1. For any given field F,\nthese transformations form a group that we call the proper Lorentz group\noverF, denoted L+F. If the field Fis not specified, it is understood to be\nR.A Lorentz transformation over a field Fis said to be orthochronous if\nit maps (1 ,0,0,0) to a vector ( t,x,y,z) wheretis a non-zero square in F.\nForF=Rorthochronous Lorentz transformations form a subgroup of L+R\ncalled the orthochronous proper Lorentz group, denoted L↑\n+R.Over finite\nfields the composition of orthochronous transformations is not ne cessarily\northochronous.\nIfGandHare two (abstract) groups and A⊆G, Y⊆Hare arbitrary\nsets of group elements, a local isomorphism between AandYis a bijective\nmapσfromA∪AAtoY∪YY(whereAA={xy:x,y∈A}) such that\nσ[A] =Yand for all x,y∈A\nσ(xy) =σ(x)σ(y)\n2It follows that if 1 G∈Athenσ(1G) = 1Hand ifx,x−1∈Athenσ(x−1) =\nσ(x)−1.If such a local isomorphism exists, the sets AandYare said to be\nlocally isomorphic. (IfAandYare subgroups they are locally isomorphic\nif and only if they are isomorphic, but otherwise the sets AandYmay be\nlocally isomorphic even if the subgroups they generate are not isomo rphic.)\nBy thenorm/bardblT/bardblof any linear transformation T:R4−→R4with stan-\ndard matrix representation ( aij)1≤i,j≤4we mean (Σ a2\ni,j)1/2, i.e. the Euclidean\n(l2) norm of the matrix. By a retraction of L↑\n+Rto a subset A⊆ L↑\n+Rwe\nmean a map f:L↑\n+R−→Asuch that f(T) =Tfor allT∈A(i.e.,such\nthatf2=f).\nTheorem 1 For every ǫ >0andM >0the orthochronous proper\nLorentz group L↑\n+Rhas a retraction f:L↑\n+R−→Ato a finite subset\nA⊆ L↑\n+Rsatisfying\n/bardblT−f(T)/bardbl< ǫ\nfor all Lorentz transformations T∈ L↑\n+Rof norm not exceeding M,and such\nthat for some prime number p≡7mod8the subset Ais locally isomorphic to\na setYof orthochronous transformations in the proper Lorentz gro upL+Fp\nover the finite field Fp.\nThe proof is provided in Section 2, together with some elaborations o f the\nmathematical aspects that are needed for the sake of precision a nd clarity.\nThese are presented as a succession of intermediate propositions . A general-\nization to the extended Lorentz group is presented in Theorem 2 of Section 3,\nwith some additional properties of the local isomorphism linking the re spec-\ntive Lorentz groups over the real numbers and over finite fields. I n Section 4\nwe briefly comment on some of the particularities of finite analogues o f the\nLorentz group.\n2 Background developments and proof of Theorem 1\n2.1 Proper Lorentz groups over fields where -1 is not a square\nMuchofthebasictheoryoffiniteLorentzgroupsthatweneedisco ntained\nin or follows from the seminal work of Dickson [D] and the more recent study\nof Beltrametti and Blasi [BB2].\n3Lorentz groups are defined according to the active perspective a s groups\nof bijective transformations, and composition of transformation s is denoted\nsimply by juxtaposition T1T2, whereT2is the transformation first applied. If\nthe standard matrix representations of these transformations areT1andT2\nthen the matrix product T1T2represents the transformation T1T2.\nAproperLorentztransformation Tover afield Fwhere−1isnotasquare\nis called a space rotation (overF) if it fixes the first standard basis vector\n(1,0,0,0)∈F4. Equivalently, space rotations over Fare the linear trans-\nformations of F4with standard matrix representation of the block diagonal\nform /parenleftbigg1\nR3/parenrightbigg\nwhereR3is a 3-by-3 orthogonal matrix (i.e. R⊤\n3=R−1\n3) with determinant\n1. There are three space rotations which permute the standard b asis vectors,\nthey are given by\nR3=\n1\n1\n1\n,R3=\n1\n1\n1\n,R3=\n1\n1\n1\n\nThese particular space rotations are called rotations of standard space axes\nand they form a 3-element subgroup of L+F. All space rotations constitute\na subgroup of L+Fover any field F.If a space rotation fixes one of the\nthreestandard space unit vectors (0,1,0,0),(0,0,1,0) or (0,0,0,1) then it is\nsaid to be an elementary space rotation (aroundthat unit vector). For each\nof these three unit vectors, elementary space rotations around that vector\nconstitute a subgroup of the group of all space rotations. Aroun d (0,1,0,0)\nthis subgroup consists of the identity, the half-turn around (0,1,0,0) given\nby\n(t,x,y,z)/mapsto→(t,x,−y−z)\nand for each α/\\e}atio\\slash= 0 inFthe transformation Rαcalled abasic rotation whose\nstandard matrix representation is\n\n1\n1\nα−α−1\nα+α−12\nα+α−1\n−2\nα+α−1α−α−1\nα+α−1\n\n4Each of the two groups of elementary space rotations around (0 ,0,1,0) and\n(0,0,0,1) is obtained from the group of elementary space rotations aroun d\n(0,1,0,0) by conjugation with space axis rotations represented by appro pri-\nate permutation matrices. All space rotations are orthochronou s, over any\nfieldF.\nFor every non-zero square αinF(which means positive αinR, quadratic\nresidue class in Z/pZ∼=Fp) thebasic boost Bαis defined as the Lorentz\ntransformation over Fwith standard matrix representation\n\nα+α−1\n2α−α−1\n2\nα−α−1\n2α+α−1\n2\n1\n1\n\nFor any space rotation Randαany non-zero square in F, the conjugate\nRBαR−1is called a boost,and for any given space rotation Rthe set of\nboosts\n{RBαR−1:α/\\e}atio\\slash= 0, αsquare in F}\nis a subgroup of L+F. If the space rotation Ris a rotation of standard space\naxes, then RBαR−1issaidtobean elementary boost . Aboostiselementary if\nandonlyifitfixestwoofthethreebasisvectors(0 ,1,0,0),(0,0,1,0),(0,0,0,1).\nAll boostsareorthochronousover R,but boostsover a finitefield donot need\nto be orthochronous: consider over F7any of the two non-trivial boosts B2\nandB4.\nOver any field F, note that composing space-time reversal\n(t,x,y,z)/mapsto→(−t,−x,−y,−z)\nwith the half-turn around (0 ,1,0,0) yields reflection in the yz plane given by\n(t,x,y,z)/mapsto→(−t,−x,y,z).\nBasicbooststogetherwiththisreflectiongeneratethesubgroup ofL+Ffixing\n(0,0,1,0) and (0 ,0,0,1). (This subgroup is also referred to as the group of\nhyperbolic isometries of the txplane.)\nSince every space rotation in L+Ris the product of 3 elementary space\nrotations, elementary space rotations and basic boosts generat eL↑\n+R, and\n5together with reflection in the yzplane they generate L+R.On the other\nhand, a classical result on linear groups over finite fields due to Dicks on\n[D] states, when particularized to the case of 4-dimensional space s overFp\nfor primes p≡7mod8, that L+Fpis generated by elementary rotations,\nelementary boosts and reflections in the yzplane. Re-stated in a slightly\nstrengthened form, based on the observation that all elementar y boosts are\nconjugates of basic boosts by axis rotations which are in turn prod ucts of\nelementary space rotations, we have the following:\nRotation-Boost Lemma over Finite Fields (from Dickson [D]) For\nany prime number p≡7mod8,the proper Lorentz group L+Fpis generated\nby basic boosts, elementary space rotations and space-time reversal. /square\nIn fact the above lemma also holds for primes p≡3mod8 with the\nexception of p= 3 (see [D]) but this fact is not needed in what follows.\nUnlike over the real number field R, over finite fields not every proper\nLorentz transformation can be represented in the form RB(orBR) with\na rotation Rand a boost B.(Any such product of a rotation and a boost\ntransforms (1 ,0,0,0) to a vector ( t,x,y,z) wherex2+y2+z2is a square\nin the field of scalars. However, over finite fields not all proper Lore ntz\ntransformations, not even the orthochronous ones, have this p roperty.) For\nthe real case see Moretti’s study [M] for a comprehensive perspec tive on\nrotation-boost decomposition.\n2.2 Lorentz transformations over local rings\nWe represent the p-element field Fpas the quotient of the localization\nring\nZ(p)=Z·(Z/downslopepZ) ={st−1:s,t∈Z, t/\\e}atio\\slash= 0,(p,t) = 1}\nby its unique maximal ideal pZ(p), an approach also taken by Ahmavaara\n[A1]. There is a unique surjective ring homomorphism from the localizat ion\nringZ(p)onto the p-element field Fp, called the canonical map (orcanonical\nsurjection ) fromZ(p)toFp. (The restriction of the canonical map to Z⊆\nZ(p)is the unique surjective ring homomorphism Z−→Fpcorresponding to\nthe usual representation of Fpas a quotient of Z.) The canonical map from\nZ(p)toFpinducesa surjective ring homomorphism from the ring of 4-by-4\nmatrices with entries in Z(p)onto the ring of 4-by-4 matrices with entries in\n6Fpand thus inducesa map from any set of linear transformations of R4with\ncoefficients in Z(p)to the set of linear transformations of F4\np\nFor any integral domain Din which −1 is not a square, those D-module\nautomorphisms LofD4whose standard 4 ×4 matrix representation has\ndeterminant 1 and which leave invariant the ( D-valued) Minkowski norm\nµ(t,x,y,z) =t2−x2−y2−z2\narecalled Lorentz transformations over Dandtheyconstituteagroup, called\ntheproper Lorentz group over D, denoted L+D.A transformation LinL+D\nis said to be orthochronous if it maps (1 ,0,0,0) to a vector ( t,x,y,z) such\nthattis a non-zero square in D. The set L↑\n+Dof such orthochronous trans-\nformations may or may not be a subgroup of L+D. AD-module automor-\nphismLofD4having determinant 1 belongs to L+Dif and only if its stan-\ndard matrix representation Land the diagonal matrix Jwith main diagonal\n(1,−1,−1,−1) satisfy the equation LJL⊤=J.For any set Cof elements\nof the integral domain D,we denote by L+DC, respectively by L↑\n+DC,the\nset of those L∈ L+D,respectively L∈ L↑\n+D,for which all entries of the\nstandard matrix representation of Lare inC. IfCis a subring of D(con-\ntaining the unit 1 of D) thenL+DCis a subgroup of L+D. and restricting\ntransformations L∈ L+DCtoC4⊆D4yields a group isomorphism, called\ncanonical isomorphism,\nρ:L+DC−→ L +C\n(Theclosure of L+DCunder inversion isdue tothefact thatforall L∈ L+D\nwith matrix Lwe haveL−1=JL⊤J.) The isomorphism ρ:L+DC−→ L +C\nobviously maps orthochronoustransformationstoorthochrono ustransforma-\ntions,ρ[L↑\n+DC] =L↑\n+C,andL↑\n+DCandL↑\n+Care locally isomorphic subsets\neven if they are not subgroups of L+DCandL+C.They are isomorphic\nsubgroups, however, in the case of particular interest, which is D=Rand\nC=Z(p)(for any p≡3mod4).The following lemma provides a represen-\ntation of the proper Lorentz group over a finite field Fpas a quotient of the\nproper Lorentzgroupover the localizationring Z(p),or equivalently, asa quo-\ntient of the group of (real) proper Lorentz transformations with coefficients\ninZ(p).\nHomomorphism Lemma Letpbe any prime number congruent to 7\nmodulo8.The canonical map Z(p)−→Fpinduces a surjective group homo-\n7morphism L+Z(p)−→ L +Fp. Pre-composing it with the canonical isomor-\nphismρyields a surjective group homomorphism L+RZ(p)−→ L +Fp.\nProofIts is obvious that the map L+Z(p)−→ L +Fpinduced by the\ncanonical map Z(p)−→Fpis a group homomorphism. In L+Fpevery basic\nboost, every elementary rotation and also the space-time revers al transfor-\nmation is in the range of this homomorphism. Its surjectivity is then a con-\nsequence of the Rotation-Boost Lemma over Finite Fields. /square\nThe two homomorphisms onto L+Fpdefined by the above lemma will be\nalso referred to as canonical .\n2.3 Local isomorphisms\nThe next few lemmas are easily verified.\nInjection Lemma 1 Ifh:G1−→G2is a group homomorphism, A⊆\nG1and the restriction hAofhtoA∪AAis injective, then hAis a local iso-\nmorphismbetween Aandh[A]. /square\nAhmavaara [A1] seems to have been aware of the fact stated in Inj ection\nLemma 1, at least in the particular case where his the canonical surjection\nZ(p)−→Fpviewed as an additive group homomorphism, while apparently\ndisregarding (or treating differently) the case where his the restriction of\nthe same canonical map to Z∗\n(p)=Z(p)\\pZ(p)and viewed as a multiplicative\ngroup homomorphism onto F∗\np=Fp\\{0}.\nForeach positive integer kconsider theset Ckofrational numbers defined\nby\nCk={st−1:s,t∈Z, t/\\e}atio\\slash= 0,|s| ≤k,|t| ≤k}\nIfpis a prime number larger than k,thenCk⊆Z(p).\nInjection Lemma 2 For any prime number p≡3mod4letkbe a\npositive integer such that 2k2< p.Then the canonical map Z(p)−→Fpis\ninjective on Ckand the canonical map L+RZ(p)−→ L +Fpis injective on\nL+RCk./square\nInjection Lemma 3 For any prime number p≡3mod4letkbe a\npositive integer such that 32k16< p.LetA=L+RCk.Then the canonical\nhomomorphism L+RZ(p)−→ L +Fpis injective on A∪AA.\n8ProofA∪AA⊆ L+RC4k8and Injection Lemma 2. /square\nThese yield the following\nLocal Isomorphism Lemma For every positive integer k and prime\nnumber p ≡3mod4 such that p >32k16,the setL+RCkof proper Lorentz\ntransformations with coefficients in Ckis locally isomorphic to a subset of\nthe finite proper Lorentz group L+Fp.A local isomorphism is provided by\nrestricting the canonical homomorphism L+RZ(p)−→ L +FptoL+RCk./square\nUsing Dirichlet’s theorem on primes in arithmetic progressions, in [K1]\nKustaanheimo proved the following:\nKustaanheimo’s Chain Theorem [K1]For every positive integer k\nthere is a prime number p > k, p ≡7mod8,such that all the non-negative\nintegers up to kare quadratic residues modulo p. There are infinitely many\nsuch primes pfor any given k, they are necessarily larger than 2k./square\nFrom this we derive a key fact about the sets Ck, noting that these are\nobtained from {1,2,...,k}by symmetrization via the adjunction of negatives\nand reciprocals:\nSymmetrized Chain Lemma For every positive integer kthere is a\nprime number p > k, p ≡7mod8,such that all the positive numbers in\nCk⊆Z(p)are quotients a/bof quadratic residues a,bmodulop,i.e. all\npositive membersof Ckare mappedto non-zerosquares in Fpby the canonical\nsurjection Z(p)−→Fp.\nProofBy the Kustaanheimo Chain Theorem there is a prime number\np > k2, p≡7mod8,such that all the non-negative integers up to k2are\nquadratic residues modulo p. /square\n2.4 Density and conclusion of proof of the theorem\nBy thecoefficients of a linear transformation of Rnwe mean the entries\nof its standard matrix representation.\nIn this Section we deal mainly with the metric space structure on the\nset of all linear transformations of R4induced by the Euclidean norm of\nstandard matrix representations as briefly introduced in Section 1 . In this\n9metric space, the distance between two transformations T,Qis the norm of\ntheir difference, /bardblT−Q/bardbl.\nInL↑\n+=L↑\n+Rthe norm of any space rotation is 2 and the norm of the\nbasic boost Bαisα+α−1. IfTis any linear transformation of R4andR\nis any space rotation, then /bardblT/bardbl=/bardblRT/bardbl=/bardblTR/bardbl.The norm of a proper\northochronous Lorentz transformation T=RSBαSinL↑\n+, whereR,Sare\nspace rotations, is α+α−1≥2,and this is also the norm of the inverse\ntransformation T−1.\nIn any metric space, a set Aof elements is said to be ǫ-dense in a set\nof elements B(whereǫis any positive real) if for every b∈Bthere is an\na∈A∩Bat distance less than ǫfromb. Note that this does not require A\nto be a subset of B, butAisǫ-dense in Bif and only if A∩Bisǫ-dense in\nB.Note that if Aisǫ-dense in a set B, and also in a set C, thenAisǫ-dense\ninB∪C, but the converse implication generally fails. Density in the usual\nsense means ǫ-density for all ǫ >0.\nApproximation Lemma 1 For every positive integer kthe setCk2is\n(1/k)-dense in the real interval [−k,k].\nProofThis follows from the inclusion\n/braceleftbig\ni/k:i∈Z−k2≤i≤k2/bracerightbig\n⊆Ck2∩[−k,k]\n/square\nApproximation Lemma 2 Letǫ >0and let K be the union of a finite\nnumber of non-trivial compact real intervals. Then there is a positive integer\nl such that for all integers k≥lthe setCkisǫ-dense in K.\nProofApproximation Lemma 1. /square\nApproximation Lemma 3 Letǫ >0and letube any of the three\nstandard space unit vectors. Then there is a positive intege rlsuch that for\nall integers k≥lthe setL+RCkisǫ-dense within the set of space rotations\naroundu.\nProofSuppose u= (0,1,0,0). In the other cases the proof follows by\nconjugation.\nThe identity transformation and the half-turn around uhave their coef-\nficients in Ck.\nThere is some M >0 such that for all basic rotations RαwithM < α\nthe distance between Rαand the identity is less than ǫ.\n10There is some 0 < µ < M such that for all basic rotations Rαwith\n0< α < µ the distance between Rαthe and the half-turn is less than ǫ.\nLetK= [−M,−µ]∪[µ,M].By uniform continuity of the map α/mapsto→Rα\non the compact set K, there is δ >0 such that for all α,γ∈K,|α−γ|< δ\nimplies/bardblRα−Rγ/bardbl< ǫ.\nBy Approximation Lemma 2, there is a positive integer lsuch that for all\nintegersk≥lthe setCkisδ-dense in K.\nLetk≥land letRαbe a basic space rotation with α∈K.Byδ-density\nthere is a γ∈Ck∩Ksuch that |α−γ|< δ. We have /bardblRα−Rγ/bardbl< ǫ.\n/square\nApproximation Lemma 4 Letǫ >0andM >0.There is a positive\nintegerlsuch that for all integers k≥lthe setL+RCkisǫ-dense within\nthe set of basic boosts of norm not exceeding M.\nProofThere is some 0 < µ < M such that all basic boosts Bαwith\nα < µhave norm greater than M.\nLetK= [µ∪M].By uniform continuity of the map α/mapsto→Bαon the\ncompact interval K, there is δ >0 such that for all α,γ∈K,|α−γ|< δ\nimplies/bardblBα−Bγ/bardbl< ǫ.\nBy Approximation Lemma 2, there is a positive integer lsuch that for all\nintegersk≥lthe setCkisδ-dense in K.\nLetk≥land letBαbe a basic boost with α∈K.Byδ-density\nthere is a γ∈Ck∩Ksuch that |α−γ|< δ. We have /bardblBα−Bγ/bardbl< ǫ.\n/square\nThe above lemmas combine to yield the following\nApproximation Lemma 5 Letǫ >0andM >0.There is a positive\nintegerlsuch that for all integers k≥lthe setL+RCkisǫ-dense within\neach of the three elementary space rotation groups around th e standard space\nunit vectors, and also within the set of basic boosts of norm n ot exceeding M.\n/square\nApproximation Lemma 6 Letǫ >0andM >0.There is a positive\nintegerlsuch that for all integers k≥lthe setL+RCkisǫ-dense within the\nset of orthochronous proper Lorentz transformations of nor m not exceeding\nM.\n11ProofBy a compactness and uniform continuity argument, there is a\nδ >0 such that for all linear transformations\nT1,...,T10,Q1,...,Q10\nofR4of norm not exceeding M,if/bardblTi−Qi/bardbl< δfor all 1≤i≤10, then we\nhave for the distance of products\n/bardblT1...T10−Q1...Q10/bardbl< ǫ\nBy Approximation Lemma 5, there is a positive integer lsuch that for all\nintegersk≥lthe setLRCkisδ-dense within each of the three elementary\nspacerotationgroupsaroundthestandardspaceunitvectors, andalsowithin\nthe group of basic boosts of norm not exceeding M.\nLetTbeanyorthochronousproperLorentztransformation, T=RSBαS−1\nwhereR,Sare space rotations, and suppose that the norm α+α−1ofTis\nat mostM.We can factorize Ras a composition R1R2R3of three elemen-\ntary space rotations, and factorize also as Sas the composition R4R5R6of\nthree elementary space rotations. For convenience write R7=R−1\n6,R8=\nR−1\n5,R9=R−1\n4, so that\nT=R1...R6BαR7R8R9\nByδ-density, there are elementary space rotations R′\n1,...,R′\n9with coefficients\ninCkand a basic boost B′also inL+RCkand of norm at most M, such that\nfor each 0 ≤i≤9 the elementary rotations RiandR′\nihave a common fixed\nstandard space unit vector, and the distances\n/bardblR′\n1−R1/bardbl,...,/bardblR′\n9−R9/bardbl,/bardblB′−Bα/bardbl\narealllessthan δ.Thenthedistanceof Tfromthecomposition R′\n1...R′\n6BαR′\n7R′\n8R′\n9\nis at most ǫ./square\nConclusion of proof of Theorem 1 Givenǫ >0 andM >0,by\nApproximation Lemma 6 we can take a positive integer ksuch that the set\nL+RCkisǫ-dense within the set of orthochronous proper Lorentz transfo r-\nmations of norm at most M.\nBy the Symmetrized Chain Lemma, there is a prime p,\np >32k16> k, p≡7mod8\n12such that the canonical surjection Z(p)−→Fpmaps all of Ckinside the group\nof non-zero squares of Fp.Then the canonical surjection L+RZ(p)−→ L +Fp\nmaps all of L↑\n+RCkinsideL↑\n+Fp.LetA=L↑\n+RCk.\nDefinetheretraction f:L↑\n+R−→Abyassociating toeachorthochronous\nproper Lorentz transformation Ta member of Aat minimum distance from\nT.Then apply the Local Isomorphism Lemma of 2.3. /square\n3 Extended Lorentz groups and line reflections\nGivenǫ >0 andM >0 the retraction of the proper orthochronous\nLorentz group L↑\n+Rprovided by Theorem 1 can in fact be defined also on the\nother three connected components of the extended Lorentz gr oup, with the\nretractAbeinglocallyisomorphictoasubset Yoftheextended Lorentz group\nLFpof all bijective linear transformations of F4\nppreserving the Minkowski\nnorm. The local isomorphism between AandYcan also be seen to preserve\nseveral other properties of transformations besides the prope rty of being or-\nthochronous.\nAs overR, over any field F(where−1 is not a square) the proper Lorentz\ngroup is an index 2 subgroup of the extended group LF, and we denote by\nL−Fthe coset consisting of the transformations of determinant −1, called\nimproper transformations. Among these we have in particular the time re-\nversaltransformation τgiven by\nτ(t,x,y,z) = (−t,x,y,z)\nSimilarly toorthochronous transformationsin L+F,atransformationin L−F\nis also called o rthochronous if it maps (1 ,0,0,0) to a vector ( t,x,y,z) where\ntis a non-zero square in F. The set of orthochronous improper Lorentz\ntransformations is denoted by L↑\n−F.\nAmongallimproperLorentztransformations, afundamentalrole isplayed\nby linereflections intime-like orspace-like lines, i.e. 1-dimensional subs paces\nofF4consisting of the scalar multiples of a vector a∈F4with Minkowski\nnormµ(a) equal to 1 or −1. In the classical case F=Rthis role was re-\ncently emphasized by Urbantke in [U]. Over any field F,reflection in the\n13line ofa= (a0,a1,a2,a3)∈F4(whereµ(a) =±1) is the improper Lorentz\ntransformation with matrix\n2µ(a)[a⊤τ(a)−I]\nwhere in the square bracket the matrices a⊤,τ(a) andIare 4×1,1×4 and\n4×4,respectively, and τis time reversal.\nTheorem 2 For every ǫ >0andM >0the extended Lorentz group\nLRhas a retraction f:LR−→A, f2=f,onto a finite subset A⊆ LR\nsatisfying\n/bardblT−f(T)/bardbl< ǫ\nfor all Lorentz transformations T∈ LRof norm not exceeding M,and such\nthat for some prime number p≡7mod8the retract Ais locally isomorphic\nto a subset of LFpvia a local isomorphism λ,where the retraction fand the\nlocal isomorphism λsatisfy the following conditions:\n(i)fmaps each of the 4connected components of LRto its intersection\nwithA,\n(ii)λmaps each of the intersection sets\nA∩L↑\n+R, A∩L↓\n+R, A∩L↑\n−R, A∩L↓\n−R\nrespectively into the corresponding cosets\nL↑\n+Fp,L↓\n+Fp,L↑\n−Fp,L↓\n−Fp\n(iii)λmaps space rotations to space rotations, boosts to boosts, l ine re-\nflections to line reflections. /square\nThe extended retraction and local isomorphism of Theorem 2 are co n-\nstructed similarly to those of Theorem 1. Generating the extended finite\nLorentz group LFpby reflections now leads to a representation of the ex-\ntended finite Lorentz group LFpas a quotient of the group of all Minkowski\nnorm preserving automorphisms of the module Z4\n(p).\n4 Remarks on finiteness and approximation\n14It has been noted (see Beltrametti and Blasi [BB2]) that, as oppos ed to\nL↑\n+RandL↑R=L↑\n+R∪ L↓\n−R,the set of orthochronous Lorentz transforma-\ntions over a finite field does not constitute a group. For example, ta ke any\nprime number p≡7mod8 and such that 3 is a quadratic residue modulo p.\nThe first positive integer qsuch that q+1 is not a quadratic residue modulo\npis even, and q/2 is then a quadratic residue. Let αandγbe non-zero\nsquares in Fpsuch that the elements of Fpcorresponding to the integers\n2 andq/2 areα2andγ2respectively. Then the basic boosts BαandBγ\nare orthochronous but BαBγ=Bαγis not. However, for all orthochronous\ntransformations T1,T2in the set Y⊆ L+Fpof Theorem 1, the construction\nof the local isomorphism between A⊆ L↑\n+RandYimplies that T1T2is also\northochronous. Thus the phenomenon of an antichronous produ ct of two\northochronous parallel boosts arises only when the factors are n ot confined\nto the set Ywhich in the Lorentz group over Fprepresents, in the sense of\nlocal isomorphism, the retract A⊆ L↑\n+R.\nIn contrast to parallel boost groups in LR,inLFpthe boost group\n{RBαR−1:α/\\e}atio\\slash= 0, αsquare in Fp}\nis cyclic for any space rotation R, and it is isomorphic to the multiplica-\ntive subgroup of non-zero squares in Fpvia the map α/mapsto→RBαR−1. The\nfinite model then involves in each space direction the existence of a b oost\nfrom which all parallel boosts in that direction are obtainable by repe ated\napplication.\nAs in the real number based model, over a finite field Fpas well the\nvelocityvαassociated with the basic boost Bαis (α−α−1)/(α+α−1) and\nfor the velocities of boosts BαandBγand of their composition Bαγwe have\nvαγ=vα+vγ\n1+vαvγ\nIn finite field models as well as over R,velocity never equals 1 .Forp≡\n7mod8,ifα2+ 1 is a square in Fp(and this is always the case if Bαis in\nthe setY⊆ L↑\n+Fplocally isomorphic to the retract A⊆ L↑\n+Ras in Theorem\n1), then 1 −vαis a square in Fpi.e. in the non-transitive order <ponFp\nconsidered by Kustaanheimo in [K1, K2] and given by\nx <py⇔y−xis a non-zero square in Fp\n15we have\n−1<pvα<p1\nVelocities vαnot constrained by these inequalities in Fpmay only arise from\nboosts in LFpthat do not correspond, via the local isomorphism of Theorem\n1, to boosts in the retract A⊆ L↑\n+Rwhich approximate all boosts of the\nreal model of norm not exceeding Mi.e. real boosts of speed not exceeding\n(1−4/M2)1/2.As the bound Mcan be as large as needed, this means that\nsuperluminal velocity (1 <pvαinFp) cannot arise from boosts in LFpthat\ncorrespond to boosts in LRwith speed effectively distinguishable from the\nspeed of light.\nReferences\n[A1] Y. Ahmavaara, Relativistic quantum theory as a group problem, II.\nWorld geometry and the elementary particles, Annales Ac. Sc. Fenn icae A\nVI Physica 95 (1962) 51pp\n[A2] Y. Ahmavaara, The structure of space andthe formalism of re lativis-\ntic quantum theory, I-IV, J. Math. Phys. 4 (1965) 87-93, 6 (196 5) 220-227,\n7 (1966) 197-201, 7 (1966) 201-204\n[BB1] E.G. Beltrametti, A.A. Blasi, Dirac spinors, covariant currents and\nthe Lorentz group over a finite field, Nuovo Cimento LVA/2 (1968) 3 01-317\n[BB2] E.G. Beltrametti, A. Blasi, Rotation and Lorentz groups in a finit e\ngeometry, J. Math. Phys. 9 (1968) 1027–1035\n[BGZG] A.A. Blasi, F. Gallone, A. Zecca, V. Gorini, A causality group\nin finite space-time, Nuovo Cimento 10A/1 (1972) 19-36\n[C] H.R. Coish, Elementary particles in a finite world geometry, Phys.\nRev. 114 - 1 (1959) 383-388\n[D]L.E.Dickson, Determinationofthestructureofalllinearhomoge neous\ngroups in a Galois field which are defined by a quadratic invariant, Amer ican\nJ. Math. 21 (1899) 193-256\n[Jo] H. Joos, Group-theoretical models of local-field theories, J. M ath.\nPhys. 5 (1964) 155-164\n16[J¨ a] G. J¨ arnefelt, Reflections on a finite approximation to euclidea n ge-\nometry. Physical and astronomical prospects. Annales Ac. Sc. F ennicae A\nI. Math.-Phys. 96 (1951) 1-43\n[JK] G. J¨ arnefelt, P. Kustaanheimo, An observation on finite geom etries,\nin Proc. Skandinaviske Matematikerkongress i Trondheim 1949, 16 6-182\n[K1] P. Kustaanheimo, A note on a finite approximation of the euclidea n\nplane geometry, Comment. Phys.-Math. Soc. Sc. Fenn. XV. 19 (19 50) 1-11\n[K2] P. Kustaanheimo, On the fundamental prime of a finite world, An -\nnales Ac. Sc. Fennicae A I. Math.-Phys. 129 (1952) 1-7\n[M] V. Moretti, The interplay ofthepolardecompositiontheorem and the\nLorentzgroup, inLectureNotesofSeminario Interdisciplinare di M atematica\n5-153 (2006) 18 pages, also arXiv:math-ph/0211047v1 at www.arx iv.org\n[N]Y.Nambu, FieldtheoryofGaloisfields, inFieldTheoryandQuantum\nStatistics, eds. J.A. Batalin et.al., Institute of Physics Publishing 198 7, pp.\n625-636\n[S] I.S. Shapiro, Weak interactions in the theory of elementary part icles\nwith finite space, Nuclear Phys. 21 (1960) 474-491\n[U] H.K. Urbantke, Lorentz transformations from reflections: so me appli-\ncations, Found. Phys. Lett. 16 (2003) 111-117\n[Y]Q.A.M.Yahia,Leptonicdecaysinfinitespace, NuovoCimentoXXIX/ 2\n(1963) 441-450\n17"    },    {        "title": "2108.07246v2.Drifting_Electrons__Nonreciprocal_Plasmonics_and_Thermal_Photonics.pdf",        "content": "Drifting electrons: Nonreciprocal plasmonics and\nthermal photonics\nS. Ali Hassani Gangaraj\u0003,yand Francesco Monticone\u0003,z\nySchool of Electrical and Computer Engineering, University of wisconsin-Madison,\nMadison, Wisconsin 53706, USA\nzSchool of Electrical and Computer Engineering, Cornell University, Ithaca, New York\n14853, USA\nE-mail: ali.gangaraj@gmail.com; francesco.monticone@cornell.edu\nAbstract\nLight propagates symmetrically in opposite directions in most materials and struc-\ntures. This fact { a consequence of the Lorentz reciprocity principle { has tremendous\nimplications for science and technology across the electromagnetic spectrum. Here, we\ninvestigate an emerging approach to break reciprocity that does not rely on magneto-\noptical e\u000bects or spacetime modulations, but is instead based on biasing a plasmonic\nmaterial with a direct electric current. Using a 3D Green function formalism and mi-\ncroscopic considerations, we elucidate the propagation properties of surface plasmon-\npolaritons (SPPs) supported by a generic nonreciprocal platform of this type, revealing\nsome previously overlooked, anomalous, wave-propagation e\u000bects. We show that SPPs\ncan propagate in the form of steerable, slow-light, unidirectional beams associated\nwith in\rexion points in the modal dispersion. We also clarify the impact of dissipa-\ntion (due to collisions and Landau damping) on nonreciprocal e\u000bects and shed light on\nthe connections between in\rexion points, exceptional points at band-edges, and com-\nplex modal transitions in leaky-wave structures. We then apply these concepts to the\n1arXiv:2108.07246v2  [physics.optics]  29 Nov 2021important area of thermal photonics, and provide the \frst theoretical demonstration\nof drift-induced nonreciprocal near-\feld radiative heat transfer between two planar\nbodies. Our \fndings may open new opportunities toward the development of nonre-\nciprocal magnet-free devices that combine the bene\fts of plasmonics and nonreciprocal\nphotonics for wave-guiding and energy applications.\nKeywords: plasmonics, nonreciprocity, surface plasmon-polaritons, nanophotonics, ther-\nmal photonics\nIntroduction\nThe Lorentz reciprocity principle limits the performance of electromagnetic and photonic\ndevices in a wide range of areas, from communication systems to quantum computing and\nthermal photonics. A reciprocal system, with its symmetric response when source and de-\ntector are interchanged, is not ideal when classical/quantum energy/information needs to\nbe emitted, transferred, and absorbed unidirectionally with high e\u000eciency, or when di\u000berent\ninformation streams need to be selectively routed or processed based on their propagation di-\nrection. Standard nonreciprocal devices, such as magneto-optical isolators or circulators, are\nroutinely used in many of these scenarios, but with the increasing demand for e\u000ecient elec-\ntromagnetic and photonic integrated systems, there has been increasing interest in di\u000berent\nsolutions that permit asymmetric or, in the extreme case, truly unidirectional and beam-like\nwave propagation at small scales. In this context, particularly intriguing opportunities may\nbe o\u000bered by combining the bene\fts of plasmonics (\feld localization and enhancement) with\nstrong nonreciprocal e\u000bects.1\nThe oldest and most common approach to break reciprocity and achieve one-way light\npropagation is based on the magneto-optical e\u000bect, which requires biasing certain mate-\nrials (plasmas, ferrites, magnetic garnets, etc.) with a static magnetic \feld.2{15However,\nthe need for a large external magnet, the weakness of the magneto-optical response at the\n2relevant frequencies, and other practical issues, make this approach less appealing to re-\nalize strong nonreciprocal e\u000bects at the (sub)wavelength scale and for on-chip integration.\nOther approaches to break reciprocity have been extensively studied in the recent past,\nrelying on nonlinear phenomena,16{22spatiotemporal modulations,23{25and multi-physics,\ne.g., optomechanical, interactions.26{28Unfortunately, however, the issues associated with\ndynamic reciprocity for nonlinear nonreciprocal devices, the typically narrowband response\nof optomechanical resonators, and the practical di\u000eculties in implementing spatiotemporal\nmodulations at high frequencies limit the applicability of these approaches.\nIn addition to unidirectional wave-guiding, another particularly important area that may\nbene\ft from strong forms of nonreciprocity is thermal photonics.30According to Kirchho\u000b's\nlaw of thermal radiation, the emissivity of a material is exactly equal to its absorptivity,\nat each frequency and angle. Rather remarkably, this law is not a consequence of the Sec-\nond Law of Thermodynamics, but it originates directly from Lorentz reciprocity29{31and is\nanalogous to other relevant symmetries in electromagnetics, such as the fact that antennas\nmade of reciprocal materials can be used in reception and transmission with exactly the same\ngain and directivity. One of the direct consequences of reciprocity in this context is that\nan e\u000ecient absorber will also be an e\u000ecient emitter. For instance, an absorber designed\nto e\u000ecently harvest solar radiation must radiate part of the energy back to the environ-\nment, which represents an intrinsic loss mechanism. This is the idea behind the Landsberg\nlimit,32which sets the upper e\u000eciency bound for harvesting incoming solar radiation. This\nlimit can only be reached with the use of nonreciprocal systems.32{35Moreover, breaking the\nsymmetry between absorptivity and emissivity may also open new opportunities for near-\n\feld heat transfer. Nonreciprocal systems have been shown to enable thermal Hall e\u000bects36\nand the emergence of persistent heat currents in systems in thermal equilibrium at a cer-\ntain temperature (analogous to persistent electrical or mass currents in superconductors or\nsuper\ruids, respectively).37Such nonreciprocal systems may become an important element\nin the quest for controlling heat \row at the nanoscale with large \rexibility and e\u000eciency.\n3As in the case of nonreciprocal wave-guiding, nonreciprocal e\u000bects in thermal photonics are\ntypically obtained by relying on magneto-optical materials,31,33,38{41which have been theo-\nretically shown to enable a near-complete violation of Kirchho\u000b's law39,40or nonreciprocal\nheat transfer between planar bodies.38Again, due to the weakness of magneto-optical ef-\nfects at optical frequencies, these platforms require a relatively strong magnetic bias, which\nmay raise questions about their practical applicability. A relevant exception is Ref.,42where\nnonreciprocal emission/absorption was realized using space-time modulations; this strategy,\nhowever, was only demonstrated at microwave frequencies, in the form of a nonreciprocal\nleaky-wave antenna, and applying it to optical frequencies involves signi\fcant challenges.\nIn contrast with these previous studies, here we focus on a drastically di\u000berent approach\nto break reciprocity in plasmonic platforms without the need for magneto-optical or dynam-\nical e\u000bects. This approach is based on biasing certain conducting materials with a direct\nelectric current, which, just like the magnetic \feld, is a quantity that is odd under time re-\nversal and, therefore, it can be used to break reciprocity.29This strategy can in principle be\napplied to either three-dimensional (3D) conducting materials (metals, degenerately doped\nsemiconductors, and plasmas) or two-dimensional (2D) media such as graphene, and it has\nbeen the subject of increasing attention in the plasmonic literature,43{53with a very recent ex-\nperimental demonstration at optical frequencies.54Strong nonreciprocal e\u000bects emerge if the\nelectron drift velocity vdis large enough to a\u000bect the dispersion of surface plasmon-polaritons\n(SPPs), which is possible in certain high-mobility conducting materials (e.g., graphene). The\ne\u000bect of drifting electrons on SPP propagation can be qualitatively explained in an intuitive\nway: surface waves on plasmonic materials involve collective charge oscillations coupled to\nelectromagnetic \felds; these propagating waves are dragged or opposed by the drifting elec-\ntrons, which implies that SPPs \\see\" di\u000berent media when propagating along or against\nthe current \row. In other words, it was argued in47that, in a current-biased 3D plasmonic\nmedium, the drifting electrons produce a Doppler shift in the isotropic material permittivity,\ni.e.,!!!\u0000k\u0001vd, wherekis the wavevector. Such a linear wavevector-dependence implies\n4that the material response becomes nonlocal (spatially dispersive), as well as nonreciprocal\nsince oppositely propagating waves see a di\u000berent permittivity. In Section 2, we clarify that,\neven though this Doppler-shift argument is strictly valid only for longitudinal modes (in\ngeneral, the permittivity becomes a spatially dispersive tensor55), under certain conditions\nit provides a good approximation of the exact results and allows for a drastic simpli\fcation\nof the analysis and simulations.\nWhile all previous works on drift-induced nonreciprocity were mostly interested in the 1D\nproblem of one-way SPP propagation along the current \row or on speci\fc 2D materials such\nas graphene, here, in the \frst part of the paper, we use a 3D Green function formalism to elu-\ncidate the di\u000berent regimes of nonreciprocal SPP propagation on the 2D surface of a generic\n3D plasmonic platform, revealing a number of anomalous and extreme wave-propagation\ne\u000bects beyond unidirectional propagation. We also clarify how the unavoidable presence of\nlosses in this plasmonic platform a\u000bects the nonreciprocal modal dispersion. Then, in the\nsecond part of this work, we investigate, for the \frst time to the best of our knowledge, the\npotential of current-biased nonreciprocal systems for thermal photonics, and we theoreti-\ncally demonstrate a clear signature of drift-induced nonreciprocity in the canonical problem\nof near-\feld radiative heat-transfer between two planar bodies.\nCurrent-Induced Nonreciprocity in Three-Dimensional\nPlasmonic Platforms\nWe consider the general problem of SPP propagation along a planar metallic-dielectric inter-\nface atz= 0, as shown in Fig. 1(a). We employ a 3D Green function analysis (see Supple-\nmental Material) to study both the dispersion properties of the supported SPP eigenmodes\nand their propagation properties in source-excited con\fgurations, for both local (non-biased,\nreciprocal) and nonlocal (current-biased, nonreciprocal) plasmonic materials. We assume a\nlaterally in\fnite interface between two di\u000berent materials with permittivity \u000f1forz <0 and\n5\u000f2forz >0 (permeability \u00161=\u00162=\u00160), and source/observation point above the interface\n(Fig. 1(a)).\nFigure 1: (a) Geometry under consideration: a plasmonic material is interfaced with a trans-\nparent dielectric. A vertically polarized point dipole source radiates close to the interface in\nthe dielectric region. The supported SPP dispersion surface is shown in (b) for a reciprocal\nisotropic plasmonic material, with no current bias, interfaced with free space, and (c) for\nthe same con\fguration but with the plasmonic material biased by a direct current with drift\nvelocity vd=\u0000c=10^x. (d,e) Source-excited in-plane distribution of the SPP electric \feld\n(time snapshot of the zcomponent) in the reciprocal and nonreciprocal cases: (d) without\nDC current at frequency !=!p= 0:6, and (e) with DC current at !=!p= 0:8. For panels\n(d) and (e) the source is located at z0=\u0015=10 above the interface, where \u0015is the free-\nspace wavelength at the radiation frequency. The black square at the center denotes the\nsource position. Animations of the calculated time-harmonic electric \feld distribution, for\nthe reciprocal and nonreciprocal cases, are included as Supplemental Material.\nFor the transparent dielectric region we consider vacuum and for the metallic/plasmonic\nregion, in the local non-biased case, we employ a standard Drude model for a free-electron\ngas with permittivity \u000f(!) = 1\u0000(!p=!)2=(1+i\u0000=!), where!pis the plasma frequency and \u0000\nis the damping rate. In the following, we \frst study the problem with \u0000 = 0 (lossless case),\nwhereas the impact of dissipation is discussed in details in Section 2.3. At a local, isotropic,\nmetal-vacuum interface, SPPs are known to exist for frequencies !\u0014!SPP=!p=p\n2. The\n63D dispersion surface of these SPP eigenmodes, calculated by tracking the poles of the Green\nfunction integrand in Eq. (33) of the Supplemental Material, is shown in Fig. 1(b) for the\nlossless case. The dispersion surface clearly exhibits a \rat dispersion asymptote at the surface\nplasmon resonance !=!SPP, as expected for a local/lossless Drude model56(see, e.g.,1,8and\nreferences therein, for a discussion of how losses and nonlocal e\u000bects modify this dispersion\ndiagram). The rotationally symmetric shape of the dispersion surface with respect to in-plane\nwavevectors indicates that the surface mode propagates reciprocally and isotropically on the\ninterface. The spatial \feld distribution of the SPPs excited by a vertically polarized source\nas in Fig. 1(a) was calculated using our 3D Green function formalism at != 0:6!p(see Fig.\n1(d)), con\frming that surface waves propagate symmetrically and omnidirectionally along\nthe interface.\nAs discussed in Ref.,1strong nonreciprocity and unidirectionality in a continuous plas-\nmonic platform of this type can be achieved if the mechanism that breaks reciprocity intro-\nduces an asymmetry in the dispersion asymptote for large wavevectors. While this is typically\nachieved with a static magnetic bias, a direct current bias, which can be directly carried by\nplasmonic materials due to their conductive nature, can also introduce a strong asymmetry,\nbut with qualitatively di\u000berent characteristics, as originally discussed in.47The presence of\na DC current means that conduction electrons move with average drift velocity vd=J0=ne\nwhereJ0is the current density, n=m!2\np=4\u0019e2is the electron density, and eis the negative\nelectron charge. Ref.47argues that the movement of electrons produces a Doppler frequency\nshift in the material permittivity, i.e., \u000f(!\u0000k\u0001vd) = 1\u0000!2\np=(!\u0000k\u0001vd)2, such that the\nmaterial response becomes spatially dispersive, with a wavevector-dependent isotropic per-\nmittivity. This point, and especially the isotropy assumption, is worth some clari\fcation. As\ndiscussed in,55,60in the presence of any non-trivial electron velocity distribution (due to a\ncurrent bias or just the thermal motion of free electrons), the permittivity of a free-electron\ngas becomes spatially dispersive (in other words, the material response is nonlocal due to the\nelectrons moving a certain distance during one period of the applied electromagnetic \feld).\n7Importantly, in the presence of spatial dispersion, the material permittivity is a tensor even\nin an isotropic medium. The general form of this wavevector-dependent permittivity tensor\ncan be written (in tensor notation in an arbitrary coordinate system) as,55\n\u000fij(!;k) =\u000fT(!;k)\u000eij+ (\u000fL(!;k)\u0000\u000fT(!;k))kikj=k2(1)\nwhere the scalar functions \u000fTand\u000fLare the transverse and longitudinal permittivities (for\nelectric \felds orthogonal or parallel to k, respectively), kis the wavevector magnitude, kikj\ndenotes a tensor product between the wavevector and itself, and \u000eijindicates the identity\nmatrix. In Supplemental Material (Notes I and II), we derive \u000fTand\u000fLstarting from the\nVlasov equation for plasmas, and we show that only the longitudinal permittivity is perfectly\nDoppler-shifted in the presence of a drift current, whereas the transverse permittivity exhibits\na di\u000berent, weaker form of spatial dispersion. Since the electric \feld of a SPP mode on a\n3D plasmonic material has both transverse and longitudinal components inside the material,\nits dispersion and propagation properties are expected to depend on both \u000fTand\u000fL. (in\na 2D electron gas, instead, it is clear that only the longitudinal conductivity is relevant\neven if the \feld has a transverse out-of-plane component). This implies that the modeling\napproach proposed in,47which uses the same equation for SPP dispersion as in the local\nunbiased case, but substituting the longitudinal Doppler-shifted permittivity, can only be\napproximately valid. To clarify this point, in the Supplemental Material (Note I), we derive\nthe exact modal solution for SPPs propagating along the electron current and compare it\nagainst the SPP dispersion obtained when only the longitudinal or transverse permittivity\nis considered. Our \fndings show that, while one has to be aware of this issue for more\naccurate calculations, the model proposed in47using only the longitudinal permittivity is\napproximately correct and captures the relevant physics especially in the large-wavevector\nregime. Based on these considerations, we can therefore safely use this approximate model to\ndrastically simplify the analysis and numerical calculations, and provide additional physical\ninsight. Furthermore, it should be mentioned that, while the nonlocal model based on\n8the Doppler e\u000bect is qualitatively valid for the 3D plasmonic materials considered here, its\nvalidity for 2D systems, such as graphene, has been the subject of debate in the literature,57\nand alternative models have been proposed in.43{45,49,58However, it has been argued that,\neven for 2D systems, this model is accurate when the electron-electron collisions predominate\nand force the electron gas to move with approximately constant velocity, which makes the\ndrift-biased medium act similarly to a moving medium.52,59\nWhile Ref.47focused on 1D propagation in the direction of the current, here we use\nour 3D Green function formalism to investigate drift-induced nonreciprocal e\u000bects over the\nentire two-dimensional interface of the three-dimensional geometry in Fig. 1(a), revealing\nanomalous and extreme forms of surface-wave propagation on this surface. The modi\fed\ndispersion and propagation properties of the SPPs supported by this nonreciprocal con\fg-\nuration can be readily analyzed using the same 3D Green function as for the reciprocal\ncase, but, as explained above, with the nonlocal, Doppler shifted, longitudinal permittivity,\n\u000fL(!;k) = 1\u0000!2\np=(!\u0000k\u0001vd)2. As a \frst example, we suppose the electron \row is along\nthe +x-axis, with drift velocity vd=\u0000c0=10 (wherec0is the speed of light in vacuum). We\nmust emphasize that this value of the drift velocity is unrealistically high and is considered\nhere just for illustrative purposes to describe the qualitative behavior of surface waves in this\nnonreciprocal platform. Later in the article, and in the Supplemental Material, we provide\nresults for much smaller values of drift velocity. The modi\fed SPP dispersion surface is\nshown in Fig. 1(c). The dispersion asymptote is now tilted as a result of the Doppler shift,\nas discussed in,47making the dispersion surface strongly asymmetric and nonreciprocal in\nthex-direction, with a frequency range where SPPs can only propagate toward the nega-\ntivex-axis. In addition, unidirectional SPPs are now supported at frequencies where SPPs\ncould not propagate at all in the reciprocal case (frequencies higher than the \rat asymptote\nin Fig. 1(b)). The \feld distribution of surface waves launched by a vertical dipole source\nis shown in Fig. 1(e) for a frequency higher than the original surface-plasmon resonance,\n!= 0:8!p>!SPP. We clearly see that, in this case, SPPs propagate preferentially toward\n9left, parallel to the electrons drift velocity, despite the homogeneity and isotropy of the sur-\nface. While this behavior is somewhat expected as an extension of the 1D results of Ref.47\nto a 2D surface, even more extreme wave-propagation e\u000bects emerge at di\u000berent frequencies\nand angles, as discussed in the next sections.\nIn\rexion Points and Slow-Light Beaming\nThe results in Fig. 1 clearly show that the dispersion properties of drift-biased SPPs strongly\ndepend on the angle \t between in-plane wavevector and current \row, as expected from the\nspatially dispersive form of the permittivity. To clarify this point further, Figure 2(a) shows\n1D dispersion curves for SPPs propagating along di\u000berent angles. SPPs propagating normal\nto the current \row (\t = 90 deg.) do not interact with the drifting electrons and the\ndispersion curve is the same as in the reciprocal case with \rat asymptotes at !=!SPP.\nConversely, SPPs propagating along the current \row (\t = 0) are maximally a\u000bected by the\npresence of the current, particularly in the large-wavevector asymptotic region where spatial\ndispersion becomes particularly strong. The tilted dispersion asymptote is associated with\nhighly localized SPPs dragged by the \row of electrons,47and its slope (group velocity) is\nindeed exactly equal to the electron drift velocity seen by the SPP mode, i.e., @!=@k =vd\u0001^uk,\nwhere ^ukis the in-plane unit wavevector. Such a tilted asymptote makes the dispersion curve\nnon-monotonic for wavevectors opposite to vd, with an in\rexion point (red dot in Figure\n2(a)) where the SPP group velocity vanishes and changes sign. The position of the in\rexion\npoint along the frequency axis depends on the direction of propagation, such that, for each\nangle \t, an in\rexion point exists at a di\u000berent frequency,\n10!inf=!pp\n2\"\n1\u00003\n2\u0012vdcos(\t)\n2c\u00132=3#\n)8\n>><\n>>:!min\ninf=!pp\n2h\n1\u00003\n2\u0000vd\n2c\u00012=3i\n;\t = 0\n!max\ninf!!pp\n2; \t!90 deg:(2)\nThus, while an in\rexion point exists for any angle \t 2[0;90) deg., its frequency is limited\nwithin the range !min\ninf\u0014!inf\u0014!max\ninf, as shown in Fig. 2(b). Minimum in\rexion-point\nfrequency occurs for SPP propagation along the current \row and maximum frequency is\nasymptotically reached for propagation normal to the current, in which case the in\rexion\npoint rapidly migrates to large values of wavevector as seen in Fig. 2(a). For any frequency\noutside this window, no in\rexion point appears, i.e., the SPP group velocity does not vanish\nalong any direction.\nThe existence of an angle-dependent in\rexion point in this frequency window leads to\nnontrivial propagation e\u000bects on the 2D surface of the considered nonreciprocal platform.\nThese e\u000bects, which have not been studied previously, can be elucidated by considering\nthe system's Green function under quasi-static approximation. Indeed, while the 3D Green\nfunction described above provides a complete formulation to study SPP propagation, it is\nusually evaluated numerically and tends to hide physical insight. A simpler, more reveal-\ning, approximate formulation can be obtained by neglecting retardation e\u000bects (quasi-static\napproximation) and assuming that the main radiation channel of the dipolar source is rep-\nresented by the excitation of a single, guided, surface mode. As was shown in Ref.,62under\nthese approximations, the source radiation intensity in a certain in-plane direction (radiated\npower per unit angle) is given by\nU(\t)\u0019!2\n16\u00191\njrkt!(kt)j1\nC(kt)j\r\u0003\u0001Ek(z0)j2; (3)\n11Figure 2: (a) Dispersion curves for SPPs propagating along di\u000berent angles with respect\nto the current \row. The angle \t is measured from the positive x-axis (see Fig. 1(a)).\nDi\u000berent curves correspond to the intersection of the dispersion surface in Fig. 1(c) with\nplanes at angle \t with respect to the kx-axis. The red dots indicate in\rexion points for\ndi\u000berent directions of propagation. (b) In\rexion-point frequencies for SPPs propagating\nalong di\u000berent angles. (c,d,e) Source-excited in-plane distribution of the SPP electric \feld\n(time snapshot of the zcomponent) for (c) !=!min\ninf, (d)!= 0:6!pand (e)!\u0019!SPP=\n!max\ninf. Thexandyaxes are normalized with respect to the free-space wavelength at the\nradiation frequency. The black squares indicate the in-plane position of the source, which\nis located in the vacuum region at z0=\u0015=10 above the interface for panels (c,d) and at\nz0=\u0015=30 for panel (e) (the source is closer to the interface in this panel to e\u000eciently excite\nthe high-wavevector components of the SPPs). The inset in panel (c) shows the amplitude\ndistribution of the z-component of the electric \feld, and the inset in panel (d) shows the real\nand imaginary parts of Ezalong the red dashed line crossing the narrow beam in the main\npanel.\nwhere ktis the in-plane wavevector, tan(\t) = k y=kx,Ek(z0) is the modal electric \feld at\nthe location z0of a source with polarization state \r,!(kt) is the dispersion relation of the\nrelevant mode (hence, rkt!(kt) is the group velocity), and C(kt) is the local curvature of the\nequifrequency contour (EFC) of the dispersion relation. Equation (3) gives the approximate\nin-plane radiation pattern of the dipole source, corresponding to the in-plane pattern of SPP\npropagation. This equation reveals that the SPP pattern can be controlled in two ways: (i)\n12by suitably selecting the polarization of the dipole source, which controls the coupling factor\nj\r\u0003\u0001Ek(z0)j2between the source and the surface mode of interest; or (ii) by engineering the\ndispersion relation of the relevant surface mode, namely, by controlling the local curvature\nC(kt) of the equifrequency contour and/or the wavevector dependence (angular dependence)\nof the group velocity, jrkt!(kt)j. In the considered current-biased plasmonic platform, the\npresence of in\rexion points where the group velocity vanishes is therefore a crucial factor\na\u000becting the in-plane propagation pattern. Indeed, at any frequency where an in\rexion point\nappears, the inverse of the group velocity in Eq. (3) diverges, 1 =jrkt!(kt)j!1 (in practice\nit becomes large but \fnite) along a speci\fc angle with respect to the current \row. Thus,\nthe in-plane propagation pattern is maximized and localized around this angle, resulting in\na peculiar \\beaming e\u000bect\" as further discussed below.\nFigure 2(c,d,e) show the in-plane \feld distributions, calculated using our Green func-\ntion formulation, for SPP propagation at three frequencies: (c) minimum and (e) maximum\nin\rexion-point frequencies and (d) another frequency in between. As shown in Fig. 2(a,b),\nminimum in\rexion-point frequency corresponds to vanishing SPP group velocity in the di-\nrection of the current \row. This leads to a diverging behavior in Eq. (3) at \t = 0, which\nimplies that the SPP \felds form a unidirectional narrow beam along the current \row, as con-\n\frmed by Fig. 2(c). Note that no energy actually propagates toward the + x-axis since the\ngroup velocity is zero (group velocity and energy velocity are virtually identical in a low-loss\nmedium even if material dispersion is high), and the narrow beam resembles a standing wave\nwith localized and enhanced \felds, as seen in the \feld amplitude distribution plotted in the\ninset of Fig. 2(c) (an animation of the calculated time-harmonic electric \feld, which further\nclari\fes this peculiar behavior, is included as Supplemental Material). Choosing a di\u000berent\nfrequency, for example != 0:6!p, an in\rexion point appears for SPPs propagating at an\nangle \t = 60 deg. with respect to the current \row (see Fig. 2(a), dashed blue line). This\nleads to narrow SPP beams along \t = \u000660 deg., as shown in Fig. 2(d). In this scenario, two\nbeams emerge because the system's response is reciprocal normal to the current, therefore\n13SPPs propagate symmetrically along the y-axis. As in the previous case, these beams are\nnarrow standing waves with enhanced and localized \felds. Interestingly, in this case the\n\feld distribution is even sharper around \t = \u000660 deg., especially toward smaller angles,\nwhere no surface-wave propagating solution exists at this frequency (see Fig. 1(c) and the\nequifrequency-contour animation in Supplemental Material). We note that, although the\n\feld pattern looks discontinuous around this angle, a closer inspection reveals that, rather\nthan an actual \feld discontinuity (which would be nonphysical on a homogeneous surface),\nthe \felds decay very sharply, within a small fraction of a wavelength, as demonstrated in\nthe inset of Fig. 2(d) which shows the real and imaginary parts of the electric \feld along the\nred dashed line. We speculate that such ultra-sharp \felds on a homogeneous surface may\nbe useful for surface-sensing applications or optical trapping near the surface. Finally, Fig.\n2(e) shows the \felds at !\u0019!SPP=!max\ninf, at which an in\rexion point appears for SPPs\npropagating close to \t = \u000690 deg. Indeed, in this case, the \feld distribution is dominated by\nthe presence of narrow beams almost normal to the current \row and with large wavenumber,\nconsistent with Fig. 2(a).\nThe emergence of these narrow, current/frequency-steerable, frozen-light beams with\nenhanced \felds is highly nontrivial, especially given the fact that the plasmonic platform\nthat supports them is homogeneous, and they may o\u000ber new opportunities for tunable,\ndirectional, enhanced, light-matter interactions on plasmonic surfaces.\nUnidirectional Propagation and Negative Group Velocity\nAs discussed above, for frequencies higher than !max\ninf=!SPPor lower than !min\ninf, no in\rexion\npoint appears, and hence the SPP group velocity never vanishes along any in-plane direc-\ntion. In these frequency regions, it is mostly the curvature of the equifrequency contour C(kt)\nthat determines, according to Eq. (3), how SPPs propagate on the two-dimensional interface.\nOther anomalous wave-propagation e\u000bects occur in this regime, qualitatively di\u000berent from\nthe behavior of SPPs in the !min\ninf<!<!min\ninfwindow. Figure 3(a) shows the dispersion sur-\n14face with two cut-planes at constant frequencies, != 0:8!p>!SPPand!= 0:52!p<!min\ninf.\nThe intersection of these planes with the dispersion surface gives the EFCs presented in Fig.\n3(b) and 3(c), respectively (an animation of how the EFC changes as a function of frequency\nis included as Supplemental Material). The colors of the density plots correspond to the\nmagnitude of the integrand in Eq. (33) of the Supplemental Material (brighter colors mean\nhigher intensity), indicating which portions of the EFC contribute more strongly to SPP ex-\ncitation for the chosen source, i.e., a dipolar emitter linearly-polarized along the z-axis and\nlocated above the interface as detailed in the caption of Fig. 1. The white arrows, normal\nto the EFC at each point, indicate the main directions of energy \row (group velocity vec-\ntors). As Fig. 3(b) suggests, at != 0:8!pSPPs are expected to propagate unidirectionally\ntoward the\u0000x-axis (propagation along the + x-axis is prohibited) with wavefront spreading\n(di\u000braction) along the y-axis. The \feld distribution in Figure 1(e) con\frms this behavior.\nFurther comments on this propagation pattern and on the impact of the source height from\nthe surface are discussed in the Supplemental Material.\nThe equifrequency contour has a more complicated non-connected shape for frequencies\nlower than !min\ninfdue to the tilted dispersion asymptote. As seen in Fig. 3(c), the contour is\nformed by a central ellipse slightly stretched toward right (in the absence of current bias, the\ncentral ellipse becomes a circle) and an approximately vertical line at larger positive values\nofkx. This vertical line corresponds to the large-wavevector SPPs dragged backward by the\ndrifting electrons, propagating with group velocity almost exactly equal to the drift velocity,\nas discussed in the previous sections. If the drift velocity is reduced, the vertical line moves\nto even larger values of kx, going asymptotically to in\fnity for vd!0. As the direction of the\nwhite arrows in Fig. 3(c) suggests, small-wavevector SPPs are allowed to propagate along\nevery in-plane direction with a slight preference toward the + x-axis (ellipse contribution),\nbut the dipole radiation may also couple to counter-propagating SPPs with positive kxbut\nnegative group velocity (vertical line contribution). Figure 3(d) shows the corresponding\nsource-excited \feld distribution at != 0:52!p, demonstrating an asymmetric SPP wave-\n15Figure 3: (a) SPP dispersion surface for vd=\u0000c=10 with two cut-planes at constant fre-\nquency,!= 0:8!pand!= 0:52!p. The cross sections correspond to the equifrequency\ncontours (EFC) shown in panels (b) and (c). EFCs are represented as density plots of the\nmagnitude of the integrand of the 3D Green function in Eq. (33) of the Supplemental Ma-\nterial. The source-excited electric \feld distribution (time snapshot of the zcomponent) at\n!= 0:8!p, corresponding to panel (b), is shown in Fig. 1(e), whereas panel (d) shows the\n\feld distribution at != 0:52!p, corresponding to the EFC in panel (c).\nfront, stronger toward the positive x-axis, and characterized by a clear interference pattern\ndue to the two contributions in Fig. 3(c). An animation of the calculated time-harmonic\nelectric \feld, corresponding to Fig. 3(d), is included as Supplemental Material. At lower\nfrequencies, the EFC and \feld distribution are qualitatively similar to those in Fig. 3(c,d),\nbut SPP propagation tends to become more symmetric and isotropic as frequency is low-\nered, consistent with the fact that the elliptical EFC becomes smaller and rounder (smaller\nwavevectors imply weaker spatial dispersion), whereas the vertical-line contribution tails o\u000b\n16as it moves to larger values of kx(hence, such large-wavevector SPPs become increasingly\ndi\u000ecult to excite).\nWe also note that, while the results in this section and the previous section have assumed\na unrealistically high value of drift velocity for illustrative purposes, the nonreciprocal e\u000bects\ndemonstrated here can be achieved at much lower drift velocities (examples are provided in\nSection 3 and in the Supplemental Material). In principle, as long as the drift velocity is\nnonzero, the plasmonic material remains nonreciprocal, but the nonreciprocal behavior may\nbecome so weak as to be overshadowed by dissipative and other e\u000bects. The non-trivial\nimpact of dissipation on the modal dispersion is discussed in the next section. Furthermore,\nother forms of nonlocality, e.g., of hydrodynamic origin, may have an impact if dissipation is\nsu\u000eciently weak. While a complete analysis of these nonlocal e\u000bects will be the subject of a\nfuture work, we expect that similar tradeo\u000bs may exist as in nonreciprocal magnetic-biased\nplasmonic systems.1,8,11,69\nModal Transitions, Exceptional Points, and Impact of Dissipation\nSince the in\rexion point described above represents the most important feature of the disper-\nsion diagram of a current-biased plasmonic system, marking the transition between di\u000berent\nwave-propagation regimes, we further investigated the local dispersion behavior around this\npoint and the impact of dissipation. Let us consider SPPs with propagation direction \fxed\nparallel to the drift current, \t = 0, i.e., with their in-plane wavevector purely in the x-\ndirection. The dispersion diagram for this case is plotted again in Fig. 4(a), but tracking\neach individual solution of the dispersion equation, including where the longitudinal modal\nwavenumber becomes complex, as frequency is varied, and assuming this is a 1D propagation\nscenario where kyis forced to be zero. Mathematically, solutions of the dispersion equation\ncannot simply disappear as frequency (assumed real) is varied; therefore, above the in\rexion\npoint, solutions still exist but they are expected to correspond to evanescent modes. In\nother words, the in\rexion point acts as a cut-o\u000b frequency or band edge and, as seen in Fig.\n174(a), the local dispersion around this point resembles the typical square-root-like bifurcation\naround an exceptional point of degeneracy (EP).66,67\nTo clarify this behavior, Fig. 4(b,c,d) show the modal solutions (SPP poles) on the\ncomplexkx-wavenumber plane as frequency is increased. At frequencies lower than the\nin\rexion point, i.e., ! < !inf, owing to the downward bending of the dispersion curve we\nhave two counter-propagating modes (opposite group velocity) on the positive side of the\nrealkxaxis (Fig. 4(b)). Right at the in\rexion point, these two modes/poles coalesce into\na single mode with vanishing group velocity (Fig. 4(c)). Finally, for frequencies higher\nthan the in\rexion-point, ! > !inf, the poles rapidly migrate away from the real axis (Fig.\n4(d)), leading to an interesting modal transition further discussed below. This behavior\nindeed suggests that the in\rexion point could also be interpreted, in this 1D propagation\nscenario, as an exceptional point of degeneracy since it marks the coalescence and branching\nof modes.66,67\nTo further con\frm this interpretation, we considered the general conditions that a solu-\ntion of the dispersion equation, in a 1D wave-guiding system, must satisfy to be identi\fed as\nan EP, as originally derived in.63{65The \frst necessary condition to obtain a \frst-order EP\nis that two \frst-order roots of the dispersion equation, D(kEP;!EP) = 0, coalesce to form a\nsecond-order root, which implies that, at the EP,\nD(kEP;!EP) =@D(k;!)\n@k\f\f\f\fk=kEP!=!EP= 0 (4)\nwherekEPand!EPare the wavenumber and angular frequency at which the degeneracy\nemerges. However, an EP should also satisfy an additional condition,\n@D(k;!)\n@!\u0001@2D(k;!)\n@k2\f\f\f\fk=kEP!=!EP6= 0: (5)\nIn fact, only if Eq. (5) is satis\fed, then the second-order root becomes a branch point (and\nnot a saddle point) where various branches of the dispersion function k(!) merge.63{65Our\n18Figure 4: (a) Dispersion diagram for SPPs propagating in the direction of the current \row\n(\t = 0), similar to Fig. 2(a), but tracking each individual solution of the dispersion equation,\nand plotting its complex in-plane modal wavenumber, as frequency is varied. Absorption\nlosses are assumed to be zero, i.e., \u0000 = 0. See Fig. 5 for the dissipative case. Solid purple\nlines indicate the light cone. (b,c,d) SPP pole constellation, on the complex kx-plane (for\n\fxedky= 0), at di\u000berent frequencies: (a) !=!p= 0:54< !min\ninf, (b)!=!p= 0:5457 =!min\ninf\nand (c)!=!p= 0:55> !min\ninf, for SPP modes propagating along the drifting electrons with\nvd=\u0000c=10. The peculiar character of the two complex modes in panel (d) is discussed in\nthe text.\nnumerical tests applied to the case of SPPs propagating along the electron current verify that\nthe local 1D dispersion function, kx(!) (forky= 0), satis\fes conditions (4) and (5) around\nthe in\rexion frequency (red point in Fig. 2(a)), hence con\frming that in this 1D scenario,\nthe in\rexion point can be interpreted as an EP (branch point) where forward-propagating\nand backward-dragged SPPs coalesce. We also note that, unlike typical EPs in open non-\nHermitian systems, the EP observed here exists in a system with no balanced distribution\n19of loss and gain. Indeed, this dispersion feature is more similar to the EPs associated with\nthe band edges of lossless periodic structures (see, for example, Refs.61,67,68); however, in the\nconsidered current-biased system, the EP exists at a sort of unidirectional band edge (the\nin\rexion point) created not by a periodic modulation, but by the Doppler shift resulting\nfrom the applied current.\nThe modal transition occurring through this EP is particularly interesting as it leads to\na rich and subtle behavior at higher frequencies, and some additional observations should\nbe made. As discussed in Ref.,1in a system with continuous translational symmetry as\nthe one considered here, a direct transition from a purely propagating solution to a purely\nevanescent one, or viceversa, can only occur at band-edges or cut-o\u000b frequencies at zero\nor in\fnite wavenumbers (where the modal wavenumber can go directly from purely real to\npurely imaginary). This is clearly not the case in Fig. 4(a), where the band-edge (exceptional\npoint) exists in the dispersion diagram at a \fnite non-zero value of wavenumber and the\nsolutions above the EP frequency are indeed complex, with the same real part but opposite\nimaginary parts. However, since we assumed that the considered structure is lossless, \u0000 = 0,\na complex wavenumber is puzzling because it implies some form of energy dissipation or\ngain as the wave propagates. One of the two complex solution branches above the EP in\nFig. 4(a) represents an exponentially growing propagating wave, which, we argue, is a non-\nphysical solution since the considered system does not provide optical gain. Yet, due to\nthe presence of the drift current, this point deserves further clari\fcation. Indeed, a direct\ncurrent bias may in principle lead to optical gain through negative Landau damping in an\notherwise passive system: if the drift velocity is comparable and greater than the wave\nphase velocity, vd> !=k , the electrons \\trapped\" inside the moving potential wells of the\nwave lose kinetic energy to the wave (the opposite process is the regular Landau damping\ndiscussed in the following).52,60However, this cannot happen in the present situation since\nthe SPP phase velocity in the region of interest of the dispersion diagram is much larger\nthanvd. In addition, it was convincingly demonstrated in Ref.46that SPP ampli\fcation and\n20instabilities are not possible in a con\fguration with a single drift-biased plasmonic medium.\nThis also makes sense physically by considering the analogy with moving media: a single\nbody moving at a constant velocity cannot lead to wave instabilities; instead, the presence\nof another body in close proximity and in relative motion is necessary, for example two\ngraphene sheets with di\u000berent bias velocities,46or a single drift-biased graphene sheet on a\nnon-biased SiC substrate.52Based on these considerations, we can safely say that the case\nconsidered in this section, with a single drift-biased material, cannot lead to ampli\fcation\nand instabilities, and any unstable pole can be considered unphysical. Indeed, the presence\nof complex poles with opposite imaginary parts is common in even simpler passive systems,\nfor example beyond a band-edge in the gapped band diagram of a periodic structure, or\nbelow the cut-o\u000b frequency of a waveguide. Mathematically, poles on both sides of the real\naxis do exist in these scenarios, but only the exponentially decaying solution is retained on\nphysical grounds (i.e., the proper branch of a multivalued dispersion relation is chosen based\non considerations of physical realizability).\nGoing back to our discussion of the dispersion diagram in Fig. 4(a), we note that the other\nsolution branch above the EP corresponds, instead, to a physical, decaying, propagating wave\nthat becomes overdamped as it crosses the light line at higher frequencies, which suggests\nthat radiation leakage plays an important role in this behavior. In fact, analogous modal\ntransitions (albeit for reciprocal systems) were observed and investigated in the context of\nleaky-wave antennas a few decades ago.70,71In the transition region between bound- and\nleaky-waves, it was similarly observed that certain propagating solutions are non-physical\n(growing in the longitudinal direction), and the unexpectedly lossy solutions below the light\nline, in an otherwise lossless structure, can be explained as the \\winding down\" of the leaky-\nwave solution right under the light cone (with a virtually negligible contribution to the total\n\felds excited by an actual source71).\nFinally, we investigated how the dispersion diagram changes when absorption losses are\nincluded in the current-biased plasmonic material. This is particularly important consid-\n21Figure 5: Impact of dissipation on nonreciprocal current-biased SPPs. (a,b) Dispersion\ncurves for SPPs propagating along the current \row, with vd=\u0000c=10 in the presence of\nintrinsic (bulk) scattering losses: (a) \u0000 = 0 :02!pand (b) \u0000 = 0 :1!p. (c,d) Dispersion curves\nfor SPPs propagating along the current \row, with vd=\u0000c=40 and (c) \u0000 = 0 :1!pand (d)\n\u0000 = 0:15!p. For the forward-propagating mode with largest wavevector, the imaginary part\nis plotted over a limited frequency range due to numerical di\u000eculties in tracking this solution\nbeyond this range. Solid black lines denote the lossless case (showing only the real branches\nfor simplicity), and dotted green and dashed red lines indicate the real and imaginary part\nof the in-plane modal wavenumber, Re( k) and Im(k), respectively. Arrows indicate which\nimaginary and real branches correspond to the same mode. Solid purple lines denote the\nlight cone. (d) Phase velocity of SPP modes propagating parallel to the drifting electrons\nwithvd=\u0000c=10, compared with a Maxwell-Boltzman velocity distribution for a typical\nplasmonic material at room temperature and with the same electron drift velocity. The\ninset shows a zoomed-in view of the distribution function.\nering the virtually unavoidable presence of dissipation mechanisms in plasmonic media. In\nparticular, here we \frst studied the impact of the intrinsic (bulk) scattering losses of the\nmaterial, modeled by an intrinsic damping rate \u0000 which is assumed to be unaltered by the\npresence of the current bias, as done for example in.52However, how this damping rate\na\u000bects the propagation and attenuation of SPPs strongly depends on the presence and ve-\nlocity of the drift current, the SPP wavevector, and the speci\fc structure. As seen in Fig.\n225(a,b), increasing \u0000 determines an increase in the imaginary part of the modal wavenumber,\nespecially in the large-wavevector regime, as expected. Non-zero dissipation also \\opens\" the\nband bifurcation at the EP, such that the dispersion curves no longer \ratten out completely.\nThis behavior is therefore qualitatively di\u000berent from the way losses a\u000bect nonreciprocal\nSPPs in magnetically biased plasmas,1,8where band edges are located at diverging values\nof wavenumber. As a result of this modi\fed dispersion diagram, the e\u000bects described in\nSection 2.1, which mostly depend on a slowing down of the group velocity at the in\rexion\npoint, are expected to be sensitive to the presence of dissipation (which may actually be\nuseful for the sensing of absorbing layers on the plasmonic platform). Figure 5(b,c) com-\npare two cases where \u0000 is kept \fxed but the drift velocity is decreased. As seen here, by\nreducing the drift velocity, the imaginary part of the modal wavenumber generally increases,\nindicating that, in this con\fguration, drift-induced nonlocality and nonreciprocity may in-\ndirectly mitigate the detrimental impact of scattering losses by modifying the dispersion\ndiagram and the group velocity of the modes. Finally, if losses are su\u000eciently large and\nthe drift velocity is low (Fig. 5(d)), dissipation starts to dominate over the drift-induced\nnonlocal and nonreciprocal e\u000bects. In particular, the backward-propagating mode now ex-\nhibits the typical \\back-bending\" of a lossy SPP (real dispersion curve bends in the opposite\ndirection) and becomes overdamped at higher frequencies (Re[ k]<Im[k]), whereas it was\nstill underdamped in Fig. 5(c). The smaller-wavevector branch of the forward-propagating\nmode shows a similar back-bending behavior (as it does in all cases in Fig. 5), whereas the\nother, \ratter branch becomes overdamped over a broad range of wavevectors (and for larger\nwavevectors it is essentially a dark mode, increasingly di\u000ecult to excite). Unidirectional\npropagation is no longer possible in this regime as the forward- and backward-propagating\nmodes are both allowed to propagate at low frequencies and, as frequency increases, they\nbecome overdamped almost simultaneously.\nIn addition to intrinsic scattering loss, another potentially important attenuation mecha-\nnism in drift-biased plasmas and plasmonic materials is bulk Landau damping, which exists\n23even in the absence of any collisions, originating instead from the kinetic energy transfer\nfrom an electromagnetic wave to the electrons \\trapped\" inside its moving potential wells.\nThis e\u000bect is possible if the velocity distribution function is not a delta function centered\nat the drift-bias velocity (cold biased plasma), but follows for example a Maxwell-Boltzman\ndistribution f0(v) for a plasma at non-zero temperature. The attenuation due to bulk Lan-\ndau damping can be shown to be proportional to the velocity derivative @vf0(v),55,60which\nindicates that this form of damping is important if the electron velocities are close to, and\nsmaller, than the wave phase velocity. For the considered drift-biased nonreciprocal con\fgu-\nration, Figure 5(d) compares the phase velocity of SPPs propagating along the current with\nthe Maxwell-Boltzman distribution function for a typical plasmonic material at 300K. With\nthe exception of the backward-propagating mode at very large frequencies, the phase velocity\nof the SPP modes lies far out on the distribution tail, where @vf0(v)'0, which implies that\nthe presence of bulk Landau damping can be safely ignored in the present case. We also\nnote that another form of Landau damping, namely, surface-collision-induced damping, as-\nsociated with the direct excitation of electron-hole pairs by the highly con\fned electric \feld\non the interface,11may be more signi\fcant in this case, but is still expected to be smaller\nthan the intrinsic (bulk) scattering losses of typical solid-state plasmonic media.\nThe analysis presented in this section shows, for the \frst time to the best of our knowl-\nedge, the impact of dissipation (due to collisions or Landau damping) on the drift-induced\nnonlocal and nonreciprocal e\u000bects in current-biased 3D plasmonic platforms. While the\npresence of losses is detrimental for wave-guiding applications, the next section discuss an\nexample of application of nonreciprocal drift-biased plasmonics where losses are actually\nnecessary, namely, radiative heat transfer mediated by SPPs.\n24Drift-Induced Nonreciprocal Radiative Heat Transfer\nWe consider the canonical problem of near-\feld radiative heat transfer between two planar\nbodies, as shown in Fig. 6(a): two plasmonic slabs, with permittivity \u000f(!) = 1\u0000(!p=!)2=(1+\ni\u0000=!), are separated by a distance dg, and backed by perfect electric conductors. A DC\nelectric current is driven in the top conductor to break reciprocity. As discussed in the\nprevious section, the current-biased plasmonic material can then be approximately described\nby a spatially and frequency dispersive Doppler-shifted permittivity function \u000f(!)!\u000f(!\u0000k\u0001\nvd). The geometry being studied and the following calculations parallel the ones in Ref.,38\nbut with the crucial di\u000berence that here we consider, for the \frst time, a direct current\nbias to realize nonreciprocal radiative heat transfer, instead of magnetized magneto-optical\nmaterials.\nBefore discussing our results in details, an important point should be clari\fed: in gen-\neral, the systems being studied here are non-equilibrium systems due the presence of drifting\nelectrons. Indeed, considering the analogy with moving media, even in the limit of zero tem-\nperature two bodies in relative motion and close proximity can exchange energy, which can\nbe explained in terms of quantum friction,72,73also related to the possible onset of wave insta-\nbilities in moving media (and negative Landau damping in drift-biased plasmas) mentioned\nin the previous section. Moreover, strictly speaking, the \ructuation-dissipation theorem74\napplies to systems at equilibrium. Fully accounting for the non-equilibrium thermodynamics\nof such systems is quite involved, as discussed for example in Ref.75Importantly, however,\nthese e\u000bects become signi\fcant only when the media/sources in relative motion are in very\nclose proximity (usually atomic distances as in72) and/or for very large relative velocities\n(not necessarily relativistic). More precisely, as shown in Ref.,75large deviations from the\nequilibrium \ructuation-dissipation theorem occur for !\u001cv=dg(i.e., 2\u0019dg=\u0015\u001cv=c), where\ndgis the separation between the moving bodies. Here, instead, we focus on a di\u000berent regime,\nwith relatively large separations and low velocities, where non-equilibrium e\u000bects are still\nsmall, but nonreciprocal properties are already clearly visible.\n25Under the assumptions discussed above, the equilibrium \ructuation-dissipation theo-\nrem74implies that the strength of the \ructuating current sources that generate thermal\nradiation is proportional to the imaginary part of the permittivity, Im ( \u000f)\u0011(\u000f\u0000\u000f\u0003)=2i.\nUnlike many areas of photonics in which reducing material absorption is an important goal,\nthermal photonic applications do require materials with non-zero losses since a lossless struc-\nture does not generate thermal emission. This relation between absorption and emission is\nhowever a subtle one in nonreciprocal media, as discussed in the following. In the setup\nin Fig. 6(a), we are interested in the radiative heat \rux density from body ito bodyj,\nSij, and vice versa, which is de\fned as the power density absorbed by body jfrom the\nradiated electric and magnetic \felds generated by the \ructuating currents in body i. The\ntotal heat \rux density from body 1 to body 2 can be expanded in terms of its time-harmonic\nplane-wave contributions, i.e., a continuous set of \\channels\" de\fned by their wavevector,\nk=\u0006(kz+kt), and frequency, !,38\nS12=Z1\n0d!\n2\u0019Zdkt\n(2\u0019)2\b(T1;!)S12(kt;!) (6)\nwhere \b(T;!) =~!h\n1\n2+1\nexp( ~!=K BT)\u00001i\nis the mean energy of photons per frequency !at\ntemperature Tand\nS12(kt;!) = Tr\u0010h\n^I\u0000^Ry\n2(kt;!)^R2(kt;!)i\n^D12(kt;!)\n\u0002h\n^I\u0000^R1(kt;!)^Ry\n1(kt;!)i\n^Dy\n12(kt;!)\u0011\n(7)\nfor propagating waves, jktj<!=c , and\nS12(kt;!) = Tr\u0010h\n^Ry\n2(kt;!)\u0000^R2(kt;!)i\n^D12(kt;!)\n\u0002h\n^R1(kt;!)\u0000^Ry\n1(kt;!)i\n^Dy\n12(kt;!)e\u00002\u000bdg\u0011\n(8)\nfor evanescent waves, jktj> !=c . The heat \rux density from body 2 to body 1 can be ob-\n26tained from Eqs. (7) and (8) by exchanging the subscripts 1 and 2 and changing the tempera-\nture in Eq. (6) from T1toT2. In the above equations, \u000bis the decay rate of evanescent waves\nalong thez-axis,Ri(kt;!) is the re\rectivity matrix of body i, whose elements are the re\rec-\ntion coe\u000ecients for light incident from vacuum, for both transverse-electric and transverse-\nmagnetic polarizations, and the parameter ^D12=h\n^I\u0000^Ry\n2(kt;!)^R1(kt;!)e\u0000i2kzdgi\nrepresents\nmultiple re\rections between the two bodies (see Ref.38for more details). This formalism al-\nlows us to calculate the near-\feld 3D heat transfer between two planar slabs with and without\na current bias. As in our 3D Green function formalism in Section 2 and Supplementary Note\nIII, the presence of a non-zero drift current modi\fes the permittivity, making it wavevector-\ndependent, and therefore changes the re\rection coe\u000ecients for plane-wave incidence.\nA nonzero net heat \row in a two-body geometry as in Fig. 6(a) requires a temper-\nature gradient, as dictated by the Second Law of Thermodynamics. In other words, if\nthe two bodies are at the same temperature, the total heat \rux densities between them\nare identical, S12=S21(integrated over all frequencies and wavevectors), independent of\nwhether the system is reciprocal or not. It is then possible to show that thermodynamics im-\nposes the same symmetry also for the individual contributions, i.e., S12(kt;!) =S21(kt;!),\nagain independent of reciprocity or the lack thereof. Interestingly, Ref.38demonstrated\nthat reciprocity imposes another symmetry constraint for opposite in-plane wavevectors:\nS12(kt;!) =S12(\u0000kt;!) =S21(\u0000kt;!). In a nonreciprocal system, this symmetry con-\nstraint can be violated since the dispersion diagram is no longer necessarily symmetric,\n!(kt)6=!(\u0000kt). As a result, S12(kt;!)6=S12(\u0000kt;!) =S21(\u0000kt;!), which implies that if\nheat transfer from body 1 to body 2 occurs through a certain wavevector-channel, k=kz+kt,\nthe transfer from body 2 to 1 does not have to occur symmetrically through the time-reversed\nchannel,kTR=\u0000kz\u0000kt=\u0000k, whose contribution can even be zero; the reverse transfer\ncan now occur entirely through a di\u000berent channel with the same kt, i.e., the specular one,\nkS=\u0000kz+kt. This results in a persistent heat current between the two bodies even when\nthey are kept at the same temperature.37,38In other words, while a nonreciprocal body may\n27emit strongly in a certain direction de\fned by k(at a certain frequency), it could be designed\nto absorb very weakly in that same direction (i.e., for \u0000k) and frequency, corresponding to\na breaking of Kirchho\u000b's law of thermal radiation (equal absorptivity and emissivity at each\ndirection and frequency).30,76{78We stress again that the existence of a persistent heat cur-\nrent for bodies at the same temperature or a violation of Kirchho\u000b's law in these systems\ndoes not break thermodynamic laws, since the total net heat \row into each body is still zero\nwhen the bodies have the same temperature. In the following, we provide the \frst theoret-\nical demonstration of these e\u000bects in a platform that does not require magneto-optical or\nspace-time-modulated media.\nFirst we consider the reciprocal scenario with no current bias in either the lower or upper\nbody. In this situation the material permittivity takes a local and isotropic form, where we\nassume the same plasma frequency, !p, and scattering rate, \u0000 = 0 :1!p, for both materials.\nFigure 6(b) shows the heat \rux density at each frequency and in-plane wavenumber. As\nexpected for reciprocal materials, the heat \rux density is symmetric with respect to in-\nplane wavenumber, S12(kt;!) =S12(\u0000kt;!), and the same plot is obtained for any in-plane\ndirection \t since the considered material is homogeneous and isotropic. We also veri\fed\nthat this wavevector- and frequency-resolved heat transfer map from body 1 to 2, S12(kt;!),\nis identical to the corresponding one from body 2 to 1, S21(kt;!), therefore respecting the\nSecond Law. We also note that the speci\fc shape of the heat \rux density map in Fig. 6(b)\nis a result of coupling between the surface waves supported by the two interfaces, which\nstrongly modi\fes their individual dispersion diagram. For subwavelength gaps between the\ntwo slabs, as in Fig. 6, the structure supports strongly con\fned SPPs, for which even\nrelatively small values of drift velocity determine large changes in their response, as seen in\nour results discussed in the following.\nWe make the system nonreciprocal by biasing the top conductor with a direct current\nwithvd=c=300 (much lower than in previous sections) along the + x-axis. The heat trans-\nfer spectra for this nonreciprocal system are shown in Figs. 6(c,d,e) for di\u000berent in-plane\n28Figure 6: (a) Illustration of the geometry considered for investigating the near-\feld radiative\nheat transfer between two planar bodies in the reciprocal and nonreciprocal (current-biased)\ncases. Two planar conducting slabs are placed at a distance dg=\u0015p=100 from each other\nand one of them is biased by a DC drift current. The thickness of the slabs is d=\u0015p, where\n\u0015pis the free-space wavelength at the plasma frequency. The two slabs are supposed to have\nidentical plasma frequency and damping rate, \u0000 = 0 :1!p. (b) Heat \rux density between the\ntwo bodies as a function of frequency and in-plane wavenumber when the current bias is set\nto zero. Identical heat \rux density maps are obtained in this case for any in-plane direction\nde\fned by the angle \t with respect to the + x-axis. (c,d,e) Heat \rux density for the same\ncon\fguration but with the top conducting body biased by a DC current with vd=\u0000c=300\nalong the + x-axis. The three panels show the heat \rux density map for di\u000berent in-plane\ndirections. While the system being studied is, strictly speaking, an active non-equilibrium\nsystem, we operate in a regime with relatively large separations and low velocities, where\nnon-equilibrium phenomena are still small, but nonreciprocal e\u000bects are clearly visible, as\nfurther discussed in the text.\nwavevector directions, de\fned by the angle \t with respect to the current \row. As seen\nfrom these results, S12(kt;!)6=S12(\u0000kt;!) for any in-plane direction except at \t = 90 deg.,\nwhich is consistent with our previous observation that there is no interaction between surface\nwaves and drifting electrons for wavevectors normal to the current \row. In this latter situa-\ntion, the heat \rux density map is exactly the same as in the reciprocal scenario in Fig. 6(b).\n29These results clearly demonstrate that such a magnet-free current-biased system exhibits a\nsignature of nonreciprocity for near-\feld radiative heat transfer, which implies, as mentioned\nabove, the presence of a persistent heat current between the two bodies even when they are\nkept at the same temperature. The asymmetric heat transfer spectra in Figs. 6(c,d) are\nsimilar, but not exactly identical, to the ones obtained with magneto-optical materials in.38\nIn particular, the heat transfer spectra in38follow the characteristic magnetic-\feld-induced\nasymmetry in the \rat dispersion asymptotes of surface magneto-plasmons, whereas our re-\nsults in Figs. 6(c,d) show an asymmetry that arises from a Doppler-shift-induced tilt of the\ndispersion asymptotes. Furthermore, we have again veri\fed that the current-biased system\nstill respects the constraint S12(kt;!) =S21(kt;!), which needs to be satis\fed regardless of\nthe local or nonlocal and reciprocal or nonreciprocal nature of the system.\nWe reiterate that the results and considerations above are valid as long as we operate\nin a regime where non-equilibrium e\u000bects due to the drifting electrons are negligible (the\nequilibrium \ructuation-dissipation theorem applies if !\u001dv=dg75). Indeed, the distance\nbetween the two slabs in Fig. 6 is much larger than the atomic distances considered in\nthe study of quantum friction in Ref.72(if graphene was used as the plasmonic material,\noperating at mid-infrared wavelengths, the distance in Fig. 6 would be on the order of\n100 nm, three orders of magnitude larger than in Ref.72). Moreover, while the two-body\nsystem being studied is active and could, in principle, exhibit optical gain, negative Landau\ndamping, and instabilities, as demonstrated for similar geometries in Refs.46,52, we have\nveri\fed that, in our case, these e\u000bects are negligible as the distance between layers is two\norders of magnitude larger ( dg\u001910\u00002\u0015for the case studied here, whereas dg\u001910\u00004\u0015in\nRef.46), for the same relative drift velocity. The analysis of small nonequilibrium corrections\nto our results and, more broadly, the full study of the nonequilibrium thermodynamics of\nsuch a nonreciprocal active system for smaller distances and higher velocities is potentially\na very interesting area of research and will be the subject of future investigations.\n30Conclusion\nThe possibility of breaking reciprocity using a direct drift-current bias in conducting materi-\nals has been known for some time, but is starting to receive more attention only recently. The\nexisting literature has been mostly focused on unidirectional propagation e\u000bects along the\ndirection of the current and/or on speci\fc platforms such as graphene. Here, instead, using a\n3D Green function formalism and microscopic considerations (starting from the Vlasov equa-\ntion, as further discussed in the Supplemental Material), we have studied current-induced\nnonreciprocal e\u000bects on the entire two-dimensional surface of a generic three-dimensional\nplasmonic platform, elucidating the dispersion and propagation properties of nonrecipro-\ncal SPPs and revealing a number of anomalous and extreme wave-propagation e\u000bects be-\nyond unidirectional propagation, including the possible excitation of frozen-light, steerable,\nsurface-wave beams with enhanced and localized \felds, and the presence of exceptional-point-\nlike modal transitions at in\rexion points. Since plasmonic media are inherently associated\nwith absorption losses, we have also clari\fed the impact of dissipation (due to collisions and\nLandau damping) on nonreciprocal current-biased SPPs and the associated tradeo\u000bs.\nWe have then focused on a particularly relevant example of application of these con-\ncepts, namely, the problem of breaking reciprocity for radiative heat transfer, which is usu-\nally achieved using magneto-optical materials. We have theoretically demonstrated a clear\nsignature of nonreciprocity in this context, namely, an asymmetry in the radiative heat \rux\ndensity between two planar bodies for opposite in-plane wavevectors, by relying, for the\n\frst time, on a direct drift-current bias instead of a magnetic bias. Our \fndings may open\nnew opportunities and directions toward the long-sought goal of controlling radiative heat\ntransfer without the constraints of time-reversal symmetry and reciprocity. Importantly,\nnonreciprocity in this context can be obtained with relatively low values of drift velocity,\nwhich could be practically realizable in certain high-mobility plasmonic media, such as high-\nmobility semiconductors and graphene in the THz, far-IR, and mid-IR spectral range. To\ndemonstrate the nonreciprocal e\u000bects discussed in this paper, these are much more promis-\n31ing material platforms than noble metals, such as gold and silver, which have relatively low\nelectron mobility. As an example, while in Ref.47a gold nanowire carrying a realistic elec-\ntric current on the order of tens of mA was predicted to support a modest drift velocity\nvd=c\u001910\u00006, producing very small nonreciprocal e\u000bects, if the same nanowire was made\nof high-mobility InSb, which exhibits plasmonic response in the low THz range,79drift ve-\nlocities up to vd=c\u001910\u00003would be achievable, comparable to the values considered in\nthis paper, for even smaller currents. Finally, thanks to its ultra-high electron mobility\nand mid-infrared plasmonic response, graphene is arguably the most interesting material\nfor drift-biased nonreciprocal plasmonics,46,50,54enabling drift velocities on the order of its\nFermi velocity, i.e., vd=c=300.\nIn summary, we believe that the rich physics of drift-biased plasmonic systems, which\nenable nonreciprocal, slow-light, active, tunable e\u000bects at the nanoscale, may open up novel\nopportunities, combining the advantages of plasmonics and nonreciprocal electrodynamics,\nfor both wave-guiding applications and thermal photonics.\nFunding\nThe authors acknowledge support from the Air Force O\u000ece of Scienti\fc Research with Grant\nNo. FA9550-19-1-0043 through Dr. Arje Nachman (whom we also thank for his suggestions\non this manuscript) and the National Science Foundation with Grant No. 1741694.\nReferences\n(1) F. 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E-mail:suoqing@shao.ac.cn\n2TAPIR & Walter Burke Institute for Theoretical Physics, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA.\n3Physics Department, University of Otago, 730 Cumberland St., Dunedin 9016, New Zealand\nABSTRACT\nWe investigate the possibility of cosmic ray (CR) confinement by charged dust grains through resonant drag instabilities\n(RDIs). We perform magnetohydrodynamic particle-in-cell simulations of magnetized gas mixed with charged dust and\ncosmic rays, with the gyro-radii of dust and GeV CRs on \u0018AUscales fully resolved. As a first study, we focus on one\ntype of RDI wherein charged grains drift super-Alfvénically, with Lorentz forces strongly dominating over drag forces.\nDust grains are unstable to the RDIs and form concentrated columns and sheets, whose scale grows until saturating\nat the simulation box size. Initially perfectly-streaming CRs are strongly scattered by RDI-excited Alfvén waves, with\nthe growth rate of the CR perpendicular velocity components equaling the growth rate of magnetic field perturbations.\nThese rates are well-predicted by analytic linear theory. CRs finally become isotropized and drift at least at \u0018𝑣Aby\nunidirectionalAlfvénwavesexcitedbytheRDIs,withauniformdistributionofthepitchanglecosine 𝜇andaflatprofile\noftheCRpitchanglediffusioncoefficient 𝐷𝜇𝜇around𝜇=0,withoutthe“ 90\u000epitchangleproblem.”WithCRfeedback\non the gas included, 𝐷𝜇𝜇decreases by a factor of a few, indicating a lower CR scattering rate, because the backreaction\non the RDI from the CR pressure adds extra wave damping, leading to lower quasi-steady-state scattering rates. Our\nstudydemonstratesthatthedust-inducedCRconfinementcanbeveryimportantundercertainconditions,e.g.,thedusty\ncircumgalactic medium around quasars or superluminous galaxies.\nKey words: cosmic rays — plasmas — methods: numerical — MHD — galaxies: active — ISM: structure\n1 INTRODUCTION\nThe transport physics of cosmic rays (CRs) has been a subject of\nactive investigation since the 1960s (Kulsrud & Pearce 1969). As\ncharged ultra-relativistic particles coupled to magnetic fields via\nLorentz forces, CRs are fundamentally governed by particle-wave-\ninteractionswithmagneticfluctuations(e.g.,Alfvénwaves).Gener-\nally speaking, when Alfvén waves are excited with wavelengths\nbroadly similar to the CR gyro radii, CRs are scattered toward\nisotropy in the wave frame from small-scale irregularities of field\nlinesandthusbecome“confined”(thenetdrift/streamingspeedrel-\native to the plasma is suppressed). The relevant confining Alfvén\nwaves can be excited by the CR streaming instability when the CR\ndrift velocity 𝑣𝐷exceeds local Alfvén velocity 𝑣A\u0011j𝑩j√︁4𝜋𝜌gas\n(Kulsrud & Pearce 1969), and/or by extrinsic turbulence (Skilling\n1971;Jokipii1966).Ontheotherhand,CRsarelessconfinedwhen\nAlfvén waves are strongly damped, via ion-neutral damping, non-\nlinear Landau damping (Lee & Völk 1973) or through the magne-\ntohydrodynamic(MHD)turbulencecascade(Yan&Lazarian2002;\nFarmer & Goldreich 2004). Therefore, studying the excitation and\ndamping mechanisms of Alfvén waves around these gyro-resonant\nscales is crucial to understanding of CR transport.\nRecently, a number of numerical studies have attempted to\nmodel the effects of CRs around energies of \u0018GeV(which dom-\ninate the CR energy density) on ISM and galactic scales. These\nstudies suggest that CRs can have a significant influence on galac-\ntic “feedback processes” regulating star and galaxy formation (e.g.,\nPakmoretal.2016;Ruszkowskietal.2017;Farberetal.2018;Chan\net al. 2019; Hopkins et al. 2019; Buck et al. 2020; Su et al. 2020)\nand the phase structure and nature of the circumgalactic medium\n(CGM) (e.g., Salem et al. 2016; Butsky & Quinn 2018; Ji et al.\n2020, 2021). Meanwhile, classical models of Galactic CR trans-\nport which compare to Solar System CR experiments (Strong &\nMoskalenko 2001; Jóhannesson et al. 2016; Evoli et al. 2017) have\nsuggested the potential for new breakthroughs in particle physics\nfromdetailedmodelingofratiosofe.g.secondary-to-primaryratiosinCRpopulations.However,inallofthesestudiesafluidorFokker-\nPlanck description of CR transport is required, usually with some\nsimple assumption that CRs have a constant “diffusion coefficient”\nor“streamingspeed”oreffectivescatteringrate.Butthisintroduces\nsignificant uncertainties, as the micro-physical behavior and even\nthequalitativephysicaloriginofthesescatteringrates(andtherefore\ntheir dependence on plasma properties) remains deeply uncertain.\nThis uncertainty is vividly demonstrated in Hopkins et al. (2020a):\ncomparing different CR transport scalings based on local plasma\npropertiesaccordingtodifferentproposedscattering-ratemodelsin\nthe literature, Hopkins et al. (2020a) showed that existing models\ncould(1)differbyfactorslargerthan \u0018106intheirpredictedeffec-\ntive transport coefficients, and (2) even models with similar mean\ndiffusion coefficients could produce (as a consequence of different\ndetaileddependenceonplasmaproperties)qualitativelydifferentCR\ntransportoutcomesandeffectsonISMandCGMproperties.There-\nfore,itisparticularlyimportanttogobeyondsimplefluidtreatments\nofCRsbyinvestigatingexplicitCRscatteringphysicsanddynamics\non the scales of the CR gyro-radius 𝑟gyrocr(\u0018AUforGeVCRs in\ntypical Solar-neighborhood ISM conditions), where the interaction\nbetween CRs and magnetic fluctuations can be fully resolved.\nOne potentially important piece of physics on these scales,\nwhich has been almost entirely neglected in the historical CR lit-\nerature, is the role of dust grains. Recently, Squire et al. (2020)\nnoted a remarkable coincidence: under a broad range of ISM con-\nditions, the gyro-radii of charged dust grains in the ISM (with sizes\n\u001810\u00003\u00001𝜇𝑚) and\u001801\u000010GeV CRs overlap. While the grains\nhave much lower charge-to-mass ratios, they also have much lower\nvelocities, giving nearly coincident gyro radii. As a result, Squire\net al. (2020) proposed that dust grains can influence \u0018GeVCR\ntransport on gyro-resonant ( \u0018AU) scales in two ways: (1) as iner-\ntial particles, dust can damp parallel Alfvén waves excited by the\nCR streaming instability and thus reduce CR confinement, and (2)\ndust can be unstable to the so-called “resonant drag instabilities”\n(RDIs; Squire & Hopkins 2018), a recently-discovered, formally\n©0000 The AuthorsarXiv:2112.00752v2  [astro-ph.HE]  22 Apr 20222Ji et al.\ninfinitely-large family of instabilities that appear in different forms,\nwhenever dust grains move through a background fluid with some\nnon-vanishing difference in the force acting on dust vs. fluid. The\nRDIs can excite small-scale parallel Alfvén waves in magnetized\ngas (Hopkins & Squire 2018; Seligman et al. 2019), which can in\nturn scatter CRs and enhance their confinement. Which of these is\nthe dominant process, or whether still different forms of the RDIs\ncan be excited, depend on the local plasma conditions, with sce-\nnario(s) being more likely in regions where the gas is relatively\ndiffuse and external acceleration of dust grains is relatively strong\n(from e.g. absorbed photon momentum around a bright source). In\nanycase,theco-existenceofCRsandchargeddustgrainsintheISM\nis self-evident. Even in the CGM far from galaxies, where Squire\netal.(2020)suggestscenario(2)wouldbemorelikely,theexistence\nof significant dust populations is theoretically and observationally\nplausible: for instance, in the cool ( \u0018104\u0000105K) or warm-hot\n(105\u0000106K) phases of the CGM where the gas number density\nis low enough ( .10\u00003cm\u00002), the hadronic and Coulomb losses of\nCRsaresmall(Guo&Oh2008),andthetemperatureanddensityare\nlowenoughthatthedustsputteringtimeislong(Tielensetal.1994).\nInfact,asignificantamountofdusthasalreadybeenobservedinthe\nCGM (Ménard et al. 2010; Peek et al. 2015). And around quasars\nor superluminous galaxies, dust grains could in principle easily be\naccelerated by radiation pressure and become unstable to the spe-\ncificRDIswhichwouldsourcestrongCRscatteringon \u0018AUscales\naccording to Squire et al. (2020). Dust clumping can scatter and\nre-emitphotonsfromQSOs(e.g.,Hennawietal.2010;Martinetal.\n2010), and thus might produce observables in absorption lines of\nthe CGM. Nevertheless, this scenario remains largely unexplored;\neven for the pure dust RDIs without CRs, the only simulation study\nthus far, in Hopkins et al. (2020b), has simulated the CGM cases\nat a resolution of\u001810 pc, far above AUscales on which CR-dust\ncoupling occurs.\nMotivated by the above, in this paper we present the very first\nnumerical study of CR-dust-MHD coupling, with resolved CR and\ndust gyro-radii. In particular, here we focus on the proposed CR\nconfinement scenario in which CRs are scattered by magnetic field\nirregularities caused by the specific forms the RDI described in\nSquire et al. (2020). The paper is organized as follows. In §2, we\ndescribe the numerical methods and setup of our simulations. §3\npresents and analyzes the simulation results. We finally summarize\nour findings in §4.\n2 METHODS & SIMULATION SETUP\n2.1 Numerical Methods\nThe numerical methods adopted here consist of three main com-\nponents, each of which has been well-studied separately: the MHD\nsolverinthecode GIZMO(Hopkins2015),dustgrainsevolvedwith\na standard super-particle method (Hopkins & Squire 2018; Selig-\nman et al. 2019), and CR particles evolved with a hybrid MHD\nparticle-in-cell (MHD-PIC) method (Bai et al. 2015, 2019). De-\ntailed implementations are described in the above-cited references,\nand we only briefly review them here.\nThe background plasma/fluid/gas and magnetic fields follow\nthe equations of ideal MHD, solved in GIZMOwith the well-tested\nconstrained-gradient meshless finite-volume Lagrangian Godunov\nmethod (Hopkins & Raives 2016; Hopkins 2016). This has been\nshown to accurately reproduce a variety of detailed MHD phenom-\nena including amplification, shocks, detailed structure of the mag-\nnetorotationalandmagnetothermalinstabilities,andmore(Hopkins\n2017; Deng et al. 2019; Grudić et al. 2020). To this we add the\n“back-reaction” or feedback force of CRs and dust on gas, detailed\nbelow.IndividualgrainsandCRsareintegratedasPIC-like“super-particles” as is standard in the literature for both (Carballido et al.\n2008;Johansenetal.2009;Bai&Stone2010;Panetal.2011;McK-\ninnon et al. 2018; Bai et al. 2015, 2019; Holcomb & Spitkovsky\n2019; van Marle et al. 2019), sampling the distribution function\nstatistically by taking each super-particle to represent an ensemble\nof identical micro-particles (individual grains or CRs). Dust grains\nobey the equation of motion\n𝑑𝒗dust\n𝑑𝑡=𝒂extdust¸𝒂gasdust (1)\nwhere 𝒗dustis the grain velocity, 𝒂extdustand𝒂gasdustare grain\naccelerationsfromexternalforces(e.g.gravity,radiation)andback-\ngroundgasandmagneticfieldsrespectively.Thelatterincludesboth\ndrag and Lorentz forces:\n𝒂gasdust=\u0000𝒗dust\u0000𝒗gas\n𝑡𝑠dust\u0000\u0000𝒗dust\u0000𝒗gas\u0001\u0002ˆ𝑩\n𝑡𝐿dust(2)\nwhere ˆ𝑩=𝑩j𝑩jis the unit vector of the magnetic field 𝑩,𝑡𝑠dust\nthe dust drag drag or stopping time and 𝑡𝐿dustthe dust Larmor or\ngyro time. Since we are interested in regimes with super-sonic drift\nand microscopic dust grains (the Epstein drag limit), the drag and\ngyro timescales are given by:\n𝑡𝑠dust\u0011√︂𝜋𝛾\n8𝜌𝑖\ndust\n𝜌gas𝜖𝑑\n𝑐𝑠 \n1¸9𝜋𝛾\n128\f\f𝒗dust\u0000𝒗gas\f\f2\n𝑐2𝑠!\u000012\n(3)\n𝑡𝐿dust\u0011𝑚dust𝑐\n𝑞dust𝑩=4𝜋𝜌𝑖\ndust𝜖3\ndust𝑐\n3𝑒j𝑍dust𝑩j (4)\nwhere𝛾istheusualadiabaticindexofthegas,and 𝜌𝑖\ndust,𝜖dust,𝑚dust\nand𝑞dust\u0011𝑍dust𝑒are the internal density, radius, mass and charge\nof dust grains respectively.\nSimilarly, the equation of motion for CRs is:\n\u0010𝑐\n˜𝑐\u0011𝑑𝒖cr\n𝑑𝑡=𝒂extcr¸𝒂gascr=𝒂extcr¸\u0000𝒗cr\u0000𝒗gas\u0001\u0002ˆ𝑩\n𝑡𝐿0cr(5)\nwhere𝑐is the speed of light (and see below for ˜𝑐), and 𝒗cris the\nvelocityoftheCRparticles,whichisrelatedtotheCRfour-velocity\n𝒖cr\u0011𝛾𝐿𝒗crvia the usual Lorentz factor 𝛾𝐿\u0011¹1\u0000j𝒗crj2𝑐2º\u000012.\nThe𝑡𝐿0cr\u00111Ωcr0is the usual non-relativistic CR gyro time\n𝑡𝐿0cr\u0011¹𝑚cr𝑐º¹𝑞cr𝑩º, where𝑚crand𝑞crare the CR mass and\ncharge.\nIn the ideal MHD approximation, the “feedback” force from\ndust grains and CRs appears simply in the gas momentum equation\nas an equal-and-opposite force as from gas onto dust+CRs:\n𝜌gas\u0012𝜕\n𝜕𝑡¸𝒗gas\u0001r\u0013\n𝒗gas=\u0000r𝑃\u0000𝑩\u0002¹r\u0002 𝑩º\n4𝜋\n\u0000∫\n𝑑3𝒗dust𝑓dust¹𝒙𝒗dustº𝒂gasdust¹𝒗dustº\n\u0000∫\n𝑑3𝒗cr𝑓cr¹𝒙𝒗crº𝒂gascr¹𝒗crº\n(6)\nwhere𝑓dust¹𝒙𝒗dustºand𝑓cr¹𝒙𝒗crºare the phase-space density\ndistributions of dust and CRs respectively. Non-Lorentz forces are\nintegrated with a semi-implicit scheme and Lorentz forces using\na Boris integrator, with the back-reaction terms implemented in a\nmanner that ensures manifest machine-accurate total momentum\nconservation. These methods have been detailed and extensively\ntested in Hopkins & Lee (2016); Lee et al. (2017); Moseley et al.\n(2019); Seligman et al. (2019); Hopkins et al. (2020b).\nNotethatuptothedetailedformofthegyroaccelerationequa-\ntion, the expressions for dust and gas evolution are functionally\nidentical–indeedwecannumericallythinkofthedustasasecond,\n“heavy” (low charge-to-mass, non-relativistic) CR species which\nMNRAS 000, 000–000 (0000)CR Confinement by Dust 3\nalsoexperiencesdrag,orthinkoftheCRsas“relativistic,drag-free\ngrains.”\nFinally,toavoidtheCourantconditionfortheCRspeed 𝑐lead-\ning to computationally impractical timesteps in our simulation, we\nadopt the reduced speed-of-light (RSOL) approximation by defin-\ning the RSOL ˜𝑐in Eq. (5) ˜𝑐  𝑐(but still keeping ˜𝑐much larger\nthan other velocities in our simulation). As shown in Ji & Hop-\nkins (2021), the particularly form of the RSOL implemented here,\nwhich simply modifies the CR acceleration by the power of ˜𝑐𝑐\nand defines the CR advection speed over the (non-relativistic) grid\nto be 𝒗advectcr\u0011𝜕𝒙cr𝜕𝑡=¹˜𝑐𝑐º𝒗cr, ensures that all steady-state\npropertiesoftheCRdistributionfunction(number,momentum,en-\nergy density, pitch-angle distribution) are mathematically invariant\nto the choice of ˜𝑐. With this new implementation, we are able to\nsimultaneouslymatchtheCRenergydensity,massdensity, andmo-\nmentumdensityofanydesiredinitialconfiguration,andaccountfor\nthe“correct”CRgyro-radiusandback-reactionforceswhicharealso\nindependentofthechoiceof ˜𝑐.Thisalsomeansthatthroughoutthis\npaperwhenwerefertoe.g.CRgyro-frequenciesandotherstandard\nquantities,theyhavetheirusualmeaningandvalues(i.e.unlessoth-\nerwise specified, the RSOL ˜𝑐does not enter our expressions here).\n2.2 Simulation Parameters & Motivation\nIn what follows, we consider a system where dust grains are accel-\nerated by some large 𝒂extdust, for example from radiation pressure\nfrome.g.aquasarorluminousstarburstintheirhostgalaxy,inlow-\ndensity magnetized gas representative of e.g. the CGM or ionized\nISM bubbles. In these cases the fastest growing of the dust RDIs\nare generally the “aligned cosmic-ray like” (so named because the\neigenmodestructurebroadlyresemblestheBell2004instability)or\n“dust gyro-resonant” RDIs, which lead to rapid growth of parallel\nAlfvén waves which can scatter CRs. For the cases of interest, the\nexternal forces on the CRs 𝒂extcrare negligible.\nBecause this is a first study, to simplify the dynamics as much\nas possible and have a well-defined, resolved CR and dust gyro-\nradius, we consider just a single species of dust and single species\nofCRs.Inrealityofcoursetherewillexistabroadspectrumofdust\nsizes, with different charge and mass, e.g., and likewise spectrum\nof CR energies; but we will focus on parameters representative of\nthe grains that contain most of the dust mass and dominate the\n“feedback”force,aswellastheCRswhichdominatetheCRenergy\ndensity/pressure. We adopt a 3D box with a side-length of 𝐿box\nandperiodicboundaryconditions,filledwithinitiallyhomogeneous\ngas, dust, and CRs, with a uniform magnetic field whose initial\ndirectiondefinesthe 𝑧axis( 𝑩0=j𝑩0jˆ𝑧)anduniformvelocityfields\n(i.e. the initial CR and dust distribution functions are taken to be\n𝛿-functions, though we discuss relaxing this below), and impose an\nisothermal¹𝛾=1)equation-of-stateonthegas(motivatedbytypical\ncooling physics in the ISM/CGM). We adopt the same particle/cell\nnumbers for gas, dust and CRs, i.e., 𝑁gas=𝑁dust=𝑁cr=𝑁3\n1D,\nwhere𝑁1Dis the 1D resolution along each sides of the simulation\nbox,𝑁1D=64in our fiducial simulations (so the box contains\n3\u0002643elements),andwealsoperformedadditionalhigh-resolution\nsimulationswith 𝑁1D=128forconvergencetests.Weinitializethe\ndust drift velocity with its homogeneous equilibrium solution (see\nHopkinsetal.2020b),thegasvelocity 𝒗gas=0,andtheCRvelocity\n𝒗cr=𝑣crˆ𝑧, defined below.\nEvenwiththesesimplifications,writingthesimulationparam-\neters in units of the initial gas density 𝜌0g, sound speed 𝑐0𝑠and\nbox size𝐿box, our simulations require we specify ten dimension-\nless numbers: (1) The plasma beta 𝛽\u0011 ¹𝑐𝑠𝑣Aº2, where𝑣A\u0011\n𝑩√︁4𝜋𝜌gasis the Alfvén velocity; (2) The box-averaged dust-to-\ngasmassratio 𝜇dust\u0011𝜌dust𝜌gas;(3)Thedustexternalacceleration\n¯𝑎dust\u0011j𝒂extdustj𝐿box¹𝑐0𝑠º2; (4) The dust “size parameter” ¯𝜖dust\u0011𝜌𝑖\ndust𝜖dust𝜌0g𝐿box, which determines the grain drag forces; (5)\nThe dust “charge parameter” ¯𝜙dust\u00113𝑍dust𝑒¹4𝜋𝑐𝜖2\ndust¹𝜌0gasº12º,\nwhichdeterminesthegraincharge-to-massratioandLorentzforces;\n(6)Theangle cos¹𝜃dustº\u0011\f\fˆ𝑩0\u0001ˆ𝒂extdust\f\fbetweeninitial 𝐵-fielddi-\nrectionandexternaldustacceleration;(7)TheCR-to-gasmassratio\n𝜇cr\u0011𝜌cr𝜌gas(orequivalentlytheCRnumberratio 𝑛cr𝑛gas,since\nwe are interested in CR protons); (8) The CR “charge parameter”\n¯𝜙cr\u0011𝑞cr¹𝑐𝑚crº𝐿box¹𝜌0gasº12, which encodes the CR charge-to-\nmass ratio; (9) The angle cos¹𝜃crº\u0011\f\fˆ𝑩0\u0001ˆ𝒗cr\f\fbetween the initial\nmagnetic fields and CR velocity. (10) The initial CR Lorentz factor\n𝛾𝐿(or equivalently initial CR momentum 𝑝cr).\nThis forms an enormous parameter space, which is impos-\nsible to explore concisely. We therefore focus on one particular\nparameter set in this first study, motivated heuristically by the sce-\nnario proposed in Squire et al. (2020) and discussed above. Con-\nsider typical interstellar/intergalactic silicate or carbonaceous dust\n(𝜌𝑖\ndust\u00181 g cm\u00003, with𝜖dust\u001801𝜇mgrains containing most of\nthe mass, a standard ISM-like dust-to-gas ratio, and obeying the\nstandard collisional+photoelectric charge law from e.g. Draine &\nSutin 1987; Tielens 2005) and typical \u00181GeV kinetic energy CRs\nwhich dominate the CR energy density ( 𝛾𝐿\u00182protons). These\nproposesinamediumwithparameterstypicalofthewarm(volume-\nfilling) CGM outside of a galaxy (gas temperature 𝑇\u0018105K, den-\nsity𝑛\u001810\u00002cm\u00003,j𝑩j \u0018 01𝜇G, and CR-to-thermal pressure\n𝑃cr𝑃therm\u0018afew;Jietal.2020).Wewishtoresolvesomenumber\nof CR gyro radii in our box, so set 𝐿box\u001810𝑟𝐿cr.1This gives\nthe numerical parameters2𝛽\u00192\u0002103,𝜇dust\u0019001,¯𝜖dust\u0019105,\n¯𝜙dust\u001916\u0002106,𝜇cr\u001936\u000210\u00007,¯𝜙cr\u001913\u0002105,𝛾𝐿\u00192.3\nWhatremainsistheexternaldustacceleration ¯𝑎dust,whichistheul-\ntimatesourceofenergyfortheRDIs:alargervalueofthisparameter\ncorrespondstomorerapidRDIgrowthrates.Squireetal.(2020)con-\nsideredsuper-sonicdustdriftvelocities( 𝑣drift\u0011j𝒗dust\u0000𝒗gasj),such\nthatcertainRDIswereexcited,sowechoose ¯𝑎dust=12\u000210\u00003such\nthattheinitialequilibriumdustdriftvelocityis 𝑣drift\u001910𝑐0𝑠(safely\nsuper-sonic).4As discussed in Hopkins & Squire (2018); Hopkins\net al. (2020b) and Squire et al. (2020), while this is intentionally a\nsomewhat extreme choice, it is plausible given the observed radia-\ntive fluxes in CGM environments around super-luminous sources\nsuch as quasars and starburst galaxies, e.g., at \u001810\u0000100kpc from\na source with luminosity \u00181012\u00001013𝐿\f.\nIt is easy to verify that under these conditions, Lorentz forces\non dust grains strongly dominate over drag forces (by a factor of\n\u0018104), as desired for our problem of interest, and that both CRs\nanddusthaveverysimilargyro-radii, \u001806 AUinthephysicalunits\nabove.\n1More precisely, we set 𝐿box=p𝑁1D𝑟𝐿crsuch that each Larmor wave-\nlengthisresolvedwith \u0018p𝑁1Delementsandtheboxcontainsp𝑁1DLarmor\nwavelengths, which provides an optimal compromise for our purposes. For\nboxes with different resolution, we rescale the numerical parameters to cor-\nrespond to fixed physicalquantities (e.g. fixed gyro radii) while rescaling\nthe box size to follow this relation. Here and throughout, we define the\ngyro-radius specifically as the gyro radius for an equivalent circular orbit,\n𝑟𝐿cr=j𝒗crj𝑡𝐿cr.\n2For our low-resolution 𝑁1D=64boxes, the parameters which depend\nexplicitly on𝐿boxrescale to ¯𝜖dust\u001915\u0002105,¯𝜙cr\u00199\u0002104,¯𝑎dust\u0019\n85\u000210\u00004.\n3We also set reduced speed of light to ˜𝑐=005𝑐\u0019230𝑐0𝑠, ensuring it is\nlarger than other speeds in the problem, but explicitly examine convergence\nwith respect to this choice below.\n4As there is no particular reason to expect alignment between 𝑩and\n𝒂extdust, we set𝜃dust=45\u000e, but this is largely a nuisance parameter.\nMNRAS 000, 000–000 (0000)4Ji et al.\n2.3 On the Initial CR Pitch Angle Distribution\nNote that our initial condition for the CRs corresponds to all CRs\nhaving pitch angle =0(𝜇=1), i.e. free-streaming directly down\nmagnetic field lines at drift velocity 𝑣0\n𝐷\u0018𝑐with the maximum-\npossible anisotropy in the CR distribution function. This is almost\ncertainly unrealistic, but our purpose is to study how CRs would\nbe scattered away from this anisotropy by dust, hence the choice.5\nHowever one consequence of this choice is that for the parameters\nabove (with¹𝑒cr𝑣0\n𝐷º¹𝑒𝐵𝑐º¡1, where𝑒crand𝑒𝐵are the CR\nand magnetic energy densities), the non-resonant Bell instability\nwouldgrowmuchfasterthantheCRresonantinstability(Bell1978;\nHaggerty et al. 2019) and much faster than the RDIs (by a factor\n\u0018¹𝜇cr𝜇dustº12¹𝑣0\n𝐷𝑣0\ndustº¹𝑡0\n𝐿cr𝑡0\n𝐿dustº\u00001\u001d1). And indeed we\nhave verified this directly in test simulations, which also allow us\nto confirm that, in code, both the non-resonant and resonant CR\ninstabilities grow initially at their expected linear growth rates.\nA more realistic initial condition would feature a close-to-\nisotropic CR distribution function, which would strongly suppress\ntheseinstabilitiesrelativetotheRDIs,andwealsoconsiderthiscase\nbelow.Howeverforafirstsimulation,inordertoseehowCRswould\nbe scattered from an arbitrarily anisotropic distribution with 𝜇=1\n(whichisparticularlyusefulforunderstandingthescatteringphysics\nthemselves),weconsiderthe“ultra-lowCRdensity”or“test-particle\nCR” limit. This amounts to to ignoring the CR feedback on the gas\n(equivalently,takingtheCRnumberorenergydensitytobenegligi-\nbly small), i.e., dropping the last term on the right-hand side in Eq.\n(6). Thus, CR scattering via self-induced instabilities cannot occur\nand we can cleanly isolate the effects of the RDIs. We study this as\na test problem in §3.1 – §3.3. After the system reaches a saturation\nstate and CRs become more isotropized with 𝑣𝐷\u001c˜𝑐(a more re-\nalistic “initial condition” for the CRs), we turn on CR feedback and\ninvestigate its consequences in §3.4.\n3 RESULTS\n3.1 Time Evolution & Saturated States\nFig. 1 and 2 shows plots of the dust grain projections, gas density\nfluctuations, gas velocity perturbations superposed with velocity\nstreamlines,magneticfieldstrengthsuperposedwithfieldlines,and\ncosmic ray particle projections, viewed along the 𝑧-axis (the di-\nrection of the initial magnetic fields) and 𝑥-axis respectively. We\nchoose the snapshots at 𝑡=978𝑡𝐿dust,1022𝑡𝐿dust,1065𝑡𝐿dust\nand1304𝑡𝐿dust, which are representative samples spanning be-\ntween the end of the linear stage, the nonlinear stage and the satu-\nration stage. Dust grains are unstable to the RDI and non-linearly\nevolve into highly-concentrated columns along the 𝑧-axis, which\nmerge and form into fewer but thicker columns or sheets with time.\nAt the saturation stage, all dust grains form into one single col-\numn in our simulation box. This is expected since all wavelengths\nare unstable to RDIs, and the RDI growth rates decrease with in-\ncreasing wavelengths. Therefore, we see the merging of structures\nuntil the box-scale mode saturates. Compared to collisionless dust\ngrains which are strongly clumped, gas is only weakly compress-\nible, with density and velocity fluctuations at levels of \u00181%and\n10%respectively. As shown in Fig. 1 and 2, since field lines are\nstrongly stretched by dust grains, magnetic fields are amplified by\nuptooneorderofmagnitudefromtheirinitialvalues,andtheregions\nof field amplification tightly trace the location of clustered dust. At\nthe saturation stage, field lines become significantly distorted and\nwraparoundcolumnsofdustgrains,withthemagnetictensionforce\n5In contrast, because of drag, the only homogeneous stable equilibrium\nsolution for the dust grains is uniform streaming at the equilibrium drift\nvelocity, as we initialize.\u0018𝑩\u0002¹r\u0002𝛿𝑩º4𝜋balancingthedrivingforcefromthedustonthe\ngas\u0018𝜌dust𝒂extdust(Seligman et al. 2019). As illustrated in Fig. 3,\nthe morphology of the gas density, dust distribution and magnetic\nfield strength are highly correlated, and the system reaches approx-\nimately dynamical equilibrium between the gas pressure 𝑃gas, the\nmagnetic pressure\u0018j𝑩j28𝜋and the dust ram pressure 𝑃dust(es-\ntimated as the dust momentum flux across the surface of the dust\nfilament𝜌dust𝑣𝑧dust𝑣𝑥𝑦dust). Starting from a random distribution\ninitially, CRs strongly react to RDI-induced magnetic field pertur-\nbations, being scattered and transiently becoming highly clustered\ntogether with the dust before scattering leads to their returning to a\nnearly random spatial distribution again at the saturation stage (but\nnowwithanearly-isotropicpitchangledistribution,asweshowbe-\nlow). Initially the CRs react coherently small-scale modes (which\nhave𝜆\u001c𝑟𝐿CR) because CRs initially have a perfectly coherent\n(𝛿function) distribution function and magnetic fields are distorted\nby the RDIs; but as the CRs scatter, these local “concentrations”\ndisperse.\nFig. 4 shows the probability density functions (PDFs) of den-\nsities (top) and velocities / Lorentz factor (bottom) for gas (left),\ndust (middle) and CRs (right). Density PDFs are obtained by map-\nping particle mass onto grids, and velocity / Lorentz factor PDFs\nare weighted directly by particle masses. Compared with gas den-\nsities which only vary by \u001820%, the PDF of grain densities spans\nover two orders of magnitude and become highly non-Gaussian at\nthe non-linear and saturation stages, with a flat tail extending to\n𝜌dust𝜌0\ndust¡10(similar to other pure-RDI simulations in, e.g.,\nSeligmanetal.2019;Hopkinsetal.2020b).Therelativelyuniform\nPDFisqualitativelymaintainedinthehigh-resolutionrun,indicating\nthatifthereisacharacteristicclumpiness,it’snotyetbeingrecovered\nbythe simulation.TheCR densityPDFs arequalitativelysimilar to\nthedustdensityPDFsatthenonlinearstage,astheinitiallycoherent\nCRs are “dragged” by the RDIs, while at the saturation stage when\nCRsbecomefullyscattered,theCRdensityPDFrecovers itsinitial\nGaussianshape(forthereasonsabove).ThegasvelocityPDFssug-\ngest that gas is significantly accelerated by dust feedback up to rms\nvelocitiesh𝑣gas𝑐0𝑠i\u001810\u00002–10\u00001. Dust grains are gently deceler-\nated with time, and the velocity PDFs do not reach an equilibrium\nstatebytheendofthesimulation.TheCRLorentzfactorPDFsgrad-\nually broaden out with time, indicating CRs are mildly accelerated\nordeceleratedby\u00183%duringtheirinteractionwithlocalmagnetic\nfields (i.e., some “diffusive re-acceleration” effects with a non-zero\nCR momentum diffusion coefficient 𝐷𝑝𝑝, as will be discussed in\n§3.3).\n3.2 Growth Rates vs. Linear Theory\nWenextexaminetheamplificationofmagneticfields j𝑩𝑥j,j𝑩𝑦jand\nj𝑩𝑧j,andthegrowthoftheCRvelocitycomponentperpendicularto\nlocalmagneticfields 𝑣?cr(normalizedtothemagnitudeoftotalCR\nvelocityj𝒗crj).Bydefiningthepitchangle 𝜃astheanglebetweenCR\nvelocitiesandlocalmagneticfieldvectors,wesee 𝑣kcrj𝒗crj=𝜇and\n𝑣?crj𝒗crj=¹1\u0000𝜇2º12,where𝜇\u0011cos𝜃isthepitchanglecosine\nand𝑣kcrthe CR velocity component parallel with local magnetic\nfields. As shown in the top and middle panels of Fig. 5, j𝑩𝑥j,j𝑩𝑦j\nand𝑣?crgrow exponentially at almost the same growth rate until\nthey reach saturation at 𝑡\u001810𝑡𝐿dust. This indicates that, starting\nfrom highly anisotropic initial conditions with 𝑣?crj𝒗crj=0, CRs\nare strongly scattered by increasingly distorted magnetic fields due\nto the development of the dust RDI.\nTo estimate the magnitude of CR bulk drift velocity, we mea-\nsure the arithmetic mean values of the CR velocity components\n(the net drift velocity) over all CR particles at the saturation stage.\nWefindthat\n𝑣f𝑥𝑦𝑧gcr\u000b\n\u0018𝛼𝑐with𝛼fluctuatingbetween \u001810\u00004–\n10\u00002withtime,whichisthushighlyisotropiccomparedwiththeCR\nMNRAS 000, 000–000 (0000)CR Confinement by Dust 5\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y9.78tL,dust\n0.50\n0.25\n0.000.250.50\nx10.22tL,dust\n0.50\n0.25\n0.000.250.50\nx10.65tL,dust\n0.50\n0.25\n0.000.250.50\nx13.04tL,dust\n123456\n100101102\n100101102\n100101\nDustParticleCount\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y9.78tL,dust\n0.50\n0.25\n0.000.250.50\nx10.22tL,dust\n0.50\n0.25\n0.000.250.50\nx10.65tL,dust\n0.50\n0.25\n0.000.250.50\nx13.04tL,dust\n-10-3010-3\n-10-3010-3\n-10-2010-2\n-10-2010-2\n(ρgas−ρ0\ngas)/ρ0\ngas\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y9.78tL,dust\n0.50\n0.25\n0.000.250.50\nx10.22tL,dust\n0.50\n0.25\n0.000.250.50\nx10.65tL,dust\n0.50\n0.25\n0.000.250.50\nx13.04tL,dust\n-10-3010-310-2\n-10-2010-2\n010-1\n010-1\n(vgas−¯vgas)/c0\ns\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y9.78tL,dust\n0.50\n0.25\n0.000.250.50\nx10.22tL,dust\n0.50\n0.25\n0.000.250.50\nx10.65tL,dust\n0.50\n0.25\n0.000.250.50\nx13.04tL,dust\n5.96.06.16.26.36.4×103\n0.60.70.80.91.0×102\n1234×102\n0.020.040.060.080.100.12\n|B|//radicalbig\n4πP0\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y9.78tL,dust\n0.50\n0.25\n0.000.250.50\nx10.22tL,dust\n0.50\n0.25\n0.000.250.50\nx10.65tL,dust\n0.50\n0.25\n0.000.250.50\nx13.04tL,dust\n100101\n100101\n100101\n100101\nCRParticleCount\nFigure1. Plotsof(fromtoptobottom)dustgrainprojections,gasdensityfluctuations,gasvelocityperturbationssuperposedwithvelocitystreamlines,magnetic\nfield strengths superposed with field lines, and CR particle projections, at (from left to right) 𝑡=978𝑡𝐿dust,1022𝑡𝐿dust,1065𝑡𝐿dustand1304𝑡𝐿dust,\nviewedalongthe 𝑧-axis(parallelwiththedirectionofinitialmagneticfields).DustgrainsareunstabletotheRDIandmodesgrowandmergeuntiltheysaturate\ninalargebox-scalesheetmode,whichsignificantlydistortsandamplifiesmagneticfields.CRsstronglyrespondtoandarescatteredbydust-inducedmagnetic\nfield irregularities.\nMNRAS 000, 000–000 (0000)6Ji et al.\n0.50\n0.25\n0.000.250.50\ny0.4\n0.2\n0.00.20.4z9.78tL,dust\n0.50\n0.25\n0.000.250.50\ny10.22tL,dust\n0.50\n0.25\n0.000.250.50\ny10.65tL,dust\n0.50\n0.25\n0.000.250.50\ny13.04tL,dust\n123456\n1234567\n100101\n100101\nDustParticleCount\n0.50\n0.25\n0.000.250.50\ny0.4\n0.2\n0.00.20.4z9.78tL,dust\n0.50\n0.25\n0.000.250.50\ny10.22tL,dust\n0.50\n0.25\n0.000.250.50\ny10.65tL,dust\n0.50\n0.25\n0.000.250.50\ny13.04tL,dust\n-10-3010-3\n-10-3010-3\n-10-20\n-10-2010-2\n(ρgas−ρ0\ngas)/ρ0\ngas\n0.50\n 0.25\n 0.00 0.25 0.500.4\n0.2\n0.00.20.49.78tL,dust\n0.50\n 0.25\n 0.00 0.25 0.5010.22tL,dust\n0.50\n 0.25\n 0.00 0.25 0.5010.65tL,dust\n0.50\n 0.25\n 0.00 0.25 0.5013.04tL,dust\n-10-3010-310-2\n-10-2010-2\n010-1\n010-1\n(vgas−¯vgas)/c0\ns\nyz\ny y y\n0.50\n0.25\n0.000.250.50\ny0.4\n0.2\n0.00.20.4z9.78tL,dust\n0.50\n0.25\n0.000.250.50\ny10.22tL,dust\n0.50\n0.25\n0.000.250.50\ny10.65tL,dust\n0.50\n0.25\n0.000.250.50\ny13.04tL,dust\n5.86.06.26.4×103\n6789×103\n1234×102\n0.020.040.060.080.100.120.14\n|B|//radicalbig\n4πP0\n0.50\n0.25\n0.000.250.50\ny0.4\n0.2\n0.00.20.4z9.78tL,dust\n0.50\n0.25\n0.000.250.50\ny10.22tL,dust\n0.50\n0.25\n0.000.250.50\ny10.65tL,dust\n0.50\n0.25\n0.000.250.50\ny13.04tL,dust\n123456\n12345678\n2468\n1234567\nCRParticleCount\nFigure 2. Plots as Fig. 1, but viewed along the 𝑥-axis (perpendicular to the direction of initial magnetic fields).\nMNRAS 000, 000–000 (0000)CR Confinement by Dust 7\n0.5\n0.00.5\ny0.4\n0.2\n0.00.20.4zPgas\n0.5\n0.00.5\nyPmag\n0.5\n0.00.5\nyPgas+Pmag\n0.5\n0.00.5\nyPdust\n0.5\n0.00.5\nyPgas+Pmag+Pdust\n0.9751.0001.0251.050\n10-310-210-1\n0.9751.0001.0251.050\n10-310-210-1\n0.9751.0001.0251.050\nFigure 3. Slice plots of (from left to right) gas thermal pressure 𝑃gas, magnetic pressure 𝑃mag, the sum of gas thermal pressure and magnetic pressure\n𝑃gas¸𝑃mag,dustrampressure 𝑃dustestimatedasdustmomentumfluxacrossthesurfaceofthedustfilament 𝜌dust𝑣𝑧dust𝑣𝑥𝑦dust,andthesumofallpressure\nterms𝑃gas¸𝑃mag¸𝑃dustin code units of 𝜌0gas¹𝑐0𝑠º2, at the saturation stage when 𝑡=1304𝑡𝐿dust. The sum of pressure terms is nearly a constant spatially,\nindicating the system is roughly in pressure balance.\n0.85 0.90 0.95 1.00 1.05\nρgas//angbracketleftbig\nρ0\ngas/angbracketrightbig10-410-310-210-1100PDF\n100101\nρdust//angbracketleftbig\nρ0\ndust/angbracketrightbig10-510-410-310-210-1PDF\n100101\nρcr//angbracketleftbig\nρ0\ncr/angbracketrightbig10-410-310-210-1PDF\n10-310-210-1100\nvgas/c0\ns10-310-210-1100101102PDF\n9.5 10.0 10.5 11.0\nvdust/c0\ns10-310-210-1100101102103PDF\n1.7 1.8 1.9 2.0 2.1 2.2 2.3\nγL10-310-210-1100101102103PDF\nt=0.0tL,dust t=9.8tL,dust t=10.2tL,dust t=10.9tL,dust t=12.0tL,dust t=13.0tL,dust\nFigure4. Thedensity(top)andvelocity/Lorentzfactor(bottom)PDFsofgas(left),dust(middle)andCRs(right).MostPDFsfeatureastrongasymmetry.The\ndensityPDFsofdustandCRsspanuptotwoordersofmagnitude,whilegasisonlyweaklycompressible.Gasanddustgrainsareacceleratedanddecelerated\nrespectively due to their momentum exchange, and the PDFs of CR Lorentz factor slightly broaden with time.\ninitial conditions of h𝑣𝑥cri0=h𝑣𝑦cri0=0andh𝑣𝑧cri0\u0018𝑐. Al-\nthoughthemagnitudeofthe“residual”driftvelocity√︃\nΣ𝑖𝑥𝑦𝑧h𝑣𝑖cri2\nis of the same order of magnitude with h𝑣Ai, given that we sample\ntheCRdistributionfunctionwith \u0018106particlesinourdefaultsim-\nulations,wecautionthatthisrangeof 𝛼ispreciselywhatwewould\nexpectfromMonteCarlosamplingnoiseforanintrinsicallyuniform\npitch-angle distribution. This sampling noise (which unfortunately\nconverges slowly, as 𝑁\u000012\ncr) dominates the “residual” drift velocity\nhere, thus we cannot draw a firm conclusion on the CR drift speed\nfromdirectmeasurement.However,thereisstillawaytoinvestigate\nthe CR drifting by Alfvén waves. As shown in the bottom panel of\nFig.5,the(negative)magnitudeofthegascrosshelicity \u0000𝛿𝒗gas\u0001𝛿𝑩,\nwhichisrelatedtotheasymmetryofAlfvénwaves,remainsasingle\nsignandgrowsexponentiallywithasquaredRDIgrowthratebefore\nsaturating. This indicates that the propagation of the Alfvén waves\nexcited by the RDIs is increasingly unidirectional (antiparallel to\nthe large-scale magnetic fields in this case when the cross helicity\nis negative); therefore, the CRs must at least drift at \u0018𝑣Aby the\nAlfvén waves all propagating in one direction.In Fig. 6, we plot the analytically predicted growth rates for\noursimulationparametersasafunctionofthewavenumber j𝒌j,for\na specific mode angle of ˆ𝒌\u0001ˆ𝒗dust=0.6We find that the growth\nratemeasuredfromoursimulationinFig.5replicatestheanalytical\nsolutionwell,withthegrowthratespeakingatwavenumbersaround\n𝑘𝐿box\u001810–40. This corresponds to a fastest growing wavelength\nof2𝜋𝑘\u0018016–06𝐿box, which is somewhat consistent with the\nstructures seen in Fig. 1 and 2.\nMNRAS 000, 000–000 (0000)8Ji et al.\n10-910-810-710-610-510-410-310-210-1100B\nexp(1.8t/tL,dust)\n|Bx|\n|By|\n|Bz|\n10-710-610-510-410-310-210-1100v,cr/|vcr|\nexp(1.8t/tL,dust)\n02468101214\nt/tL,dust10-2010-1810-1610-1410-1210-1010-810-610-410-2−δvgas·δB\nexp(3.6t/tL,dust)\nFigure 5. Time evolutions of averaged magnetic fields (top), the perpendic-\nularcomponentofCRvelocity(middle)andtheaveragednegativegascross\nhelicity (bottom). Both the magnetic fields and the CR perpendicular ve-\nlocity component follow almost the same analytically-predicted growth rate\n(and the squared growth rate for the cross helicity since it is the dot product\nof velocities and magnetic fields), indicating a strong correlation between\nmagnetic field fluctuations, Alfvén wave propagation and CR scattering.\n3.3 Pitch Angle Scattering & Transport/Scattering\nCoefficients\nFig.7showsthetimeevolutionofCRpitchanglePDFs(top)andthe\npitchanglediffusioncoefficients 𝐷𝜇𝜇(bottom),whichisdefinedas:\n𝐷𝜇𝜇¹𝜇0𝑡0º=\n¹𝜇\u0000𝜇0º2\u000b\n2¹˜𝑡\u0000˜𝑡0ºfor˜𝑡\u0000˜𝑡0=Δ˜𝑡not too large, (7)\nwhere𝜇0(˜𝑡0) and𝜇(˜𝑡) are initial and final pitch angle cosine (time)\nrespectively, Δ𝑡is the integration time, and we trace the change\nof pitch angles for all CR particles over a short time interval to\nkeep¹𝜇\u0000𝜇0ºsmall, following e.g., Beresnyak et al. (2011); Xu\n6We explicitly verified that the mode angle of ˆ𝒌\u0001ˆ𝒗dust=0, i.e., the wave\nvectoranddustvelocitiesarealignedoranti-aligned,givesthefastestgrowth\nrate,whichisconsistentwiththefindingsin(Seligmanetal.2019).Therefore,\nweonlyshowthegrowthratesforthisspecificmodeanglehere.Foradetailed\ndescription of calculating these growth rates, see Hopkins & Squire (2018).\n0.01 0.10 1 10 1000.0010.0100.100110\nk LboxIm(ωt L,dust )Figure 6. Analytical growth rates of RDI as a function of wave number\nj𝒌j, which perfectly predicts the growth rate measured directly from the\nsimulation as shown in Fig. 5.\n10-410-310-210-1100101PDF\nt=10.9tL,dust\nt=11.3tL,dustt=11.7tL,dust\nt=12.2tL,dustt=12.6tL,dust\nt=13.0tL,dust1.0\n 0.5\n 0.0 0.5 1.0\nµ10-410-310-210-1Dµµ(µ)/Ω0\ncr\nFigure7. TimeevolutionofPDFsoftheCRpitchanglecosine 𝜇(top),and\nthe CR pitch angle diffusion coefficient 𝐷𝜇𝜇(normalized by the initial CR\ngyro-frequency Ω0cr)asafunctionoftheCRpitchanglecosine(bottom).At\nthesaturationstage,CRsarefullyisotropizedwithauniformdistributionof\nthepitchanglecosine,andtheCRpitchanglediffusioncoefficientfeaturesa\nflat profile around 𝜇=0, without encountering the 90\u000epitch angle problem\npredicted by quasi-linear theories.\nMNRAS 000, 000–000 (0000)CR Confinement by Dust 9\n& Yan (2013). Here ˜𝑡\u0011¹˜𝑐𝑐º𝑡codedenotes the in-code time 𝑡code\nmultiplied by the factor ˜𝑐𝑐to correct for the reduced speed of\nlight (RSOL) approximation.7As shown in the top panel of Fig. 7,\nthe distribution of CR pitch angles becomes uniform at the satura-\ntion stage, indicating that CRs are nearly isotropized, with typical\n𝐷𝜇𝜇\u00180001\u0000001Ωcr(whereΩcr\u0011𝑞crj𝑩j¹𝛾𝐿𝑚cr𝑐ºis the\nusual relativistic CR gyro-frequency).\nForcomparison,ifweassumeisotropic/greyscatteringwiththe\nusual quasi-linear theory slab scattering expressions (Schlickeiser\n1989), then we would expect the average value of 𝐷𝜇𝜇to be given\nbyh𝐷𝜇𝜇i \u0018 ¹ 3𝜋16ºΩcrj𝛿𝑩j2j𝑩j2(Zweibel 2013), where 𝛿𝑩\nrepresentsthemagnitudeofmagneticfluctuationsongyro-resonant\nscales. We see in Fig. 7 that our typical 𝐷𝜇𝜇values correspond to\nj𝛿𝑩j\u001801j𝑩j, which is roughly what we see in Figs. 1-3 (in fact\nwe see slightly larger overall magnetic fluctuations, but what mat-\nters here is the gyro-resonant, parallel component, so the effective\nj𝛿𝑩j2entering the scattering-rate expressions we would expect to\nbe reduced by a factor of \u00182\u00003). Thus, at least to order of mag-\nnitude or better, the typical scattering rate we see is consistent with\nquasi-linear theory expectations.\nExamining𝐷𝜇𝜇¹𝜇º, we see that the CR pitch angle diffusion\ncoefficients saturates at a smooth distribution as a function of 𝜇\n(nearly𝜇-independent), which contradicts the usual prediction of\n𝐷𝜇𝜇¹𝜇=0º \u0018 0from quasi-linear theory (e.g., Skilling 1971;\nYan & Lazarian 2002). This prediction from quasi-linear theory is\nknownasthe 90\u000epitchangleproblem,since 𝜇=0correspondstoan\nextremely short resonance wave length which contains insufficient\nenergy to scatter CRs away from the 90\u000epitch angle, and thus CRs\nmight be naively expected to become “trapped” at 𝜃=90\u000ewithout\nbeing fully isotropized (e.g., Giacalone & Jokipii 1999; Felice &\nKulsrud 2001). However, in our simulation, since the RDI-induced\nmagnetic field perturbation 𝛿𝐵𝐵can grow up to a few 10\u00001, CRs\ncan be scattered across the 90\u000epitch angle and become isotropic\nwithoutanydifficulty,suggestingthedustRDIisefficientinexciting\nsmall-scaleparallelAlfvénwavesandconfineCRson \u0018AUscales.\nPrevious studies have suggested that resonance broadening due to\nnon-linear wave-particle interactions (e.g., Yan & Lazarian 2008;\nBai et al. 2019) or mirror scattering (e.g., Felice & Kulsrud 2001)\nmight help to avoid the 90\u000epitch angle problem, which may be\noccurring in our simulations as well.\nWith this, we can further estimate the CR parallel diffusion\ncoefficient𝜅kfrom our simulation in standard fashion (Earl 1974):\n𝜅k\u00191\n8∫1\n\u00001𝑑𝜇𝑣2cr¹1\u0000𝜇2º2\n𝐷𝜇𝜇¹𝜇º (8)\nCalculating this numerically in the saturation stage, we ob-\ntain𝜅k\u001817\u0002104𝑐0𝑠𝐿box\u0018700𝑐𝑟𝐿cr\u001807\u0002\n1027¹j𝑩j01𝜇Gº\u00001cm2s\u00001, i.e. a factor of\u00181000larger than\ntheBohmlimit.Thisvalueissignificantlylowerthantypicalvalues\nof𝜅\u00181029\u000030cm2s\u00001inferredfromSolarsystemmeasurementsor\n7Specifically, as shown in Ji & Hopkins (2021), the implementation of\nthe RSOL in our code is mathematically equivalent to taking the modified\nform of the general Vlasov equation for the CR distribution function to be:\n¹𝑐˜𝑐º𝜕𝑡𝑓cr¸𝒗cr\u0001r𝒙𝑓cr¸𝑭cr\u0001r𝒑𝑓cr=𝜕𝑡𝑓crjcoll, i.e. rescaling the time\nderivative of the distribution function in the simulation frame by ˜𝑐𝑐. This\nensuresthatoncetheCRdistributionfunctionreachessteady-state,alleffects\nof the choice of ˜𝑐 𝑐on its properties and on the plasma vanish, but also\nthat the CRs “respond” or evolve more slowly in time by a factor ˜𝑐𝑐to\nperturbations – effectively rescaling the units of time “as seen by” the CRs.\nThis is precisely what allows us to uniformly increase the CR timestep by\nthe factor𝑐˜𝑐, which is the purpose of the RSOL, but this means in Eq. 7,\nwe must rescale back to the “true” Δ˜𝑡=¹˜𝑐𝑐ºΔ𝑡codeto obtain the correct\n(˜𝑐-independent)valueof 𝐷𝜇𝜇.Weverifythis explicitlyinsimulationswith\nvaried ˜𝑐below.\n1.0\n 0.5\n 0.0 0.5 1.0\nµ10-410-310-210-1Dµµ(µ)/Ω0\ncr\nt=13.0tL,dust\nt=15.2tL,dustt=17.4tL,dust\nt=19.6tL,dustt=21.7tL,dustFigure 8. Time evolution of the CR pitch angle diffusion coefficient 𝐷𝜇𝜇\n(normalized by the initial CR gyro-frequency Ω0cr) as a function of the CR\npitch angle cosine 𝜇, as the bottom panel of Fig. 7 but with CR feedback\nturned on. CR feedback lowers the CR pitch angle diffusion coefficient by\nroughly one order of magnitude, and the 90\u000epitch angle problem does not\noccur either.\n𝛾-ray observations of Local Group galaxies (Blasi & Amato 2012;\nAmato & Blasi 2018; Chan et al. 2019; Hopkins et al. 2019), sug-\ngesting that dust-induced scattering near quasars or superluminous\ngalaxies can lead to strong CR confinement.\nIn Fig. 4, we clearly see that there is also some non-zero dif-\nfusion in CR momentum space. In quasi-linear theory again, the\neffectiveCRmomentum-spacediffusioncoefficient 𝐷𝑝𝑝istrivially\nrelated (to leading order in O¹𝑢𝑐º, where𝑢represents the back-\nground plasma velocities) to the pitch-angle-averaged h𝐷𝜇𝜇ias\nh𝐷𝑝𝑝i\u0018𝜒𝑝2cr𝑣2\nA\n𝑣2crh𝐷𝜇𝜇i (9)\nwhere the factor of 𝜒depends on the CR distribution function, and\nis\u001813fornearlyisotropicCRs(Hopkinsetal.2021b).Measuring\nh𝐷𝑝𝑝ieither “directly” (for an ensemble of CRs, as we estimated\n𝐷𝜇𝜇) or “indirectly” (by measuring the broadening of the PDF of\n𝑝cror𝛾𝐿inFig.4andcomparingtoananalyticdiffusionsolution),\nweconfirmthatthemomentum-spacediffusioncoefficientestimated\nnumerically is within tens of percent of the value one would infer\nfrom simply inserting our measured h𝐷𝜇𝜇iinto Eq. (9).\n3.4 Pitch Angle Scattering with CR Feedback\nWenowinvestigatehowCRfeedback(back-reactionfromCRsonto\ngas and magnetic fields) modifies the our previous findings. After\nthe CR population becomes nearly isotropized at 𝑡\u001813𝑡𝐿dust,\nwe continue evolving our simulation but now enable CR feedback.\nSinceatthisstagetheCRdriftvelocityismuchlessthanthespeedof\nlight(𝑣𝐷\u001c𝑐),andtheCR“initialconditions”aremorephysically\nrealistic, the CR non-resonant instability does not grow rapidly and\ndominatethesimulationbehavior(asitartificiallywouldifwebegan\nfrom𝑣𝐷\u0019𝑐with CR feedback included). Almost all of our quali-\ntative conclusions and the behaviors (in saturation) of gas, CR, and\ndust density fields remain qualitatively similar after turning on this\nback-reaction term, but quantitatively, there is some effect. Specif-\nically in Fig. 8, we plot the time evolution of the CR pitch angle\ndiffusioncoefficient 𝐷𝜇𝜇asafunctionoftheCRpitchanglecosine\n𝜇, after re-enabling the CR feedback. With CR feedback present,\n𝐷𝜇𝜇decreases somewhat (though it conserves the functional form\nof𝐷𝜇𝜇¹𝜇º)untilsaturatingatavaluesystematicallylowerbyafac-\ntor\u00184compared to that without CR feedback. This in turn implies\nMNRAS 000, 000–000 (0000)10Ji et al.\nfactor\u00184higher parallel diffusion coefficients. This is still more\nthan sufficient to keep the CRs isotropized and strongly-confined\n(andthe 90\u000epitch-angleproblemstilldoesnotappear),butitisnot\na negligible difference.\nPhysically, we showed in § 3.3 that the scattering rates fol-\nlowed approximately the quasi-linear theory expectation, 𝐷𝜇𝜇/\nj𝛿𝑩j2j𝑩j2. Thus a factor\u00184suppression of 𝐷𝜇𝜇by CR back-\nreaction corresponds to a factor of \u00182suppression of 𝛿𝑩on gyro-\nresonant scales. Indeed, we can directly verify that after we turn on\nthe CR feedback the fluctuations in the magnetic field are damped\nby roughly this factor. If the fluctuations are predominantly driven\nby the RDIs, then this change in their saturation amplitudes is not\nsurprising: recall from § 2.2, when we turn on back-reaction (i.e.\naccount for finite CR pressure effects on gas), we assume a fairly\nlargeCRpressurerelativetothermalpressure, 𝑃cr\u00185𝑃thermal(with\nboth𝑃crand𝑃thermalmuch larger than magnetic pressure). Thus if\nwe saturate as described in § 3.1 with the perturbations driven by\nthe RDIs (ultimately powered by the dust “ram pressure” or accel-\neration force per unit area 𝑃dust) compensated by gas thermal plus\nCR pressure, then we expect j𝛿𝑩j2to be a factor of a few lower\nin saturation (from the increase in the totalthermal+magnetic+CR\nbackgroundpressure).Effectively,thebackgroundmediumbecomes\n“stiffer” against perturbation by the dust.\nFromtheviewpointofthestandardquasi-lineartheorywherein\noneassumeslineargrowthofscatteringmodescompensatedbywave\ndamping settingthe quasi-steady-state scattering rates,with the CR\nfeedbackturnedon,thebackreactionontheRDIfromtheCRpres-\nsure plus the magnetic tension leads to stronger damping, and thus\nless CR confinement as found in the simulations. Quasi-linear the-\nories also predict that the CR feedback can generate small-scale\nAlfvén waves via the CR gyro-resonant and non-resonant instabili-\nties, i.e., the CR “self-confinement” modes which add linearly with\ntheRDIdrivenperturbationsandthus increasetheCRconfinement.\nHowever, this is in opposite to the decrease of the CR confinement\nin our simulations, because CRs are highly isotropic in the satu-\nration stage, thus the CR-excited Alfvén waves and their resulting\nCR confinement are negligible. But admittedly, it is plausible to\nimagine that if there is a continuous driving of the background CR\ngradients, the growth rates of Alfvén waves from dust and CRs do\nadd linearly when they both have drifts and small densities, and we\nmight see the enhancement of the CR confinement, as quasi-linear\nself-confinement theories predicted.\n3.5 Convergence Tests\nAlthoughpreviousstudieshaveinvestigatedhowsimulationsofjust\ntheRDIs(Moseleyetal.2019;Seligmanetal.2019;Hopkinsetal.\n2020b) or just CRs depend on resolution, those simulations did not\ncombine these physics nor investigate the same range of scales and\nparametersaswedohere.Therefore,weconsidera“high-resolution”\nsimulation with 𝑁gas=𝑁dust=𝑁cr=𝑁3\n1D=1283. Note that\nsinceweenforce 𝑟𝐿dust𝐿box=p𝑁1D,thehigh-resolutionrunwith\ndoubled 1D resolution fits a factor-of-p\n2larger number of Larmor\nwavelengthsalongeachsideofthesimulationdomain,andresolves\neach Larmor wavelength withp\n2times more elements.\nFig. 9 shows projections of dust grains, slices of magnetic\nfield strength superposed with field lines and projections of CR\nparticles along the 𝑧-axis for the high-resolution simulation. The\nhigh-resolutionrunisqualitativelysimilartothelow-resolutionrun\nshown in Fig. 1, but contains smaller-scale structures as expected.\nAt the saturation stage, the high-resolution run contains two large-\nscale coherent structures of dust columns and magnetic vortices,\nin contrast to the single sheet-like structure in the low-resolution\nrun. This is likely because the high-resolution run contains more\nLarmor wavelengths and thus it simulates a larger domain in realphysical units. The time evolution of the magnetic fields and the\nperpendicular CR velocity in the high-resolution run are shown\nin Fig. 10, where the growth rate is almost identical to the low-\nresolution run. Thus the linear growth rates of the relevant modes\nare not measurably dependent on resolution, and they agree well\nwith analytic theory; moreover the final properties (widths) of the\nPDFs are similar in both cases. Of greatest interest, the values of\n𝐷𝜇𝜇and(correspondingly) 𝜅kand𝐷𝑝𝑝differbylessthan 5%inthe\nhighresolutionrun.Thusitappearsthatthekeyresultsherearenot\nsensitivetoresolution.However,wecautionthatitisonlypossibleto\nnumerically resolve a tiny fraction of the interesting dynamic range\n(letaloneeffectsliketurbulentcascadesorfield-linewanderingfrom\nlarge-scale dynamics, which operate on orders-of-magnitude larger\nscales), so this should be taken with some caution.\nWe also examine the numerical convergence with respect\nto the RSOL by varying ˜𝑐over a factor of\u00185. For standard-\nresolution runs with ˜𝑐=¹00200501º𝑐, we find qualita-\ntively identical behavior in all properties studied here, with the\nnumerically-estimated in-saturation parallel diffusion coefficients\n𝜅k\u0018¹700700600º𝑐𝑟𝐿cr, respectively – i.e. nearly invariant to\n˜𝑐. Note there is a (very weak) hint that 𝜅kmay decrease with in-\ncreasing ˜𝑐here, perhaps owing to slightly stronger confinement at\nhigher ˜𝑐perhapsbecausewithlower ˜𝑐theCRsresponsemayslightly\nartificially lag the RDI growth rates; but if we extrapolate to ˜𝑐=𝑐\nby fitting our results, the resulting inferred 𝜅kis only decreased\nby\u001840%, a rather small correction compared to other theoretical\nuncertaintieshere(e.g.theeffectofback-reactiondiscussedbelow).\n3.6 Discussion: Damping & Saturation Scalings\nOwing to our limited numerical resolution and MHD-PIC assump-\ntions (where the plasma is treated as an MHD fluid), our sim-\nulations do not include certain plasma processes that can also\ndamp Alfvén waves, such as ion-neutral or Landau damping. Ion-\nneutral damping is not likely relevant under conditions of inter-\nest for our problem here (e.g. diffuse, warm, highly-illuminated\nCGM), as the expected neutral fractions are vanishingly small.\nBut Landau damping could be non-negligible, in principle. If\nwe consider e.g. the usual non-linear Landau damping rate with\nΓ\u0018¹p𝜋4º𝑐𝑠𝑘¹𝑘2\n?𝑘2\nkº\u0018¹p𝜋8º𝑐𝑠𝑘¹j𝛿𝑩j2j𝑩j2º, then at the\nscalewheretheRDIgrowthrateismaximized( \u0018𝑟𝐿cr)theimplied\nLandau damping rate (given the j𝛿𝑩j\u001801j𝑩jwe see) is generally\n\u001810%oftheRDIgrowthrateinFig.6.Thatsuggestsitmaynotbe\nnegligible, but it is also unlikely to qualitatively change the behav-\niors here, if included. However, the simulations do, given the finite\nresolution,havenon-zeronumericaldissipationwhichhappens(co-\nincidentally)tobesimilarinmagnitudetoLandaudamping:givena\nstandardnumericalMHDdissipationratein GIZMOwhichscalesas\n\u0018Δ𝑥𝑐𝑠𝑘2\u0018𝑐𝑠𝑘¹𝑘Δ𝑥º(whereΔ𝑥is the effective grid resolution\nfor the MHD, and the prefactor depends on the specific numerical\nproblem and details of the method, see Hopkins & Raives 2016),\nthen for𝑘\u00181𝑟𝐿cr\u00181¹10Δ𝑥ºthis is roughly similar in magni-\ntudetothephysicalLandaudamping.Ofcourse,othermechanisms\ncouldinprinciplecontributetodampingincludinginteractionswith\nextrinsic turbulence (Farmer & Goldreich 2004), which we cannot\ncaptureowingtoourlimitedrangeofscales.Dustitselfcouldactas\na damping mechanism in some circumstances (Squire et al. 2021)\nbuttheconditionswherethiswouldoccuraredramaticallydifferent\nfrom those here.\nThese limitations in physics, finite resolution, and simulation\nbox size preclude making detailed statements regarding the satura-\ntionmechanismsoftheRDIacrossabroaderparameterspace.How-\never, the fact that we see roughly isotropic 𝐷𝜇𝜇following approxi-\nmatelytheexpectedquasi-linearscalingwith j𝛿𝑩jj𝑩jgivesussome\nconfidence that the CR scattering rates induced by the RDI should\nMNRAS 000, 000–000 (0000)CR Confinement by Dust 11\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y8.92tL,dust\n0.50\n0.25\n0.000.250.50\nx9.32tL,dust\n0.50\n0.25\n0.000.250.50\nx9.72tL,dust\n0.50\n0.25\n0.000.250.50\nx11.91tL,dust\n100101102\n100101102\n100101102\n100101102\nDustParticleCount\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y8.92tL,dust\n0.50\n0.25\n0.000.250.50\nx9.32tL,dust\n0.50\n0.25\n0.000.250.50\nx9.72tL,dust\n0.50\n0.25\n0.000.250.50\nx11.91tL,dust\n5.56.06.57.0×103\n0.60.81.0×102\n12345×102\n0.020.040.060.080.100.120.14\n|B|//radicalbig\n4πP0\n0.50\n0.25\n0.000.250.50\nx0.4\n0.2\n0.00.20.4y8.92tL,dust\n0.50\n0.25\n0.000.250.50\nx9.32tL,dust\n0.50\n0.25\n0.000.250.50\nx9.72tL,dust\n0.50\n0.25\n0.000.250.50\nx11.91tL,dust\n100101\n100101\n100101102\n100101\nCRParticleCount\nFigure9. Plotsofdustgrainprojections(top),magneticfieldstrengthssuperposedwithfieldlines(middle)andCRparticleprojections(bottom),asFig.1,but\nfromthehigh-resolutionsimulation.Themorphologyofdustgrains,magneticfieldsandCRsinthehigh-resolutionrunisqualitativelysimilarwiththoseinthe\nfiducial run, but exhibits more detailed small-scale structures and contains more large-scale coherent structures (since the domain size of the high-resolution\nrun is effectively larger – see the text for a full explanation).\nindeedscalewiththesaturationamplitudeof 𝛿𝑩.Andevenifthesat-\nurationmechanismisuncertain,somebroadconclusionsarerobust.\nFor example, based on their idealized RDI-only simulations, Hop-\nkins et al. (2020b) discuss two possible saturation mechanisms, the\nfirstbeingbalancebetweenmagnetictensionanddustrampressure\nasdiscussedin§3.1,thesecondbeingascenariowherethecrossing\ntime of the RDI-generated modes ( \u0018𝛿𝒗gas𝜆) becomes faster than\nthe RDI growth time, producing non-linear dissipation. Non-linear\nLandaudampingisanotherpotentialsaturationmechanism,limiting\n𝛿𝑩whenthedampingbecomesfasterthanlinearRDIgrowthrates.\nSquireetal.(2021)consideryetanothersaturationscenario,assum-\ning a Kraichnan (1965)-like damping via self-interactions (which\ngivesadampingrateakintonon-linearLandaubutwith 𝑐𝑠replaced\nbytheAlfvénspeed).Crucially,the“driving”inallofthesescenarios\nscaleswiththedust-to-gasratio 𝑓dust\u0000gasandforce/acceleration/drift\nvelocityonthegrains(whichappearinboththedustrampressureand\nRDIgrowthrates).Thisisproportionaltotheincidentradiationflux\n𝐹rad.Combiningtheseestimatesforthesaturation 𝛿𝑩withtheusual\nscalings for gyro radii and scattering rates we obtain, for anyof the\nsaturationscenariosabove,ascatteringratewhichscalesdimension-\nallyas𝐷𝜇𝜇/𝑓dust\u0000gas𝐹07\u000015\nradj𝑩j\u0000¹08\u00003º(withaweakerresidual\ndependenceongasdensityand/ortemperature,andtheexactpower-\nlawscalingdependingonthesaturationmodel).Inotherwords,there\nisaqualitativelyrobustpredictionthatwithlower-dust-to-gas-ratiosand/or incident radiative fluxes and/or stronger magnetic fields, the\nconfinement of CRs by dust become weaker. All else equal, in or-\nder for the scattering rate from the dust RDIs to drop to below\nthat inferred for Milky Way ISM gas (and thus become relatively\nunimportant), the factor \u0018𝑓dust\u0000gas𝐹radwould need to be\u00181000\ntimessmallerthanthevalueweassumetomotivateourtests.Sofor\nthere to be “too little dust,” the metallicity or dust-to-metals ratio\nwould need to be 1000times lower than ISM values (which seems\nunlikely at least at low cosmological redshifts, even in the CGM,\ngiven the observations reviewed in § 1, which suggest this factor\nis perhaps something like \u001810times lower than in the ISM). But\nmoreplausibly,theincidentfluxcouldeasilybe 1000timessmaller,\nif for example the galaxy is a typical Milky Way-like or smaller\ndwarf galaxy with a star formation rate of \u001c10 M\fyr\u00001and has\nnegligible AGN luminosity (i.e. galaxy luminosity .1010L\f) or\nthe dust is at\u001d100kpc from the host. The magnetic field depen-\ndence is also interesting: it suggests these instabilities and ensuing\nconfinement would be easier to excite in the distant CGM or IGM\n(where nano-Gauss fields are expected), but may be suppressed in\ndenser, more highly-ionized super-bubbles near to galaxies.\n4 CONCLUSIONS\nIn this paper, we investigate the impact of the dust RDIs on cos-\nmic ray scattering, by performing the first numerical simulations of\nMNRAS 000, 000–000 (0000)12Ji et al.\n10-910-810-710-610-510-410-310-210-1100B\nexp(1.8t/tL,dust)\n|Bx|\n|By|\n|Bz|\n02468101214\nt/tL,dust10-810-710-610-510-410-310-210-1100v,cr/|vcr|\nexp(1.8t/tL,dust)\nFigure10. Timeevolutionofaveragedmagneticfields(top)andtheperpen-\ndicularcomponentofofCRvelocity,asFig.5,butfromthehigh-resolution\nsimulation.Theexponentialgrowthrateisalmostidenticaltothatinthefidu-\ncialrun(Fig.5),suggestingdifferentresolutionsdonotsignificantlyalterthe\nresults of our simulations.\nMHD-dust-CR interactions, where the charged dust and CR gyro-\nradii on\u0018AUscales are fully resolved. Since this is a first study,\nwe consider just one special case, where we might anticipate ef-\nficient dust-induced CR confinement, as compared to more typical\nSolar-neighborhood-likeISMconditions.Wefocuson oneregimeof\nthe RDIs, specifically conditions where the “cosmic ray-like” RDIs\nare rapidly growing and can produce “diffusive” dust behavior (see\nHopkins et al. 2020b), which as speculated in Squire et al. (2020)\ncould in turn lead to dust-induced CR confinement. This type of\nRDIrequiresparticularconditionstodominate: 𝑣dust\u001d𝑣A,plasma\n𝛽\u001d1,lowgasdensityandhighgraincharge,sothedustdragforce\nis substantially subdominant to the Lorentz force. These conditions\ncould arise in, e.g., the CGM around quasars or luminous galaxies.\nWe suspect that different RDIs might have different effects on CR\nscattering, which is a subject for future study. Under these condi-\ntions, we find that small-scale parallel Alfvén waves excited by the\nRDIs efficiently scatter CRs and significantly enhance CR confine-\nment.Therefore,dust-inducedCRscatteringcanpotentiallyprovide\na strong CR feedback mechanism on \u0018AUscales.\nWefirstexploretheultra-lowCRdensitylimitbyignoringCR\nfeedback to the gas. Dust quickly becomes unstable to the RDIs\nand forms high-density columns and sheets, until growth saturates\naround\u001810times the dust Larmor time 𝑡𝐿dust. The density and\nvelocity PDFs of both gas and dust grains show strong asymmetry,\nwhere the dust density spans over two orders of magnitude, while\nthe gas is only slightly compressible with \u001810%density fluctua-\ntions. Perpendicular magnetic field components are exponentially\namplified by the RDI, with the growth rates predicted by analyticalsolutions.InitiallyperfectlystreamingCRsarestronglyscatteredby\nmagnetic field fluctuations, with the growth rate of the perpendicu-\nlar CR velocity component (to local magnetic fields, 𝑣?crj𝒗crj, or√︁\n1\u0000𝜇2)equalingthegrowthrateofmagneticfields.Atthesatura-\ntionstage,theCRsareisotropizedwithanear-uniformdistributionof\nthepitchanglecosine 𝜇,andtheCRpitchanglediffusioncoefficient\n𝐷𝜇𝜇¹𝜇ºisnearlyindependentofpitch-angle 𝜇(inparticulararound\n𝜇=0).Thereisno 90\u000epitchangleprobleminoursimulations.The\nscattering rate is in order-of-magnitude agreement with the usual\nquasi-linear theory expectation 𝐷𝜇𝜇\u0018Ωcrj𝛿𝑩j2j𝑩j2, with the\nlargej𝛿𝑩j2j𝑩j2\u001810\u00003\u000010\u00002on gyro-resonant scales driven by\nthe dust RDIs (in part because the dust has broadly similar gyro-\nradiitotheCRs,underthesetypesofconditions).Thenumerically-\ncalculated CR parallel diffusion coefficient is \u0018500\u00001000times\nthe Bohm value: sufficient for strong confinement of the CRs.\nWhen the system reaches saturation and CRs become close to\nisotropic (with the CR drift velocity 𝑣𝐷\u001c˜𝑐), we turn on CR feed-\nback to the gas and study its consequences. We find that with CR\nfeedback, the CR pitch angle diffusion coefficient 𝐷𝜇𝜇decreases\nby a factor of\u00184(and thus this slightly reduces CR confinement).\nThis owes to the fact that the large assumed CR pressure (several\ntimes larger than thermal+magnetic) suppresses the saturation am-\nplitude of the magnetic field fluctuations 𝛿𝑩induced by the RDIs\nby a modest factor \u00182, and the scattering modes are thus damped\nmore by backreaction on the RDIs from the CR pressure, in ad-\ndition to the magnetic tension. Since CRs in the saturation stage\narehighlyisotropic,thequasi-linearself-confinementtheorywhich\npredictshigherscattering rates due to CR-excited Alfvén waves is\nnotapplicable here, unless there exists a continuous driving of CR\ngradients.\nWefinallystressthatseveralcaveatsapplytoourstudy.(1)Asa\nfirst experiment, we picked one particular initial condition to inves-\ntigate an interesting case, which is plausible for some conditions as\nnotedabovebutshouldnotbeconsideredtypicaleverywhere.There\nexist a variety of RDIs which can have totally different behaviors\nindifferentcircumstances(Hopkinsetal.2020b)andindeed,under\nsome conditions dust might even have the opposite effect, acting as\na wave-damping mechanism and reducing the confinement of CRs\n(Squire et al. 2020). (2) Although we perform a small resolution\nstudy, our simulations are still limited in resolution and dynamical\nrange. Even for a single grain size, the RDIs are unstable at all\nspatial wavelengths, so it is impossible to encompass their com-\npletedynamicrange(letaloneglobalscalesofstructureorextrinsic\nturbulence, which are vastly larger than our box sizes). (3) We do\nnot include any explicit wave-damping processes. While we do not\nexpect appreciable ion-neutral damping in environments of interest\n(as the neutral fractions are negligible), Landau damping could be\nimportant, as could damping from a turbulent cascade, and these\ncould reduce the efficacy of confinement. However at least for the\nextreme parameters considered here, it is unlikely these would sig-\nnificantlyreducethescatteringrates.(4)Ourperiodicboxesneglect\nlarge-scale (\u001d𝐿box) CR pressure gradients ( r𝑃cr) which act as a\nsource/driving term for super-Alfvénic CR drift. (5) For simplicity,\nweconsideronlyonegrainsize+chargeandoneCRenergy+species,\nratherthanafullspectrumofgrainsizesandCRenergies.Infuture\nwork it will be particularly interesting to see how a full spectrum\nof both modifies the dynamics here, as a broad range of gyro-radii\noverlapanddifferentgyro-resonantmodescaninteractnon-linearly\nand even linearly (when CRs+dust+gas are all combined), via their\nback-reaction on the gas.\nWith these limitations in mind, we consider this study to be a\nproofofconcept ,showingthatCRdust-gasinteractionsmightindeed\nbeveryimportantinsomeastrophysicalconditions.Forinstance,this\nmechanismmightbeabletoresolvesomeoftheincompatibilitybe-\nMNRAS 000, 000–000 (0000)CR Confinement by Dust 13\ntween CR confinement models and observations by preventing CR\n“runaway” (Hopkins et al. 2021a), at least under certain conditions\ninvestigated in this study. Considerable work remains to map out\nthe parameterspace, include additional physics,and understand the\nmacroscopic consequences of confining CR-dust interactions (for\neitherCRsthemselvesorforgalaxy/CGMevolution).Nevertheless,\nthesimulationshereclearlyarguethattheseeffectsareworthstudy-\ning in detail.\nDATA AVAILABILITY\nThe data supporting the plots within this article are available on\nreasonable request to the corresponding author. A public version\noftheGIZMOcodeisavailableat http://www.tapir.caltech.\nedu/~phopkins/Site/GIZMO.html .\nACKNOWLEDGMENTS\nSJ thanks E. Quataert for helpful discussions, and the referee\nfor constructive comments which improve this manuscript. SJ is\nsupported by a Sherman Fairchild Fellowship from Caltech, the\nNaturalScienceFoundationofChina(grants12133008,12192220,\nand 12192223) and the science research grants from the China\nMannedSpaceProject(No.CMS-CSST-2021-B02).SupportforJS\nwas provided by Rutherford Discovery Fellowship RDF-U001804\nand Marsden Fund grant UOO1727, which are managed through\nthe Royal Society Te Ap ¯arangi. Support for PFH was provided by\nNSF Research Grants 1911233 & 20009234, NSF CAREER grant\n1455342, NASA grants 80NSSC18K0562, HST-AR-15800.001-A,\nJPL 1589742. Numerical calculations were run on the Caltech\ncompute cluster “Wheeler,” allocations FTA-Hopkins/AST20016\nsupported by the NSF and TACC, and NASA HEC SMD-16-7592.\nWe have made use of NASA’s Astrophysics Data System. 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J., Casse F., Marcowith A., 2019, MNRAS, p. 2249\nMNRAS 000, 000–000 (0000)"    },    {        "title": "2202.05295v1.Non_stationary_Anderson_acceleration_with_optimized_damping.pdf",        "content": "arXiv:2202.05295v1  [math.NA]  10 Feb 2022Non-stationary Anderson acceleration with optimized\ndamping⋆\nKewang Chena,b,∗, Cornelis Vuikb\naCollege of Mathematics and Statistics, Nanjing University of Information Science and\nTechnology, Nanjing, 210044, China.\nbDelft Institute of Applied Mathematics, Delft University o f Technology, Delft, 2628XE, the\nNetherlands.\nAbstract\nAnderson acceleration (AA) has a long history of use and a strong r ecent inter-\nest due to its potential ability to dramatically improve the linear conve rgence\nof the fixed-point iteration. Most authors are simply using and analy zing the\nstationary version of Anderson acceleration (sAA) with a constan t damping\nfactor or without damping. Little attention has been paid to nonsta tionary\nalgorithms. However, damping can be useful and is sometimes crucia l for simu-\nlations in which the underlying fixed-point operator is not globally cont ractive.\nThe role of this damping factor has not been fully understood. In th e present\nwork, we consider the non-stationary Anderson acceleration algo rithm with op-\ntimized damping (AAoptD) in each iteration to further speed up linear and\nnonlinear iterations by applying one extra inexpensive optimization. W e an-\nalyze this procedure and develop an efficient and inexpensive implemen tation\nscheme. We also show that, compared with the stationary Anderso n accelera-\ntion with fixed window size sAA(m), optimizing the damping factors is related\nto dynamically packaging sAA(m) andsAA(1) in each iteration (alternating\nwindow size mis another direction of producing non-stationary AA). More-\n⋆Funding: This work was partially supported by the National Natural Sc ience Foundation\nof China [grant number 12001287]; the Startup Foundation fo r Introducing Talent of Nanjing\nUniversity of Information Science and Technology [grant nu mber 2019r106]\n∗Corresponding author\nEmail addresses: kwchen@nuist.edu.cn (Kewang Chen), c.vuik@tudelft.nl (Cornelis\nVuik)\nURL:https://homepage.tudelft.nl/d2b4e/ (Cornelis Vuik)\nPreprint submitted to Journal of Computational and Applied Mathematics.February 14, 2022over, we show by extensive numerical experiments that, in the cas e a larger\nwindow size is needed, the proposed non-stationary Anderson acc eleration with\noptimized damping procedure often converges much faster than s tationary AA\nwith constant damping or without damping. When the window size is ver y\nsmall (m≤3 was typically used, especially in the early days of application),\nAAoptD and AA are comparable. Lastly, we observed that when the system is\noverdamped (i.e. the damping factor is close to the lower bound zero ), incon-\nsistency may occur. So there is some trade-off between stability an d speed of\nconvergence. We successfully solve this problem by further restr icting damping\nfactors bound away from zero.\nKeywords: Anderson acceleration, fixed-point iteration, optimal damping.\n2010 MSC: 65H10, 65F10\n1. Introduction\nIn this part, we first give a literature review on Anderson Accelerat ion\nmethod. Thenwediscussourmainmotivationsandthe structurefo rthe present\npaper. To begin with, let us consider the nonlinearaccelerationfor t he following\ngeneral fixed-point problem\nx=g(x), g:Rn→Rn\nor its related nonlinear equations problem\nf(x) =x−g(x) = 0.\nThe associated basical fixed-point iteration is given in Algorithm 1.\nAlgorithm 1 Picard iteration\nGiven:x0.\nfork= 0,1,2,···do\nSetxk+1=g(xk).\nend for\n2The main concern related to this basic fixed-point iteration is that th e it-\nerates may not converge or may converge extremely slowly (only line ar conver-\ngent). Therefore, various acceleration methods are proposed t o alleviate this\nslow convergence problem. Among these algorithms, one popular ac celeration\nprocedure is called the Anderson acceleration method [1]. For the ab ove basic\nPicard iteration, the usual general form of Anderson acceleratio n with damping\nis given in Algorithm 2. In the above algorithm, fkis the residual for the kth\nAlgorithm 2 Anderson acceleration: AA(m)\nGiven:x0andm≥1.\nSet:x1=g(x0).\nfork= 0,1,2,···do\nSet:mk= min{m,k}.\nSet:Fk= (fk−mk,···,fk), where fi=g(xi)−xi.\nDetermine: α(k)=/parenleftig\nα(k)\n0,···,α(k)\nmk/parenrightigT\nthat solves\nmin\nα=(α0,···,αmk)T/bardblFkα/bardbl2s. t.mk/summationdisplay\ni=0αi= 1.\nSet:xk+1= (1−βk)mk/summationdisplay\ni=0α(k)\nixk−mk+i+βkmk/summationdisplay\ni=0α(k)\nig(xk−mk+i).\nend for\niteration; mis the window size which indicates how many history residuals will\nbe used in the algorithm. The value of mis typically no larger than 3 in the\nearly days of applications and now this value could be as large as up to 1 00,\nsee [2]. It is usually a fixed number during the procedure, varying mcan also\nmake the algorithm to be non-stationary. We will come back to this po int in\nsection Section 2; βk∈(0,1] is a damping factor (or a relaxation parameter) at\nkth iteration. We have, for a fixed window size m:\nβk=\n\n1, no damping,\nβ,(a constant independent of k) stationary AA,\nβk,(depending on k) non-stationary AA.\nThe constrained optimization problem can also be formulated as an eq uivalent\n3unconstrained least-squares problem [3, 4]:\nmin\n(ω1,···,ωmk)T/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublefk+mk/summationdisplay\ni=1ωi(fk−i−fk)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2(1)\nOne can easily recover the original problem by setting\nω0= 1−mk/summationdisplay\ni=1ωi.\nThis formulation of the linear least-squares problem is not optimal fo r imple-\nmentation, we will discuss this in more detail in Section 3.\nAnderson acceleration method dates back to the 1960s. In 1962, Anderson\n[1] developeda techniqueforacceleratingthe convergenceofthe Picarditeration\nassociated with a fixed-point problem which is called Extrapolation Algo rithm.\nThis technique is now called Anderson Acceleration (AA) in the applied m athe-\nmatics community and Anderson Mixing in the physics and chemistry co mmu-\nnities. This method is “essentially” (or nearly) similar to the nonlinear G MRES\nmethod or Krylov acceleration [5, 6, 7, 8] and the direct inversion on the itera-\ntive subspace method (DIIS) [9, 10, 11]. And it is also in a broad categ ory with\nmethods based on quasi-Newton updating [12, 13, 14, 15, 16]. Howe ver, unlike\nNewton-like methods, AA does not require the computation or appr oximation\nof Jacobians or Jacobian-vector products which could be an advan tage.\nAlthough the Anderson acceleration method has been around for d ecades,\nconvergence analysis has been reported in the literature only rece ntly. Fang\nand Saad [14] had clarified a remarkable relationship of AA to quasi-Ne wton\nmethods and extended it to define a broader Anderson family metho d. Later,\nWalker and Ni [17] showed that, on linear problems, AA without trunc ation\nis “essentially equivalent” in a certain sense to the GMRES method. Fo r the\nlinear case, Toth and Kelley [3] first proved the stationary version o f AA (sAA)\nwithout damping is locally r-linearly convergent if the fixed point map is a\ncontractionand the coefficients in the linear combination remain boun ded. This\nwork was later extended by Evens et al. [18] to AA with damping and t he\nauthors proved the new convergence rate is θk((1−βk−1)+βk−1κ), where κis\n4the Lipschitz constant for the function g(x) andθkis the ratio quantifying the\nconvergence gain provided by AA in step k. However, it is not clear how θk\nmay be evaluated or bounded in practice and how it may translate to im proved\nasymptotic convergence behavior in general. In 2019, Pollock et al. [19] applied\nsAA to the Picard iteration for solving steady incompressible Navier– Stokes\nequations(NSE) andprovedthat the accelerationimprovesthe co nvergencerate\nofthe Picarditeration. Then, De Sterck[20] extended the resultt o moregeneral\nfixed-point iteration x=g(x), given knowledge of the spectrum of g′(x) at\nfixed-point x∗and Wang et al. [21] extended the result to study the asymptotic\nlinear convergence speed of sAA applied to Alternating Direction Met hod of\nMultipliers (ADMM) method. Sharper local convergence results of A A remain\na hot research topic in this area. More recently, Zhang et al. [22] pr oved a\nglobal convergent result of type-I Anderson acceleration for no nsmooth fixed-\npointiterationswithoutresortingtolinesearchoranyfurtherass umptionsother\nthan nonexpansiveness. For more related results about Anderso n acceleration\nand its applications, we refer the interested readers to [2, 23, 24, 25, 26, 27, 28]\nand references therein.\nAs mentioned above, the local convergence rate θk((1−βk−1) +βk−1κ) at\nstagekis closely related to the damping factor βk−1. However, questions like\nhow to choose those damping values in each iteration [2] and how it will a ffect\nthe global convergence of the algorithm have not been deeply stud ied. Besides,\nAA is often combined with globalization methods to safeguard against erratic\nconvergence away from a fixed point by using damping. One similar idea in\nthe optimization context for nonlinear GMRES is to use line search str ategies\n[29]. This is an important strategy but not yet fully explored in the liter ature.\nMoreover, the early days of Anderson Mixing method (the 1980s, f or electronic\nstructure calculations) initially dictated the window size m≤3 due to the\nstorage limitations and costly gevaluations involving large N. However, in\nrecent years and a broad range of contexts, the window size mranging from\n20 to 100 has also been considered by many authors. For example, W alker\nand Ni [17] used m= 50 in solving the nonlinear Bratu problem. A natural\n5question will be should we try to further steep up Anderson acceler ationmethod\nor try to use a larger size of the window? No such comparison results have been\nreported. Motivated by the above works, in this paper, we propos e, analyze\nand numerically study non-stationary Anderson acceleration with o ptimized\ndamping to solve fixed-point problems. The goal of this paper is to ex plore the\nrole of damping factors in non-stationary Anderson acceleration.\nThe paper is organized as follows. Our new algorithms and analysis are in\nSection 2, the implementation ofthe new algorithmis in Section 3, expe rimental\nresults and discussion are in Section 4. Conclusions follow in Section 5.\n2. Anderson acceleration with optimized dampings\nIn this section, we focus on developing the algorithm for Anderson a cceler-\nation with optimized dampings at each iteration and studying its conve rgence\nrate explicitly.\nxk+1= (1−βk)mk/summationdisplay\ni=0α(k)\nixk−mk+i+βkmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)\n=mk/summationdisplay\ni=0α(k)\nixk−mk+i+βk/parenleftiggmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)−mk/summationdisplay\ni=0α(k)\nixk−mk+i/parenrightigg\n.(2)\nDefine the following averagesgiven by the solution αkto the optimization prob-\nlem by\nxα\nk=mk/summationdisplay\ni=0α(k)\nixk−mk+i,˜xα\nk=mk/summationdisplay\ni=0α(k)\nig(xk−mk+i). (3)\nThen (2) becomes\nxk+1=xα\nk+βk(˜xα\nk−xα\nk). (4)\nA natural way to choose “best” βkat this stage is that choosing βksuch that\nxk+1gives a minimal residual. This is similar to the original idea of Anderson\naccelerationwithwindowsizeequaltoone. Sowejustneedtosolvet hefollowing\nunconstrained optimization problem:\nmin\nβk/bardblxk+1−g(xk+1)/bardbl2= min\nβk/bardblxα\nk+βk(˜xα\nk−xα\nk)−g(xα\nk+βk(˜xα\nk−xα\nk))/bardbl2.(5)\n6Noting the fact that\ng(xα\nk+βk(˜xα\nk−xα\nk))≈g(xα\nk)+βk∂g\n∂x/vextendsingle/vextendsingle/vextendsingle\nxα\nk(˜xα\nk−xα\nk)\n≈g(xα\nk)+βk(g(˜xα\nk)−g(xα\nk)). (6)\nTherefore, (5) becomes\nmin\nβk/bardblxk+1−g(xk+1)/bardbl2\n= min\nβk/bardblxα\nk+βk(˜xα\nk−xα\nk)−g(xα\nk+βk(˜xα\nk−xα\nk))/bardbl2\n≈min\nβk/bardblxα\nk+βk(˜xα\nk−xα\nk)−[g(xα\nk)+βk(g(˜xα\nk)−g(xα\nk))]/bardbl2\n≈min\nβk/bardbl(xα\nk−g(xα\nk))−βk[(g(˜xα\nk)−g(xα\nk))−(˜xα\nk−xα\nk)]/bardbl2.(7)\nThus, we just need to calculate the projection\nβk=/vextendsingle/vextendsingle/vextendsingle(xα\nk−g(xα\nk))·[(xα\nk−g(xα\nk))−(˜xα\nk−g(˜xα\nk))]\n/bardbl[(xα\nk−g(xα\nk))−(˜xα\nk−g(˜xα\nk))]/bardbl2/vextendsingle/vextendsingle/vextendsingle. (8)\nSet\nrp= (xα\nk−g(xα\nk)), rq= (˜xα\nk−g(˜xα\nk)),\nwe have\nβk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(rp−rq)Trp\n/bardblrp−rq/bardbl2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (9)\nWe will discuss how much work is needed to calculate this βkin Section 3.\nFinally, our analysisleads to the followingnon-stationaryAnderson a cceleration\nalgorithm with optimized damping: AAoptD(m).\nRemark 2.1. As mentioned in Section 1, changing the window size mat each\niteration can also make a stationary Anderson acceleration to be non-stationary.\nComparing withthe stationary Anderson acceleration with fi xedwindow sAA(m),\nour proposed nonstationary procedure ( AAoptD(m)) of choosing optimal βkis\nsomewhat related to packaging sAA(m)andsAA(1)in each iteration in a cheap\nway. Combining sAA(m)withsAA(1)can provide really good outcomes, espe-\ncially in the case when larger mis needed. We will discuss this in detail for the\nnumerical results in Section 4.\n7Algorithm 3 Anderson acceleration with optimized dampings: AAoptD(m)\nGiven:x0andm≥1.\nSet:x1=g(x0).\nfork= 0,1,2,···do\nSet:mk= min{m,k}.\nSet:Fk= (fk−mk,···,fk), where fi=g(xi)−xi.\nDetermine: α(k)=/parenleftig\nα(k\n0,···,α(k)\nmk/parenrightigT\nthat solves\nmin\nα=(α0,···,αmk)T/bardblFkα/bardbl2s. t.mk/summationdisplay\ni=0αi= 1.\nSet:xα\nk=mk/summationdisplay\ni=0α(k)\nixk−mk+i,˜xα\nk=mk/summationdisplay\ni=0α(k)\nig(xk−mk+i).\nSet:rp= (xα\nk−g(xα\nk)), rq= (˜xα\nk−g(˜xα\nk)).\nSet: βk=(rp−rq)Trp\n/bardblrp−rq/bardbl2.\nSet:xk+1=xα\nk+βk(˜xα\nk−xα\nk).\nend for\nRemark 2.2. Here this optimized damping step is a “local optimal” strate gy at\nkth iteration. It usually will speed up the convergence rate c ompared with an\nundamped one, but not always. Because in (k+1)th iteration, it uses a combi-\nnation of all previous m history information. Moreover, whe nβkis very close\nto zero, the system is over-damped, which, sometimes, may al so slow down the\nconvergence speed. We may need to further modify our βk. See more discussion\nin our numerical results in Section 4.\nLastly, we summarize the convergence results with damping in Theor em 2.1.\nThe proof of this theorem can be found in [18].\nTheorem 2.1. [18] Assume that g:Rn→Rnis uniformly Lipschitz con-\ntinuously differentiable and there exists κ∈(0,1)such that/bardblg(y)−g(x)/bardbl2≤\nκ/bardbly−x/bardbl2for allx,y∈Rn. Suppose also that ∃Mandǫ >0such that for all\nk > m,/summationtextm−1\ni=0|αi|< Mand|αm|≥ǫ. Then\n/bardblf(xk+1)/bardbl2≤θk+1[(1−βk)+κβk]/bardblf(xk)/bardbl2+m/summationdisplay\ni=0O(/bardblf(xk−m+i)/bardbl2\n2),(10)\n8where\nθk+1=/bardbl/summationtextm\ni=0αif(xk−m+i)/bardbl2\n/bardblf(xk)/bardbl2.\n3. Implementation\nForimplementation, wemainlyfollowthepathin[4]andmodifyitasneed ed.\nWe first briefly review the implementation of AA without damping. Then we\nfocus on how to implement the optimized damping problem efficiently and ac-\ncurately.\nThe constrained linear least-squares problem in Algorithm 2 can be so lved\nin many ways. Here we rewrite it into an equivalent unconstrained for m which\ncan be solved efficiently by using QR factorizations. We define ∆ fi=fi+1−fi\nfor each iand setFk= (∆fk−mk,···,∆fk−1), then the least-squares problem\nis equivalent to\nmin\nγ=(γ0,···,γmk−1)T/bardblfk−Fkγ/bardbl2,\nwhereαandγare related by α0=γ0,αi=γi−γi−1for 1≤i≤mk−1, and\nαmk= 1−γmk−1.We assumeFhas a thin QRdecomposition i.e., Fk=QkRk\nwithQk∈Rn×mkandRk∈Rmk×mk, forwhichthesolutionoftheleast-squares\nproblem is obtained by solving the mk×mktriangular system Rkγ=QT\nkfk.\nAs the algorithm proceeds, the successive least-squares problem s can be solved\nefficiently by updating the factors in the decomposition.\nAssume that γk= (γk\n0,···,γk\nmk−1)Tis the solution to the above modified\nform of Anderson acceleration, we have\nxk+1=g(xk)−mk−1/summationdisplay\ni=0γk\ni[g(xk−mk+i+1)−g(xk−mk+i)] =g(xk)−Gkγk,\nwhereGk= (∆ggk−mk,···,∆gk−1) with ∆ gi=g(xi+1−g(xi)) for each i. For\nAnderson acceleration with damping\nxk+1= (1−βk)mk/summationdisplay\ni=0α(k)\nixk−mk+i+βkmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)\n=mk/summationdisplay\ni=0α(k)\nixk−mk+i+βk/parenleftiggmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)−mk/summationdisplay\ni=0α(k)\nixk−mk+i/parenrightigg\n.\n9Follow the idea in [4], we have\nmk/summationdisplay\ni=0α(k)\nig(xk−mk+i) =g(xk)−Gkγk, (11)\nmk/summationdisplay\ni=0α(k)\nixk−mk+i=/parenleftbig\ng(xk)−Gkγk/parenrightbig\n−/parenleftbig\nfk−Fkγk/parenrightbig\n. (12)\nThen this can be achieved equivalently using the following strategy:\nStep 1: Compute the undamped iterate xk+1=g(xk)−Gkγk.\nStep 2: Update xk+1again by\nxk+1←xk+1−(1−βk)/parenleftbig\nfk−QRγk/parenrightbig\n.\nNow we talk about how to efficiently calculate βkas described in Algorithm 3.\nTaking benefit of the QR decomposition in the first optimization proble m and\nnoting (11) and (12), we have\n˜xα\nk=mk/summationdisplay\ni=0α(k)\nig(xk−mk+i) =g(xk)−Gkγk,\nxα\nk=mk/summationdisplay\ni=0α(k)\nixk−mk+i= ˜xα\nk−/parenleftbig\nfk−Fkγk/parenrightbig\n.\nThen we could calculate optimized βkby doing two extra function evaluations\nand two dot products, which are not very expensive:\nrp= (xα\nk−g(xα\nk)), rq= (˜xα\nk−g(˜xα\nk)), βk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(rp−rq)Trp\n/bardblrp−rq/bardbl2/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nIn practice, when xkis very close to the fixed-point x∗, scientific computing\nerrors may arise in calculating these two high dimension vectors rpandrp−rq.\nThus we normalize these two vectors first, then calculate βkby simply doing a\ndot product.\n4. Experimental results and discussion\nInthissection,wenumericallycomparetheperformanceofthisnon -stationary\nAAoptD with sAA (with constant damping or without damping). The fir st part\n10containsexamples where largerwindow sizes mareneeded in orderto accelerate\nthe iteration. The second part consists of some examples where sm all window\nsizes are working very well. All these experiments are done in MATLAB 2021b\nenvironment. MATLAB codes are available upon request to the auth ors.\nThis first example is from Walker and Ni’s [17] paper, where a stationar y\nAnderson acceleration with window size m= 50 is used to solve the Bratu\nproblem. This problem has a long history, we refer the reader to Glow inski et\nal. [30] and Pernice and Walker [31], and the references in those pape rs. It is\nnot a difficult problem for Newton-like solvers.\nProblem 4.1. The Bratu problem. The Bratu problem is a nonlinear PDE\nboundary value problem as follows:\n∆u+λ eu= 0, in D = [0,1]×[0,1],\nu= 0, on ∂D.\nIn this experiment, we used a centered-difference discretization o n a 32×32,\n64×64and 128×128grid, respectively. We take λ= 6in the Bratuproblemand\nuse the zero initial approximate solution in all cases. We also applied pr econ-\nditioning such that the basic Picard iteration still works. The precon ditioning\nmatrix that we used here is the diagonal inverse of the matrix A, whereAis a\nmatrix for the discrete Laplace operator.\nThe resultsareshowninthe followingfigures. InFig.1, weplot there sultsof\napplying AA(m) andAAoptD(m) to acceleratePicarditeration with m= 5 and\nm= 10 on a grid of 32 ×32. As we see from the picture, AA(5) andAA(10) does\nnot accelerate the convergence speed very much. AAoptD(5) andAAoptD(10)\nperform much better than AA(5) andAA(10). However, we also notice that\nthere are some inconsistencies and stagnations in AAoptD(m). Thus we go\nfurther to plot the βkvalues that are used in each iteration, see Fig. 2. From\nFig. 2 we see that: for AAoptD(10), some optimized damping factors are below\n0.3(see the dashedline). As we know, the damping factor βk∈(0,1] andβk= 1\nmeans no damping. Thus small βkmay cause an over-damping phenomenon,\n110 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nAA(5)\nAA(10)\nAAoptD(5)\nAAoptD(10)\nFigure 1: Compare AA and AoptD for solving nonlinear Bratu pr oblems.\nwhich might be the reasonfor small inconsistencies observedin Fig. 1 ; Similarly,\nwe see that the residual of AAoptD(10) in Fig. 1 is not decreasing consistently\naround 10th iteration (see the read dashed square region in Fig. 1) , where the\ncorresponding βkvalues are super close to zero as shown in Fig. 2.\nTo balance the over-damping effect, we bound these βkaway from zero. The\nfirst strategy we propose is to use\nˆβk= max{βk,η}, (13)\nwhereηis a small positive number such that 0 < η <0.5. For example, to\nreduce the over-damping effect, we take η= 0.3 in (13) as a lower bound. We\nplot the new βkvalues at each iteration in Fig. 3. There are no βkvalues less\nthan 0.3 anymore. The corresponding results are in Fig. 4. Compared with t he\nresults in Fig. 1, we see that there is less stagnation (see the red da shed square\nregion in Fig. 4) and onvergence is also faster. We also note that the βkvalues\nin Fig. 3 differs a lot from the values of βkin Fig. 2. Because changing βkin\n120 10 20 30 40 50 60 70 80 90\niteration00.10.20.30.40.50.60.70.8k\nFigure 2: Optimal damping factors in each iteration for m= 10.\nprevious iterations will affect the later ones.\n0 10 20 30 40 50 60 70 80\niteration00.10.20.30.40.50.60.70.8k\nFigure 3: Modified optimal damping factors: ˆβk= max{βk,η}withη= 0.3\nAlthough the results in Fig. 4 are better than those in Fig. 1, we notic e that\nthere are still some inconsistencies in the red dashed square region . To further\nsmooth out these inconsistencies, we change these “bad” βkvalues further away\nfrom zero. Therefore, we propose our second strategy:\nˆβk=\n\nβkifβk≥η,\n1−βkifβk< η.(14)\nWe note here that there is some trade-off between stability and spe ed of conver-\ngence. This does not mean that larger βkwork better, since larger βkmay not\nspeed up the convergence if it is not appropriate. Therefore, dam ping is good,\n130 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nAA(5)\nAA(10)\nAAoptD(5)\nAAoptD(10)\nFigure 4: Solving nonlinear Bratu problems: ˆβk= max{βk,η}withη= 0.3\nbut over-damping may cause inconsistencies and stagnation. In ou r numerical\nexperiment, we take η= 0.3 in (14) as an example. The results are in Fig. 5.\nCompared with the results in Fig. 1 and Fig. 4, it becomes better. We s ee that\nthere are almost no inconsistencies and there is faster convergen ce. We also plot\nthe new βkin Fig. 5.\nTo compare with the results provided in [17], we go further to increas e the\nwindows until m= 50. Again, without bounding away from zero, there are\nsome stagnations and inconsistencies. To avoid strong over-damp ing, we apply\n(14) again with η= 0.3 and obtain our new results in Fig. 7. We easily see that\nAAoptD(20) works as well as AA(50). Moreover, to test its scaling properties,\nwe also solve the Bratu problem on larger grids. In Fig. 8, for a grid siz e 64×64,\nwe see that AAoptD(10) is already comparable with AA(60) and AAoptD(30)\nperforms better than AA(60). Similarly, for a grid size 128 ×128, Fig. 9 shows\nthatAAoptD(40) performs much better than AA(80).\n140 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nAA(5)\nAA(10)\nAAoptD(5)\nAAoptD(10)\nFigure 5: Solving nonlinear Bratu problems: ˆβk= 1−βkwhenβk<0.3.\n0 10 20 30 40 50 60 70 80 90 100\niteration0.10.20.30.40.50.60.70.8k\nFigure 6: Modified optimal damping factors: ˆβk= 1−βkwhenβk<0.3.\n150 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nPicard\nAA(10)\nAA(20)\nAA(30)\nAA(40)\nAA(50)\nAAoptD(10)\nAAoptD(20)\nFigure 7: Using larger size windows and bounding the damping factor away from zero.\n0 50 100 150 200 250 300\niteration10-910-810-710-610-510-410-310-210-1residual64x64\nPicard\nAA(10)\nAA(20)\nAA(30)\nAA(50)\nAA(60)\nAAoptD(10)\nAAoptD(20)\nAAoptD(30)\nFigure 8: Scaling: solve the Bratu problem on a 64 ×64 gird.\n160 50 100 150 200 250 300 350 400 450 500\niteration10-910-810-710-610-510-410-310-210-1residual128x128\nPicard\nAA(20)\nAA(40)\nAA(60)\nAA(80)\nAAoptD(20)\nAAoptD(40)\nFigure 9: Scaling: solve the Bratu problem on a 128 ×128 grid.\nProblem 4.2. The nonlinear convection-diffusion problem. Use AAand\nAAoptD to solve the following 2D nonlinear convection-diffu sion equation in a\nsquare region:\n(−uxx−uyy)+(ux+uy)+ku2=f(x,y),(x,y)∈D= [0,1]×[0,1]\nwith the source term\nf(x,y) = 2π2sin(πx)sin(πy)\nand zero boundary conditions: u(x,y) = 0on∂D.\nIn this numerical experiment, we use a centered-difference discre tization on\n32×32 and 64×64 grids, respectively. We take k= 3 in the above problem\nand use u0= (1,1,···,1)Tas an initial approximate solution in all cases. As\nin solving the Bratu problem, the same preconditioning strategy is us ed here so\nthat the basic Picard iteration still works. To bound βkaway from zero, we use\n(14) with η= 0.25. The results are shown in Fig. 10 and Fig. 11 for n= 32×32\n17andn= 64×64, respectively. From Fig. 10, we see that AAoptD(5) is already\nbetter than AA(15); From Fig. 11, we also observe that AAoptD(20) is better\nthanAA(50). In both cases, AAoptD(m) does a much better job than AA(m),\nwhich is consistent with our previous example.\n0 100 200 300 400 500\niteration10-1010-810-610-410-2100102residual32  32\nPicard\nAA(5)\nAA(10)\nAA(15)\nAAoptD(5)\nAAoptD(10)\nAAoptD(15)\nFigure 10: Solving the nonlinear convection-diffusion prob lem on a 32 ×32 gird.\nOur next example is about solving a linear system Ax=b. As proved by\nWalker and Ni in [17], AA without truncation is “essentially equivalent” in a\ncertain sense to the GMRES method for linear problems.\nProblem 4.3. The linear equations. Apply AA and AAoptD to solve the\nfollowing linear system Ax=b, whereAis\nA=\n2−1···0 0\n−1 2···0 0\n...............\n0 0···2−1\n0 0··· −1 2\n, A∈Rn×n\n180 100 200 300 400 500 600\niteration10-1010-810-610-410-2100102residual64  64\nPicard\nAA(10)\nAA(20)\nAA(30)\nAA(50)\nAAoptD(10)\nAAoptD(20)\nAAoptD(30)\nFigure 11: Solving the nonlinear convection-diffusion prob lem on a 64 ×64 gird.\nand\nb= (1,···,1)T.\nChoosen= 10andn= 100, respectively. Here, we choose a large nso that\na large window size mis needed in Anderson Acceleration. We also note that\nthe Picard iteration does not work for this problem.\nThe initial guess is x0= (0,···,0)T. Without bounding βkaway from zero,\nthe results are shown in Fig. 12 and Fig. 13. For small m,AA(1) does not work,\nbutAAoptD(1) works. Moreover, we obtain from Fig. 12 that AAoptD(m) still\ndoes better than AA(m). When n= 100, we need larger mvalues. In this case,\nas shown in Fig. 13, AAoptD(5) already performs much better than AA(25).\nThis example shows that AAoptD can also be used to solve linear problems.\nFinally, we consider cases where very small mworks. Our example is from\nToth and Kelley’s paper [3], where AA is applied to solve the Chandrase khar\nH-equation.\n190 50 100 150 200 250 300\niteration10-910-810-710-610-510-410-310-210-1100101residualn=10\nAA(2)\nAA(3)\nAA(4)\nAAoptD(1)\nAAoptD(2)\nAAoptD(3)\nAAoptD(4)\nFigure 12: Small m: solving a linear problem Ax=bwithn= 10.\n0 50 100 150 200 250 300 350 400 450 500\niteration10-810-710-610-510-410-310-210-1100101residualn=100\nAA(5)\nAA(10)\nAA(15)\nAA(20)\nAA(50)\nAAoptD(5)\nAAoptD(10)\nAAoptD(15)\nAAoptD(20)\nFigure 13: Large m: solving a linear problem Ax=bwithn= 100.\n20Problem 4.4. the Chandrasekhar H-equation, arising in Radiative Heat Tr ans-\nfer theory, is a nonlinear integral equation:\nH(µ) =G(H) =/parenleftbigg\n1−c\n2/integraldisplay1\n0µ\nµ+vH(v)dv/parenrightbigg−1\n,\nwherec∈[0,1)is a physical parameter.\nWe will discretize the equation with the composite midpoint rule. Here we\napproximate integrals on [0,1]by\n/integraldisplay1\n0f(µ)dµ≈1\nNN/summationdisplay\nj=1f(µj)\nwhereµj= (i−1/2)/Nfor1≤i≤N. The resulting discrete problem is\nF(x)i=xi−\n1−c\n2NN/summationdisplay\nj=1µixj\nµi+µj\n−1\n,\nwhich is a fully nonlinear system.\nIt is known [32] both for the continuous problem and its follo wing midpoint\nrule discretization, that if c <1\nρ(G′(H∗))≤1−√\n1−c <1,\nwhereρdenotes spectral radius. Hence the local convergence theor y and Picard\niteration works.\nIn our numerical experiment, we choose N= 500,c= 0.5,c= 0.99 and\nc= 1. The case c= 1 is a critical value (Picard does not work in this case, but\nAA does). The numerical results are in Fig. 14 to Fig. 16. Firstly, AA(m) and\nAAoptD(m), with very small m( ≤3) values, work for all cases including the\ncritical case c= 1 and their performances are comparable. Secondly, increasing\nmdoes not always increase the performance. Thirdly, AAoptD may no t always\nhave advantages over AA for small window size m. This result is reasonable\nsince AAoptD(m) is kind of like packaging AA(m) andAA(1). Ifmis small,\nthere is almost no difference between AA(m) andAA(1), thus packaging them\n(varying window sizes) may not give better results.\n210 1 2 3 4 5 6 7 8 9 10\niteration10-810-710-610-510-410-310-210-1100101102residualPicard\nAA(1)\nAA(2)\nAA(3)\nAAoptD(1)\nAAoptD(2)\nAAoptD(3)\nFigure 14: Solving Chandrasekhar H-equation with AA and AAo ptD:c= 0.5\n0 5 10 15 20 25 30 35 40 45 50\niteration10-1010-810-610-410-2100102residualPicard\nAA(1)\nAA(2)\nAA(3)\nAAopt(1)\nAAoptD(2)\nAAoptD(3)\nFigure 15: Solving Chandrasekhar H-equation with AA and AAo ptD:c= 0.99\n220 5 10 15 20 25 30 35 40 45 50\niteration10-810-710-610-510-410-310-210-1100101102residualPicard\nAA(1)\nAA(2)\nAAoptD(1)\nAAoptD(2)\nFigure 16: Solving Chandrasekhar H-equation with AA and AAo ptD:c= 1\n5. Conclusions\nWe proposedanon-stationaryAndersonaccelerationalgorithmwit h anopti-\nmized damping factor in each iteration to further speed up linear and nonlinear\niterations by applying one extra optimization. This procedure has a s trong con-\nnection toanotherperspective ofgeneratingnon-stationaryAA (i.e. varyingthe\nwindow size mat different iterations). It turns out that choosing optimal βk\nis somewhat similar to packaging sAA(m) and sAA(1) within a single itera tion\nin a cheap way. Moreover, by taking benefit of the QR decomposition in the\nfirst optimization problem, the calculation of optimized βkat each iteration is\ncheap if two extra function evaluations are relatively inexpensive. O ur numer-\nical results show that the gain of doing this extra optimized step on βkcould\nbe large. Moreover, damping is good but over damping is not good bec ause it\nmay slow down the convergence rate. Therefore, when the statio nary AA is not\nworking well or a larger size of the window is needed in AA, we recommen d to\n23use AAoptD proposed in the present work.\nAcknowledgments\nThis work was partially supported by the National Natural Science F ounda-\ntion of China [grant number 12001287]; the Startup Foundation for Introduc-\ning Talent of Nanjing University of Information Science and Technolo gy [grant\nnumber 2019r106]; The first author Kewang Chen also gratefully ac knowledge\nthe financial support for his doctoral study provided by the China Scholarship\nCouncil (No. 202008320191).\nReferences\n[1] D. G. Anderson, Iterative procedures for nonlinear integral e quations, J.\nAssoc. Comput. Mach. 12 (1965) 547–560. doi:10.1145/321296.321305 .\n[2] D. G. M. Anderson, Comments on “Anderson acceleration, mix-\ning and extrapolation”, Numer. Algorithms 80 (1) (2019) 135–234.\ndoi:10.1007/s11075-018-0549-4 .\n[3] A. Toth, C. T. 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He\nalso introduced the light-cone coordinate system for Loren tz boosts. Also in 1949, he stated\nthe Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations\nin the Lorentz-covariant world. It is possible to integrate these three papers to produce the\nharmonic oscillator wave function which can be Lorentz-tra nsformed. In addition, Dirac, in\n1963, considered two coupled oscillators to derive the Lie a lgebra for the generators of the\nO(3,2) de Sitter group, which has ten generators. It is proven pos sible to contract this group\nto the inhomogeneous Lorentz group with ten generators, whi ch constitute the fundamental\nsymmetry of quantum mechanics in Einstein’s Lorentz-covar iant world.\nPublished in Symmetry, Vol. 12(8), 1720 (2020).1 Introduction\nSince 1973 [1], the present authors have been publishing pap ers on the harmonic oscillator\nwave functions which can be Lorentz-boosted, leading to a nu mber of books [2, 3, 4, 5].\nWe noticed that the Gaussian form of the wave function underg oes an elliptic space-time\ndeformation when Lorentz-boosted, leading to the correct p roton form factor behavior for\nlarge momentum transfers.\nIt was then noted that the harmonic oscillator functions exh ibit the contraction and\northogonality properties quite consistent with known rule s of special relativity and quantum\nmechanics [6, 7].\nIn1977, usingtheLorentz-covariant wave functions, thepr esentauthorsshowedthat Gell-\nMann’s quark model [8] for the hadron at rest and Feynman’s pa rton model for fast-moving\nhadrons [9] are two different manifestations of one Lorentz-c ovariant entity [10, 11].\nIn 1979 [12], it was shown that the oscillator system we const ructed can be used for a\nrepresentation of the Lorentz group, particularly Wigner’ sO(3)-like little group for massive\nparticles [13]. More recently, it was shown that these oscil lator wave functions can serve as\nsqueezed states of light and Gaussian entanglement [14, 15, 16].\nIn 1983, it became known that, if the speed of a spin-1 particl e reaches that of light, the\ncomponent of spin in the direction of its momentum remains in variant as its helicity, but the\nspin components perpendicular to the momentum become one ga uge degree of freedom [17,\n18]. Indeed, this result allows us to include our oscillator -based quark-parton picture as\nfurther content of Einstein’s E=mc2[11], as shown in Table 1.\nThepurposeofthepresentpaperistoshowthattheharmonico scillator wavefunctionswe\nhave studied since 1973 can serve another purpose. It is poss ibleto obtain this covariant form\nof relativistic extended particles by integrating the thre e papers Dirac wrote, as indicated in\nTable 1.\n1. In 1927, Dirac pointed out that the time-energy uncertain ty should be considered if the\nsystem is to be Lorentz-covariant [19].\n2. In 1945, Dirac said the Gaussian form could serve as a repre sentation of the Lorentz\ngroup [20].\n3. In 1949, when Dirac introduced both his instant form of qua ntum mechanics and his\nlight-cone coordinate system [21], he clearly stated that fi nding a representation of the\ninhomogeneous Lorentz group was the task of Lorentz-covari ant quantum mechanics.\n4. In 1963, Dirac used the symmetry of two coupled oscillator s to construct the O(3,2)\ngroup [22].\nIn thefourth paperpublishedin 1963, Dirac considered two c oupled oscillators usingstep-\nup and step-down operators. He then constructed the Lie alge bra (closed set of commutation\nrelations for the generators) of thede Sitter group, also kn own asO(3,2), usingten quadratic\n2Table 1: Lorentz covariance of particles both massive and massless . The little group of Wigner\nunifies, for massive and massless particles, the internal space-tim e symmetries. The challenge for\nus is to find another unification: that which unifies, in the physics of t he high-energy realm, both\nthe quark and parton pictures. In this paper, we achieve this purp ose by integrating Dirac’s three\npapers. A similar table was published in Ref. [11].\nMassive, Slow COVARIANCE Massless, Fast\nEnergy- Einstein’s\nMomentum E=p2/2m E =/radicalig\n(cp)2+(mc2)2E=cp\nInternal S3 S3\nSpace-time Wigner’s Gauge\nSymmetry S1, S2 Little Groups Transformation\nRelativistic Integration\nExtended Quark Model of Dirac’s papers Parton Model\nParticles 1927, 1945, 1949\n3forms of those step-up and step-down operators. The harmoni c oscillator is a language of\nquantum mechanics while the de Sitter group is a language of t he Lorentz group or Einstein’s\nspecial relativity. Thus, his 1963 paper [22] provides the fi rst step toward a unified view of\nquantum mechanics and special relativity.\nIn spite of all those impressive aspects, the above-listed p apers are largely unknown in the\nphysics world. The reason is that there are soft spots in thos e papers. Dirac firmly believed\nthat one can construct physical theories only by constructi ng beautiful mathematics [23].\nHis Dirac equation is a case in point. Indeed, all of his paper s are like poems [24]. They are\nmathematical poems. However, there are things he did not do.\nFirst, his papers do not contain any graphical illustration s. For instance, his light-cone\ncoordinate system could be illustrated as a squeeze transfo rmation, but he did not draw a\npicture [21]. When he talked about the Gaussian function [20 ] in the space-time variables,\nhe could have used a circle in the two-dimensional space of th e longitudinal and time-like\nvariables.\nSecond, Dirac seldom made reference to his own earlier paper s. In his 1945 paper [20],\nthere is a distribution along the time-like variable, but he did not mention his earlier paper\nof 1927 [19] where the time-energy uncertainty was discusse d. In his 1949 paper, when he\nproposed his instant form , he eliminated all time-like excitations, but he forgot to m ention\nhis c-number time-energy uncertainty relation he formulat ed in his earlier paper of 1927 [19].\nDirac’s wife was Eugene Wigner’s younger sister. Dirac thus had many occasions to meet\nhis brother-in-law. Dirac sometimes quoted Wigner’s paper on the inhomogeneous Lorentz\ngroup [13], but without making any serious references, in sp ite of the fact that Wigner’s\nO(3)-like little group is the same as his own instant form ment ioned in his 1949 paper [21].\nPaul A. M. Dirac is an important person in the history of physi cs. It is thus important\nto examine what conclusions we can draw if we integrate all of those papers by closing up\ntheir soft spots.\nIn Section 2, we list four important papers Dirac published f rom 1927 to 1963 [19, 20, 21,\n22]. We then point out his original ideas contained therein.\nIn 1971 [25], Feynman, Kislinger, and Ravndal published a pa per saying that although\nquantum field theory works for scattering problems with runn ing waves, harmonic oscillators\nmay be useful for studying bound states in the relativistic w orld. They then formulated\na Lorentz-invariant differential equation separable into a K lein–Gordon equation for a free\nhadron, and a Lorentz-invariant oscillator equation for th e bound state of the quarks. How-\never, theirsolutionoftheoscillator equationisnotnorma lizableandisphysicallymeaningless.\nIn Section 3 we discuss their paper.\nIn Section 4, we construct normalizable harmonic oscillato r wave functions. These wave\nfunctions are not Lorentz-invariant, because the shape of t he wave function changes as it\nis Lorentz-boosted. However, the wave function is Lorentz- covariant under Lorentz trans-\nformations. It is shown further that these Lorentz-covaria nt wave functions constitute a\nrepresentation of Wigner’s O(3)-like little group for massive particles [13].\n4In Section 5, the covariant oscillator wave functions are ap plied to hadrons moving with\nrelativistic speed. It is noted that the wave function becom es squeezed along Dirac’s light-\ncone system [21]. It is shown that this squeeze property is re sponsible for the dipole cut-off\nbehavior of the proton form factor [26]. It is shown further t hat Gell-Mann’s quark model [8]\nand Feynman’s parton model [9, 27] are two limiting cases of o ne Lorentz-covariant entity,\nas in the case of Einstein’s E=mc2[10, 11].\nIn Section 6, it is shown that the two-variable covariant har monic oscillator wave function\ncan serve as the formula for two-photon coherent states comm only called squeezed states of\nlight [14]. It is then noted that Dirac, in his 1963 paper [22] , constructed the Lie algebra of\ntwo-photon states with ten generators. Dirac noted further that these generators satisfy the\nLie algebra of the O(3,2) de Sitter group.\nTheO(3,2) de Sitter group is a Lorentz group applicable to three spac e-like dimensions\nand two time-like dimensions. Thereare ten generators for t his group. If we restrict ourselves\nto one of the time-variables, it becomes the familiar Lorent z group with three rotation and\nthree boost generators. These six generators lead to the Lor entz group familiar to us. The\nremainingfourgeneratorsareforthreeLorentzboostswith respecttothesecondtimevariable\nand one rotation generator between the two time variables.\nIn Section 7, we contract these last four generators into fou r space-time translation gener-\nators. Thus, it is possible to transform Dirac’s O(3,2) group to the inhomogeneous Lorentz\ngroup [28, 29]. In this way, we show that quantum mechanics an d special relativity share the\nsame symmetry ground. Based on Dirac’s four papers listed in this section, we venture to\nsay that this was Dirac’s ultimate purpose.\n2 Dirac’sEffortstoMake QuantumMechanicsLorentz-\nCovariant\nPaul A. M. Dirac made it his lifelong effort to formulate quantu m mechanics so that it would\nbe consistent with special relativity. In this section, we r eview four of his major papers\non this subject. In each of these papers, Dirac points out fun damental difficulties in this\nproblem.\nDirac noted, in 1927 [19], that the emission of photons from a toms is a manifestation of\nthe uncertainty relation between the time and energy variab les. He also noted that, unlike\nthe uncertainty relation of Heisenberg which allows quantu m excitations, the time or energy\naxis has no excitations along it. Hence, when attempting to c ombine these two uncertainty\nrelations in the Lorentz-covariant world, there remains th e serious difficulty that the space\nand time variables are linearly mixed.\nSubsequently in 1945 [20], Dirac, using the four-dimension al harmonic oscillator at-\ntempted to construct, using the oscillator wave functions, a representation of the Lorentz\ngroup. When he did this, however, the wave functions which re sulted did not appear to be\n5Lorentz-covariant.\nUsing the ten generators of the inhomogeneous Lorentz group Dirac in 1949 [21], con-\nstructed from them three forms of relativistic dynamics. Ho wever, after imposing subsidiary\nconditions necessitated by the existing form of quantum mec hanics, he found inconsistencies\nin all three of the forms he considered.\nIn 1963 [22], Dirac constructed a representation of the O(3,2) de Sitter group. To accom-\nplish this, he used two coupled harmonic oscillators in the f orm of step-up and step-down\noperators. Thus Dirac constructed a beautiful algebra. He d id not, however, attempt to\nexploit the physical contents of his algebra.\nInspiteoftheshortcomingsmentionedabove, itisindeedre markablethatDiracworkedso\ntirelessly on this important subject. We are interested in c ombiningall of his works to achieve\nhis goal of making quantum mechanics consistent with specia l relativity. Let us review the\ncontents of these papers in detail, by transforming Dirac’s formulae into geometrical figures.\n2.1 Dirac’s C-Number Time-Energy Uncertainty Relation\nIt was Wigner who in 1972 [30] drew attention to the fact that t he time-energy uncertainty\nrelation, known from the transition time and line broadenin g in atomic spectroscopy, existed\nbefore 1927. This occurred even before the uncertainty prin ciple that Heisenberg formulated\nin 1927. Also in 1927 [19], Dirac studied the uncertainty rel ation which was applicable to\nthe time and energy variables. When the uncertainty relatio n was formulated by Heisenberg,\nDirac considered the possibility of whether a Lorentz-cova riant uncertainty relation could be\nformed out of the two uncertainty relations [19].\nDirac then noted that the time variable is a c-number and thus there are no excitations\nalong the time-like direction. However, there are excitati ons along the space-like longitudinal\ndirection starting from the position-momentum uncertaint y. Since the space and time coor-\ndinates are mixed up for moving observers, Dirac wondered ho w this space-time asymmetry\ncould be made consistent with Lorentz covariance. This was i ndeed a major difficulty.\nDirac, however, neveraddressed,eveninhislaterpapers,t heseparationintimevariableor\nthe time interval. On the other hand, the Bohr radius, which m easures the distance between\nthe proton and electron is an example of Heisenberg’s uncert ainty relation, applicable to\nspace separation variables.\nIn his 1949 paper [21] Dirac discusses his instant form of relativistic dynamics. Thus\nDirac came back to this question of the space-time asymmetry , in his 1949 paper. There\nhe addresses indirectly the possibility of freezing three o f the six parameters of the Lorentz\ngroup, and hence only working with the remaining three param eters. Wigner, in this 1939\npaper [13, 2] already presented this idea. He had observed in that paper that his little groups\nwith three independent parameters dictated the internal sp ace-time symmetries of particles.\n62.2 Dirac’s Four-Dimensional Oscillators\nSince the language of special relativity is the Lorentz grou p, and harmonic oscillators pro-\nvide a start for the present form of quantum mechanics, Dirac , duringthe second World War,\nconsidered the possibility of using harmonic oscillator wa ve functions to construct represen-\ntations of the Lorentz group [20]. He considered that, by con structing representations of\nthe Lorentz group using harmonic oscillators, he might be ab le to make quantum mechanics\nLorentz-covariant.\nThus in his 1945 paper [20], Dirac considers the Gaussian for m\nexp/parenleftbigg\n−1\n2/bracketleftig\nx2+y2+z2+t2/bracketrightig/parenrightbigg\n. (1)\nThexandyvariables can be dropped from this expression, as we are cons idering a Lorentz\nboost only along the zdirection. Therefore we can write the above equation as:\nexp/parenleftbigg\n−1\n2/bracketleftig\nz2+t2/bracketrightig/parenrightbigg\n. (2)\nSince/parenleftbigz2−t2/parenrightbigis an invariant quantity, the above expression may seem stra nge for those who\nbelieve in Lorentz invariance.\nIn his 1927 paper [19] Dirac proposed the time-energy uncert ainty relation, but observed\nthat. because time is a c-number, there are no excitations al ong the time axis. Hence the\nabove expression is consistent with this earlier paper.\nIf we look carefully at Figure 1, we see that this figure is a pic torial illustration of Dirac’s\nEquation (2). There is localization in both space and time co ordinates. Dirac’s fundamental\nquestion, illustrated in Figure 2, would then be how to make t his figure covariant. Dirac\nstops there. However, this is not the end of his story.\n2.3 Dirac’s Light-Cone Coordinate System\nThe Reviews of Modern Physics, in 1949, celebrated Einstein ’s 70th birthday by publish-\ning a special issue. In this issue was included Dirac’s paper entitledForms of Relativistic\nDynamics [21]. Here Dirac introduced his light-cone coordinate syst em. In this system a\nLorentz boost is seen to be a squeeze transformation, where o ne axis expands while the other\ncontracts in such a way that their product remains invariant as shown in Figure 3.\nWhen boosted along the zdirection, the system is transformed into the following for m:\n/parenleftbiggz′\nt′/parenrightbigg\n=/parenleftbiggcosh(η) sinh(η)\nsinh(η) cosh(η)/parenrightbigg/parenleftbiggz\nt/parenrightbigg\n. (3)\nDirac defined his light-cone variables as [21]\nz+=z+t√\n2, z −=z−t√\n2. (4)\n7Figure 1: Quantum mechanics represented in terms of space-time. As can be seen, there are no\nexcitations along the time-like direction, but quantum excitations alo ng the space-like longitudinal\ndirection are allowed.\nFigure 2: Dirac’s four-dimensional oscillators localized in a closed spac e-time region. This is not\na Lorentz-invariant concept. How about Lorentz covariance?\n8A=4u'v'u\nv\nA=4uv\n=2(t2–z2)zt Dirac 1949\nFigure 3: The light-cone coordinate system pictured with a Lorentz boost. Not only does the\nboost squeeze the square into a rectangle, but it traces a point alo ng the hyperbola.\nThen the form of the boost transformation of Equation (3) bec omes\n/parenleftbiggz′\n+\nz′\n−/parenrightbigg\n=/parenleftbiggeη0\n0e−η/parenrightbigg/parenleftbiggz+\nz−/parenrightbigg\n. (5)\nIt is then apparent that uvariable becomes expanded, but the vvariable becomes contracted.\nWe illustrate this in Figure 3. The product then becomes:\nz+z−=1\n2(z+t)(z−t) =1\n2/parenleftig\nz2−t2/parenrightig\n(6)\nwhich remains invariant. The Lorentz boost is therefore, in Dirac’s picture, a squeeze trans-\nformation.\nDirac also introduced his 1949 paper his instant form of relativistic quantum mechanics.\nThis has the condition\nx0≈0. (7)\nWhat did his approximate equality mean? In this paper, we int erpret the nature of the\ntime-energy uncertainty relation in terms of his c-number. Furthermore, it could mean that\nit is, for the massive particle is the three-dimensional rot ation group, Wigner’s little group,\nwithout the time-like direction.\nAdditionally, Diracstatedthatconstructingarepresenta tionoftheinhomogeneousLorentz\ngroup, was necessary to construct a relativistic quantum me chanics. The inhomogeneous\n9Lorentz group has ten generators, four space-time translat ion generators, three rotation gen-\nerators, and three boost generators, which satisfy a closed set of commutation relations.\nIt is now clear that Dirac was interested in using harmonic os cillators to construct a\nrepresentation of the inhomogeneous Lorentz group. In 1979 , together with another author,\nthe present authors published a paper on this oscillator-ba sed representation [12]. We regret\nthat we did not mention there Dirac’s earlier efforts along thi s line.\n3 Scattering and Bound States\nFrom the three papers written by Dirac [19, 20, 21], let us find out what he really had in\nmind. In physical systems, there are scattering and bound st ates. Throughout his papers,\nDirac did not say he was mainly interested in localized bound systems. Let us clarify this\nissue using the formalism of Feynman, Kislinger, and Ravnda l [25].\nWe are quite familiar with the Klein–Gordon equation for a fr ee particle in the Lorentz-\ncovariant world. We shall use the four-vector notations\nxµ= (x,y,z,t),andx2\nµ=x2+y2+z2−t2. (8)\nThen the Klein–Gordon equation becomes\n\n−/bracketleftigg\n∂\n∂xµ/bracketrightigg2\n+m2\nφ(x) = 0. (9)\nThe solution of this equation takes the familiar form\nexp[±i(p1x+p2y+p3z±Et)]. (10)\nIn1971, Feynmanetal.[25]consideredtwoparticlesaandbb oundtogetherbyaharmonic\noscillator potential, and wrote down the equation\n\n\n−/bracketleftigg\n∂\n∂xaµ/bracketrightigg2\n−/bracketleftigg\n∂\n∂xbµ/bracketrightigg2\n+(xaµ−xbµ)2+m2\na+m2\nb\n\nφ(xaµ,xbµ) = 0.(11)\nThe bound state of these two particles is one hadron. The constituent particles are called\nquarks. We can then define the four-coordinate vector of the hadron a s\nX=1\n2(xa+xb), (12)\nand the space-time separation four-vector between the quar ks as\nx=1\n2√\n2(xa−xb). (13)\n10Then Equation (11) becomes\n\n\n−/bracketleftigg\n∂\n∂Xµ/bracketrightigg2\n+m2\n0+\n−/bracketleftigg\n∂\n∂xµ/bracketrightigg2\n+x2\nµ\n\n\nφ(X,x) = 0. (14)\nThis differential equation can then be separated into\n\n−/bracketleftigg\n∂\n∂Xµ/bracketrightigg2\n+m2\n0\nφ(X,x) =−\n−/bracketleftigg\n∂\n∂xµ/bracketrightigg2\n+x2\nµ\nφ(X,x), (15)\nwith\nφ(X,x) =f(X)ψ(x), (16)\nwheref(X) andψ(x) satisfy their own equations:\n\n−/bracketleftigg\n∂\n∂Xµ/bracketrightigg2\n+m2\na+m2\nb+λ\nf(X) = 0 (17)\nand\n1\n2\n−/bracketleftigg\n∂\n∂xµ/bracketrightigg2\n+x2\nµ\nψ(x) =λψ(x). (18)\nHere, the wave function then takes the form\nφ(X,x) =ψ(x)exp[±i(PxX+PyY+PzZ±ET)], (19)\nwherePx,Py,Pzare for the hadronic momentum, and\nE2=P2\nx+P2\ny+P2\nz+M2,withM2=m2\na+m2\nb+λ. (20)\nHere the hadronic mass Mis determined by the parameter λ, which is the eigenvalue of the\ndifferential equation for ψ(x) given in Equation (18).\nConsidering Feynman diagrams based on the S-matrix formali sm, quantum field theory\nhas been quite successful. It is, however, only useful for ph ysical processes where, after\ninteraction, one set of free particles becomes another set o f free particles. The questions\nof localized probability distributions and their covarian ce under Lorentz transformations is\nnot addressed by quantum field theory. In order to tackle this problem and address these\nquestions, Feynman et al. suggested harmonic oscillators [ 25]. In Figure 4, we illustrate this\nidea.\nHowever, for their wave function ψ(x), Feynman et al. uses a Lorentz-invariant exponen-\ntial form\nexp/parenleftbigg\n−1\n2/bracketleftig\nx2+y2+z2−t2/bracketrightig/parenrightbigg\n. (21)\n11Figure4: Feynman, in aneffort to combine quantum mechanics with sp ecial relativity, gave us this\nroadmap. Feynman’s diagrams provide, in Einstein’s world, a satisfac tory resolution for scattering\nstates. Thus they work for running waves. Feynman suggested t hat harmonic oscillators should\nbe used as a first step for representing standing waves trapped in side an extended hadron.\nThis wave function increases as tbecomes large. This is not an acceptable wave function.\nThey overlooked the normalizable exponential form given by Dirac in Equation (1). They\nalso overlooked the form in the paper of Fujimura et al. [31] w hich was quoted in their own\npaper.\nThus, we are interested in fixing this problem and thus constr ucting Lorentz-covariant\noscillator wave functions satisfying both the rules of quan tum mechanics and the rules of\nspecial relativity.\n4 Lorentz-CovariantPictureofQuantumBoundStates\nIn 1939, Wigner considered internal space-time symmetries of particles in the Lorentz-\ncovariant world [13]. For this purpose, he considered the su bgroups of the Lorentz group\nfor a given four-momentum of the particle. For the massive pa rticle, the internal space-time\nsymmetry is defined for when the particle is at rest, and its sy mmetry is dictated by the\nthree-dimensional rotation group, which allows us to define the particle spin as a dynamical\nvariable, as indicated in Table 1.\nLet us go to the wave function of Equation (19). This wave func tionψ(x) is for the\ninternal coordinates of the hadron, and the exponential for m defines the hadron momentum.\nThus, the symmetry of Wigner’s little group is applicable to the wave function ψ(x).\nThis situation is like the case of the Dirac equation for a fre e particle. Its solution is a\nplane wave times the four-component Dirac spinor describin g the spin orientation and its\nLorentz covariance. This separation of variables for the pr esent case of relativistic extended\nparticles is illustrated in Figure 4. With this understandi nglet us write the Lorentz-invariant\n12differential equation as\n1\n2/bracketleftigg\n−∂2\n∂x2−∂2\n∂y2−∂2\n∂z2+∂2\n∂t2+/parenleftig\nx2+y2+z2−t2/parenrightig/bracketrightigg\nψ(x,y,z,t) =λψ(x,y,z,t).(22)\nHere, the variables x, y, zare for the spatial separation between the quarks, like the B ohr\nradius in the hydrogen atom. The time variable tis thetime separation between the quarks.\nThis variable is very strange, because it does not exist in th e present forms of quantum\nmechanics and quantum field theory. Paul A. M. Dirac did not me ntion this time separation\nin any of his papers quoted here. Yet, it plays the major role i n the Lorentz-covariant world,\nbecause the spatial separation (like the Bohr radius) picks up a time-like component when\nthe system is Lorentz-boosted [34].\nIn his 1927 paper [19], Dirac mentioned the c-number time-en ergy uncertainty relation,\nand he used t≈0 in his 1949 paper for his instant form of relativistic dynam ics. When he\nwrote down a Gaussian function with/parenleftbigx2+y2+z2+t2/parenrightbigas the exponent, he should have\nmeantx, yandzare for the space separation and tfor the time separation, since otherwise\nthe system becomes zero in the remote future and remote past.\nWith this understanding, we are dealing here with the soluti on of the differential equation\nof the form\nψ(x) =f(x,y,z)exp/parenleftigg\n−t2\n2/parenrightigg\n, (23)\nwheref(x,y,z) satisfy the oscillator differential equation\n1\n2/parenleftigg\n−∂2\n∂x2−∂2\n∂y2−∂2\n∂z2+x2+y2+z2/parenrightigg\nf(x,y,z) =/parenleftbigg\nλ−1\n2/parenrightbigg\nf(x,y,z).(24)\nThe form of Equation (23) tells us there are no time-like exci tations. This equation is the\nSchr¨ odinger equation for the three-dimensional harmonic oscillator and its solutions are well\nknown.\nIf we use the three-dimensional spherical coordinate syste m, the solution will give the\nspin or internal angular momentum and the orientation of the bound-state hadron [12]. This\nspherical form is for the O(3) symmetry of Wigner’s little group for massive particles . The\nLorentz-invariant Casimir operators are given in Ref. [12] .\nIf we are interested in Lorentz-boosting the wave function, we note that the original wave\nequation of Equation (22) is separable in all four variables . If the Lorentz boost is made\nalong thezdirection, the wave functions along the xandydirections remain invariant, and\nthus can be separated. We can study only the longitudinal and time-like components. Thus\nthe differential equation of Equation (22) is reduced to\n1\n2/bracketleftigg\n−∂2\n∂z2+∂2\n∂t2+/parenleftig\nz2−t2/parenrightig/bracketrightigg\nψ(z,t) =λψ(z,t). (25)\n13The solution of this differential equation takes the form\nψ(z,t) =Hn(z)exp/parenleftigg\n−/bracketleftigg\nz2+t2\n2/bracketrightigg/parenrightigg\n, (26)\nwhereHn(z) istheHermitepolynomial. Therearenoexcitations in t, thereforeitisrestricted\nto the ground state. For simplicity, we ignore the normaliza tion constant.\nIf this wave function is Lorentz-boosted along the zdirection, the zandtvariables in\nthis expression should be replaced according to\nz→(coshη)z−(sinhη)t,andt→(coshη)t−(sinhη)z. (27)\nAccording to the light-cone coordinate system introduced b y Dirac in 1949 [20], this trans-\nformation can be written as\n(z+t)→e−η(z+t),and (z−t)→eη(z−t). (28)\nThus the Lorentz-boosted wave function becomes\nψη(z,t) =Hn/parenleftbigg1√\n2/bracketleftbige−η(z+t)+eη(z−t)/bracketrightbig/parenrightbigg\nexp/parenleftbigg\n−1\n4/bracketleftig\ne−2η(z+t)2+e2η(z−t)2/bracketrightig/parenrightbigg\n,(29)\nwithout the normalization constant.\nIt is possible to write this wave function using the one-dime nsional normalized oscillator\nfunctionsφn(z) andφk(t) as\nψη(z,t) =/summationdisplay\nnkAnkφn(z)φk(t). (30)\nThis problem has been extensively discussed in the literatu re [2, 6, 7, 35].\nThe most interesting case is the expansion of the ground stat e. The normalized wave\nfunction is then\nψη(z,t) =/parenleftbigg1\nπ/parenrightbigg1/2\nexp/parenleftbigg\n−1\n4/bracketleftig\ne−2η(z+t)2+e2η(z−t)2/bracketrightig/parenrightbigg\n. (31)\nThe Lorentz-boost property of this form is illustrated in Fi gure 5. The expansion in the\nharmonic oscillator wave functions takes the form\nψη(z,t) =1\ncoshη/summationdisplay\nn(tanhη)nφn(z)φn(t). (32)\nThis expression is the key formula for two-photon coherent s tates or squeezed states of\nlight [14]. We shall return to this two-photon problem in Sec tion 6.\n14As for the wave function ψ(x), this is the localized wave function Dirac was considering\nin his papers of 1927 and 1945. If Figures 2 and 3 are combined, we end up with an ellipse\nas a squeezed circle as shown in Figure 5.\nIndeed, one of the most controversial issues in high-energy physics is explained by this\nsqueezed circle. The bound-state quantum mechanics of prot ons, which are known to be\nbound states of quarks, are often assumed to be the same as tha t of the hydrogen atom.\nThen how would the proton look to an observer on a train become s the question. According\nto Feynman [9, 27], when the speed of the train becomes close t o that of light, the proton\nappears like a collection of partons. However, the properti es of Feynman’s partons are quite\ndifferent from the properties of the quarks. This issue shall b e discussed in more detail in\nSection 5.\n5 Lorentz-Covariant Quark Model\nEarly successes in the quark model include the calculation o f the ratio of the neutron and\nproton magnetic moments [32], and the hadronic mass spectra [25, 33]. These are based on\nhadrons at rest. We are interested in this paper how the hadro ns in the quark model appear\nto observers in different Lorentz frames.\nThese days, modern particle accelerators routinely produc e protons moving with speeds\nvery closetothat of light. Thus, thequestionis thereforew hetherthecovariant wave function\ndeveloped in Section 4 can explain the observed phenomena as sociated with those protons\nmoving with relativistic speed.\nThe idea that the proton or neutron has a space-time extensio n had been developed long\nbeforeGell-Mann’s proposal for the quark model [8]. Yukawa [36] developed this idea as early\nas 1953, and his idea was followed up by Markov [37], and by Gin zburg and Man’ko [38].\nSince Einstein formulated his special relativity for point particles, it has been and still\nis a challenge to formulate a theory for particles with space -time extensions. The most\nnaive idea would be to study rigid spherical objects, and the re were many papers on this\nsubjects. But we do not know where that story stands these day s. We can however replace\nthese extended rigid bodies by extended wave packets or stan ding waves, thus by localized\nprobability entities. Then what are the constituents withi n those localized waves? The quark\nmodel gives the natural answer to this question.\nHofstadter and McAllister [39], by using electron-proton s cattering to measure the charge\ndistribution inside the proton, made the first experimental discovery of the non-zero size of\nthe proton. If the proton were a point particle, the scatteri ng amplitude would just be a\nRutherford formula. However, Hofstadter and MacAllister f ound a tangible departure from\nthis formula which can only be explained by a spread-out char ge distribution inside the\nproton.\nInthissection, weareinterestedinhowwellthebound-stat epicturedevelopedinSection4\n15Figure 5: Lorentz covariance of the internal the wave function ψη(z,t) of Equation (31). Syn-\nthesizing Figures 2 and 3, we obtain a squeezed circle as shown in this fi gure. This figure thus\nintegrates Dirac’s three papers [19, 20, 21].\n16works in explaining relativistic phenomena of those proton s. In Section 5.1 we study in detail\nhow the Lorentz squeeze discussed in Section 4 can explain th e behavior of electron-proton\nscattering as the momentum transfer becomes relativistic.\nSecond, we note that the proton is regarded as a bound state of the quarks sharing the\nsame bound-state quantum mechanics with the hydrogen atom. However, it appears as a\ncollection of Feynman’s partons. Thus, it is a great challen ge to see whether one Lorentz-\ncovariant formula can explain both the static and light-lik e protons. We shall discuss this\nissue in Section 5.2\nOne hundred years ago, Einstein and Bohr met occasionally to discuss physics. Bohr was\ninterested in how the election orbit looks and Einstein was w orrying about how things look\nto moving observers. Did they ever talk about how the hydroge n atom appears to moving\nobservers? In Section 5.3, we briefly discuss this issue.\nIt is possible to conclude from the previous discussion that the Lorentz boost might\nincrease the uncertainty. To deal with this, in Section 5.4, we address the issue of the\nuncertainty relation when the oscillator wave functions ar e Lorentz-boosted.\n5.1 Proton Form Factor\nUsing the Born approximation for non-relativistic scatter ing, we see what effect the charge\ndistribution has on the scattering amplitude. When electro ns are scattered from a fixed\ncharge distribution with a density of eρ(r), the scattering amplitude becomes:\nf(θ) =−/parenleftigg\ne2m\n2π/parenrightigg/integraldisplay\nd3xd3x′ρ(r′)\nRexp(−iQ·x). (33)\nHere we use r=|x|,R=|r−r′|,andQ=Kf−Ki,which is the momentum transfer. We\ncan reduce this amplitude to:\nf(θ) =2me2\nQ2F(Q2). (34)\nThe density function’s Fourier transform is given by F(Q2) which can be written as:\nF/parenleftig\nQ2/parenrightig\n=/integraldisplay\nd3xρ(r)exp(−iQ·x). (35)\nThis above quantity is called the form factor and it describe s the charge distribution in terms\nof the momentum transfer which can be normalized by:\n/integraldisplay\nρ(r)d3x= 1. (36)\nTherefore, from Equation (35), F(0) = 1. the scattering ampl itude of Equation (33) be-\ncomes the Rutherford formula for Coulomb scattering if the d ensity function, corresponding\n17Proton\nPElectron\nExchange PhotonK\nKif\nPf\ni\nFigure 6: Electron-proton scattering in the Breit frame. The outg oing momentum of the proton\nis opposite in sign but equal in magnitude to that of the incoming proto n.\nto a point charge, is a delta function, F/parenleftbigQ2/parenrightbig= 1, for all values of Q2. Increasing values\nofQ2, which are deviations from Rutherford scattering, give a me asure of the charge distri-\nbution. Hofstadter’s experiment, which scattered electro ns from a proton target, found this\nprecisely [39].\nWhen the energy of the incoming electron becomes higher, it i s necessary to take into\naccount the recoil effect of target proton. It then requires th at the problem be formulated\nin the Lorentz-covariant framework. It is generally agreed that quantum electrodynamics\ncan describe electrons and their electromagnetic interact ion by using Feynman diagrams for\npractical calculations. When using perturbation, a power s eries of the fine structure constant\nα.is used to expand the scattering amplitude. Therefore, the l owest order in α,can, using\nthe diagram given in Figure 6, describe the scattering of an e lectron by a proton.\nMany textbooks on elementary particle physics [40, 41] give the corresponding matrix\nelement as the form:\n¯U(Pf)Γµ(Pf,Pi)/parenleftbigg1\nQ2/parenrightbigg\n¯U(Kf)γµU(Ki), (37)\nwhere the initial and final four-momenta of the proton and ele ctron, respectively, are given\nbyPi,Pf,KiandKf. The Dirac spinor for the initial proton is U(Pi), while the (four-\nmomentum transfer)2,Q2, is\nQ2= (Pf−Pi)2= (Kf−Ki)2. (38)\nIf the proton were a point particle like the electron, the Γ µwould beγµ, but for a\nparticle, like the proton, with space-time extension, it is γµF/parenleftbigQ2/parenrightbig. The virtual photon being\n18exchanged between the electron and the proton produces the/parenleftbig1/Q2/parenrightbigfactor in Equation (37).\nFor the particles involved in the scattering process this qu antity is positive for physical values\nof the four-momenta in the metric we use.\nFrom the definition of the form factor given in Equation (35), we can make a relativistic\ncalculation of the form factor. The density function, which depends only on the target\nparticle, is proportional to ψ(x)†ψ(x). The wave function for quarks inside the proton is\nψ(x). This expression is a special case of the more general form\nρ(x) =ψ†\nf(x)ψi(x), (39)\nwhere the initial and final wave function of the target atom is given byψiandψf. The form\nfactor of Equation (35) can then be written as\nF/parenleftig\nQ2/parenrightig\n=/parenleftig\nψf(x),e−iQ·rψi(x)/parenrightig\n. (40)\nThe required Lorentz generalization, starting from this ex pression, can be made using the\nrelativistic wave functions for hadrons.\nBy replacing each quantity in the expression of Equation (35 ) by its relativistic counter-\npart we should be able to see the details of the transition to r elativistic physics. If we go\nback to the Lorentz frame in which the momenta of the incoming and outgoing nucleons have\nequal magnitude but opposite signs, we obtain\npi+pf= 0. (41)\nThis kinematical condition is illustrated in Figure 6.\nWe call the Lorentz frame in which the above condition holds t he Breit frame. As illus-\ntrated in Figure 6, there is no loss of generality if the proto n comes in along the zdirection\nbefore the collision and goes out along the negative zdirection after the scattering process.\nThe four vector, Q= (Kf−Ki) = (Pi−Pf), n this frame, has no time-like component. The\nLorentz-invariant form, Q·x, can thus replace the exponential factor Q·r. The covariant\nharmonic oscillator wave functions discussed in this paper can be used for the wave func-\ntions for the protons, assuming that the nucleons are in the g round state. Then the integral\nin the evaluation of Equation (35), which includes the time- like direction and is thus four-\ndimensional, is the only difference between the non-relativi stic and relativistic cases. The\nexponential factor, which does not depend on the time-separ ation variable, therefore, is not\naffected by the integral in the time-separation variable.\nWe can now consider the integral:\ng/parenleftig\nQ2/parenrightig\n=/integraldisplay\nd4xψ†\n−η(x)ψη(x)exp(−iQ·x), (42)\nwhere tanh ηis the velocity parameter (tanh η=v/c) for the incoming proton, and the wave\nfunctionψη, from Equation (31), takes the form:\nψη(x) =1√πexp/parenleftbigg\n−1\n4/bracketleftig\ne−2η(z+t)2+e2η(z−t)2/bracketrightig/parenrightbigg\n. (43)\n19We can, now, after the above decomposition of the wave functi ons, perform the integra-\ntions in the xandyvariables trivially. We can, after dropping these trivial f actors, write the\nproduct of the two wave functions as\nψ†\n−η(x)ψη(x) =1\nπexp/parenleftig\n−[cosh(2η)]/bracketleftig\nt2+z2/bracketrightig/parenrightig\n. (44)\nNow thezandtvariables are separated. Since the tintegral in Equation (42), as the\nexponential factor in Equation (35), does not depend on t, it can also be trivially performed.\nThen integral of Equation (42) can be written as\ng/parenleftig\nQ2/parenrightig\n=1/radicalbig\nπcosh(2η)/integraldisplay\ne−2iPzexp/parenleftig\n−cosh(2η)z2/parenrightig\ndz. (45)\nHere thezcomponent of the momentum of the incoming proton is P. Thevariable Q2, which\nis the (momentum transfer)2, now become 4 P2. The hadronic material, which is distributed\nalong the longitudinal direction, has indeed became contra cted [42].\nWe note that tanh ηcan be written as\n(tanhη)2=Q2\nQ2+4M2, (46)\nwhereMis the proton mass. This equation tells us that β= 0 whenQ2= 0, whileit becomes\none asQ2becomes infinity.\nThe evaluation of the above integral for g/parenleftbigQ2/parenrightbigin Equation (45) leads to\ng/parenleftig\nQ2/parenrightig\n=/parenleftigg\n2M2\nQ2+2M2/parenrightigg\nexp/parenleftigg\n−Q2\n2(Q2+2M2)/parenrightigg\n. (47)\nForQ2= 0,the above expression becomes 1. It decreases as\ng/parenleftig\nQ2/parenrightig\n∼1\nQ2(48)\nasQ2assumes large values.\nSo far the calculation has been performed for an oscillator b ound state of two quarks.\nThe proton, however, consists of three quarks. As shown in th e paper of Feynman et al. [25],\nthe problem becomes a product of two oscillator modes. Thus, the three-quark system is a\nstraightforward generalization of the above calculation. As a result, the form factor F/parenleftbigQ2/parenrightbig\nbecomes;\nF/parenleftig\nQ2/parenrightig\n=/parenleftigg\n2M2\nQ2+2M2/parenrightigg2\nexp/parenleftigg\n−Q2\nQ2+2M2/parenrightigg\n, (49)\n20which is 1 at Q2= 0, and decreases as\nF/parenleftig\nQ2/parenrightig\n∼/bracketleftbigg1\nQ2/bracketrightbigg2\n(50)\nasQ2assumeslargevalues. Thisformfactorfunctionhastherequ ireddipole-cut-off behavior,\nwhich has been observed in high-energy laboratories. This c alculation was carried first by\nFujimura et al. in 1970 [31].\nLet us re-examine the above calculation. If we replace βby zero in Equation (45) and\nignore the elliptic deformation of the wave functions, g/parenleftbigQ2/parenrightbigwill become\ng/parenleftig\nQ2/parenrightig\n= exp/parenleftig\n−Q2/4/parenrightig\n, (51)\nwhich will lead to an exponential cut-off of the form factor. T his is not what we observe in\nlaboratories.\nIn order to gain a deeper understanding of the above-mention ed correlation, let us study\nthe case using the momentum-energy wave functions:\nφη(q) =/parenleftbigg1\n2π/parenrightbigg2/integraldisplay\nd4xe−iq·xψη(x). (52)\nIf we ignore, as before, the transverse components, we can wr iteg/parenleftbigQ2/parenrightbigas [2]\n/integraldisplay\ndq0dqzφ∗\nη(q0,qz−P)φη(q0,qz+P). (53)\nThe above overlap integral has been sketched in Figure 7. The two wave functions overlap\ncompletely in the qzq0plane ifQ2= 0 orP= 0. The wave functions become separated when\nP increases. Because of the elliptic or squeeze deformation , seen in Figure 7, they maintain a\nsmall overlapping region. In the non-relativistic case, th ere is no overlapping region, because\nthe deformation is not taken into account, as seen in Figure 7 . The slower decrease in Q2\nis, therefore, more precisely given in the relativistic cal culation than in the non-relativistic\ncalculation.\nAlthoughourinteresthasbeeninthespace-timebehaviorof thehadronicwavefunction, it\nmust be noted that quarks are spin-1/2 particles. This fact m ust be taken into consideration.\nThis spin effect manifests itself prominently in the baryonic mass spectra. Here we are\nconcerned with the relativistic effects, therefore, it is nec essary to construct a relativistic spin\nwave function for the quarks. The result of this relativisti c spin wave function construction\nfor the quark wave function should be a hadronic spin wave fun ction. In the case of nucleons,\nthe quark spinsshould becombined in a manner to generate the form factor of Equation (44).\nNaively wecouldusefreeDiracspinorsforthequarks. Howev er, itwasshownbyLipes[43]\nthat using free-particle Dirac spinors leads to a wrong beha vior of the form factor. Lipes’\n21P\nq\nq0\nz      Wave Functions\n           Squeezed \nNot S quue zed Not Squeezed Overlap\nFigure 7: The momentum-energy wave functions are Lorentz sque ezed in the form factor calcula-\ntion. The two wave functions become separated as the momentum t ransfer increases. However,\nin the relativistic case, the wave functions maintain an overlapping re gion. In the non-relativistic\ncalculation, the wave functions become completely separated. The unacceptable behavior of the\nform factor is caused by this lack of overlapping region.\n22result, however, does not cause us any worry, as quarks in a ha dron are not free particles.\nThus we have to find suitable mechanism in which quark spins, c oupled to orbital motion,\nare taken into account. This is a difficult problem and is a nont rivial research problem, and\nfurther study is needed along this direction [44].\nIn 1960, Frazer and Fulco calculated the form factor using th e technique of dispersion\nrelations[26]. Insodoingtheyhadtoassumetheexistenceo ftheso-called ρmeson, whichwas\nlater found experimentally, and which subsequently played a pivotal role in the development\nof the quark model.\nEven these days, the form factor calculation occupies a very important place in recent\ntheoretical models, such as quantum chromodynamics (QCD) l attice theory [45] and the\nFaddeev equation [46]. However, it is still noteworthy that Dirac’s form of Lorentz-covariant\nbound states leads to the essential dipole cut-off behavior o f the proton form factor.\n5.2 Feynman’S Parton Picture\nAs we did in Section 5.1, we continue using the Gaussian form f or the wave function of the\nproton. If the proton is at rest, the zandtvariables are separable, and the time-separation\ncan be ignored, as we do in non-relativistic quantum mechani cs. If the proton moves with a\nrelativistic speed, the wave function is squeezed as descri bed in Figure 5. If the speed reaches\nthat of light, the wave function becomes concentrated along positive light cone with t=z.\nThe question then is whether this property can explain the pa rton picture of Feynman when\na proton moves with a speed close to that of light.\nIt was Feynman who, in 1969, observed that a fast-moving prot on can be regarded as\na collection of many partons. The properties of these partons appear to be quite different\nfrom those of the quarks [9, 27, 2]. For example, while the num ber of quarks inside a static\nproton is three, the number of partons appears to be infinite i n a rapidly moving proton.\nThe following systematic observations were made by Feynman :\na. When protons move with velocity close to that of light, the parton picture is valid.\nb. Partons behave as free independent particles when the int eraction time between the\nquarks becomes dilated.\nc. Partons have a widespread distribution of momentum as the proton moves quickly.\nd. There seems to be an infinite number of partons or a number mu ch larger than that of\nquarks.\nThequestion is whethertheLorentz-squeezed wave function producedin Figure5 can explain\nall of these peculiarities.\nEach of the above phenomena appears as a paradox, when the pro ton is believed to be a\nbound state of the quarks. This is especially true of (b) and ( c) together. We can ask how a\nfree particle can have a wide-spread momentum distribution .\n23To resolve this paradox, we construct the momentum-energy w ave function corresponding\nto Equation (31). We can construct two independent four-mom entum variables [25] if the\nquarks have the four-momenta paandpb.\nP=pa+pb, q=√\n2(pa−pb). (54)\nSincePis the total four-momentum, it is the four-momentum of the pr oton. The four-\nmomentumseparation between thequarksis measuredby q. We can thenwritethelight-cone\nvariables as\nq+= (q0+qz)/√\n2, q−= (q0−qz)/√\n2. (55)\nThis results in the momentum-energy wave function\nφη(qz,q0) =/parenleftbigg1\nπ/parenrightbigg1/2\nexp/braceleftbigg\n−1\n2/bracketleftig\ne−2ηq2\n++e2ηq2\n−/bracketrightig/bracerightbigg\n. (56)\nSincetheharmonicoscillator isbeingusedhere, itiseasil yseenthattheabovemomentum-\nenergy wave function has the identical mathematical form to that of the space-time wave\nfunction of Equation (31) and that these wave functions also have the same Lorentz squeeze\nproperties. Though discussed extensively in the literatur e [2, 10, 11], these mathematical\nforms and Lorentz squeeze properties are illustrated again in Figure 8 of the present paper.\nWe can see from the figure, that both wave functions behave lik e those for the static\nbound state of quarks when the proton is at rest with η= 0. However, it can also be seen\nthat asηincreases, the wave functions become concentrated along th eir respective positive\nlight-cone axes as they become continuously squeezed. If we look at the z-axis projection\nof the space-time wave function, we see that, as the proton sp eed approaches that of the\nspeed of light, the width of the quark distribution increase s. Thus, to the observer in the\nlaboratory, the position of each quark appears widespread. Thus the quarks appear like free\nparticles.\nIf we look at the momentum-energy wave function we see that it is just like the space-time\nwave function. As the proton speed approaches that of light, the longitudinal momentum\ndistribution becomes wide-spread. In non-relativistic qu antum mechanics we expect that\nthe width of the momentum distribution is inversely proport ional to that of the position\nwave function. This wide-spread longitudinal momentum dis tribution thus contradicts our\nexpectation from non-relativistic quantum mechanics. Our expectation is that free quarks\nmust have a sharply defined momenta, not a wide-spread moment um distribution.\nHowever, as the proton is boosted, the space-time width and t he momentum-energy\nwidth increase in the same direction. This is because of our L orentz-squeezed space-time and\nmomentum-energy wave functions. If we look at Figures 5 and 8 we see that is the effect\nof Lorentz covariance described in these figures. One of the q uark-parton puzzles is thus\nresolved [2, 10, 11].\n24Figure 8: Lorentz-squeezed wave functions in space-time and in mo mentum-energy variables.\nBoth wave functions become concentrated along their respective positive light-cone axes as the\nspeed of the proton approaches that of light. All the peculiarities o f Feynman’s parton picture are\npresented in these light-cone concentrations.\n25Another puzzling problem is that quarks are coherent when th e proton is at rest but the\npartons appear as incoherent particles. We could ask whethe r this means that Lorentz boost\ncoherence is destroyed. Obviously, the answer to this quest ion is NO. The resolution to this\npuzzle is given below.\nWhen the proton is boosted, its matter becomes squeezed. The result is that the wave\nfunctionfortheprotonbecomes, alongthepositivelight-c one axis, concentrated intheelliptic\nregion. The major axis is expanded in length by exp( η), and, as a consequence, the minor\naxis is contracted by exp( −η).\nTherefore we see that, among themselves, the interaction ti me of the quarks becomes\ndilated. As the wave function becomes wide-spread, the ends of the harmonic oscillator well\nincrease in distance from each other. Universally observed in high-energy experiments, it\nwas Feynman [9] who first observed this effect. The oscillation period thus increase like\neη[27, 47].\nSince the external signal, on the other hand, is moving in the direction opposite to the\ndirection of the proton, it travels along the negative light -cone axis with t=−z. As the\nprotoncontracts alongthenegativelight-coneaxis, thein teraction timedecreasesbyexp( −η).\nThen the ratio of the interaction time to the oscillator peri od becomes exp( −2η). Each\nproton, produced by the Fermilab accelerator, used to have a n energy of 900 GeV. This then\nmeans the ratio is 10−6. Because this is such small number, the external signal cann ot sense,\ninside the proton, the interaction of the quarks among thems elves.\n5.3 Historical Note\nThe hydrogen atom played a pivotal role in the development of quantum mechanics. Niels\nBohr devoted much of his life to understanding the electron o rbit of the hydrogen atom.\nBohr met Einstein occasionally to talk about physics. Einst ein’s main interest was how\nthings look to moving observers. Then, did they discuss how t he hydrogen atom looks to a\nmoving observer?\nIf they discussed this problem, there are no records. If they did not, they are excused. At\ntheir time, the hydrogen atom moving with a relativistic spe ed was not observable, and thus\nbeyond the limit of their scope. Even these days, this atom wi th total charge zero cannot be\naccelerated.\nAfter 1950, high-energy accelerators started producing pr otons moving with relativistic\nspeeds. However, the proton is not a hydrogen atom. On the oth er hand, Hofstadter’s\nexperiment [39] showed that the proton is not a point particl e. In 1964, Gell-Mann produced\nthe idea that the proton is a bound-state of more fundamental particles called quarks. It is\nassumed that the proton and the hydrogen share the same quant um mechanics in binding\ntheir constituent particles. Thus, we can study the moving h ydrogen atom by studying the\nmoving proton. This historical approach is illustrated in F igure 9.\nPaul A. M. Dirac was interested in constructing localized wa ve functions that can be\n26Figure 9: Bohr and Einstein, and then Gell-Mann and Feynman. There are no records indicat-\ning that Bohr and Einstein discussed how the hydrogen looks to movin g observers. After 1950,\nwith particle accelerators, the physics world started producing pr otons with relativistic speeds.\nFurthermore, the proton became a bound state sharing the same quantum mechanics with the\nhydrogen atom. The problem of fast-moving hydrogen became tha t of the proton. How would the\nproton appear when it moves with a speed close to that of light? This is the quark-parton puzzle.\n27Figure 10: Scattering and bound states. These days, Feynman dia grams are used for scattering\nproblems. For bound-state problems, it is possible to construct Lo rentz-covariant harmonic oscil-\nlators by integrating the papers written by Dirac. Feynman diagram s and the covariant oscillators\nare both two different representations of the inhomogeneous Lor entz group. Then is it possible\nto derive Einstein’s special relativity from the Heisenberg brackets ? This problem is addressed in\nSections 6 and 7.\nLorentz boosted. He wrote three papers [19, 20, 21]. If we int egrate his ideas, it is possible\nto construct the covariant oscillator wave function discus sed in this paper. Figure 10 tells\nwhere this integration stands in the history of physics.\nIn constructing quantum mechanics in the Lorentz-covarian t world, the present form of\nquantum field theory is quite successful in scattering probl ems where all participating parti-\ncles are free in the remote past and the remote future. The bou nd states are different, and\nFeynman suggested Lorentz-covariant harmonic oscillator s for studying this problem [25].\nYes, quantum field theory and the covariant harmonic oscilla tors use quite different math-\nematical forms. Yet, they share the same set of physical prin ciples [56]. They are both\nthe representations of the inhomogeneous Lorentz group. We note that Dirac in his 1949\npaper [21] said that we can build relativistic dynamics by co nstructing representations of\nthe inhomogeneous Lorentz group. In Figure 10, both Feynman diagrams and the covariant\noscillator (integration of Dirac’s papers) share the Lie al gebra of the inhomogeneous Lorentz\ngroup. This figure leads to the idea of whether the Lie algebra of quantum mechanics can\nlead to that of the inhomogeneous Lorentz group. We shall dis cuss this question in Section 7.\nSince the time of Bohr and Einstein, attempts have been made t o construct Lorentz-\ncovariant bound states within the framework of quantum mech anics and special relativity.\nIn recent years, there have been laudable efforts to construct a non-perturbative approach\n28to quantum field theory, where particles are in a bound state a nd thus are not free in the\nremote past and remote future [48]. If and when this approach produces localized probability\ndistributions which can Lorentz-boosted, it should explai n both the quark model (at rest)\nand its parton picture in the limit of large speed.\nIn recent years, there have been efforts to represent the obser vation of movements inside\nmaterials using Dirac electric states [49] as well as to use r elativistic methods to under-\nstand atomic and molecular structure [50]. There have also b een efforts to provide covariant\nformulation of the electrodynamics of nonlinear media [51] .\n5.4 Lorentz-Invariant Uncertainty Products\nIn theharmonicoscillator regime, theenergy-momentum wav e functionstake the same math-\nematical form, and the uncertainty relation in terms of the u ncertainty products is well un-\nderstood. However, in the present case, the oscillator wave functions are deformed when\nLorentz-boosted, as shown in Figure 8. According to this figu re, both the space-time and\nmomentum-energy wave functions become spread along their l ongitudinal directions. Does\nthis mean that the Lorentz boost increases the uncertainty?\nIn order to address this question, let us write the momentum- energy wave function as a\nFourier transformation of the space-time wave function:\nφ(qz,q0) =1\n2π/integraldisplay\nψ(z,t)exp(i[qzz−q0t])dt dz. (57)\nThe transverse xandycomponents are not included in this expression. The exponen t of this\nexpression can be written as\nqzz−q0t=q+z−+q−z+, (58)\nwith\nq±=1√\n2(qz±q0), z ±=1√\n2(z±t), (59)\nas given earlier in Equations (4) and (55).\nIn terms of these variables, the Fourier integral takes the f orm\n1\n2π/integraldisplay\nψ(z,t)exp(i[q+z−+q−z+])dt dz. (60)\nIn this case, the variable q+is conjugate to z−, andq−toz+. Let us go back to Figure 8.\nThe major (minor) axis of the space-time ellipse is conjugat e to the minor (major) axis of\nthe momentum-energy ellipse. Thus the uncertainty product s\n/angbracketleftig\nz2\n+/angbracketrightig/angbracketleftig\nq2\n−/angbracketrightig\nand/angbracketleftig\nz2\n−/angbracketrightig/angbracketleftig\nq2\n+/angbracketrightig\n(61)\nremain invariant under the Lorentz boost.\n296 O(3,2)Symmetry DerivablefromTwo-PhotonStates\nIn this section we start with the paper Dirac published in 196 3 on the symmetries from two\nharmonic oscillators [22]. Since the step-up and step-down operators in the oscillator system\nare equivalent to the creation and annihilation operators i n the system of photons, Dirac was\nworking with the system of two photons which is of current int erest [5, 52, 53, 54].\nIn the oscillator system the step-up and step-down operator s are:\na1=1√\n2(x1+iP1), a†\n1=1√\n2(x1−iP1),\na2=1√\n2(x2+iP2), a†\n2=1√\n2(x2−iP2), (62)\nwith\niPi=∂\n∂xi. (63)\nIn terms of these operators, Heisenberg’s uncertainty rela tions can be written as\n/bracketleftig\nai,a†\nj/bracketrightig\n=δij, (64)\nwith\nxi=1√\n2/parenleftig\nai+a†\ni/parenrightig\n, P i=i√\n2/parenleftig\na†\ni−ai/parenrightig\n. (65)\nWith these sets of operators, Dirac constructed three gener ators of the form\nJ1=1\n2/parenleftig\na†\n1a2+a†\n2a1/parenrightig\n, J2=1\n2i/parenleftig\na†\n1a2−a†\n2a1/parenrightig\n, J3=1\n2/parenleftig\na†\n1a1−a†\n2a2/parenrightig\n,(66)\nand three more of the form\nK1=−1\n4/parenleftig\na†\n1a†\n1+a1a1−a†\n2a†\n2−a2a2/parenrightig\n,\nK2= +i\n4/parenleftig\na†\n1a†\n1−a1a1+a†\n2a†\n2−a2a2/parenrightig\n, (67)\nK3=1\n2/parenleftig\na†\n1a†\n2+a1a2/parenrightig\n.\nTheseJiandKioperators satisfy the commutation relations\n[Ji,Jj] =iǫijkJk,[Ji,Kj] =iǫijkKk,[Ki,Kj] =−iǫijkJk. (68)\nThis set of commutation relations is identical to the Lie alg ebra of the Lorentz group\nwhereJiandKiare three rotation and three boost generators respectively . This set of\n30commutators is the Lie algebra of the Lorentz group with thre e rotation and three boost\ngenerators.\nIn addition, with the harmonic oscillators, Dirac construc ted another set consisting of\nQ1=−i\n4/parenleftig\na†\n1a†\n1−a1a1−a†\n2a†\n2+a2a2/parenrightig\n,\nQ2=−1\n4/parenleftig\na†\n1a†\n1+a1a1+a†\n2a†\n2+a2a2/parenrightig\n, (69)\nQ3=i\n2/parenleftig\na†\n1a†\n2−a1a2/parenrightig\n.\nThey then satisfy the commutation relations\n[Ji,Qj] =iǫijkQk,[Qi,Qj] =−iǫijkJk. (70)\nTogether with the relation [ Ji,Jj] =iǫijkJkgiven in Equation (68), JiandQiproduce\nanother Lie algebra of the Lorentz group. Like Ki, theQioperators act as boost generators.\nIn order to construct a closed set of commutation relations f or all the generators, Dirac\nintroduced an additional operator\nS0=1\n2/parenleftig\na†\n1a1+a2a†\n2/parenrightig\n. (71)\nThen the commutation relations are\n[Ki,Qj] =−iδijS0,[Ji,S0] = 0,[Ki,S0] =−iQi,[Qi,S0] =iKi.(72)\nIt was then noted by Dirac that the three sets of commutation r elations given in Equa-\ntions (68), (70) and (72) form the Lie algebra for the O(3,2) de Sitter group. This group\napplies to space of ( x, y, z, t, s ), which is five-dimensional. In this space, the three space- like\ncoordinates are x, y, z, and the time-like variables are given by tands. Therefore, these\ngenerators are five-by-five matrices. The three rotation gen erators for the ( x, y, z) space-like\ncoordinates are given in Table 2. Thethree boost generators with respect to the time variable\ntare given in Table 3. Table 4 contains three boost operators w ith respect to the second\ntime variable sand the rotation generator between the two time variables tands.\nIt is indeed remarkable, as Dirac stated in his paper [22], th at the space-time symmetry\nof the (3 + 2) de Sitter group is the result of this two-oscilla tor system. What is even more\nremarkable is that we can derive, from quantum optics, this t wo-oscillator system. In the\ntwo-photon system in optics, where ican be 1 or 2, aianda†\niact as the annihilation and\ncreation operators.\n31Table 2: Three generators of the rotations in the five-dimensional space of (x, y, z, t, s ). The\ntime-likesandtcoordinates are not affected by the rotations in the three-dimens ional space of\n(x, y, z).\nGenerators Differential Matrix\nJ1 −i/parenleftig\ny∂\n∂z−z∂\n∂y/parenrightig\n0 0 0 0 0\n0 0−i0 0\n0i0 0 0\n0 0 0 0 0\n0 0 0 0 0\n\nJ2 −i/parenleftig\nz∂\n∂x−x∂\n∂z/parenrightig\n0 0i0 0\n0 0 0 0 0\n−i0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n\nJ3 −i/parenleftig\nx∂\n∂y−y∂\n∂x/parenrightig\n0−i0 0 0\ni0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n\n32Table 3: Three generators of Lorentz boosts with respect the tim e variablet. Thescoordinate is\nnot affected by these boosts.\nGenerators Differential Matrix\nK1 −i/parenleftig\nx∂\n∂t+t∂\n∂x/parenrightig\n0 0 0i0\n0 0 0 0 0\n0 0 0 0 0\ni0 0 0 0\n0 0 0 0 0\n\nK2 −i/parenleftig\ny∂\n∂t+t∂\n∂y/parenrightig\n0 0 0 0 0\n0 0 0i0\n0 0 0 0 0\n0i0 0 0\n0 0 0 0 0\n\nK3 −i/parenleftig\nz∂\n∂t+t∂\n∂z/parenrightig\n0 0 0 0 0\n0 0 0 0 0\n0 0 0i0\n0 0i0 0\n0 0 0 0 0\n\n33Table 4: The O(3,2) group has four additional generators. Note that the generat ors in this table\nhave non-zero elements only in the fifth row and the fifth column. Th is is unlike those given in\nTables 2 and 3. Here the svariable is contained in every differential operator.\nGenerators Differential Matrix\nQ1 −i/parenleftig\nx∂\n∂s+s∂\n∂x/parenrightig\n0 0 0 0 i\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\ni0 0 0 0\n\nQ2 −i/parenleftig\ny∂\n∂s+s∂\n∂y/parenrightig\n0 0 0 0 0\n0 0 0 0 i\n0 0 0 0 0\n0 0 0 0 0\n0i0 0 0\n\nQ3 −i/parenleftig\nz∂\n∂s+s∂\n∂z/parenrightig\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 i\n0 0 0 0 0\n0 0i0 0\n\nS0 −i/parenleftig\nt∂\n∂s−s∂\n∂t/parenrightig\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 −i\n0 0 0i0\n\n34It is possible to construct, with these two sets of operators , two-photon states [5]. Yuen,\nas early as 1976 [14], used the two-photon state generated by\nQ3=i\n2/parenleftig\na†\n1a†\n2−a1a2/parenrightig\n. (73)\nThis leads to the two-mode coherent state. This is also known as thesqueezed state .\nIt was later that Yurke, McCall, and Klauder, in 1986 [55], in vestigated two-mode inter-\nferometers. In their study of two-mode states, they started withQ3given in Equation (73).\nThen they considered that the following two additional oper ators,\nK3=1\n2/parenleftig\na†\n1a†\n2+a1a2/parenrightig\n, S 0=1\n2/parenleftig\na†\n1a1+a2a†\n2/parenrightig\n, (74)\nwere needed in one of their interferometers. These three Her mitian operators from Equa-\ntions (73) and (74) have the following commutation relation s\n[K3,Q3] =−iS0,[Q3,S0] =iK3,[S0,K3] =iQ3. (75)\nYurke et al. called this device the SU(1,1) interferometer. Thegroup SU(1,1) is isomorphic\nto theO(2,1) group or the Lorentz group applicable to two space-like an d one time-like\ndimensions.\nIn addition, in the same paper [55], Yurke et al. discussed th e possibility of constructing\nanother interferometer exhibiting the symmetry generated by\nJ1=1\n2/parenleftig\na†\n1a2+a†\n2a1/parenrightig\n, J2=1\n2i/parenleftig\na†\n1a2−a†\n2a1/parenrightig\n, J3=1\n2/parenleftig\na†\n1a1−a†\n2a2/parenrightig\n.(76)\nThese generators satisfy the closed set of commutation rela tions\n[Ji,Jj] =iǫijkJk, (77)\ngiven in Equation (66). This is the Lie algebra for the three- dimensional rotation group.\nYurke et al. called this optical device the SU(2) interferometer.\nWe are then led to ask whether it is possible to construct a clo sed set of commutation\nrelations with the six Hermitian operators from Equations ( 75) and (76). It is not possible.\nWe have to add four additional operators, namely\nK1=−1\n4/parenleftig\na†\n1a†\n1+a1a1−a†\n2a†\n2−a2a2/parenrightig\n,\nK2= +i\n4/parenleftig\na†\n1a†\n1−a1a1+a†\n2a†\n2−a2a2/parenrightig\n, (78)\nQ1=−i\n4/parenleftig\na†\n1a†\n1−a1a1−a†\n2a†\n2+a2a2/parenrightig\n,\nQ2=−1\n4/parenleftig\na†\n1a†\n1+a1a1+a†\n2a†\n2+a2a2/parenrightig\n.\n35There are now ten operators. They are precisely those ten Dir ac constructed in his paper\nof 1963 [22].\nIt is indeed remarkable that Dirac’s O(3,2) algebra is produced by modern optics. This\nalgebra produces the Lorentz group applicable to three spac e-like and two time-like dimen-\nsions.\nThe algebra of harmonic oscillators given in Equation (64) i s Heisenberg’s uncertainty\nrelations in a two-dimensional space. The de Sitter group O(3,2) is basically a language of\nspecial relativity. Does this mean that Einstein’s special relativity can be derived from the\nHeisenberg brackets? We shall examine this problem in Secti on 7.\n7 Contraction of O(3, 2) to the Inhomogeneous\nLorentz Group\nAccording to Section 6, the group O(3,2) has anO(3,1) subgroup with six generators plus\nfour additional generators. The inhomogeneous Lorentz gro up also contains one Lorentz\ngroupO(3,1) as its subgroup plus four space-time translation generat ors. The question\narises whether the four generators in Table 4 can be converte d into the four translation\ngenerators. The purpose of this section is to prove this is po ssible according to the group\ncontraction procedure introduced first by In¨ on¨ u and Wigne r [57].\nIn their paper In¨ on¨ u and Wigner introduced the procedure f or transforming the Lorentz\ngroup into the Galilei group. This procedure is known as group contraction [57]. In this\nsection, we use the same procedure to contract the O(3,2) group into the inhomogeneous\nLorentz group which is the group O(3,1) plus four translations.\nLet us introduce the contraction matrix\nC=\n1/ǫ0 0 0 0\n0 1/ǫ0 0 0\n0 0 1/ǫ0 0\n0 0 0 1 /ǫ0\n0 0 0 0 ǫ\n. (79)\nThis matrix expands the z, y, z, t axes, and contracts saxis asǫbecomes small. Yet, the\nJiandKimatrices given in Tables 2 and 3 remain invariant:\nCJiC−1=Ji,andCKiC−1=Ki. (80)\nThese matrices have zero elements in the fifth row and fifth col umn.\nOn the other hand, the matrices in Table 4 are different. Let us c hooseQ3. The same\n36Table 5: Here the generators of translations are given in the four- dimensional Minkowski space.\nIt is of interest to convert the four generators in the O(3,2) group in Table 4 into the four\ntranslation generators.\nGenerators Differential Matrix\nQ1→P1 −i∂\n∂x\n0 0 0 0 i\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n\nQ2→P2 −i∂\n∂y\n0 0 0 0 0\n0 0 0 0 i\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n\nQ3→P3 −i∂\n∂z\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 i\n0 0 0 0 0\n0 0 0 0 0\n\nS0→P0 i∂\n∂t\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 −i\n0 0 0 0 0\n\n37algebra leads to the elements ǫ2and its inverse, as shown in this equation:\nCQ3C−1=\n1/ǫ0 0 0 0\n0 1/ǫ0 0 0\n0 0 1/ǫ0 0\n0 0 0 1 /ǫ0\n0 0 0 0 ǫ\n\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 i\n0 0 0 0 0\n0 0i0 0\n\nǫ0 0 0 0\n0ǫ0 0 0\n0 0ǫ0 0\n0 0 0ǫ0\n0 0 0 0 1 /ǫ\n(81)\n=\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 i/ǫ2\n0 0 0 0 0\n0 0 0iǫ20\n→P3=\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 i\n0 0 0 0 0\n0 0 0 0 0\n. (82)\nThe second line of this equation tells us that ǫ2becomes zero in the limit of small ǫ. If\nwe make the inverse transformation, result is P3. We can perform the same algebra to arrive\nat the the results given in Table 5. According to this table, QiandS0become contracted\nto the generators of the space-time translations. In other w ords, the de Sitter group O(3,2)\ncan be contracted to the inhomogeneous Lorentz group.\nIfthismatrixisappliedtothefive-vectorof( x, y, z, t, s ), itbecomes( x/ǫ, y/ǫ, z/ǫ, t/ǫ, sǫ ).\nIfǫbecomes very small, the saxis contracts while the four others expand. As ǫbecomes\nvery small, ǫsapproaches zero and can be replaced by ǫ, since both of them are zero. We are\nreplacing zero by another zero, as shown here:\n\nx/ǫ\ny/ǫ\nz/ǫ\nt/ǫ\nǫs\n→\nx/ǫ\ny/ǫ\nz/ǫ\nt/ǫ\nǫ\n,and\nǫ0 0 0 0\n0ǫ0 0 0\n0 0ǫ0 0\n0 0 0ǫ0\n0 0 0 0 1 /ǫ\n\nx/ǫ\ny/ǫ\nz/ǫ\nt/ǫ\nǫ\n=\nx\ny\nz\nt\n1\n.(83)\nThis contraction procedure is indicated in Figure 11.\nIndeed, the five-vector ( x, y, z, t, 1) serves as the space-time five-vector of the inhomo-\ngeneous Lorentz group. The transformation matrix applicab le to this five-vector consists of\nthat of the Lorentz group in its first four row and columns. Let us see how the generators Pi\nandS0generate translations. First of all, in terms of these gener ators, the transformation\nmatrix takes the form\nexp(−i[aPx+bPy+zPz+dPt]) =\n1 0 0 0 a\n0 1 0 0 b\n0 0 1 0 c\n0 0 0 1 d\n0 0 0 0 1\n. (84)\n38Figure 11: Contraction of O(3,2) to the inhomogeneous Lorentz group. The extra time variable\nsbecomes a constant, as shown by a flat line in this figure.\n39If this matrix is applied to the five-vector ( x, y, z, t, 1), the result is the translation:\n\n1 0 0 0 a\n0 1 0 0 b\n0 0 1 0 c\n0 0 0 1 d\n0 0 0 0 1\n\nx\ny\nz\nt\n1\n=\nx+a\ny+b\nz+c\nt+d\n1\n. (85)\nThe five-by-five matrix of Equation (84) indeed performs the t ranslations.\nLet us go to Table 5 again. The four differential forms of the tra nslation generators\ncorrespond to the four-momentum satisfying the equation E2=p2\nx+p2\ny+p2\nz+m2. This is\nof course Einstein’s E=mc2.\n8 Concluding Remarks\nSince 1973, the present authors have been publishing papers on the harmonic oscillator wave\nfunctions which can be Lorentz-boosted. The covariant harm onic oscillator plays roles in\nunderstandingsomeofthepropertiesinhigh-energypartic lephysics. 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We obtain the exact estimat ion\nof the best M-term approximations of Nikol’ski’s, Besov’s classes in th e\nLorentz space with the mixed norm.\nKeywords: Lorentz space, and Besov’s class, and approximation\nMSC:41A10 and 41A25\n1.Introduction\nLetx= (x1,...,xm)∈Tm= [0,2π)mandθj,pj∈[1,+∞),j= 1,...,m. LetL¯p,¯θ(Tm)\ndenotes the space of Lebesgue measureable functions f(¯x) defined on Rm, which have 2 π\n– period with respect to each variable such that\n/bardblf/bardblp,θ=/bardbl.../bardblf/bardblp1,θ1.../bardblpm,θm<+∞,\nwhere\n/bardblg/bardblp,θ=\n\n2π/integraldisplay\n0(g∗(t))θtθ\np−1dt\n\n1\nθ\n,\nwhereg∗a non-increasing rearrangement of the function |g|(see. [1]).\nIt is known that if θj=pj,j= 1,...,m, thenL¯p,¯θ(Tm) =L¯p(Tm) the Lebesgue measur-\nable space of functions f(¯x) defined on Rm, which have 2 π– period with respect to each\nvariable with the norm\n/bardblf/bardbl¯p=/bracketleftBigg/integraldisplay2π\n0/bracketleftbigg\n···/bracketleftbigg/integraldisplay2π\n0|f(¯x)|p1dx1/bracketrightbiggp2\np1···/bracketrightbiggpm\npm−1dxm/bracketrightBigg1\npm\n<+∞,\nwherep= (p1,...,pm),1/lessorequalslantpj<+∞, j= 1,...,m(see [2],p. 128).\nAny function f∈L1(Tm) =L(Tm) can be expanded to the Fourier series\n/summationdisplay\nn∈Zman(f)ei/angbracketleftn,x/angbracketright,\nwherean(f) Fourier coefficients of f∈L1(Tm) with respect to multiple trigonometric\nsystem{ei/angbracketleftn,x/angbracketright}¯n∈Zm,andZmis the space of points in Rmwith integer coordinates.\nFor a function f∈L(Tm) and a number s∈Z+=N∪{0}let us introduce the notation\nδ0(f,¯x) =a0(f), δs(f,x) =/summationdisplay\nn∈ρ(s)an(f)ei/angbracketleftn,x/angbracketright,\nwhere/an}bracketle{t¯y,¯x/an}bracketri}ht=m/summationtext\nj=1yjxj,\nρ(s) =/braceleftbigg\nk= (k1,...,km)∈Zm: [2s−1]/lessorequalslantmax\nj=1,...,m|kj|<2s/bracerightbigg\n,\n12 G. AKISHEV\nwhere [a] is the integer part of the number a.\nLet us consider Nikol’skii, Besov classes( see [2], [3]). Let 1 < pj<+∞,1< θj<+∞,\nj= 1,...,m, 1/lessorequalslantτ/lessorequalslant∞, andr >0\nHr\n¯p,¯θ=/braceleftbigg\nf∈L¯p,¯θ(Tm) : sup\ns∈Z+2sr/bardblδs(f)/bardbl¯p,¯θ/lessorequalslant1/bracerightbigg\n,\nBr\n¯p,¯θ,τ=\n\nf∈L¯p,¯θ(Tm) :\n/summationdisplay\ns∈Z+2srτ/bardblδs(f)/bardblτ\n¯p,¯θ\n1\nτ\n/lessorequalslant1\n\n.\nIt is known that for 1 /lessorequalslantτ/lessorequalslant∞the following holds\nBr\n¯p,¯θ,1⊂Br\n¯p,¯θ,τ⊂Br\n¯p,¯θ,∞=Hr\n¯p,¯θ.\nLetf∈L¯p,¯θ(Tm) and/braceleftbig¯k(j)/bracerightbigM\nj=1be a system of vectors ¯k(j)= (k(j)\n1,...,k(j)\nm) with integer\ncoordinates. Consider the quantity\neM(f)¯p,¯θ= inf\n¯k(j),bj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublef−M/summationdisplay\nj=1bje/angbracketlefti¯k(j),¯x/angbracketright/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n¯p,¯θ,\nwherebjare arbitrary numbers. The quantity eM(f)¯p,¯θis called the best M– term\napproximation of a function f∈L¯p,¯θ(Tm). For a given class F⊂L¯p,¯θ(Tm) let\neM(F)¯p,¯θ= sup\nf∈FeM(f)¯p,¯θ.\nThe best M– term approximation was defined by S.B.Stechkin [4]. Estimations of\nM– term approximations of different classes were provided by R.S. Ism agilov [5], E.S.\nBelinsky [6], V.E. Maiorov [7], B.S. Kashin [8], R. DeVore [9], V.N. Temlyakov [1 0], A.S.\nRomanyuk [11], Dinh Dung [12], Wang Heping and Sun Yongsheng [13], L. Q .Duan and\nG.S. Fang [14], W.Sickel and M. Hansen [15], S.A. Stasyuk [16], [17] and ot hers (see\nbibliography in [18], [19], [20]).\nFor the case p1=...=pm=pandq1=...=qm=θ1=...=θ1=qR.A. De Vore and\nV.N. Temlyakov [20] proved the following theorem.\nTheorem A. Let1/lessorequalslantp,q,τ/lessorequalslant∞andr(p,q) =m/parenleftBig\n1\np−1\nq/parenrightBig\n+if1/lessorequalslantp/lessorequalslantq/lessorequalslant2, or\n1/lessorequalslantq/lessorequalslantp <∞andr(p,q) = max/braceleftBig\nm\np,m\n2/bracerightBig\nin other cases. Then for r > r(p,q)the\nfollowing holds\neM(Br\np,τ)q≍M−r\nm+(1\np−max{1\nq,1\n2})+,\nwherea+= max{a;0}.\nMoreover, inthecaseof m(1\np−1\nq)< r <m\np, 1< p/lessorequalslant2< q <∞S.A.Stasyuk[16]proved\neM(Bp,τ)q≍M−q\n2(r\nm−(1\np−1\nq)). In the case r=m\npobtained eM(Bp,τ)q≍M−1\n2(logM)1−1\nτ\n(see [17] ).\nThe main goal of the present paper is to find the order of the quant ityeM(F)¯q,¯θfor the\nclassF=Br\n¯p,¯θ,τ.\nThe notation A(y)≍B(y) means that there exist positive constants C1, C2such that\nC1A(y)/lessorequalslantB(y)/lessorequalslantC2A(y). IfA/lessorequalslantC2BorA/greaterorequalslantC2B, then we write A << B orA >> B.ONM– TERMS APPROXIMATIONS BESOV CLASSES IN LORENTZ SPACES 3\n2.Auxiliary results\nTo prove the main results the following auxiliary propositions are used .\nTheorem B ([21] ). Letp∈(1,∞).Then there exist positive numbers C1(p),C2(p)\nsuch that for any function f∈Lp(Tm)the following inequality holds:\n/bardblf/bardblp<</vextenddouble/vextenddouble/vextenddouble/parenleftBig∞/summationdisplay\ns=0|δs(f)|2/parenrightBig1\n2/vextenddouble/vextenddouble/vextenddouble\np<</bardblf/bardblp.\nTheorem C ([22] ). Let¯n= (n1,...,nm), nj∈N,j= 1,...,mand\nT¯n(¯x) =/summationdisplay\n|kj|/lessorequalslantnj,j=1,...,mc¯kei/angbracketleft¯k,¯x/angbracketright.\nThen for 1/lessorequalslantpj< qj<∞,1/lessorequalslantθ(1)\nj,θ(2)\nj<+∞,j= 1,...mthe following inequality holds\n/bardblT¯n/bardbl¯q,¯θ(2)<<m/productdisplay\nj=1n1\npj−1\nqj\nj/bardblT¯n/bardbl¯p,¯θ(1).\nLet Ω Mbe a set containing no more than Mvectors¯k= (k1,...,km) with integer\ncoordinates, and P(ΩM,¯x) be any trigonometric polynomial, which consists of harmonics\nwith “indices” in Ω M.\nLemma 1 (see [18]). Let2< qj<+∞, j= 1,...,m.Then for any trigonometric poly-\nnomialP(ΩN)and for any natural number M < Nthere exists trigonometric polynomial\nP(ΩM),such that the following estimation holds\n/bardblP(ΩN)−P(ΩM)/bardbl¯q<<(NM−1)1\n2/bardblP(ΩN)/bardbl2,\nand, moreover, ΩM⊂ΩN.\n3.Main results\nLet us prove the main results.\nTheorem 1. . Let¯p= (p1,...,pm),¯q= (q1,...,qm),¯θ(1)= (θ(1)\n1,...,θ(1)\nm),¯θ(2)=\n(θ(2)\n1,...,θ(2)\nm),1< pj/lessorequalslant2< qj<∞,1< θ(1)\nj,θ(2)\nj<∞, j= 1,...,m,1/lessorequalslantτ/lessorequalslant∞.\n1. Ifm/summationtext\nj=1(1\npj−1\nqj)< r <m/summationtext\nj=11\npj, then\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)≍M−(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1/parenleftbigg\n1\npj−1\nqj/parenrightbigg\n)\n.\n2. Ifr=m/summationtext\nj=11\npj, then\neM/parenleftBig\nBr\n¯p,¯θ(1)τ/parenrightBig\n¯q,¯θ(2)≍M−1\n2(log2M)1−1\nτ,\nforM >1.\n3. Ifr >m/summationtext\nj=11\npj, then\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)≍M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\n.4 G. AKISHEV\nProof.Firstly, we are going to consider the upper bound in the first item. Ta king into\naccount the inclusion Br\n¯p,¯θ(1),τ⊂Hr\n¯p,¯θ(1),1/lessorequalslantτ <+∞, it suffices to prove it for the class\nHr\n¯p,¯θ(1).\nLet 1< pj/lessorequalslant2< qj<∞, j= 1,...,m,andNbe the set of natural numbers. For a\nnumberM∈Nchoose a natural number nsuch that 2nm< M/lessorequalslant2(n+1)m.For a function\nf∈Hr\n¯p,¯θ(1), it is known that\n/bardblδs(f)/bardbl¯p,¯θ(1)/lessorequalslant2−sr,1< pj<∞, j= 1,...,m.\nWe will seek an approximation polynomial P(ΩM,¯x) in the form\nP(ΩM,¯x) =n−1/summationdisplay\ns=0δs(f,¯x)+/summationdisplay\nn/lessorequalslants<αnP(ΩNs,¯x), (1)\nwhere the polynomials P(ΩNs,¯x) will be constructed for each δs(f,¯x) in accordance with\nLemma 1, and the number α >1 will be chosen during the construction.\nLetm/summationtext\nj=1(1\npj−1\nqj)< r <m/summationtext\nj=11\npj. Suppose\nNs=/bracketleftBig\n2nm2s(m/summationtext\nj=11\npj−r)\n2−nα(m/summationtext\nj=11\npj−r)/bracketrightBig\n+1,\nwhere [y] integer part of the number y.\nNow we are going to show that polynomials (1) have no more than Mharmonics (in\nterms of order). By definition of the number Ns, we have\nn−1/summationdisplay\ns=0♯{¯k= (k1,...,km) : [2s−1]/lessorequalslantmax\nj=1,...,m|kj|<2s}+/summationdisplay\nn/lessorequalslants<αnNs/lessorequalslantC2nm+\n+/summationdisplay\nn/lessorequalslants<αn/parenleftBigg\n2nm2s(m/summationtext\nj=11\npj−r)\n2−nα(m/summationtext\nj=11\npj−r)\n+1/parenrightBigg\n<<2nm+(α−1)n <<2nm≍M,\nwhere♯Adenotes the number of elements of the set A.\nNext, by the property of the norm we have\n/bardblf−P(ΩM)/bardbl¯q,¯θ(2)/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nn/lessorequalslants<αn(δs(f)−P(ΩNs))/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n¯q,¯θ(2)+\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nαn/lessorequalslants<+∞δs(f)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n¯q,¯θ(2)=J1(n)+J2(n). (2)\nLet us estimate J2(n).Applying the inequality of different metrics for trigonometric\npolynomials (see Theorem C), we can obtained\nJ2(n)/lessorequalslant/summationdisplay\nαn/lessorequalslants<+∞/bardblδs(f)/bardbl¯q,¯θ(2)<</summationdisplay\nαn/lessorequalslants<+∞2sm/summationtext\nj=1(1\npj−1\nqj)\n/bardblδs(f)/bardbl¯p,¯θ(1).\nTherefore, taking into account f∈Hr\n¯p,¯θ(1)andm/summationtext\nj=1(1\npj−1\nqj)< r, we get\nJ2(n)<</summationdisplay\nαn/lessorequalslants<+∞2−s(r−m/summationtext\nj=1(1\npj−1\nqj))\n<<2−nα(r−m/summationtext\nj=1(1\npj−1\nqj))\n. (3)ONM– TERMS APPROXIMATIONS BESOV CLASSES IN LORENTZ SPACES 5\nLet us estimate J1(n). Using the property of the quasi-norm, Lemma 1 and the inequality\nof different metrics (see Theorem C), we get\nJ1(n) =/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nn/lessorequalslants<αn(δs(f)−P(ΩNs))/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n¯q,¯θ(2)<</summationdisplay\nn/lessorequalslants<αn/bardblδs(f)−P(ΩNs)/bardbl¯q,¯θ(2)<<\n<</summationdisplay\nn/lessorequalslants<αn(N−1\ns2sm)1\n2/bardblδs(f)/bardbl2<<\n<</summationdisplay\nn/lessorequalslants<αn(N−1\ns2sm)1\n22sm/summationtext\nj=1(1\npj−1\n2)\n/bardblδs(f)/bardbl¯p,¯θ(1)<<\n<</summationdisplay\nn/lessorequalslants<αnN−1\n2s2sm/summationtext\nj=11\npj2−sr<<\n<<2−nm\n22nα\n2(m/summationtext\nj=11\npj−r)/summationdisplay\nn/lessorequalslants<αn2s(m/summationtext\nj=11\npj−r)1\n2<<2−nm\n22nα(m/summationtext\nj=11\npj−r)\n. (4)\nSuppose α=m(2m/summationtext\nj=11\nqj)−1. Then from the inequality (4), we get\nJ1(n)/lessorequalslantC2−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1\nqj))\n≍M−(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1\nqj))\n. (5)\nForα=m(2m/summationtext\nj=11\nqj)−1, using the inequality (3) and taking into account 2nm≍Mwe\nobtain\nJ2(n)<< M−(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1\nqj))\n. (6)\nBy (5) and (6), we get from the inequality (2) the following\n/bardblf−P(ΩM)/bardbl¯q,¯θ(2)<< M−(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1\nqj))\nfor any function f∈Hr\n¯p,¯θ(1)in the case ofm/summationtext\nj=1(1\npj−1\nqj)< r <m/summationtext\nj=11\npj.\nFrom the inclusion Br\n¯p,¯θ(1),τ⊂Hr\n¯p,¯θ(1)and the definition of M– term approximation, it\nfollows that\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)<< M−(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1\nqj))\nin the case ofm/summationtext\nj=1(1\npj−1\nqj)< r <m/summationtext\nj=11\npj.\nLet us consider the lower bound. We will use the well-known formula ([2 3], p.25)\neM(f)¯q,¯θ(2)= inf\nΩMsup\nP∈L⊥,/bardblP/bardbl\n¯q′,¯θ(2)′/lessorequalslant1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTmf(¯x)P(¯x)d¯x/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (7)\nwhere ¯q′= (q′\n1,...,q′\nm),¯θ(2)′\n= (θ(2)′\n1,...,θ(2)′\nm),1\nqj+1\nq′\nj= 1,1\nθ(2)\nj+1\nθ\nj(2)′= 1,j= 1,...,m,\nandL⊥\nMis the set of functions that are orthogonal to the subspace of tr igonometric\npolynomials with harmonics in the set Ω M.6 G. AKISHEV\nConsider the function\nF¯q,n(¯x) =/summationdisplay\nmax\nj=1,...,m|kj|/lessorequalslant2[nm(2m/summationtext\nj=11\nqj)−1]ei/angbracketleft¯k,¯x/angbracketright.\nLet ΩMbe a set of Mvectors with integer coordinates. Suppose\ng(¯x) =F¯q,n(¯x)−∗/summationdisplay\n¯k∈ΩMei/angbracketleft¯k,¯x/angbracketright,\nwhere the sum∗/summationtext\n¯k∈ΩMcontains those terms in the function F¯q,n(¯x) with indices only in Ω M.\nBy the inequality\n/bardbl/summationdisplay\nmax\nj=1,...,m|kj|/lessorequalslant2lei/angbracketleft¯k,¯x/angbracketright/bardbl¯p,¯θ(1)<<2lm/summationtext\nj=1(1−1\npj)\n(8)\nand the Perseval’s equality for 1 < q′\nj<2,j= 1,...,m, we obtain\n/bardblg/bardbl¯q′,¯θ(2)′/lessorequalslant/bardblF¯q,n/bardbl¯q′,¯θ(2)′+(2π)m/summationtext\nj=1(1\nqj−1\n2)\n/bardbl∗/summationdisplay\n¯k∈ΩMei/angbracketleft¯k,¯x/angbracketright/bardbl2<<\n<<(2nm\n2+M1\n2)<<2nm\n2. (9)\nNow we consider the function\nP1(¯x) =C22−nm\n2g(¯x). (10)\nThen (9) implies follows that the function P1satisfies the assumptions of the formula (7)\nfor some constant C2>0.\nConsider the function\nfn(¯x) =C32−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1))\nF¯q,n(¯x).\nBy the inequality (8), we get\n∞/summationdisplay\ns=02sr/bardblδs(fn)/bardbl¯p,¯θ(1)<<\n<<2−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1))[m(2m/summationtext\nj=11\nqj)−1]\n/summationdisplay\ns=02sr/bardblδs(F¯q,n)/bardbl¯p,¯θ(1)<<\n<<2−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1))[m(2m/summationtext\nj=11\nqj)−1]\n/summationdisplay\ns=02sr2m/summationtext\nj=1(1−1\npj)\n/lessorequalslantC3.\nHence, the function C−1\n3fn∈Br\n¯p,¯θ(1),1.\nFor the functions (10) and (11), we have by the formula (7), the f ollowing\neM(fn)¯q,¯θ(2)>>inf\nΩM/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTmfn(¯x)P1(¯x)d¯x/vextendsingle/vextendsingle/vextendsingle/vextendsingle>>\n>>2−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1))\n2−nm\n2(/bardblF¯q,n/bardbl2\n2−M)>>ONM– TERMS APPROXIMATIONS BESOV CLASSES IN LORENTZ SPACES 7\n>>2−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1))\n2−nm\n22nm(2m/summationtext\nj=11\nqj)−1\n=\n=C2−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1\nqj))\n. (12)\nHence, it follows from (12) by the inclusion Br\n¯p,¯θ(1),1⊂Br\n¯p,¯θ(1),τthat\neM(fn)¯q,¯θ(2)>>2−nm(2m/summationtext\nj=11\nqj)−1(r−m/summationtext\nj=1(1\npj−1\nqj))\nin the case ofm/summationtext\nj=1(1\npj−1\nqj)< r <m/summationtext\nj=11\npj.\nSo we have proved the first item.\nNow we consider the case r=m/summationtext\nj=11\npj. Letf∈Br\n¯p,¯θ(1),τ. Suppose\nα=m(2m/summationtext\nj=11\nqj)−1and\nNs=/bracketleftBig\n2nmn1\nτ−1/bardblδs(fn)/bardbl¯p,¯θ(1)2sr/bracketrightBig\n+1.\nThen, by definition of the numbers Nsand by Holders inequality, we obtain\nn−1/summationdisplay\ns=0♯ρ(s)+/summationdisplay\nn/lessorequalslants<αnNs<<\n<<2nm+(α−1)n+2nmn1\nτ−1/summationdisplay\nn/lessorequalslants<αn/bardblδs(fn)/bardbl¯p,¯θ(1)2sr<<\n<<2nm+(α−1)n+2nmn1\nτ−1((α−1)n)1\nτ′/parenleftBigg∞/summationdisplay\ns=0/bardblδs(fn)/bardblτ\n¯p,¯θ(1)2srτ/parenrightBigg1\nτ\n<<\n<<2nm≍M.\nTo estimate J1(n) letβ= max{q1,...,qm}.Thenβ >2 andLβ(Tm)⊂L¯q,¯θ(2)(Tm).\nTherefore by applying Theorem B and by the norm property we obta in\nJ1(n) =/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nn/lessorequalslants<αn(δs(f)−P(ΩNs))/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n¯q,¯θ(2)/lessorequalslantC/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nn/lessorequalslants<αn(δs(f)−P(ΩNs))/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nβ<<\n<</vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBigg/summationdisplay\nn/lessorequalslants<αn|δs(f)−P(ΩNs)|2/parenrightBigg1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nβ<<\n<</parenleftBigg/summationdisplay\nn/lessorequalslants<αn/bardblδs(f)−P(ΩNs)/bardbl2\nβ/parenrightBigg1\n2\n.\nIt implies by Lemma 1 and by the inequality of different metrics (see The orem C) that\nJ1(n)<</parenleftBigg/summationdisplay\nn/lessorequalslants<αnN−1\ns2sm/bardblδs(f)/bardbl2\n2/parenrightBigg1\n2\n<<\n<</parenleftBigg/summationdisplay\nn/lessorequalslants<αnN−1\ns2sm22sm/summationtext\nj=1(1\npj−1\n2)\n/bardblδs(f)/bardbl2\n¯p,¯θ(1)/parenrightBigg1\n2\n.8 G. AKISHEV\nNext, since r=m/summationtext\nj=11\npj, we have by definition of the numbers Nsand using Holders\ninequality, the following\nJ1(n)<</parenleftBigg/summationdisplay\nn/lessorequalslants<αnN−1\ns2sm22sm/summationtext\nj=1(1\npj−1\n2)\n/bardblδs(f)/bardbl2\n¯p,¯θ(1)/parenrightBigg1\n2\n<<\n<<(2−nmn1−1\nθ)1\n2/parenleftBigg/summationdisplay\nn/lessorequalslants<αn2sr/bardblδs(f)/bardbl¯p,¯θ(1)/parenrightBigg1\n2\n<<\n<<(2−nmn1−1\nθ)1\n2/parenleftBigg/summationdisplay\nn/lessorequalslants<αn2srθ/bardblδs(f)/bardblτ\n¯p,¯θ(1)/parenrightBigg1\n2τ\n(α−1)1\n2τ′=\n=C2−nm\n2n1−1\nτ≍M−1\n2(logM)1−1\nτ.\nThus\nJ1(n)<< M−1\n2(logM)1−1\nτ (13)\nin the case of r=m/summationtext\nj=11\npj.\nFor the estimation of J2(n) we apply Holders inequality and taking into account that\nr=m/summationtext\nj=11\npjandα=m(2m/summationtext\nj=11\nqj)−1, we obtain\nJ2(n)<</summationdisplay\nnα/lessorequalslants<+∞2sm/summationtext\nj=1(1\npj−1\nqj)\n/bardblδs(f)/bardbl¯p,¯θ(1)<<\n<</parenleftBigg∞/summationdisplay\ns=02srτ/bardblδs(f)/bardblτ\n¯p,¯θ(1)/parenrightBigg1\nθ/parenleftBigg/summationdisplay\nnα/lessorequalslants<+∞2−sτ′m/summationtext\nj=11\nqj/parenrightBigg1\nτ′\n<<\n<<2−nαm/summationtext\nj=11\nqj=C2−nm\n2≍M−1\n2, (14)\nwhereτ′=τ\nτ−1.\nBy (13) and (14) the inequality (2) implies that\n/bardblf−P(ΩM)/bardbl¯q,¯θ(2)<< M−1\n2(logM)1−1\nτ\nin the case r=m/summationtext\nj=11\npj. It proves the upper bound estimation in the second item.\nLetr >m/summationtext\nj=11\npj. Suppose\nNs=/bracketleftBig\n2n(r−m/summationtext\nj=1(1\npj−1))\n2−s(r−m/summationtext\nj=11\npj)/bracketrightBig\n+1.\nThen\nn−1/summationdisplay\ns=0♯ρ(s)+/summationdisplay\nn/lessorequalslants<αnNs<<\n<<2nm+(α−1)n+2n(r−m/summationtext\nj=1(1\npj−1))/summationdisplay\nn/lessorequalslants<αn2−s(r−m/summationtext\nj=11\npj)\n<<ONM– TERMS APPROXIMATIONS BESOV CLASSES IN LORENTZ SPACES 9\n<<2nm+(α−1)n <<2nm<< M.\nIff∈Hr\n¯p,¯θ(1), then, by definition of the numbers Nsandr >m/summationtext\nj=11\npjwe obtain\nJ1(n)/lessorequalslant/parenleftBigg/summationdisplay\nn/lessorequalslants<αnN−1\ns2sm22sm/summationtext\nj=1(1\npj−1\n2)\n/bardblδs(f)/bardbl2\n¯p,¯θ(1)/parenrightBigg1\n2\n<<\n<<2−n\n2(r−m/summationtext\nj=1(1\npj−1))/parenleftBigg/summationdisplay\nn/lessorequalslants<αn2s(r+m/summationtext\nj=11\npj)\n/bardblδs(f)/bardbl2\n¯p,¯θ(1)/parenrightBigg1\n2\n<<\n<<2−n\n2(r−m/summationtext\nj=1(1\npj−1))/parenleftBigg/summationdisplay\nn/lessorequalslants<αn2−s(r−m/summationtext\nj=11\npj)/parenrightBigg1\n2\n<<2−n(r+m/summationtext\nj=1(1\n2−1\npj))\n.\nThus,\nJ1(n)<< M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\n(15)\nin the case of r >m/summationtext\nj=11\npj.\nTo estimate J2(n), we suppose α= (r+m/summationtext\nj=1(1\n2−1\npj))(r+m/summationtext\nj=1(1\nqj−1\npj))−1and get\nJ2(n)<</summationdisplay\nnα/lessorequalslants<∞2sm/summationtext\nj=1(1\npj−1\nqj)\n/bardblδs(f)/bardbl¯p,¯θ(1)/lessorequalslant/summationdisplay\nnα/lessorequalslants<∞2−s(r+m/summationtext\nj=1(1\nqj−1\npj))\n<<\n<<2−nα(r+m/summationtext\nj=1(1\nqj−1\npj))\n<<2−n(r+m/summationtext\nj=1(1\n2−1\npj))\n<< M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\n(16)\nfor a function f∈Hr\n¯p,¯θ(1). By (15) and (14), it follows from (2) that\n/bardblf−P(ΩM)/bardbl¯q,¯θ(2)<< M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\nfor any function f∈Hr\n¯p,¯θ(1)in the case of r >m/summationtext\nj=11\npj.\nFromBr\n¯p,¯θ(1),τ⊂Hr\n¯p,¯θ(1)it follows that\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)<< M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\nin the case of r >m/summationtext\nj=11\npj. It proves the upper bound estimation in the item 3.\nLet us consider the lower bound estimation inthe case r=m/summationtext\nj=11\npj. Consider the function\ng1(¯x) =n/summationdisplay\ns=1/summationdisplay\n¯k∈ρ(s)m/productdisplay\nj=1k−1\njcoskjxj. (17)\nThen\nδs(g1,¯x) =/summationdisplay\n¯k∈ρ(s)m/productdisplay\nj=1k−1\njcoskjxj.10 G. AKISHEV\nIt is known that for a function ds(¯x) =/summationtext\n¯k∈ρ(s)m/producttext\nj=1coskjxjthe following relation holds\n/bardblds/bardbl¯p,¯θ(1)≍2sm/summationtext\nj=1(1−1\npj)\n,1< pj,θ(1)\nj<+∞, j= 1,...,m.\nThereforebytheinequality ofdistinct metrics (seeTheoremC) and bytheMarcinkiewiczs\ntheorem on multipliers, we have\n/bardblδs(g1)/bardbl¯p,¯θ(1)<<2−sm/bardblds/bardbl¯p,¯θ(1)/lessorequalslantC2−sm/summationtext\nj=11\npj.\nHence, since r=m/summationtext\nj=11\npjwe obtain\n/parenleftBigg∞/summationdisplay\ns=02srτ/bardblδs(g1)/bardblτ\n¯p,¯θ(1)/parenrightBigg1\nτ\n/lessorequalslantC1n1\nτ.\nTherefore the function f1(¯x) =C−1\n1n−1\nτg1(¯x) belongs to the class Br\n¯p,¯θ(1),τ,1< pj<\n+∞,j= 1,...,m.\nNow, we are going to construct a function P1, which satisfies the conditions of the\nformula (7). Let\nv1(¯x) =n/summationdisplay\ns=1/summationdisplay\n¯k∈ρ(s)m/productdisplay\nj=1coskjxj\nand Ω Mbe an arbitrary set of vectors ¯k= (k1,...,km) inMwith integer coordinates.\nConsider the function\nu1(¯x) =n/summationdisplay\ns=1/summationdisplay\n¯k∈ρ(s)∩ΩMm/productdisplay\nj=1coskjxj.\nSuppose w1(¯x) =v1(¯x)−u1(¯x). Then, since 1 < q′\nj=qj\nqj−1<2, j= 1,...,m, we obtain,\nby the Perseval‘s equality, the following\n/bardblw1/bardbl¯q′,¯θ(2)′/lessorequalslant/bardblv1/bardbl¯q′,¯θ(2)′+/bardblu1/bardbl2/lessorequalslant/bardblv1/bardbl¯q′,¯θ(2)′+CM1\n2.\nBy the property of quasi-norm and the estimation of the norm of th e Dirichlet kernel in\nthe Lorentz space, we have\n/bardblv1/bardbl¯q′,¯θ(2)′<<n/summationdisplay\ns=1/bardblδs(g1)/bardbl¯q′,¯θ(2)′<<\n<<n/summationdisplay\ns=12sm/summationtext\nj=1(1−1\nq′\nj)\n<<2nm/summationtext\nj=1(1−1\nq′\nj)\n=C2nm/summationtext\nj=11\nqj.\nTherefore, taking into account1\nqj<1\n2, j= 1,...,m, we get\n/bardblw1/bardbl¯q′,¯θ(2)′<<(2nm\n2+M1\n2)/lessorequalslantC22nm\n2.\nHence the function\nP1(¯x) =C−1\n22−nm\n2w1(¯x)ONM– TERMS APPROXIMATIONS BESOV CLASSES IN LORENTZ SPACES 11\nsatisfies the conditions of the formula (7). Then, by substituting t he functions f1andP1\ninto (7) and by orthogonally of a trigonometric system, we obtain\neM(f1)¯q,¯θ(2)>>/summationdisplay\nn1/lessorequalslants<n/summationdisplay\n¯k∈ρ(s)m/productdisplay\nj=1k−1\nj2−nm\n2n−1\nτ>>\n/greaterorequalslantC(ln2)m/summationdisplay\nn1/lessorequalslants<n2−nm\n2n−1\nτ=C(ln2)m2−nm\n2n−1\nτ(n−n1)/greaterorequalslant\n>>(ln2)m2−nm\n2n1−1\nτ≍M−1\n2(log2M)1−1\nτ,\nwheren1is a natural number such that n1<n\n2.\nSo, for the function f1∈Br\n¯p,¯θ(1),τit has been proved that\neM(f1)¯q,¯θ(2)>> M−1\n2(log2M)1−1\nτ\nin the case of r=m/summationtext\nj=11\npj. Hence\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)>> M−1\n2(log2M)1−1\nτ\nin the case of r=m/summationtext\nj=11\npj. It proves the lower bound estimation in the second item.\nLet us prove the lower bound estimation for the case r >m/summationtext\nj=11\npj. Since in this case\nan upper bound estimation of the quantity eM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)does not depend on τand\nBr\n¯p,¯θ(1),1⊂Br\n¯p,¯θ(1),τ, 1< τ <+∞, it suffices to prove the lower bound estimation for\nBr\n¯p,¯θ(1),1.\nFor a number M∈N, we choose a natural number nsuch that 2nm< M/lessorequalslant2(n+1)m\nand 2M/lessorequalslant♯ρ(n), where ♯ρ(n) denotes the number of elements in the set ρ(n).\nConsider the following function\nf3(¯x) =n−1n/summationdisplay\ns=12−sm/summationtext\nj=1(1−1\npj)/summationdisplay\n¯k∈ρ(s)m/productdisplay\nj=1k−r\nm\njcoskjxj.\nThen\n/bardblδs(f3)/bardbl¯p,¯θ(1)<<2−srn−1.\nHence\n∞/summationdisplay\ns=02sr/bardblδs(f3)/bardbl¯p,¯θ(1)/lessorequalslantC3\ni.e. the function C−1\n3f3∈Br\n¯p,¯θ(1),1.\nNext, consider the functions\nv3(¯x) =n/summationdisplay\ns=1/summationdisplay\n¯k∈ρ(s)m/productdisplay\nj=1coskjxj,\nu3(¯x) =n/summationdisplay\ns=1/summationdisplay\n¯k∈ρ(s)∩ΩMm/productdisplay\nj=1coskjxj.12 G. AKISHEV\nSuppose w3(¯x) =v3(¯x)−u3(¯x).By the Perseval‘s equality,\n/bardblu3/bardbl2/lessorequalslantM1\n2,\n/bardblv3/bardbl2= 2(n−1)m\n2.\nFrom these relations, we obtain, by the properties of the norm, th e following\n/bardblw3/bardbl2/lessorequalslant/bardblv3/bardbl2+/bardblu3/bardbl2/lessorequalslantC42nm\n2.\nTherefore the function P3(¯x) =C−1\n42−nm\n2w3(¯x) satisfies the conditions of the formula (7).\nSince 2< qjj= 1,...,m, we have eM(f3)2/lessorequalslantCeM(f3)¯q,¯θ(2). Now, by the formula (7), we\nget\neM(f3)¯q,¯θ(2)>> eM(f3)2>> n−12−nm\n2n/summationdisplay\ns=12−sm/summationtext\nj=1(1−1\npj)/summationdisplay\n¯k∈ρ(s)m/productdisplay\nj=1k−r\nm\nj>>\n>> n−12−nm\n2n/summationdisplay\ns=12−sm/summationtext\nj=1(1−1\npj)\n2s(m−r)=\n=Cn−12−nm\n2n/summationdisplay\ns=12−s(r−m/summationtext\nj=11\npj)\n>>2−n(r+m/summationtext\nj=1(1\n2−1\npj))\n.\nIt follows from the relation 2nm≍Mthat\neM(f3)¯q,¯θ(2)>> M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\nin the case r >m/summationtext\nj=11\npjfor the function C−1\n3f3∈Br\n¯p,¯θ(1),1. Hence\neM/parenleftBig\nBr\n¯p,¯θ(1),1/parenrightBig\n¯q,¯θ(2)>> M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\n.\nTherefore\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q>> M−1\nm(r+m/summationtext\nj=1(1\n2−1\npj))\nin the case r >m/summationtext\nj=11\npj. So Theorem 1 has been proved.\nTheorem 2. . Let¯p= (p1,...,pm),¯q= (q1,...,qm),1< pj< qj/lessorequalslant2,1< θ(1)\nj,θ(2)\nj<∞,\nj= 1,...,m,1/lessorequalslantτ/lessorequalslant+∞.\nIfr >m/summationtext\nj=1(1\npj−1\nqj),then\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,θ(2)≍M−1\nm(r−m/summationtext\nj=1(1\npj−1\nqj))\n.\nProof.For a number M∈Nchoose a natural number nsuch that M≍2nm. By the\ninequality of distinct metrics and by Holder‘s inequality, we have\n/bardblf−n/summationdisplay\ns=0δs(f)/bardbl¯q,¯θ(2)/lessorequalslant∞/summationdisplay\ns=n/bardblδs(f)/bardbl¯q,¯θ(2)/lessorequalslant\n/lessorequalslant/bracketleftBig∞/summationdisplay\ns=02sτr/bardblδs(f)/bardblτ\n¯q,¯θ(2)/bracketrightBig1\nτ/lessorequalslant/bracketleftBig∞/summationdisplay\ns=n2sτ′(r−m/summationtext\nj=1(1\npj−1\nqj))/bracketrightBig1\nτ′<<ONM– TERMS APPROXIMATIONS BESOV CLASSES IN LORENTZ SPACES 13\n<<2n(r−m/summationtext\nj=1(1\npj−1\nqj))\n/lessorequalslantCM−1\nm(r−m/summationtext\nj=1(1\npj−1\nqj))\nforf∈Br\n¯p,¯θ(1),τ,1\nτ+1\nτ′= 1.Therefore\neM(f)¯q,¯θ(2)/lessorequalslant/bardblf−n/summationdisplay\ns=0δs(f)/bardbl¯q,¯θ(2)<< M−1\nm(r−m/summationtext\nj=1(1\npj−1\nqj))\n.\nHence\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)<< M−1\nm(r−m/summationtext\nj=1(1\npj−1\nqj))\n.\nIt proves the upper bound estimation.\nFor the lower bound estimation, let us consider the function\nf0(¯x) =n−r+m/summationtext\nj=1(1\npj−1)\nVn(¯x),\nwhereVn(¯x) is a Valle-Poisson sum with multiplicity.\nNext, following the proof in [19] (pp. 46-47) and applying Theorem B, we obtain the\nlower bound estimation of the quantity eM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2).\nTheorem 3. . Let¯p= (p1,...,pm),¯q= (q1,...,qm),2/lessorequalslantpj< qj<∞,1< θ(1)\nj,θ(2)\nj<∞,\nj= 1,...,m,1/lessorequalslantτ/lessorequalslant+∞. Ifr >m\n2,then\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)≍M−r\nm.\nProof.By the inclusion Br\n¯p,¯θ(1),τ⊂Br\n¯2,¯θ(1),τ⊂Hr\n2,¯θ(1), we have\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)/lessorequalslanteM/parenleftBig\nBr\n2,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)/lessorequalslanteM/parenleftBig\nHr\n2,¯θ(1)/parenrightBig\n¯q,¯θ(2).\nBy Theorem 1,\neM/parenleftBig\nHr\n2,¯θ(1)/parenrightBig\n¯q,¯θ(2)<< M−r\nm.\nforpj= 2,j= 1,...,m.Hence\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)<< M−r\nm.\nit proves the upper bound estimation.\nLet us consider the lower bound estimation. Consider Rudin-Shapiro s polynomial (see\n[24], p.155) of the type\nRs(x) =2s/summationdisplay\ns=2s−1εkeikx, x∈[0,2π], εk=±1.\nit is known that /bardblRs/bardbl∞= max\nx∈[0,2π]|Rs(x)|<<2s\n2(see [24], p. 155). For a given number M\nchoose a number nsuch that M≍2nm.Now consider the function\nf0(¯x) = 2−n(m\n2+r)n/summationdisplay\ns=1m/productdisplay\nj=1Rs(xj)\nThen,by the continuity, f0∈L¯p,¯θ(1)(Tm) and\n∞/summationdisplay\ns=02sτr/bardblδs(f0)/bardblτ\n¯p,¯θ(1)= 2−n(m\n2+r)n/summationdisplay\ns=12sτr/bardblm/productdisplay\nj=1Rs(xj)/bardblτ\n¯p,¯θ(1)/lessorequalslant14 G. AKISHEV\n/lessorequalslant2−n(m\n2+r)n/summationdisplay\ns=12s(m\n2+r)τ/lessorequalslantC0.\nHence, the function C−1\n0f0∈Br\n¯p,¯θ(1),τ. Now construct a function P(¯x), which would satisfy\nthe conditions in the formula (7). Suppose\nv0(¯x) =n/summationdisplay\ns=1m/productdisplay\nj=1Rs(xj), u0(¯x) =∗/summationdisplay\nsm/productdisplay\nj=1Rs(xj),\nwhere the sign ∗means that the polynomial u0(¯x) contains only those harmonics of v0\nwhich have indices in Ω M. Suppose w0(¯x) =v0(¯x)−u0(¯x).Then, since 1 < q′\nj=qj\nqj−1<\n2, j= 1,...,m, and by the Percevals equality, we have\n/bardblw0/bardbl¯q′,¯θ(2)′/lessorequalslant/bardblw0/bardbl2/lessorequalslantC12nm\n2.\nTherefore, for the function P0(¯x) =C−1\n12−nm\n2w0(¯x), the inequality holds /bardblP0/bardbl¯q′,¯θ(2)′/lessorequalslant1.\nNow using the formula (7), we obtain\neM/parenleftBig\nBr\n¯p,¯θ(1)τ/parenrightBig\n¯q,¯θ(2)>> eM(f0)¯q,¯θ(2)>>2−n(m\n2+r)2−nm\n2(2nm−M)>>\n>>2−n(m+r)2nm>> M−r\nm.\nSo\neM/parenleftBig\nBr\n¯p,¯θ(1),τ/parenrightBig\n¯q,¯θ(2)>> M−r\nm.\nIt proves Theorem 3.\nCorollary. Let 1< p/lessorequalslant2< q <∞, 1/lessorequalslantτ/lessorequalslant∞andr=m\np. Then\neM/parenleftbig\nBr\np,τ/parenrightbig\nq≍M−1\n2(logM)1−1\nθ.\nThe proof follows from the second item of Theorem 2.1 if pj=θ(1)\nj=p, qj=θ(2)\nj=\nq,j= 1,...,m.\nRemark. In the case pj=θ(1)\nj=p, qj=θ(2)\nj=q,j= 1,...,mandr > m(1\np−1\nq),,\nthe results of R.A. DeVore and V.N. Temlyakov [20] follow from Theore m 1 - 3. If\n1< p/lessorequalslant2< q <∞andm(1\np−1\nq)< r/lessorequalslantm\np,the results of S.A. Stasyuk [16], [17] follow\nfrom the first and second items of Theorem 1.\nThe cases pj=θ(1)\nj, qj=θ(2)\nj,j= 1,...,m.of Theorem 1 - 3 were announced in [25] and\nin of Theorem 1 the first item proved [26].\nReferences\n[1] Blozinski A.P. , Multivariate rearrangements and Banach functio n spaces with mixed norms, Trans.\nAmer. Math. Soc., 263(1) (1981), 146-167 .\n[2] Nikol’ski S. M., Approximation of classes of functions of several v ariables and embeding theorems,\nMoscow, 1977 (English transl. of lst ed., 1975, Springer - Verlag, Ne w York )\n[3] Besov O.V., Investigation of on family of functional spaces in conn ection with the theorems of\nimbedding and extension, Trudy Mat. Inst. Akad. Nauk SSSR, 60 (1 961), 42 – 61 (in russain).\n[4] Stechkin S.B., On the absolute convergence of orthogonal serie s. Doklad. Akadem. Nauk SSSR ,\n102(2) (1955), 37 – 40.\n[5] IsmagilovR.S., Widths ofsetsinlinearnormedspacesandtheappro ximationoffunctions bytrigono-\nmetric polynomials, Uspehi mathem. nauk, 29(3) (1974), 161 – 178 .\n[6] Belinsky E.S., Approximation by a ’floating’ system of exponents on the classes of smooth periodic\nfunctions. Mat. sb., 132(1) (1987), 20 – 27.\n[7] MaiorovV.E., On linear widths of Sobolev classes and chains of extre mal subspaces, Mat. sb., 113(3)\n(1980), 437 – 463.ONM– TERMS APPROXIMATIONS BESOV CLASSES IN LORENTZ SPACES 15\n[8] Kashin B.S., Approximation properties of complete orthnormal sy stems , Trudy Mat. Inst. Steklov,\n172 (1985), 187 – 201.\n[9] DeVore R.A., Nonlinear approximation, Acta Numerica, 7 (1998), 51 – 150.\n[10] Temlyakov V. N., Nonlinear methods approximation, Foundations Computational Mathematics, 3\n(2003), 33–107.\n[11] Romanyuk A.S., On the best M− −term trigonometric approximations for the Besov classes of\nperiodic functions of many variables, Izv. Ros. Akad. Nauk, Ser. M at., 67(2) (2003), 61 – 100.\n[12] Dinh Dung, On asymptotic order of n - term approximations and n on-linear - n widths, Vietnam\nJournal Math., 27(4) (1999), 363 - 367.\n[13] Wang Heping, Sun Yongsheng Representation and m- term appr oximation for anisotropic classes,\nin S.M. Nikol’skii, et al. (Eds), Theory of approximation of function and applications , Institute of\nRussian Academy, (2003), P. 250 - 268.\n[14] Duan L. Q., Fang G.S. Trigonometric widths and best N– term approximations of the generalized\nperiodic Besov classes BΩ\np,θ, Journal Math. Resear. and Expos., 31(1) (2011), 129 – 141.\n[15] Hansen M., Sickel W., Best m– term approximation and Lizorkin – Triebel spaces, Journal Appro x-\nimation Theory, 163 (2011), 923 – 954.\n[16] Stasyuk S.A., Best m– term trigonometric approximation for the classes Br\np,θof functions of low\nsmoothness, Ukrain. Mathem. Journal, 62(1) (2010), 114 –122.\n[17] Stasyuk S.A., Best m– term trigonometric approximation 0f periodic function of several variables\nfrom Nikol’skii - Besov classes for small smoothness, Journal of Ap proximation Theory, 177 (2014),\n1 - 16.\n[18] Akishev G., On the exact estimations of the best M− −term approximation of the Besov class,\nSiberian Electronic Mathematical Reports , 7 (2010), 255 – 274.\n[19] Akishev G., On the order of the M−−term approximation classes in Lorentz spaces, Matematical\nJournal. Almaty , 11(1) (2011), 5 - 29.\n[20] DeVore R.A., Temlyakov V.N., Nonlinear approximation by trigonome tric sums, Jour. Fourier Anal-\nysis and applications , 2(1) (1995), 29–48\n[21] Lizorkin P.I., Generalized Holder spaces B(r)\np,θand their relations with the Sobolev spaces L(r)\np, Sib.\nMat. Zh., 9(5) (1968) , 1127 - 1152.\n[22] Akishev G., Inequalities of distinct metric of polynomials in Lorentz spaces with mixed norm, First\nErjanov reading, Pavlodar state universuty, (2004), 211-215.\n[23] Korneichuk N.P., Extreme problem in the theory of approximation , Nauka, Moskow , 1976.\n[24] Kashin B.S., Sahakyan A.A., Orthogonal series, Nauka, Moscow, 1984\n[25] Akishev G., On the M−−term approximation Besov’s classes, International Conference “ Theory of\napproximation of functions and its applications” dedicated to the 7- th anniversary of corresponding\nmember of National Academy of Ukraine, professor A.I. Stepanet s (1942 – 2007) May 28 - June 3,\n(2012), Ukraine, Kamianets-Podiisky, 12.\n[26] Akishev G. Trigonometric widths of the Nikol‘skii - Besov classes in the Lebesgue space with mixed\nnorm. Ukr. Math. Zh. , 66(6) (2014), 723-732.\nDepartment of Mathematics and Information Technology, Buk etov Karaganda State\nUniversity, Universytetskaya 28 , 100028, Karaganda , Repu blic Kazakhstan"    },    {        "title": "1009.4871v1.Spatial_Damping_of_Propagating_Kink_Waves_in_Prominence_Threads.pdf",        "content": "arXiv:1009.4871v1  [astro-ph.SR]  24 Sep 2010SPATIAL DAMPING OF PROPAGATING KINK WAVESIN\nPROMINENCE THREADS\nR. Soler, R. Oliver,and J. L.Ballester\nDepartamentdeF´ ısica,Universitatdeles IllesBalears,E -07122,PalmadeMallorca,Spain\nroberto.soler@uib.es\nABSTRACT\nTransverse oscillations and propagating waves are frequen tly observed in threads\nof solar prominences /filaments and have been interpreted as kink magnetohydrody-\nnamic (MHD) modes. We investigate the spatial damping of pro pagating kink MHD\nwavesintransverselynonuniformandpartiallyionizedpro minencethreads. Resonant\nabsorption and ion-neutral collisions (Cowling’s di ffusion) are the damping mecha-\nnismstakenintoaccount. Thedispersionrelationofresona ntkinkwavesinapartially\nionized magnetic flux tube is numerically solved by consider ing prominence condi-\ntions. Analytical expressions of the wavelength and dampin g length as functions of\nthe kink mode frequency are obtained in the Thin Tube and Thin Boundary approxi-\nmations. For typically reported periods of thread oscillat ions, resonant absorption is\nanefficientmechanismforthekinkmodespatialdamping,whileion -neutralcollisions\nhave a minor role. Cowling’s di ffusion dominates both the propagation and damping\nfor periods much shorter than those observed. Resonant abso rption may explain the\nobserved spatial damping of kink waves in prominence thread s. The transverse in-\nhomogeneity length scale of the threads can be estimated by c omparing the observed\nwavelengths and damping lengths with the theoretically pre dicted values. However,\nthe ignorance of the form of the density profile in the transve rsely nonuniform layer\nintroducesinaccuracies in thedeterminationoftheinhomo geneitylengthscale.\nSubject headings: Sun: oscillations – Sun: filaments, prominences – Sun: coron a –\nMagnetohydrodynamics(MHD)– Waves\n1. INTRODUCTION\nWaves and oscillatory motions are frequently reported in th e observations of solar promi-\nnences and filaments (see reviews by Oliver& Ballester 2002; Ballester 2006; Engvold 2008;– 2 –\nMackay et al. 2010). In high-resolution observations, the p rominence fine structures (threads) of-\ntendisplaytransverseoscillationsofsmallamplitude(e. g.,Linet al.2005,2007,2009;Okamotoet al.\n2007;Ninget al.2009),whichhavebeeninterpretedaskinkm agnetohydrodynamic(MHD)waves\n(e.g., D´ ıazet al. 2002; Dymova& Ruderman 2005; Terradas et al. 2008; Lin etal. 2009). The ob-\nserved threads in H αimages are between 3,000 km and 28,000 km long, and between 10 0 km\nand 600 km wide (Lin 2004; Linet al. 2008). The threads outlin e part of much larger magnetic\nflux tubes which are probably rooted in the solar photosphere . The majority of observed periods\nof transverse thread oscillations roughly range between 1 m in and 10 min, but a few detections\nof longer periods of about 20 min have been also informed (e.g ., Yiet al. 1991; Lin 2004). The\nwavelengths are usually between 700 km and 8,000 km, althoug h values up to 250,000 km have\nbeen reported (Okamotoet al. 2007). Recently, Soleret al. ( 2010a) pointed out that the short pe-\nriods and wavelengths are consistent with an interpretatio n in terms of propagating waves, while\nperiods larger than 10 min and wavelengths longer than 100,0 00 km could correspond to stand-\ning oscillations of the whole magnetic tube. In the case of st anding oscillations, the value of the\nwavelengthisnotstronglyinfluencedbythethreadproperti esbutismainlydeterminedbythetotal\nlength of the magnetic tube, since the fundamental kink mode wavelength is twice the length of\nthe tube, approximately (see details in Soler et al. 2010a). Although there are no direct measure-\nments of the length of prominence magnetic tubes, this param eter is estimated around 105km.\nThisroughestimationis inagreement withthewavelengthsr eported byOkamotoet al. (2007). In\naddition, a common feature of the observations is that the os cillations are strongly damped (e.g.,\nTerradas et al. 2002; Ninget al. 2009;Lin et al.2009).\nMotivatedbytheobservationalevidence,greate fforthasbeenrecentlydevotedtothetheoret-\nicalstudyofbothtemporalandspatialdampingofMHDwavesi nprominenceplasmas. Temporal\ndamping is investigated for waves with fixed wavelength, whi le spatial damping is studied for\npropagating waves with fixed frequency. Both phenomena have been extensively investigated in\nunbounded and homogeneous prominence plasmas by assuming d ifferent damping mechanisms\n(e.g.,Carbonell et al.2004,2006,2010;Forteza etal.2007 ,2008). ThereaderisreferredtoOliver\n(2009),Arregui &Ballester(2010),andreferencestherein foracompleteaccountofthetheoretical\nworks.\nInthecaseofprominencethreadoscillations,workssofarh avefocusedontemporaldamping\nbymechanismsas,e.g.,non-adiabatice ffects(Soleret al.2008),ion-neutralcollisions(Soleret a l.\n2009b), and resonantabsorption(Arreguiet al. 2008, 2010; Soleret al.2009a,c, 2010a). Thecon-\nclusions of these works indicate that resonant absorption i s efficient enough to provide realistic\nkink mode damping times consistent with the reported strong damping, whereas non-adiabatic\neffects are negligible and ion-neutral collisions are only imp ortant for shorter wavelengths than\nthoseobserved. Inthecaseofspatialdamping,P´ ecseli & En gvold(2000)studiedthee ffectofion-\nneutralcollisionsbutrestrictedthemselvestoAlfv´ enwa vesandkinkmodeswerenotinvestigated.– 3 –\nAlthough spatial damping of kink waves has been studied in th e context of coronal loops (e.g.,\nPascoeet al. 2010; Terradas etal. 2010a), to our knowledge n o detailed investigation taking into\naccount the peculiar properties of prominence threads can b e found in the existing literature. The\npurposeofthispaperistofillthisgapintheliterature,ast herecentobservationsofwavedamping\ninsolarprominencesneed tobeunderstood.\nHere, we study the spatial damping of propagating kink waves in prominence threads. Our\nmodel is composed of a cylindrical magnetic flux tube with par tially ionized prominence plasma,\nrepresenting a thread, surrounded by a fully ionized corona l environment. The thread is non-\nuniforminthetransversedirection. Resonantabsorptiona ndion-neutralcollisionsareassumedas\nthe damping mechanisms. We use the β=0 approximation, with βthe ratio of the gas pressure\nto the magnetic pressure, and the Thin Boundary approach to d escribe the effect of resonant ab-\nsorption in the Alfv´ en continuum using the connection form ulas for the perturbations across the\nresonantlayer(e.g.,Sakurai et al.1991;Goossenset al.19 92). Wedeterminethedominantdamp-\ningmechanismandobtainanalyticalexpressionsforthewav elength,thedampinglength,andtheir\nratio asfunctionsofthekinkmodefrequency.\nThis paper is organized as follows. Section 2 contains the de scription of the model configu-\nrationand thebasicEquations. First, theproblemisattack edanalyticallyinSection 3byadopting\nthethintubeapproximation. Lateron,thefulldispersionr elationisnumericallysolvedandapara-\nmetric study of the wavelength and damping length of thekink modeas functions of the period is\nperformed inSection 4. Finally,theconclusionsofthiswor k aregiveninSection 5.\n2. MODEL ANDDISPERSION RELATION\nFig. 1.—Sketch oftheprominencethread modeladoptedinthi swork.– 4 –\nThe equilibrium configuration is composed of a straight magn etic cylinder of radius aem-\nbedded in a homogeneous environment representing the coron al medium (see Figure 1). We use\ncylindrical coordinates, namely r,ϕ, andzfor the radial, azimuthal, and longitudinal coordinates,\nrespectively. The magnetic field is uniform and along the axi s of the cylinder, B=Bˆez, withB\nconstant everywhere. Hereafter, subscripts p and c denote p rominence and coronal quantities, re-\nspectively. The density within the prominence thread is den oted byρp, while the coronal density\nisρc. Bothρpandρcare homogeneous. A transverse transitional layer is includ ed in the radial\ndirection, where the density varies continuously between t he internal and external densities. We\ndo not specify the form of the density profile at this stage. Th e transverse inhomogeneous length\nscale in the transitional layer is given by the radio l/a, withlthe thickness of the layer. This ratio\nranges from l/a=0 if no transitional layer is present, to l/a=2 if the whole tube is radially\ninhomogeneous. Due to the presence of the transverse transi tional layer, the kink mode is reso-\nnantlycoupledtoAlfv´ encontinuummodes. Theresonancele adstothekinkmodedampingasthe\nenergyistransferredtoAlfv´ enmodesattheAlfv´ enresona nceposition. Thismechanismisknown\nas resonantabsorption.\nThe prominence plasma is partially ionized and we adopt the s ingle-fluid formalism (e.g.,\nBraginskii1965). Theionizationdegreeisarbitraryandis denotedherebythemeanatomicweight\nof the prominence material, ˜ µp. This parameter takes values in the range 0 .5≤˜µp≤1, where\n˜µp=0.5 corresponds to a fully ionized plasma and ˜ µp=1 to a fully neutral gas (see details\nin, e.g., Forteza etal. 2007; Soleret al. 2009b). The extern al coronal medium is assumed fully\nionized. The basic MHD Equations governing a partially ioni zed plasma can be found in, e.g.,\nFortezaet al. (2007); Pinto et al. (2008); Soler (2010). By a ssuming theβ=0 approximation and\nlinear perturbations from the equilibrium state, the basic MHD Equations discussed in this work\nare\nρ∂v\n∂t=1\nµ(∇×b)×B, (1)\n∂b\n∂t=∇×(v×B)+∇×/braceleftbiggηC\nB2[(∇×b)×B]×B/bracerightbigg\n, (2)\nwhereρisthelocaldensity,and v=/parenleftig\nvr,vϕ,vz/parenrightig\nandb=/parenleftig\nbr,bϕ,bz/parenrightig\narethevelocityandthemagnetic\nfieldperturbations,respectively. Notethat vz=0intheβ=0approximation. Inapartiallyionized\nplasma,theinductionequationcontainsatermaccountingf orCowling’sdiffusion,i.e.,thesecond\nterm on the right-hand side of Equation (2). Cowling’s di ffusion represents enhanced magnetic\ndiffusioncausedbyion-neutralcollisions,whichisseveralor dersofmagnitudemoree fficientthan\nclassical Ohm’s diffusionand is thedominante ffect in partially ionized plasmas (Cowling 1956).\nForthisreason,hereweneglectOhm’sdi ffusionandothertermsofminorimportancepresentinthe\ngeneralizedinductionequation(see,e.g.,Equation(14)o fFortezaet al.2007). Cowling’sdi ffusion\ncoefficient,ηC, depends on the ionization degree through ˜ µpas well as on the plasma physical\nconditions. The expression of ηCfor a hydrogen plasma can be found in, e.g., Pinto et al. (2008 )– 5 –\nand Soleret al. (2009b), whereas for a plasma composed of hyd rogen and helium see Soleret al.\n(2010b). Astheeffectofheliumisnegligibleforrealisticheliumabundances inprominences,here\nweconsiderapurehydrogenplasma. Thee ffectofCowling’sdi ffusionisneglectedintheexternal\nmediumbecause thecoronais assumedfullyionized.\nWe follow an approach based on normal modes. Since ϕandzare an ignorable coordinates,\nthe perturbations are expressed proportional to exp (imϕ+ikzz−iωt), whereωis the oscillatory\nfrequency, kzis the longitudinal wavenumber, and mis the azimuthal wavenumber ( m=1 for the\nkinkmode). Alternatively,theproblemcouldbeinvestigat edby meansoftime-dependentsimula-\ntionsof drivenwaves as in Pascoeet al. (2010). However, in t helinear regimethe di fferent values\nofmandkzare decoupled from each other, and a normalmodeanalysisis a simplerprocedure for\nlinear waves. If Cowling’s di ffusionis neglected, our configuration corresponds to that st udiedby\nTerradas et al. (2010a) for propagatingkinkwaves in corona l loops. Weextend theirinvestigation\nby incorporatingthee ffect ofCowling’sdi ffusionduetoion-neutralcollisions\nByusingtheThinBoundary(TB)approach(seedetailsin,e.g .,Goossenset al.2006;Goossens\n2008), theanalyticaldispersionrelationfor resonantlyd ampedkink wavespropagatingin atrans-\nversely nonuniform and partially ionized prominence threa d was obtained by Soleret al. (2009c,\nEquation (25)). If partial ionization is not considered and the effect of Cowling’s di ffusion is ab-\nsent, the dispersion relation of Soleret al. (2009c) reduce s to that investigated by Terradas et al.\n(2010a, Equation (28)) for kink waves in coronal loops. Sole ret al. (2009c) checked that the so-\nlutionsof theirdispersionrelation are in excellent agree ment with the solutionsobtained from the\nfull numerical integration of the MHD equations beyond the T B approximation. Therefore, the\ndispersion relation derived by Soleret al. (2009c) correct ly describes the kink mode behavior in\nourmodelandcomplicatednumericalintegrationsarenotne eded. Thedispersionrelationobtained\nby Soleret al.(2009c)inthecase ofa straightand homogeneo usmagneticfield is\nnc\nρc/parenleftig\nω2−k2zv2\nAc/parenrightigK′\nm(nca)\nKm(nca)−mp\nρp/parenleftig\nω2−k2zΓ2\nAp/parenrightigJ′\nm/parenleftig\nmpa/parenrightig\nJm/parenleftig\nmpa/parenrightig=−iπm2/r2\nA\nω2|∂rρ|rA, (3)\nwhereJmandKmaretheBesselfunctionandthemodifiedBesselfunctionofth efirstkindoforder\nm(Abramowitz& Stegun1972), respectively,and thequantiti esmpandncare defined as\nm2\np=/parenleftig\nω2−k2\nzΓ2\nAp/parenrightig\nΓ2\nAp,n2\nc=/parenleftig\nk2\nzv2\nAc−ω2/parenrightig\nv2\nAc, (4)\nwhereΓ2\nAp=v2\nAp−iωηCis the modified prominence Alfv´ en speed squared (Forteza et al. 2008),\nwithvAp=B√µρpandvAc=B√µρcthe prominence and coronal Alfv´ en speeds, respectively, a nd\nµ=4π×10−7N A−2the magnetic permeability. In addition, rAis the Alfv´ en resonance position,\nand|∂rρ|rAistheradialderivativeofthetransversedensityprofileat theAlfv´ enresonanceposition.– 6 –\nSoleret al.(2009c)studiedthetemporaldampingofkinkwav es,hencetheyassumedafixed,\nrealkzand solved Equation (3) to obtain the complex frequency. Her e, we investigate the spatial\ndampingandproceedtheotherwayround,i.e.,wefixareal ωandsolveEquation(3)toobtainthe\ncomplexwavenumber. Then, theperiod, P, wavelength,λ, and dampinglength, LD, are computed\nas follows,\nP=2π\nω, λ=2π\nkzR,LD=1\nkzI, (5)\nwithkzRandkzIthereal and imaginaryparts of kz, respectively.\n3. ANALYTICALAPPROXIMATIONS\nSomeanalyticalprogresscanbeperformedbeforesolvingEq uation(3)bymeansofnumerical\nmethods. To do so, we adopt the Thin Tube (TT) limit, i.e., λ/a≫1. A fist-order expansion of\nEquation(3)givesthedispersionrelationin boththeTT and TBapproximations,namely\nρp/parenleftig\nω2−k2\nzΓ2\nAp/parenrightig\n+ρc/parenleftig\nω2−k2\nzv2\nAc/parenrightig\n−iπm\nrAρpρc\n|∂rρ|rA/parenleftig\nω2−k2\nzΓ2\nAp/parenrightig/parenleftig\nω2−k2\nzv2\nAc/parenrightig\nω2=0.(6)\nIfbothCowling’sdi ffusionand resonant absorptionare omitted,thesolutiontoE quation(6)is\nk2\nz=ω2\nc2\nk, (7)\nwithc2\nk=2B2\nµ(ρp+ρc)the kink speed squared. Equation (7) corresponds to the idea l, undamped kink\nmode. The solutionsto Equation(6) considering thedi fferent dampingmechanisms are discussed\nnext.\n3.1. Damping by Cowling'sdi ffusion\nIn theabsence of transversetransitionallayer, i.e., l/a=0, resonant absorptiondoes not take\nplace and the damping is due to Cowling’sdi ffusion exclusively. In such a case, the third term on\nthe left-hand side of Equation (6) is absent. We write the wav enumber as kz=kzR+ikzIand use\nEquation(6)toobtaintheexactexpressionsfor k2\nzRandk2\nzI, namely\nk2\nzR=1\n2ω2c2\nk\nc4\nk+ω2¯η2\nC/radicaligg\n1+ω2¯η2\nC\nc4\nk+1, (8)\nk2\nzI=1\n2ω2c2\nk\nc4\nk+ω2¯η2\nC/radicaligg\n1+ω2¯η2\nC\nc4\nk−1, (9)– 7 –\nwith ¯ηC=ρp\nρp+ρcηC. By combining Equations (8) and (9), we compute the ratio of t he damping\nlengthtothewavelengthas\nLD\nλ=kzR\n2πkzI=1\n2πc2\nk+/radicalig\nc4\nk+ω2¯η2\nC\nω¯ηC. (10)\nEquations(8)–(10)areexactexpressionsthatcanbefurthe rsimplifieddependingonthevalue\noftheratioω2¯η2\nC\nc4\nk. Forω2¯η2\nC\nc4\nk≪1, i.e., in thelimitoflowfrequency ( ωsmall)and/orlargeionization\ndegree(¯ηCsmall),Equations(8)–(10)simplifyto\nk2\nzR≈ω2\nc2\nk/parenleftbigg\n1+ω2¯η2\nC\nc4\nk/parenrightbigg≈ω2\nc2\nk, (11)\nk2\nzI≈1\n4ω4¯η2\nC\nc6\nk/parenleftbigg\n1+ω2¯η2\nC\nc4\nk/parenrightbigg≈1\n4ω4¯η2\nC\nc6\nk, (12)\nLD\nλ≈1\nπc2\nk\nω¯ηC. (13)\nOn the contrary, ifω2¯η2\nC\nc4\nk≫1, i.e., high frequency and /or small ionization degree, the equivalent\nexpressionsare\nk2\nzR≈k2\nzI≈1\n2ω\nc2\nk¯ηC, (14)\nLD\nλ≈1\n2π. (15)\nThus, forω2¯η2\nC\nc4\nk≪1 the ratio of the damping length to the wavelength is inverse ly proportional to\nbothωand ¯ηC, andk2\nzRcoincides with the ideal value (Equation (7)). This case cor responds to a\nweaklydampedkinkmode. Ontheotherhand,forω2¯η2\nC\nc4\nk≫1,LD/λisindependentofωand ¯ηCand\nthe wave behavior is governed by di ffusion. By assuming typical values for the parameters in the\ncontext of oscillating prominence threads, e.g., P=3 min,B=5 G, andρp/ρc=200, we obtain\nω2¯η2\nC\nc4\nk≈6×10−17for ˜µp=0.5andω2¯η2\nC\nc4\nk≈1.6×10−4for ˜µp=0.99,meaning thatthecaseω2¯η2\nC\nc4\nk≪1\nismorerealisticin thecontextofoscillatingthreads even foran almostneutral plasma.\n3.2. Damping by resonant absorption andCowling's di ffusion\nNext,wetakethecase l/a/nequal0intoaccountandstudythecombinede ffectofresonantabsorp-\ntion and Cowling’s di ffusion. The third term on the left-hand side of Equation (6) is now present.– 8 –\nAsbefore, wewrite kz=kzR+ikzIand putthisexpressioninEquation(6). Sinceitisvery di fficult\ntogiveexactexpressionsfor kzRandkzIinthegeneralcase,wefocuson LD/λandrestrictourselves\ntoω2¯η2\nC\nc4\nk≪1. Following the procedure of Terradas et al. (2010a), we ass ume weak damping, i.e.,\nkzI≪kzR, and neglect terms with k2\nzI. The following process is long but straightforward, and we\nrefer the reader to Terradas et al. (2010a) for details. Fina lly, we arrive at the expression for the\nratio ofthedampinglengthtothewavelengthas\nLD\nλ≈/parenleftigg\nπω¯ηC\nc2\nk+m\nFl\naρp−ρc\nρp+ρc/parenrightigg−1\n, (16)\nwhere the first term within the parentheses accounts for Cowl ing’s diffusion and the second term\nfor resonant absorption. The factor Fin the second term takes di fferent values depending on\nthe density profile within the inhomogeneous layer. For exam ple,F=4/π2for a linear profile\n(Goossenset al. 2002), while F=2/πfor a sinusoidal profile with rA≈a(Ruderman &Roberts\n2002). If the term related to resonant absorption is absent, Equation (16) reverts to Equation (13).\nOn the other hand, if the term related to Cowling’s di ffusion is dropped, Equation (16) coincides\nwithEquation(13)ofTerradas et al.(2010a).\nTherelativeimportanceofthetwotermsinEquation(16)can beassessedbyperformingtheir\nratio as\nǫ≡(LD/λ)RA\n(LD/λ)C≈πFa\nlω¯ηC\nc2\nkmρp+ρc\nρp−ρc=πFa\nlωηC\nc2\nkmρp\nρp−ρc, (17)\nwhere(LD/λ)RAand(LD/λ)Cstand for the damping ratio by resonant absorption and Cowli ng’s\ndiffusion, respectively. By considering as before P=3 min,B=5 G, andρp/ρc=200, and\nadopting a linear profile with l/a=0.2, we obtainǫ≈8×10−8for ˜µp=0.5 andǫ≈0.12\nfor ˜µp=0.99, meaning that in the TT limit resonant absorption dominat es the kink mode spatial\ndamping for typical parameters of thread oscillations. Thi s result is equivalentto that obtained by\nSoleret al. (2009c) inthecaseoftemporaldamping.\n4. NUMERICALRESULTS\nNow, we solve the dispersion relation (Equation (3)) by mean s of standard numerical proce-\ndures. In the following figures, both the wavelength, λ, and the damping length, LD, are plotted\nin dimensionless form with respect to the thread mean radius ,a. The period, P, is computed in\nunits of the internal Alfv´ en travel time, τAp=a/vAp. Unless otherwise stated, the results have\nbeen computed with ρp=5×10−11kg m−3,ρp/ρc=200, and B=5 G. With these parameters,\nvAp≈63 kms−1and, fora=100km,τAp≈1.59 s.\nFigure 2 displaysλ/a,LD/a, andLD/λversusP/τApfor the case l/a=0, i.e., the damping– 9 –\nis due to Cowling’s di ffusion exclusively. We compute the results for di fferent values of ˜µp. The\nshaded areas in Figure 2 and in the other figures represent the range of observed periods of trans-\nversethread oscillations,i.e., 1min–10min,correspondi ngto40/lessorsimilarP/τAp/lessorsimilar400,approximately.\nRegardingthewavelength,weseethatthee ffectofCowling’sdi ffusionisonlyrelevantforperiods\nmuchshortedthanthoseobserved. ThisisinagreementwithE quations(11)and(14). Ontheother\nhand, an almost neutral plasma, i.e., ˜ µp→1, has to be considered to obtain an e fficient damping\nand to achieve small values of LD/λwithin the relevant range of periods. Although we do not\nknowtheexactionizationdegreein prominencethreads,suc hvery largevaluesof ˜ µpare probably\nunrealistic (see, e.g., Gouttebroze& Labrosse 2009). The a nalytical expressions for λ,LD, and\nLD/λintheTTcasegivenbyEquations(8),(9),and(10),respecti vely,areingoodagreementwith\nthefullresultsinthewholerange ofperiods(seesymbolsin Figure 2).\nNext, we study the general case l/a/nequal0. We adopt a sinusoidal density profile within the\ninhomogeneoustransitionallayer(Ruderman &Roberts2002 ). AstheAlfv´ enresonanceposition,\nrA,isneededforthecomputationsoftheresonantdamping,wef ollowatwo-stepprocedure. First,\nwe solve the dispersion relation for a fixed ωin the case l/a=0 and determine kzR. Then, we\nassume that the value of kzRis approximately the same in the case l/a/nequal0, meaning that the\nresonant condition is ω=kzRvA(rA). In the case of a sinusoidal profile, the expression of the\nresonantpositioncan beanalyticallyobtainedfrom theres onantconditionas\nrA=a+l\nπarcsinρp+ρc\nρp−ρc−2v2\nApk2\nzR\nω2ρp\nρp−ρc. (18)\nFinally, we compute |∂rρ|rAusing the previously determined rAby means of Equation (18) and\nsolvethedispersionrelationwith theseparameters to obta intheactual kzRandkzI. Figure3 shows\nthe results of these computations for di fferent values of l/awhen the ionization degree has been\nfixed to ˜µp=0.8, whereas Figure 4 displaystheequivalentcomputationsfo r different valuesof ˜µp\nwhen thetransverseinhomogeneitylengthscale hasbeen fixe d tol/a=0.2. Since thewavelength\nis not affected by the value of l/aand has the same behavior as in Figure 2(a), both Figures 3 and\n4 focus on LD/aandLD/λ. We obtain two di fferent behaviors of the solutions depending on the\nperiod. For small P/τAp, the damping length is independent of l/aand is governed by the value\nof ˜µp. On the contrary, for large P/τApthe damping length depends on l/abut is independent of\n˜µp. This result indicates that resonant absorptiondominates thedamping for large P/τAp, whereas\nCowling’sdiffusionismorerelevantforsmall P/τAp. Theapproximatetransitionalperiod,namely\nPtr,inwhichthedampinglengthbyCowling’sdi ffusionbecomessmallerthanthatduetoresonant\nabsorption can be estimated by setting ǫ≈1 in Equation (17) and writing Ptr=2π/ω. Then, one\nobtains\nPtr≈2π2Fa\nl¯ηC\nc2\nkmρp+ρc\nρp−ρc=2π2Fa\nlηC\nc2\nkmρp\nρp−ρc. (19)\nThistransitionalperiodisingoodagreementwiththenumer icalresults(seetheverticaldottedline– 10 –\nFig. 2.—Resultsforthekinkmodespatialdampinginthecase l/a=0: (a)λ/a,(b)LD/a, and(c)\nLD/λversusP/τApfor ˜µp=0.5,0.6,0.8,and0.95. Symbolsinpanels(a),(b),and(c)co rrespondto\ntheanalyticalsolutionintheTTapproximationgivenbyEqu ations(8),(9), and(10),respectively,\nwhile the horizontal dotted line in panel (c) corresponds to the limit of LD/λfor high frequencies\n(Equation(15)). Theshadedarea denotestherangeofobserv edperiodsofthread oscillations.– 11 –\nFig. 3.— Results for the kink mode spatial damping in the case l/a/nequal0: (a)LD/aand (b)LD/λ\nversusP/τApforl/a=0.05, 0.1, 0.2, and 0.4, with ˜ µp=0.8. Symbols in panel (b) correspond to\nthe analytical solution in the TT approximation given by Equ ation (16), while the vertical dotted\nline is the approximate transitional period given by Equati on (19) for l/a=0.1. The shaded area\ndenotestherangeofobservedperiodsofthread oscillation s.\nFig. 4.— Results for the kink mode spatial damping in the case l/a/nequal0: (a)LD/aand (b)LD/λ\nversusP/τApfor ˜µp=0.5, 0.6, 0.8, and 0.95, with l/a=0.2. Symbols in panel (b) correspond to\nthe analytical solution in the TT approximation given by Equ ation (16). The shaded area denotes\ntherangeofobservedperiodsofthread oscillations.– 12 –\nin Figure 3(b)). In addition, we see that Ptris much smaller than the typically observed periods,\nindicatingthat resonantabsorptionisthedominantdampin gmechanismin therelevantrange.\nFinally, we check that the analytical approximationof LD/λgivenby Equation (16) provides\nanaccuratedescriptionofthekinkmodespatialdampingint herelevantrangeofperiods(compare\nthesymbolsandthesolidlinesinFigures 3(b)and 4(b)).\n5. DISCUSSION AND CONCLUSION\nIn this paper, we have studied the spatial damping of kink wav es in prominence threads.\nResonant absorption and Cowling’s di ffusion are the damping mechanisms taken into account.\nBoth analytical expressions and numerical results indicat e that, in the range of typically observed\nperiods of prominence thread oscillations, the e ffect of Cowling’s di ffusion (and so the ionization\ndegree) is negligible. On the other hand, resonant absorpti on provides an efficient damping in\nagreement with the study of Terradas et al. (2010a) in the con text of coronal loop oscillations.\nThese conclusions are equivalent to those obtained by Soler etal. (2009c) in the case of temporal\ndamping.\nWe point out that small values of LD/λare obtained by resonant absorption in the observa-\ntionally relevant range of periods, which is consistent wit h the reported strong damping of the\noscillations. The analytical estimation of LD/λgiven by Equation (16) is very accurate in the ob-\nservationallyrelevantrangeofperiods,andthecontribut ionofCowling’sdi ffusioncanbedropped\nfromEquation(16)becausetheplasmaionizationdegreetur nsouttobeirrelevantforthedamping.\nTherefore, forkinkmodes( m=1)theradio LD/λsimplifiesto\nLD\nλ≈Fa\nlρp+ρc\nρp−ρc, (20)\nwhich coincides with the expression providedby Terradas et al. (2010a). Asρp+ρc\nρp−ρc→1 for typical\nprominenceandcoronaldensities,thisfactorcanbedroppe dfromEquation(20),meaningthatthe\nratioLD/λdepends almost exclusively on the transverse inhomogeneit y length scale, l/a, and the\nform ofthedensityprofilethrough Fas\nLD\nλ≈Fa\nl. (21)\nInthecaseofcoronallooposcillationsstudiedbyTerradas et al.(2010a),thefactorρp+ρc\nρp−ρccannotbe\ndroppedfromtheirexpressions,meaningthatincoronalloo pstheratio LD/λsignificantlydepends\non the density contrast. Therefore, information about the p arameters l/aandFin prominence\nthreads could be determined by using Equation (21) along wit h accurate measurements of the– 13 –\ndamping length and the wavelength provided from the observa tions. However, since the precise\nform of the transverse density profile in prominence threads is unknown, we have to assume an\nad hoc profile, i.e., a value of F, to infer the transverse inhomogeneity length scale from th e\nobservations,whichcan introducesomeuncertaintiesinth eestimationof l/a.\nFor example, let us assume that the ratio LD/λhas been determined from an observation of\ndampedkinkwavesinaprominencethreadandwewanttocomput ethetransverseinhomogeneity\nlength scale of the thread. For simplicity, we consider that the transverse density profile in the\ninhomogeneous layer is either linear or sinusoidal. Denoti ng as(l/a)linthe value of l/acomputed\nassuming a linear profile, and (l/a)sinthe corresponding value for a sinusoidalprofile, the relati on\nbetween bothofthemis(l/a)lin\n(l/a)sin=π\n2≈1.57, (22)\npointingoutthattherelativeuncertaintyof l/aislargerthan50%,andtheinaccuracycouldbeeven\nlargerifotherprofilesareconsidered. Thisfactshouldbet akenintoaccountinfutureseismological\ndeterminationsofthisparameter.\nThe present investigation is a first step for the study of the s patial damping of kink waves\nin prominence fine structures. Here, we have adopted a simple model of a prominence thread.\nSome effects that might influence the kink mode propagation and dampi ng are not included in\nthe present paper. Among them, plasma inhomogeneity along t he thread may affect somehow\nthe amplitude of a propagating kink mode, whereas the presen ce of flows affects the damping by\nresonantabsorption(seeTerradas et al.2010b). Theinfluen ceoftheseandothere ffectswillbethe\nsubjectofforthcomingworks.\nTheauthorsacknowledgethefinancialsupportreceivedfrom theSpanishMICINNandFEDER\nfunds (AYA2006-07637). The authors also acknowledge discu ssion within ISSI Team on Solar\nProminence Formation and Equilibrium: New data, new models . 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Phys.,132,63\nThispreprintwaspreparedwiththeAAS L ATEXmacrosv5.2."    },    {        "title": "1708.08225v2.Magnetic_reheating.pdf",        "content": "arXiv:1708.08225v2  [astro-ph.CO]  9 Dec 2017MNRAS 000, 000–000 (0000) Preprint 4 June 2022 Compiled using MNRAS L ATEX style file v3.0\nMagnetic reheating\nShohei Saga,1⋆Hiroyuki Tashiro,2and Shuichiro Yokoyama3,4\n1Yukawa Institute for Theoretical Physics, Kyoto University , Kyoto 606-8502, Japan\n2Department of Physics and Astrophysics, Nagoya University , Nagoya, 464-8602, Japan\n3Department of Physics, Rikkyo University, Tokyo 171-8501, Japan\n4Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chib a 277-8583, Japan\n4 June 2022\nABSTRACT\nWe provide a new bound on the amplitude of primordial magneti c fields (PMFs) by\nusing a novel mechanism, magnetic reheating . The damping of the MHD fluid motions\nin a primordial plasma brings the dissipation of the PMFs. In the early Universe with\nz/greaterorsimilar2×106, cosmic microwave background (CMB) photons are quickly the rmalized\nwith the dissipated energy and shift to a different Planck dis tribution with a new\ntemperature. In other words, the PMF dissipation changes th e baryon-photon number\nratio and we name such a process magnetic reheating . From the current baryon-photon\nnumber ratio obtained from the BBN and CMB observations, we p ut a strongest\nconstraint on the PMFs on small scales which CMB observation s can not access,\nB0/lessorsimilar1.0µGat the scales 104hMpc−1< k <108hMpc−1. Moreover, when the PMF\nspectrum is given in a blue power-law type, the magnetic rehe ating puts a quite strong\nconstraint, for example, B0/lessorsimilar10−17nG,10−23nG, and10−29nGat 1 comoving Mpc\nfornB= 1.0,2.0, and3.0, respectively. This constraint would give an impact on\ngeneration mechanisms of PMFs in the early Universe.\nKey words: cosmology: theory — cosmology: cosmic background radiatio n — cos-\nmology: early Universe\n1 INTRODUCTION\nMagnetic fields are ubiquitous in the universe and they have\nbeen observed in various astrophysical objects from planet s\nand stars to galaxies and galaxy clusters ( Ryu et al. 2012 ;\nWidrow et al. 2012 ). Recently there are some reports to\nsuggest the existence of magnetic fields in the intergalac-\ntic medium ( Neronov & Vovk 2010 ;Tavecchio et al. 2010 ;\nEssey et al. 2011 ;Tashiro et al. 2014 ;Chen et al. 2015 ).\nPrimordial magnetic fields (PMFs) are very attractive as\nthe origin of observed magnetic fields on cosmological\nscales. There are many proposals of the PMF generation\nmechanism in the early universe ( Durrer & Neronov 2013 ;\nSubramanian 2016 ). Therefore, observational constraints on\nPMFs can provide insights into the physics in the early uni-\nverse.\nPMFs can leave their signatures on many cosmologi-\ncal phenomena. As examples, the requirement of success-\nful Big Bang nucleosynthesis (BBN) gives a constraint on\nthe existence of PMFs at that time. The recent detailed\nstudy about the BBN with PMFs provides the upper limit\nB0<1.5µG where B0represents the present strength\nof PMFs ( Kawasaki & Kusakabe 2012 ). The measurements\n⋆shohei.saga@yukawa.kyoto-u.ac.jpof cosmic microwave background (CMB) anisotropies also\nyield constraints on PMFs, because PMFs can create\nthe additional features in the temperature and polar-\nization anisotropies of the CMB ( Subramanian & Barrow\n1998a ;Shaw & Lewis 2010 ). Recent Planck data gave\na limit on PMFs as B0/lessorsimilarO(1)nG around co-\nmoving 1Mpc, depending on the scale dependence of\nPMFs ( Planck Collaboration et al. 2016 ).\nMoreover, dissipation of PMFs in the early universe\nhas also drawn attention to obtain a constraint on PMFs\nat smaller scales, compared with the scales which could be\nconstrained from the CMB temperature and polarization\nanisotropies. Before the epoch of recombination, PMFs in-\nduce the fluid motions in a photon-baryon plasma through\nthe Lorentz force. From the viewpoint of the energy trans-\nfer, this process can be considered as a conversion of the\nmagnetic fields’ energy to the fluid kinetic energy. Since\nthere arises a viscosity in the plasma due to the finite mean\nfree path of photons, the induced motions on small scales\nare damped ( Jedamzik et al. 1998 ;Subramanian & Barrow\n1998b ). As a result, the magnetic fields’ energy on small\nscales would be dissipated into the photon-baryon plasma.\nIf the dissipation happens after the redshift z∼2×106(for\nrecent reviews see Chluba & Sunyaev 2012 ;Tashiro 2014 ),\nthese dissipated energy can be observed as CMB distortions,\nc/circlecopyrt0000 The AuthorsL2S. Saga et al.\nthat is, the spectral deviation of the CMB from the Planck\ndistribution ( Jedamzik et al. 2000 ;Miyamoto et al. 2014 ;\nKunze & Komatsu 2014 ;Ganc & Sloth 2014 ). Depending on\nthe redshift of the energy injections, the generated CMB\ndistortions are typically characterized by two parameters ,\nthe chemical potential µand the yCompton parameter.\nTherefore, the measurement of CMB distortions allows us\nto understand the thermal history of the universe from the\nredshift z∼2×106to the epoch of recombination. The\ncurrent constraint on CMB distortions has been obtained\nby COBE/FIRAS ( Fixsen et al. 1996 ). According to the\nCOBE/FIRAS constraint, the upper limit on the PMFs is\nset toB0<30 nG between comoving 400pc and0.5Mpc\n(Jedamzik et al. 2000 ).\nIn this Letter, we focus on the dissipation of the mag-\nnetic fields’ energy before the CMB distortion era ( z/greaterorsimilarzµ=\n2.0×106). Before the CMB distortion era, any kind of the\nenergy injections does not distort the energy spectrum of\nCMB photons. This is because the interaction processes for\nthe CMB thermalization, i.e.,the Compton scattering, dou-\nble Compton scattering, and bremsstrahlung have smaller\ntime scales than the cosmological time scale. In particular ,\nthe double Compton process keeps the distribution of CMB\nphotons as the Planck distribution by adjusting the number\nof CMB photons, that is, the CMB temperature. As a result,\nareheating of the CMB photons may occur.\nIf we measure the overall history of the absolute CMB\ntemperature or the number density of photons before the\nCMB distortion era, we can directly put a strong con-\nstraint on the amount of the injected energy. However, the\ncosmological observations, e.g., the measurement of CMB\nanisotropies, tell us the information only around the recom -\nbination epoch1. In this Letter, we therefore use the alter-\nnative observable, that is, a baryon-photon ratio η=nb/nγ.\nBefore the CMB distortion era, an energy injection into\nCMB photons can increase the CMB photon number due\nto the double Compton process and bremsstrahlung, while\nthe number of baryons does not change. As a result, the\nbaryon-photon number ratio decreases after the energy in-\njection occurs. Since the light element production in the\nBBN is sensitive to ηat the BBN epoch, the measurement\nof the light element amount can observationally determine\nthe value of η,ηBBN, at the BBN epoch with some errors.\nIndependently, CMB observations can provide the value of\nη,ηCMB, around the epoch of recombination with some er-\nrors. If the energy injection to the CMB photons occurred\nafter the BBN era and before the CMB distortion era, the\ndifference of the baryon-photon number ratios at between\nthe BBN era and CMB era can be reread in terms of the\ninjected energy density of photons ∆ργbetween these eras\nas\nηCMB\nηBBN= 1−3\n4∆ργ\nργ, (1)\nwhere we use the relations nγ∝T3\nγandργ∝T4\nγ.\nCombining these relations, Nakama et al. (2014) pro-\nvides a constraint on the density fraction of the injected\n1As we have mentioned, CMB distortions may tell us the infor-\nmation deeply before the recombination epoch.energy between these epochs as\n∆ργ\nργ<7.71×10−2, (2)\nwhere they adopted ηCMB,obs= (6.11−0.08)×10−10\n(Ade et al. 2014 ) andηBBN,obs= (6.19+0.21)×10−10\n(Nollett & Steigman 2014 ) as observation values for CMB\nand BBN, respectively. Here, we stress that the bound\nEq. (2) is intrinsically coming from the difference between\nthe baryon-photon number ratio at the BBN and CMB era,\nnot from the direct measurement of the CMB tempera-\nture or the CMB photon number. In order to obtain the\nabove constraint, we take into account for the negative and\npositive errors in the CMB and BBN bound, respectively,\nin order to obtain conservative limit. Note that, Eq. ( 2)\ncan be reread in terms of the CMB temperature by us-\ning the relation: ∆ργ/ργ= 4∆Tγ/Tγ+O((∆Tγ/Tγ)2), as\n∆Tγ/Tγ<1.93×10−2.\nThe dissipation of the PMFs can be considered to be\none of interesting heating sources, and we name the heat-\ning mechanism by the dissipation of PMFs magnetic re-\nheating . Before the epoch of recombination, PMFs induce\nthe magnetohydrodynamics (MHD) modes in the photon-\nbaryon plasma and these MHD modes are damped by the\nviscosity due to the diffusion process of CMB photons. In\nthis process, the energy of PMFs dissipates into the CMB\nphotons and subsequently, the baryon-photon number ratio\nchanges. Therefore, we can put a constraint on PMFs from\nEq. (2). In next section, we give the formulation for the\ninjected energy from the dissipation of PMFs and show a\nnew constraint on PMFs obtained from magnetic reheating\nprocess. The final section is devoted to the conclusion and\nsummary of our results.\n2 REHEATING BY DECAYING MAGNETIC\nFIELDS\nLet us consider the spatially-averaged injected energy int o\nthe CMB photons due to the decaying magnetic fields from\nthe BBN era to the CMB distortion era. The total injected\nenergy is represented as\n∆ργ\nργ=/integraldisplayzµ\nzidz/bracketleftbigg\n−1\nργ(z)(1+z)4\n8πd\ndz/angbracketleftbig\n|b(z,x)|2/angbracketrightbig/bracketrightbigg\n, (3)\nwhereργ(z)is the background energy density of photons\nwhich scales as ∝(1+z)4and the brackets denote ensem-\nble average. We also define b(z,x) = (1 + z)−2B(z,x)as\nthe magnetic fields without the adiabatic decay due to the\nexpansion of the universe. Here, we set zµ= 2.0×106and\nzi= 1.0×109, which eras correspond to the inefficient of\nthe double Compton scattering and the neutrino decoupling\nera, respectively ( Hu & Silk 1993 ).\nThe evolution of magnetic fields due to the MHD damp-\ning can be simply expressed in the Fourier space as\nbi(z,k) =˜bi(k)e−(k/kD(z))2, (4)\nwhere˜bi(k)denotes the initial PMFs and kD(z)represents a\nwave number of the magnetic fields which is damped at the\nredshiftz, caused by the photon viscosity, similar to the Silk\ndamping of the standard primordial plasma. The damping\nwave number can be evaluated by the mode analysis based\nMNRAS 000, 000–000 (0000)Magnetic reheating L3\non the magneto-hydrodynamics (MHD). Depending on the\nMHD modes and the scales, the damping wave number\nis different ( Jedamzik et al. 1998 ;Subramanian & Barrow\n1998b ). Here we consider the case for the fast-magnetosonic\nmode, where Alfvén and slow-magnetosonic modes are\ndamped in the photon diffusion limit in which the mode\nwavelength k−1is much larger than the photon mean free\npath. In such a case, the damping scale is similar to the Silk\ndamping scales and we adopt ( Jedamzik et al. 2000 )\nkD(z) = 7.44×10−6(1+z)3/2Mpc−1, (5)\nin the radiation dominated epoch. We will discuss the other\ncases later.\nWe assume that the initial PMFs are statistically ho-\nmogeneous and isotropic Gaussian random fields. The power\nspectrum of such fields can be written as\n∝angbracketleft˜bi(k)˜b∗\nj(k′)∝angbracketright= (2π)3δ3\nD(k−k′)δij−ˆkiˆkj\n22π2\nk3PB(k),(6)\nand then the bracket of the right-hand side in Eq. ( 3) can\nbe expressed as\n/angbracketleftbig\n|b(z,x)|2/angbracketrightbig\n=/integraldisplay\ndlnkPB(k)e−2/parenleftBig\nk\nkD(z)/parenrightBig2\n. (7)\nTherefore, Eq. ( 2) allows us to obtain the constraint on the\npower spectrum of magnetic fields. In our analysis, we em-\nploy two types of the power spectrum, one is a delta-function\ntype in logarithmic scale given by\nPB(lnk) =B2\ndeltaδD(ln(k/kp)), (8)\nwhereBdeltacorresponds to the amplitude at the peak wave\nnumberkp, and another is a power-law type given by\nPB(k) =B2/parenleftbiggk\nkn/parenrightbiggnB+3\n, (9)\nwhereBis the amplitude at the normalization wave number\nkn. In this Letter, we set the normalization scale as kn=\n1 Mpc−1.\nFirst we focus on the delta function type for the power\nspectrum. By substituting Eq. ( 8) into Eq. ( 3) with Eq. ( 7),\nwe obtain the injected energy fraction to the CMB energy\nby the decaying PMFs as\n∆ργ\nργ=B2\ndelta\n8πργ,0˜C(kp), (10)\nwhere˜C(kp)is the function of kpand which the explicit\nform is given as\n˜C(k)≡exp/bracketleftbigg\n−2k2\nk2\nD(zi)/bracketrightbigg\n−exp/bracketleftbigg\n−2k2\nk2\nD(zµ)/bracketrightbigg\n. (11)\nWhenkpis much larger than kD(zi)(> kD(zµ)), the energy\ninjection due to the decay of the PMFs becomes efficient\nbefore the BBN era. On the other hand, when kpis much\nsmaller than kD(zµ), the energy injection from the PMFs\noccurs during the CMB distortion era. We focus on the case\nwhere the energy injection from the decaying of the PMFs is\nefficient after the BBN era and before the CMB distortion\nera, that is, the case where kpis set between the two decay-\ning scales kD(zi)andkD(zµ). For such a case, we can approx-\nimately take ˜C(kp)→1(seeMiyamoto et al. 2014 ). Com-\nbining Eq. ( 2) and Eq. ( 10) yields a constraint on PMFs in\nthekp-Bdeltaplane. We present our new result obtained from101102103104\n102103104105106107108109Bdelta [nG]\nkp [Mpc-1]Magnetic reheating\nBBN\nCMB distortion\nFigure 1. The upper bound on the amplitude of the delta-\nfunction type for the power spectrum Bdelta as a func-\ntion of kpfrom the magnetic reheating. The limits from\nthe BBN ( Kawasaki & Kusakabe 2012 ) and CMB distortions\n(Jedamzik et al. 2000 ) in blue and magenta lines, respectively,\nare shown in the same plot.\nmagnetic reheating as a red line in Fig. 1. For comparison, we\nplot limits from the BBN and CMB distortions in magenta\nand blue lines, respectively. We can see that our magnetic\nreheating constraint gives a tight limit on the PMFs on small\nsales from kp= 104Mpc−1tokp= 108Mpc−1.\nNext we consider the power-law type of the power spec-\ntrum defined in Eq. ( 9). By using Eq. ( 3) with Eq. ( 7), we\nobtain\n∆ργ\nργ=B2\n8πργ,0Γ/parenleftbignB+3\n2/parenrightbig\n2(nB+5)/2/bracketleftBigg/parenleftbiggkD(zi)\nkn/parenrightbiggnB+3\n−/parenleftbiggkD(zµ)\nkn/parenrightbiggnB+3/bracketrightBigg\n.\n(12)\nHere, we focus on the magnetic reheating due to the fast-\nmagnetosonic mode given in Eq. ( 5). In Fig. 2, we plot a\nlimit for the amplitude of the power-law type as a function\nof the spectral tilt nBin a red line. We find that blue-tilted\nspectrum is strongly constrained compared with the scale-\ninvariant spectrum ( nB=−3.0). It is shown that the causal\nmechanism of PMFs generation predicts only a blue spec-\ntrum with nB/greaterorequalslant2(Durrer & Caprini 2003 ). Therefore, the\nconstraint on the blue-tilted spectrum is suggestive for su ch\na causal mechanism.\nSo far we have considered the case that the Alfvén\nand slow-magnetosonic modes are damped in the photon\ndiffusion limit. However, depending on the magnetic field\nstrength, the energy damping of these modes are insuffi-\ncient in the photon diffusion limit because the magnetic\nfield cannot accelerate the fluid efficiently. When it hap-\npens, the magnetic field can survive even below smaller scale\nthank−1\nD(z)given in Eq. ( 5). These modes are damped\nlater when the scale of the modes are smaller than the\nphoton free-streaming scale. This damping scale is roughly\ngiven by kA\nD(z)∼kD(z)/VA(z)in the radiation domi-\nnated epoch. Here VA(z)is the Alfvén velocity, VA(z) =\nBλ(z)//radicalbig\n16πργ,0/3, where we obtain the magnetic field\nBλ(z)by integrating the power spectrum over kwith a\nGaussian window function with a scale λ(z)corresponding\nMNRAS 000, 000–000 (0000)L4S. Saga et al.\n10-3010-2510-2010-1510-1010-5100105\n-3.0-2.0-1.00.01.02.03.0B [nG]\nnBFast mode\nAlfven mode\nPlanck constraint\nFigure 2. The upper bound on the amplitude of the power-\nlaw type for the power spectrum Bas the function of nBfrom\nthe magnetic reheating of the fast-magnetosonic mode (red)\nand Alfvén-magnetosonic mode (blue). The constraint from t he\nPlanck ( Planck Collaboration et al. 2016 ) is also shown. Note\nthatnB=−3.0corresponds to the scale-invariant power spec-\ntrum.\ntokA\nD(z)(Mack et al. 2002 ). Therefore, kA\nD(z)can be repre-\nsented as\nkA\nD(z)\nkn∼2π/bracketleftBigg\n3.5×105\nΓ/parenleftbignB+5\n2/parenrightbig/parenleftbiggB\n1 nG/parenrightbigg−2/parenleftbiggkD(z)\nkn/parenrightbigg2/bracketrightBigg1/(nB+5)\n.\n(13)\nReplacing kD(z)tokA\nD(z)in Eq. ( 12), we can evaluate the in-\njected energy fraction for the Alfvén and slow-magnetosoni c\nmodes in the photon free-streaming limit and plot the con-\nstraint on primordial magnetic fields as a blue line in Fig. 2.\nWe find that the magnetic reheating due to the Alfvén and\nslow-magnetosonic modes put a stronger constraint on the\namplitude of PMFs than the fast-magnetosonic mode. The\nobtained constraint can be roughly fitted as log(B/nG)/lessorsimilar\n−11−6nB(fornB/greaterorsimilar0). This constraint is tighter than\nother CMB constraints.\n3 CONCLUSION\nBefore the CMB distortion era, the additional energy injec-\ntion heats the CMB photons and due to the double Comp-\nton scattering, the number of the CMB photons increases.\nTherefore, the baryon-photon number ratio decreases by the\nreheating. By comparing the values of the baryon-photon\nnumber ratio between two epochs, we can determine the\nallowed energy injection or allowed changes of the CMB\ntemperature between them. In this Letter, we assume the\nreheating source as the decaying PMFs due to the photon\nviscosity, named the magnetic reheating. Since the baryon-\nphoton number ratio is well constrained by the current cos-\nmological observations such as the BBN and CMB, the mag-\nnetic reheating provides us a significant constraint on smal l-\nscale PMFs which we cannot access through the observa-\ntions of CMB anisotropies and distortions. From the state-\nof-the-art observations, we put the new constraint on theamplitude of small-scale PMFs as B0/lessorsimilar1µGin the range\n104hMpc−1< k <108hMpc−1.\nWe also apply the magnetic reheating constraint to\nPMFs with a power-law spectrum. Since the dissipation\nscale depends on the MHD modes, we consider two damping\nscales of not only the fast-magnetosonic mode but also the\nslow-magnetosonic and Alfvén modes. We find that for blue-\ntiled spectrum the slow-magnetosonic and Alfvén modes\ncan set stronger constraint, compared with the constraint\nfrom the fast-magnetosonic mode. In general, because the\ndamping scale of the slow-magnetosonic mode and Alfvén\nmode is smaller than that of the fast-magnetosonic mode\nthe slow-magnetosonic and Alfvén modes are sensitive to\nthe PMFs on small scales and can set a stronger constraint\nfor blue-tiled spectrum rather than the fast-magnetosonic\nmode. The obtained constraint can be roughly fitted as\nlog(B/nG)/lessorsimilar−11−6nB(fornB/greaterorsimilar0) where Bis the PMF\namplitude normalized at 1 comoving scale. This constraint\nis tighter than other CMB constraints in the case of the\nblue-tilted spectrum.\nAlthough we consider two damping scales separately in\nthis Letter, in reality, the energy dissipation of magnetic\nfields happens through the combination of these two damp-\ning scale channels. However it is not so trivial how much\npercentage of the magnetic field energy is dissipated throug h\nthe fast-, slow-magnetosonic or Alfvén modes. When the dis-\nsipated energy is in equipartition between these modes, our\nconstraint might be relaxed by a few factor. To investigate\nthe details, we will address this issues through the analysi s\nin higher order cosmological perturbations.\nAmong many mechanisms to generate PMFs in the\nearly universe, the causal mechanism can generate only blue\npower spectrum ( Durrer & Caprini 2003 ). Therefore, strong\nconstraints for the blue-tiled PMFs obtained in this Letter\nshould be a powerful tool to investigate a successful causal\nmechanism to generate PMFs in the early universe.\nACKNOWLEDGEMENTS\nThis work is supported in part by a Grant-in-Aid for JSPS\nResearch Fellow Number 17J10553 (S.S.), JSPS KAKENHI\nGrant Number 15K17646 (H.T.), 17H01110 (H.T.) and\n15K17659 (S.Y.), and MEXT KAKENHI Grant Number\n16H01103 (S.Y.).\nREFERENCES\nAde P. A. R., et al., 2014, Astron. Astrophys. , 571, A16\nChen W., Buckley J. H., Ferrer F., 2015, Phys. Rev. Lett. , 115,\n211103\nChluba J., Sunyaev R. A., 2012, MNRAS ,419, 1294\nDurrer R., Caprini C., 2003, J. Cosmology Astropart. 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Rev. ,166, 37\nMNRAS 000, 000–000 (0000)"    },    {        "title": "1505.07704v1.Damping_factors_for_head_tail_modes_at_strong_space_charge.pdf",        "content": "DAMPING FACTORS FOR HEAD-TAIL MODES  AT STRONG SPACE CHARGE Alexey Burov, Fermilab, Batavia, IL 60510 Abstract This paper suggests how feedback and Landau damping can be taken into account for transverse oscillations of bunched beam at strong space charge.  MAIN EQUATION Space charge is known to be able to change dramatically collective modes [1-3].  For transverse oscillations of bunched beams, a parameter of the space charge strength is a ratio of the maximal space charge tune shift to the synchrotron tune. When this parameter is large, the transverse oscillations are described by a one-dimensional integro-differential equation derived in Ref. [2] and reproduced here for the sake of convenience:         νy+1Qeffddτu2dydτ⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟⎟=κNˆWy+Dy()ˆWy=W(τ−s)expiζτ−s()⎡⎣⎢⎤⎦⎥−∞∞∫ρ(s)y(s)dsDy=y(τ)D(τ−s)−∞∞∫ρ(s)dsdydττ→±∞=0  (1) Here     y=y(τ) and  ν are the eigenfunction and the eigenvalue to be found for the bunch transverse oscillations,  τ and  s are longitudinal positions within the bunch, W and D are the dipole and quadrupole (or the driving and detuning) wakes, ρ is the normalized line density      ρds=1∫ , (2) N is the number of particles per bunch,     κ=r0R4πβ2γQb , (3) with  r0 as the classical radius, R as the average machine radius, β and γ as the relativistic factors and  Qb as the bare betatron tune. The parameter ζ staying in the exponents of the wake integral is a negated ratio of the chromaticity     ξ=pdQb/dp and the slippage factor    η=γt−2−γ−2, i.e.    ζ=−ξ/η. The symbol     Qeff=Qeff(τ) stays for the space charge tune shift at the given position along the bunch τ, averaged over the both transverse action, see Ref. [2]. Thus, the effective space charge tune shift is proportional to the local line density:       Qeff(τ)=Qeff(0)ρ(τ)/ρ(0).  (4)   For the transversely Gaussian bunch,       Qeff(τ)=0.52Qmax(τ)  (5) where    Qmax(τ) is the space charge tune shift at the bunch axis. The symbol     u2=u2(τ) stays for the average square of the particle longitudinal velocity     υi=dτi/dθ, with time measured as the angle θ along  the machine circumference, at the given position τ :      u2=υ2=f(υ,τ)υ2dυ∫f(υ,τ)dυ∫ , (6) where f(υ,τ) is the longitudinal distribution function. For the longitudinally Gaussian distribution with the rms bunch length  στ, the temperature function  u2 is constant along the bunch:     u2=Qs2στ2 , (7) where  Qs is the synchrotron tune.    In general, the wake term   ˆWy is a sum of single-bunch (SB), coupled-bunch (CB) wakes and the damper (G) terms:     ˆWy=ˆWSBy+ˆWCBy+ˆGy  (8) The single-bunch term   ˆWSBy is described exactly as in Eq. (1), where only    s>τcontributes due to the causality, and the integral is taken along the single-bunch interval only:     ˆWSBy=W(τ−s)expiζτ−s()⎡⎣⎢⎤⎦⎥SB∫ρ(s)y(s)ds  (9) The coupled-bunch term results from summations of the fields left by preceding passages of the bunches through the given position of the ring. This summation is especially simple when the bunches are equidistant. In this case, due to the symmetry, the offsets of the neighbor bunches, being taken at the same time, differ only by the phase factors    exp(iψµ):      y(s+s0)=exp(iψµ)y(s).  (10) For M bunches in the ring,       ψµ=2πµM;µ=0,1,...,M−1 , (11) where integer µ is a counter of the coupled-bunch modes. After that, we have to take into account that the given reference bunch sees the fields left behind by other bunches not at the same time, but certain time ago, proportional to the distance between the bunches. This leads to an additional time-related factor to be taken into account together with the space-related phase factor:      y(s+s0,θ−s0)=expiφµ()y(s,θ);φµ=ψµ+2πQbM.  (12) Remember that both time and space are measured as the angles of revolution, and the leading particles have higher coordinate than the following ones. From here, the coupled-bunch contribution in Eq. (8) follows as a single-bunch integral:     ˆWCBy=!Wµ(τ,s)expiζτ−s()⎡⎣⎢⎤⎦⎥SB∫ρ(s)y(s)ds;!Wµ(τ,s)=W(τ−s−ks0)exp(ikφµ)k=1∞∑.  (13) For those cases when the wake function of the neighbour bunch does not change much along the reference bunch, i.e. the coupled-bunch wake is flat [4],      W(τ−s−ks0)≈W(−ks0)  (14) the result of summation in the right hand side of Eq. (13) does not depend on the specific positions s, τ within the bunches; thus, the effective coupled bunch wake    !Wµ is a  constant which can be taken out of the integral:       ˆWCBy=!Wµeiζτe−iζsρ(s)y(s)dsSB∫.  (15) In principle, the damper term   ˆGy in Eq. (8) is similar to the coupled-bunch one. If the feedback bandwidth is much smaller than the inverse bunch length, the damper takes just one parameter per bunch. This parameter can be chosen as an offset of the centre of mass, and the kick can be designed to be flat along the bunch. Then the damper term is represented similar to Eq. (15):         ˆGy=!Gµeiζτe−iζsρ(s)y(s)dsSB∫ , (16) If the damper is bunch-by-bunch, there is no coupled-bunch mode dependence in the feedback factor    !Gµ, i.e.     !Gµ=!G.    Similarly to the driving wake factor, Eq. (8), there is a certain coupled-bunch contribution in the detuning wake as well:         Dy=DSBy+DCBy;DSB=ρ(s)D(τ−s)dsSB∫;DCB=ρ(s)!D(τ,s)dsSB∫;!D(τ,s)=D(τ−s−ks0)k=1∞∑.  (17) If the detuning wake is flat in the same sense as in Eq. (14), its coupled-bunch contribution is identical for all the particles, so it works as a constant quadrupole without any influence to the beam dynamics unless it leads to a dangerous resonance crossing.  SOLUTION  To find the spectrum of Eq. (1), its eigenfunctions can be expanded over its zero-wake solutions    yk0(τ) satisfying the following equation:      ν0y0+1Qeffddτu2dy0dτ⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟=0;dy0dττ→±∞=0  (18) All the eigenfunctions  are orthogonal and can be normalized, so that:       dsρ(s)yk0(s)SB∫ym0(s)=δkm.  (19) For the Gaussian distribution, the spectrum of this equation has been described in Ref.[2,3]; similarly, it can be found for any potential well and distribution function.   Expansion of the eigenfunction y(τ) over the no-wake set y0(τ),      y(τ)=Bkyk0(τ)k=0∞∑  (20) with the amplitudes B to be found, with the following multiplication of Eq.(1) on    yl0(τ)ρ(τ) and its integration over the bunch length, leads to:        κNˆW+κNˆD+Diag(ν0)⎡⎣⎢⎤⎦⎥B=νB . (21) Here  ˆW and  ˆD are the matrices of the driving and detuning wakes in the basis of the no-wake modes of Eq. (18), (19):     yk0(τ)      ˆWSB()lk=W(τ−s)ρ(τ)ρ(s)eiζ(τ−s)yl0(τ)SB∫∫yk0(s)dτds;ˆWCB()lk=!Wµ(τ,s)ρ(τ)ρ(s)eiζ(τ−s)yl0(τ)SB∫∫yk0(s)dτds;  (22)         ˆGlk=!GµIl(ζ)Ik*(ζ);Il(ζ)=ρ(τ)eiζτyl0(τ)SB∫dτ;  (23)        ˆDSB()lk=FSB(τ)SB∫ρ(τ)yl0(τ)yk0(τ)dτ;FSB(τ)=D(τ−s)SB∫ρ(s)ds;  (24)       ˆDCB()lk=FCB(τ)SB∫ρ(τ)yl0(τ)yk0(τ)dτ;FCB(τ)=!D(τ,s)SB∫ρ(s)ds.  (25)  DAMPER DETAILS    In case when the feedback takes something different from the centre of mass and its kick is not flat over the bunch, the damper matrix has to be modified with provided pickup and kicker functions    P(s),K(τ) :      ˆGlk=!GµKl(ζ)Pk*(ζ);Kl(ζ)=K(τ)ρ(τ)eiζτyl0(τ)SB∫dτ;Pk*(ζ)=P(s)ρ(s)e−iζsyl0(s)SB∫ds.  (26)  Equation (21) is a standard linear algebra eigensystem problem which solution is straightforward as soon as the wake functions, the feedback properties, the potential well, and the beam distribution functions, longitudinal and transverse, are given. This equation allows computing the instability growth rates for fairly general situations when the Landau damping can be neglected. However, without Landau damping nothing can be said about the instability threshold, so the theory is significantly incomplete.  INSTABILITY THRESHOLDS    For the strong space charge, Landau damping rates were roughly estimated in Ref. [2]. Numerical simulations give a possibility for more accurate knowledge of the damping rates, with the numerical factors to be found with a good precision. This work has been started by V. Kornilov and O. Boine-Frankenheim several years ago with their PATRIC code [5.6], has been joined recently by A. Macridin et al. with the Synergia program [7]. As soon as the Landau damping rates are reliably established, they can be introduced in Eq. (21) as externally given imaginary parts of the vector  ν0 in Eq. (21) with the following substitution:     νk0→νk0−iλk , (27) where   λk is Landau damping rate for the no-wake eigenfunction    yk0(τ). When the collective tune shifts imposed by the wakes are small compared to the synchrotron tune, their influence on the Landau damping can be neglected. Potential importance of the image charges and currents for Landau damping was shown in Ref. [5,6]. As soon as Landau damping rates are included in Eq. (21), theory of transverse stability of bunched beams with strong space charge would be complete. Hopefully, this work will be done in a reasonable future.    FNAL is operated by Fermi Research Alliance, LLC under Contract No. De–AC02–07CH11359 with the United States Department of Energy.  REFERENCES [1] M. Blaskiewicz, Phys. Rev. ST Accel. Beams 1, 044201 (1998); Phys. Rev. ST Accel. Beams 6, 014203 (2003).  [2] A. Burov, Phys. Rev. ST Accel. Beams 12, 044202 (2009).  [3] V. Balbekov, Phys. Rev. ST Accel. Beams 12, 124402 (2009)  [4] A. Burov, Phys. Rev. ST Accel. Beams 17, 021007 (2014) [5] V. Kornilov and O. Boine-Frankenheim, Phys. Rev. ST Accel. Beams 13, 114201 (2010)  [6] V. Kornilov, O. Boine-Frankenheim, D.J. Adams, B. Jones, B.G. Pine, C.M. Warsop, R.E. Williamson, “Thresholds of the Head-Tail Instability in Bunches with Space Charge”, HB’2014 workshop, East Lansing, MI, USA.  [7] A. Macridin, A. Burov, E. Stern, J. Amundson, and P. Spentzouris, “Simulation of transverse modes with intrinsic Landau damping for bunched beams with space charge”, to be published.                             "    },    {        "title": "1709.01425v1.Enhancement_of_space_charge_induced_damping_due_to_reactive_impedances_for_head_tail_modes.pdf",        "content": "arXiv:1709.01425v1  [physics.acc-ph]  5 Sep 2017Enhancement of space-charge induced damping due to reactiv e\nimpedances for head-tail modes\nV. Kornilov,\nGSI Helmholtzzentrum, Planckstr.1, Darmstadt, Germany,\nO. Boine-Frankenheim,\nGSI Helmholtzzentrum, Planckstr.1, Darmstadt, Germany,\nTU Darmstadt, Schlossgartenstr.8, Darmstadt, Germany\nOctober 9, 2018\nAbstract\nLandau damping of head-tail modes in bunches due to spreads in the tune shift can\nbe a deciding factor for beam stability. We demonstrate that the co herent tune shifts\ndue to reactive impedances can enhance the space-charge induce d damping and change\nthe stability thresholds (here, a reactive impedance implies the imagin ary part of the\nimpedance of both signs). For example, high damping rates at stron g space-charge, or\ndamping of the k= 0 mode, can be possible. It is shown and explained, how the negative\nreactive impedances (causing negative coherent tune shifts similar ly to the effect of space-\ncharge) can enhance the Landau damping, while the positive cohere nt tune shifts have\nan opposite effect. It is shown that the damping rate is a function of the coherent mode\nposition in the incoherent spectrum, in accordancewith the concep t of the interaction of a\ncollective mode with resonant particles. We present an analytical mo del, which allows for\nquantitative predictions of damping thresholds for different head- tail modes, for arbitrary\nspace-charge and coherent tune-shift conditions, as it is verified using particle tracking\nsimulations.\n1 INTRODUCTION\nThe performance of many present and future hadron ring accel erators is limited by the effects\ndue to the self-field space-charge. The transverse head-tai l modes in bunches have been\ninitially described by a theory [1, 2] without space-charge . A significant progress have been\nrecently achieved in understanding of the effects of the betat ron tune shifts due to space-\ncharge for the collective eigenmodes in bunches. Similarly to other beam dynamics issues,\nthe spread in the space-charge tune shift appeared to be very important.\nAn ”airbag” bunch model, suggested in [3], allowed for an exa ct description of the head-\ntail modes for arbitrary space-charge. This model does not t ake into account a spread in the\nspace-charge tune shift which is present in realistic bunch distributions. Nevertheless, the\npredictions of the airbag model appeared to be rather accura te for the eigenfrequencies and\n1for the eigenmodes in the realistic Gaussian bunches [4, 5]. Experimental confirmations have\nbeen demonstrated in measurements on bunches with space-ch arge in [6, 7]. Theories for\nrealistic bunch distributions have been presented in [8, 9] , where an important implication of\nthe space-charge tune spread has been identified. This is a La ndau damping of the coherent\nmodes due to interactions with the incoherent particle spec trum. In contrast to the case\nof a coasting beam, the space-charge induced tune spread pro vides damping for the bunch\neigenmodes. Damping rates due to the tune spread produced by the longitudinal bunch\ndensity variations have been demonstrated and quantified in [5].\nAnimaginaryimpedanceoftheaccelerator facility represe ntsareactiveforceandshiftsthe\ncoherent frequencies [10, 11]. In real accelerators there a re many sources of the coherent and\nincoherent tune shifts. The detuning wakes can shift the inc oherent betatron frequencies [12].\nIn this work we focus on the driving dipole wakes and on their c oherent betatron frequency\nshifts. We thus specify ∆ Qcohas the real tune shift of a coherent eigenmode due to a reactiv e\nfacility impedance. Typically, the real coherent tune shif t is negative, which is also the case\nfor the self-field space-charge tune shift ∆ Qsc, so for the both notations we use the modulus\nof the negative value. As the both shifts depress the tunes, i t makes the intersections of the\ncoherent frequencies with the incoherent spectrum possibl e. This is the main reason for the\neffect of the reactive impedances on the space-charge induced damping in bunches which we\naddress in the present paper. The combination of space-char ge with reactive impedances has\nbeen discussed in [13, 7], and the enhancement effect of the rea ctive impedances on damping\nhas been suggested in [14, 15]. In this work we present an anal ytical model and simulations\nfor a systematic description of the space-charge induced da mping with reactive impedances.\nIn Sec.2 we discuss the nature of the space-charge induced da mping and how it can\ndepend on the beam distributions, especially regarding the distribution tails. The effect of\nreactive impedancesisexplainedinSec.3, withtheanalyti cal modelforarbitraryspace-charge\nconditions and mode index number. The predictions of the ana lytical model are analysed for\ndifferent cases and verified in Sec.4. The work is concluded in S ec.5.\n2 SPACE-CHARGE INDUCED DAMPING\nTherearetwobasicsourcesfortheself-fieldspace-charge t unespreadinabunch. Thefirstone\nis due to different synchrotron amplitudes of the individual p articles, i.e. in the longitudinal\nplane. The second one is due to different betatron amplitudes, i.e. in the transverse plane.\nIn the both cases, a non-uniform density distribution cause s a tune spread.\nThe line density λ(z) variation along the bunch gives the maximal space-charge t une shift\nfor the given longitudinal position,\n∆Qsc(z) =g⊥λ(z)rpR\n4γ3β2εx, (1)\nwhere(2πR)istheringcircumference, βandγaretherelativisticparameters, rp=q2\nion/(4πǫ0mc2)\nis the classical particle radius, εxis the transverse rms emittance. This tune shift is the mod-\nulus of the negative shift. The geometry factor g⊥depends on the transverse distribution, for\n2the Gaussian profile it is g⊥= 2, for the K-V beam it is g⊥= 1. In order to characterize the\nspace-charge force in a bunch, the space-charge parameter i s used,\nq=∆Qsc(0)\nQs(2)\nwhich is the tune shift for the rms-eqivalent K-V beam ( g⊥= 1) in the peak of the line\ndensity ( λ0), normalised by the synchrotron tune Qs. It was shown in [3, 8, 9, 5, 6] that\nthis parameter is the key value to describe the space-charge conditions in a bunch. This is\nespecially true for the analytical models, as it will be also shown below. The usage of the\nrms-eqivalent K-V beam for the comparisons between different beam distributions has been\nproven to be adequate, the same is true for comparisons with t he experiments. During the\nfurther discussion we use the notation ∆ Qscfor ∆Qsc=qQsfrom Eq.(2).\nAnother reason for the space-charge tune spread is the differe nt amplitudes of the trans-\nverse betatron oscillation of theindividual particles. Th eparticles with small amplitudes have\nthe tune shift close to the local maximal space-charge tune s hift from Eq.(1). The particles\nwith large amplitudes have smaller tune shifts, technicall y down to zero, depending on the\ntransverse distribution and on its truncation. A beam with t he transverse K-V distribution\nhas no space-charge tune spread of this kind.\nDamping of the head-tail modes exclusively due to the space- charge tune spread from\nthe longitudinal plane has been studied in [5]. There, a K-V d istribution has been assumed\nfor the transverse plane. A good agreement with the results f rom [5] has been recently\nreported in [16], in simulations with a different code and differ ent methods. One might\nexpect that an additional tune spread due to the transverse p rofile should enhance damping.\nThis is confirmed in our particle tracking simulations prese nted in Fig.1, where the space-\ncharge induced damping for the Gaussian bunch in the longitu dinal and transverse planes\nis compared with the results from [5]. Additional space-cha rge tune spread provides higher\ndamping rates and larger q−regions for a chosen damping level.\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45\n 0 1 2 3 4 5 6 7103 (−Im(∆Q))\nq = ∆Qsc / QsK-V Gaussiank=1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n 0 2 4 6 8 10 12 14103 (−Im(∆Q))\nq = ∆Qsc / QsK-V Gaussiank=2\nFigure1: Space-chargeinduceddampingratesfromparticle trackingsimulationsfor k= 1and\nk= 2 head-tail modes. Solid lines correspond to the longitudi nal Gaussian and transverse K-\nV bunches, as presentedin [5]. Dashedlines arefortheGauss ian (longitudinal andtransverse)\nbunches.\n3For the particle tracking simulations we use the code PATRIC [17], similarly to the simu-\nlations presented in [5]. The code has been validated using e xact analytical prediction [4], for\nthe space-charge effects and for the coherent effects. For the tr ansverse space-charge force,\na frozen slice-to-slice electric field model is used, i.e. a fi xed potential configuration which\nfollows the center of mass for each bunch slice. This approac h is justified in the rigid-slice\nregime [18, 8]. Thus, it is physically reasonable for modera te and strong space charge. The\nrecent study [15] suggests to confirm the applicability of th e frozen space-charge model. The\nmethod to excite the head-tail eigenmodes and to determine t he damping decrements is de-\nscribed in [5]. The simulations for Fig.1 have been performe d for the chromaticity equal zero.\nThe chromaticity does not affect damping, which is also clear i n the formalism of [8], and has\nbeen verified in our simulations.\n3 ANALYTICAL MODEL\nThe basic mechanism of Landau damping is the interaction bet ween a collective wave and the\nresonant particles. In the case of the head-tail modes in bun ches these roles are played by a\ncoherent buncheigenmodeandtheindividualparticlebetat ronoscillations. Thedampingrate\nis then proportional to the number of the resonant particles . It can be especially clear in the\nsimulations for different longitudinal and transverse distr ibutions in a bunch. Allowing more\nparticlesinthedistributiontails, bothlongitudinaland transverse, providesstrongerdamping.\nIn the experimental conditions, the bunch distributions ca n be very complicated, the tail\ntruncations depend on the transverse acceptance and on the m omentum acceptance of the\nmachine. It should be however possible to find general finding s for different distributions and\ntail truncations. In our simulations we consider the exampl e with the Gaussian distribution,\nand with the truncations at 2.5 of the standard deviation.\nA certain damping rate can be reached if the coherent oscilla tion interacts with a large\nenough number of resonant particles. The particle incohere nt spectrum is provided by tune\nshifts of different nature. In the case of space-charge in a Gau ssian bunch, this distribution\nis non-zero between ( −2qQs)<∆Q <0. The particles with tune shifts just below zero\n(∆Q <0), are in the distribution tails, which correspond to parti cles with large amplitudes.\nA growing beam intensity with a fixed beam distribution means a growing space-charge tune\nshift. If we consider resonant particles with the fixed oscil lation amplitudes, longitudinal\nand transverse, we see that the incoherent shift of these par ticles increases linearly with the\nintensity parameter,\n∆Qη=−η∆Qsc. (3)\nHere, the parameter ηis introduced in order to relate the coherent lines to the inc oherent\nspectrum in the upcoming discussion, and to indicate the fac t that space-charge tune shifts\ngrow linearly with the beam intensity (for a fixed emittance) . The value of ηin a specific\nbunch depends on the chosen damping rate and on the particle d istribution in the bunch.\nThe coherent tune shift dueto space-charge can be analytica lly estimated using the airbag\n4model [3],\n∆Qk\nQs=−q\n2±/radicalbigg\nq2\n4+k2, (4)\nwhere ”+” is for modes k≥0. As we discuss in the Introduction, this expression provid es a\nrather accurate prediction for the eigenfrequencies even i n the Gaussian bunches, especially\nfor strong space-charge. This has been confirmed in simulati ons, experiments, and theoretical\nstudies [8, 9, 5, 6].\nThe head-tail modes drop in frequency for stronger space-ch arge. Thus the coherent mode\ncan cross the border of the damping with a chosen damping rate . Since the head-tail modes\nwithk >0 are close to the upper tail of the incoherent spectrum (clos e to the bare tune\nQ0), there is a top border for the chosen damping. If the coheren t mode has a frequency\nbelow this border the damping rate will exceed the border dec rement. The conditions for a\nchosen damping rate should be estimated by the relation of th e border Eq.(3) to the coherent\nk−mode frequency which is modulated by the harmonics of the syn chrotron frequency [9, 5],\n∆Qk−kQs, (5)\nwhere ∆Qk=Qk−Q0is the coherent frequency of the mode k. The coherent line shifts are\npositive (∆ Qk>0) fork >0, and the incoherent tune shifts due to space-charge are neg ative.\nThis damping model thus also explains why damping is still po ssible.\nFigure2 illustrates the above discussion and shows the cohe rent head-tail frequencies\nmodulated by the harmonics of the synchrotron frequency for the modes k= 1 and k= 2.\nFrom the simulations with a Gaussian bunch, see Fig.1, we hav e determined η= 0.24, which\nis shown by the dashed line in Fig.2.\n-2-1.5-1-0.5 0\n 0  2  4  6  8  10Re(∆Qk)/Qs − k\nq = ∆Qsc / Qsk=0\nk=1\nk=2\nFigure 2: Positioning of the head-tail modes from Eqs.(4, 5) with respect to the incoherent\nspectrum border for the damping analysis. The dashed line in dicates the border Eq.(3) of\nthe chosen damping η= 0.24.\nThereis avalueofthe q−parameter wherethemodecrosses thedampingborder. Itcan b e\npredicted using the expression for the coherent head-tail e igenfrequency Eq.(4). Comparing\n5(∆Qk−kQs) with the damping border Eq.(3), we come to the conclusion th at a mode crosses\nthe border at q= 0, and at\nq=k1−2η\nη(1−η). (6)\nThis gives a prediction for the range of the space-charge par ameterqfor a chosen damping\nrate for arbitrary mode index. Figure2 illustrates this: fo r very small qeach head-tail mode\nenters the incoherent particle spectrum and thus experienc es a strong damping (see Fig.1).\nAs the coherent line crosses the damping border again, the da mping becomes weaker due to\na smaller number of the resonant particles. There is a smooth transition between damping\nregimes as the number of resonant particles changes. Estima tions for qwhere the modes cross\nthe damping borders Eq.(6) provide q≈2.8 fork= 1 and q≈5.7 fork= 2 (here η= 0.24),\nas we can also observe in Fig.2. All these observations are su pported by particle tracking\nresults for Gaussian bunches in Fig.1.\nThe damping rates decrease at strong space-charge ( q > k). For Gaussian bunches, the\nscaling Im(∆ Q)∼ −1/q3has been predicted in [12]. The drop of the damping rate for\nhigherqis in agreement with the concept of the coherent frequency sh ift Eq.(5) towards\nlower density in the incoherent spectrum. The fast decrease of the damping rate reflects the\nfact that the coherent lines are shifted further into the tai ls of the incoherent distribution\n. For realistic bunches, the specific scaling depends on the b unch distributions and the tail\ntruncations. Below we should demonstrate that the effect of re active impedances can lead to\nhigh damping rates also for strong space-charge.\n4 DAMPINGWITH REACTIVEIMPEDANCES: ANALYT-\nICAL MODEL\nThe additional effect of a coherent tune shift has been include d into the airbag theory in [13],\n∆Qk=−∆Qsc+∆Qcoh\n2±/radicalbigg\n(∆Qsc−∆Qcoh)2\n4+k2Q2s, (7)\nwhere ”+” is for modes k≥0. The coherent tune shift ∆ Qcohof the head-tail mode k\nis the result of the interaction with the imaginary part of th e facility impedance. It can\nbe calculated as an integral over the bunch spectrum with the frequency-dependent reactive\nimpedance[1,10,11]. For abroadband-typeimpedance(with respecttothebunchspectrum),\nthe neighboring modes have very close ∆ Qcoh, for example for the image charges. Figure3\nshows the combined effect of space-charge with the coherent tu ne shift in this case. For a\nnarrowband impedance (with respect to the bunch spectrum), the coherent tune shifts can be\nvery different and can contribute to the excitation of the tran sverse mode coupling instability\n[3, 8, 11] which is not in the scope of this paper.\nThe difference between the effect of an external impedance and sp ace-charge on head-\ntail modes is especially obvious for the k= 0 mode. Space-charge alone does not affect the\nk= 0 mode, ∆ Qk=0= 0. In contrast, a facility impedance alone shifts the k= 0 mode,\n∆Qk=0=−∆Qcoh.\n6-5-4.5-4-3.5-3-2.5-2-1.5-1-0.5 0\n 0 0.5  1 1.5  2 2.5  3Re(∆Qk)/Qs − k\n∆Qcoh / Qsk=0\nk=1\nk=2\nk=3\nFigure 3: Combined effect of a reactive impedances and space-c harge on the head-tail mode\npositioning according to Eq.(7) for q= 10. The dashed line indicates the damping border\n∆Qη.\nThere is an important implication from Eq.(7). For increasi ng ∆Qcoh(and a fixed ∆ Qsc),\neach head-tail mode crosses the damping border and enters th e regime with a stronger damp-\ning, see Fig.3. Some of the modes are damped shortly above ∆ Qcoh= 0, for example k= 3\nin Fig.3. Even the mode k= 0 is affected by the space-charge induced damping after certa in\n∆Qcoh. This implies that the coherent effects can change strongly th e damping conditions.\nHigh damping rates at strong space-charge, damping of the k= 0 mode, etc., are possible\nwith reactive impedances, in contrast to the previously dis cussed [8, 5] case of the self-field\nspace-charge only.\nThe coherent tune shift at which a head-tail mode Eq.(7) cros ses the damping border can\nbe determined using Eq.(3). The corresponding solution is\n∆Qcoh= ∆Qscη−kQs1−η\nk/q+1−η. (8)\nFor thek= 0 mode, this expression is\n∆Qcoh= ∆Qscη . (9)\nFor strong space-charge q≫kEq.(8) takes a simpler form,\n∆Qcoh= ∆Qscη−kQs. (10)\nThis means that the damping border grows linearly with ∆ Qsc(orq). For the higher-order\nhead-tail modes, the necessary ∆ Qcohis equidistantly shifted by kQs. Figure4 demonstrates\nthat the coherent tune shift needed for the chosen damping ra te can be easily predicted for\narbitrary space-charge conditions.\n7 0 2 4 6 8 10 12 14 16 18 20\n 0 10 20 30 40 50 60 70 80∆Qcoh / Qs\nq = ∆Qsc / Qsk=0\nk=1\nk=2\nFigure 4: Coherent tune shift ∆ Qcohneeded for a chosen damping rate for different head-tail\nmodes and for different strength of space-charge in bunches as predicted by Eq.(8).\n5 COMPARISONS BETWEEN THEORY AND SIMULA-\nTIONS\nWe verify the basic predictions of the analytical model Eq.( 8) using particle tracking simu-\nlations. Figure5 shows the positioning of the head-tail mod es with respect to the damping\nborder for η= 0.24. The damping conditions are compared for the moderate spa ce-charge\nq= 2 (left plot) and for the stronger space-charge q= 6 (right plot). This discussion\ncorresponds to a beam with a fixed intensity, but with an incre asing transverse imaginary\nimpedance. The mode k= 2 is always damped in the both plots of Fig.5. The mode k= 1 is\nalways damped for q= 2, but has a region outside the strong damping for q= 6. The mode\nk= 0 experiences no damping without coherent tune shifts, but it is affected by damping\nfor increasing ∆ Qcoh. Forq= 6, the border of the chosen damping is higher in ∆ Qcohthan\nforq= 2. The mode k= 1 forq= 6 has only a weak damping without coherent effect, and\naround ∆ Qcoh= 0.5Qsthe damping becomes stronger. Results of the simulations in Fig.6\nconfirm these observations.\nIn the next case we discuss an increasing ∆ Qscwith a fixed ∆ Qcoh/∆Qsc−ratio. This\ncorresponds to beams of an increasing intensity and with a co nstant transverse emittance.\nThe imaginary impedance is also fixed, for example, the vacuu m pipe. Figure7 shows the\npredictionoftheanalytical modelforthemode k= 1withthedifferentvaluesoftheparameter\nα= ∆Qcoh/∆Qsc. The mode crosses the damping boundary at higher q−values for larger\nα−ratios. The condition for this crossing for arbitrary kcan also be obtained from Eqs.(3,7),\nq=kα+1−2η\n(η−α)(1−η). (11)\nThis applies for α < ηand is plotted in Fig.8 for η= 0.24. Theq−value of the transfer across\nthe damping border goes to infinity as αapproaches η.\nPredictions of the analytical model in Figs.7,8 are compare d with the simulation results\ninFig.9. For thetwo cases ∆ Qcoh= 0and∆ Qcoh= 0.1∆Qsc, thereisaregion withthestrong\n8-2.5-2-1.5-1-0.5 0\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Re(∆Qk)/Qs − k\n∆Qcoh / Qsk=0\nk=1\nk=2\n-2.5-2-1.5-1-0.5 0\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Re(∆Qk)/Qs − k\n∆Qcoh / Qsk=0\nk=1\nk=2\nFigure 5: Effect of the reactive impedances on damping for differ ent space-charge tune shifts\naccording to Eqs.(3, 7). Left plot: q= ∆Qsc/Qs= 2, right plot: q= 6. The dashed lines\nindicate the damping border.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0  0.2  0.4  0.6  0.8  1103 (−Im(∆Q))\n∆Qcoh / Qsk = 0\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n 0  0.5  1  1.5103 (−Im(∆Q))\n∆Qcoh / Qsk = 1\nk = 0\nFigure 6: Effect of the reactive impedances on damping obtaine d from the particle tracking\nsimulations. Left plot: q= ∆Qsc/Qs= 2, right plot: q= 6. Note that without reactive\nimpedances the k= 0 mode experiences no effect of space charge and no damping.\ndamping which transfers to a weaker damping by higher q. In the case of ∆ Qcoh= 0.1∆Qsc\nthe damping is stronger and has a larger q−region. Within the considered parameters, the\nmode stays in the strong damping for ∆ Qcoh= 0.2∆Qsc. Correspondingly, the coherent\nline remains below the damping border in Fig.7. Finally, a ne gative coherent shift ∆ Qcoh\npushes the head-tail mode out of the damping region and makes the damping weaker, and\nthe damping region smaller. This can be seen for the case with ∆Qcoh=−0.1∆Qscin\nthe analytical model Figs.7,8, and it is in a good agreement w ith the simulations in Fig.9.\nFigure9 also illustrates the enhancement of Landau damping due to an external impedance,\nespecially for stronger space-charge.\nUsing Fig.9 we can test the suggestion that the space-charge induced damping is directly\nrelated to the position of the coherent line with respect to t he damping border in the incoher-\nent spectrum. For every q−value and ∆ Qcohthe distance between the modulated theoretical\n9-4-3-2-1 0 1\n 0 2 4 6 8 10 12 14Re(∆Q/Qs − k\nq = ∆Qsc / Qs∆Qcoh = 0∆Qcoh = −0.1 ∆Qsc\n∆Qcoh = 0.1 ∆Qsc\n∆Qcoh = 0.2 ∆Qsc\nFigure 7: Effect of the reactive impedances on the k= 1 head-tail mode according to Eq.(7)\nfor different coherent tune shift in the case of a fixed α= ∆Qcoh/∆Qsc. The dashed line\nindicates the damping border Eq.(3) for η= 0.24.\n 0 5 10 15 20\n-0.1-0.05 0 0.05 0.1 0.15 0.2q\nα = ∆Qcoh / ∆Qsc\nFigure 8: The space-charge parameter at which the head-tail modek= 1 crosses the damping\nborder as given by Eq.(11) with η= 0.24.\ncoherent frequency Eq.(7) and the damping border Eq.(3),\n∆Qdamp= ∆Qk−kQs−∆Qη, (12)\ncanbecalculated andplottedasthehorizontal axis, seeFig .10. Thevalues∆ Qdamp<0imply\nthe position of the mode beyond the spectrum part ηsuch that the resulting damping rate\nshould behigher than that at the dampingborder. The points a tq= 0 in Fig.9 correspondto\n∆Qdamp= 0 in Fig.10. As we observe from Fig.7, the coherent lines div e below the damping\nborderfor small q, which correspondsto ∆ Qdamp<0 in Fig.10. With increasing q, the modes\ncome back to the damping border, and, in fact, the decrements return on the close path back\nto ∆Qdamp= 0 (see Fig.10). The exception is ∆ Qcoh= 0.2∆Qsc(blue), which stays well\n10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0 1 2 3 4 5 6 7 8103 (−Im(∆Q))\nq = ∆Qsc / Qs∆Qcoh = 0∆Qcoh = 0.1 ∆Qsc∆Qcoh = 0.2 ∆Qsc\n∆Qcoh = −0.1 ∆Qsc\nFigure 9: Damping decrement of the k= 1 head-tail mode obtained from the particle tracking\nsimulations for a Gaussian bunch, Qs= 0.01, for different coherent tune shift due to reactive\nimpedances. The chosen parameters correspond to the analyt ical model predictions in Fig.7.\nbelow the damping border, its ∆ Qdampremain strongly negative and the decrements stay\nhigh. For the cases ∆ Qcoh= 0 (black) and ∆ Qcoh=−0.1∆Qsc(green), the modes gain\nlarge ∆Qdamp>0 and the damping becomes weak. All these damping rates are lo cated\nfairly close to a universal line of the space-charge induced damping. This demonstrates that\nthe damping rate is a monotonic function of the coherent mode position in the incoherent\nspectrum, irrespective the space-charge and ∆ Qcohconditions.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n-0.6-0.4-0.2 0 0.2 0.4 0.6103 (−Im(∆Q))\n∆Qdamp∆Qcoh = 0∆Qcoh = 0.1 ∆Qsc∆Qcoh = 0.2 ∆Qsc\n∆Qcoh = −0.1 ∆Qsc\nFigure 10: Damping rates from Fig.9 vs. the mode-spectrum di stance Eq.(12). The plot\ndemonstrates that the space-charge induced damping depend s directly on the coherent mode\nposition with respect to the incoherent spectrum, irrespec tive the space-charge and ∆ Qcoh\nconditions.\n116 CONCLUSIONS\nSpace-charge inducedLandau damping in bunches has been stu died usingan analytical model\nand particle tracking simulations. It has been demonstrate d that two sources of the space-\ncharge tune spread, longitudinal and transverse, contribu te to the resulting damping. A\nspecific damping rate (for example, strong enough damping to stabilise an instability) can be\nconsidered as the damping border. The parameter ηfor the shift of the coherent mode into\nthe incoherent spectrum Eq.(3) can then be defined, dependin g on the distribution in bunch\ntails and on the chosen damping border.\nA model for the space-charge induced damping in bunches is pr esented. Using the airbag-\nbased analytical model [3] for the head-tail model with arbi trary space-charge Eq.(4), and the\nborder of the incoherent spectrum for a chosen damping Eq.(3 ), theq−region of the space-\ncharge induced damping can be predicted by Eq.(6). This resu lt is confirmed by the particle\ntracking simulations. Enhancement of Landau damping is dem onstrated for all considered\nhead-tail modes.\nWith the help of the analytical expression [Eq.(7)] from [13 ], a model for the damping\nborder has been suggested, Eq.(8). For the mode k= 0 and for strong space-charge, there\nare simple relations Eqs.(9,10). This model gives a univers al prediction for the coherent tune\nshift needed for the damping, Fig.4, for arbitrary space-ch arge conditions and for different\nhead-tail modes k.\nIt has been discussed [8, 9, 5, 6] that in the case of self-field space-charge only the mode\nk= 0cannotbeaffectedbyspace-chargeandbyrelateddamping. H erewehavedemonstrated\nthat theanalytical model(Fig.5)andthesimulations (Fig. 6) consistently show theconditions\nwhere the mode k= 0 can be damped.\nEffects of reactive impedances on different modes kare discussed for a fixed q= ∆Qsc/Qs\n(constant beam intensity and growing reactive impedance) a nd for a fixed α= ∆Qcoh/∆Qsc\n(constant reactive impedance and growing beam intensity). The both cases reveal different\ninteresting properties in the space-charge induced dampin g with reactive impedances. The\nanalytical model Eqs.(6,8,11) provide specific prediction s for the transition across the damp-\ning border for different modes k, which are confirmed by the particle tracking simulations.\nThe analysis of the simulation results has also demonstrate d (see Fig.10) that the space-\ncharge induced damping depends directly on the coherent mod e position with respect to the\nincoherent spectrum, irrespective the space-charge and th e reactive impedance conditions.\nIt has been shown that the coherent tune shifts can have a stro ng effect on the damping\ndue to space-charge in bunches. Landau damping can be dramat ically enhanced due to the\neffect of reactive impedances. Thus the reactive impedances a nd other ∆ Qcoh−sources should\nbetaken intoaccount inaninstability thresholdanalysis. Ontheother hand,thespace-charge\ninduceddampinginbunchescanbeenhancedbyemployingaddi tional reactive impedances, in\na passive (dedicated hardware elements) or in an active (a re active feedback [10, 19]) manner.\n12References\n[1] F.Sacherer, Proc. First Int. School of Particle Acceler ators, Erice, p.198 (1976)\n[2] F.Sacherer, CERN Report CERN/SI-BR/72-5 (1972)\n[3] M.Blaskiewicz, Phys. Rev. ST Accel. Beams 1, 044201 (1998)\n[4] V. Kornilov and O. Boine-Frankenheim, Proc. of ICAP2009 , San Francisco, USA, Aug\n31 - Sep 4 (2009)\n[5] V.Kornilov and O.Boine-Frankenheim, Phys. Rev. ST Acce l. Beams 13, 114201 (2010)\n[6] V.Kornilov and O.Boine-Frankenheim, Phys. Rev. ST Acce l. Beams 15, 114201 (2012)\n[7] R.Singh, et.al., Phys. Rev. ST Accel. Beams 16, 034201 (2013)\n[8] A.Burov, Phys. Rev. ST Accel. Beams 12, 044202 (2009); A.Burov, Phys. Rev. ST\nAccel. Beams 12, 109901(E) (2009)\n[9] V.Balbekov, Phys. Rev. ST Accel. Beams 12, 124402 (2009)\n[10] K.Y.Ng, Physics of Intensity Dependent Beam Instabili ties (World Scientific 2006)\n[11] A.Chao, Physics of Collective Beam Instabilities in Hi gh Energy Accelerators (John\nWiley Sons, Inc., New York, 1993).\n[12] A.Burov, V.Danilov, Phys. Rev. Letters 82, 2286 (1999)\n[13] O. Boine-Frankenheim, V. Kornilov, Phys. Rev. ST Accel . Beams, 12, 114201 (2009)\n[14] V.Kornilov, et.al., Proc. of HB2014, East Lansing, USA , November 10-14 (2014)\n[15] V.Kornilov, O.Boine-Frankenheim, Space-Charge 2015 Workshop, Oxford, March 23-27,\nhttps://www.cockcroft.ac.uk/events/SpaceCharge15/in dex.html (2015)\n[16] A.Macridin, A.Burov, E.Stern, J.Amundson, P.Spentzo uris, Phys. Rev. ST Accel.\nBeams18, 074401 (2015)\n[17] O.Boine-Frankenheim, V.Kornilov, Proc. of ICAP2006, 2-6 Oct., Chamonix Mont-\nBlanc, (2006)\n[18] A.Burov and V.Lebedev, Phys. Rev. ST Accel. Beams 12, 034201 (2009)\n[19] R.Ruth, CERN Report LEP-TH/83-22 (1983)\n13"    },    {        "title": "2209.14120v1.Tunable_nonlinear_damping_in_parametric_regime.pdf",        "content": "Tunable nonlinear damping  in parametric regime  \nParmeshwar Prasad*, Nishta Arora, and A. K. Naik*  \nCentre for Nano  Science and Engineering,  \nIndian Institute of Science, Bangalore, India, 560012  \n \nNonlinear damping plays a significant  role in several area of physics and it is becoming \nincreasingly important to understand its underlying mechanism . However, microscopic origin \nof nonlinear damping is still a debatable topic. Here , we probe and report  nonlinear  damping \nin a highly tunable MoS 2 nano mechanical drum resonator  using electrical homodyne \nactuation and detection technique . In our experiment, we achieve 2:1 internal resonance by \ntuning  resonance frequency and observe enhanced non -linear damping . We probe the effect of \nnon-linear damping by characterizing parametric gain.  Geometry  and tunability  of the device \nallow  us to reduce the effect of other prominent Duffing non -linearity to probe the non -linear \ndamping effectively.  The enhanced non -linear damping in the vicinity of internal resonance is \nalso observed in direct drive , supporting  possible origin of non -linear damping . Our \nexperiment  demonstrate s, a highly tunable 2D material based nano resonator offer s an \nexcellent platform to study the nonlinear  physics  and exploit nonlinear  damping in parametric \nregime.  \n \nDynamical systems in nature show ri ch physics demonstrating energy exchange.  The systems \nfind its equilibrium m ediated by energy exchange. There are several ways through which a \nsystem accomplishes  energy exchange with its surroundings. The energy exchange could be \ndissipative in nature and  can be modelled as linear damping in the simplest case. However, \nthere are system which shows diverse  nonlinear  dissipative phenomena1–3. For example , nano \nmechanical systems  (NEMS)  offer  such a system  where  vibration amplitude can be  tuned to an \nextent to  witness  nonlinear  dissipation2–5. In general,  vibration amplitude comparable to the \nlength scale  of its thickness  drives  the system in to nonlinear  regime. In this regime, the \ndynamical systems  offer rich physics dealing with  nonlinear  dynamics.  As the nonlinear ity in \nthe system increases the presence of dissipative mechanics play an important role in dynamics \nof the nano -mechanical system s.  \nFigure (1) Characterisation of the device: (a) Inset: SEM image of the device, the two thick \nyellow lines are source and dr ain electrodes, the suspended region is 3 𝜇𝑚 in diameter. The \ngate is 300 𝑛𝑚 below the suspended membrane . Schematic of the measurement setup : \nHomodyne capacitive actuation and detection technique.  Membrane frequency is tuned using \nDC gate voltage 𝑉𝑔𝑑𝑐, actuated  directly using 𝑉𝑔𝑎𝑐at frequency 𝜔 and parametrically at 2𝜔 \nusing 𝑉𝑝𝑎𝑐. AC voltages are combined using RF power combine r, further  DC is combined with \nthe help of a bias-tee. Readout is amplified using low noise amplifier  and measured  at lock-in \namplifier locked at  𝜔. Frequency dispersion with 𝑉𝑔𝑑𝑐, two modes 𝜔1 (b) and 𝜔2 (c) with \nfrequency tuning > 1 MHz/V  and 𝜔2~2 𝜔1.  \nNonlinear dissipation has been of interest due to its presence in the applicat ion in biology6, \nmagnetization7, quantum optics8 and quantum information technologies9. However, source of \ntheir origin remains  unclear  and remains an important area to explore especially in low \ndimensional nanoresonators . Advancement in the nano -electromechanical system  (NEMS) in \nthe last decade  has led to explor ation of  nonlinear  dissipation mechanisms  2–4,10–12. It has been \nshown that the effect of nonlinear  damping increases with vibration amplitude13. The nonlinear  \nterm can be incorporated in the force equation as ∝𝜂𝑥2𝑧̇ where 𝜂 is the nonlinear  damping \ncoefficient , 𝑧 is the displacement and 𝑧̇ is the velocity13. Though there are several models  \nexplaining the microscopic origin of the nonlinear  damping, the precise  mechanism is not \nunderstood  10,14,15. This is partly due to the  challeng es involved  in isolat ing and decoupl ing the \nmechanism responsible for the nonli near damping from other nonlinear effects . Nonlinear \nphenomena are  easily achievable in low -dimensional materials due to their extreme sensitiv ity \nto the external perturbation.  In this context, 2D materials such graphene and transition metal \ndi-chalcogenid e (TMDC) provide  an excellent platform to probe  nonlinear damping.  Recent ly, \nwork by Keskekler et. al4. shed light on the possible mechanism of the nonlinear  damping \nthrough internal resonance  (IR) in graphene using the opt ical actuation and detection method. \nThough optical method provides a sensitive detection techni que, it lacks the tunabi lity of the \nresonance frequency without heating the device . Heating is one of most common way to \ndissipate energy.  \n \nFigure (2)  (a) Amplitude response curve of mode1  (𝜔1) at multiple gate voltages. 𝑉𝑔𝑑𝑐×𝑉𝑔𝑎𝑐=\n0.2 𝑉2 is constant for the experiments.  Shaded region shows the variation of amplitude. \nAmplitude  is minimum at  𝑉𝑔𝑑𝑐=22 𝑉 (b) Inverse of quality factor at different 𝑉𝑔𝑑𝑐, it quantifies \nthe damping in the system. Damping is maximum  at  𝑉𝑔𝑑𝑐=22 𝑉. \nIn this work, we demonstrat e the nonlinear  damping mechanism in a highly tunable  \nmolybdenum disulphide ( MoS2) device using the electrostatic  homodyne capacitive  actuation \nand detection method.  The method provides a clean , fast and simple platform to study the \nnonlinear  system16,17. Our device shows two  prominent modes 𝜔1 and 𝜔2 the two modes are  \n𝜔2∼2𝜔1. The modes provide favourable scenario to study the nonlinear  damping in the \nparametric regime  and probe the dissipation mechanism . In our experiment , we are able to tune \nthe nonlinear  damping by changing t he applied gate voltage. We find the effect of nonlinear  \ndamping  considerably increases when the two modes are commensurate with each other  in 2:1 \nratio. To probe the nonlinear  damping , we use parametric pumping technique.  Examining  the \nparametric  gain,  we find that the effect of nonlinear  damping reaches maximum in the vicinity \nof IR. Parametric gain decreases by as low as 8 3% when the nonlinear  damping is maximum.  \nWe observe , the enhancement  in nonlinear  damping  is due to strong coupling between the two  \nmodes near internal resonance . The experiment shed light into the mechanism of the nonlinear \ndamping in to nano mechanical system  using a highly tunable NEMS . \nOur device is ~ 10 layer thick MoS2 drum resonator suspended over 3  μm diameter trench. The \ndevice is strained and actuated by applying voltage s (𝑉𝑔) at gate electro de. The gate electrode \nis 300 nm below the suspended membrane.  Figure 1 a shows the schematic of the measurement \nused to measure the vibration of the membrane and for applying param etric pumping . The \nmembrane is strained by applying a DC gate voltage ( 𝑉𝑔𝑑𝑐) and actuated with an AC voltage \n(𝑉𝑔𝑎𝑐). The DC and the AC is combined using a bias -tee before applying to the gate electrode . \nApplied potential difference exerts f orce on the suspended membrane there by displacing it \nfrom the equilibrium position. The displacement of the membrane modulates the geometric \ncapacitance between gate and the membrane, leading to the change in voltage at the drain. The \nchange in drain volt age is amplifie d using a low -noise amplifier  before measuring at the lock -\nin amplifier. Figure 1 b and c show the dispersion of the frequency with applied DC gate \nvoltage . We observe two distinct modes having similar tuning with each other. The two \nfrequenc ies are 𝜔1~ 56 MHz  and 𝜔2 ~111  MHz   at around  𝑉𝑔𝑑𝑐=23 V . The modes are \ninteresting due to 2:1  (i.e. 𝜔2~2𝜔1) ratio of the frequenc ies, which enables us to achieve \ninternal resonance by driving the fundamental mode in parametric regi me. We can achieve this \nIR ratio by tuning the gate voltage.  \nTo obtain the elementary  characteristics of the device, we drive it with fixed force  (𝐹∝\n𝑉𝑔𝑑𝑐 𝑉𝑔𝑎𝑐) over a range of gate voltages ( 𝑉𝑔𝑑𝑐). A constant force allows us to observe and \ncompare the variation of amplitude response  curve at different resonant frequency . We drive \nthe resonator with small force such that the response does not show nonlinear  behaviour . The linear response  can be quantified by the symmetry  of the pe ak about the resonance frequency . \nIn a nonlinear  regime a hysteresis in the frequency response curve  is observed . Figure 2a shows \namplitude response with frequency at different gate voltages.  We observe that amplitude \ndecreases with increase in gate voltage a nd reaches minimum value around 𝑉𝑔𝑑𝑐∼22 V. In a \nlinear system, a constant force results in a constant amplitude and linewidth due to linear \ndissipation. Reduction in amplitude for a constant applied force indicates enhanced  dissipation. \nThe line wi dth of the response curve is shown in the figure 2b , it shows  damping is maximum \nat 22 V. To probe the enhancement in damping , we perform parametric  pumping  by tuning \nstiffness of the membrane . Parametric pumping  offers  advantage in probing  a resonance mod e \nby applying a force away from the resonant frequency  and study the coupling of modes.  \nParametric pump  can be used alone or in a combination with direct drive . Parametric pump \nworks in a principle that it provides energy to overcome the linear damping of the system by \nmodulating the stiffness.  \nWe use the following equation  of motion  to describe the parametric amplification in our \nsystem13: \n𝑚 𝑧̈+𝛾 𝑧̇+𝑚 𝜔02 𝑧+𝜂 𝑧̇ 𝑧2+𝛼𝑒𝑓𝑓 𝑧3=𝐹(𝜔 𝑡)+𝐹𝑝(2 𝜔 𝑡) 𝑧 (1) \nWhere m is mass of the system 𝛾 is the linear damping, 𝜔0 is the resonance frequency,  𝜂 is the \nnonlinear  damping coefficient, 𝛼𝑒𝑓𝑓 is the effe ctive Duffing nonlinear ity coefficient associated \nwith geometric nonlinear ity. 𝐹 is the direct force applied at the frequency 𝜔, 𝐹𝑝 is the \nparametric force acting at the system at the frequency 2𝜔 and 𝑧 is the vibration amplitude.  \nTo study parame tric amplification, we apply an alternating voltage  (𝑉𝑝𝑎𝑐) at twice the resonance \nfrequency of the fundamental mode at the gate in addition to 𝑉𝑔𝑎𝑐 (see Figure 1 a). In a \nparametric system, there exists a critical pumping force  beyond which  the system overcomes \nlinear damping and  reaches instability region also known as Mathew’s tongue13,18. The critical \npumping force can be measured by sweeping 𝑉𝑝𝑎𝑐 at 2𝜔. Figure 3 shows amplitude response \nusing parametric drive beyond critical pump voltage ( 𝑉𝑝𝑎𝑐0) for different 𝑉𝑔𝑑𝑐. Beyond the \ncritical pumping , the amplitude response is governed by the nonlinear  parameters and offers \nan excellent regime to probe the nonlinear  coefficients.  There are two dominant nonlinear ities \npresent in our device  namely Duffing nonline arity ( 𝛼𝑒𝑓𝑓) and nonlinear  damping ( 𝜂). We \nobserve  that 𝛼𝑒𝑓𝑓 changes from negative to positive , it changes sig n around 𝑉𝑔𝑑𝑐~24 V. It is \ninteresting for the fact that this allows us  to probe the effect of nonlinear  damping by minimi zing the effect of the prominent geometric nonlinear ity. Figure 3d shows the computed \nvalue of 𝛼𝑒𝑓𝑓 using continuum model  (see supplementary for derivation and computation ). The \nvariation of  computed  𝛼𝑒𝑓𝑓 matches well with our experimental fi ndings.  Figure 3 a also shows \ndecreases  in amplitude  response at around 𝑉𝑔𝑑𝑐~24 V. The variation in amplitude response \nsimilar to  the experimental findings  shown in  figure 2a. In the past, i t has been shown that the \namplitudes in parametric regime are saturated by nonlinear ities present in the system19. The \ncurrent device provides favourable platform to investigate the eff ect of  the nonlinear  damping \nwhile suppressing the effect of 𝛼𝑒𝑓𝑓. To probe the nonlinear  damping, we perform parametric \namplification and observe the amplitude gain in the system.  \n \nFigure 3 Tuning of geometric nonlinear ity (a) Amplitude response with upward frequency \nsweep for different 𝑉𝑔𝑑𝑐 with  𝑉𝑝𝑎𝑐=400  𝑚𝑉 and 𝑉𝑔𝑎𝑐=0. Change in the s hape of amplitude \nresponse curve  shows variation in  geometric nonlinear ity. Amplitude response at (b)  𝑉𝑔𝑑𝑐=\n16 V and (c) 𝑉𝑔𝑑𝑐=28 V. (d) Blue line is effective Duffing nonlinearity  (𝛼𝑒𝑓𝑓) calculated \nusing continuum model, it crosses the 𝛼𝑒𝑓𝑓=0 (red line) around 𝑉𝑔𝑑𝑐=24 V, which matches \nthe experimental observation of in (a) . \nIn parametric amplification , the gain ( 𝐺=𝑧𝑜𝑛\n𝑧𝑜𝑓𝑓) is defined as the ratio of amplitude when 𝑉𝑝𝑎𝑐 \nis ON to OFF. Figure 4 a shows parametric gain with 𝑉𝑝𝑎𝑐 for different 𝑉𝑔𝑑𝑐. It shows that gain \nincreases steadily and get saturated after the critical pumping voltage. The maximum gain  𝐺𝑚𝑎𝑥 \nobtained before the critical  pumping  is shown in the figure 4b. It shows that the 𝐺𝑚𝑎𝑥  decreases \nsignificantly in the region  𝑉𝑔𝑑𝑐= 22 V to 𝑉𝑔𝑑𝑐= 26 V. The decrease in the 𝐺𝑚𝑎𝑥 is as low as \n83% as compared the 𝐺𝑚𝑎𝑥 at 𝑉𝑔𝑑𝑐= 20 V. To find the nonlinear  damping coefficient  we use \nthe relation 𝜂=2(𝐹𝑝𝑄−2)\n𝑧𝑚𝑎𝑥2  , where 𝐹𝑝 is parametric force, 𝑄 quality factor and 𝑧 is the maximum \namplitude13. The relation allows us to estimate the coefficient of nonlinear damping \nindependent of any nonlinear  parameter  in the lowest order . Figure 4e shows the coefficient of \nthe nonlinear  damping extra cted for different gate voltages.  \n \nFigure (4) Parametric amplification: (a) Amplitude gain  (𝐺) vs applied pump strength 𝑉𝑝𝑎𝑐 for \nmultiple 𝑉𝑔𝑑𝑐. Gain increases as pump  strength increases, eventually saturating due to \nnonlinear  effects.  (b) Maximum gain (𝐺𝑚𝑎𝑥) at the start of instability  tongue  vs 𝑉𝑔𝑑𝑐. 𝐺𝑚𝑎𝑥 \ndecreases in the region 𝑉𝑔𝑑𝑐=22 𝑉 to 𝑉𝑔𝑑𝑐=26 V. (c)(d)  Self oscillation of mode1 , response  \nmeasured at 𝜔 is plotted against pump  𝑉𝑝𝑎𝑐 applied at frequency  2𝜔. (c) and (d) are upward \nand downward frequency sweep respectively. Dashed rectangle shows the region of frequency \nlocking, resulting from 1:2 internal resonance . (e) Coefficient of nonlinear  damping 𝜂 \ncalculated using the parametric model using the equation 1 . Higher value of 𝜂 is observed in \nthe region 𝑉𝑔𝑑𝑐=22 𝑉 to 𝑉𝑔𝑑𝑐=26 V.  It explains the low gain observed in (b) and matches \nthe experimental findings.  \n \nWe observe that the nonlinear  damping increases significantly around  𝑉𝑔𝑑𝑐=24 𝑉. Around the \ngate voltage , we observe the minima of the amplitude  and increased damping in case of direct \ndriving  (See figure  2a, b). Enhancement of nonlinear  damping is evident , as suggested by \ndecrease in amplitude during direct drive and the gain during parametric amplification . Figure \n4c,d shows instability region, 𝑉𝑝𝑎𝑐 is swept at frequency 2𝜔 and amplitude response is \nmeasured at 𝜔. As we increase the parametric pump strength, the direction of peak shifts from \nlower frequency and to higher frequency, indicating change in the value of 𝛼𝑒𝑓𝑓, which \nmatches our theoretical prediction using continuum model of the drum resonator.  Upon further \nincreasing the pump, we observe the frequency locking , we note that this is in the vicinity of \nfavourable  2:1 internal resonance.  In the internal resonance, the two m odes are coupled to each \nother, it makes  pathway to exchange the energy. In the vicinity  of IR, the mode 1 dissipates \nenergy to mode2 enhancing the nonlinear  damping.  \nIn summary, we study  rich nonlinear  physics in MoS 2 drum resonator  in parametric regime . \nWe observe two prominent tunable modes in 50 MHz to 200 MHz range  and they are coupled \nto each other  through tension in the membrane . Dominant nonlinear  coefficients  𝛼𝑒𝑓𝑓 and 𝜂 \nare tunable in our device  using  DC gate voltage . We tune the modes  such that it facilitates 2:1 \ninternal resonance  and exchange energy with each other.  We observe that the nonlinear  \ndamping increases significantly in the vicinity  of internal resonance , 𝜂 reaching as high as \n60×1015 kg m−2s−1 in MoS 2 drum resonator . In our experiment, minimizing the effect  of \n𝛼𝑒𝑓𝑓 helps us to probe nonlinear  damping efficiently.  A highly tunable  NEMS  device offers \nversatile control on the nonlinear  coefficient s and in probing them.  Our experiment show s \nevidence of the microscopi c origin  of nonlinear  damping and shed light in understan ding the \nunderlying physics . Parametric regime provides excellent approach to probe the nonlinear  \nphysics  by tuning the fundamental parameters such as stiffness.   \n Acknowledgements:  \nWe acknowledge fu nding support from Nano Mission,  Department of Science and Technology \n(DST), India through  Grants SR/NM/NS -1157/2015(G) and SR/NMITP -62/2016(G) and from \nBoard of Research in Nuclear Sciences  (BRNS), India through Grant 37(3)/14/25/2016 -BRNS. \nP.P. acknowled ges scholarship support from CSIR, India. N.A.  acknowledges fellowship \nsupport under Visvesvaraya Ph.D.Scheme, Ministry of Electronics and Information \nTechnology(MeitY), India. We also acknowledge funding from MHRD,  MeitY, and DST \nNano Mission for supporti ng the facilities at  CeNSE.  We gratefully acknowledge the  usage of \nthe National Nanofabrication Facility (NNfC) and the  Micro and Nano Characterization \nFacility (MNCF) at CeNSE,  IISc, Bengaluru . \n \nReferences : \n1 A. Eichle r, J. Moser, J. Chaste, M. Zdrojek, I. Wilson -Rae, and A. Bachtold, Nat. \nNanotechnol. 6, 339 (2011).  \n2 J. Güttinger, A. Noury, P. Weber, A.M. Eriksson, C. Lagoin, J. Moser, C. Eichler, A. \nWallraff, A. Isacsson, and A. Bachtold, Nat. Nanotechnol. 12, 631 (2 017).  \n3 A. Eichler, J. Chaste, J. Moser, and A. Bachtold, Nano Lett. 11, 2699 (2011).  \n4 A. Keşkekler, O. Shoshani, M. Lee, H.S.J. van der Zant, P.G. Steeneken, and F. Alijani, \nNat. Commun. 12, 1099 (2021).  \n5 J. Atalaya, T.W. Kenny, M.L. Roukes, and M.I. Dykman, Phys. Rev. B 94, 195440 (2016).  \n6 M. Amabili, P. Balasubramanian, I. Bozzo, I.D. Breslavsky, G. Ferrari, G. Franchini, F. \nGiovanniello, and C. Pogue, Phys. Rev. X 10, 011015 (2020).  \n7 B. Divinskiy, S. Urazhdin, S.O. Demokritov, and V.E. Demidov, Nat. Commun. 10, 5211 \n(2019).  \n8 L. Mandel and E. Wolf, Optical Coherence and Quantum Optics  (Cambridge University \nPress, 1995).  \n9 Z. Leghtas, S. Touzard, I.M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K.M. Sliwa, A. Narla, \nS. Shankar, M.J. Hatridge, M. Reagor, L. Frunzio, R.J. Schoelkopf, M. Mirrahimi, and M.H. \nDevoret, Science  347, 853 ( 2015).  10 S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, Nonlinear Dyn. 67, 859 (2012).  \n11 V. Singh, O. Shevchuk, Y.M. Blanter, and G.A. Steele, Phys. Rev. B 93, 245407 (2016).  \n12 M. Imboden, O. Williams, and P. Mohanty, Appl. Phys. Lett. 102, 103502  (2013).  \n13 R. Lifshitz and M.C. Cross, Reviews of Nonlinear Dynamics and Complexity  (Wiley -VCH \nVerlag GmbH & Co. KGaA, Weinheim, Germany, 2008).  \n14 X. Dong, M.I. Dykman, and H.B. Chan, Nat. Commun. 9, 3241 (2018).  \n15 A. Croy, D. Midtvedt, A. Isacsson, and  J.M. Kinaret, Phys. Rev. B 86, 235435 (2012).  \n16 P. Prasad, N. Arora, and A.K. Naik, Nano Lett. 19, 5862 (2019).  \n17 J.P. Mathew, R.N. Patel, A. Borah, R. Vijay, and M.M. Deshmukh, Nat. Nanotechnol. 11, \n747 (2016).  \n18 A.H. Nayfeh and D.T. Mook, Nonlinear O scillations  (Wiley -VCH Verlag GmbH, \nWeinheim, Germany, 1995).  \n19 R. Lifshitz and M.C. Cross, Phys. Rev. B 67, 134302 (2003).  \n \n \n \n \n \n \n \n \n \n \n \n Supplementary Information  \nTunable nonlinear damping in parametric regime  \n \nCalculation of Effective Duffing Non -linearit y (𝜶𝒆𝒇𝒇): \nFigure S1 describes the MoS 2 resonator clamed at the edge of a circular trench. An electrostatic \npotential ( 𝑉𝑔𝑑𝑐) applied at gate bends the membrane and tune the frequency. The applied strain \nusing the electrostatic force also modif ies the non -linear coefficients. The gate voltage can be \nused as a knob to control the non -linear coefficient which is discussed the following section.  \n \n.  \nFigure S 1: MoS 2 NEMS: A DC voltage 𝑉𝑔𝑑𝑐 is applied at the gate to de form the membrane. \nDistance between the trench and the gate is 𝑧0.  \n \nWe consider continuum model of the drum resonator1: \n𝜕𝜏2𝜁𝛼+Λ𝛼𝜁𝛼+∑∑𝑄𝛽𝛾𝛼𝜁𝛽𝜁𝛾∞\n𝛾≥𝛽∞\n𝛽=1+∑∑∑𝐶𝛽𝛾𝜂𝛼𝜁𝛽𝜁𝛾𝜁𝜂∞\n𝜂≥𝛾∞\n𝛾≥𝛽∞\n𝛽=1=𝑓𝛼(𝜏) (𝑆1.1) \nWhere, 𝜁 are the coordinates, 𝑄𝛽𝛾𝛼 and 𝐶𝛽𝛾𝜂𝛼 are the coupling constants, 𝑓𝛼(𝜏) is the applied \nforce and Λ𝛼 are the eigen frequencies squared.  \nThe maximum scaled deflection ( 𝜁0) at the center of the resonator is given by:  \n𝜁0=𝑏 (2\n3𝜃)1\n3\n−(𝜃\n18𝑐3)1\n3\n(𝑆1.2) \n \n𝜃=√3(4(𝑏 𝑐)3+27𝑎2𝑐4)12⁄−9𝑎 𝑐2(𝑆1.3) \nWhere 𝑎= −𝑅 𝜖0 𝑉𝑔𝑑𝑐2\n4𝑌ℎ 𝑑 𝑧02, 𝑏=2𝑇0\n𝑌ℎ−𝑅2𝜖0 𝑉𝑔𝑑𝑐2\n3𝑌 ℎ 𝑧03 and 𝑐=7−𝜈\n6(1−𝜈) \nWhich translates the Stress ( 𝑇) in the membrane as:  \n𝑇=𝑇0(1+𝑧2\n4) (𝑆1.4) \nWhere 𝑧 is scaled static center deflection and given by:  \n𝑧=𝜁0(𝑌 ℎ\n𝑇0)1\n2√(3−𝜈\n1−𝜈) (𝑆1.5) \nThe above information is used to find the non -linear coefficients.  \n𝛼3=𝐶1𝑇\n𝑅4𝜌0 (𝑆1.6) \n \n𝛼2=𝑄1𝑇\n𝑅3𝜌0 (𝑆1.7) \n  \nWhere 𝐶1 and 𝑄1 are the coupling constant determined as follows:  \n𝐶1=𝑌 ℎ\n𝑇(3.92+3.68 𝛿𝜈) (𝑆1.8) And \n𝑄1=𝜁0𝑌 ℎ\n𝑇(11.7+11.3 𝛿𝜈) (𝑆1.9) \nAfter the determination of the 𝛼2 and 𝛼3, 𝛼𝑒𝑓𝑓 can be defined as:2 \n𝛼𝑒𝑓𝑓=𝛼3−10\n9(𝛼2\n𝜔0)2\n(𝑆1.10) \nWe used the following parameters for the calculation of 𝛼𝑒𝑓𝑓 \nParameters  Value  \nYoung’s modulus  0.33×1012  Pa \nMass density of MoS2 (ρ) 5.06×103 kg.m−3 \nPoisson ratio ( ν) 0.27 \n \n   \n(a) (b) (c) \n \n(d) \nFigure S3: Variation (a) Static deflection (b) DC force (c) AC force (d) 𝛼𝑒𝑓𝑓 with applied \nDC gate voltage 𝑉𝑔𝑑𝑐. \n \nCalculat ion of non -linear damping coefficient:  \nWe consider the following equation to describe the parametric drive3: \n𝑚 𝑧̈+𝛾 𝑧̇+𝑚 𝜔02 𝑧+𝜂 𝑧̇ 𝑧2+𝛼𝑒𝑓𝑓 𝑧3=𝐹(𝜔 𝑡)+𝐹𝑝(2 𝜔 𝑡) 𝑧 (𝑆2.1) \nEquation S2.1 describes nano mechanical Duffing resonator with linear and non -linear \ndamping driven by a direct force 𝐹 and a parametric force 𝐹𝑝 acting at 𝜔 and 2𝜔 respectively. \nThe resonator has mass 𝑚 and resonance frequency 𝜔0. 𝛼𝑒𝑓𝑓, 𝛾, and 𝜂 are Duffing, linear and \nnon-linear damping coefficients respectively. Displacement amplitude ( 𝑧) is a time ( 𝑡) varying \nfunction and dot represents derivative with  respect to time.  \nCoefficients of non -linear damping ( 𝜂) can be derived from equation S1 and written as follows:  \n𝜂=2 𝐹𝑝 𝑄−4\n𝑧2(𝑆2.2) \nWhere the vibration amplitude ( 𝑧) is scaled as follows:  \n(𝑚 𝜔02\n𝛼̃)1\n2\n𝑧=𝑧̃ (𝑆2.3) \n𝜂̃𝜔0\n𝛼= 2 𝐹𝑝 𝑄−4\n𝑧̃2𝛼\n𝑚 𝜔02(𝑆2.4) \n𝜂̃=(2 𝐹𝑝 𝑄−4) 𝑚 𝜔0\n𝑧̃2(𝑆2.5) \nWhere:  \n𝐹𝑝=2 𝐴 𝜖0 𝑉𝑔𝑑𝑐 𝑉𝑝𝑎𝑐\n𝑧̃03(𝑆2.6) \n \nNon-linear damping constant can be estimated using equation S2.5.  \n \nReferences:  \n1 A.M. Eriksson, D. Midtvedt, A. Croy, and A. Isacsson, Nanotechnology 24, 395702 (2013).  2 A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations  (Wiley -VCH Verlag GmbH, Weinheim, \nGermany, 1995 ). \n3 R. Lifshitz and M.C. Cross, Reviews of Nonlinear Dynamics and Complexity  (Wiley -VCH \nVerlag GmbH & Co. KGaA, Weinheim, Germany, 2008).  \n \n "    },    {        "title": "0710.4610v1.Damping_of_Condensate_Oscillation_of_a_Trapped_Bose_Gas_in_a_One_Dimensional_Optical_Lattice_at_Finite_Temperatures.pdf",        "content": "arXiv:0710.4610v1  [cond-mat.other]  25 Oct 2007Damping of Condensate Oscillation of a Trapped Bose Gas in a\nOne-Dimensional Optical Lattice at Finite Temperatures\nEmiko Arahata and Tetsuro Nikuni\nDepartment physics, Faculty of science, Tokyo University o f Science,\n1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan\n(Dated: November 5, 2018)\nAbstract\nWe study damping of a dipole oscillation in a Bose-Condensed gas in a combined cigar-shaped\nharmonic trap and one-dimensional (1D) optical lattice pot ential at finite temperatures. In order\nto include the effect of thermal excitations in the radial dire ction, we derive a quasi-1D model of\nthe Gross-Pitaeavskii equation and the Bogoliubov equatio ns. We use the Popov approximation to\ncalculate the temperature dependence of the condensate fra ction with varying lattice depth. We\nthen calculate the Landau damping rate of a dipole oscillati on as a function of the lattice depth\nand temperature. The damping rate increases with increasin g lattice depth, which is consistent\nwith experimental observations. The magnitude of the dampi ng rate is in reasonable agreement\nwith experimental data. We also find that the damping rate has a strong temperature dependence,\nshowing a sharp increase with increasing temperature. Fina lly, we emphasize the importance of\nthe radial thermal excitations in both equilibrium propert ies and the Landau damping.\n1I. INTRODUCTION\nRecently the dynamics of ultracold atomic gases in optical lattices ha ve attracted atten-\ntion both theoretically and experimentally [1]. In particular, center- of-mass dipole oscilla-\ntions of Bose-Einstein condensates in a combined cigar shaped trap and one dimensional\n(1D) optical lattice potential have been experimentally studied in de tail [2, 3, 4]. In the\npresence of the periodic lattice potential, a decrease of the dipole m ode frequency was ob-\nserved [3]. This decrease can be understood in terms of the increas e of the effective mass\ndue to the lattice potential [5]. On the other hand, at finite tempera tures where an ap-\npreciable number of atoms are thermally excited out of the condens ate, strong damping of\nthe dipole oscillation of the condensate was observed in the presenc e of the lattice potential\n[4]. In a pure harmonic potential, a Bose gas exhibits undamped dipole o scillations even at\nfinite temperatures, since the condensate and noncondensate a toms oscillate with the same\nfrequency without changing their density profiles [6]. In contrast, in the presence of the\nperiodic lattice potential, only the condensate component can cohe rently tunnel through\nthe potential barriers, while the thermal component is locked by th e lattice potential [4]. It\nis clear that this incoherent thermal conponent gives rise to the da mping of the condensate\noscillations. In Bose-condensed gases trapped in harmonic potent ials, Landau damping is\nknown to be the dominant contribution to damping of the condensat e oscillations in the\ncollisionless regime [7, 8, 9, 10]. In the analysis of damping of dipole oscilla tions in opti-\ncal lattice, the authors of Ref. [4] gave a rough estimate of the La ndau damping rate and\ncompared it with their experimental data. However, quantitative c alculations of the Landau\ndamping rate of dipole condensate oscillations in optical lattices have not given in any detail\nso far. In fact, even equilibrium properties of a Bose gas in an optica l lattice at finite tem-\nperatures in connection with the experimentas of Ref. [4] have not been studied in detail.\nAlthough several papers have discussed finite-temperature pro perties of ultracold atoms in a\n1D optical lattice, most theoretical studies have concentrated o n the first Bloch band using\nthe Bose-Hubbard model, and have ignored the effect of radial exc itations [11, 12, 13]. This\napproximation is only valid when kBT≪ERandkBT≪¯hω⊥, whereERis the recoil energy\nthat is roughly the highest energy in the first Bloch band, and ¯ hω⊥is the first excitation\nenergy in the radial direction. In order to obtain more quantitative results that is applicable\nto the experiments of Refs. [3, 4], however, it is important to consid er thermal excitations in\n2the higher bloch bands as well asa the radial direction, since the exp eriments do not always\nsatisfykBT≪ERandkBT≪¯hω⊥.\nIn this paper, we study the Landau damping of condensate oscillatio ns of a trapped Bose\ngas in a 1D optical lattice, with explicitly including the effect of the radia l excitations. In\nSec. II, we derive a quasi-1D model of the Gross-Pitaeavskii equa tion and the Bogoliubov\nequations that include the effect of the excitations in the radial dire ction. Using the Hatree-\nFock-Bogoliubov-Popov (HFB-Popov) approximation, we solve the se equations to calculate\nthe temperature dependence of the condensate fraction.\nIn Sec. III, we calculate the Landau damping of the dipole condensa te oscillation in an\noptical lattice. We calculate the damping rate as a function of the lat tice depth with a fixed\ntemperature, and compare it with the experimental data [4]. The ma gnitude of the damping\nrate is found to be in reasonable agreement of the experimental da ta [4]. The damping rate\nincreases with increasing lattice depth, which is consistent with the e xperimental result [4].\nWe also calculate the temperature dependence of damping rate with a fixed lattice depth.\nWefindthatthedampinghasastrongtemperature, showing ashar pincreasewithincreasing\ntemperature.\nII. QUASI 1D MODELING OF A TRAPPED BOSE GAS\nWe consider a Bose condensed gas in a combined potential of highly-e longated harmonic\ntrap and 1D optical lattice. Our system is described by the following H amiltonian:\nˆH=/integraldisplay\ndr/braceleftBigg\nˆψ†(r)/bracketleftBigg\n−¯h2\n2m∇2+Vext(r)/bracketrightBigg\nˆψ(r)\n+g\n2ˆψ†(r)ˆψ†(r)ˆψ(r)ˆψ(r)/bracerightBigg\n, (1)\nwhereg=4π¯h2a\nmis the coupling constant determined by the s-wave scattering length\na. The external potential Vextis given by Vext(r) =Vtrap(r) +Vop(z), whereVtrap(r) =\nm\n2[ω2\n⊥(x2+y2)+ω2\nzz2] is an anisotropic harmonic potential and Vop(z) =sERcos2(kz) is an\noptical lattice potential. Here sis a dimensionless parameter describing the intensity of the\nlaser beam creating the 1D lattice in units of the recoil energy ER≡¯h2k2\n2m, wherek=2π\nλis\nfixed by the wavelength λof the laser beam.\nInthis paper we consider ahighly-anisotropic cigar shaped harmonic trappotential ω⊥≫\nωz. In order to take into account this quasi-1D situation, we expand t he field operator in\n3terms of the radial wave function [14]:\nˆψ(r) =/summationdisplay\nαˆψα(z)φα(x,y), (2)\nwhereφα(x,y) is the eigenfunction of the radial part of the single-particle Hamilto nian,\n/bracketleftBigg\n−¯h2\n2m∇2\n⊥+m\n2ω2\n⊥(x2+y2)/bracketrightBigg\nφα(x,y)\n=ǫαφα(x,y), (3)\nwhich satisfy the orthonormality condition/integraltextdxdy φ∗\nα(x,y)φβ(x,y) =δαβ. Hereα= (nx,ny)\nis the index of the single-particle state with the eigenvalue ǫ(nx,ny)= ¯hω⊥(nx+ny+ 1).\nInserting Eq. (2) into Eq. (1) and using Eq. (3), we obtain\nˆH=/summationdisplay\nα/integraldisplay\ndzˆψ†\nα(z)/bracketleftBigg\n−¯h2\n2m∂2\n∂z2+m\n2ω2\nzz2+Vop(z)+ǫα/bracketrightBigg\nˆψα(z)\n+/summationdisplay\nαα′ββ′gαα′ββ′\n2/integraldisplay\ndzˆψ†\nαˆψ†\nβˆψβ′ˆψα′, (4)\nwhere the renormalized coupling constant is defined by\ngαα′ββ′≡g/integraldisplay\ndxdyφ∗\nαφ∗\nβφβ′φα′. (5)\nFollowing the procedure described in Ref. [15] , we separate out the condensate wavefunc-\ntionfromthe field operatoras ˆψα=/an}b∇acketle{tˆψα/an}b∇acket∇i}ht+˜ψα≡Φα+˜ψα, where Φ α=/an}b∇acketle{tˆψα/an}b∇acket∇i}htis the condensate\nwavefunction and ˜ψαis the noncondensate field operator. Within the HFB-Popov approx i-\nmation, we obtain the generalized Gross-Pitaevskii (GP) equation,\nµΦα=/bracketleftBigg\n−¯h2\n2m∂2\n∂z2+m\n2ω2\nzz2+Vop+ǫα/bracketrightBigg\nΦα\n+/summationdisplay\nα′ββ′gαα′ββ′/parenleftBig\nΦ∗\nβΦβ′Φα′+2Φβ′/angbracketleftBig˜ψ†\nβ˜ψα′/angbracketrightBig/parenrightBig\n. (6)\nThePopovapproximationneglectstheanomalouscorrelation/angbracketleftBig˜ψ˜ψ/angbracketrightBig\n[15]. Fromthenumerical\nsolutions of the GP equation (6) using the trap frequencies relevan t to the experiment [4],\nwe find that |Φα|2/|Φ0|2≪10−6(α/ne}ationslash= 0), where we have denoted the lowest radial state\nasα= 0 = (0,0), and thus the contribution from higher radial modes to the cond ensate\nwavefunction isnegligiblesmall. Forthisreason, wewill henceforthap proximateΦ α≈Φδα,0.\nTaking the usual Bogoliubov transformations for the noncondens ate,\n˜ψα(z) =/summationdisplay\nj/parenleftBig\nujαˆαj−v∗\njαˆα†\nj/parenrightBig\n, (7)\n4we obtain the coupled Bogoliubov equations,\nˆLαujα+/summationtext\nα′/bracketleftBigg\n(2gα\nα′n0+gαα′ββ′˜nββ′)ujα′−gα\nα′n0vjα′/bracketrightBigg\n=Ejujα, (8)\nˆLαvjα+/summationtext\nα′/bracketleftBigg\n(2gα\nα′n0+gαα′ββ′˜nββ′)vjα′−gα\nα′n0ujα′/bracketrightBigg\n=−Ejujα, (9)\nwhere we have introduce the operator\nˆLα≡ −¯h2\n2m∂2\n∂z2+m\n2ω2\nzz2+Vop(z)+ǫα−µ. (10)\nWehavealsointroducethesimplified notations n0(z) =|Φ(z)|2,gα\nα′=gαα′00,˜nββ′=/angbracketleftBig˜ψ†\nβ˜ψβ′/angbracketrightBig\n.\nAs noted above, we only include the lowest mode ( α= 0) in the condensation wavefunction\nΦ. Sums over the repeated indices β,β′are implied in Eqs. (8), and (9). These equations\ndefine the quasi-particle excitation energies Ejand the quasi-particle amplitudes ujαand\nujα. The orthonormality of the quasi-particle amplitudes is specified by t he relation\n/integraldisplay\ndz/parenleftBig\nu∗\njαuiα−uiαu∗\njα/parenrightBig\n=δij (11)\nUsing the solutions of Eqs. (8) and (9), one can obtain the noncond ensate density from\n˜n=/summationtext\nα˜nαα, where\n˜nαβ=/summationdisplay\nj/bracketleftbigg\n(ujαujβ+vjαvjβ)N(Ej)+vjαvjβ/bracketrightbigg\n(12)\nwithN(Ej) = 1/[exp(βEj)−1].\nSolving the coupled equations (6), (8) and (9), we self-consistent ly determine the exci-\ntations spectrum Ejand the condensate fraction at finite temperatures. Our calculat ion\nprocedure is summarized as follows. Eq. (6) is first solved self-cons istently for µand Φ ne-\nglecting the interaction terms. Once Φ is known, Ej,ujαandujβare obtained from Eqs. (8)\nand (9) with ˜ nββ′set to zero. This is inserted into Eq. (6) and the process is repeate d until\nconvergence is reached. At each step, we define the normalization of the condensate wave-\nfunctionsby/integraltextdz|Φ(z)|2=N−˜N, where˜N=/integraltextdz˜n(z)isthetotalnumber ofnoncondensate\natoms.\nThroughout this paper we use the following parameters of the expe riment of Ref. [4]:\nm(87Rb)=1.44 ×10−25kg,ωz/2π= 9.0 Hz,ω⊥/2π= 92 Hz, scattering length a= 5.82 nm\n5and the wavelength of the optical lattice λ=795 nm. We fix the total number of atoms as\nN= 4×105. InFig.1, weplotthecondensatefraction Nc/Nasafunctionofthetemperature\nfor various values of the lattice depth sby solving GP equation Eq. (6) and Bogolibov\nequations Eqs. (8) and (9). It can be seen that each line falls to zer o at approximately\nT=140 nK, which is close to the semiclassical prediction of the BEC tran sition temperature\nof an ideal Bose gas in a 3D harmonic trap T0\nc= 0.94¯h(ω2\n⊥ωz)1/3N1/3/kB= 141 nK [17]. In\ncontrast,Tcof an ideal Bose gas in a 1D harmonic trap is Tc∼¯hωzN\nkBlnN=13.4µK [17].\nThis crearly shows that one must explicitly take into account the exc itations in the radial\ndirection in order to obtain the correct thermodynamic behavior at finite temperatures\n[14]. The transition temperature deceases with increasing lattice de pth, but there is no\nsignificant change in the temperature dependence of the condens ate fraction, as long as the\nlattice potential is not so deep, i. e., s≤2,\nFIG. 1: The condensate fraction Nc/Nas a function of the temperature. “ s=0” represents the\nideal Bose gas result without a lattice potential Nc/N= 1−(T/T0\nc)3, whereT0\nc=141 nK.\nWe turn to the detailed structure of the excitation spectrum Ej. From Eqs. (8) and\n(9), one sees that the different radial modes are coupled due to int eractions. However the\ncoupling is so small that there are still distinct branches correspon ding the radial modes.\nWe thus label each branch with the index α.For example, the lowest branch ( α= 0) can be\nidentified with the branch corresponding to the lowest radial mode. In Fig 2, we plot the\nexcitation spectrum. Frequency of the condensate collective mod e is related the excitation\nenergy through Ej= ¯hωj. In particular, the dipole mode frequency can be identified with\nthe lowest frequency. In Fig. 3, we plot the dipole-mode frequency as a function of the\nlattice depth with a fixed temperature T= 40nK. One can see the decrease of the dipole-\n6FIG. 2: The Bogoiubov excitation energy spectrum for s=1.2, where iis the energy label for each\nbranch\nmode frequency with increasing lattice depth. The same behavior wa s also observed in\nthe experiment [3]. The negative energy shift also found for other excitation modes. The\ndecrease of Tccan be attributed to these negative shifts.\nFIG. 3: The dipole mode frequency as a function of the lattice depthswith a fixed temperature\nT= 40nK.\nIII. LANDAU DAMPING RATE OF DIPOLE MODE\nIn this Section, we calculate Landau damping of the dipole mode. Land au damping is\nthe dominant damping mechanism for low-lying collective modes in trapp ed Bose-condensed\ngases in the collisionless regime at finite temperatures [8]. Landau da mping originates from\nthe coupling between single-particle excitations and the collective os cillations. Damping\noccurs because the thermal bath of the elementary excitations c an absorb quanta of the\n7collective oscillations. A general expression for the Landau damping rate is given in Refs. [7,\n8, 10, 17]. In our quasi-1D model, Landau damping rate can be expre ssed in terms of the\nquasi-particle excitation energies Ejand quasi-particle amplitudes ujαandvjαcalculated in\nthe previous Section:\nγL= 4π/summationdisplay\nα,α′gαα′00/summationdisplay\ni/negationslash=j|Aαα′\nij|2/bracketleftBig\nN(Ei)−N(Ej)/bracketrightBig\nδ(¯hω+Ei−Ej), (13)\nwhereωis the eigenfrequency of a condensate dipole oscillation. The matrix e lementAαα′\nij\nis defined by,\nAαα′\nij≡/integraldisplay\ndzΦ{u10[u∗\niαujα′+v∗\niαvjα′−v∗\niαujα′]\n−v10[u∗\niαujα′+v∗\niαvjα′−u∗\niαvjα′]} (14)\nwhereu10andv10are the quasi-particle amplitudes corresponding the dipole mode. On e\ndifficulty in calculating the damping rate in Eq. (13) using the discrete e xcitation spectrum\nEjis that it involves the energy-conserving delta functions. This difficu lty can be overcome\nby replacing each delta function by a function with a finite width. In th is paper, we use the\nfollowing replacement:\nδ(¯hω+Ei−Ej)→1\n2∆Θ/parenleftBig/vextendsingle/vextendsingle/vextendsingle∆−¯hω+Ei−Ej/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n. (15)\nThe width factor ∆ is somewhat arbitrary, and the result for γLwill vary with ∆. In Fig. 4,\nFIG. 4: The variation of the Landau damping rate γL(arbitrary unit) with the width factor ∆\nfor lattice depth s= 1.0 and temperature T= 120 nK. To capture the ∆ dependence of γLwhile\nsaving computational time, we take the matrix element Aαα′\nijas a constant in this figure.\nwe plot a typical behavior of the ∆ dependence of γL. One can see that the variation of\n8γLis very weak when ∆ /¯hωzlies between 1 ×10−4and 3×10−4. The same behaviors are\nalso found for other temperatures and lattice heights. We thus ta ke ∆/¯hωz= 2.5×10−4in\ncalculating the damping rate γL.\nFIG. 5: The damping rate as a function of lattice depth swith a fixed temperature T= 120nK.\nFIG. 6: The damping rate as a function of temperature with a fix ed lattice depth s= 1.0\nInFig.5,weplotthedampingrateasafunctionoflatticedepth swithafixedtemperature\nT= 120nK calculated from Eq. (13). We see that the magnitude of the damping rate is\n∼1s−1, which is in reasonable agreement with the experimental data [4]. We also see\nthat the damping rate increases with increasing lattice depth, which is consistent with the\nexperimental result [4]. This increase of the damping rate can be att ributed to the increase\nof the number of the elementary excitations because of a reductio n of the excitation energy\nwith increasing lattice depth.\nWe next investigate the temperature dependence of the damping r ate. In Fig. 6, we plot\nthedampingrateasafunctionoftemperaturewithafixedlatticede pths= 1.0. Wefindthat\n9damping rate decreases rapidly with decreasing temperature. This is due to the reduction\nof thermal excitations with decreasing temperature. Because of this strong temperature\ndependence of the damping rate, it is very difficult to make a quantita tive comparison with\nthe experimental data without precise knowledge of the experimen tal temperature.\nHerewecommentontheeffectofadiabaticloadingofanopticallattic eonthetemperature\nof a gas. The usual path for preparing for condensed Bose gases in an optical lattice consists\nof first forming a ultracold bosons in a weak magnetic trap, to which a 1D lattice potential\nis adiabatically applied by slowly rating up the light field intensity. In this c ase, the initial\nand final temperature are not usually equal since the energy spec trum changes during lattice\nloading [16]. We used the entropy-temperature curves to consider the effect of the adiabatic\nloading into a lattice. For the initial ( s=0) temperature T=120nK, the temperature shift in\nthe final state s= 1.6 is only about 2nK. Thus, one can ignore the effect of the adiabatic\nloading in shallow lattice regime.\nWe note that the radial excitations are important in Landau damping . If we calculated\nthe the damping rate ignoring the radial thermal excitations, the d amping rate would be\norder of magnitude smaller. This means that the radial thermal exc itations make significant\ncontributions to the Landau damping.\nIV. CONCLUSION\nIn this paper, we studied the Landau damping of dipole oscillations of B ose-condensed\ngases in a combined potential of highly-elongated harmonic trap and 1D optical lattice, with\nexplicitly including the effect of the radial excitations. While we treate d the condensate\nwavefunction only with the lowest radial mode, we took into account the radial excitations\nfor thermal cloud.\nFirst, we studied equilibrium properties of Bose-condensed gases in a combined harmonic\ntrap and 1D optical lattice potentials. We have presented a detailed calculation of the\ncondensate fraction in a 1D optical lattice at finite temperatures. We find the negative shift\noftheexcitationenergieswithincreasinglatticedepth. Weobtainth edipole-modefrequency\nas a function of the lattice depth. The negative shift of the dipole-m ode frequency was also\nobserved in the experiment.\nSecond, we calculated Landau damping rate of dipole-modes with var ying lattice depth\n10and temperature. Result for the damping rate is consistent with ex perimented data. There-\nfore, the experimentally observed damping can be understood as L andau damping. We also\nshowed that the radial thermal excitations are important in both e quilibrium condensate\nfractions and Landau damping rate.\nACKNOWLEDGMENTS\nThis research was support by Academic Frontier Project (2005) o f MEXT.\n[1] For a recent review, see for example O. Morsch and M. Obert haler Rev. Mod. Phys. 78, 179\n(2006)\n[2]S. Burger et al., Phys. Rev. Lett. 86, 4447(2001)\n[3]F. S. Cataliotti et al., Science 293, 843(2001)\n[4]F. Ferlaino et al., Phys. Rev. A 66, 011604(2002)\n[5]M. Kr¨ amer et al., Eur. Phys. J. D 27, 247 (2003)\n[6]E. Zaremba, T. Nikuni and A. Griffin, J. Low. Temp. Phys. 116277 (1999)\n[7]S.Giorgini, Phys. Rev. A 57, 2949(1998)\n[8]Pitaevskii,L.P.and Stringari,S. Phys. Rev. Lett. 235, 398 (1997)\n[9]M. Guilleumas and L. P. Pitaevskii, Phys. Rev. A 61013602 (2000)\n[10]B. Jackson and E. Zaremba New J. Phys. 588(2003)\n[11]S. Tsuchiya and A. Griffin, Phys. Rev. A 70, 023611(2004)\n[12]A. M. Rey, et al., Phys. Rev. A 72, 033616(2005)\n[13]B. G. Wild, et al., Phys. Rev. A 73, 023604(2006)\n[14]E. Arahata and T. Nikuni J. Low Temp. Phys. 148, 345 (2007)\n[15]A. Griffin, Phys. Rev. B 53, 9341(1995)\n[16]P.B.Blakie and J.V.Porto, Phys. Rev. A 69, 013603(2004)\n[17]C. J. Pethich and H. Smith, Bose-Einstein Condensation in Dilu te Bose Gass (UNIVERSITY\nPRESS CAMBRIDGE)\n11"    },    {        "title": "0709.0539v2.Causal_sets_and_conservation_laws_in_tests_of_Lorentz_symmetry.pdf",        "content": "arXiv:0709.0539v2  [gr-qc]  24 Jun 2008Causal sets and conservation laws in tests of Lorentz symmet ry\nDavid Mattingly\nUniversity of New Hampshire∗\nMany of the most important astrophysical tests of Lorentz sy mmetry also assume that energy-\nmomentum of the observed particles is exactly conserved. In the causal set approach to quantum\ngravity a particular kind of Lorentz symmetry holds but ener gy-momentum conservation may be\nviolated. We show that incorrectly assuming exact conserva tion can give rise to a spurious signal\nof Lorentz symmetry violation for a causal set. However, the size of this spurious signal is much\nsmaller than can be currently detected and hence astrophysi cal Lorentz symmetry tests as currently\nperformed are safe from causal set induced violations of ene rgy-momentum conservation.\nI. INTRODUCTION\nThe search for a complete and experimentally veri-\nfied theory of quantum quantum gravity is one of the\nmost important open questions in physics today. Unfor-\ntunately, despite the efforts of numerous eminent physi-\ncists, we do not yet have a theoretically complete model\nfor quantum gravity. Since the natural energy scale of\nquantum gravity is the Planck scale ( MP= 1019GeV)\nit is also extremely difficult to perform direct experi-\nments that support one candidate model for quantum\ngravity over another. Fortunately, various ideas about\nquantum gravity have suggested that the defining sym-\nmetry of special relativity, Lorentz symmetry, is not an\nexact symmetry even at low energies. In string theory\nthis can occur perhaps by tensor VEV’s [1] or noncritical\nstrings [2, 3] while in loop quantum gravity the ultimate\nfate of Lorentz symmetry and how it’s implemented is an\nopenquestion(c.f. [4, 5, 6, 7]). Emergentspacetimemod-\nels, for example analog spacetime [8], contain Lorentz\nsymmetry violation (LV) at high energies. Canonical\nnoncommutative field theories also contain Lorentz vi-\nolation [9, 10, 11] although they are invariant under the\ntwisted Poincare group [12]. Note that there are also\nnoncommutative models where Lorentz invariance is de-\nformed rather than explicitly violated (see [13, 14] and\nreferences therein).\nOf course any violation of Lorentz invariance must be\nvery, very small and therefore for any model with LV\nthere is a severe hierarchy problem. For example, in an\neffective field theory context, mass dimension three op-\nerators that generate Lorentz symmetry violation must\nbe less than 10−32GeV while the dimension four oper-\nators are constrained at the level of 10−28[15]. These\nextreme bounds mean that any Lorentz violating theory\nmust answer the question of why we have such good ap-\nproximate low energy Lorentz invariance. One could fine\ntune operators, but this is unnatural. In an attempt to\navoidthis, manyauthorshavelookedathigherdimension\noperators where the magnitude of the operator is sup-\npressed by MPand so is “hidden”. However, standard\n∗Electronic address: davidmmattingly@comcast.neteffective field theory arguments require that the higher\ndimension operators dimensionally transmute into lower\ndimension ones [16]. This means that dimension five and\nsix operators are constrained at the same order as the\ndimension three and four operators and so LV alone is\nextremely unlikely. It is possible to naturally suppress\nthe renormalizable operators by introducing new physics\nat scales just above that currently accessible by particle\ndetectors. For example, imposing supersymmetry [17]\nas well as Lorentz violation stabilizes the generation of\nlower dimension operators at the SUSY breaking scale.\nIf the SUSY breaking scale is given by mSUSY, and the\noriginal higher dimension operators are suppressed by\nMP, then SUSY will naturally suppress the dimension\nthree and four operators to the level of m2\nSUSY/MPand\nm2\nSUSY/M2\nPrespectively. For mSUSY<100 TeV, this\ngives a suppression to a level of approximately 10−9GeV\nand 10−28. The dimension three operators are still in\nconflictwithobservationalbounds, butthesecanbeelim-\ninated by further imposing CPT invariance. This would\nmake a model with low energy SUSY and CPT invari-\nance and LV sourced by Planck scale quantum gravity\nviable, but requires significant new low energy physics as\nwell (in this case SUSY and the assumption of CPT).\nThere is, of course, nothing a priori wrong with intro-\nducing new low energy physics and SUSY certainly has\na number of other compelling features unrelated to LV.\nIf we restrict ourselves just to the question of LV, how-\never, then Occam’s razor suggests that adding twonew\nphysical ideas is disfavored. Hence, instead of either fine\ntuning or imposing new low energy physics in addition\nto LV, it would be better perhaps to have a quantum\ngravity theory that respects Lorentz invariance exactly.\nThe discrete model we will discuss for the rest of this\npaper, causal sets, is singled out among discrete models\nas it is constructed to be Lorentz invariant by definition.\nRecently, Sorkin, Bombelli, and Henson [18] proved that\na causal set is Lorentz invariant for an abstract operator\nthat represents a measurement of a preferred frame for a\nsection of the causal set (the proof is that such operators\ncannot exist). In this workwe arguethat this operator D\ndoes not quite reflect the way that we currently analyze\nmany real Lorentz violating experiments and that such\nexperiments (in particular astrophysical tests) may the-\noretically show “spurious” Lorentz violating effects if the2\nunderlying spacetime is a causal set. However, we also\nshow that any effect is much smaller than our current\nexperimental sensitivities and can be safely ignored.\nThe essence of our argument is that swerves, a hypo-\nthetical effect on particle propagation in causal set the-\nory [19], can mimic a signal of LV in time of flight or\nthreshold astrophysics experiments where the expected\nflux of incoming high energy particles is very low. This\nmay seem strange, as causal sets are supposed to not\ngive any LV signal. However, swerves violate energy-\nmomentum conservation, which we usually assume holds\nexactly in the vacuum when we analyze LV experiments.\nThis mismatch between theory and assumption leads to\na spurious LV signal. Violations of energy-momentum\nconservation are tightly constrained from cosmology and\nthe constraints are strong enough that this effect is irrel-\nevant at current sensitivities in LV experiments but may\nnot be in the future. Furthermore, other ideas about\nquantum gravity have raised the spectre of a small vi-\nolation of energy-momentum conservation for particles\ntraveling in the vacuum(c.f [20, 21, 22]), so exploring\nhow this changes LV experiments is required if we are to\nanalyze the experiments properly.\nAn added benefit to this study is that, for a population\nof particles, the violation of energy-momentum conserva-\ntion predicted from causal sets satisfies a low energy dif-\nfusion equation in energy-momentum space. This equa-\ntion is unique, and therefore our result implies that we\ncan rather generically neglect the effects of any stochas-\ntic, Lorentzinvariantviolationofenergy-momentumcon-\nservation in Lorentz symmetry tests, whether or not we\nbelievein causalsets. Hencewhile causalsetsarethe mo-\ntivation, the results apply more broadly (although causal\nsets are the only model at this point that hypothesizes\nsuch an effect). To put it differently, assuming energy-\nmomentum conservation is not a priori warranted when\nsearching for quantum gravity induced LV. However, the\nconstraints on Lorentz invariant conservation violation\nare so tight that we can safely ignore it at our current\nlevel of sensitivity in LV tests, independent of whatever\nthe fundamental theory turns out to be.\nThroughout the following discussion we choose metric\nsignature −2 and units such that /planckover2pi1=c= 1.\nII. CAUSAL SETS AND THE POINCARE\nGROUP\nBefore we can discuss the fate of the Poincare group\nand conservation laws in the context of causal sets, we\nneed to know what exactly a causal set is. At its basic\nlevel, a causal set is a partially ordered set (a poset),\nconsisting of ‘points’ x,y,...and relations x≺ywhich\nencode causal ordering, i.e. x≺yimpliesxis to the\npast ofy. The ordering obeys two other rules, x≺y,\ny≺z⇒x≺zandx/ne}ationslash≺x. The first is simple transitivity\nsuch that if xis in the past of yandyis in the past\nofz,xhad better also be in the past of z. The othercondition forbids closed causal curves. Causal sets are\nusually considered to be finite, so for any ordered pair\n{x,y}withx≺y, there are a finite number of points z\nthat satisfy x≺zandz≺y. While it is certainly not\nnecessaryfor a spacetime to be present for a causal set to\nbe defined, a useful picture is to think of a causal set as\na random lattice “sprinkled” in a Lorentzian spacetime\nwhere there is an average of one point per every volume\nV0(which will be assumed here to be a Planck volume\nL4\nPl). The causal set sprinkling therefore approximates\nM. For more information on causal sets, see [23] and\nreferences therein.\nIt has been argued that an individual causal set is\nLorentz invariant [18], even locally, in a particular sense:\nthere is no experiment that can assign an intrinsic vio-\nlation of Lorentz invariance to a causal set at any point.\nHere “intrinsic violation” means a frame that depends\nonly on the sprinkling. We now briefly outline the proof\nin [18], restricting ourselves to the simplest type of LV -\na preferred frame. The question is then, can a causal set\ndefine such a frame? Certainly a regular discrete struc-\nture for spacetime does, but a causal set is a random\nstructure where the points are distributed in spacetime\nfrom a Lorentz invariant probability distribution. A pre-\nferred frame is specified by the existence, in vacuum, of a\nunit future pointing timelike vector field uaeverywhere\non the spacetime manifold M. Consider now a subset\nΩ of a causal set sprinkling that approximates M. An\nexperiment that assigns a LV preferred frame can be rep-\nresented as an operator Dthat maps Ω to the unit hy-\nperboloid of future timelike vectors H, i.e.ua=D◦Ω.\nuatransforms as an ordinary four vector under a Lorentz\ntransform Λ. Dmust not posses any intrinsic preferred\ndirection(to ensurethat anyLVcomesfrom Ωitself) and\nthereforeDmust commute with Λ so that DΛΩ = Λua.\nThe proof is then that no such operator Dcan exist and\nthere is no experiment that can assign a preferred frame\nto a local patch Ω of a causal set. The implication is that\nno LV experiment can ever assign a preferred frame to a\ncausal set intrinsically.\nThis result, on its face, is incompatible with another\nhypothesized effect of causal sets, a violation of transla-\ntion invariance due to the so-called “swerve” effect [19].\nThe swerve effect, which we discuss in more detail in\nthe next section, manifests itself at low energies via a\nLorentz invariant diffusion equation in momentum space.\nAn initial collection of particles with some momentum\ndistribution ρ(p) therefore evolves into a different state\nover (proper) time, which obviously violates translation\ninvariance. The immediate question is then, how does\nthis not directly yield LV? After all, translation violation\nallows us to immediately define a frame by taking, for\nexample, the gradient of the evolving quantity as the op-\neratorD. The answer is that the frame so defined is not\nintrinsic tothecausalset, butalsoinvolvestheinitialdis-\ntributionρ(p). In particular, if ρ(p) is a Lorentz invari-\nant function (which is unrealistic physically but useful\nfor this discussion) it is preserved by the diffusion equa-3\ntion. Hence there is neither a violation of translation or\nLorentz invariance in this case. If, instead, ρ(p) is not\nLorentz invariant then there can be translation invari-\nance violation, but both it and the corresponding LV are\nfunctions ofthe causalset andthe initial LVdistribution.\nThere is a “signal” of LV, but it is not the intrinsic LV\nwhich is forbidden by the proof outlined above because\nour operator Dcontains a preferred direction, which vi-\nolates one of the assumptions in the proof.\nThe difference between these two types of LV signal -\na real signal intrinsic to the underlying spacetime vs. a\nspurious signal due to the combination of the preferred\nframe of an experiment with another property of space-\ntime (in this case swerves) is a possibility which has not\narisen before in analysis of LV experiments. Since all of\nour observations and experiments that search for LV are\nnot LI in and of themselves (as they are local and hap-\npen in a particular frame) we must consider it. Hence\nthe question of LV in causal sets for a practical experi-\nment is not quite as straightforward as presented in [18].\nThe particular example we examine in this paper is how\ntheassumption of translation invariance, which is cus-\ntomary in LV experiments, can generate a spurious LV\nsignal from a causal set.\nTranslation invariance is a bad assumption due to the\nswerve effect, as we a) single out a preferred Killing vec-\ntor which is not an intrinsic property of the causal set\nitself and b) neglect momentum space diffusion due to\nswerves. The most common astrophysical tests of LV\nare time of flight tests, where we compare the arrival\ntime of two high energy particles, and anomalous scat-\ntering/decay phenomena, where we look for a modifica-\ntion to the scattering amplitude caused by LV dispersion\nrelations for the in/out free particle states. As will be-\ncome clear below, when analyzing these phenomena we\nmustassume something about translationinvariance and\nconservation of energy and momentum in order to put\nconstraints on any possible LV. Usually, translation in-\nvariance has been assumed to hold exactly in flat space,\nasthis is most consistentwith astraightforwardfield the-\noretical approach if the background LV tensor fields are\nconstant.1With causal sets, this assumption isn’t right\nand we must verify that a LV signal is really due to LV\nand not the swerve effect.\n1There are two major exceptions to this, the doubly special re la-\ntivity program (see [13] for an introduction) which deforms both\nthe translational and Lorentz subgroups of the Poincare gro up,\nand spacetime foam ideas which couple LV to a fluctuating dis-\npersion term [24, 25]. These approaches, however, have a sig nif-\nicant a priori modification of bothLorentz invariance and trans-\nlation invariance which makes them distinct from the causal set\napproach.A. Swerves\nConsider the intuitive picture for swerves in [19], that\nof a classical particle propagating on a random space-\ntime lattice with mass mand velocity v. The particle\nis constrained to move from point to point, which poses\na problem as generically there is no probability of a lat-\ntice point lying on the future worldline of the particle.\nThe particle must therefore “swerve” slightly so that it\nremains on the lattice and slightly change its velocity. A\nchange in velocity is equivalent to the particle moving\nto a different point on its mass shell, which is assumed\nto be unchanged since causal sets are Lorentz invariant.\nThe net result of swerving is that a collection of parti-\ncles initially with an energy-momentum distribution ρ(p)\nwill diffuse in momentum space along their mass shell\naccording to the unique Lorentz invariant diffusion equa-\ntion [19, 26],\n∂ρ\n∂τ=k∇2\nPρ−m−1pµ∂µρ. (1)\nHerekis the diffusion constant, ∇2\nPis the Laplacian\nin momentum space on the mass shell of the particle, τ\nis the proper time, ∂µis an ordinary spacetime deriva-\ntive, andmis the mass. While the underlying classi-\ncal picture is almost certainly incorrect,(1) is the unique\nLorentz invariant diffusion equation. Therefore if there\nis any type of stochastic violation of Poincare invariance\ndue to a random discrete structure underlying spacetime\na la causal sets, at lowest order it should be described by\n(1). There is a very tight limit on kfor neutrinos from\ncosmology[27], k<10−61GeV3, which comes from limits\non the amount relic neutrinos can contribute to hot dark\nmatter (and hence how much energy they can gain from\nswerves). While theoretically each particle species could\nhave a different k, it would be rather unnatural if they\nwere too far apart, especially as the source of the diffu-\nsion is supposed the same discrete spacetime structure.\nTherefore we shall take k<10−61GeV3to be our rough\nconstraint for all particles. As it turns out, any value\nofkeven close to 10−61GeV3makes energy-momentum\nconservation violation irrelevant for LV searches, so this\nis a perfectly safe assumption.\nEnergy is bounded below and so a particle initially at\nrest will increase its kinetic energy due to swerves. Since\nkis solow, we canmake asimplifying assumptionfor any\ncollection of particles that are not of cosmological age.\nIn the initial rest frame of the particle the swerves can\nfirst be treated in the non-relativistic limit for a certain\nperiod of time that depends on k. This is obvious as over\ntime energy is being added to the particle via swerves,\nbut as long as the total energy is less than the rest mass\nthe appropriate limit is the non-relativistic one. In the\nnon-relativistic limit, (1) simplifies to\n∂ρ\n∂t=k∇2\nPρ (2)\nwheretis now the coordinate time and ∇2\nPis the stan-4\ndard Laplacian operator on momentum space R3. The\nsolution for a collection of particles all initially at rest\nhas been derived in [27] and is given by\nρ(p) = (4πkt)−3/2e−p2\n4kt (3)\nwhich is the thermal distribution for a non-relativistic\ngas at temperature T= 2kt/m.\nWe can now ask how long a collection of initially cold\nparticles must exist for the non-relativistic approxima-\ntion to break down. This happens when the temperature\nis roughly equal to the mass, which gives t≈m2/(2k).\nFor a neutrino of mass 10−1eV, the approximation\nbreaks down after 1017seconds, while for electrons and\nprotons the approximation is always good as the break-\ndown time is far longer than the age of the universe.\nAfter a population becomes relativistic it will still gain\nenergybut notasquicklysincethepropertime isshorter.\nTherefore we know that the energy gained (per neutrino)\nby a population of initially cold neutrinos over the life-\ntime of the universe (also approximately 1017seconds) is\nno more than about the rest massof the neutrino and the\nenergy gain for other species is far less. This neglects the\neffects of cooling due to expansion, etc. which are dealt\nwith in [27], however we can ignore these secondary ef-\nfects for our purposes. Now consider a population of\nparticles with a high gamma factor γthat have traveled\nto earth from a source at one Gpc. The lifetime of the\nparticle in our frame Ois 109years = 1016seconds. How-\never, the time in the (initial) rest frame O′of the particle\nis much shorter, 1016/γseconds. Since any particle can\ngain no more than mνin energy over 1017seconds and\nthe propagation time in O′is much shorter, very little\nenergy is gained in O′due to swerves. Hence in O′the\nparticles are all still very non-relativistic once we include\nthe swerveeffect aslongas γ≫1and the non-relativistic\napproximation is valid for their entire lifetime.\nOf particular interest is the averagedeviation from the\ninitial energy Eiof a particle. If we define ∆ E/Ei=\n|(Ef−Ei)|/Ei, whereEfis the energy at time of mea-\nsurement, then from the discussion above we know that\n∆E/Eiis at least less than γ−1(on average) for any\nspecies. We can see this more explicitly by noting from\nabove that for a neutrino with a lifetime of the age of\nthe universe at rest in our frame, the total energy gain is\nat best the mass of the neutrino. If instead the neutrino\nis boosted with respect to our frame, the proper time is\nreduced by a factor of γ−1and so the total energy gain\nis reduced also by γ−1from its initial energy Ei=γm.\nThereforeforneutrinos the ratio∆ E/Ei≤γ−1. ForTeV\nand above astrophysical neutrinos, which we are primar-\nily interested in, ∆ E/Ei≤γ−1= 10−13at worst. The\nswerverateisslowerforotherspecies, sothisboundholds\nforthemaswell. Thisgainisverysmall,howeverLVtests\ncan be sensitive to fractional changes in energy of order\n10−28so it is not a priori obvious that swerves are irrele-\nvant. We now turn to how this diffusion affects these LV\ntests.III. SWERVES AND ASTROPHYSICAL TESTS\nOF LV\nA. Time of flight\nTime offlight tests areperhaps the simplest type ofLV\nexperiment. In these experiments one looks for delays in\nthe arrival time of high energy particles from distant as-\ntrophysical events. A time of flight experiment compares\nthe arrival time of at least two high energy particles and\nthe time delay between arrivals can be caused by three\ndistinct effects. The first is source effects - the particles\nare not produced at the same time or location in the as-\ntrophysical event. The source delay can be quite long, in\nthe case of neutrinos from gamma ray bursts the delay\ntime of a neutrino associated with the burst can be days.\nThe second type of delay is detector response. These de-\nlaysareusuallysmallandknownandwewill notconsider\nthem further. Finally, an unexplained time delay is usu-\nally considered evidence that the speed of propagation\nof the high energy particles is different than the speed\nof light. This is the “interesting” signal of a violation of\nLorentz invariance.\n1. Time of flight in field theory\nIt will be useful to discuss in detail how these experi-\nments work in a veryconcrete and established framework\nfirst, before we consider the causal set scenario, so let us\nanalyze time of flight in a field theory context first. LV\noccurs when fields are coupled in vacuum to a non-zero\ntensor field other than the metric. If there exists a pre-\nferred frame in nature, the fields couple to a unit time-\nlike vectoruawhich describes the preferred frame. There\nare many ways a field could couple in both the matter\nand gravitational sectors [28, 29, 30, 31], for a review of\nboth the renormalizable and non-renormalizable opera-\ntors see [32]. The free field mass dimension five couplings\nand below are already tightly constrained, so we consider\nhere an unconstrained operator - that of a dimension six\nCPT even operator. While terms like this may be able\nto be tested in ultrahigh energy neutrino observatories in\nthe future and hence are intrinsically relevant, we choose\nthis term for another reason: if the swerve effect is ir-\nrelevant for tests of this term it is certainly irrelevant\nfor any LV time of flight test we could conceivably per-\nform in the near future. The dimension six CPT even LV\nmodification to the kinetic term for a fermion is\nLf=ψ(i/∂−m)ψ−i\nE2\nPlψ(u·∂)3(u·γ)(αLPL+αRPR)ψ\n(4)\nwherePR,Lare the usual right and left chiral projec-\ntion operators and αR,Lare coefficients. Usually αR,L\nare assumed to be O(1). We choose the operator to be\nsuppressed by the Planck scale EPlas this would be the\nnatural scale if the term was generated by some theory5\nof quantum gravity at EPl.\nThe Hamiltonian corresponding to (4) commutes with\nthe helicity operator, hence the eigenspinors of the modi-\nfied Dirac equation will also be helicity eigenspinors. We\nnow solve the free field equations for the positive fre-\nquency eigenspinor ψ. Assume the eigenspinor is of the\nformψse−ip·xwhereψsis a constant four spinor and\ns=±1 denotes positive and negative helicity. Then the\nDirac equation becomes the matrix equation\n/parenleftBigg\n−m E −sp−α(6)\nRE3\nE2\nPl\nE+sp−α(6)\nLE3\nE2\nPl−m/parenrightBigg\nψs= 0 (5)\nThe dispersion relation, given by the determinant of (5),\nis\nE2−(α(6)\nRE3)(E+sp)\n−(α(6)\nLE3)(E−sp) =p2+m2(6)\nwhere we have dropped terms quadratic in α(6)\nR,Las they\nare small relative to the first order corrections for those\nterms.\nAtE≫m, as appropriate for high energy astrophys-\nical particles, the helicity states are almost chiral, with\nmixing due solely to the particle mass. Note also that at\nenergiesE >>m , we can replace Ebypat lowest order,\nwhich yields the approximate dispersion relation\nE2=p2+m2+2αR,Lp4\nE2\nPl. (7)\nPositive coefficients correspond to superluminal propa-\ngation, i.e. v=∂E/∂p > 1, while negative coefficients\ngive subluminal propagation. In either case, the speed\nof astrophysical particles does not asymptote to cas the\nenergy increases.\nThe group velocity ∂E/∂pis\nv= 1−m2\n2p2+3αR,Lp2\nE2\nPl= 1+∆v. (8)\nFor a source at distance dfrom earth the arrival time\ntaof a high energy particle is ta=d/v. The difference\n∆tLVbetween a light pulse emitted from the source at\nthe same time as the particle is\n∆tLV=tlight−ta=d−d\nv=d(v−1\nv)≈d∆v(9)\n=d/parenleftbig\n−m2\n2E2+3αR,LE2\nE2\nPl/parenrightbig\nwhere we have replaced pbyEin the high energy limit.\nTheαR,Lterm in (9) grows with energy. In order to\nestablish the best possible constraints we therefore need\nto look at the highest energy particles. The best chance\nwehaveinatimeofflightexperimenttoseeLVatthis or-\nder is in the comparisonofthe arrivaltime ofhigh energy\nneutrinos produced by GRB’s with the prompt emissionarrival. The flux at these energies is very low which has\nboth positive and negative ramifications. On the neg-\native side we require large detectors like ICECUBE to\nsee any appreciable flux of ultra high energy GRB neu-\ntrinos. However the background flux is also very low.\nRecently, it has been proposed by Jacob and Piran [33]\nthat since the background is so low even the detection of\na single neutrino event days or weeks after a GRB can be\nassociated with the GRB and used to bound αR,Lin a\ntime of flight experiment. While this approach has other\nproblems, primarily long source delays [34] due to the\nGRB fireball mechanics, it raises an interesting question\nwith regards to causal sets - what happens in the swerve\npicture when there are very, very low statistics?\n2. Time of flight with swerves\nIn time of flight experiments with large fluxes, swerves\naren’t a problem. For a strong multiparticle signal the\naveragearrival time is still that as predicted by special\nrelativity. However, for a single particle signal with mea-\nsured energy Ethere is no concept of averaging and the\narrival time will not be that predicted by special rela-\ntivity. The reason is simple - during propagation the\nparticle’s energy is not Eand the particle’s velocity is\nnotv= 1−m2/(2E2). Therefore to conclusively ascribe\na time delay to a LV dispersion as in (9) we need to make\nsure the same delay cannot be due to swerves.\nWe can overestimate the time of flight delay for a typi-\ncal particle compared to special relativity by considering\na particle propagating with energy Ef+∆E, whereEf\nis the measured (final) energy and ∆ Eis the average\ndeviation for a single particle introduced previously. It\nis an overestimate since we apply the energy difference\nover the entire propagation of the particle. The disper-\nsion relation is simply the relativistic dispersion relation,\nso ∆tSis\n∆tS=d/parenleftbigg\n−m2\n2(Ef+∆E)2/parenrightbigg\n=d/parenleftBigg\n−m2\n2E2\nf(1+∆E/Ef)2/parenrightBigg\n(10)\n∆E/Ef<γ−1≪1 and we therefore have\n∆tS=d/parenleftBigg\n−m2\n2E2\nf(1−2∆E\nEf)/parenrightBigg\n. (11)\nComparing(11)with (9)weseethatanexperimentsensi-\ntive to LV at our chosen order is also sensitive to swerves\nif\n3αR,LE2\nf\nE2\nPl≈m2\nE2\nf∆E\nEf. (12)\nRewriting this equation for ∆ E/Efwe have\n∆E\nEf= 3αR,LE4\nf\nm2E2\nPl. (13)6\nThere is no intrinsic size to the LV term in (13) and\ndepending on how accurate a LV time of flight measure-\nment is, it could theoretically also probe swerves. How-\never, the duration of a long GRB can be of order 1000\nseconds and it is unclear when in the burst emission the\nneutrinos will occur. Therefore there is an intrinsic un-\nknown source delay of at least 1000 seconds2. Any sig-\nnificant time delay must be greater than this value and\nhence there is a lower limit on a meaningful ∆ t(swerve\nor LV induced). For a GRB with this value of ∆ tand a\ndistance of 1 Gpc, ∆ t/d≈10−14seconds. With (9) this\nestablishes a lower limit that the LV dispersion term of\n|αR,LE2\nf\nE2\nPl| ≥10−14(14)\nwhich must be satisfied if we are to see anything mean-\ningful in a time of flight GRB experiment. If we are con-\nservative and take our high energy neutrinos to be above\n1 TeV (actual proposed energies are much higher) and a\nneutrino mass of approximately 0.1 eV, the gamma fac-\ntor is 1013. From (13), ∆ E/Effrom swerves must then\nbe greater than 1012, which is 25 orders of magnitude\nlargerthanthe upper limit forswervingneutrinos. Hence\nswervesare completely and totally irrelevant. This result\ncan easily be generalized to other forms of LV dispersion\nand in none are astrophysical time of flight experiments\nsensitive to swerves by many orders of magnitude.\nB. Anomalous particle interactions\nAnomalous particle interactions are much more sensi-\ntive tests ofLV than time offlight tests and alsoof course\nrequire assumptions about energy-momentum conserva-\ntion. Therefore they will also be sensitive to the swerve\neffect at some level. There are two types of anomalous\ninteractions. The first type is when the interaction oc-\ncurs only in the LV model. The second type is when the\ninteraction begins to occur at a certain energy and this\nenergy is different for Lorentz invariant versus LV mod-\nels. We deal with each type of interaction separately as\nthe effect of swerves is different.3\n2The actual value is of course dependent on the exact mechanic s\nof the GRB and can be shorter. We use this value for illustrati ve\npurposes as an order of magnitude estimate.\n3We are not considering particle creation due to the time depe n-\ndence of the spectrum of an initial flux of particles due to swe rves\nor how to correctly define initial/final states in a model with out\nasymptotic translation invariance. These questions must a lso be\nanswered in regards to the swerve effect but are outside the sc ope\nof this paper.1. New particle interactions\nConsider a proton with dispersion relation like that\ngiven in (7). If αR,L>0 then at high energies protons\nbecome unstable and emit photons in what is known as\nthe “vacuum Cerenkov effect”. They rapidly lose energy\nvia this process and hence the existence of ultra high en-\nergy cosmic ray protons (for which this process cannot\nbe occurring) limits how positive αR,Lcan be. This is\nan example of a test that uses a particle interaction com-\npletely absent in usual Lorentz invariant physics to limit\npossible LV dispersion relations. The key to these tests\nis the energy-momentum conservation equations, which\ntell us how much parameter space is available for the\nreaction - zero in the Lorentz invariant case and non-\nzero with LV. Naively then, it seems reasonable that a\nfluctuating energy-momentum of the initial and/or final\nstates may allow for new reactions to also occur. We now\nshow that in the case of causal sets, this is not true. If\nwe consider just a swerve induced change in the energy-\nmomentum of the initial and final states then if a reac-\ntion does not occurin straightforwardspecial relativityit\ndoes not occur in causal sets. Note that in the following\ndiscussion we have implicitly assumed that the swerve\neffect for massless particles yields diffusion in a null cone\nin energy-momentum space (the m→0 limit of a mass\nshell).\nThe idea is very simple. Let us consider a generic\nmulti-particle reaction i1+i2+...→f1+f2+...whereia\nare the incoming particles and fbare the outgoing parti-\ncles. If the reaction does not occur in Lorentz invariant\nphysics it means that there is no solution to the conser-\nvation equation\npµ\n1+pµ\n2+...=qµ\n1+qµ\n2+... (15)\nwherepµ\na(theincoming4-momenta)and qµ\nb(theoutgoing\n4-momenta) are subject to the on-shell constraints p2\na=\nm2\na,q2\nb=m2\nb. In a LV theory the on-shell constraints\nchange, in causal sets they do not. In a causal set, the\nincoming and outgoing momenta are, however, modified\nby a fluctuation term ∆ pa,∆qbas the particle will swerve\nduringthe courseofthe interaction. We thereforerewrite\nthe conservation equation in the causal set approach as\npµ\n1+∆pµ\n1+pµ\n2+∆pµ\n2+...=qµ\n1+∆qµ\n1+qµ\n2+∆qµ\n2...(16)\nsubject to the on-shell constraints ( pa+ ∆pa)2=\nm2\na,(qb+ ∆qb)2=m2\nbsince the fluctuations must keep\nall particles on-shell. This though is just a relabeling of\nmomenta and doesn’t change any physics - we can define\nnew momenta ¯ pµ\na=pµ\na+∆pµ\naetc. and the conservation\nand constraint equations take the exact same form as be-\nfore. Therefore there is still no solution and the reaction\ndoesn’t happen with swerves either.\nThere is a caveat to the above argument. In keeping\nwith other LV threshold tests, where one looks at only\ninitial and final states, we have not considered the ef-\nfect of swerves in the interaction region. This is more7\ndangerous for causal sets, as field theory on causal sets\nis not well developed enough to know what the swerve\neffect might do to virtual states that are not necessarily\non-shell. Hence this conclusion can not be considered ab-\nsolutely concrete until we understand more of quantum\nfield theory on a causal set. However, since no LV reac-\ntions have been seen to date it is likely that causal set\nQFT does respect LI to a very good approximation even\nwhen quantum effects are taken into account.\n2. Modification of existing energy thresholds\nThe situation is different for reactions where there is\na Lorentz invariant solution. Here, the swerve effect can\nchange the energy the reaction occurs at, although the\nshift is tiny. Let us take a specific reaction, pion pro-\nduction by proton-photon scattering, p+γ→p+π0.\nThis reaction is important in LV tests as the high en-\nergy cosmic ray spectrum should exhibit a cutoff (the\nGreisen-Zapsetin-Kuzmin cutoff) around 5 ×1019eV due\nto the scattering of cosmic ray protons off the cosmic mi-\ncrowave background. LV tests can shift this cutoff, and\nthe recent confirmation of the GZK cutoff by HiRes [35]\nand the Pierre Auger observatory [36] constrains some\nLV models. Again, we have a limited number of events\n(although with Auger the statistics are getting rapidly\nbetter) and so one might wonder if the random nature of\nswerves can cause a spurious signal. Theoretically this is\ntrue, but the effect of swerves on GZK protons is negli-\ngible, even if we wildly overestimate the length of time\nswerves have to act. A GZK proton has a gamma fac-\ntor of near 1010, which means that its maximum proper\nlifetime is 1017/γ= 107seconds if it was generated very\nearly in the universe. The maximum amount of energy\ntheprotoncangaininitsinitialrestframe,usingourcon-\nstraintfork, is 10−30GeV, whichin turn implies that the\nshift in energy possible for a GZK proton in our frame is\n10−20GeV. The GZK cutoff will hence be broadened byat best the insignificant amount of 10−20GeV and there-\nfore swerves are irrelevant in the GZK reaction. This\ntype of analysis can be done for many different thresh-\nold reactions, for example the scattering of high energy\nTeV photons off the infrared background. In all cases\nthe limits on kare strong enough that swerves have no\nappreciable effect.\nIV. CONCLUSION\nLV searches in astrophysics rely not only on assump-\ntions about the nature of the LV model being explored,\nbut also on energy-momentum conservation of the parti-\ncles involved. In this paper we have argued that causal\nsets, while intrinsically Lorentz invariant, can technically\nintroducespuriousLVsignalsifwedealwith practicalex-\nperiments and make the usual assumption that energy-\nmomentum is conserved due to the swerve effect. How-\never, we have also shown that existing limits on swerves\nimply that any errors introduced are negligible at cur-\nrent experimental sensitivity. A bonus is that the dif-\nfusion equation (1) that describes the swerve effect is\nthe unique low energy Lorentz invariant diffusion equa-\ntion. Any statistical process that respects Lorentzinvari-\nancebut causesfluctuations in energy-momentumshould\nthereforebe described bythis equation andwe canrather\ngenerically conclude that any signal of LV in a time of\nflight or anomalous particle interaction experiment can-\nnot be due instead to a Lorentz invariant modification of\ntranslation invariance.\nV. ACKNOWLEDGEMENTS\nWe thank Stefano Liberati for useful comments on a\ndraft of this paper.\n[1] V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683\n(1989).\n[2] J. R. Ellis, N. E. Mavromatos and D. V. Nanopoulos,\narXiv:gr-qc/9909085.\n[3] N. E. Mavromatos, arXiv:0708.2250 [hep-th].\n[4] R. Gambini and J. Pullin, Phys. Rev. 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Yamamoto [Pierre Auger Collaboration],\narXiv:0707.2638 [astro-ph]."    },    {        "title": "1010.3990v1.Lorentz_Symmetry_and_Matter_Gravity_Couplings.pdf",        "content": "arXiv:1010.3990v1  [hep-ph]  19 Oct 20101\nLORENTZ SYMMETRY AND MATTER-GRAVITY\nCOUPLINGS\nJAY D. TASSON\nDepartment of Physics, Whitman College\nWalla Walla, WA 99362, USA\nE-mail: tassonjd@whitman.edu\nThis proceedings contribution summarizes recent investig ations of Lorentz vi-\nolation in matter-gravity couplings.\n1. Introduction\nIn spite of the many high-sensitivity investigations of Lorentz violat ion1\nperformed in the context of the fermion sector of the minimal Stan dard-\nModel Extension (SME) in Minkowski spacetime,2only about half of the\ncoefficientsforLorentzviolationinthatsectorhavebeeninvestiga tedexper-\nimentally. Reference 3 establishes a methodology for obtaining sens itivities\nto some of these open parameters by considering gravitational co uplings in\nthe fermions sector of the SME,4extending pure gravity work.5Of partic-\nular interest are the aµcoefficients for baryons and charged leptons, which\nare unobservable in principle in Minkowski spacetime, but could be rela -\ntively large due to gravitational countershading.6\nThe first half of Ref. 3 develops the necessary theoretical result s for\nthe analysis of Lorentz violation in matter-gravity couplings. Those results\nare summarized in Sec. 2 below, while Sec. 3 summarizes the experimen tal\npredictions provided in the second half of that work.\n2. Theory\nThe theoretical portion of Ref. 3 addresses a number of useful c onceptual\npoints prior to developing the necessary results for experimental analysis.\nThis includes a discussion of the circumstances under which relevant types\nof Lorentz violation are observable in principle. It turns out that th eaµco-\nefficient,which canbe removedfromthe singlefermiontheoryinMinko wski2\nspacetime via a spinor redefinition cannot typically be removed in the p res-\nenceofgravity.4Thismakesit aninterestingcaseforstudyin theremainder\nof Ref. 3. A coordinate choice that can be used to fix the sector of the the-\nory that defines isotropy is also discussed and ultimately used to tak e the\nphoton sector to have ηµνas the background metric.\nAnother issue is the development of general perturbative techniq ues\nto treat the fluctuations in the coefficient fields in the context of ma tter-\ngravity couplings. Two notions of perturbative order are introduc ed. One,\ndenoted O( m,n), tracks the orders in Lorentz violation and in gravity,\nwhere the first entry represents the order in the coefficients for Lorentz\nviolation and the second represents the order in the metric fluctua tionhµν.\nThe secondary notion of perturbative order, denoted PNO( p), tracks the\npost-newtonianorder.ThegoalofRef. 3istoinvestigatedominan tLorentz-\nviolating implications in matter-gravity couplings, which are at O(1,1).\nReference 3 provides the necessary results to analyze experimen ts at a\nvariety of levels while workingtoward the classical nonrelativistic equ ations\nof motion, which are most relevant for many of the experiments to b e con-\nsidered. Development of the quantum theory of the gravity-matt er system\nprovides the first step. Starting from the field-theoretic action, the rela-\ntivistic quantum mechanicsinthe presenceofgravitationalfluctua tionsand\nLorentzviolationisestablishedafterinvestigatingmethods ofident ifying an\nappropriatehamiltonianinthe presenceofaneffectiveinversevierb einE0\nµ.\nThe explicit form of the relativistic hamiltonian involving all coefficients f or\nLorentz violation in the minimal QED extension is provided. Attention is\nsubsequently specialized to the study of spin-independent Lorent z-violating\neffects, which are governed by the coefficient fields ( aeff)µ,cµνand the met-\nric fluctuation hµν. Analysis then proceeds to the nonrelativistic quantum\nhamiltonian via the standard Foldy-Wouthuysen procedure.\nWhile the quantum mechanics above is useful for analysis of quan-\ntum experiments, most measurements of gravity-matter coupling s are per-\nformed at the classical level. Thus the classical theory7associated with\nthe quantum-mechanical dynamics involving nonzero ( aeff)µ,cµν, andhµν\nis provided at leading order in Lorentz violation both for the case of t he\nfundamental particles appearing in QED and for bodies involving many\nsuch particles. These results enable the derivation of the modified E instein\nequation and the equation for the trajectory of a classical test p article.\nSolving for the trajectory requires knowledge of the coefficient an d metric\nfluctuations. A systematic methodology for calculating this informa tion is\nprovided, and general expressionsfor the coefficient and metric fl uctuations3\nto O(1,1) in terms of various gravitational potentials and the backg round\ncoefficient values ( aeff)µandcµνare obtained. Bumblebee models are con-\nsidered as an illustration of the general results.\n3. Experiments\nA major class of experiments that can achieve sensitivity to coefficie nts\n(aeff)µandcµνinvolve laboratory tests with ordinary neutral matter. Tests\nof this type are analyzed via the PNO(3) lagrangian describing the dy nam-\nics of a test body moving near the surface of the Earth in the prese nce of\nLorentz violation. The analysis reveals that the gravitational forc e acquires\ntiny corrections both along and perpendicular to the usual free-f all trajec-\ntory near the surface of the Earth, and the effective inertial mas s of a test\nbody becomes a direction-dependent quantity. Numerous laborat ory exper-\nimentssensitiveto theseeffects areconsidered.The tests canbe classifiedas\neither gravimeter or Weak Equivalence Principle (WEP) experiments a nd\nas either force-comparison or free-fall experiments for a total of 4 classes.\nFree-fall gravimeter tests monitor the acceleration of freely fallin g ob-\njects and search for the characteristic time dependence associa ted with\nLorentz violation. Falling corner cubes8and matter interferometry9,10pro-\nvide examples of such experiments and are discussed in Ref. 3. Forc e-\ncomparison gravimeter tests using equipment such as supercondu cting\ngravimeters are also studied.11Note that the distinction, force comparison\nversesfree fall, is importantdue to the potentialLorentz-violatin gmisalign-\nment of force and acceleration. Making direct use of the flavor dep endence\nassociated with Lorentz-violating effects implies signals in WEP tests. A\nvariety of free-fall WEP tests are considered including those using falling\ncornercubes,12atom interferometers,10,13tossed masses,14balloondrops,15\ndrop towers,16and sounding rockets,17along with force-comparison WEP\ntests with a torsion pendulum.18For all of the tests considered, the pos-\nsible signals for Lorentz violation are decomposed according to their time\ndependence, and estimates of the attainable sensitivities are obta ined.\nSatellite-based WEP tests,19which offer interesting prospects for im-\nproved sensitivities to Lorentz violation, are also discussed in detail. The\nsignal is decomposed by frequency and estimated sensitivities are o btained.\nThe experimental implications of Lorentz violation in the gravitationa l\ncouplingsofchargedparticles,antimatter,andsecond-andthird -generation\nparticles are also studied. These tests are experimentally challengin g, but\ncan yield sensitivities to Lorentz and CPT violation that are otherwise dif-\nficult or impossible to achieve. Possibilities including charged-particle in -4\nterferometry,20ballistic tests with charged particles,21gravitational experi-\nmentswithantihydrogen,22andsignalsinmuoniumfreefall23arediscussed.\nSimple toymodelsareusedtoillustratesomefeaturesofantihydrog entests.\nSolar-system tests of gravity including lunar and satellite laser rang ing\ntests24and measurements of the precession of the perihelion of orbiting\nbodies25are also considered. The established advance of the perihelion for\nMercury and for the Earth is used to obtain constraints on combina tions of\n(aeff)µ,cµν, andsµν, which provides the best current sensitivity to ( aeff)J.\nA final class of tests involves the interaction of photons with gravit y.\nSignals arising in measurements of the time delay, gravitational Dopp ler\nshift, and gravitational redshift, are considered along with compa risons of\nthe behaviors of photons and massive bodies. Implications for a var iety of\nexisting and proposed experiments and space missions are consider ed.26\nExisting and expected sensitivities from the experiments and obser va-\ntionssummarizedabovearecollectedinTablesXIVandXVofRef.3.T hese\nsensitivities reveal excellent prospects for using matter-gravity couplings to\nseek Lorentz violation. The opportunities for measuring the count ershaded\ncoefficients ( aeff)µare particularly interesting in light of the fact that these\ncoefficients typically cannot be detected in nongravitational searc hes. Thus\nthe tests proposed in Ref. 3 offer promising new opportunities to se arch for\nsignals of new physics, potentially of Planck-scale origin.\nReferences\n1.Data Tables for Lorentz and CPT Violation, 2010 edition, V.A. Kosteleck´ y\nand N. Russell, arXiv:0801.0287v3.\n2. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n3. V.A. Kosteleck´ y and J.D. Tasson, arXiv:1006.4106.\n4. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n5. Q.G. Bailey and V.A. Kosteleck´ y, Phys. Rev. D 74, 045001 (2006); Q.G.\nBailey, Phys. Rev. D 80, 044004 (2009); arXiv:1005.1435.\n6. V.A. Kosteleck´ y and J.D. Tasson, Phys. Rev. Lett. 102, 010402 (2009).\n7. V.A. Kosteleck´ y and N. Russell, Phys. Lett. B 693, 443 (2010); N. Russell,\nthese proceedings, arXiv:1009.1383.\n8. I. Marson and J.E. Faller, J. Phys. E 19, 22 (1986).\n9. A. Peters, K.Y. Chung, and S. Chu, Nature 400, 849 (1999); J.M. McGuirk\net al., Phys. Rev. A 65, 033608 (2002); N. Yu et al., Appl. Phys. B 84, 647\n(2006); B. Canuel et al., Phys. Rev. Lett. 97, 010402 (2006); H. Kaiser et al.,\nPhysica B 385-386 , 1384 (2006).\n10. S. Dimopoulos et al., Phys. Rev. D 78, 042003 (2008).\n11. R.J. Warburton and J.M. Goodkind, Astrophys. J. 208, 881 (1976); S. Sh-\niomi, arXiv:0902.4081; L. Carbone et al., in T. Damour, R.T. Jantzen, and5\nR. Ruffini, eds., Proceedings of the Twelfth Marcel Grossmann Meeting on\nGeneral Relativity , World Scientific, Singapore, 2010.\n12. K. Kuroda and N. Mio, Phys. Rev. D 42, 3903 (1990); T.M. Niebauer, M.P.\nMcHugh, and J.E. Faller, Phys. Rev. Lett. 59, 609 (1987).\n13. S. Fray et al., Phys. Rev. Lett. 93, 240404 (2004).\n14. R.D. Reasenberg, in V.A. Kosteleck´ y, ed., CPT and Lorentz Symmetry II ,\nWorld Scientific, Singapore, 2005.\n15. V. Iafolla et al., Class. Quantum Grav. 17, 2327 (2000).\n16. H. Dittus and C. Mehls, Class. Quantum Grav. 18, 2417 (2001).\n17. R.D. Reasenberg and J.D. Phillips, Class. Q. Grav. 27, 095005 (2010).\n18. Y. Su et al., Phys. Rev. D 50, 3614 (1994); S. Schlamminger et al., Phys.\nRev. Lett. 100, 041101 (2008).\n19. P. Touboul et al., Comptes Rendus de l’Acad´ emie des Sciences, Series IV,\n2, 1271 (2001); T.J. Sumner et al., Adv. Space Res. 39, 254 (2007); A.M.\nNobiliet al., Exp. Astron. 23, 689 (2009); G. Amelino-Camelia et al., Exp.\nAstron.23, 549 (2009); P. Worden, these proceedings.\n20. F. Hasselbach and M. Nicklaus, Phys. Rev. A 48, 143 (1993); B. Neyenhuis,\nD. Christensen, and D.S. Durfee, Phys. Rev. Lett. 99, 200401 (2007).\n21. F.S. Witteborn and W.M. Fairbank, Phys. Rev. Lett. 19, 1049 (1967).\n22. G. Gabrielse, Hyperfine Int. 44, 349 (1988); N. Beverini et al., Hyperfine\nInt.44, 357 (1988); R. Poggiani, Hyperfine Int. 76, 371 (1993); T.J. Phillips,\nHyperfineInt. 109,357(1997); AGECollaboration, A.D.Cronin et al.,Letter\nof Intent: Antimatter Gravity Experiment (AGE) at Fermilab ,February2009;\nD. Kaplan, these proceedings, arXiv:1007.4956; J. Walz and T.W. H¨ ansch,\nGen. Rel. Grav. 36, 561 (2004); P. P´ erez et al.,Letter of Intent to the CERN-\nSPSC,November 2007; F.M. Huber, E.W. Messerschmid, and G.A. Smit h,\nClass. QuantumGrav. 18,2457 (2001); AEGISCollaboration, A.Kellerbauer\net al., Nucl. Instr. Meth. B 266, 351 (2008).\n23. K. Kirch, arXiv:physics/0702143; B. Lesche, Gen. Rel. G rav.21, 623 (1989).\n24. J.G. Williams, S.G. Turyshev, and H.D. Boggs, Phys. Rev. Lett.93, 261101\n(2004); T.W. Murphy et al., Pub. Astron. Soc. Pac. 120, 20 (2008).\n25. C.M. Will, Living Rev. Relativity 4, 4 (2001).\n26. B. Bertotti, L. Iess, and P. Tortora, Nature 425, 374 (2003); T. Appour-\nchauxet al., Exp. Astron. 23, 491 (2009); L. Iess and S. Asmar, Int. J. Mod.\nPhys. D 16, 2117 (2007); P. Wolf et al., Exp. Astron. 23, 651 (2009); B.\nChristophe et al., Exper. Astron. 23, 529 (2009); S.G. Turyshev et al., Int.\nJ. Mod. Phys. D 18, 1025 (2009); S.B. Lambert and C. Le Poncin-Lafitte,\nAstron. Astrophys. 499, 331 (2009); R. Byer, Space-Time Asymmetry Re-\nsearch, Stanford University proposal, January 2008; L. Cacciapuo ti and C.\nSalomon, Eur. Phys. J. Spec. Top. 172, 57 (2009); S.G. Turyshev and M.\nShao, Int. J. Mod. Phys. D 16, 2191 (2007); S.C. Unwin et al., Pub. Astron.\nSoc. Pacific 120, 38 (2008)."    },    {        "title": "1212.4448v4.Coupled_scalar_fields_Oscillons_and_Breathers_in_some_Lorentz_Violating_Scenarios.pdf",        "content": "arXiv:1212.4448v4  [hep-th]  27 Feb 2015Coupled scalar fields Oscillons and Breathers in some Lorent z Violating Scenarios\nR. A. C. Correa1,∗and A. de Souza Dutra2,†\n1CCNH, Universidade Federal do ABC, 09210-580, Santo Andr´ e , SP, Brazil\n2UNESP-Campus de Guaratinguet´ a, 12516-410, Guaratinguet ´ a, SP, Brazil\n(Dated: February 23, 2015)\nIn this work we discuss the impact of the breaking of the Loren tz symmetry on the usual oscillons,\nthe so-called flat-top oscillons, and the breathers. Our ana lysis is performed by using a Lorentz vio-\nlation scenario rigorously derived in the literature. We sh ow that the Lorentz violation is responsible\nfor the origin of a kind of deformation of the configuration, w here the field configuration becomes\noscillatory in a localized region near its maximum value. Fu rthermore, we show that the Lorentz\nbreaking symmetry produces a displacement of the oscillon a long the spatial direction, the same\nfeature is present in the case of breathers. We also show that the effect of a Lorentz violation in\nthe flat-top oscillon solution is responsible by the shrinki ng of the flat-top. Furthermore, we find\nanalytically the outgoing radiation, this result indicate s that the amplitude of the outgoing radia-\ntion is controlled by the Lorentz breaking parameter, in suc h away that this oscillon becomes more\nunstable than its symmetric counterpart, however, it still has a long living nature.\nKeywords: oscillons, breathers, Lorentz, symmetry, break ing, violation\n1. INTRODUCTION\nThe study of nonlinear systems is becoming an area of\nincreasing interest along the last few decades [ 1,2]. In\nfact, such nonlinear behavior of physical systems is found\nin a broad part of physical systems nowadays. This in-\ncludescondensedmatter systems, field theoreticalmodels,\nmodern cosmology and a large number of other domains\nof the physical science [ 3]-[28]. One of the reasons of this\nincreasing interest is the fact that many of those systems\npresent a countable number of distinct degenerate mini-\nmal energy configurations. In many cases that degenerate\nstructure can be studied through simple models of scalar\nfields possessing a potential with two or more degenerate\nminima. For instance, in two or more spatial dimensions,\none can describe the so called domain walls [ 4] connecting\ndifferent portions of the space were the field is at differ-\nent values of the degenerate minima of the field poten-\ntial. In other words, the field configuration interpolates\nbetween two of those potential minima. At this point, it\nis important to remark that a powerful insight to solve\nnonlinear problems analytically was introduced by Bogo-\nmolnyi, Prasad and Sommerfield [ 5,6]. In this case, the\nmethod shown by Bogomolnyi, Prasad and Sommerfield\nis now called of BPS approach, and it is based in ob-\ntaining a first-order differential equation from the energy\nfunctional. By using this method, it is possible to find so-\nlutions that minimize the energy of the configuration and\nthat ensure their stability.\nIn the context of the field theory it is quite common the\nappearance of solitons [ 7], which are field configurations\npresenting a localized and shape-invariant aspect, having\na finite energy density as well as being capable of keep-\n∗rafael.couceiro@ufabc.edu.br\n†dutra@feg.unesp.bring their shape unaltered after a collision with another\nsolitons. The presence of those configurations is nowa-\ndays well understood in a wide class of models, presenting\nor not topological nature. As examples one can cite the\nmonopoles, textures, strings and kinks [ 8].\nAn important feature of a large number of interesting\nnonlinear models is the presence of topologically stable\nconfigurations, which prevents them from decaying due\nto small perturbations. Among other types of nonlinear\nfield configurations, there is a specially important class of\ntime-dependent stable solutions, the breathers appearing\nin the Sine-Gordon like models. Another time-dependent\nfield configuration whose stability is granted for by charge\nconservation are the Q-balls as baptized by Coleman [ 9]\nor nontopological solitons [ 10]. However, considering the\nfact that many physical systems interestingly may present\na metastable behavior, a further class of nonlinear sys-\ntems may present a very long-living configuration, usually\nknown as oscillons. This class of solutions was discovered\nin the seventies of the last century by Bogolyubsky and\nMakhankov [ 29], and rediscovered posteriorly by Gleiser\n[30]. Thosesolutions,appearedinthestudyofthedynam-\nics of first-order phase transitions and bubble nucleation.\nSince then, more and more works were dedicated to the\nstudy of these objects [ 30]-[48].\nOscillons are quite general configurations and are found\nin inflationary cosmological models [ 30], in the Abelian-\nHiggsU(1) models [ 31], in the standard model SU(2)×\nU(1) [32], in axion models [ 33], in expanding universe sce-\nnarios [34] and in systems involving phase transitions [ 35].\nTheusualoscillonaspectistypicallythatofabellshape\nwhich oscillates sinusoidally in time. Recently, Amin and\nShirokoff [ 36] have shown that depending on the intensity\nof the coupling constant of the self-interacting scalar field,\nit is possible to observe oscillons with a kind of plateau\nat its top. In fact, they have shown that these new oscil-\nlons are more robust against collapse instabilities in three\nspatial dimensions.2\nAt this point it is interesting to remark that Segur and\nKruskal[ 37] haveshownthat the asymptoticexpansiondo\nnot represent in general an exact solution for the scalar\nfield, in other words, it simply represents an asymptotic\nexpansion of first order in ǫ, and it is not valid at all\norders of the expansion. They have also shown that in\none spatial dimension they radiate [ 37]. In a recent work,\nthe computation of the emitted radiation of the oscillons\nwas extended to the case of two and three spatial dimen-\nsions [38]. Another important result was put forward by\nHertzberg [ 39]. In that work he was able to compute the\ndecayingrateofquantizedoscillons,anditwasshownthat\nits quantum rate decayis verydistinct of the classicalone.\nOn the other hand, some years ago, Kostelecky and\nSamuel [ 49] started to study the problem of the Lorentz\nandCPT(charge conjugation-parity-time reversal) sym-\nmetry breaking. This was motivated by the fact that\nthe superstring theories suggest that Lorentz symmetry\nshould be violated at higher energies. After that seminal\nwork, a theoretical framework about Lorentz and CPT\nsymmetry breaking has been rigorously developed. As an\nexample, the effects on the standard model due to the\nCPTviolation and Lorentz breaking were presented by\nColladay and Kostelecky [ 50]. Recently, a large amount\nof works considering the impact of some kind of Lorentz\nsymmetry breaking have appeared in the literature [ 51]-\n[66]. As one another example, recently Belich et. al.[52]\nstudied the Aharonov-Bohm-Casher problem with a non-\nminimal Lorentz-violating coupling. In that reference the\nauthorshaveshown that the Lorentz-violationis responsi-\nblebytheliftingoftheoriginaldegeneraciesintheabsence\nof magnetic field, even for a neutral particle.\nBy introducing a dimensional reduction procedure to\n(1+2) dimensions presented in Ref. [ 53], Casana, Car-\nvalho and Ferreira applied the approach to investigate the\ndimensional reduction of the CPT-even electromagnetic\nsector of the standard model extension. Another impor-\ntant work was presented by Boldo et al.[54], where the\nproblem of Lorentz symmetry violation gauge theories in\nconnection with gravity models was analyzed. In a very\nrecentwork, Kosteleckyand Mewes[ 55], alsoanalyzedthe\neffects of Lorentz violation in neutrinos.\nIn recent years, investigations about topological defects\nin the presence of Lorentz symmetry violation have been\naddressed in the literature [ 56]-[58]. Works have also been\ndone on monopole and vortices in Lorentz violation sce-\nnarios [59]. For instance, in Ref. [ 59], a question about\nthe Lorentz symmetry violation on BPS vortices was in-\nvestigated. In that paper, the Lorentz violation allows a\ncontrol of the radial extension and of the magnetic field\namplitude of the Abrikosov-Nielsen-Olesen vortices.\nIn fact, Lorentz invariance is the most fundamental\nsymmetry of the standard model of particle physics and\nthey have been very well verified in several experiments.\nBut, it is important to remark that we can not be sure\nthat this, or any other, symmetry is exact apart from\nan experimental accuracy. This affirmation is encouraged\ndue to the fact that there exists some experimental testsof the Lorentz invariance being carried in low energies, in\nother words, energies smaller than 14 Tev. Thus, from\nthis fact, we can suspect that at high energies the Lorentz\ninvariance could not be preserved. As an example, in the\nstring theory there is a possibility that we could be living\nin an Universe which is governed by noncommutative co-\nordinates [ 67]. In this scenario it was shown in Ref. [ 68]\nthat the Lorentz invariance is broken.\nFurthermore, in a cosmologicalscenario, the occurrence\nof high energy cosmic rays above the Greisen-Zatsepin-\nKuzmin (GZK) cutoff [ 69] or super GZK events, has been\nfound in astrophysical data [ 70]. This event indicate the\npossibility of a Lorentz violation [ 71].\nThe impact of Lorentz violation on the cosmological\nscenario is very important, because several of its weak-\nnesses could be easily explained by the Lorentz violation.\nFor instance, it was shown by Bekenstein [ 72] that the\nproblem of the dark matter is associated with the Lorentz\nviolating gravity and in Ref.[ 73] Lorentz violation also is\nused to clarify the dark energy problem. Nowadays, the\nbreaking of the Lorentz symmetry is a fabulous mecha-\nnism for description of several problems and conflicts in\ncosmology, such as the baryogenesis, primordial magnetic\nfield, nucleosynthesis and cosmic rays [ 74].\nIn the inflationary scenario with Lorentz violation,\nKannoandSoda[ 75]haveshownthatLorentzviolationaf-\nfects the dynamics of the inflationary model. In this case,\nthatauthorsshowedthat, usingascalar-vector-tensorthe-\nory with Lorentz violation, the exact Lorentz violation\ninflationary solutions are found in the absence of the in-\nflaton potential. Therefore, the inflation can be connected\nwith the Lorentz violation.\nHere, it is convenient to us to emphasize that the in-\nflation is the fundamental ingredient to solve both the\nhorizon as the flatness problems of the standard model\nof the very early universe. Approximately 10−33seconds\nafter the inflation, the inflaton decays to radiation, where\nquarks, leptons and photons were coupled to each other.\nInthiscase,thebaryonicmatterwaspreventedfromform-\ning. Therefore, approximately 1 .388×1012seconds after\nthe Big Bang, the universe has cooled enough to allow\nphotons to freely travel through the universe. After that,\nmatter has became dominant in the universe.\nAt this point, it is important to remark that the post-\ninflationaryuniverseisgovernedbyrealscalarfieldswhere\nnonlinear interactions are present. Thus, it was shown\nin Ref. [ 76] that oscillons can easily dominate the post-\ninflationary universe. In that work, it was demonstrated\nthatthe post-inflationaryuniversecancontainaneffective\nmatter-dominated phase, during which it is dominated by\nlocalized concentrations of scalar field matter. Further-\nmore, in a very recent work [ 77], a class of inflationary\nmodels was introduced, giving rise to oscillons configu-\nrations. In this case, it was argued that these oscillons,\ncould dominate the matter density of the universe for a\ngiven time. Thus, one could naturally wonder about the\neffect of Lorentz violation over this scenario.\nThus, in this work we are interested in answer the fol-3\nlowing issues: Can oscillons and breathers exist in sce-\nnarios with Lorentz violation symmetry? If oscillons and\nbreathers exists in these scenarios how their profile is\nchanged? Furthermore, what happens with the lifetime\nof the oscillons?\nTherefore, in this paper, we will show that oscillons\nand breathers can be found in Lorentz violation scenarios,\nour study is performed by using Lorentz violation theories\nrigorously derived in the literature [ 50,78]. As a conse-\nquence, the principal goal here is to analyze the case of\ntwo nonlinearly coupled scalar fields case. However, we\nuse a constructive approach, so that we start by studying\nthe cases of one scalar field models and, then, use those\nresults in the study we are primarily interested in.\nThis paper is organized as follows. In section 2 we\npresent the description of the Lagrangian density for a\nrealscalarfield in presenceofa Lorentzviolationscenario.\nIn section 3 we calculate the respective commutation rela-\ntions of the Poincar´ egroup in the Standard-Model Exten-\nsion (SME) in a 1+1 dimensional flat Minkowski space-\ntime. The approach of the equation of motion is given in\nsection4. Usualoscillonsinthe backgroundoftheLorentz\nviolation is analyzed in section 5. In section 6 we will find\nthe flat-top oscillons which violates the Lorentz symme-\ntry. The breathers solutions are presented in section 7.\nWe discuss the outgoing radiation by oscillons in section\n8. In the section 9 we will present the oscillons in a two\nscalar field theory. Finally, we summarize our conclusions\nin section 10.\n2. STANDARD-MODEL EXTENSION\nLAGRANGIAN\nIn this section, we present a scalar field theory in a\n3+1-dimensional flat Minkowski space-time, but here we\nconsider a break of the Lorentz symmetry. In low energy,\nLorentz and CPTsymmetries the standard model (SM)\nofparticlephysicsisexperimentallywellsupported, but in\nhighenergiesthesuperstringtheoriessuggestthatLorentz\nsymmetry should be violated, in this context, the frame-\nwork to study Lorentz and CPTviolation is the so-called\nstandard-model extension. In the description of the SME,\nthe Lagrangian density for a real scalar field containing\nLorentz violation (LV), which can be read as a simplified\nversion of the Higgs model, is given by [ 50,78]\nL=1\n2∂µϕ∂µϕ+1\n2kµν∂µϕ∂νϕ−V(ϕ),(1)\nwhereϕis a real scalar field, kµνis a dimensionless tensor\nwhich controls the degree of Lorentz violation and V(ϕ)\nis the self-interaction potential. It is important to remark\nthat, some years ago [ 56], this Lagrangian density was\nused to study defect structures in Lorentz and CPTvio-\nlating scenarios. In that case the authors showed that the\nviolation of Lorentz and CPTsymmetries is responsible\nby the appearance of an asymmetry between defects andantidefects. This was generalized in [ 56]. Furthermore,\none similar Lagrangian density have been applied in the\nstudy on the renormalization of the scalar and Yukawa\nfield theories with Lorentz violation. In that case, it was\nshown that a LV theory with Nscalar fields, interacting\nthrough a φ4interaction, can be written as\nLK=1\n2(∂µϕi)(∂µϕi)+1\n2N/summationdisplay\ni=1Ki\nµν∂µϕi∂µϕi−1\n2λ2ϕ2\ni\n+N/summationdisplay\ni=1uβ\niϕi∂βϕi+N/summationdisplay\nj=1ϕ2\nivβ\nj∂βϕj−g\n4!(ϕ2\ni)2.(2)\nAs a simple example, that authors showed for Ki\nµν=\nKi\n00δ0\nµδ0\nνthat the dispersion relation is given by E=/radicalbig\np2−Ki\n00(p0)2+λ2, which implies in a LV. Therefore,\nusing explicit calculations, the quantum correctionsin the\nabove LV theory was studied, and these results show that\nthe theory is renormalizable.\nNow, returning to the equation ( 1), we can write the\nLagrangian density in the form\nL=1\n2(ηµν+kµν)∂µϕ∂νϕ−V(ϕ). (3)\nIn this case, the Minkowsky metric is modified from\ngµνtoηµν+kµν, which is responsible for the breaking\nof the Lorentz symmetry [ 50,78,79]. At this point it\nis possible to apply an appropriate linear transformation\nof the space-time variable xµ, in order to map the above\nLagrangiandensity into a Lorentz-likecovariantform, but\nthisleadstochangesinthefieldsandcouplingconstantsof\nthe potential. Thus, the coupling constants and the fields\nare rescaled in function of the kµνparameters.\nClearly, asafinalproduct, theLVandLorentzinvariant\nLagrangians have the same equation of motion. The fun-\ndamental difference between these two equations comes\nfrom the fact that the new variables xµcarry informa-\ntion of the Lorentz violations through of the kµνparam-\neters. In other words, in the transformed variables, the\nsystem looks to be covariant (under boosts of the trans-\nformed space-time variables). However, as a consequence\nof the fact that the resulting couplings become not invari-\nant when one changes from a reference frame to another,\nthere is no real Lorentz invariance. For instance, such be-\nhavior would be analogous to a change of the value of the\nelectrical charge when one moves from an inertial refer-\nence frame to another one, which is forbidden.\nIn the Lagrangian density ( 1),kµνis a constant tensor\nrepresented by a 4 ×4 matrix. It is the term which can\nbe responsible for the breaking of the Lorentz symmetry.\nThus, we write the tensor kµνin the form\nkµν=\nk00k01k02k03\nk10k11k12k13\nk20k21k22k23\nk30k31k32k33\n, (4)4\nIngeneral kµνhasarbitraryparameters,butit isimpor-\ntant to remark that if this matrix is real, symmetric, and\ntraceless, the CPTsymmetry is kept [ 50,78]. Here, we\ncomment that under CPToperation, ∂µ→ −∂µ, the term\nkµν∂µϕ∂νϕgoes askµν∂µϕ∂νϕ→+kµν∂µϕ∂νϕ. Thus,\none notices that kµνis always CPT-even, regardless its\nproperties. Furthermore, the tensor kµνshould be sym-\nmetric in order to avoid a vanishing contribution.\nIn a recent work, Anacleto et al.[80] also analyzed\na similar process to break the Lorentz symmetry, where\nthe tensor kµνwas used to study the problem of acous-\ntic black holes in the Abelian Higgs model with Lorentz\nsymmetry breaking. In another work by Anacleto et al.\n[80] the tensor kµνwas used to study the superresonance\neffect from a rotating acoustic black hole with Lorentz\nsymmetry breaking. Finally, in a very recent work [ 57],\nit was introduced a generalized two-fields model in 1+1\ndimensions which presents a constant tensor and vector\nfunctions. In that case, it was found a class of traveling\nsolitons in Lorentz and CPTbreaking systems.\nHowever,wecantofindsystemswithLorentzsymmetry\nbreak which has an additional scalar field [ 79]\nL=1\n2∂µϕ1∂µϕ1+1\n2∂µϕ2∂µϕ2+1\n2kµν∂µϕ1∂νϕ1(5)\n−m1ϕ2\n1\n2−m2ϕ2\n2\n2−V(ϕ1,ϕ2).\nIn the above Lagrangian density, we have a different\ncoefficient correcting the metric, but the coefficients for\nLorentz violation cannot be removed from the Lagrangian\ndensity using variables or fields redefinitions and observ-\nable effects of the Lorentz symmetry break can be de-\ntected in the above theory. Therefore, theories with fewer\nfields and fewer interactions allow more redefinitions and\nobservable effects.\n3. SME LAGRANGIAN: ONE FIELD THEORY\n(OFT)\nIn this section, we will work in a 1 + 1-dimensional\nMinkowski space-time. Here, we study a scalar field the-\nory in the presence of a Lorentz violating scenario. The\ntheory that we will study is given by the Lagrangian den-\nsity (1). Thus, in this case, the corresponding Lagrangian\ndensity must\nL1+1=1\n2α1(∂tϕ)2−1\n2α2(∂xϕ)2+1\n2α3∂tϕ∂xϕ−V(ϕ),\n(6)\nwhere\nα1≡(1+k00), α2≡(1−k11), α3≡(k01+k10),\n(7)\n∂t≡∂/∂t, ∂ x≡∂/∂x.\nAt this point, it is important to remark that the La-\ngrangiandensity clearly has not manifest covariance. Fur-\nthermore, it is possible to observe that the covariance isrecovered by choosing k00=k11= 0 and k01=−k10\n(ork01=k10= 0). Another possibilities that does\nnot represent a LV are k00=−k11andk01=−k10(or\nk01=k10= 0).\nNow, from the above, we can easily construct the cor-\nresponding Hamiltonian density\nH=β1Π2+β2(∂xϕ)2+β3Π(∂xϕ)+V(ϕ),(8)\nwhereβ1= 1/(2α1),β2= [2α1α2+α3(α3−1)]/(4α1),\nβ3=−α3/(2α1) and Π is the conjugate momentum,\nwhich is given by\nΠ =α1∂tϕ+(α3/2)∂xϕ. (9)\nLet us now see how the Poincar` e algebra is modified in\nthisscenario. Theideaofthepresentanalysisistoseehow\nthe Poincar´ e invariance is broken. In other words, ver-\nify how this scenario has the Lorentz symmetry violated.\nTherefore, for this we write down the three Poincar` e gen-\nerators, the Hamiltonian H, the total momentum Pand\nthe Lorentz boost M\nH=/integraldisplay\ndxH, (10)\nP=/integraldisplay\ndx/bracketleftbiggΠ(∂xϕ)\nα1−α3(∂xϕ)2\n2α1/bracketrightbigg\n, (11)\nM=/integraldisplay\ndx/braceleftbigg\nt/bracketleftbiggΠ(∂xϕ)\nα1−α3(∂xϕ)2\n2α1/bracketrightbigg\n−xH/bracerightbigg\n.(12)\nWith this, we can calculate the commutation relations\nof the Poincar` e group. Thus, after straightforward calcu-\nlations of the usual commutation relations, it is not diffi-\ncult to conclude that\n[H,P] =−i/parenleftbiggα3\nα2\n1/parenrightbigg/integraldisplay\ndx(∂xϕ)(∂xΠ), (13)\n[M,H] =−i/integraldisplay\ndx/parenleftbig\n(4β1β2+β2\n3)Π(∂xϕ) (14)\n−α3\nα2\n1(∂xϕ)(∂xΠ)+2β2β3(∂xϕ)2+2β1β3(Π)2/parenrightbigg\n,\n[M,P] =−iH\nα1+iα3\n2α2\n1/integraldisplay\ndx(Π(∂xϕ) (15)\n+x(∂xϕ)(∂xΠ)+β3\n4β1(∂xϕ)2/parenrightbigg\n.\nFrom the above relations, we can see that the Poincar` e\nalgebra is not closed, since that the usual commutations\nare not recovered. As a consequence, in this scenario we\nhaveoneviolationoftheLorentzsymmetry. However,itis\npossible torecoverthe complete commutation relationsby\ntakingk00=k11= 0 and k01=−k10(ork01=k10= 0).\nFor instance, making k00=k11= 0 and k01=−k10we\nhave\n[H,P] = 0,[M,H] =−iP,[M,P] =−iH. (16)5\nAt this pointwe canverifythat, forthe case k00=−k11\nandk01=−k10(ork01=k10= 0), the commutation\nrelations ( 13)-(15) lead to\n[H,P] = 0,[M,H] =−iα1P,[M,P] =−iH/α1.(17)\nHowever, in the above case, we can recover the usual\nPoincar` e algebra using the re-scale P=˜P/α1. Thus,\nsuch commutation relations indicates that there is no LV\nin this tensor configuration.\nIn summary, the Lagrangian density ( 6) has explicit\ndependenceontheparameters k00,k11,k01andk10, which\nis responsible for the violation of the Lorentz symmetry.\nThis happens due to the fact that the Poincar´ einvariance\nis not preserved, as one can see from ( 13)-(15).\n4. EQUATION OF MOTION IN LORENTZ\nVIOLATION SCENARIOS: OFT\nIn this section, we will study the equation of motion in\nthe presence of the scenario with Lorentz violation of the\nprevious section. Here, our aim is to study the case in\nthe 1+1 -dimensional Minkowski space-time. As a conse-\nquence, we will study the theory that is governed by the\nLagrangian density ( 6). Consequently, the corresponding\nclassical equation of motion can be written as\nα1∂2ϕ(x,t)\n∂t2−α2∂2ϕ(x,t)\n∂x2+α3∂2ϕ(x,t)\n∂x∂t+Vϕ= 0,(18)\nwhereVϕ≡∂V/∂ϕ. Note that the above equation is\ncarrying information about the symmetry breaking of the\ntheory.\nHere, if one applies the transformation involving the\nLorentz boost in the above equation of motion, one gets\nq1∂ϕ2(x,,t,)\n∂t,2−q2∂ϕ2(x,,t,)\n∂x,2+q3∂ϕ2(x,,t,)\n∂x,∂t,+Vϕ= 0,\n(19)\nwhere\nx,=γ(x−vt),t,=γ(t−vx/c2),γ= 1//radicalbig\n1−(v/c)2,(20)\nand\nq1=γ2/parenleftbiggα1c2−α2v2−α3cv\nc4/parenrightbigg\n,\nq2=γ2/parenleftbigg−α1v2+α2c2+α3cv\nc2/parenrightbigg\n, (21)\nq3=γ2/parenleftbigg−2vα1c+2cα2v−α3(c2+v2)\nc3/parenrightbigg\n.\nFollowing the above demonstration, we can see clearly\nthatthisequationisnotinvariantunderboosttransforma-\ntions. For instance, we can conclude that the possibilities\n[k00=−k11,k01=−k10] or [k00=−k11,k01=k10= 0]leads to the equations\nα1\nc2∂ϕ2(x,t)\n∂t2−α1∂ϕ2(x,t)\n∂x2+Vϕ= 0,(22)\nα1\nc2∂ϕ2(x,,t,)\n∂t,2−α1∂ϕ2(x,,t,)\n∂x,2+Vϕ= 0.(23)\nNote that there is no modification of the equations, in\nother words, the possibilities [ k00=−k11,k01=−k10] or\n[k00=−k11,k01=k10= 0] does not represent a genuine\nfactor for LV.\nIn order to solve analytically the differential equation\n(18) and simultaneously keep the breaking of the Lorentz\nsymmetry ,we must decouple the equation. For this, we\napply the rotation\n/parenleftbiggx\nt/parenrightbigg\n=/parenleftbiggcos(θ)−sin(θ)\nsin(θ) cos(θ)/parenrightbigg/parenleftbiggX\nT/parenrightbigg\n,(24)\nwhereθis an arbitrary rotation angle. Thus, the equation\n(18) in the new variables is rewritten as\nh1∂2ϕ(X,T)\n∂T2−h2∂2ϕ(X,T)\n∂X2+Vϕ= 0,(25)\nwith the definitions\nθ≡ −1\n2arctan/parenleftbiggα3\nα1+α2/parenrightbigg\n,\nh1≡α2\n1−α2\n2+[α2\n3+(α1+α2)2]cos(2θ)\n2(α1+α2),(26)\nh2≡α2\n2−α2\n1+[α2\n3+(α1+α2)2]cos(2θ)\n2(α1+α2).\nNote that the rotation angle θhas been chosen in or-\nder to eliminate the dependence in the term ∂2ϕ/∂X∂T .\nNow, performingthedilations T=√h1ΥandX=√h2Z,\none gets\n∂2ϕ(Z,Υ)\n∂Υ2−∂2ϕ(Z,Υ)\n∂Z2+Vϕ= 0.(27)\nFrom now on we will use the above equation to de-\nscribe the profile of oscillons and breathers. It is of great\nimportance to remark that the above equation has all the\ninformation about the violation of the Lorentz symmetry.\nIn fact, the field ϕ(Z,Υ) carries on the dependence of the\nparameters that break the Lorentz symmetry, this infor-\nmation arises from the fact that new variables Zand Υ\nhave explicit dependence on the kµνelements.\n5. USUAL OSCILLONS WITH LORENTZ\nVIOLATION: OFT\nNow, we study the case of a scalar field theory which\nsupports usual oscillons in the presence of Lorentz violat-\ning scenarios. The profile of the usual oscillons is one in\nwhich the spatial structure is localized in the space and,6\nin the most cases, is governed by a function of the type\nsech(x). On the other hand, the temporal structure is like\ncos(t), which is periodic. The theory that we will study\nis given by the Lagrangian density ( 6). In this case, we\nshowed in the last section that the corresponding classical\nequation of motion, after some manipulations, can be rep-\nresented by the equation ( 27). Thus, in order to analyze\nusual oscillons in this situation, we choose the potential\nthat was used in [ 36], which is written as\nV(ϕ) =1\n2ϕ2−1\n4ϕ4+g\n6ϕ6, (28)\nwheregrepresents a free coupling constant and we will\nconsider a regime where g >>1.\nSince our primordial interest is to find periodic and lo-\ncalized solutions, it is useful, as usual in the study of the\noscillons, to introduce the following scale transformations\nintandx\nτ=ωΥ, y=ǫZ, (29)\nwithω=√\n1−ǫ2. Thus, the equation of the motion ( 27)\nbecomes\nω2∂2ϕ(y,τ)\n∂τ2−ǫ2∂2ϕ(y,τ)\n∂y2+ϕ−ϕ3+gϕ5= 0.(30)\nNow we are in a position to investigate the usual oscil-\nlons. But it is important to remark that the fundamental\npoint is that here we have the effects of the Lorentz sym-\nmetry breaking. We can see this by inspecting the above\nequation of motion, which is carrying information about\nthe terms of the Lorentz breaking through the variables\nyandτ. We observe that it is possible to recover the\noriginal equation of motion for usual oscillons choosing\nk00=k11= 0 and k01=−k10(ork01=k10= 0). In this\ncase the Lorentz symmetry is recovered.\nNext we expand ϕas\nϕ(y,τ) =ǫϕ1(y,τ)+ǫ3ϕ3(y,τ)+ǫ5ϕ5(y,τ)+....(31)\nNote that the above expansion has only odd powers of\nǫ, this occurs because the equation is odd in ϕ. Let us\nnow substitute this expansion of the scalar field into the\nequation of motion ( 30). This leads to\n∂2ϕ1\n∂τ2+ϕ1= 0, (32)\n∂2ϕ3\n∂τ2+ϕ3−∂2ϕ1\n∂τ2−∂2ϕ1\n∂y2−ϕ3\n1= 0.(33)\nTherefore, the solution of equation ( 32) is of the form\nϕ1(y,τ) = Φ(y)cos(τ), (34)\nHere we call attention to the fact that the solution must\nbe smooth at the origin and vanishing when ybecomes\ninfinitely large.In order to find the solution of Φ( y), let us substitute\nthe solution obtained for ϕ1(y,τ) into the equation ( 33).\nThus, it is not hard to conclude that\n∂2ϕ3\n∂τ2+ϕ3=/parenleftbiggd2Φ\ndy2−Φ+3\n4Φ3/parenrightbigg\ncos(τ)+1\n4Φ3cos(3τ).\n(35)\nSolving the above equation we find a term which is\nlinear in the time-like variable τ, resulting into a non-\nperiodical solution, and we are interested in solutions\nwhich are periodical in time. Then to avoid this we shall\nimpose that\nd2Φ\ndy2−Φ+3\n4Φ3= 0. (36)\nAt this point, one can verify that the above equation\ncan be integrated to give\n/parenleftbiggdΦ\ndy/parenrightbigg2\n+U(Φ) =E, (37)\nwhereU(Φ) =−Φ2+ (3/8)Φ4. Note that in the above\nequation, the arbitrary constant Eshould be set to zero\nin order to get solitonic solution. This condition allows\nthe fieldconfigurationtogoasymptoticallytothevacuaof\nthe field potential U(Φ). Now, we must solvethe equation\n(37) withE= 0. In this case one gets\nΦ(y) =4/radicalbigg\n8\n3[sech(y)]1/2. (38)\nAs one can see, up to the order O(ǫ), the corresponding\nsolution for the field in the original variables is given by\nϕosc(x,t) =ǫ4/radicalbigg\n8\n3/parenleftigg/radicaligg\nsech/bracketleftbiggǫ[xcos(θ)+tsin(θ)√h2/bracketrightbigg/parenrightigg\n(39)\n×cos/bracketleftbiggω[−xsin(θ)+tcos(θ)]√h1/bracketrightbigg\n+O(ǫ3).\nThe profile of the above solution is plotted in Fig. 1\nfor some values of the kµνparameters. In the Figure 1\nwe see the profile of the usual oscillon in the presence of\nthe backgroundofthe Lorentzbreakingsymmetry. In this\ncase, one can check that the dependence ofthe solution on\nthe Lorentz breaking parameters is responsible for a kind\nofdeformation ofthe configuration, where the field config-\nuration becomes oscillatory in a localized region near its\nmaximumvalue. Furthermore,inthecourseofthetime, it\nis possible to observethat the Lorentz breakingsymmetry\nproduces a displacement of the oscillon along the spatial\ndirection. In this case we will call these configurations as\n”envelopedoscillons”, since in t= 0 the new configuration\nis enveloped by the oscillon with Lorentz symmetry.\nMoreover, one can note that if one wants to recover the\nLorentz symmetry, it is necessary to impose that k00=\nk11= 0 and k01=−k10(ork01=k10= 0).7\n6. FLAT-TOP OSCILLONS WITH LORENTZ\nVIOLATION: OFT\nSome years ago, a new class of oscillons, which is char-\nacterized by a kind of plateau at its top, was presented by\nAmin and Shirokoff [ 36]. In that work, the authors have\nshown that this configuration has an important impact\non an expanding universe. Thus, in this section, we will\ndescribe the impacts of the Lorentz violation overthe flat-\ntop oscillons. We will study the case in 1+1-dimensional\nMinkowski space-time where the classical equation of mo-\ntion is given by ( 27). Also, in order to analyze the flat-top\noscillons in this scenario, we choose the potential that was\nused in [36], which is represented in ( 28).\nNow, we begin a direct attack to the problem of find-\ning the flat-top oscillons. Likewise to the procedure pre-\nsented in [ 36], we introduce a re-scaled scalar field by\nϕ(Z,Υ) =φ(y,τ)/√g, where Z=√gy,τ=̟Υ and\n̟=/radicalbig\n1−α2/g. It is important to remark that the con-\nstantα2is responsible by the change in the frequency,\nits presence comes from the nonlinear potential. Thus, it\nis not difficult to conclude that the classical equation of\nmotion can be rewritten as\n(∂2\nτφ+φ)+g−1[−α2∂2\nτφ−∂2\nyφ−φ3+φ5] = 0.(40)\nSo, we are in a position to investigate the so-called flat-\ntop oscillons. But it is important to remark that the fun-\ndamental point is that all the effects of the Lorentz sym-\nmetry breaking are present implicitly in the classical field.\nOf course, it is possible to recover the original equation\nof motion presented by Mustafa [ 36] through a suitable\nchoice of kµν.\nLet us go further on our search for flat-top oscillons.\nFor this, we expand φas\nφ(y,τ) =φ1(y,τ)+g−1φ3(y,τ)+.... (41)\nIf we substitute the above expansion of the scalar field\ninto the equation of motion ( 40), and collect the terms in\norderO(1) andO(g−1), we find\n∂2φ1\n∂τ2+φ1= 0, (42)\n∂2φ3\n∂τ2+φ3−α2∂2φ1\n∂τ2−∂2φ1\n∂y2−φ3\n1+φ5\n1= 0.(43)\nTherefore, the solution of equation ( 42) is of the form\nφ1(y,τ) = Ψ(y)cos(τ), (44)\nIn order to find the solution of Ψ( y) let us substitute\nthe solution obtained for φ1(y,τ) into the equation ( 43).\nThus, it is not hard to conclude that\n∂2φ3\n∂τ2+φ3=/parenleftbiggd2Ψ\ndy2−α2Ψ+3\n4Ψ3−5\n8Ψ5/parenrightbigg\ncos(τ)\n(45)\n+/parenleftbigg3\n4Ψ3−5\n16Ψ5/parenrightbigg\ncos(3τ)−Ψ5\n16cos(5τ).whose solution can be written as\nφ3(y,τ) =1\n8[4G(y)−2H(y)+8c1]cos(τ)\n−H(y) cos(3τ)+4[G(y)τ+2c2]sin(τ),(46)\nwhere we defined that G(y) ≡/parenleftig\nd2Ψ\ndy2−α2Ψ+3\n4Ψ3−5\n8Ψ5/parenrightig\nandH(y)≡/parenleftbig3\n4Ψ3−5\n16Ψ5/parenrightbig\n.\nFurthermore, c1andc2are arbitrary integration con-\nstants.\nSince that the solution of the function φ3has a term\nwhich is linear in the variable τ, resulting into a non-\nperiodical solution, and we are interested in solutions\nwhich are periodical in time, we shall impose that G(y)\nvanishes. As a consequence we get\nd2Ψ\ndy2=/parenleftbigg\nα2Ψ−3\n4Ψ3+5\n8Ψ5/parenrightbigg\n, (47)\nAt this point, one can verify that the above equation\nhas the same profile ofthe equationpresented in Ref. [ 36].\nTherefore, this equation can be integrated to give\n1\n2/parenleftbiggdΨ\ndy/parenrightbigg2\n+U(Ψ) =E, (48)\nwhereU(Ψ) =−(1/2)α2Ψ2+(3/16)Ψ4−(5/48)Ψ6. Note\nthat in the above equation, the arbitrary constant E\nshould be set to zero in order to get solitonic solution.\nThis condition allows the field configuration to go asymp-\ntotically to the vacua of the field potential U(Ψ). On\nthe other hand, it is usual to impose that the profile of\nΨ(y) be smooth at y= 0, then it is necessary to make\ndΨ(0)/dy= 0. As a consequence E=U(Ψ0) = 0, which\nimplies\nα2=3\n8Φ2\n0−5\n24Φ4\n0, (49)\nwith Ψ 0≡Ψ(0). Thus, solving the above equation in Ψ 0,\nwe have a critical value a≤αc=/radicalbig\n27/160. Above this\ncritical value, Ψ 0becomes imaginary.\nNow, we must solve the equation ( 48) withE= 0. In\nthis case, we have\ndΨ/radicalig\nα2Ψ2−3\n8Ψ4+5\n24Ψ6=dy. (50)\nFrom this it follows that\nΨ(y) =(u4√\n4vu)/radicalig\n2√v+cosh[2y/radicalbig\nuv(α2c−α2)],(51)\nwherev= 27/[160(α2\nc−α2)] andu= (v−1)/v.8\nAs one can see, up to the order O(1), the corresponding\nsolution for the field in the original variables is given by\nϕFT(x,t) = (52)\nu4√\n4vu/radicaligg\n2g√v+gcosh/braceleftbigg\n2[xcos(θ)+tsin(θ)]√\nuv(α2c−α2)√gh2/bracerightbigg\n×cos/braceleftbigg̟[−xsin(θ)+tcos(θ)]√h1/bracerightbigg\n+O(g−3/2).\nThe profile of the above solution is plotted in Fig. 2.\nIn the Figure 2 we see the profile of the flat-top oscillon\nin the presence of the background of the Lorentz breaking\nsymmetry. Inthiscase,onecancheckthatthedependence\nof the solution on the Lorentz breaking parameters is re-\nsponsible for a control of the size of the oscillon plateau.\nThus, by measuring the width of the oscillon one could be\nable to verify the existence and the degree of the breaking\nof the symmetry. In Fig. 3 we see the typical profile of\nthe flat-top oscillon.\nThere one can note that the effect of the Lorentz break-\ning over the energy density, it is to becoming it more and\nmore localized around the origin.\n7. BREATHERS WITH LORENTZ VIOLATION:\nOFT\nWe will now construct the profile ofa breather in a 1+1\ndimensionalMinkowskispace-time. Again, we will usethe\nclassical equation of motion ( 27). The breather solutions\narise from the sine-Gordon model\nV(ϕ) =γ\nβ[1−cos(βϕ]. (53)\nThe sine-Gordonmodel is invariantunder ϕ→ϕ+2nπ,\nwherenis an integer number. In this case, the classical\nequation of motion is\n∂2ϕ(Z,Υ)\n∂Υ2−∂2ϕ(Z,Υ)\n∂Z2+γsin(βϕ) = 0.(54)\nThe above equation can be solved by the inverse-\nscattering method [ 81]. Thus, after straightforward cal-\nculations we conclude that the breather solution is given\nby\nϕB(Z,Υ) =4\nβarctan/bracketleftigg/radicalbig\nγ−w2sin(wΥ)\nwcosh(Z/radicalbig\nγ−w2)/bracketrightigg\n,(55)\nwherewis the frequency of oscillation and describe dif-\nferent breathers. In Figures 4 and 5 we show the behavior\nof the above solution.8. RADIATION OF OSCILLONS WITH LORENTZ\nVIOLATION SYMMETRY: OFT\nAn important characteristic of the oscillons is its radi-\nation emission. In a seminal work by Segur and Kruskal\n[37] it was shown that oscillons in one spatial dimension\ndecay emitting radiation. Recently, the computation of\nthe emitted radiation in two and three spatial dimensions\nwas did in [ 38]. On the other hand, in a recent paper by\nHertzberg [ 39], it was found that the quantum radiation\nis very distinct of the classic one. It is important to re-\nmark that the author has shown that the amplitude of the\nclassical radiation emitted can be found using the ampli-\ntude of the Fourier transform of the spatial structure of\nthe oscillon.\nThus, inthis section, wedescribetheoutgoingradiation\nin scenarios with Lorentz violation symmetry. Here, we\nwill establish a method in 1 + 1 dimensional Minkowski\nspace-time that allows to compute the classical radiation\nof oscillons in scenarios with Lorentz symmetry breaking.\nThis is done by following the method presented in [ 39].\nThis method suggests that we can write the solution of\nthe classical equation of motion in the following form\nϕsol(x,t) =ϕosc(x,t)+η(x,t), (56)\nwhereϕosc(x,t) is the oscillon solution and η(x,t) repre-\nsents a small correction. Let us substitute this decompo-\nsition of the scalar field into the equation of motion ( 18).\nThis leads to\nα1∂2ϕosc\n∂t2−α2∂2ϕosc\n∂x2+α3∂2ϕosc\n∂x∂t+α1∂2η\n∂t2\n(57)\n−α2∂2η\n∂x2+α3∂2η\n∂x∂t+U(ϕosc,η) = 0,\nwhereU(ϕosc,η) is a function which depends on the form\nofVϕsol(ϕsol). In order to decouple the above equation we\napply the rotation ( 24) and the dilations T=√h1Υ and\nX=√h2Z. Thus, we find\n∂2ϕosc(Z,Υ)\n∂Υ2−∂2ϕosc(Z,Υ)\n∂Z2+∂2η(Z,Υ)\n∂Υ2(58)\n−∂2η(Z,Υ)\n∂Z2+U(ϕosc,η) = 0.\nFrom the above equation it is possible to find the so-\nlution for η(Z,Υ) which carries the dependence on the\nparameters that break the Lorentz symmetry. We want\nto investigate the model given by ( 28), then we have\nU(ϕosc,η) =ϕosc+η−ϕ3\nosc−η3+3ϕ2\noscη+3ϕoscη2\n(59)\n+g(ϕ5\nosc+η5+10ϕ2\noscη3+10ϕ3\noscη2\n+5ϕoscη4+5ϕ4\noscη).9\nAsηrepresents a small correction, we assume that the\nnonlinear terms η2,η3,η4,η5and the parametric driving\nterms 3ηϕ2\nosc, 5gηϕ4\nosccan be neglected. At this point, it\nis important to remark that the parametric driven terms\nwere not considered because we are working in an asymp-\ntotic regime where ϕoscis also small. In this case, the\nequation ( 58) takes the form\n∂2η(Z,Υ)\n∂Υ2−∂2η(Z,Υ)\n∂Z2+η(Z,Υ) =−J(Z,Υ),(60)\nwhere\nJ(Z,Υ) =∂2ϕosc(Z,Υ)\n∂Υ2−∂2ϕosc(Z,Υ)\n∂Z2(61)\n+ϕosc(Z,Υ)−ϕ3\nosc(Z,Υ)+gϕ5\nosc(Z,Υ).\nWe can use the Fourier transform for solving the differ-\nential equation ( 60) whereJ(Z,Υ) acts as a source. With\nthis in mind, we write down the Fourier integral trans-\nforms\nη(R,w) =1√\n2π/integraldisplay\ndZdΥη(Z,Υ) (62)\n×exp[−i(RZ−wΥ)],\nJ(R,w) =1√\n2π/integraldisplay\ndZdΥJ(Z,Υ) (63)\n×exp[−i(RZ−wΥ)].\nThen, we have the corresponding solution\nη(Z,Υ) =1√\n2π/integraldisplay\ndRdwη(R,w)(64)\n×exp[i(RZ−wΥ)],\nwhere\nη(R,w) =−J(R,w)\nR2−(w2+1). (65)\nFrom the above approach it is possible to find the ra-\ndiation field for the oscillons. As a consequence of the\nmethod, the oscillons expansion must be truncated.\n8.1. SME Usual Oscillons Radiation: OFT\nIn this subsection we will study the outgoing radiation\nof the usual oscillons in a Lorentz violation scenario. In\nthis case, the oscillon expansion truncated in order Nis\ngiven by\nϕ(y,τ) =ǫϕ1(y,τ)+ǫ3ϕ3(y,τ)+ǫ5ϕ5(y,τ)(66)\n+...+ǫNϕN(y,τ).As an example, we will consider N= 1. This is the case\nwhere the field configuration corresponds to the oscillon\nϕosc(y,τ) =ǫϕ1(y,τ). (67)\nSubstituting ( 67) in (61), we obtain\nJ(Z,Υ) =/parenleftigg\n4/radicalbigg\n8\n3/parenrightigg\nǫ3[sech(ǫZ)]3/2cos(3ωΥ).(68)\nThus, for N= 1 we can solve easily the integral ( 64)\nwhich allows to find η(Z,Υ). Therefore, we can gener-\nalize the result to Nsubstituting the expansion ( 66) in\n(61), and using the differential equation ( 30). After the\ncalculations, the result is\nJ(Z,Υ) =CNǫN+2[sech(ǫZ)]N+1/2cos(¯nωΥ)+...,(69)\nwhereCNare constant coefficients. For instance, for N=\n1 we have C1=4/radicalbig\n8/3. Next we calculate η(Z,Υ) as\ngiven by ( 64). After straightforward computations, one\ncan conclude that\nη(Z,Υ) =π√πCNǫN+2\nkradcos(ωradΥ) (70)\n×sin(kradZ)/integraldisplay\ndZsech(ǫZ)]N+1/2cos(kradZ).\nwhere\nωrad= ¯nω, k rad=/radicalig\nω2\nrad−1. (71)\nOn the expression ( 70), we note that there is an out-\ngoing radiation which has an amplitude described by the\nintegral\nA(krad) =π√πCNǫN+2\nkrad(72)\n×/integraldisplay\ndZsech(ǫZ)]N+1/2cos(kradZ),\nwe also note that the radiation has frequency ωradand\nwave number krad. We can make use of the above gen-\neralization to calculate the amplitude of radiation of the\nusual oscillons in Lorentz violation scenario. For instance,\nforN= 1, we have\nA(krad) =4π√\n2πC1ǫ3\nkrad(73)\n×[b1F(a1,a2,a3,−1)+b∗\n1F(a1,a∗\n2,a∗\n3,−1)],\nwhereF(a1,a2,a3,−1) andF(a1,a∗\n2,a∗\n3,−1) are hyperge-\nometric functions with\nb1=1\n3ǫ−2ikrad, a1=3\n2, (74)\na2=3\n4−ikrad\n2ǫ, a3=7\n4−ikrad\n2ǫ.10\nIn Fig. 6 we see how the amplitude of the outgoing\nradiation changes with the parameters of kµν. From that\nFigure one can see that the amplitude of the outgoing ra-\ndiation of the oscillons is controlled by the terms of the\nLorentz breaking of the model, in such way that the radi-\nation amplitude will decay faster when the Lorentz break-\ning increases.\n8.2. SME Flat-top oscillons radiation: OFT\nWe will now present the outgoing radiation by the Flat-\ntop oscillons in Lorentz violation scenario. Here, the as-\nsociated oscillon expansion truncated in Nis defined as\nϕ(y,τ) =ϕ1(y,τ)+1\ngϕ3(y,τ) (75)\n+1\ng2ϕ5(y,τ)+...+1\ngN−1ϕ2N−1(y,τ).\nSubstituting the above expansion in ( 61), we have that\nJ(Z,Υ) =¯CN (76)\nׯCN\n\n(u4√\n4vu)/radicalig\n2g√v+gcosh[2Z/radicalbig\nuv(α2c−α2)/√g]\n\nN+2\n×cos(¯n¯ωΥ)+...,\nwhere¯CNare constant coefficients. Now we calculate\nη(Z,Υ) as given by ( 64). After straightforward compu-\ntations, one can conclude that\nη(Z,Υ) =π√π¯CN\n¯kradcos(¯ωradΥ)sin(¯kradZ) (77)\n×/integraldisplay\nd˜Z\n\n(u4√\n4vu)/radicalig\n2g√v+gcosh[2˜Z/radicalbig\nuvg(α2c−α2)]\n\nN+2\n×cos(¯krad˜Z).\nwhere\n¯ωrad= ¯n¯ω,¯krad=/radicalig\n¯ω2\nrad−1. (78)\nFrom the above expression, we see that there is an out-\ngoing radiation which has its amplitude described by the\nintegral\nA(krad) =π√π¯CN\n¯krad/integraldisplay\ndZcos(¯kradZ) (79)\n×\n\n(u4√\n4vu)/radicalig\n2√v+cosh[2Z/radicalbig\nuv(α2c−α2)/√g]\n\nN+2\n.We can make use the above generalization to calcu-\nlate the amplitude of radiation of the Flat-top oscillons\nin Lorentz violation scenario. For instance, for N= 1, we\nhave\nA(krad) =4π¯CN\nA0¯krad/parenleftigg\nu4√\n4vu√g/parenrightigg3\n(ξ1Fa+ξ∗\n1Fb),(80)\nwhereFa=F(Ω1;Ω2;Ω2;Ω3,Ω4,Ω5) andFb=\nF(Ω∗\n1;Ω2;Ω2;Ω∗\n3,Ω4,Ω5) are the Appell hypergeometric\nfunctions of two variables, and\nA0= 2/radicaligg\nuv(α2c−α2)√g, ξ1= 3+2ikrad\nA0,\nΩ1=3\n2−ikrad\nA0,Ω2=3\n2,Ω3=5\n2−ikrad\nA0,(81)\nΩ4=/radicalig\nA2\n0−1−A0,Ω5=1/radicalbig\nA2\n0−1−A0.\nIn this case we see that the amplitude of the outgoing\nradiation changes with the parameters kµν. We can see\nthat the amplitude of the outgoing radiation of the oscil-\nlons is controlled by the terms of the Lorentz breaking of\nthe model, in such way that the radiation amplitude will\ndecay faster when the Lorentz breaking increases.\n9. OSCILLONS WITH LV: TWO FIELD THEORY\n(TFT)\nWe have seen in section 2 that the most important sce-\nnariowithLVisthatdescribedbyatheorywithtwoscalar\nfields, because it is possible to find observable effects of\nthe LV. Then, in this section, we study a two scalar field\ntheory in the presence of a LV scenario. The theory that\nwewillstudyissimilartothatgivenbyPotting[ 79]. Here,\nwe will work with the corresponding Lagrangian density\nL=1\n2∂µϕ1∂µϕ1+1\n2∂µϕ2∂µϕ2 (82)\n+1\n2kµν∂µϕ1∂νϕ2−V(ϕ1,ϕ2).\nwhereV(ϕ1,ϕ2)istheinteractionpotential. Forexample,\nin order to find oscillons solutions, we can to choose the\npotential in the form\nV(ϕ1,ϕ2) =g\n3/parenleftbig\nϕ6\n1+ϕ6\n2/parenrightbig\n−1\n2/parenleftbig\nϕ4\n1+ϕ4\n2/parenrightbig\n(83)\n+ϕ2\n1+ϕ2\n2+5g/parenleftbig\nϕ4\n1ϕ2\n2+ϕ2\n1ϕ4\n2/parenrightbig\n−3ϕ2\n1ϕ2\n2.\nIn order to decouple the Lagrangian density ( 82), we\napply the rotation\n/parenleftbigg\nϕ1\nϕ2/parenrightbigg\n=1\n2/parenleftbigg\n1 1\n1−1/parenrightbigg/parenleftbigg\nσ1\nσ2/parenrightbigg\n. (84)11\nAfter straightforward computations, one can conclude\nthat\nL=1\n2∂µσ1∂µσ1+1\n2kµν\n1∂µσ1∂νσ1 (85)\n+1\n2∂µσ2∂µσ2+1\n2kµν\n2∂µσ2∂νσ2−V(σ1,σ2),\nwhere\nkµν\n1=1\n4kµν,kµν\n2=−1\n4kµν, (86)\nand the potential is\nV(σ1,σ2) =V(σ1)+V(σ2), (87)\nwith\nV(σi) =g\n6σ6\ni−1\n4σ4\ni+1\n2σ2\ni, i= 1,2.(88)\nIt is important to note that applying the rotations in\nthe fields, the Lagrangian density was decoupled into two\nindependent Lagrangians L=2/summationtext\ni=1Li, where\nLi=1\n2∂µσi∂µσi+1\n2kµν\ni∂µσi∂νσi−V(σi),(89)\nWe can see that all the preceding approaches and re-\nsults can be used here to find the fields σ1andσ2. An-\nother important point that it is convenient to remark at\nthis point, comes from the fact that any variable xµredef-\ninition will carry information of the parameter kµνwhich\nit is responsible by LV.\nAs we are working in 1+1-dimensions, the Lagrangians\n(89) become\nL(1+1)\ni=1\n2ai(∂tσi)2−1\n2bi(∂xσi)2(90)\n+1\n2di∂tσi∂xσi−V(σi),i= 1,2.\nIn this case, we have\nai≡(1+k00\ni), bi≡(1−k11\ni), di≡(k01\ni+k10\ni).(91)\nNow it is quite clear why the Lagrangian density ( 82)\nis more important and general than the one described\nby (1). First, because the commutation relations of the\nPoincar` e group is not closed, indicating a Lorentz Viola-\ntion. Second, because it is impossible to perform coor-\ndinate changes to eliminate the LV parameters in ( 85),\nbecause if we apply a coordinate change in order to write\nthe Lagrangian in an covariant form, only one of the sec-\ntors will stay invariant.Now, by using the approaches described in section 4,\nwe find the equations\n∂2σi(Zi,Υi)\n∂Υ2\ni−∂2σi(Zi,Υi)\n∂Z2\ni+Vσi= 0,(92)\nwhere\nZi=xcos(θi)+tsin(θi)√Li, (93)\nΥi=−xsin(θi)+tcos(θi)√Hi. (94)\nwith the set\nθi=−1\n2arctan/parenleftbiggdi\nai+bi/parenrightbigg\n, (95)\nLi=b2\ni−a2\ni+[d2\ni+(ai+bi)2]cos(2θi)\n2(ai+bi),(96)\nHi=a2\ni−b2\ni+[d2\ni+(ai+bi)2]cos(2θi)\n2(ai+bi).(97)\nFortunately, we can find periodical solutions for the\nfieldsσ1andσ2from the equation ( 92). In this case,\nwe are looking oscillons-like solutions. These solutions\nwere presented in the sections 5 and 6. Thus, from those\nsections we can show that\nσ(USUAL )\ni(x,t) = (98)\nǫi4/radicalbigg\n8\n3/parenleftigg/radicaligg\nsech/braceleftbiggǫi[xcos(θi)+tsin(θi)]√Li/bracerightbigg/parenrightigg\n×cos/braceleftbiggωi[−xsin(θi)+tcos(θi)]√Hi/bracerightbigg\n+O(ǫ3\ni),\nand\nσ(FLAT−TOP)\ni (x,t) = (99)\nui4√4viui/radicaligg\n2g√vi+gcosh/braceleftbigg\n2[xcos(θi)+tsin(θi)]√\nuivi(α2c−α2\ni)√gLi/bracerightbigg\n×cos/braceleftbigg̟i[−xsin(θi)+tcos(θi)]√Hi/bracerightbigg\n+O(g−3/2).\nIn the above solutions σ(USUAL )\ni represents the usual\noscillons and σ(FLAT−TOP)\ni are the Flat-Top ones. Fur-\nthermore, we have\nωi=/radicalig\n1−ǫ2\ni,̟i=/radicalig\n1−α2\ni/g,\n(100)\nvi= 27/[160(α2\nc−α2\ni)],ui= (vi−1)/vi.12\nAs above asserted, the original scalar fields ϕ1andϕ2\nare obtained from the fields σ1andσ2in the following\nform\nϕ1=σ1+σ2\n2,ϕ2=σ1−σ2\n2. (101)\nIt is important to remarkthat the resulting solutionsdo\nnot present merely algebraic relation between σiand the\noriginal parameters of the theory, but essentially lead to\nphysical consequences. As one can see, there are two kind\nof frequencies which can be combined for each scalar field\nϕi. This means that their solutions can be considered\nas a superposition of two independent fields and, as a\nconsequence, we can have an interference phenomena in\nthe structure of the oscillon.\nAn important question concerns the stability of the so-\nlutions, given that each field ϕiis a combination of the\nfieldsσi, the stability and longevity of the oscillons are\nguaranteed. From a mathematical point of view, one can\nthink that the originalfieldsconsistoflinearcombinations\nofσi. The same occurs when we calculated the outgoing\nradiation, in that case we have two radiation fields η1and\nη2, which are independent solutions with small resulting\namplitudes. As a consequence, their linear combinations,\n¯η1=η1+η2and ¯η2=η1−η2, will give the radiation field\nof solutions ϕi. Therefore, as ηiare very small solutions,\nwe still have the stability and longevity of the solutions\nguaranteed.\n10. CONCLUSIONS\nIn this work we have investigated the so-called flat-top\noscillons in the case of Lorentz breaking scenarios. We\nhave shown that the Lorentz violation symmetry is re-\nsponsible for the appearance of a kind of deformation of\nthe configuration. On the order hand, from inspection\nof the results coming from the flat-top oscillons in 1+1-\ndimensions with Lorentz breaking in comparison with the\nflat-topgivenin[ 36], onecanseethatthe oscillonsarecar-\nryinginformationabout the termsofthe Lorentzbreaking\nof the model, in this case by taking k00=k11= 0 and\nk01=−k10(ork01=k10= 0) one recovers the solution\npresented in Ref. [ 36]. Furthermore, this can lead one\nto obtain the degree of symmetry breaking by measuring\nthe width of the oscillon in 1 + 1 dimensions. One im-\nportant question about the non-linear solution is related\nto its stability. Thus, we studied the solutions found here\nby using the procedure introduced by Hertzberg [ 36,39].\nWe concluded that the radiation emitted by these oscil-\nlons is controlled by the terms of the Lorentz breaking\nof the model, in such way that the radiation will decay\nmore quickly as the terms become larger. Finally, all the\nresults obtained for the case of one scalar field models are\npromptlyextendedforthe caseofdoubletsofnounlinearly\ncoupled scalar fields.\nMoreover, it is important to highlight that the bounds\nin Lorentz violation theories in the Standard Model arevery small, and are compatible with the stability observed\nfor the oscillons here introduced. On the other hand, ob-\nservableeffects ofthese oscillonsinthe realworld, arepos-\nsible, for instance, in a cosmologicalcontext. In that case,\nthe life time of these oscillons can be decisive in the gener-\nation of coherent structures after cosmic inflation [ 82,83],\nwhere it was shown that oscillons can contribute up to\n20% of the energy density of the Universe. Thus, in this\nscenario, one should find bounds on the Lorentz violation\nwhich will open a new window to detect observable effects\nof breaking Lorentz symmetry. This possibility is encour-\naged by the fact that the break of the Lorentz symmetry\ninduces a kind of beat phenomenon in the structure of\nthe outgoing radiation, in contrast with the Lorentz in-\nvariant case (see Fig. 6). In this way, in a real world, one\ncan detect the difference in the frequency of the outgoing\nradiation, effect that would indicate the presence of a vi-\nolation of the Lorentz symmetry. Therefore, in order to\ndealing with these questions, we are presently working in\na future workwhere oscillonsin cosmologicalbackgrounds\nwith Lorentz symmetry breaking are presented.\nAcknowledgments\nThe authors thank Professor D. Dalmazi for useful dis-\ncussions. R.A.C.CorreathankP.H.R.S.Moraesfordis-\ncussions regarding cosmology. R. A. C. Correa also thank\nAlan Kostelecky for helpful discussions about Lorentz\nbreaking symmetry. The authors also thank CNPq and\nCAPES for partial financial support. Furthermore, the\nauthors also are grateful to the anonymous referees for\nthe comments that lead to improving the results and con-\nclusions of this work.\n[1]G. B. Whitham, Linear and Non-Linear Waves , John Wi-\nley and Sons, New York, (1974).\n[2]A.C. Scott, F. Y. F. Chiu, and D. W. Mclaughlin, Proc.\nI.E.E.E. 61, 1443 (1973).\n[3]R. Rajaraman and E. J. Weinberg, Phys. Rev. D 11, 2950\n(1975).\n[4]T. 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The thin line corresponds to the case with k00= 0.12,k11= 0.30,k01= 0.27 andk10= 0.21 and the thick line to\nthe case with kµν= 0.\n/Minus60/Minus40/Minus20 204060x\n/Minus0.50.5/CurlyPΗi/LParen1x,0/RParen1\n/Minus100/Minus50 50 100x\n/Minus1.0/Minus0.50.51.0/CurlyPΗi/LParen1x,200/RParen1\nFIG. 2: Profile of the Flat-Top oscillons in 1+1-dimensions w ith Lorentz symmetry breaking for t= 0 (left) and t= 200 (right)\nwithg= 5. The thin line corresponds to the case with k00= 0.12,k11= 0.30,k01= 0.27 andk10= 0.21 and the thick line to\nthe case with kµν= 0.\nxt/CurlyPhi/LParen1x,t/RParen1\nx xt/CurlyPhi/LParen1x,t/RParen1\nx\nFIG. 3: Typical profile of the Flat-Top oscillon. The left-ha nd figure corresponds to the case with Lorentz breaking symme try\nand the right-hand figure to the one with Lorentz symmetry.15\n/Minus6/Minus4/Minus2 246x\n/Minus0.50.51.01.52.02.53.0/CurlyPΗi/LParen1x,0/RParen1\n/Minus6/Minus4/Minus2 2 4 6x\n/Minus3/Minus2/Minus112/CurlyPΗi/LParen1x,10/RParen1\nFIG. 4: Profile of the Breathers 1+1-dimensions with Lorentz symmetry breaking for t= 0 (left) and t= 10 (right) with v= 2,\nw= 1,β= 1. The thin line corresponds to the case with k00= 0.28,k11= 0.30,k01= 0.27 andk10= 0.37 and the thick line to\nthe case with kµν= 0.\n-2024\nx05101520t\n-4-2024\nx05101520t\nFIG. 5: Density plot of a Breather. Solution with Lorentz sym metry breaking (left) and to the one Lorentz symmetry (right ).\n50100150200250300350x\n/Minus4./Multiply10/Minus9/Minus2./Multiply10/Minus92./Multiply10/Minus94./Multiply10/Minus9Η/LParen1x,t/RParen1\n1020304050x\n/Minus6./Multiply10/Minus13/Minus4./Multiply10/Minus13/Minus2./Multiply10/Minus132./Multiply10/Minus134./Multiply10/Minus136./Multiply10/Minus13Η/LParen1x,t/RParen1\nFIG. 6: Amplitude of the outgoing radiation determined by th e Fourier transform. The left-hand figure corresponds to the case\nwith Lorentz breaking symmetry and the right-hand to the cas e with Lorentz symmetry."    },    {        "title": "1508.05265v1.Radiative_damping_in_wave_guide_based_FMR_measured_via_analysis_of_perpendicular_standing_spin_waves_in_sputtered_Permalloy_films.pdf",        "content": "Radiative damping in wave guide based FMR measured via\nanalysis of perpendicular standing spin waves in sputtered\nPermalloy \flms\nMartin A. W. Schoen,1Justin M. Shaw,2Hans T.\nNembach,2Mathias Weiler,3and Thomas J. Silva2\n1Institute of Experimental and Applied Physics,\nUniversity of Regensburg, 93053 Regensburg, Germany\n2Electromagnetics Division, National Institute of\nStandards and Technology, Boulder, CO, 80305\n3Walther-Meiner-Institut, Bayerische Akademie\nder Wissenschaften, D-85748 Garching, Germany\n(Dated: August 20, 2021)\nAbstract\nThe damping \u000bof the spinwave resonances in 75 nm, 120 nm, and 200 nm -thick Permalloy\n\flms is measured via vector-network-analyzer ferromagnetic-resonance (VNA-FMR) in the out-of-\nplane geometry. Inductive coupling between the sample and the waveguide leads to an additional\nradiative damping term. The radiative contribution to the over-all damping is determined by\nmeasuring perpendicular standing spin waves (PSSWs) in the Permalloy \flms, and the results are\ncompared to a simple analytical model. The damping of the PSSWs can be fully explained by\nthree contributions to the damping: The intrinsic damping, the eddy-current damping, and the\nradiative damping. No other contributions were observed. Furthermore, a method to determine\nthe radiative damping in FMR measurements with a single resonance is suggested.\nContribution of NIST, not subject to copyright\n1arXiv:1508.05265v1  [cond-mat.mtrl-sci]  21 Aug 2015I. INTRODUCTION\nExcited magnetic moments relax towards their equilibrium orientation due to damp-\ning. Several physical mechanism can cause damping. Many mechanisms, like eddy current\ndamping1in conducting ferromagnets, were already identi\fed in the 1950s. More recently,\nenhanced damping due to spin pumping2from a ferromagnet into an adjacent metallic layer\nwas identi\fed, and remains a topic of ongoing investigation3{6. Furthermore, wavenumber-\ndependent contributions to the damping caused by intralayer spin pumping have been theo-\nretically predicted7,8and currently are the subject of experimental investigation9,10. Another\ndamping process, referred to as radiative damping11,12, has been known to exist since the\n1970s and is purely due to inductive coupling between the sample and the waveguide in ferro-\nmagnetic resonance (FMR) experiments. More recently, this phenomenon has been further\ninvestigated in the context of strong magnon-photon coupling experiments, with possible\napplications in quantum information processing13. In these quantum-coherent experiments,\nradiative damping was identi\fed as a manifestation of non-resonant magnon-photon cou-\npling, and it was determined that such coupling14is indeed a source of extrinsic line width\nin cavity-based FMR studies. In a radiative damping process, the time-varying magnetic\n\rux associated with the dynamic magnetization generates microwave-frequency currents in\nthe proximate conductor of a microwave waveguide that carries the resultant power away\nfrom the sample. This process is similar to that exploited in eddy-current brakes and can\nbe seen as a non-local counterpart to the eddy-current damping in conductive ferromagnets.\nTo determine the magnitude of radiative damping in magnetic thin \flms, we used broad-\nband vector-network-analyzer ferromagnetic resonance (VNA-FMR) to measure damping in\nNi0:8Fe0:2Permalloy (Py) \flms with thicknesses \u000evarying between 70 nm and 200 nm. By\nuse of the geometry sketched in Fig. 1(a), we determine the total damping for each mode\n\u000bnas a sum of intrinsic damping \u000bint, eddy-current damping \u000beddyand radiative damping\n\u000brad\nn. We then perform a quantitative analysis of the PSSW resonance \felds, amplitudes and\ndamping to extract the di\u000berent contributions to \u000bn. We \fnd that eddy current damping is\nonly signi\fcant for the lowest order mode, the radiative damping strongly a\u000bects the \frst\n\fve modes, and no additional contributions to the damping are detectable for spin waves up\ntok= 1:75\u0002106cm\u00001. This last \fnding is in contrast with reports of exchange mediated\ndamping in both nanostructures10and thin \flms9.\n2II. DAMPING MODELS\nAccording to Faraday's Law, the time-varying \rux of a precessing magnetic moment\ngenerates an ac voltage in any conducting material that passes through the \rux. As shown\nin Fig. 1, spin wave precession in a conducting ferromagnet on top of a coplanar waveguide\n(CPW) induces ac currents both in the ferromagnet and the CPW. The dissipation of these\neddy currents in the sample, and the \row of energy away in the CPW give rise to two\ncontributions to magnetic damping. Historically, the damping caused by eddy currents in\nthe ferromagnet \u000beddyis called eddy current damping, while the damping caused by the eddy\ncurrents in the waveguide is called radiative damping \u000brad\nn.\nEddy current damping has been recognized since the 1950s1,15. For the lowest order mode\nin FMR,\n\u000beddy=C\n16\r\u00162\n0Ms\u000e2\n\u001a; (1)\nwith the resistivity \u001a, saturation magnetization Ms, the vacuum permeability \u00160, the gyro-\nmagnetic ratio \r, and the sample thickness \u000e(see derivation in Appendix Sec. B). We\nintroduce a correction factor Cto account for details of the eddy current spatial pro\fle. As\nshown in a later section, \u000beddyfor all higher order PSSW modes investigated in this study\nis much smaller than that of the lowest order mode.\nWe now turn to the radiative damping \u000brad\nn. We consider the experimental geometry\nsketched in Fig. 1 (a). A ferromagnetic sample with thickness \u000eand length lis placed on\ntop of the center conductor of a coplanar waveguide with width W. The sample dimension\nalongxis much larger than W. The sample and CPW are separated by a gap of height d.\nAn external dc magnetic \feld H0is applied perpendicular to the sample plane, and the spin\nwave resonances (SWR) are driven by microwaves in the CPW at resonance frequency f. A\nfraction of the ac magnetic induction Bdue to the dynamic component of the magnetization\nmx\nn(H0;I;z) wraps around the center conductor.\nTo derive a quantitative expression for \u000brad\nn, we start by calculating hx(I;x;z), the x\ncomponent of the driving \feld hmwthat is generated by an excitation current Iin the\ncenter conductor. We assume hx(I;x;z) is uniform along y, but we allow for variation along\nxandz. To estimate hx(I;x;z) we use the Karlqvist equation16\nhx(I;x;z) =I\n2\u0019W\u0014\narctan\u0012x+W=2\nz\u0013\n\u0000arctan\u0012x\u0000W=2\nz\u0013\u0015\n: (2)\n3n=0 123\nxz\nwdδ\nq31 -1z\nH\n0Mm\nx\nBdl\nδxyz\nCPW\nw(a)\n(b)h\nmwFigure 1: Schematic of the radiative damping process. (a) Mis the dynamic magnetization, H0\nthe applied external \feld and Bis the magnetic inductance due to the x-component mxof the\ndynamic magnetization. Wis the width of the center conductor, lthe length of the sample on the\nwaveguide, \u000ethe thickness of the sample and dthe spacing between sample and wave guide. (b)\nSimpli\fed depiction of the PSSW eigenfunctions qnfor mode numbers n= 0, 1, 2, 3. We exemplarily\nused boundary conditions that are completely pinned on one side and completely un-pinned on the\nother side. The origin of the coordinate system is indicated.\n4This microwave \feld can excite PSSWs in the sample. Schematic mode pro\fles for the\nfundamental mode ( n= 0) and the \frst three PSSW modes are shown in Fig. 1(b), where\nwe use unpinned boundary conditions at the top surface and pinned boundary conditions at\nthe bottom surface. As shown in Fig. 1(b), the mode pro\fles describe a z-dependence of the\ndynamic magnetization components mxandmy. In the perpendicular geometry used here,\njmxj=jmyjeverywhere, i.e. the precession is circular. In what follows, we will only discuss\nmx, the dynamics of which are inductively detected in the measurement. For a PSSW with\nmode number n,emx\nn(x;z) =qn(z)\u001fnhqn(z)hx(I;x;z)iwherehidenotes spatial averaging\ninxandzdirections, as de\fned in the Appendix, \u001fn=\u001fxx\nnis the diagonal component of\nthe magnetic susceptibility of the n-th order mode, and \u00001\u0014qn(z)\u00141 is the normalized\nmode pro\fle (eigenmode), an example of which is sketched in Fig. 1 (b) for n= 3. The\nmode inductance Lnis given by Ln=\u001fneLn, where, as detailed in the Appendix, we de\fne\na normalized mode inductance eLnfor thenth PSSW mode,\neLn=\u00160l\nI2hqn(z)hx(I;x;z)i2W\u000e: (3)\neLn, as explained in the appendix, no longer has any dependence on magnetic \feld or\nexcitation frequency. In the simplest case of a uniform magnetization pro\fle q0(z) = 1\n(FMR-mode) and uniform excitation \feld hx(I;x;z) =hx(I; 0;0) =I=(2W), the normalized\ninductance is eLn=\u00160\u000el=(4W).\nThex-component of the dynamic magnetization mx\nn(x;z) produces a net \rux \b n=\u001fnIeLn\nthat threads around CPW center conductor, leading to a power dissipation\nPn=!2\n2Z0\u0010\n\u001fnIeLn\u00112\n; (4)\nwhereZ0is the waveguide impedance (in our case Z0= 50 \n) and !is the angular frequency\nof the magnetization precession. With Eq. (4), the power dissipation rate\u0010\n1\nT1\u0011\nn=Pn=En\ncan be calculated, where Enis the energy of the dynamic component of the magnetization\nderived in the Appendix. This power \row from the sample to the waveguide leads to the\nradiative damping contribution\n\u000brad\nn=1\n2!\u00121\nT1\u0013\nn=\u0011\r\u0016 0MseLn\nZ0;\n(5)\n5where\u0011=\u000e=\u0010\n4R\u000e\n0dzjqn(z)j2\u0011\nis a dimensionless parameter that accounts for the actual\nmode pro\fle in the sample, see Appendix Sec. A. In the case of sinusoidal PSSWs, \u0011= 1=2,\nand for a completely uniform mode pro\fle, i.e. qn(z) = 1,\u0011= 1=4. From Eq. (5), it is\nevident that \u000brad\nnis proportional to eLnforn > 0. In the simplest case of uniform driving\n\feldhx=I=(2W), the radiative contribution is given by\n\u000brad\n0=\u0011\r\u0016 0Ms\nZ0eLn\u0018=\u0011\rM s\u00162\n0\u000el\n2Z0W: (6)\nNote that the radiative damping thus depends on the sample and waveguide dimensions, in\nparticular linearly on the sample thickness. Unlike eddy-current damping, \u000brad\nnis indepen-\ndent of the conductivity of the ferromagnet, hence this damping mechanism is also operative\nin ferromagnetic insulators.\nIII. SAMPLES AND METHOD\nWe deposit Ta(3)/Py( \u000e)/Si 3N4(3), Ta(3)/Py( \u000e)/Ta(5), Ta(3)/Py( \u000e) and Py(\u000e) layers on\n100\u0016m thick glass substrates by DC magnetron sputtering at a Ar pressure of 0 :7 Pa (\u0019\n5\u000110\u00003Torr) in a chamber with a base-pressure of less than 5 \u000110\u00006Pa (\u00194\u000110\u00008Torr);\nwhere\u000e= 75 nm, 120 nm and, 200 nm is the Permalloy thickness. The Py thickness was\ncalibrated by x-ray re\rectivity. We estimate that the damping enhancement due to spin\npumping into the Ta layer is two orders of magnitude smaller than the intrinsic damping of\nthe Permalloy layer for Permalloy samples of these thicknesses. The various combinations of\ncapping and seed layers are chosen to determine the sensitivity of our results on the spinwave\nboundary conditions and the resultant mode pro\fles. Prior to deposition, the substrates are\ncleaned by Ar plasma sputtering. The samples are coated with approximately 150 nm of\nPMMA in order to avoid electrical shorting when samples are placed directly on the CPW.\nThe CPW has a center conductor width of W= 100 \u0016m. The SWR are characterized using\n\feld-swept VNA-FMR17{19in the out-of-plane geometry (see Fig. 1) with an external static\nmagnetic \feld H0applied perpendicular to the sample plane. The excitation microwave \feld\nhx(x;y) is applied over a frequency range of 10 GHz to 30 GHz. A VNA is used to measure\nthe complex S21transmission parameter (ratio of voltage applied at one end of the CPW\nto voltage measured at the other end) for the waveguide/sample combination. The change\ninS21due to the FMR of the sample is then \ftted with a linear superposition of complex\n6susceptibility tensor components \u001fn,\n\u0001S21\nn(H0) =NX\nn=0An\u001fn(H0)ei\u001en+ linear background (7)\nwith the mode number n, phase\u001en, and dimensionless mode amplitude An, as de\fned in the\nAppendix. A complex linear background and o\u000bset is included in the \ft. The susceptibility\ncomponents are derived from the Landau-Lifshitz equation for the perpendicular geometry;\nin the \fxed-frequency, swept-\feld con\fguration, we obtain20\n\u001f(H0)n=Ms(H0\u0000Me\u000b\nn\u0000Hex\nn)\n(H0\u0000Me\u000b\nn\u0000Hex\nn)2\u0000(He\u000b)2\u0000i\u0001Hn(H0\u0000Me\u000b\nn\u0000Hex\nn)(8)\nwithHe\u000b=!=(\r\u00160) andMe\u000b\nn=Ms\u0000Hk, whereHkis the perpendicular anisotropy, Hex\nnis\nthe exchange \feld (de\fned below), and \u0001 Hnis the linewidth. An example of the resulting\n\fts for the complex S21data is shown in Fig. 2 (a) and (b).\nIV. EXPERIMENT\nWe detect both even and odd PSSW modes. If we assume a uniform excitation \feld\nand Dirichlet boundary conditions (completely pinned), only odd modes would be detected.\nAlternatively if we assume Neumann boundary conditions (completely unpinned), only the\nfundamental mode would be detected.\nTwo e\u000bects can contribute to our ability to detect all the PSSW modes. First, the\nexcitation \feld pro\fle might not be uniform due to eddy current shielding21,22. Second, the\ninterfacial boundary conditions might be asymmetrical, as alluded to above. According to\nthe criterion in Ref. [ 22], the threshold sheet resistance for the onset of eddy current shielding\nat 20 GHz is 0 :065 \n=\u0003. We estimate that the sheet resistance for our 200 nm is in excess of\n0:345 \n=\u0003, so we conclude that the eddy current shielding is relatively weak for our samples.\nOn the other hand, all modes are in principle detectable if we assume asymmetric in-\nterfacial anisotropy. For the sake of simplicity of the analysis, we will assume interfa-\ncial anisotropy for a single interface, then use an optimization approach to determine the\nwavenumber of the modes that is consistent with such a hypothesis. However we must\nemphasize that this approach does not provide a unique \ft for the measured distribution\nof resonance \felds for the PSSW spectrum, but simply allows us to accommodate for the\nwavenumber values required to be consistent with the measured spectrum. As such, the\n71.51 .61 .7-0.20-0.15-0.10-0.05-0.45-0.40-0.35-0.300\n1x1062x1060.00.51.0(\nb)4\n235 10  \nIm(S21)µ\n0H0 [T] 53412 \nRe(S21)0\n(a) \n(c)µ0H exn\n[T] \nk\nn [cm-1]Figure 2: Measured S21transmission parameter (black circles) at 20 GHz and the multi-peak\n-susceptibility \ft (red line) for the (a) real part and (b) imaginary part obtained with the Ta(3)-\nPy(200)-Si 3N4(3) sample. The \frst 6 modes are shown. (c) The exchange \feld Hex\nn(black squares)\nand exchange \feld \ft, from Eqs. (9) and (10) (red crosses) for all 13 detected modes plotted as a\nfunction of the \ftted wave numbers kn.\n\ftted value for Ksis to be interpreted as no more than a self-consistent value associated\nwith only one of many possible scenarios.\nIf we assume negligible magnetocrystalline perpendicular anisotropy Hk,Hres\nnis related\nto the exchange \feld via\nHres\nn=Hex\nn+Ms;\nwithHex\nn=2Aex\n\u00160Msk2\nn:(9)\nHere,knis the spinwave wavevector, and Aexis the exchange energy that is related to the\nspinwave sti\u000bness DviaD=2Aexg\u0016B\nMs. On the other hand, if we want to include interfacial\nanisotropy for a single interface in our analysis, we can numerically solve the transcendental\nequation23\u0012\n\u00001\n2kna+Ks\n2Aexkn+ 1\u0013\ntan(kn\u000e) =Ks\n2Aexkn; (10)\n8whereKsis the interfacial anisotropy, and a= 0:3547 nm is the lattice constant24. We\nminimize the residue of the \ft of Eq. (9) to Hres\nnwith the \ftting parameters Ms,Aex, and\nKsfrom Eq. (10) by use of a Levenberg-Marquardt optimization algorithm. This yields the\npairs (kn,Hex\nn) shown in Fig.2 (c) for all modes.\nFrom the \ft, we obtain a saturation magnetization of \u00160Ms= 1:02\u00060:01 T, in agree-\nment with that determined by magnetometry. The exchange sti\u000bness constant of D=\n3:22\u00060:04 meVnm2is close to a value of D\u00193:1 meVnm2reported by Maeda et al.25.\nThe exchange \ft also yields a single surface anisotropy Ksthat depends on the\ncap and seed layer con\fgurations. For the Ta(3)-Py( \u000e)-Si 3N4(3) sample series,\nKs= (5:1\u00060:8)\u000210\u00004J=m2, while all the other samples have a higher Ksof\n(7\u00061)\u000210\u00004J=m2. All values for Ksare in the range of other reported interface\nanisotropies for Permalloy layers of these thicknesses26.\nWe now turn to the linewidth \u0001 Hnand the amplitude Anfor the individual modes. The\nGilbert damping parameter \u000bnis extracted from the slope of the linewidth vs. frequency f\nplot10shown in Fig. 3(a) via\n\u0001Hn=4\u0019\u000bnf\nj\rj\u00160+ \u0001H0\nn; (11)\nwhere \u0001H0\nnis the inhomogeneous broadening that gives rise to a nonzero linewidth in the\nlimit of zero frequency excitation. The normalized inductance of the modes eLnis extracted\nin a similar fashion from the dependence of the mode amplitude Anon the frequency f, see\nFig. 3(b) and Eq. (38) in the Appendix\nAn= 2\u0019feLn\nZ0+A0\nn; (12)\nwhereA0\nnis an o\u000bset for each mode. A0\nnis a phenomenological \ftting parameter, which is\nnot yet fully understood.\nWe plot\u000bnandeLnas a function of mode number nin Fig. 3 (c). The damping and\nthe normalized mode inductance are found to be proportional. In order to explore this\ncorrelation, we plot \u000bnvs.eLnin Fig. 4(a). Here, the data for \u000bnvs.eLnare linearly\ncorrelated for all modes except for n= 0, as seen by the linear \ft (line) to the data for\nn\u00151. This is as expected for the radiative damping model, as summarized in Eq. (5). The\nadditional damping of the fundamental mode is interpreted as the result of eddy current\ndamping, as quanti\fed in Eq. (1). In Fig. 4(b), we plot the residual \u0001 \u000bnof the linear \ft\n901 2 3 4 0.0070.0080.0090.0100.011~ α( c)Ln [pH]αnm\node number n0n0\n123 \nLn  \n01 02 03 0010203040n4213\nµ0ΔH [mT]f\n [GHz](a)0\n4\n213\n0\n1 02 03 0-0.0020.0000.0020.0040.0060.0080.010 (b) \n f\n [GHz]AnFigure 3: Parameter extraction for the \frst 5 PSSWs of the Ta(3)-Py(200)-Si 3N4(3) sample. (a)\nExtraction of \u000bfrom the linewidth \u00160\u0001H(data points) via linear \fts (lines); staggered for display.\n(b) Extraction of the normalized mode inductance eLn(data points) from the resonance amplitude\nAn(f) via linear \fts (lines). (c) \u000bn(black squares) and eLn(red crosses) for each PSSW.\nshown in Fig. 4(a) for all modes. \u0001 \u000bnis negligible for all modes except for n= 0. We\nextract \u0001\u000bn=0for all the samples and plot \u0001 \u000bn=0vs.\u000e2in Fig. 4(c).\nIt appears that \u0001 \u000bn=0for all the samples scales linearly with \u000e2, as expected from Eq. (1)\nfor eddy current damping. Simultaneous weighted \fts of all the data to Eq. (1) yields\nC= 0:4\u00060:1. This value suggests a localization of eddy currents, since Ccorrects for the\n1001 2 3 4 0.00000.00030.00060.00090.00.51.01.52.02.53.00.0070.0080.0090.0100.0110\n1 x1042x1043x1044x1040.00000.00030.00060.0009 \n Δαnn\n(b)~4321n =0 \nαnL\nn[pH](a) \n Δαn=0δ\n2[nm2](c)Figure 4: Damping \u000bnand inductance eLnfor the Ta(3)-Py(200)-Si 3N4(3) sample. (a) Linear \ft\nof\u000bto Eq. (5) where the \ft is constrained to the n= 1, 2, 3, 4 modes. (b) The residual of the\nlinear \ft, showing enhanced damping for the 0-th order mode (black). We attribute the enhanced\ndamping to an eddy current contribution. (c) Enhanced 0-th order mode damping for all samples.\nThe red line is a \ft of the data points to the eddy current damping model from Eq. (1).\neddy current distribution in the sample.\nFor then\u00151 modes, it can be shown27that\u000beddy\nn/1=k2\nn. The calculated wavevectors\nfrom Eq. (10) for the n= 1 mode of all the samples is at least a factor three larger than that\nof then= 0 mode and, therefore, the eddy current damping of the n= 1 mode is predicted\nto be approximately one order of magnitude smaller than the eddy current damping of the\nn= 0 mode. Thus, the eddy current damping of the n\u00151 modes is negligible to within\n11the error bars, i.e., \u000beddy\nn\u00190 forn\u00151. This supports the analysis of the data in Ref. [ 9],\nwhich also neglects the eddy current damping in higher order modes.\nV. EXTRACTION OF THE RADIATIVE CONTRIBUTION TO THE DAMPING\nBy use of Eq. (1) and our \ftted value of C= 0:4, we subtract the eddy current contri-\nbution to the damping of all the n= 0 modes to obtain a corrected damping value \u000b0\nn=0,\nwhere\u000b0\nn=0=\u000bn=0\u0000\u000beddy(C= 0:4). The corrected data for all the modes are plotted in\nFig. 5.\nFigures 5 (a) to (c) group all data obtained for a set of samples with identical Py thickness\n\u000e. The lines are linear \fts to Eq. (5). For each thickness \u000e, we observe a signi\fcant correlation\nof\u000bnandeLnfor all seed and cap layer con\fgurations, as expected for a radiative damping\nmechanism.\nFurthermore, by use of Eq. (6) for the n= 0 mode of the 75 nm thick sample, using a\nvalue of\u0011\u00190:46 as determined in the Appendix, we estimate \u000brad\n0\u00190:00023\nThe experimentally determined value is \u000brad\n0\u00190:00035\u00060:0001. The deviance from\nthe calculated value is possibly due to non-uniformities of both the excitation \feld and\nmagnetization pro\fle in Eq. (6), that requires the solution of the integral in Eq. (25).\nNevertheless the estimated value for \u000brad\n0is of the correct order of magnitude.\nWe determine the intrinsic damping \u000bintfrom theeLn= 0 intercept of the linear \fts in\nFig. 5. We plot \u000bintfor the three values of \u000ein Fig. 6 (right scale). We \fnd that \u000bintis\napproximately constant to within \u00065% for all samples. In addition the average value over\nall the \flm thicknesses is in reasonable agreement to the previously reported value of \u000bint=\n0.006 (dotted red line)28.\nThe other \ftting parameter \u0011, extracted from the slope of \u000bnvs.eLn, is also plotted in\nFig. 6 (left scale). For anti-symmetric boundary conditions, \u0011= 1=2 is expected, whereas\nfor the uniform mode, \u0011= 1=4.\nWe see that the \ftted values lie exclusively within these extremes, within error bars.\nThere have been recent reports of a non-zero, wavenumber-dependent component for\ndamping for both localized eigenmodes in magnetic nanostructures10and PSSWs in thick\n120.00.51.01.52.02.53.00.0060.0070.0080.0090.010η\n=0.24~\n(c)δ\n = 75 nmαnαn \n L\nn[pH]0.0060.0070.0080.0090.010η\n=0.36(b)δ\n = 120 nm \n 0.0060.0080.0100.012(\na)δ\n = 200 nmαn \nη=0.45Figure 5: Dependence of damping \u000bon the normalized mode inductance eLnafter correction for\nthe eddy current damping \u0001 \u000b0of the fundamental mode, plotted for all sample con\fgurations\nand di\u000berent thicknesses: (a) \u000e= 200 nm, (b) \u000e= 120 nm and (c) \u000e= 75 nm. The red lines are\nweighted linear \fts to the data by use of Eq. (5), that describes the radiative component of the\ndamping.\n13501 001 502 000.00.10.20.30.40.50.6u\nniform ηδ\n [nm]αintsinusoidal0\n.0000.0020.0040.0060.0080.010 \nFigure 6: Mode pro\fle parameter \u0011(black squares, left axis) and intrinsic damping \u000bint(red circles,\nright axis) as a function of Py thickness \u000e. The mode pro\fle parameter \u0011lies between the value\nof 1=2 for sinusoidal PSSWs with anti-symmetric boundary conditions (dashed black line) and the\nvalue of 1=4 for the uniform mode (both dashed black lines). The intrinsic damping is close to\n\u000bint= 0:006 (dotted red line).\nPermalloy \flms9. Such exchange-mediated damping of the form \u000bex:=Aexk2was originally\npredicted by Baryakhtar based on symmetry alone8. Nembach, et al.10, obtained a value of\nAex= 1:4 nm\u00002, whereas Li, et al.8, found a much smaller value of 0 :09 nm\u00002. To determine\nwhether wavenumber-dependent damping is apparent in our data, we examined the residual\ndamping after subtraction of both the intrinsic damping \u000bintand the radiative damping \u000brad\nn\nfrom all the modes, as well as subtraction of the eddy current damping from the n= 0 mode.\nThe residual damping \u000bresis plotted in Fig. 7 (b). Within the scatter of \u0018=\u00060:001,\u000bresdoes\nnot have any clear dependence on k. Thus, we obtain an upper bound of Aex\u00140:045 nm\u00002\nfor this particular system, given the sensitivity of our measurements. For comparison and to\nensure that the subtraction of \u000bint,\u000brad\nn, and\u000beddydid not hide a potential k2contribution\nthe measured damping of the Ta(3)-Py(200)-Si 3N4(3) sample up to the n= 10 mode is\nshown in Fig. 7 (a). For n\u00155 the measured damping scatters around the for the 200 nm\nsamples determined intrinsic damping \u000bintand no trend for higher mode numbers (larger k\nvalues) is discernible.\n14Tserkovnyak, et al., calculated the damping coe\u000ecient Aexin terms of a microscopic\nmodel for the di\u000busive transport of dissipative transverse spin current within a ferromagnetic\nmetal2. The theory in Ref. [ 2] framed the exchange-mediated damping in terms of a so-called\ntransverse spin conductivity \u001b?,\nAex=\u0012\r\nMs\u0013\u0012~\n2e\u00132\n\u001b?; (13)\nwhere\n\u001b?:=\u0010\u001b\n\u001c\u0011 \n\u001c?\u0000\n1 + (!ex\u001c?)2\u0001!\n; (14)\nwith the exchange splitting ~!ex, the conductivity \u001b, the spin scattering time \u001c, and\ntransverse spin scattering time \u001c?. Given that ~!ex\u00191 eV for Permalloy, the maximum\nvalue forAexpredicted by the transverse spin current theory is 0 :001 nm\u00002. Insofar as\nwe are not able to observe any such wavenumber-dependent damping down to the level of\n0:045 nm\u00002, our results are consistent with the predictions of the microscopic theory.\nWhile the theory in Ref. [ 9] is speci\fc to the microscopic mechanism of transverse spin\naccumulation in a metallic ferromagnet, the phenomenology of exchange mediated damping,\nas described in Ref. [ 8], is not limited to such a microscopic mechanism. As such, it remains\nplausible that extrinsic material-speci\fc parameters that have not yet been identi\fed could\nbe responsible for the previously reported values for k2damping. For example the presence\nof anti-symmetric exchange at interfaces, i.e. the Dzyaloshinskii-Moriya interaction (DMI)\ncould enhance the coupling between magnons and Stoner-excitations insofar as the DMI\ngives rise to exotic spin textures29with nanometer length scales, that are comparable to\nthe wavelength of low energy Stoner-excitations30. Thus, the results of Ref. [ 10] could be\na manifestation of interfacial enhancement for Aex\nn, insofar as the magnetic \flms used in\nRef. [ 10] are only 10 nm thick.\nIn another experiment, we further validate the presence of radiative damping and demon-\nstrate an alternative method to determine \u000brad\nnby varying the distance din Eq. (2) between\nthe sample and waveguide. To this end, we insert a d= 200 \u0016m glass spacer between the\nsample and waveguide. By comparing h(0;0) toh(0;200\u0016m) via Eq. (2), we estimate that\nthe insertion of the spacer decreases the microwave magnetic \feld by about a factor of 6.25.\nReferring to Eq. (3), the normalized mode inductance eLndecreases by a factor of \u0018=40. To\n1502 4 6 8 1 00.0050.0060.0070.0080.0090.0100.0110\n1 x1062x106-0.0010.0000.0010.002( b) \n αresn\nk\n [cm-1] \nαnn\nαint(a)Figure 7: (a) The measured damping for the \frst 11 PSSW modes of the Ta(3)-Py(200)-Si 3N4(3)\nsample. The enhanced damping due to inductive coupling to the waveguide and eddy currents\nin the sample only a\u000bects the \frst \fve modes at wavevectors \u00147\u0002105cm\u00001. (b) The residual\ndamping for all detected modes for all samples is plotted against their respective wave vector k.\nWithin the scatter, no dependence of the residual damping on kis observed.\ndetermine the e\u000bect of the reduced inductive coupling on the radiative damping, we used\nVNA-FMR to measure the \frst 4 modes for the Ta(3)-Py(120) sample with and without\nthe spacer. The e\u000bect of the spacer can be seen in the raw data, reducing the linewidth\nof the \frst two modes measured at 10 GHz in the 120 nm samples by approximately 6 Oe,\nwell outside error bars. The \ftted values of eLnare shown in Fig. 8 (a). Indeed, eLnde-\ncreases on average for all modes by a factor of \u0018=50 after inserting the spacer, in good\n16agreement with the predictions of Eq. (2) and (3). Thus, we will assume that \u000brad\nnis neg-\nligible when the spacer is used. The data for the damping \u000bnof the \frst four modes, both\nwith and without the spacer, are plotted in Fig. 8 (b). Indeed, the damping determined\nfrom the measurement with the spacer layer (circles) is consistently lower than that found\nwithout the spacer layer (squares). The line in Fig. 8 (b) is the previously determined in-\ntrinsic damping. Under the assumption that the radiative damping contribution is given by\n\u000brad\nn=\u000bn(d= 0)\u0000\u000bn(d= 200 \u0016m), we plot \u000brad\nnvseLn(d= 0) in Fig. 8 (c). The line is\nthe calculated \u000brad\nn, where we used Eq. (5) with \u0011= 0:35 and\u000e= 120 nm, as determined\nfrom the \fts in Fig. 5. Good agreement between the calculated and measured values for \u000brad\nn\nare obtained, which demonstrates the self-consistency of our analysis. Of great importance\nis that the spacer-layer approach can also be used to determine the radiative contribution\nto the damping in the absence of PSSWs (single resonance). By measuring \u000bfor varying\ndistancedbetween sample and waveguide and extrapolating \u000btod!1 , both the intrinsic\nvalue for the damping and the radiative contribution can be determined, under conditions\nwhere eddy current damping is negligible.\nVI. SUMMARY\nIn summary, we identi\fed three contributions to the damping in PSSWs: Intrinsic damp-\ning\u000bint, eddy current damping \u000beddy, and radiative damping \u000brad\nn. The latter exhibits a\nlinear dependence on the normalized sample inductance eLnin a waveguide based FMR\nmeasurement. We attribute this linear dependence to radiative losses that stem from the\ninductive coupling between the sample and the waveguide. The radiative damping term is\ninherent to the measurement process and is thus present in all FMR measurements. The\nradiative damping constitutes up to 40 % of the total damping of the spin wave modes\nin our 200 nm thick Permalloy \flms. Furthermore, the radiative damping can be already\nimportant for much lower \flm thicknesses, in materials with small intrinsic damping.\nAs an example, the radiative damping calculated from Eq. (6) for a 20 nm thick and\n1 cm long sample of Yttrium-Iron-Garnet (YIG), measured on a 100 \u0016m wide wave guide is\n\u000brad\n0\u00191:26\u000110\u00004. When compared to the reported value for the damping of \u000b= 2:3\u000110\u00004,31\nwe see, that the radiative part of the damping, among others32, can substantially in\ruence\nthe determination of \u000bint. As such, careful analysis of \u000bvs. inductance is required to isolate\n1701 2 3 1E-30.010.110\n.00 .51 .01 .52 .00.0000.0010.0020.00301 2 3 0.0050.0060.0070.0080.009~d=200µm \nLn[pH]n\n(a)d=0αradn\n~\n \n L\nn(d=0)[pH](c)αint(b) αn n\nFigure 8: Measurement of the \frst four PSSWs of the Ta(3)-Py(120) sample with and without a\nspacer inserted between sample and CPW. (a) Inductance eLndetermined for the sample directly\non the CPW (black squares) and for a 200 \u0016m spacer between sample and CPW (red circles). (b)\nThe resulting damping constants for both measurements (same symbols and colors). The red line is\nthe previously extracted intrinsic damping \u000bint. (c) The di\u000berence between the damping with and\nwithout the spacer (black squares) is in good agreement with the radiative damping from Fig. 5(b)\n(gray line).\nthe radiative damping contribution.\nVII. AKNOWLEDGEMENT\nThe authors are grateful for the assistance of Mikhail Kostylev in the development of our\ntheoretical analysis.\n18VIII. APPENDIX\nA. Derivation of \u000brad\nn\nIn this section, we derive model equations for the normalized mode inductance eLn, mode\namplitudeAn, and the radiative damping \u000brad\nn. We are restricting our analysis to the case\nof ideal perpendicular standing spin wave modes that only vary through the \flm thickness\nwithout any lateral variation. It is assumed that the excitations are in the perpendicular\ngeometry with the magnetization saturated out of the \flm plane. As such, the response to\nthe z-coordinate component of the microwave excitation \feld above the waveguide can be\nneglected. In addition, the magnetization precession is always circular. As such, the magne-\ntization dynamics in the x- and y-coordinates in response to the microwave \feld generated\nby the waveguide are degenerate, outside of a phase factor of \u0019=2. This eliminates the need\nto explicitly consider the full Polder susceptibility tensor in the calculation of the sample\nresponse to the excitation \feld. The sample dimensions are lalong the waveguide direction,\n\u000ein thickness, but in\fnite in the lateral direction.\nWe begin by introducing the concepts of a spin wave mode susceptibility \u001fn, and the di-\nmensionless, normalized spin wave amplitude qn(z) for the nth spin wave mode, such that\nthe magnetic excitation of amplitude in x-direction ~mx\nn(H0;I;z) that results from the ap-\nplication of a microwave magnetic \feld of amplitude hx(I;x;z), given in Eq. (2), driven by\nan ac current I=Vin=Z0in an applied \feld H0is given by\n~mx\nn(H0;I;z) := ~mx\nn(z) =qn(z)\u001fn(H0)hqn(z)hx(I;x;z)i (15)\nwhere the quantity in brackets is simply the overlap integral of the excitation \feld and\nthe spatial pro\fle of the nth spin wave mode. The magnetic excitation of amplitude in\ny-direction ~my\nn(z) can be written in a similar way. In the trivial case of a uniform excitation\n\feld and uniform spin wave mode, we recover the usual relation between the excitation\n\feld and the magnetization dynamics via the Polder susceptibility tensor component, \u001fxx.\nHowever, if the product of the mode pro\fle and excitation \feld has odd spatial symmetry,\ndynamics are not excited, as we expect. The overlap integral is nothing more than the\nspatial average of the mode/excitation product:\nhqn(z)hx(I;x;z)i=1\nW\u000eZ1\n\u00001dxZ\u000e+d\nddzqn(z)hx(I;x;z): (16)\n19First, the power transferred to the waveguide via inductive coupling with the spin wave\ndynamics is given by\nPn=j@t\bn(H0;I)j2\n2Z0; (17)\nwhere\n@t\bn(H0;I) =\u00160`1Z\n\u00001dx\u000e+dZ\nddz(@tmx\nn(z))~hx(x;z); (18)\nwith ~hx(x;z) =hx(I;x;z)=I.\nIt is important to recognize at this point that the power dissipation is not constant with\ntime, given that Pnis proportional only to @tmx\nn. As such, the damping associated with the\nre-radiation of the microwave energy back into the waveguide is best characterized with an\nanisotropic damping tensor, to be elaborated upon more fully later in this Appendix. To\ncalculate the energy of the spin wave mode, we start by de\fning a spatially averaged spin\nwave excitation density33,\n\n\u00162\nn(H0;I)\u000b\n=R1\n\u00001dxR\u000e+d\nddz[(@tmx\nn(z)) (my\nn(z))\u0003\u0000(@tmy\nn(z)) (mx\nn(z))\u0003]\n4!\u000eW: (19)\nWe can then calculate the magnon density Nnassociated with the nth spin wave excitation\nas\nNn=h\u00162\nn(H0;I)i\n2g\u0016BMs(20)\nThe total energy associated with the spin wave mode is given by\nEn=!h\u00162\nn(H0;I)i\n\rMs\u000e`W (21)\nThe energy dissipation rate (1 =T1)nfor the nth mode is therefore\n\u00121\nT1\u0013\nn=Pn\nEn=2\u00160`!M\nZ0\f\f\f\f1R\n\u00001dx\u000e+dR\nddz(@tmx\nn(z))~hx(x;z)\f\f\f\f2\n1R\n\u00001dx\u000e+dR\nddz\u0002\n(@tmx\nn(z)) (my\nn(z))\u0003\u0000(@tmy\nn(z)) (mx\nn(z))\u0003\u0003;(22)\n20where!M=\r\u00160Ms. We then apply the Fourier transform to move into the frequency do-\nmain, where @tmx\nn(H0;I;z)$i!~mx\nn(H0;I;z), such that the energy relaxation rate (1 =T1)x\nn\nfor magnetization oscillations along the x-axis is\n\u00121\nT1\u0013x\nn=!\u00160`!M\nZ0Kn; (23)\nwhere\nKn:=\f\f\f\f1R\n\u00001dx\u000e+dR\nddz( ~mx\nn(z))~hx(x;z)\f\f\f\f2\n1R\n\u00001dx\u000e+dR\nddzIm [ ~mx\nn(z) ( ~mx\nn(z))\u0003](24)\nis a dimensionless inductive coupling parameter. In the limiting case of the n= 0 (i.e.,\nuniform) mode with a uniform excitation \feld due to current \rowing only through the\nwaveguide center conductor, and an in\fnitesimal spacing between the waveguide and the\nsample, we have K0=\u000e=4w. Substituting Eq. (15) into Eq. (24), we obtain the general\nresult\nKn=\f\f\f\f1R\n\u00001dx\u000e+dR\nddzqn(z)~hx(x;z)\f\f\f\f2\n\u000f\u000e+dR\nddzjqn(z)j2; (25)\nwith\u000f=j~mz\nnj=j~mx\nnj.\nSince the energy dissipation rate for the case of radiative damping is anisotropic, it must be\ngenerally treated in the damping tensor formalism, where the Gilbert damping torque ~Tis\ngiven by\nTk=\"ijk\u000bij^mi(@t^m)j: (26)\nThe equation of motion is\n@t^m=\u0000\r\u00160^m\u0002~H+~T (27)\nand ^m=~M=Msis the normalized magnetization. For the coordinates in Fig 1, the\nonly nonzero radiative damping tensor components are \u000bzxand\u000byx. For the perpendicular\n21FMR geometry, the relationship between the energy relaxation rate and the Gilbert damping\ncomponents is\n\u00121\nT1\u0013x\n=\u000bzx!x; (28)\nand\n\u00121\nT1\u0013y\n=\u000bzy!y; (29)\nwhere!xand!yare the respective sti\u000bness frequencies, de\fned as\n!i:=\r\nMs@2Um\n@^mi(30)\nandUmis the magnetic free energy function. The frequency-swept linewidth \u0001 !=\n\r\u00160\u0001H, where \u0001His the \feld-swept linewidth in Eq. (11), is given by\n\u0001!=\u0010\n1\nT1\u0011x\n+\u0010\n1\nT1\u0011y\n2(31)\n=\u000bzx!x+\u000bzy!y (32)\nFor perpendicular FMR, !x=!y=!, and the speci\fc case of anisotropic radiative\ndamping,\u000bzx=\u000brad\nn,\u000bzy= 0, and we obtain\n\u000brad\nn=1\n2!\u00121\nT1\u0013x\nn=\u00160l!M\n2Z0Kn (33)\nand \u0001!rad\nn=\u000brad\nn!. This is in contrast to the case of isotropic damping processes, such\nas eddy currents and intrinsic damping, where we obtain \u0001 !iso\nn= 2\u000biso\nn!instead. Thus,\nthe net damping due to the sum of anisotropic radiative damping, and any other isotropic\nprocesses, is given by\n\u000bn=\u000bint+\u000beddy\nn+\u000brad\nn\n2(34)\nwhere\u000bnis the damping parameter in Eq. (11) for the \feld-swept linewidth.\nWe use a vector network analyzer (VNA) to measure the two-port S-parameter matrix\nelement for the nth spin wave mode, \u0001 S21\nn. The matrix element is de\fned as the ratio of the\nvoltage induced in the waveguide by the nth spin wave mode Vn(H0) in an applied magnetic\n\feldH0, and the excitation voltage Vin,\n22\u0001S21\nn:=Vn(H0)\nVin: (35)\nIf we model the reactance of the nth spin wave mode as nothing more than a purely induc-\ntive element of inductance Lnin series with an impedance matched transmission line, and\nif we assume the sample inductance is much smaller than the transmission line impedance,\nwe can approximate \u0001 S21\nnas\n\u0001S21\nn(H0)\u0018=\u0000i!Ln(H0)\nZ0; (36)\nwhereLn(H0) = \bn(H0;I)=I.\nWe de\fne a normalized, \feld-independent mode-inductance eLnas\neLn:=Ln(H0)\n\u001fn(H0); (37)\nand a dimensionless, \feld-independent mode-amplitude An,\nAn:=i!eLn\nZ0: (38)\nsuch that\n\u0001S21\nn(H0) =\u0000An\u001fn(H0): (39)\nThus,Anis the dimensionless amplitude parameter that we obtain when \ftting data for\n\u0001S21\nn(H0). By use of Eqs. 18, 38, and 37, we can rewrite the mode-amplitude as\nAn:=i!\u00160l\nW\u000eZ 0\u0012Z1\n\u00001dxZ\u000e+d\nddzqn(z)~hx(x;z)\u00132\n: (40)\nRemembering that the normalized mode inductance has a factor identical to the numer-\nator of Eq. (25), we can rewrite the radiative damping in terms of the normalized mode\ninductance,\n\u000brad\nn\neLn=!M\u0011n\nZ0; (41)\nwhere\n\u0011n:=\u000e\n4R\u000e+d\nddzjqn(z)j2(42)\n23We emphasize that Eq. (41) is a very general result, regardless of the details of the\nexcitation \feld pro\fle . Thus, even if the \feld pro\fle is highly non-uniform due to the\ncombination of eddy current and capacitive coupling e\u000bects22,34, there should still be a \fxed\nscaling between the radiative damping and the normalized inductance.\nIn the case of the uniform mode, \u0011= 1=4 and Eq. (41) reduces to\n\u000brad\nn\neLn=!M\n4Z0: (43)\nHowever, for a sinusoidal mode of the form\nqn(z) = cos\u0012(2n+ 1)\u0019z\n2\u000e\u0013\n(44)\nthat is expected in the case of a pinned boundary condition at one interface and an open\nboundary condition at the other interface, as shown in Fig. 1 (b), we obtain \u0011= 1=2 and\n\u000brad\nn\neLn=!M\n2Z0; (45)\nFor the case of the wavenumber values extracted from the data shown in Fig. 2 (c) for a\n200 nm Py \flm, we can determine the value for \u0011nand the degree to which it can vary with\nmode number. We use the following form for the spin wave pro\fle:\nqn(z) = cos (knz) (46)\nconsistent with our assumption, when extracting knfrom our PSSW data, that an un-\npinned boundary condition applies to only one of the interfaces, i.e. at z= 0. Using these\nextracted values for the wavenumber, we obtain values for \u0011nshown in Fig. 9.\nWe see in Fig. 9 that the variation in \u0011nwith varying mode number is less than 10%.\nThus, to within \frst order, we can treat \u0011nas a constant for the purposes of \ftting our data,\ni.e.\u0011n\u0018=\u0011.\nB. Derivation of \u000beddy\nFor the derivation of the eddy current damping \u000beddyuniform magnetization dynamics\nare assumed. The notation stays the same as for the radiative damping.\nThen the total \rux passing through the magnetic \flm is\n2402 4 6 8 10120.00.20.40.60.81.0ηnn\nFigure 9: Dependence of calculated values for \u0011non mode number for the case of the spectral data\npresented in Fig. 2 (c) and 5 (a). Wavenumbers are extracted from those data via the procedure\noutlined in the main text of the paper, based upon a model with a single surface with interfacial\nanisotropy, and an unpinned boundary condition at the other interface. Within the context of that\nparticular model, and the expected quadratic dependence of spin wave resonance frequency with\nwavenumber that it produces, we see that \u0011nhas a weak dependence on mode number, justifying\nour presumption that \u0011ncan be treated as a constant for the purposes of \ftting the damping data.\n@t\b =\u00160`\u000e(@t~ m)x; (47)\nwhere (@t~ m)x= ^x\u0001@t~ m. The electrical power dissipated by the eddy-currents is\nPind=1\n2j@t\bj2\n\u0010\n2\u001a`\n\u000ee\u000bW\u0011 (48)\n=C\n8\u00162\n0\u000e3`W\n\u001aj(@t~ m)xj2; (49)\nwith\u000ee\u000b:=C\u000e=2, where 0\u0014C\u00141 is a phenomenological parameter that accounts for\ndetails of the non-uniform eddy-current distribution in the ferromagnet. Analogous to the\nderivation of the radiative damping, we now need the energy of the magnetic excitations.\nThe number of magnons in the system is given by\nNmag=Ms\ng\u0016B!2j@t~ mj2: (50)\n25Thus, the total magnon energy is\nEmag=~!NmagW`\u000e (51)\n=Ms\n\r!j@t~ mj2W`\u000e: (52)\nThe rate of energy dissipation is then given by\n1\nT1=Pind\nEmag=C\n8\r!\u00162\n0Ms\u000e2\n\u001aj(@t~ m)xj2\nj@t~ mj2: (53)\nThe maximum energy decay rate occurs when ( @t^m)x=j@t^mj, in which case\n\u00121\nT1\u0013x\n=C\n8\r!\u00162\n0Ms\u000e2\n\u001a; (54)\nwhere the superscript indicates that this is the maximum decay rate for magnetization\noscillations along the x-axis. For the case of a perpendicular applied \feld su\u000ecient to\nsaturate the static magnetization out of the \flm plane, the damping process is isotropic,\ni.e.,\n\u00121\nT1\u0013y\n=C\n8\r!\u00162\n0Ms\u000e2\n\u001a: (55)\nTherefore, analogous to Eq. 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Vieira,1\n1Universidade Federal do Triˆ angulo Mineiro - Campus Ituram a,\nIturama, MG 38280-000, Brasil\nIn this work, we show that surface terms, which map dependenc e on regular-\nization, can be fixed requiring momentum routing invariance of tadpoles or\ndiagrams with more external legs. This condition makes the L orentz-violating\nterms induced by quantum corrections determined and unique s.\nLorentz and CPT violating operators reappear in quantum correct ions\nand are called induced terms. If the coupling generates a loop with a fi nite\nintegral, the induced term is well defined1. On the other hand, if the loop\ncorrection is divergent, it requires a regularization and a renormaliz ation\nscheme. Thechoiceoftheformermightleadtospurioussymmetryb reaking\ntermsandtoambiguousinducedterms. Sincethefinitepiecesofamp litudes\nof the Standard Model Extension (SME) are in principle observable, they\nshould not depend on this choice. However, symmetries of the built m odel\nare used to decide what kind of induced term is allowed and we can also\nfind out if spurious terms appeared because of the choice of regula rization2.\nThe question if a symmetry of the classical theory is also a symmetry of\nthe quantum theory is non-trivial. The breaking of a classical symme try at\nthe quantum level is called an anomaly and the ideal regularization sho uld\npreserve all symmetries so that it can tell us if the anomaly is physica l\nor spurious. In the SME, this question is even more interesting beca use\nLorentz or CPT symmetries are broken at the classical theory by o ne of the\nSME coefficients but the same coefficient might not appear at the qua ntum\nlevel or it may be a correction to a tree level term from a different se ctor.\nFor instance, the bµcoefficient of the fermion sector appears as a correction\nto the CS-like term ( kAF)µfrom the photon sector.\nThere is no such ideal regularization but the spurious breaking of th e\nsymmetries is usual related to the assumption of a regulator. If it is not\nassumed, the so called implicit regularization3, the breaking of the symme-\ntries is mapped on surface terms. These surface terms manifest t hemselves\nas differences between two infinities. They can be any number and de pendProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n2\non the regularization scheme. They are born zero in dimensional reg ular-\nization and if explicit computed, with a hard cutoff for instance, furn ish a\nfinite ora divergentresult. There are severalexamples where the y show up,\nespecially when computing the breaking of a classical symmetry by qu an-\ntum corrections. Let is consider the QED vacuum polarization tenso r as\nan example:\nΠµν(p) =4\n3(p2ηµν−pµpν)Ilog(m2)−4υ2ηµν−\n+4\n3(p2ηµν−pµpν)υ0−4\n3(p2ηµν+2pµpν)(ξ0−2υ0)−\n−8i\n(4π)2(p2ηµν−pµpν)/integraltext1\n0x(1−x)logm2−p2x(1−x)\nm2, (1)\nwhereIlog(m2) =/integraltextd4k\n(2π)41\n(k2−m2)2is a logarithmic basic divergent integral,\nυ2is a quadratic surface term, υ0andξ0arelogarithmic surface terms. The\nbasic divergent integrals are defined as\nIµ1···µ2n\nlog(m2)≡/integraldisplayd4k\n(2π)4kµ1···kµ2n\n(k2−m2)2+n(2)\nand\nIµ1···µ2n\nquad(m2)≡/integraldisplayd4k\n(2π)4kµ1···kµ2n\n(k2−m2)1+n, (3)\nwhile the surface terms are defined according to the relations∗\nυ2wgµν=gµνI2w(m2)−2(2−w)Iµν\n2w(m2),\nξ2wg{µνgαβ}=g{µνgαβ}I2w(m2)−4(3−w)(2−w)Iµναβ\n2w(m2),\nσ2wg{µνgαβgγδ}=g{µνgαβgγδ}I2w(m2)−\n−8(4−w)(3−w)(2−w)Iµναβγδ\n2w(m2), (4)\nwhere 2wis the degree of divergence and we substitute the subscripts log\nandquadby 0 and 2, respectively.\nThe Ward-Takahashi identity ( pµΠµν= 0) we get with equation (1)\nreveals the quadratic surface term υ2is zero and the relation ξ0= 2υ0.\nThus, it is not always possible to fix all the surface terms requiring a s ym-\nmetry. Another example similar to this one is the induced CS-like term\nin the massless case. The result is Πµν\n5(p) = 4iυ0bαpβǫναµβ4, where we\ncan see the logarithmic surface term υ0makes the induced term undeter-\nmined. Also, the Ward-Takahashi identity is compatible with any value of\nthis surface term.\n∗The curly brackets stand forpermutation of indices, A{µνBαβ}=AµνBαβ+AµαBνβ+\nAµβBνα, for instance.Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n3\nAt the same time, there is a freedom of choosing the momentum rout ing\nin the internal lines of loop diagrams. If we set an arbitraryrouting in these\nlines, constrained by momentum conservation at the vertices, this arbitrary\nrouting is always accompanied by a surface term. For example, in the\nequation Πµν(p,l) = Πµν(p,l′), where landl′are arbitrary routing. A\ntextbook example where we can apply that is the axial anomaly. The u sual\nprocedure in its diagrammatic computation is choosing the routing so as to\nfulfill the gauge Ward-Takahashi identities and break the axial cur rent by\nthe known finite amount. However, this does not necessarily mean t hat the\nanomaly depends on the routing. It is possible to show in a diagrammat ic\ncomputationthatthisresultisvalidforanarbitraryrouting5. Furthermore,\nthere is a one-to-one diagrammatic relation between gauge symmet ry and\nmomentum routing invariance in abelian and non-abelian theories. Figu re\n1 shows the pictorial representation of a gauge Ward-Takahashi identity\nwhere we can see an external photon leg being inserted wherever is possible\nina2−pointfunctiondiagram,generatingadifferencebetweentwo1 −point\nfunction diagramswith differentrouting. Asimilarrelationcanbe obta ined\nbeyond one-loop and for diagrams with more external legs.\nFig. 1. The gauge and momentum routing invariance relation f or a 2-point function\ndiagram. The dot refers to the QED extension matrix Γν=γν+cµνγµ+dµνγ5γµ+\neν+ifνγ5+1\n2gλµνσλµ.\nThe gauge and momentum routing invariance relation can be compute d\nwith implicit regularization and reveals a set of equations for the surf ace\nterms. In the QED extension this result yields for the cµνcoefficient:\npµΠµν(p) =τν(p)−τν(0) =−16(3pνcpp+p2cνp+p2cpν)(ξ0−υ0)+\n+4(p2meν+2mpνe·p)(2υ0−ξ0)+4σ0(2pνcpp+p2cνp+p2cpν)+\n+4(ξ2−2υ2)(cνp+cpν) = 0, (5)\nwhereτν(p) is the tadpole for the full fermion propagator and cpν≡pµcµν.Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n4\nWe see in equation (5) that the only possible solution is to make all\nsurface terms zero. Thus, it is possible to fix the arbitrariness req uir-\ning momentum routing invariance and this automatically fulfill the gaug e\nWard-Takahashi identity. If the surface terms were computed w ith an ex-\nplicit regulator like a hard cutoff, we would find spurious breaking of ga uge\ninvariance at one-loop. However, QED extension is gauge invariant b eyond\ntree level6. In this sense, we have determined induced terms. Some ex-\namples of the one-loop vacuum polarization tensor of the QED exten sion\nare\nΠµν\nb(p) =/parenleftbigg\n4ie2υ0+e2m2\nπ2ι0/parenrightbigg\nbαpβǫαβνµ=e2m2\nπ2ι0bαpβǫαβνµ\nΠµν\nd(p) = 2ie2(dλα+dαλ)ǫλµνβpβpα(υ0−ξ0) = 0\nΠµν\ne,a(p) =−4e2(ηµν(me·p−a·p)+pµ(meν−aν)+\n+pν(meµ−aµ))(ξ0−2υ0) = 0, (6)\nwhereι0=/integraltext1\n0dx(1−x)\nm2−p2x(1−x).\nWe see in eq. (6) that we do not expect the induction of dµν,eµor\naµcoefficients and the mass term in the induced CS-like term avoids the\ninfrared divergence. This is in agreement with recent results7where only\ngµναandbµcoefficients appear in the low-energy limit of the one-loop\nvacuum polarization tensor.\nReferences\n1. F. A. Brito, J. R. Nascimento, E. Passos, and A. Yu. Petrov, Phys. Lett. B\n664, 112 (2008).\n2. R. Jackiw, Int. J. of Mod. Phys. B 14, 2011 (2000); B. Altschul, Phys. Rev.\nD.99, 125009 (2019).\n3. O.A. Battistel, A.L. Mota, M.C. Nemes, Mod. Phys. Lett. A 13, 1597 (1998).\n4. J.C.C. Felippe, A.R. Vieira, A.L. Cherchiglia, A.P.B. Sc arpelli and M. Sam-\npaio, Phys. Rev. D 89, 105034 (2014).\n5. A.C.D. Viglioni, A.L. Cherchiglia, A.R. Vieira, B. Hille r and M. Sampaio,\nPhys. Rev. D 94, 065023 (2016).\n6. T.R.S. Santos, R.F. Sobreiro, Phys. Rev. D 94, 125020 (2016); A.R. Vieira,\nA.L. Cherchiglia and M. Sampaio, Phys. Rev. D 93, 025029 (2016).\n7. V.A. Kostelecky, R. Lehnert, N. McGinnis, M. Schreck, B. S eradjeh, Phys.\nRev. Res. 4, 023106 (2022)."    },    {        "title": "2104.04596v2.New_binary_pulsar_constraints_on_Einstein_æther_theory_after_GW170817.pdf",        "content": "New Binary Pulsar Constraints on Einstein-æther\nTheory after GW170817\nToral Gupta1, Mario Herrero-Valea2;3, Diego Blas4, Enrico\nBarausse2;3, Neil Cornish1, Kent Yagi5and Nicolás Yunes6\n1eXtreme Gravity Institute, Department of Physics, Montana State University,\nBozeman, Montana 59717, USA.\n2SISSA, Via Bonomea 265, 34136 Trieste, Italy & INFN, Sezione di Trieste.\n3IFPU - Institute for Fundamental Physics of the Universe,\nVia Beirut 2, 34014 Trieste, Italy.\n4Theoretical Particle Physics and Cosmology Group, Department of Physics, King’s\nCollege London, Strand, London WC2R 2LS, UK.\n5Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA.\n6Illinois Center for Advanced Studies of the Universe, Department of Physics,\nUniversity of Illinois at Urbana-Champaign, Urbana, Illinois, 61820 USA.\nE-mail: toralgupta@montana.edu\nAbstract. The timing of millisecond pulsars has long been used as an exquisitely\nprecise tool for testing the building blocks of general relativity, including the strong\nequivalence principle and Lorentz symmetry. Observations of binary systems involving\nat least one millisecond pulsar have been used to place bounds on the parameters\nof Einstein-æther theory, a gravitational theory that violates Lorentz symmetry\nat low energies via a preferred and dynamical time threading of the spacetime\nmanifold. However, these studies did not cover the region of parameter space that\nis still viable after the recent bounds on the speed of gravitational waves from\nGW170817/GRB170817A. The restricted coverage was due to limitations in the\nmethods used to compute the pulsar “sensitivities”, which parameterize violations of\nthe strong-equivalence principle in these systems. We extend here the calculation\nof pulsar sensitivities to the parameter space of Einstein-æther theory that remains\nviable after GW170817/GRB170817A. We show that observations of the damping of\nthe period of quasi-circular binary pulsars and of the triple system PSR J0337+1715\nfurther constrain the viable parameter space by about an order of magnitude over\nprevious constraints.\nPACS numbers: 04.30Db,04.50Kd,04.25Nx,97.60Jd\nSubmitted to: Class. Quantum Grav.arXiv:2104.04596v2  [gr-qc]  27 Aug 2021Updated Binary Pulsar Constraints on Einstein-æther theory 2\n1. Introduction\nLorentz symmetry has been the foundation of the magnificent edifice of theoretical\nphysics for more than a century, playing a central role in special and general relativity\n(GR), as well as in the quantum theory of fields. Because of its special status,\nLorentz invariance has been tested to exquisite precision in the matter sector via\nparticle physics experiments [49, 50, 54, 45]. More recently, this experimental program\nhas been extended to the matter-gravity [48], dark matter [16, 13], and pure-gravity\nsectors [42, 53], where bounds on Lorentz violations (LVs) have been historically looser\n(because of the intrinsic weakness of the gravitational interaction).\nCompelling theoretical reasons to seriously consider the possibility of LVs in the purely\ngravitational sector were provided by the realization that they could generate a better\nbehavior in the ultraviolet (UV) limit. In particular, P. Hořava [39] showed that\nby allowing for a non-isotropic scaling between space and time, one can construct\na theory that is power-counting renormalizable in the UV. Renormalizability beyond\npower counting (i.e. pertubative renormalizability) in special (“projectable”) versions of\nHořava gravity has also been proven [10].\nThe low-energy limit of Hořava gravity reduces to “khronometric theory” [15, 43], which\nconsists of GR plus an additional hypersurface-orthogonal and timelike vector field,\noften referred to as the “æther”. Because this vector field is hypersurface orthogonal,\nit selects a preferred spacetime foliation, which makes LVs manifest. A more general\nboost-violating low-energy gravitational theory, however, can be obtained by relaxing\nthe assumption that the æther be hypersurface-orthogonal, in which case it selects a\npreferredtime threading ofthespacetimeratherthanapreferredfoliation. Theresulting\ntheory is known as Einstein-æther theory [46].\nDespite allowing for an improved UV behavior, LVs in gravity face long-standing\nexperimental challenges, particularly when it comes to their percolation into the matter\nsector, where particle physics experiments are in excellent agreement with Lorentz\nsymmetry. While some degree of percolation is inevitable, because of the coupling\nbetweenmatterandgravity, mechanismssuppressingithavebeenputforward, including\nsuppression by a large energy scale [58], or the effective emergence of Lorentz symmetry\nat low energies as a result of renormalization group flows [23, 12, 11] or accidental\nsymmetries [38].\nAt the same time, purely gravitational bounds on LVs are becoming increasingly\ncompelling. The parameters (“coupling constants”) of both Einstein-æther and\nkhronometric theory have been historically constrained by theoretical considerations\n(absence of ghosts and gradient instabilities [18, 41, 37], well-posedness of the\nCauchy problem [60]), by the absence of vacuum Cherenkov cascades in cosmic-\nray experiments [30]), by solar-system tests [69, 34, 18, 19, 55], by observations ofUpdated Binary Pulsar Constraints on Einstein-æther theory 3\nthe primordial abundances of elements from Big-Bang nucleosynthesis [22], by other\ncosmological tests [8], and by precision timing of binary pulsars (where LVs generically\npredict violations of the strong equivalence principle) [33, 32, 73, 74, 9]. More recently,\nthe coincident detection [3, 2] of gravitational waves (GW170817) and gamma rays\n(GRB170817A) emitted by the coalescence of two neutron stars and the subsequent\nkilonova explosion has allowed extremely strong constraints on the propagation speed\nof gravitational waves, which must equal that of light to within z10\u000015[1], which in turn\nplaces even more stringent bounds on the couplings of both theories [31, 59, 60, 56].\nThe bounds from the coincident GW170817/GRB170817A observations force us to\nrethink the parameter spaces of both Einstein-æther and khronometric theory, as the\nonly currently allowed regions appear to be ones that were previously thought to be\nof little interest, and which were not explored extensively. In the case of khronometric\ntheory, Refs. [59, 9] found that the couplings that remain viable after GW170817 and\nGRB170817 produce exactly no deviations away from the predictions of GR, not only\nin the solar system, but also in binary systems of compact objects, be they black holes\n(BHs) or neutron stars (NSs), to leading post-Newtonian (PN) order. Reference [35]\nextended this result to the quasinormal modes of spherically symmetric black holes and\nto fully non-linear (spherical) gravitational collapse, where again no deviations from\nthe GR predictions are found. It would therefore seem that the most promising avenue\nto further test khronometric theory may be provided by cosmological observables (e.g.\nBig-Bang nucleosynthesis abundances or CMB physics), where the viable couplings do\nproduce non-vanishing deviations away from the GR phenomenology.\nLike for khronometric theory, the parameter space where detailed predictions for\nisolated/binary pulsars were obtained in Einstein-æther theory [73, 74] does notinclude\nthe region singled out by the combination of the GW170817/GRB170817A bound and\nexisting solar-system constraints (see Ref. [60] for a discussion). The goal of this paper\nis therefore to extend the previous analysis of binary/isolated-pulsar data by some of\nus [73, 74] to this region of parameter space. This will require a significant modification\nof the formalism that Refs. [73, 74] utilized to calculate pulsar “sensitivities”, i.e. the\nparameters that quantify violations of the strong-equivalence principle in these systems.\nMoreover, we will extend our analysis to include additional data over that considered\nin Refs. [73, 74], namely the triple system PSR J0337+1715 [7]. Overall, we find that\nobservations of the damping of the period of quasi-circular binary pulsars, and that of\nthe triple system PSR J0337+1715, reduce the viable parameter space of Einstein-æther\ntheory by about an order of magnitude over previous constraints.\nWe will also amend an error (originally pointed out in Ref. [70]) in the calculation of\nthe strong-field preferred-frame parameters ^\u000b1and^\u000b2for isolated pulsars, which were\npresented in Refs. [73, 74]. While we have checked that this error does not impact the\nbounds presented in Refs. [73, 74], we present in Appendix A a detailed derivation of\nzSee also [24] for looser bounds coming from mergers of black holes.Updated Binary Pulsar Constraints on Einstein-æther theory 4\n^\u000b1and ^\u000b2for possible future applications, also correcting a few typos present in the\noriginal calculation of Ref. [70].\nThis paper is organized as follows. In Sec. 2 we give a succinct introduction to Einstein–\næther theory, including the modified field equations and the current observational\nbounds on the coupling constants. In Sec. 3 we introduce the concept of stellar\nsensitivities as parameters regulating violations of the strong equivalence principle.\nSolutionsdescribingslowlymovingstarsarederivedinSec.4, andtheyareusedinSec.5\nto compute the sensitivities. Section 6 uses the sensitivities to obtain the constraints on\nEinstein–æther theory resulting from observations of binary and triple pulsar systems.\nWe summarize our conclusions in Sec. 7. Appendix A contains a calculation of the\nstrong-field preferred-frame parameters ^\u000b1and ^\u000b2in Einstein-æther theory, fixing an\noversight in [73], which was pointed out by [70], and correcting also a few typos present\nin [70] itself. We will adopt units where c= 1and a signature +\u0000\u0000\u0000, in accordance\nwith most of the literature on Einstein-æther theory.\n2. Einstein æther theory\nIn order to break boost (and thus Lorentz) symmetry, Einstein-æther theory introduces\na dynamical threading of the spacetime by a unit-norm, time-like vector field U. This\nvector field, often referred to as the æther, physically represents a preferred “time\ndirection” at each spacetime event. Requiring the action to also include the usual\nspin-2 graviton of GR, to be quadratic in the æther derivatives, and to feature no direct\ncoupling between the matter and the æther (so as to enforce the weak equivalence\nprinciple, i.e. the universality of free fall, and the absence of matter LVs at tree level),\none obtains the action [46, 44]\nS=\u00001\n16\u0019GZh\nR+1\n3c\u0012\u00122+c\u001b\u001b\u0016\u0017\u001b\u0016\u0017+c!!\u0016\u0017!\u0016\u0017+caA\u0016A\u0016\n+\u0015(U\u0016U\u0016\u00001)ip\u0000gd4x+Smat( ;g\u0016\u0017); (1)\nwhereRis the four-dimensional Ricci scalar, gthe determinant of the metric, G\nthe bare gravitational constant (related to the value GNmeasured locally by GN=\nG=(1\u0000ca=2)[22, 42]), collectively denotes the matter degrees of freedom, \u0015is a\nLagrange multiplier enforcing the æther’s unit norm, c\u0012,c\u001b,c!andcaare dimensionless\nconstantsx, and we have decomposed the æther congruence into the expansion \u0012, the\nxNote that much of the earlier literature on Einstein-æther theory uses a different set of coupling\nconstantsci(i= 1;:::; 4), which are related to our parameters by c1= (c!+c\u001b)=2,c2= (c\u0012\u0000c\u001b)=3,\nc3= (c\u001b\u0000c!)=2andc4=ca\u0000(c\u001b+c!)=2.Updated Binary Pulsar Constraints on Einstein-æther theory 5\nshear\u001b\u0016\u0017, the vorticity !\u0016\u0017and the acceleration A\u0016as follows:\nA\u0016=U\u0017r\u0017U\u0016; (2)\n\u0012=r\u0016U\u0016; (3)\n\u001b\u0016\u0017=r(\u0017U\u0016)+A(\u0016U\u0017)\u00001\n3\u0012h\u0016\u0017; (4)\n!\u0016\u0017=r[\u0017U\u0016]+A[\u0016U\u0017]; (5)\nwithh\u0016\u0017=g\u0016\u0017\u0000U\u0016U\u0017the projector onto the hyperspace orthogonal to U.\nBy varying the action with respect to the metric, the æther and the Lagrange multiplier,\nand by eliminating the latter from the equations, one obtains the generalized Einstein\nequations\nE\u000b\f\u0011G\u000b\f\u0000TÆ\n\u000b\f\u00008\u0019GTmat\n\u000b\f= 0 (6)\nand the æther equations\nÆ\u0016=\u0014\nr\u000bJ\u000b\u0017\u0000\u0012\nca\u0000c\u001b+c!\n2\u0013\nA\u000br\u0017U\u000b\u0015\nh\u0016\u0017= 0; (7)\nwhereG\u000b\fis the Einstein tensor, the æther stress-energy tensor is\nTÆ\n\u000b\f=r\u0016\u0000\nJ(\u000b\u0016U\f)\u0000J\u0016\n(\u000bU\f)\u0000J(\u000b\f)U\u0016\u0001\n+c!+c\u001b\n2[(r\u0016U\u000b)(r\u0016U\f)\u0000(r\u000bU\u0016)(r\fU\u0016)]\n+U\u0017(r\u0016J\u0016\u0017)U\u000bU\f\u0000\u0012\nca\u0000c\u001b+c!\n2\u0013\u0002\nA2U\u000bU\f\u0000A\u000bA\f\u0003\n+1\n2M\u001b\u001a\n\u0016\u0017r\u001bU\u0016r\u001aU\u0017g\u000b\f;\n(8)\nwith\nJ\u000b\n\u0016\u0011M\u000b\f\n\u0016\u0017r\fU\u0017;\nM\u000b\f\n\u0016\u0017=\u0012c\u001b+c!\n2\u0013\nh\u000b\fg\u0016\u0017+\u0012c\u0012\u0000c\u001b\n3\u0013\n\u000e\u000b\n\u0016\u000e\f\n\u0017+\u0012c\u001b\u0000c!\n2\u0013\n\u000e\u000b\n\u0017\u000e\f\n\u0016+caU\u000bU\fg\u0016\u0017;\nand the matter stress-energy tensor is defined as usual by\nT\u000b\f\nmat\u0011\u00002p\u0000g\u000eSmat\n\u000eg\u000b\f: (9)\nAs already mentioned, a number of experimental and theoretical results constrain\nEinstein-æthertheoryandthecouplings ci. Inmoredetail,perturbingthefieldequations\nabout Minkowski space yields propagation equations for spin-0 (i.e. scalar), spin-1\n(i.e. vector) and spin-2 (i.e. tensor gravitons) modes. Their propagation speeds are\nrespectively given by [41]\nc2\nT=1\n1\u0000c\u001b; (10)\nc2\nV=c\u001b+c!\u0000c\u001bc!\n2ca(1\u0000c\u001b); (11)\nc2\nS=(c\u0012+ 2c\u001b)(1\u0000ca=2)\n3ca(1\u0000c\u001b)(1 +c\u0012=2): (12)Updated Binary Pulsar Constraints on Einstein-æther theory 6\nIn order to ensure stability at the classical level (i.e. no gradient instabilities) and at\nthe quantum level (i.e. no ghosts) one needs to have c2\nT>0,c2\nV>0andc2\nS>0[41, 37].\nIf we also require the modes to carry positive energy, we get ca>0andc!>0[29].\nFurthermore, significantly subluminal graviton propagation would cause ultrarelativistic\nmatter to lose energy to gravitons via a Cherenkov-like process [30]. Since this effect\nis not observed e.g. in ultrahigh energy cosmic rays, one must have c2\nI&1\u0000O(10\u000015)\n(withI=T;V;S). More recently, the coincident detection of a neutron-star merger in\nGW170817 (gravitational waves) and GRB170817A (gamma rays) had led to the bound\n\u00003\u000210\u000015<cT\u00001<7\u000210\u000016[1].\nExpanding the field equations through 1PN order leads to the conclusion that\nthe 1PN dynamics is well described (like in GR) by the parametrized PN (PPN)\nexpansion [69, 55]. However, unlike in GR, the preferred frame parameters \u000b1and\n\u000b2appearing in the PPN expansion do not vanish, but are given by [34]\n\u000b1= 4c!(ca\u00002c\u001b) +cac\u001b\nc!(c\u001b\u00001)\u0000c\u001b; (13)\n\u000b2=\u000b1\n2+3(ca\u00002c\u001b)(c\u0012+ca)\n(2\u0000ca)(c\u0012+ 2c\u001b): (14)\nSolar system experiments require j\u000b1j.10\u00004andj\u000b2j.10\u00007[69, 55]. By saturating\nthese bounds (i.e. requiring in particular that j\u000b1j.10\u00004butnotj\u000b1j \u001c 10\u00004)\nand combining them with the constraints on the propagation speeds, one finds c\u001b\u0019\nO(10\u000015),ca\u0019 O (10\u00004), andc\u0012\u00193ca[1 +O(10\u00003)]. The resulting experimentally\nviable parameter space, therefore, is effectively (i.e. to within a fractional width of\n10\u00004or better in the parameters) one-dimensional: c\u001b; ca; c\u0012\u00190, butc!is essentially\nunconstrained [60].\nAnother viable region of the parameter space can be obtained by notsaturating the\nPPN constraints [60]. In more detail, one may require j\u000b1jbe much smaller than its\nupper limit, so as to automatically satisfy the bound on \u000b2(since\u000b2/\u000b1ifc\u001b\u00190,\nas imposed by GW170817 and GRB170817A). This leads to jcaj.10\u00007and thus to an\neffectively two-dimensional experimentally viable parameter space (c\u0012;c!), with the only\nadditional requirement that jc\u0012j.0:3to ensure that the production of light elements\nduring Big Bang Nucleosynthesis gives predictions in agreement with observations [22].\nBoth of the viable regions of parameter space identified above were notconsidered\nin Refs. [73, 74], where neutron-star sensitivities in Einstein-æther theory were first\ncomputed. This is because back when Refs. [73, 74] were written, the strongest\nconstraints available were the solar system ones (since the GW170817/GRB170817A\nconstraint was not yet available). Refs. [73, 74] solved for caandc\u0012in terms of c\u001b,\nc!,\u000b1and\u000b2, and fixed the latter two to their largest allowed values (respectively\n\u000b1= 10\u00004and\u000b2= 10\u00007). Therefore, (i)this does not include the first region listed\nabove [ (c\u0012;c\u001b;ca).10\u00004withc!kept free], which is obtained by choosing \u000b1/\u000b2; andUpdated Binary Pulsar Constraints on Einstein-æther theory 7\n(ii)Refs. [73, 74] did not sample accurately enough the sub-region c\u001b\u00190,\u000b1.10\u00004,\n\u000b2.10\u00007, since there was no reason to do so at that time and because the techniques\nemployed broke down in that sub-region (as we will show in the following).\n3. Strong-equivalence principle violations and sensitivities\nMost theories extending/modifying GR involve additional degrees of freedom besides\nthe massless tensor graviton of GR. These additional gravitational polarizations cannot\ndirectly couple with matter significantly, to avoid introducing unwanted fifth forces\nin particle physics experiments, and to prevent violations of the weak equivalence\nprinciple(andparticularlyviolationsoftheuniversalityoffreefallforweaklygravitating\nobjects). Nevertheless, effective couplings between the extra gravitons and matter may\nbe mediated by the metric perturbations (i.e. by the tensor gravitons present also in\nGR), which are typically coupled non-minimally to the extra gravitational degrees of\nfreedom. These effective couplings become important when the metric perturbations\nare “large”, which is the case for strongly gravitating systems such as those involving\nNSs and/or BHs.\nA useful way to parametrize this effective coupling is provided by the sensitivity\nparameters. Because of the aforementioned effective couplings, the mass of strongly\ngravitating objects will be comprised not only of the contributions from matter and\nthe metric (like in GR), but it will also generally depend on the additional gravitational\nfields. We can thus describe isolated objects, and members of a widely separated binary,\nby a point particle model (like in GR), but with a non-constant mass depending on the\nextrafields. Becausethemassisascalarquantity,itmustdependonascalarconstructed\nfrom the æther field U, the simplest of which is the Lorentz factor \r\u0011u\u0001U, where u\nis the particle’s (i.e. the body’s) four-velocity.\nIn many practical situations (including the long inspiral of a binary system of compact\nobjects) one may assume that the relative speed between the æther and the object is\nsmall compared to the speed of light, and thus Taylor-expand the mass \u0016(\r)around\n\r= 1:\n\u0016(\r) = ~m\u0014\n1 +\u001b(1\u0000\r) +1\n2\u001b0(1\u0000\r)2+:::\u0015\n(15)\nwhere ~m,\u001band\u001b0are constant parameters. In particular, the latter two are often\nreferred to as the “sensitivities” and their derivatives:\n\u001b\u0011\u0000d ln\u0016(\r)\nd ln\r\f\f\f\n\r=1; (16)\n\u001b0\u0011\u001b+\u001b2+d2ln\u0016(\r)\nd(ln\r)2\f\f\f\n\r=1: (17)\nInordertounderstandtheeffectofthesensitivitiesandtheirderivativesonthedynamicsUpdated Binary Pulsar Constraints on Einstein-æther theory 8\nof binary systems, one can derive the equations of motion simply by varying the point\nparticle action\nSpp=\u0000X\nAZ\n\u0016A(\rA)d\u001cA; (18)\nwhereAis an index identifying the objects, and \u001cis the proper time. This yields the\nequation of motion\n[\u0016A(\rA)\u0000\u00160\nA(\rA)\rA]aA\n\f=\u0000\u00160\nA(\rA)(\u0000u\u0016\nAr\fUA\n\u0016+u\u0016\nAr\u0016UA\n\f)\u0000\u001600\nA(\rA) _\rA(UA\n\f\u0000\rAuA\n\f);\n(19)\nwhere again the index Aidentifies the particle under consideration (when used in \u0016,\r\nandu) or at which position the æther field Uand its acceleration Aare to be computed,\nthe prime denotes a derivative with respect to the function’s argument, and the overdot\nrepresents a derivative along u(i.e. with respect to the proper time).\nReinstating the dependence on the speed of light cand expanding in PN orders (i.e. for\nc!1), one obtains the 1PN equations of motion for a binary as\ndvA\ndt=\u0000mBn\nr2n\nGAB\u0000(3GABBAB+DABB)mB\nr\n\u00001\n2\u0002\n2G2\nAB+ 6GABB(AB)+ 2DBAA+GAB(CAB+EAB)\u0003mA\nr\n+1\n2[3BAB\u0000GAB(1 +AA)]v2\nA+1\n2(3B21+GAB+EAB)v2\nB\n\u00001\n2\u0000\n6B(AB)+ 2GAB+CAB+EAB\u0001\nvA\u0001vB\u00003\n2(GAB+EAB) (n\u0001vB)2o\n+mBvA\nr2n\u0001f[3BAB+GAB(1 +AA)]vA\u00003BABvBg\n\u00001\n2mBvB\nr2n\u0001\u0002\u0000\n6B(AB)+ 2GAB+CAB+EAB\u0001\nvA\u0000\u0000\n6B(AB)+CAB\u0000EAB\u0001\nvB\u0003\n\u00001\n2mBw\nr2n\u0001\u0002\u0000\nCAB\u00006B[AB]+EAB\u00002GABAA\u0001\nvA\u0000\u0000\nCAB\u00006B[AB]\u0000EAB\u0001\nvB\u0003\n;\n(20)\nwhere indices A6=B,r\u0011jxA\u0000xBj,n\u0011(xA\u0000xB)=r,vA\u0011dx=dt, and\nGAB =GN\n(1 +\u001bA)(1 +\u001bB);\nAA=\u0000\u001b0\nA\n1 +\u001bA;\nBAB =GAB(1 +\u001bA);\nDABB=GAB2(1 +\u001bA);\nCAB =GAB[\u000b1\u0000\u000b2\u00003 (\u001bA+\u001bB)\u0000QAB\u0000RAB];\nEAB =GAB[\u000b2+QAB\u0000RAB];\nQAB=\u00001\n2\u00122\u0000ca\n2c\u001b\u0000ca\u0013\n(\u000b1\u00002\u000b2)(\u001bA+\u001bB) + 3\u00122\u0000ca\n2c\u001b+c\u0012\u0013\n\u001bA\u001bB;\nRAB=1\n2\u00128 +\u000b1\nc!+c\u001b\u0013\n[\u0000c!(\u001bA+\u001bB) + (1\u0000c!)\u001bA\u001bB]: (21)Updated Binary Pulsar Constraints on Einstein-æther theory 9\nNote that we have defined the “active” masses mA\u0011~mA(1 +\u001bA), in terms of which\nthe Newtonian acceleration matches the GR result, albeit with a rescaled gravitational\nconstantGAB. To derive Eq. (20) we have also used the PN-expanded solutions for the\nmetric and æther found in [33, 73] (dropping divergent terms due to the point particle\napproximation, as usual in PN calculations):\ng00= 1\u00002GN~m1\nr1c2+1\nc4\u00142G2\nN~m2\n1\nr2\n1+2G2\nN~m1~m2\nr1r2+2G2\nN~m1~m2\nr1r12\u00003GN~m1\nr1v2\n1(1 +\u001b1)\u0015\n+ 1$2 +O(1=c6); (22)\ng0i=\u00001\nc3\u0014\nB\u0000\n1GN~m1\nr1vi\n1+B+\n1GN~m1\nr1vj\n1nj\n1ni\n1\u0015\n+ 1$2 +O(1=c4); (23)\ngij=\u0000\u0012\n1 +1\nc22GN~m1\nr1\u0013\n\u000eij+ 1$2 +O(1=c4); (24)\nU0= 1 +1\nc2GN~m1\nr1+ 1$2 +O(1=c4); (25)\nUi=1\nc3GN~m1\nr1\u0000\nC\u0000\n1vi\n1+C+\n1vj\n1nj\n1ni\n1\u0001\n+ 1$2 +O(1=c5); (26)\nB\u0006\nA\u0011\u00063\n2\u00002\u00061\n4(\u000b1\u00002\u000b2)\u0012\n1 +2\u0000ca\n2c\u001b\u0000ca\u001bA\u0013\n\u00002c!\nc!+c\u001b\u001bA\u00001\n4\u000b1\u0012\n1 +c!\nc!+c\u001b\u001bA\u0013\n;\n(27)\nC\u0006\nA\u00111\n4\u00128 +\u000b1\nc!+c\u001b\u0013\n[c!\u0000(1\u0000c!)\u001bA]\u00062\u0000ca\n2\u00122\u000b2\u0000\u000b1\n2(2c\u001b\u0000ca)+3\u001bA\n2c\u001b+c\u0012\u0013\n; (28)\nwhererA\u0011jx\u0000xAjandnA\u0011(x\u0000xA)=rA.\nNote that the æther solution (25)–(26) has space components Uivanishing at large\ndistances from the binary, i.e. the equations of motion are valid in a preferred reference\nframe in which the æther is asymptotically at rest. The dependence of the dynamics on\nthe velocity wof the binary’s center of mass with respect to the preferred frame can beUpdated Binary Pulsar Constraints on Einstein-æther theory 10\nreinstated by performing a boost. If w\u001cc, one then obtains [70]\ndvA\ndt=\u0000mBn\nr2n\nGAB\u0000(3GABBAB+DABB)mB\nr\n\u00001\n2\u0002\n2G2\nAB+ 6GABB(AB)+ 2DBAA+GAB(CAB+EAB)\u0003mA\nr\n+1\n2[3BAB\u0000GAB(1 +AA)]v2\nA+1\n2(3B21+GAB+EAB)v2\nB\n\u00001\n2\u0000\n6B(AB)+ 2GAB+CAB+EAB\u0001\nvA\u0001vB\u00003\n2(GAB+EAB) (n\u0001vB)2\n+1\n2(CAB+GABAA)w2+1\n2\u0000\nCAB\u00006B[AB]+EAB+ 2GABAA\u0001\nvA\u0001w\n+1\n2\u0000\nCAB+ 6B[AB]\u0000EAB\u0001\nvB\u0001w\n+3\n2EAB\u0002\n(w\u0001n)2+ 2(w\u0001n)(vB\u0001n)\u0003\u001b\n+mBvA\nr2n\u0001f[3BAB+GAB(1 +AA)]vA\u00003BABvB+GABAAwg\n\u00001\n2mBvB\nr2n\u0001\b\u0000\n6B(AB)+ 2GAB+CAB+EAB\u0001\nvA\n\u0000\u0000\n6B(AB)+CAB\u0000EAB\u0001\nvB+ 2EABw\t\n\u00001\n2mBw\nr2n\u0001\b\u0000\nCAB\u00006B[AB]+EAB\u00002GABAA\u0001\nvA\n\u0000\u0000\nCAB\u00006B[AB]\u0000EAB\u0001\nvB\u00002 (GABAA\u0000EAB)w\t\n;\n(29)\nwith which Eq. (20) of course agrees for w= 0.\nThe sensitivities and their derivatives enter into the conservative dynamics of the binary\nsystem at Newtonian and 1PN order, as can be seen explicitly in Eqs. (20) and (29). In\nAppendix A we will use these equations as starting point for studying in more detail the\n1PN dynamics of binaries in Einstein-æther theory. In doing so, we will also amend the\ncalculation of the strong-field preferred-frame parameters ^\u000b1and^\u000b2performed in [73],\nfixing an oversight pointed out by [70] and correcting also a few typos present in [70]\nitselfk.\nThe sensitivities and their derivatives, however, also enter the dissipative dynamics. In\nmore detail the total energy emitted in GWs (including not only tensor but also scalar\nand vector æther modes) by a binary in quasi-circular orbits was derived in [33, 73] via\nkThe corrections to ^\u000b1and^\u000b2do not significantly affect the final results of [73], since the strong-field\npreferred-frame parameters did not play a crucial role in constraining the parameter space of Lorentz-\nviolating gravity in that paper. The measurements of ^\u000b1and^\u000b2were used mainly to constrain c!, but\nsimilar bounds on that coupling constant can be obtained from the measurement of the damping of the\nperiod of PSR J0348+0432 (see Fig. 7 of [73]).Updated Binary Pulsar Constraints on Einstein-æther theory 11\na standard multipole expansion and reads\n_Eb\nEb= 2\u0012G12Gm 1m2\nr3\u0013(\n32\n5(\t1+S\t2+S2\t3)v2\n21\n+ (s1\u0000s2)2\"\n\u00102+ 2\u00103[w2\u0000(w\u0001n)2] +18\n5\t3w2+\u00126\n5\t3+ 36\u00101\u0013\n(w\u0001n)2#)\n;\n(30)\nwheresA\u0011\u001bA=(1 +\u001bA)is the rescaled sensitivity for the A-th body; v21=v2\u0000v1is\nthe relative velocity of the two bodies; the total (potential and kinetic) energy of the\nbinary is\nEb=\u0000G12m1m2\n2r; (31)\nwis the velocity of the binary’s center of mass with respect to the preferred frame; and\nwe have introduced the definitions\n\t1\u00111\ncT+2cac2\n\u001b\n(c\u001b+c!\u0000c\u001bc!)2cV+3ca(Z\u00001)2\n2cS(2\u0000ca); (32)\n\t2\u00112(Z\u00001)\n(ca\u00002)c3\nS\u00002c\u001b\n(c\u001b+c!\u0000c\u001bc!)c3\nV; (33)\n\t3\u00111\n2c5\nVca+2\n3ca(2\u0000ca)c5\nS; \u0010 1\u00111\n9cac5\nS(2\u0000ca); (34)\n\u00102\u00114\n3c3\nSca(2\u0000ca)+4\n3cac3\nV; \u0010 3\u00111\n6c5\nVca; (35)\nZ\u0011(\u000b1\u00002\u000b2)(1\u0000c\u001b)\n3(2c\u001b\u0000ca);S\u0011mBsA+mAsB\nmA+mB: (36)\nNote that the dipole flux is proportional to \u00102and to (s1\u0000s2)2(just like in scalar-tensor\ntheories of the Fierz-Jordan-Brans-Dicke type [28, 26, 72]). Therefore, it may dominate\nover GR’s quadrupole emission at low frequencies, depending on the sensitivities and\nthe coupling parameters of the theory [76].\n4. Solutions for slowly moving stars\nIn order to compute the sensitivities, we start from the observation that the metric and\næther solutions for a single point particle [Eqs. (22)–(26) with ~m2= 0] depend on the\nsensitivity\u001balready at linear order in the particle’s velocity. Moreover, \u001bregulates the\ndecay of the metric and æther components at large radii and enters already at O(1=r).\nThe sensitivity is of course a free parameter in the metric and æther solutions for a\npoint particle, but it can be determined by replacing the point particle with a body of\nfinite size. Once a fully non-linear solution for such a body (e.g., in our case, a NS) has\nbeen obtained, one can extract its sensitivity from the asymptotic fall-off of the metricUpdated Binary Pulsar Constraints on Einstein-æther theory 12\nand æther fields. Obviously, since \u001bappears at linear order in velocity, the NS must be\nmoving relative to the preferred foliation. Here, we follow [73] and consider a star in\nslow motion with respect to the æther, solve the field equations through linear order in\nthe star’s velocity, and extract the sensitivities from the asymptotic decay of the fields.\n4.1. Metric Ansatz\nHere, we consider the case of a non-spinning NS at rest with a background æther field\nmoving relative to it. The system is in a stationary regime, i.e, there is no dependence\nof the metric and the æther field on the time coordinate. For this configuration, letting\nvibe the velocity of the star relative to the æther, we consider the following ansatz for\nthe metric and the æther\nds2=e\u0017(r)dt2\u0000\u0012\n1\u00002M(r)\nr\u0013\u00001\ndr2\u0000r2(d\u00122+ sin2\u0012d\u001e2)\n+ 2vV(r;\u0012)dtdr + 2vS(r;\u0012)dtd\u0012 +O(v2); (37)\nU\u0016=e\u0017(r)=2\u000et\n\u0016+vW(r;\u0012)\u000er\n\u0016+vQ(r;\u0012)\u000e\u0012\n\u0016+O(v2); (38)\nand we will set GN= 1from here on. Note that M(r)has dimensions of length and\nM(r)!M?asr!1, withM?thus being the measured mass of the star. Here we\nhave adopted a coordinate system that is comoving with the fluid elements of the NS,\nby aligning the time coordinate vector to the fluid 4-velocity u\u0016[73]. In these comoving\ncoordinates, thefluidelementsareatrestwhiletheætherismoving. Thefluid4-velocity\nfield is\nu\u0016=e\u0000\u0017=2\u000e\u0016\nt: (39)\nThe ansatz of Eqs. (37)–(39) depends on M(r)and\u0017(r)atO(v0)and on four potentials\nV(r;\u0012),S(r;\u0012),W(r;\u0012)andQ(r;\u0012)atO(v). However, one can perform a coordinate\ntransformation of the form\nt0=t+vH(r;\u0012); (40)\nwhich allows for any one of the four potentials to be set to zero while keeping the ansatz\nvalid atO(v). Here we choose to set Q= 0without loss of generality.\n4.2. Zeroth order in velocity\nFrom Eqs. (6)–(7) let us derive the field equations at zeroth order in velocity,\nwhich we will solve to construct the background NS solution. The (t;t),(r;r)and\n(\u0012;\u0012)components of the field equations are the only non-trivial ones and give threeUpdated Binary Pulsar Constraints on Einstein-æther theory 13\nindependent equations [73]\n16dM\ndr\u00004car(r\u00002M)d2\u0017\ndr2\u0000car(r\u00002M)\u0012d\u0017\ndr\u00132\n+ 4ca\u0012\nrdM\ndr\u00002r+ 3M\u0013d\u0017\ndr= 64\u0019\u001ar2; (41)\ncar2(r\u00002M)\u0012d\u0017\ndr\u00132\n+ 8r(r\u00002M)d\u0017\ndr\u000016M= 64\u0019r3P; (42)\n4r2(r\u00002M)d2\u0017\ndr2\u0000(ca\u00002)r2(r\u00002M)\u0012d\u0017\ndr\u00132\n\u00004r\u0012\nrdM\ndr\u0000r+M\u0013d\u0017\ndr\u00008rdM\ndr+ 8M= 64\u0019r3P; (43)\nwhereP(r)and\u001a(r)are rescaled NS pressure and density respectively (rescaled because\nGN=2G\n2\u0000ca). These can be expressed as\nP\u00112\u0000ca\n2~P; \u001a\u00112\u0000ca\n2~\u001a; (44)\nwith ~Pand ~\u001arepresenting the pressure and energy density that enter directly in the\nstress-energy tensor for the matter field, which we take to be of a perfect fluid form\nTmat\n\u0016\u0017=\u0010\n~\u001a+~P\u0011\nu\u0016u\u0017\u0000~Pg\u0016\u0017+O(v2): (45)\nNote, that there is a bijective correspondence between the original parametrization\n(ca;c\u0012;c!;c\u001b)and (\u000b1;\u000b2;c!;c\u001b)that can be derived from Eqs. (13) and (14). As\ndiscussed in Sec. 2, c\u001b\u001c10\u000015from gravitational wave observations; we thus rewrite\nEqs. (41)–(43) in terms of ( \u000b1,\u000b2,c\u001b,c!). This is justified because the bound on\nc\u001bis much stronger than those on the other parameters, which are constrained by\nsolar-system and stability requirements (absence of gradient instabilities and vacuum\nCherenkov radiation, and positive energy) to satisfy\n\u000b1<8\u000b2<0; c!>\u0000\u000b1\n2; (46)\nwithj\u000b1j.10\u00004andj\u000b2j.10\u00007, in the limit c\u001b!0. Upon simplification, we get the\nmodified Tolman-Oppenheimer-Volkoff (TOV) equations\ndM\ndr=1\n\u000b1(\u000b1+ 8)rn\n\u00004p\nr\u00002M(\u000b1+ 8)p\n(\u0000\u000b1+ 8)M\u00004P\u000b1\u0019r3+ 4r\n\u0000\u0000\n\u000b2\n1+ 24\u000b1+ 128\u0001\nM\u000016r\u0010\n\u000b1\u0019r2(\u000b1+ 2)P\u00002\u0019r2\u000b1\u001a\u0000\u000b1\n2\u00004\u0011o\n;(47)\nd\u0017\ndr=1\n\u000b1(r\u00002M)rh\n\u00008p\nr\u00002Mp\n(\u0000\u000b1\u00008)M\u00004P\u000b1\u0019r3+ 4r+ 16r\u000032Mi\n;\n(48)\ndp\ndr=1\n\u000b1(r\u00002M)r4(P+\u001a)hp\nr\u00002Mp\n(\u0000\u000b1\u00008)M\u00004P\u000b1\u0019r3+ 4r\u00002r+ 4Mi\n:\n(49)Updated Binary Pulsar Constraints on Einstein-æther theory 14\nModifications to the GR TOV equations can be singled out by expanding the above\nequations(41)–(43)inasmallcouplingapproximation,i.e., ca\u001c1or\u000b1\u001c1[66,57,73].\n4.3. First order in velocity\nWe derive field equations at first order in velocity from Eqs. (6) and (7), which include\nthe potentials as functions of rand\u0012, at first order in velocity. We can separate variables\ninrand\u0012using a Legendre decomposition [73] to obtain\nV(r;\u0012) =X\nnKn(r)Pn(cos\u0012); (50)\nS(r;\u0012) =X\nnSn(r)Pn(cos\u0012)\nd\u0012; (51)\nW(r;\u0012) =X\nnWn(r)Pn(cos\u0012); (52)\nwherePnis the Legendre polynomial of order n. More details on tensor harmonic\ndecomposition can be found in [65]. By separation of variables, we arrive at O(v)\nequations, where only the (t;r)and(t;\u0012)components of the modified Einstein equations\nand therand\u0012components of the æther field equations are non-trivial. We are\nonly interested in n= 1component of Legendre decomposition since these functions\ndetermine sensitivities and consequently the change in orbital period.\nSincec\u001b\u001c10\u000015(c.f. Sec. 2), we proceed with calculations in the limit c\u001b!0to\nobtain [73]\ndS1\ndr=1\n\u000b1r(r\u00002M)n\n\u00002S1p\nr\u00002M(\u000b1+ 4)p\n(\u0000\u000b1\u00008)M\u00004P\u000b1\u0019r3+ 4r\u0000\n(r\u00002M)\u0000\nJ1e\u0017=2c!\u000b1\u0000(3\u000b1+ 16)S1+K1\u000b1(c!\u00001)\u0001\t\n; (53)\ndK1\ndr=1\nc!(r\u00002M)(\u000b2\n1+ (2\u00002\u000b2)\u000b1\u000016\u000b2)\u000b1(\u000b1+ 8)r\b\n2\u0002\n((c!+ 1)\u000b1+ 2c!)J1e\u0017=2\n\u0000(6\u000b1+ 32)S1+c!K1\u000b1] (\u000b2\n1+ (2\u00002\u000b2)\u000b1\u000016\u000b2)(\u000b1+ 8)p\nr\u00002M\n\u0002p\n(\u0000\u000b1\u00008)M\u00004P\u000b1\u0019r3+ 4r+ [\u0000(\u000b1+ 8)(r\u00002M)\u000b1r\n((c!+ 1)\u000b2\n1\u00002c!(\u000b2\u00001)\u000b1\u000016\u000b2c!)\u0012@J1\n@r\u0013\n\u000016\u0012\n\u00001\n8\u0012\n3\u0012\u0012\nc!+5\n3\u0013\n\u000b3\n1\n+\u0012\u0012\n\u00002\u000b2+14\n3\u0013\nc!\u00008\u000b2\n3+8\n3\u0013\n\u000b2\n1+\u0012\u0012\n\u000064\u000b2\n3+16\n3\u0013\nc!\u000064\u000b2\n3\u0013\n\u000b1\n\u0000128\u000b2c!\n3\u0013\n(\u000b1+ 8)M\u0013\n+\u0000\n\u0019(\u000b1+ 2)\u000b1r2((c!+ 1)\u000b2\n1\u00002c!(\u000b2\u00001)\u000b1\n\u000016\u000b2c!)P\u00002\u0019\u000b1r2((c!+ 1)\u000b2\n1\u00002c!(\u000b2\u00001)\u000b1\u000016\u000b2c!)\u001a\n+1\n4\u0012\n(\u000b1+ 8)\u0012\u0012\nc!+3\n2\u0013\n\u000b3\n1+ ((\u00002\u000b2+ 4)c!\u00002\u000b2+ 2)\u000b2\n1Updated Binary Pulsar Constraints on Einstein-æther theory 15\n+((\u000020\u000b2+ 4)c!\u000016\u000b2)\u000b1\u000032\u000b2c!)))r)J1]e\u0017=2\n+ 6\u0012\n\u00002(\u000b1+ 8)2S1\n3+c!K1\u000b1\u0013\n(\u000b1+ 8)(\u000b2\n1+ (2\u00002\u000b2)\u000b1\u000016\u000b2)M\n\u000016\u0014\u0012\n\u0019(\u000b2\n1+ (2\u00002\u000b2)\u000b1\u000016\u000b2)(\u000b1+ 4)\u000b1r2P\u000011\u000b3\n2\n8\n+(3\u000b2\u000011)\u000b2\n1+ (40\u000b2\u000016)\u000b1+ 128\u000b2\u0001\n(\u000b1+ 8)S1+c!K1(\u000b2\n1\n+(2\u00002\u000b2)\u000b1\u000016\u000b2)\u000b1\u0000\n\u0019r2(\u000b1+ 2)P\u00002\u0019r2\u001a+\u000b1=4 + 2\u0001\u0003\nr\t\n; (54)\nd2J1\ndr2=1p\n(\u0000\u000b1\u00008)M\u00004P\u000b1\u0019r3+ 4r(r\u00002M)3=2\u000b3\n1r2(\u000b1+ 8)(4f[(\u000b1+ 8)\n\u0012\n(\u000b2\n1\u00004\u000b1\u000b2\u000032\u000b2)S1+\u0012\n\u0000\u000b2\n1\n2+c!(\u000b2\u00001)\u000b1+ 8\u000b2c!\u0013\nK1\u0013\n\u000b1re\u0000\u0017=2\n\u000012\u000b2\n1\u0012\u00121\n24\u000b2\n1+\u000b1+16\n3\u0013\nM+\u0012\n\u000b1\u0019r2(\u000b1+ 2)P\u00002\u0019r2\u000b1\u001a+\u000b2\n1\n24\u00008\n3\u0013\nr\u0013\n\u0002r\u0012@J1\n@r\u0013\n+\u0012\n8\u0012@\u001a\n@r\u0013\n\u000b3\n1\u0019r4+1\n2\u0000\n(\u000b1+ 8)\u0000\n\u000b3\n1+ (\u00008\u000b2+ 56)\u000b2\n1\n+ (\u0000192\u000b2+ 128)\u000b1\u00001024\u000b2)M) +\u0000\n8\u0019(\u000b3\n1+ (\u00005\u000b2+ 14)\u000b2\n1\n+ (\u000056\u000b2+ 24)\u000b1\u0000128\u000b2)\u000b1r2P+ 4\u0019\u000b2\n1r2\u0000\n\u000b2\n1+ (\u00002\u000b2+ 16)\u000b1\u000016\u000b2\n+ 16)\u001a+\u0012\u000b3\n1\n2+ (\u000b2c!\u0000c!\u000012)\u000b2\n1+ (\u000032 + (8c!+ 32)\u000b2)\u000b1+ 256\u000b2\u0013\n\u0002(\u000b1+ 8)r)J1]p\nr\u00002Mp\n(\u0000\u000b1\u00008)M\u00004P\u000b1\u0019r3+ 4r\n+ 64\u0010\u0010\u000b1\n4+ 2\u0011\nM+P\u000b1\u0019r3\u0000r\u0011\n\u0002\u00141\n8\u0012\n\u0000S1\u000b1\u000b2r(\u000b1+ 8)2e\u0000\u0017=2+\u000b2\n1r(\u000b1+ 8)(r\u00002M)@J1\n@r\u0013\n+J1\u00123(\u000b1+ 8)\n4\n\u0002\u0012\n\u000b2\n1+\u0012\u00008\u000b2\n3+8\n3\u0013\n\u000b1\u000064\u000b2\n3\u0013\nM+\u0000\nr2\u0019\u000b2\n1(\u000b1+ 2)P\n+\u000b2\n1\u0019r2(\u000b1+ 2)\u001a\u00003\u000b3\n1\n8+ (\u000b2\u00004)\u000b2\n1+ (16\u000b2\u00008)\u000b1+ 64\u000b2\u0013\nr\u0013\u0015\u001b\u0013\n;(55)\nwhere we have defined Jn=Wn+e\u0000\u0017=2Kn[73]. With the above set of equations at\nhand, the next section describes the methods of solving these equations at each order\nin velocity.\n5. The calculation of the sensitivities\nThe sensitivities are calculated by solving the coupled differential equations in Eqs. (47)-\n(49) and Eqs. (53)-(55), which are obtained from the modified Einstein and the æther\nfield equations in a v\u001c1expansion atO(v0)andO(v)respectively [73]. In Secs. 5.1\nand 5.2 we describe and apply two methods to solve these equations and find the NS\nsensitivities. The first method, outlined in Sec. 5.1, was used previously in Ref. [73],Updated Binary Pulsar Constraints on Einstein-æther theory 16\nbut we will explain how it leads to unstable solutions in particular regions of parameter\nspace. A second method outlined in Sec. 5.2 provides stable results in all regions of\nparameter space.\nTheO(v0)solutions are common to both methods, as they both involve solving O(v0)\ndifferential equations (47)–(49) numerically once in the interior and then in the exterior\nof the NS. The initial and boundary conditions to it are obtained by imposing regularity\nat the NS center, while imposing asymptotic flatness at spatial infinity respectively. The\ndifferential equations are solved from a core radius (i.e. some small initial radius) to the\nstellar surface radius, where the pressure goes to zero. These numerical solutions at the\nNS surface are now used as initial conditions to solve the exterior evolution equations\nfrom the stellar surface to an extraction radius rb. Using continuity and differentiability\nofthesolutions, theasymptoticsolutionsatspatialinfinityarematchedtothenumerical\nsolutions evaluated at rb. This gives the observed mass of the NS and the integration\nconstantcorrespondingto \u0017(0)(obtainedbysolvingthe O(v0)differentialequations[73])\nwhich will be used in solving the O(v)equations discussed further.\n5.1. Method 1: Direct Numerical Solutions\nIn this method, the aforementioned O(v)differential equations are solved in two regions,\nthe interior of the star, and the exterior. The initial conditions at O(v)are obtained\nby solving the corresponding differential equations asymptotically about a core radius,\nwhile imposing regularity at the core, and asymptotically about spatial infinity, while\nimposing asymptotic flatness [73]. In both cases, the solutions depend on integration\nconstants – ~Cand ~Din the interior asymptotic solution and ~Aand ~Bin the exterior\nasymptotic solution – that must be chosen so as to guarantee that the numerical interior\nand exterior solutions are continuous and differentiable at the stellar surface, where\npressure becomes significantly smaller than their core values.\nAs defined above, the global solution reduces to finding the right constants (~A;~B;~C;~D),\nwhich in turn is a shooting problem. In practice, Ref. [73] solved this shooting prob-\nlem by first picking two sets of values for interior constants, ~ c(1)= (~C(1);~D(1))and\n~ c(2)= (~C(2);~D(2)), andthensolvingtheinteriorequationstwicefromthecoreradius rcto\nthe NS surface R?to find the solutions ~fint\n(1)(r) = [S(1;int)\n1(r);K(1;int)\n1(r);J(1;int)\n1;J0(1;int)\n1 (r)]\nand~fint\n2(r) = [S(2;int)\n1(r);K(2;int)\n1(r);J(2;int)\n1(r);J0(2;int)\n1 ]. Then, each interior nu-\nmerical solution is evaluated at the stellar surface and used as initial con-\nditions for a numerical evolution in the exterior, leading to two exte-\nrior solutions ~fext\n1(r) = [S(1;ext)\n1(r);K(1;ext)\n1 (r);J(1;ext)\n1 (r);J0(1;ext)\n1 (r)]and~fext\n2(r) =\n[S(2;ext)\n1(r);K(2;ext)\n1 (r);J(2;ext)\n1 (r);J0(2;ext)\n1 (r)].\nThe global solutions ~fglo\n1;2(r) =~fint\n1;2(r)[~fext\n1;2(r)are then automatically continuous and\ndifferentiable at the surface, but in general they will not satisfy the boundary conditionsUpdated Binary Pulsar Constraints on Einstein-æther theory 17\nat spatial infinity. Because of the linear and homogeneous structure of the differential\nsystem, one can find the correct global solution through linear superposition\n~fglo(r;C0;D0) =C0~fglo\n1(r) +D0~fglo\n2(r); (56)\nwhereC0andD0are new constants, chosen to guarantee that ~fglosatisfies the correct\nasymptotic conditions near spatial infinity, which in turn depend on (~A;~B), i.e.\n~fglo(rb;C0;D0) =~fglo;1(rb;~A;~B); (57)\nwhererb\u001dR?is the matching radius, ~fglo(rb;C0;D0)is given by Eq. (56) evaluated at\nr=rb(which depends on (C0;D0)) and~fglo;1(rb;~A;~B)is the asymptotic solution to\nthe differential equations near spatial infinity evaluated at the matching radius (which\ndepends on (~A;~B)).\nWith this at hand, one can calculate the NS sensitivities via [73]\n\u001b= 2~A\u000b1\n\u000b1+ 8; (58)\nwhere ~Ais the coefficient of 1=rin the near-spatial infinity asymptotic solution of Wext\n1\nsuch that\nWext\n1(r) =~AM?\nr+O\u0012M2\n?\nr2\u0013\n; (59)\nwhile we recall that \u000b1.10\u00004(c.f. Sec. 2). Because of the latter constraint, it is\nobvious that the sensitivities are essentially controlled by \u001b\u0019~A\u000b1, so the numerical\nstability of its calculation relies entirely on the numerical stability of the calculation of\nthis coefficient. Unfortunately, as we show below, this calculation is not numerically\nstable in the region of parameter space we are interested in.\nFigure 1 shows S1as a function of radius, assuming (\u000b1;\u000b2;c!) = (10\u00004;4\u000210\u00007;\u00000:1),\nand setting (rc;rb) = (102;2\u0002107)cm. Observe that both S(1;glo)\n1andS(2;glo)\n1diverge at\nspatial infinity, so in order to find an Sglo\n1that is finite at spatial infinity, a very delicate\ncancellation of large numbers needs to take place. This cancellation needs to lead to\n~A\u000b1\u00190but in general ~A\u000b16= 0, since\u001b\u0019~A\u000b1=4\u001c16= 0, and precisely by how\nmuch ~A\u000b1deviates from 0is what determines the value of the sensitivity. We find in\npractice that \u001bis highly sensitive to the accuracy of the numerical algorithm used to\nsolve for~fglo\n(1;2), as well as the choice of rc,rband the value of p(R?)that defines the\nstellar surface. Figure 1 is in the parameter region that is outside of interest but it\nindicates how sensitive the calculations are to the aforementioned cancellation, making\nit difficult to find numerically stable solutions.\n5.2. Method 2: Post-Minkowskian Approach\nGiven that the first method does not allow us to robustly compute the sensitivities in\nthe regime ofinterest, we developed a newpost-Minkowskian method, which we describeUpdated Binary Pulsar Constraints on Einstein-æther theory 18\nFigure 1: The metric function | S1| is plotted against the radius in the entire numerical\ndomain, where the radius of the star is at 11.1 km (vertical dashed line). Observe that\nboth of the trial solutions S(1;glo)\n1andS(2;glo)\n1diverge at spatial infinity. Hence, the global\nlinearly combined solution S(glo)\n1shows a diverging behaviour representing numerical\ninstabilities in the calculation of sensitivities.\nhere. In this method, the background O(v0)equations are solved by direct integration,\nas done in method 1. The differential equations at O(v), however, are expanded in\ncompactnessCand solved order by order. This is a post-Minkowskian approximation\nbecause the compactness always appears multiplied by G=c2, so in this sense it is a\nweak-field expansion. NSs are not weak-field objects, but their compactness is always\nsmaller than\u00181=3(and usually between [0:1;0:3]), so provided enough terms are kept\nin the series, this approximation has the potential to be valid. Moreover, and perhaps\nmore importantly, we will show below that such a perturbative scheme stabilizes the\nnumerical solution for the NS sensitivities.\nThe procedure presented above is not technically a standard post-Minkowskian series\nsolution because the background equations (or their solutions) are not expanded in\npowers ofC. Had we expanded in powers of Ceverywhere, we would have encountered\nterms in the differential equations with derivatives of the equation of state (EoS). Such\nderivatives would introduce numerical noise because “realistic” (tabulated) EoSs are not\nusually smooth functions, potentially introducing steep jumps, see, e.g., [75]. By not\nexpanding theO(v0)differential equations, we are implicitly adding higher order terms\nin compactness, so this procedure could be seen as a resummation technique.Updated Binary Pulsar Constraints on Einstein-æther theory 19\nThrough this approach, the differential equations at O(v1)turn intonsets of differential\nequations for an expansion carried out to O(Cn), withntherefore labeling the\ncompactness order. In order to derive these equations, however, one must first establish\nthe order of the background solutions, which can be shown to satisfy\nM(r) =O(C); P (r) =O(C2); (60)\n\u001a(r) =O(C);and\u0017(r) =O(C); (61)\nby looking at the differential equations these functions obey at O(v0). The metric\nperturbation functions at O(v1)are then expanded in powers of compactness through\nYi(r) =nX\nj=1Yij\u000fj; (62)\nwhereYi\u0011(S1;K1;W1),\u000fis a bookkeeping parameter of O(C)andjindicates the order\nofCto be summed over. We work with W1(r)instead ofJ1(r)to avoid introducing\nnumerical error during the conversion between these two functions.\nUsing these expansions in the differential equations at O(v), and re-expanding them in\npowersofcompactness,onefinds nsetsofdifferentialequations. At O(C),thedifferential\nsystem becomes\ndS11\ndr=2rK11\u00002r(S11+\u000b1W11)\u0000(4 +\u000b1)M\n2r2; (63)\ndK11\ndr=\u00001\n2c!r2\u000b1\u0002\n8c!\u0019r3\u000b1\u001a+ 4c!r\u000b1(K11\u0000S11)\u00004r\u000b2\n1W11\n+c!(8\u000b1+\u000b2\n1\u000016\u000b2\u00002\u000b1\u000b2)M\u00004c!r2\u000b1W0\n11\n\u00002r2(\u000b2\n1+c!\u000b2\n1\u000016c!\u000b2\u00002c!\u000b1\u000b2)W0\n11\u0003\n; (64)\nd2W11\ndr2=2W11\nr2; (65)\nwhereS11(r),K11(r)andW11(r)are metric functions at O(C), and recall that the\ndensity\u001a(r)is related to pressure through the EoS as \u001a(r)=\u001a(P(r)). We note that\nW11(r)is decoupled from S11(r)andK11(r), and so we can solve Eq. (65) separately and\nanalytically in the regions r\u0014rbandr\u0015rb. The solutions to them are W11(r) =~D1r2\nforr\u0014rbandW1\n11(r) = ~A1=rforr\u0015rb, where ~A1and ~D1are integration constants.\nBy requiring continuity and differentiability of metric functions W11(r)andW0\n11(r), we\nmatch the solutions at the extraction radius rb. This fixes the values of two integration\nconstants ~A1= 0and ~D1= 0.\nThe remaining two equations, namely Eqs. (63) and (64), are solved numerically with\ninitial conditions obtained using regularity at the NS center and asymptotic flatness at\nspatial infinity. At the center, we have\nS11(r) =~C1\u00001\n120\u000b1\u0019\u0002\n(240\u000b1+ 40\u000b2\n1\u0000128\u000b2\u000016\u000b1\u000b2)\u001acUpdated Binary Pulsar Constraints on Einstein-æther theory 20\n+(48\u000b1\u000b2\u000072\u000b2\n1\u000015\u000b3\n1+ 6\u000b2\n1\u000b2)pc\u0003\nr2\n+1\n1260\u00192\u0002\n(\u0000160\u000b1\u000035\u000b2\n1+ 80\u000b2+ 10\u000b1\u000b2)\u001a2\nc\n+ (288\u000b2\u0000576\u000b1\u0000126\u000b2\n1+ 36\u000b1\u000b2)p2\nc\n+(336\u000b2\u0000672\u000b1\u0000147\u000b2\n1+ 42\u000b1\u000b2)\u001acpc\u0003\nr4+O(r6); (66)\nK11(r) =~C1\u00001\n120\u000b1\u0019\u0002\n(400\u000b1+ 40\u000b2\n1\u0000384\u000b2\u000048\u000b1\u000b2)\u001ac\n+(144\u000b1\u000b2\u000096\u000b2\n1\u000015\u000b3\n1+ 18\u000b2\n1\u000b2)pc\u0003\nr2\n+1\n1260\u00192\u0002\n(400\u000b2\u0000240\u000b1\u000035\u000b2\n1+ 50\u000b1\u000b2)\u001a2\nc\n+ (1440\u000b2\u0000864\u000b1\u0000126\u000b2\n1+ 180\u000b1\u000b2)p2\nc\n+(1680\u000b2\u00001088\u000b1\u0000147\u000b2\n1+ 210\u000b1\u000b2)\u001acpc\u0003\nr4+O(r6);(67)\nwhere ~C1is an integration constant P. At spatial infinity, we have\nS1\n11(r) =\u0000~B1\n2r3\u00001\n2c!r\u000b1h\n~A1(\u000b2\n1\u00002c!\u000b1\u00002\u000b2\n1c!+ 16c!\u000b2+ 2c!\u000b1\u000b2)\n+c!(8\u000b2\u00006\u000b1\u0000\u000b2\n1+\u000b1\u000b2)M?\u0003\n+ (16\u000b2\u00008\u000b1\u0000\u000b2\n1+ 2\u000b1\u000b2)M2\n?\n64r2\n+ (16\u000b2\u000016\u000b1\u00003\u000b2\n1+ 2\u000b1\u000b2)M3\n?\n192r3\n+ (8\u000b2\u00002\u000b1+\u000b1\u000b2) ln\u0012r\nM?\u0013M3\n?\n48r3+O\u0012M4\n?\nr4\u0013\n; (68)\nK1\n11(r) =~B1\nr3+4M?+ 2~A1\u000b1+M?\u000b1\n2r\u0000(\u000b2\n1+ 16\u000b2+ 2\u000b1\u000b2)M2\n?\n64r2\n+ (2\u000b1\u00008\u000b2\u0000\u000b1\u000b2) ln\u0012r\nM?\u0013M3\n?\n24r3+O\u0012M4\n?\nr4\u0013\n(69)\nwhere ~B1is an integration constant and M?is the mass of the star.\nWenextexplainhowtosolveEqs.(63)and(64)toconstructthesolutionfor S11andK11.\nFirst, homogeneous solutions are given by Shom\n11=Khom\n11=~C1. Next, one can construct\nparticular solutions Spart\n11andKpart\n11by setting ~C1= 0and numerically integrate the\nequations from rctoR?. We then use the numerically calculated interior solutions,\nevaluated at R?, as the initial conditions to solve the exterior evolution equations with\nzero pressure and density from R?torb. The true solutions are simply the sum of the\nhomogeneous and particular solutions, namely\nS11(r) =~C1+Spart\n11; (70)\nK11(r) =~C1+Kpart\n11: (71)\nPEquations (66) and (67) (and also Eqs. (68) and (69)) contain terms higher than O(C)because the\nbackground functions are not expanded in a series of Cand thus contain higher order contributions.Updated Binary Pulsar Constraints on Einstein-æther theory 21\nBy requiring continuity and differentiability of all metric functions, we match the true\nnumerical solution to the analytic asymptotic solution in Eqs. (68) and (69) at rb.\nApplying this matching condition gives the values of ~B1and ~C1.\nNow let us focus on the solution to O(C2)differential equations. The equation for the\nmetric function W12(r)is\nd2W12\ndr2=2W12\nr2+ 4\u0019rW 11\u001a0\u00002\u0019\u001a\u0012\n\u00002K11+ 2S11+ (4 +\u000b1+2\u000b1\nc!)W11\u00006rW0\n11\u0013\n\u00002\u0019\u001a(6 +\u000b1)M\nr\u0000W0\n11\u0012\n4 +\u000b2+8\u000b2\n\u000b1\u0013M\nr2\n\u0000\u0012\n3K11\u00003S11\u000010W11\u00002\u000b1W11\u00002\u000b1W11\nc!\u0013M\nr3\n+\u0012\n5 +\u000b1+\u000b2\n2+4\u000b2\n\u000b1\u0013M2\nr4; (72)\nwhere\u001a0(r)is the derivative of \u001a(P(r))obtained from the EoS. This equation is\ndecoupled from the remaining metric functions at O(C2),S12andK12, and can be\nsolved numerically on its own. The initial condition obtained at the center of the NS is\nW12(r) =~D2r2+1\n529200\u000b1\u00192r4\u0002\n560\u001a3\nc\u0019r2\u000b1(\u000b1+ 8)(5\u000b1+ 4\u000b2)\n\u000027p2\nc\u000b2\n1(70(8\u0019pcr2\u00007)\u000b2\n1\u00008\u0019pcr2\u000b1(\u000b2\u0000422) + 49\u000b1(\u000b2\u000062)\n+ 8(49\u00008\u0019pcr2)\u000b2)\u0000252\u001acpc\u000b1\b\n70pc\u0019r2\u000b3\n1+\u000b2\n1(70\u0000pc\u0019r2(\u000b2\u0000382))\n+64(7\u00004\u0019pcr2)\u000b2\u00008\u000b1(5\u0019pcr2(\u000b2+ 8)\u00007(\u000b2+ 10))\t\n\u000012\u001a2\nc\u0000\n350\u0019pcr2\u000b4\n1\u00005\u0019pcr2\u000b3\n1(\u0000226 +\u000b2)\u000031360\u000b2\n\u0000784\u000b1(\u000040 + (5 + 8\u0019pcr2)\u000b2)\u00008\u000b2\n1(\u0000490 +\u0019pcr2(980 + 103\u000b2))\u0001\u0003\n+O(r6);\n(73)\nwhere ~D2is an integration constant. The boundary condition to Eq. (72) at spatial\ninfinity is\nW1\n12(r) =~A2\nr+1\n320r3\u000b1c!h\n~A1M?(40r\u0000M?\u000b1)(\u000b2\n1+c!(4\u000b2\n1\n+\u000b1(34\u00004\u000b2)\u000032\u000b2)) +c!(80M2\n?r(\u000b1+ 8)(\u000b1\u0000\u000b2)\n+M3\n?\u000b1(\u00004\u000b2\n1+ 56\u000b2+\u000b1(\u000038 + 7\u000b2)))\u0003\n+O\u0012M4\n?\nr4\u0013\n; (74)\nwhere ~A2is an integration constant. To construct the solution, we first note that the\nhomogeneous solution is given by Whom\n12=~D2r2. Next, we set ~D2= 0and find the\nparticular solution Wpart\n12(r)numerically in the interior of the NS by solving Eq. (72).\nThis interior solution evaluated at the NS surface now serves as initial conditions to\nsolve the differential equations in the exterior up to the boundary radius rb. The correct\nsolution in the entire numerical domain is then\nW12(r) =~D2r2+Wpart\n12(r); (75)Updated Binary Pulsar Constraints on Einstein-æther theory 22\nwhere the values of ~A2and ~D2are obtained using the matching condition at rb. The\nequations for S12andK12are solved similar to the way Eqs. (63) and (64) are solved,\nso we omit a more detailed description here for brevity. We can use the above method\nto solve differential equations at higher order in C.\n5.3. Comparison between numerical and analytical approaches\n5.3.1. Tabulated APR4 EoS The sensitivity in the æther theory \u001bfor an isolated NS\ndepends on the EoS chosen, and here we perform the calculations of the previous section\nfor the APR4 tabulated EoS [4]. The results are representative of what one finds with\nother EoSs.\nEq. (58) gives the expression of sensitivity in terms of the integration constant ~A[73]\nwhere ~Acan be expressed as\n~A\u00111\nM?nX\nj=2~Aj\u000fj; (76)\nwherenis the order of the compactness expansion, with ~A1= 0. The coefficients ~Aj\ncan be calculated numerically as described in the previous subsection. Notice that the\nleading contribution to ~A(and hence to the sensitivities) is of O(\u000f), sinceM?=O(\u000f).\nThe calculation of the sensitivity as described above requires one to choose the\ntruncation order nof the post-Minkowskian expansion. We will choose nby the\nsensitivities computed from methods 1 and 2 in a regime of parameter space where\nmethod 1 yields stable results [73]. In particular, we will focus on the choice\n(\u000b1;\u000b2;c!;c\u001b) = (10\u00004;4\u000210\u00007;10\u00004;0). Figure 2 compares the sensitivities computed\nwith the two methods with this parameter choice. Observe that as the order of post-\nMinkowskian approximation increases (i.e. as nincreases), the curves approach the\nmethod 1 solution, but in an oscillatory manner. The bottom panel of Fig 2 shows the\nstability of post-Minkowskian method at order n= 3.\nIn the weak field limit, the sensitivity can be well-approximated as the ratio of the\nbinding energy to the NS mass (\n=M\u0003)through [33]\nswf=\u0012\n\u000b1\u00002\n3\u000b2\u0013\nM?; (77)\nwhere the stellar binding energy \nis [32]\n\n =\u00001\n2Z\nd3x\u001a(r)Z\nd3x0\u001a(r0)\njx\u0000x0j; (78)\nwithr=jxjandr0=jx0j. We can use a Legendre expansion of the Green’s function to\nevaluate this integral, and to leading order in C, we find a result that is identical to thatUpdated Binary Pulsar Constraints on Einstein-æther theory 23\nFigure 2: (Top) Sensitivity as a function of compactness using the APR4 EoS, for\nvarious post-Minkowskian truncation orders at (\u000b1;\u000b2;c!;c\u001b) = (10\u00004;4\u000210\u00007;10\u00004;0).\nAt leading order in C, the sensitivity curve overlaps with that computed analytically\nin the weak field limit in [32] (Eq. (77)). As the compactness order is increased, the\nsensitivity curve starts to converge toward the solution found with method 1. (Bottom)\nFractional difference between the sensitivities at different order of compactness and\nthose found from method 1. Observe that when n= 3, the truncated post-Minkowskian\nseries is already an excellent approximation. The vertical dashed line corresponds to\nthe compactness of a 1M\fNS.\ncomputed in the weak field limit by [32]. This can also be seen numerically in Fig. 2,\nwhere the weak field curve coincides with O(C1)post-Minkowskian approximation.\n5.3.2. Tolman VII EoS We now focus on the sensitivity of a NS using the Tolman VII\nEoS. The latter is an analytic model that accurately describes non-rotating NSs [66] by\nthe energy density profile\n\u001a(r) =\u001ac\u0012\n1\u0000r2\nR2\n?\u0013\n: (79)Updated Binary Pulsar Constraints on Einstein-æther theory 24\nThe advantage of using the Tolman VII EoS is that the background solution is known\nanalytically in GR [66, 47]. We expand analytically both the O(v0)andO(v)equations\norder by order in compactness. The sensitivity obtained is then\ns=5\n21C(\u00003\u000b1+ 2\u000b2)\n+ 5\u0012573\u000b3\n1+\u000b2\n1(67669\u0000764\u000b2) + 96416\u000b2\n2+ 68\u000b1\u000b2(\u00002632 + 9\u000b2)\n252252\u000b1\u0013\nC2\n+1\n1801079280 c!\u000b2\n1n\n(4\u000b1)2(8 +\u000b1)(36773030\u000b2\n1\u000039543679\u000b1\u000b2\n+ 11403314 \u000b2\n2) +c!\u0002\n\u00001970100\u000b5\n1+ 13995878400 \u000b3\n2\n+ 640\u000b1\u000b2\n2(\u000049528371 + 345040 \u000b2) + 5\u000b4\n1(\u000019596941 + 788040 \u000b2)\n+\u000b3\n1(\u00002699192440 + 440184934 \u000b2\u00005974000\u000b2\n2)\n16\u000b2\n1\u000b2(1294533212\u000029152855\u000b2+ 212350\u000b2\n2)\u0003o\nC3+O(C4): (80)\nNote that the above expression is not regular in the limit of \u000b1!0while keeping \u000b2\nfinite orc!!0while keeping \u000b1or\u000b2finite. This is a known feature of Einstein-æther\ntheory, which recovers GR only when a certain combination of coupling constants is\ntaken to zero at a specific rate.\nWith this EoS, the compactness can be expressed as a function on \n=M?as\nC=\u00007\n5M?+35819\u000b1\n3\n85800M3\n?+O\u0012\n4\nM4\n?\u0013\n: (81)\nHereCandM?are the observed values with æther corrections included. With this at\nhand, we can rewrite the sensitivity as a function of \nto find\ns=(3\u000b1+ 2\u000b2)\n3\nM?\n+\u0012573\u000b3\n1+\u000b2\n1(67669\u0000764\u000b2) + 96416\u000b2\n2+ 68\u000b1\u000b2(9\u000b2\u00002632)\n25740\u000b1\u0013\n2\nM2\n?\n+1\n656370000c!\u000b2\n1n\n\u00004\u000b2\n1(\u000b1+ 8)\u0002\n36773030\u000b2\n1\u000039543679\u000b1\u000b2\n+11403314\u000b2\n2\u0003\n+c!\u0002\n1970100\u000b5\n1\u000013995878400 \u000b3\n2\n\u0000640\u000b1\u000b2\n2(\u000049528371 + 345040 \u000b2)\u00005\u000b4\n1(19548109 + 788040 \u000b2)\n\u000016\u000b2\n1\u000b2(1294533212\u000029152855\u000b2+ 212350\u000b2\n2)\n+\u000b3\n1(2699192440\u0000309701434\u000b2+ 5974000\u000b2\n2)\u0003o\n3\nM3\n?+O\u0012\n4\nM4\n?\u0013\n:(82)\nNote that this expression matches identically to that of [33] when working to leading\norder in the binding energy.\nOne may wonder whether the above analytic expression is capable of approximating\nthe sensitivity when using other EoSs. Figure 3 shows the absolute magnitude ofUpdated Binary Pulsar Constraints on Einstein-æther theory 25\nFigure 3: Top panel shows the plot of sensitivity as a function of binding energy for\ndifferent EoS including Tolman VII (Eq. (82)) valid to O(C3). The bottom panel shows\nthe relative fractional difference between the EoS from data and the Tolman case, which\nrepresents the EoS variation in the relations. Observe that the universality holds to\nbetter than 3%.\nthe sensitivity as a function of \ncomputed analytically with Eq. (82), as well as\nnumerically with six other EoSs. Here we have chosen to work in a different region\nof parameter space, namely (\u000b1;\u000b2;c!) = (\u000010\u00004;\u00004\u000210\u00007;10\u00003), where we obtain a\nstable smoothly varying sensitivity curve. Observe that the sensitivities differ by less\nthan 3 %, exhibiting an approximate universality already discovered in [73] as a function\nof compactness. Given these results, in all future calculations we will use the analytic\nsensitivities computed with the Tolman VII EoS.\n6. Constraints from binary pulsars and triple systems\nThe majority of millisecond pulsars are found in binary and triple systems. The orbital\ndynamics of these systems modulate the time of arrival of radio waves and allow for\nprecise measurements of the orbital parameters [27, 40, 63, 64]. In this section, weUpdated Binary Pulsar Constraints on Einstein-æther theory 26\ndiscuss the use of precise orbital parameter data to place constraints on the Lorentz-\nviolating Einstein-æther theory. In GR energy is carried away at quadrupolar order\ndue to propagation of tensor modes whereas in this theory (and many of other modified\ntheories of gravity), one usually finds radiation from extra scalar and vector modes\nwhich are responsible for energy loss at dipole order, i.e., \u00001PN order (c.f. the term\nproportional toEin Eq. (30)) as compared to GR. Hence, energy is radiated faster than\nwhat is predicted in GR. This results in a decrease in the orbital separation and orbital\nperiod (Pb) of the binary. The modified orbital period decay rate ( _Pb) relates to the\ntotal energy of the binary, i.e., Eq. (30) via\n_Pb\nPb=\u00003\n2_Eb\nEb; (83)\nsuggesting a strong dependence of _Pbon the sensitivity of the NS [17]. Since the GR\npredictions agree with the observed value of _Pbwithin observational uncertainty, this\nallows for stringent constraints to be placed on æther theory.\n6.1. Observations\nFollowing from Eqs. (30) (with LV terms set to zero) and (83), one can relate the\npost-Keplerian parameter _Pbin GR to the Keplerian parameter Pbvia [73]\n _Pb\nPb!\nGR=\u0000384\u0019\n522=3\u0012\u0019(m1+m2)\nPb\u00135=3m1m2\n(m1+m2)21\nPb: (84)\nHerem1andm2are the masses in the binary system. In principle, if we can measure the\nmasses, orbitalperiodsandtheorbitalperioddecayrateswithsomeuncertaintyandfind\nthat they are consistent with GR predictions, then we can place constraints on Einstein-\næther theory. In this section we focus on data from the measurements of Keplerian\nand post Keplerian parameters of four different pulsar systems PSR J1738+0333, PSR\nJ0348+0432, PSR J1012+5307 and PSR J0737-3039 (Table 1) and a stellar triple\nsystem [67]. The first three are pulsar-white dwarf binaries in orbits with O(10\u00007)\neccentricity and 8.5-hour period, 0.17 eccentricity and 4.74-hour period and O(10\u00007)\neccentricity and 14.5-hour period respectively. The fourth is the double pulsar binary\nsystem with 0.088 eccentricity and 2.45-hour period. Because of the small eccentricity\nof these systems, we will ignore it in the following, i.e. we will consider quasi-circular\nbinaries.\n6.2. Parameter Estimation and Bayesian Analysis\nOur goal is to constrain the theory parameters using measurements of _Pb. We discuss\nbriefly the Bayesian formalism with Markov-Chain Monte-Carlo (MCMC) explorationUpdated Binary Pulsar Constraints on Einstein-æther theory 27\nTable 1: Orbital parameters as measured for the binary systems studied in this paper.\nTable shows the estimated values of the parameters and the 1- \u001buncertainty in the last\ndigits in parentheses. Here, _Pbobsis the observed value of _Pb.\nPulsar System m1(M\f)m2(M\f) Pb(days) _Pbobs\nPSR J1738+0333[36] 1:46+0:06\n\u00000:05 0:181+0:008\n\u00000:007 0:3547907398724(13) \u000025:9(3:2)\u000210\u000015\nPSR J0348+0432 [6] 2.01(4) 0.172(3) 0:102424062722(7) \u00000:273(45)\u000210\u000012\nPSR J1012+5307 [21] [52] 1:64(0:22) 0:16(0:02) 0:60467272355(3) \u00001:5(1:5)\u000210\u000014\nPSR J0737-3039 [51] 1.3381(7) 1.2489(7) 0.10225256248(5) \u00001:252(17)\u000210\u000012\nused to calculate the posteriors on the model parameters [c.f. Sec. 6.2.1] and derive\nrobust constraints.\nFor the parameter estimation, we need the expression for the orbital period decay, which\ndepends on both the relative velocity of the binary constituents v21and the center-of-\nmass velocity wof the binary’s center of mass with respect to the æther field. A natural\nchoice for the æther field direction is provided by the cosmic microwave background (i.e.\nthe æther is expected to be approximately aligned with the cosmological background\ntime direction). In this case, a typical value for the center-of-mass velocity is w\u0018\n10\u00003[33], which for binary pulsar observations is of the same order as v21. If so, the w-\ndependent corrections in the rate of change of the binding energy [Eq. (30)] and orbital\nperiod [Eq. (85)] enter at the same PN order (0PN) as the quadrupole emission terms\nof GR, but multiplied by either (s1\u0000s2)or powers of it. As such, these w-dependent\ncorrections are negligible for both white dwarf-pulsar systems (for which the dominant\nterm is the\u00001PN dipole emission) and also for the relativistic double pulsar system\n(for which s1\u0000s2\u00190as a result of the similar pulsar masses, which kills both the\ndipole emission and the w-dependent corrections to quadrupole emission). Therefore,\nour results are independent of the exact value of was long as that is of order w\u001810\u00003\nor smaller [73]\nFrom the above assumption and using Eqs. (83) and (30), the orbital period decay rate\nin Einstein-æther theory is a function of the individual masses (m1;m2)+, pulsar radii\n(R?;1;R?;2), orbital period ( Pb) and coupling constants (\u000b1;\u000b2;c!)as shown below\n_Pb\nPb=1\n5(m1+m2)4Pbq\n\u000b1\n(\u000b1\u00008\u000b2)q\n\u0000c!\n\u000b1\u000b3\n1c2\n!(\n3 28=3\u0019 \n\u0019(m1+m2)\nPb!5=3\nm1m2\u0014\n\u00001\n12 \n5\u000b1c!(m1+m2)2(s1\u0000s2)2\n+Thesearetheactivemasses, whosefractionaldifferencefromthe“real” masses ( ~m1;~m2)isoftheorder\nof the sensitivities and thus negligible. In the following we will therefore typically identify (m1;m2)and\n( ~m1;~m2). Note that this could however introduce correlations not captured by our sufficient statistics\napproach, but as we show, even large correlations would have little impact on the results.Updated Binary Pulsar Constraints on Einstein-æther theory 28\n \n25=6\u000b2\n1(\u000b1+ 8)r\u000b1\n(\u000b1\u00008\u000b2)\u0000213=3r\u0000c!\n\u000b1c!(\u000b1\u00008\u000b2)!\n \nPb\n\u0019m!2=3!\n+ \n(s1\u00001)(s2\u00001)!2=3 \n(\u000b1+ 8)\u000b3\n1\n \n\u00004c2\n!(m1+m2)2r\u0000c!\n\u000b1+p\n2\u000b1(m1s2+m2s1)2!r\u000b1\n(\u000b1\u00008\u000b2)+\n2q\n\u0000c!\n\u000b1(\u000b1\u00008\u000b2)2c2\n!((m1+m2)\u000b1+ 8m1s2+ 8m2s1)2\n3!#)\n; (85)\nwheres1ands2arefunctionsofCandcouplingconstants. Onemayworrythattheterms\ninside the square roots in the above expression may be negative, leading to a complex\norbital decay rate, but this is not the case because when \u000b1>0, thenc!\u0014\u0000\u000b1=2, while\nwhen\u000b1<0, thenc!\u0015\u0000\u000b1=2. As noted in Fig. 3, sensitivities are independent of the\nEoS. Here we choose to work with the Tolman VII EoS since it gives stable analytic\nsolutions for the sensitivities.\nThere are some phenomenological constraints on the æther coupling constants as\ndiscussed in Sec. 2, i.e., j\u000b1j.10\u00004,j\u000b2j.10\u00007(Solar system constraints), \u000b1<0,\n\u000b1<8\u000b2<0andc!>\u0000\u000b1=2(positive energy, absence of vacuum Cherenkov\nradiation and gradient instabilities) and c\u001b.10\u000015(GW constraint).\u0003Using these\npre-existing constraints and by determining if the estimated value of _Pblies within the\nrange _Pbobs\u0006\u000e_Pbobs(Table 1), we determine the consistency of points in the parameter\nspace with observations.\nOne important point is that we are not using the pulsar timing data directly [5], but\ninstead we are using existing constraints on _Pb; m 1andm2derived from the primary\npulsar timing data as a sufficient statistic . Unfortunately, the published results only\nquote values for the individual parameters and their uncertainties, so we do not have\naccess to the joint posterior distributions. For simplicity we assume that the parameter\ncorrelations are negligible. To check the impact of this assumption, we compared results\nwith zero correlations with a case with 90% correlation between the parameters, and\nfound that it only changed the results by a maximum of 17%.\n6.2.1. Bayesian Analysis We are interested in constructing a posterior distribution on\na set of model parameters ~\u0015= (m1,m2,Pb,R?;1,R?;2,\u000b1,\u000b2,c!) and using an MCMC\nalgorithm to explore the parameter space. According to Bayes’ theorem, the probability\n\u0003Note that we cannot use existing bounds on ^\u000b1[62, 69] and ^\u000b2[61, 62, 69] as priors. This is because\nthose quantities depend on the derivatives of the sensitivities (c.f. Appendix A), which are currently\nunknown.Updated Binary Pulsar Constraints on Einstein-æther theory 29\ndensity for parameters ~\u0015given data D and hypothesis H (the theory) is\nP(~\u0015jD;H) =P(Dj~\u0015;H)P(~\u0015jH)\nP(DjH); (86)\nwhereP(~\u0015jH)is called the prior which represents the state of knowledge about the\nparametersbeforeweanalyzethedata. P(Dj~\u0015;H)iscalledthelikelihoodwhichdescribes\nthe probability of measuring data D given the model H and a set of parameters ~\u0015.\nP(DjH)is called the model evidence which represents the overall normalization factor.\nIn practice it is better to work with log probability densities to better cover the dynamic\nrange of the densities.\nWe assumed uniform priors on \u000b2andc!such that\u00004\u000210\u00007\u0014\u000b2\u00144\u000210\u00007and\n\u0000105\u0014c!\u0014105and a Gaussian prior for \u000b1,m1,m2,_Pbwith mean and standard\ndeviation given by the existing bounds listed in Table 1. We use Gaussian priors on\nR?;1andR?;2with mean and standard deviation given 12:4\u00061:1km based on LIGO\nand NICER measurements [25]. While these bounds are derived assuming GR, the\ncorrections due to LV effects are sub-dominant compared to those impacting _Pb(c.f. e.g.\nfootnoteP). Using lunar laser ranging experiments, the bounds on \u000b1were obtained to\nbe\u000b1= (\u00000:7\u00060:9)\u000210\u00004[55] (c.f. also Sec. 2).\nThe log likelihood function is\nln(P(Dj~\u0015;H))/\u00001\n2\u0012\u0010\n_Pb=Pb\u0011obs\n\u0000\u0010\n_Pb=Pb\u0011th\u00132\n\u001b2\n(_Pb=Pb); (87)\nwhere (_Pb=Pb)this the theoretically predicted value of (_Pb=Pb)from the model. With the\nlikelihood and the priors in place, we can find the posterior using a MCMC algorithm.\nWestarttheMCMCsimulationnearthemeanvaluesforthemodelparameters,calculate\nthe posterior and iterate through these steps. Model parameters are allowed to explore\nthe entire range of parameter space and that gives the joint posterior distribution on all\nparameters ~\u0015. For the proposal distribution we use the prior distribution for a certain\nset of parameters, and a relative jump from the current position for the remaining.\nProposed jumps are accepted or rejected based on the Metropolis-Hastings acceptance\nprobability\nH=min \nP(~\u0015newjH)P(Dj~\u0015new;H)Q(~\u0015oldj~\u0015new)\nP(~\u0015oldjH)P(Dj~\u0015old;H)Q(~\u0015newj~\u0015old);1!\n: (88)\nA random number u\u0018U[0;1]is drawn, and if H > uthe proposed jump is accepted,\notherwise it is rejected. This process is repeated multiple times to ensure convergence.\nWe begin by considering a single observation from the pulsar-white dwarf system PSR\nJ1738+0333. Since the sensitivity of a white dwarf (WD) is negligible compared to theUpdated Binary Pulsar Constraints on Einstein-æther theory 30\nFigure4: Priorandposteriordistributiononthemodelparameters ~\u0015from _Pbconstraints\nfor PSR J1738+0333. The pre-existing constraints from solar system, Big-Bang\nnucleosynthesis and stability requirements are applied to uniform priors shown in blue\nwhere, for \u000b1negative it results in a negative \u000b2and positive c!. The posterior\ndistributions depend on the _Pbconstraints for PSR J1738+0333. The three shades\nof contours in the prior and posterior distribution in the off-diagonal cross-correlation\npanels represent 1- \u001b, 2-\u001band 3-\u001buncertainty on model parameters starting from the\ncenter(weonlyshow1- \u001bshadedregionsfortheone-dimensionalmarginaldistributions).\nThere is a small dip at very small magnitudes of \u000b1, as that is the region where the\npre-existing constraints come into play, while for larger magnitudes the constraints on\n\u000b1are automatically satisfied. Observe that the value of \u000b1is further constrained by a\nfactor of 2 compared to existing solar system constraints (prior).Updated Binary Pulsar Constraints on Einstein-æther theory 31\nFigure 5: Joint prior and posterior distribution on \u000b1,\u000b2andc!from _Pbconstraints for\nall four pulsars listed in (Table 1) The constraint on \u000b1is improved by a factor of 2.\nNS we can set s2=sWD= 0(thusR?;2is excluded from ~\u0015) but for a double pulsar binary\nwe should have s26= 0. Figure 4 shows the prior and posterior distribution on the model\nparameters. We are recovering our priors on the masses and radii, given that these are\nwell constrained as can be noted from Table 1. The distribution on \u000b1and\u000b2is such\nthat\u000b1<0and\u000b2<0from existing constraints. This pulsar system further constrains\nthe value of parameter \u000b1by approximately a factor of 2 while the coupling constants\n\u000b2andc!remain unconstrained. Notice that the posteriors on \u000b2andc!are flat and\nvery similar to the priors. Therefore, one cannot model them as Gaussian and construct\nconfidence region, as no information is gained for the values of these parameters.\nWe then consider constraints on the coupling parameters by stacking all four different\nbinary systems from Table 1 and computing the joint constraints (Fig. 5). These\njoint constraints also restrict the region of \u000b1by a factor of 2 better than the existing\nconstraints.\nIn the coming years we expect to have more observations, as the sensitivities of radio\ntelescopes will improve as a result of larger collecting areas (e.g. the Square Kilometre\nArray (SKA) project [68, 20]), which will allow for discovering more pulsars. Moreover\nthelongerobservationtime( T)willreducetheerrorinmeasurementsof _PbbyT\u00005=2[14],\nallowing for more precise measurements of the orbital parameters. One may wonder,Updated Binary Pulsar Constraints on Einstein-æther theory 32\nFigure 6: Prior and posterior distribution on the model parameters \u000b1,\u000b2andc!from\n_Pbconstraints for PSR J0348+0432, in a scenario where the observational uncertainties\ntighten by a factor of 10 and the value of _Pbmatches the GR prediction. It shows that\nthe GR values are favoured and the value of \u000b1is very closely centered around zero.\nwhether we can get tighter constraints from finding Nsimilar systems or a single system\nmeasuredwithhigherSNR(signal-to-noiseratio). TheSNR2growslinearlywithnumber\nof sourcesNand the observing time T, and quadratically with the effective collecting\narea of the radio telescope. Significant improvements in the measurement sensitivity are\nmore likely to come from some combination of larger telescopes and additional observing\ntime than from discovering large numbers of systems similar to those known. Figure 6\nillustrates the kind of bounds we will get for a PSR J0348+0432 system if _Pbobsmatches\nthe GR prediction _PbGRand the uncertainties are tightened by a factor of 10. The\nimproved constraints on _Pbtranslate directly into similarly improved bounds on \u000b1. We\nalso considered an alternative scenario, in which the uncertainties in _Pbimproved by a\nfactor of 10, but stayed centered on the current observed value. As shown in Figure 7,\nthis leads to a value for \u000b1bounded away from zero. In other words, in a scenario where\nthe observed period derivative stays at the current value while the uncertainty drops by\na factor of ten, we would find that Einstein-æther theory would be favored over GR!Updated Binary Pulsar Constraints on Einstein-æther theory 33\nFigure7: Priorandposteriordistributiononthemodelparameters \u000b1,\u000b2andc!from _Pb\nconstraints for PSR J0348+0432, in a scenario where the uncertainties in measurements\nare reduced by a factor of 10, and the value of _Pbstays at the currently observed value.\nThe 1-\u001buncertainty shows that \u000b1= 0, i.e., the GR value is disfavoured. It can also be\nnoted from Table 2 that posterior does not include \u000b1= 0:\n6.3. Constraints from the triple system\nNextwehaveconstraintscomingfromapulsarinastellartriplesystemPSRJ0337+1715\nconsisting of an inner millisecond pulsar-white dwarf binary and a second white dwarf\n(WD)inanouterorbit[7]. DuetothegravitationalpulloftheouterWD,thepulsarand\nthe inner WD experience accelerations that differ fractionally. If the strong equivalence\nprinciple is violated (as a result of the sensitivities), the triple system constrains the\nfractional acceleration difference parameter \u000eato(+0:5\u00061:8)\u000210\u00006[67]. The relation\nbetween\u000eaand the sensitivity parameter \u001bpulsar(before rescaling) in Einstein-æther\ntheory is [70, 9]\nj\u000eaj=\f\f\f\f\u001bpulsar\n1 +\u001bpulsar=2\f\f\f\f\u0019j\u001bpulsarj; (89)\nas can be obtained directly from Eq. (29) (in the Newtonian limit).\nWe use MCMC simulations in Bayesian analysis similar to that for the _PbconstraintUpdated Binary Pulsar Constraints on Einstein-æther theory 34\nFigure 8: Joint prior and posterior distribution on the model parameters ~\u0015from _Pb\nconstraints for all four pulsars listed in (Table 1) and stellar triple system. Observe\nthat inclusion of stellar triple system improves the constraints and parameter \u000b1is now\nconstrained by a factor of 10 better than lunar laser ranging experiments.\nand with the likelihood\nP(Dj~\u0015;H)/exp \n\u00001\n2\u0000\n\u001bobs\npulsar\u0000\u001bth\npulsar\u00012\n\u001b2\n(\u001bpulsar )!\n; (90)\nwhere\u001bobs\npulsar = (+0:5\u00061:8)\u000210\u00006from Eq. (89) and \u001bth\npulsaris given by Eq. (58),\nto constrain the model parameters. Figure 8 shows the joint pulsar and triple system\nconstraints on the model parameters ~\u0015assuming uniform distribution in \u000b2andc!.\nThe 95% upper limit on \u000b1, which was\u00002:4\u000210\u00004(from the prior constraints) has\nnow shifted to \u000b1=\u00002:4\u000210\u00005. It shows that the preferred frame parameter \u000b1is\nconstrained by a factor of 10 better than the lunar laser ranging experiments.\nTable 2 shows bounds on \u000b1from binary and triple systems mentioned in this paper.\nThe data from joint binary + triple system allows us to put a stringent constraint on\n\u000b1, which is an order of magnitude stronger than the bounds from lunar laser ranging\nexperiments [71, 55].Updated Binary Pulsar Constraints on Einstein-æther theory 35\nTable 2: Our bounds on \u000b1from different pulsar systems shown in Figs. 4–8 with 1- \u001b\nuncertainity. The first half shows the bounds from existing measurements, while the\nsecond half shows projected future bounds assuming that the measurement error on\n_Pobs\nbreduces by a factor of 10 with the central value of _Pobs\nbat the GR predicted value\n(\u000027:3\u000210\u000014) and at the current measured value ( \u000025:3\u000210\u000014).\nPulsar System \u000b1\nPSR J1738+0333 (\u00003:975\u00062:968)\u000210\u00005\nJoint binary system (\u00004:073\u00062.936)\u000210\u00005\nJoint binary + triple system (\u00001:111\u00060.674)\u000210\u00005\nPSR J0348+0432 ( _Pbobs=\u000027:3\u000210\u000014)(\u00008:119\u00064.622)\u000210\u00005\nPSR J0348+0432 ( _Pbobs=\u000025:3\u000210\u000014)(\u00001:729\u00061.805)\u000210\u00005\n7. Conclusions\nWe have investigated Einstein-æther theory in the context of binary pulsars and NSs.\nWe have recalculated the sensitivities in the regime of coupling parameter space that\nstill survives after the recent measurement of the speed of GWs. This required the\ndevelopment of a new post-Minkowskian approach that allows for stable numerical\nevaluation of the sensitivities, in addition to the derivation a closed form analytic\nsolution for the Tolman VII EoS. We used these results to place a constraint on certain\ncoupling constants of Einstein-æther theory using Bayesian analysis of binary pulsar\nobservations, including recent observations on the triple system. We find that these\ndata allows for constraints on a certain combination of the coupling constants, \u000b1, of\nO(10\u00005), improving current Solar System constraints by one order of magnitude.\nThe work carried out here opens the door to several avenues for future research. One\nsuch avenue is to use gravitational wave data directly to place constraints on Einstein-\næther theory, now that the sensitivites have been analytically calculated. This can be\ndone today to leading post-Newtonian order in the inspiral, and it remains to be seen\nwhether it is enough to lead to interesting constraints. To include the very late inspiral\nand merger phase, numerical simulations of coalescing NSs would have to be carried out\nin Einstein-æther theory. However, since the parameter space of the theory is already\nquite well constrained, it is not clear whether stronger bounds can be achieved with\ngravitational wave data.\nAnotheravenueforfutureresearchconcernscomputingsensitivitiesforblackholes. This\nhas been done in khronometric theory [59] but not yet in Einstein-æther theory. Once\nthe black hole sensitivities are in hand, and assuming they do not vanish, one could useUpdated Binary Pulsar Constraints on Einstein-æther theory 36\nthe existing GW data for binary black hole mergers to constrain Einstein-æther theory,\nincluding the dipole radiation effect in the gravitational waveform.\nOne more avenue for future work would be along the lines of improving the analysis\nin this paper by directly analyzing binary pulsar data and carrying out a parameter\nestimation and model selection study with a GR and a non-GR timing model. For this,\nit would be ideal to compute the derivative of the NS sensitivities that enter in the\nconservative post-Keplerian parameters, such as the periastron precession and Shapiro\ntime delay.\nAcknowledgements\nWe would like to thank Clifford Will and Ted Jacobson for many insightful discussions,\nwhich motivated part of this work. TG and NJC acknowlege the support of NASA\nEPSCoR grant MT-80NSSC17M0041 and NSF grant PHY-1912053. KY acknowledges\nsupport from NSF Grant PHY-1806776, NASA Grant No. 80NSSC20K0523, a Sloan\nFoundation Research Fellowship and the Owens Family Foundation. KY would\nlike to also thank the support by the COST Action GWverse CA16104 and JSPS\nKAKENHI Grants No. JP17H06358. NY acknowledges support from NASA grant\nNo. NNX16AB98G, No. 80NSSC17M0041, No. 80NSSC18K1352 and NSF grant PHY-\n1759615. EB and MHV acknowledge financial support provided under the European\nUnion’s H2020 ERC Consolidator Grant “GRavity from Astrophysical to Microscopic\nScales” grant agreement no. GRAMS-815673.\nAppendix A. Modified EIH technique\nIn this Appendix, we start from the 1PN acceleration (29) and analyze the effect of the\n1PN conservative dynamics on the orbital parameters of a binary of compact objects.\nWe will follow the osculating-orbits technique of [70], which will lead us to amend the\ncalculation of the preferred frame parameters ^\u000b1and^\u000b2presented in [73]. In doing so,\nwe will also correct a few typos that we found in the expressions of [70]. ]\nThe relative acceleration between the two gravitating bodies is obtained by simply\nletting\na=dv1\ndt\u0000dv2\ndt; (A.1)\nwhile the position of the center of mass is not accelerated. Thus, we can set _X=X= 0\nat Newtonian order without any loss of generality, getting\nx1=\u0010m2\nm+O(\u000f)\u0011\nx; (A.2)\n]We have double checked the correctness of our expressions with the authors of [70].Updated Binary Pulsar Constraints on Einstein-æther theory 37\nx2=\u0000\u0010m1\nm+O(\u000f)\u0011\nx; (A.3)\nwhere\u000f\u0018m=r\u0018v2\n21is a book-keeping parameter that counts PN order.\nHere the acceleration of every individual body is given by (29). Hereinafter we will\nborrow the notation of [70] and thus we define\nm=m1+m2; \u0011 =m1m2\nm2;\u0001 =m2\u0000m1\nm; (A.4)\nand the functions of the sensitivities\nG=G12;B+=B(12);B\u0000=B[12];D=m2\nmD122+m1\nmD211;\nE=E12;A(n)=\u0010m2\nm\u0011n\nA1\u0000\u0010\n\u0000m1\nm\u0011n\nA2: (A.5)\nUsing this, the relative acceleration can be written in a compact form\na=aL+aPF; (A.6)\nwhere we have separated the purely local contributions and those coming from preferred\nframe effects. The former reads\naL=m\nr2h\nn\u0010\n^A1v2\n21+^A2_r2+^A3m\nr\u0011\n+ _r^Bv21i\n; (A.7)\nwhere\nv21=_x2\u0000_x1; (A.8)\n^A1=1\n2\u0002\nG(1\u00006\u0011)\u00003B+\u00003\u0001B\u0000\u0000\u0011(C12+ 2E) +GA(3)\u0003\n; (A.9)\n^A2=3\u0011\n2(G+E); (A.10)\n^A3=D+G[2\u0011G+ 3B++\u0011(C12+E) + 3\u0001B\u0000]; (A.11)\n^B=G(1\u00002\u0011) + 3B++ 3\u0001B\u0000+\u0011G+GA(3): (A.12)\nThese expressions agree with those of [70]. However, we find a difference in the\nacceleration due to preferred frame effects\naPF=m\nr2\u001a\n\u0000n\u0014\u0012^\u000b1\n2+ 2GA(2)\u0013\n(!\u0001v21) +3\n2\u0000\n^\u000b2+GA(1)\u0001\n(!\u0001n)2\u0015\n(A.13)\n\u0000!\u0014^\u000b1\n2(n\u0001v21) + ^\u000b2(n\u0001!)\u0015\n+GA(2)v21(n\u0001!)\u001b\n\u0000m!2\n2r2\u0000\nC12+GA(1)\u0001\nn:\nwhere we have already specified the generic boost velocity win (29) to match the\nvelocity of the preferred frame !.Updated Binary Pulsar Constraints on Einstein-æther theory 38\nThis differs from the result of [70] in a sign multiplying the first whole line, as well as in\nthe last term, which is absent in [70]. Here we have defined the following compact-body\neffective PPN parameters\n^\u000b1= \u0001(C12+E)\u00006B\u0000\u00002GA(2); ^\u000b2=E\u0000GA(1): (A.14)\nThese are the strong field versions of the parameters \u000b1and\u000b2, which are contained\ninside the definition of the calligraphic objects in (A.14), so that ^\u000b1and^\u000b2are implicit\nfunctions of them. They are directly proportional to each other only in the case in which\nthe sensitivities vanish exactly.\nIn the absence of PN corrections, the motion of the two-body system describes a\nKeplerian orbit, parametrized by x=rnwith\nn= [\u0000cos \n cos(!+f)\u0000cos\u0013sin \n sin(!+f)]eX+ [sin \n cos( !+f)\n+ cos\u0013cos \n sin(!+f)]eY+ sin\u0013sin(!+f)eZ; (A.15)\nwhere the orbital elements are: inclination \u0013, longitude of the ascending node \nand\npericenter angle !. The element f=!\u0000\u001eis the true anomaly, with \u001ethe orbital phase\nmeasured from the ascending node. The reference vectors eiform an orthonormal basis.\nWhen the extra force (A.6) is included, Keplerian orbits are not solutions to the\nequations of motion anymore. However, provided that the force is small enough relative\nto the Newtonian force, we can use perturbation theory and translate the dependence\non time of the motion to the orbital parameters. This is the method of osculating orbits\ndescribed in [71], which leads to a secular variation of the orbital elements under the\neffect of a. In order to parametrize this change in terms of the velocity vector of the\npreferred frame, we decompose the latter by projecting it onto the orbital plane by\ndefining\n!P=!\u0001eP; !Q=!\u0001eQ; !Z=!\u0001z; (A.16)\nas well as onto the angular momentum vector\n!h=!\u0001h=!Zp\nGmp; (A.17)\nFollowing the computation in [71], we thus find that the local terms in aLinduce a\nchange only on the pericenter angle !, which in an orbit changes by\n\u0001L!=6\u0019m\nGp\u0014\nGB++1\n6\u0000\nG2\u0000D\u0001\n+1\n6G\u0000\n6\u0001B\u0000+\u0011(2C12+E) +GA(3)\u0001\u0015\n;(A.18)\nandisofcourseindependentof !. Again,thisagreeswith[70]uptoatypographicalerror\nin their result. The rest of secular changes vanish, either because they are identically\nzero or because they compensate along the orbit.Updated Binary Pulsar Constraints on Einstein-æther theory 39\nOn the other hand, the force induced by preferred frame effects produces a secular\nchange in all orbital parameters\n\u0001PFa=2\u0019e!P\n(1\u0000e2)2\u0012mp\nG\u00131\n2\u0000\n^\u000b1+ 4A(2)G\u0001\n; (A.19)\n\u0001PF\u0013=\u0019^\u000b1\u0012m\nGp\u00131\n2\n!hsin(!)eF(e)\u00002\u0019^\u000b2!h!RF(e)\nGp\n1\u0000e2; (A.20)\n\u0001PF\n =\u0000\u0019^\u000b1\u0012m\nGp\u00131\n2!h\nsin(\u0013)cos(!)eF(e)\u0000)2\u0019^\u000b2!h!SF(e)\nGsin(\u0013)p\n1\u0000e2; (A.21)\n\u0001PF$=\u0000\u0019^\u000b1\u0012m\nGp\u00131\n2\n!Qp\n1\u0000e2F(e)\ne\u0000\u0019^\u000b2(!2\nP!2\nQ)F(e)2\n+\u0019!Q\ne\u0012m\nGp\u00131\n2\n(^\u000b1+ 4A(2)G); (A.22)\n\u0001PFe=\u0000\u0019^\u000b1\u0012m\nGp\u00131\n2\n!P(1\u0000e2)F(e) + 2\u0019^\u000b2!P!Qep\n1\u0000e2F(e)2\n+\u0019!P\u0012m\nGp\u00131\n2\n(^\u000b1+ 4A(2)G); (A.23)\nwhere \u0001PF$= \u0001 PF!+ cos(\u0013)\u0001PF\nand\nF(e) =1\n1 +p\n1\u0000e2; (A.24)\n!R=!Pcos!\u0000!Qp\n1\u0000e2sin!; (A.25)\n!S=!Psin!+!Qp\n1\u0000e2cos!: (A.26)\nOut of these deviations, the most relevant one is the variation of the semimajor axis,\nwhich can be related to the change in the period of the orbit by using Kepler’s third law\n\u0001T\nT=3\n2\u0001a\na: (A.27)\nNote however that this change is sub-leading with respect to the change expected from\nemission of gravitational radiation in a binary system like the one considered throughout\nthis paper [c.f. 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D , 100:083012, Oct\n2019."    },    {        "title": "1909.08360v1.Global_smooth_solutions_of_the_damped_Boussinesq_equations_with_a_class_of_large_initial_data.pdf",        "content": "arXiv:1909.08360v1  [math.AP]  18 Sep 2019Global smooth solutions of the damped Boussinesq\nequations with a class of large initial data\nJinlu Li1∗, Xing Wu2†, Weipeng Zhu3‡\n1School of Mathematics and Computer Sciences, Gannan Normal U niversity,\nGanzhou, Jiangxi, 341000, China\n2College of Information and Management Science, Henan Agricul tural University,\nZhengzhou, Henan, 450002, China\n3School of Mathematics and Information Science, Guangzhou Un iversity, Guangzhou 510006, China\nAbstract: The globalregularityproblemconcerning theinviscid Boussinesq equ ationsremains\nan open problem. In an attempt to understand this problem, we exa mine the damped Boussinesq\nequations and study how damping affects the regularity of solutions . In this paper, we consider the\nglobal existence tothedampedBoussinesq equations withaclass of largeinitial data, whose Bs\np,ror\n˙Bs\np,rnorms can be arbitrarily large. The idea is splitting the linear Boussines q equations from the\ndampedBoussinesq equations, theexponentially decaying solutiono ftheformerequationstogether\nwith the structure of the Boussinesq equations help us to obtain th e global smooth solutions.\nKeywords: Boussinesq equations; Global existence; Large initial data.\nMSC (2010): 35Q35; 76B03\n1 Introduction\nThis paper considers the global smooth solutions for the incompres sible Boussinesq equations with\ndamping\n\n\n∂tu+u·∇u+νu+∇p=θed, x∈Rd,t >0,\n∂tθ+u·∇θ+λθ= 0, x ∈Rd,t >0,\ndivu= 0, x ∈Rd,t≥0,\n(u,θ)|t=0= (u0,θ0), x ∈Rd,(1.1)\nwhereuis the velocity vector, pis the pressure, θdenotes the temperature or density which is a\nscalar function. ed= (0,0,...,0,1)Tandν,λare two positive parameters, standing for kinematic\nviscosity and thermal diffusivity, respectively.\nThe Boussinesq equations model large scale atmospheric and ocean ic flows that are responsible\nfor cold fronts and the jet stream [ 12,14] and mathematically has received significant attention,\n∗E-mail: lijinlu@gnnu.cn\n†E-mail: ny2008wx@163.com\n‡E-mail: mathzwp2010@163.com\n1since it has a vortex stretching effect similar to that in the 3D incompr essible flow. When νu\nis replaced by −ν∆u,λθby−λ∆θ, (1.1) becomes the standard viscous Boussinesq equations,\nthe global in time regularity in two dimension is well understood even in t he zero diffusivity\n(ν >0,λ= 0) or the zero viscosity case( ν= 0,λ >0) [1,3,4,5,8,9,10], however the global\nregularity in dimension three appears to be out of reach. while ν= 0 and λ= 0, (1.1) is reduced\nto the inviscid Boussinesq equations, due to the absence of dissipat ive terms, the global solution or\nfinite-time singularity evoluting from general initial data remains uns olved in spite of the progress\non the local well-posedness and regularity criteria [ 6,7,11,15,16,17]. Recently, following the\nconvex integration method, Tao and Zhang [ 16] obtained the H¨ older continuous solution with\ncompact support both in space and time for inviscid 2D Boussinesq eq uations.\nWhen adding velocity damping term νuand temperature damping term λθ, Adhikar et al. [ 2]\nproved (1.1) admits a unique global small solution with the initial data satisfying\n/ba∇dbl∇u0/ba∇dbl˙B0\n∞,1<min{ν\n2C,λ\nC},/ba∇dbl∇θ0/ba∇dbl˙B0\n∞,1<ν\n2C/ba∇dbl∇u0/ba∇dbl˙B0\n∞,1,\nwhere˙B0\n∞,1is the homogeneous Besov space. We are interested in that whethe r or nor ( 1.1)\npossesses a global solution without the smallness assumption. If no t, it may be helpful to obtain\nthe global solutions for a class of large initial data. Our effort in this p aper is precisely based on\nthis. In what follows, let ( U,Θ) be the solution of the ”linearized damped Boussinesq equation”,\n\n\n∂tU+νU+∇p= Θed,\n∂tΘ+λΘ = 0,\ndivU= 0,\n(U,Θ)|t=0= (U0,Θ0).(1.2)\nThe second equation in ( 1.2) satisfied by Θ is a simple linear equation, and has the solution\nΘ =e−λtΘ0. If we resort to the equations ( 1.2) of the vorticity W=∇×U, thenWsatisfys\n2D:∂tW+νW=∂1Θ,\n3D:∂tW+νW= (∂2Θ,−∂1Θ,0)T.\nTherefor both Wand Θ have explicit exponential decay expressions, so does for U. By virtue of\nthe good exponential decay property of Uand Θ, we obtain the global solution of ( 1.1) which is\nconstructed with the large initial data.\nWe first recall the definition of the Besov spaces. Choose a radial, n on-negative, smooth and\nradially decreasing function χ:Rd→[0,1] such that it is supported in {ξ∈Rd:|ξ| ≤4\n3}and\nχ≡1 for|ξ| ≤3\n4. Letϕ(ξ) =χ(ξ\n2)−χ(ξ). Thenϕis supported in the ring {ξ∈Rd:3\n4≤ |ξ| ≤8\n3}\nandϕ≡1 for4\n3≤ |ξ| ≤3\n2. Foru∈ S′,q∈Z, we define the Littlewood-Paley operators:\n˙∆qu=F−1(ϕ(2−q·)Fu), ∆qu=˙∆quforq≥0, ∆qu= 0 forq≤ −2 and ∆ −1u=F−1(χFu), and\nSqu=F−1/parenleftbig\nχ(2−qξ)Fu/parenrightbig\n. Here we use F(f) or/hatwidefto denote the Fourier transform of f.\nThe standard vector-valued functions u:Rd→Rdin Besov spaces Bs\np,rand˙Bs\np,rcan be defined\nby\n/ba∇dblu/ba∇dblBsp,r/defines/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2js/ba∇dbl∆ju/ba∇dblLp)j∈Z/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nℓr<+∞,\n/ba∇dblu/ba∇dbl˙Bsp,r/defines/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2js/ba∇dbl˙∆ju/ba∇dblLp)j∈Z/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nℓr<+∞.\nObviously, if supp ˆ u∈ {ξ:4\n3≤ |ξ| ≤3\n2}, then we have\n||u||Lp=||u||Bsp,r=||u||˙Bsp,r.\nOur main result is stated as follows.\n2Theorem 1.1 Letd= 2,3. Assume that the initial data fulfills divu0= 0and\nu0=U0+v0andθ0= Θ0+ϑ0\nwith\nsuppˆU0(ξ),ˆΘ0(ξ)⊂ C:=/braceleftBig\nξ/vextendsingle/vextendsingle4\n3≤ |ξ| ≤3\n2/bracerightBig\n. (1.3)\nDenote\nE0=/integraldisplay∞\n0(||U·∇U||H3+||U·∇Θ||H3)dt, F0=/integraldisplay∞\n0||(U,Θ)||L∞dt.\nThere exists a sufficiently small positive constant δ, and a universal constant Csuch that if\n/parenleftBig\n||v0||2\nH3+||ϑ0||2\nH3+E0/parenrightBig\n·exp/parenleftBig\nCF0+CE0/parenrightBig\n≤δ, (1.4)\nthen the system (1.1)has a unique global solution.\nCorollary 1.1 Ford= 2, we letv0=ϑ0= 0and\nU0=∇⊥a0=/parenleftbigg∂2a0\n−∂1a0/parenrightbigg\n,Θ0=a0,\nwhere\nsupp ˆa0(ξ)⊂/tildewideC:=/braceleftBig\nξ/vextendsingle/vextendsingle|ξ1−ξ2| ≤ε,4\n3≤ |ξ| ≤3\n2/bracerightBig\n.\nThen, direct calculations show that the left side of ( 1.4) becomes\nCε||a0||L2||ˆa0||L1exp/parenleftBig\nCε||a0||L2||ˆa0||L1+C||ˆa0||L1/parenrightBig\n.\nFord= 3, we letv0=ϑ0= 0and\nU0=\n∂2a0\n−∂1a0,\n0\n,Θ0=a0,\nwhere\nsupp ˆa0(ξ)⊂/tildewideC0:=/braceleftBig\nξ/vextendsingle/vextendsingle|ξ1−ξ2| ≤ε,41\n30≤ |ξh| ≤22\n15, ε≤ |ξ3| ≤2ε/bracerightBig\n.\nThen, direct calculations show that the left side of ( 1.4) becomes\nCε||a0||L2||ˆa0||L1exp/parenleftBig\nCε||a0||L2||ˆa0||L1+C||ˆa0||L1/parenrightBig\n.\nRemark 1.1 Ford= 2, we set\na0(x1,x2) =ε−1/parenleftBig\nloglog1\nε/parenrightBig1\n2χ(x1,x2),\n3where the smooth functions χsatisfying ˆχ(−ξ1,−ξ2) = ˆχ(ξ1,ξ2),\nsuppˆχ⊂/tildewideC,ˆχ(ξ)∈[0,1]andˆχ(ξ) = 1forξ∈/tildewideC1,\nwhere\n/tildewideC1/defines/braceleftBig\nξ∈R2:|ξ1−ξ2| ≤1\n2ε,25\n18≤ |ξ| ≤13\n9/bracerightBig\n.\nThen, direct calculations show that the left side of (1.4)becomes\nCε1\n2/parenleftBig\nloglog1\nε/parenrightBig\nexp/parenleftBig\nCloglog1\nε/parenrightBig\n.\nIn fact, one has\n||ˆa0||L1≈/parenleftBig\nloglog1\nε/parenrightBig1\n2and||a0||L2≈ε−1\n2/parenleftBig\nloglog1\nε/parenrightBig1\n2.\nTherefore, choosing εsmall enough, we deduce that the system ( 1.1) has a unique global solution.\nMoreover, we can show that\n||u0||L∞/greaterorsimilar/parenleftBig\nloglog1\nε/parenrightBig1\n2,||θ0||L∞/greaterorsimilar/parenleftBig\nloglog1\nε/parenrightBig1\n2.\nRemark 1.2 Ford= 3and1< p <∞we set\na0(x1,x2,x3) =ε−2(p−1)\np(loglog1\nε)1\n2χ(x1,x2)φ(x3).\nwhere the smooth functions χ,φsatisfying\nsuppˆχ∈/tildewideC,ˆχ(ξ)∈[0,1]andˆχ(ξ) = 1forξ∈/tildewideC1,\nand\nsuppˆφ(ξ′)∈[ε,2ε],ˆφ(ξ′)∈[0,1]andˆφ(ξ) = 1forξ′∈[5\n4ε,7\n4ε].\nThen, direct calculations show that the left side of (1.4)can be bounded by\nCε4\nploglog1\nεeCloglog1\nε.\nTherefore, choosing εsmall enough, we deduce that the system ( 1.1) has a global solution.\n||u0||Lp/greaterorsimilar/parenleftBig\nloglog1\nε/parenrightBig1\n2,||θ0||Lp/greaterorsimilar/parenleftBig\nloglog1\nε/parenrightBig1\n2.\nNotations : Letβ= (β1,β2,β3)∈N3be a multi-index and Dβ=∂|β|/∂β1x1∂β2x2∂β3x3with|β|=\nβ1+β2+β3. For the sake of simplicity, a/lessorsimilarbmeans that a≤Cbfor some “harmless” positive\nconstant Cwhich may vary from line to line. [ A,B] stands for the commutator operator AB−BA,\nwhereAandBare any pair of operators on some Banach space X. We also use the notation\n||f1,···,fn||X/defines||f1||X+···+||fn||X.\n42 Proof of Theorem 1.1\nIn this section, we give the proof of Theorem 1.1. Before giving the proof, we present some\nestimates which will be used in the proof of Theorem 1.1.\nProof of Theorem 1.1Denoting v=u−Uandϑ=θ−Θ, we can reformulate the system\n(1.1) and (1.2) equivalently as\n\n\n∂tv+v·∇v+U·∇v+v·∇U+νv+∇p′−ϑed=−U·∇U:=f,\n∂tϑ+v·∇ϑ+U·∇ϑ+v·∇Θ+λϑ=−U·∇Θ :=g,\ndivv= 0,\n(v,c)|t=0= (v0,c0).(2.5)\nApplying Dβto (2.5)1and (2.5)2respectively and taking the scalar product with σDβvandDβϑ\nrespectively, adding them together and then summing the result ov er|β| ≤3, we get\n1\n2d\ndt/parenleftBig\nσ||v||2\nH3+||ϑ||2\nH3/parenrightBig\n+σν||v||2\nH3+λ||ϑ||2\nH3−σ(ϑ,vd)H3/defines4/summationdisplay\ni=1Ii, (2.6)\nwhere\nI1=−σ/summationdisplay\n0<|β|≤3/integraldisplay\nR3[Dβ,v·]∇v·Dβvdx−/summationdisplay\n0<|β|≤3/integraldisplay\nR3[Dβ,v·]∇ϑ·Dβϑdx,\nI2=−σ/summationdisplay\n0<|β|≤3/integraldisplay\nR3Dβ(U·∇v)·Dβvdx−/summationdisplay\n0<|β|≤3/integraldisplay\nR3Dβ(U·∇ϑ)·Dβϑdx,\nI3=−σ/summationdisplay\n0≤|β|≤3/integraldisplay\nR3Dβ(v·∇U)·Dβvdx−/summationdisplay\n0≤|β|≤3/integraldisplay\nR3Dβ(v·∇Θ)·Dβϑdx,\nI4=−σ/summationdisplay\n0≤|β|≤3/integraldisplay\nR3Dβ(U·∇U)·Dβvdx−/summationdisplay\n0≤|β|≤3/integraldisplay\nR3Dβ(U·∇Θ)·Dβϑdx.\nNext, we need to estimate the above terms one by one.\nAccording to the commutate estimate (See [ 13]),\n/summationdisplay\n|α|≤m||[Dα,g]f||L2≤C(||f||Hm−1||∇g||L∞+||f||L∞||g||Hm), (2.7)\nwe obtain\nI1≤σ/summationdisplay\n0<|β|≤3||[Dβ,v·]∇v||L2||∇v||H2+/summationdisplay\n0<|β|≤3||[Dβ,v·]∇ϑ||L2||∇ϑ||H2\n≤C||∇v||L∞||v||H3||∇v||H2+C(||∇v||L∞||∇ϑ||H2+||v||H3||∇ϑ||L∞)||∇ϑ||H2\n≤C||v||H3/parenleftBig\n||∇v||2\nH2+||∇ϑ||2\nH2/parenrightBig\n≤C||v||H3/parenleftBig\n||v||2\nH3+||ϑ||2\nH3/parenrightBig\n. (2.8)\nInvoking the following calculus inequality which is just a consequence o f Leibniz’s formula,\n/summationdisplay\n|β|≤3||[Dβ,g]f||L2≤C(||∇g||L∞+||∇3g||L∞)||f||H2,\n5we obtain\nI2≤σ/summationdisplay\n0<|β|≤3||[Dβ,U·]∇v||L2||∇v||H2+/summationdisplay\n0<|β|≤3||[Dβ,U·]∇ϑ||L2||∇ϑ||H2\n≤C/parenleftBig\n||∇U||L∞+||∇3U||L∞/parenrightBig/parenleftBig\n||v||2\nH3+||ϑ||2\nH3/parenrightBig\n≤C||U||L∞/parenleftBig\n||v||2\nH3+||ϑ||2\nH3/parenrightBig\n,(2.9)\nwhere we have used the fact that supp ˆU(ξ),suppˆΘ(ξ)⊂suppˆU0(ξ)∪suppˆΘ0(ξ)⊂ C. By\nLeibniz’s formula and H¨ older’s inequality, one has\nI3≤σ||v·∇U||H3||v||H3+||v·∇Θ||H3||ϑ||H3\n≤C/parenleftBig\n||∇(U,Θ)||L∞+||∇4(U,Θ)||L∞/parenrightBig/parenleftBig\n||v||2\nH3+||ϑ||2\nH3/parenrightBig\n≤C||(U,Θ)||L∞/parenleftBig\n||v||2\nH3+||ϑ||2\nH3/parenrightBig\n. (2.10)\nUsing H¨ older’s inequality and Young inequality, we deduce\nI4≤C(||U·∇U||H3+||U·∇Θ||H3)(||v||H3+||ϑ||H3)\n≤C||(U·∇U,U·∇Θ)||H3+C||(U·∇U,U·∇Θ)||H3(||v||2\nH3+||ϑ||2\nH3).(2.11)\nPutting all the estimates ( 2.8)–(2.11) into (2.6), we obtain\nd\ndt/parenleftBig\nσ||v||2\nH3+||ϑ||2\nH3/parenrightBig\n+σν||v||2\nH3+λ||ϑ||2\nH3−σ(ϑ,vd)H3\n/lessorsimilar||v||H3/parenleftBig\n||v||2\nH3+||ϑ||2\nH3/parenrightBig\n+/parenleftBig\n||(U,Θ)||L∞+||(U·∇U,U·∇Θ)||H3/parenrightBig/parenleftBig\n||v||2\nH3+||ϑ||2\nH3/parenrightBig\n+||(U·∇U,U·∇Θ)||H3. (2.12)\nAccording to the inequality\nσ(ϑ,vd)H3≤Cσ3\n2||v||2\nH3+Cσ1\n2||ϑ||2\nH3,\nthen we can choose σsmall enough such that\nσν||v||2\nH3+λ||ϑ||2\nH3−σ(ϑ,vd)H3≈σν||v||2\nH3+λ||ϑ||2\nH3.\nFor simplicity, we denote\nA(t) =σ||v||2\nH3+||ϑ||2\nH3, B(t) =σν||v||2\nH3+λ||ϑ||2\nH3,\nthen (2.12) can be rewritten as\nd\ndtA(t)+B(t)≤CA1\n2(t)B(t)+C/parenleftBig\n||(U,Θ)||L∞+||(U·∇U,U·∇Θ)||H3/parenrightBig\nA(t)\n+C||(U·∇U,U·∇Θ)||H3. (2.13)\nNow, we define\nΓ/definessup{t∈[0,T∗) : sup\nτ∈[0,t]A(τ)≤η},\n6whereηis a small enough positive constant which will be determined later. Ass ume that Γ < T∗.\nFor allt∈[0,Γ], we obtain from ( 2.13) that\nd\ndtA(t)+B(t)≤C/parenleftBig\n||(U,Θ)||L∞+||(U·∇U,U·∇Θ)||H3/parenrightBig\nA(t)+C||(U·∇U,U·∇Θ)||H3.\n(2.14)\nwhich together with the assumption ( 1.4) yields that\nA(t)≤C/parenleftBig\n||v0||2\nH3+||ϑ0||2\nH3+E0/parenrightBig\n·exp/parenleftBig\nCE0+CF0/parenrightBig\n≤Cδ.\nChoosing η= 2Cδ, thus we can get supτ∈[0,t]A(τ)≤η\n2fort≤Γ. So if Γ < T∗, due to the\ncontinuity of the solutions, we can obtain there exists 0 < ǫ≪1 such that supτ∈[0,t]A(τ)≤ηfor\nt≤Γ+ǫ < T∗, which contradicts with the definition of Γ. Thus, we can conclude Γ = T∗and\nsup\nτ∈[0,t]/parenleftBig\n||v(τ)||2\nH3+||ϑ(τ)||2\nH3/parenrightBig\n≤C <∞for allt∈(0,T∗),\nwhich implies that T∗= +∞. This completes the proof of Theorem 1.1./square\n3 Proof of Corollary 1.1\nIn this section, we will give the proof of Corollary 1.1.\nCase 1: d=2. Firstly, Wsatisfys\n∂tW+νW=∂1Θ =∂1Θ0e−λt, W|t=0=W0=∇×U0.\nFormally,\nW=/braceleftBigg\ne−νtW0+1\nν−λ∂1Θ0(e−λt−e−νt), ν/ne}ationslash=λ,\ne−νtW0+te−νt∂1Θ0, ν=λ.\nTherefore, we can deduce that\nU= (−∆)−1∇⊥W=/braceleftBigg\ne−νtU0+1\nν−λ∂1(−∆)−1∇⊥Θ0(e−λt−e−νt), ν/ne}ationslash=λ,\ne−νtU0+te−νt∂1(−∆)−1∇⊥Θ0, ν=λ\nwhere∇⊥= (∂2,−∂1)T.\nLemma 3.1 Letd= 2. For small enough ε, under the assumptions of Theorem 1.1, the following\nestimates hold\nE0≤Cε||a0||L2||ˆa0||L1, F 0≤C||ˆa0||L1. (3.15)\nProof of Lemma 3.1With the above expressions of Uand Θ, we can show that\nU·∇Θ =/braceleftBigg\ne−(ν+λ)tU0·∇Θ0+1\nν−λ(e−2λt−e−(ν+λ)t)∂1(−∆)−1∇⊥Θ0·∇Θ0, ν/ne}ationslash=λ\ne−2νtU0·∇Θ0+te−2νt∂1(−∆)−1∇⊥Θ0·∇Θ0, ν=λ.\n7Notice that U0·∇Θ0= 0 and\n∂1(−∆)−1∇⊥Θ0·∇Θ0= (∂2−∂1)∂1(−∆)−1a0∂1a0+∂2\n1(−∆)−1a0(∂1−∂2)a0.\nUsing the classical Kato-Ponce product estimates and the fact th e Fourier transform of a distribute\nbelonging to L1lies inL∞, by a simple calculation, we obtain\n||∂1(−∆)−1∇⊥Θ0·∇Θ0||H3≤C||(∂2−∂1)a0||L∞||a0||L2≤Cε||a0||L2||ˆa0||L1,\nwhich implies\n||U·∇Θ||H3≤Ce−λtε||a0||L2||ˆa0||L1.\nTake the similar argument as the term U·∇Θ, we also have\n||U·∇U||H3≤Ce−min{ν,λ}tε||a0||L2||ˆa0||L1.\nThus, we complete the proof of Lemma 3.1./square\nCase 2: d=3. Firstly, W=∇×Usatisfys\n∂tW+νW= (∂2Θ,−∂1Θ,0)T= (∂2Θ0,−∂1Θ0,0)Te−λt, Wt=0=W0=∇×U0.\nFormally, we have\nW=/braceleftBigg\ne−νtW0+1\nν−λ(∂2Θ0,−∂1Θ0,0)T(e−λt−e−νt), ν/ne}ationslash=λ,\ne−νtW0+te−νt(∂2Θ0,−∂1Θ0,0)T, ν=λ.\nTherefore, we can deduce that\nU= (−∆)−1∇×W\n=/braceleftBigg\ne−νtU0+1\nν−λ(−∆)−1/parenleftbig\n∂1∂3Θ0,∂2∂3Θ0,−(∂2\n1+∂2\n2)Θ0/parenrightbigT(e−λt−e−νt), ν/ne}ationslash=λ,\ne−νtU0+te−νt(−∆)−1/parenleftbig\n∂1∂3Θ0,∂2∂3Θ0,−(∂2\n1+∂2\n2)Θ0/parenrightbigT, ν=λ.\nLemma 3.2 Letd= 3. For small enough ε, under the assumptions of Theorem 1.1, the following\nestimates hold\nE0≤Cε||a0||L2||ˆa0||L1, F 0≤C||ˆa0||L1. (3.16)\nProof of Lemma 3.2For the term U·∇Θ, we can show that when ν/ne}ationslash=λ,\nU·∇Θ =e−(ν+λ)tU0·∇Θ0+1\nν−λ(e−2λt−e−(ν+λ)t)(−∆)−1/parenleftbig\n∂1∂3Θ0,∂2∂3Θ0,−(∂2\n1+∂2\n2)Θ0/parenrightbigT·∇Θ0,\nwhileν=λ,\nU·∇Θ =e−2νtU0·∇Θ0+te−2νt(−∆)−1/parenleftbig\n∂1∂3Θ0,∂2∂3Θ0,−(∂2\n1+∂2\n2)Θ0/parenrightbigT·∇Θ0.\nNotice that U0·∇Θ0= 0 and\n(−∆)−1/parenleftbig\n∂1∂3Θ0,∂2∂3Θ0,−(∂2\n1+∂2\n2)Θ0/parenrightbigT·∇Θ0\n= (−∆)−1∂1∂3Θ0∂1Θ0+(−∆)−1∂2∂3Θ0∂2Θ0−(−∆)−1(∂2\n1+∂2\n2)Θ0∂3Θ0.\n8Using the classical Kato-Ponce product estimates and the fact th e Fourier transform of a distribute\nbelonging to L1lies inL∞, after a simple calculation, we obtain\n||(−∆)−1/parenleftbig\n∂1∂3Θ0,∂2∂3Θ0,−(∂2\n1+∂2\n2)Θ0/parenrightbigT·∇Θ0||H3\n≤C||∂3a0||L∞||a0||L2≤Cε||a0||L2||ˆa0||L1,\nwhich implies\n||U·∇Θ||H3≤Ce−λtε||a0||L2||ˆa0||L1.\nTake a similar argument as the term U·∇Θ, we also have\n||U·∇U||H3≤Ce−min{ν,λ}tε||a0||L2||ˆa0||L1.\nThus, we complete the proof of Lemma 3.2.\nNow Corollary 1.1follows immediately from Lemma 3.1and Lemma 3.2. We complete the\nproof of Corollary 1.1/square\nAcknowledgments\nThe work of Jinlu Li is supported by the National Natural Science Fo undation of China (Grant\nNo.11801090). The work of Weipeng Zhu is partially supported by the China National Natural\nScience Foundation under grant number 11901092 and Guangdong Natural Science Foundation\nunder grant number 2017A030310634.\nReferences\n[1] H. Abidi, T. Hmidi, On the global well-posedness for Boussinesq sys tem, J. Differential Equa-\ntions 233 (2007) 199-220.\n[2] D. Adhikar, C. Cao, J. Wu, X. Xu, Small global solutions to the dam ped two-dimensional\nBoussinesq equations, J. Differential Equations, 256 (2014) 3594 -3613.\n[3] D. Chae, Global regularity for the 2D Boussinesq equations with p artial viscosity terms, Adv.\nMath. 203 (2006) 497-513.\n[4] D. Chae, H.S. Nam, Local existence and blow-up criterion for the Boussinesq equations. Proc.\nR. Soc. Edinb. A 127(5) (1997) 935-946.\n[5] C. Chen, J. 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Zhang, Local well-posedness and blow up crit erion of the Boussinesq\nequations in critical Besov spaces, J. Math. Fluid Mech. 12 (2010) 2 80-292.\n[12] A. Majda, Introduction to PDEs and Waves for the Atmospher e and Ocean, Courant Lecture\nNotes in Mathematics, vol. 9, AMS/CIMS, 2003.\n[13] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Camb ridge University Press, Cam-\nbridge, 2001.\n[14] J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1 987.\n[15] Y. Taniuchi, A note on the blow-up criterion for the inviscid 2-D Bo ussinesq equations, In:\nThe NavierCStokes Equations: Theory and Numerical Methods (Ed . Salvi, R.). Lecture Notes\nin Pure and Applied Mathematics, Vol. 223 (2002) 131-140.\n[16] T. Tao, L. Zhang, H¨ older continuous solutions of Boussinesq e quation with compact support,\nJournal of Functional Analysis, 272 (2017) 4334-4402.\n[17] R. Wan, J. Chen, Global well-posedness for the 2D dispersive SQ G equation and inviscid\nBoussinesq equations, Z. Angew. Math. Phys. (2016) 67:104.\n10"    },    {        "title": "0708.1643v1.Local_Lorentz_Transformation_and__Lorentz_Violation_.pdf",        "content": "arXiv:0708.1643v1  [physics.gen-ph]  13 Aug 2007Local Lorentz Transformation and\n“Lorentz Violation”\nYing-Qiu Gu∗\nSchool of Mathematical Science, Fudan University, Shangha i 200433, China\nAbstract\nSome solutions to the anomalies of ultra high energy cosmic- ray(UHECR)\nand TeV γ-rays require disturbed non-quadratic dispersion relatio ns, which sug-\ngest the Lorentz violation. Also, some new theories such as q uantum gravity,\nstring theory and the standard model extension imply the Lor entz violation. In\nthis paper, we derive some transformation laws for the class ical parameters of\nnonlinear field system, and then provide their dispersion re lations. These dis-\npersion relations also have non-quadratic form but keep the Lorentz invariance.\nThe analysis of this paper may be helpful for understanding t he quantum theory\nand the plausible Lorentz violation.\nPACS numbers: 04.20.Gz, 04.20.Cv, 03.50.Kk, 03.65.Pm\nKey Words: Lorentz violation, Lorentz transformation, dis-\npersion, mass energy relation, GZK cutoff\n1 Introduction\nRecently the “Lorentz violation” becomes a hot topic. This situation arises from both\nexperimental and theoretical developments. One is the astronom ical observation of the\nanomalies of ultra high energy cosmic-ray(UHECR) and the TeV γ-rays. The other\nis the implication of some new physics such as the extended gauge the ory, quantum\ngravity and string theory. In observational aspect, the UHECR p rotons with energy\n∗email: yqgu@fudan.edu.cn\n1>1020eVwereobserved[1,2,3]. Theenergyoftheseprotonsarebeyon dtheGZKcutoff\n5×1019eV, which was derived independently by Greisen[4] and Zatsepin , Ku zmin[5]\nshortly after the discovery of the cosmic microwave background r adiation(CMBR).\nAccording to the particle theory, the cosmic-ray nucleons with suffi cient energy will\ninelastic collide with photons of CMBR to produce baryons or pions as f ollows\nP+γ(CMB)→∆, P+γ(CMB)→p+Nπ, (1.1)\nso the nucleons with energies beyond the GZK limit cannot reach us fr om a source\nfurther than a few dozen Mpc. It should be mentioned that, this GZ K limit and the\nphysical process are well understood and measured in the laborat ory[6].\nAnother similar paradox is the TeV γ-rays. Two photons with energy over 2 mec2\ncan produce an electron-positron pair γ+γ→e−+e+. Photons of 10 TeV are most\nsensitive to 30 µminfrared photons, so a photon with enough energy propagating in t he\nintergalacticspacewillinteractwithinfraredbackgroundphotons , andbeexponentially\nsuppressed. However the 10 TeV photons from Mkn501, a BL Lac o bjects at distance\nabout 150Mpc[7, 8], could reach the Earth.\nTo explain the above anomalies, numerous solutions have been propo sed. Most of\nthese solutions are related to slight violation of Lorentz invariance t o get a shift of the\nthresholds of the energy. Glashow believe that the limit speed of a pa rticle depends\non its species and should be the eigenvalue of the velocity eigenstate s[9]-[15]. Then for\nparticlePthe dispersion relation becomes\nE2=p2c2\nP+m2\nPc4\nP. (1.2)\nIn [16], the authors suggested a non-quadratic dispersion relation for photons\np2c2=E2(1+f(δ)), δ=E\nEQG→0 (1.3)\nwhereEQG∼1019GeV is an energy scale caused by quantum gravity effect. Then the\nlight speed is perturbed by E as\n/tildewidec=∂E\n∂p≈c(1+kδ). (1.4)\nThis assumption causes the Lorentz violation and deformed special relativity, which is\nrelatedto the κ-Poincar´ esuperalgebra[17]. The Casimir of the κ-Poincar´ e superalgebra\nhas a structure similar to (1.3). Nowadays, the deformed relativity or noncommutative\ngeometry is greatly developed[18]-[26].\nIn theoretical aspect[27, 28, 29], the Standard Model Extension and quantum grav-\nity suggest that Lorentz invariance may not be an exact symmetry . The possibility\n2Lorentz violation has been investigated in different quantum gravity models, including\nstring theory[30, 31], warped brane worlds[32], and loop quantum gr avity[33]. These\nmodels adopt the Lagrangian like the following[34]\nL=¯ψ/parenleftbigg1\n2eµ\naΓa↔\n∂µ−M/parenrightbigg\nψ, (1.5)\nwhereψis Dirac spinor, eµ\nais the vierbein, and ea\nµis the inverse,\nΓa=γa−cµνeνaeµ\nbγb−fµeµa−ikµeµaγ5+···, (1.6)\nM=m+iµγ5+aµeµ\naγa+bµeµ\naγaγ5+···. (1.7)\nThe first right terms of (1.6) and (1.7) correspond to the normal L orentz invariant\nkinetic term and mass for the Dirac spinor. But the other coefficient s are Lorentz\nviolating coefficients arising from nonzero vacuum expectation value s of the coupling\ntensor fields, which seem to be introduced quite arbitrarily.\nIn contrast with the above theories of Lorentz violation, the tors ion theory is the\nmost natural one in logic[35, 36], which is derived from the fact that t he connection\ncan compatibly introduce an antisymmetric part, namely, the torsio n.\n/tildewideΓµ\nαβ= Γµ\nαβ+Tµ\nαβ, (1.8)\nwhere Γµ\nαβis the Christoffel symbol, and Tµ\nαβ=−Tµ\nβαis the torsion of the spacetime.\nDifferent from all matter fields like electromagnetic field, torsion is a g eometrical inter-\naction similar to gravity, which uniformly interacts with all matter in an accumulating\nmanner. So to test torsion, it seems more effective to measure the movement of heaven\nbody rather than atoms.\nHowever, in contrast with the natural essence and deep philosoph ical meanings\nof the Lorentz invariance, the violation theories seem to be someho w artificial[37].\nSome experiments have been elaborated to test the Lorentz violat ion, but all results\ngave negative answers in high accuracy[38]-[44], so one should take a little conservative\nattitudetowards theLorentz violation. Then how toexplain thethr eshold anomalies of\nUHECR andTeV γ-rays? Herewepresent anotherscenario basedonthenonlinearfi eld\ntheory, which also provides non-quadratic mass-energy relations or dispersion relations\nsimilar to (1.3), but strictly keeps theLorentz invariance[45, 46, 47 ]. In what follows we\nexamine the local Lorentz transformation for some classical para meters defined from\nnonlinear fields and establish their relations.\n32 Local Lorentz transformation and non-quadratic\ndispersion relation\n2.1 Local Lorentz transformation for classical parameters\nTakingtheMinkowski metricas ηµν= diag[1,−1,−1,−1], weconsider thefieldsystems\nof nonlinear spinor ψand scalarφ. For the nonlinear spinor ψ, the dynamic equation\nis given by[48]-[52]\nαµ(¯hi∂µ−eAµ)ψ= (µ−F′)γψ, (2.1)\nwhere the 4 ×4 Hermitian matrices are defined by\nαµ=\n\n\nI0\n0I\n,\n0/vector σ\n/vector σ0\n\n\n, γ=\nI0\n0−I\n (2.2)\nwith Pauli matrices\n/vector σ= (σk) =/braceleftBigg/parenleftBigg0 1\n1 0/parenrightBigg\n,/parenleftBigg0−i\ni0/parenrightBigg\n,/parenleftBigg1 0\n0−1/parenrightBigg/bracerightBigg\n. (2.3)\nF=F(ˇγ) is a positive function of the quadratic scalar ˇ γ≡ψ+γψ.\nFor scalar field φ(xµ), the dynamic equation is given by\n∂µ∂µφ=Kφ, (2.4)\nwhereK=K(|φ|2,∂µφ+∂µφ) is a smooth real function.\nFor both spinor ψand scalarφ, the current conservation law holds due to the gauge\ninvariance of their dynamic equations,\n∂µρµ= 0, (2.5)\nwhere the current is defined respectively by\nρµ=\n\nψ+αµψ for spinor,\niκℑ(φ+∂µφ) for scalar .(2.6)\nκis a normalizing constant. By (2.5) we have the normalizing condition\n/integraldisplay\nR3ρ0d3x= 1. (2.7)\nFor the nonlinear equations (2.1) and (2.4), their solutions have par ticle-wave\nduality[48]-[52]. In [46, 47, 53, 54], the local Lorentz transformatio ns were widely\nused for the classical parameters without proof. Here we set the transformations on\na solid base at first, and then derive the non-quadratic dispersion r elations for some\ncases. The conditions for these results are helpful to understan d the relation between\n4classical mechanics and quantum theory. To clarify the status of t he field system, we\ndefined\nDefinition 1 For the field system ψorφ, we define the central coordinate /vectorX\nand drifting speed/vector vrespectively by\n/vectorX(t) =/integraldisplay\nR3/vector xρ0d3x, /vector v=d\ndt/vectorX, (2.8)\nwheret=x0. The coordinate system with central coordinate /vectorX= 0 is called the\ncentral coordinate system of the field.\nDefinition 2 If a field is a localized wave pack drifting smoothly without emitting\nand absorbing energy quantum, that is, it is at the energy eigensta te in its central\ncoordinate system, we call it at the particle state . Otherwise, the field is in the\nprocess ofexchanging energy quantum with itsenvironment, we ca ll it inthe quantum\nprocess.\nBy the current conservation law (2.5), we have\n/vector v=/integraldisplay\nR3/vector x∂0ρ0d3x=−/integraldisplay\nR3/vector x∇·/vector ρd3x=/integraldisplay\nR3/vector ρd3x. (2.9)\nFor the field at particle state with mean radius much less than the cha racteristic length\nof its environment, by (2.8) and (2.9) we have the classical approxim ation\nρµ→uµ√\n1−v2δ(/vector x−/vectorX), (2.10)\nwhere\nuµ≡(ξ,ξ/vector v), ξ=1√\n1−v2. (2.11)\n(2.10) is the precondition for validity of classical mechanics[46, 47, 5 3]. Forsuch system\nat particle state, we can clearly define the classical parameters su ch as “momentum”,\n“energy” and “mass”, and derive the classical mechanics. From [46 , 47], we learn that\na system at particle state can be described by classical mechanics in high accuracy,\nwhereas for the system in the quantum process, we must describe it by quantum theory\nor by the original equation (2.1) or (2.4). The quantum process is an unstable state,\nwhich is usually completed in a very short time.\nIn what follows, we examine the local Lorentz transformation for t he classical pa-\nrameters. Since the rotation transformation is trivial, we only cons ider the boost one.\nFor the case of flat spacetime, assume xµis the Cartesian coordinate. Consider the\ncentral coordinate system of the field with coordinate ¯Xµ, which moves along x1at\nspeedvwith¯Xk(k/negationslash= 0) parallel to xk, and¯Xk= 0 corresponds to the central coordi-\nnate of the field Xk(t). Then the Lorentz transformation between xµand¯Xµin the\nform of matrix is given by\nx=L(v)¯X,¯X=L(v)−1x=L(−v)x (2.12)\n5wherex= (t,x1,x2,x3)T,¯X= (¯X0,¯X1,¯X2,¯X3)Tand\nL(v) = diag/bracketleftBigg/parenleftBiggξ ξv\nξv ξ/parenrightBigg\n,1,1/bracketrightBigg\n= (Lµ\nν). (2.13)\nAssumeS,PµandTµνareanyscalar, vectorandtensordefinedbytherealfunctions\nofφorψand their derivatives, such as S=|φ|2,Tµν=ℜ(ψ+αµ∂νψ). For the field\nat particle state, all these functions are independent of proper t ime¯X0. Thus in the\ncentral coordinate system, the spatial integrals of these funct ions define the proper\nclassical parameters of the field, and these proper parameters a re all constants. Their\nLorentz transformation laws are given by\nTheorem 1 For a field system at particle state, the integrals of covaria nt functions\nS,PµandTµνsatisfy the following instantaneous Lorentz transformati on laws under\nthe boost transformation (2.12) between xµand¯Xµatdt= 0,\nI≡/integraldisplay\nR3S(x)d3x=√\n1−v2¯I. (2.14)\nIµ≡/integraldisplay\nR3Pµ(x)d3x=√\n1−v2Lµ\nν¯Iν. (2.15)\nIµν≡/integraldisplay\nR3Tµν(x)d3x=√\n1−v2Lµ\nαLν\nβ¯Iαβ. (2.16)\nWhere¯I,¯Iµ,¯Iµνare the proper parameters defined in the central system\n¯I=/integraldisplay\nR3S(¯X)d3¯X,¯Iµ=/integraldisplay\nR3¯Pµd3¯X,¯Iµν=/integraldisplay\nR3¯Tµνd3¯X. (2.17)\nProofWe take (2.15) as example to show therelations. Forthe field at the p article\nstate, by the transformation law of vector, we have\nPµ(x) =Lµ\nν¯Pν(¯X) =Lµ\nν¯Pν(¯X1,¯X2,¯X3) =Pµ(ξ(x1−vt),x2,x3).(2.18)\nSo the integral can be calculated as follows\nIµ=/integraldisplay\nR3Pµ(x)d3x|dt=0\n=/integraldisplay\nR3Pµ(ξ(x1−vt),x2,x3)√\n1−v2d(ξ(x1−vt))dx2dx3(2.19)\n=/integraldisplay\nR3Lµ\nν¯Pν(¯X)√\n1−v2d3¯X=√\n1−v2Lµ\nν¯Iν.\nThe proof is finished.\nRemarks 1 The Lorentz transformation laws (2.14), (2.15) and (2.16) a re valid\nfor the varying speed v(t), because the integrals are only related to the simultaneous\nconditiondx0=dt= 0, and the relations only related to algebraic calculations.\nRemarks 2 When the field is not at the particle state, the covariant inte grands\nwill depend on the proper time ¯X0, then the calculation (2.19) can not pass through,\n6and the relations (2.14)-(2.16) are usually invalid, excep t the integrand satisfies some\nconservation law similar to (2.5).\nIn some text books, the relations (2.14)-(2.16) are directly derive d via Lorentz\ntransformation of the integrands and volume element relation\nd3x=√\n1−v2d3¯X. (2.20)\nAs mentioned by remark 2, the calculation will provides wrong result if the field is not\nat the particle state, because (2.20) is just spatial volume, it suffe rs from the problem\nof simultaneity.\nUsually, the proper parameters have very simple form. For any tru e vector, it\nalways takes ¯Iµ= (¯I0,0,0,0), then\nIµ=√\n1−v2¯I0uµ. (2.21)\nFor some cases of tensor Iµν, it can be expressed as\nIµν=√\n1−v2/parenleftBig¯Kuµuν+¯Jηµν/parenrightBig\n, (2.22)\nwhere¯K,¯Jare constants.\nIn the curved spacetime with orthogonal time coordinate, if the ra dius of curvature\ninthe neighborhoodof the center of the field is much larger thanthe mean radius of the\nfield, the above calculations and relations can be parallel transform ed into the curved\nspacetime. In this case, let ¯Xµbe the central Cartesian coordinate of the tangent\nspacetime at the center of the field. Xµis an inertial coordinate system in the tangent\nspacetime fixed with the curved spacetime. ¯Xµinstantaneously moves along X1at\nspeedv. Then it is easy to check the local Lorentz transformation laws (2.1 4), (2.15)\nand (2.16) also hold, as long as the field is at the particle state.\n2.2 Non-quadratic dispersion relation for spinor\nFor the dark spinor, e= 0 in (2.1). From [46, 47], we get the momentum and mass-\nenergy relation of the spinor as follows\npµ=/parenleftBigg\nm0+Wln1√\n1−v2/parenrightBigg\nuµ, (2.23)\nE=m0√\n1−v2+W/parenleftBigg1√\n1−v2ln1√\n1−v2+√\n1−v2/parenrightBigg\n, (2.24)\nwhereW≪m0is the proper energy provided by the nonlinear potential. The mass-\nenergy relation or dispersion relation in the usual form is given by\n/parenleftBig\nE−W√\n1−v2/parenrightBig2=/vector p2+/parenleftBigg\nm0+Wln1√\n1−v2/parenrightBigg2\n. (2.25)\n7Referring to electron, by estimation we haveW\nm0∼10−6. For a spinor moving at speed\n1−10−n, forn>1 we have\nln1√\n1−v2≈1.15n−0.35. (2.26)\nThe Lagrangian of the particle becomes\nL=−\n(m0+W)+Wln1/radicalBig\ngµν˙xµ˙xν\n/radicalBig\ngµν˙xµ˙xν, (2.27)\nwhere ˙xµ=d\ndtxµ. So the nonlinear term leads to a tiny departure from the geodesic.\nFor a spinor with interaction like electromagnetic field, the mass-ene rgy relation of\nthe particle will be much complicated. More generally, we consider the system with\nfollowing Lagrangian[47]\nL=ψ+αµ(i∂µ−eAµ)ψ−µˇγ+F(ˇγ)−sˇγG\n−1\n2κ(∂µAν∂µAν−a2AµAµ)−1\n2λ(∂µG∂µG−b2G2),(2.28)\nwhereAµandGare potentials produced by ψitself,κ=±1 andλ=±1 stand for\nthe repulsive or attractive self interaction. Then the 4-dimensiona l momentum pµand\nenergyEof the system at particle state are respectively defined by\npµ≡/integraldisplay\nR3ψ+\nk(i∂µ−eAµ)ψd3x, (2.29)\nE≡/integraldisplay\nR3\n/summationdisplay\n∀f∂L\n∂(∂tf)∂tf−L\nd3x=p0+EF+EA+EG,(2.30)\nwhere the classical parameters are given by\npµ=/parenleftBigg\nm0+sWγG+WFln1√\n1−v2/parenrightBigg\nuµ, (2.31)\nEF=WF√\n1−v2, (2.32)\nEA=WA1√\n1−v2/parenleftBigg\n1+v2\n3/parenrightBigg\n+Wav2\n√\n1−v2, (2.33)\nEG=WG1√\n1−v2/parenleftBigg\n1−v2\n3/parenrightBigg\n−Wbv2\n√\n1−v2, (2.34)\nin which the proper parameters are calculated by\nWF=/integraldisplay\nR3(F′ˇγ−F)d3¯x>0, Wγ=/integraldisplay\nR3ˇγd3¯x, (2.35)\nWA=κe2\n8π/integraldisplay\nR6e−ar\nr/parenleftBig\n|ψ(¯x)|2|ψ(¯y)|2+/vector ρ(¯x)·/vector ρ(¯y)/parenrightBig\nd3¯xd3¯y, (2.36)\nWG=−λs2\n8π/integraldisplay\nR6e−br\nrˇγ(¯x)ˇγ(¯y)d3¯xd3¯y, (2.37)\nWa=κ\n3/parenleftbiggae\n4π/parenrightbigg2/integraldisplay\nR3/parenleftBigg/integraldisplay\nR3e−ar\nr|ψ(¯y)|2d3¯y/parenrightBigg2\nd3¯x, (2.38)\nWb=λ\n3/parenleftBiggbs\n4π/parenrightBigg2/integraldisplay\nR3/parenleftBigg/integraldisplay\nR3e−br\nrˇγ(¯y)d3¯y/parenrightBigg2\nd3¯x, (2.39)\n8andr=|¯x−¯y|. By (2.30)-(2.34) we get the dispersion relation as follows\n(E−EF−EA−EG)2=/vector p2+/parenleftBigg\nm0+sWγG+WFln1√\n1−v2/parenrightBigg2\n.(2.40)\nBy (2.25) and (2.40), we find that the interaction term can result in v ery complicated\ndispersion relation. How to use these relation to explain the thresho ld paradoxes of the\nhigh energy comic rays is involved in the interpretation of parameter s, which will be\ndiscussed elsewhere. By the way, the scalar interaction may be abs ent in the nature,\nbecause it manifestly appears in proper mass of feimions, see (2.31) and (2.40).\n3 Discussion and conclusion\nFrom the above analysis, we find that nonlinear field theories do includ e the non-\nquadraticdispersion relationsuch as(2.25)and(2.40). Sotheexpla nationforthreshold\nparadoxes of UHECR and TeV γ-rays by dispersion relation does not definitely require\nLorentz violation. It should be mentioned that, all interactions are actually related to\nnonlinearity, for instance, the charge density of the electromagn etic interaction ψ+αµψ\nisnonlinear, whichcontributesproperenergy WAfortheparticleasdescribedby(2.36).\nThis part of energy satisfy the energy-speed relation (2.33).\nBy (2.31)-(2.34), we learn different interaction term leads to differe nt energy-speed\nrelation. So the experiments towards such relations should bring us important in-\nformation from each interaction, and the high energy cosmic rays m ay be the useful\nmaterials. On the other hand, the disturbance of nonlinear effect m ay influence our\nastronomical observation. For the movement of a heaven body, b y (2.27), we learn\nthat the order of the relative deviation from geodesic is Err=Wv2\nmc2, whereWis the\nenergy contribution of all interactions, mis the usual mass. The typical values for a\nproton are W∼1MeV and mp∼103MeV, so we haveW\nm∼10−3. In a galaxy, the\ntypical speed of heaven bodies relative to the CMB is 300km/s[55], so we have the\ntypical nonlinear deviation for galactic system Err|galaxy∼10−9. In the solar system,\nthe typical speed of the planets is about v∼30km/s, so we have the typical nonlinear\ndeviation for solar system Err|planet∼10−11. Thus, before the nonlinear effects are\nclearly worked out, an astronomical measurement with relative err or less than Erris\ndifficult. The precession of the perihelion of Mercury is 43 seconds of arc per century\n43/(100×360×3600)≈3.3×10−7, so the nonlinear effects have not influence on this\nresult.\nLorentz invariance includes two fold meanings: One is the covariance of the uni-\nversal physical laws, which is a problem of philosophy. The other is pr operty of the\nspacetime, namely the measurement of line element. Whether the sp acetime manifold\n9is measurable is also a philosophical problem, but how to measure the d istances is a\nproblem of geometry. The philosophical problem involves the most fu ndamental and\nuniversal postulates which could only be acceptable as faith, becau se we can neither\ntest all particles whether they satisfy the covariant equation eve rywhere and every\ntime, nor can we check whether all parts of the spacetime are meas urable, including\nthe singularity inside a black hole and each points at a line of Planck lengt h. Even\nthough the Lorentz violation theories can not manifestly violate the covariance, see\nthe forms of (1.5)-(1.7) as example. One may argue the thermodyn amics is not covari-\nant, the answer is that it is just a conditional theory but not a unive rsal one, thus to\nabuse its concepts and laws, such as entropy and the second law, w ithout restriction is\ninadequate and leads to confusion.\nWe once measured the spacetime with rule dst=|dt|,ds2\ns=dx2+dy2+dz2\nand solved infinite practical problems. Today we measure the univer se withds2=\ngµνdxµdxνthen we achieved beauty and harmony. There are infinite rules to me asure\nlength consistently, but only the spacetime with rule of quadratic fo rm has wonderful\nproperties and potential. If accepting the quadratic rule, the loca l Lorentz invariance\ncertainly holds for the spacetime.\nThere are also fields satisfy the covariance but violate the Lorentz invariance. Their\nLagrangian always includes term as follows[27]\nLLV= (gµν+τµν)∂µ∂νφ+ψ+(αµ+aµ)∂µψ+···.\nFor such fields, the propagating speed is not pure geometrical, whic h depends on fields\nτµνandaµ. Similar to the case of Navier-Stockes equation in fluid mechanics, th e\nnonlinearity is much worse than that of (2.28). Although for adequa tely small value\nτµνandaµ, the solutions to the dynamical equation will also be finite, but the wo rld\nincluding such fields will become a mess, because each atom of the sam e element have\ndifferent spectrum depending on coordinates, crystal lattices ar e distorted, the solar\nsystem has turbulence and chaos, and the double-helix of DNA has d isordered knots.\nSo‘the coefficients of the highest order derivatives in the Lagr angian must\nbe constants’ should be a fundamental postulate. This postulate is related to the\nquaternion structure of the world, only of such excellent structu re the world becomes\nso luxuriant but so harmonious. In percipience of such opinion, the m odification of\ngeneral relativity with terms φR,f(R) is also doubtful.\nSomeconfusionsinphysicsarecausedbyambiguousconceptsorcir cularrelations[56,\n57]. For example, to the relation between classical mechanics(CM) a nd quantum me-\nchanics(QM), the common answer must be that: “transform the c lassical parameters,\nHamiltonian and the energy equation of CM into operators, we get QM . Contrarily,\nthe limit of QM as ¯ h→0 provides CM.”\n10At first, by the answer, it seems that both CM and QM are equivalent theories\nin logic. In fact they are different theory suitable for different stat us of a system.\nThe basic concepts such as ‘coordinate’, ‘momentum’ have different meanings in each\nmechanics, although they have close relations. The uncertainty re lation is the typical\nplausible laws making puzzles and paradoxes[56, 57]. Clearly they sho uld be unified\nin a higher level theory[45, 46, 50, 52]. secondly, ¯ his a universal constant acting as a\nunit to measure other physical parameters, we have not a concep t ¯h→0? Whether\na physical system should be described by CM or QM is obviously not det ermined by\nexternal conditions such as ¯ h→0 and CM or QM, but determined by the status of\nthe system itself. This is the meaning of the Definition 2 .\nSpacetimeandfieldsaredifferentcomponentsoftheworld. Theypa lydifferent roles\nwith different characteristics, and satisfy completely different pos tulates and measure-\nment rules. So it is hard to understand the motivation of the quantu m gravity. Why\nwe should modify a well defined and graceful theory, without any de finite violation of\nexperiments, by an ambiguous and incomplete theory? Why not modif y the quantum\nfield theory by general relativity? The mission of a physical theory is to find out the\nintrinsic truth and beauty and harmony of the nature. But at its be st, besides some\nill defined concepts as foam, wormhole, Lorentz violation, what can quantum gravity\nactually provide us?\nInthespinortheoryofgeneralrelativitycontext, thereexistst hepseudo-violation\nof Lorentz invariance[58, 59], which is caused by the derivatives of t he vierbein or local\nframe. The vierbein is defined in the tangent spacetime of a fixed poin t in spacetime\nmanifold, and the Lorentz transformation is just an algebraic oper ation in this fixed\ntangent spacetime. Whereas the derivatives of the vierbein must in volve the tangent\nspacetimes of different points in some sense, so it violates the local L orentz invariance.\nIn strict sense, the equivalence principle only holds for the linear ten sors, where the in-\nfluences of vierbein andnonlinear fields are absent. 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Gu, Simplification of the covariant derivatives of spinors , gr-qc/0610001\n15"    },    {        "title": "1707.05192v2.Damping_of_gravitational_waves_by_matter.pdf",        "content": "arXiv:1707.05192v2  [gr-qc]  17 Oct 2017Damping of gravitational waves by matter\nGordon Baym,a,bSubodh P. Patil,band C. J. Pethickb,c\naDepartment of Physics, University of Illinois, 1110 W. Gree n Street, Urbana, IL 61801-3080\nbThe Niels Bohr International Academy, The Niels Bohr Instit ute,\nUniversity of Copenhagen, Blegdamsvej 17, DK-2100 Copenha gen Ø, Denmark\ncNORDITA, KTH Royal Institute of Technology and Stockholm Un iversity,\nRoslagstullsbacken 23, SE-10691 Stockholm, Sweden\n(Dated: October 18, 2017)\nWe develop a unified description, via the Boltzmann equation , of damping of gravitational waves\nby matter, incorporating collisions. We identify two physi cally distinct damping mechanisms –\ncollisional and Landau damping. We first consider damping in flat spacetime, and then generalize\nthe results to allow for cosmological expansion. In the first regime, maximal collisional damping of a\ngravitational wave, independentofthedetails ofthecolli sions inthematteris, as weshow, significant\nonly when its wavelength is comparable to the size of the hori zon. Thus damping by intergalactic or\ninterstellar matterfor all butprimordial gravitational r adiationcanbeneglected. Althoughcollisions\nin matter lead to a shear viscosity, they also act to erase ani sotropic stresses, thus suppressing\nthe damping of gravitational waves. Damping of primordial g ravitational waves remains possible.\nWe generalize Weinberg’s calculation of gravitational wav e damping, now including collisions and\nparticles of finite mass, and interpret the collisionless li mit in terms of Landau damping. While\nLandau damping of gravitational waves cannot occur in flat sp acetime, the expansion of the universe\nallows such damping by spreading the frequency of a gravitat ional wave of given wavevector.\nPACS numbers:\nI. INTRODUCTION\nThe opening of a new window on the universe through\nthe ongoing observations of gravitational waves [ 1] un-\nderlines the importance of reexamining how they prop-\nagate through the matter in the universe, and asking\nwhat gravitational wave measurements can teach one\nabout this matter. Half a century ago, Hawking showed\nthat if matter could be treated in the hydrodynamic\nlimit the damping rate of a gravitational wave would be\nγ= 16πGη, whereGis Newton’s gravitational constant\nandηthe viscosity of the matter [ 2,3]. Using this result,\nGoswami et al. [ 4] argued that gravitational wave obser-\nvations could be used to constrain the viscosity of dark\nmatter between the source and Earth. But, as Hawking\nfirst pointed out, there are in general too few collisions\nin matter for hydrodynamics to be valid, and the damp-\ning would be less than the hydrodynamic result. Ref-\nerence [5] estimated damping in the almost collisionless\nlimit by investigating the response of individual particles\nto a gravitational wave and found that the damping rate\nof the wave by nonrelativistic particles is\nγ∼Gnm\nω2/parenleftBig¯v\nc/parenrightBig21\nτ. (1)\nHereωisthefrequencyofthewave, ntheparticledensity,\nmthe particle mass, ¯ vthe typical particle velocity, and\nτthe particle-particle collision time; the damping is ∼\n1/(ωτ)2smaller than the viscous result.\nIn additiontodamping bycollisionsin matter, gravita-\ntional waves can also be attenuated by Landau damping,\nin which particles surf the gravitational wave and ex-\ntract its energy, first proposed for gravitational waves inRef.[6]. Thiseffectwasoriginallyinvestigatedinthecon-\ntext of plasma physics [ 7], then in galactic dynamics [ 8],\nand later in quantum chromodynamic plasmas [ 9]. In a\nstaticflat universemassiveparticlescannotproduceLan-\ndaudampingsincetheymovemoreslowlythan agravita-\ntional wave. In an expanding universe, however, Landau\ndamping becomes possible, as we show, since the expan-\nsion in the presence of matter effectively spreads the fre-\nquency of a gravitational wave. Indeed the damping of\ncosmological gravitational waves by non-interacting neu-\ntrinos, as first proposed by Weinberg [ 10] and expanded\nupon in Refs. [ 11,12], can in fact be understood in terms\nof Landau damping, as we indicate below.\nOur aim in this paper is to present a unified treatment\nof the damping of gravitational waves by matter, for ar-\nbitrary collision rates, thus encompassing the hydrody-\nnamic and nearly collisionless limits studied earlier, as\nwell as cosmological expansion. We begin, in Sec. II,\nby considering a weak gravitational wave propagating\nthrough a dilute gas of colliding particles of arbitrary\nmass in an otherwise flat spacetime, and calculate, in\nSec.III, the response of the matter to the wave using\nthe Boltzmann equation. For simplicity we work in the\nrelaxation or collision time approximation.\nAs we show, in Sec. IV, the maximum damping of\nthe amplitude of a gravitational wave with frequency\nωis or order 1 /(ωτU) where τUis the age of the uni-\nverse. Thus collisional damping by matter of gravita-\ntional waves generated by astrophysical sources cannot\nprovide useful information about the nature of matter in\nthe universe. Furthermore, damping by dense environ-\nments surrounded localized sources of gravitational ra-\ndiation is, as we estimate, insignificant. After a general2\ndiscussion of Landau damping in Sec. VI, we generalize\nthe Boltzmann equation results in Sec. VIIto describe\ncollisional damping by particles of arbitrary mass in the\npresence of an expanding cosmological background.\nII. STATIC SPACETIME\nInitially, we do not include the expansion of the uni-\nverse, and consider rather the Minkowski space metric\nwith a gravitational wave superimposed:\nds2=−dt2+gijdxidxj, (2)\nwhere\ngij=δij+hij(/vector r,t), (3)\nwithhijthe weak metric perturbation caused by a grav-\nitational wave. We work with hijin the transverse–\ntraceless gauge, and generally set c= 1.\nThe effects on a gravitational wave passing through\nmatter are given in terms of the gravitational wave equa-\ntion in the transverse traceless gauge\n∂µ∂µhij=/parenleftbigg\n−∂2\n∂t2+∇2/parenrightbigg\nhij=−16πGπij,(4)\nwhereπijis the transverse traceless part of the matter\nstress tensor, Tij,M, defined by\nπi\nj≡Ti\nj,M−δi\nj\n33/summationdisplay\nk=1Tk\nk,M. (5)\nIn equilibrium,1\n3/summationtext3\nk=1Tk\nk,Mis simply the pressure Pof\nthe matter.\nThe effect of a gravitational wave on a particle is given\nin terms of the dispersion relation,\npµpνgµν+m2= 0, (6)\nwhich in the present case implies that the particle energy\nǫis given by\nǫ2=gijpipj+m2. (7)\nThus a weak gravitational wave changes the particle dis-\npersion relation from ǫ0=/radicalbig\np2+m2to\nǫ=ǫ0+δǫ. (8)\nTo first order in hij\nδǫ=1\n2hijpipj\nǫ0=−1\n2hijpipj\nǫ0, (9)\nsince to this order, hij=−hij.III. BOLTZMANN EQUATION\nWe treat the matter as a dilute gas and calculate πij\nfrom the Boltzmann equation for the matter. We first\nwrite the non-linear Boltzmann equation for the particle\ndistributionfunction f(ri,pj)asafunctionoftheparticle\npositions and canonical momenta,\n/parenleftbigg∂\n∂t+/vector∇pǫ·/vector∇r−/vector∇rǫ·/vector∇p/parenrightbigg\nf=C,(10)\nwhereCis the collision term. Here position gradients\nare taken with respect to ri, and momentum gradients\nwith respect to pi. This form of the equation is valid for\nrelativistic as well as non-relativistic particles.\nThe conservation laws of energy and momentum are\nfound by taking the moments of Eq. ( 10) with respect\ntopiandǫ; assuming that collisions conserve the total\nenergy and momentum of the particles, we find (as in\nstandard Fermi liquid theory)\n∂\n∂t/integraldisplay\npǫf+∇i/integraldisplay\npǫvif=/integraldisplay\np∂ǫ\n∂tf, (11)\nand\n∂\n∂t/integraldisplay\nppif+∇jTj\ni,M=−/integraldisplay\np(∇riǫ)f, (12)\nwhere/integraltext\np≡g/integraltext\nd3p/(2π)3, withgthe number of internal\nstates, e.g., spin, and\nTj\ni,M=/integraldisplay\nppivjf=gjk/integraldisplay\nppipk\nǫf (13)\nis the matter stress tensor.\nWith Eq. ( 9) the right side of Eq. ( 11) becomes\n/integraldisplay\np∂ǫ\n∂tf=1\n2∂hij\n∂t/integraldisplay\nppipj\nǫf=1\n2∂hij\n∂tTij,M,(14)\nso that the change in energy of the matter is given by\n∂E\n∂t=1\n2/integraldisplay\nd3r∂hij\n∂tπij. (15)\nOnly the transverse-traceless part of the stress tensor,\nEq. (5), enters Eq. ( 14).\nNote that, for a wave of the form H(z−ct), say, the\nright side of Eq. ( 12) becomes\n−/integraldisplay\np∂ǫ\n∂zf=1\n2∂hij\n∂tπij, (16)\nindicating that as momentum qis transferred from the\ngravitational wave, energy qis also transferred. Since\nπijis itself at least of first order in hijthe energy and\nmomentum transfers are second order and higher in the\namplitude of the gravitational wave.3\nWe turn now to calculating the transverse-traceless\npart of the matter stress tensor, Eq. ( 13); to linear order\ninhij,\nδTij,M=/integraldisplay\nppipj\nǫ0/bracketleftbigg\nδf−δǫ\nǫ0f0/bracketrightbigg\n(17)\nwhereδf=f−f0,withf0thedistributionfunctioninthe\nabsence of hij. Theδǫterm arises from the dependence\nofǫin the denominator on hij, Eq. (9). Subtracting out\nthe trace, we find, after using the vanishing trace of hij\nand writing the equilibrium pressure of the matter as/integraltext\np(p2/3ǫ0)f0, that\nπij=/integraldisplay\nppipj\nǫ0/bracketleftbigg\nδf−δǫ\nǫ0f0/bracketrightbigg\n+hij/integraldisplay\npp2\n3ǫ0f0.(18)\nThesecondtermofthisexpressionismanifestlytraceless;\nthe trace of the first term vanishes since the integrations\nover both δfandδǫ, being symmetric in angles, vanish.\nThe latter two terms that contain f0can be sim-\nply combined, with an integration by parts using the\ntransverse-traceless structure of hij, into a term propor-\ntional to ∂f0/∂ǫso that Eq. ( 18) becomes,\nπij=/integraldisplay\nppipj\nǫ0/bracketleftbigg\nδf−δǫ∂f0\n∂ǫ/bracketrightbigg\n. (19)\nThis combination of terms falls out naturally, as we shall\nsee, from the Boltzmann equation.\nCollisions between the particles, prior to freeze-out,\ntend to bring the distribution function into a local equi-\nlibrium in the presence of hij:\nf→fh=1\neβ(ǫ−µ)∓1, (20)\nwhereǫ, givenby Eq.( 7), depends on hij;βis the inverse\ntemperature, and µthe particle chemical potential. Note\nthat to first order in hij,\nfh=f0+δǫ∂f\n∂ǫ. (21)\nFor simplicity we employ a collision time approximation.\nSince the only disturbances relevant here involve spher-\nical harmonics of degree greater than one, we can write\nthe collision term as\nC=−f−fh\nτ=−1\nτ/parenleftbigg\nδf−δǫ∂f\n∂ǫ/parenrightbigg\n,(22)\nwhereτis the collision time, and δf−δǫ∂f/∂ǫ is the\ndeviation of the distribution from local equilibrium; the\nadditional terms commonly introduced to ensure conser-\nvation of particle number and total momentum (which\ninvolve spherical harmonics of degree zero and one) are\nnot relevant [ 7]. The linearized Boltzmann equation then\nreduces to\n/parenleftbigg∂\n∂t+1\nτ+/vector v·/vector∇/vector r/parenrightbigg\nδf=/parenleftbigg\n/vector v·∇rδǫ+1\nτδǫ/parenrightbigg∂f0\n∂ǫ.\n(23)Withhij(/vector r,t) =ei(/vector q·/vector r−ωt)hij, and Fourier transforming\nin space and time we find the solution of Eq. ( 23),\nδf=∂f0\n∂ǫ/parenleftbigg−/vector q·/vector v+i/τ\nω−/vector q·/vector v+i/τ/parenrightbigg\nδǫ. (24)\nThe deviation from local equilibrium is thus given by\nδf−∂f0\n∂ǫδǫ=−/parenleftbiggω\nω−/vector q·/vector v+i/τ/parenrightbigg∂f0\n∂ǫδǫ.(25)\nWe then find the general result,\nπij(q,ω) =/integraldisplay\nppipj\nǫ0δǫ/parenleftbiggω\n/vector q·/vector v−ω−i/τ/parenrightbigg∂f0\n∂ǫ.\n(26)\nFourier transformed back to time, the stress tensor is\nπij(q,t) = (27)\n−/integraldisplay\nppipj\nǫ0/integraldisplayt\n−∞dt′e−(iq·v+1/τ)(t−t′)∂f0\n∂ǫ˙δǫ(t′).\nThe response can be written in the more general form\nπij(q,ω) =−ωhij/integraldisplaydω′\n2πA(q,ω′)\nω−ω′+iξ,(28)\nwhere the spectral function is\nA(q,ω) =−2/integraldisplay\np/parenleftbiggpxpy\nǫ0/parenrightbigg21/τ\n(ω−/vector q·/vector v)2+1/τ2∂f0\n∂ǫ,\n(29)\nandξis a positive infinitesimal. In the collisionless limit,\n1/τ→0, and for relativistic particles,\nA(q,ω) =πρ∝angb∇acketleft(1−ζ2)2δ(ω−qζ)∝angb∇acket∇ight, (30)\nwhereρis the energy density of the excitations, and the\nangularbracketsdenotetheaverageover ζ≡ˆq·ˆv. Fourier\ntransformed back to time,\nπij(q,t) =−/integraldisplayt\n−∞dt′/integraldisplaydω\n2πe−iω(t−t′)A(q,ω)˙hij(t′).\n(31)\nThe damping of gravitational waves is governed by the\nimaginary part of the response,\nℑ/parenleftbiggπij\nhij/parenrightbigg\n=−ω/integraldisplay\np/parenleftBigpipj\nǫ/parenrightBig2 1/τ\n(ω−/vector q·/vector v)2+1/τ2∂f0\n∂ǫ.(32)\nIn the collision-dominated regime, τ≪1/ω; doing the\nangular averages in the integrals we have\nπij=iτω/integraldisplay\nppipj\nǫ0δǫ∂f0\n∂ǫ=−iτω\n15/integraldisplay\npp4\nǫ2\n0∂f0\n∂ǫhij\n=−η˙hij. (33)4\nThe viscosity calculated in the relaxation time approxi-\nmation is\nη=−/integraldisplay\np/parenleftbiggpipj\nǫ0/parenrightbigg2∂f0\n∂ǫ0τ, (34)\nwithi∝negationslash=j. In this limit the damping rate of a gravita-\ntional wave, from Eq. ( 4), is 16πGη, in agreement with\nearlier hydrodynamic treatments [ 2,3].\nFor non-relativistic matter, T≪mc2, the/vector q·/vector vin the\ndenominator of Acan be neglected, and we have\nπij≃ω\nω+i/τP hij. (35)\nFrom the imaginary part of Eq. ( 4), the dispersion re-\nlation of gravitational waves is\nω≃q+8πG\nωπij\nhij, (36)\nso that the damping of a wave is given by\nℑω=8πG\nωℑ/parenleftbiggπij\nhij/parenrightbigg\n. (37)\nFor fully relativistic matter, in the nearly collisionless\nlimit, to first order in 1 /τω,\nπij=−P/parenleftbigg\n1−2i\nτω/parenrightbigg\nhij, (38)\nwhile in the collision-dominated regime, πijis given by\nEq. (33).\nIV. MAXIMUM COLLISIONAL DAMPING\nAs one can see from Eq. ( 32),ℑ(πxy/hxy) has its max-\nimum magnitude for ωτ∼1. The possible damping is\nthus limited by\nMax(|ℑω|)/lessorsimilar−8πG\nω/integraldisplay\npp4\n15ǫ2\n0∂f0\n∂ǫ≤8πGP\nω,(39)\nwherePis the total local pressure of the matter under\nconsideration, which gives rise to the damping. It is in-\nstructive to write this bound in terms of the age of the\nuniverse, defined by\n1\nτ2\nU=8πG\n3¯ρ=/parenleftbigg˙a\na/parenrightbigg2\n, (40)\nwhereais the cosmological scale parameter and ¯ ρthe\nmean mass density of the universe. Since the mean pres-\nsure obeys ¯P≤¯ρ/3, we find\nMax(|ℑω|)/lessorsimilarP\n¯P1\nωτ2\nU. (41)A wave traversing matter will average the local pressure,\nand thus we can conclude,\nMax(|ℑω|)/lessorsimilar1\nωτ2\nU, (42)\nindicating that damping of a gravitational wave by mat-\nter in the universe can only be significant for a wave of\nfrequency of order 1 /τU. This bound includes all contri-\nbutions from dark matter particles as well [ 4].\nTo express the result ( 42) in another way, the colli-\nsionaldamping ofagravitationalwavewithin the charac-\nteristic expansion time of the universe is of order 1 /ωτU.\nForω∼103s−1, as in the recent gravitational wave de-\ntections [ 1], andτU∼1018s, this ratio is ∼10−21. Col-\nlisional damping in intergalactic or interstellar matter of\ngravitational waves produced by astrophysical sources is\nnot useful to determine the nature of matter in the uni-\nverse. This result is valid for any particle-like form of\ndark matter, including that in a possible shadow uni-\nverse [13] or matter that only interacts with gravitation-\nallysuppressedinteractions[ 14]. Furthermore, collisional\ndamping in locally high dense environments, e.g., in the\nneighborhood of mergers of black holes or neutron stars,\nis also negligible for gravitational waves produced by as-\ntrophysical sources, as we argue in the next section. On\nthe other hand for primordial gravitational waves with\nω∼10−16−10−15, one has 1 /ωτU∼10−3−10−2, an\neffect that could play a role in interpretation of future\nprecision measurements of the spectrum of primordial\ngravitational radiation.1\nAs we discuss in the following sections, Landau damp-\ning cannot occur in a flat spacetime. Even in an expand-\ning universe, the Landau damping rate is ∼1/ω2t3\nU, so\nthat the total damping within the expansion time of the\nuniverse is ∼1/(ωtU)2, a factor 1 /ωtUsmaller than the\nmaximum collision damping.\nOne can write the contribution to the damping from a\nparticular component, s, of the matter, e.g., neutrinos or\ndark matter, in the form\n|ℑω|s≡nsσGW,s, (43)\nwhereσGW,sis the graviton scattering cross section on\nparticles of species s, andnsis the number density of\nspeciess. The ratio of the cross section to the Planck\nlength,ℓPl, squared is essentially bounded above by\nσGW,s\nℓ2\nPl/lessorsimilar∝angb∇acketleftpv∝angb∇acket∇ights\nω(44)\nwhere∝angb∇acketleftpv∝angb∇acket∇ightsis the mean product of the particle momen-\ntum and velocity of species s, which is of order the tem-\nperature for a species in thermal equilibrium, or the tem-\nperature at which the species froze out. Thus in general,\nσGW,i\nℓ2\nPl/lessorsimilarT\nω, (45)\n1We thank Vicky Kalogera and Chris Pankow for this observatio n.5\nwith the above understanding of T.\nV. MAXIMAL COLLISIONAL DAMPING IN\nDENSE ENVIRONMENTS\nWe looknow at the damping ofgravitationalwavepro-\nduced by binary astrophysical sources, as the waves pass\nthrough the dense medium surroundingthe sources. Col-\nlisional damping is limited by |ℑω|< γmax, where from\nEq. (39),\nγmax∼1\nM2\nplP\nω=w\nM2\nplρ\nω; (46)\nwe have introduced the Planck mass M−2\npl= 8πGand\nwritten the relation between the pressure and the energy\ndensity by the equation of state parameter w=P/ρ.\nAssuming the gravitational wave source to a binary\nsystem inside a regionsurroundedby matter with a given\ndensity profile with equation of state parameter w, we\nfind collisional damping along a line of sight to be signif-\nicant if\n/integraldisplayR\n0drγmax∼1 (47)\nwhereRis a physical radius enclosing the ambient mat-\nter. A reasonable first estimate is simply to associate\nthe integral with the characteristic size Rcof the dense\nregion:\n/integraldisplayR\n0drγmax∼w\nM2\nplρ\nωRc∼w\nM2\nplR2cM\nω(48)\nwhereMis the total mass in the region with character-\nistic size Rc. To go beyond this estimate, one could take\nthe density profile from detailed calculations, e.g. the\nprofile of a typical dark matter halo (determined via phe-\nnomenological models that fit N-body simulations such\nas the Navarro-Frenk-White or Einasto density profiles\n[15]), and find numerical factors that little affect the con-\nclusion. In “natural” units, one solar mass M⊙∼1066\neV, 1 Hz ∼10−15eV,M−1\npl=ℓPl∼10−35m, and 1 kpc\n∼1019m, one has\n/integraldisplayR\n0drγmax∼w10−27\n(Rc/kpc)2/parenleftbiggM\nM⊙/parenrightbigg1\n(ν/Hz),(49)\nwhereν=ω/2π. We first consider a typical galactic halo\nsurrounding a binary system source of the gravitational\nwave. Here, typically M∼1012M⊙,Rc∼100 kpc, so\nthat\n/integraldisplayR\n0drγmax∼w10−19\n(ν/Hz)(50)\nwhich is feeble for all astrophysical sources within the\nhalo.We next consider the ambient region surrounding a bi-\nnarysystem similar to that which gaveriseto GW150914\n–containingadensedistributionof(notnecessarilydark)\nmatter. If the ambient matter has a mass comparable\nto that of the binary system localized within some re-\ngionRc≫Rs, the Schwarzschild radius associated with\nthe total mass of the binary system, we find that at\nthe lowest frequencies of the binary system, the factor\nM/M⊙×(ν/Hz)−1isO(1); furthermore for the expected\nnon-relativisticlow pressure surroundingmatter, w∼c2\ns,\nthe square of the adiabatic sound velocity. Thus\n/integraldisplayR\n0drγmax∼c2\ns1011\n(Rc/m)2. (51)\nwhich can only be of order unity if the ambient matter\nis localized to within a radius Rc/lessorsimilarcs×300km around\nthe binary system, which even for mildly non-relativistic\nambientmatter(e.g. cs∼ O(10−1))wouldrequireamass\ncomparable to that of the binary system to be crammed\ninto a region comparable to the Schwarzschild radius of\nthe final merged black hole ( ∼70 km∼Rs); such a high\ndensity is contrary to our initial assumption that Rc≫\nRs. Requiring that this matter be distributed within\na region an order of magnitude larger than the binary\nsystem yields/integraltextR\n0drγmax∼c2\ns≪1. We conclude that\na distribution of non-relativistic matter of high density\nsurrounding the source of gravitational radiation is not\ncapable of significantly damping gravitational radiation.\nMore realistically, one is in general far from the con-\ndition of maximal collisional damping, that the collision\nrateτ−1, be comparable to the frequency of the gravita-\ntional wave. Maximal collisional damping is a highly un-\nlikely prospect even for relativistic matter jets and lobes\nclose to the mergerof neutron star/blackhole binary sys-\ntems [16,17]. To see this we write roughly, τ−1=nσv,\nwherenis the density of particles, σis a particle-particle\nscattering cross section, and va mean particle velocity.\nIn terms of the mass, M, and characteristic radius, Rc,\nof the dense environment,\n1\nτ∼M\nM⊙1035\n(Rc/m)3/parenleftbiggσ\nfm2/parenrightbiggv\ncs−1, (52)\nClearly, for a typical nuclear or particle physics cross sec-\ntion, the above is much larger than the typical frequency\nofanastrophysicalbinarysystembymanyordersofmag-\nnitude, so that ωτ≪1.\nVI. LANDAU DAMPING: GENERAL\nCONSIDERATIONS\nIn flat space in the collisionless limit ( τ→ ∞) Eq. (32)\nreduces to\nℑ/parenleftbiggπij\nhij/parenrightbigg\n=πω/integraldisplay\np/parenleftbiggpipj\nǫ0/parenrightbigg2\nδ(ω−/vector q·/vector v)∂f0\n∂ǫ,(53)6\na result describing Landau damping, the decay of the\nmode into a single particle–hole pair.2The particle-hole\nexcitationsarespacelike. Foragravitationalwave, ω=q,\nthe integral vanishes except possibly for massless parti-\ncles moving in the same direction, say ˆ z, as the grav-\nitational wave. However, for such particles, the factor\np2\nip2\nj→p2\nxp2\nyvanishes; Landau damping is forbidden in\nthe absence of cosmologicalexpansion. Following general\nremarks on Landau damping in this section we show in\nthe following section how the collisionless damping pro-\ncess described by Weinberg [ 10] can be understood as a\ngeneralization of Landau damping, driven by the expan-\nsion of the universe.\nIn an expanding universe, the gravitational wave en-\nergy changes during expansion, i.e., the frequency of the\ngravitational wave is not constant, since the expansion\nabsorbs energy from the wave. This energy loss is differ-\nent from Landau damping by the matter traversedby the\nwave. When the phase velocity of the wave is different\nfrom the group velocity of the excitations in the matter,\nenergy in flat spacetime is pumped to and fro between\nthe wave and the matter, but the net rate of transfer\nis zero because the energy transferred in one half-cycle\nof the wave is exactly cancelled by the loss in the other\nhalf-cycle. In an expanding universe, however, the can-\ncellation is incomplete.\nWe recall the energy loss caused by expansion. A weak\ngravitational wave of period small compared with the\nage of the universe behaves in the absence of matter as\nhij=χ(u)e−iqu, whereuis conformal time, related to\ncoordinate time by du=dt/a(t); as we see in the next\nsection, χ(u)∝1/a(u). This structure is expected on\nthe basis of simple arguments: the energy of a gravita-\ntional wave packet is proportional to the energy density\nin the wave packet times the volume of the packet. The\nenergy density of the wave varies as gii(∂χ/∂xi)2∼a−4\nand the volume of the packet varies as a3, so the total\nenergydecreasesas 1 /a. This result alsoagreeswith sim-\nple redshift arguments: The energy of a massless particle\nvaries as 1+ z∼1/aowing to the expansion of the uni-\nverse, and thus the energy density measured in locally\nMinkowskian spacetime ( ds2=−dt2+ (d/vector r)2) varies as\n1/(1+z)4. For example, the redshift of the first LIGO\nevent GW150914 was z= 0.09+0.03\n−0.04[1], leading to an en-\nergy density reduction by a factor ≃1−1/(1.09)4≈30%\nfrom cosmological expansion. By comparison, even were\nthe intervening matter collisionless, Landau damping of\nthe wave would be totally negligible, ∼1/(ωtU)2.\n2In the language of quantum mechanics, the damping may be re-\ngarded as the creation, with amplitude ∝1/(ω−/vector q·/vector vs), of a\nvirtual single particle–hole pair, which subsequently dec ays into\ntwo real particle–hole pairs. Equation ( 32) can be understood,\nwhen 1/τ→0, as this amplitude squared, summed over all mo-\nmenta.VII. GRAVITATIONAL WAVE DAMPING\nWITH COSMOLOGICAL EXPANSION\nWe turn now to relate the gravitational radiation\ndamping derived by Weinberg [ 10] to the calculations\nabove, and to Landau damping in collisionless plasmas,\ndriven by the expansion of the universe. We first gener-\nalize the treatment of [ 10] to allow for massive particles\nand collisions, working in conformal time, u, related to\ncoordinate time by dt=adu, whereais the scale pa-\nrameter of the expansion. The metric in the presence of\nexpansion and a gravity wave is given by\nds2=a(u)2[−du2+(δij+hij)dxidxj].(54)\nOwing to the explicit a2, upper and lower components of\nvectors are related by xµ=a2xµto zeroth order in h.\nIn addition, the energy of a particle in the metric ( 54) is\ngiven, in the absence of hij, by\nǫ2\n0=p2\ni+a2m2. (55)\nWe study, following [ 10], the evolution of the coupled\ngravitational wave – matter system, after an initial time\nu0at which the matter distribution function is given as\nf0, essentially the fhin Eq. (20). Since f0includes the\nmetric perturbations hij(u0) at that time, the additional\nperturbations of the energy that modify the distribution\nfunction by δf(u) at later time depend only on the devi-\nation from hij(u0), that is,\nδǫ=−pipj\n2ǫ0[hij(u)−hij(u0)]. (56)\nThe Boltzmann equation in conformal time, Fourier\ntransformed in space (cf. Eq. ( 23)) is\n/parenleftbigg∂\n∂u+1\nτc+i/vector q·/vector v/parenrightbigg\nδf=∂f0\n∂ǫ/parenleftbigg1\nτc+i/vector q·/vector v/parenrightbigg\nδǫ,(57)\nThe particle velocity, v, the distribution function f0, and\nthe conformalcollision time τc=τ/a, aredirectly depen-\ndent on the scale factor. (More generally τwill depend\non the cosmological epoch and thus contain further de-\npendence on the scalefactor, a question wedo not pursue\nhere.) In particular\nvi=∂ǫ0/∂pi=pi/radicalbig\npjpj+m2a(u)2. (58)\nSimilarly, an equilibrium distribution function,\nf0=1\neǫ0/T0(u)∓1(59)\n(with∓for bosons or fermions) depends on athrough\nthe term m2a2inǫ, andT0(u)/a(u) is the temperature\nofthe darkmatter. Formasslessparticles, T0is constant.\nIn addition τcis a function of the ambient density along\nthe trajectoryofthe gravitationalwave, andso in general7\ndepends on time through its dependence on the scale fac-\ntor, aswell as throughthe evolvingparticle distributions.\nEquation ( 57) has the general solution\nδf(u,p) =/integraldisplayu\nu0du′∂e−Φ(u,u′)\n∂u′∂f0\n∂ǫ(u′)δǫ(u′); (60)\nwhere we write\nΦ(u,u′)≡/integraldisplayu\nu′du′′/parenleftbigg1\nτc(u′′)+i/vector q·/vector v(u′′)/parenrightbigg\n=ℓ(u,u′)+iˆq·ˆps(u,u′), (61)\nin terms of\nℓ(u,u′) =/integraldisplayu\nu′du′′\nτc(u′′), (62)\nand\ns(u,u′) =q/integraldisplayu\nu′du′′v(u′′), (63)\nwhich is the displacement of the particle in the interval\nu′toutimes the wavevector. The deviation from local\nequilibrium in ( 65) is therefore\nδf−∂f\n∂ǫδǫ=−/integraldisplayu\nu0du′e−Φ(u,u′)∂\n∂u′/bracketleftbigg∂f0\n∂ǫδǫ(u′)/bracketrightbigg\n.\n(64)\nOn a cosmological background,\nπij=/integraldisplay\nppipj√−gǫ0/bracketleftbigg\nδf−∂f\n∂ǫδǫ/bracketrightbigg\n, (65)\nthe generalization of Eq. ( 19), where in the absence of a\ngravitational wave,√−g=a4. With Eq. ( 64), we then\nhave\nπij(q,u) =−/integraldisplay\nppipj\na(u)4ǫ0/integraldisplayu\nu0du′e−Φ(u,u′)\n×∂\n∂u′/bracketleftbigg∂f0\n∂ǫδǫ(u′)/bracketrightbigg\n. (66)\nThis equation is the direct generalization of Eq. ( 28) to\nan expanding spacetime.\nUsingǫdǫ=pdpwe have\nπij=1\na(u)4/integraldisplay\nppipjpkpl\n2pǫ0(u)/integraldisplayu\nu0du′e−Φ(u,u′)\n×∂\n∂u′/bracketleftbigg∂f0\n∂p(hkl(u′)−hkl(u0))/bracketrightbigg\n.(67)\nSincek∝negationslash=lthe angular average above has the form\n(δikδjl+δilδjk)K(s) where in terms of spherical Bessel\nfunctions K(s) =j2(s)/s2, and explicitly,\nK(s)=/integraldisplaydΩ\n4πe−iζ s(1−ζ2)2sin2ϕcos2ϕ\n=−sins\ns3−3coss\ns4+3sins\ns5, (68)withζ= cosθ; thus\nπij(q,u) =1\na(u)4/integraldisplay\npp3\nǫ0(u)/integraldisplayu\nu0du′e−ℓ(u,u′)K(s)\n×∂\n∂u′/bracketleftbigg∂f0\n∂p(hij(u′)−hij(u0))/bracketrightbigg\n.(69)\nIn the masslesslimit, we integratethe momentum deriva-\ntive by parts, using ǫdǫ=pdp, and noting that s→\nq(u−u′), to obtain\nπij=−4¯ρ/integraldisplayu\nu0du′e−ℓ(u,u′)K(q(u−u′))h′\nij(u′),(70)\nwhere the prime denotes d/du, and\n¯ρ=1\na4/integraldisplay\nppf0 (71)\nis the mass density of the matter. Away from the mass-\nless limit, generalizing Ref. [ 10], we find extra contribu-\ntions from the pdependence of sinK. We see from Eq.\n(69) or (70), that, as expected, the net effect ofcollisional\ninteractionsistoefficientlyeraseanisotropicstresses,and\nhence limit their ability to damp gravitational waves.\nSince astrophysical sources of gravitationalwaves have\ncharacteristic frequencies much greater than the inverse\nHubble scale at late times, we can expand the time de-\npendence of the mode functions in powers of a′/(aq). For\nsuch a wave, the spatial Fourier component /vector qobeys the\nequation of motion,\nh′′\nij+2a′\nah′\nij+q2hij= 16πGa2πij. (72)\nThe solution for hij(q,u) in the absence of matter is, to\nlowest order in a′/(aq),\nhij(q,u)∝e−iqu\na(u)(73)\n(during radiation domination, this result is exact). We\nassume that the gravitational wave is in the form of a\nwavepacket for which\nhij(/vector r,u) =/integraldisplayd3q\n(2π)3e−iqu\na(u)F(q), (74)\nwhereF(q) is localized about a wave vector /vector q0. We con-\nsider the absorption by a region of matter much smaller\nthan the horizon size.\nTo see the mechanism of Landau damping in the ab-\nsence of collisions, we calculate the damping of the wave\ndirectly in terms of the energy transfer to the matter,\nwriting, from Eq. ( 15), using conformal time\n∂E\n∂u=−1\n2/integraldisplay\nd3rh′\nij(/vector r,u)πij(/vector r,u)\n=−1\n2ℜ/integraldisplayd3q\n(2π)3h′\nij∗(q,u)πij(q,u). (75)8\nWe work in the massless limit, in order to illustrate the\nphysics with the fewest complications. Then f0does not\ndepend on a, and Φ(u,u′)→iqζ(u−u′), and one has,\n∂E\n∂u=−1\n4a(u)4ℜ/integraldisplayd3q\n(2π)3h′\nij∗(q,u)/integraldisplay\nppipjpkpl\np2\n×/integraldisplayu\nu0du′e−i/vector q·ˆp(u−u′)∂f0\n∂ph′\nkl(q,u′)\n=−1\n16a(u)4ℜ/integraldisplayd3q\n(2π)3h′\nij∗(q,u)/integraldisplay\npp2∂f0\n∂p\n×(1−ζ2)2/integraldisplayu\nu0du′e−iqζ(u−u′)h′\nij(q,u′).(76)\nFrom Eq. ( 73), we see that h′\nij(u) =−(iq+\nH(u))hij(u), where H ≡a′(u)/a(u). The explicit H(u),\nwhichissmallrelativeto the qtermandforanastrophys-\nical gravitational wave produces only a small correction\nto the Landau damping, can be neglected. Over the time\nspan of a gravitational wavepacket transversing a given\nregion of matter, the scale factor a(u′) inhij(u′) can be\nexpanded as a(u′) =a(u)+a′(u)(u′−u)≃a(u)eH(u′−u).\nThus the u′integral can be written as\n−iqe−iquF(q)\na(u)/integraldisplayu\nu0du′e(iq(1−ζ)+H(u))(u−u′)\n≃1\nH+iq(1−ζ)/parenleftBig\nh′\nij(q,u)−e−iqζ(u−u0)h′\nij(q,u0)/parenrightBig\n.\n(77)\nSince the characteristic frequencies are large compared\nwith 1/(u−u0) the phase factor in the final term will\naverage to zero inside the qandζintegrals in Eq. ( 76).\nWe find then\n∂E\n∂u=¯ρ\n4/integraldisplayd3q\n(2π)3|h′\nij(q,u)|2\n×/integraldisplay1\n−1dζ\n2H(1−ζ2)2\nH2+q2(1−ζ)2.(78)\nWe see here how expansion of the universe introduces\na spread in frequencies ∼ ±Haboutq, thus allowing\nLandau damping; in the absence of expansion, H= 0,\nand Landau damping vanishes.\nTo lowest order in H/qthe integral is simply 4/3, so\nthat\n∂E\n∂u=¯ρ\n3a′\na/integraldisplayd3q\n(2π)3|h′\nij(q,u)|2\nq2. (79)\nThe energy density of the gravitational wave is\nEgw=/integraldisplayd3q\n(2π)3|h′\nij(q,t)|2\n32πG, (80)\nso that for a wavepacket centered about a frequency ¯ q\n∂E\n∂u=32πG¯ρ\n3¯q2HEgw. (81)Finally we note that 8 πG¯ρ/3∼(a′/a2)2and thus\n∂E\n∂u∼4H3\n(a¯q)2Egw. (82)\nThe characteristic absorption time via Landau damping\nis thus∼ω2t3\nU(withω= ¯q), which is thoroughly negligi-\nble. The corresponding fractional change in energy over\nan expansion time of the universe is ∼1/(ωtU)2.\nVIII. CONCLUDING REMARKS\nIn this paper we have laid out a framework for eval-\nuating the damping of gravitational radiation by matter\nwith arbitrary mass particles and collision strengths. By\nconsidering the damping of gravitational waves in both\nflat spacetime and in an expanding universe, we identify\ntwo distinct mechanisms through with damping can oc-\ncur – the first in which collisions produce the damping,\nand the second, via Landau damping.\nWhen the expansion of spacetime can be neglected,\nthe damping of a wave of a given frequency, propor-\ntionaltothe relaxation rate 1/τinthecollisionlessregime\n(ωτ≫1) and to the collision time ,τ, in the hydro-\ndynamic regime ( ωτ≫1), is maximal when ωτ≃1.\nFor the frequencies to which LIGO is sensitive and for\nplausible models of dark matter, calculations of damping\nbased on hydrodynamical considerations are gross over-\nestimates, and we conclude that it is impossible from\ncurrent observations of gravitational waves to put use-\nful bounds on the properties of dark matter. Landau\ndamping in this case is not possible because particles\nhave velocities less than c. As we estimate in Sec. V,\ncollisional damping of gravitational waves of frequencies\nproduced by astrophysical binary systems, propagating\nthrough dense local environments, is also insignificant.\nCollisionless damping is possible in an expanding uni-\nverse since the frequency of the gravitational wave and\nthe energies of particles depend on time. Damping of\ngravitational waves by free-streaming relativistic parti-\ncles, proposed by Weinberg [ 10], may, as we have shown,\nbe regarded as a generalization of Landau damping; we\nhave also generalized Weinberg’s formalism to allow for\ncollisions in the matter. We note in passing that one can\nstraightforwardlyextend the present framework to incor-\nporate scenarios of non-thermal dark matter, since it was\nnot essential to assume a specific functional form for the\ndistribution, see, e.g., Eq. ( 20).\nIn the future we will apply the present framework to\nstudy the processing of stochastic gravitational waves of\nprimordial origin, e.g., from (first order) phase transi-\ntions in the early universe during matter domination, in\nscenarios of ultralight dark matter. Such scenarios are\nsimilar to damping by neutrinos during radiation dom-\nination [10,11], and might describe damping by axions\n[18–20].9\nAcknowledgments\nWe are grateful to Stu Shapiro and Subir Sarkar\nfor very helpful remarks. The research of author GB\nwas supported in part by NSF Grant PHY1305891 and\nPHY1714042. He is grateful to the Aspen Center forPhysics, supported in part by NSF Grants PHY1066292\nand PHY1607611, and the Niels Bohr International\nAcademy where parts of this research were carried out.\nAuthor SP is supported by funds from DanmarksGrund-\nforskningsfond under Grant No. 1041811001.\n[1] B. P. Abbott et al. (LIGO Scientific Collab. and Virgo\nCollab.) Phys. Rev. Lett. 116, 061102 (2016); ibid. 116,\n241103 (2016); arXiv:1706.01812.\n[2] S. W. Hawking, Astrophys. J. 145, 544 (1966).\n[3] S. Weinberg, Gravitation and Cosmology: Principles and\nApplications of the General Theory of Relativity , (Wiley,\nNY, 1972). Ch. X.\n[4] G. Goswami, G. K. Chakravarty, S. Mohanty, and\nA. R. Prasanna, Phys. Rev. D 95, 103509 (2017).\n[5] A. P. Lightman, W. H. Press, R. H. Price, and S. A.\nTeukolsky, Problem Book in Relativity and Gravitation\n(Princeton Univ. Press, 1975), Problem 18.15.\n[6] S.GayerandC. F.Kennel, Phys.Rev.D 19, 1070 (1979).\n[7] A. A. Abrikosov and I. M. Khalatnikov, Rep. Prog. Phys.\n22, 329 (1959), Eq. (10.1).\n[8] D. Lynden-Bell, MNRAS 124, 279 (1962).\n[9] G. Baym, H. Monien, C. J. Pethick and D. G. Ravenhall,\nPhys. Rev. Letters 64, 1867 (1990).\n[10] S. Weinberg, Phys. Rev D 69, 023503 (2004).\n[11] T. Watanabe and E. Komatsu, Phys. Rev. D 73, 123515(2006).\n[12] B. A. Stefanek and W. W. Repko, Phys. Rev. D 88,\n083536 (2013).\n[13] K. Nishijima and M. H. Saffouri, Phys. Rev. Lett. 14,\n205 (1965).\n[14] M. Garny, M. Sandora and M. S. Sloth, Phys. Rev. Lett.\n116, 101302 (2016)\n[15] A. W. Graham, D. Merritt, B. Moore, J. Diemand and\nB. Terzic, Astron. J. 132, 2685 (2006).\n[16] V. Paschalidis, M. Ruiz and S. L. Shapiro, Astrophys. J.\nLetters806, L14:1-5, (2015).\n[17] M. Ruiz, R. Lang, V. Paschalids and S. L. Shapiro, As-\ntrophys. J. Letters 824, L1:1-5 (2016).\n[18] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Phys.\nRev. D95, 043541 (2017).\n[19] N. Banik and P. Sikivie, in Universal Themes of Bose-\nEinstein Condensation, (eds. D. Snoke, N. Proukakis and\nP. Littlewood, Cambridge Univ. Press, 2017).\n[20] D. J. E. Marsh, Phys. Repts. 6431-79 (2016)."    },    {        "title": "1805.08022v1.Critical_damping_in_nonviscously_damped_linear_systems.pdf",        "content": "arXiv:1805.08022v1  [math.DS]  21 May 2018Critical damping in nonviscously damped linear systems\nMario L´ azaro1\n1Dep. of Continuum Mechanics and Theory of Structures. Universit ` at Politecnica de\nVal` encia. 46022, Valencia, Spain\nAugust 25, 2021\nAbstract\nIn structural dynamics, energy dissipative mechanisms with non-v iscous damping are character-\nized by their dependence on the time-history of the response veloc ity, mathematically represented\nby convolution integrals involving hereditary functions. Combination of damping parameters in\nthe dissipative model can lead the system to be overdamped in some ( or all) modes. In the domain\nof the damping parameters, the thresholds between induced oscilla tory and non–oscillatory motion\nare called critical damping surfaces (or manifolds, since we can have a lot of parameters). In this\npaper a general method to obtain critical damping surfaces for no nviscously damped systems is\nproposed. The approach is based on transforming the algebraic eq uations which defined implicitly\nthe critical curves into a system of differential equations. The der ivations are validated with three\nnumerical methods covering single and multiple degree of freedom sy stems.\n1 Introduction\nIn this paper, nonviscously damped linear systems are under consideration. Nonviscous (also named\nby viscoelastic) materials have been widely used for vibrat ing control in mechanical, aerospace, au-\ntomotive and civil engineering applications. This paper de als precisely with those applications where\nvibrations are tried to be disappeared, that is, designing o f damping devices which are able to avoid\noscillatory motion at dynamical systems. In nonviscous mod els, damping forces are assumed to be\ndependent on the history of the response velocity via kernel time functions. As far as the motion\nequations concerned, this fact is represented by convoluti on integrals involving the velocities of the\ndegrees–of–freedom (dof) and affected by the hereditary kern els. Denoting by u(t)∈Rnto the array\nwith the degrees of freedom of the system, this vector verifie s the dynamicequilibrium equations which\nin turn has an integro-differential form\nM¨u+/integraldisplayt\n−∞G(t−τ)˙udτ+Ku=f(t) (1)\nwhereM,K∈Rn×nare the mass and stiffness matrices assumed to be positive defin ite and positive\nsemidefinite, respectively; G(t)∈Rn×nis thenonviscous dampingmatrix in the time domain, assumed\nsymmetric, which satisfies the necessary conditions of Goll a and Hughes [1] for a strictly dissipative\nbehavior. As known, the viscous damping is just a particular case of Eq. (1) with G(t)≡Cδ(t),\nwhereCis the viscous damping matrix and δ(t) the Dirac’s delta function. The time–domain system\nof motion equations are then reduced to the well known expres sions\nM¨u+C˙u+Ku=f(t) (2)\nConsidering now the free motion case f(t)≡0in Eq. (1), we test exponential solutions of the type\nu(t) =uest, withuandsto be found. Then, the classical nonlinear eigenvalue probl em associated\nto viscoelastic vibrating structures yields\n/bracketleftbig\ns2M+sG(s)+K/bracketrightbig\nu≡D(s)u=0 (3)\niwhereG(s) =L{G(t)} ∈CN×Nis the damping matrix in the Laplace domain and D(s) is the dy-\nnamical stiffness matrix or transcendental matrix.\nResponse of Eqs. (1) is closely related to the eigensolution s of the eigenvalue problem (3). Ad-\nhikari [2] derived modal relationships and closed form expr essions for the transfer function in the\nLaplace domain. Due to the non–linearity, induced by a frequ ency-dependent damping matrix, the\nsearch eigensolutions is in general much more computationa lly expensive than that of classical viscous\ndamping [3]. A survey of the different viscoelastic models can be found on the references [4, 5] al-\nthough as far as this paper concerned we work with hereditary damping models based on exponential\nkernels [6].\nSince the present paper is devoted on critical damping and th is field has been deeply studied in\nthe bibliography for viscously damped systems, we consider relevant to review the main works on\nit. Duffin [7] defined an overdamped system in terms of the quadr atic forms of the coefficient ma-\ntrices. Nicholson [8] obtained eigenvalue bounds for free v ibration of damped linear systems. Based\non these bounds, sufficient condition for subcritical dampin g were derived. M¨ uller [9] characterized\nan underdamped system in similar terms to Duffin’s work derivi ng a sufficient condition expressed\nas function of the definiteness of the system matrices. Inman and Andry [10] proposed sufficient\nconditions for underdamped, overdamped and critically dam ped motions in terms of the definiteness\nof the system matrices. These conditions are valid for class ically damped systems although Inman\nand Andry shown that they also could work for non–classical s ystems. Inman and Orabi [11] and\nGray and Andry [12] proposed more efficient method for computi ng the critical damping condition.\nHowever, Barkwell and Lancaster [13] pointed out some deffici encies in the Inman and Andry criterion\nof ref. [10] presenting a counterexample and they provided s ome reasons explaining why this crite-\nrion had been usually adopted to check criticality in damped systems. Additionally, Barkwell and\nLancaster [13] obtained necessary and sufficient conditions for overdamping in gyroscopic vibrating\nsytems. Bhaskar [14] presented a more complete overdamping condition which somehow corrected\nthat of Inman and Andry [10] giving a generalization to a clas s of non–conservative systems. Beskos\nand Boley [15] established conditions for finding critical d amping surfaces from the determinant of\nthe system and its derivative. They proved that a critically damped eigenvalue was simultaneously\nroot of the characteristic equation and its derivative, som ething that can be used to detect critical\ndamping surfaces. In the work [16] the same authors studied c onditions for critical damping in con-\ntinuous systems. Beskou and Beskos [17] presented an approx imate method computationally efficient\nto find critical damping surfaces separating overdamping or partially overdamping regions from those\nof underdamping for viscously damped systems.\nAs far as nonviscous systems concerned, research on critica l damping has not been as exhaustive as\nthat ofviscousdamping. Mainly, investigations havebeenf ocusedonsingledegree–of–freedom systems\non the discussion of the type of response attending at the dam ping parameters of a single–exponential\nhereditary kernel. Muravyov and Hutton [18] and Adhikari [1 9] analyzed the conditions under which\nsingle degree–of–freedom nonviscously damped systems by o ne exponential kernel becomes critically\ndamped. He carried out an exhaustive analysis of the roots na ture of the resulting third order char-\nacteristic polynomial. Adhikari [20] studied the dynamic r esponse of nonviscously damped oscillators\nand discussed the effect of the damping parameters on the frequ ency response function. M¨ uller [21]\nperformed a detailed analysis on the nature of the eigenmoti ons of a single degree of freedom Zener\n3–parameter viscoelastic model. Muravyov [18] obtained cl osed–form solutions for forced nonviscoulsy\ndamped beams studying the conditions for overdamping or und erdamping time response. As known,\nviscoelastic systems modeled by hereditary exponential fu nctions are characterized by having extra\nreal overdamped modes associated to those kernels. As far as these type of modes concerned the\nreferences [22, 23, 24] provide a mathematical characteriz ation and some numerical methods to their\nevaluation.L´ azaro [5] observed that certain recursive me thod to obtain eigenvalues in proportionally\ndamped viscoelastic systems always converges under a linea r rate except just in the critical surfaces\nwhere the scheme is underlineal.\niiIn this paper, critical damping surfaces of nonviscously da mped linear systems are presented.\nCritical damping is refereed to the set of damping parameter s within the threshold between induced\noscillatory and non–oscillatory motion (for all or for some modes). The general procedure to extract\nthese manifolds in the domain of the damping parameters is to eliminate a parameter of a system of\ntwo algebraical equations. Encouraged by the fact that this elimination is not possible for polynomials\nwith order greater than four, a new method to construct criti cal curves is developed. This method is\nbased on to transform the algebraical equations into two ord inary differential equations. The method\nis validated with three numerical examples. The two first are devoted on single degree–of–freedom\nsystems with one and two hereditary exponential kernesl, re spectively. The application of the current\napproach for multiple degree–of–freedom sytems is present ed in the third example.\n2 Conditions of criticality in terms of determinant of the sy stem\nIn this section we will extend the main results derived by Bes kos and Boley [15] on critical viscous\ndamping to nonviscously damped systems. In order to establi sh the basis of our work, we will describe\nthe type of damping model adopted in its most general form. We will consider a damping matrix\nbased on hereditary Biot’s exponential kernels. Mathemati cally, this model adopts the following form\nin time and in frequency domain\nG(t) =N/summationdisplay\nk=1Ckµke−µkt,G(s) =L{G(t)}=N/summationdisplay\nk=1µk\ns+µkCk (4)\nwhereµk>0, 1≤k≤Nrepresent the relaxation or also called nonviscous coefficie nts andCk∈Rn×n\nare the (symmetric) matrices of the limit viscous damping mo del, defined as the limit\nN/summationdisplay\nk=1Ck= lim\nµ1...µN→∞G(s) (5)\nCoefficients µkcontrol the time and frequency dependence of the damping mod el while the spatial\nlocation andthelevelofdampingarecontrolledbycoefficien tswithinmatrices Ck. Itisstraightforward\nthat the following relationships hold\nN/summationdisplay\nk=1Ck=/integraldisplay∞\n0G(t)dt=G(0) (6)\nHenceforth, we will consider the damping matrix, and for ext ension the transcendental matrix, as\ndepending not only on the frequency via s, but also on set of parameters controlling the dissipative\nbehavior. In the most general case, let us see that the symmet ric damping model presented in (4)\ndepends on pmax=N+Nn(n+1)/2 parameters. Indeed, Nnonviscous coefficients µ1,...,µ Nplus\nn(n+1)/2 possible independent entrees in every symmetric matrix Ck, with 1 ≤k≤N. Thus, the\ncomplete set of parameters can be listed as\nµ1,...,µ N,C111,...,C 1nn,...,C N11,...,C Nnn (7)\nwhereCkij=Ckjiis the entree ijof matrix Ck. Real applications depend in general on much less\nparameters, say p << p max. In the sake of clarity, we will denote by θ={θ1,...,θp}the set of\nindependent damping parameters and consequently we can wri te the damping matrix as G(s,θ).\nAccording to the said above, we can denote as D(s,θ) = det[D(s)] the determinant associated to\nthe nonlinear eigenvalue problem (3). Eigenvalues are then roots of the equation\nD(s,θ) = det/bracketleftBigg\ns2M+sN/summationdisplay\nk=1µk\ns+µkCk+K/bracketrightBigg\n= 0 (8)\niiiAttending to the values of θ∈Rp, the algebraic structure of the spectrum of this problem can vary.\nIf the level of damping induced by matrix G(s,θ) is light, we will have 2 ncomplex eigenvalues with\noscillatory nature and rreal eigenvalues with non–oscillatory nature and associat ed to the nonviscous\nhereditary kernels (hence they are also usually named as non viscous eigenvalues). Furthermore, the\ntotal number of these real eigenvalues is [3]\nr=r1+···+rN=N/summationdisplay\nj=1rank(Cj) (9)\nAs longas thereexist 2 ncomplex eigenvalues and rnonviscous eigenvalues, wewill say that thesystem\nis completely underdamped. As thedampinglevel increases, the real part of eigenvalues (not necessary\nall) becomes higher (in absolute value) and the imaginary pa rt decreases. For certain value of the\ndamping parameters a conjugate–complex pair could merge in to a double real negative root. The set\nof damping parameters is said then to be on a critical surface , which in turn represents the threshold\nbetween underdamping and overdamping. If oscillatory mode s coexist with those non–oscillatory,\nthen we say the system is partially overdamped (or mixed over damping). The system is said to be\ncompletely overdamped if all modes are so. For mixed or compl ete overdamping, some (or maybe all)\nof the roots of (8) are negative real numbers, say s=λ, withλ <0 so that\nD(λ,θ) = det/bracketleftbig\nλ2M+λG(λ,θ)+K/bracketrightbig\n= 0 (10)\nFor each value of λ, Eq. (10) defines a p–dimensional surface in the space where the parameters arra y\nθcan take values. Since for light damping we have initially npairs of conjugate–complex eigenvalues,\nwe will have as much as ncritical surfaces because, as Beskos and Boley [15] point ou t: “there are at\nmost as many partial critical damping possibilities as the n umber of the pairs in (8) of roots swith\nzero imaginary part”. The mathematical principle which cha racterizes a critical damping surfaces can\nbe extrapolated to non–vicous damping and therefore these o nes can be found imposing a minimum\namong all possible values of λin Eq. (10), that is\n∂\n∂λD(λ,θ) =∂\n∂λdet[D(λ,θ)] = 0 (11)\nThis condition is consistent with the fact that a critical ro ot is double just under critical condition.\nTherefore, Eqs. (8) and (11) define a set of critical surfaces resulting after eliminating parameter λ\nfrom both equations. This process, although well defined fro m a theoretical point of view, can only be\ncarried out if an analytical closed–form of D(λ,θ) is provided something that only occurs for small to\nmoderately sized systems. For nonvisocusly damped systems , this procedure has not been used yet,\nto the authors knowledge. Instead, they have been found for s ingle degree–of–freedom systems and\nforN= 1 kernels since this particular problem leads to a three ord er polynomial, which as known\nallows radicals–based analytical solution (Cardano’s for mulas). Additionally, in the present paper we\nalso attempt to improve the numerical evaluation of critica l damping surfaces proposing a numerical\nmethod which will be described in the following paragraphs.\nNumerical evaluation of critical surfaces consists in solv ing Eqs. (8) and (11) simultaneously for\na prefixed range of values of damping parameters. This proces s becomes in general computationally\ninefficient sincefor each value oftheprefixedparameters, as ystem of two non–linear equations must be\nsolved. Attempting to improve the numerical procedure for c onstructing critical surfaces we propose\na method valid to build critical curves formed by two paramet ers, assuming as fixed the rest of them.\nThe method is able to find critical overdamped regions in two d imensional cross sections of the p–\ndimensional real critical manifolds. Thus, from the comple te set of parameters θ={θ1,...,θp}, we\nchose two of them, which will be named as design parameters. W ithout loss of generality, we can take\nθ1andθ2while the rest of parameters remain fixed, say θ30,...,θp0. The challenge is to draw the\ncritical damping curves in the plane ( θ1,θ2). For a sake of clarity in the notation, we will denote by\np=θ1andq=θ2and will assume then that the critical curve(s) are function s of the form q=q(p).\nFor each value of p, both equations\nD(λ,p,q) = 0,∂\n∂λD(λ,p,q) = 0 (12)\nivallow to find a pair ( λ,q) (or several, since λis within a polynomial). Let us consider a point p0for\nwhichq0andλ0are solutions of Eqs. (12) and let us assume around p=p0the functions q(p) andλ(p)\nexist. The three numbers ( p0,q0,λ0) form a initial point of the proposed approach. The derivati ves\nλ′(p) = dλ/dpandq′(p) == dq/dpcan be evaluated just applying the chain rule in Eqs. (12). In deed,\nD,λλ′(p)+D,qq′(p)+D,p= 0 (13)\nD,λλλ′(p)+D,λqq′(p)+D,λp= 0 (14)\nwhere subscripts denoting partial derivatives. Since from Eq. (12), we have D,λ= 0, then we can solve\nforλ′(p) andq′(p)\nq′(p) =−D,p\nD,q\nλ′(p) =D,pD,λq\nD,qD,λλ−D,λp\nD,λλ(15)\nThese two equations form a system of two ordinary differential equations whose solutions are be well\ndefinedprovided that thederivatives D,λλandD,qdonot vanishat ( p0,q0,λ0). Existence of thecritical\ncurve beyond a close interval around the initial point will b e subordinate to the existence of those\nderivatives along the curve. Critical curves arise now as th e numerical solution of a system of ordinary\ndifferential equations, forwhichRunge–Kuttabasedmethods canbeused. Before, themethodrequires\nsolving the system of two algebraical equations (12) and two unkowns, say λ0, q0, which in general\nresults in several solutions because of the polynomial form . Pairs ( λ0,q0) both reals and verifying\nλ0<0 andq0≥0 (we will assume a positive range for parameters) will be app ropriate solutions lying\non a critical surface. Taking derivatives repeatedly respe ctpin (14) also lead us to obtain higher order\nderivatives, allowing to find the Taylor expansion of the cri tical curve around p=p0. This procedure\nis used to find an approximation of a critical curve in the one– kernel single–dof numerical example.\n3 Numerical examples\n3.1 Single degree of freedom systems, N= 1exponential kernel PSfrag replacements\nF(t) F(t)R(t)\nG(t)m mku(t)\nFigure 1: A single degree–of–freedom viscoelastic oscilla tor\nFirst, wewillconsiderthesingledofnonviscous systemwit honehereditaryexponential kernel. The\ndof represents the displacement of certain mas mattached to ground by the viscoelastic constraint.\nFig. 1 shows the schematic configuration mass–spring–viscoelastic damper and the corresponding free\nbody diagram. Hence, the internal force is related to the dis placement by\nR(t) =/integraldisplayt\n−∞G(t−τ)˙u(τ)dτ+ku(t) (16)\nkis the constant of the linear–elastic spring and G(t) is the dissipative kernel or damping function\nwith the general form, both in time and frequency domain\nG(t) =cµe−µt, G(s) =µc\ns+µ(17)\nvwhereµandcare respectively is the nonviscous and the viscous coefficien ts. The free motion equation\ncan be deduced from the dynamic equilibrium for F(t)≡0\nm¨u+/integraldisplayt\n−∞G(t−τ)˙u(τ)dτ+ku(t) = 0 (18)\nAnd the associated characteristic equation\nms2+sG(s)+k=ms2+sµc\ns+µ+k= 0 (19)\nIt is quite appropriate to board this problem using dimensio nless variables in order to compare with\nexisting results presented in the bibliography. Thus, we de fine the following non–dimensional variables\nx=s\nωn, ν=ωn\nµ, ζ=c\n2mωn(20)\nwhereωn=/radicalbig\nk/mis he natural frequency of the undamped system. After straig ht operations and\nmultiplying Eq. (19) bythe denominator of thedampingfunct ionwe obtain the characteristic equation\nin non–dimensional form as the third order polynomial\nD(x,ν,ζ) = (1+ νx)(x2+1)+2xζ=νx3+x2+(ν+2ζ)x+1 = 0 (21)\nAs known, the three roots are available as function of the coe fficients so that a detailed discussion of\nthe nature of the three roots can be addressed as function of t he values of ζ >0 andν >0. This work\nwas carried out by Adhikari in the references [19, 20] where c losed form expressions of the critical\ncurves enclosing the overdamped region were derived. For th e sake of our exposition we consider\ninteresting transcript here the Adhikari’s results of the c ritical curves, since later we will present also\napproximations of them. Thus, the overdamped region can be d efined as the set\n/braceleftbig\n(ζ,ν)∈R+:ζL(ν)≤ζ≤ζU(ν)/bracerightbig\n(22)\nwhere the critical damping curves ζL(ν), ζU(ν) are\nζL(ν) =1\n24ν/bracketleftbigg\n1−12ν2+2/radicalbig\n1+216ν2+cos/parenleftbigg4π+θ\n3/parenrightbigg/bracketrightbigg\nζU(ν) =1\n24ν/bracketleftbigg\n1−12ν2+2/radicalbig\n1+216ν2+cos/parenleftbiggθ\n3/parenrightbigg/bracketrightbigg\n(23)\nwith\nθ= arccos/bracketleftBigg\n−5832ν4+540ν2−1\n(1+216ν2)3/2/bracketrightBigg\n(24)\nLet us apply the proposed method to find critical damping curv es based on the solution of the\nsystem of differential equations (15). According to the theor etical derivations, the critical surfaces\narises from eliminating xfrom the two following equations\nD(x,ν,ζ) =νx3+x2+(ν+2ζ)x+1 = 0\n∂D\n∂x= 3νx2+2x+ν+2ζ= 0 (25)\nFrom the second equation we can obtain the two roots x1,2= (−1±/radicalbig\n−6ζν−3ν2+1)/3νand then\nplug them into the first one. After some simplifications, we ob tain the critical surfaces in implicit form\n8ζ3ν+12ζ2ν2−ζ2+6ζν3−10ζν+ν4+2ν2+1 = 0 (26)\nwhich coincides with the third order polynomial obtained by Adhikari [19].\nviWe will attempt now to obtain curves of the form ζ=ζ(ν), therefore our independent parameter\nisp=νand the dependent variable is q=ζ. We need to find the partial derivatives of D(x,ν,ζ) and\nD,x(x,ν,ζ) respect to x,ζandν, obtaining\nD,ν=x+x3,D,ζ= 2x\nD,xν= 1+3 x2,D,xζ= 2,D,xx= 2+6νx (27)\nAfter some math, the two differential equations are set as\nζ′(ν) =−2\n1+x2\nx′(ν) =2x2\n(1+x2)(1+3νx)(28)\nThese equations must becompleted with initial conditions. Takingζ0= 1 equations (25) can besolved\nobtaining four pairs of roots\n(x=−1, ν= 0) ; ( x=−3.38298, ν= 0.134884) ; ( x= 0.191±0.508i, ν=−3.067±2.327i)\n(29)\nOnly real solutions with x <0, ν≥0 are of interest as initial values. The first pair results\nin the initial values ν0= 0,ζ0= 1,x0=−1 of the critical curve ζL(ν) while the second pair\nν0= 0.134884,ζ0= 1,x0=−3.38298 gives as a result ζU(ν). Both curves have been plotted in\nFig. (2). Results fit perfectly with those of exact solutions of Eqs. (23). Although from an analytical\n0.000.050.100.150.200.25\n0.0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0\nViscous damping factor, ζNon–viscous damping factor, ν\nζL\n≈\n1−\nν−\nν2/2ζU≈1\n8ν−ν\n2xL≈ −1−ν−5ν2/2\nxU≈ −1\n2νOverdamped region\nExact critical curves\nProposed approximations\nFigure2: Example1: singledegree–of–freedom with N= 1exponential kerkel. Exactandapproximate\n(proposed) overdamped region. xLandxUare the overdamped critical eigenvalues along the critical\n(approximate) curves\npoint of view, the problem of determining critical curves in this case is solved, we will go further\nproposing two closed–form approximations of ζL(ν) andζU(ν). We consider these derivations of in-\nterest on one hand, by their simplicity respect to those of th e exact expressions. And, on the other\nhand, by the procedure to deduce them.\nEncouragedby thefact that theinitial point of thecritical curveζL(ν) isas simpleas ζL(0) = 1and\nalso by its regularity and low curvature (information alrea dy known since the exact result is available\nin Fig. 2, we think that the Taylor series expansion around ν0= 0 can provide accurate results and\nviiin turn simple in form. Indeed, the first derivative can be det ermined just from Eq. (28) for x0=−1\nandν0= 0, resulting\nζ′\nL(0) =−1, x′(0) =−1\nNow, taking again derivatives respect to νin Eqs. (27) and after some operations, second derivatives\nζ′′\nL(0) andx′′(0) can be found, so that\nζ′′\nL(0) =−1, x′′(0) =−5\nHence, Taylor series expansions up to the second order of the critical damping curve ζL(ν), and its\nassociated critical eigenvalue xL(ν) are then\nζL(ν)≈1−ν−ν2/2 (30)\nxL(ν)≈ −1−ν−5ν2/2 (31)\nSimilar procedure could be followed to find a Taylor based app roximation around upper critical\ncurveζU, however, this function presents higher changes of curvatu res and a wider domain of ζ(in fact\nan infinite range). In is expected that a polynomial based app roximation only will work around the\ninitial point and of course it will not be able to represent th e asymptotic behavior. To undertake this\napproach a recent result on asymptotic behavior of polynomi al roots proposed by L´ azaro et al. [25]\nwill be used. Given a polynomial\na0+a1X+···+an−2Xn−2+an−1Xn−1+Xn(32)\nthen the numbers (called polynomial pivots)\n−an−1\n2±/radicalbigg/parenleftBigan−1\n2/parenrightBig2\n−an−2 (33)\npresent the property of lying close to one (or two) roots prov ided that they are not relatively smaller\nthan the rest of the polynomial coefficients. The exact mathem atical conditions describing this state-\nment are given in form of several theorems in the reference [2 5]. We check if the so defined pivots\nof the third order polynomial D(x,ν,ζ) can give us valuable information respect to the nature of th e\nroots. Thus, the pivots are\n−a2\n2±/radicalbigg/parenleftBiga2\n2/parenrightBig2\n−a1=−1\n2ν±/radicalbigg\n1\n4ν2−2ζ\nν−1 (34)\nLooking for critical damping curves, we know that along them , the roots have double multiplicity\n(double roots). Therefore, if we admint that the pivots are c lose to two roots of the problem and we\nforce the discriminant of Eq. (34) to be zero, then the obtain ed root will be double and we will lie on\na critical damping curve. Hence, to vanish the discriminant results in the approximation of the upper\ncurveζU(ν). Indeed,\n1\n4ν2−2ζ\nν−1 = 0→ζU(ν)≈1\n8ν−ν\n2(35)\nThe associated damped eigenvalue can be approximated by\nxU(ν)≈ −1\n2ν(36)\nAccording to the results of [25] the bigger the pivots the clo ser to the roots of the polynomial. There-\nfore, we can predict that the lower the nonviscous parameter νthe more accurate the results, as can\nbe appreciated in the Fig. 2.\nviii3.2 Single degree of freedom systems, N= 2exponential kernels\nNow we will attempt to find the critical damping surfaces of a s ingle dof system with N= 2 hereditary\nkernels. Theapproachcaneasilybeextrapolatedtothegene ralcaseof Nkernels. AccordingtoEq.(4),\nthe damping function is\nG(t) =c1µ1e−µ1t+c2µ2e−µ2t, G(s) =L{G(t)}=µ1c1\ns+µ1+µ2c2\ns+µ2(37)\nAnd the characteristic equation yields\nms2+sG(s)+k=ms2+s/parenleftbiggµ1c1\ns+µ1+µ2c2\ns+µ2/parenrightbigg\n+k= 0 (38)\nAs noticed, the dissipative model has four parameters, c1,c2,µ1,µ2. Our procedure allows to draw\ncritical curves of two parameters, hence before finding the s olution of the proposed differential equa-\ntions two parameters must be fixed. For a sake of the solution r epresentation, we fusion the damping\ncoefficients into only one, considering the case c1=c2=c. Let us define the following dimensionless\nparameters\nx=s\nωn, ν1=ωn\nµ1, ν2=ωn\nµ2, ζ=c\nmωn(39)\nWhereωn=/radicalbig\nk/mis the natural frequency of the system. The Eq. (38) can be exp ressed now in\ndimensionless form\nx2+xζ/parenleftbigg1\n1+ν1x+1\n1+ν2x/parenrightbigg\n+1 = 0 (40)\nNote that ν1=ν2= 0 yields the particular case of viscous damping leading to ζcr= 1. Multiplying\nEq. (40) by (1+ ν1x)(1+ν2x) we transform the characteristic equation into a four order polynomial\nequation\nD(x,ν1,ν2,ζ) = (1+ ν1x)(1+ν2x)(x2+1)+xζ(2+ν1x+ν2x) = 0 (41)\nwhich together with\nD,x(x,ν1,ν2,ζ) = 2ζ+2x+2xν1ν2(1+2x3)+(1+2 xζ+3x3)(ν1+ν2) (42)\nallow to find the critical surfaces after eliminating x. Observe that the last equation is a third order\npolynomial, therefore the Cardano formulas can be used to ob tain the three roots. Plugging them into\nEq. (41) would lead the exact critical curves. These derivat ions will not be carried out here since the\nresulting expressions would be hardly handled and together with the difficulty to follow properly the\nexposition. On the other hand, they can easily be programmed in a symbolic software and results be\ncompared to those of the present method.\nAccording to the proposed approach, the critical surfaces a re defined in terms of two parameters,\nleaving fixed the rest. For the current example we will consid er critical curves in the plane ( ζ,ν2) (i.e.\nfunctions ν2=f(ζ) withν1fixed) and also in the plane ( ν1,ν2) (i.e. functions ν2=f(ν1) withζfixed).\nTwo systems of differential equations must be assembled, one i nvolving the functions x(ζ),ν2(ζ) and\nthe other one the functions x(ν1),ν2(ν1). From Eq. (14) the two problems can be written in matrix\nform as\n•Critical curves in plane ( ζ,ν2). Parameter ν1fixed.\n/bracketleftbigg0D,ν2\nD,xxD,xν2/bracketrightbigg/braceleftbiggx′(ζ)\nν′\n2(ζ)/bracerightbigg\n=−/braceleftbiggD,ζ\nD,xζ/bracerightbigg\n,/braceleftbiggx(ζ0)\nν2(ζ0)/bracerightbigg\n=/braceleftbiggx0\nν20/bracerightbigg\n(43)\n•Critical curves in plane ( ν1,ν2). Parameter ζfixed.\n/bracketleftbigg0D,ν2\nD,xxD,xν2/bracketrightbigg/braceleftbiggx′(ν1)\nν′\n2(ν1)/bracerightbigg\n=−/braceleftbiggD,ν1\nD,xν1/bracerightbigg\n,/braceleftbiggx(ν10)\nν2(ν10)/bracerightbigg\n=/braceleftbiggx0\nν20/bracerightbigg\n(44)\nixwhere\nD,ν1=x/bracketleftbig\n1+x(x+ζ+ν2+x2ν2)/bracketrightbig\nD,ν2=x/bracketleftbig\n1+x(x+ζ+ν1+x2ν1)/bracketrightbig\nD,ζ=x[2+x(ν1+ν2)]\nD,xν1= 1+x[2(ζ+ν2)+x(3+4xν2)]D,xν2= 1+x[2(ζ+ν1)+x(3+4xν1)]D,xζ= 2[1+x(ν1+ν2)]\nThe initial conditions come from solving Eqs. (41) and (42) f or prescribed values of two of the pa-\nrameters. Table 3.2 lists a complete set of initial values wh ich allow to address the solution of the\ndifferential equations. From the solution of the aforementio ned algebraic equations one can find simul-\ntaneously two different initial conditions. Thus, for instan ce, for the case ν10= 0.00 andζ0= 1.00,\namong other complex solutions we find ( ν20,x0) = (0,−1) and (ν20,x0) = (0.1916,−2.7693) as the\ninitial conditions of the two first curves shown in Table 3.2. Notice that this fact also takes place for\nother cases in the table, always if we somewhat are fortunate in our election of the prescribed pair of\nparameters. Otherwise, we could not obtain satisfactory so lutions, for example taking ν10= 0.25 and\nζ0= 5.00 we only find complex solutions to the system (41) and (42).\nINITIAL VALUES\nValue of the fixed parameter Curve ζ0ν20 x0\nν1: fixed ν1= 0.00 C1 1.00000 0.00000 -1.00000\nC2 1.00000 0.19160 -2.76929\nC3 4.00000 1.80565 -2.22076\nν1= 0.05 C1 0.95000 0.04736 -1.05573\nC2 0.95000 0.19197 -2.72450\nC3 5.20000 0.79903 -4.84245\nC4 5.20000 1.93559 -9.57166\nν1= 1.50 C1 4.00000 0.03207 -2.61688\nC2 4.00000 0.06738 -6.74685\nValue of the fixed parameter ν10 ν20 x0\nζ: fixed ζ= 0.90 C1 0.00000 0.17053 -1.14479\nC2 0.00000 0.22155 -2.26820\nζ= 5.00 C1 0.00000 0.03443 -16.97040\nC2 0.00000 1.32591 -2.83001\nC3 0.05200 2.06935 -9.20408\nC4 1.20000 0.01599 -3.10458\nC5 1.20000 0.05344 -8.61426\nζ= 8.00 C1 0.00000 0.02148 -27.25100\nC2 0.00000 0.76381 -4.62400\nC3 0.03400 0.56548 -13.0963\nC4 0.70000 0.00952 -5.03922\nC5 0.70000 0.03329 -13.89100\nTable 1: Initial conditions used to the computation of the cr itical damping curves shown in Fig. 3\nxIn Fig. 3, the different obtained critical curves have been plo tted. If we read overdamped regions\nas certain volumes enclosed by critical surfaces in the para metric space ( ζ,ν1,ν2), then the left curves\nare cross sections of these volumes for certain values of the damping ratios (in Fig. 3 left cases\nζ= 0.9,5.0,8.0 are shown). On the other hand, right plots have the same inte rpretation but as\ncross sections of the planes ν1= 0.0,0.054,1.50. The critical curves are in correspondence with\nthe notation used in Table 3.2. As expected overdamped regio n in the plane ( ν1,ν2) are symmetric\nrespect to line ν1=ν2since the physical model has inherently this symmetry. In th e plane ( ν2,ζ)\nforν1= 0 we observe an overdamped region for values ν2≪1 very similar to that obtained in the\nfirst example, see Fig. 2. Now, due to the commented symmetry o f the problem respect to ν1and\nν2, this long–triangle–like of the top–right plot is in fact a s ection of a conoid–like form in the space\n(ζ,ν1,ν2). Actually, it is a quarter of conoid because ν1,ν2≥0. Another section of this conoid is\nobtained just shifting the cross section to the value ν1= 0.05 (middle–right plot). It is also interesting\nthe new overdamped region arising in the corner of the right– top plot (section ν1= 0). Presumably\nhigh values of ζandν2(simultaneously) lead the system to non–oscillatory motio n. Let us see that\nthe damper model in this case is formed by two dampers in paral lel, one viscous and the other one\nnonviscous with parameter µ1. Indeed, for ν1= 0 (which is equivalent to µ1→ ∞) the damping\nfunction is transformed into\nlim\nµ1→∞G(t) =c1δ(t)+c2µ2e−µ2tlim\nµ1→∞G(s) =c1+µ2c2\ns+µ2(45)\nSomehow, we can interpret this overdamped region as the effect producedin the nature of the response\nof both the nonviscous parameter µ2and the viscous coefficcient c1. This is the reason because such\ncritical surface did not appear in the example 1. Again, from the symmetry respect to the nonviscous\nparameters, this overdamped region also appears in the the p lane (ζ,ν1) forν2= 0. Furthermore,\nthis effect is extended as an narrow volume in the approximate r ange 0≤ν1≤0.07 (respectively for\nsymmetry in 0 ≤ν2≤0.05), see middle–left and bottom–left plots. It seems clear t hat adding new\ndamping parameters will make more difficult how to read into th e form of the overdamped manifolds,\nspecially since they do not follow regular geometrical stru ctures, as seen in this example. However,\nthe proposed method could be applied sequentially to extrac t those more interesting curves for our\nanalysis, for instance in artificial dampers design problem s. Let us see now in a final example how to\nextract critical damping curves for a multiple dof system.\n3.3 Multiple degree of freedom systems\nm\nk\nmk\nmk\nmk\nG (t) G (t)u1u2u3u4\nFigure 4: Example 3: The four degrees–of–freedom discrete s ystem.G(t) represents the hereditary\nfunction of nonviscous dampers\nIn order to validate the proposed approach to find critical da mping curves for multiple dof systems a\ndiscrete lumped mass dynamical system with four dof is analy zed. The Fig. 4 represents the distri-\nbution of masses m, rigidities kand viscoelastic dampers with a hereditary function G(t). The mass\nmatrix of the system is M=mI4while, according to the rigidities and dampers distributio n, stiffness\nximatrix yields\nK=k\n2−1 0 0\n−1 2−1 0\n0−1 2−1\n0 0−1 2\n=kK (46)\nWe will assume a dampingfunction formed by one hereditary ex ponential kernel of dampingcoefficient\ncand nonviscous parameter µ. Hence, te damping matrix can be expressed as G(t) =µCe−µtwhere\nC=c\n0 0 0 0\n0 1−1 0\n0−1 2−1\n0 0 1 1\n≡cC (47)\nWith help of these dimensionless matrices, say KandC, we can express the non–linear eigenvalue\nproblem associated to this problem under dimensionless for m as\n/bracketleftbigg\nx2I4+2xζ\n1+νxC+K/bracketrightbigg\nu=0 (48)\nwhere\nx=s\nω0, ν=ω0\nµ, ζ=c\n2mω0, ω0=/radicalbig\nk/m (49)\nSincer= rank(C) = 2, the system has 2 n+r= 10 eigenvalues. Therefore, the determinant of the\ntranscendental matrix can be transformed into a 10th order p olynomial multiplying by the factor\n(1+νx)2. We define then our function D(x,ν,ζ) as\nD(x,ν,ζ) = (1+ νx)2det/bracketleftbigg\nx2I4+2xζ\n1+νxC+K/bracketrightbigg\n= 5+2(8 ζ+5ν)x+[(2ζ+ν)(6ζ+5ν)+20]x2+(54ζ+40ν)x3\n+/parenleftbig\n32ζ2+54ζν+20ν2+21/parenrightbig\nx4+(40ζ+42ν)x5+/parenleftbig\n12ζ2+40ζν+21ν2+8/parenrightbig\nx6\n+8(ζ+2ν)x7+/parenleftbig\n1+8νζ+8ν2/parenrightbig\nx8+2νx9+νx10(50)\nWe are looking for overdamped regions enclosed by critical c urves of type ν=ν(ζ), therefore we con-\nstruct our system of differential equations following the met hodology described in Section 2, resulting\n/bracketleftbigg0D,ν\nD,xxD,xν/bracketrightbigg/braceleftbiggx′(ζ)\nν′(ζ)/bracerightbigg\n=−/braceleftbiggD,ζ\nD,xζ/bracerightbigg\n,/braceleftbiggx(ζ0)\nν(ζ0)/bracerightbigg\n=/braceleftbiggx0\nν0/bracerightbigg\n(51)\nFor a sake of clarity in the exposition the expressions of the partial derivatives will not be written.\nInitial conditions can be found solving the system of algebr aical equations for a particular value of ν0\nandζ0\nD(x0,ν0,ζ0) = 0,D,x(x0,ν0,ζ0) = 0 (52)\nTesting for ν0= 0.06 we obtain four different pairs ( ζ0,x0), listed in Table 2. After solving Eqs. (51)\nINITIAL VALUES\nCurve ζ0 ν0 x0\nC1 0.53258 0.06000 -2.05512\nC2 0.72949 0.06000 -7.88861\nC3 1.25218 0.06000 -1.45645\nC4 2.14493 0.06000 -8.07994\nTable 2: Initial conditions used for the critical damping cu rves shown in Fig. 5\nwe plot the solutions in Fig. 5. The four found curves enclose two overdamped regions which in turn\nxiiintersect each other. The solid–filled region represents th e set of values ζ,νwhich lead the fourth\nmode to overdamping. On the other hand, lines–filled area cor responds to the overdamped region\nof the second mode. This can be checked following a root–locu s plot varying parameters ζ,νfrom\nundamping to overdamping. Moreover, 2nd and 4th mode are pre cisely those modes for which degrees\nof freedom linked to the viscoelastic dampers are most activ ated. Obviously, it follows then that the\noverlapping zone (with both types of shading in Fig. 5) corre sponds with the overdamping of both\nmodes, simultaneously.\n0.000.050.100.150.20\n0.0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0\nViscous damping factor, ζNon–viscous damping factor, νOverdamped Region (4th mode)\nOverdamped Region (2nd mode)\nCurve C1\nCurve C2\nCurve C3\nCurve C4\nInitial values\nFigure 5: Example 3: Critical damping curves and the corresp onding enclosing overdamped regions\nfor the four degrees–of–freedom system\nA deeper inspection of Fig. 5 leads us to ask ourselves about t he existence of singularities in the\ndomain ( ζ,ν) which can impede the application of our approach. Well, it i s known that a system\nof differential equations like that shown in (51) has solution (and it is unique) provided that the\ndeterminantofthematrixdoesnotvanishinaneighborhooda roundtheinitialvalue. Thisdeterminant\nis equal to −D,νD,xx, whence it follows that these two system of algebraic equati ons\nS1:\n\nD= 0\nD,x= 0\nD,ν= 0S2:\n\nD= 0\nD,x= 0\nD,xx= 0(53)\nallow us to find singularities. For every solution of the first system S1, some of the variables, x,ζor\nνis within the complex plain. On the other hand, we do find valid solutions of system S2, verifying\nx <0, ζ,ν≥0, say\nx=−3.1059 ζ= 0.46473 ν= 0.10658\nx=−2.3221 ζ= 1.06113 ν= 0.14088 (54)\nThese two points are precisely the vertexes of the two overda mped regions, namely, intersection points\nof curves C1–C2 and C3–C4, respectively. These points can no t be used as initial points since ac-\ncording to the implicit function theorem, equations D= 0 and D,x= 0 do not define x(ζ) andν(ζ)\nunequivocally. Furthermore, both points verifies D,xx= 0, hence they are triple roots.\nWe wonder now how does the solution behave in the intersectio n point between curves C2 and C3,\nlocated approximately at ζ= 1.30465, ν= 0.03239. This point satisfies −D,νD,xx/ne}ationslash= 0, therefore it\nshould be valid as initial value of our method. However, it be longs to both curves simultaneously, so\nxiiithat at a first sight the solution would not seem to be well defin ed. However, in this point we find\ntwo solutions for the variable x, sayx=−1.38042 and x=−15.2148, which leads to two different\ninitial values. Somehow this point does not correspond with a intersection point of the curves in the\n3D domain ( x,ν,ζ), while the tripleroots of (54) do.\nAs far as multiple degree-of-freedom systems concern, the s uccess of the method lies on the avail-\nability of the transcendental matrix determinant and their derivatives. Therefore, from a numerical\npoint of view, large systems will require high computationa l effort which limits the range of applica-\nbility for small or moderate order systems. Currently, our e fforts are addressed to find out numerical\nprocedures allowing to construct approximate critical cur ves but for larger systems, something that is\nunder research.\n4 Conclusions\nIn this paper critical damping of nonviscously damped linea r systems is studied. Nonviscous or vis-\ncoelastic vibrating structures are characterized by dissi pative mechanisms depending on the history\nof response through hereditary functions. For certain valu es of the damping parameters, the response\ncan become non–oscillatory. It is said then that some (or all ) modes are overdamped. Particular\nvalues of the damping parameters which establish the limit b etween oscillatory and non–oscillatory\nmotion are said to be within a critical surface (or critical m anifold). In the present paper a general\nprocedure to build critical damping surfaces is developed. In addition, a numerical method based on\nthe transformation of the algebraical equations into a syst em of two ordinary differential equations is\nproposed. This approach allow to find critical curves of two p arameters for certain fixed values of the\nrest of parameters.\nTo validate the theoretical results three numerical exampl es are analyzed. In the first example,\nthe well–known overdamped region of a single degree–of–fre edom system with one exponential kernel\nis resolved, shown perfect fitting between our curves, obtai ned from the differential equations, and\nthose of the analytical expressions. In order to give added v alue to this problem, we propose sim-\nplified approximate expressions for the critical curves. Th e second example is devoted to construct\noverdamped regions for a two exponential kernels based damp ing function. This problem involves\nthree parameters, so that the critical damping curves are pl otted along several cross sections defined\nby the free parameter. The third example shows how the method can be applied for multiple degrees\nof freedoms systems deriving overdamping regions for differe nt modes and interpreting the obtained\noverlapping regions. Since this method is based on the evalu ation and derivation of the determinant\nof the transcendental matrix, its range of validity is reduc ed to small or moderately sized systems.\nEncouraged by this limitation, the author is currently inve stigating how to extrapolate this method\nfor larger systems.\nReferences\n[1] D. Golla, P. Hughes, Dynamics of Viscoelastic Structure s - A Time-domain, Finite-element For-\nmulation, Journal of Applied Mechanics-Transactions of th e ASME 52 (4) (1985) 897–906.\n[2] S. Adhikari, Dynamics of Non-viscously Damped Linear Sy stems, Journal of Engineering Me-\nchanics 128 (3) (2002) 328–339.\n[3] N. Wagner, S. Adhikari, Symmetric state-space method fo r a class of nonviscously damped sys-\ntems, AIAA Journal 41 (5) (2003) 951–956.\n[4] S. Adhikari, A Reduced Second-Order Approach for Linear Viscoelastic Oscillators, Journal of\nApplied Mechanics-Transactions of the ASME 77 (4) (2010) 1– 8.\nxiv[5] M. L´ azaro, J. L. P´ erez-Aparicio, M. Epstein, Computat ion of eigenvalues in proportionally\ndampedviscoelastic structuresbased onthefixed-pointite ration, AppliedMathematics andCom-\nputation 219 (8) (2012) 3511–3529.\n[6] M. Biot, Variational Principles in Irreversible Thermo dynamics with Application to Viscoelastic-\nity, Physical Review 97 (6) (1955) 1463–1469.\n[7] R. Duffin, A MINIMAX THEORY FOR OVERDAMPED NETWORKS, JOURN AL OF RA-\nTIONAL MECHANICS AND ANALYSIS 4 (2) (1955) 221–233.\n[8] D. Nicholson, EIGENVALUE BOUNDS FOR DAMPED LINEAR-SYST EMS, MECHANICS\nRESEARCH COMMUNICATIONS 5 (3) (1978) 147–152.\n[9] P. Muller, OSCILLATORY DAMPED LINEAR-SYSTEMS, MECHANI CS RESEARCH COM-\nMUNICATIONS 6 (2) (1979) 81–85.\n[10] D. Inman, A. Andry, SOME RESULTS ON THE NATURE OF EIGENVA LUES OF DISCRETE\nDAMPEDLINEAR-SYSTEMS,JOURNALOFAPPLIEDMECHANICS-TRAN SACTIONSOF\nTHE ASME 47 (4) (1980) 927–930.\n[11] D. Inman, I. Orabi, AN EFFICIENT METHOD FOR COMPUTING TH E CRITICAL DAMP-\nING CONDITION,JOURNALOFAPPLIEDMECHANICS-TRANSACTIONS OFTHEASME\n50 (3) (1983) 679–682.\n[12] J. Gray, A. Andry, A SIMPLE CALCULATION FOR THE CRITICAL DAMPING MATRIX\nOF A LINEAR MULTIDEGREE OF FREEDOM SYSTEM, MECHANICS RESEAR CH COM-\nMUNICATIONS 9 (6) (1982) 379–380.\n[13] L. Barkwell, P. Lancaster, Overdamped and Gyroscopic V ibrating Systems, Journal of Applied\nMechanics-Transactions of The ASME 59 (1) (1992) 176–181.\n[14] A. Bhaskar, Criticality of damping in multi-degree-of -freedom systems, Journal of Applied Me-\nchanics, Transactions ASME 64 (2) (1997) 387–393, cited By 7 .\n[15] D. BESKOS, B. BOLEY, CRITICAL DAMPING IN LINEAR DISCRET E DYNAMIC-\nSYSTEMS, JOURNAL OF APPLIED MECHANICS-TRANSACTIONS OF THE ASME 47 (3)\n(1980) 627–630.\n[16] D. BESKOS, B. BOLEY, CRITICAL DAMPING IN CERTAIN LINEAR CONTINUOUS\nDYNAMIC-SYSTEMS, INTERNATIONAL JOURNAL OF SOLIDS AND STRU CTURES\n17 (6) (1981) 575–588.\n[17] S. Papargyri-Beskou, D. Beskos, On critical viscous da mping determination in linear discrete\ndynamic systems, ACTA MECHANICA 153 (1-2) (2002) 33–45.\n[18] A. Muravyov, S. Hutton, Free vibration response charac teristics of a simple elasto-hereditary\nsystem, Journal of Vibration and Acoustics-Transactions o f the ASME 120 (2) (1998) 628–632.\n[19] S.Adhikari, Qualitative dynamiccharacteristics ofa non-viscouslydampedoscillator, Proceedings\nof the Royal Society A-Mathematical Physical and Engineeri ng Sciences 461 (2059) (2005) 2269–\n2288.\n[20] S. Adhikari, Dynamic response characteristics of a non viscously damped oscillator, Journal of\nApplied Mechanics-Transactions of the ASME 75 (1) (2008) 01 1003.01–011003.12.\n[21] P. Muller, Are the eigensolutions of a l-d.o.f. system w ith viscoelastic dampingoscillatory or not?,\nJournal of Sound and Vibration 285 (1-2) (2005) 501–509.\nxv[22] M. L´ azaro, J. L. P´ erez-Aparicio, Characterization o f real eigenvalues in linear viscoelastic oscilla-\ntors and the non-viscous set, Journal of Applied Mechanics ( Transactions of ASME) 81 (2) (2014)\nArt. 021016–(14pp).\n[23] M. L´ azaro, Nonviscous modes of nonproportionally dam ped viscoelastic systems, Journal of Ap-\nplied Mechanics (Transactions of ASME) 82 (12) (2015) Art. 1 21011 (9 pp).\n[24] M. L´ azaro, C. F. Casanova, C. L´ azaro, Nonviscous Mode s of Viscoelastically Damped Vibrating\nSystems, InTech, 2016, Ch. Viscoelastic and Viscoplastic M aterials, pp. 165–187.\n[25] M. L´ azaro, P. Mart´ ın, A. Ag¨ uero, I. Ferrer, The polyn omial pivots as initial values for a new\nroot-finding iterative method, Journal of Applied Mathemat ics Vol. 2014 (Special Issue: Iterative\nMethods and Applications) (2015) Article ID 413816, 14 page s.\nxviOverdamped regions for ζ≡const Overdamped regions for ν1≡const\n0.000.250.500.751.001.251.500.00 0.25 0.50 0.75 1.00 1.25 1.50Non–viscous parameter, ν2Cross section\nζ= 0.90\nCurve C1\nCurve C2\n0.000.250.500.751.001.251.500.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0\nCross section\nν1= 0.00\nCurve C1\nCurve C2\nCurve C3\n0.000.250.500.751.001.251.50Non–viscous parameter, ν2Cross section\nζ= 5.00\nCurve C1\nCurve C2\nCurve C3\nCurve C4\nCurve C5\n0.000.250.500.751.001.251.50\nCross section\nν1= 0.05\nCurve C1\nCurve C2\nCurve C3\nCurve C4\n0.000.250.500.751.001.251.50\n0.00 0.25 0.50 0.75 1.00 1.25 1.50\nNon–viscous parameter, ν1Non–viscous parameter, ν2Cross section\nζ= 8.00\nCurve C1\nCurve C2\nCurve C3\nCurve C4\nCurve C5\n0.000.250.500.751.001.251.50\n0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0\nViscous damping factor, ζCross section\nν1= 1.50\nCurve C1\nCurve C2\nFigure 3: Example 2: Single degree–of freedom nonviscous sy stem with N= 2 kernels. Critical\ndamping curves and overdamped regions\nxvii"    },    {        "title": "1206.1695v2.A_realisation_of_Lorentz_algebra_in_Lorentz_violating_theory.pdf",        "content": "arXiv:1206.1695v2  [hep-th]  14 Nov 2012A realisation of Lorentz algebra in Lorentz violating theor y\nOindrila Ganguly∗\nS. N. Bose National Centre for Basic Sciences, Kolkata 70009 8, India\nJune 16, 2018\nAbstract\nA Lorentz non-invariant higher derivative effective action in flat spacetime, characterised by a\nconstant vector, can be made invariant under infinitesimal L orentz transformations by restricting\nthe allowed field configurations. These restricted fields are defined as functions of the background\nvector in such a way that background dependance of the dynami cs of the physical system is no longer\nmanifest. We show here that they also provide a field basis for the realisation of Lorentz algebra and\nallow the construction of a Poincar´ e invariant symplectic two form on the covariant phase space of\nthe theory.\n1 Introduction\nAny departure from exact Lorentz symmetry is expected to be a t elltale footprint of quantum gravity,\nthe theory of physics beyond the Planck scale (denoted by Planck m assMPl). This possibility is tremen-\ndouslyexcitingowingtothefactthatquantumgravityeffectsfallla rgelyoutsidethedomainofourcurrent\nexperiments and observations. Moreover, it is always challenging to find the limits of validity of any sym-\nmetry. These factorshavestimulated a lot of workon the theoret ical, phenomenologicaland experimental\naspects of Lorentz violation since the last couple of decades [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].\nIn the present report, we shall focus on a low energy effective field theory of scalar, vector and spinor\nfields developed by Myers and Pospelov [10] that incorporates devia tion from Lorentz symmetry through\nthe inclusion of an extra term. This new term contains a Planck suppr essed dimension five operator with\na constant background vector nthat essentially assigns a preferred direction to spacetime. Howev er, in\n[15], the authors demonstrate that there exist quite general field configurations which can restore Lorentz\nsymmetryin thelimit ofinfinitesimal transformationsin modified action of[10]. TheseLorentzpreserving\nfields, as they have been named in [15], locally conserve the N¨ other current for Lorentz transformations.\nNotice though that the Lorentz violation that had been woven into t he Myers Pospelov theory is not\nlost. It manifests itself through the dependance of the Lorentz p reserving fields on the directionality of\nthe background i.e. on n. Given this scenario, a natural question to ask is whether Lorentz Lie algebra\ncan be realised on a basis of such special field configurations.\nWe shall consider the case of the Lorentz preserving scalar fields [1 5] for the modified action of [10] as\nan illustrative example. The extended action of a complex scalar field φproposed by Myers and Pospelov\nis,\nSMPφ=/integraldisplay\nd4xLMPφ,\n=/integraldisplay\nd4x/bracketleftbig\n|∂φ|2−m2|φ|2/bracketrightbig\n+/integraldisplay\nd4xiκ\nMPlφ∗∂3\nnφ . (1)\n∗oindrila@bose.res.in\n1ThefirstintegralcontainstheusualLagrangiandensityofacomp lexscalarfield withmass mwhilethe\nterm under the second integral is the Lorentz violating contributio n.κis a real, dimensionless parameter\nandn·∂≡∂n,nbeing a constant four vector. According to [15], this action shall be symmetric under\ninfinitesimal Lorentz boosts or rotations if the complex scalar field c an be split as,\nφ(x) =φ/bardbl(x/bardbl)+φ⊥(x⊥) (2)\nwhere,φ/bardblandφ⊥are arbitrary functions of their respective arguments x/bardblandx⊥defined by n.\nx/bardbl≡x·n\nn2nandn·x⊥= 0, so that x=x/bardbl+x⊥. So, the derivative operator can be written as\n∂=∂x/bardbl+∂x⊥=∂/bardbl+∂⊥where the notation ∂/bardbl≡∂x/bardbland∂⊥≡∂x⊥.\nA key aspect of the additional term in (1) is that it contains third ord er derivatives of the field unlike\nstandard first order Lagrangians. But higher derivative Lagrang ians are not new to physics [16]. Back\nin 1961, Ostrogradskii [17] had developed a canonical formalism for dealing with them. Reviews and\nmodifications of his technique [18, 19, 20, 21] study mostly systems having finite number of degrees of\nfreedom and higher time derivatives of the generalised coordinates . An extension to special relativistic\ncontinuous systems is presented in [22], though it relies on the exist ence of Lorentz symmetry. In the\nnext section, we shall briefly review Ostrogradskii’s original constr uction and its generalisation to field\nsystems, tailored to suit the Lagrangian (1) with fields that decoup le as (2)\nUsually, the canonical formalism is understood to be non-covariant because it involves the choice of a\nspacelikehypersurfaceand its orthogonaltime direction. In fact , all the peculiaritiesofahigherderivative\ntheory are due only to higher order time derivatives. Spatial deriva tives are quite benign, staying within\nthe scope of standard first order canonical approach. So, in sec tion 3, the study of the modified scalar\nfield theory will be split up into two cases distinguished by n2being timelike and spacelike. We will not\ncomment on n2= 0 in this paper. We shall also take the liberty of working in certain Lor entz frames\nthat simplify calculations.\nHowever, a covariant framework for the canonical formulation of a relativistic theory may also be\ndeveloped [23, 24, 25, 26, 27] through the construction of a cova riant phase space and a symplectic two-\nform on it. Reference [28] has presented a natural extension of t his technique to higher derivative field\ntheories. Section 4 will contain a summary of the main features of th e covariant phase space for first\norder as well as higher order derivative field systems. We shall conc lude by showing how the Lorentz\npreserving fields facilitate the construction of a covariant phase s pace structure for the Lorentz violating\neffective field theory under consideration.\n2 Canonical formalism in the presence of higher derivatives\n2.1 Non-relativistic systems with finite degrees of freedom\nLet us consider a system described by the Lagrangian L(qa,dtqa,...,dl\ntqa) which is a function not only of\nthe generalised coordinate qa(t) and the velocity dtqa(t) but of all derivatives dtqa(t),d2\ntqa(t),...,dl\ntqa(t)\nupto order l. Herealabels the different degrees of fredom and we adopt the notation qa(j)≡dj\ntqa, j=\n0,...,lso thatqa(j+1)= ˙qa(j). Each of these qa(j)uptoj=l−1 will have a conjugate momentum\npa(j). Remember though that the superscript jofpa(j)only denotes that it is conjugate to qa(j).\nThe system is now specified by a point in the phase space spanned by qa(j),pa(j);j= 0,...,l−1. We\nhave here assumed that the highest derivative qa(l)can be written as a function of the other variables\nqa(l)(qa(0),pa(0);...;qa(l−1),pa(l−1)). The condition for extremisation of the action S[q(t)] =/integraltext\ndt Lis,\nδqS= 0 =/integraldisplay\ndtl/summationdisplay\nj=0(−dt)j∂L\n∂qa(j)δqa+/integraldisplay\ndtdtl/summationdisplay\ni=j+1l−1/summationdisplay\nj=0(−dt)i−(j+1)∂L\n∂qa(i)δqa(j).\nIf the boundaries are such that the variations of qaand its derivatives upto order l−1 vanish, the\nsecond integral will go to zero and the equation of motion will be\n2(−dt)j∂L\n∂qa(j)= 0. (3)\nOn the other hand, if this system undergoes a symmetry transfor mation,δS= 0 and substitution of\nequation of motion (3) yields,\ndt\nl/summationdisplay\ni=j+1l−1/summationdisplay\nj=0(−dt)i−(j+1)∂L\n∂qa(i)δqa(j)\n= 0.\nThe quantity within the square brackets is the conserved N¨ other currentJ. From its structure, we\nmay read off the conjugate momenta\np(j)\na≡l/summationdisplay\ni=j+1(−dt)i−(j+1)∂L\n∂qa(i), j= 0,...,l−1. (4)\nThe N¨ other current may then be cast into the standard form J=l−1/summationtext\nj=0p(j)\naδqa(j).\n2.2 Relativistic continuous systems\nHere, in place of the generalised coordinates, we shall be working wit h special relativistic fields hav-\ning infinite number of degrees of freedom. Let us consider a system of scalar fields φa(x). If\nL(φa,φa,ρ1,φa,ρ1ρ2,...,φa,ρ1...ρl) be the Lagrangian density (where φa,ρ1...ρj≡∂ρ1...∂ρjφa), then by ex-\ntremising the action S[φ(x)] =/integraltext\nd4xLwith respect to the fields φa(x), we get the equation of motion\nl/summationdisplay\nj=0(−1)j∂ρ1...∂ρj∂L\n∂φa,ρ1...ρj= 0. (5)\nThe necessary boundary conditions are that the fields and their de rivatives upto φρ1...ρl−1fall off at\ninfinity. It would be convenient if we could write down a general form o f the N¨ other current for such\nhigher derivative systems. This is achieved by applying a symmetry tr ansformation to the fields and\nconsequently employing the equation of motion (5) to get the N¨ oth er current,\nJρ1=l/summationdisplay\ni=j+1l−1/summationdisplay\nj=0(−1)i−(j+1)∂ρj+2...∂ρi∂L\n∂φa,ρ1...ρiδφρ2...ρj+1, (6)\nwhich is locally conserved i.e. ∂ρ1Jρ1= 0. It may so happen that instead of δLbeing zero, the\nLagrangian density varies by a total derivative. This would then con tribute to the N¨ other current.\nHowever, as we have already mentioned, a canonical formalism of re lativistic field theories requires\nus to foliate spacetime into spacelike hypersurfaces. This in turn inv olves the separation of temporal and\nspatial derivatives of the fields1. It is most often possible to arrange terms in the Lagrangian such t hat\nmixed derivatives of the fields as in ∂2\nt∂kφdo not survive. This is true not only when the Lagrangian is\nLorentz invariant [22] but also when the Lorentz violating action (1) is written in terms of the Lorentz\n1At this stage, it is imperative that we sort the indices. Lati n letters from the middle of the alphabet set viz. i,j,k,l are\nbeing used as summation indices while those from the end like r,s,...,z will denote spatial components. Different fields w ill\nbe labelled by the alphabets a,b,c,d. Greek letters are rese rved for spacetime indices.\n3preserving fields, as will be illustrated in the next section. In such sit uations, the canonical momenta will\nbe given by\nπa(j)≡l/summationdisplay\ni=j+1(−dt)i−(j+1)∂L\n∂φa(i), j= 0,...,l−1. (7)\nThe canonical variables will satisfy the Poisson bracket\n{φa(i)(t,/vector x),πb(j)(t,/vector x′)}=δb\naδj\niδ(3)(/vector x−/vector x′). (8)\n3 Myers Pospelov theory\n3.1 Constant timelike background vector\nWithout loss of generality, we are going to work in a Lorentz frame de fined by n= (1,/vector0). Then the\nLorentz preserving fields φ/bardbl(t),φ⊥(/vector x) become spatially homogeneous and static, spatially inhomogeneous\nrespectively. This greatly simplifies the Lagrangian density:\nLMPφ=˙φ∗\n/bardbl˙φ/bardbl−/vector∇φ∗\n⊥·/vector∇φ⊥+iκ\nMPl(φ∗\n/bardbl+φ∗\n⊥)...φ/bardbl, (9)\n=LMPφ(φ/bardbl,˙φ/bardbl,¨φ/bardbl,...φ/bardbl,φ∗\n/bardbl,˙φ∗\n/bardbl,φ⊥,/vector∇φ⊥,φ∗\n⊥,/vector∇φ∗\n⊥). (10)\nHere, we have neglected the masses of the fields as we are interest ed in behaviour of the system at\nenergiesmuch higher than the field masses. It is now evident why our chosenLorentz frame is particularly\nuseful. Eq.(9) has only higher order time derivatives of φ/bardbl(t) while/vector∇φ/bardbl= 0 =∂tφ⊥. Thus all mixed\nderivatives in the sense described above will vanish. This permits us t o safely use eq.(7) to determine the\ncanonical momenta. We list them in the following table.\nGeneralised Generalised\ncoordinate momentum\nφ/bardbl(0)=φ/bardblπ/bardbl(0)=˙φ∗\n/bardbl+iκ\nMPl¨φ∗\n/bardbl\nφ/bardbl(1)=˙φ/bardblπ/bardbl(1)=−iκ\nMPl˙φ∗\n/bardbl\nφ/bardbl(2)=¨φ/bardblπ/bardbl(2)=iκ\nMPl(φ∗\n/bardbl+φ∗\n⊥)\nφ∗\n/bardbl(0)=φ∗\n/bardblπ/bardbl∗(0)=˙φ/bardbl\nφ⊥(0)=φ⊥π⊥(0)= 0\nφ∗\n⊥(0)=φ∗\n⊥π⊥∗(0)= 0\nTable 1: Canonically conjugate phase space variables\nNote that π⊥(0)= 0 and π⊥∗(0)= 0 are constraint relations that will be imposed weakly. The equal\ntime Poisson Bracket, eq. (8), will hold for the canonical variables.\nUnder an infinitesimal Lorentz transformation, δαβφ=x[α∂β]φand the Lagrangian density being a\nscalar function, also changes δαβL=x[α∂β]L=∂ρ1(x[αδρ1\nβ]L). The Lorentz preserving fields conserve the\nN¨ other current Jµ\nαβ[15]. The N¨ other charge is given by\nQαβ=/integraldisplay\nΣd3σρ1Jρ1\nαβ.\n4Here, Σ is a three dimensional hypersurface. If we orient it orthog onal to the time axis then,\nQαβ=/integraldisplay\nΣd3/vector x/parenleftBig\nπ(0)\n/bardblδαβφ/bardbl(0)+π(1)\n/bardblδαβφ/bardbl(1)+π(2)\n/bardblδαβφ/bardbl(2)+π/bardbl∗δαβφ∗\n/bardbl−x[αδ0\nβ]L/parenrightBig\n,\n=/integraldisplay\nΣd3/vector x/parenleftBig\nπ(j)\naδαβφa(j)−x[αδ0\nβ]L/parenrightBig\n. (11)\nHereφa=φ/bardbl,φ∗\n/bardbl. The other variables do not contribute. From the structure of Qαβwe can deduce\nthat\n{Qαβ(t),φb(k)(t,/vector x)}=−δαβφb(k)(t,/vector x), for φ b=φ/bardbl,φ∗\n/bardbl;\n= 0, for φ b=φ⊥,φ∗\n⊥. (12)\nThe final step is to evaluate the algebra of the charges. A simple calc ulation using eq.,(11), the basic\nPoisson bracket (8) and the relation δαβφ=x[α∂β]φgives,\n{Qαβ(t),Qρσ(t)}=ηασQβρ(t)−ηαρQβσ(t)+ηβρQασ(t)−ηβσQαρ(t). (13)\nEqs. (12), (13) confirm that N¨ other charges defined in terms of the Lorentz preserving fields are\ngenerators of Lorentz transformation and satisfy the standar d Lorentz algebra. Thus, the special field\nconfigurations provide a valid basis for the realisation of Lorentz Lie algebra which determines the local\nstructure of Lorentz group near the identity.\n3.2 Constant spacelike background vector\nNext, we go over to the Lorentz frame where n= (0,/vector1). Then the Lorentz preserving fields will be\nφ/bardbl(/vector x),φ⊥(t) and the Lagrangian will take the form\nLMPφ=˙φ∗\n⊥˙φ⊥−/vector∇φ∗\n/bardbl·/vector∇φ/bardbl+iκ\nMPl(φ∗\n/bardbl+φ∗\n⊥)|/vector∇|3φ/bardbl. (14)\nThis Lagrangian density contains only first order time derivatives of the fields. Thus, no extra phase\nspace variables will be required. The table below lists the pairs of cano nical coordinates and momenta.\nGeneralised Generalised\ncoordinate momentum\nφ/bardblπ/bardbl= 0\nφ∗\n/bardblπ/bardbl∗= 0\nφ⊥ π⊥=˙φ∗\n⊥\nφ∗\n⊥ π⊥∗=˙φ⊥\nTable 2: Canonically conjugate phase space variables for n= (0,/vector1)\nThe fundamental Poisson Bracket is,\n{φa(t,/vector x),πb(t,/vector x′)}=δabδ(3)(/vector x−/vector x′). (15)\n5Integration of the zeroth component of the N¨ other current ov er an appropriately chosen three dimen-\nsional spatial slice normal to the time axis gives the N¨ other charge for Lorentz transformation,\nQαβ=/integraldisplay\nd3/vector x/parenleftBig\nπ⊥δαβφ⊥+π⊥∗δαβφ∗\n⊥−x[αδ0\nβ]L/parenrightBig\n. (16)\nThis is of the same form as the standard N¨ other current obtained for a first order system of scalar\nfields. Hence, the N¨ other charges would generate infinitesimal Lo rentz transformation and satisfy the\nLorentz algebra.\n4 Overview of covariant phase space formulation\nThe covariant construction of phase space keeps intact Poincar´ e invariance of a physical system [23, 24,\n25, 26, 27]. As opposed to the standard decomposition of phase sp ace in the 3+1 framework, the classical\ncovariant phase space Zof a physical theory is defined as the space of classical solutions of the dynamical\nequations of the theory. Functions φ(x), tangent vectors δφ, one forms δφ(x) and exterior derivatives δ\ncan be defined on Zfollowing [25, 26].\nThe phase space Zis naturally endowed with a closed, non-degenerate two-form ˜ ω( ˜ωis non-\ndegenerate provided the one-form ˜ ω(V\n/tildewide) = 0 if and only if V\n/tildewide= 0) called a symplectic structure. It\ncan be written as an integral of some closed, conserved symplectic two-form current ˜ ωµ(note that ˜ ωµis\na two-form in phase space Zand a vector current in spacetime; ∂µ˜ωµ= 0,δ˜ωµ= 0) over a hypersurface\nΣ,\n˜ω=/integraldisplay\nΣdσµ˜ωµ(17)\nHence, the task now is to find a suitable symplectic current, given an y Lagrangian. Owing to our\npresent interest in the covariant description of canonical formalis m for higher derivative theories [28], we\nshall straight away take the general example of a Lagrangian dens ityL(φa,φa,ρ1,φa,ρ1ρ2,...,φa,ρ1...ρl).\nForφabelonging to Zand an arbitrary linear transformation δφa(x) that takes φa(x)→φa(x)+δφa(x)\non the phase space, we have\nδL=∂µl/summationdisplay\ni=j+1l−1/summationdisplay\nj=0(−1)i−(j+1)∂ρj+2...∂ρi∂L\n∂φa,µρ2...ρiδφρ2...ρj+1\n=∂µjµ, (18)\nafter substituting the Euler Lagrange equation of motion (5). jµis interpreted as a pre-symplectic\ncurrent because it is used to define the symplectic current ˜ ωµ=δjµ. From eq. (18) one can see that\nit is obviously closed in phase space and conserved through its depen dance on spacetime. Hence, the\nsymplectic structure\n˜ω=/integraldisplay\nΣdσµ˜ωµ=δ/integraldisplay\nΣdσµjµ(19)\nis not only closed but also exact. Moreover, the local conservation of the symplectic current in\nspacetime guarantees that ˜ ωwill not change with the choice of the surface of integration Σ in (19) and\nin particular, will be Poincar´ e invariant [26].\n65 Symplectic structure with Lorentz preserving fields\nThecovariantversionofthe canonicalformalismismeaningful onlyw hen thephysicaltheoryhasPoincar´ e\ninvariance. The original Myers Pospelov Lorentz violating model doe sn’t meet this requirement. But\nwhen the allowed field configurations are restricted to the Lorentz preserving fields, the dynamics of\nthe theory becomes invariant under infinitesimal Lorentz transfo rmations enabling us to construct its\ncovariant phase space. Only in this context, eq. (18) holds for LMPφ.\nOne must have observed that the presymplectic current (18) and the N¨ other current (6) have identical\nforms. Infact, with theinterpretationof δφa(x) asaone-formonphasespace,the N¨ othercurrentbecomes\nthepre-symplecticcurrentoneform. Itmustalsobestressedth attheentireconstructionofthe symplectic\nstructure (19) follows from (18), the validity of which is ensured by the Lorentz preserving fields. This is\nturn guarantees the existence of a Lorentz invariant (and Poinca r´ e invariant) symplectic stucture on the\ncovariant phase space of Lorentz preserving solutions of the equ ation of motion of Myers Pospelov action\n[15].\nAcknowledgements\nI am grateful to D. Gangopadhyay and P. Majumder for suggestin g the problem and having numerous\ninsightful discussions. I also thank A. V. Sleptsov and P. I. Dunin-B arkowski for pointing out reference\n[29] to us.\nReferences\n[1]S. M. Carroll ,G. B. Field , andR. Jackiw ,Phys.Rev. D41, 1231 (1990).\n[2]J. I. Latorre ,P. 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Majumdar ,Europhys.Lett. 96, 61001 (2011).\n[16]B. Podolsky andP. Schwed ,Rev. Mod. Phys. 20, 40 (1948).\n7[17]M. V. Ostrogradskii ,Complete Collected Works , volume 2, Akad. Nauk Ukrain, SSR, Kiev,\n1961, in Russian.\n[18]D. Musicki ,J.Phys.A A11, 39 (1978).\n[19]J. Barcelos-Neto andN. R. Braga ,Acta Phys.Polon. B20, 205 (1989).\n[20]J. Z. Simon ,Phys.Rev. D41, 3720 (1990).\n[21]A. Morozov ,Theor.Math.Phys. 157, 1542 (2008).\n[22]F. de Urries andJ. Julve ,J.Phys.A A31, 6949 (1998).\n[23]G. J. Zuckerman ,Conf. Proc. C8607214 , 259 (1986).\n[24]E. Witten ,Nucl.Phys. B276, 291 (1986).\n[25]C. Crnkovic andE. Witten ,Covariant description of canonical formalism in geometric al theories ,\nchapter 16, pp. 676–684, Cambridge University Press, Cambridge , 1987.\n[26]C. Crnkovic ,Class.Quant.Grav. 5, 1557 (1988).\n[27]A. Ashtekar ,L. Bombelli , andO. Reula ,The covariant phase space of asymptotically flat\ngravitational fields , pp. 417–450, Elsevier Science Publishers B. V., North-Holland, Ams terdam\n(Netherlands), 1991.\n[28]V. Aldaya ,J. Navarro-Salas , andM. Navarro ,Phys.Lett. B287, 109 (1992).\n[29]P. Dunin-Barkowski andA. Sleptsov ,Theor.Math.Phys. 158, 61 (2009).\n8"    },    {        "title": "1507.04509v1.Lorentz_Dispersion_Law_from_classical_Hydrogen_electron_orbits_in_AC_electric_field_via_geometric_algebra.pdf",        "content": "Lorentz Dispersion Law from classical Hydrogen electron orbits\nin AC electric \feld via geometric algebra\nUzziel Perez\nNational Institute of Physics, University of the Philippines,\nDiliman, Quezon City, Philippines\nAngeleene S. Ang\nAteneo de Manila University, Department of Physics,\nLoyola Heights,Quezon City, Philippines 1108\u0003\nQuirino M. Sugon, Jr. and Daniel J. McNamara\nManila Observatory, Upper Atmosphere Division,\nAteneo de Manila University Campusy\nAkimasa Yoshikawa\nDepartment of Earth and Planetary Sciences, Faculty of Sciences,\nKyushu University, Fukuoka, Japan\n(Dated: July 2, 2018)\nWe studied the orbit of an electron revolving around an in\fnitely massive nucleus of a large\nclassical Hydrogen atom subject to an AC electric \feld oscillating perpendicular to the electron's\ncircular orbit. Using perturbation theory in geometric algebra, we show that the equation of motion\nof the electron perpendicular to the unperturbed orbital plane satis\fes a forced simple harmonic\noscillator equation found in Lorentz dispersion law in Optics. We show that even though we did\nnot introduce a damping term, the initial orbital position and velocity of the electron results to a\nsolution whose absorbed energies are \fnite at the dominant resonant frequency !=!0; the electron\nslowly increases its amplitude of oscillation until it becomes ionized. We computed the average power\nabsorbed by the electron both at the perturbing frequency and at the electron's orbital frequency.\nWe graphed the trace of the angular momentum vector at di\u000berent frequencies. We showed that at\ndi\u000berent perturbing frequencies, the angular momentum vector traces epicyclical patterns.\nPACS numbers: 45.10.Hj, 45.10.Na, 37.10.Vz\nI. INTRODUCTION\nIn standard optics texts, the position xof an electron\nof chargeqand massmunder the time-varying electric\n\feldE=Eei!tof light is given by[1{3]\nx+ \u0000 _x+!2\n0x=q\n2mE; (1)\nwhere \u0000 is the damping coe\u000ecient, !0is the natural fre-\nquency of oscillation of the electron. The complex solu-\ntion ~xto this equation is shown by Akhmanov and Nikitin\n[4] to be\n~x=q\nm1\n!2\n0\u0000!2+i!\u0000E; (2)\nso that the power absorbed by the atom is\nhPi=hqE_xi=q2\n2m!2\u0000\n(!2\n0\u0000!2)2+!2\u00002jEj2;(3)\n\u0003Electronic address: angeleene.ang@gmail.com\nyAlso at Ateneo de Manila University, Department of Physics,\nLoyola Heights,Quezon City, Philippines 1108which in complex space becomes\nP=e\n4i!\u0000\nE\u0003~x+E~xe2i!t\u0001\n: (4)\nNotice that the damping term \u0000 makes the power ab-\nsorbed \fnite at the resonant frequency !=!0.\nAt present it is still not clear why an atom can be\ndescribed as a simple harmonic oscillator subject to si-\nnusoidal electric \feld of light, e.g. what is responsible\nfor the restoring force constant k=m!2\n0and what is the\ncause of damping force \u0000 _ x?\nIn this paper we wish to show that a forced harmonic\noscillator equation can arise for a large Hydrogen atom\nwith a circular orbital of frequency !0subject to a lin-\nearly polarized light of frequency !whose corresponding\nwavelength 2 \u0019c=! is much larger than the electron's or-\nbital radius r0, e.g. microwave frequencies, so that the\nphase of light is approximately the same at any point in\nthe orbital path within a certain time period. That is, if\nthe electron is initially in circular orbit in the xy-plane,\nthe position sof the electron along the z-axis is given by\ns+!2\nos=\u0000qE\nmcos (!t); (5)\nas similarly given in Born and Wolf [5]. Even though\nthis equation does not have a damping term, we shallarXiv:1507.04509v1  [physics.atom-ph]  16 Jul 20152\nshow that the energy absorbed at the resonant frequency\n!=!0remains \fnite, provided we take into account the\nposition rof the electron in 3D and use the vector form\nof the electrical energy dissipation expression [1]:\nP=qE\u0001_ r: (6)\nWe shall show that hPiis \fnite at the resonant frequency\n!=!0even though there is no damping.\nIn 1974, Bay\feld and Koch experimentally studied the\nionization of hydrogen atoms under microwave frequen-\ncies [6]. Since then, many tried to study the interaction of\nmicrowave radiation with classical hydrogen atom within\nthe context of Rydberg atoms.[7] Some authors stud-\nied the interaction with circularly polarized light [8{12],\nwhile others such as Leopold [13], Grosfeld and Friedland\n[14] and Neishtadt [15] focused on the linearly polarized\ncase.\nFor Leopold, his Hamiltonian is of the form:\nH(~ r;~ p) =1\n2p2\u0000r\u00001+zFmaxcos!t; (7)\nand the equations are solved using Monte-Carlo tech-\nniques. For Grosfeld and Friedland, their Hamiltonian\nis of the form:\nH=1\n2p2\u0000r\u00001+Z\u0016cos \t; (8)\nwhere the frequency !(t) =d\t=dtand the authors used\naction angle variables. This Hamiltonian is the same one\nused by Neishtadt and Vasiliev, except that the latter\nauthors used Delaunay elements.\nIn our work, we shall not use the Hamiltonian ap-\nproach. Instead, we shall use the force equation\nr=\u0000kq2\nmr\njrj3\u0000\u0015e3q\nmE0cos(!t+'); (9)\nand use linear perturbation theory to simplify the equa-\ntion to a simple harmonic oscillator equation in (5) for a\nmotion perpendicular to the initial circular orbital plane\nof the electron. This method is simpler than those of the\nprevious authors because the solution to the simple har-\nmonic oscillator equation is well-known. Just as in the\noptical dispersion theory, we computed for the average\npower absorption by the atom and showed that it only\ndepends on the z-coordinate as in the standard theory.\nhPi\u001c=1\n\u001cZ\u001c\n0Pdt: (10)\nWe shall show that the orbit of the electrons at integral\nfrequency ratios are similar to De Broglie waves, except\nthat the oscillation is perpendicular to the electron's or-\nbital plane.\nWe shall divide the paper into six sections. Section 1\nis Introduction. In Section 2, we shall discuss the Ge-\nometric Algebra formalism applied to planar rotations.In Section 3, we shall describe the unperturbed circu-\nlar orbit of the electron around the nucleus. After this,\nwe shall introduce an oscillating electric \feld perturba-\ntion and derive the equations of motion of the electron's\noscillation perpendicular to its orbital plane, using the\ngeometric algebra framework in our previous paper on\nCopernican epicyclical orbits[16]. In Section 4, we shall\ncompute the electron's orbital angular momentum and\ndetermine its limiting form at the resonant frequency. In\nSection 5, we shall compute the electrical power dissi-\npation of the electron and plot the results for di\u000berent\nvalues of the ratio between the orbital and light frequen-\ncies. We shall show that the average power, either over\nthe perturbing or orbital period, is approximately simi-\nlar to the standard absorption resonance curve with \fnite\npeak. Finally, we graph the angular momentum of the\nelectron at di\u000berent frequency ratios and show that the\nangular momentum vector traces epicyclical patterns.\nII. GEOMETRIC ALGEBRA\nA. Scalars, Vectors, Bivectors, and Trivectors\nIn Cli\u000bord (Geometric) Algebra Cl3;0, also known as\nthe Pauli Algebra, the product of the three unit vectors\ne1,e2, ande3satis\fes the orthonormality relation [17{19]\nejek+ekej= 2\u000ejk; (11)\nwhere\u000ejkis the Kronecker delta function. In other words,\nthe square of the length of the vectors is equal to one and\nthe product of two perpendicular vectors anticommute.\nLetaandbbe two vectors spanned by e1,e2, and\ne3. We can show that their product satis\fes the Pauli\nidentity[20, 21]\nab=a\u0001b+i(a\u0002b); (12)\nwherei=e1e2e3is the unit trivector which behaves like\nan imaginary scalar that transforms vectors to bivectors.\nThe Pauli identity states that the geometric product of\ntwo vectors is equal to the sum of their scalar dot product\nand their imaginary cross product.\nB. Exponential Function and Rotations\nLetie3\u0012be the product of a bivector ie3=e1e2with\nthe scalar\u0012. Since the square of ie3\u0012is negative, then\nthe exponential of ie3\u0012is given by Euler's theorem\neie3\u0012= cos\u0012+ie3sin\u0012: (13)\nFrom this we can see that\ncos\u0012=1\n2(eie3\u0012+e\u0000ie3\u0012); (14a)\nsin\u0012=1\n2ie3(eie3\u0012\u0000e\u0000ie3\u0012); (14b)3\ne1eie3\u0012e2eie3\u0012e2\ne1\u0012\u0012\nFIG. 1: Rotation of e1ande2about e3counterclockwise by\nan angle\u0012\nwhich are the known exponential de\fnitions of cosine and\nsine functions.\nMultiplying Eq. (13) by e1,e2, and e3, we obtain\ne1eie3\u0012=e1cos\u0012+e2sin\u0012=e\u0000ie3\u0012e1; (15a)\ne2eie3\u0012=e2cos\u0012\u0000e1sin\u0012=e\u0000ie3\u0012e2; (15b)\ne3eie3\u0012=e3cos\u0012+e3ie3sin\u0012=eie3\u0012e3: (15c)\nNotice that e1eie3\u0012is a rotation of e1counterclockwise\nabout e3by an angle \u0012, while e2eie3\u0012is a rotation of e2\ncounterclockwise about the same direction and the same\nangle. Notice, too, that the argument of the exponen-\ntial changes sign when e1ore2trades places with the\nexponential, while e3commutes with the exponential.\nA vector ain 2D can be expressed in both rectangular\nand polar forms:\na=axe1+aye2=ae1eie3\u0012: (16)\nExpanding the exponential using Eq. (15a) and separat-\ning the e1ande2components, we arrive at the standard\ntransformation equations for polar to rectangular coor-\ndinates:\nx=acos\u0012; (17a)\ny=asin\u0012: (17b)\nWe may also factor out e1in Eq. (16) either to the left\nor to the right to get\na=e1^a=e1(x+ie3y) =e1aeie3\u0012; (18a)\na= ^a\u0003e1= (x\u0000ie3y)e1=ae\u0000ie3\u0012e1: (18b)\nFactoring out e1yields the de\fnition of the complex num-\nber ^aand that of its complex conjugate ^ a\u0003:\n^a=ax+ie3ay=aeie3\u0012; (19a)\n^a\u0003=ax\u0000ie3ay=ae\u0000ie3\u0012: (19b)In general, we have the following relations:\ne1^a= ^a\u0003e1; (20a)\ne2^a= ^a\u0003e2; (20b)\ne3^a= ^ae3: (20c)\nThat is, e1ande2both changes the complex number ^ a\nto its conjugate ^ a\u0003after commutation, while e3simply\ncommutes with ^ a[17{19, 22].\nIII. LIGHT-ATOM INTERACTION\nA. Unperturbed Electron Orbit\nClassically, the position rof an electron of mass m\nand charge\u0000qas it revolves around a massive proton of\nchargeqis given by Coulomb's law:\nr=\u0000kq2\nmr\njrj3; (21)\nwherekis the electrostatic force constant. We claim that\na solution to Eq. (21) is given by\nr=r0=e1^r0^ 0; (22)\nwhere\n^r0=r0eie3'0(23a)\n^ 0=eie3!0t(23b)\nare the complex amplitude and the rotation operator,\nrespectively. Substituting these back to Eq. (22), we get\nr0=e1r0eie3(!0t+'0); (24)\nwhich yields\nr0=e1r0cos(!0t+'0) +e2r0sin(!0t+'0);(25)\nafter expanding the exponential and distributing e1.\nEquation (22) states that the electron moving around the\nproton in circular orbit of radius r0with angular velocity\n!0and rotational phase angle '0.\nTo verify that Eq. (22) is indeed a solution to the\nCoulomb's law in Eq. (21), we \frst compute the \frst\nand second time derivatives of Eq. (22):\n_r=e1^r0ie3!o^ 0; (26a)\nr=\u0000e1^r0!2\n0^ 0: (26b)\nNow, substituting Eqs. (22) and (26b) to the Coulomb's\nlaw in Eq. (21), we obtain\n!2\no=kq2\nmr3\n0; (27)\nafter cancelling out e1^r0^ 0. Equation (27) is the familiar\ncircular orbit condition.4\nr0e1eie3('+!t)\nr0e1eie3'\nr0e1'!t\nFIG. 2: Uniform circular motion of an electron with a distance\nr0from the nucleus. The orbital angular frequency is !and\nthe phase angle is '.\n+jqj\n\u0000jqjFcFp\nv0r0\nFIG. 3: The Coulomb force Fcand the perturbing force Fp\non an electron moving with velocity v0and radius r0\nB. Perturbation by an Oscillating Field\nSuppose that the electron is subject not only to the\nCoulomb force\u0000qEcdue to the proton, but also to the\nforce\u0000qEpdue to an oscillating perturbing \feld. The\nequation of motion of the electron then becomes\nmr=\u0000qEc\u0000\u0015qEp; (28)\nwhere\u0015is a perturbation parameter that shall later be\nset equal to unity. More speci\fcally, we write\nr=\u0000kq2\nmr\njrj3\u0000\u0015e3q\nmE0cos(!t+'); (29)\nwhereE0,!, and'are the amplitude, angular frequency,\nand phase of the perturbing electric \feld. Our aim is\nto determine the position rof the electron that satis\fes\nEq. (29).\nTo \fnd the solution to the perturbed orbit equation\nin Eq. (29), we assume that the solution is a sum of the\nelectron's unperturbed circular orbit in Eq. (22) and a\nslight perturbation ^ sperpendicular to this orbit. So we\nwrite\nr=r0+\u0015r1=e1^r0^ 0+\u0015e3^s: (30)The \frst and second time derivatives of rare\n_r=_r0+\u0015_r1=e1^r0ie3!0^ 0+\u0015e3_^s; (31a)\n r= r0+\u0015 r1=\u0000e1^r0!2\no^ 0+\u0015e3^s: (31b)\nEquation (31b) shall take care of the left side of Eq. (29).\nTo expand the right-hand side, we need \frst to take\nthe square of the position vector rin Eq. (30) and retain\nonly the terms up to \frst order in \u0015:\nr2=r2\n0+ 2\u0015(r0\u0001r1): (32)\nSince r0lies on the unperturbed orbital plane of the elec-\ntronxyplane and r1=e3is perpendicular to this plane,\nthenr0\u0001r1= 0, so that Eq. (32) reduces to\nr2=r2\n0=e1^r0^ 0e1^r0^ 0=e1e1^r\u0003\n0^ \u0003\n0^r0^ 0\n= ^r\u0003\n0^r0=r2\n0; (33)\nwhere we used the de\fnitions of ^ r0and ^ 0in Eqs. (23a)\nand (23b). Thus, jrj=r0, so that\nr\njrj3=1\nr3\n0e1^r0^ 0: (34)\nEquation (34) shall take care of the Coulomb term on the\nright side of Eq. (29).\nNow, substituting Eqs. (31b) and (34) back to equation\nof motion in Eq. (29), we obtain\n\u0000e1^r0!2\no^ 0+\u0015e3s=\u0000!2\n0(e1^r0^ 0+\u0015e3^s)\n\u0000\u0015e3q\nmE0cos(!t+');(35)\nwhere we used the circular orbit condition in Eq. (27).\nThe term zeroth order in \u0015cancels out, so we are left\nwith the term \frst order in \u0015. Hence,\n^s+!2\n0^s=\u0000q\nmE0cos(!t+'); (36)\nafter rearranging the terms. Notice that Eq. (36) is a\nsimple harmonic oscillator equation with sinusoidal forc-\ning, which is the standard model for classical light-atom\ninteraction.\nC. Solving the Forced Harmonic Oscillator\nEquation\nLet ^shand ^spbe the homogeneous and particular so-\nlutions of the Eq. (36). That is,\n^s= ^sh+ ^sp; (37)\nand\n^sh+!2\n0^sh= 0; (38)\n^sp+!2\n0^sp=\u0000q\nmE0cos(!t+'): (39)5\nThe solution to the homogenous equation in Eq. (38)\nis a sum of a sines and cosines:\n^sh=ch1cos(!0t) +ch2sin(!0t); (40)\nwherech1andch2are scalar constants that will be de-\ntermined from the boundary conditions. On the other\nhand, the solution to the particular equation in Eq. (39)\nis of the same form as the perturbing \feld:\n^sp=cpcos(!t+'); (41)\nwherecpis a scalar constant. Substituting Eq. (41) back\nto the particular equation in Eq. (39) and solving for cp,\nwe get\ncp=qE0\nm!2\n01\n(\u000b2\u00001); (42)\nwhere\n\u000b=!\n!0(43)\nis the ratio of the perturbing frequency !to the electron's\norbital frequency !0. Hence,\n^sp=q\nm!2\n01\n(\u000b2\u00001)E0cos(\u000b!0t+'): (44)\nAdding the homogenous solution ^ shin Eq. (40) to the\nparticular solution ^ spin Eq. (41) yields the total solution:\n^s=ch1cos(!0t) +ch2sin(!0t)\n+q\nm!2\n01\n(\u000b2\u00001)E0cos(\u000b!0t+'): (45)\nIts time derivative is\n_^s=\u0000ch1!0sin(!0t) +ch2!0cos(!0t)\n\u0000q\nm! 0\u000b\n(\u000b2\u00001)E0sin(\u000b!0t+'): (46)\nTo determine the unknown constants ch1andch2, we\n\frst substitute the expressions for sand _sin Eqs. (45)\nand (46) back to the expressions for the position rand\nvelocity _rin Eqs. (30) and (31a) to get\nr=e1^r0^ 0+e3(ch1cos(!0t) +ch2sin(!0t))\n+e3q\nm!2\n01\n(\u000b2\u00001)E0cos(\u000b!0t+'); (47a)\n_r=e1^r0ie3!0^ 0+e3(\u0000ch1!0sin(!0t) +ch2!0cos(!0t))\n\u0000e3q\nm! 0\u000b\n(\u000b2\u00001)E0sin(\u000b!0t+'); (47b)\nafter setting the perturbation parameter \u0015= 1. If we\nassume that at t= 0, the electron is in its unperturbed\ncircular orbit around the nucleus, then\nr(0) = e1^r0; (48a)\n_r(0) = e1^r0ie3!0: (48b)Substituting these to Eqs. (47a) and (47b), and setting\nt= 0, we arrive at the expressions for the parameters ch1\nandch2:\nch1=\u0000q\nm!2\n01\n(\u000b2\u00001)E0cos'; (49a)\nch2=q\nm!2\n01\n(\u000b2\u00001)E0sin': (49b)\nSubstituting Eqs. (49a) and (49b) back to the expres-\nsion for the position rin Eq. (47a), we get\nr=e1^r0^ 0+e3qE0\nm!2\n01\n(\u000b2\u00001)(\u0000cos'cos(!0t)\n+ sin'sin(!0t) + cos(\u000b!0t+')): (50)\nUsing the identity for the cosine of a sum of two angles,\nEq. (50) reduces to\nr=e1^r0^ 0+e3qE0\nm!2\n01\n(\u000b2\u00001)\u0002\n(cos(\u000b!0t+')\u0000cos(!0t+')): (51)\nIts time derivative is\n_r=e1^r0ie3!0^ 0+e3qE0\nm! 01\n(\u000b2\u00001)\u0002\n(\u0000\u000bsin(\u000b!0t+') + sin(!0t+'); (52)\nwhere we used the de\fnition of \u000b=!=! 0. Equations (51)\nand (52) are the position and velocity of the electron\ninitially orbiting at radius r0, angular frequency !0, and\nphase'0, and perturbed by an oscillating electric \feld\nwith amplitude E0, frequency !, and phase '.\nD. Orbit Equations and Limiting Conditions\nTo convert Eqs. (51) and (52) into rectangular coor-\ndinates, we use the expansions in Eq. (15a) and (15b),\ntogether with the identity e1ie3=e2to arrive at\nx=r0cos(!0t+'0); (53a)\ny=r0sin(!0t+'0); (53b)\nz=qE0\nm!2\n01\n(\u000b2\u00001)(cos(\u000b!0t+')\u0000cos(!0t+'));\n(53c)\nand\n_x=\u0000r0!0sin(!0t+'0); (54a)\n_y=r0!0cos(!0t+'0); (54b)\n_z=qE0\nm! 01\n(\u000b2\u00001)(\u0000\u000bsin(\u000b!0t+') + sin(!0t+')):\n(54c)\nEquations (53a) to (53c) are the equations for plotting\nthe orbit of the electron as a function of time. Equa-\ntions (54a) to (54c) are for plotting the corresponding\nvelocities.6\nFIG. 4: The unperturbed orbit lies \rat along the e1\u0000e2plane. When \u000b= 0, or when the electric \feld is constant in time, the\norbit slants. When \u000b!1 , more waves are observed.\nWhen the perturbing frequency != 0, corresponding\nto\u000b= 0, the expressions for zand _zin Eqs. (53c) and\n(54c) reduces to\nz=\u0000qE0\nm!2\n0(cos'\u0000cos(!0t+')); (55a)\n_z=\u0000qE0\nm! 0sin(!0t+'): (55b)\nIf'= 0, the perturbing \feld is E=E0e3, so that\nz=\u0000qE0\nm!2\n0(1\u0000cos(!0t)); (56a)\n_z=\u0000qE0\nm! 0sin(!0t): (56b)\nOn the other hand, if '=\u0019, the perturbing \feld is E=\nE0e3, so that\nz=qE0\nm!2\n0(1 + cos(!0t)); (57a)\n_z=qE0\nm! 0sin(!0t): (57b)\nThese are the behavior of the electron's orbit along the\nz\u0000direction when the perturbing electric \feld is constant,\nalso known as the DC electric \feld.\nNow, when the perturbing frequency !=!0, corre-\nsponding to \u000b= 1, the \feld resonates with the electron'sorbit. The only terms a\u000bected are zand _zin Eqs. (53c)\nand (54c). Since both their numerators and denomina-\ntors approach zero as \u000b!1, we apply L'hopital's rule by\ndi\u000berentiating the numerators and denominators prior to\nevaluation of the limits:\nlim\n\u000b!1z=qE0\nm!2\n0lim\n\u000b!1\u0012\n\u0000!0tsin(\u000b!0t)\n2\u000b\u0013\n; (58a)\nlim\n\u000b!1_z=\u0000qE0\nm! 0lim\n\u000b!1\u0012sin(\u000b!0t+')\n2\u000b\u0013\n\u0000qE0\nm! 0lim\n\u000b!1\u0012\u000b!0tcos(!t+')\n2\u000b\u0013\n: (58b)\nHence,\nlim\n\u000b!1z=\u0000qE0\n2m! 0tsin(\u000b!0t); (59a)\nlim\n\u000b!1_z=\u0000qE0\n2m! 0(sin(!0t+') +!0tcos(!0t+')):\n(59b)\nNotice that the amplitude of the oscillations along zand\nits corresponding velocity are linearly increasing in time.\nOnce the amplitudes of the oscillations becomes so large,\nour perturbation approximations breaks down. Thus, our\ntheory cannot really say what happens during ionization\nor whether ionization will really happen at all at the res-\nonant frequency. (See Figs. 4 and 5)7\n\u000b= 1\n\u000b= 0:5\n\u000b= 2\ntz\nFIG. 5: Height zof the electron from its unperturbed circular\norbit with respect to time.\nIV. ANGULAR MOMENTUM\nA. Product Form\nLet us compute the product of the position rin\nEq. (30) and its velocity _rin Eq. (31a):\nr_r= (e1^r0^ 0+\u0015e3^s)(e1^r0ie3!0^ 0+\u0015e3_^s): (60)\nDistributing the terms, we get\nr_r=e1e1^r\u0003\n0^ \u0003\n0^r0ie3!0^ 0\n+e3e1(^s^r0ie3!0^ 0\u0000_^s^r\u0003\n0^ \u0003\n0) +e3e3s_^s; (61)\nafter setting the perturbation parameter \u0015= 1. Since\ni=e1e2e3andie3=e1e2, then Eq. (61) reduces to\nr_r=ie3!0r2\n0\u0000ie1s^r0!0^ 0\u0000ie2s^r\u0003\n0^ \u0003\n0+ ^s_^s: (62)\nSeparating the scalar and bivector parts of Eq. (62), we\narrive at\nr\u0001_r= ^s_^s; (63a)\nr\u0002_r=e3!0r2\n0\u0000e1s^r0!0^ 0\u0000e2s^r\u0003\n0^ \u0003\n0; (63b)\nafter factoring out the trivector iin the second equation.\nMultiplying Eq. (63b) by the electron's mass myields\nthe the electron's angular momentum:\nL=mr\u0002_r=e3!0r2\n0\u0000e1s^r0!0^ 0\u0000e2s^r\u0003\n0^ \u0003\n0;(64)\nwheresand _sare the e3components of the electron's\nposition rand velocity _rin Eqs. (51) and (52):\ns=qE0\nm!2\n01\n(\u000b2\u00001)(cos(\u000b!0t+')\u0000cos(!0t+'));\n(65a)\n_s=qE0\nm! 01\n(\u000b2\u00001)(\u0000\u000bsin(\u000b!0t+') + sin(!0t+'):\n(65b)Using the de\fnitions ^ r0=r0eie3'and ^ 0=eie3!0tin\nEq. (64), and separating the e1,e2, and e3components,\nwe arrive at\nL1=\u0000m! 0sr0cos(!0t+'0)\u0000m_sr0sin(!0t+'0);\n(66a)\nL2=\u0000m! 0sr0sin(!0t+'0)\u0000m_sr0cos(!0t+'0);\n(66b)\nL3=m! 0r2\n0; (66c)\nwhich are the parametric expressions for the angular mo-\nmentum in rectangular coordinates.\nB. Harmonic Form\nThe vertical oscillation sand its derivative _ sin\nEqs. (65a) and (65b) may be expressed in exponential\nforms:\ns=qE0\n2m!2\n01\n(\u000b2\u00001)(eie3(\u000b!0t+')+e\u0000ie3(\u000b!0t+')\n\u0000eie3(!0t+')\u0000e\u0000ie3(!0t+'));(67a)\n_s=qE0\n2m! 0\u0000ie3\n(\u000b2\u00001)(\u0000\u000beie3(\u000b!0t+')+\u000be\u0000ie3(\u000b!0t+')\n+eie3(!0t+')\u0000e\u0000ie3(!0t+')):(67b)\nThese may be rewritten as\ns=qE0\n2m!2\n01\n(\u000b2\u00001)(^\u0011^ \u000b\n0+ ^\u0011\u0003^ \u0000\u000b\n0\n\u0000^\u0011^ 0\u0000^\u0011\u0003^ \u00001\n0); (68a)\n_s=qE0\n2m! 0\u0000ie3\n(\u000b2\u00001)(\u0000\u000b^\u0011^ \u000b\n0+\u000b^\u0011\u0003^ \u0000\u000b\n0\n+ ^\u0011^ 0\u0000^\u0011\u0003^ \u00001\n0); (68b)\nwhere\n^\u0011=eie3': (69)\nSubstituting Eqs. (68a) and (68b) back to Eq. (64) and\nnoting that e2ie3=\u0000e1, we obtain\nL=e3m! 0r2\n0\u0000e1qE0\n2!01\n(\u000b2\u00001)^ L; (70)\nwhere\n^ L= ^\u0011^r0^ \u000b+1\n0+ ^\u0011\u0003^r0^ \u0000\u000b+1\n0\u0000^\u0011^r0^ 2\n0\u0000^\u0011\u0003^r0\n\u0000\u000b^\u0011^r\u0003\n0^ \u000b\u00001\n0+\u000b^\u0011\u0003^r\u0003\n0^ \u0000\u000b\u00001\n0 + ^\u0011^r\u0003\n0\u0000^\u0011\u0003^r\u0003\n0^ \u00002\n0:\n(71)\nExpanding the terms of ^ Linto exponential form, we get\n^ L=r0(eie3((\u000b+1)!0t+'+'0)+eie3((\u0000\u000b+1)!0t\u0000'+'0)\n\u0000eie3(2!0t+'+'0)\u0000eie3(\u0000'+'0)\n\u0000\u000beie3((\u000b\u00001)!0t+'\u0000'0)+\u000beie3(\u0000(\u000b+1)!0t\u0000'\u0000'0)\n+eie3('\u0000'0)\u0000eie3(\u00002!0t\u0000'\u0000'0)); (72)8\nSubstituting the result back to Eq. (70) and separating\nthee1,e2, and e3components, we arrive at\nL1=\u0000qE0r0\n2!01\n(\u000b2\u00001)\r1; (73a)\nL2=\u0000qE0r0\n2!01\n(\u000b2\u00001)\r2; (73b)\nL3=m! 0r2\n0; (73c)\nwhere\n\r1= (1 +\u000b) cos((\u000b+ 1)!0t+'+'0)\n+ (1\u0000\u000b) cos((\u000b\u00001)!0t+'\u0000'0)\n\u00002 cos(2!0t+'+'0); (74a)\n\r2= (1\u0000\u000b) sin((\u000b+ 1)!0t+'+'0)\n\u0000(1 +\u000b) sin((\u000b\u00001)!0t+'\u0000'0)\n+ 2 sin('\u0000'0): (74b)\nThus, since \u000b=!=! 0, we see that the orbit of the tip of\nthe angular momentum vector Lis a linear combination\nof circular motions with the following orbital frequencies:\n!L=f\u0006(!+!0);\u0006(!\u0000!0);\u00062!0;0g: (75)\nC. Limiting Conditions\nIn the DC \feld limit, \u000b!0, so that\nlim\n\u000b!0L1=qE0r0\n2!0(cos(!0t+'+'0)\n+ cos(\u0000!0t+'\u0000'0)\n\u00002 cos(2!0t+'+'0)); (76a)\nlim\n\u000b!0L2=qE0r0\n2!0(sin(!0t+'+'0)\n\u0000sin(\u0000!0t+'\u0000'0)\n+ 2 sin('\u0000'0)): (76b)\nOn the other hand, in the resonance frequency limit, \u000b!\n1, both the numerators and denominators of L1andL2\napproach zero, so that we apply the L'hopital's rule:\nlim\n\u000b!0L1=\u0000qE0r0\n4!0(cos(2!0t+'+'0)\n\u00002!0tsin(2!0t+'+'0)\u0000cos('\u0000'0));\n(77a)\nlim\n\u000b!0L2=\u0000qE0r0\n4!0(\u0000sin(2!0t+'+'0)\n\u0000sin('\u0000'0)\u00002!0tcos('\u0000'0)):(77b)\nNotice that at the resonant frequency !=!0, theL1\nandL2components of the angular momentum increases\nin time; in our perturbative approximation, the atom will\nbe ionized.V. POWER AND ENERGY ABSORPTION\nA. Work-Energy Theorem\nThe work-energy theorem states that\nZr\nr0F\u0001dr=1\n2mv2\u00001\n2mv02: (78)\nThis may be rewritten as\nZt\n0Pdt =1\n2mv2\u00001\n2mv02; (79)\nwhere the power Pis de\fned as\nP=F\u0001_r: (80)\nThat is, the integral of the power Pexpended by a force\nFacting to move a mass mfrom timet= 0 totis equal\nto the change in the mass's kinetic energy between these\ntimes.\nIn our model, there are two forces acting on the elec-\ntron: the Coulomb force Fcand the perturbing force Fp.\nBut since the sum of these two forces is mr, as given in\nEq. (28), then the left side of Eq. (79) becomes\nP=mr\u0001_r: (81)\nWe shall use this equation to compute the power ab-\nsorbed by the atom.\nB. Power: Product Form\nTo evaluate the left side of Eq. (81), we \frst multiply\nthe expressions for rand_rin Eqs. (31b) and (31a):\nr_r= (\u0000e1^r0!2\n0^ 0+\u0015e3s)(e1^r0ie3!0^ 0+\u0015e3_s) (82)\nDistributing the terms, we get\nr_r=\u0000e1e1^r\u0003\n0!2\n0^ \u0003\n0^r0ie3!0^ 0+e3e1s^r0ie3!0^ 0\n\u0000e1e3^r0!2\n0^ 0_s+e3e3s_s; (83)\nafter setting \u0015= 1. Since i=e1e2e3andie3=e1e2,\nthen Eq. (83) reduces to\nr_r=\u0000ie3r3\n0!3\n0\u0000ie1_s^r0!0^ 0+ie2_s^r0!2\n0^ 0+ s_s:(84)\nSeparating the scalar and bivector parts, we get\nr\u0001_r= s_s; (85a)\nr\u0002_r=\u0000e3r3\n0!3\n0\u0000e1_s^r0!0^ 0+e2_s^r0!2\n0^ 0;(85b)\nafter factoring out iin the second equation.\nEquation (85a) leads to a very simple expression for\nthe power absorbed by the atom:\nP=mr\u0001_r=ms_s: (86)9\nFIG. 6: Projection of the tip of the angular momentum vector Lin thexyplane for di\u000berent values of \u000b.\nTaking the time derivative of _ sin Eq. (65b),\ns=qE0\nm1\n(\u000b2\u00001)(\u0000\u000b2cos(\u000b!0t+') + cos(!0t+'));\n(87)\nand substituting this and that of _ sto Eq. (86), we get\nP=q2E2\n0\n2m!2\n01\n(\u000b2\u00001)2\u0002\n(\u0000\u000b2cos(\u000b!0t+') + cos(!0t+'))\u0002\n(\u0000\u000bsin(\u000b!0t+') + sin(!0t+')): (88)\nWe can show that this is equivalent to\nP=q2E2\n0\n2m! 01\n(\u000b2\u00001)2\u0000\n\u000b3sin(2\u000b!0t+ 2')\n\u0000(\u000b2+\u000b) sin((\u000b+ 1)!0t+ 2')\n+ (\u000b2\u0000\u000b) sin((\u000b\u00001)!0t)\n+ + sin(2!0t+ 2')); (89)\nwhich is the desired harmonic form of the power Pab-\nsorbed by the atom. Notice that power is not constant\nbut \ructuating in time.\nhPi\u001c\n\u000bhPi\nFIG. 7: This is the graph of the average power over the per-\nturbing period10\nC. Average Power over Perturbing Period\nLet us de\fne the average power over the perturbing\nperiod as\nhPi\u001c=1\n\u001cZ\u001c\n0Pdt; (90)\nwhere\n\u001c=2\u0019\n!=2\u0019\n\u000b!0: (91)\nNow, let us take the time average of the power Pin\nEq. (98) over the perturbing period:\nhPi\u001c=q2E2\n0\n2m! 01\n(\u000b2\u00001)2\u0000\n\u000b3hsin(2\u000b!0t+ 2')i\u001c\n\u0000(\u000b2+\u000b)hsin((\u000b+ 1)!0t+ 2')i\u001c\n+ (\u000b2\u0000\u000b)hsin((\u000b\u00001)!0t)i\u001c\n+hsin(2!0t+ 2')i\u001c); (92)\nwhere\nhsin(2\u000b!0t+ 2')i\u001c\n= 0; (93a)\nhsin((\u000b+ 1)!0t+ 2')i\u001c\n=\u00001\n2\u0019\u000b\n(\u000b+ 1)(cos((1 + 1=\u000b)2\u0019+ 2')\u0000cos(2'))\n(93b)\nhsin((\u000b\u00001)!0t)i\u001c\n=\u00001\n2\u0019\u000b\n(\u000b\u00001)(cos((1\u00001=\u000b)2\u0019)\u00001); (93c)\nhsin(2!0t+ 2')i\u001c\n=1\n2\u0019\u000b(cos(4\u0019=\u000b+ 2')\u0000cos(2')): (93d)\nSubstituting these back to Eq. (98), we get\nhPi\u001c=q2E2\n0\n4\u0019m! 01\n(\u000b2\u00001)2\u0002\n(\u0000\u000b2(cos((1 + 1=\u000b)2\u0019+ 2')\u0000cos(2'))\n\u0000\u000b2(cos((1\u00001=\u000b)2\u0019)\u00001)\n+\u000b(cos(4\u0019=\u000b+ 2')\u0000cos(2')): (94)\nIf'= 0, we have\nhPi\u001c=q2E2\n0\n4\u0019m! 01\n(\u000b2\u00001)2\u0002\n(\u0000\u000b2(cos((1 + 1=\u000b)2\u0019)\u00001)\n\u0000\u000b2(cos((1\u00001=\u000b)2\u0019)\u00001)\n+\u000b(cos(4\u0019=\u000b)\u00001)): (95)\nEquation (95) is graphed in Fig. 7. Notice that despite\nthe small oscillations in the interval \u000b= 0 and\u000b= 1,\nthe power absorption curve is similar to that in Lorentz\ndispersion theory.\nhPi\u001c0\n\u000bhPi\nFIG. 8: This is the graph of the average power over the orbital\nperiod\nD. Average Power over Orbital Period\nLet us de\fne the average power over the perturbing\nperiod as\nhPi\u001c0=1\n\u001c0Z\u001c0\n0Pdt; (96)\nwhere\n\u001c0=2\u0019\n!0: (97)\nNow, let us take the time average of the power Pin\nEq. (98) over the orbital period:\nhPi\u001c0=q2E2\n0\n2m! 01\n(\u000b2\u00001)2(\u000b3hsin(2\u000b!0t+ 2')i\u001c0\n\u0000(\u000b2+\u000b)hsin((\u000b+ 1)!0t+ 2')i\u001c0\n+ (\u000b2\u0000\u000b)hsin((\u000b\u00001)!0t)i\u001c0\n+hsin(2!0t+ 2')i\u001c0); (98)\nwhere\nhsin(2\u000b!0t+ 2')i\u001c0\n=\u00001\n2\u0019cos(4\u0019\u000b+ 2')\u0000cos(2')\n2\u000b; (99a)\nhsin((\u000b+ 1)!0t+ 2')i\u001c0\n=\u00001\n2\u0019cos(2\u0019(\u000b+ 1))\u0000cos(2')\n\u000b+ 1; (99b)\nhsin((\u000b\u00001)!0t)i\u001c0\n=\u00001\n2\u0019cos(2\u0019\u000b)\u00001\n(\u000b\u00001); (99c)\nhsin(2!0t+ 2')i\u001c0\n= 0: (99d)11\nSubstituting these back to Eq. (98), we get\nhPi\u001c0=q2E2\n0\n8\u0019m! 01\n(\u000b2\u00001)2\u0002\n(\u0000\u000b2(cos(4\u0019\u000b+ 2')\u0000cos(2'))\n\u00002\u000b(cos(2\u0019(\u000b+ 1))\u0000cos(2'))\n\u00002\u000b(cos(2\u0019\u000b)\u00001)): (100)\nIf'= 0, we have\nhPi\u001c0=q2E2\n0\n8\u0019m! 0\u00001\n(\u000b2\u00001)2(\u000b2(cos(4\u0019\u000b)\u00001)\n\u00002\u000b(cos(2\u0019(\u000b+ 1))\u00001)\u00002\u000b(cos(2\u0019\u000b)\u00001)):\n(101)\nEquation (101) is graphed in Fig. 8. Notice that unlike\nin Fig. 7, there are periodic oscillations after \u000b= 2.\nVI. CONCLUSION AND RECOMMENDATION\nWe modelled the classical Hydrogen atom as an elec-\ntron revolving in circular orbit around an immovable pro-\nton subject to the Coulomb force. We subjected this\natom to a perturbing oscillating electric \feld perpendicu-\nlar to the electron's initial orbital plane. We showed that\nthe resulting equations of motion of the electron along\nthe axis of the perturbing electric \feld is similar to that\nof a simple harmonic oscillator with sinusoidal forcing.\nFurthermore, the absorbed energy averaged over the pe-\nriod of the perturbing \feld or over the orbital frequencyof the electron is approximately similar to a resonance\ncurve with one dominant frequency with \fnite peak at\n!=!o; other small resonance peaks occur to the left or\nto the right of the major resonant frequency.\nThe Lorentz dispersion model of the light-atom inter-\naction assumes that the electron is subject to Hooke's\nforce and the force due to the oscillating electric \feld\nof the light. Interestingly, even if our initial assumption\nis an electron in circular orbit around the nucleus, we\nstill obtained the same forced harmonic oscillator equa-\ntion as that of the standard model. We also obtained\nthe same resonant frequency, though the actual peak is\nat a frequency slightly smaller than !0. But what is new\nis that even though we did not put a damping term in\nour harmonic oscillator equation, we still obtained a \f-\nnite energy absorption at the resonant frequency !. We\nalso computed the electron's angular momentum vector\nand showed that its tip traces rosette patterns similar to\nepicycles[23].\nIn the future work, we shall extend our work to the in-\nteraction of the hydrogen atom with elliptically polarized\nradiation.\nAcknowledgements\nThis work was supported by the Loyola Schools Schol-\narly Work Faculty Grants of the Ateneo de Manila Uni-\nversity.\n[1] J. Jackson, Classical Electrodynamics (Wiley, 1975).\n[2] A. Zangwill, Modern Electrodynamics (Cambridge Uni-\nversity Press, 2013).\n[3] M. Klein and T. Furtak, Optics (John Wiley and Sons,\n1986).\n[4] S. Akhmanov and S. Nikitin, Physical Optics (Oxford\nUniversity Press, 1997).\n[5] M.Born and E. Wolf, Principles of Optics (Oxford Uni-\nversity Press, 1964).\n[6] J. Bay\feld and P. Koch, Physical Review Letters 33, 258\n(1974).\n[7] H. Haken and H. C. Wolf, The Physics of Atoms\nand Quanta: Introduction to Experiments and Theory\n(Springer-Verlag, 2000).\n[8] D. Cole and Y. Zou, Journal of Scienti\fc Computing 21,\n145 (2004).\n[9] A. Brunello, D. Farelly, and T. Uzer, Physical Review A\n55, 3730 (1996).\n[10] D. Farelly and T. Uzer, Physical Review 74, 1720 (1995).\n[11] D. Farelly and J. Gri\u000eths, Physical Review A 45, R2678\n(1992).\n[12] M. Gajda, B. Piraux, and K. Rzazweski, Physical Review\nA50, 2528 (1994).\n[13] J. Leopold and I. Percival, Physical Review Letters 41,\n944 (1978).[14] E. Grosfeld and L. Friedland, Physical Review Letters E\n65, 1 (2002).\n[15] A. Neishtadt and A. Vasiliev, Physical Review Letters E\n71, 1 (2005).\n[16] Q. M. Sugon Jr., S. Bragais, and D. J. McNamara, Coper-\nnicus's epicycles from newton's gravitational force law via\nlinear perturbation theory in geometric algebra (2008),\nmath/0807.2708v1.\n[17] T. Vold, Am. J. Phys. 61, 505 (1993).\n[18] D. Hestenes, Am. J. Phys. 71, 104 (2003).\n[19] Q. M. Sugon Jr. and D. J. McNamara, Am. J. Phys. 72,\n104 (2004).\n[20] W. E. Baylis, Cli\u000bord (Geometric) Algebras with Ap-\nplications in Physics, Mathematics, and Engineering\n(Birkh auser, 1996).\n[21] P. Lounesto, Cli\u000bord Algebra and Spinors (Cambridge\nUniversity Press, 2001).\n[22] B. Jancewicz, Multivectors and Cli\u000bord Algebras in Elec-\ntrodynamics (World Scienti\fc, 1989).\n[23] G. Gallavotti, ATTI-Accademia Nazionale Dei Lincei\nRendiconti Lincei Classe di Scienze Fisiche Matemaiche\ne Naturali Serie 9 Matematica e Applicazioni 12, 125\n(2001)."    },    {        "title": "1211.4781v1.Damping_rates_of_surface_plasmons_for_particles_of_size_from_nano__to_micrometers__reduction_of_the_nonradiative_decay.pdf",        "content": "arXiv:1211.4781v1  [physics.optics]  20 Nov 2012Damping rates of surface plasmons for particles of size from nano- to\nmicrometers; reduction of the nonradiative decay\nK. Kolwas∗∗, A. Derkachova∗\nInstitute of Physics, Polish Academy of Sciences, Al. Lotni k´ ow 32/46, Warszawa, Poland\nAbstract\nDamping rates of multipolar, localized surface plasmons (SP) of gold a nd silver nanospheres\nof radii up to 1000 nmwere found with the tools of classical electrodynamics. The significa nt\nincrease in damping rates followed by noteworthy decrease for larg er particles takes place along\nwith substantial red-shift of plasmon resonance frequencies as a function of particle size. We\nalso introduced interface damping into our modeling, which substant ially modifies the plasmon\ndamping rates of smaller particles. We demonstrate unexpected re duction of the multipolar SP\ndamping rates in certain size ranges. This effect can be explained by t he suppression of the\nnonradiative decay channel as a result of the lost competition with t he radiative channel. We\nshow that experimental dipole damping rates [H. Baida, et al., Nano Le tt. 9(10) (2009) 3463,\nand C. S¨ onnichsen, et al., Phys. Rev. Lett. 88 (2002) 077402], an d the resulting resonance\nquality factors can be described in a consistent and straightforwa rd way within our modeling\nextended to particle sizes still unavailable experimentally.\nKeywords : plasmon damping, radiation damping, interface damping, surface p lasmon res-\nonance, multipolar plasmons, multipolar plasmon modes, Mie theory, g old nanoparticles, sil-\nver nanoparticles, nanosphere, nanoantenna, nanophotonics, plasmonics, size dependent optical\nproperties, SERS, SP.\n1. Introduction\nThe response of noble-metal nanoparticles to electromagnetic (E M) excitation is dominated\nbyresonantexcitationsoflocalizedsurfaceplasmons(SPs)(see: [1,2,3,4,5]forreviews). When\na metal particle is illuminated at corresponding resonance frequenc y, notably strong surface-\nconfined optical fields can be generated. This property is applied in s urface-enhanced Raman\n∗Corresponding author\n∗∗Principal corresponding author\nEmail addresses: Krystyna.Kolwas@ifpan.edu.pl (K. Kolwas), Anastasiya.Derkachova@ifpan.edu.pl\n(A. Derkachova)\nPreprint submitted to Elsevier November 21, 2012spectroscopy (SERS) [6, 7, 8], high-resolution microscopy [9] or imp rovement of plasmonic solar\ncells [10, 11, 12, 13, 14, 15]. Excitation of SPs at optical frequencie s, non-diffraction-limited\nguidingofthem(e.g. viaalinearchainofgoldnanospheres[8,16,17, 18,19,20])andtransferring\nthem back into freely propagating light are processes of great impo rtance in many applications.\nGold and silver nanoparticles are important components for subwav elength integrated optics\nand high-sensitivity biosensors [4, 21, 22] due to their chemical iner tness and unique optical\nproperties in the visible to near-infrared spectral range.\nMetal nanospheres are the simplest and the most fundamental st ructures for studying the\nbasis of plasmon phenomena. Understanding the resonant interac tion of light with plasmonic\nnanoparticles is also essential for applications, e.g. for designing us eful photonic devices. The\nmodeling of plasmon properties under the simplest electrostatic (qu asistatic) approximation is\nvalid only for particles much smaller than the wavelength of light; than size dependence of\nplasmon properties can be neglected. If a nanosphere is larger, th e properties of the surface\nplasmons supported by this structure become dependent on the s ize of the particle [1, 23, 24,\n25, 26, 27]. Size dependence of the retarded electronic response of a plasmonic particle is\ndetermined by the properties of metal at the SP resonance wavele ngth and the presence of the\nmetal-dielectric interface.\nNanoparticles much smaller than optical wavelength first of all exhib it dipolar surface plas-\nmon oscillations. Higher order multipolar resonances appear as new f eatures in the optical\nspectra, at frequencies higher than that of the dipolar plasmon fr equency. The spectral sig-\nnatures of these higher order plasmon resonances were observe d in elongated gold and silver\nnanoparticles [28, 29, 30] first. Fewer experimental investigation s directly demonstrated the\nmultipolar character of the observed resonances in spherical par ticles [26, 31, 32].\nControlling the spectral properties of a plasmonic nanosphere for technological or diagnostic\napplications is not possible without knowing the direct dependence of plasmon properties on\nparticlesize. Stillitisbelieved(e.g. [5,25])thatexistingtheoriesdono tallowtherigorousdirect\ncalculation of the frequencies of the multipolar SP modes. However, apart from the plasmon\nresonance frequencies, also the plasmon decay rates (damping tim es) are essential in controlling\nthespectralresponseoftheplasmonicparticle. Totaldampingra tesofmultipolarSPs[27]define\nabsorbing and emitting properties of a plasmonic nanoantennas whic h can be tuned by particle\nsize. Nanoantenna of the right size can serve as an effective trans itional structure which is able\nto absorb or to transmit electromagnetic radiation in plasmonic mech anism. The knowledge\nand modeling of plasmon damping effect as a function of particle size ar e thus of key interest for\nthe development of plasmonic nanosystems. The central feature here is the actual enhancement\nof the electromagnetic field. It served as a stimulus for us for deve loping a strict and direct\nsize characterization of multipolar plasmons resonance frequencie s and plasmon damping rates.\n2Our extended modeling up to the radius R= 1000nm, and up to SP multipolarity l= 10\nallows us to determine the intrinsic properties of plasmonic spheres in the size range which has\nnever been studied before. We do not apply any restrictions on the particle radius in relation\nto the wavelength of light; our study goes far beyond the particle s ize for which the quasistatic\napproximation is justified (e.g. [2, 5]).\nThe main reason of size sensitivity of SP which we discuss (usually refe rred to as the retar-\ndation effect ([33] and references therein)) is responsible for impo rtant red-shift of the plasmon\nresonance frequency for larger nanoparticles. It is accompanied by the significant increase in\ndamping rate followed by noteworthy decrease for larger particles . In addition, we discuss the\neffect called the surface electron scattering effect [1] or interfac e damping [34] which causes the\nsubstantial modificationoftheplasmondamping ratesinparticles of sizes comparableorsmaller\nthan the mean electron free path. Sometimes called the ”intrinsic siz e effect”, it is described by\nthe 1/Rdependence added to the electron relaxation rate γof the dielectric function [1, 5, 35].\nWe perform both types of modeling including or neglecting the effect o f surface scattering, in\norder to examine the causes of the size-dependent plasmon featu res.\nOur extended modeling for large particles reveals new features of t he total plasmon damping\nrate. We demonstrated, for the first time in spherical particles, t he effect of reducing of mul-\ntipolar SP damping rates below its low size limit which is equal to the nonra diative damping\nrate. Suppression of the total damping rates proven to exist in ce rtain size ranges allowed a new\ninsight into the role of radiation damping in the plasmon decay mechanis m. We also describe\nchanges of the quality Q-factor of SP multipolar resonances with pa rticle size. Q-factor is used\nfor determination of the local field enhancement [34] and effective s usceptibility in nonlinear\noptical processes [36].\nWe also confront the size characteristics resulting from our modelin g with the experimental\nresults obtained for the dipole plasmon for gold [32, 34] and silver [37 ] particles of different sizes.\nWe show that the measured dipole damping rates and quality factors [32, 34] can be described\nin a consistent and straightforward way within our modeling which deliv er also predictions for\nparticle of size range still unavailable experimentally.\n2. Size dependence of SP parameters derived from optical spe ctra\nIt is widely accepted, that excitation of surface plasmons (collectiv e free-electron oscilla-\ntions) is responsible for the commonly observed pronounced peaks in the optical absorption or\nextinction spectra. The spectra collected for nanoparticles of va rious sizes are used as a source\nof data allowing to reconstruct the change of the resonance (usu ally dipole only) position with\nsize (e.g. [1, 34, 38]. The width of the peak is related to the (dipole) pla smon damping rate\n[32, 34, 37, 39, 40, 41].\n3Mie theory delivers an indispensable formalism enabling the description of scattering of a\nplane monochromatic wave by a homogeneous sphere of known radiu s, surround by a homoge-\nneous medium [42, 43, 44, 45]. Mie formulas allow to predict the intensit ies of light scattered in\na given direction or the absorption and scattering cross-section s pectra for particles of a chosen\nsize. The Mie spectra can be compared with the spectra (intensities of a bsorbed or scattered\nlight) measured experimentally for particles of the same size .\n/s49/s44/s50 /s49/s44/s54 /s50/s44/s48 /s50/s44/s52 /s50/s44/s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s49/s44/s50 /s49/s44/s54 /s50/s44/s48 /s50/s44/s52 /s50/s44/s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s32/s32\n/s115/s99/s97/s116\n/s32/s32\n/s97/s98/s115\n/s39\n/s108/s61 /s51/s39\n/s108/s61/s50/s39\n/s108/s61 /s49\n/s98/s41\n/s115/s99/s97/s116/s44/s32/s32\n/s97/s98/s115/s82/s32 /s61/s32/s56/s32/s110/s109/s97/s41\n/s32/s91/s101/s86/s93/s32\n/s115/s99/s97/s116/s44\n/s97/s98/s115\n/s39\n/s108/s61 /s52/s39\n/s108/s61 /s51/s39\n/s108/s61 /s49/s39\n/s108/s61 /s50/s82/s32 /s61/s32/s56/s48/s32/s110/s109\nFigure 1: Absorption Cabs(ω) and scattering Cscat(ω) cross-sectionsfor nanospheres ofradii a) 8nm and b) 80nm\n(Mie theory). Figures illustrate suppression of absorption in larger particles. Broadening, deformation and shift\nof the maxima in the scattering spectra of larger particles in compar ison with the SP resonance eigenfrequencies\nω′\nlis also shown.\nHowever, predicting the size dependence of multipolar plasmon resonance frequencies, Mie\nscattering theory is an inconvenient tool. SP size dependence can b e determined only indirectly,\nby laborious derivation of positions of maxima in the consecutive spec tra collected for various\nparticleradii. Thesameappliestoderivationofthesizedependence o fthe(dipole)SPfrequency\n4corresponding to the peak position from experimental data. In ad dition, the maximum in\nthe spectrum ascribed to SP resonance can be broadened, defor med and shifted in respect\nto the frequency of plasmon oscillation mode [26, 27], as illustrated in F igure 1. That can\naffect the plasmon resonance position obtained from the experimen tal or calculated scattering\nspectra. It is even more difficult to determine the SP damping times fr om the width of the SP\nresonances, manifested in the optical spectra of larger particles , and to understand the derived\nsize dependence then (e.g. [34, 37, 38]). The processes leading to plasmon damping are the\nsubject of extensive research and debate (e.g. [1, 34, 39, 40, 46 , 47, 48]. The homogeneous\nlinewidth of the SP resonance was connected to the SP damping time. The properties of the\nSP are strongly influenced by this parameter. Mie scattering theor y is not sufficient to define\na general rule describing the size dependence of SP parameters; t he damping rates and the\nresonance frequencies are not parameters of this theory.\n3. Direct size characterization of multipolar SP modes\nIn order to derive parameters of SP as a function of particle size, w e use an accurate elec-\ntrodynamic approach based on [49], and described in more details in e .g. [50]. The formalism\nof Mie theory is used. However, the problem is formulated in absence of the illuminating light\nfield; we are interested in intrinsic eigenproperties of the spherical particle (the analogy to the\ncavity eigenproblem) [26, 27]. In the present paper we extend the n umerical calculations up to\nthe radius of 1000 nmand plasmon polarity from l= 1 up tol= 10. That allowed us to explore\nnew features of localized SP excitations and reconsider the results of [27].\nWe consider continuity relations at the spherical metal/dielectric int erface for the electro-\nmagnetic (EM) fields, which arethe solutions of the Helmholtz equatio n in spherical coordinates\n(r,θ,φ) inside and outside the sphere. The transverse magnetic (TM) mod es of EM waves lo-\ncalized on the interface possess a nonzero component of the elect ric field normal to the surface\nEr, which can couple with free-electron charge oscillations at the boun daryr=R. The condi-\ntions for the nontrivial solutions of the continuity relations for TM m ode define the dispersion\nrelation:\n/radicalBig\nεin(ω)ξ′\nl(kout(ω)R)ψl(kin(ω)R)−/radicalBig\nεout(ω)ξl(kout(ω)R)ψ′\nl(kin(ω)R) = 0 (1)\nwhich is fulfilled for the complex eigenfrequencies of the field Ω lin successive multipolar\nmodesl= 1,2,3,... . ψl(z) andξl(z) are Riccati-Bessel spherical functions, the prime symbol\n(′) indicates differentiation with respect to the argument, kin=ω\nc/radicalBig\nεin(ω) andkout=ω\nc√εout,\nεin(ω) andεoutare the dielectric functions of a metal and of the particle dielectric s urrounding,\nrespectively. The dispersion relation (Equation(1)) is solved numer ically for complex values of\nΩl(R) =ω′\nl(R) +iω′′\nl(R) (ω′′\nl(R)<0) forlin the range of 1 ÷10. We solved the problem\n5for successive Rvalues from 1 to 1000 nmin 1nmsteps. The fsolvefunction of the MatLab\nprogram, utilizing the Trust-region dogleg algorithm was used.\n3.1. Material properties of plasmonic particles\nThe simplest analytic function often used to describe a wavelength d ependence of optical\nproperties of metals like gold or silver [51, 52, 53, 54, 55] results fro m the Drude-Lorentz-\nSommerfeld model:\nεD(ω) =ε0−ω2\np/(ω2+iγω) (2)\nwithε0= 1 (e.g. [1, 5]). The effective parameter ε0>1 describes contribution of bound\nelectrons,ωpis the effective bulk plasma frequency which is associated with effectiv e concentra-\ntion of free-electrons, γis the phenomenological damping constant of electron motion. For b ulk\nmetalsγ=γbulkis related to the electrical resistivity of the metal and is supposed t o include all\nmicroscopic damping processes due to photons, phonons, impuritie s and electron-electron inter-\nactions. In the present modeling we accept the following effective pa rameters of the dielectric\nfunction of the bulk gold: ε0= 9,84,ωp= 9,010eV,γbulk= 0,072eV.\nThe collision time 1 /γbulkdetermines the electron mean free path in bulk metals. At room\ntemperatures the electron mean free path in gold is 42 nm[1]. It can be comparable or larger\nthan a dimension of a particle. An additional relaxation term added to the relaxation rate γ\naccounts for the effect of scattering of free electrons by the su rface [1, 35, 41, 56, 57, 58]:\nγR(R) =γbulk+AvF\nR(3)\nwherevFis the Fermi velocity ( vF= 1.4·10−6m/sin gold), and Ais the theory dependent\nquantity of the order of 1 [1]. We accept the value A= 0.33 in our modeling, according to [57].\nThen the size-adopted bulk dielectric function is:\nεγ(R)(ω,R) =ε0−ω2\np/(ω2+iγR(R)ω) (4)\nThe real part of εγ(R)is not modified by surface scattering. Frequency (and radius) dep en-\ndence of the dielectric function (Equation 2 or 4) couples to the ove rall frequency (and radius)\ndependence of the plasmon dispersion relation (Equation(1)).\nTo underline the crucial role of interface damping in smaller particles, we will compare the\nresults obtained with the dielectric function εin=εD(ω) given by Equation (2) (with γ=γbulk,\nsurface scattering neglected) and εin=εγ(R)(ω,R) given by Equation (4) (with γ=γR(R)).\nThe dielectric function of the particle surrounding (vacuum/air) is a ssumed to be εout= 1, or\nis chosen to reflect the index of refraction of the particle environm ent, as described in Section\n7, where we compare the results of our modeling with some experimen tal results of [34, 37].\n63.2. Resonance frequencies and damping rates vs radius; the role of the interface damping\nThe solutions of the dispersion relation (Equation(1)) define the siz e dependence of multipo-\nlarplasmonoscillationfrequencies ω′\nl(R) =Re(Ωl(R))anddampingrates |ω′′\nl(R)|=|Im(Ωl(R))|\n(and SP damping times Tl(R) = ¯h/|ω′′\nl(R)|) of the EM surface modes oscillations. We express\nω′\nl,ω′′\nl,ωandγin electronvolts (eV) for convenience.\nThe size dependence of ω′\nl(R) and|ω′′\nl(R)|of SP modes for the first ten multipolar plasmons\nfor gold spheres in vacuum/air ( εout= 1,γ=γbulk= 0.072eV, the radius up to 1000 nm) is\npresented in Figure 2. Black line ( l= 1) represents the dipole resonance frequency. Resonant\nexcitation of SP oscillations takes place when the frequency of incom ing light field ωapproaches\nthe eigenfrequency of a plasmonic nanoantenna of a given radius R:ω=ω′\nl(R),l= 1,2,3,.... If\nexcited, plasmon oscillations are damped at corresponding rates |ω′′\nl(R)|(Figure 2b)). Damping\nof surface plasmon oscillations is the inherent property of SP modes that defines absorbing and\nscattering properties of the plasmonic particle as a function of size .\n/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48/s51/s44/s53\n/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52/s48/s44/s53/s48/s44/s54/s48/s44/s55/s48/s44/s56\n/s48 /s51/s48/s48 /s54/s48/s48/s48/s44/s48/s51/s50/s48/s44/s48/s51/s52/s48/s44/s48/s51/s54/s39\n/s108/s32/s32/s91/s101/s86/s93/s32\n/s82 /s32/s91/s110/s109/s93/s97/s41 /s98/s41\n/s47/s50/s39/s39\n/s108/s124/s32/s32/s91/s101/s86/s93/s32\n/s82 /s32/s91/s110/s109/s93/s32/s108/s32/s61/s32/s49\n/s32/s108/s32/s61/s32/s50\n/s32/s108/s32/s61/s32/s51\n/s32/s108/s32/s61/s32/s52\n/s32/s108/s32/s61/s32/s53\n/s32/s108/s32/s61/s32/s54\n/s32/s108/s32/s61/s32/s55\n/s32/s108/s32/s61/s32/s56\n/s32/s108/s32/s61/s32/s57\n/s32/s108/s32/s61/s32/s49/s48/s47/s50/s124\n/s108/s39/s39/s124/s32/s91/s101/s86/s93/s32\n/s32/s82/s32 /s91/s110/s109/s93\n/s32/s32\n/s32/s84\n/s108/s32/s32/s91/s102/s115/s93\n/s49/s56/s44/s51/s54/s44/s54/s51/s44/s51/s50/s44/s50/s49/s44/s54/s49/s44/s51/s49/s44/s49/s48/s44/s57/s48/s44/s56\nFigure 2: a) Multipolar plasmon resonance frequencies ω′\nl(R) and b) plasmon damping rates |ω′′\nl(R)|(left axis)\nand corresponding damping times Tl(right axis) vs particle radius R. Reduction of the plasmon damping rates\n|ω′′\nl(R)|below the nonradiative limit γ/2) is demonstrated in the inset.\nThe SP frequencies ω′\nl(R) decrease with size monotonically, as illustrated in Figure 2a). For\ngiven particle radius R,ω′\nl(R) increase with plasmon polarity l. The multipolar SP resonances\nfor larger particles are better spectrally resolved than those for the smaller ones.\nIn the limit of small size, the analytic expressions for ω′\nl(R) andω′′\nl(R) can be found from\nthe relation (1) after applying the power series expansion of the sp herical Bessel and Hankel\nfunctions. Keeping only the first terms of the power series one can get:\n7ω′\n0,l=/bracketleftBiggω2\np\nε0+l+1\nlεout−/parenleftbiggγ\n2/parenrightbigg2/bracketrightBigg1/2\n, (5)\nω′′\n0,l=−γ\n2. (6)\nFor a perfect free-electron metal ( ε0= 1,γ= 0) andεout= 1, Equation (5) leads to\nthe well-known plasmon frequencies within the ”quasistatic approxim ation” [1, 50, 59]: ω′\n0,l=\nωp//radicalBig\n1+εout(l+1)/l, and in particular to the giant Mie resonance frequency ω′\n0,l=1=ωp/√\n3\nforl= 1.\nIn the smallest particles, the plasmon damping rates |ω′′\nl(R)|fall to pure nonradiative damp-\ning rates |ω′′\n0,l|=γnrad=γ/2, as demonstrated in Figure 2b). The values of |ω′′\n0,l|are the same\nfor all multipolar plasmon modes l= 1,2,3...(Equation (6)). The nonradiative damping rates\nγnradare due to the ohmic losses if interface damping is neglected ( γ=γbulk), or are equal to\nγR/2 (Equation 3) if interface damping is included in the modeling.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56\n/s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s44/s48/s51/s53/s48/s44/s48/s52/s48/s48/s44/s48/s52/s53/s48/s44/s48/s53/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s50/s44/s53/s53/s50/s44/s54/s48/s50/s44/s54/s53/s50/s44/s55/s48\n/s32/s108/s32/s61/s32/s49\n/s32/s108/s32/s61/s32/s50\n/s32/s108/s32/s61/s32/s51\n/s32/s108/s32/s61/s32/s49\n/s32/s108/s32/s61/s32/s50\n/s32/s108/s32/s61/s32/s51\n/s40/s82 /s41 /s48/s32/s45/s32/s50\n/s112/s47/s40/s50\n/s32/s45/s32/s105\n/s82/s40/s82 /s41 /s41/s68 /s48/s32/s45/s32/s50\n/s112/s47/s40/s50\n/s32/s45/s32/s105 /s41/s108/s39/s32/s91/s101/s86/s93/s32\n/s82 /s32/s91/s110/s109/s93/s97/s41 /s98/s41\n/s47/s50/s108/s39/s39/s124/s32/s91/s101/s86/s93/s32\n/s82 /s32/s91/s110/s109/s93/s47/s50/s32/s124\n/s108/s39/s39/s124/s32/s91/s101/s86/s93\n/s32/s82 /s32/s91/s110/s109/s93/s82/s40/s82 /s41/s47/s50\n/s49/s56/s44/s51\n/s32/s84\n/s108/s32/s91/s102/s115/s93\n/s54/s44/s54/s51/s44/s51/s50/s44/s50/s49/s44/s54/s49/s44/s51/s49/s44/s49/s48/s44/s57/s48/s44/s56/s108/s39/s32/s91/s101/s86/s93 /s32\n/s82 /s32/s91/s110/s109/s93/s97/s41\nFigure 3: a) Dipole, quadrupole and hexapole plasmon resonance fre quencies ω′\nl(R) and b) corresponding plas-\nmon damping rates |ω′′\nl(R)|calculatedwithout (solid lines) and with (lines with hollow circles)interfa ce damping\ntaken into account. Insets are magnifications of figures a) and b) for particles of small radii.\nInterface damping practically does not influence the size dependen ce ofω′\nl(R) as illustrated\nin Figure 3a). The minute red shift for the smallest sizes due to interf ace damping is shown\nin inset of Figure 3a). A similar finding results from the experimental o bservations [32, 34].\nTherefore, the red shift of the (dipole) SP resonance frequency sometimes observed in small\n8particles versus decreasing size must result from some other phen omena due to the complex\nchemical effects and uncertainties in experimental samples [60].\nIn extended size range that we study, plasmon damping rates |ω′′\nl(R)|are not simple mono-\ntonicfunctionsoftheradius(Figure2b)). Theinitialfastincrease of|ω′′\nl(R)|withsizeisfollowed\nby gradual decrease for sufficiently large radii R. The increase in |ω′′\nl+1(R)|in the subsequent\nl+1 SP modes is followed by the decrease in the SP damping rates |ω′′\nl(R)|of lower polarity\nmodes.\nThe surface scattering effect affects the total damping rate |ω′′\nl(R)|, as illustrated in Figure\n3b) and in the magnification of the part of this graph presented in Fig ure 4. The surface\nscattering contribution to the total relaxation rate γR(R) looses its importance for particles of\nradius larger than ∼300nm(using the criterion ( γR−γ)/γR≈1%).\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s44/s48/s51/s53/s48/s44/s48/s52/s48/s48/s44/s48/s52/s53/s48/s44/s48/s53/s48\n/s32/s84\n/s108/s32/s91/s102/s115/s93\n/s82/s124\n/s108/s39/s39/s124/s32/s91/s101/s86/s93\n/s82 /s32/s91/s110/s109/s93/s68 /s48/s32/s45/s32/s50\n/s112/s47/s40/s50\n/s32/s45/s32/s105 /s41/s32/s32/s32/s32\n/s40/s82 /s41 /s48/s32/s45/s32/s50\n/s112/s47/s40/s50\n/s32/s45/s32/s105\n/s82/s40/s82 /s41 /s41/s32\n/s32/s32/s32 /s32/s108/s32/s61/s32/s49 /s32/s32/s32 /s32/s108/s32/s61/s32/s49\n/s32/s32/s32 /s32/s108/s32/s61/s32/s50 /s32/s32/s32 /s32/s108/s32/s61/s32/s50\n/s32/s32/s32 /s32/s108/s32/s61/s32/s51 /s32/s32/s32 /s32/s108/s32/s61/s32/s51\n/s32/s32/s32 /s32/s108/s32/s61/s32/s52 /s32/s32/s32 /s32/s108/s32/s61/s32/s52\n/s32/s32/s32 /s32/s108/s32/s61/s32/s53 /s32/s32/s32 /s32/s108/s32/s61/s32/s53\n/s49/s54/s44/s53/s49/s52/s44/s54/s49/s51/s44/s50\n/s49/s56/s44/s56\nFigure 4: Demonstration of the reduction effect of plasmon damping rates|ω′′\nl(R)|calculated without ( γ=γbulk,\nsolid lines) and with ( γ=γR(R), lines with hollow circles) interface damping; the magnification of a pa rt of\nFigure 3.\n4. Radiative and nonradiative contributions to the total SP damping rates\nThe processes leading to damping of SP have been the subject of wid e discussion and ex-\ntensive studies, e.g. [1, 32, 34, 61, 62, 63] which, however, were lim ited to the dipole case. It\nwas accepted that (dipole) SP in metal nanoparticles decays throu gh both inelastic processes\nand elastic dephasing process which were usually neglected. Inelast ic processes can be further\ndivided into radiative and non-radiative decay processes [33, 34, 61 ]. The SP damping time\nTwas determined from the homogeneous linewidth Γ = 2¯ h/Tof a maximum in the spectrum\n9due to the SP dipole resonance. The observed spectral broadenin g of the maxima with size due\nto radiation is sometimes expected to be proportional to the partic le volume [34, 57, 64]. Our\nresults show (Figure 2b)), that the multipolar plasmon damping rate s|ω′′\nl(R)|are not simple,\nmonotonic functions of particle radius. The initial fast increase of t he total SP damping rate\n|ω′′\nl(R)|with size is followed by gradual decrease for sufficiently large radii fo r the given mode\nl. Our modeling suggest that the proportionality of the multipolar rad iation damping rate to\nR3is not a general rule (see Figure 2b)).\nOur extended modeling allows us to consider the higher order multipola r damping rates\n|ω′′\nl(R)|l=1,2...10, as well. The total multipolar damping times ¯ h/Tlcan be related to the\ncorresponding damping rates (homogeneous linewidths of the abso rption [27] spectra Γ l(R))\nand their size dependencies in a natural way:\nΓl(R) = 2|ω′′\nl(R)|= 2¯h/Tl (7)\nand\n|ω′′\nl|=γrad\nl+γnrad(8)\nwhereγnrad=γ/2. The assumption, that the nonradiative and radiative processes (in\ndipoleplasmon mode) areadditive andareindependent, is commonly ac cepted (see for example:\n[27, 41, 57, 58, 61]). However, expectation that the size depende nce of the total damping rate\n|ω′′\nl(R)|is due to the size dependence of the radiative damping rate γrad\nl(R) only (|ω′′\nl(R)|=\nγrad\nl(R)+γnrad, [27]) must be reconsidered (see Section 5 bellow).\nIf the surface scattering effect is included, |ω′′\nl(R)|becomes a decreasing function of size,\nstarting fromsmaller particles (see Figure3b) and4, lines with symbo ls). The total SPdamping\nrates follow the size dependence of nonradiative damping: γR(R)/2:|ω′′\nl(R)| ≈γR(R)/2 =\nγnrad(R) in the range of radii which extends to larger Rfor growing SP polarity l. After\nreaching theminimum, |ω′′\nl(R)|tendstofollowtheradiusdependence unaffectedbytheinterface\ndamping.\nA contribution of the radiation damping γrad\nl(R) to the total damping rate |ω′′\nl(R)|can be\ntreated as a measure of the ability of the particle to couple to the inc oming field and to emit\nlight in plasmonic mechanism. As long as |ω′′\nl(R)| ≃γnrad, the SP mode lcan only weakly\ncouple with the incoming radiation and has weak radiative abilities. Cons equently, the weakly\nradiative plasmons appear in the absorption spectra with a smaller am plitude and in scattering\nspectra (see Figure 1) they hardly manifest. With increasing l, SP plasmons gain the radiative\ncharacter starting from larger particles due to the fact that γrad\nl>1(R)>γrad\nl−1(R). Therefore, the\nhigher order SP modes can be excited (and appear in the scattering spectra) for larger particles.\n10This fact is known from Mie scattering theory, but its physical sens e have not been explained\nby Mie solutions.\n5. Effect of reduction of multipolar plasmon damping rates\nIn the easier case (interface damping omitted in the modeling), redu ction of the total SP\ndumping rates γnradbellow the γ/2 value (see the inset in Figure 2b)) manifests for plasmon\nmodes with l>1. This reduction effect can be clearly distinguished from other size d ependent\nprocesses in some size ranges which grow with increasing l; for the quadrupole ( l= 2) plasmon\nmode, the reduction effect extends up to R≃40nm, for the hexapole ( l= 3) plasmon mode up\ntoR≃92nm, forl= 4 up toR≃155nmand so on.\nWe have checked, the reduction of the total damping rate is not pr esent ifγ= 0 (2). On\nthe contrary, it does not disappear if γ/negationslash= 0 andε0= 1. That suggests, that if the additive form\nof the Expression (8) holds, reduction of the total plasmon dampin g rate is due to the decrease\nof nonradiative decay γnradas compared with its low size limit γnrad=γ/2 resulting from\nthe energy dissipation of meatal (absorption). Consequently, th e radiative and nonradiative\ncontributions have to be coupled by their size dependence. The mod el with interface damping\nincluded, reproduces the SP rate reduction below γnrad=γR/2 (see Figure 4), as well.\nWe can conclude, that reduction of |ω′′\nl(R)|below theγ/2 =γbulk/2 (orγ/2 =γR/2) value\ntakes place in the regions of sizes where the nonradiative damping is s till not dominated by the\nfast radiative damping. Reduction in the total plasmon damping rate must be connected with\nsuppression of the nonradiative decay channel. It can be inferred then, that the radiative and\nnonradiativeprocessesarenotindependent. Theeffective rateo fthenonradiativedampingmust\nbe size dependent: γnrad=γnrad(R) with the value for the small particle limit: γnrad\n0=γ/2.\nSuppression of the nonradiative damping is not restricted to the siz e ranges for which the effect\nwas demonstrated, but influences the optical properties of plasm onic particles in a large range\nof sizes. Reduction of γnrad(R) manifests in the absorption spectrum of particles; absorptive\nabilities of large particles are poor as compared with small particles. T his fact is described by\nsolutions of Mie scattering theory, but its physical meaning have no t been explained.\nThe effect of the total damping rate reduction with particle dimensio n was observed for\nthe first time in the experiment with gold nanorods [34]. The size depe ndence of the dipole\nplasmon damping rate was deduced from the homogeneous linewidth Γ = 2¯h/Tof the maxima\nin the scattered intensities both for nanorods with various aspect ratios and spherical nanopar-\nticles with radii in the range from 10 to 75 nm. For nanorods, (dipole) plasmon damping rates\ndecreased significantly for lower dipole plasmon oscillation frequencie s. However, the decrease\nin the damping rate was not ascribed to the radiative processes. Th e authors of explain this\n11effect as reduction nonradiative plasmon decay by the fact that int erband excitations in gold\nrequire a threshold energy of about 1 .8eVand expect, that suppression of the damping rate\nin such mechanism would also be present in gold spheres, but for plasm on resonance energies\nbelow 1.8eV. Suppression of the plasmon damping rates in spheres had not been observed\nexperimentally, as far as we know; the conclusive experimental dat a are limited to the plas-\nmon dipole only, while according to our study the effect manifests for l >1. Importance of\nthe threshold energy at 1 ,8eVin gold and exclusion of the radiative damping as a reducing\nmechanism was not confirmed by our study. On the contrary, the r eduction of multipolar SP\nnonradiative damping rates results from competition between radia tive damping γrad\nl(R) and\nall other damping processes included in γnrad(R) and is present in plasmonic particles of any\nmaterial.\n6. Quality factor of multipolar plasmon resonances\nEnhancement of optical response in resonance is usually described by the quality factor\nQ, defined as the product of the resonance center frequency and the bandwidth. In case of\nplasmonic particles the quality factor is interpreted as a measure of the local field enhancement\n[34] and is expected to define the effective susceptibility in nonlinear o ptical processes [36].\n/s48/s44/s48 /s48/s44/s56 /s49/s44/s54 /s50/s44/s52/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54\n/s48/s44/s48 /s48/s44/s56 /s49/s44/s54 /s50/s44/s52/s48/s49/s48/s50/s48/s51/s48/s52/s48/s50/s44/s53 /s50/s44/s54 /s50/s44/s55/s48/s44/s48/s51/s50/s48/s44/s48/s51/s52/s48/s44/s48/s51/s54/s48/s44/s48/s51/s56/s48/s44/s48/s52/s48\n/s68 /s48/s32/s45/s32/s50\n/s112/s47/s40/s50\n/s32/s45/s32/s105 /s41/s32/s32/s32/s32 /s32/s108/s32/s61/s32/s49\n/s32/s108/s32/s61/s32/s50\n/s32/s108/s32/s61/s32/s51\n/s32/s108/s32/s61/s32/s52\n/s32/s108/s32/s61/s32/s53\n/s32/s108/s32/s61/s32/s54\n/s32/s108/s32/s61/s32/s55\n/s32/s108/s32/s61/s32/s56\n/s32/s108/s32/s61/s32/s57\n/s32/s108/s32/s61/s32/s49/s48/s110\n/s111/s117/s116/s61 /s49/s124\n/s108/s39/s39/s124/s32/s91/s101/s86/s93\n/s32/s84\n/s108/s32/s91/s102/s115/s93\n/s49/s56/s44/s51\n/s49/s57/s44/s52\n/s50/s48/s44/s54\n/s108/s39/s32/s91/s101/s86/s93\n/s108/s39/s32/s91/s101/s86/s93/s81\n/s108/s61\n/s108/s39/s47/s40/s50/s124\n/s108/s39/s39/s124/s41/s32\n/s50/s44/s53 /s50/s44/s54 /s50/s44/s55/s51/s48/s51/s51/s51/s54/s51/s57/s52/s50/s32/s97/s41\n/s98/s41\nFigure 5: a) Damping rates |ω′′\nl|vsω′\nlfor successive radii Rand b) quality factors Ql=ω′\nl(R)/(2|ω′′\nl(R)|) for\nmultipolar plasmon modes of gold nanoparticles ( nout= 1, surface scattering neglected). Graphs on the right\nare magnifications of circled regions of the graphs on the left.\nThe dependence of the ω′′\nlvsω′\nlfor successive radii Rand the resulting quality factors\n12Ql(R) =ω′\nl(R)/(2|ω′′\nl(R)|) for multipolar plasmon modes of gold nanoparticles ( nout= 1) are\npresented in Figures 5 (surface scattering neglected) and 6 (sur face scattering included). The\ngraphs on the right are the magnifications of the graphes (circled r egions) on the left.\n/s48/s44/s48 /s48/s44/s56 /s49/s44/s54 /s50/s44/s52/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54\n/s48/s44/s48 /s48/s44/s56 /s49/s44/s54 /s50/s44/s52/s48/s49/s48/s50/s48/s51/s48/s52/s48/s50/s44/s52 /s50/s44/s53 /s50/s44/s54 /s50/s44/s55/s48/s44/s48/s53/s48/s44/s49/s48/s48/s44/s49/s53/s48/s44/s50/s48/s124\n/s108/s39/s39/s124/s32/s91/s101/s86/s93\n/s32/s84\n/s108/s32/s91/s102/s115/s93\n/s52/s44/s52\n/s54/s44/s54\n/s49/s51/s44/s50/s110\n/s111/s117/s116/s61 /s49\n/s40/s82 /s41 /s48/s32/s45/s32/s50\n/s112/s47/s40/s50\n/s32/s45/s32/s105\n/s82/s40/s82 /s41 /s41/s32\n/s108/s39/s32/s91/s101/s86/s93\n/s108/s39/s32/s91/s101/s86/s93/s81\n/s108/s61\n/s108/s39/s47/s40/s50/s124\n/s108/s39/s39/s124/s41/s32\n/s32/s108/s32/s61/s32/s49\n/s32/s108/s32/s61/s32/s50\n/s32/s108/s32/s61/s32/s51\n/s32/s108/s32/s61/s32/s52\n/s32/s108/s32/s61/s32/s53\n/s50/s44/s52 /s50/s44/s53 /s50/s44/s54 /s50/s44/s55/s49/s48/s50/s48/s51/s48/s52/s48/s32/s98/s41/s97/s41\nFigure 6: a) Damping rates |ω′′\nl|vsω′\nlfor successive radii Rand b) quality factors Ql=ω′\nl(R)/(2|ω′′\nl(R)|) for\nmultipolar plasmon modes of gold nanoparticles ( nout= 1, surface scattering included). Graphs on the right are\nmagnifications of circled regions of the graphs on the left.\nFigure 5a) illustrates the effect of the total plasmon damping rate r eduction to the value\nbellowγnrad= 0,032eVfor modes with l>1. Figure 5b) illustrates the resulting quality factors\nQlfor successive SP modes which we use as a measure of energy store d in the SP oscillation of\nplasmon mode lat the resonance frequency ω′\nl(R). In some ranges of SP resonance frequencies,\nthe higher polarity plasmon modes ( l>1) are more efficient in storing the SP oscillation energy.\nThe similar conclusion comes from Figure 6b) (interface damping includ ed). Quality factor of\nthe dipole plasmon resonance reaches maximum value Ql=1≈29 forω′\nl(R)≈2,606eVfor gold\nnanoparticle of the radius R≈16nm(see Figure 7) and is the fast decreasing function of size\nfor both smaller and larger particles. If a gold particle is embedded in a medium of higher\noptical density (see Figure 9, nout= 1,5), the maximum quality factor of the dipole plasmon\nresonance is even smaller, Ql=1≈26. Such value corresponds to the resonance frequency of a\ngold particle of radius R≈11nm.\n13/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s32/s108/s32/s61/s32/s49\n/s32/s108/s32/s61/s32/s50\n/s32/s108/s32/s61/s32/s51\n/s32/s108/s32/s61/s32/s52\n/s32/s108/s32/s61/s32/s53/s40/s82 /s41 /s48/s32/s45/s32/s50\n/s112/s47/s40/s50\n/s32/s45/s32/s105\n/s82/s40/s82 /s41 /s41/s32/s81\n/s108/s61\n/s108/s39/s47/s40/s50/s124\n/s108/s39/s39/s124/s41/s32\n/s82 /s32/s91/s110/s109/s93\nFigure 7: Quality factors Qlvs radius Rfor multipolar plasmon modes of gold nanoparticles ( nout= 1, surface\nscattering included).\n7. Dipole plasmon damping rates vs size: comparison with exp erimental results for\nsilver and gold nanoparticles [34, 37]\nExperimental investigations of spectral properties of plasmonic p articles are usually per-\nformed on large particle ensembles, where inevitable variations in size , shape and surface prop-\nerties tend to mask the spectral properties of the individual part icles. The effects of surface\nchemistry (especially in Ag particles) and uncertainties in sample para meters affect linewidths\nand maxima positions which define damping times and resonance frequ encies of SP. We have\nchosen two experiments [34, 37], which provide the spectroscopic d ata for individual spherical\nparticles of silver and gold. The experiments were performed in well-c ontrolled particle environ-\nments as a function of size for nanoparticles up to R= 25nmfor silica-coated silver (Ag@SiO 2)\nand for gold nanoparticles immersed in a index matching fluid ( nout= 1.5) for relatively large\nrange of radii (up to R= 75nm). However, well resolved data was reported only for the dipole\nplasmon resonance. In Figures8 and9 we compare theexperimenta l results presented in[34, 37]\nwith the results of our modeling.\nFigure 8 illustrates the spectral width Γ measured in different single A g@SiO 2nanoparticles\nvs the inverse of their equivalent diameter Deq, optically determined by fitting the extinction\nspectra (see Figure 4 in [37]). Dashed line represents γR((2R)−1) dependence (Equation (3) for\nA= 0.7 andγbulk= 0.125eV). These parameters [37] result from a linear fit to the experimenta l\ndata. Solid line results from our modeling in the extended range of size s forωp= 9.10eV,\n14/s65/s103 /s32/s32/s32 /s110\n/s111/s117/s116/s32/s61/s32/s49/s44/s52/s53/s32/s40/s83/s105/s79\n/s50/s41\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32 /s115/s112/s101/s99/s116/s114/s97/s108/s32/s119/s105/s100/s116/s104/s32 /s32/s91/s51/s55/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32\n/s108/s32/s61/s49/s50 /s82 /s61/s32/s50/s124\n/s108/s32/s61/s49/s39/s39/s40 /s82 /s41/s124/s32/s40/s116/s104/s105/s115/s32/s119/s111/s114/s107/s41 /s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32\n/s82/s40/s82 /s41/s32/s61/s32\n/s98/s117/s108/s107/s43 /s50 /s65/s118\n/s70/s47/s50 /s82\n/s48/s44/s48/s48 /s48/s44/s48/s50 /s48/s44/s48/s52 /s48/s44/s48/s54 /s48/s44/s48/s56 /s48/s44/s49/s48 /s48/s44/s49/s50/s48/s44/s48/s48/s48/s44/s50/s53/s48/s44/s53/s48/s48/s44/s55/s53/s49/s44/s48/s48/s49/s44/s50/s53/s32/s91/s101/s86/s93\n/s40/s50 /s82 /s41/s45/s49\n/s32/s91/s110/s109/s45/s49\n/s93\nFigure 8: Spectral width Γ of plasmon resonances measured [37] in single spherical Ag@SiO 2particles vs the\ninverse of their equivalent diameter. Dashed line represents γR((2R)−1) dependence (equation (3)), with the\nparameters A= 0.7 andγbulk= 0.125eVresulting from linear fit the line to the experimental data [37]. Solid\nline results from our model with the parameters ωp= 9.10,ε0= 4eV, γbulk= 0.125eVandA= 0.7.\nε0= 4eV,γbulk= 0.125eVandA= 0.7. Our model with such input parameters reproduces the\nexperimental data for smallest particles perfectly and describes t he departure of ωl=1((2R)−1)\ndependence from linear. The ωl(R) model functions describe consistently the size dependence\nof the SP damping rates for experimentally available particles and pre dicts damping rates (and\nthe resulting damping times) for larger sizes (experimentally unavaila ble so for) and for higher\nplasmon multipolarities.\nFigure 9a) illustrates spectral widths Γ vs resonance energy ω′\nl=1(R) measured for different\nsingle Au nanoparticles (up to 2 R= 150nm) (Figure 4 of [34]). The experimental data seems\nto suggest that Γ decays linearly with the SP resonance frequency . However, it is not the case.\nThe dependence of Γ l=1(ωl=1(R)) = 2|ω′′\nl=1(R)|(line in figure 9a)) reproduces the experimental\ndata and predicts the decrease in Γ l=1for particles larger than those studied experimentally.\nSuch decrease in damping rate |ω′′\nl=1(R)|of the dipole SP is accompanied by the increase in\nthe damping rate of the quadrupole |ω′′\nl=2(R)|SP and of the following higher polarity SPs, as\ndiscussed in Section 5 (see Figure 2b)).\nFigure 9b) illustrates the quality factor of the dipole resonance Ql=1=ω′\nl=1(R)/2|ω′′\nl=1(R)|\nvs resonance energy ω′\nl=1(R) (solid lines). Our modeling shows that interface damping (dashed\nline) substantially reduces the quality factor for particles from the smallest size range. The\nmaximum factor Ql=1≈26 is found in nanoparticles of radii R= 12nm. The agreement with\n15the experimental data of [34] is very good.\n/s48/s44/s53 /s49/s44/s48 /s49/s44/s53 /s50/s44/s48 /s50/s44/s53/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48\n/s48/s44/s53 /s49/s44/s48 /s49/s44/s53 /s50/s44/s48 /s50/s44/s53/s48/s49/s48/s50/s48/s51/s48/s52/s48/s32/s91/s101/s86/s93\n/s32\n/s108/s61/s49/s32/s61/s32/s50/s124 \n/s108/s61/s49/s34 /s124/s32/s40 /s116/s104/s105/s115/s32/s119/s111/s114/s107 /s44/s32\n/s98/s117/s108/s107/s41\n/s32\n/s108/s61/s49/s32/s61/s32/s50/s124 \n/s108/s61/s49/s34 /s124/s32/s40 /s116/s104/s105/s115/s32/s119/s111/s114/s107 /s44/s32\n/s98/s117/s108/s107/s82 /s41/s41\n/s32/s49/s48/s32/s110/s109\n/s32/s50/s48/s32/s110/s109\n/s32/s51/s48/s32/s110/s109\n/s32/s52/s48/s32/s110/s109\n/s32/s53/s48/s32/s110/s109\n/s32/s55/s53/s32/s110/s109/s65/s117\n/s110\n/s111/s117 /s116/s61/s49/s46/s53/s97/s41\n/s98/s41\n/s81/s117/s97/s108/s105/s116/s121/s32/s102/s97/s99/s116/s111/s114\n/s108/s61 /s49/s39/s40 /s82 /s41/s32/s91/s101/s86/s93\nFigure 9: a) Spectral widths Γ of plasmon resonances vs resonanc e energy measured [34] in single spheri-\ncal Au particles. b) Quality factor resulting from the experimental data. Lines represent the spectral width\nΓl=1(ωl=1(R)) = 2|ω′′\nl=1(R)|and the quality factor Ql=1=ω′\nl=1(R)/2|ω′′\nl=1(R)|vsω′\nl=1(R) for the dipole mode\nobtained from our modeling with surface damping neglected (dashed line) and included (solid line).\n8. Conclusions\nThedependence oftheSPresonancefrequencies ω′\nl(R)anddampingrates |ω′′\nl(R)|onparticle\nradius determine the intrinsic optical properties of plasmonic spher es (illuminated or not). Our\nstudy provides direct, accurate size characteristics for a broad range of particle radii (up to\nR= 1000nm) and plasmon polarities (up to l= 10). At present, this exceeds the range\nexperimentally explored.\nSize dependence of SP damping rates |ω′′\nl(R)|allows to distinguish the size ranges in which\nefficient transfer of radiation energy into heat takes place (large c ontribution of the nonradiative\ndecay) and those in which the particles are effective radiating anten nas (dominant contribution\nof the radiative damping). As long as the contribution of the radiativ e decay is negligible, the\nparticle is not able to couple to the incoming field effectively and has wea k radiative abilities.\nIf the contribution of the radiative damping prevails, the particle is a ble to emit light within\n16plasmonic mechanism efficiently. Effective radiating enables efficient int eraction of SP near-\nfield with other structures at a desired resonance frequency ω=ω′\nl(R). The knowledge of\nsize dependence of both: the multipolar SP resonance frequencies ω′\nl(R) and the corresponding\ndamping rates |ω′′\nl(R)|= ¯h/Tl(R) is indispensable to shape the particle plasmonic features\neffectively.\nOur study, extended toward large particle sizes and plasmon multipo larities, revealed new\nfeatures of the total plasmon damping rates. In certain ranges o f radii, reduction of the multi-\npolar SP damping rates, as compared with its small size limit, takes plac e. The small size limit\nis equal to the nonradiative damping rate resulting from absorption and heat dissipation. The\nsuppression of nonradiative damping is not limited to the size range, in which the effect was\ndirectly demonstrated. It is present when the radiative damping br ings the dominant contribu-\ntion to the total plasmon damping. The reduction of γnrad(R) with the growing contribution of\nradiation damping is revealed in the absorption spectra of particles; absorptive abilities of large\nparticles are poor. This fact is described by solutions of Mie scatter ing theory, but its physical\nbackground have not been explained, as far as we know. Our study led us to conclusion, that\nthe reduction of multipolar SP nonradiative damping rates results fr om competition between\nradiative damping and all other damping processes included in γnrad(R). As far as we know,\nsuch hypothesis has not been proposed before. Our study provid es a new starting point for\nbetter understanding of rules that define contributions of plasmo n radiative and nonradiative\ndecay ratesto thetotalSP decay rateasafunction ofparticle siz e. The independent modeling is\nnecessary for better understanding of the role of SP radiative an d nonradiative decay channels.\nThe proposed SP characteristics vs particle size provide a consiste nt, uniform description\nof the experimental SP damping rates measured for single spherica l particles in [34] and [37].\nNot only dipole damping rates (found experimentally) but also multipole damping rates (and\nresulting damping times) in a broad range of sizes can be predicted.\nThe derived size dependence of plasmon decay rates |ω′′\nl(R)|at resonance frequencies ω′\nl(R\n), and the quality factors Ql(R) of SP multipolar modes define not only the spectral scattering\nabilities of the plasmonic spheres, but also reflect changes in the str ength of coupling of SP\nmodeswiththe external field. 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We find the impact of the Lorentz-violating term for the free en ergy and carry out a numerical\nestimation for the Lorentz-breaking parameter.\n∗Electronic address: mgomes,ajsilva@fma.if.usp.br\n†Electronic address: tmariz@if.ufal.br\n‡Electronic address: alesandro,jroberto,petrov@fisica.u fpb.br\n1I. INTRODUCTION\nNowadays the Lorentz symmetry breaking is treated as an impo rtant ingredient of field theory\nmodels for probing quantum gravity phenomena. In fact, Lore ntz-violating theories have been\nstudied in various contexts, such as string theory [1], nonc ommutative field theory [2], and, more\nrecently, Horava-Lifshitz gravity [3].\nMany of the effects observed for the Lorentz-breaking field the ories at zero temperature are\nknown to persist also at finite temperatures. For example, th e dependence of the loop corrections\non the regularization scheme in the finite temperature case w as shown to arise in the one-loop order\nin the Lorentz-breaking quantum electrodynamics (QED) [4] . Analogous situation takes place in\nthe Lorentz-breaking Yang-Mills theory [5]. Different issue s related to the Lorentz-violating QED\nin the finite temperature case were considered in [6]. In [7] t he finite temperature properties in the\nCPT-odd Lorentz-breaking extension of QED for a purely spac elike background were studied, and\nin [8] these properties were analysed for the CPT-even Loren tz-breaking extension.\nMost of the previous studies of the Lorentz-breaking theori es, including those described in [4, 5],\nwere based on the use of couplings which violate not only Lore ntz symmetry but also the CPT\nsymmetry, although CPT-even Lorentz breaking interaction s are also possibleand certainly require\nmore detailed study. In this paper, we will see that, at the hi gh temperature regime, the linear\ncontribution in the Lorentz-breaking parameter may arise.\nThe aim of this paper is the study of one and two-loop correcti ons to the free energy (second\norder in the coupling constant e) of QED in the presence of Lorentz breaking terms, at finite\ntemperature. As it is known thefree energy provides an impor tant information to different physical\nissues such as plasma behavior, solar interior and Big Bang n ucleosynthesis (BBN) (see, [9–11]). In\nthis context, some aspects of thecorrections to the freeene rgy in theories without Lorentz-breaking\nare discussed in [12, 13].\nFor estimating bounds of the Lorentz-breaking parameter, w e use information of the BBN (for\na review, see [14]), which is one of the observational pillar s of the standard cosmology. Note that\nthe Lorentz-violating parameter can explain for the light e lements abundance. In particular, the\ndifference between the theoretical and observational result s can be understood as come from a\ncontribution of the Lorentz-breaking parameter to the prim ordial helium abundance.\nThe structure of the paper is as follows. In Sec. II we present the basic features of the Lorentz-\nviolating QED in the regime of high temperature. In Sec. III w e calculate the one and two-loop\ncontributions to the free energy for QED involving the Loren tz-breaking fermion coupling. In\n2Sec. IV, by using information about the primordial helium ab undance, we obtain a numerical\nestimation for the Lorentz violation parameter. A summary i s presented in Sec. V.\nII. LORENTZ-VIOLATING QED AT HIGH TEMPERATURE\nLet us start by considering the Lagrangian of the Lorentz- an d CPT-violating QED extension\n[15]\nL=−1\n4FµνFµν−1\n4(kF)µνλρFµνFλρ+1\n2(kAF)µǫµνλρAνFλρ+¯ψ(iΓµDµ−M)ψ+Lgf+Lgh,(1)\nwhere Γµ=γµ+Γµ\n1,M=m+M1, with\nΓµ\n1=cµνγν+dµνγ5γν+eµ+ifµγ5+1\n2gλνµσλν (2)\nM1=aµγµ+bµγ5γµ+1\n2Hµνσµν, (3)\nandDµ=∂µ+ieAµ.Lgfis the gauge fixing term and Lghis the ghost field term, which decouples\nfrom the rest of the Lagrangian. The coefficients carrying an o dd (even) number of Lorentz indices\nare CPT-odd (-even).\nThe two Lorentz-violating terms of the photon sector, the Ch ern-Simons-like (CPT-odd) and\n(kF)µνλρFµνFλρ(CPT-even) terms, can be induced by radiative corrections f rom the terms with\nthe coefficients bµandcµνof the fermion sector, respectively, so that ( kAF)µ∝bµ[16] and [17]\n(kF)µνλρ∝1\n2gµλ(cνρ+cρν)+1\n2gνρ(cµλ+cλµ)−1\n2gµρ(cνλ+cλν)−1\n2gνλ(cµρ+cρµ).(4)\nThe coefficients aµ,bµ,Hµν, and (kAF)µhave dimensions of mass, while cµν,dµν,eµ,fµ,\ngλνµ, and (kF)µνλρare dimensionless. In the high temperature regime ( T≫M) the dimensionful\ncoefficients may be neglected. For instance, the bµ-corrections to the free energy are observed to\nbe proportional to b2T2, similarly to the scenario occurring with the corrections s temming from\nthe Chern-Simions-like term [7]. Thus both bµand (kAF)µare negligible at high temperature, as\nwell asaµandHµν.\nAmong the dimensionless coefficients, eµ,fµ, andgλνµare expected to bemuch smaller than the\nother, because their terms cannot be obtained directly from the standard model extension [15] (for\nmore details, see also [18]). Moreover, if we require that th e theory at high temperature is invariant\nunder chiral transformations, these terms are ruled out, si nce{γ5,eµ+ifµγ5+1\n2gλνµσλν} ∝ne}ationslash= 0.\nTherefore, the remaining coefficients are cµν,dµν, and (kF)µνλρ. Now, in order to get a Clifford\nalgebra for Γµ=γµ+cµνγν+dµνγ5γν, we must choose dµν=Q(δµν+cµν), whereQis a constant\n3[19]. With this assumptions, the theory (1) becomes\nLhigh=−1\n4FµνFµν−1\n4(kF)µνλρFµνFλρ+¯ψ[i∂µ(gµν+cµν)˜γν−eAµ(gµν+cµν)˜γν]ψ\n+Lgf+Lgh, (5)\nwhere ˜γµ= (1+Qγ5)γµ.\nWe now assume rotational invariance, such that the coefficien ts in (5) may be reduced to\nproducts of a given unit timelike vector uµ, which describes the preferred frame (see [20], for more\ndetails). Proceeding in this way, we write cµν=κuµuν, and\n(kF)µνλρ= ˜κ(gµλuνuρ+gνρuµuλ−gµρuνuλ−gνλuµuρ), (6)\nsee Eq. (4), where uµ= (1,0,0,0) and now κand ˜κare the coefficients that determine the scale of\nLorentz violation. By choosing α= 1 (Feynman gauge) and ˜ κ= (1+κ\n2)κ, which is convenient to\nkeep the Lagrangian (5) formally covariant, we get\nL=−1\n4˜Fµν˜Fµν+i¯ψ˜∂µ˜γµψ−e¯ψ˜Aµ˜γµψ+1\n2(˜∂µ˜Aµ)2+(˜∂µ¯C)(˜∂µC), (7)\nwhere˜Fµν=˜∂µ˜Aν−˜∂ν˜Aµ, with˜∂µ= ((1+κ)∂0,∂i) and˜Aµ= ((1+κ)A0,Ai). The corresponding\nFeynman rules are shown in Fig. 1.\nFIG. 1: Feynman Rules. Continuous, wavy, and dashed lines repres ent the fermion, photon, and ghost\npropagators, respectively, with momenta ˜ pµ= ((1+κ)p0,pi). The fermion-photon vertex is the usual one,\n−ie˜γµ.\nIII. THE FREE ENERGY\nLet us now compute the free energy per unit of volume (pressur e) as a function of the temper-\natureTand of the coupling constant e, in the regime of high temperature. We shall calculate the\nexpression for the pressure to order e2, which has the form\nP=TlnZ\nV=P0+P2, (8)\nwhereP0is the zero order contribution in the coupling constant and P2is the second order one.\nThe free energy density is minus the expression (24).\n4A. One-loop contribution\nThe lowest-order contributions are given by the three one-l oop vacuum diagrams, displayed in\nFig. 2, and written in the imaginary time formalism as\nP0= tr/summationtext/integraldisplay\n{dp}ln∝ne}ationslash˜p+4(−1\n2)/summationtext/integraldisplay\ndpln˜p2+2(1\n2)/summationtext/integraldisplay\ndpln ˜p2, (9)\nrespectively, where we have introduced the shorthand notat ion\n/summationtext/integraldisplay\n{dp}=T/summationdisplay\np0=2π(n+1\n2)T/integraldisplayd3p\n(2π)3(10)\nfor fermionic loop momenta and\n/summationtext/integraldisplay\ndp=T/summationdisplay\np0=2πnT/integraldisplayd3p\n(2π)3(11)\nfor bosonic loop momenta, and ∝ne}ationslash˜p= ˜pµ˜γµ.\nAs usual, the bosonic contributions of Eq. (9) have four degr ees of freedom for the gauge field\n(Fig. 2(b)) and two degrees of freedom for the ghost field (Fig . 2(c)). The fermionic contribution,\n(a) ( b) ( c)\nFIG. 2: One loop vacuum diagrams.\nafter the calculation of the trace,\nPf\n0= 2/summationtext/integraldisplay\n{dp}ln[(1−Q2)(p2\n0(1+κ)2+p2)], (12)\nhas the same form as the bosonic contributions,\nPb\n0=−/summationtext/integraldisplay\ndpln(p2\n0(1+κ)2+p2). (13)\nIn order to evaluate these expressions we proceed similarly to [21], such that\nP0=/integraldisplayd3p\n(2π)3/bracketleftbigg|p|\n1+κ+4\nβln/parenleftbigg\n1+e−|p|β\n1+κ/parenrightbigg\n−2\nβln/parenleftbigg\n1−e−|p|β\n1+κ/parenrightbigg/bracketrightbigg\n, (14)\nwhich is independent of the parameter Q. The zero temperature (divergent) part is absorbed in a\nrenormalization of the vacuum energy [12]. Therefore, afte r integrating in the angular variables,\nwe get\nP0=2\nπ2β/integraldisplay∞\n0drr2ln/parenleftBig\n1+e−rβ\n1+κ/parenrightBig\n−1\nπ2β/integraldisplay∞\n0drr2ln/parenleftBig\n1−e−rβ\n1+κ/parenrightBig\n, (15)\n5and finally, we obtain\nP0=11π2\n180T4(1+κ)3. (16)\nTherefore we conclude that the only impact of the Lorentz sym metry breaking in the case of\nthe CPT-even Lorentz-breaking parameter consists in the mo dification of a constant factor.\nB. Two-loop contribution\nIn this subsection we study the two-loop contributions to th e free energy. The Lorentz violating\ncontribution to free energy up to the two-loop order (which c orresponds to the second order in e,\nis given by Fig. 3), which can be written as\nFIG. 3: Two-loop diagram.\nP2=1\n2e2/summationtext/integraldisplay\n{dp}/summationtext/integraldisplay\n{dq}tr/bracketleftbigg\n˜γµ1\n∝ne}ationslash˜p˜γµ1\n∝ne}ationslash˜q1\n(˜p+ ˜q)2/bracketrightbigg\n. (17)\nNote that, due to the fact that\n˜γµ1\n˜pα˜γα=γµ1\n˜pαγα, (18)\nthe above second order contribution is also independent of t he parameter Q, and so on for the\nother contributions. Thus, we can rewrite (17) as\nP2=1\n2e2/summationtext/integraldisplay\n{dp}/summationtext/integraldisplay\n{dq}tr/bracketleftbigg\nγµ˜pαγα\n˜p2γµ˜qβγβ\n˜q21\n(˜p+ ˜q)2/bracketrightbigg\n, (19)\nso that, after calculating the trace, we get\nP2= 2e2/summationtext/integraldisplay\n{dp}/summationtext/integraldisplay\n{dq}/bracketleftbigg\n−1\n˜p2˜q2+1\n˜q2(˜p+ ˜q)2+1\n˜p2(˜p+ ˜q)2/bracketrightbigg\n. (20)\nUsing the results for the sum-integrals,\n/summationtext/integraldisplay\n{dp}/summationtext/integraldisplay\n{dq}1\n˜p2˜q2=1\n576T4(1+κ)2, (21)\n/summationtext/integraldisplay\n{dp}/summationtext/integraldisplay\n{dq}1\n˜q2(˜p+ ˜q)2=−1\n288T4(1+κ)2, (22)\n6we arrive at the following Lorentz violating contribution t o the free energy,\nP2=−5\n288e2T4(1+κ)2. (23)\nThus, the modification in the free energy due to the Lorentz-b reaking parameter for the two-loop\norder consists only in multiplying by a constant, just as in t he one-loop order.\nTherefore, the expression for the pressure to order e2in the presence of Lorentz symmetry\nbreaking looks like\nP=π2T4\n9/bracketleftbigg11\n20(1+κ)3−5e2\n32π2(1+κ)2/bracketrightbigg\n. (24)\nIV. NUMERICAL ESTIMATIONS\nIn order to estimate a bound for the Lorentz violating parame terκ, we use the theoretical\npredictions of the primordial helium abundance Ydeveloped in the references [9–11]. A way to\ndetermineYconsists in the analysis of the change in thermodynamics qua ntities such as the energy\ndensityρ, the pressure Pand the neutrino temperature Tν. Using the result (24), let us now study\nthese changes in the presence of the Lorentz-breaking.\nThe contribution to the energy density can be found from the s tandard thermodynamic relation\nρ=−P+T(∂P/∂T), so that\nρ=π2T4\n15(N+δN), (25)\nwhereN=11\n4andδN≈ −0.007+8.236κ. Thus, we obtain\n∆ρ\nρ≈ −2.5×10−3+2.994κ. (26)\nThe fraction ∆ Yis affected by QED in several ways, as can be seen in [10]. The tot al effect is\napproximately\n∆Y≈2.9×10−4+0.15∆Tν\nTν+0.07∆ρ\nρ, (27)\nwhereTνdepends on energy density ρ(for more details, see [22]).\nThe usual theoretical result, without parameter κ, is ∆Y≈10−4, whereas the experimental one\nis ∆Y≈10−3[14]. Therefore, an upper boundfor κnecessary for the coincidence of the theoretical\nand experimental results must be κ∼10−2. This result agrees with the value found in [23, 24].\n7V. SUMMARY\nWe have calculated the contributions to the free energy in th e rotationally invariant Lorentz-\nviolating QED in one- and two-loop approximations at high te mperature. The corresponding\ncorrection to the pressure was then determined.\nWe also observe that Lorentz violation can be used to explain the difference between theoretical\nand experimental predictions of the primordial helium abun dance. By matching these predictions,\nwe have estimated the Lorentz-breaking parameter κ∼10−2, which agrees with that obtained in\n[23].\nAcknowledgements. This work was partially supported by Conselho Nacional de De sen-\nvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq) and Funda¸ c˜ a o de Amparo ` a Pesquisa do Estado de\nS˜ ao Paulo (FAPESP), Coordena¸ c˜ ao de Aperfei¸ coamento do Pessoal do Nivel Superior (CAPES:\nAUX-PE-PROCAD 579/2008) and CNPq/PRONEX/FAPESQ.\n[1] V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).\n[2] S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. 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Rev.\nD26, 2694 (1982).\n[10] A. Heckler, Phys. Rev. D 49, 611 (1994).\n[11] R. E. Lopez and M. S. Turner, Phys. Rev. D 59, 103502 (1999).\n[12] J. Kapusta, ”Finite-temperature field theory”, Cambridge Un iv. Press, 1989.\n[13] J. O. Andersen, ”On effective field theories at finite temperatu re”, arXiv:hep-ph/9709331.\n8[14] F. Iocco, G. Mangano, G. Miele, O. Pisanti and P. D. Serpico, Ph ys. Rept. 472, 1 (2009)\n[arXiv:0809.0631 [astro-ph]].\n[15] D. Colladay and V. A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997) [arXiv:hep-ph/9703464]; Phys. Rev.\nD58, 116002 (1998) [arXiv:hep-ph/9809521].\n[16] R. Jackiw and V. A. Kosteleck´ y, Phys. Rev. Lett. 82, 3572 (1999) [arXiv:hep-ph/9901358].\n[17] V. A. Kosteleck´ y, C. Lane, A. Pickering, Phys. Rev. D 65, 056006 (2002) [arXiv:hep-th/0111123].\n[18] D. Mattingly, Living Rev. Rel. 8, 5 (2005) [arXiv:gr-qc/0502097].\n[19] P. Arias, H. Falomir, J. Gamboa, F. Mendez and F. A. Schaposnik , Phys. Rev. D 76, 025019 (2007)\n[arXiv:0705.3263 [hep-th]].\n[20] T. Jacobson, S. Liberati and D. Mattingly, Ann. Phys. 321, 150 (2006) [arXiv:astro-ph/0505267].\n[21] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).\n[22] S. Weinberg, Gravitation and Cosmology , J. Wiley, New York, 1972.\n[23] C. D. Lane, Phys. Rev. D 72, 016005 (2005) [arXiv:hep-th/0505130].\n[24] V. A. Kostelecky, N. Russell, “ Data tables for Lorentz and CPT violation”, arXiv:0801.0287 [hep-ph].\n9"    },    {        "title": "1009.4717v1.Lorentz_violation_in_solar_neutrino_oscillations.pdf",        "content": "arXiv:1009.4717v1  [hep-ph]  23 Sep 2010November 9, 2018 5:7 WSPC - Proceedings Trim Size: 9in x 6in pr oceedings\n1\nLORENTZ VIOLATION IN\nSOLAR-NEUTRINO OSCILLATIONS\nJONAH E. BERNHARD\nDepartment of Physics and Astronomy, Swarthmore College,\nSwarthmore, PA 19081, United States\nE-mail: jbernha2@swarthmore.edu\nSolar-neutrino oscillations are considered using a massiv e model with pertur-\nbative Lorentz violation. The adiabatic approximation is u sed to calculate the\neffects of Lorentz violation to leading order. The results ar e more compact\nthan previous work involving vacuum oscillations, and are a ccurate for small\nLorentz-violating coefficients.\n1. Introduction\nNeutrinos are convenient for studying Lorentz violation due to the ir small\nmass and high velocities. Previous research has considered neutrin o oscilla-\ntions with short1and long2baselines, but mostly zero or constant matter\neffects. Very little has been done with variable matter effects, such as those\nin the Sun. We attempt to develop a perturbative technique for eas ily cal-\nculating Lorentz-violating effects in solar-neutrino oscillations.\n2. Theory\nWe assume neutrinos have the standard mass differences, and tre at Lorentz\nviolation as a perturbative effect. The neutrino Hamiltonian thus has three\nterms\nH=1\n2EU†∆m2U+V+δH, (1)\nwhere the first term on the right-hand side is the standard vacuum Hamil-\ntonian,Vis the Sun’s matter potential, and δHis a small Lorentz-violating\nHamiltonian. The diagonal mass squared matrix ∆ m2and the unitary mix-November 9, 2018 5:7 WSPC - Proceedings Trim Size: 9in x 6in pr oceedings\n2\ningmatrix Utakethestandardform,usingtheapproximateobservedvalues\n∆m2\n⊙≃8.0×10−17MeV2,∆m2\natm≃2.5×10−15MeV2,\nθ12≃34◦, θ23≃45◦, θ13≃0◦, δ≃0.\nSinceθ23≃45◦andθ13≃0◦, muon and tau neutrinos mix equally and\nPeµ=Peτ. Given this similarity, it is convenient to group muon and tau\ntogether and consider only two flavors. In this case, there are on ly two\nparameters: a mixing angle θand a mass difference ∆ m2, equivalent to\nθ12and ∆m2\n⊙, respectively, in the three-generation case. This significantly\nsimplifies many calculations. The two-generationcase will be consider ed for\nthe remainder of this work.\nThe Sun produces very large quantities of electron neutrinos in its c ore.\nAs they propagate to the surface, they interact with matter. Th is effect can\nbe described by the Sun’s matter potential V, given approximately by\nV=/parenleftbigg√\n2GFne0\n0 0/parenrightbigg\n, GFne≃1.32×10−17e−10.54R/R⊙MeV,(2)\nwhereGFis the Fermi coupling constant, neis the number density of\nelectrons in the Sun, and R⊙is the solar radius.3\nThe variable matter potential precludes an exact solution; howeve r,pro-\nvided the Hamiltonian changes sufficiently slowly, approximate analytic ex-\npressions can be obtained using the adiabatic approximation. This ap proxi-\nmation assumes energy eigenstates change slowly enough that neu trinos do\nnot transition between them as they propagate. High-frequency oscillations\nbetween flavors can then be averaged away, resulting in simpler exp ressions\nthan in previous work, such as reference 2. The averages can be c alculated\nfrom\n/angbracketleftP(0)\neb/angbracketright=/angbracketleftbig\n|νb|2/angbracketrightbig\n=/summationdisplay\na/vextendsingle/vextendsingle(U∗)ab(U0)a1/vextendsingle/vextendsingle2(3)\nwhere the superscript (0) indicates that these are zeroth-orde r (Lorentz-\ninvariant) probabilities, Uis the vacuum mixing matrix, and U0is the\neffective mixing matrix at R= 0. This evaluates to\n/angbracketleftP(0)\nee/angbracketright=s2\nθs2\nθ0+c2\nθc2\nθ0,/angbracketleftP(0)\neµ/angbracketright=s2\nθc2\nθ0+c2\nθs2\nθ0, (4)\nwheresθ≡sinθ,cθ≡cosθ,θis the vacuum mixing angle, and θ0is the\neffective mixing angle at R= 0. Note that the matter potential only needs\nto be known at the center and surface of the Sun.\nFigure 1 shows the electron neutrino survival probability as a funct ion\nof energy. The adiabatic approximation agrees with the numerical s olution\nalmost exactly.November 9, 2018 5:7 WSPC - Proceedings Trim Size: 9in x 6in pr oceedings\n3\nFig. 1. Average electron (left) and muon (right) neutrino su rvival probability (solid)\nand envelope function (dashed, given by 2 sθsθ0cθcθ0) as a function of energy in the\nadiabatic limit. For comparison, the probability was calcu lated numerically at 10000\nlogarithmically spaced energies from 0.1 to 100 MeV (grey).\n3. Lorentz violation\nLorentz violation is a natural extension to the theory. Assuming Lo rentz\nviolation is sufficiently small, it can be treated as a first-order pertur bation\nto the zeroth-order probabilities from Eq. (4).\nThe Lorentz-violating term has the form4\nδHab=1\nE/bracketleftbig\n(aL)αpα−(cL)αβpαpβ/bracketrightbig\nab(5)\nwhere (aL)α\nab, (cL)αβ\nabare complex Lorentz-violating coefficients and pαis\nthe neutrino energy-momentum four-vector, pα= (E,−/vector p)≃E(1,−ˆp). The\ncoefficients ( aL)T\nab, (cL)TT\nabare isotropic and hence introduce changes to\nenergy dependence, while the remaining coefficients are anisotropic and\ncause annual variations. The CPT-odd ( aL)α\nabare constant with energy;\nthe CPT-even ( cL)αβ\nabare linear.\nThe probabilities from the adiabatic approximation become\n/angbracketleftPeb/angbracketright=/angbracketleftP(0)\neb/angbracketright+2E\n∆m2Mij\nebδHij, (6)\nwhereMij\nebis a unitless, order one effective scale factor for δH, used to\ncalculate perturbed probabilities via simple linear combinations of coeffi -\ncients. It is determined by expanding the Lorentz-violating sine and cosine\nto leading order, which gives\nMij\nee=∆m2\nE∆λ/parenleftbigg−1\n4/parenleftbig\ns2\n2θc2θ0+s2\n2θ0c2θ/parenrightbig\n−sθc3\nθc2θ0−sθ0c3\nθ0c2θ\ns3\nθcθc2θ0+s3\nθ0cθ0c2θ1\n4/parenleftbig\ns2\n2θc2θ0+s2\n2θ0c2θ/parenrightbig/parenrightbigg\n(7)\nandMij\neµ=−Mij\nee, where ∆ λis the difference between the two eigenvalues\natR= 0.November 9, 2018 5:7 WSPC - Proceedings Trim Size: 9in x 6in pr oceedings\n4\nFig. 2. Comparison of the energy dependence of electron neut rino survival probability\nwith no Lorentz violation (solid), first-order expansion (d ashed, calculated from Mij\nee),\nand exact (dotted, calculated numerically from the adiabat ic approximation). The values\nused are ( aL)T\n11= 2×10−19MeV in the left plot, and ( cL)TT\n11= 2×10−19on the right.\nFigure 2 shows the effects of the isotropic coefficients on the energ y de-\npendence of the electron neutrino survival probability. The first- order ex-\npansion is effective with nonzero ( aL)T\n11, especiallyat lowenergies. It begins\nto fail with nonzero ( cL)TT\n11at high energies, since these effects grow with\nenergy. However, the Sun does not produce neutrinos with energ ies higher\nthan approximately 10 MeV, so this shortcoming is of limited relevance .\n4. Discussion\nSolar neutrinos are convenient for the study of Lorentz violation d ue to\nthe large quantities and significantly lower energy and longer baseline than\naccelerator experiments. The adiabatic approximation allows simple a nd\naccurate calculations of Lorentz-violating effects in solar-neutrin o oscilla-\ntions.\nFurther work will generalize the analysis to the three-generation c ase\nand explore additional consequences of Lorentz violation, such as an-\nnual variations caused by nonzero anisotropic coefficients and neu trino-\nantineutrino mixing.\nReferences\n1. V. A. Kostelecky and M. Mewes, Phys. Rev. D 70, 076002 (2004).\n2. J. S. Diaz, V. A. Kostelecky, and M. Mewes, Phys. Rev. D 80, 076007 (2009).\n3. J. N. Bahcall, M. H. Pinsonneault and S. Basu, Astrophys. J .555, 990 (2001).\n4. V. A. Kostelecky and M. Mewes, Phys. Rev. D 69, 016005 (2004)."    },    {        "title": "1106.6317v2.An_Osserman_type_condition_on__g_f_f__manifolds_with_Lorentz_metric.pdf",        "content": "An Osserman-type condition on g:f:f-manifolds\nwith Lorentz metric\u0003\nLetizia Brunetti\nAbstract\nA condition of Osserman type, called '-null Osserman condition, is introduced and stud-\nied in the context of Lorentz globally framed f-manifolds. An explicit example shows the\nnaturalness of this condition in the setting of Lorentz S-manifolds. We prove that a Lorentz\nS-manifold with constant '-sectional curvature is '-null Osserman, extending a result stated\nfor Lorentz Sasaki space forms. Then we state a characterization for a particular class of\n'-null Osserman S-manifolds. Finally, some examples are examined.\n2000 Mathematics Subject Classi\fcation. 53C25, 53C50, 53B30.\nKeywords and phrases. Lorentz metrics, Osserman condition, g:f:f -structure.\n1 Introduction\nThe study of the behaviour of the Jacobi operators is an important topic in Riemannian and, more\ngenerally, in semi-Riemannian geometry. More precisely, let ( M;g) be a Riemannian manifold with\ncurvature tensor Rand consider a point pinM. For any unit vector X2TpM, the symmetric\nendomorphism RX=Rp(\u0001;X)X:X?!X?is called the Jacobi operator with respect to X.\nIf the eigenvalues of RXare independent of the choices of Xandp, one says that ( M;g) is an\nOsserman manifold ([15]).\nSeveral results have been obtained looking for the solution of the Osserman Conjecture ([8, 22]),\nwhich states that an Osserman manifold is \rat or it is locally a rank-one symmetric space ([8, 9,\n10, 18, 19, 20]). Osserman manifolds have been studied in the Lorentzian context ([4, 13, 14]),\nwhere a complete solution for the Osserman conjecture has been found. Recently, in [1], Atindogbe\nand Duggal have introduced and studied suitable operators of Jacobi type associated with a semi-\nRiemannian degenerate metric.\nIn ([14]) the authors de\fned the Jacobi operator \u0016Ru,ubeing a null (or lightlike) vector tangent\nto a Lorentz manifold M. Given a unit timelike vector ztangent to M, they introduced and\ninvestigated the so-called null Osserman condition with respect to z(see also [15]).\nObviously, Lorentz almost contact manifolds are studied in this context. In particular, a Lorentz\nSasaki space form, whose characteristic vector \feld \u0018is timelike, is globally null Osserman with\nrespect to\u0018([15]). This result does not hold in the context of Lorentz globally framed f-manifolds\n(M2n+s;';\u0018\u000b;\u0011\u000b;g),s\u00152, as we will see with a counterexample.\nThis motivates the introduction of a more general condition of Osserman type, which we will\ncall'-null Osserman condition .\nThe main results of this paper state the links between the '-null Osserman condition and\nthe behaviour of the '-sectional curvature in Lorentz S-manifolds. After a preliminary section,\n\u0003The author wishes to express her thanks to professors A.M. Pastore and M. Falcitelli for helpful comments and\nfor many stimulating conversations. The work was supported by the Research Program n. 01.08 of University of\nBari\n1arXiv:1106.6317v2  [math.DG]  27 Sep 2012where we gather some facts about g:f:f -manifolds, needed in the rest of the paper, in Section 3\nwe discuss the relationship between the null Osserman condition and the Lorentz S-structures,\ngiving an example of Lorentz S-space form which does not satisfy the null Osserman conditions.\nWe endow the compact Lie group U(2) with a Lorentz S-structure of rank 2. This manifold is an\nS-space form with two characteristic vector \felds \u00181and\u00182,\u00181timelike, that does not satisfy the\nnull Osserman condition with respect to \u00181.\nIn Section 4 we introduce the notion of '-null Osserman manifold, and we state that a Lorentz\nS-manifold with constant '-sectional curvature is '-null Osserman with respect to the timelike\ncharacteristic vector \feld. We prove, in Section 5, an algebraic characterization for the Riemannian\ncurvature tensor \feld in a particular class of '-null Osserman Lorentz S-manifolds. Namely, we\ndivide this section in two parts. In the \frst subsection we deal with technical results which are\nvery useful in the second subsection where we state the main result. Moreover, we look at the\nbehaviour of the '-sectional curvature when the number of the eigenvalue of the Jacobi operator\nis one.\nIn particular, it is interesting to note that the existence of the only eigenvalue 1 of the Jacobi\noperator is related to the '-sectional \ratness of the manifold.\nFinally in the case of 4-dimensional '-null Osserman manifolds we \fnd a compact example,\nusing the Lie group U(2), and also a non compact example.\nAll manifolds, tensor \felds and maps are assumed to be smooth, moreover we suppose all\nmanifolds are connected. We will use the Einstein convention omitting the sum symbol for repeated\nindexes. Following the notations of S. Kobayashi and K. Nomizu ([12]), for the curvature tensor\nRwe haveR(X;Y )Z=rXrYZ\u0000rYrXZ\u0000r [X;Y]Z, andR(X;Y;Z;W ) =g(R(Z;W )Y;X),\nfor anyX;Y;Z;W2X(M). The sectional curvature Kp(\u0019) atpof a non-degenerate 2-plane\n\u0019=spanfX;Ygis given by\nKp(\u0019) =Kp(X;Y ) =Rp(X;Y;X;Y )\n\u0001(\u0019)=gp(Rp(X;Y )Y;X)\n\u0001(\u0019);\nwhere \u0001(\u0019) =g(X;X )g(Y;Y)\u0000g(X;Y )26= 0.\n2 Preliminaries\nFollowing [3, 6, 23], we recall some de\fnitions. An almost contact manifold is a (2 n+1)-dimensional\nmanifoldMendowed with an almost contact structure, i.e. M2n+1has a (1;1)-tensor \feld fsuch\nthatrank (f) = 2n, a 1-form\u0011and a vector \feld \u0018satisfyingf2(X) =\u0000X+\u0011(X)\u0018and\u0011(\u0018) = 1.\nMoreover, if gis a semi-Riemannian metric on M2n+1such that, for any X;Y2X(M2n+1),\ng(fX;fY ) =g(X;Y )\u0000\"\u0011(X)\u0011(Y);\nwhere\"=\u00061 according to the causal character of \u0018, thenM2n+1is called an inde\fnite almost\ncontact manifold. Such a manifold is said to be an inde\fnite contact manifold if d\u0011= \b, where\n\b is de\fned by \b( X;Y ) =g(X;fY ). Furthermore, if the structure ( f;\u0018;\u0011 ) is normal, that is\nN= [f;f] + 2d\u0011\n\u0018= 0, then the inde\fnite contact structure is called an inde\fnite Sasaki\nstructure and, in this case, the manifold ( M2n+1;f;\u0018;\u0011;g ) is called inde\fnite Sasaki.\nIn the Riemannian case a generalization of these structures have been studied by Blair in [2],\nby Goldberg and Yano in [17]. In [6] we studied such structures in semi-Riemannian context.\nA manifold Mis called a globally framed f-manifold (brie\ry g:f:f -manifold) if it is endowed\nwith a nowhere-vanishing (1 ;1)-tensor \feld 'of constant rank, such that ker 'is parallelizable i.e.\nthere exist global vector \felds \u0018\u000b,\u000b2f1;:::;sg, and 1-forms \u0011\u000b, satisfying\n'2=\u0000I+\u0011\u000b\n\u0018\u000band\u0011\u000b(\u0018\f) =\u000e\u000b\n\f:\n2Ag:f:f -manifold ( M2n+s;';\u0018\u000b;\u0011\u000b),\u000b2f1;:::;sg, is said to be an inde\fnite g:f:f -manifold\nifgis a semi-Riemannian metric satisfying the following compatibility condition\ng('X;'Y ) =g(X;Y )\u0000\"\u000b\u0011\u000b(X)\u0011\u000b(Y)\nfor any vector \felds X;Y , being\"\u000b=\u00061 according to whether \u0018\u000bis spacelike or timelike. Then,\nfor any\u000b2f1;:::;sgandX2X(M2n+s), one has\u0011\u000b(X) =\"\u000bg(X;\u0018\u000b).\nAn inde\fnite g:f:f -manifold is an inde\fnite S-manifold if it is normal and d\u0011\u000b= \b, for any\n\u000b2f1;:::;sg, where \b(X;Y ) =g(X;'Y ) for anyX;Y2X(M2n+s). The normality condition is\nexpressed by the vanishing of the tensor \feld N=N'+ 2d\u0011\u000b\n\u0018\u000b,N'being the Nijenhuis torsion\nof'.\nFurthermore, as proved in [6], the Levi-Civita connection of an inde\fnite S-manifold satis\fes:\n(rX')Y=g('X;'Y )e\u0018+e\u0011(Y)'2(X);\nwheree\u0018=Ps\n\u000b=1\u0018\u000bande\u0011=\"\u000b\u0011\u000b. Note that, for s= 1, we reobtain the notion of inde\fnite Sasaki\nmanifold.\nWe recall thatrX\u0018\u000b=\u0000\"\u000b'Xand ker'is an integrable \rat distribution since r\u0018\u000b\u0018\f= 0, for\nany\u000b;\f2f1;:::;sg. Anyway, an inde\fnite S-manifold is never \rat and it is never a real space\nform since, for example, K(X;\u0018\u000b) =\"\u000bfor any non lightlike X2Im'p.\nFor more details we refer to [6], where we describe three examples of non compact inde\fnite S-\nmanifolds. More precisely we construct two di\u000berent inde\fnite S-structures with metrics of index\n\u0017= 2 on R6and an inde\fnite S-structure with Lorentz metric on R4. Moreover, in [7] we give\nexplicit examples of compact inde\fnite g:f:f -manifolds and inde\fnite S-manifolds.\nWe also remark that every g:f:f -manifold is subject to the following topological condition: it\nhas to be either non compact or compact with vanishing Euler characteristic, since it admits never\nvanishing vector \felds. This implies that such a g:f:f -manifold always admits Lorentz metrics.\nLet us \fx few notation about curvature tensor \feld. As usual, a 2-plane \u0019=spanfX;'Xgin\nTpM, withp2MandX2Im'p, is said to be a '-plane and the sectional curvature at pof such\na plane, with Xa non lightlike vector, is called the '-sectional curvature atpand is denoted by\nHp(X).\nAn inde\fniteS-manifold ( M;';\u0018\u000b;\u0011\u000b;g) is said to be an inde\fnite S-space form if the '-\nsectional curvature Hp(X) is constant, for any point and any '-plane. In particular, in [6] it is\nproved that an inde\fnite S-manifold (M;';\u0018\u000b;\u0011\u000b;g) is an inde\fniteS-space form with Hp(X) =c\nif and only if the Riemannian (0 ;4)-type curvature tensor \feld Ris given by\nR(X;Y;Z;W ) =\u0000c+ 3\"\n4fg('Y;'Z )g('X;'W )\u0000g('X;'Z )g('Y;'W )g (1)\n\u0000c\u0000\"\n4f\b(W;X )\b(Z;Y)\u0000\b(Z;X)\b(W;Y )\n+ 2\b(X;Y )\b(W;Z )g\u0000fe\u0011(W)e\u0011(X)g('Z;'Y )\n\u0000e\u0011(W)e\u0011(Y)g('Z;'X ) +e\u0011(Y)e\u0011(Z)g('W;'X )\n\u0000e\u0011(Z)e\u0011(X)g('W;'Y )g;\nfor any vector \felds X,Y,ZandWonM, where\"=Ps\n\u000b=1\"\u000b.\nIn regard to the curvature tensor of an inde\fnite S-manifold, it is important to recall the\n3following formulas, for any X;Y;Z;W2Im'and any\u000b;\f;\r;\u000e2f1;:::;sg:\nR(X;\u0018\u000b;X;Y ) =\"\u000bg(X;X )g(e\u0018;Y) = 0;\nR(\u0018\u000b;X;\u0018\f;Y) =\"\u000b\"\fg(X;Y );\nR(\u0018\u000b;X;\u0018\f;\u0018\r) =\"\u000b\"\fg(X;\u0018\r) = 0; (2)\nR(\u0018\u000b;\u0018\u000e;\u0018\f;\u0018\r) = 0;\nR(X;Y;'Z;W ) +R(X;Y;Z;'W ) =\"P(X;Y ;Z;W );\nwhereP(X;Y ;Z;W ) = \b(X;Z)g(Y;W )\u0000\b(X;W )g(Y;Z)\u0000\b(Y;Z)g(X;W ) + \b(Y;W )g(X;Z).\nFinally, we recall some useful properties for a curvature-like algebraic tensor. Let ( V;g) be a\npseudo-Euclidean real vector space of index \u0017, 0<\u0017 < dimV. A multilinear map F:V4!Ris\ncalled a curvature-like map (or curvature-like algebraic tensor) if it satis\fes the following conditions\nF(y;x;z;w ) =\u0000F(x;y;z;w );\nF(z;w;x;y ) =F(x;y;z;w );\nF(x;y;z;w ) +F(x;z;w;y ) +F(x;w;y;z ) = 0:\nFor any non-degenerate 2-plane \u0019=spanfz;wginVit is possible to de\fne the number\nk(z;w) =F(z;w;z;w )\n\u0001(\u0019):\nIfk(z;w) is constant for any non-degenerate 2-plane and k(z;w) =kthen one gets F(x;y;z;w ) =\nk(g(x;z)g(y;w)\u0000g(y;z)g(x;w)). Now, arguments similar to those in Proposition 28 ([21, page\n229]), can be used to prove the following result.\nLemma 2.1. Let(V;g)be a Lorentz real vector space and F:V4!Ra curvature-like map. Then\nthe following conditions are equivalent.\na)F(x;y;z;w ) =k(g(x;z)g(y;w)\u0000g(y;z)g(x;w)),\nb)F(x;y;y;x ) = 0 for any degenerate plane \u0019=spanfx;yginV.\n3 Null Osserman condition and Lorentz S-manifolds\nIt is well-known that a Lorentz manifold has constant sectional curvature at a point pif and only\nif it satis\fes the Osserman condition at p.\nContrary to this, no Lorentz S-manifold can satisfy the Osserman condition since, as remarked\nin Section 2, a Lorentz S-manifold can not have constant sectional curvature.\nIn [14] the authors introduce another Osserman condition, named the null Osserman condition.\nNamely, let ( M;g) be a Lorentz manifold, p2Mandua null vector in TpM. Then the orthogonal\ncomplement u?ofuis a degenerate vector space since spanfug \u001au?. So one considers the\nquotient space \u0016 u?=u?=spanfugand the canonical projection \u0019:u?!\u0016u?. It is possible to\nde\fne a positive de\fnite inner product \u0016 gon \u0016u?putting\n\u0016g(\u0016x;\u0016y) =g(x;y);\nwhere, for any x;y2u?, \u0016x=\u0019(x) and \u0016y=\u0019(y).\nFrom now on, every bar-object will stand for geometrical objects related to \u0016 u?. So, \fxed a\nnull vector u2TpM, the Jacobi operator with respect to ucan be de\fned by the linear map\n\u0016Ru: \u0016u?!\u0016u?such that \u0016Ru\u0016x=\u0019(R(x;u)u) ([14] and De\fnition 3.2.1 in [15]).\n4Clearly, \u0016Ruis self-adjoint with respect to \u0016 g, hence \u0016Ruis diagonalizable.\nIn Lorentzian geometry it is well-known that a null vector uand a timelike vector zare never\northogonal. Hence, in a Lorentz manifold ( M;g), the null congruence set determined by a timelike\nvectorz2TpMatp, denoted by N(z), is de\fned by\nN(z) =fu2TpMjg(u;u) = 0; g(u;z) =\u00001g:\nA Lorentz manifold ( M;g) is called null Osserman with respect to a unit timelike vector z2TpM\nat a pointpif the characteristic polynomial of \u0016Ruis independent of u2N(z). LetLbe a timelike\nline subbundle of TM. If (M;g) is null Osserman with respect to each unit timelike vector z2L,\nthen (M;g) is called pointwise null Osserman with respect to L. Moreover, if ( M;g) is pointwise\nnull Osserman with respect to Land the characteristic polynomial of \u0016Ruis independent of the\nchoice of a unit z2L, then (M;g) is said to be globally null Osserman with respect to L.\nAnother set associated to a unit timelike vector zinTpMis the celestial sphere S(z) ofzgiven\nby\nS(z) =fx2z?jg(x;x) = 1g:\nAccording to a result in [15], using the celestial sphere of z, one can obtain all the elements of\nN(z). In fact one has\n8u2N(z)9jx2S(z) such that u=z+x:\nIt is very natural to use this de\fnition in the context of Lorentz contact manifolds. In particular, as\nstated in [15], Lorentz Sasaki space forms are globally null Osserman with respect to the timelike\ncharacteristic vector \feld. An easy example shows that in a Lorentz S-space form the null Osserman\ncondition with respect to a timelike characteristic vector does not hold.\nIndeed, considering the 4-dimensional manifold U(2) and the Lie algebra u(2), we denote by\n\u00181;\u00182;X;Y the left-invariant vector \felds on U(2), determined, in the same order, by the basis\nf{E11;\u0000{E22;E12\u0000E21;{(E12+E21)gofu(2), where ( Eij)i;j2f1;2gis the canonical basis of gl(2;C).\nThen, we get:\n[X;Y ] = 2\u00181+ 2\u00182;[X;\u0018\u000b] =\u0000Y;[Y;\u0018\u000b] =X; [\u0018\u000b;\u0018\f] = 0\nfor any\u000b;\f2f1;2g. Let us consider the left-invariant 1-forms \u00111and\u00112determined by the dual\n1-forms of{E11and\u0000{E22, respectively, and the left-invariant tensor \feld 'such that'(X) =Y,\n'(Y) =\u0000Xand'(\u00181) ='(\u00182) = 0. The manifold U(2) is compact, connected, with Euler number\n\u001f(U(2)) = 0, thus we can de\fne a left-invariant Lorentz metric gsuch that the vector \felds \u00181,\u00182,\nXandYform an orthonormal basis with g(\u00181;\u00181) =\u00001. Such a structure on U(2) is constructed\nin the Riemannian context ([11]) and then it is adapted to the Lorentzian case ([7]).\nThis structure is a normal inde\fnite g:f:f -structure and its associated Sasaki 2-form \b veri\fes\n\b =d\u0011\u000b, for any\u000b2f1;2g, so that it turns out to be a Lorentz S-structure on U(2). Moreover,\none sees at once that U(2) has constant '-sectional curvature 4. We see that U(2) does not verify\nthe null Osserman condition with respect to ( \u00181)p, for anyp2U(2). In fact, \fxing p2U(2) and\nputting\nu1=Xp+ (\u00181)p; u2=Yp+ (\u00181)p; u3= (\u00182)p+ (\u00181)p;\none hasu1;u2;u32N((\u00181)p). By (1), we have\nR(Yp;u1)u1=Yp+ 3g(Yp;'u 1)'u1+e\u0011(u1)e\u0011(u1)Yp= 5Yp;\nR((\u00182)p;u1)u1=2X\n\u000b=1(\u0018\u000b)p+Xp= (\u00182)p+u1:\n5Analogously, for u2, we obtain\nR(Xp;u2)u2=Xp+ 3Xp+Xp= 5Xp;\nR((\u00182)p;u2)u2=2X\n\u000b=1(\u0018\u000b)p+Yp= (\u00182)p+u2:\nFor anyz2u?\n3, we have\nR(z;u3)u3=\u0000e\u0011(u3)e\u0011(u3)'2z= 0;\nsincee\u0011(u3) = 0.\nThen it is evident that the eigenvalues of \u0016Ru1and \u0016Ru2are 5 and 1 whereas \u0016Ru3= 0.\n4 The '-Null Osserman Condition\nIn this section, inspired by the example of U(2), we introduce a new Osserman condition that will\nbe applied to Lorentz g:f:f -manifolds.\nLet (M;';\u0018\u000b;\u0011\u000b;g),\u000b2f1;:::;sg, be a Lorentz g:f:f -manifold, it is easy to check that the\ntimelike vector \feld must be a characteristic vector \feld. Without loss of generality we can assume\nthat\u00181is the timelike vector \feld.\nTaking in mind the example in Section 3, we claim that, if s\u00152, then the \ratness of ker '\nin\ruences the behaviour of the Jacobi operators \u0016Ru\u000bwithu\u000b= (\u00181)p+(\u0018\u000b)p, for any\u000b2f2;:::;sg\nandp2M. Since the matter is related to the null vector u\u000b, we give the following Osserman\ncondition.\nGiven a point pofM, the set\nS'((\u00181)p) =S((\u00181)p)\\Im'p;\nis called the '-celestial sphere of (\u00181)patp. We de\fne the analogous of the null congruence set,\ncalled the'-null congruence set, denoted by N'((\u00181)p), putting\nN'((\u00181)p) =fu2TpMju= (\u00181)p+x; x2S'((\u00181)p)g:\nNow, we are ready to state the de\fnition of '-null Osserman condition with respect to the\ntimelike vector ( \u00181)pat a pointp2M.\nDe\fnition 4.1. Let (M;';\u0018\u000b;\u0011\u000b;g) be a Lorentz g:f:f -manifold, dim M= 2n+s,n;s\u00151, with\ntimelike vector \feld \u00181and consider p2M.Mis called'-null Osserman with respect to ( \u00181)pat\na pointp2Mif the characteristic polynomial of \u0016Ruis independent of u2N'((\u00181)p), that is the\neigenvalues of \u0016Ruare independent of u2N'((\u00181)p).\nRemark 4.2. If (M;';\u0018;\u0011;g ) is a Lorentz almost contact manifold, then it can be considered as a\nLorentzg:f:f -manifold with s= 1. Obviously one has S((\u0018)p) =S'((\u0018)p) andN((\u0018)p) =N'((\u0018)p),\nfor anyp2M. It follows that the null Osserman condition with respect to \u0018pat a pointpcoincides\nwith the'-null Osserman condition at the same point.\nIt is clear that U(2) veri\fes the '-null Osserman condition with respect to ( \u00181)pat a point\np2U(2). In fact, we consider an arbitrary unit vector zof Im'pputtingz=aXp+bYp. Setting\nu4=z+ (\u00181)p, we haveu42N'((\u00181)p) and\nu?\n4=spanfXp+a(\u00181)p;Yp+b(\u00181)p;(\u00182)pg=spanf'u4;u4;(\u00182)pg:\n6Then, we get\nR('u4;u4)u4='u4+ 3'u4+'u4= 5'u4;\nR((\u00182)p;u4)u4=2X\n\u000b=1(\u0018\u000b)p\u0000'2u4= (\u00182)p+ (\u00181)p+z= (\u00182)p+u4:\nIt follows that, for any u=z+ (\u00181)pinN'((\u00181)p) withz2Im'pandg(z;z) = 1, the eigenvalues\nof\u0016Ruare 5 and 1, hence the eigenvalues of \u0016Ruare independent of the choice of u2N'((\u00181)p).\nTaking into account the classical de\fnitions for the Osserman manifolds, we introduce the\nglobally'-null Osserman condition.\nDe\fnition 4.3. Let (M;';\u0018\u000b;\u0011\u000b;g) be a Lorentz g:f:f -manifold, dim M= 2n+s,n;s\u00151, with\ntimelike vector \feld \u00181. IfMis'-null Osserman with respect to ( \u00181)p, for anyp2M, and the\ncharacteristic polynomial of \u0016Ruis independent of the choice of p2M, thenMis said to be globally\n'-null Osserman with respect to \u00181.\nLooking again at the example of U(2) one can see at once that it is a globally '-null Osserman\nmanifold with rispect to \u00181. In fact, it is clear that the eigenvalues of \u0016Ruare independent of the\npointp.\nIn the next theorem we prove, more generally, that each Lorentz S-space form satis\fes the\n'-null Osserman condition.\nTheorem 4.4. Let(M;';\u0018\u000b;\u0011\u000b;g),dimM= 2n+s, be a LorentzS-manifold with \u00181timelike\nand constant '-sectional curvature, Let p2M. ThenMveri\fes the '-null Osserman condition\nwith respect to the timelike characteristic vector at a point p.\nProof. Letp2M. Denoting by cthe'-sectional curvature, (1) holds with \"=s\u00002.\nLetube a vector in N'((\u00181)p), henceu= (\u00181)p+x1withx12S'((\u00181)p), and consider x2u?.\nWe have:\ng('u;'u ) =g(u;u)\u0000sX\n\u000b=1\"\u000b\u0011\u000b(u)\u0011\u000b(u) =\u00111(u)\u00111(u) = 1; (3)\ng('x;'u ) =g(x;u)\u0000sX\n\u000b=1\"\u000b\u0011\u000b(x)\u0011\u000b(u) =\u00111(x): (4)\nBy (1), (3) and (4) we compute R(x;u;u;w ) for anyw2TpM, obtaining\nRp(x;u;u;w ) =\u0000c+ 3(s\u00002)\n4fg('x;'w )\u0000\u00111(x)g('u;'w )g (5)\n\u00003\n4(c\u0000s+ 2)g(x;'u )g(w;'u )\n\u0000fe\u0011(w)e\u0011(x) +e\u0011(w)\u00111(x) +g('w;'x ) +e\u0011(x)g('w;'u )g:\nNow, being dim M= 2n+s, we considerfx1;'x 1;x3;:::;x 2ngas an orthonormal base of\nIm'p, which determines the bases B=fu;'x 1;(\u00182)p;:::; (\u0018s)p;x3;:::;x 2ngofu?andB=\nf'x1;(\u0016\u00182)p;:::; (\u0016\u0018s)p;\u0016x3;:::; \u0016x2ngof \u0016u?. For brevity, we also denote them by B=feig1\u0014i\u0014m,\n\u0016B=f\u0016eig1\u0014i\u0014m\u00001, beingm= 2n+s\u00001. In general, for any x2u?, one has\n\u0016Ru(\u0016x) =\u0000m\u00001X\ni=1Rp(x;u;u;ei)\u0016ei: (6)\n7By (5) and (6) we obtain\n\u0016Ru('x1) =fc+ 3(s\u00002)\n4+3\n4(c\u0000s+ 2)g'x1+'x1= (c+ 1)'x1;\n\u0016Ru(\u0016xj) =c+ 3(s\u00002)\n4\u0016xj+ \u0016xj=c+ 3s\u00002\n4\u0016xj;8j2f2;:::; 2ng\n\u0016Ru((\u0016\u0018\f)p) =sX\n\r=2e\u0011((\u0018\f)p)e\u0011((\u0018\r)p)(\u0016\u0018\r)p=sX\n\r=2(\u0016\u0018\r)p;8\f2f2;:::;sg:\nIt follows that the representation matrix of \u0016Ruwith respect to \u0016Bis independent of the choice of\nu2N'((\u00181)p). In particular, it is easy to compute that the other eigenvalues are 0 and s\u00001,\nhaving eigenvectors \u0016 x\u000b= (\u0016\u00182)p\u0000(\u0016\u0018\u000b)p,\u000b2f3;:::;sg, and \u0016x=Ps\n\f=2(\u0016\u0018\f)p, respectively. This\ncompletes the proof.\nBy the above proof we note that, as for U(2), each Lorentz S-manifold ( M;';\u0018\u000b;\u0011\u000b;g),\ndimM= 2n+s, with constant '-sectional curvature is globally '-null Osserman with respect\nto\u00181.\nFrom now on, since the Osserman conditions are formulated pointwise, to simplify the notation\nwe omit any reference to the point, there is no ambiguity.\n5 The '-null Osserman condition on Lorentz S-manifolds\nwith additional assumptions\nIn this section we proceed with the study of '-null Osserman manifolds and we will \fnd an expres-\nsion for the curvature tensor \feld of a '-null Osserman Lorentz S-manifold with two characteristic\nvector \felds, using a suitable expression for null vectors. An analogous statement can be found in\ndi\u000berent contexts ([15]). In the \frst part of this section we collect the technical issues needed for\nthe main result, which will be provided in the second subsection.\n5.1 Technical results\nIn [16] the authors have given the esplicit construction of a complex structure on a (4 m+ 2)-\ndimensional globally Osserman manifold with exactly two distinct eigenvalues of the Jacobi opera-\ntors with multiplicities 1 and 4 m(see also [15]). We will use such a construction, adapting it when\nthe manifold veri\fes the '-null Osserman condition at a point.\nFollowing [14, 15], we recall that if ( M;g) is a Lorentz manifold and uis a null vector of\nTpMthen a non-degenerate subspace W\u001au?such that dim W= dim \u0016u?is called a geometric\nrealization of \u0016u?. Let\u0019jW: (W;g)!(\u0016u?;g) be an isometry where, to simplify, we use the same\nlettergfor non-degenerate metrics on Wand \u0016u?. A vector x2Wis said to be a geometrically\nrealized eigenvector of\u0016RuinWcorresponding to an eigenvalue \u0015if\u0019jW(x) = \u0016xis an eigenvector\nof\u0016Ruwith eigenvalue \u0015([15]).\nRemark 5.1. Let (M;';\u0018\u000b;\u0011\u000b;g) be a (2n+s)-dimensional '-null Osserman Lorentz S-manifold\nat a pointp2Mandu2N'(\u00181). We suppose that the Jacobi operator \u0016Ru, restricted to u?\\Im',\nhas exactly two eigenvalues, c1andc2, with multiplicities 1 and 2 n\u00002.\nSinceu=\u00181+x,x2S'(\u00181), using (2), it is easy to see that the eigenvalues and the eigenvectors\nof the Jacobi operator \u0016Ruare connected with those of Rxjx?\\Im'. Namely, one can prove that\nv2x?\\Im'is an eigenvector of Rxrelated to the eigenvalue \u0015if and only if it is a geometrically\nrealized eigenvector of \u0016Rurelated to the eigenvalue \u0015+ 1 ([5]).\n8Now, let us \fx p2Mand, following [16], identify S'(\u00181)\u0018=S2n\u00001. For anyx2S2n\u00001consider\nthe operator Rx:x?\\Im'!x?\\Im'and the line bundle over the sphere S2n\u00001, de\fned by\nthe eigenspace corresponding to the eigenvalue c1\u00001 ofRx. Since any line bundle over a sphere\nis trivial, we have a map J:S'(\u00181)!S'(\u00181) such that Jx=vxfor anyx2S'(\u00181), wherevis a\nglobal unit section of the line bundle. To simplify the writing, we put \u0015=c1\u00001 and\u0016=c2\u00001.\nThen, with the following sequence of claims, we proceed along the same lines as the authors made\nin [16], which the reader is referred to for details.\nClaim (a). The mapJsatis\fesJ2(x) =\u0000xandJ(\u0000x) =\u0000J(x) for anyx2S'(\u00181).\nConsidered the 2-plane Vx=spanfx;Jxg, ifwis a unit vector in Vxthen there exists \u00122[0;2\u0019[\nsuch thatw= cos(\u0012)x+ sin(\u0012)Jx. De\fning z(w) =\u0000sin(\u0012)x+ cos(\u0012)Jx, one proves that z(w)\nis eigenvector of Rwcorresponding to \u0015, thenz(w) =\u0006Jw. Using this last formula, it follows\nJ2(x) =\u0000xandJ(\u0000x) =\u0000J(x) for anyx2S'(\u00181).\nClaim (b). J: Im'!Im'is linear.\nThe mapJis extended to Im 'puttingJ(ax) =aJ(x), wherea2R. Assuming that J(cos(\u0012)x+\nsin(\u0012)y) = cos(\u0012)Jx+sin(\u0012)Jyfor all angles \u0012and anyx,yunit vectors such that y?Vx, we obtain\nJ(x0+y0) =J(x0) +J(y0), for anyx0andy0such thaty0?V0\nx, which implies the claim.\nClaim (c). J(cos(\u0012)x+ sin(\u0012)y) = cos(\u0012)Jx+ sin(\u0012)Jyfor all angles \u0012and anyx,yunit vectors\nsuch thaty?Vx.\nLet us de\fne J0=\u0006Jand consider A\u0012= cos(\u0012)x+ sin(\u0012)y,B\u0012= cos(\u0012)Jx+ sin(\u0012)J0y. Assuming\nthatB\u0012is an eigenvector of RA\u0012, then one has B\u0012=\u0006JA\u0012, for any angle \u0012. For\u0012= 0 one has\nB\u0012=JA\u0012then the plus sign occurs. For \u0012=\u0019\n2it followsJ0y=B\u0012=JA\u0012=Jy, i.e.J0=J, that\nimplies the claim.\nClaim (d). RA\u0012(B\u0012) =\u0015B\u0012.\nNote that the claim is equivalent to proving that R(B\u0012;A\u0012;A\u0012;B\u0012) =\u0000\u0015. Expanding this last\nformula in term of x,Jx,yandJ0yone \fndsR(Jx;x;y;J0y) +R(J0y;x;y;Jx ) =\u0016\u0000\u0015. After the\nfollowing two technical lemmas one obtains the claim.\nLemma 5.2 ([16]).(1)R(z;v)w=\u0000R(z;w)vwhenv,wandzare unit vectors such that v?w\nandw;v?Vz.\n(2)R(z;v)w= 0whenv,wandzare unit vectors such that z?Vvandz;v?Vw.\n(3)2R(x;y;J0y;Jx ) =R(Jx;x;y;J0y).\n(4)2R(J0y;x;y;Jx ) =R(Jx;x;y;J0y).\nLemma 5.3 ([16]).The curvature tensor satis\fes R(Jx;x;y;J0y) =\u00062(\u0016\u0000\u0015)\n3.\nNow we give some remarks about a null vector of a Lorentz S-manifold with two characteristic\nvector \felds and then we prove a lemma.\nRemark 5.4. Let (M;';\u0018\u000b;\u0011\u000b;g),\u000b2f1;2g, be a LorentzS-manifold with timelike vector \feld\n\u00181andua null vector in TpM,p2M. SinceTM= Im'\bker', one can write uin the following\nway\nu=\u0015x+a\u00181+b\u00182;\nwherex2Im'withg(x;x) = 1. Being ua null vector, we have \u00152+b2=a2therefore there exists\n\u00122[0;2\u0019[ such that ucan be written as follows\nu=a(cos\u0012x+\u00181+ sin\u0012\u00182);\n9and it is not a restriction to use\nu= cos\u0012x+\u00181+ sin\u0012\u00182: (7)\nFor cos\u00126= 0 consider the vector w= tan\u0012\u00181+1\ncos\u0012\u00182. It is easy to check that wis a unit vector\northogonal to u, therefore\nu?=spanfu;'x;x 2;'x 2;:::xn;'xn;wg:\nAnyy2u?can be written as\ny=\u001au+\u0017y0+\u0014w; (8)\nwherey02spanf'x;x 2;'x 2;:::xn;'xng\u001aIm'p\\u?and\u001a;\u0014;\u00172R.\nWe need to de\fne two (1 ;3)-type tensors S\u0003andS\u0003putting\nS\u0003(x;y)v=e\u0011(y)e\u0011(v)x\u0000e\u0011(x)e\u0011(v)y+g(y;v)e\u0011(x)e\u0018\u0000g(x;v)e\u0011(y)e\u0018;\nS\u0003(x;y)v=\u0000g('y;'v )'2x+g('x;'v )'2y:\nRemark 5.5. Ifu2N'(\u00181) andy2Im'\\u?, then\ng(S\u0003(u;y)u;y)\u0000g(S\u0003(u;y)u;y) = 0:\nThe following lemma allows to state the expression of a curvature-like map FwhenFvanishes\non a particular type of degenerate 2-plane and it has a suitable behaviour with respect to the\ncharacteristic vector \felds.\nLemma 5.6. Let(M;';\u0018\u000b;\u0011\u000b;g),\u000b2f1;2g, be a Lorentz g:f:f -manifold with timelike vector\n\feld\u00181. Fixed a point p2M, letF: (TpM)4!Rbe a curvature-like map such that, for any\nx;y;v2Im'and any\u000b;\f;\r2f1;2g,\nF(x;\u0018\u000b;y;v) = 0; F(\u0018\u000b;x;\u0018\f;y) =\"\u000b\"\fg(x;y); F(\u0018\u000b;x;\u0018\f;\u0018\r) = 0; F(\u00181;\u00182;\u00181;\u00182) = 0:(9)\nThen the following statements are equivalent.\na)Fvanishes on any degenerate 2-plane\u0019=spanfu;yg, withu2N'(\u00181)andy2u?\\Im',\nb)F(x;y;v;z ) =g(S\u0003(x;y)v;z)\u0000g(S\u0003(x;y)v;z).\nProof. An easy computation, using Remark 5.5, shows that b))a).\nConversely, \fx p2Mand consider the curvature-like map Hsuch that, for any x;y;z;v2TpM,\nH(x;y;v;z ) =F(x;y;z;w )\u0000g(S\u0003(x;y)v;z) +g(S\u0003(x;y)v;z): (10)\nConditiona) and Remark 5.5 imply that Hvanishes on any degenerate 2-plane spanfu;yg, for\nanyu2N'(\u00181) andy2u?\\Im'. We start proving that Hvanishes on any degenerate 2-plane.\nTo see this, let ube a null vector of TpMas in (7) such that cos \u00126= 0. By the hypotheses and\nusing (8), for any y2u?we have\ng(S\u0003(u;y)u;y) = (\u001ag('u;'u ) +\u0017g('u;'y0))2\u0000g('u;'u )\u0000\n\u001a2g('u;'u ) +\u00172g('y0;'y0)\u0001\n=\u001a2g('u;'u )2\u0000\u001a2g('u;'u )2\u0000\u00172g('y0;'y0)g('u;'u ) =\u0000\u00172g(y0;y0)g('u;'u );\ng(S\u0003(u;y)u;y) =\u0000e\u0011(u)e\u0011(u)g(y;y);\nF(u;y;u;y ) =\u00172F(u;y0;u;y0) + 2\u0014\u0017F(u;y0;u;w ) +\u00142F(u;w;u;w ) =\u00172cos2\u0012F(x;y0;x;y0)\n+ (1\u0000sin\u0012)2(\u00172g(y0;y0) +\u00142) =\u00172g('u;'u )F(x;y0;x;y0) +e\u0011(u)e\u0011(u)g(y;y)\n=\u00172g('u;'u )F(u0;y0;u0;y0)\u0000\u00172g('u;'u )g(y0;y0) +e\u0011(u)e\u0011(u)g(y;y);\n10whereu0=x+\u00181which belongs to N'(\u00181). Hence one obtains\nH(u;y;u;y ) =\u0017g('u;'u )F(u0;y0;u0;y0); (11)\nwithu0=x+\u00181andy2u?\\Im'.\nIf cos\u0012= 0, thenu=\u00181\u0006\u00182andu?=spanfug\bIm'. By direct computation, it is easy to\nverify that\nH(u;y;u;y ) = 0; (12)\nfor anyy2u?.\nEquations (11) and (12) clearly imply that Hvanishes on any degenerate 2-plane. Applying\nLemma 2.1 to Hone has\nF(x;y;v;z ) =k(g(x;v)g(y;z)\u0000g(y;v)g(x;z)) +g(S\u0003(x;y)v;z)\u0000g(S\u0003(x;y)v;z): (13)\nBy de\fnition of k, using the hypotheses and (10), we deduce\nk=H(\u0018\u000b;x;\u0018\u000b;x)\n\"\u000bg(x;x)=F(\u0018\u000b;x;\u0018\u000b;x)\u0000g(x;x)\n\"\u000bg(x;x)= 0:\nThen, substituting in (13), we obtain our assertion.\n5.2 Main results\nNow, we consider the following two standard tensor \felds of type (1 ;3), evaluating them at the\npointp:\nR0(x;y)v=g(\u0019I(y);\u0019I(v))\u0019I(x)\u0000g(\u0019I(x);\u0019I(v))\u0019I(y);\nRJ(x;y)v=g\u0000\nJ(\u0019I(y));\u0019I(v)\u0001\nJ(\u0019I(x))\u0000g\u0000\nJ(\u0019I(x));\u0019I(v)\u0001\nJ(\u0019I(y))\n+ 2g\u0000\n\u0019I(x);J(\u0019I(y))\u0001\nJ(\u0019I(v));\nwhere\u0019I:TpM!Im'is the projection on Im 'andJis an almost Hermitian structure on Im '.\nIt is useful to note that RJandR0vanish on the triplets containing a characteristic vector and\nthat they are orthogonal to \u00181and\u00182, for anyx;y;v2TpM.\nNow we are ready to prove the following result.\nTheorem 5.7. Let(M;';\u0018\u000b;\u0011\u000b;g),\u000b2f1;2gandn > 1, be a (2n+ 2) -dimensional Lorentz\nS-manifold with timelike vector \feld \u00181. The following three statements are equivalent.\na)Mis'-null Osserman with respect to \u00181and for any u2N'(\u00181)the Jacobi operator\n\u0016RujIm'\\u?has exactly two distinct eigenvalues c1andc2with multiplicities 1and2(n\u00001),\nrespectively.\nb) There exist an almost complex structure JonIm'pandc1;c22Rsuch that, for any x;y;v2\nTpM,\nR(x;y)v=S\u0003(x;y)v\u0000S\u0003(x;y)v+c2R0(x;y)v+c1\u0000c2\n3RJ(x;y)v:\nc) 1. For any v2spanf\u00181g,x2\u0018?\n1we have\nR(x;v)v= (\u00111(v))2(x\u0000e\u0011(x)\u00182):\n112. There exist an almost complex structure JonIm'pandc1;c22Rsuch that, for any\nv;y;x2\u0018?\n1, we have\nR(x;y)v=\u00112(v)\u0000\n\u00112(y)x\u0000\u00112(x)y\u0001\n+\u0000\ng(y;v)\u00112(x)\u0000g(x;v)\u00112(y)\u0001e\u0018\n+g('y;'v )'2x\u0000g('x;'v )'2y+c2R0(x;y)v+c1\u0000c2\n3RJ(x;y)v:\nProof. We begin proving a))b). Under the assumption a)by Remark 5.1 we know that Im 'pis\nendowed with an almost complex structure Jsuch thatJxis an eigenvector of \u0016Rurelated to the\neigenvaluec1. To prove b), we consider the curvature-like map FonTpMgiven by\nF(x;y;v;z ) =R(x;y;v;z ) +\u0016g(R0(x;y)v;z) +\u001cg(RJ(x;y)v;z); (14)\nwhere\u0016;\u001c2R.\nWe want to apply Lemma 5.6 to F. About the hypotheses of Lemma 5.6, we see at once that F\nsatis\fes (9) since F=Rif one of its four arguments is a characteristic vector and (2) hold. Thus\nwe must only compute F(u;y;u;y ), for any degenerate vector u2N'(\u00181) andy2u?\\Im'.\nNamely, considering a null vector u2N'(\u00181) and a vector y2u?\\Im', we \fnd the suitable\nvalues of\u0016and\u001cinRfor whichFvanishes on degenerate 2-plane \u0019=spanfu;yg.\nPuttingy1=Jx12u?, one computes\nF(y1;u;u;y 1) =\u0000g(R(y1;u)u;y1) +\u0016g(R0(y1;u)u;y1) +\u001cg(RJ(y1;u)u;y1) (15)\n=\u0000c1+\u0016+ 3\u001c:\nAnalogously, if y2andy0\n2are orthonormal eigenvectors of \u0016Ruwith respect to the eigenvalue c2,\nthen we have\nF(y2;u;u;y 2) =\u0000g(R(y2;u)u;y2) +\u0016g(R0(y2;u)u;y2) +\u001cg(RJ(y2;u)u;y2)) (16)\n= (\u0000c2+\u0016);\nF(y2;u;u;y0\n2) =\u0000g(R(y2;u)u;y0\n2) +\u0016g(R0(y2;u)u;y0\n2) +\u001cg(RJ(y2;u)u;y0\n2)) = 0; (17)\nF(y2;u;u;y 1) =\u0000g(R(y2;u)u;y1) +\u0016g(R0(y2;u)u;y1) +\u001cg(RJ(y2;u)u;y1)) = 0: (18)\nNow, imposing F= 0, we get\n\u0016=c2and\u001c=c1\u0000c2\n3: (19)\nSo, since a vector yofu?\\Im'can be written as y=ay1+bjyj\n2, wherey1andyj\n2are eigenvectors\nof\u0016Ruinu?\\\u0018?\n1corresponding to c1andc2, respectively. By (15), (16), (17) and (18) we have\nF(y;u;u;y ) =a2F(y1;u;u;y 1) +abjF(y1;u;u;yj\n2) +abkF(yk\n2;u;u;y 1)\n+bkbjF(yk\n2;u;u;yj\n2) = 0:\nTherefore, applying Lemma 5.6, we obtain F(x;y;v;z ) =g(S\u0003(x;y)v;z)\u0000g(S\u0003(x;y)v;z), for any\nx;y;v;z2TpM. Then, by (14) and (19), we get\nR(x;y;v;z ) =g(S\u0003(x;y)v;z)\u0000g(S\u0003(x;y)v;z)\u0000c2g(R0(x;y)v;z)\u0000c1\u0000c2\n3g(RJ(x;y)v;z):\nThus, one obtains\nR(x;y)v=\u0000S\u0003(x;y)v+S\u0003(x;y)v+c2R0(x;y)v+c1\u0000c2\n3RJ(x;y)v:\n12The proofb))c) is straightforward. In fact, for any v2spanf\u00181g,x2\u0018?\n1, we have\nR(x;v)v=S\u0003(x;v)v= (\u00111(v))2(x+e\u0011(x)\u00181+\"1e\u0011(x)e\u0018) = (\u00111(v))2(x\u0000e\u0011(x)\u00182);\nso obtaining c)1.\nFor anyv;y;x2\u0018?\n1, by b)one gets\nR(x;y)v=\u00112(y)\u00112(v)x\u0000\u00112(x)\u00112(v)y+ (g(y;v)\u00112(x)\u0000g(x;v)\u00112(y))e\u0018\n+ (\u0000S\u0003+c2R0+c1\u0000c2\n3RJ)(x;y)v;\nthat is c)2.\nFinally, we prove c))a). Consider u2N(\u00181),u=\u00181+x1and puty1=Jx1. One has\nR(y1;u)u=R(y1;\u00181)\u00181+R(y1;x1)\u00181+R(y1;\u00181)x1+R(y1;x1)x1:\nSo, using c)1. and c)2., we have\nR(y1;\u00181)\u00181=y1andR(y1;x1)x1= (c1\u00001)y1:\nByc)2., for anyv2\u0018?\n1, it is clear that\ng(R(y1;x1)\u00181;v) =\u0000g(R(y1;x1)v;\u00181) = 0; g(R(y1;\u00181)x1;v) =g(R(x1;v)y1;\u00181) = 0:\nOn the other hand, if v=\u00181, then\ng(R(y1;x1)\u00181;\u00181) = 0; g(R(y1;\u00181)x1;\u00181) =\u0000g(y1;x1) = 0:\nHence, \u0016Ru(y1) =c1y1.\nAnalogously, considering y22(spanfx1;y1g)?\\Im', then\nR(y2;u)u=R(y2;\u00181)\u00181+R(y2;x1)\u00181+R(y2;\u00181)x1+R(y2;x1)x1:\nAs fory1, using c), it is easy to check that R(x1;v)y2= 0 andR(y2;x1)v= 0. Moreover, applying\nc)1., we get\nR(y2;\u00181)\u00181=y2:\nThe relation c)2. implies\nR(y2;x1)x1= (c2\u00001)y2:\nTherefore we have \u0016Ru(y2) =c2y2.\nFinally, to prove the '-null Osserman condition, we have to check that every eigenvalue does\nnot depend on u2N'(\u00181). In fact, by c)we \fnd\nR(\u00182;\u00181)\u00181= 0;\nR(\u00182;x1)x1=g(x1;x1)e\u0018=\u00181+\u00182:\nIt is easy to see that, for any v2\u0018?\n1\ng(R(\u00182;\u00181)x1;v) +g(R(\u00182;x1)\u00181;v) =\u00002g(R(\u00182;x1)v;\u00181) +g(R(\u00182;v)x1;\u00181)\n= 2g(x1;v)\u0000g(x1;v) =g(x1;v):\nMoreover, since\ng(R(\u00182;\u00181)x1;\u00181) +g(R(\u00182;x1)\u00181;\u00181) =\u0000g(R(\u00182;\u00181)\u00181;x1) = 0;\none obtains R(\u00182;\u00181)x1+R(\u00182;x1)\u00181=x1. Then one gets R(\u00182;u)u=\u00182+\u00181+x1=\u00182+u, so\n\u0016Ru(\u00182) =\u00182. This proves a).\nThis concludes the proof.\n13Remark 5.8. SinceRhas to satisfy the last formula in (2), for any x;y;v;z2Im'one gets\n(1\u0000c2)P(x;y;v;z) +c1\u0000c2\n3\u0000\ng(RJ(x;y)'v;z ) +g(RJ(x;y)v;'z )\u0001\n= 0: (20)\nIf'x1is an eigenvector of \u0016Ru, withu=\u00181+x12N'(\u00181), related to the eigenvalue c1, then\n'=\u0006Jand (20) yields c1\u00004c2+ 3 = 0.\nBy Theorem 4.4, it is a simple matter to prove the following result in the particular case of the\nJacobi operator with exactly one eigenvalue.\nProposition 5.9. Let(M;';\u0018\u000b;\u0011\u000b;g),\u000b2f1;2g,n > 1be a (2n+ 2) -dimensional Lorentz S-\nmanifold with timelike vector \feld \u00181. ThenMis'-null Osserman with respect to \u00181, and the\nJacobi operator \u0016Ruju?\\Im'has a single eigenvalue \u0015, if and only if it is a Lorentz S-space form\nwith'-sectional curvature c= 0. Moreover, \u0015= 1.\nNow we end dealing with the case n= 1, which is a special case because it is clear that any\n4-dimensional Lorentz g:f:f -manifold is '-null Osserman with respect to \u00181. More precisely, for\nanyu=\u00181+x12N'(\u00181) the only eigenvector of the Jacobi operator \u0016Ruju?\\Im'is realized\ngeometrically by 'x1inu?\\\u0018?\n1. Unlike before, the eigenvalue of the Jacobi operator does not\nnecessarily have to be one, as in the case of U(2), but, when it is one, the '-sectional curvature\nwill be zero. About this case we have a non compact example. It is carried out by R4endowed\nwith the Lorentz S-structure, constructed as follows ([6]). Denoting the standard coordinates with\nfx;y;z1;z2g, we de\fne on R4two vector \felds and two 1-forms putting\n\u0018\u000b=@\n@z\u000b; \u0011\u000b=dz\u000b+ydx;\nfor any\u000b2f1;2g. The tensor \felds 'andgare given in the standard basis by\nF:=0\nBB@0\u00001 0 0\n1 0 0 0\n0y0 0\n0y0 01\nCCAG:=0\nBB@1\n20\u0000y y\n01\n20 0\n\u0000y0\u00001 0\ny0 0 11\nCCA\nrespectively. It is easy to check that ( R4;';\u0018\u000b;\u0011\u000b;g),\u000b2f1;2g, is a LorentzS-manifold with\ndi\u000berent causal type of the characteristic vector \felds. Moreover it is a Lorentz space form with\n'-sectional curvature c= 0. Therefore, by (1), one obtains\nR(X;Y;V ) =e\u0011(X)g('V;'Y )2X\n\u000b=1\u0018\u000b\u0000e\u0011(Y)g('V;'X )2X\n\u000b=1\u0018\u000b\u0000e\u0011(Y)e\u0011(V)'2X\n+e\u0011(V)e\u0011(X)'2Y;\nfor anyX;Y;V2X(R4). Since Im '=hX;YiwhereX=p\n2(@\n@x\u0000y\u00181\u0000y\u00182) andY=p\n2@\n@y, one\nhas\n\u0016Ru'Z='Z; \u0016Ru\u00182=\u00182;\nfor anyZ=aX+bYandu=\u00181+Zwherea2+b2= 1. Then the only eigenvalue of \u0016Ru,\nu2N'(\u00181), is 1.\n14References\n[1] C. Atindogbe and K.L. 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(Debrecen, 1994), 201{211, Math. Appl., 350, Kluwer Acad. Publ., Dordrecht,\n1996.\n[14] E. Garc\u0013 \u0010a-R\u0013 \u0010o, D. Kupeli and M.E. V\u0013 azquez-Abal, One problem of Osserman in Lorentzian\ngeometry, Di\u000berential Geom. Appl. 7(1997), 85{100.\n[15] E. Garc\u0013 \u0010a-R\u0013 \u0010o, D. Kupeli and R. V\u0013 azquez-Lorenzo, Osserman manifolds in semi-Riemannian\ngeometry , Lecture Notes in Mathematics, Vol. 1777, Springer-Verlag, Berlin, 2002.\n[16] P. Gilkey, A. Swann and L. Vanhecke, Isoparametric geodesic spheres and a conjecture of\nOsserman concerning the Jacobi operator, Quart. J. Math. Oxford 46(1995), 299{320.\n[17] S.I. Goldberg and K. Yano, On normal globally framed f-manifolds, T^ ohoku Math. J. 22\n(1970), 362{370.\n[18] Y. Nikolayevsky, Osserman manifolds of dimension 8, Manuscripta Math. 115(2004), 31{53.\n15[19] Y. Nikolayevsky, Osserman Conjecture in dimension n6= 8;16, Math. Ann. 331(2005), 505{\n522.\n[20] Y. Nikolayevsky, On Osserman manifolds of dimension 16, Contemporary Geometry and Re-\nlated Topics, Proc. Conf. Belgrade, 2005 (2006), 379-398.\n[21] B. O'Neill, Semi-Riemannian geometry , Academic Press, New York, 1983.\n[22] R. Osserman, Curvature in the eighties, Amer. Math. Monthly 97(1990), 731-756\n[23] T. Takahashi, Sasakian manifold with pseudo-Riemannian metric, T^ ohoku Math. J. (2) 21\n(1969), 271{290.\n16"    },    {        "title": "1905.11562v1.Lorentz_Violation_and_Riemann_Finsler_Geometry.pdf",        "content": "arXiv:1905.11562v1  [hep-ph]  26 May 2019Proceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n1\nLorentz Violation and Riemann-Finsler Geometry\nBenjamin R. Edwards\nPhysics Department, Indiana University,\nBloomington, Indiana 47405, USA\nThe general charge-conserving effective scalar field theory incorporating viola-\ntions of Lorentz symmetry is presented. The dispersion rela tion is used to infer\nthe effect of spin-independent Lorentz violation on point pa rticle motion. A\nlarge class of associated Finsler spaces is derived, and the properties of these\nspaces are explored.\n1. Introduction\nConnections between Riemann-Finsler spaces and theories with Lor entz\nviolation have recently been uncovered.1A lack of physical examples is\nan obstacle on the path toward developing a strong intuition about F insler\nspaces. Inthe firstsection, thegeneraleffectivequadraticsca larfieldtheory\nincorporating violation of Lorentz symmetry will be developed. In th e next\nsection, a method to generate the lagrangian describing the motion of an\nanalogue point particle experiencing spin-independent Lorentz viola tion is\nderived. The last section explores the properties of these Finsler s paces.\nThese proceedings are based on results in Ref. 2.\n2. Field theory\nFor a complex scalar field φpropagating in an n-dimensional Minkowski\nspacetime with metric ηµν, the quadratic Lagrange density incorporating\nLorentz violation is\nL(φ,φ†) =∂µφ†∂µφ−m2φ†φ+1\n2/bracketleftbig\n∂µφ†(/hatwidekc)µν∂νφ−iφ†(/hatwideka)µ∂µφ+ h.c./bracketrightbig\n.(1)\nThe Lorentz violation is realized by the CPT-odd operator ( /hatwideka)µ, and the\nCPT-evenoperator( /hatwidekc)µν, eachofwhichcanincludecoefficientsforLorentz\nviolation associated with operators of arbitrarily large mass dimensio nd.\nThe hermiticity of Limplies these operators are themselves hermitian. InProceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n2\nthe special case of hermitian scalar fields, the term involving ( /hatwideka)µis pro-\nportional to a total derivative. It follows that CPT symmetry is gua ranteed\nwhenφ=φ†.\nField redefinitions can eliminate any traces present in the coefficients\nfor Lorentz violation by absorbing them into the terms with lower mas s\ndimension. We can therefore take them to be traceless without loss of\ngenerality. The commutativity of derivatives implies that they are to tally\nsymmetric in all their indices. From these considerations, it is found t hat\nthe coefficients contain (2 d−n+2)(d−1)!/(d−n+2)(n−2)!independent\ncomponents.\nThe dispersion relation for this theory is found to be\np2−m2+(/hatwidekc)µνpµpν−(/hatwideka)µpµ= 0, (2)\nwhere the operators ( /hatwidekc)µνand (/hatwideka)µare expressed in momentum space as\n(/hatwidekc)µν=∞/summationdisplay\nd=n(k(d)\nc)µνα1α2···αd−npα1pα2···pαd−n,\n(/hatwideka)µ=/summationdisplay\nd=n−1(k(d)\na)µα1···αd−n+1pα1pα2···pαd−n+1, (3)\nwith the sums running over even powers of p. For brevity, both types of\ncoefficients will be expressed without the aorcsubscripts in what follows,\nand the appropriate sign difference will be absorbed into the k(d)coefficient\nwhere the CPT properties will be determined by the mass dimension d.\n3. Classical kinematics\nA method has been developed to extract point particle lagrangians f rom a\ngiven field theory.3Using the three equations\nR(p) = 0, (4)\n∂p0\n∂Pj=−uj\nu0, (5)\nL=−uµpµ, (6)\nthe idea is to identify the centroid of a localized wave packet with the\npoint particle. The method starts with the dispersion relation R(p). Next,\nthe components of the classical velocity uµof the particle are related to\nthe group velocity of the corresponding wave packet. The last equ ation is\nrequired by translation invariance of L, along with the requirement that L\nbe one-homogeneousin the velocity, L(λu) =λL(u). The first two relationsProceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n3\ncan then be used to eliminate the components of the n-momentum to write\nLonly as a function of the velocity uµ.\nThese equations can be combined to produce a quadratic polynomial in\nL. For the case d=n, solving this quadratic leads to the exact lagrangian.\nFor the nonminimal cases d≥5, corrections to the usual lagrangian can\nbe determined through an iterative procedure. The process begin s with\nan expansion in ( k(d)) of the roots of the quadratic. Call this root L=\nL(uµ,pµ,(k(d))). Define L0=L(uµ,pµ,0) =−√uµηµνuν, and then Lj=\nL(uµ,p(j−1)\nµ(uµ),(k(d))) where p(j)\nµ=−∂Lj/∂uµ. This leads to\nL(d)\n3=L(d)\n0/bracketleftbig\n1−1\n2/tildewidek(d)−1\n8(d−n+1)2(/tildewidek(d))2\n+1\n8(d−n+2)2/tildewidek(d)\nα/tildewidek(d)α−1\n16(d−n+1)4(/tildewidek(d))3\n+1\n16(d−n+1)(d−n+2)2(2d−2n+1)/tildewidek(d)/tildewidek(d)α/tildewidek(d)α\n−1\n16(d−n+1)(d−n+2)3/tildewidek(d)α/tildewidek(d)αβ/tildewidek(d)β/bracketrightbig\n, (7)\nwhere the\n/tildewidek(d)=mn−d(k(d))α1α2···αd−n+2ˆuα1ˆuα2···ˆuαd−n+2,\n/tildewidek(d)\nα1···αl=mn−d(k(d))α1···αlαl+1···αd−n+2ˆuαl+1···ˆuαd−n+2,(8)\ncontain the directional dependence ˆ uµ=uµ/u,u=√uµηµνuν. This La-\ngrangian matches the first order correction found by Reis and Sch reck for\nthe nonminimal fermion sector using an ansatz-based technique.4\n4. Finsler geometry\nThe Finsler structure (or Finsler norm) plays a central role in the st udy of\nFinsler spaces. Classical lagrangians satisfy many of the requireme nts in\nthe definition of Finsler structures. Though there are important d ifferences\npreventing the lagrangians derived above from fulfilling the definition of a\nFinsler structure, a method exists to generate associated Finsler structures\nfrom a given lagrangian.5\nFor the lagrangians developed from the scalar field theories discuss ed\nabove, the prescription to generate a Finsler structure in this cas e is given\nbypx(u)→(−i)npx(y),k(d)x→ink(d)x,L→ −F=−y·p,ux→(i)nyx.\nAs a demonstration of the kinds of geometrical quantities one can c alcu-\nlate in these spaces, we use the Finsler space associated with the fir st order\nlimit of the lagrangian given in Eq. (7). The Finsler structure associat ed\nwith this lagrangian is\nF(d)=y−1\n2y/tildewidek(d). (9)Proceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n4\nThe fundamental tensor of a Finsler space determines the metric a nd\ntherefore also the geodesics. The definition of this tensor is gjk=\n1\n2∂2F2/∂yj∂yk. For the limit under consideration, the fundamental ten-\nsor is given by\ng(d)\njk=rjk(1+1\n2(d−n)/tildewidek(d))−1\n2(d−n+2)(d−n+1)/tildewidek(d)\njk\n+1\n2(d−n)(d−n+2)[/tildewidek(d)jˆyk+/tildewidek(d)kˆyj−/tildewidek(d)ˆyjˆyk].(10)\nInspection shows gjkreduces to a purely riemannian one for the cases d=n\nandd=n−2. This is consistent with the fact that the d=ncoefficient\ncan be absorbed into the metric at the level of the field theory, while a\nd=n−2 coefficient would correspond to a rescaling of the mass term.\nThe situation is not as straightforward for other values of mass dim en-\nsion. It has been demonstrated that the nonvanishing of the Cart an torsion\nimplies non-riemmannian nature of the underlying space.6Calculation of\nthis tensor shows it vanishes for d=nandd=n−2, and also in the case\nofn= 1, which represents a Riemann curve, but is nonvanishing in other\ncases. Calculation of the Matsumoto torsion7shows only d=n−1 reduces\nto a Randers metric.\nThe geodesics are obtained from the geodesic equation Fd\ndλ(yj/F) +\nGj= 0. A calculation shows the geodesic spray coefficients Gjare\n1\ny2Gj=1\n2/tildewideDj/tildewidek(d)+1\n2(d−n)ˆyj/tildewideD•/tildewidek(d)\n−1\n2(d−n+2)rjl/tildewideD•/tildewidek(d)\nl+/tildewideγj\n••, (11)\nwhere a•subscript denotes contraction of ˆ yjwith a lower jindex, with all\ncontractions exterior to any derivatives.\nIt is clear from this expression that if the backgroundfield is covaria ntly\nconstant with respect to the riemannian metric, /tildewideDj/tildewidek(d)l= 0, then the\ngeodesics are unaffected. This situation was conjectured to hold in general\nand is confirmed here, but remains unproved.\nReferences\n1. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n2. B.R. Edwards and V.A. Kosteleck´ y, Phys. Lett. B 786, 319 (2018).\n3. V.A. Kosteleck´ y and N. Russell, Phys. Lett. B 693, 443 (2010).\n4. J.A.A.S. Reis and M. Schreck, Phys. Rev. D 97, 065019 (2018).\n5. V.A. Kosteleck´ y, Phys. Lett. B 701, 137 (2011).\n6. A. Deicke, Arch. Math. 4, 45 (1953).\n7. M. Matsumoto, Tensor 24, 29 (1972),\nM. Matsumoto and S. H¯ oj¯ o , Tensor 32, 225 (1978)."    },    {        "title": "1807.09730v1.Regularity_and_asymptotic_behaviour_for_a_damped_plate_membrane_transmission_problem.pdf",        "content": "arXiv:1807.09730v1  [math.AP]  25 Jul 2018REGULARITY AND ASYMPTOTIC BEHAVIOUR FOR A\nDAMPED PLATE-MEMBRANE TRANSMISSION PROBLEM\nBIENVENIDO BARRAZA MART ´INEZ, ROBERT DENK,\nJAIRO HERN ´ANDEZ MONZ ´ON, FELIX KAMMERLANDER, AND MAX NENDEL\nAbstract. We consider a transmission problem where a structurally\ndamped plate equation is coupled with a damped or undamped wa ve equa-\ntion by transmission conditions. We show that exponential s tability holds\nin the damped-damped situation and polynomial stability (b ut no exponen-\ntial stability) holds in the damped-undamped case. Additio nally, we show\nthat the solutions first defined by the weak formulation, in fa ct have higher\nSobolev space regularity.\n1.Introduction\nIn this paper, we study a coupled plate-membrane system, whe re we assume\nstructural damping for the plate and damping / no damping for the wave equa-\ntion. More precisely, we consider the following geometric s ituation: Let Ω ⊂R2\nbe a bounded C4-domain with boundary Γ, and let Ω 2⊂Ω be a non-empty\nboundedC4-domain satisfying Ω2⊂Ω. We set Ω 1:= Ω\\Ω2andI:=∂Ω2. Then\nIis the interface between Ω 1and Ω2(see Figure 1for the geometric situation).\nByν, we denote the outer unit normal with respect to Ω 1both on Γ and on I.\nIn Ω1∪Ω2, we consider the plate-membrane (plate-wave) system\nutt+∆2u−ρ∆ut= 0 in(0,∞)×Ω1, (1.1)\nwtt−∆w+βwt= 0 in(0,∞)×Ω2, (1.2)\nwhereρ≥0 andβ≥0 are fixed constants. For ρ >0, we have structural\ndamping for the plate equation ( 1.1), whereas the coefficient β≥0 describes\nthe damping (or the absence of damping) for the wave equation (1.2). On the\nouter boundary Γ, we impose clamped (Dirichlet) boundary co nditions\n(1.3) u=∂νu= 0 on(0,∞)×Γ.\nOn the interface I, we have transmission conditions of the form\nu=won(0,∞)×I, (1.4)\nB1u= 0 on(0 ,∞)×I, (1.5)\nFinancial Support through DAAD, COLCIENCIAS via Project 12 1571250194 and the\nGermanResearchFoundationviaCRC1283“TamingUncertaint y”isgratefullyacknowledged.\nDate: July 26, 2018.\n2010Mathematics Subject Classification. 74K20; 74H40; 35B40; 35Q74.\nKey words and phrases. Plate-membrane equation, transmission problem, asymptot ic\nbehaviour.\n12 B. BARRAZA MART ´INEZ ET AL.\nνν\nΩ2Ω1\nΓ\nI\nFigure 1. The set Ω = Ω 1∪I∪Ω2.\nB2u−ρ∂νut=−∂νwon(0,∞)×I (1.6)\nwith\nB1u:= ∆u+(1−µ)B1uand B2u:=∂ν∆u+(1−µ)∂τB2u,\nwhere\nB1u:=−/a\\}b∇acketle{tτ,(∇2u)τ/a\\}b∇acket∇i}htandB2u:=/a\\}b∇acketle{tτ,(∇2u)ν/a\\}b∇acket∇i}ht.\nHere,µ∈/parenleftbig\n0,1\n2/parenrightbig\nis Poisson’s ratio and τ:= (−ν2,ν1)⊤. As we have a coupling of\na fourth-order equation with a second-order equation, we ha ve two transmission\nconditions (( 1.4) and (1.6)) and one boundary condition ( 1.5) on the interface\nI.\nFinally, the boundary-transmission problem ( 1.1)–(1.6) is endowed with ini-\ntial conditions of the form\nu|t=0=u0, ut|t=0=u1in Ω1, (1.7)\nw|t=0=w0, wt|t=0=w1in Ω2. (1.8)\nThe aim of the present paper is to investigate well-posednes s as well as reg-\nularity and stability of the solution of ( 1.1)–(1.8). Note that we omitted all\nphysical constants for simplicity. Concerning the modelli ng of plate-membrane\nsystems and more detailed models including physical consta nts, we refer to,\ne.g., [6], [15], and [19].\nIt is well known that the structurally damped plate equation itself has expo-\nnential stability and leads to the generation of an analytic C0-semigroup even in\ntheLp-setting, see [ 12] and the references therein. Due to the hyperbolic struc-\nture of the wave equation ( 1.2),Lp-theory is not feasible for the coupled system,\nand we will consider the plate-membrane system in an L2-framework. It is not\nhard to see the for all ρ≥0 andβ≥0 we have well-posedness, i.e. generation\nof aC0-semigroup in the corresponding L2-Sobolev spaces (see Theorem 2.2\nbelow). The main results of the present paper state that we ha ve exponential\nstability if both dampings are present ( ρ >0 andβ >0) but no exponential\nstability if the wave equation is undamped ( β= 0), see Theorems 3.1and3.2.\nIn the case of a structurally damped plate equation and an und amped wave\nequation (ρ >0 andβ= 0) we obtain polynomial stability (Theorem 5.2).\nMoreover, the “good” parabolic structure of the damped plat e equation implies\nhigh elliptic regularity for uandw(Theorem 4.5). In particular, the transmis-\nsion conditions ( 1.4)–(1.6) hold in the sense of boundary traces.A PLATE-MEMBRANE TRANSMISSION PROBLEM 3\nThere is a huge amount of literature on transmission problem s for elastic\nsystems, most of them dealing with wave-wave systems. For wa ve-plate trans-\nmission problems, we mention [ 17], where Kelvin-Voigt damping for the plate\nequation is considered (see also [ 18] for the one-dimensional case). In [ 5] expo-\nnential stability was obtained for a damped wave / damped pla te transmission\nproblem under some geometric condition which leads to a flat i nterface. This\nwas generalized in [ 26] to a model with curved middle surface by virtue of geo-\nmetric multiplier method. In [ 16], stabilization of a damped wave / damped\nplate system with variable coefficients is studied by means of a Riemannian\ngeometrical approach. For stability of coupled wave-plate systems within the\nsame domain, we mention, e.g., [ 21].\nWhereas the above mentioned results show exponential stabi lity for many\ncases of damped-damped systems, this cannot be expected in t he damped-\nundamped situation where we have, from a mathematical point of view, a\nparabolic-hyperbolic coupled system (see, e.g., [ 7], [8], [13] for heat-wave sys-\ntems).\nFor transmission problems in (thermo-)viscoelasticity, w e mention, e.g., [ 3],\n[4], [14], [23], and [24]. In particular, in [ 24] polynomial stability for a (thermo-)\nviscoelastic damped-undamped system with Kelvin-Voigt da mping has been\nshown. The proof is based on an extended version of a characte rization of\npolynomial stability due to Borichev and Tomilov [ 9]. It turns out that some\narguments in [ 24] can be adapted to the plate-wave situation considered in\nthe present paper to show that the system is not exponentiall y but polyno-\nmially stable (Section 5). We remark that our proof of polyno mial stability is\nbased on rather general methods which should be applicable f or other transmis-\nsion problems. However, by this method we do not obtain optim al polynomial\nrates. The proof of higher regularity (Section 4) uses argum ents similar to [ 11]\nwheredampedplate/undampedplatetransmissionproblemsw ereinvestigated.\nIn particular, we apply the classical theory of parameter-d ependent boundary\nvalue problems (see [ 2]) to obtain sufficiently good estimates in the damped\npart.\nThe structure of the paper is as follows: In Section 2, we defin e the basic\nspacesandoperatorsandshowthegenerationofa C0-semigroupofcontractions.\nExponential stability for ρ >0 andβ >0 and non-exponential stability for\nβ= 0 is shown in Section 3, whereas the proof of higher regulari ty based on\nparameter-elliptic theorycanbefoundinSection4. Finall y, polynomialstability\nforρ>0 andβ= 0 is proven in Section 5.\n2.Well-posedness\nWe denote by H2\nΓ(Ω1) the space of all u∈H2(Ω1) withu|Γ=∂νu|Γ= 0. On\nH2\nΓ(Ω1) we consider the inner product\n/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}htH2\nΓ(Ω1):=/integraldisplay\nΩ1∇2u:∇2v+µ[u,v]dx,\nwhere\n∇2u:∇2v:=ux1x1vx1x1+ux2x2vx2x2+2ux1x2vx1x24 B. BARRAZA MART ´INEZ ET AL.\nand\n[u,v] :=ux1x1vx2x2+ux2x2vx1x1−2ux1x2vx1x2\nfor allu,v∈H2\nΓ(Ω1). We thus have that\n/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}htH2\nΓ(Ω1)=µ/a\\}b∇acketle{t∆u,∆v/a\\}b∇acket∇i}htL2(Ω1)+(1−µ)/a\\}b∇acketle{t∇2u,∇2v/a\\}b∇acket∇i}htL2(Ω1)4\nfor allu,v∈H2\nΓ(Ω1). By Poincar´ e’s inequality, we have that\n/ba∇dblu/ba∇dbl2\nH2(Ω1)≤C/parenleftbig\n/ba∇dbl∇u/ba∇dbl2\nL2(Ω1)2+/ba∇dbl∇2u/ba∇dbl2\nL2(Ω1)4/parenrightbig\n=C/parenleftbig\n/ba∇dblux1/ba∇dbl2\nL2(Ω1)+/ba∇dblux2/ba∇dbl2\nL2(Ω1)+/ba∇dbl∇2u/ba∇dbl2\nL2(Ω1)4/parenrightbig\n≤C/parenleftbig\n/ba∇dbl∇ux1/ba∇dbl2\nL2(Ω1)2+/ba∇dbl∇ux2/ba∇dbl2\nL2(Ω1)2+/ba∇dbl∇2u/ba∇dbl2\nL2(Ω1)4/parenrightbig\n≤C/ba∇dbl∇2u/ba∇dbl2\nL2(Ω1)4\nforu∈H2\nΓ(Ω1). Here and in the following, Cdenotes a generic constant which\nmay change at each appearance. The above estimate shows that /ba∇dbl· /ba∇dblH2\nΓ(Ω1)is\nequivalent to the H2(Ω1)-norm onH2\nΓ(Ω1). In particular,/parenleftbig\nH2\nΓ(Ω1),/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htH2\nΓ(Ω1)/parenrightbig\nis a Hilbert space.\nWe will also use the following result on integration by parts .\nLemma 2.1. (See[10], p. 27.) For u∈H4(Ω1)∩H2\nΓ(Ω1)andv∈H2\nΓ(Ω1)it\nholds\n(2.1) /a\\}b∇acketle{t∆2u,v/a\\}b∇acket∇i}htL2(Ω)=/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}htH2\nΓ(Ω1)−/a\\}b∇acketle{tB1u,∂νv/a\\}b∇acket∇i}htL2(I)+/a\\}b∇acketle{tB2u,v/a\\}b∇acket∇i}htL2(I).\nLet\nH:=/braceleftbig\nU= (u1,v1,u2,v2)⊤∈H2\nΓ(Ω1)×L2(Ω1)×H1(Ω2)×L2(Ω2):u1|I=u2|I/bracerightbig\nbe endowed with the inner product\n/a\\}b∇acketle{tU,/tildewideU/a\\}b∇acket∇i}htH:=/a\\}b∇acketle{tu1,/tildewideu1/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{tv1,/tildewidev1/a\\}b∇acket∇i}htL2(Ω1)\n+/a\\}b∇acketle{t∇u2,∇/tildewideu2/a\\}b∇acket∇i}htL2(Ω2)2+/a\\}b∇acketle{tv2,/tildewidev2/a\\}b∇acket∇i}htL2(Ω2)\nforU,/tildewideU∈H. Then ( H,/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htH) is a Hilbert space. Note that we can omit the\nterm/a\\}b∇acketle{tu2,/tildewideu2/a\\}b∇acket∇i}htL2(Ω2)by Poincar´ e’s inequality as u1χΩ1+u2χΩ2∈H1\n0(Ω). Here,\nχΩjstands for the characteristic function of Ω j.\nWe introduce the operator matrix Agiven by\nA:=\n0 1 0 0\n−∆2ρ∆ 0 0\n0 0 0 1\n0 0 ∆ −β\n.\nBy (2.1), we have that\n/a\\}b∇acketle{tAU,/tildewideU/a\\}b∇acket∇i}htH=/a\\}b∇acketle{tv1,/tildewideu1/a\\}b∇acket∇i}htH2\nΓ(Ω1)−/a\\}b∇acketle{t∆2u1−ρ∆v1,/tildewidev1/a\\}b∇acket∇i}htL2(Ω1)\n+/a\\}b∇acketle{t∇v2,∇/tildewideu2/a\\}b∇acket∇i}htL2(Ω2)2+/a\\}b∇acketle{t∆u2−βv2,/tildewidev2/a\\}b∇acket∇i}htL2(Ω2)\n=/a\\}b∇acketle{tv1,/tildewideu1/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{t∇v2,∇/tildewideu2/a\\}b∇acket∇i}htL2(Ω2)2−/a\\}b∇acketle{tu1,/tildewidev1/a\\}b∇acket∇i}htH2\nΓ(Ω1)\n−/a\\}b∇acketle{tB2u1,/tildewidev1/a\\}b∇acket∇i}htL2(∂Ω1)+/a\\}b∇acketle{tB1u1,∂ν/tildewidev1/a\\}b∇acket∇i}htL2(∂Ω1)\n−ρ/a\\}b∇acketle{t∇v1,∇/tildewidev1/a\\}b∇acket∇i}htL2(Ω1)2+ρ/a\\}b∇acketle{t∂νv1,/tildewidev1/a\\}b∇acket∇i}htL2(∂Ω1)A PLATE-MEMBRANE TRANSMISSION PROBLEM 5\n−/a\\}b∇acketle{t∇u2,∇/tildewidev2/a\\}b∇acket∇i}htL2(Ω2)2−β/a\\}b∇acketle{tv2,/tildewidev2/a\\}b∇acket∇i}htL2(Ω2)−/a\\}b∇acketle{t∂νu2,/tildewidev2/a\\}b∇acket∇i}htL2(I)\nfor all sufficiently smooth U,/tildewideU. This leads us to the following interpretation of\nthe transmssion conditions ( 1.5) and (1.6): we say that Usatisfies the trans-\nmission conditions (1.5)and(1.6)weaklyif the equality\n(2.2)/a\\}b∇acketle{tAU,Φ/a\\}b∇acket∇i}htH=/a\\}b∇acketle{tv1,ϕ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{t∇v2,∇ϕ2/a\\}b∇acket∇i}htL2(Ω2)2−/a\\}b∇acketle{tu1,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)\n−ρ/a\\}b∇acketle{t∇v1,∇ψ1/a\\}b∇acket∇i}htL2(Ω1)2−/a\\}b∇acketle{t∇u2,∇ψ2/a\\}b∇acket∇i}htL2(Ω2)2−β/a\\}b∇acketle{tv2,ψ2/a\\}b∇acket∇i}htL2(Ω2)\nholds true for all Φ = ( ϕ1,ψ1,ϕ2,ψ2)⊤∈H2\nΓ(Ω1)×H2\nΓ(Ω1)×H1(Ω2)×H1(Ω2)\nsatisfyingϕ1=ϕ2andψ1=ψ2onI.\nNow, we consider the linear operator A:D(A)⊂H→H, U/mapsto→AUwith\nD(A) :=/braceleftbig\nU∈H:v1∈H2\nΓ(Ω1),v2∈H1(Ω2),∆2u1∈L2(Ω1),∆u2∈L2(Ω2),\nv1=v2onIand (1.5),(1.6) are weakly satisfied/bracerightbig\n.\nAs\n(2.3) Re /a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−ρ/ba∇dbl∇v1/ba∇dbl2\nL2(Ω1)2−β/ba∇dblv2/ba∇dbl2\nL2(Ω2)≤0\nfor allU∈D(A), the operator Ais dissipative. The same argument shows\nthat for any smooth solution ( u,w) of (1.1)-(1.6), the energy\nE(t) :=1\n2/integraldisplay\nΩ1µ|∆u(t)|2+(1−µ)|∇2u(t)|2+|ut(t)|2dx\n+1\n2/integraldisplay\nΩ2|∇w(t)|2+|wt(t)|2dx\nis decreasing and the dissipation is caused by the damping bo th in Ω 1and Ω2.\nMoreover, the system is still dissipative if only one of the d amping terms is\nactive (ρ+β >0) and the system is conservative if there is no damping at all\n(ρ=β= 0).\nIn what follows, we show that the system ( 1.1)-(1.6) is well-posed for any\nchoice ofρ≥0 andβ≥0.\nTheorem 2.2. The operator A:H⊃D(A)→Hgenerates a strongly con-\ntinuous semigroup (S(t))t≥0of contractions on H.\nProof.First, we show that 1 −Ais surjective. Let F= (f1,g1,f2,g2)⊤∈H.\nWe need to show that there exists a U= (u1,v1,u2,v2)⊤∈D(A) such that\n(1−A)U=F,i.e.\nu1−v1=f1,\nv1+∆2u1−ρ∆v1=g1,\nu2−v2=f2,\nv2−∆u2+βv2=g2.\nPlugging in vi=ui−fifori= 1,2, we have to solve\nu1+∆2u1−ρ∆u1=f1+g1−ρ∆f1, (2.4)\nu2−∆u2+βu2=f2+g2+βf2. (2.5)6 B. BARRAZA MART ´INEZ ET AL.\nMotivated by the notion of the weak transmission conditions , we introduce the\nspace\nV:={u= (u1,u2)⊤∈H2\nΓ(Ω1)×H1(Ω2) :u1=u2onI}.\nEndowed with the scalar product\n/a\\}b∇acketle{tu,/tildewideu/a\\}b∇acket∇i}htV=/a\\}b∇acketle{tu1,/tildewideu1/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{t∇u2,∇/tildewideu2/a\\}b∇acket∇i}htL2(Ω2)2(u,/tildewideu∈ V),\n(V,/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htV) becomes a Hilbert space.\nIn order to solve ( 2.4), (2.5), we will use the theorem of Lax-Milgram in the\nHilbert space V.Letb:V ×V → Rbe defined by\nb(u,ϕ) :=/a\\}b∇acketle{tu1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{tu1,ϕ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)+ρ/a\\}b∇acketle{t∇u1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1)2\n+(1+β)/a\\}b∇acketle{tu2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)+/a\\}b∇acketle{t∇u2,∇ϕ2/a\\}b∇acket∇i}htL2(Ω2)2.\nObviously, bis bilinear and continuous. Since\nb(u,u) =/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu1/ba∇dbl2\nH2\nΓ(Ω1)+ρ/ba∇dbl∇u1/ba∇dbl2\nL2(Ω1)2\n+(1+β)/ba∇dblu2/ba∇dbl2\nL2(Ω2)+/ba∇dbl∇u2/ba∇dbl2\nL2(Ω2)2\n≥ /ba∇dblu1/ba∇dbl2\nH2\nΓ(Ω1)+/ba∇dbl∇u2/ba∇dbl2\nL2(Ω2)2\nholds for all u∈ V,the bilinear form bis coercive on V.Hence, there exists a\nuniqueu∈ Vsatisfying\nb(u,ϕ) = Λ(ϕ) (2.6)\nfor allϕ∈ V, where the linear functional Λ: V →Ris given by\nΛ(ϕ) :=/a\\}b∇acketle{tf1+g1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+ρ/a\\}b∇acketle{t∇f1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1)2\n+/a\\}b∇acketle{tg2+(1+β)f2,ϕ2/a\\}b∇acket∇i}htL2(Ω2).\nNote that for ϕ1∈C∞\n0(Ω1) we have\n/a\\}b∇acketle{tu1,ϕ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)=/a\\}b∇acketle{tu1,∆2ϕ1/a\\}b∇acket∇i}htL2(Ω1)−/a\\}b∇acketle{tu1,B2ϕ1/a\\}b∇acket∇i}htL2(∂Ω1)+/a\\}b∇acketle{t∂νu1,B1ϕ1/a\\}b∇acket∇i}htL2(∂Ω1)\n=/a\\}b∇acketle{t∆u1,∆ϕ1/a\\}b∇acket∇i}htL2(Ω1).\nIn particular, for any ( ϕ1,ϕ2)∈C∞\n0(Ω1)×C∞\n0(Ω2)⊂ V, we have that ( 2.4)\nand (2.5) are satisfied in L2(Ω1) andL2(Ω2), respectively. This implies that\n∆2u1∈L2(Ω1) and ∆u2∈L2(Ω2).We set\nU:=\nu1\nu1−f1\nu2\nu2−f2\n∈H.\nFinally, using ( 2.4), (2.5) and (2.6), we calculate\n/a\\}b∇acketle{tAU,Φ/a\\}b∇acket∇i}htH=/a\\}b∇acketle{tv1,ϕ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)−/a\\}b∇acketle{t∆2u1−ρ∆v1,ψ1/a\\}b∇acket∇i}htL2(Ω1)\n+/a\\}b∇acketle{t∇v2,∇ϕ2/a\\}b∇acket∇i}htL2(Ω2)2+/a\\}b∇acketle{t∆u2−βv2,ψ2/a\\}b∇acket∇i}htL2(Ω2)\n=/a\\}b∇acketle{tv1,ϕ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)−/a\\}b∇acketle{tg1+f1,ψ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{tu1,ψ1/a\\}b∇acket∇i}htL2(Ω1)\n+/a\\}b∇acketle{t∇v2,∇ϕ2/a\\}b∇acket∇i}htL2(Ω2)2−/a\\}b∇acketle{tf2+g2,ψ2/a\\}b∇acket∇i}htL2(Ω2)\n=/a\\}b∇acketle{tv1,ϕ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{t∇v2,∇ϕ2/a\\}b∇acket∇i}htL2(Ω2)2A PLATE-MEMBRANE TRANSMISSION PROBLEM 7\n−ρ/a\\}b∇acketle{t∇v1,∇ψ1/a\\}b∇acket∇i}htL2(Ω1)2−β/a\\}b∇acketle{tv2,ψ2/a\\}b∇acket∇i}htL2(Ω2)\n−/a\\}b∇acketle{tu1,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)−/a\\}b∇acketle{t∇u2,∇ψ2/a\\}b∇acket∇i}htL2(Ω2)2\nfor any Φ = ( ϕ1,ψ1,ϕ2,ψ2)⊤∈H2\nΓ(Ω1)×H2\nΓ(Ω1)×H1(Ω2)×H1(Ω2) satisfying\nϕ1=ϕ2andψ1=ψ2onI.Therefore,Usatisfies the transmission conditions\nweakly. Hence, U∈D(A) and (1 −A)U=F.\nAsAis dissipative and 1 −Ais surjective, Agenerates a C0-semigroup of\ncontractions by the Lumer-Phillips Theorem. /square\nRemark 2.3. In the same way as in the previous proof, one can show that the\noperator Ais continuously invertible, i.e. 0 belongs to the resolvent setρ(A).\nTo show this, we now have to consider\n∆2u1=g1−ρ∆f1, (2.7)\n−∆u2=g2+βf2 (2.8)\ninstead of ( 2.4) and (2.5). The sesquilinear form Band the functional Λ are\nnow defined by B(u,ϕ) :=/a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}htVand\nΛ(ϕ) :=/a\\}b∇acketle{tg1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+ρ/a\\}b∇acketle{t∇f1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1)2+/a\\}b∇acketle{tg2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)+β/a\\}b∇acketle{tf2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)\nforu= (u1,u2),ϕ= (ϕ1,ϕ2)∈ V. The Riesz Representation Theorem implies\nthat there exists a unique solution u= (u1,u2)∈ Vsatisfying\n(2.9) B(u,ϕ) = Λ(ϕ) for all ϕ∈ V.\nIn particular, choosing ( ϕ1,ϕ2)∈C∞\n0(Ω1)×C∞\n0(Ω2)⊂ Vwe see that ( 2.7)\nand (2.8) hold in the sense of distributions in Ω 1and Ω2,respectively. As the\nright-hand side of ( 2.7) belongs to L2(Ω1), the same holds for the left-hand\nside, i.e. ∆2u1∈L2(Ω1). In the same way, we see that ( 2.8) holds as equality\ninL2(Ω2) and therefore ∆ u2∈L2(Ω2). Now, set\n(2.10) vi:=−fifori= 1,2.\nThenU:= (u1,v1,u2,v2)⊤∈Hand\n(2.11) ∆2u1−ρ∆v1=g1,−∆u2+βv2=g2.\nIn the same way as in the proof of Theorem 2.2, one sees that Usatisfies the\ntransmission conditions weakly. Therefore Ubeolongs to D(A) and satisfies\n−AU=F.\nOn other hand, if /tildewideU∈D(A) solves −A/tildewideU=F, thenB(/tildewideu,ϕ) = Λ(ϕ)\nholds for all ϕ∈ Vdue to the definition of D(A) and the weak transmission\nconditions. Therefore U=/tildewideU, and Ais a bijection. Since Ais the generator of\naC0−semigroup by Theorem 2.2,Ais closed and hence 0 ∈ρ(A).\n3.Results on exponential stability\nIn this section, we study exponential stability of the semig roup (S(t))t≥0\ngenerated by A. First, we consider the case where we have damping in both\nsub-domains, i.e., ρ>0 andβ >0. Itis nosurprisethat in this case exponential\nstability holds.8 B. BARRAZA MART ´INEZ ET AL.\nTheorem 3.1. Letρ >0andβ >0. Then the semigroup (S(t))t≥0is expo-\nnentially stable, i.e., for any U0∈D(A)andU(t) :=S(t)U0(t≥0)we have\nE(t)≤Ce−κtE(0)with positive constants Candκ, whereE(t) :=1\n2/ba∇dblU(t)/ba∇dbl2\nH.\nProof.LetU(t) = (u1(t),v1(t),u2(t),v2(t))⊤=S(t)U0withU0∈D(A). For\nthe energy E(t) we obtain\n(3.1)E′(t) = Re/a\\}b∇acketle{tAU(t),U(t)/a\\}b∇acket∇i}htH=−ρ/ba∇dbl∇v1(t)/ba∇dbl2\nL2(Ω1)2−β/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2).\nWe defineF(t) :=/a\\}b∇acketle{tu1(t),v1(t)/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{tu2(t),v2(t)/a\\}b∇acket∇i}htL2(Ω2)fort≥0. Then\n|F(t)| ≤1\n2/parenleftig\n/ba∇dblu1(t)/ba∇dbl2\nL2(Ω1)+/ba∇dblv1(t)/ba∇dbl2\nL2(Ω1)+/ba∇dblu2(t)/ba∇dbl2\nL2(Ω2)+/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2)/parenrightig\n.\nBy definition of H, we haveu1(t) =u2(t) on the interface I, and therefore\nthe function u(t) :=u1(t)χΩ1+u2(t)χΩ2belongs to H1\n0(Ω) for all t≥0. An\napplication of Poincar´ e’s inequality yields\n/ba∇dblu1(t)/ba∇dbl2\nL2(Ω1)+/ba∇dblu2(t)/ba∇dbl2\nL2(Ω2)=/ba∇dblu(t)/ba∇dbl2\nL2(Ω)≤C/ba∇dbl∇u(t)/ba∇dbl2\nL2(Ω)2\n=C/parenleftig\n/ba∇dbl∇u1(t)/ba∇dbl2\nL2(Ω1)2+/ba∇dbl∇u2(t)/ba∇dbl2\nL2(Ω2)2/parenrightig\n≤C/parenleftig\n/ba∇dblu1(t)/ba∇dbl2\nH2\nΓ(Ω1)+/ba∇dbl∇u2(t)/ba∇dbl2\nL2(Ω2)2/parenrightig\n.\nTherefore, for some constant c1>0 we get\n(3.2) |F(t)| ≤c1\n2/ba∇dblU(t)/ba∇dbl2\nH=c1E(t).\nUsingU′(t) =AU(t), we obtain\nF′(t) =/a\\}b∇acketle{tu′\n1(t),v1(t)/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{tu1(t),v′\n1(t)/a\\}b∇acket∇i}htL2(Ω1)\n+/a\\}b∇acketle{tu′\n2(t),v2(t)/a\\}b∇acket∇i}htL2(Ω2)+/a\\}b∇acketle{tu2(t),v′\n2(t)/a\\}b∇acket∇i}htL2(Ω2)\n=/ba∇dblv1(t)/ba∇dbl2\nL2(Ω1)−/a\\}b∇acketle{tu1(t),∆2u1(t)−ρ∆v1(t)/a\\}b∇acket∇i}htL2(Ω1)\n+/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2)+/a\\}b∇acketle{tu2(t),∆u2(t)−βv2(t)/a\\}b∇acket∇i}htL2(Ω2).\nNow we use the fact that U(t)∈D(A) and take Φ := (0 ,u1(t),0,u2(t))⊤in\nthe weak transmission conditions ( 2.2). We obtain\nF′(t) =/ba∇dblv1(t)/ba∇dbl2\nL2(Ω1)+/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2)+/a\\}b∇acketle{tΦ,AU(t)/a\\}b∇acket∇i}htH\n=/ba∇dblv1(t)/ba∇dbl2\nL2(Ω1)+/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2)−/ba∇dblu1(t)/ba∇dbl2\nH2\nΓ(Ω1)−/ba∇dbl∇u2(t)/ba∇dbl2\nL2(Ω2)2\n−ρ/a\\}b∇acketle{t∇u1(t),∇v1(t)/a\\}b∇acket∇i}htL2(Ω1)2−β/a\\}b∇acketle{tu2(t),v2(t)/a\\}b∇acket∇i}htL2(Ω2).\nBy Young’s inequality and Poincar´ e’s inequality in Ω 1, for every δ >0 there\nexists aCδ>0 such that\n−ρ/a\\}b∇acketle{t∇u1(t),∇v1(t)/a\\}b∇acket∇i}htL2(Ω1)2≤ρδ/ba∇dbl∇u1(t)/ba∇dbl2\nL2(Ω1)2+ρCδ/ba∇dbl∇v1(t)/ba∇dbl2\nL2(Ω1)2\n≤c2ρδ/ba∇dbl∇2u1(t)/ba∇dbl2\nL2(Ω1)2+ρCδ/ba∇dbl∇v1(t)/ba∇dbl2\nL2(Ω1)2\n≤c3ρδ/ba∇dblu1(t)/ba∇dbl2\nH2\nΓ(Ω1)+ρCδ/ba∇dbl∇v1(t)/ba∇dbl2\nL2(Ω1)2. (3.3)A PLATE-MEMBRANE TRANSMISSION PROBLEM 9\nIn the same way, using Poincar´ e’s inequality in Ω,\n−β/a\\}b∇acketle{tu2(t),v2(t)/a\\}b∇acket∇i}htL2(Ω2)≤βδ/ba∇dblu2(t)/ba∇dbl2\nL2(Ω2)+βCδ/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2)\n≤/tildewidec3βδ/parenleftbig\n/ba∇dblu1(t)/ba∇dbl2\nH2\nΓ(Ω1)+/ba∇dbl∇u2(t)/ba∇dbl2\nL2(Ω2)2/parenrightbig\n+βCδ/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2).(3.4)\nChoosingδsmall enough such that ( c3ρ+/tildewidec3β)δ≤1\n2, we get from ( 3.3) and\n(3.4) (again using Poincar´ e’s inequality for v1(t) in Ω1)\nF′(t)≤c4/parenleftbig\n/ba∇dbl∇v1(t)/ba∇dbl2\nL2(Ω1)2+/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2)/parenrightbig\n−1\n2/parenleftbig\n/ba∇dblu1(t)/ba∇dbl2\nH2\nΓ(Ω1)+/ba∇dbl∇u2(t)/ba∇dbl2\nL2(Ω2)2/parenrightbig\n. (3.5)\nNow letL(t) :=c5E(t) +F(t), where the constant c5satisfiesc5≥2c1and\nmin{ρ,β}c5≥c4+1\n2. By (3.1) and (3.5) we see that\n(3.6)L′(t)≤ −1\n2/parenleftig\n/ba∇dblu1(t)/ba∇dbl2\nH2\nΓ(Ω1)+/ba∇dbl∇v1(t)/ba∇dbl2\nL2(Ω1)2\n+/ba∇dbl∇u2(t)/ba∇dbl2\nL2(Ω2)2+/ba∇dblv2(t)/ba∇dbl2\nL2(Ω2)/parenrightig\n≤ −CE(t).\nAs|F(t)| ≤c1E(t)≤c5\n2E(t), we obtain\nc5\n2E(t)≤L(t)≤3c5\n2E(t).\nTherefore, ( 3.6) yieldsL′(t)≤ −κL(t) with some positive constant κ. By Gron-\nwall’s lemma, L(t)≤e−κtL(0) which yields\nE(t)≤CL(t)≤Ce−κtL(0)≤Ce−κtE(0). /square\nNow let us consider the case where the membrane is not damped, i.e.,β= 0.\nIn this situation, we show that the system is not exponential ly stable, nomatter\nifρ >0 orρ= 0. The proof of the following theorem follows an idea of [ 24,\nTheorem 3.5].\nTheorem 3.2. Forβ= 0andρ≥0, the system is not exponentially stable.\nProof.We consider the closed subspace\nH0:={0}×{0}×H1\n0(Ω2)×L2(Ω2)\nofH. OnH0we consider the C0-semigroup ( /tildewideS(t))t≥0with the generator\n/tildewiderA:H0⊃D(/tildewiderA)→H0, U/mapsto→\n1 0 0 0\n0 1 0 0\n0 0 0 1\n0 0 ∆ 0\nU\nwhereD(/tildewiderA) :={0} × {0} ×(H2(Ω2)∩H1\n0(Ω2))×H1\n0(Ω2). In the sequel, we\nwill show that S(t)−/tildewideS(t):H0→His compact. For U0∈H0, we consider\nE(t) :=1\n2/vextenddouble/vextenddoubleS(t)U0−/tildewideS(t)U0/vextenddouble/vextenddouble2\nH10 B. BARRAZA MART ´INEZ ET AL.\nfort≥0. Then, we denote by ( u,ut,w,wt)⊤:=S(t)U0the solution of the\ntransmission problem ( 1.1)–(1.6) and (0,0,/tildewidew,/tildewidewt)⊤:=/tildewideS(t)U0the solution of\nthewave equationinΩ 2withhomogeneousDirichlet boundaryconditions.Then\nz:=w−/tildewidewsolves the wave equation ztt−∆z= 0 in Ω 2withz|I=w|I=u|I.\nTherefore, applying the weak transmission conditions to\n/a\\}b∇acketle{tAS(t)U0,S(t)U0−/tildewideS(t)U0/a\\}b∇acket∇i}htH\nand using integration by parts for /a\\}b∇acketle{t∆/tildewidew(t),zt(t)/a\\}b∇acket∇i}htL2(Ω2), we obtain\nE′(t) = Re/parenleftig\n/a\\}b∇acketle{tu(t),ut(t)/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{tut(t),utt(t)/a\\}b∇acket∇i}htL2(Ω1)\n+/a\\}b∇acketle{t∇z(t),∇zt(t)/a\\}b∇acket∇i}htL2(Ω2)2+/a\\}b∇acketle{tzt(t),ztt(t)/a\\}b∇acket∇i}htL2(Ω2)/parenrightig\n= Re/parenleftig\n/a\\}b∇acketle{tu(t),ut(t)/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{tut(t),−∆2u(t)+ρ∆ut(t)/a\\}b∇acket∇i}htL2(Ω1)\n−/a\\}b∇acketle{t∂νz(t),zt(t)/a\\}b∇acket∇i}htL2(I)+/a\\}b∇acketle{tB2u(t)+ρ∂νut(t),ut(t)/a\\}b∇acket∇i}htL2(I)/parenrightig\n=−ρ/ba∇dbl∇ut(t)/ba∇dbl2\nL2(Ω1)2+Re(/a\\}b∇acketle{t∂ν/tildewidew(t),ut(t)/a\\}b∇acket∇i}htL2(I)).\nThis implies that\n(3.7)E(t)+/integraldisplayt\n0ρ/ba∇dbl∇ut(s)/ba∇dbl2\nL2(Ω1)2ds=/integraldisplayt\n0Re(/a\\}b∇acketle{t∂ν/tildewidew(s),ut(s)/a\\}b∇acket∇i}htL2(I))ds.\nNow, let (Uk\n0)k∈N⊂H0beaboundedsequence. We define /tildewidewkandukas/tildewidewandu\nbutwithU0beingreplaced by Uk\n0fork∈N.Then,asthesequence/parenleftbig\n∂ν/tildewidewk/parenrightbig\nk∈N⊂\nL2/parenleftbig\n[0,t];L2(I)/parenrightbig\nis uniformly bounded, there exists a subsequence of ( /tildewidewk)k∈N\nwhich will again be denoted by ( /tildewidewk)k∈Nsuch that/parenleftbig\n∂ν/tildewidewk/parenrightbig\nk∈Nconverges weakly\ninL2/parenleftbig\n[0,t];L2(I)/parenrightbig\n. Moreover, the sequences ( uk\nt)k∈N⊂L2/parenleftbig\n[0,t];H2(Ω1)/parenrightbig\nand\n(uk\ntt)k∈N⊂L2/parenleftbig\n[0,t];L2(Ω1)/parenrightbig\nare both uniformly bounded. By the Aubin-Lions\nLemma, there exists a subsequence of ( uk)k∈N, which will again be denoted by\n(uk)k∈Nsuch that (uk\nt)k∈N⊂L2/parenleftbig\n[0,t];H1(Ω1)/parenrightbig\nconverges. As the trace\nH1(Ω1)→L2(I), v/mapsto→v|I\nis continuous, we obtain that ( uk\nt)k∈N⊂L2/parenleftbig\n[0,t];L2(I)/parenrightbig\nis convergent. For\nk,l∈Nwe now denote by\nEkl(t) :=1\n2/vextenddouble/vextenddoubleS(t)(Uk\n0−Ul\n0)−/tildewideS(t)(Uk\n0−Ul\n0)/vextenddouble/vextenddouble2\nH.\nThen, by ( 3.7) we get that\nEkl(t)≤/integraldisplayt\n0/angbracketleftbig\n∂ν/tildewidewkl(s),ukl\nt(s)/angbracketrightbig\nL2(I)ds=/angbracketleftbig\n∂ν/tildewidewkl,ukl\nt/angbracketrightbig\nL2/parenleftbig\n[0,t];L2(I)/parenrightbig→0\nask,l→ ∞,where/tildewidewklanduklaredefinedas /tildewidewandubutwithU0beingreplaced\nbyUk\n0−Ul\n0fork,l∈N. Therefore, (( S(t)−/tildewideS(t))Uk\n0)k∈Nis a Cauchy sequence in\nHand thus convergent. This shows the compactness of S(t)−/tildewideS(t):H0→H.\nAs/tildewideS(t) is the semigroup related to the wave equation, its essentia l spectral\nradius equals 1. An application of [ 24, Theorem 3.3] gives that the essentialA PLATE-MEMBRANE TRANSMISSION PROBLEM 11\nspectral radius of S(t) equals 1, too, and thus ( S(t))t≥0is not exponentially\nstable. /square\n4.Higher regularity\nIn this section, we show that the functions in the domain of Ahave higher\nregularity, which implies that the transmission condition s hold in the strong\nsense of traces. For this, we need some results from the theor y of parameter-\nelliptic boundary value problems developed in the 1960’s ([ 2], see also [ 1]).\nLet Ω⊂R2be a domain, and let A(D) =/summationtext\n|α|≤2maα∂αbe a linear differential\noperatorinΩoforder2 m.ThenA(D)iscalledparameter-ellipticiftheprincipal\nsymbolA(iξ) :=/summationtext\n|α|=2maα(iξ)αsatisfies\nλ−A(iξ)/\\e}atio\\slash= 0 (Reλ≥0, ξ∈R2,(λ,ξ)/\\e}atio\\slash= 0).\nLetB1(D),...,B m(D) be linear boundary operators on ∂Ω of the form\nBj(D) =/summationtext\n|β|≤mjbjβ∂βof ordermj<2mwith principal symbols Bj(iξ) :=/summationtext\n|β|=mjbjβ(iξ)β. Then we say that the boundary value problem is parameter-\nelliptic ifA(D) is parameter-elliptic and if the following Shapiro-Lopat inskii\ncondition holds:\n(SL)Letx0∈∂Ω, and rewrite the boundary value problem ( A(D),\nB1(D),...,B m(D)) in the coordinate system associated with x0, which is ob-\ntained from the original one by a rotation after which the pos itivex2-axis has\nthe direction of the interior normal to ∂Ω atx0. Then the trivial solution w= 0\nis the only stable solution of the ordinary differential equat ion on the half-line\n(λ−A(iξ1,∂2))w(x2) = 0 (x2∈(0,∞)),\nBj(iξ1,∂2)w(0) = 0 (j= 1,...,m)\nfor allξ1∈Rand Reλ≥0 with (ξ1,λ)/\\e}atio\\slash= 0.\nIt was shown in [ 2] that the operator corresponding to a parameter-elliptic\nboundary value problem generates an analytic C0-semigroup in L2(Ω). We will\napply these results to ∆2and ∆ in Ω 1and Ω 2, respectively, with different\nboundary operators.\nLemma 4.1. The operator −∆2inΩ1, supplemented with the boundary oper-\nators B1andB2on∂Ω1, is parameter-elliptic. The same holds for −∆2with\nclamped boundary conditions u=∂νu= 0on∂Ω1and for−∆2with boundary\nconditionsu=B1u= 0on∂Ω1.\nProof.Obviously, the operator −∆2with symbol −(ξ2\n1+ξ2\n2)2is parameter-\nelliptic. Let x0∈∂Ω1, and choose a coordinate system associated with x0.\nThen thex1-axis is in tangential direction, while the positive x2-axis coincides\nwith the inner normal direction. In these coordinates, we ha ve to solve the\nordinary differential equation\n(4.1)/parenleftbig\nλ+(∂2\n2−ξ2\n1)2/parenrightbig\nw(x2) = 0 (x2∈(0,∞)),\n(B1(iξ1,∂2)w)(0) = 0,\n(B2(iξ1,∂2)w)(0) = 0.12 B. BARRAZA MART ´INEZ ET AL.\nBy the definition of the boundary operators B1andB2, we obtain the local\nsymbols B1(iξ1,∂2)w= (∂2\n2−µξ2\n1)wandB2(iξ1,∂2)w=/parenleftbig\n−∂3\n2+(2−µ)ξ2\n1∂2/parenrightbig\nw.\nNow we use the following identity for w∈H2((0,∞)), which is obtained by\nintegration by parts in (0 ,∞):\n(4.2)/a\\}b∇acketle{t(∂2\n2−ξ2\n1)2w,w/a\\}b∇acket∇i}htL2((0,∞))=µ/ba∇dbl(∂2\n2−ξ2\n1)w/ba∇dbl2\nL2((0,∞))\n+(1−µ)/parenleftig\n/ba∇dblξ2\n1w/ba∇dbl2\nL2((0,∞))+/ba∇dbl∂2\n2w/ba∇dbl2\nL2((0,∞))+2/ba∇dblξ1∂2w/ba∇dbl2\nL2((0,∞))/parenrightig\n+/parenleftbig\nB1(iξ1,∂2)w/parenrightbig\n(0)∂2w(0)+/parenleftbig\nB2(iξ1,∂2)w/parenrightbig\n(0)w(0).\nNote that this can be seen as a localized version of ( 2.1).\nLetwbe a stable solution of ( 4.1). We multiply the firstline in ( 4.1) byw(x2)\nand integrate over x2∈(0,∞). Due to the boundary conditions, all boundary\nterms in ( 4.2) disappear, and we obtain\n0 =/a\\}b∇acketle{t(λ+(∂2\n2−ξ2\n1)2)w,w/a\\}b∇acket∇i}htL2((0,∞))\n=λ/ba∇dblw/ba∇dbl2\nL2((0,∞))+µ/ba∇dbl(∂2\n2−ξ2\n1)w/ba∇dbl2\nL2((0,∞))\n+(1−µ)/parenleftig\n/ba∇dblξ2\n1w/ba∇dbl2\nL2((0,∞))+/ba∇dbl∂2\n2w/ba∇dbl2\nL2((0,∞))+2/ba∇dblξ1∂2w/ba∇dbl2\nL2((0,∞))/parenrightig\n.\nAs Reλ≥0 andµ∈(0,1), we can take the real part and obtain\n/ba∇dblξ2\n1w/ba∇dblL2((0,∞))= 0 and therefore w= 0 in the case ξ1/\\e}atio\\slash= 0. Ifξ1= 0, then\nλ/\\e}atio\\slash= 0, and we obtain λ/ba∇dblw/ba∇dbl2L2((0,∞))= 0 which again implies w= 0. Therefore,\nthe Shapiro-Lopatinskii condition (SL) holds.\nThe statement for the other combinations of boundary condit ions follows\nexactly in the same way, as in all cases the boundary terms in ( 4.2) disappear.\n/square\nWe will apply parameter-elliptic theory to a boundary value problem in Ω 1\nwith clamped boundary conditions on Γ and free boundary cond itions onI. In\nthe next lemma, we show that the resolvent of such boundary va lue problems\nwith ‘mixed’ boundary conditions exists and satisfies a unif orm estimate.\nLemma 4.2. Consider the boundary value problem\n(4.3)(λ+∆2)u=finΩ1,\nu=∂νu= 0onΓ,\nB1u=B2u= 0onI.\nThen there exists a λ0>0such that for all λ≥λ0and for all f∈L2(Ω1)\nthere exists a unique solution u∈H4(Ω1)of(4.3). Moreover, for all λ≥λ0\nthe uniform a priori-estimate\n(4.4) /ba∇dblu/ba∇dblH4(Ω1)+λ/ba∇dblu/ba∇dblL2(Ω1)≤C1/ba∇dblf/ba∇dblL2(Ω1)\nholds with a constant C1depending on λ0but not on λorf.\nProof.(i)We first show the existence of a solution. Let f∈L2(Ω1). We choose\nϕ1∈C∞(Ω1) with 0 ≤ϕ1≤1,ϕ1= 1 in a neighbourhood of Γ, and supp ϕ1∩\nI=∅. We setϕ2:= 1−ϕ1onΩ1. Further, let ψj∈C∞(Ω1),j= 1,2, with\n0≤ψj≤1,ψj= 1 on supp ϕj, suppψ1∩I=∅, and suppψ2∩Γ =∅.A PLATE-MEMBRANE TRANSMISSION PROBLEM 13\nBy Lemma 4.1, the boundary value problem given by −∆2and clamped\nboundary conditions is parameter-elliptic. Therefore (se e [2, Theorem 5.1]) for\nλ≥λ0with sufficiently large λ0there exists a unique solution u(1)=R1(λ)ψ1f\nof\n(λ+∆2)u(1)=ψ1fin Ω1,\nu(1)=∂νu(1)= 0 on∂Ω1.\nIn the same way, using parameter-ellipticity of the boundar y value problem\n(−∆2,B1,B2), there exists a unique solution u(2)=R2(λ)ψ2fof\n(λ+∆2)u(2)=ψ2fin Ω1,\nB1u(2)=B2u(2)= 0 on∂Ω1.\nMoreover, the a priori-estimate\n(4.5) /ba∇dblu(j)/ba∇dblH4(Ω1)+λ/ba∇dblu(j)/ba∇dblL2(Ω1)≤c2/ba∇dblψjf/ba∇dblL2(Ω1)\nholds for all λ≥λ0with a constant c2independent of λandf(see [2, Theo-\nrem 4.1]).\nForλ≥λ0, we define\nR(λ)f:=ϕ1R1(λ)ψ1f+ϕ2R2(λ)ψ2f.\nBy the product rule,\n(λ+∆2)R(λ)f=ϕ1(λ+∆2)R1(λ)ψ1f+ϕ2(λ+∆2)R2(λ)ψ2f\n+S1(D)R1(λ)ψ1f+S2(D)R2(λ)ψ2f,\nwhereS1(D) andS2(D) are linear partial differential operators of order 3 de-\npending on the choice of ϕ1, but not on λorf. As (λ+ ∆2)Rj(λ)ψjf=ψjf\nandϕjψj=ϕj,j= 1,2, we obtain\n(4.6) ( λ+∆2)R(λ)f= (1+T(λ))f\nwithT(λ)f:=S1(D)R1(λ)ψ1f+S2(D)R2(λ)ψ2f. AsSj(D) are bounded linear\noperators from H3(Ω1) toL2(Ω1), we can estimate\n/ba∇dblSj(D)Rj(λ)ψjf/ba∇dblL2(Ω1)≤C/ba∇dblRj(λ)ψjf/ba∇dblH3(Ω1)\n≤δ/ba∇dblRj(λ)ψjf/ba∇dblH4(Ω1)+Cδ/ba∇dblRj(λ)ψjf/ba∇dblL2(Ω1)\n≤c2δ/ba∇dblf/ba∇dblL2(Ω1)+c2\nλCδ/ba∇dblf/ba∇dblL2(Ω1).\nHere we used the interpolation inequality and ( 4.5). Now we first choose δ>0\nsmall enough such that c2δ≤1\n4and thenλ≥λ0withλ0large enough such\nthatc2\nλCδ≤1\n4. Therefore, the norm of T(λ) as a bounded operator in L2(Ω1)\nis not larger than1\n2, and 1+T(λ) is invertible. So we can define\nu:=R(λ)(1+T(λ))−1f∈H4(Ω1).\nFrom (4.6) we see (λ+∆2)u=f, and by definition of R(λ) we have\nu|Γ= (R1(λ)ψ1f)|Γ= 0, ∂νu|Γ= 0\nas well as\nBju|I=Bj/parenleftbig\nϕ2R2(λ)ψ2f/parenrightbig\n|I=BjR2(λ)ψ2f|I= 0 (j= 1,2).14 B. BARRAZA MART ´INEZ ET AL.\nTherefore,uis a solution of the boundary value problem ( 4.3).\n(ii)Now we show that every solution of ( 4.3) satisfies the a priori-estimate\n(4.4). Letu∈H4(Ω1) be a solution of ( 4.3). Thenu(1):=uϕ1is a solution of\nthe boundary value problem\n(λ+∆2)u(1)=ϕ1f+/tildewideS1(D)uin Ω1,\nu(1)=∂νu(1)= 0 on∂Ω1,\nwhere/tildewideS1(D) is a linear partial differential operator of order 3. By param eter-\nelliptic theory [ 2, Theorem 4.1], u(1)satisfies\n/ba∇dblu(1)/ba∇dblH4(Ω1)+λ/ba∇dblu(1)/ba∇dblL2(Ω1)≤C/parenleftbig\n/ba∇dblf/ba∇dblL2(Ω1)+/ba∇dblu/ba∇dblH3(Ω1)/parenrightbig\n.\nThe same holds for u(2):=ϕ2uby parameter-ellipticity of ( −∆2,B1,B2). For\nthe sumu=u(1)+u(2), we get\n/ba∇dblu/ba∇dblH4(Ω1)+λ/ba∇dblu/ba∇dblL2(Ω1)≤C/parenleftbig\n/ba∇dblf/ba∇dblL2(Ω1)+/ba∇dblu/ba∇dblH3(Ω1)/parenrightbig\n.\nNow, by interpolation inequality again, we can estimate\n/ba∇dblu/ba∇dblH4(Ω1)+λ/ba∇dblu/ba∇dblL2(Ω1)≤C/ba∇dblf/ba∇dblL2(Ω1)+δ/ba∇dblu/ba∇dblH4(Ω1)+Cδ/ba∇dblu/ba∇dblL2(Ω1).\nChoosingδ≤1\n2and thenλ0>2Cδ, we can absorb the u-dependent terms on\nthe right-hand side and obtain\n/ba∇dblu/ba∇dblH4(Ω1)+λ/ba∇dblu/ba∇dblL2(Ω1)≤C/ba∇dblf/ba∇dblL2(Ω1)(λ≥λ0).\nThis also yields uniqueness of the solution. /square\nCorollary 4.3. Letf∈L2(Ω1),g1∈H7/2(Γ),g2∈H5/2(Γ),h1∈H3/2(I),\nandh2∈H1/2(I). Then for sufficiently large λ0>0, the boundary value prob-\nlem\n(4.7)(λ0+∆2)u=finΩ1,\nu=g1onΓ,\n∂νu=g2onΓ,\nB1u=h1onI,\nB2u=h2onI\nhas a unique solution u∈H4(Ω1). Moreover, the a priori-estimate\n(4.8)/ba∇dblu/ba∇dblH4(Ω1)≤C2/parenleftig\n/ba∇dblf/ba∇dblL2(Ω1)+/ba∇dblg1/ba∇dblH7/2(Γ)+/ba∇dblg2/ba∇dblH5/2(Γ)\n+/ba∇dblh1/ba∇dblH3/2(I)+/ba∇dblh2/ba∇dblH1/2(I)/parenrightig\nholds with a constant C2>0which depends on λ0but not onuor on the data.\nProof.We defineG:= (g1,g2,0,0)⊤on Γ andH:= (0,0,h1,h2−(divν)h1)⊤\nonI. By [25], Section 4.7.1, p. 330, the map\nR:u/mapsto→/parenleftbig\nu|∂Ω1,∂νu|∂Ω1,∂2\nνu|∂Ω1,∂3\nνu|∂Ω1,/parenrightbig⊤\nisaretractionfrom H4(Ω1)to/producttext3\nj=0H4−j−1/2(∂Ω1).LetEdenoteacoretraction\ntoR, and set\nu(1):=E/parenleftbig\nχΓG+χIH/parenrightbig\n∈H4(Ω1).A PLATE-MEMBRANE TRANSMISSION PROBLEM 15\nThe boundary operators B1andB2can be expressed in terms of normal and\ntangential derivatives as (see [ 20], Propositions 3C.7 and 3C.11)\nB1u(1)=∂2\nνu(1)+µ∂2\nτu(1)+µ(divν)∂νu(1),\nB2u(1)=∂3\nνu(1)+∂ν∂2\nτu(1)+(1−µ)∂τ∂ν∂τu(1)+∂ν[(divν)∂νu(1)].\nAsu(1)=∂νu(1)= 0 onIdue to the definition of u(1), we obtain ∂k\nτu(1)=\n∂k\nτ∂νu(1)= 0 onIfor allk∈N. Moreover, applying the identity\n∂ν∂τw=∂τ∂νw−(divν)∂τw\n(see [20, Corollary 3C.10]) to w:=u(1)and tow:=∂τu(1), respectively, we see\nthat\n∂ν∂2\nτu(1)=∂τ∂ν∂τu(1)= 0 onI.\nTherefore,\nB1u(1)=∂2\nνu(1)=h1,\nB2u(1)=∂3\nνu(1)+(divν)∂2\nνu(1)=h2\nonI. By continuity of E, we have\n(4.9)/ba∇dbl(λ0+∆2)u(1)/ba∇dblL2(Ω1)≤C/ba∇dblu(1)/ba∇dblH4(Ω1)\n≤C/parenleftig\n/ba∇dblg1/ba∇dblH7/2(Γ)+/ba∇dblg2/ba∇dblH5/2(Γ)+/ba∇dblh1/ba∇dblH3/2(I)+/ba∇dblh2/ba∇dblH1/2(I)/parenrightig\nwithCdepending only on λ0.\nConsidering u(2):=u−u(1), we see that usolves (4.7) if and only if u(2)\nsolves the boundary value problem\n(λ0+∆2)u(2)=/tildewidefin Ω1,\nu(2)=∂νu(2)= 0 on Γ ,\nB1u(2)=B2u(2)= 0 onI.\nHere,/tildewidef:=f−(λ0+ ∆2)u(1). By Lemma 4.2, this is uniquely solvable, and\nthe a priori estimate /ba∇dblu(2)/ba∇dblH4(Ω1)≤C/ba∇dbl/tildewidef/ba∇dblL2(Ω1)in connection with ( 4.9) yields\n(4.8). /square\nRemark 4.4. a) The statement and the proof of Lemma 4.2and Corollary 4.3\nare independent of the particular equation. We have shown un ique solvabil-\nity and uniform a priori-estimates for boundary value probl ems where we have\ndifferent boundary operators on disjoint and not connected pa rts of the bound-\nary, given that on each part of the boundary the Shapiro-Lopa tinskii condition\nholds.\nb) From elliptic theory, it is well known that the analog stat ement of Corol-\nlary4.3also holds (with λ0= 0) in the much easier situation of the Dirich-\nlet Laplacian in Ω 2: For every f∈L2(Ω2) andg∈H3/2(I) there exists\na uniqueu∈H2(Ω2) with ∆u=fin Ω2andu|I=g, and/ba∇dblu/ba∇dblH2(Ω2)≤\nC(/ba∇dblf/ba∇dblL2(Ω2)+/ba∇dblg/ba∇dblH3/2(I)).\nThe elliptic regularity results above are the key for the str ong solvability of\nthe transmission problem, i.e. for higher regularity of the weak solution.16 B. BARRAZA MART ´INEZ ET AL.\nTheorem 4.5. LetU= (u1,v1,u2,v2)⊤∈D(A). Thenu1∈H4(Ω1)and\nu2∈H2(Ω2). In particular, the transmission conditions hold in the stro ng sense\nof traces on the interface I.\nProof.LetU∈D(A) andF= (f1,g1,f2,g2)⊤:=AU. Thenv1=f1∈\nH2\nΓ(Ω1),v2=f2∈H1(Ω2), ∆u2=g2+βf2, and ∆2u1=ρ∆f1−g1.\nBy Remark 4.4b), there exists a unique /tildewideu2∈H2(Ω2) such that ∆ /tildewideu2=\ng2+βf2in Ω1and/tildewideu2|I=u1|I. Asu2−/tildewideu2belongs to H1\n0(Ω2) and is a weak\nsolution of ∆( u2−/tildewideu2) = 0, we immediately obtain /tildewideu2=u2which already yields\nu2∈H2(Ω2).\nSimilarly, by Corollary 4.3there exists a unique solution /tildewideu1∈H4(Ω1) of the\nboundary value problem\n(4.10)(λ0+∆2)/tildewideu1=λ0u1+ρ∆f1−g1in Ω1,\n/tildewideu1=∂ν/tildewideu1= 0 on Γ ,\nB1/tildewideu1= 0,B2/tildewideu1=ρ∂νf1−∂νu2onI.\nNote here that λ0u1+ρ∆f1−g1∈L2(Ω1) andρ∂νf1+∂νu2∈H1/2(I), and\nthat all boundary conditions hold in the trace sense.\nLetψ1∈H2\nΓ(Ω1). Then ( 2.1) in combination with the boundary conditions\nabove yields\n(4.11)/a\\}b∇acketle{t(λ0+∆2)/tildewideu1,ψ1/a\\}b∇acket∇i}htL2(Ω1)=λ0/a\\}b∇acketle{t/tildewideu1,ψ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t/tildewideu1,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)\n+/a\\}b∇acketle{tρ∂νf1−∂νu2,ψ1/a\\}b∇acket∇i}htL2(I).\nWe compare /tildewideu1with the weak solution u1. For this, we consider Φ :=\n(0,ψ1,0,ψ2)⊤withψ1∈H2\nΓ(Ω1),ψ2∈H1(Ω2), andψ1=ψ2onI. By defi-\nnition ofD(A), we obtain\n/a\\}b∇acketle{tAU,Φ/a\\}b∇acket∇i}htH=/a\\}b∇acketle{t−∆2u1+ρ∆v1,ψ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆u2−βv2,ψ2/a\\}b∇acket∇i}htL2(Ω2)\n=−/a\\}b∇acketle{tu1,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)−ρ/a\\}b∇acketle{t∇v1,∇ψ1/a\\}b∇acket∇i}htL2(Ω1)2\n−/a\\}b∇acketle{t∇u2,∇ψ2/a\\}b∇acket∇i}htL2(Ω2)2−β/a\\}b∇acketle{tv2,ψ2/a\\}b∇acket∇i}htL2(Ω2).\nFrom this,v1=f1and integration by parts (as we already know u2∈H2(Ω2)),\nwe see that\n/a\\}b∇acketle{t∆2u1,ψ1/a\\}b∇acket∇i}htL2(Ω1)=/a\\}b∇acketle{tρ∆v1,ψ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆u2,ψ2/a\\}b∇acket∇i}htL2(Ω2)\n+/a\\}b∇acketle{tu1,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)+/a\\}b∇acketle{tρ∇v1,∇ψ1/a\\}b∇acket∇i}htL2(Ω1)2+/a\\}b∇acketle{t∇u2,∇ψ2/a\\}b∇acket∇i}htL2(Ω2)2\n=/a\\}b∇acketle{tρ∂νf1,ψ1/a\\}b∇acket∇i}htL2(I)−/a\\}b∇acketle{t∂νu2,ψ2/a\\}b∇acket∇i}htL2(I)+/a\\}b∇acketle{tu1,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)\n=/a\\}b∇acketle{tρ∂νf1−∂νu2,ψ1/a\\}b∇acket∇i}htL2(I)+/a\\}b∇acketle{tu1,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1).\nIn the last step we used ψ1=ψ2onI. Therefore, ( 4.11) also holds with /tildewideu1\nbeing replaced by u1.\nBy definition of /tildewideu1, we have (λ0+∆2)/tildewideu1= (λ0+∆2)u1=λ0u1+ρ∆f1−g1.\nTherefore, we can insert the difference w:=/tildewideu1−u1into (4.11) and obtain\n0 =λ0/a\\}b∇acketle{tw,ψ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{tw,ψ1/a\\}b∇acket∇i}htH2\nΓ(Ω1)for allψ1∈H2\nΓ(Ω1). But by construction\nw∈H2\nΓ(Ω1), so we can set ψ1:=wand getw= 0, i.e.,u1=/tildewideu1∈H4(Ω1)./squareA PLATE-MEMBRANE TRANSMISSION PROBLEM 17\n5.Polynomial stability\nAs we saw in Section 3, the system is not exponentially stable when β= 0.\nWhenβ=ρ= 0, (2.3) shows that the system is conservative. In this section\nwe consider the case β= 0 andρ>0 and show that polynomial decay is still\nguaranteed under certain geometrical conditions. More pre cisely, we assume\nthat there exists some x0∈R2such that\nq·ν=q⊤ν≤0 (5.1)\nonI, whereq(x) :=x−x0. Note that νis the inner normal w.r.t. to Ω 2, which\nis why we require q·ν≤0 instead of q·ν≥0. In order to prove the polynomial\nstability, we use the following result by Borichov and Tomil ov (Theorem 2.4 in\n[9])\nTheorem 5.1. Let(T(t))t≥0be a bounded C0-semigroup on a Hilbert space\nHwith generator Asuch thatiR⊂ρ(A). Then, for fixed α >0the following\nconditions are equivalent:\n(i)There exist C >0andλ0>0such that for all λ∈Rwith|λ|> λ0and\nallF∈Hit holds\n/ba∇dbl(iλ−A)−1F/ba∇dbl ≤C|λ|α/ba∇dblF/ba∇dbl.\n(ii)There exists some C >0such that for all t >0and allU0∈D(A)it\nholds\n/ba∇dblT(t)U0/ba∇dbl ≤Ct−1\nα/ba∇dblAU0/ba∇dbl.\nWe now state the main result of this section: we show polynomi al stability for\nthe transmission problem in the case where only the plate equ ation is damped\nbut the wave equation is undamped. Using rather general meth ods, it is very\nlikely that the rate of decay is not optimal. On the other hand , the approach\nmight be versatile enough to be applicable to different transm ission problems\nof a similar form, i.e. transmission problems where the equa tion in the outer\ndomain is parameter-elliptic, whereas the equation in the i nner domain simply\nis of lower order.\nTheorem 5.2. Letβ= 0andρ>0and assume that the geometrical condtion\n(5.1)is satisfied. Then the semigroup (S(t))t≥0decays polynomially of order at\nleast1/30, i.e. there exists some constant C >0such that\n/ba∇dblS(t)/ba∇dblH≤Ct−1\n30/ba∇dblAU0/ba∇dblH\nfor allt>0andU0∈D(A).\nThroughout the remainder of this section, let λ0>0,λ∈Rwith|λ|>\nλ0,F= (f1,g1,f2,g2)⊤∈HandU= (u1,v1,u2,v2)⊤∈D(A) such that\n(iλ−A)U=F. We first observe that ( iλ−A)U=Fimplies\nv1=iλu1−f1, (5.2)\n−λ2u1+∆2u1−iλρ∆u1=g1+iλf1−ρ∆f1, (5.3)\nv2=iλu2−f2, (5.4)\n−λ2u2−∆u2=g2+iλf2. (5.5)18 B. BARRAZA MART ´INEZ ET AL.\nMultiplying ( 5.3) by−u1and (5.5) by−u2, integrating and adding yields\nλ2/parenleftbig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightbig\n−/a\\}b∇acketle{t∆2u1,u1/a\\}b∇acket∇i}htL2(Ω1)\n+iλρ/a\\}b∇acketle{t∆u1,u1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆u2,u2/a\\}b∇acket∇i}htL2(Ω2)\n=−/a\\}b∇acketle{tg1+iλf1−ρ∆f1,u1/a\\}b∇acket∇i}htL2(Ω1)−/a\\}b∇acketle{tg2+iλf2,u2/a\\}b∇acket∇i}htL2(Ω2).\nUsing Lemma 2.1, integration by parts and plugging in theboundaryand trans -\nmission conditions we obtain\nλ2/parenleftbig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightbig\n−/ba∇dblu1/ba∇dbl2\nH2\nΓ(Ω1)−iλρ/ba∇dbl∇u1/ba∇dbl2\nL2(Ω1)2−/ba∇dbl∇u2/ba∇dbl2\nL2(Ω2)2\n=−ρ/a\\}b∇acketle{t∇f1,∇u1/a\\}b∇acket∇i}htL2(Ω1)2−/a\\}b∇acketle{tg1+iλf1,u1/a\\}b∇acket∇i}htL2(Ω1)−/a\\}b∇acketle{tg2+iλf2,u2/a\\}b∇acket∇i}htL2(Ω2).\nTaking the real part in the above equality we see that\n/ba∇dblu1/ba∇dbl2\nH2\nΓ(Ω1)+/ba∇dbl∇u2/ba∇dbl2\nL2(Ω2)2≤λ2/parenleftbig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightbig\n+/parenleftbig\n|λ|/ba∇dblf1/ba∇dblL2(Ω1)+/ba∇dblg1/ba∇dblL2(Ω1)/parenrightbig\n/ba∇dblu1/ba∇dblL2(Ω1)\n+/parenleftbig\n|λ|/ba∇dblf2/ba∇dblL2(Ω2)+/ba∇dblg2/ba∇dblL2(Ω1)/parenrightbig\n/ba∇dblu2/ba∇dblL2(Ω2)\n+ρ/ba∇dbl∇f1/ba∇dblL2(Ω1)2/ba∇dbl∇u1/ba∇dblL2(Ω1)2\n≤λ2/parenleftbig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightbig\n+C|λ|/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH,\nwhere we used the fact that |λ| ≥λ0. Moreover, due to ( 5.2) and (5.4), we have\nthat\n/ba∇dblvj/ba∇dbl2\nL2(Ωj)=/ba∇dbliλuj−fj/ba∇dbl2\nL2(Ωj)≤2/parenleftbig\nλ2/ba∇dbluj/ba∇dbl2\nL2(Ωj)+/ba∇dblfj/ba∇dbl2\nL2(Ωj)/parenrightbig\nforj= 1,2. Hence,\n(5.6)/ba∇dblv1/ba∇dbl2\nL2(Ω1)+/ba∇dblv2/ba∇dbl2\nL2(Ω2)≤C/parenleftig\nλ2/parenleftbig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightbig\n+/ba∇dblF/ba∇dbl2\nH/parenrightig\nand therefore, combining ( 5.6) with the estimate for u1andu2, we get that\n/ba∇dblU/ba∇dbl2\nH≤C/parenleftig\nλ2/parenleftbig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightbig\n+/parenleftbig\n|λ|/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH+/ba∇dblF/ba∇dbl2\nH/parenrightbig/parenrightig\n.\nIt remains to estimate /ba∇dblu1/ba∇dbl2\nL2(Ω1)and/ba∇dblu2/ba∇dbl2\nL2(Ω2). In order to estimate\n/ba∇dblu1/ba∇dbl2\nL2(Ω1), we observe that, due to ( 2.3), it holds\n/ba∇dbl∇v1/ba∇dbl2\nL2(Ω1)2≤1\nρ/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH,\nand therefore, using Poincar´ e’s inequality, we obtain tha t\nλ2/ba∇dblu1/ba∇dbl2\nH1(Ω1)=/ba∇dblf1+v1/ba∇dbl2\nH1(Ω1)≤2/parenleftbig\n/ba∇dblf1/ba∇dbl2\nH1(Ω1)+/ba∇dblv1/ba∇dbl2\nH1(Ω1)/parenrightbig\n≤C/parenleftbig\n/ba∇dblF/ba∇dbl2\nH+/ba∇dbl∇v1/ba∇dbl2\nL2(Ω1)2/parenrightbig\n≤C/parenleftbig\n/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH+/ba∇dblF/ba∇dbl2\nH/parenrightbig\n. (5.7)\nUsing the fact that |λ| ≥λ0and/ba∇dblu1/ba∇dblL2(Ω1)≤ /ba∇dblu1/ba∇dblH1(Ω1), we thus get that\n(5.8) /ba∇dblU/ba∇dbl2\nH≤C/parenleftig\nλ2/ba∇dblu2/ba∇dbl2\nL2(Ω2)+/parenleftbig\n|λ|/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH+/ba∇dblF/ba∇dbl2\nH/parenrightbig/parenrightig\nand it remains to estimate /ba∇dblu2/ba∇dbl2\nL2(Ω2), which will be done in the following lem-\nmas.A PLATE-MEMBRANE TRANSMISSION PROBLEM 19\nLemma 5.3. It holds\nλ2/ba∇dblu2/ba∇dbl2\nL2(Ω2)≤C/parenleftbig\n|λ|/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH+/ba∇dblF/ba∇dbl2\nH/parenrightbig\n+/integraldisplay\nI/vextendsingle/vextendsingle∂νu2(q∇u2)/vextendsingle/vextendsingledS.\nProof.Using Rellich’s identity (cf. [ 22], Eq. (2.5)), we have that\n(5.9) Re/integraldisplay\nΩ2∆u2(q∇u2)dx=−Re/integraldisplay\nI∂νu2(q∇u2)−1\n2(q·ν)|∇u2|2dS.\nWe multiply ( 5.5) byq∇u2, integrate over Ω 2, take the real part and use ( 5.9)\nin order to obtain that\nRe/parenleftbigg\n−λ2/integraldisplay\nΩ2u2(q∇u2)dx/parenrightbigg\n+Re/parenleftbigg/integraldisplay\nI∂νu2(q∇u2)−1\n2(q·ν)|∇u2|2dS/parenrightbigg\n= Re/integraldisplay\nΩ2(g2+iλf2)(q∇u2)dx.\nAsq∇u2= div(qu2)−2u2, integration by parts and taking the real part shows\nRe/integraldisplay\nΩ2u2(q∇u2)dx=−/ba∇dblu2/ba∇dbl2\nL2(Ω2)−1\n2/integraldisplay\nI(qν)|u2|2dS\nand we obtain\nλ2/ba∇dblu2/ba∇dbl2\nL2(Ω2)=−λ2\n2/integraldisplay\nI(q·ν)|u2|2dS+1\n2/integraldisplay\nI(q·ν)|∇u2|2dS\n−Re/integraldisplay\nI∂νu2(q∇u2)dS+Re/integraldisplay\nΩ2(g2+iλf2)(q∇u2)dx.\nSinceq·ν≤0 onIandu1=u2onI, we arrive at\nλ2/ba∇dblu2/ba∇dbl2\nL2(Ω2)≤C/parenleftbig\nλ2/ba∇dblu1/ba∇dbl2\nH1(Ω1)+|λ|/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH/parenrightbig\n+/integraldisplay\nI/vextendsingle/vextendsingle∂νu2(q∇u2)/vextendsingle/vextendsingledS\n≤C/parenleftbig\n|λ|/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH+/ba∇dblF/ba∇dbl2\nH/parenrightbig\n+/integraldisplay\nI/vextendsingle/vextendsingle∂νu2(q∇u2)/vextendsingle/vextendsingledS,\nwhere in the first step we used the trace theorem and in the last step we used\n(5.7) as well as |λ| ≥λ0. /square\nLemma 5.4. For anyε>0, there exists a constant C(ε)>0such that/integraldisplay\nI/vextendsingle/vextendsingle∂νu2(q∇u2)/vextendsingle/vextendsingledS≤ε/ba∇dblU/ba∇dbl2\nH+C(ε)|λ|60/ba∇dblF/ba∇dbl2\nH.\nProof.Using the transmission conditions, we can estimate/integraldisplay\nI/vextendsingle/vextendsingle∂νu2(q∇u2)/vextendsingle/vextendsingledS=/integraldisplay\nI/vextendsingle/vextendsingleB2u1−iλρ∂νu1+ρ∂νf1/vextendsingle/vextendsingle/vextendsingle/vextendsingleq∇u2/vextendsingle/vextendsingledS\n≤C/ba∇dblB2u1−iλρ∂νu1+ρ∂νf1/ba∇dblL2(I)/ba∇dbl∇u2/ba∇dblL2(I)2\n≤C/parenleftbig\n/ba∇dblu1/ba∇dblH7/2(Ω1)+|λ|/ba∇dblu1/ba∇dblH3/2(Ω1)+/ba∇dblF/ba∇dblH/parenrightbig\n/ba∇dblu2/ba∇dblH3/2(Ω2).\nIn order to estimate the terms on the right-hand side, we will use interpolation\ntheory for both the terms /ba∇dblu1/ba∇dblH7/2(Ω1)and/ba∇dblu2/ba∇dblH3/2(Ω2). Hence, we start with\nan estimate for /ba∇dblu2/ba∇dblH2(Ω2).\nBy (5.5),u2satisfies the equation\n∆u2=−(λ2u2+g2+iλf2).20 B. BARRAZA MART ´INEZ ET AL.\nTherefore, Remark 4.4b) andu1=u2onIyield the estimate\n/ba∇dblu2/ba∇dblH2(Ω2)≤C/parenleftbig\n/ba∇dblλ2u2+g2+iλf2/ba∇dblL2(Ω2)+/ba∇dblu1/ba∇dblH3/2(I)/parenrightbig\n≤C/parenleftbig\nλ2/ba∇dblu2/ba∇dblL2(Ω2)+|λ|/ba∇dblF/ba∇dblH+/ba∇dblu1/ba∇dblH2\nΓ(Ω1)/parenrightbig\n≤C|λ|/parenleftbig\n|λ|/ba∇dblU/ba∇dblH+/ba∇dblF/ba∇dblH/parenrightbig\n. (5.10)\nUsing interpolation inequality and the equivalence of the p-norms on R2, we get\nthat\n/ba∇dblu2/ba∇dblH3/2(Ω2)≤C/ba∇dblu2/ba∇dbl1/2\nH2(Ω2)/ba∇dblu2/ba∇dbl1/2\nH1(Ω2)≤C/ba∇dblu2/ba∇dbl1/2\nH2(Ω2)/ba∇dblU/ba∇dbl1/2\nH\n≤C/parenleftbig\n|λ|/ba∇dblU/ba∇dblH+|λ|1/2/ba∇dblU/ba∇dbl1/2\nH/ba∇dblF/ba∇dbl1/2\nH/parenrightbig\n. (5.11)\nIn the next step, we will estimate the term /ba∇dblu1/ba∇dblH7/2(Ω1). By (5.3),u1satisfies\nthe equation\n(λ+∆2)u1=λu1+λ2u1+iλρ∆u1+g1+iλf1−ρ∆f1.\nHence, Corollary 4.3states\n/ba∇dblu1/ba∇dblH4(Ω1)≤C/parenleftbig\n/ba∇dblλu1+λ2u1+iλρ∆u1+g1+iλf1−ρ∆f1/ba∇dblL2(Ω1)\n+/ba∇dblB1u1/ba∇dblH7/2(I)+/ba∇dblB2u1/ba∇dblH1/2(I)/parenrightbig\ndue to the homogeneous boundary conditions on Γ .Using the trace theorem,\nthe transmission conditions, ( 5.2) and (5.3) as well as ( 5.10), we obtain\n/ba∇dblu1/ba∇dblH4(Ω1)≤C/parenleftig\n|λ|/parenleftbig\n|λ|/ba∇dblU/ba∇dblH+/ba∇dblF/ba∇dblH/parenrightbig\n+/ba∇dbl∂νu2/ba∇dblH1/2(I)/parenrightig\n≤C|λ|/parenleftbig\n|λ|/ba∇dblU/ba∇dblH+/ba∇dblF/ba∇dblH/parenrightbig\n.\nMoreover, note that ( 5.7) reformulates to\n/ba∇dblu1/ba∇dblH1(Ω1)≤C\n|λ|/parenleftbig\n/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH+/ba∇dblF/ba∇dbl2\nH/parenrightbig1/2.\nAgain, by interpolation inequality and the equivalence of t hep-norms on R2,\nwe thus get that\n/ba∇dblu1/ba∇dblH7/2(Ω1)≤C/ba∇dblu1/ba∇dbl5/6\nH4(Ω1)/ba∇dblu1/ba∇dbl1/6\nH1(Ω1)\n≤C|λ|5/6/parenleftbig\n|λ|/ba∇dblU/ba∇dblH+/ba∇dblF/ba∇dblH/parenrightbig5/6|λ|−1/6/parenleftbig\n/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH+/ba∇dblF/ba∇dbl2\nH)1/12\n≤C|λ|2/3/parenleftbig\n|λ|5/6/ba∇dblU/ba∇dbl5/6\nH+/ba∇dblF/ba∇dbl5/6\nH/parenrightbig/parenleftbig\n/ba∇dblU/ba∇dbl1/12\nH/ba∇dblF/ba∇dbl1/12\nH+/ba∇dblF/ba∇dbl1/6\nH/parenrightbig\n≤C/parenleftbigg\n|λ|3/2/parenleftbig\n/ba∇dblU/ba∇dbl11/12\nH/ba∇dblF/ba∇dbl1/12\nH+/ba∇dblU/ba∇dbl5/6\nH/ba∇dblF/ba∇dbl1/6\nH/parenrightbig\n+|λ|2/3/parenleftbig\n/ba∇dblU/ba∇dbl1/12\nH/ba∇dblF/ba∇dbl11/12\nH+/ba∇dblF/ba∇dblH/parenrightbig/parenrightbigg\n. (5.12)\nYoung’s inequality\na2−αbα≤εa2+C(ε)b2\nfor fixedα∈(0,2) andε>0 arbitrary, yields\n|λ|5/2/ba∇dblU/ba∇dbl23/12\nH/ba∇dblF/ba∇dbl1/12\nH=/ba∇dblU/ba∇dbl23/12\nH/parenleftbig\n|λ|30/ba∇dblF/ba∇dblH/parenrightbig1/12≤ε/ba∇dblU/ba∇dbl2\nH+C(ε)|λ|60/ba∇dblF/ba∇dbl2\nH.A PLATE-MEMBRANE TRANSMISSION PROBLEM 21\nConsidering the powers of |λ|, this is the worst term appearing in the estimate\nof/parenleftbig\n/ba∇dblu1/ba∇dblH7/2(Ω1)+|λ|/ba∇dblu1/ba∇dblH3/2(Ω1)+/ba∇dblF/ba∇dblH/parenrightbig\n/ba∇dblu2/ba∇dblH3/2(Ω2). This is due to the fact\nthat in any other term appearing, the power of /ba∇dblU/ba∇dblHis less than23\n12which\nresults in lower powers of |λ|after applying Young’s inequality. Now, using\n|λ|>λ0, we can conclude\n/integraldisplay\nI/vextendsingle/vextendsingle∂νu2(q∇u2)/vextendsingle/vextendsingledS≤C/parenleftbig\n/ba∇dblu1/ba∇dblH7/2(Ω1)+|λ|/ba∇dblu1/ba∇dblH3/2(Ω1)+/ba∇dblF/ba∇dblH/parenrightbig\n/ba∇dblu2/ba∇dblH3/2(Ω2)\n≤ε/ba∇dblU/ba∇dbl2\nH+C(ε)|λ|60/ba∇dblF/ba∇dbl2\nH,\nwhereε>0 is arbitrary and C(ε)>0 is a constant only depending on ε.\n/square\nWe are now able to finish the proof of Theorem 5.2.\nProof of Theorem 5.2.By (5.8), Lemma 5.3and Lemma 5.4together with\nYoung’s inequality applied to the term |λ|/ba∇dblU/ba∇dblH/ba∇dblF/ba∇dblH,we get\n/ba∇dblU/ba∇dbl2\nH≤ε/ba∇dblU/ba∇dbl2\nH+C(ε)|λ|60/ba∇dblF/ba∇dbl2\nH\nfor anyε>0 and a constant C(ε)>0 only depending on ε.This shows\n/ba∇dblU/ba∇dblH≤C|λ|30/ba∇dblF/ba∇dblH.\nTakingF= 0,this estimate also shows that iR∩σp(A) =∅.Since A−1is\ncompact, the spectrum σ(A) ofAcoincides with the point spectrum σp(A)\nofAand we may conclude that iR⊂ρ(A).Now, the assertion follows from\nTheorem 5.1. /square\nReferences\n[1] M.Agranovich,R.Denk,andM. Faierman.Weaklysmoothno nselfadjoint spectral elliptic\nboundary problems. In Spectral theory, microlocal analysis, singular manifolds , volume 14\nofMath. Top. , pages 138–199. Akademie Verlag, Berlin, 1997.\n[2] M. S. Agranovich and M. I. Vishik. Elliptic problems with a parameter and parabolic\nproblems of general type. Russian Math. Surveys , 19(3):53–157, 1964.\n[3] M. Alves, J. Mu˜ noz Rivera, M. Sep´ ulveda, O. Vera Villag r´ an, and M. a. Zegarra Garay.\nThe asymptotic behavior of the linear transmission problem in viscoelasticity. Math.\nNachr., 287(5-6):483–497, 2014.\n[4] M. Alves, J. Mu˜ noz Rivera, M. Sep´ ulveda, and O. V. Villa gr´ an. The lack of exponential\nstability in certain transmission problems with localized Kelvin-Voigt dissipation. SIAM\nJ. Appl. Math. , 74(2):345–365, 2014.\n[5] K. s. Ammari and S. Nicaise. Stabilization of a transmiss ion wave/plate equation. J.\nDifferential Equations , 249(3):707–727, 2010.\n[6] J. A. Arango, L. P. Lebedev, and I. I. Vorovich. Some bound ary value problems and\nmodels for coupled elastic bodies. Quart. Appl. Math. , 56(1):157–172, 1998.\n[7] G. Avalos, I. Lasiecka, and R. Triggiani. Heat-wave inte raction in 2–3 dimensions: optimal\nrational decay rate. J. Math. Anal. Appl. , 437(2):782–815, 2016.\n[8] C. Batty,L.Paunonen,andD.Seifert. Optimal energydec ayin aone-dimensional coupled\nwave-heat system. J. Evol. Equ. , 16(3):649–664, 2016.\n[9] A. Borichev and Y. Tomilov. Optimal polynomial decay of f unctions and operator semi-\ngroups.Math. Ann. , 347(2):455–478, 2010.\n[10] I. Chueshov and I. Lasiecka. Von Karman evolution equations . Springer Monographs in\nMathematics. Springer, New York, 2010. Well-posedness and long-time dynamics.22 B. BARRAZA MART ´INEZ ET AL.\n[11] R. Denk and F. Kammerlander. Exponential stability for a coupled system of damped-\nundamped plate equations. IMA J. Appl. Math. , 83:302–322, 2018.\n[12] R. Denk and R. Schnaubelt. A structurally damped plate e quation with Dirichlet-\nNeumann boundary conditions. J. Differential Equations , 259(4):1323–1353, 2015.\n[13] T. Duyckaerts. Optimal decay rates of the energy of a hyp erbolic-parabolic system cou-\npled by an interface. Asymptot. Anal. , 51(1):17–45, 2007.\n[14] H. D. Fern´ andez Sare and J. E. Mu˜ noz Rivera. Analytici ty of transmission problem to\nthermoelastic plates. Quart. Appl. Math. , 69(1):1–13, 2011.\n[15] R. K. Gazizullin and V. N. Paimushin. The transmission o f an acoustic wave through a\nrectangular plate between barriers. J. Appl. Math. Mech. , 80(5):421–432, 2016.\n[16] B. Gong, F. Yang, and X. Zhao. Stabilization of the trans mission wave/plate equation\nwith variable coefficients. J. Math. Anal. Appl. , 455(2):947–962, 2017.\n[17] F. Hassine. Asymptotic behavior of the transmission Eu ler-Bernoulli plate and wave\nequation with a localized Kelvin-Voigt damping. Discrete Contin. Dyn. Syst. Ser. B ,\n21(6):1757–1774, 2016.\n[18] F. Hassine. Energy decay estimates of elastic transmis sion wave/beam systems with a\nlocal Kelvin-Voigt damping. Internat. J. Control , 89(10):1933–1950, 2016.\n[19] J. Hern´ andez Monz´ on. A system of semilinear evolutio n equations with homogeneous\nboundary conditions for thin plates coupled with membranes . InProceedings of the 2003\nColloquium on Differential Equations and Applications , volume 13 of Electron. J. Differ.\nEqu. Conf. , pages 35–47. Southwest Texas State Univ., San Marcos, TX, 2 005.\n[20] I. Lasiecka and R. Triggiani. Control theory for partial differential equations: continuo us\nand approximation theories. I, Abstract parabolic systems , volume 74 of Encyclopedia of\nMathematics and its Applications . Cambridge University Press, Cambridge, 2000.\n[21] H. Liu and N. Su. Existence and uniform decay of solution s for a class of generalized\nplate-membrane-like systems. Int. J. Math. Math. Sci. , pages Art. ID 83931, 24, 2006.\n[22] E. Mitidieri. A Rellich type identity and applications .Comm. Partial Differential Equa-\ntions, 18(1-2):125–151, 1993.\n[23] J. E. Mu˜ noz Rivera and H. Portillo Oquendo. A transmiss ion problem for thermoelastic\nplates.Quart. Appl. Math. , 62(2):273–293, 2004.\n[24] J. E. Mu˜ noz Rivera and R. Racke. Transmission problems in (thermo)viscoelasticity with\nKelvin-Voigt damping: nonexponential, strong, and polyno mial stability. SIAM J. Math.\nAnal., 49(5):3741–3765, 2017.\n[25] H. Triebel. Interpolation theory, function spaces, differential operat ors, volume 18 of\nNorth-Holland Mathematical Library . North-Holland Publishing Co., Amsterdam-New\nYork, 1978.\n[26] W. Zhang and Z. Zhang. Stabilization of transmission co upled wave and Euler-Bernoulli\nequations on Riemannian manifolds by nonlinear feedbacks. J. Math. Anal. Appl. ,\n422(2):1504–1526, 2015.A PLATE-MEMBRANE TRANSMISSION PROBLEM 23\nB. Barraza Mart ´ınez, Universidad del Norte, Departamento de Matem ´aticas y\nEstad´ıstica, Barranquilla, Colombia\nE-mail address :bbarraza@uninorte.edu.co\nR. Denk, Universit ¨at Konstanz, Fachbereich f ¨ur Mathematik und Statistik,\nKonstanz, Germany\nE-mail address :robert.denk@uni-konstanz.de\nJ. Hern ´andez Monz ´on, Universidad del Norte, Departamento de Matem ´aticas\ny Estad ´ıstica, Barranquilla, Colombia\nE-mail address :jahernan@uninorte.edu.co\nF. Kammerlander, Universit ¨at Konstanz, Fachbereich f ¨ur Mathematik und\nStatistik, Konstanz, Germany\nE-mail address :felix.kammerlander@uni-konstanz.de\nM. Nendel, Universit ¨at Bielefeld, Institut f ¨ur Mathematische Wirtschafts-\nforschung, Bielefeld, Germany\nE-mail address :max.nendel@uni-bielefeld.de"    },    {        "title": "2402.08519v2.Hyperballistic_transport_in_dense_ionized_matter_under_external_AC_electric_fields.pdf",        "content": "Hyperballistic transport in dense ionized matter under external AC electric fields\nDaniele Gamba1, Bingyu Cui2,3, Alessio Zaccone1,4∗\n1Department of Physics “A. Pontremoli”, University of Milan, via Celoria 16, 20133 Milan, Italy\n2School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, P. R. China\n3Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA and\n4Institute for Theoretical Physics, University of G¨ ottingen,\nFriedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany.\n(Dated: February 21, 2024)\nThe Langevin equation is ubiquitously employed to numerically simulate plasmas and dusty plas-\nmas. However, the usual assumption of white noise becomes untenable when the system is subject\nto an external AC electric field. This is because the charged particles in the plasma, which provide\nthe thermal bath for the particle transport, become themselves responsive to the AC field and the\nthermal noise is field-dependent and non-Markovian. We theoretically study the particle diffusiv-\nity in a Langevin transport model for a tagged charged particle immersed in a dense plasma of\ncharged particles that act as the thermal bath, under an external AC electric field, by properly\naccounting for the effects of the AC field on the thermal bath statistics. We analytically derive the\ntime-dependent generalized diffusivity D(t) for different initial conditions. The generalized diffusiv-\nity exhibits damped oscillatory-like behaviour with initial very large peaks, where the generalized\ndiffusion coefficient is enhanced by orders of magnitude with respect to the infinite-time steady-state\nvalue. The latter coincides with the Stokes-Einstein diffusivity in the absence of external field. For\ninitial conditions where the external field is already on at t= 0 and the system is thermalized under\nDC conditions for t≤0, the short-time behaviour is hyperballistic, MSD∼t4(where MSD is the\nmean-squared displacement), leading to giant enhancement of the particle transport. Finally, the\ntheory elucidates the role of medium polarization on the local Lorentz field, and allows for estimates\nof the effective electric charge due to polarization by the surrounding charges.\nI. INTRODUCTION\nThe Langevin equation has become one of the most\nwidely employed mathematical tools to numerically sim-\nulate plasmas and dusty plasmas, let alone colloidal sys-\ntems and the molecular dynamics of liquids and solids\n[1]. The Langevin equation in its standard form reads\nas:\nmdv\ndt=−γ0v− ∇V(x) +Fp(t) (1)\nwhere vandmare the particle’s velocity and mass. re-\nspectively, γ0is a viscous friction coefficient, Vis a con-\nservative field, and Fpis a stochastic force. The latter\nis always assumed to obey the statistics of a white noise,\nas given by the fluctuation-dissipation theorem (FDT):\n⟨Fi,p(t)Fj,p(t′)⟩= 2γ0kBTδi,jδ(t−t′) (2)\nwhere i, jdenote Cartesian components and δ(t−t′) is\na Dirac delta function.\nMorfill and co-workers [2] established a computational\nscheme to simulate plasmas and dusty plasmas via the\nLangevin equation, with a delta-correlated noise as in\nthe above Eq. (2). All subsequent studies in the area\nof plasmas and dusty plasmas have adopted the same\nscheme [3–5], including dusty plasmas in oscillating (AC)\nexternal fields [6–9], lane and band formation in dusty\n∗alessio.zaccone@unimi.it; daniele.gamba1@studenti.unimi.itplasmas under AC fields [10–12], and machine learning\nfor searching phase transitions in dusty plasmas [13].\nIn Ref. [14], it was mathematically proved that, for\nsystems of charged particles in an external AC electric\nfield, the (generalized) Langevin equation is intrinsically\nnon-Markovian. This is reflected in the fact that the\nfluctuation-dissipation theorem (i.e. the statistics of the\nthermal noise) is given (in 1D) by\n⟨Fp(t)Fp(t′)⟩=mkBTK(t−t′) +Q2E(t)E(t′) (3)\nwhere E(t) represents the external AC electric field, e\nis the electron charge and Qis a renormalized effective\ncharge related to FPvia⟨FP⟩=QE(t).\nThis modified FDT has an extra term ∼E(t)E(t′) in\naddition to the usual term K(t−t′). While the latter can,\nunder certain conditions, be reduced to a Dirac delta as\ndiscussed e.g. in [15], the term ∼E(t)E(t′) obviously\ncannot be reduced to a Dirac delta under any conditions,\nbecause E(t)∼sin(Ω t) is a sinusoidal function. This\neffect, therefore, arises only in presence of an external\ntime-dependent field, whereas, if the external field is DC,\nthe effect is not there and Eq. (2) remains valid.\nTherefore, the correct FDT for a plasma (and any\nother system of charged particles such as e.g. liquid elec-\ntrolytes [16, 17]) is always non-Markovian and is given\nby Eq. (3). This fact is new and has been systematically\noverlooked in the literature on Langevin simulations of\nplasmas, where the standard Markovian FDT Eq. (2) is\nused even when the plasma is under an AC electric or\nelectromagnetic field.\nWe present a mathematical model of particle transport\nin dense ionized matter under an external AC electricarXiv:2402.08519v2  [physics.plasm-ph]  20 Feb 20242\nfield that is relevant to dense plasmas and dusty plasmas,\nhigh energy density matter, solute colloidal systems, liq-\nuid metals and ferrofluids, and liquid electrolytes.\nIn this model, the generalized Langevin equation of\nmotion for the charged particle is derived from a mi-\ncroscopic Caldeira-Leggett particle-bath Hamiltonian,\nwhere, crucially, the bath is formed of charged harmonic\noscillators which also respond to the external AC field.\nThis is a trick to effectively decompose the thermal kicks\non the tagged particle due to innumerable collisions by\ndecomposing the corresponding thermal bath into the vi-\nbrational eigenmodes of the system.\nIn this way, upon setting suitable initial conditions, we\nanalytically derive the time-dependent particle diffusiv-\nityD(t), which we study as a function of the AC elec-\ntric field parameters, in particular the field amplitude E0\nand frequency Ω. The magnitude of the effective polar-\nization charge is estimated, and a giant enhancement of\nthe polarization field is found when the field frequency\napproaches the characteristic vibration frequency of the\ncharged particles forming the bath. Predictions concern-\ning the behaviour of the generalized diffusion coefficient\nD(t) are obtained including the prediction of superdiffu-\nsive transport when the external field frequency becomes\nvery small. In particular, for the case of initial conditions\nwhere the AC field is switched on at t= 0, this short-\ntime superdiffusive regime is simply ballistic. Instead,\nif the system has thermalized under DC conditions for\ntimes t≤0 and then the field becomes AC for t >0,\na hyperballistic superdiffusive regime is predicted. It is\nalso established that D(t→ ∞ ) coincides with the par-\nticle diffusivity in the absence of the electric field under\nall initial conditions, which is a non-trivial result.\nII. MODEL OF PARTICLE-BATH SYSTEM\nWe consider the same model as in Ref. [14], in which an\nexternal, time-dependent electric field, E(t), is applied to\na dense medium. The motion of the tagged particle of\nmass mis embedded in a dense medium (thermal bath)\nof other particles. In order to effectively decompose the\neffect of the thermal bath onto the normal modes of the\nsystem, we use the Caldeira-Leggett trick of representing\nthe thermal environment as fictitious harmonic oscilla-\ntors with natural frequencies corresponding to the eigen-\nmodes of the medium. In our theory, these fictitious har-\nmonic oscillators are electrically charged, and thus are\nsubject to the effect of the external electric field via the\nLorentz force. The latter acts on all the charged parti-\ncles which, in reality, form the medium and the thermal\nbath. Hence, the thermal bath responds to the external\nfield and its statistics is modified by the external field,\nas it should be, in the physical reality. We shall neglect\nmagnetic forces throughout.\nThe Hamiltonian H(in 1D) of the total system consistsof the Hamiltonian of the tagged particle\nHp=p2\n2m−q x E (t), (4)\nwhere pis the momentum, xthe position and qthe\nparticle’s charge, plus the Hamiltonian Hbof the bath.\nThe latter is effectively modeled by Nindependent\nelectrically-charged harmonic oscillators of coordinates\n{xi, pi}, with mass mi, electric charge qi, and oscillating\nat some eigenfrequency ωi[15, 18]. Since each oscillator\nicarries a charge qi, it thus feels the electric force due\nto the external AC field, and the bath as a whole is re-\nsponsive to the AC field. The Hamiltonian of the bath\noscillators can be written as [14]\nHb=1\n2NX\ni=1 \np2\ni\nmi+miω2\ni\u0012\nxi−ν2\ni\nω2\nix\u00132\n−2qixiE(t)!\n=H0\nb−1\n2E(t)NX\ni=1qixi, (5)\nwith\nH0\nb=1\n2NX\ni=1 \np2\ni\nmi+miω2\ni\u0012\nxi−ν2\ni\nω2\nix\u00132!\n. (6)\nNote that we have completed the square to put the ex-\npression in a more convenient form for future calcula-\ntions, and we have also supposed that displacements x\nandxiare small so that the i-th oscillator couples to\nthe Brownian particle linearly. The coupling strength is\ndenoted as ν2\ni.\nThe external electric field oscillates with frequency Ω\nand amplitude E0. We consider two different initial con-\nditions (ICs): field-off initial conditions , given by\nE(t) =(\n0, t < 0\nE0sin(Ω t), t≥0,, (7)\nandfield-on initial conditions , given by\nE(t) =(\nE0, t < 0\nE0cos(Ω t), t≥0, (8)\nThe field behavior in the two cases is shown in Fig. 1.\nFollowing the canonical formalism, the derivation of\nRef. [14] gives the equation of motion of the tagged par-\nticle as\ndp\ndt=−Zt\n0K(t′)p(t−t′)dt′−∇V(x)+qE(t)+Fp(t),(9)\nwhere\nK(t) =NX\ni=1miν4\ni\nmω2\nicos(ωit) (10)3\n0 1 2 3 4-1.5-1.0-0.50.00.51.01.5\ntE(t)/E0\n(a)\n0 1 2 3 4-1.5-1.0-0.50.00.51.01.5\ntE(t)/E0\n(b)\nFigure 1: Different initial conditions for the\nparticle-bath Hamiltonian (5) under the action of an\nexternal AC electric field E(t). (a): Field-off initial\nconditions, where the field is switched on at t= 0\n(Eq. (7)). (b): Field-on initial conditions, where the\nfield is already on at t= 0 (Eq. (8)).\nis the memory kernel describing the time accumulation\neffect on the tagged particle due to the coupling to the\nbath, and\nFp(t) =NX\ni=0\"\nmiν2\ni\u0012\nxi(0)−ν2\ni\nω2\nix(0)\u0013\ncos(ωit) +\n+ν2\ni\nωipi(0) sin( ωit) +qiE′\ni(ωi, t)#\n(11)\nwhere\nE′\ni(ωi, t) =ν2\ni\nωiZt\n0E(t′) sin(ωi(t−t′))dt′. (12)\nThe “noise” (11) depends on the distribution of the\nbath oscillators and on the electric field, and is a well-\ndefined function of time. It describes the effect of a huge\nnumber of irregular kicks on the tagged particle due to\nthe collisions with the particles of the bath. The forcePN\ni=1qiE′\ni(t) acting on the Brownian particle is due to\nthe internal polarization of the medium under the exter-\nnal AC field E(t). The function E′\n0(ω0, t) (Eq. (12)) isplotted in Fig. 2 as a function of time, and in Fig. 3 as\na funcion of frequency.\nIn most cases, the medium contains a large number\nof oscillating modes, so we can regard the collection of\noscillating eigenfrequencies as continuous. Therefore, the\nsum might be replaced with the integralR\ng(ω)dω, where\ng(ω) =P\niδ(ω−ωi) is the density of vibrational states.\nThus, in the continuum limit, the friction kernel (10) may\nbe written as\nK(t) =m0\nmZωD\n0ν(ω)4\nω2cos(ωt)g(ω)dω, (13)\nwhere we have set mi=m0for all bath oscillators i.\nFollowing Zwanzig [15], we assume a Debye form for the\ndensity of states of the bath:\ng(ω) =(\ng0ω2,0< ω < ω D,\n0, otherwise,(14)\nwhere g0is a constant. This choice is motivated by the\nfact that the main low-energy excitations in the plasma\nare sound waves with some dispersion relation ω(k) =ck,\nwhere cis the sound speed. The corresponding density\nof states of these low-energy eigenmodes ωis thus given\nby the quadratic Debye form [19].\nEq. (13) becomes\nK(t) =g0m0\nmZωD\n0ν(ω)4cos(ωt)dω (15)\nWe also introduce, for later convenience, the renormal-\nized effective charge\nQ=NX\ni=1qiν2\ni\nω2\ni=g0q0ZωD\n0ν(ω)2dω≡g0q0f(0),(16)\nwhere we have set qi=q0for all oscillators i, and\nf(t) =ZωD\n0ν(ω)2cos(ωt)dω. (17)\nThe average noise Fpand its autocorrelation function can\nbe computed explicitly in the two situations described\nbelow.\nA. Pseudo-resonance of the polarization function\nConsider the situation described by Eq. (7), where the\nexternal electric field Eis switched on at time t= 0.\nHowever, for very short times, the bath doesn’t have time\nto instantaneously respond to the change, and is still in\nthe same equilibrium distribution ∼e−H0\nb/kBTas at t= 0\nand earlier times. Therefore, the average noise becomes\n⟨Fp(t)⟩=NX\ni=0qiν2\niE0\nωiωisin(Ω t)−Ω sin( ωit)\nω2\ni−Ω2.(18)4\n0 5 10 15 20 25 30-2-1012\ntE'0(t)\n(a)\n0 5 10 15 20 25 30-2-1012\ntE'0(t)\n(b)\nFigure 2: Panel (a): field-off ICs. Panel (b): field-on\nICs. Plot of the effective (Lorentz) field as a function of\ntime. Blue, yellow and green lines correspond to values\nΩ/ω0= 0.8,0.99,1.13: the oscillations are giantly\nenhanced at the resonant frequency Ω ≃ω0, and their\namplitude increases with time (Eq. (12)).\nFor the large values of Ω, that are used for instance in\nTHz spectroscopy, it can become comparable with the\nsmallest of the ωivalues, say ω0, then\n⟨Fp(t)⟩ ∼q0E′\n0(ω0, t) =q0ν2\n0E0\n2ω0\u0014sin(ω0t)\nω0−tcos(ω0t)\u0015\n(19)\nand the tagged particle is subject to an oscillating force\nwhose amplitude grows linearly in time.\nThe trend of E′\n0(ω0, t) (Eq. (12)) is shown in Figs. 2, 3,\nfor both the field-off and the field-on ICs.\nNow we consider the situation described by Eq. (8) in\nwhich at t= 0, the electric field has already been on for a\ntime sufficiently long for the bath to equilibrate under the\naction of the external field. The usual relaxation time for\natomic/molecular systems is of order 10−13seconds, so\npossibly much shorter then the time of observation. The\nstate of the medium is therefore thermalized according\n0 2 4 6 8 10-2-1012\nω0E'0(ω0)(a)\n0 2 4 6 8 10-2-1012\nω0E'0(ω0)\n(b)\nFigure 3: Panel (a): field-off ICs. Panel (b): field-on\nICs. Plot of the effective (Lorentz) field as a function of\nthe resonant frequency ω0, fort= 7. Blue, yellow and\ngreen lines correspond to values Ω /ω0= 0.8,0.99,1.13.\nOne can see peaks where the value of Ω is very close to\nω0(Eq. (12)).\nto the distribution ∼e−Hb(t=0)/kBT, with which we find\n⟨Fp(t)⟩=NX\ni=1qiν2\niE0(\ncos(ωit)\nω2\ni+cos(Ω t)−cos(ωit)\nω2\ni−Ω2)\n.\n(20)\nAgain, if Ω is comparable with some resonant frequency\nω0, then\n⟨Fp(t)⟩ ∼q0E′\n0(ω, t) =q0ν2\n0E0\n2sin(ω0t)\nω0t, (21)\nand we have an oscillating force whose amplitude in-\ncreases linearly in time.\nAs we see, Eqs. (18, 20) exhibit a pole in coincidence\nof Ω≃ω0. As explained in [20], this pole represents a\nresonance peak, when the experimental frequency Ω is\nprobing exactly a characteristic frequency ω0of the ma-\nterial medium, and this gives rise to spectra of emission\nand absorption. One usually solves the problem of inte-\ngrating the singularity, by displacing the pole by a little\namount in the complex plane: this physically corresponds5\nto recognizing that each mode ωialso suffers some dis-\nsipation. In that case however, the resonance is made\npossible when the probing oscillating field has been on\nfor a long amount of time. In our case, the divergence\nis avoided because we have been probing the medium on\nits resonance frequency only for a time t.\nThese frequency sums can be evaluated analytically.\nOne should note that the sum contains diverging quan-\ntities, which, however, when summed together give rise\nto a finite quantity, due to a negative interference effect.\nThe result is a quantity that is oscillating in time, and\nthe giant amplitude enhancement that grows linear in t\nhas no effect on the quantity integrated in time, because\nall the giant oscillating terms average to quantities of or-\nder 1. Interesting effects might appear if we consider the\nprobing frequency Ω comparable with the highest eigen-\nfrequency ωDof the medium, but that goes beyond the\nscope of the present paper and may be the subject for\nfuture work.\nHaving made the above considerations, we can keep\nworking in the limit Ω ≪ωifor every mode i, and in this\nlimit the sum in Eq. (18) can be evaluated and gives\n⟨Fp(t)⟩=QE0sin(Ω t) (22)\nfor field-off ICs; similarly Eq. (20) gives\n⟨Fp(t)⟩=QE0cos(Ω t) (23)\nfor field-on ICs, where\nQ=g0q0ν2\n0ωD (24)\n(see App. A for the detailed derivation of\nEqs. (22, 23, 24)). So, as we can see, the quantity\nQhas the role of an effective or renormalized charge ac-\nquired by the tagged particle, due to the polarization of\nthe surrounding medium. The physical picture is that, in\ndense or supercooled liquid ionized matter, each particle\nis trapped in a cage by the surrounding particles. If the\noscillatory AC field drives the particle in one direction\nand opposite charges in the opposite direction, then we\nwill have a resonance effect with the normal mode of the\nparticle in the cage. The effect adds up to the electric\nfield on the particle, because in general the positive ions\ndraw the particle in the opposite direction of the field,\nand if q=−Qthey would completely screen the effect\nof the external AC field. See Fig. 4 for a schematic\nvisualization of this effect. It is also important to note\nthat results (22, 23) show insensitivity of the system\nto the choice of initial conditions, i.e. whether field-off\nor field-on, because memory of the initial conditions is\nlost in a time proportional to ω−1\nD, which is a very short\ntime, e.g. if ωDcoincides with the plasma frequency.\nIndeed, in (20), in the limit Ω ≪ωD, the first term,\nintegrated on the frequency, is proportional tosin(ωDt)\nt,\nbecomes negligible in comparison with the second term,\nproportional to ωD, because the time of observation is\nt≫ω−1\nD. Hence, the extra term due to field-on initial\nconditions becomes negligible after very short times, and\nthe dependence on the initial conditions is lost.\n+\n+++\n+\n------------------\n-+\n+\n+\n+\n+\n+\n+\n+\n+\n+V(x)\n-+Figure 4: Schematic depiction of the physical system\n(dense ionized matter) considered in the theoretical\nmodel. The tagged particle (green) moves in a thermal\nbath (dense medium) of other charges schematically\nrepresented as charged harmonic oscillators in the\ntheoretical model. The frequencies of these fictitious\noscillators schematically represent the eigenfrequencies\nof the sound modes of the system. These oscillators are\ncoupled to the tagged particle by different coupling\nstrengths, represented as red lines. The effective charge\nof the tagged particle, Q(represented as a shaded area),\naccounts for the local charge distribution and polarizing\nLorentz field around the tagged particle. The system is\nunder an AC electric field (depicted in figure at a\ncertain instant of time) exerted by the electrodes\n(represented by two opposite charged plates in the\ngraph) and might be subjected to an overall static field\nV(x), which is not explicitly taken into account in our\nderivations but the presence of which would not\nqualitatively change our results.\nB. Autocorrelation functions\nThe influence of the external field is also exhibited in\nthe autocorrelation function of the random force,\n⟨Fp(t)Fp(t′)⟩=mkBTK(t−t′) + (QE0)2sin(Ω t) sin(Ω t′),\n(25)\nas has been proposed in [14], for field-off initial condi-\ntions, and\n⟨Fp(t)Fp(t′)⟩=mkBTK(t−t′) + (QE0)2cos(Ω t) cos(Ω t′)\n(26)\nfor field-on initial conditions (here derived for the first\ntime; the details of the calculation are presented in Ap-\npendix A). In contrast to the previous case of the field-off\nICs, the effective Lorentz field arising inside the medium\nis phase-shifted according to the external electric field.6\n0 5 10 15 2002468\ntD˜(t)\n(a)\n0 5 10 15 2002468\ntD˜(t)\n(b)\nFigure 5: Time evolution of the generalized diffusivity\n˜D=D2mγ0\nkBTfor (a) field-off (32) and (b) field-on (34)\ninitial conditions, obtained with different field\namplitudes E0(Eq. (33)), for Ω = 1. Yellow, green and\norange lines correspond to E0= 1,2,3, whereas the blue\nline is the Stokes-Einstein result obtained for E0= 0.\nWe can also calculate the momentum autocorrelation\nfrom the force correlation in the overdamped Brownian\nlimit, i.e., when γ0≫m(see Eq. (1)). Setting ν(ω) =ν0,\nEq. (15) becomes K(t) =γ0δ(t) with\nγ0=πg0m0\nmν4\n0. (27)\nWe find\n⟨p(t1)p(t2)⟩=1\nγ2\n0⟨F(t1)F(t2)⟩, (28)\nwith\nF(t) =Fp(t) +qE(t). (29)\n0 100 200 300 400 500 600020406080\ntD˜(t)(a)\n0 100 200 300 400 500 600020406080\ntD˜(t)\n(b)\nFigure 6: Time evolution of diffusivity ˜D=D2mγ0\nkBTfor\n(a) field-off (32) and (b) field-on (34) initial conditions\nfor different Ω values, comparable to the vibrational\nfrequencies of the bath oscillators. Yellow, green and\norange lines correspond to Ω /ωD= 0.06,0.03,0.01, for\nE0= 1, whereas the blue line is the Einstein result\nobtained for E0= 0. For small Ω the diffusivity is\ngiantly augmented in comparison with the DC\ndiffusivity, which coincides with the Stokes-Einstein\nresult (blue line).\nIII. PARTICLE DIFFUSIVITY UNDER\nEXTERNAL AC FIELD\nUsing the time correlation of the velocity v=p/m, we\nare able to calculate the diffusivity\nD(t) =1\n2t⟨{x(t)−x0}2⟩, (30)\nwhere x0is the position of the Brownian particle at t= 0,\nsince\n⟨{x(t)−x0}2⟩=Zt\n0dt1Zt\n0dt2⟨v(t1)v(t2)⟩. (31)7\nFor the field-off ICs, the diffusivity becomes\nD(t) =kBT\n2mγ0\u0012\n1 +E2\n0(cos(Ω t)−1)2\nΩt\u0013\n(32)\nwith\nE0=(q+Q)E0√mkBTγ0Ω, (33)\nwhile for the field-on ICs, we find\nD(t) =kBT\n2mγ0\u0012\n1 +E2\n0sin2(Ωt)\nΩt\u0013\n(34)\n(see App. B for a detailed derivation).\nInterestingly, we shall note that both the above expres-\nsions for D(t) asymptotically recover the particle diffu-\nsivity in the absence of the external field in the limit\nt→ ∞ :\nlim\nt→∞D(t) =kBT\n2mγ0, (35)\nwhich is the standard Stokes-Einstein result, independent\nof the AC field parameters. Recalling the discussion fol-\nlowing Eq. (21), the generalized diffusivity D(t) doesn’t\nshow a divergent behavior for frequencies near the reso-\nnant frequencies of the system: in fact, in Eqs. (32, 34),\nthe resonant frequencies of the system affect only the pa-\nrameter E0, trough the renormalized charge Q.\nThe non-dimensional diffusivity ˜D(t) =2mγ0\nkBTD(t) as-\nsociated with the motion of the tagged particle under\nthe field-off ICs is shown in Figs. 5 and 6. In Fig. 5, we\ncompare D(t) for different values of the non-dimensional\nfield amplitude Eq. (33). In Fig. 6, we compare D(t) for\ndifferent values of the probing frequency Ω.\nOverall, the oscillatory diffusivity ultimately decays to-\nwards the t→ ∞ Stokes-Einsten value (35) as if there\nwas no field applied. The effect of the AC field under the\nfield-off ICs is more significant at short times, resulting\nin a first peak of D(t) which is larger for larger values of\nE0. Similar behaviour is also reported for the response\nof a colloidal particle to a time-dependent quadratic po-\ntential due to an optical trap being dragged through the\nfluid [21]. In our case, these oscillations are augmented\nby a factor Qdue to the polarization of the surrounding\nmedium (Eq. (33)). If the external frequency Ω is very\nsmall, in the case of field-on ICs we recover for times\nt <Ω−1the limit of ballistic motion under a constant\nfield, that is, the limit for small Ω of (34), which is\nD(t)≈kBT\n2mγ0\u0000\n1 +E2\n0Ωt\u0001\n(36)\nThis asymptotic behavior for small tis visible in the ini-\ntial slope in Figs. 5b and 6b.\nEquation (34) shows that the first peak in D(t) is at\nΩt= 1.16:\nDmax≃D(1.16/Ω) =kBT\n2mγ0\u0000\n1 + 0 .72E2\n0\u0001\n. (37)SinceE2\n0∝1/Ω, when Ω is decreased, the peak is shifted\nforward in time and amplified. In the DC limit of con-\nstant field, i.e. for small Ω, and for finite times, Eq. (36)\nreduces to the Stokes-Einstein value, plus a standard drift\ncontribution, linear in time, which contributes to the\ndressed or generalized diffusivity D(t), while the naked\ndiffusivity is given simply by the Stokes-Einstein value.\nIn the case of field-off ICs, instead, the short-time\nbehaviour is given by taking the limit of small Ω tin\nEq. (32), which gives:\nD(t)≈kBT\n2mγ0\u0000\n1 +E2\n0Ω3t3\u0001\n(38)\nand, in this case, there is an inflection point at t= 0.\nImportantly, in this case, a hyperballistic behaviour is\nobserved at short times, D(t)∼t3, giving a faster and\nlarger growth of the first peak in D(t) compared to the\nprevious case. This corresponds to a mean squared dis-\nplacement (MSD) that grows as ⟨x(t)−x02⟩ ∼t4, i.e.\nwith the fourth power of time.\nThis is a consequence of the AC field ∼sin(Ω t) being\nswitched on at t= 0 following thermalization under DC\nconditions. In the overdamped regime, this gives a par-\nticle displacement cos(Ω t)−1∼t2. Eq. (32) shows that\nthe first peak of D(t) is at Ω t= 2.78:\nDmax≃D(2.78/Ω) =kBT\n2mγ0\u0000\n1 + 1 .35E2\n0\u0001\n. (39)\nWhen Ω is decreased, the peak is shifted forward in time\nand amplified. In the Ω t≪1 limit, Eq. (39) shows that,\nfor finite times, the Stokes-Einstein value (blue line in\nFig. 6a) is recovered. This is the first prediction, to our\nknowledge, of hyperballistic superdiffusive transport in\ndense systems of charged particles under an AC electric\nfield.\nConsistent with results in Ref. [22], while the charge is\naccelerating under the external field, large velocities are\nsmeared out by the damping γ0. Charges moving faster\nwill be dragged by the friction to a larger extent. For\na lump of charge density moving in the field direction,\nthe charges in the front are faster and undergo major\ndamping, while those on the rear do not undergo much\nfrictional resistance and do not lose much momentum.\nAs a result, charges accumulate in the front, increasing\nthe local charge density. The overall effect is that the\npulse gets compressed in the direction of the field, which\nreduces diffusivity parallel to the field. On the other\nhand, the transverse diffusivity might increase. In the\nexample of diffusion-convection of a solute in a liquid,\ncfr. Ref. [23], there is complete uniformity in Taylor dis-\npersion transverse to the applied field, so there is no rise\nin gradients of velocity or concentration. However, in this\npaper, we are only considering the longitudinal case for\nthe 1D Langevin equation along the field direction. The\ntransverse equation might be coupled to what happens\nin the longitudinal one, but this would require a full 3D\ntreatment which may not be analytically tractable within\nthe Langevin equation framework.8\nAs we have seen from the above theory, the correction\nto the (naked) diffusivity coefficient due to the external\nAC electric field is proportional to ωD(this is because\nE0∝Q∝ωD, see Eqs. (33, 24)). Since this is the highest\nfrequency of the bath, it can be taken as the plasma\nfrequency, and hence,\nωD∝1√m(40)\nwhere mis the mass of the electron, or in general, of the\nlightest charged particle in the medium. This implies\nthat, in general, the diffusivity enhancement effect pre-\ndicted by the above theory will be larger the smaller the\nmass of the lightest constituent present in the medium.\nIV. CONCLUSIONS\nWe studied a Caldeira-Leggett particle-bath model\nwhere both the tagged particle and the oscillators form-\ning the bath are electrically charged and respond to an\nexternal AC electric field. Since the bath is responding to\nthe external AC field, the thermal noise is no longer white\nnoise, but is non-Markovian and strongly influenced by\nthe external time-dependent electric field. Based on this\nmodel, we analytically derived results for the autocorre-\nlation functions of force and momentum, and used the\nlatter to evaluate the time-dependent generalized parti-\ncle diffusivity D(t). Analytical close-formed expressions\nforD(t) as a function of the physical parameters of the\nsystem are obtained for two different initial conditions\n(AC field off and AC field on at t= 0, respectively),\nand are given by Eq. (32) and (34). We found that the\ngeneralized diffusivity in the asymptotic t→ ∞ steady\nstate is the same as in the absence of the external field\nand is given by the Stokes-Einstein formula, which is a\nrather remarkable result. Also, the generalized diffusiv-\nity exhibits time-dependent damped fluctuations in the\ntransient regime which, for low AC field frequency Ω,\nproduce a transient giant enhancement of the general-\nized diffusivity with respect to the steady-state value.\nFor low enough frequency of the external AC field, the\nenhancement can be up to few orders of magnitude with\nrespect to the steady-state value. In particular, our the-\nory predicts: (i) ballistic superdiffusion for the field-off\ninitial conditions (the AC field is switched on at t= 0\nfollowing equilibration at zero external field), and (ii) hy-\nperballistic superdiffusion with quartic power of the mean\nsquared displacement, MSD ∼t4, for the case of field-\non initial conditions (the AC field is switch on at t= 0\nfollowing thermalization under DC field).\nThis effect, predicted here for the first time, can be\ninterpreted as the result of a memory accumulation pro-\ncess, due to the non-Markovianity of the noise (Eqs. (25)\nand (26)). This is made evident if we take the DC limit of\nconstant field (Ω →0): in this case, both the naked diffu-\nsivities corresponding to (36) and (38) are reduced to the\nStokes-Einstein value. From a physical point of view, thismemory accumulation means that the stochastic force\nacting on the particle, representing collisions with other\nparticles, retains memory of previous collisions which in\nthe field-on ICs case are continuously enhanced by the\ndrift over past times, and hence grows as a function of\ntime. Also, the enhancement of the transient generalized\ndiffusivity is predicted to be inversely proportional to the\nsquare root of the lightest particle in the medium (e.g.\nthe electron mass in the case of plasmas).\nThe theory also allows one to evaluate the local\n(Lorentz) field acting on the tagged particle due to po-\nlarization in the surrounding medium. Interestingly, a\nresonance-type behaviour of the local effective field is un-\nveiled corresponding to a frequency value of the external\nAC field that matches the value of a pole in one of the\nterms constituting the average random force ⟨Fp(t)⟩act-\ning on the tagged particle. Also interestingly, however,\na destructive interference with other oscillating terms in\n⟨Fp(t)⟩conspires to remove this resonance in the overall\nbehaviour of ⟨Fp(t)⟩which turns out to be given by the\nsame sinusoidal function of the external AC field, with\na renormalized charge due to charge renormalization by\nthe surrounding charged particles.\nSince the diffusivity is a key parameter in plasma\nphysics, e.g. it is a control parameter for the formation of\nplasma blobs in tokamaks [24, 25] and plasma turbulence\n[26], the current predictions may be useful for optimiz-\ning transport phenomena and overall efficiency of fusion\nreactors.\nFrom the fundamental point of view of nonequilibrium\nstatistical mechanics, we have discovered and predicted\na new physical effect which represents a new example\nof anomalous diffusion [27, 28], namely of hyperballis-\ntic superdiffusion [29, 30]. In particular, we showed that\ndynamic accumulation of the stochastic force, which rep-\nresents random collisions enhanced by an external drift,\nleads to hyperballistic transport [31]. Our theoretical\nfindings may connect to previous experimental reports of\nhyperballistic transport in photons propagating through\ndynamic disorder [32, 33]. It may also connect to the\nphenomenon of secondary Fermi acceleration, proposed\nto explain the origin and acceleration of cosmic rays [34].\nThis is the mechanism by which a plasma is heated and\naccelerated by time-dependent stochastic driving fields.\nIn the original setup, this was a model for cosmic rays be-\ning scattered by magnetized interstellar clouds that move\nrandomly and act as magnetic mirrors.\nAll in all, the hyperballistic dynamics under an exter-\nnal AC field discovered here could potentially lead to a\nnew controlled setup for the acceleration of ionized mat-\nter [35]. Future extensions of the current theory will in-\nclude the effect of magnetic fields for magnetized plasmas\nand possibly magneto-hydrodynamic effects. Also, the\npresent theory may represent the starting point of linear\nresponse theories to predict material response properties\nunder external fields [36–38] or within molecular simula-\ntions [1].9\nAppendix A: Average of noise and its\nautocorrelations\nThe average force fluctuations on the particle is the\nsum of all the incoherent contributions on some finite\ntime span T,\n⟨Fp(t)⟩=1\nTZt+T\ntFp(t′)dt′. (A1)\nSimilarly, the autocorrelation of Fv, i.e. the second mo-\nment, is expressed as\n⟨Fp(t)Fp(t+τ)⟩=1\nTZt+T\ntFp(t′)Fp(t′+τ)dt′.(A2)\nIf the bath is in thermal equilibrium at t= 0, since\nthe equilibrium system is ergodic, the average, Eq. (A1),\nis equivalent to a Boltzmann average on every possible\nbath configuration {x1, .., x N, p1, .., p N}:\n⟨Fp(t)⟩H0\nb=NY\ni=1Z+∞\n−∞dpie−p2\ni\n2mikBT×\n×Z+∞\n−∞dxie−miω2\ni\n2kBT\u0012\nxi−ν2\ni\nω2\nix\u00132\nFv(t).(A3)\nCompleting the squares in the Gaussian integrals, we ob-tain\n⟨pi⟩= 0, (A4a)\n⟨p2\ni⟩=mikBT, (A4b)*\nxi−ν2\ni\nω2\nix+\n= 0, (A4c)\n*\u0012\nxi−ν2\ni\nω2\nix\u00132+\n=kBT\nmiω2\ni. (A4d)\nWhen the bath oscillators are subject to the action of\nthe external field, they respond and relax fast (on times\nγ−1\n0≃10−13s, much shorter than the time-scale Ω−1on\nwhich the transient is observed) relative to the dynamics\nof the tagged particle, so the oscillatory field can be ap-\nproximated with its value in a neighborhood of t= 0 [15]:\nHb(t= 0)≈\n1\n2NX\ni=1 \np2\ni\nmi+miω2\ni\u0012\nxi−ν2\ni\nω2\nix−qi\nmiω2\niE0\u00132!\n(A5)\nplus a constant term that depends on x. Since the latter\nremains inert in the computation of bath averages, it is\nirrelevant for our results. Then, the average force is\n⟨Fp(t)⟩=NY\ni=1Z+∞\n−∞dxie−miω2\ni\n2kBT\u0012\nxi−ν2\ni\nω2\nix−qi\nmiω2\niE0\u00132\nFv(t).\n(A6)\nFurther, we have\n*\nxi−ν2\ni\nω2\nix−qi\nmiω2\niE0+\n= 0 (A7)\nand\n*\u0012\nxi−ν2\ni\nω2\nix−qi\nmiω2\niE0\u00132+\n=kBT\nmiω2\ni. (A8)\nTherefore, referring to field-on (7) and field-off (8) ICs,\nwe find the effective field (12) is equal to\nE′\ni(ωi, t) = ν2\niE0(1\nωiωisin(Ω t)−Ω sin( ωit)\nω2\ni−Ω2 ,off,\ncos(Ω t)−cos(ωit)\nω2\ni−Ω2 , on.(A9)\nthen for the average force we have\n⟨Fp(t)⟩=NY\ni=1Z+∞\n−∞dxiFp(t)\n\nexp\"\n−miω2\ni\n2kBT \nxi−ν2\ni\nω2\nix!2#\n, off,\nexp\"\n−miω2\ni\n2kBT \nxi−ν2\ni\nω2\nix−qiE0\nmiω2\ni!2#\n,on.(A10)10\nUsing formula (11) for the noise,\n⟨Fp(t)⟩=NX\ni=1(\nqiE′\ni(t), off,\nmiν2\niqiE0\nmiω2\nicos(ωit) +qiE′\ni(t),on,\n=NX\ni=0qiν2\niE0(1\nωiωisin(Ω t)−Ω sin( ωit)\nω2\ni−Ω2 , off,\ncos(ωit)\nω2\ni+cos(Ω t)−cos(ωit)\nω2\ni−Ω2 ,on,\n=q0g0ν2\n0E0ZωD\n0dω(\nωωsin(Ω t)−Ω sin( ωt)\nω2−Ω2 , off,\ncos(ωt) +ω2cos(Ω t)−cos(ωt)\nω2−Ω2,on,(A11)\nand in the small Ω limit,\n⟨Fp(t)⟩=(\nQE0sin(Ω t),off,\nQE0cos(Ω t),on,(A12)\nwhich gives Eqs. (22, 23) of the main text. Moreover,\nsetting ν(ω) =ν0in Eq. (16), one finds\nQ=g0q0ν2\n0ωD (A13)\nwhich is Eq. (24) of the main text.\nThe autocorrelation function can be similarly evalu-\nated, leading to Eqs. (25, 26) of the main text, which for\nE0= 0 reduce to Kubo’s result [39]:\n⟨Fp(t)Fp(t′)⟩=mkBTK(t−t′). (A14)\nAppendix B: Diffusivity\nThe mean square displacement of the Brownian parti-\ncle is\n⟨{x(t)−x0}2⟩=Zt\n0dt1Zt\n0dt2⟨v(t1)v(t2)⟩. (B1)\nUnder field-off ICs, with a Markovian kernel, we get\n⟨{x(t)−x0}2⟩=mkBT\nγ0+\u0012(q+Q)E0\nΩγ0\u00132\n(cos(Ω t)−1)2.\n(B2)The second term is a new contribution due to the po-\nlarization of the bath, which may be observable even for\nlarge times tif the field’s intensity E2\n0is large enough.\nThe diffusivity, D(t) =⟨{x(t)−x0}2⟩/(2t), is\nD(t) =mkBT\nγ0\u0012\n1 +E2\n0(cos(Ω t)−1)2\nΩt\u0013\n(B3)\nwhere we have introduced the dimensionless quantity\nE0=(q+Q)E0√γ0mΩkBT. (B4)\nEqs. (B3, B4) are Eqs. 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Kubo, Reports on Progress in Physics 29, 255 (1966)."    },    {        "title": "1908.06862v1.Spectral_determinant_for_the_damped_wave_equation_on_an_interval.pdf",        "content": "arXiv:1908.06862v1  [math-ph]  19 Aug 2019SPECTRAL DETERMINANT FOR THE DAMPED WAVE\nEQUATION ON AN INTERVAL\nPEDRO FREITAS\nDepartamento de Matem´ atica, Instituto Superior T´ ecnico , Universidade de Lisboa,\nAv. Rovisco Pais 1, P-1049-001 Lisboa, Portugal andGrupo de F´ ısica Matem´ atica,\nFaculdade de Ciˆ encias, Universidade de Lisboa, Campo Gran de, Edif´ ıcio C6,\nP-1749-016 Lisboa, Portugal\nJIˇR´I LIPOVSK ´Y\nDepartment of Physics, Faculty of Science, University of Hr adec Kr´ alov´ e,\nRokitansk´ eho 62, 50003 Hradec Kr´ alov´ e, Czechia\nAbstract. We evaluate the spectral determinant for the damped wave equat ion on\nan interval of length Twith Dirichlet boundary conditions, proving that it does not\ndepend on the damping. This is achieved by analysing the square of th e damped\nwave operator using the general result by Burghelea, Friedlander , and Kappeler on\nthe determinant for a differential operator with matrix coefficients .\nPACS: 46.40.Ff, 03.65.Ge\n1.Introduction\nWe consider the simple mathematical model of wave propagationon a damped string\nfixed at both ends given by\n∂2v(t,x)\n∂t2+2a(x)∂v(t,x)\n∂t=∂2v(t,x)\n∂x2, (1.1)\nwith the space variable xon an interval [0 ,T],v(0) =v(T) = 0 and a(x)∈C([0,T]).\nDespite its apparent simplicity, the problem is nontrivial and interest ing and has re-\nceived much attention over the last two decades – see, for instanc e, [GH11, FL17,\nCFN+91, CZ94, BF09].\nThe operator associated with (1.1) is non-selfadjoint and the asym ptotical location\nof its eigenvalues was determined to first order in [CFN+91, CZ94], wh ere it was shown\nthat eigenvalues λconverge to the vertical line Re λ=−/a\\}bracketle{ta/a\\}bracketri}htas their imaginary part\ngoes to±∞, where/a\\}bracketle{ta/a\\}bracketri}htdenotes the average of the damping function. The general\nasymptotic behaviour was analysed in [BF09], where further spectr al invariants were\ndetermined.\nComingfromothersourcesintheliterature, thenotionofdetermin antofamatrixhas\nbeen generalised to operators. In this analogy, we would like to obta in a regularisation\ncorresponding to the product of eigenvalues of the given operato r. If the considered\nE-mail addresses :psfreitas@fc.ul.pt, jiri.lipovsky@uhk.cz .\n12 DETERMINANT FOR THE DAMPED WAVE EQUATION\noperator Shas eigenvalues {λj}∞\nj=1, in agreement with [RS71] (see also [GY60, MP49])\nwe define the generalised zeta function associated with the operat orSby\nζS(s) =∞/summationdisplay\nj=1λ−s\nj,\nfor complex sin a half-plane such that the above Dirichlet series converges. The\nspectral determinant may then be defined by the formula\nDetS= e−ζ′\nS(0), (1.2)\nwhere the prime denotes the derivative with respect to the variable s. Note that the\nseries defining the zeta function will not, in general, be convergent fors= 0. We use\nthe definition of ζSfor the real part of slarge enough and understand the formula in\nthe sense of the analytic continuation of the generalized zeta func tion to the complex\nplane.\nThe spectral determinant was computed for the Sturm-Liouville op erator in [LS77],\nwhere an elegant expression using the solution of a corresponding C auchy problem was\npresented. This was extended to the case of quantum graphs in [AC D+00, Fri06]. We\npoint out that spectral determinants have several applications e .g. in quantum field\ntheory [Dun08].\nTo the best of our knowledge the spectral determinant for the da mped wave equation\nhad not been studied previously, so in this note we bring together th ese two topics and\nevaluate this object. From a mathematical perspecive there is also what we believe to\nbe the interesting feature of applying the concept of the determin ant of an operator to\na non-selfadjoint operator. Furthermore, and as we will see, the determinant does not,\nin fact, depend on the damping. This may be expected form a formal analysis, and\nour purpose is to give a rigorous justification of this fact.\nThis note is structured as follows. In the next section, we ellaborat e on the math-\nematical description of the model and state the main result. In Sec tion 3 we then\naddress the problem for the case without damping, as this already d isplays some of the\nimportant features which we will need to consider later, namely, the fact that the asso-\nciated zeta function will depend on the branch cut which is chosen fo r the logarithm.\nIn Section 4 we recall the general result of Burghelea, Friedlander , and Kappeler. In\nSection 5 we apply this result to the square of our operator, since a direct application\nis not possible. Finally, we find the sought determinant for the dampe d wave equation\nin Section 6.\n2.Basic setting and formulation of the main result\nEquation (1.1) may be written in a different form, namely,\n∂\n∂t/parenleftbigg\nv0(t,x)\nv1(t,x)/parenrightbigg\n=/parenleftbigg0 1\n∂2\n∂x2−2a(x)/parenrightbigg/parenleftbigg\nv0(t,x)\nv1(t,x)/parenrightbigg\nwhich will prove to be more convenient for our purposes. Using the a nsatzv0(t,x) =\neλtu0(x),v1(t,x) = eλtu1(x), we can translate the initial value problem into the follow-\ning spectral problem\nH/parenleftbigg\nu0(x)\nu1(x)/parenrightbigg\n=λ/parenleftbigg\nu0(x)\nu1(x)/parenrightbiggDETERMINANT FOR THE DAMPED WAVE EQUATION 3\nwhereHdenotes the matrix operator\nH=/parenleftbigg0 1\n∂2\n∂x2−2a(x)/parenrightbigg\n.\nThe domain of this operator consists of functions u(x) =/parenleftbigg\nu0(x)\nu1(x)/parenrightbigg\nwith components\nin the Sobolev spaces uj(x)∈W2,2([0,T]),j= 0,1 satisfying the Dirichlet boundary\nconditions\nuj(0) =uj(T) = 0, j= 0,1.\nOur main result is the following.\nTheorem 2.1. Assumea(x)∈C([0,T]), and let εbe a positive number such that there\nare no eigenvalues with phase on the interval [π−ε,π), Then the spectral determinant\nof the operator Hdoes not depend on the damping and equals ±2T, where the plus and\nminus signs correspond to whether we define λ−s\nj= e−slogλjin such a way that the\nbranch cut of the logarithm is λ=tei(π−ε), orλ=tei(2π−ε),t∈[0,∞), respectively.\nRemark 2.2. Note that a value of εas above always exists, since on any compact set\nthere are only a finite number of eigenvalues.\nRemark 2.3. The case of the damped wave equation where a potential is adde d to\nthe right-hand side of (1.1)may be treated in a similar fashion and the corresponding\ndeterminant also turns out to be independent of the damping t erm. We discuss this\nsituation in Remark 6.2.\n3.The case of a(x) = 0\nWe begin by considering the case without damping, and denote the co rresponding\noperator by H0. It is a simple exercise that its eigenvalues are of the form λj=ijπ\nT,\nj∈Z\\{0}. To obtain the spectral determinant in this instance, we start fro m the zeta\nfunction resulting from the definition (1.2). However, one must pro ceed carefully here,\nas the result depends on the definition of λ−s\njand, in particular, on which branch of\nthe logarithm we use when defining i−sand (−i)−s.\nFirst, we consider that the logarithm has the cut in the negative rea l axis, i.e. the\neigenvalues of H0in the upper half-plane are λj=jπ\nTeiπ\n2,j∈Nand the eigenvalues in\nthe lower half-plane are λj=jπ\nTe−iπ\n2,j∈N.\nThe generalized zeta function for this operator is\nζH0(s) =∞/summationdisplay\nj=1/bracketleftBigg/parenleftbiggjπ\nTeiπ\n2/parenrightbigg−s\n+/parenleftbiggjπ\nTe−iπ\n2/parenrightbigg−s/bracketrightBigg\n=∞/summationdisplay\nj=1/parenleftBig\ne−iπ\n2s+eiπ\n2s/parenrightBig/parenleftbiggjπ\nT/parenrightbigg−s\n= 2eslogT\nπcos/parenleftBigπs\n2/parenrightBig\nζR(s),\nwhereζR(s) =∞/summationdisplay\nj=1j−sis the Riemann zeta function. We obtain\n−ζ′\nH0(0) =−2logT\nπζR(0)−2ζ′\nR(0) = logT\nπ+log(2π) = log(2 T),4 DETERMINANT FOR THE DAMPED WAVE EQUATION\nwhere we have used ζR(0) =−1\n2andζ′\nR(0) =−1\n2log(2π). Hence the spectral determi-\nnant for the operator H0is given by\nDetH0= e−ζ′\nH0(0)= 2T .\nNow we are going to compute the determinant in the case where we ch oose the cut\nto be the positive real axis. The eigenvalues are λj=jπ\nTeiπ\n2,j∈N, for the upper half-\nplane and λj=jπ\nTe3iπ\n2,j∈Nfor the lower half-plane. The generalized zeta function is\nnow\nζH0(s) =∞/summationdisplay\nj=1/bracketleftBigg/parenleftbiggjπ\nTeiπ\n2/parenrightbigg−s\n+/parenleftbiggjπ\nTe3iπ\n2/parenrightbigg−s/bracketrightBigg\n= e−iπs∞/summationdisplay\nj=1/parenleftBig\neiπ\n2s+e−iπ\n2s/parenrightBig/parenleftbiggjπ\nT/parenrightbigg−s\n= 2e−iπseslogT\nπcos/parenleftBigπs\n2/parenrightBig\nζR(s).\nHence we have\n−ζ′\nH0(0) =−2logT\nπζR(0)+2iπζR(0)−2ζ′\nR(0) = logT\nπ−iπ+log(2π) =−iπ+log(2T).\nThe spectral determinant for the operator H0is\nDetH0= e−ζ′\nH0(0)=−2T .\n4.A general result\nThe starting point for finding the determinant for the damped wave equation is a\ngeneral result by Burghelea, Friedlander and Kappeler [BFK95]. Th is result gives a\nformula for the determinant of a more general matrix-valued oper ator on an interval.\nFor convenience, we state this result here, together with the nec essary definitions.\nDefinition 4.1. Let us for n∈Ndefine the operator A=2n/summationdisplay\nk=0ak(x)(−i)kdk\ndxk, where\nakarer×rmatrices, in general smoothly dependent on x∈[0,T]. We assume that\nthe leading term a2nis nonsingular and that there exist an angle θso thatspeca2n∩\n{ρeiθ,0≤ρ <∞}=∅. We assume the following boundary conditions at the end poin ts\nof the interval.\nαj/summationdisplay\nk=0bjku(k)(T) = 0,βj/summationdisplay\nk=0cjku(k)(0) = 0,1≤j≤n.\nHere,bjkandcjkare for each j,kconstant r×rmatrices and bjαj=cjβj=I(Idenotes\nther×ridentity matrix). The integer numbers αjandβjsatistfy\n0≤α1< α2<···< αn<2n−1,\n0≤β1< β2<···< βn<2n−1.\nMoreover, we define |α|=/summationtextn\nj=1αj,|β|=/summationtextn\nj=1βj. We define the 2n×2nmatrices\nB= (Bjk)andC= (Cjk), whose entries are r×rmatrices. Here 1≤j≤2n,DETERMINANT FOR THE DAMPED WAVE EQUATION 5\n0≤k≤2n−1and\nBjk:=/braceleftbigg\nbjkfor 1≤j≤nand 0≤k≤αj\n0 otherwise,\nCjk:=/braceleftbigg\nbj−n,kforn+1≤j≤2nand 0≤k≤αj−n\n0 otherwise.\nWe define a 2n×2nmatrixY(x) = (ykℓ(x)),0≤k,ℓ≤2n−1whose entries are\nr×rmatrices ykℓ(x)defined by\nykℓ(x) :=dkyℓ(x)\ndxk,\nwhereyℓ(x)is the solution of the Cauchy problem Ayℓ(x) = 0with the initial conditions\nykℓ(0) =δkℓI. We are interested in the value of the matrix Yat the point T.\nFinally, we introduce\ngα:=1\n2/parenleftbigg|α|\nn−n+1\n2/parenrightbigg\n,\nhα:= det\nwα1\n1... wα1n.........\nwαn\n1... wαnn\n,\nwherewk= exp/parenleftbig2k−n−1\n2nπi/parenrightbig\n. Similarly, we define gβandhβ. We denote by γj,j=\n1,...,rthe eigenvalues of the matrix a2nand define\n(deta2n)gα\nθ:=r/productdisplay\nj=1|γj|gαexp(igαarg(γj))\nwithθ−2π <argγj< θ.\nTheorem 4.2. (Burghelea, Friedlander and Kappeler)\nThe spectral determinant for the operator Ais\nDetA=Kθexp/parenleftbiggi\n2/integraldisplayT\n0Tr(a2n(x)−1a2n−1(x))dx/parenrightbigg\ndet(BY(T)−C),\nwhere\nKθ= [(−1)|β|(2n)nh−1\nαh−1\nβ]r(deta2n(0))gβ\nθ(deta2n(T))gα\nθ.\n5.The square of the operator H\nOurpurposeistouseTheorem4.2toobtainthespectraldetermina ntfortheoperator\nH. However, a direct application of the theorem is not possible since th e highest order\nderivative is only present in one of the entries of the matrix which give s the operator\nH. This would contradict the assumption that the matrix a2nis nonsingular. In order\nto overcome this difficuly, we shall consider the operator A=H◦H=H2for which it\nis then possible to apply Theorem 4.2.\nHowever, we have to be careful when computing the spectral det erminant of Hfrom\nthe spectral determinant of A, as the determinant of the composition of two operators\ndoes not necessarily have to equal the product of their determina nts. This is known in\nthe literature as the multiplicative anomaly and, as has been shown in [BS03, Woj01],\neven the determinant of the square of an operator is not always th e square of the\ndeterminant.6 DETERMINANT FOR THE DAMPED WAVE EQUATION\nA direct calculation yields\nA=/parenleftbigg0 1\n∂2\n∂x2−2a(x)/parenrightbigg2\n=/parenleftbigg∂2\n∂x2 −2a(x)\n−2a(x)∂2\n∂x2∂2\n∂x2+4a2(x)/parenrightbigg\n=/parenleftbigg\n−1 0\n2a(x)−1/parenrightbigg/parenleftbigg\n−i∂\n∂x/parenrightbigg2\n+/parenleftbigg\n0−2a(x)\n0 4a2(x)/parenrightbigg/parenleftbigg\n−i∂\n∂x/parenrightbigg0\n.\nHence we have n= 1,\na2(x) =/parenleftbigg\n−1 0\n2a(x)−1/parenrightbigg\n, a1=/parenleftbigg\n0 0\n0 0/parenrightbigg\n, a0=/parenleftbigg\n0−2a(x)\n0 4a2(x)/parenrightbigg\n.\nWe deal with matrices 2 ×2, sor= 2, and have α1=β1= 0 and hence |α|=|β|= 0.\nThe boundary conditions according to Definition 4.1 are\nb10u(T)+b11u′(T) = 0, c10u(0)+c11u′(0) = 0.\nWe have the Dirichlet boundary conditions u(T) =u(0) =0, and thus the matrices\nbjkandcjkmust be chosen as\nb10=c10=I , b 11=c11= 0,\nwhereIis the 2×2 identity matrix.\nThe matrices BandCare 2×2 matrices with the entries being 2 ×2 matrices. Their\nrows are indexed by 1 and 2, their columns by 0 and 1. According to De finition 4.1 we\nhave\nB10=b10=I, B 11=B20=B21= 0,\nC20=c10=I , C 11=C10=C21= 0.\nHence we have\nB=/parenleftbigg\nI0\n0 0/parenrightbigg\n, C=/parenleftbigg\n0 0\nI0/parenrightbigg\n.\nThe matrix a2n=a2clearly has eigenvalues γ1=γ2=−1. Moreover, we have\ngα=gβ=1\n2/parenleftbigg0\n2−1+1\n2/parenrightbigg\n=−1\n4, hα=hβ=w0\n1= 1.\nand\n(deta2n)gα\nθ=2/productdisplay\nj=11−1/4ei(−1/4)(−π)=i\nTo sum up,\nKθ= [(−1)0(2·1)11−2]2i2=−4.\nNow, we are going to compute the matrix Y(x). By its definition, we have\nY(x) =/parenleftbigg\ny00(x)y01(x)\ny10(x)y11(x)/parenrightbigg\n=/parenleftbigg\ny0(x)y1(x)\ny′\n0(x)y′\n1(x)/parenrightbigg\n,\nwhere the entries of this matrix are 2 ×2 matrices with the following boundary condi-\ntions atx= 0\ny0(0) =I, y′\n0(0) = 0, y1(0) = 0, y′\n1(0) =I . (5.1)DETERMINANT FOR THE DAMPED WAVE EQUATION 7\nFor the matrix in the formula in Theorem 4.2 we have\ndet(BY(T)−C) = det/bracketleftbigg/parenleftbigg\nI0\n0 0/parenrightbigg/parenleftbigg\ny0(T)y1(T)\ny′\n0(T)y′\n1(T)/parenrightbigg\n−/parenleftbigg\n0 0\nI0/parenrightbigg/bracketrightbigg\n= det/parenleftbigg\ny0(T)y1(T)\n−I0/parenrightbigg\n= dety1(T).\nNow, we will find the solutions of the Cauchy problem, i.e. the equation\nH2/parenleftbigg\nu0(x)\nu1(x)/parenrightbigg\n= 0\nwith the boundary conditions (5.1). We obtain\nu′′\n0(x)−2a(x)u1(x) = 0, (5.2)\n−2a(x)u′′\n0(x)+u′′\n1(x)+4a2(x)u1(x) = 0. (5.3)\nMultiplying (5.2) by 2 a(x) and adding the result to (5.3) we obtain\nu′′\n1(x) = 0. (5.4)\nThe boundary conditions are y(k)\nl(0) =δklI. Hence to obtain the matrix y1(x), we have\nto find two vectors/parenleftbigg\nu01(x)\nu11(x)/parenrightbigg\nand/parenleftbigg\nu02(x)\nu12(x)/parenrightbigg\n, for which\ny1(0) =/parenleftbigg\nu01(0)u02(0)\nu11(0)u12(0)/parenrightbigg\n= 0, y′\n1(0) =/parenleftbigg\nu′\n01(0)u′\n02(0)\nu′\n11(0)u′\n12(0)/parenrightbigg\n=I.(5.5)\nThe general form of the solution of (5.4) is u1(x) =ax+b, and from the first condition\nin (5.5) we have b= 0. The second condition in (5.5) yields\nu11(x) = 0, u12(x) =x.\nHence in the first case we have from (5.2) u′′\n01(x) = 0, hence we have u01(x) =cx+d.\nBy the first condition in (5.5) we have d= 0, by the second one u01(x) =x.\nIn the second case ( u12(x) =x) we obtain from (5.2) u′′\n02(x) = 2a(x)x. Hence with\nthe use of the second condition in (5.5) we have u′\n02(x) =/integraltextx\n02a(s)sdsand with the\nuse of the first condition in (5.5) we have u02(x) =/integraltextx\n0/integraltextq\n02a(s)sdsdq.\nTo sum up,\ny1(x) =/parenleftbigg\nx/integraltextx\n0/integraltextq\n02a(s)sdsdq\n0 x/parenrightbigg\nand hence det y1(T) =T2. We obtained\nDetA=−4T2.\n6.The determinant of H\nLet us consider the operator Hin the general case. In each horizontal strip |Imλ|<\nKthere are finitely many eigenvalues. Hence there exists a positive εso that there\nare no eigenvalues whose arguments are in the intervals (0 ,ε], [π−ε,π), (π,π+ε],\nand [2π−ε,2π). We will choose the cut of the logarithm as the half-line λ=tei(π−ε),\nt∈[0,∞). We denote the eigenvalues of Hin the upper half-plane by ˜ µj, and their\ncomplex conjugates by ¯˜µj, the positive real eigenvalues by ˜ νj,j∈I1(hereI1is a finite8 DETERMINANT FOR THE DAMPED WAVE EQUATION\nindex set) and the negative real eigenvalues as −˜ωj,j∈I2(hereI2is a finite index\nset). It can be easily proven that there are no zero eigenvalues of H.\nFor the operator A, we denote the eigenvalues as µ2\nj, ¯µ2\nj,ν2\njandω2\nj. We consider the\nphases of all these eigenvalues in the interval [0 ,2π−2ε), the cut of the logarithm for\nthe operator Ais the half-line λ=te−2iε. Hence one can easily obtain\nlog ˜µj=1\n2logµ2\nj,log¯˜µj=1\n2log ¯µ2\nj−πi,\nlog˜νj=1\n2logν2\nj,log(−˜ωj) =1\n2logω2\nj−πi.\nThe zeta functions for these operators are\nζA(s) =∞/summationdisplay\nj=1[(µ2\nj)−s+(¯µ2\nj)−s]+/summationdisplay\nj∈I1(ν2\nj)−s+/summationdisplay\nj∈I2((−ωj)2)−s\n=∞/summationdisplay\nj=1(e−slogµ2\nj+e−slog ¯µ2\nj)+/summationdisplay\nj∈I1e−slogν2\nj+/summationdisplay\nj∈I2e−slogω2\nj.\nζH(s) =∞/summationdisplay\nj=1[˜µ−s\nj+¯˜µj−s]+/summationdisplay\nj∈I1˜ν−s\nj+/summationdisplay\nj∈I2(−˜ωj)−s\n=∞/summationdisplay\nj=1(e−slog ˜µj+e−slog¯˜µj)+/summationdisplay\nj∈I1e−slog ˜νj+/summationdisplay\nj∈I2e−slog(−˜ωj)\n=∞/summationdisplay\nj=1(e−1\n2slogµ2\nj+e−1\n2slog ¯µ2\njeπis)+/summationdisplay\nj∈I1e−1\n2slogν2\nj+/summationdisplay\nj∈I2e−1\n2slogω2\njeπis.\nHence we have\nζH(s)−ζA/parenleftBigs\n2/parenrightBig\n= (eπis−1)/parenleftBigg∞/summationdisplay\nj=1e−1\n2slog ¯µ2\nj+/summationdisplay\nj∈I2e−1\n2slogω2\nj/parenrightBigg\n= 2ieπis\n2sinπs\n2/parenleftBigg∞/summationdisplay\nj=1e−1\n2slog ¯µ2\nj+/summationdisplay\nj∈I2e−1\n2slogω2\nj/parenrightBigg\n.\nFor the derivatives at zero we obtain, using the fact that the sine w ill vanish in that\ncase,\nζ′\nH(0)−1\n2ζ′\nA(0) =iπlim\ns→0∞/summationdisplay\nj=1e−1\n2slog ¯µ2\nj+iπcardI2\n=iπcardI2+iπlim\ns→0∞/summationdisplay\nj=1e−iπs\n2eslogT\nπe−slogj\n+iπlim\ns→0∞/summationdisplay\nj=1/parenleftBig\ne−1\n2slog ¯µ2\nj−e−iπs\n2eslogT\nπe−slogj/parenrightBig\n.\nWewillprove thatthelasttermiszero. Wewillusetheasymptotics (s eee.g. [BF09])\n¯µ2\nj=−j2π2\nT2/parenleftbigg\n1+O/parenleftbigg1\nj/parenrightbigg/parenrightbiggDETERMINANT FOR THE DAMPED WAVE EQUATION 9\nand by our assumption ¯ µ2\njis not close to zero. We have\nlim\ns→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1(e−1\n2slog ¯µ2\nj−e−iπs\n2eslogT\nπe−slogj)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim\ns→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1e−iπs\n2eslogT\nπe−slogj\n×(e−1\n2slog(1+O(1 /j))−1)/vextendsingle/vextendsingle/vextendsingle\n≤lim\ns→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1e−slogj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|eslogT\nπ||e−sC−1|\n= lim\ns→0|ζR(s)||eslogT\nπ||e−sC−1|\n= 0.\nwhereCis a real constant and ζR(0) =−1\n2. Hence we have\nζ′\nH(0)−1\n2ζ′\nA(0) =iπcardI2+iπζR(0) =iπcardI2−iπ\n2. (6.1)\nThen the determinant is equal to\nDetH= e−ζ′\nH(0)= e−1\n2ζ′\nA(0)eiπ\n2e−iπcardI2=∓i√\nDetA=∓i√\n−4T2=±2T .\nThe number of negative real eigenvalues card I2is always even, so this term does\nnot influence the result. The aim of the rest of our analysis is to find t he sign of the\ndeterminant for the case of the cut we chose. First, we will follow an approach different\nfrom that which was used in Section 3 to find the determinant for the operator without\ndamping. The zeta function for the operator A0is\nζA0(s) = 2∞/summationdisplay\nj=1/parenleftbiggjπ\nT/parenrightbigg−2s\n(−1)−s= 2/parenleftbiggT\nπ/parenrightbigg2s\ne−slog(−1)∞/summationdisplay\nj=1j−2s= 2e−iπs/parenleftbiggT\nπ/parenrightbigg2s\nζR(2s).\nThen we have\n−ζ′\nA0(0) = 2πiζR(0)−4logT\nπζR(0)−4ζ′\nR(0) =−πi+2log(2 T).\nUsing equation (6.1) we obtain\nDetH0= e−ζH′\n0(0)= e−1\n2ζA′\n0(0)eiπ\n2= e−iπ\n2eiπ\n2elog(2T)= 2T .\nNow, we are going to generalize this approach to the operator with d amping. Our\naim is to find the imaginary part of ζ′\nH0(0). We denote the absolute value of µ2\njbyrj\nand its phase by π−ϕj. We have\nµ2\nj=rjei(π−ϕj),¯µ2\nj=rjei(π+ϕj).\nWe obtain\n(µ2\nj)−s+(¯µ2\nj)−s=r−s\nje−iπs(eiϕjs+e−iϕjs) = 2e−iπsr−s\njcos(ϕjs)\nand since r−s\njcos(ϕjs),ν2\njandω2\njare real\nImd\nds/braceleftBigg∞/summationdisplay\nj=1/bracketleftbig\n(µ2\nj)−s+(¯µ2\nj)−s/bracketrightbig\n+/summationdisplay\nj∈I1(ν2\nj)−s+/summationdisplay\nj∈I1(ω2\nj)−s/bracerightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0=−2πlim\ns→0∞/summationdisplay\nj=0r−s\njcos(ϕjs)\n=−πlim\ns→0∞/summationdisplay\nj=1/bracketleftbig\n(µ2\nj)−s+(¯µ2\nj)−s/bracketrightbig10 DETERMINANT FOR THE DAMPED WAVE EQUATION\nWe will use the asymptotic behaviour of the eigenvalues of the opera torH(see\n[BF09])\nµj=jπ\nTi−/a\\}bracketle{ta/a\\}bracketri}ht−i/a\\}bracketle{ta2/a\\}bracketri}htT\n2πj+O/parenleftbigg1\nj2/parenrightbigg\n(here/a\\}bracketle{t·/a\\}bracketri}htdenotestheaverageofthefunctionontheinterval)whichleadsto thefollowing\nbehaviour of the eigenvalues of the operator A\nµ2\nj=−j2π2\nT2−2jπ/a\\}bracketle{ta/a\\}bracketri}ht\nTi+O(1).\nHence we obtain\n(µ2\nj)−s= (−1)−sj−2sT2s\nπ2s/parenleftbigg\n1−2T/a\\}bracketle{ta/a\\}bracketri}hts\nπji+O(j−2)/parenrightbigg\n.\nSince ¯µ2\njis the complex conjugate of µ2\nj, we have\n∞/summationdisplay\nj=1/bracketleftbig\n(µ2\nj)−s+(¯µ2\nj)−s/bracketrightbig\n= 2T2s\nπ2s(−1)−s∞/summationdisplay\nj=0j−2s+∞/summationdisplay\nj=1O(j−2s−2).\nThesecondseriesisabsolutelyconvergent downtoRe s= 0,andhencewecanexchange\nthe sum and the limit. Moreover, the term under the sum goes to zer o ass→0, as it\nis multiplied by s. The first term can be written as 2e−iπsT2s\nπ2sζR(2s), hence its limit is\nequal to −1. We conclude that\n−Imζ′\nA(0) =−π.\nUsing equation (6.1) similarly to the case of no damping leads to the pos itive sign for\nthe determinant with the cut of the logarithm taken just above the negative real axis.\nThis proves Theorem 2.1.\nRemark 6.1. It is possible prove in the similar manner that if one moves th e cut so\nthat it passes finitely many eigenvalues of H, the determinant does not change.\nRemark 6.2. If one considers the dampedwave equation with a potential te rm, namely,\n∂2v(t,x)\n∂t2+2a(x)∂v(t,x)\n∂t=∂2v(t,x)\n∂x2+b(x)v(t,x),\nthen it is possible to use a similar approach to the above. If t he operator on the right-\nhand side has only negative eigenvalues, the negative eigen values of the damped wave\nequation always appear in pairs, as in the case without the po tential. The only difference\nis in the solution of the Cauchy problem. We define y(x)satisfying y′′(x)+b(x)y(x) = 0,\ny(0) = 0,y′(0) = 1. The result for the determinant is DetH= 2y(T)for the cut of\nthe logarithm just above the negative real axis, and −2y(T)for the opposite case. If the\noperator on the right-hand side also has positive eigenvalu es (but no zero eigenvalue),\nthe phase (−1)cardI2appears multiplying the determinant, changing the sign acc ording\nto whether the number of negative eigenvalues of the damped w ave equation is odd or\neven.DETERMINANT FOR THE DAMPED WAVE EQUATION 11\nAcknowledgements\nP.F. was partially supported by the Funda¸ c˜ ao para a Ciˆ encia e a T ecnologia, Portu-\ngal, through project PTDC/MAT-CAL/4334/2014. J.L. was suppo rted by the project\n“International mobilities for research activities of the University o f Hradec Kr´ alov´ e”\nCZ.02.2.69/0.0/0.0/16 027/0008487. J.L. thanks the University of Lisbon for its hos-\npitality during his stay in Lisbon.\nReferences\n[ACD+00] E. Akkermans, A. Comtet, J. Desbois, G. Montambaux, C . Texier, Spectral determinant\non quantum graphs, Annals of Physics 284(2000), 10–51. DOI: 10.1006/aphy.2000.6056.\n[BF09] D. Borisov, P. Freitas, Eigenvalue asymptotics, inverse pro blems and a trace for-\nmula for the linear damped wave equation, J. Diff. Eq. 247(2009), 3028–3039. DOI:\n10.1016/j.jde.2009.07.029.\n[BFK95] D. Burghelea, L. Friedlander, T. Kappeler, On the determin ant of elliptic boundary\nvalue problems on a line segment, Proc. Amer. Math. Soc. 123(1995), 3027–3038. DOI\n10.2307/2160656.\n[BS03] C. B¨ ar, S. Schopka, The Dirac Determinant of Spherical Sp ace Forms, in Hildebrandt\nS., Karcher H. (eds), Geometric Analysis and Nonlinear Part ial Differential Equations .\nSpringer, Berlin, Heidelberg (2003). DOI: 10.1007/978-3-642-55 627-23.\n[CFN+91] G. Chen, S.A. Fulling, F.J. Narcowich, S. 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DOI: 10.1142/9789812810571 0004."    },    {        "title": "1409.6268v1.On_the_Origin_of_a_Sunquake_during_the_29_March_2014_X1_Flare.pdf",        "content": "arXiv:1409.6268v1  [astro-ph.SR]  22 Sep 2014On the Origin of a Sunquake during the 29 March 2014 X1 Flare\nPhilip G. Judge\nHigh Altitude Observatory, National Center for Atmospheri c Research1, P.O. Box 3000,\nBoulder CO 80307-3000, USA; judge@ucar.edu\nLucia Kleint\nInstitute of 4D Technologies, University of Applied Scienc es and Arts Northwestern\nSwitzerland, 5210 Windisch, Switzerland; lucia.kleint@f hnw.ch\nAlina Donea\nCenter for Astrophysics, School of Mathematical Science, M onash University, Victoria\n3800, Australia; alina.donea@monash.edu\nAlberto Sainz Dalda\nStanford-Lockheed Institute for Space Research, Stanford University, HEPL, 466 Via\nOrtega, Stanford, CA 94305, USA; asdalda@stanford.edu\nand\nLyndsay Fletcher\nSUPA, School of Physics and Astronomy, University of Glasgo w, Glasgow G12 8QQ, UK;\nlyndsay.fletcher@glasgow.ac.uk\nABSTRACT\nHelioseismic data from the HMI instrument have revealed a sunquake associ-\nated with the X1 flare SOL2014-03-29T17:48 in active region NOAA 12 017. We\ntry to discover if acoustic-like impulses or actions of the Lorentz fo rce caused\nthe sunquake. We analyze spectropolarimetric data obtained with t he Facility\nInfrared Spectrometer (FIRS) at the Dunn Solar Telescope (DST ). Fortuitously\nthe FIRS slit crossed the flare kernel close to the acoustic source , during the im-\npulsive phase. The infrared FIRS data remain unsaturated throug hout the flare.\nStokes profiles of lines of Si I 1082.7nm and He I 1083.0nm are analyze d. At the\nflare footpoint, the Si I1082.7 nm core intensity increases by a factor of several,\nthe IR continuum increases by 4 ±1%. Remarkably, the Si Icore resembles the\nclassical Ca IIKline’s self-reversed profile. With nLTE radiative models of H,– 2 –\nC, Si and Fe these properties set the penetration depth of flare h eating to 100\n±100 km, i.e. photospheric layers. Estimates of the non-magnetic en ergy flux\nareat leasta factor of two less than the sunquake energy flux. Milne-Eddingto n\ninversions of the Si Iline show that the local magnetic energy changes are also\ntoo small to drive the acoustic pulse. Our work raises several ques tions: Have\nwe “missed” the signature of downward energy propagation? Is it in termittent\nin time and/or non-local? Does the 1-2 s photospheric radiative damp ing time\ndiscount compressive modes?\nSubject headings: Sun: atmosphere - Sun: chromosphere - Sun: corona - Sun:\nsurface magnetic fields - Sun: flares\n1. Introduction\nFlares are among the most energetic phenomena in the solar system , with well-known\nimpactsontheEarth. Beginninginthe1960s, itbecameclearthatth eonlyoptionforstoring\nthe large amount of energy for sudden release is the free energy a ssociated with the mag-\nnetic field threading the Sun’s atmosphere. According to the stand ard model (Carmichael\n1964; Sturrock 1966; Hirayama 1974; Kopp and Pneuman 1976), fl ares start by magnetic\nreconnection in the tenuous coronal plasma. Only here is the Alfv´ e n speed sufficiently high\nto permit rapid evolution. Subsequently, downward directed energ y in the forms of accel-\nerated particles, magneto-plasma waves, radiation and thermal c onduction deposit energy\nfrom above leading to bright ribbon-like structures in the chromosp here and, during strong\nflares, in the photosphere. Such temporarily heated structures (durations of minutes) then\nevaporate plasma into the corona, leading to post-flare loops that are bright in soft X rays\nand UV radiation on time scales of hours.\nLocal helioseismology has revealed flares which are accompanied by a coustic pulses\n(“sunquakes”) propagating below the visible surface. The mechan isms by which the flare\ndisturbance, originatinghighinthesolaratmosphere, couplestoint eriormodesisnotknown,\nthere being several challenges. Firstly, flares are difficult to obser ve, generally speaking, at\nboth the necessarily small time and length scales associated with the initial energy release\n(impulsive phase). Secondly, most acoustic sources preferentially occur in the magnetically\ncomplex penumbrae of sunspots (e.g. Fletcher et al.2011, section 3.6). Thirdly, the energy\n1The National Center for Atmospheric Research is sponsored by th e National Science Foundation– 3 –\nhas to propagate through the 9 pressure scale heights of the poo rly-constrained chromo-\nsphere. The chromosphere of active regions appears to be as com plex as sunspot penumbrae\n(Judge 2010). Measurements of magnetic fields there are difficult a nd rare (Navarro 2005a,b;\nUitenbroek 2011).\nProgress on sunquakes has been significant. For example, Donea a nd Lindsey (2005)\nhave demonstrated that only a small fraction,∼<10−3of the flare energy, is needed to\ntrigger a seismic transient in the photosphere. How this happens it is not yet understood\n(see, e.g., discussions by Donea 2011; Kosovichev 2014). Recent s pace missions have vastly\nimproved our ability to understand the evolving photospheric magne tic field, and significant\nsteps have been taken towards understanding changing magnetic fields and flares. The lower\nsolar atmosphere can show stepwise changes in line-of-sight (LOS) magnetic field strength\n(Kosovichev and Zharkova 1999) and shear (Wang 1992) during fla res. Sudol and Harvey\n(2005) observed a LOS field change in 15 X-class flares with a median o f 90 G. Recent\nobservers have found photospheric field and inclination changes ev en during small B1 flares\n(Murray et al.2012), using vector spectro-polarimetric data from the SP instru ment on the\nHinode spacecraft.\nIn this paper we relate acoustic sources found by Donea and other s (2014) from data\nfrom the Helioseismic Magnetic Imager (HMI) on the Solar Dynamics Ob servatory (SDO)\nspacecraft, to measurements from the Facility Infrared Spectr ometer (“FIRS” Jaeggli 2011)\nat the Dunn Solar Telescope of the National Solar Observatory in Su nspot, New Mexico.\nTable 1 lists some properties of the flare and acoustic source from D onea and others (2014).\nWhile many acoustic sources are present on the Sun with this intensit y, the spatial and tem-\nporal characteristics of this particular source mark it as generat ed by the flare. Changes in\nthe thermal and magnetic structure in the atmosphere are reflec ted in our Stokes polarime-\nter data through a photospheric Si Iline at 1082.7 nm and in the He I1083.0 nm multiplet\nformed near the top of the chromosphere.\n2. Observations\nWemadespectropolarimetricobservationsonMarch292014withFI RSandtheImaging\nBIdimensionalSpectropolarimeter(IBIS)instrumentsattheDST .Thelatterwillbereported\nelsewhere. In addition, we acquired, every 60 seconds, bursts of data in G-band (430.5 nm)\nand Ca II (393.3 nm) narrowband filters for speckle reconstructio n, and a white light camera\nacquired rapid cadence images. FIRS was used in a single slit, dual-bea m mode with a 40\nmicron wide slit, subtending an angular width of 0 .′′30, oriented close to the E-W line on the\nSun.– 4 –\nThe polarization modulation scheme was a four-state balanced sche me with 125 ms ex-\nposures and a full cycle of 1.2 s, with 10 such cycles co-added by the instrument at each\nscan position on the Sun. This relatively slow modulation, set by the ne ed for the liquid\ncrystal variable retarders to relax in response to voltage change s, runs the risk of encoding\nlight variations entering the polarimeter due to residual seeing motio n and/or solar evolution\ninto systematic errors called “crosstalk” (Lites 1987; Judge et al.2004; Casini et al.2012).\nCrosstalk appears to be important in the He Iline during the flare as the tenuous chro-\nmospheric plasma radiating the helium emission evolves on timescales co mparable or faster\nthan the 1.2s modulation cycle. No evidence is seen for such crossta lk in the photospheric\nSiIline.\nThroughout all our observations the count rate remained in the line ar regime of the IR\ndetector (below 8000 ADU). The solar image was scanned from S to N across the FIRS slit\nin 100 or 120 steps of 0 .′′3 to produce images in four spectropolarimetric states Si(linear\ncombinations of I,Q,UandV), covering a spectral range from 1081.93 to 1085.01nm, and\na spatial area of 30′′or 36′′×75′′for all scans reported here. The images have bin sizes of\n0.3′′. Five scans of the slit across NOAA 12017 were begun at 16:29:26, 16 :55:58, 17:40:06,\n18:01:55, 18:30:13 UT. The seeing was good enough for the adaptive o ptics system (AO) to\nmaintain a lock on the sunspot during the observing run. The peak fla re emission is seen in\nthe FIRS data during the third scan begun at 17:40:06 UT.\nFigure 1 shows a G-bandimage with contours superposed, showing ( black) the egression\npower from the acoustic holography reported by Donea and other s (2014), and (white) the\ncore intensity of the Si Iline at 1082.7 nm. The G-band image was aligned with a continuum\nimage from the HMI instrument on the SDO spacecraft obtained at 1 7:45:00 UT by eye,\nco-alignment uncertainties are at most one arcsecond. (The co-a lignment accuracy is limited\nby the fact that the FIRS scan was obtained under varying seeing c onditions and over a\n20 minute scanning period). Black contours show the dominant local sources of power for\nwaves traveling downintothesolarinterior. Thewhitecontourssho wtheinfluence ofheating\nprocesses from the flare on the regions of formation of the Si Iline in the Sun’s atmosphere.\nThe G-band image is a composite, speckle reconstructed image obta ined at 17:46:44 UT.\nThe image shows a diffuse brightening at these wavelengths centere d near X=522, Y=260,\nwhich is real flare emission, with an amplitude of ≈1.2−1.4 times the non-flaring intensity,\nperhaps a component of the still poorly understood white light emiss ion. None of these data\nare strictly contemporaneous, the FIRS data shown were built of a raster scan that began at\nY=254.5 at time 17:40:12 UT, and ended at Y=284.2 at 18:01:39 UT. The h orizontal dashed\nlines show the positions of the FIRS slit at 17:45 and 17:50 UT.– 5 –\n2.1. Data Reduction\nThe FIRS data were reduced using software originally developed by J aeggli (2011) and\nmodified by Tom Schad (private communication 2013). The reduction s followed standard\nprocedures: correction for detector non-linearities; subtract ion of dark frames; division by\nflat fields; co-registration of the two beams (including corrections for image rotation); polar-\nization calibration; de-modulation (conversion of linear combinations of Stokes parameters\nto individual Stokes parameters). Since the required polarization s ensitivity is very high\nin chromospheric lines (e.g. Uitenbroek 2011), special care is neede d in handling calibra-\ntions. Usual dark frames and flat fields were acquired, and a gain line arization correction\nwas applied to the data using a curve from Jaeggli (2011). We used fl ats obtained with a\ncalibration lamp which is vignetted across the detector frame, in pre ference to solar flats in\nwhich photospheric spectral lines are always present. This is becau se we analyze below the\ndetailed profiles of the Si Iline at 1082.7 nm.\nResidual fringes and some detector artifacts are present in thes e data. Fringes are a\nsource of systematic error. Using careful corrections for flat fi elds we have reduced fringing\nto∼<2×10−3Ic(peak-to-trough) where ICis the continuum intensity, which is defined\nusing wavelengths for each individual scan shown in Figure 2. The wav elength scales of the\nspectra are determined using solar flat-field scans and solar photo spheric absorption lines.\nThe spectrograph was stable at the level of 0.2 pixels in wavelength ( 0.004 nm) during the\nobservations, equivalent to a Doppler shift of 0.2 kms−1.\nIt isimportanttonotethatatinfraredwavelengths, theenhance ment ofintensity during\nflares is moderate, quite unlike the well-known enormous UV and X-ra y enhancements. All\nof the FIRS data were obtained in the linear regime.\n2.2. Stokes line profiles\nFigure 2 shows the mean intensity spectrum with annotated spectr al features and, in\nthe lower panels, Stokes profiles for I,Q,UandVfrom left to right. The particular data\nshown in the lower panels are from the 29th scan obtained through t he flare footpoint be-\nginning at 17:46:29 UT, just as the flare was in the impulsive phase as fo und from RHESSI\ndata analysis (Donea and others 2014). The line profiles of the phot ospheric Si Iline are\nessentially consistent with polarization induced by the Zeeman effect , with the possible ex-\nception of those seen inthe flare kernel. On theother hand, while th eHeIlinear polarization\n(Q,U) profiles might initially appear to be of solar origin, the result of atomic alignment,\nthe presence of linear polarization in the J= 0 upper level to J= 1 lower level transition– 6 –\nat 1082.9 nm must be due to systematic errors. No atomic alignment is possible with these\nquantum numbers, nor is it possible for levels involving hyperfine stru cture of any3He nuclei\nthat might be present.\nIn our figures, all data are taken from the dual-beam system, but we also examined\nsingle-beam data. The signals in the two beams are very similar for all w avelengths outside\noftheheliumlines, butthatsignificant differencesarepresent inthe polarizedheliumprofiles.\nThis is a clear sign of I−(QUV) crosstalk in the helium lines. The dual-beam corrections\nare clearly doing an excellent job at other wavelengths (for example , those in the Si Iline\ncore). We surmise that the helium lines are evolving in intensity at least as rapidly as the 1.2\ns cadence of the modulation cycle, at the level of a few percent, th ereby producing spurious\nsignals. Faster modulation seems appropriate for flare observatio ns of the chromosphere. We\ndefer further analysis of the flare kernel data for He Ito later work.\nThe noise levels of these Stokes Q,U,Vdata are close to 8 ×10−4IC. The largest fringes\nremaining are in Stokes Vat the level of 2 ×10−3IC. The noise levels are more than adequate\nfor us to attempt inversions of the Si Iline data.\n2.3. Quick look parameters\nWe used the Stokes profiles to derive several simpler quantities. Fr om Stokes I(inten-\nsity), we compute the n= 0,1,2 moments M(n)of the line profiles weighted by wavelength\nfrom line center. We define the continuum-subtracted line intensity as\nI′\nx=IC−Ix,\nwhere the Doppler shift xis defined as c(λ/λLAB−1) withcin kms−1. Then we define\nM(n)=/integraldisplay\nI′\nxxndx.\nThe “Doppler shift” of the line is v=M(1)/M(0)kms−1, the line width (not shown here) is\nw=/radicalbig\nM(2)/M(0)kms−1. We also computed “quick-look” quantities from the IQUVStokes\nparameters. These include a LOS “magnetogram” which is simply the m edian of the ratio\nof Stokes Vto the first derivative of Stokes Iwith respect to wavelength over wavelengths of\nsignificant line absorption or emission. The other magnetic paramete r is the field “azimuth”\nwhich is1\n2arctan(U/Q) where again median values of the ratio U/Qare taken across both\nthe SiIline and He Imultiplet. The relationship of these quantities to physical paramete rs\nin the Sun arises only when the polarization is dominated by the Zeeman effect (see, e.g.\nJefferieset al.1989) and only when the polarization Q2+U2+V2≪1, and when the Stokes– 7 –\nprofiles all originate from the same physical volumes underlying a give n pixel. Nevertheless,\nsuch quantities as LOS magnetograms are very familiar to solar phys icists which, with care,\nillustrate properties of the solar magnetic field.\nFigures 3 and 4 show some of the quick look parameters and continuu m intensity from\nthe three scans obtained before, during and after the impulsive ph ase. These we label phases\n“bef”,“dur” and “aft”. The white contours of Figure 1 are from th e Si core data shown in\nthe second row of Figure 3. Figure 4 shows parameters related to t he magnetic field. Salient\nfeatures of these plots include:\n•The IR continuum shows only a weak brightening during scan “dur”.\n•Boththephotospheric2SiIandchromospheric He Ilinesshowconsiderable brightening\nduring scan “dur”, in the line cores.\n•The filament seen in the He Iline core images, lying roughly along the neutral line\nseen in “bef” scan magnetograms, disappears by scan “aft”.\n•The He Iabsorbers in the filament are seen moving upwards by between 5 and 10\nkms−1in scan “bef”. During scan “aft”, the filament is replaced by a diffuse region of\nHeIline absorption.\n•Magnetograms show only subtle changes from scans “bef” to “dur ” and “aft”.\n•The photospheric magnetic azimuthal angles show systematic chan ges as the flare\nprogresses.\n•The He IU/Qsignals during scan “dur” have coherent structure whose origin inc ludes\nat least some I→(QU) crosstalk. The spectral profiles show that they certainly\ncannot be interpreted as a traditional Zeeman-induced linear polar ization.\nDuring scan “dur” (the impulsive phase), magnetic quick-look data f or the He Iline can-\nnot be trusted to represent magnetic fields because of cross-ta lk. We will attempt to correct\nfor this crosstalk in a future publication, since even in the presence of atomic polarization,\nthe 1083 nm multiplet still retains accurate information on field azimut hal angles, for field\nstrengths >10G (the strong field limit of the Hanle effect). To quantify the magne tic field\nchanges we turn to inversions for the Si IIQUVdata.\n2Other photospheric lines, not shown, also show emission cores in the FIRS flare footpoint spectra:\n1081.83 nm (Fe I), 1083.91 nm (Ca I), 1084.40 nm (Si I?).– 8 –\nFirst we will use properties of the Si Iline and continuum to constrain the depth of\npenetration of flare energy that is sufficient to change the temper ature structure in the\ndeep chromosphere and photosphere. At a first glance the contin uum intensity appears\nto change very little, if at all. But both p modes and granulation modify the continuum\nintensity at a level of a few percent at angular resolutions similar to t hose of FIRS (e.g.\nS´ anchez Cuberes et al.2000), making relatively small changes difficult to see in slit rasters.\nCareful examination of the FIRS spectra obtained beginning at 17:4 6:16 UT, show a signif-\nicant brightening of 4 ±1% above levels in the neighboring spectra taken 13 seconds before\nand after, close to the flare kernel observed in RHESSI and line dat a. Evidence for this is\nshown in Figure 5. These data include variations in the transparency and seeing conditions\nof the atmosphere and hence vary significantly from row to row in th e figure. As is obvious\nin the figure, transparency variations were strongest in the first scan, getting progressively\nweaker in the second and third scans. However, relative intensities alongeach row in each\npanel can be fairly compared. The uncertainty quoted above is a 1 σstatistical variation of\nthe detrended intensity measured along the rows immediately adjac ent to the row containing\nthe flare.\nThe coherent streak of brightness in continuum data from 17:46:16 has all the charac-\nteristics of a genuine brightness increase associated with a white ligh t flare. This picture is\nsupported by HMI continuum data from the SDO spacecraft, which shows a≈5% increase\nin continuum intensity during the flare. Its appearance only in one FI RS scan indicates a\nvery rapid evolution, characteristic of an origin from dense photos pheric material which has\na radiative relaxation time of 1-2 seconds (Spiegel 1957).\n3. Analysis\nThe spatio-temporal behavior of the flare as obtained by FIRS is su mmarized in Fig-\nures 3 and 4. Remarkably, the slit happened to scan across the flar e footpoint ribbons at\nthe flare peak, 17:46 UT (Table 1). Also shown on the figure is the cor e of the flare-related\nacoustic source (Donea and others 2014). It is clear that FIRS ma naged to capture those\nlocations in solar- yheliographic coordinate that correspond to the time and place of th e\nacoustic source.– 9 –\n3.1. Helioseismic holography\nSeismic transients from solar flares can be detected by pre-proce ssing solar data and\napplying the analytical technique of helioseismic holography to Dopple r measurement of\nthe active region hosting the solar flare. Donea and others (2014) analyzed Doppler maps\nfrom the Helioseismic and Magnetic Imager instrument (HMI; Schou et al.2012) on board\nthe Solar Dynamics Observatory satellite (SDO). HMI measures pro perties of photospheric\ndynamics and magnetic fields every 45 seconds. Donea and others ( 2014) generated Postel\nprojection maps of the seismic emission of NOAA 12017. We refer disc ussion of the prin-\nciples of seismic holography to Section 4 of Lindsey and Braun (2000) , with application to\nflare observations to Donea et al.(1999). Briefly, the seismic responses to the flare pertur-\nbations are identified through an excess of the emission power, |H+(r,t)|2. Each pixel in\nan image of |H+(r,t)|2is a representation of the coherent acoustic power for waves tha t\nhave propagated downward from the focus, traveled thousands of kilometers beneath the\nsolar surface, and re-emerged into a pupil a significant distance fr om the focus. With this\ntechnique Donea and others (2014) uncovered a weak but significa nt seismic source at the\nfootpoint shown in Figure 1. Hard X-ray emission, magnetic transien ts and strong UV foot-\npoint emission were analyzed by Donea and others(2014), confirmin g thattheseismic source\nis indeed associated with the flare.\n3.2. The depth of penetration of flare energy\nBy comparing the brightness of models of continuum and Si I1082.7 nm line to ob-\nservations, we can in principle constrain the depth in the atmospher e to which significant\nheating from above can penetrate. Our approach is simple. We ask: what are the deepest\nand shallowest layers in the atmosphere heated by the flare that ar e compatible with the\ndata?\nTo preface the model calculations below, we note that the flare Si Iprofile (Figure 2)\nresembles classical Ca IIHandKself-reversed profiles (Linsky and Avrett 1970), but with\nfar weaker line absorption wings. The Ca IIline cores, much more opaque than the line\nof silicon, form in the chromosphere with a source function dominate d by scattering. The\nsimple observation of a self-reversed profile of Si Iimplies a significant column mass, much\nhigher than that for the calcium line. The breadth of a Doppler-broa dened, self-reversed\nline is larger than an optically thin line formed under the same conditions by the factor\n≈√lnτ0whereτ0is the line center optical depth. For τ0= 100 this factor is over 2.1. The\nself-reversal is also very narrow (FWHM ≈0.015 nm, see Figure 2), indicating turbulent\nspeeds of FWHM/1.66 ≈2.5 kms−1where the line core forms. A profile averaged along the– 10 –\nregion with obvious Si Iemission in Figure 2 is shown in Figure 6. The averaging washes\nout the self-reversal in the latter plot.\nWe model the Si Iline and the neighboring IR continuum both during and outside\nof the impulsive phase. We performed nLTE radiative transfer calcu lations, in several 1D\nmodels of the solar atmosphere, following the tradition of Vernazza et al.(1973, 1976, 1981,\nhenceforth VAL81). We solved nLTE statistical equilibrium equation s for atoms of H, C,\nSi and Fe using the program RH (Uitenbroek 2000). These atoms we re chosen because\nUV radiation controlling the Si Ispectral line at 1082.7 nm is dependent on the nLTE\nsolutions of these abundant elements. We considered using one of s everal flare models (e.g.\nMachado et al.1989). However, these models were constructed to try to identif y the origin\nof white light emission in flares. Our goal is different, to try to see if mo deling can provide a\ndepth of penetration of flare energy into the photosphere. Ther efore we adopted a different,\nmore straightforward strategy. We started with the model “C” o f VAL81 and explored the\neffects of introducing temperatures plateaus of the form\nTe=T0−T′\n1logm, m 2> m,\nwheremis the column mass of the atmosphere, T0,T′\n1are non-negative constants, and m2\nis a column mass above which temperatures are changed. We have th ree free parameters,\nand so our results will not be unique. But such plateaus, with small gr adientsT′\n1, have\njustification at least during some phases of flaring plasmas seen in ra diation hydrodynamic\ncalculations (see the 50s panel of Figure 3 of Allred et al.2005, for example). The main\nsensitivity of the emerging spectra is to the two parameters aandm2. Given an estimate of\nm2the height of the energy penetration follows from the m(z) relationship for the model.\nWe made calculations in two limits: in the calculations shown in the Figures below\nwe allowed the atmosphere to relax to a state of hydrostatic equilibr ium; in the other limit\nwe merely solved the statistical equilibrium equations with no such adj ustment. The sound\ncrossing time of the photosphere is on the order of a few scale heigh ts divided by 7 kms−1, a\nminute or so, comparable to the duration of the flare impulsive phase . These limits probably\nspan the behavior of intensities from an evolving atmosphere. The d ifferences between the\ncalculationsaresmallinphotosphericlayersbutaresignificantforr egionsandspectraformed\nabove 600 km above the photosphere. Such differences do not affe ct our conclusions which\ndepend only on the photospheric Si Iline.\nThese calculations are not state-of-the art in terms of dynamics, our focus is instead on\na careful treatment of the formation of the Si I1082.7 nm line and of the continua formed\nbetween 125 and 180 nm for later comparisons with SDO/AIA data. W e therefore took\ncare to use modern and complete atomic data for the Si and Fe neut rals. We used atomic\nenergy levels and transition probabilities from NIST up to and including the 4plevels in– 11 –\nSi, and we used photoionization cross sections from the OPACITY pr oject (Seaton 1987),\ntreated as outlined in Judge (2007). Collisions with electrons were tr eated using the impact\napproximation for permitted transitions (Seaton 1962), Seaton’s semi empirical formula for\ndirect ionization (Allen 1973), and a collision strength of 0.1 for forbid den transitions.\nFigure 6 shows, in the right panels, computed and observed profiles of SiI1082.7 nm,\nwith all intensities normalized to quiet Sun values. These calculations a re representative of\ntwo limits of the value of m2– and hence height of penetration – used in the models.\nThe first class (upper panel) allows penetration of energy and enha nced temperatures\ndown to photospheric layers - we allowed temperatures to rise down to 0 km height by adding\nvarious plateaus at such depths. Remarkably, the model shown pr oduces an acceptable\nmatch to the observed profiles and continuum (the He Iline is not modeled here). Exploring\ndifferent temperature plateaus we determined that a reasonable a greement with the line and\ncontinuum observations requires the flare energy to penetrate a nd heat down to a height of\n∼>100±100 km above the photosphere. The “error bar” comes from the n eed to produce\nthe 4% enhancement in continuum emission ( <200 km) with temperatures that can match\nthe SiIprofile, both features spanning the region between 0 and 700 km.\nThe second limiting case is one where flare energy penetrates only to the mid-upper\nchromosphere. Downward propagating radiation enhances the co res of lines, a typical calcu-\nlation is shown in the lower panels. The line width is very narrow even tho ugh we adopted\nnon-thermal speeds (microturbulence) of up to 8 km/s in the middle chromosphere (close\nto the sound speed). The continuum, formed predominantly in the p hotosphere with a tiny\ncontribution from optically thin emission in the plateau, is close to the p re-flare level. The\ncomputed continuum includes thermal photospheric emission as well as hydrogen recombi-\nnation from the plateaus. These two contributions have been discu ssed by Machado et al.\n(1989); Kerr and Fletcher (2014), among others. The contribut ion from the latter is small\nin our calculations, the Balmer continuum originating from an optically t hick layer near 350\nkm and the longer-wavelength ( >364 nm) H−and Paschen continua near 0 km.\nThe core of the Si Iline during the flare is broad compared with a thermal width near\n1.8 kms−1(Figure 6), and like the well-studied Ca IIHandKlines the origin of this width\nappears most naturally explained through scattering (see above) . Some decades ago there\nwas a discussion of the Wilson-Bappu effect, an empirical relationship between the width\nof the core of the Ca IIlines and stellar luminosity, in favor of line formation in terms of\nscattering (Ayres 1979) and not optically thin micro- or macro-tur bulence (Fosbury 1973).\nThe presence of the narrow self-reversed core seems irrefutab le evidence for the presence\nof scattering and argues strongly for a deep formation of the cor e. Only calculations of\npenetration of flare energy to the photosphere produce lines bro adened by scattering and– 12 –\nself-reversals, the latter happen to be weak in the case shown in Fig ure 6, but not in obvious\ndisagreement with the observed profile.\nWe stress that the detailed structure of our calculations is not uniq ue and should only\nbe viewed as an attempt to find the depth of penetration of significa nt heating during the\nimpulsive phase of the flare. Overall, our comparisons with observat ions of the Si I1082.7\nnm line, and taking into consideration the difficulties of tying down the c ontinuum intensity\nduring the flare, we conclude that heating sufficient to change detectably the photospheric\ntemperature occurs at least to about 100 ±100km above the visible photosphere . Based on an\nexploration of values of T0,m2in our model, we believe that this aspect of our calculations\nis robust.\n3.3. Inversions\nWe used the code MELANIE (Socas-Navarro 2003) to invert the Si IStokesIQUV\nprofiles to derive the vector magnetic field in the photosphere. This was done only for\nscans obtained before and after the impulsive phase. Codes exist f or inversion of the He I\nmultiplet (e.g. L´ opez Ariste and Casini 2002; Lagg et al.2004; Asensio Ramos et al.2008),\nbutwehavenotattemptedsuchinversions yetbecausewemustde alwithsignificant crosstalk\nin the He IQUprofiles during the flare, and because outside of the flare these pr ofiles are\nmostly of low signal-to-noise ratio.\nThe observed Si Iline – 3p4s3Po\n2−3p4p3Pe\n2(lower and upper levels respectively) –\nformsbetween ≈100km (wings) and600km (core) above thephotosphere inour 1Dm odels.\nMELANIEsolvesforasolutiontotheMilne-Eddingtonequations(sou rcefunctionlinearwith\noptical depth) for lines with Zeeman-induced polarization, minimizing d ifferences between\nobserved and computed profiles. The solution includes the vector m agnetic field (with its\n180◦ambiguity), opacity, Doppler width and shift, damping parameter, n on-magnetic filling\nfactor. The Milne-Eddington approximation is a simplification that sur ely is invalid during\nthe flare itself. But before and after the flare its use appears rea sonable, outside of bright\nflare ribbons and below say 600 km in the atmosphere. Our conclusion s will be based only\non the non-flaring atmosphere.\nWe inverted all five scans. We set the statistical uncertainty of ea ch data point to\n10−3ICto evaluate values of χ2, estimated using the measured fluctuations in QUVat\ntypical continuum wavelengths. For comparison, some of the best vector polarimetric data,\nthe “deep mode high S/N” observations from the SP instrument on t he Hinode spacecraft\nhave rms noise of 3 ×10−4ICin the 630 nm region, for integrations of 67 s (Lites et al.2008).– 13 –\nOutside of the flare scan, the distribution of χ2peaks near 40, showing that systematic errors\nare large and/or the model parameterization is poor. Given the res idual fringing and other\nartifacts evident in the data, this does not by necessity imply that t he model is poor. The\nreproducibility of the inversions was tested by initializing the same dat aset with two different\nrandom initial guesses. The resulting rms variations in the magnetic fi eld strength Bare\n140 G, inclination 18◦, azimuth 41◦, and the LOS B30 G.\nFigures 7 and 8 show results of inversions of the scans obtained bef ore and after the\nflare, begun at 16:29:26, 16:55:58, and 18:01:55 UT. No attempt at a r esolution of the 180◦\nambiguity in the field azimuth has been made, and the angles are define d relative to the local\nvertical3(inclination) and in the plane of sky (azimuth, zero and 180◦being along the E-W\ndirection). Circles show the location of the center of the acoustic s ource. Figure 7 shows\nmeasured changes in magnetic parameters in the two scans obtaine d before the flare. There\nare detectable differences across the bulk of the field of view in all ma gnetic parameters.\nFocusing on data in the circled region of the acoustic source, we see a significant increase\nin the field strength in this region, accompanied by becoming more inclin ed to the vertical\ndirection (data shown in the first two rows of the Figure). Note tha t the circled region is\nsome 5′′from the polarity inversion line. The maps of Bsuggest that a channel of weak\nfield is moved to the west by an arcsecond. Initially the field is inclined at some 130◦to\nthe vertical. But by 17:05 UT two bands of field connected in a “Y-sha pe” on its side in\nthe image appear more inclined to the vertical. The field azimuth in the “ Y” shape departs\nsignificantly from initially E-W to more N-S. The LOS field within at the circ le’s center\nshows an increase that results from increases in Bdespite the decrease in inclination. It\nis unclear from our data if these changing fields arise from motions of field vertically (flux\nemergence) or horizontally (flows). There is little evidence for vert ical motions from the LOS\nvelocity measurements shown in Figure 3, but the inversion data (no t shown) reveal a very\nsmall (-0.3 kms−1) blue-shift pattern in the Si Idata in the 10:55:58 UT scan that might\nconceivably be associated with the upper part only of the “Y” patte rn seen in the magnetic\ndata.\nThe scans upon which the inversions are based are26 minutes apart . The above changes\nare unremarkable when compared to the larger field of view, except that they are within the\ncircle encompassing the acoustic source and they are significant in a ll magnetic parameters.\nFigure8showsmeasuredchangesbeforeandaftertheflareitself , scansbegun66minutes\napart. The difference panels show again an increase of Band azimuth, and a weak reduction\nof inclination, in a band in the E-W direction cutting through the circled region, flanked\n3The vertical direction of center of the region is rotated 39.6◦E-W and 15.8◦S-N relative to the LOS.– 14 –\nby regions of increased inclination just to the S and N. This sheared r egion (differentially\nchanging field inclinations with time) appears aligned with the bright foo tpoint emission\nseen in the core intensities of the Si Iand HeIlines. The data are noisy, however.\nThus, our analysis hints that magnetic fields associated with the par ticular acoustic\nsource evolve to become more sheared (i.e. inclination angles divergin g in time), stronger\n(perhaps due to flux emergence) and rotated relative to the EW dir ection, during the flare.\nThese results appear to correspond to a mixture of earlier results . Wang et al.(2012a)\nfound penumbral fields which became more vertical after flaring. I n contrast, for some flares\nMart´ ınez-Oliveros et al.(2008); Wang et al.(2012b) reported field lines highly inclined to\nthe vertical after a flare-associated seismic transient. We note t hat the seismic source we\nhave analyzed is unusual. It is found near a magnetic pore, emerging from a magnetically\nquieter area somewhere between the main two sunspots of the AR1 2017 (Donea and others\n2014), instead of in a penumbra.\nLastly, if flux emergence were responsible for these measured cha nges in magnetic field,\nin 1 hour the plasma and magnetic field moving vertically through the co mpressible sub-\nphotosphere with a surface velocity∼<0.3 kms−1could have emerged from depths no deeper\nthan≈200 km. If advected by granules with 1 kms−1speeds, the flux could have emerged\nfrom no deeper than 600km. It is interesting to consider how such c hanges to the immediate\nsubsurface structure might or might not affect the generation of sunquakes.\n3.4. The mode of transfer of flare energy down through the atmo sphere\nArmed with a unique dataset, we have studied the depth of penetra tion of flare energy\ndown into the solar photosphere. We have shown that the detecte d changes in thermal\nstructure in the atmosphere reach the photospheric level, but ba rely. Here we examine\npossible modes by which the energy might be transported through t he photosphere into the\ndeeper solar layers, thereby exciting the sunquake.\nThe power in the main kernel of the acoustic source measured using seismology from\nHMI is (Donea and others 2014):\nP= 1.3±0.05×1026erg s−1.\nThis power is distributed over an area including the main kernel cente red at (X,Y) =\n(518,264) (see Figure 1), the source just to the SW requiring an addition al 1.0×1026erg s−1.\nThe mainsource’s spatialdistribution isnearly bi-Gaussianwith ageom etricmeanfull width\nat half-maximum (FWHM) of w= 4.2 HMI pixels, w≡1.5×108cm. The peak of the power– 15 –\nper unit area is F=P/(A=πa2) witha=w/2√\nln2, or\nF= 5×109erg cm−2s−1.\nThis should be regarded as a lower limit since both the holographic tech nique and HMI have\nnon-negligible angular resolutions. The area A= 2.6×1016cm2is strictly an upper limit\nfor the same reasons.\nLet us consider first “non-magnetic mechanisms” by which energy is transported to the\nacoustic source. In this picture the changing magnetic field genera tes thermal perturbations\nindirectly via the end product of large coronal magnetic restructu ring (conduction, particles,\nlocal downward radiative heating), channeling some flare energy int o the photosphere. We\ncan estimate energy fluxes into the acoustic source region that ar e compatible with our\nobservations in several ways. First, we note that the excess the rmal energy radiated from\nthe photosphere during the few minutes of the rise phase is roughly 4-5% (i.e. the measured\ncontinuum enhancement) of the unperturbed solar radiative flux d ensityF⊙= 6.33×1010\nergscm−2s−1:\nPRAD≈0.04F⊙A∼<7×1025erg s−1.\nThe radiative cooling time of photospheric plasma is 1-2 s (Spiegel 195 7). Curiously then,\nalthough PRAD∼P, this excess thermal energy is simply radiated into space on such\ntimescales, and is unavailable to contribute to P. We can look at the enthalpy fluxFenth\nassociated with bulk flows into the photosphere, for this we need a m easurement of plasma\nmotions and we turn to the Si Iline core emission which forms between 200 and 500 km in\nour models. We use pressures p= 2×104dyne cm−2and densities ρ= 4×10−8g cm−2.\ncorresponding to 300 km height. These are conservatively high valu es for average ther-\nmal properties of the plasma where this line is formed, favoring highe r estimates of energy\ntransport.\nA careful comparison of the flare emission core and the pre-flare a bsorption profile of\nthe SiIline reveals an upper limit to differential flows of roughly 0.5 wavelength pixels,\n0.5 kms−1. This is equivalent to 2.5 σwhereσis the sensitivity of the Doppler shifts\nfrom our FIRS spectra. A Doppler photospheric signature of the fl are is present in HMI\ndata at the location of the seismic source with a shift equivalent to u≈0.3−0.5 kms−1\n(Donea and others 2014). However, such filtergram data, scann ing wavelengths in time,\ncannot be trusted during flaring and so we adopt the upper limit abov e. We then find an\nupper limit to the enthalpy energy flux of\nFenth∼<5\n2puA≈6×1025erg s−1.\nThe close agreement of the upper limit for FenthwithFradmeans that the excess energy\nradiated by the photosphere during the flare can be supplied by a bu lk flow of energy– 16 –\nassociated with a subsonic downflow of 0.5 kms−1induced (somehow) by the flare. The\npower in the acoustic pulse is a factor of at least 2 larger than our op timistic estimate of\nFenth.\nIf however the pressure pulse involves high frequency phenomena (ν > cS/H≈60 mHz,\nwhereHis the pressure scale height and cSthe sound speed), the pulse would be invisible\nto observation except as a broadening of spectral lines to at most the sound speed (for linear\nwaves), the lines being formed over a length ≈Hin a stratified atmosphere. The WKB\nexpression for the energy flux density (propagating both upward s and downwards) at the\nsound speed is\nFwave=ρcS/angbracketleftξ2/angbracketrighterg cm−2s−1,\nwhereξis the velocity amplitude of the wave. We can set limits on ξthrough the measured\nline broadening and line profiles during the flare itself. Before and aft er the flare, the\ninversions yield ξ∼<2.8 kms−1. During the flare the measured line wings are similar in\nshape to the pre- and post- flare profiles. The emission core of the profile has a FWHM of\n0.05 nm (Figures 2 and 6). Treated as optically thin emission, this FWHM is equivalent\nto an e-folding Doppler broadening speed of 8 kms−1. In the presence of scattering this\nis a strict upper limit. To estimate the energy flux available in such mode s we again use\nρ= 4×10−8g cm−2, and the upper limit of 8 kms−1. Assuming that only half of the waves\nare emitted downwards, we find\nFwaveA∼<2.5×1026erg s−1.\nBut this, we believe, is a gross over-estimate. Firstly, the scatter ing leads to emission profiles\na factor of ≈√lnτ0broader than mere Doppler broadening where τ0is the line center optical\ndepth. We are able to reproduce the core Si Iemission using microturbulent speeds of 1-\n2 kms−1and the full Voigt profile, reducing the above estimate Fwaveby a factor of at\nleast 16! Secondly, the line profile shows, within a broad emission core , a very narrow self-\nreversal during the flare (lower left panel of Figure 2), indicating b oth scattering-induced\nline broadened profiles and values of ξin the line core far smaller than those adopted above.\nLastly, any high frequency waves with frequencies of a few Hz or les s are rapidly damped\nin the photosphere by the continuum radiative exchange processe s first modeled by Spiegel\n(1957). The thermal perturbations associated with high frequen cy wave energy rapidly\nradiatethis energy fromthephotosphere ona timescale of1-2s. O nlywaves withfrequencies\nin excess of several Hz could propagate down into the interior unmo dified by radiation\ndamping. All things considered, it seems unlikely that the power of th e sunquake can be\nprovided by such high frequency waves.\nWe conclude that non-magnetic modes of energy transport int o the interior are very\nunlikely to be sound waves . More likely is a coherent downward-moving plug of plasma– 17 –\ncarrying enthalpy of almost the right magnitude, but we have above already set an upper\nlimit to this process that is optimistically a factor of two smaller than ne eded.\nConsider in turn the energetics of the Lorentz force picture. The field strength from\ninversions from the Si Iline, before and after the flare, is of order 800 G from which the\nmagneticenergydensityis B2/8π= 2.5×104ergcm−3,about1/3ofthephotosphericthermal\nenergy density,3\n2pph(the latter is a lower limit since we neglect latent heat of ionization).\nThe Alfv´ en speed cAfor a photospheric density of 2 .6×10−7g cm−3is 4.3 kms−1, so the\nlocal magnetic energy flux is at most\nFMA∼<cAB2\n8πA= 2.4×1026erg s−1,\nscaling as B3/√ρ. This estimate is entirely a local one, it does not take into account\nthe connections of the magnetic field throughout the entire flux sy stem and the fact that\nmomentum andenergy is readily imparted to localitiesfroma far larger reservoir ofmagnetic\nenergy encompassing the entire volume of the active region. Thus t here is naturally sufficient\nenergy inthemagnetic field oftheentire active regionto account fo r theacoustic source. But\nit should be remembered also that only a fraction of the total magne tic energy is available\nas free energy, only a fraction will penetrate into the interior, and measured changes in\nmagnetic fields before and after flares are small relative to the amb ient field. Estimating the\nfree energy change over the entire active region is not a simple task , fraught with problems\n(De Rosa et al.2009) and so is not attempted here.\nInstead let us assume that a magnetic force is responsible for the im pulse into and below\nthephotosphere attheacousticsource. The sameargument for theenthalpy flux applieshere\nno matter if a Lorentz-forced flow is responsible, and the same inad equate energy flux results\nfrom the stringent limit on flows found from the spectra of the Si Iline. The reason is simple:\nmagnetic fields are frozen to plasma under conditions of high magnet ic Reynolds number,\nmotions of plasma mustaccompany any forcing no matter its origin. The collision time for\na proton to exchange its momentum with a hydrogen atom is of order 4×109/√\nTnHsec.\nWithT= 6000 K, nH≈1017cm−3, this time is far smaller than any dynamical time scale\n(Gilbert et al.2002). The partially ionized plasma behaves dynamically as a fully ionized\nplasmawiththesamechargedparticlesbutwiththeadditionalneutr almass. Againtherefore\nwe must appeal to unresolved motions, i.e. waves. But the fastest magnetosonic mode is the\nfield-aligned modified sound wave when the plasma β >1, so that the same conclusions are\ndrawn as for the non-magnetic case. The magnetic forces are either (a) incompatible with\nthe observations of line profiles revealing down-flowing mat erial with insufficient energy flux\nto account for the acoustic source, or (b) are in the form of mo dified high frequency sound\nwaves with the same problems that the sound waves have.– 18 –\n4. Discussion and Conclusions\nWehave reportedsome unusual spectral andpolarimetricprofiles oflines ofSi IandHeI\nobtained at a flare footpoint at infrared wavelengths. Several ph otospheric lines revealed line\ncore emission in excess of non-flaring conditions, and the helium 1083 nm multiplet almost\ndoubled in brightness. Using the enhanced brightness of the Si I1082.7 nm line and the\nneighboring continuum, we have demonstrated via 1D radiative tran sfer models that flare-\nrelated heating can be detected all the way to the photosphere, t o 100±100 km. This is\ndeeperbyseveralscaleheightsthantheexpecteddepthofpene trationofhardXrayemission.\nOur models merely show the depth to which heating is observed to occ ur through the flare\nmechanism(s), they shed no light on the nature of this transport f rom the corona into the\nSun’s deeper atmosphere. It could be a direct mechanical effect or a two-stage mechanism\nof mechanical transport followed by radiative back warming (Macha doet al.1989). Using\ndynamical models in a stratified atmosphere, Allred et al.(2005) found the flare energy\npenetrated only to the mid chromosphere (800 km). Mart´ ınez Olive roset al.(2012) found\nheights of 305 ±170 km and 195 ±70 km, respectively, for the centroids of the hard X ray\nand white light footpoint sources of a flare observed stereoscopic ally.\nHow robust is our new result? We can produce significant emission in th e core of\nthe SiIline using both deep and shallow penetration models (enhanced tempe ratures only\nabove 500 km). But the shallow models can be eliminated: the continuu m intensities are\nunchanged from pre-flare conditions; the Si Iline cores are far too narrow; also, only models\nwith energy penetrating down into the photosphere have sufficient opacity to produce the\nopacity broadening and subtle narrow self-reversal observed in t he very center of the line.\nThe combination of broad, self reversed line emission and a brighter c ontinuum appears to\nbe a clear signature of flare heating down to 100 ±100 km. This picture also eliminates\nthe possibility that hydrogen recombination radiation contributes t o the visible and infrared\ncontinuum emission for this flare (Kerr and Fletcher 2014). One sho uld expect that our\nconclusions depend strongly on the adopted model of the photosp here/chromosphere: surely\n1D models are inadequate for studies of the Sun’s atmosphere in gen eral? It is sometimes\nforgotten that the photosphere/low chromosphere are charac terized by subsonic motion, and\nthat therefore the atmosphere is strongly stratified4. It is this essential property of these\nplasmas, in which the Si Idiagnostic line we have used is formed, that is most critical for\ndetermining the depth of penetration of flare energy. There appe ars to be no credible way to\nreconcile the salient line profile and continuum observations with heat ing occurring purely\n4Very dynamic phenomena seen at UV wavelengths, for example with t he HRTS or IRIS instruments, are\nformed mostly in less dense structures above the stratified layer t hat is optically thick to most UV radiation.– 19 –\nabove 200 km. Whatever model we would choose to use, we would still require continuum\nand line formation within the photosphere, not the chromosphere.\nWe found significant pre- and post-flare changes in the magnetic co nditions within and\noutside of the acoustic source region. Within the source region, th ese include an increase in\nmagnetic field strength, a rotation of the magnetic azimuth and a re duction of inclination\nwith respect to the solar vertical. The further interpretation of t hese changes, part of the\nevolution of the entire active region, is beyond the scope of this pap er.\nWe detected no signature of downward energy transport capable of carrying 2 .3×1026\nerg s−1needed to account for the acoustic source, that is compatible with our ground-based\nobservations. Curiously, both line and continuum emission is present along an extended\nribbon, but the acoustic emission is confined to a smaller region only as sociated with the\nbrightest parts of the ribbon. At some future time, the flux of ene rgy insomething propa-\ngating downwards through the Sun’s atmosphere must be detecte d. We have shown using\ncurrent spectroscopic capabilities that macroscopic motions drive n magnetically or otherwise\ncarry insufficient energy by at least an order of magnitude, and tha t high frequency acoustic\nand magnetic waves are also are very likely to fall short. Spectropo larimetric analysis reveals\nmagnetic fields that are too small at the acoustic source for magne tic wave modes to carry\nthe energy ( cA∼<cS), and if a Lorentz force is responsible for the impulse, we must at so me\npoint also observe either a Doppler shifted downflow of plasma or lines broadened enough for\nthe fluxρcAξ2to carry sufficient energy. If we have captured the essential dyn amics of this\nparticular flare in our observations, it seems that we have ruled out two important classes\nof models for the forcing of acoustic sources under the photosph ere. Anything that relies on\ntransport of energy (radiation, conduction) more than momentu m seems to be incompatible\nwith our data.\nOur work therefore raises a problem, namely that the energy is tra nsported downwards\nin a fashion that is somehow invisible to our observations. Yet the latt er include the infrared\ncontinuum and lines of Si Iand HeIand they span the entire photosphere and chromosphere\nof the Sun. Is it possible that we have missed the region of “action”? We do not believe so\nsince our slit passed right across the bright flare kernel accompan ying the acoustic source at\nthe right time. Perhaps the energy is transmitted in intense but ver y short ( <1 s) bursts\nwhich, when integrated over the 13 s dwell time at each slit position, m ight contribute only\nto very broad spectral lines that are washed out spectrally and/o r temporally in our data.\nSuch variations might be absent in the continuum and line emission that we can measure.\nOr perhaps the energy is transmitted to the source over a larger a rea of the visible surface,\nand focused somehow into the source itself. Our estimate of A∼<2.6×1016cm2is our\nbest estimate based upon the acoustic wave analysis of Donea and o thers (2014). Other– 20 –\nalternatives might appeal to high energy particles accelerated in th e corona. The power\ndelivered by electrons of energy sufficient to reach altitudes of 100 km above the photosphere\ncan be estimated using the RHESSI quick-look spectral fits to obta in photon spectral index\nand intensity, which is related straightforwardly to electron power using the collisional thick\ntarget approximation (see e.g. Fletcher et al.2007) and the approximation that to traverse\na column of Ncm−2requires an electron of energy E2=N/(1017µo), where µois the pitch-\nangle cosine of the electron (set equal to 1 here) and Eis measured in keV. In the VAL-C\nmodel, 100 km above the photosphere corresponds to N= 1.3×1024cm−2(assuming a\npure hydrogen target) meaning that electrons of 3.6 MeV are requ ired. The HXR spectrum\nis hardest and most intense at 17:46:00-17:47:00, during which time th e photon spectral\nindex at high energies is γ= 3.23, and the intensity at 50 keV is approximately 1 photon\ns−1cm−2keV−1. Using these data in equations (1) and (2) from Fletcher et al.(2007) gives\na power in electrons above 3.6 keV of 4 .4×1024erg s−1. Thus the electrons appear to fall\nshort by a factor of 30 from directly depositing the needed energy to the 100 km depth.\nWe can of course speculate, along with many previous workers, tha t high energy protons\nmight provide the needed energy flux deep in the solar atmosphere. Protons require energies/radicalbig\nmp/mehigher, i.e. inexcessof160MeV.Suchprotonsaresometimesseenin interplanetary\nspace, associated with flares. But to remain invisible to our observa tions, such beams must\npenetrate and dump their energy and momentum belowthe photosphere, bumping up the\nenergy requirements through the column mass by a factor of thre e or more. We see no\nevidence for an up-welling of energy or plasma in response to such en ergy deposition below\nthe surface in our two scans obtained after the flare itself.\nOur spectra serve to remind us of difficulties in measuring magnetic an d velocity field\nchanges during flares. The Si Iline profiles bear no resemblance to what is usually assumed\nto “invert” photospheric lines, and the usual magnetograms and v elocities render spurious\nresultsduringawhite-lightflare. Suchphotosphericchangeswillsim ilarlyaltertheformation\nof the Ni I676.8 nm or Fe I617.3 nm and other lines routinely used on spacecraft.\nIn future, several promising lines of research should be pursued f or this uniquely well-\nobserved flare. We will analyze further the FIRS and IBIS data, ex ploring the magnetic field\nchanges across the atmosphere using lines formed at a variety of d epths from photosphere to\nchromosphere. 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Phys. 140, 85\nWang, H., Deng, N., and Liu, C.: 2012a, Astrophys. J. 748, 76– 24 –\nWang, S., Liu, C., Liu, R., Deng, N., Liu, Y., and Wang, H.: 2012b, Astrophys. J. Lett. 745,\nL17\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 25 –\nFig. 1.— A G-band image centered near 430.5 nm from the Dunn Solar Te lescope is shown\nalong with two sets of contours. The black contours are of egress ion power, |H+(r,t)|2,\nreferred to in the text. The white contours are of core intensity o f the 1082.7 nm line of\nSiI. The FIRS data were acquired between 17:40:12 and 18:01:39 UT by st epping the slit\n(horizontal in the figure) in the Y-direction (S-N). The positions of the slit during the flare\nare shown as dashed lines. The absolute heliographic coordinates of our ground-based data\nare accurate to 1 second of arc, as obtained by co-alignment with a continuum image from\nHMI obtained at 17:50:00 UT.– 26 –\n1082.0 1082.5 1083.0 1083.5 1084.0\nwavelength nm100200300400countsCr IH2O\nSi IHe I Ju=0\n    He I Ju=1,2\nH2OCa I\nH2ONa IH2OH2OCa I\nI\n-0.20.00.20.4\nWavelength from Si I nm500520540560solar X for Y=263.5\n-0.20.00.20.4500520540560Q\n U\n V\nFig. 2.— The upper panel shows the median intensity spectrum from t he spectral scan\nobtained by FIRS started at 17:40:06 UT. Line identifications are mar ked, the H 2O features\nare all telluric and should be polarization-free. Those wavelengths u sed to derive the con-\ntinuum intensity are marked on the spectrum with symbols. The four lower panels show\nStokesI,Q,UandVspectra taken from the 30th scan obtained through the flare foo tpoint\nbeginning at 17:46:29 UT, as a function of wavelength and position on t he Sun. The dotted\nlines show the wavelengths plotted on the Uimage. The images for the QUVimages are\nclipped at ±2×10−2IC. The Si Iline is in emission near X≈530. The QUVprofiles of this\nline are as expected from Zeeman-induced polarization, even when t he line is in emission.\nThe He IQUandVprofiles are peculiar and to some degree reflect the systematic err ors\nof “crosstalk” ( QUhave some characteristics of IandV). Some spectral fringes are visible,\nparticularly in the QandVimages.– 27 –\nI continuum\n     260262264266268270\n0.70.80.91.01.11.2\nI Si core\n     260262264266268270\n0.40.60.81.01.21.41.6\nI He core\n     260262264266268270\n0.40.60.81.01.21.41.6\nv He\n510515520525530\nSolar X arcsec260262264266268270Solar Y arcsec\n-6-4-20246\nbefore\nI continuum\n           \nI Si core\n           \nI He core\n           \nv He\n510515520525530      \nduring flare\nI continuum\n           \nI Si core\n           \nI He core\n           \nv He\n510515520525530      \nafter\nFig. 3.— “Quick-look” raster images of thermal data obtained using t he FIRS instrument.\nThe columns, from left to right, show data from 16:29:26, 17:40:06 (d uring the impulsive\nphase of the flare) and 18:30:13 respectively. The top panels show c ontinuum intensity\naveraged over 30 pixels on the blue side of the Si I 1028.7 nm line. The n ext two rows shows\nthe intensity at the cores of the Si I and He I 1083.0 nm lines. Row 4 sh ows Doppler shifts of\nthe HeImultiplet in units of kms−1. The flare began around 17:45 UT, at about the time\ntheFIRSslitcrossed theflareribbonseenintheSiIandHeIcoreint ensities (middlecolumn,\nrefer also to Figure 1). The circle shows the location of the “acoust ic source” associated with\nthe flare in the field of view shown. The arrows mark the erupting filam ent seen in He I.– 28 –\nI He core\n     260262264266268270\n0.40.60.81.01.21.41.6\nSi \"magnetogram\"\n     260262264266268270\n-0.08-0.06-0.04-0.020.000.020.04\nSi \"azimuth\"\n     260262264266268270\n-1.5-1.0-0.50.00.51.01.5\nHe \"magnetogram\"\n510 515 520 525 530\nSolar X arcsec260262264266268270Solar Y arcsec\n-0.08-0.06-0.04-0.020.000.020.04\nbefore\nI He core\n           \nSi \"magnetogram\"\n           \nSi \"azimuth\"\n           \nHe \"magnetogram\"\n510 515 520 525 530      \nduring flare\nI He core\n           \nSi \"magnetogram\"\n           \nSi \"azimuth\"\n           \nHe \"magnetogram\"\n510 515 520 525 530      \nafter\nFig. 4.— Quick-look raster images relevant to magnetic fields obtained using the FIRS\ninstrument. The top panels show the intensity near the core of the He I 1083.0 nm lines\nfrom Figure 3. Rows 2 and 3 show “magnetograms” and “azimuths” o f the magnetic field\n(see text), of the Si Iline. Rows 4 and 5 show similar data but for the He Imultiplet. Care\nshould be taken not to interpret the crude magnetic data in terms o f magnetic field especially\nduring the flare, which occurred during the scan shown in the secon d column.– 29 –\nTable 1. Properties of the acoustic source and X1 flare of 29 March 2014\nProperty Instrument timing UT position\nImpulsive phase start RHESSI 30-70 keV 17:45\nImpulsive phase peak RHESSI 30-70 keV 17:47:16 519.7, 263.2\nAcoustic source peak HMI 5.5 mHz 17:48 518.5, 264.0\nHMI 6 mHz 17:51\nIR continuum peak FIRS ≈17:46:10 ≈520, 263\nIR SiIcore peak FIRS ≈17:46:37 ≈519.7, 263.5\nIR HeIcore peak FIRS ≈17:46:10 ≈521, 263\nNote. — Note that FIRS is not an imaging instrument, the slit was scan ned across the\nsolar surface (see Figure 1).\nbefore\n515 520 525\nX (arcsecs)260265270Y (arcsecs)\n0.80.91.01.11.2\nflare\n515 520 525\nX (arcsecs)   Y (arcsecs)\nafter\n515 520 525\nX (arcsecs)   Y (arcsecs)\nFig. 5.— Continuum intensity data are shown from the three scans imm ediately before,\nduring and after the X1 flare impulsive phase. The contours show th e intensity from the\ncore of the Si Iline. Enhanced intensity in the continuum is seen as the central part of one\nrow in the image the middle panel, marked by an arrow. Its amplitude is 4 ±1% of the\nneighboring intensities.– 30 –\n-5000500100015002000\nHeight km05.0·1031.0·1041.5·1042.0·104Temperature (K)Te\n91011121314\nLog10 Ne (cm-3)ne \npre-flareFlare\n1082.5 1082.6 1082.7 1082.8 1082.9\nwavelength (nm)0.20.40.60.81.01.21.4I/IC(quiet)He I\nSi I\nModeledObserved\n-5000500100015002000\nHeight km05.0·1031.0·1041.5·1042.0·104Temperature (K)\n91011121314\nLog10 Ne (cm-3)\n1082.5 1082.6 1082.7 1082.8 1082.9\nwavelength (nm)0.20.40.60.81.01.21.4I/IC(quiet)\nFig. 6.— The results of nLTE calculations are shown. The atmospheric structure is shown\nin the left panels, emergent intensities are shown in the right panels, for a heliocentric angle\nwith cosine of 0.95. Two “flare models” are shown, exploring the dept hs and magnitudes\nof enhanced temperatures in the flaring atmosphere. Typical obs erved profiles of the flare\nribbon and a non-flaring region from the same exposure are shown a s thin solid and dotted\nlines respectively. The upper panels show a “deep penetration” mod el. The lower panels\nallow penetration to a depth near 600 km such that the Si Iline is in emission, but is a\nshallow penetration model. Notice that the shallow model predicts no detectable increase in\ncontinuum emission and it produces a line emission core that is in qualitat ive disagreement\nwith the data.– 31 –\nB 16:29:26 UT\n      258260262264266268270272\ninclination 16:29:26 UT\n      258260262264266268270272\nazimuth 16:29:26 UT\n      258260262264266268270272\nBLOS  16:29:26 UT\n505510515520525530\nSolar X arcsec258260262264266268270272Solar Y arcsec\nB 16:55:58 UT\n      258260262264266268270272\n02004006008001000\ninclination 16:55:58 UT\n      258260262264266268270272\n406080100120140\nazimuth 16:55:58 UT\n      258260262264266268270272\n-150-100-50050100150\nBLOS  16:55:58 UT\n505510515520525530258260262264266268270272\n-400-2000200400\nB difference\n      258260262264266268270272\n-400-2000200\ninclination difference\n      258260262264266268270272\n-60-40-200204060\nazimuth difference\n      258260262264266268270272\n-150-100-50050\nBLOS  difference\n505510515520525530258260262264266268270272\n-1000100200\nFig. 7.— Magnetic field properties determined from the scans obtaine d before the flare,\nbegun at 16:29:26 and 16:55:58 UT, using the code MELANIE. The azimu th is measured\nin the E-W direction, the sign of the azimuth is not determined. The an gles are measured\nin the local solar frame (E-W and local vertical reference direction s). The circle shows the\ncenter of the acoustic source. The rightmost column shows differe nced data.– 32 –\nB 16:55:58 UT\n      258260262264266268270272\ninclination 16:55:58 UT\n      258260262264266268270272\nazimuth 16:55:58 UT\n      258260262264266268270272\nBLOS  16:55:58 UT\n505510515520525530\nSolar X arcsec258260262264266268270272Solar Y arcsec\nB 18:01:55 UT\n      258260262264266268270272\n02004006008001000\ninclination 18:01:55 UT\n      258260262264266268270272\n20406080100120140\nazimuth 18:01:55 UT\n      258260262264266268270272\n-150-100-50050100150\nBLOS  18:01:55 UT\n505510515520525530258260262264266268270272\n-600-400-2000200400\nB difference\n      258260262264266268270272\n-400-2000200400\ninclination difference\n      258260262264266268270272\n-60-40-20020406080\nazimuth difference\n      258260262264266268270272\n-100-50050100150\nBLOS  difference\n505510515520525530258260262264266268270272\n-300-200-1000100200\nFig. 8.— Magnetic field properties determined from the scans obtaine d before (16:55:58 UT)\nand after (18:01:55 UT) the flare, shown as in Figure 7."    },    {        "title": "1106.2035v2.On_Lorentz_Violating_Supersymmetric_Quantum_Field_Theories.pdf",        "content": "arXiv:1106.2035v2  [hep-th]  21 Dec 2011On Lorentz-Violating Supersymmetric Quantum Field Theori es\nDiego Redigolo1\n1Université Libre de Bruxelles (ULB) & International Solvay Institutes (ISI),\nCampus Plaine CP231, B-1050, Belgium\ndredigol@ulb.ac.be\nAbstract\nWe study the possibility of constructing Lorentz-violatin g supersymmetric quantum field theories\nunder the assumption that these theories have to be describe d by lagrangians which are renor-\nmalizable by weighted power counting. Our investigation st arts from the observation that at high\nenergies Lorentz-violation and the usual supersymmetry al gebra are algebraically compatible. De-\nmanding linearity of the supercharges we see that the requir ement of renormalizability drastically\nrestricts the set of possible Lorentz-violating supersymm etric theories. In particular, in the case\nof supersymmetric gauge theories the weighted power counti ng has to coincide with the usual one\nand the only Lorentz-violating operators are introduced by some weighted constant cthat explic-\nitly appears in the supersymmetry algebra. This parameter d oes not renormalize and has to be\nvery close to the speed of light at low energies in order to sat isfy the strict experimental bounds on\nLorentz violation. The only possible models with non trivia l Lorentz-violating operators involve\nneutral chiral superfields and do not have a gauge invariant e xtension. We conclude that, under\nthe assumption that high-energy physics can be described by a renormalizable Lorentz-violating\nextension of the Standard Model, the Lorentz fine tuning prob lem does not seem solvable by the\nrequirement of supersymmetry.Contents\n1 Introduction 2\n2 SUSY algebra vs Lorentz symmetry breaking 4\n2.1 Berger-Kosetelcký construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.2 Higher-momenta superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n3 Renormalizable Lorentz-Violating Theories for Chiral Sup erfields 7\n3.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3.2 Homogeneous Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n3.3 Classification of neutral chiral superfield’s models . . . . . . . . . . . . . . . . . . . 13\n4 Gauge invariant theories 15\n5 Low-energy limit and Lorentz symmetry recovery 17\n6 Conclusion 20\n1 Introduction\nLorentz symmetry is a basic ingredient of the Standard Model and it has been verified with great\nprecision by different experiments [ 1]. However, from a theoretical point of view it would be pos-\nsible to violate Lorentz invariance at very high energies re quiring this symmetry to be restored at\nlow energies [ 2]. Moreover, it has been shown that Lorentz violation could e merge as a preferred-\nframe effect both in string theory [ 3,4] and in loop quantum gravity [ 5] .\nIn our approach we want to preserve the requirement of renorm alizability as an a priori se-\nlection criterion for physical theories at high energies an d at the same time enlarge the class of\nrenormalizable theories relaxing the hypothesis of Lorent z invariance but preserving both locality\nand unitarity.\nThe issue of renormalizability in the context of Lorentz-vi olating theories has been defined\nand widely studied for both matter and gauge fields [ 6,7,8,9,10,11]. The basic idea is that\none can break Lorentz symmetry at high energies introducing a weighted power counting that\nweights differently time and space coordinates. Thanks to th is modified power counting we can\nimprove the UV behavior of propagators by introducing highe r space derivatives terms in the\nkinetic lagrangian. The number of higher space derivatives in a given theory is parametrized by\nsome integer n. This procedure preserves perturbatively the unitarity of the theories because the\nweighted power counting coincides with the usual one on the t ime coordinate. Lorentz-violating\ntheories become less divergent and contain new interaction s which are non-renormalizable by the\nusual power counting but renormalizable by the weighted one . We assume that Lorentz-violating\nterms arise at energies greater than a breaking energy scale ΛL.\nThe implications of Lorentz violation on physics beyond the Standard Model and in particular\nits consequences on low-energy phenomenology has been wide ly explored in the literature [ 12,13].\nThere are several purposes of possible Lorentz-violating e xtensions of the Standard Model that\nanswer some questions about neutrino physics such as neutri no oscillations and neutrino masses\n2[14] and give an alternative framework which concerns the origi n of the mass of elementary par-\nticles [ 15,16,17].\nThe recovery of Lorentz invariance in the low-energy limit i s not automatic and depends mostly\non the running of the coupling constants associated to non-c ovariant operators that are renormal-\nizable in the usual sense and for this reason are not suppress ed by any powers of ΛL. It has been\nshown that these coupling constants go to zero in the IR limit for CPT-preserving operators in\nthe context of pure Yang-Mills theories, but grow for CPT-vi olating ones [ 18]. However, even if\nwe assume CPT invariance we need a strong fine tuning on dimens ion 2 and dimension 3 opera-\ntors at low energies to be in agreement with the strict constr aints on Lorentz violation [ 19,20].\nThis problem is known in the literature as the Lorentz fine tun ing problem [ 21]. One of the best\ncandidates for trying to solve this new naturalness problem in the context of Lorentz-violating\ntheories seems to be supersymmetry [ 22,23].\nOn this basis it can be interesting to study quantum field theo ries that are exact represen-\ntations of the SUSY algebra and contain Lorentz-violating t erms. The idea of combining super-\nsymmetry and Lorentz violation arises from a simple observa tion: even though the SUSY algebra\nis constructed as the graded extension of the Poincaré algeb ra, Lorentz violation and the SUSY\nalgebra are not incompatible. In fact, if we weight different ly space and time we have to eliminate\nfrom the Poincaré algebra the generators of the boosts, but t hese generators do not appear in the\nanticommutator of two supercharges. Hence we can eliminate all the relations that contain them\nobtaining again a closed algebra [ 22].\nFrom this algebraic compatibility we can start to construct N= 1supersymmetric field the-\nories with Lorentz violation. The classification of Lorentz -violating effective field theories that\nare exact representations of the SUSY algebra and the study o f their quantum corrections have\nalready been the subject of careful investigations in the li terature [ 24,25]. However, in this work\nwe want to focus our attention on the possibility of construc ting Lorentz-violating supersymmet-\nric theories which are renormalizable by weighted power cou nting. If we require the existence\nof superspace and the linearity of supercharges in the momen ta we find, using weighted power\ncounting, that the Kähler potential must have the same form a s in the Lorentz invariant case and\nin particular it cannot contain higher space derivatives. T his happens because the Grassmannian\nspinor variables associated with the supercharges are not a ffected by the weighted power counting.\nHowever, for theories with even nwe can introduce an even number of higher space derivatives a s\n“mass” terms in the superpotential, preserving the rotatio nal invariance in the spatial submanifold\n¯M¯d. In this approach we can classify all the possible theories f or chiral superfields and derive the\ncorresponding Feynman rules. The structure of Feynman grap hs in the Grassman variables is not\nmodified so that the non-renormalization theorem works as in the Lorentz invariant case.\nUsing a result of [ 24], we show that it is not possible to construct a gauged versio n of the terms\nwith higher space derivatives in the superpotential. Moreo ver, by looking at the gauge sector we\nfind that a generic Lorentz-violating gauge theory, defined o nce we have assigned weights to the\nfields, admits a supersymmetric extension if and only if the w eighted power counting coincides\nwith the usual one. Therefore, the requirement of supersymm etry rules out the possibility of\nhaving high energy renormalizable extensions of the Standa rd Model with higher dimensional\noperators. The only Lorentz-violating operators in the the ory have to be renormalizable in the\nusual sense and their coupling constants need to fulfill the s trict experimental bounds on Lorentz\nviolation. In the supersymmetric case the Lorentz invarian ce recovery is regulated by the constant\n3cin the spatial part of the kinetic term that does not renormal ize if we assume that supersym-\nmetry is softly broken at a scale µs<<ΛL. Therefore, the experimental bounds at low energies\ndetermine the values of the Lorentz-violating parameter al so at high energies and the Lorentz vi-\nolation effects will be irrelevant. The only non trivial high energy theories that we can construct\ninvolve neutral superfields which are singlets with respect to the Standard Model gauge group\nand interact through Landau-Ginzburg vertices ΦN.\nHowever, the problem of constructing Lorentz-violating su persymmetric field theories remains\nan open field of research because in principle we could consid er various modified supersymmetry\nalgebras in which the anticommutator of two conjugate super charges is linear in the time mo-\nmentum but non linear in the space momenta. These structures are still compatible with the\nweighted power counting and the Coleman-Mandula theorem [ 26,27] and in the limit ΛL→ ∞\nthey approach the usual supersymmetry algebra. It is straig htforward to find free lagrangians\nthat are invariant under the action of the new supercharges b ut, however, the non-linearity in the\nspatial momenta makes the problem of constructing interact ing theories a very complicated one.\nThis paper is organized as follows. In section 2we study the possible supersymmetry algebras\nin the Lorentz-violating case and discuss the problem of con structing interacting theories when\nthe supercharges are non-linear in the momenta. In particul ar we explain on which point we\ndisagree with the results of [ 28]. In section 3, assuming linearity of supercharges in the momenta,\nwe construct the most general renormalizable Lorentz-viol ating theory for chiral superfields in\nfour dimensions. We study the renormalization properties o f these theories and show that the\nusual non-renormalization theorem for supersymmetric the ories has a trivial extension to the\nLorentz-violating case. We also give a complete classificat ion of the possible supersymmetric\nLorentz-violating theories for neutral chiral superfields . In section 4we study the problem of\nconstructing Lorentz-violating gauge theories. We apply a result of [ 24] to show that the theories\nof section 3are not generalizable to the case of charged chiral superfiel ds. Finally, we show that\nif we demand supersymmetry for gauge theories the weighted p ower counting has to coincide\nwith the usual one. In section 5we discuss the low-energy limit of our theories and the recov ery\nof Lorentz invariance. In the superfield formalism it is almo st trivial to show that the Lorentz-\nviolating parameter cin the kinetic lagrangian does not renormalize to all orders of perturbation\ntheory. However, we will explicitly see in components how th e requirement of supersymmetry\nremarkably changes the one loop renormalization group equa tions for Lorentz-violating theories\nat low energies. Moreover, we discuss whether or not the devi ation from the speed of light is\nphysically observable in the Lorentz-violating supersymm etric theories. Section 6contains our\nconclusions.\n2 SUSY algebra vs Lorentz symmetry breaking\n2.1 Berger-Kosetelcký construction\nWe consider the spacetime manifold Mdand we take d= 4. Defining the weighted power counting,\nwe splitMdinto the product of two submanifolds Mˆd×M¯dand so the complete Lorentz symmetry\nSO(1,3)breaks into the residual symmetry SO(1,ˆd−1)×SO(¯d). The weighted power counting\nis defined once we assign the scaling laws, or equivalently th e weights of coordinates or momenta\n4[9]:\nˆx→ˆxe−Ω,¯x→e−Ω/n,[ˆp] = 1,[¯p] =1\nn. (1)\nSince we are breaking Lorentz symmetry, we have to discard th e boost generator Jˆµ¯νbetweenMˆd\nandM¯dfrom the Poincaré algebra p. However, Jˆµ¯νappears only on the left of the SUSY algebra\ncommutation relations, so we can eliminate from the SUSY alg ebraspthe relations that contain\nJˆµ¯νobtaining again a closed superalgebra sp′. In general the anticommutator between the two\nconjugate supercharges will be:\n/braceleftbig\nQα,¯Q˙α/bracerightbig\n= 2aσˆµ\nα˙αPˆµ+2bσ¯µ\nα˙αP¯µ. (2)\nEvaluating ( 2) on a generic physical state |ψ/an}bracketri}htand contracting the spinor indices we obtain\n0</an}bracketle{tψ|θα{Qα,Q˙α}θ˙α|ψ/an}bracketri}ht= 2/an}bracketle{tψ|θα/parenleftBig\naσˆµ\nα˙αPˆµ+bσ¯µ\nα˙αP¯µ/parenrightBig\nθ˙α|ψ/an}bracketri}ht.\nThe operator σµPµis positive definite on the physical states and this implies t hata >0∀bfor\nboth massive and massless particles. We can rescale the supe rcharges defining Q′\nα=√aQαand\ndropping the primes we obtain\n/braceleftbig\nQα,¯Q˙α/bracerightbig\n= 2σˆµ\nα˙αPˆµ+2cσ¯µ\nα˙αP¯µ. (3)\nTo be consistent with the definition of weighted power counti ng (1) the constant chas to be\ndimensionless and to weight [c] = 1−1/n. The weight of supercharges is fixed by the commutation\nrelations and is equal to their dimension:\n[Q] =/bracketleftbig¯Q/bracketrightbig\n=1\n2.\nWe can reabsorb the modification of the SUSY algebra by redefin ing the Minkowski metric ηµν\nand theσmatrices [ 22]:\nσ′ˆµ=σˆµ, σ′¯µ=cσ′¯µ,\n/parenleftbig\nσ′µ¯σ′ν+σ′ν¯σ′µ/parenrightbigα\nβ=−2δα\nβη′µν=−2δα\nβ/parenleftbig\nηˆµˆν+c2η¯µ¯ν/parenrightbig\n,\n✷′=η′µν∂µ∂ν=∂ˆµ∂ˆµ+c2∂¯µ∂¯µ,\np′2=η′µνpµpν=pˆµpˆµ+c2p¯µp¯µ (4)\nand after these redefinitions the algebra ( 3) looks like the usual one. Therefore the structure of\nthe superspace is identical to the Lorentz invariant case as was observed in [ 29]. In particular we\ncan define, as usual, the covariant derivatives so that {D,Q}=/braceleftbig\nD,¯Q/bracerightbig\n= 0. In our construction\nthe crucial requirement is the linearity of the supercharge s in the space momenta P¯µ. This\nassumption makes possible the usual definitions of the N= 1superspace as a coset space defined\nby the set of variables zπ= (ˆx,¯x,θ,¯θ). The operator U(ˆy,¯y,η,¯η) =ei(ˆyˆP+¯y¯P+iηQ+i¯η¯Q)is a\nwell defined translation in superspace and we can define a scal ar superfield S(ˆx,¯x,θ,¯θ)so that\nδUS=−iˆyˆP−i¯y¯P+ηQS+¯η¯QS. From the Leibniz rule for the supercharges and the definitio n of\ncovariant derivatives it follows that every polynomial in t he superfields and its covariant derivatives\nis still a superfield.\n52.2 Higher-momenta superalgebras\nThe Lorentz-violating case admits a larger class of superal gebras compatible with the Coleman-\nMandula theorem and the weighted power counting. The antico mmutator {Qα,¯Q˙α}is in the\n(1/2,1/2)representation of the Poincaré algebra and has dimension 1. In the Lorentz-invariant\ncase this implies that it has to be proportional to σµPµ, which is the only operator of dimension\n1in the same representation. In the Lorentz-violating case w e can construct new operators ¯O¯µof\nweight[¯O] = 1and dimension 1in the vector representation of the so(¯d)algebra. These operators\nare simply weighted polynomials of odd degree in the momentu m¯P\n¯O¯µ=/summationdisplay\nkak¯P2k¯P¯µ\nΛ2k\nL,withk≤/bracketleftbiggn−1\n2/bracketrightbigg\n,\nwhere the constants akweight[ak] = 1−(2k+1)/nand[ak]≥0. The parameter nshould be\nunderstood as the highest power of ¯Pthat appears in the quadratic terms of the lagrangian, as\nexplained in [ 9]. Therefore we can construct new superalgebras allowed by t he Coleman-Mandula\ntheorem and by the weighted power counting:\n/braceleftbig\nQα,¯Q˙α/bracerightbig\n= 2σ′µ\nα˙αPµ+2σ¯µ\nα˙α¯O¯µ,/braceleftbig\nDα,¯D˙α/bracerightbig\n=−2σ′µ\nα˙αPµ−2σ¯µ\nα˙α¯O¯µ. (5)\nThe new operators ¯O¯µhave the same commutation rules as ¯Pµin the SUSY algebra because,\nremovingJˆµ¯νfrom the original super-algebra sp,¯P2becomes a Casimir operator for the new\nsuper-algebra sp′. If we take the low-energy limit ΛL→ ∞ in (5) we obtain again ( 3). In\nprinciple sp′has the same superspace as spbecause we have not introduced new supercharges,\nbut now our supercharges will be non-linear operators in ¯P:\nQα=∂\n∂θα−iσ′µ\nα˙α¯θ˙α∂µ−iσ¯µ\nα˙α¯θ˙α¯O¯µ, (6)\nso the operator U(ˆy,¯y,η,¯η) =ei(ˆyˆP+¯y¯P+iηQ+i¯η¯Q)is not a simple translation in superspace any-\nmore. We can still define a superfield Sso thatδUS=−iˆyˆP−i¯y¯P+ηQS+ ¯η¯QS, but now it\nis in general not true that every polynomial in Sis still a superfield because the supercharges\n(6) do not respect Leibniz rule: δU(S1S2)/ne}ationslash=δUS1S2+S1δUS2. For this reason the superfield\nformalism loses its usefulness and it is not possible to prom ote directly the usual supersymmetric\nlagrangians for matter and gauge fields to lagrangians which are symmetric under the modified\nSUSY agebra ( 5). The construction of a recent publication [ 28] is completely invalidated by this\nobservation. Actually, that construction works only for fr ee theories because the Leibniz rule\nfor the supercharges ( 6) is fulfilled up to a total derivative if we consider only bili nears in the\nsuperfields. As an example of our general argument we conside r the same theory as [ 28], namely\nfor (1,3) splitting we take the superalgebra ( 5) forn= 3. For the chiral supermultiplet we can\nwrite a kinetic lagrangian:\nLkin=ˆ∂φ†ˆ∂φ+(c1¯∂+¯∂2¯∂\nΛ2\nL)φ†(c1¯∂+¯∂2¯∂\nΛ2\nL)φ+χ†(iˆ/∂+c1i¯/∂+i¯/∂3\nΛ2\nL)χ+F†F . (7)\n6The lagrangian is invariant under the transformations\nδηφ=√\n2ηχ ,\nδηχ=i√\n2(σˆµ¯η∂ˆµ+σ¯µ¯η(c1∂¯µ+¯∂2∂¯µ\nΛ2\nL))φ+√\n2ηF ,\nδηF=i√\n2¯η(¯σˆµ∂ˆµ+c1¯σ¯µ(c1∂¯µ+∂2∂¯µ\nΛ2\nL))χ , (8)\nwhich are clearly generated by supercharges that satisfy th e superalgebra ( 5) forn= 3. The\nauthor [ 28] claims that if we want to add interactions invariant under ( 8) to this theory it is\nsufficient to rephrase the usual superfield formalism for the n ew supercharges. Following his\nderivation, from the definition of the chiral superfield ¯D˙αΦ = 0 we obtain:\nΦ(x,θ) =φ(x)+iθσˆµ¯θ∂ˆµφ(x)−iθσ¯µ¯θ(c1∂¯µ+¯∂2∂¯µ\nΛ2\nL)φ(x)+1\n4θ2¯θ2(ˆ∂2+2c1¯∂4\nΛ2\nL+¯∂6\nΛ4\nL)φ(x)+\n+√\n2θχ(x)+i√\n2θ2¯θ(¯σˆµ∂ˆµ+ ¯σ¯µ(c1∂¯µ+¯∂2∂¯µ\nΛ2\nL))χ(x)+θ2F(x).(9)\nIf the superfield formalism worked, then each holomorphic fu nction of the superfield ( 9) would be\ninvariant under ( 8). As an example of a possible interaction lagrangian let us c onsider the usual\ncubic interaction in the superfields:\nLint=g\n3/integraldisplay\nd2θΦ3+h.c=gFφ2−gφψψ+h.c. .\nNow varying Lintwith respect to ( 8) we obtain\nδLint=i√\n2¯η[∂ˆµ(¯σˆµχφ2)+∂¯µ(c1¯σ¯µχφ2)+∂¯µ\nΛ2\nL(¯∂2χφ2−2¯σ¯ν∂¯µχ∂¯νφφ+2¯σ¯ν(∂¯µ∂¯νφφ+∂¯µφ∂¯νφ))]\n+i2√\n2¯η¯σ¯µχ∂¯ν∂¯µφ∂¯νφ .\nBecause of the last term in the previous equation δLint/ne}ationslash=∂ˆµAˆµ+∂¯µB¯µand therefore the action\nis not invariant under the transformations ( 8). This was expected, since the supercharges are\nnon-linear in the spatial derivatives.\nClearly, the possibility of constructing interacting quan tum field theories which are invariant\nunder the super-algebra ( 5) is not ruled out by our observation and could be an interesti ng open\nfield of research which, however, needs more involved constr uctions.\nIn the rest of this paper we will restrict ourselves to the cas e in which the supercharges are\nassumed to be linear in the spatial momenta and we will work wi th the superymmetric algebra\n(3).\n3 Renormalizable Lorentz-Violating Theories for Chiral Su per-\nfields\n3.1 General Discussion\nThe integration measure in four dimensions for a general spl itting4 =ˆd+¯dweights−đ≡ −(ˆd+¯d\nn),\nso a renormalizable lagrangian has to be a weighted polynomi al in the momenta of weight [L] =đ\n7and dimension 4, as we want to keep the action dimensionless and weightless.\nConsidering the super-algebra ( 3) we take a chiral superfield Φ(ˆx,¯x,θ,¯θ)in theN= 1super-\nspace defined by the constraint ¯DΦ = 0 :\nΦ(ˆx,¯x,θ,¯θ) =φ+iθσ′µ¯θ∂µφ+θ2¯θ2\n4✷′φ+√\n2θχ+i√\n2θ2¯θ¯σ′µ∂µχ+θ2F .\nThe complex scalar field φ(x)and the Weyl spinor χ(x)are propagating quantum fields, and we\ncan derive their weights from their kinetic lagrangian [ 9]. The chiral superfield then weights:\n[Φ] = [φ] =đ−2\n2= [θ]+[χ] = [θ]+đ−1\n2. (10)\nThe equation ( 10) fixes the weight of the spinor coordinates to [θ] =−1/2∀n, which is equal\nto the difference of weight between scalars and fermions and i s constant with respect to n. The\nspinor coordinates in the superspace do not rescale with nand their weights are equal to their\ndimensions so that, as in the Lorentz invariant case,/bracketleftbig\nd4θ/bracketrightbig\n= 2and/bracketleftbig\nd2θ/bracketrightbig\n=/bracketleftbig\nd2¯θ/bracketrightbig\n= 1∀n.\nWe want to construct the most general Lorentz-violating lag rangian for Mdifferent chiral\nsuperfields Φiwithi= 1...M. The Kähler potential K[Φ,¯Φ]weights[K] =đ−2and the\nsuperpotential f[Φ]weights[f] =đ−1. If we demand polynomialilty of the lagrangian, [Φ]>0,\nwe obtain:\nL(ˆd,¯d)=/integraldisplay\nd4θ¯ΦiΦi+/summationdisplay\nα,N/integraldisplay\nd2θλN,α\nNΛN(d−2)/2+p1+p2−3\nL/bracketleftBig\nˆ∂p1¯∂p2ΦN/bracketrightBig\nα+h.c., (11)\nwhereαlabels possible different derivative structures and a combi natorical factor can appear in\nthe denominator if there are identical superfields. The most general Kähler potential which is\nrenormalizable by weighted power counting has the same form as the Lorentz invariant one. This\nobservation severely restricts the possibility of constru cting supersymmetric Lorentz-violating\nmodels and at the same time will have important implications in the study of the RG flow at low\nenergies of the cparameter. In the superpotential, the derivative structur e of the vertex defines\na monomial in the superfield momenta of weight δα\nN=p1+p2/n. If we want to preserve CPT\ninvariance and symmetry under rotation in the submanifold M¯dwe have to assume that p1and\np2are even numbers. The coupling constant λα,Nassociated to a vertex with Nsuperfields is a\nsymmetric tensor with Ninternal indices i1...iN, whereik= 1...M∀k, and by power counting\nit has to weight\n[λN,α] =đ−1−N[Φ]−δα\nN. (12)\nWe will call a vertex weighted marginal when its coupling con stant weights [λN,α] = 0, weighted\nrelevant when [λN,α]>0and weighted irrelevant when [λN,α]<0. As it has been shown in\n[9], the renormalization rules in the Lorentz-violating case work as in the Lorentz invariant one\nafter the substitution d→đ, so that we can express the renormalizabilty condition imp osing that\nthe weight of the coupling constant has to be greater or equal to zero,[λN,α]≥0. As we need\n[Φ]>0for polynomiality, taking N= 2in this inequality we can derive an upper bound on\nthe weight of the monomial in the momenta δα\nN≤1, that ensures perturbative unitarity of the\ntheory and forbids the presence of terms with time derivativ es in the superpotential. We write\n8L(ˆd,¯d)=Lkin+Lint, where the kinetic term of the lagrangian ( 11) is\nLkin=/integraldisplay\nd4θ¯ΦiΦi+/integraldisplay\nd2θ\n/summationdisplay\nl≤[n/2](al)ij\n2Λ2l−1\nL¯∂lΦi¯∂lΦj\n+h.c.. (13)\nOnly terms with an even number of space derivatives are allow ed and the index lin the sum is\nan integer that goes from zero to the integer part of the ratio [n/2]. The higher space derivatives\nterms generalize the mass term in the Wess-Zumino model [ 30] and are regulated by the coupling\nconstants (al)ij, which are M×Msymmetric matrix of weight [al] = 1−2l/n. The diagonal\nterms ofalbehave as Majorana mass terms, whereas the off-diagonal term s behave as Dirac mass\nterms. In order to simplify the notation we omit the internal indicex structure and take the free\npartition function of a theory with only one chiral superfiel d, in which the coupling constant al\nbecomes a coefficient and we can construct only Majorana mass t erms.\nZ0/bracketleftbig\nJ,¯J/bracketrightbig\n=/integraldisplay\nDΦD¯Φexp−/braceleftBigg/integraldisplay\nd8z/bracketleftBigg\n1\n2/parenleftbig\nΦ¯Φ/parenrightbig\nA/parenleftbiggΦ\n¯Φ/parenrightbigg\n−/parenleftbig\nΦ¯Φ/parenrightbig/parenleftBigg\nD2\n4✷′J\n¯D2\n4✷′¯J/parenrightBigg/bracketrightBigg/bracerightBigg\nandA=/parenleftBigg\nA11D2\n4✷′1\n1A22¯D2\n4✷′/parenrightBigg\nwhereA11=A22=\n/summationdisplay\nl≤[n/2](−)lal\nΛ2l−1\nL¯∂2l\n.\nFrom the partition function we can derive the propagators fo r the chiral superfield using the\nmethods of [ 31]:\n/angbracketleftbig\nΦ(1)¯Φ(2)/angbracketrightbig\nJ=0=δ12\np′2+/parenleftbigg/summationtext\nl≤[n/2]al\n2Λ2l−1\nL(¯p2)l/parenrightbigg2, (14)\n/an}bracketle{tΦ(1)Φ(2) /an}bracketri}htJ=0=D2\n4\n/summationdisplay\nl≤[n/2]al(¯p2)l\nΛ2l−1\nL\nδ12\np′2/parenleftBigg\np′2+/parenleftbigg/summationtext\nl≤[n/2]al(¯p2)l\nΛ2l−1\nL/parenrightbigg2/parenrightBigg, (15)\nwhere we have reabsorbed the −D2\n4factors in the Feynman rules for the vertices and p′2is defined\nin (4). If we differentiate the propagators ( 14) and ( 15) with respect to any coefficient alwith\nl<[n/2]the weight of the denominator increases by 1−2l/nand differentiating with respect to\ncincreases by 1−1/n. Hence we can make any Feynman graph convergent by differenti ating it\na suitable number of times with respect to the coefficients alorc, and the counterterms will be\npolynomials in alandc. We can consider the super-renormalizable operators assoc iated with the\ncoefficients alandcas vertices with two external lines and treat them perturbat ively. Doing that,\nwe can study the UV behavior of the Lorentz-violating theori es keeping in the propagators ( 14)\n9and (15) only the terms with the maximum number of spatial derivativ es:\nPΦ¯Φ=δ12\nˆp2+a2\n[n/2](¯p2)2[n/2]\nΛ4[n/2]−2\nL, (16)\nPΦΦ=D2\n4a[n/2](¯p2)[n/2]\nΛ2[n/2]−1\nLδ12\np′2/parenleftbigg\nˆp2+a2\n[n/2](¯p2)2[n/2]\nΛ4[n/2]−2\nL/parenrightbigg. (17)\nThe weight of the coefficient a[n/2]is zero for even nand strictly positive for odd n, and in fact\nthe operators/parenleftbig¯∂[n/2]Φ/parenrightbig2in the free lagrangian ( 13) are strictly renormalizable for even nand\nsuper-renormalizable for odd n. Therefore, for odd nwe cannot construct propagators which\nare the inverse of homogeneous polynomials of weight 2 and th is fact completely invalidate our\nconstruction. To understand what does not work in the odd ncase we compare the kinetic terms\nof the fermionic and the scalar lagranians in the non-supers ymmetric case for an arbitrary n[9]:\nLs=1\n2(ˆ∂φ)2−c2\n2(¯∂φ)2−n/summationdisplay\nl=2a2\nl\n2Λ2l−2\nL(∂lφ)2−m2\n2φ2, (18)\nLf=¯ψ(iˆ/∂+vi¯/∂+n/summationdisplay\nloddb′\nl\nΛl−1\nL(i¯/∂)l+n/summationdisplay\nlevenbl\nΛl−1\nL(i¯/∂)l−M)ψ . (19)\nFrom the equations of motion associated with the lagrangian s (18) and ( 19) we derive the corre-\nsponding dispersion relations1:\nE2\ns(¯p) =c2¯p2+n/summationdisplay\nl=2a2\nl¯p2l\nΛ2l−2\nL+m2, (20)\nE2\nf(¯p) = ¯p2(v+n/summationdisplay\nloddb′\nl\nΛl−1\nL¯pl−1)2+(M+n/summationdisplay\nlevenbl\nΛl−1\nL¯pl)2. (21)\nIn the Lorentz invariant case the dispersion relation among energy and spatial momentum is\nuniversal for all particles E2(¯p) =c2¯p2+m2. Conversely, we see from ( 21) that in the Lorentz-\nviolating case the dispersion relation for fermions contai ns two different kind of contributions\nthat are related respectively to terms with an even or an odd n umber of derivatives in the kinetic\nlagrangian ( 19). This happens because the terms with an even number of deriv atives behave like\nmass terms from the point of view of the spin 1/2representation of the Lorentz group, while\nthe terms with an odd number of derivatives behave like ¯/p. A necessary condition for the theory\nto be supersymmetric is that all the dispersion relations of particles in the same supermultiplet\nhave to be the same. We have shown that all the higher spatial d erivatives terms that behave\nlike masses can be supersymmetrized by adding appropriate F -terms to the superpotential. On\nthe contrary, all terms with an odd number of higher spatial d erivatives would correspond to\n1For simplicity we write the dispersion relations in the case of(1,3)splitting, in which there is a natural\nidentification of the energy with the only component of the fo ur momentum whose weighted dimension coincides\nwith the usual one. The extension to different splittings is s traightforward.\n10modifications of the Kälher potential. However, a Kähler pot ential which is renormalizable by\nweighted power counting should have the same form as the Lore ntz invariant one and therefore\nwe cannot construct a supersymmetric version of a theory wit h an odd number of higher spatial\nderivatives in the fermionic kinetic term. If nis odd this means that it is not possible to construct\na supersymmetric version of a free theory with scalars and fe rmions. For even nwe can construct\nsupersymmetric theories in which dispersion relations for fermions will be of the form ( 21) with\nb′\nl= 0∀l.\nThe Feynman rules for the vertices remain unchanged with res pect to the Lorentz invariant\ncase [31] because the Lorentz-violating terms do not modify the θ-structure of Feynman graphs.\nTherefore the divergent contributions to the effective ener gy are polynomials in the external\nmomenta of the form\nΓ∞=/integraldisplay\nd4xd4θF(Φ,¯Φ,DΦ...,¯∂Φ,...). (22)\nBy power counting we obtain [F] =đ−2, so that the non-renormalization theorem [ 32] for the\ndivergent contributions still works in the Lorentz-violat ing case and the divergent contributions\naffect only the Kähler potential. We can calculate the superfi cial divergence for a generic Feynman\ngraphGatLloops, with Ipropagators, Vvertices and Eexternal lines2\nω(G) = (đ−2)L−2I−E+/summationdisplay\nN,αvN(N−1+δα\nN)\n=d(E)−2−/summationdisplay\nN,αvα\nN[λα,N],whered(E) =đ−E[Φ](23)\nand the weights of the chiral superfield Φand of the coupling constant are defined in ( 10) and ( 12).\nFor renormalizable theories, taking E= 2yields an upper bound for the superficial divergence,\nω(G)≤0, that is in agreement with the result of the non-renormaliza tion theorem ( 22). Therefore\nin a supersymmetric Lorentz-violating theory the Kähler po tential can receive radiative corrections\nwith logarithmic divergences if and only if there are strict ly renormalizable interactions. If the\ntheory contains only super-renormalizable interactions, then the theory is finite. In the Lorentz\ninvariant Wess-Zumino model, the non-renormalization the orem ensures that the behavior of the\ntheory at different energies is regulated only by the wave fun ction renormalization constant. In our\nmodels this is not true anymore. However, we can derive relat ions among different renormalization\nconstants. The most general renormalizable interaction la grangian contains two different kinds of\ncomposite operators:\nLint⊃/integraldisplay\nd2θ/braceleftBigg\nλk\nΛk−3\nLΦk+λ′\nk,l,α\nΛk+2l−3\nL/bracketleftBig\n¯∂2lΦk/bracketrightBig\nα/bracerightBigg\n+h.c..\nDemanding the renormalizability of the theory it is easy to s ee thatkis an integer number\n2<k≤¯N, where ¯Nis the maximum number of legs for the chiral vertex in a Lorent z-violating\n2The covariant derivatives algebra contains the positive we ighted constant c, so that their commutation rules\ngenerate non-homogeneous polynomials of degree 1in the momenta. A chiral vertex/integraltext\nd2θΦNyeldsN−1D2\nfactors to the numerators, that correspond to a non-homogen eous polynomial of maximum degree N−1in the\nmomenta. In the computation of the superficial divergence we consider the term of maximum degree for every\npolynomial generated by the covariant derivatives commuta tion rules. The other terms will have a better UV\nbehavior.\n11theory with fixed n,\n¯N=/bracketleftbigg\n2đ−1\nđ−2/bracketrightbigg\n, (24)\nand we indicate with the squared brackets the integer part of the enclosed ratio.\nConsequently n/2≤l<n, where in our case n/2is alway an integer because our theories are\ndefined for even nonly. We will classify all the possible models for different s plittings in section\n(3.3). The renormalization constant of operators like Φkis simply the product of ktimes the\nrenormalization constant of the superfield Φ, whereas composite operators like Ok,l,α≡/bracketleftbig¯∂2lΦk/bracketrightbig\nα\nrenormalize in a non-trivial way because of the space deriva tives in the vertices. Rewriting the\nlagrangian respect to the barequantities we get\n(λk)B=µǫ(k/2−1)Zλkλk,ΦB=Z1/2\nΦΦ,(al)B=Zalal, cB=c+∆c,\n(λ′\nk,l,α)B=µǫ(k/2−1)Zλ′\nk,l,αλ′\nk,l,α,(Ok,l,α)B=ZOk,l,αOk,l,α,ΛLB=ZΛLΛL.\nFrom the non-renormalization of the superpotential we then obtain:\nZk/2\nΦZλk\nZk−3\nΛL= 1,ZOk,l,αZλ′\nk,l,α\nZk+2l−3\nΛL= 1. (25)\nOn the other hand the Kähler potential is renormalized by a si ngle superfield wave function\nrenormalization KB[¯Φ,Φ] =ZΦK[¯Φ,Φ], so that:\n∆c= 0,Zal\nZΛL= 1. (26)\nThis last result implies that the deviation from the speed of light of a supermultiplet does not\nrenormalize in a supersymmetric theory. We will discuss in m ore detail the physical consequences\nof this result in section ( 5).\n3.2 Homogeneous Theories\nWe call homogeneous a theory whose vertices are all marginal . This kind of theories is invariant\nat the classical level under the weighted dilatations ( 1), while at the quantum level the weighted\nscale invariance is anomalous and related to the renormaliz ation group [ 9]. Moreover in the non-\nsupersymmetric case homogeneous models for scalars and spi nors were classified in [ 9]. In this\nsection we want to study the possibility of constructing hom ogeneous Lorentz-violating supersym-\nmetric theories. To analyze this problem it is important to e mphasize that in the supercharges\nalgebra ( 3) Lorentz violation is introduced by means of the weighted co nstantc. Therefore if\n[c]>0andc/ne}ationslash= 0the kinetic term in the Kähler potential always introduces a non-homogeneous\nterm in the propagator, which breaks the weighted scale inva riance.\nWe can define Lorentz-violating homogeneous theories if [c] = 0 andn= 1. In this case we\nobtain the model proposed in [ 22], in which the interaction sector is equal to the Wess-Zumin o\nmodel. However, we can obtain another class of homogeneous t heories by taking c= 0. In this\ncase the supercharges algebra ( 3) becomes:\n/braceleftbig\nQα,¯Q˙α/bracerightbig\n= 2σˆµ\nα˙αPˆµ. (27)\n12The resulting algebra ( 27) is the usual N= 1supersymmetry algebra in d= 4, but projected\non the submanifold Mˆd. It is clear that the Kähler potential for the chiral superfie lds associated\nwith this algebra does not introduce non-homogeneous terms in the propagators. Therefore we\ncan define a class of free homogeneous lagrangians for every e ven value of n, inserting in the\nsuperpotential ( 13) only the bilinear with the maximum number of spatial deriva tives, regulated\nby the weightless constant an/2. The propagators are the inverse of homogeneous polynomial s of\nweight2:\n/an}bracketle{tΦ(1)¯Φ(2)/an}bracketri}ht=δ12\nˆp2+a2\nn/2¯p2n\nΛ2n−2\nL,\n/an}bracketle{tΦ(1)Φ(2) /an}bracketri}ht=D2\n4an/2(¯p2)n/2\nΛn−1\nLδ12\nˆp2/parenleftBig\nˆp2+a2\nn/2(¯p2)n\nΛ2n−2\nL/parenrightBig. (28)\nFrom these free theories we can construct interacting lagra ngians by adding all the renormalizable\nterms in the superpotential. When c= 0, if we consider only the strictly renormalizable interac-\ntions in the superpotential, we then obtain homogenous inte racting theories. At the quantum level\nthe fixed points of the renormalization group for these theor ies are still invariant under weighted\nscale transformations, but far from the fixed points the symm etry is anomalous. In the low-energy\nlimit we cannot restore the usual N= 1supersymmetry algebra in d= 4and we cannot obtain a\nLorentz-invariant theory with usual propagators because o f the weighted scale invariance; in fact\nsuper-renormalizable terms cannot be produced by renormal ization because they would break the\nweighetd scale invariance. In the IR limit the propagators ( 28) become\nlim\nΛL→∞/an}bracketle{tΦ(1)¯Φ(2)/an}bracketri}ht=δ12\nˆp2,lim\nΛL→∞/an}bracketle{tΦ(1)Φ(2) /an}bracketri}ht= 0. (29)\nThe propagators ( 29) do not depend on ¯p, so that all the diagrams constructed with these propa-\ngators are not computable, because they contain divergence s that no counterterms can eliminate.\nHence the IR limit is singular. From homogenous theories we c an construct non-homogeneous the-\nories invariant under the algebra ( 27), by adding to the superpotential all the super-renormaliz able\nterms. Doing that, we are breaking the weighted scale invari ance, but terms fundamental for the\nLorentz symmetry recovery such as φ∗∂¯µ∂¯µφori∂¯µ¯ψ¯σ¯µψare not generated by renormalization\nbecause they break the symmetry of the lagrangian under supe rsymmetry transformations ( 27).\nThis means that for c= 0Lorentz symmetry cannot be restored and the IR limit of these theories\nis still singular.\n3.3 Classification of neutral chiral superfield’s models\nWe want to classify all the possible theories invariant unde r the superalgebra ( 3) for all possible\nsplittings of the four dimensional spacetime manifold. The basic ingredient of such classification\nwill be the maximum number of legs for a chiral vertex defined i n (24).\n1.For splitting (0,4) we have ¯N=/bracketleftBig\n24−n\n4−2n/bracketrightBig\n.The only solution is the n= 1and¯N= 3and\nwe thus recover the Wess-Zumino model.\n132.For (1,3) splitting we obtain ¯N=/bracketleftBig\n23\n3−n/bracketrightBig\n. Forn= 1we find again the Lorentz-violating\nWess-Zumino model proposed in [ 22,29]. Takingn= 2we find ¯N= 6, which is the only\nnon-trivial solution of the condition ( 24). Hence, for (1,3)splitting the only theory with\nstrictly renormalizable interactions has n= 2and đ= 5/2. The Lagrangian, requiring\nsymmetry under the transformation Φ→ −Φ, will be\nL=/integraldisplay\nd4θ¯ΦΦ+/integraldisplay\nd2θ/bracketleftbigg(¯∂Φ)2\n2ΛL+mΦ2\n2/bracketrightbigg\n+/integraldisplay\nd2θ/bracketleftbigg\nλ4Φ4\n4!ΛL+λ6Φ6\n6!Λ3\nL/bracketrightbigg\n+h.c..(30)\nWe can derive the propagators for the superfields and expand t hem for high momenta in\norder to study the UV behavior of the theory:\nPΦ¯Φ=δ12\nˆp2+(c2+2m\nΛL)¯p2+¯p4\nΛ2\nL+m2≃δ12\nˆp2+¯p4\nΛ2\nL, (31)\nPΦΦ=D2\n4¯p2\nΛLδ12\nˆp2+(c2+2m\nΛL)¯p2+¯p4\nΛ2\nL+m2≃D2\n4¯p2\nΛLδ12\np′2/parenleftBig\nˆp2+¯p4\nΛ2\nL/parenrightBig. (32)\nThe first divergent radiative correction to the Kähler poten tial is at 4loops:\nΦ ¯Φ=λ2\n6\n5!/integraldisplaydˆk\n2πd3¯k\n(2π)3/integraldisplay\nd4θ1d4θ2Φ(−k,θ1)Φ(k,θ2)BD.\nThe covariant derivatives algebra is easy to compute if we re call the usual identities\nD=δ12D2¯D2\n16δ12¯D2D2\n16δ12D2¯D2\n16δ12D2¯D2\n16δ12=δ12.\nTherefore, the superfields computation is reduced to the com putation of the bosonic integral\nB, which is not easily computable because of the modified form o f the propagators ( 31) and\n(32).\n3.For(2,2)splitting ¯N=n+ 2and we can construct theories with strictly renormalizable\ninteractions for any even n. For example we can choose n= 2and write the complete\ntheory:\nL=/integraldisplay\nd4θ¯ΦΦ+/integraldisplay\nd2θ/bracketleftbigg(¯∂Φ)2\n2ΛL+m\n2Φ2/bracketrightbigg\n+/integraldisplay\nd2θ/bracketleftbiggλ3\n3!Φ3+λ4\n4!ΛLΦ4/bracketrightbigg\n.\nThe kinetic term is the same of the theory n= 2for(1,3)splitting and the propagators are\n(31) and ( 32).\n14The first divergent radiative correction to the Kähler poten tial is at 2loops:\nΦ ¯Φ=λ2\n4\n3!/integraldisplayd2ˆk\n(2π)2d2¯k\n(2π)2/integraldisplay\nd4θ1d4θ2Φ(−k,θ1)Φ(k,θ2)BD\nAgain, for this graph D=δ12but the bosonic integral Bis again very hard to compute.\n4.In the(3,1)case we obtain ¯N=/bracketleftBig\n22n+1\nn+1/bracketrightBig\nthat has only one integer solution for n= 1that\nis the trivial one. Therefore we can construct only super-re normalizable theories.\n5.For splitting (4,0)we obtain the Lorentz invariant Wess-Zumino model.\n4 Gauge invariant theories\nWe want to study the problem of finding a gauge invariant versi on of the Lorentz-violating su-\npersymmetric theories that we have found in section 3. First of all we apply a result of [ 24] to\nshow that it is not possible to generalize the theories of sec tion 3 to the case of charged chiral\nsuperfields. The basic observation in our construction was t hat it is possible to insert higher space\nderivatives as mass terms in the superpotential in ( 13) because they preserve the chirality of Φ,\nwhich is clear recalling that [Dα,∂¯µ] = 0. Charged chiral superfields transform with a phase under\nthe action of a general SU(N)gauge group\nΦi→eΛΦi,\nwhereΛis a chiral superfield. We can define a gauge invariant version of the supersymmetric\ncovariant derivative:\nDαΦi=e−VDα(eVΦi),\nwhereV(ˆx,¯x,θ,¯θ)is the vector superfield with its usual gauge trasformation l aw:eV→e¯ΛeVe−Λ.\nIt is clear that explicit spatial derivatives in the action b reak gauge invariance. In principle, as was\nobserved in [ 24], we could still introduce higher spatial derivatives in th e superpotential promoting\n∂¯µto a covariant derivative:\nD¯µΦ =−i¯σ˙αα\n¯µ\n4¯D˙αDαΦ. (33)\nThe problem, however, is that D¯µdoes not preserve the chirality condition. In fact, as it was\nchecked in [ 24]:\n¯D˙α¯D˙β/parenleftbig\ne−VDαeVΦi/parenrightbig\n= 2ε˙β˙αWαΦ/ne}ationslash= 0. (34)\nThis argument shows that the theories that we have construct ed are not generalizable to the case\nof charged chiral superfields.\nNow we will see directly in the gauge sector that, requiring g auge invariance and supersym-\nmetry, the weighted power counting has to coincide with the u sual one. We briefly review the\n15derivation of the weights for the gauge fields referring to [ 7,8] for a complete study of Lorentz-\nviolating gauge theories. In the Lorentz-violating case th e gauge field Aµhas to be decomposed\nas all the other vectors into time and space components A= (ˆA,¯A). Therefore the covariant\nderivative is decomposed as\nD= (ˆD,¯D) = (ˆ∂+eˆA,¯∂+e¯A), (35)\nwhereeis the gauge coupling constant. To be consistent with the defi nition of covariant derivative\n(35) we have to weight [eˆA′] = [ˆ∂] = 1and[e¯A] = [¯∂] = 1/n. The field strength is split into three\nparts:\nˆFµν≡Fˆµˆν,˜Fµν≡Fˆµ¯ν,¯Fµν≡F¯µ¯ν.\nThe kinetic lagrangian has to contain (ˆ∂ˆA)2, so we can obtain the weight of ˆA, that is equal to the\nweight of the scalar field, and from the definition of covarian t derivatives we derive the weights\nfor¯Aand[e] = 2−đ/2. Summarizing we have:\n[ˆA] =đ−2\n2,[¯A] =đ\n2−2+1\nn,[ˆF] =đ\n2,[˜F] =đ\n2−1+1\nn,[¯F] =đ\n2−2+2\nn.(36)\nThe requirement of absence of spurious subdivergences [ 7] implies that\nˆd= 1,đ<2+2\nn, d=even, n=odd,\nand the only acceptable splitting is (1,3). In this case we have ˆF= 0so it is possible to rearrange\nthe weights of the gauge field and the gauge coupling so that th e producteAmaintains the same\nweight and at the same time [˜F] =đ/2:\n[ˆA] =đ\n2−1\nn,[¯A] =đ\n2−1,[˜F] =đ\n2,[¯F] =đ\n2−1+1\nn,[e] = 1+1\nn−đ\n2.(37)\nIn the supersymmetric case both weight assignments ( 36) and ( 37) have to be consistent with\nthe relations among the weights of the fields imposed by the su persymmetric transformations\ngenerated by the supercharges ( 3). For the vector multiplet the supersymmetric transformat ion\nare:\nδηAµ= ¯η¯σµ′λ+¯λ¯σµ′η ,\nδηλ=i\n2σµ′¯σν′ηFµν+ηD ,\nδηD= ¯η¯σµ′∂µλ−∂µ¯λ¯σµ′η , (38)\nwhere the gaugino λis a propagating Majorana fermion, Dis an auxiliary field and ηis the\nspinorial parameter of the supersymmetry transformation. Since we know the weights of λ,ηand\nof the weighted constant c, we can obtain the weights of the other fields of the supermult iplet,\napplying the weighted power counting to the relations ( 38) yelds\n[ˆA] =đ−2\n2,[¯A] =đ\n2−1\nn,[D] =đ\n2,\n[ˆF] =đ\n2,[˜F] =đ\n2−(1−1\nn),[¯F] =đ\n2−2(1−1\nn). (39)\n16Any gauge theory that has a supersymmetric extension invari ant under the supersymmetry algebra\n(3) has to satisfy the constraint on the weight of the fields ( 39). Therefore, we can take the two\npossible weight assignments for Lorentz-violating gauge fi eld theories ( 36) and ( 37) and check for\nwhich value of nthese theories can admit a supersymmetric extension. Doing that we found that\nthe only possible value is n= 1in both cases and hence, as long as we require supersymmetry a nd\ngauge invariance, we have to weight time and space in the same way, regardless of the condition\nof absence of spurious divergences. The only Lorentz-viola ting operators are introduced by the\nweighted constant c. These operators are renormalizable in the usual sense and c orrespond to the\nCPT preserving operator in the gauge sector of the SME [ 13]. In particular they can be expressed\nintroducing a twisted derivative ˜∂µ=∂µ+kν\nµ∂ν[38], where in our case kµν= (c−1)δ¯µ¯ν. We will\nshow in the next section that the constant cdoes not renormalize in the supersymmetric case. As\nwe need to fine tune cin order to recover Lorentz symmetry at low energies, the Lor entz-violating\nparameters will be extremely small also at high energies. Th is argument shows that demanding\nrenormalizability and gauge invariance for supersymmetri c theories, the Lorentz invariance follows\nas a consequence.\n5 Low-energy limit and Lorentz symmetry recovery\nIf we consider Lorentz-violating theories as candidates to describe high-energy physics, then\nLorentz invariant theories are effective field theories whic h describe physics at low energies with\nrespect to ΛL. In the framework of renormalizable Lorentz-violating the ories it is still true that\nthe renormalization procedure commutes with the low-energ y limit. Therefore it is a general re-\nsult that a Lorentz-violating high-energy theory renormal izable by weighted power counting tend\nto a low-energy theory which is renormalizable by usual powe r counting and contains Lorentz-\nviolating parameters. The low-energy theory is then obtain ed from the high-energy one simply\nby taking the limit ΛL→ ∞. The recovery of Lorentz invariance at low energies is regul ated by\nthe RG behavior of the Lorentz-violating parameters at low e nergies that correspond to operators\nof dimension less than or equal to four, which are not suppres sed by any power of ΛL.\nIn this section we want to study how the Lorentz recovery prob lem at low energies is modified\nin presence of supersymmetry. If supersymmetry is an exact s ymmetry of nature at high energies\nthen in the low-energy limit supersymmetry has to be broken f or several phenomenological reasons\n[33]. Ignoring the exact mechanism of SUSY breaking we can param etrize the supersymmetry\nbreaking at low energies by adding explicit breaking terms t o the supersymmetric lagrangian.\nWe require the breaking to be soft, in the sense that the super symmetry breaking terms should\nnot generate quadratic divergences. It has been pointed out in [34] that the natural setting for\nstudying the low-energy limit of supersymmetric theories i n presence of soft breaking terms is the\nsuperfield formalism. The soft breaking terms are parametri zed as couplings among dynamical\nsuperfields and external spurion superfields. The possible t ypes of spurion superfields which break\nsupersymmetry softly for Lorentz invariant theories are cl assified in [ 34].\nOn this basis we can compute the low-energy limit for the gene ral supersymmetric Lorentz-\nviolating lagrangian ( 11). Assuming that the supersymmetry breaking soft terms are g enerated\n17at energyµs<<ΛLwe obtain the standard Lorentz-invariant soft terms:\nL=/integraldisplay\nd4θ¯ΦiΦi+/integraldisplay\nd2θ(mij\n2ΦiΦj+λijk\n3ΦiΦjΦk)+h.c.\n+/integraldisplay\nd4θUij¯ΦiΦj+/integraldisplay\nd2θij(χΦiΦj+ηijkΦiΦjΦk)+h.c.,(40)\nwhereUij=µ2\nsuijθ2¯θ2,χij=µ2\nsxijθ2andηijk=µsnijkθ2are the soft breaking spurion superfields\nandu, x, n are dimensionless matrices in the generations indices. We c an write the resulting\nsoft-breaking terms in components:\nLbreak=µ2((u+x)ijAiAj+(u−x)ijBiBj)+µnijk(AiAjAk−3AiBjBk). (41)\nAll the terms in ( 41) are super-renormalizable and they introduce new divergen ces in addition to\nthe usual wave function renormalization of the Wess-Zumino model, which are studied in [ 34,35].\nSince they all are super-renormalizable, these terms do not affect the divergent part of the wave\nfunction renormalization. Therefore, even for softly brok en supersymmetry, it is easy to see that\nthe Lorentz-violating parameter cdoes not renormalize, because it does not appear explicitly in\nthe superfield lagrangian. In the IR limit we need to fine tune caccording to the experimental\nbounds on Lorentz violation, but the low-energy value of cwill be also the value of the constant c\nat high energy, because cdoes not renormalize. Hence the deviation from the speed of l ight of the\nlimiting speed of elementary particle is negligible for Lor entz-violating supersymmetric models.\nWe will consider an explicit model in components in order to u nderstand how the behavior\nof the renormalization group equations is modified by the fac t that the effective theory at low\nenergies is the low-energy limit of a supersymmetric Lorent z-violating theory with soft breaking.\nFor this specific model we will explicitly show at one loop how the renormalization properties\nof a softly broken supersymmetric theory imply the non renor malization of c. Let us consider\nthe most general low-energy limit of a renormalizable super symmetric Lorentz-violating theory\nfor an interacting chiral multiplet. Since supersymmetry i s softly broken at low-energy, from the\nprevious discussion it is clear that we can parametrize this breaking considering as independent the\ndimensionfull parameters of the lagrangian. Therefore the low-energy lagrangian will be described\nby six independent parameters:\nL=(ˆ∂A)2\n2+c2(¯∂A)2\n2+(ˆ∂B)2\n2+c2(¯∂B)2\n2+m2A2\n2+m′2B2\n2+1\n2¯ψ(ˆ/∂+v¯/∂+M)ψ\n+λ3\n3!A3+λ3\n2′\nAB2+g2\n4!(A4+B4+6A2B2)+gA¯ψψ+igB¯ψγ5ψ .\nIn the approximation of small deviations from the speed of li ght we can put c2≃1 +δc2and\nv≃1+δv. The bare quantities are defined as:\nAb=Z1\n2\nAA , Bb=Z1\n2\nBB , ψb=Z1\n2\nψψ , δc2b=δc2+∆δc2, δc2b=δv+∆δv,\nm2\nb=m2+∆m2, m′2\nb=m′2+∆m′2\nB, Mb=M+∆M ,\nλ3b=µǫ\n2(λ3+∆λ3), λ′\n3b=µǫ\n2(λ′\n3+∆λ′\n3), gb=µǫ(g+∆g).\n18Recalling the Feynman rules for Majorana fermions [ 36] we obtain at one loop:\nZA=ZB= 1−g2\n2π2ǫ(1−¯dδv), Zψ= 1−g2\n2π2ǫ(1−¯d\n3δc2−¯d\n3δv),\n∆δc2=g2\n2π2ǫ(δc2\n2−δv),∆δv=g2\n2π2ǫ(2δv−δc2).\nIf supersymmetry is softly broken the divergent part of the w ave function renormalization has to\nbe the same for all particles in the same supermultiplet. The refore, imposing Zψ=ZAwe obtain\n2δv=δc2, which implies ∆δc2= ∆δv= 0. We thus conclude that the Lorentz-violating parameter\ndoes not renormalize. The only renormalization constant le ft is a common renormalization for the\nwave functions ZA=ZB=Zψ= 1−g2\n2π2ǫ(1−¯dδv), that at the zeroth order in δvis in agreement\nwith the well known result for the Wess-Zumino model.\nAn interesting problem to address concerns the nature of the constant which parametrizes\nthe deviation from the speed of light in supersymmetric Lore ntz-violating theories. In particular\nwe want to understand whether or not the weighted constant cis physically observable. As was\nalready noted in [ 24], if we consider a supersymmetric Lorentz-violating theor y with one single\nsector the parameter cappears both in the kinetic lagrangian ( 13) and in the supersymmetry\ntransformations ( 3) and can therefore be reabsorbed by a rescaling of the spatia l coordinates\nx′\n¯µ=cx¯µ. Now, as far as supersymmetry is an exact symmetry of our theo ry, all interacting\nsupermultiplets have the same limiting speed c, because this parameter explicitly appears in the\nsupersymmetry transformations. In this type of theories th e parameter cis physically unobserv-\nable because we can always set it to one by suitably choosing t he length units. However, as was\nsuggested in [ 37], we can construct more complicated situations with two or m ore sectors which\nare separately invariant under supersymmetry transformat ions (3) with different limiting speeds\nci, where the lower index labels the number of sectors. For exam ple let us consider two different\nsectorsS1andS2, separately invariant under the supersymmetry transforma tions\n/braceleftbig\nQα,¯Q˙α/bracerightbig\n= 2σˆµ\nα˙αPˆµ+2c1σ¯µ\nα˙αP¯µ, (42)\n/braceleftbig\nQα,¯Q˙α/bracerightbig\n= 2σˆµ\nα˙αPˆµ+2c2σ¯µ\nα˙αP¯µ. (43)\nIfS1andS2are completely decoupled, the supersymmetry algebras ( 42,43) are exactly realized\nin their respective sectors.3We can still rescale the spatial coordinates in order to reab sorbc1or\nc2but, after the rescaling, we will have a Lorentz invariant se ctorS1withc1/ne}ationslash= 1and an Lorentz\nviolating hidden sector S2completely decoupled with c2/ne}ationslash= 0. Moreover, we can make S1andS2\ninteracting by adding super-renormalizable interactions which will softly break supersymmetry in\nboth sectors. The deviation from the speed of light in S2then becomes experimentally observable\nbecause any rescaling performed in order to remove the c2factor will produce Lorentz-violating\neffects inS1, as was already pointed out in [ 37].\nIn conclusion, there is always the possibility to set one lim iting speed to one by rescaling the\nspatial coordinates. This fact can make the Lorentz-violat ing supersymmetric models presented\nin [22,37] physically equivalent to the Lorentz invariant ones if we r estrict our attention to models\n3The Lorentz-violating theories that we are considering are rigid supersymmetric theories with very good ap-\nproximation. Indeed the scale ΛLhas to be around 1014GeV in order to explain the neutrino masses [ 14]. Therefore\nwe can neglect gravitational effects and consider the two sec torsS1andS2completely decoupled.\n19with a single sector. Besides that we have shown that the param etercdoes not renormalize in\nany softly broken supersymmetric theory, so that even if the deviation from the the speed of light\nis indeed observable it would be extremely small at any energ y scale.\n6 Conclusion\nIn this paper we have investigated the possibility of constr ucting supersymmetric Lorentz-violating\ntheories that can be renormalized by a weighted power counti ng. Our analysis starts from the\nobservation that supersymmetry and Lorentz violation are c ompatible at the level of the alge-\nbra [22].\nMoreover, we have shown that in the Lorentz-violating case i t is possible to construct new\nsuperalgebras with supercharges non-linear in the spatial momenta. However, the non-linearity\nof the supercharges makes the problem of finding interacting theories invariant under the new\nsuperalgebras very involved, because the superfield formal ism looses its usefulness. As an exam-\nple of this general difficulty we have shown that the interacti ng theory constructed in [ 28] is not\ninvariant under this new class of superalgebras.\nAssuming linearity of the supercharges in the spatial momen ta we find renormalizable super-\nsymmetric Lorentz-violating models for even nand classify them. It is straightforward to verify\nin the superspace formalism that the non-renormalization t heorem is still valid in the Lorentz-\nviolating case and as a consequence our models exhibit the im proved ultraviolet behavior typical\nof supersymmetric theories. Moreover, the weighted consta ntcappearing in the supercharges\nalgebra which parametrizes the limiting speed of the multip let does not renormalize at high ener-\ngies because the Kähler potential of a renormalizable Loren tz-violating theory has the same form\nas in the Lorentz invariant case. Furthermore, the low-ener gy recovery of Lorentz symmetry is\nparametrized only by this parameter c, which does not renormalize even at low energies if we\nassume supersymmetry to be broken softly.\nIn the case of gauge theories we show that demanding supersym metry implies that the weighted\npower counting has to coincide with the usual one. The only Lo rentz-violating operators are then\nintroduced by the weighted constant c, which does not renormalize and has to be very close to\nthe speed of light at low energies in order to satisfy the expe rimental bounds on Lorentz violation\n[1]. Therefore, if we demand renormalizabilty, gauge invaria nce and supersymmetry, the Lorentz\ninvariance follows as a consequence.\nOur analysis agrees with the conjecture that supersymmetry can solve the Lorentz fine tuning\nproblem for Lorentz-violating theories, but at the same tim e it reveals that the requirement of su-\npersymmetry restricts drastically the possibility of cons tructing renormalizable Lorentz-violating\ntheories at high energies. Indeed, the final picture which em erges from our investigation is that the\nonly possible models with non trivial Lorentz-violating op erators involve neutral chiral superfields\nand do not have a gauge invariant extension. Therefore, if we want to construct renormalizable\nLorentz-violating extensions of the Standard Model which h ave new interesting phenomenological\nconsequences, the Lorentz fine tuning problem [ 21,23] does not seem solvable by the requirement\nof supersymmetry.\n20Acknowledgements\nThe author is grateful to D. Anselmi for suggesting the probl em and for useful discussions. He also\nthanks R. Argurio for useful comments. The research of D.R. i s supported by the “Communauté\nFrançaise de Belgique” through the ARC program, and also in pa rt by IISN-Belgium (conventions\n4.4511.06, 4.4505.86 and 4.4514.08) and by the Belgian Feder al Science Policy Office through the\nInteruniversity Attraction Pole IAP VI/11.\nReferences\n[1]V. Kostelecky and N. Russell, “Data Tables for Lorentz and CP T Violation,”\nRev.Mod.Phys. 83(2011) 11, arXiv:0801.0287 [hep-ph] . —2,20\n[2]S. Chadha and H. B. Nielsen, “Lorentz Invariance As a Low-Ener gy Phenomenon,”\nNucl.Phys. B217 (1983) 125 . —2\n[3]V. 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Anselmi, “Standard Model Without Elementary Scalars And High Energy Lorentz\nViolation,” Eur.Phys.J. C65(2010) 523–536 ,arXiv:0904.1849 [hep-ph] . —3\n[17]D. Anselmi and E. Ciuffoli, “Low-energy Phenomenology Of Sca larless Standard-Model\nExtensions With High-Energy Lorentz Violation,” Phys.Rev. D83(2011) 056005 ,\narXiv:1101.2014 [hep-ph] . —3\n[18]D. Colladay and P. McDonald, “One-Loop Renormalization of P ure Yang-Mills with\nLorentz Violation,” Phys.Rev. D75(2007) 105002 ,arXiv:hep-ph/0609084 [hep-ph] . —3\n[19]S. R. Coleman and S. L. Glashow, “High-energy tests of Lorent z invariance,”\nPhys.Rev. D59(1999) 116008 ,arXiv:hep-ph/9812418 [hep-ph] . —3\n[20]R. Iengo, J. G. Russo, and M. Serone, “Renormalization group in Lifshitz-type theories,”\nJHEP 0911 (2009) 020 ,arXiv:0906.3477 [hep-th] . —3\n[21]J. Collins, A. Perez, D. Sudarsky, L. Urrutia, and H. Vucetic h, “Lorentz invariance and\nquantum gravity: an additional fine-tuning problem?,” Phys.Rev.Lett. 93(2004) 191301 ,\narXiv:gr-qc/0403053 [gr-qc] . —3,20\n[22]M. Berger and V. Kostelecky, “Supersymmetry and Lorentz viol ation,”\nPhys.Rev. D65(2002) 091701 ,arXiv:hep-th/0112243 [hep-th] . —3,5,12,14,19,20\n[23]P. Jain and J. P. Ralston, “Supersymmetry and the Lorentz fine tuning problem,”\nPhys.Lett. B621 (2005) 213–218 ,arXiv:hep-ph/0502106 [hep-ph] . —3,20\n[24]S. Groot Nibbelink and M. Pospelov, “Lorentz violation in su persymmetric field theories,”\nPhys.Rev.Lett. 94(2005) 081601 ,arXiv:hep-ph/0404271 [hep-ph] . —3,4,15,19\n[25]P. A. Bolokhov, S. G. Nibbelink, and M. Pospelov, “Lorentz vio lating supersymmetric\nquantum electrodynamics,” Phys.Rev. D72(2005) 015013 ,\narXiv:hep-ph/0505029 [hep-ph] . —3\n[26]S. R. Coleman and J. Mandula, “All Possible Symmetries Of The S Matrix,”\nPhys.Rev. 159(1967) 1251–1256 . —4\n[27]R. Haag, J. T. Lopuszanski, and M. Sohnius, “All Possible Gen erators of Supersymmetries\nof the s Matrix,” Nucl.Phys. B88(1975) 257 . —4\n[28]W. Xue, “Non-relativistic Supersymmetry,” arXiv:1008.5102 [hep-th] . —4,6,7,20\n22[29]M. Berger, “Superfield realizations of Lorentz and CPT violat ion,”\nPhys.Rev. D68(2003) 115005 ,arXiv:hep-th/0308036 [hep-th] . —5,14\n[30]J. Wess and B. Zumino, “A Lagrangian Model Invariant Under Sup ergauge\nTransformations,” Phys.Lett. B49(1974) 52 . —9\n[31]M. T. Grisaru, W. Siegel, and M. Rocek, “Improved Methods for Supergraphs,”\nNucl.Phys. B159 (1979) 429 . —9,11\n[32]S. Gates, M. T. Grisaru, M. Rocek, and W. Siegel, “Superspace Or One Thousand and One\nLessons in Supersymmetry,” Front.Phys. 58(1983) 1–548,\narXiv:hep-th/0108200 [hep-th] . —11\n[33]S. P. Martin, “A Supersymmetry primer,” arXiv:hep-ph/9709356 [hep-ph] . —17\n[34]L. Girardello and M. T. Grisaru, “Soft Breaking of Supersymme try,”\nNucl.Phys. B194 (1982) 65 . —17,18\n[35]J. Iliopoulos and B. Zumino, “Broken Supergauge Symmetry and R enormalization,”\nNucl.Phys. B76(1974) 310 . —18\n[36]A. Denner, H. Eck, O. Hahn, and J. Kublbeck, “Compact Feynman rules for Majorana\nfermions,” Phys.Lett. B291 (1992) 278–280 . —19\n[37]D. Colladay and P. McDonald, “Vector Superfields and Lorentz Violation,”\nPhys.Rev. D83(2011) 025021 ,arXiv:1010.1781 [hep-ph] . —19\n[38]D. Colladay and P. McDonald, “Lorentz Violation and Extende d Supersymmetry,”\narXiv:1008.1279 [hep-ph] . —17\n23"    },    {        "title": "1801.08838v1.Ads_spacetime_in_Lorentz_covariant_gauges.pdf",        "content": "arXiv:1801.08838v1  [gr-qc]  26 Jan 2018AdS spacetime in Lorentz covariant gauges\nP. Valtancoli\nDipartimento di Fisica, Polo Scientifico Universit´ a di Fir enze\nand INFN, Sezione di Firenze (Italy)\nVia G. Sansone 1, 50019 Sesto Fiorentino, Italy\nAbstract\nWe show how to generate the AdS spacetime metric in general Lo rentz covariant\ngauges. In particular we propose an iterative method for sol ving the Lorentz gauge.1 Introduction\nRecently there has been a renewed interest in the cosmological con stant of general relativity,\nsince it could be useful as a cheap explanation for the antigravity fo rce which drives the\naccelerated expansion of the universe, better known as dark ene rgy [1]-[2]. It is now widely\naccepted that the cosmological constant is a very small paramete r, analogously to what\nhappens for the neutrino mass, and related to the vacuum energy of the quantum fields [3].\nTheAdSspace-time [4] is usually represented in a Lorentz non-covariant fo rm, but there\nare physical applications like for example the study of cosmological g ravitational waves [5],\nin which it is useful to reformulate it in a Lorentz covariant form. It is the purpose of this\npaper to generate all the Lorentz covariant solutions for AdSspace-time. We start from\nsolving perturbatively the Einstein equations up to the second orde r in Λ in the Lorentz\ngauge, which turns out to be a tedious calculation.\nTo simplify the discussion we are going to present a general 5-dimens ional representation\nof theAdSspace-time in terms of an arbitrary function f(Λx2), where x2=ηµνxµxν. We\nthen show that the Lorentz gauge fixes this arbitrary function wit h a recursive method and\nthe result agrees with the direct perturbative solution of the Einst ein equations.\n2 Perturbative calculation of the Einstein equations\nWe are going to discuss all the aspects of the AdS space-time in the L orentz gauge. We\nstart performing the perturbative calculation directly using the Ein stein equations, and only\nafterwards we will introduce a faster method. We recall the known perturbative solution of\ntheAdSspace-time at the first order in Λ\ngµν=ηµν+h(1)Λ\nµν+O(Λ2)\nh(1)Λ\nµν=−Λ\n9(xµxν+2ηµνx2) (2.1)\nIt is straightward to show that eq. (2.1) satisfies the Lorentz gau ge:\nηµµ′∂µ′h(1)Λ\nµν=1\n2∂ν(ηµµ′h(1)Λ\nµµ′) (2.2)\nThe associated connection is, at this order, given by:\nΓµ(1)Λ\nαβ=−Λ\n9( 2δµ\nαxβ+2δµ\nβxα−ηαβxµ) (2.3)\n1We can easily compute the curvature tensor, limited at the linear ter m in the connection\nRα(1)Λ\nµβν=Λ\n3(ηβνδα\nµ−ηβµδα\nν) (2.4)\nThe corresponding Ricci tensor and the curvature are then\nR(1)Λ\nβν= ΛηβνR(1)Λ=ηβνR(1)Λ\nβν= 4Λ (2.5)\nsolving perturbatively the Einstein equations.\nNow we are going to compute the perturbative metric at the second order in Λ\ngµν=ηµν+h(1)Λ\nµν+h(2)Λ\nµν+O(Λ3) (2.6)\nWe can again impose the Lorentz gauge\nηµµ′∂µ′h(2)Λ\nµν=1\n2∂ν(ηµµ′h(2)Λ\nµµ′) (2.7)\nWe must be careful that the connection is made by several pieces\nΓµ(2)Λ\nαβ= Γµ(2)ΛI\nαβ+ Γµ(2)ΛII\nαβ\nΓµ(2)ΛI\nαβ=1\n2ηµµ′(∂αh(2)Λ\nµ′β+∂βh(2)Λ\nµ′α−∂µ′h(2)Λ\nαβ)\nΓµ(2)ΛII\nαβ=−1\n2hµµ′\n(1)Λ(∂αh(1)Λ\nµ′β+∂βh(1)Λ\nµ′α−∂µ′h(1)Λ\nαβ) (2.8)\nwherehµν\n(1)Λ=ηµµ′ηνν′h(1)Λ\nµ′ν′.\nLet us work out the second term\nΓα(2)ΛII\nβν=−Λ2\n81( 4xαxβxν−3ηβνxαx2+4δα\nβxνx2+4δα\nνxβx2) (2.9)\nIts contribution to the curvature tensor is given by\nRα(2)ΛII\nβµν =∂µΓα(2)ΛII\nβν−(µ↔ν) =\n=−Λ2\n81[ 10ηβµxνxα−10ηβνxµxα+4δα\nνxβxµ\n−4δα\nµxβxν+7δα\nνηβµx2−7δα\nµηβνx2] (2.10)\n2The Ricci tensor\nR(2)ΛII\nβν=δµ\nαRα(2)ΛII\nβµν=Λ2\n81( 2xβxν+31ηβνx2) (2.11)\nand its contribution to the Einstein equation is\nR(2)ΛII\nβν−1\n2R(2)ΛIIηβν=2Λ2\n81(xβxν−16ηβνx2) (2.12)\nThe first term of the connection produces the following curvature tensor Γµ(2)ΛI\nαβ\nR(2)ΛI\nβν=−1\n2ηρρ′∂ρ∂ρ′h(2)Λ\nβν+ gauge terms (2.13)\ngiving rise to the another contribution to the Einstein equations\nR(2)ΛI\nβν−1\n2R(2)ΛIηβν=−1\n2ηρρ′∂ρ∂ρ′/parenleftbigg\nh(2)Λ\nβν−1\n2ηβνh(2)Λ/parenrightbigg\n(2.14)\nTill now we have worked out in the curvature tensor only linear terms in the connections.\nLet us add now the non linear ones:\nRα(2)ΛIII\nβµν = Γα(1)Λ\nµα′Γα′(1)Λ\nβν−(µ↔ν) =\n=Λ2\n81[4δα\nµxνxβ−2δα\nµηβνx2−4δα\nνxµxβ+\n+2δα\nνηβµx2−ηβµxαxν+ηβνxαxµ] (2.15)\nThe associated Ricci tensor is\nR(2)ΛIII\nβν=Λ2\n81( 11xνxβ−5ηβνx2) (2.16)\ngiving rise to the third contibution to the Einstein equations\nR(2)ΛIII\nβν−1\n2R(2)ΛIIIηβν=Λ2\n81( 11xνxβ−1\n2ηβνx2) (2.17)\nWe have not yet finished. We must add some residual extra contribu tions, given by\n1\n2R(2)ΛIV=h(1)ΛβνR(1)Λ\nβν→ −1\n2R(2)ΛIVηβν=−Λ2x2\n2ηβν(2.18)\n3and finally\n−1\n2R(1)Λh(1)Λ\nβν+Λh(1)Λ\nβν=−Λh(1)Λ\nβν (2.19)\nBy collecting all the various ( five ) terms\n−1\n2ηρρ′∂ρ∂ρ′/parenleftbigg\nh(2)Λ\nβν−1\n2ηβνh(2)Λ/parenrightbigg\n+2Λ2\n81(xβxν−16ηβνx2)\n+Λ2\n81( 11xνxβ−1\n2ηβνx2)−Λ2x2\n2ηβν+Λ2\n9(xβxν+2ηβνx2) = 0 (2.20)\nwe arrive at the final equation\nηρρ′∂ρ∂ρ′/parenleftbigg\nh(2)Λ\nβν−1\n2ηβνh(2)Λ/parenrightbigg\n=22\n81(2xβxν−5ηβνx2) (2.21)\nThe compatibility between this equation and the Lorentz gauge is str aightforward since\napplying the operator ∂β( or∂ν) to both sides of this equation we simply get zero.\nFinally we obtain the solution of this equation at the second order in Λ:\nh(2)Λ\nβν=11\n16·81Λ2( 4xβxνx2+5ηβνx4)\nh(2)Λ=ηβνh(2)Λ\nβν=11\n54Λ2x4x4≡(x2)2(2.22)\nWe have just learned that the Lorentz gauge imposes strong cons traints on the general\nsolution. Let us suppose to make the perturbative calculation at th e generic order n:\ngµν=ηµν+h(1)Λ\nµν+....+h(n)Λ\nµν+.... (2.23)\nthe final equation, similar to eq. ( 2.21 ), at the order nis of the type\nηρρ′∂ρ∂ρ′/parenleftbigg\nh(n)Λ\nβν−1\n2ηβνh(n)Λ/parenrightbigg\n= (Anxβxνx2(n−2)+Bnηβνx2(n−1)) (2.24)\nLet us apply the operator ∂β( or∂ν) to both sides of this equation\n∂β(...) = 0 →Bn=−2n+1\n2(n−1)An (2.25)\n4the Lorentz gauge implies the constraint\nηρρ′∂ρ∂ρ′/parenleftbigg\nh(n)Λ\nβν−1\n2ηβνh(n)Λ/parenrightbigg\n=cnΛn[ 2(n−1)xβxνx2(n−2)−(2n+1)ηβνx2(n−1)] (2.26)\nTherefore the solution at the order nis of the type\nh(n)Λ\nβν=cnΛn/bracketleftbiggxβxνx2(n−1)\n2(n+2)+(n+3)ηβνx2n\n4n(n+2)/bracketrightbigg\nh(n)Λ=ηβνh(n)Λ\nβν=3\n2ncnΛnx2nx2n≡(x2)n(2.27)\nUnfortunately the cncoefficients cannot be computed only from the Lorentz gauge and\nmust be determined by solving the Einstein equation.\n3 General solution of the Einstein equations\nTo solve the Einstein equation, we are going to embed the AdSspace-time in the following\n5dspace\nds2=dX2\n0+dX2\n1−dX2\n2−dX2\n3−dX2\n4 (3.1)\nsubject to the constraint\nX2\n0+ηijXiXj=3\nΛi= 1,...,4 (3.2)\nWecanautomaticallygeneratesolutionsoftheEinstein equationssim ply determining the\nmapping X0=X0(xµ),Xi=Xi(xµ). In this choice we must respect the Lorentz covariance\nof the 4dspace-time and we are forced to choose this mapping\nX0=/radicalbigg\n3\nΛ/radicalbigg\n1−Λ\n3x2f2(Λx2)\nXi=xif(Λx2) (3.3)\nwith the 4 dmetric defined as usual by the formula\n5gµν=ηabdXa\ndxµdXb\ndxν(3.4)\nAll these mappings produce solutions of the Einstein equation in a cov ariant gauge and\ndepend on an arbitrary function f(Λx2). First let us study the basic solution with f(Λx2) =\n1:\nX0=/radicalbigg\n3\nΛ/radicalbigg\n1−Λ\n3x2Xi=xi\ng(0)\nµν=ηµν+Λ\n3xµxν\n1−Λ\n3x2(3.5)\nWe can easily verify that it solves the Einstein equations exactly. The inverse metric is given\nby\ngµν(0)=ηµν−Λ\n3xµxν(3.6)\nWe can compute the associated connection\nΓµ(0)\nαβ=Λ\n3xµg(0)\nαβ (3.7)\nand the complete curvature tensor\nRα(0)\nβµν=∂µΓα(0)\nβν+ Γα(0)\nσµΓσ(0)\nβν−(µ↔ν) =\n=Λ\n3(δα\nµg(0)\nβν−δα\nνg(0)\nβµ) (3.8)\nAnalogously the Ricci tensor and the curvature have simple forms\nR(0)\nβν= Λg(0)\nβνR(0)= 4Λ (3.9)\nsolving exactly the Einstein equations. Let us note that this solution ( similarly to those\nwith an arbitrary function f(Λx2) ) is singular for large events of the order\nx2≃1\nΛ(3.10)\nThis singularity is present both for positive and negative cosmologica l constant.\n6It is possibile fixing the arbitrary function f(Λx2) adding into this scheme the Lorentz\ngauge condition\nηµµ′∂µ′hµν=1\n2∂ν(ηµµ′hµµ′) (3.11)\nBy substituting the definition of the metric in terms of the mapping Xa=Xa(xi) we get\nηabηµµ′∂µ∂µ′Xa·∂νXb= 0 (3.12)\nThe variables Xaare constrained leading to the property ηabXa·∂νXb= 0, therefore\nwe must impose that\nηµµ′∂µ∂µ′Xa=λ(Λx2)Xa (3.13)\nwhere we have introduced a second function λ(Λx2). We are going to show that this\nequation (3.13) determines univocally both functions f(Λx2) andλ(Λx2).\nLet us introduce the following power series\nλ(Λx2) =∞/summationdisplay\nn=1λnΛnx2(n−1)\nf(Λx2) = 1 +∞/summationdisplay\nn=1fnΛnx2n(3.14)\nThe coefficients λnare determined by the eq. for X0while the coefficients fnare deter-\nmined by the eq. for Xiin a recursive scheme. We start from the lowest order\nX0≃/radicalbigg\n3\nΛ/parenleftbigg\n1−Λ\n6x2/parenrightbigg\n→λ1=−4\n3(3.15)\nThe coefficient λ1allows to compute the coefficient f1by using the equation for Xi\nηρρ′∂ρ∂ρ′Xi≃ηρρ′∂ρ∂ρ′(xi·f1Λx2) =−4\n3Λxi (3.16)\nfrom which we obtain\nf1=−1\n9→f(Λx2) = 1−Λ\n9x2+O(Λ2x4) (3.17)\n7Going back to the eq. for X0at the next order in Λ, we can use f1to determine λ2as\nfollows\nX0≃/radicalbigg\n3\nΛ/parenleftbigg\n1−Λ\n6x2+5\n9·24Λ2x4+O(Λ3x6)/parenrightbigg\n(3.18)\nfrom which we compute λ2=1\n3and therefore\nλ(Λx2) =−4\n3Λ+1\n3Λ2x2+O(Λ3x4) (3.19)\nRepeating the iteration, the coefficient λ2determines f2at the next order in Λ by using\nthe equation for Xi\nf2=13\n27·32(3.20)\nfrom which we know Xiup to the second order in Λ\nXi≃xi/parenleftbigg\n1−Λ\n9x2+13\n27·32Λ2x4+O(Λ3x6)/parenrightbigg\n(3.21)\nBy substituting these findings in the equation (3.4), defining the met ric in terms of the\nmapping Xa(xi)\ngµν=ηabdXa\ndxµdXb\ndxν=ηµν+h(1)Λ\nµν+h(2)Λ\nµν+.... (3.22)\nwe recover the perturbative solutions (2.1) e (2.22)\nh(1)Λ\nµν=−Λ\n9(xµxν+2ηµνx2)\nh(2)Λ\nµν=11\n16·81Λ2( 4xµxνx2+5ηµνx4) (3.23)\nobtained with a very laborious calculation from the Einstein equations .\n4 Conclusion\nTheAdSspace-time is an example of curved space-time in the absence of mat ter. In this\npaper, we have searched how to describe it with a Lorentz covarian t coordinate system,\n8which is better suited for discussing the propagation of gravitation al waves in such a curved\nspace.\nWe first built the perturbative solution of the Einstein equations with a cosmological\nconstant Λ intheLorentz gauge, andthenintroduced amoresyst ematic methodtoconstruct\nthe solution by embedding the AdSspace-time into a 5 dspace with a quadratic constraint\nbetween the coordinates.\nWe have succeeded to identify the right mapping Xa=Xa(xi) which allows us to reach\nthe Lorentz gauge and constructed an iterative method to calcula te theAdSsolution to the\nvarious perturbative orders in Λ. This method is the easiest way to s olve the Lorentz gauge\ncondition if compared to the direct perturbative calculation of the E instein equations.\nOnce the structure of the AdSspace-time has been accurately identified in a Lorentz\ncovariant coordinate system, we believe that it will be easier to stud y the propagation of a\ngravitational wave in the presence of such a background.\nReferences\n[1] Planck collaboration, Astronomy and Astrophysics 594(2016) A13,\narXiv:1502.01589v3.\n[2] Frieman J. A., Turner M. S.and Huterer D., Annal Review of Astron omy and Astro-\nphyics46(2008) 385, arXiv:0803.0982.\n[3] Weinberg S., Phys. Rev. Lett. 59(1987) 2607.\n[4] Ellis G. F. R., Hawking S. W. (1973) ”The large scale structure of sp ace-time”, Cam-\nbridge University Press pp. 124-134.\n[5] Bernabeu J., Espriu D. and Puigdomenech D., Phys. Rev. D 84(2012) 063523,\narXiv:1106.4511 [hep-th].\n9"    },    {        "title": "1502.00431v1.Evolution_of_Higgs_mode_in_a_Fermion_Superfluid_with_Tunable_Interactions.pdf",        "content": "arXiv:1502.00431v1  [cond-mat.quant-gas]  2 Feb 2015Evolution of Higgs mode in a Fermion Superfluid with Tunable I nteractions\nBoyang Liu,1Hui Zhai,1and Shizhong Zhang2\n1Institute for Advanced Study, Tsinghua University, Beijin g, 100084, China\n2Department of Physics and Center of Theoretical and Computa tional Physics,\nThe University of Hong Kong, Hong Kong, China\n(Dated: October 4, 2018)\nIn this letter we present a coherent picture for the evolutio n of Higgs mode in both neutral\nand charged s-wave fermion superfluids, as the strength of attractive int eraction between fermions\nincreases from the BCS to the BEC regime. In the case of neutra l fermionic superfluid, such as\nultracold fermions, the Higgs mode is pushed to higher energ y while at the same time, gradually\nloses its spectral weight as interaction strength increase s toward the BEC regime, because the\nsystem is further tuned away from Lorentz invariance. On the other hand, when damping is taken\ninto account, Higgs mode is significantly broadened due to co upling to phase mode in the whole\nBEC-BCS crossover. Inthechargedcase ofelectron supercon ductor, theAnderson-Higgsmechanism\ngaps out the phase mode and suppresses the coupling between t he Higgs and the phase modes, and\nconsequently, stabilizes the Higgs mode.\nThe experimental search for Higgs boson in particle\nphysics has made remarkable progresses [1, 2]. On the\nother hand, Higgs mode has also generated considerable\ninterest in condensed matter and cold atom systems.\nEarly in 1980s’, Raman scattering experiment has re-\nvealed an unexpected peak in a superconducting charge\ndensity wave compound NbSe 2[3], which was later at-\ntributed to the Higgs mode [4, 5]. Signal of Higgs mode\nhas also been observed in antiferromagnet TlCuCl 3by\nthe neutron scattering [6], and recently in superconduct-\ning NbN sample by terahertz pump probe spectroscopy\nin a nonadiabatic excitation regime [7, 8]. In cold atom\nsystem, Higgs mode has been observed near the super-\nfluid to Mott insulator phase transition of bosonic atoms\nin optical lattices at integer filling [9, 10].\nTheoretically, the simplest field theory where Higgs\nmode emerges is a relativistic U(1) field theory with\nLorentz invariance in the symmetry broken phase. This\noccurs, for example, in the weak coupling BCS super-\nconductor [11, 12] or in the Mott-superfluid transition of\nBose-Hubbard model at integer filling [13, 14]. However,\nin most condensed matter systems, Lorentz invariance\nonly emerges with fine tuning and the generic symmetry\nis usually Galilean [15]. Thus, it is an interesting ques-\ntion to investigate how the Higgs mode evolves as the\nsystem is tuned away from the Lorentz invariance point.\nMoreover, in condensed matter systems, further compli-\ncations often occur because the Higgs mode is usually\ncoupled to other elementary excitations which leads to\nits damping [16–19]. In this Letter, we investigate these\nissues in the context of the BEC-BCS crossover model.\nIn the BCS limit, the system obeys approximate Lorentz\nsymmetry due to particle-hole symmetry and is expected\nto host Higgs mode. In the BEC limit, it is a condensate\nof molecular bosons and obeys the Galilean invariance.\nIt thus provides a unique system to describe the fate of\nHiggs mode as the system is tuned away from Lorentz\ninvariant limit. In addition, due to tunable interactions/Minus4/Minus3/Minus2/Minus10120100200300400500600700\nΖ/Equal1/Slash1kFas/Minus4/Minus3/Minus2/Minus1000.050.1\nΖ/Equal1/Slash1kFasv''/CapDelta0/Slash1u''\nu''/Slash1u'v'/CapDelta0/Slash1u'\nFIG. 1: (Color online) v′∆0/u′andu′′/u′as functions of the\nscattering length ζ= 1/kFas. In the inset we show v′′∆0/u′′\nas a function of ζ.\nin the BEC-BCS crossover, it also provides a great plat-\nform to investigate the interaction effects on the Higgs\nmode due to coupling to collective and quasi-particle ex-\ncitations [20].\nWe investigate these questions based on the time-\ndependent Ginzburg-Landau formulation of the BEC-\nBCS crossover,\nS=/integraldisplay\ndtd3x/bracketleftbig\nφ∗(−iu∂t+v∂2\nt−∇2\n2m∗−r)φ+b\n2|φ|4/bracketrightbig\n,(1)\nwhereφis the Ginzburg-Landau order parameter. The\nvarious parameters u,v,r,b andm∗can be computed\nalongBEC-BCScrossoverintermsofthechemicalpoten-\ntialµ, temperature Tandζ= 1/(kFaS), whereaSis the\ns-wave scattering length. Within the Nozi` eres-Schmit-\nRink [21] framework, this can be calculated as detailed\nin the supplementary material [22]. The coefficients of\nthe time derivative terms u=u′+iu′′andv=v′+iv′′\nare complex in general. The real parts u′andv′describe2\nthe propagating behavior of the cooper pair field, while\nthe imaginary parts u′′andv′′describe its damping due\nto coupling to the fermionic quasi-particles. A plot of\nvarious parameters are given in Fig.1. We note the fol-\nlowing features.\n(i) Consider the real parts u′andv′in the BEC-BCS\ncrossover. In the BCS limit, u′/v′∆0→0 because of\nthe approximate particle-hole symmetry in the weak-\ncoupling BCS theory while ∆ 0=/radicalbig\nr/bis the mean field\nvalue of order parameter. As a result, the system ac-\nquires an emergent Lorentz invariance, and one expects\nthe emergenceofHiggs mode, togetherwith the standard\nAnderson-Bogoliubov mode for neutral fermion super-\nfluid. In the BEC limit, however, v′∆0/u′∼∆0/|µ| ≪1,\nand we can neglect the v′-term. This leads to a Galilean\ninvariantneutralbosontheory,forwhichonlyBogoliubov\nmode exists.\n(ii) The damping terms ( u′′) becomes important as\none moves to the BCS side, because of the decreasing\nfermionic excitation gap and as a result, a stronger cou-\npling of the pairing field to the quasi-particle excitations.\nThis corresponds to finite lifetime of Cooper pairs at fi-\nnite temperature. We will show that the damping u′′-\nterm itself will generate considerable effect for the ap-\npearance of the Higgs mode different from that in a pure\nLorentz invariance theory. In the BEC limit, the imagi-\nnary parts vanishes within NSR. On the other hand, we\nfind that whenever they are nonzero, v′′∆0/u′′≪1 for\nthe entire crossover regime and we shall thus neglect v′′-\nterm altogether in the following discussion.\nSpectral Weight Transfer without Damping. To inves-\ntigate the evolution of Higgs mode as the system is tuned\ngradually from its Lorentz-invariant BCS limit towards\nthe Galilean invariant BEC limit, we shall first neglect\nthedampingtermsinEq.4andstudythetransferofspec-\ntral weight between the Higgs and Goldstone modes. In\nthe symmetry broken state, we can write the order pa-\nrameterφ= ∆0+δa+iδp, whereδaandδpdescribe\namplitude and phase fluctuations, respectively. In terms\nofδaandδpand withu′′=v′′= 0, we can write the\naction Eq. 4 in the Fourier space as\nS=/integraldisplaydω\n2πd3k\n(2π)3¯Φ(−ω,−k)G−1Φ(ω,k),(2)\nwith¯Φ(ω,k) = (δa(ω,k),δp(ω,k)) and the kernel Gis\ngiven by\nG−1=/parenleftbigg−v′ω2+ξk+2r iu′ω\n−iu′ω−v′ω2+ξk/parenrightbigg\n,(3)\nwithk=|k|andξk=k2/2m∗. Two branches of spec-\ntrum can be identified, with mode frequencies given by\nω2\n±=ξk+r\nv′+u′2\n2v′2±/radicalbigg\nr2\nv′2+u′4\n4v′4+u′2\nv′3(ξk+r).(4)\nIn the BCS limit, v′∆0≫u′and solutions can be writ-\nten asω−(k) =k/√\n2m∗v′andω+(k) =/radicalbig\n(ξk+2r)/v′;\nFIG. 2: (Color online) Spectral function Aaa(k,ω) in the ab-\nsence of damping term. Aaa(k,ω) as a function of k(in unit\nof 1/ξ) andω(in unit of ∆ 0) for three different interaction\nstrength ζ=−1/(kFas),ζ=−7 for (a), ζ=−3 for (b)\nandζ=−1 for (c), corresponding to different gaps ∆ 0/EF=\n10−5, ∆0/EF= 4×10−3and ∆ 0/EF= 7×10−2, respectively.\n(a2-c2): Aaa(k,ω) as a function of ωfork= 0.1/ξ(purple\ndashed line) and k= 0.01/ξ(blue solid line). T/Tc= 0.9 and\nδis taken as 10−4∆0.\nthe first being the Goldstone mode with linear dis-\npersion and the second Higgs mode, with Higgs gap\nω+(0) =/radicalbig\n2r/v′=/radicalbig\n2b∆2\n0/v′. Using the facts that\nv′= 7β2ζ(3)ν0/16π2andb= 7β2ζ(3)ν0/8π2in the BCS\nlimit, one finds ω+(0) = 2∆ 0, as expected for a Lorentz-\ninvariant theory. Here β= 1/kBTis the inverse temper-\nature,ζ(n) is the Riemann-Zeta function and ν0is the\ndensity of state at the Fermi energy ǫF.\nIn the BEC limit, u′≫v′∆0, we find ω−(k) =/radicalbig\nξk(ξk+2r)/u′is the Bogoliubov mode while the other\nmodeω+(k) =/radicalbig\n2ξk/v′+2r/v′+(u′/v′)2has a gap\n∼ |µ|, of order of binding energy of the molecule in the\nBEC limit. The existence of the gapped mode is a reflec-\ntion of the fact that our bosonic field φis a composite of\ntwo fermions and disappears in the infinite binding limit\nwhere only Bogoliubov mode exists as it should.\nIn between these two limits, Lorentz invariance is bro-\nken and the coupling between the amplitude and phase\ndegrees of freedom becomes stronger, as characterized\nby the off-diagonal term iu′ω. We note that for low en-\nergy Bogoliubov excitations, such coupling is small, but\nfor gapped Higgs mode, it provides significant mixing of\nthe amplitude and phase. To characterize such mixing,\nwe calculate the spectral function for the amplitude δa,\ngiven byAaa(k,ω) =−1\nπImGaa(k,ω+iδ). Explicitly,\nthis can be written as\nAaa(k,ω) =A+(k)δ(ω−ω+(k))+A−(k)δ(ω−ω−(k))\n(5)\nwithA+(k) andA−(k) being the spectral weight den-3\nsities associated with two modes ω−andω+[22]. In\nFig.2 (a,b,c), we plot the spectral function Aaa(k,ω) for\nthree representative values of ζ(corresponding to differ-\nent ∆0/EF). Two features can be noticed immediately.\nFirst, the Higgs gap increases beyond 2∆ 0of the BCS\nlimit as interaction strength increases. Secondly, there is\nincreasingspectralweighttransferfromthegappedHiggs\nmode to the gapless mode. One can show explicitly that\nA+/A−= 4v′r2(u′2/radicalbig\nk2/2m∗(k2/2m∗+2r))−1, which\nindicates the gradual increasing of the mixing between\nphase and amplitude degrees of freedom.\nIncluding Damping Term. Due to the presence of\ndamping term, the time-dependent Ginzburg-Landau\ntheory is not a pure Lorentzinvariant U(1) theory. Thus,\natanyfinitetemperature, evenintheBCSlimit, thepeak\nof Higgs excitation ω+(k) will not as sharp as discussed\nabove. To calculate the equilibrium spectral weight in\nthe presence of damping, we need to introduce the so-\ncalled Langevin force η(t,x), which satisfies the follow-\ning conditions, /angb∇acketleftη(t′,x′)η(t,x)/angb∇acket∇ight=/angb∇acketleftη∗(t′,x′)η∗(t,x)/angb∇acket∇ight= 0,\nand/angb∇acketleftη∗(t′,x′)η(t,x)/angb∇acket∇ight= 2u′′kBTδ(t−t′)δ(x−x′). In-\ncluding the corresponding term in the action as SL=/integraltextdtd3x(φ∗η+φη∗), we obtain the equations of mo-\ntion forδaandδp, by setting ∂(S+SL)/∂δa= 0 and\n∂(S+SL)/∂δp= 0,\n/parenleftbig\n−vω2+ξk+2r/parenrightbig\nδa−iuωδp+η′= 0,(6)\n/parenleftbig\n−vω2+ξk/parenrightbig\nδp+iuωδa+η′′= 0,(7)\nwhereη′andη′′are the real and imaginary parts of the\nLangevin force η, respectively. The spectral functions for\nthe amplitude fluctuation is given by, using fluctuation\ndissipation theorem,\nAaa=u′′ω\n2|−vω2+ξk|2+|uω|2\n|−(uω)2+(−vω2+ξk)(−vω2+ξk+2r)|2.\n(8)\nBy comparing Fig. 3 with Fig. 2, one can see three\nimportantfeaturesbroughtaboutbyincludingthedamp-\ning term. First, the spectral weight transfer is enhanced.\nFor instance, for ζ=−7, there is almost no spectral\nweight transfer in the absence of damping (Fig. 2(a))\nwhile in the presence ofdamping, for verysmall k≪1/ξ,\nAaa(k,ω) exhibits a clear peak at the energy of Bogoli-\nubov mode, with a weight proportional to u′′[22]. Sim-\nilar enhancement of spectral weight transfer can also be\neasily seen in Fig. 3(b) for ζ=−3. Secondly, also for\nk≪1/ξ, in the BCS limit, the location of Higgs peak is\nsubstantially reduced from/radicalbig\n2r/v′to/radicalbig\n2r/v′−u′′2/v′2,\nas shown for ζ=−7 and−3 in Fig. 3(a) and (b), re-\nspectively [22]. Thirdly, as kstarts to derivate from zero,\nthe Higgs mode quickly loses its identity, due to strong\nhybridization with the Bogoliubov mode. For instance,\neven fork= 0.1/ξ, as displayedby the purple dashed line\nin Fig. 3(a2-c2), no feature of sharp peak is observed in\nFIG. 3: (Color online) Spectral function Aaa(k,ω)in presence\nof damping term. Aaa(k,ω) as a function of k(in unit of 1 /ξ)\nandω(in unit of ∆ 0) for three different interaction strength\nζ=−1/(kFas),ζ=−7for (a), ζ=−3for (b) and ζ=−1for\n(c), corresponding to different ∆ 0/EF= 10−5, ∆0/EF= 4×\n10−3and ∆ 0/EF= 7×10−2, respectively. (a2-c2): Aaa(k,ω)\nas a function of ωfork= 0.1/ξ(purple dashed line) and\nk= 0.01/ξ(blue solid line). T/Tc= 0.2 andδis taken as\n10−4∆0.\nAaa(k,ω). And for ζ=−1, no sharp peak exists even\nfork= 0.01/ξ.\nEffects of Coupling to External Gauge Fields . Now we\nunderstand that, in the weakly interacting BCS side of a\nneutral superfluid, the appearance of Higgs mode suffers\nsignificant broadeningdue to finite u′′-term at finite tem-\nperature, which couples the Higgs mode to the collective\nBogoliubov excitations. Therefore, if we further consider\nthe presence of coupling to external electromagnetic field\nfor the case of charged fermions, the Bogoliubov mode\nis gapped out by the Anderson-Higgs mechanism. Thus,\nwe expect that the Higgs mode is easier to observe in\nthe charged case. To incorporate this effect of external\nelectromagnetic field, we introduce the gauge potential\nϕ(t,x) and extend the action as\nSc=/integraldisplay\ndtd3x/braceleftbig\nφ∗[−iu(∂t−2eϕ)+v(∂t−2eϕ)2(9)\n−∇2\n2m∗−r]φ+b\n2|φ|4−1\n8πϕ∇2ϕ/bracerightbig\n,\nwhereeis the charge of the electron. Following the same\nprocedure as before, we find that the coupling between\nδaandδpis modified and is now proportional to k2\ni2uωk2\nk2+32πve2∆2\n0δa(ω,k)δp(−ω,−k).(10)\nAs a result, the original gapless phase mode is gapped to\na finite frequency, ω(k) =/radicalbig\nξk/v′+16πe2∆2\n0/m∗, which\nis known as the Anderson-Higgs mechanism. Thus, the4\nFIG. 4: (Color online) Spectral function Aaa(k,ω) for the\ncharged case. Aaa(k,ω) as a function of k(in unit of 1 /ξ)\nandω(in unit of ∆ 0) for three different interaction strength\nζ=−1/(kFas),ζ=−7 for (a), ζ=−3 for (b) and ζ=−1for\n(c), corresponding to different ∆ 0/EF= 10−5, ∆0/EF= 4×\n10−3and ∆ 0/EF= 7×10−2, respectively. (a2-c2): Aaa(k,ω)\nas a function of ωfork= 0.1/ξ(purple dashed line) and\nk= 0.01/ξ(blue solid line). T/Tc= 0.9 andδis taken as\n10−4∆0.\nlarge energy separation between this gapped phase mode\nand Higgs mode strongly suppresses their coupling. A\nfurther consequence of the modification is that at long\nwave length k→0, the coupling between phase and\namplitude mode becomes small. The spectral function\nAaa(k,ω) for charged case is plotted in Fig. 4 [22]. In\nsharp contrast to neutral case Fig. 3, the presence of\ndamping term has almost no effect on Higgs mode, and\nthere is always a peak located at ω= 2∆0. In this case,\nas attractive interaction increases and the system gradu-\nally loses its Lorentz invariance, the peak becomes more\nand more broad.\nConclusion. Insummary, wehaveinvestigatedtheevo-\nlution of Higgs mode in the BEC-BCS crossover for both\nneutral and charged Fermi superfluid. Our main con-\nclusions include: i) Towards the BEC side, as the sys-\ntem gradually loses the Lorentz invariance, the Higgs\nmode is pushed to very high energy and the spectral\nweight is transferred to Bogoliubov mode. ii) In the BCS\nside, damping terms arises in the Ginzburg-Landau the-\nory, due to coupling between Cooper pair field and the\nfermionic quasi-particles, and strongly couples the Higgs\nmodetothegaplessphasemodein theneutralsuperfluid,\nwhich enhances the spectral weight transfer and washes\nout features of Higgs mode at finite momentum. (iii)\nFor the charged case, the phase mode is gapped out by\ncoupling to external electromagnetic field, and the Higgs\nmode becomes much more stable.\nOur results also deepen our understandings of Higgs\nmode in superconductor. The physical picture behindthe observation of Higgs mode in a BCS superconduc-\ntor is much more subtle and its observability is not\nmerely guaranteed by Lorentz symmetry. While the\ndamping terms broadens the Higgs peak, the Anderson-\nHiggs mechanism alleviate the coupling between Higgs\nand phase mode and as a result, Higgs mode remains\nat energy 2∆ 0. As for cold atom system, because of the\ncooling limit, so far we can not reachFermi superfluid for\nζ <−1. However, our results show no Higgs feature in\nspectralfunction for ζ >−1. On the otherhand, with re-\ncentdevelopmentofsyntheticgaugefield, therearemany\nproposals to generate a synthetic dynamic gauge field in\ncold atom system [29]. If such a dynamic gauge field can\nbe experimentally realized and coupled to fermions, the\nAnderson-Higgsmechanism will be activated and a Higgs\nmode will be observed. This can be used as a way to test\nour theory.\nAcknowledgements. BY and HZ are supported by\nTsinghua University Initiative Scientific Research Pro-\ngram, NSFC Grant No. 11174176, No. 11325418 and\nNKBRSFC under Grant No. 2011CB921500. SZ is sup-\nportedbyastart-upgrantfromUniversityofHongKong,\nthe Collaborative Research Fund HKUST3/CRF/13G\nandrgc-grf 17306414. HZwouldliketothankHongKong\nUniversity for hospitality where this work was initiated.\n[1] CMS collaboration, Phys. Lett. B 716, 30 (2012).\n[2] ATLAS collaboration, Phys. Lett. B 716, 1 (2012).\n[3] R. Sooryakumar, and M. V. Klein, Phys. Rev. Lett. 45,\n660 (1980).\n[4] P.B. Littlewood and C.M. Varma, Phys. Rev. B 26, 4883\n(1982).\n[5] P.B. Littlewood and C.M. Varma, Phys. Rev. Lett. 47,\n811 (1981).\n[6] Ch. R¨ uegg, B. Normand, M. Matsumoto, A. Furrer, D.F.\nMcMorrow, K.W. Kramer, H.U. Gudel, S.N. Gvasaliya,\nH. Mutka, and M. Boehm, Phys. Rev. Lett. 100, 205701\n(2008).\n[7] R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H.\nTerai, Z. Wang, and R. Shimano, Phys. Rev. Lett. 111,\n057002 (2013).\n[8] R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise,\nY. Uzawa, H. Terai, Z. Wang, H. Aoki and R. Shimano,\nScience345, 1145 (2014).\n[9] U. 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Juzeli¯ unas, P. ¨Ohberg and I B Spielman,\nReports on Progress in Physics, 77126401 (2014)\nSUPPLEMENTARY MATERIALS\nTime-dependent Ginzburg-Landau theory of BEC-BCS crossov er\nA time-dependent Ginzburg-Landau theory can be constructed f or the entire BEC-BCS crossover in the vicinity of\nTc[1]. The partition function takes the form Z=/integraltext\nD[¯ψσ,ψσ]e−S[¯ψσ,ψσ], with\nS[¯ψσ,ψσ] =/integraldisplay\ndτd3x/braceleftBig\n¯ψσ(∂τ−∇2\n2m−µ)ψσ−g¯ψ↑¯ψ↓ψ↓ψ↑/bracerightBig\n, (1)\nwhereψσare Grassman fields and gis the contact interaction between fermions of opposite spins. µis the chemical\npotential which is determined by requiring the number density to be e qual ton. To investigate the fluctuation effects\nin the Cooper channel, we use a Hubbard-Stratonovich transform ation to decouple the interaction term in the Cooper\nchannel and then integrating out the fermions. We obtain an effect ive theory for the bosonic field ∆( τ,x), which\nrepresents the cooper pair field. Straightforward calculations yie ld the partition function in terms of field ∆ as\nZ=/integraldisplay\nD(¯∆,∆)exp/bracketleftBig\n−1\ng/integraldisplay\ndτdx|∆|2+lndet ˆG−1/bracketrightBig\n, (2)\nwhere\nˆG−1=/parenleftBigg\n−∂τ+∇2\n2m+µ ∆\n¯∆ −∂τ−∇2\n2m−µ/parenrightBigg\n(3)\nis the Gor’kov Green function.\nIn the vicinity of the phase transition the gap parameter ∆ is small an d an expansion in terms of ∆ becomes\npossible. Including both the spatial and time derivatives (after Wick rotation) and retaining the parameter ∆ up to\nthe forth order we obtain an effective action as\nS[¯∆,∆] =/integraldisplay\ndtd3x/braceleftBig\n¯∆/bracketleftbig\n−iu∂t+v∂2\nt−∇2\n2m∗−r/bracketrightbig\n∆+b\n2¯∆¯∆∆∆/bracerightBig\n, (4)\nwhereu=u′+iu′′andv=v′+iv′′are complex in general and all the parameters can be expressed in t erms of2\nmicroscopic parameters as\nu′=(2m)3/2\n16π2/bracketleftBigg\n2√\n2β/radicalbig\n|µ|\nπ∞/summationdisplay\nn=0/radicalbigg/radicalBig\n1+(2n+1)2(π\nβµ)2−sgn(µ)\n(2n+1)2−πβ\n2/radicalbig\n|µ|θ(−µ)/bracketrightBigg\n, (5)\nu′′=m3/2\n8√\n2πβ/radicalbig\n|µ|Θ(µ), (6)\nv′=(2m)3/2\n32π2/bracketleftBigg\n2√\n2β2/radicalbig\n|µ|\nπ2∞/summationdisplay\nn=0/radicalbigg/radicalBig\n1+(2n+1)2(π\nβµ)2+sgn(µ)\n(2n+1)3−πβ\n4/radicalbig\n|µ|θ(−µ)/bracketrightBigg\n, (7)\nv′′=−m3/2\n32√\n2πβ/radicalbig\n|µ|Θ(µ), (8)\n1\n2m∗=1\n2m/integraldisplayd3k\n(2π)3/braceleftBigg\n1−2N(ξk)\n8ξ2\nk+∂N(ξk)\n∂ξk\n4ξk+∂2N(ξk)\n∂ξ2\nk·k2\n2m\n6ξk/bracerightBigg\n, (9)\nr=m\n4πa+/integraldisplayd3k\n(2π)3/braceleftBigg\n1−2N(ξk)\n2ξk−1\n2ǫk/bracerightBigg\n, (10)\nb=/integraldisplayd3k\n(2π)3/braceleftBigg\n1−2N(ξk)\n4ξ3\nk+βN(ξk)[N(ξk)−1]\n2ξ2\nk/bracerightBigg\n. (11)\nIn the above equations, N(ξk) = 1/(exp(βξk)+1) is the Fermi distribution function and ξk=ǫk−µwithǫk=k2/2m.\nFunction Θ(2 µ) is the heaviside step function. Explicitly, the parameter bis the result of one-loop calculation with\nfour fermion propagators\nb=−1\nβ2/summationdisplay\nωn/integraldisplayd3k\n(2π)31\n(−iωn+k2/2m−µ)21\n(iωn+k2/2m−µ)2. (12)\nThe other parameters u,v,1\n2m∗andrare all derived from the inverse vertex function Γ−1(ωn,k), which after the\nstandard renormalization by replacing gwith the two-body scattering length as, is given by\nΓ−1(ωn,k) =−m\n4πas−/integraldisplayd3k\n(2π)3/braceleftbigg1−N(ǫk−µ)−N(ǫk−q−µ)\n−iωn+ǫk+ǫk−q−2µ−1\n2ǫk/bracerightbigg\n. (13)\nTo derive the time-dependent Ginzburg-Landau equation, we first analytically continue vertex function to real fre-\nquencyiωn→ω+i0+. This procedure generates a time-dependent term with paramete ruandv. The detailed\nderivation is as following.\nThe frequency dependent part of Γ−1(ω,k) is\nΓ−1(ω,0)−Γ−1(0,0) =−m\n4πa−/integraldisplayd3k\n(2π)3/braceleftBigg\n1−2N(ǫk−µ)\n−ω−iη+2ǫk−2µ−1\n2ǫk/bracerightBigg\n−Γ−1(0,0). (14)\nThen we expand it in series of small ωas\nΓ−1(ω,0)−Γ−1(0,0)≃ −ω·/integraldisplayd3k\n(2π)31−2N(ǫk−µ)\n(2ǫk−2µ−iη)2−ω2·/integraldisplayd3k\n(2π)31−2N(ǫk−µ)\n(2ǫk−2µ−iη)3. (15)\nWe define the parameters as u≡/integraltextd3k\n(2π)31−2N(ǫk−µ)\n(2ǫk−2µ−iη)2andv≡/integraltextd3k\n(2π)31−2N(ǫk−µ)\n(2ǫk−2µ−iη)3. They both can be calculated by\ncontour integration.\nu≡/integraldisplayd3k\n(2π)31−2N(ǫk−µ)\n(2ǫk−2µ−iη)2\n=ν(ǫF)\n4√ǫF/integraldisplay∞\n0dǫ√ǫ1−2N(ǫ−µ)\n(ǫ−µ−iη)23\n=ν(ǫF)\n4√ǫF·1\n2/integraldisplay\ncdz√z1−2N(z−µ)\n(z−µ−iη)2, (16)\nwherecdenotes the contour in Fig. 1. There are infinite first-order poles zn=µ+(2n+1)πi\nβand one second order\npolezη=µ+ηi. The contour integration can be evaluated in the summation of the r esiduals as\nFIG. 1: The contour “c” in the calculation of Eq. (16). The dot s “·” denote the first-order poles zn=µ+(2n+1)πi\nβand the\ncross “×” denotes the second order pole zη=µ+ηi.\n/integraldisplay\ncdz√z1−2N(z−µ)\n(z−µ−iη)2\n= 2πi/bracketleftBigg\nlim\nz→zn2√z/β\n(z−µ−ηi)2+ lim\nz→zηd\ndz/bracketleftBig√z/parenleftbig\n1−2\nexp(β(z−µ))+1/parenrightbig/bracketrightBig/bracketrightBigg\n= 2πi/bracketleftBigg\n−2β√µ\nπ2+∞/summationdisplay\nn=−∞/radicalbig\n1+(2n+1)πi/βµ\n(2n+1)2+√µβ\n2/bracketrightBigg\n. (17)\nCalculation shows that/summationtext+∞\nn=−∞√\n1+(2n+1)πi/βµ\n(2n+1)2is pure imaginary due to the symmetry of the znpole\nlocations with respect to the horizontal axes. Then it can be writte n as/summationtext+∞\nn=−∞√\n1+(2n+1)πi/βµ\n(2n+1)2=\n√\n2i/summationtext+∞\nn=0/radicalBig√\n1+((2n+1)π/βµ)2−sgn(µ)\n(2n+1)2.Hence, the parameter uis calculated as\nu=(2m)3/2\n16π2/bracketleftBigg\n2√\n2β/radicalbig\n|µ|\nπ∞/summationdisplay\nn=0/radicalbigg/radicalBig\n1+(2n+1)2(π\nβµ)2−sgn(µ)\n(2n+1)2−πβ\n2/radicalbig\n|µ|θ(−µ)+iπβ\n2/radicalbig\n|µ|θ(µ)/bracketrightBigg\n.(18)\nIn the same manner, the parameter vcan also be calculated as shown in Eq. (6) and (7). We should note tha t while\nthe expressions for uand others look different from the standard expression, as given in ref. [1], they in fact reduce\nto the same expressions. We found that this form is more convenien t to use the above expression when dealing with\nhigher order time-derivative terms.\nIn the BCS and BEC limits all the parameters can be analytically derived as shown in Table I.\nSpectral weight function in the case without damping term\nIf we ignore the damping term by taking u′′= 0 the action can be written as\nS=/integraldisplaydω\n2πd3k\n(2π)3¯Φ(−ω,−k)G−1Φ(ω,k), (19)4\nParameters BCS limit BEC limit\nu′0πν(ǫF)\n8√\nǫF|µ|\nu′′ν(ǫF)·π\n8kBT0\nv′ 7ν(ǫF)\n16π2(kBT)2·ζ(3)πν(ǫF)\n64√ǫF|µ|3/2\nv′′−ν(ǫF)·π\n32kBTǫF0\n1\n2m∗1\n2m·7ν(ǫF)ǫF\n12π2(kBT)2ζ(3)1\n2m·πν(ǫF)\n16√\nǫF|µ|\nr ν(ǫF)lnTc\nTπν(ǫF)\n2√\n2√ǫF·(1√mas−/radicalbig\n2|µ|)\nb7ν(ǫF)\n8π2(kBT)2·ζ(3)πν(ǫF)\n32√ǫF|µ|3/2\nTABLE I: Asymptotic behaviors of the parameters in the time- dependent Ginzburg-Landau theory in the BCS and BEC limits.\nwith¯Φ(ω,k) = (δa(ω,k),δp(ω,k)) and the kernel Gis given by\nG−1=/parenleftbigg\n−v′ω2+ξk+2r iu′ω\n−iu′ω−v′ω2+ξk/parenrightbigg\n, (20)\nwithk=|k|andξk=k2/2m∗. Then the amplitude-amplitude correlation function can be easily calc ulated as\nGaa(ω,k) =−v′ω2+ξk\n−u′2ω2+(−v′ω2+ξk)(−v′ω2+ξk+2r). (21)\nStraight forward calculation yields the spectral function as\nAaa(ω,k) =−1\nπImGaa(ω+iδ,k)\n=A+(k)δ(ω−ω+(k))+A−(k)δ(ω−ω−(k)), (22)\nwhere the mode frequencies are given as\nω2\n±=ξk+r\nv′+u′2\n2v′2±/radicalbigg\nr2\nv′2+u′4\n4v′4+u′2\nv′3(ξk+r) (23)\nand the spectra weight density\nA+(k) =v′ω2\n+−ξk\n2v′2ω+(ω2\n+−ω2\n−),\nA−(k) =−v′ω2\n−+ξk\n2v′2ω−(ω2\n+−ω2\n−). (24)\nAt BCS limit the ratio of the two spectral weight densities can be appr oximately calculated as\nA−(k)\nA+(k)=u′2/radicalbig\nk2/2m∗(k2/2m∗+2r)\n4v′r2. (25)\nAt BCS limit we have u′/v′→0, this ratio vanishes. This spectral weight transfer is shown in Fig. 2 in the main\ntext.\nSpectral weight function in the case with damping term\nThe spectral weight function of the amplitude mode in the case with d amping term u′′is\nAaa=u′′ω\n2·|−vω2+k2\n2m∗|2+|uω|2\n|−(uω)2+(−vω2+k2\n2m∗)(−vω2+k2\n2m∗+2r)|2\n=u′′ω\n2·|−vω2+k2\n2m∗|2+|uω|2\n|v′2(ω2−˜ω2\n+)(ω2−˜ω2\n−)−2iu′u′′ω2|2, (26)5\nwhere the eigen mode frequencies are\n˜ω2\n±=ξk+r\nv′+u′2−u′′2\n2v′2±/radicalbigg\nr2\nv′2+(u′2−u′′2)2\n4v′4+u′2−u′′2\nv′3(ξk+r). (27)\nFor small momentum they can be approximated as\n˜ω−=/radicalbigg\n2rξk\n2v′r+u′2−u′′2,\n˜ω+=/radicalBigg\n2v′r+u′2−u′′2\nv′2+2v′r+2u′2−2u′′2\nv′(2v′r+u′2−u′′2)ξk. (28)\nCompared with the case without damping term we see that the gap of the Higgs mode is reduced from/radicalbig\n2r/v′to/radicalbig\n2r/v′−u′′2/v′2at BCS limit.\nFor smallξkthe spectral weight on the Goldstone mode can be calculated as\nAaa(˜ω−,k) =u′′\n8u′2˜ω−. (29)\nDifferent from the case without damping term, we see that in the cas e with damping term the spectral function has\na weight proportional to u′′on the Goldstone mode.\nThe spectral weight function in the case with Coulomb intera ction\nA time-dependent Ginzburg-Landau theory with Coulomb interactio n can be cast as [2]\nF=/integraldisplay\ndtd3x/braceleftBig\n−1\n8πφ∇2φ+¯∆/parenleftBig\n−iu(∂t−2eφ)+v(∂t−2eφ)2−∇2\n2m∗−r/parenrightBig\n∆+b\n2¯∆¯∆∆∆/bracerightBig\n, (30)\nwhereeis the electric charge and φ(t,x) is the electric field. By taking a symmetry breaking ∆ →∆0+δa+iδpwe\ncan have a free energy for the low energy excitations in the moment um space as\nF=/integraldisplaydω\n2πd3k\n(2π)3/braceleftBigg\n2uiωδa(ω,k)δp(−ω,−k)+δa(−ω,−k)(−vω2+k2\n2m∗+2r)δa(ω,k)+δp(−ω,−k)\n(−vω2+k2\n2m∗)δp(ω,k)+4iue∆0φ(−ω,−k)δa(ω,k)−4veω∆0φ(−ω,−k)δp(ω,k)+4ve2∆2\n0φ(−ω,−k)φ(ω,k)/bracerightBigg\n.\n(31)\nWe integrate out the electric field φand obtain\nF=/integraldisplaydω\n2πd3k\n(2π)3/braceleftBigg\n2uiωk2/8π\nk2/8π+4ve2∆2\n0δa(ω,k)δp(−ω,−k)+δp(−ω,−k)(−vω2k2/8π\nk2/8π+4ve2∆2\n0+k2\n2m∗)δp(ω,k)/bracerightBigg\n+δa(−ω,−k)(−vω2+k2\n2m∗+2r)δa(ω,k). (32)\nThen the spectral functions can be calculated as\nImχaa=u′′ω\n2·(−vω2k2/8π\nk2/8π+4ve2∆2\n0+k2\n2m∗)2+|uωk2/8π\nk2/8π+4ve2∆2\n0|2\n|−(uωk2/8π\nk2/8π+4ve2∆2\n0)2+(−vω2k2/8π\nk2/8π+4ve2∆2\n0+k2\n2m∗)(−vω2+k2\n2m∗+2r)|2,\nImχpp=u′′ω\n2·(−vω2+k2\n2m∗+2r)2+|uωk2/8π\nk2/8π+4ve2∆2\n0|2\n|−(uωk2/8π\nk2/8π+4ve2∆2\n0)2+(−vω2k2/8π\nk2/8π+4ve2∆2\n0+k2\n2m∗)(−vω2+k2\n2m∗+2r)|2. (33)\n[1] C. A. R. S´ a de Melo, M. Randeria, and J. R. Engelbrecht, Ph ys. Rev. Lett. 71, 3202 (1993).\n[2] Adriaan M. J. Schakel, Boulevard of Broken Symmetries: Effective Field Theories of C ondensed Matter , World Scientific,\nSingapore, 2008."    },    {        "title": "1309.2741v1.Initial_versus_tangent_stiffness_based_Rayleigh_damping_in_inelastic_time_history_seismic_analyses.pdf",        "content": "arXiv:1309.2741v1  [physics.class-ph]  11 Sep 2013INITIAL VS. TANGENT STIFFNESS-BASED RAYLEIGH\nDAMPING IN INELASTIC TIME HISTORY SEISMIC\nANALYSES\nP. Jehel12, P. L´ eger3and A. Ibrahimbegovic4\npierre.jehel[at]ecp.fr\nPaper accepted for publication in Earthquake Engineering and Structural Dynamics\nPublished online (wileyonlinelibrary.com). DOI: 10.1002 /eqe.2357\nAbstract. In the inelastic time history analyses of structures in seismic mo-\ntion, part of the seismic energy that is imparted to the structure is absorbed\nby the inelastic structural model, and Rayleigh damping is commonly us ed in\npractice as an additional energy dissipation source. It has been ac knowledged\nthat Rayleigh damping models lack physical consistency and that, in t urn, it\nmust be carefully used to avoid encountering unintended conseque nces as the\nappearance of artificial damping. There are concerns raised by th e mass pro-\nportional part of Rayleigh damping, but they are not considered in t his paper.\nAs far as the stiffness proportional part of Rayleigh damping is conc erned, ei-\nther the initial structural stiffness or the updated tangent stiffn ess can be used.\nThe objective of this paper is to provide a comprehensive compariso n of these\ntwo types of Rayleigh damping models so that a practitioner i) can obj ectively\nchoose the type of Rayleigh damping model that best fits her/his ne eds and ii)\nis provided with useful analytical tools to design Rayleigh damping mod el with\ngood control on the damping ratios throughout inelastic analysis. T o that end,\na review of the literature dedicated to Rayleigh damping within these la st two\ndecades is first presented; then, practical tools to control the modal damping\nratios throughout the time history analysis are developed; a simple e xample is\nfinally used to illustrate the differences resulting from the use of eith er initial or\ntangent stiffness-based Rayleigh damping model.\nkeywords: Rayleigh damping, inelastic structure, modal analysis, damping\nratio time history, upper and lower bounds for damping ratios.\n1Laboratoire MSSMat /CNRS-UMR 8579, ´Ecole Centrale Paris, Grande voie des Vignes,\n92295 Chˆ atenay-Malabry Cedex, France\n2Department of Civil Engineering and Engineering Mechanics, Columbia University, 630 SW\nMudd, 500 West 120th Street, New York, NY, 10027, USA\n3Department of Civil Engineering, ´Ecole Polytechnique de Montr´ eal, P.O. Box 6079, Station\nCV, Montreal, QC, H3C 3A7, Canada\n4LMT-Cachan(ENS Cachan/CNRS/UPMC/PRESUniverSud Paris), 61 avenue du Pr´ esident\nWilson, 94235 Cachan Cedex, France\n12 RAYLEIGH DAMPING IN INELASTIC ANALYSES\n1.Introduction\nSeismic analyses were first developed for elastic structures. An ela stic struc-\nture does not absorb energy and, therefore, damping was added to represent all\nthe energy dissipation sources at the boundary conditions or at sm all scales not\nexplicitly considered for civil engineering applications (a list of such en ergy dissi-\npating phenomena can be found in [ 15]). For this purpose, Rayleigh damping is\nmathematicallyveryconvenient because, onceprojectedontoth eundampedmodal\nbasis, the set of equations of motion of the discrete structures a re uncoupled. This\ntype of damping is herein referred to as elastic damping . Then, to account for\nyielding mechanisms in the overall structural response, elastic visc ous-equivalent\nseismic analyses appeared [ 7, p.74]. In this case, the structural model still is elastic\nbut the damping properties are enhanced so as to account for bot h elastic damp-\ning and energy dissipation due to the actual, but not explicitly modeled , yielding\nresponse. This latter type of damping is herein referred to as hysteretic damping .\nHowever, inelastic timehistoryanalysis(ITHA), whicharebasedont hebuilding\nof an inelastic structural model, is the most appropriate way to pro perly account\nfor hysteretic mechanisms in seismic analyses. In this case, the str uctural modes\nare not uncoupled anymore in the modal basis. The modal basis could be updated\nafter each inelastic event to maintain uncoupled modes, but the com putational\nbenefit would be counterbalanced by the additional cost resulting f rom the succes-\nsive modal analyses required. Consequently, there is few mathema tical advantage\nin using Rayleigh damping for ITHA (the main advantage is that there is no need\nto explicitly build and store a damping matrix because mass and stiffnes s matrices\nalready are stored for other purposes). In spite of this, Rayleigh damping still is\ncommonly used in ITHAs. Ideally, Rayleigh damping should be added to m odel\nelastic damping only, and hysteretic damping should arise from the ex plicit mod-\neling of the energy dissipation mechanisms in the inelastic structural model. In\npractice, this can hardly be achieved because, on the one hand, co ntrolling the\namount of elastic damping in ITHA is challenging due to the intrinsic natu re of\nRayleigh damping models and, on the other hand, inelastic structura l models only\nprovide an approximation of the numerous inelastic phenomena that actually con-\ntribute to the seismic energy absorption in the structure. This pap er only focusses\non the issue of maintaining good control on elastic Rayleigh damping th roughout\ninelastic analysis.\nIn its most general form, Rayleigh damping consists in adding viscous forces of\nthe form fD(t) =C(t)˙u(t) in the discrete structural equations of motion, where\n˙uis the displacements rate and the damping matrix C(t) is built as a linear\ncombination of the structural mass and stiffness matrices MandK:\n(1) C(t) =α(t)M+β(t)K(t).\nThe Rayleigh coefficients αandβare computed so that the critical damping ratios\nξAandξBareobservedatthefrequencies ωAandωB.αandβcanbeeithersetonceRAYLEIGH DAMPING IN INELASTIC ANALYSES 3\nand then frozen or updated throughout ITHA. The stiffness matr ix is commonly\nbuilteitherfromtheinitialorfromtheupdatedtangentstiffnesses . Bothmass-and\nstiffness-proportional terms of Rayleigh damping models can gener ate difficulties\nin controlling elastic damping throughout an inelastic analysis. Contra ry to the\nstiffness-proportional term, which is difficult to control in inelastic a nalyses only,\nproblems can arise from the mass-proportional component no mat ter whether it\nis an elastic(-equivalent) or an inelastic time-history analysis. Indee d, assuming\nthat the mass matrix is diagonal, the physical counterpart of a mas s-proportional\nterm is a viscous damper connecting a structural degree of freed om to the frame\nwhich contains the reference point for measuring displacements in t he structure.\nThen, the mass-proportional term can lead to unrealistically high da mping forces\nwhenever the whole or parts of the structure behave like a rigid bod y. This issue\nencountered with the mass-proportional term has already been w ell explained and\nillustrated [ 10]. Consequently, in this paper, although both mass- and stiffness-\nproportional terms are used to construct the damping matrix, on ly the stiffness-\nproportional component is focussed on.\nHow to avoid spurious damping forces andimprove control on dampin g through-\nout inelastic analyses has been widely studied and led to practical rec ommenda-\ntions (e.g.[10,14,5]). However, when it comes to dealing with the stiffness-\nproportional term, one can find in the literature differing viewpoints on whether\nto use initial or tangent stiffness. The objective of this paper is to p rovide a\ncomprehensive comparison of Rayleigh damping based on either initial or tangent\nstiffness, and to answer the question: “which of initial and tangent stiffness-based\nRayleigh damping provides better control on damping ratio through out ITHA?”\n(the mass-proportional term being also present in the Rayleigh dam ping models).\nTo that aim, we first present in the next section a review of the litera ture where\nthe different strategies that have been proposed to build the Rayle igh damping\nstiffness-proportional term are exposed. Then, in section 3, mat hematical de-\nvelopments lead to formulas that allow quantifying damping ratios shif ts due to\nstiffness degradations. These relations are useful to design a Ray leigh damping\nmodel before running an analysis and, once the analysis has been ru n, to assess\nthe validity of the elastic damping modeling. To the best of our knowled ge, this is\nthe first time such formulas are provided for initial stiffness-based Rayleigh damp-\ning. For the sake of completeness, we already mention here that th e formula we\nderive in section 3 are only valid when the two following assumptions are made: i)\nthe damping ratios ξAandξBchosen to identify the Rayleigh coefficients αandβ\nare taken as equal ( ξA=ξB), and ii) the structural equations of motion are uncou-\npled when expressed in modal coordinates, which is in most of the cas es only an\napproximation for inelastic structures with initial stiffness-based R ayleigh damp-\ning. Those two assumptions are common for the type of problems we consider in\nthis work and they do not reduce the contributions of this paper co mparing to4 RAYLEIGH DAMPING IN INELASTIC ANALYSES\nprevious publications on the subject. Before closing the paper with some conclu-\nsions, we compare the performances of initial and tangent stiffnes s-based Rayleigh\ndamping models in the analysis of a simple structure with stiffness degr adations.\n2.In the literature\nAccording to Charney [ 5], one can go back to the eighties to find the first papers\ndedicated to problems pertaining to modeling damping in inelastic struc tures with\nthe work of Chrisp in 1980 [ 6] and Shing and Mahin in 1987 [ 16].\nIn the nineties, L´ eger and Dussault [ 14] investigate, in the case of inelastic frame\nstructures in seismic loading, the variation of nonlinear response ind icators (aver-\nage ductility, hysteretic-to-input energy ratio, average number of yield incursions)\naccording to the additional damping model used. A bilinear hysteres is model is\nassigned to the structural elements. Amongst other mass- or st iffness-proportional\ndamping models, Rayleigh damping models either with elastic or tangent stiffness\nand either with frozen or updated coefficients ( α,β) are considered. The authors\nhave developed a computer program to update αandβat each time step. This\nallows maintaining constant critical damping ratio throughout the se ismic analy-\nses for the two frequencies used to identify the Rayleigh coefficient s, which are, in\nthis work, the first natural frequency and the frequency for wh ich 95% of effective\nmodalmass isrepresented by thetruncated eigenbasis. Notetha t these frequencies\nare not the initial elastic ones but that they also are updated at eac h time step.\nL´ eger and Dussault show that, while having little effects on the amou nt of energy\nimparted to a structure by an earthquake, the choice of an additio nal damping\nmodel significantly influences the amount of hysteretic energy due to damage in\nthe structure. The term “additional damping” is used here to refe r to a source of\ndamping that comes in addition to the damping resulting from the abso rption of\nhysteretic energy in the structural model. L´ eger and Dussault u se for instance the\ndisplacement ductility averaged over all the stories as an indicator o f the inelastic\nstructural response and, for the El Centro ground motion cons idered, variations\nof this latter indicator of up to 40% from one damping model to the ot her are\nobserved. Considering the Rayleigh model with updated coefficients as a base-\nline, recommendations on which additional damping model to use given the elastic\nfundamental period of the structure close the paper.\nFrom the statement that the number of degrees of freedom (DOF s) needed to\nassemble the stiffness matrix is often much larger than that needed for building\nan adequate mass matrix, Bernal [ 3] shows that spurious damping forces are likely\nto arise from the presence of massless – or with relatively small inert ia – DOFs in\ninelastic structural systems. Indeed, massless DOFs have the te ndency to undergo\nabrupt changes in velocity when stiffness changes, leading to unrea listically large\nviscous damping forces. In this work, Bernal adds proportional d amping in the\nequilibrium equations using the Caughey series. It is stated that the most impor-\ntant effect of spurious damping forces is found in the distortion the y introduceRAYLEIGH DAMPING IN INELASTIC ANALYSES 5\nin the maximum internal forces rather than the displacements. As a solution to\nprevent spurious damping due to massless DOFs, it is suggested to a ssemble the\ndamping matrix using the stiffness matrix condensed to the size of th e DOFs with\nmass, followed by expansion to the full set of coordinates with colum ns and rows\nof zeros. This procedure is equivalent to what is proposed in other w orks where\nit is expressed in the other terms: assembling the damping matrix, ele ment by\nelement, with zero damping assigned to the DOFs where local abrupt changes of\nstiffness can occur (see e.g.the procedure advocated in [ 8] for structures with fiber\nelements, or the issues raised by local stiffness changes in [ 5]).\nIn [10], Hall focuses on practical situations where the use of Rayleigh dam ping\ncan lead to damping forces that are unrealistically large compared to the restoring\nforce, and then proposes a capped viscous damping formulation to overcome some\nof the problems pointed out. It is stated in this paper that the tang ent stiffness\nshould not beused to build theRayleigh damping matrix, especially beca use ofthe\nconvergence difficulties it can cause. To illustrate this assertion, th e author gives\nthe example of the local damping stress, resulting from the stiffnes s-proportional\npart ofRayleigh damping, that canjump fromzero toa possibly large value incase\nof crack closing as soonascontact ismade. Then, Hall distinguishes between prob-\nlems arising from the mass- and stiffness-proportional part of Ray leigh damping\nand quantifies the likely undesirable effects. The mass-proportiona l term can lead\nto unrealistically high damping forces whenever the whole or parts of the structure\nbehave like a rigid body: formulation of an earthquake analysis in term s of total\nmotion, superstructure on a relatively flexible base, portions of a s tructure that\nbreak loose like in a dam undergoing sliding at its base,... As far as the stiff ness-\nproportional term of Rayleigh damping is concerned, very large dam ping forces\ncan be generated when Rayleigh damping is based on the initial stiffnes s, while\nstructural elements yield, leading to an increase in the velocity grad ient: gravity\ndam undergoing cracking, presence of penalty elements,... Finally, as a remedy\nto the problems listed in the paper, the author proposes a capped v iscous damp-\ning formulation in which the mass-proportional contribution is eliminat ed and the\nstiffness-proportional contribution is limited by bounds defined in ac cordance with\nthe actual physical mechanism that limits the structural restorin g forces.\nIn [5], Charney first investigates the effects of global stiffness change s on the\nseismic response of a 5-story structure when Rayleigh damping is us ed. Two cases\nare considered: a reduction of the story stiffness by the same fac tor for each story,\nwhich doesnotchangethemodeshapes, andanonuniformstoryst iffness reduction\nalong the height, which leads to mode shapes that are different from those of the\ninitial structure. In both cases, it is shown that, when structure yields, i) artificial\ndamping is generated when the Rayleigh damping matrix is computed ac cording\nto the initial structural properties; ii) significant but reduced art ificial damping is\ngeneratedwhenRayleighdampingisbuiltwithtangentstiffnessandfix edRayleigh\ncoefficients computed fromtheinitial stiffness; iii) virtually no artificia l damping is6 RAYLEIGH DAMPING IN INELASTIC ANALYSES\ngenerated when Rayleigh damping is based on both tangent stiffness and updated\ncoefficients from the tangent stiffness.\nThen, Charney [ 5] also investigates the effects of local stiffness changes on\nthe seismic structural time-history response when initial stiffness -based Rayleigh\ndamping is used. The same 5-story structure as previously is consid ered, but this\ntime with inelastic rotational springs at beam-to-column connection s, which rep-\nresent local yielding mechanisms. There is no rotational mass and ze ro hardening\nafteryielding. Under these conditions, threedifferent pairsofbea ms/columns stiff-\nness and connections elastic stiffness are defined so that, in each c ase, the overall\nstiffness matrix of the structure be unchanged. The response his tories are identi-\ncal when a linear analysis is performed and when the damping matrix is b uilt so\nthat the rows and columns pertaining to the rotational DOFs are fille d with zeros.\nHowever, when the Rayleigh damping matrix is built with non-zero initial stiffness\nassociated to the massless DOFs, artificial viscous damping forces develop. These\nforces can be extremely high, especially as the initial spring stiffness is large. Yet\nthey are not easy to detect.\nTo Charney, the best strategy to model damping would be of cours e to eliminate\nthe use of viscous damping, because it is not physically sound, and re place it by\nfrictional or hysteretic devices. Nevertheless, amongst other r ecommendations,\nCharney advocates that Rayleigh damping based on tangent stiffne ss be used.\nIndeed, with this method, the problems associated with local stiffne ss changes are\nalways eliminated, and reduced artificial damping is generated becau se of global\nstiffness changes, which, moreover, can still be reduced by anticip ating the shift\nof the structural natural frequencies. If elastic stiffness-bas ed Rayleigh damping\nonly is implemented in the computer program used, the damping matrix should\nbe computed with zero stiffness associated to the elements that ha ve large initial\nstiffness and that are likely to yield.\nThe more recent papers which also focus on modeling structural da mping in in-\nelastic time history seismic analyses are those of Zareian and Medina [ 17] and Er-\nduran [9]. In these papers, practical strategies are presented to cope w ith the\nproblems encountered with Rayleigh damping in the context of perfo rmance-based\nseismic design. In the approach presented in [ 17], each structural element is mod-\neled with an equivalent combination of an elastic element with initial stiffn ess-\nproportional damping and yielding springs at the two ends without st iffness-\nproportional damping. With this strategy, i) numerical solution inst abilities when\nsignificant changes in stiffness values occur are avoided because init ial stiffness\nmatrix is used in the damping model, and, also, no local spurious dampin g forces\nare generated in the structural parts that yield because there is no stiffness-\nproportional part in the damping model pertaining to these parts.\nIn [9], Erduran recalls that it has been shown that designing Rayleigh damp ing\nmodels based on the initial stiffness matrix results in unreasonably hig h dampingRAYLEIGH DAMPING IN INELASTIC ANALYSES 7\nforces after yielding and consequently decides to exclusively use ta ngent stiffness-\nbased Rayleigh damping models in his study. This study consists in analy zing the\nstory drift ratios, floor accelerations and damping forces in two 3- and 9-story steel\nmoment-resisting frame buildings for three seismic hazard levels. It is concluded\nin [9] that, as long as they are designed according to reduced modal fr equencies\n(comparing to the elastic properties), Rayleigh damping models with b oth mass-\nand tangent stiffness-proportional components lead to reasona ble damping forces\nand floor acceleration demands without suppressing higher modes e ffects in the\n9-story building.\nEither following or initiating these research efforts, there are comp uter program\nuser manuals and technical reports who directly provide practition ers with ad-\nvanced tools and guidelines for proper implementation of Rayleigh dam ping in\ninelastic time history seismic analyses. For instance: eight types of d amping mod-\nels are implemented in the computer program RUAUMOKO, whose user manual\nalso comes with comprehensive discussion on damping [ 4, pp. 9-10]; visualization\ntools for damping effects in the simulations are available in the compute r program\nPERFORM-3D, and an entire chapter is dedicated to modeling damping in the\naccompanying user guide [ 8,§18]; a large share is dedicated to viscous damping for\ninelastic seismic analyses in the technical reports [ 2,§2.4.2] and, to a more limited\nextent, in [ 1,§6.4.4].\nFrom the review of the literature above, it is obvious that some auth ors recom-\nmend using initial stiffness-based Rayleigh damping models while others recom-\nmend using tangent stiffness. In the following section, we provide ma thematical\ndevelopment for a rational comparison of the two approaches, re lying on an anal-\nysis of the damping ratios time history throughout inelastic analyses .\n3.Mathematical developments\n3.1.Modal analysis for inelastic structures. The equilibrium equations of a\nstructure in seismic loading, with additional viscous damping, discret ized in space,\nand transformed to undamped modal coordinates at time t, read:\n(2)ΦT\ntMΦt¨U(t)+ΦT\ntC(t)Φt˙U(t)+ΦT\ntK(t)ΦtU(t) =ΦT\nt/parenleftbig\nFsta(t)+Fsei(t)/parenrightbig\nwhereU(t) ={Um(t)}m=1,..,Nmis the undamped modal coordinates vector, Fsta(t)\nandFsei(t) are the pseudo-static and seismic forces (the static force is app lied step\nby step prior to the application of the seismic load to grasp possible no nlinear\nmechanisms), Φt= (φ1(t)... φNm(t)) is the matrix composed by the undamped\nmodal shape vectors, and /squareTdenotes the transpose operator. Mode shapes can\nalso be computed from the damped system, but the undamped assu mption is\ncommonly retained for systems with low damping ratios, as it is the cas e in this\nwork. Although cumbersome, the explicit dependence of every qua ntity on time\ntis indicated ( /square(t) or/squaret) to emphasize that each of these quantities can change\nwithin the inelastic time history of the structure.8 RAYLEIGH DAMPING IN INELASTIC ANALYSES\nThere are two ways to compute the viscous damping matrix. If modal damping\nis used, the modal damping ratios ξmare chosen at time twhen the modal analysis\nis performed, and the damping matrix is built as diagonal:\n(3) ΦT\ntC(t)Φt= diag(Cm(t)) with Cm(t) = 2ξm(t)ωm(t)Mm(t).\nIn this relation, we have introduced the undamped modal circular fr equencies ωm\nasω2\nm(t) =Km(t)/Mm(t), along with the notation Am=φT\nmAφmwhereA=M,\nCorK. IfRayleigh damping is used, a set of coefficients ( α,β) is computed, and\nthe damping matrix is built as:\n(4)C(t) =α(t)M+β(t)K(t)⇒ C m(t) =α(t)Mm(t)+β(t)Km(t).\nFrom equations ( 3) and (4), we thus have the relation\n(5)α(t)φT\nm(t)Mφm(t)+β(t)φT\nm(t)K(t)φm(t) = 2ξm(t)ωm(t)φT\nm(t)Mφm(t)\nbetween – right-hand side – the modal quantities that would arise fr om a modal\nanalysis at any time tin thehistory of thestructure and– left-handside – what ac-\ntually results from the use of Rayleigh damping at this specific time t. Relation ( 5)\nprovides a definition of the modal damping ratio.\nIn equation ( 2), bothΦT\ntMΦtandΦT\ntK(t)Φtare diagonal matrices. ΦT\ntC(t)Φt\nhowever can be non-diagonal, which implies that ( 2) is a set of coupled equa-\ntions. A damping model that is not diagonalized in the modal basis is qua lified as\nnon-classical ornon-proportional . As shown in relation ( 3), modal damping inher-\nently is classical. Rayleigh damping is however not necessarily classical: if initial\nstiffness is used, off-diagonal terms appear in ΦT\ntCΦtin most of the cases when\nstructure yields. Relation ( 5) provides exact values of ξm(t) forclassical damping\nonly; otherwise, not all the damping energy is included in the damping r atios.\nConsequently, in the presence of non-classical damping, the defin ition ofξmin (5)\nis only valid if the off-diagonal terms in ΦT\ntCΦtcan be neglected. For systems\nwith low damping ratios (a few percents of critical), as it is the case in t his work,\nthe assumption that the equations in ( 2) can approximated as decoupled in the\npresence of non-classical damping generally is, explicitly or implicitly, r egarded as\nvalid in the literature ( e.g.in [10,5]). Accordingly, we will hereafter retain this\nassumption.\n3.2.Implementing Rayleigh damping. In practice, there are several methods\nto build a Rayleigh damping matrix, which all have different consequenc es on the\ntime history of the damping ratios throughout the inelastic analysis:\na. The Rayleigh coefficients, as well as the stiffness matrix, are set on ce for all\nbefore the beginning of the dynamic analysis. We refer to these qua ntities\nas (α0,β0) andK0. These quantities are not necessarily identified consid-\nering the state of the structure at the beginning of the dynamic an alysis:\nbased on an approximation of the reduced stiffness, the state of t he struc-\nture at any time within or at the end of the seismic loading can be used f orRAYLEIGH DAMPING IN INELASTIC ANALYSES 9\nthe identification of the Rayleigh coefficients before running the dyn amic\nanalysis. At a given time tin the structure time history, equation ( 5) thus\ntakes the following expression:\n(6) ξa\nm(t) =1\n2ωm(t)/parenleftbigg\nα0+β0φT\nm(t)K0φm(t)\nMm(t)/parenrightbigg\nwhich we rewrite as\n(7)ξa\nm(t) =1\n2/parenleftbiggα0\nωm(t)+β0ha\nm(t)ωm(t)/parenrightbigg\nwithha\nm(t) =φT\nm(t)K0φm(t)\nKm(t).\nNotethat ΦT\ntK0Φtgenerallyisnotdiagonalandthattheoff-diagonalterms\nare neglected in this definition of ξa\nm(t).\nb. TheRayleighcoefficientsaresetonceforallatthebeginningofthe dynamic\nanalysis, asincase a,andthetangentstiffnessmatrix K(t),updatedateach\ntime step, is used to build C(t). With this definition, Rayleigh damping is\nclassical throughout the analysis and we have:\n(8) ξb\nm(t) =1\n2/parenleftbiggα0\nωm(t)+β0ωm(t)/parenrightbigg\n.\nc. Both the Rayleigh coefficients and the stiffness matrix are updated at each\ntime step in the numerical analysis. In this case, the modal damping r atio\nξc\nm(t) can be controlled throughout the analysis, e.g.maintained constant\nby updating ( α,β) accordingly:\n(9) ξc\nm(t) =1\n2/parenleftbiggα(t)\nωm(t)+β(t)ωm(t)/parenrightbigg\n.\nd. There is no risk of generating spurious local damping forces when m eth-\nods based on the updated tangent stiffness are used to implement R ayleigh\ndamping [ 5]. However, this risk arises when, as in case a, the Rayleigh\ncoefficients along with the stiffness matrix are set once for all befor e the\nbeginning of the dynamic analysis, and there are some elements in the\nstructure with artificially large initial stiffness (plastic hinges, gap or con-\ntact elements). It is then advocated, e.g.in [5], to associate reduced β0\nfactors to this latter type of element ( βe\n0< β0). However, this method can\npotentially generate the same type of global effects as method a. Indeed,\ndefining a reduced initial stiffness matrix Kr\n0as:\n(10) C=Ae(α0Me+βeKe\n0) =α0M+β0Kr\n0withβ0Kr\n0=Aeβe\n0Ke\n0,\nwhereAedenotes the finite element assembly procedure [ 11], it comes from\nequation ( 5):\n(11)ξd\nm(t) =1\n2/parenleftbiggα0\nωm(t)+β0hd\nm(t)ωm(t)/parenrightbigg\nwithhd\nm(t) =φT\nm(t)Kr\n0φm(t)\nKm(t).10 RAYLEIGH DAMPING IN INELASTIC ANALYSES\nIf the reduced initial stiffness Kr\n0better approximates the tangent stiffness\nmatrix than the initial stiffness K0, the global effects with method dare re-\nduced comparing to method a. Also, using this method, off-diagonal terms\ncan appear in the generalized damping matrix; those terms are negle cted\nin this definition of ξd\nm(t).\nFor methods bandc, the updated stiffness matrix could loose its positiveness if\nsignificant softening or second-order effects would occur. Mathe matically, it would\nimply that the modal frequencies be null or imaginary ( ω2≤0), which, physically,\nwould not make any sense. However, this would only happen if the str ucture\nbecame unstable and, consequently, this is an ultimate potential pr oblem that we\nlet out of the scope of this work.\nMethodcis rarely used in practice because a modal analysis has to be carried\nout after each inelastic event. Therefore, hereafter, we do not consider case cany-\nmore and the Rayleigh coefficients ( α0,β0) are set once for all before the dynamic\nanalysis.\n3.3.Rayleigh coefficients computation. For method i=a,bord, we have,\naccording to what precedes:\n(12) ξi\nm(t) =1\n2/parenleftbiggα0\nωm(t)+β0hi\nm(t)ωm(t)/parenrightbigg\nwithhb\nm(t) = 1,∀m ,∀t .\nTwo independent equations can be written from equation ( 12) with two different\npairs of damping ratios and circular frequencies hereafter referr ed to as ( ξA,ωA)\nand (ξB,ωB). The resulting set of two equations can then be solved to obtain α0\nandβ0. For inelastic analyses, the practitioner can choose the pairs ( ξA,ωA) and\n(ξB,ωB) either for two different modes mAandmBat the same time t=tA=tB,\nor for two different modes mAandmBat two different times tAandtB, or for the\nsame mode m=mA=mBat two different times tAandtB.AandBthus refer to\na mode and a particular time in the history of the structure and, in th e following,\nAhas to be understood as the pair ( t=tA,m=mA), and so for B.\nFor the sake of conciseness, we abandon the superscript ithat explicitly refers\nto method a,bord. Yet, one should keep in mind that the quantities ξA/B,ωA/B,\nhA/B,α0andβ0all depend on the method used. Rewriting equation ( 12) forA\nandB, we then have:\n(13)/braceleftbigg\nξA\nξB/bracerightbigg\n=1\n2/parenleftbigg\n1/ωAhAωA\n1/ωBhBωB/parenrightbigg/braceleftbigg\nα0\nβ0/bracerightbigg\n.\nTheRayleighcoefficientscanthenbecomputedbyinvertingrelation( 13),which,\nas long as hAω2\nA/ne}ationslash=hBω2\nB, is always possible:\n(14)/braceleftbigg\nα0\nβ0/bracerightbigg\n=2ωAωB\nhBω2\nB−hAω2\nA/parenleftbigg\nhBωB−hAωA\n−1/ωB1/ωA/parenrightbigg/braceleftbigg\nξA\nξB/bracerightbigg\n.RAYLEIGH DAMPING IN INELASTIC ANALYSES 11\nWe assume that ξA=ξB=ξ0, which will simplify the following developments.\nBecause this assumption is commonly done in practice, it does not red uce the\ncontributions of the present work. We then have:\n(15)α0=2ωAωB(hBωB−hAωA)\nhBω2\nB−hAω2\nAξ0andβ0=2(ωB−ωA)\nhBω2\nB−hAω2\nAξ0.\nNote that the coefficients hgenerally are absent in the definition of the Rayleigh\ncoefficients. Common definitions for α0andβ0indeed are:\n(16) α0=2ωAωB\nωA+ωBξ0andβ0=2\nωA+ωBξ0,\nbut this actually is only exact for case bwherehb\nm(t) = 1 for all tandm. There\nis another important notion that is generally not explicitly mentioned in the def-\ninition of the Rayleigh coefficients in ITHA and that we recall here: the indexA\n(respectively B) does not only refer to a specific mode but to a mode at a given\ntime in the history of the structure.\n3.4.Upper and lower bounds for the damping ratios. We now seek to\npredict the shift of the damping ratiospertaining to modal frequen cies that remain\nin a pre-determined range, throughout the inelastic time history an alysis. In other\nwords, we seek an answer to the problem illustrated in figure 1: How to compute\n∆>0 so that, for ωA,ωB, and a targeted damping ratio ˆξgiven, for all tand for\nallm:\n(17) ωA≤ωm(t)≤ωB⇒ˆξ−∆< ξm(t)<ˆξ+∆ ?\nWe denote ξmax=ˆξ+ ∆ and ξmin=ˆξ−∆. We moreover set ξmax=ξ0(=\nξA=ξB) and introduce R >1 so that ωB=R×ωA. With these notations and\nconsidering the expression of the Rayleigh coefficients in equations ( 15), we rewrite\nrelation ( 12) as:\n(18)ξm(t)\nξmax=1\nR2hB−hA/parenleftBigg\nR(R hB−hA)1\nωm(t)\nωA+(R−1)hm(t)ωm(t)\nωA/parenrightBigg\n.\nTo guarantee ξm(t)>0, withhm(t)>0,ωm(t)/ωA>0, andR >1, we require\nthathA>0 andhB>0 be chosen so that they satisfy the condition:\n(19) R hB−hA≥0,\nwhich implies that R2hB−hA≥0 because R≥1.\nWe now study relation ( 18) for methods a,banddexplicitly. Because method\nbis the easiest to deal with, we start with it and methods aanddare considered\ntogether right after.12 RAYLEIGH DAMPING IN INELASTIC ANALYSES\nFigure 1. We seek to characterize the grey box within which the\ndampingratiosremainallalongtheinelasticanalysis( t∈[0,T]).ˆξis\nthe targeted damping ratio in the range [ ωA,ωB], ∆ is the deviation\nof the actual damping ratio from the targeted one. The black point s\nindicate two possible points to identify the Rayleigh coefficients\n(α0,β0).\nMethod b:hA=hB= 1 and hm(t) = 1 for all tandm, leading to the function\n(20)ξm(t)\nξmax=1\n1+R/parenleftBigg\nR1\nωm(t)\nωA+ωm(t)\nωA/parenrightBigg\n,\nwhich we write asξm(t)\nξmax=f/parenleftBig\nωm(t)\nωA/parenrightBig\n. To study the variation of ξm(t) with respect\ntoωm(t) in the range [ ωA,ωB], it is useful to compute the derivative f′of the\nfunction fand to analyze its sign:\n(21) f′/parenleftbiggωm(t)\nωA/parenrightbigg\n=d(ξm(t)/ξmax)\nd(ωm(t)/ωA)=1\n1+R/parenleftBigg\n1−R1/parenleftbigωm(t)\nωA/parenrightbig2/parenrightBigg\n,\nwhich implies that f′/parenleftBig√\nR/parenrightBig\n= 0 and also that f′/parenleftBig\nωm(t)\nωA/parenrightBig\n<0 forωA≤ωm(t)<\n√\nR ωA, andf′/parenleftBig\nωm(t)\nωA/parenrightBig\n>0 for√\nR ωA< ωm(t)≤ωB, because R >1. Conse-\nquently, ξm(t) decreases for ωm(t)∈[ωA,√\nR ωA] an then increases for ωm(t)∈\n[√\nR ωA,ωB] after reaching the minimum:\n(22)ξm(t)\nξmax/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nmin=ξm(t)\nξmax/vextendsingle/vextendsingle/vextendsingle/vextendsingleωm(t)\nωA=√\nR=2√\nR\n1+R.\nBecause stiffness degradations lead to a decrease of the modal fr equencies, the\ndamping ratios in the frequency range [ ωA,√\nR ωA] (respectively [√\nR ωA,ωB])\nwill necessarily increase (respectively necessarily decrease) durin g the inelastic\nanalysis. As will be illustrated in the applications that follow this section , this\nresult is useful to design the damping model because it provides info rmation onRAYLEIGH DAMPING IN INELASTIC ANALYSES 13\nhow to choose the modes and times AandBfor which the maximum damping\nratioξmaxare observed.\nThen, one can compute ∆bas follows. By definition, 2∆b=ξmax−ξmin, with\n(see equation ( 22))ξmin=2√\nR\n1+Rξmaxandξmax=ˆξ+∆b, which gives:\n(23) ∆b=ˆξ1+R−2√\nR\n1+R+2√\nR.\nThis result is mode- and time-independent. It is a useful measure of the damping\nratios drifts in the range [ ωA,ωB], throughout ITHA. Being able to guarantee\nlow ∆ for large Rwould be an ideal situation to provide good control on the\ndamping ratios. It is however not possible for Rayleigh damping model because\n∆bmonotonically grows with R(d∆b\ndR=2(R−1)ˆξ\n(1+R+2√\nR)2≥0).\nAn analogous development is presented in [ 10] where the same result is given for\n∆ (equation (5) in [ 10]). But, it should be noted that it is only valid for method\nb, not for methods aord.\nMethods aandd:Finding a mode- and time-independent expression for ∆ is in\nthese cases not as straightforward as for method bbecause the value of hm(t) is not\nknowna priori. First, we define Has a mode- andtime-independent variable, such\nthathm(t)≥H≥1, which, assuming that the stiffness matrix remains positive,\nis true for all mandt(see equations ( 7) and (11)). Then, replacing hm(t) byH,\nwe can rewrite equation ( 18) as:\n(24)\nξm(t)\nξmax≥1\nR2hB−hA/parenleftBigg\nR(R hB−hA)1\nωm(t)\nωA+(R−1)Hωm(t)\nωA/parenrightBigg\n=g/parenleftbiggωm(t)\nωA/parenrightbigg\nandstudythevariationsofthefunction gwithrespectto ωm(t). Fromananalogous\nprocedure as for method b, we conclude that gdecreases for ωm(t)< G ω A, is\nminimum for ωm(t) =G ωA, and then increases for ωm(t)> G ω A, whereG=/radicalBig\nR(RhB−hA)\n(R−1)H. Consequently:\n(25)ξm(t)\nξmax≥g(G) =2/radicalbig\nR(R−1)(R hB−hA)H\nR2hB−hA.\nBecause H≥1, it follows that a mode- and time-independent lower bound for\nξminis:\n(26) ξmin≥ξmax2/radicalbig\nR(R−1)(R hB−hA)\nR2hB−hA.\nIn contrast to what was found for method b, we only have here an upper bound\nforξmin. Besides, for method b, it is certain that the minimum of ξm(t) is attained\nforωm(t)∈[ωA,ωB] whereas, for methods aandd, we cannot determine a priori\nwhere the minimum will be observed. Therefore, when designing the d amping14 RAYLEIGH DAMPING IN INELASTIC ANALYSES\nmodel, there is in these cases no indication on how to efficiently choose the modes\nand times AandBfor which ξmaxis reached.\nConsidering that AandBhave been chosen, and using the same procedure as\nfor method b, we obtain the following upper bound for ∆a,d:\n(27) ∆a,d≤∆a,d\nmax=ˆξR2hB−hA−2/radicalbig\nR(R−1)(RhB−hA)\nR2hB−hA+2/radicalbig\nR(R−1)(RhB−hA).\nNote that for hA=hB= 1, we recover ∆a,d\nmax= ∆b.\n3.5.Is∆a,d\nmax≤∆b?If this inequality were true, ∆a,dwould necessarily be lower\nthan ∆bforRandωAgiven and no matter what hAandhBwould be equal\nto. Unfortunately, we show in figure 2that, for ha,d\nA≥1 andha,d\nB≥1, we have\n∆a,d\nmax≥∆b, with ∆a,d\nmax= ∆bforha,d\nA=ha,d\nB= 1. It is thus impossible to con-\nclude anything about whether using initial stiffness-based Rayleigh d amping can\nprovide better control on the damping ratios than tangent stiffne ss-based Rayleigh\ndamping. Besides, one can see in figure 2that, in certain area of the ( ha,d\nA,ha,d\nB)\nplane, ∆a,d\nmaxcan be large comparing to ∆b.\nFigure 2. ∆a,d\nmaxwith respect to ( ha,d\nA,ha,d\nB). ∆a,d\nmax= ∆bfor\n(ha,d\nA,ha,d\nB) = (1,1). For the purpose of illustration, we use here\nR= 10 and ˆξ= 2%.\n3.6.Summary. In table1, we summarize the formula we derived in this section\nto calculate the damping ratios time histories and to evaluate their ma ximum\ndrift. Obviously, Rayleigh damping models designed according to meth odbhave\ntwo advantages over models designed with methods aord:RAYLEIGH DAMPING IN INELASTIC ANALYSES 15\n(i) The sole quantities that are a priori unknown with method bareωAand\nR, which is much easier to approximate with fair accuracy with some ex-\nperience or simplified analyses than the quantities hwith methods aor\nd;\n(ii) We have access to the exact value of ∆b, whereas we can only calculate\nan upper bound for ∆a,dand it is not straightforward to identify the pair\n(tB,mB) that defines ξB=ξmax.\nNevertheless, we recall that damping models designed from method bcan cause\nsolution convergence problems, which is reported in the review of th e literature\nabove. It is important to stress here that those models only have t he potential\nto provide practitioners with a better control on the damping ratio s time history\nthroughout inelastic seismic analysis. Indeed, as will be illustrated in t he next\nsection, it does not imply that a damping model designed from method aordis\nnecessarily poor.\nTable 1. Initial vs. tangent stiffness-based Rayleigh damping:\nwhat the calculus provides. ξmax=ξA=ξBwithξA/B=\nξm=mA/B(t=tA/B) andξmax=ˆξ+ ∆ with ˆξthe targeted damp-\ning ratio in [ ωA,R×ωA].\ni ξi\nm(t)/ξmax ∆i/ˆξ Comments\na1\n1+R/parenleftbigg\nR\nωm(t)\nωA+ha\nm(t)ωm(t)\nωA/parenrightbigg\n≤R2ha\nB−ha\nA−2√\nR(R−1)(Rha\nB−ha\nA)\nR2ha\nB−ha\nA+2√\nR(R−1)(Rha\nB−ha\nA)·ha\nm(t) =φT\nm(t)K0φm(t)\nφT\nm(t)K(t)φm(t)\n·AandBa prioriunknown\nb1\n1+R/parenleftbigg\nR\nωm(t)\nωA+ωm(t)\nωA/parenrightbigg\n=1+R−2√\nR\n1+R+2√\nR/\nd1\n1+R/parenleftbigg\nR\nωm(t)\nωA+hd\nm(t)ωm(t)\nωA/parenrightbigg\n≤R2hd\nB−hd\nA−2√\nR(R−1)(Rhd\nB−hd\nA)\nR2hd\nB−hd\nA+2√\nR(R−1)(Rhd\nB−hd\nA)·hd\nm(t) =φT\nm(t)Kr\n0φm(t)\nφT\nm(t)K(t)φm(t)\n·AandBa prioriunknown\n4.Illustrative applications\n4.1.Example 1: Rayleighdamping designed from elastic structur al prop-\nerties.The inelastic structural models considered in this example are the sa me\nas in Charney’s work [ 5], which we briefly present here. The structure is a five-\nstory building modeled as a system of five DOFs – the horizontal displa cements\n– connected by inelastic columns which all have the same elastic prope rties. At\neach DOF same mass is lumped. Concerning the inelastic response, tw o different\nyielding scenarios are imagined to occur during a hypothetical ITHA:\n(i) The entire stiffness matrix is assumed to uniformly reduce to 50% o f its\noriginal value;16 RAYLEIGH DAMPING IN INELASTIC ANALYSES\n(ii) Thestructuralelements nonuniformlyyieldalongthebuildingheigh t:Nth-\nstory stiffness is reduced to 10%+( N−1)×20% of its original value, that\nis 10% for the the 1ststory, 30% for the 2nd,..., 90% for the 5th.\nFor this first example, we also design the same Rayleigh damping models as in\nCharney’s work [ 5], including with method cfor a complete comparison. Accord-\ningly, Rayleigh damping models are designed so that a critical damping r atio of\n2% is observed for the modes 1 and 3 of the structure in its initial sta te (t= 0).\nAdopting theanalytical frameworkpresented intheprevious sect ion, itmeans that\nwe design damping models with the following parameters: ωA=ω1(0),ωB=ω3(0)\nandξ0=ξA=ξB=ˆξ= 2%. To clearly show the time history of the damping\nratios, we arbitrarily set the total duration of the inelastic analysis T= 1 s and\ndivide the analysis into 5 time steps (0 ,0.2 s,...,1 s). Each time step corresponds\nto immediate degradations in the structure: e.g., in the case of the uniform yield-\ning scenario described above (scenario 1), at t= 0 the structural stiffness matrix\nKis equal to the initial stiffness matrix K0, then at t= 0.2 s the stiffness matrix\nsuddenly changes to K= 90%×K0, att= 0.4 s there is another sudden degrada-\ntion toK= 80%×K0,..., finally, at t= 1 s,K= 50%×K0. The time histories\nof the damping ratios for methods a,bandcare shown in figure 3for uniform\nstiffness degradations (yielding scenario 1) and in figure 4for nonuniform stiffness\ndegradations (yielding scenario 2).\nFrom figures 3and4, the best control on the damping ratios time histories is\nobtainedformethod c,althoughthedampingratioofmode5significantlyincreases\nfor the structure with nonuniform stiffness degradations. In par ticular, and as\nexpected for method c, modes 1 and 3 are perfectly controlled: in the ( ξ,t)-plane\nin figures 3[bottom, right] and 4[bottom, right], ξ1(t) andξ3(t) both describe the\nsame straight line ξ1(t) =ξ3(t) =ξ0. We recall that method cis rarely used in\npractice. Itisalsoobviousthat, inthisparticularcasewheretheRa yleighdamping\nmodels are designed according to the initial structural modal prop erties, method\nbprovides better control on damping than method a. For both methods aand\nb, the damping models designed lead to overdamped first modes espec ially when\nnonuniform stiffness degradations are assumed. This is not accept able because\nfirst mode generally is important to reconstruct the overall struc tural behavior. In\nthe case of uniform stiffness degradations, one can also observe in figure3[top,\nright] that ξ1(t) andξ3(t) describe the same curve and consequently have the same\ntime history.\nIn table2, we gather the time history of the modal properties of the struct ure\nwith nonuniform stiffness degradations. In particular, this table sh ows that the\nfactorha\nmcan become significantly large comparing to their initial unit value.\n4.2.Example 2: Design of optimal Rayleigh damping models. In this ex-\nample, thesamestructureasforexample1isconsidered, butonlyw ithnonuniform\nstiffness degradations (yielding scenario 2), which is the more realist ic case.RAYLEIGH DAMPING IN INELASTIC ANALYSES 17\n051015202530354000.511.522.5\nωξa/ξ0\n0 0.2 0.4 0.6 0.8 100.511.522.5\ntξa/ξ0\n051015202530354000.511.522.5\nωξb/ξ0\n0 0.2 0.4 0.6 0.8 100.511.522.5\ntξb/ξ0\n051015202530354000.511.522.5\nωξc/ξ0\n0 0.2 0.4 0.6 0.8 100.511.522.5\ntξc/ξ0\nFigure 3. Time histories of the damping ratios associated to the\nfive modes of the structure with uniform stiffness degradations (s ce-\nnario 1). ×is used for mode 1, ◦for mode 2, /squarefor mode 3, ⋆for\nmode 4, and ⋄for mode 5. Time histories are illustrated in ( ξ,ω)- and\n(ξ,t)-planes. For method b[center], the curve ξb(ω) is the same for all\nmodes. Conversely, for methods a[top] and c[bottom], ξ(ω) depends\non the mode (the ×’s,◦’s,...all belong to a different trajectory in the\n(ξ,ω)-plane). The curves plotted in the ( ξ,ω)-plane are commonly used\nin the literature to illustrate Rayleigh damping. The dashe d lines are\nplotted at t= 0.\nWhen choosing ωA,ωB,ξAandξB, the practitioner has to seek for the best con-\ntrol on all the damping ratiospertaining to the most important mode s, throughout\nthe ITHA. When structure yields, the modal frequencies decreas e, which can lead\nto overdamping modes, especially for the fundamental one which ca n increase a\nlot from its initial value to its actual value after yielding (see figure 4where it\nincreases by a factor of up to 2 .5). In this second example, we present how to18 RAYLEIGH DAMPING IN INELASTIC ANALYSES\n051015202530354000.511.522.53\nωξa/ξ0\n0 0.2 0.4 0.6 0.8 100.511.522.53\ntξa/ξ0\n051015202530354000.511.522.53\nωξb/ξ0\n0 0.2 0.4 0.6 0.8 100.511.522.53\ntξb/ξ0\n051015202530354000.511.522.53\nωξc/ξ0\n0 0.2 0.4 0.6 0.8 100.511.522.53\ntξc/ξ0\nFigure 4. Time histories of the damping ratios associated to the\nfive modes of the structure with nonuniform stiffness degradation s\n(scenario 2). Further details provided in the caption of figure 3also\napply here.\ndesign optimal Rayleigh damping models with methods a(ord) andbto provide\nthe best control possible on the damping ratios time histories.\nTo that purpose, and according to the analytical developments pr esented in the\nprevious section, we now design Rayleigh damping models accounting f or stiffness\ndegradations:\n(i) As above, we consider that assigning a damping ratio to modes 1 an d 3\nis relevant to control the damping ratio of the most important mode s and\nthus setmA= 1andmB= 3. Weremarkthat ω1andω3necessarily remain\nin the range [ ω1(T),ω3(0)] throughout ITHA because ω1(T)< ω1(0) and\nω3(0)> ω3(T) due to stiffness degradations, where Tis the total duration\nof the simulation. Note that the quantities at time Tarea priori not\nknown and need some preliminary analysis results to be efficiently chos en;RAYLEIGH DAMPING IN INELASTIC ANALYSES 19\nTable 2. Time history of the eigenfrequencies [rad.s−1] andha\nm(t)\n[-] factors for the structure with nonuniform stiffness degradat ions\n(see also figure 4).\ntω1ha\n1ω2ha\n2ω3ha\n3ω4ha\n4ω5ha\n5\n0.05.56 1.00 16.23 1.00 25.58 1.00 32.87 1.00 37.49 1.00\n0.25.17 1.16 15.42 1.11 24.34 1.11 31.27 1.11 35.87 1.09\n0.44.72 1.41 14.49 1.28 22.90 1.27 29.45 1.26 34.42 1.16\n0.64.19 1.84 13.37 1.56 21.18 1.54 27.42 1.46 33.15 1.22\n0.83.51 2.85 11.94 2.13 19.05 2.00 25.29 1.68 32.02 1.27\n1.02.39 8.10 9.81 3.82 16.41 2.75 23.18 1.89 31.00 1.31\nhereafter, wewillusetheresultspreviously obtainedwithexample1 . Then,\nthe choice of tAandtBdepends on the method used:\n–Formethod a: accordingtofigure 4,ξa\n1(t)andξa\n3(t)constantlyincrease\nthroughout the analysis. Consequently, the maximum damping ratio s\nfor modes 1 and 3 will be observed at time t=Tand this is why we\nsettA=tB=T. Then, ωA=ω1(T) = 2.39 rad.s−1,ωB=ω3(T) =\n16.41 rad.s−1(R= 6.87),hA=h1(T) = 8.10, andhB=h3(T) = 2.75;\n–For method b: we set tA=TandtB= 0, because, according to\nfigure4,ξ1,max=ξ1(T) andξ3,max=ξ3(0). Then, ωA=ω1(T) =\n2.39 rad.s−1andωB=ω3(0) = 25.58 rad.s−1(R= 10.70).\n(ii) With a targeted damping ratio in the range [ ωA,R ωA] (R ωA=ωB) of\nˆξ= 2%, we compute:\n–For method a: ∆a\nmax= 1.06% from equation ( 27). ∆a\nmaxis an upper\nbound and a lower value is therefore likely to be more appropriate.\nNevertheless, we take ξmax=ξ0=ξA=ξB= 2% + 1 .06% = 3 .06%,\nwhich has to be validated once the analysis has been run;\n–For method b: ∆b= 0.57% from equations ( 23) and we set ξmax=\nξ0=ξA=ξB= 2%+0 .57% = 2.57%.\nThe histories of the damping ratios resulting from the Rayleigh dampin g models\ndesigned according to the procedure presented just above are s hown in figure 5\nwhen method ais used and in figure 6for method b. For the sake of comparison,\nthedampingratiostimehistoriesshowninfigure 4,viz.whenstiffnessdegradations\nare not accounted for in the damping models design, are reproduce d in the top\nhalf of figures 5and6.\nAccording to figure 5, one can make the following observations for method a:\n(i) The damping ratios pertaining to the first three modes remain in th e range\n[1.11%,2.98%] which is, as expected after the analytical developments in\nthe previous section, included in the range [0 .94%,3.06%] = [ ˆξ−∆a\nmax,ˆξ+\n∆a\nmax]. The choice of ξa\nmax=ξa\n1(T) =ξa\n3(T) thus is validated.20 RAYLEIGH DAMPING IN INELASTIC ANALYSES\n(ii)ˆξ= 2% is well centered in [1 .11%,2.98%], which means that the choice of\nξa\nmax=ξa\nA=ξa\nB=ˆξ+∆a\nmaxwas satisfying.\n(iii) Control on the damping ratios is significantly improved when stiffne ss\ndegradations are anticipated, especially for the first mode.\n(iv) Althoughmodes4and5areoutoftheselectedcontrolrange[ ω1(T),ω3(T)],\nthey are less overdamped than when the damping model does not ac count\nfor stiffness degradations.\n051015202530354000.511.522.5No anticipation of stiffness degradations\nωξa/ˆξ\n0 0.2 0.4 0.6 0.8 100.511.522.5\ntξa/ˆξ\n051015202530354000.511.522.5With anticipation of stiffness degradations\nωξa/ˆξ\n0 0.2 0.4 0.6 0.8 100.511.522.5\ntξa/ˆξ\nFigure 5. Structure with nonuniform stiffness degradations. [bot-\ntom] Rayleigh damping model of type awith anticipation of stiffness\ndegradations. [top]Forthesakeofcomparison, thefiguresrepr oduce\nthe results shown in figure 4where stiffness degradations are not ac-\ncounted for in the design of the damping model.\nLooking now at figure 6, we can remark the following for method b:\n(i) The damping ratios pertaining to the first three modes remain in th e range\n[1.47%,2.57%]whichis, asexpected, includedintherange[1 .43%,2.57%] =\n[ˆξ−∆b,ˆξ+∆b]. Wecoulddecrease ξb\nmaxof0.02%tohave ˆξperfectlycentered\nin [1.47%−0.02%,2.57%−0.02%] = [1 .45%,2.55%]: the correction isminorRAYLEIGH DAMPING IN INELASTIC ANALYSES 21\nin this example because the minimum of the ξb(ω) curve is almost reached\nbyξb\n3(T) (see figure 6[bottom])5.\n(ii) Control on the first damping ratio is significantly improved when st iffness\ndegradations are anticipated.\n(iii) Higher modes are overdamped. We could increase Rto have a better\ncontrol on these modes too, but this would increase ∆band consequently\nalter the control on the first three modes.\n051015202530354000.511.522.5No anticipation of stiffness degradations\nωξb/ˆξ\n0 0.2 0.4 0.6 0.8 100.511.522.5\ntξb/ˆξ\n051015202530354000.511.522.5With anticipation of stiffness degradations\nωξb/ˆξ\n0 0.2 0.4 0.6 0.8 100.511.522.5\ntξb/ˆξ\nFigure 6. Structure with nonuniform stiffness degradations. [bot-\ntom] Rayleigh damping model of type bwith anticipation of stiffness\ndegradations. [top]Forthesakeofcomparison, thefiguresrepr oduce\nthe results shown in figure 4where stiffness degradations are not ac-\ncounted for in the design of the damping model.\nFinally, comparing figure 5[bottom] to 6[bottom], we remark that:\n(i) A Rayleigh damping model designed with method awith anticipation of\nstiffness degradations can be more efficient than designed with meth odb\nwithout accounting for stiffness degradations. In particular, it av oids the\nstrong increase of the damping ratio pertaining to the first mode.\n5Such a simple a posteriori correction of ξmaxis possible here because hA=hB= 1 and\nξA=ξB.22 RAYLEIGH DAMPING IN INELASTIC ANALYSES\n(ii) As far as the first three modes are concerned, the control pr ovided by\nmethodbis better: [1 .43%,2.57%]⊂[1.11%,2.98%].\n(iii) Better control on the modes 4 and 5 is observed with method a. However,\nRayleigh damping models were not designed to control those two mod es.\nFormethod b, onecouldhavebettercontrolonmodes4and5, byincreasing\nωB, and soR, until ∆breaches the actual ∆a= 0.98%.\n4.3.Some additional guidelines for practical use. As illustrated above, it is\nnecessary to have some hint of the overall nonlinear structural b ehavior to design a\nRayleighdampingmodelthatefficientlycontrolsthedampingratiotime historyfor\nthe structural modes of interest. This issue is inherent to every R ayleigh damping\nmodel in ITHA. In practice, the time history of the modal propertie s are however\nnot known a priori and designing optimal Rayleigh damping model is thus not\nstraightforward, especially with method a, and might need some iterations. The\nfollowing practical guidelines can therefore be useful:\n(i) It isrecommended in[ 14]thatωAbethefirstmodeand ωBthelowest mode\nfor which the cumulative effective mass exceeds 90%-95%of the tot al mass.\n(ii) The following possible methods are recommended in the user manua l of\nthe computer program PERFORM-3D [ 8,§18.2] to define ωAandωB:\n(28) ( i)/braceleftbigg\nωA= 1.10ω1(0)\nωB= 4.00ω1(0)or (ii)/braceleftbigg\nωA= 1.10ω1(0)/√µ\nωB= 0.85ω1(0),\nwhereµis the ductility of the structure. Method (i) is the same as for\na linear analysis and method (ii) is particularly effective for structure s\ndominated by their first mode.\n5.Conclusions\nFrom the review of the literature proposed in the second section of this paper, it\nis obvious that some researchers or practitioners advocate using the initial stiffness\nmatrix in the design of Rayleigh damping models, whereas others advo cate using\nthe tangent stiffness matrix. Controlling the damping ratios throug hout inelastic\ntime history analyses is an important issue to avoid the appearance o f spurious\ndamping forces. To that purpose, useful analytical formulas are developed in\nthe third section of this paper for both Rayleigh models based on initia l and\ntangent stiffness. Whereas there exists a simple relation that allows controlling\nthe damping ratios time histories when tangent stiffness is used, the re is no such\nrelation when initial stiffness is used and controlling damping ratios is, a lthough\nachievable, not straightforward.\nFrom these latter analytical relations and from the examples shown in section\n4, we can conclude that it is easier to design a Rayleigh damping model w ith well-\ncontrolled damping ratiostime histories throughout the inelastic ana lysis when the\ntangent stiffness is used than with the initial stiffness. However, co ntrolling theRAYLEIGH DAMPING IN INELASTIC ANALYSES 23\ndamping ratios with initial stiffness-based damping models can be achie ved. That\nis why, when convergence difficulties are experienced with the solutio n algorithm,\ninitialstiffnesscanbeusedwithoutnecessarily leadingtoveryhighda mpingratios.\nWhether initial ortangent stiffness isused inthedesignoftheRayleig hdamping\nmodel used for inelastic time history seismic analysis, we advocate th at:\n(i) Figures like figures 5and6in this paper be plotted to keep control on\nthe damping ratios time histories. To that purpose, the required an alytical\nrelations are summarized in table 1of this paper. These later relations also\nhelp improving control on the damping ratios;\n(ii) Thedampingforcesbecomputedandcomparedtotheotherres istingforces\nso as to guarantee there are no spurious damping forces generat ed in the\nsystem. Indeed, even if the damping ratios are well-controlled thro ughout\nthe inelastic time history seismic analysis, Rayleigh damping still lacks\nphysical evidence and there is no guarantee that the actual damp ing forces\nare properly modeled.\nConsidering the difficulty to control Rayleigh damping in inelastic struc tures\nthat is illustrated in this paper, strategies that rely on using models w ith nonlin-\nearities expected to develop in clearly identified structural parts w ith the other\nparts remaining elastic appear as promising. Such strategies are me ntioned in the\nsecond section of this paper but are out of the scope of the develo pments that\nfollow in sections 3 and 4. However, there are some obvious argumen ts that would\nmotivate further investigations for this recent class of methods in future work: i)\nin elastic parts, there is no concerns about damping ratio shift; ii) a r educed – in\ncomparison to the number of structural DOFs – eigenbasis could be defined ( e.g.\nwith the procedures proposed in [ 13] and [12]), which would increase computa-\ntional efficiency; iii) this would simplify the damping model design becaus e the\nnumber of modes, which a proper damping ratio has to be associated to, would be\nreduced; iv) the inelastic parts could be separately treated with pr actical methods\nspecifically developed to avoid the problems likely to be encountered w hen using\nRayleigh damping.\nAcknowledgements This research was supported by a Marie Curie Interna-\ntional Outgoing Fellowship within the 7th European Community Framew ork Pro-\ngramme (proposal No. 275928). The financial support provided b y the Fond\nQu´ eb´ ecois de Recherche sur la Nature et les Technologies (FQRNT ), the Natural\nScience and Engineering Research Council of Canada (NSERC), and the ENS-\nCachan Invited Professor Program also are gratefully acknowledg ed.\nReferences\n[1] Quantification of building seismic performance factors. Technica l Report FEMA P695, Fed-\neral Emergency Management Agency, Washington, DC, June 2009 .24 RAYLEIGH DAMPING IN INELASTIC ANALYSES\n[2] Modeling and acceptance criteria for seismic design and analysis of tall buildings. Techni-\ncal Report PEER 2010/111 or PEER/ATC-72-1, Pacific Earthquak e Engineering Research\nCenter, Richmond (CA), October 2010.\n[3] Dionisio Bernal. Viscous damping in inelastic structural response. ASCE Journal of Struc-\ntural Engineering , 120(4):1240–1254, April 1994.\n[4] Athol J. Carr. Ruaumoko manual, Volume 2: User manual for the 2-dimensional version\nRuaumoko2D. Technical report, University of Canterburry, Chr istchurch, New Zealand,\nNovember 2008.\n[5] Finley A Charney. Unintended consequences of modeling damping in structures. Journal of\nStructural Engineering , 134(4):581–592, April 2008.\n[6] D.J. Chrisp. Damping models for inelastic structures. Master’s th esis, University of Canter-\nbury, Christchurch, New Zealand, 1980.\n[7] Ray W Clough and Joseph Penzien. Dynamics of structures . McGraw-Hill, Inc., 1975.\n[8] CSI. Perform3D User’s manual. Technical report, California, 20 07.\n[9] Emrah Erduran. Evaluation of Rayleigh damping and its influence on engineering demand\nparameterestimates. Earthquake Engineering and Structural Dynamics , 41:1405–1919,2012.\n[10] J F Hall. Problems encountered from the use (or misuse) of Rayle igh damping. Earthquake\nEngineering and Structural Dynamics , 35:525–545, 2006.\n[11] Adnan Ibrahimbegovic. Non-linear solid mechanics: theoretical formulations and finite ele-\nment solution methods . Springer, Berlin, 2009.\n[12] Adnan Ibrahimbegovicand EdwardL Wilson. Simple numerical algor ithms for the mode su-\nperposition analysis of linear structural systems with non-propor tionaldamping. Computers\nand Structures , 33(2):523–531, 1989.\n[13] Adnan Ibrahimbegovic and Edward L Wilson. Efficients solution pro cedures for systems\nwith local non-linearities. Engineering Computations , 9:385–398, 1992.\n[14] Pierre L´ eger and Serge Dussault. Seismic-Energy Dissipation in MDOF Structures. ASCE\nJournal of Structural Engineering , 118(6):1251–1267, May 1992.\n[15] A. D. Nashif, D. I. G. Jones, and J. P. Henderson. Vibration damping . John Wiley Inter-\nscience Publications, New York, 1985.\n[16] P. B. Shing and S. A. Mahin. Elimination of spurious higher-mode re sponse in pseudody-\nnamic tests. Earthquake Engineering and Structural Dynamics , 15:425–445, 1987.\n[17] Farzin Zareian and Ricardo A Medina. A practical method for pro per modeling of structural\ndamping in inelastic plane structural systems. Computers and Structures , 88:45–53, 2010."    },    {        "title": "2311.12357v2.Electrodynamics_with_violations_of_Lorentz_and_U_1__gauge_symmetries_and_their_Hamiltonian_structure.pdf",        "content": "arXiv:2311.12357v2  [hep-th]  10 Apr 2024Electrodynamics with violations of Lorentz and U(1) gauge s ymmetries and their\nHamiltonian structure\nXiu-Peng Yanga,b,1,∗Bao-Fei Lia,b,1,†and Tao Zhua,b1,‡\n1aInstitute for Theoretical Physics and Cosmology,\nZhejiang University of Technology, Hangzhou, 310032, China\nbUnited Center for Gravitational Wave Physics (UCGWP),\nZhejiang University of Technology, Hangzhou, 310032, China\nThis article aims to study the Lorentz/U(1) gauge symmetry- breaking electrodynamics in the\nframework of the Standard-Model Extension and analyze the H amiltonian structure for the the-\nory with a specific dimension d≤4 of Lorentz breaking operators. For this purpose, we con-\nsider a general quadratic action of the modified electrodyna mics with Lorentz/gauge-breaking\noperators and calculate the number of independent componen ts of the operators at different di-\nmensions in gauge invariance and breaking. With this genera l action, we then analyze how the\nLorentz/gauge symmetry-breaking can change the Hamiltoni an structure of the theories by con-\nsidering Lorentz/gauge-breaking operators with dimensio nd≤4 as examples. We show that the\nLorentz-breaking operators with gauge invariance do not ch ange the classes of the constraints of the\ntheory and the number of the physical degrees of freedom of th e standard Maxwell’s electrodynam-\nics. When the U(1) gauge symmetry-breaking operators are pr esent, the theories in general lack\nfirst-class constraint and have one additional physical deg ree of freedom, compared to the standard\nMaxwell’s electrodynamics.\nI.Introduction\nThe Standard Model (SM) successfully describes the\nfundamental constituents of matter using quarks, lep-\ntons, gauge bosons, and Higgs bosons. It effectively\nexplains phenomena involving elementary particles and\ntheir interactions. The last predicted elementary parti-\ncle of the SM, the Higgs boson, was experimentally con-\nfirmed in 2012 [ 1,2], giving a happy ending to the de-\nvelopment of this great theory. General Relativity (GR)\nlinks gravity with the curvature of spacetime and pro-\nvides a highly successful description of gravity-related\nphenomena. Since itsinception, it hasbeen recognizedas\na great theory and continues to be corroborated through\nvarious experiments. Its basic principles and predictions\nhave been confirmed through multiple observations and\nmeasurements, establishing its foundational position in\nmodern physics. In 2016, gravitational waves were ob-\nserved in experiments [ 3], which established the great\nposition of general relativity. These two theories are ex-\npected to be unified at the Planck scale and potentially\nexhibit observable quantum gravity effects at accessible\nlow-energy scales. This signal may be related to Lorentz\nsymmetry breaking and can be described by an effective\nfield theory [ 4].\nTo construct a consistent effective field theory that in-\ncorporates both GR and SM, Kosteleck´ y and Colladay\nproposed an effective field theory called the Standard\nModel Extension (SME) [ 5,6], which features general\nLorentz and CPT violations. The measured and derived\n∗youngxp@zjut.edu.cn\n†libaofei@zjut.edu.cn\n‡corresponding author: zhut05@zjut.edu.cnvalues of coefficients for Lorentz and CPT violations in\nthe SME can be found in the data table organized by\nKosteleck´ y and Russell[7]. In recent years, research\non the Cosmic Microwave Background (CMB) radiation\nand the ultra high energy cosmic rays (UHECR) have\nprovided new opportunities for studying the pure pho-\nton sector of the SME. One reason is that any predic-\ntion involving the pure photon part that deviates from\nSM could potentially indicate Lorentz violation originat-\ning from the pure photon sector of the SME. A large\namount of research has been conducted in this area [ 8–\n23]. Compared to conventional Maxwell electrodynam-\nics, the pure-photon sector of the SME includes addi-\ntional Lorentz-breaking terms, which can be classified\nas CPT-even and CPT-odd. The inclusion of these\nterms leads to the emergence of new effects, which has\nspurred extensive research in this field. Studying the\ngeneral aspects of the pure photon sector is a challeng-\ning task, Ref. [ 24] proposed a general electrodynamics\nextension theory with a quadratic action, which can be\nused to describe many related studies , including pho-\nton interactions [ 25], optical activity of media [ 26], the\nLorentz-invariance-Violating term [ 27,28], the Chern-\nSimons term [ 29], nonminimal SME [ 30], and other re-\nlated phenomena involving photons [ 31]. Theories with\nLorentz-breaking operators of dimension 4 have received\nconsiderable attention, including the Lorentz-invariance-\nviolating(LIV)[ 32], Carroll-Field-Jackiw(CFJ)[ 33], and\nProca electrodynamics [ 34], with the first two being U(1)\ngauge invariant theories and the last one involving U(1)\ngauge symmetry breaking. In Refs. [ 27,28,35,36], they\nconducted Hamiltonian analyses of the aforementioned\nthree theories and identified the constraint structures of\nthe theories, which motivates us to study the constraint\nstructure of more general theories with a specific dimen-\nsiond≤4.2\nThe Lagrangian density of quadratic electrodynamics,\ngiven in [ 24], is a quadratic polynomial in the photon\nfieldAα1and its higher-order derivatives ∂α3...∂αdAα2\nwithd≥2.Since the constant coefficients Kα1α2α3...αd\n(d),\nwhich are contracted with Aα1∂α3...∂αdAα2, remain in-\nvariant under coordinate transformations, it leads to a\nviolation of the Lorentz symmetry of the theory . These\nconstant coefficients can be regarded as originating from\nthe vacuum expectation value of an operator in the un-\nderlying theory, or the dominant component of dynamic\nbackground fields, or an averaged effect. Through di-\nmensional analysis, it can be found that the constants\nKα1α2α3...αd\n(d)associated with Aα1∂α3...∂αdAα2which is\nof dimension dmust have dimension 4 −d. Some re-\nsearchers believe that the theories with power series that\nare renormalizable have mass dimension d≤4 [24,37],\nand it is these theories with mass dimension d≤4 that\nare mainly studied in this work. Usually they are also\nthose theories that contain only the first time deriva-\ntive of the field Aµ, which makes the theories healthy\nas this avoids the potential Ostrogradsky instability that\nthe theory could possess [38,39].\nIn this work, we will extend the U(1) gauge-invariant\nLagrangian density of quadratic electrodynamics de-\nscribed in [ 24] to the one that includes U(1) gauge sym-\nmetry breaking terms. We also perform Hamiltonian\nanalysis for renormalizable specific dimension cases of\nd≤4. Our purpose is to clarify how the U(1) gauge\nbreaking terms affect the constraint structure and phys-\nical degrees of freedom of the theory.\nOur results indicate that the Lagrangian density of\ngeneral quadratic electrodynamics is the combination of\nfive terms. One corresponds to the Lagrangian density\nofstandard electrodynamics, one is U(1) gaugeinvariant,\nonecontainsbothU(1)gaugeinvariantandbreaking,and\nthe remaining two are U(1) gauge breaking. The Hamil-\ntoniananalysisofthegeneralgauge-breakingsystemwith\na specific dimension d≤4 reveals that the system with\nd= 4 requires additional conditions to become a con-\nstrained system, while systems with d= 2 and d= 3 do\nnot. It also shows that the Lorentz-breaking operators\nwith gauge invariance do not change the classes of the\nconstraints of the theory and the number of the phys-\nical degrees of freedom of the standard Maxwell’s elec-\ntrodynamics. When the U(1) gauge symmetry-breaking\noperators are presented, the theories in general lack first-\nclass constraint and have one additional physical degree\nof freedom, compared to the standard Maxwell’s electro-\ndynamics.\nThestructureofthis paperis asfollows. The basicthe-\nory is discussed in Sec. II, where we give the number of\nindependent components of the Lorentz-breaking opera-\ntors with U(1) gauge violation and extend it to the gen-\neral Lagrangian density containing U(1) gauge-breaking\nterms. Sec. IIIis about the Hamiltonian structure and\ndegrees of freedom of theories with a specific dimension\nd≤4. In Sec. IV, We apply the obtained results to some\nspecific models and derive the results for LIV, CFJ, andProca electrodynamics. Our summary is in Sec. V.\nFor the sake of clarity and conciseness, we will use two\nconventions:\n1.The Greek indices range from 0 to 3 and the Latin\nindices run from 1 to 3. The metric of the back-\nground spacetime ηµν≡(1,−1,−1,−1);\n2.The time argument of the vector field Aµis\nsuppressed throughout the manuscript, namely\nAµ(x)≡Aµ(t,x).\nII.Electrodynamics with violations of Lorentz and\nU(1) gauge symmetry\nIn this section, we present a brief introduction of the\nelectrodynamics with quadratic action in the pure pho-\nton sector in the framework of SME with both Lorentz\nand U(1) gauge symmetry breaking. This represents\nan extension of the Lorentz-violating modified electro-\ndynamics with U(1) gauge invariance in [ 24] by includ-\ning the U(1) gauge symmetry-breaking operators in the\nquadratic action. In the construction of this theory, we\nalso analyze the properties of the coefficients for Lorentz-\nand U(1) gauge-violating operators.\nForourpurpose,followingthesimilarconstructionper-\nformed in [ 24], let us start with the general quadratic\naction for the Lorentz-violating electrodynamics in the\npure photon sector, which can be written as [ 24]\nS=/integraldisplay\nd4xL (1)\nwith\nL=−1\n4FµνFµν+∞/summationdisplay\nd=2Kα1α2α3...αd\n(d)Aα1∂α3...∂αdAα2,\n(2)\nwhereFµν≡∂µAν−∂νAµandKα1α2α3...αd\n(d)are con-\nstant coefficients with mass dimension 4 −d. One possi-\nbleexplanationforthe coefficients Kα1α2α3...αd\n(d)originates\nfrom non-zero vacuum expectation values to the Lorentz-\ntensorfields. Note that each term associatedwith the co-\nefficientKα1α2α3...αd\n(d)violates CPTif dis odd orpreserves\nCPT ifdis even.\nThe symmetry among indices {α3,...,α d}of tensor\n∂α3...∂αdand the use of integration by parts results in\ntwo properties of the coefficients Kα1α2α3...αd\n(d). One is the\ntotal symmetry in the d−2 indices {α3,···,αd}, another\nis the symmetry of the two indices {α1,α2}whendis\neven and antisymmetry when dis odd. Depending on the\nspecific intrinsic symmetries of the Lorentz-violating op-\nerators, the coefficients Kα1α2α3...αd\n(d)can be decomposed\ninto five representations [ 24]. When one imposes the\nconditions of the U(1) gauge invariance on the Lorentz-\nviolatingoperators,thesefiverepresentationsarereduced3\ninto two representations, one corresponds to the CPT-\neven coefficient and the other corresponds to CPT-odd\ncoefficient [ 24].\nA.Lorentz-violating electrodynamics with U(1)\ngauge invariance\nLet us first consider the Lorentz-violating electrody-\nnamics with U(1) gauge invariance [ 24]. TheU(1) gauge\ninvariance is a symmetry of the theory under the U(1)\ngauge transformation\nAµ→˜Aµ=Aµ+∂µΛ, (3)\nwhere Λ is an arbitrary function. The variation of the\naction (1) under this gauge transformation reads\nδgS=−∞/summationdisplay\nd=2/integraldisplay\nd4xKα1α2α3···αd\n(d)Λ\n×∂α3···∂αd/parenleftbigg\n∂[α1Aα2]±+1\n2∂[α1∂α2]±Λ/parenrightbigg\n= 0,\n(4)\nwhere “ + /−” corresponds to even/odd dimension,\nand the brackets [] +and []−indicate symmetrization and\nantisymmetrization respectively . Specifically, the terms\nin the big bracket of the above expression can be written\nas\n∂[α1Aα2]±+1\n2∂[α1∂α2]±Λ\n=/braceleftBigg\n∂α1Aα2+∂α2Aα1+∂α1∂α2Λ, dis even,\n∂α1Aα2−∂α2Aα1, d is odd.(5)\nTheU(1) gauge invariance requires the variation of the\naction in ( 4) vanishes. When dis even, since the two first\nindicesα1andα2are symmetric, in order to make ( 4)\nvanishes, one has to require both indices α1andα2are\nantisymmetric with one of {α3,α4,···,αd}[24]. With\nthese properties, it is straightforward to infer that all\nthe CPT-even operators with d= 2 are gauge-violating\nand the gauge invariance requires d≥4. By using these\nproperties, one canalsocountthe number ofindependent\ncomponents of the CPT-even operators, which leads to\nNF= (d+1)d(d−3). (6)\nWhendis odd, similarly, the first two indices α1andα2\nare antisymmetric, and the vanishing of ( 4) require α1\nandα2are antisymmetric with one of {α3,α4,···,αd}.\nFor this case, only operators with d≥3 are allowed and\nthe number of independent components of the CPT-odd\noperators is\nNAF=1\n2(d+1)(d−1)(d−2). (7)\nNote that in Table. I, we summarize the number of in-\ndependent components for the CPT-odd and CPT-even\noperators.B.Lorentz-violating electromagnetics with the\nviolations of U(1)gauge symmetry\nNow let us turn to consider the case with the break-\ning of the U(1) gauge symmetry. When the U(1) gauge\nsymmetry is violated, one does not require the variation\nof the action in ( 4) vanishing. For this case, one does\nnot need to impose extra conditions on the coefficients\nKα1α2α3...αd\n(d)to ensure the gauge invariance of the theory.\nFor CPT-even operators without gauge invariance, as\nwe mentioned before, while the indices {α3,α4,···,αd}\nin the CPT-even coefficients Kα1α2α3...αd\n(d)are symmetric,\nfirst two indices α1andα2are also symmetric. Simi-\nlarly, for CPT-odd operators without gauge invariance,\nthe indices {α3,α4,···,αd}are symmetric, while the\nfirst two indices α1andα2are antisymmetric. One can\nthen count the numberofindependent components ofthe\nCPT-evenand CPT-oddoperatorswith a specific dimen-\nsiond, which are summarized in Table. I.\nTABLE I: The number of independent components of\nthe Lorentz-violatingoperatorswith/without U(1) gauge\ninvariance.\ndCPTGauge Number of\ninvariance independent components\n2 even no 10\neven,\n≥4evenyes ( d+1)d(d−3)\nno2\n3(d+2)(d+1)d\nodd,\n≥3oddyes1\n2(d+1)(d−1)(d−2)\nno1\n2(d+2)(d+1)(d−1)\nFor later convenience, one can decompose the La-\ngrangian density ( 2) into two parts, the U(1) gauge in-\nvariance part and the gauge-violating part. Specifically,\nthis involves rewriting the Lorentz-breaking coefficients\nin five different forms with distinct symmetries. The spe-\ncific representation decomposition of these five forms can\nbe found in [ 24]. Then, the Lagrangian density ( 2) can4\nbe rewritten as\nL=−1\n4FανFαν\n+∞/summationdisplay\nevend=2K(1)α1α2α3...αd\n(d)Aα1∂α3...∂αdAα2\n+∞/summationdisplay\nd=3K(2)α1µνα2...αd−2\n(d)Aα1∂ν∂α2...∂αd−2Aµ\n+∞/summationdisplay\nd=3K(3)µα1να2...αd−2\n(d)Aµ∂ν∂α2...∂αd−2Aα1\n+∞/summationdisplay\nd=4K(4)µνρσα 1...αd−4\n(d)Aµ∂ρ∂σ∂α1...∂αd−4Aν\n+∞/summationdisplay\noddd=3K(5)µνρα1...αd−3\n(d)Aµ∂ρ∂α1...∂αd−3Aν,\n(8)\nwhere the five coefficients K(i)α1α2α3···αd\n(d)withi=\n1,2,3,4,5 are distinguished by their distinct symmetries\namong the indices {α1,α2,α3,···,αd}. The coefficients\nK(1)α1α2α3···αd\n(d)are all CPT-even coefficients with d≥2,\nwhiletheirindices {α1,α2,α3,···,αd}areallsymmetric.\nIt is obvious to infer from ( 4) that these coefficients vio-\nlate theU(1) gauge symmetry of the theory. For the co-\nefficients K(2)α1µνα2···αd−2\n(d), they are be either CPT-even\nor CPT-odd with d≥3. Except the two indices µand\nνare antisymmetric, the rest indices {α1,α2,···,αd−2}\nare all symmetric. Note that these coefficients are also\ngauge-violating. Similarly, for K(3)µα1να2···αd−2\n(d), they can\nalso be either CPT-even or CPT-odd with d≥3. The\ntwo indices of this coefficient µandνare antisymmetric\nandtherestindices {α1,α2,···,αd−2}areallsymmetric.\nOne can check that these coefficients also break the U(1)\ngauge symmetry of the theory. Then for the indices in\nthe coefficients K(4)µνρσα 1...αd−4\n(d), the two indices µand\nρ, and the two indices νandσ, are both antisymmet-\nric. We also note that these coefficients are symmetric\nupon interchange of the two pairs of indices ( µ,ρ) and\n(ν,σ). By inspecting the variation of the action ( 4) with\nthese coefficients, one can infer that the CPT-odd oper-\nators with K(4)µνρα1···αd−4\n(d)violates the gauge invariance,\nwhile the CPT-even ones are gauge invariance. The last\ncoefficients K(5)µνρα1...αd−3\n(d)are CPT-odd coefficient with\nd≥3, andtheir threeindices {µ,ν,ρ}areantisymmetric.\nThese coefficients preserve the U(1) gauge symmetry of\nthe theory.\nTo simplify the laterhandling of the Hamiltonian anal-\nysis of the theory, one can rewrite the Lagrange density\nof the theory in a compact form of\nL=−1\n4FανFαν+Aα1ˆK(1)α1α2Aα2+1\n2Aα1ˆK(2,3)µνα1Fνµ\n−1\n4FµρˆK(4)µρνσFνσ+1\n2ǫκµνρAµˆK(5)\nκFνρ, (9)where\nˆK(1)α1α2≡/summationdisplay\nd=evenK(1)α1α2α3...αd\n(d)∂α3...∂αd,(10)\nˆK(4)µρνσ≡∞/summationdisplay\nd=4K(4)µνρσα 1...αd−4\n(d)∂α1...∂αd−4,(11)\nˆK(5)\nκ≡ǫκµνρ\n6/summationdisplay\nd=oddK(5)µνρα1...αd−3\n(d)∂α1...∂αd−3,\n(12)\nand\nˆK(2,3)µνα1≡\n∞/summationdisplay\nd=3/bracketleftBig\nK(2)α1µνα2...αd−2\n(d)+(−1)dK(3)µα1να2...αd−2\n(d)/bracketrightBig\n×∂α2...∂αd−2. (13)\nThe indice symmetries of the operators in ( 9) are shown\nin Table. II. The indices enclosed in the same braces\n{}of the second and third columns represent symmetry\nand antisymmetry between them, respectively. {µρ,νσ}\nin the second column indicates that the corresponding\noperators are symmetry when the two pairs ( µ,ρ) and\n(ν,σ) are exchanged. The fourth column displays the\nconditions under which each class of operators appears.\nTABLE II: Symmetries of the indices of the coefficients\nof operators in ( 9).\nCoefficient Symmetry Antisymmetry d\nˆK(1)α1α2{α1,α2} ··· even,≥2\nˆK(2,3)µνα1 ··· { µ,ν} ≥ 3\nˆK(4)µρνσ{µρ,νσ} { µ,ρ},{ν,σ} ≥ 4\nˆK(5)\nκ ··· ··· odd,≥3\nIII.Hamiltonian structure of the theory\nInthissection, weperformHamiltoniananalysisonthe\nLorentz-violatingelectromagneticwith and without U(1)\ngaugeinvariancebyusingthe Dirac-Bergmannprocedure\n[40–44]. For simplicity, we will focus on the theory with\nLagrangian densities with a specific dimension d≤4 of\nLorentz/gauge-breaking operators.5\nA.d= 4\nWe start the Hamiltonian analysis for d= 4. For this\ncase, the Lagrangian density of the theory reduces to\nL(4)=−1\n4FµνFµν−1\n2Uµνρσ∂µAν∂ρAσ\n−1\n2VµνρσFνµ∂σAρ−1\n4WµρνσFµρFνσ,(14)\nwhere\nUµνρσ= 2K(1)µνρσ\n(4), (15)\nVµνρσ=K(2)ρµνσ\n(4)+K(3)µρνσ\n(4), (16)\nWµρνσ=K(4)µνρσ\n(4). (17)\nThe terms in L(4)with coefficients UµνρσandVµνρσ\nbreak the U(1) gauge symmetry. The four indices\n{µ,ν,ρ,σ}inUµνρσare total symmetric, while Vµνρσ\nareantisymmetricin {µ,ν}andsymmetricin {ρ,σ}. The\ntermswithcoefficients Wµρνσaregaugeinvariantandthe\ncorresponding indices of the coefficients Wµρνσhave the\nsame symmetric properties of the Riemann tensor, i.e.\nthe first pair {µρ}and the last pair {νσ}ofWµρνσare\nboth antisymmetric, and symmetric upon interchange of\nthe two pairs. Variation the action ( 1) with the above\nLagrangian with respect to the field Aµ, one obtains the\nequation of motion of the electromagnetic, i.e.,\n∂µ/parenleftBig\nFµν+Uµνρσ∂ρAσ+Vνµρσ∂σAρ\n+1\n2VρσνµFσρ+WρσµνFρσ/parenrightBig\n= 0.(18)\nNow to perform the Hamiltonian analysis, it is conve-\nnient to define the conjugate momentum\nπµ≡∂L(4)\n∂˙Aµ=−F0µ−U0µνρ∂νAρ−Vµ0νρ∂ρAν\n−1\n2Vνρµ0Fρν−W0µνρFνρ,(19)which makes the fundamental Poisson brackets (PB) as\n{Aµ(x),πν(y)}=δν\nµδ(x−y). (20)\nFrom the conjugate momentum, the canonical Hamilto-\nnian of the system can be expressed as\nH(4)≡˙Aµπµ−L(4). (21)\nIn Hamiltonian analysis, a significant portion of the\nwork involves computing the Poisson bracket between\nthe Hamiltonian and functions in phase space. In this\ncontext, it is important to express the Hamiltonian as a\nfunction of the conjugate momenta and coordinates. To\ndo so, let’s write the time and spatial components of the\nconjugate momentum for ( 19),\nπ0=−U00µν∂µAν−1\n2Vµν00Fνµ,(22)\nπk=DkiF0i+Nk, (23)\nwhere the matrices DkiandNkare defined as\nDki≡ −δki−Vi0k0−Vk00i−2W0k0i−U0k0i,(24)\nNk≡ −/parenleftBig1\n2Vjik0+W0kij/parenrightBig\nFij−(U0k00+Vk000)∂0A0\n−2(U0k0i+Vk00i)∂iA0−(U0kij+Vk0ij)∂iAj.\n(25)\nFrom the expression of Dki, it is easy to get that Dki=\nDik. Further analysis is required to determine the con-\nstraint structure of the system. For this purpose, we\nassume that Dkiis a non-degenerate matrix, just like in\ngauge invariance [ 27,28], such that\nF0i= (D−1)i\nk(πk−Nk). (26)\nAfter some manipulations, one can finally express the\ncanonical Hamiltonian density as a function of the field\nquantity and its conjugate momentum, namely\nH(4)=πk∂kA0+1\n2(D−1)jl(πl−Nl)/bracketleftBig\nπj−Nj+(V0j0k−Vk0j0)(D−1)k\ni(πi−Ni)−2(U0j00+Vj000)∂0A0/bracketrightBig\n+1\n2Fij/parenleftbigg1\n2Fij+1\n2WijklFkl+2Vji0k∂kA0+Vjikl∂lAk/parenrightbigg\n+/parenleftbigg1\n2Uijkl∂iAj+2U0jkl∂jA0/parenrightbigg\n∂kAl\n+2U00kl∂kA0∂lA0−1\n2U0000∂0A0∂0A0, (27)\nwhere we keep the term of ∂0A0explicitly for the reason\nthat when studying the constrained system with d= 4 in\nthe following we have to impose some constraints on the\nLorentz-breaking coefficients and one of our choice here\nis exactly U0000= 0 that can be seen as throwing awaythe term with respect to ∂0A0.\nFollowing the Dirac-Bergmann algorithm, our next\nstep is to write the total Hamiltonian density of the sys-\ntem, which is composed of the canonical Hamiltonian\ndensity and primary constraints. However, we will see6\nthat for the system in d= 4, the existence of primary\nconstraints requires certain additional conditions.\n1.Conditions for the existence of the constraints\nLet us look at what conditions the system described\nby (14) contains constraints. According to the Dirac-\nBergmann procedure, the presence of constraints in the\nsystem with Lagrangian density L(4)can be determined\nbycheckingwhethertheHessianmatrix∂2L\n∂˙Aµ∂˙Aνisdegen-\nerate. Therefore, for making the system with Lagrangian\ndensityL(4)a constrained system, it is sufficient to pro-\nvide the condition that the matrix∂2L(4)\n∂˙Aµ∂˙Aνis degenerate.\nThe Hessian matrix takes the form,\n∂2L(4)\n∂˙Aµ∂˙Aν=−/parenleftbig\nηµν−η0µδ0ν/parenrightbig\n−2W0µ0ν−U0µ0ν\n−Vµ0ν0−Vν0µ0. (28)\nWhen the Hessian matrix is non-degenerate, the system\ndoesnotpossessanyconstraintthatisnot ofourinterest.\nIt is easy to observe that for gauge invariant case since\none has\nUµνρσ= 0 =Vµνρσ, (29)\nthe Hessian matrix is identically degenerate. This indi-\ncates that the theory with U(1) gauge symmetry always\nhas constraints. However, when the U(1) gauge symme-\ntry is broken, whether the theory possesses constraints\ndepends on the specific forms of the gauge-breaking co-\nefficients UµνρσandVµνρσ.\nFor our purpose, we intend to consider the case when\nthe Hessian matrix is degenerate, either with gauge in-\nvariance or gauge breaking. Considering the complexity\nof the Lagrangian density for d= 4 case, for simplicity,\nlet us only focus on the following simple case to make the\nHessian matrix degenerate,\nU0000= 0, U000i=−Vi000. (30)\nFrom the above equation, it can be seen that this con-\nstraint condition only restricts the gauge-breaking coef-ficients and the first row and column of Hessian matrix\nthat are 0, which allows the subsequent conclusions to be\nfully applicable to gauge-invariant systems. For the dis-\ncussion clarity, similar to gauge invariance case [ 27,28],\nwe will assume that the 3 ×3 matrix∂2L(4)\n∂˙Ai∂˙Ajis non-\ndegenerate.\nThus, for d= 4 case, hereafter we will analyze the\nsystem with\nL(4)=−1\n4FµνFµν−1\n2Uµνρσ∂µAν∂ρAσ\n−1\n2VµνρσFνµ∂σAρ−1\n4WµρνσFµρFνσ,(31)\nwith conditions U0000= 0 and U000i=−Vi000.\n2.Canonical analysis\nAfter obtaining the constrained system with a La-\ngrangiandensityof( 31), thissubsectionanalyzesthecon-\nstraint structure of the system. Under the condition of\n(30), the conjugate momentum in ( 23) can be written as\nπ0=−(U00ij+Vji00)∂iAj−2U000i∂iA0,(32)\nπk=Dk\niF0i+Nk, (33)\nwhere\nNk=−(1\n2Vjik0+W0kij)Fij−2(U0k0i+Vk00i)∂iA0\n−(U0kij+Vk0ij)∂iAj. (34)\nThe rank of matrix∂2L(4)\n∂˙Aµ∂˙Aνis 3, giving rise to a unique\nprimary constraint\nφ(4)\n1=π0+(U00ij+Vji00)∂iAj+2U000i∂iA0≈0.\n(35)\nThesymbol” ≈”iscalledweakequalitysymbolwhichim-\nplies the equation only holds on the constraint surfaces\nbut not throughout phase space. With the expressions\nof (30), the canonical Hamiltonian density ( 27) also be-\ncomes\nH(4)=πk∂kA0+1\n2(D−1)jl(πl−Nl)/parenleftBig\nπj−Nj+(V0j0k−Vk0j0)(D−1)k\ni(πi−Ni)/parenrightBig\n+1\n2Fij/parenleftBig1\n2Fij+1\n2WijklFkl+2Vji0k∂kA0+Vjikl∂lAk/parenrightBig\n+/parenleftbigg1\n2Uijkl∂iAj+2U0jkl∂jA0/parenrightbigg\n∂kAl+2U00kl∂kA0∂lA0, (36)\nand the total Hamiltonian density can be written as\nH(4)T=H(4)+u(4)φ(4)\n1, (37)which gives the total Hamiltonian in the form of\nH(4)T=/integraldisplay\nH(4)T(x)d3x, (38)7\nwhereu(4)is an arbitrary Lagrangian multiplier. Note\nthat∂tA0(x) ={A0(x),H(4)T}=u(4)(x), because the\nPoisson bracket between A0(x) andH(4)(y) is zero, and\nbetween A0(x) and other terms in φ(4)\n1(y), except for\nπ(y), is also zero. This gives meaning to the coefficient\nu1(x): the time derivative of A0(x).\nFollowingthe standardDirac-Bergmannprocedure, we\nthen analyze the requirement for the preservation of the\nprimary constraint. Such a requirement is also known\nas the consistency condition of the primary constraint,\nwhich requires that the time derivative of this constraint\nalso vanishes. The consistency condition of the primary\nconstraint, i.e.,\n˙φ(4)\n1(x) ={φ(4)\n1(x),H(4)T} ≈0, (39)\ngives rise to a secondary constraint of the system,\nφ(4)\n2=∂kπk+Mmi(∂mπi−∂mNi)\n+(5U00ij+Vji00)∂i∂jA0\n+Vji0k∂kFij+2U0jkl∂j∂kAl≈0,(40)\nwhere\nMmi=1\n2/bracketleftBig\n(D−1)ij+(D−1)ji\n+2(V0l\n0k−Vk0l0)(D−1)li(D−1)k\nj/bracketrightBig\n×/parenleftbig\n3U0j0m+2Vj00m+Vjm00/parenrightbig\n,(41)\nwhich indicates that the structure of Gauss’s law is in-\nfluenced by the gauge-breaking term.\nSimilar to the primary constraint, the secondary con-\nstraintφ2≈0, being a constraint itself, also has a corre-\nsponding consistency condition, which is\n˙φ(4)\n2(x) ={φ(4)\n2(x),H(4)T} ≈0. (42)\nThis condition then leads to\nOik\n1j∂i∂k(πj−Nj)+Oijk\n2∂i∂j∂kA0+Ojilk\n3∂j∂i∂lAk\n+Oklij\n4∂k∂lFij+Tij∂i∂ju(4)≈0, (43)where\nOjilk\n3≡(δjm+Mjm)(1\n2Vmikl+Uimlk), (44)\nOklij\n4≡1\n2/parenleftbig\nδkm+Mkm/parenrightbig\n(ηliηmj+Wijlm+Vjiml),(45)\nOik\n1j≡1\n2/bracketleftBig\n(δim+Mim)/parenleftbig\nVmkl0+2W0lkm\n+U0lkm+Vl0km/parenrightbig\n+Mi\nm(Vlkm0+2W0mkl+U0mkl+Vm0kl)\n+2(Vli0k+U0ikl)/bracketrightBig\n×/bracketleftBig\n(D−1)jl+(D−1)lj\n+2(V0i10k1−Vk10i10)(D−1)i1j(D−1)k1\nl/bracketrightBig\n,\n(46)\nOijk\n2≡(δil+Mil)(Vlj0k+2U0jkl)\n+Mil(U0ljk+Vl0jk)+2U0ijk, (47)\nand\nTij≡6U00ij+(3U00jk+Vkj00+2Vk00j)Mi\nk.\n(48)\nForgaugeinvariancecase, one has Tij= 0, one can check\nthat the condition ( 43) satisfies identically. When the\nU(1) gaugesymmetry of the theory is violated, Tijare in\ngeneralnonzeros,then( 43)representsarestrictiononthe\nLagrange multiplier u(4). One special case is that Tij=\n0 for some specific combination of the gauge breaking\ncoefficients, for which a new constraint arises, which is\nφ(4)\n3=Oik\n1j(∂i∂kπj−∂i∂kNj)+Oijk\n2∂i∂j∂kA0\n+Ojilk\n3∂j∂i∂lAk+Oklij\n4∂k∂lFij≈0.(49)\nWith this constraint, repeating the above procedure, its\nconsistency condition may produce more constraints un-\nder certain conditions. It is worth mentioning that ac-\ncording to Table. 1, the number of independent coeffi-\ncients is finite, and the emergence of new constraints will\ngive a set of limiting equations about the coefficients.\nJust as in the presence of φ(4)\n3≈0,Tij= 0 gives 9\nequations between the coefficients, so that the number of\nindependent coefficients will be reduced. Detailed analy-\nsis will show that each time a new constraint is present,\nthe number of limiting equations for the coefficients will\nincrease quickly as compared to the previous constraint,\neventuallystoppingthe generationofpossibleconstraints\nat a certain step, thus making the Poisson bracket closed\nand the number of constraints limited. According to the\nanalysis in [ 45,46], such constraint may also lead to the\nunphysical halfdegreeof freedom. However, this requires\na very special choice of gauge-breaking coefficients. For\nsimplicity, we will not explore these specific cases in de-\ntail in this paper.\nBefore we go further, we would like to summarize the\nmain results we have gotten so far for the d= 4 case. For8\nboth the gauge invariance case and gauge breaking case\nwith the degenerate condition ( 30), the theory can have\none primaryconstraint φ(2)\n1and one secondaryconstraint\nφ(4)\n2.\n3.Counting the degrees of freedom\nAfter obtaining all the primary and secondary con-\nstraints of the system, let us turn to identify the first-\nand second-class constraints of the system by analyzing\nthe Poisson bracket of the constraints. In general, the\nfirst-class constraints are associated with the gauge sym-\nmetry of the theory, they are gauge generators, which\ngenerategaugetransformationsthat don’talterthephys-\nical state. The second-class constraints can not gener-\nate gauge transformations since the transformation gen-\nerated by a second-class does not preserve all the con-\nstraints, whichviolatestheconsistencycondition, butare\nimportant in the definition of Dirac bracket, which plays\na key role in the transition from classical theory to quan-\ntum theory [ 43]. Specifically, the first-class constraints\nare those constraints whose Poisson bracket with every\nconstraintvanishesweakly,otherwisearethesecond-class\nof constraint.\nThe Poissonbracketof the primaryconstraint φ(4)\n1and\nthe secondary constraint φ(4)\n2gives\n{φ(4)\n2(y),φ(4)\n1(x)}= (C1)ij∂′i∂jδ(x−y)\n+(C2)ij∂′\ni∂′\njδ(x−y).(50)\nwhere∂′denotes the partial derivative with respect to y,\nand\n(C1)ij=−/parenleftbig\nU00ji+(U00jk+Vkj00)Mik/parenrightbig\n,(51)\n(C2)ij= 2Mik(U0k0j+Vk00j)+5U00ij.(52)\nIt is easy to see that for the gauge invariance case,\nCij\n1= 0 =Cij\n2, therefore, φ(4)\n1andφ(4)\n2are both first-\nclass constraints. For the case without the gauge sym-\nmetry, the gauge breaking coefficients UµνρσandVµνρσ\nare in general nonzero, thus the theory does not possess\nany first-class constraint. In this case, φ(4)\n1andφ(4)\n2are\nboth second-class constraints. This result is also what\nwe expect since the first-class constraint can only exist\nwhen the theory has gauge symmetry, and thus a the-\nory without gauge symmetry should not have first-class\nconstraint.\nGetting all first- and second-class constraints, one can\ncount the number of physical degrees of freedom ( NDOF)\nby using the following formula [ 47],\nNDOF=1\n2(Nvar−2NOF−NOS), (53)\nwhere “Nvar” means the total number of canonical vari-\nables, “NOF” is the number of constraints of the first\nclass, and “NOS” represents the number of second-classconstraints. Thus, for the gauge invariance case, the\nnumber of degrees of freedom is\nNDOF=1\n2(8−2×2) = 2, (54)\nwhich is the same as that in the standard Maxwell’s elec-\ntrodynamics, while in the case of gauge violation, one\nhas\nNDOF=1\n2(8−2) = 3. (55)\nTherefore, the violation of the U(1) gauge symmetry in-\nduces one additional physical degree of freedom, com-\nparedtothe standardMaxwell’selectrodynamicsandthe\ntheory with gauge invariance. This additional physical\ndegree of freedom results in a third state of polarization,\ncorresponding to a new particle called longitudinal pho-\nton [48]. This representsa new electromagneticradiation\nwhich may alter the radiation spectra of a lot of sources\nwith nonzerotemperature [ 48]. However, its phenomeno-\nlogical effects in both experiments and astrophysical ob-\nservations are expected to be too small to be detected up\nto now [48–50].\nB.d= 3\nIn this subsection, let’s turn to analyze the constraint\nstructure of the theory with Lorentz/gauge-breaking op-\nerators with dimension d= 3. The Lagrangian density\nof the considered theory can be written in the form of\nL(3)=−1\n4FανFαν+1\n2SµνρAρFνµ\n+1\n2ǫκµνρ(kAF)κAµFνρ. (56)\nwhere\n(kAF)κ=1\n3!ǫκµνρK(5)µνρ\n(3), (57)\nSµνρ=K(2)ρµν\n(3)−K(3)µρν\n(3). (58)\nNote that the terms with ( kAF)κare gauge-invariantand\nthe terms with Sµνρare gauge-breaking. Variation of the\naction with respect to Aµ, one obtains field equation\nǫκνµρ(kAF)κFµρ+1\n2SµρνFρµ\n+∂µ(Fµν−SνµρAρ) = 0. (59)\nThecorrespondingconjugatemomentumofthistheory\nread,\nπµ=∂L(3)\n∂˙Aµ=−F0µ+/parenleftbig\nǫβν0µkAFβ+Sµ0ν/parenrightbig\nAν.(60)\nTo analyze the constraint structure of the theory, it is\nconvenient to write down the time and spatial compo-\nnents of the above conjugate momentum ( 60) as\nπ0= 0, (61)\nπk=−F0k+/parenleftbig\nǫjµ0kkAFj+Sk0µ/parenrightbig\nAµ.(62)9\nOne can conclude that in the specific dimension d= 3,\nthegauge-breakingandgauge-invariantLorentz-breaking\nelectrodynamics have the same form for the conjugate\nmomentum π0of the photon field A0. From these prop-\nerties, the only primary constraint of the system is\nφ(3)\n1=π0≈0. (63)\nThen, the canonical Hamiltonian density and total\nHamiltonian density are given by\nH(3)=πk˙Ak−L(3), (64)\nand\n(H(3))T=H(3)+u(3)φ(3)\n1. (65)\nHere, similar to the case of d= 4, the coefficient u(3)is\nalso the time derivative of A0.\nOnce again, the consistency condition of the primary\nconstraint gives a secondary constraint\nφ(3)\n2=˙φ(3)\n1\n=∂kπk+1\n2Fik/parenleftBig\nǫj0ik(kAF)j+Ski0/parenrightBig\n+Sk00(ǫjµ0k(kAF)j+Sk0µ)Aµ−πkSk00\n≈0. (66)\nSimilarly, this means that the form of Gauss’s law needs\nto be modified when considering the presence of the\ngauge-breaking term. The consistency condition of φ(3)\n2\nimpose a restriction on the u(3):\n˙φ(3)\n2=Sl00Sl00u(3)−/parenleftbig\nSi0k+Sik0/parenrightbig\n∂kπi\n+Sk00/parenleftbig\n2ǫjk0i(kAF)j+Si0k−Sk0i/parenrightbig\nπi\n+Si00/bracketleftbig\n2ǫl0ik(kAF)l+Si0k−Sk0i/bracketrightbig\n×[ǫjn0k(kAF)j+Sk0n]An\n+1\n2/parenleftbig\n−2Sj00ηki+Sjik)∂kFij\n−1\n2(ǫ0kij(kAF)0+Sjik)Sk00Fij\n+2Sl00(Sl0k+Slk0)∂kA0\n+Si00/bracketleftbig\n2ǫl0i\nk(kAF)l+Si0\nk−Sk0i/bracketrightbig\nSk00A0\n+/bracketleftBig\nSk00(ǫ0jik(kAF)0+Skij)\n+(Sk0i+Ski0)/bracketleftbig\nǫlj0k(kAF)l+Sk0j/bracketrightbig/bracketrightBig\n∂iAj\n≈0. (67)\nFor gauge invariance case, all the gauge breaking coef-\nficientsSµνρ= 0, and thus the above consistency con-\ndition satisfies identically. In this case, the theory only\nhas two constraints, the primary constraint φ(3)\n1and the\nsecondary constraint φ(3)\n2.\nWhen the gauge symmetry is violated, the gauge\nbreaking coefficients Sµνρare in general nonzeros. Forthis case, the above consistency condition leads to a spe-\ncific form of u(3),\nu(3)≈ −1\nSm00Sm00(68)\n×/bracketleftBigg\nSi00/bracketleftbig\n2ǫl0ik(kAF)l+Si0k−Sk0i/bracketrightbig\n×[ǫjn0k(kAF)j+Sk0n]An\n−/parenleftbig\nSi0k+Sik0/parenrightbig\n∂kπi\n+Sk00/parenleftbig\n2ǫjk0i(kAF)j+Si0k−Sk0i/parenrightbig\nπi\n+1\n2/parenleftbig\n−2Sj00ηki+Sjik)∂kFij\n−1\n2(ǫ0kij(kAF)0+Sjik)Sk00Fij\n+2Sl00(Sl0k+Slk0)∂kA0\n+Si00(2ǫl0i\nk(kAF)l+Si0\nk−Sk0i)Sk00A0\n+/bracketleftBig\nSk00(ǫ0jik(kAF)0+Skij)\n+(Sk0i+Ski0)/bracketleftbig\nǫlj0k(kAF)l+Sk0j/bracketrightbig/bracketrightBig\n∂iAj/bracketrightBigg\n.\n(69)\nThis indicates that the theory for this case does not have\nany additional constraints. Similar to the gauge invari-\nance case, it only has two constraints, one primary con-\nstraintφ(3)\n1and one secondary constraint φ(3)\n2. Here we\nwould like to mention that, under certain conditions on\nthe gauge breaking coefficients such that S00\nlSl00= 0,\nthe theorymay produceadditional constraints. However,\nthis requires a very special choice of gauge-breaking co-\nefficients. For simplicity, we will not explore this specific\ncase in detail in this paper and focus on the case with\nS00\nlSl00/ne}ationslash= 0 when gauge symmetry is violated.\nThen, let us consider the Poisson bracket of the pri-\nmary constraint φ(3)\n1and the secondary constraint φ(3)\n2,\nwhich is\n{φ(3)\n1(x),φ(3)\n2(y)}=−Sk00Sk00δ(x−y).(70)\nIt is obvious that for gauge invariance case, since Sµνρ=\n0, the abovePoissonbracketvanishes and the constraints\nφ(3)\n1andφ(3)\n2are both first-class. Thus, the number of\nthe physical degrees of freedom is\nNDOF=1\n2(8−2×2) = 2, (71)\nwhich is the same as that in the standard Maxwell’s elec-\ntrodynamics.\nFor the case with gauge violation, since in general\nSk00Sk00/ne}ationslash= 0,φ(3)\n1,φ(3)\n2arebothsecond-classconstraints.\nThus, the number of the physical degrees of freedom is\nNDOF=1\n2(8−2) = 3. (72)10\nSimilar to the case with d= 4, the violation of the U(1)\ngauge symmetry induces one extra physical degree of\nfreedom, compared to the standard Maxwell’s electro-\ndynamics and the case with gauge invariance.\nC.d= 2\nAfter completing the constraint structure analysis of a\nspecific dimension d= 3 andd= 4, let us turn to analyze\nthe structure of d= 2, which Lagrangian density is\nL(2)=−1\n4FµνFµν+1\n2UµνAµAν, (73)\nwith\nUµν= 2K(1)µν\n(2), (74)\nwhich are gauge-breaking terms. Variation of the action\nwith respect to Aµ, one obtains\nUνµAµ+∂µFµν= 0 (75)\nThe conjugate momenta now read\nπµ=∂L(2)\n∂˙Aµ=−F0µ. (76)\nOne difference from d= 3 and d= 4 is that at this point\nthe conjugate momenta πµare not affected by gauge-\nbreaking coefficients, Uµν, so they are the same as in\nMaxwell electrodynamics since the time derivative of the\nphotofield Aµonlyappearsinthe Maxwellterm, the first\nterm in ( 73). Therefore, the only primary constraint is\nφ(2)\n1=π0≈0. (77)\nThe canonical Hamiltonian density just differs by one\nterm ofUµνfrom that in Maxwell electrodynamics:\nH(2)=πk∂kA0−1\n2πkπk+1\n4FikFik−1\n2UµνAµAν.\n(78)\nThis gives the total Hamiltonian density:\nH(2)T=H(2)+u(2)φ(2)\n1. (79)\nThe influence of gauge-breaking terms are reflected in\nsecondary constraint because it originates from the Pois-\nson bracket between primary constraint and total Hamil-\ntonian which now is gauge-breaking. It reads\nφ(2)\n2=∂kπk+U0µAµ≈0. (80)\nThe consistency condition of φ2is\n˙φ(2)\n2=Uµk∂kAµ+U0k∂kA0−U0kπk+u(2)U00≈0,\n(81)which provides the constraint on u(2)whenU00/ne}ationslash= 0:\nu(2)≈(U00)−1/bracketleftbig\nU0k(πk−∂kA0)−Uµk∂kAµ/bracketrightbig\n.(82)\nNoticing the absence of π0inH(2), we get ˙A0(x) =\n{A0(x),H(2)T}=u(2)(x). As a consequence, the mean-\ning ofu(2)is the time derivative of A0. Combining ( 82),\nthe time component A0of the photon field will be deter-\nmined by the first-order differential equation.\nFinally, the Poisson bracket of φ(2)\n1andφ(2)\n2is\n{φ(2)\n1(x),φ(2)\n2(y)}=−U00δ(x−y). (83)\nWhen the gauge breaking coefficients Uµνare nonzeros,\nφ(2)\n1,φ(2)\n2areboth secondclassconstraints. Similartothe\ncases ofd= 4 andd= 3, we will not explore in detail the\ncase with U00= 0, which may induce more constraint.\nThen the number of the physical degree of freedom is\nNDOF=1\n2(8−2) = 3. (84)\nAgain, comparedto the standardMaxwellelectrodynam-\nics, since d= 2 operators always break the U(1) gauge\nsymmetry, it induces one additional physical degree of\nfreedom.\nIV.Map to several specific models\nThe Lorentz-violating electrodynamics presented in\nthis paper provide a unifying framework for describing\npossible violations of Lorentz and U(1) gauge symme-\ntries in the electromagnetic interaction. In this section,\nwe present several specific modified electrodynamics by\nwriting their actions in the form of ( 1) and summa-\nrize their Hamiltonian structure from our general anal-\nysis. We consider three specific theories, the Lorentz-\ninvariance-Violating (LIV) electrodynamics, the Carroll-\nField-Jackiw (CFJ) electrodynamics, and the Proca elec-\ntrodynamics. The first two theories only break the\nLorentz symmetry of the theory, while the last one only\nbreaksU(1) gauge symmetry.\nA.LIV electrodynamics\nWe first consider LIV electrodynamics, which is pro-\nposed in [ 27,28] and the Lagrangian density is given by\n[27,28]\nLLIV=−1\n4FµνFµν−1\n4WµνρσFµνFρσ,(85)\nwhich corresponds to the case in our model where the\ngauge-breaking coefficients UµνρσandVµνρσare set to\nbe zeros for d= 4. Using the result from ( 35) and (40),\nwe obtain the constraint structure in this case:\n(φ1)LIV=π0≈0, (86)\n(φ2)LIV=∂kπk≈0. (87)11\nTABLE III: The number of the first-class and second-class const raints, and the number of the physical degrees of\nfreedom for each case with a specific dimension d= 2,d= 3, and d= 4.\ndgauge invariance number of first-class constraint number of first-class constraint number of DOF\n2 no 0 2 3\n3yes 2 0 2\nno 0 2 3\n4yes 2 0 2\nno 0 2 3\nAnd we need verify that the consistency of ( φ2)LIVgives\nno new constraints. Noticing that Tij=Mij= 0 by\nusing (41) and (48) sinceVµνρσ=Uµνρσ= 0, then we\nonly need to check whether φ(4)\n3in (49) is identically zero\nthroughout the phase space, not just on the constraint\nsurface. In this case, the only possible non-zero coeffi-\ncients remaining in ( 49) areOik\n1jandOklij\n4. According\n(33), as a result, φ(4)\n3now changes to\nW0lik((D−1)jl+(D−1)lj)∂k∂i(πj−Nj)\n+1\n2(ηliηkj+Wijlk)∂k∂lFij. (88)\nSincethelasttwoindicesof Wijklareantisymmetric, and\ni,jinηliηkj∂k∂lare symmetric, ( 88) is equal to 0. This\nproves that ( φ2)LIVdoes not give new constraints. In\naddition, since the theory has gauge symmetry, both the\nconstraints ( φ1)LIVand (φ2)LIVare first-class, thus the\ntheory has two degrees of freedom, the same as that in\nthe standard Maxwell electrodynamics. Our result here\nis consistent with the results in LIV [ 27,28].\nB.CFJ electrodynamics\nFor CFJ electrodynamics, its Lagrangian density is\ngiven by [ 35]:\nLCFJ=−1\n4FµνFµν+1\n2ǫκµνρ(kAF)κAµFνρ,(89)\nwith the caveat that the coefficients ( kAF)κdiffers by a\nfactor of −1\n2compared to [ 35]. The Lagrangian density\n(89) can be obtained in our model by setting Sµνρ= 0 to\nzero ind= 3. Similarly, according to the results in ( 63)\nand (66), we have two constraints for this theory,\n(φ1)CFJ=π0=F00≈0, (90)\n(φ2)CFJ=∂kπk+1\n2Fikǫj0ik(kAF)j≈0,(91)\nwhere (φ1)CFJis the primary constraint and ( φ2)CFJ\nis the secondary constraint. According to ( 67), it canbe concluded that in this case, ( ˙φ2)CFJis always zero\nin phase space, which does not generate new constraints.\nThis result confirms [ 35], with the additional point to\nnote that the coefficient kAFhere differs by a factor of\n−1\n2compared to [ 35]. Note that both the ( φ1)CFJand\n(φ2)CFJare first-class and the number of degrees of free-\ndom is the same as that in the standard Maxwell electro-\ndynamics.\nC.The Proca electrodynamics\nWhend= 2, if one sets Uµν=m2ηµν, then the La-\ngrangian density L(2)will return to the case of Proca\nelectrodynamics [ 36]:\nLProca=−1\n4FανFαν+1\n2m2AµAµ,(92)\nwheremis the mass of the photon. Because of this mass\nterm, the Proca electrodynamics breaks the U(1) gauge\nsymmetry of the theory. Replacing the constant coeffi-\ncientsUµνwith the product of metric tensor ηµνandm2\nis the reason why Proca electrodynamics doesn’t break\nLorentz symmetry as the product is also a tensor. Same\nreason as d= 3,4, at this point, the theory has two con-\nstraints as well,\n(φ1)Proca=π0≈0, (93)\n(φ2)Proca=∂kπk+m2A0≈0.(94)\nAnd (φ2)Procagives no new constraints. They are\nboth second-class since {(φ1)Proca(x),(φ2)Proca(y)}=\nm2δ(x−y). Thus this theory propagates three physical\ndegrees of freedom, which is different from the two de-\ngrees of freedom in the standard Maxwell electrodynam-\nics and the cases with gauge invariance. These results\nare consistent with [ 36], with the only difference being\nthat their coefficient m2differs from ours by a factor of\n1\n2.12\nV.Summary and discussion\nIn this paper, we perform an extended analysis of\nthe modified electrodynamics with the violations of both\nLorentz symmetry and U(1) gauge symmetry in the\nframework of the Standard-Model Extension. This rep-\nresents an extension of the previous construction of the\nLorentz-violating electrodynamics with gauge invariance\n[24]. For our purpose, by following the procedure in\n[24], we construct the quadratic Lagrangian density of\nelectrodynamics by allowing the violations of both the\nLorentz and U(1) gauge symmetries. The Lorentz- and\ngauge-violating effects in the quadratic Lagrangian are\nrepresented by the new operators with specific dimen-\nsiond≤4. With the constructed quadratic Lagrangian,\nwe calculate in detail the number of independent com-\nponents of the Lorentz-violating operators at different\ndimensions with both cases of the gauge invariance and\ngauge violation cases.\nWe then perform the Hamiltonian analysis of the gen-\neral theory by considering Lorentz and gauge-breaking\noperators with dimension d≤4 as examples. Specifi-\ncally, weperformthe analysisfor d= 4,d= 3, and d= 2,\nrespectively. It is shown that the Lorentz-breaking op-\nerators with gauge invariance do not change the classes\nof the constraints of the theory and have the same num-\nber of physical degrees of freedom as that in the stan-\ndard Maxwell’s electrodynamics. When the U(1) gauge\nsymmetry-breaking operators are presented, the theo-ries in general lack first-class constraint and have one\nadditional physical degree of freedom, compared to the\nstandard Maxwell’s electrodynamics. The results of the\nHamiltonian structure and their corresponding number\nof degrees of freedom are presented in Table. III.\nFinally,wealsomapourgeneralanalysistoseveralspe-\ncific modified electrodynamics, including LIV electrody-\nnamics, CFJ electrodynamics, and Proca electrodynam-\nics. While the former two theories represent two specific\nexamples of Lorentz-violating theory with gauge invari-\nance, the latter one is a theory that breaks U(1) gauge\nsymmetrybutstillkeepsthe Lorentzsymmetry. 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Gen. 6L12 (1973) ."    },    {        "title": "2106.11212v1.Gagliardo_Nirenberg_inequalities_in_Lorentz_type_spaces_and_energy_equality_for_the_Navier_Stokes_system.pdf",        "content": "arXiv:2106.11212v1  [math.AP]  21 Jun 2021Gagliardo-Nirenberg inequalities in Lorentz type spaces a nd\nenergy equality for the Navier-Stokes system\nYanqing Wang∗, Wei Wei†and Yulin Ye‡\nAbstract\nIn this paper, we derive some new Gagliardo-Nirenberg type inequalit ies in Lorentz\ntype spaces without restrictions on the second index of Lorentz n orms, which generalize\nalmost all known corresponding results. Our proof mainly relies on th e Bernstein in-\nequalities in Lorentz spaces, the embedding relation among various L orentz type spaces,\nand Littlewood-Paley decomposition techniques. In addition, we est ablish several novel\ncriteria in terms of the velocity or the gradient of the velocity in Lore ntz spaces for\nenergy conservation of the 3D Navier-Stokes equations. Particu larly, we improve the\nclassical Shinbrot’s condition for energy balance to allow both the sp ace-time directions\nof the velocity to be in Lorentz spaces.\nMSC(2020): 35A23, 42B35, 35L65, 76D03\nKeywords: Gagliardo-Nirenberg inequality; Lorentz spaces; energy e quality; Navier-\nStokes equations; Littlewood-Paley decomposition\n1 Introduction\n1.1 Gagliardo-Nirenberg inequality\nAn important way to study well-posedness of partial different ial equations is via the re-\nsearch of their weak solutions. It is well known that Gagliar do-Nirenberg inequality is a\nfundamental tool to improve the regularity of weak solution s, which has gained widespread\napplications such as in Hilbert’s 19th problem [15], Caffarel li-Kohn-Nirenberg theorem [8]\nfor the 3D Navier-Stokes equations and the critical quasi-g eostrophic equations [9]. The\nclassical integer version of Gagliardo-Nirenberg inequal ity is the generalization of Sobolev\nembedding theorem and was discovered independently by Gagl iardo [19] and Nirenberg [33]\nas follows: for all smooth functions uinRnwith compact support, there holds\n/ba∇dblDju/ba∇dblLp(Rn)≤C/ba∇dblDmu/ba∇dblθ\nLr(Rn)/ba∇dblu/ba∇dbl1−θ\nLq(Rn), (1.1)\n∗Department of Mathematics and Information Science, Zhengz hou University of Light Industry,\nZhengzhou, Henan 450002, P. R. China Email: wangyanqing200 56@gmail.com\n†Corresponding author. Center for Nonlinear Studies, Schoo l of Mathematics, Northwest University,\nXi’an, Shaanxi 710127, P. R. China Email: ww5998198@126.co m\n‡School of Mathematics and Statistics, Henan University, Ka ifeng, Henan 475004, P. R. China. Email:\nylye@vip.henu.edu.cn\n1wherej,mare any integers satisfying 0 ≤j <m, 1≤q,r≤ ∞,and\n1\np−j\nn=θ(1\nr−m\nn)+(1−θ)1\nq\nfor allθ∈[j\nm,1], unless 1 <r<∞andm−j−n\nris a nonnegative integer.\nUp to now, there have been extensive investigations on Gagli ardo-Nirenberg inequalities\ninvolving fractional derivatives and various function spa ces (see [7, 13, 14, 22, 26, 30, 31,\n34, 36, 40–42]). On one hand, the most general form of fractio nal Gagliardo-Nirenberg\ninequality in Lebesgue spaces was recently presented by Haj aiej-Molinet-Ozawa-Wang [21]\nand by Chikami [13]: for 0 ≤σ<s<∞and 1≤q,r≤ ∞,there holds\n/ba∇dblΛσu/ba∇dblLp(Rn)≤C/ba∇dblu/ba∇dblθ\nLq(Rn)/ba∇dblΛsu/ba∇dbl1−θ\nLr(Rn)(1.2)\nwithn\np−σ=θn\nq+(1−θ)/parenleftBign\nr−s/parenrightBig\n,\nwhere 0≤θ≤1−σ\ns(θ/\\e}atio\\slash= 0 ifs−σ≥n\nr/parenrightbig\nand Λs= (−∆)s\n2is defined via /hatwidestΛsf(ξ) =|ξ|sˆf(ξ).\nFor the fractional Gagliardo-Nirenberg inequality in the f ramework of nonhomogeneous\nSobolev spaces, Besov spaces and Fourier-Herz spaces, we re fer the reader to [7, 13, 21] and\nreferences therein. On the other hand, recent progress on Ga gliardo-Nirenberg inequality\nin Lorentz spaces for some special cases was made in [14, 22, 3 1]. In particular, Hajaiej-\nYu-Zhai [22] established the Gagliardo-Nirenberg inequal ity for Lorentz spaces below: for\n1≤p,p2,q,q1,q2<∞,0<α<q, 0<s<n and 1<p1<n/s,\n/ba∇dblu/ba∇dblLp,q(Rn)≤C/ba∇dblΛsu/ba∇dblα\nq\nLp1,q1(Rn)/ba∇dblu/ba∇dblq−α\nq\nLp2,q2(Rn), (1.3)\nwith\nα\nq1+q−α\nq2= 1 andα/parenleftbigg1\np1−s\nn/parenrightbigg\n+(q−α)1\np2=q\np.\nIn [31], McCormick-Robinson-Rodrigo proved the following inequality\n/ba∇dblf/ba∇dblLp(Rn)≤C/ba∇dblf/ba∇dblθ\nLq,∞(Rn)/ba∇dblΛsf/ba∇dbl1−θ\nL2(Rn), (1.4)\nwhere\n1\np=θ\nq+(1−θ)/parenleftbigg1\n2−s\nn/parenrightbigg\n, n/parenleftbigg1\n2−1\np/parenrightbigg\n<s,1≤q<p<∞ands≥0.\nSubsequently, this result was improved by Dao-D´ ıaz-Nguye n [14] as follows: for any α>0,\n/ba∇dblf/ba∇dblLp,α(Rn)≤C/ba∇dblf/ba∇dblθ\nLq,∞(Rn)/ba∇dblΛsf/ba∇dbl1−θ\nL2(Rn)(1.5)\nwith\n1\np=θ\nq+(1−θ)/parenleftbigg1\n2−s\nn/parenrightbigg\n, n/parenleftbigg1\n2−1\np/parenrightbigg\n<s,1≤q<p<∞ands≥0.\nOne of main purposes of this paper is to establish the most gen eral form of Gagliardo-\nNirenberg inequality in Lorentz spaces, parallel to the cla ssical version (1.1) and the frac-\ntional case (1.2).\n2Theorem 1.1. Suppose that u∈Lq,∞(Rn) and Λsu∈Lr,∞(Rn). Let 0≤σ<s< ∞and\n1<q,r≤ ∞.Then there exists a positive constant C=C(n,q,p,r,s,σ ) such that\n/ba∇dblΛσu/ba∇dblLp,1(Rn)≤C/ba∇dblu/ba∇dblθ\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn)(1.6)\nwithn\np−σ=θn\nq+(1−θ)/parenleftBign\nr−s/parenrightBig\n,\nwhere 0<θ<1−σ\nsands−n\nr/\\e}atio\\slash=σ−n\np.\nRemark 1.1.The motivation of this theorem is twofold. On one hand, it is w orth noting\nthat Gagliardo-Nirenberg inequalities (1.3)-(1.5) in Lor entz spaces mentioned above are\nonly concerned with the subcritical case or the case that q < p, and the left-hand side\nof estimates (1.3)-(1.5) is without higher derivatives. On the other hand, since there are\nsome extra restrictions on the second index of Lorentz norms in (1.3)-(1.5) and that in the\nresults of [26], it seems that their versions of Gagliardo-N irenberg inequalities for Lorentz\nspaces are not easily applicable in the field of partial differe ntial equations. We emphasize\nhere that Theorem 1.1 removes all the aforementioned restri ctions and covers most of the\npossible range of exponents.\nRemark 1.2.It is essential to make an exception that s−n\nr/\\e}atio\\slash=σ−n\npin this theorem,\notherwise the generalized Gagliardo-Nirenberg inequalit y (1.6) is invalid for u(x) =|x|s−n\nr.\nWe also remark that the restriction that q,r>1 results from an application of Lemma 3.1.\nRemark 1.3.A special case of (1.6) is that\n/ba∇dblu/ba∇dblLp,p1(Rn)≤C/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn)/ba∇dblu/ba∇dblθ\nLq,∞(Rn),1<q,r≤ ∞,1≤p1≤ ∞, s>0,\nwheren\np= (1−θ)/parenleftBign\nr−s/parenrightBig\n+nθ\nq,0<θ<1.\nThis result still extends the aforementioned inequalities (1.3)-(1.5) for Lorentz spaces.\nInspiredbythework[13], weshalldividetheproofofTheore m1.1intothreecases bythe\nrelationship between s−σandn/r. Firstly, we focus on the subcritical case s−σ <n/r.\nRough speaking, we observe that there exist at least four kin ds of equivalent definitions\nof Lorentz norms (see (2.3) for details). This together with the pointwise interpolation\nestimate for derivatives (3.5) given in [1] enables us to pro ve that\n/ba∇dblΛσu/ba∇dblLp,∞(Rn)≤C/ba∇dblu/ba∇dbl1−σ\ns\nLq,∞(Rn)/ba∇dblΛsu/ba∇dblσ\ns\nLr,∞(Rn),with1\np=/parenleftBig\n1−σ\ns/parenrightBig1\nq+σ\nsr.(1.7)\nThen, making use of the interpolation characteristic (2.4) and Sobolev inequality in Lorentz\nspaces, we can obtain the desired estimates in this case. The second case is devoted to\ndealing with the critical case s−σ=n/r. To this end, in the spirit of [13, 41, 42], we\nestablish the following estimate in the critical Besov-Lor entz spaces\n/ba∇dblu/ba∇dblLq,l(Rn)≤C/ba∇dblu/ba∇dblp\nq\nLp,∞(Rn)/ba∇dblu/ba∇dbl1−p\nq\n˙Bn\nrr,∞,∞,withp<q. (1.8)\nTo achieve this, various Bernstein inequalities (2.12)-(2 .15) in Lorentz spaces are derived.\nThe new Bernstein inequality (2.15) allows us to get the fact that\n/ba∇dblf/ba∇dbl˙Bsp,∞,∞≤C/ba∇dblΛsf/ba∇dblLp,∞(Rn). (1.9)\n3Combining (1.7), (1.8) and (1.9), we may prove the critical c ase. For the third case, we\nproceed with the case s−σ>n/r by setting up the following key estimate\n/ba∇dblu/ba∇dblL∞(Rn)≤C/ba∇dblu/ba∇dblθ\nLp,∞(Rn)/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞,with 0 =θn\np+(1−θ)/parenleftBign\nr−s/parenrightBig\n,0<θ≤1.\nBy a similar argument used in the previous two cases, this yie lds the generalized Gagliardo-\nNirenberg inequality (1.6) under the supcritical case, whi ch concludes Theorem 1.1.\nFurthermore, it should be stated that the Bernstein inequal ities (2.13)-(2.14) together\nwith low-high frequency techniques as in [13] also guarante e the following generalized\nGagliardo-Nirenberg inequality in the framework of Besov- Lorentz spaces, which extends\nthe corresponding results in [13, 21].\nTheorem 1.2. Assume that u∈˙Bs\nr,∞,∞(Rn)∩˙B0\nq,∞,∞(Rn) with 1< q,r≤ ∞and\n0≤σ<s<∞.Then there exists a positive constant C=C(n,q,p,r,s,σ ) such that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞, (1.10)\nwithn\np−σ=θn\nq+(1−θ)/parenleftBign\nr−s/parenrightBig\n,0<θ<1−σ\ns, s−n\nr/\\e}atio\\slash=σ−n\np.\nRemark1.4.Thistheorem implies several versions of Gagliardo-Nirenb erg inequalities, such\nas Theorem 1.1 for Lorentz spaces and [13, Theorem 5.3] for Be sov spaces. Due to [21], it\nis necessary to make an assumption that s−n\nr/\\e}atio\\slash=σ−n\npin Theorem 1.2.\nIt is worth pointing out that there has existed an extensive s tudy on Besov-Lorentz\nspaces (see [24, 35, 42, 44] and references therein). In [42] , Wadade presented the critical\nBesov-Lorentz inequality (1.8) under the case that q>max{p,r}, and he also proved that\n/ba∇dblu/ba∇dblLq1,q2(Rn)≤C/ba∇dblu/ba∇dblp1\nq1\nLp1,p2(Rn)/vextenddouble/vextenddouble/vextenddoubleΛn\np1u/vextenddouble/vextenddouble/vextenddouble1−p1\nq1\nLp1,p2(Rn),with 1<p1≤q1<∞,1≤p2≤q2≤ ∞,\n(1.11)\nwhich improves the classical result below due to Ozawa [34]\n/ba∇dblu/ba∇dblLq(Rn)≤C/ba∇dblu/ba∇dblp\nq\nLp(Rn)/vextenddouble/vextenddouble/vextenddoubleΛn\npu/vextenddouble/vextenddouble/vextenddouble1−p\nq\nLp(Rn),with 1<p≤q<∞. (1.12)\nCombining the Besov-Lorentz inequality (1.8) and the embed ding relation in Lemma 2.5,\nwe have the following corollary.\nCorollary 1.3. For1< p < q < ∞,1< r <∞and1≤l≤ ∞, there exists a positive\nconstantC=C(n,q,p,r,l )such that\n/ba∇dblu/ba∇dblLq,l(Rn)≤C/ba∇dblu/ba∇dblp\nq\nLp,∞(Rn)/ba∇dblΛn\nru/ba∇dbl1−p\nq\nLr,∞(Rn), (1.13)\n/ba∇dblu/ba∇dblLq,l(Rn)≤C/ba∇dblu/ba∇dblp\nq\nLp,∞(Rn)/ba∇dblu/ba∇dbl1−p\nq\n˙Fn\nrr,∞,∞, (1.14)\n/ba∇dblu/ba∇dblLq,l(Rn)≤C/ba∇dblu/ba∇dblp\nq\nLp,∞(Rn)/ba∇dblu/ba∇dbl1−p\nq\n˙Bn\nrr,∞,∞. (1.15)\nRemark 1.5.This corollary generalizes the critical interpolation ine qualities (1.11) and\n(1.12). The results obtained here can be applied to deduce th e Trudinger-Moser type\ninequality as in [34, 41, 42].\n4Next, as an application of the generalized Gagliardo-Niren berg inequalities (1.6) and\n(1.10) in Lorentz type spaces, we shall derive some new condi tions for energy equality\nof the Leray-Hopf weak solutions to the 3D Navier-Stokes equ ations. For the study of\nincompressiblehydrodynamicsequations in Lorentz spaces , werefer the reader to[2, 10, 25].\n1.2 Energy conservation in the Navier-Stokes system\nThe 3D incompressible Navier-Stokes system can be written a s\n\n\nvt−∆v+v·∇v+∇Π = 0,\ndivv= 0,\nv|t=0=v0,(1.16)\nwhere the unknown vector v=v(x,t) describes the flow velocity field, and the scalar\nfunctionΠrepresentsthepressure. Theinitialdatum v0isgivenandsatisfiesthedivergence-\nfree condition. It is known that regular solutions to the 3D N avier-Stokes equations (1.16)\nsatisfy the energy equality\n/ba∇dblv(T)/ba∇dbl2\nL2(R3)+2/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl2\nL2(R3)ds=/ba∇dblv0/ba∇dbl2\nL2(R3).\nHowever, the global Leray-Hopf weak solutions of the 3D Navi er-Stokes equations (1.16)\njust obey the energy inequality\n/ba∇dblv(T)/ba∇dbl2\nL2(R3)+2/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl2\nL2(R3)ds≤ /ba∇dblv0/ba∇dbl2\nL2(R3).\nThe criteria for energy equality of the Leray-Hopf weak solu tions have been established by\nseveral authors (see e.g. [3, 5, 11, 11, 12, 28, 37, 38, 45] and references therein). We list some\nknown results in this direction below. A Leray-Hopf weak sol utionvto the Navier-Stokes\nequations (1.16) satisfies the energy equality if one of the f ollowing conditions holds\n•Lions [28]: v∈L4(0,T;L4(R3));\n•Shinbrot [37]:\nv∈Lp(0,T;Lq(R3)),with2\np+2\nq= 1 andq≥4; (1.17)\n•Taniuchi [38], Beirao da Veiga-Yang [3]: v∈Lp(0,T;Lq(R3)),with2\np+2\nq= 1 andq≥\n4,or1\np+3\nq= 1 and 3<q<4;\n•Cheskidov-Constantin-Friedlander-Shvydkoy [11]: v∈L3(0,T;B1\n3\n3,∞(R3));\n•Cheskidov-Luo[12]: v∈Lβ,∞(0,T;B2\nβ+2\np−1\np,∞(R3)),with2\np+1\nβ<1and1≤β <p≤ ∞;\n•Berselli-Chiodaroli [5], Zhang [45]:\n∇v∈Lp/parenleftbig\n0,T;Lq/parenleftbig\nR3/parenrightbig/parenrightbig\n,with1\np+3\nq= 2 and3\n2<q<9\n5,or1\np+6\n5q= 1 andq≥9\n5.\n(1.18)\n5From the work of Cheskidov-Luo [12], we see that for Shinbrot ’s condition (1.17) the\nLebesgue spaces in time direction can be replaced by Lorentz spaces. Since the Lorentz\nspacesLr,∞are larger than the Lebesgue spaces Lrin general, a natural question arises\nwhether energy equality holds for the Leray-Hopf weak solut ionvwhose space direction\nbelongs to Lorentz spaces. Our next main result gives a parti ally affirmative answer.\nTheorem 1.4. Theenergy equality of Leray-Hopf weak solutions vto the3D Navier-Stokes\nequations (1.16) is valid if one of the following five conditi ons is satisfied\n(1)v∈L4(0,T;L4,∞(R3)); (1.19)\n(2)v∈Lp,∞(0,T;Lq,∞(R3)),with2\np+2\nq= 1, q>4; (1.20)\n(3)v∈Lp(0,T;Lq,∞(R3)),with1\np+3\nq= 1,3<q<4; (1.21)\n(4)∇v∈Lp/parenleftbig\n0,T;Lq,∞/parenleftbig\nR3/parenrightbig/parenrightbig\n,with1\np+3\nq= 2,3\n2<q<9\n5; (1.22)\n(5) Λsv∈Lp/parenleftbig\n0,T;Lq,∞/parenleftbig\nR3/parenrightbig/parenrightbig\n,with1\np+6\n5q=2s\n5+3\n5, s>1, q>1,1\ns<p<3.\n(1.23)\nRemark 1.6.Theorem 1.4 here refines the corresponding results in [3, 5, 3 7, 38, 45]. It\nis worth pointing out that the proof of (1.21) and (1.23) stro ngly rests on the Gagliardo-\nNirenberg type inequality (1.6) without restrictions on th e second index of Lorentz norms.\nRemark 1.7.Following the path in the proof of (1.23) and using (1.10), on e can prove the\nfollowing condition via Besov-Lorentz spaces for energy eq uality\nv∈Lp(0,T;˙B5\n2p+3\nq−3\n2q,∞,∞(R3)),withq>1,0<p<3 and5\n2p+3\nq−3\n2>max{1,1\np}.\nWe remark that even the stronger version of this condition, w ith space direction in Besov\nspaces˙B5\n2p+3\nq−3\n2q,∞(R3),still improves the results involving nonhomogeneous Besov spaces in\n[12].\nRemark 1.8.According to the boundedness of Riesz transform on Lorentz s paces,∇vin\n(1.22) can be replaced by its symmetrical part vorticity cur lvor its antisymmetric part\n1\n2(∇v−∇vT).\nRemark 1.9.To the best of authors’ knowledge, it remains an open problem to show that\nenergy equality can be derived from the following condition\nv∈L4,∞(0,T;L4,∞(R3)).\nRemark 1.10.Eventually, we would like to mention that an energy conserva tion criterion\nvia a combination of velocity and its gradient for the equati ons (1.16) in T3was recently\nestablished in [43].\nThe rest of this paper is organized as follows. In Section 2, w e recall some basic ma-\nterials of various Lorentz type spaces and present embeddin g relation among these spaces.\nThe generalized Young inequality and Bernstein inequaliti es for Lorentz spaces are also\nestablished in this section. Section 3 is devoted to the proo f of Theorem 1.1 and Theorem\n1.2. Finally, as an application of the above two theorems, we prove Theorem 1.4 in Section\n4, which gives several new criteria for energy conservation of 3D Navier-Stokes equations in\nLorentz spaces.\n62 Notations and key auxiliary lemmas\n2.1 Lorentz spaces and generalized Bernstein inequality\nThroughout this paper, we will use the summation convention on repeated indices. Cwill\ndenote positive absolute constants which may be different fro m line to line unless otherwise\nstated in this paper. a≈bmeans that C−1b≤a≤Cbfor some constant C >1.χΩ\nstands for the characteristic function of a set Ω ⊂Rn.|E|represents the n-dimensional\nLebesgue measure of a set E⊂Rn. LetMbe the Hardy-Littlewood maximal operator and\nits definition is given by\nMf(x) = sup\nr>01\n|B(r)|/integraldisplay\nB(r)|f(x−y)|dy,\nwherefis any locally integrable function on Rn, andB(r) is the open ball centered at the\norigin with radius r>0.\nNext, we present some basic facts on Lorentz spaces. Recall t hat the distribution func-\ntion of a measurable function fon Ω is the function f∗defined on [0 ,∞) by\nf∗(α) =|{x∈Ω :|f(x)|>α}|.\nThe decreasing rearrangement of fis the function f∗defined on [0 ,∞) by\nf∗(t) = inf{α>0 :f∗(α)≤t}.\nForp,q∈(0,∞], we define\n/ba∇dblf/ba∇dblLp,q(Ω)=\n\n/parenleftBig/integraldisplay∞\n0/parenleftBig\nt1\npf∗(t)/parenrightBigqdt\nt/parenrightBig1\nq,ifq<∞,\nsup\nt>0t1\npf∗(t),ifq=∞.\nFurthermore,\nLp,q(Ω) =/braceleftbig\nf:fis a measurable function on Ω and /ba∇dblf/ba∇dblLp,q(Ω)<∞/bracerightbig\n,\nwhich implies that L∞,∞=L∞,Lq,q=LqandL∞,q={0}for 0<q<∞.\nNotice that identity definition of Lorentz norm can be found i n [20, 29]. Indeed, for\n0<p≤ ∞and 0<q≤ ∞, there holds\n/ba∇dblf/ba∇dblLp,q(Ω)=p1\nq/ba∇dbl/ba∇dblαχ(α,∞)(|f(·)|)/vextenddouble/vextenddoubleLp(Ω)/vextenddouble/vextenddouble\nLq(R+,dα\nα)=\n\n/parenleftBig\np/integraldisplay∞\n0αqf∗(α)q\npdα\nα/parenrightBig1\nq,ifq<∞,\nsup\nα>0αf∗(α)1\np,ifq=∞.\nSimilarly, one can define Lorentz spaces Lp,q(0,T;X) in time for 0 < p,q≤ ∞.f∈\nLp,q(0,T;X) means that /ba∇dblf/ba∇dblLp,q(0,T;X)<∞, where\n/ba∇dblf/ba∇dblLp,q(0,T;X)=\n\n/parenleftBig\np/integraldisplay∞\n0αq|{t∈[0,T) :/ba∇dblf(t)/ba∇dblX>α}|q\npdα\nα/parenrightBig1\nq,ifq<∞,\nsup\nα>0α|{t∈[0,T) :/ba∇dblf(t)/ba∇dblX>α}|1\np,ifq=∞.\n7Note that the triangle inequality is not valid for /ba∇dbl·/ba∇dblLp,q(Rn).Another equivalent norm\nin Lorentz spaces is defined as\n/ba∇dblf/ba∇dbl∗\nLp,q(Rn)=\n\n/parenleftbigg/integraldisplay∞\n0/parenleftBig\nt1\npf∗∗(t)/parenrightBigqdt\nt/parenrightbigg1\nq\n,if 1<p<∞,1≤q<∞,\nsup\nt>0t1\npf∗∗(t),if 1<p≤ ∞,q=∞,(2.1)\nwhere\nf∗∗(t) =1\nt/integraldisplayt\n0f∗(s)ds= sup\n|E|≥t/parenleftbigg1\n|E|/integraldisplay\nE|f(x)|dx/parenrightbigg\n, t>0.\nIn addition, Lorentz spaces endowed with the norm /ba∇dbl·/ba∇dbl∗\nLp,qare Banach spaces, and there\nholds\n/ba∇dblf/ba∇dblLp,q(Rn)≤ /ba∇dblf/ba∇dbl∗\nLp,q(Rn)≤p\np−1/ba∇dblf/ba∇dblLp,q(Rn). (2.2)\nMost of the above statement is borrowed from [6, 10, 20].\nSubsequently, we present norm-equivalence concerning Lor entz spaces.\nLemma 2.1. Letfbe inLp,q(Rn)with1<p≤ ∞and1≤q≤ ∞. Then there holds\n/ba∇dblf/ba∇dblLp,q(Rn)≤C1/ba∇dblf/ba∇dbl∗\nLp,q(Rn)≤C2/ba∇dblMf/ba∇dblLp,q(Rn)≤C3/ba∇dblf/ba∇dbl∗\nLp,q(Rn)≤C4/ba∇dblf/ba∇dblLp,q(Rn),(2.3)\nwhereC1,C2,C3andC4are positive constants depending only on p,qandn.\nProof.Sincef∈Lp,q(Rn) with 1<p≤ ∞and 1≤q≤ ∞, it follows from (2.7) that fis a\nlocally integrablefunctionon Rn. RecallthepointwiseinequalityinvolvingHardy-littlew ood\nmaximal operator (see [16, p.41] and [4, Chapter 3]) below,\nc1(Mf)∗(t)≤f∗∗(t)≤c2(Mf)∗(t), t>0,\nwherec1andc2are positive constants depending only on n. This together with (2.1) means\n/ba∇dblf/ba∇dbl∗\nLp,q(Rn)≤C/ba∇dblMf/ba∇dblLp,q(Rn)≤C/ba∇dblf/ba∇dbl∗\nLp,q(Rn).\nThe conclusion is a straightforward consequence of the latt er and (2.2).\nRemark 2.1.Even if 0< q <1, the equivalent relation /ba∇dblMf/ba∇dblLp,q(Rn)≈/ba∇dblf/ba∇dblLp,q(Rn)still\nholds for all functions f∈Lp,q(Rn) with 1< p≤ ∞. Indeed, thanks to Marcinkiewicz’s\ninterpolation theorem for Lorentz spaces [20, Theorem 1.4. 19], it follows from the fact that\nHardy-littlewood maximal operator Mis a sublinear operator of both weak type (1 ,1) and\nstrong type ( ∞,∞) thatMis also bounded on Lp,q(Rn) for anyp∈(1,∞] andq∈(0,∞],\nwhich yields that /ba∇dblMf/ba∇dblLp,q(Rn)≤C(n,p,q)/ba∇dblf/ba∇dblLp,q(Rn)for all functions f∈Lp,q(Rn). On\nthe other hand, Lebesgue’s differentiation theorem implies t hat|f(x)| ≤ Mf(x) for almost\nallx∈Rn, which yields that f∗≤(Mf)∗and/ba∇dblf/ba∇dblLp,q(Rn)≤ /ba∇dblMf/ba∇dblLp,q(Rn)for 1<p≤ ∞\nand 0<q≤ ∞.\nWe list the properties of Lorentz spaces as follows.\n8•Interpolation characteristic of Lorentz spaces [6, 14]\n/ba∇dblf/ba∇dblLp,p1(Rn)≤/bracketleftbigg(r−q)p2\n(r−p)(p−q)p1/bracketrightbigg1\np1/ba∇dblf/ba∇dblα\nLq,∞(Rn)/ba∇dblf/ba∇dbl1−α\nLr,∞(Rn),\nwith1\np=α\nq+1−α\nr,0<α<1,0<q<p<r ≤ ∞and 0<p1≤ ∞.(2.4)\n•\n/ba∇dbl|f|λ/ba∇dblLp,q(Rn)=/ba∇dblf/ba∇dblλ\nLλp,λq(Rn),with 0<λ<∞and 0<p,q≤ ∞.\n•H¨ older’s inequality in Lorentz spaces [32]\n/ba∇dblfg/ba∇dblLr,s(Ω)≤C(r1,r2,s1,s2)/ba∇dblf/ba∇dblLr1,s1(Ω)/ba∇dblg/ba∇dblLr2,s2(Ω),\nwith1\nr=1\nr1+1\nr2,1\ns=1\ns1+1\ns2,0<r1,r2,s1,s2≤ ∞.(2.5)\n•The Lorentz spaces increase as the exponent qincreases [20, 29]\nFor 0<p≤ ∞and 0<q1<q2≤ ∞,\n/ba∇dblf/ba∇dblLp,q2(Ω)≤/parenleftBigq1\np/parenrightBig1\nq1−1\nq2/ba∇dblf/ba∇dblLp,q1(Ω). (2.6)\n•Inclusion in Lorentz spaces on bounded domains [20, 29]\nFor any 1 ≤m<M≤ ∞, 1≤r,q≤ ∞,\n/ba∇dblf/ba∇dblLm,r(Ω)≤/parenleftBig1\nm/parenrightBigr−1\nr/parenleftBigq\nM/parenrightBig1\nq|Ω|1\nm−1\nM\n1\nm−1\nM/ba∇dblf/ba∇dblLM,q(Ω). (2.7)\n•Sobolev inequality in Lorentz spaces [32, 40]\n/ba∇dblf/ba∇dbl\nLnp\nn−p,p(Rn)≤C(n,p)/ba∇dbl∇f/ba∇dblLp(Rn)with 1≤p<n. (2.8)\n•Young inequality in Lorentz spaces [32]\nLet 1<p,q,r< ∞, 0<s1,s2≤ ∞,1\np+1\nq=1\nr+1, and1\ns=1\ns1+1\ns2. Then there holds\n/ba∇dblf∗g/ba∇dblLr,s(Rn)≤C(p,q,s1,s2)/ba∇dblf/ba∇dblLp,s1(Rn)/ba∇dblg/ba∇dblLq,s2(Rn). (2.9)\nIt should be mentioned that the classical Young inequality ( 2.9) due to O’Neil requires\nthe first index of every Lorentz norm is larger than 1. Grafako s [20] improved O’Neil’s\nresult and showed that\n/ba∇dblf∗g/ba∇dblLq,∞(Rn)≤C/ba∇dblf/ba∇dblLr,∞(Rn)/ba∇dblg/ba∇dblLp(Rn),with1\nq+1 =1\np+1\nr,1≤p<∞and 1<q,r< ∞.\nThe following lemma extends it to a more general version, whi ch we shall give a different\nproof from that of [20, Theorem 1.2.13].\n9Lemma 2.2. Suppose that 0<l≤s≤ ∞,1≤r <∞and1<p,q < ∞. Iff∈Lq,l(Rn)\nandg∈Lr(Rn)with\n1\np+1 =1\nq+1\nr, (2.10)\nthenf∗g∈Lp,s(Rn), and there exists a positive constant Cdepending only on r,q,sandl\nsuch that\n/ba∇dblf∗g/ba∇dblLp,s(Rn)≤C/ba∇dblf/ba∇dblLq,l(Rn)/ba∇dblg/ba∇dblLr(Rn). (2.11)\nRemark 2.2.In general, (2.11) fails when l > s. Indeed, here is a counterexample as\nfollows: for any f∈Lq,l(Rn) with 0< s < l ≤ ∞and 1< q <∞, it follows from\n(2.7) and (2.6) that fis a locally integrable function on Rn. Takeg=χQ, whereQ=\n{(y1,y2,...,yn)∈Rn: maxi|yi| ≤1/2}is a cube in Rn. Let (χQ)r(x) =r−nχQ(x/r) for\nallx∈Rnandr >0. Then Lebesgue’s differentiation theorem guarantees that l im\nm→∞f∗\n(χQ)1\nm(x) =f(x) for almost all x∈Rn, which together with [20, Proposition 1.4.5] implies\nthatf∗≤liminf\nm→∞/parenleftBig\nf∗(χQ)1\nm/parenrightBig∗\n. Hence we may apply Fatou’s lemma and (2.11) with r= 1\nto derive that for 0 <s<l≤ ∞and 1<q<∞,\n/ba∇dblf/ba∇dblLq,s(Rn)≤liminf\nm→∞/ba∇dblf∗(χQ)1\nm/ba∇dblLq,s(Rn)≤C/ba∇dblf/ba∇dblLq,l(Rn)/ba∇dblχQ/ba∇dblL1(Rn)=C/ba∇dblf/ba∇dblLq,l(Rn).\nThis contradicts the fact that Lq,s(Rn)/subse≈orno≈dbleqlLq,l(Rn). Additionally, we remark that necessity\nof the condition (2.10) results from dilation structure of t he convolution f∗gin (2.11),\nand the exclusion of endpoint cases for the three indices q, randpis due to [20, Example\n1.2.14].\nThe proof of this lemma relies on Marcinkiewicz’s interpola tion theorem for Lorentz\nspaces as follows, which is given in [20, Theorem 1.4.19].\nLemma 2.3. Let0< r≤ ∞,0< p0/\\e}atio\\slash=p1≤ ∞,and0< q0/\\e}atio\\slash=q1≤ ∞and let(X,µ)\nand(Y,v)be two measure spaces. Let Tbe either a quasilinear operator with some constant\nK >0defined on Lp0(X) +Lp1(X)and taking values in the set of measurable functions\nonYor a linear operator defined on the set of simple functions on Xand taking values as\nbefore. Assume that for some M0,M1<∞the following (restricted) weak type estimates\nhold:\n/ba∇dblT(χA)/ba∇dblLq0,∞≤M0µ(A)1/p0,\n/ba∇dblT(χA)/ba∇dblLq1,∞≤M1µ(A)1/p1,\nfor all measurable subsets AofXwithµ(A)<∞.Fix0<θ<1and let\n1\np=1−θ\np0+θ\np1and1\nq=1−θ\nq0+θ\nq1.\nThen there exists a positive constant C,which depends only on K,p0,p1,q0,q1,randθ,such\nthat for all functions fin the domain of Tand inLp,r(X)we have\n/ba∇dblT(f)/ba∇dblLq,r≤C(M0+M1)/ba∇dblf/ba∇dblLp,r.\nNow we continue with the proof of Lemma 2.2.\n10Proof of Lemma 2.2. Since (2.6) implies Lp,l(Rn)֒→Lp,s(Rn), it suffices to prove (2.11) for\nthe case when s=l.\nTo this end, fix g∈Lr(Rn). LetT(f) =f∗g, thenTis a linear operator defined on the\nset of simple functions on Rn. For all measurable subsets AofRnwith|A|<∞, it follows\nfrom (2.6) and Young’s inequality for Lebesgue spaces that\n/ba∇dblT(χA)/ba∇dblLr,∞(Rn)≤ /ba∇dblT(χA)/ba∇dblLr(Rn)≤ /ba∇dblg/ba∇dblLr(Rn)/ba∇dblχA/ba∇dblL1(Rn)=/ba∇dblg/ba∇dblLr(Rn)|A|,\n/ba∇dblT(χA)/ba∇dblL∞,∞(Rn)=/ba∇dblT(χA)/ba∇dblL∞(Rn)≤ /ba∇dblg/ba∇dblLr(Rn)/ba∇dblχA/ba∇dblLr′(Rn)=/ba∇dblg/ba∇dblLr(Rn)|A|1/r′,\nwhere 1<r′≤ ∞and 1/r+1/r′= 1.\nTakeθ= 1−r/p=r(1−1/q)∈(0,1). Then the hypotheses on the indices imply that\n1\nq=1−θ\n1+θ\nr′and1\np=1−θ\nr+θ\n∞.\nWith the help of Lemma 2.3, we obtain that for all functions f∈Lq,l(Rn) withl∈(0,∞]\nthere holds\n/ba∇dblf∗g/ba∇dblLp,l(Rn)=/ba∇dblT(f)/ba∇dblLp,l(Rn)≤C/ba∇dblg/ba∇dblLr(Rn)/ba∇dblf/ba∇dblLq,l(Rn).\nHereC >0 depends only on r,qandl. This completes the proof.\nAs an application of this lemma, we may derive the generalize d Bernstein inequality for\nLorentz spaces as follows.\nLemma 2.4. Let a ballB={ξ∈Rn:|ξ| ≤R}with0< R <∞and an annulus C=\n{ξ∈Rn:r1≤ |ξ| ≤r2}with0<r1<r2<∞. Then a positive constant Cexists such that\nfor any nonnegative integer k,any couple (p,q)with1<p<q < ∞,anyλ∈(0,∞), and\nany function uinLp,∞(Rn)or inLq,l(Rn)with0<l≤ ∞, there hold\nsup\n|α|=k/ba∇dbl∂αu/ba∇dblL∞(Rn)≤Cλk+n\np/ba∇dblu/ba∇dblLp,∞(Rn)withsupp/hatwideu⊂λBand1<p≤ ∞; (2.12)\nsup\n|α|=k/ba∇dbl∂αu/ba∇dblLq,1(Rn)≤Cλk+n/parenleftBig\n1\np−1\nq/parenrightBig\n/ba∇dblu/ba∇dblLp,∞(Rn)withsupp/hatwideu⊂λB; (2.13)\nsup\n|α|=k/ba∇dbl∂αu/ba∇dblLq,l(Rn)≤Cλk/ba∇dblu/ba∇dblLq,l(Rn)withsupp/hatwideu⊂λB; (2.14)\nC−1λk/ba∇dblu/ba∇dblLq,l(Rn)≤sup\n|α|=k/ba∇dbl∂αu/ba∇dblLq,l(Rn)≤Cλk/ba∇dblu/ba∇dblLq,l(Rn)withsupp/hatwideu⊂λC.(2.15)\nHereλB={ξ∈Rn:|ξ| ≤λR}andλC={ξ∈Rn:λr1≤ |ξ| ≤λr2}.\nRemark 2.3.This lemma extends the following result due to McCormick-Ro binson-Rodrigo\nin [31], for supp /hatwidef⊂λB,\n/ba∇dblf/ba∇dblLp(Rn)≤cλn(1/q−1/p)/ba∇dblf/ba∇dblLq,∞(Rn),1<q<p≤ ∞.\nRemark2.4.Inthis lemma, thenecessity of theassumption that p,q>1results from Young\ninequality (2.11) and (2.9) for Lorentz spaces.\n11Proof.(1) Letψbe a Schwartz function on Rnsuch thatχB≤ˆψ≤χ2B. Since/hatwideu(ξ) =\nˆψ(ξ/λ)/hatwideu(ξ) when supp /hatwideu⊂λB, we have\n∂αu=i|α|F−1(ξαˆu) =i|α|F−1(ξαˆψ(ξ/λ)/hatwideu(ξ)) =λ|α|(∂αψ)λ∗u.\nHere (∂αψ)λ(x) =λn∂αψ(λx) for allx∈Rn.\nFix|α|=k. From the H¨ older inequality (2.5) for Lorentz spaces, we in fer that\n/ba∇dbl∂αu/ba∇dblL∞(Rn)≤λksup\nx∈Rn/integraldisplay\nRn|(∂αψ)λ(x−y)||u(y)|dy\n≤Cλksup\nx∈Rn/ba∇dbl(∂αψ)λ(x−·)/ba∇dbl\nLp\np−1,1(Rn)/ba∇dblu/ba∇dblLp,∞(Rn)\n=Cλk+n\np/ba∇dbl∂αψ/ba∇dbl\nLp\np−1,1(Rn)/ba∇dblu/ba∇dblLp,∞(Rn),\nwhereC=C(p)>0. Note that ∂αψ∈ S(Rn), then for any ε>0, there exists a constant\nC=C(ε,n,∂αψ)>0 such that 0 ≤(∂αψ)∗(s)≤Cs−εfor alls>0. This yields that\n/ba∇dbl∂αψ/ba∇dbl\nLp\np−1,1(Rn)=p\np−1/integraldisplay∞\n0(∂αψ)p−1\np\n∗(s)ds≤C(/integraldisplay1\n0s−1\n2ds+/integraldisplay∞\n1s−2ds)<∞,\nwhich implies (2.12).\n(2) Take 1/r= 1+1/q−1/p, then the hypotheses on the indices imply that 1 < r <\nq<∞. In light of (2.9), we see that there exists a positive consta ntC=C(p,q) such that\n/ba∇dbl∂αu/ba∇dblLq,1(Rn)=λk/ba∇dbl(∂αψ)λ∗u/ba∇dblLq,1(Rn)\n≤Cλk/ba∇dbl(∂αψ)λ/ba∇dblLr,1(Rn)/ba∇dblu/ba∇dblLp,∞(Rn)\n=Cλk+n(1−1\nr)/ba∇dbl∂αψ/ba∇dblLr,1(Rn)/ba∇dblu/ba∇dblLp,∞(Rn)\n=Cλk+n/parenleftBig\n1\np−1\nq/parenrightBig\n/ba∇dbl∂αψ/ba∇dblLr,1(Rn)/ba∇dblu/ba∇dblLp,∞(Rn).\nBy a similar argument used in the proof of (2.12), we may deriv e that/ba∇dbl∂αψ/ba∇dblLr,1(Rn)<∞.\nThis implies (2.13).\n(3) As∂αψ∈ S(Rn)⊂L1(Rn), Lemma 2.2 enable us to deduce that\n/ba∇dbl∂αu/ba∇dblLq,l(Rn)=λk/ba∇dbl(∂αψ)λ∗u/ba∇dblLq,l(Rn)\n≤Cλk/ba∇dbl(∂αψ)λ/ba∇dblL1(Rn)/ba∇dblu/ba∇dblLq,l(Rn)\n=Cλk/ba∇dbl∂αψ/ba∇dblL1(Rn)/ba∇dblu/ba∇dblLq,l(Rn),\nwhereC=C(q,l)>0. This implies (2.14).\n(4) Observe that (2.14) implies the second inequality of (2. 15), it suffices to show the\nfirst inequality in (2.15). Let ηbe a Schwartz function on Rnsuch thatχC≤ˆη≤χ˜C, where\n˜C={ξ∈Rn:r1/2≤ |ξ| ≤2r2}. It follows from suppˆ u⊂λCthat for all ξ∈Rn,\nˆu(ξ) =/summationdisplay\n|α|=k(−iξ)α\n|ξ|2kˆη(ξ/λ)(iξ)αˆu(ξ) =λ−k/summationdisplay\n|α|=k(−iξ/λ)α\n|ξ/λ|2kˆη(ξ/λ)F(∂αu)(ξ).\n12Therefore, we may write\nu=λ−k/summationdisplay\n|α|=k(gα)λ∗∂αu,\nwhere (gα)λ(x) =λngα(λx) for allx∈Rn, and\ngα=F−1/parenleftbigg(−iξ)α\n|ξ|2kˆη(ξ)/parenrightbigg\n∈ S(Rn)⊂L1(Rn).\nThis together with Lemma 2.2 yields that a constant C=C(n,q,l,k)>0 exists such that\n/ba∇dblu/ba∇dblLq,l(Rn)≤Cλ−k/summationdisplay\n|α|=k/ba∇dbl(gα)λ/ba∇dblL1(Rn)/ba∇dbl∂αu/ba∇dblLq,l(Rn)\n≤Cλ−k\n/summationdisplay\n|α|=k/ba∇dblgα/ba∇dblL1(Rn)\nsup\n|α|=k/ba∇dbl∂αu/ba∇dblLq,l(Rn).\nThis concludes (2.15) and the proof is complete.\n2.2 Besov-Lorentz spaces, Sobolev-Lorentz spaces and Trie bel-Lizorkin-\nLorentz spaces\nSdenotes the Schwartz class of rapidly decreasing functions ,S′the space of tempered\ndistributions, S′/Pthequotient spaceof tempered distributionswhich modulop olynomials.\nWe useFfor/hatwidefto denote the Fourier transform of a tempered distribution f. To define\nBesov-Lorentz spaces, we need the following dyadic unity pa rtition (see e.g. [1]). Choose\ntwo nonnegative radial functions ̺,ϕ∈C∞(Rn) be supported respectively in the ball\n{ξ∈Rn:|ξ| ≤4\n3}and the shell {ξ∈Rn:3\n4≤ |ξ| ≤8\n3}such that\n̺(ξ)+/summationdisplay\nj≥0ϕ(2−jξ) = 1,∀ξ∈Rn;/summationdisplay\nj∈Zϕ(2−jξ) = 1,∀ξ/\\e}atio\\slash= 0.\nThe nonhomogeneous dyadic blocks ∆ jare defined by\n∆ju:= 0 ifj≤ −2,∆−1u:=̺(D)u,∆ju:=ϕ/parenleftbig\n2−jD/parenrightbig\nuifj≥0, Sju:=/summationdisplay\nk≤j−1∆ku.\nThe homogeneous Littlewood-Paley operators are defined as f ollows\n∀j∈Z,˙∆jf:=ϕ(2−jD)fand˙Sjf:=/summationdisplay\nk≤j−1˙∆kf.\nThe homogeneous Besov-Lorentz space ˙Bs\np,q,r(Rn) is the set of f∈ S′(Rn)/P(Rn) such that\n/ba∇dblf/ba∇dbl˙Bsp,q,r:=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/braceleftbigg\n2js/vextenddouble/vextenddouble/vextenddouble˙∆jf/vextenddouble/vextenddouble/vextenddouble\nLp,q(Rn)/bracerightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nℓr(Z)<∞.\nHereℓr(Z) represents the set of sequences with summable r-th powers. The homogeneous\nSobolev-Lorentz norm /ba∇dbl·/ba∇dbl˙Hsp,p1(Rn)isdefinedas /ba∇dblf/ba∇dbl˙Hsp,p1(Rn)=/ba∇dblΛsf/ba∇dblLp,p1(Rn).Whenp1=p,\nthe Sobolev-Lorentz spaces reduce to the classical Sobolev spaces˙Hs\np(Rn).\n13Forp,q,r∈(0,∞] ands∈R,the homogeneous Triebel-Lizorkin-Lorentz space\n˙Fs\np,q,r(Rn) is defined by\n˙Fs\np,q,r(Rn) =/braceleftBig\nf∈ S′(Rn)/P(Rn) :/ba∇dblf/ba∇dbl˙Fsp,q,r<∞/bracerightBig\n.\nHere\n/ba∇dblf/ba∇dbl˙Fsp,q,r:=\n\n/ba∇dbl{∞/summationdisplay\nj=−∞(2js|˙∆jf|)r}1\nr/ba∇dblLp,q(Rn),0<r<∞,\n/ba∇dblsup\nj∈Z2js|˙∆jf|/ba∇dblLp,q(Rn), r=∞.\nLemma 2.5. Suppose that f∈˙Hs\np,∞(Rn)withs∈Randp∈(1,∞]. Then there holds\n/ba∇dblf/ba∇dbl˙Bsp,∞,∞≤ /ba∇dblf/ba∇dbl˙Fsp,∞,∞≤C/ba∇dblf/ba∇dbl˙Hsp,∞(Rn), (2.16)\nwhereC >0depends only on n,s,pandϕ.\nRemark 2.5.It is worth remarking that the nonhomogeneous embedding Hs\np,∞֒→Fs\np,∞,∞\nwas recently proved by Ko and Lee in [24] and they mentioned th at the results hold for\nthe homogeneous case. One can modify the argument in [24] to g et the homogeneous case\n˙Hs\np,∞֒→˙Fs\np,∞,∞. Here, we shall present a different proof via the Hardy-Little wood maximal\nfunction.\nProof.Since it is obvious that\n/ba∇dblf/ba∇dbl˙Bsp,∞,∞= sup\nj∈Z/ba∇dbl2js|˙∆jf|/ba∇dblLp,∞(Rn)≤ /ba∇dblsup\nj∈Z2js|˙∆jf|/ba∇dblLp,∞(Rn)=/ba∇dblf/ba∇dbl˙Fsp,∞,∞,\nwe focus on the proof\n/ba∇dblf/ba∇dbl˙Fsp,∞,∞≤C/ba∇dblf/ba∇dbl˙Hsp,∞(Rn).\nIt suffices to show that for all functions g∈Lp,∞(Rn),\n/ba∇dblsup\nj∈Z|φj∗g|/ba∇dblLp,∞(Rn)≤C/ba∇dblg/ba∇dblLp,∞(Rn),\nwhereφ= (|ξ|−sϕ(ξ))∨andφj(x) = 2jnφ(2jx) for allx∈Rn.\nTo this end, observe that φis a Schwartz function on Rnand there exists a positive\nconstantCsuch that for all x∈Rn,|φ(x)| ≤C(1+|x|)−n−1.This yields that for all j∈Z\nandx∈Rn,\n|φj∗g(x)| ≤C2jn/integraldisplay\nRn|g(x−y)|(1+|2jy|)−n−1dy\n=C∞/summationdisplay\nk=−∞2jn/integraldisplay\n2k<|y|≤2k+1|g(x−y)|(1+|2jy|)−n−1dy\n≤C∞/summationdisplay\nk=−∞2jn(1+2j+k)−n−1/integraldisplay\n2k<|y|≤2k+1|g(x−y)|dy\n≤C∞/summationdisplay\nk=−∞2(j+k)n(1+2j+k)−n−1Mg(x)\n14=C∞/summationdisplay\nk=−∞2kn(1+2k)−n−1Mg(x)\n≤CMg(x)(−1/summationdisplay\nk=−∞2kn+∞/summationdisplay\nk=02−k)\n≤CMg(x),\nwhich implies that there exists a positive constant C=C(n,φ) such that for all x∈Rn,\nsup\nj∈Z|φj∗g(x)| ≤CMg(x). (2.17)\nThen it follows from (2.3) that\n/ba∇dblsup\nj∈Z|φj∗g|/ba∇dblLp,∞(Rn)≤C/ba∇dblMg/ba∇dblLp,∞(Rn)≤C/ba∇dblg/ba∇dblLp,∞(Rn),\nwhereC >0 depends only on n,pandφ. This concludes the proof.\n3 Proof of Theorem 1.1 and 1.2\n3.1 Gagliardo-Nirenberg inequalities in Lorentz spaces\nThe goal of this subsection is to prove Theorem 1.1 involving Gagliardo-Nirenberg inequal-\nities in Lorentz spaces. Firstly, we establish three key ine qualities, which play an important\nrole in the proof of three cases in Theorem 1.1. Secondly, in v iew of a pointwise interpo-\nlation estimate for derivatives found in [1] and equivalent norms of Lorentz spaces, we get\nProposition 3.2. Finally, we are in a position to show Theore m 1.1.\nLemma 3.1. (1) Suppose that u∈˙Hs\nr,p1(Rn)with1<r<∞,0≤s<n/r and0<p1≤\n∞. Then there exists a positive constant C=C(n,s,p,r,p 1)such that\n/ba∇dblu/ba∇dblLp,p1(Rn)≤C/ba∇dblΛsu/ba∇dblLr,p1(Rn)withn\np=n\nr−s. (3.1)\n(2) Let1<q,p,r < ∞,1≤l≤ ∞ands=n/r.Then there exists a positive constant C\nsuch that for all functions u∈Lp,∞(Rn)∩˙Bs\nr,∞,∞(Rn), there holds\n/ba∇dblu/ba∇dblLq,l(Rn)≤C/ba∇dblu/ba∇dblp\nq\nLp,∞(Rn)/ba∇dblu/ba∇dbl1−p\nq\n˙Bsr,∞,∞withp<q.\n(3) Letu∈Lp,∞(Rn)∩˙Bs\nr,∞,∞(Rn)withs > n/r and1< p,r≤ ∞.Then there exists a\npositive constant C=C(n,s,p,r)such that\n/ba∇dblu/ba∇dblL∞(Rn)≤C/ba∇dblu/ba∇dblθ\nLp,∞(Rn)/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞,\nwhere\n0 =θn\np+(1−θ)/parenleftBign\nr−s/parenrightBig\n,0<θ≤1.\n15Remark 3.1.As a corollary of this lemma and imbedding relation in Lemma 2 .5, we have\n/ba∇dblΛσu/ba∇dblLq,l(Rn)≤C/ba∇dblΛσu/ba∇dblp\nq\nLp,∞(Rn)/ba∇dblΛσ+n\nru/ba∇dbl1−p\nq\nLr,∞(Rn)(3.2)\nwith 1<p<q< ∞,1<r<∞and 1≤l≤ ∞, and\n/ba∇dblΛσu/ba∇dblL∞(Rn)≤C/ba∇dblΛσu/ba∇dblθ\nLp,∞(Rn)/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn)(3.3)\nwith 0 =θn/p+(1−θ)(n/r−s+σ), 0<θ≤1,s−σ>n/r and 1<p,r≤ ∞.\nRemark 3.2.In this lemma, the necessity of the condition that p,r>1 results from Lemma\n2.2, Young inequality (2.9) and Bernstein inequality (2.13 ) for Lorentz spaces. Due to (2.2),\nit is also essential to make the assumption that q >1 andl≥1 to ensure that the space\nLq,l(Rn) is normable.\nProof of Lemma 3.1. (1) Thanks to Fourier transform, there exists a positive con stantC=\nC(n,s) such that\nf=F−1/parenleftBig1\n|ξ|s|ξ|sˆf(ξ)/parenrightBig\n=F−1/parenleftBig1\n|ξ|s/parenrightBig\n∗Λsf=C|·|s−n∗Λsf.\nWiththehelpoftheYounginequality (2.9)inLorentzspaces andthefactthat |x|−1∈Ln,∞,\nwe see that\n/ba∇dblf/ba∇dblLp,p1(Rn)≤C/ba∇dbl|·|s−n∗Λsf/ba∇dblLp,p1(Rn)\n≤C/ba∇dbl|·|s−n/ba∇dblLn\nn−s,∞(Rn)/ba∇dblΛsf/ba∇dblLr,p1(Rn)\n≤C/ba∇dblΛsf/ba∇dblLr,p1(Rn).\n(2) By means of the low and high frequencies, it follows from ( 2.2) that\n/ba∇dblu/ba∇dblLq,l(Rn)≤ /ba∇dblu/ba∇dbl∗\nLq,l(Rn)≤ /ba∇dbl˙Sj0u/ba∇dbl∗\nLq,l(Rn)+/summationdisplay\nj≥j0/ba∇dbl˙∆ju/ba∇dbl∗\nLq,l(Rn)\n≤q\nq−1/ba∇dbl˙Sj0u/ba∇dblLq,l(Rn)+q\nq−1/summationdisplay\nj≥j0/ba∇dbl˙∆ju/ba∇dblLq,l(Rn).(3.4)\nHerej0is an integer to be chosen later. Observe that\n˙Sj0u=/summationdisplay\nk≤j0−1˙∆ku=hj0∗u,\nwherehj0(x) = 2j0nh(2j0x) for allx∈Rn, and\nh=F−1/parenleftBig/summationdisplay\nk≤−1ϕ(2−kξ)/parenrightBig\n∈ S(Rn).\nOwing to Bernstein’s inequality (2.13) and (2.6), we may app ly Lemma 2.2 to infer that\nthere exists a positive constant Cindependent of j0such that\n/ba∇dbl˙Sj0u/ba∇dblLq,l(Rn)≤C2j0n(1\np−1\nq)/ba∇dbl˙Sj0u/ba∇dblLp,∞(Rn)\n=C2j0n(1\np−1\nq)/ba∇dblhj0∗u/ba∇dblLp,∞(Rn)\n≤C2j0n(1\np−1\nq)/ba∇dblhj0/ba∇dblL1(Rn)/ba∇dblu/ba∇dblLp,∞(Rn)\n16=C2j0n(1\np−1\nq)/ba∇dblh/ba∇dblL1(Rn)/ba∇dblu/ba∇dblLp,∞(Rn).\nFor the high-frequency part, it follows from (2.4), (2.6), ( 2.13) and Lemma 2.2 that\n/summationdisplay\nj≥j0/ba∇dbl˙∆ju/ba∇dblLq,l(Rn)≤C/summationdisplay\nj≥j0/ba∇dbl˙∆ju/ba∇dbl1−α\nLp,∞(Rn)/ba∇dbl˙∆ju/ba∇dblα\nLq+r,∞(Rn)\n≤C/summationdisplay\nj≥j02jnα(1\nr−1\nq+r)/ba∇dbl˙∆ju/ba∇dblα\nLr,∞(Rn)/ba∇dblϕj∗u/ba∇dbl1−α\nLp,∞(Rn)\n≤C/summationdisplay\nj≥j02−jnα\nq+r/ba∇dblu/ba∇dblα\n˙Bsr,∞,∞/ba∇dblϕj/ba∇dbl1−α\nL1(Rn)/ba∇dblu/ba∇dbl1−α\nLp,∞(Rn)\n=C2−j0nα\nq+r/ba∇dblϕ/ba∇dbl1−α\nL1(Rn)/ba∇dblu/ba∇dbl1−α\nLp,∞(Rn)/ba∇dblu/ba∇dblα\n˙Bsr,∞,∞.\nHeres=n/r, 1/q= (1−α)/p+α/(q+r) with 0<α<1, andϕj(x) = 2jnϕ(2jx) for all\nx∈Rn. It turns out that\n/ba∇dblu/ba∇dblLq,l(Rn)≤C2j0n(1\np−1\nq)/ba∇dblu/ba∇dblLp,∞(Rn)+C2−j0nα\nq+r/ba∇dblu/ba∇dbl1−α\nLp,∞(Rn)/ba∇dblu/ba∇dblα\n˙Bsr,∞,∞,\nwhere the positive constant Cis independent of j0. Since 1/p−1/q+α/(q+r) =α/p, by\nchoosingj0such that 2j0n(1\np−1\nq)/ba∇dblu/ba∇dblLp,∞(Rn)≈2−j0nα\nq+r/ba∇dblu/ba∇dbl1−α\nLp,∞(Rn)/ba∇dblu/ba∇dblα\n˙Bsr,∞,∞, we may derive\nthat\n/ba∇dblu/ba∇dblLq,l(Rn)≤C/ba∇dblu/ba∇dblp\nq\nLp,∞(Rn)/ba∇dblu/ba∇dbl1−p\nq\n˙Bsr,∞,∞.\n(3) Ifs>n/r, we takeq=l=∞in (3.4). Using Bernstein’s inequality (2.12) and Young\ninequality for Lorentz spaces, we find that\n/ba∇dbl˙Sj0u/ba∇dblL∞(Rn)≤C2j0n\np/ba∇dblu/ba∇dblLp,∞(Rn)and/summationdisplay\nj≥j0/ba∇dbl˙∆ju/ba∇dblL∞(Rn)≤C2j0n(1\nr−s\nn)/ba∇dblu/ba∇dbl˙Bsr,∞,∞,\nwhere the positive constant Cis independent of j0. As the derivation of the above, we may\nchoosej0appropriately to conclude that\n/ba∇dblu/ba∇dblL∞(Rn)≤C2j0n\np/ba∇dblu/ba∇dblLp,∞(Rn)+C2j0n(1\nr−s\nn)/ba∇dblu/ba∇dbl˙Bsr,∞,∞\n≤C/ba∇dblu/ba∇dbl1−1\np\n1\np−1\nr+s\nn\nLp,∞(Rn)/ba∇dblu/ba∇dbl1\np\n1\np−1\nr+s\nn\n˙Bsr,∞,∞.\nThis completes the proof of this lemma.\nProposition 3.2. Assume that u∈Lq,q1(Rn)∩˙Hs\nr,r1(Rn)with1< q,r≤ ∞and1≤\nq1,r1≤ ∞. Then there holds for 0<σ<s< ∞,\n/ba∇dblΛσu/ba∇dblLp,p1(Rn)≤C/ba∇dblu/ba∇dbl1−σ\ns\nLq,q1(Rn)/ba∇dblΛsu/ba∇dblσ\ns\nLr,r1(Rn),\nwith1\np=/parenleftBig\n1−σ\ns/parenrightBig1\nq+σ\nsr,1\np1=/parenleftBig\n1−σ\ns/parenrightBig1\nq1+σ\nsr1.\nHereCis a positive constant depending only on q,r,q1,r1,s,σandn.\n17Remark 3.3.As a special case of this proposition, there holds the follow ing estimate\n/ba∇dblΛσu/ba∇dblLp,∞(Rn)≤C/ba∇dblu/ba∇dbl1−σ\ns\nLq,∞(Rn)/ba∇dblΛsu/ba∇dblσ\ns\nLr,∞(Rn),\nwhereq,r,p,sandσsatisfy the same conditions as in Proposition 3.2. This ineq uality will\nbe used in the proof of Theorem 1.1.\nRemark 3.4.Very recently, by means of Maz’ya-Shaposhnikova pointwise estimates in [30],\nFiorenza-Formica-Rosaria-Soudsky [17] showed the follow ing estimate\n/vextenddouble/vextenddouble∇ju/vextenddouble/vextenddouble\nLp,p1(Rn)≤C/vextenddouble/vextenddouble/vextenddouble∇ku/vextenddouble/vextenddouble/vextenddoublej\nk\nLr,r1(Rn)/ba∇dblu/ba∇dbl1−j\nk\nLq,q1(Rn),\nwherej,k∈N,1≤j <k,1<q,r≤ ∞,1≤q1,r1≤ ∞and\n1\np=j\nk\nr+1−j\nk\nq,1\np1=j\nk\nr1+1−j\nk\nq1.\nProposition3.2extendstheaforementioned integercaseso fGagliardo-Nirenberginequalities\nin [17] to the fractional cases.\nProof.Thankstothedecompositionoflowandhighfrequencies, one canderivethefollowing\npointwise estimate\n|Λσu(x)| ≤C(Mu(x))1−σ\ns(MΛsu(x))σ\ns, (3.5)\nwhose proof can be found in [1, p.84].\nAs a consequence, it follows from the H¨ older’s inequality f or Lorentz spaces that\n/ba∇dblΛσu/ba∇dblLp,p1(Rn)≤C/ba∇dbl(Mu)1−σ\ns(MΛsu)σ\ns/ba∇dblLp,p1(Rn)\n≤C/ba∇dbl(Mu)1−σ\ns/ba∇dbl\nLsq\ns−σ,sq1s−σ(Rn)/ba∇dbl(MΛsu)σ\ns/ba∇dblLsr\nσ,sr1σ(Rn)\n≤C/ba∇dblMu/ba∇dbl1−σ\ns\nLq,q1(Rn)/ba∇dblM(Λsu)/ba∇dblσ\ns\nLr,r1(Rn).\nAccording to (2.3), we conclude the desired estimate.\nNow, at this stage, we can prove Theorem 1.1.\nProof of Theorem 1.1. (I) First, we consider (1.6) under the hypothesis that 0 <s−σ<n\nr.\n(I1) Ifσ= 0, it follows from the interpolation characteristic (2.4) of Lorentz spaces that\n/ba∇dblu/ba∇dblLp,1(Rn)≤C/ba∇dblu/ba∇dbl1−θ\nL˜p,∞(Rn)/ba∇dblu/ba∇dblθ\nLq,∞(Rn),with1\np=1−θ\n˜p+θ\nq,0<θ<1,(3.6)\nwhere we have used the fact that1\n˜p=1\nr−s\nn/\\e}atio\\slash=1\nq. This together with the Sobolev inequality\n(3.1) ensures that\n/ba∇dblu/ba∇dblLp,1(Rn)≤C/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn)/ba∇dblu/ba∇dblθ\nLq,∞(Rn),\nwith1\np= (1−θ)(1\nr−s\nn)+θ\nq,0<θ<1.\n18(I2) Ifσ>0, the Sobolev embedding (3.1) yields\n/ba∇dblΛσu/ba∇dblLr∗,∞(Rn)≤C/ba∇dblΛsu/ba∇dblLr,∞(Rn), (3.7)\nwith1\nr∗=1\nr−s−σ\nn.\nIt follows from Proposition 3.2 that for 1 <q,r≤ ∞,\n/ba∇dblΛσu/ba∇dblL˜p,∞(Rn)≤C/ba∇dblu/ba∇dbl1−σ\ns\nLq,∞(Rn)/ba∇dblΛsu/ba∇dblσ\ns\nLr,∞(Rn)(3.8)\nwith1\n/tildewidep=/parenleftBig\n1−σ\ns/parenrightBig1\nq+σ\nsr.\nThen the hypothesis on the indices that\nn\np−σ=θn\nq+(1−θ)/parenleftBign\nr−s/parenrightBig\n,0<θ<1−σ\ns,\nimply the following relation\n1\np=α\n/tildewidep+1−α\nr∗, α=θ\n1−σ\ns∈(0,1).\nObserve that the condition s−n\nr/\\e}atio\\slash=σ−n\npguarantees /tildewidep/\\e}atio\\slash=r∗. From the interpolation\ncharacteristic (2.4) of Lorentz spaces, we see that\n/ba∇dblΛσu/ba∇dblLp,1(Rn)≤C/ba∇dblΛσu/ba∇dblα\nL˜p,∞(Rn)/ba∇dblΛσu/ba∇dbl1−α\nLr∗,∞(Rn). (3.9)\nPlugging (3.7) and (3.8) into (3.9), we get\n/ba∇dblΛσu/ba∇dblLp,1(Rn)≤C/parenleftBig\n/ba∇dblu/ba∇dbl1−σ\ns\nLq,∞(Rn)/ba∇dblΛsu/ba∇dblσ\ns\nLr,∞(Rn)/parenrightBigα\n/ba∇dblΛsu/ba∇dbl1−α\nLr,∞(Rn)\n=C/ba∇dblu/ba∇dbl(1−σ\ns)α\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−α(1−σ\ns)\nLr,∞(Rn)\n=C/ba∇dblu/ba∇dblθ\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn).\n(II) Next, we turn our attention to the case that s−σ=n/rin (1.6). By virtue of\n(3.2), we have proved (1.6) with s=n/randσ= 0. In the following, we assume that σ>0.\nAccording to Proposition 3.2, we arrive at that for1\n˜p=/parenleftbig\n1−σ\ns/parenrightbig1\nq+σ\nsr=/parenleftbig\n1−σ\ns/parenrightbig/parenleftBig\n1\nq+σ\nn/parenrightBig\n,\n/ba∇dblΛσu/ba∇dblL˜p,∞(Rn)≤C/ba∇dblu/ba∇dbl1−σ\ns\nLq,∞(Rn)/ba∇dblΛsu/ba∇dblσ\ns\nLr,∞(Rn). (3.10)\nSince1\np=θ/parenleftBig\n1\nq+σ\nn/parenrightBig\n</parenleftbig\n1−σ\ns/parenrightbig/parenleftBig\n1\nq+σ\nn/parenrightBig\n=1\n˜p, it follows from (3.2) that\n/ba∇dblΛσu/ba∇dblLp,1(Rn)≤C/ba∇dblΛσu/ba∇dbl˜p\np\nL˜p,∞(Rn)/ba∇dblΛsu/ba∇dbl1−˜p\np\nLr,∞(Rn). (3.11)\nSubstituting (3.10) into (3.11), we obtain\n/ba∇dblΛσu/ba∇dblLp,1(Rn)≤C/ba∇dblu/ba∇dbl˜p\np(1−σ\ns)\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−˜p\np(1−σ\ns)\nLr,∞(Rn)=C/ba∇dblu/ba∇dblθ\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn).\n19(III) Finally, it is enough to show (1.6) under the hypothesis tha ts−σ>n/r.\n(III1) Ifσ>0, we conclude from Proposition 3.2 that for 1 <q,r≤ ∞,\n/ba∇dblΛσu/ba∇dblL˜p,∞(Rn)≤C/ba∇dblu/ba∇dbl1−σ\ns\nLq,∞(Rn)/ba∇dblΛsu/ba∇dblσ\ns\nLr,∞(Rn), (3.12)\nwhere1\n/tildewidep=/parenleftBig\n1−σ\ns/parenrightBig1\nq+σ\nsr.\nObserve that\n1\np=θ/parenleftbigg1\nq+s\nn−1\nr/parenrightbigg\n+/parenleftbigg1\nr−s\nn+σ\nn/parenrightbigg\n</parenleftBig\n1−σ\ns/parenrightBig/parenleftbigg1\nq+s\nn−1\nr/parenrightbigg\n+/parenleftbigg1\nr−s\nn+σ\nn/parenrightbigg\n=1\n/tildewidep.\nIn view of the interpolation characteristic (2.4) of Lorent z spaces, we see that\n/ba∇dblΛσu/ba∇dblLp,1(Rn)≤C/ba∇dblΛσu/ba∇dblα\nL˜p,∞(Rn)/ba∇dblΛσu/ba∇dbl1−α\nL∞(Rn)(3.13)\nwith1\np=α\n/tildewidep+1−α\n∞,0<α<1.\nFurthermore (3.3) ensures that\n/ba∇dblΛσu/ba∇dblL∞(Rn)≤C/ba∇dblΛσu/ba∇dblβ\nL˜p,∞(Rn)/ba∇dblΛsu/ba∇dbl1−β\nLr,∞(Rn), (3.14)\nwhere\n0 =βn\n˜p+(1−β)/parenleftBign\nr−s+σ/parenrightBig\n,0<β≤1.\nInserting (3.14) into (3.13) and using (3.12), we have\n/ba∇dblΛσu/ba∇dblLp,1(Rn)≤C/ba∇dblu/ba∇dbl(1−σ\ns)[α+(1−α)β]\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl(1−β)(1−α)+σ\ns[α+(1−α)β]\nLr,∞(Rn)\n=C/ba∇dblu/ba∇dblθ\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn).\n(III2) Ifσ= 0, we note that\n1\np=θ\nq+(1−θ)/parenleftbigg1\nr−s\nn/parenrightbigg\n<1\nq.\nThen it follows from the interpolation characteristic (2.4 ) of Lorentz spaces that\n/ba∇dblu/ba∇dblLp,1(Rn)≤C/ba∇dblu/ba∇dblτ\nLq,∞(Rn)/ba∇dblu/ba∇dbl1−τ\nL∞(Rn),with1\np=τ\nq+1−τ\n∞and 0<τ <1.(3.15)\nIn view of (3.3), we obtain\n/ba∇dblu/ba∇dblL∞(Rn)≤C/ba∇dblu/ba∇dblλ\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−λ\nLr,∞(Rn),with 0 =λ\nq+(1−λ)/parenleftbigg1\nr−s\nn/parenrightbigg\nand 0<λ≤1.\nThis together with (3.15) yields that\n/ba∇dblu/ba∇dblLp,1(Rn)≤C/ba∇dblu/ba∇dblτ+λ(1−τ)\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl(1−λ)(1−τ)\nLr,∞(Rn)=C/ba∇dblu/ba∇dblθ\nLq,∞(Rn)/ba∇dblΛsu/ba∇dbl1−θ\nLr,∞(Rn).\nWe complete the proof of Theorem 1.1.\n203.2 Gagliardo-Nirenberg inequalities in Besov-Lorentz sp aces\nIn this subsection, by means of the Littlewood-Paley decomp osition and generalized Bern-\nstein inequalities in Lemma 2.4, we shall prove the Gagliard o-Nirenberg inequality (1.10)\nin Besov-Lorentz spaces.\nProof of Theorem 1.2. First, we assert that q≤p,σ >0 andn\np−σ=θn\nq+(1−θ)(n\nr−s)\nimply that\ns−σ−n\nr+n\np>0, (3.16)\nwhich will be frequently used later. Indeed, thanks to q≤pandσ>0, we see that\nn\np−(1−θ)σ>n\np−σ=θn\nq+(1−θ)(n\nr−s)≥θn\np+(1−θ)(n\nr−s),\nthat is,\ns−σ−n\nr+n\np>0.\nThe assertion follows. Similarly, we also assert that q<pandσ= 0 yield (3.16).\n(I) We next consider the case q=rof (1.10). Note that\n1\np=θ\nq+1−θ\nr−1\nn[(1−θ)s−σ]<θ\nq+1−θ\nr=1\nq.\nTherefore, we see that r=q < pand (3.16) are valid. With the help of the Bernstein\ninequality (2.13) in Lorentz spaces, we infer that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1=/summationdisplay\nj≤k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)+/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)\n≤C/summationdisplay\nj≤k2j[σ+n(1\nr−1\np)]/ba∇dbl˙∆ju/ba∇dblLr,∞(Rn)+C/summationdisplay\nj>k2−j[s−σ−n(1\nr−1\np)]2js/ba∇dbl˙∆ju/ba∇dblLr,∞(Rn)\n≤C2k[σ+n(1\nr−1\np)]\n1−2−[σ+n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙B0r,∞,∞+C2−k[s−σ−n(1\nr−1\np)]\n1−2−[s−σ−n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙Bsr,∞,∞,\n(3.17)\nwhere we have used σ+n(1\nr−1\np)>0 and (3.16).\nAs a consequence, by choosing the integer kappropriately such that\n2k[σ+n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙B0r,∞,∞≈2−k[s−σ−n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙Bsr,∞,∞, we further get\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0r,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞.\n(II) We turn our attention to the case q<r. In order to get (3.16), we check that q<p\nvia the following straightforward calculation\nn\np= (1−σ\ns)n\nq+σ\nsn\nr+(1−σ\ns−θ)(n\nr−n\nq−s)\n<(1−σ\ns)n\nq+σ\nsn\nq\n=n\nq.\n21We also see that1\np<(1−σ\ns)1\nq+σ\ns1\nr. (3.18)\nThe following discussion will be divided into three subcase s thatq <r<p ,q <p<r and\nq<p=r.\n(II1) We examine the case q<r<p . As the derivation of (3.17), we find that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/summationdisplay\nj≤k2j[σ+n(1\nq−1\np)]/ba∇dbl˙∆ju/ba∇dblLq,∞+C/summationdisplay\nj>k2−j[s−σ−n(1\nr−1\np)]2js/ba∇dbl˙∆ju/ba∇dblLr,∞\n≤C2k[σ+n(1\nq−1\np)]\n1−2−[σ+n(1\nq−1\np)]/ba∇dblu/ba∇dbl˙B0q,∞,∞+C2−k[s−σ−n(1\nr−1\np)]\n1−2−[s−σ−n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙Bsr,∞,∞,\nwhere we have used σ+n(1\nq−1\np)>0 and (3.16).\nHence, we conclude that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞.\n(II2) We deal with the case q < p < r . By means of the interpolation characteristic\n(2.4) of Lorentz spaces, we observe that\n/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)≤C/ba∇dbl˙∆ju/ba∇dblα\nLq,∞(Rn)/ba∇dbl˙∆ju/ba∇dbl1−α\nLr,∞(Rn),1\np=α\nq+1−α\nr.\nCombining this, the Bernstein inequality (2.13) in Lorentz spaces and (3.16), we know that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1=/summationdisplay\nj≤k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)+/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)\n≤C/summationdisplay\nj≤k2j[σ+n(1\nq−1\np)]/ba∇dbl˙∆ju/ba∇dblLq,∞(Rn)+C/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dblα\nLq,∞(Rn)/ba∇dbl˙∆ju/ba∇dbl1−α\nLr,∞(Rn)\n≤C2k[σ+n(1\nq−1\np)]\n1−2−[σ+n(1\nq−1\np)]/ba∇dblu/ba∇dbl˙B0q,∞,∞+C2−k[s(1−α)−σ]\n1−2−[s(1−α)−σ]/ba∇dblu/ba∇dblα\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−α\n˙Bsr,∞,∞,\n(3.19)\nwhere we have used σ+n(1\nq−1\np)>0 ands(1−α)−σ >0 which is derived from (3.18).\nChoosingtheinteger ksuch that 2k[σ+n(1\nq−1\np)]/ba∇dblu/ba∇dbl˙B0q,∞,∞≈2−k[s(1−α)−σ]/ba∇dblu/ba∇dblα\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−α\n˙Bsr,∞,∞,\nwe also have\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞.\n(II3) We treat the case q<p=r. It follows from the interpolation characteristic (2.4)\nof Lorentz spaces that\n/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)≤C/ba∇dbl˙∆ju/ba∇dbl1−α\nLq,∞(Rn)/ba∇dbl˙∆ju/ba∇dblα\nL(1+ε)p,∞(Rn),1\np=1−α\nq+α\n(1+ε)p,(3.20)\nwhereε>0 will be determined later. We derive from this, the Bernstei n inequality (2.13)\n22in Lorentz spaces and (3.16) that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1=/summationdisplay\nj≤k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)+/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)\n≤C/summationdisplay\nj≤k2j[σ+n(1\nq−1\np)]/ba∇dbl˙∆ju/ba∇dblLq,∞(Rn)+C/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dbl1−α\nLq,∞(Rn)/ba∇dbl˙∆ju/ba∇dblα\nL(1+ε)p,∞(Rn)\n≤C2k[σ+n(1\nq−1\np)]\n1−2−[σ+n(1\nq−1\np)]/ba∇dblu/ba∇dbl˙B0q,∞,∞+C/summationdisplay\nj>k2−j[sα−nεα\n(1+ε)p−σ]/ba∇dblu/ba∇dbl1−α\n˙B0q,∞,∞/ba∇dblu/ba∇dblα\n˙Bsr,∞,∞,\n(3.21)\nwhere we have used the fact that σ+n(1\nq−1\np)>0.\nDenoteδ(ε) =sα−nεα\n(1+ε)p−σ. From (3.20), we see that\nδ(ε) =ps(1+ε)(p−q)−σp[(1+ε)p−q]−εn(p−q)\np[(1+ε)p−q],\nandδ(ε) is a continuous function on a neighborhood of 0. Since δ(0)>0, there exists a\nsufficiently small ε>0 such that δ(ε)>0. It follows from (3.21) that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C2k[σ+n(1\nq−1\np)]\n1−2−[σ+n(1\nq−1\np)]/ba∇dblu/ba∇dbl˙B0q,∞,∞+C2−k[sα−nεα\n(1+ε)p−σ]\n1−2−[sα−nεα\n(1+ε)p−σ]/ba∇dblu/ba∇dbl1−α\n˙B0q,∞,∞/ba∇dblu/ba∇dblα\n˙Bsr,∞,∞,\n(3.22)\nwhich also yields that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞.\n(III) Finally, it remains to show (1.10) under the case that q>r.\n(III1) We first consider (1.10) under the hypothesis that r<q≤p. We divide this case\ninto two subcases that r<q<p andr<q=p.\n(III11) We handle with the case r <q <p . In view of the Bernstein inequality (2.13),\nwe see that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/summationdisplay\nj≤k2j[σ+n(1\nq−1\np)]/ba∇dbl˙∆ju/ba∇dblLq,∞(Rn)+C/summationdisplay\nj>k2−j[s−σ−n(1\nr−1\np)]2js/ba∇dbl˙∆ju/ba∇dblLr,∞(Rn)\n≤C2k[σ+n(1\nq−1\np)]\n1−2−[σ+n(1\nq−1\np)]/ba∇dblu/ba∇dbl˙B0q,∞,∞+C2−k[s−σ−n(1\nr−1\np)]\n1−2−[s−σ−n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙Bsr,∞,∞,\nwhere we have used σ+n(1\nq−1\np)>0 and (3.16).\nTherefore, we conclude that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞.\n(III12) We need to show (1.10) under the hypothesis that r<q=p. Observe that the\nconditions−n\nr/\\e}atio\\slash=σ−n\npimplies that σ>0 in this case. In the same manner as (3.20), we\nsee that\n/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)≤C/ba∇dbl˙∆ju/ba∇dbl1−α\nLr,∞(Rn)/ba∇dbl˙∆ju/ba∇dblα\nL(1+ε)p,∞(Rn),1\np=1−α\nr+α\n(1+ε)p,(3.23)\n23whereε>0 will be determined later. Then we obtain\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/summationdisplay\nj≤k2jσ/ba∇dbl˙∆ju/ba∇dbl1−α\nLr,∞(Rn)/ba∇dbl˙∆ju/ba∇dblα\nL(1+ε)p,∞(Rn)+C/summationdisplay\nj>k2−j[s−σ−n(1\nr−1\np)]2js/ba∇dbl˙∆ju/ba∇dblLr,∞(Rn)\n≤C/summationdisplay\nj≤k2j[σ+nεα\np(1+ε)−s(1−α)]/ba∇dblu/ba∇dblα\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−α\n˙Bsr,∞,∞+C2−k[s−σ−n(1\nr−1\np)]\n1−2−[s−σ−n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙Bsr,∞,∞,\nAs the arguments in ( II3), we can choose ε>0 sufficiently small to ensure that σ+nεα\np(1+ε)−\ns(1−α)>0. This yields the desired inequality (1.10). We omit the det ails.\n(III2) Letr<qandp<q. It is clear that\n1\np=/parenleftBig1−σ\ns−θ\n1−σ\ns/parenrightBig/parenleftBig1\nr−s−σ\nn/parenrightBig\n+/parenleftBigθ\n1−σ\ns/parenrightBig/parenleftBig\n(1−σ\ns)1\nq+σ\nsr/parenrightBig\n, (3.24)\nwhich yields that r<pin this case. Additionally, the condition s−n\nr/\\e}atio\\slash=σ−n\npguarantees\nthat1\np/\\e}atio\\slash= (1−σ\ns)1\nq+σ\ns1\nr.\n(III21) We assume that r<q, p<q and1\np<(1−σ\ns)1\nq+σ\ns1\nr. Then (3.16) follows from\n(3.24). Note that r<p<q , a slight modification of the proof of (3.19) together with (3 .16)\nimplies that\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1=/summationdisplay\nj≤k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)+/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)\n≤C/summationdisplay\nj≤k2jσ/ba∇dbl˙∆ju/ba∇dblα\nLq,∞(Rn)/ba∇dbl˙∆ju/ba∇dbl1−α\nLr,∞(Rn)+C/summationdisplay\nj>k2−j[s−σ−n(1\nr−1\np)]2js/ba∇dbl˙∆ju/ba∇dblLr,∞(Rn)\n≤C2k[σ−s(1−α)]\n1−2−[σ−s(1−α)]/ba∇dblu/ba∇dblα\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−α\n˙Bsr,∞,∞+C2−k[s−σ−n(1\nr−1\np)]\n1−2−[s−σ−n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙Bsr,∞,∞.\nHere we need the fact that σ−s(1−α)>0 with 1/p=α/q+(1−α)/r, which is derived\nfrom the hypothesis that1\np<(1−σ\ns)1\nq+σ\ns1\nr. Therefore, we obtain the desired estimate\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞.\n(III22) Now, it remains to show (1.10) with r<qand1\np>(1−σ\ns)1\nq+σ\ns1\nr, which imply\nthat\np<q, s (1−α)−σ>0 and1\np=α\nq+1−α\nr. (3.25)\nIt follows from (3.24) and1\np>(1−σ\ns)1\nq+σ\ns1\nrthat\n1\np<1\nr−s−σ\nn<1\nr,\nwhich together with p<qenables us to derive that\nσ−s+n(1\nr−1\np)>0 and 0<α<1. (3.26)\n24Making use of the Bernstein inequality (2.13) for Lorentz sp aces, (2.4), (3.25) and (3.26),\nwe arrive at\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1=/summationdisplay\nj≤k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)+/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dblLp,1(Rn)\n≤C/summationdisplay\nj≤k2j[σ−s+n(1\nr−1\np)]2js/ba∇dbl˙∆ju/ba∇dblLr,∞(Rn)+C/summationdisplay\nj>k2jσ/ba∇dbl˙∆ju/ba∇dblα\nLq,∞(Rn)/ba∇dbl˙∆ju/ba∇dbl1−α\nLr,∞(Rn)\n≤C2k[σ−s+n(1\nr−1\np)]\n1−2−[σ−s+n(1\nr−1\np)]/ba∇dblu/ba∇dbl˙Bsr,∞,∞+C2−k[s(1−α)−σ]\n1−2−[s(1−α)−σ]/ba∇dblu/ba∇dblα\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−α\n˙Bsr,∞,∞.\nWe thereby deduce the inequality\n/ba∇dblu/ba∇dbl˙Bσ\np,1,1≤C/ba∇dblu/ba∇dblθ\n˙B0q,∞,∞/ba∇dblu/ba∇dbl1−θ\n˙Bsr,∞,∞.\nThe proof of this theorem is completed.\n4 Proof of Theorem 1.4\nThissection is concernedwiththeapplication ofGagliardo -Nirenberg inequalities inLorentz\ntype spaces to the energy conservation of 3D Navier-Stokes e quations. We shall follow the\npath of [12] to prove (1.20).\nProof of Theorem 1.4. As in [11, 12], since Leray-Hopf weak solutions vsatisfy (1.16) in\nthe sense of distributions, there holds for any Q∈Z,\n1\n2/ba∇dblSQv(T)/ba∇dbl2\nL2(R3)+/integraldisplayT\n0/ba∇dbl∇SQv/ba∇dbl2\nL2(R3)ds=1\n2/ba∇dblSQv0/ba∇dbl2\nL2(R3)+/integraldisplayT\n0/integraldisplay\nTr(SQ(v⊗v)·∇SQv)dxds.\nIn order to get the energy equality, it is enough to show/integraltextT\n0/integraltext\nTr(SQ(v⊗v)·∇SQv)dxds→0\nasQ→ ∞. To this end, we recall the following estimates proved in [11 , 12]\n/integraldisplayT\n0/integraldisplay\n|Tr(SQ(v⊗v)·∇SQv)|dxds\n≤C/integraldisplayT\n0\n/summationdisplay\nk<Q22k\n3/ba∇dbl∆kv/ba∇dbl2\nL3(R3)2−4|r−Q|\n3\n3\n2\nds+C/integraldisplayT\n0\n/summationdisplay\nk≥Q22k\n3/ba∇dbl∆kv/ba∇dbl2\nL3(R3)2−2|k−Q|\n3\n3\n2\nds\n≤C/integraldisplayT\n0/summationdisplay\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds.\n(4.1)\n(1) In view of the interpolation inequality (2.4) and Young i nequality (2.11) for Lorentz\nspaces, we know that\n/ba∇dbl∆kv/ba∇dblL3(R3)≤C/ba∇dbl∆kv/ba∇dbl1\n3\nL2(R3)/ba∇dbl∆kv/ba∇dbl2\n3\nL4,∞(R3)\n≤C/ba∇dbl∆kv/ba∇dbl1\n3\nL2(R3)/ba∇dblv/ba∇dbl2\n3\nL4,∞(R3).(4.2)\n25Then we may derive from (4.2) that\n/integraldisplayT\n0/summationdisplay\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds\n≤C/integraldisplayT\n0/summationdisplay\nk2−2|k−Q|\n32k/ba∇dbl∆kv/ba∇dblL2(R3)/ba∇dblv/ba∇dbl2\nL4,∞(R3)ds\n≤C/summationdisplay\nk2−2|k−Q|\n3/parenleftBig/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2/parenleftBig/integraldisplayT\n0/ba∇dblv/ba∇dbl4\nL4,∞(R3)ds/parenrightBig1\n2.(4.3)\nNext, we show/summationtext\nk2−2|k−Q|\n3/parenleftBig/integraltextT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2→0 asQ→ ∞. Indeed, since the\nenergy inequality for Leray-Hopf weak solutions guarantee s thatv∈L∞(0,T;L2(R3)) and\n∇v∈L2(0,T;L2(R3)),it follows from Parseval’s identity of the Fourier transfor m and the\nfact thatϕ∈C∞(R3) is supported in the shell {ξ∈R3:3\n4≤ |ξ| ≤8\n3}that\n∞/summationdisplay\nk=0/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds=∞/summationdisplay\nk=0/integraldisplayT\n0/vextenddouble/vextenddouble/vextenddouble/bracketleftBig/vextendsingle/vextendsingle/vextendsingle·\n2k/vextendsingle/vextendsingle/vextendsingle−1\nϕ/parenleftBig·\n2k/parenrightBig/bracketrightBig\nF/parenleftBig\n|∇v|/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\nL2(R3)ds\n≤16\n9∞/summationdisplay\nk=0/integraldisplayT\n0/vextenddouble/vextenddouble/vextenddoubleϕ/parenleftBig·\n2k/parenrightBig\nF/parenleftBig\n|∇v|/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\nL2(R3)ds\n≤16\n9/integraldisplayT\n0/integraldisplay\nR3/parenleftBig∞/summationdisplay\nk=0ϕ/parenleftBig\n2−kξ/parenrightBig/parenrightBig/vextendsingle/vextendsingle/vextendsingleF/parenleftBig\n|∇v|/parenrightBig\n(ξ)/vextendsingle/vextendsingle/vextendsingle2\ndξds\n≤16\n9/integraldisplayT\n0/vextenddouble/vextenddouble/vextenddoubleF/parenleftBig\n|∇v|/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\nL2(R3)ds=16\n9/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl2\nL2(R3)ds<∞,\nwhich together with the classical H¨ older inequality yield s that\n/summationdisplay\nk>Q\n22−2|k−Q|\n3/parenleftBig/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2\n≤/parenleftBig/summationdisplay\nk>Q\n22−4|k−Q|\n3/parenrightBig1\n2/parenleftBig/summationdisplay\nk>Q\n2/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2\n≤/parenleftBig\n2∞/summationdisplay\nk=Q2−4|k−Q|\n3/parenrightBig1\n2/parenleftBig/summationdisplay\nk>Q\n2/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2\n≤2/parenleftBig/summationdisplay\nk>Q\n2/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2→0,asQ→ ∞.(4.4)\nOn the other hand, we observe that\n/summationdisplay\nk≤Q\n22−2|k−Q|\n3/parenleftBig/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2\n≤/parenleftBig1\n2/parenrightBig5\n3+2Q\n3/parenleftBig/integraldisplayT\n0/ba∇dbl∆−1v/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2+4\n3/summationdisplay\n0≤k≤Q\n22−2|k−Q|\n3/parenleftBig/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2(4.5)\n26≤/parenleftBig1\n2/parenrightBig5\n3+2Q\n3/parenleftBig/integraldisplayT\n0/ba∇dblv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2+/parenleftBig1\n2/parenrightBigQ\n3−8\n3/parenleftBig/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2→0,asQ→ ∞.\nHere we have used the fact that\n/integraldisplayT\n0/ba∇dbl∆−1v/ba∇dbl2\nL2(R3)ds=/integraldisplayT\n0/ba∇dbl̺ˆv/ba∇dbl2\nL2(R3)ds≤/integraldisplayT\n0/ba∇dblˆv/ba∇dbl2\nL2(R3)ds=/integraldisplayT\n0/ba∇dblv/ba∇dbl2\nL2(R3)ds<∞.\nCombining (4.4) and (4.5), we thereby conclude that\n/summationdisplay\nk2−2|k−Q|\n3/parenleftBig/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n2→0,asQ→ ∞. (4.6)\n(2) Letq>4. Before going further, we write\nIQ=:{s∈[0,T] :/ba∇dblv(s)/ba∇dblLq,∞(R3)>2Q(q−2)\nq}\nandIc\nQ=: [0,T]\\IQ.\nAccording to the interpolation inequality (2.4) and (2.11) , we infer that\n/ba∇dbl∆kv/ba∇dblL3(R3)≤C/ba∇dbl∆kv/ba∇dbl2q−6\n3(q−2)\nL2(R3)/ba∇dbl∆kv/ba∇dblq\n3(q−2)\nLq,∞(R3)\n≤C/ba∇dbl∆kv/ba∇dbl2q−6\n3(q−2)\nL2(R3)/ba∇dblv/ba∇dblq\n3(q−2)\nLq,∞(R3).(4.7)\nPlugging this into (4.1) and using the H¨ older inequality, w e observe that\n/integraldisplay\nIQ/summationdisplay\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds\n≤C/integraldisplay\nIQ/summationdisplay\nk2k2−2|k−Q|\n3/ba∇dbl∆kv/ba∇dbl2q−6\nq−2\nL2(R3)/ba∇dblv/ba∇dblq\nq−2\nLq,∞(R3)ds\n≤C/summationdisplay\nksup\nt∈[0,T]/ba∇dbl∆kv/ba∇dbl4q−14\n3(q−2)\nL2(R3)/integraldisplay\nIQ2−2|k−Q|\n322k\n3/ba∇dbl∆kv/ba∇dbl2\n3\nL2(R3)2k\n3/ba∇dblv/ba∇dblq\nq−2\nLq,∞(R3)ds\n≤Csup\nt∈[0,T]/ba∇dblv/ba∇dbl4q−14\n3(q−2)\nL2(R3)/summationdisplay\nk2−2|k−Q|\n3/parenleftBig/integraldisplay\nIQ22k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n32k\n3/parenleftBig/integraldisplay\nIQ/ba∇dblv/ba∇dbl3q\n2(q−2)\nLq,∞(R3)ds/parenrightBig2\n3.(4.8)\nThe hypothesis (1.20) enables us to obtain\nf∗(λ) =|{s∈[0,T] :/ba∇dblv(s)/ba∇dblLq,∞(R3)>λ}| ≤Cλ−2q\nq−2. (4.9)\nThis yields that\n2k\n3/parenleftBig/integraldisplay\nIQ/ba∇dblv/ba∇dbl3q\n2(q−2)\nLq,∞(R3)ds/parenrightBig2\n3=2k\n3/parenleftBig3q\n2q−4/parenrightBig2\n3/parenleftBig/integraldisplay∞\n2Q(q−2)\nqλ3q\n2(q−2)−1f∗(λ)dλ+/integraldisplay2Q(q−2)\nq\n0λ3q\n2(q−2)−1|IQ|dλ/parenrightBig2\n3\n≤C2k\n3/parenleftBig/integraldisplay∞\n2Q(q−2)\nqλ3q\n2(q−2)−1−2q\nq−2dλ+23Q\n2f∗(2Q(q−2)\nq)/parenrightBig2\n3\n≤C2k\n3/parenleftBig/integraldisplay∞\n2Q(q−2)\nqλ−q\n2(q−2)−1dλ+2−Q\n2/parenrightBig2\n3\n≤C2k−Q\n3.\n(4.10)\n27Inserting the latter inequality into (4.8), we get\n/integraldisplay\nIQ/summationdisplay\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds\n≤C(/ba∇dblv0/ba∇dblL2(R3))/summationdisplay\nk2−|k−Q|\n3/parenleftBig/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n3.(4.11)\nIn the same manner as derivation of (4.6), we arrive at\n/summationdisplay\nk2−|k−Q|\n3/parenleftBig/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1\n3→0,asQ→ ∞.\nNow, it suffices to show/integraltext\nIc\nQ/summationtext\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds→0 asQ→ ∞.\nA slight modification of deduction of (4.8) ensures that for a nyε∈(0, q−4],\n/integraldisplay\nIc\nQ/summationdisplay\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds\n≤C/integraldisplay\nIc\nQ/summationdisplay\nk2k2−2|k−Q|\n3/ba∇dbl∆kv/ba∇dbl2q−6\nq−2\nL2(R3)/ba∇dblv/ba∇dblq\nq−2\nLq,∞(R3)ds\n≤C/summationdisplay\nksup\nt∈[0,T]/ba∇dbl∆kv/ba∇dbl2q−8−2ε\n(2+ε)(q−2)\nL2(R3)/integraldisplay\nIc\nQ2−2|k−Q|\n322(1+ε)k\n2+ε/ba∇dbl∆kv/ba∇dbl2(1+ε)\n2+ε\nL2(R3)2−εk\n2+ε/ba∇dblv/ba∇dblq\nq−2\nLq,∞(R3)ds\n≤Csup\nt∈[0,T]/ba∇dblv/ba∇dbl2q−8−2ε\n(2+ε)(q−2)\nL2(R3)/summationdisplay\nk2−2|k−Q|\n3/parenleftBig/integraldisplay\nIc\nQ22k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1+ε\n2+ε2−εk\n2+ε/parenleftBig/integraldisplay\nIc\nQ/ba∇dblv/ba∇dblq(2+ε)\nq−2\nLq,∞(R3)ds/parenrightBig1\n2+ε.\n(4.12)\nThen it follows from (4.9) that\n2−εk\n2+ε/parenleftBig/integraldisplay\nIc\nQ/ba∇dblv/ba∇dblq(2+ε)\nq−2\nLq,∞(R3)ds/parenrightBig1\n2+ε≤2−εk\n2+ε/parenleftBigq(2+ε)\nq−2/integraldisplay2Q(q−2)\nq\n0λq(2+ε)\nq−2−1f∗(λ)dλ/parenrightBig1\n2+ε\n≤C2−εk\n2+ε/parenleftBig/integraldisplay2Q(q−2)\nq\n0λq(2+ε)\nq−2−1−2q\nq−2dλ/parenrightBig1\n2+ε\n≤C2−εk\n2+ε/parenleftBig/integraldisplay2Q(q−2)\nq\n0λqε\nq−2−1dλ/parenrightBig1\n2+ε\n≤C2ε(Q−k)\n2+ε.\nInserting the latter inequality into (4.12), we get\n/integraldisplay\nIc\nQ/summationdisplay\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds\n≤C(/ba∇dblv0/ba∇dblL2(R3))/summationdisplay\nk2−(4−ε)|k−Q|\n3(2+ε)/parenleftBig/integraldisplayT\n022k/ba∇dbl∆kv/ba∇dbl2\nL2(R3)ds/parenrightBig1+ε\n2+ε.(4.13)\nTake the positive constant ε <min{4, q−4}. By a similar argument like (4.6), we also\nhave/integraltext\nIc\nQ/summationtext\nk2k/ba∇dbl∆kv/ba∇dbl3\nL3(R3)2−2|k−Q|\n3ds→0 asQ→ ∞.\n28(3) In light of the Gagliardo-Nirenberg inequality (1.6) fo r Lorentz spaces and (2.6), we\nobtain\n/ba∇dblv/ba∇dblL4(R3)≤C/ba∇dbl∇v/ba∇dbl3(4−q)\n2(6−q)\nL2(R3)/ba∇dblv/ba∇dblq\n12−2q\nLq,∞(R3),with 3<q<4.\nHence, the H¨ older inequality entails that\n/integraldisplayT\n0/ba∇dblv/ba∇dbl4\nL4(R3)ds≤C/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl6(4−q)\n6−q\nL2(R3)/ba∇dblv/ba∇dbl2q\n6−q\nLq,∞(R3)ds\n≤C/parenleftBig/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl2\nL2(R3)ds/parenrightBig3(4−q)\n6−q/parenleftBig/integraldisplayT\n0/ba∇dblv/ba∇dblq\nq−3\nLq,∞(R3)ds/parenrightBig2(q−3)\n6−q.\nThis means that v∈L4(0,T;L4(R3)),which helps us to get energy conservation.\n(4) Since 3 /2< q <9/5, we may choose an index ˜ q∈(q,9/5) such that 1 /˜q=\n(1−α)/2+α/qwith someα∈(0,1).Let 1/˜p+3/˜q= 2.Then it follows from q>3/2 that\n1/˜p= 2−3(1−α)/2−3α/q>(1−α)/2.\nThanks to the interpolation characteristic (2.4) of Lorent z spaces, (2.6) and the H¨ older\ninequality, we arrive at\n/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl˜p\nL˜q(R3)ds≤C/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl(1−α)˜p\nL2(R3)/ba∇dbl∇v/ba∇dblα˜p\nLq,∞(R3)ds\n≤C/parenleftBig/integraldisplayT\n0/ba∇dbl∇v/ba∇dbl2\nL2(R3)ds/parenrightBig(1−α)˜p\n2/parenleftBig/integraldisplayT\n0/ba∇dbl∇v/ba∇dblp\nLq,∞(R3)ds/parenrightBig1−(1−α)˜p\n2,\nwhere we have used the fact that\n2α˜p\n2−(1−α)˜p=p,and 0<(1−α)˜p\n2<1.\nHence, we conclude the desired energy equality from the know n result (1.18).\n(5) Note that 1 /s < p < 3,we may choose an index p1such that max {1,p}< p1<\nmin{3,sp}.Takeθ= 1−p/p1,then 0<θ<1−1/s.Let 1/p1+6/(5q1) = 1,which implies\nthat 9/5<q1<∞and\n3\nq1−1 =3θ\n2+(1−θ)/parenleftbigg3\n2−5\n2p/parenrightbigg\n=3θ\n2+(1−θ)/parenleftbigg3\nq−s/parenrightbigg\n.\nBy the Gagliardo-Nirenberg inequality (1.10) for Besov-Lo rentz spaces and (2.6), we find\nthat /integraldisplayT\n0/ba∇dbl∇v/ba∇dblp1\nLq1(R3)ds≤C/integraldisplayT\n0/ba∇dblv/ba∇dblθp1\nL2(R3)/ba∇dblv/ba∇dbl(1−θ)p1\n˙Bsq,∞,∞ds\n≤C/ba∇dblv/ba∇dblp1−p\nL∞(0,T;L2(R3))/integraldisplayT\n0/ba∇dblv/ba∇dblp\n˙Bsq,∞,∞ds.\nThis together with the known result (1.18) yields the energy equality.\nConsequently, we complete the proof of this theorem.\n29Acknowledgements\nThe authors would like to express their sincere gratitude to Dr. Xiaoxin Zheng at Bei-\nhang University, for a lot of useful discussion on this topic . Wang was partially supported\nby the National Natural Science Foundation of China under gr ant (No. 11971446, No.\n12071113 and No. 11601492). Wei was partially supported by t he National Natural Science\nFoundation of China under grant (No. 11601423, No. 11771352 , No. 11871057). Ye was\npartially supported by the National Natural Science Founda tion of China under grant (No.\n11701145) and China Postdoctoral Science Foundation (No. 2 020M672196).\nReferences\n[1] H. Bahouri, J. Chemin and R. Danchin,, Fourier analysis a nd nonlinear partial dif-\nferential equations. Grundlehren der Mathematischen Wiss enschaften 343. Springer,\nHeidelberg, 2011.\n[2] O. A. Barraza, Self-similar solutions in weak Lp-spaces of the Navier-Stokes equations.\nRev. Mat. Iberoamericana 12 (1996), 411–439.\n[3] H. Beirao da Veiga and J. 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Anal.\nAppl. 480 (2019), no. 2, 123443, 9 pp.\n32"    },    {        "title": "2307.11676v1.Ascent_and_Descent_of_Composition_Operators_on_Orlicz_lorentz_space.pdf",        "content": "arXiv:2307.11676v1  [math.FA]  21 Jul 2023ASCENT AND DESCENT OF COMPOSITION OPERATORS ON\nORLICZ-LORENTZ SPACES\nNEHA BHATIA AND ANURADHA GUPTA\nAbstract. The aim of this paper is to discuss the characterizations for the\ncomposition operator on Orlicz-Lorentz spaces Lφ,wto have\nfinite ascent (descent).\n1.Introduction\nLetVbe a vector space and Tbe a linear operator on V. ThenN(T) andR(T)\ndenotes the null space and the range space of Tgiven by\nN(T) ={v∈V:T(v) = 0}\nand\nR(T) ={T(v) :v∈V}.\nForm≥0, the null space and range space of Tmsatisfies\nN(T)⊆N(T2)⊆N(T3)⊆ ···N(Tm)⊆N(Tm+1)···,\nR(T)⊆R(T2)⊆R(T3)··· ⊆R(Tm)⊆R(Tm+1)···\nDefinition 1. If there is an integer m >0such that N(Tm) =N(Tm+1), the\nsmallest such integer is called the ascent of Tand we denote it by A(T). If no such\nintegermexists such that N(Tm) =N(Tm+1), we say that ascent of Tis infinite\ni.e.,A(T) =∞.\nDefinition 2. If there is an integer m >0such that R(Tm+1) =R(Tm), the\nsmallest such integer is called the descent of T and is denote d byD(T). If there\nis no such integer msuch that R(Tm+1) =R(Tm), we say that descent of Tis\ninfinite i.e., D(T) =∞.\nMany authors have studied the characterizations of composition a nd weighted\ncomposition operators with ascent and descent on different funct ion spaces such\nasLpspace,lpspace, Orlicz space, Lorentz space [[2], [4], [5], [6]]. Motivated\nby their work, we have discussed the characterization of composit ion operators on\nOrlicz-Lorentz spaces L(φ,ω)to have finite ascent and descent.\n2010Mathematics Subject Classification. 47B33, 47B38, 46E30.\nKey words and phrases. Ascent,composition operators, descent, Orlicz-Lorentz s paces, non-\nsingular transformation, Radon-Nikodym derivative.\nSubmitted Month Day, 20xx. Published Month Day, 20xx.\n12 NEHA BHATIA AND ANURADHA GUPTA\n2.Composition Operators on Orlicz-Lorentz Spaces\nLet(E,E,ν)beaσ-finitemeasurespaceand fbeanycomplex-valuedmeasurable\nfunction on the space.\nFors≥0, define νfthedistribution function as\nνf(s) =ν{x∈E:|f(x)|> s}\nClearly,νfis decreasing. By f∗we mean the non-increasing rearrangement off\ngiven as\nf∗(t) = inf{s >0 :µf(s)≤t}, t≥0\nf∗is a non-negative and decreasing function.\nBy a weight function ω, we mean ω:J→JwhereJ= (0∞) a non-increasing\nlocally integrable function such that/integraltext∞\n0ω(t)dt=∞.\nLetφ: [0∞)→[0∞) be a convex function such that\nφ(x) = 0⇔x= 0,\nlim\nx→∞φ(x) = +∞.\nSuch a function is called a young function . The young function is said to satisfy\nthe ∆2-condition if for some k >0,\nφ(2x)≤kφ(x),for allx >0.\nIfφsatisfies ∆ 2-condition, then we define the space Lφ,ωas\n/braceleftbigg\nf:E→Cmeasurable :/integraldisplay∞\n0φ(αf∗(t))ω(t)dt <∞for some α >0/bracerightbigg\nThe space L(φ,ω)is called an Orlicz-Lorentz space and is a Banach space with\nrespect to the Luxemburg norm given by\n/bardblf/bardblφ,ω= inf/braceleftbigg\nǫ >0 :/integraldisplay∞\n0φ(|f∗(t)|/ǫ)ω(t)dt≤1/bracerightbigg\nOrlicz-Lorentzspaceis commongeneralizationofOrliczspaceand Lo rentzspace.\nIfw(t) = 1 then it becomes an Orlicz space. If φ(x) =xp, 1≤p <∞, then it\nbecomes a Lorentz space L(p,q). LetB(L(φ,ω)) denote the Banach algebra of all\nbounded linear operators on L(φ,ω). For more details on Orlicz-Lorentz spaces, we\nrefer to [1], [7].\nLetΨ :E→Ebeameasurablenon-singulartransformation(i.e., ν(Ψ−1(S)) = 0\nwhenever ν(S) = 0 for S∈ E).\nIf Ψ is non-singular, then we say that νΨ−1is absolutely continuous with re-\nspect to ν. Hence, by Radon-Nikodym theorem, there exists a unique non-ne gative\nmeasurable function gsuch that\n(νΨ−1)(S) =/integraldisplay\nSgdν,forS∈ E.\ngis called the Radon-Nikodym derivative and is denoted bydνΨ−1\ndν. Form≥2,\nwe observe that νm\nΨ≪νm−1\nΨ≪ ···νΨ. Hence\nνm\nΨ=/integraldisplay\nSfΨdν for S ∈ E.\nDefinition 3. Measures ν1andν2are said to be equivalent if ν2≪ν1≪ν2.ASCENT AND DESCENT OF COMPOSITION OPERATORS ON ORLICZ-LORE NTZ SPACES3\nDefinition 4. A measurable transformation Ψis said to be measure preserving if\nit preserves the measure i.e., ν(Ψ−1(A)) =ν(A)for allA∈ E.\nDefinition 5. Let(E,E,ν)be a measure space. A measurable transformation Ψ :\nE→Eis said to be pre-positive if it satisfies the condition ν(Ψ−1(A))>0whenever\nν(A)>0.\nLet Ψ be a measurable transformation then the composition operat orCΨ[8]\ninduced by Ψ is given by\nCΨ=f◦Ψ,for every f∈L(φ,w).\nLet Ψ :E→Eis a non- singular measurable transformation, then Ψmis also\na non-singular measurable transformation for every non-negativ e integer mwith\nrespect to the measure ν. Thus, we can define the composition operator CΨmon\nOrlicz-Lorentz space L(φ,w), such that Cm\nΨ(fΨ) =fΨoΨm=CΨm(fΨ) for every\nmeasurable function fΨof the Orlicz-Lorentz space. Also, the measure νoΨ−mis\ndefined as\nνoΨ−m(S) =νoΨ−(m−1)(Ψ−1(S)) =νoΨ−1(Ψ−(m−1)(S))for S∈ E.\nThen\n...≪ν oΨ−(m+1)≪ν oΨ−m≪νΨ−(m−1)≪...≪ν oΨ−1≪ν.(2.1)\nIf we put νm=ν oΨ−m, then by Radon - Nikodyn theorem, there exists a\nnon-negative locally integrable function fΨmonEsatisfying\nνm(S) =/integraldisplay\nSfΨm(x)dν(x)for S∈ E. (2.2)\nHere,fΨm/parenleftbig\n=dνm\ndν/parenrightbig\nis called the Radon -Nikodym derivative of νmwith respect\ntoν.\nIn [1], the necessary and sufficient condition for the boundedness o f the com-\nposition operators on Orlicz-Lorentz spaces Lφ,ware discussed and given by the\nfollowing theorem:\nTheorem 2.1. LetΨ :E→Ebe a non-singular measurable transformation. Then\nthe composition operator CΨis bounded on Lφ,wif and only if there exists a constant\nK >0such that\nνΨ−1(A)≤Kν(A)for A∈ E.\nAlso, the measurable transformation Ψ is said to be bounded away fr om zero if\nthere exists a positive real number ǫ, such that f(x)≥ǫfor almost all x∈Swhere\nS={x∈E:f(x)/ne}ationslash= 0}.\n3.Ascent of the Composition Operators\nTheorem 3.1. LetCΨbe a composition operator on L(φ,w). Then N(Cm\nΨ) =\nL(φ,w)(Em)form≥0i.e., the kernel is the collection of all the measurable func tions\nfΨinL(φ,w)satisfying fΨ(x) = 0forx∈E/Em, whereEm={x∈E:fΨm(x) =\n0}.\nProof.Letfbe an element of L(φ,w)(Em). Then\nν{x∈E:f oΨm(x)/ne}ationslash= 0} ≤ν oΨ−m(Em) =/integraldisplay\nEmfΨm(x)dν(x) = 0.4 NEHA BHATIA AND ANURADHA GUPTA\nThis implies that f◦Ψm= 0 a.e. This gives that fΨ∈ N(Cm\nΨ). This implies that\nL(φ,w)(Em)⊆ N(Cm\nΨ). (3.1)\nAlso, let fΨ∈ N(Cm\nΨ), thenν oΨ−m{x∈E:fΨ(x)/ne}ationslash= 0}= 0.TakeF={x∈\nE/Em:fΨ(x)/ne}ationslash= 0}andG={x∈Em:fΨ(x)/ne}ationslash= 0}. From (2.2), we have\n0 =/integraldisplay\nFfΨm(x)dν(x) +/integraldisplay\nGfΨm(x)dν(x)\n=/integraldisplay\nFhΨm(x)dν(x)\n≥1\nn/integraldisplay\nFn∩Fdν\n=1\nnν(Fn∩F)\nfor each nwhereFn={x∈E/Em:fΨ>1\nn} ⊆F. SinceFn∩F=F, this gives\nthatν(Fn∩F) =ν(F) for each n. Therefore ν(F) = 0. This implies that f= 0\na.e. onE/Em. Hencef∈L(φ,w)(Em). This gives that\nN(Cm\nΨ)⊆L(φ,w)(Em). (3.2)\nFrom (3.1) and (3.2), we obtain\nN(Cm\nΨ) =L(φ,w)(Em).\n/square\nLemma 3.2. LetΨbe a non-singular measurable transformation on the measure\nspace(E,E,ν)that induces the composition operator CΨon the Orlicz-Lorentz space\nL(φ,w). ThenN(Cm\nΨ) =N(Cm+1\nΨ)if and only if νmandνm+1are equivalent.\nProof.LetCΨ∈ B(L(φ,w)). Suppose that νmandνm+1are equivalent. Then\nνm+1≪νm≪νm+1. Sofrom(2 .1), wehave νm≪νm+1≪νandνm+1≪νm≪ν,\nand hence, by the chain rule,\nfΨm(x) =dνm\ndνm+1(x).fΨm+1(x)\nfΨm+1(x) =dνm+1\ndνm(x).fΨm(x)\nThis gives that Em=Em+1. SinceN(Cm\nΨ) =L(φ,w)(Em) form≥0, therefore\nN(Cm\nΨ) =L(φ,w)(Em) =L(φ,w)(Em+1) =N(Cm+1\nΨ).\nConversely,supposethat N(Cm\nΨ) =N(Cm+1\nΨ). Thisgives Lφ,w(Em) =Lφ,w(Em+1).\nWe first claim that ν(Em/Em+1) = 0. The assumption of ν(Em/Em+1)>0 pro-\nvides a set Fn={x∈Em:fΨm+1(x)>1\nn,n∈N}of non-zero finite measure.\nCharacteristic functions χSare dense in Orlicz-Lorentz space for S∈ E. Now,\n/bardblχFn/bardbl=1\nφ−1(1\nν(Fn)).\nThus,χFn∈L(φ,w)(Em) =L(φ,w)(Em+1) i.e.,χFnvanishes outside Em+1which\ngives that Fn⊆Em+1. Therefore,\n0≤1\nn/integraldisplay\nFnχFndν≤/integraldisplay\nFnfΨm+1dν= 0.ASCENT AND DESCENT OF COMPOSITION OPERATORS ON ORLICZ-LORE NTZ SPACES5\nThis implies that ν(Fn) = 0 which is a contradiction to our assumption. Similarly,\nν(Em+1/Em) = 0. To show that νmandνm+1are equivalent, it is suffices to show\nthatνm≪νm+1. Suppose that νm+1(A) = 0 for A∈ E. This implies that for each\nsubsetBofA, we have\n/integraldisplay\nBfΨm+1dν≤/integraldisplay\nAfΨm+1dν=νm+1(A) = 0\n⇒ν{x∈A:fΨm(x)/ne}ationslash= 0,fΨm+1(x)/ne}ationslash= 0}= 0.\nAlso,ν{x∈A:fΨm(x)/ne}ationslash= 0,fΨm+1(x) = 0}= 0 as it is a subset of Em+1/Em.\nTherefore,\nνm(A) =/integraldisplay\nCfΨmdν=/integraldisplay\nDfΨmdν= 0\nwhereC={x∈A:fΨm= 0}andD={x∈A:fΨm/ne}ationslash= 0}. /square\nTheorem 3.3. LetΨbe a non-singular measurable transformation on the measure\nspaceEinducing the composition operator CΨon the Orlicz-Lorentz space L(φ,w).\nA necessary and sufficient condition for A(CΨ) =mis thatmis the least non-\nnegative integer such that the measures νmandνm+1are equivalent.\nCorollary 3.4. LetCΨ∈ B(L(φ,w)). Then the A(CΨ) = 0in each of the following\nsituations:\n(1)Ψis a measure preserving.\n(2)Ψis surjective.\nTheorem 3.3 can be restated in the following manner:\nTheorem 3.5. The necessary and sufficient condition for a composition oper ator\nCΨon Orlicz-Lorentz space L(φ,w)to haveA(CΨ) =∞is that the measures νm\nandνm+1can never be equivalent for any natural number m.\nTheorem 3.6. A sufficient condition for an operator CΨ∈ B(L(φ,w)), induced by\na measurable transformation Ψ, to have A(CΨ)<∞is thatΨis a pre-positive\nmeasurable transformation.\nProof.Let Ψ be a pre-positive measurable transformation inducing CΨonL(φ,w).\nLet us suppose that A(CΨ)≮∞, then for each natural number m, there exists\nfm∈ N(Cm\nΨ) such that\nν/braceleftbig\nx∈E:fψm◦Ψm−1(x)/ne}ationslash= 0/bracerightbig\n>0.\nSince Ψ is pre-positive, so ν(Ψ−1(Em)>0 where\nν(Ψ−1(Em) =ν({x∈E: (fΨm◦Ψm)(x)/ne}ationslash= 0}),\nwhich is a contradiction to the fact that fΨm∈ N(Cm\nΨ). Therefore, it follows that\nA(CΨ)<∞. /square\nTheorem 3.7. If the measurable transformation Ψinducing CΨ∈ B(L(φ,w)), is\npre-positive, then A(CΨ) = 0.\nTheorem 3.8. LetCΨ∈ B(L(φ,w)). If there exists a sequence {A}mof mea-\nsurable sets such that for each m,0< ν(Am)<∞,ν(Ψ−m(Am)) = 0 and\nν(Ψ−(m−1)(Am))/ne}ationslash= 0, thenA(CΨ) =∞.6 NEHA BHATIA AND ANURADHA GUPTA\nProof.Let{Am}m≥1be a sequence of measurable sets satisfying 0 < ν(Am)<∞,\n(Ψ−m(Am)) = 0 and ν(Ψ−(m−1)(Am))/ne}ationslash= 0 for each m. For each measurable set S,\nwe know that\n/bardblχS/bardbl=1\nφ−1(1\nν(S)).\nHence, for each natural number m, the characteristics function XAm∈L(φ,w)\nandχAm∈(N(Cm\nΨ)/N(Cm−1\nΨ)). Therefore, A(CΨ) =∞. /square\nTheorem 3.9. Let the measurable transformation ΨonEinducing CΨonL(φ,w)\nbe such that the image of each measurable set is measurable. I fA(CΨ)≮∞on\nL(φ,w), then there exists a sequence of subsets AmofEsuch that for all m >1,\n(1)0< ν(Am)<∞,\n(2)Am⊆Ψm−1(B)for some B∈ E,\n(3)Am/∈ {Ψm(S) :S∈ Eand ν(S)>0}.\nProof.LetA(CΨ) =∞. Then, for each positive integer m, we have N(Cm−1\nΨ)⊂\nN(Cm\nΨ). This implies that there exists fΨm∈ N(Cm\nΨ) such that fΨm/∈ N(Cm−1\nΨ).\nTakeEm=/braceleftbig\nx∈E:fΨm◦Ψm−1(x)/ne}ationslash= 0/bracerightbig\n. Clearly, ν(Em)>0 and Ψm−1(Em)\nis measurable. Now, we show that ν(Ψm−1(Em))>0. Let us suppose that\nν(Ψm−1(Em)) = 0. Since Ψ is non-singular and Em⊆Ψ−(m−1)(Ψm−1(Em)),\nwe get\nν(Em)≤ν(Ψ−(m−1)(Ψm−1(Em))) = 0,\nwhich is a contradiction. Hence, ν(Ψm−1(Em))>0.\nNow, since the measure νisσ-finite, we can choose a measurable subset Am\nof Ψm−1(Em) such that 0 < ν(Am)<∞wherex∈Amsuch that fΨm(x)/ne}ationslash= 0.\nNext, let if possible that Am= Ψm(S) for some measurable set Shaving positive\nmeasure, then\nν({x∈S:fΨm◦Ψm(x) = 0}) = 0.\nSincefΨm∈ N(Cm\nΨ),ν({x∈S: (fΨm◦Ψm)(x)/ne}ationslash= 0}) = 0. The above two sets each\nhaving measure zero mean that ν(S) = 0, this contradicts the fact that ν(S)>0.\nHence the proof. /square\nIf we put E=N,E= 2Nandνis the counting measure, then the correspond-\ning Orlicz-Lorentz space is called as Orlicz-Lorentz sequence space and is denoted\nbyl(φ,ω)[3]. The following theorem gives the characterization of the composit ion\noperators to have infinite ascent on the Orlicz-Lorentz sequence spacesl(φ,ω).\nTheorem 3.10. A necessary and sufficient condition for the composition oper ator\nCΨon the Orlicz-Lorentz sequence space l(φ,ω)induced by Ψ :N→Nto have\nA(CΨ) =∞is that there exists a sequence of distinct natural numbers /an}bracketle{tnm/an}bracketri}htsuch\nthatnm/ne}ationslash∈Ψm(N)butnm∈Ψm−1(N)for each m≥1.\nProof.The sufficient part is a direct conclusion. For the necessary part, u sing\nTheorem3.9, wegetasequenceofnonemptymeasurablesubsets AmofNsatisfying\nAm⊆Ψm−1(N) andAm/notsubseteqlΨm(N)⊆Ψm−1(N). Hence we can take a sequence of\ndistinct natural number nmsatisfying nm∈Ψm−1(N) andnm/∈Ψm(N). /squareASCENT AND DESCENT OF COMPOSITION OPERATORS ON ORLICZ-LORE NTZ SPACES7\n4.Descent of composition operators\nDefinition 6. A measure space (E,E,ν)is said to be separable if for every distinct\npair of points xandyinE, we can find disjoint positive measurable sets E1and\nE2such that x∈E1andy∈E2.\nTheorem 4.1. The composition operator CΨon the Orlicz-Lorentz space L(φ,w)\nhasD(CΨ) = 0ifCΨis bounded away from zero on its support and Ψ−1(E) =E.\nTheorem 4.2. Suppose that (E,E,ν)is a separable σ-finite measure space. A\nnecessary condition for the composition operator CΨonL(φ,w)to haveD(CΨ)<∞\nis that ˆΨmis injective for some non negative integer m, where ˆΨm= Ψ|R(Ψm):\nR(Ψm)→ R(Ψm).\nProof.Suppose that D(CΨ)<∞onL(φ,w). Let if possible foreach m, the mapping\nˆΨmis not injective, then there exist x1,x2, inEsuch that\nΨm(x1)/ne}ationslash= Ψm(x2) and\nΨm+1(x1) = Ψm+1(x2).\nAlso, since ( E,E,ν) is separable and σ-finite, therefore there exists two disjoint\nmeasurable sets E1andE2with non-zero finite measures containing x(= Ψm(x1))\nandy(= Ψm(x2)), respectively.\nHenceχE1,χE2∈L(φ,ω). Now consider the element f1ofL(φ,ω)given by\nf1=XX1−XX2.\nThenf2=Cm\nΨf1∈ R(Cm\nΨ). Butf2/∈ R(Cm+1\nΨ) because if f2=Cm+1\nΨf3for\nsomef3∈L(φ,ω)(E),\n1 =f1(x)\n=f1(Ψmx1)\n=Cm+1\nΨfΨ(x1)\n=Cm+1\nΨfΨ(x2)\n=f2(x2)\n=Cm\nΨf1(x2)\n=f1(y)\n=−1.\nThis implies that R(Cm+1\nΨ)⊂ R(Cm\nΨ) for each non negative integer m. Hence\nD(CΨ) =∞. /square\nTheorem 4.3. Suppose that (E,E,ν)is a separable σ-finite measure space. Then a\nnecessary condition for the composition operator CΨonL(φ,w), to have the descent\nat mostmis thatΨmis injective.\nCorollary 4.4. Let(E,E,ν)be the measure space such that every singleton set has\npositive measure and CΨ∈ B(L(φ,ω)). ThenD(CΨ)> mif the mapping Ψ|R(Ψm):\nR(Ψm)→ R(Ψm)is not injective.\nProof.Let the measure space ( E,E,ν) beσ-finite and separable and Ψ |R(Ψm)is not\ninjective, then (as in Theorem 4.2), there exists measurable sets E1andE2con-\ntainingx(= Ψm(x1)) andy(= Ψm(x2)) as the singleton sets x1andx2respectively,\nwe obtain that D(CΨ)> m. /square8 NEHA BHATIA AND ANURADHA GUPTA\nReferences\n[1] S.C. Arora, G. Datt, S. Verma, Multiplication and compos ition operators on Orlicz-Lorentz\nspaces,Int. J. Math. Anal. (Ruse) 1no. 25 (2007), 1227–1234.\n[2] D.S.Bajaj, G. Datt, Ascent and Descent of Composition op erators on Lorentz Spaces, Com-\nmun. Korean Math. Soc. 37(2022), no. 1, 195-205.\n[3] P. Bala, A. Gupta, N. Bhatia, Multiplication Operators o n Orlicz-Lorentz Sequence Spaces,\nInt. Journal of Math. Analysis 7(2013), no. 30, 1461-1469.\n[4] H. Chandra, P. Kumar, Ascent and Descent of Composition O perators on lpspaces,Demon-\nstratio Math. 43(2010), no. 1, 161-165.\n[5] R. K. Giri, S. Pradhan, On the properties of ascent and des cent of composition operator on\nOrlicz spaces, Math. Sci. Appl. E-Notes 5(2017), no. 1, 70-76.\n[6] R. Kumar, Ascent and Descent of Weighted Composition Ope rators on Lp-spaces, Mathe-\nmaticki Vensik ,60(2008), no. 1, 47-51.\n[7] S.J. Montgomery-Smith, Orlicz-Lorentz spaces, Procee dings of the Orlicz Memorial Confer-\nence, Oxford, Mississippi, 1991.\n[8] R. K. Singh, J. S. Manhas, Composition operators on funct ion spaces, North-Holland Mathe-\nmatics Studies, 179, North-Holland Publishing Co., Amster dam, 1993.\nNeha Bhatia\nDepartment of Mathematics, University of Delhi, Delhi 11000 7, India\nEmail address :nehaphd@yahoo.com\nAnuradha Gupta\nDepartment of Mathematics, Delhi College of Arts and Commerc e, University of Delhi,\nDelhi 110023, India\nEmail address :dishna2@yahoo.in"    },    {        "title": "1008.1279v1.Lorentz_Violation_and_Extended_Supersymmetry.pdf",        "content": "arXiv:1008.1279v1  [hep-ph]  6 Aug 2010November 12, 2018 15:41 WSPC - Proceedings Trim Size: 9in x 6i n patsusyproc\n1\nLORENTZ VIOLATION AND EXTENDED\nSUPERSYMMETRY\nDON COLLADAY and PATRICK MCDONALD∗\nDivision of Natural Science, New College of Florida\nSarasota, FL 34234, USA\n∗E-mail: mcdonald@ncf.edu\nWe construct a collection of Lorentz violating Yang-Mills t heories exhibiting\nsupersymmetry.\n1. Introduction and background\nSymmetryhasplayedafundamentalroleintheconstructionoffield theories\npurportingto describefundamental physics.Nowhereis this more true than\nin the construction of the Minimal Supersymmetric Standard Model where\nsymmetries mixing bosonic and fermionic states lead to tightly constr ained\ntheories often exhibiting remarkable properties. This is particularly true for\nN= 4 extended supersymmetric Yang-Mills theories,1which are known to\nbe finite.\nIn this note we construct field theories which exhibit N= 4 ex-\ntended supersymmetry and Lorentz violation. To do so we combine id eas of\nBergerandKosteleck´ y2involvingsupersymmetricscalartheoriesexhibiting\nLorentz violation, and well-known constructions of extended supe rsymmet-\nric theories involving dimensional reduction (nicely described in the wo rk\nof Brink, Schwartz and Scherk1). We begin by establishing notation in the\ncontext of the standard construction.\nConsider a gauge theory involving a single fermion λand lagrangian\nL=−1\n4F2+i\n2¯λ/ne}ationslash∂λ, (1)\nwhereFis the field strength, Fµν= [Dµ,Dν]/igandDis the covariant\nderivative,\nDµ=∂µ+igAµ. (2)November 12, 2018 15:41 WSPC - Proceedings Trim Size: 9in x 6i n patsusyproc\n2\nTo simplify notation, we will first consider the abelian case. To implemen t\nsupersymmetry we introduce a supercharge, Q,satisfying\n[Pµ,Q] = 0, {Q,¯Q}= 2γµPµ, (3)\nwhereγµare the standard Dirac matrices\n{γµ,γν}= 2gµν, (4)\nand the energy momentum 4-vector Pµgenerates spacetime translations.\nThe construction of a supercharge is elegantly carried out in the co ntext\nofsuperspace. More precisely, introduce four independent anticommuting\nvariables, θ,and consider the general vector superfield\nV(x,θ) =C(x)+i¯θγ5w(x)−i\n2¯θγ5θM(x)−1\n2¯θθN(x)+1\n2¯θγ5γµθAµ\n−i¯θγ5θ¯θ[λ+i\n2/ne}ationslash∂w(x)]+1\n2(¯θθ)2(D(x)−1\n2∂µ∂µC(x)) (5)\nand theQoperator\nQ=−i∂¯θ−γµθ∂µ. (6)\nFixing an arbitrary spinor α,standard analysis3of the operator δQV=\n−i¯αQVleads to the supersymmetry transformations defining an N= 1\nsupersymmetric theory.\n2.N= 1 supersymmetry\nFollowing Berger and Kosteleck´ y,2we introduce Lorentz violation by defin-\ning a twisted derivative:\n˜∂µ=∂µ+kµν∂ν, (7)\nwherekµνis a symmetric, traceless, dimensionless tensor parametrizing\nLorentz violation. To obtain a gauge invariant theory, we also twist t he\nunderlying connection:\n˜Aµ=Aµ+kµνAν. (8)\nThese perturbations lead to a general vector superfield\n˜V(x,θ) =C(x)+i¯θγ5w(x)−i\n2¯θγ5θM(x)−1\n2¯θθN(x)+1\n2¯θγ5γµθ˜Aµ\n−i¯θγ5θ¯θ[λ+i\n2/ne}ationslash˜∂w(x)]+1\n2(¯θθ)2(D(x)−1\n2˜∂µ˜∂µC(x)) (9)\nand perturbed Qoperator\n˜Q=−i∂¯θ−γµθ˜∂µ. (10)November 12, 2018 15:41 WSPC - Proceedings Trim Size: 9in x 6i n patsusyproc\n3\nUsing Wess-Zumino gauge we obtain a lagrangian\n˜L=1\n4˜F2+i\n2¯λ/ne}ationslash˜∂λ+1\n2D2, (11)\nwhereDis an auxiliary chiral field and ˜Fis the twisted field strength,\n˜Fµν=˜∂µ˜Aν−˜∂ν˜Aµ.The twisted field strength can be written in terms of\nthe standard SME parameters: ˜F2=F2+kµναβ\nFFµνFαβ, where\nkµναβ\nF= 2(2kαµ+(k2)αµ)gβν+4(kµα+(k2)αµ)kνβ+(k2)αµ(k2)βν.(12)\nDirect calculation confirms that the action is invariant under the sup ersym-\nmetry transformations\nδ˜Aµ=−i¯αγµλ,\nδλ=i\n2σµν˜Fµνα−γ5Dα,\nδD= ¯α/ne}ationslash∂γ5λ. (13)\nThis defines an N= 1 supersymmetric theory with Lorentz violation.\n3.N= 4 supersymmetry\nTo build an N= 4 supersymmetric theory we work in 4 + 6-dimensional\nspacetime. We represent the 32 ×32 gamma matrices via Γµ=γµ⊗I8\nwhereI8is the 8×8 identity matrix and µ= 0,1,2,3,and\nΓ4= Γ14+Γ23,Γ6= Γ34+Γ12, iΓ8= Γ24+Γ13,\nΓ5= Γ24−Γ13, iΓ7= Γ14−Γ23, iΓ9= Γ34−Γ12,(14)\nwhere\nΓij=γ5⊗/parenleftbigg0ρij\nρij0/parenrightbigg\n(15)\nand the 4 ×4 matrices ρare defined by\n(ρij)kl=δikδjl−δjkδil,\n(ρij)kl=1\n2ǫijmn(ρmn)kl=ǫijkl. (16)\nWe consider the lagrangian\nL=−1\n4˜F2+i\n2¯λ˜/ne}ationslash∂λ, (17)\nwhere˜/ne}ationslash∂= Γµ(∂µ+kµν∂ν) is the twisted derivative with kµνparametrizing\nSO(1,9) violation and ˜Fis the corresponding perturbed field strength.November 12, 2018 15:41 WSPC - Proceedings Trim Size: 9in x 6i n patsusyproc\n4\nImposingboththeWeylandtheMajoranaconditionandcompactify ing,\nthe fermion λsatisfies\nλ=/parenleftbiggLχ\nR˜χ/parenrightbigg\n, (18)\nwhereLandRdenote left and right projection operators, respectively,\nthe spinor χtransforms as a 4 of SU(4) and the (independent) spinor ˜ χ\ntransforms as a ¯4 ofSU(4).\nChoosing the Lorentz violating parameters with care leads to Loren tz\nviolating extended supersymmetric theories which are easy to desc ribe. For\nexample, taking the kµνto vanish in the compactified directions leads to a\nlagrangian of the form\nL=−1\n4˜F2+i¯χ˜/ne}ationslash∂Lχ+1\n4˜∂µφij˜∂µφij, (19)\nwhere the complex scalar fields φijtransform as a 6 of SU(4) and are given\nby\nφi4=1√\n2(Ai+3+iAi+6),\nφjk=1\n2ǫjklmφlm= (φjk)∗. (20)\nThe associated action is invariant under the supersymmetry trans forma-\ntions\nδ˜Aµ=−i(¯αiγµLχi−¯χiγµLαi),\nδφij=−i√\n2(¯αjR˜χi−¯αiR˜χj+ǫijkl¯˜αkLχl),\nδLχi=i\n2σµν˜FµνLαi−√\n2γµ˜∂µφijR˜αj,\nδR˜χi=i\n2σµν˜FµνR˜αi+√\n2γµ˜∂µφijLαj. (21)\nSimilarly, choosing the kµνparameters to vanish in the spacetime direc-\ntionsµ= 0,1,2,3,we obtain a Lagrangian of the form\nL=−1\n4F2+i¯χ/ne}ationslash∂Lχ+1\n4∂µ˜φij∂µ˜φij, (22)\nwhere˜φij=φij+ Λijklφklwith the matrix Λ ijklcontaining the effect of\nLorentz violation in the compactified directions. The associated act ion is\ninvariant under the supersymmetry transformationsNovember 12, 2018 15:41 WSPC - Proceedings Trim Size: 9in x 6i n patsusyproc\n5\nδAµ=−i(¯αiγµLχi−¯χiγµLαi),\nδ˜φij=−i√\n2(¯αjR˜χi−¯αiR˜χj+ǫijkl¯˜αkLχl),\nδLχi=i\n2σµνFµνLαi−√\n2γµ∂µ˜φijR˜αj,\nδR˜χi=i\n2σµνFµνR˜αi+√\n2γµ∂µ˜φijLαj. (23)\nNote that if the scalars ˜φijare identified with physical scalars φijwe\nrestoreSU(4) symmetry and remove any Lorentz violating effects. If, how-\never, the φijcouple to other sectors, Lorentz effects may reappear in these\nsectors.\n4. Extensions and clarifications\nThe above results warrant a number of additional comments:\n•Theseconstructionscanbecarriedoutinthenonabeliancasewher e\nthey yield supersymmetric theories which exhibit Lorentz viola-\ntion.4\n•The same techniques can be applied to obtain an N= 2 supersym-\nmetric theory with Lorentz violation. The construction proceeds\nby working in 4+2-dimensional spacetime and using dimensional\nreduction.4\n•Because these constructions involve changing the structure of t he\nunderlying superalgebra,4,5the no-go results of Nibbelink and\nPospelov6do not apply.\nReferences\n1. L. Brink, J. Schwartz and J. Scherk, Nucl. Phys. B 121, 77 (1977).\n2. M. Berger and V.A. Kosteleck´ y, Phys. Rev. D. 65, 091701 (2002).\n3. M. Srednicki, Quantum Field Theory. Cambridge University Press, Cam-\nbridge, 2007.\n4. D. Colladay and P. McDonald, in preparation.\n5. D. Colladay and P. McDonald, J. Math. Phys. 43, 3554 (2002).\n6. S. Nibbelink and M. Pospelov, Phys. Rev. Lett. bf 94, 08160 1 (2005)."    },    {        "title": "1110.5258v1.Magnetohydrodynamics_dynamical_relaxation_of_coronal_magnetic_fields__I__Parallel_untwisted_magnetic_fields_in_2D.pdf",        "content": "arXiv:1110.5258v1  [astro-ph.SR]  24 Oct 2011Astronomy& Astrophysics manuscriptno.13902 c/ci∇cleco√y∇tESO 2018\nNovember20,2018\nMHDDynamical RelaxationofCoronal Magnetic Fields\nI.ParallelUntwisted MagneticFields in 2D\nJorge Fuentes-Fern´ andez, ClareE.Parnell and Alan W.Hood\nSchool of Mathematics andStatistics,Universityof StAndr ews, NorthHaugh, StAndrews, Fife,KY16 9SS,Scotland\nPreprintonline version: November 20, 2018\nABSTRACT\nContext. For the last thirtyyears, most of the studies on the relaxati on of stressed magnetic fields in the solar environment have o nly\nconsidered the Lorentz force, neglecting plasma contribut ions, andtherefore, limitingeveryequilibrium tothat of a force-free field.\nAims.Here we begin a study of the non-resistive evolution of finite beta plasmas and their relaxation to magnetohydrostatic st ates,\nwhere magnetic forces are balanced by plasma-pressure grad ients, by using a simple 2D scenario involving a hydromagnet ic dis-\nturbance to a uniform magnetic field. The final equilibrium st ate is predicted as a function of the initial disturbances, w ith aims to\ndemonstrate what happens tothe plasma during the relaxatio n process and tosee what e ffects ithas on the finalequilibrium state.\nMethods. A set of numerical experiments are run using a full MHD code, w ith the relaxation driven by magnetoacoustic waves\ndamped by viscous e ffects. The numerical results are compared with analytical ca lculations made within the linear regime, in which\nthe whole process must remainadiabatic. Particular attent ionis paidtothe thermodynamic behaviour of the plasma duri ng the relax-\nation.\nResults.The analyticalpredictions forthe finalnonforce-freeequi librium depend onlyontheinitialperturbations andthetot alpres-\nsure of the system. It is found that these predictions hold su rprisingly well even for amplitudes of the perturbation far outside the\nlinear regime.\nConclusions. Includingtheeffectsofafiniteplasmabetainrelaxationexperimentsleadst osignificantdifferencesfromtheforce-free\ncase.\nKey words. Magnetohydrodynamics (MHD) –Sun:corona – Magnetic fields\n1. Introduction\nThe magnetic field of the solar corona is believed to evolve\nthrough a series of force-free states (Heyvaerts&Priest 19 84).\nSince the solar corona involves a low-beta plasma in which\nmagnetic forces dominate over plasma forces, this is not\nan unreasonable assumption, and so, most of the recent\nstudies on the relaxation of coronal magnetic fields (e.g.\nMackay&vanBallegooijen 2006; Ji et al. 2007; Inoueet al.\n2008; Janse &Low 2009; Milleret al. 2009; Pontinet al. 2009)\nhave been done by considering the approximation of an ex-\ntremelytenuousplasma,forwhichtheplasmapressuredoesn ot\nplay an important role, and the persistent hydromagnetic st ruc-\nturesofthe solarcoronaareassumedto bein magneticbalanc e,\nwith zeropressuregradients.On the otherhand,onlywithin the\npast few years, Ruanet al. (2008) and Gary (2009) have starte d\nto consider the reconstruction of the global coronal magnet ic\nfield including a finite Lorentz force balanced by magnetic an d\ngravityforces.\nIn addition, there are many codes available to calculate\nthose force-free fields from the observed magnetic field in th e\nphotosphere (Amariet al. 1998; Wiegelmannetal. 2006, 2008 ;\nSchrijveret al. 2006; Metcalfetal. 2008). These codes have\nbeenusedwithvaryingdegreesofsuccesstodeterminethema g-\nnetic field of solar flares and active regions (e.g. DeRosa et a l.\n2009;Schrijveretal.2008;R´ egnier2008;Wheatland& R´ eg nier\n2009).However,problemsremainwiththeseapproaches.Inp ar-\nticular, a non-linear force-free field determined from a lin e-of-\nsight photosphericmagnetic field is not unique,but is one of aninfinite number of possible solutions. This fact is well know n\nandhasbeendiscussedbyseveralauthors(seeLow2006).\nAlso, the low beta plasma assumption is only valid for the\nsolar corona, but is not valid in the photosphere, chromosph ere\nor much of the transition region where plasma e ffects become\nmore important. Furthermore, in the solar corona, the tenuo us\nplasma will be able to create a high beta in the surroundingso f\na magnetic null point (McLaughlin& Hood 2006). Moreover,\neven where the plasma has a low beta, there are still some ef-\nfectsonthe finalequilibriumofthe magneticfield that will l ead\ntoenergeticconsequencesasthefieldisrelaxingtoitsnewe qui-\nlibriumstate.\nInparticular,relaxationcaninvolvethemagneticfieldevo lv-\ning from a stressed state to a force-free field with a lower\nenergy, and for a consistent scenario where the energy can-\nnot escape the system, conservation of energy implies that t he\nlosses of magnetic energy must be converted into something\nelse. Browninget al. (2008) and Hoodet al. (2009) investiga ted\nTaylor relaxation (Taylor 1974) through a series of non-lin ear\n3Dsimulations,initiatedbyan idealMHD instability.Alth ough\nthe initial state was force-free, the final state involved a h igh\ntemperatureplasmawith a significantvalueforthe plasmabe ta.\nPlacing aside some extra contributions such as radiative lo sses,\nifasubstantialfractionofthemagneticenergyreleasedgo esinto\ntheinternalenergy,thentheplasmabetacannotbesmall.He nce,\nconsidering the behaviour of the plasma is important even if it\nhas little effect on the final magnetic equilibrium. On the other\nhand, Gary (2001) suggested the possibility that there is hi gh\nbetaplasmainthe solarcoronaaboveactiveregions.2 Jorge Fuentes-Fern´ andez, ClareE.Parnell andAlanW.Hoo d: MHD Dynamical Relaxation of Coronal Magnetic Fields\nIn thispaper,we will start byconsideringa simple scenario ,\nwithauniform2Dmagneticfield.Therewillbenocurrentshee t\nformation and no changes in the magnetic topology. The aim\nof the paper is to investigate the relaxation of the hydromag -\nnetic fluid for various di fferent values of the plasma beta. To\nperturb the system, we introducea local small enhancement ( or\ndeficit) in the plasma pressure (or in the magnetic field), un-\nderthefrozen-incondition.Thesubsequentrelaxationlea dstoa\nnew magnetohydrodynamic equilibrium in which the perturba -\ntionhasbeendissipatedbyviscousforcesactingontheflow.\nIt is worthwhile mentioning that relaxation via Ohmic dis-\nsipation, due to the e ffect of resistivity, or magnetic di ffusiv-\nity, represents a substantially di fferent problem; While viscos-\nity dissipates the plasma velocity, di ffusivity tends to eliminate\nthe electric current density, and such a relaxed state can on ly\ninvolve potential fields, which are mathematically well defi ned\nandareuniquelydeterminedbythecomponentsofthemagneti c\nfield normal to the boundaries.Furthermore,the time-scale s for\nan Ohmic relaxation in very high magnetic Reynolds number\nenvironments, such as the solar corona, are in general proba bly\nlarger than the age of the Sun itself, outwith regions with ve ry\nsmall lengthscales(seePriest 1982).\nBy considering the e ffects of the plasma pressure in the re-\nlaxation, we are facing a totally di fferent problem from that of\nthe force-free relaxation studied by many others. In our non -\nresistive MHD relaxation, the plasma displacements driven by\nthe initial pressure enhancement will carry the magnetic fie ld\nwith them, generating an electric current and a magnetic pre s-\nsure.Hence,theresultingequilibriumwillhavetoinvolve abal-\nance between the Lorentz force and the plasma-pressure grad i-\nent.\nThe effectsof includinga finite plasma beta are relevantnot\nonlyinthehighplasmabetaregionsofthesolaratmospheres uch\nas the photosphere and chromosphere, but will also be releva nt\nin the solar corona. Obvious regions where the plasma beta is\nlikely to have a significant e ffect are in the vicinities of mag-\nnetic neutral points, where the magnetic field vanishes. The se\nconfigurations will be the subject of a further paper, while i n\nthe present paper we study the plasma beta e ffects on the sim-\nplestmagneticconfiguration,auniformmagneticfield,whic his\nabsolutely general and might be compared with di fferent solar\nenvironments such as a region in a coronal prominence or part\nofacoronalloop.\nIn Section 2, we first present the setup of the two-\ndimensional linear problem. In Sections 2.1 and 2.2, we solv e\nthe equationsfor a one-dimensionalvertical and horizonta lper-\nturbation, respectively, and in 2.3 we obtain the whole solu tion\nfrom a general two-dimensional linear perturbation, produ cing\na practical and precise analytical solution to the problem. In\nSection 3, we present a set of numerical experiments run us-\ning the Lare2D code (Arberet al. 2001), and we compare these\nwiththepreviousanalyticalresults.Thee ffectsofthenon-linear\nterms are considered when the magnitude of the perturbation is\nincreased,inSection4,followedbythesummaryandsomecon -\nclusionswhichare presentedinSection5.\n2. Linear2DEquations\nThe initial set up involves a uniform magnetic field pointing in\nthe vertical y-direction, B0=B0ˆey, and a background plasma\nof a constant gas pressure, density and temperature. The ini -\ntial disturbances are supposed to be small, in order to stay i n\nthe linear regime. Expressing each quantity q(x,y,t) as the sum\nof a background constant value plus a perturbation, q(x,y,t)=q0+q1(x,y,t), and substituting those expressions into the vis-\ncous, non-resistive, two-dimensional MHD equations, with no\ngravityforceforsimplicity,andneglectingtermsinvolvi ngprod-\nucts of perturbations, we obtain the following set of linear ized\nequations,inscalarform:\n∂ρ1\n∂t=−ρ0∇·v1, (1)\nρ0∂vx\n∂t=−∂p1\n∂x−B0\nµ0∂By\n∂x+B0\nµ0∂Bx\n∂y+ρ0ν/parenleftigg\n∇2vx+1\n3∂\n∂x(∇·v1)/parenrightigg\n,(2)\nρ0∂vy\n∂t=−∂p1\n∂y+ρ0ν/parenleftigg\n∇2vy+1\n3∂\n∂y(∇·v1)/parenrightigg\n, (3)\n∂p1\n∂t=−γp0∇·v1, (4)\n∂Bx\n∂t=B0∂vx\n∂y, (5)\n∂By\n∂t=−B0∂vx\n∂x, (6)\nwhereρ0,p0andB0are the constant background plasma den-\nsity, plasma pressure and magnetic field, νis the kinematic vis-\ncosity,ρ1,p1andv1are the perturbations on plasma density,\nplasmapressureandvelocity,and vx,vy,BxandByarethexand\nycomponentsof the perturbedvelocity and perturbed magneti c\nfield, respectively. Also, the plasma pressure, p, density,ρ, and\ninternalenergy,ǫ, arerelatedbytheperfectgaslaw,foranideal\npolytropicgas,\np=ρǫ(γ−1), (7)\nwhereγistheratioofspecificheats,oftenassumedtobe5 /3for\na highlyionisedhydrogenplasma.\nIn a general scheme, viscosity would add a heating term to\nthe energy equation, but this term is second order, and so, th e\nprocessisadiabaticwithinthelinearregime,andthereisn oheat-\ning of any kind taking place. Hence, the entropy per unit mass ,\nS=p/ργ, is conserved, for each single fluid element, and for\ntheentirebox.\nFrom the conservation of entropy, a relation between the\nplasma pressure and density perturbations may be obtained,\nwithin first order (i.e. neglecting the terms involving prod ucts\nofperturbations):\np0+p1\n(ρ0+ρ1)γ=p0\nργ\n0/parenleftigg\n1+p1\np0−γρ1\nρ0/parenrightigg\n=constant.\nHence,\n∆p1\np0−γ∆ρ1\nρ0=0,\nwhere∆indicatesthedifferencebetweenfinalandinitialstateof\ntheperturbation,suchthat\n∆p1=c2\ns∆ρ1, (8)\nwherecs=/radicalbig\nγp0/ρ0is thesoundspeed.\nTo solve these equations, we first determine the solution for\na one-dimensionalperturbation, which depends only on x(Sec.\n2.1), then only on y(Sec. 2.2), and finally, using the 1D results,\nwe derive the solution for a more general 2D perturbation(Se c.\n2.3).Jorge Fuentes-Fern´ andez, ClareE.Parnell andAlanW.Hood : MHD Dynamical Relaxation of Coronal Magnetic Fields 3\n2.1. Perturbationdependentonx\nConsidera perturbationvaryingonlyinthe directionperpe ndic-\nulartothemagneticfieldlines, x.Themagneticfieldvectorwill\nhave a non-zero y-component, B1(x,t)=By(x,t)ˆey, while the\nvelocitywill have a non-zero x-component, v1(x,t)=vx(x,t)ˆex.\nEquations(1)to (6)reduceto\n∂ρ1\n∂t=−ρ0∂vx\n∂x, (9)\nρ0∂vx\n∂t=−∂pT\n∂x+ρ0ν′∂2vx\n∂x2, (10)\n∂p1\n∂t=−γp0∂vx\n∂x, (11)\n∂By\n∂t=−B0∂vx\n∂x, (12)\nwithν′=4ν/3,whereνisthe kinematicviscosity,and pTisthe\nperturbedtotalpressure,givenby\npT=p1+B0By\nµ0. (13)\nThe equation governing the final equilibrium state can be ob-\ntained using Eq. (10). At equilibrium,the time dependence d is-\nappears,andthevelocityiszero,thus,theequilibriummus thave\nconstanttotal pressure:\n∂pT\n∂x=0. (14)\nCombiningEquations(11) and (12), we get the evolutionof th e\ntotalpressureas\n1\nρ0∂pT\n∂t=−(c2\ns+c2\nA)∂vx\n∂x, (15)\nwherecsisthesoundspeed,definedabove,and cA=B0/√µ0ρ0\nistheAfv´ enspeed.BytheappropiatecombinationofEquati ons\n(10)and(15)wegetthewaveequationforfastmagneto-acous tic\nwavesforthetotal pressure:\n∂2pT\n∂t2=(c2\ns+c2\nA)∂2pT\n∂x2+ν′∂\n∂t/parenleftigg∂2pT\n∂x2/parenrightigg\n. (16)\nAssuming that the total pressure can be considered as a conti n-\nuous, periodic function, the solution of the last equation c an be\nexpressedasasuperpositionofplanewaves,suchas\npT(x,t)=Re/summationdisplay\nkϕkei(kx−ωt), (17)\nwhereeachwavenumber kcorrespondstoadi fferentoscillation\nmodeandisassociatedwitha complexfrequency ω(k)givenby\nthedispersionrelation,\nω2+ik2ν′ω−(c2\ns+c2\nA)k2=0.\nThe dispersion relation has the solution ω=a−bi, whereais\ntherealfrequencyofthewave,and bisthedampingterm:\na=k\n2/radicalig\n4(c2s+c2\nA)−k2ν′2,\nb=1\n2k2ν′.\nIn orderto havea harmonicmode,the wave number kmust sat-\nisfyk2ν′2<4(c2\ns+c2\nA), and higher modeswill be dampedwith-\nout anytypeof oscillation.On the otherhand,notice that ω=0whenk=0.Theundampedmode k=0 correspondstothe con-\nstantFouriercoefficientintheexpansionofEq.(17),whichdoes\nnotchangeintime,andisgivenbyFourieranalysisasthehom o-\ngeneous redistribution of the initial total pressure to its average\nvalue. As t→∞, it is only this constant that remains, as all the\nother terms are proportional to e−bt. Hence, this homogeneous\nredistributionisexactlytheconstantperturbedtotalpre ssurethat\ndefinesourfinal equilibriumstate:\npT(∞)=1\nLx/integraldisplay\nx/parenleftigg\np1(x,0)+B0By(x,0)\nµ0/parenrightigg\ndx, (18)\nwhereLxisthelengthofthe x-domain.\nFrom the solution of Eq. (16), and Eq. (15), we obtain an\nexpression for v1(x,t), which, after substitution into Equations\n(9),(11)and(12),gives\n∂ρ1\n∂t=1\nc2s+c2\nA∂pT\n∂t, (19)\n∂p1\n∂t=c2\ns\nc2s+c2\nA∂pT\n∂t, (20)\n∂By\n∂t=B0\nρ0(c2s+c2\nA)∂pT\n∂t. (21)\nIntegrating now from t=0 tot=∞, we obtain the perturbed\nquantitiesforthe final equilibriumstate, as functionsof t heper-\nturbedtotalpressure,to beaddedtothebackgroundvalues:\nρ1(x)=ρ1(x,0)+1\nc2s+c2\nA(pT(∞)−pT(x,0)), (22)\np1(x)=p1(x,0)+c2\ns\nc2s+c2\nA(pT(∞)−pT(x,0)), (23)\nBy(x)=By(x,0)+B0\nρ0(c2s+c2\nA)(pT(∞)−pT(x,0)).(24)\nEquations (22), (23) and (24) state that no matter how we\nset ourinitial disturbance,the final equilibriumdistribu tionsare\ncompletelydeterminedbytheinitialandthefinaltotalpres sures\nofthesystem,whicharegivenbythesolutionofthewaveequa -\ntion. Note, that also the adiabatic equation for the linear r egime\ngiveninEq.(8) issatisfied.\n2.2. Perturbationdependentony\nIn the same way as above, we can get the solution for a pertur-\nbation varying only along the field. This time, we are dealing\nwith a purely non-magneticevolution,that will lead to a hom o-\ngeneousredistribution of the plasma pressure all along the field\nlines. Theequationsto solveare the y-componentsofEquations\n(1),(3) and(4).Theequilibriumisnowgivenby\n∂p1\n∂y=0, (25)\nandthesolutionfortheperturbedquantitiesis\np1(∞)=1\nLy/integraldisplay\nyp1(y,0)dy, (26)\nρ1(y)=ρ1(y,0)+1\nc2s(p1(∞)−p1(y,0)), (27)\nwhereLyis thelengthofthe y-domain.4 Jorge Fuentes-Fern´ andez, ClareE.Parnell andAlanW.Hoo d: MHD Dynamical Relaxation of Coronal Magnetic Fields\n2.3. Two-dimensionalperturbation\nIn this section, we study the relaxation of a general individ -\nualtwo-dimensionalperturbationwithinthelinearregime .Once\nagain,settingthevelocitiestozero,wegettheequationsg overn-\ningthefinal2D equilibrium:\n∂\n∂x/parenleftigg\np1+B0By\nµ0/parenrightigg\n−B0\nµ0∂Bx\n∂y=0, (28)\n∂p1\n∂y=0. (29)\nEquation (29) tells us that the final plasma pressure cannot\ndepend on y, so the solution for the pressure must remain one-\ndimensional,asbefore.Ontheotherhand,Eq.(28)doesnoth ave\na direct interpretation, as both spatial derivatives are in volved.\nThe term p1+B0By/µ0represents the perturbed total pressure\nfrom Sec. 2.1, and the new term B0Bx/µ0represents the mag-\nnetic tension due to the curvature of the field lines, which wa s\nzerointhe1Dcases.Asusual,whenlookingforaperiodicsol u-\ntion,Fourieranalysingmakeslifemuchsimpler:Expressin gour\nvariablesas functionsof ei(kx+ly−ωt), where each pair ( k,l) repre-\nsentsonesinglemodeofoscillationintheglobaltimeevolu tion,\nEquations(28) and(29) canberewritenas\nk/parenleftigg\np1+B0By\nµ0/parenrightigg\n−lB0Bx\nµ0=0,\nlp1=0,\nwhere:\ni. The mode k=0,l=0 represents the unperturbed back-\ngroundvalues.\nii. Fork/nequal0,l=0,theequationoftheequilibriumis\n∂\n∂x(p1+B0By)=0. (30)\nThese modes only depend on x, and represent the homoge-\nneousredistributionofthetotalpressurestudiedinSec.2 .1.\niii. Fork=0,l/nequal0we get\n∂p1\n∂y=0, (31)\n∂Bx\n∂y=0. (32)\nThesemodesdonotmodify By,insteadtheysimplyremove\nboth the vertical gradients of magnetic tension and plasma\npressureasinSec.2.2.Eachofthemistreatedindividually ,\nastheyarenotcoupledintheequations.\niv. Finally,forthosemodeswith k/nequal0,l/nequal0,we get\nkBy−lBx=0,\nwhich can be combined with the solenoidal condition for\nthemagneticfield, ∇·B=0,or,withinourfouriernotation,\nkBy+lBx=0.\nFrom these equations, we can conclude that, in the final\nequilibrium, the existance of a variation of Byin thex-\ndirection is totally incompatible with a variation of Bxin\nthey-direction. Hence, the modes with both wave numbers\nkandlnon-zeromayappearinthedynamicalevolution,but\nnotinthefinal equilibriumdistributions.Therefore, with our uniform background magnetic field\npointingstraightinthevertical y-direction,thefinal equilibrium\nstate is a combination of the backgroundvalues ( k=0,l=0),\nplus the vertical non-magnetic evolution to a state with pla sma\npressure that is constant along y, and/or the smoothing of the\nhorizontal component of the magnetic field ( k=0), plus the\none-dimensional hydromagnetic evolution across the field l ines\n(l=0). Note, that the perturbed Bxin the vertical direction (i.e.\ncurved magnetic field) is not coupled with either p1orBy, so\nthe final magnetic field remains as straight lines, and Bxis not\ninvolvedinthe evolutionofthetotal pressureofthesystem .\nHence, we calculate the analytical 2D equilibrium in two\nsteps: Firstly, the non-magnetic evolution in the vertical direc-\ntion,\np∗\n1(x)=1\nLy/integraldisplay\nyp1(x,y,0)dy, (33)\nρ∗\n1(x,y)=ρ1(x,y,0)+1\nc2s(p∗\n1(x)−p1(x,y,0)), (34)\nand secondly, the hydromagneticevolution in the horizonta l di-\nrection,acrossthefield,\nρ1(x,y)=ρ∗\n1(x,y)+1\nc2s+c2\nA(pT(∞)−p∗\nT(x)), (35)\np1(x,y)=p∗\n1(x)+c2\ns\nc2s+c2\nA(pT(∞)−p∗\nT(x)), (36)\nBy(x,y)=By(x,0)+B0\nρ0(c2s+c2\nA)(pT(∞)−p∗\nT(x)).(37)\nInEquations(33)to (37)the quantitieswitha superscriptr epre-\nsent the state after the vertical evolution, with p∗\nT(x)=p∗\n1(x)+\nB0By(x,0)/µ0.\nLooking back at the equations, we see a well known result\nfrom magnetohydrostatics, namely: In equilibrium, and in t he\nabsence of gravity, the plasma pressure must be constant alo ng\nthe field lines. The constant plasma pressure in the y-direction\ngiven by Eq. (29) is aligned with the straight magnetic config u-\nration,with nomagnetictension,forthe finalequilibriums tate.\nWhen analyzing the validity of the results above for a non-\nidealexperiment,itisimportanttorememberthatEquation s(34)\nto (37) come from the linear approximation, but Eq. (33) does\nnot. Hence, we expect our analytical calculations for the pr es-\nsure to hold for much larger initial perturbations than the o nes\nfor the density. If the initial pressure disturbance is not s mall,\nbut the linear expression for the plasma pressure is still va lid,\nthe adiabatic condition (i.e. conservation of entropy), wh ich is\nmore robust than the linear calculations, gives us a better a p-\nproximationfor the final equilibriumplasma density, calcu lated\nas\nρ(x,y,t→∞)=/parenleftiggp(x,y,t→∞)ργ(x,y,t=0)\np(x,y,t=0)/parenrightigg1/γ\n.(38)\n3. NumericalExperiments\nTo investigate the validity of the analytical results, we ha ve\nusedLare2D,astaggeredLagrangian-remapcodewithuserco n-\ntrolled viscosity, to solve the full MHD equations, with the re-\nsistivity set to zero (for further details, see Arberetal. 2 001).\nThe numerical domain is a square box with a uniform grid of\n256×256 points. The background magnetic field is pointing in\nthe vertical y-directionand all the perturbationsdependon bothJorge Fuentes-Fern´ andez, ClareE.Parnell andAlanW.Hood : MHD Dynamical Relaxation of Coronal Magnetic Fields 5\nxandy. The top andbottomboundariesof the boxare periodic,\nsothatthefieldlinesarenotline-tied.Theboundariesonth eleft\nandrightsidesareclosed.Thechoiceofeitherclosedorper iodic\nside boundariesmakesno di fferenceforour experiments.There\nisneithermassnorenergyflowingacrossthe sideboundaries .\nThe numerical code uses the normalised MHD equations,\nwherethenormalisedmagneticfield,densityandlengths,\nx=Lˆx,y=Lˆy,B=BnˆB, ρ=ρnˆρ,\nimply that the normalising constants for pressure, interna l en-\nergyandcurrentdensityare\npn=B2\nn\nµ0, ǫn=B2\nn\nµ0ρnandjn=Bn\nµ0L.\nWehaveusedherethesubscript nforthenormalisingconstants,\ninstead of 0, to avoid confusion with the initial background\nvalues. The hatquantities are the dimensionless variables with\nwhich the code works. The expression for the plasma beta can\nbeobtainedfromthisnormalisationas\nβ=2ˆp\nˆB2=2ˆρˆǫ(γ−1)\nˆB2. (39)\nFig.1.Time evolution of the energies for the case where\nmax(p1)=p0=0.1B2\nn/µ0andβ=0.2,integratedoverthewhole\n2Dbox.Thefourevolutionshavebeenshiftedinthevertical axis\nbysubstractingtheconstantvalues0,0.5,0.15and0.65,re spec-\ntively, for kinetic, magnetic, internal and total energy, b ut their\nrelative amplitudes are not scaled. The final losses of inter nal\nenergyare entirelybalancedwith a netincrease ofmagnetic en-\nergy.The top figure is logarithmicin time and coversthe whol e\nrelaxation. The bottom figure is linear in time and only cover s\nthe first part of the relaxation. From this graph, we can appre -\nciate the complex oscillation periods, which are products o f the\nsumofthedifferentplanewavesthat drivetherelaxation.The initial disturbance consists of an internal energy and\npressure enhancement, leaving the initial plasma density a nd\nmagneticfieldundisturbed.Theperturbationistakenasasi ngle\ntwo-dimensionalGaussiancenteredinthe middleofthebox:\nǫ1(x,0)=aexp/parenleftigg\n−(x−b)2\n2c2/parenrightigg\nexp/parenleftigg\n−(y−b)2\n2c2/parenrightigg\n, (40)\nwithb=0.5Landc=0.05L. As there is no initial perturbation\nof the magnetic field, the initial perturbed total pressure i s just\ntheperturbedplasmapressure,andthefinalanalyticalpert urbed\ntotalpressureistheaveragevalueofthatinitialperturbe dplasma\npressuredistribution,so that\npT(∞)−p∗\nT(x)=/integraldisplay\nx/parenleftigg/integraldisplay\nyp1(x,y,0)dy/parenrightigg\ndx−/integraldisplay\nyp1(x,y,0)dy.\nFor the experiment shown here, we have chosen a rather\nlarge perturbation, of the same order as the background valu e,\ni.e.a=ǫ0, for which one may expectlinear theory notto be\napplicable.Thebackgroundplasmabetais0 .2.\nFigure1 showsthe time evolutionof the variousenergiesof\nthe system, integrated over the whole box. Kinetic energy (i n\ngreen) grows quickly from zero to its maximum value, and is\nsubsequently damped to zero in the final equilibrium. A small\nfractionoftheinternalenergyisconvertedintomagnetice nergy\nat the new equilibrium. For this particular set up, in which t he\nperturbationhasbeenintroducedintheplasmapressure,it isthe\nplasmathatlosessomeofitsinitialinternalenergy,trans feringit\ntothemagneticfield.Note,thattheamountofenergytransfe red\nisdirectlyproportionaltothemagnitudeoftheperturbati on.The\ntotal energy (i.e. the sum of the magnetic, kinetic and inter nal\nenergies)isconservedtoanaccuracyof ∼10−7.\nThe 2D contour plots of the normalised plasma pressure,\ndensityandperpendicularcurrentdensityfortheinitiala ndfinal\nequilibriumstateareshowninFig.2forthiscase.Theiniti alin-\ncreaseofplasmapressurecreatesalocalizeddecreaseinpl asma\ndensity. The displacement of the magnetic field lines is hard to\nappreciate from the field lines themselves, but is clear from the\nnon-zeroperpendicularelectric current density, jz, that exists in\nthefinal equilibrium.\nFigures3 and4showverticalcutsat x=L/2andhorizontal\ncutsaty=L/2,respectively,throughthecontourplotsinFig.2,\nfor the relevant quantities for the initial and final state. I n these\nplots, a detailed comparisonof the numerical and analytica l so-\nlutionscanbemade.Theanalyticalpredictionsfortotalpr essure\nare very good, even though the amplitude of the perturbation is\nrelatively large ( a=ǫ0) and so, strictly speaking, our analytical\napproximationshould not hold.However,the linear approxi ma-\ntionfortheplasmadensity(redcrossesin Figures3and4)do es\nnot fit well. Instead, if we use Eq. (38), a much better fit for\nthe plasmadensityis found(bluecrossesin theplots), impl ying\nthat the processis approximatelyadiabatic. Since the nume rical\nexperiments have been performed using a full MHD code that\nsolves the non-linearequations, the processis not entirel y adia-\nbatic, but must have a finite amount of viscous heating that wi ll\nbecomeimportantasthe initialperturbationisincreased.\n4. Importanceofnon-lineareffects\nTo study how non-linearity a ffects the results as the magnitude\nof the initial perturbation increases, we focus again on the total\npressure. The total pressure of the final numerical equilibr ium\nmust be constant, whether the relaxation remains in the line ar6 Jorge Fuentes-Fern´ andez, ClareE.Parnell andAlanW.Hoo d: MHD Dynamical Relaxation of Coronal Magnetic Fields\nFig.2.Two-dimensionalcontourplotsof plasma pressure (top),\nplasmadensity(center)andcurrentdensity(bottom)inthe initial\nstate (left column) and final equilibrium (right column), fo r the\nsameexperimentasFig. 1.\nFig.3.Vertical cuts for the plasma pressure (left) and plasma\ndensity (right), for the same experiment as Fig. 1 and Fig. 2.\nInitial perturbed state (dashed) is compared with the final e qui-\nlibrium,asfoundbythefullMHDnumericalsimulations(sol id)\nand predicted by the linear analysis (red crosses). For the d en-\nsity predictions, the blue crosses represent predictions f rom the\nadiabaticconditiongivenbyEq.(38).\nregimeor not.On the otherhand,the analyticaldefinitionof to-\ntal pressure given by (13) is an approximation from the linea r\nanalysis, and will become less valid as the non-linear terms be-\ncome more important. We perform a series of experiments for\nvarious plasma beta values in which the relative amplitude o f\nthe initial perturbationis changedfrom a very small value, well\nwithin the linear regime, to a large value way outside it. Usi ng\ntheseexperiments,weinvestigatehowthefinaltotalpressu rede-\npartsfromthelinearpredictionsfordi fferentbackgroundplasma\nbetavalues.\nBut first, we recall that the 2D relaxation may be separated\ninto a verticalnon-magneticevolution(vertical redistri butionof\nplasma pressure)and a horizontalevolution(horizontalre distri-\nbution of total pressure), in which the total pressure in the ver-Fig.4.Horizontalcutsfortheplasmapressure(topleft),plasma\ndensity(topright),totalpressure(bottomleft)andmagne ticfield\nstrength (bottom left), for the same experiment as Figures 1 , 2\nand3.\ntical case is notdeterminedbythe linearanalysis.Thissuggests\nthat,effectively,inordertofindasignificanterrorinthefinalto-\ntal pressure for the 2D experiment,we will need very large va l-\nues of the initial two-dimensional perturbation. Hence, th e fol-\nlowing experimentshave been made for just a one-dimensiona l\nperturbationacrossthefieldlines.Theseresultsmaybemap ped\nontothoseforourintital 2Dperturbation,usingthefollow ing:\n/parenleftiggmax(p1)\np0/parenrightigg\n2D=L/integraltext\nyexp/parenleftbig−(y−b)2/2c2/parenrightbigdy/parenleftiggmax(p1)\np0/parenrightigg\n1D.\nFigure 5 shows the relative error of the linear aproximation\nin both 1D and 2D for the total pressure, as a function of the\namplitude of the initial perturbation, for five di fferent values of\nthe plasma beta (β=0.05,β=0.1,β=0.2,β=1.3andβ=2).\nThe bottom x-axis showsthe magnitudeof the one-dimensional\nperturbation,and the top x-axis shows the magnitudeof the ini-\ntial two-dimensional perturbation before its vertical exp ansion.\nThe erroron the y-axisis calculated as the maximumdi fference\nbetween the linear prediction and the numerical results for the\ntotalpressure,\nmax(|plin\nT−pnum\nT|)\npnum\nT,\nwherepnum\nTisthe final constant total pressureobtainedfromthe\nnumerical simulations, and, for our non-magneticinitial p ertur-\nbation,\nplin\nT(x)=p0+p1(x,∞)+B2\n0\n2µ0+B0By(x,∞)\nµ0,\nwithp1(x,∞) andBy(x,∞) being the final perturbed plasma\npressureand perturbedmagneticfield fromthe numericalsim u-\nlations.\nAsβ→∞, we expect the relative error of the linear anal-\nysis to tend to zero, independently of the perturbation, as i n\nthis case, the magnetice ffects dissapear, and the initial pressure\nperturbation completely redistributes to a well defined con stant\nvalue in the whole box. On the other hand, if β≪1, then the\nmagneticfield will dominateoverthe plasma contributions, andJorge Fuentes-Fern´ andez, ClareE.Parnell andAlanW.Hood : MHD Dynamical Relaxation of Coronal Magnetic Fields 7\nFig.5.Relative error in the linear prediction of the total pres-\nsure against the magnitude of the initial pressure perturba tion,\nforfivedifferentvaluesoftheplasmabeta.Theslowgrowthrate\noftheerror(non-lineare ffects)indicatesthevalidityofthelinear\nanalysisstudiedinthispaper.\nmuch larger values for the relativeinitial perturbation will be\nneeded for strictly leaving the linear regime. These two beh av-\niorscanbeseeninFig.5,wheretheplotsforlargeplasma-be tas\ntend to a smaller error, while the plots for small plasma-bet as\ntake longer to reach significant errors, i.e. to escape from t he\nlinear regime.Furthermore,we must not forgetthat we are he re\nonlytalkingabouttheinitialbackgroundplasmabeta,so al arge\nbackgroundbeta combined with a large initial perturbation will\nmakethefinalplasmabetaevenhigher.Thus,max( p1)/p0→∞\nwill implyβ→∞for the final equilibrium, so we expect the\ncurves of the relative error of the linear analysis to turn ba ck to\nzero as the initial perturbation is greatly increased. In te rms of\nenergyconservation,asthevelocityiszeroattheinitiala ndfinal\nstates,theintegraloverthewholedomainofinternalenerg yplus\nmagneticenergymustbeconserved:If β→∞,thentheinternal\nenergyismuchlargerthanthemagneticenergy,andwilljust re-\ndistribute the plasma pressure, without transferring any e nergy\nintothemagneticfield.\nOn the contrary, the final plasma density is entirely deter-\nmined by the linear analysis, in both the vertical and the hor i-\nzontal evolutions along and across the field lines, or in a bet ter\napproximation,bytheadiabaticcondition.Hence,thenon- linear\neffectsfortheplasmadensitywill growmuchquicker,asshown\nin Fig. 6. These last numericalexperimentshave been made fo r\nthe original two-dimensional Gaussian perturbation. The e rror\nonthey-axisisgivenby\nmax(ρad−ρnum)\nρnum,\nwhereρadis the plasma density given by Eq. (38), and ρnumis\nthefinal densityobtainedwiththenumericalexperiments.\nThe relative error in the plasma density is considerably big -\nger than the relative error in pressure, and so, for only a sma ll\nchangein p1/p0inthe2Dcase,wefindalargeerrorin ρ.Asthis\nerror quickly reaches significant values, the plasma beta pl ays\nmuch less of a role for the non-lineare ffects in the plasma den-\nsity thanin theabovetotal pressure.Fig.6.Relative error of the density predicted assuming an adi-\nabatic evolution, with Eq. 38, against the magnitude of the 2 D\ninitial pressureperturbation,forthreevaluesoftheplas mabeta.\nNote, that the x-axis in this plot is to be compared with the top\nx-axisin Fig.5.\n5. Summaryand Conclusions\nWe have presentedanalyticaland numericalcalculationsfo r the\n2D magnetohydrodynamicrelaxation of an untwisted perturb ed\nmagnetic system embedded in a plasma with beta of any size,\nandfindafinalequlibriumstatewhichdi fferssubstantiallyfrom\nthe initial background configuration. The equilibrium reac hed\nis a non force-free state in which the plasma pressure gradie nts\nare balanced by the magnetic Lorentz force. For a set of speci -\nfiedboundaries,allthehydromagneticquantitiesarefully deter-\nminedbytheinitial perturbedstate.\nThe initial disturbance evolves into the final relaxed state\nby different families of magnetoacoustic waves, dissipated via\nviscousdamping.Fast magnetoacousticwavespropagateacr oss\nthe field lines in the horizontal direction, slow magnetoaco ustic\nwaves redistribute the thermodynamicquantities along the field\nin the vertical direction, and an extra contributionof slow mag-\nnetoacousticwavespropagatesalongthemagneticfieldline sin-\ntroducingamagnetictensionterm.Nevertheless,theselas t slow\nmagnetoacoustic waves dissipate the magnetic tension in su ch\na way that it is totally unimportant when determining the fi-\nnal equilibrium distributions. The vertical redistributi on of the\nplasmapressureto a homogeneousvaluedemandsthe magnetic\ntensiontodissapearcompletely,soboththeplasmapressur eand\ntotalpressureareone-dimensionalat theendofthe process .\nWithin the linear regime, the final distributions are com-\npletely independent of the viscosity, even though it is requ ired\ntopermittherelaxationtoocurr,asitistheonlydampingme ch-\nanism of the waves. An increase in the viscosity enhances the\ndiffusive term in the wave equation,and so, acceleratesthe pro-\ncess, but the final distribution is not modified. Instead, in t he\nfinal equilibrium, all the quantities are simply determined by\nthe behaviour of the final equilibrium total pressure, invol ving\nplasma andmagnetic e ffects. Hence,the final equilibriumstates\nforplasmapressureandmagneticfielddonotdi fferifthe initial\nperturbationisofthedensity,temperatureorinternalene rgy.\nFinally,byinvestigatingthelinearregime,wehavebeenab le\nto make analytical predictions for the final MHS equilibrium ,\neven when the regime is far from linear. The linear predictio ns\nremain remarkably valid even outside the linear regime, as t he8 Jorge Fuentes-Fern´ andez, ClareE.Parnell andAlanW.Hoo d: MHD Dynamical Relaxation of Coronal Magnetic Fields\ngrowth rate of the non-ideal e ffects is very small, compared to\ntheinitial perturbations.\nIn this paper, the introduction of plasma e ffects in the re-\nlaxation of hydromagnetic systems have been studied in sim-\nple schemes, producing a series of analytical predictions w hich\nhave been confirmed by our numerical results. By starting wit h\na uniform magnetic field, we have reached a state in which the\nmagnetic field itself remains almost uniform, but where some\ncurrent density has been built. Also, even in this simple con -\nfiguration, a non-neglectible amount of energy has been tran s-\nferredfromtheplasmatothemagneticfield.Theseimplicati ons\nofenergytransferduringtherelaxationprocessindicatet hatthis\nprocess will be of importance in the solar corona, specially in\nthe study of magnetic null points and their surroundings. Nu ll\npoints have been found to have a reasonable populationdensi ty\nin the Solar Corona by Longcope&Parnell (2009). In further\nstudies, we will consider more complex scenarios, introduc ing\ntwo-dimensional null points and starting to consider the im pli-\ncationsofnon-zeroplasmabetasin three-dimensionalmagn etic\nenvironments.\nReferences\nAmari, T.,Boulmezaoud, T.Z.,&Maday, Y.1998, A&A,339, 252\nArber, T. D., Longbottom, A. W., Gerrard, C. L., & Milne, A. 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Phys.,247, 249"    },    {        "title": "2401.09803v1.Multithermal_apparent_damping_of_slow_waves_due_to_strands_with_a_Gaussian_temperature_distribution.pdf",        "content": "Astronomy &Astrophysics manuscript no. multiT_dampingslow_for_arxiv ©ESO 2024\nJanuary 19, 2024\nMultithermal apparent damping of slow waves due to strands with\na Gaussian temperature distribution\nT. Van Doorsselaere1, S. Krishna Prasad2, V . Pant2, D. Banerjee2,3,4, and A. Hood5\n1Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven,\nBelgium\ne-mail: tom.vandoorsselaere@kuleuven.be\n2Aryabhatta Research Institute of Observational Sciences, Nainital, India\n3Indian Institute of Astrophysics, Koramangala, 560034, Bengaluru, India\n4Center of Excellence in Space Sciences, IISER, 741246, Kolkata, India\n5School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, UK\nReceived ; accepted\nABSTRACT\nContext. Slow waves in solar coronal loops are strongly damped. The current theory of damping by thermal conduction cannot explain\nsome observational features.\nAims. We investigate the propagation of slow waves in a coronal loop built up from strands of di fferent temperatures.\nMethods. We consider the loop to have a multithermal, Gaussian temperature distribution. The di fferent propagation speeds in di ffer-\nent strands lead to an multithermal apparent damping of the wave, similar to observational phase mixing. We use an analytical model\nto predict the damping length and propagation speed for the slow waves, including in imaging with filter telescopes.\nResults. We compare the damping length due to this multithermal apparent damping with damping due to thermal conduction and find\nthat the multithermal apparent damping is more important for shorter period slow waves. We have found the influence of instrument\nfilters on the wave’s propagation speed and damping. This allows us to compare our analytical theory to forward models of numerical\nsimulations.\nConclusions. We find that our analytical model matches the numerical simulations very well. Moreover, we o ffer an outlook for using\nthe slow wave properties to infer the loop’s thermal properties.\nKey words. Magnetohydrodynamics (MHD) – Plasmas – Waves – Methods: analytical – Methods: numerical – Sun: oscillations\n1. Introduction\nSince the turn of the century, slow waves in coronal loops are\nregularly observed through high resolution space observations\n(Berghmans & Clette 1999). These waves are seen as propagat-\ning intensity disturbances along open magnetic field or in the\nfootpoint of loops (De Moortel et al. 2002; Krishna Prasad et al.\n2012). In previous years, there was a debate on their interpreta-\ntion in terms of slow waves or periodic flows (e.g. De Moortel\net al. 2015), but for coronal loops or fans rooted in sunspots,\nthere is a consensus that these are definitely slow waves driven\nby p-mode wave leakage in the sunspot (Banerjee et al. 2021).\nSlow waves in coronal loops are observed to be very heav-\nily damped. Traditionally, it has been thought that the damp-\ning is caused by thermal conduction, after an extensive review\nof damping mechanisms by De Moortel & Hood (2003, 2004).\nStill, some of the observed properties of the damping may not be\nexplained adequately by this traditional approach. For instance,\nin closed-field regions, it was found that the damping scales\nwith the period with a positive coe fficient (Krishna Prasad et al.\n2014), which is still more-or-less compatible with the damp-\ning by thermal conduction (Mandal et al. 2016). However, for\nopen-field regions in coronal holes, it was found that the damp-\ning scales with the period with a negative coe fficient (Krishna\nPrasad et al. 2014, 2017), which is incompatible with that damp-\ning theory. Gupta (2014) finds di fferent damping behaviour inslow waves at di fferent heights. At larger heights (10-70 Mm),\nthey find shorter damping lengths for short period waves (as ex-\npected due to thermal conduction) but closer to the limb ( <10\nMm), the long period waves ( >6 min) appear to damp faster. Ad-\nditionally, Mandal et al. (2018) have shown, through a statistical\nstudy, that the damping length of slow waves in polar regions, in-\ndeed display a negative dependence on oscillation period. More-\nover, Krishna Prasad et al. (2019) find that the observed damp-\ning lengths of slow waves are much shorter than those expected\nfrom the theory of thermal conduction. Thus, it seems that other\ndamping mechanisms are also at work.\nAnother point to consider is that the propagation speeds of\nslow modes in a loop seems to depend on the filter passband of\nthe spacecraft that is used (King et al. 2003; Kiddie et al. 2012;\nUritsky et al. 2013). This, too, seems to be incompatible with a\nslow mode wave propagating through a monolithic loop, subject\nto damping by thermal conduction. It was originally attributed\nto the fact that two adjacent loops (or aligned along the line-of-\nsight) would be observed in di fferent filters. This may be correct,\nbut it leaves the question why the slow waves are so coherently\nin phase between the observed structures.\nAnother reason for disagreeing with the damping of slow\nwaves by thermal conduction should be that it was previously\nobserved that slow waves are damped with a Gaussian envelope\n(Krishna Prasad et al. 2014). This feature could not be explained\nby damping by thermal conduction, nor resonant damping of\nArticle number, page 1 of 9arXiv:2401.09803v1  [astro-ph.SR]  18 Jan 2024A&A proofs: manuscript no. multiT_dampingslow_for_arxiv\nslow waves in the cusp continuum (Yu et al. 2017b,a; Geeraerts\net al. 2022). The latter is despite the fact that resonant absorp-\ntion of kink modes in the Alfvén continuum has been convinc-\ningly shown to result in a Gaussian damping profile (Pascoe et al.\n2012, 2017, 2022). However, the mechanism at work for res-\nonant absorption in the Alfvén continuum (Hood et al. 2013)\ndoes not seem to carry over to slow waves (Hood, 2015, private\ncommunication).\nFurthermore, it was argued by Wang et al. (2015) that the\nthermal conduction coe fficient is significantly suppressed in ob-\nserved coronal loops. Despite the suppression of the thermal con-\nduction, they still find a strong damping of slow waves. In their\npaper, they argue that this damping is caused by an enhanced\ncompressive viscosity (Wang & Ofman 2019). However, here\nwe offer a suggestion that may also explain the observed strong\ndamping of slow waves, without invoking unrealistically high\nviscosity coe fficients (a factor 10 higher than normally consid-\nered, Wang & Ofman 2019). The model we propose below, does\nnot need either thermal conduction or viscosity to result in ap-\nparent damping.\nIn DC heating models for the corona, it is thought that the\nloops are built out of isolated strands (e.g. Aschwanden et al.\n2000), which are each individually heated by nanoflares after\nwhich the higher temperature is redistributed by thermal con-\nduction only along the magnetic field, but inhibited across (e.g.\nWilliams et al. 2021, and references therein). They are conse-\nquently modelled as a collection of 1D field lines and their ther-\nmodynamic evolution. This is in contrast to numerical evidence\nthat loop strands have a short life time, because of the mixing by\ntransverse waves Magyar & Van Doorsselaere (2016). The latter\nwould result in a more continuous transverse temperature profile\nperpendicular to the magnetic field (Judge 2023).\nIn models of AC heating, transverse waves lead to turbulent\nbehaviour in the loop boundary and entire cross-section (Karam-\npelas & Van Doorsselaere 2018), leading to patchy heating in\nthe cross-section, or in the turbulent layers (Van Doorsselaere\net al. 2018; Shi et al. 2021). Despite the di fferences between AC\nand DC heating models, it is safe to say that coronal loops do\nnot have a uniform temperature across their cross-section. This\nwill have a major impact on the propagation properties of slow\nwaves in those non-uniform temperature profiles. Here we will\nshow that this leads to extra apparent damping (which we call\n“multithermal apparent damping” or MAD) and di fferent propa-\ngation speeds in di fferent filter channels. In the future, this will\nallow us to infer the thermal structure of coronal loops from the\npropagation and damping behaviour. Aside from this potential\nfor probing the coronal thermal structure, nowadays slow waves\nare also considered for their ability to diagnose the coronal heat-\ning function (Kolotkov et al. 2019) through their perturbation of\nthe energy balance equation for the background corona. Thus, it\nseems that slow waves are the optimal MHD waves for seismol-\nogy of thermal e ffects in the solar corona (perhaps aside from\nthe entropy mode).\nEven though our results were derived independently, it was\npointed out in discussions at conferences that the physical e ffect\nwe consider is the same as was considered by V oitenko et al.\n(2005)1. They modelled the propagation of sound waves in a\nmulti-stranded loop with strands drawn from a uniform distri-\nbution, as seen in a top-hat shaped instrumental filter. Here we\ngo much beyond that initial description of this phenomena.\n1In fact, given their pioneering idea on this, as much as 18 years be-\nfore this manuscript, we may consider naming the multithermal appar-\nent damping as V oitenko-damping of slow waves.2. Results – theoretical models\nWe model a coronal loop as a superposition of strands, each with\ntheir own temperature and associated sound speed vs. However,\nthe model also carries over to a loop with a temperature continu-\nously varying in its cross-section. Both of these models have no\ntemperature variation along the magnetic field. In these models,\na sound wave front is launched at the footpoint. We describe the\npropagation of the sound wave in the multithermal plasma.\n2.1. Intuition\nWe take the z-direction along the uniform magnetic field, and\nonly consider the hydrodynamic behaviour along the magnetic\nfield lines (e.g. De Moortel & Hood 2003; V oitenko et al. 2005;\nMandal et al. 2016). We first think about the position zpof the\npeak perturbation on the strands. We have that\nzp=z0+vst, (1)\nwhere z0is the height of the initial excitation of the wave, in a\nstrand with sound speed vs. We consider the initial position of\nthe peak z0to be independent of the strand, mimicking a joint\nimpulsive excitation of the pulse low down in the atmosphere. In\nthis paper, we consider a loop for which the strands have a sound\nspeed that is randomly drawn from a normal distribution centred\non ¯vand standard deviation σv\nvs∼N(¯v,σ2\nv). (2)\nThe distribution of the temperature in these strands is tightly\nrelated to the heating mechanism, which is currently not well\nunderstood (e.g. Van Doorsselaere et al. 2020). Thus, this as-\nsumption of a Gaussian distribution of strand’s sound speed is\nan ad-hoc assumption in this paper. A sketch of the considered\nconfiguration is included in Fig. 1.\nWith such a Gaussian distribution of the strands, we then find\nthat the peak positions zpare also a Gaussian distribution. Fol-\nlowing well-known rules for transforming random variables in\nstatistics, we find that its distribution is\nzp∼N(z0+¯vt,t2σ2\nv). (3)\nWe then consider all pulse perturbations on each strand to have\nthe same amplitude. The line-of-sight integration over all these\nstrands will then result in an intensity variation that is modelled\nwell by this Eq. 3. This shows that the peak position distribution\n(and integrated intensity signal) propagates up with the average\nsound speed ¯ vin the loop bundle. It also shows that the peak\nposition distribution steadily widens linearly in time, because its\nstandard deviation goes as tσv.\nThe crucial realisation for understanding the multithermal\napparent damping of sound waves is that the normalising factor\nof the Gaussian distribution with a certain σis given as 1 /σ√\n2π.\nApplying this for the distribution of zp, we find that its peak value\nwill vary over time as\n1\ntσv√\n2π, (4)\nand thus the wave will have a multithermal apparent damping\nthat is proportional to t−1. Indeed, given that there is no dis-\nsipation (all the wave energyR\nvsR\nzρv2\nzdvsdzis still in the in-\nfinitely long system), the damping is only apparent, because of\nthe spreading of the wave front over time. This is because of\nthe di fferent propagation speeds in each strand, leading to an in-\ncreasing spread in z. Thus, it is very similar to the process of\nphase mixing.\nArticle number, page 2 of 9T. Van Doorsselaere et al.: Multithermal apparent damping of slow waves\n/vectorB\nz0\nFig. 1. A schematic representation of the considered configuration.\nThree magnetic strands are shown. The cyan Gaussian pulses are ex-\ncited at time t=0 at z=z0on all strands simultaneously. On each\nstrand they propagate with a di fferent speed, first to the blue line and\nthen the purple line. The resultant observed intensity, as integrated over\nthe di fferent strands, is given by the red line, which shows the multi-\nthermal apparent damping and broadening.\n2.2. Gaussian pulses\nLet us now build on this intuition to describe a system with an\ninitial Gaussian pulse in the density perturbation, as could be the\nresult of an impulsive excitation at the footpoint of the loop be-\ncause of (e.g.) granular bu ffeting or a reconnection event. We\nimagine a group of strands all simultaneously excited with a\npulse W(z,0) of the form\nW(z,0)=aexp \n−(z−z0)2\n2w2!\n(5)\nat position z0, with pulse width wand amplitude a. Because of\nthe propagation of the pulse on each individual strand, at a later\ntime, it will appear as\nW(z,t)=aexp \n−(z−(z0+vst))2\n2w2!\n, (6)\nin which vsdiffers from strand to strand. A sketch of the config-\nuration is shown in Fig. 1.\nNow we look at the integrated signal for a Gaussian strand\ndistribution for which vs∼N (¯v,σ2\nv) as before. We consider the\nintegral of the di fferent wavepackets (Eq. 6), with the distribu-\ntion of the strand’s sound speeds as weight function. So, in a\nsense, the integral is computing the intensity of a line-of-sight\nintegration through the multistranded loop with a Gaussian dis-\ntribution in the DEM (di fferential emission measure, see e.g. VanDoorsselaere et al. 2018). The emission measure is defined asR\nzn2dz(with electron density n), which linearises to 2R\nzn0n1dz\nfor background density n0and density perturbation n1. The in-\ntegral over the line-of-sight covers all strands with density n0in\nour loop model, and thus the wave perturbation W(z,t) has to be\nmultiplied with the Gaussian strand distribution. Moreover, the\nintegral over z(which traverses the entire loop system) is equiv-\nalent to integrating over all strands (in sound speed space).\nWith this reasoning, the total signal S(z,t) is then given as\nS(z,t)=Z\nvs1\nσv√\n2πexp \n−(vs−¯v)2\n2σ2v!\naexp \n−(z−(z0+vst))2\n2w2!\ndvs,\n(7)\nor\nS(z,t)=a\nσv√\n2πZ\nvsexp \n−1\n2\"(vs−¯v)2\nσ2v+(z−(z0+vst))2\nw2#!\ndvs.\n(8)\nCompleting the square, the evaluation of the integral is given as\nS(z,t)=awp\nw2+σ2vt2exp \n−(z−(z0+¯vt))2\n2(w2+σ2vt2)!\n. (9)\nThis is a signal that peaks at z0+¯vtand thus propagates with the\naverage sound speed upwards. Its Gaussian width (as a function\nofz) is given byp\nw2+σ2vt2, showing that it steadily increases\nin a hyperbolic fashion. For large t, the width increases approxi-\nmately linearly.\nAs in Subsect. 2.1, we also note here that the amplitude of\nthe peak signal (at z=z0+¯vt, i.e. co-propagating with the wave)\ndecreases steadily over time. Its decay d(t) from its initial ampli-\ntude is given as\nd(t)=wp\nw2+σ2vt2=1q\n1+σ2vt2\nw2. (10)\nFor large time t, this scales thus as t−1, recovering the results of\nSubsect. 2.1. These latter results may also be recovered consid-\nering the limit of w→0, corresponding to an initial δ-function\nperturbation.\nIn Fig. 2, we display the predicted damping envelope of\nEq. 10, and compare it to a Monte Carlo simulation of a Gaus-\nsian wave packet on di fferent strands. For the Monte Carlo\nsimulation, we have drawn 1000 vsfrom the normal distribu-\ntionN(¯v,σ2\nv), with ¯ v=152km /s (corresponding to 1MK) and\nσv=26.4km/s (for a motivation of these particular values, see\nSubsect. 3.2). The correspondence between the analytical solu-\ntion and the Monte Carlo simulation is excellent.\nWe may calculate the damping time τas the e-folding time\nof this damping profile d(t). We have then that\ne−1=d(τ)=1q\n1+σ2vτ2\nw2(11)\nresulting in\nσvτ\nw=√\ne2−1≈2.53 orτ=w\nσv√\ne2−1. (12)\nWith a substitution ∆z≡z−z0=¯vt, we may transform Eq. 10\nto a damping profile as a function of ∆z. We find that\nd(∆z)=1q\n1+σ2v∆z2\nw2¯v2. (13)\nArticle number, page 3 of 9A&A proofs: manuscript no. multiT_dampingslow_for_arxiv\nFig. 2. Comparison of Monte Carlo simulation with the analytical result.\nThe analytically predicted envelope (Eq. 10) is drawn with the dark\nblue line. The evolution of the Monte Carlo wave packet is drawn with\nthe light blue to pink colour, progressively in time. The mean sound\nspeed was taken as ¯ v=152km /s and the spread in sound speed as σv=\n26.4km/s\nMaking the same reasoning as in the derivation of the damping\ntimeτ, we may also derive the damping length Ld.\ne−1=d(Ld)=1q\n1+σ2vL2\nd\nw2¯v2orLd=w¯v\nσv√\ne2−1=¯vτ. (14)\n2.3. Driven waves\nNow we consider the case of driven, sinusoidal waves. At a cer-\ntain height z=0, a periodic driver is inserted, resulting in propa-\ngating waves asin(kz−ωt), with amplitude a, frequency ωand\nwavenumber k. The resulting intensity signal of the loop bundle\nis then given as an integral (similar to Eq. 7)\nS(z,t)=Z\nvs1\nσv√\n2πexp \n−(vs−¯v)2\n2σ2v!\nasin(kz−ωt)dvs\n=a\nσv√\n2πZ\nvsexp \n−(vs−¯v)2\n2σ2v!\nsin ωz\nvs−ωt!\ndvs, (15)\nwhere we have used the dispersion relation k=ω/vs. The latter\nintegral is not analytically solvable. We can still compare it to a\nMonte Carlo simulation with a 1000 vsdrawn from aN(¯v,σ2\nv)\ndistribution and summed up (see Fig. 3). The Monte Carlo simu-\nlation is shown with the blue line, while the full integral in Eq. 15\nis shown with the light orange line. The two lines closely match,\nand the deviation is due to the finite number of drawn vsvalues.\nFor a higher number of draws, the two lines converge.\nFurther analytical progress is possible by making a Taylor\napproximation of the denominator in the sine:\nsin ωz\nvs−ωt!\n=sin\u0012ωz\n¯v+δv−ωt\u0013\n(16)\n≈sin\u0012ωz\n¯v(1−δv\n¯v)−ωt\u0013\n. (17)\nThis approximation is valid if δv≪¯v. Since 95% of the contri-\nbution to the full integral (Eq. 15) is for |δv|≤3σv, the Taylor\napproximation is reasonably satisfied for our considered param-\neters of ¯ v=152km /s andσv=26.4km/s, for which we sub-\nsequently have|δv| ≤ 3σv=79.2km/s⪉¯v=152km /s. So,\nFig. 3. Comparison of Monte Carlo simulation of driven sine functions\n(blue line) with the full integral (light orange) and the approximations\nin Eq. 19 (green), all normalised to the starting value of 1. The expected\nGaussian damping envelope (Eq. 26) is shown in red. The time is ar-\nbitrarily chosen to be t=0, and the mean sound speed 152km /s and\nspread in sound speed σv=26.4km/s was chosen as before.\nthe assumption δv≪¯vseems to be su fficiently well satisfied in\nloops that are not too extremely multithermal (i.e. with σv≲¯v).\nThis Taylor approximation allows to rewrite S(z,t) as\nS(z,t)≈a\nσv√\n2π(\nsin\u0012ωz\n¯v−ωt\u0013Z\nδvexp \n−δv2\n2σ2v!\ncos\u0012ωzδv\n¯v2\u0013\ndδv\n−cos\u0012ωz\n¯v−ωt\u0013Z\nδvexp \n−δv2\n2σ2v!\nsin\u0012ωzδv\n¯v2\u0013\ndδv)\n.(18)\nIt turns out that the rightmost integral in this expression is exactly\n0, because its integrand is an odd function in δv. Thus, we have\nthat\nS(z,t)≈a\nσv√\n2πsin\u0012ωz\n¯v−ωt\u0013Z\nδvexp \n−δv2\n2σ2v!\ncos\u0012ωzδv\n¯v2\u0013\ndδv.\n(19)\nThe numerically calculated result of Eq. 19 is shown in Fig. 3\nwith the green line. It matches the Monte Carlo simulations\n(blue) and full integral (light orange) reasonably well. The in-\ntegral in Eq. 19 can be calculated analytically by writing it as a\ncomplex function:\nZ\nδvexp \n−δv2\n2σ2v!\ncos\u0012ωzδv\n¯v2\u0013\ndδv=ℜZ\nδvexp \n−δv2\n2σ2v+ıωzδv\n¯v2!\ndδv.\n(20)\nWe subsequently have\nℜZ\nδvexp \n−δv2\n2σ2v+ıωzδv\n¯v2!\ndδv (21)\n=ℜZ\nδvexp−δv√\n2σv−ı√\n2σvωz\n2¯v22\n−σ2\nvω2\n2¯v4z2dδv (22)\n=exp \n−σ2\nvω2\n2¯v4z2!\nℜZ\nδvexp−δv√\n2σv−ı√\n2σvωz\n2¯v22dδv\n(23)\n=√\n2πσvexp \n−σ2\nvω2\n2¯v4z2!\n. (24)\nArticle number, page 4 of 9T. Van Doorsselaere et al.: Multithermal apparent damping of slow waves\nFig. 4. Here we show the expected damping lengths (in Mm) as a func-\ntion of period (in s), for both multithermal apparent damping (Eq. 26)\nand thermal conduction (Eq. 3 in Mandal et al. 2016). The density was\ntaken to be 109cm−3, the mean temperature as 106K, and the spread in\ntemperature as σv=26.4km/s.\nInserting this in Eq. 19, we find as end result\nS(z,t)≈aexp \n−σ2\nvω2\n2¯v4z2!\nsin\u0012ωz\n¯v−ωt\u0013\n. (25)\nThe wave is thus propagating with the average sound speed, and\nhas additionally a Gaussian damping envelope with a Gaussian\ndamping length (keeping the traditional factor 2 in the denomi-\nnator, Pascoe et al. 2017)\nLG=¯v2\nσvω. (26)\nFor the values considered in this paper (¯ v=152km /s,σv=\n26.4km/s,ω=2π/180s), this reduces to a damping length\nLG=25.1Mm.\nThe formula shows that the damping length is inversely pro-\nportional to the frequency. This is a di fferent dependence than\nthe thermal conduction damping length, which is proportional to\nω−2. In Fig. 4, we compare the multithermal apparent damping\nto the damping by thermal conduction. For the latter, we have\ntaken the results in Mandal et al. (2016), and these are shown\nwith the blue line. The other, light orange line corresponds to\nEq. 26. The graph shows that for intermediate periods (i.e. be-\ntween 300s and 1000s), the damping by thermal conduction is\ncomparable. However, for shorter or longer periods, the mul-\ntithermal apparent damping becomes more significant. Caution\nis appropriate here, because the multithermal apparent damping\nhas a Gaussian damping profile which is compared in this graph\nto the exponential damping profile of the thermal conduction.\n3. Results – loops in filter images\n3.1. Influence of a finite filter in imaging observations\nLet us consider the influence of a filter on the observability and\nmultithermal apparent damping of slow waves. For a filter Fde-\nscribed by a Gaussian function in vs-space as\nF(vs)=aFexp−(vs−vF)2\n2σ2\nF (27)with amplitude aF, mean vFand widthσF, the resulting observed\nsignal (equivalent to Eq. 7) would be\nS(z,t)=Z\nvs1\nσv√\n2πexp \n−(vs−¯v)2\n2σ2v!\naFexp−(vs−vF)2\n2σ2\nF\naexp \n−(z−(z0+vst))2\n2w2!\ndvs.(28)\nThe first two Gaussian distributions may be combined by realis-\ning that\nexp \n−(vs−¯v)2\n2σ2v!\naFexp−(vs−vF)2\n2σ2\nF\n=aFexp−(vF−¯v)2\n2(σ2\nF+σ2v)exp \n−(vs−V)2\n2Σ2!\n,(29)\nwhere we have introduced the notation\n1\nΣ2=1\nσ2\nF+1\nσ2vV=σ2\nvvF+σ2\nF¯v\nσ2\nF+σ2v(30)\nfor the width Σand average Vof the resulting Gaussian. This\nmeans that the width of the Gaussian is always decreased, due to\nthe harmonic average.\nThese expressions for ΣandVmay then be inserted in Eq. 9,\nwhile also remembering to also take the extra factors of Eq. 29\nalong and incorporate them with a. This will result in\nS(z,t)=aaFw√\nw2+ Σ2t2exp−(vF−¯v)2\n2(σ2\nF+σ2v)exp \n−(z−(z0+Vt))2\n2(w2+ Σ2t2)!\n.\n(31)\nAs before, the damping (following a wave packet, at a ray of\nz=z0+Vt) will be as\nd(t)=w√\nw2+ Σ2t2. (32)\nThis is weaker damping than in the non-filtered case, because\nΣ<σ v.\nLikewise, we may also insert ΣandVforσvand ¯vrespectively\nin the Gaussian damping lengths (Eq. 26):\nLG=V2\nΣω. (33)\nIt is also possible to use the propagation speed in di fferent\nfilters to estimate the temperature spread σvand the mean tem-\nperature ¯ v. This model naturally explains the di fferent propaga-\ntion speeds in di fferent filter channels and this di fference may be\nused to measure the loop’s fundamental thermal properties and\nquantify its DEM. As in Eq. 30, we see that σFandvFare known\nfor each filter. Then the propagation speed Vmay be measured\nin different filters, allowing us to estimate ¯ vandσvthrough least\nsquares fitting.\nMoreover, the e ffective filter width σFis much larger for imag-\ning observations than spectral observations. Thus, it is to be ex-\npected that imaging observations are much more profoundly im-\npacted by the e ffect of multithermal apparent damping. Spectral\nobservations will experience very little damping from the multi-\nthermal apparent damping e ffect, given the narrowness of the ef-\nfective filter. Thus, we may use the combination and contrast be-\ntween imaging and spectral observations to disentangle the real\ndamping mechanism from the multithermal apparent damping.\nThat opens exciting prospects for seismology of thermal proper-\nties of coronal loops.\nArticle number, page 5 of 9A&A proofs: manuscript no. multiT_dampingslow_for_arxiv\n3.2. Comparison with simulations\nHere, we verify the analytical results described in the previous\nsections using a 3D MHD multithermal loop model similar to\nthe one presented in Krishna Prasad & Van Doorsselaere (2023).\nThey solve the ideal MHD equations with MPI-AMRV AC (Porth\net al. 2014), where only numerical di ffusion is present and no\nexplicit di ffusive terms. They consider a bundle of 33 vertical\nstrands with randomly assigned plasma temperatures and den-\nsities to represent a coronal loop, similar to the setup we con-\nsider in this paper. To elaborate, the plasma temperature T(num-\nber density n) for each strand is selected from a random nor-\nmal distribution whose peak value corresponds to log T=6.0\n(logn=9.2), with a standard width of 0.15 (0.10). The plasma\ntemperature outside the strands and that outside the loop are kept\nat the same value, 1 MK. The corresponding number densities\nare fixed at 5×108cm−3. The magnetic field is vertical and par-\nallel to the axis of the loop. For further details on the simulation\nsetup, we refer the interested reader to Krishna Prasad & Van\nDoorsselaere (2023).\nConsidering the peak ( µlogT=6.0) and the width ( σlogT=\n0.15) values of temperature distribution in the simulations, and\nassuming that the resulting distribution of sound speeds is su ffi-\nciently normal (so that the theory in Sec. 2 applies), we can es-\ntimate the sound speed distribution properties from the temper-\nature distribution. For that, we calculate that log T=6.0±0.15\ncorresponds to a sound speed value of vs=152+28\n−24km/s, by\ncalculating the sound speed for µlogTandµlogT±σlogTsepa-\nrately. So, in what follows we take the following parameters:\n¯v=152km/s andσv=26.4km/s. Since the period of the driver\nin the simulation was 180s, this ¯ vis expected to result in a wave-\nlength of 27.4Mm. The wavelength values λobtained in Fig. 5\nindicate a good agreement with this.\nIn their model, Krishna Prasad & Van Doorsselaere excite\nslow magneto-acoustic waves within the loop by periodically\n(period≈180 s) driving the vertical velocity ( vz) at the bottom\nboundary of the loop, with an amplitude of ≈7.6 km s−1which\nis approximately 5% of the sound speed at 1MK. This driving\namplitude was chosen to be small, because we wanted to avoid\nany damping caused by non-linear e ffects of the waves. We use\nthe same driver in this study, however, to highlight the multi-\nthermal e ffects we restrict the spatial location of the driver to\nthe positions of the strands. In other words, the amplitude of the\ndriver is zero outside the strand locations and consequently the\noscillations are restricted to the strands. Once the generated slow\nwaves start approaching the top boundary of the loop, we com-\npute the mean density and vertical velocity ( vz) across the loop\nas a function of distance along the loop to analyse how the oscil-\nlation amplitudes evolve. Fig. 5 displays the density and vzpro-\nfiles along the loop in the top and bottom panels respectively. As\ncan be seen, the oscillations appear to damp very quickly. For a\nproper quantitative assessment, we measure the damping lengths\nby fitting the data with the following damped sinusoid function\nf(z)=A0exp−z2\n2L2\ndsin 2πz\nλ+ϕ!\n+b1z+b0. (34)\nHere zis the coordinate along the loop axis, A0is the maximum\namplitude, Ldis the damping length, λis the wavelength, ϕis the\ninitial phase, and b1andb0are appropriate constants. It may be\nnoted that this function describes a Gaussian damped sine wave\nsimilar to that described in Pascoe et al. (2016). Although an\nexponential damping is generally considered for slow waves, as\ndescribed in Sec. 2.3, the multithermal apparent damping is ex-\npected to produce a Gaussian damping which justifies our choice\nFig. 5. The average density (top) and vertical velocity, vz(bottom) pro-\nfiles along the simulated multithermal loop. The solid lines represent a\nGaussian damped sinusoid fit to the data following the function given in\nEq. 34. The obtained wavelength ( λ) and damping length Ldvalues are\nlisted in the plot.\nhere. The solid light orange lines in Fig. 5 represent the obtained\nfits to the data. The damping lengths obtained from the fits are\n20±0.3 Mm, and 21.2±0.3 Mm for the density and vertical ve-\nlocity respectively. These values are within 20% of the expected\nvalue of 25Mm (see Sec. 2.3, Eq. 26), and are thus a reasonable\nmatch. The deviation may be due to (1) the approximation of\nthe full integral (Eq. 15 by Eq. 19), or (2) the “small” number\nof strands (only 33) in the simulations of Krishna Prasad & Van\nDoorsselaere (2023), insu fficient to fully cover the continuous,\nGaussian DEM which is modelled in Eq. 15 due to the finite\nsample size. Because of the chosen small driver amplitude, non-\nlinear e ffects do not play a role in the damping of these waves.\nNumerical di ffusion could play a role, but we have verified that\nthe increasing the numerical resolution has no e ffect on the mea-\nsured damping lengths.\nFor a direct comparison with observations, we also forward\nmodel the data using the FoMo code (Van Doorsselaere et al.\n2016). In particular, we generate synthetic images in the 6 coro-\nnal channels of SDO /AIA, namely, 94 Å, 131 Å, 171 Å, 193 Å,\n211 Å, and 335 Å. As described in Krishna Prasad & Van Doors-\nselaere (2023), we add appropriate data noise (following Yuan\n& Nakariakov 2012) and subsequently build time-distance maps\nto study the evolution of oscillations along the loop. The oscil-\nlations are found only in three channels, the 171 Å, 193 Å, and\n211 Å. The propagation properties are exactly the same as those\nfound in Krishna Prasad & Van Doorsselaere (2023) so we are\nnot going to discuss them in detail here. Just highlighting one\ncrucial point, Krishna Prasad & Van Doorsselaere found that the\nforward modelled propagation speeds in the 211Å and 193Å fil-\nter were very close to each other, despite their di fference in tem-\nperature. Here we explain quantitatively this phenomena: as can\nbe seen in Table 1, the predicted propagation speeds Vare indeed\nvery close to each other for these filters. We propose that the\nvariation of phase speeds in di fferent observational filters (e.g.\nKing et al. 2003) may be quantitatively explained through the\nproposed formula in Eq. 30: indeed, that equation shows that the\nobserved phase speed Vis influenced by the specific filter’s σF\nandvF.\nLet us now focus on the damping properties of the slow\nwaves. Fig. 6 displays the spatial intensity profiles at a particular\nArticle number, page 6 of 9T. Van Doorsselaere et al.: Multithermal apparent damping of slow waves\nFig. 6. The spatial intensity profiles at a particular instant along the\nloop obtained from the synthetic data corresponding to the AIA 171,Å,\n193,Å, and 211,Å filters. The solid lines represent a Gaussian damped\nsinusoid fit to the data following the function given in Eq. 34. The ob-\ntained wavelength ( λ) and damping length Ldvalues are listed in the\nplot.\ninstant along the loop for the three AIA channels. The solid lines\nin the figure correspond to the fitted profiles following Eq. 34.\nWe notice that the fitted curve in AIA 171 has a significant\ndeviation beyond a distance of 40 Mm. Imposing tighter con-\nstraints on the fitting function did not improve the results much.\nHowever, the larger uncertainty obtained on the corresponding\ndamping length should have incorporated this. The resulting as-\nsociated damping lengths are 31.8 ±1.3 Mm, 36.8±0.9 Mm, and\n34.7±0.9 Mm, for the 171 Å, 193 Å, and 211 Å channels, respec-\ntively. As may be noted these values are di fferent from those ob-\ntained for the density and velocity parameters (see Fig. 5). This\nis due to the temperature response of the observing filter, which\nalso has an influence on this multithermal apparent damping, as\ndescribed in Sec. 3.1.\nIn order to quantitatively assess the e ffect of SDO /AIA fil-\nters, we fit the temperature response curves (version 9) for each\ncoronal filter with a Gaussian function and estimate their stan-\ndard width. The temperature response curves and the fitted pro-\nfiles are plotted in Fig. 7. For the curves with multiple peaks, we\nchoose the peak that is closer to log T=6.0, which is the charac-\nteristic temperature in our simulations. In each of the panels, the\ndashed line shows the full response curve, the light orange line\ndenotes the fitted segment, and the green line shows the fitted\nfunction. The obtained widths σlogTare 0.22±0.01, 0.17±0.00,\n0.13±0.00, 0.12±0.01, 0.12±0.01, and 0.31±0.01, for the 94 Å,\n131 Å, 171 Å, 193 Å, 211 Å, and 335 Å channels, respectively.\nThese values along with the respective peak locations are listed\nin Table 1. From these fitted filter curves in log T-space, we have\ncomputed, for each filter, the corresponding peak propagation\nspeed vFfrom the sound speed of the fitted peak temperature.\nThen, we have calculated σF, as the average of sound speeds be-longing to the peak temperature plus and minus the filter peak\nwidth. Thus, we have assumed that the filter is symmetric in ve-\nlocity space. The results of these calculations are listed in table 1.\nThen we have used Eqs. 30 to compute ΣandV, and we have also\nlisted the obtained values in table 1.\nFor the density, we have calculated the predicted damping\ntime with Eq. 26. For predicting the damping times observed in\nthe filters, we have used the values for ΣandVand inserted them\nin Eq. 33. The predicted damping values are listed in table 2,\nalong with the measured damping values from the simulations.\nThe predicted damping times match reasonable well with the\nmodelled damping times (with a maximum deviation of 30%).\nAs before, we think that this deviation between the numerical\ndamping lengths and the predicted damping lengths is due to\nthe finite number of strands of which the loop consists in the\nsimulation. Thus, the number of strands is insu fficient to fill\nthe entire Gaussian DEM. In essence, there are insu fficient\nMonte Carlo realisations of the strands to completely cover the\nexpected Gaussian DEM distribution.\n4. Conclusions and discussion\nIn this paper, we have considered the apparent damping of slow\nwaves (which we call “multithermal apparent damping” (MAD)\nor “V oitenko-damping”) due to a di fferent propagation speed in\ncoronal loop strands. We have considered a superposition of δ-\nfunction impulses, Gaussian pulses or driven waves. All of these\nmodels led to the multithermal apparent damping of slow waves\ndue to observational phase mixing. We should stress that the\ndamping is indeed only apparent, and that no wave energy was\nharmed dissipated during the production of this paper. This mul-\ntithermal apparent damping of the slow waves is expected to be\nstronger than damping by thermal conduction for short periods\n(less than 200s), and comparable for longer periods. We have\nfound that the case of driven slow waves leads to a predicted\nGaussian damping profile, with a predicted damping length LG\nof\nLG=¯v2\nσvω,\nwhere ¯ vis the average sound speed in the loop, σvis the spread in\nthe sound speed, and ωis the frequency. The resulting, predicted\nvalue of the damping length matches reasonably well with the\none found in the simulations of Krishna Prasad & Van Doorsse-\nlaere (2023). The predicted damping length scales linearly with\nthe period of the wave. This is compatible with the observational\nsynthesis made by Cho et al. (2016), who observed a unified pic-\nture of solar and stellar quasi-periodic pulsations with a damp-\ning time scaling linearly with the period. Moreover, this di fferent\nscaling of the multithermal apparent damping time with period\nfrom thermal conduction may explain the di fference in damp-\ning scalings in open-field or closed-field regions (e.g. Krishna\nPrasad et al. 2014, and follow-up works), but also for di fferent\ndamping regimes at di fferent heights (Gupta 2014). These dif-\nferent damping regimes could then be associated with di fferent\nlevels of multi-thermal structuring of loops /plumes and the rel-\native importance of thermal conduction damping and multither-\nmal apparent damping.\nIn the second part of the paper, we have considered the e ffect\nof a finite filter width in imaging instruments such as SDO /AIA.\nWe have found that the finite filter has as e ffect that the waves\nhave a di fferent propagation speed Vand damping length LGin\nArticle number, page 7 of 9A&A proofs: manuscript no. multiT_dampingslow_for_arxiv\nFig. 7. SDO/AIA temperature response curves for the 6 coronal channels as listed. In each of the panels, the blue dashed line represents the full\nresponse curve, the light orange solid line represents the segment fitted with a Gaussian function, and the green solid line represents the fitted\nfunction. The obtained standard widths are listed in the plot.\nTable 1. Properties of AIA filter response curves.\nChannel name AIA 94 AIA 131 AIA 171 AIA 193 AIA 211 AIA 335\nPeak temperature µlogT(logT) 6.02±0.01 5.75±0.0 5.90±0.0 6.14±0.01 6.24±0.01 5.91±0.00\nvF(km/s) 135 179 200\nPeak Width σlogT(logT) 0.22±0.01 0.17±0.0 0.13±0.0 0.12±0.01 0.12±0.01 0.31±0.01\nσF(km/s) 20.3 24.7 27.8\nΣ(km/s) 16.1 18.0 19.1\nV(km/s) 141.6 166.1 174.9\nTable 2. Gaussian damping lengths in Mm for various quantities.\nDensity Velocity AIA 171 AIA 193 AIA 211\nNumerical model 20.0 ±0.3 21.2±0.3 31.8±1.3 36.8±0.9 34.7±0.9\nPredicted damping 25.1 35.7 43.8 45.8\neach filter. These are given by\nV=σ2\nvvF+σ2\nF¯v\nσ2\nF+σ2v,LG=V2\nΣω,1\nΣ2=1\nσ2\nF+1\nσ2v\nwhere vFis the central sound speed of the filter, and σFis the\nwidth of the filter. This explains two phenomena: (1) the ob-\nserved phase speed in di fferent filters depends on the thermal\nproperties of the loop, and (2) the damping in each filter is also\ndifferent. We have once again checked these formulas against the\ndamping in forward models of the simulations of Krishna Prasad\n& Van Doorsselaere (2023). We found that our predictions matchreasonably well with the simulated values (within 30%). We sus-\npect that the deviation is mostly caused by the small number\nof strands in the simulation, which compares to our continuous\nDEM distribution that we considered in this paper.\nWe expect that these results may be used in the future to\nperform MHD seismology (Nakariakov & Verwichte 2005) of\ncoronal loops with slow waves. With the above formulas, it is\npossible to fit the loop’s DEM properties of central temperature\n(through the average sound speed ¯ v) and spread in temperature\n(through the value of the spread in sound speed σv). These DEM\nproperties of the loops are only sensitive to the loop itself in\nwhich the slow wave propagates. This is in contrast to the cur-\nArticle number, page 8 of 9T. Van Doorsselaere et al.: Multithermal apparent damping of slow waves\nrently used method of DEM inversion (e.g. Hannah & Kontar\n2012; Cheung et al. 2015; Krishna Prasad et al. 2018), which\nis very sensitive to the careful background subtraction from the\nloop’s emission. This proposed method will at least allow to re-\nmove this sensitivity, and perhaps reveal more detailed thermal\nproperties of loops.\nAlso the combination of spectral observations with imaging ob-\nservations is an interesting avenue to consider, because the spec-\ntral observations are much less impacted by the multithermal ap-\nparent damping, and the combination of this with the imaging\nobservations would allow to disentangle physical damping from\nthe multithermal apparent damping.\nTopics for future research as a follow-up to this work would\nbe (1) to consider the e ffect of a combination of multithermal ap-\nparent damping and thermal conduction in a multistranded loop\nsystem, (2) to model the e ffect of multithermal apparent damp-\ning on standing sound waves in e.g. flaring loops (Wang 2011;\nCho et al. 2016), and (3) investigation of the usage of di fferent\nlines-of-sight from di fferent spacecraft (e.g. Solar Orbiter and\nSDO) to probe the inner multithermal structure of loops using\nmultithermal apparent damping properties.\nAcknowledgements. TVD was supported by the European Research Council\n(ERC) under the European Union’s Horizon 2020 research and innovation pro-\ngramme (grant agreement No 724326), the C1 grant TRACEspace of Internal\nFunds KU Leuven, and a Senior Research Project (G088021N) of the FWO\nVlaanderen. The research benefited greatly from discussions at ISSI. TVD would\nlike to thank Dipankar Banerjee, Vaibhav Pant and Krishna Prasad for their hos-\npitality when visiting ARIES, Nainital in spring 2023. TVD would like to thank\nRoberto Soler for referring to the V oitenko et al. (2005) paper at the AGU Chap-\nman conference in Berlin 2023. VP is supported by SERB start-up research\ngrant (File no. SRG /2022/001687). AWH acknowledges the financial support\nof the Science and Technology Facilities Council (STFC) through Consolidated\nGrants ST /S000402 /1 and ST /W001195 /1 to the University of St Andrews and\nsupport from the European Research Council (ERC) Synergy grant ‘The Whole\nSun’ (810218). 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M. 2012, A&A, 543, A9\nArticle number, page 9 of 9"    },    {        "title": "1704.06505v1.Magnetic_field_influence_on_the_early_time_dynamics_of_heavy_ion_collisions.pdf",        "content": "Magnetic \feld in\ruence on the early time dynamics of heavy-ion collisions\nMoritz Greif,1,\u0003Carsten Greiner,1and Zhe Xu2, 3\n1Institut f ur Theoretische Physik, Johann Wolfgang Goethe-Universit at,\nMax-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany\n2Department of Physics, Tsinghua University, Beijing 100084, China\n3Collaborative Innovation Center of Quantum Matter, Beijing, China\n(Dated: April 24, 2017)\nIn high energy heavy-ion collisions the magnetic \feld is very strong right after the nuclei penetrate\neach other and a non-equilibrium system of quarks and gluons builds up. Even though quarks might\nnot be very abundant initially, their dynamics must necessarily be in\ruenced by the Lorentz force.\nEmploying the 3+1d partonic cascade BAMPS we show that the circular Larmor movement of the\nquarks leads to a strong positive anisotropic \row of quarks at very soft transverse momenta. We\nexplore the regions where the e\u000bect is visible, and explicitly show how collisions damp the e\u000bect.\nAs a possible application we look at photon production from the \rowing non-equilibrium medium.\nI. INTRODUCTION.\nShortly after the collision of heavy nuclei in experi-\nments at the LHC at CERN or RHIC at BNL the en-\nergy density is high enough that a so called quark-gluon\nplasma (QGP) is formed [1{4]. Due to the high velocity\nand small distances of the nucleons passing each other at\nthe moment of the collision, the magnetic \feld for non-\ncentral collisions in the center of the reaction at, e.g., top\nRHIC/LHC energies is very high. Indeed, those \felds can\nbe expected to be among the largest \feld strengths in the\nuniverse [5]. The \felds will decay very fast, even assum-\ning a conducting medium which, by Maxwells equations,\nslows down the decay of the magnetic \rux [6]. Strong\nenough magnetic \felds can generate several novel e\u000bects,\nsuch as the chiral magnetic e\u000bect, the chiral vortical ef-\nfect [7{10], the chiral separation e\u000bect [7], or a chiral\nmagnetic wave [11, 12]. In this article, we turn to a fun-\ndamental question which has not gained much attention\nin literature: will there be directly measurable e\u000bects of\nthe electromagnetic Lorentz force in heavy-ion collisions?\nEarly attempts [13] to model a hadron gas under the in-\n\ruence of magnetic \felds did not show strong e\u000bects.\nThe authors of Ref. [14] have studied the charge de-\npendent directed \row of pions and protons in a simpli\fed\nanalytic model, taking electric and magnetic \felds into\naccount. They \fnd a very small signal, owing mainly to\nthe currents induced by electric \felds generated by the\nfast decaying magnetic \felds (Faraday e\u000bect). For very\nlowpThowever, they see a strong in\ruence of the mag-\nnetic \feld itself (dubbed as \\Hall\" e\u000bect). In particular,\nthe magnetic e\u000bect becomes important for the directed\n\row at transverse momenta pT.0:25 GeV for RHIC and\nLHC. In this paper, we will also see the growing in\ruence\nof the magnetic e\u000bect at low pT.\nHydrodynamic calculations including magnetic \felds\nare rare, and still under development [15, 16]. Recently,\nit has been found that the directed \row of charm quarks\n\u0003greif@th.physik.uni-frankfurt.deis very sensitive to the magnetic and electric \feld [17].\nFurthermore, it has been proposed, that the J=\t forma-\ntion becomes anisotropic which leads, e.g., to a sizable\nelliptic \row at high transverse momenta [18].\nWe attempt an exploratory study of the early-time\nnon-equilibrium dynamics of decon\fned quarks and glu-\nons including simple parametrizations of an external\nmagnetic \feld. We \fnd that the quark momenta ro-\ntate parallel to the event plane and develop a surprisingly\nlarge momentum anisotropy at mid- and forward rapidity\nfor very low transverse momenta. This is roughly rem-\niniscent to the Hall e\u000bect. The particle velocity stems\nmainly from the large boosts in beam direction (longitu-\ndinal expansion), whereas the magnetic \feld comes from\nthe charged spectator nucleons.\nWe see furthermore that the spectra are also slightly\nenhanced at early times and we show explicitly how col-\nlisions damp both the e\u000bect of the \row and the spectra.\nThis paper is organized as follows. In Sec. II we\nexplain the model setup, including the magnetic \feld\nparametrizations. In Sec. III we show the collisionless\nresult which is purely due to the Larmor movement of\nquarks in the magnetic \feld, before we turn in Sec. IV to\nthe result with collisions and conclude in Sec. V.\nFigure 1. Geometry of our model. The magnetic \feld is\nconstant and homogeneous.arXiv:1704.06505v1  [hep-ph]  21 Apr 20172\nII. SIMPLE MODEL WITHOUT COLLISIONS\nWe investigate how strong the e\u000bect of the Lorentz\nforce alone can be on the particle distributions. To this\nend, we outline a simple model for the heavy-ion collision,\nneglecting collisions in a \frst step. We consider the colli-\nsion of two heavy nuclei along the z-axis. For simplicity,\nwe assume the magnetic \feld ~Bto be constant and homo-\ngeneous, pointing in y-direction, ~B\u0011By~ ey. The situa-\ntion is depicted in Fig. 1. Here we neglect electrodynami-\ncal induction e\u000bects. We assume that all events are sym-\nmetric and the impact parameter points in x-direction.\nIn this geometry elliptic \row can be seen as an average\n(h\u0001i) momentum asymmetry\n(p2\nx\u0000p2\ny)=p2\nT\u000b\n\u0011v2, where\npT=q\np2x+p2y.\nA. Initial state and formation time\nIn this simple model setup we do not consider space-\ndependent e\u000bects, thus we sample only four-momenta of\nthe particles. All particles are assumed to be massless.\nThepTdistribution is sampled according to a power law,\ndN=dpT= \nn\u00001\np1\u0000n\nT;min!\np\u0000n\nT; n = 2;3;4: (1)\nWe choose a minimal value pT;min= 0:01 GeV. For all\nthe following results we assume a constant distribution\nin rapidity, y= 1=2 log(E+pz)=(E\u0000pz),\ndN=dy = const:; pz=pTsinhy: (2)\nWe \fnd that the results are not dependent on the rapid-\nity window in which we initialize the particles, as long\nas it is larger than the observed rapidity bins. For most\nstudies,\u00003< y < 3 is su\u000ecient. It is possible to use\na formation time \u0001 tf= cosh(y)=pTduring which par-\nticles are still o\u000b-shell and do not interact, but propa-\ngate freely. This formation time has been used earlier\nin transport approaches using the Minijet model for the\ninitial condition [19{21]. We can assume that the mag-\nnetic \feld will also not in\ruence the partons within their\nformation time. However, as quarks carry their electric\ncharge even o\u000b-shell, their classical interaction with mag-\nnetic \felds is arguable, and the formation time could be\nirrelevant. As this point is conceptionally uncertain, we\nshow results for both options, assuming the particle-\feld\ninteraction to be switched on immediately (no formation\ntime), or, only after \u0001 tf, respectively. In this simpli\fed\ncollisionless scenario we only initialize quarks, carrying\nthe electric charge q=e=3 orq= 2e=3, respectively.\nThe exact quark and gluon content in the early phase of\nheavy-ion collisions is under debate. It is clear, that the\nmore gluon dominated the system is, the less pronounced\nsuch electromagnetic e\u000bects will be.B. Magnetic \feld parametrizations\nThe external magnetic \feld present at t= 0+after\nthe collision is still subject to active research, and de-\npends strongly on the geometric modeling of the nuclei\nas well as the electric conductivity and also possible non-\nequilibrium e\u000bects. Common to all the results in litera-\nture is the dominant By-component, perpendicular to the\nevent plane, which is about an order of magnitude larger\nthan theBx-component, the Bz-component is nearly ab-\nsent. The authors of Ref. [22] look explicitly at \ruc-\ntuations of the direction of the magnetic \feld and \fnd\nthat for middle central collisions the \feld \ructuates less\naround the direction perpendicular to the event plane\nthan for near central or very peripheral collisions. For\nthe qualitative understanding of the dynamical e\u000bects to\nthe quark momenta, we adopt several simpli\fed scenar-\nios for the \feld strength By, and setBx=Bz= 0. In\nRef. [13] it was found that the spatial dependence over\nthe overlap region is mild, so that we restrict ourselves\nhere to a homogeneous \feld in space, parametrized as\nparam 1:eBy(t) = 4m2\n\u0019\u0002(0:3 fm=c\u0000t)\nparam 2:eBy(t) =eBt=0\ny(1 +t2=t2\nc)\u00003=2withtc=\n0:065 fm=c.\neBy [mπ2]\nt [fm/c]param 1\nparam 2\n024689 \n 0  0.1  0.2  0.3  0.4  0.5\nFigure 2. The two simple parametrizations of the homoge-\nneous magnetic \feld. Param 2 follows Ref. [23].\nParam 2 is the parametrization of the results of Ref. [23]\nas given in Ref. [5]. We use it with parameters\ncorresponding to RHIC collisions (Au+Au,psNN =\n200 GeV) at typical impact parameters of \u00188 fm\ncorresponding to 20 \u000040% centrality (see also, e.g.,\nRefs. [24, 25] for typical \feld strengths). In the very early\nstage, the medium is assumed to be gluon dominated,\nsuch that the electric conductivity can be neglected [5]\n(being roughly proportional to the sum of the electric\ncharges squared, weighted by the densities of the charge-\ncarrying species [26, 27]). The authors of Ref. [5, 23]\napproximate the total magnetic \feld thus by the exter-\nnal component produced by the charged nucleons passing3\neach other. The full solution of the Maxwell- and Boltz-\nmann equation will slow down the decay of the magnetic\n\feld, but so far, only little is known about the precise\nevolution. Parametrization 1 is an optimistic imitation\nof a strongly conducting medium, which would keep the\nmagnetic \feld present for some time. We have tried even\nhigher or longer \feld parametrizations, but for simplicity\nwe restrict ourselves to an optimistic, and a realistic one.\nC. Larmor movement\nThe magnetic \feld changes the direction of velocity of\nthe particles by the Lorentz force, ~FL=q~ v\u0002~B. In our\ngeometry, particles will move in a circle around the y-\ndirection, clockwise or anticlockwise depending on their\nchargeq. Thereby, any momentum in z-direction will\nincrease or decrease the momentum in x-direction, px!\npx+ \u0001px.\nTo analytically estimate the e\u000bect of increasing px\ncomponents, we note that by symmetry hpxi= 0;hpyi=\n0. We consider two particles with opposite pxmomentum\ncomponents, px;1=\u0000px;2as representer of the particle\nensemble. Their pymomenta are equal, and chosen in\na way, that the initial momentum asymmetry v2takes a\ngiven value. The change px!px+ \u0001pxon thev2of the\nwhole particle ensemble can then, in a simpli\fed fashion,\nbe estimated by\nv2(\u0001px)\n=1\n2 \n(px+ \u0001px)2\u0000p2\ny\n(px+ \u0001px)2+p2y+(\u0000px+ \u0001px)2\u0000p2\ny\n(\u0000px+ \u0001px)2+p2y!\n:(3)\nIn Fig. 3 we show this momentum asymmetry for three\nchoices of the initial v2, positive, zero and negative.\nClearly, for zero initial v2, the increase of \u0001 pxmust be\nlarger than pTin order to enhance the asymmetry. All\ncases show a minimum in v2for \u0001px< pT, which can\nbe strongly negative. The radius of de\rection due to the\nmagnetic \feld is\nrLarmor =p\np2x+p2z\nqBy; (4)\nand the angle of the circular movement of time tis\n\u000bLarmor =t=rLarmor . The value of \u0001 pxdepends on the\nmomentapxandpzof each particle, for a given magnetic\n\feld times its duration, Byt.\nIn this study, we do not include electromagnetic e\u000bects\nother than the Larmor movement. The reason is outlined\nin the following. The Faraday e\u000bect due to the time\ndependent magnetic \rux '=ByAthrough surface A\ngenerates an electric \feld,\n\u0000@'\n@t=I\nd~ r\u0001~E: (5)\nThis electric \feld accelerates charges in the opposite di-\nrection than the Lorentz force ~FL=qv\u0002~B. Assuming\n-1-0.5 0 0.5 1\n 0  1  2  3  4  5v2(∆px)\n∆px /pTpx=py=pT/√ 2\npx=2py\npx=py/2Figure 3. After the increase of px, the momentum asymme-\ntryv2is in all cases positive for \u0001 px> pT. The result is\nsymmetric in \u0001 px.\nfor a moment, that ~FL\u00110, the electric current due to the\nforceq~Ewill generate a magnetic \feld component Bind\ncounter balancing the decay of the \feld, ~Bind\u0018@~B=@t ,\ndepending on the electric conductivity. On top of these\ne\u000bects, the electric \felds generated by the spectators, al-\nbeit small in magnitude, has also an x-component [28],\nwhich is positive Espec\nx>0 forx > 0 and negative\nEspec\nx<0 forx < 0. All these 3 e\u000bects can cancel\nor enhance each other, and depend crucially on the as-\nsumed electric conductivity and parametrization of the\nbare spectator induced \felds. Furthermore, the calcu-\nlation of the magnetic \rux as well as the electric \felds\nwould require a full space-time dependent (propagating)\nsolution of the electromagnetic \felds. This is why we re-\nstrict ourselves to show what maximum e\u000bect on the par-\nticle dynamics is expected from the magnetic \feld only.\nIII. RESULTS OF THE COLLISIONLESS\nMODEL\nFirst we show how pT-spectra of quarks are in\ruenced\nin the early time by a magnetic \feld in Fig. 4. Here,\nfor parametrization 1 of the magnetic \feld, the spectrum\nis enhanced for 0 :02< pT=GeV<0:1 due to the Lar-\nmor turn of pz-momenta. In Fig. 3 we explained that\nthe \fnal momentum asymmetry depends strongly on the\nadditional \u0001 px. We explore which momentum space re-\ngion (regions in rapidity) is necessary to gain su\u000eciently\nlarge values of \u0001 pxfor thev2to change visibly. Here\nwe di\u000berentiate between initial quantities, and those af-\nter the circular Larmor-movement has been applied to\nthe particle. For this purpose we show in Fig. 5 the \f-\nnalv2(after the Larmor movement had been applied for\na timet) as function of initial rapidity yinitial . We split\nthis up in a soft region, for \fnal pT<0:3 GeV, where\nthe averaged v2reaches large values, and the region of\n\fnalpT>0:3 GeV, where the v2is consistent with zero.4\n1021041061081010\n 0.01  0.1  1dNquarks/(2πpTdpTdy) [GeV-2]\npT [GeV]dashed: initial|y|<0.5\neBy=4 mπ2\nt=0.3 fm/c\nno collisions\narbitrary normalisationn=2\nn=3\nn=4\nFigure 4. For three di\u000berent initial pT-distributions (power-\nlaw exponent n) we show how the spectra change after\n0:3 fm=c under the in\ruence of a magnetic \feld. Here\nwe use an arbitrary number of particles and magnetic \feld\nparametrization 1.\nThis ultrasoft pTrange can already be expected from the\nspectra, Fig. 4. Note that the saturation in Fig. 5 is due\nto the cuts in \fnalpT, which means, that, in the curve\nfor \fnalpT<0:3 GeV, the larger sinh( y), the smaller\nthe values of pTwhich contribute.\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-3 -2 -1  0  1  2  3final v2\nyinitialeB=4 mπ2\nt=0.3 fm/cinitial state: \nn=2, |y| < 3all final pTfinal pT > 0.3 GeV\nfinal pT < 0.3 GeV\nFigure 5. Final average v2per particle for three di\u000berent pT\nranges as function of initial rapidity yinitial of the particle.\nHere we use an initial state with power law exponent n= 2\nand magnetic \feld parametrization 1. The result for n= 3;4\nlooks very similar, only the maximal value of the \fnal v2\nincreases by up to 25%.\nClearly, momentum rapidities y > 1 are responsible\nfor \u0001px&pTand the average momentum asymmetries\nlarger than zero. The three initial pT-distributions show\nsimilar behavior, only the maximal value of v2increases\nwith increasing n. Finally we turn to the di\u000berential v2.\nUsing magnetic \feld parametrization 1, we show in Fig. 6\nthe resulting v2(pT) without the use of the formation\ntime, for mid- and forward rapidity and all three initial\nstate parametrizations. The v2can be (temporary) up\n-0.2 0 0.2 0.4 0.6 0.8 1\n 0  0.1  0.2  0.3  0.4  0.5v2(pT)\npT [GeV]eB=4 mπ2\nt=0.3 fm/c\nno formation timesn=2:  |y|<0.5\n1<|y|<3\nn=3:  |y|<0.5\n1<|y|<3\nn=4:  |y|<0.5\n1<|y|<3Figure 6. The pT-di\u000berential v2in a collisionless toy model for\ninitial power law spectra with exponent n= 2;3;4. Here we\ndo not assign formation times. We show results for forward-\nand midrapidity with magnetic \feld parametrization 1.\n-0.2 0 0.2 0.4 0.6 0.8 1\n 0  1  2  3  4  5v2(pT)\npT [GeV]eB=4 mπ2\nt=0.3 fm/c\nformation timesn=2:  |y|<0.5\n1<|y|<3\nn=3:  |y|<0.5\n1<|y|<3\nFigure 7. Same as Fig. 6, but here formation times had been\nassigned. For the shown snapshot at t= 0:3 fm=c, the mini-\nmum occurring momentum at midrapidity is pT\u00190:66 GeV.\nto 80 %. It is larger for forward rapidity. In Fig. 7 we\nshow the result when the formation time was taken into\naccount. This results in deleting all relevant interactions\namong the \feld and the particles, and the v2remains\nzero.\nIV. THE EFFECT OF COLLISIONS\nNext we want to consider the e\u000bect of particle colli-\nsions. To this end we employ the 3+1-dimensional trans-\nport approach BAMPS (Boltzmann Approach to Multi-\nParton Scatterings), which solves the relativistic Boltz-\nmann equation by Monte-Carlo techniques [29, 30] for\nmassless on-shell quarks and gluons1. The Boltzmann\n1corresponding to an ideal equation of state5\n101102103104105106\n 0.01  0.1dNquarks/(2πpTdpTdy) [GeV-2]\npT [GeV]|y|<0.5eBy=4 mπ2\narbitrary normalisationinitial, n=2\nno collisions, t=0.3 fm/c\ncollisions, t=0.3 fm/c\nno field, collisions, t=0.3 fm/c\nFigure 8. Transverse momentum spectra of quarks under the\nin\ruence of a magnetic \feld in a free streaming and a col-\nlisional medium. We use an arbitrary number of particles\nand magnetic \feld parametrization 1. Particles collide with\nconstant isotropic cross sections, \u001btot= 10 mb.\nequation is ideally suited to study thermalization and\nisotropization processes [29, 31] and the electromagnetic\n\felds enter by an external force term. With the phase-\nspace distribution function fi(x;k)\u0011fi\nkfor particle\nspeciesi, the Boltzmann equation reads\nk\u0016@\n@x\u0016fi\nk+k\u0017qiF\u0016\u0017@\n@k\u0016fi\nk=NspeciesX\nj=1Cij(x\u0016;k\u0016);(6)\nwhereCijis the collision term, and qithe electric charge.\nThe \feld strength tensor F\u0016\u0017=E\u0016u\u0017\u0000E\u0017u\u0016\u0000B\u0016\u0017,\nwithB\u00160=B0\u0017= 0,Bij=\u0000\u000fijkBkandE\u0016= (0;~E),\nintroduces the electromagnetic forces to the charged par-\nticles [32]. For the BAMPS simulations we include 3\n\ravors of light quarks, antiquarks and gluons. Space is\ndiscretized in small cells with volume \u0001 Vand particles\nscatter and propagate within timesteps \u0001 t. In each cell,\nthe probability for binary/inelastic scattering is\nP22=23=\u001btot;22=23(s)\nNtestvrel\u0001t\n\u0001V; (7)\nwhere\u001btot(s) is the (in general Mandelstam sdependent)\ntotal cross section and vrelthe relative velocity. The in-\nelastic backreaction works similar. In the simplest case,\nwe employ constant and isotropic cross sections, how-\never, BAMPS features binary and radiative perturbative\nQuantum Chromo Dynamic (pQCD) cross sections (see,\ne.g., Ref. [33, 34]) and running coupling \u000bs(Q2), which\nis evaluated at the momentum transfer Q2of the respec-\ntive scattering process [35]. For the purpose of this study\nhere, there is no qualitative di\u000berence when employing\npQCD cross sections, so we restrict ourselves to constant\nand isotropic scattering. As a new feature, we include\nthe electromagnetic force, which within the Monte-Carlo\nframework reduces to the additional change of the parti-\n-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6\n0.01  0.1  0.2  0.3  0.4  0.5quark v2\npT [GeV]σtot=10 mb, |y|<0.5no field, t=0.3 fm/c\nno field, t=2 fm/c   \nparam 1, t=0.3 fm/c\nparam 1, t=2 fm/c   \nPHENIXFigure 9. The pT-di\u000berential v2from BAMPS for the ultrasoft\npTrange with and without magnetic \feld. The initial state\nis equivalent to Sec. II A, with exponent n= 2. The initial\ngeometry is equivalent to an impact parameter of b= 8:5fm.\nWe ignore formation times here. For a rough comparison we\nshow data from PHENIX [37] (unidenti\fed charged hadrons,psNN= 200 GeV, 20% \u000060% centrality,j\u0011j<0:35).\ncle momenta (for every computational timestep) by\nd~ki= \u0001tFLorenz = \u0001tqi\u0010\n~E+~ v\u0002~B\u0011\n: (8)\nAs mentioned before, we set ~E= 0. It is clear that\nthe propagation of \felds (by retarded Li\u0013 enard-Wiechert\npotentials) generated by moving quarks would re\fne\nthe picture, this will be done in a forthcoming publi-\ncation. Nevertheless, electric currents appear by default\nin BAMPS, and the electric conductivity of the matter\nis built in naturally [26]. We use the same initial state\nin momentum space in BAMPS as in the simple model\nfrom Sec. II, and use smooth a Glauber Monte Carlo dis-\ntribution of particle positions. Here we use an impact pa-\nrameter ofb= 8:5 fm. The particle numbers are roughly\nequal to simulations performed in earlier studies using\nBAMPS for Au+Au collisions atpsNN= 200 GeV [34{\n36]. Flavors for gluons and quarks ( Nf= 3) are sampled\nrandomly with probabilities Pg= 16=52;Pq= 36=52.\nWe note that this setup is certainly rough, but it should\nsu\u000ece for our purpose of an optimistic upper estimate\nof the \\Hall current\" to the anisotropic \row. We\nshow in Fig. 8, how the spectra are a\u000bected by colli-\nsions. Here we see, that in the viscous case (including\ncollisions,\u001btot= 10 mb), the spectra in\ruenced by the\nmagnetic \feld for momenta pT&1 GeV are very close to\nthe \feld free case. Without \felds, the medium thermal-\nizes at timescales of 0 :5\u00181 fm=c (see Ref.[29]). Again,\nin the region of 0 :01.pT=GeV.0:1 the spectra are\nenhanced compared to the \feld free spectra. The green\ndashed line shows the collisionless result, which is close\nto the initial power law for larger momenta, pT&1 GeV,\nand enhanced in the soft region.\nIn Fig. 9 we show the pT-di\u000berential v2of light quarks\nfrom BAMPS, and turn the magnetic \feld on and o\u000b.6\n-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6\n0.01  0.1  0.2  0.3  0.4  0.5quark v2\npT [GeV]σtot=10 mb, |y|<0.5param 1, t=0.3 fm/c\nparam 1, t=2 fm/c   \nparam 2, t=0.3 fm/c\nparam 2, t=2 fm/c   \nPHENIX\nFigure 10. Same as Fig. 9, but here we compare the magnetic\n\feld parametrizations 1 and 2.\nClearly, the \feld causes a strong momentum anisotropy\nbelowpT\u00180:1 GeV, but has hardly any e\u000bect above\nthis softpT-range. For comparison we plot the softest\npoints of experimentally measured unidenti\fed charged\nparticle \row from PHENIX [37]. Unfortunately they are\nstill measured at such high transverse momenta, that a\ndetection of the presented magnetic \feld e\u000bect is unlikely\nat present.\nWe see in comparison with Fig. 6, which shows the col-\nlisionless result, that the collisions damp the v2(about\n20% lower v2at aroundpT= 40\u000060 MeV). Here we\nshow results with parametrization 1, which switches o\u000b\nthe \feld at t= 0:3 fm=c. After that time, the collisions\nisotropize this initial \row, such that after t= 2 fm=c it\nis around 0:34 and the maximum is pushed to even lower\npT. In Fig. 10 we compare the e\u000bect of the two mag-\nnetic \feld parametrizations. Parametrization 2, probably\nmore realistic, has a weaker e\u000bect than parametrization\n1. The maximal \row is still about 30%, but, more impor-\ntantly, it is shifted to much lower transverse momenta.\nWe need to recall at this point, that all strong elliptic \row\nsignals appear only, when the formation time of quarks\nis neglected for the interaction among the \feld and the\nparticles. This issue must be further addressed in future.\nWe note, that we ignore for simplicity hadronization and\nthe subsequent hadron gas evolution here, nevertheless,\nelliptic \row on the order of 30% is likely to survive to\nsome degree. This remains subject for future work.\nPhotons are an ideal probe to test e\u000bects throughout\nthe spacetime evolution of the medium, such as magnetic\n\feld induced \row, as they leave the \freball nearly undis-\nturbed, once produced. In an earlier study [38] we have\nimplemented photon production in BAMPS, consistent\nwith leading order rates. Here, we make use of the 2 $2\nphoton production method from Ref. [38]. At very low\npT(where all interesting magnetic e\u000bects happen), the\nmicroscopic photon production processes for collisions of\ntwo lowpTpartons will have typical Mandelstam vari-\nables at magnitudes, where the concept of perturbative\n-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n0.01  0.1  0.2v2\npT [GeV]|y|<0.5, Param 1\nn=3, σtot=10 mbQuarks\nPhotonsFigure 11. Photon v2as function of pTcompared to quark v2\nfor magnetic \feld parametrization 1. Here we use an initial\nstate with power law exponent n= 3. Photons and quarks\nare evaluated at midrapidity at t= 2 fm=c.\nQCD methods is questionable ( s.\u00032\nQCD). Neverthe-\nless, we allow photons to be produced as we are mainly\ninterested in the non-equilibrium e\u000bect of photon pro-\nduction from a \rowing quark medium. In Fig. 9 we show\nthev2(pT) of produced photons with that of the quarks\nat timet= 2 fm=c. Photons are produced during the\nwhole collision, and observables are thus always space-\ntime averaged, weighted by the production yield. In the\nbeginning of the collision, the medium is dense and the\nenergydensity is high, so many photons are produced, but\n(in our simpli\fed initial state) the \row is zero. Later,\nthe photons inherit some of the \row, but their rate de-\ncreases steadily. This is the reason, why the observed\nphoton \row is smaller than the pure quark \row. Below\npT.0:1 GeV the photon \row is enhanced. It will be\nchallenging to measure such an e\u000bect, considering that\nrecent measurements of (direct) photon \row [39, 40] ex-\ntend down to pT= 0:4 GeV (PHENIX)/ pT= 1 GeV\n(ALICE).\nV. SUMMARY\nWe have shown how the Lorentz force in heavy-ion\ncollisions can a\u000bect observables. To this end, we have\nassumed two simple parametrizations for an external,\nhomogeneous magnetic \feld, which is produced by the\nfast spectator nucleons. We investigate a free streaming,\nand a viscous medium (with collisions), employing the\npartonic transport simulation BAMPS. We use a sim-\nple boost invariant initial state, assuming a power-law\nin the transverse momentum distribution, and a periph-\neral Monte-Carlo Glauber geometry of the overlap zone.\nWe have shown that the magnetic \feld will generate a\nstrong elliptic \row only at very small transverse momenta\ndue to the Larmor movement of the charged particles.\nIn this very soft region, also the transverse momentum7\nspectra are enhanced. We show that collisions will wash\nout both the enhancement of the spectra and the elliptic\n\row. However, the \row is still quite large, such that it\ncould be measured, if experiments had access to ultrasoft\ntransverse momenta. Assuming an initial formation time\nof the particles, within which the magnetic \feld can not\nact, all strong e\u000bects are deleted. The interaction of clas-\nsical \felds and unformed particles is however a di\u000ecult\ntheoretical problem and must be clari\fed further. This\nstudy can be extended in several ways. Apart from other\nobservables, electric \felds, stemming from the Faraday\nlaw, might also play a role. A logical next step would be\na realistic space dependence of the external \felds, and,\nin the long run, a full spacetime evolution of retarded\n\felds including induction e\u000bects (similar to Ref. [13]).\nWe emphasize, that the present study should only give\nan order-of-magnitude estimate of what can be expected\nfrom the magnetic Lorentz force (\\Hall e\u000bect\") for light\nquarks. Apart of the experimental challenge, there might\nbe other consequences. Especially \fnal spectra of tomo-graphic probes like photons or dileptons will inherit in-\nformation of this strongly \rowing but ultrasoft region.\nTo get an idea of this e\u000bect we have shown that photons\ninherit a fraction of the elliptic \row from the quarks at\nnearly the same ultralow transverse momenta.\nACKNOWLEDGEMENTS\nM.G. is grateful to Tsinghua university in Beijing for\ntheir hospitality and acknowledges the support from the\n\\Helmhotz Graduate School for Heavy Ion research\".\nThe authors are grateful to the Center for Scienti\fc Com-\nputing (CSC) Frankfurt for the computing resources.\nThis work was supported by the Helmholtz Interna-\ntional Center for FAIR within the framework of the\nLOEWE program launched by the State of Hesse. XZ\nwas supported by the MOST, the NSFC under Grants\nNo. 2014CB845400, No. 11275103, No. 11335005, No.\n11575092.\n[1] I. Arsene et al. (BRAHMS), Nucl. Phys. A757 , 1 (2005),\narXiv:nucl-ex/0410020.\n[2] K. Adcox et al. 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The observed high-frequency instability is mitigated using methods\nincluding an electromagnetic solver with tunable coe \u000ecients, its extension to accomodate Perfectly Matched Layers\nand Friedman’s damping algorithms, as well as an e \u000ecient large bandwidth digital filter. It is shown that choosing\nthe frame of the wake as the frame of reference allows for higher levels of filtering and damping than is possible in\nother frames for the same accuracy. Detailed testing also revealed serendipitously the existence of a singular time step\nat which the instability level is minimized, independently of numerical dispersion, thus indicating that the observed\ninstability may not be due primarily to Numerical Cerenkov as has been conjectured. The techniques developed\nfor Cerenkov mitigation prove nonetheless to be very e \u000ecient at controlling the instability. Using these techniques,\nagreement at the percentage level is demonstrated between simulations using di \u000berent frames of reference, with\nspeedups reaching two orders of magnitude for a 0.1 GeV class stages. The method then allows direct and e \u000ecient\nfull-scale modeling of deeply depleted laser-plasma stages of 10 GeV-1 TeV for the first time, verifying the scaling of\nplasma accelerators to very high energies. Over 4, 5 and 6 orders of magnitude speedup is achieved for the modeling\nof 10 GeV , 100 GeV and 1 TeV class stages, respectively.\nKeywords: laser wakefield acceleration, particle-in-cell, plasma simulation, special relativity, frame of reference,\nboosted frame\nOctober 30, 2018arXiv:1009.2727v2  [physics.acc-ph]  15 Sep 2010Contents\n1 Introduction 2\n2 Theoretical speedup dependency with the frame boost 4\n2.1 Estimated speedup for 0.1-100 GeV stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n3 Input and output to and from a boosted frame simulation 6\n3.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3.1.1 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3.1.2 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.2 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n4 High frequency instability and Numerical Cerenkov 9\n4.1 Wideband lowpass digital filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n4.2 Tunable solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n4.2.1 Numerical dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n4.2.2 Current deposition and Gauss’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n4.3 Friedman adjustable damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n5 Application to the modeling of laser wakefield acceleration 19\n5.1 Scaled 10 GeV stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n5.1.1 Using standard numerical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n5.1.2 E \u000bect of filtering, solver with adjustable dispersion and damping . . . . . . . . . . . . . . . . 26\n5.2 Full scale 10 GeV class stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n5.2.1 Simulations in 2-1 /2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n5.2.2 Simulations in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n5.3 E \u000bects of numerical parameters on the observed instability . . . . . . . . . . . . . . . . . . . . . . . 31\n5.3.1 E \u000bects of spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\n5.3.2 E \u000bects of time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\n5.3.3 E \u000bects of field gathering procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\n5.4 Full scale 100 GeV - 1 TeV class stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37\n6 Conclusion and outlook 38\n7 Appendix I: One dimensional analysis of the CK solver 38\n8 Appendix II: Perfectly Matched Layer 39\n9 Acknowledgments 41\n1. Introduction\nLaser driven plasma waves o \u000ber orders of magnitude increases in accelerating gradient over standard accelerating\nstructures [2] (which are limited by electrical breakdown), thus holding the promise of much shorter particle accelera-\ntors [3]. High quality electron beams of energy up-to 1 GeV have been produced in just a few centimeters [4, 5, 6, 7],\nwith 10 GeV stages being planned as modules of a high energy collider [8].\nAs the laser propagates through a plasma, it displaces electrons while ions remain essentially static, creating a\npocket of positive charges that the displaced electrons rush to fill. The resulting coherent periodic motion of the\nelectrons oscillating around their original position creates a wake with periodic structure following the laser. The\nalternate concentration of positive and negative charges in the wake creates very intense electric fields. An electron (or\npositron) beam injected with the right phase can be accelerated by those fields to high energy in a much shorter distance\nthan is possible in conventional particle accelerators. The e \u000eciency and quality of the acceleration is governed by\n2several factors which require precise three-dimensional shaping of the plasma column, as well as the laser and particle\nbeams, and understanding of their evolution.\nComputer simulations have had a profound impact on the design and understanding of past and present exper-\niments [9], with accurate modeling of the wake formation and beam acceleration requiring fully kinetic methods\n(usually Particle-In-Cell) with large computational resources due to the wide range of space and time scales involved\n[10, 11]. For example, modeling 10 GeV stages for the LOASIS (LBNL) BELLA proposal [12] demanded as many\nas 5,000 processor hours for a one-dimension simulation on a NERSC supercomputer [13]. Various reduced models\nhave been developed to allow multidimensional simulations at manageable computational costs: fluid approximation\n[14], quasistatic approximation [15, 16, 17], laser envelope models [16], scaled parameters [18, 19]. However, the\nvarious approximations that they require result in a narrower range of applicability. As a result, even using several\nmodels concurrently does not usually provide a complete description. For example, scaled simulations of 10 GeV\nLPA stages do not capture correctly some essential transverse physics, e.g. the laser and beam betatron motion, which\ncan lead to inaccurate beam emittance (a measure of the beam quality). An envelope description can capture these\ne\u000bects correctly at full scale for the early propagation through the plasma but can fail as the laser spectrum broadens\ndue to energy depletion as it propagates further in the plasma.\nAn alternative approach allows for orders of magnitude speedup of simulations, whether at full or reduced scale,\nvia the proper choice of a reference frame moving near the speed of light in the direction of the laser [1]. It does\nso without alteration to the fundamental equations of particle motion or electrodynamics, provided that the high-\nfrequency part of the light emitted counter to the direction of propagation of the beam can be neglected. This approach\nexploits the properties of space and time dilation and contraction associated with the Lorentz transformation. As\nshown in [1], the ratio of longest to shortest space and time scales of a system of two or more components crossing at\nrelativistic velocities is not invariant under such a transformation (a laser crossing a plasma is just such a relativistic\ncrossing). Since the number of computer operations (e.g., time steps), for simulations based on formulations from first\nprinciples, is proportional to the ratio of the longest to shortest time scale of interest, it follows that such simulations\nwill eventually have di \u000berent computer runtimes, yet equivalent accuracy, depending solely upon the choice of frame\nof reference.\nThe procedure appears straightforward: identify the frame of reference which will minimize the range of space\nand/or time scales and perform the calculation in this frame. However, several practical complications arise. First,\nthe input and output data are usually known from, or compared to, experimental data. Thus, calculating in a frame\nother than the laboratory entails transformations of the data between the calculation frame and the laboratory frame.\nSecond, while the fundamental equations of electrodynamics and particle motion are written in a covariant form, the\nnumerical algorithms that are derived from them may not retain this property, and calculations in frames moving at\ndi\u000berent velocities may not be successfully conducted with the use of the exact same algorithms. For example, it\nwas shown in [20] that calculating the propagation of ultra-relativistic charged particle beams in an accelerator using\nstandard Particle-In-Cell techniques lead to large numerical errors, which were fixed by developing a new particle\npusher. The modeling of a laser plasma accelerator (LPA) stage in a boosted frame involves the fully electromagnetic\nmodeling of a plasma propagating at near the speed of light, for which Numerical Cerenkov [21, 22] is a potential\nissue. Third, electromagnetic calculations that include wave propagation will include waves propagating forward and\nbackward in any direction. For a frame of reference moving in the direction of the accelerated beam (or equivalently\nthe wake of the laser), waves emitted by the plasma in the forward direction expand while the ones emitted in the\nbackward direction contract, following the properties of the Lorentz transformation. If one is to resolve both forward\nand backward propagating waves emitted from the plasma, there is no gain in selecting a frame di \u000berent from the\nlaboratory frame. However, the physics of interest for a laser wakefield is the laser driving the wake, the wake, and\nthe accelerated beam. Backscatter is weak in the short-pulse regime, and does not interact as strongly with the beam\nas do the forward propagating waves which stay in phase for a long period. It is thus often assumed that the backward\npropagating waves can be neglected in the modeling of LPA stages. The accuracy of this assumption is shown by\ncomparison between explicit codes which include both forward and backward waves and envelope or quasistatic\ncodes which neglect backward waves [10, 19, 23].\nAfter the idea and basic scaling for performing simulations of LPA in a Lorentz boosted frame were published\nin [1], there have been several reports of the application of the technique to various regimes of LPA [13, 24, 25, 26,\n11, 27, 28]. Speedups varying between several to a few thousands were reported with various levels of accuracy in\nagreement between simulations performed in a Lorentz boosted frames and in a laboratory frame. High-frequency\n3instabilities were reported to develop in 2D or 3D calculations, that were limiting the velocity of the boosted frame\nand thus the attainable speedup [29, 27, 28].\nIn this paper, we present numerical techniques that were implemented in the Particle-In-Cell code Warp [30]\nfor mitigating the numerical Cerenkov instability, including a solver with tunable coe \u000ecients, and show that these\ntechniques are e \u000bective for suppressing the high frequency instability observed in boosted frame simulations. A\ndetailed study of the application of these techniques to the simulations of downscaled LPA stages reveals that choosing\nthe frame of the wakefield as the reference frame allows for more aggressive application of the standard techniques\nmitigating numerical Cerenkov, than is possible in laboratory frame simulations. It is shown that the instability that\ndevelops with high-boost frames is well controlled, allowing for the first time 2D and 3D simulations of LPA in the\nwakefield frame, for 100 GeV and 1 TeV class stages, achieving the maximum theoretical speedups of over 105and\n106respectively.\nThis paper is organized as follows. The theoretical speedup expected for performing the modeling of a LPA stage\nin a boosted frame is derived in Section 2. Section 3 addresses the issue of input and output of data in a boosted frame.\nHigh frequency instability issues and remedies are presented in Section 4. These techniques enable accurate modeling\nof 0.1 GeV-1 TeV LPA stages. Stage modeling results are presented in section 5, and observed speedup is contrasted\nto the theoretical speedup of section 2.\n2. Theoretical speedup dependency with the frame boost\nThe obtainable speedup is derived as an extension of the formula that was derived in [1], taking in addition into\naccount the group velocity of the laser as it traverses the plasma. In [1], the laser was assumed to propagate at the\nvelocity of light in vacuum during the entire process, which is a good approximation when the relativistic factor of\nthe frame boost \ris small compared to the relativistic factor of the laser wake \rwin the plasma. The expression is\ngeneralized here to higher values of \r, for which the actual group velocity of the wake in the plasma must be taken\ninto account. We shall show that for a 10 GeV class LPA stage, the maximum attainable speedup is above four orders\nof magnitude.\nAssuming that the simulation box is a fixed number of plasma periods long, which implies the use (which is\nstandard) of a moving window following the wake and accelerated beam, the speedup is given by the ratio of the time\ntaken by the laser pulse and the plasma to cross each other, divided by the shortest time scale of interest, that is the\nlaser period. Assuming for simplicity that the wake propagates at the group velocity of plane waves in a uniform\nplasma of density ne, the group velocity of the wake is given by\nvw=c=\fw=0BBBBB@1+!2\np\n!21CCCCCA\u00001=2\n(1)\nwhere!p=p\n(nee2)=(\u000f0me) is the plasma frequency, !=2\u0019c=\u0015is the laser frequency, \u000f0is the permittivity of\nvacuum, cis the speed of light in vacuum, and eandmeare respectively the charge and mass of the electron.\nIn the simulations presented herein, the runs are stopped when the last electron beam macro-particle exits the\nplasma, and a measure of the total time of the simulation is given by\nT=L+\u0011\u0015p\nvw\u0000vp(2)\nwhere\u0015p\u00192\u0019c=!pis the wake wavelength, Lis the plasma length, vwandvp=\fpcare respectively the velocity\nof the wake and of the plasma relative to the frame of reference, and \u0011is an adjustable parameter for taking into\naccount the fraction of the wake which exited the plasma at the end of the simulation. The numerical cost Rtscales as\nthe ratio of the total time to the shortest timescale of interest, which is the inverse of the laser frequency, and is thus\ngiven by\nRt=Tc\n\u0015=\u0010\nL+\u0011\u0015p\u0011\n\u0010\n\fw\u0000\fp\u0011\n\u0015(3)\n4In the laboratory, vp=0 and the expression simplifies to\nRlab=Tc\n\u0015=\u0010\nL+\u0011\u0015p\u0011\n\fw\u0015(4)\nIn a frame moving at \fc, the quantities become\n\u0015\u0003\np=\u0015p=\u0002\r(1\u0000\fw\f)\u0003(5)\nL\u0003=L=\r (6)\n\u0015\u0003=\r(1+\f)\u0015 (7)\n\f\u0003\nw=(\fw\u0000\f)=(1\u0000\fw\f) (8)\nv\u0003\np=\u0000\fc (9)\nT\u0003=L\u0003+\u0011\u0015\u0003\np\nv\u0003w\u0000v\u0003p(10)\nR\u0003\nt=T\u0003c\n\u0015\u0003=\u0010\nL\u0003+\u0011\u0015\u0003\np\u0011\n\u0000\f\u0003w+\f\u0001\u0015\u0003(11)\nwhere\r=1=p\n1\u0000\f2.\nThe expected speedup from performing the simulation in a boosted frame is given by the ratio of RlabandR\u0003\nt\nS=Rlab\nR\u0003\nt=(1+\f)\u0010\nL+\u0011\u0015p\u0011\n(1\u0000\f\fw)L+\u0011\u0015p(12)\nAssuming that \r<<\r w, and that\fw\u00191 (which is a valid approximation for most practical cases of interest), this\nexpression is consistent with the expression derived in [1] for the LPA case which states that R\u0003\nt=\u000bRt=(1+\f)with\n\u000b=(1\u0000\f+l=L)=(1+l=L), where lis the laser length which is generally proportional to \u0011\u0015p, and S=Rt=R\u0003\nT.\nThe linear theory predicts that for the intense lasers (a &1) typically used for acceleration, the laser depletes\nits energy over approximately the same length Ld=\u00153\np=2\u00152over which the particles dephase from the wake [2].\nAcceleration is compromised beyond Ldand in practice, the plasma length is proportional to the dephasing length, i.e.\nL=\u0018Ld. In most cases, \r2\nw>>1, which allows the approximations \fw\u00191\u0000\u00152=2\u00152\np, and L=\u0018\u00153\np=2\u00152\u0019\u0018\r2\nw\u0015p=2>>\n\u0011\u0015p, so that Eq.(12) becomes\nS=(1+\f)2\r2 \u0018\r2\nw\n\u0018\r2w+(1+\f)\r2(\u0018\f=2+2\u0011)(13)\nFor low values of \r, i.e. when\r<<\r w, Eq.(13) reduces to\nS\r<<\r w=(1+\f)2\r2(14)\nConversely, if \r!1 , Eq.(13) becomes\nS\r!1=4\n1+4\u0011=\u0018\r2\nw (15)\nFinally, in the frame of the wake, i.e. when \r=\rw, assuming that \fw\u00191, Eq.(13) gives\nS\r=\rw\u00192\n1+2\u0011=\u0018\r2\nw (16)\nSince\u0011and\u0018are of order unity, and the practical regimes of most interest satisfy \r2\nw>>1, the speedup that is\nobtained by using the frame of the wake will be near the maximum obtainable value given by Eq.(15).\nNote that without the use of a moving window, the relativistic e \u000bects that are at play in the time domain would\nalso be at play in the spatial domain, as shown in [1], and the \r2scaling would transform to \r4. In the frame of the\n5wake, there is no need of the moving window, thus simplifying the procedure, while in a frame traveling faster than\nthe wake in the laboratory, a moving window propagating in the backward direction is needed. However, the scaling\nshows that there would be very little gain in doing the latter.\n2.1. Estimated speedup for 0.1-100 GeV stages\nFormula (13) is used to estimate the speedup for the calculations of 100 MeV , 1 GeV , 10 GeV and 100 GeV class\nstages, assuming a laser wavelength \u0015=0:8\u0016m. Using parameters and scaling laws from [18], the corresponding\ninitial plasma densities neare respectively 1019cc, 1018cc, 1017cc and 1016cc, while the plasma lengths Lare 1.5 mm,\n4.74 cm, 1.5 m, and 47.4 m, with \u0018\u00191:63. For these values, the wake wavelengths \u0015pare respectively 10 :6\u0016m,\n33:4\u0016m, 106:\u0016m, 334:\u0016m, and relativistic factors \rware 13:2, 41:7, 132 and 417 :In the simulations presented in this\npaper, the beam is injected near the end of the wake period (first ”bucket”). In first approximation, the beam has\npropagated through about half a wake period to reach full acceleration, and we set \u0011\u00190:5. For a beam injected into\nthenthbucket,\u0011would be set to n\u00001=2. If positrons were considered, they would be injected half a wake period\nahead of the location of the electrons injection position for a given period, and one would have \u0011=n\u00001. For the\nparameters considered here, L\u0019\u0015p=\r2\nw, and (15) gives S\r!1\u00192\r2\nw.\nFigure 1: Speedup versus relativistic factor of the boosted frame from Eq.(13) for 100 MeV - 100 GeV LPA class stages.\nThe speedup versus the relativistic factor of the boosted frame \ris plotted in Fig. 1. As expected, for low values\nof\r, the speedup scales as (14), and asymptotes to a value slightly lower than 2 \r2\nwfor large values of \r. It is of interest\nto note that the qualitative behavior is identical to the one obtained in [1] (see Fig. 1 and accompanying analysis) in\nthe analysis of the crossing of two rigid identical beams, confirming the generality of the generic analysis presented\nin [1]. For a 100 GeV class stage, the maximum estimated speedup is as large as 300,000.\n3. Input and output to and from a boosted frame simulation\nThis section describes the procedures that have been implemented in the Particle-In-Cell framework Warp [30] to\nhandle the input and output of data between the frame of calculation and the laboratory frame. Simultaneity of events\nbetween two frames is valid only for a plane that is perpendicular to the relative motion of the frame. As a result, the\ninput/output processes involve the input of data (particles or fields) through a plane, as well as output through a series\nof planes, all of which are perpendicular to the direction of the relative velocity between the frame of calculation and\nthe other frame of choice.\n63.1. Input\n3.1.1. Particles\nParticles are launched through a plane using a technique which applies to many calculations in a boosted frame, in-\ncluding LPA, and is illustrated using the case of a positively charged particle beam propagating through a background\nof cold electrons in an assumed continuous transverse focusing system, leading to a growing transverse instability [1].\nIn the laboratory frame, the electron background is initially at rest and a moving window is used to follow the beam\nprogression. Traditionally, the beam macroparticles are initialized all at once in the window, while background elec-\ntron macroparticles are created continuously in front of the beam on a plane that is perpendicular to the beam velocity.\nIn a frame moving at some fraction of the beam velocity in the laboratory frame, the beam initial conditions at a given\ntime in the calculation frame are generally unknown and one must initialize the beam di \u000berently. However, it can be\ntaken advantage of the fact that the beam initial conditions are often known for a given plane in the laboratory, either\ndirectly, or via simple calculation or projection from the conditions at a given time. Given the position and velocity\nfx;y;z;vx;vy;vzgfor each beam macroparticle at time t=0 for a beam moving at the average velocity vb=\fbc(where\ncis the speed of light) in the laboratory, and using the standard synchronization ( z=z0=0 att=t0=0) between the\nlaboratory and the calculation frames, the procedure for transforming the beam quantities for injection in a boosted\nframe moving at velocity \fcin the laboratory is as follows (the superscript0relates to quantities known in the boosted\nframe while the superscript\u0003relates to quantities that are know at a given longitudinal position z\u0003but di \u000berent times\nof arrival):\n1. project positions at z\u0003=0 assuming ballistic propagation\nt\u0003=(z\u0000¯z)=vz (17)\nx\u0003=x\u0000vxt\u0003(18)\ny\u0003=y\u0000vyt\u0003(19)\nz\u0003=0 (20)\nthe velocity components being left unchanged,\n2. apply Lorentz transformation from laboratory frame to boosted frame\nt0\u0003=\u0000\rt\u0003(21)\nx0\u0003=x\u0003(22)\ny0\u0003=y\u0003(23)\nz0\u0003=\r\fct\u0003(24)\nv0\u0003\nx=v\u0003\nx\n\r(1\u0000\f\fb)(25)\nv0\u0003\ny=v\u0003\ny\n\r(1\u0000\f\fb)(26)\nv0\u0003\nz=v\u0003\nz\u0000\fc\n1\u0000\f\fb(27)\nwhere\r=1=p\n1\u0000\f2. With the knowledge of the time at which each beam macroparticle crosses the plane into\nconsideration, one can inject each beam macroparticle in the simulation at the appropriate location and time.\n3. synchronize macroparticles in boosted frame, obtaining their positions at a fixed t0(=0) which is before any\nparticle is injected\nz0=z0\u0003\u0000¯v0\u0003\nzt0\u0003(28)\nThis additional step is needed for setting the electrostatic or electromagnetic fields at the plane of injection.\nIn a Particle-In-Cell code, the three-dimensional fields are calculated by solving the Maxwell equations (or\nstatic approximation like Poisson, Darwin or other [20]) on a grid on which the source term is obtained from\n7the macroparticles distribution. This requires generation of a three-dimensional representation of the beam\ndistribution of macroparticles at a given time before they cross the injection plane at z0\u0003. This is accomplished\nby expanding the beam distribution longitudinally such that all macroparticles (so far known at di \u000berent times\nof arrival at the injection plane) are synchronized to the same time in the boosted frame. To keep the beam\nshape constant, the particles are ”frozen” until they cross that plane: the three velocity components and the\ntwo position components perpendicular to the boosted frame velocity are fixed, while the remaining position\ncomponent is advanced at the average beam velocity. As particles cross the plane of injection, they become\nregular ”active” particles with full 6-D dynamics.\nFigure 2: (top) Snapshot of a particle beam “frozen” (grey spheres) and “active” (colored spheres) macroparticles traversing the injection plane (red\nrectangle). (bottom) Snapshot of the beam macroparticles (colored spheres) passing through the background of electrons (dark brown streamlines)\nand the diagnostic stations (red rectangles). The electrons, the injection plane and the diagnostic stations are fixed in the laboratory plane, and are\nthus counterpropagating to the beam in a boosted frame.\nFigure 2 (top) shows a snapshot of a beam that has passed partly through the injection plane. As the frozen beam\nmacroparticles pass through the injection plane (which moves opposite to the beam in the boosted frame), they are\nconverted to “active” macroparticles. The charge or current density is accumulated from the active and the frozen\nparticles, thus ensuring that the fields at the plane of injection are consistent.\n3.1.2. Laser\nSimilarly to the particle beam, the laser is injected through a plane perpendicular to the axis of propagation of the\nlaser (by default z). The electric field E?that is to be emitted is given by the formula\nE?(x;y;t)=E0f(x;y;t)sin\u0002!t+\u001e(x;y;!)\u0003(29)\nwhere E0is the amplitude of the laser electric field, f(x;y;t)is the laser envelope, !is the laser frequency, \u001e(x;y;!)\nis a phase function to account for focusing, defocusing or injection at an angle, and tis time. By default, the laser\nenvelope is a three dimensional gaussian of the form\nf(x;y;t)=e\u0000(x2=2\u001b2\nx+y2=2\u001b2\ny+c2t2=2\u001b2\nz)(30)\nwhere\u001bx,\u001byand\u001bzare the dimensions of the laser pulse; or it can be defined arbitrarily by the user at runtime. If\n\u001e(x;y;!)=1, the laser is injected at a waist and parallel to the axis z.\n8If, for convenience, the injection plane is moving at constant velocity \fsc, the formula is modified to take the\nDoppler e \u000bect on frequency and amplitude into account and becomes\nE?(x;y;t)=(1\u0000\fs)E0f(x;y;t)sin\u0002(1\u0000\fs)!t+\u001e(x;y;!)\u0003: (31)\nThe injection of a laser of frequency !is considered for a simulation using a boosted frame moving at \fcwith\nrespect to the laboratory. Assuming that the laser is injected at a plane that is fixed in the laboratory, and thus moving\nat\fs=\u0000\fin the boosted frame, the injection in the boosted frame is given by\nE?\u0000x0;y0;t0\u0001=(1\u0000\fs)E0\n0f\u0000x0;y0;t0\u0001sin\u0002(1\u0000\fs)!0t0+\u001e\u0000x0;y0;!0\u0001\u0003(32)\n=(E0=\r)f\u0000x0;y0;t0\u0001sin\u0002!t0=\r+\u001e\u0000x0;y0;!0\u0001\u0003(33)\nsince E0\n0=E0=!0=!=1=(1+\f)\r.\nThe electric field is then converted into currents that get injected via two dual 2-D arrays of macro-particles, with\none positive and one negative macro-particle per cell in the plane of injection, whose weights and motion are governed\nbyE?(x0;y0;t0). Injecting using these dual arrays of macroparticles o \u000ber the advantages of automatically including\nthe longitudinal component which arise from emitting into a boosted frame, and to verify the discrete Gauss law\nthanks to the use of the Esirkepov current deposition scheme [31].\nThe technique implemented in Warp presents several advantage over other procedures that have been proposed\nelsewhere [13, 28]. In [28], the laser beam is initialized entirely in the computational box, leading to larger boxes\ntransversely in a boosted frame, as the Rayleigh length of the laser shortens and the overall laser pulse radius rises,\neventually o \u000bsetting the benefits of the boosted frame. The transverse broadening of the box is avoided in [13] at the\ncost of a more complicated injection scheme, requiring to launch the laser from all but one faces of the simulation\nbox. The method presented here avoids the caveat of the broadening and retains simplicity with a standard injection\ntechnique through one plane.\n3.2. Output\nSome quantities, e.g. charge, are Lorentz invariant, while others, like dimensions perpendicular to the boost\nvelocity, are the same in the laboratory frame. Those quantities are thus readily available from standard diagnostics\nin the boosted frame calculations. Quantities which do not fall in this category are recorded at a number of regularly\nspaced “stations”, immobile in the laboratory frame, at a succession of time intervals to record data history, or averaged\nover time. A visual example is given on Fig. 2 (bottom). Since the space-time locations of the diagnostic grids in the\nlaboratory frame generally do not coincide with the space-time positions of the macroparticles and grid nodes used for\nthe calculation in a boosted frame, some interpolation is performed at runtime during the data collection process. As\na complement or an alternative, selected particle or field quantities are dumped at regular interval for post-processing.\nThe choice of the methods depends on the requirements of the diagnostics and particular implementations.\n4. High frequency instability and Numerical Cerenkov\nAs reported in [27] and [28], for simulations using a boosted frame at \r\u001510\u000020 (depending on parameters),\na fast growing short wavelength instability was observed developing at the front of the plasma (see Fig. 3). The\npresence and growth rate of the instability was observed to be very sensitive to the resolution (slower growth rate at\nhigher resolution), choice of field solver, and to the amount of damping of high frequencies and smoothing of short\nwavelengths. The instability is always propagating at some angle from the longitudinal axis, and is observed in 2D\nand 3D runs but was never observed in any of the 1D runs performed by the authors. When modeling an LPA setup in\na relativistically boosted frame, the background plasma is traveling near the speed of light and it has been conjectured\n[28] that he observed instability might be caused by numerical Cerenkov. We investigate in this paper whether the\ninstability that is observed in boosted frame simulations of LPA is indeed of numerical Cerenkov type and if the cures\naimed at mitigating numerical Cerenkov are e \u000bective.\nDue to spatial and time discretization of the Maxwell equations, numerical light waves may travel faster or slower\non the computational grid than the actual speed of light in vacuum c, with the magnitude of the e \u000bect being larger at\nshort wavelength, where discretization errors are the largest. When the numerical speed is lower than c, it is possible\n9Figure 3: Snapshot of a surface plot of the longitudinal field from a 2-1 /2D simulation of a full scale 10GeV LPA in a boosted frame at \r=130\n(elevation is proportional to the magnitude of the electric field). The laser is propagating from left to right and the plasma from right to left. A fast\ngrowing short wavelength instability is developing at the front of the plasma.\nfor fast macro-particles to travel faster than the wave modes, leading to numerical Cerenkov e \u000bects that may result\nin instabilities [21, 22, 32, 33, 34]. The e \u000bect was studied analytically and numerically in detail for one-dimensional\nsystems in [32, 33]. Several solutions were proposed: smoothing the current deposited by the macro-particles [21, 32],\ndamping the electromagnetic field [34, 35, 36], solving the Maxwell equations in Fourier space [22], or using a field\nsolver with a larger stencil to provide lower numerical dispersion [34].\nSeveral of the abovementioned techniques to mitigate numerical Cerenkov and high frequency errors have been\nimplemented in Warp. All the simulations presented in this paper employed cubic splines for current deposition and\nelectromagnetic force gathering between the macro-particles and the grid [37], whose beneficial e \u000bects on standard\nLPA PIC simulations have been demonstrated in [38]. In addition, a Maxwell solver with tunable coe \u000ecients was\nimplemented, as well as a damping scheme, and filtering of the deposited current and gathered electromagnetic fields,\nwhich are described in this section. The use of Fourier based Maxwell solvers is not considered in this paper.\n4.1. Wideband lowpass digital filtering\nIt is common practice to apply digital filtering to the charge or current density in Particle-In-Cell simulations, for\nsmoothing or compensation purpose [47]. The most commonly used filter is the three points filter\n\u001ef\nj=\u000b\u001ej+(1\u0000\u000b)\u001ej\u00001+\u001ej+1\n2(34)\nwhere\u001efis the filtered quantity. This filter is called a binomial filter when \u000b=0:5. Assuming \u001e=ejkxand\n\u001ef=g(\u000b;k)ejkx, where gis the filter gain, which is function of the filtering coe \u000ecient\u000band the wavenumber k, we\nfind from (34) that\ng(\u000b;k)=\u000b+(1\u0000\u000b)cos(k\u000ex) (35)\n\u00191\u0000(1\u0000\u000b)(k\u000ex)2\n2+O\u0010\nk4\u0011\n(36)\nFornsuccessive applications of filters of coe \u000ecients\u000b1...\u000bn, the total attenuation Gis given by\nG=nY\ni=1g(\u000bi;k) (37)\n\u00191\u00000BBBBB@n\u0000nX\ni=1\u000bi1CCCCCA(k\u000ex)2\n2+O\u0010\nk4\u0011\n(38)\nIf\u000bn=n\u0000Pn\u00001\ni=1\u000bithen G\u00191+O\u0010\nk4\u0011\n, providing a sharper cuto \u000binkspace. Such step is called a compensation\nstep [47]. For the bilinear filter ( \u000b=1=2), the compensation factor is \u000bc=2\u00001=2=3=2. For a succession of n\napplications of the bilinear factor, it is \u000bc=n=2+1. The gain versus wavelength is plotted in Fig. 4 for the bilinear\n10Figure 4: Gain versus wavelength for the bilinear filter without compensation ( g=g(1=2;k)), with compensation ( g\u0001c3=2=g(1=2;k)\u0001g(3=2;k)),\nand n-pass bilinear filters with compensation ( gn\u0001c\u000bc=g(1=2;k)n\u0001g(\u000bc;k)) for n=4, 20, 50 and 80.\nfilter without compensation ( G=g(1=2;k)), with compensation ( G=g(1=2;k)\u0001g(3=2;k)), and four n-pass bilinear\nfilters with compensation ( G=g(1=2;k)n\u0001g(3=2;k)) for n=4, 20, 50 and 80.\nThe bilinear filter provides complete suppression of the signal at the grid Nyquist wavelength (twice the grid cell\nsize). Suppression of the signal at integers of the Nyquist wavelength can be obtained by using a stride sin the filter\n\u001ef\nj=\u000b\u001ej+(1\u0000\u000b)\u001ej\u0000s+\u001ej+s\n2(39)\nfor which the gain is given by\ng(s;\u000b;k)=\u000b+(1\u0000\u000b)cos(sk\u000ex) (40)\n\u00191\u0000(1\u0000\u000b)(sk\u000ex)2\n2+O\u0010\nk4\u0011\n(41)\nThe gain is plotted in Fig. 5 (top) for four passes bilinear filters with compensation ( G=g(s;1=2;k)4\u0001g(s;3=2;k))\nfor strides s =1 to 4. For a given stride, the gain is given by the gain of the bilinear filter shifted in k space, with the\npole g=0 shifted from \u0015=2=\u000exto\u0015=2s=\u000ex, with additional poles, as given by\nsk\u000ex=arccos\u0012\u000b\n\u000b\u00001\u0013\n(mod 2\u0019) (42)\nThe resulting filter is pass band between the poles, but since the poles are spread at di \u000berent integer values in k space,\na wide band low pass filter can be constructed by combining filters at di \u000berent strides. Examples are given in Fig. 5\n(bottom) for combinations of the filter with stride 1 to 4.\nThe combined filters with strides 2, 3 and 4 have nearly equivalent fall-o \u000bs in gain (in linear scale) to the 20, 50\nand 80 passes of the bilinear filter (see Fig. 6). Yet, the filters with stride need respectively 10, 15 and 15 passes of\na three-point filter while the n-pass bilinear filer need respectively 21, 51 and 81 passes, giving gains of respectively\n2.1, 3.4 and 5.4 in number of operations in favor of the filters with stride.\n11Figure 5: (top) gain for four passes bilinear filters with compensation ( Gs=g(s;1=2;k)4\u0001g(s;3=2;k)) for strides s =1 to 4 linear with (left)\nlinear ordinate (right) logarithmic ordinate; (bottom) gain for four low pass filters combining the G1toG4filters with (left) linear ordinate (right)\nlogarithmic ordinate.\nFigure 6: Comparison between filters with stride and filter s20-80 with (left) linear ordinate (right) logarithmic ordinate.\n124.2. Tunable solver\nIn [39] and [40], Cole introduced an implementation of the source-free Maxwell’s wave equations for narrow-\nband applications based on non-standard finite-di \u000berences (NSFD). In [41], Karkkainen et al adapted it for wideband\napplications. At the Courant limit for the time step and for a given set of parameters, the stencil proposed in [41]\nhas no numerical dispersion along the principal axes, provided that the cell size is the same along each dimension\n(i.e. cubic cells in 3D). The solver from [41] was modified to be consistent with the Particle-In-Cell methodology and\nimplemented in the code Warp, with the ability given to the user of setting the solver adjustable coe \u000ecients, providing\ntunability of the numerical properties of the solver to better fit the requirements of a particular application.\nThe ”Cole-Karkkainnen”’s solver [41] uses a non-standard finite di \u000berence formulation (extended stencil) of the\nMaxwell-Ampere equation. For implementation into a Particle-In-Cell code, the formulation must introduce the\nsource term into Cole-Karkkainen’s source free formulation in a consistent manner. However, modifying the NSFD\nformulation of the Maxwell-Ampere equation so that it includes the source term in a way that is consistent with the cur-\nrent deposition scheme is challenging. To circumvent this problem, Warp implementation departs from Karkkainen’s\nby applying the enlarged stencil on the Maxwell-Faraday equations, which does not contain any source term but\nis formally equivalent to the source-free Maxwell-Ampere equation. Consequently, in Warp’s implementation, the\ndiscretized Maxwell-Ampere equation is the same as in the Yee scheme, and the discretized Maxwell’s equations\nread:\n\u0001tB=\u0000r\u0003\u0002E (43)\n\u0001tE=c2r\u0002B\u0000J\n\u000f0(44)\nr\u0001E=\u001a\n\u000f0(45)\nr\u0003\u0001B=0 (46)\nwhere\u000f0is the permittivity of vacuum, and Eq. 45 and 46 not being solved explicitly but verified via appropriate\ninitial conditions and current deposition procedure. The di \u000berential operators are defined as\nr= \u0001 xˆx+ \u0001 yˆy+ \u0001 zˆz (47)\nr\u0003= \u0001\u0003\nxˆx+ \u0001\u0003\nyˆy+ \u0001\u0003\nzˆz; (48)\nthe finite di \u000berences and sums operators being\n\u0001tGjn\ni;j;k=Gjn+1=2\ni;j;k\u0000Gjn\u00001=2\ni;j;k\n\u000et(49)\n\u0001xGjn\ni;j;k=Gjn\ni+1=2;j;k\u0000Gjn\ni\u00001=2;j;k\n\u000ex(50)\n\u0001\u0003\nx=\u0010\n\u000b+\fS1\nx+\rS2\nx\u0011\n\u0001x (51)\nwith\nS1\nxGjn\ni;j;k=Gjn\ni;j+1=2;k+Gjn\ni;j\u00001=2;k\n+Gjn\ni;j;k+1=2+Gjn\ni;j;k\u00001=2 (52)\nS2\nxGjn\ni;j;k=Gjn\ni;j+1=2;k+1=2+Gjn\ni;j\u00001=2;k+1=2\n+Gjn\ni;j+1=2;k\u00001=2+Gjn\ni;j\u00001=2;k\u00001=2 (53)\nThe quantity Gis a sample vector component, \u000etand\u000exare respectively the time step and the grid cell size along\nx, while\u000b,\fand\rare constant scalars verifying \u000b+4\f+4\r=1. The operators along yandz, i.e.\u0001y,\u0001z,\u0001\u0003\ny,\u0001\u0003\nz,S1\ny,\nS1\nz,S2\ny, and S2\nz, are obtained by circular permutation of the indices.\n13In 2D, assuming the plane ( x;z), the enlarged finite operators simplify to\n\u0001\u0003\nx=\u0010\n\u000b+\fS1\nx\u0011\n\u0001x (54)\nS1\nxGjn\ni;j;k=Gjn\ni;j+1=2;k+Gjn\ni;j\u00001=2;k: (55)\nAn extension of this algorithm for non-cubic cells provided by Cowan in [43] is not considered in this paper. How-\never, all considerations given here for the solver implemented in Warp apply readily to the solver developed by Cowan.\n4.2.1. Numerical dispersion\nThe dispersion relation of the solver is given by\n0BBBB@sin!\u000et\n2\nc\u000et1CCCCA2\n=Cx0BBBBB@sinkx\u000ex\n2\n\u000ex1CCCCCA2\n+Cy0BBBBBB@sinky\u000ey\n2\n\u000ey1CCCCCCA2\n+Cz0BBBBB@sinkz\u000ez\n2\n\u000ez1CCCCCA2\n(56)\nwith\nCx=\u000b+2\f(cy+cz)+4\rcycz (57)\nCy=\u000b+2\f(cz+cx)+4\rczcx (58)\nCz=\u000b+2\f(cx+cy)+4\rcxcy (59)\nand\ncx=cos(kx\u000ex) (60)\ncy=cos\u0010\nky\u000ey\u0011\n(61)\ncz=cos(kz\u000ez) (62)\nThe Courant-Friedrichs-Lewy condition (CFL) is given by\nc\u000etc\u0014min[\u000ex;\u000ey;\u000ez;\n1=q\n(\u000b\u00004\r)maxh\n\u0014x+\u0014y;\u0014x+\u0014z;\u0014y+\u0014zi\n;\n1=q\n(\u000b\u00004\f+4\r)\u0010\n\u0014x+\u0014y+\u0014z\u0011\n] (63)\nwhere\u0014x=1=\u000ex2,\u0014y=1=\u000ey2and\u0014z=1=\u000ez2.\nAssuming cubic cells ( \u000ex=\u000ey=\u000ez), the coe \u000ecients given in [41] ( \u000b=7=12,\f=1=12 and\r=1=48) allow\nc\u000et=\u000ex, and thus no dispersion along the principal axes.\nIt is of interest to note that (56) can be rewritten\n0BBBB@sin!\u000et\n2\nc\u000et1CCCCA2\n=\u0010\ns2\nx+s2\ny+s2\nz\u0011\n+\f0\u0010\ns2\nxs2\ny+s2\nxs2\nz+s2\nys2\nz\u0011\n+\r0\u0010\ns2\nxs2\nys2\nz\u0011\n(64)\nwith sx=sin(kx\u000ex=2),sy=sin\u0010\nky\u000ey=2\u0011\n,sz=sin(kz\u000ez=2),\f0=\u00008\f\u000016\rand\r0=48\r, for which the coe \u000ecients\nfrom [41] take the nice values \f0=\u00001 and\r0=1.\nSets of possible coe \u000ecients and the corresponding CFL condition, assuming cubic cells, are given in Table 1. The\nnumerical dispersion using those coe \u000ecients are plotted in figure 7 along the principal axes and diagonals for cubic\ncells (\u000ex=\u000ey=\u000ez) and contrasted with the one of the Yee solver (all taken at each solver’s CFL time step limit). At\nthe CFL limit, the Yee algorithm o \u000bers no numerical dispersion along the 3D diagonal, but relatively large numerical\ndispersion at the Nyquist frequency along the main axes. Conversely, the Cole-Karkkainen solver (CK) o \u000bers no\nnumerical dispersion along the main axes but significant dispersion along the diagonals. The CK solver also allows\n14Yee CK CK2 CK3 CK4 CK5\n\f00\u00001\u00001=2 0\u00001=2\u00009=10\n\r00 1 1=2\u00001 0 9=10\n\u000b 1 7=12 19=24 11=12 3=4 5=8\n\f 0 1=12 1=24 1=24 1=16 3=40\n\r 0 1=48 1=96\u00001=48 0 3=160\nc\u000et=\u000ex1=p\n3 1 1=p\n2 1=p\n2p\n2=p\n3p\n5=p\n6\nTable 1: List of coe \u000ecients\nlarger time steps than the Yee solver by almost a factor of two in 3D. The solver labeled ”CK2” o \u000bers numerical\ndispersion that is intermediate between the Yee solver and the CK solver along the main axes and the 3D diagonal, but\nslightly degraded along the 2D diagonal. Conversely, while solver CK3 also o \u000bers intermediate numerical dispersion\nalong the main axes and the 3D diagonal, it o \u000bers no numerical dispersion along the 2D diagonal. Solver CK4\nimproves slightly the numerical dispersion along the main axes over CK2 and CK3 at the expense of the dispersion\nalong the diagonals. Finally, CK5 o \u000bers the highest level of isotropy. The CFL time steps of solvers CK2, 3, 4 and\n5 are intermediate between the Yee and the CK CFL time steps. This provides solvers with a range of numerical\ndispersion among which some may be more favorable with regard to the mitigation of numerical instabilities for a\ngiven application.\nTo reduce numerical dispersion to its lowest level, it is desirable to operate the CK solver as close as possible to\nthe CFL limit c\u000et=\u000ex. However, an instability (other than numerical Cerenkov) arises at the Nyquist frequency in\nsuch a case. The analysis is given in 1D in Appendix I, as well as its mitigation using digital filtering. Since for the\nCK solver, the CFL limit is independent of dimensionality, the analysis and mitigation apply readily to 2D and 3D\nsimulations.\nFor absorption of outgoing waves at the computational box boundaries, the extension of the solver to a Perfectly\nMatched Layer [44] is given in Appendix II.\n15Yee CK\nCK2 CK3\nCK4 CK5\nFigure 7: Numerical dispersion along the principal axis and diagonals for cubic cells ( \u000ex=\u000ey=\u000ez) at the Courant limit for the solver with\nadjustable numerical dispersion using the parameters from Table 1.\n164.2.2. Current deposition and Gauss’ Law\nIn most applications, it is essential to prevent accumulations of errors to the discretized Gauss’ Law. This is\naccomplished by providing a method for depositing the current from the particles to the grid which is compatible with\nthe discretized Gauss’ Law, or by providing a mechanism for ”divergence cleaning” [47, 48, 49, 50]. For the former,\nschemes which allow a deposition of the current that is exact when combined with the Yee solver is given in [51] for\nlinear form factors and in [31] for higher order form factors. Since the discretized Gauss’ Law and Maxwell-Faraday\nequation are the same in our implementation as in the Yee solver, charge conservation is readily verified using the\ncurrent deposition procedures from [51] and [31], and this was verified numerically. Hence divergence cleaning is not\nnecessary.\n4.3. Friedman adjustable damping\nThe tunable damping scheme developed by Friedman [36] was shown to be the most potent practical method for\nmitigating the numerical Cerenkov instability in [34], among the selected methods that were considered. It is readily\napplicable to the solver presented above by modifying (43) to\nBn+3=2=Bn+1=2\u0000\u000etr\u0003\u0002\"\u0012\n1+\u0012\n4\u0013\nEn+1\u00001\n2En+ 1\n2\u0000\u0012\n4!\n¯En\u00001#\n(65)\nwith\n¯En\u00001=\u0012\n1\u0000\u0012\n2\u0013\nEn+\u0012\n2¯En\u00002(66)\nwhere 0\u0014\u0012\u00141 is the damping factor. The numerical dispersion becomes\n0BBBB@sin!\u000et\n2\nc\u000et1CCCCA2\n=F\n2(67)\nwhere\nF=1\u00002\u0012sin2(!\u000et=2)\n2e\u0000i!\u000et\u0000\u0012(68)\nand\n\n2=266666664Cx0BBBBB@sinkx\u000ex\n2\n\u000ex1CCCCCA2\n+Cy0BBBBBB@sinky\u000ey\n2\n\u000ey1CCCCCCA2\n+Cz0BBBBB@sinkz\u000ez\n2\n\u000ez1CCCCCA2377777775(69)\nThe CFL is given by\nc\u000et\u0003\nc=c\u000etcr\n2+\u0012\n2+3\u0012(70)\nwhere\u000etcis the critical time step of the numerical scheme without damping ( \u0012=0), as given by (63).\nThe numerical dispersion of the Cole-Karkkainen-Friedman (CKF) solver (using the coe \u000ecients from the CK\nsolver in Table 1) is plotted in figure 8 along the principal axis and diagonals for cubic cells ( \u000ex=\u000ey=\u000ez) and\ncontrasted with the one of the Yee-Friedman (YF) solver (both taken at the Courant time step limit). The amount\nof phase error rises with the value of the damping parameter \u0012(partly due to the slightly more constraining limit on\nthe critical time step). However, it was shown in [34] that the amount of damping provided by the YF solver was\nsu\u000ecient to counteract the slight degradation of numerical dispersion with raising \u0012, reducing the numerical Cerenkov\ne\u000bects to an acceptable level for the problem that was considered. The damping is very isotropic with the CKF\nsolver but not with the YF one. The YF implementation o \u000bers a higher level of damping of the shortest wavelengths\nalong the 3D diagonals, while the CKF o \u000bers more damping along the axes, and the amount of damping along the 2D\ndiagonals are similar. In summary, the YF implementation delivers respectively the highest /lowest level of damping in\nthe direction of lowest /highest numerical dispersion, while the CKF implementation delivers a proportionally higher\nlevel of dispersion than the YF implementation along the direction of highest numerical dispersion. Thus it may be\nexpected that the CKF implementation will be more e \u000ecient in reducing numerical Cerenkov e \u000bects.\n17Yee-Friedman Cole-Karkkainen-Friedman\nFigure 8: Numerical dispersion along the principal axis and diagonals for cubic cells ( \u000ex=\u000ey=\u000ez) at the Courant limit for: (left) the Yee-Friedman\nsolver; (right) the Cole-Karkkainen-Friedman solver. The real part (phase) and the imaginary part (amplitude) are plotted respectively in the top\nand bottom rows.\n185. Application to the modeling of laser wakefield acceleration\nThis section presents applications of the methods to the modeling of 10 GeV LPA stages at full scale in 2-1 /2D\nand 3D, which has not been done fully self-consistently with other methods. It has been shown that many parameters\nof high energy LPA stages can be accurately simulated at reduced cost by simulating stages of lower energy gain, with\nhigher density and shorter acceleration distance, by scaling the physical quantities relative to the plasma wavelength,\nand this has been applied to design of 10 GeV LPA stages [18, 19]. The number of oscillations of a mismatched laser\npulse in the plasma channel however depends on stage energy and does not scale, though this e \u000bect is minimized for\na channel guided stage as considered in [18, 19]. The number of betatron oscillations of the trapped electron bunch\nwill also depend on the stage energy, and may a \u000bect quantities like the emittance of the beam. For these reasons, and\nto prove validity of scaled designs of other parameters, it is necessary to perform full scale simulations, which is only\npossible by using reduced models or simulations in the boosted frame.\nAs a benchmarking exercise, we first perform scaled simulations similar to the ones performed in [18], at a\ndensity of ne=1019cm\u00003, using various values of the boosted frame relativistic factor \rto show the accuracy and\nconvergence of the technique. These stages were shown to e \u000eciently accelerate both electrons and positrons with low\nenergy spread, and the scaled simulations predicted acceleration of hundreds of pC to 10 GeV energies using a 40\nJ laser. The accuracy of the technique is evaluated by modeling scaled stages [18, 19] at 0.1 GeV , which allows for\na detailed comparison of simulations using a reference frame ranging from the laboratory frame to the frame of the\nwake. Excellent agreement is obtained on wakefield histories on axis, beam average energy history and momentum\nspread at peak energy, with speedup over a hundred, in agreement with the theoretical estimates from Section 2. The\ndownscaled simulations are also used for an in-depth exploration of the e \u000bects of the methods presented in Sections 3\nand 4, and evaluation of which techniques are required to permit maximum \rboost while maintaining high accuracy.\nWe then apply the boosted frame technique to provide full scale simulation of high e \u000eciency quasilinear LPA stages\nat higher energy, verifying the scaling laws in the 10 GeV-1 TeV range.\n5.1. Scaled 10 GeV stages\nThe parameters were chosen to be close (though not identical) to the case where kpL=2 in [18] where kpis\nthe plasma wavenumber and Lis the laser pulse length. In the cases considered in this paper, the beam is composed\nof test particles only, with the goal of testing the fidelity of the algorithm in modeling laser propagation and wake\ngeneration. The results from simulations of LPA in a boosted frame where beam loading is present will be presented\nelsewhere. These simulations are scaled replicas of 10 GeV stages that would operate at actual densities of 1017cm\u00003\n[18, 19] and allow short run times to permit e \u000bective benchmarking between the algorithms. The main physical and\nnumerical parameters of the simulation are given in Table 2. Unless noted otherwise, in all the simulations presented\nherein, the field is gathered from the grid onto the particles directly from the Yee mesh locations, i.e. using the ’energy\nconserving’ procedure (see [47], chapter 10).\n5.1.1. Using standard numerical techniques\nThese runs were done using the standard Yee solver with no damping, and with the 4-pass stride-1 filter plus\ncompensation, similarly to the simulations reported in [18]. No signs of detrimental numerical instabilities were\nobserved at the resolutions reported here with these settings.\nThe approximate relativistic factor of the wake that is formed by the laser traveling in the plasma is given, ac-\ncording to linear theory, by \rw=2\u0019c=\u0015! pwhere!p=p\nnee2=\u000f0meis the electron plasma frequency. For the given\nparameters, \rw\u001913:2. Thus, Warp simulations were performed using reference frames moving between \r=1 (lab-\noratory frame) and 13. For a boosted frame associated with a value of \rapproaching \rwin the laboratory, the wake\nis expected to travel at low velocity in this boosted frame, and the physics to appear somewhat di \u000berent from the one\nobserved in the laboratory frame, in accordance to the properties of the Lorentz transformation. Figure 9 and 10 show\nsurface renderings of the transverse and longitudinal electric fields respectively, as the beam enters its early stage of\nacceleration by the plasma wake, from a calculation in the laboratory frame and another in the frame at \r=13. The\ntwo snapshots o \u000ber strikingly di \u000berent views of the same physical processes: in the laboratory frame, the wake is fully\nformed before the beam undergoes any significant acceleration and the imprint of the laser is clearly visible ahead of\nthe wake; while in the boosted frame calculation, the beam is accelerated as the plasma wake develops, and the laser\n19Table 2: List of parameters for scaled 10GeV class LPA stage simulation.\nbeam radius Rb 82:5 nm\nbeam length Lb 85:nm\nbeam transverse profile exp\u0010\n\u0000r2=8R2\nb\u0011\nbeam longitudinal profile exp\u0010\n\u0000z2=2L2\nb\u0011\nlaser wavelength \u0015 0:8\u0016m\nlaser length (FWHM) L 10:08\u0016m\nnormalized vector potential a0 1\nlaser longitudinal profile sin (\u0019z=L)\nplasma density on axis ne 1019cm\u00003\nplasma longitudinal profile flat\nplasma length L 1:5 mm\nplasma entrance ramp profile half sinus\nplasma entrance ramp length 4 \u0016m\nnumber of cells in x Nx 75\nnumber of cells in z Nz860 (\r=13)-1691 (\r=1)\ncell size in x \u000ex 0:65\u0016m\ncell size in z \u000ez \u0015=32\ntime step \u000et at CFL limit\nparticle deposition order cubic\n# of plasma particles /cell 1 macro-e\u0000+1 macro-p+\nimprint is not visible on the snapshot. Close examination reveals that the short spatial variations which make the laser\nimprint in front of the wake are transformed into time variations in the boosted frame of \r=13.\nHistories of the perpendicular and longitudinal electric fields recorded at a number of stations at fixed locations in\nthe laboratory o \u000ber direct comparison between the simulations in the laboratory frame ( \r=1) and boosted frames at\n\r=2, 5, 10 and 13. Figure 11 and 12 show respectively the transverse and longitudinal electric fields collected at the\npositions z=0:3 mm and z=1:05 mm (in the laboratory frame) on axis ( x=y=0). The agreement is excellent and\nconfirms that despite the apparent di \u000berences from snapshots taken from simulations in di \u000berent reference frames,\nthe same physics was recovered. This is further confirmed by the plot of the average scaled beam energy gain as a\nfunction of position in the laboratory frame, and of relative longitudinal momentum dispersion at peak energy (Fig.\n13). The small di \u000berences observed on the mean beam energy histories and on the longitudinal momentum spread are\nattributed to a lack of convergence at the resolution that was chosen. The beam was launched with the same phase in\nthe 2-1 /2D and the 3D simulations, resulting in lower energy gain in 3D, due to proportionally larger laser depletion\ne\u000bects in 3D than in 2-1 /2D.\nThe CPU time recorded as a function of the average beam position in the laboratory frame (Fig. 13-middle)\nindicates that the simulation in the frame of \r=13 took\u001925 s in 2-1 /2D and\u0019150 s in 3D versus \u00195;000 s in\n2-1/2D and\u001920;000 s in 3D in the laboratory frame, demonstrating speedups of \u0019200 in 2-1 /2D and\u0019130 in 3D,\nbetween calculations in a boosted frame at \r=13 and the laboratory frame.\nAll the simulations presented so far in this section were using the Yee solver, for which the Courant condition is\ngiven by c\u000et<\u0010\n1=\u000ex2+1=\u000ez2\u0011\u00001=2in 2D and c\u000et<\u0010\n1=\u000ex2+1=\u000ey2+1=\u000ez2\u0011\u00001=2in 3D where \u000etis the time step and\n\u000ex,\u000eyand\u000ezare the computational grid cell sizes in x,yandz. As\rwas varied, the transverse resolution was kept\nconstant, while the longitudinal resolution was kept at a constant fraction of the incident laser wavelength \u000ez=\u0010\u0015,\nsuch that in a boosted frame, \u000ez\u0003=\u0010\u0015\u0003=\u0010(1+\f)\r\u0015. As a result, the speedup becomes, when using the Yee solver\nSyee2D=S\u000ezp\n1=\u000ex2+1=\u000ez2\n\u000e\u0003zp\n1=\u000ex2+1=\u000ez\u00032(71)\n20Figure 9: Colored surface rendering of the transverse electric field from a 2-1 /2D Warp simulation of a laser wakefield acceleration stage in the\nlaboratory frame (top) and a boosted frame at \r=13 (bottom), with the beam (white) in its early phase of acceleration. The laser and the beam are\npropagating from left to right.\nin 2D and\nSyee3D=S\u000ezp\n1=\u000ex2+1=\u000ey2+1=\u000ez2\n\u000e\u0003zp\n1=\u000ex2+1=\u000ey2+1=\u000ez\u00032(72)\nin 3D where Sis given by Eq. (13).\nThe speedup versus relativistic factor of the reference frame is plotted in Fig. 14, from (13), (71) and (72),\nand contrasted with measured speedups from 1D, 2-1 /2D and 3D Warp simulations, confirming the scaling obtained\nanalytically.\n21Figure 10: Colored surface rendering of the longitudinal electric field from a 2-1 /2D Warp simulation of a laser wakefield acceleration stage in the\nlaboratory frame (top) and a boosted frame at \r=13 (bottom), with the beam (white) in its early phase of acceleration. The laser and the beam are\npropagating from left to right.\n2-1/2D 3-D\nFigure 11: History of transverse electric field at the position x=y=0,z=0:3 mm and z=1:05 mm (in the laboratory frame) from simulations in\nthe laboratory frame ( \r=1) and boosted frames at \r=2, 5, 10 and 13.\n222-1/2D 3-D\nFigure 12: History of longitudinal electric field at the position x=y=0,z=0:3 mm and z=1:05 mm (in the laboratory frame) from simulations\nin the laboratory frame ( \r=1) and boosted frames at \r=2, 5, 10 and 13.\n232-1/2D 3-D\nFigure 13: (top) Average scaled beam energy gain and (middle) CPU time, versus longitudinal position in the laboratory frame from simulations;\n(bottom) distribution of relative longitudinal momentum dispersion at peak energy, in the laboratory frame ( \r=1) and boosted frames at \r=2, 5,\n10 and 13.\n24Figure 14: Speedup versus relativistic factor of the boosted frame from Eq. (13), (71), (72), and Warp simulations.\n255.1.2. E \u000bect of filtering, solver with adjustable dispersion and damping\nThe modeling of full scale stages, which allow for higher values of \rfor the reference frame, is more prone to\nthe high frequency instability that was mentioned in a previous section, as we will show below. In anticipation of the\napplication of the method presented above to mitigate the instability, simulations of the scaled stage were conducted\nusing the Yee solver with digital filter S(1:2:4) as described above (Fig. 15), the Cole-Karkkainen solver (Fig. 16) or\nthe Yee-Friedman solver (Fig. 17).\nSmoothing with the wideband filter S(1:2:4) did not produce significant degradations for the calculation in the\nwake frame ( \r=13) but did otherwise. The calculations with the Yee solver and the Cole-Karkkainen solver gave\nidentical results, validating our implementation of the CK solver. Despite the more expensive stencil, the run with the\nCK solver was almost 40% faster, due to a time step larger byp\n2. Similarly to filtering, damping aggressively did\nnot degrade the result in the range 10 \u0014\r\u001413 but did significantly in the range 1 \u0014\r\u00145. Comparing the timings\nwith those of Fig. 13 (middle-left) shows that the smoothing and the damping added less than a factor of two of total\nruntime to the simulations.\nFigure 15: ( (left) Average scaled beam energy gain and (right) CPU time, versus longitudinal position in the laboratory frame from simulations\nin the laboratory frame ( \r=1) and boosted frames at \r=2, 5, 10 and 13, using the Yee solver with digital filter S(1:2:4) (grey cross is reference\nfrom run with filter S(1)).\nFigure 16: ( (left) Average scaled beam energy gain and (right) CPU time, versus longitudinal position in the laboratory frame from simulations in\nthe laboratory frame ( \r=1) and boosted frames at \r=2, 5, 10 and 13, using the Cole-Karkkainen solver with filter S(1) (red curve is reference\nfrom calculation with Yee solver and filter S(1)).\nThose results lead to several observations: (i) while the grid dimensions and number of cells were chosen such that\nsquare cells were obtained for \r=13, meaning a larger dispersion in the longitudinal direction with the Yee solver\nthan with the Cole-Karkkainen solver, both gave the same result. This is significant since for simulations of LPA in\nthe laboratory frame reported in the literature, the need to have nearly perfect numerical dispersion in the longitudinal\ndirection imposes a constraint on the cell aspect ratio and thus on resolution [45, 46]. This constraint is removed when\nsimulating in the frame of the wake ( \r=13\u0019\rw); (ii) damping of high frequencies with the Yee-Friedman solver\nor wideband smoothing of short wavelength have a negligible e \u000bect on accuracy for simulations in the frame of the\n26Figure 17: ( (left) Average scaled beam energy gain and (right) CPU time, versus longitudinal position in the laboratory frame from simulations in\nthe laboratory frame ( \r=1) and boosted frames at \r=2, 5, 10 and 13, using the Yee-Friedman solver with \u0012=1 (grey cross is reference from run\nwith no damping).\nwake, but degrade the accuracy very significantly for slower moving reference frames. The dependency of the e \u000bect of\ndamping and smoothing with \rboost has two causes. First, simulations with a boost \r\u0019\rwrequire fewer time steps\nthan simulations using a lower value of \r. Thus, for a given value of the damping coe \u000ecient\u0012, the integrated amount\nof damping will be lower for the simulations with \r\u0019\rw. Second, as mentioned above in the discussion of the surface\nrenderings shown in Fig. 9 and 10, a large fraction of the short wavelength content that is present in the simulations\nin the laboratory frame is transformed into time oscillations in simulations in the wake frame. Hence, filtering short\nwavelength has less e \u000bect on the physics when calculating in the wake frame than when calculating in the laboratory\nframe; (iii) the cost of using even the most aggressive damping or smoothing is low, especially considering that the\nsimulations presented here were using only two plasma macro-particles per cell.\nIn summary, calculating in a boosted frame near the frame following the wake ( \r\u0019\rw) relaxes the constraint on\nthe numerical dispersion in the direction of propagation of the laser (which is essential in simulations in the laboratory\nframe), and allows for more aggressive damping of high frequencies and smoothing of short wavelengths than is\npossible in standard laboratory frame calculations.\n5.2. Full scale 10 GeV class stages\nAs noted in [13], full scale simulations using the laboratory frame of 10 GeV stages at plasma densities of 1017\ncm\u00003are not practical on present computers in 2D and 3D. At this density, the wake relativistic factor \rw\u0019132, and\n2-1/2D and 3D simulations were done in boosted frames up to \r=130.\nFigure 18: (left) Average beam energy gain versus longitudinal position (in the laboratory frame), (right) Fourier Transform of the longitudinal\nelectric field at t =40 ps, averaged over whole domain, from 2D-1 /2 simulations of a full scale 10GeV LPA in a boosted frame at \r=130, using the\nYee solver and various digital filter kernels. Square cells ( \u000ex=\u000ez=6:5\u0016m) and the CFL time step (c \u000et=\u000ez=1=p\n2) were used.\n27Figure 19: Average beam energy gain versus longitudinal position (in the laboratory frame) from 2D-1 /2 simulations of a full scale 10GeV LPA in\na boosted frame at \r=30, 60 and 130, using the Yee solver.\n5.2.1. Simulations in 2-1 /2D\nFig. 18 shows the average beam energy gain versus longitudinal position and the averaged Fourier Transform of\nthe longitudinal electric field taken at t =40 ps, from 2D-1 /2 simulations of a full scale 10GeV LPA in a boosted frame\nat\r=130, using the Yee solver and various smoothing kernels. Fig. 19 shows the average beam energy gain versus\nlongitudinal position from simulations in boosted frames at \r=30, 60 and 130. All runs gave the same beam energy\nhistory within a few percents, and no sign of instability is detected in the Fourier transform plot of the longitudinal\nelectric field. The average energy gain peaks around 8 GeV , in agreement with the scaled simulations (see Fig. 13).\n5.2.2. Simulations in 3D\nIn 3D, all simulations at \r=130 using the Yee solver (using cubic cells and a time step at the CFL limit) developed\nthe instability and loss of the beam, regardless of the amount of filtering or damping that has been tried. The failure of\nthe 3D simulations using the Yee solver motivated use of the Cole-Karkkainen-Friedman (CKF) solver, with various\nlevels of filtering and damping. Data from 3D simulations using the CKF solver and various smoothing kernels are\nplotted in Fig. 20. Stability is attained when using a su \u000ecient level of filtering. Damping is detrimental to stability at\nlow levels (\u0012=0:1) but is beneficial at a higher level ( \u0012=0:5).\nNext, simulations using the solver coe \u000ecients CK2-5 from Table 1 were performed, with the time step set at their\nrespective CFL limit. The best results were obtained using solvers CK2 and CK3, while CK4 and CK5 did not o \u000ber\nsubstantial improvement over the CK solver. The results from the runs using CK2 and CK3 were nearly identical and\nhence only thoses from CK2 are reported in Fig. 21, which show very consistent beam energy gain histories, and no\nsign of instability in the Fourier Transform plot of the longitudinal electric field at t =40 ps (closer inspection revealed\nthat when using the lowest level of filtering S(1), a mild instability was developing but it was not a \u000becting the average\nbeam energy gain history). As shown on Fig. 22, the results at \r=30\u0000125 are in excellent agreement while the run\nat\r=130 predicts a slightly lower energy gain, all within 10 percent of the maximum energy gain predicted around\n5.7 GeV by the scaled simulations shown on Fig. 13 (top-right).\nIn summary, the full scale 6-7 GeV simulations using the frame of the wake performed in this subsection show:\n(i) 2-1 /2D simulations using the Yee solver at the CFL limit (with square cells) were free of instability; (ii) 3D\nsimulations using the CK solver developed moderately strong instabilities that were mitigated using moderate to high\nlevels of damping and /or filtering, the latter being the most e \u000bective; (iii) 3D simulations using the CK2 (or CK3)\nsolver developed very mild instabilities that were mitigated with a low level of filtering.\n28\u0012=0\n \u0012=0:1\n\u0012=0:5\nFigure 20: (left) Average beam energy gain versus longitudinal position (in the laboratory frame); (right) Fourier Transform of the longitudinal\nelectric field at t =40 ps, averaged over plane on axis perpendicular to laser polarization, from 3D simulations of a full scale 10GeV LPA in a\nboosted frame at \r=130, using the Cole-Karkkainen-Friedman solver and various smoothing kernels, with (top) no numerical damping ( \u0012=0),\n(middle) damping with \u0012=0:1 and (bottom) \u0012=0:5.\n29Figure 21: (left) Average beam energy gain versus longitudinal position (in the laboratory frame), (right) Fourier Transform of the longitudinal\nelectric field at t =40 ps, averaged over whole domain, from 3D simulations of a full scale 10GeV LPA in a boosted frame at \r=130, using the\nCK2 solver and various digital filter kernels.\nFigure 22: Average beam energy gain versus longitudinal position (in the laboratory frame) from 3D simulations of a full scale 10GeV LPA in a\nboosted frame at \r=30, 60, 120, 125 and 130, using the Yee solver ( \r=30 and 60) and the CK2 solver ( \r=120\u0000130), with digital filter S(1)\nand with the time step set by c \u000et=\u000ez=1=p\n2 for stability (see discussion below) .\n305.3. E \u000bects of numerical parameters on the observed instability\nFigure 23: Fourier Transform of the longitudinal electric field at t =40 ps, averaged over plane on axis perpendicular to laser polarization, from\n(left) 3D and (right) 2D-1 /2 simulations of a full scale 10GeV LPA in a boosted frame at \r=130, using the Yee solver and various smoothing\nkernels. The same time step at the 3D CFL limit c \u000et=\u000ex=p\n3 was used for both simulations.\nThe Fourier transform of the longitudinal electric field averaged over the whole domain at t =40 ps, from 3D\nsimulations using the Yee solver, is given in Fig. 23 (left). It is contrasted to the same data taken from 2-1 /2D\nsimulations (right). Both simulations used the same time step at the 3D CFL limit c \u000et=\u000ez=p\n3. The similarity of the\ntwo plots indicates that the degradation of the numerical dispersion that resulted from going from the 2D to the 3D\nCFL limit is the cause of the failure of the 3D runs using the Yee solver. Taking advantage of this observation, we\nstudy in this section the instability arising in 2-1 /2D simulations using a time step at the 3D CFL limit.\n5.3.1. E \u000bects of spatial resolution\nSnapshots of the longitudinal electric field at the front of the plasma taken at t=12:5 ps, and their corresponding\nFourier transform, are given in Fig. 24, from 2-1 /2D simulations using the Yee solver with the time step at the 3D\nCFL limit c \u000et=\u000ez=p\n3. Three resolutions were considered: (a) \u000ex=\u000ez=13\u0016m, (b)\u000ex=\u000ez=6:5\u0016m, and (c)\n\u000ex=\u000ez=3:25\u0016m. The amplitude of the instability is roughly inversely proportional to the resolution. For this\nconfiguration, the instability exhibits two primary modes at various relative levels, both at a fixed number of grid\ncells in the longitudinal direction, but at a fixed absolute length in the transverse direction. This indicates that the\ntransverse part of the modes is governed by the physical geometry of the problem while the longitudinal part is\ngoverned by numerical resolution.\nResults from 2-1 /2D simulation using the CK solver at the 3D CFL limit c \u000et=\u000ez=1=p\n3 at the resolution \u000ex=\n\u000ez=6:5\u0016m are given in Fig. 25. The same two modes that were observed in the plots from the equivalent simulation\nusing the Yee solver (see Fig. 24-middle), are present, and the overall amplitude of the instability is similar. These\nsimilarities on the details of the instability between the Yee and CK solvers indicate that the di \u000berences in numerical\ndispersion of the solvers do not constitute a key factor a \u000becting the instability.\n5.3.2. E \u000bects of time step\nIt is striking that all the solvers that lead to the lowest levels of instability had the same CFL time step c \u000etCFL=\n\u000ez=p\n2. For checking whether this is coincidental, simulations were performed using the CK solver, scanning the time\nstep between c \u000et=\u000ez=0:5 and c\u000et=\u000ez=1. The Fourier Transform of the longitudinal field averaged over the entire\ndomain taken at t=40 ps, is given in Fig. 26, exhibiting a sharp reduction of the instability level in a narrow band\naround c\u000et=\u000ez=p\n2. Since the numerical dispersion degrades in all directions when the time step diminishes, this\nindicates that the value of the time step value is of more importance than the numerical dispersion of the solver being\nused.\nSimulations using the Yee or the CK solver with the singular time step c\u000et=\u000ez=p\n2 were performed and produced\nlevels of instabilities that were much reduced (and delayed) compared to the 3D CFL time step (not shown). The\nsnapshot of the electric field and its Fourier Transform taken at t=49 ps are given in Fig. 27. The Fourier spectrum\nis very similar in each case, although the instability is slightly stronger with the CK solver than with the Yee solver.\nIn both cases, the instability is easily removed by using the S(1:2) filter (see Fig. 28).\n31As mentioned in the previous section, the solvers CK, CK4 and CK5, which all have a CFL time step above the\nsingular time step c \u000et=\u000ez=p\n2, produced significant levels of instability when run at their CFL limit. It was verified\nthat using those solvers in 3D at the time step c\u000et=\u000ez=p\n2 resulted in greatly reduced levels of instability. It was also\nobserved that running the Yee solver using non-cubic cells, e.g. with lower resolution transversely such as \u000ex=2\u000ezat\n\r=130, or\u000ex=2:6\u000ezat\r=50, produced the same pattern: a significant instability was present when using the CFL\ntime step and was greatly reduced by using c\u000et=\u000ez=p\n2. Hence for the suppression of the instability, the choice of the\nsolver seems to depends solely on whether its CFL condition allows stability at the special time step c\u000et=\u000ez=p\n2 for\na given grid cell aspect ratio, but not significantly on its numerical dispersion nor on the value of the grid cell aspect\nratio.\n5.3.3. E \u000bects of field gathering procedure\nThe scan of time step was repeated using the ’momentum conserving’ procedure [47] , in which the field values\nare interpolated at the grid nodes before being gathered onto the particles. The result is given in Fig. 29. With the\nmomentum conserving procedure, the level of instability is consistently high and independent of the time step. Since\nthe numerical dispersion of the solver varies substantially with the time step, this result supports the conclusion that\nthe instability may not be of numerical Cerenkov nature. The identification of the nature of the instability and the\nexplanation of the singular time step c\u000etScall for a multidimensional (no instability was observed in 1D regardless of\nthe field gathering method) analysis of the discretized Vlasov algorithm that was employed, which is left for future\nwork.\nThe results that were obtained lead to the following conclusions: (i) the time step c \u000etS=\u000ez=p\n2 consistently pro-\nduces the lowest levels of instability, regardless of dimensionality (2D vs 3D), the field solver being used, resolution,\naspect ratio of cells (within the range of the finite number of cases that were experimented); (ii) the main advantage\nof the tunable field solver resides in allowing access to the singular time step c\u000etSrather than providing improved\nnumerical dispersion, which consequently do not appear to be a primary driver of the instability; (iii) the instability is\nnot completely removed at c\u000etSand filtering is still needed, albeit at lower levels; (iv) the field gathering procedure\nis key, as the existence of a singular time step at which the instability is greatly reduced is observed using an ’energy\nconserving’ procedure, but not using a ’momentum conserving’ procedure. These results indicate that the instability\nthat is being observed may not be a type of numerical Cerenkov instability, as originally conjectured.\n32E== FFTE==\nFigure 24: (left) Snapshot of the longitudinal electric field ( E==) at the front of the plasma at t=12:5 ps; (right) Fourier Transform of the longitudinal\nelectric field, from 2-1 /2D simulations of a full scale 10GeV LPA in a boosted frame at \r=130, using the Yee solver, for (top) \u000ex=\u000ez=13\u0016m;\n(middle)\u000ex=\u000ez=6:5\u0016m; (bottom)\u000ex=\u000ez=3:25\u0016m. The time step at the 3D CFL limit c \u000et=\u000ez=p\n3 was used for all three simulations.\n33Figure 25: (left) Snapshot of the longitudinal electric field ( E==) at the front of the plasma at t=12:5 ps; (right) Fourier Transform of the longitudinal\nelectric field, from 2-1 /2D simulations of a full scale 10GeV LPA in a boosted frame at \r=130, with the CK solver, using \u000ex=\u000ez=6:5\u0016m, and\nthe time step at the 3D CFL limit c \u000et=\u000ex=p\n3.\nFigure 26: Fourier Transform of the longitudinal electric field at t =40 ps, averaged over the whole domain, from 2-1 /2D simulations of a full scale\n10GeV LPA in a boosted frame at \r=130, using the CK solver, for time steps between c\u000et=\u000ez=0:5 and c\u000et=\u000ez=1, versus\u0015=\u000ez(left) and at\n\u0015=\u000ez=4 (right).\n34Figure 27: (left) Snapshot of the longitudinal electric field ( E==) at the front of the plasma at t=49 ps; (right) Fourier Transform of the longitudinal\nelectric field, from 2-1 /2D simulations of a full scale 10GeV LPA in a boosted frame at \r=130, using\u000ex=\u000ez=6:5\u0016m, and the time step at the\n2D CFL limit c\u000et=\u000ez=p\n2, for (top) the Yee solver; (bottom) the CK solver.\nFigure 28: Snapshot of the longitudinal electric field ( E==) at the front of the plasma at t=49 ps from 2-1 /2D simulations of a full scale 10GeV\nLPA in a boosted frame at \r=130, using\u000ex=\u000ez=6:5\u0016m, and the time step at the 2D CFL limit c\u000et=\u000ez=p\n2, for (left) the Yee solver; (right) the\nCK solver. The filter S(1:2) was used to remove the instability that is visible in Fig. 27. The remaining feature is the wake.\n35Figure 29: Fourier Transform of the longitudinal electric field at t =40 ps, averaged over the whole domain, from 2-1 /2D simulations of a full\nscale 10GeV LPA in a boosted frame at \r=130, using the CK solver, for time steps between c\u000et=\u000ez=0:5 and c\u000et=\u000ez=1, using a ’momentum\nconserving’ field gathering scheme.\n365.4. Full scale 100 GeV - 1 TeV class stages\nFigure 30: Average beam energy gain versus longitudinal position (in the laboratory frame) for simulations at ne=1019cc down to 1015cc, using\nframes of reference between \r=12 and\r=1300, in 2-1 /2D (left) and 3D (right).\nUsing the knowledge acquired from the 10 GeV class study, simulations of stages in the range of 0.1 GeV-1 TeV\nwere performed in 2-1 /2D and in the range of 0.1-100 GeV in 3D. The plasma density nescales inversely to the energy\ngain, from 1019cc down to 1015cc in the 0.1 GeV-1 TeV range. These simulations used the parameters given in Table\n2 scaled appropriately, and used the high speed of the boosted simulations to allow fast-turnaround improvement of\nthe stage design [18, 19]. Scaled energy gain was increased by adjusting the phase of the beam injection behind the\nlaser by\u001812% in 3D and 7% in 2D, with respect to the results presented in the preceding section. The 5% level\ndi\u000berence between the 2D and 3D beam phases is likely due to small di \u000berences in wake structure, laser depletion,\nand the small number of betatron oscillations of the laser. To minimize beam loss, the beam dimensions were reduced\nby a factor of 3 in each dimension. Simulations showing performance of this design in 2-1 /2D were performed using\nthe Yee solver with filter S(1) for the 0.1-10 GeV runs, S(1:2) for the 100 GeV and S(1:2:3) for the 1 TeV ones. The\n3D simulations were performed using the CK2 solver with filter S(1) for the 0.1-1 GeV runs, and S(1:2) for the 10-100\nGeV ones. The average beam energy gain history is plotted in Fig. 30, scaling the 0.1-100 GeV runs to the 1 TeV\nrange in 2-1 /2D, and the 0.1-10 GeV runs to the 100 GeV range in 3D. The results exhibit an excellent agreement on\nthe peak scaled beam energy gain between 0.1-100 GeV runs, and on the scaled beam energy gain histories between\nthe 1-100 GeV runs. A higher level of smoothing was needed for the 1TeV case, explaining the deviation past 1 km.\nThis deviation is of little importance in practice, where one is mostly interested in the beam evolution up-to the peak\nenergy point. The di \u000berences at 1019on the scaled beam energy gain history can be attributed to the e \u000bects from\nhaving only a few laser oscillations per pulse.\nUsing (13), the speedup of the full scale 100 GeV class run, which used a boosted frame of \r=400 as frame of\nreference, is estimated to be over 100,000, as compared to a run using the laboratory frame. Assuming the use of a\nfew thousands of CPUs, a simulation that would require several decades to complete using standard PIC techniques\nin the laboratory frame, was completed in four hours using 2016 CPUs of the Cray system at NERSC. With the same\nanalysis, the speedup of the 2-1 /2D 1 TeV stage is estimated to be over a million.\n376. Conclusion and outlook\nThe technique proposed in [1] was applied successfully to speedup by orders of magnitude calculations of laser-\nplasma accelerators from first principles. The theoretical speedup estimate from [1] was improved, while complica-\ntions associated with the handling of input and output data between a boosted frame and the laboratory frame were\ndiscussed. Practical solutions were presented, including a technique for injecting the laser that is simpler and more\ne\u000ecient than methods proposed previously.\nControl of an instability that was limiting the speedup of such calculations in previous work is demonstrated, via\nthe use of a field solver with tunable coe \u000ecients and digital filtering. The tunable solver was shown to be compatible\nwith existing ”exact” current deposition techniques for conservation of Gauss Law, and accommodates Perfectly\nMatched Layers for e \u000ecient absorption of outgoing waves.\nExtensive testing of the methods presented for numerical Cerenkov mitigation reveals that choosing the frame of\nthe wake as the frame of reference allows for higher levels of filtering and damping than is possible in other frames with\nthe same accuracy. It also revealed that there exists a singular time step for which the level of instability is minimal,\nindependently of other numerical parameters, especially the numerical dispersion of the solver. This indicates that\nthe observed instability may not be caused by numerical Cerenkov e \u000bects. Analysis of the nature of the instability is\nunderway, but regardless of cause, the methods presented mitigate it e \u000bectively. The tunability of the field solver is\nkey in providing stability in 3D at the singular time step, which is not attainable by the standard Yee solver.\nThe use of those techniques permitted the first calculations in the optimal frame of 10 GeV , 100 GeV and 1\nTeV class stages, with speedups over 4, 5 and 6 orders of magnitude respectively over what would be required by\n”standard” laboratory frame calculations, which are impractical for such stages due to computational requirements.\nThese results show that the technique can be applied to the modeling of 10 GeV stages, and future work will\ninclude the e \u000bects of beam loading, plasma density ramps, as well as particle trapping in the near future. Future work\non the numerical methods include a comprehensive analysis of the instability and the existence of a singular time\nstep under certain conditions, as well as the local application of filtering, smoothing and /or mesh refinement [57, 58]\naround the front of the plasma, where the instability develops. The latter is expected to provide mitigation of the\ninstability while preserving accuracy in the core of the simulation.\n7. Appendix I: One dimensional analysis of the CK solver\nAlthough the most interesting applications of the CK solver require two or three dimensions, analysis of the\nmethod in one dimension reveals a potential issue when c\u000et=\u000ex. In one dimension (choosing x), Equations (43)-(44)\nreduce to\nByjn+1=2\ni+1=2=Byjn\u00001=2\ni+1=2+\u000et\n\u000ex\u0010\nEzjn\ni+1\u0000Ezjn\ni\u0011\n(73)\nEzjn+1\ni =Ezjn\ni+c2\u000et\n\u000ex\u0010\nByjn+1=2\ni+1=2\u0000Byjn+1=2\ni\u00001=2\u0011\n\u0000Jn\ni\n\u000f0(74)\nDue to uniform time discretization and linearity, the response of the system (73)-(74) to arbitrary distributions and\nevolutions of sources (i.e. macro-particles) can be written as the sum of its response to the excitation from a Heaviside\nfunction in time, at one location in the grid. Assuming a source term of the form Jjn\ni=H(t) where His the Heaviside\nfunction, and setting the time step at the Courant limit c\u000et=\u000ex, the system (73)-(74) produces a spurious ”odd-even”\noscillations at the Nyquist frequency, as shown in Fig. 31 (middle-left). If a sinusoidal signal oscillating at the Nyquist\nfrequency is added to the source term, the amplitude of the spurious oscillation grows linearly with time, as shown in\nFig. 31 (middle-right). The spurious oscillation is e \u000bectively suppressed in both cases by the application of a ”1-2-1”\nbilinear digital filter, as shown in Fig. 31 (bottom) . These types of filtering are of common use in Particle-In-Cell\ncodes, often repeated a prescribed number of times and followed by a compensation stage to avoid excessive damping\nof long wavelengths [47].\nThe impact of the spurious oscillations and the e \u000bectiveness of the bilinear filtering at suppressing it in actual\nsimulations was tested on a 1D simulation of a scaled wakefield acceleration stage. The physical and numerical\nparameters of the simulation are given in table 3. Snapshots of the transverse electric field (aligned with the laser\n38Heaviside step excitation oscillatory excitation\nFigure 31: (top) time history (in time steps) of the current source for (left) a Heaviside step (right) a heaviside step modulated by a sinusoidal\noscillation at the Nyquist frequency; (middle) response of the system of equations (73)-(74) via a snapshot of the electric field after 10 time steps,\nwithout filtering of the source term; (bottom) response of the system of equations (73)-(74) with application of bilinear digital filter of the source\nterm in space. A time step of c \u000et=\u000exwas used in all runs and scaled constants c=\u000f0=1 were assumed.\npolarization) and the plasma electron phase space, taken once the laser has propagated about half way through the\nplasma (after\u001820,000 time steps) are given in Fig. 32. Without filtering of the current density, an instability develops\nat the grid Nyquist frequency, severely disrupting the plasma wake, despite the fact that cubic splines were used to de-\nposit current from macro-particles to the grid and gather the electromagnetic field from the grid to the macro-particles.\nOne application of the bilinear filtering (without compensation) is su \u000ecient to suppress the spurious instability and\nproduce a steady and clean wake.\n8. Appendix II: Perfectly Matched Layer\nThe split form of Perfectly Matched Layer (PML) [52] framework applies readily to Eqs (43)-(44). The equations\non the component along zof the magnetic field are given by\n(\u0001t+\u001bx)Bzx=\u0000\u0001\u0003\nxEy (75)\u0010\n\u0001t+\u001by\u0011\nBzy= \u0001\u0003\nyEx (76)\n(\u0001t+\u001bx)Ey=\u0000c2\u0001x\u0010\nBzx+Bzy\u0011\n(77)\n\u0010\n\u0001t+\u001by\u0011\nEx=c2\u0001y\u0010\nBzx+Bzy\u0011\n(78)\nwhere\u001bxand\u001byare the absorbing layer coe \u000ecients along xandyrespectively. The equations for the other compo-\nnents of the magnetic field and for the electric field are obtained similarly, applying the standard di \u000berence operator\non the spatial derivatives of the electric field and the enlarged di \u000berence operator on the spatial derivatives of the\nmagnetic field. The formula to update the fields is obtained by solving the finite-di \u000berence equations or by integrating\nover one time step, giving\nBzxjn+1=2\ni+1=2;j+1=2;k=\u0018xBzxjn\u00001=2\ni+1=2;j+1=2;k\u00001\u0000\u0018x\n\u001bx\u0001\u0003\nxEyjn\ni+1=2;j+1=2;k (79)\n39Table 3: List of parameters for scaled 10GeV class LPA stage simulation.\nbeam length Lb 85 nm\nbeam peak density nb 1014cm\u00003\nbeam longitudinal profile exp\u0010\n\u0000z2=2L2\nb\u0011\nlaser wavelength \u0015 0:8\u0016m\nlaser length (FWHM) L 10:08\u0016m\nnormalized vector potential a0 1\nlaser longitudinal profile sin (\u0019z=L)\nplasma density on axis ne 1019cm\u00003\nplasma longitudinal profile flat\nplasma length L 1:5 mm\nplasma entrance ramp profile half sinus\nplasma entrance ramp length 4 \u0016m\nnumber of cells Nz 952\ncell size \u000ez\u0015=24\ntime step \u000et\u000ez=c\nparticle deposition order cubic\n# of plasma particles /cell 10\nBzyjn+1=2\ni+1=2;j+1=2;k=\u0018yBzyjn\u00001=2\ni+1=2;j+1=2;k+1\u0000\u0018y\n\u001by\u0001\u0003\nyExjn\ni+1=2;j+1=2;k (80)\nEyjn+1\ni;j+1=2;k=\u0018xEyjn\ni;j+1=2;k\u0000c21\u0000\u0018x\n\u001bx\u0001x\u0010\nBzx+Bzy\u0011\njn+1=2\ni;j+1=2;k(81)\nExjn+1\ni+1=2;j;k=\u0018yExjn\ni+1=2;j;k+c21\u0000\u0018y\n\u001by\u0001y\u0010\nBzx+Bzy\u0011\njn+1=2\ni+1=2;j;k(82)\nwhere\u0018=(1\u0000\u001b\u000et=2)=(1+\u001b\u000et=2)via direct solve, or \u0018=e\u0000\u001b\u000etvia time integration (note that in our tests, both\nimplementations gave nearly identical results).\nThe PML using the stencil given by (82) was tested and compared to the standard Yee implementation in 2D and\n3D. Fig. 33 snapshots from 2D simulations of the reflected residue from a PML layer of a pulse with amplitude given\nby the Harris function (10 \u000015\u0003cos(2\u0019ct=L)+6\u0003cos(4\u0019ct=L)\u0000cos(6\u0019ct=L))=32 where tis time, cis the speed\nof light and L=50\u000exis the pulse length in cell size units. A grid of 400x400 cells was used with \u000ex=\u000ey. The\nabsorbing layer was 8 cells deep and the dependency of the PML coe \u000ecients with the index position iin the layer\nwas\u001bi=\u001bm(i\u000ex=\u0001)nwith\u001bm=4=\u000ex,\u0001 = 5\u000exandn=2. The alternative prescription for the coe \u000ecients given in\n[53, 54], which reads \u001b\u0003\ni=\u0000\u0018i+1=2\u00001=\u0018i\u0001=\u000exwith\u0018i=e\u0000\u001bi\u000etand\u001bi=\u001bm(i\u000ex=\u0001)n, was also tested.\nFor the generic test case that has been considered, the new implementation exhibited a very low residue of re-\nflections from the PML layer, which are qualitatively and quantitatively very similar to the residue obtained with a\nstandard PML implementation. In agreement with results from [53, 54], the use of the modified coe \u000ecients\u001b\u0003led to\nan order of magnitude improvement over the use of the standard coe \u000ecients.\nThe 3D tests gave similar absorption e \u000eciency between the Yee and the new solver implementations of the PML,\nfor all the CK solver coe \u000ecients given in Table 1.\nIt was shown in [53, 54] that the e \u000eciency of the layer can be improved further for the standard PML by augment-\ning the equations with additional terms. However, a similar extension may not be readily available when using the\nCole-Karkkainen stencil and is not considered here.\n40No filtering bilinear filtering\nFigure 32: Snapshots from transverse electric field (normalized to maximum laser amplitude E0) and plasma electrons longitudinal phase space\nprojection, from a 1D simulation of a laser wakefield acceleration stage the CFL limits ( c\u000et=\u000ex) with (left) no filtering of current density; (right)\napplication of a bilinear digital filter to the current density.\n9. Acknowledgments\nWe are thankful to D. L. Bruhwiler, J. R. Cary, B. Cowan, E. Esarey, A. Friedman, C. Huang, S. F. Martins, W. B.\nMori, B. A. Shadwick, and C. B. Schroeder for insightful discussions.\n41Yee Cole-Karkkainen\n\u001b \u001b\u0003\n400 \n300 \n200 \n100 \n0 Y \n80 40 0\nX-50 -45 -40 -35 -30 dB \n-50 -45 -40 -35 -30 \n400 \n300 \n200 \n100 \n0 Y \n80 40 0\nX-50 -45 -40 -35 -30 dB \n-50 -45 -40 -35 -30 \u001b \u001b\u0003\n400 \n300 \n200 \n100 \n0 Y \n80 40 0\nX-50 -45 -40 -35 -30 dB \n-50 -45 -40 -35 -30 \n400 \n300 \n200 \n100 \n0 Y \n80 40 0\nX-50 -45 -40 -35 -30 dB \n-50 -45 -40 -35 -30 \nFigure 33: Reflected signal (in dB) from a PML layer using the Yee or the Cole-Karkkainen solver. Each simulation was run for the time step set\nat the Courant limit.\n42References\n[1] J.-L. Vay, Phys. Rev. Lett. 98(2007) 130405.\n[2] T. Tajima, J. M. Dawson, Phys. Rev. Lett. ,43(1979) 267.\n[3] E. Esarey, et al. ,Rev. Modern Phys. 81, 252 (2009) 1229.\n[4] C. G. R. Geddes, et al. ,Nature 431, 538 (2004).\n[5] S. P. D. Mangles, et al. ,Nature 431, 535 (2004).\n[6] J. Faure, et al. ,Nature 431, 541 (2004).\n[7] W. P. Leemans, et al. ,Nature Physics 2, 696 (2006).\n[8] C. B. Schroeder, et al. Proc. 13th Advanced Accelerator Concepts Workshop , Santa Cruz, CA (2008) 208.\n[9] C. G. R. Geddes, et al. , ”Laser Plasma Particle Accelerators: Large Fields for Smaller Facility Sources,” SciDAC Review 13(2009) 13.\n[10] C. G. R. Geddes et al. ,J. Phys. Conf. Series V 125(2008) 12002 /1-11.\n[11] C. Huang et al. ,J. Phys. Conf. Series 180(2009) 12005\n[12] http: //loasis.lbl.gov\n[13] D. L Bruhwiler et al. ,Proc. 13th Advanced Accelerator Concepts Workshop , Santa Cruz, CA (2008) 29.\n[14] B. A Shadwick, C. B. Schroeder, E. Esarey, Phys. Plasmas 16(2009) 056704\n[15] P. Sprangle, E. Esarey, and A. Ting, Phys. Rev. Letters 64(1990) 2011-2014.\n[16] C. Huang et al. ,J. of Comput. Phys. 217(2006) 658-679.\n[17] B. Feng, C. Huang, V . Decyk, W.B. Mori, P. Muggli, T. Katsouleas, J. Comput. Phys. 228(2009) 5340.\n[18] E. Cormier-Michel, et al. Proc. 13th Advanced Accelerator Concepts Workshop , Santa Cruz, CA (2008) 297.\n[19] C. G. R. Geddes et al. ,Proc. Particle Accelerator Conference , Vancouver, Canada (2009) WE6RFP075.\n[20] J.-L. Vay, Phys. Plasmas ,15(2008) 056701.\n[21] J. P. Boris, J. Comput. Phys. 12(1973) 131-136.\n[22] I. Haber, R. Lee, H. H. Klein, J. P. Boris, Proc. Sixth Conf. Num. Sim. Plasmas , Berkeley, CA (1973) 46-48.\n[23] B. Cowan, et al. Proc. 13th Advanced Accelerator Concepts Workshop , Santa Cruz, CA (2008) 309.\n[24] J.-L. Vay et al. ,Proc. Particle Accelerator Conference , Vancouver, Canada (2009) TU1PBI04.\n[25] S. F. Martins, Proc. Particle Accelerator Conference , Vancouver, Canada (2009) TH4GBC05.\n[26] J.-L. Vay et al. ,J. Phys. Conf. Series 180(2009) 12006\n[27] J. -L. Vay, W. M. Fawley, C. G. Geddes, E. Cormier-Michel, D. P. Grote, arXiv:0909.5603 (Sept. 2009)\n[28] S. F. Martins, R. A. Fonseca, L. O. Silva, W. Lu, W. B. Mori, Comput. Phys. Comm. 182(2010) 869-875.\n[29] D. L. Bruhwiler, Private Communication .\n[30] D. P. Grote, A. Friedman, J.-L. Vay, I. Haber, AIP Conf. Proc. 749(2005) 55.\n[31] T. Esirkepov, Comput. Phys. Comm. 135(2001) 144-53.\n[32] B. B. Godfrey, J. Comput. Phys. 15(1974) 504-521.\n[33] B. B. Godfrey, J. Comput. Phys. 19(1975) 58-76.\n[34] A. D. Greenwood, K. L. Cartwright, J. W. Luginsland, E. A. Baca, J. Comp. Phys. 201(2004) 665-684.\n[35] B. B. Godfrey, Proc. Ninth Conf. on Num. Sim. of Plasmas (1980).\n[36] A. Friedman, J. Comput. Phys. 90(1990) 292.\n[37] H. Abe, N. Sakairi, R. Itatani, H. Okuda, J. Comput. Phys. 63(1986) 247-267.\n[38] E. Cormier-Michel, B. A. Shadwick, C. G. R. Geddes, E. Esarey, C. B. Schroeder, W. P. Leemans, Phys. Rev. E 78(2008) 016404.\n[39] J. B. Cole, IEEE Trans. Microw. Theory Tech. ,45(1997) 991996.\n[40] J. B. Cole, IEEE Trans. Antennas Prop. ,50(2002) 11851191.\n[41] M. Karkkainen, E. Gjonaj, T. Lau, T. Weiland, Proc. International Computational Accelerator Physics Conference , Chamonix, France (2006).\n[42] K. S. Yee, IEEE Trans. Ant. Prop. 14(1966) 302-307\n[43] B. Cowan, Proc. 10thInternat. Comput. Accel. Phys. Conf. , San Francisco, CA (2009).\n[44] J.-P. B ´erenger, J. Comput. Phys. 114(1994) 185.\n[45] F. S. Tsung, et al. ,Phys. Plasmas 13(2006) 056708.\n[46] B. Cowan, Private communication .\n[47] C. K. Birdsall and A. B. Langdon, Plasma Physics Via Computer Simulation (Adam-Hilger, 1991).\n[48] A. B. Langdon, Comput. Phys. Comm. 70(1992) 447.\n[49] B. Marder, J. Comput. Phys. 68(1987) 48.\n[50] J.-L. Vay, C. Deutsch, Phys. Plasmas 5(1998) 1190.\n[51] J. Villasenor, O. Buneman, Comput. Phys. Comm. 69(1992) 306.\n[52] J.-P. Berenger, J. Comput. Phys. 114(1994) 185.\n[53] J.-L. Vay, J. Comput. Phys. 165(2000) 511.\n[54] J.-L. Vay, J. Comput. Phys. 183(2002) 367.\n[55] J.-L. Vay, et al. ,in preparation.\n[56] A. B. Langdon, C. K. Birdsall, Phys. Fluids 13(1970) 2115.\n[57] J.-L. Vay et al. ,Phys. Plasmas 11(2004) 2928.\n[58] J.-L. Vay, J.-C. Adam, A. H ´eron, Comput. Phys. Comm. 164(2004) 171.\n43"    },    {        "title": "1608.08373v1.Astroparticles_and_tests_of_Lorentz_invariance.pdf",        "content": "arXiv:1608.08373v1  [hep-ph]  30 Aug 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nAstroparticles and Tests of Lorentz Invariance\nJ.S. D´ ıaz\nInstitute for Theoretical Physics, Karlsruhe Institute of Technology\n76128 Karlsruhe, Germany\nSearches for violations of Lorentz invariance using cosmic rays, gamma rays,\nand astrophysical neutrinos and the prospects for future te sts using cosmic-ray\nshowers are presented.\n1. Introduction\nThe study of energetic particles bombarding Earth from distant as trophys-\nical sources has led to the development of a new discipline: astropar ticle\nphysics. The high energy and the long propagation distance of thes e as-\ntroparticles can serve as a sensitive tools to search for new physic s, as\nminute unconventional effects can get enhanced by the energy an d the path\nlength.\n2. Cosmic rays and gamma rays\nA simple modification of quantum electrodynamics (QED) is obtained by\nincorporating a Lorentz-violatingoperator that preserves CPT, coordinate,\nand gauge invariance in the form1\nL=−1\n4FµνFµν+ψ/bracketleftbig\nγµ(i∂µ−eAµ)−m/bracketrightbig\nψ−1\n4(kF)µνρσFµνFρσ,(1)\nwhere the first two terms correspond to conventional QED, while t he last\nterm is a dimension-four operator for Lorentz violation in the Stand ard-\nModel Extension (SME).1The nine independent components of the tensor\n(kF)µνρσthat produce nonbirefringent effects have been constrained usin g\nthe observation of high-energy cosmic rays.2,3In particular, the isotropic\nlimit is controlled by a single parameter κ, whose upper limit has been\nconstrained by the observation of high-energy cosmic rays, wher eas a lower\nlimit has been obtained from energetic gamma rays.4\nForκ>0,aneffectiverefractiveindexisproduced,whichallowsthepro-\nduction of Cherenkov radiation in vacuum. This is an efficient energy- lossProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\nmechanism for electrically charged fermions above a threshold ener gyEth.\nCosmic-ray primaries would rapidly lose energy and fall below the thre sh-\nold, so that the emission of Cherenkov photons would rapidly stop. H ence,\ncosmic rays reaching Earth will always have energies below the thres hold\nE <E th, which allows constraining the positive range of values of κ. The\nmost stringent limit is κ<6×10−19(98% C.L.).4Treating cosmic-ray pri-\nmaries as point particles described in QED is a restrictive approximatio n\nbecause hadronic cosmic-ray primaries are composite particles. In order to\nperform a more realistic description of proton primaries, we conside red the\nCherenkov photon to be emitted by the charged constituents of t he proton\ninstead.5The calculation involves the determination of the power radiated\nby the quarks in the proton, which is performed in the conventional way\nand then folded with the corresponding parton distribution functio ns. A\ndecrease in the power radiated is expected compared to the descr iption of\nthe full proton as a Dirac fermion because of the smaller electric cha rge\nof quarks and also due to the fact that charged partons carry on ly about\nhalf of the proton energy (the other half is carried by gluons). Our results\nagree with these expected features; nonetheless, the total po wer radiated\ndecreasesonlybyoneorderofmagnitude. Given theastronomical distances\ntraveled by cosmic rays, the limits on κremain unaffected even in a realistic\ntreatment of proton primaries.5\nForκ <0, a photon becomes unstable and rapidly decays into an\nelectron-positron pair. This process corresponds to an efficient e xtinction\nmechanism for photons above a threshold energy ωthbecause astrophysical\ngamma rays with energies above the threshold would rapidly decay int o\nlepton pairs. Hence, gamma rays reaching Earth will always have ene rgies\nbelow the threshold ω<ωth, which allows constraining the negative range\nof values of κ. The most stringent limit is κ >−9×10−16(98% C.L.).4\nProspects for potential improvement on this lower limit could make us e of\ncosmic rays by dedicated studies of the development of extensive a ir show-\ners in the atmosphere, whose maximum is very sensitive to the value o f\nκ.6\n3. Astrophysical neutrinos\nLorentz-violating neutrinos and antineutrinos in the SME are effect ively\ndescribed by a 6 ×6 hamiltonian of the form7,8\nH=|ppp|/parenleftbigg10\n01/parenrightbigg\n+1\n2|ppp|/parenleftbiggm20\n0m2∗/parenrightbigg\n+1\n|ppp|/parenleftBigg\nˆaaaeff−ˆccceff−ˆgggeff+ˆHHHeff\n−ˆggg†\neff+ˆHHH†\neff−ˆaaaT\neff−ˆcccT\neff/parenrightBigg\n,(2)Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\nwhereeachblockisa3 ×3matrixinflavorspace. Thefirsttwotermsappear\nin the Lorentz-invariant description of massive neutrinos of momen tumppp\ncharacterized by a mass-squared matrix m2. The last term includes all the\nmodifications introduced by Lorentz violation. Each matrix compone nt\ndepends in general on the neutrino momentum, direction of propag ation,\nand the relevant coefficients controlling Lorentz violation.8In particular,\ncoefficients arising from a dimension-three operator lead to CPT viola tion\nthat can mimic nonstandard interactions9and even the cosmic neutrino\nbackground.10\nThe effective hamiltonian (2) has been used for the formulation of\nnovel models describing global neutrino-oscillation data.11This hamilto-\nnian also has served for implementing generic searches of the key sig natures\nof Lorentz violation in experiments12using neutrino oscillations,13–23beta\ndecay,24,25and double beta decay.26,27\nAstrophysical neutrinos of very high energy have been observed by Ice-\nCube,28which can be used to determine stringent limits on coefficients for\nLorentz violation that modify the kinematics of neutrinos but are un ob-\nservable in oscillation experiments. These oscillation-free coefficient s can\nbe constrained using an approach similar to the one presented in the pre-\nvious section for cosmic rays and gamma rays. For isotropic operat ors\nof arbitrary dimension d, the relevant coefficients are denoted by ˚ c(d)\nof.8\nFor ˚c(d)\nof<0, an effective refractive index is produced for neutrinos. This\nallows the Cherenkov production of Zbosons, which rapidly decay into\nelectron-positron pairs. This is an efficient energy-loss mechanism f or neu-\ntrinos above a threshold energy Eth. Astrophysical neutrinos would rapidly\nlose energy and fall below the threshold, so that the Cherenkov em ission\nwould rapidly stop. Hence, high-energy astrophysical neutrinos r eaching\nEarth will always have energies below the threshold E <E th, which allows\nconstraining the negative range of values of ˚ c(d)\nof.29,30The observation of\nmultiple events distributed in the sky allows also the study of anisotro pic\noperators.29Furthermore, flavor-mixing operators could be studied by sen-\nsitive measurements of the flavor ratios of astrophysical neutrin os.31\nAt low energies, antineutrinos from the supernova SN1987A have b een\nused to constrain dispersion effects produced by oscillation-free o perators\nof dimension d>4.8Regarding the sensitive interferometric measurements\nof neutrino oscillations, the absence of an antineutrino component in the\nflux of solar neutrinos can be used to determine the most stringent limits\non Majorana couplings for CPT violation in the neutrino sector of the SME\nthat would produce neutrino-antineutrino oscillations.32Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n4\nAcknowledgments\nThis work was supported in part by the German Research Foundatio n\n(DFG) under Grant No. KL 1103/4-1.\nReferences\n1. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n2. F.R. Klinkhamer and M. Risse, Phys. Rev. D 77, 016002 (2008).\n3. F.R. Klinkhamer and M. Risse, Phys. Rev. D 77, 117901 (2008).\n4. F.R. Klinkhamer and M. Schreck, Phys. Rev. D 78, 085026 (2008).\n5. J.S. D´ ıaz and F.R. Klinkhamer, Phys. Rev. D 92, 025007 (2015).\n6. J.S. D´ ıaz, F.R. Klinkhamer, and M. Risse, arXiv:1607.02 099.\n7. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 69, 016005 (2004).\n8. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 85, 096005 (2012).\n9. J.S. D´ ıaz, arXiv:1506.01936.\n10. J.S. D´ ıaz and F.R. Klinkhamer, Phys. Rev. D 93, 053004 (2016).\n11. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 70, 031902 (2004); T. Katori\net al., Phys. Rev. D 74, 105009 (2006); V. Barger et al., Phys. Lett. B 653,\n267 (2007); J.S. D´ ıaz and V.A. Kosteleck´ y, Phys. Lett. B 700, 25 (2011);\nPhys. Rev. D 85, 016013 (2012).\n12.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2016 edition, arXiv:0801.0287v9.\n13. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 70, 076002 (2004).\n14. J.S. D´ ıaz, V.A. Kosteleck´ y, and M. Mewes, Phys. Rev. D 80, 076007 (2009).\n15. L.B. Auerbach et al., Phys. Rev. D 72, 076004 (2005).\n16. P. Adamson et al., Phys. Rev. Lett. 101, 151601 (2008); Phys. Rev. Lett.\n105, 151601 (2010); Phys. Rev. D 85, 031101 (2012).\n17. R. Abbasi et al., Phys. Rev. D 82, 112003 (2010).\n18. T. Katori, Mod. Phys. Lett. A 27, 1230024 (2012).\n19. Y. Abe et al., Phys. Rev. D 86, 112009 (2012).\n20. A.A. Aguilar-Arevalo et al., Phys. Lett. B 718, 1303 (2013).\n21. B. Rebel and S. Mufson, Astropart. Phys. 48, 78 (2013).\n22. J.S. D´ ıaz et al., Phys. Lett. B 727, 412 (2013).\n23. K. Abe et al., Phys. Rev. D 91, 052003 (2015).\n24. J.S. D´ ıaz, V.A. Kosteleck´ y, and R. Lehnert, Phys. Rev. D88, 071902 (2013).\n25. J.S. D´ ıaz, Adv. High Energy Phys. 2014, 305298 (2014).\n26. J.S. D´ ıaz, Phys. Rev. D 89, 036002 (2014).\n27. J.B. Albert et al., Phys. Rev. D 93, 072001 (2016).\n28. M.G. Aartsen et al., Science 342, 1242856 (2013).\n29. J.S. D´ ıaz, V.A. Kosteleck´ y, and M. Mewes, Phys. Rev. D 89, 043005 (2014).\n30. J.S. D´ ıaz, Adv. High Energy Phys. 2014, 962410 (2014).\n31. C.A. Arg¨ uelles et al., Phys. Rev. Lett. 115, 161303 (2015).\n32. J.S. D´ ıaz and T. Schwetz, Phys. Rev. D 93, 093004 (2016)."    },    {        "title": "1202.3131v1.Lorentz_violating_vs_ghost_gravitons__the_example_of_Weyl_gravity.pdf",        "content": "arXiv:1202.3131v1  [gr-qc]  14 Feb 2012YITP-12-3\nLorentz-violating vsghost gravitons: the example of Weyl gravity\nNathalie Deruelle,1Misao Sasaki,2Yuuiti Sendouda,3,1,2and Ahmed Youssef4\n1APC, CNRS-Universit´ e Paris 7, 75205 Paris CEDEX 13, France\n2Yukawa Institute for Theoretical Physics, Kyoto Universit y, Kyoto 606-8502, Japan\n3Graduate School of Science and Technology, Hirosaki Univer sity, Hirosaki, Aomori 036-8561, Japan\n4Institut f¨ ur Mathematik und Institut f¨ ur Physik,\nHumboldt-Universit¨ at zu Berlin, 12489 Berlin, Germany\n(Dated: November 2, 2018)\nWe show that the ghost degrees of freedom of Einstein gravity with a Weyl term can be eliminated\nby a simple mechanism that invokes local Lorentz symmetry br eaking. We demonstrate how the\nmechanism works in a cosmological setting. The presence of t he Weyl term forces a redefinition of\nthe quantum vacuum state of the tensor perturbations. As a co nsequence the amplitude of their\nspectrum blows upwhen the Lorentz-violatingscale becomes comparable tothe Hubbleradius. Such\na behaviour is in sharp contrast to what happens in standard W eyl gravity where the gravitational\nghosts smoothly damp out the spectrum of primordial gravita tional waves.\nPACS numbers: 04.50.Kd, 04.62.+v, 04.30.-w, 98.80.Cq\nI. INTRODUCTION\nFour-dimensional gravity theories based on action integrals which a re non-linear in the curvature invariants have\nbeen part of the landscape of fundamental physics since Weyl [1] in troduced them, soon after Einstein’s invention of\nGeneral Relativity.\nSuch theories possess, apart from the einsteinian ones, new degr ees of freedom (dofs) because the action contains\nterms which are non-linear in the second derivatives of the metric. I t is commonly believed that, except in f(R)\ntheories, see e.g. [2], these new dofs will render even flat spacetime unstable, because they are ghosts, that is, dofs\nwith kinetic terms with the “wrong” sign, which implies that the phase s pace of the whole system is no longer finite\nand its energy is unbounded from below. The reason why that is cons idered dangerous is that any kind of interaction\nterm in the action is expected to yield equations of motion whose solut ions are erratic, unstable and eventually diverge\n[3].\nThe simplest representative of such pathological theories is Weyl g ravity whose action is Einstein-Hilbert’s supple-\nmented by a Weyl-squared term:1\nS[gab] =1\n2κ/integraldisplay\nd4x√−g/parenleftBig\nR−γ\n2CabcdCabcd/parenrightBig\n. (1.1)\nIt was analysed by Stelle [4], who showed the existence of ghosts (no n-tachyonicif γ >0) but did not exhibit explicitly\ntheir malignancy since he worked at linear level, where the dofs do not interact.\nIn [5], we analysed linear cosmological perturbations generated dur ing inflation in Weyl gravity, restricting however\nour attention, as in [6], to the case when the Weyl length scale√γis shorter than the Hubble scale. We found that,\ndespite the fact that no interactions between the dofs were take n into account at this linear level, still the scalarmodes\ndiverge in the newtonian gauge, although they remain bounded in the comoving slicing. It therefore seems that the\nmalignancy of ghosts shows up already at linear level, when the backg round is richer than Minkowski spacetime.\nRealising a potential difficulty associated with Weyl gravity already at linear level, we present in the present paper\na drastic way out of the potential “horrors” of its ghosts. Followin g the lines of [7] and [8] in their covariant version\nof Hoˇ rava gravity [9] we introduce a scalar field that breaks local L orentz covariance (in defining a preferred time\ndirection) and couple it to the Weyl tensor in a way that eliminates its g hosts. However Lorentz-violation will imply a\nmodification of the dispersion relation of the remaining, einsteinian, d ofs. Specialising to an inflationary background\nwe study tensor perturbations and their spectrum (vector and s calar perturbations are not affected by the presence\nof our Weyl term) that we compare and contrast to the behaviour of tensor modes and their spectrum in standard\nWeyl gravity (1.1).\n1Units:c= 1;κ= 8πG,γhas dimension length2,κhas dimension length /mass,/planckover2pi1=κM2\nPl. Indices a,b,···run from 0 to 3; i,j,···\nrun from 1 to 3; metric gabwith inverse gab, determinant gand signature ( −,+,+,+).Rabcd=∂cΓabd−∂dΓabc+···with Γabcthe\nChristoffel symbols; Rbd=Rabad;R=gabRab;Gab=Rab−1\n2gabRandCabcd=Rabcd−1\n2(gacGbd−gadGbc−gbcGad+gbdGac)−\nR\n3(gacgbd−gadgbc).2\nThe organisation of the paper is as follows. In section II, we presen t our model. In section III we show, in a\ncosmological setting, how it eliminates the Weyl ghosts. Section IV d escribes the behaviour of tensor perturbations\non a de Sitter cosmological background. In section V, we quantise t hese tensor perturbations to obtain the spectrum\nof primordial gravitational waves. Section VI develops the analysis of [6] of tensor perturbations in standard Weyl\ngravity when ghost dofs are present. Section VII discusses and s ummarises our results.\nII. THE MODEL\nOur gravity model is specified by the action\nS[gab,χ] =1\n2κ/integraldisplay\nd4x√−g/parenleftbig\nR+2γCabcdCefghγaeγbfγcguduh/parenrightbig\n+Sχ[gab,χ], (2.1)\nwhere\nua≡∂aχ√−∂aχ∂aχandγab≡gab+uaub. (2.2)\nNote that the Weyl-term is conformally invariant, see appendix A, wh ere its relation to the Weyl-square scalar is also\ngiven. For our purpose the specific form of Sχ[gab,χ] for the scalar field χneed not be specified, but is requested to be\nsuch that it yields solutions in which the gradient vector ∂aχis everywhere timelike and future-directed. The vector\nfielduathen determines a preferred time direction and that necessarily imp lies that the theory breaks local Lorentz\ncovariance. Indeed, the spacetime solution is then preferentially f oliated by a family of spacelike hypersurfaces {Σχ}\non each of which χtakes a constant value. Viewed geometrically, uais the future-directed unit normal to Σ χandγab\nis the induced metric on Σ χ. The scalar field χcan be called a chronon as in [7]. Note that it can act as an inflaton\nand drive cosmological inflation.\nThe equations of motion (eoms) for the metric extremise the action with respect to perturbations of the metric and\nread\nGab−γBab=κTχ\nab, (2.3)\nwhereGabis the Einstein tensor, Babis an analogue of the Bach tensor whose expression is given in append ix B, and\nTab\nχ≡2(−g)−1/2δSχ/δgabis the energy-momentum tensor of χ. The eom for χ, on the other hand, is\n1√−gδSχ\nδχ−γ\nκ∇aWa= 0, (2.4)\nwhere∇aWa, the variational derivative with respect to χof the Weyl part of the action, is also given in appendix B.\nAs one can see from the expression of Babin appendix B, the Einstein equation (2.3) contains the derivatives of\nthe metric up to fourth order. One may hence worry that the theo ry should be pathological because higher order\nderivatives theories a priori induce ghosts.\nHowever, as a closer examination shows, see its expression in appen dix B,Babcontains only time derivatives up to\nsecond order if uais timelike. Since ghosts are induced by the presence of timederivatives higher than the second in\nthe eoms, our theory may therefore be free of ghosts. Note tha t our Weyl action does not give rise either to timelike\nderivatives higher than second in the chronon eom (2.4) because th e divergence term only contains spacelike derivative\nof the spacelike vector Wa. The reason why the field equations are not fourth order in time der ivatives is due to\nthe presence of the timelike vector ua, which allows to distinguish time and space derivatives, and thus brea ks local\nLorentz covariance.\nIII. GHOST ELIMINATION BY LORENTZ VIOLATION\nLet us first show explicitly and in a specific case that the eoms are inde ed second order in time.\nWeshallconsidertheexamplewhentheconstant χsurfacesareintrinsicallyflatandtakeaflatFriedmann-Lemaˆ ıtre-\nRobertson-Walker (FLRW) spacetime\ngabdxadxb=a2(η)(−dη2+δijdxidxj) (3.1)\nas a background solution, where ηis conformal time.3\nNote that because of conformal flatness of FLRW spacetimes the Weyl-squared term in the action does not affect\nthe background Friedmann equation for the scale factor a(η) since, then, BabandWavanish, see appendix B; its\nevolution is therefore the same as in Einstein gravity coupled to a sca lar fieldχ=χ(η).\nMetric perturbations about a flat FLRW spacetime are decomposed into scalar, vector and tensor parts as\nδgabdxadxb=a2/bracketleftbig\n−2Adη2+2(∂iB+Bi)dηdxi+(2Cδij+2∂i∂jE+∂iEj+∂jEi+hij)dxidxj/bracketrightbig\n,(3.2)\nwhere∂iBi=∂iEi= 0 and where ∂ihij=hii= 0 (all spatial indices being raised with δij). The following variables\nare gauge-invariant [10]:\nΨ≡A+(B−E′)′+H(B−E′),Φ≡C+H(B−E′),Ψi≡Bi−E′\ni, hij, (3.3)\nwhere the Hubble parameter is defined by H ≡a′/aand where hereafter a prime denotes derivative with respect to\nη.\nThe perturbation to second-order of the action of General Relat ivity with a scalar field can be expressed in terms of\ngauge-invariant perturbations of scalar, vector, and tensor ty pes [10]. The perturbation of our higher-curvature term\nis found to depend on the gauge-invariant tensor and vector varia bles,hijand Ψ i, but not on the scalar variables. It\nreads, see [5] and appendix A,\n(2)SW2[hij,Ψi] =/integraldisplay\ndηd3x√−g(1)Cabcd(1)Cefghγaeγbfγcguduh\n=/integraldisplay\ndηd3x(1)Cijk0(1)Cijk\n0|a(η)=1\n=1\n2/integraldisplay\ndηd3x/parenleftbigg\n∂kh′\nij∂kh′ij+1\n2△Ψi△Ψi/parenrightbigg\n,(3.4)\nwhere△ ≡∂i∂i.\nThis is where the role of the chronon is unveiled; it was so coupled to th e Weyl tensors that the perturbation of the\naction only contains first order time derivatives, whereas the othe r possible combinations, CabcdCefghγaeγbfγcgγdh\nandCabcdCefghγaeubufγcguduh, are quadratic in second order time derivatives, see appendix A.\nIt is clear that the vector perturbations are nondynamical, as no t ime derivatives of Ψ iappear in (3.4). Moreover\nthey are constrained to vanish just as in pure Einstein gravity.\nThe total action for the remaining, tensor, perturbations is (see [10, 11] for the obtention of the Einstein contribu-\ntion):\nST[hij] =1\n8κ/integraldisplay\ndηd3x/bracketleftbig\na2(h′\nijh′ij−∂khij∂khij)+4γ∂kh′\nij∂kh′ij/bracketrightbig\n, (3.5)\nand the corresponding eoms are\n/parenleftbigg\n1−4γ△\na2/parenrightbigg\nh′′\nij+2Hh′\nij−△hij= 0. (3.6)\nWe hence see explicitly that, asannounced, this action containsonly Einstein’s gravitonsas dofs and that the resulting\neoms are second order in time derivatives.\nLet us now check that these tensorial perturbations are not gho sts. To do so we decompose them, as usual, into\ntwo orthogonal polarisations and go to Fourier space:\nhij(η,/vector x) =/summationdisplay\nλ=1,2/integraldisplayd3k\n(2π)3/2eλ\nij(/vectork)hλ\n/vectork(η)ei/vectork·/vector x, (3.7)\nwhere the two polarisation tensors eλ\nij(/vectork), such that eλ\nij(/vectork)eij\nλ′∗(/vectork) =δλ\nλ′, are transverse and traceless, kjeλ\nij(/vectork) =\neλii(/vectork) = 0, and where the reality conditions,\neλ\nij(/vectork) =eλ\nij∗(−/vectork) and hλ\n/vectork(η) =hλ\n−/vectork∗(η), (3.8)\nare imposed, a star denoting complex conjugation. Then the action (3.5) reads, in Fourier space\nST[{hλ\n/vectork}] =1\n8κ/summationdisplay\nλ=1,2/integraldisplay\ndηd3k/bracketleftBig\n(a2+4γk2)|h′λ\n/vectork|2−a2k2|hλ\n/vectork|2/bracketrightBig\n, (3.9)4\nwherek=|/vectork|. The gravitoncan thus be viewed as a collection of non-interacting s calarfields as in General Relativity.\nThe important point to note here is that γmust be positive, otherwise the graviton modes would be tachyonic g hosts\non Minkowski spacetime when a(η) = 1. Therefore ( a2+4γk2) is always positive, so that the kinetic terms |h′λ\n/vectork|2in\n(3.9) are always positive, and the graviton never becomes a ghost o n a FLRW background.\nIV. EVOLUTION OF TENSOR PERTURBATIONS IN INFLATIONARY COSM OLOGY\nWhat is to be checked now is whether or not the Weyl term modifies dr astically the time evolution of the modes.\nThe eom in Fourier space deduced from (3.6) and (3.9) is:\nχ′′\n/vectork+Ω2\nkχ/vectork= 0 with χ/vectork(η)≡/radicalbig\na2+4γk2h/vectork(η), (4.1)\nwhere from now on we omit the index λand where the pulsation Ω k(η) is given by2\nΩ2\nk(η)≡a2/bracketleftbiggk2−H2−H′\na2+4γk2−4γk2H2\n(a2+4γk2)2/bracketrightbigg\n. (4.2)\nWe shall first discuss the time evolution of the modes on a de Sitter ba ckground. Setting\na=1\n−Hηandz≡ −kη, (4.3)\nHbeing a constant, the eom reduces to\nd2χ/vectork\ndz2+z2−2+4ǫ2z2(z2−3)\nz2(1+4ǫ2z2)2χ/vectork= 0 with χ/vectork=k√\n1+4ǫ2z2\nH zh/vectork, (4.4)\nwhere we have introduced the dimensionless parameter\nǫ≡√γH . (4.5)\nEquation (4.4) can be solved exactly in terms of hypergeometric fun ctions, see next section. It is however enlightening\nto study first the qualitative behaviour of the modes.\nAs in General Relativity, a mode with wave number kbecomes larger than the horizon at late times, when z≪1,\nthat is when its physical wavelength a/k= 1/(Hz) becomes much larger than the Hubble radius H−1. As for the\nLorentz violating regime it holds at early times when ǫz≫1, that is when the physical wavelength a/kof the mode\nkis still much shorter than the length scale√γset by the Weyl term.\nLet us first consider the case ǫ≪1, when the length scale√γon which the Lorentz-violating Weyl term operates\nis much shorter than the Hubble radius H−1. (This is the case if the Weyl correction is viewed as a low energy limit\nof some Planck scale quantum theory of gravity, inflation happening at, say, the grand unified theory (GUT) scale.)\nThe modes χ/vectork, which solve (4.4), will then go through three different regimes: the early, Lorentz violating regime,\nwhenǫz≫1; an intermediate regime, 1 ≫ǫz≫ǫwhen they have left the Lorentz violating period but are still\nsub-horizon; and the late stage, z≪1 when they have exited the Hubble scale.\nIn the late and intermediary regimes, when ǫz≪1, which are Lorentz symmetric, equation (4.4) is the same as in\nGeneral Relativity:\nd2χ/vectork\ndz2+/parenleftbigg\n1−2\nz2/parenrightbigg\nχ/vectork≃0, (4.6)\nwhose two independent solutions are well-known: after horizon cro ssing, that is for z≪1, we have χ(g)\n/vectork∝1/zand\nχ(d)\n/vectork∝z2, so that the growing tensor perturbation behaves as h/vectork∝zχ(g)\n/vectork→const., i.e., “freezes out”; and on\nsub-horizon scales z≫1 (butz≪1/ǫ) the modes χ/vectorkoscillate as sin zand cosz.\n2The analysis of the modes can also be performed in terms of cos mic time tsuch that dt=a(η)dη, in which case the eom reads\n¨fk+ω2\nkfk= 0 with fk≡a3/2/radicalbig\n1+4γ(k/a)2hk, where a dot denotes derivative with respect to cosmic time, whereH≡˙a/aand\nwhere\nω2\nk=−1\n4(H2+2˙H)+(k/a)2−(2H2+˙H)\n1+4γ(k/a)2−4γ(k/a)2H2\n[1+4γ(k/a)2]2.5\n510 50100 500z\n/Minus4/Minus2246810Χk/OverRVector/LParen1g/RParen1\n2Εz\n510 50100 500z\n/Minus4/Minus2246810Χk/OverRVector/LParen1d/RParen1\n2Εz\nFIG. 1. The growing and decaying mode functions χ(g)\n/vectork(z) andχ(d)\n/vectork(z) on a de Sitter background, with z=−kη=k/(aH), for\nǫ≡√γH≪1, that is, when the energy scale κM2\nPl/√γof the Weyl-term is much higher than the inflationary scale κM2\nPlH.\nIn the above figures, where ǫ= 0.01 and time increases from right to left, the Lorentz violati ng period ends at around z= 100\nand Hubble radius crossing occurs at around z= 1.\nIn the, early, Lorentz-violating regime, ǫz≫1,χ/vectork∝h/vectork, and the eom (4.4) reduces to\nd2χ/vectork\ndz2+χ/vectork\n4ǫ2z2≃0, (4.7)\nwhose solutions oscillate as χ/vectork∝z1/2±iν/2, withν≡/radicalbig\n1/ǫ2−1 real since ǫ <1. See figure 1.\nLet us now consider the case ǫ≫1, when the length scale on which the Lorentz-violating Weyl term op erates is\nmuchlongerthan the Hubble radius H−1. (This would be the case if the Weyl correction was emerging from so me\neffective theory operating below the inflationary, GUT, energy sca le.)\nAt late times, ǫz≪1≪ǫthe modes behave as before and as in General Relativity: Ω2\nk≃ −2/z2so thatχ(g)\n/vectork∝1/z\nandχ(d)\n/vectork∝z2. In the intermediary stage on the other hand, when 1 ≪ǫz≪ǫ, they are already in the Lorentz-\nviolating stage and since, then, Ω2\nk≃ −3/(4ǫ2z4), they behave as χ/vectork∝ze±√\n3/(2ǫz)and do not oscillate. But the\nreally important difference with the previous caseoccurs at veryea rly times when ǫz≫ǫ≫1. We still have then that\nΩ2\nk≃1/(4ǫ2z2) but the modes, instead of oscillating, behave as χ/vectork∝z1/2±¯ν/2with ¯ν=/radicalbig\n1−1/ǫ2real. Because they\nnever oscillate we can qualify these modes as “rampant”, that is, as “flourishing and spreading unchecked” (Oxford\ndictionary).\nWe have up to now approximated inflation by a de Sitter stage. In the case of slow-roll inflation, the Hubble\nparameter His no longer constant but slowly decreases with cosmic time, i.e. with de creasing z. The previous results\nthen hold when the parameter ǫ=√γHslowly decreases.\nThus, if the energy scale at which inflation starts is lower than the en ergy scale set by the Weyl term, in other\nwords if the Lorentz-violating length scale is always shorter than th e inflationary Hubble radius ( ǫ≡√γH <1), then\nall modes behave as in figure 1. If now inflation starts when the Hubb le radius is shorter than the Lorentz-violating\nscale, then the modes whose wavelengths are big enough to exit the Hubble scale when ǫis still bigger than 1 will\nnever oscillate. On the other hand shorter wavelengths modes, wh ich will still be within the Hubble radius when ǫ\ngoes through the critical value 1 will start in a non-oscillatory way at early times and then will behave qualitatively\nas in figure 1, once ǫbecomes smaller than 1.\nLet us summarise sections III and IV: we confirmed that our Loren tz-violating Weyl action (2.1), when considered\nin a cosmological setting, indeed yields second order equations of mo tion for the tensor perturbations (which are\nthe only ones to be modified by the presence of the Weyl term), see (3.6); we confirmed also that, for γ >0, the\nperturbations neverbecome ghost-likedespite the modification of the coefficient ofthe kinetic term of the gravitondue\nto the Lorentz-violatingWeyl term, see (3.9); finally we studied the perturbation modes, saw that they differ markedly\nfrom standard Einstein-de Sitter gravitons in the distant past, an d distinguished two different early time behaviours\naccording to whether the Lorentz-violating scale is shorter or long er than the Hubble scale. These modifications\nof the early time behaviour of the modes will induce a significant chang e in the spectrum of quantized primordial\ngravitational waves, as we shall now see.6\nV. QUANTISATION OF TENSOR PERTURBATIONS AND PRIMORDIAL GRA VITATIONAL WAVE\nSPECTRUM\nLet us for clarity write again the action (3.5), for the tensor pertu rbations on a FLRW background with scale factor\na(η):\nST[hij] =1\n8κ/integraldisplay\ndηd3x/bracketleftbig\na2(h′\nijh′ij−∂khij∂khij)+4γ∂kh′\nij∂kh′ij/bracketrightbig\n.\nThe momentum conjugate to hijis\nπij=1\n4κ(a2h′ij−4γ△h′ij). (5.1)\nCanonical quantisation turns hijandπijinto operators satisfying the standard commutation relation\n[ˆhij(η,/vector x1),ˆπij(η,/vector x2)] = 2i/planckover2pi1δ(/vector x1−/vector x2), (5.2)\nall other commutators being zero (the factor 2 comes from the fa ct thatˆhijis the sum of two independent dofs).\nExpanding ˆhijin Fourier modes as in (3.7), we further decompose ˆhλ\n/vectorkas\nˆhλ\n/vectork(η) = ˆaλ\n/vectorkhk(η)+ˆ¯aλ\n−/vectork¯hk(η), (5.3)\nwhere ˆaλ\n/vectorkandˆ¯aλ\n/vectorkare arbitrary operators (we introduce ˆ¯aλ\n−/vectorkfor later convenience) and where hk(η) and¯hk(η) are two\nindependent solutions, depending on k=|/vectork|only, of the second order eoms in Fourier space deduced from (3.6) , that\nis:\n/parenleftbigg\n1+4γk2\na2/parenrightbigg\nh′′\nk+2Hh′\nk+k2hk= 0,/parenleftbigg\n1+4γk2\na2/parenrightbigg\n¯h′′\nk+2H¯h′\nk+k2¯hk= 0. (5.4)\nIt follows from the eoms (5.4) that the Wronskian of hk(η) and¯hk(η),\nW(hk,¯hk)≡hk(a2+4γk2)¯h′\nk−¯hk(a2+4γk2)h′\nk, (5.5)\nis a constant ∝ne}ationslash= 0 ifhk(η) and¯hk(η) are independent solutions of (5.4). Hence\nˆhij(η,/vector x) =/summationdisplay\nλ=1,2/integraldisplayd3k\n(2π)3/2/bracketleftBig\neλ\nij(/vectork)ˆaλ\n/vectorkhk(η)ei/vectork·/vector x+eλ\nij∗(/vectork)ˆ¯aλ\n/vectork¯hk(η)e−i/vectork·/vector x/bracketrightBig\n, (5.6)\nwhere we used the reality condition on the polarisation tensors (3.8) . (Note that we have not yet implemented the\nhermiticity condition on ˆhλ\n/vectork(η).)\nPlugging the expansion (5.6) into (5.1), we find that the commutation relations (5.2) are equivalent to\n[ˆaλ\n/vectork1,ˆ¯aλ\n/vectork2] =δ(/vectork1−/vectork2) (5.7)\n(all other commutators being zero) if the Wronskian of the two inde pendent mode functions hkand¯hkis normalised\nto\nW(hk,¯hk)≡hk(a2+4γk2)¯h′\nk−¯hk(a2+4γk2)h′\nk= 4κ/planckover2pi1i. (5.8)\nIn Minkowski space we have a= 1 and the eoms (5.4) for the two independent solutions hk(η) and¯hk(η) reduce to\n¨hk+ω2\nMkhk= 0,¨¯hk+ω2\nMk¯hk= 0 with ω2\nMk=k2\n1+4γk2, (5.9)\nwhereω2\nMkis positive and where, for clarity, a dot denotes derivation with resp ect to cosmic time tsuch that\ndt=a(η)dη.\nThe “positive frequency” modes, which will define the vacuum |0∝an}b∇acket∇i}htsuch that ˆ a/vectork|0∝an}b∇acket∇i}ht= 0 and ˆ¯a†\n/vectork|0∝an}b∇acket∇i}ht= 0, are chosen\nto be\nhk=nke−iωMkt,¯hk= ¯nke+iωMkt, (5.10)7\nwhere the coefficients nkand ¯nkmust be such that the hermiticity and Wronskian conditions, see (3.8 ) and (5.8), are\nsatisfied, that is, such that\nˆaλ\n/vectorknk=ˆ¯aλ\n/vectork†¯n∗\nk, nk¯nk=2κ/planckover2pi1ωMk\nk2, (5.11)\nwhich impose (up to irrelevant constants)\nˆaλ\n/vectork=ˆ¯aλ\n/vectork†, nk= ¯n∗\nk=√2κ/planckover2pi1ωMk\nk. (5.12)\nTherefore, the choice for the modes is\nhk=¯h∗\nk=√2κ/planckover2pi1ωMk\nke−iωMkt. (5.13)\nNote for further reference that in the short wavelength limit, γk2→ ∞, the pulsation becomes independent of k:\nωMk→(4γ)−1/2, so that we have\nhk→√\nκ/planckover2pi1\nkγ1/4e−it/(2√γ). (5.14)\nNext we take accelerated cosmological expansion into account, wh ere another scale H, the Hubble parameter,\ncomes into play.\nIn the de Sitter case when a= 1/(−Hη) withH= const. and ηconformal time, the eoms for the modes as given\nin (5.4) read\n(1+4y2)d2hk\ndy2−2\nydhk\ndy+hk\nǫ2= 0,(1+4y2)d2¯hk\ndy2−2\nyd¯hk\ndy+¯hk\nǫ2= 0, (5.15)\nwhere\nǫ≡√γHandy≡ −ǫkη. (5.16)\nAs for the Wronskian normalisation condition (5.8) it becomes\nhkd¯hk\ndy−¯hkdhk\ndy=−i4κ/planckover2pi1H2\nk3ǫ3y2\n1+4y2. (5.17)\nNow, two independent solutions of (5.15) are\nh(g)(y) =1\n2F/parenleftbigg−1−iν\n4,−1+iν\n4,−1\n2;−4y2/parenrightbigg\n, h(d)(y) =32\n3y3F/parenleftbigg5+iν\n4,5−iν\n4,5\n2;−4y2/parenrightbigg\n(5.18)\nwith\nν≡/braceleftBigg/radicalbig\n1/ǫ2−1 if 0 < ǫ <1\ni/radicalbig\n1−1/ǫ2ifǫ >1. (5.19)\nNote that the hypergeometric functions introduced in (5.18) are r eal, whether νis real or imaginary, and that\nh(g)(0) =1\n2, h(d)(0) = 0. (5.20)\nIn order now to determine which combinations of h(g)andh(d)we must choose as our independent modes hkand\n¯hkwe have to study the early-time, large klimit.\nWheny→ ∞we have\nh(g)(y)→c−\n(g)(2y)1/2+iν/2+c+\n(g)(2y)1/2−iν/2, h(d)(y)→c−\n(d)(2y)1/2+iν/2+c+\n(d)(2y)1/2−iν/2(5.21)\nwith, when νis real,\nc+\n(g)=c−\n(g)∗=−√πΓ(−iν/2)\nΓ2(−1/4−iν/4), c+\n(d)=c−\n(d)∗=√πΓ(−iν/2)\nΓ2(5/4−iν/4). (5.22)\n(Whenνis imaginary then the coefficients are real but unequal.)8\nIncosmictime t, suchthat dt=adη, wehave a= eH tandH y=kǫe−H t, sothat, for νreal(i.e.for0 < γH2<1):\ny1/2+iν/2∝e−H t/2e−iωdStwith ωdS=/radicalbig\n1−γH2\n2√γ, (5.23)\nwhich, when γH2≪1, identifies with the Minkowski positive high frequency modes chose n in (5.10):\ny1/2+iν/2∝e−it/(2√γ). (5.24)\nWe recover here what we have discussed in the previous section, to wit that the early time behaviour of the modes\ndiffers drastically depending on the value of γH2: they oscillate if νis real (γH2<1), and are “rampant” if νis\nimaginary ( γH2>1). It is clear that quantisation will make (easy) sense only when mod es can qualify as “positive\nfrequency ”, which implies that they must oscillate. We shall therefore suppose from now on that\n0< γH2<1⇐⇒0< ǫ≡√γH <1⇐⇒ν≡/radicalbig\n1/ǫ2−1 real. (5.25)\nIn keeping to what we chose when the background is Minkowski spac etime, we shall hence impose that the two\nindependent solutions hkand¯hkbehave, for large y, as\nhk(y)→nk(2y)1/2+iν/2,¯hk(y)→¯nk(2y)1/2−iν/2. (5.26)\nAs for the values of the coefficients nkand ¯nkthey follow from the large ylimit of Wronskian normalisation condi-\ntion (5.17) as well as the hermiticity condition, see (3.8), which impose s, up to irrelevant constant that\nˆaλ\n/vectork=ˆ¯aλ\n/vectork†, nk= ¯n∗\nk=/radicalbigg\nκ/planckover2pi1\n2νk3ǫ3H . (5.27)\nThus, all in all, the hermitian operator ˆhijreads\nˆhij(η,/vector x) =/summationdisplay\nλ=1,2/integraldisplayd3k\n(2π)3/2/bracketleftBig\neλ\nij(/vectork)ˆaλ\n/vectorkhk(η)ei/vectork·/vector x+h.c./bracketrightBig\n, (5.28)\nwhere the modes are:\nhk=¯h∗\nk=−i/radicalbigg\nκ/planckover2pi1ν\n8k3ǫ3H(c+\n(d)h(g)−c+\n(g)h(d)), (5.29)\nwhere the functions h(g)andh(d)and the coefficients c(g)andc(d)are defined in (5.18) and (5.22) (and where we used\nthe fact that c+\n(d)c−\n(g)−c−\n(d)c+\n(g)= 2i/ν). Finally the “Bunch-Davies” vacuum |0∝an}b∇acket∇i}htis defined as usual by\nˆaλ\n/vectork|0∝an}b∇acket∇i}ht= 0. (5.30)\nThe power spectrum P(k;η) of the gravitational waves hijis now defined as\n∝an}b∇acketle{t0|ˆhij(η,/vector x1)ˆhij(η,/vector x2)|0∝an}b∇acket∇i}ht=/integraldisplay\nd3kP(k;η)\n4πk3ei/vectork·(/vector x1−/vector x2)(5.31)\nand is given by, using the commutation rule (5.7):\nP(k;η) =k3\nπ2|hk(η)|2. (5.32)\nAfter horizon crossing, that is in the limit η→0, we have hk∝k−3/2, see (5.29), and the power spectrum is scale\ninvariant just like the power spectrum of ordinary gravitational wa ves in Einstein theory on a de Sitter background.\nHowever its amplitude depends on ν=/radicalbig\n1/ǫ2−1 and is, using the late time limit of h(g)andh(d), see (5.20)\nP(k;η→0)≡2κH2\nπ2Ξ,where Ξ =cosh(πν/2) coth(πν/2)Γ2(−1/4+iν/4)Γ2(−1/4−iν/4)\n128π2ǫ3,(5.33)\nwhere we used the relations Γ( −iν/2)Γ(iν/2) = 2π/[νsinh(πν/2)] and Γ(5 /4−iν/4)Γ(5/4 + iν/4)Γ(−1/4−\niν/4)Γ(−1/4+iν/4) = 2π2/cosh2(πν/2), see figure 2.9\n0.0 0.2 0.4 0.6 0.8 1.0Ε0.51.01.52.0/CapXi\nFIG. 2. The modification factor of the power spectrum of the te nsor perturbations on a de Sitter background as a function of\nǫ=γH2.\nThis is the main result of the paper: In our ghost-free but Lorentz violating Weyl theory of gravity the spectrum\nof gravitational waves, when evaluated on a de Sitter background , is scale-invariant, as in General Relativity, but its\namplitude in the late time limit varies with ǫ≡√γH, where√γis the length scale on which the Weyl term operates\nandH−1is the Hubble radius. For small ǫ, Ξ≈1: the spectrum of primordial gravitational waves is almost the\nsame as in General Relativity [11] because the Lorentz symmetric re gime before horizon crossing lasts long (it spans\naz-interval much bigger than 1); as ǫincreases, that is as the Lorentz-violating length scale approache s the Hubble\nradius, its amplitude decreasesup to a factor 65%, before eventu ally blowing up as ǫgrowsfurther and approachesthe\ncritical value 1. For ǫ >1 the spectrum, as we saw, cannot be defined as the vacuum expec tation value of quantized\nmodes since they do not oscillate at early time. Another criterion mus t then be chosen to define it in this regime.\nInanycase, thespectrumofprimordialgravitationalwavesisgre atlymodified, iftheLorentz-violatingscalehappens\nto be of the same order of magnitude or larger than the Hubble radiu s. But, again, it must be stressed that only\nwhen the Hubble radius is considerably larger than the Weyl-correct ion characteristic length scale ( ǫ≪1) do the high\nfrequency modes become those of flat spacetime and it is only then t hat there is a natural definition of the vacuum\nstate.\nIn slow-roll inflation now, the Hubble parameter His no longer constant but becomes a slowly decreasing function\nof cosmic time. The previous results therefore still hold but H(and hence ǫ=√γHandν=/radicalbig\n1/ǫ2−1) in the\nexpression (5.33) of the spectrum may be evaluated around horizo n crossing when H(t) =k/a(t). This allows to\nexpresst, and therefore H,ǫandνin terms of k. Sinceǫ(k) is a decreasing function of k, the qualitative behaviour\nof gravitational wave spectrum P(k;t|H=k/a) is then read off from the right to left in figure 2 when kincreases. What\nhappens when ǫ(k) crosses the critical value 1 during inflation remains to be investigat ed.\nVI. PRIMORDIAL GRAVITATIONAL WAVES IN EINSTEIN PLUS PURE WE YL SQUARE GRAVITY\nIn order to compare and contrast the ghost-free but Lorentz- violating Weyl theory of gravity that we have studied\nin this paper with ordinary “ghastly”3Weyl theory, we summarise and develop here the results of [6] to ob tain the\nexact spectrum of gravitational waves on a de Sitter background in Weyl gravity.\nWhen expanded to second orderin the tensorialperturbations ar ounda FLRW background,see (3.2) for definitions,\nthe action of Weyl gravity (1.1) becomes (see see [5] or appendix A)\nS[hij] =1\n8κ/integraldisplay\ndηd3x/bracketleftbig\na2(h′\nijh′ij−∂khij∂khij)−γ(h′′\nijh′′ij−2∂kh′\nij∂kh′ij+∂klhij∂klhij)/bracketrightbig\n,(6.1)\nwhich must be compared to (3.5). (Recall, see [4], that γmust be positive to avoid the ghost dofs to be tachyonic on\nMinkowski spacetime.) Proceeding as in [6] “` a la” Ostrogradsky we in troduce a new variable Qij≡h′\nijas well as a\n3i.e. “causing great horror or fear” (Oxford dictionary)10\nLagrange multiplier λij(both of them transverse traceless) and consider the equivalent action\nS[hij,Qij,λij] =1\n8κ/integraldisplay\ndηd3x/bracketleftbig\na2(h′\nijh′ij−∂khij∂khij)\n−γ(Q′\nijQ′ij−2∂kh′\nij∂kh′ij+∂klhij∂klhij)+2λij(Qij−h′\nij)/bracketrightbig\n.(6.2)\nThe conjugate momenta are\nπij\nh=1\n4κ(a2h′ij−2γ△h′ij−λij), πij\nQ=−γ\n4κQ′ij. (6.3)\nThe eoms obtained by extremisation of the action with respect to λijandQijare\nQij=h′\nij, γQ′′\nij+λij= 0 (6.4)\nand allow to express the momenta in terms of hijalone as\nπij\nh=1\n4κ(a2h′ij−2γ△h′ij+γh′′′ij), πij\nQ=−γ\n4κh′′ij. (6.5)\nQuantization is implemented by imposing the commutation relations\n[ˆhij(η,/vector x1),ˆπij\nh(η,/vector x2)] = 2i/planckover2pi1δ(/vector x1−/vector x2),[ˆQij(η,/vector x1),ˆπij\nQ(η,/vector x2)] = 2i/planckover2pi1δ(/vector x1−/vector x2), (6.6)\nall other commutators being zero and the factor 2 coming from the fact that both hijandQijrepresent two dofs.\nWe now expand ˆhijin Fourier modes\nˆhij(η,/vector x) =/integraldisplayd3k\n(2π)3/2/bracketleftbigg/parenleftbigg/summationdisplay\nλ=1,2eλ\nij(/vectork)ˆaλ\n/vectorkh(1)\nk(η)+/summationdisplay\nλ=1,2eλ\nij(/vectork)ˆbλ\n/vectorkh(2)\nk(η)/parenrightbigg\nei/vectork·/vector x+h.c./bracketrightbigg\n. (6.7)\nPlugging this expansion into the definitions of the momenta (6.5), we fi nd that the commutation relations (6.6) are\nequivalent to imposing (omitting the index λ)\n[ˆa/vectork1,ˆa†\n/vectork2] =δ(/vectork1−/vectork2),[ˆb/vectork1,ˆb†\n/vectork2] =−δ(/vectork1−/vectork2) (6.8)\n(all other commutators vanishing) if the mode functions satisfy th e following Wronskian conditions\nh(1)\nk[(a2+2γk2)h′(1)\nk∗+γh′′′(1)\nk∗]−h(2)\nk[(a2+2γk2)h′(2)\nk∗+γh′′′(2)\nk∗]−c.c.= 4κi/planckover2pi1,\nh′(1)\nkh′′(1)\nk∗−h′(2)\nkh′′(2)\nk∗−c.c.=−4κi/planckover2pi1\nγ.(6.9)\nThe constancy of the Wronskians is ensured if the mode functions h(1)\nkandh(2)\nkand their complex conjugates are four\nindependent solutions of the eom for hijwhich reads, in Fourier space\n(a2h′\nk)′+a2k2hk+γ(h′′′′\nk+2k2h′′\nk+k4hk) = 0. (6.10)\nLet us now specialise to a de Sitter background where a= 1/(−Hη),Hbeing a constant and ηconformal time.\nThen, as shown in [6] the eom (6.10) factorises neatly: if h(1)\nk≡zµ(1)\nkandh(2)\nk≡zµ(2)\nkare taken to solve, respectively,\n(withz≡ −kη)\nd2µ(1)\nk\ndz2+/parenleftbigg\n1−2\nz2/parenrightbigg\nµ(1)\nk= 0,\nd2µ(2)\nk\ndz2+/parenleftbigg\n1+1\nγH2z2/parenrightbigg\nµ(2)\nk= 0,(6.11)\nthen, as can be shown explicitly, they satisfy the original eom (6.10) and, also, the Wronskian conditions (6.9) if\nh(1)\nkdh(1)\nk∗\ndz−c.c.=h(2)\nkdh(2)\nk∗\ndz−c.c.=−4κi/planckover2pi1z2\nγk3[2+1/(γH2)]. (6.12)11\nThe first equation (6.11) can be seen as the eom for the usual Einst ein gravitonand the second for the Weyl ghostdofs.\nWhat remains to be done is to choose “positive frequency” modes wh ich define a “Bunch-Davies” vacuum state |0∝an}b∇acket∇i}ht\nsuch that ˆ ak|0∝an}b∇acket∇i}ht=ˆbk|0∝an}b∇acket∇i}ht= 0. This is easy in this case since there is no violation of Lorentz covar iance, so that the\neoms (6.11) both reduce to the standard Minkowski eoms for mass less fields for z≫1 andz≫1/(γH2), that is in\nthe remote past when the physical wavelengths a/kof the modes are much shorter than both the Hubble radius scale\nH−1and the length scale√γset by the Weyl term. We therefore impose that for z→ ∞,\nh(1)\nk∼h(2)\nk∼/radicalbigg\n2κ\nk3H/radicalbig\n1+2γH2zeiz. (6.13)\nThe solutions of the eoms (6.11) having this early time/short wavelen gth behaviour are, up to a phase\nh(1)\nk(z) =/radicalbigg\n2κ\nk3H/radicalbig\n1+2γH2(1+iz)eiz,\nh(2)\nk(z) =/radicalbigg\n2κ\nk3H/radicalbig\n1+2γH2/radicalbiggπ\n2eiπ¯ν/2z3/2H(1)\n¯ν(z)(6.14)\nwith ¯ν≡(1/2)/radicalbig\n1−4/(γH2).\nThe power spectrum P(k;z) of the gravitational waves hijis again defined as\n∝an}b∇acketle{t0|ˆhij(η,/vector x1)ˆhij(η,/vector x2)|0∝an}b∇acket∇i}ht=/integraldisplay\nd3kP(k;η)\n4πk3ei/vectork·(/vector x1−/vector x2)(6.15)\nand is given by\nP(k;z) =k3\nπ2/parenleftBig\n|h(1)\nk(z)|2−|h(2)\nk(z)|2/parenrightBig\n=2κH2\nπ21\n1+2γH2/parenleftBig\n1+z2−π\n2z3|eiπ¯ν/2H(1)\n¯ν(z)|2/parenrightBig\n.(6.16)\nWhatever the value of γH2, that is whatever the sign of ¯ ν2=1\n4[1−4/(γH2)], the last two terms do not contribute\nto the power spectrum when z→0, that is, at late time when the modes have exited the Hubble radius, and we have\nthat\nP(k;z→0)≡2κH2\nπ2ΞW,where Ξ W=1\n1+2γH2. (6.17)\n(This generalises [6], where the spectrum was computed for small va lues of the parameter γH2only.)\nThis de Sitter spectrum of primordial gravitational waves at late tim es, obtained in Einstein plus pure Weyl-square\ngravity based on the action (1.1), and that obtained in (5.33) and fig ure 2 in the ghost-free but Lorentz-violating\nWeyl gravity theory based on the action (2.1), are very different: contrarily to Ξ, Ξ Wnever blows up and, as γH2\ntends to infinity, goes to zero as ( γH2)−1.\nVII. CONCLUSION\nWe have proposed and investigated a mechanism to eliminate the ghos t degrees of freedom of higher-curvature\ngravity theories. The mechanism invokes the gradient of a scalar fie ld which is timelike and therefore implies a\nbreakdown of Lorentz covariance. Although we only discussed her e the particular example of Weyl gravity, the\nmechanism appears generic enough to eliminate any higher-derivativ e ghosts. We expect that a canonical way of\nanalysing generic higher-curvature gravity, developed by the aut hors [12], will be useful in studies in this direction.\nWe investigated quantisation of the inflationary tensor perturbat ions in our ghost-free Weyl gravity model and\ndefined the “positive frequency” modes as those reducing to flat s pacetime, Lorentz-violating, positive frequency\nmodes. Such a modification of the quantum vacuum state may offer a n observable signature of Lorentz violation at\nshort wavelengths. Indeed, we found that in de Sitter inflation this modification of the vacuum state gives rise to an\nextra overall factor for the super-horizon power spectrum of g ravitational-wave background generated from quantum\nfluctuations.\nSince, from a phenomenological point of view, the energy scale of Lo rentz violation κM2\nPl/√γcan be as low as the\nHubble parameter during inflation we expect that inflationary cosmo logical perturbations may open a useful window\nto the physics of Lorentz violation.12\nACKNOWLEDGMENTS\nND thanks YITP for its enduring and generous hospitality. She is also grateful to Hirosaki University where this\nwork was completed. YS thanks ASC at LMU for hospitality, where pa rt of this work was done. He also thanks\nShunichiro Kinoshita for useful conversations. This work was supp orted in part by MEXT thorough Grant-in-Aid\nfor Scientific Research (A) No. 21244033 and Grant-in-Aid for Cre ative Scientific Research No. 19GS0219 (MS) and\nby JSPS through Postdoctoral Fellowship for Research Abroad an d Grant-in-Aid for JSPS Fellows (YS). This work\nwas also supported in part by MEXT through Grant-in-Aid for the Glo bal COE Program “The Next Generation of\nPhysics, Spun from Universality and Emergence” at Kyoto Universit y.\nAppendix A: Weyl-squared actions\nThe Weyl-squared action SW2=/integraltextd4x√−gCabcdCefghγaeγbfγcguduhdiscussed in the main text is one of the\nthree invariants consisting of quadratic Weyl tensor fully contrac ted with γabandua; the other two are\nSW0=/integraldisplay\nd4x√−gCabcdCefghγaeγbfγcgγdh, SW4=/integraldisplay\nd4x√−gCabcdCefghγaeubufγcguduh.(A1)\nThey are such that\n−4SW2+SW0+4SW4=/integraldisplay\nd4x√−gCabcdCabcd, (A2)\nhence the coefficient −γ/2 in (1.1) and +2 γin (2.1).\nA useful fact is that these actions are invariant under a conforma l transformation, gab= Ω2˜gab: the Weyl tensor\nis invariant, Cabcd=˜Cabcd, whereas the geometrical quantities associated with the chronon field are transformed\nasua= Ω˜uaandγab= Ω2˜γab, respectively. Then it can be checked that SW0,SW2andSW4are all conformally\ninvariant.\nThe expansion of these actions to second order in gauge-invariant perturbations around FLRW spacetimes are (see\nmain text for definitions and with W≡Ψ−Φ and [5]):\n(2)SW2=−1\n4/integraldisplay\ndηd3x(−2∂kh′\nij∂kh′ij−△Ψi△Ψi),\n(2)SW0= 4(2)SW4\n=/integraldisplay\ndηd3x/bracketleftbigg1\n4h′′\nijh′′ij+1\n4△hij△hij+1\n2∂kh′\nij∂kh′ij+1\n2∂iΨ′\nj∂iΨ′j+2\n3(△W)2/bracketrightbigg\n.(A3)\nAppendix B: Derivation of the equations of motion\nTocompute thevariationofourghost-freeWeylactionwith respe cttothe metric, it isuseful toemployanADM-like\nformalism. To begin with, we express the action as\nSW2=/integraldisplay\ndDx√−gCabcdCefghγaeγbfγcguduh≡/integraldisplay\ndDx√−gWabcWabc, (B1)\nwhere\nua≡N ∂aχ, N≡1√σ∂aχ∂aχ, γab≡gab−σuaub, σ≡uaua, Wabc≡γadγbeγcfugCdefg.(B2)\nNote that W[ab]c=Wabc,uaWabc=uaWbca= 0, and Wabb= 0. We assume σnever vanishes so that the lapse\nfunction Nremains finite. We also define the extrinsic curvature of the constant- χsurfaces by, ∇being the covariant\nderivative,\nKab≡γacγbd∇duc. (B3)\nThe Codazzi relation gives\nWabc=γadγbeγcf/bracketleftbigg\n2∇[dKe]f−2\nD−2∇[d(Ke]f−Kγe]f)/bracketrightbigg\n, (B4)13\nwhereK≡γabKab. In order to avoid copious appearances of γabandua, we introduce the following notations:\nγcdγaeγfb···∇dTe···f···→DcTa···b···, udγaeγfb···∇dTe···f···→ ∇uTa···b···, uaTa···b···→Tu···b···(B5)\nand so on. With this, we can write\nKab=Daub, Wabc= 2D[aKb]c−2\nD−2D[a(Kb]c−Kγb]c). (B6)\nIt should be noted that the tensor Wabconly contains one derivative along the direction of ua.\nLet us concentrate on the variation with respect to the metric ten sor,gab→gab+δgab. Define the variables\nA≡σ\n2uaubδgab, Ba≡γabucδgbc, Cab≡1\n2γacγbdδgcd. (B7)\nThen we have e.g.\nδgab= 2Cab+2σu(aBb)+2σAuaub, δ√−g=C+A, (B8)\nwhereC≡γabCab. We also have\nδua=Aua, δua=−Aua−Ba, δγab= 2Cab+2σu(aBb), δγab=σuaBb, δγab=−2Cab,(B9)\nand\nδKab=∇uCab+2K(acCb)c−N−1D(a(N Bb))−KabA+2σu(aKb)cBc. (B10)\nThe variation of the Weyl action with respect to the metric can henc e be computed, up to divergences, as\nδSW2=1\n2/integraldisplay\ndDx√−gBabδgab, (B11)\nwhere\nBab≡4∇u[N−1Dc(N Wc(ab))]+4N−1Dc(N Wc(ab))K−4N−1Dc(N Wcd(a)Kdb)−4N−1Dc(N Wc(a|d|)Kdb)\n−4Dd[N−1Dc(N Wcd(a)]ub)−4Dd[N−1Dc(N Wc(a|d|)]ub)+4σN−1Dc(N Wcde)Kdeuaub\n+4N−1Dc(N W(a|dc|Kb)d)+4N−1Dc(N Wcd(aKb)d)+4N−1Dc(N Wd(ab)Kcd)\n+8\nD−2Wc(ab)Dd(Kcd−Kγcd)−4WacdWbcd−2WcdaWcdb+WcdeWcdegab.\n(B12)\nIt can be shown that Babtensor is traceless and free from the trace part of Kab.\nA most important property of Babis that it contains derivatives ∇ualonguaup to second order only (in the first\nterm which is of the type ∇uDWwithW∼D∇uγ); hence the eoms remain second order in timelike derivatives\nwhenever uais timelike. Note also that the other fourth order terms in Babare all of the type DDWwhich are\nfirst order in time and third order in space when uais timelike. Because of this distinction between space and time\nderivatives Babis not Lorentz covariant.\nThe variations of the quantities with respect to χare given by\nδua=Va, δγab=−2σu(aVb)whereVa≡N γab∂bδχ. (B13)\nThe variation of the Weyl action with respect to χis then computed, up to divergence, as\nδSW2=−/integraldisplay\ndDx√−g∇aWaδχ, (B14)\nwhere\nWa≡4σWabcCbucu+2γabWcdeCbcde. (B15)\nNote that we may rewrite as ∇aWa=N−1Da(N Wa).14\n[1] H. Weyl, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) , 465 (1918); Annalen Phys. (Leipzig) 59, 101 (1919);\nRaum - Zeit - Materie , 5th ed. (Springer-Verlag, Berlin, 1923) Chap. IV, [Englis h translation Space, Time, Matter (fourth\nedition, Dover Publications, NewYork, USA, 1952)].\n[2] A. De Felice and S. Tsujikawa, Living Rev. Rel. 13, 3 (2010), arXiv:1002.4928 [gr-qc].\n[3] A. Pais and G. E. Uhlenbeck, Phys. Rev. 79, 145 (1950); J. M. Cline, S. Jeon, and G. D. Moore,\nPhys. Rev. D70, 043543 (2004), arXiv:hep-ph/0311312 [hep-ph]; R. Woodar d, Lect. Notes Phys. 720, 403 (2007),\narXiv:astro-ph/0601672 [astro-ph]; A. V. Smilga, Nucl. Ph ys.B706, 598 (2005), arXiv:hep-th/0407231 [hep-th];\nSIGMA 5, 017 (2009), arXiv:0808.0139 [quant-ph].\n[4] K. S. Stelle, Gen. Rel. Grav. 9, 353 (1978).\n[5] N. Deruelle, M. Sasaki, Y. Sendouda, and A. Youssef, J. Co smol. Astropart. Phys. 1103, 040 (2011),\narXiv:1012.5202 [gr-qc].\n[6] T. Clunan and M. Sasaki, Class. Quant. Grav. 27, 165014 (2010), arXiv:0907.3868 [hep-th].\n[7] D. Blas, O. Pujolas, and S. Sibiryakov, J. High Energy Phy s.10, 029 (2009), arXiv:0906.3046 [hep-th].\n[8] C. Germani, A. Kehagias, and K. Sfetsos, J. High Energy Ph ys.0909, 060 (2009), arXiv:0906.1201 [hep-th].\n[9] P. Horava, Phys.Rev. D79, 084008 (2009), arXiv:0901.3775 [hep-th].\n[10] J. M. Bardeen, Phys. Rev. D22, 1882 (1980); H. Kodama and M. Sasaki, Prog. Theor. Phys. Sup pl.78, 1 (1984); V. F.\nMukhanov, H. A. Feldman, and R. H. Brandenberger, Phys. Rept .215, 203 (1992).\n[11] L. P. Grishchuk, J. Exp. Theor. Phys. 40, 409 (1975); A. A. Starobinskii, J. Exp. Theor. Phys. Lett. 30, 682 (1979).\n[12] N. Deruelle, M. Sasaki, Y. Sendouda, and D. Yamauchi, Pr og. Theor. Phys. 123, 169 (2010), arXiv:0908.0679 [hep-th]."    },    {        "title": "0707.3247v1.Cosmological_Particle_Creation_in_the_Presence_of_Lorentz_Violation.pdf",        "content": "arXiv:0707.3247v1  [hep-ph]  22 Jul 2007Cosmological Particle Creation in the Presence of Lorentz V iolation\nE. Khajeh,∗N. Khosravi,†and H. Salehi‡\nDepartment of Physics, Shahid Beheshti University, Evin, T ehran 19839, Iran\nIn recent years, the effects of Lorentz symmetry breaking in c osmology has attracted considerable\namount of attention. In cosmological context several topic s can be affected by Lorentz violation,e.g.,\ninflationary scenario, CMB, dark energy problem and barryog enesis. In this paper we consider the\ncosmological particle creation due to Lorentz violation (L V). We consider an exactly solvable model\nfor finding the spectral properties of particle creation in a n expanding space-time exhibiting Lorentz\nviolation. In this model we calculate the spectrum and its va riations with respect to the rate and\nthe amount of space-time expansion.\nPACS numbers: 04.62.+v, 11.30.Cp, 98.80.-k, 98.80.Jk\nI. INTRODUCTION\nParticle creation in the cosmological context is one of\nthe most interesting feature of QFT in curved space time\n[1]. In this process, a quantum field propagates on an ex-\npanding space-time and the field quanta (particles) are\ngenerated through an impulse of the space-time expan-\nsion. The density and the rate of particle production\ndepend on the vigor of the expansion.\nFurthermore, it is well known that if a conformally in-\nvariant field propagates in a space-time which is confor-\nmally equivalent to Minkowski (conformal triviality) no\nparticle production occurs. Thus, we expect that mass-\nless quanta of Maxwell and Dirac fields do not arise from\nthe expansion of Universe. But if a term is added to the\nfield equation such that conformal invariance is broken\nthen these particles are created [1, 2].\nQFT in curved background is based on the hypothesis\nthatthe fieldequationsarelocallyLorentzinvariant. But\nin recent years there have been many attentions to the\npossibility of Lorentz symmetry breaking at high ener-\ngies. The initial motivation came from string theory [3],\nand more recently this symmetry breaking has been dis-\ncussed in the context of noncommutative geometry [4, 5].\nAlso, there has been evidence that this symmetry may be\nbroken in at least three different phenomena:\ni) Observation of ultra-high energy cosmic rays with\nenergies beyond the so-called GZK cutoff, EGZK⋍\n1019eV[6, 7].\nii) Events involving gamma radiation with energies be-\nyond 20TeV from distant sources such as Markarian 421\nand Markarian 501 blazers [8].\niii)Studies of the evolution of air showers produced by\nultra high-energy hadronic particles suggest that pions\nlive longer than expected [9]. These observations can\nbe explained via the breaking of Lorentz symmetry [10,\n11]. Now, if we want to consider the effect of LV in\nQFT in curved space-time, an important question for\n∗e-khajeh@sbu.ac.ir\n†n-khosravi@sbu.ac.ir\n‡h-salehi@sbu.ac.irinvestigation is: How dose Lorentz violation affect the\nparticle creation in an expanding space-time?\nThe possible effects of LV physics in inflationary cos-\nmology has been studied in [12, 13, 14, 15]. The au-\nthors found that the spectrum of fluctuations in infla-\ntionary cosmology and so the spectrum of temperature\nanisotropiesintheCosmicMicrowaveBackgroundcanbe\naffected by LV. Also it is shown that Lorentz violation\nis relevant to the dark energy problem and barryogene-\nsis [16, 17, 18, 19]. In the present work we look for the\neffects of LV in particle creation process in an expanding\nspace-time. Our interest is to study the form and prop-\nerties of particle creation spectrum of a scalar field. For\nthis purpose we choose an exactly solvable model and we\nfind the characteristics of the spectrum.\nWe benefit from the model that has been introduced\nin [20, 21] for LV to study this subject. In this model\nthe usual dispersion relation is modified by adding a\ntermα2k4. The modified dispersion relation preserves\nrotational invariance but violates the boost invariance of\nLorentzsymmetry. It has been shown that addinga term\nα2k4to the dispersion relation is equivalent to putting a\nvector field which is coupled with matter field in the La-\ngrangian of the model. This vector field can physically\nbe assumed as the four velocity of a preferred inertial\nobserver. The additional term in the Lagrangian which\nenforces the LV, breaks the conformal invariance. There-\nfore we expect that the massless field quanta are created\nduring the space-time expansion.\nThe organization is as follows: in section II we quan-\ntize the Lorentz violation model introduced in [20] on\nMinkowski space-time. In section III we define an ex-\npanding cosmological model and consider the quantiza-\ntion of the LV model on this space-time. Finally, we ob-\ntain and discuss the spectrum characteristics of created\nmassless particles in details.\nII. LORENTZ VIOLATION MODEL AND\nQUANTIZATION IN MINKOWSKI SPACE-TIME\nAs was mentioned before, in the model introduced for\nLV in [20] the dispersion relation is modified by an ad-\nditional term α2k4and the modified dispersion relation2\nreads as follows\nω2(/vectork) =/vextendsingle/vextendsingle/vextendsingle/vectork/vextendsingle/vextendsingle/vextendsingle2\n−α2/vextendsingle/vextendsingle/vextendsingle/vectork/vextendsingle/vextendsingle/vextendsingle4\n. (1)\nA Lagrangian that can lead the above dispersion relation\nfor a scalar field φis\nL=1\n2(∂µϕ∂µϕ+α2(D2ϕ)2), (2)\nwhereαis a constant (of order the Planck energy) that\nsets the scale of the Lorentz violation and D2is the spa-\ntial Laplacian, that is\nD2ϕ=−DµDµϕ=−qµν∂ν(qτ\nµ∂τϕ),(3)\nwhereqµνis the (positive definite) spatial metric orthog-\nonal to the unit timelike vector uµ\nqµν=−ηµν+uµuν, ηµνuµuν= 1.(4)\nThe vector field uµcan physically be interpreted as the\nfour velocity of a preferred inertial observer. The rest\nframe of this preferred observer may be called aether. In\nthis rest frame we have uµ=(1,0,0,0).\nFor the subsequent consideration in this chapter we\nshall take a coordinate system in which uµis constant\n(The aether is a particular example for such a coordinate\nsystem). With this assumption the equation of motion\nfor the Lagrangian (2) is\n[/square−α2qµνqγδ∂µ∂ν∂γ∂δ]ϕ= 0. (5)\nThe complex mode solution of (5) are taken as\nu/vectork∝e−ikµxµ, (6)\nwhere\nω2(/vectork) =|/vectork|2−α2qµνqγδkµkνkγkδ. (7)\nIn general the relation (7) admits imaginary frequen-\ncies. These frequencies lead to instability and unbound-\nedness [2]. It is necessary in this context to restrict our-\nselves to ordinary solutions with real frequencies. In this\ncase the mode solutions (6) become positive frequency\nmode solutions.\nWe define the scalar product for two solutions ϕ1and\nϕ2of Eq.(5) as follows\n(ϕ1,ϕ2) =−i/integraldisplay\nΣϕ1(x)(← →∂µ−α2qµνqγδ←−−→∂ν∂γ∂δ)ϕ∗\n2(x)dΣµ,\n(8)\nwheredΣµ=nµdΣ, withnµafuture-directedunit vector\northogonalto the space like hypersurface Σ and dΣ is the\nvolumeelementinΣ. ThehypersurfaceΣistakentobe a\nCauchy surface in the space-time and we can show, using\nGauss’ theorem, that the value of ( ϕ1,ϕ2) is independent\nof Σ. The notation←−−→∂ν∂γ∂δin (8) is defined by\nϕ1←−−→∂ν∂γ∂δϕ2=\n+ϕ1∂ν∂γ∂δϕ2−ϕ2∂ν∂γ∂δϕ1\n−∂νϕ1∂γ∂δϕ2+∂νϕ2∂γ∂δϕ1\n+∂γϕ1∂ν∂δϕ2−∂γϕ2∂ν∂δϕ1\n−∂δϕ1∂γ∂νϕ2+∂δϕ2∂γ∂νϕ1.(9)The ordinary modes u/vectorkare orthogonal\n(u/vectork,u/vectork′) = 0,/vectork∝negationslash=/vectork′, (10)\nand if we choose\nu/vectork= [2(2π)3(ω−2α2q0νqγδkνkγkδ)]−1\n2e−ikµxµ,(11)\nthese ordinary modes are normalized in the sense of the\nscalar product (8). An ordinary solution of Eq.(5) may\nbeexpandedin termofthe ordinarymodes(11)andtheir\ncomplex conjugates\nϕ(t,/vector x) =/summationdisplay\n/vectorka/vectorku/vectork(t,/vector x)+a†\n/vectorku∗\n/vectork(t,/vector x)ω∈R,(12)\nandthe system isquantized in the canonicalquantization\nscheme by imposing the following commutation relations\n[a/vectork,a/vectork′] = 0 [ a†\n/vectork,a†\n/vectork′] = 0 [ a/vectork,a†\n/vectork′] =δ/vectork/vectork′.(13)\nWith these commutation relations, a/vectork’s anda†\n/vectork’s are an-\nnihilation and creation operators and the vacuum of the\nLorentz violation model in Minkowski space-time is de-\nfined by\na/vectork|0M\nLV∝angb∇acket∇ight= 0, ω∈R. (14)\nThe above vacuum was defined with respect to a coor-\ndinate system in which the components of uµwere con-\nstant. We shall work in a coordinate system which corre-\nsponds to the rest frame of a preferred inertial observer\n(aether). In this case uµtakes the form (1 ,0,0,0).With\nrespect to the aether the dispersion relation (7) and the\nnormalization constant in (11) are transformed to Eq.(1)\nandω−1/2, respectively.\nItisimportanttonotethatinwhatsenseistheLorentz\ninvariance violated. The Lagrangian (2) does not ex-\nhibit the invariance under particle boost transformations\n[22, 23]. These are transformations of a physical system\n(particles or localized fields) within a fixed coordinate\nsystem. It is in this sense that Lorentz invariance is vi-\nolated. Therefore the above vacuum may not be consid-\nered as an invariant state under a particle Lorentz trans-\nformation transforming a physical system within the rest\nframe of the preferred observer (aether).\nIII. COSMOLOGICAL MODEL AND PARTICLE\nCREATION\nTo show how particle creation occurs in the cosmologi-\ncal context, we use a two-dimensional Robertson-Walker\nspace-time with the line element\nds2=dt2−a2(t)dx2, (15)\nand consider the generalization of the Lagrangian (2) to\nthis cosmological space-time, namely\nL(ϕ,uµ,λ) =1\n2√−g(gµν∇µϕ∇νϕ−ξRϕ2+α2(D2ϕ)2\n+λ(1−uµuµ)), (16)3\nwhereD2ϕis the covariant spatial Laplacian [20](the co-\nvariant analogue of (3))\nD2ϕ=−DµDµϕ=−qµν∇ν(qτ\nµ∇τϕ),(17)\nandqµν=−gµν+uµuνwithgµνcorrespondingto (15). ξ\nis a coupling constant between the scalar field and scalar\ncurvature. The vector field uµis as a non-dynamical\nvectorfieldtobespecifiedbytheconditionsofthetheory.\nBy introducing the conformal time η, defined by dη=\ndt/a(t), the metric (15) takes the form\nds2=c(η)(dη2−dx2), c(η) =a2(t).(18)\nThe Lagrange multiplier λin (16) imposes the follow-\ning constraint on uµ\ngµνuµuν= 1. (19)\nIn the context of the homogeneous and isotopic cosmo-\nlogicalmetric(18) thevectorfield uµistakenastosatisfy\nthe isotropic property of cosmological metric and equa-\ntion (19). This leads to\nuµ≡(/radicalbig\nc(η),0). (20)\nThusqµνis\nq00=q01=q10= 0and q 11=c(η).(21)\nNow setting the variation of the action S=/integraltext\nL(ϕ)d4x\nwith respect to ϕequals to zero yields the equations of\nmotion for ϕin the metric (18) as follows\n/squareϕ+ξRϕ−α2\nc2(η)∂4ϕ\n∂x4= 0. (22)/s32/s32/s67/s40 /s41\n/s61/s48/s65/s43/s66\n/s65/s45/s66/s65/s45 /s66/s65/s43 /s66\n/s65\nFIG. 1: The figure shows c(η) with respect to η. It shows\nc(η=±∞) =A±Bsoa(η=±∞) =√\nA±B. Also it\nshows the time interval ∆ η=η2−η1that the proportion\nof total expansion that occurs in this time interval is ε. We\nchooseε= 0.99.We want to quantize ϕin equation (22) when ξ= 0\n(it is the conformal coupling case in two dimension in\nabsence of the Lorentz violation term). For this purpose\nwe will obtain the mode solutions uk(x) of equation (22).\nIn order to avoid the well known ambiguities in the\nparticle concept in curved space-time [1], we suppose\nthat the space-time can be treated as asymptotically\nMinkowskian in the remote past and future. We refer\nto the remote past and future as the inandoutregions\n, respectively. We take c(η) as\nc(η) =A+Btanh(ρη), (23)\nwhereA,Bandρare some constants such that 2 Bandρ\nrepresentthe amountandthe rateofexpansionin confor-\nmal time η, respectively [see fig.(1)]. Then in the remote\npast and future the space-time becomes Minkowskian\nsince\nc(η)→A±B, η→±∞. (24)\nThis kind of conformal scale factor first introduced by\nBernard and Duncan in [24] and then has been used in\nmany works [25, 26, 27, 28] for considering the cosmolog-\nical particle creation.\nWe can solve Eq.(22) and obtain uk(x) by the method\nof separation of variables. The mode solution of this\nequation is\nuk(η,x) = (2π)−1\n2eik.xχk(η), (25)\nsuch that χk(η) satisfies the following equation\nd2\ndη2χk(η)+(k2−α2\nc(η)k4)χk(η) = 0.(26)\nThis equation can be solved in terms of hypergeometric\nfunctions. The normalized modes which behave like the\npositive frequency Minkowski space modes in the remote\npast (η−→−∞) are\nukin(η,x) = (4πωin)−1\n2exp(ikx−iω+η (27)\n−iω−\nρln[(A+B)eρη+(A−B)e−ρη])\n×F(1+iω−/ρ,iω−/ρ;1−iωin/ρ;z),\nand the modes which behave like positive frequency\nMinkowski modes in the outregion as η−→+∞are\nfound to be\nuout\nk(η,x) = (4πωin)−1\n2exp(ikx−iω+η (28)\n−iω−\nρln[(A+B)eρη+(A−B)e−ρη])\n×F(1+iω−/ρ,iω−/ρ;1−iωout/ρ;1−z),\nwhere\nz=1\n2(A+B)tanh(ρη)+1\nA+Btanh(ρη), (29)4\nand\nωin=k(1−α2\nA−Bk2)1\n2\nωout=k(1−α2\nA+Bk2)1\n2\nω±=1\n2(ωout±ωin).(30)\nThe energies ωinandωoutcan be imaginary. But\nwe limit ourselves to the modes with real frequencies\n(the ordinary modes). The ordinary modes ukin(η,x)/parenleftBig\nuout\nk(η,x)/parenrightBig\nare orthonormal in the following conserved\ninner product\n(ϕ1,ϕ2) = (31)\n−i/integraldisplay\nΣ√−gϕ1(x)(← →∂µ−α2qµνqγδ←−−→∂ν∂γ∂δ)ϕ∗\n2(x)dΣµ,\nwhere it is the generalization of inner product (8) to\ncurved space-time and←−−→∂ν∂γ∂δis defined in (9) .Thus we\nmayexpandoneordinarysolutionofEq.(22)withrespect\ntoukin(η,x)/parenleftBig\nuout\nk(η,x)/parenrightBig\nas\nϕ(t,x) =/summationtext\nkain\nkuin\nk(t,x)+a†in\nku∗in\nk(t,x)ωin∈R\nϕ(t,x) =/summationtext\nkaout\nkuout\nk(t,x)+a†out\nku∗out\nk(t,x)ωout∈R\nand the second quantization is implemented in the same\nway as the Minkowski space-time by the following com-\nmutation relations\n[ain\nk,a†in\nk′] =δkk′[aout\nk,a†\nk′out] =δkk′.(32)\nAs we mentioned before we have chosen a conformally\nflat expanding space-time which is in the inandoutre-\ngions Minkowskian. With using the Eqs.(20) and (23) we\ndefine the aether in the inandoutregions as a frame in\nwhichuµis (√\nA−B,0) and (√\nA+B,0),respectively.\nThe vacuums with respect to the aether in the inand\noutregions are defined by\nain\n/vectork|0M\nLV∝angb∇acket∇ightin= 0 ωin∈R\naout\n/vectork|0M\nLV∝angb∇acket∇ightout= 0 ωout∈R.(33)\nSuppose the quantum field in the remote past resides\nin the state|0M\nLV∝angb∇acket∇ightinand suppose we are working in the\nHeisenberg picture. Thus in the outregion, the quantum\nfield is also in the state |0M\nLV∝angb∇acket∇ightin. But this state is not\nregarded by the aether in the outregion as the physical\nvacuum, this role being reserved for the state |0M\nLV∝angb∇acket∇ightout.\nWe want to calculate the number of particles detected\nin theoutregion in the state |0M\nLV∝angb∇acket∇ightin. If we use the\nlinear transformation properties of hypergeometric func-\ntions, we expand uin\nkwith respect to uout\nkand obtain the\nfollowing relation\nukin(η,x) =αkuout\nk(η,x)+βkuout\n−k(η,x),(34)/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s48/s46/s48/s53/s46/s48/s120/s49/s48/s45/s57\n/s32/s50\n/s40/s107/s41\n/s107/s40/s69\n/s80/s41/s40/s98/s41\n/s107\n/s109/s97/s120\nFIG. 2: The figure shows the spectrum of created particles\nwhere√\n2B+1 = 103andρf= 10−5soλρf= 0.032≪1 .\n(Noteα= 1−,ε= 0.99 andA−B= 1)\nwhere\nαk= (ωout\nωin)1\n2Γ(1−(iωin/ρ))Γ(−iωout/ρ)\nΓ(−iω+/ρ)Γ(1−iω+/ρ)\nβk= (ωout\nωin)1\n2Γ(1−(iωin/ρ))Γ(iωout/ρ)\nΓ(iω−/ρ)Γ(1+iω−/ρ).(35)\nasiswellknown, αkandβkaretheBogolubovcoefficients\nand|βk|2is equal to the number of particles per mode\nkthat is detected in the outregion (created particles),\n|βk|2=sinh2(πω−/ρ)\nsinh(πωin/ρ)sinh(πωout/ρ).(36)\nIn the above relation |βk|2is dependent on the param-\netersα,A,Bandρ. We set the first parameter αin nat-\nural units ( /planckover2pi1=c= 1) asα= 1−[29] because we assume\nthat LV occurs in the scale of Planck energy. The three\nlast parameters are related to the conformal scalar factor\nc(η) by (23). These parameters determine the spectrum\nof created particles with respect to the coordinates ( η,x)\nand metric (18) through |βk|2in (36).\nIt is instructive to relate |βk|2to the physical param-\neters in the free-fall frame which is determined by the\ncoordinates ( t,x) and the metric (15). To do this we first\nfind the analogue of the parameters A,Bandρin the\nfree-fall frame. The parameters, AandBhave a good\nmeaning in the free-fall frame because as we have shown\nin fig.(1),√\nA−Band√\nA+Bare initial and final scale\nfactors in the free fall frame, respectively. For some ad-\nvantages we put A−B= 1, it implies that the conformal\ntime is the same as the proper time in the remote past\nalso the initial scale factor is 1 and the final scale factor\nis√\n2B+1. Now let us consider the analogue of ρ. This\nparameter is proportional to the inverse of time inter-\nval of expansion with respect to the conformal time. We\nshow this by the following argument. We define an effec-\ntive time interval of expansion, ∆ η.As we have shown5\n/s48/s46/s48\n/s52/s46/s48/s120/s49/s48/s51\n/s56/s46/s48/s120/s49/s48/s51\n/s49/s46/s50/s120/s49/s48/s52/s48/s46/s48/s48/s49/s46/s50/s48/s120/s49/s48/s45/s52\n/s40/s97/s41/s32\n/s32/s32/s40/s107\n/s109/s97/s120/s41\n/s40/s50/s66/s43/s49/s41/s49/s47/s50\n/s48/s46/s48\n/s52/s46/s48/s120/s49/s48/s51\n/s56/s46/s48/s120/s49/s48/s51\n/s49/s46/s50/s120/s49/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s98/s41\n/s32/s32/s107\n/s109/s97/s120/s40/s69\n/s80/s41\n/s40/s50/s66/s43/s49/s41/s49/s47/s50\n/s48/s46/s48\n/s52/s46/s48/s120/s49/s48/s45/s53\n/s56/s46/s48/s120/s49/s48/s45/s53\n/s49/s46/s50/s120/s49/s48/s45/s52/s48/s46/s48/s48/s49/s46/s53/s48/s120/s49/s48/s45/s52\n/s32/s32/s50\n/s40/s107\n/s109/s97/s120/s41\n/s32\n/s102/s32/s40/s49/s47/s116\n/s80/s41/s40/s99/s41\n/s48/s46/s48\n/s52/s46/s48/s120/s49/s48/s45/s53\n/s56/s46/s48/s120/s49/s48/s45/s53\n/s49/s46/s50/s120/s49/s48/s45/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s100/s41\n/s32/s32/s107\n/s109/s97/s120/s40/s69\n/s80/s41\n/s102/s32/s40/s49/s47/s116\n/s80/s41\nFIG. 3: The abovefiguresshowtheproperties ofthespectrum\nof created particles. (a) and (b) show β2(kmax) andkmax\nversus the final amount of the scale factor ,√\n2B+1, where\nρf= 10−5. (c) and (d) show β2(kmax) andkmaxversus the\nrate ofexpansionin free fall frame, ρf, where√\n2B+1 = 103.\n(Noteα= 1−,ε= 0.99 andA−B= 1)in fig.(1), ∆ η(=η2−η1) is taken as a time interval such\nthat the proportion εof the total expansion is done in\nthis interval. We set ε= 0.99 that means 99% of total\nexpansion occurs in ∆ η. By this definition one can find\n,from (23), η2=−η1= tanh−1(ε)/ρand so\nρ=2tanh−1(ε)\nη2−η1, (37)\nwhereη1andη2arethe initialandthefinaltime oftheef-\nfective expansion respectively, and tanh−1(0.99) = 2.64.\nThe Eq.(37) shows ρ∝1/∆ηand the proportional con-\nstant is of order one. This relation makes it clear that ρ\nis proportional to the inverse of the time interval of ex-\npansion so it can be interpreted as the rate of expansion.\nIn analogy to ρwe can define ρfin free falling frame\nsuch that it is the inverse of expansion time interval in\nfree falling frame, namely\nρf=1\nt(η2)−t(η1). (38)\nFor evaluating ρfin the above relation we find the rela-\ntionship between tandηfrom (23) and dt=/radicalbig\nc(η)dη.\nWe get\nt(η) =√\n2B+1\nρtanh−1/parenleftBigg/radicalbig\nB[1+tanh( ρη)]+1√2B+1/parenrightBigg\n+1\n2ρln/parenleftBigg/radicalbig\nB[1+tanh( ρη)]+1−1/radicalbig\nB[1+tanh( ρη)]+1+1/parenrightBigg\n+C,(39)\nwhereCis a constant that sets the initial values of ηand\nt. From (39) together with (38) we find ρf=ρ/λwhere\nλis a function of Bandε, namely\nλ=√\n2B+1tanh−1/parenleftBigg/radicalbig\nB(1+ε)+1√\n2B+1/parenrightBigg\n−√\n2B+1tanh−1/parenleftBigg/radicalbig\nB(1−ε)+1√2B+1/parenrightBigg\n+1\n2ln/parenleftBigg/radicalbig\nB(1+ε)+1−1/radicalbig\nB(1+ε)+1+1/parenrightBigg\n−1\n2ln/parenleftBigg/radicalbig\nB(1−ε)+1−1/radicalbig\nB(1−ε)+1+1/parenrightBigg\n. (40)\nFrom (40) we find that for sufficiently large amount of B,\nλcan be approximated by (2 B)1\n2. Substituting ρbyλρf\nin (36) we get|βk|2in term of the physical parameter ρf\nin the free falling frame. We can approximate (36) in the\nlimitλρf≫1 by\n|βk|2=(ωout−ωin)2\n4ωinωout, (41)\nwhere in the region B≫1 andk∝negationslash= 0 the above relation\ntakes the form\n|βk|2=/parenleftbig\n1−√\n1−α2k2/parenrightbig2\n4√\n1−α2k2, (42)6\nwhich implies that |βk|2is independent of λρf.\nIn the opposite limit where λρfis small the spectrum\nshows an interesting behavior. The figure(2) shows the\nspectrum of created particles for√2B+1 = 103and\nρf= 10−5soλρf= 0.032. It shows that particles are\nnot created at low and high momenta. Most particles\nare produced with momenta around kmax. In figure (3)\nwe show the properties of this spectrum. Figures (3,a)\nand (3,c) show the variations of the number of created\nparticles at momentum kmaxwith respect to the amount\n,√\n2B+1, and the rate , ρf,of the space-time expansion\nrespectively. With increasing the amount and the rate of\nexpansion the number of created particles increases. Fig-\nures (3,b) and (3,d) shows kmaxwith respect to√\n2B+1andρf. We see that with increasing the amount and the\nrate of the expansion, the particles with higher momenta\nare created.\nAcknowledgments\nE. Khajehand N. Khosravithank H. Salehi forencour-\nagements. E. Khajeh thanks S. Jalalzadeh for useful dis-\ncussionsandresearchofficeofShahidBeheshtiUniversity\nfor financial supports. The authors thank an unknown\nreferee for useful comments.\n[1] N. D. Birrell and P. C. W. Davies, ”Quantum Fields In\nCurved Space”, (Cambridge University Press) (1982).\n[2] S. A. Fulling, ”Aspects of Quantum Field Theory\nin Curved Space-Time”, (Cambridge University Press)\n(1989).\n[3] V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683\n(1989).\n[4] S. Carroll et. al., Phys. Rev. Lett. 87, 141601 (2001).\n[5] F. Lizzi, G. Mangano, G. Miele and M. Peloso, JHEP\n0206, 049 (2002), [hep-th/0203099].\n[6] K. Greisen, Phys. Rev. Lett. 16, 748 (1966).\n[7] M. Takeda et. al., (AGASA Collab), Phys. Rev. Lett. 81,\n1163 (1998).\n[8] F. Krennrich et. al., Ap. J. 560, L45 (2001).\n[9] E.E. Antonov, et. al., Pisma ZhETF 73, 506 (2001).\n[10] H. Sato, [astro-ph/0005218].\n[11] S. R. Coleman and S. L. Glashow, Phys. Rev. D 59,\n116008 (1999), [ hep-ph/9812418].\n[12] R. H. Brandenberger and J. Martin, Int. J. Mod. Phys.\nA17 (2002) 3663-3680, [hep-th/0202142]; Phys. 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D 58,\n116002 (1998).\n[24] C. Bernard and A. Duncan, Ann. Phys. (NY) 107 (1977)\n201.\n[25] M. R. Setare, Int.J.Theor.Phys. 43 (2004) 2237-2242,\n[hep-th/0203126].\n[26] Andrew J. Tolley, Daniel H. Wesley, Phys.Rev. D72\n(2005) 124009, [hep-th/0509151].\n[27] Alex J. Hamilton, Daniel Kabat, Maulik K. Parikh,\nJHEP 0407 (2004) 024, [hep-th/0311180].\n[28] Don Koks, B. L. Hu, Andrew Matacz, Alpan Raval,\nPhys.Rev. D56 (1997) 4905-4915, [gr-qc/9704074].\n[29] The minus in 1−means approaching 1 from the left."    },    {        "title": "2007.14480v5.Complete_complementarity_relations_and_its_Lorentz_invariance.pdf",        "content": "Complete complementarity relations and its Lorentz invariance\nMarcos L. W. Basso1,\u0003and Jonas Maziero1,y\n1Departamento de Física, Centro de Ciências Naturais e Exatas,\nUniversidade Federal de Santa Maria, Avenida Roraima 1000,\nSanta Maria, Rio Grande do Sul, 97105-900, Brazil\nIt is well known that entanglement under Lorentz boosts is highly dependent on the boost scenario\nin question. For single particle states, a spin-momentum product state can be transformed into an\nentangled state. However, entanglement is just one of the aspects that completely characterizes\na quantum system. The other two are known as the wave-particle duality. Although the entan-\nglement entropy does not remain invariant under Lorentz boosts, and neither do the measures of\npredictability and coherence, we show here that these three measures taken together, in a complete\ncomplementarityrelation(CCR),areLorentzinvariant. Peresetal., in[Phys. Rev. Lett. 88, 230402\n(2002)], realized that even though it is possible to formally define spin in any Lorentz frame, there is\nno relationship between the observable expectation values in different Lorentz frames. Analogously,\none can, in principle, define complementary relations in any Lorentz frame, but there is no obvious\ntransformation law relating complementary relations in different frames. However, our result shows\nthatthe CCRhave thesame valuein any Lorentz frame, i.e., there’sa transformationlawconnecting\nthe complete complementarity relations. In addition, we explore relativistic scenarios for single and\ntwo particle states, which helps in understanding the exchange of different aspects of a quantum\nsystem under Lorentz boosts.\nKeywords: Complete complementarity relations; Lorentz boosts; Relativistic scenarios\nI. INTRODUCTION\nEntanglement is one of the most intriguing characteristics that turns apart the quantum world from the classical\nworld. Its fundamental importance in quantum foundations [1, 2], together with its application in several areas, such\nas quantum information and quantum computation [3–5], has made the entanglement theory achieve great progress\nin recent decades. Moreover, there has been more and more interest in how entanglement behaves under relativistic\nsettings [6]. For instance, in Ref. [7] the author considered the relativistic version of the famous Einstein-Podolsky-\nRosen experiment with massive spin-1/2 particles. Czachor argued that the degree of violation of the Bell inequality\nis dependent on the velocity of the particles, leading to implications for quantum cryptography. A few years later,\nthe authors in Refs. [8, 9] showed that the entanglement of Bell states depends on the velocity of an observer. On\nthe other hand, the authors of Ref. [10] argued that the entanglement fidelity of a Bell state remains invariant for\na Lorentz boosted observer. However, in the same year, it was demonstrated by Peres et al. [11] that the entropy\nof a single massive spin-1/2 particle does not remain invariant under Lorentz boosts. Thereafter, the behavior of\nentanglement under Lorentz boosts has been receiving a lot of attention by researchers [12–20].\nAs pointed out by Palge and Dunningham in Ref. [21], the main aspect to be noticed here is that many of these\napparently conflicting results involve systems containing different particle states and boost geometries. Therefore,\nentanglement under Lorentz boosts is highly dependent on the boost scenario in question [22]. For single particle\nstates, a spin-momentum product state can be transformed into an entangled state. Beyond that, Lorentz boosts\ncan be regarded as controlled quantum operations where momentum plays the role of the control system, whereas\nthe spin can be taken as the target qubit, as argued in Ref. [17]. This implies that Lorentz boosts perform global\ntransformations on single particle systems. As in Refs. [16, 18, 21], by using discrete momentum states, in this article\nwe discuss the fact that for a spin-momentum product state be transformed into a entangled state it needs coherence\nbetween themomentumstates. Otherwise, ifthemomentumstateiscompletelypredictable, thespin-momentumstate\nremains separable, and the Lorentz boost will at most generate superposition between the spin states. In addition,\nwe discuss similar results for the two-particle states under Lorentz boosts. As already noticed in Refs. [8, 21], the\nstate and entanglement changes of the different degrees of freedom depend considerably on the initial states involved,\nas well as on the geometry of the boost scenario. Whereas some states and geometries leave the overall entanglement\ninvariant, others create entanglement.\n\u0003Electronic address: marcoslwbasso@mail.ufsm.br\nyElectronic address: jonas.maziero@ufsm.brarXiv:2007.14480v5  [quant-ph]  31 Aug 20212\nBesides, it is known that entanglement is just one of the aspects that completely characterizes a quanton [24]. The\nother two, which also are intriguing characteristics that turn apart the quantum world from the classical world, are\nknown as the wave-particle duality. This distinguished aspect is generally captured, in a qualitative way, by Bohr’s\ncomplementarity principle [23]. For instance, in the Mach-Zehnder interferometer or in the double-slit interferometer,\nthe wave aspect is characterized by interference fringes visibility, meanwhile the particle nature is given by the which-\nway information of the path along the interferometer. A quantitative version of the wave-particle duality was first\ninvestigated by Wooters and Zurek [25], and later captured by a complementarity inequality in Refs. [26, 27]:\nP2+V2\u00141; (1)\nwherePis an a priori predictability, for which the particle aspect is inferred once the quanton is more likely to\nfollow one path than the other and it is directly related to the probability distribution given by the diagonal elements\nof the density operator as we will discuss in further section. Besides, Vis the visibility of the interference pattern.\nRecently, several steps have been taken towards the quantification of the wave-particle duality, with the establishment\nof minimal and reasonable conditions that visibility and predictability measures should satisfy [29, 30]. As well, with\nthe development of the field of quantum information, it was suggested that the quantum coherence [31] would be\na good generalization of the visibility measure [32–35]. Until now, many approaches were taken for quantifying the\nwave-particle properties of a quantum system [36–40]. As pointed out by Qian et al. [41], complementarity relations\nlike Eq. (1) do not really predict a balanced exchange between PandVsimply because the inequality permits a\ndecrease of PandVtogether, or an increase by both. It even allows the extreme case P=V= 0to occur (neither\nwave or particle) while, in an experimental setup, we still have a quanton on hands. Such a quanton cannot be\nnothing. Thus, one can see that something must be missing from Eq. (1). As noticed by Jakob and Bergou [42], this\nlack of knowledge about the system is due to entanglement, or, more generally, to quantum correlations [43]. This\nmeans that the information is being shared with another system and this kind of quantum correlation can be seen as\nresponsible for the loss of purity of each subsystem such that, for pure maximally entangled states, it is not possible\nto obtain information about the local properties of the subsystems. Hence, to completely quantify a quanton, one has\nalso to regard its correlations with other systems, such that the entire system is pure.\nIn this paper, we study how these different aspects of a quanton behave under Lorentz boosts. Even though\nentanglement entropy does not remain invariant under Lorentz boosts, and neither do measures of predictability and\ncoherence, we show that these three measures together, in what is known as a complete complementarity relation\n(CCR), are Lorentz invariant. In Ref. [11], the authors showed that, even though it is possible to formally define spin\nin any Lorentz frame, there is no relationship between the observable expectation values in different Lorentz frames.\nHere the situation is different. First, one can define complementary relations in any Lorentz frame, but there is no\nobvious transformation law relating complementary relations in different frames. However since the purity of a state\nis preserved under transformations between inertial frames, the complementary relations have the same value in any\nLorentz frame, i.e., there is a transformation law connecting the complete complementarity relations. In addition, we\nexplore several relativistic scenarios for single and two particle states, what helps in understanding the exchange of\nthese different aspects of a quanton under Lorentz boosts.\nThe organization of this article is as follows. In Sec. II, we discuss the representations of the Poincaré group in\nthe Hilbert space, as well as the Wigner’s little group, by focusing in spin- 1=2massive particles. In Sec. III, we\nobtain complete complementarity relations for multipartite pure quantum systems, and show that CCR are Lorentz\ninvariant. Thereafter, in Sec. IV, we turn to the study of the behavior of CCR in relativistic scenarios for several\nsingle and two particle states. Lastly, in Sec. V, we give our conclusions.\nII. REPRESENTATIONS OF THE POINCARÉ GROUP IN THE HILBERT SPACE\nOne of the fundamental questions when studying the relativistic formulation of the quantum theory is how quantum\nstates behave under Lorentz boosts. In the language of group theory, we are seeking to represent an element of the\nLorentz group by a unitary operator on the Hilbert space that the quantum states belongs to. More specifically, single\nparticle quantum states are classified by their transformation under the inhomogeneous Lorentz group, or Poincaré\ngroup, which consists of homogeneous Lorentz transformations \u0003and translations a[44]. For our discussion, we adopt\nthe following notation: Greek indices run over the 4-spacetime coordinate labels f0;1;2;3g; Latin indices run over the\nthree spacial coordinates labels f1;2;3g; the Minkowski metric \u0011\u0016\u0017is diagonal with elements f\u00001;1;1;1g;4-vectors\nare in un-boldfaced type while spacial vectors are represented by an arrow. For instance, the 4-momentum for a\nparticle with mass mis given byp= (p0;p1;p2;p3) = (p0;~ p), with norm p2:=p\u0016p\u0016=\u0011\u0016;\u0017p\u0017p\u0016=\u0000(p0)2+~ p2=\u0000m2,\nwhere we use natural units, i.e., c=~= 1.\nAn inertial reference frame Ois related to another inertial frame O0via a Poincaré transformation\nx0\u0016:=T(\u0003;a)x\u0017= \u0003\u0016\n\u0017x\u0017+a\u0016; (2)3\nwithx= (x0;~ x)being the coordinates of O, and similarly for O0. ThenT(\u0003;a)induces a unitary transformation on\nquantum states characterized by\nj\ti!U(\u0003;a)j\ti; (3)\nwhich satisfies the same composition rule of T(\u0003;a):\nU(\u00031;a1)U(\u00032;a1) =U(\u00031\u00032;\u00031a2+a1): (4)\nSingle particle quantum states can be denoted by jpi\nj\u001bi:=jp;\u001bi, whereplabels the 4-momenta and \u001blabels the\nspin for massive particles. The quantum states jp;\u001biare eigenvectors of the momentum operator P\u0016with eigenvalues\np\u0016, i.e.,P\u0016jp;\u001bi=p\u0016jp;\u001bi. This corresponds to a basis of plane waves and, thus, transforms under translations as\nU(I;a)jp;\u001bi:=U(a)jp;\u001bi=e\u0000ipajp;\u001bi, wherepa:=p\u0016a\u0016=p\u0016a\u0016. Meanwhile, a general Lorentz boost \u0003takes\nthe eigenvalue p\u0016!\u0003\u0016\n\u0017p\u0017, and therefore U(\u0003;0)jp;\u001bi:=U(\u0003)jp;\u001bimust be a linear combination of all states with\nmomentum \u0003p, i.e.,\nU(\u0003)jp;\u001bi=X\n\u0015D\u0015;\u001b(\u0003;p)j\u0003p;\u0015i: (5)\nAsU(\u0003)is a representation, it preserves the group structure, and imposes conditions on the values of D\u0015;\u001b. To see\nthis, let us recall that U(\u0003)leavesp2:=p\u0016p\u0016=~ p\u0001~ p\u0000E2=\u0000m2and the sign of p0=Eunchanged for a particle\nwith massm. Hence, we can use these two invariant quantities to classify states into specific classes. For each value\nofp2and for each sign(p0), it is possible to choose a ‘standard’ 4-momentum kthat identifies a specific class of\nquantum states [45]. For massive particles, we can fix the standard momentum kto be the particle’s momentum in\nthe rest frame, i.e., k= (m;0;0;0). Then, any momenta pcan be expressed in terms of the standard momentum, i.e.,\np\u0016= (L(p)k)\u0016=L(p)\u0016\n\u0017k\u0017, whereL(p)is a Lorentz transformation which depends on pand takesk!p. Therefore,\nquantum statesjp;\u001bican be defined in terms of the standard momentum state jk;\u001bi:\njp;\u001bi=U(L(p))jk;\u001bi: (6)\nNow, if we apply a Lorentz boost \u0003onjp;\u001bi, then\nU(\u0003)jp;\u001bi=U(\u0003)U(L(p))jk;\u001bi (7)\n=U(I)U(\u0003L(p))jk;\u001bi (8)\n=U(L(\u0003p)L\u00001(\u0003p))U(\u0003L(p))jk;\u001bi (9)\n=U(L(\u0003p))U(L\u00001(\u0003p)\u0003L(p))jk;\u001bi (10)\n=U(L(\u0003p))U(W(\u0003;p))jk;\u001bi; (11)\nwhereW(\u0003;p) =L\u00001(\u0003p)\u0003L(p)is called Wigner rotation, which leaves the standard momentum kinvariant, and\nonly acts on the internal degrees of freedom of jk;\u001bi:kL\u0000 !p\u0003\u0000 !\u0003pL\u00001\n\u0000\u0000\u0000!k. Hence, the final momentum in the rest\nframe is different from the original one by a Wigner rotation, i.e., U(W(\u0003;p))jk;\u001bi=P\n\u0015D\u0015;\u001b(W(\u0003;p))jk;\u0015i. On\nthe other hand, U(L(\u0003p))takesk!\u0003pwithout affecting the spin, by definition. Therefore,\nU(\u0003)jp;\u001bi=U(L(\u0003p))U(W(\u0003;p))jk;\u001bi (12)\n=U(L(\u0003p))X\n\u0015D\u0015;\u001bW(\u0003;p))jk;\u0015i (13)\n=X\n\u0015D\u0015;\u001b(W(\u0003;p))j\u0003p;\u0015i: (14)\nIt is worth mentioning that the subscripts of D\u0015;\u001b(W(\u0003;p))can be suppressed, and we can write U(\u0003)jp;\u001bi=\nj\u0003pi\nD(W(\u0003;p))j\u001bi. The set of Wigner rotations forms a group known as the little group , which is a subgroup of\nthe Poincaré group [46]. In other words, under a Lorentz transformation \u0003, the momenta pgoes to \u0003p, and the spin\ntransforms under the representation D(W(\u0003;p))of the little group W. For massive particles, the little group is the\nwell known group of rotations in three dimensions, SO(3). However, it is also known that SO(3)is homomorphic to\nSU(2), and the irreducible unitary representations of SU(2)span a Hilbert space of 2j+ 1dimensions, with j=n=2,\nwherenis an integer [47, 48]. The value of jis what we usually refer to as the spin of the massive particle. In this\narticle, we will be interested in spin- 1=2particles, hence the representation of the Wigner rotation is given by [49, 50]\nD(W(\u0003;p)) =(p0+m) cosh(!=2)I2\u00022+ (~ p\u0001^e) sinh(!=2)\u0000isinh(!=2)~ \u001b\u0001(~ p\u0002^e)p\n(p0+m)((\u0003p)0+m)(15)\n= cos\u001e\n2I2\u00022+isin\u001e\n2(~ \u001b\u0001^n); (16)4\nwithI2\u00022being the identity matrix, meanwhile ~ \u001bare the Pauli matrices, and\ncos\u001e\n2=cosh(!=2) cosh(\u000b=2) + sinh(!=2) sinh(\u000b=2)(^e\u0001^p)q\n1\n2(1 + cosh!cosh\u000b+ sinh!sinh\u000b(^e\u0001^p)); (17)\nsin\u001e\n2^n=sinh(!=2) sinh(\u000b=2)(^e\u0002^p)q\n1\n2(1 + cosh!cosh\u000b+ sinh!sinh\u000b(^e\u0001^p)); (18)\nwhere cosh\u000b=p0=m,!= tanh\u00001vis the rapidity of the boost [51], ^eis the unit vector pointing in the direction of\nthe boost,pis the 4-momenta of the particle in O, and \u0003pis the 4-momenta of the particle in O0. As an example, if\nthe momentum is in the x-direction of the referece frame Oand the boost is given in the z-axis, then\nD(W(\u0003;p)) = cos\u001e\n2I2\u00022\u0000isin\u001e\n2\u001by=\u0012\ncos\u001e\n2\u0000sin\u001e\n2\nsin\u001e\n2cos\u001e\n2;\u0013\n(19)\nwhere the Wigner angle \u001eis given by\ncos\u001e\n2=cosh(!=2) cosh(\u000b=2)q\n1\n2(1 + cosh!cosh\u000b); (20)\nsin\u001e\n2^n=sinh(!=2) sinh(\u000b=2)^yq\n1\n2(1 + cosh!cosh\u000b); (21)\ntan\u001e=sinh!sinh\u000b\ncosh!+ cosh\u000b; (22)\nwhich implies that \u001e2[0;\u0019=2]. Hence, the transformation law for the spin- 1=2particle with momentum ~ palong the\nx-axis ofOis given by\nU(\u0003)jp;0i=j\u0003pi\n(cos\u001e\n2j0i+ sin\u001e\n2j1i); (23)\nU(\u0003)jp;1i=j\u0003pi\n(\u0000sin\u001e\n2j0i+ cos\u001e\n2j1i); (24)\nwherej0imeans spin ‘up’ and j1istands for spin ‘down’ along the z-axis. Therefore, as one can see, for separable\nand completely predictable states, a Lorentz boost will only generate superposition between the possible states of the\nspin of the particle, as already noticed in Ref. [45].\nIII. THE LORENTZ INVARIANCE OF COMPLETE COMPLEMENTARITY RELATIONS\nIn Ref. [43], we developed a general framework to obtain complete complementarity relations for a subsystem that\nbelongs to an arbitrary multipartite pure quantum system, just by exploring the purity of the multipartite quantum\nsystem. To make our investigation easier, we begin by assuming that momenta can be treated as discrete variables\n[16, 18, 21]. This can be justified once we can consider narrow distributions centered around different momentum\nvalues such that it is possible to represent them by orthogonal state vectors, i.e., hpijpji=\u000ei;j. Although narrow\nmomenta are an idealization, it is a system worth studying since it helps to understand more realistic systems, and,\nalso, it is possible to approximate continuous momenta as a finite (but large) number of discrete momenta. Also,\nthroughout this article, we will consider only massive particles of spin 1=2. By doing this, we are considering a\nparticular representation of the Wigner little group. However, the result obtained in this section does not depend on\nthe particular choice of representation, once the representation is unitary.\nSo, let us consider nmassive quantons with spin 1=2in a pure state described by j\tiA1;:::;A 2n2H 1\n:::\nH 2n\nwith dimension d=dA1dA2:::dA2n, in the reference frame O. For instance, A1,A2are referred as the momentum and\nspin of the first quanton, and so on. By defining a local orthonormal basis for each degree of freedom (DOF) Am,\nfjimiAmgdm\u00001\ni=0,m= 1;:::;2n, the state of the multipartite quantum system can be written as [52]\n\u001aA1;:::;A 2n=j\tiA1;:::;A 2nh\tj=X\ni1;:::;i 2nX\nj1;:::;j 2n\u001ai1:::i2n;j1:::j2nji1;:::;i 2niA1;:::;A 2nhj1;:::;j 2nj: (25)5\nWithout loss of generality, let us consider the state of the DOF A1, which is obtained by tracing over the other\nsubsystems,\n\u001aA1=X\ni1;j1\u001aA1\ni1;j1ji1iA1hj1j=X\ni1;j1X\ni2;:::;j 2n\u001ai1i2:::i2n;j1i2:::i2nji1iA1hj1j; (26)\nforwhichtheHilbert-Schmidtquantumcoherenceandthecorrespondingpredictabilitymeasureintermsofthedensity\nmatrix elements are given by\nChs(\u001aA1) =X\ni16=j1\f\f\f\u001aA1\ni1;j1\f\f\f2\n=X\ni16=j1\f\f\f\f\f\fX\ni2;:::;i 2n\u001ai1i2:::i2n;j1i2:::i2n\f\f\f\f\f\f2\n; (27)\nPl(\u001aA1) =X\ni1(\u001aA1\ni1;i1)2\u00001=dA1=X\ni1(X\ni2;:::;i 2n\u001ai1i2:::i2n;i1i2:::i2n)2\u00001=dA1: (28)\nBesides, such predictability measure can be first defined as Pl(\u001aA1) :=Smax\nl\u0000Sl(\u001aA1diag)[40], where \u001aA1diagcor-\nresponds to the diagonal elements of \u001aA1andSl(\u001a) := 1\u0000Tr\u001a2is the linear entropy. To get more intuition about\nthis predictability measure, let us consider the projector onto the state index i1:\u0005i1:=ji1ihi1j, which can be\none of the paths of a Mach-Zehnder interferometer. Now, the uncertainty of the state i1is given by its variance\nV(\u001aA1;\u0005i1) =\n\u00052\ni1\u000b\n\u0000h\u0005i1i2=\u001aA1\ni1i1\u0000(\u001aA1\ni1i1)2such that sum of the uncertainties of all the possible states (or paths)\nis given byP\ni1V(\u001aA1;\u0005i1) = 1\u0000P\nj(\u001aA1\ni1i1)2;which represents the total uncertainty of all states. One can easily see\nthatP\njV(\u001aA1;\u0005j)is exactly the linear entropy of \u001aA1diag:Sl(\u001aA1diag) = 1\u0000Tr\u001a2\nA1diag. Thus, after repeating the\nsame experiment several times, we obtain a probability distribution given by \u001aA1\n00;:::;\u001aA1\ndA1\u00001dA1\u00001;which represents\na probability of the quanton being measured in the state j0i;:::;jdA1\u00001i. From this, it is possible to calculate the\nuncertainty about the paths through Sl(\u001adiag)such thatPl(\u001aA1diag) :=Smax\nl\u0000Sl(\u001aA1diag)offers a measure of the\ncapability to predict what outcome will be obtained in the next run of the experiment. For instance, if we obtain a\nuniform probability distribution, i.e., f\u001aA1\ni1i1= 1=dA1gdA1\u00001\ni1=0, after repeating the experiment several times, our ability\nto make a prediction is null. On the other hand, if \u001aA1diag6=IA1=dA1, whereIA1is the identity operator, then\nPhs(\u001aA1)6= 0. Therefore, it is possible to see that Eq.(28) is a way of quantifying how much the probability distri-\nbution expressed by \u001aA1diagdiffers from the uniform probability distribution. While the Hilbert-Schmidt quantum\ncoherence can be defined as Chs(\u001aA1) := min\u00132Ijj\u001aA1\u0000\u0013jj2\nhs, whereIis the set of all incoherent states (diagonal\ndensity operators), and the Hilbert-Schmidt’s norm of a matrix M2Cd\u0002dis defined askMkhs:=qP\nj;kjMjkj2. The\nminimization procedure yields Eq.(27). Therefore, the Hilbert-Schmidt quantum coherence is measuring how distant\nthe density operator \u001aA1is in comparison with its closest incoherent state, which in this case is \u001aA1diag, given the\nHilbert-Schmidt’s norm. Loosely speaking, the quantum coherence is measuring “how much” orthogonal superposi-\ntion is encoded in a given density operator \u001aA1and it is directly related to the off-diagonal elements of \u001aA1. Besides,\nwe showed in Ref. [40] that these are bona-fide measures of visibility and predictability, respectively. From these\nequations, an incomplete complementarity relation, Phs(\u001aA1) +Chs(\u001aA1)\u0014(dA1\u00001)=dA1, is obtained by exploring\nthe mixedness of \u001aA1, i.e., 1\u0000Tr\u001a2\nA1\u00150.\nNow, since \u001aA1;:::;A 2nis a pure quantum system, then 1\u0000Tr\u001a2\nA1;:::;A 2n= 0, or equivalently,\n1\u0000\u0010X\n(i1;:::;i 2n)=(j1;:::;j 2n)+X\n(i1;:::;i 2n)6=(j1;:::;j 2n)\u0011\nj\u001ai1i2:::i2n;j1j2:::j2nj2= 0; (29)\nwhere\nX\n(i1;:::;i 2n)6=(j1;:::;j 2n)=X\ni16=j1\ni2=j2\n...\ni2n=j2n+X\ni1=j1\ni26=j2\n...\ni2n=j2n+:::+X\ni1=j1\ni2=j2\n...\ni2n6=j2n+X\ni16=j1\ni26=j2\n...\ni2n=j2n+:::+X\ni16=j1\ni2=j2\n...\ni2n6=j2n+:::+X\ni16=j1\ni26=j2\n...\ni2n6=j2n: (30)\nThe purity condition (29) can be rewritten as a CCR:\nPl(\u001aA1) +Chs(\u001aA1) +Sl(\u001aA1) =dA1\u00001\ndA1; (31)6\nwithSl(\u001aA1)being the linear entropy of the subsystem A1given by\nSl(\u001aA1) :=X\ni16=j1X\n(i2;:::;i 2n)6=(j2;:::;j 2n)\u0010\nj\u001ai1i2:::i2n;j1j2:::j2nj2\u0000\u001ai1i2:::i2n;j1i2:::i2n\u001a\u0003\ni1j2:::j2n;j1j2:::j2n\u0011\n:(32)\nIt is worthwhile mentioning the CCR in Eq. (31) is a natural generalization of the complementarity relation obtained\nby Jakob and Bergou [53, 54] for bipartite pure quantum systems. More generally, E=p\n2Sl(\u001aA1), whereEis the\ngeneralized concurrence obtained in Ref. [55] for multi-particle pure states. Now, for the boosted observer O’ of Sec.\nII, the same nmassive quantons system is described by j\t\u0003iA1;:::;A 2n=U(\u0003)j\tiA1;:::;A 2n, and the density matrix of\nthe multipartite pure quantum system can be written as [56, 57]\n\u001a\u0003\nA1;:::;A 2n=j\t\u0003iA1;:::;A 2nh\t\u0003j=U(\u0003)\u001aA1;:::;A 2nUy(\u0003); (33)\nwhich implies that Tr\u0000\n\u001a\u0003\nA1;:::;A 2n\u00012= Tr(\u001aA1;:::;A 2n)2, and the whole system remains pure under the Lorentz\nboost. As we used the purity of the density matrix to obtain the complete complementarity relation, then, from\n1\u0000Tr\u0000\n\u001a\u0003\nA1;:::;A 2n\u00012= 0, we can obtain\nPl(\u001a\u0003\nA1) +Chs(\u001a\u0003\nA1) +Sl(\u001a\u0003\nA1) =dA1\u00001\ndA1: (34)\nThis proves our claim that CCR are invariant under Lorentz transformations. For continuous momenta, the result\nshowed here remains valid if applied to the discrete degrees of freedom with the continuous momenta been traced\nout, since we defined Pl;ChsandSlonly for the discrete degrees of freedom. Therefore, one can see that there is a\ntransformation law connecting the CCRs for different Lorentz frames.\nIV. RELATIVISTIC SETTINGS\nA. Single-particle system scenarios\nWe begin by considering three different single-particle states where the particle is moving in two opposing directions\nalong theyaxis and the spins are aligned with the zaxis irrespective of the direction of the boost in the reference\nframeO:\nj\ti=1p\n2(jpi+j\u0000pi)\nj0i; (35)\nj\u0004i=1p\n2(jp;0i+j\u0000p;1i); (36)\nj\bi=1\n2(jpi+j\u0000pi)\n(j0i+j1i); (37)\nwherej0imeans spin ‘up’, and j1ispin ‘down’. In addition, j\u0000piis describing the state whose spatial momentum has\nopposite direction in comparison with jpi. It is worthwhile mentioning that the states j\ti;j\biare separable states,\nwhilej\u0004iis a maximal entangled state. Moreover, the state j\tihas maximal coherence in the momentum degree of\nfreedom, and maximal predictability in the spin degree of freedom. Meanwhile j\biis maximally coherent in both\ndegrees of freedom, and j\u0004ihas no local properties. Now, let us consider an observer O’ boosted with velocity vin\na direction orthogonal to the momentum of the particle in the frame O, i.e., in the x\u0000zplane, making an angle\n\u00122[0;\u0019=2]with thex-axis. Hence, the direction of boost is given by ^e= cos\u0012^x+ sin\u0012^z, and the Wigner rotation\nfollows directly:\nD(W(\u0003;\u0006p)) = cos\u001e\n2I2\u00022+isin\u001e\n2(\u0007sin\u0012\u001bx\u0006cos\u0012\u001bz) (38)\n=\u0012\ncos\u001e\n2\u0006isin\u001e\n2cos\u0012\u0007isin\u001e\n2sin\u0012\n\u0007isin\u001e\n2sin\u0012 cos\u001e\n2\u0007isin\u001e\n2cos\u0012\u0013\n; (39)7\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Sl\n(a)Sl(\u001a\u0003s) =Sl(\u001a\u0003p)as a function of the\nWigner angle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Pl\n=0\n=/8\n=/4\n=/3\n=/2\n(b)Pl(\u001a\u0003s)as a function of the Wigner\nangle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.000.020.040.060.080.100.12Chs(c)Chs(\u001a\u0003s)as a function of the Wigner\nangle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Chs\n(d)Chs(\u001a\u0003p)as a function of the Wigner\nangle.\nFigure 1: The different aspects of the degrees of freedom in the state j\t\u0003ifor different values of \u0012.\nsince\u0006^p=\u0006^y. Therefore, the observer in O’ assigns in general a different state to the same system. For instance,\nthe state given by Eq. (35) in O’ is described by\nj\t\u0003i=U(\u0003)j\ti=1p\n2(j\u0003pi\nD(W(\u0003;p))j0i+j\u0000\u0003pi\nD(W(\u0003;\u0000p))j0i) (40)\n=1p\n2\u0010\nj\u0003pi[(cos\u001e\n2+isin\u001e\n2cos\u0012)j0i\u0000isin\u001e\n2sin\u0012j1i] +j\u0000\u0003pi[(cos\u001e\n2\u0000isin\u001e\n2cos\u0012)j0i+isin\u001e\n2sin\u0012j1i]\u0011\n;\n(41)\nwhich in general is an entangled state. The reduced density matrix of the spin (momentum) is obtained by tracing\nout the momentum (spin) states:\n\u001a\u0003s= Tr \u0003pj\t\u0003ih\t\u0003j=\u0012\ncos2\u001e\n2+ sin2\u001e\n2cos2\u0012\u0000sin2\u001e\n2sin\u0012cos\u0012\n\u0000sin2\u001e\n2sin\u0012cos\u0012 sin2\u001e\n2sin2\u0012\u0013\n; (42)\n\u001a\u0003p= Tr \u0003sj\t\u0003ih\t\u0003j=\u00121\n21\n2(cos\u001e+isin\u001ecos\u0012)\n1\n2(cos\u001e\u0000isin\u001ecos\u0012)1\n2\u0013\n: (43)\nIn Fig. 1, we plotted the different aspects of the degrees of freedom of the quanton for different values of \u0012. For\ninstance, if there is no boost, i.e., \u001e= 0, the state remains unchanged, regardless the direction of the boost. Also, if\nthe boost is along the x-axis,\u0012= 0, the state remains the same. Now, if the boost is along the z-axis, the entanglement\nbetween the moment and the spin of the particle increases with the increase of the Wigner angle. In exchange, the\ncoherence of the momentum and the predictability of the spin decrease with \u001e. Beyond that, for any \u0012;\u001e2[0;\u0019=2],\nthe complete complementarity relation Phs+Chs+Sl= 1=2is always satisfied.\nNow, the statej\u0004igiven by Eq. (36) is described in O0as\nj\u0004\u0003i=1p\n2\u0010\nj\u0003pi[(cos\u001e\n2+isin\u001e\n2cos\u0012)j0i\u0000isin\u001e\n2sin\u0012j1i] +j\u0000\u0003pi[isin\u001e\n2sin\u0012j0i+ (cos\u001e\n2+isin\u001e\n2cos\u0012)j1i]\u0011\n;\n(44)\nwhile the reduced density matrices are given by\n\u001a\u0003s=\u001ay\n\u0003p=\u00121\n2icos\u001e\n2sin\u001e\n2sin\u0012\n\u0000icos\u001e\n2sin\u001e\n2sin\u00121\n2\u0013\n: (45)8\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Sl\n=0\n=/8\n=/4\n=/3\n=/2\n(a)Sl(\u001a\u0003j),j=s;p, as a function of the\nWigner angle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Chs(b)Chs(\u001a\u0003j),j=s;p, as a function of the\nWigner angle.\nFigure 2: The different aspects of the degrees of freedom of the quanton in state the j\u0004\u0003ifor different values of \u0012.\nIn this example, by inspecting Fig. 2, if there is no boost, i.e., \u001e= 0the state remains unchanged, regardless\nthe direction of the boost. Also, if the boost is along the x-axis,\u0012= 0, the state remains the same. However, for\n\u00122(0;\u0019=2]and\u001e6= 0, there is an increase of the coherence of both degrees of freedom, in exchange of the consumption\nof the entanglement between the momenta and spin of the particle. In the extreme case where \u0012=\u0019=2and\u001e!\u0019=2,\nboth degrees of freedom have maximal coherence and the state j\u0004\u0003ibecomes separable\nj\u0004\u0003i\u001e=\u0012=\u0019=2=1\n2(j\u0003pi+ij\u0000\u0003pi)\n(j0i\u0000ij1i): (46)\nLastly, in the boosted frame O0, the statej\bigiven by Eq. (37) is described by\nj\b\u0003i=1\n2\u0010\nj\u0003pif[cos\u001e\n2+isin\u001e\n2(cos\u0012\u0000sin\u0012)]j0i+ [cos\u001e\n2\u0000isin\u001e\n2(cos\u0012+ sin\u0012)]j1ig (47)\n+j\u0000\u0003pif[cos\u001e\n2\u0000isin\u001e\n2(cos\u0012\u0000sin\u0012)]j0i+ [cos\u001e\n2+isin\u001e\n2(cos\u0012+ sin\u0012)]j1ig\u0011\n; (48)\nwith the reduced density matrices being\n\u001a\u0003s=\u00121\n21\n2(cos2\u001e\n2\u0000sin2\u001e\n2cos 2\u0012)\n1\n2(cos2\u001e\n2\u0000sin2\u001e\n2cos 2\u0012)1\n2\u0013\n; (49)\n\u001a\u0003p=\u00121\n21\n2(cos\u001e\u0000isin\u001esin\u0012)\n1\n2(cos\u001e+isin\u001esin\u0012)1\n2\u0013\n: (50)\nIn contrast with the second example, here the entanglement between momentum and spin increases with the Wigner\nangle, in exchange of the consumption of the coherence of both degrees of freedom. However, for \u0012=\u0019=2the state is\nseparable:\nj\b\u0003i\u001e=\u0019=2=1\n2(Aj\u0003pi+A\u0003j\u0000\u0003pi)\n(j0i+j1i); (51)\nwhereA=cos\u001e\n2\u0000isin\u001e\n2andA\u0003is the complex conjugate of A. The coherence and entropy of the momentum has\nthe same qualitative behavior as for the spin, which is plotted in Fig. 3.\nFrom these examples, we can see if the state of the system is separable, and has no superposition between the mo-\nmentum states in the reference frame O, then a Lorentz boost cannot generate entanglement between the momentum\nand spin degrees of freedom. This result helps to explain the reported generation of entanglement between momenta\nand spin in one of the first studies of single-particle systems carried out by Peres et al. [11], where they considered\na particle with mass mwhose momentum wave function in the rest frame is given by  (~ p) = (2\u0019)\u00003=4w3=2e\u0000~ p2=2w2,\nwithwbeing the width of the wave packet. This is a Gaussian state of minimum uncertainty, but still represents a\ncontinuous superposition. Therefore, in this case it is possible to generate entanglement via a Lorentz transformation.\nB. Two-particle system scenarios\nAs showed in Sec. IVA, if the momentum states have no coherence, then it is not possible to generate entanglement\nbetween the momenta and spin degrees of freedom. In this section, we begin by discussing this issue for the two-9\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Sl\n(a)Sl(\u001a\u0003s)as a function of the Wigner\nangle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.000.020.040.060.080.100.12Pl=0\n=/8\n=/4\n=/3\n=/2\n(b)Pl(\u001a\u0003s)as a function of the Wigner\nangle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Chs(c)Chs(\u001a\u0003s)as a function of the Wigner\nangle.\nFigure 3: The different aspects of the spin of the quanton in the state j\b\u0003ifor different values of \u0012.\nparticle case and end this section giving two examples. Now, let us consider a two-particle state described in O\nas\nj\tiA;B=X\n\u001b;\u0015 \u001b \u0015jp;qiA;B\nj\u001b;\u0015iA;B; (52)\nwhereP\njj jj2= 1forj=\u001b;\u0015. In addition,jp;qiA;B=jpiA\njqiBdenotes the momentum state of particle A and\nB, respectively, meanwhile j\u001b;\u0015iA;B=j\u001biA\nj\u0015iBrepresents the state of the spins of the particle A and B. The state\nj\tiA;Bis separable and has no coherence between momentum states in the reference frame O. Now, in the boosted\nframeO0, we havej\t\u0003iA;B=U(\u0003)j\tiA;B, i.e.,\nj\t\u0003iA;B=X\n\u001b;\u0015 \u001b \u0015j\u0003p;\u0003qiA;B\nD(W(\u0003;p))j\u001biA\nD(W(\u0003;q))j\u0015iB(53)\n=j\u0003p;\u0003qiA;B\nX\n\u001b \u001bD(W(\u0003;p))j\u001biA\nX\n\u0015 \u0015D(W(\u0003;q))j\u0015iB; (54)\nwhich is also separable. In this case, the Wigner rotation will only change the coherences of the spin states of the\nparticles A and B. Now, let’s consider a state in Owith superposition in the momentum states of the particle A\nj\biA;B=X\np (p)jp;qiA;B\nj\u001b;\u0015iA;B; (55)\nwithP\npj (p)j2= 1. Then, a Lorentz boost can generate entanglement between the momentum and spin of the\nparticle A\nj\b\u0003iA;B=X\np \u001b(p)j\u0003piA\nD(W(\u0003;p))j\u001biA\nj\u0003qiB\nD(W(\u0003;q))j\u0015iB: (56)\nHowever, there is no entanglement between particles A and B. Similarly, if we consider that the state in Ohas\ncoherence in the momentum states of A and B, there will be no entanglement between particles A and B. To obtain\nan entangled state of the whole system in O0, we have to consider a state in Oalready entangled in the momentum\ndegrees of freedom, i.e.,\nj\u0004iA;B=X\np;q (p;q)jp;qiA;B\nj\u001b;\u0015iA;B; (57)\nwithP\np;qj (p;q)j2= 1. Hence, in boosted frame O0we have\nj\u0004\u0003iA;B=X\np;q (p;q)j\u0003p;\u0003qiA;B\nD(W(\u0003;p))j\u001biA\nD(W(\u0003;q))j\u0015iB: (58)\nFor instance, if we consider the particles moving in two opposing directions along the y axis and the spins are\naligned with the zaxis irrespective of the direction of the boost in the reference frame O,\nj\u0004iA;B=1p\n2(jp;\u0000pi+j\u0000p;pi)\nj0;0i; (59)10\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Sl\n(a)Sl(\u001a\u0003s)as a function of the Wigner\nangle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Pl\n=0\n=/8\n=/4\n=/3\n=/2\n(b)Pl(\u001a\u0003s)as a function of the Wigner\nangle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.000.020.040.060.080.100.12Chs(c)Chs(\u001a\u0003s)as a function of the Wigner\nangle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Chs,Sl\nChs\nSl\n(d)Chs(\u001a\u0003p\u0003p)as a function of the\nWigner angle.\nFigure 4: The different aspects of j\u0004\u0003ifor different values of \u0012.\nas before, the observer O’ is boosted with velocity vin a direction orthogonal to the momentum of the particle in the\nframeO, i.e., in the x\u0000zplane, making an angle \u00122[0;\u0019=2]with thex-axis. Then, the Wigner rotation is given by\nEq. (39) and the state in the boosted frame is described by\nj\u0004\u0003iA;B=1p\n2(j\u0003p;\u0000\u0003pi+j\u0000\u0003p;\u0003pi)\n[(cos2\u001e\n2+ sin2\u001e\n2cos2\u0012)j0;0i+ sin2\u001e\n2sin2\u0012j1;1i\n\u0000sin2\u001e\n2sin\u0012cos\u0012(j0;1i+j1;0i)] +icos\u001e\n2sin\u001e\n2sin\u0012p\n2(j\u0003p;\u0000\u0003pi\u0000j\u0000 \u0003p;\u0003pi)\n(j0;1i\u0000j1;0i);(60)\nwhich is an entangled state between all degrees of freedom. In this case, the resource consumed to generate entangle-\nment of the spins of the particles is the bipartite coherence of the reduced momentum-momentum density matrix of\nthe particles A and B\n\u001a\u0003p;\u0003p=1\n2(j\u0003p;\u0000\u0003pih\u0003p;\u0000\u0003pj+j\u0000\u0003p;\u0003pih\u0000\u0003p;\u0003pj) +1\n2(cos4\u001e\n2+ sin4\u001e\n2)(j\u0003p;\u0000\u0003pih\u0000\u0003p;\u0003pj+t:c:);(61)\nwhere t.c. stands of transpose conjugated. In Fig. 4(d), we plotted the coherence and the linear entropy of \u001a\u0003p;\u0003p\nas a function of \u001e, whereSl(\u001a\u0003p;\u0003p)is measuring the entanglement of \u001a\u0003p;\u0003pas a whole with rest of the degrees of\nfreedom. Meanwhile, the concurrence measure [58] Eof\u001a\u0003p;\u0003pdecreases monotonically with Wigner angle, which\nmeans the momentum-momentum entanglement decreases with \u001e, onceE(\u001a\u0003p\u0003p) =p\n2Chs(\u001a\u0003p\u0003p). In addition,\nFigs. 4(a), 4(b) and 4(c) represent the behavior of the different aspects of the spin of particle A. The aspects of the\nspin of particle B display similar behavior. It is worth emphasizing that it is not the entanglement between spin-spin\nthat increases, but the entanglement of the spin of one of the particles with all the other degrees of freedom. In Ref.\n[8], the authors discussed the generation of spin-spin entanglement for two particles under Lorentz boosts. Meanwhile,\nthe momentum state of particles A and B are given by \u001aA\n\u0003p=\u001aB\n\u0003p=1\n2(j\u0003pih\u0003pj+j\u0000\u0003pih\u0000\u0003pj), which implies that\nSl(\u001aj\n\u0003p) = 1=2,j=A;B, and the overall entanglement between the momentum of particle A (B) with rest of the\nsystem does not change under the Lorentz boost, even though the entanglement of momentum-momentum decreases.\nHence, in this case, the entanglement of the momentum of particle A (B) is redistributed among the others degrees\nof freedom. For \u001e= 0(no boost), just the momentum of the particles are entangled. In the limit \u001e=\u0019=2, the\nmomentum of the particles are entangled with the spins, therefore the entanglement of momentum-momentum has to\ndecrease.11\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Chs,Sl\nChs(s)\nSl(s)\n(a)Chs(\u001a\u0003s);Sl(\u001a\u0003s)as a function of the\nWigner angle.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\n0.00.10.20.30.40.5Chs,SlChs(pp)\nSl(pp)\n(b)Chs(\u001a\u0003p\u0003p);Sl(\u001a\u0003p\u0003p)as a function of\nthe Wigner angle.\nFigure 5: The different aspects of j\u0007\u0003i.\nIn contrast, we also can redistribute entanglement to generate coherence in the spin states of the particles A and\nB. For instance, let us consider the following two-particle states in O\nj\u0007iA;B=1p\n2(jp;0iA\nj\u0000p;1iB+j\u0000p;1iA\njp;0iB) =1p\n2(jp;\u0000piA;B\nj0;1iA;B+j\u0000p;piA;B\nj1;0iA;B);(62)\nwith the momentum of the particles along the y-axis. Now, for a boosted frame O’ along the z-axis, the Wigner\nrotation is given by Eq. (39) imposing \u0012=\u0019=2. Hence,\nj\u0007\u0003iA;B=1p\n2icos\u001e\n2sin\u001e\n2(j\u0003p;\u0000\u0003pi+j\u0000\u0003p;\u0003pi)\n(j0;0i\u0000j1;1i) +1p\n2j\u0003p;\u0000\u0003pi\n(cos2\u001e\n2j0;1i\n+ sin2\u001e\n2j1;0i) +1p\n2j\u0000\u0003p;\u0003pi\n(sin2\u001e\n2j0;1i+ cos2\u001e\n2j1;0i): (63)\nThe reduced spin density matrices of each particle are given by\n\u001aA\n\u0003s=\u001aB\n\u0003s=\u00121\n2icos\u001e\n2sin\u001e\n2\n\u0000icos\u001e\n2sin\u001e\n21\n2\u0013\n; (64)\nand\u001aA\n\u0003p=\u001aB\n\u0003p=1\n2I2\u00022, whereI2\u00022is the identity matrix. The entanglement of the spin of the particle A (B) with\nthe rest of the system decreases with the Wigner angle. In exchange, the coherence of the spin of particle A (B)\nincreases, as shown in Fig. 5(a). In addition, the entanglement of \u001a\u0003p\u0003pas a whole with the spins of the particles also\ndecreases with \u001e. From Fig. 5(b), the bipartite coherence increases, since it is related to the momentum-momentum\nentanglement of particle A and B, once E(\u001a\u0003p\u0003p) =p\n2Chs(\u001a\u0003p\u0003p), as we also can see from\n\u001a\u0003p\u0003p=1\n2(j\u0003p;\u0000\u0003pih\u0003p;\u0000\u0003pj+j\u0000\u0003p;\u0003pih\u0000\u0003p;\u0003pj) +\u0012\n2 cos2\u001e\n2sin2\u001e\n2j\u0003p;\u0000\u0003pih\u0000\u0003p;\u0003pj+t:c:\u0013\n:(65)\nHence, the entanglement of the momentum of the particle A (B) with rest of the degrees of the system remains\nthe same under the Lorentz boost, although it is shuffled around among the degrees of freedom. For instance, when\n\u001e= 0(no boost), the momentum of particle A is entangled with all the others degrees of freedom. However, in the\nlimit\u001e=\u0019=2, the momentum of the particle A is entangled just with the momentum of the particle B, since\n\f\f\u0007\u0003\u001e=\u0019=2\u000b\nA;B=1p\n2(j\u0003p;\u0000\u0003piA;B+j\u0000\u0003p;\u0003piA;B)\n1p\n2(ij0iA+j1iA)\n1p\n2(j0iB\u0000ij1iB);(66)\nandSl(\u001a\u0003p\u0003p) = 0for\u001e=\u0019=2.\nV. CONCLUSIONS\nAlthough the entanglement entropy does not remain invariant under Lorentz boosts, and neither do the measures\nof predictability and coherence, we showed in this work that these three measures taken together, in a complete12\ncomplementarity relation (CCR), are Lorentz invariant. Even though it is possible to formally define spin in any\nLorentzframe, thereisnorelationshipbetweentheobservableexpectationvaluesindifferentLorentzframes, according\nto Peres et. al. [11]. Here the situation is quite different. First, it is possible to formally define complementarity in any\nLorentz frame and, in principle, there is no relationship between the complementarity relations in different Lorentz\nframes. However, our results showed that it is possible indeed to connect complete complementarity relations in\ndifferent Lorentz frames. Therefore, we showed how the connection between the CCR defined in different Lorentzian\nframes is possible, and disclosed interesting aspects of the redistribution of quantum features for the relativistic\ndynamics of one- and two-particle states and how the role of CCRs helps to keep tracking of how this redistribution\nof quantum features is done.\nAcknowledgments\nThis work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), process\n88882.427924/2019-01, and by the Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ),\nprocess 465469/2014-0.\n[1] E. 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The theory describes Lorentz covariant electrodynamics\nof superconductors where Anderson-Higgs mechanism occurs, at the same time the dynamics of\nconduction electrons remains non-relativistic. It is demonstrated that Goldstone oscillations cannot\nbe accompanied by oscillations of charge density and they generate the transverse \feld only. In\naddition, we consider Goldstone modes and features of Anderson-Higgs mechanism in two-band\nsuperconductors. We study dissipative processes, which are caused by movement of the normal\ncomponent of electron liquid and violate the Lorentz covariance, on the examples of the damped\noscillations of the order parameter and the skin-e\u000bect for electromagnetic waves. An experimental\nconsequence of the extended time-dependent Ginzburg-Landau theory regarding the penetration of\nthe electromagnetic \feld into a superconductor is proposed.\nPACS numbers: 74.20.De, 74.25.-q, 74.25.N-, 74.25.Nf\nKeywords: Ginzburg-Landau theory, Lorentz covariance, gauge invariance, Anderson-Higgs mechanism, Lon-\ndon penetration depth, wave skin-e\u000bect\nI. INTRODUCTION\nThe system undergoing the second-order phase transition (on example of superconductivity) is described with a\nLandau functional\nF=Z\nd3rF=Z\nd3r\u0014~2\n4m(r\t)\u0000\nr\t+\u0001\n+aj\tj2+b\n2j\tj4\u0015\n; (1)\nwhereFis density of free energy, a=\u000b(T\u0000Tc), \t =j\t(r)jei\u0012(r)is a two-component order parameter (the wave\nfunction of condensate of Cooper pairs) so that ns= 2j\tj2is density of superconducting (SC) electrons, a term\n~2\n4mjr\tj2can be understood as density of kinetic energy of Cooper pairs of mass 2 meach. Con\fguration of the \feld\n\t(r) which minimizes the functional (1) is obtained from equation:\n\u000eF\n\u000e\t+=@F\n@\t+\u0000r@F\n@(r\t+)= 0)~2\n4m\u0001\t\u0000a\t\u0000bj\tj2\t = 0: (2)\nThis con\fguration is shown in Fig.1a: they say that the \feld is a string laying in a valley of the potential aj\tj2+b\n2j\tj4.\nAt the same time, out the equilibrium the \feld \t depends on time. For small deviations from the equilibrium\nit is natural to assume [1, 2], that the time derivative @\t=@tis proportional to the variational derivative of the\nfree energy functional \u000eF=\u000e \t+which is equal to zero at the equilibrium. Thus, one can write the time-dependent\nGinzburg-Landau equation [1{10] (TDGL equation) in a form:\n~2\n4mD@\t\n@t=\u0000\u000eF\n\u000e\t+)\u001c@ \n@t=\u00182\u0001 + \u0000j j2 ; (3)\nwhere = \t=\t0is the dimensionless order parameter, \t 0=p\n\u0000a=bis an equilibrium value of the spatial homo-\ngeneous order parameter, Dis a di\u000busion coe\u000ecient of electrons in the normal state. The temperature-dependent\ncoherence length \u0018(T) =q\n~2\n4mja(T)jand the temperature-dependent relaxation time \u001c(T) =~2\n4mDja(T)jcan be found\nfrom the microscopic theory for the case of a gapless SC alloy containing a high concentration of paramagnetic im-\npurities [3, 4] and for dirty superconductors in the Ginzburg-Landau (GL) regime jTc\u0000Tj<<Tc[2, 5]. However, in\n∗Electronic address: konst.phys@gmail.comarXiv:1906.02097v8  [cond-mat.supr-con]  27 May 20212\nFigure 1: Variation of Landau free energy Fs\u0000Fn(for example, SC state sversus normal state n) with the order parameter\n\t. (a) - equilibrium state of the \feld, (b) - monotonous relaxation to the equilibrium, (c) - relaxation to the equilibrium with\ndamped oscillation, (d) - forced oscillation under the action of an external \feld.\nthe general case, we have to treat these parameters phenomenologically. For T >Tcthe TDGL equation has a form\n\u001c0@\t\n@t=\u00182\u0001\t\u0000\t, where\u001c0=\u0019~\n8(T\u0000Tc)is the relaxation time for a homogeneous mode (here and further kB= 1) [6].\nIn this case the equilibrium value is h\ti= 0 but\n\t2\u000b\n6= 0. The corresponding relaxation processes are illustrated in\nFig.1b.\nThus, Eq.(3) describes relaxation of the order parameter at small deviations from the equilibrium. At the same\ntime, in the system undergoing the second-order phase transition the collective excitations can exist which are resonant\noscillations. When a continuous symmetry is spontaneously broken, there emerge two types of collective modes in\ngeneral: massive Higgs mode, which is oscillation of modulus j\tjof the order parameter, and Goldstone mode, which\nis oscillation of the phase \u0012. As it has been demonstrated in [11{13] in collisionless approximation the Higgs mode in\nSC system can exist as perturbation of amplitude of the gap \u0001 which takes the form of oscillations having a frequency\n\u0018j\u0001j. At the same time, the oscillations of order parameter damp in some time \u001c. Thus, the system is out from\nthe equilibrium then the relaxation process depends on relation between period of the eigen oscillations \u00181=!and\nthe damping time: if 1 =!\u001d\u001cthen aperiodic relaxation occurs as shown in Fig.1b and it is described with TDGL\nequation (3). In general case the parameters 1 =!and\u001ccan be in arbitrary relation, hence the relaxation process can\nhave more complicated form, for example, if 1 =!\u001c\u001cthen an oscillatory process with small damping occurs as shown\nin Fig.1c. Moreover, an external \feld can swing the system, that is the undamped oscillations occur, while heat is\nreleased (for example, electromagnetic wave, falling on a superconductor, induces Foucault currents both normal jn\nand superconducting js). Such situation is shown in Fig.1d.\nIn presence of electric 'and magnetic Apotentials the replacements\n@\n@t!@\n@t+i2e\n~';r!r\u0000i2e\nc~A (4)\nmust be done in free energy functional (1) and in Eqs.(2,3) for a gauge invariance. Moreover, the total current is a sum\nof normal current and supercurrent: j=\u001b\u0000\n\u0000r'\u00001\nc@A\n@t\u0001\n+js, where\u001bis conductivity, js=\u0000ie~\n2m(\t+r\t\u0000\tr\t+)\u00003\n2e2\nmcj\tj2A. Thus, the TDGL equations determine dynamics of both order parameter  and electromagnetic \feld\nA\u0016\u0011(';A). On the other hand, equations for electromagnetic \feld should be Lorentz covariant both in vacuum and\nwithin any medium, despite dynamics of particles (medium) is non-relativistic, for example, Maxwell equations in\ndielectrics. However the light speed in the medium is less than the speed in vacuum cand is determined with dynamic\nproperties of the system. At the same time, the dissipative mechanisms give terms which violate Lorentz covariance\nsince the dissipation distinguishes a time direction, i.e., violates the time symmetry t$\u0000twhich is symmetry of the\nLorentz boost. For example, Maxwell equation in medium with conductivity \u001bhas a form:\ncurlH=1\nc@D\n@t+4\u0019\nc\u001bE: (5)\nHere, the \frst two terms are an equation from Lorentz covariant Maxwell equations, the last term4\u0019\nc\u001bEis a dissipative\npart. Depending on material and processes occurring in it, some terms in the equations can be neglected. So, in\nmetals the dissipative term dominates, for example, the strength of electrostatic \feld must be E= 0 inside metal:\nthe nonzero strength would lead to a current, meanwhile the propagation of the current is associated with energy\ndissipation according to Joule-Lenz law Q=j2=\u001b=\u001bE2and, therefore, it cannot be supported in an equilibrium\nstate by itself [14]. For not very large frequencies of electromagnetic waves (less than plasma frequency) a condition\n\u001b\n!\u001d\"(!) is satis\fed for good metals, then the conduction current \u0000\u001b1\nc@A\n@tgives main contribution in electromagnetic\nresponse that gives the skin-e\u000bect, at the same time the displacement current\"\n4\u0019@E\n@tcan be neglected [14, 15]. On the\ncontrary, in dielectrics the dissipative processes can be neglected in the \frst approximation (since conductivity is zero);\nhence, Lorentz covariant electrodynamics remains only. In bad conductors (semiconductors, weak electrolytes, weakly\nionized plasma) both conduction and displacement currents can be important. Analogously to dielectrics and metals\nelectrodynamics of superconductors must be composed of both Lorentz covariant part and dissipative part (due to\nfriction of normal electrons, damping of collective excitations and breaking of Cooper pairs). Which term is dominant\nis determined with physical conditions and processes. Unlike the normal metal phase, in the SC phase the conductivity\nessentially depends on temperature due to density of normal electrons nn:\u001b(T) =\u001cph\nme2nn(T))\u001b(T!0)!0\n(here\u001cphis the mean free time of electron caused by electron-phonon interaction), that is at low temperatures\nthe contribution of the dissipative part decreases. It should be noted that superconductivity is a thermodynamic\ne\u000bect and is not electrodynamic one (superconductor is not ideal conductor). Hence equations for electromagnetic\n\feld in superconductor are result of variation of some functional of action (or free energy functional for equilibrium\ncase):\u000eS\n\u000eA\u0016= 0. The action S[\t(r;t);A\u0016(r;t)] must be Lorentz invariant in order to ensure Lorentz covariant\nelectrodynamics of super\ruid component. The dissipative terms are introduced in the equations of motion by means\nof Rayleigh dissipation function. At the same time, the Lorentz invariance of the action should have consequences for\ndynamics of the scalar \feld \t: the collective pseudo-relativistic excitations (Higgs mode and Goldstone mode) occur.\nIn [2, 5] the general equations for the dynamic behavior of dirty superconductors in GL regime jTc\u0000Tj<< Tcare\nderived from microscopic theory. The local equilibrium approximation leads to a simple generalized TDGL equation\ndescribing relaxation of the order parameter. As indicated above, in this nonequilibrium regime the dissipative\nprocesses dominant, hence the Lorentz covariation can be neglected and a relaxation equation of type TDGL (3) is\nvalid in mean \feld approximation.\nThe dynamic extension of GL theory has been proposed in [16] as a time-dependent nonlinear Schr odinger La-\ngrangian:\nL=i~\t+@\t\n@t\u0000~2\n4mr\t+r\t\u0000V(j\tj); (6)\nwhich describes the low-frequency, long-wavelength dynamics of the pair \feld \t( r;t) for a BCS-type s-wave super-\nconductor at T= 0,Vis potential leading to spontaneously broken U(1) symmetry and is assumed to be a function\nofj\tjonly. We can see that Lagrangian (6) is Galilean invariant. In this Lagrangian the electromagnetic \feld\nA\u0016= (';A) must be included by replacement (4) based on gauge invariance, and Lagrangian of electromagnetic\n\feld\u0000\nE2\u0000H2\u0001\n=8\u0019should be added. Varying the corresponding action:\u000eS(\t;A\u0016)\n\u000eA\u0016= 0 we will \fnd equations for\nthe electromagnetic \feld A\u0016(r;t) in SC medium. Obviously, these equations will not be Lorentz covariant, since\ninitial Lagrangian (6) is not Lorentz invariant. However, as mentioned above, equations for electromagnetic \feld in\nnondissipative medium must be Lorentz covariant like the Maxwell equations in vacuum.\nIt is believed that superconductors cannot contain macroscopic electric \felds in static con\fgurations. This fact is\ndirectly based on the \frst of the London equations:\ndjs\ndt=nse2\nmE (7)\ncurljs=\u0000nse2\nmcH)\u0001H=1\n\u00152H; (8)4\nwhere\u0015=q\nmc2\n4\u0019nse2is the London penetration depth, nsis density of SC electrons, js=nsevis supercurrent density,\nandE,Hare electric and magnetic \felds respectively. Indeed, unlike normal metals, in superconductors due to zero\nresistance to support the direct current jthe presence of the electric \feld Eis not necessary. This means that in the\nstationary regimedj\ndt= 0 we have E= 0 inside superconductor. However, it should be noted, that the second London\nequation (8) is a result of minimization of free energy of the superconductor:1\n8\u0019R\u0002\nH2+\u00152(curlH)2\u0003\ndV, where the\n\frst term is energy of the magnetic \feld, the second term is kinetic energy of the supercurrent. At the same time, the\n\frst London equation is not result of minimization of the free energy of superconductor: it is suggested that motion\nof SC electrons is not accompanied with friction hence they are accelerated by electric \feld E, i.e., it is just the\nsecond Newton law. This means that the \frst London equation (7)is equation of an ideal conductor that discussed\nin Appendix A. Superconductivity is the thermodynamically steady state: con\fguration of both the electric \feld E\nand the magnetic \feld Hmust be found from minimum of some free energy functional F(\t;\t+;';A).\nFrom Eqs.(7,8) we can see that the coupling of the supercurrent density to the electromagnetic \felds through the\nLondon and GL equations is not space-time covariant : under Lorentz boost, the supercurrent density jin the presence\nof the electric \feld Eought to transform as the space components of a 4-vector whose time component would then\nplay role of some supercharge density, while the electric Eand magnetic H\felds transform as components of the two\nindex antisymmetric \feld strength tensor F\u0016\u0017. Thus, it would seem, the Lorentz covariance requires the possibility of\nelectric \felds on the same footing as magnetic ones within superconductors, with an electric penetration depth equal\nto the familiar magnetic one. In works [17, 18] it has been proposed the natural covariant extension of the GL free\nenergy (1) in terms of the Higgs Lagrangian for spontaneous U(1) gauge symmetry breaking in the vacuum (in SI\nunits):\nL=1\n2\"0c2\u0012~\n2e\u0015\u00132(\f\f\f\f\u0012\n@\u0016+i2e\n~A\u0016\u0013\n \f\f\f\f2\n\u00001\n2\u00182(j j2\u00001)2)\n\u00001\n4\"0c2F\u0016\u0017F\u0016\u0017; (9)\nwhere the order parameter  is normalized to the density of electron pairs in a bulk sample  (x) = \t(x)=\t0\n(\t0=p\n\u0000a=b) in the absence of any electromagnetic \feld A\u0016= ('=c;\u0000A), the tensor F\u0016\u0017=@\u0016A\u0017\u0000@\u0017A\u0016is the\nelectromagnetic \feld strength with @\u0016=@=@x\u0016being the space-time gradients for the coordinates x\u0016= (ct;r),\u0015is\nthe magnetic penetration depth. The dynamics is determined from the Lorentz invariant action S=R\nLd4xthrough\nthe variational principle provides the equations of motion for the scalar \feld  and the vector \feld A\u0016respectively.\nMain result of this model is the e\u000bect of electrostatic \feld on a superconductor: the \feld Eas well as the \feld Bcan\ndestroy the SC state. Namely, the critical values of the \felds satisfy the relation:\n\u0012B\nBcm\u00132\n+\u0012E=c\nBcm\u0013\n'1; (10)\nwhereBcmis a thermodynamical magnetic \feld (at B > Bcma type-I superconductor goes into the normal state\nby the \frst-order phase transition). Thus, we can see that at electric \feld E'cBcmsuperconductivity should be\ndestroyed at absence of magnetic \feld. The analysis of experimental data presented in [19]suggests that an external\nelectric \feld does not a\u000bect signi\fcantly the superconducting state . This conclusion is in total contradiction with the\nexpected behavior based on the covariant Lagrangian (9). It has been suggested that such negative experimental\nresult can be explained with that some non-paired normal electrons could play a crucial role against the electric \feld,\nwhereas such contributions are not included in the model.\nIn the work [20] relativistically covariant London equations have been proposed, and they can be understood as\narising from the \"rigidity\" of the super\ruid wave function in a relativistically covariant microscopic theory. They\npredict that for slowly varying electric \felds, both longitudinal and transverse, and assuming no charge density in the\nsuperconductor:\nr2E=1\n\u00152E: (11)\nwhich implies that an electric \feld penetrates a distance \u0015, as a magnetic \feld does. Thus, the screening length of\nthe electric \feld in the SC state is equal to the London penetration depth, not the Thomas-Fermi screening length\n\u0015TF. The associated longitudinal dielectric function is obtained as\n\"(q;!!0) = 1 +1\n\u00152q2(12)\nunlike the dielectric function for normal metal \"(q;!!0) = 1 +1\n\u00152\nTFq2. This model remains debatable [21, 22] and\nrequires experimental veri\fcation [23, 24].5\nIn the Lorentz covariant models the Higgs mode (spectrum with energy gap) and the Goldstone mode (acoustic\nspectrum) should arise. However, unlike the Higgs mode, the Goldstone mode is unobservable that can be explained\nwith the Anderson-Higgs mechanism: oscillations of the phase \u0012are absorbed into the gauge \feld A\u0016. At the same\ntime, the above-mentioned models essentially di\u000ber from the results of the theory of gauge-invariant response of\nsuperconductors to external electromagnetic \feld presented in [25]. In this model the Coulomb interaction \"pushes\"\nthe frequency of the acoustic oscillations to the plasma frequency !p. Thus, Goldstone mode becomes unobservable\nin itself since it turns to plasma oscillations. In addition, in such model the electric \feld is screened with Thomas-\nFermi length \u0015TFlike in the normal metal phase. However, it should be noted, that the phase oscillations \u0012(r;t)\nare oscillations of the order parameter j\tjei\u0012, i.e., they are speci\fc for the SC state. At the same time, the plasma\noscillations exist unchanged both in SC phase and in the normal metal phase, i.e., they are unrelated to the SC\nordering and are not speci\fc for the SC state. Thus, if the plasma oscillations were the phase oscillations of the order\nparameter, then this would be a\u000bected them at the transition point Tcnecessarily; however, the spectrum of the\nplasma oscillations does not depend on temperature.\nProceeding from aforesaid, we are aimed to generalize the GL theory for the nonstationary regimes Fig.1(b-d):\nthe theory should describe the relaxation of the system with accounting of eigen oscillations of internal degrees of\nfreedom, and also forced oscillations under the action of external \feld. Thus, this theory includes the GL theory and\nthe TDGL equation as special cases, and we will call it as the extended TDGL theory . Moreover, this theory must be\nLorentz covariant without accounting of dissipative processes, since it includes Lorentz covariant electrodynamics, at\nthe same time the dynamics of conduction electrons remains non-relativistic. Accounting of dissipative mechanisms\nshould violate the Lorentz covariance of the theory and should lead to relaxation processes in SC system. Thus, our\npaper is organized by the following way. In Sect.II we generalize GL free energy functional to relativistic-like action\nby phenomenological approach. Using this action we study the possible eigen oscillations of the order parameter\n\t(t;r): Higgs mode and Goldstone mode. Moreover, we obtain this relativistic-like GL functional by microscopical\napproach also. In Sect.III using the gauge symmetry we formulate electrodynamics of superconductors in the sense\nof Lorentz covariant equations for 4D electromagnetic potential A\u0016\u0011(';A). The equations describe propagate of\nelectromagnetic \feld in SC medium where Anderson-Higgs mechanism (absorption of the Goldstone bosons into the\ngauge \feld A\u0016and the gaining of mass by the gauge \feld) takes place. In Sect.IV we consider features of Anderson-\nHiggs mechanism in two-band superconductors and occurrence of Leggett's mode. In Sect.V we study in\ruence of the\nmotion of normal component of electron liquid which causes the damping of oscillations both order parameter and\nelectromagnetic \feld. As example we consider the wave skin-e\u000bect, eigen electromagnetic oscillations and relaxation\nof \ructuation of the order parameter. We propose an experimental consequence of the extended TDGL theory\nregarding the penetration of the electromagnetic \feld in a superconductor. Besides we demonstrate that the London\nelectrodynamics and the TDGL equation (3) are limit cases of the extended TDGL theory.\nII. NORMAL MODES AND PSEUDO-RELATIVISTIC COLLECTIVE EXCITATIONS\nA. Phenomenological approach\nIn general case the SC order parameter \t is both spatially inhomogeneous and it can change over time: \t = \t( r;t).\nThe order parameter is a complex scalar \feld which is equivalent to two real \felds: modulus j\t(r;t)jand phase\u0012(r;t)\n(the modulus-phase representation):\n\t(r;t) =j\t(r;t)jei\u0012(r;t): (13)\nFor stationary case \t = \t( r) the free energy functional (1) exists and the steady con\fguration of the \feld \t( r)\nminimizes this functional. However for the nonstationary case \t( r;t) the minimizing procedure loses any sense.\nThus, it is necessary to \fnd an equation determining evolution of the order parameter in time. Our method for\nsolving this problem is as follows. The parameter t- the time can be turned into a coordinate t!\u001dtin some\n4D Minkowski space f\u001dt;rg, where\u001dis an parameter of dimension of speed (like the light speed) which must be\ndetermined with dynamical properties of the system . At the same time, the dynamics of conduction electrons remains\nnon-relativistic. Then the two-component scalar \feld \t( r;t) minimizes some action S(like in the relativistic \feld\ntheory [26]) in the Minkowski space:\nS=1\n\u001dZ\nL(\t;\t+)d\n; (14)\nwhereLis some Lagrangian (density of Lagrange function L=R\nLd3r),d\n\u0011\u001ddtd3ris an element of the 4D\nMinkowski space. The Lagrangian can be built by generalizing the density of free energy Fin Eq.(1) to \"relativistic\"6\ninvariant form by substitution of covariant and contravariant di\u000berential operators\ne@\u0016\u0011\u00121\n\u001d@\n@t;r\u0013\n;e@\u0016\u0011\u00121\n\u001d@\n@t;\u0000r\u0013\n(15)\ninstead the gradient operators: r\t!e@\u0016\t;r\t+!e@\u0016\t+. Then the required Lagrangian takes a form:\nL=~2\n4m\u0010\ne@\u0016\t\u0011\u0010\ne@\u0016\t+\u0011\n\u0000aj\tj2\u0000b\n2j\tj4=~2\n4m1\n\u001d2\u0012@\t\n@t\u0013\u0012@\t+\n@t\u0013\n\u0000~2\n4m(r\t)\u0000\nr\t+\u0001\n\u0000aj\tj2\u0000b\n2j\tj4\u0011T\u0000F:(16)\nHere, as we can see from Eq.(1), the spatial terms of the Lagrangian (free energy F) play role of \"potential\" energy,\nand the time term plays role of \"kinetic\" energy T. Lagrange equation for functional (14) is\ne@\u0016@L\n@(e@\u0016\t+)\u0000@L\n@\t+= 0)~2\n4me@\u0016e@\u0016\t +a\t +bj\tj2\t = 0; (17)\nwhere\ne@\u0016e@\u0016=e@\u0016e@\u0016=1\n\u001d2@2\n@t2\u0000\u0001: (18)\nSince the \feld \t is complex and Lagrangian (16) has U(1) symmetry then the conserving charge takes place. Using\nEq.(17) and its complex conjugate equation we can obtain the continuity condition in a form:\ne@\u0016j\u0016=@\u001a\n@t+ divjs= 0; (19)\nwherej\u0016is a 4D supercurrent:\nj\u0016\u0011(\u001d\u001as;\u0000js) =ie~\n2m\u0010\n\t+e@\u0016\t\u0000\te@\u0016\t+\u0011\n=\u0012\n\u001die~\n2m\u001d2\u0012\n\t+@\t\n@t\u0000\t@\t+\n@t\u0013\n;ie~\n2m\u0000\n\t+r\t\u0000\tr\t+\u0001\u0013\n=\u0012\n\u0000\u001de~\nm\u001d2j\tj2@\u0012\n@t;\u0000e~\nmj\tj2r\u0012\u0013\n: (20)\nIt should be noted that the correct de\fnition of the current j\u0016is possible only with using of gauge invariance, which\nis done in Sect.III.\nSubstituting representation (13) in the Lagrangian (16) we obtain:\nL=~2\n4me@\u0016j\tje@\u0016j\tj+~2\n4mj\tj2e@\u0016\u0012e@\u0016\u0012\u0000aj\tj2\u0000b\n2j\tj4: (21)\nLet us consider small variations of modulus of order parameter from its equilibrium value: j\tj=p\u0000a\nb+\u001e\u0011\t0+\u001e,\nwherej\u001ej\u001c\t0. Then atT <Tcthe Lagrangian takes a form:\nL=~2\n4me@\u0016\u001ee@\u0016\u001e\u00002jaj\u001e2+~2\n4m\t2\n0e@\u0016\u0012e@\u0016\u0012+1\n2a2\nb: (22)\nCorresponding Lagrange equations are\n~2\n4me@\u0016e@\u0016\u001e+ 2jaj\u001e\u0011~2\n4m\u00121\n\u001d2@2\u001e\n@t2\u0000\u0001\u001e\u0013\n+ 2jaj\u001e= 0; (23)\ne@\u0016e@\u0016\u0012\u00111\n\u001d2@2\u0012\n@t2\u0000\u0001\u0012= 0: (24)\nWe can see that modulus and phase variables are separated. Thus, the \feld coordinates j\t(r;t)jand\u0012(r;t) are normal\ncoordinates, and their small oscillations are normal oscillations. From Eqs.(23,24) we obtain dispersion relations for\nharmonic oscillations \u001e(r;t) =\u001e0ei(qr\u0000!t)and\u0012(r;t) =\u00120ei(qr\u0000!t)accordingly:\n(~!)2= 8jajm\u001d2+ (~q)2\u001d2\u0000 Higgs mode ; (25)\n(~!)2= (~q)2\u001d2\u0000 Goldstone mode : (26)7\nThus, we have two types of collective excitations: with an energy gapp\n8jajm\u001d2- Higgs mode and with acoustic\nspectrum!=\u001dq- Goldstone mode illustrated in Fig.2. We can see that for these dispersion relations minh\n!(q)\nqi\n=\n\u001d>0, that is the Landau criterion for super\ruidity is satis\fed. Since the Higgs mode is oscillations of modulus of the\norder parameterj\tjwhich determines the density of SC electrons as ns= 2j\tj2atT!Tc, then these oscillations\nare accompanied by changes of SC density. At the same time, the total electron density must be n=ns+nn= const,\nbecause otherwise the oscillations of charge (plasmons) should take place (the plasmons exist in normal metal too,\nhence these oscillations are not speci\fc for SC state). Thus, the oscillation of nswhenn= const and q6= 0 can be\npresented as counter\rows of SC and normal components so that nsvs+nnvn= 0 - Fig.3. Hence the Higgs mode\ncan be considered as sound in the gas of above-condensate quasiparticles (with spectrump\nj\u0001j2+v2\nF(p\u0000pF)2) -\nthe second sound. It should be noted that since the normal component nn=n\u00002j\tj2is gas of excitations above\ncondensate of the Cooper pairs (moreover, the size of a pair is much more than average distance between electrons),\nthen separation of electrons into SC and normal is some conditionality: in reality each electron makes SC and normal\nmovements simultaneously.\nThe speed\u001dcan be found with the following way. Let us consider the long-wave limit q!0 for Higgs mode, that\nis the whole system oscillates in a phase. To change the SC density and, hence, the normal density, one Cooper pair\nmust be broken as minimum. For this the energy 2 j\u0001jmust be spent as minimum (SC energy gap). In turn, from\nEq.(25) it can see that minimal energy to excite one Higgs boson isp\n8jajm\u001d2, then\np\n8jajm\u001d2= 2j\u0001j)\u001d=vFp\n3; (27)\nwherevFis Fermi velocity, we have used j\u0001j=Tc\u0010\n8\u00192\n7\u0010(3)\u00111=2\u0010\n1\u0000T\nTc\u00111=2\n,a=6\u00192T2\nc\n7\u0010(3)\"F\u0010\nT\nTc\u00001\u0011\nfor pure supercon-\nductor [27]. Thus, the speed \u001ddoes not depend on temperature (at T!Tc) and is\u0018106m=s\u001cc. Physical sense\nof this speed will be de\fned in the next section. Using Eq.(27) we can see that a value ~!= 2j\u0001jis achieved in the\nGoldstone mode (26) at q=p\n2\n\u0018, where\u0018=q\n~2\n4mjajis a temperature-dependent coherence length. Noteworthy that,\nif we setq=1\n\u0015, then ~!(q)>2j\u0001jbe for type-I superconductors, and ~!(q)<2j\u0001jbe for type-II superconductors -\nFig.2, since \u0014\u0011\u0015\n\u0018<1p\n2for the type-I and \u0014>1p\n2for the type-II.\nFigure 2: Higgs oscillations with spectrum (25) - blue line, and Goldstone oscillations with spectrum (26) - red line. For q=p\n2\n\u0018\nwe have ~!= 2j\u0001jin the Goldstone mode. If to set q=1\n\u0015, then ~!(q)>2j\u0001jfor type-I superconductors and ~!(q)<2j\u0001j\nfor type-II superconductors. The region where the pair breaking occurs, because E > 2j\u0001j, is shaded in gray. The free Higgs\nmode lies entirely in this region, hence this mode is unstable due to decay to the above-condensate quasiparticles.\nThe dispersion relation (25) can be rewritten in the following relativistic-like form:\nE2=em2\u001d4+p2\u001d2; (28)8\nFigure 3: Higgs oscillations with spectrum (25) at q6= 0. This mode is oscillations of modulus of order parameter j\tj, the\noscillations are accompanied by changes of density of SC electrons ns= 2j\tj2. At the same time, the total electron density\nmust ben=ns+nn= const, hence the oscillations can be presented as counter\rows of SC and normal components so that\nnsvs+nnvn= 0.\nwhere\nem\u0011p\n8jajm\n\u001d=p\n2~\n\u0018\u001d/(Tc\u0000T)1=2(29)\nis the mass of a Higgs boson. The mass emdetermines the scale of spatial inhomogeneities in a superconductor:\nletE= 0 (that is we consider a stationary con\fguration: \t( t) = const) then p2=\u0000em2\u001d2)p=i~p\n2\n\u0018, hence\n\t = \t 0\u0000\u001e0eip\n~x= \t 0\u0000\u001e0e\u0000p\n2\n\u0018x(for example, proximity e\u000bect, where a superconductor occupies half-space x>0),\nthat is the order parameter is recovered on length\u0018p\n2=~\nem\u001d. It should be noticed since in the normal phase (that is at\nT >TcorH >Hcwherens= 0,nn=n) equilibrium value of the order parameter is \t = 0 then the Goldstone and\nHiggs oscillations loss any sense. In the normal phase the relaxation of the nonequilibrium order parameter \t 6= 0\nto the equilibrium one \t = 0 takes place only. In the normal phase the speed \u001dlosses physical sense and the correct\nlimit transition to the normal state in expressions, which depend on the factor c=\u001d, will be formulated in Sect.III.\nAs seen from Eqs.(25,27) the energy of Higgs boson is E\u00152j\u0001j, that is this mode exists in the free quasiparticle\ncontinuum. Hence the free Higgs oscillation decays to quasiparticles with energyp\nj\u0001j2+v2\nF(p\u0000pF)2each. Thus,\nthe free Higgs mode in a pure superconductor is unstable , hence its observation is problematical. So, investigations of\nresonant excitations in cuprate superconductors using THz pulse in [28] exhibits that in this nonlinear optical process\nthe light-induced excitation of Cooper pairs is dominated, while the collective amplitude (Higgs) \ructuations of the\nSC order parameter give, in general, a negligible contribution. At the same time, Higgs mode is a scalar excitation\nof the order parameter, distinct from charge or spin \ructuations, and thus does not couple to electromagnetic \felds\nlinearly. It makes possible to study the Higgs mode through the nonlinear light{Higgs coupling which manifests itself,\nfor example, in the third harmonic generation and pump-probe spectroscopy mediated by the Higgs mode [29]. It\nshould be noted that the impurity scattering drastically enhances the light{Higgs coupling: in the pure limit the\nHiggs-mode contribution is subleading to quasiparticles, whereas in the dirty regime it becomes comparable with or\neven larger than the quasiparticle contribution. The precise ratio between the Higgs and quasiparticle contributions\nmay depend on details of the system.\nThe Goldstone mode (24) is eddy (Foucault) currents: alternating current j=e~\nmj\tj2r\u0012(since\u0012/ei!t) generates\nalternating magnetic \feld curl H(t) =4\u0019\ncj(t), hence the electric \feld curl E=1\nc@H\n@tis induced, which drives the\neddy currents in turn. In the long wave limit ( q!0) the energy of the Goldstone mode is E= 0, that means\nthe passing of nondissipative direct current. These processes will be detail considered in Sects.III,V. Substituting the\nsupercurrent (20) in the continuity condition (19) we obtain the equation for Goldstone mode (24). Thus, the equation\nfor Goldstone mode (24) is the continuity condition for the corresponding eddy currents . In the Sect.III we will reveal\nthat Goldstone oscillations are not accompanied by the charge oscillations, i.e.,@\u001as\n@t= 0, hence these supercurrents\nare closured div js= 0.\nAt the same time, in this Section we neglected by the normal component nn=n\u00002j\tj2with corresponding friction\n\u0000m\n\u001cphv, where\u001cphis the mean free path time of electrons caused by electron-phonon interaction (or by scattering on\nlattice defects). The friction causes damping of oscillations of the order parameter and can lead to the overdamped\nregime when monotonic relaxation of a \ructuation occurs. These processes are considered in Sect.V.9\nB. Microscopic derivation\nFollowing [30] let us consider electrons in a normal metal ( T >Tc) propagating in a random \"\feld\" of thermodynamic\n\ructuations of the SC order parameter \u0001 (however h\u0001i= 0) which are static and \"smooth\" enough in space. Then\nwe can write down the following Hamiltonian for interaction of an electron with these \ructuations:\nbHint=X\nkh\n\u0001a+\nk\"a+\n\u0000k#+ \u0001a\u0000k#ak\"i\n: (30)\nCorrection to thermodynamic potential (free energy) due to (30) is\nFs\u0000Fn=\u0000j\u0001j2TX\nkX\nnG(k;\"n)G(\u0000k;\u0000\"n) +j\u0001j2TcX\nkX\nnG(k;\"n)G(\u0000k;\u0000\"n)jT=Tc\n+T\n2j\u0001j4X\nkX\nnG2(k;\"n)G2(\u0000k;\u0000\"n) +:::; (31)\nwhereG=1\ni\"n\u0000\u0018is an electron propagator for normal metal, \"n=\u0019T(2n+ 1),\u0018=~\u001dF(k\u0000kF) is energy of an\nelectron near Fermi surface. The \frst term (which is /j\u0001j2T) diverges, but it is compensated by the second term\n(which is/j\u0001j2Tc), the third term is \fnite and it can be taken at T=TcnearTc. The terms from Eq.(31) can be\nrepresented by diagrams shown in Fig.4. Calculation of the coe\u000ecients at j\u0001j2,j\u0001j4gives:\nFs\u0000Fn=V\u0017FT\u0000Tc\nTcj\u0001j2+V\u0017F7\u0010(3)\n16\u00192T2cj\u0001j4+:::; (32)\nwhere\u0017F=mkF\n2\u00192~2is density of state on Fermi surface per spin, Vis volume of the system,P\nk!V\n(2\u0019)2R\nd3k. Thus,\nwe have microscopic method of derivation of GL expansion of the free energy functional which will be generalized as\nfollows.\nLet us consider the \frst diagram in Fig.4 and let us give an additional momentum qand an additional energy\nparameter!m=\u0019T(2m+ 1) to one side of the loop as illustrated in Fig.5. This means that the order parameter will\nbe function of these parameters \u0001( q;!m), and note \u0018k+q\u0019\u0018k+~2kq\nm2wherejkj=kF. Then instead the \frst term\nin the expansion (31) we have:\n\u0000j\u0001j2TX\nkX\nnG(k+q;\"n+!m)G(\u0000k;\u0000\"n) =\u0000j\u0001j2TX\nkX\nn1\ni\"n\u0000\u0018+\u0010\ni!m\u0000~2kq\nm\u00111\n\u0000i\"n\u0000\u0018\n\u0019\u0000j \u0001j2TV\u0017FZ+1\n\u00001d\u0018X\nn1\n\"2n+\u00182+j\u0001j2TcV\u0017F\n2Z+1\n\u00001d\u0018Z1\n\u00001d(cos\u0012)X\nn\"2\nn\u0000\u00182\n(\"2n+\u00182)3\u0014\n(i!m)2+~4k2\nF\nm2q2cos2\u0012\u0015\n=\u0000j\u0001j2TV\u0017FZ+1\n\u00001d\u0018X\nn1\n\"2n+\u00182+j\u0001j2V\u0017F1\n4\u00192T2c7\u0010(3)\n4(i!m)2+j\u0001j2V\u0017F~2\u001d2\nF\n12\u00192T2c7\u0010(3)\n4q2(33)\nup to the second-order terms in the additional parameters qand!m.\nFigure 4: Diagrammatic representation of Ginzburg-Landau expansion from [30].10\nFigure 5: The \frst diagram from Fig.4 which accounts spatial and time inhomogeneities of the order parameter.\nAt \frst let us consider the expansion in the parameter qonly:\nFq=V\u0017F~2\u001d2\nF\n12\u00192T2c7\u0010(3)\n4q2j\u0001j2+V\u0017FT\u0000Tc\nTcj\u0001j2+V\u0017F7\u0010(3)\n16\u00192T2cj\u0001j4: (34)\nLet us introduce \"wave function\" of Cooper pairs which determines density of SC component ns= 2j\tj2[30, 31] so\nthat to obtain the term~2\n4mq2j\tj2(\"kinetic energy\" of Cooper pairs) instead the \frst term in Eq.(34):\n\t =(14\u0010(3)n)1=2\n4\u0019Tc\u0001; (35)\nwheren=k3\nF\n3\u00192is the total electron density. Then\nFq=V~2\n4mq2j\tj2+Vaj\tj2+Vb\n2j\tj4; (36)\nwhere\na=\u0017F(4\u0019Tc)2\n14\u0010(3)nT\u0000Tc\nTc; b =\u0017F(4\u0019Tc)2\n14\u0010(3)n2; (37)\nso thatj\tq=0j2=\u0000a\nb=nTc\u0000T\nTc)ns\nn= 2\u0010\n1\u0000T\nTc\u0011\n. The function Fqshould be understood as free energy per a\nwave-vector q. Corresponding free energy functional is\nF=X\nqFq=VX\nq\u0014~2\n4mq2j\tqj2+aj\tqj2+b\n2j\tqj4\u0015\n\u0019Z\u0014~2\n4mjr\t(r)j2+aj\t(r)j2+b\n2j\t(r)j4\u0015\nd3r; (38)\nwhere\n\t(q) =1\nVZ\n\t(r)e\u0000iqrd3r;\t(r) =X\nq\t(q)eiqr=VZ\n\t(q)eiqrd3q\n(2\u0019)3: (39)\nThus, we have standard GL expansion.\nNow let us consider the expansion with a term with ( i!m)2in Eq.(33). Then we have an expression:\neFq;!m=V1\n4m(i!m)2\n(\u001dF=p\n3)2j\tq;!mj2+V~2\n4mq2j\tq;!mj2+Vaj\tq;!mj2+Vb\n2j\tq;!mj4: (40)\nTo perform analytic continuation of this expression to the real axis we have to make a substitution i!m!~!+i\u000e\nwhere\u000e!0 [30]. Then we obtain:\neFq;!=V~2\n4m\u001d2!2j\tq;!j2+V~2\n4mq2j\tq;!j2+Vaj\tq;!j2+Vb\n2j\tq;!j4\u0011Tq+Fq: (41)11\nHere, we have noted:\n\u001d\u0011\u001dFp\n3; (42)\nthat coincides with a \"light\" speed (27). Presence of the frequency !means dependence of the order parameter on\ntime since\n\t(t) =TZ\n\t(!)e\u0000i!td!\n2\u0019;\t(!) =1\nTZ\n\t(t)ei!tdt; (43)\nwhere T\u00183p\nV=\u001d is some time interval introduced for the same dimensions of \t( !) and \t(t). The function eFq;!does\nnot have sense of free energy. From Eq.(41) we can see that the \frst term plays role of \"kinetic\" energy Tqand the\nremaining terms Fqplays role of \"potential\" energy. Therefore corresponding Lagrangian has a form:\nLq;!=Tq\u0000Fq=V~2\n4m\u001d2!2j\tq;!j2\u0000V~2\n4mq2j\tq;!j2\u0000Vaj\tq;!j2\u0000Vb\n2j\tq;!j4: (44)\nThen an action takes place (like a free energy functional (38)):\nS=T2X\nqZd!\n2\u0019Lq;!=Z\nL[\t(r;t)]d3rdt=1\n\u001dZ\nL[\t(r;t)]d\n; (45)\nwhere the Lagrangian is\nL[\t(r;t)]\u0019~2\n4m1\n\u001d2\u0012@\t\n@t\u0013\u0012@\t+\n@t\u0013\n\u0000~2\n4m(r\t)\u0000\nr\t+\u0001\n\u0000aj\tj2\u0000b\n2j\tj4\n\u0011~2\n4m\u0010\ne@\u0016\t\u0011\u0010\ne@\u0016\t+\u0011\n\u0000aj\tj2\u0000b\n2j\tj4; (46)\nwhich coincides with phenomenological Lagrangian (16) in the extended TDGL theory. Here, e@\u0016ande@\u0016are covariant\nand contravariant di\u000berential operators (15) accordingly. We can see that Lagrangian (46) is Lorentz invariant in\nsome 4D Minkowski space f\u001dt;rg, where\u001dis an parameter of dimension of speed (like the light speed) which is\ndetermined by dynamical properties of the system. At the same time, the dynamics of conduction electrons remains\nnon-relativistic.\nIII. ELECTRODYNAMICS\nA. Gauge invariance\nLet a superconductor be in electromagnetic \feld A\u0016= (';\u0000A) (or contravariant vector A\u0016= (';A)). In order to\nensure the gauge invariance the di\u000berential operation @\u0016\t must be changed as follows:\n@\u0016\t!\u0012\n@\u0016+i2e\nc~A\u0016\u0013\n\t =ei\u0012\u0014\n@\u0016j\tj+ij\tj\u0012\n@\u0016\u0012+2e\nc~A\u0016\u0013\u0015\n: (47)\nIndeed, making a gauge transformation\n\u0012=\u00120+2e\nc~\u001f; A\u0016=A0\n\u0016\u0000@\u0016\u001f=\u001a\n'='0\u00001\nc@\u001f\n@t\nA=A0+r\u001f\u001b\n; (48)\nwe have:\n@\u0016\u0012+2e\nc~A\u0016=@\u0016\u00120+2e\nc~A0\n\u0016: (49)\nHowever in Lagrangian (16) the di\u000berential operators (15) take place:\ne@\u0016\u0011\u00121\n\u001d@\n@t;r\u0013\n6=@\u0016\u0011\u00121\nc@\n@t;r\u0013\n: (50)12\nAt the same time, in order to ensure the gauge invariance of Maxwell equations the \feld A\u0016must be transformed\nwith transformation (48) only. Hence the equality (49) with the di\u000berential operator e@\u0016cannot be satis\fed:\ne@\u0016\u0012+2e\nc~A\u00166=e@\u0016\u00120+2e\nc~A0\n\u0016; (51)\ntherefore the gauge invariance of the Lagrangian is violated. This fact is consequence of that the \felds \t and A\u0016\nmove in di\u000berent Minkowski spaces: with the limit speeds \u001dandcaccordingly.\nIn order to ensure the gauge invariance of the Lagrangian (16) with electromagnetic \feld we should consider a \feld\neA\u0016=\u0010c\n\u001d';\u0000A\u0011\n\u0011(e';\u0000A): (52)\nThen from the transformation (48) we obtain the gauge transformations for the \feld eA\u0016:\n\u0012=\u00120+2e\nc~\u001f;eA\u0016=eA0\n\u0016\u0000e@\u0016\u001f=\u001a\ne'=e'0\u00001\n\u001d@\u001f\n@t\nA=A0+r\u001f\u001b\n: (53)\nAs a result we have\ne@\u0016\u0012+2e\nc~eA\u0016=e@\u0016\u00120+2e\nc~eA0\n\u0016: (54)\nIt should be noticed the following property:\ne@\u0016eA\u0016\u0011\u00121\n\u001d@e'\n@t;rA\u0013\n=\u00121\nc@'\n@t;rA\u0013\n\u0011@\u0016A\u0016: (55)\nInteraction of a charge ewith electromagnetic \feld A\u0016is described with an action\nSint=\u0000Ze\ncA\u0016dx\u0016: (56)\nTaking into account A\u0016dx\u0016='cdt\u0000Adr=e'\u001ddt\u0000Adr=eA\u0016dex\u0016and de\fning the new charge eease\nc=ee\n\u001dwe have\nSint=\u0000Ze\ncA\u0016dx\u0016=\u0000Zee\n\u001deA\u0016dex\u0016: (57)\nThus, interaction of the new charges with the new \felds does not change, for example e'=eee'. Moreover, the\nmagnetic \rux quantum does not change also: \b 0=\u0019~c\ne=\u0019~\u001d\nee. Then the action for a charged particle in the \feld is\nSp+int=Z\u0012mv2\n2+ee\n\u001dAv\u0000eee'\u0013\ndt: (58)\nTherefore equation of motion is\nmdv\ndt=\u0000ee\n\u001d@A\n@t\u0000eere'+ee\n\u001dv\u0002curlA\u0011eeeE+ee\n\u001dv\u0002H=eE+e\ncv\u0002H: (59)\nHere, the electric \feld eEis such that\neE=c\n\u001dE;eeeE=eE: (60)\nFrom the de\fnition of eEandHin Eq.(59) the \frst pair of Maxwell equations follows:\ncurleE=\u00001\n\u001d@H\n@t; (61)\ndivH= 0: (62)\nThen corresponding action for the electromagnetic \feld eA\u0016should be\nSf=\u00001\n16\u0019\u001dZ\neF\u0016\u0017eF\u0016\u0017d\n =1\n8\u0019Z\u0010\neE2\u0000H2\u0011\ndVdt; (63)13\nwhered\n =\u001ddtdV is an element of the 4D Minkowski space, eF\u0016\u0017=e@\u0016eA\u0017\u0000e@\u0017eA\u0016\u0011\u0010\neE;H\u0011\nis Faraday tensor. The\naction (57) can be written as\nSint=\u00001\n\u001dZ\ne\u001aeA\u0016dex\u0016dV=\u00001\n\u001d2Z\neA\u0016ej\u0016d\n; (64)\nwhere the 4D current is\nej\u0016=e\u001adex\u0016\ndt= (\u001de\u001a;e\u001av) =\u0010\n\u001de\u001a;ej\u0011\n; (65)\nhere the new charge and current are\ne\u001a=\u001d\nc\u001a;ej=\u001d\ncj: (66)\nThen the action describing the electromagnetic \feld and interaction of the current with the \feld is\nSint+f=Z\u0014\n\u00001\n\u001d2eA\u0016ej\u0016\u00001\n16\u0019\u001deF\u0016\u0017eF\u0016\u0017\u0015\nd\n: (67)\nVariation of the action \u000eSint+fgives the second pair of Maxwell equations:\ncurlH=1\n\u001d@eE\n@t+4\u0019\n\u001dej; (68)\ndiveE= 4\u0019e\u001a; (69)\nfrom where we can obtain conservation of charges both e\u001aand\u001ain a form:\ne@\u0016ej\u0016= 0)@e\u001a\n@t+ divej= 0)@\u001a\n@t+ divj= 0: (70)\nIn order to \fgure out the physical sense of \feld eE, chargee\u001aand speed \u001dlet us consider Maxwell equations for\nelectromagnetic \feld in a dielectric:\ncurlE=\u00001\nc@H\n@t\ndivH= 0\ncurlH=1\nc@D\n@t+4\u0019\ncj\ndivD= 4\u0019\u001af; (71)\nhereD=\"Eis electric displacement, \"is electric permittivity, \u001afis density of free charges, j=\u001afvis conduction\ncurrent. Then for a case \"= const we can write:\ncurlE=\u00001\nc@H\n@t\ndivH= 0\ncurlH=\"\nc@E\n@t+4\u0019\ncj\ndivE= 4\u0019\u001af\n\"\u0011curl (p\"E) =\u0000p\"\nc@H\n@t\ndivH= 0\ncurlH=p\"\nc@(p\"E)\n@t+4\u0019p\"\ncjp\"\ndiv (p\"E) = 4\u0019\u001afp\"\u0011curleE=\u00001\n\u001d@H\n@t\ndivH= 0\ncurlH=1\n\u001d@eE\n@t+4\u0019\n\u001dej\ndiveE= 4\u0019e\u001af; (72)\nwhere we have de\fned eE\u0011p\"E,e\u001af\u0011\u001af=p\",ej\u0011j=p\",\u001d\u0011c=p\". Comparing Eqs.(72) with Eqs.(61,62,68,69) we\nconclude that superconductor is equivalent to dielectric (in some e\u000bective sense, not in conductivity) with permittivity\n\"=c2\n\u001d2\u0018105; (73)\nand the speed\u001dis the light speed in SC medium if there were no the skin-e\u000bect and Meissner e\u000bect . We can see that\nthis permittivity is giant compared to the permittivity of true dielectrics (maximum \u0018103in segnetoelectrics). In\nvacuum\u001d=c, that is\"= 1 andeA\u0016\u0011A\u0016.\nIt should be noted that the dielectric permittivity is \"=c2=\u001d2in the long wave limit q<1=\u0018only. If frequency of\nelectromagnetic wave is such that ~!\u00152j\u0001j, then a photon can break a Cooper pair with transfer of its constituents\nin the free quasiparticle states. Hence in this area the strong absorption of the waves takes place. Thus, we can\nsuppose the permittivity \"is equal toc2=\u001d2= const in a frequency interval ~!<2j\u0001jonly, at ~!\u001d2j\u0001jwe suppose14\n\"!\"n(!), where\"n(!) is the dielectric function of normal metal. As we transit into the normal phase, where \u0001 = 0,\nthe permittivity c2=\u001d2losses sense, then \"must be replaced by the dielectric function of normal metal \"n(!) with the\nfollowing properties[14, 15]:\n\"n(!6= 0)\u001c4\u0019\u001b\n!; \"n(0) =1; (74)\nwhere the conductivity \u001bcan be assumed constant at low frequencies. The case \"(!= 0) is special and it will be\nconsider in next subsection.\nIn these terms, energy of the electric \feld Ueland density of the \rux of electromagnetic energy Selare\nUel=ED\n8\u0019=\"E2\n8\u0019=eE2\n8\u0019(75)\nSel=c\n4\u0019E\u0002H=\u001d\n4\u0019eE\u0002H; (76)\nso that div Sel=\u0000@Uel\n@t(here,Uelshould be understood as energy of the electromagnetic \feldeE2\n8\u0019+H2\n8\u0019). The\nelectromagnetic \feld tensors in a dielectric are F\u0016\u0017= (E;H) andH\u0016\u0017= (\u0000D;H) with an invariant1\n2F\u0016\u0017H\u0016\u0017=\nH2\u0000ED =H2\u0000\"E2=H2\u0000eE2. In new representation the tensors can be written in a symmetrical form:\neF\u0016\u0017=\u0010\neE;H\u0011\nandeF\u0016\u0017=\u0010\n\u0000eE;H\u0011\nwith the corresponding invariant1\n2eF\u0016\u0017eF\u0016\u0017=H2\u0000eE2=1\n2F\u0016\u0017H\u0016\u0017.\nAs a result of the above reasoning we can write the Lorentz-invariant gauge invariant Lagrangian:\nL=~2\n4m\u0012\ne@\u0016+i2ee\n\u001d~eA\u0016\u0013\n\t\u0012\ne@\u0016\u0000i2ee\n\u001d~eA\u0016\u0013\n\t+\u0000aj\tj2\u0000b\n2j\tj4\u00001\n16\u0019eF\u0016\u0017eF\u0016\u0017(77)\nfor the action\nS=1\n\u001dZ\nL(\t;\t+;eA\u0016;eA\u0016)\u001ddtd3r: (78)\nThe Lagrangian (77) is a sum of energy of spatial-time inhomogeneity (the \frst term), potential energy of the \feld \t\nand self-action (the second and third terms), and the last term is Lagrangian of the electromagnetic \feld eA\u0016which\nis a sum of the external \feld and the self-consistent internal \feld.\nB. Anderson-Higgs mechanism\nThe modulus-phase representation (13) can be considered as a local gauge U(1) transformation [26]:\n\t =j\tjei2ee\n~\u001d\u001f; (79)\nso that the gauge \feld is transformed as\neA0\n\u0016=eA\u0016+e@\u0016\u001f: (80)\nAfter transformations (79,80) the Lagrangian (77) takes a form:\nL=~2\n4m\u0012\ne@\u0016+i2ee\n~\u001deA\u0016\u0013\nj\tj\u0012\ne@\u0016\u0000i2ee\n~\u001deA\u0016\u0013\nj\tj\u0000aj\tj2\u0000b\n2j\tj4\u00001\n16\u0019eF\u0016\u0017eF\u0016\u0017\n\u0019~2\n4me@\u0016\u001ee@\u0016\u001e\u00002jaj\u001e2+1\n2a2\nb+~2\n4m\u00122ee\n~\u001d\u00132\n\t2\n0eA\u0016eA\u0016\u00001\n16\u0019eF\u0016\u0017eF\u0016\u0017; (81)\nwhere we have redesignated eA0\n\u0016!eA\u0016after the gauge transformation. We have neglected the nonlinear term\n~2\n2m\u00002ee\n~\u001d\u00012\t0\u001eeA\u0016eA\u0016which describes coupling between a Higgs boson \u001eand two photons, since in the pure limit\nthe Higgs-mode contribution is subleading to quasiparticles due to instability of the Higgs boson [29]. Compar-\ning this Lagrangian with Eq.(22) we can see that the Goldstone boson \u0012is absorbed into the gauge \feld eA\u0016, i.e.,\nAnderson-Higgs mechanism occurs. From this Lagrangian the equation for the \feld eA\u0016can be obtained as\ne@\u0016@L\n@\u0010\ne@\u0016eA\u0017\u0011\u0000@L\n@eA\u0017= 0)e@\u0016eF\u0016\u0017+1\n\u00152eA\u0017= 0; (82)15\nwhere we denoted\n\u00152\u0011m\u001d2\n8\u0019ee2\t2\n0=mc2\n8\u0019e2\t2\n0: (83)\nUsing the Lorentz gauge e@\u0016eA\u0016= 0, Eq.(82) is reduced to\ne@\u0016e@\u0016eA\u0017+1\n\u00152eA\u0017= 0)1\n\u001d2@2A\n@t2\u0000\u0001A+1\n\u00152A= 0\n1\n\u001d2@2e'\n@t2\u0000\u0001e'+1\n\u00152e'= 0: (84)\nTaking the \felds as harmonic modes A=A0ei(qr\u0000!t)ande'=e'0ei(qr\u0000!t)we obtain dispersion relation for photons\nin a superconductor:\n!2=\u001d2q2+\u001d2\n\u00152)E2=\u001d2p2+m2\nA\u001d4; (85)\nwhere\nmA=~\n\u0015\u001d/(Tc\u0000T)1=2(86)\nis mass of a photon, i.e., the Higgs mechanism takes place. It is noteworthy that the mass of a Higgs boson (29)\nem=p\n2~\n\u0018\u001dand the mass of a photon (86) are related as\nem(T)\nmA(T)=p\n2\u0014; (87)\nwhere\u0014=\u0015=\u0018is GL parameter. That is for type-I superconductors em<mA, for type-II superconductors em>mA.\nThe dispersion relation (85) can be rewritten in a form:\nq2=\u00001\n\u00152+!2\n\u001d2; (88)\nthat determines the penetration of electromagnetic \feld in a superconductor. Let us consider stationary case != 0,\nthenq2=\u00001\n\u00152, hence A=A0eiqx=A0e\u0000x=\u0015. We can see that Anderson-Higgs mechanism manifests itself in that the\nelectromagnetic (magnetic) \feld penetrates a superconductor in the depth \u0015(83), which is London penetration depth .\nAt the same time, for the nonstationary case !6= 0 the penetration depth is obtained as \u0015\u0010\n1\u0000\u00152!2\n\u001d2\u0011\u00001=2\n> \u0015. If\n!\u0015!c, where\n!c=\u001d\n\u0015=mA\u001d2\n~/(Tc\u0000T)1=2; (89)\nwe haveq2\u00150, that is electromagnetic \feld penetrates superconductor through entire its depth. Thus, the increase\nof the penetration depth with frequency and existence of the critical frequency (89), when the depth becomes in\fnitely\nlarge, is principal result of the extended TDGL theory . The London screening is caused by SC electrons with density\nns. However in a superconductor the normal electrons with density nnexist even at T= 0 due to impurities [30, 31],\natT6= 0 the normal component is thermally excited quasiparticles. The normal component causes absorption of\nelectromagnetic waves and the skin-e\u000bect. These processes smear the penetration e\u000bect, that will be considered in\nSect.V. Moreover, if the frequency ~!\u00152j\u0001j, then intensive absorption of the electromagnetic waves occurs due to\nthe breaking of Cooper pairs, hence in order to observe the penetration e\u000bect it must be ~!c<2j\u0001j. Using Eq.(27)\nin a form ~\u001dp\n2\n\u0018= 2j\u0001j, where\u0018=~p\n4mjajis coherence length, we can represent !cin another form:\n~!c= 2j\u0001j1p\n2\u0014: (90)\nThus, the condition ~!c<2j\u0001jcan be satis\fed in type-II superconductors only (where \u0014 > 1=p\n2). It should be\nnoted that this result has been obtained only for s-wave superconductors.16\nThe physical cause of the threshold !cis illustrated in Fig.6 and it is as follows. Electromagnetic \feld A\u0016is\nscreened in the depth \u0015by the induced supercurrent js=\u0000c\n4\u0019\u00152Aaccording to Higgs mechanism. Thus, the system\ncan respond to the external \feld A\u0016in the minimal length\u0015. For example, if we have a thin plate d\u001c\u0015, then\nmagnetic \feld penetrates it completely and does not a\u000bect its state [6, 32]. Let, at \frst, the electromagnetic wave\nwith wavelength \u0003 =2\u0019\u001d\n!\u001d\u0015fall on a superconductor. Then the depth \u0015is enough to screen this \feld. Now, let\nthe wavelength be \u0003 \u001c\u0015, then the \feld essentially changes within the length \u0015- it changes the sign as illustrated in\nFig.6. In this situation the SC system should screen the oppositely directed \felds in the length which is much less\nthan\u0015, that cannot be done. Hence the \felds with wavelength \u0003 .\u0015(that is!\u0015!c) cannot be screened by the\nsupercurrent. It should be noted that, according to our model, the speed of light in a superconductor is \u001d\u001cc, hence\nat given frequency !the wavelength in the superconductor is much less than in vacuum: \u0003 =2\u0019=\u001d\n!\u001cc\n!, precisely\nbecause of this property the screening e\u000bect can be observable: at the frequencies ~! < 2j\u0001jwe can get into the\ninterval \u0003 .\u0015. In turn, in\ruence of the Anderson-Higgs mechanism on the wave with !\u0015!cis reduced to increasing\nof the wavelength as \u0003 ( !) =\u0015\u0010\n!2\n\u001d2\u00152\u00001\u0011\u00001=2\n, so that \u0003 ( !c) =1.\nFigure 6: long-wave \feld (blue line) \u0003 \u001d\u0015and short-wave \feld (red line) \u0003 \u001c\u0015fall on surface of a superconductor. The\nlong-wave \feld can be screened by SC component in the London depth \u0015(solid blue line). The short-wave oscillation changes\nits sign within the depth \u0015(dash red line), hence the system cannot screen this \feld, because the screening supercurrent would\nhave to turn over many times within the minimal length \u0015. In\ruence of the Anderson-Higgs mechanism on this wave is reduced\nto increasing of the wavelength (solid red line). The screening by the normal component (skin e\u000bect) is not considered here.\nIf we suppose q= 0 in Eq.(85) then we obtain homogeneous oscillation (i.e., all points of a sample oscillate in a\nphase) with frequency !c(89). AtT= 0 we have\n!c(0) =\u001d\n\u0015(0)=\u001dFp\n3cr\n4\u0019e2n\nm\u0011\u001dFp\n3c!p\u001c!p; (91)\nwhere!pis the plasma frequency in metal. We can see this frequency is much lower than the ordinary plasma frequency.\nAs have been demonstrated before these oscillations are consequence of Anderson-Higgs mechanism: Goldstone boson\nis absorbed into gauge \feld and the electromagnetic oscillation mode (84) with spectrum (85) appears. As will\nbe demonstrated below Goldstone oscillations are not accompanied by longitudinal electric \feld (by oscillations of\ncharge density) and they are eddy Meissner currents which generate the transverse \feld div A= 0 only. Thus,\nthese electromagnetic oscillations are eigen oscillations of the SC system instead the phase oscillations. The critical\nfrequency!cis a minimal limit of frequencies which can propagate through the system , since for frequencies ! < !c\nwe haveq2<0. If electromagnetic wave with frequency !\u0015!cfalls on superconductor then the wave is carried by\nthese eigen oscillations hence the superconductor becomes transparent for such wave, at the same time for the wave\nwith frequencies !<!cthe carrier is absent hence such wave re\rects from the surface of superconductor.\nIt should be noted that electromagnetic oscillations (84) with spectrum (85) and critical frequency !c(89) is\nanalogous to propagation of electromagnetic wave (which can be presented as phase oscillations) in plane of Josephson\njunction [6]. Such Josephson plasma frequency exists !J=c\n\u0015J\u001c!p(wherec\u001ccis speed of the wave, \u0015Jis\nmagnetic penetration depth in the Josephson junction) which is the lowest limit of frequencies able to propagate in\nthe junction plane and it has been observed in experiment [34]. Analogy between above-described electrodynamics of\nbulk superconductors and the phase wave in the plane of Josephson junction is presented in a Table I.17\nbulk superconductor Josephson junction at \u0012\u001c\u0019\nwave equation \u0001A\u00001\n\u001d2@2A\n@t2=1\n\u00152A@2\u0012\n@y2\u00001\nc2@2\u0012\n@t2=1\n\u00152\nJ\u0012\nmagnetic penetration depth \u0015=q\nmc2\n8\u0019e2j\tj2\u0015J=q\nc\b0\n8\u00192J0(2\u0015+d)\u0019r\nmc2\n8\u0019e2j\tj2(1+2\u0015\nd)\nlight speed \u001d=cp\"=\u001dFp\n3\u001ccc=cp\"J\u0000\n1 +2\u0015\nd\u0001\u00001\n2\u001cc\ncritical frequency !c=\u001d\n\u0015!J=c\n\u0015J\nTable I: Analogy between the electromagnetic wave in a bulk superconductor and the phase wave in the plane of Josephson\njunction [6] when phase di\u000berent is small \u0012\u001c\u0019. Here,J0is a maximal current so that J=J0sin\u0012(we can suppose J0=e~j\tj2\nmd),\n\b0=\u0019~c\neis the magnetic \rux quantum, d\u001c\u0015is thickness of the junction, \"Jis dielectric constant of material of the junction.\nIn order to obtain equations for the \felds \t and eA\u0016we have to variate an action (78):\nD\u0016@L\n@(D\u0016\t)+\u0000@L\n@\t+= 0)~2\n4m\u0012\ne@\u0016+i2ee\n~\u001deA\u0016\u0013\u0012\ne@\u0016+i2ee\n~\u001deA\u0016\u0013\n\t +a\t +bj\tj2\t = 0; (92)\ne@\u0017@L\n@\u0010\ne@\u0017eA\u0016\u0011\u0000@L\n@eA\u0016= 0)e@\u0017eF\u0016\u0017=\u0000i2\u0019ee~\nm\u001d\u0014\n\t+\u0012\ne@\u0016+i2ee\n\u001d~eA\u0016\u0013\n\t\u0000\t\u0012\ne@\u0016\u0000i2ee\n\u001d~eA\u0016\u0013\n\t+\u0015\n; (93)\nwhere we have denoted D\u0016\u0011e@\u0016+i2ee\n~\u001deA\u0016. If we neglect space-time variation of \t, i.e., e@\u0016\t = 0, then we obtain the\n\feld equation (82). In Eq.(93) we can extract the current ej\u0016using Maxwell equation e@\u0017eF\u0016\u0017=\u00004\u0019\n\u001dej\u0016:\nej\u0016=iee~\n2m\u0014\n\t+\u0012\ne@\u0016+i2ee\n\u001d~eA\u0016\u0013\n\t\u0000\t\u0012\ne@\u0016\u0000i2ee\n\u001d~eA\u0016\u0013\n\t+\u0015\n: (94)\nThe relevant boundary conditions are\n\u0012\ne@\u0016+i2ee\n~\u001deA\u0016\u0013\n\tn\u0016= 0; (95)\nwheren\u0016= (n0;n) is a normal to some hypersurface. If the superconductor is bordered by vacuum or dielectric then\nthe current jcannot \row in or out it [32]:\n\u0012\nr\u0000i2e\n~cA\u0013\n\tn= 0; (96)\nwhere nis a normal to the surface of superconductor. The vector nis determined by the surface of SC sample, i.e.,\nthis vector is given, at the same time the time component n0of the normal is arbitrary, then in order to satisfy the\nboundary condition (95) we should set\n@\u0012\n@t+2ee\n~e'= 0; (97)\nif@j\tj\n@t= 0. Relation (97) determines change of the phase \u0012in time in the presence of the scalar potential 'that, in\nparticular, gives the AC Josephson e\u000bect [33] if 'is understood as the potential of a voltage source.\nIf to exclude the phase \u0012by means of gauge transformations (53) with \u001f=~\u001d\n2ee\u0012then the current (94) can be\nrepresented in a form:\nej\u0016\u0011(\u001de\u001as;ejs) =\u0014\n\u0000\u001dee~\nm\u001d2j\tj2\u0012@\u0012\n@t+2ee\n~e'\u0013\n;ee~\nmj\tj2\u0012\nr\u0012\u00002ee\n\u001d~A\u0013\u0015\n!\u0000\u001d\n4\u0019\u00152(e';A)\u0011\u0000\u001d\n4\u0019\u00152eA\u0016: (98)\nFrom this equation we obtain the ordinary London equation\nejs=\u0000\u001d\n4\u0019\u00152A)js=\u0000c\n4\u0019\u00152A (99)18\nand the London-like equation for charge \u001asand scalar potential ':\ne\u001as=\u00001\n4\u0019\u00152e')\u001as=\u0000\"\n4\u0019\u00152': (100)\nUsing Eq.(98) we can turn the continuity condition into the Lorentz gauge:\n@e\u001a\n@t+ divej= 0)1\n\u001d@e'\n@t+ divA= 0: (101)\nwhich is another manifestation of Anderson-Higgs mechanism. However we must apply the boundary condition (97)\nin 4D London equation (98). Then we obtain \u001as= 0, hence, from electrostatic \"London equation\" (100), we have\n'= 0 (in the London gauge). This means that inside superconductor we have '= const , i.e., the potential electric\n\feld is absent E=\u0000r'= 0like in normal metals . From Lorentz gauge1\n\u001d@e'\n@t+ divA= 0 we obtain div A= 0, hence\ntransverse \felds take place only.\nIt should be noted that the obtained results essentially di\u000ber from the results of the theory of gauge-invariant\nresponse of superconductors to external electromagnetic \feld presented in [25]. In this model the Coulomb interaction\n\"pushes\" the frequency of oscillations of the phase \u0012from acoustic spectrum to plasma frequency !2\np= 4\u0019ne2=m.\nThe Goldstone mode is related to the appearance of oscillations in the order parameter phase, necessarily causing\noscillations of the current j=e~\nmj\tj2r\u0012, which, in turn, give rise to oscillations in the electron number density. Any\nchange in the charge density in metals generates strong longitudinal electric \felds, which results in oscillations at the\nplasma frequency. Thus, Goldstone mode becomes unobservable in itself since it turns to plasma oscillations. However\nin our model the unobservability of Goldstone mode is explained with the Anderson-Higgs mechanism: the phase \u0012\nis absorbed into the gauge \feld eA\u0016. This makes impossible to accompany the phase oscillations by charge density\noscillations. Indeed, according to the continuity condition@\u001a\n@t+ divj= 0, presence of such current that div j6= 0\nchanges the charge density \u001a. Using equation for Goldstone mode (24) we obtain div j=e~\nmj\tj2\u0001\u0012=e~\nmj\tj21\n\u001d2@\n@t@\u0012\n@t.\nHowever we must use the gauge invariant expression@\u0012\n@t+2ee\n~e'instead@\u0012\n@t. Then the continuity condition takes a form:\n@e\u001a\n@t+ee~\nmj\tj21\n\u001d2@\n@t\u0012@\u0012\n@t+2ee\n~e'\u0013\n= 0)@e\u001a\n@t= 0; (102)\nwhere we have used the boundary condition (97). Thus, the phase oscillations are not accompanied by oscillations of\nthe charge density and the condition div j= 0 takes place.\nFrom other hand the current generates magnetic \feld as curl H=4\u0019\ncj, thus the continuity condition can be written\nas follows:\n@e\u001a\n@t+ee~\nmj\tj2div\u0012\nr\u0012\u00002ee\n\u001d~A\u0013\n= 0!@e\u001a\n@t\u0000ee~\nm2ee\n\u001d~j\tj2divA)divA= 0; (103)\nwhere we have excluded the phase \u0012by gauge transformation A=A0+r\u001fwith\u001f=~\u001d\n2ee\u0012and we have used Eq.(102).\nThus, Goldstone oscillations generate the transverse \feld divA= 0 only as result of the boundary condition (97),\nthereby remaining themselves as low-energy excitations ~!(q!0)!0 (acoustic spectrum !=\u001dq) unlike plasmons\n~!p\u001dj\u0001j. For this reason in the London equation j=\u0000c\n4\u0019\u00152Athe transverse \feld div A= 0 takes place only, since\nthis excitation corresponds to Goldstone mode with the lowest energy ~!(q!0)!0, that ensures the closed current\ndivj= 0. Thereby a superconductor is characterized with the dielectric permittivity \"=c2\n\u001d2for the induced electric\n\feldE=\u00001\nc@A\n@tonly (for frequencies 0 <~!<2j\u0001j) and the speed \u001dis the light speed in SC medium. For electrostatic\n\feldE=\u0000r'the permittivity is \"(!= 0;q= 0) =1like in metals. This fact explains experimental results in\n[19] which illustrate that the external electric \feld does not a\u000bect signi\fcantly the SC state unlike predictions of the\ncovariant extension of GL theory [17, 18], since in our model a superconductor screens electrostatic \feld by metallic\nmechanism in the microscopic Thomas-Fermi length, unlike the screening of magnetic \felds by SC electrons in London\ndepth. Thus, unlike models [17{20], electrostatics and magnetostatics have di\u000berent form despite Lorentz covariance\nof the model. It should be noted that the \fxing of the transverse gauge divA= 0is result of the spontaneously broken\ngauge symmetry in SC phase .\nThus, from Eq.(102) we can see that the oscillations of phase (Goldstone mode) cannot change the charge density \u001a,\nhence the phase oscillations are not a\u000eliated with the plasmons. Indeed, the phase oscillations \u0012(r;t) are oscillations\nof the order parameter j\tjei\u0012, i.e., are speci\fc for the SC state. At the same time, the plasma oscillations exist\nunchanged both in SC phase and in the normal metal phase, i.e., they are unrelated to the SC ordering and are not\nspeci\fc for the SC state. Therefore if the plasma oscillations were the phase oscillations of the order parameter, then\nthis would be a\u000bected them at the transition point Tcnecessarily, however the spectrum of the plasma oscillations\ndoes not depend on temperature.19\nSince inside superconductor the electrostatic \feld is absent E= 0 as consequence of boundary condition (97), then\nin order to screen the \feld the electron liquid must become spatial inhomogeneous in the Thomas-Fermi length \u0015TF\n(screening charge is created in this region). However the SC phase is characterized by spatial scales: London depth\n\u0015(T) and coherence length \u0018(T) which are \u0015;\u0018\u001d\u0015TFeven atT= 0. Analogously, there is a time scale \u001c\u0018~=j\u0001j,\nwhich is such that j\u0001j\u001c ~!p, where!pis the plasma frequency. At variations over shorter distances l\u001c\u0018and\ntime intervals \u000et\u001c\u001cthe SC system does not have reserve of the free energya2\n2b\u0018~2\nmj\tj2\n\u00182\u0018~2\nm\u001d2j\tj2\n\u001c2(where the\nscales\u0018and\u001care related as \u0018=\u001c\u0018\u001d) to compensate the spatial-time variation energies \u0018~2\nm1\nl2j\tj2,\u0018~2\nm\u001d21\n(\u000et)2j\tj2.\nThus, variation of order parameter is impossible on distances \u0018\u0015TFand time intervals \u00181=!p. Therefore within\nthese spatial-time scales (TF length and plasma frequency) the SC order parameter \t (or \u0001) losses any sense and\na superconductor should behave like ordinary metal. Thus, the plasma (Thomas-Fermi) screening and the plasma\noscillations exist in metal independently on its state.\nLet us di\u000berentiate Eq.(84) with respect to time and apply the operation curl supposing e'= 0 and div A= 0, then\nwe obtain:\n1\n\u001d2@2eE\n@t2\u0000\u0001eE+1\n\u00152eE= 0 (104)\n1\n\u001d2@2H\n@t2\u0000\u0001H+1\n\u00152H= 0; (105)\nwhere we have used eE=\u00001\n\u001d@A\n@tandH= curl A. Using Maxwell equations curl eE=\u00001\n\u001d@H\n@tand curl H=1\n\u001d@eE\n@t+4\u0019\n\u001dej\nwe obtain accordingly:\n@js\n@t=c2\n4\u0019\u00152E (106)\ncurljs=\u0000c\n4\u0019\u00152H; (107)\nwhich are the \frst and the second London equations. Since Goldstone oscillations generate the transverse \feld only\nA\u0011A?(here div A?= 0) then the electric \feld Eis transverse too: E?=\u00001\nc@A?\n@t. However Eqs.(106,107) are not\nequivalent to Eqs.(104,105) (or to Eq.(84)) since we have extracted the current jfrom the \feld equations by Maxwell\nequations (61,62,68,69) which have unbroken gauge symmetry . The current is Goldstone mode which are absorbed\ninto the gauge \feld Aas we have seen before. Thus, the extraction of jviolates Anderson-Higgs mechanism. As a\nresult we obtain the \frst London equation which is the second Newton law for SC electrons (charges are accelerated\nby the electric \feld Ebut the friction is absent), that is Eq.(106) is equation for ideal conductor. As a result we\nlose the term1\n\u001d2@2A\n@t2which plays role at relatively high frequencies !\u0018!c, and the frequency dependence is caused\nby the complex conductivity \u001bs=\u0000ic2\n4\u0019\u00152!only that provides the low-frequency regime such that the penetration\ndepth does not depend on frequency: L(!) =\u0015[6]. Thus, the \feld equations (84,104,105) are more general than\nLondon equations (106,107). In the ordinary GL theory the extraction of the current jsby Maxwell (Poisson) equation\ncurlH=\u0000\u0001A=4\u0019\ncejfrom the \feld equation\n\u0001A=1\n\u00152A)js=\u0000c\n4\u0019\u00152A (108)\ndoes not lead to the losses in dynamics because GL theory is stationary (equilibrium) theory. At the same time, as\nit has been demonstrated in Appendix A, the \frst London equation (106) is a consequence of the second London\nequation (107). This confusion is caused by the violation of Anderson-Higgs mechanism in the following way. The\nsecond equation has a form without the current as \u0001 H=1\n\u00152Hwhich is a result of minimization of free energy of\na superconductor1\n8\u0019R\u0002\nH2+\u00152(curlH)2\u0003\ndV. As described above, the applying of Maxwell equation curl H=4\u0019\n\u001dej\nextracts the current jwhich had to be absorbed into the gauge \feld A. This violation results in the appearance of\nequation for ideal conductor (the \frst London equation) from the equation for superconductor (the second London\nequation).\nIV. GOLDSTONE MODES IN A TWO-BAND SUPERCONDUCTOR\nIn previous section we could observe Anderson-Higgs mechanism: the Goldstone boson \u0012is absorbed into the gauge\n\feldeA\u0016- Eq.(81). Thus, the Goldstone mode is not observable in single-band superconductors. Let us consider this\nproblem in two-band superconductors. In presence of two order parameters in a bulk isotropic s-wave superconductor20\nthe GL free energy functional (at A= 0) can be written as [35{37]:\nF=Z\nd3r[~2\n4m1jr\t1j2+~2\n4m2jr\t2j2\n+a1j\t1j2+a2j\t2j2+b1\n2j\t1j4+b2\n2j\t2j4+\u000f\u0000\n\t+\n1\t2+ \t 1\t+\n2\u0001\n]; (109)\nwherem1;2denote the e\u000bective mass of carriers in the correspond band, the coe\u000ecients a1;2are given as ai=\ri(T\u0000Tci)\nwhere\riare some constants, the coe\u000ecients b1;2are independent on temperature, the quantity \u000fdescribes interband\nmixing of the two order parameters (proximity e\u000bect). If we switch o\u000b the interband interaction \u000f= 0, then we\nwill have two independent superconductors with di\u000berent critical temperatures Tc1andTc2because the intraband\ninteractions can be di\u000berent. Interaction between gradients of the order parameters (drag e\u000bect), which is considered\nin [38], is omitted for easing of consideration.\nMinimization of the free energy functional with respect to the order parameters, if r\t1;2= 0, gives\n\u001a\na1\t1+\u000f\t2+b1\t3\n1= 0\na2\t2+\u000f\t1+b2\t3\n2= 0\u001b\n: (110)\nNear critical temperature Tcwe have \t3\n1;2!0, hence we can \fnd the critical temperature as a solvability condition\nof the linearized system (110):\na1a2\u0000\u000f2=\r1\r2(Tc\u0000Tc1)(Tc\u0000Tc2)\u0000\u000f2= 0: (111)\nSolving this equation we \fnd Tc> Tc1;Tc2, moreover, the solution does not depend on the sign of \u000f. The sign\ndetermines the phase di\u000berence of the order parameters j\t1jei\u00121andj\t2jei\u00122:\ncos(\u00121\u0000\u00122) = 1 if\u000f<0\ncos(\u00121\u0000\u00122) =\u00001 if\u000f>0; (112)\nthat follows from Eq.(110). The case \u000f<0 corresponds to attractive interband interaction, the case \u000f>0 corresponds\nto repulsive interband interaction.\nAccording to the method described in Sect.II the two-component scalar \felds \t 1;2(r;t) should minimize an action\nSin the Minkowski space:\nS=1\n\u001dZ\nL(\t1;\t2;\t+\n1;\t+\n2)\u001ddtd3r: (113)\nThe LagrangianLis built by generalizing the density of free energy in Eq.(109) to the \"relativistic\" invariant form\nby substitution of covariant and contravariant di\u000berential operators (15) instead the gradient operator:\nL=~2\n4m1\u0010\ne@\u0016\t1\u0011\u0010\ne@\u0016\t+\n1\u0011\n+~2\n4m2\u0010\ne@\u0016\t2\u0011\u0010\ne@\u0016\t+\n2\u0011\n\u0000a1j\t1j2\u0000b1\n2j\t1j4\u0000a2j\t2j2\u0000b2\n2j\t2j4\u0000\u000f\u0000\n\t+\n1\t2+ \t 1\t+\n2\u0001\n; (114)\nwhere for \t 1and \t 2with the masses m1andm2accordingly the same speed \u001dis used. According to [38] the theory\nof a two-band superconductor can be reduced to GL theory of a single-band superconductor for equilibrium values\nof the orders parameters (time independent). In this model the orders parameters are related as \t 2=C(T)\t1at\nT!Tc;T >Tc1;Tc2, where the coe\u000ecient Cis\nC=q\na1\na2;if\u000f<0\nC=\u0000q\na1\na2;if\u000f>0: (115)\nThen the relation (27) can be written in a form:\np\n8jAjM\u001d2= 2j\u0001j; (116)\nwhereA=a1+a2C2+2\u000fC,B=b1+b2C4,M\u00001=1\nm1+C2\nm2andj\u0001j/\t0=p\n\u0000A=B is an e\u000bective gap. Obviously,\n\u001d\u0018vF1;vF2like in Eq.(27).21\nLet us consider movement of the phases only. Using the modulus-phase representation and assuming j\t1j= const\nandj\t2j= const Lagrangian (114) takes a form:\nL=~2\n4m1j\t1j2\u0010\ne@\u0016\u00121\u0011\u0010\ne@\u0016\u00121\u0011\n+~2\n4m2j\t2j2\u0010\ne@\u0016\u00122\u0011\u0010\ne@\u0016\u00122\u0011\n\u00002j\t1jj\t2j\u000fcos (\u00121\u0000\u00122) +L(j\t1j;j\t2j):(117)\nCorresponding Lagrange equations are\n~2\n4m1j\t1j2e@\u0016e@\u0016\u00121\u0000j\t1jj\t2j\u000fsin (\u00121\u0000\u00122) = 0 (118)\n~2\n4m2j\t2j2e@\u0016e@\u0016\u00122+j\t1jj\t2j\u000fsin (\u00121\u0000\u00122) = 0: (119)\nThe phases can be written in a form of oscillations:\n\u00121=\u00120\n1+Aei(qr\u0000!t)\n\u00122=\u00120\n2+Bei(qr\u0000!t); (120)\nwhere equilibrium phases \u00120\n1;2satisfy the relation (112). Substituting the phases in Eqs.(118,119) and linearizing\n(using the relation (112) in a form \u000fcos(\u00120\n1\u0000\u00120\n2) =\u0000j\u000fjand sin(\u00120\n1\u0000\u00120\n2) = 0) we obtain the following dispersion\nrelations:\n!2=q2\u001d2; (121)\nwhereinA=B, and\n(~!)2= 4j\u000fjj\t1j2m2+j\t2j2m1\nj\t1jj\t2j\u001d2+ (~q)2\u001d2; (122)\nwherein\nA\nB=\u0000m1\nm2j\t2j2\nj\t1j2: (123)\nFor symmetrical bands m1=m2\u0011mandj\t1j=j\t2jwe obtain\n(~!)2= 8j\u000fjm\u001d2+ (~q)2\u001d2; A =\u0000B: (124)\nThus, we can see that in two-band superconductors there are two modes of the phase oscillations: the common mode\noscillations with spectrum (121) like Goldstone mode (26) in single-band superconductors, and the oscillations of the\nrelative phase between two SC condensates (122,124) which can be identi\fed as Leggett's mode [39{41]. In a two\nband superconductor a current (\row) takes the following form:\nj=e~\u0012j\t1j2\nm1r\u00121+j\t2j2\nm2r\u00122\u0013\n=iei(qr\u0000!t)e~\u0012j\t1j2\nm1A+j\t2j2\nm2B\u0013\nq; (125)\nfrom where we can see that for Leggett's mode (123) the current is j= 0. Thus, unlike the Goldstone mode (121)\nwhich is the eddy current, Leggett's oscillations are not accompanied by any currents.\nLet us consider the Anderson-Higgs mechanism in a two-band superconductor. The gauge invariant form ( U(1)\u0002\nU(1) symmetry) of Lagrangian (114) is [35{38]:\nL=~2\n4m1\u0012\ne@\u0016+i2ee\n\u001d~eA\u0016\u0013\n\t1\u0012\ne@\u0016\u0000i2ee\n\u001d~eA\u0016\u0013\n\t+\n1+~2\n4m2\u0012\ne@\u0016+i2ee\n\u001d~eA\u0016\u0013\n\t2\u0012\ne@\u0016\u0000i2ee\n\u001d~eA\u0016\u0013\n\t+\n2\n\u0000a1j\t1j2\u0000b1\n2j\t1j4\u0000a2j\t2j2\u0000b2\n2j\t2j4\u0000\u000f\u0000\n\t+\n1\t2+ \t 1\t+\n2\u0001\n\u00001\n16\u0019eF\u0016\u0017eF\u0016\u0017: (126)\nAs in previous consideration the modulus-phase representation (79) can be considered as local gauge U(1) transfor-\nmation:\n\t1=j\t1jei2ee\n~\u001d\u001f1;\t2=j\t2jei2ee\n~\u001d\u001f2: (127)22\nThen the gauge \feld should be transformed as\neA0\n\u0016=eA\u0016+\u000be@\u0016\u001f1+\fe@\u0016\u001f2; (128)\nwhere\n\u000b=j\t1j2\nm1\nj\t1j2\nm1+j\t2j2\nm2; \f =j\t2j2\nm2\nj\t1j2\nm1+j\t2j2\nm2; (129)\nso that\n\u000b+\f= 1;j\t1j2\nm1\f=j\t2j2\nm2\u000b: (130)\nThe transformation (128) excludes the phases \u00121and\u00122from Lagrangian (126) individually leaving only their di\u000ber-\nence:\nL=~2\n4m1\u0012\ne@\u0016+i2ee\n\u001d~eA\u0016\u0013\nj\t1j\u0012\ne@\u0016\u0000i2ee\n\u001d~eA\u0016\u0013\nj\t1j+~2\n4m2\u0012\ne@\u0016+i2ee\n\u001d~eA\u0016\u0013\nj\t2j\u0012\ne@\u0016\u0000i2ee\n\u001d~eA\u0016\u0013\nj\t2j\n+~2\n4j\t1j2j\t2j2\nj\t1j2m2+j\t2j2m1e@\u0016(\u00121\u0000\u00122)e@\u0016(\u00121\u0000\u00122)\u00002\u000fj\t1jj\t2jcos (\u00121\u0000\u00122) +L\u0010\nj\t1j;j\t2j;eF\u0016\u0017eF\u0016\u0017\u0011\n:(131)\nThus, the gauge \feld eA\u0016absorbs the Goldstone bosons \u00121;2so that the Lagrangian becomes dependent on the di\u000berence\n\u00121\u0000\u00122only. The equation for \u00121\u0000\u00122has a form:\ne@\u0016e@\u0016(\u00121\u0000\u00122)\u00004\n~2j\t1j2m2+j\t2j2m1\nj\t1jj\t2j\u000fsin (\u00121\u0000\u00122) = 0: (132)\nThis equation is similar to the sin-Gordon equation, but the coe\u000ecient \u000fis a function of di\u000berence of the equilibrium\nphases\f\f\u00120\n1\u0000\u00120\n2\f\f= 0;\u0019according to Eq.(112). Eq.(132) can be linearized using \u000fcos(\u00120\n1\u0000\u00120\n2) =\u0000j\u000fj, sin(\u00120\n1\u0000\u00120\n2) = 0\nand small oscillations (120), that gives the following spectrum:\n(~!)2= 4j\u000fjj\t1j2m2+j\t2j2m1\nj\t1jj\t2j\u001d2+ (~q)2\u001d2;\nwhich coincides with Leggett's mode spectrum (122). Thus, we can see that in two-band superconductors the common\nmode oscillations with the spectrum (121) are absorbed into the gauge \feld eA\u0016like in single-band superconductors.\nAt the same time, the oscillations of the relative phase between two SC condensates (Leggett's mode) \"survive\" due to\nthat these oscillations are not accompanied by current j= 0 - Eqs.(123,125). Hence Leggett's mode can be observable,\nthat is con\frmed in experiment [42].\nV. DAMPING AND RELAXATION\nIn the previous sections we have considered eigen harmonic oscillations of the order parameter - Higgs mode and\nGoldstone modes absorbed into the gauge \feld except Leggett's mode (in multi-band systems). At the same time,\nmovement of the normal component is accompanied with friction and generation of heat by Joule-Lenz law Q=j2\nn=\u001b,\nwhere jn=ennvnis normal current, nn=n\u0000ns=n\u00002j\tj2is density of the normal component, vnis its speed, \u001b\nis conductivity. Thus, to support the normal movement some electric \feld Emust act, the current and the \feld are\nconnected with Ohm's law: jn=\u001bE. Then, instead the harmonic oscillations, we will have situations illustrated in\nFig.1b - relaxation, Fig.1c - damping oscillations, Fig.1d - forced (undamped) oscillations under the action of external\n\feld.\nThe energy dissipation is accounted with the Rayleigh dissipation function Rwhich determines speed of change of\nthe energy of the system WasdW=dt =\u00002R, that isQ= 2R. As a rule, the dissipative force (friction) is proportional\nto generalized velocity _ q:F=\u0000k_q(kis a friction coe\u000ecient), then R=1\n2k_q2andF=\u0000dR\nd_q. Corresponding equation\nof motion is\nd\ndt@L\n@_q\u0000@L\n@q+@R\n@_q= 0: (133)\nLet us consider several important examples using Lagrangian (77).23\nA. Wave skin-e\u000bect in superconductors\nLet monochromatic electromagnetic wave eA\u0016= (0;A) (in a gauge '= 0, div A= 0) with frequency ~! < 2j\u0001j\nfall on a superconductor perpendicularly to its surface. The wave induces eddy currents both superconducting jsand\nnormal jn. For normal electrons an equation of motion is m_v=eE(t)\u0000m\n\u001cphv, where E(t) =\u00001\nc@A\n@tis the electric\n\feld,\u001cphis the mean free path time of electrons ( \u001c\u00001\nphis frequency of collisions of electrons with phonons or with the\nlattice defects). Then equation for the normal current is\nm\nenndjn\ndt+m\nennjn\n\u001cE=eE(t): (134)\nFor the harmonic \feld E=E0ei(qr\u0000!t)and current jn=jn0ei(qr\u0000!t)we have the Ohm's law in a form jn=\n\u001b1+i!\u001cph\n1+(!\u001cph)2E, where\u001b=\u001cph\nme2nnis conductivity of the normal component. We will consider regime of normal skin-\ne\u000bect only, that is when !\u001cph\u001c1. The Joule-Lenz heat is Q=\u001b!E2, where\u001b!=\u001b\n1+(!\u001cph)2. Respectively, the\nRayleigh dissipation function is\nR=1\n2\u001b!(1 +i!\u001cph)E2=1\n2e\u001b!(1 +i!\u001cph)eE2: (135)\nHere,eE\u0011\u00001\n\u001d@A\n@t\u0011\u0000e@0eA\u0016=e@0eA\u0016, so that\u001bE2=e\u001beE2wheree\u001b=\u001d2\nc2\u001b. Unlike the Joule-Lenz law the Rayleigh\ndissipation function has both an active part and a reactive part determined by the term i!\u001cphfor agreement with the\nresults of London theory [32]. The active part determines dissipation of the electromagnetic energy, the reactive part\ndetermines the phase shift of the current jnrelatively to the \feld Edue to inertia of the system. Then an analog of\nEq.(133) using Eq.(82) can be written:\ne@\u0016@L\n@\u0010\ne@\u0016eA\u0017\u0011\u0000@L\n@eA\u0017+1\n\u001d@R\n@\u0010\ne@0eA\u0017\u0011= 0: (136)\nObviously, this equation is not Lorentz covariant due to the dissipative term. Dissipation distinguishes a time direction,\ni.e., violates the time symmetry t$\u0000twhich is symmetry for the Lorentz boost. Using the gauge '= 0, div A= 0\nEq.(136) is reduced to\n1\n\u001d2@2A\n@t2\u0000\u0001A+1\n\u00152A+4\u0019e\u001b!(1 +i!\u001cph)\n\u001d2@A\n@t= 0: (137)\nTaking the \feld as harmonic mode A=A0ei(qr\u0000!t)we obtain a dispersion relation for photons in a superconductor:\n!2=\u001d2q2+\u001d2\n\u00152!\u0000i4\u0019e\u001b!!)q2=!2\n\u001d2\u00001\n\u00152!+i4\u0019\nc2\u001b!!: (138)\nHere, we have denoted:\n1\n\u00152!\u00111\n\u00152+4\u0019\nc2\u001b!!2\u001cph=1\n\u00152\u0012\n1 +nn\nns(!\u001cph)2\n1 + (!\u001cph)2\u0013\n: (139)\nObviously, \u0015!\u0014\u0015, atT= 0 in pure metal nn= 0 takes place [30, 31] hence we have \u0015!=\u0015in this case. The\nobtained expression (138) di\u000bers from the result of London theory q2=\u00001\n\u00152!+i4\u0019\nc2\u001b!!by a term!2\n\u001d2. However, at\nsmall frequencies ~!\u001cj\u0001j,!2\n\u001d2\u001c1\n\u00152this term can be omitted, main contribution is given by the Meissner e\u000bect \u00001\n\u00152\nand by the skin-e\u000bect of the normal component i4\u0019\nc2\u001b!!. At frequencies !\u0018!c=\u001d\n\u0015the term!2\n\u001d2becomes important,\nthat is discussed in Sect.III. The term!2\n\u001d2has the following nature. The Maxwell equations for the electromagnetic\n\feld in normal metal are\n8\n<\n:curlH=4\u0019\nc\u001bE+1\nc@D\n@t\ncurlE=\u00001\nc@H\n@t9\n=\n;)\u0001H=4\u0019\u001b\nc@H\n@t+\"n(!)\nc2@2H\n@t2; (140)\nwhere D=\"nEhas been used. Taking the \feld as harmonic mode H=H0ei(qr\u0000!t)we obtain the dispersion relation\nfor photons in metal:\nq2=i4\u0019\nc2\u001b!+\"n(!)\nc2!2: (141)24\nIn good metals4\u0019\u001b\n!\u001d\"n(!) (if!6= 0), then the second term in Eq.(141) can be neglected [14, 15]. Hence we obtain\nusual expression for the skin-e\u000bect: q2=i4\u0019\nc2\u001b!. In superconductors the conductivity is determined with the normal\ncomponent \u001b=\u001cph\nme2nnwhich is\u001b!0 in pure system at small temperatures T\u001cTc, so that\u001b\n!.\"can be. Thus,\nfor SC materials the induction term1\nc@D\n@tis as important as the ohmic term \u001bE, unlike normal metals where the\ninduction term can be neglected for quasistationary \felds.\nIt should be noted that the accounting of the complex conductivity from the \frst London equation js=\u0000ic2\n4\u0019\u00152!E\n(i.e.,\u001bs=\u0000ic2\n4\u0019\u00152!) is unnecessary because the induced electric \feld \u00001\nc@A\n@tdrives the eddy currents which are\nGoldstone oscillations, which, in turn, are absorbed into the gauge \feld A. Thus, this complex conductivity has\nalready hidden in the nondissipative part of the \feld equation (137). As we have seen before, the London equations\n(106,107) are not equivalent to Eqs.(84,104,105) since the extraction of current jviolates Anderson-Higgs mechanism.\nAs a result of the extraction we obtain the \frst London equation (106) which is the second Newton law for SC\nelectrons, i.e., it is equation for an ideal conductor. Since we lose a term1\n\u001d2@2eA\n@t2then the frequency dependence\nis caused by the complex conductivity \u001bsonly, that provides the low-frequency regime such that the penetration\ndepth does not depend on frequency: L(!) =\u0015[6] (if we exclude the wave skin-e\u000bect due to normal electrons where\nL(!) =q\nc2\n2\u0019\u001b!).\nFigure 7: Schematic image of dependence of the penetration depth Lon the frequency !for di\u000berent densities of the normal\ncomponent nnand for the London theory. At !!0 the penetration depth is equal to the London depth \u0015for all regimes. If\nnn= 0 then for !\u0015!cSC material becomes transparent i.e., L!1 , however at frequencies ~!\u00152j\u0001jstrong absorption\ntakes place (the dashed lines). For large normal density (low SC density) nn.nthe result for Lis close to the London theory\n(Lweakly depends on frequency). It should be noted that for di\u000berent normal densities nnwe have di\u000berent parameters !c,\nj\u0001jand\u0015.\nFrom Eq.(138) we obtain:\nq=4s\n1\n\u00154!\u0012!2\n\u001d2\u00152!\u00001\u00132\n+\u00124\u0019\nc2\u001b!!\u00132\u0010\ncos'\n2+isin'\n2\u0011\n; (142)\nwhere\n'= arccotRe\u0000\nq2\u0001\nIm (q2)= arccot\u00001\n\u00152!\u0010\n1\u0000!2\n\u001d2\u00152\n!\u0011\n4\u0019\nc2\u001b!!; (143)\nthat corresponds to attenuation of the wave A=A0ei(qx\u0000!t)in the depth of the superconductor which occupies\nhalf-spacex>0. Then substituting q(142) in this \feld Awe obtain the penetration depth:\nL=\u0015!\n4q\u0000!2\n\u001d2\u00152!\u00001\u00012+\u00004\u0019\nc2\u00152!\u001b!!\u000121\nsin'\n2; (144)25\nin a sense A=A0e\u0000x=Lei(Re(q)x\u0000!t). Let us consider the following limit cases:\n1.!= 0. Then '=\u0019and from Eq.(144) we have L=\u0015. i.e., low-frequency \feld is screened like static \feld.\n2. The frequency is equal to the critical frequency (89): !=!c\u0011\u001d\n\u0015!(it should be noted that at nn!0 we have\n\u0015!=\u0015). In this case we have '=\u0019\n2, hence\nL=1q\n2\u0019e2\nmc2nn!c\u001cph\n1+(!c\u001cph)2: (145)\nWe can see that at nn!0 we have result of Sect.III, where the penetration depth becomes in\fnitely large\nL\u001d\u0015. Formally we can suppose \u001cph= 0, thenL!1 too, however the condition \u001cph= 0 means full blocking\nof movement of normal electrons i.e., the substance ceases to be conductor.\n3.! > !c. Then from Eq.(143) we can see 0 \u0014' <\u0019\n2. Ifnn!0 then'!0, hence from Eq.(144) we have\nL!1 , i.e., the superconductor becomes transparent for such electromagnetic waves.\n4.!<!c. Then from Eq.(143) we can see\u0019\n2<'\u0014\u0019. If evennn!0 then'!\u0019, hence from Eq.(144) we have\nL<1, i.e., the superconductor re\rects such electromagnetic waves.\n5.nn=nhencens= 0,\u0015=1,c2\n\u001d2!\"n(!)\u001c4\u0019\u001b\n!. Then from Eq.(143) we can see '!\u0019\n2, and then, we obtain\nL(!) =q\nc2\n2\u0019\u001b!, i.e., the wave skin-e\u000bect in normal metal takes place.\nAt the same time, according to the London theory:\nq2=\u00001\n\u00152!+i4\u0019\nc2\u001b!!)LLondon =\u0015!\n4q\n1 +\u00004\u0019\nc2\u00152!\u001b!!\u000121\nsin'\n2; ' = arccot\u00001\n\u00152!\n4\u0019\nc2\u001b!!: (146)\nWe can see that at !!0 we haveLLondon!\u0015like in Eq.(144). At the same time, even if nn= 0 (hence \u001b= 0),\nthen we have LLondon (!) =\u0015also, at that for any frequency.\nThe total result is shown in Fig.7 schematically. We can see that at !!0 bothLLondon!\u0015andL!\u0015. At large\nfrequencies !\u0018!cthe results of the extended TDGL theory and the London theory can be essentially di\u000berent. So, at\nnn!n(i.e.,ns!0) the penetration depth Lis close toLLondon . However at nn!0 (i.e.,ns!n) we obtain L\u001d\u0015\ninside an interval !c\u0014!<2j\u0001j=~(only for type-II superconductors - Eq.(90)), that corresponds to result of Sect.III\nwhere superconductor becomes transparent for electromagnetic waves with frequencies from this interval. Photons\nwith frequency ~!\u00152j\u0001jbreak Cooper pairs, hence intensive absorption of the electromagnetic waves takes place.\nThus, atT= 0pure (in order to ensure the condition nn!0) type-II (in order to ensure ~!c<2j\u0001j) superconductor\nshould be transparent for electromagnetic waves with frequencies !c\u0014!<2j\u0001j=~, that can be subject for experimental\nveri\fcation . However we can see from Eq.(145) the penetration depth Lis very sensitive to the normal density nn, in\naddition, for clear observation of this e\u000bect the substance should not absorb in this frequency range in itself, therefore\nobservation of this e\u000bect can be di\u000ecult. It should be noted that this problem has been considered only for s-wave\nsuperconductors.\nResults of calculation of the frequency interval \u0017c(0)< \u0017 <2j\u0001(0)j\nh(here\u0017= 2\u0019!,h= 2\u0019~) for pure elemental\ntype-II superconductors niobium Nb, technetium Tcand vanadium VatT= 0 are presented in a Table II. Parameters\nTcand\u0014has been taken from [43{45]. We can see that the transparency interval is in a terahertz range.\nTc, K2j\u0001(0)j= 3:52Tc, K\u0014\u0017c=kB\nh2j\u0001(0)jp\n2\u0014, THzkB\nh2j\u0001(0)j, THz\nNb9.25 32.56 1:43 0.34 0.68\nTc7.77 27.35 0.92 0.44 0.57\nV5.43 19.11 0.85 0.33 0.40\nTable II: Transparency intervals \u0017c(0)\u0014\u0017 <kB\nh2j\u0001(0)jfor pure elemental type-II superconductors niobium, technetium and\nvanadium in THz. The energy gap j\u0001(0)jis measured in Kelvins, \u0014=\u0015\n\u0018=Hc2p\n2Hcmis a GL parameter.26\nB. Eigen electromagnetic oscillations\nIn the Sect.III B we have demonstrated that Goldstone boson is absorbed into gauge \feld, and instead the phase\noscillations the electromagnetic oscillation mode (84) with spectrum (85) appears. These electromagnetic oscillations\nare speci\fc for SC system, and they are eddy Meissner current. The transparency of superconductor for electromag-\nnetic wave with frequency !\u0015!cis caused by the eigen electromagnetic oscillation of SC medium with minimal\nfrequency!c. However, at T >0 and for dirty superconductors for T= 0 even [31] the normal electrons are present.\nInduced electric \feld E=\u00001\nc@A\n@tin such oscillations causes movement of the normal electrons according to Ohm's law\njn=\u001bEand dissipation in a form of Joule-Lenz heat Q=j2\nn=\u001b=\u001bE2occurs. Let us consider a homogeneous mode,\ni.e.,q= 0 is supposed in Eq.(138), and \u0015!=\u0015,\u001b!=\u001bare supposed for simplicity. Then we obtain an equation for\nfrequency:\n!2+i4\u0019\u001d2\nc2\u001b!\u0000\u001d2\n\u00152= 0; (147)\nwhose solution is\n2!=\u0000i4\u0019\u001d2\nc2\u001b\u0006s\n4\u001d2\n\u00152\u0000\u00124\u0019\u001d2\nc2\u001b\u00132\n: (148)\nThis solution determines oscillations of electromagnetic \feld A=A0e\u0000i!t=A0e\u0000t=\u001ce\u0000iRe(!)t, where 1=\u001c=\u0000Im(!)\nis the decay time. We can see that the oscillations are sensitive to the normal density nnvia conductivity \u001b=\u001cph\nme2nn.\nLet us consider the following limit cases:\n1.T!Tc. Then\u0015(T)!1 andnn\u0019n, hence!=\u0000i4\u0019\u001d2\nc2\u001b, that is\u001c=c2\n4\u0019\u001d2\u001band Re(!) = 0. Thus, monotonic\nattenuation of the electromagnetic excitation occurs.\n2.nn= 0, that takes place at T= 0 in pure material [31]. Then !=\u001d\n\u0015which coincides with the critical frequency\n!c(89), and Im( !) = 0. Thus, harmonic oscillations of the electromagnetic \feld in superconductor occur.\nObviously, if ~!c\u00152j\u0001j, then a quant of these oscillations breaks Cooper pair, hence intensive absorption of these\noscillations takes place. As we can see from Eq.(148) the frequency !is sensitive to the normal density nnin the\nsense that the normal electrons caused attenuation of the oscillations. The crossover between overdamped (at high\nT) and underdamped (low T) regimes can be found from Eq.(148) in a form:\nc\n\u0015(T)=2\u0019\u001d\nc\u001b(T): (149)\nIf take\u001cph\u001810\u000010c,n\u00181022cm\u00003,\u001d\u0018108cm=c,\u0015=\u0015(0)\u0012\n1\u0000\u0010\nT\nTc\u00114\u0013\u00001=2\nandnn=n\u0010\nT\nTc\u00114\n, then we can\nevaluate the temperature of crossover:T\nTc\u00180:5. However these oscillations are bulk oscillations, and, as we could see\nin previous subsection, propagation of \ructuations of electromagnetic \feld (with q6= 0) is blocked by the skin-e\u000bect\nifnn6= 0. That is any \feld, by which such eigen oscillations are tried to excite, will be re\rected from surface of a\nsuperconductor. Thus, experimental observation of such eigen electromagnetic oscillations is problematically. Only\nif normal electrons are absent nn= 0, such \ructuations can show themselves by the transparency of superconductor\nfor electromagnetic wave with frequency !c<!< 2j\u0001j=~.\nSimilar frequency (148) and crossover (149) have been obtained in [20]. However in our model these oscillations are\nGoldstone mode absorbed into the gauge \feld, and at that@\u001a\n@t= 0 (hence div js= 0), div A= 0. At the same time,\nin model [20] the analogous oscillations are oscillations of charge density \u001aand longitudinal electric \feld div E= 4\u0019\u001a\nrespectively. In our model at T > Tcthe eigen electromagnetic oscillations do not turn into relaxation of charge\ndensity with relaxation time1\n4\u0019\u001b. According to the rulec2\n\u001d2!\"n(!) (either supposing q= 0 in Eq.(141)) we have\n\u001c(T >Tc) =\"n(!)\n4\u0019\u001b, which is relaxation time for eddy current in metal.\nC. Relaxation of a \ructuation\nLet a \ructuation is formed in some region so that j\u0001(x)j>j\u0001j, wherej\u0001jis the equilibrium value - Fig.8, but\nn=nn+ns= const. Let us consider relaxation of this bubble which can be both damped oscillation and monotonous27\nrelaxation to the equilibrium - Fig.1b,c. After the \ructuation is formed the system tends to the equilibrium: the \row\nof the normal component is directed to the bubble, at the same time the \row of the super\ruid component is directed\nfrom the bubble so that the total \row is j=nsvs+nnvn= 0. Then div js= div(nsvs) =\u0000div(nnvn)\u0019\u00001\n\u0018nnvn,\nbecause the changes of super\ruid and normal components occur in the coherence length \u0018(T)/(Tc\u0000T)\u00001=2. The\ndensity of the super\ruid component is ns= 2j\tj2. The normal motion is accompanied with friction f=\u0000m\n\u001cphvn,\ncorresponding Rayleigh dissipation function is R=m\n2\u001cphnnv2\nn=1\n2\u001b\ne2v2\nn. Using the continuity equation@ns\n@t=\u0000divjs\nandnn\u0019nthe dissipation function takes a form:\nR=\u001b\ne22\u00182\nn2\u0012\n\t@\t+\n@t+ \t+@\t\n@t\u00132\n=\u001b\ne22\u00182\u001d2\nn2\u0010\n\te@0\t++ \t+e@0\t\u00112\n: (150)\nThen equation for the \feld \t is\ne@\u0016@L\n@\u0010\ne@\u0016\t+\u0011\u0000@L\n@\t++1\n\u001d@R\n@\u0010\ne@0\t+\u0011= 0)~2\n4me@\u0016e@\u0016\t +a\t +bj\tj2\t +\u001b\ne24\u00182\u001d\nn2\u0010\n\te@0\t++ \t+e@0\t\u0011\n\t = 0:(151)\nAnalogously to previously considered problem about the skin-e\u000bect this equation is not Lorentz covariant due to the\ndissipative term. Using the modulus-phase representation (13) and linearizing with help j\tj=p\u0000a\nb+\u001e\u0011\t0+\u001e\n(wherej\u001ej\u001c\t0) we obtain an equation for small deviations of the super\ruid density \u001eand for the phase \u0012accordingly:\n~2\n4me@\u0016e@\u0016\u001e+ 2jaj\u001e+\u001b\ne28\u00182\u001d\nn2\t2\n0e@0\u001e= 0; (152)\ne@\u0016e@\u0016\u0012= 0: (153)\nThe oscillations of phase remains without damping in this approximation, since, as has been demonstrated in Sect.II,\nthe Goldstone mode is the eddy supercurrent. For harmonic oscillations \u001e(r;t) =\u001e0ei(qr\u0000!t)we obtain their dispersion\nrelation in a form:\n!2\u0000q2\u001d2\u0000!2\n0+i2\r!= 0; (154)\nwhere!2\n0=8jajm\u001d2\n~2=4j\u0001j2\n~2is a characteristic frequency of the system and \r=\u001b\ne216\u00182m\u001d2\n~2n2\t2\n0is an attenuation\nconstant. We can see that !0/(Tc\u0000T)1=2, at the same time \r/\u00182\t2\n0= const at T!Tc, i.e.,!0\u001c\r. Thus, the\nregime of overdamped oscillation takes place - the \feld \t( r;t) is monotonically relaxing to the thermodynamically\nsteady state \t 0. Then evolution of the \ructuation is \u001e(t)/e\u0000t=\u001c, where\n\u001c=2\r\n!2\n0/(Tc\u0000T)\u00001(155)\nis the relaxation time for a homogeneous ( q= 0) mode in the limit !0\u001c\r. Another solution with \u001c= 1=2\rcan\nbe omitted because this mode decays much faster. The relaxation time is lifetime of the \ructuation: the closer\ntemperature to Tcthe larger size of a \ructuation \u0018(T!Tc)!1 and it lives longer \u001c(T!Tc)!1 . The new phase\nis a \ructuation of in\fnite size (\flls the entire system) and in\fnite lifetime. The relaxation time (155) corresponds to\nthe reduced equation describing the relaxation only:\nq2\u001d2+!2\n0\u0000i2\r!= 0)\u0000~2\n4m\u0001\t +a\t +bj\tj2\t + 2\r~2\n4m\u001d2@\t\n@t= 0: (156)\nThis equation can be made dimensionless using a dimensionless order parameter  = \t=\t0:\n\u001c@ \n@t=\u00182\u0001 + \u0000j j2 ; (157)\nwhere\u0018(T) =q\n~2\n4mja(T)jis temperature-dependent coherence length and \u001c(T) =~2\r\n4mja(T)j\u001d2=\u00182(T)\r\n\u001d2is the\ntemperature-dependent relaxation time. Thus, due to the strong damping at T!Tcthe equation (151) is re-\nduced to Eq.(157) which is analogous to TDGL equation (3). Thus, the TDGL equation is a limit case of the extended\nTDGL theory proposed here . We can see that in consequence of the strong damping the monotonous relaxation of\n\ructuation with the relaxation time (155) takes place. Moreover, this means strong damping of the free Higgs mode,\nso that observation of these oscillations is problematical at T!Tc. On the other hand at T!0 we can have such\nsituation that !0>\rthen the Higgs mode can be more clearly observable.28\nFigure 8: relaxation of a \ructuation of modulus of the order parameter j\tj/j\u0001j. The \ructuation is accompanied by changes\nof density of SC electrons ns= 2j\tj2and normal electrons nnso thatns+nn=n= const. The relaxation is carried out by\ncounter\rows of SC and normal components so that nsvs+nnvn= 0. The dissipation is caused by the friction f=\u0000m\n\u001cphvnof\nthe normal component.\nVI. RESULTS\nIn this work we have formulated the extended TDGL theory which is generalization of the GL theory for the\nnonstationary regimes: damped eigen oscillations (including relaxation) and forced oscillations of the order parameter\n\t(r;t) under the action of external \feld. In this theory, instead the GL functional (1), we propose action (78)\nwith Lorentz invariant Lagrangian (77) for the complex scalar \feld \t = j\tjei\u0012and the gauge \feld A\u0016= (';A) in\nsome 4D Minkowski space f\u001dt;rg, where speed \u001dis determined with dynamical properties of the system. At the\nsame time, the dynamics of conduction electrons remains non-relativistic. For accounting of movement of the normal\ncomponent, which is accompanied by friction, we have used approach with the Rayleigh dissipation function which\ndetermines speed of the dissipation. This makes the theory is not Lorentz covariant since dissipation distinguishes\na time direction, i.e., violates the time symmetry t$\u0000twhich is symmetry of the Lorentz boost. Our results are\nfollows:\n1) The SC system has two types of collective excitations: with an energy gap (quasi-relativistic spectrum) E2=\nem2\u001d4+p2\u001d2(whereemis the mass of a Higgs boson, so that em\u001d2= 2j\u0001j) - free Higgs mode, and with acoustic\n(ultrarelativistic) spectrum E=p\u001d- Goldstone mode, which are oscillations of the order parameter \t( t;r). The\nlight speed \u001dis determined with dynamical properties of the system (27), and it is much less than the vacuum\nlight speed: \u001d=vF=p\n3\u001cc. The Higgs mode is oscillation of modulus of the order parameter j\t(t;r)jand it\ncan be considered as sound in the gas of above-condensate quasiparticles. It should be noted that the free Higgs\nmode in a pure superconductor is unstable due to both strong damping of these oscillations at T!Tc, so that\naperiodic relaxation takes place and decay into above-condensate quasiparticles since E(q)\u00152j\u0001j. At the same time,\nvarious manifestations of Higgs mechanism plays important role in the dynamics of SC system. The Goldstone mode\nis oscillations of the phase \u0012which are eddy current and they are absorbed into the gauge \feld A\u0016according to\nAnderson-Higgs mechanism.\n2) In two-band superconductors the Goldstone mode splits into two branches: common mode oscillations r\u00121=r\u00122\nwith the acoustic spectrum, and the oscillations of the relative phase \u00121\u0000\u00122between two SC condensates (for symmet-\nrical condensates we have r\u00121=\u0000r\u00122) with energy gap in spectrum determined by interband coupling - Leggett's\nmode. The common mode oscillations are absorbed into the gauge \feld A\u0016like in single-band superconductors, at\nthe same time Leggett's mode \"survives\" due to these oscillations are not accompanied by current. Hence Leggett's\nmode can be observable that is con\frmed in experiment [42].\n3) From the gauge invariance of Lagrangian (77) it follows that superconductor is equivalent to dielectric (in some\ne\u000bective sense, not in conductivity) with permittivity \"=c2\n\u001d2\u0018105. At the same time, inside superconductor the\npotential electric \feld is absent E=\u0000r'= 0 as consequence of boundary condition (97). Hence the dielectric\npermittivity \"=c2\n\u001d2is correct for the induced electric \feld E=\u00001\nc@A\n@tonly, and the speed \u001dis the speed of light\nin SC medium if there were no skin-e\u000bect and Meissner e\u000bect. For electrostatic \feld E=\u0000r'the permittivity is\n\"(!= 0;q= 0) =1like in metals. This fact explains experimental results in [19] which illustrate that the external\nelectric \feld does not a\u000bect signi\fcantly the SC state unlike predictions of the covariant extension of GL theory\n[17, 18], since in our model a superconductor screens electrostatic \feld by metallic mechanism in the microscopic\nThomas-Fermi length, unlike the screening of magnetic \felds by SC electrons in the London depth. Thus, unlike the\nmodels [17{20], electrostatics and magnetostatics take di\u000berent forms despite Lorentz covariance of the model. It\nshould be noted a photon with frequency ~!\u00152j\u0001jcan break a Cooper pair with transfer of its constituents in the\nfree quasiparticle states. Hence in this frequency region the strong absorption of the waves takes place. Thus, the\npermittivity \"is equal to c2=\u001d2only when 0 <~! <2j\u0001j. At ~!\u001d2j\u0001jwe can suppose \"=\"n(!), where\"n(!) is29\nthe dielectric function of normal metal.\n4) Unlike popular opinion [25] the Goldstone mode cannot be associated with the plasmon mode. We have demon-\nstrated - Eqs.(102,103), that Goldstone oscillations cannot be accompanied by oscillations of charge density, they\ngenerate the transverse \feld div A= 0 only and they are currents for which div j= 0 as result of the boundary\ncondition (97). This is expressed in that the Anderson-Higgs mechanism takes place: the oscillations of the phase \u0012\nare absorbed into the gauge \feld A\u0016, hence the pure Goldstone oscillations become unobservable. It should be noted\nthat the \fxing of the transverse gauge div A= 0 is result of the spontaneously broken gauge symmetry in SC phase.\nVariation of order parameter is impossible in distances \u0018\u0015TFand time intervals \u00181=!p, since the SC system does\nnot have reserve of the free energy to compensate the spatial-time variation energies. Therefore on these spatial-time\nscales (TF length and plasma frequency) the SC order parameter \t (or \u0001) losses any sense and a superconductor\nshould behave like ordinary metal. Thus, the plasma (Thomas-Fermi) screening and the plasma oscillations exist in\nmetal independently on its state.\n5) As result of interaction of the gauge \feld A\u0016with the scalar \feld \t with spontaneously broken U(1) symmetry a\nphoton obtains mass mA=~\n\u0015\u001din a superconductor, i.e., the Anderson-Higgs mechanism takes place, which manifests\nitself as the penetration depth L(0) =\u0015. The penetration depth depends on frequency: the depthL(!)increases\nwith frequency and such frequency exists !c=\u001d\n\u0015, that the depth becomes in\fnitely large: L(!\u0015!c) =1. This\nis principal result of the extended TDGL theory. It should be noted that ~!c<2j\u0001jfor type-II superconductors\nonly. However the normal component causes absorption of electromagnetic waves and the wave skin-e\u000bect occurs,\nas a result the penetration depth Lis very sensitive to the normal density nn, that is for observation of the e\u000bect\nL(!\u0015!c)!1 the normal electrons should be absent nn= 0. Thus, we have shown that atT= 0 pure type-\nII superconductors (in understanding of a monocrystalline sample without defects and impurities) should become\ntransparent for electromagnetic waves with frequency !from interval !c\u0014! < 2j\u0001j=~, that can be subject for\nexperimental veri\fcation , that illustrated in Fig.7. Thereby observation of this e\u000bect can be di\u000ecult. At the other\nhand, when Goldstone mode is absorbed into gauge \feld, the electromagnetic oscillation mode (84) with spectrum\n(85) appears instead the phase oscillations. These electromagnetic oscillations are eigen oscillations of the SC system\ninstead phase oscillations and they are eddy Meissner current. The critical frequency !cis a minimal limit of\nfrequencies which can propagate through the system. And besides this frequency is much lower than the ordinary\nplasma frequency: !c(T= 0) =\u001d\nc!p\u001c!p. If electromagnetic wave with frequency !\u0015!cfalls on superconductor\nthen the wave is carried by these eigen oscillations hence the superconductor becomes transparent for such wave, at\nthe same time for the wave with frequencies ! < !cthe carrier is absent hence such wave re\rects from the surface\nof superconductor. Moreover, the analogy between above-described electrodynamics of bulk superconductors and\nthe phase wave in a plane of Josephson junction exists that is presented in a Table I. Results of calculation of the\ntransparency intervals \u0017c\u0014\u0017 <2j\u0001j\nhfor pure elemental type-II superconductors niobium Nb, technetium Tcand\nvanadium VatT= 0 are presented in a Table II. The interval is in terahertz range.\n6) The London electrodynamics is a low frequency limit of the extended TDGL theory, i.e., the term1\n\u001d2@2A\n@t2can be\nneglected in the equation for the \feld (137), but the Rayleigh dissipation function (135), which gives the term \u0018\u001b@A\n@t,\nhas to be accounted. At high frequencies !\u0018!c, we should use the extended TDGL theory. At the same time, at\nthe large normal density (low SC density) nn!nthe results are close to the London theory at high frequencies too.\nMoreover, the TDGL equation (3) is a limit case of the extended TDGL theory at T!Tc: due to strong damping\nof oscillations of j\tjthe monotonous relaxation of a \ructuation to the equilibrium state takes place, the process is\ndescribed with Eq.(157) which is analogous to the TDGL equation.\nIn conclusion, it should be noted that the extended TDGL theory has a form of a scalar \feld theory with sponta-\nneously broken gauge symmetry interacting with a gauge \feld. However, the boundary condition (95) violates this\nanalogy due to the fact that, unlike \felds, a superconductor is \fnite system with some surface on which corresponding\nboundary conditions must be set by determining of current through the surface. Consequently, electrostatics and\nmagnetostatics take di\u000berent forms despite Lorentz covariance of the model. In addition, it should be noted, that\nGL theory is two-liquid approximation (the total electron density is sum of the SC density and the normal den-\nsityn=ns+nn). At the same time, in dynamics and electrodynamics of superconductors the coherency factors\nuk0uk\u0006vk0vkanduk0vk\u0006vk0ukplay some role [6]: they give corrections at temperature slightly less than Tc, i.e.,\nwhen there are many quasiparticles. At low temperatures T\u001cTc, there are few quasiparticles, hence the a\u000bect of\nthe coherency factors is negligible.30\nAppendix A: \"Derivation\" of the second London equation from the \frst London equation and vice versa\nFollowing [46] let us write the Newton equation for SC electrons (they do not experience friction) in the electric\n\feldE:mdv\ndt=eE, which can be given a form:\ndjs\ndt=nse2\nmE; (A1)\nwhere js=nsevis supercurrent, nsis density of SC electrons. Eq.(A1) is the \frst London equation (7). Making the\noperation curl for both sides of the equation and taking into account the Maxwell equation curl E=\u00001\nc@H\n@twe obtain:\nd\ndt(curljs) =\u0000nse2\nmc@H\n@t: (A2)\nIntegrating over time we obtain:\ncurljs=\u0000nse2\nmc(H\u0000H0); (A3)\nwhere H0- a constant of integration which does not depend on time, but it can be function of coordinates. Thus,\nby setting the \feld H0we set the initial condition, i.e., con\fguration of the \feld His determined by both response\nof the medium and the initial \feld . Supposing H0= 0 (a sample is introduced into magnetic \feld) we obtain the\nsecond London equation (8). However supposing H06= 0 (a sample in the normal state is in the magnetic \feld H0,\nthen it is cooled below the transition temperature Tc), we obtain the freezing of the magnetic \feld inside the sample.\nThus, the \feld H0is a constant of motion, that characterizes an ideal conductor: curl E=\u00001\nc@B\n@t)B= const\nsince we have inside the sample E=\u001aj= 0 due to ideal conductivity \u001a= 0. Therefore Eq.(A3) does not describe\nthermodynamically steady state, this equation describes the ideal conductor which pushes out or freezes magnetic\n\feld due to the electromagnetic induction and Lenz's rule.\nFrom other hand, following [6] we can di\u000berentiate the second London equation in a form js=\u0000c\n4\u0019\u00152A, where\ndivA= 0 (i.e., A=A?), with respect to time:\n@js\n@t=\u0000c\n4\u0019\u00152@A\n@t=c2\n4\u0019\u00152E=nse2\nmE: (A4)\nThus, we obtain the \frst London equation (A1) which is the second Newton law for SC electrons, i.e., it is equation\nfor an ideal conductor. At the same time, the second London equation is result of minimization of the free energy\nfunctional, i.e., it is equation for a superconductor. This contradiction is resolved in Subsection III B within the\nAnderson-Higgs mechanism. It should be noticed that in Eq.(A4) the \feld Eis transverse \feld E=E?=\u00001\nc@A?\n@t\nonly. 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Using this natural equation we prove a fundamental theorem of Bonnet type\nfor Lorentz surfaces of general type. We consider the special cases of Lorentz surfaces of\nconstant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples\nof Lorentz surfaces illustrating the developed theory.\nContents\n1. Introduction 1\n2. Preliminaries 3\n3. Canonical isotropic coordinates of Lorentz surfaces in R3\n1 4\n4. Natural equation of Lorentz surfaces of general type in R3\n1 7\n5. Examples 10\nReferences 12\n1.Introduction\nThe question of describing surfaces with prescribed mean or Gauss curvature in the Eu-\nclidean 3-space R3and also in the other Riemannian space forms have been subject of an\nintensive study. Especially, the geometry of spacelike or timelike surfaces in the Minkowski\n3-space R3\n1has been of wide interest. For example, a Kenmotsu-type representation formula\nfor spacelike surfaces with prescribed mean curvature was obtained by K. Akutagawa and\nS. Nishikawa in [1]. In [7], G´ alvez et al. obtained a representation for spacelike surfaces\ninR3\n1using the Gauss map and the conformal structure given by the second fundamental\nform. M. A. Magid proved that the Gauss map and the mean curvature of a timelike sur-\nface satisfy a system of partial differential equations and found a Weierstrass representation\nformula for timelike surfaces in R3\n1[13]. Timelike surfaces in R3\n1with prescribed Gauss cur-\nvature and Gauss map are studied in [2] where a Kenmotsu-type representation for such\nsurfaces is given. This representation is used to classify the complete timelike surfaces with\npositive constant Gaussian curvature in terms of harmonic diffeomorphisms between simply\nconnected Lorentz surfaces and the universal covering of the de Sitter Space.\n2010Mathematics Subject Classification. Primary 53B30; Secondary 53A35.\nKey words and phrases. Lorentz surfaces, pseudo-Euclidean space, canonical coordinates, natural\nequations.\n1arXiv:2111.10599v1  [math.DG]  20 Nov 20212 KRASIMIR KANCHEV, OGNIAN KASSABOV, AND VELICHKA MILOUSHEVA\nOn the other hand, it is known that the minimal Lorentz surfaces in R3\n1,R4\n1, andR4\n2\ncan be parametrized by special isothermal coordinates, called canonical , such that the main\ninvariants (the Gauss curvature and the normal curvature) of the surface satisfy a system\nof partial differential equations called a system of natural PDEs . The geometry of the\ncorresponding minimal surface is determined by the solution of this system of natural PDEs.\nIn [10], canonical coordinates for the class of minimal Lorentz surfaces in the Minkowski\nspaceR4\n1are introduced and the following system of natural PDEs is obtained:\n(1.1)4q\nK2+{2\u0001hln4q\nK2+{2= 2K;\n4q\nK2+{2\u0001harctan{\nK= 2{;K2+{26= 0;\nwhereKis the Gauss curvature, {is the curvature of the normal connection (the normal\ncurvature), and \u0001his the hyperbolic Laplace operator in R2\n1.\nSimilar results are obtained for minimal Lorentz surfaces in the pseudo-Euclidean space\nwith neutral metric R4\n2in [3] and [9]. The corresponding system of PDEs has the following\nform:\n(1.2)4q\f\fK2\u0000{2\f\f\u0001hln4q\f\fK2\u0000{2\f\f= 2K;\n4q\f\fK2\u0000{2\f\f\u0001hln\f\f\f\fK+{\nK\u0000{\f\f\f\f= 4{;K2\u0000{26= 0:\nThe minimal Lorentz surfaces in R3\n1can also be considered as surfaces in R4\n1orR4\n2, in\nwhich cases {= 0. So, systems (1.1) and (1.2) are reduced to one PDE:\n(1.3)p\njKj\u0001hlnp\njKj= 2K;K6= 0;\nwhich is the natural equation of minimal Lorentz surfaces in R3\n1. Of course, the results in\nthis case can be directly obtained. In [8], canonical coordinates are introduced for minimal\nLorentz surfaces in R3\n1and equations equivalent to (1.3) are derived.\nThus the following natural question arises: How to generalize the concepts of canonical\ncoordinates and natural equation for a wider class of Lorentz surfaces in R3\n1than that of the\nminimal ones? The class of Weingarten Lorentz surfaces in R3\n1with different real principal\ncurvatures (which is equivalent to H2\u0000K > 0, whereKandHare the Gauss curvature\nand the mean curvature, respectively) is considered in [14]. Canonical principal coordinates\nare introduced for this class of surfaces and a natural non-linear partial differential equation\nis derived, which is equivalent to (1.3) in the case of a minimal surface.\nIn the present paper, we propose an alternative approach. We consider Lorentz surfaces\ninR3\n1satisfyingH2\u0000K6= 0and call them surfaces of general type . We introduce special\nisotropiccoordinates(whichwecall canonical )forthesesurfacesandobtaina natural integro-\ndifferential equation . The natural equation for the class of minimal Lorentz surfaces is given\nbyp\njKj\u0000\nlnp\njKj\u0001\nuv=K;K6= 0:\nIt can be reduced to (1.3) by changing the isotropic coordinates with isothermal ones. This\nshows, that the newly obtained results for an arbitrary Lorentz surface of general type in R3\n1\ngeneralize the known results for the case of a minimal Lorentz surface.\nIn Section 2, we give some basic formulas for Lorentz surfaces in R3\n1parametrized by\narbitrary isotropic coordinates. We present the Gauss and Codazzi equations in terms of\nthese coordinates and formulate the fundamental theorem of Bonnet type.CANONICAL COORDINATES AND NATURAL EQUATION FOR LORENTZ SURFACES IN R3\n1 3\nIn Section 3, we introduce the notion of canonical isotropic coordinates for the class of\nLorentz surfaces of general type ( H2\u0000K6= 0) inR3\n1. We prove existence and uniqueness\ntheorems for these coordinates and give the relation between the canonical coordinates and\nthe natural parameters of the isotropic curves of the surface.\nIn Section 4, we consider Lorentz surfaces of general type parametrized by canonical\ncoordinates and show that the coefficient Fof the first fundamental form and the mean\ncurvatureHof such surface satisfy the following integro-differential equation\nFF uv\u0000FuFv\nF=\u0010\n\"1+Rv\nv0F(u;s)Hu(u;s)ds\u0011\u0010\n\"2+Ru\nu0F(s;v)Hv(s;v)ds\u0011\n\u0000F2H2;\n\"1=\u00061;\"2=\u00061. We call it the natural equation of the Lorentz surfaces in R3\n1and prove\na fundamental theorem of Bonnet type. We consider in detail the special cases of a Lorentz\nsurface with non-zero constant mean curvature and a minimal Lorentz surface.\nIn Section 5, we give examples of different types of Lorentz surfaces and their canonical\ncoordinates in R3\n1.\n2.Preliminaries\nLetR3\n1be the standard three-dimensional pseudo-Euclidean space in which the indefinite\ninner scalar product is given by the formula:\nha;bi=\u0000a1b1+a2b2+a3b3:\nLetM= (D;x)be a Lorentz surface in R3\n1, whereD\u001aR2andx :D!R3\n1is an immersion.\nThe coefficients of the first fundamental form of Mare denoted as usually by E;F;Gand\nL;M;N denote the coefficients of the second fundamental form. Then, the Gauss curvature\nKand the mean curvature HofMare given by the formulas (see [4, 12]):\nK=LN\u0000M2\nEG\u0000F2;H=EN\u00002FM +GL\n2(EG\u0000F2):\nIn a neighbourhood of each point of Mthere exist isotropic coordinates (u;v)such that\nE=G= 0[4]. Such parameters are also called null coordinates [11]. It can easily be seen\nthat if (u;v)and(~u;~v)are two different pairs of isotropic coordinates in a neighbourhood of\na fixed point, then they are related either by u=u(~u),v=v(~v), oru=u(~v),v=v(~u).\nFurther, we consider a surface Mparametrized by isotropic coordinates and without loss\nof generality we assume that F > 0. Then, the formulas for KandHtake the following\nform:\n(2.1) K=M2\u0000LN\nF2;H=M\nF:\nConsider the tangent vector fields X = x u;Y = x vand denote by lthe unit normal vector\nfieldl=xu\u0002xv\njxu\u0002xvjsuch thatfX;Y;lgbe a positively oriented frame field in R3\n1. SinceMis\nparametrized by isotropic coordinates, we have:\nX2= Y2= 0;l2= 1;hX;Yi=F;hX;li=hY;li= 0:\nHence, we get\nXv= Y u;hXu;Xi=hXv;Xi=hYu;Yi=hYv;Yi=hlu;li=hlv;li= 0:4 KRASIMIR KANCHEV, OGNIAN KASSABOV, AND VELICHKA MILOUSHEVA\nUsing the last equalities we obtain the following Frenet-type formulas for the frame field\nfX;Y;lg:\n(2.2)\f\f\f\f\f\f\f\f\f\fXu=Fu\nFX + Ll;\nYu= Ml;\nlu=\u0000M\nFX\u0000L\nFY;\f\f\f\f\f\f\f\f\f\fXv= Ml;\nYv=Fv\nFY +Nl;\nlv=\u0000N\nFX\u0000M\nFY:\nThe integrability conditions of (2.2), considered as a system of PDE for the triple (X;Y;l),\nimply the following Gauss equation\n(2.3)FF uv\u0000FuFv\nF=LN\u0000M2\nand the Codazzi equations\n(2.4) Lv=F\u0012M\nF\u0013\nu;Nu=F\u0012M\nF\u0013\nv:\nNote that (2.3) and (2.1) imply K=\u00001\nF(lnF)uv, which is the Gauss’s Theorema Egregium\nin the case of isotropic coordinates.\nAs it is well known, the Gauss and Codazzi equations are not only necessary, but also\nsufficient conditions for the existence of a solution to the PDE system (2.2). This gives us\na fundamental Bonnet-type theorem for Lorentz surfaces in R3\n1. The proof is analogous to\nthat of the classical theorem for surface in R3(see ([5]).\nTheorem 2.1. LetMbe a Lorentz surface in R3\n1parametrized by isotropic coordinates.\nThen, the coefficients F,L,M,Nof the first and the second fundamental form of Mgive\na solution to the Gauss and Codazzi equations (2.3)and(2.4). If ^Mis obtained fromMby\na proper motion in R3\n1, then ^Mgenerates the same solution to (2.3)and(2.4).\nConversely, if the functions F,L,M,Nsatisfy equations (2.3)and(2.4), then, at least\nlocally, there exists a unique (up to a proper motion in R3\n1) Lorentz surface Mparametrized\nby isotropic coordinates, such that the given functions are the coefficients of the first and the\nsecond fundamental form of M.\n3.Canonical isotropic coordinates of Lorentz surfaces in R3\n1\nIn the present section, we will show that the Lorentz surfaces satisfying H2\u0000K6= 0\nadmit special isotropic coordinates which we will call canonical. It follows from the Codazzi\nequations (2.4) and the second equality of (2.1) that\n(3.1) Lv=FH u;Nu=FH v:\nIntegrating the last equalities, we obtain\n(3.2)L=L(u;v 0) +Zv\nv0F(u;s)Hu(u;s)ds;N=N(u0;v) +Zu\nu0F(s;v)Hv(s;v)ds:\nNow, we will try to choose the isotropic coordinates in such a way that L(u;v 0)and\nN(u0;v)to have the simplest form.CANONICAL COORDINATES AND NATURAL EQUATION FOR LORENTZ SURFACES IN R3\n1 5\nFirst, let’s find the transformation formulas for the coefficients of the first and the second\nfundamental form under changes of the isotropic coordinates. Consider the following change\nof the isotropic coordinates (it preserves the numeration):\nu=u(~u);v=v(~v):\nThen, we have ~F=Fu0v0, which implies u0v0>0, since we have assumed at the beginning\nthat ~F > 0andF > 0. In this case, the orientation of the surface does not change, i.e. ~l=l.\nThen, for the coefficients ~F,~L,~M,~Nwe have\n(3.3) ~F=Fu0v0; ~L=Lu02; ~M=Mu0v0; ~N=Nv02:\nIn the case of changing the numeration, it is sufficient to consider only the change of\ncoordinates:\nu= ~v;v= ~u;\nsince the general case is reduced to this and the previous one. In this case, the orientation\nof the surface changes, i.e. ~l=\u0000land we have\n(3.4) ~F=F; ~L=\u0000N; ~M=\u0000M; ~N=\u0000L:\nThe transformation formulas (3.3) and (3.4) show that, if L= 0orN= 0for some\nisotropic coordinates, then ~L= 0or~N= 0for any isotropic coordinates. Further, we\nconsider surfaces satisfying L6= 0andN6= 0at least locally. It follows from (2.1) that\n(3.5) H2\u0000K=LN\nF2;\nwhich implies that the conditions L6= 0andN6= 0are equivalent to H2\u0000K6= 0.\nWe give the following definition.\nDefinition 3.1. A Lorentz surface MinR3\n1is said to be of general type , ifH2\u0000K6= 0.\nThe Lorentz surfaces of general type are naturally divided into two subclasses.\nDefinition 3.2. A Lorentz surface of general type in R3\n1is said to be of first kind (resp.\nsecond kind ), ifH2\u0000K > 0(respH2\u0000K < 0).\nRemark 3.1.SinceKandHare invariants ofM, the property of a surface to be of general\ntype, as well as the kind of the surface, are geometric – they do not depend on the local\nparametrization and are invariant under motions in R3\n1. It is known that the discriminant\nof the characteristic polynomial of the Weingarten map has the form D= 4(H2\u0000K)[12].\nHence, the surfaces of general type are those surfaces for which the Weingarten map has two\ndifferent eigenvalues. Moreover, the surfaces of first kind are those with real eigenvalues, the\nsurfaces of second kind are those with complex eigenvalues.\nNow, we will introduce special isotropic coordinates by the following\nDefinition 3.3. LetM= (D;x)be a Lorentz surface of general type in R3\n1parametrized by\nisotropic coordinates (u;v), such that F > 0, and x0= x(u0;v0)be a point ofM. We call\n(u;v)canonical coordinates with initial point x0, if the coefficients of the second fundamental\nform satisfy the conditions\n(3.6) L(u;v 0) =\"1;N(u0;v) =\"2;\nwhere\"1=\u00061and\"2=\u00061.6 KRASIMIR KANCHEV, OGNIAN KASSABOV, AND VELICHKA MILOUSHEVA\nRemark 3.2.In the general case, condition (3.6) for the coordinates to be canonical depends\non the choice of the initial point (u0;v0). If(u1;v1)is another point in the same neighbour-\nhood, from (3.2) it is obvious that L(u;v 0)6=L(u;v 1)andN(u0;v)6=N(u1;v), in general.\nIn the special case of a surface with constant mean curvature H,Lis a function of uand\nNis a function of v, because of (3.1). Hence, for the class of surfaces with constant mean\ncurvature the canonicity of the coordinates does not depend on the choice of the initial point\n(u0;v0).\nTheorem 3.1. IfM= (D;x)is a Lorentz surface of general type in R3\n1andx0is a fixed\npoint, then we can introduce canonical coordinates with initial point x0.\nProof.Letx0be a fixed point of Mand(u;v)be isotropic coordinates in a neighbourhood\nofx0such thatF > 0andx0= x(u0;v0). Consider the change of the coordinates u=u(~u)\nandv=v(~v). According to (3.3) and Definition 3.3, the new coordinates (~u;~v)are canonical\nif and only if\n(3.7) L(u;v 0)u02=~L(~u;~v0) =\u00061;N(u0;v)v02=~N(~u0;~v) =\u00061:\nThe signs in the right-hand sides of (3.7) are chosen to coincide with the signs of Land\nN, respectively. Thus we obtain ordinary differential equations for u(~u)andv(~v)whose\nsolutions have the following form:\n(3.8) ~u= ~u0+Zu\nu0p\njL(s;v 0)jds; ~v= ~v0+Zv\nv0p\njN(u0;s)jds;\nwhere (~u0;~v0)is arbitrary chosen. It follows from (3.5) that L6= 0andN6= 0, which\nimply ~u0>0and ~v0>0. Hence, equalities (3.8) define new isotropic coordinates (~u;~v)\nsatisfying the condition ~F > 0. Since (3.8) is equivalent to (3.7), then ~L(~u;~v0) =\u00061and\n~N(~u0;~v) =\u00061. So, (~u;~v)are canonical coordinates of Mwith initial point x0. \u0003\nRemark 3.3.IfMis a surface of first kind according to Definition 3.2, then (3.5) implies\nLN > 0. It follows from (3.4) that a change in the coordinates numeration leads to a change\nin the signs of LandN. Hence, surfaces of first kind admit both canonical coordinates for\nwhichL=N= 1and canonical coordinates for which L=N=\u00001.\nIfMis of second kind, then LN < 0, and hence, (3.3) and (3.4) show that the signs of\nLandNdo not change under changes of the isotropic coordinates. Therefore, the surfaces\nof second kind can be divided into two subclasses: surfaces with canonical coordinates such\nthatL= 1andN=\u00001, and surfaces with canonical coordinates such that L=\u00001and\nN= 1.\nNow, we will discuss the question of uniqueness of the canonical coordinates.\nTheorem 3.2. LetMbe a Lorentz surface of general type in R3\n1and(u;v)be canonical\ncoordinates with initial point x0=x(u0;v0)ofM. Then (~u;~v)is another pair of canonical\ncoordinates with the same initial point x0if and only if\n(3.9)u=\u000e~u+c1;\nv=\u000e~v+c2;oru=\u000e~v+c1;\nv=\u000e~u+c2;\nwhere\u000e=\u00061,c1andc2are constants.\nProof.First, we consider the case u=u(~u)andv=v(~v). Equalities (3.3) imply that (~u;~v)\nare canonical coordinates if and only if u02= 1,v02= 1, andu0v0>0. The last conditions\nare equivalent to the first pair of equalities in (3.9)\nThe caseu=u(~v)andv=v(~u)reduces to the previous one by means of (3.4). \u0003CANONICAL COORDINATES AND NATURAL EQUATION FOR LORENTZ SURFACES IN R3\n1 7\nThe meaning of the last theorem is that the canonical coordinates are uniquely determined\nup to a numeration, a sign and an additive constant.\nAt the end of this section we will characterize the canonical coordinates in terms of the\nnull curves lying on the considered surfaces. Recall that, if \u000bis a null curve ( \u000b02= 0) in\nR3\n1, then\u000b002\u00150. The null curves satisfying \u000b002>0are called non-degenerate . It is known\nthat these curves admit a parametrization such that \u000b002= 1[12]. Such parameter is known\nin the literature as a natural parameter orpseudo arc-length parameter , since it plays a role\nsimilar to the role of the arc-length parameter for non-null curves [6].\nTheorem 3.3. LetM= (D;x)be a Lorentz surface in R3\n1parametrized by isotropic coor-\ndinates (u;v), such that F > 0, and x0= x(u0;v0)be a point ofM. Then,Mis of general\ntype if and only if the null curves lying on Mare non-degenerate. The coordinates (u;v)of\nMare canonical with initial point x0if and only if they are natural parameters of the null\ncurves onMpassing through x0.\nProof.The Frenet formulas (2.2) imply x2\nuu= X2\nu=L2andx2\nvv= Y2\nv=N2. Hence, the\nu-lines andv-lines are non-degenerate if and only if L6= 0andN6= 0, which is equivalent\ntoMbeing of general type. Moreover, uis a natural parameter ( x2\nuu= 1) of the null curve\nx(u;v 0)passing through x0if and only if L2(u;v 0) = 1. Analogously, vis a natural parameter\n(x2\nvv= 1) of the null curve x(u0;v)passing through x0if and only if N2(u0;v) = 1. The\nlast conditions are equivalent to L(u;v 0) =\u00061andN(u0;v) =\u00061, which means that the\ncoordinates (u;v)ofMare canonical with initial point x0. \u0003\n4.Natural equation of Lorentz surfaces of general type in R3\n1\nIn this section, we will consider the Gauss and Codazzi equations of a Lorentz surface of\ngeneral typeM= (D;x)inR3\n1parametrized by canonical isotropic coordinates (u;v)with\ninitial point x0= x(u0;v0)2M. In such case, the coefficients of the second fundamental\nform are expressed by the coefficient Fof the first fundamental form, the mean curvature\nH, and the constants \"1and\"2(see Definition 3.3). It follows from (2.1), (3.2), and (3.6)\nthat:\n(4.1)L=\"1+Rv\nv0F(u;s)Hu(u;s)ds;M=FH;N=\"2+Ru\nu0F(s;v)Hv(s;v)ds;\nwhere\"1=\u00061;\"2=\u00061, the signs depending on the kind of the surface. Substituting these\nexpressions in the Gauss equation (2.3), we obtain:\n(4.2)FF uv\u0000FuFv\nF=\u0010\n\"1+Rv\nv0F(u;s)Hu(u;s)ds\u0011\u0010\n\"2+Ru\nu0F(s;v)Hv(s;v)ds\u0011\n\u0000F2H2:\nConsequently, FandHgive a solution to the integro-differential equation (4.2), which we\ncall the natural equation of the Lorentz surfaces of general type in R3\n1. The converse is\nalso true. Namely, the following Bonnet-type theorem holds:\nTheorem 4.1. LetM= (D;x)be a Lorentz surface of general type in R3\n1and(u;v)be\ncanonical isotropic coordinates with initial point x0= x(u0;v0)2M. Then, the coefficient\nFof the first fundamental form and the mean curvature HofMgive a solution to the natural\nequation (4.2). If ^Mis obtained fromMby a proper motion in R3\n1, then ^Mgenerates the\nsame solution to (4.2).\nConversely, let F > 0andHbe functions of (u;v)defined in a neighbourhood of (u0;v0)\nand satisfying the natural equation (4.2), where\"1=\u00061and\"2=\u00061. Then, at least locally,\nthere exists a unique (up to a proper motion in R3\n1) Lorentz surface of general type in R3\n1\ndefined byx=x(u;v)in canonical isotropic coordinates with initial point x0=x(u0;v0),8 KRASIMIR KANCHEV, OGNIAN KASSABOV, AND VELICHKA MILOUSHEVA\nsuch that the given functions FandHare the non-zero coefficient of the first fundamental\nform and the mean curvature, respectively, and the signs of the corresponding coefficients L\nandNof the second fundamental form coincide with the signs of \"1and\"2.\nProof.We have already seen that the coefficient Fand the mean curvature Hof a surface\nwith the given properties satisfy the natural equation. Now we will prove the converse.\nGiven the functions FandHsatisfying (4.2), we define functions L,M, andNby equal-\nities (4.1). Then, (4.2) implies that the quadruple F,L,M,Nsatisfies the Gauss equation\n(2.3). Differentiating (4.1) we get that the Codazzi equations (2.4) are also fulfilled. Apply-\ning Theorem 2.1 we get a Lorentz surface Mparametrized by isotopic coordinates whose\ncoefficient of the first fundamental form is the given function Fand the coefficients of the\nsecond fundamental form are the functions L,M,N, defined by (4.1). Comparing (3.2) with\n(4.1) we obtain L(u;v 0) =\"1,N(u0;v) =\"2, which means that Mis of general type and\n(u;v)are canonical isotropic coordinates. Comparing (2.1) with (4.1) we see that the mean\ncurvature ofMis the given function H. Hence, the surface Mhas the necessary properties.\nMoreover, if ^Mis another surface with the same properties, then (4.1) is also valid for ^M.\nSo,Mand ^Mhave one and the same coefficients of the first and second fundamental form.\nHence, according to Theorem 2.1, ^Mis obtained fromMby a proper motion in R3\n1.\u0003\nNow, we will consider the natural equation (4.2) in the case of a surface with constant\nmean curvature H. In this case, equation (4.2) takes the form:\n(4.3)FF uv\u0000FuFv\nF=\"1\"2\u0000F2H2;\nand (4.1) implies L=\"1,N=\"2. Then, it follows from (3.5) that\n(4.4) H2\u0000K=\"1\"2\nF2;F=1p\njH2\u0000Kj:\nIf we rewrite (4.3) in the form\n(4.5)1\nF(lnF)uv=\"1\"2\nF2\u0000H2\nthen by use of (4.4) we obtain\n(4.6)p\njH2\u0000Kj\u0000\nlnp\njH2\u0000Kj\u0001\nuv=K;H2\u0000K6= 0:\nWe call (4.6) the natural equation of constant mean curvature Lorenz surfaces in R3\n1.\nSo, we can formulate the following Bonnet-type theorem for Lorentz surfaces of constant\nnon-zero mean curvature.\nTheorem 4.2. LetM= (D;x)be a Lorentz surface of general type in R3\n1with constant non-\nzero mean curvature Hparametrized by canonical isotropic coordinates. Then, the Gauss\ncurvatureKsatisfies the natural equation (4.6). If ^Mis obtained from Mby a proper\nmotion in R3\n1, then ^Mgenerates the same solution to (4.6).\nConversely, let Hbe a non-zero constant and Kbe a function of (u;v)satisfying the natural\nequation (4.6). Then, at least locally, there exist (up to a proper motion in R3\n1) exactly two\nLorentz surfaces of general type in R3\n1parametrized by canonical isotropic coordinates, with\nthe constant Has non-zero mean curvature and the function Kas Gauss curvature.\nProof.We have already seen that the mean curvature Hand the Gauss curvature Kof a\nsurface with the given properties satisfy equation (4.6). Now we will prove the converse.\nGiven the constant Hand the function Ksatisfying (4.6), we define a function Fand\nconstants\"1=\u00061,\"2=\u00061such that equalities (4.4) hold true. Then equality (4.6) impliesCANONICAL COORDINATES AND NATURAL EQUATION FOR LORENTZ SURFACES IN R3\n1 9\n(4.5), which is equivalent to (4.3), the latter being the natural equation (4.2) in the case H\nis constant. Note that the function Fis determined uniquely by (4.4), while for the choice\nof\"1and\"2we have two different options depending on the choice of signs. This means that\naccording to Theorem 4.1 we obtain two different Lorentz surfaces M1andM2parametrized\nby canonical isotropic coordinates, whose mean curvature is the given constant Hand the\ncoefficient of the first fundamental form is the given function F. Equalities (4.4) hold true\nfor bothM1andM2, and hence, Kis determined uniquely by FandH. Consequently, the\nGauss curvature of both M1andM2is the given function K.\nThe surfaceM2cannot be obtained from M1by a proper motion in R3\n1, since the proper\nmotions preserve the signs of \"1and\"2, and the signs of these constants are different for M1\nandM2. If ^Misanothersurfacewiththesameproperties, thenequalities(4.4)holdtruealso\nfor^M. Hence,M1,M2, and ^Mhave one and the same coefficients of the first fundamental\nform and equal mean curvatures. Moreover, the constants \"1and\"2for ^Mcoincide with\nthe constants for one of the two surfaces M1orM2. So, according to Theorem 4.1, ^Mcan\nbe obtained from one of the two surfaces M1orM2by a proper motion in R3\n1. \u0003\nRemark 4.1.The two surfaces obtained in the last theorem are really different. We have\nalready seen that M2cannot be obtained from M1by a proper motion in R3\n1. Furthermore,\nM2cannot be obtained from M1by coordinate change of the form u=u(~u)andv=v(~v),\nsince such a change preserves the signs of \"1and\"2, according to (3.3). M2cannot be\nobtained fromM1also by coordinate change of the form u=u(~v)andv=v(~u), since in\nsuch case the sign of Hchanges, according to (2.1) and (3.4), but the surfaces M1andM2\nhave equal mean curvatures. Such a pair of surfaces is presented in Examples 5.4 and 5.5.\nNow we will consider the case of a surface Mwith zero mean curvature H, i.e.Mis a\nminimal surface in R3\n1. In this case, equality (4.6) takes the form:\n(4.7)p\njKj\u0000\nlnp\njKj\u0001\nuv=K;K6= 0:\nLet us point out that the Gauss curvature Kand the canonical coordinates are invariant\nunder non-proper motions in R3\n1. Hence, in this case the surface is determined uniquely by\nthe solution of (4.7) up to an arbitrary motion. We have the following Bonnet-type theorem\nfor minimal Lorentz surfaces in R3\n1.\nTheorem4.3. LetM= (D;x)be a minimal Lorentz surface of general type in R3\n1parametrized\nby canonical isotropic coordinates. Then, the Gauss curvature KofMsatisfies the natural\nequation (4.7). If ^Mis obtained fromMby a motion in R3\n1, then ^Mgenerates the same\nsolution to (4.6).\nConversely, given a function Kof(u;v)satisfying the natural equation (4.7), there exists,\nat least locally, a unique (up to a motion in R3\n1) minimal Lorentz surface of general type\nparametrized by canonical isotropic coordinates, such that its Gauss curvature is the given\nfunctionK.\nProof.TheproofissimilartotheproofofTheorem4.2, thereforewewillnotgiveitindetails,\nwe will only point out the difference. Again, given the function K, we obtain two different\nminimal surfacesM1andM2parametrized by canonical isotropic coordinates, having Kas\nthe Gauss curvature and having different signs of \"1and\"2. Let ~M1be a surface obtained\nfromM1by a non-proper motion in R3\n1. Then, the Gauss curvature Kand the signs of \"1\nand\"2of~M1are the same as those of M2. So, equalities (4.4) imply that ~M1andM2have\none and the same coefficient Fof the first fundamental form. Hence, according to Theorem\n4.1,M2is obtained from ~M1by a proper motion in R3\n1. Consequently,M2is obtained from\nM1by a non-proper motion in R3\n1. \u000310 KRASIMIR KANCHEV, OGNIAN KASSABOV, AND VELICHKA MILOUSHEVA\n5.Examples\nIn this section, we will consider examples of Lorentz surfaces of general type in R3\n1, illus-\ntrating the developed theory. First, we give an example of a minimal Lorentz surface.\nExample 5.1.Let us consider the surface in R3\n1, determined by the following parametrization:\n(5.1) x =1\n6(u3\u0000v3+ 3u\u00003v;\u0000u3+v3+ 3u\u00003v;3u2\u00003v2):\nThe coefficients of the first and the second fundamental form are:\nE=G= 0;F=1\n2(u\u0000v)2;L= 1;M= 0;N= 1:\nThe Gauss curvature and the mean curvature of the surface defined above are expressed as\nfollows:\n(5.2) K=\u00004\n(u\u0000v)4;H= 0:\nHence,thesurfacedefinedby(5.1)isaminimalLorentzsurface(ofEnneper-type)parametrized\nby isotropic coordinates. Moreover, the coordinates (u;v)are canonical, since L=N= 1.\nThe Gauss curvature Kis negative, so the surface is of first kind according to Definition 3.2.\nThe function Kgiven in (5.2) is a solution to the natural equation (4.7).\nExample 5.2.Let us consider the surface in R3\n1, defined by\n(5.3) x =1\n6(u3\u0000v3+ 3u\u00003v;\u0000u3+v3+ 3u\u00003v;3u2+ 3v2):\nThe coefficients of the first and the second fundamental form are:\nE=G= 0;F=1\n2(u+v)2;L= 1;M= 0;N=\u00001:\nThe Gauss curvature and the mean curvature are expressed as follows:\n(5.4) K=4\n(u+v)4;H= 0:\nAs in the previous example, (5.3) defines a minimal Lorentz surface of Enneper-type\nparametrized by canonical isotropic coordinates. In this example, the Gauss curvature Kis\npositive and hence, the surface is of second kind according to Definition 3.2. The function\nKgiven in (5.4) is also a solution to the natural equation (4.7).\nNow, we will give examples of surfaces with non-zero constant mean curvature.\nExample 5.3.We consider the Lorentz sphere in R3\n1parametrized by isothermal coordinates\n(t;s)as follows:\nx = (sinhtsechs;coshtsechs;tanhs):\nChanging the coordinates with isotropic ones, we obtain:\n(5.5) x = (sinh(u\u0000v) sech(u+v);cosh(u\u0000v) sech(u+v);tanh(u+v)):\nThe coefficients of the first and the second fundamental form are:\nE=G= 0;F= 2 sech2(u+v);L= 0;M= 2 sech2(u+v);N= 0:\nThe Gauss curvature and the mean curvature are given by:\nK= 1;H= 1:\nHence, the surface defined by (5.5) is a Lorentz surface with non-zero constant mean\ncurvature parametrized by isotropic coordinates. In this example, H2\u0000K= 0which meansCANONICAL COORDINATES AND NATURAL EQUATION FOR LORENTZ SURFACES IN R3\n1 11\nthat the surface is not of general type within the meaning of Definition 3.1. In this case, we\ncannot introduce canonical coordinates in the sense of Definition 3.3.\nExample 5.4.Let us consider the cylinder in R3\n1, parametrized by isothermal coordinates\n(t;s)as follows:\nx = (t;coss;sins):\nChanging the coordinates with isotropic ones, we obtain:\nx = (u\u0000v;cos(u+v);sin(u+v)):\nThe coefficients of the first and the second fundamental form are:\nE=G= 0;F= 2;L= 1;M= 1;N= 1:\nThe Gauss curvature and the mean curvature are given by:\nK= 0;H=1\n2:\nThisisanexampleofaLorentzsurfacewithnon-zeroconstantmeancurvatureparametrized\nby canonical isotropic coordinates. It corresponds to the trivial (zero) solution to equation\n(4.6).\nExample 5.5.Now, let us consider the hyperbolic Lorentz cylinder in R3\n1, parametrized by\nisothermal coordinates (t;s)as follows:\nx = (sinhs;coshs; t):\nChanging the coordinates with isotropic ones, we obtain:\nx = (sinh(u\u0000v);cosh(u\u0000v); u+v):\nThe coefficients of the first and the second fundamental form are:\nE=G= 0;F= 2;L=\u00001;M= 1;N=\u00001:\nThe Gauss curvature and the mean curvature are given by:\nK= 0;H=1\n2:\nThis is also an example of a Lorentz surface with non-zero constant mean curvature\nparametrized by canonical isotropic coordinates. It also corresponds to the trivial (zero)\nsolution to equation (4.6).\nComparing the results of the last two examples, we see that the two cylinders have equal\nconstant mean curvatures and equal Gauss curvatures. Hence, they give one and the same\nsolution to the natural equation (4.6). But there is a difference in the signs of \"1=Land\n\"2=N. These two cylinders form a pair of surfaces M1andM2as the ones described in\nthe proof of Theorem 4.2.\nFinally, we will consider a surface with non-constant mean curvature.\nExample 5.6.Let us consider the hyperbolic Lorentz cone MinR3\n1, parametrized by isother-\nmal coordinates (t;s)as follows:\nx =\u0000\net\n2sinhs;p\n3 et\n2;et\n2coshs\u0001\n:\nChanging the coordinates with isotropic ones, we obtain:\nx =\u0000\neu+v\n2sinh(u\u0000v);p\n3 eu+v\n2;eu+v\n2cosh(u\u0000v)\u0001\n:12 KRASIMIR KANCHEV, OGNIAN KASSABOV, AND VELICHKA MILOUSHEVA\nThe coefficients of the first and the second fundamental form are:\nE=G= 0;F= 2eu+v;L=p\n3\n2eu+v\n2;M=\u0000p\n3\n2eu+v\n2;N=p\n3\n2eu+v\n2:\nThe Gauss curvature and the mean curvature are given by:\nK= 0;H=\u0000p\n3\n4e\u0000u+v\n2:\nHence,inthisexample, MisaLorentzsurfacewithnon-constantmeancurvatureparametrized\nby isotropic coordinates. The coordinates (u;v)are not canonical, since L(u;v 0)6=\u00061and\nN(u0;v)6=\u00061.\nWe will introduce canonical isotropic coordinates (~u;~v)with initial point (u0;v0) = (0;0).\nUsing formulas (3.8) we get:\n~u= ~u0+ 2p\n24p\n3(eu\n4\u00001); ~v= ~v0+ 2p\n24p\n3(ev\n4\u00001):\nTo simplify the formulas we choose ~u0= ~v0= 2p\n24p\n3. Then,\n~u= 2p\n24p\n3 eu\n4; ~v= 2p\n24p\n3 ev\n4;u= 4 ln~u\n2p\n24p\n3;v= 4 ln~v\n2p\n24p\n3:\nUsing the last equalities and formulas (3.3), we express the coefficient of the first funda-\nmental form ~Fand the mean curvature ~Hin terms of the canonical coordinates (~u;~v)as\nfollows:\n(5.6) ~F=~u3~v3\n1152; ~H=\u000048p\n3\n~u2~v2:\nAccording to Theorem 4.1, the functions ~Fand ~Hgiven by (5.6) give a solution to the\nnatural equation (4.2) in the case \"1=\"2= 1.\nAcknowledgments: The third author is partially supported by the National Science\nFund, Ministry of Education and Science of Bulgaria under contract DN 12/2.\nReferences\n[1] K. Akutagawa and S. Nishikawa. The Gauss map and spacelike surfaces with prescribed\nmean curvature in Minkowski 3-space. Tohoku Math. J. (2) , 42(1):67–82, 1990. doi:\n10.2748/tmj/1178227694.\n[2] J. A. Aledo, J. M. Espinar, and J. A. G´ alvez. Timelike surfaces in the\nLorentz–Minkowski space with prescribed Gaussian curvature and Gauss map. Journal\nof Geometry and Physics , 56(8):1357–1369, 2006. doi: 10.1016/j.geomphys.2005.07.004.\n[3] Y. Aleksieva and V. Milousheva. Minimal Lorentz surfaces in pseudo-Euclidean 4-space\nwith neutral metric. Journal of Geometry and Physics , 142:240–253, 2019. doi:\n10.1016/j.geomphys.2019.04.008.\n[4] H. Anciaux. Minimal Submanifolds in Pseudo-Riemannian Geometry . World Scientific\nPublishing Co. Pte. Ltd., 2011. ISBN 978-981-4291-24-8.\n[5] M. P. do Carmo. Differential Geometry of Curves and Surfaces . Dover Publications,\nInc. Mineola, New York, 2 edition, 2016. ISBN 978-0-486-80699-0.\n[6] K. Duggal and D. H. Jin. Null curves and hypersurfaces of semi-Riemannian manifolds .\nWorld Scientific, 2007. ISBN 978-981-3106-97-0.\n[7] J. A. G´ alvez, A. Mart´ ınez, and A. Mil´ an. Complete constant Gaussian curvature sur-\nfaces in the Minkowski space and harmonic diffeomorphisms onto the hyperbolic plane.\nTohoku Math. J. (2) , 55(4):467–476, 2003. doi: 10.2748/tmj/1113247124.CANONICAL COORDINATES AND NATURAL EQUATION FOR LORENTZ SURFACES IN R3\n1 13\n[8] G. Ganchev. Canonical Weierstrass representation of minimal and maximal surfaces\nin the three-dimensional Minkowski space, 2008. URL https://arxiv.org/abs/0802.\n2632.\n[9] G. Ganchev and K. Kanchev. Canonical coordinates and natural equations for mini-\nmal time-like surfaces in R4\n2.Kodai Mathematical Journal , 43(3):524–572, 2020. doi:\n10.2996/kmj/1605063628.\n[10] G. Ganchev and V. Milousheva. Timelike surfaces with zero mean curvature in\nMinkowski 4-space. Isr. J. Math. , 196:413–433, 2013. doi: 10.1007/s11856-012-0169-y.\n[11] J. Inoguchi. Timelike surfaces of constant mean curvature in Minkowski 3-space. Tokyo\nJ. Math., 21(1):141–152, 1998. doi: 10.3836/tjm/1270041992.\n[12] R. L´ opez. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int.\nElectron. J. Geom. , 7(1):44–107, 2014. ISSN 1307-5624. doi: 10.36890/iejg.594497.\n[13] M. A. Magid. Timelike surfaces in Lorentz 3-space with prescribed mean curvature and\nGauss map. Hokkaido Math. J. , 20(3):447–464, 1991. doi: 10.14492/hokmj/1381413979.\n[14] V. Mihova and G. Ganchev. Partial differential equations of time-like Wein-\ngarten surfaces in the three-dimensional Minkowski space. Ann. Sofia Univ., Fac-\nulty Math. and Inf. , 101:143–165, 2013. URL https://www.fmi.uni-sofia.bg/en/\nannuaire-by-tome/101 .\nDepartment of Mathematics and Informatics, Todor Kableshkov University of Trans-\nport, 158 Geo Milev Str., 1574, Sofia, Bulgaria\nEmail address :kbkanchev@yahoo.com\nInstitute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.\nBonchev Str. bl. 8, 1113, Sofia, Bulgaria\nEmail address :okassabov@math.bas.bg\nInstitute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.\nBonchev Str. bl. 8, 1113, Sofia, Bulgaria\nEmail address :vmil@math.bas.bg"    },    {        "title": "1303.2057v1.Polariton_excitation_in_epsilon_near_zero_slabs__transient_trapping_of_slow_light.pdf",        "content": "arXiv:1303.2057v1  [physics.optics]  8 Mar 2013Polariton excitation in epsilon-near-zero slabs: transie nt trapping of slow light\nAlessandro Ciattoni1, Andrea Marini2, Carlo Rizza1,3, Michael Scalora4and Fabio Biancalana2,5\n1Consiglio Nazionale delle Ricerche, CNR-SPIN 67100 L’Aqui la,\nItaly and Dipartimento di Fisica, Universit` a dell’Aquila , 67100 L’Aquila, Italy\n2Max Planck Institute for the Science of Light, Guenther-Sch arowsky-Straße 1, 91058 Erlangen, Germany\n3Dipartimento di Fisica e Matematica, Universit` a dell’Ins ubria, Via Valleggio 11, 22100 Como, Italy\n4Charles M. Bowden Research Center RDMR-WDS-WO,\nRDECOM, Redstone Arsenal, Alabama 35898-5000, USA and\n5School of Engineering & Physical Sciences, Heriot-Watt Uni versity, Edinburgh, EH14 4AS, United Kingdom\n(Dated: October 1, 2018)\nWe numerically investigate the propagation of a spatially l ocalized and quasi-monochromatic\nelectromagnetic pulsethroughaslab withLorentzdielectr ic response intheepsilon-near-zeroregime,\nwheretherealpartofthepermittivityvanishesatthepulse carrier frequency. Weshowthatthepulse\nis able to excite a set of virtual polariton modes supported b y the slab, the excitation undergoing\na generally slow damping due to absorption and radiation lea kage. Our numerical and analytical\napproaches indicate that in its transient dynamics the elec tromagnetic field displays the very same\nenhancement of the field component perpendicular to the slab , as in the monochromatic regime.\nThe transient trapping is inherently accompanied by a signi ficantly reduced group velocity ensuing\nfrom the small dielectric permittivity, thus providing a no vel platform for achieving control and\nmanipulation of slow light.\nI. INTRODUCTION\nPhysical mechanisms driving to slow and fast light\nhave attracted considerable attention from the scien-\ntific community in the last decades [1–3]. The inher-\nent interest in slow light comes from the long matter-\nradiation interaction time, which can lead to consider-\nable enhancement of all nonlinear processes that in turn\nmay be exploited for active functionalities [4], e.g. all-\noptical switching and modulation [5, 6]. The nonlin-\nearity may also be enhanced by reducing the effective\narea in subwavelength silicon on insulator and plasmonic\nwaveguides [7, 8], where tight confinement opens up pos-\nsibilities for miniaturized nonlinear applications [9–11].\nAlternatively, extreme nonlinear dynamics [12, 13], en-\nhanced second and third harmonic generation [14, 15]\nis predicted in epsilon-near-zero (ENZ) metamaterials,\nwhere the linear susceptibility is tailored in such a way\nthat its modulus becomes comparable with the nonlinear\ncounterpart. BoostingthenonlinearityofENZplasmonic\nchannels can also lead to active control of tunneling [16],\nswitching and bistable response [17]. ENZ metamateri-\nals have also been used for directive emission [18, 19],\ncloaking [20], energy squeezing in narrow channels [21]\nand subwavelength imaging [22, 23].\nInalloftheabovementionedmechanisms,thetrade-off\nneeded to achieve enhanced active functionalities is paid\nin terms of increased losses. As a result, the ENZ regime\none usually invokes refers to the case where the real part\nof the susceptibility becomes very small, while its imag-\ninary part remains finite. Indeed, due to the stringent\nphysical requirement of causality, Kramers-Kronig rela-\ntions impose that dispersion be inherently accompanied\nby loss and the dielectric susceptibility can not become\nrigorously null. The residual loss either limits or even\npreventsgiant enhancement ofcoherent mechanisms, e.g.in second and third harmonic generation setups [14, 15].\nRecently, in the context of surface plasmon polaritons,\na method has been proposed to overcome the loss bar-\nrier for superlensing applications by loading the effect\nof loss into the time domain [24]. In our analytical cal-\nculations, we will use a similar approach to study the\nbehavior of an electromagnetic pulse that scatters from\na slab having a Lorentz dielectric response. Indeed, by\nconsidering non-monochromatic virtual modes with com-\nplex frequency [25], it is possible to drop off the effect of\nloss on the temporal dependence of the “mode” itself. In\nthis complex frequency approach, it is possible to achieve\nthe condition where the dielectric susceptibility exactly\nvanishes. Our formalism treats the dielectric polariza-\ntion of the medium as a generic Lorentz oscillator that,\nin the epsilon-equal-to-zero condition, encompasses lon-\ngitudinal collective oscillations of both electrons (volume\nplasmons) and ions (longitudinal phonons) that can not\nbe excited by light [26]. Recently, the question whether\nor not volume plasmons can be excited ornot by classical\nlight has been revived [27–31]. Some studies on Mie ex-\ntinctionefficienciesrevealamaximumaroundthecharac-\nteristic frequency where the dielectric susceptibility van-\nishes, attributing the enhanced extinction to the excita-\ntion of volume plasmons [28, 30]. Conversely, other sim-\nilar studies identify the physical origin of the enhanced\nextinction in the excitation of leaky modes [27, 29]. The\nlatter interpretation is also supported by studies of the\nexcitation ofsurface phonon polaritons in ENZ slabs [32–\n35].\nIn this manuscript we numerically investigate and ana-\nlyticallyinterpretthe scatteringofaspatiallyandtempo-\nrally localized optical pulse from a dielectric slab in the\nENZ regime. We used a finite difference time domain\n(FDTD) algorithm to solve the full vectorial Maxwell\nequations coupled to the Lorentz oscillator equation for2\nthe dielectric polarizationofthe slab. We find that, if the\ncarrier frequency of the optical pulse matches the ENZ\ncondition, electromagnetic quasi-trapping occurs within\nthe Lorentz slab since, after the pulse has passed through\nit, an elecromagnetic-polarization (polariton) oscillation\npersists and generally slowly damps out. We demon-\nstrate that non-trivial ENZ features like the enhance-\nment of the longitudinal electric field component are still\nobservable in the time-domain. We also find that the\nabove mentioned phenomenology is not observed for op-\ntical pulses with carrier frequencies far from the ENZ\ncondition. Thus, in order to grasp the underpinning\nphysical mechanisms responsible for transient trapping\nin the ENZ regime, we analytically investigate the scat-\nteringfeaturesofthe Lorentzslabbystudying the virtual\nleaky modes of the structure. We recognize that a set of\npolariton modes with reduced transverse group velocity\n(vg≃c/100, where cis the speed of light in vacuum)\nis excited. Indeed, the plasma frequency plays the role\nof a cut-off frequency and polaritons in the ENZ regime\nare intrinsically characterized by a reduced group veloc-\nity. Thus, we are able to interpret the transient trapping\nby means of the excitation of slow polariton modes that\nare damped off due to medium absorption and radiation\nleakage in the outer medium.\nThe paper is organized as follows. In Sec.II we re-\nport the results of numerical finite difference time do-\nmain (FDTD) simulations, comparing the distinct phe-\nnomenologies occuring in ENZ and standard dielectric\nregimes. In Sec.III we analytical investigate the virtual\nleaky modes of the structure, we address their properties\nand we discuss their role in the interpretation of numeri-\ncal results are developed in section III. In Sec.IVwe draw\nour conclusions.\nII. FINITE DIFFERENCE TIME DOMAIN\nANALYSIS OF THE TIME-DOMAIN ENZ\nREGIME\nA. Pulse scattering by a Lorentz slab\nLet us consider the scattering interaction sketched in\nFig.1(a), where an electromagnetic pulse is launched\nalong the z-axis in vacuum and orthogonally impinges\non the surface of a dielectric slab. The pulse is a Trans-\nverse Magnetic (TM) excitation, with electric Ex(x,z,t),\nEz(x,z,t) and magnetic Hy(x,z,t) field components.\nThe initial profile profile of the transverse electric com-\nponent is\nEx(x,z,t= 0) =E0e−x2\nw2xe−(z−z0)2\nw2zsin/bracketleftBig¯ω\nc(z−z0)/bracketrightBig\n.(1)\nThis field is spatially confined both along the x- and\nz- axis,wxandwzbeing its transverse and longitu-\ndinal widths, respectively, it is centered at the point\n(x,z) = (0,z0) and it is longitudinally modulated with\nperiod¯λ= 2πc/¯ω. Hereafter we will focus on very\nFIG. 1: (a) Interaction geometry of the pulse colliding onto\nthe dielectric slab. (b) Real and imaginary part of the slab\ndielectric permittivity for the Lorentz parameters used in the\nFDTD analysis as a function of the wavelength.\nlong pulses such that wz≫c/(¯ω). Thus, the quasi-\nmonochromatic condition δω/¯ω≪1 is satisfied, where\n¯ωis the pulse central frequency and δω≃c/wzis the\nspectral width.\nThe dielectric slab has width L, it is centered at\n(x,z) = (0,0) and we assume that, in the presence of\nthe external electric field E, the dynamics of its dielectric\npolarization is governed by the Lorentz oscillator model\nd2P\ndt2+γdP\ndt+ω2\neP=ǫ0feE, (2)\nwhereωeis the resonant angular frequency, γis the\ndamping constant, feis the oscillator strength and ǫ0\nis the dielectric permittivity of vacuum. It is well\nknown that Eq.(2) leads, in the frequency domain, to\nthe constitutive relation ˜Dω=ǫ0ǫ(ω)˜Eωwhere˜fω=/integraltext+∞\n−∞dteiωtf(t) is the Fourier transform of f(t),D=\nǫ0E+Pis the displacement field vector and ǫ= 1 +\nfe/(ω2\ne−iγω−ω2) is the frequency dependent medium\ndielectric permittivity. The realistic model of Eq.(2) is\nparticularlyaccuratefordescribingthemediumdielectric\nresponse to fields with frequencies close to the resonant\nfrequency ωe(so that contributions due to other reso-\nnances can be neglected). Therefore, the Lorentz model\nis particularly suitable for our analysis since we are here\nconcerned with quasi-monochromatic pulses whose car-\nrier frequency ¯ ωcoincides with (or is close to) the fre-\nquency\nω0=/radicalbigg\n1\n2/bracketleftBig\n(2ω2e−γ2+fe)+/radicalbig\nf2e+γ2(γ2−4ω2e−2fe)/bracketrightBig\n(3)\nwhere the real part of the permittivity vanishes, i.e.\nRe[ǫ(ω0)] = 0, the so called epsilon-near-zero (ENZ)\nregime.\nWehaveperformedthe numericalanalysisofthepulse-\nslab collision by means of a Finite Difference Time Do-\nmain (FDTD) scheme where the polarization dynamics\nof Eq.(2) are coupled to Maxwell equations for the TM3\nFIG. 2: Results of FDTD simulation pertaining the interacti on of the pulse 0 (with ¯ ω=ω0) with the slab. The fields Ex,Ez\nand the Fourier transform F[Ex] are captured at three different time steps corresponding to the three rows. In the first and\nthe second time steps the pulse peak is just behind and beyond the slab, respectively, whereas in the third time step the pu lse\nhas completely left the slab. Note the large Ezcomponent within the slab (signature of the epsilon near zer o regime) and the\npersisting and pulse-free polariton oscillation in the thi rd time step.\nfield. Specifically, in order to isolate the relevant phe-\nnomenology characterizing the ENZ regime we have ana-\nlyzed through FDTD simulations two different situations\nwherethesamedielectricslabishitbytwospatiallyequal\npulseswith differentcarrierfrequencies ¯ ω: the first(pulse\n0) is such that ¯ ω=ω0so that it is suitable to scan the\nslab behavior in its ENZ regime; the second (pulse 1) has\n¯ω=ω1for which Re[ ǫ(ω1)]>0 so that it experiences\nstandard dielectric behavior.\nIn view of the generality and ubiquity of the Lorentz\nmodel of Eq.(2), we have chosen for our numerical sim-\nulations a realistic medium with Lorentz parameters\nωe= 3.75·1015Hz,γ= 1.50·1012sec−1andfe=\n2.25·1030sec−2in order to deal with optical pulses in the\nvisible spectrum, the resonant frequency corresponding\nto the wavelength λe= 2πc/ωe= 0.502µm. For such\nparameters Eq.(3) yields ω0= 4.03·1015Hzand we have\nsetω1= 4.23·1015Hz(for which Re[ ǫ(ω1)] = 0.42),\nthe two frequencies corresponding to the wavelengths\nλ0= 2πc/ω0= 0.466µmandλ1= 2πc/ω1= 0.445µm\nrespectively. In Fig.1(b) we plot the real and imaginary\nparts of the dielectric permittivity for the chosen Lorentz\nparameters as functions of the wavelength λ= 2πc/ω,\nindicating the carrier wavelengths λ0andλ1that char-\nacterize the two pulses. We have chosen a slab width\nL= 0.41µmto minimize the pulse propagation features\nand to effectively highlight the impact of the medium\npolarization on field dynamics. We set z0=−150µm,so that the pulse peak reaches the slab 500 fsafter it\nhas been launched. Pulse widths are chosen so that\nwx= 1µmandwz= 60µm: the former is compa-\nrable with the central wavelength of the pulse in order to\nprovide the optical beam with a non negligible longitu-\ndinal field component Ez(see below) whereas the latter\ncorresponds to a temporal width T=wz/c= 200fs\nand a spectral width δω≃1/T= 5·1012Hzso that the\npulses are in the quasi-monochromatic regime.\nB. Pulse 0scattering\nIn Fig.2 we report the main results of the FDTD sim-\nulation dealing with the interaction of the pulse 0 with\ncarrier frequency ¯ ω=ω0with the Lorentz slab repre-\nsented in the figure by the semitransparent rectangu-\nlar blocks. The first two columns of the figure con-\ntain the plots of Ex(x,z,t),Ez(x,z,t) as functions of\n(x,z) whereas the third contains the Fourier transform\nF[Ex](kx,z,t) =/integraltext+∞\n−∞dxeikxxEx(x,z,t) as a function of\n(kx,z); each row of the figure corresponds to a selected\nsimulation time step. At the first time step (first row of\nFig.(2)), t= 371fs, the pulse is fully interacting with\nthe slab (the pulse peak being about to hit the slab at\nt= 500fs) and the electromagnetic field is character-\nized by standard reflection/transmission features; in par-\nticular both the reflected and transmitted pulses have a4\nFIG. 3: Results of FDTD simulation pertaining the interacti on of the pulse 1 (with ¯ ω=ω1) with the slab. The fields Ex,Ez\nand the Fourier transform F[Ex], for comparison purposes, are captured at same time steps c onsidered in Fig.2. Note that\nbothExandF[Ex] are bell-shaped, that magnitudes of Ezwithin the slab and in vacuum are comparable and that no resid ual\npolariton oscillation has been produced by pulse passage.\ntransverse bell-shaped spatial profile and accordingly the\nFourier transform F[Ex] is peaked as well. Note however\nthat, even in this early transient stage of the interac-\ntion, within the slab the longitudinal component Ezis\ncomparable with Exand much greater than its vacuum\ncounterpart. Such an enhancement of the electric field\ncomponent perpendicular to the slab is a feature typi-\ncally associated with monochromatic ENZ regime, aris-\ningasaconsequenceofthecontinuityofthedisplacement\nfield component perpendicular to the interface [14, 15].\nTherefore this is the first evidence that the ENZ regime\ncaneffectivelybeobservedinthoroughlyrealisticLorentz\nslabs by means of an equally realistic scattering interac-\ntion configuration. The second row of Fig.2 corresponds\nto the time step t= 767fsa time when the incoming\npulse (if freely propagating) would have passed behind\nthe slab (its temporal width being 200 fs). Note that\nthe longitudinal component Ezis even greater than the\nprevious time step, testimony to the fact that the ENZ\nregime also occurs in the time-domain. The transverse\ncomponent Exshows novel spatial features, even more\nevidently displayed by its Fourier transform F[Ex] which\nis no longer bell-shaped and characterized by a complex\nmulti-structured profile. The third row of Fig.2 considers\na later time step t= 941fsmuch longer than the time\nspentbythepulsetofullytravelintotheslabandleaveit.\nAt this time step, the longitudinal component Ezisstill\nvery large within the slab and the transverse component\nExdisplays novel and unexpected features: it is symmet-ric under the reflection z→ −z, it is not transversally\nbell-shaped and its Fourier transform F[Ex] displays two\npeaks at the sides of kx= 0. Such phenomenology can\nbe interpreted only by assuming that the interaction of\npulse 0 with the slab is accompanied by the excitation\nof a polariton mode whose oscillation lasts a time much\nlonger than the pulse-slab interaction time.\nC. Pulse 1scattering\nIn order to appreciate the novelty of the above dis-\ncussed time-domain ENZ phenomenology, we now dis-\ncuss the interaction of pulse 1 with the slab, its carrier\nfrequency being associated to standard slab dielectric be-\nhavior. In Fig.3 we report the results of the FDTD sim-\nulation relative to pulse 1 and, for comparison purposes,\nwe have given Fig.3 the same structure as Fig.2 with the\nsame fields at same time steps. Remarkably, both Ex\nandF[Ex] are everywhere and always bell-shaped, while\nthe magnitude of Ezwithin the slab is comparable with\nits vacuum magnitude. At the last time step the slab\nhosts no residual polariton oscillation resulting from the\npulse passage. This is precisely the standard expected\nphenomenology of the reflection and transmission of the\npulsebyadielectricslab, andthe comparisonwiththere-\nsults of Fig.2 proves that the phenomenology it contains\nis a manifestation of the time-domain ENZ regime5\nFIG. 4: FDTD predictions about the fields ExandEz, for both pulse 0 and pulse 1 as function of ( z,t) at a fixed plane\nx=xp= 1.76µ. Pulse 0 triggers a novel mechanism of metastable light trap ping since its passage produces a strong and\ndamped polariton oscillation which is absent in the case of p ulse 1.\nD. Transient trapping in the time-domain epsilon\nnear zero regime\nIn addition to the remarkable fact that the same fea-\ntures of the monochromatic ENZ regime characterize its\ntime domain counterpart (e.g. the slab hosts a pro-\nnounced enhancement of the field Ez), the results dis-\ncussed in the previous sections also clearly reveal that\nthe scattering situation leads to the unique excitation of\na polariton mode. In order to show more explicitly such\na phenomenology, in Fig.4 we have plotted the fields Ex\nandEz, for both pulse 0 and pulse 1 as functions of ( z,t)\nat a fixed plane x=xp= 1.76µm. The evident fea-\nture that emerges is that pulse 0 (see subplots (a) and\n(b) of Fig.4) produces a strong and damped electromag-\nnetic self-oscillation persisting a time (about 5500 fs)\nmuch longer than the probing pulse duration (200 fs),\nself-oscillation which is conversely not produced by pulse\n1 (see subplots (c) and (d) of Fig.4), whose electromag-\nnetic track fades within the slab just after it has left the\nmedium (at about t= 1000fs). We conclude that, in\nthe ENZ regime, the pulse travelling through the slab\ntriggers a novel mechanism of transient light trapping.\nIII. THEORETICAL ANALYSIS OF\nTIME-DOMAIN ENZ REGIME\nA. Polariton virtual modes analysis\nFrom the above discussed phenomenology, it is evi-\ndent that a quasi-monochromatic pulse with a spectrum\ncentered at the zero of the real part of the slab permit-\ntivity excites a polariton mode that lasts a time much\nlonger than the pulse-slab interaction time. In order to\nrigorously prove this statement and gain deeper under-\nstanding of the underpinning physical mechanisms thatsupport the time-domain ENZ regime, in this section we\nanalyze the exact quasi-steady modes (virtual modes) of\nthe slab. In ouranalysiswefully takeinto accountdamp-\ning processes, which include medium absorbtion and ra-\ndiation leakage in vacuum, adopting the complex fre-\nquency approach [25]. We start our analysis from the\ncurl Maxwell equations for TM fields\n−∂Ez\n∂x+∂Ex\n∂z=−µ0∂Hy\n∂t,\n−∂Hy\n∂z=ǫ0∂Ex\n∂t+∂Px\n∂t, (4)\n∂Hy\n∂x=ǫ0∂Ez\n∂t+∂Pz\n∂t,\nwhere the polarization P(x,z,t) =Px(x,z,t)ˆex+\nPz(x,z,t)ˆezsatisfies Eq.(2) within the slab ( |z|< L/2)\nand it vanishes outside the slab ( |z|> L/2). We take the\nAnsatzAj(x,z,t) = Re/bracketleftbig\naj(z)ei(kxx−Ωt)/bracketrightbig\nfor every field\ncomponent( A=E,H,P,a=e,h,pandj=x,z), where\nkxis the (real) transverse wavevector and Ω = ω−iΓ is\nthe complex angular frequency with Γ >0 so that only\ndamping modes are considered. Owing to the mutual\ntemporal evolution of the electromagnetic field ( E,H)\nand of the polarization field P, the Ansatz effectively\namounts to considering polariton virtual modes. The\nmagnetic field can be expressed in terms of the electric\nfield components hy=−(kxez+i∂zex)/(µ0Ω) so that\nMaxwell’s equations reduce to\nez=ikx\nΩ2\nc2˜ǫ(Ω,z)−k2\nxdex\ndz, (5)\nd2ex\ndz2+/bracketleftbiggΩ2\nc2˜ǫ(Ω,z)−k2\nx/bracketrightbigg\nex= 0, (6)\nwhere ˜ǫ(Ω,z) =ǫ(Ω)θ(L/2− |z|) +θ(|z| −L/2) (θ(z)\nbeing the Heaviside step function) is the z-dependent di-\nelectric profile. It is worth stressing that the permittivity6\nǫis evaluated at the complex frequency Ω. The general\nsolution of Eqs.(5,6) is explicitly given by\nex(z) =C\n\nΘe−iΞK(z+L\n2)z <−L\n2,\neikz+Θe−ikz\neikL\n2+Θe−ikL\n2−L\n2≤z≤L\n2,\neiΞK(z−L/2)z >L\n2,\n(7)\nez(z) =C\n\nΘΞkx\nKe−iΞK(z+L\n2)z <−L\n2,\nkx\nk−eikz+Θe−ikz\neikL\n2+Θe−ikL\n2−L\n2≤z≤L\n2,\n−Ξkx\nKeiΞK(z−L\n2)z >L\n2,\nwhereK=/radicalBig\nΩ2\nc2−k2x,k=/radicalBig\nΩ2\nc2ǫ(Ω)−k2x,Cis the arbi-\ntrary mode amplitude, Θ = ±1 is a parameter that dis-\ntinguishes the symmetry of the solutions and Ξ = ±1 is\nanother parameter selecting the sign of the exponentials\nin vacuum. By construction, the modal fields in Eqs.(7)\nalready satisfy the continuity of the field component par-\nallel to the slab surface ( ex) at the interfaces x=±L/2.\nThe boundary conditions (BCs) for the continuity of the\ndisplacement field component (˜ ǫez)perpendicular to the\ninterfaces x=±L/2 yield the dispersion relation\n(Kǫ−Ξk)eikL= Θ(Kǫ+Ξk), (8)\nwhich provides the complex frequency Ω for every given\nslab thickness Land transverse wave vector kx. We have\nsolved Eq.(8) numerically and obtained the allowed Ω\ncorresponding to different values of kxusing the same\nslab thickness and Lorentz dispersive parameters of the\nslab considered in Section II. In Fig.5 we plot the results\nforthe caseΘ = −1in the complex planeΩ parametrized\nthrough the wavelength λand the damping constant Γ\n(i.e. Ω = 2 πc/λ−iΓ), using circles and stars for the\nΞ =−1 and Ξ = 1 modes, respectively, and using the\nmarker color to label the value of the corresponding kx.\nNote that the Ξ = −1 and Ξ = 1 modes belongs to two\ndifferent branches which are characterized by the fact\nthat the Ξ = −1 modes have damping constant greater\nthan the Ξ = +1 modes. This property can be easily\nunderstood by considering the z-component of the non-\noscillatory part of the Poynting vector S=E×Hfor\nz > L/2:\n/angbracketleftSz/angbracketright= Ξe−2[ΞIm(K)(z−L\n2)+Γt]1\n2ǫ0Re/parenleftbiggΩ\nK/parenrightbigg\n|C|2.(9)\nFor Ξ = −1, the energy outflows from the slab and the\ndamping of the virtual mode is more rapid since it loses\nenergy through both medium absorption and radiation\nleakage. Conversely, for Ξ = 1 the electromagnetic en-\nergyisdraggedintothe slab, thus partiallycompensating\nfor the medium absorption and consequently decreasing\nthe virtual mode damping time (note that there is also a\npoint where Γ = 0 on the Ξ = 1 branch corresponding to0.4655 0.4660 0.46650123456x 10−3Γ (fs−1)\n  \n00.511.522.53Θ = −1 kx (µm−1)\n: Ξ = −1\n: Ξ = 1°\n∗\nλ (µm)ΩPδλ\nλ0\nFIG. 5: Virtual modes of the slab considered in Sec.II for\nthe symmetry Θ = −1 in the complex plane Ω = 2 πc/λ−\niΓ. Circles and stars label the Ξ = −1 and Ξ = 1 modes\nwhereas the marker color labels the corresponding kxvalue.\nThe complex frequency Ω Pis such that ǫ(ΩP) = 0. The\nthin continuous line is the temporal spectrum of the incomin g\npulse reported in arbitrary units.\nthe exact balance between medium absorption and radi-\nation drag). It is remarkable that the Ξ = −1 and Ξ = 1\nbranches intersect each other at point Ω Pforkx→0. In\nthis limit the dispersion relation of Eq.(8) (for Θ = −1)\nreduces to\neiΩP\nc√\nǫ(ΩP)=Ξ−/radicalbig\nǫ(ΩP)\nΞ+/radicalbig\nǫ(ΩP), (10)\nand is satisfied only if ǫ(ΩP) = 0. Starting from the\nLorentzmodel, itisstraightforwardtoprovethattheper-\nmittivity vanishes at Ω P=/radicalbig\nfe+ω2e−γ2/4−iγ/2that,\nfor the above used Lorentz dispersive parameters, yields\nλP= 2πc/Re(ΩP) = 0.4667µmand Γ P=−Im(ΩP) =\n8.3·10−3fs−1, precisely matching the point of Fig.5\nwhere the two branches Ξ = ±1 intersect each other. In\nturn, the plasmonic ΩPat which the permittivity van-\nishes plays a central role in the analysis of the virtual\nmodes. For the complex frequencies Ω reported in Fig.5,\n|ǫ(Ω)|<0.06 and therefore, all the obtained modes with\nsymmetry Θ = −1 imply the time-domain ENZ regime.\nIn the same portion of the complex plane Ω we have\nnumerically found no allowed modes for the symmetry\nΘ = 1. This can be grasped by expanding both sides\nof Eq.(8) in Taylor series of ǫ(since|ǫ(Ω)| ≪1); at the\nzeroth order we readily obtain e−kxL=−Θ which is not\nconsistent if Θ = 1 (and which, on the other hand, yields\nkx= 0 for Θ = −1).\nB. Interpretation of FDTD results in terms of\nvirtual polariton modes and slow-light regime\nUsually, within the standard real frequency approach,\nthe solutions for the slab modes with Ξ = −1 are dis-7\nregarded since they are considered unphysical. Indeed,\nif|kx|> ω/c, the solutions with Ξ = +1 represent con-\nfined modes propagating along the x-direction, while so-\nlutions with Ξ = −1 are unbound modes that diverge\natz→ ±∞. In addition, the introduction of the com-\nplex frequency introduces an inherent field singularity\nin the far past t→ −∞. Due to such intrinsic singu-\nlarities, it is strictly impossible to rigorously excite a\nsingle virtual mode, its global existence on the whole\nspace-time being unphysical. However, both singular-\nities occur asymptotically and therefore virtual modes\nprovide a very adequate description of the transient ENZ\nslabbehavioroccurringwithinaspatiallyboundedregion\nand through a finite time lapse. In order to prove this\nstatement and to basically provide a theoretical analyt-\nical description of the transient light trapping discussed\nin Sec.2D, in Fig.5 we have superimposed the temporal\nspectrum profile of the incoming pulse of Eq.(1) (using\nthe thin continuous line) on the complex-plane virtual\nmodal structure. Note that, due to its wavelength band-\nwidthδλ=/parenleftbig\n2πc/ω2\n0/parenrightbig\nδω= 5.8·10−4µm, the pulse spec-\ntrum centered at λ0overlaps a limited portion of the\nconsidered complex frequency plane so that, specifically,\nthe sole virtual modes with |λ−λ0|< δλ/2 are actu-\nally excited by the considered pulse 0. From Fig.5 it\nis evident that the excited virtual modes are character-\nized by the transverse wavevector kxspanning the range\n1.1µm−1< kx<2.7µm−1and that, due to the fi-\nnite bandwidth of the impinging pulse, the excited vir-\ntual modes with largest amplitude are those around the\ncentral transverse wavevector kx= 1.9µm, which cor-\nresponds to λ= 2πc/Re(Ω) close to λ0= 0.466. This\nobservation is in striking agreement with the results con-\ntained in Fig.2, where one can see that the transverse\nFourier transform of the field has, at the latest time step,\ntwo peaks centered at kx≃1.9µmandkx≃ −1.9µm\nwhose width is of the order of δkx≃0.8µm. There-\nfore, when the pulse 0 impinges onto the slab, it ex-\ncites precisely the virtual modes analytically predicted\nin Sec.IIIA, which are compatible with its spectral struc-\nture. As a further validation of this statement, note that\nthe virtual modes excited by the pulse 0 ( |λ−λ0|<\nδλ/2) have a damping constant Γ spanning the range\n1.2·10−3fs−1<Γ<4.9·10−3fs−1(see Fig.5), which\ncorresponds to the extinction time τ= 3/Γ spanning the\nrange 609 fs < τ < 2604fs. Also this prediction based\non the above virtual mode analysis is in striking agree-\nment with the FDTD results since, by looking at panels\n(a) and (b) of Fig.4, one can see that the electromag-\nnetic excitationpersists for a time ofthe orderof3000 fs,\nwhichiscompatiblewiththemaximumextinctiontimeof\nthe excited virtual modes. In addition, the spatial sym-\nmetry of the Θ = −1 virtual polariton modes matches\nthe numerical results displayed in Fig.4: the transverse\nfield component ( ex) is antisymmetricwith respectto the\nz= 0 axis, whereas the longitudinal field component ( ez)\nis symmetric.\nThe final ingredient needed to thoroughlyinterpret thetransient light trapping observed in FDTD simulations\nis related to the intrinsic slow-light nature of the phe-\nnomenon, which may be preliminarily grasped by con-\nsidering a bulk Lorentz medium, where transverse plane\nwaves satisfy the dispersion relation k(ω) = (ω/c)/radicalbig\nǫ(ω).\nNeglecting medium absorption ( γ≃0), one finds that\nthe phase velocity is vf=ω/kand the group velocity\nvg= dω/dkis\nvg(ω) =c/bracketleftBigg\n/radicalbig\nǫ(ω)+feω2\n/radicalbig\nǫ(ω)(ω2e−ω2)2/bracketrightBigg−1\n.(11)\nThus, in the ENZ regime the phase velocity diverges\nvf→ ∞whereasthe groupvelocitytends to zero vg→0.\nEven though the subwavelength Lorentz slab used in our\nFDTD simulations is not a bulk medium and absorp-\ntion has not been neglected, the rough argument above\nstillpredictsthecorrectoutcome. Indeed, bynumerically\nsolving the dispersion relation of Eq.(8) without neglect-\ninglossesonefindsthat, attheopticalwavelength λ0, the\ntransversephasevelocityofvirtualpolaritonmodesis su-\nperluminal vf=ω/kx≃10c, while the transverse group\nvelocity is extremely reduced vg= dω/dkx≃c/100.\nFor this reason, it is now clear how the virtual polari-\nton modes, once excited, do not disperse quickly in the\nx-direction and remain quasi-trappedwithin the slab ow-\ning to the tremendously reduced temporal dynamics. We\nconclude that the above described transient light trap-\nping can be fully interpreted and physically understood\nby means of slow polariton modes supported by the slab.\nC. Volume plasmons\nAlthough the above discussed numerical and analyti-\ncal analysis of the transient light trapping characteriz-\ning the time-domain ENZ regime is quite exhaustive, we\nnow discuss its connection with the purely longitudinal\nmodes, either volume plasmons (collective oscillations of\nelectrons) or volume phonons (collective oscillations of\nions), which the Lorentz medium can support. Hereafter\nwe focus on volume plasmons, considering an unbounded\nbulk Lorentzmedium where the TM electromagneticand\npolarizationdynamicsaredescribedbyEqs.(2,4). Forthe\nplane-wave Ansatz Aj(x,z,t) = Re/bracketleftbig\najei(kxx+kzz−Ωt)/bracketrightbig\n,\nwhereA=E,H,P,a=e,h,p,j=x,z,kx,kzare\nthe (real) wavevector components and Ω is the generally\ncomplex frequency one gets\nµ0Ωhy=−kxez+kzex,\nkzhy= Ωǫ0ǫ(Ω)ex, (12)\nkxhy=−Ωǫ0ǫ(Ω)ez,\nwhereǫ(Ω) is the dielectric permittivity with complex\nfrequency Ω. Volume plasmons are purely longitudinal\nelectric oscillations owing to to the collective motion of\nelectrons and are not accompanied by the generation of\nmagnetic field, a feature that for plane waves amounts8\nto the collinearity of the wave vector k=kxˆex+kzˆez\nand the electric field E=kxˆEx+kzˆEz. Therefore, im-\nposing the condition k×E= 0, i.e. −kxez+kzex= 0,\nEqs.(12) readily yield hy= 0 and ǫ(ΩP) = 0. Thus, vol-\nume plasmonsareinherentlyinvolvedin the time-domain\nENZ regime we are considering in this paper. However,\nit is worth noting that the virtual modes of the Lorentz\nslab are polaritons, entities fundamentally different from\nvolume plasmons (or volume phonons). Indeed, a vol-\nume plasmon is strictly characterized by the condition\nǫ(Ω) = 0 that implies the severe dispersion Ω = Ω Pand,\nin the presence of the slab boundaries at z=±L/2, in-\nevitably leads to the inconsistency Ez→ ∞within the\nslab unless Ez= 0 in the outer medium. This is consis-\ntent with the well-known impossibility to excite volume\nplasmons by means of light. On the other hand, from\nEq.(7) one can see that the virtual polariton mode com-\nponentEzneither vanishes outside the slab nor diverges\nwithin it. This is because for polaritons the dispersion\nrelation of Eq.(8) is not as stiff as the volume plasmon\ndispersion Ω = Ω Pand is satisfied also for Ω /negationslash= ΩP. In\naddition the volume plasmon is a purely electric oscilla-\ntion with strictly null magnetic field, whereas the consid-\nered virtual polariton modes are accompanied by a mag-\nneticfield. Inturn, eventhoughTMpolaritonmodesand\nvolume plasmons occur in the same spectral region and\nare accidentally connected by the fact that in the limit\nL >> λ the Lorentz slab is almost equivalent to a bulk\nmedium, conceptually they are very distinct entities. In\nview of this, we conclude remarking that volume plas-\nmons can not be excited by classical light and that the\nabsorption peak observed in experiments [28, 30] is dueto the excitation of virtual polariton modes, confirming\nthe results given in Refs. [27, 29].\nIV. CONCLUSIONS\nIn conclusion we have investigated both numerically\nand analytically the properties of the time-domain ENZ\nregime. Specifically we have considered a dielectric slab\nwhose polarization dynamics has been described through\nthe realistic and ubiquitous Lorentz model and we have\nanalyzed its interaction with quasi-monochromatic and\nspatially confined pulses with carrier frequencies close\nto the crossing point of the permittivity real part. The\nFDTD analysis has shown that the pulse is able to ex-\ncite a polarization-electromagnetic(polariton) oscillation\nwhich is damped and persists for a time generally longer\nthan the effective time required by the pulse for passing\nthrough the slab. 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Gref-\nfet, Physical Review Letters 109, 237401 (2012)."    },    {        "title": "2109.07597v1.Penning_Trap_Searches_for_Lorentz_and_CPT_Violation.pdf",        "content": "arXiv:2109.07597v1  [hep-ph]  15 Sep 2021\nArticle\nPenning-Trap Searches for Lorentz and CPT Violation\nYunhua Ding1,2,*, Teague D. Olewiler2and Mohammad Farhan Rawnak2\n1W.M. Keck Science Department, Claremont McKenna, Pitzer, a nd Scripps Colleges, Claremont, CA 91711,\nUSA; yding@kecksci.claremont.edu (Y.D.)\n2Department of Physics, Gettysburg College, Gettysburg, PA 17325, USA;\nolewte01@gettysburg.edu(T.D.O.); rawnmo01@gettysburg .edu(M.F.R.)\n*Correspondence: yding@kecksci.claremont.edu\nVersion September 17, 2021 submitted to Symmetry\nAbstract: An overview of recent progress on testing Lorentz and CPT sym metry using Penning\ntraps is presented. The theory of quantum electrodynamics w ith Lorentz-violating operators of mass\ndimensions up to six is summarized. Dominant shifts in the cy clotron and anomaly frequencies of\nthe confined particles and antiparticles due to Lorentz and C PT violation are derived. Existing\nresults of the comparisons of charge-to-mass ratios and mag netic moments involving protons,\nantiprotons, electrons, and positrons are used to constrai n various coefficients for Lorentz violation.\nKeywords: Lorentz and CPT violation; Penning trap; Standard-Model Ex tension\n1. Introduction\nPrecision experiments involving Penning traps have in rece nt years achieved impressive\nsensitivities to properties of fundamental particles. For example, the magnetic moment of electrons\nhas been measured to a record precision of 0.28 ppt [ 1]. The high precision offered by Penning-trap\nexperiments provides excellent opportunities to test fund amental symmetries. This includes the\nLorentz symmetry, one of the foundations of both general rel ativity and the Standard Model of\nparticle physics. It has been recently shown that tiny viola tions of Lorentz symmetry could naturally\narise in a fundamental theory that unifies gravity with quant um physics at the Plank scale MP∼1019\nGeV , such as string theory [ 2,3]. Since in any effective field theory, violations of CPT symm etry\nalso break Lorentz symmetry [ 4–6], testing Lorentz symmetry also includes CPT tests. In rece nt\nyears, searches for Lorentz and CPT violation in precision e xperiments have been performed across\nmany subfields of physics [ 7], including Penning-trap experiments. Here, we provide an overview\nof the recent progress on testing Lorentz and CPT violation i n Penning-trap experiments measuring\ncharge-to-mass ratios and magnetic moments of protons, ant iprotons, electrons, and positrons.\nIn the context of effective field theory, the comprehensive f ramework that describes all possible\nLorentz violation is the Standard-Model Extension (SME) [ 4,5,8]. The Lagrange density of the SME is\nconstructed from general relativity and the Standard Model by adding all possible Lorentz-violating\nterms. Each of such terms is formed from a coordinate-indepe ndent contraction of a general\nLorentz-violating operator with a corresponding coefficie nt. The subset of the SME with operators\nof mass dimensions d≤4 is called the minimal SME, which is power-counting renorma lizable.\nFor the nonminimal SME, it restricts attention to operators of mass dimensions d>4, which\nis viewed to produce higher-order effects. Study of the nonm inimal SME serves as a basis\nfor further investigations of many aspects of Lorentz and CP T violation, such the causality and\nstability [ 9,10], Lorentz-violating models in supersymmetry [ 11], noncommutative Lorentz-violating\nquantum electrodynamics [ 12–14], and the underlying pseudo-Riemann-Finsler geometry [ 15–18].\nSubmitted to Symmetry , pages 1 – 20 www.mdpi.com/journal/symmetryVersion September 17, 2021 submitted to Symmetry 2 of 20\nFor Penning-trap experiments measuring charge-to-mass ra tios and magnetic moments of\nconfined particles or antiparticles, both the minimal and no nminimal SME can produce various\nmeasurable Lorentz- and CPT-violating effects via shifts i n the cyclotron and anomaly frequencies.\nThese effects in general can depend on sidereal time and diff er between particles and antiparticles. In\nthe minimal SME, Refs. [ 19,20] present the first theoretical analysis to study Lorentz and CPT violation\nin Penning traps. An extension to the nonminimal SME by inclu ding Lorentz-violating operators\nof mass dimensions up to six was recently made in Ref. [ 21], in which analysis of the magnetic\nmoment comparisons between particles and antiparticles us ing Penning traps was also performed.\nA similar application to charge-to-ratio comparisons is pr esented in Ref. [ 22]. For the effects arising\nfrom sidereal variations due to the Earth’s rotation, the re lated discussions are given in Ref. [ 23–25].\nIn this work, we provide an overview of recent progress on sea rching for Lorentz- and\nCPT-violating signals using Penning traps and provide the m ost updated constraints on the\ncoefficients for Lorentz violation that are relevant to thes e experiments. The results provided in\nthis work are complementary to these from the studies of Lore ntz and CPT violation in experiments\ninvolving measurements of the muon’s anomalous magnetic mo ment [ 26,27], the spectroscopic\nanalysis of hydrogen, antihydrogen, and other related syst ems [ 28], and clock comparisons [ 29].\nThis work is organized as follows. We start in Section 2with the related theory, where we\npresent in Subsection 2.1the theory of Lorentz-violating quantum electrodynamics w ith operators\nof mass dimensions up to six. The perturbative energy shifts to the confined particles or antiparticles\nare obtained in Subsection 2.2using perturbation theory. Subsection 2.3gives the corresponding\nshifts in the cyclotron and anomaly frequencies due to Loren tz and CPT violation. The discussion\nof sidereal variations and rotation matrices are given in Su bsection 2.4. Then we turn in Section 3\nto experimental applications to various Penning-trap expe riments and present the constraints on the\ncoefficients for Lorentz violation. The applications to cha rge-to-mass ratio comparisons are treated in\nSubsection 3.1, with Subsection 3.1.1 focusing on the proton sector, and Subsection 3.1.2 discussing the\nelectron sector, respectively. The resulting constraints on the coefficients for Lorentz violation from\nthe reported experimental results are summarized in Table 1, Table 2, and Table 3. Subsection 3.2\ndiscusses the applications to magnetic moment comparisons , with Subsection 3.2.1 for the proton\nsector, and Subsection 3.2.2 for the electron sector, respectively. The corresponding l imits on the\ncoefficients for Lorentz violation are listed in Table 4and Table 5. The summary of this work is\ngiven in Section 4. Finally, Appendix Alists the explicit expressions of the transformations of th e\nrelevant coefficients into different frames. Throughout th is work, we follow the same notation used\nin Refs. [ 21,22] and adopt natural units with ¯ h=c=1.\n2. Theory\nIn this section, we summarize the theory developed in Ref. [ 21] of Lorentz-violating quantum\nelectrodynamics with operators of mass dimension d≤6 and derive the energy shifts for particles\nand antiparticles confined in Penning traps due to Lorentz an d CPT violation.\n2.1. Lagrange density\nFor a single Dirac fermion field ψof charge qand mass mψ, the general Lorentz-violating\nLagrange density Lψcan be constructed by adding a general Lorentz-violating op erator/hatwideQto the\nconventional Lagrange density,\nLψ=1\n2ψ(γµiDµ−mψ+/hatwideQ)ψ+H.c., (1)\nwhere Dµ= (∂µ+iqA µ)is the covariant derivative given by the minimal coupling wi thAµbeing\nthe electromagnetic four-potential. H.c. means Hermitian conjugate. The general Lorentz-violating\noperator /hatwideQin the Lagrange density ( 1) is a 4×4 spinor matrix that contains terms formed by\nthe contraction of a generic coefficient for Lorentz violati on, the covariant derivative iDµ, theVersion September 17, 2021 submitted to Symmetry 3 of 20\nantisymmetric electromagnetic field tensor Fαβ≡∂αAβ−∂βAα, and one of the 16 Dirac matrix\nbases. For example, one of the dimension-five operators invo lving the F-type coefficients for Lorentz\nviolation takes the form b(5)µαβ\nFFαβγ5γµ. For mass dimension d≤6, a full list of the relevant\ncoefficients for Lorentz violation and their properties are given in Table I in Ref. [ 21]. Note that\nthe hermiticity of the Lagrange density ( 1) indicates that the operator /hatwideQsatisfies the condition\n/hatwideQ=γ0/hatwideQ†γ0. In the free-fermion limit where Aα=0, the explicit expression of the Lagrange density\n(1) at arbitrary mass dimension has been studied in Ref. [ 30]. For the interaction case where Aα/negationslash=0,\nRef. [ 21] developed a theory for operators of mass dimensions up to si x. An extension of the theory\nto include operators of arbitrary mass dimension was recent ly given in Ref. [ 31]. Similar analysis\nhas also been performed for other SME sectors, including the se for photon [ 32], neutrino [ 33] and\ngravity [ 34].\nDue to the existence of the general operator /hatwideQin the Lagrange density ( 1), the conventional Dirac\nequation for a fermion in electromagnetic fields is modified t o\n(p·γ−mψ+/hatwideQ)ψ=0, (2)\nwhere we have chosen the momentum space for convenience, wit h the identification pα↔iDα. Given\nthe fact that no Lorentz-violating signals have been observ ed so far, any such signal must be tiny\ncompared to the energy scale of the system of interest. There fore, we can treat the corrections due to\nLorentz and CPT violation to the conventional Hamiltonian a s perturbative and apply perturbation\ntheory to obtain the dominant shifts to the energy levels of t he confined particles and antiparticles.\nFrom Equation ( 2), the exact Hamiltonian Hcan be defined as\nHψ≡p0ψ=γ0(p·γ+mψ−/hatwideQ)ψ= (H0+δH)ψ, (3)\nwhere p0is the exact energy of the system of interest, including all c ontributions from Lorentz and\nCPT violation,H0is the conventional Hamiltonian for a fermion in an electrom agnetic field, and\nδH=−γ0/hatwideQis the exact perturbative Hamiltonian.\nTo construct δH, we note that operator /hatwideQin general contains terms that are of powers of p0,\nwhich corresponds to the exact Hamiltonian Hitself. In certain simple cases, one can apply a field\nredefinition to remove the additional time derivatives and a dopt the standard procedure involving\ntime translation on wave functions to obtain the exact pertu rbative Hamiltonian δH[20]. In more\ngeneral cases, it is challenging to directly construct δHdue to the existence of powers of time\nderivatives. However, we notice that any contributions to δHdue to the exact Hamiltonian Hare at\nsecond or higher orders in the coefficients for Lorentz viola tion. Therefore, to obtain the leading-order\nresults, one can apply the following substitution [ 21,33],\nδH≈− γ0/hatwideQ|p0→E0, (4)\nwhere E0is the unperturbative eigenvalue, which can be obtained by s olving the conventional Dirac\nequation for a fermion in an electromagnetic field.\n2.2. Perturbative energy shifts\nWith the perturbative Hamiltonian δHdetermined by Equation ( 4), the shifts in the energy levels\nof a confined particle can be obtained by applying perturbati on theory,\nδEn,±=/angbracketleftχn,±|δH|χn,±/angbracketright, (5)\nwhere χn,±are the unperturbative stationary eigenstates with nbeing the level number and ±\ndenoting the spin for a positive-energy fermion. δEn,±are the perturbative corrections to the energy\nlevels due to Lorentz and CPT violation.Version September 17, 2021 submitted to Symmetry 4 of 20\nTo derive the expressions for δEn,±, we can take an idealized Penning trap where a constant\nuniform magnetic field is applied to confine the particle’s ra dial motion and a quadrupole electric\nfield provides the axial confinement. The dominant effects in the unperturbative energy levels are\ndue to the interactions between the confined particle and the magnetic field. The quadruple electric\nfield generates effects suppressed by a factor of E/B≈10−5in natural units for a typical Penning\ntrap field configuration with E≈20 kV/m and B≈5 T. Therefore, the leading-order results in\nδEn,±can be obtained by further idealizing the trap as a pure unifo rm magnetic field in which a\nquantum fermion moves. Following the above discussion, for a spin-1/2 fermion, the leading-order\nperturbative energy shifts due to Lorentz and CPT violation are found to be [ 22]\nδEw\nn,±1=/tildewidea0\nw∓σ/tildewideb3\nw−/tildewidem3\nF,wB±σ/tildewideb33\nF,wB+/parenleftbig±σ/tildewideb′3\nw−mw[/tildewidec00\nw+(/tildewidec11\nw+/tildewidec22\nw)s]/parenrightbig(2n+1∓σ)|qB|\n2m2w\n+/parenleftbig∓σ(/tildewideb311\nw+/tildewideb322\nw)−1\nmw(/tildewidec11\nw+/tildewidec22\nw)s¬/parenrightbig(2n+1)|qB|\n2, (6)\nwhere σis the charge sign of the fermion, given by q≡σ|q|. The subscript windicates the fermion\nspecies. For example, for electrons and protons, w=eand w=p, respectively. The tilde coefficients\nare defined by\n/tildewidea0\nw=a0\nw−mwc00\nw−mwe0\nw+m2\nwm(5)00\nw+m2\nwa(5)000\nw−m3\nwc(6)0000\nw−m3\nwe(6)000\nw ,\n/tildewideb3\nw=b3\nw+H12\nw−mwd30\nw−mwg120\nw+m2\nwb(5)300\nw+m2\nwH(5)1200\nw−m3\nwd(6)3000\nw−m3\nwg(6)12000\nw ,\n/tildewidem3\nF,w=m(5)12\nF,w+a(5)012\nF,w−mwc(6)0012\nF,w−mwe(6)012\nF,w,\n/tildewideb33\nF,w=b(5)312\nF,w+H(5)1212\nF,w−mwd(6)3012\nF,w−mwg(6)12012\nF,w,\n/tildewideb′3\nw=b3\nw+mw(g120\nw−g012\nw+g021\nw)−m2\nwb(5)300\nw−2m2\nw(H(5)1200\nw−H(5)0102\nw+H(5)0201\nw)\n+2m3\nwd(6)3000\nw+3m3\nw(g(6)12000\nw−g(6)01002\nw+g(6)02001\nw),\n/tildewidec00\nw=c00\nw−mwm(5)00\nw−2mwa(5)000\nw+3m2\nwc(6)0000\nw+2m2\nwe(6)000\nw , (7)\nand the “11+22\" types of tilde coefficients are defined by\n(/tildewidecjj\nw)s=cjj\nw−2mwa(5)j0j\nw+3m2\nwc(6)j00j\nw ,\n(/tildewidecjj\nw)s¬=−mwa(5)0jj\nw−mwm(5)jj\nw+3m2\nwc(6)00jj\nw+3m2\nwe(6)0jj\nw ,\n/tildewideb3jj\nw=b(5)3jj\nw+H(5)12jj\nw−3mwd(6)30jj\nw−3mwg(6)120jj\nw , (8)\nwith j=1 or 2 only. The subscripts sand s¬in the/tildewidecjj\nwtilde coefficients in the energy shifts ( 6) show\nthat(/tildewidecjj\nw)sgive both spin-dependent and spin-independent energy shif ts, while (/tildewidecjj\nw)s¬produce only\nspin-independent ones, as evident from the corresponding p roportional factors 2 n+1∓σand 2 n+1.\nWe note that the tilde coefficients /tildewidea0\nw,/tildewideb3\nw,/tildewidem3\nF,w, and/tildewideb33\nF,win the energy shifts ( 6) produce effects that\nare independent of the level number n.\nFollowing a similar analysis, the corresponding leading-o rder shifts to the energy levels of\nantifermions can be determined by using\nδEc\nn,±=/angbracketleftχc\nn,±|δHc|χc\nn,±/angbracketright, (9)\nwhere χc\nn,±are the positive-energy antifermion eigenstates, obtaine d by applying charge conjugation\non the negative-energy fermion solutions χn,±.δHcis the antifermion perturbative Hamiltonian,Version September 17, 2021 submitted to Symmetry 5 of 20\nwhich can be obtained from δHby reversing the charge sign σand changing the signs for all CPT-odd\ncoefficients in operator /hatwideQ. Using Equation ( 9), the antifermion results are given by [ 22]\nδEw\nn,±1=−/tildewidea∗0\nw±σ/tildewideb∗3\nw−/tildewidem∗3\nF,wB∓σ/tildewideb∗33\nF,wB+/parenleftbig∓σ/tildewideb′∗3\nw−mw[/tildewidec00\nw+(/tildewidec11\nw+/tildewidec22\nw)s]/parenrightbig(2n+1∓σ)|qB|\n2m2w\n+/parenleftbig±σ(/tildewideb∗311\nw+/tildewideb∗322\nw)−1\nmw(/tildewidec11\nw+/tildewidec22\nw)s¬/parenrightbig(2n+1)|qB|\n2, (10)\nwhere the tilde quantities with a star subscript are given by\n/tildewidea∗0\nw=a0\nw+mwc00\nw−mwe0\nw−m2\nwm(5)00\nw+m2\nwa(5)000\nw+m3\nwc(6)0000\nw−m3\nwe(6)000\nw ,\n/tildewideb∗3\nw=b3\nw−H12\nw+mwd30\nw−mwg120\nw+m2\nwb(5)300\nw−m2\nwH(5)1200\nw+m3\nwd(6)3000\nw−m3\nwg(6)12000\nw ,\n/tildewidem∗3\nF,w=m(5)12\nF,w−a(5)012\nF,w−mwc(6)0012\nF,w+mwe(6)012\nF,w,\n/tildewideb∗33\nF,w=b(5)312\nF,w−H(5)1212\nF,w+mwd(6)3012\nF,w−mwg(6)12012\nF,w,\n/tildewideb′∗3\nw=b3\nw+mw(g120\nw−g012\nw+g021\nw)−m2\nwb(5)300\nw+2m2\nw(H(5)1200\nw−H(5)0102\nw+H(5)0201\nw)\n−2m3\nwd(6)3000\nw+3m3\nw(g(6)12000\nw−g(6)01002\nw+g(6)02001\nw),\n/tildewidec∗00\nw=c00\nw−mwm(5)00\nw+2mwa(5)000\nw+3m2\nwc(6)0000\nw−2m2\nwe(6)000\nw , (11)\nand the corresponding “11+22\" starred tilde quantities are defined by\n(/tildewidec∗jj\nw)s=cjj\nw+2mwa(5)j0j\nw+3m2\nwc(6)j00j\nw ,\n(/tildewidec∗jj\nw)s¬=mwa(5)0jj\nw−mwm(5)jj\nw+3m2\nwc(6)00jj\nw−3m2\nwe(6)0jj\nw ,\n/tildewideb∗3jj\nw=b(5)3jj\nw−H(5)12jj\nw+3mwd(6)30jj\nw−3mwg(6)120jj\nw . (12)\nIn the result ( 10), the charge sign σis understood to be reversed for the antifermion. Comparing\nthe fermion and antifermion energy shifts results ( 6) and ( 10),δEw\nn,±1can also been obtained from\nδEw\nn,±1by reversing the charge sign σ, the spin orientation, and the signs of all CPT-odd coefficie nts\nfor Lorentz violation, as expected.\nWe remark in passing that the rotation properties of the coef ficients for Lorentz violation\nappearing in results ( 6) and ( 10) are represented by their indices. For example, the index pa ir “12\" on\nthe right sides of the definitions ( 7) and ( 11) is antisymmetric. This implies that these coefficients for\nLorentz violation transform like a single “3\" index under ro tations, while coefficients with an index\n“0\" or an index pair “00\" are invariant under rotations. The c ylindrical symmetry of the Penning\ntrap is correctly reflected by the fact that the results ( 6) and ( 10) only depend on index “0\", “3\", and\n“11+22\".\n2.3. Cyclotron and anomaly frequencies\nThe primary observables of interest in a Penning-trap exper iment are frequencies. Two key\nfrequencies are the cyclotron frequency νc≡ωc/2πand the Larmor spin-precession frequency νL≡\nωL/2π. The difference of the two frequencies gives the anomaly fre quency νL−νc=νa≡ωa/2π\n[35]. For a confined fermion of flavor win a Penning trap, the cyclotron and anomaly frequencies are\ndefined as the energy difference between the following energ y levels [ 20,21],\nωw\nc=Ew\n1,σ−Ew\n0,σ, ωw\na=Ew\n0,−σ−Ew\n1,σ. (13)\nFor an antifermion of flavor w, the corresponding definitions for the cyclotron and anomal y\nfrequencies are given by [ 20,21]\nωw\nc=Ew\n1,σ−Ew\n0,σ, ωw\na=Ew\n0,−σ−Ew\n1,σ, (14)Version September 17, 2021 submitted to Symmetry 6 of 20\nwith the understanding that the charge signs σin definitions ( 14) are reversed compared to these in\ndefinitions ( 13).\nIn a Lorentz-invariant scenario, the charge-to-mass ratio and the gfactor of a particle or an\nantiparticle confined in a Penning trap with a magnetic field s trength Bare related to the above\ncyclotron and anomaly frequencies by\n|q|\nm=ωc\nB, (15)\nand\ng\n2=ωL\nωc=1+ωa\nωc, (16)\nrespectively. For Penning-trap experiments comparing the charge-to-mass ratios or the gfactors\nbetween a particle of species wand its corresponding antiparticle w, the CPT theorem guarantees\nthat the following differences must be zero,\n(|q|/m)w\n(|q|/m)w−1=ωwc\nωwc−1=0, (17)\nand\n1\n2(gw−gw) =ωwa\nωwc−ωwa\nωwc=0, (18)\nwhere in Equation ( 17), for simplicity, we have assumed the same magnetic field is u sed in the\ncomparison. When different magnetic fields are used, the exp ression can be easily obtained by\nsubstituting ωw\nc/ωw\ncin the middle term of Equation ( 17) by(ωw\nc/Bw)(ωw\nc/Bw), where Bwand Bw\nare the strengths of the magnetic fields used to confine the par ticle and antiparticle, respectively.\nHowever, this assumption is not required to derive Equation (18) as the ratios ωw\na/ωw\ncand ωw\na/ωw\nc\ndon’t depend on the magnetic field used in the trap.\nHowever, in the presence of Lorentz and CPT violation, the pi cture changes dramatically as\nboth the cyclotron and anomaly frequencies for fermions and antifermions can be shifted due to the\nLorentz- and CPT-violating corrections to the energy level s,\nδωw\nc=δEw\n1,σ−δEw\n0,σ, δωw\na=δEw\n0,−σ−δEw\n1,σ,\nδωw\nc=δEw\n1,σ−δEw\n0,σ δωw\na=δEw\n0,−σ−δEw\n1,σ. (19)\nApplying energy shift results ( 6) and ( 10), the corrections to the cyclotron frequencies for a fermio n\nand antifermion are found to be [ 22]\nδωw\nc=/parenleftbigg1\nm2w/tildewideb′3\nw−1\nmw(/tildewidec00\nw+/tildewidec11\nw+/tildewidec22\nw)−(/tildewideb311\nw+/tildewideb322\nw)/parenrightbigg\neB,\nδωw\nc=/parenleftbigg\n−1\nm2w/tildewideb′∗3\nw−1\nmw(/tildewidec∗00\nw+/tildewidec∗11\nw+/tildewidec∗22\nw)+(/tildewideb∗311\nw+/tildewideb∗322\nw)/parenrightbigg\neB, (20)\nwhere the tilde and starred tilde coefficients are given by de finitions ( 7) and ( 11). The/tildewidecjj\nwand/tildewidec∗jj\nwtilde\ncoefficients with jtaking values of 1 or 2 are the sum of the two quantities given i n definitions ( 8)\nand ( 12),\n/tildewidecjj\nw= (/tildewidecjj\nw)s+(/tildewidecjj\nw)s¬\n=cjj\nw−2mwa(5)j0j\nw+3m2\nwc(6)j00j\nw−mwa(5)0jj\nw−mwm(5)jj\nw+3m2\nwc(6)00jj\nw+3m2\nwe(6)0jj\nw ,\n/tildewidec∗jj\nw= (/tildewidec∗jj\nw)s+(/tildewidec∗jj\nw)s¬\n=cjj\nw+2mwa(5)j0j\nw+3m2\nwc(6)j00j\nw+mwa(5)0jj\nw−mwm(5)jj\nw+3m2\nwc(6)00jj\nw−3m2\nwe(6)0jj\nw . (21)Version September 17, 2021 submitted to Symmetry 7 of 20\nFor the shifts in the anomaly frequencies, the results are fo und to be [ 21]\nδωw\na=2/tildewideb3\nw−2/tildewideb33\nF,wB,\nδωw\na=−2/tildewideb∗3\nw+2/tildewideb∗33\nF,wB, (22)\nwhere various tilde and starred tilde coefficients are given by definitions ( 7), (11), (8) and ( 12). We\nnote in passing that the shifts in the cyclotron and anomaly f requencies for an antifermion in results\n(20) and ( 22) reveals that they can be obtained from these for the fermion by changing the signs of all\nthe basic coefficients for Lorentz violation that control CP T-odd effects, as might be expected.\nResult ( 20) shows that the cyclotron frequency shifts due to Lorentz an d CPT violation for a\nfermion are different from these for its corresponding anti fermion. The same conclusion holds for\nthe anomaly frequency shifts from the result ( 22). This implies that in the presence of Lorentz and\nCPT violation, the differences ( 15) and ( 17) do not vanish in general. For the charge-to-mass ratio\ncomparisons, the result becomes\n(|q|/m)w\n(|q|/m)w−1←→ωw\nc\nωwc−1=δωw\nc−δωw\nc\nωwc, (23)\nwhere the Lorentz- and CPT-invariant pieces in the measured cyclotron frequencies are exactly\ncanceled by the CPT theorem if the same magnetic field is used. The notation←→ indicates\nthe correspondence between the experimental interpreted c harge-to-mass ratio comparison and the\nmeasured frequency difference ωwc/ωwc−1. For the gfactor comparison between a fermion and\nantifermion, the related ratio comes to be\n1\n2(gw−gw)←→ωw\na\nωwc−ωw\na\nωwc=δωw\na\nωwc−δωw\na\nωwc, (24)\nwhere again all Lorentz- and CPT-invariant contributions a re canceled out on the right side. We\nnote that in the above expression ( 24), we only keep shifts in the anomaly frequencies δωw\naand δωw\na\ndue to Lorentz and CPT violation, as contributions from the c yclotron frequencies δωwcand δωwcare\nsuppressed by factors of eB/m2\nw, as evident from results ( 20) and ( 22). Even for a comparatively large\nmagnetic field of B≈5 T in a Penning trap, these factors are at orders of eB/m2\ne≈10−9for electrons\nor positrons, and eB/m2\np≈10−16for protons or antiprotons, which can be ignored in Equation (24).\n2.4. Sidereal variations\nThe cyclotron and anomaly frequency shifts ( 20) and ( 22), which also appear in comparisons ( 23)\nand ( 24), are obtained in a particular apparatus frame xa≡(x1,x2,x3), where the positive ˆx3axis\nis aligned with the applied magnetic field in the trap. Howeve r, as Earth rotates about its axis, this\napparatus frame is not inertial. The standard canonical fra me that is adopted in the literature to\ncompare results from different experiments searching for L orentz and CPT violation is called the\nSun-centered frame XJ≡(X,Y,Z)[36,37]. In this frame, the Zaxis is defined to be aligned along\nthe Earth’s rotation axis, the Xaxis points from the Earth to the Sun, and the Yaxis completes a\nright-handed coordinate system. The time origin of this coo rdinate system is chosen to be at the\nvernal equinox 2000. The coefficients for Lorentz violation in this frame are assumed to be constants\nin time and space [ 4,8].\nTo explicitly express the relationship of the coefficients f or Lorentz violation between the\nSun-centered frame XJ≡(X,Y,Z)and the apparatus frame xa≡(x1,x2,x3), it is convenient to\nintroduce a third frame called the standard laboratory fram exj≡(x,y,z), with the zaxis pointing\nto the local zenith, the xaxis aligned with the local south, and the yaxis completing a right-handed\ncoordinate system. To relate the coordinates of these three frames, we define two rotation matrices RajVersion September 17, 2021 submitted to Symmetry 8 of 20\nand RjJ[36,37], with Rajconnecting xj≡(x,y,z)toxa≡(x1,x2,x3)byxa=Rajxj, and RjJrelating\nXJ≡(X,Y,Z)to(x,y,z)byxj=RjJXJ. The expressions for these two rotation matrices are given b y\nRjJ=\ncosχcosω⊕T⊕cosχsinω⊕T⊕−sinχ\n−sinω⊕T⊕ cosω⊕T⊕ 0\nsinχcosω⊕T⊕sinχsinω⊕T⊕ cosχ\n, (25)\nand\nRaj=\ncosγ sinγ0\n−sinγcosγ0\n0 0 1\n×\ncosβ0−sinβ\n0 1 0\nsinβ0 cos β\n×\ncosα sinα0\n−sinαcosα0\n0 0 1\n, (26)\nwhere ω⊕≈2π/(23 h 56 min )is the sidereal frequency of the Earth’s rotation, T⊕denotes the\nlocal sidereal time, χspecifies the colatitude of the laboratory, and (α,β,γ)are the Euler angles\nin the convenient “ y-convention\" of the rotation. The coordinates of the appara tus frame and the\nSun-centered frame can then be related by using the followin g expression,\nxa=Rajxj=RajRjJxJ. (27)\nThe relationship between the coefficients for Lorentz viola tion in these two frames can also be derived\nfrom this result. We note that in the case that a vertical upwa rd magnetic field is used in the trap, the\nEuler angles become (α,β,γ) = ( 0, 0, 0)and the rotation matrix Rajreduces to the identity matrix.\nThe above transformation ( 27) shows that in general the coefficients for Lorentz violatio n and\nthus the cyclotron and anomaly frequency shifts ( 20) and ( 22) determined in the apparatus frame\ndepend on the sidereal time and the geometric location of the laboratory. As a result, signals\nobserved in Earth-based experiments, including the above c omparisons ( 23) and ( 24), can oscillate\nat harmonics of the Earth’s sidereal frequency ω⊕, with the amplitudes depending on the laboratory\ncolatitude χ. To explicitly illustrate this, here we consider a Penning- trap experiment located\nat colatitude χand assume the magnetic field is aligned with the zaxis. We give two explicit\nexamples of the transformation of coefficients for Lorentz v iolation using Equation ( 27). We first\nfocus on the single-index laboratory-frame coefficient /tildewideb′3\nw, which appears in the shifts of the cyclotron\nfrequency ( 20). Applying the transformation ( 27) and taking (α,β,γ) = ( 0, 0, 0)imply\n/tildewideb′3\nw=cosω⊕T⊕/tildewideb′X\nwsinχ+sinω⊕T⊕/tildewideb′Y\nwsinχ+/tildewideb′Z\nwcosχ. (28)\nThis result relates the tilde coefficients /tildewideb′3\nwobserved in the noninertial apparatus frame to the constant\ncoefficients /tildewidebJ\nwwith J=X,Y,Zin the canonical inertial Sun-centered frame. The expressi on contains\nterms proportional to the first harmonic in the Earth’s sider eal frequency and a constant. The\ncolatitude dependence is evident from the factors sin χand cos χappearing above.\nA slightly more complicated result can also be obtained for t he sum of the two-index tilde\nquantities /tildewidec11\nw+/tildewidec22\nwin an analogous fashion. Applying the rotation ( 27) for each index with Rajbeing\nthe identity matrix yields\n/tildewidec11\nw+/tildewidec22\nw=cos 2 ω⊕T⊕/parenleftbig−1\n2(/tildewidecXX\nw−/tildewidecYY\nw)sin2χ/parenrightbig+sin 2 ω⊕T⊕(−/tildewidec(XY)\nw sin2χ)\n+cosω⊕T⊕(−/tildewidec(XZ)\nw sin 2 χ)+sinω⊕T⊕(−/tildewidec(YZ)\nw sin 2 χ)\n+1\n4(/tildewidecXX\nw+/tildewidecYY\nw)(3+cos 2 χ)+/tildewidecZZ\nwsin2χ, (29)\nwhere a parenthes on two indices of the coefficients indicate s symmetrization with a factor of 1/2. For\nexample, /tildewidec(XY)\nw= (/tildewidecXY\nw+/tildewidecYX\nw)/2. It is revealed from the result ( 29) that the sum of the tilde coefficients\n/tildewidec11\nw+/tildewidec22\nwin the apparatus frame depends on six independent quantitie s/tildewidec(JK)\nw with J,K=X,Y,ZinVersion September 17, 2021 submitted to Symmetry 9 of 20\nthe Sun-centered frame, producing up to second harmonics in the sidereal frequency of the Earth’s\nrotation. For both examples ( 28) and ( 29), if the applied magnetic field points along a generic direct ion,\ntrigonometric functions of the Euler angles ( α,β,γ) appear as well.\nAt the end of this subsection, we note that the revolution of t he Earth about the Sun can generate\nadditional types of time variations for the coefficients for Lorentz violation, such as the boost of the\nEarth relative to the Sun β⊕≈10−4, and the boost of the laboratory due to the Earth’s rotation\nβL≈10−6. However, as studied in the literature [ 27,28,38–40], such boost effects are suppressed by\none or more powers of their boost factors β⊕and βLcompared to these from rotations. Therefore, we\ntreat them as negligible effects in the present work.\n3. Experiments\nIn this section, we analyze existing Penning-trap experime nts that compare the charge-to-mass\nratios and the gfactors between protons, antiprotons, electrons, and posi trons. We use reported\nresults for the comparisons ( 23) and ( 24) from these experiments to constrain the relevant\nSun-centered frame tilde coefficients for Lorentz violatio n.\n3.1. The charge-to-mass ratios\nFor charge-to-mass ratio comparisons between a particle an d its corresponding antiparticle, the\nresult ( 23) implies that the relevant tilde coefficients for Lorentz vi olation are /tildewideb′3\nw,/tildewidec11\nw+/tildewidec22\nw,/tildewideb311\nw+/tildewideb322\nw,\n/tildewideb′∗3\nw,/tildewidec∗11\nw+/tildewidec∗22\nw, and/tildewideb∗311\nw+/tildewideb∗322\nwin the apparatus frame, given in the result ( 20). The transformations\nof these tilde coefficients into the Sun-centered frame depe nd on the field configuration in the\ntrap. For a typical Penning-trap experiment, the magnetic fi eld is oriented either horizontally or\nvertically. Ref. [ 22] presents a complete list of transformation results for the se tilde coefficients for\nthe above two field orientations. Here, for completeness, we include them in Appendix A. From\nthese transformation results, it shows that for a given ferm ion of species w, the relevant quantities\nin the Sun-centered frame that are related to the charge-to- mass comparisons in a Penning trap\nare the following 54 independent tilde coefficients, /tildewideb′J\nw,/tildewidec(JK)\nw,/tildewidebJ(KL)\nw ,/tildewideb′∗J\nw,/tildewidec∗(JK)\nw , and/tildewideb∗J(KL)\nw . In the\nfollowing subsections, we apply the reported precisions fo r the charge-to-mass ratio comparisons\nfrom Penning-trap experiments to set bounds on the relevant tilde coefficients for Lorentz violation.\n3.1.1. The proton sector\nWe start the analysis with the charge-to-mass comparisons b etween protons and antiprotons.\nIn a Penning-trap experiment located at CERN by the ATRAP col laboration, Gabrielse and his\ngroup achieved a precision of 90 ppt for the proton-antiprot on charge-to-mass ratio comparison [ 41].\nThe experiment used a trap with a vertical uniform magnetic fi eldB=5.85 T. Recently, another\nPenning-trap experiment at CERN by the BASE collaboration l ed by Ulmer improved the comparison\nto the record sensitivity of 69 ppt [ 42], by applying a horizontal magnetic field B=1.946 T which\nis oriented 60◦east of north. For the measurement of the charge-to-mass rat io of a proton, both\nexperiments used a trapped hydrogen ion (H−) as a proxy for the proton to eliminate systematic\nshifts caused by polarity switching of the trapping voltage s. The charge-to-mass ratio comparison\nbetween an antiproton and a proton is then related to that bet ween an antiproton and a hydrogen ion\nby\n(|q|/m)¯p\n(|q|/m)p−1=(|q|/m)¯p\nR(|q|/m)H−−1←→δω¯p\nc−RδωH−\nc\nRωH−\nc, (30)\nwhere R=mH−/mp=1.001089218754 is the ratio of the mass between a hydrogen io n\nand a proton [ 42], and ωH−\ncis the cyclotron frequency for the hydrogen ion, with δωH−\ncbeing\nthe corresponding shifts. To obtain δωH−\nc, one can take w=H−in the expression ( 20) and\nthe related tilde coefficients for Lorentz violation become the effective ones for a hydrogen ion.\nExpressing these effective coefficients in terms of the corr esponding fundamental coefficients forVersion September 17, 2021 submitted to Symmetry 10 of 20\nthe hydrogen ion constituents, which are the coefficients fo r electrons and protons, is challenging\ndue to nonperturbative issues including binding effects in the composite hydrogen ion. However,\nan approximation to these coefficient relations can be obtai ned by treating the wave function of the\nhydrogen ion as a product of the wave functions of a proton and two electrons. Applying perturbation\ntheory at the lowest order and ignoring the related binding e nergies, the cyclotron frequency shifts\nδωH−\ncof the hydrogen ion due to Lorentz and CPT violation can then b e approximated as the sum of\nthese for its constituents, δωH−\nc≈δωp\nc+2δωe−\nc. Substituting this into the result ( 30) yields\n(|q|/m)¯p\n(|q|/m)p−1←→δω¯p\nc−Rδωp\nc−2Rδωe−\nc\nRωH−\nc. (31)\nAs shown from the above result, the choice of using a hydrogen ion as a proxy for the proton in\nthe Penning trap provides sensitivities not only to the coef ficients for Lorentz violation in the proton\nsector, but also introduces additional sensitivities to th ese for electrons. Putting the above discussion\ntogether, the related Sun-centered frame tilde coefficient s for Lorentz violation that are sensitive to\nPenning-trap experiments comparing the charge-to-mass ra tios between protons and antiprotons are\nthe following 81 independent tilde quantities, /tildewideb′J\np,/tildewidec(JK)\np,/tildewidebJ(KL)\np ,/tildewideb′∗J\np,/tildewidec∗(JK)\np ,/tildewideb∗J(KL)\np ,/tildewideb′J\ne,/tildewidec(JK)\ne, and/tildewidebJ(KL)\ne .\nThe published results for the comparison ( 23) from both the ATRAP and the BASE experiments can\nbe adopted to set bounds on these tilde coefficients for Loren tz violation.\nFor the ATRAP experiment, the reported precision was obtain ed by analyzing the measurements\nof the cyclotron frequencies in a time-averaged way, so any o scillating effects in the difference ( 31)\naveraged out. This implies that only the constant terms that appear in the transformation results in the\nfirst half of Appendix Acan be constrained using the published precision. Applying expression ( 31)\nby taking the reported precision of 90 ppt for (|q|/m)¯p/(|q|/m)p−1 and identifying ωH−\nc=2π×89.3\nMHz given in the ATRAP experiment, the following limit can be obtained,\n|δω¯p\nc−1.001 δωp\nc−2.002 δωe−\nc|const∼<3.33×10−26GeV, (32)\nwhere the subscript “const\" indicates that only the constan t terms in the transformations results are\nrelevant to the limit. However, future sidereal-variation analysis of the measurements of the cyclotron\nfrequencies can provide constraints on the non-constant te rms that appear in the harmonics in the\ntransformation results.\nFor the experiment carried out by the BASE collaboration, si nce the magnetic field was oriented\nat 60◦east of north, both of the transformation matrices ( 25) and ( 26) are required to relate the\ntilde coefficients for Lorentz violation in the apparatus fr ame to these in the Sun-centered frame.\nFor a general horizontal magnetic field with an angle θmeasured from the local south in the\ncounterclockwise direction, the corresponding Euler angl es are found to be (α,β,γ) = ( θ,π/2, 0).\nThe second half of Appendix Alists the full expressions of the transformations for the re lated tilde\ncoefficients for Lorentz violation. The BASE experiment ana lyzed the data of the charge-to-mass ratio\ncomparisons to search for both time-averaged effects and si dereal variations in the first harmonics of\nthe Earth’s rotation frequency. Therefore, the reported re sults can be taken to set bounds on not only\nthe constant terms but also on the terms proportional to the fi rst harmonics in the transformation\nresults given in Appendix A. Using the reported 69 ppt for the time-averaged precision a nd 720 ppt\nfor the limit of the first harmonic amplitude for the comparis on (31), and taking ωH−\nc=2π×29.635\nMHz for the BASE experiment, the following limits are obtain ed,\n|δω¯p\nc−1.001 δωp\nc−2.002 δωe−\nc|const∼<8.46×10−27GeV (33)\nand\n|δω¯p\nc−1.001 δωp\nc−2.002 δωe−\nc|1st∼<8.83×10−26GeV, (34)Version September 17, 2021 submitted to Symmetry 11 of 20\nwhere the subscript “const\" in the limit ( 33) has similar meaning as the one in ( 32), while the subscript\n“1st\" in the limit ( 34) specifies only the amplitude of the first harmonics in the sid ereal variation.\nThe limit ( 32) for the ATRAP experiment and limits ( 33) and ( 34) for the BASE experiment set\nconstraints on a combination of the Sun-centered frame tild e coefficients /tildewideb′J\np,/tildewidec(JK)\np,/tildewidebJ(KL)\np ,/tildewideb′∗J\np,/tildewidec∗(JK)\np ,\n/tildewideb∗J(KL)\np ,/tildewideb′J\ne,/tildewidec(JK)\ne, and/tildewidebJ(KL)\ne . These constraints can be obtained by substituting result ( 20) in limit ( 32)\nand applying the corresponding transformations given in Ap pendix Awith χ=43.8◦for the ATRAP\nexperiment, and substituting result ( 20) in both limits ( 33) and ( 34) and identifying θ=2π/3 and\nχ=43.8◦in the transformation expressions in Appendix Afor the BASE experiment. To get some\nintuition about the scope of the constraints on the individu al component of the tilde coefficients,\nwe can take a common practice that is adopted in many subfields searching for Lorentz and CPT\nviolation [ 7] which assumes that only one individual tilde coefficient is nonzero at a time. From the\nATRAP limit ( 32), constraints on 27 independent tilde coefficients for Lore ntz violation are obtained,\nwhile from the BASE limits ( 33) and ( 34), a total of 69 independent tilde coefficients for Lorentz\nviolation are constrained. We summarize in Table 1and Table 2the constraints in the proton sector\nand the electron sector, respectively. In both tables, the fi rst column lists the individual components,\nthe second column presents the corresponding constraint on the modulus of the component, the third\ncolumn specifies the related experiment, and the related ref erence is given in the final column. We\nnote that some of the tilde coefficients for Lorentz violatio n are constrained by both the ATRAP and\nthe BASE experiments. To keep the results clean, we only keep the more stringent ones in both tables.\nTable 1. Constraints on tilde coefficients for Lorentz violation in t he proton sector using the\ncharge-to-mass ratio comparisons from the ATRAP and the BAS E experiments.\nCoefficient Constraint Experiment Reference\n|/tildewideb′Zp|,|/tildewideb′∗Zp| <1×10−10GeV ATRAP [ 41]\n|/tildewidecXXp|,|/tildewidec∗XXp| <1×10−10ATRAP [ 41]\n|/tildewidecYYp|,|/tildewidec∗YYp| <1×10−10ATRAP [ 41]\n|/tildewidecZZp|,|/tildewidec∗ZZp| <8×10−11BASE [ 42]\n|/tildewidebX(XZ)\np|,|/tildewideb∗X(XZ)\np|<2×10−10GeV−1BASE [ 42]\n|/tildewidebY(YZ)\np|,|/tildewideb∗Y(YZ)\np|<2×10−10GeV−1BASE [ 42]\n|/tildewidebZZZp|,|/tildewideb∗ZZZp|<2×10−10GeV−1BASE [ 42]\n|/tildewidebZXXp|,|/tildewideb∗ZXXp|<2×10−10GeV−1ATRAP [ 41]\n|/tildewidebZYYp|,|/tildewideb∗ZYYp|<2×10−10GeV−1ATRAP [ 41]\n|/tildewideb′Xp|,|/tildewideb′∗Xp| <7×10−10GeV BASE [ 42]\n|/tildewideb′Yp|,|/tildewideb′∗Yp| <7×10−10GeV BASE [ 42]\n|/tildewidec(XZ)\np|,|/tildewidec∗(XZ)\np|<1×10−9BASE [ 42]\n|/tildewidec(YZ)\np|,|/tildewidec∗(YZ)\np|<1×10−9BASE [ 42]\n|/tildewidebXXXp|,|/tildewideb∗XXXp|<2×10−9GeV−1BASE [ 42]\n|/tildewidebX(XY)\np|,|/tildewideb∗X(XY)\np|<2×10−9GeV−1BASE [ 42]\n|/tildewidebXYYp|,|/tildewideb∗XYYp|<1×10−9GeV−1BASE [ 42]\n|/tildewidebXZZp|,|/tildewideb∗XZZp|<9×10−10GeV−1BASE [ 42]\n|/tildewidebYXXp|,|/tildewideb∗YXXp|<1×10−9GeV−1BASE [ 42]\n|/tildewidebY(XY)\np|,|/tildewideb∗Y(XY)\np|<2×10−9GeV−1BASE [ 42]\n|/tildewidebYYYp|,|/tildewideb∗YYYp|<2×10−9GeV−1BASE [ 42]\n|/tildewidebYZZp|,|/tildewideb∗YZZp|<9×10−10GeV−1BASE [ 42]\n|/tildewidebZ(XZ)\np|,|/tildewideb∗Z(XZ)\np|<3×10−9GeV−1BASE [ 42]\n|/tildewidebZ(YZ)\np|,|/tildewideb∗Z(YZ)\np|<3×10−9GeV−1BASE [ 42]Version September 17, 2021 submitted to Symmetry 12 of 20\nTable 2. Constraints on tilde coefficients for Lorentz violation in t he electron sector using the\ncharge-to-mass ratio comparisons from the ATRAP and the BAS E experiments.\nCoefficient Constraint Experiment Reference\n|/tildewideb′Ze| <2×10−17GeV ATRAP [ 41]\n|/tildewidecXXe|<3×10−14ATRAP [ 41]\n|/tildewidecYYe| <3×10−14ATRAP [ 41]\n|/tildewidecZZe| <2×10−14BASE [ 42]\n|/tildewidebX(XZ)\ne|<1×10−10GeV−1BASE [ 42]\n|/tildewidebY(YZ)\ne|<1×10−10GeV−1BASE [ 42]\n|/tildewidebZZZe|<1×10−10GeV−1BASE [ 42]\n|/tildewidebZXXe|<9×10−11GeV−1ATRAP [ 41]\n|/tildewidebZYYe|<9×10−11GeV−1ATRAP [ 41]\n|/tildewideb′Xe| <1×10−16GeV BASE [ 42]\n|/tildewideb′Ye| <1×10−16GeV BASE [ 42]\n|/tildewidec(XZ)\ne|<3×10−13BASE [ 42]\n|/tildewidec(YZ)\ne|<3×10−13BASE [ 42]\n|/tildewidebXXXe|<1×10−9GeV−1BASE [ 42]\n|/tildewidebX(XY)\ne|<9×10−10GeV−1BASE [ 42]\n|/tildewidebXYYe|<5×10−10GeV−1BASE [ 42]\n|/tildewidebXZZe|<5×10−10GeV−1BASE [ 42]\n|/tildewidebYXXe|<5×10−10GeV−1BASE [ 42]\n|/tildewidebY(XY)\ne|<9×10−10GeV−1BASE [ 42]\n|/tildewidebYYYe|<1×10−9GeV−1BASE [ 42]\n|/tildewidebYZZe|<5×10−10GeV−1BASE [ 42]\n|/tildewidebZ(XZ)\ne|<2×10−9GeV−1BASE [ 42]\n|/tildewidebZ(YZ)\ne|<2×10−9GeV−1BASE [ 42]\n3.1.2. The electron sector\nFor the comparison of the charge-to-mass ratios between an e lectron and a positron, the current\nmost accurate result was made in an experiment at the Univers ity of Washington, with a precision at\n130 ppb [ 43]. From the comparison ( 23), this time-average result gives the following limit,\n|δωe−\nc−δωe+\nc|const∼<7.66×10−20GeV. (35)\nFollowing a similar analysis as the one used for the proton se ctor in the previous subsection, one can\nuse the colatitude and magnetic field that are relevant to thi s experiment χ=42.5◦and B=5.1 T\nupward, together with the transformation expressions give n in Appendix A, to obtain the constraints\non the tilde coefficients for Lorentz violation /tildewideb′J\ne,/tildewidec(JK)\ne,/tildewidebJ(KL)\ne ,/tildewideb′∗J\ne,/tildewidec∗(JK)\ne , and/tildewideb∗J(KL)\ne . We summarize\nthe results in Table 3, which is organized in the same way as Table 1and Table 2. We note that some of\nthe constraints of the tilde coefficients from the electron- positron charge-to-mass ratio comparison are\nnot comparable to these given in Table 2from the proton-antiproton comparison, so we don’t include\nthem in Table 3. We also note that the Penning trap experiment at the Univers ity of Washington used\na radioactive source of positrons that requires special pre cautions. Efforts in applying a safe source\nand ensuring efficient positron accumulation are currently being made at both Harvard University\nand Northwestern University [ 44,45], providing great potential for improving the current boun ds for\nthe tilde coefficients listed in Table 2and Table 3.Version September 17, 2021 submitted to Symmetry 13 of 20\nTable 3. Constraints on tilde coefficients for Lorentz violation in t he electron sector using the\ncharge-to-mass ratio comparisons from the experiment at th e University of Washington.\nCoefficient Constraint Experiment Reference\n|/tildewideb′∗Ze|<9×10−11GeV Washington [ 43]\n|/tildewidec∗XXe|<2×10−7Washington [ 43]\n|/tildewidec∗YYe|<2×10−7Washington [ 43]\n|/tildewidec∗ZZe|<3×10−7Washington [ 43]\n|/tildewideb∗X(XZ)\ne|<8×10−4GeV−1Washington [ 43]\n|/tildewideb∗Y(YZ)\ne|<8×10−4GeV−1Washington [ 43]\n|/tildewideb∗ZZZe|<8×10−4GeV−1Washington [ 43]\n|/tildewideb∗ZXXe|<4×10−4GeV−1Washington [ 43]\n|/tildewideb∗ZYYe|<4×10−4GeV−1Washington [ 43]\n3.2. The g factors and magnetic moments\nFor the gfactor and magnetic moment comparisons between particles a nd antiparticles using\nPenning traps, the result ( 24) implies that the relevant coefficients for Lorentz violati on in the\napparatus frame are these given in Equation ( 22),/tildewideb3\nw,/tildewideb∗3\nw,/tildewideb33\nF,w, and/tildewideb∗33\nF,w. For a Penning trap\nwith a vertical or horizontal magnetic field, the expression s of the transformation for these tilde\ncoefficients into the Sun-centered frame are also given in Ap pendix A. The results show that there\nare 18 independent components of the tilde coefficients for e ach fermion species w, given by /tildewidebJ\nw,/tildewideb∗J\nw,\n/tildewideb(JK)\nF,w, and/tildewideb∗(JK)\nF,w. In the subsection follows, we list the related Penning-tra p experiments and use\nthe reported comparison results to set bounds on the corresp onding tilde coefficients for Lorentz\nviolation.\n3.2.1. The proton sector\nIn the proton sector, the current best measurements of the ma gnetic moments for both a\nproton and an antiproton were achieved by the BASE collabora tion. The proton magnetic moment\nmeasurement has a sensitivity of 0.3 ppb using a Penning trap located at Mainz [ 46], improving their\nprevious best measurement [ 47] by a factor of 11. The antiproton magnetic moment measureme nt\nwas measured to a precision of 1.5 ppb with a similar Penning t rap located at CERN [ 48]. Combining\nthe reported 0.3 ppb and 1.5 ppb precisions for the time-aver aged measurements, and identifying\nωp\nc=2π×28.96 MHz and ωp\nc=2π×29.66 MHz for each experiment, comparison ( 24) yields\n|δωp\na−0.98δωp\na|const∼<9.53×10−25GeV, (36)\nwhere same subscript “const\" is used to specify only the cons tant terms in the transformation are\nrelevant to the above limit.\nThe corresponding constraints for the combinations of the t ilde coefficients in the Sun-centered\nframe can be obtained by applying the corresponding transfo rmation results given in Appendix A\nthat are related to the specific field configuration in the trap and substituting the numerical values\nof the laboratory quantities for both experiments in limit ( 36). For the proton magnetic moment\nmeasurement at Mainz, the colatitude is χ=40.0◦and the magnetic field B=1.9 T points θ=18◦\nfrom local south in the counterclockwise direction [ 46]. For the experiment measuring the antiproton\nmagnetic moment at CERN, the trap is located at a slight diffe rent colatitude χ=43.8◦and the\nmagnetic field B=1.95 T points θ=120◦in the same convention as above [ 48]. Adopting the same\nassumption that only one tilde coefficient is nonzero at a tim e, the constraint on each independent\ntilde coefficient can be obtained. We summarize them in Table 4in the same fashion as before.Version September 17, 2021 submitted to Symmetry 14 of 20\nTable 4. Constraints on tilde coefficients for Lorentz violation in t he proton sector using the gfactor\ncomparison from the BASE experiments at Mainz and CERN.\nCoefficient Constraint Experiment Reference\n|/tildewidebZp| <8×10−25GeV BASE [ 46,48]\n|/tildewideb∗Zp| <1×10−24GeV BASE [ 46,48]\n|/tildewidebXX\nF,p+/tildewidebYY\nF,p|<4×10−9GeV−1BASE [ 46,48]\n|/tildewidebZZ\nF,p| <3×10−9GeV−1BASE [ 46,48]\n|/tildewideb∗XX\nF,p+/tildewideb∗YY\nF,p|<3×10−9GeV−1BASE [ 46,48]\n|/tildewideb∗ZZ\nF,p| <1×10−8GeV−1BASE [ 46,48]\nAt the end of this subsection, we point out that a sidereal-va riation analysis of the anomaly\nfrequencies for both protons and antiprotons is currently b eing carried out by the BASE collaboration.\nThis could in principle set bounds on other components of the tilde coefficients /tildewidebJ\np,/tildewideb(JK)\nF,p,/tildewideb∗J\npand\n/tildewideb∗(JK)\nF,pthat haven’t been constrained before. The BASE collaborati on is also developing a quantum\nlogic readout system to allow more rapid measurements of the anomaly frequencies in the trap [ 49],\nwhich will offer excellent opportunities to perform a sider eal-variation analysis of the experimental\ndata, with great potential to set more stringent limits on th e tilde coefficients for Lorentz violation.\n3.2.2. The electron sector\nIn the electron sector, the comparison of the anomaly freque ncies between electrons and\npositrons were performed in a Penning-trap experiment at th e University of Washington, with a\nprecision of about 2 ppt [ 50]. The experimental data were analyzed in a time-averaged wa y to obtain\na constraint of b∼<50 rad/s using the notation given in Ref. [ 50]. Translating to the notation in this\nwork yields\n|δωe−\na−δωe+\na|const∼<2.09×10−23GeV. (37)\nTaking the experimental quantities χ=42.5◦and B=5.85 T upward, as well as the transformation\npresented in Appendix A, the limit ( 37) can be converted to constraints on the following Sun-cente red\nframe tilde coefficients, /tildewidebZ\ne,/tildewideb∗Z\ne,/tildewidebXX\nF,e+/tildewidebYY\nF,e,/tildewideb∗XX\nF,e+/tildewideb∗YY\nF,e,/tildewidebZZ\nF,e, and/tildewideb∗ZZ\nF,e.\nThe measurement of the gfactor for electrons has reached a record precision of 0.28 p pt at\nHarvard University [ 1]. A sidereal-variation analysis of the anomaly frequencie s was performed\nat the frequencies of ω⊕and 2 ω⊕, yielding the same limit on the amplitudes of both the first an d\nthe second harmonics in the oscillation, |δωea|1st/2nd∼<2π×0.05 Hz [ 24]. Converting the results in\nnatural units gives the following limits,\n|δωe−\na|1st∼<2.07×10−25GeV, (38)\nand\n|δωe−\na|2nd∼<2.07×10−25GeV. (39)\nTaking the magnetic field adopted in the experiment as B=5.36 T in the local upward direction\nwith the geometrical colatitude χ=47.6◦and applying the transformation results in Appendix A,\nconstraints on the following additional components of the t ilde coefficients, /tildewidebX\ne,/tildewidebY\ne,/tildewideb(XY)\nF,e,/tildewideb(XZ)\nF,e,/tildewideb(YZ)\nF,e,\nand/tildewidebXX\nF,e−/tildewidebYY\nF,eare obtained. The results from the University of Washington and Harvard University\nare summarized in Table 5.Version September 17, 2021 submitted to Symmetry 15 of 20\nTable 5. Constraints on tilde coefficients for Lorentz violation in t he electron sector using the gfactor\nmeasurements from the experiments at the University of Wash ington and Harvard University.\nCoefficient Constraint Experiment Reference\n|/tildewidebXe| <1×10−25GeV Harvard [ 24]\n|/tildewidebYe| <1×10−25GeV Harvard [ 24]\n|/tildewidebZe| <7×10−24GeV Washington [ 50]\n|/tildewideb∗Ze| <7×10−24GeV Washington [ 50]\n|/tildewidebXX\nF,e+/tildewidebYY\nF,e|<2×10−8GeV−1Washington [ 50]\n|/tildewidebZZ\nF,e| <8×10−9GeV−1Washington [ 50]\n|/tildewideb(XY)\nF,e| <2×10−10GeV−1Harvard [ 24]\n|/tildewideb(XZ)\nF,e| <1×10−10GeV−1Harvard [ 24]\n|/tildewideb(YZ)\nF,e| <1×10−10GeV−1Harvard [ 24]\n|/tildewideb∗XX\nF,e+/tildewideb∗YY\nF,e|<2×10−8GeV−1Washington [ 50]\n|/tildewideb∗XX\nF,e−/tildewideb∗YY\nF,e|<4×10−10GeV−1Harvard [ 24]\n|/tildewideb∗ZZ\nF,e| <8×10−9GeV−1Washington [ 50]\nThe measurement of the magnetic moment for positrons are cur rently under development at\nboth Harvard University and Northwestern University [ 44,45]. A comparison with the results for\nthat of electrons would offer opportunities to extract the c oefficients that control only CPT-odd effects\nfrom the combination of coefficients in the difference ( 24). A sidereal-variation analysis of the positron\nanomaly frequencies would offer limits on the starred tilde coefficients /tildewideb∗J\ne,/tildewideb∗(JK)\nF,eas well.\nAt the end of this subsection, we note that from Table 4and Table 5, for the 18 independent\ncomponents of the tilde coefficients for Lorentz violation t hat are relevant to the gfactor\nmeasurements in Penning-trap experiments, only 6 of them in the proton sector and 12 of them in\nthe electron sector have been constrained so far. Performin g a full sidereal-variation analysis for the\nmeasurements data would permit access to the other componen ts of the tilde coefficients.\n4. Summary\nIn this work, we provide an overview of recent progress on sea rching for Lorentz- and\nCPT-violating signals using measurements of charge-to-ma ss ratios and magnetic moments in\nPenning-trap experiments. We first revisit the theory of Lor entz-violating quantum electrodynamics\nwith operators of mass dimensions d≤6. The explicit expressions of the Lagrange density ( 1)\nare given in Ref. [ 21]. Perturbation theory is then applied to obtain the dominan t energy shifts\n(6) and ( 10) for a confine particle and antiparticle due to Lorentz and CP T violation. This leads\nto the corresponding contributions to the cyclotron and ano maly frequencies in Equations ( 20)\nand ( 22). The results are then used to relate the coefficients for Lor entz violation to the experimental\ninterpreted charge-to-mass ratio comparisons ( 23) between a particle and an antiparticle, as well as\nthe magnetic moment comparisons ( 24). The general transformation of the related coefficients fo r\nLorentz violation into different frames is performed using Equation ( 27). The explicit expressions\nrelating the coefficients in the apparatus frame to the Sun-c entered frame are given in Appendix A.\nThe results show that for the charge-to-mass ratio comparis ons between particles and antiparticles in\nPenning-trap experiments, the related coefficients for Lor entz violation in the Sun-centered frame are\n/tildewideb′J\nw,/tildewidec(JK)\nw,/tildewidebJ(KL)\nw ,/tildewideb′∗J\nw,/tildewidec∗(JK)\nw , and/tildewideb∗J(KL)\nw . For experiments involving magnetic moment comparisons,\nthe corresponding coefficients are /tildewidebJ\nw,/tildewideb∗J\nw,/tildewideb(JK)\nF,w, and/tildewideb∗(JK)\nF,w. Using published results from existing\nPenning-trap experiments, constraints on various compone nts of the coefficients for Lorentz violation\nare obtained. They are summarized in Tables 1-5. In conclusion, the high-precision measurementsVersion September 17, 2021 submitted to Symmetry 16 of 20\nand excellent coverage of the coefficients for Lorentz viola tion offered by Penning-trap experiments\nprovide strong motivations to continue the searches for pos sible Lorentz- and CPT-violating signals.\nAppendix A Transformations\nIn this Appendix, we list the explicit expressions of the tra nsformation results for the tilde\ncoefficients for Lorentz violation that are relevant to the c yclotron and anomaly frequency shifts.\nThese coefficients are /tildewideb′3w,/tildewidec11w+/tildewidec22w,/tildewideb311w+/tildewideb322w,/tildewideb3w,/tildewideb3\nF,w,/tildewideb′∗3w,/tildewidec∗11w+/tildewidec∗22w,/tildewideb∗311w+/tildewideb∗322w,/tildewideb∗3w, and/tildewideb∗3\nF,w\nin the apparatus frame.\nWe first consider Penning-trap experiments that use a vertic al upward magnetic field. Taking\nRajas the identity matrix and applying transformation ( 27) yield\n/tildewideb′3\nw=cosω⊕T⊕/tildewideb′X\nwsinχ+sinω⊕T⊕/tildewideb′Y\nwsinχ+/tildewideb′Z\nwcosχ, (A1)\n/tildewidec11\nw+/tildewidec22\nw=cos 2 ω⊕T⊕/parenleftig\n−1\n2(/tildewidecXX\nw−/tildewidecYY\nw)sin2χ/parenrightig\n+sin 2 ω⊕T⊕/parenleftig\n−/tildewidec(XY)\nw sin2χ/parenrightig\n+cosω⊕T⊕/parenleftig\n−/tildewidec(XZ)\nw sin 2 χ/parenrightig\n+sinω⊕T⊕/parenleftig\n−/tildewidec(YZ)\nw sin 2 χ/parenrightig\n+1\n4(/tildewidecXX\nw+/tildewidecYY\nw)(3+cos 2 χ)+/tildewidecZZ\nwsin2χ, (A2)\n/tildewideb311\nw+/tildewideb322\nw=cos 3 ω⊕T⊕/parenleftig\n[−1\n4(/tildewidebXXX\nw−/tildewidebXYY\nw)+1\n2/tildewidebY(XY)\nw]sin3χ/parenrightig\n+sin 3 ω⊕T⊕/parenleftig\n[−1\n2/tildewidebX(XY)\nw−1\n4(/tildewidebYXX\nw−/tildewidebYYY\nw)]sin3χ/parenrightig\n+cos 2 ω⊕T⊕/parenleftig\n[−/tildewidebX(XZ)\nw+/tildewidebY(YZ)\nw−1\n2(/tildewidebZXX\nw−/tildewidebZYY\nw)]cosχsin2χ/parenrightig\n+sin 2 ω⊕T⊕/parenleftig\n[−/tildewidebX(YZ)\nw−/tildewidebY(XZ)\nw−/tildewidebZ(XY)\nw]cosχsin2χ/parenrightig\n+cosω⊕T⊕/parenleftig\n1\n8/tildewidebXXX\nw(5+3 cos 2 χ)sinχ+1\n8/tildewidebXYY\nw(7+cos 2 χ)sinχ+/tildewidebXZZ\nw sin3χ\n−1\n2/tildewidebY(XY)\nw sin3χ−2/tildewidebZ(XZ)\nw cos2χsinχ/parenrightig\n+sinω⊕T⊕/parenleftig\n−1\n2/tildewidebX(XY)\nw sin3χ+1\n8/tildewidebYXX\nw(7+cos 2 χ)sinχ+1\n8/tildewidebYYY\nw(5+3 cos 2 χ)sinχ\n+/tildewidebYZZ\nwsin3χ−2/tildewidebZ(YZ)\nw cos2χsinχ/parenrightig\n−(/tildewidebX(XZ)\nw+/tildewidebY(YZ)\nw−/tildewidebZZZ\nw)cosχsin2χ+(/tildewidebZXX\nw+/tildewidebZYY\nw)cosχcos 2 χ, (A3)\n/tildewideb3\nw=cosω⊕T⊕/tildewidebX\nwsinχ+sinω⊕T⊕/tildewidebY\nwsinχ+/tildewidebZ\nwcosχ, (A4)\nand\n/tildewideb33\nF,w=cos 2 ω⊕T⊕/parenleftig\n1\n2(/tildewidebXX\nF,w−/tildewidebYY\nF,w)sin2χ/parenrightig\n+sin 2 ω⊕T⊕/tildewideb(XY)\nF,wsin2χ/parenrightig\n+cosω⊕T⊕/tildewideb(XZ)\nF,wsin 2 χ+sinω⊕T⊕/tildewideb(YZ)\nF,wsin 2 χ\n+1\n2(/tildewidebXX\nF,w+/tildewidebYY\nF,w−2/tildewidebZZ\nF,w)sin2χ+/tildewidebZZ\nF,w. (A5)Version September 17, 2021 submitted to Symmetry 17 of 20\nWe then switch to the case where a horizontal magnetic field is used in the trap, directing at\nan angle θfrom the local south in the counterclockwise direction. Thi s implies that the Euler angles\n(α,β,γ) = ( θ,π/2, 0). Applying the transformation ( 27) we have\n/tildewideb′3\nw=cosω⊕T⊕/parenleftig\n/tildewideb′X\nwcosθcosχ+/tildewideb′Y\nwsinθ/parenrightig\n+sinω⊕T⊕/parenleftig\n−/tildewideb′X\nwsinθ+/tildewideb′Y\nwcosθcosχ/parenrightig\n−/tildewideb′Z\nwcosθsinχ, (A6)\n/tildewidec11\nw+/tildewidec22\nw=cos 2 ω⊕T⊕/parenleftig\n1\n8(/tildewidecXX\nw−/tildewidecYY\nw)(1−3 cos 2 θ−2 cos2θcos 2 χ)−/tildewidec(XY)\nw cosχsin 2 θ/parenrightig\n+sin 2 ω⊕T⊕/parenleftig\n1\n2(/tildewidecXX\nw−/tildewidecYY\nw)cosχsin 2 θ+1\n4/tildewidec(XY)\nw(1−3 cos 2 θ−2 cos2θcos 2 χ)/parenrightig\n+cosω⊕T⊕/parenleftig\n/tildewidec(XZ)\nw cos2θsin 2 χ+/tildewidec(YZ)\nw sin 2 θsinχ/parenrightig\n+sinω⊕T⊕/parenleftig\n−/tildewidec(XZ)\nw sin 2 θsinχ+/tildewidec(YZ)\nw cos2θsin 2 χ/parenrightig\n+1\n2(/tildewidecXX\nw+/tildewidecYY\nw)(cos2θ+cos2χsin2θ+sin2χ)+/tildewidecZZ\nw(cos2χ+sin2θsin2χ), (A7)\n/tildewideb311\nw+/tildewideb322\nw\n=cos 3 ω⊕T⊕/parenleftig\n[1\n64(/tildewidebXXX\nw−/tildewidebXYY\nw)−1\n32/tildewidebY(XY)\nw][3(cosθ−5 cos 3 θ)cosχ−4 cos3θcos 3 χ]\n+[1\n16/tildewidebX(XY)\nw+1\n32(/tildewidebYXX\nw−/tildewidebYYY\nw)][3 sin θ(1−4 cos2θcos 2 χ)−5 sin 3 θ]/parenrightig\n+sin 3 ω⊕T⊕/parenleftig\n[−1\n32(/tildewidebXXX\nw−/tildewidebXYY\nw)+1\n16/tildewidebY(XY)\nw][3 sin θ(1−4 cos2θcos 2 χ)−5 sin 3 θ]\n+[1\n32/tildewidebX(XY)\nw+1\n64(/tildewidebYXX\nw−/tildewidebYYY\nw)][3(cosθ−5 cos 3 θ)cosχ−4 cos3θcos 3 χ]/parenrightig\n+cos 2 ω⊕T⊕/parenleftig\n[1\n16/tildewidebX(XZ)\nw−1\n16/tildewidebY(YZ)\nw+1\n32(/tildewidebZXX\nw−/tildewidebZYY\nw)][(5 cos 3 θ−cosθsinχ+4 cos3θsin 3 χ]\n+(/tildewidebX(YZ)\nw+/tildewidebY(XZ)\nw+/tildewidebZ(XY)\nw)sinθcos2θsin 2 χ/parenrightig\n+sin 2 ω⊕T⊕/parenleftig\n[−/tildewidebX(XZ)\nw+/tildewidebY(YZ)\nw−1\n2(/tildewidebZXX\nw−/tildewidebZYY\nw)]sinθcos2θsin 2 χ\n−(1\n16/tildewidebX(YZ)\nw+1\n16/tildewidebY(XZ)\nw+1\n16/tildewidebZ(XY)\nw)[(cosθ−5 cos 3 θ)sinχ−4 cos3θsin 3 χ]/parenrightig\n+cosω⊕T⊕/parenleftig\n1\n16/tildewidebXXX\nw cosθcosχ(−6 cos2θcos 2 χ+3 cos 2 θ+7)\n−1\n2/tildewidebX(XY)\nw sinθ(cos2θcos 2 χ+sin2θcos2χ+sin2χ)\n+1\n16/tildewidebXYY\nwcosθcosχ(−2 cos2θcos 2 χ+cos 2 θ+13)\n+/tildewidebXZZ\nw cosθcosχ(sin2θsin2χ+cos2χ)\n+1\n32/tildewidebYXX\nw[sinθ(25−4 cos2θcos 2 χ)+sin 3 θ]\n+1\n16/tildewidebY(XY)\nw cosθ[(cos 2 θ−7)cosχ−2 cos2θcos 3 χ]\n+1\n32/tildewidebYYY\nw[sinθ(11−12 cos2θcos 2 χ)+3 sin 3 θ]\n+/tildewidebYZZ\nwsinθ(sin2θsin2χ+cos2χ)\n−2/tildewidebZ(XZ)\nw cos3θsin2χcosχ−2/tildewidebZ(YZ)\nw sinθcos2θsin2χ/parenrightig\n+sinω⊕T⊕/parenleftig\n1\n32/tildewidebXXX\nw[sinθ(12 cos2θcos 2 χ−11)−3 sin 3 θ]\n+1\n16/tildewidebX(XY)\nw cosθ[(cos 2 θ−7)cosχ−2 cos2θcos 3 χ]\n+1\n32/tildewidebXYY\nw[sinθ(4 cos2θcos 2 χ−25)−sin 3 θ]Version September 17, 2021 submitted to Symmetry 18 of 20\n−/tildewidebXZZ\nw sinθ(sin2θsin2χ+cos2χ)\n+1\n16/tildewidebYXX\nw cosθcosχ(−2 cos2θcos 2 χ+cos 2 θ+13)\n+1\n2/tildewidebY(XY)\nw sinθ(cos2θcos 2 χ+sin2θcos2χ+sin2χ)\n+1\n16/tildewidebYYY\nwcosθcosχ(−6 cos2θcos 2 χ+3 cos 2 θ+7)\n+/tildewidebYZZ\nwcosθcosχ(sin2θsin2χ+cos2χ)\n+2/tildewidebZ(XZ)\nw sinθcos2θsin2χ−2/tildewidebZ(YZ)\nw cos3θsin2χcosχ/parenrightig\n+1\n8(/tildewidebX(XZ)\nw+/tildewidebY(YZ)\nw−/tildewidebZZZ\nw)cosθ(−4 cos 2 θsin3χ+5 sin χ+sin 3 χ)\n+1\n16(/tildewidebZXX\nw+/tildewidebZYY\nw)[2 cos3θsin 3 χ−cosθsinχ(3 cos 2 θ+11)], (A8)\n/tildewideb3\nw=cosω⊕T⊕/parenleftig\n/tildewidebX\nwcosθcosχ+/tildewidebY\nwsinθ/parenrightig\n+sinω⊕T⊕/parenleftig\n−/tildewidebX\nwsinθ+/tildewidebY\nwcosθcosχ/parenrightig\n−/tildewidebZ\nwcosθsinχ, (A9)\nand\n/tildewideb33\nF,w=cos 2 ω⊕T⊕/parenleftig\n1\n2(/tildewidebXX\nF,w−/tildewidebYY\nF,w)(cos2θcos2χ−sin2θ)+/tildewideb(XY)\nF,wsin 2 θcosχ/parenrightig\n+sin 2 ω⊕T⊕/parenleftig\n1\n4/tildewideb(XY)\nF,w(2 cos2θcos 2 χ+3 cos 2 θ−1)−(/tildewidebXX\nF,w−/tildewidebYY\nF,w)sin 2 θcosχ/parenrightig\n+cosω⊕T⊕/parenleftig\n−2/tildewideb(XZ)\nF,wcos2θcosχ−/tildewideb(YZ)\nF,wsin 2 θsinχ/parenrightig\n+sinω⊕T⊕/parenleftig\n/tildewideb(XZ)\nF,wsin 2 θsinχ−2/tildewideb(YZ)\nF,wcos2θcosχ/parenrightig\n+1\n2(/tildewidebXX\nF,w+/tildewidebYY\nF,w)(cos2θcos2χ+sin2θ)+/tildewidebZZ\nF,wcos2θsin2χ. (A10)\nThe corresponding transformations for the starred tilde qu antities/tildewideb′∗3\nw,/tildewidec∗11\nw+/tildewidec∗22\nw,/tildewideb∗311\nw+/tildewideb∗322\nw,\n/tildewideb∗3w, and/tildewideb33\nF,wtake the same forms as these given above, with the substituti ons of the usual tilde\ncoefficients by the starred ones.\nFunding: This work was supported in part by the Gettysburg College Res earch and Professional Development\nGrant, the Gettysburg College Cross-Disciplinary Science Institute (X-SIG), and the Keck Grant from the W.M.\nKeck Science Department at Claremont McKenna, Pitzer, and S cripps Colleges.\nAcknowledgments: The authors would like to thank Matthew Mewes for the invitat ion and Lisa Portmess for\nreading the manuscript.\nConflicts of Interest: The authors declare no conflict of interest.\nReferences\n1. Hanneke, D.; Fogwell, S.; Gabrielse, G. New measurement o f the electron magnetic moment and the fine\nstructure constant. Phys. Rev. Lett. 2008 ,100, 120801.\n2. 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Towards sympathet ic laser cooling and detection of single\n(anti-)protons. in Kostelecký, V .A. ed., Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry, World\nScientific, Singapore, 2017.\n50. Dehmelt, H.G.; Mittleman, R.K.; Van Dyck R.S.; Schwinbe rg, P . Past electron-positron g−2 experiments\nyielded sharpest bound on CPT violation for point particles .Phys. Rev. Lett. 1999 ,83, 4694.\n© 2021 by the authors. Submitted to Symmetry for possible open access publication\nunder the terms and conditions of the Creative Commons Attri bution (CC BY) license\n(http://creativecommons.org/licenses/by/4.0/)."    },    {        "title": "2106.11782v3.Sharp_decay_rate_for_the_damped_wave_equation_with_convex_shaped_damping.pdf",        "content": "arXiv:2106.11782v3  [math.AP]  5 Jan 2022SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH\nCONVEX-SHAPED DAMPING\nCHENMIN SUN\nAbstract. We revisit the damped wave equation on two-dimensional toru s where the damped region\ndoes not satisfy the geometric control condition. It was sho wn in [AL14] that, for sufficiently regular\ndamping, the damped wave equation is stale at a rate sufficient ly close to t−1. We show that if the\ndamping vanishes like a H¨ older function |x|β, and in addition, the boundary of the damped region is\nlocally strictly convex with positive curvature, the wave i s stable at rate t−1+2\n2β+7, which is better\nthan the known optimal decay rate t−1+1\nβ+3for strip-shaped dampings of the same H¨ older regularity.\nMoreover, we show by example that the decay rate is optimal. T his illustrates the fact that the sharp\nenergy decay rate depends not only on the order of vanishing o f the damping, but also on the shape of\nthe damped region. The main ingredient of the proof is the ave raging method (normal form reduction)\ndeveloped by Hitrik and Sj¨ ostrand ([ Hi1][Sj]).\n1.Introduction\n1.1.Background. Let(M,g)beacompact RiemannianmanifoldwiththeBeltrami-Laplac e operator\n∆. Consider the damped wave equation\n/braceleftBigg\n∂2\ntu−∆u+a(z)∂tu= 0,inR+×M,\n(u,∂tu)|t=0= (u0,u1),inM,(1.1)\nwherea(z)≥0 is the damping. The well-posedness of ( 1.1) is a consequence of the Lumer-Philips\ntheorem and the maximal dissipative property of the generat or\nL=/parenleftbigg0 Id\n∆−a(z)/parenrightbigg\n(1.2)\non the Hilbert space H:=H1(M)×L2(M). For a solution ( u,∂tu)∈H1(M)×L2(M), the energy\ndefined by\nE[u](t) :=1\n2/ba∇dbl∇u(t)/ba∇dbl2\nL2(M)+1\n2/ba∇dbl∂tu(t)/ba∇dbl2\nL2(M)\nis decreasing in time:\nd\ndtE[u](t) =−/integraldisplay\nMa|∂tu|2≤0.\nA basic question is the decay rate of the energy as t→+∞.\nIt was proved by Rauch-Taylor [ RaT] (∂M=∅) and by Bardos-Lebeau-Rauch [ BLR] (∂M/\\e}atio\\slash=∅)\nthat, for continuous damping a∈C(M), if the set ω={a>0}verifies the geometric control condition\n(GCC), then there exists α0>0 such that the uniform stabilization holds:\nE[u](t)≤E[u](0)e−α0t,∀t≥0. (1.3)\n12 CHENMIN SUN\nIf (GCC) for ω={a >0}is not satisfied, there are very few cases that the uniform sta bilization\n(1.3) holds (see [ BG17] and [Zh])1. Lebeau [ Le93] constructed examples with arbitrary slowly decaying\ninitial data in the energy space H1(M)×L2(M). Nevertheless, if the initial data is more regular,\nsay inH2(M)×H1(M), the uniform decay rate 1 /log(1 +t) holds ([ Le93]). Since then, intensive\nresearch activities focus on possible improvement of the lo garithmic decay rate for regular initial data,\nin special geometric settings.\nBeyond(GCC),thedeterminationofbetterdecay rateforspe cial manifolds Mandspecialdampings\ndepends on at least the following factors from the existing l iterature:\n(a) The dynamical properties for the geodesic flow of the unde rlying manifold M.\n(b) The dimension of the trapped rays as well as relative posi tions between trapped rays and the\nboundary∂{a>0}of the damped region.\n(c) Regularity and the vanishing properties of the damping anear∂{a>0}.\nIt is known that the energy decay rate is linked to the average d function along the geodesic flow ϕt:\nρ/ma√sto→ AT(a)(ρ) :=1\nT/integraldisplayT\n0a◦ϕt(ρ)dt, ρ∈T∗M.\nIndeed, (GCC) is equivalent to the lower bound AT(ρ)≥c0>0 for some T >0 large enough on the\nsphere bundle S∗M. Roughly speaking, when the geodesic flow is “unstable”, one may improve the\nenergy decay rate (see [ No] for more detailed explanation and references therein). As an illustration\nof (a), when Mis a compact hyperbolic surface, Jin [ Ji] shows the exponential energy decay rate for\nregular data living in H2(M)×H1(M). In this direction, we refer also [ BuC],[Ch],[CSVW],[Riv] and\nreferences therein.\nThepolynomial decay rate is theintermediate situation bet ween thelogarithmic decay rates andthe\nexponential decay rates, exhibitedinless chaotic geometr y liketheflattorusandboundeddomains(see\n[LiR][BH05][Ph07][AL14] and references therein), where the generalized geodesic fl ows are unstable.\nWe refer [ LLe],[BZu] for other situations of polynomial stabilization, where t he undamped region is a\nsubmanifold.\nWe point out that the factor (a) is almost decisive for the obs ervability (and exact controllability)\nof wave and Schr¨ odinger equations. Comparing with the obse rvability for the wave equation where\n(GCC) is the only criteria (see [ BLR] [BG97]), the stabilization problem is more complicated. Indeed,\nit was shown in [ AL14] (Theorem 2.3) that the observability for the Schr¨ odinger semigroup in some\ntimeT >0 implies automatically that the dampedwave is stable at rat et−1\n2. However, this decay rate\nis not optimal in general. On the two-dimensional torus, if t he damping function is regular enough\nand vanishing nicely, the decay rate can be very close to t−1( [BH05][AL14]). Even when the damping\nis the indicator of a vertical (or horizontal) strip, the opt imal decay rate is known to be t−2\n3(the lower\nbound was obtained by Nonnenmacher in [ AL14] and the upper bound was obtained in [ St]). These\nresults provide evidences of factors (b) and (c) mentioned p reviously. As explained in [ AL14], the\nsignificant difference to the controllability problem is that , there is no general monotonicity property\n1If the trapped rays are all grazing on the boundary ∂{a >0}, the uniform stabilization ( 1.3) can be characterized\n(see [BG17] and [Zh]) by relative positions of the grazing trapped rays and a >0.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 3\nof the type: a1≤a2implies the decay rate associated to a2is better (or worse) than the decay rate\nassociated to a1.\nIn this article, we revisit the polynomial stabilization fo r wave equations on flat torus. Our main\nresult reveals that, with the same vanishingorder, the curv ature of the boundaryof the damped region\nalso affects the energy decay rate of damped wave equations.\n1.2.The main result. We concern the polynomial decay rate for ( 1.1) on the two-dimensional flat\ntorusM=T2:=R2/(2πZ)2:\n/braceleftBigg\n∂2\ntu−∆u+a(z)∂tu= 0,inR+×T2,\n(u,∂tu)|t=0= (u0,u1),inT2.(1.4)\nTo present the main result, we introduce some definitions.\nDefinition 1.1. We say that (1.4)isstable at rate t−α, if there exists C >0, such that all the\nsolutionuwith initial data (u0,u1)∈ H2:=H2(T2)×H1(T2)satisfies\n(E[u](t))1\n2≤Ct−α/ba∇dbl(u0,u1)/ba∇dblH2.\nWe say that the rate t−αis optimal, if moreover\nlimsup\nt→+∞tαsup\n0/negationslash=(u0,u1)∈H2(E[u](t))1\n2\n/ba∇dbl(u0,u1)/ba∇dblH2>0.\nNext we introduce the class of damping that we will consider:\nDefinition 1.2. Letm,k∈N,σ>0andkσ<1. The function class Dm,k,σ(Td)is defined by:\nDm,k,σ:={f∈Cm(Td) :|∂αf|/lessorsimilarα,σ|f|1−|α|σ,∀|α| ≤k}.\nNote that Dm,k,σ1⊂ Dm,k,σ2, ifσ1< σ2andkσ2<1. This class contains non-negative functions\nwhich vanish like H¨ older functions. One typical example is\na1(z) =b(z)(max{0,0.1−|z|})1\nσ∈ Dm,m,σ(Td),\nwhereσ <1\nm,b∈Cm(Td) and infTdb>0. The associated damped region is {z∈Td:a1(z)>0}=\n{z∈Td:|z|<0.1}is a disc. Another example is the strip-shaped damping a2(z) =a2(x) such that\n{z∈T2:a2(z)>0}:= (−0.1,0.1)x×Tyand for some m≥4,\ndm\ndxma2(x)≤0 nearx= 0.1 anddm\ndxma2(x)≥0 nearx=−0.1.\nIt was shown in Lemma 3.1 of [ BH05] thata2∈ Dm,m,1\nm(T2).\nFora∈ Dm,k,σ, we denote by Σ a:=∂{a(z)>0}. LetT2\nA,B:=R2/(2πA×2πB) be a general flat\ntorus defined via the covering map πA,B:R2/ma√sto→T2\nA,B.\nDefinition 1.3. An open set ω⊂T2\nA,Bis said to be locally strictly convex with positive curvature , if\nthe boundary of each component of π−1\nA,B(ω)⊂R2isC2and has strictly positive curvature, as a closed\ncurve in R2. Sometimes, we also say that the boundary is locally strictl y convex.\nOur main result is the following:4 CHENMIN SUN\nTheorem 1.1. Letβ >4,m≥10. Assume that a≥0, a∈ Dm,2,1\nβand the open set ω:={z∈T2:\na(z)>0}is locally strictly convex with positive curvature. Assume thata(z)is locally H¨ older of order\nβnear∂ω, in the sense that there exists R0>1, such that\n1\nR0dist(z,∂ω)β≤a(z)≤R0dist(z,∂ω)β,forz∈ωnear∂ω.\nThen the damped wave equation (1.4)is stable at rate t−1+2\n2β+7.Moreover, the decay rate t−1+2\n2β+7is\noptimal in the following sense: there exists a damping a0(z), satisfying all the hypothesis above with\nβ=m≥10, such that the associated damped wave equation (1.4)cannot be stable at rate t−1+2\n2β+7−ǫ,\nfor anyǫ>0.\nRemark 1.4. As a comparison, if a(z) =a(x) depends only on one direction (supported on the\nvertical strip ω) and is locally H¨ older of order βnear∂ω, the optimal stable rate is t−1+1\nβ+3(see\n[Kl][DKl]) which is worse than t−1+2\n2β+7. Our result provides examples that, with the same local\nH¨ older regularity, smaller damped regions better stabili ze the wave equation. To the best knowledge\nof the author, Theorem 1.1also provides the first example where not only the vanishing o rder of the\ndamping can affect the stable rate, but also the shape of the bou ndary of the damped region.\nω2\nω1\na1(z) = (0.1−|z|)β\n+, a2(z) = (0.5−|x|)β\n+T2=R2/(2πZ)2\nThe damping a1generates better decay rate than a2\nRemark 1.5. In the case of rectangular H¨ older regular damping a(z) =a(x), the optimal decay\nrate is shown to be t−1+1\nβ+3([Kl][DKl]) for allβ≥0. In our result, the regularity assumptions\nβ >4,m≥10 in the first part and β≥10 in the second part of Theorem 1.1might be relaxed, as\nfar as tools of microlocal analysis can still be applied (for example, the paradifferential calculus for\nlow-regularity symbols). We leave open the problem to deter mine whether the optimal decay rate\nt−1+2\n2β+7in Theorem 1.1can be extended to rough dampings (i.e. small β≥0), as well as the case\nwherea(z) is an indicator function for a strictly convex subset.\nRemark 1.6. Itwas shown in [ AL14] that, whenthe dampingsatisfies |∇a| ≤Ca1−1\nβfor large enough\nβ, then (1.4) is stable at rate t−1+4\nβ+4(Theorem 2.6 of [ AL14]). With an additional assumption on\n|∇2a| ≤Ca1−2\nβ, our proof of Theorem 1.1essentially provides an alternative proof of this rougher\nstable rate. Indeed, we need one more condition for |∇2a|only to perform the normal form reduction\nin the Section 5. Once reduced to the one-dimensional setting, we are able to apply the same argument\nof Burq-Hitrik [ BH05].\nRemark 1.7. For the reason of exhibition, we have used a contradiction ar gument and the notion of\nsemiclassical defect measures in the proof of Theorem 1.1. Comparing to the argument of [ AL14], we\ndo not make use of second semiclassical measures.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 5\nFinally we give some microlocal interpretation of Theorem 1.1. It is known that the decay rate of\nthe damped wave equation is related to the time average along geodesics (see [ No]). As the damping\nhas conormal singularities at the boundary ∂ωof the damped region, the decay rate depends more\nprecisely on the reflected and transmitted energy of waves co ncentrated on trapped rays. If along\nthe conormal direction, the damping is more regular, its int eraction with transversal free waves is\nweaker (analog to the high-low frequency interaction), hen ce the transmission effect is stronger than\nthe reflection, and consequently, the decay rate is better. W hen the boundary ∂ωof the damped\nregion is convex, the average of the damping along any direct ion gains1\n2local H¨ older regularity, near\nthe vanishing points along the transversal direction (see P roposition 4.4for details). This heuristic\nindicates that, with the same local H¨ older regularity near the boundary (of the damped region), the\nconvex-shaped dampinghasbetterstablerate for( 1.4), thanstrip-shapeddampings(that areinvariant\nalong one direction).\n1.3.Resolvent estimate. The proof of Theorem 1.1relies on Borichev-Tomilov’s criteria of the\npolynomial semi-group decay rate and the corresponding res olvent estimate for Lgiven in ( 1.2):\nProposition 1.8 ([BoT10]).We have Spec (L)∩iR=∅. Then, the following statements are equivalent:\n(a)/vextenddouble/vextenddouble(iλ−L)−1/vextenddouble/vextenddouble\nL(H)≤C|λ|1\nαfor allλ∈R,|λ| ≥1;\n(b)The damped wave equation (1.1)is stable at rate t−α.\nBy Proposition 1.8and Proposition 2.4 of [ AL14], the proof of Theorem ( 1.1) can be reduced to\nthe following semiclassical resolvent estimate:\nTheorem 1.2. Letβ >4,m≥10. Assume that a≥0, a∈ Dm,2,1\nβand the open set ω:={z∈T2:\na(z)>0}is locally strictly convex with positive curvature. Assume thata(z)is locally H¨ older of order\nβnear∂ω, in the sense that there exists R0>1, such that\n1\nR0dist(z,∂ω)β≤a(z)≤R0dist(z,∂ω)β,forz∈ωnear∂ω.\nThen there exist h0∈(0,1)andC0>0, such that for all 0<h≤h0,\n/ba∇dbl(−h2∆−1+iha(z))−1/ba∇dblL(L2(T2))≤C0h−2−2\n2β+5.\nMoreover, there exists a damping function a0(z), satisfying the hypothesis above with β=m≥10,\nsuch that\n/ba∇dbl(−h2∆−1+iha0(z))−1/ba∇dblL(L2(T2))≥Ch−2−2\n2β+5.\nAs a comparison, let us recall themain resolvent estimate in [DKl][Kl], correspondingto the optimal\nenergy decay rate t−1+1\nβ+3when the damping a(z) depends only on xvariable and is locally H¨ older\nof orderγ:\nTheorem 1.3 ([DKl][Kl]).Letγ≥0. Assume that W=W(x)≥0and{W >0}is disjoint unions\nof vertical strips (αj,βj)x×Tyand{W≥0} /\\e}atio\\slash=T2. Assume moreover that for each j∈ {1,···,l},\nC1Vj(x)≤W(x)≤C2Vj(x)on(αj,βj),6 CHENMIN SUN\nwhereVj(x)>0are continuous functions on (αj,βj), satisfying\nVj(x) =\n\n(x−αj)γ, αj<x<3αj+βj\n4,\n(βj−x)γ,αj+3βj\n4<x<βj.(1.5)\nThen there exist h0∈(0,1)andC >0such that for all 0<h<h 0,\n/ba∇dbl(−h2∆−1+ihW(x))−1/ba∇dblL(L2)≤Ch−2−1\nγ+2.\nFurthermore, the above resolvent estimate is optimal, in th e sense that there exists quasi-modes\n(uh)0<h≤h0, such that\n/ba∇dbluh/ba∇dblL2= 1,/ba∇dbl(−h2∆−1+ihW(x))uh/ba∇dblL2(T2)=O(h2+1\nγ+2).\nIntherestof thearticle, wewill proveTheorem 1.2. Theproofis basedonacontradiction argument.\nThis leads to the fact that the semiclassical measure associ ated to quasi-modes ( uh)h>0of order\no(h2+2\n2β+5) is non-zero along finitely many closed trajectories with pe riodic directions on the phase\nspace. On the other hand, it turns out that the restriction of semiclassical measure to any periodic\ndirection is zero, which leads to a contradiction. This anal ysis follows from a second microlocalization\nprocedure and will be achieved in three major steps:\n•In Section 3, using the positive commutator method, we show that the semi classical measure\ncorresponding to the transversal high frequency part of sca le/greaterorsimilarh−1\n2−1\n2β+5is zero.\n•In Section 5, we deal with scales for transversal low frequencies. Using the averaging method,\nwe transfer quasi-modes ( uh)h>0to new quasi-modes ( vh)h>0, satisfying new equations that\ncommute with the vertical derivative. This allows us to redu ce the problem to the one-\ndimensional setting. This is the key part of the proof , for which we need several elementary\nproperties of the averaging operator, presented in Section 4.\n•In Section 6, we prove the reduced one-dimensional resolvent estimate ( Proposition 6.1).\nFurthermore, we prove in Section 7the lower bound in Theorem 1.2. At the end of this article, we add\ntwo appendices. In Appendix A, we reproduce the proof of Theo rem1.3in order to be self-contained\nand to fix some gaps in the paper of [ DKl]. In Appendix B, we review several technical results about\nthe semiclassical pseudo-differential calculus, needed in S ection5.\nAcknowledgment. The author is supported by the program: “Initiative d’excel lence Paris Seine”\nof CY Cergy-Paris Universit´ e and the ANR grant ODA (ANR-18- CE40- 0020-01). The author would\nlike to thank anonymous referee’s suggestions that help to i mprove this article.\n2.Contradiction argument and the first microlocalization\n2.1.A priori estimate and the contradiction argument. We will adapt basic conventions for\nnotations in the semiclassical analysis ([ Zw12]). Denote by Ph=−h2∆−1 and we fix the parameter\nσ=1\nβthroughout this article. We denote by δha small parameter such that δh→0 ash→0. We\ndenoteby /planckover2pi1=h1\n2δ1\n2\nhasecond semiclassical parameter. For theproofof Theorem 1.2, we fixδh=h2\n2β+5.\nTheorem 1.2is the consequence of the following key proposition:SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 7\nProposition 2.1. Letuhbe a sequence of quasi-modes of width h2δhwithδh=h2\n2β+5i.e.\n(Ph+iha)uh=fh=oL2(h2δh).\nThen ifuh=OL2(1), we haveuh=oL2(1).\nThe proof of Proposition 2.1will occupy the rest of this article. We argue by contradicti on. First\nwe delete all the zero elements in a given sequence of uh. Then, up to extracting a subsequence and\nrenormalization, we may assume that\n/ba∇dbluh/ba∇dblL2(T2)= 1. (2.1)\nThe following a priori estimate is simple:\nLemma 2.2. We have the following a priori estimates:\n(a)/ba∇dbla1/2uh/ba∇dblL2=o(h1\n2δ1\n2\nh) =o(/planckover2pi1).\n(b)/ba∇dblh∇uh/ba∇dbl2\nL2−/ba∇dbluh/ba∇dbl2\nL2=o(h2δh).\nProof.Multiplying the equation ( Ph+iha)uh=fhbyuh, and integrating by part, we get\n/ba∇dblh∇uh/ba∇dbl2\nL2−/ba∇dbluh/ba∇dbl2\nL2+ih(auh,uh)L2= (fh,uh)L2.\nTaking the imaginary part and the real part, we obtain (a) and (b), with respectively. /square\nSince the sequence ( uh)h>0is bounded L2(T2), there exist a subsequence, still denoted as ( uh)h>0,\nand a Radon measure µonT∗T2, such that for any symbol a∈C∞\nc(T∗T2), there holds\nlim\nh→0(Oph(a)uh,uh)L2=/a\\}b∇acketle{tµ,a/a\\}b∇acket∇i}ht. (2.2)\nBelow, we will denote µthe semiclassical defect measure associated to this subseq uence (uh)h>0. For\nthe proof of this existence of semi-classical measure, one m ay consult Chapter 5 of [ Zw12].\nLemma 2.3. we have\nsupp(µ){(z,ζ)∈T∗T2:|ζ|= 1}andµ|ω×R2= 0,\nwhereω={z∈T2:a>0}.\nProof.This property follows from the standard elliptic regularit y which only requires quasi-mode for\nPhof orderOL2(h). The damping term ihauhcan be roughly treated as an error OL2(h). For example,\none can consult Theorem 5.4 of [ Zw12] for a proof. /square\nLetϕtbe the geodesic flow on T∗T2. We recall the following invariant property of the semiclas sical\nmeasure:\nLemma 2.4. The semiclassical measure µis invariant by the flow ϕt, i.e.\nϕ∗\ntµ=µ.\nProof.This property holds true for quasi-mode of Phof orderoL2(h). From (a) of Lemma 2.2, we\nhavefh−ihauh=oL2(h). The proof then follows from a standard propagation argume nt (see for\nexample Theorem 5.5 of [ Zw12]). /square8 CHENMIN SUN\n2.2.Reducing to periodic trapped directions. Inordertoperformthefineranalysisneartrapped\nrays, we need to do a change of coordinate, following [ BZ12] and [AL14]. The spirit of the second-\nmicrolocalization here is also close to the work [ AM14].\nBy identifying T2=R2/(2πZ)2, we decompose S1as rational directions\nQ:={ζ∈S1:ζ=(p,q)/radicalbig\np2+q2,(p,q)∈Z2,gcd(p,q) = 1}\nand irrational directions R:=S1\\Q. Since the orbit of an irrational direction is dense, by Lemm a\n2.3and Lemma 2.4, we have\nµ=µ|T2×Q=/summationdisplay\nζ0∈Qµζ0.\nIt suffices to show that, for each ζ0=(p0,q0)√\np2\n0+q2\n0∈ Q, the restricted measure µζ0is zero2. Denote by Λ 0,\nthe rank 1 submodule of Z2generated by Ξ 0= (p0,q0). Denote by\nΛ⊥\n0:={ζ∈R2:ζ·Ξ0= 0}\nthe dual of the submodule Λ 0. Denote by\nT2\nΞ0:= (RΛ0/(2πΛ0))×(Λ⊥\n0/(2πZ)2∩Λ⊥\n0).\nThen we have a natural smooth covering map πΞ0:T2\nΞ0/ma√sto→T2of degreep2\n0+q2\n0. The pullback of a\n2π×2πperiodic function fsatisfies\n(π∗\nΞ0f)(X+kτ,Y+lτ) = (π∗\nΞ0f)(X,Y), k,l∈Z,(X,Y)∈R2,\nwhereτ= 2π/radicalbig\np2\n0+q2\n0.\nT2\nΞ0\nΞ0= (3,−2)\nΛ0Λ⊥\n0\nBy pulling back to the torus T2\nΞ0, we can identify the sequence ( uh)⊂L2(T2) as (π∗\nΞ0uh)⊂L2(T2\nΞ0)\nand in this new coordinate system, ζ0=Ξ0\n|Ξ0|= (0,1). The semi-classical defect measure µonT∗T2is\nthe pushforward of the semi-classical measure associated t o (π∗\nΞ0uh). Since the period of the torus T2\nΞ0\n2In fact, we only need to consider finitely many ζ0∈ Q, since when p2\n0+q2\n0is large enough, the associated periodic\ndirection is close to an irrational direction and the trajec tory will eventually enter ω.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 9\nhas no influence of the analysis in the sequel3, we will still use the notation T2to stand for T2\nΞ0, the\nvariablesz= (x,y),ζ= (ξ,η) to stand for variables Z= (X,Y),Ξ onT∗T2\nΞ0, and assuming the period\nto be 2πfor simplicity. The only thing that will change is that the pr e-image of the damping π−1\nΞ0(ω)\nis now a disjoint union of p2\n0+q2\n0copies ofωonT2\nΞ0, and each component is still strictly convex with\npositive curvature. For this reason, in the hypothesis of Pr oposition 4.4, we assume that the boundary\nof{a>0}is made of disjoint unions of strictly convex curves.\n3.Analysis of the transversal high frequencies\nRecall that ζ0= (0,1), and our goal is to show that µ1ζ=ζ0= 0. In this section, we deal with the\ntransversal high frequencies of size O(/planckover2pi1−1) and use the positive commutator method to show these\nportions are propagated into the flowout of the damped region ω.\nFor the geodesic flow ϕtonT∗T2andζ∈S1, we denote by ϕt(·,ζ) :T2→T2the projection of the\nflow mapϕt. By shifting the coordinate, we may assume\nω0:=I0×Ty⊂/uniondisplay\nt∈[0,2π]ϕt(·,ζ0)(ω),\nwhere\nI0= (−σ0,σ0)⊂π1({z:a(z)≥c0>0}),for someσ0<π\n100\nandπ1:T2/ma√sto→Txthe canonical projection. Therefore, there exist ǫ0>0,c0>0 sufficiently small and\nT0>0, such that for any |ζ|= 1,z0∈ω0,|ζ−ζ0| ≤ǫ0,\n/integraldisplayT0\n0(a◦ϕt)(z0,ζ)dt≥c0>0. (3.1)\nω0\nsupp(a)ζ0= (0,1)\nTo microlocalize the solution near ζ0, we pickψ0∈C∞\nc(R) and consider u1\nh:=ψ0/parenleftbighDx\nǫ0/parenrightbig\nuh, then\n(Ph+iha)u1\nh=f1\nh:=ψ0/parenleftbighDx\nǫ0/parenrightbig\nfh−ih/bracketleftbig\nψ0/parenleftbighDx\nǫ0/parenrightbig\n,a/bracketrightbig\nuh.\nLetµ1be the semiclassical measure of u1\nh, thenµ1=|ψ/parenleftbigξ\nǫ0/parenrightbig\n|2µ.\nLemma 3.1. We have\n/ba∇dbla1\n2u1\nh/ba∇dblL2=o(h1\n2δ1\n2\nh),/ba∇dblf1\nh/ba∇dblL2=o(h2δh).\n3As we fix one periodic direction ζ0and consider the semi-classical limit h→0, one does not need to worry about\nthe fact that the period 2 π/radicalbig\np2\n0+q2\n0may be very large.10 CHENMIN SUN\nProof.It suffices to show that f1\nh=oL2(h2δh), and the first assertion follows from the same proof of\n(a) of Lemma 2.2. By the symbolic calculus,\ni/bracketleftbig\nψ0/parenleftbighDx\nǫ0/parenrightbig\n,a/bracketrightbig\n=ǫ−1\n0hOph/parenleftbig\nψ′\n0/parenleftbigξ\nǫ0/parenrightbig\n∂xa/parenrightbig\n+OL2(ǫ−2\n0h2).\nFrom the pointwise inequality for the non-negative functio na:\n|∇a(x)|2≤2/ba∇dbla/ba∇dblW2,∞a(x), (3.2)\nwe have, for some C >0,\nCa−/vextendsingle/vextendsingleǫ−1\n0ψ′\n0/parenleftbigξ\nǫ0/parenrightbig\n∂xa/vextendsingle/vextendsingle2≥0 onT∗T2.\nTherefore, by the sharp G˚ arding inequality, we get\n/vextenddouble/vextenddoubleǫ−1\n0Oph/parenleftbig\nψ′\n0/parenleftbigξ\nǫ0/parenrightbig\n∂xa/parenrightbig\nuh/vextenddouble/vextenddouble\nL2≤C|(auh,uh)L2|1\n2+Ch1\n2/ba∇dbluh/ba∇dblL2.\nTogether with (a) of Lemma 2.2, this implies that\n/vextenddouble/vextenddoubleih/bracketleftbig\nψ0/parenleftbighDx\nǫ0/parenrightbig\n,a/bracketrightbig\nuh/vextenddouble/vextenddouble\nL2=o(h5\n2) =o(h2δh).\nThe proof of Lemma 3.1is complete. /square\nRecall that /planckover2pi1=h1\n2δ1\n2\nh. Letψ∈C∞\nc(R), and consider\nvh=ψ(/planckover2pi1Dx)u1\nh, wh= (1−ψ(/planckover2pi1Dx))u1\nh. (3.3)\nIn this decomposition, whcorresponds to the transversal high frequency part, while vhcorresponds to\nthe transversal low frequency part for which will be treated in next sections. Note that\n(Ph+iha)vh=ψ(/planckover2pi1Dx)f1\nh−ih[ψ(/planckover2pi1Dx),a]u1\nh=:r1,h (3.4)\nand\n(Ph+iha)wh= (1−ψ(/planckover2pi1Dx))f1\nh+ih[ψ(/planckover2pi1Dx),a]u1\nh=:r2,h. (3.5)\nWe need to show that the commutator term h[ψ(/planckover2pi1Dx),a]u1\nhcan be viewed as the remainder:\nLemma 3.2. We have\n/ba∇dblr1,h/ba∇dblL2+/ba∇dblr2,h/ba∇dblL2=o(h2δh) =o(h/planckover2pi12).\nConsequently, from Lemma 2.2,\n/ba∇dbla1\n2vh/ba∇dblL2+/ba∇dbla1\n2wh/ba∇dblL2=o(h1\n2δ1\n2\nh) =o(/planckover2pi1).\nProof.According to the symbolic calculus,\ni[ψ(/planckover2pi1Dx),a] =/planckover2pi1Op/planckover2pi1(ψ′(ξ)∂xa)+C/planckover2pi12Op/planckover2pi1(∂2\nξψ·∂2\nxa)+OL(L2)(/planckover2pi13).\nUsing the fact that a∈ Dm,2,σandσ <1\n4, we havea1\n2∈C2\nc(T2). Applying the special symbolic\ncalculus Lemma B.3(b) withκ=∂x(a1\n2),b2=a1\n2andϕ=ψ′, we have\n1\n2Op/planckover2pi1(ψ′(ξ)∂xa) =Op/planckover2pi1(ψ′(ξ)∂x(a1\n2))a1\n2−1\ni/planckover2pi1Op/planckover2pi1(ψ′′(ξ)∂x(a1\n2)·∂x(a1\n2))+OL(L2)(/planckover2pi12).SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 11\nApplying Lemma B.3(a) withκ=b1=∂x(a1\n2),ϕ=ψ′′, we have\n−/planckover2pi1\niOp/planckover2pi1(ψ′′(ξ)∂x(a1\n2)·∂x(a1\n2)) =−/planckover2pi1\niOp/planckover2pi1(ψ′′(ξ)∂x(a1\n2))∂x(a1\n2)+OL(L2)(/planckover2pi12).\nSinceψis only a function of ξanda−1\n2∂xa∈L∞, we have\n/vextenddouble/vextenddoubleOp/planckover2pi1/parenleftbig\nψ′(ξ)∂xa\na1\n2/parenrightbig\na1\n2u1\nh/vextenddouble/vextenddouble\nL2(T2)=/vextenddouble/vextenddouble(a−1\n2∂xa)ψ′(hDx)(a1\n2u1\nh)/vextenddouble/vextenddouble\nL2(T2)≤C/ba∇dbla1\n2u1\nh/ba∇dblL2=o(/planckover2pi1).\nFor the term Op/planckover2pi1(∂2\nξψ∂2\nxa), since|∂2\nxa|/lessorsimilara1−2σ/lessorsimilara1\n2, by the sharp G˚ arding inequality,\nRe/parenleftbig\nOp/planckover2pi1(Ca−|ψ′′(ξ)∂2\nxa|2)u1\nh,u1\nh/parenrightbig\nL2≥ −C/planckover2pi1/ba∇dblu1\nh/ba∇dbl2\nL2.\nThus/ba∇dbl/planckover2pi12Op/planckover2pi1(ψ′′(ξ)∂2\nxa)u1\nh/ba∇dblL2=O(/planckover2pi15\n2). Therefore, by Calder´ on-Vaillancourt (Theorem B.1), we can\nwrite\nih[ψ(/planckover2pi1Dx),a] =h/planckover2pi1A/planckover2pi1a1\n2+h/planckover2pi12B/planckover2pi1∂x(a1\n2)+OL(L2)(h/planckover2pi13),\nwithA/planckover2pi1,B/planckover2pi1bounded operators on L2, uniformly in /planckover2pi1. Since|∂x(a1\n2)|/lessorsimilara1\n2−σ, by (a) of Lemma 2.2\nand the interpolation, we get\n/ba∇dblih[ψ(/planckover2pi1Dx),a]u1\nh/ba∇dblL2=o(h/planckover2pi12)+O(h/planckover2pi13).\nThis completes the proof of Lemma 3.2. /square\nRemark 3.3. Compared to [ AL14] where the damping only satisfies a∈Wk0,∞(T2) and|∇a|/lessorsimilara1−σ,\nour assumption a∈ Dm,2,σis slightly stronger, in order to ensure the commutator of th e damping\nterm is still a remainder. Indeed, here we chose /planckover2pi1=h1\n2δ1\n2\nh≪h1\n2while in [ AL14], the authors chose\n/planckover2pi1=hα,α<1\n3(see the sentence after Proposition 7.2 of [ AL14]). The use of the sharp G˚ arding as in\n[AL14] would not get o(h/planckover2pi12) for the remainders r1,h,r2,h. We also remark that a direct application of\ntheCalder´ on-Vaillancourt theorem inthesymbolic calcul us requires a1\n2∈W3,∞. Sinceweassumeonly\na∈ Dm,2,σ, (thusa1\n2∈W2,∞), we need to exploit the special structure of the commutator [ψ(/planckover2pi1Dx),a]\nand apply the special symbolic calculus Lemma B.3.\nRecall that ω0=I0×Ty.\nLemma 3.4. We have\n/ba∇dblwh1ω0/ba∇dblL2+/ba∇dblh∇wh1ω0/ba∇dblL2=O(h1\n2),ash→0.\nProof.The proof follows from the classical propagation argument, using the geometric control condi-\ntion. Take small intervals I′\n0⊂Tx,I1= (σ1,σ2)⊂Ty, such that I0⊂I′\n0andω1=I′\n0×I1⊂ {a≥δ0}\nfor someδ0>0. For any z0= (x0,y0)∈ω0, by the geometric control condition, there exist\nT1>0,δ1>0,δ2>0, and the small neighborhood U= (x0−δ1,x0+δ1)×(y0−δ1,y0+δ1) of\nz0, such that for all |ζ−ζ0| ≤ǫ0,z∈U, we have\nz+sζ∈ω1, s∈[T1−δ2,T1+δ2].\nIn particular, a(z+sζ)≥δ0. Without loss of generality, we assume that π > σ2> σ1> y0+δ1>\ny0−δ1>−π. Pick two cutoffs χ1(x),χ2(y)≥0, supported in ( x0−δ1,x0+δ1), (y0−δ1,y0+δ1) and12 CHENMIN SUN\nequal to 1 on ( x0−δ1/2,x0+δ1/2),(y0−δ1/2,y0+δ1/2), respectively. Let χ0∈C∞\nc(R) be a cutoff\nnear|ζ−ζ0| ≤ǫ0. For anys≥0, define the symbol\nbs(z,ζ) :=χ0(ζ)·(χ1⊗χ2)◦ϕ−s(z,ζ) =χ0(ζ)χ1(x−sξ)χ2(y−sη).\nω0\nω1\nUϕT0(·,ζ)(U)⊂ω1\nDirect computation yields\nd\nds(Oph(bs)wh,wh)L2=(Oph(∂sbs)wh,wh)L2=−(Oph(ζ·∇zbs)wh,wh)L2.\nIntegrating this equality from s= 0 tos=T0,\n(Oph(bT1)wh,wh)L2−(Oph(b0)wh,wh)L2=−/integraldisplayT1\n0(Oph(ζ·∇zbs)wh,wh)L2ds. (3.6)\nNote that for fixed s∈[0,T1],\ni\nh[Ph,Oph(bs)] = 2Oph(ζ·∇zbs)+OL(L2)(h),\nwe have\n(Oph(b0)wh,wh)L2=(Oph(bT1)wh,wh)L2+i\n2h/integraldisplayT1\n0([Ph,Oph(bs)]wh,wh)L2+O(h).(3.7)\nUsing the equation\nPhwh=r2,h−ihawh,\nwe have\n1\nh([Ph,Oph(bs)]wh,wh)L2=2i\nhIm(Oph(bs)wh,r2,h−ihawh)L2\n=o(h1+δ)+O(1)/ba∇dbla1\n2wh/ba∇dblL2/ba∇dbla1\n2Oph(bs)wh/ba∇dblL2\n=O(h), (3.8)\nwhere to the last step, we write\na1\n2Oph(bs) = Oph(bs)a1\n2+[a1\n2,Oph(bs)]\nand use the last assertion of Lemma 3.2, as well as the symbolic calculus.\nFinally, from the support property of b0(z) =χ0(ζ)χ1(x)χ2(y), we have\na(z)χ0(ζ)≥δ0bT1(z,ζ).\nThus by the sharp G˚ arding inequality,\n|(Oph(bT1)wh,wh)L2| ≤δ−1\n0|(Oph(a(z)χ0(ζ))wh,wh)L2|+O(h)≤Cδ−1\n0/ba∇dbla1\n2wh/ba∇dbl2\nL2+O(h).SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 13\nCombining this with ( 3.7),(3.8) and the last assertion of Lemma 3.2, we deduce that /ba∇dblwh1U/ba∇dblL2=\nO(h1\n2),/ba∇dblh∇wh1U/ba∇dblL2=O(h1\n2). By the partition of unity of ω0, we complete the proof of Lemma\n3.4. /square\nNow we are ready to prove the main result in this section, that is the transversal high frequency\npart is of order oL2(1):\nProposition 3.5. We have /ba∇dblwh/ba∇dblL2=O(δh) =O(/planckover2pi1h−1\n2), ash→0.\nProof.We use the positive commutator method to detect the transver sal propagation, similarly as in\n[BS19]. Recall that ω0=I0×TandI0= (−σ0,σ0),σ0<π\n100. Takeφ=φ(x)∈C∞(Tx;[0,1]) such\nthat:\nsupp(1−φ)⊂I0, φ≡0 near[−σ0\n2,σ0\n2],supp(φ′)⊂I0.\nDenote by\nX(x) := (x+π)1−π≤x<−σ0\n2+(x−π)1σ0\n2≤x<π,\nthenφ(x)X∂xis well-defined smooth vector field on T2. We now compute the inner product\n([Ph,φ(x)X∂x]wh,wh)L2\nin two ways. On the one hand, from the commutator relation\n[Ph,φ(x)X∂x] =−2(φX)′h2∂2\nx−h2(φX)′′∂x,\nwe have\n([Ph,φ(x)X∂x]wh,wh)L2≥2(φ(x)h∂xwh,h∂xwh)L2−C/ba∇dbl(φ′(x))1/2h∂xwh/ba∇dbl2\nL2\n−Ch/ba∇dblh∂xwh/ba∇dblL2/ba∇dblwh/ba∇dblL2. (3.9)\nOn the other hand, using the equation ( 3.5), we have\n([Ph,φ(x)X∂x]wh,wh)L2=(φ(x)X∂xwh,r2,h−ihawh)L2−(φ(x)X∂x(r2,h−ihawh),wh)L2\n≤C\nh(/ba∇dblh∂xwh/ba∇dblL2+h/ba∇dblwh/ba∇dblL2)/ba∇dblr2,h/ba∇dblL2+C/ba∇dbla1/2h∂xwh/ba∇dblL2/ba∇dbla1\n2wh/ba∇dblL2+C/ba∇dbla1\n2wh/ba∇dbl2\nL2.\n(3.10)\nCombining ( 3.9) and (3.10) and Lemma 3.4, we get\n/ba∇dblh∂xwh/ba∇dbl2\nL2≤C/ba∇dblh∂xwh1ω0/ba∇dbl2\nL2+C/ba∇dbla1/2h∂xwh/ba∇dbl2\nL2+C/ba∇dbla1\n2wh/ba∇dbl2\nL2+C\nh2/ba∇dblr2,h/ba∇dbl2\nL2+Ch2/ba∇dblwh/ba∇dbl2\nL2\n≤O(h)+o(hδh).\nIn particular, by the definition of wh, we have\nC(h/planckover2pi1−1)/ba∇dblwh/ba∇dblL2≤ /ba∇dblh∂xwh/ba∇dblL2≤O(h1\n2),\nand this completes the proof of Proposition 3.5. /square14 CHENMIN SUN\n4.The averaging properties of functions\nIn order to treat the transversal low frequency part vh=ψ(/planckover2pi1Dx)u1\nhof (3.3), we will adapt the\naveraging argument of Sj¨ ostrand ([ Sj]) and Hitrik ([ Hi1]) to average the operator Ph+ihaalong the\ntrapped direction (it worth also mentioning a series work of Hitrik-Sj¨ ostrand ([ HS1][HS2][HS3]) that\na related averaging procedure was used to describe the spect rum of the damped wave operator). This\namounts to understand the regularity of averaged functions . The goal of this section is to establish\nseveral properties of the averaging operator which will be u sed in Section 5for normal form reductions.\nMore importantly, we will prove the key geometric propositi on (Proposition 4.4) for convex-shaped\ndamping that is responsible for the improvement of the resol vent estimate.\nGiven a direction v∈S1, we say that visperiodic, ifv= (ξ,η) andξ,ηareQ-linearly dependent.\nOtherwise, we say that visergodic4. We define the averaging operator along v:\nf/ma√sto→ A(f)v(z) := lim\nT→∞1\nT/integraldisplayT\n0f(z+tv)dt,\nwhere the limit exists, thanks to Weyl’s equidistribution t heorem.\nLemma 4.1. Assume that v∈S1. Then for any non-negative function f∈ Dm,k,σ(T2),m≥10, the\naveraged function A(f)v∈ Dm,k,σ(T2).\nProof.First we assume that v= (ξ,η) is periodic. This implies that the orbit z/ma√sto→z+tvis periodic,\nandTvis the period. Clearly, for any f∈ Dm,k,σ,\nA(f)v(z) =1\nTv/integraldisplayTv\n0f(z+tv)dt.\nSince the function s/ma√sto→ |s|1−|α|σis concave, by Jensen’s inequality we have\n1\nTv/integraldisplayTv\n0|f(z+tv)|1−|α|σdt≤/parenleftBig1\nTv/integraldisplayTv\n0f(z+tv)dt/parenrightBig1−|α|σ\n. (4.1)\nIndeed, if/integraltextTv\n0f(z+tv)dt= 0, thenf(z+tv)≡0 for allt∈[0,Tv], and the inequality ( 4.1) is trivial.\nAssume now that X0=1\nTv/integraltextTv\n0f(z+tv)dt>0, then for any X≥0, we have\nX1−|α|σ≤X1−|α|σ\n0+(1−|α|σ)X−|α|σ\n0(X−X0).\nReplacing the inequality above by X=f(z+tv) and averaging over t∈[0,Tv], we obtain ( 4.1). Since\nby definition, |∂αf|/lessorsimilarα,σ|f|1−|α|σfor all|α| ≤k, we get\n|∂α(A(f)v)(z)|/lessorsimilarα,σ|A(f)v(z)|1−|α|σ.\nNext we assume that v= (ξ,η) is ergodic. In this case, the orbit z/ma√sto→z+tvis ergodic, then by\nWeyl’s equidistribution theorem, we have\nAv(f)(z) =/hatwidef(0) =1\n(2π)2/integraldisplay\nT2f(z′)dz′. (4.2)\n4These definitions stemmed from the fact that we are on the two- dimensional torus T2.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 15\nTo prove this, by the assumption of v, we havek·v/\\e}atio\\slash= 0 for allk∈Z2\\{0}. Moreover, since f∈Cm,\nwe can write\n1\nT/integraldisplayT\n0f(z+tv)dt=1\nT/integraldisplayT\n0/summationdisplay\nk∈Z2/hatwidef(k)eik·(z+tv)dt=/hatwidef(0)+/summationdisplay\nk/negationslash=0/hatwidef(k)eik·z·eiTk·v−1\niT(k·v).\nAs|eiTk·v−1| ≤2|iTk·v|for allk∈Z2and\nlim\nT→∞eiTk·v−1\niTk·v= 0,\nby the dominated convergence theorem, we get ( 4.2), which means that Av(f) is a constant function.\nClearly,A(f)v(z)∈ Dm,k,σ(T2). The proof of Lemma 4.1is now complete. /square\nBy the triangle inequality, the following Lemma is immediat e:\nLemma 4.2. Letv∈S1. For any function fonT2, there holds\n|A(f)v| ≤ A(|f|)v.\nMoreover, if f1,f2are two non-negative functions such that f1≤f2, we have\nA(f1)v≤ A(f2)v.\nLemma 4.3. Assume that f∈ Dm,k,σ(T2)andf≥0. Denote by\nF(x,y) :=/integraldisplayy\n−π/parenleftbig\nf(x,y′)−A(f)e2(x)/parenrightbig\ndy′,−π<y<π,\nwheree2= (0,1). Then for 0≤j≤k, we have\n|F(x,y)| ≤4πA(f)e2(x),|∂j\nxF(x,y)| ≤4πA(f)1−jσ\ne2.\nMoreover, for all j1≥0,1≤j2≤k−j1, we have\n|∂j1x∂j2yF(x,y)| ≤ A(f)1−(j1+j2−1)σ\ne2(x)+f(x,y)1−(j1+j2−1)σ.\nProof.Sincef≥0, the bound for |F(x,y)|is trivial. Taking derivatives, we get\n∂j1x∂j2yF(x,y) =∂j2y/integraldisplayy\n−π(∂j1xf(x,y′)−∂j1xA(f)e2(x))dy′.\nWhenj2= 0, the absolute vaule of the above quantity can be bounded by\n2πA(f)1−j1σ\ne2(x)+2π∂j1xA(f)e2(x)≤4π(A(f)e2)1−j1σ(x),\nthanks to Lemma 4.1and Lemma 4.2and Jensen’s inequality ( 4.1). Whenj2≥1, by definition, we\nhave\n∂j1x∂j2yF(x,y) =∂j2−1\ny∂j1xf(x,y)−∂j2−1\ny∂j1xA(f)e2(x).\nTaking the absolute value and the proof of Lemma 4.3is complete. /square\nFinally, we prove the key geometric proposition, allowing u s to improve the local H¨ older regularity\nfor averaged damping functions, provided that the original damped region is locally strictly convex\nwith positive curvature:16 CHENMIN SUN\nProposition 4.4. Assume that a∈ Dm,k,σ(T2)such thata(z)≥0and the damping boundary Σa=\n∂{z:a(z)>0}is a disjoint union of strictly convex curves with positive c urvature. Assume moreover\nthat there exists R>0such that for every z∈ {a>0}nearΣa,\nR−1·dist(z,Σa)1\nσ≤a(z)≤R·dist(z,Σa)1\nσ. (4.3)\nThen for any periodic direction v∈S1, we have A(a)v∈ Dm,k,2σ\nσ+2, as a one-dimensional periodic\nfunction. Furthermore, there exists Rv>0, such that for every z∈ {A(a)v(z)>0}nearΣA(a)v, we\nhave\nR−1\nvdist(z,ΣA(a)v)1\nσ+1\n2≤ A(a)v(z)≤Rvdist(z,ΣA(a)v)1\nσ+1\n2. (4.4)\nProof.Since the vector vis periodic, we can find p0,q0∈Z, gcd(p0,q0) = 1, such that v=(p0,q0)√\np2\n0+q2\n0.\nNow we perform a change of coordinate as described in Section 2.2. Recall that we have a covering\nmapπv:T2\nv/ma√sto→T2of degreep2\n0+q2\n0that lifts every 2 πperiodic function to a 2 π/radicalbig\np2\n0+q2\n0-periodic\nfunction. Moreover, the change of coordinate system z/ma√sto→(X,Y) is given by z=Xv⊥+Yvlocally.\nAsπvis locally isometric, each component of the pre-image of the damped region π−1\nv(ω) is still\nstrictly convex with positive curvature, so the inequality (4.3) is preserved, near the boundary of each\ncomponent. Denote by τ= 2π/radicalbig\np2\n0+q2\n0. We define the averaging operator on the new torus T2\nvas\n/tildewideA(F)(X,Y) :=1\nτ/integraldisplayτ/2\n−τ/2F(X,Y+t)dt.\nthen by definition\nπ∗(A(a)v)(X,Y) =A(a)v(π(Xv⊥+Y)) =1\nτ/integraldisplayτ\n0a◦π(Xv⊥+(Y+t)v)dt=1\nτ/integraldisplayτ\n0(π∗a)(X,Y+t)dt.\nThusπ∗\nv/parenleftbig\nA(a)v/parenrightbig\n=/tildewideA(π∗\nva).Therefore, if we are able prove ( 4.4) for the lifted averaging function\n/tildewideA(π∗\nv(a)), we obtain automatically ( 4.4) by projection.\nInsummary, from theargument above, withoutloss of general ity, weassumethat v= e2andassume\nthat Ω := {a>0}hasl=l(v) connected components Ω 1,···,Ωlsuch that the boundary Σ a,jof each\nΩjhas positive curvature. We first consider the situation wher el= 1. By translation invariance, we\nmay assume that Ω 1={a1>0}is contained in the fundamental domain ( −Kπ,Kπ)x×(−Mπ,Mπ )y.\nThen the function A(a)e2can be identified as a function on Rx,\nSince Ω 1={a1>0}is strictly convex, each line PxofR2, passing through ( x,0) and parallel to e 2\ncan intersect at most 2 points of the curve Σ a,1. Consider the function x/ma√sto→P(x) := mes( Px∩Ω1).\nThis function is continuous and is supported on a single inte rvalI= (α,β)⊂(−Kπ,Kπ). Since\nA(a1)(x) = 0 ifP(x) = 0, the vanishing behavior of A(a1) is determined when xis close toαandβ.\nBelow we only analyze A(a1)(x) forx∈[α,α+ǫ), since the analysis is similar for xnearβ. First we\nobserve that Pαmust be tangent at a point z0:= (α,y0) to the curve Σ a,1. For sufficiently small ǫ>0,\nwe may parametrize the curve Σ a,1nearz0by the function x=α+g(y) withg(y0) =g′(y0) = 0 and\ng′′(y0) =c0>0, thanks to the fact that the curvature is strictly positive . Therefore, there exists a C1\ndiffeomorphism Y= Φ(y) from a neighborhood of y0to a neighborhood of Y= 0 such that Φ( y0) = 0\nandg(y) =Y2. For eachx∈(α,α+ǫ),Px∩Σa,1={(x,l−(x)),(x,l+(x))}. We have\nl+(x) = Φ−1(√\nx−α), l−(x) = Φ−1(−√\nx−α).SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 17\nΩ1\nxz0\nαα+ǫ(x,l−(x))(x,l−(x))Px\nAveraging improves the local H¨ older regularity\nSince near z0, we havea(z)∼dist(z,Σa)1\nσ= dist(z,Σa,1)1\nσ∼(x−(α+g(y)))1\nσ\n+. Forx∈(α,α+ǫ),\nwe have\nA(a1)(x) =1\n2Kπ/integraldisplayKπ\n−Kπa1(x,y)dy=1\n2Kπ/integraldisplayl+(x)\nl−(x)a1(x,y)dy\n=1\n2Kπ/integraldisplay√x−α\n−√x−α|x−α−Y2|1\nσ|(Φ−1)′(Y)|dY∼Φ,K,σ|x−α|1\nσ+1\n2.\nHere the implicit constant depends on Φ ,K,σ, hence it depends on the periodic direction vand the\ndampinga(z).\nFinally, since a1∈ Dm,k,σ, forj≤k,x∈(α,α+ǫ),\n|∂j\nxA(a1)(x)| ≤1\n2Kπ/integraldisplayKπ\n−Kπ|∂j1xa1(x,y)|dy/lessorsimilarK/integraldisplayl+(x)\nl−(x)a1−jσ\n1(x,y)dy\n/lessorsimilar|x−α|1−jσ\nσ+1\n2∼(A(a1)(x))1−2σj\nσ+2.\nThis implies that A(a1)∈ Dm,k,2σ\nσ+2.\nTo complete the proof of Proposition 4.4, we need to deal with the situation where l >1. In this\ncase, the supports of A(aj) may overlap. By linearity and the inequality\n|∂j′\nxA(a1+···al)| ≤l/summationdisplay\nj=1|∂j′\nxA(aj)| ≤C(a1,···,al)A(a1+···+al)1−2σj′\nσ+2,\nwe deduce that A(a)∈ Dm,k,2σ\nσ+2. It remains to show ( 4.4). We define\nS+:=/braceleftbig\nj∈ {1,···,l}:/integraldisplayx0+ǫ\nx0A(aj)(x)dx>0,∀ǫ>0/bracerightbig\n,\nS−:=/braceleftbig\nj∈ {1,···,l}:/integraldisplayx0\nx0−ǫA(aj)(x)dx>0,∀ǫ>0/bracerightbig\n.\nObserve that x0∈ΣA(a)if and only if A(aj)(x0) = 0 for all j∈ {1,···,l}andS+∪S−/\\e}atio\\slash=∅.Note that\nforj∈S±, we have\ndist(x,ΣA(a))1\n2+1\nσ= dist(x,ΣA(aj))1\n2+1\nσ∼ A(aj)(x),∀x∓x0>0 nearx0,\nandA(aj) = 0, in a neighborhood of x0, for allj /∈S+∪S−. Summing over j∈S+∪S−, we obtain\n(4.4). The proof of Proposition 4.4is now complete. /square18 CHENMIN SUN\n5.Normal form reductions\nNow we treat the transversal low frequency part vh=ψ(/planckover2pi1Dx)u1\nh, defined in ( 3.3). We want to\naverage the operator Ph+ihaalong the direction e 2= (0,1). Recall that A(a) is the averaging of a\nalong the vertical direction. Recall that vhsatisfies the equation\n(Ph+iha)vh=r1,h=oL2(h2δh) =oL2(h/planckover2pi12).\nWe will apply successively two normal form reductions. The fi rst reduction replaces ihabyihA(a),\nwith aO(h2) anti-selfadjoint remainder which cannot be absorbed dire ctly as a remainder of size\nO(h2δh). Weneedtoperformasecondnormalformreductiontoaverag etheanti-selfadjoint remainder.\n5.1.The first averaging. Throughout this section, we denote by\nA(x,y) :=/integraldisplayy\n−π(a(x,y′)−A(a)(x))dy′.\nWe will also fix a cutoff ψ1∈C∞\nc(R), supported on |η±1| ≤1\n2andψ1(η) = 1 if|η±1| ≤1\n4. We need\nthe following basic lemma for exponentials of bounded linea r operators:\nLemma 5.1. Let(Gh)0<h<1be a family of h-dependent, uniformly bounded operators on a Hilbert\nspaceH. Defining the exponential via\nesGh:=∞/summationdisplay\nk=0(sGh)k\nk!, s∈R.\nThen the operator esGhis invertible with inverse e−sGh. Moreover,\n/ba∇dblesGh/ba∇dblL(H)≤e|s|/bardblGh/bardblL(H)<∞.\nFor any linear operator B,\nd\nds(esGhBe−sGh) =esGhadGh(B)e−sGh, (5.1)\nwhereadA(B) := [A,B]. Consequently, for any bounded operator BonH,\n/ba∇dbl[esGh,B]/ba∇dblL(H)≤ |s|e3|s|/bardblGh/bardblL(H)/ba∇dbladGh(B)/ba∇dblL(H). (5.2)\nFinally, we have the Taylor expansion\nesGhBe−sGh=N−1/summationdisplay\nk=0sk\nk!adk\nGh(B)+1\n(N−1)!/integraldisplay1\n0(1−s)N−1esGhadGh(B)e−sGhds,∀N∈N,(5.3)\nwhereadk\nA(B) = adA(adk−1\nA(B)),ad0\nA(B) =B.\nProof.Assume that /ba∇dblGh/ba∇dblL(H)≤Mfor allh∈(0,1). The series\n∞/summationdisplay\nk=0/ba∇dbl(sGh)k/ba∇dblL(H)\nk!≤∞/summationdisplay\nk=0|s|kMk\nk!=e|s|M\nconverges absolutely. Therefore,\n/ba∇dblesGh/ba∇dblL(H)≤e|s|/bardblGh/bardblL(H).SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 19\nTo show that esGhis invertible, again, by the absolute convergence of the ser ies/summationtext\nk≥0/bardbl(sGh)k/bardblL(H)\nk!, we\ndeduce that\ne−sGhesGh=∞/summationdisplay\nk=0skGk\nh\nk!/summationdisplay\nk1+k2=k(−1)k1·k!\nk1!k2!= Id.\nTherefore,esGhis invertible with inverse e−sGh. One easily verifies that Ghcommutes with esGhand\nd\ndsesGh=GhesGh=esGhGh.This implies ( 5.1). To prove ( 5.2), we remark that\n[esGh,B] =/parenleftBig/integraldisplays\n0es′GhadGh(B)e−s′Ghds′/parenrightBig\nesGh. (5.4)\nTaking the operator norm we obtain ( 5.2). The last identity ( 5.3) follows directly from the Taylor\nexpansion with integral remainders. The proof of Lemma 5.1is complete. /square\nOur goal is to find an exponential (elliptic) eGhfor someGh= Opw\nh(g) withg∈S0(Ty×Rη),\ndepending smoothly in xsuch that\neGh(Ph+iha)e−Gh=Ph+ihA(a)+lower orders .\nTo find the operator Gh, we consider the conjugate operator\nFh(s) :=esGh(Ph+iha)e−sGh\nUsing the Taylor expansion ( 5.3) up to order N= 2 in Lemma 5.1and the symbolic calculus, we get\nFh(1) =Ph+iha−ih·i\nh[Gh,Ph+iha]+1\n2[Gh,[Gh,Ph+iha]]+oL(L2)(h/planckover2pi12).\nTo average the leading order of the anti-selfadjoint part iha, we expect the principal symbol of a−\ni\nh[Gh,h2D2\ny] to beA(a). To this end, we need to solve the cohomological equation\na+H|η|2(g) =A(a).\nSet\ng(x,y,η) =−ψ1(η)\n2η/integraldisplayy\n−π(a(x,y′)−A(a)(x))dy′=−ψ1(η)\n2η/integraldisplayy\n−π(a(x,y′)−A(a)(x))dy′A(x,y),\nwhereψ1,A(x,y) are defined at the beginning of Subsection 5.1. We can indeed define explicitly the\nquantization of gas\nGh= Opw\nh(g) =−ψ1(hDy)\n4hDyA(x,y)−A(x,y)ψ1(hDy)\n4hDy.\nThenGhis self-adjoint and is O(1)h-semiclassical of order 0 and O(h−1) classical of order −1,\nsmoothly depending on x∈T. In particular, Ghis uniformly bounded on L(L2(T2)). Therefore, by\nLemma5.1, the operators\nesGh:=∞/summationdisplay\nn=0snGn\nh\nn!, e−sGh=∞/summationdisplay\nn=0(−1)nsnGn\nh\nn!.\nare well-defined and invertible on L2(T2), andesGhe−sGh=e−sGhesGh= Id,for alls∈R.\nWe now state and prove the main result in this subsection:20 CHENMIN SUN\nProposition 5.2. Giveng(x,y,η) =−ψ1(η)\n2ηA(x,y)andGh= Opw\nh(g). Letv(1)\nh:=eGhvh. Then\nPhv(1)\nh+ihA(a)v(1)\nh−[h2D2\nx,Gh]v(1)\nh=/tildewiderh=oL2(h/planckover2pi12).\nMoreover,v(1)\nhsatisfies\n(a)/ba∇dbla1\n2v(1)\nh/ba∇dblL2+/ba∇dblA(a)1\n2v(1)\nh/ba∇dblL2=o(/planckover2pi1).\n(b) WFm\nh(v(1)\nh)⊂WFh(vh)⊂ {(z,ζ) :ζ=ζ0= (0,1)}.\n(c)For any /tildewideψ∈C∞\nc(R;[0,1])such that /tildewideψ(ξ) = 1on the support of ψwhich defines vhin(3.3),\nwe have\n(1−/tildewideψ(/planckover2pi1Dx))v(1)\nh=OL2(hm\n2).\nNote that the definition of semiclassical wavefront sets WFm(·) and WF h(·) are recalled at the end\nof Appendix B.\nFrom Proposition 5.2, one may deduce from Lemma 4.3that the anti-selfadjoint remainder satisfies\n[Gh,h2D2\nx]v(1)\nh=oL2(h2).\nThough we can not absorb it directly as an error of order o(h/planckover2pi12), its principal part is non self-\nadjoint and can be viewed as an lower order perturbation of th e averaged damping ihA(a). In the\nnext subsection, we will perform a second normal form to aver age the operator [ Gh,h2D2\nx] so that it\nbecomes independent of the variable y.\nProof.To simplify the notation, we will use Rhto denote operators of size at most oL(L2)(h/planckover2pi12) and\nrhto denote errors of size oL2(h/planckover2pi12). Both of them may change from line to line.\nForFh(s) =esGh(Ph+iha)e−sGh, using the equation for vh, we have\nFh(s)v(1)\nh=esGh(Ph+iha)vh=rh.\nAs motivated before, we write down the full conjugate operat or:\nFh(1) =eGh(Ph+iha)e−Gh=Ph+ihA(a)+ih(a−A(a))−[h2D2\ny,Gh]\n−[h2D2\nx,Gh]+ih[Gh,a]+1\n2[Gh,[Gh,Ph]]+Rh. (5.5)\nHere the lower order operators\nih(a−A(a))−[h2D2\ny,Gh], ih[Gh,a],[Gh,[Gh,Ph]]\nare not in priorly negligible, since they are merely of order OL(L2)(h2). Nevertheless, it turns out that\nthey are negligible when acting on the function v(1)\nh:=eGhvh. We will prove this fact through several\nlemmas. First we show that the normal form transform eGhvhdoes not alter WF h(vh). In particular,\nwe prove (b), (c) of Proposition 5.5:\nLemma 5.3. We have\nWFm\nh(v(1)\nh)⊂WFh(vh).\nMoreover, for any /tildewideψ∈C∞\nc(R;[0,1])such that /tildewideψ(ξ) = 1on the support of ψwhich defined vhin(3.3),\nwe have\n/ba∇dbl(1−/tildewideψ(/planckover2pi1Dx))v(1)\nh/ba∇dblL2=O(/planckover2pi1m).SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 21\nProof.Letl(z,ζ) be a symbol supported on a compact set of T∗T2\\WFh(vh) andLh= Oph(l), it\nsuffices to show that\nLheGhvh=OL2(hN),\nfor anyN≤m. Recall that by definition of WFm\nh(vh) (see the end of Appendix B), for any h-\npseudodifferential operator Qhwith the principal symbol supported away from WF h(vh),\nQhvh=OL2(hm).\nUsing again ( 5.3), we write\nLheGhvh=eGh(e−GhLheGh)vh\n=eGh/parenleftBigN−1/summationdisplay\nn=0(−s)n\nn!adn\nGh(Lh)vh/parenrightBig\n−1\n(N−1)!eGh/integraldisplay1\n0(1−s)N−1e−sGhadN\nGh(Lh)esGhvhds.\nBy the symbolic calculus, the last term is OL2(hN), and for each n≤N−1, adn\nGh(Lh) is ah-\npseudodifferential operator with symbol supported away from WFh(vh), thus ad Gh(Lh)vh=OL2(hN).\nThis shows that WFm\nh(v(1)\nh)⊂WFh(vh).\nFor the second assertion, we first note that, since v(1)\nh=ψ(/planckover2pi1Dx)u1\nh, (1−/tildewideψ(/planckover2pi1Dx))vh=OL2(hm).\nWe observe also that since Ghis of the form\nGh=b(hDy)A(x,y)+A(x,y)b(hDy)\nand the Fourier multiplier b(hDy) commutes with /tildewideψ(/planckover2pi1Dx), we have\n[Gh,/tildewideψ(/planckover2pi1Dx)]v(1)\nh=b(hDy)[A,/tildewideψ(/planckover2pi1Dx)]v(1)\nh+[A,/tildewideψ(/planckover2pi1Dx)]b(hDy)v(1)\nh=OL2(/planckover2pi1m) =oL2(hm\n2),\nsince [A,/tildewideψ(/planckover2pi1Dx)] is a/planckover2pi1-pseudodifferential operator with symbol supported away fro m the support of\nψ(ξ), thanks to the fact that /tildewideψ≡1 on supp(ψ). More generally, for any other cutoff χ1such that\nχ1= 1 on supp( ψ), we always have\n(1−χ1(/planckover2pi1Dx))Ghv(1)\nh=OL2(/planckover2pi1m).\nSince\nadn+1\nGh(/tildewideψ(/planckover2pi1Dx))v(1)\nh=Ghadn\nGh(/tildewideψ(/planckover2pi1Dx))v(1)\nh−adn\nGh(/tildewideψ(/planckover2pi1Dx))Ghv(1)\nh.\nBy writing Ghv(1)\nh= (1−χn(/planckover2pi1Dx))Ghv(1)\nh+χn(/planckover2pi1Dx)Ghv(1)\nhfor some cutoffs χn, 1≤n≤m, such that\n/tildewideψ= 1 on supp( χn) andχn+1= 1 on supp( χn). Therefore, by induction, we deduce that for every\n1≤n≤m,\nadn\nGh(/tildewideψ(/planckover2pi1Dx))v(1)\nh=OL2(/planckover2pi1m).\nBy Taylor expansion, this shows that e−Gh(1−/tildewideψ(/planckover2pi1Dx))eGhvh=OL2(/planckover2pi1m). The proof of Lemma 5.3\nis now complete. /square\nNext we show that ih(a−A(a))v(1)\nh−[h2D2\ny,Gh]v(1)\nhandih[Gh,a]v(1)\nhare indeed remainders.\nLemma 5.4. We have\n/ba∇dbla1\n2v(1)\nh/ba∇dblL2=o(/planckover2pi1), ih/ba∇dbl[Gh,a]v(1)\nh/ba∇dblL2=O(h2/planckover2pi1)\nand\n/ba∇dblih(a−A(a))v(1)\nh−[h2D2\ny,Gh]v(1)\nh/ba∇dblL2=o(h2/planckover2pi1).22 CHENMIN SUN\nProof.Sincea1\n2∈W1,∞, by Lemma B.2,e−Gha1\n2eGh=a1\n2+OL(L2)(h) anda1\n2vh=oL2(/planckover2pi1), we have\na1\n2v(1)\nh=eGh(a1\n2vh+OL2(h)) =oL2(/planckover2pi1).\nNote that\n[Gh,a] =a1\n2[Gh,a1\n2]+[Gh,a1\n2]a1\n2= 2[Gh,a1\n2]a1\n2+[a1\n2,[Gh,a1\n2]],\nSincea1\n2∈W1,∞, from Corollary B.2, we have\nih/ba∇dbl[Gh,a]v(1)\nh/ba∇dblL2≤Ch2/ba∇dbla1\n2v(1)\nh/ba∇dblL2+Ch3/ba∇dblv(1)\nh/ba∇dblL2=o(h2/planckover2pi1).\nFor the last assertion, denote by b(η) =−ψ1(η)\n4η, thenGh=b(hDy)A+Ab(hDy), whereA=A(x,y).\nDirect computation yields\n[h2D2\ny,Gh] =ih\n2/parenleftbig\nψ1(hDy)∂yA+∂yAψ1(hDy)/parenrightbig\n−h2/parenleftbig\n∂2\nyAb(hDy)−b(hDy)∂2\nyA/parenrightbig\n.\nSince\n∂yA= (a−A(a)), ∂2\nyA=∂ya.\nUsing the symbolic calculus, we are able to write\nih(a−A(a))v(1)\nh−[h2D2\ny,Gh]v(1)\nh=ih(1−ψ1(hDy))(a−A(a))v(1)\nh+ih\n2[ψ1(hDy),a]v(1)\nh\n+h2Bh∂yav(1)\nh+OL2(h3),\nwhereBh=OL2(1), uniformly in 0 < h <1. Note that supp(1 −ψ1(η))∩WFm\nh(vh) =∅, hence by\nthe first assertion of Lemma 5.3, supp(1 −ψ1(η))∩WFm\nh(v(1)\nh) =∅, the first term on the right side\nisOL2(h3), say. Next, writing [ ψ1(hDy),a] as 2[ψ1(hDy),a1\n2]a1\n2+ [a1\n2,[ψ1(hDy),a1\n2]], using the first\ninequality of Lemma 5.4and Corollary B.2, we get\nh[ψ1(hDy),a]v(1)\nh=oL2(h2/planckover2pi1).\nFinally, since |∂ya|/lessorsimilara1\n2, we deduce that h2Bh∂yav(1)\nh=oL2(h2/planckover2pi1). The proof of Lemma 5.4is now\ncomplete. /square\nOur next goal is to show that [ Gh,[Gh,Ph]]v(1)\nhis a remainder. The argument is slightly more\ntricky. To this end, we need to exploit an extra smallness fro m the operator Gh. This in turns requires\nto show that /ba∇dblA(a)1\n2v(1)\nh/ba∇dblL2has the same order of /ba∇dbla1\n2v(1)\nh/ba∇dblL2. The key observation is that, modulo\nOL(L2)(h3), the operator [ Gh,[Gh,Ph]] is self-adjoint, thus we can perform the same energy estim ate\nfor the anti-selfadjoint part only, as in the proof of (a) in L emma2.2.\nLemma 5.5. Ifhδ−1\n3\nh=o(1), we have\n/ba∇dblA(a)1\n2v(1)\nh/ba∇dblL2=o(/planckover2pi1).\nProof.Recall the notation that rhrepresents the error terms of size oL2(h/planckover2pi12), from Lemma 5.4and\n(5.5), we have the equation\n/parenleftbig\nPh+ihQh+1\n2[Gh,[Gh,Ph]]/parenrightbig\nv(1)\nh=rh=oL2(h/planckover2pi12), (5.6)\nwhere\nQh=A(a)+i\nh[h2D2\nx,Gh].SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 23\nMultiplying by v1\nh, integrating and taking the imaginary part, we have\nh(Qhv(1)\nh,v(1)\nh)L2= Im(v(1)\nh,rh)L2−1\n2Im([Gh,[Gh,Ph]]v(1)\nh,v(1)\nh)L2.\nSince ([Gh,[Gh,Ph]])∗−[Gh,[Gh,Ph]] =OL2(h3), we have\n|(Qhv(1)\nh,v(1)\nh)L2| ≤o(/planckover2pi12). (5.7)\nTo conclude, we need to estimate |(i\nh[h2D2\nx,Gh]v(1)\nh,v(1)\nh)L2|. Note that,\ni\nh[h2D2\nx,Gh] = 2(∂xGh)hDx−ih(∂2\nxGh). (5.8)\nRecall that with b(η) =−χ(η)\n4η, we have∂j\nxGh=b(hDy)(∂j\nxA)+(∂j\nxA)b(hDy) forj= 1,2. By Lemma\n4.3,|∂j\nxA| ≤4πA(a)1−jσ.Thus forj= 1,2, we can write5\n∂j\nxGh=∂j\nxGh\nA(a)1−jσA(a)1−jσ.\nForj= 1,2, the operator∂j\nxGh\nA(a)1−jσis bounded on L2(T2), sinceσ<1\n4, we have\n|h(∂2\nxGhv(1)\nh,v(1)\nh)L2|/lessorsimilarh/ba∇dblA(a)1−2σv(1)\nh/ba∇dblL2/ba∇dblv(1)\nh/ba∇dblL2/lessorsimilarh/ba∇dblA(a)1\n2v(1)\nh/ba∇dblL2/ba∇dblv(1)\nh/ba∇dblL2.(5.9)\nDenote by /tildewideGh=∂xGh\nA(a)1−σ, which is self-adjoint, uniformly bounded on L2and commutes with A(a),\nwe have\n|(∂xGhhDxv(1)\nh,v(1)\nh)L2|=|(/tildewideGhA(a)1\n2A(a)1\n2−σhDxv(1)\nh,v(1)\nh)L2|/lessorsimilar/ba∇dblA(a)1\n2hDxv(1)\nh/ba∇dblL2/ba∇dblA(a)1\n2−σv(1)\nh/ba∇dblL2.\n(5.10)\nFromLemma 5.3, moduloanacceptableerror OL2(h3),say, wemayreplace hDxv(1)\nhbyh/planckover2pi1−1b1(/planckover2pi1Dx)v(1)\nh,\nwithb1(ξ) =ξ/tildewideψ(ξ). By the Corollary B.2and the fact that A(a)1\n2∈W1,∞(sinceA(a)∈ Dm,2,σ), we\nhave\n/ba∇dblA(a)1\n2hDxv(1)\nh/ba∇dblL2≤h/planckover2pi1−1/ba∇dbl[A(a)1\n2,b1(/planckover2pi1Dx)]v(1)\nh/ba∇dblL2+h/planckover2pi1−1/ba∇dblb1(/planckover2pi1Dx)A(a)1\n2v(1)\nh/ba∇dblL2+O(h3)\n/lessorsimilarh/ba∇dblv(1)\nh/ba∇dblL2+h/planckover2pi1−1/ba∇dblA(a)1\n2v(1)\nh/ba∇dblL2+O(h3).\nCombining this with the interpolation:\n/ba∇dblA(a)1\n2−σv(1)\nh/ba∇dblL2≤ /ba∇dblv(1)\nh/ba∇dbl2σ\nL2/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl1−2σ\nL2,\nwe get from ( 5.10) that\n|(∂xGhhDxv(1)\nh,v(1)\nh)L2|/lessorsimilarh/ba∇dblv(1)\nh/ba∇dbl1+2σ\nL2/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl1−2σ\nL2+h/planckover2pi1−1/ba∇dblv(1)\nh/ba∇dbl2σ\nL2/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl2−2σ\nL2+O(h3).\nTogether with ( 5.7) and (5.9), we finally get\n/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl2\nL2/lessorsimilarh/ba∇dblA(a)1\n2v(1)\nh/ba∇dblL2/ba∇dblv(1)\nh/ba∇dblL2+h/ba∇dblv(1)\nh/ba∇dbl1+2σ\nL2/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl1−2σ\nL2\n+h/planckover2pi1−1/ba∇dblv(1)\nh/ba∇dbl2σ\nL2/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl2−2σ\nL2+o(/planckover2pi12).\nBy Young’s inequality\nK1K2≤ǫKp\n1+CǫKp′\n2,1\np+1\np′= 1, ǫ>0,\n5Note that A(a) commutes with ∂j\nxGh.24 CHENMIN SUN\nwe deduce that\n/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl2\nL2≤1\n2/ba∇dblA(a)1\n2v(1)\nh/ba∇dbl2\nL2+Cmax{h2,(h/planckover2pi1−1)1\nσ,h2\n1+2σ}/ba∇dblv(1)\nh/ba∇dbl2\nL2+o(/planckover2pi12)\nNote thatσ<1\n4sinceβ≥4, andh3δ−1\nh=o(1), the second term on the right hand side is o(/planckover2pi12). This\ncompletes the proof of Lemma 5.5. /square\nFinally, we note that the principal symbol of −1\nh2[Gh,[Gh,Ph]] is\nq0(x,y,ξ,η) =−∂ηg∂y(2ξ∂xg+2η∂yg)+∂xg∂ξ(2ξ∂xg+2η∂yg)+∂yg∂η(2ξ∂xg+2η∂yg).\nSinceg(x,y,η) = 2b(η)A(x,y) = 2b(η)/integraltexty\n−π(a(x,y′)− A(a)(x))dy′, we deduce from Lemma 4.3, the\nfact thata,A(a)∈ Dm,2,σ,σ<1\n4, that\n|q0(x,y,ξ,η)|2≤Ca(x,y)+CA(a)(x)+q1(x,y,ξ,η),\nwhere supp( q1)∩WFm\nh(v(1)\nh) =∅. Therefore, by the sharp G˚ arding inequality,\n/ba∇dbl1\nh2[Gh,[Gh,Ph]]v(1)\nh/ba∇dblL2≤C/ba∇dbla1\n2v(1)\nh/ba∇dblL2+C/ba∇dblA(a)1\n2v(1)\nh/ba∇dblL2+O(h1\n2) =o(δh).\nHence [Gh,[Gh,Ph]]v(1)\nh=oL2(h2δh) =oL2(h/planckover2pi12).The proof of Proposition 5.2is now complete. /square\n5.2.The second averaging. In this subsection, we prove the following proposition of th e second\nnormal form reduction. Unlike the first normal form which is l ess perturbative (the operator eGhis\nnot close to the identity), we are able to make the normal form transform close to the identity, in the\nspirit of [ BZ12] (see also [ BS19],[LeS] for related applications to the Bouendi-Grushin operator s):\nProposition 5.6. There exist a real-valued symbol g1(x,y,η)inS0and the associated pseudo-differential\noperatorG1,h= Oph(g1), a function κ(x)∈W1,∞(Tx), the Fourier multiplier b(hDy), such that\nv(2)\nh:= (Id−G1,hhDx)−1v(1)\nhsatisfies the equation\n(Ph+ihA(a))v(2)\nh+ihκ(x)A(a)1\n2b(hDy)hDxv(2)\nh=r4,h=oL2(h/planckover2pi12).\nMoreover,\n/ba∇dblv(2)\nh−v(1)\nh/ba∇dblL2=O(h/planckover2pi1−1),/ba∇dblA(a)1\n2v(2)\nh/ba∇dblL2=o(/planckover2pi1).\nThe importance of the above proposition is that it makes poss ible to take the Fourier transform\ninyvariable and reduce the equation of v(2)\nhmode-by-mode to one-dimensional ordinary differential\nequations.\nTo prove Proposition 5.6, we want to average the non self-adjoint part [ h2D2\nx,Gh] in the equation\n(Ph+iA(a))v(1)\nh−[h2D2\nx,Gh]v(1)\nh=/tildewiderh=oL2(h/planckover2pi12).\nWe first identify the principal part of this lower order non-s elfadjoint part:\nLemma 5.7. We have\n[h2D2\nx,Gh]v(1)\nh=−4ih2/planckover2pi1−1(∂xA)b(hDy)/planckover2pi1Dxv(1)\nh+oL2(h/planckover2pi12).\nMoreover,\n[h2D2\nx,Gh]v(1)\nh=oL2(h2).SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 25\nProof.Recall that Gh=b(hDy)A+Ab(hDy), hence\n[h2D2\nx,Gh] = [[h2D2\nx,A],b(hDy)]+2b(hDy)[h2D2\nx,A].\nThe principal symbol of −1\nh2[[h2D2\nx,A],b(hDy)] is\nq2(x,y,ξ,η) =−2ξb′(η)(∂xa−A′(a)(x)).\nThus from Lemma 4.3,|∂xa|+|A′(a)| ≤Ca1\n2+CA(a)1\n2, thus\n|q2(x,y,ξ,η)|2≤Ca(x,y)+CA(a)(x)+q3(x,y,ξ,η),\nwhere supp( q3)∩WFm\nh(v1\nh) =∅. By the sharp G˚ arding inequality (Theorem B.2),\n/ba∇dbl[[h2D2\nx,A],b(hDy)]v(1)\nh/ba∇dblL2≤O(h5\n2) =o(h/planckover2pi12).\nIt remains to treat b(hDy)[h2D2\nx,A]. Note that [ h2D2\nx,A] =h2(D2\nxA)+2h(DxA)hDx.From Lemma\n5.3we may replace v(1)\nhby/tildewideψ(/planckover2pi1Dx)v(1)\nh. Therefore,\n[h2D2\nx,A]v(1)\nh=h2(D2\nxA)v(1)\nh+2h2/planckover2pi1−1b(hDy)(DxA)/planckover2pi1Dx/tildewideψ(/planckover2pi1Dx)v(1)\nh+OL2(h3).\nFrom Lemma 4.3,|Dj\nxA|/lessorsimilarA(a)1−jσ≤ A(a)1\n2forj= 1,2, we have\nh2b(hDy)(D2\nxA)v(1)\nh=oL2(h2/planckover2pi1).\nNext, we write\nhb(hDy)(DxA)hDx=h2/planckover2pi1−1[b(hDy),DxA]/planckover2pi1Dx+h2/planckover2pi1−1(DxA)b(hDy)/planckover2pi1Dx.\nBy the symbolic calculus, h2/planckover2pi1[b(hDy),DxA]/planckover2pi1Dx/tildewideψ(/planckover2pi1Dx)v(1)\nh=OL2(h3/planckover2pi1−1).This completes the proof\nof Lemma 5.7. /square\nProof of Proposition 5.7.Thanks to Lemma 5.7, we can write\n(Ph+ihA(a))v(1)\nh+ihQ1,hhDxv(1)\nh=r2,h=oL2(h/planckover2pi12),\nwhere\nQ1,h=−4(∂xA)b(hDy), b(η) =−χ(η)\n4η.\nNote that the principal symbol of Q1,his independent of ξvariable. Now we perform a second normal\nform transform. Recall that\nA(x,y) =/integraldisplayy\n−π(a(x,y′)−A(a)(x))dy′.\nConsider the ansatz v(1)\nh= (1−G1,hhDx)v(2)\nh, whereG1,hwill be chosen such that G1,hhDx=\nOL(L2)(h/planckover2pi1−1). The new quasi-modes v(2)\nhsatisfy the equation\n(Ph+ihA(a)+ihQ1,hhDx)v(2)\nh−[h2D2\ny,G1,hhDx]v(2)\nh=r3,h\nwhere\nr3,h=(1−G1,hhDx)−1r2,h+(1−G1,hhDx)−1[h2D2\nx+ihA(a)+ihQ1,hhDx,G1,hhDx]v(2)\nh\n+/parenleftbig\n(1−G1,hhDx)−1−Id/parenrightbig\n[h2D2\ny,G1,hhDx]v(2)\nh. (5.11)26 CHENMIN SUN\nNote that if G1,hhDx=OL(L2)(h/planckover2pi1−1), the operator (1 −G1,hhDx) is invertible for sufficiently small h\n(thus/planckover2pi1). In particular,\nv(2)\nh= (1−G1,hhDx)−1v(1)\nh=∞/summationdisplay\nn=0(G1,hhDx)nv(1)\nh=9/summationdisplay\nn=0(G1,hhDx)nv(1)\nh+OL2(h10/planckover2pi1−10),\nwhere the last error term is oL2(h/planckover2pi12). Since from Lemma 5.3, we may replace v(1)\nhby/tildewideψ(/planckover2pi1Dx)v(1)\nh,\nmodulo an error of OL2(hN\n2) for anyN≤m, we may also replace v(2)\nhby/tildewideψ(/planckover2pi1Dx)v(2)\nhimplicitly in the\nargument below. Therefore, with G1,h= Oph(g1)/tildewideψ(/planckover2pi1Dx), we have\n(Ph+ihA(a))v(2)\nh+ihQ1,hhDxv(2)\nh+ihOph({η2,g1})hDxv(2)\nh+h2ChhDxv(2)\nh=r3,h,\nwhereChis uniformly bounded on L2(T2). Note that the last term on the right hand size is of size\nOL2(h3/planckover2pi1−1). Now we set6\ng1(x,y,η) =2b(η)\nη/integraldisplayy\n−π/parenleftbig\n∂xA(x,y′)−A(∂xA)(x)/parenrightbig\ndy′.\nThen we have 2 η∂yg1−4(∂xA)b(η) =−4A(∂xA)(x)b(η). By the symbolic calculus, modulo an error of\nsizeOL2(h3/planckover2pi1−1), wecanreplace ih(Q1,h+Oph({η2,g1}))hDxv(2)\nhby−4ihA(∂xA)b(hDy)/tildewideψ(/planckover2pi1Dx)hDxv(2)\nh.\nTherefore, the equation of v(2)\nhbecomes\n(Ph+ihA(a)−4ihA(∂xA)b(hDy)hDx)v(2)\nh=r4,h,\nwhere\nr4,h=r3,h+OL2(h3/planckover2pi1−1) =r3,h+oL2(h/planckover2pi12).\nIt is clear that /ba∇dblv(2)\nh−v(1)\nh/ba∇dblL2=O(h/planckover2pi1−1) =o(1),and from the relation\nv(2)\nh=v(1)\nh+G1,hhDxv(1)\nh+OL2(h2/planckover2pi1−2),\nwe deduce that A(a)1\n2v(2)\nh=oL2(/planckover2pi1).\nSet\nκ:=−4A(∂xA)\nA(a)1\n2,\nto complete the proof, we need to verify that\n(i)κ∈W1,∞(Tx);\n(ii)r3,h=oL2(h/planckover2pi12).\nTo verify (i), observe that A′(f)(x) =A(∂xf)(x). By Lemma 4.2, Lemma 4.3and the fact that σ<1\n4,\nwe have\n|A(∂j\nxA)| ≤ A(|∂j\nxA|)≤CA(a)1−jσ≤CA(a)1\n2,∀j= 1,2.\nThis shows that κ,κ′are bounded.\nIt remains to prove (ii). Recall ( 5.11) and the fact that (1 −G1,hhDx)−1−Id =OL(L2)(h/planckover2pi1−1), it\nsuffices to show that\n[h2D2\nx+ihA(a)+ihQ1,hhDx,G1,hhDx]v(2)\nh=oL2(h/planckover2pi12) (5.12)\n6Recall that the support of b(η) is away from η= 0.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 27\nand\n[h2D2\ny,G1,hhDx]v(2)\nh=oL2(/planckover2pi13). (5.13)\nDenote by\nA2(x,y) =/integraldisplayy\n−π/parenleftbig\n∂xA(x,y′)−A(∂xA)(x)/parenrightbig\ndy′.\nPointwise, we have\n|A2|/lessorsimilarA(|∂xA|)/lessorsimilarA(a)1−σ,|∂xA2|/lessorsimilarA(|∂2\nxA|)/lessorsimilarA(a)1−2σ\nand\n|∂yA2|/lessorsimilar|∂xA|+A(|∂xA|)/lessorsimilarA(a)1−σ,\nthanks to Lemma 4.3. Therefore,\n|∇jA2|v(2)\nh=oL2(/planckover2pi1), j= 0,1. (5.14)\nNote that by the symbolic calculus,\nih[Q1,hhDx,G1,hhDx] =ih(h/planckover2pi1−1)2[Q1,h/planckover2pi1Dx,G1,h/planckover2pi1Dx] =OL(L2)(h3/planckover2pi1−1),\nwhich isoL(L2)(h/planckover2pi12) sinceh2=o(/planckover2pi13). For the other terms, if we only apply the symbolic calculus , we\nwill gain only O(h2)+O(h3/planckover2pi1−2) for (5.12) andO(h2/planckover2pi1−1) for (5.13), which are not enough to conclude.\nWe need to open the definition of G1,h. Sinceh2=o(/planckover2pi13), it suffices to take into account the principal\npart ofG1,h. Therefore, without loss of generality, we assume that G1,h=A2(x,y)b1(hDy)/tildewideψ(/planckover2pi1Dx),\nwithb1(η) =2b(η)\nη. Note that any commutator will generate at least one more /planckover2pi1, the main contribu-\ntion of [h2D2\nx,G1,hhDx] is−2ihb1(hDy)/tildewideψ(/planckover2pi1Dx)h2D2\nx(∂xA2)v(2)\nhwhoseL2norm iso(h3/planckover2pi1−1) =o(h/planckover2pi12),\nthanks to ( 5.14). Similarly, modulo acceptable errors from the commutator s, the main contribution\nofih[A(a),G1,hhDx]v(2)\nhis\nih2/planckover2pi1−1b1(hDy)[A(a),/tildewideψ(/planckover2pi1Dx)/planckover2pi1Dx]A2v(2)\nh,\nwhoseL2norm is, thanks to ( 5.14), bounded by O(h2)/ba∇dblA2v(2)\nh/ba∇dblL2=o(h/planckover2pi12). This verifies ( 5.12). By\nthe same argument, to verify ( 5.13), we note that, modulo acceptable errors from commutators, the\nmain contribution of [ h2D2\ny,G1,hhDx]v(2)\nhis\n−ih/planckover2pi1−12h·hDyb1(hDy)/tildewideψ(/planckover2pi1Dx)/planckover2pi1Dx(∂yA2)v(2)\nh,\nwhich is of size oL2(h2) =oL2(/planckover2pi13), by (5.14). This verifies ( 5.13) and the proof of Proposition 5.6is\nnow complete. /square\n6.One-dimensional resolvent estimate\nFrom Proposition 5.6,v(2)\nhsatisfies the equation\n(Ph+ihA(a))v(2)\nh+ihκ(x)A(a)1\n2b(hDy)hDxv(2)\nh=r4,h=oL2(h/planckover2pi12), (6.1)\nand/ba∇dblv(2)\nh/ba∇dblL2=O(1),/ba∇dblA(a)1\n2v(2)\nh/ba∇dblL2=o(/planckover2pi1). In this section, we are going to show that /ba∇dblv(2)\nh/ba∇dblL2=o(1).\nSince the left hand side of ( 6.1) commutes with Dy, by taking the Fourier transform in y, are can\nreduce the analysis to a sequence of one-dimensional proble ms.28 CHENMIN SUN\n6.1.1D resolvent estimate for the H¨ older damping. In order to finish the proof of Theorem 1.2,\nit remains to prove a one-dimensional resolvent estimate. B elow we establish a slightly more general\nversion. By abusing a bit the notation, we denote by vh,E∈H2(Tx), solutions of equations\n−h2∂2\nxvh,E−Evh,E+ihW(x)vh,E+h2κh,E(x)W(x)1\n2∂xvh,E=rh,E. (6.2)\nWe assume that ( κh,E)h>0,E∈Ris a uniform bounded family in W1,∞(T;R).\nProposition 6.1. Assume that W∈ Dm,2,θ(Tx),θ≤1\n4be a non-negative function such that the set\n{W(x)>0}is a disjoint unions of finitely many intervals Ij= (αj,βj)⊂Tx,j= 1,···,land\nC−1(x−αj)1\nθ\n+≤W(x)≤C(x−αj)1\nθ\n+inIjnearαj (6.3)\nand\nC−1(βj−x)1\nθ\n+≤W(x)≤C(βj−x)1\nθ\n+inIjnearβj, (6.4)\nfor allj∈ {1,···,l}. Then there exists h0∈(0,1)andC0>0, such that for all h∈(0,h0)and all\nE∈R, the solutions vh,Eof(6.2)satisfy the uniform estimate\n/ba∇dblvh,E/ba∇dblL2≤C0h−2−θ\n2θ+1/ba∇dblrh,E/ba∇dblL2+C0h−3θ+1\n2(2θ+1)/ba∇dblW1\n2vh,E/ba∇dblL2. (6.5)\nRemark 6.2. We will reduce the proof, in the low-energy hyperbolic regim e to a known one-\ndimensional resolvent estimate (Proposition 6.9), which is the main result of [ DKl]. However, in\nthe paper of [ DKl], the final gluing argument is not clear to the author. For thi s reason as well as\nself-containedness, we will reprove Proposition 6.9(thus Theorem 1.3) in Appendix A.\nWe postpone the proof of Proposition 6.1for the moment and proceed on proving Theorem 1.2.\nLetθ=2\n2β+1andk= 2m,δ=θ\n2θ+1, and/planckover2pi1=h1+δ\n2=h1\n2δ1\n2\nh=h3θ+1\n2(2θ+1). LetW(x) =A(a)(x). By\nProposition 4.4,W∈ Dm,2,θ(Tx) andWsatisfies ( 6.3), (6.4) near the vanishing points inside the\ndamped region. Take the Fourier transform in yfor (6.1) and denote by v(2)\nh,n(x) =Fy(v(2)\nh)(x,n), we\nhave\n(−h2∂2\nx+h2n2−1+ihW(x)+h2κ(x)W(x)1\n2b(hn)∂x)v(2)\nh,n=Fyr4,h.\nRecall that /ba∇dblr4,h/ba∇dblL2(T2)=o(h/planckover2pi12) =o(h2+δ) and/ba∇dblW1\n2v(2)\nh/ba∇dblL2(T2)=o(/planckover2pi1) =o(h1+δ\n2). LetE= 1−h2n2\nandκh,E(x) =κ(x)b(hn) which is uniformly bounded in W1,∞(T) with respect to handn. Applying\nProposition 6.1for each fixed n∈Zand then taking the l2\nnnorm, by Plancherel we get\n/ba∇dblv(2)\nh/ba∇dblL2/lessorsimilarh−2−δ/ba∇dblr4,h/ba∇dblL2(T2)+h−1+δ\n2/ba∇dblW1\n2v(2)\nh/ba∇dblL2(T2)=o(1).\nRecall that from Proposition 5.2and Proposition 5.6,\nv(2)\nh= (Id−G1,hhDx)−1v(1)\nh= (Id−G1,hhDx)−1◦eGhvh.\nThus/ba∇dblvh/ba∇dblL2=o(1), and this contradicts to ( 2.1). The proof of Proposition 2.1(as well as Theorem\n1.2) is now complete.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 29\nNow we prove Proposition 6.1. In what follows, we note that δ=θ\n2θ+1≤1\n6. We argue by\ncontradiction. If ( 6.5) is untrue, by normalization, we may assume that there exist sequenceshn→\n0,(En)n∈N⊂Rand (vhn,En)n∈N⊂L2, (rhn,En)n∈N⊂L2, such that\n(−h2\nn∂2\nx−En+ihnW(x)+h2\nnκhn,En(x)W(x)1\n2∂x)vhn,En=rhn,En (6.6)\nand\n/ba∇dblvhn,En/ba∇dblL2= 1,/ba∇dblrhn,En/ba∇dblL2=o(h2+δ\nn),/ba∇dblW1\n2vhn,En/ba∇dblL2=o(h1+δ\n2n). (6.7)\nIn what follows, when we use the asymptotic notations as smal loandO, we mean a limit (or bound)\nindependent of the sequences hn→0 andEn, asn→ ∞. To simplify the notation, we will sometimes\nomit the subindex nin the sequel. For a function f, sometimes we denote by f′=∂xf. Also, when\nwe write /lessorsimilar,/greaterorsimilar, the implicit bounds are independent ofhandE.\nWe record an elementary weighted energy identity which allo ws us to deal with the elliptic regime\nwhereE≪h2.\nLemma 6.3 (Weighted energy identity) .Letw∈C2(T;R), then\n/integraldisplay\nTw(x)|h∂xvh,E|2dx+/integraldisplay\nT(−1\n2h2∂2\nxw−Ew)|vh,E|2dx−h2\n2/integraldisplay\nT(wκh,EW1\n2)′|vh,E|2dx= Re/integraldisplay\nTwrh,Evh,Edx.\n(6.8)\nProof.Multiplying ( 6.6) bywvh,Eand integrating over T, taking the real part and using the relation\n(|vh,E|2)′= 2Re(v′\nh,Evh,E), we get\nRe/integraldisplay\nTh2(wvh)′v′\nh,Edx−/integraldisplay\nTEw|vh,E|2dx−h2\n2/integraldisplay\nT(wκh,EW1\n2)′|vh,E|2dx= Re/integraldisplay\nTwrh,Evh,Edx.\nTo finish the proof, we just write Re( w′vh,Ev′\nh,E) =1\n2w′(|vh,E|2)′and integrate by part. /square\nBy choosing w= 1 and using the fact that κ′\nh,Eis uniformly bounded in L∞(T) and (W1\n2)′/lessorsimilarW1\n4,\nwe have\nh2|((κh,EW1\n2)′,|vh,E|2)L2|/lessorsimilarh2/ba∇dblW1\n8vh,E/ba∇dbl2\nL2≤h2/ba∇dblvh,E/ba∇dbl3\n2\nL2/ba∇dblW1\n2vh,E/ba∇dbl1\n2\nL2=o(h9+δ\n4).\nSinceδ≤1\n6, we have:\nCorollary 6.4 (Energy identity) .There holds\n/ba∇dblh∂xvh,E/ba∇dbl2\nL2−E/ba∇dblvh,E/ba∇dbl2\nL2=o(h2+δ).\nThe proof of Proposition 6.1will be divided into several steps, according to the range of E.\n•(A) Elliptic regime E≪h2:Recall that Wis supported on disjoint intervals Ij= (αj,βj)⊂\n(−π,π),j= 1,···,l. Therefore, we are able to construct a weight w∈C2(T;R) such that\nw≥c0>0, w′′<0,in a neighborhood of T\\∪l\nj=1Ij.\nTherefore, there exists c1>0, sufficiently small, such that\n−1\n2w′′(x)−c1w>0,in a neighborhood of T\\∪l\nj=1Ij.30 CHENMIN SUN\nLemma 6.5. IfE≤c1h2, the solution vhn,Ensatisfies\n/ba∇dblvh,E/ba∇dblL2/lessorsimilarh−2/ba∇dblrh,E/ba∇dblL2+/ba∇dblW1\n2vh,E/ba∇dblL2.\nProof.SinceE≤c1h2, we have1\n2h2w′′+Ew <0 in a neighborhood of T\\∪l\nj=1Ij.Thus there exists\na compact set K⊂ ∪l\nj=1Ijsuch that\n/integraldisplay\nT(1\n2h2∂2\nxw+Ew)|vh,E|2dx≤/integraldisplay\nK(1\n2h2∂2\nxw+Ew)|vh,E|2dx/lessorsimilarh2/integraldisplay\nTW(x)|vh,E|2dx.\nThen applying Lemma 6.3, we have\nc0/ba∇dblh∂xvh,E/ba∇dbl2\nL2≤/integraldisplay\nTw(x)|h∂xvh,E|2dx\n≤Re/integraldisplay\nTwrh,Evh,Edx+h2/integraldisplay\nTW(x)|vh,E|2dx+Ch2\n2/integraldisplay\nT(wκh,EW1\n2)′|vh,E|2dx.\nNote that |(wκh,EW1\n2)′|/lessorsimilarW1\n4(x), by interpolation\n/ba∇dblW1\n8vh,E/ba∇dbl2\nL2≤ /ba∇dblW1\n2vh,E/ba∇dbl1\n2\nL2/ba∇dblvh,E/ba∇dbl3\n2\nL2\nand Young’s inequality, we deduce that\n/ba∇dblv′\nh,E/ba∇dblL2≤Ch−1/ba∇dblvh,E/ba∇dbl1\n2\nL2/ba∇dblrh,E/ba∇dbl1\n2\nL2+Cǫ/ba∇dblW1\n2vh,E/ba∇dblL2+ǫ/ba∇dblvh,E/ba∇dblL2,∀ǫ>0.\nBy the Poincar´ e-Wirtinger inequality,\n/vextenddouble/vextenddoublevh,E−/hatwidevh,E(0)/vextenddouble/vextenddouble\nL2(T)≤C/ba∇dblv′\nh,E/ba∇dblL2,\nwhere/hatwidevh,E(0) =1\n2π/integraltext\nTvh,E. Combining with the fact that/integraltext\nTW >0 and the elementary inequality\n/parenleftBig/integraldisplay\nTWdx/parenrightBig\n|/hatwidevh,E(0)|2≤C/integraldisplay\nTW(x)|vh,E(x)|2dx+C/integraldisplay\nTW(x)|vh,E(x)−/hatwidevh,E(0)|2dx,\nwe deduce that\n/ba∇dblvh,E/ba∇dblL2+/ba∇dblv′\nh,E/ba∇dblL2/lessorsimilarh−1/ba∇dblwvh,E/ba∇dbl1\n2\nL2/ba∇dblrh,E/ba∇dbl1\n2\nL2+/ba∇dblW1\n2vh,E/ba∇dblL2.\nUsingYoung’s inequality again to absorb /ba∇dblvh,E/ba∇dblL2to theleft, wecomplete theproofof Lemma 6.5./square\n•(B) High energy hyperbolic regime E > h1+δ:In this regime, we put the damping terms to\nthe right as remainders and use the estimate from the geometr ic control as a black box. Let us recall:\nLemma 6.6. LetI⊂Tbe a non-empty open set. Then there exists C=CI>0, such that for any\nv∈L2(T),f1∈L2(T),f2∈H−1(T),λ≥1, if\n(−∂2\nx−λ2)v=f1+f2,\nwe have\n/ba∇dblv/ba∇dblL2(T)≤Cλ−1/ba∇dblf1/ba∇dblL2(T)+C/ba∇dblf2/ba∇dblH−1(T)+C/ba∇dblv/ba∇dblL2(I).\nThe proof is standard and can be found, for example in [ Bu19] (Proposition 4.2). In the one-\ndimensional setting, a straightforward proof using the mul tiplier method is also available. Conse-\nquently, we have:SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 31\nCorollary 6.7. IfE >h1+δ, then\n/ba∇dblvh,E/ba∇dblL2(T)/lessorsimilarh−3+δ\n2/ba∇dblrh,E/ba∇dblL2(T)+h−1+δ\n2/ba∇dblWvh,E/ba∇dblL2(T)+/ba∇dblW1\n2vh,E/ba∇dblL2.\nProof.Letλ=h−1E1\n2(≥h−1−δ\n2), then\n(−∂2\nx−λ2)vh,E=h−2rh,E−ih−1Wvh,E−κh,EW1\n2∂xvh,E.\nApplying Lemma 6.6tov=vh,E,f1=h−2rh,E−ih−1Wvh,E,f2=−κh,EW1\n2v′\nh,EwithI= (α1+\nǫ0,β1−ǫ0) for someǫ0<β1−α1\n2, we get\n/ba∇dblvh,E/ba∇dblL2(T)/lessorsimilarǫ0λ−1/ba∇dblh−2rh,E−ih−1Wvh,E/ba∇dblL2(T)+/ba∇dblκh,EW1\n2v′\nh,E/ba∇dblH−1(T)+/ba∇dblvh,E/ba∇dblL2(I)\n/lessorsimilarh−3+δ\n2/ba∇dblrh,E/ba∇dblL2+h−1+δ\n2/ba∇dblWvh,E/ba∇dblL2+/ba∇dbl(κh,EW1\n2vh,E)′−(κh,EW1\n2)′vh,E/ba∇dblH−1(T).\nSince (κh,EW1\n2)′/lessorsimilarW1\n4, the last term on the right hand side is bounded by\nC/ba∇dblW1\n4vh,E/ba∇dblL2≤C/ba∇dblW1\n2vh,E/ba∇dbl1\n2\nL2/ba∇dblvh,E/ba∇dbl1\n2\nL2.\nBy Young’s inequality, we obtain the desired estimate. /square\n•(C) Low energy hyperbolic regime: c1h2<E≤h1+δ\nAgain we denote by λ=h−1E1\n2, thenλ≤h−1−δ\n2. In this situation, the non self-adjoint term\nh2κh,EW1\n2v′\nh,Ecan be absorb to the right as a remainder:\nLemma 6.8. Assume that E≤h1+δ, then\n/ba∇dblW1\n2v′\nh,E/ba∇dblL2(T)/lessorsimilarh−2/ba∇dblrh,E/ba∇dblL2+h−1−δ\n2/ba∇dblW1\n2vh,E/ba∇dblL2+/ba∇dblW1\n2vh,E/ba∇dbl1\n2\nL2/ba∇dblvh,E/ba∇dbl1\n2\nL2.\nProof.With the notation λ=h−1E1\n2,vh,Esolves the equation\n−v′′\nh,E−λ2vh,E+ih−1Wvh,E=h−2rh,E−κh,EW1\n2v′\nh,E. (6.9)\nDoing the integration by part and inserting the equation ( 6.9), we have\nRe/integraldisplay\nTWv′\nh,Ev′\nh,Edx=−Re/integraldisplay\nTW′v′\nh,Evh,Edx−Re/integraldisplay\nTWv′′\nh,Evh,Edx\n=−Re/integraldisplay\nTW′v′\nh,Evh,Edx\n−Re/integraldisplay\nTWvh,E(−λ2vh,E+ih−1Wvh,E−h−2rh,E+κh,EW1\n2v′\nh,E)dx.\nBy writing Re( v′\nh,Evh,E) =1\n2(|vh,E|2)′, we have\n−Re/integraldisplay\nTW′v′\nh,Evh,Edx=1\n2/integraldisplay\nTW′′|vh,E|2dx/lessorsimilar/ba∇dblW1\n4vh,E/ba∇dbl2\nL2≤ /ba∇dblW1\n2vh,E/ba∇dblL2/ba∇dblvh,E/ba∇dblL2,\nwhere we used |W′′|/lessorsimilarW1\n2. Writing\nRe/integraldisplay\nTWvh,Eκh,EW1\n2v′\nh,Edx=1\n2/integraldisplay\nTW3\n2κh,E(|vh,E|2)′dx=−1\n2/integraldisplay\nT(κh,EW3\n2)′|vh,E|2dx,32 CHENMIN SUN\none verifies that/vextendsingle/vextendsingle/vextendsingleRe/integraldisplay\nTWvh,E·κh,EW1\n2v′\nh,Edx/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dblW1\n2vh,E/ba∇dbl2\nL2.\nSinceλ2≤h−(1−δ), we have\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTκh,EW3\n2λ2|vh,E|2dx/vextendsingle/vextendsingle/vextendsingle/lessorsimilarh−(1−δ)/ba∇dblW1\n2vh,E/ba∇dbl2\nL2.\nThe last term\nRe/integraldisplay\nTWvh,E(−ih−1Wvh,E−h−2rh,E)dx=h−2Re/integraldisplay\nTWvh,Erh,Edx,\nand it can be easily controlled by h−2/ba∇dblrh,E/ba∇dblL2/ba∇dblW1\n2vh,E/ba∇dblL2.Putting the bounds together, we complete\nthe proof of Lemma 6.8. /square\nThe importance of Lemma 6.8is that, in the low energy hyperbolic regime, the term h2κh,EW1\n2v′\nh,E\nhas the same size o(h2+δ) inL2, and thus can be absorbed as a remainder.\nAt this state, we are able to apply the following 1D resolvent estimate for the H¨ older-like damping\nin [DKl]:\nProposition 6.9 ([DKl]).Letγ≥0. Assume that W=W(x)≥0and{W >0}is disjoint unions\nof intervals Ij= (αj,βj),j= 1,2,···,land that for each j∈ {1,···,l},\nC1Vj(x)≤W(x)≤C2Vj(x)on(αj,βj),\nwhereVj(x)>0are continuous functions on (αj,βj)such that\nVj(x) =\n\n(x−αj)γ, αj<x<3αj+βj\n4\n(βj−x)γ,αj+3βj\n4<x<βj.(6.10)\nThen there exist h0>0,c1>0,C >0, such that for all 0< h < h 0,√c1≤λ≤h−1−δ\n2and all\nsolutionsvh,λof the equation\n−v′′\nh,λ−λ2vh,λ+ih−1W(x)vh,λ=rh,λ,\nwe have\n/ba∇dblvh,λ/ba∇dblL2≤Ch−1\nγ+2/ba∇dblrh,λ/ba∇dblL2. (6.11)\nThe proof of Proposition 6.9will be given in Appendix A.\nNow applying Proposition 6.9forγ=1\nθ(then1\nγ+2=δ=θ\n2θ+1),vh,λ=vh,Eandrh,λ=h−2rh,E−\nκh,EW1\n2v′\nh,Ein our previous setting, combining with Lemma 3.2, we deduce that when c1h2≤E <\nh1+δ,\n/ba∇dblvh,E/ba∇dblL2≤Ch−2−δ/ba∇dblrh,E/ba∇dblL2+h−1+δ\n2/ba∇dblW1\n2vh,E/ba∇dblL2+h−δ/ba∇dblW1\n2vh,E/ba∇dbl1\n2\nL2/ba∇dblvh,E/ba∇dbl1\n2\nL2.(6.12)\nFinally, to get a contradiction, we denote three index sets f or three regimes:\nEA:={n:En≤c1h2\nn},EB:={n:En>h1+δ\nn},EC:={n:c1h2\nn<En≤h1+δ\nn}.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 33\nClearly,N=EA∪EB∪EC. From Lemma 6.5, Corollary 6.7and (6.12), we deduce that\nlimn→∞\nn∈S/ba∇dblvhn,En/ba∇dblL2= 0,∀S∈ {A,B,C},\nand this implies that /ba∇dblvhn,En/ba∇dblL2=o(1), a contradiction to ( 6.7). This finishes the proof of Proposition\n6.1. In summary, the proof of Theorem 1.2is now complete.\n7.Optimality\nLetr0∈(0,π) andβ≥10. Consider the function\na0(x,y) =cβ(r2\n0−x2−y2)β\n+\n(r0+|x|)β+1\n2, cβ= 2π/parenleftBig/integraldisplay1\n−1(1−ρ2)β/parenrightBig−1\n.\nObviously, the damped region of a0is the disc ω0:={|(x,y)|< r0}, which is strictly convex with\npositive curvature. Moreover, close to |(x,y)|=r0,a0(x,y) vanishes exactly of order ( r0−|(x,y)|)β\n+.\nTherefore,a0verifies all the hypothesis of the damping function in Theore m1.2.\nThe averaged damping of a0along e 2is given by\nb0(x) :=1\n2π/integraldisplayπ\n−πa0(x,y)dy,\nand direct computation yields\nb0(x) = (r0−|x|)β+1\n2\n+,as|x|close tor0.\nThe goal of this section is to show that\n/ba∇dbl(−h2∆−1+iha0)−1/ba∇dblL(L2)/greaterorsimilarh−2−1\nγ+2,withγ=β+1\n2. (7.1)\nThe idea is to use the quasimodes constructed in [ Kl] that saturates the lower bound of\n/ba∇dbl(−h2∆−1+ihb0(x))−1/ba∇dblL(L2)\nand use the averaging argument to transform back to obtain th e desired quasimodes for the operator\n−h2∆−1+iha0(z). In order to control remainders appearing in the normal for m analysis, rather than\nusing the construction in [ Kl] as a blackbox, we should keep track of the regularity (aniso tropic) of\nthe quasimodes. For this reason, we will briefly review the co nstruction of [ Kl] and prove some extra\nestimates in the next subsection.\n7.1.Estimates for the T2quasimodes. First we review the construction of quasimodes of −h2∆−\n1 +ihb0(x) in [Kl]. The original idea for the construction dates back to the ap pendix in [ AL14] by\nNonnenmacher.\nRecall that b0(x) = (r0−|x|)γ\n+, we consider the ansatz\nuk(x,y) := eikyvhk(x),\nwith 0< hk/lessorsimilar1\n|k|to be specified later. We fix δ=1\nγ+2throughout this section. As before, we will\ndrop the dependence in kand write simply h=hk,vh=vhk. Plugging into the equation\n(−h2∆−1+ihb0(x))uh=OL2(h2+δ),34 CHENMIN SUN\nwe would like vhto satisfy\n−h2∂2\nxvh+ihb0(x)vh= (1−h2k2)vh+OL2(h2+δ),/ba∇dblvh/ba∇dblL2∼1. (7.2)\nThis amounts to solve the eigenvalue problem:\n−h2∂2\nxvh+ihb0(x)vh=λ2\nhvh (7.3)\nwithλh=Ch+O(h1+δ). We consider the even eigenfunction vh(x) =vh(|x|) with\nvh(x) =/braceleftBigg\nvh,l(x),0≤x<r0,\nvh,r(x), r0≤x<π\nwherevh,r(x) = cos/parenleftbigλh\nhx/parenrightbig\n.The left function vh,lshould satisfy the equation\n−h2∂2\nxvh,l+ih(r0−x)γ\n+vh,l=λ2\nhvh,l,0≤x<r0. (7.4)\nIn order to ensure vh∈H2, the function vh,lshould satisfy the compatibility condition\nvh,l(r0) = cos/parenleftbigλhr0\nh/parenrightbig\n, v′\nh,l(r0) =−λh\nhsin/parenleftbigλhr0\nh/parenrightbig\n. (7.5)\nSince the mass in the damped region is very small compared wit h the total mass, and the amplitude\nof the transmitted mass should be the same size as the reflecte d mass, so we expect that |v′\nh,l(r0)| ≫\n|vh,l(r0)|. Asλh\nhis of sizeO(1), the principal part of the argumentλhr0\nhmust belong to π/parenleftbig\nl+1\n2/parenrightbig\n,l∈Z.\nTherefore, we take the following ansatz for λh:\nλh=π/parenleftbig\nl+1\n2/parenrightbig\nh\ny0+O(h1+δ). (7.6)\nNext, denote by F(x;θ) theH1(R+) solution of the Neumann problem (with a parameter θ∈C)\n/braceleftBigg\n−F′′(x)+ixγF−θF= 0, x>0\nF′(0) = 1,(7.7)\nthenvh,l(x) takes the form hδαhF/parenleftbigr0−x\nhδ;θh/parenrightbig\n, withθh=h−2(γ+1)\nγ+2λ2\nh, and a constant O(1) =αh∈Cto\nbe determined in order to match the compatibility condition (7.5).\nDenote byµ0be the lowest Neumann eigenvalue of the operator −∂2\ny+xγonL2(R+) andF0the\nDirichelet trace F(x;0)|x=0. It was shown that µ0>0 (Lemma 4.1 of [ Kl]). Moreover, by using the\nimplicit function theory (Lemma 4.2 of [ Kl]), there exists a uniform constant C0>0, such that for all\n|θ| ≤µ0\n2, the (unique) solution F(y;θ) of the Neumann problem ( 7.7) satisfies\n1\nC0≤ |F(0;θ)| ≤C0. (7.8)\nIn particular, F0=F(0;0)/\\e}atio\\slash= 0.\nNow we are ready to solve ( 7.4) with the compatibility condition ( 7.5). In view of ( 7.6), we precise\nthe ansatz of λhas\nλh=π/parenleftbig\nl+1\n2/parenrightbig\nh\nr0+γhh1+δ, O(1) =γh∈C.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 35\nNowαh,γhare the parameters to be determined. Plugging into ( 7.5) with\nvh,l(x) =hδαhF/parenleftbigr0−x\nhδ;θh/parenrightbig\n,\nwe obtain a system\nαhF(0;θh) = (−1)l+1h−δsin(γhr0hδ),\nαh= (−1)l/bracketleftbigπ/parenleftbig\nl+1\n2/parenrightbig\nr0+γhhδ/bracketrightbig\ncos(γhr0hδ). (7.9)\nSince|θh| ∼h2\nγ+2, by Taylor expansion of sin and cos, the leading term of γhshould beγ0:=π/parenleftbig\nl+1\n2/parenrightbig\nF0\nr2\n0\nand the leading term for αhshould beα0= (−1)lπ/parenleftbig\nl+1\n2/parenrightbig\ny0. Fix the number l, by using the implicit\nfunction theorem, the solution ( αh,γh) to (7.9) exists for 0 ≤h≪1 and\n|(αh,γh)−(α0,γ0)|/lessorsimilarhδ.\nFor the detailed argument, we refer Lemma 4.3 and Lemma 4.4 of [Kl].\nFinally, we take h=hk=r0 /radicalBig\nk2r2\n0+π2(l+1\n2)2, then 1−h2\nkk2=λ2\nhk+O(h2+δ\nk), hence\nuh(x,y) =eikyvh(x) =eiky/bracketleftbig\ncos/parenleftbigλh|x|\nh/parenrightbig\n1r0<|x|≤π+hδαhF/parenleftbigr0−|x|\nhδ;ηh/parenrightbig\n1|x|≤r0/bracketrightbig\n(7.10)\nare the desired T2quasimodes, satisfying\n−h2∆uh−uh+ihb0(x)uh=OL2(h2+δ).\nWe are going to prove more estimates on uhas well as its transformations:\nProposition 7.1. Letuhis given by (7.10)andψ∈C∞\nc(R)be a bump function supported near 0.\nThen/tildewideuh:=ψ(h1+δ\n2Dx)uhsatisfies\n(−h2∆−1+ihb0(x))/tildewideuh=OL2(h2+δ).\nMoreover, there exist uniform constants h0>0,C1>0, such that for all 0<h<h 0:\n(a)1\nC1≤ /ba∇dbl/tildewideuh/ba∇dblL2≤C1,/ba∇dblb0(x)1\n2/tildewideuh/ba∇dblL2≤C1h1+δ\n2.\n(b)/ba∇dblhj∂j\nx/tildewideuh/ba∇dblL2≤C1hj(1−δ)+δ\n2and/ba∇dblhj∂j\ny/tildewideuh/ba∇dblL2≤C1, forj= 1,2.\n(c)/ba∇dblb′\n0(x)∂x/tildewideuh/ba∇dblL2≤C1hδ,/vextenddouble/vextenddouble/vextenddouble/parenleftBig/integraltexty\n−π(∂xa0(x,y′)−b′\n0(x))dy′/parenrightBig\n∂x/tildewideuh/vextenddouble/vextenddouble/vextenddouble\nL2≤C1hδ.\nMoreover, for any S0symbolg(x,y,η)in(y,η), smoothly depending on x, if/tildewidevh=e−Opw\nh(g)/tildewideuh, the\nabove estimates (a),(b),(c)still hold, up to some different uniform constant C2>0.\nWe need a Lemma:\nLemma 7.2. Letµ0be the least eigenvalue of the operator Aγ=−∂2\nx+xγonL2(R+)associated\nto the Neumann boundary condition. Then there exists a unifor m constant C >0, such that for all\n|θ| ≤µ0\n4, the solution F(x;θ)of(7.7)satisfies\n/ba∇dblF/ba∇dblH2(R+)≤C.36 CHENMIN SUN\nProof.TakeH∈C∞\nc([0,∞)) such that H(0) = 0 and H′(0) = 1. Consider /tildewideF:=F−H, then\n−/tildewideF′′+ixγ/tildewideF−θ/tildewideF=W,\nwithW=H′′−ixγH+θH. Multiplying by /tildewideFand doing the integration by part, we get/integraldisplay∞\n0|/tildewideF′(x)|2dx+i/integraldisplay∞\n0xγ|/tildewideF(x)|2dx−θ/integraldisplay∞\n0|/tildewideF(x)|2dx=/integraldisplay∞\n0W(x)/tildewideF(x)dx.\nTaking the real part and imaginary part, we get\n/ba∇dbl/tildewideF′/ba∇dbl2\nL2(R+)≤ |Reθ|/ba∇dbl/tildewideF/ba∇dbl2\nL2(R+)+/ba∇dblW/ba∇dblL2(R+)/ba∇dbl/tildewideF/ba∇dblL2(R+),\nand\n/ba∇dblxγ\n2/tildewideF/ba∇dbl2\nL2(R+)≤ |Imθ|/ba∇dbl/tildewideF/ba∇dbl2\nL2(R+)+/ba∇dblW/ba∇dblL2(R+)/ba∇dbl/tildewideF/ba∇dblL2(R+).\nAdding two inequalities above, using Aγ−µ0≥0 and the fact that |θ| ≤µ0\n4, we get\nµ0/ba∇dbl/tildewideF/ba∇dbl2\nL2(R+)≤µ0\n2/ba∇dbl/tildewideF/ba∇dbl2\nL2(R+)+2/ba∇dblW/ba∇dblL2(R+)/ba∇dbl/tildewideF/ba∇dblL2(R+).\nThis proves the boundedness of /ba∇dbl/tildewideF/ba∇dblL2(R+)+/ba∇dbl∂x/tildewideF/ba∇dblL2(R+)+/ba∇dblxγ\n2/tildewideF/ba∇dblL2(R+)in terms of /ba∇dblW/ba∇dblL2(R+).\nReplacing /tildewideFby/tildewideF′, the same argument yields the boundedness of /ba∇dbl/tildewideF′′/ba∇dblL2(R+)in terms of /ba∇dblW/ba∇dblH1(R+).\nSince theH1bound forWis uniform with respect to |θ| ≤µ0\n4, the proof of Lemma 7.2is complete.\n/square\nProof of Proposition 7.1.Denote by /planckover2pi1=h1+δ\n2as before. The fact that /tildewideuh=ψ(/planckover2pi1Dx)uhsatisfies the\nequation of quasimodes is clear from Lemma 3.2, up to changing oL2(h/planckover2pi12) there toOL2(h/planckover2pi12). We need\nto prove estimates only.\nAt the first step, we prove the same estimates (a),(b),(c) for uh. The inequality (a) is clear by the\nconstruction and Lemma 2.2. For (b), since |k| ∼1\nh, we have /ba∇dblhj∂j\nyuh/ba∇dblL2/lessorsimilar1 for allj∈N. The\nderivatives of uhinxsatisfy\n|∂j\nxuh|/lessorsimilar1+1|x|≤r0h−(j−1)δ|(∂j\nxF)/parenleftbigr0−|x|\nhδ;θh/parenrightbig\n|,\nby Lemma 7.2, we deduce that /ba∇dblhj∂j\nxuh/ba∇dblL2/lessorsimilarhj−(j−1\n2)δ, forj= 1,2. Finally, since ∂xuh=\nF′/parenleftbigr0−|x|\nhδ;θh/parenrightbig\n,on supp(b′\n0), we have\n/ba∇dblb′\n0(x)∂xuh/ba∇dblL2/lessorsimilar/vextenddouble/vextenddouble(r0−|x|)γ−1\n+F′/parenleftbigr0−|x|\nhδ;θh/parenrightbig/vextenddouble/vextenddouble\nL2/lessorsimilarhδ\n2·hδ(γ−1)=hγ−1/2\nγ+2=h1−5δ\n2≤hδ,(7.11)\nthanks toδ=1\nγ+2andγ >2. Next,\n/vextenddouble/vextenddouble/vextenddouble/parenleftBig/integraldisplayy\n−π(∂xa0(x,y′)−b′\n0(x))dy′/parenrightBig\n∂xuh/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/parenleftBig/integraldisplayπ\n−π|∂xa0(x,y′)|dy′/parenrightBig\n∂xuh/vextenddouble/vextenddouble/vextenddouble\nL2(|x|≤r0)+/ba∇dbl|b′\n0(x)|∂xuh/ba∇dblL2(|x|≤r0).\nSince for |x| ≤r0,\n/integraldisplayπ\n−π|∂xa0(x,y′)|dy′=/integraldisplay√\nr2\n0−x2\n−√\nr2\n0−x2|∂xa0(x,y′)|dy′/lessorsimilar(r0−|x|)γ−1\n+,\nthe same computation as ( 7.11) yields\n/vextenddouble/vextenddouble/vextenddouble/parenleftBig/integraldisplayy\n−π/parenleftbig\n∂ya0(x,y′)−b′\n0(x)/parenrightbig\ndy′/parenrightBig\n∂xuh/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilarh1−5δ\n2≤hδ.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 37\nAs the second step, we deal with /tildewideuh. As the Fourier multiplier ψ(/planckover2pi1Dx) commutes with derivatives,\nthe upper bounds for /ba∇dbl/tildewideuh/ba∇dblL2,/ba∇dblhj∂j\nx/tildewideuh/ba∇dblL2and/ba∇dblhj∂j\ny/tildewideuh/ba∇dblL2follow directly from the estimates for uh,\nhence (b) holds for /tildewideuh. To prove the lower bound of /ba∇dbl/tildewideuh/ba∇dblL2, we take a smooth function χ∈C∞(R)\nsuch thatχ≡1 on the support of 1 −ψ. Then we write\nuh−/tildewideuh=/parenleftbig\n∂−1\nxχ(/planckover2pi1Dx)/parenrightbig\n·∂x(1−ψ(/planckover2pi1Dx))uh.\nSince/ba∇dbl∂xuh/ba∇dblL2/lessorsimilarh−δ/2and/ba∇dbl∂−1\nxχ(/planckover2pi1Dx)/ba∇dblL(L2)/lessorsimilar/planckover2pi1, we obtain that /ba∇dbluh−/tildewideuh/ba∇dblL2/lessorsimilar/planckover2pi1h−δ=o(1).\nTherefore, the assertions (a) follows for /tildewideuh. By the commutator estimate\n/ba∇dbl[f,ψ(/planckover2pi1Dx)]/ba∇dblL(L2)/lessorsimilar/planckover2pi1,\nforf∈W1,∞and the fact that /ba∇dbl∂xuh/ba∇dblL2/lessorsimilarh−δ\n2, we deduce that\n/vextenddouble/vextenddouble/vextenddouble/parenleftBig/integraldisplayy\n−π(∂xa0(x,y′)−b′\n0(x))dy′/parenrightBig\n∂x/tildewideuh/vextenddouble/vextenddouble/vextenddouble\nL2+/ba∇dblb′\n0(x)∂x/tildewideuh/ba∇dblL2/lessorsimilar/planckover2pi1·h−δ\n2+hδ/lessorsimilarhδ.\nThe last step is to prove estimates (a),(b),(c) for /tildewidevh=e−Gh/tildewideuhwhereGh= Opw\nh(g). Sincee−Gh\nis invertible and is uniformly bounded, we get /ba∇dbl/tildewidevh/ba∇dblL2∼1. Moreover, since Ghcommutes with the\nmultiplication by functions depending only in x, we have /ba∇dblb0(x)1\n2/tildewidevh/ba∇dblL2/lessorsimilarh1+δ\n2. So (a) holds for /tildewidevh.\nTo prove (b),(c) for /tildewidevh, from the same estimates for /tildewideuh, it suffices to control the commutator terms\ninvolving the operator e−Gh. Applying the formula\n[B2,e−Gh] = 2[B,e−Gh]B+[B,[B,e−Gh]],\ntoB=h∂x,h∂yand using ( 5.2) and (5.4), we deduce that for j= 1,2,\n/ba∇dbl[hj∂j\nx,e−Gh]/tildewideuh/ba∇dblL2/lessorsimilarhj,/ba∇dbl[hj∂j\ny,e−Gh]/tildewideuh/ba∇dblL2/lessorsimilarhj,\nhence (b) follows for /tildewidevh.\nTo estimate /ba∇dbl[b′\n0(x)∂x,e−Gh]/tildewideuh/ba∇dblL2, sinceGh,esGhboth commute with b′\n0(x), by formula ( 5.4), we\nhave\nb′\n0(x)[∂x,e−Gh] = [∂x,e−Gh]b′\n0(x).\nFrom (5.2),/ba∇dbl[∂x,e−Gh]/ba∇dblL(L2)/lessorsimilar1 and|b′\n0|/lessorsimilarb1/2\n0, we deduce that\n/ba∇dbl[b′\n0(x)∂x,e−Gh]/tildewideuh/ba∇dblL2/lessorsimilarh1+δ\n2.\nFinally, to estimate /ba∇dbl[(∂xA)∂x,e−Gh]/tildewideuh/ba∇dblL2, where\nA(x,y) :=/integraldisplayy\n−π(a0(x,y′)−b0(x))dy′,\nwe write\n[(∂xA)∂x,e−Gh]/tildewideuh= [(∂xA),e−Gh]∂x/tildewideuh+(∂xA)[∂x,e−Gh]/tildewideuh.\nSince/ba∇dbl[(∂xA),e−Gh]/ba∇dblL(L2)/lessorsimilarh, together with the estimate (b) for /tildewideuh, we have\n/ba∇dbl[(∂xA),e−Gh]∂x/tildewideuh/ba∇dblL2/lessorsimilarh1−δ\n2/lessorsimilarhδ.\nFurther commuting ∂xAand [∂x,e−Gh], by (5.4), we can write\n(∂xA)[∂x,e−Gh] = [∂x,e−Gh](∂xA)+OL(L2)(h).38 CHENMIN SUN\nTherefore,\n/ba∇dbl(∂xA)[∂x,e−Gh]/tildewideuh/ba∇dblL2/lessorsimilar/ba∇dbl(∂xA)/tildewideuh/ba∇dblL2+O(h).\nBy Lemma 4.3, together with the fact that |∇a0|/lessorsimilara1\n2\n0,|b′\n0|/lessorsimilarb1/2\n0and Jensen’s inequality, we have\n|(∂xA)(x,y)|/lessorsimilarb0(x)1\n2.Therefore, we obtain that /ba∇dbl(∂xA)/tildewideuh/ba∇dblL2/lessorsimilarh1+δ\n2≤hδ, hence (c) follows.\nThe proof of Proposition 7.1is now complete. /square\n7.2.Proof of the optimality. Take quasimodes /tildewideuhin Proposition 7.1. Define /tildewidevh=e−Gh/tildewideuh, we are\ngoing to show that /tildewidevhare the desired quasimodes, satisfying\n(−h2∆−1+iha0)/tildewidevh=OL2(h2+δ),/ba∇dbl/tildewidevh/ba∇dblL2∼1. (7.12)\nThen (7.12) implies ( 7.1).\nRecall the notations b0(x) =A(a0)(x),Gh= Opw\nh(g(x,y,η)), where\ng(x,y,η) =−ψ1(η)\n2η/integraldisplayy\n−π(a0(x,y′)−b0(x))dy′.\nFirst, we prove:\nProposition 7.3. The quasimodes /tildewidevh=e−Gh/tildewideuhsatisfy the equation\n(−h2∆−1+iha0)/tildewidevh+[h2D2\nx,Gh]/tildewidevh=OL2(h/planckover2pi12),\nwhere/planckover2pi1=h1+δ\n2. Moreover, /tildewidevhverifies properties (a),(b),(c)in Proposition 7.1and\n/ba∇dbla1\n2\n0/tildewidevh/ba∇dblL2+/ba∇dblb1\n2\n0/tildewidevh/ba∇dblL2=O(/planckover2pi1).\nProof.Theproof is very similar to the proof of Proposition 5.2, but much simpler, since we have better\nestimates for /tildewidevh=e−Gh/tildewideuh, thanks to Proposition 7.1. Consider the conjugate operator\n/tildewideFh(s) :=e−sGh(Ph+ihb0(x))esGh,\nwherePh=−h2∆−1. Similar to ( 5.5), we obtain a simpler formula\n/tildewideFh(1) =e−Gh(Ph+ihb0(x))eGh=Ph+ih/parenleftbig\nb0−i\nh[h2D2\ny,Gh]/parenrightbig\n+[h2D2\nx,Gh]+1\n2[Gh,[Gh,Ph]]+OL(L2)(h2+δ). (7.13)\nHere we use the fact that [ Gh,b0(x)] = 0. As the principal symbol ofi\nh[h2D2\ny,Gh] isψ1(η)(b0−a0),\nwe have\n(Ph+iha0+[h2D2\nx,Gh])/tildewidevh=fh+OL2(h2+δ), (7.14)\nwhere\nfh=ih/parenleftbig\nb0−a0−i\nh[h2D2\ny,Gh]/parenrightbig\n/tildewidevh−1\n2[Gh,[Gh,Ph]]/tildewidevh. (7.15)\nIt remains to show that fh=OL2(h2+δ).\nFirst of all, the analogue of Lemma 5.3holds with the same proof:SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 39\nLemma 7.4. Let/tildewideψbe any cutoff such that /tildewideψ≡1on the support of ψthat defined /tildewideuhin Proposition\n7.1. Then\nWF10\nh(/tildewidevh)⊂WFh(/tildewideuh)\nand\n/ba∇dbl(1−/tildewideψ(/planckover2pi1Dx))/tildewidevh/ba∇dblL2=O(/planckover2pi14) =O(h2+2δ).\nSimilarly, the analogue of Lemma 5.4holds. Therefore, we have\n(Ph+iha0+[h2D2\nx,Gh])/tildewidevh+1\n2[Gh,[Gh,Ph]]/tildewidevh=OL2(h2+δ).\nNext, we claim that\n/ba∇dbla1\n2\n0/tildewidevh/ba∇dblL2=O(/planckover2pi1) =O(h1+δ\n2). (7.16)\nThis follows by mimicking the proof of Lemma 5.5with the new operator\nQh=a0−i\nh[h2D2\nx,Gh].\nIndeed, by multiplying by /tildewidevhto the equation and taking the imaginary part, we have\n(Qh/tildewidevh,/tildewidevh)L2=O(/planckover2pi12).\nHence\n/ba∇dbla1\n2\n0/tildewidevh/ba∇dbl2\nL2+i\nh([h2D2\nx,Gh]/tildewidevh,/tildewidevh) =O(/planckover2pi12).\nBy (5.8), (5.9) and (5.10), we have\n/vextendsingle/vextendsinglei\nh([h2D2\nx,Gh]/tildewidevh,/tildewidevh)/vextendsingle/vextendsingle/lessorsimilarh/ba∇dblb0(x)1\n2/tildewidevh/ba∇dblL2/ba∇dbl/tildewidevh/ba∇dblL2+h/ba∇dblb0(x)1\n2hDx/tildewidevh/ba∇dblL2/ba∇dblb0(x)1\n2−σ/tildewidevh/ba∇dblL2,\nwhereσ<1\n4. Note that by Lemma 7.4, we can write /tildewidevh=ψ(/planckover2pi1Dx)/tildewidevh+OL2(h2+2δ), hence\n/ba∇dblb0(x)1\n2hDx/tildewidevh/ba∇dblL2≤h/planckover2pi1−1/ba∇dblb0(x)1\n2/planckover2pi1Dxψ(/planckover2pi1Dx)/tildewidevh/ba∇dblL2+O(h2+2δ)≤h/planckover2pi1−1/ba∇dbl[b1\n2\n0,/planckover2pi1Dxψ(/planckover2pi1Dx)]/tildewidevh/ba∇dblL2+O(h)\n≤O(h),\nthanks to the symbolic calculus. Therefore,\n/vextendsingle/vextendsinglei\nh([h2D2\nx,Gh]/tildewidevh,/tildewidevh)/vextendsingle/vextendsingle≤O(h2).\nIn particular, we obtain ( 7.16). Finally, the estimate\n/ba∇dbl[Gh,[Gh,Ph]]/tildewidevh/ba∇dblL2=O(h2+δ)\nfollows from the same argument in the last paragraph of the pr oof of Proposition 5.2. The proof of\nProposition 7.3is now complete. /square\nIn view of Proposition 7.1and Proposition 7.3, to complete the proof of ( 7.12), it remains to show\nthat\n/ba∇dbl[h2D2\nx,Gh]/tildewidevh/ba∇dblL2=O(h/planckover2pi12). (7.17)\nBy the same argument as in the proof of Lemma 5.7, we have\n[h2D2\nx,Gh]/tildewidevh=−4ih2/planckover2pi1−1(∂xA)b(hDy)/planckover2pi1Dx/tildewidevh+OL2(h/planckover2pi12),40 CHENMIN SUN\nwhere\nA(x,y) =/integraldisplayy\n−π(a0(x,y′)−b0(x))dy′, b(hDy) =−χ(hDy)\n4hDy.\nSince/ba∇dbl[(∂xA),b(hDy)]/ba∇dblL(L2)=O(h), from (c) of Proposition 7.1, we deduce that\n/ba∇dbl4ih2/planckover2pi1−1(∂xA)b(hDy)/planckover2pi1Dx/tildewidevh/ba∇dblL2≤O(h3/planckover2pi1−1)+O(h2+δ) =O(h2+δ),\nthanks to the fact that δ<1\n3. This proves ( 7.17).\nAppendix A.Proof of Proposition 6.9\nIn this appendix, we prove Proposition 6.9. Note that the proof works also for γ= 0, thus covering\nthe main result in [ St] for the piecewise constant rectangular damping. Without l oss of generality,\nwe assume that ∪l\nj=1Ij/\\e}atio\\slash=T, otherwise, we can apply Theorem 1.7 of [ LLe] and the corresponding\nresolvent estimate h2\nγ+2is much better than ( 6.11).\nRecall the numerology: δ=1\nγ+2.To simplify the notation in the exposition, we argue by contr adic-\ntion. We assume that there exists a sequence hn→0 andλn⊂Rsuch thatc1\n2\n1≤λn≤h−1−δ\n2n, such\nthat\n−v′′\nhn,λn−λ2\nnvhn,λn+ih−1W(x)vhn,λn=rhn,λn (A.1)\nand\n/ba∇dblvhn,λn/ba∇dblL2= 1,/ba∇dblrhn,λn/ba∇dblL2=o(hδ\nn).\nFor simplicity, we will ignore the subindex nforhn,λnand write simply v=vh,λ,r=rh,λsometimes\nwithout displaying their dependence in handλ. First we record the a priori estimate, for which the\nproof is a direct consequence of integration by part (see the proof of Lemma 2.2)\nLemma A.1.\n(a)/ba∇dblv′/ba∇dbl2\nL2−λ2/ba∇dblv/ba∇dbl2\nL2=o(hδ).\n(b)/ba∇dblW1\n2v/ba∇dblL2=o(h1+δ\n2).\nPickχ∈C∞(R;[0,1]) such that χ(s)≡1 whens≤1 andχ(s)≡0 whens >2. Denote by\nIj= (αj,βj) and without loss of generality, we assume that\n−π<α1<β1≤α2<β2≤ ··· ≤αl<βl≤π.\nDefine the function\nV0(x) :=l/summationdisplay\nj=1max{0,(x−αj)(βj−x)}.\nNotethatV0(x) = (x−αj)(βj−x)whenx∈Ijforsomej∈ {1,···,l}andV0(x) = 0whenever W(x) =\n0. Defineχh(x) :=χ/parenleftBig\nV0(x)3\nh3δ/parenrightBig\n. Note that χh∈C2. Denote by Ij,h= (αj,αj+2πhδ)∪(βj−2πhδ,βj)\nforj∈ {1,···,n}and/tildewideIj,h= (αj+σhδ,αj+2πhδ)∪(βj−2πhδ,βj−σhδ). Here the constant σ<2πSHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 41\nis chosen so that χh|Ijis constant on I\\/tildewideIj,h. Hence supp( χ′\nh), supp(χ′′\nh) are all contained in ∪l\nj=1/tildewideIj,h,\nand\n|χ′\nh|/lessorsimilarh−δl/summationdisplay\nj=11/tildewideIj,h,|χ′′\nh|/lessorsimilarh−2δ/summationdisplay\nj=11/tildewideIj,h. (A.2)\nWe decompose\nv=v1+v2, v1=χhv, v2= (1−χh)v.\nWhenγ >0,v2is supported in the damped region W/greaterorsimilarhαwhilev1is supported on W/lessorsimilarhα, where\nα=γ\nγ+2.\nLemma A.2. Forv2, we have\n/ba∇dblv2/ba∇dblL2=o(h3\n2(γ+2)).\nProof.First we assume that γ >0, then from (b) of Lemma A.1,\n/ba∇dblv2/ba∇dblL2≤h−α\n2/ba∇dblW1\n2v/ba∇dblL2≤o(h1+δ−α\n2) =o(h3\n2(γ+2)).\nWhenγ= 0,δ=1\n2, by Lemma 5.5,/ba∇dblv2/ba∇dblL2≤ /ba∇dblW1\n2v/ba∇dblL2≤o(h3\n4). /square\nIt remains to estimate v1. We see that v1solves the equation\n−v′′\n1−λ2v1+ih−1Wv1=/tildewider:=χhr−2(χ′\nhv)′+χ′′\nhv. (A.3)\nRemark A.3. IfWisC2, the choice of cutoff χhis the same as χ(h−αW). Note that the parameter\nαis chosen so that the size of ih−1Wvandχ′′\nhvare the same in L2. This choice of cutoff is more\naccurate than the choice χ(h−1W) in [BH05], in order to balance the size of ih−1Wvandχ′′\nhvcoming\nfrom the commutator on the right hand side of ( A.3).\nIfλis relatively large, we are still able to apply the estimate f rom the geometric control:\nLemma A.4. Ifλ≥h−δ\n2, we have\n/ba∇dblv1/ba∇dblL2/lessorsimilarhδ\n2/ba∇dblr/ba∇dblL2+h−1+δ\n2/ba∇dblW1\n2v/ba∇dblL2.\nProof.By Lemma 6.6, we have\n/ba∇dblv1/ba∇dblL2/lessorsimilarλ−1/ba∇dblχhr−ih−1Wv1+χ′′\nhv/ba∇dblL2+/ba∇dbl(χ′\nhv)′/ba∇dblH−1+/ba∇dblWv1/ba∇dblL2\n/lessorsimilarhδ\n2/ba∇dblr/ba∇dblL2+h−1+α+δ\n2/ba∇dblW1\n2v/ba∇dblL2+h−3δ\n2l/summationdisplay\nj=1/ba∇dblv/ba∇dblL2(/tildewideIj,h)+hα\n2/ba∇dblW1\n2v/ba∇dblL2,\nwhere we use the fact that W/lessorsimilarhαon supp(χh). SinceW∼hαon/tildewideIj,handα+2δ= 1, we have\n/ba∇dblv/ba∇dblL2(/tildewideIj,h)/lessorsimilarh−α\n2/ba∇dblW1\n2v/ba∇dblL2/lessorsimilarh−1\n2+δ/ba∇dblW1\n2v/ba∇dblL2.\nThis completes the proof of Lemma A.4. /square42 CHENMIN SUN\nIt remains to deal with the regime where c1\n2\n1≤λ < h−δ\n2, the key point is to exclude the possible\nconcentration of the energy density\ne0(x) :=|v′\n1(x)|2+λ2|v1(x)|2\nin the damped shell of size hδnear the interface where W= 0. The tool used in [ DKl] is a Morawetz\ntype inequality introduced:\nLemma A.5 (Morawetz type inequality) .LetΦ∈C0(T)be a piece-wise C1function on T, then there\nexists a uniform constant C >0, such that/integraldisplay\nTΦ′(x)e0(x)dx≤Ch−1/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTΦ(x)W(x)v1v′\n1dx/vextendsingle/vextendsingle/vextendsingle\n+C/vextendsingle/vextendsingle/vextendsingleRe/integraldisplay\nTΦ(x)v′\n1·(χhr−2(χ′\nhv)′+χ′′\nhv)dx/vextendsingle/vextendsingle/vextendsingle. (A.4)\nProof.Direct computation yields\n(Φe0)′= Φ′e0+2Re(Φv′′\n1v′\n1+λ2Φ(|v1|2)′).\nUsing the equation ( A.3), we have\n(Φe0)′=Φ′e0+λ2Φ∂x(|v1|2)−2λ2Re(Φv′\n1v1)+2h−1Re(iΦv′\n1·Wv1)\n−2Re[Φv′\n1(χhr−2(χ′\nhv)′+χ′′\nhv)].\nSince 2λ2Re(Φv′\n1v1) =λ2Φ·∂x(|v1|2), integrating the above identity, we obtain the desired est imate\n(A.4). /square\nLetǫj<βj−αj\n2,j∈ {1,···,n}. Define\nΨh(x) :=\n\nh−δ, x∈ ∪l\nj=1Ij,h;\n1, x∈ ∪l\nj=1(αh+2πhδ,αj+ǫj)∪(βj−ǫj,βj−2πhδ);\n−M, x∈ ∪l\nj=1Ij\\((αj,αj+ǫj)∪(βj−ǫj,βj));\n1, x∈T\\∪l\nj=1Ij\nwhere the constant M >0 (independent of h) is chosen such that/integraltext\nTΨh(x)dx= 0. Then the primitive\nfunction Φ h∈C0(T) is well-defined, piecewise smooth and Φ′\nh(x) = Ψh(x). Since Φ his unique up to\na constant, we choose Φ hsuch that Φ h(0) = 0, then /ba∇dblΦh/ba∇dblL∞(T)≤CM. Define\nΘ(x) = Φ′\nh(x)1Φ′\nh>0,\nsince supp( e0)⊂supp(χh), we have Φ′\nh(x)e0(x) = Θ(x)e0(x). From Lemma A.5andv′\n1=χ′\nhv+χhv′,\nwe have/integraldisplay\nTΘ(x)e0(x)dx≤Ch−1/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTW(x)(χ2\nhvv′+2χhχ′\nh|v|2)dx/vextendsingle/vextendsingle/vextendsingle\n+C/integraldisplay\nT|χhr(χ′\nhv+χhv′)|dx+C/integraldisplay\nT|χ′′\nhv(χ′\nhv+χhv′)|dx\n+C/vextendsingle/vextendsingle/vextendsingleRe/integraldisplay\nTχ′\nhvΦh(x)χhv′′dx/vextendsingle/vextendsingle/vextendsingle+C/integraldisplay\nT|χ′\nh(Φhχh)′vv′|dx+C/integraldisplay\nT|χ′\nhv(Φh(x)χ′\nhv)′|dx,\n(A.5)SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 43\nwhere the terms in the third line of the right side is obtained by integration by part of\n/vextendsingle/vextendsingle/vextendsingleRe/integraldisplay\nTΦh(x)v′\n1(χ′\nhv)′dx/vextendsingle/vextendsingle/vextendsingle.\nHere we keep the real part for this term in order to perform som e cancellation when replacing v′′\nby−λ2v+ih−1Wv−rlater on, after using the equation ( A.1) ofv. Denote by /tildewideIh=∪l\nj=1/tildewideIj,hand\nIh=∪l\nj=1Ij,h. Using ( A.2) and the facts that λ2≤h−δ,W∼hαon/tildewideIh, we can control terms on the\nright hand side of ( A.5) by the sum of following types:\nType I : I = h−δ/ba∇dblv1/tildewideIh/ba∇dblL2/ba∇dblr/ba∇dblL2+/ba∇dblr/ba∇dblL2/ba∇dblv′χh/ba∇dblL2;\nType II : II = h−1/integraldisplay\nTW|vv′|1/tildewideIhdx+h−1/integraldisplay\nTWχ2\nh|vv′|dx;\nType III : III = h−(1+δ)/ba∇dblW1\n2v1/tildewideIh/ba∇dbl2\nL2.\nWe analyze the type II term. By Cauchy-Schwarz,\nII/lessorsimilarh−1/ba∇dblW1\n2v1Ih/ba∇dblL2/ba∇dblW1\n21Ihv′/ba∇dblL2. (A.6)\nNote that on Ih, Θ(x) =h−δ, we have\nII/lessorsimilarh−1+α\n2/ba∇dblW1\n2v/ba∇dblL2/ba∇dbl1Ihv′/ba∇dblL2≤h−1+α\n2+δ\n2/ba∇dblW1\n2v/ba∇dblL2/parenleftBig/integraldisplay\nTΘ(x)e0(x)dx/parenrightBig1\n2.\nSinceα+2δ= 1, using Young’s inequality, we have\nII≤ǫ/integraldisplay\nTΘ(x)e0(x)dx+Cǫh−(1+δ)/ba∇dblW1\n2v/ba∇dbl2\nL2,∀ǫ>0.\nPlugging into ( A.5), we get\n/ba∇dblv1/ba∇dbl2\nL2≤/integraldisplay\nTΘ(x)e0(x)dx/lessorsimilarh−(1+δ)/ba∇dblW1\n2v/ba∇dbl2\nL2+h−δ/ba∇dblv/ba∇dblL2/ba∇dblr/ba∇dblL2+/ba∇dblr/ba∇dblL2/ba∇dblv′/ba∇dblL2.\nSinceλ≤h−δ\n2, by (a) of Lemma A.1,\n/ba∇dblv′/ba∇dbl2\nL2≤o(hδ)+λ2/ba∇dblv/ba∇dbl2\nL2≤o(h−δ).\nRecall that /ba∇dblr/ba∇dblL2=o(hδ) and/ba∇dblW1\n2v/ba∇dblL2=o(h1+δ\n2), we get\n/ba∇dblv1/ba∇dbl2\nL2/lessorsimilar/integraldisplay\nTΘ(x)e0(x)dx/lessorsimilaro(1).\nSince we have already shown that /ba∇dblv2/ba∇dblL2=o(1), this is a contradiction to our assumption that\n/ba∇dblv/ba∇dblL2= 1. The proof of Proposition 6.9is now complete.\nAppendix B.Semiclassical pseudo-differential calculus\nForm∈R, the symbol class Sm(T∗Rd) consists of smooth functions c(z,ζ) such that\n|∂α\nz∂β\nζc(z,ζ)| ≤Cα,β/a\\}b∇acketle{tζ/a\\}b∇acket∇i}htm−|α|.\nGiven a symbol c(z,ζ), we associate it with the Weyl quantization Opw\nh(c):\nf∈ S(Rd)/ma√sto→Opw\nh(f)(z) :=1\n(2πh)d/integraldisplay/integraldisplay\nR2dc/parenleftbigz+z′\n2,ζ/parenrightbig\nei(z−z′)·ζ\nhf(z′)dz′dζ.44 CHENMIN SUN\nMost of the time we will use the standard quantization Oph(c):\nOph(c)(f)(z) :=1\n(2πh)d/integraldisplay/integraldisplay\nR2dc(z,ζ)ei(z−z′)·ζ\nhf(z′)dz′dζ.\nAn important mapping property is the following theorem due t o Calder´ on-Vaillancourt:\nTheorem B.1. There exists a constant C >0such that for any function conT∗Rdwith uniformly\nbounded derivatives up to order d, we have\n/ba∇dblOph(c)/ba∇dblL(L2(Rd))+/ba∇dblOpw\nh(c)/ba∇dblL(L2(Rd))≤C/summationdisplay\n|α|,|β|≤dh|β|/ba∇dbl∂α\nz∂β\nζc/ba∇dblL∞(T∗Rd).\nWe use frequently the sharp G˚ arding inequality:\nTheorem B.2 (Sharp G˚ arding’s inequality) .Assume that c∈S0(T∗Td)andc(z,ζ)≥0for all\n(z,ζ)∈T∗Td. Then there exist C >0andh0>0such that for all 0<h≤h0andf∈L2(Td),\nRe(Oph(c)f,f)L2≥ −Ch/ba∇dblf/ba∇dbl2\nL2(Td).\nSince we deal with symbols (damping) of limited regularity i n this article, we need additional\nestimates. First we recall the following boundedness prope rty onL2:\nLemma B.1. Assume that b(x,y,ξ)∈L∞(R3d)and there exists µ0>0, such that for all |α| ≤d+1,\n|∂α\nξb(x,y,ξ)| ≤Cα/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−(|α|+µ0).\nThen the operator Thassociated with the Schwartz kernel\nKh(x,y) =1\n(2πh)d/integraldisplay\nRdb(x,y,ξ)ei(x−y)·ξ\nhdξ\nis bounded on L2(Rd), uniformly in h∈(0,1].\nProof.The proof with µ0= 1 can be found in Lemma A.1 of [ BS20]. The same argument works for\nallµ0>0. /square\nA direct consequence is the following commutator estimates for Lipschitz functions:\nCorollary B.2. Assume that κ∈W1,∞(Rd)andb∈S0(T∗Rd), then there exists C >0such that\n/ba∇dbl[κ,Oph(b)]/ba∇dblL(L2(Rd))+/ba∇dbl[κ,Opw\nh(b)]/ba∇dblL(L2(Rd))≤Ch.\nMoreover generally, for some m≥1, we have\n/ba∇dbladm\nκ(Oph(b))/ba∇dblL(L2(Rd))+/ba∇dbladm\nκ(Opw\nh(b))/ba∇dblL(L2(Rd))≤Chm.\nProof.We do the proof for the standard quantization. The same proof applied to the Weyl quantiza-\ntion. Denote by Tj= adj\nκ(Oph(b)) andKj(x,y) the Schwartz kernel of Tj. Then\nKj(x,y) = (κ(x)−κ(y))jK0(x,y),\nwhere\nK0(x,y) =1\n(2πh)d/integraldisplay\nRdb(x,ξ)ei(x−y)·ξ\nhdξSHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 45\nis the Schwartz kernel of Oph(b). Sinceκ∈W1,∞(Rd), there exists Ψ ∈L∞(R2d;Rd), such that\nκ(x)−κ(y) = (x−y)·Ψ(x,y) =d/summationdisplay\nl=1(xl−yl)Ψ(l)(x,y),\nwhere Ψ(l)is thel-th component of Ψ. Thus by integration by part,\nKm(x,y) =/summationdisplay\nj1,···,jmhm\n(2πh)d/integraldisplay\nRdei(x−y)·ξ\nh(Dξ1···Dξmb)(x,ξ)m/productdisplay\nk=1Ψ(jk)(x,y)dξ.\nApplying Lemma B.1, we deduce that /ba∇dblTm/ba∇dblL(L2(Rd))≤Chm. /square\nWe also need a special version of symbolic calculus, used in t he proof of Lemma 3.2:\nLemma B.3. Letκ∈C1\nc(R2),ϕ=ϕ(ξ)∈C∞\nc(R)andb2∈C1\nc(R2),b2∈C2\nc(R2). Denote by\nc(x,y,ξ) =κ(x,y)ϕ(ξ). Then\n(a)/ba∇dblOph(c)b1−Oph(cb1)/ba∇dblL(L2(R2))≤C1h;\n(b)/vextenddouble/vextenddoubleOph(c)b2−Oph(cb2)−h\niOph(∂ξc·∂xb2)/vextenddouble/vextenddouble\nL(L2(R2))≤C2h2,\nwhere the constants C1depend only on /ba∇dblκ/ba∇dblW1,∞,/ba∇dblb1/ba∇dblW1,∞andC2depend only on /ba∇dblκ/ba∇dblW1,∞and\n/ba∇dblb2/ba∇dblW2,∞.\nProof.Sincecdoes not depend on η, by viewing yas a parameter as in the proof of Lemma B.1,\nit suffices to view Oph(c),bas operators acting on L2(Rx) and prove the one-dimensional estimate.\nHence we will not display the dependence in yin the analysis below.\nForj= 1,2, note that the symbol of operators Oph(c)bjare given by\ncj(x,ξ) =1\n2πh/integraldisplay/integraldisplay\nR2e−ix1·ξ1\nhc(x,ξ+ξ1)bj(x+x1)dx1dξ1=1\n2π/integraldisplay\nReixξ′c(x,ξ+hξ′)/hatwidebj(ξ′)dξ′.\nNote that the Fourier transform makes sense since the functi onbjhas compact support. By the Taylor\nexpansions up to order 1 and 2:\nc(x,ξ+hξ′) =c(x,ξ)+hξ′/integraldisplay1\n0∂ξc(x,ξ+thξ′)dt,\nc(x,ξ+hξ′) =c(x,ξ)+hξ′∂ξc(x,ξ)+h2ξ′2/integraldisplay1\n0(1−t)∂2\nξc(x,ξ+thξ′)dt,\nwe have\nc1(x,ξ) =c(x,ξ)b1(x)+h/integraldisplay1\n0Φ1,t(x,ξ)dt\nand\nc2(x,ξ) =c(x,ξ)b2(x)+h\ni∂ξc(x,ξ)∂xb2(x)+h2/integraldisplay1\n0(1−t)Φ2,t(x,ξ)dt,\nwhere\nΦ1,t(x,ξ) =1\n2πi/integraldisplay\nReixξ′∂ξc(x,ξ+thξ′)/hatwidest∂xb1(ξ′)dξ′46 CHENMIN SUN\nand\nΦ2,t(x,ξ) =−1\n2π/integraldisplay\nReixξ′∂2\nξc(x,ξ+thξ′)/hatwidest∂2xb2(ξ′)dξ′.\nTherefore, the symbol of the operator Oph(c)b1−Oph(cb1) ish/integraltext1\n0Φ1,t(x,ξ)dtand the symbol of the\noperator Oph(c)b2−Oph(cb2)−h\niOph(∂ξc∂xb2) ish2/integraltext1\n0(1−t)Φt(x,ξ)dt. It suffices to show that\noperatorsTj,twith Schwartz kernels\nKj,t(x,x′) :=1\n2πh/integraldisplay\nRei(x−x′)ξ\nhΦj,t(x,ξ)dξ, j= 1,2\nare uniformly bounded on L2(R) with respect to t∈(0,1) andh∈(0,1]. Sincec(x,ξ) =κ(x)ϕ(ξ),\nexplicit computation yields\nKj,t(x,x′) =1\nijh/hatwidestϕ(j)/parenleftbigx′−x\nh/parenrightbig\nκ(x)·(∂j\nxbj)((1−t)x+tx′), j= 1,2.\nSince|Kj,t(x,x′)| ≤C1\nh/vextendsingle/vextendsingle/hatwiderϕ′′/parenleftbigx′−x\nh/parenrightbig/vextendsingle/vextendsingle. Finally, by Young’s convolution inequality, we obtain tha t\n/ba∇dblTj,tf/ba∇dblL2(R)≤C/ba∇dblf/ba∇dblL2(R)\nforj= 1,2. This completes the proof of Lemma B.3. /square\nThe above estimates can be generalized on to symbols and func tions on compact manifolds. In\nthe special situation where the manifold is Td, we still have explicit formulas. Indeed, following\n[Zw12] (Chapter 5), for a symbol c(z,ζ) onT∗Td, by periodicity c(z+ 2πk,ξ) =c(z,ζ),k∈Zd, the\nquantization is explicitly given by\nOph(c)f(z) =/summationdisplay\nk∈ZdCkf(z), Ckf(z) :=1\n(2πh)d/integraldisplay\nRd/integraldisplay\nTdc(z,ζ)ei(z−z′+2πk)·ζ\nhf(z′)dz′dζ.\nThenCk=1Tdτ−2πkOph(a)1Tdwhereτz0f(z) :=f(z−z0). By the stationary phase analysis, for\nsymbolsc∈S0(T∗Td) and for |k|>2,\n/ba∇dblCk/ba∇dblL(L2(Td))=O(hN/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht−N),∀N∈N.\nThese facts imply that Lemma B.1, Lemma B.2and Lemma B.3still hold by replacing Rd,R2,Rto\nTd,T2,T, respectively7.\nFinally, we recall the definition of the semiclassical wavef ront set, following Chapter 8 of [ Zw12].\nThesemiclassical wavefront set WFh(u) associated with a h-tempered family u= (uh)0<h≤h0is\nthe complement of the set of points ( z0,ζ0)∈T∗Tdfor which there exists a symbol c∈S0such that\na(z0,ζ0)/\\e}atio\\slash= 0 and Oph(c)uh=OL2(hN) for allN. Thesemiclassical wavefront set of order m, WFm\nh(u)\nis a modification of the definition WF h(u). It is the complement of the set of points ( z0,ζ0)∈T∗Td\nfor which there exists c∈S0withc(z0,ζ0)/\\e}atio\\slash= 0 and Oph(c)uh=OL2(hm).\n7One can also argue in local coordinate and use the partition o f unity.SHARP DECAY RATE FOR THE DAMPED WAVE EQUATION WITH CONVEX-SH APED DAMPING 47\nReferences\n[AL14] N. Anantharaman, M. L´ eautaud, Sharp polynomial decay rates for the damped wave equation on the torus ,\nAnalysis and PDE, Vol. 7, No. 1, 2014.\n[AM14] N. Anantharaman, F. Maci` a, Semiclassical measures for the Schr¨ odinger equation on th e torus, J. Euro. Math.\nSoc., 16(6), p. 1253–1288 (2014).\n[BLR] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, an d stabilization of\nwaves from the boundary , SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065.\n[BaD] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach sp aces, J. Evol. Equ.,\n8(4):765–780, 2008.\n[BoT10] A. Borichev, Y. 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PDE.\n[BS20] N. Burq, C. Sun Decay rates for Kelvin-Voigt damped wave equations II: the g eometric control condition , to\nappear in Proc. AMS.\n[BZu] N. Burq, C. Zuily, Concentration of Laplace Eigenfunctions and Stabilizatio n of Weakly Damped Wave Equation ,\nCommu. Math. Phys. volume 345, 1055—1076 (2016).\n[BZ04] N. Burq and M. Zworski, Control in the presence of a black box , J. Amer. Math. Soc. 17 (2004), 443–471.\n[BZ12] N. Burq and M. Zworski. Control for Schr¨ odinger equations on tori , Math. Research Letters, 19 (2012), 309-324.\n[Ch] H. Christianson, Semiclassical non–concentration near hyperbolic orbits , J. Funct. Anal. 246(2007), 145–195.\n[CSVW] H. Christianson, E. Schenck, A. Vasy, J. Wunsch, From resolvent estimates to damped waves , J. Anal. Math.\n122(2014), 143–162.\n[PG90] P. G´ erard, Microlocal defect measure , Commu. PDEs, 16(11): 1761—1794, 1991.\n[DKl] K. Datchev, P. Kleinhenz, Sharp polynomial decay rates for the damped wave equation wi th H¨ older-like damping ,\nProc. AMS. Vol. 148, No. 8, 3417—3425, 2020.\n[Hi1] M. Hitrik, Eigenfrequencies for damped wave equations on Zoll manifold s. Asymptot. Anal. 31 (2002), no. 3-4,\n265—277.\n[HS1] M. Hitrik, J. Sj¨ ostrand, Non-Selfadjoint Perturbations of Selfadjoint Operators i n 2 Dimensions I, Ann. Henri\nPoincar´ e, 5 (2004), 1-73.\n[HS2] M. Hitrik, J. Sj¨ ostrand, Non-Selfadjoint Perturbations of Selfadjoint Operators i n 2 Dimensions II: Vanishing\nAverages, Comm. Partial Differential Equations, 30 (2005), 1065-1106 .\n[HS3] M. Hitrik, J. Sj¨ ostrand, Non-Selfadjoint Perturbations of Selfadjoint Operators i n 2 Dimensions IIIa: One Branch-\ning Point Can. J. Math., 60 (2008), 572-657.\n[LeS] C. Letrouit, C. Sun Observability of Bouendi-Grushin type equations through r esolvent estimates . to appear in J.\nInst. Math. Jussieu, doi:10.1017/S1474748021000207.\n[Ji] L. Jin, Damped Wave Equations on Compact Hyperbolic Surfaces , Commu. Math. Phys. volume 373, 771—794\n(2020).\n[Kl] P. Kleinhenz, Stabilization rates for the damped wave equation with H¨ old er-regular damping , Commu. Math. Phys.\n369(3):1187-1205, 2019.\n[Kl2] P. Kleinhenz, Decay rates for the damped wave equation with finite regulari ty damping , to appear in Math Research\nLetters (2021).48 CHENMIN SUN\n[Le93] G. Lebeau, Equation des ondes amorties Algebraic and geometric methods in mathematical physics (K aciveli,\n1993), Math. Phys. Stud., vol. 19, Kluwer Acad. Publ., Dordr echt, 1996, pp. 73–109.\n[LLe] M. L´ eautaud, N. Lerner, Energy decay for a locally undamped wave equation , Ann. Fac. Sci. Toulouse Math. (6)\n26 (2017), no. 1, 157–205,\n[LiR] Z. Liu, B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation , Z. Angew.\nMath. Phys. 56, No.4, (2005) 630-644.\n[No] S. Nonnenmacher, Spectral theory of damped quantum chaotic systems , J.´Equ. D´ eriv. Partielles, 2011 (2011), 1–23.\n2011.\n[Ph07] K.-D. Phung. Polynomial decay rate for the dissipati ve wave equation. J. Differential Equations, 240(1):92–124 ,\n2007.\n[RaT] J. Rauch, M. Taylor, Exponential decay of solutions to hyperbolic equations in b ounded domains , Indiana Univ.\nMath. J. 24 (1974), 79–86,\n[Riv] G. Rivi` ere, with an appendix by St´ ephane Nonnenmach er and Gabriel Rivi` ere, Eigenmodes of the damped wave\nequation and small hyperbolic subsets , Ann. Inst. Fourier (Grenoble) 64(2014), 1229-1267.\n[Sj] J. Sj¨ ostrand, Asymptotic distribution of eigenfrequencies for damped wa ve equations , Publ. RIMS, Kyoto Univ. 36\n(2000), 573-611.\n[St] R. Stath, Optimal decay rate for the wave equation on a square with cons tant damping on a strip , Zeit. Angew.\nMath. Phys., 68 (2): 36, 2017.\n[Zh] H. Zhu, Stabilization of damped waves on spheres and Zoll surfaces of revolution , ESAIM: Control, Optimisation\nand Calculus of Variations 24 (4), 1759-1788.\n[Zw12] M. Zworski, Semiclassical analysis , Graduate Studies in Mathematics, (2012) AMS.\nCY Cergy-Paris Universit ´e, Laboratoire de Math ´ematiques AGM, UMR 8088 du CNRS, 2 av. Adolphe\nChauvin 95302 Cergy-Pontoise Cedex, France\nEmail address :chenmin.sun@cyu.fr"    },    {        "title": "1706.04670v2.Temperature_dependent_Gilbert_damping_of_Co2FeAl_thin_films_with_different_degree_of_atomic_order.pdf",        "content": "1 \n Temperature -dependent Gilbert damping of Co 2FeAl thin films with different degree of \natomic order  \nAnkit Kumar1*, Fan  Pan2,3, Sajid  Husain4, Serkan  Akansel1, Rimantas  Brucas1, Lars \nBergqvist2,3, Sujeet Chaudhary4, and Peter  Svedlindh1# \n \n1Department of Engineering Sciences, Uppsala University, Box 534, SE -751 21 Uppsala, \nSweden \n2Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of \nTechnology, Electrum 229, SE -16440 Kista, Sweden \n3Swedish e -Science Research Center, KTH Roy al Institute of Technology, SE -10044 \nStockholm, Sweden \n4Department of Physics, Indian Institute of Technology Delhi, New Delhi -110016, India  \n \nABSTRACT  \nHalf-metallicity and low magnetic damping are perpetually sought for in spintronics materials \nand full He usler alloys in this respect provide outstanding properties . However, it is \nchallenging to obtain the well -ordered half-metallic phase in as -deposited full Heusler alloys \nthin films and theory has struggled to establish  a fundamentals understanding of the \ntemperature dependent Gilbert damping in these systems. Here we present a study of the \ntemperature dependent Gilbert damping of differently ordered as -deposited Co 2FeAl full \nHeusler alloy thin films. The sum of inter - and intraband electron scattering in conjunction \nwith the finite electron lifetime in Bloch states govern the Gilbert damping for the well -\nordered phase in contrast to the damping of partially -ordered and disordered phases which is \ngoverned by interband electronic scat tering alone. These results, especially the ultralow room \ntemperature intrinsic damping observed for the well -ordered phase provide new fundamental \ninsight s to the physical origin of the Gilbert damping in full Heusler alloy thin films.   \n 2 \n INTRODUCTION \nThe Co-based full Heusler alloys have gained massive attention over the last decade due to \ntheir high Curie temperature and half-metallicity; 100% spin polarization of the density of \nstates at the Fermi level [1 -2]. The room temperature half- metallicity and lo w Gilbert \ndamping make them ideal candidates for magnetoresistive and thermoelectric spintronic \ndevices [3]. Co2FeAl (CFA), which is one of the most studied Co-based Heusler alloys , \nbelongs to the 𝐹𝐹𝐹𝐹 3𝐹𝐹 space group, exhibits half-metallicity  and a high C urie temperature \n(1000 K) [2, 4]. In CFA, half-metallicity is the result of hybridization between the d orbitals \nof Co and Fe. The d orbitals of Co hybridize resulting in bonding (2e g and 3t 2g) and non-\nbonding hybrids (2e u and 3t 1u). The bonding hybrids of Co further hybridise with the d \norbitals of Fe yielding bonding and anti -bonding hybrids. However, the non-bonding hybrids \nof Co cannot hybridise with the d orbitals of Fe. The half-metallic gap arises from the \nseparation of non-bonding states, i.e. the conduction band of e u hybrids and the valence band \nof t 1u hybrids [5, 6]. However, chemical or atomic disorder modifies the band hybridization \nand results in a reduc ed half-metallicity in CFA. The ordered phase of CFA is the L2 1 phase, \nwhich is half -metallic [7]. The partially ordered B2 phase forms when the Fe and Al atoms \nrandomly share their sites, while the disordered phase forms when Co, Fe, and Al atoms \nrandomly share all the sites [5-8]. These chemical disorders strongly influence the physical \nproperties and result in additional states at the Fermi level therefore reducing the half-\nmetallicity or spin polarization [7, 8]. It is challenging to obtain the ordered L2 1 phase of \nHeusler alloys in as-deposited films, which is expecte d to possess the lowest Gilbert damping \nas compared to the other phases [4, 9-11]. Therefore, in the last decade several attempts have \nbeen made to grow the ordered phase of CFA thin films employing different methods [4, 9-\n13]. The most successful attempts  used post -deposition annealing to reduce the anti -site \ndisorder by a thermal activation process [4]. The observed value of the Gilbert damping for \nordered thin films was found to lie in the range of 0.001-0.004 [7-13]. However, the \nrequirement of post -deposition annealing might not be compatible with the process constraints \nof spintronics and CMOS devices. The annealing treatment requirement for the formation of \nthe ordered phase can be circumvented by employing energy enhanced growth mechanisms \nsuch as io n beam sputtering where the sputtered species carry substantially larger  energy, ~20 \neV , compared to other deposition techniques [14, 15]. This higher energy of the sputtered \nspecies enhances the ad -atom mobility during coalescence of nuclei in the initial  stage of the \nthin film growth, therefore enabling the formation of the ordered phase. Recently we have 3 \n reported growth of the ordered CFA phase on potentially advantageous Si substrate  using ion \nbeam sputtering. The samples deposited in the range of 300°C to 500°C substrate temperature \nexhibited nearly equivalent I(002)/I(004) Bragg diffraction intensity peak ratio, which \nconfirms at least B2 order ed phase as it is difficult to identify  the formation of the L2 1 phase \nonly by X -ray diffraction analysis [16] .  \nDifferent theoretical approaches have been employed to calculate the Gilbert damping in Co -\nbased full Heusler alloys, including first principle calculations on the ba sis of (i) the torque \ncorrelation  model [17], (ii) the fully relativistic Korringa -Kohn-Rostoker model in \nconjunction with the coherent potential approximation and the linear response formalism [8] , \nand (iii) an approach considering different exchange correlation effects using both the local \nspin density approximation including the Hubbard  U and the local spin density approximation \nplus the dynamical mean field theory approximation [7]. However, very little is known about \nthe temperature dependence of the Gilbert damping in differently ordered Co-based Heusler \nalloys and a unifying conse nsus between theoretical and experimental results is still lacking. \nIn this study we report the growth of differently ordered  phases, varying from disordered to \nwell-ordered phases, of as -deposited CFA thin films grown on Si employing ion beam \nsputtering a nd subsequently the detailed temperature dependent measurements of the Gilbert \ndamping. The observed increase in intrinsic Gilbert damping with decreasing temperature in \nthe well -ordered sample is in contrast to the continuous decrease in intrinsic Gilbert  damping \nwith decreasing temperature observed for partially ordered and disordered phases. These \nresults are satisfactorily explained by employing spin polarized relativistic Korringa -Kohn-\nRostoker band structure calculations in combination with the local spin density \napproximation.  \nSAMPLES & METHODS  \nThin films of CFA were deposited on Si substrates at various growth temperatures using ion \nbeam sputtering system operating at 75W RF ion-source power ( 𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖). Details of the \ndeposition process as well as st ructural and magnetic properties of the films have been \nreported elsewhere [16]. In the present work to study the temperature dependent Gilbert \ndamping of differently ordered phases (L2 1 and B2) we have chosen CFA thin films deposited \nat 573K, 673K and 773K substrate temperature ( 𝑇𝑇𝑆𝑆) and the corresponding samples are named \nas LP573K, LP673K and LP773K, respectively. The sample thickness was kept constant at 50 \nnm and the samples were capped with a 4 nm thick Al layer. The capping layer protects the \nfilms by forming a 1.5 nm thin protective layer of Al 2O3. To obtain the A2 disordered CFA 4 \n phase, the thin film was deposited at 300K on Si employing 100W ion-source power, this \nsample is referred to  as HP300K. Structural and magnetic properties of this film are presented \nin Ref. [18]. The absence of the (200) diffraction peak in the HP300K sample [18] reveals that \nthis sample exhibits the A2 disordered structure. The appearance of the (200) pea k in the LP \nseries samples clearly indicates at least formation of B2 order [16]. Employing the Webster model along with the analysis approach developed by Takakura et al. [19] we have calculated \nthe degree of B2 ordering in the samples, S\nB2= �I200 I220⁄\nI200full orderI220full order�� , where \nI200 I220⁄  is the experimentally obtained intensity ratio of the (200) and (220) diffractions and \nI200full orderI220full order⁄  is the theoretically calculated intensity ratio for fully ordered B2 structure \nin polycrystalline films  [20]. The estimated values of SB2 for the LP573, LP673, and LP773 \nsamples are found to be ∼  90 %, 90% and 100%, respectively , as presented in Ref.  [20]. The \nI200 I400⁄  ratio of the (200) and (400) diffraction peaks for all LP series samples is ∼  30 %, \nwhich compares well with  the theoretical value for perfect B2 order [21, 22]. Here it is \nimportant to note that the L21 ordering parameter, SL21, will take different values depending \non the degree of B2 ordering. S L21 can be calculated from the I111 I220⁄  peak ratio in \nconjunction with the SB2 ordering parameter [19]. However, in the recorded grazing incident \nXRD spectra on the polycrystalline LP samples (see Fig. 1 of Ref. [16]) we did not observe \nthe (111) peak. This could be attributed to the fact t hat theoretical intensity of this peak is only \naround two percent of the (220) principal peak. The appearance of this peak is typically \nobserved in textured/columnar thicker films [19, 23 ]. Therefore, here using the experimental \nresults of the Gilbert damping, Curie temperature and saturation magnetization, in particular employing the temperature dependence of the Gilbert damping that is very sensitive to the \namount of site disorder in CFA films, and comparing with corresponding results obtained \nfrom first principle calculations, we provide a novel method for determining the type of \ncrystallographic ordering in full Heusler alloy thin films.  \nThe observed values of the saturation magnetization ( µ0MS) and coercivity ( µ0Hci), taken \nfrom Refs. [16, 18] are presented in Table I. The temperature dependence of the magnetization \nwas recorded in the high temperature region (300–1000K) using a vibrating sample \nmagnetometer i n an external magnetic field of 𝜇𝜇 0𝐻𝐻=20 mT. An ELEXSYS EPR \nspectrometer from Bruker equipped with an X -band resonant cavity was used for angle \ndependent in-plane ferromagnetic resonance (FMR) measurements . For studying the 5 \n temperature dependent spin dynamics in the magnetic thin films, an in-house built out -of-\nplane FMR setup was used. The set up, using a Quantum Design Physical Properties \nMeasurement System covers the temperature range 4 – 350 K and the magnetic field range \n±9T. The system employs an Agilent N5227A PNA network analyser covering the frequency \nrange 1 – 67 GHz and an in-house made coplanar waveguide. The layout of the system is \nshown in Fig. 1. The complex transmission coefficient ( 𝑆𝑆21) was recorded as a function of \nmagnetic field for different frequencies in the range 9-20 GHz and different temperatures in \nthe range 50-300 K.  All FMR measurements were  recorded keeping constant 5 dB power.  \nTo calculate the Gilbert damping, we have the used the torque –torque correlation model [7, \n24], which includes both intra - and interband transitions. The electronic structure was \nobtained from the spin polarized relativistic Korringa -Kohn-Rostoker (SPR- KKR) band \nstructure method [24, 25] and the local spin density approximation (LSDA) [26] was used for \nthe exchange correlation potential. Relativistic effects were taken into account by solving the Dirac equation for the electronic states, and the atomic sphere approximation (ASA) was employed for the shape of potentials. The experimental bulk value of the lattice constant [27] \nwas used. The angular momentum cut -off of 𝑙𝑙\n𝑚𝑚𝑚𝑚𝑚𝑚 =4 was used in the mu ltiple -scattering \nexpansion.  A k-point grid consisting of ~1600 points in the irreducible Brillouin zone was \nemploye d in the self -consistent calculation while a substantially more dense grid of ~60000  \npoints was employe d for the Gilbert damping calculation.  The exchange parameters 𝐽𝐽 𝑖𝑖𝑖𝑖 \nbetween the atomic magnetic moments were calculated using the magnetic force theorem \nimplemented in the Liechtenstein -Katsnelson -Antropov-Gubanov (LKAG) formalism [28, 29] \nin order to construct a parametrized mod el Hamiltonian. For the B2 and L2 1 structures, the \ndominating exchange interactions were found to be between the Co and Fe atoms, while in A2 the Co-Fe and Fe -Fe interactions are of similar size. Finite temperature properties such as the \ntemperature dependent magnetization was obtained by performing Metropolis Monte Carlo \n(MC) simulations [30] as implemented in the UppASD software [31,  32] using the \nparametrized Hamiltoni an. The coherent potential approximation (CPA) [33, 34] was ap plied \nnot only for the treatment of the chemical disorder of the system, but also used to include the \neffects of quasi -static lattice displacement and spin fluctuations in the calculation of the \ntemperature dependent Gilbert damping [35–37] on the basis of linear response theory [38].  \nRESULTS & DISCUSSION  \nA. Magnetization vs. temperature measurements  6 \n Magnetization measurements were performed with the ambition to extract values for the \nCurie temperature ( 𝑇𝑇𝐶𝐶) of CFA films with different degree of atomic order; the results a re \nshown in Fig. 2(a). Defining 𝑇𝑇𝐶𝐶 as the inflection point in the magnetization vs. temperature \ncurve, the observed values are found to be 810  K, 890 K and 900 K for the LP573K, LP773K \nand LP673K samples, respectively. The 𝑇𝑇𝐶𝐶 value for the HP300K sample is similar to the \nvalue obtained for LP573. Using the theoretically calculated exchange interactions, 𝑇𝑇𝐶𝐶 for \ndifferent degree of atomic order in CFA varying from B2 to L2 1 can be calculated using MC \nsimul ations. The volume was kept fixed as the degree of order varied between B2 and L2 1 \nand the data presented here represent the effects of differently ordered CFA phases. To obtain \n𝑇𝑇𝐶𝐶 for the different phases, the occupancy of Fe atoms on the Heusler alloy 4a sites was varied \nfrom 50% to 100%, corresponding to changing the structure from B2 to L2 1. The estimated \n𝑇𝑇𝐶𝐶 values , cf. Fig. 2 (b), monotonously increases from 𝑇𝑇 𝐶𝐶=810 K (B2) to 𝑇𝑇𝐶𝐶=950 K \n(L2 1). A direct comparison between experimental and calculated  𝑇𝑇𝐶𝐶 values is hampered by the \nhigh temperature (beyond 800K) induced structural transition from well -ordered to partially -\nordered CFA phase which interferes with the magnetic transition [39, 40]. The irreversible \nnature of the recorded magnetization vs . temperature curve  indicates a distortion of structure \nfor the ordered phase during measurement , even though interface alloying at elevated \ntemperature cannot be ruled out . The experimentally observe d 𝑇𝑇𝐶𝐶 values are presented in \nTable I.  \nB. In-plane angle dependent FMR measurements  \nIn-plane angle dependent FMR measurements were performed at 9.8 GHz frequency for all \nsamples; the resonance field 𝐻𝐻𝑟𝑟 vs. in -plane angle  𝜙𝜙𝐻𝐻 of the applied magnetic field is plotted \nin Fig. 3. The  experimental results have been fitted using the expression [41],  \n𝑓𝑓=\n𝑔𝑔∥𝜇𝜇𝐵𝐵𝜇𝜇0\nℎ��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−𝜙𝜙𝑀𝑀)+2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−\n𝜙𝜙𝑀𝑀)+ 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒+𝐾𝐾𝑐𝑐\n2𝜇𝜇0𝑀𝑀𝑠𝑠(3+cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2 (𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��12�\n, (1) \nwhere 𝑓𝑓 is resonance frequency , 𝜇𝜇𝐵𝐵 is the Bohr magneton  and ℎ is Planck constant . 𝜙𝜙𝑀𝑀, 𝜙𝜙𝑢𝑢 \nand 𝜙𝜙𝑐𝑐 are the in -plane directions of the magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively , with respect to the [100] direction of the Si substrate . 𝐻𝐻𝑢𝑢=2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠 and \n𝐻𝐻𝑐𝑐=2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠 are the in-plane uniaxial and cubic anisotropy fields , respectively, and 𝐾𝐾𝑢𝑢 and 𝐾𝐾𝑐𝑐 7 \n are the uniaxial and cubic magnetic anisotrop y constant s, respectively, 𝑀𝑀𝑠𝑠 is the saturation \nmagnetization and 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 is the effective magnetization . By considering 𝜙𝜙 𝐻𝐻 ∼ 𝜙𝜙𝑀𝑀, 𝐻𝐻𝑢𝑢 and 𝐻𝐻𝑐𝑐 \n<<𝐻𝐻𝑟𝑟<< 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, equation (1) can be simplified as:  \n𝐻𝐻𝑟𝑟=�ℎ𝑒𝑒\n𝜇𝜇0𝑔𝑔∥𝜇𝜇𝐵𝐵�21\n𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒−2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝐻𝐻−𝜙𝜙𝑐𝑐)−2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝐻𝐻−𝜙𝜙𝑢𝑢) . (2) \nThe extracted cubic anisotropy fields µ 0Hc ≤ 0.22mT are negligible for all the samples. The \nextracted in -plane Landé splitting factors g∥ and the uniaxial anisotropy fields  µ0Hu are \npresented in T able I. The purpose of the angle dependent FMR measurements was only to \ninvestigate the symmetry of the in -plane magnetic anisotropy. Therefore, care was not taken \nto have the same in -plane orientation of the samples during angle dependent FMR \nmeasurements, which explains why the maxima appear at diffe rent angles for the different \nsamples.  \nC. Out-of-plane FMR measurements  \nField -sweep out -of-plane FMR measurements were performed at different constant \ntemperatures in the range 50K – 300K and at different constant frequencies in the range of 9-\n20 GHz. Figure 1(b) shows the amplitude  of the complex transmission coefficient 𝑆𝑆21(10 \nGHz) vs. field measured for the LP673K thin film at different temperatures. The recorded \nFMR spectra were fitted using the equation [42],  \n𝑆𝑆21=𝑆𝑆�∆𝐻𝐻2��2\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐴𝐴�∆𝐻𝐻2��(𝐻𝐻−𝐻𝐻𝑟𝑟)\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐷𝐷∙𝑡𝑡,  (3) \nwhere 𝑆𝑆 represents the coefficient describing the transmitted microwave power, 𝐴𝐴 is used to \ndescribe a waveguide induced phase shift  contribution  which is, however, minute , 𝐻𝐻 is \napplied magnetic field, ∆𝐻𝐻 is the full-width of half maxim um, and 𝐷𝐷∙𝑡𝑡 describes the linear \ndrift in time  (𝑡𝑡) of the recorded signal. The extracted ∆ 𝐻𝐻 vs. frequency at different constant  \ntemperatures  are shown in Fig. 4 for all the samples. For brevity only data at a few \ntemperatures are plotted. The Gilbert damping was estimated using the equation [42 ],  \n∆𝐻𝐻=∆𝐻𝐻0+2ℎ𝛼𝛼𝑒𝑒\n𝑔𝑔⊥𝜇𝜇𝐵𝐵𝜇𝜇0   (4) \nwhere ∆𝐻𝐻0 is the inhomogeneous line -width broadening, 𝛼𝛼 is the experimental Gilbert \ndamping constant , and 𝑔𝑔⊥ is the Landé  splitting factor measured employing out -of-plane \nFMR. The insets in the figures show the temperature dependence of  𝛼𝛼. The effective 8 \n magnetization ( 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) was estimated  from the 𝑓𝑓 vs. 𝐻𝐻𝑟𝑟 curves using out -of-plane Kittel’s \nequation [43],  \n𝑓𝑓=𝑔𝑔⊥𝜇𝜇0𝜇𝜇𝐵𝐵\nℎ�𝐻𝐻𝑟𝑟−𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒�,     (5) \nas shown in Fig. 5 . The temperature dependence of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 and 𝜇𝜇0∆𝐻𝐻0 are shown as  insets  in \neach figure . The observed room temperature values of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 are closely equal  to the  𝜇𝜇0𝑀𝑀𝑠𝑠 \nvalues obtained  from static magnetizat ion measurements, presented in T able I. The extracted \nvalues of g⊥ at different temperatures are within error limits  constant for all samples. \nHowever , the difference between estimated values of g ∥ and g⊥ is ≤ 3%. This difference c ould \nstem from the limited frequency range  used since  these values are quite sensitive to the value \nof 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, and even a minute uncertainty in this quantity can result in the observed small \ndifference between  the g∥ and g⊥ values.  \nTo obtain  the intrinsic Gilbert damping (𝛼𝛼 𝑖𝑖𝑖𝑖𝑖𝑖) all extrinsic contributions to the experimental 𝛼𝛼 \nvalue  need to be  subtracted. In metallic ferromagnets , the intrinsic Gilbert damping is mostly \ncaused by electron magnon scattering, but  several other extrinsic co ntributions can also \ncontribute to the experimental value of the damping constant. One  contribution is two -\nmagnon scattering which is however minimized for the  perpendicular geometry used in this \nstudy and therefore this contribution is disregarded [44].  Another contribution is spin-\npumping into the  capping layer as the LP573K, LP673K and LP773K samples are capped \nwith 4 nm of Al that naturally forms a thin top layer consisting of  Al2O3. Since spin pumping \nin low spin-orbit coupling materials with  thickness less than the spin-diffusion length is quite \nsmall this contribution is also disregarded in all samples. However, the HP300K sample is \ncapped with Ta and therefore a spin-pumping contribution have been subtracted from the \nexperimental 𝛼𝛼 value ; 𝛼𝛼𝑠𝑠𝑠𝑠= 𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(with  Ta capping )−𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(without capping )≈\n1×10−3. The third contribution arises  from the inductive coupling between  the precessi ng \nmagnetization and the CPW , a reciprocal phenomenon of FMR, known as radiative damping \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 [45]. This damping is directly proportional to the magnetization and thickness of the thin \nfilms samples and therefore usually dominates in thicker and/or high magnetization samples. \nThe l ast contribution is eddy current damping ( αeddy) caused  by eddy current s in metallic \nferromagnetic  thin films [ 45, 46]. As per Faraday’s law the time varying magnetic flux density  \ngenerate s an AC voltage in the metallic ferromagnetic  layer and therefore result s in the eddy 9 \n current damping . Thi s damping is directly proportional to the square of the film thickness  and \ninversely proportional to the resistivity of the sample [ 45].  \nIn contrast to eddy -current damping, 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 is independent of the conductivity of the \nferromagnetic layer, hence this damping mechanism is also operati ve in ferromagnetic \ninsulators. Assuming a uniform magnetization of the sample the radiative damping can be \nexpressed as [45],  \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 𝜂𝜂𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿𝛿𝛿\n2 𝑍𝑍0𝑤𝑤 ,   (6) \nwhere 𝛾𝛾=𝑔𝑔𝜇𝜇𝐵𝐵ℏ� is the gyromagnetic ratio, 𝑍𝑍0 = 50 Ω is the waveguide impedance, 𝑤𝑤 = 240 \nµm is the width of the waveguide, 𝜂𝜂 is a dimensionless parameter which accounts for the \nFMR mode profile and depends on boundary conditions, and 𝛿𝛿 and 𝑙𝑙 are the thickness and \nlength of the sample on the waveguide, respectively. The strength of this inductive coupling \ndepends on the inductance of the FMR mode which is determined by the waveguide width, \nsample length over waveguide, sample saturation magnetization and sample thickness.  The \ndimensions of the LP573K, LP673K  and LP773K samples were 6.3×6.3 mm2, while the \ndimensions of the HP300K sample were  4×4 mm2. Th e 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 damping was estimated  \nexperimentally as explained by Schoen et al.  [45] by  placing a 200 µm thick glass spacer \nbetween  the waveguide and the sample , which decreases  the radiative damping by more than \none order magnitude as shown in Fig. 6(a). The measured radiative damping by placing the \nspacer  between the waveguide and the LP773 sample, \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟=𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ𝑖𝑖𝑢𝑢𝑖𝑖  𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟  −𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟≈ (2.36 ±0.10×10−3) − (1.57 ±0.20×10−3)=\n0.79±0.22×10−3. The estimated value matches  well with the calculated value using Eq. \n(6); 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 0.78 ×10−3. Our results are also analogous to previously reported results  on \nradiative damping [45].  The estimated temperature dependent radiative damping values for all \nsamples are shown in Fig. 6(b).   \nSpin wave precession in ferromagnetic layers induces an AC current in the conducting \nferromagnetic layer which results in  eddy current damping. It can be  expressed as [45, 46], \n𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 = 𝐶𝐶𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿2\n16 𝜌𝜌 ,   (7) \nwhere 𝜌𝜌 is the resistivity of the sample and 𝐶𝐶 accounts for the eddy current distribution in the \nsample ; the smaller the value of 𝐶𝐶 the larger is the localization of eddy currents in the sample. \nThe measured  resistivity values between 300  K to 50 K temperature range fall in the ranges \n1.175 – 1.145  µΩ-m, 1 .055 – 1.034 µΩ -m, 1 .035 – 1.00 µΩ -m, and 1. 45 – 1.41 µΩ -m for the \nLP573K, LP673 , LP773 and HP300K samples, respect ively. The parameter 𝐶𝐶 was obtained 10 \n from thickness dependent experimental Gilbert damping constants measured for B2 ordered  \nfilms,  by line ar fitting of 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 vs. 𝛿𝛿2 keeping other parameters  constant  (cf. Fig. \n6(c)). The fit to the data yield ed 𝐶𝐶 ≈ 0.5±0.1. These results are concurrent to those  \nobtained for permalloy  thin films [45]. Since the variations of the resistivity and \nmagnetization for the samples  are small , we have used the same 𝐶𝐶 value  for the estimation of \nthe eddy current damping in all the samples. The estimated temperature dependent values of \nthe eddy current damping are presented in Fig. 6(d).  \nAll these contributions have  been subtracted from the experimentally observed values of 𝛼𝛼. \nThe estimated intrinsic Gilbert damping 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values so obtained are plotted in Fig. 7(a) for all \nsamples.   \nD. Theoretical results: first principle calculations  \nThe calculated temperature dependent intrinsic Gilbert damping for Co 2FeAl phases with \ndifferent degree of atomic order are shown in Fig. 7(b). The temperature dependent Gilbert \ndamping indicates that the lattice displacements and spin fluctuations contribute differently in \nthe A2, B2 and L2 1 phases. The torque correlation model [47, 48] describes qualitatively two \ncontributions to the Gilbert damping. The first one is the intraband scattering where the band \nindex is always conserved. Since it has a linear dependence on the electron lifetime, in the \nlow temperature regime this term increases rapid ly, it is also known as the conductivity like \nscattering. The second mechanism is due to interband transitions where the scattering occurs \nbetween bands with different indices. Opposite to the intraband scattering, the resistivity like \ninterband scattering with an inverse depe ndence on the electron lifetime  increases with \nincreas ing temperature. The sum of the intra - and interband electron scattering contributions \ngives rise to a non-monotonic dependence of the Gilbert damping on temperature for the L2 1 \nstructure. In contrast to the case for L2 1, only interband scattering is present in the A2 and B2 \nphases, which results in a monotonic increase of the intrinsic Gi lbert damping with increas ing \ntemperature. This fact is also supported by a previous study [37 ] which showed that even a \nminute chemical disorder can inhibit the intraband scattering of the system. Our theoretical results manifest that the L2\n1 phase has the lowest Gilbert damping around 4.6 × 10−4 at 300 \nK, and that the value for the B2 phase is only slightly larger at room temperature. According \nto the torque correlation model, the two main contributions to damping are the spin orbit \ncoupling and the density of states (DOS) at the Fermi level [47 , 48]. Since the spin orbit \nstrength is the same for the different phases it is enough to focus the discussion on the DOS 11 \n that provide s a qualitative explanation  why damping is found lower in B2 and L2 1 structure s \ncompare d to A2 structure. The DOS at the Fermi level of the B2 phase (24.1 states/Ry/f.u;  f.u \n= formula unit ) is only slightly  larger to that of the L2 1 phase (20.2 states/Ry/f.u.) , but both \nare significantly smaller than for the A2 phase (59.6 states/Ry/f.u.) as shown in Fig. 8. The \ngap in the mi nority spin channel of the DOS for the B2 and L2 1 phases indicate half-\nmeta llicity, while the A2 phase is metallic. The atomically resolved spin polarized DOS \nindica tes that the Fermi -level states mostly have  contr ibutions from Co and Fe atoms. For \ntransition elements such as Fe and Ni, it has been reported that  the intrinsic Gilbert damping \nincreases significantly below 100K with decreas ing temperature [37]. The present electronic \nstructure calculations were performed using Green’s functions, which do rely on a \nphenomenological relaxation time parameter,  on the expense that the different contributions to \ndamping cannot  be separated eas ily. The reported results in Ref. [37] are by some means \nsimilar to our findings of the temperature dependent Gilbert damping in full Heusler alloy films with different degr ee of atomic order. The intermediate states of B2 and L2\n1 are more \nclose to the trend of B2 than L2 1, which indicates that even a tiny atomic orde r induced by the \nFe and Al site disorder will inhibit the conductivity -like channel in  the low temperature \nregion. The theoretically calculated Gilbert damping constants are matching qualitatively with \nthe experimentally observed  𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values as shown in Fig. 7. However, the theoretically \ncalculated 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 for the L2 1 phase increases rapidly below 100K, in co ntrast to the \nexperimental results for the well -ordered CFA thin film (LP673K ) indicating that  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 saturates at low temperature. This discrepancy between the theoretical and experimental \nresults can be understood taking into account the low temperature behaviour of the life time τ \nof Bloch states. The present theoretical model assum ed that the Gilbert damping has a linear \ndependence on the electron lifetime in intraband transitions which is however correct only in \nthe limit of small lifetime, i.e., 𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏≪1, where q is the magnon wave vector and 𝑣𝑣𝐹𝐹 is the \nelectron Fermi velocity. However, in the low temperature limit the lifetime  𝜏𝜏 increases and as \na result of the anomalous skin effect the intrinsic Gilbert damping saturates  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖∝ tan−1𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏𝑞𝑞𝑣𝑣𝐹𝐹�  at low temperature [37], which is evident from our experimental \nresults.  \nRemaining discrepancies between theoretical and experimental values of the intrinsic Gilbert \ndamping might stem from the fact that the samples used in the present study are 12 \n polycrystalline and because of sample imperfections these fil ms exhibit significant \ninhomogenous line -width broadening due to superposition of local resonance fields.  \nCONCLUSION  \nIn summary , we report temperature dependent FMR measurements on as -deposited Co 2FeAl \nthin films with different degree of atomic order. The degree of atomic ordering is established \nby comparing experimental and theoretical results for the temperature dependent intrinsic \nGilbert damping constant. It is evidenced that the experimentally observed intrinsic Gilbert \ndamping in samples with atomic  disorder (A2 and B2 phase samples) decreases with \ndecreasing temperature. In contrast, the atomically well -ordered sample, which we identify at \nleast partial L21 phase, exhibits an intrinsic Gilbert damping constant that increases with \ndecreasing temperat ure. These temperature dependent results are explained employing the \ntorque correction model including interband transitions and both interband as well as \nintraband transitions for samples with atomic disorder and atomically ordered phases, \nrespectively.  \nACKNOWLEDGEMENT \nThis work is supported by the Knut and Alice Wallenberg (KAW) Foundation,  Grant No.  \nKAW 2012.0031  and from Göran Gustafssons Foundation (GGS), Grant No.  GGS1403A. The \ncomputations were performed on resources provided by SNIC (Swedish National \nInfrastructure for Computing) at NSC (National Supercomputer Centre) in Linköping, \nSweden.  S. 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Lett. 99 , 027204 (2007) .  \n  15 \n Table I P arameters describing magnetic properties of the different CFA samples.  \nSample  𝜇𝜇0𝑀𝑀𝑆𝑆(𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) \n(T) 𝜇𝜇0𝐻𝐻𝑐𝑐𝑖𝑖 \n(mT)  𝑔𝑔∥(𝑔𝑔⊥) \n 𝜇𝜇0𝐻𝐻𝑢𝑢 \n(mT)  𝑇𝑇𝐶𝐶 \n(K) 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 \n(× 10-3) \nLP573K  1.2±0.1 (1.09 1±0.003)  0.75 2.06 (2.0)  1.56 810 2.56 \nLP673K  1.2±0.1 (1.11 0±0.002)  0.57 2.05 (2.0)  1.97 >900  0.76 \nLP773K  1.2±0.1 (1.081 ±0.002)  0.46 2.05 (2.0)  1.78 890 1.46 \nHP300K  0.9±0.1 (1.066 ±0.002)  1.32 2.01 (2.0)  3.12 -- 3.22  \n 16 \n Figure 1  \nFig. 1.  (a) Layout of the in -house made VNA-based out -of-plane ferromagnetic resonance  \nsetup . (b) Out -plane ferromagnetic resonance spectra recorded for the well -ordered LP673K \nsample at different temperatures  𝑓𝑓=10 GHz . \n  \n \n17 \n Figure 2  \nFig. 2. (a) Magnetization vs. temperature plots measured on the CFA films with different \ndegree of atomic order. (b) Theoretically calculated magnetization vs. temperature curves for \nCFA phases with different degree of atomic order, where 50 % (100 %) Fe atoms on Heusler \nalloy 4a sites indicate B2 (L2 1) ordered phase, and the rest are intermediate B2 & L2 1 mixed \nordered  phases.  \n  \n \n \n18 \n Figure 3  \nFig. 3.  Resonance field vs. in -plane orientation of the applied magnetic field of (a) 𝑇𝑇𝑆𝑆=\n300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇 𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited films. Red lines \ncorrespond to fits to the data using Eq. (1).  \n    \n \n \n19 \n  \nFigure 4  \nFig. 4.   Line-width vs. frequency of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to fits to the data to extract the \nexperimental Gilbert damping constant and inhomogeneous line -width. Respective insets \nshow the experimentally determined temperature dependent Gilbert damping constants.  \n  \n  \n \n \n20 \n Figure 5  \nFig. 5.  Frequency vs. applied field of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=\n400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to Kittel’s fits to the data. Respective \ninsets show the temperature dependent effective magnetization a nd inhomogeneous line -width \nbroadening values.  \n  \n  \n \n21 \n Figure 6  \nFig. 6.  (a) Linewidth vs. frequency with and without a glass spacer between the waveguide \nand the sample.  Red lines correspond to fits using Eq. (4).  (b) Temperature dependent values \nof the radiative damping using Eq.  (6). The lines are guide to the eye. (c) 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 \nvs 𝛿𝛿2. The red line corresponds to a fit  using Eq. (7) to extract the value of the correction \nfactor  𝐶𝐶. (d) Temperature dependent values of eddy current dampi ng using Eq. (7). The lines \nare guide to the eye.  \n  \n  \n \n \n22 \n Figure 7  \nFig. 7.  Experimental (a) and theoretical (b) results for the temperature dependent intrinsic \nGilbert damping constant for CFA samples with different degree of atomic order . The B2 & \nL21 mixed phase corresponds to the 75 % occupancy of Fe atoms on the Heusler alloy 4a \nsites. The lines are guide to the eye.  \n  \n \n \n23 \n Figure 8  \nFig. 8.  Total and atom -resolved spin polarized density of states plots for various \ncompositional CFA phases; (a) A2, (b) B2 and (c) L2 1. \n  \n \n \n"    },    {        "title": "2009.11947v1.Squeezing_the_Parameter_Space_for_Lorentz_Violation_in_the_Neutrino_Sector_by_Additional_Decay_Channels.pdf",        "content": "arXiv:2009.11947v1  [hep-ph]  22 Sep 2020Squeezing the Parameter Space for Lorentz Violation in\nthe Neutrino Sector by Additional Decay Channels\nUlrich D. Jentschura\nDepartment of Physics, Missouri University\nof Science and Technology, Rolla, Missouri 65409, USA; ulj@mst.edu\nMTA–DE Particle Physics Research Group,\nP.O. Box 51, H–4001 Debrecen, Hungary\nMTA Atomki, P.O. Box 51, H–4001 Debrecen, Hungary\nSeptember 28, 2020\nAbstract\nThe hypothesis of Lorentz violation in the neutrino sector has intrig ued\nscientists for the last two to three decades. A number of theoret ical argu-\nments support the emergence of such violations first and foremos t for neutrinos,\nwhich constitute the “most elusive” and “least interacting” particle s known to\nmankind. It is of obvious interest to place stringent bounds on the L orentz-\nviolating parameters in the neutrino sector. In the past, the most stringent\nbounds have been placed by calculating the probability of neutrino de cay into\na lepton pair, a process made kinematically feasible by Lorentz violatio n in\nthe neutrino sector, above a certain threshold. However, even m ore stringent\nbounds can be placed on the Lorentz-violating parameters if one ta kes into\naccount, additionally, the possibility of neutrino splitting, i.e., of neut rino de-\ncay into a neutrino of lower energy, accompanied by “neutrino-pair ˘Cerenkov\nradiation”. This process has negligible threshold and can be used to im prove\nthe bounds on Lorentz-violating parameters in the neutrino secto r. Finally, we\ntake the opportunity to discuss the relation of Lorentz and gauge symmetry\nbreaking, with a special emphasis on the theoretical models employe d in our\ncalculations.\nKeywords: Lorentz Violation, Neutrinos, Gauge Invariance , Mass Mixing, IceCube\nDetector; Physics beyond the Standard Models\n1 Introduction\nNeutrinos are the most elusive particles within the Standar d Model of Elemen-\ntary Interactions. Speculation about their tachyonic natu re started with Ref. [1],\nand has led to the development of a few interesting scenarios [2]. Within\nthe Lorentz-violating scenarios [3–13], many different tens or structures involving\nLorentz-violating parameters can be pursued. Here, we assu me that, in a preferred\n1particular (observer) Lorentz frame, an isotropic dispers ion relation of the form\nE=/radicalbig\n/vector p2v2+m2withv >1 holds. (In this article, we use physical units with\n/planckover2pi1=ǫ0=c= 1.) Formalizing the Lorentz-violating ideas, the Lorentz –Violating\nExtension of the Standard Model (SME) was developed with a st rong inspiration\nfrom string theory [14,15].\nKinematically, decay among neutrino mass eigenstates is al lowed due to their mass\ndifferences, while decay rates for “ordinary” neutrinos with in the Standard Model\nformalism (for both Dirac as well as Majorana) exceed the lif etime of Universe by\norders of magnitude. Lorentz-violating neutrinos undergo stronger decay and energy\nloss mechanisms than “ordinary” neutrinos because of their dispersion relation E=/radicalbig\n/vector p2v2+m2≈ |/vector p|v(at high energy), which makes a number of decay channels\n(without GIM suppression, see Refs. [16,17]) kinematicall y possible.\nSome remarks on the origin of modified dispersion relations o f the form E=/radicalbig\n/vector p2v2+m2≈ |/vector p|vmight be in order. Modified dispersion relations could in pri n-\nciple be induced by modified theories of gravity, and modifica tions of the Einstein\nequivalence principle. In Ref. [18], it has been suggested t hat the invariant arena\nfor nonquantum physics is a phase space rather than spacetim e, and the locality\nof an even in space-time is replaced by relative locality in which different observers\nsee different spacetimes, and the spacetimes they observe are energy and momen-\ntum dependent. This hypothesis can lead to modified dispersi on relation of the\nkind investigated here. Effects due to quantum gravity may als o induce modified\ndispersion relations [19,20]. In a different context, Lorent z breaking induced at\nthe Planck scale may also induce such relations [21–23], in t he sense of “doubly\nspecial relativity” which works on the assumption that dyna mics are governed by\ntwo observer-independent quantities, the speed of light an d an additional constant\nenergy scale, which could be the Planck energy scale (see als o Ref. [24]). The phe-\nnomenological consequences of tiny Lorentz violations, wh ich are rotationally and\ntranslationally invariant in a preferred frame, and are ren ormalizable while preserv-\ning anomaly cancellation and gauge invariance under the Sta ndard Model gauge\ngroupSU(3)⊗SU(2)⊗U(1), have been analyzed in Ref. [25].\nThere are a number of phenomena which could direct us to have a look at the neu-\ntrino sector for Lorentz violation. Namely, for example, th e early arrival of neutrinos\nfrom the1987 supernovastill inspires(some) physicists. S pecifically, undertheMont\nBlanc, in the early morning hours of February 23, 1987, a show er of neutrinos of in-\nterstellar origin arrived about 6 hours earlier then the vis ible light from the Siderius\nNuntius SN1987A supernova. This event has been recorded in R ef. [26], and it was\nasserted that such an event could happen by accident once in a bout 1000 years.\nDirect measurements of neutrino velocities have given resu lts that are consistent\nwith the speed of light within experimental uncertainty, bu t with the experimental\nresult being a littler larger than the speed of light. For exa mple, the MINOS exper-\niment [27] has measured superluminal neutrino propagation velocities which differ\nfrom the speed of light by a relative factor of (5 .1±2.9)×10−5at an energy of about\nEν≈3GeV, compatible with an earlier FERMILAB experiment [28]. Furthermore,\nneutrinos cannot be used to transmit information (at least n ot easily) because of\ntheir small interaction cross sections. Superluminality o f neutrinos would thus not\n2necessarily lead to violation of causality at a macroscopic level, as demonstrated in\nAppendix A.2 of Ref. [29]. Similar arguments have been made i n Ref. [30], where\nit was shown that problems with microcausality, in Lorentz- violating theories, are\nalleviated for small Lorentz-violating parameters and in s o-calledconcordant frames\nwhere the boost velocities are not too large. For neutrinos, corresponding prob-\nlems are further alleviated by the fact that their interacti on cross sections are small;\nhence, it becomes very hard to transport information superl uminally even if the\ndispersion relation indicates such effects (see also Appendi x A.2 of Ref. [29]).\nWe should also note that, when Lorentz invariance is violate d, superluminality does\nnot necessarily lead to problems. Under certain additional assumptions, causality\narguments related to superluminal signal propagation simp ly do not apply in this\ncase. For example, if photons themselves propagate faster t hanc(the limiting\nvelocity of massive standard fermions) in some Lorentz-vio lating theories, then, as\nlongasinformationpropagates alongorinsidethemodifiedl ightcone(as definedbya\nmodified effective metric), it can propagate faster than cwithout implying causality\nissues (see, e.g., the paragraph around Eq. (9) in Ref. [31]) . Further information\non this point can be found in Refs. [32–36]. E.g., Ref. [32] pr ovides information\non (micro)causality problems for the purely timelike case o f Maxwell-Chern-Simons\ntheory (operator of dimension 3 for photons).\nSome more explanatory remarks on gauge invariance and Loren tz violation are prob-\nably in order. One might argue that gauge invariance should b e seen as the guiding\nprinciple to make consistent a nonabelian gauge interactio n mediated by spin-one\nmassive particles through the (experimentally confirmed) H iggs mechanism, and\nthat gauge invariance should be retained at all cost, even if Lorentz symmetry is\nbroken. However, this demand overlooks two aspects. The firs t is that the origin of\nthe Lorentz-violating terms could be assumed to be commensu rate with the Planck\nscale [14,21,24], in which case it is questionable if our usu al concepts of gauge in-\nvariance could be transported without changes to the extrem e energy scales; in any\ncase, we would assume the Standard Model gauge group, at the e nergy scale relevant\nto Lorentz violation, to be replaced by a unified gauge group, e.g., SO(1,13) (see\nRefs. [37–41]). Further fundamental modifications of the ga uge principle at extreme\nenergyscalesarealsoconceivable. Thesecondandmoreimpo rtantoverlooked aspect\nis that Lorentz violation constitutes a form of gauge invariance violation. Namely,\nEinstein’s theory of general relativity isthe classical, gauged theory of gravitation,\nin which the global Lorentz symmetry is elevated to a gauged, local symmetry [42].\nThis observation offers the construction principle for the sp in connection of Dirac\nfields in curved space-times, which has been explored in rece nt monographs [42] and\npapers [43,44]. To reemphasize the point, we recall that Lor entz invariance violation\nisgauge invariance violation within General Relativity [42] .\nWe organize the paper as follows. After a consideration of th e calculation of the\nthreshold for the superluminal decay processes (Sec. 2), we outline the calculation\nof the decay and energy loss rates in Sec. 3, before discussin g an attractive scenario\nfor Lorentz violation in theneutrinosector, andits signat ures, in Sec. 4. Conclusions\nare reserved for Sec. 5.\n3e\nν(m)\niν(m)\ni\nZ0e−p2 p4\np1p3\n(a)ν(m)\nf\nν(m)\niν(m)\ni\nZ0ν(m)\nf−p2 p4\np1p3\n(b)\nFigure 1: In the lepton-pair ˘Cerenkov radiation process (a), an on-\ncoming Lorentz-violating initial neutrino mass eigenstat eν(m)\nide-\ncays, underemission ofavirtual Z0boson, intoanelectron-positron\npair. The sum of the outgoing pair momenta is p2+p4; one ob-\nserves the inverted direction of the fermionic antiparticl e line. The\narrow of time is from bottom to top. The (blue) bosonic line ca r-\nries the four-momentum q. Diagram (b) describes the neutrino-pair\n˘Cerenkov radiation process, with a final neutrino mass eigen state\nν(m)\nf.\n2 Threshold Considerations\nWe refer to the lepton-pair ˘Cerenkov radiation process (LPCR) in Fig. 1(a) and\nthe neutrino-pair ˘Cerenkov radiation process (NPCR) depicted in Fig. 1(b). In\norder to make neutrino decay kinematically possible, it is n ecessary to fulfill certain\nthreshold conditions. Let us denote the outgoing fermions i n the generic decay\nprocesses depicted in Fig. 1 by\nν→ν+f+¯f, (1)\nwith apair of amassive fermion fandits antiparticle ¯fbeingemitted in theprocess.\nEnergy-momentum conservation implies that (in the notatio n of Fig. 1)\n(p1−p3)2=q2= (p2+p4)2. (2)\nLet us first consider the case of a massive outgoing pair 2 + 4, w ith rest mass\nmf, and vanishing Lorentz-violating parameter. Threshold is reached for collinear\nemission geometry. The incoming four-momentum is p1= (E1,/vector p1), whereE1=/radicalBig\n/vector p2\n1v2\ni+m2ν≈ |/vector p1|vi, andmνis theneutrinomass. Assumingan incoming neutrino\nenergy well above its rest mass, we can do this approximation . By contrast, one has\np3= (0,/vector0), so that the total four-momentum transfer qgoes into the pair.\nA few remarkson thedispersionrelations usedinthecurrent paper, areinorder. We\nobservethat therelation E1=/radicalBig\n/vector p2\n1v2\ni+m2νis isotropic, andit inducessuperluminal\n4velocities. However, it should be noted that the modified dis persion relation applied\nto neutrino sector can introduce a rich phenomenology even i n other sectors, as\noscillations, without requiring superluminal velocities forvi>1. Many models can\nbe explored (see Refs. [5–7,9,45–48]). As explained above, we here assume a simple\nform for the modified dispersion relation, applicable at lea st in a preferred observer\nLorentz frame.\nReturning to the discussion of the threshold, we observe tha t, for collinear geometry,\none haspµ\n2=pµ\n4= (Ef,/vector pf), whereEf=/radicalBig\n/vector p2\nf+m2. Under these assumptions,\np2+p4= (2/radicalBig\n/vector p2\nf+m2,2/vector pf), so that ( p2+p4)2= 4m2\nf. The threshold condition\nbecomes\np2\n1(v2\ni−1)≥4m2\nf, p 1≈E1≥2mf/radicalBig\nv2\ni−1=2mf√δi. (3)\nHere,\nvi=/radicalbig\n1+δi. (4)\nThe threshold condition Eth= 2mf/√δihas been used extensively in Refs. [49–51].\nThe formula (3) implies that the threshold for NPCR is lower b y at a least six orders\nof magnitude as compared to LPCR.\nThe kinematic considerations are very different in the high-e nergy regime, when\nboththe incoming (decaying) particle as well as the outgoing par ticles are Lorentz\nviolating. Masses can be neglected. In this case, one has at t hresholdp2=p4=\n(Ef,/vector pf), whereEf=|/vector pf|vf, so that at threshold\np2\n1(v2\ni−1)≥4p2\nf(v2\nf−1), (5)\nDue to equipartition of the energy among each particle of the outgoing pair at\nthreshold, one has pf≈Ef=E1/2≈p1/2. In this case, the threshold condition\nreduces to\n(v2\ni−1)≥(v2\nf−1), δi>δf. (6)\nHere,vf=/radicalbig\n1+δf. Forδi=δf, no phase space is available in order to accom-\nmodate for the decay. This consideration explains why all re sults communicated in\nRef. [51] display a factor δi−δf; decay takes place from “faster” to “slower” mass\neigenstates.\n3 Outline of the Calculation\nTheunderstandingof decay processes involving Lorentz-vi olation has beenadvanced\nthrough Refs. [49–51]. Let us briefly recall elements of the d erivation given in\nRef. [51]. One particular question is how to express the deca y rate for an (initially)\nflavor-eigenstate neutrino (the electroweak Lagrangian is flavor-diagonal) in terms\nof mass eigenstates. We have, in the same obvious notation as used in Ref. [51],\nν(f)\nk=/summationdisplay\nℓUkℓν(m)\nℓ, (7)\n5with the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix Ukℓ. The interaction\ninteraction LWwith theZ0boson in the flavor basis is\nLW=−gw\n4 cosθW/summationdisplay\nkν(f)\nkγµ(1−γ5)ν(f)\nkZµ. (8)\nHere,gwis the weak coupling constant, and θWis the Weinberg angle. A unitary\ntransformation leads to\nL=−gw\n4 cosθW/summationdisplay\nk,ℓ,ℓ′U+\nℓkUkℓ′ν(m)\nℓγµ(1−γ5)ν(m)\nℓ′Zµ. (9)\nThe interaction with the Z0boson in the mass eigenstate basis therefore reads as\nfollows,\nL=−gw\n4 cosθW/summationdisplay\nℓν(m)\nℓγµ(1−γ5)ν(m)\nℓZµ. (10)\nIn order to model the free Lorentz-violating neutrino Lagra ngian, one introduces an\neffective metric with a tilde:\nL=/summationdisplay\nℓiν(m)\nℓγµ(1−γ5)˜gµν(vℓ)∂νν(m)\nℓ. (11)\nThe modified metric defines a modified light cone according to ˜ gµν(vℓ)kµkν= 0.\nHere,\n˜gµν(vℓ) = diag(1,−vℓ,−vℓ,−vℓ). (12)\nThe dispersion relation\nEℓ=|/vector p|vℓ (13)\nfollows as the massless limit of Eℓ=/radicalBig\n(|/vector p|vℓ)2+m2\nℓ. For neutrinos, we know that\nthemℓterms are different. So, there is reason to assume that the δℓ=/radicalBig\nv2\nℓ−1\nterms are also different among mass (flavor) eigenstates, if th ey are nonvanishing.\nOne defines parameters vintandδintby the relation\nvint=/radicalbig\n1+δint (14)\nfortheunifieddescriptionofLPCRandNPCR;theeffectivefour -fermionLagrangian\nfor the process reads as\nLint=feGF\n2√\n2ν(m)\niγλ(1−γ5)ν(m)\ni˜gλσ(vint)¯ψfγσ(cV−cAγ5)ψf.(15)\nCohen and Glashow [49] set vint= 1. (In Ref. [51], on a number of occasions, the\nparameter usedin Ref. [49] had beeninadvertently indicate d asvint= 0, which is not\nthecase. We take theopportunity to point out that of course, theparameter vint= 1\nimplies that δint= 0, which was the intended statement in Ref. [49].) Bezrukov and\nLee [50] use the parameters vint= 1 (“model I”) and vint=vi(“model II”). In\nRef. [51], the parameter vintis kept as a variable. As explained in detail in Ref. [52],\n“gauge invariance” (with respect to a restricted subgroup o f the electroweak sector)\n6can be restored if one uses the value vint=vivf. Both Cohen and Glashow [49],\nas well as Bezrukov and Lee [50], assume that δf= 0 for LPCR. The parameter fe\ncharacterizes the process:\nfe=/braceleftBigg\n1, ψ f=ν(m)\nf\n2, ψ f=e. (16)\nApproximately, one has\n(cV,cA) =/braceleftBigg\n(1,1) ψf=ν(m)\nf\n(0,−1\n2), ψ f=e. (17)\nThe characteristic matrix element is\nM=feGF\n2√\n2/bracketleftBig\n¯ui(p3)γλ(1−γ5)ui(p1)/bracketrightBig\n˜gλσ(vint)/bracketleftbig\n¯uf(p4)(cVγσ−cAγσγ5)vf(p2)/bracketrightbig\n.\n(18)\nKey to the calculation is the fact that one can split the three -particle outgoing phase\nspace\nΓ =1\n2E1/integraldisplay\ndφ3(p2,p3,p4;p1)1\nns/summationdisplay\nspins|M|2\n=1\n2E1/integraldisplayM2\nmax\nM2\nmindM2\n2πdφ2(p3,p24;p1)dφ2(p2,p4;p24)1\nns/summationdisplay\nspins|M|2.(19)\nwith appropriate limits for M2\nminandM2\nmaxbeing given as follows,\nM2\nmin=δf(|/vector p2|+|/vector p4|)2, M2\nmax=δi(|/vector p1|−|/vector p3|)2. (20)\nThe following splitting relation for the phase space is cruc ial to a simplification of\nthe integrations [for details, see Ref. [53] and Eq. (43) of R ef. [51]],\ndφ3(p2,p3,p4;p1)\n=/integraldisplaydM2\n2πd4p3\n(2π)3δ+(p2\n3−δik2\n3)d4p24\n(2π)3δ+(p2\n24−M2)(2π)4δ(4)(p1−p3−p24)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=dφ2(p3,p24;p1)\n×d4p2\n(2π)3δ+(p2\n2−δfk2\n2)d4p4\n(2π)3δ+(p2\n4−δfk2\n4)(2π)4δ(4)(p24−p2−p4)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=dφ2(p2,p4;p24)\n=/integraldisplaydM2\n2πdφ2(p3,p24;p1)dφ2(p2,p4;p24).(21)\nHere,p24=p2+p4, andwedenotethespatialpartofthefour-vector pµ\nias/vectorkiwithi=\n1,2,3,4, so thatpµ\ni= (Ei,/vectorki), and since Ei= (1+δi)|/vectorki|, one hasgµν˜gµα˜gνβpα\nipβ\ni=\ngµνpµ\nipν\ni−δik2\ni=p2\ni−δik2\ni[see also Eq. (40) of Ref. [51]]. This relation explains\n7the argument of some of the Dirac- δfunctions in Eq. (21). The general result for\nthe decay rate, unifying both processes depicted in Fig. 1, r eads as follows,\nΓνi→νiψf¯ψf=G2\nFk5\n1\n192π3f2\nec2\nV+c2\nA\n420ns(δi−δf)/bracketleftbigg\n(60−43σi)(δi−δf)2\n+2(50−32σi−25σf+7σiσf)(δi−δf)δf\n+7(4−3σi−3σf+2σiσf)δ2\nf+7δ2\nint/bracketrightbigg\n.(22)\nThisresultvanishesfor δi=δf(seethediscussioninSec.2). CohenandGlashow[49]\nhavens= 2 active spin states for the (initial) neutrino, while Bezr ukov and Lee [50]\ncalculate with ns= 1, implying that the authors of Ref. [50] assume that only on e\nspin state exists in nature. This assumption affects the avera ging over the initial\nquantum states involved in the process. The σparameters depend on the way in\nwhich spin polarization sums are carried out,\nσi=/braceleftbigg0,CG spin sum for νi\n1,BL spin sum for νi, σf=/braceleftbigg0,CG spin sum for ψf\n1,BL spin sum for ψf.(23)\nInRef.[49], theCohen–Glashow (CG) spinsum(“polarizatio n sum”)issimplytaken\nas the standard spin sum for massless fermions,\n/summationdisplay\nsνℓ,s⊗¯νℓ,s=pµgµνγν. (24)\nInRef. [50], theBezrukov–Lee (BL) spinsumis based on asome whatmoreadvanced\ntreatment of the eigenspinors of superluminal neutrino mas s eigenstates and reads\nas /summationdisplay\nsνℓ,s⊗¯νℓ,s=pµ˜gµν(vℓ)γν. (25)\nThe general result for the energy loss rate, applicable to bo th processes in Fig. 1,\nreads as\ndEνi→νiψf¯ψf\ndx=−G2\nFk6\n1\n192π3f2\nec2\nV+c2\nA\n672ns(δi−δf)\n×/bracketleftbigg\n(75−53σi)(δi−δf)2+(122−77σi−61σf+16σiσf)(δi−δf)δf\n+8(4−3σi−3σf+2σiσf)δ2\nf+8δ2\nint/bracketrightbigg\n.(26)\nInRef.[51], wehaveverifiedandchecked compatibility with all formulascontainedin\nRefs. [49] and [50]. This is important because it confirms tha t the model dependence\nof the results is only contained in the numerical prefactors , but not in the overall\nscaling of the results.\nAs outlined in Ref. [51], one can parameterize the results fo r NPCR as follows,\nΓνi→νiνf¯νf=bG2\nF\n192π3k5\n1,dEνi→νiνf¯νf\ndx=−b′G2\nF\n192π3k6\n1. (27)\n8For the CG spin sum, one obtains the following bparameters,\nbCG=1\n7(δi−δf)/bracketleftbigg\n(δi−δf)2+5\n3δf(δi−δf)+7\n15δ2\nf/bracketrightbigg\n, (28a)\nb′\nCG=25\n224(δi−δf)/bracketleftbigg\n(δi−δf)2+112\n75δf(δi−δf)+32\n75δ2\nf/bracketrightbigg\n.(28b)\nFor the BL spin sum, one obtains\nbBL=17\n210(δi−δf)/bracketleftbigg\n(δi−δf)2+7\n17δ2\nint/bracketrightbigg\n, (29a)\nb′\nBL=11\n168(δi−δf)/bracketleftbigg\n(δi−δf)2+4\n11δ2\nint/bracketrightbigg\n. (29b)\nTypically, one finds [51] numerical prefactors in these form ulas are larger than those\nfor LPCR by a factor of four or five. Also, NPCR has negligible t hreshold.\nIn papers of Stecker and Scully [12,54,55], the following bo und is derived for the\nLorentz-violating parameter of the electron-positron fiel d alone (watch out for a\ndifference in the conventions used for defining the δeparameter):\nδe≤1.04×10−20. (30)\nWe should stress that this bound concerns oncoming electron s (not neutrinos!) and\nhas nothing to do with the processes studied here.\nThe observation of very-high-energy neutrinos by IceCube, taking into consideration\nthe LPCR process (but not NPCR!), implies that the Lorentz-v iolating parameter\nfor neutrinos cannot be larger than (Ref. [12])\nδν≤2.0×10−20. (31)\nThis bound is based on the assumption that δeandδνare different. Colloquially\nspeaking, we can say that, if δνwere larger, then “Big Bird” (the 2PeV specimen\nfound in IceCube, see Refs. [56,57]) would have already deca yed before it arrived at\nthe IceCube detector. However, the full analysis requires M onte Carlo simulations\ninvolving astrophysical data and is much more involved [12, 54,55].\nProvidedtheLorentz-violating parametersforthedifferent neutrinomasseigenstates\naredifferent, low-energy neutrinosareaffectedbythedecayan denergylossprocesses\nconnected with NPCR, in view of a negligible threshold for NP CR. As already\nemphasized, typical numerical coefficients forNPCR areafac tor of fourorfivelarger\nthan for LPCR, depending on the model used for the spin sums. T his enhances the\nimportance of the NPCR effect. Inspired by Eq. (31), we thus con jecture here that\na full analysis of astrophysical data, using the NPCR proces s as a limiting factor for\nthe observation of high-energy neutrinos, should yield a bo und on the order of\n|δi−δf| ≤1\n51/3×2.0×10−20∼1.2×10−21, (32)\nwhere the prefactor takes into account the scaling of the effec t with theδparameter.\nSpecifically, the decay and energy loss rates typically scal e with the factor ( δi−δf)3.\nIt would be very fruitful if this conjecture were to be checke d against astrophysical\ndata in an independent investigation.\n94 An Attractive Scenario\nAt first, one might see a dilemma: Within a fully SU(2)Lgauge-invariant theory,\nwith uniform Lorentz-violating parameters over all partic le generations, one nec-\nessarily has δν=δe(see Ref. [52] for a detailed discussion), and so, the bound\nδν≤2.0×10−20given in Eq. (31) is not applicable, because the LPCR process\ndoes not exist. But then, one has to acknowledge that the boun dδe≤1.04×10−20\ngiven in Eq. (30), which is originally derived for electrons , based on other physical\nprocesses, automatically also applies to the neutrino sect or.\nSo, the dilemma is that either, one has to give up gauge invari ance and use dif-\nferent Lorentz-violating parameters for neutrinos as oppo sed to charged leptons\nwithin the same particle generation, or, if one insists on ga uge invariance, or, as-\nsume different Lorentz-violating parameters for different gen erations. Otherwise,\nthe insistence on gauge invariance would defeat part of the p urpose of looking at\nthe neutrino sector for Lorentz violation. This is because i n the latter case, for\nuniform Lorentz-violating parameters among all three gene rations, because, by as-\nsumption, the Lorentz-violating parameters for neutrinos and charged left-handed\nleptons within the same SU(2)Ldoublet are necessarily the same, the tight bounds\non Lorentz-violating parameters in the charged-fermion se ctor automatically apply\nto the neutrino sector as well1. This observation has important consequences when\nexamining the first-generation SU(2)Ldoublet, consisting of ( νe,eL). Electrons and\npositrons are stable particles, and small violations of Lor entz invariance would lead\nto violations of causality on a macroscopic level (see Appen dix A.2 of Ref. [29])2.\nConversely, if we had to carry over all restrictions on Loren tz-violating electron pa-\nrameters to the electron neutrino sector, then this would nu llify all the motivations\nlisted in Sec. 1 for investigating the first-generation neut rino sector.\nOnthecontrary, If oneaccepts thenecessity that different Lo rentz-violating parame-\nters should be used for each of the three known particle gener ations, then one needs\nto acknowledge that the parameter space for differential Lore ntz-violation among\nneutrino mass eigenstates is restricted by additional cons traints due to the NPCR\nprocess [51]. An attractive gauge-invariant scenario coul d still be found, as follows.\nNamely, one might observe that, as per the discussion in Appe ndix A.2 of Ref. [29],\ncausality violations due to Lorentz violation are less seve re for unstable particles,\nwhich decay and therefore are not amenable to the reliable tr ansport of informa-\ntion. Part of the above sketched dilemma could thus be avoide d as follows. One\nfirst observes that, as per the above argument, problems with respect to causality\nare less severe in the second-generation SU(2)Ldoublet (νµ,µL) and also in the\nthird-generation SU(2)Ldoublet (ντ,τL), which are composed entirely of unstable\nparticles. Full gauge invariance can be retained if we assum e generation-dependent\nLorentz-violating parameters δe,δµ, andδτ, for the three SU(2)Ldoublets, which\n1We here ignore the somewhat remote possibility of different L orentz violating parameters for\nthe right-handed and left-handed sectors of one and the same generation.\n2Note, also, that this statement does not hold if the limiting velocity for fermions turns out\nto be smaller as opposed to larger than the speed of light. Fur thermore, there are conceivable\nmodifications of Maxwell theory, as already discussed in Sec . 1, where causality violations are\navoided due to modified light cones [31–36].\n10could be encoded in modified Dirac matrices /tildewideγi=vfγiwithf=e,µ,τ[see Eq. (5)\nof Ref. [52]]. In the charged-fermion sector, we have nearly no mixing of mass and\ncharge eigenstates. Let us then go into the high-energy regi me where, where mass\nand flavor eigenstates, under the assumptions\nδµ,δτ>0, δµ/negationslash=δτ, δe= 0, (33)\nbecome equal. In this case, at high energy, one would have two neutrino mass eigen-\nstates, which asymptotically approach the muon neutrino an d tau neutrino flavor\neigenstates at very high energy, decay into electron-posit ron pairs and (asymptoti-\ncally) electron neutrinos, via LPCR and NPCR.\nOf course, other scenarios and flavor and mass mixing phenome nologies are also\npossible, as discussed in Sec. IV B of Ref. [51]. In general, o ne could interpret\nthe emergence of a specific predominant flavor composition of incoming super-high-\nenergy cosmic neutrinos, consistent with one, and only one, specific mass eigenstate,\nas a signature of Lorentz violation. This is because a single , defined, oncoming mass\neigenstate would be consistent with the two other mass eigen states being “faster”\nand thus decaying into the single “slow” eigenstate.\nIn all discussed scenarios, one might find a conceivable expl anation for the apparent\ncutoff in the cosmic neutrino spectrum at about 2PeV, at the ex pense of reducing\nthe allowed regime of Lorentz-violating δparameters to the range of about 10−20.\nIn our “attractive scenario”, one retains gauge invariance as outlined in Sec. 4 of\nRef. [52] and still is able to account for a super-high-energ y cutoff of the cosmic\nneutrino spectrum. Experimental confirmation or dismissal of this hypothesis will\nrequire better cosmic neutrino statistics at very high ener gies.\n5 Conclusions\nThe existence of the NPCR process [see Fig. 1(b)] reveals a ce rtain dilemma for\nLorentz-violating neutrinos (provided the Lorentz violat ing parameters indicate su-\nperluminality). Namely, under the hypothesis of a nonvanis hing Lorentz-violating\nparameterδ, given as in Eq. (4), the virtuality\nE2−p2=p2(v2−1)≈E2(v2−1) =E2δ (34)\nof a neutrino becomes large for large energy, rendering a num ber of decay pro-\ncesses kinematically possible. Conversely, based on high- energy astrophysical obser-\nvations, very strict bounds can be imposed on the Lorentz-vi olating parameters [see\nEqs. (30), (31), and (32)].\nDeep connections exist between Lorentz violation and gauge invariance. In Ref. [58],\nit is shown that spontaneous Lorentz violation can lead to an effective low-energy\nfield theory with both Lorentz-breaking as well as gauge-inv ariance breaking terms.\nAccording to Refs. [58–69], even the photon could potential ly be formulated as the\nNambu-Goldstone boson linked to spontaneous Lorentz invar iance violation. (This\nansatzwas originally formulated before electroweak unification. ) For a broader view\n11of this point, we refer to Appendix A of Ref. [52]. If one insis ts on the persistence\nof gauge invariance within the electroweak sector, then one has to acknowledge that\nbounds on Lorentz-violating parameters for charged lepton s [e.g., Eq. (30)] also ap-\nply to the neutrino sector [thus lowering the bound otherwis e given in Eq. (31) by a\nfactortwo, andfurtherrestrictingtheavailableparamete rspaceforLorentz-violating\nparameters in the neutrino sector]. Also, the assumption th atδν=δewould defeat\nthe purpose of looking at neutrinos for Lorentz violation. 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B 821, 65–73 (2009).\n16"    },    {        "title": "2210.02114v1.Searching_for_Lorentz_violating_Signatures_from_Astrophysical_Photon_Observations.pdf",        "content": "arXiv:2210.02114v1  [astro-ph.HE]  5 Oct 2022Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n1\nSearching for Lorentz-violating Signatures from Astrophy sical\nPhoton Observations\nJun-Jie Wei1,2,3\n1Purple Mountain Observatory, Chinese Academy of Sciences,\nNanjing 210023, China\n2School of Astronomy and Space Sciences, University of Scien ce and Technology of\nChina, Hefei 230026, China\n3Guangxi Key Laboratory for Relativistic Astrophysics,\nNanning 530004, China\nAs a basic symmetry of Einstein’s theory of special relativi ty, Lorentz invari-\nance has withstood very strict tests. But there are still mot ivations for such\ntests. Firstly, many theories of quantum gravity suggest vi olations of Lorentz\ninvariance at the Planck energy scale. Secondly, even minut e deviations from\nLorentz symmetry can accumulate as particle travel across l arge distances,\nleading to detectable effects at attainable energies. Thank s to their long base-\nlines and high-energy emission, astrophysical observatio ns provide sensitive\ntests of Lorentz invariance in the photon sector. In this pap er, I briefly in-\ntroduce astrophysical methods that we adopted to search for Lorentz-violating\nsignatures, including vacuum dispersion and vacuum birefr ingence.\n1. Introduction\nAlthough experimental tests of Lorentz invariance violation (LIV) have\nbeen performed in a wide range of systems (see Ref. 1 for a compilat ion of\nresults), there are still motivations for such tests. From theore tical consid-\neration, establishing a quantum theory that includes gravity is being hailed\nas the Holy Grail of modern physics. However, theories of quantum gravity\n(QG) predict that Lorentz invariance may be violated at the Planck e nergy\nscaleEPl. From experimental feasibility, even very tiny deviations from\nLorentz symmetry can become measurable at attainable energies ≪EPl,\nsince Lorentz-violating effects gradually accumulate over large pro pagation\ndistances. In brief, both the prospect of relativity violations arisin g in a\ngrand unified theory and the feasibility of discovering LIV have attr acted\nphysicists to constantly work on experimental searches.Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n2\nAstrophysical observations involving long baselines are very suitab le to\nsearch for LIV effects. In vacuum, the LIV-induced modifications to the\nphoton dispersion relation can produce rich and detectable astrop hysical\nphenomena. With a slight modification, the speed of light will no longer\nbe independent of frequency and polarization. If the modified dispe rsion\nrelation is related to the frequency of photons, then we may obser ver a\nfrequency-dependent vacuum dispersion of light. If the modificat ion is re-\nlated to the circular polarization state of photons, the photons wit h right-\nand left-handed polarization states have different velocities, then we may\nobserver vacuum birefringence. Vacuum dispersion can be tested by mea-\nsuringthe arrival-timedifferencesofphotonswith differentfreque nciesorig-\ninating from the same astrophysical source (see, e.g., Ref. 2). Va cuum\nbirefringence results in a frequency-dependent rotation of the p olarization\nvector of linearly polarized light, which can be tested by astrophysic alspec-\ntropolarimetric measurements (see, e.g., Refs. 3, 4).\nIn this paper, we present some recent tests of nonbirefringent L IV using\nspectral-lag transitions of gamma-ray bursts (GRBs),5,6and some searches\nfor LIV using polarization measurements of GRBs and blazars.7,8\n2. Vacuum dispersion\nVacuum dispersion is a potential LIV signature. The arrival time dela y\nbetween photons with different energies ( Eh> El) emitted simultaneously\nfrom the source at redshift zcan be derived by introducing the LIV terms\nin a Taylor series:9\n∆t=s±1+n\n2H0En\nh−En\nl\nEn\nQG,n/integraldisplayz\n0(1+z′)ndz′\n/radicalbig\nΩm(1+z′)3+ΩΛ, (1)\nwheres±=±1 is the sign of LIV, corresponding to the “subluminal”\n(s±= +1) or “superluminal” ( s±=−1) scenarios. EQGdenotes the\nQG energy scale at which LIV effects become significant, n= 1 (n= 2)\ncorrespondsto the linear (quadratic)energydependence, and( H0, Ωm, ΩΛ)\nare the cosmological parameters of the standard flat ΛCDM model.\nGRBs are ideal astrophysical phenomena that one can used to per form\ntime-of-flight tests because they are the most distant transient sources in-\nvolving a wide range of photon energies. However, a key challenge in s uch\ntests is to distinguish an intrinsic astrophysical time delay at the sou rce\nfrom a time delay induced by LIV. We proposed that GRB 160625B, th e\nburst having an apparent transition from positive to negative spec tral lags,Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n3\nprovides a good opportunity to disentangle the intrinsic time delay pr ob-\nlem.5Spectral lag, the arrivaltime difference between high- and low-ene rgy\nphotons, is conventionally defined to be positive if the high-energy p hotons\nprecede the low-energy ones. In the subluminal case ( s±= +1), photons\nwith higher energies would arrive at the observer after those with lo wer\nones, implying a negative spectral lag due to LIV. Assuming the sour ce-\nintrinsic time lag to have a positive dependence on the photon energy ,\nand considering the contributions to the spectral lag from both th e in-\ntrinsic positive lag and LIV-related negative lag, we derived new limits o n\nlinear and quadratic leading-order Lorentz-violating vacuum disper sion by\ndirectly fitting the spectral lag behavior of GRB 160625B.Recently , similar\ntime-of-flight tests were carried out by analyzing the spectral-lag transition\nof GRB 190114C.6\n3. Vacuum birefringence\nVacuum birefringence is another potential LIV signature. For a so urce at\nredshiftz, the LIV-induced rotation angle of the polarization vector of a\nlinearly polarized wave is4\n∆φ(E)≃ηE2\n/planckover2pi1EPlH0/integraldisplayz\n0(1+z′)dz′\n/radicalbig\nΩm(1+z′)3+ΩΛ, (2)\nwhereEis the energy of the observed light and ηis a dimensionless pa-\nrameter characterizing the degree of LIV effects.\nIf the rotation angles of photons with different energies differ by mo re\nthanπ/2 over an energy band ( E1< E < E 2), significant depletion of\nthe initial polarization of the signal is expected. The detection of hig h\npolarization can therefore set upper limits on the birefringent para meterη.\nWe gave a detailed calculation on the GRB polarization evolution arising\nfrom the birefringent effect, and confirmed that the initial polariza tion is\nnot significantly depleted even if the differential rotation angle |∆φ(E2)−\n∆φ(E1)|is as large as π/2.7Applying our formulate for calculating LIV-\ninduced polarization evolution to the gamma-ray polarimetric data of 12\nGRBs, we improved existing bounds on the birefringent parameter ηby\nfactors ranging from 2 to 10.\nIf all photons in the observed energy range are assumed to be emit ted\nwith the same intrinsic polarization angle, we expect to observe vacu um\nbirefringence as an energy-dependent linear polarization vector. We tried\nto search for an energy-dependent change of the linear polarizat ion angle\nin the spectropolarimetric data of 5 blazars.8At the 2 σconfidence level,Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n4\nthe absence of the birefringent effect was limited to be in the range o f\n−9.63×10−8< η <6.55×10−6. As might be expected, opticalpolarimetric\ndata of blazars pose less stringent constraints on ηas compared to gamma-\nray polarizations of GRBs.\n4. Summary\nGRBs are promising astrophysical sources for searching for LIV- induced\nvacuum dispersion and vacuum birefringence. Future detections o f\nextremely-high-energy emission from GRBs with LHAASO, MAGIC,\nHAWC, and the future international Cherenkov Telescope Array c ould im-\nprove the limits on LIV using vacuum dispersion (time-of-flight) test s. Fu-\nture X-ray/gamma-ray polarization measurements of GRBs with PO LAR-\nII, TSUBAME, COSI, and GRAPE could also improve the limits on LIV\nusing vacuum birefringence tests.\nAcknowledgments\nThis work is partially supported by the National Natural Science Fou nda-\ntion of China (grant Nos. 11725314 and 12041306), the Key Resea rch Pro-\ngram of Frontier Sciences (grant No. ZDBS-LY-7014) of Chinese A cademy\nof Sciences, the Major Science and Technology Project of Qinghai Province\n(2019-ZJ-A10), the Natural Science Foundation of Jiangsu Prov ince (grant\nNo. BK20221562), the China Manned Space Project (CMS-CSST-2 021-\nB11), and the Guangxi Key Laboratory for Relativistic Astrophys ics.\nReferences\n1.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2022 edition, arXiv:0801.0287v15.\n2. G. Amelino-Camelia et al., Nature 393, 763 (1998); A.A. Abdo et al., Nature\n462, 331 (2009); V. Vasileou et al., Phys. Rev. D 82, 122001 (2013).\n3. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); V.A.\nKosteleck´ y and M. Mewes, Phys. Rev. Lett. 110, 201601 (2013).\n4. P. Laurent et al., Phys. Rev. D 83, 121301 (2011); K. Toma et al., Phys.\nRev. Lett. 109, 241104 (2012).\n5. J.-J. Wei et al., Astrophys. J. Lett. 834, L13 (2017).\n6. S.-S. Du et al., Astrophys. J. 906, 8 (2021).\n7. J.-J. Wei, Mon. Not. Royal Astron. Soc. 485, 2401 (2019).\n8. Q.-Q. Zhou et al., Galaxies 9, 44 (2021).\n9. U. Jacob and T. Piran, J. Cosmol. Astropart. Phys. 01, 031 (2008)."    },    {        "title": "1109.6624v4.Superluminal_neutrino_and_spontaneous_breaking_of_Lorentz_invariance.pdf",        "content": "arXiv:1109.6624v4  [hep-ph]  20 Dec 2011JETP Lett. 94 (2011) 673 arXiv:1109.6624\nSuperluminal neutrino and spontaneous breaking of Lorentz invariance\nF.R. Klinkhamer#1)and G.E. Volovik∗+1)\n#Institute for Theoretical Physics, University of Karlsruh e, Karlsruhe Institute of Technology, 76128 Karlsruhe, Ger many\n∗Low Temperature Laboratory, Aalto University P.O. Box 1510 0, FI-00076 AALTO, Finland\n+L.D. Landau Institute for Theoretical Physics, Russian Aca demy of Sciences, Kosygina 2, 119334 Moscow, Russia\nSubmitted October 10, 2011\nGenerally speaking, the existence of a superluminal neutri no can be attributed either to re-entrant Lorentz\nviolation at ultralow energy from intrinsic Lorentz violat ion at ultrahigh energy or to spontaneous breaking\nof fundamental Lorentz invariance (possibly by the formati on of a fermionic condensate). Re-entrant Lorentz\nviolation in the neutrino sector has been discussed elsewhe re. Here, the focus is on mechanisms of spontaneous\nsymmetry breaking.\nPACS: 11.30.Cp, 11.30.Qc, 14.60.St\nIt is possible that OPERA’s claimed discovery [1] of\na superluminal muon-type neutrino does not come from\nthe violation of Lorentz invariance but from unknown\nfactorsin the clock-synchronizationprocess[2]orfroma\npurely statistical effect [3]. In fact, it has been shown[4]\nthat OPERA’s claimed value ( vνµ−c)/c∼10−5is ruled\nout by the expected but unobserved energy losses from\nelectron-positron-pair emission ( νµ→νµ+e−+e+), at\nleast, as long as there exists a preferred frame from the\nLorentz violation.\nStill, the claim by OPERA has provided new impe-\ntus for the discussion on the possible sources of Lorentz\nviolation. Inordertoengageinthisdiscussion, let usas-\nsume that OPERA’s result is correct qualitatively (ex-\nistence of a superluminal muon-neutrino) even if not\nquantitatively (most likely, |vνµ−c|/c≪10−5).\nCondensed-matter physics, which possesses an ana-\nlog of Lorentz invariance (LI), now suggests several dif-\nferent scenarios of Lorentz violation (LV). Among them\nare:\n(1a) LI is not a fundamental symmetry but an approxi-\nmatesymmetrywhichemergesatlowenergiesand\nis violated at ultrahigh energies (cf. [5]).\n1)e-mail: frans.klinkhamer@kit.edu, volovik@boojum.hut. fi(1b) Intrinsic LV at ultrahigh energies gives an emer-\ngent Lorentz-invariant theory at lower energies\nbut ultimately, at or below an ultralow energy\nscale,inducesare-entrantviolationofLI(see, e.g.,\nSec. 12.4 of [6]).\n(2) LI is fundamental but broken spontaneously (see,\ne.g., [7, 8] and references therein).\nIn this Letter, we discuss the spontaneous break-\ning of Lorentz invariance (SBLI), that is, the sponta-\nneous appearance of a preferred frame in the vacuum,\nwhich can be derived from Lorentz-invariant physical\nlaws. The order parameter of SBLI can be a vector\nfieldbα(for example, the vector field of Fermi-point\nsplitting [9, 10] or an aether-type velocity field [11]), an\nemergent tetrad-type field eα\na[12, 13, 14, 15], or any\nother field which is covariant but not invariant under\nLorentz transformations.\nIf SBLI occurs only in the neutrino sector, which\ninteracts weakly with the charged-matter sector, then\nSBLI has no direct impact on this other matter (cer-\ntain indirect quantum-loop effects can be suppressed by\nnear-zero mixing angles). The non-neutrino matter es-\nsentially does not feel the existence of the preferred ref-\nerence frame. In fact, it is very well possible that SBLI\noccurs only for the neutrino field, because the other\nfermions have already experienced electroweak symme-\n12 F.R. Klinkhamer, G.E.Volovik\ntry breakingphase transitions and aretoo heavyfor any\nfurther type of symmetry breaking.\nIn condensed-matter physics, the re-entrant viola-\ntion of LI, as well as Fermi-point splitting (FPS), fol-\nlow from general topological properties of the vacuum\n(ground state) in 3–momentum space. Condensed mat-\nter provides many examples of homogeneous vacuum\nstates, which have nontrivial topology in 3–momentum\nspace [6]. Among them is a class of vacua which have\nFermi points, exceptional points in 3–momentum space\nwhere the energy of fermionic excitations is nullified.\nSuch a Fermi point (alternatively called Dirac or Weyl\npoint) has a topological invariant. The existence of\nthe Fermi point is thus protected by topology, or by\nthe combined action of topology and symmetry. The\nFermi point is robust to small perturbations of the sys-\ntem. In turn, different Fermi points may collide, an-\nnihilate, and split again, but their total topological\ncharge is conserved. In this respect, Fermi points in\n3–momentum space behave as topologically-charged ’t\nHooft–Polyakovmagnetic monopoles in real space. The\nsplitting or recombination of Fermi points represents\na topological quantum phase transition. This type of\nquantum phase transition takes place, for example, in\ngraphene, graphite, etc. (see, e.g., Figs. 4 and 6 in [16]\nfor the splitting of a degenerate Fermi point into 3 and\n4 elementary points, respectively).\nFigure 8 in the review [10] (which elaborates on the\ndiscussion of the original research paper [9]) illustrates\nthespecialroleofneutrinos. Fromthemomentum-space\ntopology of Fermi points, it follows that the phase tran-\nsition awayfrom the symmetric vacuum ofthe Standard\nModel with massless fermions may occur in two ways:\neither coinciding Fermi points with opposite topologi-\ncal charge annihilate each other, giving rise to a Dirac\nmass (the process commonly known as the Higgs mech-\nanism), orcoinciding Fermi points do not annihilate but\nsplit in momentum space, giving rise to Lorentz viola-\ntion [9]. As mentioned above, it is possible that this\nFPS process only occurs for neutrinos, since all other\nparticles have already obtained Dirac masses via the\nHiggs mechanism. After this splitting, the energy spec-tra of the left- and right-handed neutrino are given by\nthe following expressions (the small neutrino mass can\nbe neglected for the conditions relevant to the OPERA\nexperiment):\ngαβ/parenleftBig\ncpα−/tildewidebL\nα/parenrightBig/parenleftBig\ncpβ−/tildewidebL\nβ/parenrightBig\n= 0, (1a)\ngαβ/parenleftBig\ncpα−/tildewidebR\nα/parenrightBig/parenleftBig\ncpβ−/tildewidebR\nβ/parenrightBig\n= 0, (1b)\nwithcthe velocity of light in vacuo . In (1), we have\nput a tilde on the dimensional vector /tildewidebαin order to dis-\ntinguish it from the dimensionless vector bαappearing\nbelow and we allow for /tildewidebL\nα∝negationslash=/tildewidebR\nα.\nThe possible role of FPS for the (qualitative)\nOPERA result has already been discussed in [17]. Here,\nwe considertwoscenariosforthe spontaneousformation\nof a preferred reference frame. The first scenario corre-\nsponds to the appearance of a dimensionless 4–vector\n(bα) = (b0,b) in the neutrino vacuum. This 4–vector bα\ninteracts with the neutrino Dirac field in a way which\ndoes not violate the fundamental laws of special relativ-\nity:\nS=/integraldisplay\nd3xcdtψbα(−i∇α)ψ. (2)\nThis action term corresponds to a momentum-\ndependent mass term,2)M=pαbα/c, which modifies\nthe spectrum of the neutrino as follows:\npαpα≡E2−c2|p|2=/parenleftbig\ncpαbα/parenrightbig2.(3)\nLet us, for example, take bαto be a timelike vector,\nhaving (b0)2− |b|2>0 , and consider the particular\nreference frame with b= 0. Assume |b0|<1. Then,\nthe neutrino energy spectrum becomes\nE2/bracketleftbig\n1−(b0)2/bracketrightbig\n=c2|p|2, (4)\nwhich is superluminal for b0∝negationslash= 0. The same happens\nfor a spacelike vector bα. In the reference frame with\nb0= 0, the neutrino energy spectrum is given by\nE2=c2|p|2+c2(b·p)2, (5)\n2)At this stage, it is clear that the same procedure can be fol-\nlowed with a Majorana mass term ψc\nLmψLin the action density,\nsimply replacing the Majorana mass mby−ibα∇α/c.Superluminal neutrino and spontaneous breaking of Lorentz invar iance 3\nwhich is both anisotropic and superluminal for b∝negationslash= 0.\nThe 4–vector field bαmay emerge as the order pa-\nrameter of the neutrino condensate,\nbα∝gαβ/angbracketleftbig\nψ(−i∇β)ψ/angbracketrightbig\n, (6)\nin a theory with 4–fermionormulti-fermion interactions\nof the following type:\nSint=/integraldisplay\nd3xcdt f(X), (7a)\nX=−gαβ/parenleftbig\nψ∇αψ/parenrightbig /parenleftbig\nψ∇βψ/parenrightbig\n, (7b)\nwith appropriate dimensional constants entering the\nfunctionf. This scenario gives a possible realization\nof the phenomenological Coleman–Glashow model [18]\nin terms of fermionic condensates [7, 8].\nThe second scenario involves another type of neu-\ntrino condensate, which also leads to SBLI. Specifi-\ncally, this neutrino condensate gives rise to a tetrad-like\nfield [12, 13, 14, 15]:\nea\nα∝/angbracketleftbig\nψγa(−i∇α)ψ/angbracketrightbig\n. (8)\nThe induced tetrad field ea\nαmust be added to the orig-\ninal fundamental tetrad E(0)α\na= diag(−1,c, c, c), and\nthe fermionic action becomes\nS=/integraldisplay\nd3xcdt Eα\naψγa(−i∇α)ψ , (9a)\nEα\na=E(0)α\na+eα\na. (9b)\nUsing the induced tetrad field ( eα\na) = diag(b0,0,0,0)\nas an example, one obtains, for 0 < b0<1, the super-\nluminal neutrino velocity vν=c/(1−b0).\nThe tetrad-type neutrino condensate (8) may pro-\nvide a realization of the hypothetical spin-2 field dis-\ncussed in [19]. A recent paper [20] presents another\nmodel, where a scalar-field composite plays a similar\nrole as our condensate eα\na. Also related may be a geo-\nmetric model [21], based on a particular class of Finsler-\nspacetime backgrounds, which essentially modifies the\neffective metric entering particle dispersion relations.\nThiscompletesourdiscussionoftwopossible scenar-\nios of spontaneous symmetry breaking to explain a su-\nperluminal neutrino. Spontaneous breaking of Lorentzinvariance in the neutrino sector corresponds to the ap-\npearance of a preferred frame for the relevant neutrino:\nLI is violated if the neutrino momentum pαis trans-\nformed but not the vacuum field bα(orea\nα) which is\nkept at a fixed value. Still, LI remains an exact sym-\nmetry of the physical laws: the invariance holds if both\nexcitations and vacuum are transformed, that is, if the\nLorentz transformation acts simultaneously on pαand\nbα(orea\nα).\nIn these SBLI scenarios, as well as in the FPS sce-\nnario [17], the vacuum remains homogeneous, which is\nthereasonwhyconservationofenergyandmomentumis\nexact. But the energy spectrum of the neutrino is mod-\nified, which must certainly have consequences for reac-\ntions involving neutrinos. Hence, there must be experi-\nmental constraints on bαorea\nα. Alternatively, the study\nof neutrino-interaction processes may provide valuable\ninformation on mechanisms proposed to explain non-\nstandard (e.g., superluminal) propagation properties of\nthe neutrinos.\nThe advantage of the spontaneous-symmetry-\nbreaking scenario is that it stays fully within the realm\nof standard physics, which obeys special relativity. The\nmulti-fermion interaction (7) can, in principle, origi-\nnate from trans-Planckian physics, but we now have\nbounds [22, 23] indicating that Lorentz invariance holds\nfar above the Planck energy scale, i.e., ELV≫EPlanck.\nThis suggests that, if a neutrino has superluminal mo-\ntion, it can be attributed either to re-entrant Lorentz\nviolation at ultralow energy due to intrinsic (built-in)\nLorentz violation at ultrahigh trans-Planckian ener-\ngies [presumably with a re-entrance energy of order\n(EPlanck/ELV)nEPlanckforn≥1] or to spontaneous\nbreaking of fundamental Lorentz invariance [possibly\nby the formation of a fermionic condensate].\n1. T. Adam et al.[OPERA Collaboration], “Measurement\nof the neutrino velocity with the OPERA detector in\nthe CNGS beam,” arXiv:1109.4897v1.\n2. C.R. Contaldi, “The OPERA neutrino velocity result\nand the synchronisation of clocks,” arXiv:1109.6160.4 F.R. Klinkhamer, G.E.Volovik\n3. R. Alicki, “A possible statistical mechanism of\nanomalous neutrino velocity in OPERA experiment?,”\narXiv:1109.5727.\n4. A.G.CohenandS.L.Glashow, “Newconstraints onneu-\ntrino velocities,” arXiv:1109.6562.\n5. S. Chadha and H.B. Nielsen, “Lorentz invariance as\na low-energy phenomenon,” Nucl. Phys. B 217, 125\n(1983).\n6. G.E. Volovik, The Universe in a Helium Droplet ,\nClarendon Press, Oxford (2003).\n7. J.D. Bjorken, “A dynamical origin for the electromag-\nnetic field,” Annals Phys. 24, 174 (1963).\n8. A. Jenkins, “Spontaneous breaking of Lorentz\ninvariance,” Phys. Rev. D 69, 105007 (2004),\narXiv:hep-th/0311127.\n9. F.R. Klinkhamer and G.E. Volovik, “Emergent CPT vi-\nolation from the splitting of Fermi points,” Int. J. Mod.\nPhys. A 20, 2795 (2005), arXiv:hep-th/0403037.\n10. G.E. Volovik, “Quantum phase transitions from topol-\nogy in momentum space,” Lect. Notes Phys. 718, 31\n(2007), arXiv:cond-mat/0601372.\n11. T. Jacobson, “Einstein–aether gravity: A status re-\nport,” PoS QG-PH , 020 (2007), arXiv:0801.1547.\n12. K. Akama, “An attempt at pregeometry – gravity with\ncomposite metric,” Prog. Theor. Phys. 60, 1900 (1978).\n13. G.E. Volovik, “Superfluid3He–B and gravity,” Physica\nB162, 222 (1990).14. C. Wetterich, “Gravity from spinors,” Phys. Rev. D 70,\n105004 (2004), arXiv:hep-th/0307145.\n15. D. Diakonov, “Towards lattice-regularized quantum\ngravity,” arXiv:1109.0091.\n16. T.T. Heikkil¨ a and G.E. Volovik, “Fermions with cu-\nbic and quartic spectrum,” JETP Lett. 92, 681 (2010),\narXiv:1010.0393.\n17. F.R. Klinkhamer, “Superluminal muon-neutrino veloc-\nity from a Fermi-point-splitting model of Lorentz viola-\ntion,” arXiv:1109.5671.\n18. S. R. Coleman and S. L. Glashow, “Cosmic ray and neu-\ntrino tests of special relativity,” Phys. Lett. B 405, 249\n(1997), arXiv:hep-ph/9703240; S. R. Coleman and S.\nL. Glashow, “High-energy tests of Lorentz invariance,”\nPhys. Rev.D 59, 116008 (1999), arXiv:hep-ph/9812418.\n19. G. Dvali and A. Vikman, “Price for environmental\nneutrino-superluminality,” arXiv:1109.5685.\n20. A. Kehagias, “Relativistic superluminal neutrinos,”\narXiv:1109.6312.\n21. C. Pfeifer and M.N.R. Wohlfarth, “Beyond the speed of\nlight on Finsler spacetimes,” arXiv:1109.6005.\n22. O. Gagnon and G.D. Moore, “Limits on Lorentz viola-\ntion from the highest energy cosmic rays,” Phys. Rev.\nD70, 065002 (2004), arXiv:hep-ph/0404196.\n23. S. Bernadotte and F.R. Klinkhamer, “Bounds on length\nscales of classical spacetime foam models,” Phys. Rev.\nD75, 024028 (2007), arXiv:hep-ph/0610216."    },    {        "title": "1909.01990v1.Probing_Lorentz_Invariance_With_Top_Pair_Production_at_the_LHC_and_Future_Colliders.pdf",        "content": "Proceedings of the Eighth Meeting on CPT and Lorentz Symmetry (CPT'19), Indiana University, Bloomington, May 12{16, 2019\n1\nProbing Lorentz Invariance\nWith Top Pair Production at the LHC and Future Colliders\nA. Carle, N. Chanon, and S. Perri\u0012 es\nUniversit\u0013 e de Lyon, Universit\u0013 e Claude Bernard Lyon 1,\nCNRS-IN2P3, Institut de Physique Nucl\u0013 eaire de Lyon,\nVilleurbanne 69622, France\nThis article presents prospects for Lorentz-violation searches with t\u0016tat the\nLHC and future colliders. After a short presentation of the Standard-Model\nExtension as a Lorentz-symmetry-breaking e\u000bective \feld theory, we will focus\nont\u0016tproduction. We study the impact of Lorentz violation as a function of\ncenter-of-mass energy and evaluate the sensitivity of collider experiments to\nthis signal.\n1. Introduction\nThe top-quark sector of Standard-Model Extension (SME) is weakly con-\nstrained. Since the LHC is a top factory, it provides a unique opportunity\nto search for Lorentz violation (LV). The SME is an e\u000bective \feld theory\nincluding all LV operators. Here, we consider the LV CPT-even part of the\nlagrangian modifying the top-quark kinematics:1\nLSME\u001bi\n2(cL)\u0016\u0017\u0016Qt\r\u0016$\nD\u0017Qt+i\n2(cR)\u0016\u0017\u0016Ut\r\u0016$\nD\u0017Ut; (1)\nwhereQtandUtdenote the left- and right-handed top-quark spinors, re-\nspectively. The c\u0016\u0017coe\u000ecients are constant in an inertial frame, taken to\nbe the Sun-centered frame. We aim at measuring the constant coe\u000ecients:2\nc\u0016\u0017=1\n2[(cL)\u0016\u0017+ (cR)\u0016\u0017]; d\u0016\u0017=1\n2[(cL)\u0016\u0017\u0000(cR)\u0016\u0017]: (2)\nExpressions for these coe\u000ecients in a laboratory frame on Earth will intro-\nduce a time dependence of the cross section for t\u0016tproduction owing to the\nEarth's rotation around its axis. This time dependence can be exploited to\nsearch for LV at hadron colliders.\nTo express the c\u0016\u0017coe\u000ecients in the reference frame of a hadron circular\ncollider, we need:arXiv:1909.01990v1  [hep-ph]  4 Sep 2019Proceedings of the Eighth Meeting on CPT and Lorentz Symmetry (CPT'19), Indiana University, Bloomington, May 12{16, 2019\n2\n\u000fthe latitude \u0015, i.e., the angle between the equator and the poles,\n\u000fthe azimuth \u0012,3i.e., the angle between the Greenwich tangent vec-\ntor and the clockwise ring collider tangent vector,\n\u000fthe longitude impacts only the phase of the signal because of the\nEarth's rotation around its axis, and\n\u000fthe Earth's angular velocity \n.\n2. Modulation of the t\u0016tcross section\nThe analysis aims at measuring the time dependence of the t\u0016tcross section\n\u001bSME= [1 +f(t)]\u001bSM: (3)\nA \frst analysis of this kind was performed with the D0 detector at the\nTevatron.4We use here the same benchmarks. We analyze Wilson's coe\u000e-\ncients for a couple of non-null c\u0016\u0017:cXX=\u0000cYY,cXY=cYX,cXZ=cZX\norcYZ=cZY. Each of these scenarios generates an oscillating behavior\nof the amplitude. The latitude \u0015and the azimuth \u0012a\u000bect the amplitude\nwhile the Earth's angular velocity \n a\u000bects the frequency. In the case of\ncXX=\u0000cYYandcXY=cYX,f(t) has a period of one sidereal day. On\nthe other hand, in the cXZ=cZXandcYZ=cZYcase, the amplitude has\na period of one half of a sideral day. More detailed expressions are given in\nRefs. 2, 4.\n3. Expected sensitivity\nIn this work, samples of t\u0016twith dilepton decay were generated with\nMadGraph-aMC@NLO 2.6. It was found that the amplitude of the LV\nt\u0016tsignal is increasing with the center-of-mass energy. The signal ampli-\ntude as a function of the center-of-mass energy in p{pcollisions (with CMS\nor ATLAS as the laboratory frame) increases from 0 :001 at D0 (in the\ncXY=cYX= 0:01 scenario) to 0 :045 at the LHC Run II (13 TeV) and to\n0:055 at the Future Circular Collider (FCC, 100 TeV).\nWe evaluate the expected sensitivity to the signal for each benchmark.5\nAs a consequence of the increase in luminosity, the increase in cross section,\nand the increase in the amplitude of the LV signal, we \fnd the following\nexpected sensitivities to the SME coe\u000ecient c\u0016\u0017in thecXX=\u0000cYYcase:\n\u000f\u0001c= 7\u000210\u00001: D0 (ps= 1:96 TeV,L= 5:3 fb\u00001),\n\u000f\u0001c= 1\u000210\u00003: LHC Run II (ps= 13 TeV,L= 150 fb\u00001),\n\u000f\u0001c= 2\u000210\u00004: HL-LHC (ps= 14 TeV,L= 3000 fb\u00001),Proceedings of the Eighth Meeting on CPT and Lorentz Symmetry (CPT'19), Indiana University, Bloomington, May 12{16, 2019\n3\n\u000f\u0001c= 3\u000210\u00005: HE-LHC (ps= 27 TeV,L= 15 ab\u00001),\n\u000f\u0001c= 9\u000210\u00006: FCC (ps= 100 TeV,L= 15 ab\u00001).\n4. Signal amplitude at hadron colliders\nA noticeable fact is the dependence of the signal amplitude on the lati-\ntude and azimuth of the collider experiment on Earth. This dependence\nis presented in Fig. 1. We \fnd that performing such an experiment at the\nAmplitude fSME (/g79,/g84) fXX Azimuth /g84 (in rad) CMS\n0000.20.40.60.8\n0.5 111.21.4\n1.5 -0.5 -1 -1.5123456\nATLAS \nD0 \nLatitude /g79 (in rad)\nFig. 1. Amplitude of f(\u0015;\u0012) as a function of latitude and azimuth for the XX,YY,\nandXYbenchmarks.\nLHC would increase the sensitivity to SME coe\u000ecients in the top sector\nby two orders of magnitude. Further improvements are expected at future\ncolliders.\nAcknowledgments\nThanks to Alan Kosteleck\u0013 y and Ralf Lehnert for giving me the opportunity\nto expose my work in front of a benevolent and hard-working community.\nReferences\n1. D. Colladay and V.A. Kosteleck\u0013 y, Phys. Rev. D 93, 036005 (2016).\n2. M.S. Berger, V.A. Kosteleck\u0013 y, and Z. Liu, Phys. Rev. D 93, 036005 (2016).\n3. M. Jones, Activity Report, EDMS, 322747 (2005).\n4. D0 Collaboration, V.M. Abazov et al. , Phys. Rev. Lett. 108, 261603 (2012).\n5. A. Carle, N. Chanon, and S. Perri\u0012 es, arXiv:1908.11256 [hep-ph]"    },    {        "title": "1412.2574v3.Bi___cal_PT___symmetry_in_nonlinearly_damped_dynamical_systems_and_tailoring___cal_PT___regions_with_position_dependent_loss_gain_profiles.pdf",        "content": "arXiv:1412.2574v3  [nlin.PS]  15 Dec 2015Bi-PTsymmetry in nonlinearly damped dynamical systems and\ntailoring PTregions with position dependent loss-gain profiles\nS. Karthiga1, V.K. Chandrasekar2, M. Senthilvelan1, M. Lakshmanan1\n1Centre for Nonlinear Dynamics, School of Physics,\nBharathidasan University, Tiruchirappalli - 620 024, Tami l Nadu, India.\n2Centre for Nonlinear Science & Engineering, School of Elect rical & Electronics Engineering,\nSASTRA University, Thanjavur -613 401, Tamil Nadu, India.\nWe investigate the remarkable role of position dependent da mping in determining the parametric\nregions of symmetry breaking in nonlinear PT-symmetric systems. We illustrate the nature of\nPT-symmetry preservation and breaking with reference to a rem arkable integrable scalar nonlinear\nsystem. In the two dimensional cases of such position depend ent damped systems, we unveil the\nexistence of a class of novel bi- PT-symmetric systems which have two fold PTsymmetries. We\nanalyze the dynamics of these systems and show how symmetry b reaking occurs, that is whether the\nsymmetry breaking of the two PTsymmetries occurs in pair or occurs one by one. The addition o f\nlinear damping in these nonlinearly damped systems induces competition between the two types of\ndamping. This competition results in a PTphase transition in which the PTsymmetry is broken\nfor lower loss/gain strength and is restored by increasing t he loss/gain strength. We also show that\nby properly designing the form of the position dependent dam ping, we can tailor the PT-symmetric\nregions of the system.\nPACS numbers: 11.30.Er, 05.45.-a, 11.30.Qc\nI. INTRODUCTION\nIn recent times considerable interest has been shown\nin investigating systems which do not show parity ( P)\nand time reversal ( T) symmetries separately but which\nexhibit a combined PTsymmetry. These PT-symmetric\nsystemshaveseveralintriguingfeaturessuchaspoweros-\ncillations [1], absorption enhanced transmission [2], dou-\nble refraction, and non-reciprocity of light propagation\n[1]. Thus, these systems open up novel applications in\noptics [1], quantum optics [3, 4], solid state physics [5],\nmetamaterials [6, 7], optomechanical systems [8, 9], etc.\nThe understanding of PT-symmetric systems as non-\nisolated systems with balanced loss and gain has led to\ntheexplorationofthesesystemsinmechanicsaswellasin\nelectronics. Suchobservationsof PT-symmetricmechan-\nical and electronic systems provide the simplest ground\nto experiment on these PT-symmetric systems [10–14].\nA.Bi-PT symmetry The above oscillator based PT-\nsymmetric systems are generically constructed by cou-\npling an oscillator with linear loss to an oscillator with\nequal amount of linear gain [11–14]. Apart from the\nabove type of systems, there exists a class of interest-\ning dynamical systems with position dependent damp-\ning (or position dependent loss-gain profile) where the\namount of damping depends on its displacement. Conse-\nquently one can have PT-symmetric systems even with\na single degree of freedom. In this case, the systems are\ninvariant with respect to the PToperation defined by P:\nx→ −x,T:t→ −t, so that PT:x→ −x,t→ −twhich\nwe denote as the PT−1 operation. As the position de-\npendent damping term is found to be a nonlinear term in\nthe evolution equation, we call this damping as nonlinear\ndamping for simplicity. The main aim of this paper is to\ninvestigate the dynamics and underlying novel structuresin these systems in comparison with the standard ones.\nThe recent explorations on the damping in systems\nwith one or more atomic-scale dimensions have unveiled\nthat the damping present in these systems is strongly po-\nsition dependent [15–17]. Ref. [15] shows that this type\nof damping in mechanical resonators enhances the fig-\nure of merit of the system tremendously. In particular,\nwith this type of damping, a quality factor of 100 ,000\nhas been achieved with graphene resonators. In addi-\ntion, such systems are found to play an important role in\nmany areas of physics, biology and engineering [18] and\nthey are typically called Li´ enard systems or Li´ enard os-\ncillators. Recently, a class of chemical and biochemical\noscillations which are governed by two-variable kinetic\nequations are shown to be reducible to Li´ enard systems\nby linear transformations. As the nonlinear damping\nterm in the Li´ enard systems can act as a damping term\nor a pumping term depending on the amplitude of the\noscillation, through an internal energy source, it gives\nrise to self sustained oscillations. The above property\nenables one to understand and to control several chem-\nical and biochemical oscillations which are discussed in\n[19]. Li´ enard systems are also found to be paradigmatic\nmodels in the biological regulatory systems [20]. For ex-\nample, they have been used to model the heart and res-\npiratory systems (van-der Pol equation [21, 22]) and the\nnerve impulse (FitzHugh-Nagumo equations [23]). The\nLi´ enard equation with a cubic polynomial potential has\nbeen used to describe the isotropic turbulence [24]. One\ncan also find the appearance ofthese systems in reaction-\ndiffusion systems [25].\nConcerning the importance of the above type of non-\nlinearly damped systems, we here focus on the PT-\nsymmetriccasesofthis category. The Hamiltonian struc-\nture [26] and quantization [27, 28] of some of the nonlin-2\nearly damped PT-symmetric systems with single degree\nof freedom have been studied recently, which show in-\nteresting symmetry breaking in these systems (see also\nSection III below).\nA proper coupling of two scalar nonlinearly damped\nPT-symmetric systems can yield novel bi- PT-symmetric\nsystems which are invariant with respect to the PT −1\n(x→ −x,y→ −y,t→ −t) operation as well as with\nthePT −2 operation which is defined as PT −2:x→\n−y,y→ −x,t→ −t. Such type of studies on the\nsystems with multiple PTsymmetries is interesting, for\nexample one can see an earlier paper on such multiple\nPTsymmetriccases[29]. Inthispaper,wepointoutthat\nthe study of bi- PTsymmetries in such coupled nonlinear\ndamped systems can lead to interesting novel dynamical\nstates of PTsymmetry preserving and breaking types,\nbesides oscillation death and bistable states.\nB. Spontaneous symmetry breaking: An interesting\nmechanism that is found to arise in the PTsymmetric\nsystems is the spontaneous symmetry breaking, where\nthe system in the symmetric state transits to an asym-\nmetric state by the variation of certain parameters. In\nclassical systems, the simplest state of broken symme-\ntry is the equilibrium state which may correspond to the\nminimum of the potential but which does not possess\nall the symmetries underlying the dynamical equation.\nLetGbe the transformation under which the dynamical\nequation is invariant. Then a symmetric state u=us\ncorresponds to the state which remains invariant under\nthe transformation us=Gus. But an asymmetric or\nsymmetry broken state ua(that may also correspond to\nthe minimum of the potential) is the one that gets trans-\nformed into another asymmetric state ui=Guaunder\nthe transformation G. Here the transformed state ui\nalso corresponds to an equilibrium of the system. A typ-\nical example is the reflection symmetry in a double well\nquartic anharmonic oscillator. From a dynamical point\nof view the spontaneous breaking of symmetries is also\nmanifested in the stability nature of the fixed points and\nthe trajectories around it in the phase space and nature\nof bifurcations as a system parameter is varied, again\nas in the case of the double well quartic oscillator un-\ndergoing spontaneous P-symmetry breaking. In this pa-\nper, we also show that the above existence of symmetry\npreserving/breaking equilibrium states can be identified\nwith the existence or nonexistence of the general solu-\ntion of the initial value problem underlying the dynam-\nical system satisfying the symmetry and the system can\nadmit more general classes of solution corresponding to\nsymmetry preservation/breaking.\nA universalfeature ofthe standard PT-symmetricsys-\ntems is that the PTsymmetry is broken by increas-\ning the loss/gain strength and is restored by reducing\nit [11, 12]. In contrast to this behavior, Liang et al. [30]\nhaveobservedareverse PTphasetransitionphenomenon\nin a lattice model known as PT-symmetric Aubry-Andre\nmodel[31], inwhichthe PTsymmetryisbrokenforlower\nloss/gain strength and is restored for higher loss/gainstrength. Theyobservedthisphenomenononlywhentwo\nlattice potentials that introduce loss/gain in the system\nare applied simultaneously (which is not observed when\na single lattice potential is present). This type of inverse\nPTphase transition arises as a result of the competition\nbetween the two lattice potentials. Similarly, Mirosh-\nnichenko et al. [32] have studied the competing effect of\nlinearand nonlinearloss-gainprofilein discretenonlinear\nSchr¨ odinger system. The observation of PTrestoration\nat higher loss-gain strengths also attracted wide inter-\nests and the recent studies show that it could happen\neven through an interplay of kinematical and dynamical\nnonlocalities [33].\nC. Nonlinear damping and PTsymmetry: From a dif-\nferent point of view, in the present work, we add a linear\ndamping in addition to the nonlinear damping and study\nthe competing effects of the linear and nonlinear damp-\ning forces. With a single nonlinear damping, our sys-\ntem shows PTsymmetry breakinglike the standard PT-\nsymmetric systems, but as soon we add the linear damp-\ning to the nonlinear damping, we observe PTrestora-\ntion at higher loss/gain strength similar to the case of\nAubry-Andre model. Importantly, we illustrate that this\ncompetition among the damping terms in addition to the\npositiondependent natureofdampingaidin tailoringthe\nPTregions of the system.\nThe organization of the paper is as follows, in section\nII, we discuss the loss-gain profiles of the scalar PT-\nsymmetric and non- PT-symmetric nonlinearly damped\nsystems. In section III, we consider a specific model\nof scalar PTsymmetric nonlinear damped oscillator,\nnamely the modified Emden equation. Analyzing the\ninitial value problem of an integrable case explicitly,\nwe greatly clarify the nature of PTsymmetry preser-\nvation/breaking. In section IV, we consider a coupled\nsystem with a simple nonlinear damping h(x,˙x) =x˙x,\nwhich is also a bi- PT-symmetric system. In section V,\nin addition to the nonlinear damping, we introduce a lin-\near damping in the system and show the occurrence of\nPTrestoration at higher values of loss/gain strength. In\nsection VI, we consider a general coupled system with\nlinear and nonlinear damping and show the tailoring of\nPTregions in the system. In section VII, we summa-\nrize the results of our work. In Appendix A, we consider\nthe initial value problem of a double-well oscillator and\ndiscuss the spontaneous P-symmetrybreakingfrom solu-\ntion point of view. In Appendix B we consider non- PT\nsymmetric scalar systems. In Appendices C, D and E,\nwe have presented the eigenvalues obtained through the\nlinear stability analysis for the systems we considered.\nII. NONLINEARLY DAMPED\nSYSTEMS-REVISITED\nTo start with, we analyze the loss-gain profiles of posi-\ntion dependent scalar nonlinearly damped systems. For\nthispurpose,wefirstconsiderasystemwhichisdescribed3\nby the second order nonlinear differential equation\n¨x+h(x,˙x)+g(x) = 0./parenleftbigg\n˙=d\ndt/parenrightbigg\n(1)\nHere,h(x,˙x) =f(x)˙xis the position dependent damp-\ning which we call for simplicity as the nonlinear damping\nterm. Also, f(x) is taken as a non-constant function in\nx. The above equation can be considered as a dynamical\nsystemonitsownmerit, oftenwithanonstandardHamil-\ntonian description [26], or as a conservative nonlinear os-\ncillator perturbed by a nonlinear damping force h(x,˙x)\nwhich supplies or absorbs energy at different points in\nthe (x,˙x) phase space,\n¨x+g(x) =−h(x,˙x) =−f(x)˙x. (2)\nThekinetic andthe potential energiesofthe unperturbed\nparticle are given respectively by\nT(˙x) =1\n2˙x2;V(x) =/integraldisplay\ng(x)dx. (3)\nThusthetotalenergyoftheparticleinthepotential V(x)\nwhenh(x,˙x) = 0 is\nE=1\n2˙x2+/integraldisplay\ng(x)dx. (4)\nThe rate of change of energy of the particle is\ndE\ndt= ˙x(¨x+g(x)). (5)\nFrom Eq. (1), we can write\ndE\ndt=−˙xh(x,˙x) =−f(x)˙x2. (6)\nIf the quantitydE\ndt<0 (or ˙xh(x,˙x)>0) in a region\nin (x,˙x) phase space, then the energy is withdrawn from\nthe system for the states lying in this region and the role\nofh(x,˙x) is like a damping or loss term and ifdE\ndt>0\n(or ˙xh(x,˙x)<0), then in the corresponding region the\neffect of h(x,˙x) is like negative-damping or gain.\nThe above type of nonlinearly damped systems can\nbe classified as ( i)PT-symmetric systems and ( ii) non-\nPT-symmetric systemsdepending on the form of h(x,˙x),\nwhereas all linearly damped systems are always non- PT-\nsymmetric. Here, the PT-symmetric systems are those\nsystems that are invariant under the combined operation\nofPT(and not individual operation of PorT):x→\n−x,t→ −t. We denote this as PT −1 symmetry (in\norder to distinguish it from the additional PTsymmetry\nin two dimensional systems). Then PT −1 symmetric\nsystems belonging to (1) are those systems where h(x,˙x)\nis a nonlinear function in x, ˙xthat is odd in xas well\nas ˙x. In this article, we focus our attention towards the\nsystems with h(x,˙x) =f(x)˙x, wheref(x) andg(x) in\n(1) are odd functions. Systems of the form (1) which do\nFIG. 1: (Color online) Loss-gain profilesdE\ndtof the systems\ngiven by (a) Eq. (7), (b) Eq. (8) and (c) Eq. (9) in the ( x,˙x)\nspace: The pink shaded regions in the figures correspond to\nthe regions in whichdE\ndtis positive (or it denotes the region\nin which gain is present). Similarly, the gray shaded region s\ndenote the regions in whichdE\ndtis negative.\nnotmeetthis requirementarenon- PT-symmetric. These\nnon-PT-symmetricsystemsaretypicallyoftwotypes, ( i)\nsystems exhibiting damped oscillations and ( ii) systems\nadmitting limit cycle oscillations. In the following we\npresent specific examples of these three cases:\n1.PT-symmetric conservativesystem - Modified Em-\nden Equation (MEE)[26, 34]:\n¨x+αx˙x+βx3+ω2\n0x= 0 (7)\n2. Non-PT-symmetric damped system[35]:\n¨x+αx2˙x+βx3+ω2\n0x= 0 (8)\n3. Limit cycle oscillator (van der Pol oscillator) [36]:\n¨x+(x2−1)˙x+ω2\n0x= 0. (9)\nThe system (7) is known as the modified Emden equa-\ntion and is obviously invariant under the PT −1 opera-\ntion. The PT-symmetric nature of this system [26] and\nits quantization [27] have been studied for the specific\ncaseβ=α2\n9which admits symmetry breaking states for\nλ <0. A critical analysis of the PT- symmetry of (7)\nis given in section III. The systems given in Eqs. (8)\nand (9) are examples of non- PT-symmetric ones, as the\ndamping term in these cases are found to be even func-\ntions ofx. The system (8) admits damped oscillations,\nwhile the system (9) (the famous van der Pol oscillator)\nis found to have self sustained oscillations which is also\nnoted in Appendix B.4\nFigure 1 shows the loss-gain profiles corresponding to\nEqs. (7), (8) and (9), which are obtained by substitut-\ning the corresponding forms of f(x) in Eq. (6) . From\nthe loss-gain profile (shown in Fig.1(a)) corresponding to\nthePT −1 symmetric case (7), we can find that we have\nvarying loss along the positive x−axis and varying gain\nalong the negative x−axis. The amount of gain present\nforx <0 is balanced by the amount of loss present for\nx >0. Then from Figs. 1(b) and 1(c), we can see that\nin the case of non- PT-symmetric systems, the loss and\ngain will not be balanced. In the case of the non- PT\ndamped oscillator (8), from Fig. 1(b) we can find that\nloss is present everywhere in space. In the case of limit\ncycle oscillator (9), from Fig. 1(c), we can find that gain\nexists in the region |x|<1 and loss exists in the region\n|x|>1. This clearly shows that in this case, the amount\nof loss present in the ( x,˙x) space is not balanced by an\nequal amount of gain.\ntx\n30 15 00.7\n0\n-0.7\ntdE\ndt\n30 15 00.8\n0\n-0.8\ntx\n30 15 00.6\n0\n-0.6\ntdE\ndt\n30 15 00.07\n0\n-0.07\n-0.14\ntx\n30 15 02.5\n0\n-2.5\ntdE\ndt\n30 15 06.5\n0\n-6.5(a) (b)\n(c) (d)\n(e) (f)\nFIG. 2: (Color online) Figures ( a), (c) and (e) depict the\nsolution of Eqs. (7), (8) and (9), respectively, for two dif-\nferent initial conditions. Figures ( b), (d) and (f) show the\ncorresponding ratesdE\ndtas a function of time.\nFrom Fig. 2, we can see that in the PT-symmetric and\nlimit cycle oscillator cases, there exists periodic and selfsustained oscillations (Figs. 2(a), 2(e)), respectively, and\nin the non- PT-symmetric damped oscillator case (Fig.\n2(c)), we have damped oscillations. The corresponding\nrates of change of energydE\ndtprofiles are shown in Figs.\n2(b), 2(d) and 2(f), respectively.\nComparing the periodic oscillations (Figs. 2(a) and\n2(e)) corresponding to the PT-symmetric oscillator case\n(Eq. (7)) and the limit cycle oscillator case (Eq. (9)),\nwe can find that the PT-symmetric system takes up dif-\nferent paths for different initial conditions but the limit\ncycle oscillator for different initial conditions tends to a\nparticular path as time t→ ∞. The reason is that the\nbalanced loss-gain profile (shown in Fig. 1(a)) of the\nPT-symmetric system allows it to have multiple paths\nalong which netdE\ndtis zero. But in the case of limit cy-\ncle oscillator, Fig. 1(c) shows that the loss and gain are\nnot balanced in the ( x,˙x) space. Thus the paths along\nwhich totaldE\ndtis zero are limited in this case. Conse-\nquently the phase space of limit cycle oscillators contains\nisolated paths only.\nNow let us consider a system of coupled nonlinear\ndamped oscillators (for simplicity we consider a linear\ncoupling)\n¨x+h1(x,˙x)+h2(x,˙x)+g(x)+κy= 0,\n¨y+h1(y,˙y)−h2(y,˙y)+g(y)+κx= 0,(10)\nwhereh1(x,˙x) =f1(x)˙xandh2(x,˙x) =f2(x)˙xare the\ntwo position dependent nonlinear damping terms. Here,\nthe functions f1(x) andf2(x) are chosen to be odd and\neven functions in x, respectively, and also the function\ng(x) is chosen as odd. Consequently, the system becomes\nsymmetric with respect to the PT −2 operation (which\nis defined as PT −2:x→ −y,y→ −x,t→ −t). Now,\nby making f2(x) to be zero, the system is symmetric\nwithrespecttoboth PT −1andPT −2operations. (Here\nPT −1 corresponds to the operation x→ −x,y→ −y,\nt→ −t.) Thus the system is bi- PT-symmetric in this\ncase.\nSimilar to the scalar case, we can consider the above\nsystem as a system of two coupled oscillators\n¨x+g(x)+κy= 0,\n¨y+g(y)+κx= 0, (11)\nacted upon by additional external forces h1(x,˙x) and\nh2(x,˙x). The total energy of the system (in the absence\nof nonlinear damping) is given by\nE=1\n2˙x2+/integraldisplay\ng(x)dx+1\n2˙y2+/integraldisplay\ng(y)dy+κxy.(12)\nThe rate of change of energy in the system due to the\nweak nonlinear damping term as specified by Eq. (10) is\ngiven by\ndE\ndt=−˙x[h1(x,˙x)+h2(x,˙x)]−˙y[h1(y,˙y)−h2(y,˙y)].(13)\nThe above expression shows that similar to the scalar\ncase, thecoupledsystem(10)alsohaspositiondependent5\nloss-gain profile. Further, the question whether a non-\nstandard Hamiltonian description similar to the scalar\ncase (Section III) exists for (10) has not yet been an-\nswered in the literature as far as the knowledge of the\nauthors goes, though a class of such systems has recently\nbeen identified [37, 38].\nIII.PTSYMMETRY BREAKING IN THE\nMODIFIED EMDEN EQUATION\nThe system mentioned in Eq. (7), namely\n¨x+αx˙x+βx3+λx= 0, λ=ω2\n0,(14)\nis the simplest example for PT −1 symmetric system.\nThe reversible nature of the system has been studied and\nthis equation is used as a normal form for describing the\nsymmetry breaking bifurcation in certain reversible sys-\ntems which includes an externally injected class B laser\nsystem [39]. This x˙xtype damping has been found to ap-\npear in many chemically relevant kinetic equations [19].\nThe model is found to be useful in fluid mechanics where\nthe linearly forced isotropic turbulence [24] can be de-\nscribed in terms of a cubic Li´ enard equation which is of\nthe form similar to (14). This system is also found to\nappear in some important astrophysical phenomena and\nit occurs in the study of equilibrium configurations of a\nspherical cloud acting under the mutual attraction of its\nmolecules and is subject to the thermodynamic laws [40].\nEq. (14) is known to admit a nonstandard conservative\nHamiltonian description [34] and interesting dynamical\nproperties [35]. In particular, the specific choice β=α2\n9\nadmitsisochronousproperties[26](seebelow)andcanbe\neven quantized in momentum space, exhibiting PTsym-\nmetry and broken PTsymmetry as shown by Chithiika\nRuby et al [27] recently, see subsection IIIB below.\nA. Linear stability analysis\nLet us analyze the dynamical behavior of the system\n(14) qualitatively through a linear stability analysis. Eq.\n(14) can be rewritten as\n˙x=x1\n˙x1=−αxx1−βx3−λx. (15)\nThissystemhasatrivialequilibriumpoint E0: (x∗,x∗\n1) =\n(0,0) and a pair of non-trivial equilibrium points sym-\nmetrically positioned along x-axis about x= 0,E1,2:\n(±/radicalBig\n−λ\nβ,0) (which exist only if λ <0 orβ <0). In our\nfollowing analysis, we take β >0 and so E1,2exist only\nforλ <0. The Jacobian matrix corresponding to the\nsystem (15) is given by\nJ=/bracketleftbigg0 1\n−αx∗\n1−3βx∗2−ω2\n0−αx∗/bracketrightbigg\n(16)The eigenvalues of Jcorresponding to the equilibrium\npointE0areµ(0)\n1,2=±i√\nλ. Similarly, the eigenvalues of\nJcorresponding to E1andE2areµ(1)\n1,2=1\n2√β(−α√\n−λ±\n/radicalbig\n−λ(α2−8β)),µ(2)\n1,2=1\n2√β(α√\n−λ±/radicalbig\n−λ(α2−8β)).\nThe real part of the eigenvalues of Jassociated with\nthe above equilibrium points are given in Fig. 3. The fig-\nure shows that in the region λ >0, the equilibrium point\nE0alone exists and all the eigenvalues of E0are found to\nbe pure imaginary(or Re[µ] = 0). So in the region λ >0,\nperiodic oscillations exist in the system corresponding to\nwhichthephasetrajectoriesaroundtheequilibriumpoint\nE0preserve their structure under PToperation, where\nE0itself remains invariant: PT[E0]=E0. Thus PT-\nsymmetry is unbroken while λ >0. But by varying λ\ntoλ <0, a pair of equilibrium points ( E1andE2) with\nopposite stabilities arise, where E1is stable (as all the\neigenvalues have Re[µ]<0) while E2is unstable (as all\neigenvalues have Re[µ]>0). In this region E0becomes\na saddle (as one of the eigenvalues of E0hasRe[µ]>0\nand the other eigenvalue has Re[µ]<0 ). Under the\nPToperation, E1gets transformed to E2and vice-versa:\nPT[E1]=E2andPT[E2]=E1so that the PTsymmetry\ngets broken. Correspondingly the trajectories around E1\nget transformed to trajectories around E2and vice-versa\nunder the PToperation. Note that the above kind of\nbifurcations fall within the scope of Thom’s catastrophe\ntheory [41].\n−3 −1 1 3\nλ−3.0−1.50.01.53.0Re[µ]E0\nE1\nE2\nUnbroken PTregion BrokenPTregion\nE0(us),E1(s),E2(us) E0(ns)(s): stable\n(us): unstable\n(ns): neutrally stable\nFIG. 3: (Color online) Plot of the real part of the eigenvalue s\nofJassociated with the equilibrium point E0,E1,E2of the\nsystem (14) for the values of α= 2 and β= 1.\nTo appreciate these aspects more clearly, we plot the\nphase portraits of the system for the explicitly integrable\ncaseβ=α2\n9, obtained from the exact solutions of the\nsystem [26]. A qualitatively similar set of phase portraits\nresults for the general case β/negationslash=α2\n9, which can be drawn\nthrough a numerical analysis.6➤ ➤➤ ➤\n➤ ➤E0➤\n➤➤➤➤➤\n➤\n➤➤\nx(t)˙x(t)\n1.5 0 -1.54\n2\n0\n-2\nFIG. 4: (Color online) Phase portrait of the system (14) for\nλ= 1,α= 3 and β= 1. The green colored diamond in the\nfigure denotes the position of the neutrally stable equilibr ium\npointE0.\n➤\nE0➤ ➤➤ ➤ ➤ ➤➤➤➤➤\n➤\nx(t)˙x(t)\n2 0 -23\n0\n-3\nFIG. 5: (Color online) Phase portrait of the system (14) at\nthe bifurcation point λ= 0 with α= 3 and β= 1.\nB. The exactly integrable case: β=α2\n9\nWe consider the specific case β=α2\n9of Eq. (14),\nnamely\n¨x+αx˙x+α2\n9x3+λx= 0 (17)\nor equivalently\n˙x=y,\n˙y=−αxy−α2\n9x3−λx. (18)\nEq. (17) or (18) admits a nonstandard Lagrangian/ con-\nservative Hamiltonian description [26] with\nL=27λ3\n2α2/parenleftBigg\n1\nα˙x+α2\n3x2+3λ/parenrightBigg\n+3λ\n2α˙x−9λ2\n2α2.(19)➤ ➤➤ ➤➤➤ ➤➤➤➤➤\n➤➤➤\n➤➤\n➤\n➤➤ ➤➤ ➤E1 E2E0\nx(t)˙x(t)\n2 0 -22\n0\n-2\nFIG. 6: (Color online) Phase portrait of the system (14) for\nλ=−1,α= 3 and β= 1. The green circle denotes the sta-\nble node type equilibrium point E1, red colored triangle and\nsquare correspond to the saddle type equilibrium point ( E0)\nand unstable node type of equilibrium point E2respectively.\nThe continuous line denotes the orbits corresponding to sym -\nmetric solutions and the dashed lines corresponds to that of\nasymmetric solutions.\nThen the canonically conjugate momentum is\np=−27λ3\n2α/parenleftBigg\n1\n(α˙x+α2\n3x2+3λ)2/parenrightBigg\n+3λ\n2α,(20)\nso that the Hamiltonian H\nH=9λ2\n2/parenleftBigg\n((˙x+α\n3x2)2+λx2)\n(α˙x+α2\n3x2+3λ)2/parenrightBigg\n=9λ2\n2α2/bracketleftBigg\n2−2/parenleftbigg\n1−2αp\n3λ/parenrightbigg1\n2\n+α2x2\n9λ−2αp\n3λ−2α3x2p\n27λ2/bracketrightBigg\n(21)\nwhich is a conserved quantity and we may call it as the\n’energy’E.\nNow the exact solution of (17) for the three cases\nλ >0,λ= 0 and λ <0 are as follows [26]:\n(i) Case-1: λ >0:Here one has periodic solutions of\n(17) or (18) as\nx(t) =Asin(ω0t+δ)\n1−Aα\n3ω0cos(ω0t+δ), ω0=√\nλ(22a)\n˙x(t) =Aω0cos(ω0t+δ)\n1−Aα\n3ω0cos(ω0t+δ)−α\n3x2(t),(22b)\nwhere,Aandδare constants. Note that the solution is\nperiodic and bounded for 0 ≤A <3ω0\nα. ForA≥3ω0\nα, the\nsolution is singular and periodic. Also one can evaluate\nfrom (21) using (22) the ’energy’ in this case as\nH=E=1\n2ω2\n0A2. (23)7\n(ii) Case-2: λ= 0:One has a decaying type or front\nlike solution in this case as\nx(t) =I1+t\nαt2\n6+I1αt\n3+I2, (24a)\nand\n˙x(t) =1\nαt2\n6+I1αt\n3+I2−α\n3x2(t),(24b)\nsuch that\nH=E= 0, (25)\nwhereI1andI2are arbitrary constants.\n(iii) Case-3: λ <0:Here we have the general solution\nx(t) =3/radicalbig\n|λ|(I1e√\n|λ|t−e−√\n|λ|t)\nα(I1I2+I1e√\n|λ|t+e−√\n|λ|t)(26a)\n˙x(t) =3|λ|(I1e√\n|λ|t+e−√\n|λ|t)\nα(I1I2+I1e√\n|λ|t+e−√\n|λ|t)−α\n3x2(t).(26b)\nwith\nH=E=18|λ|2\nα21\nI1I2\n2, (27)\nwhereI1andI2are arbitrary constants.\nNow treating the nonlinear differential equation (17)\nor (18) as a dynamical system, we shall consider the so-\nlution of its initial value problem (IVP) admitting the\nPTsymmetry. Since we require the PTsymmetry to be\nvalid for the entire duration of evolution, starting from\ntheinitialreferencetimewhichmaybetakenwithoutloss\nof generality as t= 0, we require the initial values of the\ndynamical variables corresponding to a definite ’energy’\nsatisfy the PT-symmetry conditions ( x(t)→ −x(−t),\nt→ −t, ˙x(t)→˙x(−t)):\nx(0) =c1=−x(0)\n˙x(0) =c2= ˙x(0) (28)\nwherec1andc2are arbitrary constants. Then one can\nidentify two possibilities.\n(i)PT-symmetric solution:\nc1= 0, c2=c (29)\nsuch that\nPT[x(t)] =−x(−t) =x(t),\nPT[˙x(t)] = ˙x(−t) = ˙x(t),for allt≥0,(30)\n(ii)PT-asymmetric solution:\nOne can consider two distinct values\nx1(0) =c1, x2(0) =−c1, c1/negationslash= 0 (31)such that for t >0, one can have a disjoint set of two\ndisconnected solutions/trajectories for a given E:\nPT[x1(t)] =−x1(−t) =x2(t)/negationslash=x1(t),\nPT[˙x1(t)] = ˙x1(−t) = ˙x2(t)/negationslash= ˙x1(t),(32)\nand\nPT[x2(t)] =−x2(−t) =x1(t)/negationslash=x2(t),\nPT[˙x2(t)] = ˙x2(−t) = ˙x1(t)/negationslash= ˙x2(t),for allt≥0.(33)\nassociated with the same energy value E. Sincex1(t) and\nx2(t) correspondto two distinct unconnected trajectories\nin phase space but with the same ’energy’ value, they\nrepresent solutions of broken PT- symmetry.\nWe now point out explicitly the abovetype ofsolutions\nfor the system (17) or (18) in the following, depending on\nthe sign of λ. We also demonstrate in Appendix A that a\nsimilar type of consideration exists for the P-symmetric\nsystem also, for example in the case of the double well\ncubic anharmonic oscillator.\nC. Observation of symmetry breaking from the\nsolution point of view\nCase-1: λ >0:PTinvariant solutions\nConsidering the general solution (22) for λ >0, with-\nout loss of generality we consider the solution of the ini-\ntial value problem with\nx(0) = 0,˙x(0) =B=3Aω2\n0\n3ω0−Aα(34)\nwhich is itself PTinvariant. This fixes δ= 0 in the solu-\ntion (22). Then the resultant general solution (22) with\nδ= 0oftheinitialvalueproblemisfully PT-invariantfor\nallt≥0, that satisfies (30). The ’energy’ associated with\nthe solution is again E=1\n2ω2\n0A2as given in (23). Note\nthat the above solution includes the equilibrium point\nE0= (0,0) when A= 0 with the energy Etaking the\nminimum value. The corresponding phase trajectories in\n(x,˙x) space are plotted in Fig. 4 which form concentric\nclosed curves around E0as long as A <3ω0\nα, so that it\nis a centre type equilibrium point. The associated eigen-\nvalues of the equilibrium point E0are±i√\nλ(as shown\nin Sec. IIIA above). Note that for A≥3ω0\nα, the solution\nbecomes singular at finite times giving rise to open tra-\njectories in the phase space Fig. 4 but which shall show\nPTsymmetry.\nOne can also observe that the phase trajectories are\ninvariant under time translation. Consequently, the so-\nlution corresponding to any other initial condition ob-\ntainable from (22) also follows an identical phase trajec-\ntory for a given Aand so a given value of ’energy’ Eas\nit is obtained by a time translation which is an allowed\nsymmetry of the original dynamical system (14). Hence\nthese solutions may not be treated as distinct from the8\none corresponding to (34), if time translation symmetry\nis also included, along with PTsymmetry. Due to the\nreason,nosymmetrybreakingasymmetricsolutionexists\nhere.\nCase-2: λ= 0- Bifurcation point\nHere again the solutions of the initial value problem\nwithx(0) = 0, ˙x(0) =1\nI2deduced from (24) correspond-\ning toE= 0, satisfy the PTsymmetry as shown with the\nphase trajectories in Fig. 5.\nCase-3: λ <0-PTsymmetry breaking\nIn this case one can identify three distinct classes of\nsolutionsfrom the generalsolution (26) of(17) or(18) for\nλ <0, namely x0(t),x1(t) andx2(t). Among them x0(t)\nforms the symmetric solution satisfying (30) and the set\nx(t) = (x1(t),x2(t)) satisfying(32)and (33) constitutes a\nspontaneously symmetry breaking set of solutions which\nare discussed below.\n(a) Symmetric solution:\nThe explicit form of the solution satisfying the initial\nconditions x0(0) = 0, ˙ x0(0) = constant turns out to be\nthe following:\nx0(t) =3/radicalbig\n|λ|(e√\n|λ|t−e−√\n|λ|t)\nα(I2+e√\n|λ|t+e−√\n|λ|t)\n˙x0(t) =3|λ|(e√\n|λ|t+e−√\n|λ|t)\nα(I2+e√\n|λ|t+e−√\n|λ|t)−α\n3x2\n0(t),(35)\nas can be deduced from the general solution (26). Here\nI2is an arbitrary constant. Note that the solution\n(35) satisfies the PTsymmetry PT(x0(t),˙x0(t)) =\n(x0(t),˙x0(t)) and that ( x0,˙x0) = (0,0) =E0in the\nlimitI2→ ∞. Also, we observe that asymptotically,\nast→ ∞, (x0(t),˙x0(t))−→\nt→∞(3√\n|λ|\nα,0) =E1. That is\nall the nonsingular trajectories approach the fixed point\nE1, exceptE0, so that E0is a saddle.\n(b) Asymmetric solution:\nNext we have the other two distinct solutions which\nbreak the PTsymmetry. The first one is given by\nx1(t) =3/radicalbig\n|λ|(I1e√\n|λ|t−e−√\n|λ|t)\nα(−2+I1e√\n|λ|t+e−√\n|λ|t), I1<0,\n˙x1(t) =3|λ|(I1e√\n|λ|t+e−√\n|λ|t)\nα(−2+I1e√\n|λ|t+e−√\n|λ|t)−α\n3x2\n1(t).(36)\nNote that ( x1(0),˙x1(0)) = (3√\n|λ|\nα,6|λ|\nα(I1−1)) and asymp-\ntotically ( x1(∞),˙x1(∞)) = (3√\n|λ|\nα,0) =E1. Also whenI1→ ∞, (x1(0),˙x1(0)) tends to E1. Again all the non-\nsingular trajectories approach E1asymptotically.\nSimilarly, we have the other distinct set of trajectories\nx2(t) =3/radicalbig\n|λ|(e√\n|λ|t−I1e−√\n|λ|t)\nα(−2+e√\n|λ|t+I1e−√\n|λ|t), I1<0,\n˙x2(t) =3|λ|(e√\n|λ|t+I1e−√\n|λ|t)\nα(−2+e√\n|λ|t+I1e−√\n|λ|t)−α\n3x2\n2(t).(37)\nNote that ( x2(0),˙x2(0)) = ( −3√\n|λ|\nα,6|λ|\nα1\nI1−1). In\nthe limit I1→ −∞ this approaches the equilibrium\npointE2= (−3√\n|λ|\nα,0). Interestingly, these trajec-\ntories (except E2) also approach E1asymptotically:\n(x2(∞),˙x2(∞))=(3√\n|λ|\nα,0). Notethatintheaboveeach\ndistinct trajectory corresponds to the invariant ’energy’\nE=18|λ|2\nα21\nI1I2\n2.\nThe above facts are illustrated by the corresponding\nphase trajectories for the case λ <0 in Fig. 6. In the\ncase 0< I1<∞(but not equal to 1), the evolution\ncorrespondingto x1(t)andx2(t)frominitialtime t= 0to\n∞lie along the same path, the trajectory corresponding\ntox1(t) is found to be a part of the trajectory of x2(t)\n(=PT[x1(t)]) (forI1>1) as well as that of x0(t) or the\ntrajectory corresponding to x2(t) is found to be a part\nof the trajectory of x1(t) (=PT[x2(t)]) for 0 < I1<1\nas well as that of x0(t). Under time translation these\ntrajectories may be mapped onto each other and so may\nbe considered equivalent to the symmetrical trajectories\nx0(t). These are not shown explicitly in Fig. 6.\nBut the most important fact is that for I1<0,x1(t)\nandx2(t) are truely asymmetric and so break the PT\nsymmetry. Consequently the solutions x1(t) andx2(t)\ngiveriseto distinct trajectories, depending on the choices\nof the arbitrary constants I1andI2.\nThus the above detailed analysis of the completely in-\ntegrable nonlinear damped system (17) or (18) estab-\nlishes the fact that a necessary and sufficient condition\nfor the preservation of PT −1 symmetry is the existence\nof a single fixed point which is of PT −1 invariant center\ntype (that is neturally stable fixed point associated with\nimaginary eigenvalues of the linearized equation). Note\nthat this requirement demands the existence of a single\nwell potential and rules out cases like three well potential\nforPT −1 symmetry preservation. Also the origin has\nto be necessarily the fixed point for PT−1 invariance,\nx→ −x,t→ −t. The above requirement allows the ex-\nistence of PT-symmetric non-isolated periodic solutions\naroundthe fixed point correspondingto concentricclosed\ncurves as trajectories as shown in Fig.4. Otherwise the\nPTsymmetry is broken as confirmed for the λ <0 case.\nThe above discussion also confirms that the existence of\nPTsymmetric fixed point and PT-symmetric solutions\nnear it alone does not imply PTsymmetry of the full\nsystem if the fixed point is not of centre type as seen in\nthe case of λ <0. Now we can use the above criteria as\nthe basis for PTinvariance for our further studies.9\nWe also note that the above results hold good for\nthe case of standard Hamiltonian type complex classi-\ncalPTsymmetric systems also, where one can find that\nthe symmetry implies x(t) =−x∗(−t) which implies\nRe[x(t)] =xR(t) =−xR(−t), Im[x(t)] =xI(t) =xI(−t),\nRe[p(t)] =pR(t) = ˙xR(t) = ˙xR(−t) =pR(−t) and\nIm[p(t)] =pI(t) = ˙xI(t) =−˙xI(−t) =−pI(−t). Thus\nin these cases the PTpreserving fixed point will be of\nthe form ( xR(t),xI(t),pR(t),pI(t)) = (0,c1,c2,0), where\nc1andc2are arbitrary constants. The studies on the\nclassical trajectories of complex PTsymmetric systems\nshow the existence of regular periodic orbits (possibly\nwith some unbounded orbits) in the unbroken PTre-\ngions and non-periodic or open and irregular trajectories\nin the case of broken PTregions [42–44]. In addition, in\n[42, 44] one can also note that the closed orbits are cen-\ntered around the PTpreserving fixed point as discussed\nabove which confirms our results.\nIV. A BI- PT-SYMMETRIC SYSTEM\nAs a simple case of the coupled nonlinear damped sys-\ntem (10), we consider a system of coupled modified Em-\nden equations (MEE)\n¨x+αx˙x+βx3+ω2\n0x+κy= 0,\n¨y+αy˙y+βy3+ω2\n0y+κx= 0. (38)\nHere,αis the nonlinear damping coefficient, κis the\ncoupling strength and ω0is the natural frequency of the\nsystem when ω2\n0>0. However, we will also consider the\ncaseω2\n0<0 corresponding to the double well potential.\nIt is obvious that the system (38) admits a bi- PTsym-\nmetry. ( i) It is invariant under the PT −1 symmetry:\nx→ −x,y→ −yandt→ −t. Eq. (38) is also invari-\nant under ( ii)PT −2 symmetry: x→ −y,y→ −xand\nt→ −t. Note that the above two symmetries also imply\nthe symmetry x(t)→y(t).\nEq (38) can be rewritten as\n˙x=x1,\n˙x1=−αxx1−βx3−ω2\n0x−κy,\n˙y=y1,\n˙y1=−αyy1−βy3−ω2\n0y−κx. (39)\nThe above set of dynamical equations (39) admit five\nsymmetrical equilibrium points, e0,e1,e2,e3ande4:\n(i) The trivial equilibrium point e0: (x∗,x∗\n1,y∗,y∗\n1) =\n(0,0,0,0).\n(ii) A symmetric pair of non-zero equilibrium points\ne1,2: (x∗,x∗\n1,y∗,y∗\n1)=(±a∗\n1,0,∓a∗\n1,0), where a∗\n1=/radicalBig\nκ−ω2\n0\nβ.\n(iii) Another pair of symmetric non-zero equilibrium\npointse3,4: (x∗,x∗\n1,y∗,y∗\n1) = (±a∗\n2,0,±a∗\n2,0), where\na∗\n2=/radicalBig\n−κ−ω2\n0\nβ.Besides the above five fixed points there exist four more\nasymmetric fixed points which turn out to be unstable\nin the parametric range of our interest. So we do not\nconsider them in this paper further.\nA. Case: Ω =ω2\n0>0\nIn analyzing (38), we first consider the case where\nΩ =ω2\n0>0. The existence of the above mentioned\nequilibrium points in different regions in the parametric\nspace for this case is indicated in Table I (for our further\nstudies we let β >0 in Eq. (38) or (39)).\nBefore entering into the classification of unbroken and\nbrokenPTregions of the system, we note here that the\nequilibrium points are also playing a key role in iden-\ntifying symmetry breaking as shown in the scalar case\nin the previous section. In this connection, we classify\nthePT −1 andPT −2 invaraint fixed points of the sys-\ntem (38), which can be identified by looking for the fixed\npoints which satisfy PT −k[ei]=ei, wherek= 1,2 and\ni= 0,1,2,3,4. Using this, one can find that the fixed\npointe0alone isPT −1 invariant (that is PT −1[e0]=e0),\nwhile the three fixed points e0,e1ande2arePT −2 in-\nvarantandthefixedpoints e3ande4areinvariantneither\nunderPT −1 symmetry nor under PT −2 symmetry.\nGeneralizing the discussion in the previous section, we\ncan identify the following two criteria on the fixed points\nof the coupled system of the type (38) or (39) for the\ninvariance of PT −1 andPT −2 symmetries:\n(i) For the preservation of PT −1 symmetry again one\nrequires the existence of a single fixed point at the\noriginwhichisofneutrallystabletype. Therequire-\nment that for PT −1 symmetry x→ −x,y→ −y\nt→ −tdemands the exclusion of any other fixed\npointand that the originwill be the solefixed point.\n(ii) For the preservation of PT −2 symmetry which de-\nmandsx→ −y,y→ −x,t→ −t, the criterion is\nthe existence of one or more fixed points which are\nallPT −2 invariant out of which atleast one should\nbe neutrally stable type. For example, in the above\nsystem (39) as well as (47) below besides the origin\ne0, the fixed points e1ande2are also PT −2 in-\nvaraint and it is sufficient that atleast one of them\nis neutrally stable for preservation of PT −2 sym-\nmetry (see Figs. 7 and 11 below). A specific case is\nillustrated in Fig. 8 below.\n1. Linear Stability Analysis\nNow, to explore the regions in which PTsymmetries\nare found to be broken and unbroken, we first deduce\nthe Jacobian matrix obtained from the linear stability10\nκ <−ω2\n0−ω2\n0≤κ≤ω2\n0κ > ω2\n0\nΩ =ω2\n0>0e0,e3,e4 e0 e0,e1,e2\nΩ =ω2\n0= 0 e0,e3,e4 e0 e0,e1,e2\nκ < ω2\n0ω2\n0≤κ≤ −ω2\n0κ >−ω2\n0\nΩ =ω2\n0<0e0,e3,e4e0,e1,e2e0,e1,e2\ne3,e4\nTABLE I: Symmetric equilibrium points of (39) in different\nregions of the ( κ,ω0) parametric space with β >0 and Ω =\nω2\n0>0,ω2\n0= 0 and ω2\n0<0.\nanalysis of the above system. It is given by\nJ=\n0 1 0 0\nc21−αx∗−κ0\n0 0 0 1\n−κ0c43−αy∗\n, (40)\nwherec21=−αx∗\n1−3βx∗2−ω2\n0,c43=−αy∗\n1−3βy∗2−ω2\n0\nand (x∗,x∗\n1,y∗,y∗\n1) are the equilibrium points of (39).\nThe eigenvalues of the above matrix determine the dy-\nnamical behavior of the system in the neighborhood of\nthe equilibrium points qualitatively and the results will\nbe helpful in identifying the broken and unbroken PT-\nsymmetric regions of the system. In the unbroken PT\nregion, the trajectories of the system, in addition to the\nevolution equation, replicate the full symmetry of the\nsystem, while in the symmetry broken region it does not.\nIn order that the trajectories of the system to be sym-\nmetric under PToperation, it should havea non-isolated\nperiodic nature (due to the presence of the time reversal\noperator TinthePToperator). Thus,welookforthere-\ngions of the system parameters for which the equilibrium\npoint isneutrally stable , that is the eigenvalues of the Ja-\ncobian matrix corresponding to the equilibrium point are\npure imaginary. These regions give rise to unbroken PT-\nsymmetric ranges. The eigenvalues of the linear stability\nmatrixJcorresponding to different equilibrium points of\nthe system are presented in the Appendix C, where the\nranges of linear stability are also discussed.\nFixing the parameters α,ω0,βasα= 1.0,ω0= 1.0\nandβ= 1.0, Fig. 7 shows the real parts of the eigenval-\nues of the equilibrium points e0,e1,2ande3,4(given in\nAppendix C, Eqs. (C1), (C2) and (C8)) under the varia-\ntionofκ. Whenevertherealpartsofalltheeigenvaluesof\nJ(Re[µ]) corresponding to an equilibrium point become\nzero, the eigenvalues are purely imaginary and the latter\nis said to be neutrally stable. On the other hand, when\nallRe[µ]’s corresponding to an equilibrium point are less\nthen zero, it is said to be stable, while the equilibrium\npoint is unstable in all the other cases. From the forms of\nthe fixed points and the nature of their stability proper-\nties, we can identify four separateregions R1,R2,R3and\nR4in the (κ,Re[µ]) plane, as follows: (i) R1:κ <−ω2\n0,\n(ii)R2:−ω2\n0< κ < ω2\n0, (iii)R3:ω2\n0< κ < cω2\n0, wherec−3 −1 1 3 5\nκ−2−1012Re[µ]e0\ne1,2e3\ne4\nR1 R2 R3 R4\nPT-1Broken\nPT-2BrokenPT-1Unbroken\nPT-2UnbrokenPT-1\nBroken\nPT-2\nunbrokenPT −1Broken\nPT −2Broken\ne0(us),e3(s)\ne4(us)e0(ns)e0(us)\ne1,2(ns)e0,1,2(us)(s)-stable\n(us)-unstable\n(ns)-neutrally stable\nFIG. 7: (Color online) Linear stability of equilibrium poin ts\nof (39) for Ω = ω2\n0>0 given in Table. I. Real parts of\neigenvalues of Jgiven by Eq. (40) are plotted as a function\nofκfor the parameters α= 1.0,β= 1.0 andω0= 1.0.\nis given in Eq. (C5) in Appendix C, (iv) R4:κ > cω2\n0.\nNote that ω2\n0= 1.0 in Fig.3. Then, using the criteria\ndiscussed above, we can identify the following facts, as\ndepicted in Fig. 7.\n•In the region R1(denoted in Fig. 7), where κ <\n−ω2\n0=−1.0, one can see that three branches ap-\npear fore0, and a single branch appears each for e3\nande4. Among the four eigenvalues of e0(see Eq.\n(C1)), two are found to be pure imaginary, while\nthe third one has a positive real part and the other\nhas a negative real part. Thus, in the region R1,\nthere are three branches corresponding to e0. In\neach of the cases of e3ande4, all the eigenvalues\nhave the same real parts (as seen from Eq. (C8)).\nThus,e3ande4havea single branch each in Fig. 7.\nFrom the values of Re[µ] in the region R1, we can\nfind that among the equilibrium points e0,e3and\ne4, onlye3is found to be stable. The stabilization\nofe3in the region gives rise to oscillation death .\nHere oscillation death in a system of coupled os-\ncillators denotes the stabilization of the system to\na non-trivial steady state due to the interaction of\noscillators in the system. We can also note that\nthe equilibrium points e3ande4get transformed\nto one another by both PT −1 andPT −2 opera-\ntions(that is PT −1[e3]=e4,PT −2[e3]=e4andvice\nversa) and the symmetry preserving equilibrium\nstatee0(that isPT −1[e0]=e0andPT −2[e0]=e0)\nis unstable. Thus both the PT −1 andPT −2\nsymmetries are broken in this region.11\n•In the region R2, where −ω2\n0≤κ≤ω2\n0(that is\nregion−1≤κ≤1), the equilibrium points e3and\ne4disappear, and e0alone exists. The eigenvalues\nof the equilibrium point e0in this region are found\nto be pure imaginary (see also Eq. (C1)). The\nneutral stability of the symmetric state e0signals\nthat in this region R2both the PT−1 andPT −2\nsymmetries are unbroken.\n•Forκ > ω2\n0= 1, in the region R3, (defined\nby Eq. (C5)), e0loses its stability and gives\nrise to two new equilibrium points e1ande2.\nThese new equilibrium points are found to\nbe neutrally stable. Further, they also get\ntransformed to each other by PT −1 operation:\nPT−1[e1]⇒ PT − 1[(a∗\n1,0,−a∗\n1,0)]=(−a∗\n1,0,a∗\n1,0)\n=e2and similarly PT −1[e2] =e1. How-\never, the equilibrium points show invari-\nance under PT −2 operation: PT −2[e1]⇒\nPT−2[(a∗\n1,0,−a∗\n1,0)]=(a∗\n1,0,−a∗\n1,0)=e1and\nsimilarly PT −2[e2] =e2. The invariance of the\nequilibrium points e1ande2withPT −2 operation\nis also illustrated in terms of the phase portraits\nin Fig. 8 obtained by numerical analysis of (39).\nAs the fixed point preserving PT −1 symmetry\n(e0) is not of neutrally stable type and due to the\ncoexistence of PT −1 violating fixed points e1and\ne2, thePT −1 symmetry is broken in the region.\nIn the case of PT −2 symmetry, all the fixed points\n(e0,e1ande2) preserve the symmetry and also two\nof them ( e1ande2) are neutrally stable. Thus the\nPT−2 symmetry is unbroken, as demonstrated in\nFig. 8.\n•For values of κin the region R4(beyond R3), all\nthe equilibrium points e0,e1ande2are found to\nbe unstable. Thus, both the PT −1 andPT −2\nsymmetries are found to be broken in the region.\n2. Dynamics in the (κ,α)parametric space\nNext, weextendourstudyasafunctionofthedamping\nparameter αalso. Fig. 9 shows the broken and unbro-\nkenPT-symmetric regions corresponding to system (38)\nin the (κ,α) parametric space. It shows that oscillation\ndeath appears in the region κ <−1 due to the stabi-\nlization of e3as seen earlier in Fig. 7 (as can be seen\nfrom Eq. (C8) in Appendix C). Looking at the region\n−1≤κ≤1 in Fig. 9, we can observe that the cou-\npled nonlinearly damped system (38) like the scalar case\n(7) (see Sec. III), does not show any symmetry break-\ning on increasing α(see Eq. (C1) in Appendix C). This\nis in contrast to the systems with linear damping which\nshow symmetry breaking when the loss/gain strength is\nincreased [11]. As mentioned in the previous subsection,\nin this region (that is the region R2seen in Fig. 7), both\nPT −1 andPT−2 symmetries are unbroken. Increasingx(t)˙x(t)\n0.9 0 -0.90.3\n0\n-0.3\ny(t)˙y(t)\n0.9 0 -0.90.3\n0\n-0.3PT-1 PT-1(a) (b)\nPT-2\nPT-2\nFIG. 8: (Color online) Illustration of broken PT −1 and un-\nbrokenPT −2 in region R3: (a) (x−˙x) (b) (y−˙y) projections\nshow the oscillations about the equilibrium points e1ande2\nin the region R3forκ= 1.5,α= 1.0,β= 1.0 andω2\n0= 1.0\nobtained by solving Eq. (39) numerically (The trajectories\naway from e1ande2are not shown here). The filled square\nand the circle represents the position of e1ande2, respec-\ntively. By PT −1 operation on e1we transit to e2and so\nPT −1 symmetry is broken. But, on the operation of PT −2\none1the equilibrium point remains unchanged thereby the\nsymmetry remains unbroken.\nOscillation \nDeathOscillations \nabout e0Oscillations \nabout e1,2PT−  1 Broken\nPT−  2  BrokenPT−  1 Unbroken\nPT−  2  UnbrokenPT−  1 Broken \nPT−  2  UnbrokenPT−  1, PT−  2 \nBroken\nBistable\n/Minus3 0 3012\nΚΑ\nFIG. 9: (Color online) Unbroken and broken PTregions in\nthe parametric space of ( κ,α) for Ω = ω2\n0>0 = 1.0 and\nβ= 1.0. Here the light-gray shaded region denotes the re-\ngion where both PTsymmetries are unbroken and the dark-\ngray shaded region denotes unbroken PT −2 symmetric re-\ngion. The dark gray shaded regions are denoted as bistable\nregions in the sense that the equilibrium points e1ande2are\nneutrally stable in that region. The light blue shaded regio ns\ncorrespond to the oscillation death regions.\nκfurther ( κ >1), the system shows breaking of PT −112\nsymmetry (for the values of κ >1 or in the region R3in\nFig. 7) through a pitchfork bifurcation. In this region,\nPT −2 symmetry alone is unbroken. Fig. 9 shows that\nthePT −2 symmetry is unbroken only if αis small (from\nEq. (C6) in Appendix C) and it is broken for increased\nα(Note that this type of symmetry breaking at higher\nvalues of loss/gain strength is a universal feature of all\nthePT-symmetric systems [11]). On further increasing\nκ, Fig. 9 shows that the PTregions with respect to α\nget reduced.\n3. Rotating wave approximation\nIn this section, we analyze the stability of the symmet-\nric orbits centered around e0in the region R2using the\nwell known rotating wave approximation. We consider\nperiodic solutions for the system in the region R2to be\nof the form\nx(t) =R1(t)eiωt+R∗\n1(t)e−iωt,\ny(t) =R2(t)eiωt+R∗\n2(t)e−iωt, (41)\nwhereω=ω0−∆ω, and ∆ωis a small deviation. Here,\nR1(t) andR2(t) are the slowly varying amplitudes with\nrespect to a slow time variable. Substituting (41) in (38),\nand by rotating wave approximation, we obtain\n˙R1=1\n2iω(−3β|R1|2R1+(ω2−ω2\n0)R1−κR2),(42)\n˙R2=1\n2iω(−3β|R2|2R2+(ω2−ω2\n0)R2−κR1).(43)\nNow, we separate the real and imaginary parts of\nthe equation as R1=a1+ib1,R2=a2+ib2. We\nhave steady periodic solutions when ˙ ai=˙bi= 0,\ni= 1,2. Thus, the equilibrium points of the\nsystem represent steady periodic solutions. The\nsystem has five symmetric equilibrium points rep-\nresenting symmetric orbits, which are E0:(0,0,0,0),\nE1,2:(0,±b∗\n11,0,∓b∗\n11),E3,4:(0,±b∗\n22,0,±b∗\n22), where\nb∗\n11=/radicalBig\nκ+ω2−ω2\n0\n3βandb∗\n22=/radicalBig\n−κ+ω2−ω2\n0\n3β. The system\nalso has asymmetric equilibrium points, which are E5,6:\n(0,±b∗\n33,0,±2κ\n6βb∗2\n44b∗\n33),E7,8: (0,±b∗\n44,0,±2κ\n6βb∗2\n33b∗\n44),\nwhereb∗\n33=/radicalbigg\n(ω2−ω2\n0)+√\n−4κ2+(ω2−ω2\n0)2\n6βand,b∗\n44\n=/radicalbigg\n(ω2−ω2\n0)−√\n−4κ2+(ω2−ω2\n0)2\n6β. Asω=ω0−∆ωand ∆ω\nis a small deviation, ω2−ω2\n0is also small. Thus, b∗\n22,b∗\n33\nandb∗\n44cannot be real and the equilibrium points E3,4,\nE5,6will not exist. So, we confine our attention to the\nequilibrium points E0,E1andE2.\nNow, in order to investigate the stability of the above\nperiodic solutions through a linear stability analysis, we\nobtain the eigenvalue equation as Aχj=λjχj, whereχj= [ξ1η1ξ2η2]Tand\nA=\n−3βa∗\n1b∗\n1\nωc110−κ\n2ω\nc123βa∗\n1b∗\n1\nωκ\n2ω0\n0−κ\n2ω−3βa∗\n2b∗\n2\nωc21\nκ\n2ω0c223βa∗\n2b∗\n2\nω\n.(44)\nHereci1=−3β(a∗\ni2+3b∗\ni2)\n2ω+ω2−ω2\n0\n2ω,ci2=3β(3a∗\ni2+b∗\ni2)\n2ω−\nω2−ω2\n0\n2ω,i= 1,2.λj,χj(j= 1,2,3,4) are the eigenvalues\nandeigenfunctionsofthe aboveeigenvalueequation. The\neigenvalues of Acorresponding to the equilibrium point\nE0: (0,0,0,0) are\nλj=±i(−κ+(ω2−ω2\n0))\n2ω,±i(κ+(ω2−ω2\n0))\n2ω.(45)\nThe eigenvalues of Acorresponding to E1andE2are\nλj=±/radicalbigg\n−2κ2−κ(ω2−ω2\n0)\nω,0,0. (46)\nThe eigenvalues of Acorresponding to E0are found to\nbe neutrally stable always, whereas two of the eigenval-\nues associated with E1andE2are pure imaginary when\n2κ2+κ(ω2−ω2\n0)>0. Whenalltheeigenvaluesof Acorre-\nsponding to an equilibrium point are pure imaginary, the\nneutral stability of the equilibrium point will make the\noscillation with frequency ωto be modulated by a slowly\nvarying periodic amplitude. It indicates that the system\nshows beats type oscillations. As the equilibrium point\nE0is always neutrally stable, we have stable beats type\nperiodic oscillations in the complete region R2. However,\nthe equilibrium points E1,2have two of their eigenvalues\nas zero, and so one needs to include higher order correc-\ntions to conclusively decide about their stability.\nB. Case: Ω =ω2\n0= 0\nIn this case, the existence of equilibrium points for dif-\nferent values of κis demonstrated in Table. I. The eigen-\nvalues of Jwith respect to e0(Eq. (C1)) clearly show\nthat it is always unstable. The equilibrium points e1and\ne2are found to be neutrally stable for κ >0 and for the\nvalues of αspecified in (C6). The equilibrium points e3\nore4stabilize for κ <0 and give rise to oscillation death.\nC. Case Ω =ω2\n0≤0:\nNext, wewishtoshowtheunbrokenandbroken PTre-\ngions corresponding to the system (38) with Ω = ω2\n0<0\nor the double well potential case. The equilibrium points\nat different values of κfor this case are also given in Ta-\nble. I. From the table, we can note that in contrastto the\nprevious cases, in the region −ω2\n0≤κ≤ω2\n0, the equilib-\nrium points e3,4coexist with e1,2. From the results of the\nlinear stability analysis of this case (where Ω = ω2\n0<0),13\nR0\nOscillations about e1,2PT−  1 Broken\nPT−  2  Unbroken\nOscillation DeathPT−  1 Broken\nPT−  2  BrokenPT−  1, PT−  2  Broken\ne3stable\n/Minus3 0 301\nΚΑ\nFIG. 10: (Color online) Phase diagram of (38) in ( κ,α) para-\nmetric space for Ω = ω2\n0<0. Figure is plotted for Ω = −1.0,\nβ= 1.0, which shows the regions in oscillations about e1and\ne2exists (dark gray shaded region) and the region where os-\ncillation death (light blue shaded region) occurs. One can\nclearly note from the figure that the PT −1 symmetry is bro-\nkeneverywhere. Intheregion denotedby R0(Regionoutlined\nby thick black line), we can find that there exists oscillatio ns\naboute1,2and oscillation death occurs about e3, thusPT −2\nsymmetry is broken in the region. The gray shaded region\nexcluding R0region gives rise to unbroken PT −2 region.\nwe can find that the equilibrium point e0(see Eq. (C1)\nAppendix C) completely loses its stability. Thus, when\nΩ<0, as in the scalar case, PT −1 symmetry is always\nbroken. The symmetric pair of equilibrium points e1,e2\nande3,e4are still found to be stable in some regions in\nthe (κ,α) parametric space. The region in which they\nare found to be neutrally stable or stable is given by Eqs.\n(C6) and (C8) and are shown by Fig.10. From the fig-\nure, we can observe that the PT −1 symmetry is broken\neverywhere in the parametric space.\nRegarding the PT −2 symmetry, Fig. 10 shows the re-\ngionin whichthe PT −2preservingfixed points e1ande2\nare neutrally stable (gray shaded region) and the region\nin which PT −2 violating fixed point e3is stable (Light\nblue shaded regions). All the regionsin which e3is stable\nobviously correspond to the broken PT −2 region. Inter-\nestingly, in this case, there exists a region denoted by\nR0in Fig. 10, in which the stable region of e1ande2\noverlaps with the oscillation death region (stable region\nof thePT−2 violating fixed point e3). Due to such co-\nexistence, PT −2 symmetry is broken in the region R0.\nThus the PT −2 symmetry is unbroken only in the gray\nshaded region excluding R0.V. NONLINEAR PLUS LINEAR DAMPING\nNext we wish to investigate the effect of the introduc-\ntion of a linear damping on the dynamics of the non-\nlinearly damped system (38). For this purpose, let us\nintroduce the linear damping terms in addition to the\nnonlinear damping introduced in Eq. (38). Now, the\nsystem takes the form\n¨x+γ˙x+αx˙x+βx3+ω2\n0x+κy= 0,\n¨y−γ˙y+αy˙y+βy3+ω2\n0y+κx= 0,(47)\nwhereγis the linear loss/gain strength. Obviously, the\nadded linear damping term in (47) breaks the PT −1\nsymmetry. Thus the system is only symmetric with re-\nspect to the PT −2 operation. Note that the equilibrium\npoints of this system are the same as that of (38). The\nstability determining Jacobian matrix in this case be-\ncomes\nJ=\n0 1 0 0\nc21−γ−αx∗−κ0\n0 0 0 1\n−κ0c43γ−αy∗\n, (48)\nwhere,c21=−αx∗\n1−3βx∗2−ω2\n0,c43=−αy∗\n1−3βy∗2−\nω2\n0. The eigenvalues of this Jacobian matix for different\nequilibrium points are given in Appendix D. For simplic-\nity, we take β= 1 for further studies. As in Sec. III, we\nlook forPTregions of (47) for the cases Ω = ω2\n0>0 and\nΩ =ω2\n0≤0 respectively.\nA. Case: Ω =ω2\n0>0\nTo begin, we look for the PTregions of the system\nwith respect to κfor the case Ω = ω2\n0>0. By fixing all\nthe other parameters of the system as α= 1.0,γ= 0.5,\nβ= 1.0 andω2\n0= 1.0 in (47), Fig. 11 shows the plot\nof the real part of eigenvalues of Jcorresponding to the\nequilibriumpoints e0,e1,e2,e3ande4asκisvaried. It is\ndivided into seven regions S1,S2, ...,S7along the κ-axis.\nFor the system (47), PT −2 symmetry alone exists and\nthePTregions correspond to the regions in which the\nPT −2 symmetry is unbroken. The details are as follows.\n•In the region S1ofFig. 11, where κ <−ω2\n0=−1.0,\nwe can find that among the equilibrium points e0,\ne3ande4, onlye3is found to be stable which leads\nto oscillation death. As mentioned in the previous\ncase, the PT −2 symmetry is broken in this region.\n•The region corresponding to the values of κbe-\ntween−ω2\n0< κ < ω2\n0(−1< κ <1) is now divided\ninto three regions, namely S2,S3andS4. In these\nregions as mentioned in Table. I, the equilibrium\npointe0alone exists.\n(a) In the region S2, where −ω2\n0< κ≤\n−/radicalBig\n4ω4\n0−(2ω2\n0−γ2)2\n4(that is−1≤κ≤ −0.484),14\n−2 0 2 4\nκ−1.50.01.5Re[µ]S1 S2 S3S4S5 S6 S7e0 e1 e2 Broken PTregions - S1,S3,S7\nUnbroken PTregions - S2,S4,S5,S6e3 e4\ne0,4(us)\ne3(s)e0\n(ns)e0\n(us)e0\n(ns)e0\n(us)\ne1,2\n(ns)e0,1(us)\ne2(ns)e0,1,2\n(us)(s)-stable\n(us)-unstable\n(ns)-neutrally stable\nFIG. 11: (Color online) Linear stability of equilibrium poi nts of (47) for Ω = ω2\n0>0 given in Table. I. Real parts of eigenvalues\nofJgiven by Eq. (48) are plotted as a function of κfor the parameters γ= 0.5,α= 1.0,β= 1.0 andω0= 1.0.\ntx(t)\n60 30 00.3\n0\n-0.3\nty(t)\n40 20 015\n0\n-15a) b)\nFIG. 12: (Color online) Broken PTsymmetry in the region\nS3: Figures ( a) and (b) are plotted for κ= 0.01,γ= 0.2,\nα= 1.0,ω0= 1.0 andβ= 1.0 that show the time series plots\nofxandy. The damped and growing oscillations of x(t) and\ny(t) indicate that for finite values of κ, thePT −2 symmetry\nis broken.\nwe can note that the equilibrium point e0is\nfound to be neutrally stable (which can also\nbe seen from Eq. (D2)) and gives rise to an\nunbroken PTregion.\n(b) In the region S3, whereκtakes smaller values,\n−/radicalBig\n4ω4\n0−(2ω2\n0−γ2)2\n4≤κ≤/radicalBig\n4ω4\n0−(2ω2\n0−γ2)2\n4\n(that is−0.484≤κ≤0.484), we can see that\nthe equilibrium point e0loses its stability (can\nbe seen also from Eq. (D2)) and the PT −2symmetry is broken now. As this PT −2 sym-\nmetry appears because of coupling (that is,\nthePT−2 symmetry disappears when κ= 0)\nit will not be preserved for smaller values of\nκ. Figs. 12(a) and 12(b) are plotted in the\nregion, which shows the damped oscillation in\nxand grow up oscillation ywhich shows the\nunbalanced energy between the xandyoscil-\nlators.\n(c) Now increasing κ, in the region S4, for/radicalBig\n4ω4\n0−(2ω2\n0−γ2)2\n4≤κ < ω2\n0,e0again becomes\nneutrallystableandgivesrisetounbroken PT\nregion.\n•Forκ > ω2\n0= 1, there exists three regions which\nare designatedas S5,S6andS7, identified from Eq.\n(D4). In these regions, the equilibrium points e0,\ne1ande2are found to exist (see Table-I).\n(a) In the region S5, (1.0< κ <1.28), the equi-\nlibrium point e0is found to be unstable, but\ne1ande2are found to be neutrally stable (can\nbe seen also from Eq. (D4)). As these equi-\nlibrium points traces itself upon PT −2 oper-\nation (that is PT −2[e1]=e1), thePT −2 sym-\nmetry in the region is said to be unbroken.\n(b) In the region S6, (1.28< κ <4.4), in addition\ntoe0,e1also loses its stability (can be seen15\nFIG. 13: (Color online) Broken and unbroken PTregions cor-\nresponding to the system (47) in the ( κ,γ) parametric space\nfor Ω = ω2\n0>0, which is plotted for α= 1.0,ω0=β= 1.0.\nThe light blue shaded region corresponds to the oscillation\ndeath region. The light and dark gray shaded regions denote\nthe unbroken PT −2 region. The dark gray shaded region\ncorresponds to the bistable region in the sense that the equi -\nlibrium points e1ande2are neutrally stable in the region.\nalso from Eq. (D4)). But e2is still neutrally\nstable, thus the region again corresponds to\nan unbroken PTregion.\n(c) On further increasing κ, forκ >4.4, in the\nregionS7, all the equilibrium points e0,e1and\ne2become unstable. Thus PTis broken for\nhigher values of κ.\nForα= 1.0,ω0= 1.0, andβ= 1.0, the broken and\nunbroken regions in the ( κ,γ) parametric space of (47)\nare indicated in Fig. 13. By comparing Fig. 13 with\nFig. 9, we can find the appearance of oscillation death\nfor the values κ <−1 as in the previous case (38). But\nin contrast to the previous case, the oscillation death\nregime disappears with an increase of γ. By increasing\nκ, the unbroken PTregion appears in the range −ω2\n0≤\nκ≤ω2\n0(whereω0= 1.0). In this region by increasing γ,\nthe system shows symmetry breaking (see Eqs. (D1) in\nAppendix D). But in the previous case (38), we cannot\nfind this type of behavior, where the PTsymmetry is\nnever broken by increasing the loss/gain strength α(see\nFig. 9 and Eq. (C1)).\nForκ >1 (the region in which e1ande2appear), Fig.\n13 indicates that when κis smaller than ≈2.2, thePT\nsymmetry of the system is preserved for lower values of\nγand it is broken for higher values of γ. Increasing κ\nbeyond≈2.2, thePTsymmetry of the system is brokenfor lowervalues of γ, and on increasing γthePTsymme-\ntry is restored or it becomes unbroken for the values of γ\nmentionedinEq. (D6). Onfurtherincreasing γ,thesym-\nmetry is again broken. Generally, in the standard type of\nPT-symmetric systems, PTis unbroken for lower values\nofγand broken for higher values of γ. Thus, this type\nofPTrestoration with the increase of loss/gain strength\nis unusual compared to the general PT-symmetric sys-\ntems, except for the case of Aubry- Andre model with\ntwo lattice potentials [30, 31]. As mentioned in the in-\ntroduction, the latter model is a lattice model in which\nthe lattice potential is applied in such a way that each\nelement of the lattice has different amount of loss and\ngain that makes the loss and gain present in the lattice\nto be position dependent. Then, the phenomenon of PT\nrestorationat higher values of loss/gainstrength appears\nonly when two such lattice potentials are applied simul-\ntaneously. The reason for this type of PTrestoration\nis the competition between the two applied potentials\nwhich introduces loss and gain in the system [30].\nSimilarly, in our case if a single damping is present in\nthe system (38), we cannot observe such PTrestoration\nat higher loss/gain strength (see Fig. 9). But when two\nor more types of damping present in the system, as in\nthe caseof(47) (where linearand nonlineardampingsare\npresent in the system) we can observe this type of PT\nrestoration(see Fig. 13). The above point will be further\ndiscussed in detail in the next section, where we will also\nshow that by properly choosing the form of nonlinear\ndamping, we can also tailor the PTregions of the system\nin the parametric space. Fig. 13 shows that there exists\nbistable regions for finite values of γand by increasing\nthe coupling strength κthe bistable region disappears.\nB. Rotating wave approximation\nNow, we look for the stability of the periodic orbits\naboute0in the region −ω2\n0≤κ≤ω2\n0. As we did in the\nprevious case (38), we find that the amplitude equations\nare\n˙R1=1\n2iω(−iγωR1−3β|R1|2R1+(ω2−ω2\n0)R1−κR2),\n˙R2=1\n2iω(iγωR2−3β|R2|2R2+(ω2−ω2\n0)R2−κR1).(49)\nNow, separating the real and imaginary parts of the\nequation as R1=a1+ib1,R2=a2+ib2, and from the\nlinear stability analysis of the above equation, we can\nfind that the system has an equilibrium point (0 ,0,0,0),\nwhose eigenvalues are\nλ=±1\n2ω/radicalBig\n(−(κ2−γ2ω2)−(ω2−ω2\n0)2±2√c1) (50)\nwherec1= (κ2−γ2ω2)(ω2−ω2\n0)2. The equilibrium\npoints are found to be neutrally stable for −/radicalbigκ\nω≤γ\n≤/radicalbigκ\nω. The linear stability discussed in the previous\nsection tells that the equilibrium point e0can become\nneutrally stable in the region given by Eq. (D3) (see16\nS0\nPT−  2  Broken  PT−  2 Broken  \nOscillation DeathPT −  2  Unbroken\n/Minus3 0 3012\nΚΓ\nFIG. 14: (Color online) Broken and unbroken PTregions cor-\nresponding to the system (47) in the ( κ,γ) parametric space\nfor Ω = ω2\n0<0, which is plotted for α= 1.0, Ω = −1.0,\nβ= 1.0. The light blue and light gray shaded regions corre-\nspond to oscillation death (stable region of e3) and neutrally\nstable region of e2respectively. In the region S0, the region of\nstable region of e2coexists with the stable region of e3, thus\nPTsymmetry is broken in the region. The unbroken region\ncorresponds to the gray shaded region excluding S0.\nAppendix D) and the above stability analysis of periodic\norbits in the region shows that the oscillations are found\nto be stable only for the values of γmentioned above.\nC. Case: Ω =ω2\n0≤0\nBy taking Ω = ω2\n0≤0, the equilibrium point e0loses\nits stability (see Eq. D1). The equilibrium points e1,2\nande3,4alone are found to be stable and the stable re-\ngions of these equilibrium points are given in Appendix\nD.Similartothepreviouscase, wehaveobservedaregion\ndenoted by S0in Fig. 14, in which a neutrally stable PT\npreserving fixed point ( e2) coexists with PTviolating\nfixed points. Thus this region S0corresponds to broken\nPTregion. The gray shaded region excluding S0alone\ncorresponds to the unbroken PTregion. As in the case\nwhereΩ >0, herealso PTrestorationathigherloss/gain\noccurs.\nVI. GENERAL CASE\nIn this section, we consider a more general coupled\nPT-symmetric cubic anharmonic oscillator system with\nnonlineardamping. Here, we take the nonlinear damping\ntermh(x,˙x) to be of the form f(x)˙xso that the equationof motion will take the form\n¨x+γ˙x+(−1)nαf(x)˙x+βx3+ω2\n0x+κy= 0,\n¨y−γ˙y+αf(y)˙y+βy3+ω2\n0y+κx= 0,(51)\nwheren= 0iff(x) isanoddfunction and n= 1iff(x)is\neven. Thus the system is PT-symmetric with respect to\nthePT−2 operation. The novel bi- PT-symmetric case\narises when f(x) is odd and γ= 0. For all forms of\nf(x), the equilibrium points are found to be the same as\nthat of (38). Now through the linear stability analysis let\nus find the unbroken and broken PT-symmetric regions.\nThe Jacobian matrix corresponding to (51) is\nJ=\n0 1 0 0\nc21−γ−(−1)nαf(x∗)−κ0\n0 0 0 1\n−κ 0 c43γ−αf(y∗)\n,(52)\nwherec21=−αf′(x∗)x∗\n1−3βx∗2−ω2\n0,c43=\n−αf′(y∗)y∗\n1−3βy∗2−ω2\n0. For simplicity, we consider\nthe case of ω0= 1,β= 1. The eigenvalues of this Ja-\ncobian matrix corresponding to odd and even f(x) cases\nof the system (51) about various equilibrium points are\ngiven in Appendix E.\nA. Case: f(x)is odd\nConsidering the case where f(x) is an odd function, in\nthe region −1≤κ≤1 (see Table I), in which the equilib-\nrium point e0alone exists, the corresponding eigenvalues\nofJare the same as in (D1). In this region, we can find\nthat the eigenvalues do not depend on αbut depends on\nγ(see Eq. (D1)). The region of unbroken PTsymmetry\nis confined to\n−/radicalBig\n2−2/radicalbig\n1−κ2≤γ≤/radicalBig\n2−2/radicalbig\n1−κ2.(53)\nFrom the above, it is clear that when γ= 0 the PT\nis always unbroken for all the values of αin the region\n−1≤κ≤1. This indicates that in a purely nonlinearly\ndamped system, we cannot observeany symmetry break-\ning while varying the nonlinear damping strength ( α) in\nthis region. By varying γ, we observesymmetry breaking\nfor higher values of |γ|>/radicalbig\n2−2√\n1−κ2.\nIn the region κ >1, where the non-trivial equilibrium\npointse1ande2come into action, we will show that by\nproperly choosing the nonlinear damping we can tailor\nthePTregions. In this regime, forthe casein which f(x)\nis an odd function, the unbroken PTregion lies within\nthe range of γspecified by (see Eq. (E3) in Appendix E)\n±αf(√\nκ−1)−√a1≤γ≤ ±αf(√\nκ−1)+√a1,(54)\nwherea1= (6κ−4)−4/radicalbig\n(2κ−1)(κ−1). The presence\nof the term αf(√κ−1) in the above equation is found\nto be important. Because considering the case where\nαf(√κ−1) = 0, the PTsymmetry is unbroken for lower17\nUnbroken PT region\nBistableBroken PT region\nPT revival\n0 5 1 0 1 5 2 00123\nΚ/DoubleGamma\nUnbroken PT region\nBistable/DoubleGammac\nΑcBroken PT region\n0 1 2 30123\nΑ/DoubleGammaBroken\nPT Region\nBistableUnbroken \nPT region\n0 1 2 3012\nΚ/DoubleGammaa) b) c)\nFIG. 15: (Color online) Phase diagram in ( κ,γ) space: ( a) and (b) denote broken (white), unbroken (gray) and bistable (dark\ngray) regions with f(x) =x3andf(x) =sinx, respectively, for α= 1.5. (c): Phase diagram in ( α,γ) space corresponding to\nf(x) =sinxandκ= 1.5\nvalues of γspecified by |γ|<√a1and is broken for the\nhigher values of γspecified by |γ|>√a1. But, in the\ncasewhere αf(√κ−1)/negationslash= 0, forthe valuesof γdefined by\n0<|γ|< αf(√κ−1)−√a1, thePTsymmetryis broken\nwhile it is unbroken for the values of γdefined by (54).\nThus, here the PTsymmetry breaking occurs at lower\nvalues of γand the restoration of symmetry occurs by\nincreasing γ. We can also note that the term αf(√κ−1)\ndepends on the form of f(x), which helps in tailoring PT\nregions of the system.\nIn Fig. 15, we have presented the PTregions of the\nsystem for the cases f(x) =x3andf(x) =sinx, which\nclearlyshowthat the PTregionscanbe tailoredwith the\nsystems of the type (51) by properly choosing the form\noff(x). From Fig. 15(b), we can note that by choosing\nf(x) to be a periodic one, we can observe PTrevivals.\nIn Fig. 15(c), we have shown the PTregions of the\nsystem in the ( γ,α) parametric space corresponding to\nthef(x) =sinxcase, while the figure looks qualitatively\nthe same for f(x) =x3. The figure indicates that in-\ncreasing γ(orα) beyond a critical value, denoted as γc\n(orαc), the unbroken PTregion appears only when α\n(orγ) is also sufficiently large.\nB. Case: f(x)is even\nThe case of even f(x) can again be divided into two\nsub-cases: (i) f(0) = 0and(ii) f(0) =anonzeroconstant\nsay, 1 (For the odd f(x) case,f(0) = 0 always and so\nthere are no sub-cases.)\nCase (i) f(0) = 0:Considering the case of f(x)\nwithf(0) = 0 (Example: f(x) =x2), in the region\n−1≤κ≤1 where the equilibrium point e0alone exists\n(see Table-I), the corresponding eigenvalues of J(given\nin (52)) are found to be the same as in (D1) and the\nunbroken PTregions of the system are also the same as\nthat of (53).Case (ii) f(0) = 1:In this case, for example f(x) =\ncosxore−x2, the eigenvalues of Jare different from case\n(i) and they aregiven in (E4). In contrastto the previous\ncases, the eigenvalues of Jcorresponding to e0are found\nto depend on α, see Eq. (E4), and the PTunbroken\nregion can be given in terms of γas\nα−√a2≤γ≤α+√a2. (55)\nwherea2= 2ω2\n0−/radicalbig\n4(1−κ2). This equation indicates\nthat thePTsymmetryis found to be brokenfor valuesof\nγoutside the range specified by (55) and PTsymmetry\nbecomes unbroken by choosing γwithin the range given\nin(55). Thusthe PTsymmetryisbrokenforlowervalues\nofγ,γ < α−√a2and restored at higher γ, as in Eq.\n(55).\nAsPTis broken for γ < α−√a2, forα >0 the\nPTregions preferentially exist for γ >0 and found to\nbe scarce for γ <0. In other words, the unbroken PT\nregions are abundant, if the loss due to the linear (or\nnonlinear) damping is introduced in the x-oscillator and\nthe loss due to the nonlinear (or linear) damping is intro-\nduced in the y- oscillator. When loss (or also gain) due\nto both the linear and nonlinear damping is introduced\nin the same oscillator, the unbroken PTregions become\nscarce.\nNow considering the region ( κ >1), where the non-\ntrivial equilibrium points exist (see Table-I), the dynam-\nics corresponding to the two sub-cases (case (i) and case\n(ii)) are the same. The eigenvalues of e1ande2are given\nin (E6), which become purely imaginary in the region\nαf(√\nκ−1)−√a1≤γ≤αf(√\nκ−1)+√a1,(56)\nwherea1= (6κ−4)−4/radicalbig\n(2κ−1)(κ−1). Comparing\nthe above with the one corresponding to the f(x) odd\ncase (see Eqs. (54) and (56)), we can find that in this\ncase the unbroken PTregions are scarce for γ <0. The\npresence of the term αf(√κ−1) indicates that the PT\nrestoration can occur at higher values of loss/gain, which\nconfirms that the PTregions can be tailored by a proper\nchoice of f(x).18\nUnbroken \nPT regionBroken PT \nregionBistable\n0 1 2 30123\nΚ/DoubleGamma\nBroken PT region\nUnbroken PT region\n0 5 1 0 1 50123\nΚ/DoubleGamma\nΓc\nΑcBroken PT region\nUnbroken PT region\n0 1 2 30123\nΑ/DoubleGammaa)\nUnbroken PT region\nBroken PT region\n0 0 .5 10123\nΚ/DoubleGammab) c)\nFIG. 16: (Color online) Phase diagram in ( κ,γ) space: ( a) and (b) denote broken (white), unbroken (gray) and bistable (dark\ngray) regions with f(x) =x2andf(x) =cosx(whereα= 1.5). The inset in Fig. (b) shows the PTregions corresponding to\nthe case f(x) =cosxfor values of κbetween 0 < κ <1 in (κ,γ) space. (c): Phase diagram in ( α,γ) space corresponding to\nf(x) =cosxandκ= 1.5\nFig. 16(a) shows the PTregions of the system (51)\nfor the choice of f(x) =x2which corresponds to the sub-\ncase (i)f(0) = 0. Fig. 16(b) is plotted for f(x) =cosx,\ncorrespondingtothe sub-case(ii), namely, f(0) = 1. The\ninset in the figure clearly shows that in this system even\nforκ <1 thePTrestoration at higher loss/gainstrength\noccurs. Figs. 16(a) and 16(b) clearly show that the PT\nregions can be tailored by the proper choice of f(x). Fig.\n16(c) show the PTregions in the ( γ,α) parametric space\nfor the choice f(x) = cosx, which shows the existence of\ncritical values γcandαcabove which the PTis unbroken\nfor higher loss/gain strength.\nVII. CONCLUSION\nIn this work, we have brought out the nature of the\nnovel bi-PTsymmetry of certain nonlinear systems with\nposition dependent loss-gain profiles. We have pointed\nout that the PT-symmetric cases of this type of non-\nlinear systems with position dependent loss-gain profile\noccur even with a single degree of freedom. These scalar\nnonlinear PT-symmetric systems are also found to show\nPTsymmetry breaking. We have demonstrated the na-\nture ofPT-symmetry preservation and breaking with an\ninteresting integrable example of damped nonlinear sys-\ntem. By coupling two such scalar PT-symmetric sys-\ntems in a proper way, we have shown the existence of the\nnovel bi- PT-symmetric systems in two dimensions. We\nhavealsoillustratedthe phenomenonofsymmetrybreak-\ning of the two PTsymmetries in this bi- PT-symmetric\nsystem. When this system is acted upon by a single\nnonlinear damping, we observed that for smaller cou-\npling strengths, the coupled system shows no symme-\ntry breaking while varying nonlinear loss/gain strength,\nwhereas the coupled PT-symmetric system with a lin-\near damping [11] shows symmetry breaking by increasing\nloss/gain strength. By strengthening the coupling, this\nnonlinearlydampedsystemshowssymmetrybreakingforhigher loss/gain strength. Then, by applying the linear\ndamping in addition to the nonlinear damping in a com-\npetingway,ourresultsshowthatasinthe PT-symmetric\nAubry-Andre model, PTrestoration at higher values of\nloss/gain strength occurs. The advantage of having po-\nsition dependent nonlinear damping with a competing\nlinear damping is to help to tailor the PTregions of the\nsystem according to the needs by properly designing the\nnonlinear loss and gain profile. We have also observed\nPTrevivals in the systems which have loss and gain pe-\nriodically in space.\nAcknowledgement\nSK thanks the Department of Science and Technology\n(DST), Government of India, for providing a INSPIRE\nFellowship. The work of VKC forms part of a research\nproject sponsored by INSA Young Scientist Project. The\nwork of MS forms part of a research project sponsored\nby Department of Science and Technology, Government\nof India. The work forms part of an IRHPA project of\nML, sponsored by the Department of Science Technology\n(DST), Government of India, who is also supported by a\nDAE Raja Ramanna Fellowship.\nAppendix A: Symmetry breaking in a P- symmetric\ncubic anharmonic oscillator\nHere, we demonstrate the P-symmetry breaking in a\ncubic anharmonic oscillator through the solution of its\nIVP. Let us consider the cubic oscillator equation\n¨x+λx+βx3= 0, λ=ω2\n0. (A1)\nFor simplicity, we consider β >0 for further discussions.\nThePsymmetry breakingin such a system is well known\nin the literature. For λ >0, this system has an equilib-\nrium point e0:(0,0) and the equilibrium point is found to19\nbe neutrally stable. As P[(0,0)]=(0,0), thePsymmetry\nin the region is unbroken. By decreasing λtoλ <0,\nthe equilibrium point e0loses its stability and gives birth\nto two new neutrally stable equilibrium points which are\ne1,2:(±/radicalBig\n−λ\nβ,0). In fact e0is a saddle and e1,2are cen-\ntre type equilibrium points. But these new equilibrium\npointse1ande2do not preservesymmetry as P(e1) =e2\nand vice versa. Thus Psymmetry is broken while λ <0.\nAll the stable equilibrium points correspond to minimum\nenergy values.\nThe system is an integrable one and its exact solu-\ntion is also available in the literature [45, 46]. Now, we\ndemonstrate the above Psymmetry breaking from the\nsolution of the IVP of the system.\nHere the general solution of the system is given as\nfollows\nCase-1:λ >0:\nx(t) =Acn[Ωt+δ,k] (A2)\nwhere Ω =/radicalbig\nω2\n0+βA2, the square of the modulus\nk2=βA2\n2(ω2\n0+βA2)andδis a constant. The associated\nenergy integral is E=H=1\n2˙x2+1\n2ω2\n0x2+1\n4βx4\n=1\n2ω2\n0A2+1\n4βA4. Then considering without loss of\ngenerality the IVP, x(0) =A, ˙x(0) = 0, in order that\nPx(0) =x(0),P˙x(0) = ˙x(0)⇒A=−Awhich is\npossible only if A= 0. Further since one requires\nP[x(t)] =−x(t)⇒x(t) =−x(t). From (A2), only the\npossibility A= 0⇒x(t) = 0, ˙x(t) = 0 for all t≥0 is\nthe admissible solution of the IVP which preserves P\nsymmetry. The corresponding energy E= 0 has the\nminimum value. The excited states of the system may\nsaidtobe Psymmetricifthetimetranslationisincluded.\nCase-2:λ <0:\nOn the other hand one finds the following general so-\nlutions for the case λ <0 in Eq. (A1)\n(i) 0≤A≤/radicalBig\n|λ|\nβ:\nIn this range only the trivial solution exists.\nx(t) = 0,˙x= 0 (A3)\n(ii)/radicalBig\n|λ|\nβ≤A≤/radicalBig\n2|λ|\nβ:\nIn this region we have the following two distinct peri-\nodic solutions in the two wells\nx(t) =±Adn(Ωt+δ,k) (A4)\n˙x(t) =∓AΩk2sn(Ωt+δ,k)cn(Ωt+δ,k) (A5)\nwhereΩ2=βA2\n2andk2=2(βA2−|λ|)\nβA2, andδis aconstant.\n(iii)A≥/radicalBig\n2|λ|\nβ:\nIn this region, one has the solution\nx(t) =Acn(Ωt+δ), (A6)\n˙x(t) =−AΩsn(Ωt+δ,k)dn(Ωt+δ,k),(A7)\nΩ =/radicalbig\n−|λ|+βA2, k2=βA2\n2(−|λ|+βA2).Considering the IVP x(0) =A, ˙x(0) = 0, one again finds\nx(t) = 0, ˙x(t) = 0 is the only possible P-symmetric\nsolution, existing when A </radicalBig\n|λ|\nβ. But in the region/radicalBig\n|λ|\nβ< A </radicalBig\n2|λ|\nβ, one also has the non-trivial distinct\nset of solutions\nx1(t) = +Adn(Ωt,k), (A8)\nx2(t) =−Adn(Ωt,k), (A9)\nsuch that\nPx1(t) =x2(t) andPx2(t) =x1(t),(A10)\nand so also\nP˙x1(t) = ˙x2(t) andP˙x2(t) = ˙x1(t).(A11)\nNote that the value of the corresponding energy in-\ntegralE=−1\n2|λ|A2+1\n4βA4and its minimum value\nE=Emin=−1\n4|λ|2\nβis attained when the amplitude\nA=/radicalBig\n|λ|\nβ. In this case, the square of the modulus k= 0,\nΩ = 0 and so dn(u,0) = 1 and\nx(t) =±/radicalBigg\n|λ|\nβ,˙x(0) = 0. (A12)\nNote that the solution (A8, A9), including the limit-\ning case, all correspond to energies lower than the P-\nsymmetric state E0= (0,0) and break the Psymmetry.\nFinally in the region A≥/radicalBig\n2|λ|\nβ, there exists no P-\nsymmetric solution, unless time translation and time re-\nversal symmetries are also allowed in which the cases the\nphase trajectories are closed with A≥/radicalBig\n2|λ|\nβ. The asso-\nciated phase trajectories are presented in Fig. 17.\nAppendix B: Non- PT-symmetric oscillator\nThe non- PT-symmetric oscillator given in Eq. (8),\nshows damped oscillations as given in Fig. 18(a). But\nthe linear stability analysis of this system indicates a dif-\nferent dynamical behavior. Note that this system has the\nsame equilibrium points as that of (14). The eigenvalues\nassociated with the equilibrium point E0, namely ±i√\nλ,\nshow that it has periodic oscillations. But the numer-\nical results show that it has damped oscillations. The\napparent ambiguity can be removed using its amplitude\nequation. We assume\nx(t) =R(t)eiω0t+R∗(t)e−iω0t(B1)\nwhereR(t) =r(t)eiδ(t),r(t) andδ(t) are slowly varying\namplitude and phase. By differentiating we have\n˙x(t) = (˙R(t)+iω0R(t))eiω0t+c.c.,\n¨x(t) = (¨R(t)+2iω0˙R(t)−ω2\n0R(t))eiω0t+c.c.,(B2)20\nxV(x)\n2 1 0 -1 -22\n1\n0(a)\nx(t)˙x(t)\n2.5 0 -2.53.5\n0\n-3.5(b)\nxV(x)\n1.5 0 -1.50.2\n0\n-0.2\nx(t)˙x(t)\n1.8 0 -1.81\n0\n-1(c) (d)\nFIG. 17: (Color online) (a) and (b) Single well: Potential\nenergy curve and phase portrait of the system (A1) for λ= 1\nandβ= 1. (c) and (d) Double well: potential energy and\nphase portrait of the system (A1) for λ=−1 andβ= 1.\nwherec.c.denotescomplexconjugate. As R(t)is aslowly\nvarying quantity, ˙R(t)<< ω0R(t) and¨R(t)<< ω2\n0˙R(t).\nThus, we use approximations like\n˙x(t) =iω0R(t)eiω0t+c.c.,\n¨x(t) = (2iω0˙R(t)−ω2\n0R(t))eiω0t+c.c.(B3)\nSubstituting (B3) and(B1)in(8), wegetfortheequation\nfor amplitude ( r(t)),\n˙r=−αr3(t)\n2. (B4)\nBy solving the above, we get\nr(t) =1/radicalbig\nα(t−t0), t0,constant (B5)\nThis indicates that the amplitude of oscillation decreases\ndue to the introduced nonlinear term. This is the reason\nwhy the system in (8) has damped oscillations.\nOn the other hand, the amplitude equation associated\nwithE0corresponding to MEE (14) is found to be\n˙r= 0. (B6)\nThus,r(t) = constant, in the case of MEE. Thus, it has\nperiodic oscillations with constant amplitude.\nNow consideringthe non- PT-symmetric limit cycle os-\ncillator equation given in (9), we see that it has an equi-\nlibrium point E0at (0,0). The associatedeigenvaluesare1±√\n1−4ω2\n0\n2. This shows that the system is unstable. But\nthe amplitude equation of the system (obtained as in the\nprevious case) is\n˙r=−r3−r\n2(B7)\nindicates that ˙ r= 0 forr= 1. Thus the system exhibits\nlimit cycle oscillations. Fig. 18(b) shows the limit cycle\noscillation of (9).\ntx(t)\n100 50 00.6\n0\n-0.6\ntx(t)\n40 20 03\n0\n-3(a) (b)\nFIG. 18: Temporal behavior of (a) non- PT-symmetric\ndamped oscillator Eq. (8) and (b) the limit cycle oscillator\nEq. (9).\nAppendix C: Eigenvalues of Eq. (38)\nIn this section, we present the eigenvalues of the Jaco-\nbian matrix J(given in (40)) associated with the various\nequilibrium points. The eigenvalues of Jcorresponding\nto the system (38) for the equilibrium point e0are\nµ(0)\nj=±i/radicalBig\nω2\n0±κ, j = 1,2,3,4.(C1)\nThe eigenvalues are found to be pure imaginary when\nω2\n0≤κ≤ −ω2\n0. In this range, for all values of the nonlin-\near damping coefficient α, the eigenvalues are pure imag-\ninary. This indicates that there is no symmetry breaking\nwhile increasing α.\nNow, we consider the equilibrium points e1,2which exist\nonly for κ > ω2\n0. The eigenvalues of (40) corresponding\ntoe1ande2are the same and they are given by\nµ(1,2)\nj=±/radicalBigg\nb1±/radicalbig\nb2\n1−b2\n2;j= 1,2,3,4,(C2)\nwhere,\nb1=\nα/radicalBigg\nκ−ω2\n0\nβ\n2\n−(6κ−4ω2\n0),(C3)\nb2= 16(2κ−ω2\n0)(κ−ω2\n0). (C4)\nFor fixed values of αandβ, these eigenvalues are found\nto be pure imaginary for the values of κin the range\nω2\n0< κ≤(α2−6β)(α2−4β)−24β2−4αβ√2β\n((α2−6β)2−32β2)ω2\n0.(C5)21\nSimilarly, for a particular value of κin the range κ >\nω2\n0, the range of values of αfor which the eigenvalues will\nbe pure imaginary is given below,\n−/radicalBigg\nβ\nκ−ω2\n0b3≤α≤/radicalBigg\nβ\nκ−ω2\n0b3 (C6)\nwhere\nb3=/radicalBig\n6κ−4ω2\n0−√\nb2 (C7)\nwith the values of κ≥ω2\n0.\nThe eigenvalues of Jcorresponding to the equilibrium\npointe3(which exists when κ < ω2\n0) are\nµ(3)\n1,2=−α/radicalbig\n−(κ+ω2\n0)±/radicalbig\n(−α2+8β)(κ+ω2\n0)\n2√β\nµ(3)\n3,4=−α/radicalbig\n−(κ+ω2\n0)±/radicalbig\n−α2(κ+ω2\n0)+8β(2κ+ω2\n0)\n2√β(C8)\nWe can find from the above equation that these eigen-\nvalues can never be pure imaginary if α/negationslash= 0. The equi-\nlibrium point e3is found to be stable and gives rise to\noscillation death when α >0. The eigenvalues of Jcor-\nresponding to e4can be obtained by simply changing\nα→ −αin Eq. (C8). One can check that its eigenvalues\ncan never be pure imaginary for α/negationslash= 0 and that they can\nbecome stable when α <0.\nAppendix D: Eigenvalues of Eq. (47)\nIn this appendix, we present the eigenvalues of Jgiven\nin(48) fortheequilibriumpointsofthesystem(47). This\nsystem has the same set of equilibrium points as that of\n(38). The eigenvalues of Jfore0are\nµ(0)\nj=±/radicalBigg\n−(2ω2\n0−γ2)±/radicalbig\n(2ω2\n0−γ2)2+4(κ2−ω4\n0)\n2.(D1)\nFor the values of γin the range −/radicalbig\n2ω2\n0< γ </radicalbig\n2ω2\n0,\nthe eigenvalues are easily seen to be pure imaginary only\nfor the values of κin the range\n−ω2\n0< κ≤ −/radicalbigg\n4ω4\n0−(2ω2\n0−γ2)2\n4and\nω2\n0> κ≥/radicalbigg\n4ω4\n0−(2ω2\n0−γ2)2\n4. (D2)\nFor a particular values of κin the region −ω2\n0≤κ≤\nω2\n0,e0is neutrally stable for the values of γdefined by\n−/radicalbigg\n2ω2\n0−2/radicalBig\nω4\n0−κ2≤γ≤/radicalbigg\n2ω2\n0−2/radicalBig\nω4\n0−κ2.(D3)\nFrom the above relations, one can see that the increase\ninγbeyond this range causes symmetry breaking in the\nsystem (in the region −ω2\n0≤κ≤ω2\n0).Then, the eigenvalues of Jfor the equilibrium point e1\nare\nµ(1)\nj=±/radicalBigg\nb′1±/radicalbig\n(b′2\n1−b2)\n2(D4)\nwhere\nb′\n1=\nα/radicalBigg\nκ−ω2\n0\nβ+γ\n2\n−(6κ−4ω2\n0),(D5)\nandb2is given in (C4). The equilibrium point e1exists\nonly when κ > ω2\n0, and the associated eigenvalues are\npure imaginary when\n−α/radicalBigg\nκ−ω2\n0\nβ−b3≤γ≤ −α/radicalBigg\nκ−ω2\n0\nβ+b3(D6)\nwhereb3is given in (C7). Thus, the PTsymmetry is\nunbroken in the region given above. Similarly, the eigen-\nvalues of Jwith respect to e2and the regions in which\nthey take pure imaginary eigenvalues can be obtained by\nreplacing αbe−αin (D4) and (D6).\nThen, considering the equilibrium point e3(which exist\nforκ≤ −ω2\n0), its eigenvalues are the roots of the alge-\nbraic equation\nµ(3)4+2α/radicalBigg\n−(κ+ω2\n0)\nβµ(3)3+(−α2(κ+ω2\n0)\nβ−γ2\n−(6κ+4ω2\n0))µ(3)2+α/radicalBigg\n−(κ+ω2\n0)\nβ(6κ+4ω2\n0)µ(3)\n+4(2κ−ω2\n0)(κ−ω2\n0) = 0.(D7)\nAs the coefficients of µ(3)3andµ(3)are non-zero for\nα/negationslash= 0,β/negationslash= 0, the eigenvalues of the equilibrium point\ncannot take pure imaginary values. Similarly, the eigen-\nvalue equation corresponding to the equilibrium point e4\ncan be obtained by changing α→ −αin (D7).\nAppendix E: Eigenvalues of Eq. (51)\nNowweconsiderthe generalcaseofEq. (51), wherewe\ncan choose f(x) to be an odd or an even function. In this\nsection, depending on the nature of f(x) (odd or even),\nwe have presented their corresponding eigenvalues.\n1. Case: f(x)- odd\nIn this case, the eigenvalues of Jcorresponding to the\nequilibrium point e0are found to be the same as in (D1).\nThe eigenvalues about the equilibrium point e1ande2\nare\nµ(1,2)\nj=±/radicaltp/radicalvertex/radicalvertex/radicalbt˜b(1,2)\n1±/radicalBig\n((˜b(1,2)\n1)2−b2)\n2(E1)22\nwhere\n˜b(1)\n1=\n+αf\n/radicalBigg\nκ−ω2\n0\nβ\n+γ\n2\n−(6κ−4ω2\n0),\n˜b(2)\n1=\n−αf\n/radicalBigg\nκ−ω2\n0\nβ\n+γ\n2\n−(6κ−4ω2\n0),(E2)\nandb2is as given in (C4). The regions in which the\neigenvalues of e1ande2are found to be pure imaginary\nare given respectively by\n−αf\n/radicalBigg\nκ−ω2\n0\nβ\n−b3≤γ≤ −αf\n/radicalBigg\nκ−ω2\n0\nβ\n+b3and\n+αf\n/radicalBigg\nκ−ω2\n0\nβ\n−b3≤γ≤+αf\n/radicalBigg\nκ−ω2\n0\nβ\n+b3.(E3)\nwhereb3is given in (C7).\n2. Case: f(x)- even\nConsidering the case of even f(x), the eigenvalues of\ne0are\nµ(0)\nj=±/radicalBigg\n−c±/radicalbig\nc2−4(ω4\n0−κ2)\n2. (E4)wherec= (2ω2\n0−(γ−αf(0))2). The eigenvalues in (E4)\nare found to be same as that of (D1) when f(0) = 0. In\nthe case f(0) = 1, thus the eigenvalues given in (E4) are\ndifferent from that of (D1). In contrast to the previous\ncases(Eq. (C1) and(D1)), the eigenvaluescorresponding\ntoe0are found to depend on αand the region in which\nthe eigenvalues given in (E4) take pure imaginary values\nis\nα−/radicalbigg\n2ω2\n0−/radicalBig\n4(ω4\n0−κ2)≤γ≤α+/radicalbigg\n2ω2\n0−/radicalBig\n4(ω4\n0−κ2).(E5)\nThe eigenvalues corresponding to both e1ande2are\nfound to be the same and they are\nµ(1,2)\nj=−/radicaltp/radicalvertex/radicalvertex/radicalbt˜b(2)\n1±/radicalBig\n(˜b(2)\n1)2−b2\n2. 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Breaking this assumption is expected\nto open up novel possibilities and result in exceeding conventional limitations. However, to explore\nthe \feld of time-varying electromagnetic structures, we primarily need to contemplate the funda-\nmental principles and concepts from a nonstationarity perspective. Here, we revisit one of those\nkey concepts: The polarizability of a small particle, assuming that its properties vary in time. We\ndescribe the creation of induced dipole moment by external \felds in a nonstationary, causal way,\nand introduce a complex-valued function, called temporal complex polarizability, for elucidating a\nnonstationary Hertzian dipole under time-harmonic illumination. This approach can be extended\nto any subwavelength particle exhibiting electric response. In addition, we also study the classical\nmodel of the polarizability of an oscillating electron using the equation of motion whose damping\ncoe\u000ecient and natural frequency are changing in time. Next, we theoretically derive the e\u000bective\npermittivity corresponding to time-varying media (comprising free or bound electrons, or dipolar\nmeta-atoms) and explicitly show the di\u000berences with the conventional macroscopic Drude-Lorentz\nmodel. This paper will hopefully pave the road towards better understanding of nonstationary\nscattering from small particles and homogenization of time-varying materials, metamaterials, and\nmetasurfaces.\nI. INTRODUCTION\nTemporal modulation [1] in electromagnetic systems\n(e.g., Refs. [2{11]) is an e\u000ecient technique to achieve\nexotic wave phenomena and intriguing functionalities.\nNonreciprocity and isolation [12{18], frequency con-\nversion and generation of higher-order frequency har-\nmonics [19{21], wavefront engineering [21{23], one-way\nbeam splitting [24], extreme accumulation of energy [25],\nparametric ampli\fcation [2, 3, 26, 27], and wideband\nimpedance matching [28] are some of those functionalities\nthat have been reported in the past. One possibility that\ntime modulation can provide is to instantaneously con-\ntrol the radiation from subwavelength particles [29, 30].\nThis is due to the fact that electric and magnetic dipole\nmoments, p(t) and m(t), induced in a particle under il-\nlumination, can be temporally engineered in a desired\nfashion. In general, both the geometry of the particle\nand the optical properties of the material from which the\nparticle is made can be properly modulated in time by\nan external force.\nFrom the stationary perspective, it is assumed that the\nparticle is static, and its characteristics do not vary in\ntime. As a consequence, the induced dipole moments are\nconventionally described in the frequency domain simply\nthrough the complex dyadic electric and magnetic polar-\nizabilities [31]:\np=\u000bee(!)\u0001E+\u000bem(!)\u0001H;\nm=\u000bme(!)\u0001E+\u000bmm(!)\u0001H:\n\u0003mohammad.mirmoosa@aalto.\f\nyThese authors jointly supervised this work.Here, EandHare the Fourier transforms of the ex-\nternal electric and magnetic \felds, respectively. How-\never, the above equations cannot be generally applied\nfor a dynamic particle, because the very de\fnition of\nfrequency-domain parameters is based on the assump-\ntion that the particle is stationary. In fact, concerning\na time-varying particle, we need to return to the time\ndomain, and, subsequently, revisit the description of the\ninstantaneously induced dipole moments in terms of the\ndyadic polarizabilities model. An appropriate description\nshould explicitly indicate the nonstationary characteris-\ntic of the problem, along with the linearity and memory\n(frequency dispersion). The importance of this study is\nnot limited only to the understanding and engineering\nof instantaneous radiation, but it is also important for\nthe proper characterization and realistic implementation\nof time-varying metamaterials and metasurfaces [32, 33],\nbecause they are formed by time-varying meta-atoms.\nTherefore, having a clear picture about the polarizabil-\nity of meta-atoms paves the road towards homogenization\nmodels [34, 35] taking into account nonstationarity, and\nits interplay with dispersion phenomena.\nIn this paper, we thoroughly scrutinize the concept of\npolarizability associated with a particle which is vary-\ning in time. For simplicity, we assume that the parti-\ncle has only isotropic electric response. We analytically\nstudy how to determine the polarizability of such a par-\nticle when it is located in free space. For this study,\nwe employ the Hertzian dipole model which is a conven-\ntional model to describe a stationary resonant particle\nwith electric response. Determining the polarizability,\nwe also explain the interaction of nonstationary dipoles\nwith waves in terms of the particle polarizability. Fur-\nthermore, we move forward and consider the particle asarXiv:2002.12297v3  [physics.app-ph]  5 Oct 20212\na constituent of a time-varying material. Regarding this\nscenario, we focus on the classical bound electron (as the\nparticle under study) and derive the corresponding po-\nlarizability by assuming a time-dependent damping coef-\n\fcient and natural frequency in the equation of motion.\nAccordingly, we obtain the nonstationary Drude-Lorentz\nmodel for an e\u000bective medium and show how fundamen-\ntally di\u000berent this new model is from the conventional\nmodel written for a stationary medium.\nThe paper { as a foundational step towards under-\nstanding of nonstationary scattering from small particles\nand time-varying (arti\fcial) media { is organized as fol-\nlows: In Section II, we give a fundamental description\nof polarization of an arbitrary time-varying dipolar par-\nticle as a response to the excitation \feld by using the\nconcept of electric polarizability. Since this description\n(initially inspired by the analysis and synthesis of linear\ntime-varying systems in communications and control en-\ngineering [36, 37]) is not well covered in the literature and\nis missing in the classical electrodynamics textbooks [38{\n40], it helps the reader to obtain a proper perspective and\nis essential for understanding of the other sections of the\npaper. In the same section, we additionally discuss causal\nmodels of e\u000bective material parameters of time-varying\nmedia, following the same principles as for a single time-\nvarying particle. Next, in Sections III and IV, under\nnonstationary conditions, we treat small particles and\nclassical electrons based on their corresponding polariz-\nabilities. Section IV is also devoted to e\u000bective material\nmodels of nonstationary media. Finally, Section V con-\ncludes the paper.\nII. BASIC CONCEPTS\nA. Polarizability kernel\nFor a linear and stationary subwavelength particle with\nelectric response, there is a temporally nonlocal connec-\ntion between the instantaneous electric dipole moment\np(t) and the exciting electric \feld E(t). This connection\nis described by a convolution integral as\np(t) =Z1\n0\u000b(\r)E(t\u0000\r)d\r; (1)\nwhere\u000b(\r) is a time-dependent function called electric\npolarizability kernel (here, we assume that the dipole\nand the \feld are parallel and there is no bianisotropy).\nThe above equation illustrates two notable characteris-\ntics. The \frst is that if the electric \feld is temporally\nshifted bytsh, the dipole moment will be also shifted by\nthe same time tshdue to the stationarity of the particle.\nIn other words,\np(t\u0000tsh) =Z1\n0\u000b(\r)E(t\u0000\r\u0000tsh)d\r: (2)\nThe second characteristic, associated with causality,\nstates that the instantaneous dipole moment at a certaintime depends on the \feld at that time and the evolution-\nary progress over past times.\nThe situation is very di\u000berent if the particle under\nstudy is changing in time. Causality is certainly a fun-\ndamental concept in nature which should be scrutinised\ncarefully. However, the \frst characteristic, having to do\nwith invariance with respect to translations in time, is\nnot true anymore. For interactions of nonstationary par-\nticles with \felds, a temporal shift of the electric \feld does\nnot result in an equivalent temporal shift of the induced\ndipole moment. We should use a more general linear and\ncausal relation between the induced dipole moment and\nthe exciting \feld, which we write as\np(t) =Z+1\n0\u000b(\r;t)E(t\u0000\r)d\r: (3)\nHere, in contrast to Eq. (1), the polarizability kernel \u000bis\nnot only a function of the delay time between the action\nand reaction ( \r), but it also depends on the observation\ntime (t). In other words, this formula means that at\nevery moment of time t, we deal with a di\u000berent particle\nand with a di\u000berent frequency dispersion rule (de\fned by\nthe integral kernel as a function of \r). As a consequence\nof that, the instantaneous value of the dipole moment\ndepends not only on the past and present values of the\nexciting \feld, but also on the whole history of evolution\nof the particle properties.\nBased on Eq. (3), let us discuss the physical mean-\ning of the polarizability of a nonstationary particle. If\nthe electric \feld is chosen to be the Dirac delta function\nE(t) =\u000e(t\u0000t0)u(uis a unit vector), the dipole moment\nequals\np(t) =\u000b(t\u0000t0;t)u: (4)\nIn other words, the polarizability \u000bis the impulse re-\nsponse of the dipole. As we see, in the nonstationary\nsituation, the impulse response depends, as in usual sta-\ntionary linear systems, on how much time has passed\nsince the pulse excitation was applied, but also on time\nexplicitly. This property clearly manifests the fact that\nthe particle responds di\u000berently at di\u000berent moments of\ntime.\nBeside using Eq. (3), it is sometimes convenient to ap-\nply an alternative integral form to describe the dipole\nmoment (for example, see Sections III and IV). Let us\nconsider the following independent variable: \u001c=t\u0000\r.\nBy changing variable in Eq. (3) and de\fning\nh(t;\u001c) =\u000b(\r;t)j\r=t\u0000\u001c; (5)\nwe can equivalently write\np(t) =Zt\n\u00001h(t;\u001c)E(\u001c)d\u001c: (6)\nIn this alternative representation of causal linear rela-\ntions, the\u001cvariable has the meaning of time moments\nin the past, and the chosen integration limits ensure that3\nthe induced dipole does not depend on the \feld values in\nthe future. Also, in this form, assuming delta-function\nexcitation E(\u001c) =\u000e(\u001c\u0000t0)u, we \fnd the impulse re-\nsponse in general form p(t) =h(t;t0)u. Notice that in\nthe stationary scenario, the function h(t;\u001c) dependents\nonly on the time di\u000berence between the observation time\nand a time moment in the past: h(t;\u001c) =h(t\u0000\u001c). Conse-\nquently, the integral expressed by Eq. (6) becomes simply\na convolution, and the dependency of the polarizability\nkernel on the observation time tvanishes, i.e., the polar-\nizability kernel is only written in terms of the variable\n\r:\u000b(\r;t) =h(t;t\u0000\r) =h(\r). Accordingly, we obtain\nEq. (1) which was explained in the beginning of this sec-\ntion.\nAt this point, where the functions \u000b(\r;t) andh(t;\u001c)\nand the corresponding relation between them have been\ndiscussed, it is better to build a general consensus that\nwe refer only to the function \u000b(\r;t) as the polarizability\nkernel. This agreement will help us to avoid any con-\nfusion throughout the paper. Therefore, when working\nwith Eq. (6) and the function h(t;\u001c), remember that we\nneed to do a simple algebraic manipulation and deter-\nmine\u000b(\r;t) in order to present the polarizability kernel.\nB. Temporal complex polarizability\n1. De\fnition\nLet us consider a time-harmonic excitation by a given\nelectric \feld E(t) = Reh\nE0exp(j!t)i\n. Here, E0denotes\nthe complex amplitude, !is the angular frequency, and\nRe means the real part of the expression inside the brack-\nets. The reason for choosing the time-harmonic exci-\ntation is the fact that we want to concentrate on un-\nderstanding the e\u000bects of time variations of the particle\nitself, and it is convenient to use the simplest possible\nexciting \felds. Since the particle is linear, response to\narbitrary excitation can be found using the Fourier ex-\npansion of the incident \feld. Therefore, it is logical to\ncreate a model for time-harmonic excitation.\nSince from the beginning we have assumed that the\n\feld and the dipole moment are parallel, the polarizabil-\nity kernel is a scalar value. Substituting the electric \feld\ninto Eq. (3), we \fnd that\np(t) = Reh\n\u000bT(!;t)\u0001E0exp(j!t)i\n; (7)\nwhere\n\u000bT(!;t) =Z+1\n0\u000b(\r;t) exp(\u0000j!\r)d\r: (8)\nThe instantaneous electric dipole moment is the real part\nof a complex-valued function which is multiplied by the\ncomplex amplitude of the time-harmonic electric \feld\nE0exp(j!t). This is in clear analogy with the conven-\ntional stationary case in which the instantaneous dipole\n(a))\n(b))\n(c))\n𝐩𝑡=0+∞𝛼𝛾,𝑡𝐄𝑡−𝛾𝑑𝑡)\n𝐩𝑡=Re𝛼T𝜔,𝑡𝐄0exp𝑗𝜔𝑡 )\nഥ𝐩Ω=1\n2𝜋−∞+∞ത𝛼T𝜔,Ω−𝜔ത𝐄𝜔𝑑𝜔 )ഥ𝐩𝜔,Ω=𝐄0ത𝛼T𝜔,Ω−𝜔+𝐄0∗ത𝛼T∗𝜔,−Ω−𝜔/2\n𝐩𝑡=1\n2𝜋−∞+∞𝛼T𝜔,tത𝐄𝜔exp𝑗𝜔𝑡𝑑𝜔 )FIG. 1. Schematic view of wave scattering by a time-varying\nspherical particle. We assume that, for example, the optical\nproperties of the particle is changing in time. (a){The par-\nticle under time-harmonic illumination. We use the general\nconcept of polarizability kernel \u000b(\r;t) in order to study this\nscattering problem. (b){The particle under the same time-\nharmonic illumination. However, we employ the particular\nconcept of temporal complex polarizability \u000bT(!;t) and its\nFourier transform \u000bT(!;\n) for studying the corresponding\nproblem. (c){The particle under arbitrary signal illumina-\ntion. Here, temporal complex polarizability and its Fourier\ntransform are still valid to be used.\nmoment is the real part of the complex-valued, frequency-\ndomain electric polarizability multiplied by the electric\n\feld amplitude and the time-harmonic exponential fac-\ntor. However, here, the complex function \u000bTdepends\non the time variable t. Thus, we name such function as\n\\temporal complex polarizability\". The index \\T\", re-\nminding \\temporal\", distinguishes the function \u000bTfrom\nthe polarizability kernel \u000b.\nRecall that the above de\fnition is for the case of time-\nharmonic excitation (which is also indicated in Figure 1).\nAs one can realize in accordance with Eq. (7), by using\nthis de\fnition, the complexity of the problem dramati-\ncally reduces, and the induced dipole moment is simply\ndescribed based on the temporal complex polarizability\n\u000bT(!;t). The result of such simplicity is clear, for ex-\nample, in the next section { Section III A { where we\ndiscuss the interaction of the point electric dipole with\ntime-harmonic incident electric \felds. We will see how\nthe instantaneous power exerted on the dipole and the in-\nstantaneous power scattered from the dipole are elegantly\nwritten in terms of the temporal complex polarizability\n(without making any integration). Furthermore, the im-\nportance and advantage of the aforementioned simplicity\ncan be also understood in more complicated problems\nsuch as homogenization of time-varying arti\fcial media\n(metamaterials), in which the e\u000bective macroscopic pa-\nrameters should be derived, and, subsequently, the corre-\nsponding dispersion relations need to be calculated. Re-4\ngarding those problems, therefore, it is quite reasonable\nto employ the concept of the temporal complex polariz-\nability\u000bT(!;t) instead of the polarizability kernel \u000b(\r;t)\nin the time domain. We emphasize that due to the linear-\nity, the temporal complex polarizability can be used for\nnonharmonic excitations as well by applying the Fourier\nexpansion (see Figure 1).\n2. Properties\nLet us describe some features which are inferred from\nEqs. (7) and (8). By contemplating Eq. (8), we \frstly see\nthat the temporal complex polarizability is the Fourier\ntransform of the polarizability kernel with respect to the\ntemporal variable \r. Secondly, because the functions\np(t),E(t), and\u000b(\r;t) are real-valued, based on Eq. (8)\nand for real angular frequencies, we deduce that\n\u000b\u0003\nT(!;t) =\u000bT(\u0000!;t); (9)\nin which\u0003represents the complex conjugate. Further-\nmore, similar to the stationary scenario, in Eq. (8), the\nintegration is over the positive half-axis of \r, which re-\n\rects causality of the system and indicates that the tem-\nporal complex polarizability obeys Kramers-Kronig rela-\ntions [38].\nAbout Eq. (7), it explicitly con\frms the expectations\nthat the dipole moment induced by time-harmonic \felds\nis not necessarily time-harmonic. In Ref. [29], the authors\nhave recently discussed this fact without studying the po-\nlarizability. Importantly, the temporal variations of the\ndipole moment can be in principle fully engineered (while\nthe excitation \feld is time-harmonic) only by choosing\nthe proper temporal variation of the particle modula-\ntion. Beside this property, it is instructive to take the\nFourier transform of Eq. (7). For that, this equation can\nbe rewritten as\np(t) =1\n2\"\nE0\u000bT(!;t) exp(j!t) +E\u0003\n0\u000b\u0003\nT(!;t) exp(\u0000j!t)#\n:\n(10)\nNow, by de\fning the usual Fourier transform of\nan arbitrary temporal function g(t) asg(\n) =R1\n\u00001g(t) exp(\u0000j\nt)dtand applying this operation to\nEq. (10), we obtain\np(!;\n) =E0\u000bT(!;\n\u0000!) +E\u0003\n0\u000b\u0003\nT(!;\u0000\n\u0000!)\n2:(11)\nThe \frst argument of the Fourier transform of p(t) in-\ndicates that this expression is derived for excitation by\na time-harmonic electric \feld at frequency !. For exci-\ntations at other frequencies, the time dependence of the\ninduced dipole moment p(t) will be di\u000berent. The de-\npendence on two frequency variables is intriguing: The\n\frst frequency argument, !, corresponds to the Fourier\ntransform with respect to the variable \r, and the second\nangular frequency, \n, is due to the Fourier transform withrespect to the variable t. Notice that since the dipole mo-\nment p(t) is real-valued, we have p\u0003(!;\n) = p(!;\u0000\n).\nThis relation can be readily proved by using Eq. (11). It\nis worth noting that \u000bT(!;t) does not obey this relation,\nbecause\u000bT(!;t) is not necessarily a real-valued function.\nTherefore, in general,\n\u000b\u0003\nT(!;\n)6=\u000bT(!;\u0000\n): (12)\n3. Temporal complex susceptibility and permittivity\nBefore moving to the next section, it is worth noting\nthat in analogy with the dipole moments, a similar time-\ndomain description should be used for the electric and\nmagnetic \rux densities D(t) andB(t) as linear and causal\nfunctions of E(t) and H(t). If a medium is stationary, we\nreadily write [31]\nD=\u000f(!)\u0001E+\u0018(!)\u0001H;\nB=\u0010(!)\u0001E+\u0016(!)\u0001H;(13)\nin which\u000f,\u0016,\u0018,\u0010are the frequency-domain material pa-\nrameters. However, for a medium which is not stationary\nand its properties are time-variant, we need to express the\nconstitutive relations which concurrently respect nonsta-\ntionarity and memory. In the literature, assuming a time-\nvarying dielectric isotropic medium ( \u0016=\u0018=\u0010= 0), that\nconstitutive relation is often given by (e.g., Refs. [41{43])\nD(t) =\u000f(t)E(t): (14)\nThis model of a time-varying dielectric medium is based\non a very dramatic approximation of instantaneous re-\nsponse of matter, which is not consistent with the tem-\nporal dispersion naturally present in materials. There-\nfore, more complete and rigorous de\fnitions need to be\nintroduced and applied. Indeed, for any point in space\nwe should write (for an isotropic time-varying dielectric\nmedium)\nD(t) =\u000f0E(t) +Z+1\n0\u000f0\u001f(\r;t)E(t\u0000\r)d\r; (15)\nin which\u001f(\r;t) is the electric susceptibility kernel (which\nmay depend also on spatial coordinates). In general,\ntime-varying \felds can be expressed as an inverse Fourier\ntransform\nE(t) =1\n2\u0019Z+1\n\u00001E(!) exp(j!t)d!: (16)\nSubstituting the above expression in Eq. (15), we deduce\nthat\nD(t) =\u000f0\n2\u0019Z+1\n\u00001\u000fT(!;t)E(!) exp(j!t)d!; (17)\nwhere the temporal complex relative permittivity equals\n\u000fT(!;t) = 1 +\u001fT(!;t) (18)5\nwith\n\u001fT(!;t) =Z+1\n0\u001f(\r;t) exp(\u0000j!\r)d\r: (19)\nThis is similar to the de\fnition used by N. S. Stepanov\nin Ref. [44] for describing macroscopic susceptibility of\ntime-varying plasma. Here, the indices \\T\" are used\nto discern the temporal complex functions, \u000fT(!;t) and\n\u001fT(!;t), from the relative permittivity and susceptibil-\nity kernels, respectively (index \\T\" refers to \\temporal\").\nBased on the above derivations and explanations, what\nwe unequivocally perceive is the fact that the temporal\ncomplex susceptibility and relative permittivity are gen-\neral concepts which are valid and useful even if the \feld\nis nonharmonic. Later, in the last part of Section IV,\nwe employ these important equations and de\fnitions,\nEqs. (15){(19), to calculate the e\u000bective macroscopic pa-\nrameters of time-varying materials.\nIn the end of this discussion, we should mention that in\nanalogue to Eq. (11), we can take the Fourier transform of\nthe electric \rux density given by Eq. (17). This operation\nsigni\fcantly helps, allowing us to study and solve the\nMaxwell equations in the frequency domain (\n domain).\nKeeping in mind that \u000fT(!;\n) is the Fourier transform\nof\u000fT(!;t) with respect to the time variable t, we simply\nconclude that\nD(\n) =\u000f0\n2\u0019Z+1\n\u00001\u000fT(!;\n\u0000!)E(!)d!: (20)\nTo appraise this relation, we carefully look at two par-\nticular instances. The \frst one is if the medium is im-\nmutable and stationary, but dispersive. Thus, the rela-\ntive permittivity kernel depends on only one time vari-\nable and is given by \u000f(\r;t) =\u000f(\r). Consequently, the\ntemporal complex function and its Fourier transform are\nexpressed as \u000fT(!;t) =\u000f(!) and\u000fT(!;\n) = 2\u0019\u000f(!)\u000e(\n),\nrespectively, in which \u000e(\n) is the one-dimensional Dirac\ndelta function. By plugging this result in Eq. (20) and\nusing the property thatR\nf(x)\u000e(x\u0000x0)dx=f(x0), we\nobtain D(\n) =\u000f0\u000f(\n)E(\n) which is the conventional\nexpression for modeling dispersive time-invariant media.\nFor the second case, we assume that the medium pos-\nsesses instantaneous response and varies in time. As we\nwrote earlier, this is what a multitude of research works\nhave assumed in their studies of interactions of waves\nwith time-varying media. Regarding this case, the rela-\ntive permittivity kernel is \u000f(\r;t) =\u000e(\r)\u000fI(t) which results\nin\u000fT(!;t) =\u000fI(t) and\u000fT(!;\n) =\u000fI(\n). The initial ob-\nservation, based on Eq. (17), is that D(t) =\u000f0\u000fI(t)E(t),\nand the next observation, in accordance with Eq. (20),\nexplains the fact that D(\n) is the convolution of \u000fI(\n)\nand the \feld E(\n). Both these two observations are quite\nexpected for the nondispersive model of time-varying me-\ndia.III. INDIVIDUAL TIME-VARYING PARTICLE\nIN FREE SPACE\nBased on the fundamental notions introduced and dis-\ncussed above, we can address an important question on\nhow to determine the polarizability kernel \u000b(\r;t) and\nthe temporal complex polarizability \u000bT(!;t) for a single\ntime-varying particle located in free space. To answer\nthis central question, \frst, we need to write a linear di\u000ber-\nential equation which describes the polarization dynam-\nics of time-varying particles. Studying this equation and\nusing the fundamentals explained before, we will develop\na systematic method to \fnd the particle polarizability.\nHere, let us focus on the canonical example of a non-\nstationary Hertzian dipole. The understanding of this\nbasic scatterer can be extended to small time-varying\ninclusions which have electric response. We model a\nHertzian dipole as a short wire antenna of length l, as-\nsuming that the current is uniform along the wire. The\nantenna parameters are its e\u000bective inductance L, capaci-\ntanceC, and a resistive load R(that accounts for dissipa-\ntion losses). Parameters LandCmeasure the magnetic\nand electric energies stored in the reactive \felds around\nthe dipole. Accordingly, for such a Hertzian dipole, a\nthird-order di\u000berential equation for the oscillating charge\nQis expressed as\n\u0000Zd3Q\ndt3+Ld2Q\ndt2+RdQ\ndt+1\nCQ=lE: (21)\nThis is the \\R udenberg equation\" which was initially\nwritten in 1907 for elucidating the Hertzian dipole an-\ntenna in the receiving regime [45]. On the right side, E\ndenotes the incident electric \feld at the dipole location,\nparallel to the dipole. Notice that the \frst term in the\nabove equation on the left side, which is proportional to\nthe third time derivative of the electric charge, is associ-\nated with the radiation of the dipole and is linked with\nthe radiation reaction (see, e.g., Ref. [30]). The param-\neterZ=l2\u00160=(6\u0019c) determines what R udenberg calls\n\\Strahlungswiderstand\" or radiation resistance (here, \u00160\nis the free-space permeability, and cdenotes the speed\nof light). Including this \frst term in Eq. (21), we see\nthat the R udenberg equation is in fact analogous to the\nAbraham-Lorentz equation [46] that also includes the ra-\ndiation reaction term. Since the electric dipole moment\nis the multiplication of the electric charge and the dipole\nlength, therefore, we readily modify the authentic version\nof R udenberg equation and write\n\u0000l\u00160\n6\u0019cd3p\ndt3+L\nld2p\ndt2+R\nldp\ndt+1\nlCp=lE: (22)\nTill this point, we have assumed that the Hertzian dipole\nis stationary and the resistance, inductance, and the ca-\npacitance are immutable and constant in time. To re-\nalize a nonstationary Hertzian dipole, it is su\u000ecient to\nmake those parameters time-dependent in the di\u000beren-\ntial equation. For instance, suppose that we add a time-\nvarying capacitance as an extra load which is connected6\nin series with a dynamic load resistance R(t). For such\ntime-varying dipoles, the di\u000berential equation, Eq. (22),\nis rewritten as\n\u0000l2\nL\u00160\n6\u0019cd3p\ndt3+d2p\ndt2+R(t)\nLdp\ndt+1\nLC(t)p=l2\nLE:(23)\nNote thatC(t) is the total capacitance which contains\nboth the e\u000bective capacitance due to the stored electric\nenergy around the dipole and the time-modulated load\ncapacitance. Also, note that in Eq. (23), if the \frst term\n(the radiation term) on the left side is neglected (by as-\nsuming that it is much smaller than the last term (the\nrestoring-force term)), we achieve the di\u000berential equa-\ntion that is utterly similar to the equation of motion for\nbound electrons in matter, discussed in the next section\n(see Eq. (38)). Here, however, we study a dipole in free\nspace and keep the radiation term.\nEquation (23) relates the instantaneous dipole moment\nand the excitation \feld. On the other hand, in Sec-\ntion II, we related the instantaneous dipole moment and\nthe \feld in terms of the polarizability kernel. There-\nfore, by bringing these two models together, we can \fnd\nthe polarizability kernel (and the temporal complex po-\nlarizability) in terms of the time-varying resistance and\ncapacitance. We start from Eq. (6), which is an alterna-\ntive integral form to describe the dipole moment pof a\ntime-varying particle excited by an external electric \feld\nE:p(t) =Rt\n\u00001h(t;\u001c)E(\u001c)d\u001c. Next, we need to replace\nthis description in Eq. (23). For replacing, however, we\nhave to calculate the \frst, second, and third time deriva-\ntives of the dipole moment. To \fnd these derivatives, we\nuse the chain rule and the Leibniz integral rule which say\nthat for any integrable function f(x;y), we can write\nd\ndxZb(x)\na(x)f(x;y)dy=\nf(x;b(x))db(x)\ndx\u0000f(x;a(x))da(x)\ndx+Zb(x)\na(x)@\n@xf(x;y)dy;\n(24)\nwherea(x) andb(x) denote the lower and upper limits,\nrespectively. By employing Eq. (24) and doing careful al-\ngebraic manipulations, the \frst, second, and third deriva-\ntives of the dipole moment are written in terms of h(t;\u001c)\nand the corresponding partial derivatives of h(t;\u001c) as\ndp\ndt=h(t;\u001c)j\u001c=tE+Zt\n\u00001@h(t;\u001c)\n@tE(\u001c)d\u001c;\nd2p\ndt2=\"\n2@h(t;\u001c)\n@tj\u001c=t+@h(t;\u001c)\n@\u001cj\u001c=t#\nE\n+h(t;\u001c)j\u001c=tdE\ndt+Zt\n\u00001@2h(t;\u001c)\n@t2E(\u001c)d\u001c;(25)and\nd3p\ndt3=\"\n3@2h(t;\u001c)\n@t2+ 3@2h(t;\u001c)\n@t@\u001c+@2h(t;\u001c)\n@\u001c2#\n\u001c=tE+\n\"\n3@h(t;\u001c)\n@t+ 2@h(t;\u001c)\n@\u001c#\n\u001c=tdE\ndt+h(t;\u001c)j\u001c=td2E\ndt2+\nZt\n\u00001@3h(t;\u001c)\n@t3E(\u001c)d\u001c:\n(26)\nUltimately, by substituting the results of Eqs. (25) and\n(26) into Eq. (23), we derive four expressions (one char-\nacteristic equation and three initial conditions) which de-\n\fne the function h(t;\u001c). These four expressions read\n1)R@3h(t;\u001c)\n@t3+@2h(t;\u001c)\n@t2+R\nL@h(t;\u001c)\n@t+\n1\nLC(t)h(t;\u001c) = 0;\n2) 3R@2h(t;\u001c)\n@t2j\u001c=t+ 3R@2h(t;\u001c)\n@t@\u001cj\u001c=t+\nR@2h(t;\u001c)\n@\u001c2j\u001c=t+ 2@h(t;\u001c)\n@tj\u001c=t+@h(t;\u001c)\n@\u001cj\u001c=t=l2\nL;\n3) 3@h(t;\u001c)\n@tj\u001c=t+ 2@h(t;\u001c)\n@\u001cj\u001c=t= 0;\n4)h(t;\u001c)j\u001c=t= 0;\n(27)\nin which R=\u0000(l2=L)(\u00160=6\u0019c).\nAt this point, one may ask what will happen if the\ndipole is loaded with a time-varying inductance rather\nthan a time-varying capacitance. How do the above ex-\npressions change? This is a valid and intriguing question.\nIn fact, we should \frst revise the third-order di\u000beren-\ntial equation which describes the nonstationary dipole\nconnected to time-varying inductance. For that, as one\nmay expect, we start from the primary version of the\nR udenberg equation, Eq. (21), which is based on the\nelectromotive force, and we rewrite the voltage over the\ntime-varying inductance in this equation. Subsequently,\nafter a simple modi\fcation, we express the desired di\u000ber-\nential equation that is similar to Eq. (23). Therefore, we\nhave\n\u0000l2\nL(t)\u00160\n6\u0019cd3p\ndt3+d2p\ndt2+\u0000(t)dp\ndt+1\nL(t)Cp=l2\nL(t)E;(28)\nwhere \u0000(t) = [1=L(t)][R+dL(t)=dt] is a function that is\nrelated to the time derivative of the inductance. Conse-\nquently, this function becomes a constant if the deriva-\ntive of the inductance vanishes. Here, we do not continue\nwith the derivations of the polarizability kernel because\nthe method for deriving them is quite clear for the readers\n(Eqs. (25) and (26) should be substituted into Eq. (28)).\nTherefore, we pass those derivations and calculations on\nto the interested reader.7\nThe above theory, which we developed for determining\nthe polarizability kernel, will be complemented by deriv-\ning another partial di\u000berential equation that \\directly\"\ndescribes the temporal complex polarizability when the\nincident electric \feld is time-harmonic. Similar to what\nwe did for the polarizability kernel in Eq. (23), it is su\u000e-\ncient to supersede the dipole moment by its de\fnition in\nterms of\u000bT(!;t) (i.e., p(t) = Re\u0002\n\u000bT(!;t)\u0001E0exp(j!t)\u0003\n)\nin order to derive such a partial di\u000berential equation. Af-\nter doing simpli\fcations, we write that\nR@3\u000bT(!;t)\n@3t+\u0012\n1 +j3!R\u0013@2\u000bT(!;t)\n@2t+\n\u0012R\nL+j2!\u00003!2R\u0013@\u000bT(!;t)\n@t+\n\u00121\nLC(t)\u0000!2+j!R\nL\u0000j!3R\u0013\n\u000bT(!;t) =l2\nL:(29)\nNotice that here we assumed that the nonstationary\ndipole is loaded by a time-varying capacitance. How-\never, one can repeat the same procedure when the dipole\nload is a time-varying inductance (see Eq. (28)).\nA. Dipole interaction with incident waves\nThe analytical approach for determining the polariz-\nability kernel \u000b(\r;t) and the temporal complex polariz-\nability\u000bT(!;t) of nonstationary dipoles was explained in\nthe previous part of this section. At the following step, we\nscrutinize classical interactions of nonstationary dipoles\nwith external \felds based on the notion of polarizability,\nand study instantaneous powers exerted on and radiated\nby the dipole under illumination.\n1. Instantaneous power exerted on dipole\nLet us assume that a Hertzian dipole is located at\nthe origin of the Cartesian coordinate system, directed\nalong thezaxis, and illuminated by a time-harmonic\nelectric \feld. Also, let us assume that the nonstationary\ncharacteristic of the dipole can be realized, for example,\nby loading the dipole with a time-varying lumped ele-\nment [29] or varying the dipole length in time. For the\n\frst case, in which the e\u000bective length of the Hertzian\ndipole is \fxed and the dipole is loaded by a time-varying\nlumped element, we can simply employ the concept of in-\nduced electromotive force and calculate the total instan-\ntaneous power exerted on the dipole. From the basics,\nwe know that the electric dipole moment is the multipli-\ncation of the electric charge Q(t) and the dipole length\nl:p(t) =lQ(t). Since the time derivative of the electric\ncharge is identical with the electric current, as a result,\nthe time derivative of the dipole moment becomes equal\nto the length of the dipole lmultiplied by the electric\ncurrenti(t) carried by the dipole: dp(t)=dt=l\u0001i(t). We\nstress that in this simple calculation, the length of thedipole is not changing in time, and the temporal vari-\nation of the dipole moment is only due to the tempo-\nral variation in the electric charge. On the other hand,\nthe induced electromotive force corresponding to e\u000bective\nlengthlis expressed as v(t) =l\u0001E(t), in whichE(t) is the\ncomponent of the excitation \feld parallel to the Hertzian\ndipole. Using the above two equations, representing the\ninduced electromotive force and the time derivative of\nthe dipole moment, the instantaneous power is obtained\nfromS(t) =v(t)\u0001i(t), which \fnally gives\nS(t) =E(t)\u0001dp(t)\ndt: (30)\nRegarding the second case, where the length of the dipole\nis also changing in time, and, therefore, the temporal\nvariation of the dipole moment is not only due to the\nelectric charge Q(t), but it is also due to the length vari-\nationsl(t), it may initially seem to be complicated to \fnd\nthe instantaneous power. However, in general, the elec-\ntric current density corresponding to the Hertzian dipole\nis written as J(r;t) =dp(t)\ndt\u000e3(r), and the instantaneous\npowerS(t) is expressed as\nS(t) =Z\nVJ\u0001Ed3r; (31)\nwhereVdenotes the volume occupied by the dipole.\nBased on Eq. (31) and by substituting the electric cur-\nrent density, we explicitly achieve the same expression as\ngiven by Eq. (30). Therefore, we conclude that Eq. (30)\nis valid even for the case when the dipole length is chang-\ning in time. Notice that Eq. (30) is also true for any time\nvariation of the dipole moment, including the stationary\nscenario.\nAccording to Eq. (7), the dipole moment p(t) is de-\nscribed in terms of the temporal complex polarizability\n\u000bT(!;t). Therefore, this equation can be substituted into\nEq. (30) in order to present the power S(t) in terms on\nthe polarizability \u000bT(!;t). Before we proceed and do the\nsubstitution, let us note that due to the single-frequency\nexcitation, for brevity, we can drop the \frst argument of\n\u000bT(!;t) (i.e. the angular frequency of the incident \feld).\nIn addition, for simplicity, we also de\fne the following\ncomplex-valued function:\n\u0010(t) =\u000bT(t) +1\nj!d\u000bT(t)\ndt; (32)\nwhich is associated with the temporal complex polariz-\nability and its time derivative. Now, by substituting the\ntemporal complex polarizability and using this auxiliary\nfunction de\fnition, Eq. (32), the extracted power is sim-\npli\fed to\nS(t) =E(t)\u0001Reh\nj!\u0010(t)\u0001E0exp(j!t)i\n: (33)\nWriting the real part as Re[ x] = (1=2)(x+x\u0003), \fnally,\nthe extracted power reduces to\nS(t) =\u0000!\n2Im\u0002\n\u0010(t)\u0003\njE0j2\u0000!\n2Im\u0014\n\u0010(t)E2\n0exp(j2!t)\u0015\n;\n(34)8\nin which Im[ ] denotes the imaginary part.\nLet us check this equation for the special case of a\nstationary dipole. In this case the time derivative of\n\u000bT(t) vanishes, and \u0010becomes a complex constant which\nis equal to \u000bT. As a consequence, the time-averaged\nvalue of the second term in the above equation be-\ncomes zero and the time-averaged power extracted by\nthe dipole from the incident \feld is simply Sstationary =\n\u0000!\n2Im[\u0010]jE0j2=\u0000!\n2Im[\u000bT]jE0j2, which is the same rela-\ntion as we know from the literature (see e.g. Ref. [47]).\nHere, we stress that for stationary dipoles the expression\nin Eq. (34) is also valid in the time domain.\nAnother special case is the case when \u0010(t) = 0. From\nEq. (34) we see that if \u0010(t) = 0, the extracted power S(t)\nis zero meaning that the dipole does not interact with\nthe incident \feld. According to Eq. (32), the condition\n\u0010(t) = 0 corresponds to \u000bT(t) =Ae\u0000j!twhereAis a\nconstant coe\u000ecient. Substituting this result into Eq. (7),\nwe see that the dipole moment is constant over time. In\nother words, we have a static dipole moment whose time\nderivative is zero. Consequently, there should not be any\ninteraction with the incident \feld.\nConsidering Eq. (34), it is intriguing to assume a\nperiodic function \u0010(t). This is because periodicity al-\nlows us to employ simple time averaging. Based on the\nFourier series written for a periodic function, depending\non the period and the complex Fourier coe\u000ecients, the\ntime-averaged value associated with the second term in\nEq. (34) is not zero, and it can signi\fcantly contribute to\nthe time-averaged total power. For example, it is clearly\nseen that if the period is equal to the excitation period\nT= 2\u0019=!, the second-order term in the Fourier series\nn= 2 can produce a nonzero averaged value (in contrast\nwith the stationary case, in which the average is zero).\n2. Instantaneous power radiated by dipole\nSome part of the extracted power is re-radiated to the\nbackground medium (here, free space). The instanta-\nneous power which is re-radiated by the dipole is propor-\ntional to the \frst and the third time derivatives of the\ndipole moment [29, 30]:\nSrad(t) =\u0000\u00160\n6\u0019cdp(t)\ndt\u0001d3p(t)\ndt3: (35)\nSimilarly to what we did for S(t), we substitute the tem-\nporal complex polarizability in the above equation for\nthe re-radiated power. In accordance with Eqs. (7) and\n(32), the \frst and the third time derivatives of the dipole\nmoment are given by\ndp(t)\ndt= Re\"\nj!\u0010(t)\u0001E0exp(j!t)#\n;\nd3p(t)\ndt3= Re\"\nj!\u0010d2\u0010\ndt2+j2!d\u0010\ndt\u0000!2\u0010\u0011\n\u0001E0exp(j!t)#\n:\n(36)Subsequently, by using these equations and after doing\nsome algebraic manipulations, we \fnd the re-radiated\npower (in Eq. (35)) as\nSrad(t) =\u00160!4\n12\u0019cj\u0010j2jE0j2+\n\u00160!4\n12\u0019cjE0j2Re\"\n\u0010\u00121\n!2d2\u0010\ndt2+j2\n!d\u0010\ndt\u0000\u0010\u0011E2\n0\njE0j2exp(j2!t)\n\u0000\u0010\u00121\n!2d2\u0010\ndt2+j2\n!d\u0010\ndt\u0013\u0003#\n:\n(37)\nIf the dipole is stationary, all the time derivatives in the\nabove equation become zero, and the time-averaged scat-\ntered power is simpli\fed to Sstationary\nrad=\u00160!4\n12\u0019cj\u0010j2jE0j2=\n\u00160!4\n12\u0019cj\u000bTj2jE0j2, which can be conveniently found in the\nliterature (e.g. Ref. [47]).\nTo summarize, for a nonstationary electric dipole un-\nder illumination, we \frst derived the corresponding dif-\nferential equations (with the corresponding initial con-\nditions) for obtaining the polarizability. Afterwards, we\nutilized the introduced notion of the temporal complex\npolarizability to \fnd the total instantaneous extracted\npower and the instantaneous scattered power, which are\nnot necessarily equal to each other (see Appendix B for\nmore information).\nIV. NONSTATIONARY PARTICLE AS A\nCONSTITUENT OF A TIME-VARYING\nMATERIAL\nIn contrast to the previous section, where the time-\nvarying particle is located in free space, in this section, we\nassume that the particle is a constituent of a time-varying\nmaterial, and, accordingly, we investigate the polarizabil-\nity of such a particle. This investigation is important\nfor understanding the e\u000bective macroscopic parameters\nsuch as susceptibility and permittivity of dynamic ma-\nterials. To determine the polarizability kernel and the\ntemporal complex polarizability, we need to study the\ncorresponding di\u000berential equation. Since the particle is\nimmersed in a time-varying material composed of many\nidentical particles, the radiated power is compensated by\nthe power received from other particles. This cancel-\nlation ensures that in the absence of dissipation in the\nparticles and power exchange with the external devices\nthat modulate the particles, the e\u000bective medium param-\neters correspond to a lossless material. In this case, the\norder of the di\u000berential equation is two because the ra-\ndiation reaction term that was proportional to the third\nderivative of the dipole moment vanishes.9\nA. Di\u000berential equations for the dipole moment\nand the polarizability kernel\nLet us concentrate on the classical model of a bound\nelectron as our particle under study. In this case, the\ndi\u000berential equation needed for description of the elec-\ntric dipole moment is given by the classical equation of\nmotion. Based upon this equation, we model external\ntime modulations of the system by assuming that the\ndamping coe\u000ecient \u0000 Dand the natural frequency !nare\nvarying in time. Therefore, the second-order di\u000berential\nequation describing the electron motion is expressed as\nd2x(t)\ndt2+ \u0000D(t)dx(t)\ndt+!2\nn(t)x(t) =e\nmE(t), in which mde-\nnotes the electron mass, erepresents the electron charge,\nandx(t) is the displacement. Since the dipole moment\nis the multiplication of the electron charge and the dis-\nplacement, we can consequently write\nd2p\ndt2+ \u0000D(t)dp\ndt+!2\nn(t)p=e2\nmE: (38)\nHere, it is worth noting that one can entitle Eq. (38)\nas the Lorentz equation which results in the Lorentz\nmodel for the e\u000bective macroscopic parameters of dielec-\ntric materials. This model reduces to the Drude model\nin the limit of vanishing natural frequency ( !n= 0),\nwhich means that there is no restoring force and the\nelectron is not bound. Therefore, in general, we use the\nterm \\Drude-Lorentz\" to entitle Eq. (38). By employing\nEqs. (6) and (24), we already derived the \frst and sec-\nond time derivatives of the electric dipole moment and\ndemonstrated them based on the function h(t;\u001c) and its\npartial derivatives. Thus, by inserting those derivations\n(written in Eq. (25)) into Eq. (38) (the Drude-Lorentz\nequation), we arrive to three crucial expressions which\ndetermine the polarizability:\n1)@2h(t;\u001c)\n@t2+ \u0000D(t)@h(t;\u001c)\n@t+!2\nn(t)h(t;\u001c) = 0;\n2) 2@h(t;\u001c)\n@tj\u001c=t+@h(t;\u001c)\n@\u001cj\u001c=t=e2\nm;\n3)h(t;\u001c)j\u001c=t= 0:(39)\nWe stress that the function h(t;\u001c) is not the polariz-\nability kernel. Indeed, the polarizability kernel that we\nintroduced above is \u000b(\r;t) =h(t;\u001c) when\u001c=t\u0000\r. As\na consequence, \u001c=tin the second and third expressions\nrefers to\r= 0, and Eq. (39) de\fnes two initial condi-\ntions at\r= 0 for\u000b(\r;t). Depending on the temporal\nfunctions of the damping coe\u000ecient and the natural fre-\nquency, these three expressions in Eq. (39) give a speci\fc\nfunction for the polarizability.\nAs a check, let us examine the results by considering\n\frst the stationary scenario, assuming that the damping\ncoe\u000ecient and the natural frequency are not varying in\ntime. By remembering that h(t;\u001c) =h(t\u0000\u001c) in this\nscenario and solving Eq. (39), the polarizability kernel isderived as\n\u000b(\r;t) =h(t;t\u0000\r) =\ne2\nmq\n!2n\u0000\u00002\nD\n4exp\u0000\n\u0000\u0000D\n2\r\u0001\nsin\u0000r\n!2n\u0000\u00002\nD\n4\r\u0001\n:(40)\nNotice that the full derivation is explained in the Ap-\npendix of the paper. In the above equation, as it is ex-\nplicitly seen, the polarizability kernel depends on only\none time parameter, \r. Therefore, its Fourier transform\nis only a function of the frequency and gives the known\nDrude-Lorentz dispersion in the frequency domain (e.g.,\n[40]).\nRegarding the nonstationary scenario, in which the\ndamping coe\u000ecient or the natural frequency depends on\ntime, we will give a complete example in the last part\nof this section, comprehensively calculating the polar-\nizability kernel (see Eqs. (52){(57) and the correspond-\ning explanations). Also, subsequently, we will compare\nthe obtained results with the ones known for the con-\nventional stationary scenario. In this example, we will\nassume that the damping coe\u000ecient is temporally mod-\nulated as \u0000 D(t) = 2\u0000 0=(1 + \u0000 0t) (where \u0000 0corresponds\nto the damping coe\u000ecient at t= 0) and the natural fre-\nquency is zero.\n1. Noncausal interpretation\nPrior to studying the temporal complex polarizabil-\nity, here, we would like to have a brief discussion around\ncausality of time-modulated particles. If we do not re-\nspect causality, the dipole moment can also depend on\nthe electric \feld in the future, and, therefore, p(t) =R+1\n\u00001h(t;\u001c)E(\u001c)d\u001c. Such interpretation a\u000bects strikingly\nthe results of Leibniz integral expressions. In fact, by\napplying Eq. (24), the \frst and second derivatives of the\ndipole moment are modi\fed as\ndp\ndt=Z1\n\u00001@h(t;\u001c)\n@tE(\u001c)d\u001c;\nd2p\ndt2=Z1\n\u00001@2h(t;\u001c)\n@t2E(\u001c)d\u001c:(41)\nWe can compare these equations with the expressions\nfor the causal interpretation (Eq. (25)), in order to un-\nderstand the fundamental di\u000berence between them. By\nusing Eq. (41) and considering Eq. (38), which is the\nDrude-Lorentz equation, we write\nZ1\n\u00001\"\n@2h(t;\u001c)\n@t2+ \u0000D(t)@h(t;\u001c)\n@t+!2\nn(t)h(t;\u001c)#\nE(\u001c)d\u001c\n=e2\nmE(t):\n(42)10\nAs we explicitly see from the above, since the electric\n\feld is simultaneously attending both sides of the equal-\nity and it is inside the integral on the left side, we can con-\nclude that the whole expression within the square brack-\nets should be equal to the Dirac delta function. In other\nwords,\n@2h(t;\u001c)\n@t2+ \u0000D(t)@h(t;\u001c)\n@t+!2\nn(t)h(t;\u001c) =e2\nm\u000e(\u001c\u0000t):\n(43)\nEquation (43) is the key equation for calculating the po-\nlarizability kernel describing a noncausal response. Solv-\ning this equation and \fnding a solution for the function\nh(t;\u001c) may not be straightforward (and even possible)\ndue to the presence of the Dirac delta function on the\nright side. However, if there is a nonzero solution, notice\nthat it must be a real solution in time because the func-\ntionh(t;\u001c) is indeed a real-valued function. Here, we also\npoint out an intriguing issue which in\ruences Eq. (43).\nIn the expression p(t) =R+1\n\u00001h(t;\u001c)E(\u001c)d\u001cwritten ini-\ntially, although we chose the upper limit of the integral\nas in\fnity, we can de\fnitely assume a \fnite upper limit\nand still describe a noncausal response. The only condi-\ntion is that the \fnite upper limit should be larger than\nthe observation time t, i.e. upper limit = t+\fwhere\f\nis real and \f > 0. In accordance with this assumption,\np(t) =Rt+\f\n\u00001h(t;\u001c)E(\u001c)d\u001c, subsequently, we can simply\nrewrite Eq. (43) by using the Leibniz integral rule and\nDrude-Lorentz equation. In this paper, since our focus\nis only on the causal response, we leave such derivations\nfor the interested reader.\nB. Di\u000berential equation for \fnding the temporal\ncomplex polarizability\nWe have hitherto discussed how to derive the polariz-\nability kernel of the electron based on the classical model.\nHere, our aim is to introduce a linear di\u000berential equation\nwhose solution gives the temporal complex polarizability.\nWe already know that, for time-harmonic excitation, the\ninduced electric dipole moment is expressed in terms of\n\u000bT(!;t) (see Eq. (7)). Ergo, by substituting that expres-\nsion into the Drude-Lorentz equation (see Eq. (38)), we\ncome to the desired di\u000berential equation for the temporal\ncomplex polarizability, which is written as\n@2\u000bT(!;t)\n@t2+\u0010\n\u0000D(t) +j2!\u0011@\u000bT(!;t)\n@t+\n\u0010\n!2\nn(t)\u0000!2+j!\u0000D(t)\u0011\n\u000bT(!;t) =e2\nm:\n(44)\nEquation (44) is a second-order di\u000berential equation\nwith time-dependent coe\u000ecients which allows us to \fnd\n\u000bT(!;t) for arbitrary time variations of the particle pa-\nrameters. Certainly, this equation should be comple-\nmented by initial conditions in order to obtain a speci\fcsolution for \u000bT(!;t). These initial conditions are given\nby Eq. (39) in which h(t;\u001c) and the polarizability ker-\nnel are present. Therefore, after solving Eq. (44) and\n\fnding\u000bT(!;t), one needs to \frstly make the inverse\nFourier transform to calculate the polarizability kernel\n\u000b(\r;t) and subsequently h(t;\u001c). As a result, by having\nthe function h(t;\u001c), the initial conditions expressed in\nEq. (39) (the second and third expressions) can be read-\nily checked to achieve the speci\fc solution.\nIn order to check Eq. (44), we \frst make \u0000 D(t) and\n!n(t) time-invariant. Since \u000bT(!;t) does not depend on\ntime in this case, the corresponding time derivatives in\nEq. (44) become zero, and we instantly see that the so-\nlution is the usual Lorentz dispersion rule:\n\u000bT(!;t) =e2\nm\n!2n\u0000!2+j!\u0000D: (45)\nAs a second check, we consider a time-variant \u0000 D(t) or\n!n(t). For this, let us take the same example as in the\nnext subsection of the paper, where we will assume that\nthe damping coe\u000ecient is \u0000 D(t) = 2\u0000 0=(1 + \u0000 0t) and\nthe natural frequency is zero. For this example, we ap-\nply Eq. (39) and derive the polarizability kernel which\nis given by Eq. (57) in the next subsection. Therefore,\nsince we know \u000b(\r;t), we can obtain the temporal com-\nplex polarizability \u000bT(!;t) by simply taking the Fourier\ntransform with respect to \r, see Eq. (8). After some\nalgebraic manipulations, \u000bT(!;t) is expressed as\n\u000bT(!;t) =\u0000e2\nm\u00101\n!2+j2\n!3\u00000\n1 + \u0000 0t\u0011\n: (46)\nNow, as one can expect, this expression in Eq. (46) should\nde\fnitely satisfy the second-order di\u000berential equation\nEq. (44). If we substitute \u000bT(!;t) (written above) into\nEq. (44), we observe that Eq. (44) indeed holds. Notice\nthat Eq. (46) is not calculated based on Eq. (44), but\nit is obtained by employing the comprehensive Eq. (39).\nTherefore,\u000bT(!;t) in Eq. (46) represents the general so-\nlution of the second-order di\u000berential equation, which\nconsists of both the complementary function (the so-\nlution of the homogeneous equation) and the particu-\nlar integral solution (the solution of the inhomogeneous\nequation). This general solution also respects the ini-\ntial conditions expressed in Eq. (39). From this point\nof view, such general solution can be \fnally considered\nas the unique solution to Eq. (44) with !n(t) = 0 and\n\u0000D(t) = 2\u0000 0=(1 + \u0000 0t).\nC. On the Drude-Lorentz model of time-varying\ndielectrics and plasma\nNext, we use the above theoretical results to analyse\napproximate models of e\u000bective parameters of Lorentzian\ndielectrics and electron plasma. The dipole moment of\neach electron is governed by Eq. (38), where the param-\neters may depend on time due to changing environment11\nwhere the charges are located. However, apart from\nEq. (38), the electron density (i.e. the number of elec-\ntrons per unit volume N(t)) can also depend on time.\nIn consequence, in the following, we will consider two\ndi\u000berent cases: A particular case in which only the elec-\ntron density is time variant and the parameters in the\nequation of motion (Eq. (38)) do not change in time, and\na more general case in which those parameters are also\ntime dependent in addition to the electron density. Both\naforementioned cases de\fnitely result in nonstationary\nmodels. The reason for having a discussion on the for-\nmer case is that while the polarizability kernel of the\nsingle electron should be the same as the one for the sta-\ntionary scenario, because the damping coe\u000ecient and the\nnatural frequency are assumed to be time invariant, and,\ntherefore, the polarizability kernel is only a function of\n\r(i.e.\u000b(\r;t) =\u000b(\r)), we will soon show that the corre-\nsponding e\u000bective permittivity kernel becomes a function\nof both\randt(i.e.\u000f(\r;t)).\nAccordingly, let us start from the \frst case which is\npossibly the simplest case. Since \u0000 Dand!nare constant\nin time and only the density N(t) varies, we can see this\nas a low-density approximation where we assume that\nthe electrons interact very weakly and, as a result, the\ncharacteristics of movement of a single electron do not\ndepend on the electron density. Under these assump-\ntions, the volume density of electric dipole moment or\npolarization density is written as\nP(t) =N(t)p(t) =Z+1\n0N(t)\u000b(\r;t)E(t\u0000\r)d\r=\nZ+1\n0\"0\u001f(\r;t)E(t\u0000\r)d\r;(47)\nin which the electric susceptibility kernel equals\n\u001f(\r;t) =N(t)\n\u000f0\u000b(\r;t) =\nN(t)\n\u000f0e2\nmq\n!2n\u0000\u00002\nD\n4exp\u0000\n\u0000\u0000D\n2\r\u0001\nsin\u0000r\n!2n\u0000\u00002\nD\n4\r\u0001\n:\n(48)\nHere, in Eq. (48), note that we take the polarizabil-\nity kernel from Eq. (40). Before proceeding, we draw\nthe attention to Eq. (47) which describes the polariza-\ntion density. In this equation, we simply wrote P(t)\nas the multiplication of the electric dipole moment and\nthe electron density. To do that, \frstly, we should as-\nsume that by varying the number of electrons (per unit\nvolume) in time, the time-varying material remains ho-\nmogeneous meaning that the electric susceptibility (or\npermittivity) does not depend on the position vector r:\n\u001f(r;\r;t) =\u001f(\r;t). Secondly, we should also suppose\nthat the process of changing the electron density in time\ndoes not a\u000bect the velocities of electrons. Within these\nassumptions, Eq. (47) is valid, and, in fact, we can write\nP(t) =N(t)p(t). Now, by knowing the electric suscep-\ntibility kernel from Eq. (48), we readily \fnd the relativepermittivity kernel of the e\u000bective medium as\n\u000f(\r;t) =\u000e(\r)+\nN(t)\n\u000f0e2\nmq\n!2n\u0000\u00002\nD\n4exp\u0000\n\u0000\u0000D\n2\r\u0001\nsin\u0000r\n!2n\u0000\u00002\nD\n4\r\u0001\n;\n(49)\nwhere\u000e(\r) is the one-dimensional Dirac delta function.\nSince we have the kernels from the above equations,\nEqs. (48) and (49), we can apply the Fourier transform\nEq. (19) and calculate the temporal complex relative per-\nmittivity de\fned in Eqs. (17) and (18). The result reads\n\u000fT(!;t) = 1 +!2\np(t)\n!2n\u0000!2+j\u0000D!; (50)\nin which\n!2\np(t) =e2\n\u000f0mN(t) (51)\nis the time-dependent plasma frequency. The expression\nin Eq. (50) is complex-valued and explicitly depends on\ntime indicating the nonstationarity characteristic. Sub-\nstituting!n= 0 (free-electron plasma) we arrive to the\nconventionally used expression for the e\u000bective permit-\ntivity of plasma with varying electron density, e.g. [48].\nThe only di\u000berence with the stationary case is that in the\nformula the plasma frequency explicitly depends on time.\nThe reason is due to the low-density approximation that\nwe have made. Within this approximation, the damping\ncoe\u000ecient and the natural frequency are considered to be\nconstant in time. Therefore, the polarizability kernel is\nthe same as the one written for the stationary scenario,\nand consequently the same kind of dispersion is observed.\nLet us consider a more general case when the damp-\ning coe\u000ecient and the natural frequency also change in\ntime. In this case, \u000fT(!;t) can be dramatically di\u000berent.\nSpeci\fc dependencies of the e\u000bective parameters in (50)\nare determined by the plasma structure and can be set\nas empirical parameters. As a particular example, here\nwe assume that h(t;\u001c) is a product of two functions K(t)\nandL(\u001c) which depend on single independent variables\ntand\u001c, respectively. Since h(t;t) = 0, according to the\ninitial condition in Eq. (39), one also assumes a multiplier\nin form (t\u0000\u001c)n. Thus, we consider the time-varying dis-\npersion kernel in form\nh(t;\u001c) = (t\u0000\u001c)nK(t)L(\u001c): (52)\nContemplating the second expression in Eq. (39), we \fnd\nthat\nn(t\u0000\u001c)n\u00001K(\u001c)L(\u001c) =e2=m; (53)\nin whichtand\u001cmust be equal. Due to this feature\n(t=\u001c), we can conclude that if n6= 1, the above iden-\ntity does not hold for nonzero functions of K(t) andL(\u001c).12\nTherefore, the only condition for holding the identity oc-\ncurs when nbecomes equal to unity, and, as a result,\nonly under this condition there is a nonzero solution for\nh(t;\u001c). Now, in the case of n= 1, satisfying the second\ninitial condition determines the function L(\u001c) as inversely\nproportional to K(\u001c) such that L(\u001c) =e2=[mK(\u001c)]. Us-\ning the partial di\u000berential equation (the \frst expression)\nin Eq. (39) and substituting\nh(t;\u001c) =e2\nm(t\u0000\u001c)K(t)\nK(\u001c); (54)\nwe \fnd the corresponding function K(t):\nK(t) = exp\u0010\n\u0000Z\u0000D(t)\n2dt\u0011\n; (55)\nwith an important constraint:\n!2\nn(t) =1\n4h\n\u00002\nD(t) + 2d\u0000D(t)\ndti\n: (56)\nThis equation shows that in this case the temporal vari-\nation of the natural frequency fully depends on the tem-\nporal variation of the damping coe\u000ecient.\nNext, let us assume a free-electron model so that !n=\n0. Based on Eq. (56), this assumption forces the damping\ncoe\u000ecient to vary homographically as \u0000 D(t) = 2\u0000 0=(1 +\n\u00000t), which results in K(t) = 1=(1 + \u0000 0t), according to\nEq. (55). With h(t;\u001c) = (e2=m)(t\u0000\u001c)(1+\u0000 0\u001c)=(1+\u0000 0t),\nthe electric polarizability kernel is given by\n\u000b(\r;t) =e2\nm\r\u0010\n1\u0000\u00000\r\n1 + \u0000 0t\u0011\n: (57)\nHaving the polarizability kernel from Eq. (57), and after\nsome algebraic manipulations, the relative permittivity\nkernel is expressed as\n\u000f(\r;t) =\u000e(\r) +N(t)\n\u000f0e2\nm\r\u0010\n1\u0000\u00000\r\n1 + \u0000 0t\u0011\n: (58)\nAs a sanity check, if \u0000 0= 0 andN(t) are time-invariant,\nwe obtain the conventional stationary lossless Drude\nmodel:\n\u000f(\r) =\u000e(\r) +N\n\u000f0e2\nm\r: (59)\nLet us again apply the Fourier transform (19) and \fnd\nthe temporal complex relative permittivity which corre-\nsponds to kernel (58). In accordance with the properties\nof Fourier transform, since\nZ+1\n0(\u0000j\r)nexp(\u0000j!\r)d\r=dn\nd!n\u00101\nj!\u0011\n; (60)\nwe \fnd that\n\u000fT(!;t) = 1\u0000!2\np(t)\n!2\u0000j2!2\np(t)\u00000\n!3(1 + \u0000 0t);\nor\n\u000fT(!;t) = 1\u0000!2\np(t)\n!2\u0000j!2\np(t)\n!3\u0000D(t):(61)The temporal permittivity has the imaginary part which\nis time-dependent, and is negative indicating that the\nmedium is lossy. Comparing the above expression with\nthe conventional stationary Drude model\n\u000fDrude (!) = 1\u0000!2\np\n!21\n1 + (\u0000D\n!)2\u0000j!2\np\n!3\u0000D\n1 + (\u0000D\n!)2;(62)\nwe explicitly observe how the e\u000bective permittivity of\nplasma with a time-varying damping coe\u000ecient cannot\nbe found by simply assuming that \u0000 Ddepends on time in\nthe conventional Drude formula: Time variations of the\ndamping coe\u000ecient \u0000 D(t) modi\fes the real and imagi-\nnary parts of the relative permittivity in a di\u000berent way,\nas is seen from Eq. (61).\nFinally, before \fnishing this section, we point out an\nissue about numerical simulations. The time-invariant\n(stationary) frequency-dispersive media have been thor-\noughly studied, and there are well developed time-\ndomain full-wave electromagnetic simulation tools such\nas Ansys HFSS, CST Microwave Studio, and COMSOL\nMultiphysics. However, up to our knowledge, there are\nno numerical tools that could \\properly\" simulate disper-\nsive time-varying (nonstationary) media. We hope that\nthis work will help developing such numerical methods\nwhich would allow studying electromagnetic processes in\ndispersive and time-varying media (including time-space\nmodulated metamaterials).\nV. CONCLUSIONS\nWe have theoretically studied the fundamental prin-\nciples related to the electric polarizability of arbitrary\ndipolar linear particles whose characteristics are varying\nin time due to some external force. Since at every mo-\nment of time the time-varying particle is di\u000berent, the po-\nlarizability additionally depends on the observation time\n(the one at which we measure the dipole moment). Im-\nportantly, this time-dependent polarizability is not fully\ndetermined by the particle parameters at the observation\nmoment, this polarizability is a causal-response parame-\nter which depends on the whole history of the particle.\nThis is in contrast with a stationary particle whose po-\nlarizability depends only on the delay time between the\nexcitation and observation moments.\nFor time-harmonic excitations, we demonstrated that\nthe instantaneous dipole moment is found as the real part\nof a complex-valued temporal function that is multiplied\nby the complex amplitude of the \feld and the time-\nharmonic exponential factor. This temporal response\nfunction is the Fourier transform of the polarizability ker-\nnel with respect to the delay time ( \r!!). We named\nthis function as temporal complex polarizability and ex-\nplained some of its salient properties. Importantly, using\nthe notion of temporal complex polarizability, we intro-\nduced the second Fourier transform that is with respect\nto the observation time ( t!\n). By this way, we could13\ndescribe the dipole moment completely in the frequency\ndomain.\nNext, we considered a nonstationary particle that is\nlocated in free space. By employing the R udenberg equa-\ntion (the Hertzian dipole model), we presented a method-\nical approach to determine the polarizability kernel and\nthe temporal complex polarizability. Having the polariz-\nability, we studied the classical interaction of the dipole\nwith the incident time-harmonic electromagnetic wave.\nTherefore, in terms of the temporal complex polarizabil-\nity, we contemplated the instantaneous powers that are\nextracted by the dipole and scattered from the dipole.\nFinally, we took one step forward and considered the\ndipole particle as a constituent of a time-varying ma-\nterial. This time, we focused on the classical bound\nelectron model and derived the corresponding equations\nfor \fnding the polarizability. To do that, we used the\nequation of motion (or the Drude-Lorentz equation) and\nassumed that the damping coe\u000ecient and the natural\nfrequency are varying in time. Afterwards, we moved\ntowards e\u000bective material parameters, and for particular\nexample cases, we derived the e\u000bective permittivity of the\ntime-varying medium comprising bound or free electrons.\nIt is observed that this model for describing the e\u000bective\npermittivity is signi\fcantly di\u000berent from the conven-\ntional Drude-Lorentz formula with time-dependent pa-\nrameters.\nACKNOWLEDGMENTS\nThis work was supported by the Academy of Finland\nunder grant 330260. M.S.M. wishes to acknowledge the\nsupport of Ulla Tuominen Foundation. Also, the authors\nthank V. Asadchy and A. Sihvola for their invaluable\ncomments. In addition, M.S.M. thanks X. Wang for help-\ning to prepare the \fgure of the paper.\nAppendix A: Polarizability of a classical bound\nelectron for the stationary scenario\nLet us present the rigorous derivation of Eq. (40). The\n\frst expression in Eq. (39) results in\nd2h(u)\ndu2+ \u0000Ddh(u)\ndu+!2\nnh(u) = 0!h(u) = exp(\u0000\u0000D\n2u)\n\u0002\u0010\nH1cos(r\n!2n\u0000\u00002\nD\n4u) +H2sin(r\n!2n\u0000\u00002\nD\n4u)\u0011\n:\n(A1)\nHere, there are two unknown coe\u000ecients, H1andH2,\nwhich should be determined from the other two remain-\ning expressions in Eq. (39). In principle, the second and\nthe third expressions, as mentioned, are the initial condi-\ntions for determining a speci\fc solution for h(t;\u001c). Fromthe second expression, we deduce that\ndh(u)\nduju=0=e2\nm!\u0000\u0000D\n2H1+r\n!2n\u0000\u00002\nD\n4H2=e2\nm;\n(A2)\nand from the last expression in Eq. (39), we conclude\nthat\nh(0) = 0!H1= 0: (A3)\nFinally, by combining the above results, h(t;\u001c) and the\npolarizability are given by\nh(t;\u001c) =h(t\u0000\u001c) =\ne2\nmq\n!2n\u0000\u00002\nD\n4exp\u0000\n\u0000\u0000D\n2(t\u0000\u001c)\u0001\nsin\u0000r\n!2n\u0000\u00002\nD\n4(t\u0000\u001c)\u0001\n;\n(A4)\nand\n\u000b(\r;t) =h(t;t\u0000\r) =\ne2\nmq\n!2n\u0000\u00002\nD\n4exp\u0000\n\u0000\u0000D\n2\r\u0001\nsin\u0000r\n!2n\u0000\u00002\nD\n4\r\u0001\n;(A5)\nrespectively.\nAppendix B: Inverse polarizability and\ninstantaneous reactive powers\nUtilizing the introduced notion of the temporal com-\nplex polarizability, we can \fnd the instantaneous scat-\ntered power and the total extracted power. Here, we\nremind an important relation for the imaginary part of\nthe inverse polarizability which is known for stationary\ndipoles. In the lossless regime and in the time-averaged\nperspective, by writing that Sstationary =Sstationary\nrad, we\nsimply derive the following expression:\nImh1\n\u000bTi\n=k3\n0\n6\u0019\u000f0; (B1)\nwherek0and\u000f0are the free-space wavenumber and per-\nmittivity, respectively. However, under nonstationary\nconditions, the above equation is not true, and \fnding\na similar relation is not straightforward. This is due to\nthe fact that the conservation of instantaneous power is\nnotsimplyS(t) =Srad(t) even for the stationary dipole.\nThe stored reactive electric and magnetic energy per unit\ntime should be also taken into account. In other words,\nby neglecting Ohmic losses, S(t) =Srad(t) +Sreactive (t),\nwhereSreactive (t) =Selectric (t)+Smagnetic (t). To perceive\nthis expression, as an example, let us consider a nonsta-\ntionary Hertzian dipole loaded with a reactive element\n(e.g., an inductance Lload) changing in time. By using the\ncircuit theory, we can describe the instantaneous reactive\npowers,Selectric (t) andSmagnetic (t), through the e\u000bective14\nparameters of the dipole and the temporally modulated\nreactive load. Those e\u000bective parameters, which play\nan important role in the interaction of the dipole with\nthe incident \feld, are an e\u000bective capacitance Cand an\ne\u000bective inductance L. As it can be expected, the ef-\nfective capacitance Ccorresponds to the stored electric\nenergy near to the dipole, and the stored magnetic energy\naround the dipole is measured by the e\u000bective inductance\nL. Based on this explanation, one can conclude that the\ntotal electric energy per unit time is equal to Selectric (t) =\n(Q(t)=C)i(t) in whichQ(t) is the electric charge and i(t)\nis the induced electric current in the dipole. 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Lozovik3, 1\n1National Research University Higher School of Economics, 109028 Moscow, Russia\n2Institute of Microelectronics Technology and High Purity Materials,\nRussian Academy of Sciences, Chernogolovka 142432, Russia\n3Institute for Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow, Russia\nTwo-component systems consisting of mutually interacting particles can demonstrate both in-\ntracomponent transport effects and intercomponent entrainment (or drag) effects. In the presence\nof superfluidity, the intracomponent transport is characterized by dissipative conductivity and su-\nperfluid weight in the framework of two-fluid model, and intercomponent entrainment gives rise to\nnormal and nondissipative drag effects. We present unified treatment of all these effects for spatially\nhomogeneous two-component atomic Bose-Einstein condensates based on the Bogoliubov theory,\nfocusing specifically on the drag effects. Calculating finite-frequency intra- and intercomponent\nconductivities with taking into account quasiparticle damping, we derive and numerically check an-\nalytical Drude-like approximations applicable at low frequencies, and Lorentz-like approximations\napplicable at higher frequencies in vicinity of the resonant energy of spin-to-density Bogoliubov\nquasiparticle conversion. As possible physical realizations of two-component atomic systems, we\nconsider three-dimensional Bose-Bose mixtures and closely spaced two-layered systems of magnetic\ndipolar atoms.\nI. INTRODUCTION\nUnderstanding of many-body phenomena in ultracold\natomic gases helps to shed light on the properties of con-\ndensed matter systems. For example, studying Bose-\nEinstein condensation (BEC) of ultracold atomic gases\nprovides deeper insight into physics of superconductors,\nsuperfluids, and strongly correlated systems [1–4]. One of\nsuch phenomena, which might occur in semiconducting,\nsuperconducting, and ultracold atomic systems, is drag\neffect, the transport phenomenon which reveals both\nsingle-particle and many-body physics.\nThe Coulomb drag effect in closely spaced two-layer\nsystems, which is caused by frictional entrainment of par-\nticles in one layer in response to a current in the other\nlayer, is extensively studied in solid-state electronic sys-\ntems [5]. Experimentally, this effect is detected by mea-\nsuring nonlocal transresistance between layers. In super-\nfluid or superconducting two-component systems, a non-\ndissipative counterpart of the drag, or Andreev-Bashkin\neffect, can also emerge, when superfluid or superconduct-\ning components of the constituents entrain each other\nwithout dissipation. This effect was predicted for3He-\n4He mixtures [6], superconducting systems [7], ultracold\natomic gases [8], superfluid mixtures of nucleons in the\ncores of neutron stars [9], and for superconducting layers\ninteracting with polaritons [10].\nBoth Coulomb and Andreev-Bashkin drag effects are\nconventionally studied in the DC regime (at ω= 0).\nRecently there appeared an interest in studying the AC\n(ω >0) drag effect [11]: an alternating force at nonzero\n∗afaminov@hse.ru\n†asokolik@hse.ru\nL(b) (a)z\njbja Fa\njbja FaFIG. 1. Schematic depiction of systems considered. (a) 3D\natomic mixture. Alternating force Faimposed on the con-\nstituent agives rise to both intracomponent ja∼σaaFaand\nintercomponent jb∼σabFaresponse currents. (b) Dipolar\natomic quasi-2D system. Atoms in two pancake-like Bose-\ncondensed clouds have their dipole moments aligned with the\nzaxis, and the drag effects are induced by long-range inter-\naction across the interlayer distance L.\nfrequency acts upon one component, and the alternat-\ning current of the other component is detected. This\neffect is described by the conductivity matrix σij(ω) re-\nsolved over the components i, j=a, b. The key feature\nof AC drag effects in a superfluid system is interplay of\ndissipative and non-dissipative current responses, when\nthe Coulomb drag and Andreev-Bashkin effects in their\npure DC form can be extracted from analysis of the low-\nfrequency limit of AC drag conductivity σab(ω).\nIn this paper we calculate, using many-body theory,\nthe AC mass conductivities σij(ω) of a two-component\natomic BEC at nonzero temperature, considered as a\nhomogeneous 3D mixture [see Fig. 1(a)]. We analyze\nboth intracomponent conductivities (or intraconductivi-\nties)σaa,σbb, which characterize normal and superfluid\nresponses of each component, and intercomponent con-\nductivity (or transconductivity )σab, which is responsi-arXiv:2402.11606v1  [cond-mat.quant-gas]  18 Feb 20242\nble for normal and superfluid drag effects. In contrast\nto Ref. [11], we consider generally non-symmetric two-\ncomponent system with different masses and densities of\nconstituents, and assume nonzero damping γof the Bo-\ngoliubov excitations, which can arise from interaction of\natoms with the external disordered potentials, conden-\nsate inhomogeneities, and non-condensed particles [12–\n14].\nBesides three-dimensional mixtures, we study spatially\nseparated magnetic dipolar atomic gases [see Fig. 1(b)],\nwhere the interlayer drag can appear due to long-range\ndipole-dipole interaction. Such systems are gaining pop-\nularity nowadays: BEC of dipolar atomic gases was re-\nalised in recent experiments [15], and mutual friction (i.e.\nnormal drag effect) in a non-condensed phase was de-\ntected in the two-layered geometry [16].\nWe show that at large enough temperatures the dissi-\npative and non-dissipative response currents may be of\nthe same order, leading to non-trivial phase shifts (be-\nsides 0 and π/2) between currents and driving forces.\nAt low frequencies both intra- and transconductivities\nσij(ω) are well approximated analytically by a kind of\ntwo-fluid Drude model [17, 18] with a mixture of nondis-\nsipative response (giving rise to superfluidity of each com-\nponent and Andreev-Bashkin effect between the com-\nponents) and dissipative response caused by quasipar-\nticle decay (which gives rise to a normal conductivity\nof each component and normal drag between the compo-\nnents). At higher frequencies of the order of atomic chem-\nical potentials, σij(ω) in certain conditions can reveal\nthe Lorentz-type resonance originating from interconver-\nsion between spin and density Bogoliubov quasiparti-\ncles, which is also analytically approximated. For dipolar\natoms with interlayer interaction, we predict similar be-\nhavior of transconductivity, although the resonance fre-\nquency may be tuned by changing the interlayer distance.\nThe paper is structured as follows. In Sec. II the\noutline of the theory is presented, providing the general\nexpressions for conductivity calculations and parameters\nof the atomic systems we consider. Then in Sec. III we\nderive analytic approximations for AC conductivities in\nthe Drude (low-frequency) and Lorentz (high-frequency)\nregimes, followed by Sec. IV, where the results of numer-\nical calculations are presented and compared with the\nanalytical approximations. Sec. V concludes the paper\nwith discussion. Appendices A, B, C, D present details\nof calculations.\nII. THEORY\nA. Intra- and transconductivities\nSuperfluid, dissipative and drag transport effects in a\ntwo-component system are characterized by AC intracon-\nductivities σaa(ω),σbb(ω), and transconductivity σab(ω),\nwhich relate the force Fje−iωtimposed on the component\njto the current density (or flux density of particles) in-duced in the component i:\nji(t) =σij(ω)Fje−iωt. (1)\nSuch conductivities have dimensionality of ℏ−1cm−1\n(ℏ−1) for 3D (2D) system. In experiments on ultra-\ncold atomic gases, they can be determined by measuring\nvelocities and coordinates of atoms using time-of-flight\nexpansion imaging [19] or temperature change due to\ndissipation-induced heating [20].\nIn the linear response theory, the AC conductivities\ncan be related to the retarded correlation functions of\ncurrents [11, 21]\nσij(ω) =i\nω\u0014δijni\nmi+ lim\nq→0χT\nij(q, ω)\u0015\n. (2)\nHere the first term is diamagnetic response present only\nin the intracomponent channel i=j, with niandmi\nbeing the atomic density and mass of the ith compo-\nnent; χT\nijis transverse part of the retarded paramagnetic\ncurrent response tensor. In Matsubara representation at\nnonzero temperature T, this tensor is given by\nχνη\nij(q, iω) =−1\nA1/TZ\n0dτ eiωτ\nTτjν\ni(q, τ)jη\nj(−q,0)\u000b\n,(3)\nwhere jν\ni(q, τ) is the Heisenberg-evolved (in imaginary\ntime) operator of the qth spatial harmonic of the ith\ncomponent current density along the axis ν,Ais the\nsystem volume (area) in the case of 3D (2D) geometry,\nand hereafter ℏ= 1 is assumed in the formulas. The\nretarded correlation function of currents entering Eq. (2)\ncan be obtained from the Matsubara one (3) by taking\ntransverse tensorial part over ν,η, and performing an-\nalytic continuation iω→ω+i0 from the upper half of\nthe complex plane. Since we are interested in response\nof currents on a homogeneous force Fi, we take q= 0 in\nEq. (3). Note, however, that the ω→0 limit has to be\ntaken carefully in DC conductivity calculations. It can be\nshown [22], that in order to correctly calculate the super-\nfluid drag density [23] in a system without quasiparticle\ndamping, the DC limit ω→0 has to be taken before the\nq→0 limit, although in the presence of damping these\ntwo limits commute, which will be used below.\nThe induced current can be divided into the in-phase\n(with respect to the driving force) part, which is responsi-\nble for dissipation, and π/2 phase-delayed part, which is\nnon-dissipative. The latter can be additionally divided\ninto the diamagnetic and paramagnetic contributions.\nBy this reason, imaginary part of each conductivity\nσs\nij(ω)≡Imσij(ω) =δijni\nmiω+σsp\nij(ω), (4)\nwhich consists of dia- ( δijni/mi) and paramagnetic ( σsp\nij)\nparts, will be referred to as superfluid conductivity, and\nthe real part\nσn\nij(ω)≡Reσij(ω) (5)3\nwill be referred to as normal conductivity. Note that the\ndistinction between superfluid and normal responses is\nstrictly defined only in the DC limit ω= 0 [17, 18, 24],\nwhere the 1 /ωsingularity of σs\nijindicates superfluidity\nin both inter- and intracomponent channels. In particu-\nlar, the theory of DC superfluid drag [23] deals with the\nsuperfluid drag mass density\nρdr=mamblim\nω→0ωσs\nab(ω), (6)\nand the intracomponent superfluid mass density is re-\nlated to the low-frequency divergence of the intraconduc-\ntivity\nρs\ni=m2\nilim\nω→0ωσs\nii(ω). (7)\nIn contrast, σn\nij(ω) tends to a constant in the limit ω→0,\nand its intracomponent part σn\nii(0) provides dissipative\nDC conductivity, while the intercomponent part σn\nab(0)\nis related to the DC drag coefficient, or transresistiv-\nityσn\nab(0)/[σn\naa(0)σn\nbb(0)−σn\nab(0)2], which is usually mea-\nsured in drag experiments [5].\nAtω >0, the strict distinction between superfluid and\nnormal responses becomes elusive because even in normal\nsystems both σn\nijandσs\nijare finite and nonzero. Detect-\ning the dissipative response of a normal current against\nthe superfluid background is harder task for Bose systems\nthan for conventional s-wave superconductors, where the\ndissipative part of conductivity is suppressed at ω <2∆\n[25, 26]. In contrast, Bose-condensed systems lack gap in\nthe quasiparticle spectrum, so the normal conductivity is\ngenerally nonzero at any ω >0 [26]. This is why analy-\nsis of AC conductivities σij(ω) provides unified and more\ndetailed information about both the superfluid entraint-\nment (Andreev-Bashkin) effect and normal drag effect, as\nwell as about normal (dissipative) and superfluid (nondis-\nsipative) responses of each component, than conventional\nDC calculations commonly accepted in the theories of\ndrag and superfluidity.\nAs specific examples of 3D mixtures [Fig. 1(a)], we\nconsider spinor atomic BECs: the symmetric mixtures\nof87Rb and23Na in atomic states F= 1, mF=±1\n[27], and the non-symmetric mixture of39K in the states\nF= 1, mF= 1 and F= 1, mF= 0 [28]; here mFare\nmagnetic sublevels of hyperfine state with total angular\nmomentum F. Besides, we consider the mixture of atoms\nwith different masses,174Yb-133Cs [29]. As the spatially\nseparated quasi-2D dipolar system [Fig. 1(b)], we con-\nsider pairs of parallel clouds of either52Cr or168Er atoms\nwith the long-range magnetic dipole interaction [30].\nThe realistic parameters used in numerical calculations\nare listed in the Table I. Each intra- or intercomponent\ninteraction constant gij= 2πas\nij(1/mi+ 1/mj) is related\nto the s-wave scattering length as\nij, and we neglect the\nprocesses which permit population transfer between dif-\nferent magnetic sublevels mFof the state with total angu-\nlar momentum F. In the case of spatially separated dipo-\nlar atomic clouds, we take into account both s-wave scat-\ntering and dipole-dipole interaction within each cloud,Non-dipolar atoms\nParameter87Rb23Na39K174Yb-133Cs\nas\nii(a0) 100 55 30, 100 105, 150\nas\nab(a0) 95 51 −50 −75\nTc(nK) 170 1000 150 460, 200P\niµi/2π(kHz) 2.7 10.8 1 12.7\nDipolar atoms\nParameter52Cr168Er\ndi(µB) 6 7\nas\nii(a0) 103 137\nTc(nK) 700 410\nTABLE I. Upper table: Parameters for 3D spinor mixtures,\nnamely intra- as\niiand intercomponent as\nabscattering lengths\nin the units of the Bohr radius a0≈0.529˚A, critical tem-\nperatures Tc, and sums of chemical potentials µiof the com-\nponents at T= 0 (separated by a comma for non-symmetric\nmixtures). Lower table: parameters for dipolar atoms includ-\ning magnetic dipole moments diin Bohr magneton µBunits.\nand only the dipole interaction between atoms from dif-\nferent clouds (see details in Appendix D). In accordance\nwith the recent experiment [16], the thickness of both\nclouds is assumed to be wz= 20 nm, and the distance\nbetween clouds is L= 60 nm.\nIn this paper we consider systems with large con-\ndensate fraction, when the temperature is much lower\nthan the BEC critical temperatures of both constituents,\nT≪Ti\nc, but nonzero, since we are interested in the nor-\nmal drag effect as well. For 3D homogeneous atomic\ngases, the condensate densities n0\niare found with tak-\ning into account their thermal and quantum depletions\nfrom the system of equations n0\ni(T) =ni−nnc\ni(n0\na, n0\nb, T),\ni=a, b, where nnc\niis a density of non-condensed fraction\ngiven by Eq. (A8), niis the total density of the icompo-\nnent, which is assumed to be temperature-independent\nand estimated from the experimental critical tempera-\nture as ni=ζ(3/2)\u0000\nmiTi\nc/2π\u00013/2. The sums of chemical\npotentialsP\niµi=P\nigiin0\nilisted in Table I, which pro-\nvide characteristic energy scales of excitation energies,\nare taken at zero temperature.\nB. Current response function\nIn order to calculate the current response function (3),\nwe use diagrammatic technique to express it in the single-\nloop approximation through the intra- and intercompo-\nnent matrix Green functions ˆGij. Explicit formulas for\nthe Green functions are provided in Appendix A, and\ncalculation details for the current response are given in\nAppendix B.\nIt is known [23, 31], that in a two-component BEC\ntwo types of quasiparticles emerge, which correspond to\ndensity and spin collective modes, with dispersions Ed(q)\nandEs(q), respectively [see Fig. 2(a-b)]. Calculating the4\ntotal current response function χµν\nij, we express it through\nthe response functions S(Eα, Eβ) resolved over the quasi-\nparticle branches α, β= d,s and weighted with uandv\nBogoliubov coefficients. For the transverse part of the\ncurrent response tensor (3), we obtain\nχT\nij(q= 0, iω) =X\npp2\n2Adm imjX\nα1α2s1s2s1s2\n×\u0000\nus1\niα1us2\niα2−u−s1\niα1u−s2\niα2\u0001\u0000\nus1\njα1us2\njα2−u−s1\njα1u−s2\njα2\u0001\n×S(s1Eα1, s2Eα2).(8)\nHere the sums are taken over d-dimensional momentum\np, positive and negative energy indices s1,2=±, and\nquasiparticle branches α1,2= d,s. The response function\nS(Eα, Eβ) =−TX\niωn1\n(iωn−Eα+iω)(iωn−Eβ)(9)\ncorresponds to the loop-diagram constructed from two\nMatsubara Green functions 1 /(iωn−Eα) of Bogoliubov\nquasiparticles, which have infinite lifetime. Since we aim\nto analyze both normal and superfluid drag effects, we\nought to account for their non-zero damping γby re-\nplacing the quasiparticle Green functions 1 /(iωn−Eα)\nwith the broadened onesR\ndx ρ α(x)/(iωn−x), where\nρα(x) = (γ/π)[(x−Eα)2+γ2]−1is the Lorentzian spec-\ntral function. For simplicity of the forthcoming analytical\ncalculations, we assume γto be momentum- and energy-\nindependent and to be the same for both spin and density\nmodes. In this approximation the sum over Matsubara\nfrequencies in Eq. (9) can be taken analytically:\nS(Eα, Eβ) =Z\ndxdx′ρα(x)ρβ(x′)nB(x′)−nB(x)\niω+x′−x,\n(10)\nwhere nB(x) = (ex/T−1)−1is the Bose-Einstein distribu-\ntion function. To approximate this integral, we perform\nTaylor expansion of nB(x) and nB(x′) near the max-\nimax=Eα, x′=Eβof the spectral functions. After\nthat, integration over x, x′and analytical continuation\niω→ω+i0 yield the approximate retarded S-function\nS(Eα, Eβ) =nB(Eα)−nB(Eβ)−iγ[n′\nB(Eα) +n′\nB(Eβ)]\nEα−Eβ−ω−2iγ,\n(11)\nwhich will be used in the following. Numerical verifi-\ncation of this approximation proves its accuracy in the\nconsidered parameter ranges for α̸=β. In contrast, at\nα=βthis approximation lacks quantitative accuracy\nin the Hagen-Rubens regime ω≪γ, although it pro-\nvides qualitatively correct results and becomes exact in\nthe clean DC limit γ= 0, ω→0 (assumed, e.g., in the\nsuperfluid drag calculations in Ref. [23]).\nWe will limit ourselves to the case of relatively weak\ndamping γto maintain applicability of the quasiparticle\ndescription. Similarly to the Mott-Ioffe-Regel bound [32],\nvalidity of quasiparticle description requires the mean\n(c)\n(e)\n(f)\n                       \n                     ωdd\n                       \n                     \n                       \n                     \nωsd\n                       \n                     \n                       \n                     \nωsd\n                       \n                     \n≈ μa+μb\n∝ p2(d)\n                       \n                     \nωs\n                       \n                     s(a)\n(b)Δ\nΔFIG. 2. Left panels: Bogoliubov quasiparticle dispersions\nEd,s(p) in the cases of close (a) and distant (b) atomic masses.\nGreen arrows indicate energy differences Ed(p)−Es(p) at the\nrelevant momenta p∼¯p. Right panels (c-f): excitations of\npairs of the Bogoliubov quasiparticles contributing to conduc-\ntivities.\nfree path l=ci/γof quasiparticles (with their character-\nistic velocities ci=p\nµi/mi) being larger than the mean\ninterparticle distance n1/3\ni. Expressing nithrough the Ti\nc,\nwe obtain restriction for the damping rate γ≪p\nµiTic.\nFortunately this condition allows us to consider the sys-\ntem in ballistic regime and neglect the vertex corrections,\nbecause the ballistic approximation is appropriate when-\never ¯pl > 1 [33], where ¯ pis the characteristic momentum\nof quasiparticles defined in the next section. Roughly es-\ntimating this momentum as ¯ p∼√\nmT(see Appendix C),\nwe obtain ¯ pl∼√µiT/γ. At low enough damping rate\nassumed above, we obtain ¯ plmuch larger than the ratio p\nT/Tic, which is expected to be of the order of unity\nat moderate temperatures T∼1\n3Ti\nctaken in our cal-\nculations. Thus our neglect of the vertex corrections is\nconsistent in the assumed range of parameters.\nIII. ANALYTICAL APPROXIMATIONS\nA. Contributions of quasiparticle branches\nIn this section we develop analytical approximations\nfor frequency-dependent conductivities σn\nij(ω), σs\nij(ω),\nwhich resemble the familiar Drude and Lorentz mod-\nels. Our analysis is applicable to 3D atomic mixtures\nwith short-range interactions in the temperature range\nµi< T≪Tc. Inserting the approximate S-function (11)\ninto Eq. (8) and performing momentum integration, we\nobtain\nσij(ω)≈i\nω\u001aδijni\nmi−D0\nij\n−(−1)δij\nmimj\u0002\nΛ+(ω) + Λ−(ω)\u0003\u001b\n+iD0\nij\nω+ 2iγ.(12)5\nHere the D0\nijterms describe processes where quasiparti-\ncles are scattered from one branch into the same branch\n[density-to-density and spin-to-spin, see Figs. 2(c-d)],\nand the corresponding conductivity weights are defined\nas\nD0\nij=−X\npp2\n2Adm imj[PidPjdn′\nB(Ed) +PisPjsn′\nB(Es)].\n(13)\nThe coefficients Piα, quantifying contribution of the ith\ncomponent to Bogoliubov excitation branch α, are de-\nfined by Eq. (A5). The expression (13) is the counterpart\nof conventional Landau formula for density of the normal\ncomponent [34, 35] generalized for a two-component su-\nperfluid system.\nTwo other terms Λ±(ω) depend on frequency and can-\nnot be calculated analytically, so we derive approxima-\ntions for them. The function Λ+(ω) is responsible for the\nprocesses of quasiparticle scattering with interconversion\nfrom one branch to the distinct one [spin-to-density and\nvice versa, see Fig. 2(e)]. The second function Λ−(ω)\ncorresponds to creation or annihilation of two quasipar-\nticles of different branches [Fig. 2(f)]; note that similar\nsame-branch processes are forbidden at q= 0. These\nfunctions can be written as momentum integrals\nΛ±(ω) =∞Z\n0dp\u0002\nf±(p)R±(ω, p) +f∗\n±(p)R∗\n±(−ω, p)\u0003\n,\n(14)\nwhere f±(p) are defined in Appendix B and will be called\nenvelope functions , while the resonant functions are de-\nfined as\nR±(ω, p) =1\nEd(p)∓Es(p)−ω−2iγ. (15)\nThe envelope functions f±(p) endure power-law increase\nat low momenta and decrease exponentially at Ed,s≳T\nthanks to the Bose-Einstein distribution functions, so\nthey have extrema at some momentum ¯ pwhere the quasi-\nparticle energies match the temperature. Therefore it\nis convenient to define the characteristic momentum ¯ p,\nwhose neighbourhood provides the major contribution to\nthe integral, as solution of equation Ed(¯p) +Es(¯p) = 2 T\n(see more detailed discussion in Appendix C). Thus the\ncharacteristic sum of quasiparticle energies entering R−\nis of the order of T. The characteristic difference of en-\nergies entering R+is the important energy parameter\n∆ =Ed(¯p)−Es(¯p), (16)\nwhich has a meaning of resonance frequency for quasi-\nparticle inter-conversion processes [Fig. 2(e)] giving rise\nto the Lorentz-type response at moderately high ω. De-\npending on relationship between ωand ∆, we can sepa-\nrate the Drude and Lorentz regimes.\nγ <<T, μi\n10\n1\n0.1Lorentz \nregime\nω  Δ << T\nωγ\nμa+μbDrude  \nregime\nω << Δ,T ~\n                       \n                     \n                       \n                     κF(p)\np\nF(p)\npF(p)\n|f + (p)|\n|F(p)||R+(ω,p)|F(p)\np(a)\n(c) (d)Strong -damping \ncase,  κ>1∞\np(b)FIG. 3. Schematic depiction of different regimes on the\nω,γplane: the Drude regime at low frequencies and the\nLorentz regime at higher frequencies. According to the val-\nues of κshown by color, we separate the Lorentz regime\ninto weak- ( κ < 1) and strong-damping ( κ > 1) cases. In-\nsets (a-d) show how f+(p),R+(p), and the total integrand\nF(p) =f±(p)R±(ω, p) +f∗\n±(p)R∗\n±(−ω, p) in Eq. (14) behave,\nby absolute value, as functions of p.\nB. Drude regime\nThe Drude regime occurs when ωis far lower than the\nresonance frequencies Ed±Esin denominators of R±.\nAccording to the estimates above, it corresponds to the\nfrequency range ω≪∆, Tshown by green shading in\nFig. 3. In this limit we assume R±(ω, p)≈R±(0, p) in\nthe integrals (14), so the functions Λ±(ω) become almost\nfrequency-independent, and we obtain the simple expres-\nsion for conductivities in the Drude regime:\nσij(ω)≈i\nω\u001aδijni\nmi−D0\nij+D+\nij+D−\nij\u001b\n+iD0\nij\nω+ 2iγ.(17)\nHere\nD±\nij=−(−1)δij\nmimjΛ±(0)\n=∓(−1)δij\nmimj∞Z\n0dpRe2f±(p)\nEd∓Es−2iγ. (18)\nNote that the terms D±\nij, being almost reals, contribute\nmainly to the nondissipative part of the conductivities\n(17), because low frequencies ωare far off-resonant from\nthe absorption processes, corresponding to these terms\nand depicted in Fig. 2(e-f). Totally, the diamagnetic\nni/miωand paramagnetic −D0\nij+D+\nij+D−\nijterms in the6\nFIG. 4. Temperature dependencies of the Drude (left panel)\nand superfluid (right panel) weights for Yb-Cs mixture at\nγ/2π= 1 Hz. Solid lines correspond to the intracomponent,\nand black dashed line to the intercomponent conductivities.\nThin red line shows the approximation Ds\nab≈πρdr/mamb,\nwhere ρdris the drag density calculated at γ= 0 [23].\nbraces of Eq. (17) do not cancel each other in the Bose-\ncondensed regime giving rise to the uncompensated i/ω\nsingularity of both intra- and transconductivities in the\nDC limit ω→0. The superfluid weights corresponding to\nsuch singularities are Ds\nij=π(δijni/mi−D0\nij+D+\nij+D−\nij),\nand the Drude weights, corresponding to the dissipative\nresponse and defined as 2R∞\n0dωReσij(ω) [22], equal to\nDn\nij=πD0\nij.\nThe example of temperature dependencies of Drude\nand superfluid weights is shown in Fig. 4 for the mass-\nimbalanced Yb-Cs mixture. As expected, Drude weights\nvanish at T= 0, when the mixture is fully in superfluid\nstate, and superfluid weights involving Cs subsystem van-\nish at T=TCs\ncwhen it becomes normal. We also notice\nthat our results at low γare in agreement with the theory\nof DC superfluid drag developed for clean systems (red\nthin line) by Fil and Shevchenko [23]. The recession of\nthe intercomponent Drude weight Dn\nabdown to zero near\nT=TCs\ncis the artefact of our one-loop approximation,\nwhich neglects more complicated diagrams contributing\nto drag in the normal state [33]. However, they can be\nneglected at low enough temperatures T≲Ti\nc[36].\nC. Lorentz regime\nThe Lorentz regime occurs when ωis close to the res-\nonant energy ∆ of the spin-to-density quasiparticle con-\nversion. In this regime only R+demonstrates a reso-\nnance behavior and becomes dominant, and the other\nfunction R−can be neglected, because Ed+Es≫Ed−Es\nat typical momentum ¯ p. Also we may notice that the\nf+(p)R+(ω, p) term in the integral (14) is dominant over\nthe off-resonant term f∗\n+(p)R∗\n+(−ω, p).\nWe subdivide the Lorentz regime into weak- and\nstrong-damping cases, depending on the dimensionless\nparameter κ= ∆p/¯p, defined as the related to ¯ pmo-\nmentum width ∆ p≈4γ/[E′\nd(p+)−E′\ns(p+)] of the reso-\nnant function R+(ω, p) around its maximum at p=p+,which characterizes both the maximum and the typi-\ncal width of the envelope function f+(p). As shown in\nFig. 3, the weak-damping case κ≪1 [Fig. 3(d)] means\nthatR+(ω, p) is very narrow along the momentum axis,\nin comparison with f+(p). In the strong-damping case\nκ≫1, as depicted in Figs. 3(a-c), the situation is oppo-\nsite. We assign the frequency region ω > µ a+µb, where\nR+is never resonant and monotonously increases (so ∆ p\nis undefined), to the strong-damping case as well, setting\nformally κ=∞in this region.\n1. Weak-damping case\nIn the weak-damping case, when R±(ω, p) is very nar-\nrow, we can bring γin its denominator to zero and find\nΛ+(ω) analytically by integration of the resulting Dirac\ndelta function in Eq. (14) to obtain\nRe Λ+(ω)≈2∞Z\n0dp f +(p)Ed(p)−Es(p)\n[Ed(p)−Es(p)]2−ω2,(19)\nIm Λ+(ω)≈ −πf+(p+)\nE′\nd(p+)−E′s(p+), (20)\nwhere p+is solution of equation Ed−Es=ωdependent\nonω, i.e. the momentum where |R+(p, ω)|attains sharp\nmaximum; we also assume γ= 0 in the expression (B6)\nforf+(p). The result (20) for the function Im Λ+(ω), re-\nlated to the dissipation spectrum Re σij(ω), may be inter-\npreted as the sharp resonant function R+(ω, p) scanning\nthe broad envelope function f+(p) when ωis changed.\nThe derivative dp+/dω = [E′\nd(p+)−E′\ns(p+)]−1deter-\nmines the scanning speed along the paxis and hence\nmagnitude of Im Λ+(ω). The resonance maximum of\nReσij(ω) is located at p+= ¯p, where the maxima of\ntwo functions R+(ω, p) and f+(p) coincide. Shape of this\nresonance depends on the ratio of atomic masses ma,mb\nof two components. As discussed in more detail in Ap-\npendix C, we outline two cases: when atomic masses are\nclose to each other (and, in particular, equal in the case of\nspin mixtures), and when they are distant. These cases\nare distinguished by how the energy difference Ed−Es\ndepends on pin the relevant range of momenta p∼¯p.\nIn case of close masses this difference is almost constant\n[Fig. 2(a)], Ed−Es≈µa+µb, thus the scanning speed\ndp+/dωis high, and the resonance is sharp (see Fig. 5\nin the next section). In the case of distant masses the\nenergy difference retains an essential momentum depen-\ndence, Ed−Es∝p2[Fig. 2(b)], so dp+/dωis low, and\nthe resonance becomes strongly smeared or vanishes com-\npletely (see results for Yb-Cs mixture in Fig. 6 below).\n2. Strong-damping case\nIn the strong-damping case, the resonant function\nR+(ω, p) is much wider than the envelope f+(p) [κ≫1,7\nsee Figs. 3(a-c)], so we approximate R+(ω, p) byR+(ω,¯p)\nto obtain\nΛ+(ω)≈\u00121\n∆−ω−2iγ+1\n∆ +ω+ 2iγ\u0013∞Z\n0dp f +(p).\n(21)\nHere we neglected the γn′\nBterms in the f+(p) function\n(B6), since their order is γ/T≪1 (however in the Drude\nregime these terms should be retained, see Appendix B).\nThe conductivity in this case is predominantly deter-\nmined by the first term in the parentheses of Eq. (21)\nwhich is resonant near ω= ∆ with the width 2 γ. The\nrole of the second non-resonant term is to red-shift and\nbroaden this resonance.\nIV. NUMERICAL CALCULATIONS\nIn this section we present numerical results for the con-\nductivities for various systems and compare them with\nanalytical approximations. The numerically calculated\nconductivities are found using Eqs. (2), (8) with the ap-\nproximation (11) for the S-function, which proves to be\nquite accurate in the considered range of parameters.\nThe analytical approximations are given by Eq. (17) in\nthe Drude regime and Eq. (12) in the Lorentz regime,\nwith Λ−= 0 and Λ+given by Eqs. (19)–(20) in the\nweak-damping case ( κ <1) and by Eq. (21) in the strong-\ndamping case ( κ >1). For each atomic mixture, we take\nthe temperature T=1\n3min[Ta\nc, Tb\nc], which is low enough\nfor the Bogoliubov approximation to be applicable yet\nstill experimentally feasible.\nIn Fig. 5 we show the trans- and intraconductivities for\nthe symmetric Rb-Rb mixture at weak [Fig. 5(a,c)] and\nstrong [Fig. 5(b,d)] damping. At low frequencies ω≪γ,\nthe superfluid conductivities σs\nijare positive and diverge\nas 1/ω(in the intracomponent channel i=jthe pos-\nitive diamagnetic part na/maωdominates the negative\nparamagnetic part σsp\naa). It is a signature of nonzero and\npositive drag (6) and superfluid (7) densities. The nor-\nmal conductivities σn\nijtend to constants in DC limit in\nconformity with traditional normal Coulomb drag effect\nand Drude theory of conductivity, although in the weak-\ndamping case [Fig. 5(a,c)] their levelling off at ω→0\nis not visible at the chosen scale, because the Drude\npeaks are much higher than the Lorentz-regime features\nwhich we are concentrating on. At higher frequencies\nnear ω= ∆ (which is ∆ ≈µa+µbwhen ma=mb),\nabsolute value of the normal conductivity σn\nijexhibits\nabsorption peak, while the superfluid transconductivity\nσs\naband paramagnetic part σsp\naain the intracomponent\nchannel change sign. Such features resemble resonant\nbehaviour of the Lorentz model [Fig. 5(c,d)], although in\nthe intracomponent channel this resonant-like behavior\nof the paramagnetic superfluid conductivity (4) is masked\nby the large and monotonously decreasing diamagnetic\nterm.\nγ/2π = 1 Hz γ/2π = 300 Hz\nγ/2π = 300 Hz γ/2π = 1 HzRb-Rb(a) (b)\n(c) (d)FIG. 5. Trans- (a,b) and intercondictivity (c,d) for Rb-Rb\nspinor mixture at weak (left panels) and strong (right panels)\ndamping γ. Solid and dashed lines show numerical calcula-\ntions, and symbols show analytical approximations for ap-\npropriate regimes depicted by the same color shadings as in\nFig. 3: Drude regime (squares, green), weak-damping Lorentz\nregime (circles, blue), and strong-damping Lorentz regime\n(triangles, red). Vertical dashed lines indicate the frequency\nω=µa+µb, which is close to the resonance frequency ∆. In\nthe intracomponent channel (c,d) the superfluid conductivity\nσs\naais separated into dia- ( na/maω) and paramagnetic ( σsp\naa)\nparts.\nAt frequencies near the resonance ∆, the conduc-\ntivities are related to each other via m2\niσsp,n\nii(ω)≈\n−mambσs,n\nab(ω). This is evident in Fig. 5 where σsp\naa(ω)≈\n−σs\nab(ω) and σn\naa(ω)≈ −σn\nab(ω) near the resonance, since\nma=mb. This feature follows from Eq. (12) if we omit\nthe terms iD0\nij/(ω+ 2iγ) and −iD0\nij/ω, whose contribu-\ntion is diminished at large frequencies.\nTo get more insight into behavior of transconductivi-\nties, in Fig. 6 we show them for equal-mass K-K, Na-Na\nmixtures and for mass-imbalanced mixture Yb-Cs. It can\nbe seen that the transconductivity of the non-symmetric\nspin mixture K-K with relatively low resonance energy ∆\nexhibits the same features as for Rb-Rb mixture: Drude\npeak at low frequencies and resonance at ω≈∆. In the\ncase of symmetric spin mixture Na-Na, the resonance\nfrequency ∆ is higher (more than 10 kHz) and presum-\nably out of reach of present experiments capabilities. For\nthe mass-imbalanced mixture Yb-Cs, the resonance fre-\nquency ∆ turns out to be much lower than µa+µb, but\nthe resonance itself is degraded in both weak- and strong-\ndamping cases by the reasons discussed in Sec. III C 1.\nIn Fig. 7 we present numerically calculated transcon-\nductivities for the pairs of dipolar atomic gases Er-Er and\nCr-Cr arranged into quasi-2D two-layered systems [see\nFig. 1(b)]. The analytical approximations are not ap-8\nYb-Csγ/2π= 1 Hz\nγ/2π= 1 HzK-K\nNa-Naγ\n/2π\u0003= 600\u0003Hz\nγ/2π= 600 Hz\nγ/2π= 1 Hz γ/2π= 100 Hz\nFIG. 6. Transconductivities of K-K, Na-Na, and Yb-Cs mix-\ntures, each calculated at two values of γ, where the weak- or\nstrong-damping cases develop in the Lorentz regime. Desig-\nnations of curves and symbols are the same as in Fig. 5.\nplied in this case due to different form of intercomponent\ndipole-dipole interaction which retains essential momen-\ntum dependence, as discussed in Appendix D. In contrast\nto 3D mixtures with short-range interactions, here we can\ntune the resonance frequency ∆ by varying the interlayer\ndistance L. This frequency, found as the maximum of the\nquasiparticle energy difference ∆ = max[ Ed(p)−Es(p)],\napproximately follows the ∆ ∝L−1trend, as shown in\nEr-Er\nγ/2π = 10 Hz\nCr-Cr\nγ/2π = 1 Hz γ/2π = 100 Hz\nγ/2π = 300 Hz\nFIG. 7. Transconductivity between spatially separated clouds\nof dipolar atoms Er-Er and Cr-Cr in the systems at weak (left\npanels) and strong (right panels) damping with L= 60 nm.\nVertical dashed lines indicate the resonance frequency ω= ∆.\nInsets show dependence of ∆ on interlayer distance L.\nthe insets in Fig. 7. The intraconductivities in this case\nare not shown, since the relation between total niand\ncondensate n0\nidensities, needed to describe partial com-\npensation of the diamagnetic term with quantitative ac-\ncuracy, is not well-defined in 2D systems in the frame-\nwork of Bogoliubov theory, and more complicated ap-\nproaches, such as quasicondensate analysis [37], should\nbe applied, which is beyond the scope of our paper.\nV. DISCUSSION\nIn this paper, we studied intra- and transconductivi-\ntiesσij(ω) of a homogeneous two-component superfluid\nBose-condensed systems at nonzero frequencies ω. We\ncalculated the conductivities in one-loop approximation\nusing the Bogoliubov theory of a two-component BEC at\nfinite temperature and with taking into account the phe-\nnomenological damping γof spin and density quasipar-\nticle modes. Two possible setups of the two-component\natomic system are considered: 3D spinor atomic mix-\ntures (Rb-Rb, K-K, Na-Na, Yb-Cs) and spatially sep-\narated two-layered systems with magnetic dipole-dipole\ninteractions (Er-Er and Cr-Cr).\nWe separate each conductivity σij(ω) into the real part\nσn\nij(ω), which is responsible for dissipative response (cur-\nrent in phase with a driving force), and imaginary part\nσs\nij(ω), which corresponds to non-dissipative response\nwith the π/2 phase delay. Our analysis shows that, at\nfrequencies much lower than the characteristic energy\ngap ∆ between spin and density quasiparticle modes, the\nconductivities are well described by the two-fluid Drude9\nmodel [17, 18] where σn\nij(ω) exhibits the Drude peak\n∝[ω2+ 4γ2]−1like the normal metallic conductivity (in\nthe intracomponent channel i=j) or normal drag ef-\nfect (in the intercomponent channel i̸=j), while σs\nij(ω)\ndemonstrates the 1 /ωsingularity indicating superfluid-\nity (at i=j) or superfluid drag effect (at i̸=j). Thus\nour theory describes dissipative conductivity, superfluid-\nity, as well as normal and superfluid drag effects on equal\nfooting.\nAt higher frequencies near ω∼∆ the dissipative\npart of conductivity σn\nij(ω) exhibits peak, while non-\ndissipative part σsp\nij(ω) changes sign, which is qualita-\ntively similar to the Lorentz model of resonant response.\nHowever in our case the resonance shape is asymmetric\nand can essentially differ depending on the damping rate\nγand whether the atomic masses in a mixture are close\nto each other or distant enough. In a symmetric mixture\nwith equal masses, ∆ is close to the sum µa+µbof atomic\nchemical potentials, and the general case is considered in\nAppendix C. For two-layered quasi-2D system of dipolar\natoms, ∆ can be tuned by varying the interlayer separa-\ntionL. For 3D mixtures [Fig.1(a)], we derive the ana-\nlytical formulas which approximate the conductivities in\nboth Drude and Lorentz regimes rather accurately.\nThe drag effects predicted in our paper can be ob-\nserved in experiments with two-component or two-\nlayered atomic BECs by detecting currents arising in re-\nsponse to an alternating force, which selectively drives\none of the components (or drives them in opposite di-\nrections). The currents can be determined by measur-\ning atomic velocities via atomic cloud imaging after trap\nrelease or by time-of-flight measurements. The driving\nforce can be imposed by magnetic field gradients [38], op-\ntical lattices [39], magnetic trap shaking [20], or sudden\ndisplacement of optical trap [16]. Such methods can pro-\nvide oscillation frequencies up to several kHz, and achiev-\nable frequency ranges are often dictated by properties of\nthe atoms themselves [26].\nLet us estimate a magnetic field gradient required to in-\nduce strong enough oscillations, which could be observed\nby standard atomic cloud imaging. Consider the Yb-\nCs atomic mixture [29], where174Yb lacks magnetic mo-\nment, so its Lande factor is zero ( gF= 0), and thus\nonly133Cs is affected by magnetic field ( gF=−0.25\n[40]). In a homogeneous system the field gradient, re-\nquired to induce oscillations of the133Cs atomic cloud\nwith the amplitude xCsand frequency ω, is|∇B|=\nmCsω2xCs/mFgFµB, where mCsis the mass of133Cs\natom, mFis the magnetic sublevel of hyperfine state\nwith angular momentum F. Assuming the detectable\namplitude xCs∼10µm and using parameters from [29],\nwe obtain |∇B|(G/cm)≈[(ω/2π)(Hz)]2×10−4. The\ngradients up to 3000 G/cm used in experiments [41] are\nsufficient to create oscillations in both Drude ( ω/2π∼\n100 Hz, |∇B| ∼1 G/cm) and Lorentz ( ω/2π∼5000 Hz,\n|∇B| ∼2500 G/cm) regimes. The presence of harmonic\ntrap alters relationship between xCsand∇B, and we may\nhope to use the mechanical resonance effects to enhancethe oscillation amplitude even more.\nFor reliable detection of the drag effects, we need to\nachieve large enough amplitude xYbof oscillating motion\nof the174Yb atomic cloud in response to the magnetic\nfield gradient force applied to133Cs atoms. The ratio of\noscillation amplitudes can be estimated as xYb/xCs=\njYbnCs/jCsnYb= (nCs/nYb)× |σab(ω)/σaa(ω)|. At\nplausibly low damping rate γ/2π= 1 Hz, we obtain\nxYb/xCs∼0.01 at ω/2π= 100 Hz and xYb/xCs∼0.05\natω/2π= 5000 Hz. Such ratios are not restrictingly\nsmall, so we may hope to detect oscillations of the pas-\nsive174Yb component at high enough oscillating force\nand large enough oscillation amplitudes xCsof the active\ncomponent. For Rb-Rb mixture (see Fig. 5) this ratio\nis generally larger: xa/xb∼0.06 at low frequencies and\nxa/xb∼0.4 at the Lorentz-like resonance.\nPlane-parallel systems of magnetic dipole atoms [15,\n16] possess several additional controllable parameters:\nthickness of the clouds wz, intercloud separation L, and\ndipole moments orientation. The theory presented in our\npaper allows us to calculate the transconductivity be-\ntween dipolar atomic clouds in the setup of Ref. [16].\nHowever, direct comparison of our calculations with re-\nsults of this experiment is hindered because atomic gases\nin Ref. [16] were not Bose-condensed, and harmonic traps\nused to hold them made the atomic clouds inhomoge-\nneous and prone to mean-field repulsion not described by\nour theory. It is of interest to extend our approach to take\ninto account the normal-state drag diagrams [5, 33, 36]\nwhich would allow to describe the AC drag in wide tem-\nperature range both below and above Tc.\nAn alternative way to infer information about trans-\nand intraconductivities can rely on measuring tempera-\nture changes after several oscillations [20]. Mutual en-\ntrainment of two components can also affect dispersions\nand damping rates of first and second sounds in the two-\ncomponent BECs, which can be detected in sound veloc-\nity measurements [42]. Our approach of conductivity cal-\nculations is aimed on homogeneous systems correspond-\ning to flat traps [42–44]. In harmonic traps the resonance\nin center-of-mass motion of atomic clouds alters the be-\nhaviour of conductivity [20], and the mean-field repulsion\neffects mimicking intrinsic interlayer conductivity can ap-\npear [29, 45–47], so the problem of mutual entrainment\nbecomes more complicated.\nTo conclude, the theory of conductivities of Bose-\ncondensed two-component systems developed in this\npaper unifies calculations of the normal drag effect,\nAndreev-Bashkin effect, as well as intracomponent DC\nconductivity and superfluid density. Investigation of fre-\nquency dependencies of the conductivity tensor allows\nus to study interplay of dissipative and nondissipative\ncurrent responses. Our approach can be generalized for\nspin conductivity calculations [11, 26] and for coupled 1D\natomic gases [4]. Besides, similar AC entrainment effects,\nboth dissipative and nondissipative, can be expected in\nFermi-atom and condensed-matter superconducting sys-\ntems.10\nACKNOWLEDGMENTS\nThe work on analytical calculation and approximation\nof conductivities was done as a part of research Project\nNo. FFUU-2024-0003 of the Institute for Spectroscopy of\nthe Russian Academy of Sciences. The work on numeri-\ncal calculations was supported by the Program of Basic\nResearch of the Higher School of Economics.\nAppendix A: Green functions\nThe Hamiltonian of homogeneous two-component\natomic system is\nˆH=X\nipϵipa†\nipaip\n+1\n2AX\nijpp′qVij(q)a†\ni,p+qa†\nj,p′−qajp′aip, (A1)\nwhere aipis the destruction operator of the atomic par-\nticle of the component i=a, bwith momentum p,\nϵip=p2/2miis atomic dispersion, and Vij(q) is the\nFourier transform of the interaction between particles i\nandj. Replacing each zero-momentum operator ai,p=0\nby square root of the number of condensate particles\n(An0\ni)1/2, we obtain the mean-field Bogoliubov Hamil-\ntonian, which can be further diagonalized by the trans-\nformation\naip=u+\nidBdp+u−\nidB†\nd,−p+u+\nisBsp+u−\nisB†\ns,−p(A2)\ninto usual formP\np(EdB†\ndpBdp+EsB†\nspBsp), where Bdp,\nBspare destruction operators of density and spin quasi-\nparticles. Their energies read\nE2\nd,s=E2\na+E2\nb\n2±s\u0012E2a−E2\nb\n2\u00132\n+ 4ϵaϵbn0an0\nb|Vab|2,\n(A3)\nandEiis the energy of Bogoliubov excitation of isolated\nith component: Ei=p\nϵi(ϵi+ 2n0\niVii). The Bogoliubov\ntransformation coefficients are\nuζ\naα=ϵa+ζEα\n2√ϵaEαp\nPaα, uζ\nbα=±ϵb+ζEα\n2√ϵbEαp\nPbα,\n(A4)\nwhere\nPiα=±4n0\nan0\nb|Vab|2\n(E2\nd−E2s)(E2α−E2\ni)(A5)\nis the positive weight fraction of the ith component in\ntheαth quasiparticle mode. The upper and lower signs\nin Eqs. (A4)–(A5) correspond, respectively, to the den-\nsity ( α= d) and spin ( α= s) modes. To approximate the\ninteraction potentials Vij(q) in the case of 3D mixtures,\nwe assume the momentum-independent contact interac-\ntions gijrelated to s-wave scattering lengths.We define the matrix Green functions in the imaginary-\ntime domain as\nˆGij(p, τ) =−⟨Tτ \naip(τ)\na†\ni,−p(τ)!\u0010\na†\njp(0)aj,−p(0)\u0011\n⟩.\n(A6)\nIn a two-component Bose-condensed system, these func-\ntions can be found from the Dyson-Beliaev equations [48],\nand in the frequency domain they can be written as com-\nbinations\nˆGij(p, iωn) =X\nα=d,sX\ns=±s\niωn−sEα \nus\niα\nu−s\niα!\u0010\nus\njαu−s\njα\u0011\n(A7)\nof positive- and negative-frequency Green functions\n1/(iωn∓Eα) of Bogoliubov quasiparticles weighted with\nthe transformation coefficients (A4).\nThe density of non-condensate fraction of the ith com-\nponent can be calculated as nnc\ni=A−1P\np̸=0⟨a†\nipaip⟩.\nUsing the Bogoliubov transformation (A2) and taking\nthe thermal averages, we obtain\nnnc\ni=1\nAX\npX\nα=d,s\u001a\n(u−\niα)2+(u+\niα)2+ (u−\niα)2\neEα/T−1\u001b\n.(A8)\nAppendix B: Current response function\nWe define the Fourier harmonic operator of current as\nji(q) =m−1\niP\np(p+1\n2q)a†\nipai,p+q. In the simplest one-\nloop approximation, which is also used by other authors\nto describe the conductivity and superfluid drag effect in\nmulti-component ballistic systems [11, 21], the transverse\npart of the current response tensor (3) in Matsubara rep-\nresentation at q= 0 reads:\nχT\nij(0, iω) =−TX\npωnp2\n2Adm imj\n×Tr [σzˆGij(p, iωn+iω)σzˆGji(p, iωn)].(B1)\nUsing here the Green functions (A7), we obtain Eqs. (8)–\n(9) for the current response function. Separating terms\nwith α1=α2andα1̸=α2, we obtain\nχT\nij(0, iω) = Υ ij(iω)−(−1)δij\nmimj\u0002\nΛ+(iω) + Λ−(iω)\u0003\n.\n(B2)\nThe function Υ ij(ω), responsible for intra-branch scat-\ntering processes [see Fig. 2(c,d)], is defined as\nΥij(iω) =X\npX\nα=d,sp2PiαPjα\n2Adm imj\n×[S(Eα, Eα) +S(−Eα,−Eα)]. (B3)\nAfter introducing the quasiparticle damping and per-\nforming analytical continuation iω→ω+i0, we obtain11\nS(Eα, Eα) = 2 iγn′\nB(Eα)/(ω+ 2iγ) from Eq. (11), so this\nfunction can be written as\nΥij(ω) =X\npiγp2\nAdm imjPidPjdn′\nB(Ed) +PisPjsn′\nB(Es)\nω+ 2iγ\n=−2iγD0\nij\nω+ 2iγ, (B4)\nwhere we defined the conductivity weight (13).\nThe functions Λ+and Λ−are defined as\nΛ±(iω) =±X\npp2\n8Adp\nPidPisPjdPjs(Ed±Es)2\nEdEs\n×X\nα=d,s[S(Eα,±E˜α) +S(−Eα,∓E˜α)],(B5)\nwhere ˜d = s and ˜ s = d. Using the identity PadPas=\nPbdPbsfor the weight factors (A5), we obtain for 3D sys-\ntems the final expression (14) with the envelope functions\nf±(p) =±p4PidPis\n8π2d(Ed±Es)2\nEdEs\n× {nB(Ed)−nB(±Es)−iγ[n′\nB(Ed) +n′\nB(Es)]}.(B6)\nIn the Lorentz regime we omit the terms γn′\nB, because\nγn′\nB∼γ/T, which is much smaller than 1 in realistic\nsystems (since 1 nK ≈2π×138 Hz, so Tc∼102−103nK\ncorresponds to ∼105Hz). However, these terms should\nbe taken into account in the Drude regime: the con-\nductivity weights (13), (18) should be calculated as ac-\ncurately as possible, because their combined contribu-\ntion to the intercomponent superfluid weight Ds\nab=\nπ(−D0\nab+D+\nab+D−\nab) can be close to zero due to almost\ncomplete canceling of intra- and inter-Bogoliubov branch\nexcitation processes. For instance, for the Rb-Rb mixture\nwith γ/2π= 300 Hz and T=1\n3Tc, the Drude weights are\nD+\nab≈1.02D0\nabandD−\nab≈0.04D0\nab, and, consequently,\nDs\nab≈0.06πD0\nab. Therefore, even small errors in calcula-\ntions of D0,±\nijcan significantly affect conductivity in the\nlow-frequency limit.\nAppendix C: Approximations for ∆\nThe envelope functions (B6) increase at low momenta,\nwhen Ed,s< T, due to the power-law factor p4, and then\nexponentially decrease at large momenta, when Ed,s>\nT, thanks to the Bose distribution functions. Therefore,\nf±(p) reach maxima near some intermediate momentum\n¯pwhere Ed,s∼T. We restrict ourselves to the case when\nEd−Es≪Ed+Esand hence Ed≈Esnear p= ¯p,\nso that we are able to formally define ¯ pas a solution of\nequation Ed(¯p) +Es(¯p) = 2 T. In the parameter range\nwe consider, when T > µ a,b, the dispersions Ed,sare\nalmost quadratic near the momentum ¯ p. Using Eq. (A3)\nin the this quadratic regime, we can approximate it as\n¯p≈p\n2Tmamb/(ma+mb).To comprehend behaviour of the integrands in\nEq. (14), we should consider Ed+EsandEd−Esin\ndenominators of the R±functions (15) near p= ¯p.\nThe sum of energies, by definition, is about Ed(¯p) +\nEs(¯p) = 2 Tnear this momentum. The difference of\nenergies, denoted as ∆ = Ed(¯p)−Es(¯p), can be es-\ntimated using quadratic approximation of dispersions\n(A3): ∆ ≈p\n(E2a−E2\nb)2+ 16r2ϵaϵbµaµb/(ϵa+ϵb), where\nr2=g2\nab/gaagbbshould be less than 1 for stability of\nthe two-component BEC [23]. The first term under the\nsquare root can be rewritten as E2\na−E2\nb= (ϵ2\na−ϵ2\nb) +\n(2ϵaµa−2ϵbµb). It is straightforward to show, that ∆ ≈\n|ϵa−ϵb|when|ϵ2\na−ϵ2\nb| ≫ϵiµiatp= ¯p, which happens\nwhen masses of atoms are distant enough: such condition\ncan be written as |ma−mb|/mamb≳µi/Tm i. Other-\nwise, when masses are close to each other, |ϵ2\na−ϵ2\nb| ≪ϵiµi,\nwe obtain ∆ ≈p\nµ2a+µ2\nb+ 2(2−r2)µaµb≈µa+µb.\nOverall, near p= ¯p, where the envelope functions\nf±(p) attain the maximum, we obtain the following esti-\nmates for sum and difference of the quasiparticle energies:\nEd+Es∼2T, E d−Es∼∆, (C1)\nwhere\n∆∼\n\nT|ma−mb|\nma+mbif|ma−ma|\nmamb≳µi\nTmi\nµa+µb if|ma−ma|\nmamb≪µi\nTmi.(C2)\nThe first and second lines in Eq. (C2) correspond to the\ncase of distant and close masses, respectively. The sym-\nmetric mixtures ma=mbare obviously related to the\nsecond case. The aforementioned condition Ed−Es≪\nEd+Es, taken at the most relevant momenta p≈¯p, re-\nduces to ∆ ≪2T. In the case of distant masses it reads\n|ma−mb| ≪ma+mb(thus implying that the mass dif-\nference in this case is bounded both above and below),\nand in the case of close masses it is µa+µb≪T(which\nis fulfilled in the parameter ranges we consider).\nAppendix D: Interactions between dipolar atoms\nIn a system of magnetic dipolar atoms, total inter-\natomic interaction\nVij(r) =gijδ(r) +Vdd\nij(r), (D1)\nconsists of conventional isotropic interaction due to short-\nrange atomic scattering gijδ(r) and long-range magnetic\ndipole-dipole interaction\nVdd\nij(r) =didj1−3 cos2θ\n|r|3, (D2)\nwhere diis a magnetic dipole moment of the ith atomic\nspecie, and θis the angle between rand magnetic dipole\nmoments of all atoms which are assumed to be directed\nalong the zaxis.12\nWe consider quasi-two-dimensional atomic clouds with\nan effective thickness wz. In this case 2D Fourier trans-\nform of the full intracomponent (i.e. in the same planar\ncloud) interaction (D1) can be approximated as [49, 50]\nVii(q) =gii(1−Ci|q|), (D3)\nwhere Ci= 2πd2\niwz/gii. This interaction potential is\nevaluated with assumption r∗q≪1, where r∗=mid2\ni\nis the characteristic range of dipole-dipole interaction.\nThis assumption is valid for the parameters used in our\ncalculations: r∗= 12 nm and 2 .6 nm for168Er and52Cr\natoms respectively is much smaller than interlayer dis-\ntance L= 60 nm, which determines the scale of inverse\nmomentum q−1.\nThe Fourier transform of interaction between particles\nin different spatially separated atomic clouds is found as\nfollows. First, we rewrite the interaction (D2) for the\ntwo-layer geometry:\nVdd\nab(ρ, z−z′) =dadbr2−3(L+z−z′)2\nr5\n×\u0012π\n2wz\u00132\ncos\u0012πz\nwz\u0013\ncos\u0012πz′\nwz\u0013\n,(D4)\nwhere ρand L+z−z′are in-plane and out-\nof-plane distances between two atoms, while r=p\n(L+z−z′)2+ρ2is the total distance; zandz′are\ntheir vertical coordinates relative to the cloud centers\nranging from −wz/2 to wz/2. The cosine functions\nmodel atomic density profiles in the z-axis direction, and\n(π/2wz)2is normalization factor. 2D Fourier transform\nof Eq. (D4) in the xyplane reads\nVdd\nab(q, z, z′) =−2πdadbqe−q(L+z−z′)\n×\u0012π\n2wz\u00132\ncos\u0012πz\nwz\u0013\ncos\u0012πz′\nwz\u0013\n.(D5)\nWe assume thinness of atomic clouds, wz≪L(which was\nachieved in the recent experiment [16]), so out-of-plane\nmomenta of interacting particles are almost unchanged\nby the interlayer interaction. 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Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F\u0013 \u0010sica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nMagnetic damping is a key metric for emerging technologies based on magnetic nanoparticles,\nsuch as spin torque memory and high-resolution biomagnetic imaging. Despite its importance,\nunderstanding of magnetic dissipation in nanoscale ferromagnets remains elusive, and the damping\nis often treated as a phenomenological constant. Here we report the discovery of a giant frequency-\ndependent nonlinear damping that strongly alters the response of a nanoscale ferromagnet to spin\ntorque and microwave magnetic \feld. This novel damping mechanism originates from three-magnon\nscattering that is strongly enhanced by geometric con\fnement of magnons in the nanomagnet. We\nshow that the giant nonlinear damping can invert the e\u000bect of spin torque on a nanomagnet leading\nto a surprising current-induced enhancement of damping by an antidamping torque. Our work\nadvances understanding of magnetic dynamics in nanoscale ferromagnets and spin torque devices.\nI. INTRODUCTION\nNanoscale magnetic particles are the core components\nof several emerging technologies such as nonvolatile spin\ntorque memory [1], spin torque oscillators [2{7], targeted\ndrug delivery, and high-resolution biomagnetic imaging\n[8{11]. Control of magnetic damping holds the key to\nimproving the performance of many nanomagnet-based\npractical applications. In biomagnetic characterization\ntechniques such as magnetic resonance imaging [12], re-\nlaxometry [13], and magnetic particle imaging [14, 15],\nmagnetic damping a\u000bects nanoparticles relaxation times\nand image resolution. In spin torque memory and oscil-\nlators, magnetic damping determines the electrical cur-\nrent necessary for magnetic switching [1] and generation\nof auto-oscillations [16] and thereby determines energy-\ne\u000eciency of these technologies. The performance of\nnanomagnet-based microwave detectors is also directly\na\u000bected by the damping [17{19]. Despite its impor-\ntance across multiple disciplines, magnetic damping in\nnanoparticles is poorly understood and is usually mod-\neled as a phenomenological constant [6, 16].\nIn this article, we experimentally demonstrate that a\nferromagnetic nanoparticle can exhibit dynamics quali-\ntatively di\u000berent from those predicted by the constant\ndamping model. We show that nonlinear contributions\nto the damping can be unusually strong and the damp-\ning parameter itself can exhibit resonant frequency de-\npendence. Our work demonstrates that nonlinear damp-\ning in nanomagnets is qualitatively di\u000berent from that in\nbulk ferromagnets and requires a new theoretical frame-\nwork for its description. We show both experimentally\nand theoretically that such resonant nonlinear damping\noriginates from multi-magnon scattering in a magnetic\n\u0003igorb@ucr.edusystem with a discrete spectrum of magnons induced by\ngeometric con\fnement.\nWe also discover that the resonant nonlinear damping\ndramatically alters the response of a nanomagnet to spin\ntorque. Spin torque arising from injection of spin cur-\nrents polarized opposite to the direction of magnetization\nacts as negative damping [2]. We \fnd, however, that the\ne\u000bect of such antidamping spin torque is reversed, lead-\ning to an enhanced dissipation due to the nonlinear res-\nonant scattering. This counterintuitive behavior should\nhave signi\fcant impact on the operation of spin torque\nbased memory [1], oscillators [2{7] and microwave detec-\ntors [17{19].\nII. RESULTS\nA. Spin wave spectroscopy\nWe study nonlinear spin wave dynamics in nanoscale\nelliptical magnetic tunnel junctions (MTJs) that consist\nof a CoFeB free layer (FL), an MgO tunnel barrier, and a\nsynthetic antiferromagnet (SAF) pinned layer [20]. Spec-\ntral properties of the FL spin wave modes are studied in a\nvariety of MTJs with both in-plane and perpendicular-to-\nplane equilibrium orientations of the FL and SAF magne-\ntization. We observe strong resonant nonlinear damping\nin both the in-plane and the perpendicular MTJs, which\npoints to the universality of the e\u000bect.\nWe employ spin torque ferromagnetic resonance (ST-\nFMR) to measure magnetic damping of the FL spin wave\nmodes. In this technique, a microwave drive current\nIacsin(2\u0019ft) applied to the MTJ excites oscillations of\nmagnetization at the drive frequency f. The resulting\nmagnetoresistance oscillations Racsin(2\u0019ft+\u001e) generate\na direct voltage Vmix. Peaks in ST-FMR spectra Vmix(f)\narise from resonant excitation of spin wave eigenmodes\nof the MTJ [21{28]. To improve signal-to-noise ratio,arXiv:1803.10925v1  [cond-mat.mes-hall]  29 Mar 20182\n0 0.5 1 1.5 2H1H2\n36912\nField (kOe)0 0.5 1 1.5 2H2H1\n00.20.4\nField (kOe)Linewidth (GHz)Experiment\nSimulationFrequency (GHz)\n+1\n1a b\n<Vmix>~\nFIG. 1. Spin wave spectra in a nanoscale MTJ. (a) Normalized ST-FMR spectra h~Vmix(f)iof spin wave eigenmodes in a\nperpendicular MTJ device (Sample 1) measured as a function of out-of-plane magnetic \feld. Resonance peaks arising from\nthree low frequency modes of the MTJ free layer j0i,j1i, and j2iare observed. (b) Spectral linewidth of the quasi-uniform\nj0ispin wave mode as a function of out-of-plane magnetic \feld. Strong linewidth enhancement is observed in the resonant\nthree-magnon regime at H1andH2.\nthe magnitude of external magnetic \feld Happlied par-\nallel to the free layer magnetization is modulated, and\na \feld-derivative signal ~Vmix(f) = dVmix(f)=dHis mea-\nsured via lock-in detection technique [20]. Vmix(f) can\nthen be obtained via numerical integration (Supplemen-\ntal Material).\nFigure 1(a) shows ST-FMR spectra ~Vmix(f) measured\nas a function of out-of-plane magnetic \feld Hfor an el-\nliptical 52 nm\u000262 nm perpendicular MTJ device (Sam-\nple 1). Three spin wave eigenmodes with nearly linear\nfrequency-\feld relation fn(H) are clearly visible in the\nspectra. Micromagnetic simulations (Supplemental Ma-\nterial) reveal that these modes are three lowest frequency\nspin wave eigenmodes of the FL (Supplemental Material).\nThe lowest frequency (quasi-uniform) mode j0iis node-\nless and has spatially uniform phase. Each of the two\nhigher-order modes jni(n= 1;2) has a single node at\nthe FL center that is either perpendicular ( n= 1) or\nparallel (n= 2) to the ellipse long axis.\nThe spectral linewidth of the resonances in Fig. 1(a)\ncan be used for evaluation of the mode damping. The\nquasi-uniform mode j0iresonance visibly broadens at\ntwo magnetic \feld values: H1= 0:74 kOe (4 GHz) and\nH2= 1:34 kOe (6 GHz). Near H1, the modej1iresonance\nalso broadens and exhibits splitting, same behavior is ob-\nserved for the mode j2iatH2. At these \felds, the higher-\norder mode frequency is twice that of the quasi-uniform\nmodefn= 2f0. This shows that three-magnon con\ru-\nence [29{33] is the mechanism of the quasi-uniform mode\ndamping increase: two magnons of the quasi-uniform\nmodej0imerge into a single magnon of the higher-ordermodejni.\nThe most striking feature of the quasi-uniform mode\nresonance near H1is its split-peak shape with a local min-\nimum at the resonance frequency. Such a lineshape can-\nnot be \ft by the standard Lorentzian curve with symmet-\nric and antisymmetric components [20]. We therefore use\na double-peak \ftting function (Supplemental Material)\nto quantify the e\u000bective linewidth \u0001 f0of the resonance\npro\fle. For applied \felds su\u000eciently far from H1, the\nST-FMR curve recovers its single-peak shape and \u0001 f0\nis determined as half width of the standard Lorentzian\n\ftting function [20]. Figure 1(b) shows \u0001 f0as a function\nofHand demonstrates a large increase of the linewidth\nnear the \felds of the resonant three-magnon regime H1\nandH2. The stepwise increase of \u0001 f0nearH1is a result\nof the ST-FMR curve transition between the split-peak\nand single-peak shapes. For \felds near H2, the resonance\npro\fle broadens but does not develop a visible split-peak\nlineshape. As a result, \u0001 f0(H) is a smooth function in\nthe vicinity of H2.\nB. E\u000bect of spin torque\nIn MTJs, direct bias current Idcapplied across the\njunction exerts spin torque on the FL magnetization, act-\ning as antidamping for Idc>0 and as positive damping\nforIdc<0 [22, 34]. The antidamping spin torque in-\ncreases the amplitude of the FL spin wave modes [22, 35]\nand decreases their spectral linewidth [36]. We can em-\nploy spin torque from Idcto control the amplitude of spin3\nFIG. 2. E\u000bect of spin torque on spin wave resonance lineshape. (a)-(b) Spin wave resonance lineshapes in the nonresonant\nregime at H > H 1for di\u000berent values of direct bias current Idc. (c)-(d) Spin wave resonance lineshapes in the resonant three-\nmagnon regime at H=H1. (a), (c) Measured ST-FMR spectra (Sample 2). (b), (d) Solutions of Eqs. (3) and (4). Identical\nbias current values Idc(displayed in (a) are used in (a)-(d).\nwave eigenmodes excited in ST-FMR measurements, and\nthereby study the crossover between linear and nonlinear\nregimes of spin wave resonance.\nFigure 2 shows the dependence of ST-FMR resonance\ncurve of thej0imodeVmix(f) onIdcfor a 50 nm\u0002110 nm\nelliptical in-plane MTJ (Sample 2). For in-plane mag-\nnetic \feld values far from the three-magnon resonance\n\feldsHn, the amplitude of ST-FMR resonance curve\nVmix(f) shown in Fig. 2(a) monotonically increases with\nincreasing antidamping spin torque, as expected. At\nH=H1, the antidamping spin torque has a radically\ndi\u000berent and rather surprising e\u000bect on the resonance\ncurve. As illustrated in Fig. 2(c), increasing antidamp-\ning spin torque \frst broadens the resonance at H=H1\nand then transforms a single-peak resonance lineshape\ninto a split-peak lineshape with a local minimum at the\nresonance frequency f0. The data in Fig. 2 demonstrate\nthat the unusual split-peak lineshape of the resonance is\nonly observed when (i) the three-magnon scattering of\nthe quasi-uniform mode is allowed by the conservation of\nenergy and (ii) the amplitude of the mode is su\u000eciently\nhigh, con\frming that the observed e\u000bect is resonant and\nnonlinear in nature.\nFig. 2(c) reveals that antidamping spin torque can in-\ncrease the spectral linewidth and the e\u000bective damping\nof the quasi-uniform spin mode if the mode undergoes\nresonant three-magnon scattering. Figure 3 further illus-\ntrates this counterintuitive e\u000bect. It shows the linewidth\nof the quasi-uniform mode of a 50 nm \u0002110 nm elliptical\nin-plane MTJ (Sample 3) measured as a function of bias\ncurrent. In Fig. 3, blue symbols show the linewidth mea-\nsured at an in-plane magnetic \feld su\u000eciently far fromthe three-magnon resonance \felds Hn. At this \feld, the\nexpected quasi-linear dependence of the linewidth on Idc\nis observed for currents well below the critical current\nfor the excitation of auto-oscillatory magnetic dynamics.\nNear the critical current, the linewidth increases due to\na combination of the fold-over e\u000bect [37{39] and ther-\nmally activated switching between the large- and small-\namplitude oscillatory states of the fold-over regime [22].\nThe red symbols in Fig. 3 show the linewidth measured\nin the resonant three-magnon regime at H=H1. In con-\ntrast to the nonresonant regime, the linewidth increases\nwith increasingjIdcjfor both current polarities. Fur-\nthermore, the maximum linewidth is measured for the\nantidamping current polarity.\nIII. THEORETICAL MODEL\nNonlinear interactions among spin wave eigenmodes\nof a ferromagnet give rise to a number of spectacu-\nlar magneto-dynamic phenomena such as Suhl instabil-\nity of the uniform precession of magnetization [40, 41],\nspin wave self-focusing [42] and magnetic soliton forma-\ntion [43{45]. In bulk ferromagnets, nonlinear interac-\ntions generally couple each spin wave eigenmode to a\ncontinuum of other modes via energy- and momentum-\nconserving multi-magnon scattering [40]. This kinemat-\nically allowed scattering limits the achievable amplitude\nof spin wave modes and leads to broadening of the spin\nwave resonance. These processes lead to a resonance\nbroadening [40, 46{48] and cannot explain the observed\nsplit-peak lineshape of the resonance. In nanoscale ferro-4\nmagnets, geometric con\fnement discretizes the spin wave\nspectrum and thereby generally eliminates the kinemati-\ncally allowed multi-magnon scattering. This suppression\nof nonlinear scattering enables persistent excitation of\nspin waves with very large amplitudes [49] as observed in\nnanomagnet-based spin torque oscillators [2, 50]. Tun-\nability of the spin wave spectrum by external magnetic\n\feld, however, can lead to a resonant restoration of the\nenergy-conserving scattering [31]. The description of\nnonlinear spin wave resonance in the nanoscale ferromag-\nnet geometry therefore requires a new theoretical frame-\nwork. To derive the theory of resonant nonlinear damp-\ning in a nanomagnet, we start with a model Hamilto-\nnian that explicitly takes into account resonant nonlinear\nscattering between the quasi-uniform mode and a higher-\norder spin wave mode (in reduced units with ~\u00111):\nH=!0aya+!nbyb+\t0\n2ayayaa+\tn\n2bybybb (1)\n+( naaby+ \u0003\nnayayb)\n+\u0010\b\nexp(\u0000i!t)ay+ exp(i!t)a\t\nwhereay,aandby,bare the magnon creation and an-\nnihilation operators for the quasi-uniform mode j0iwith\nfrequency!0and for the higher-order spin wave mode\njnimode with frequency !n, respectively. The non-\nlinear mode coupling term proportional to the coupling\nstrength parameter  ndescribes the annihilation of two\nj0imagnons and creation of one jnimagnon, as well as\nthe inverse process. The Hamiltonian is written in the\nresonant approximation, where small nonresonant terms\nsuch asaab,aaayare neglected. The terms proportional\nto \t 0and \t ndescribe the intrinsic nonlinear frequency\nshifts [51] of the modes j0iandjni. The last term de-\nscribes the excitation of the quasi-uniform mode by an\nexternal ac drive with the amplitude \u0010and frequency !.\nWe further de\fne classically a dissipation function Q,\nwhere\u000b0and\u000bnare the intrinsic linear damping param-\neters of the modes j0iandjni[52{54]:\nQ=day\ndtda\ndt(\u000b0+\u00110aya) +dby\ndtdb\ndt(\u000bn+\u0011nbyb) (2)\nFor generality, Eq. (2) includes intrinsic nonlinear\ndamping [16] of the modes j0iandjnidescribed by the\nnonlinearity parameters \u00110and\u0011n. However, our analy-\nsis below shows that the split-peak resonance lineshape\nis predicted by our theory even if \u00110and\u0011nare set equal\nto zero.\nEquations describing the nonlinear dynamics of the\ntwo coupled spin wave modes of the system follow from\nEq. (1) and Eq. (2):\nida\ndt=@H\n@ay+@Q\n@(day=dt)(3)\nidb\ndt=@H\n@by+@Q\n@(dby=dt)(4)\nIt can be shown (Supplemental Material) that these\nequations have a periodic solution a= \u0016aexp (\u0000i!t) and\n100 50 0 50 10000.10.20.30.40.5\nI (A)f0 (GHz)Resonant regime\nNonresonant regimeFIG. 3. E\u000bect of spin torque on linewidth. Linewidth of the\nquasi-uniform spin wave mode as a function of the applied\ndirect bias current (Sample 3): blue symbols { in the non-\nresonant regime H6=H1and red symbols { in the resonant\nthree-magnon regime H=H1. Lines are numerical \fts using\nEqs. (3) and (4).\nb=\u0016bexp (\u0000i2!t), where \u0016a,\u0016bare the complex spin wave\nmode amplitudes. For such periodic solution, Eqs. (3)\nand (4) are reduced to a set of two nonlinear algebraic\nequations for absolute values of the spin wave mode am-\nplitudesj\u0016ajandj\u0016bj, which can be solved numerically.\nSince the ST-FMR signal is proportional to j\u0016aj2(Supple-\nmental Material), the calculated j\u0016aj2(!) function can be\ndirectly compared to the measured ST-FMR resonance\nlineshape.\nWe employ the solution of Eqs. (3) and (4) to \ft the\n\feld dependence of the quasi-uniform mode linewidth in\nFig. 1(b). In this \ftting procedure, the resonance line-\nshapej\u0016aj2(!) is calculated, and its spectral linewidth\n\u0001!0is found numerically. The resonance frequencies !0\nand!nare directly determined from the ST-FMR data\nin Fig. 1(a). The intrinsic damping parameters \u000b0and\n\u000bnnearH1andH2are found from linear interpolations\nof the ST-FMR linewidths \u0001 f0and \u0001fnmeasured at\n\felds far from H1andH2. We \fnd that \u0001 !0weakly\ndepends on the nonlinearity parameters \t and \u0011, and\nthus these parameters are set to zero (Supplemental Ma-\nterial). We also \fnd that the calculated linewidth \u0001 !0\ndepends on the product of the drive amplitude \u0010and\nmode coupling strength  n, but is nearly insensitive to\nthe individual values of \u0010and nas long as\u0010\u0001 n= const\n(Supplemental Material). Therefore, we use \u0010\u0001 nas a\nsingle \ftting parameter in this \ftting procedure. Solid\nline in Fig. 1(b) shows the calculated \feld dependence\nof the quasi-uniform mode linewidth on magnetic \feld.\nThe agreement of this single-parameter \ft with the ex-\nperiment is excellent.\nFigures 2(b) and 2(d) illustrate that Eqs. (3) and\n(4) not only describe the \feld dependence of ST-FMR\nlinewidth but also qualitatively reproduce the spectral5\nlineshapes of the measured ST-FMR resonances as well\nas the e\u000bect of the antidamping spin torque on the line-\nshapes. Fig. 2(b) shows the dependence of the calculated\nlineshapej\u0016aj2(!) on antidamping spin torque for a mag-\nnetic \feldHfar from the three-magnon resonance \felds\nHn. At this nonresonant \feld, increasing antidamping\nspin torque induces the fold-over of the resonance curve\n[37] without resonance peak splitting. The dependence of\nj\u0016aj2(!) on antidamping spin torque for H=H1is shown\nin Fig. 2(d). At this \feld, the resonance peak in j\u0016aj2(!)\n\frst broadens with increasing antidamping spin torque\nand then splits, in qualitative agreement with the ex-\nperimental ST-FMR data in Fig. 2(c). Our calculations\n(Supplemental Material) reveal that while the nonlinear-\nity parameters \t 0,\u00110, \tnand\u0011nhave little e\u000bect on\nthe linewidth \u0001 !0, they modify the lineshape of the res-\nonance. Given that the nonlinearity parameter values\nare not well known for the systems studied here, we do\nnot attempt to quantitatively \ft the measured ST-FMR\nlineshapes.\nEquations (3) and (4) also quantitatively explain\nthe observed dependence of the quasi-uniform mode\nlinewidth \u0001 !0on direct bias current Idc. Assuming an-\ntidamping spin torque linear in bias current [36, 55, 56]:\n\u000b0!\u000b0(1\u0000Idc=Ij0i\nc),\u000bn!\u000bn(1\u0000Idc=Ijni\nc), where\nIjni\nc>Ij0i\ncare the critical currents, we \ft the measured\nbias dependence of ST-FMR linewidth in Fig. 3 by solv-\ning Eqs. (3) and (4). The solid lines in Fig. 3 are the best\nnumerical \fts, where \u0010\u0001 nandIcare used as indepen-\ndent \ftting parameters. The rest of the parameters in\nEqs. (3) and (4) are directly determined from the experi-\nment following the procedure used for \ftting the data in\nFig. 1(b). Theoretical curves in Fig. 3 capture the main\nfeature of the data at the three-magnon resonance \feld\nH1{ increase of the linewidth with increasing antidamp-\ning spin torque.\nIV. DISCUSSION\nFurther insight into the mechanisms of the nonlinear\nspin wave resonance peak splitting and broadening by an-\ntidamping spin torque can be gained by neglecting the in-\ntrinsic nonlinearities \t nand\u0011nof the higher-order mode\njni. Setting \t n= 0 and\u0011n= 0 in Eqs. (3) and (4) allows\nus to reduce the equation of motion for the quasi-uniform\nmode amplitudej\u0016ajto the standard equation for a single-\nmode damped driven oscillator (Supplemental Material)\nwhere a constant damping parameter \u000b0is replaced by\nan e\u000bective frequency-dependent nonlinear damping pa-\nrameter\u000be\u000b\n0:\n\u000be\u000b\n0=\u000b0+\u0014\n\u00110+4\u000bn 2\nn\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(5)and the resonance frequency is replaced by an e\u000bective\nresonance frequency:\n!e\u000b\n0=!0+\u0014\n\t0+2j nj2(2!\u0000!n)\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(6)\nEquation (5) clearly shows that the damping parame-\nter of the quasi-uniform mode itself becomes a resonant\nfunction of the drive frequency with a maximum at half\nthe frequency of the higher order mode ( !=1\n2!n). The\namplitude and the width of this resonance in \u000be\u000b\n0(!) are\ndetermined by the intrinsic damping parameter \u000bnof\nthe higher-order mode jni. If\u000bnis su\u000eciently small,\nthe quasi-uniform mode damping is strongly enhanced\nat!=1\n2!n, which leads to a decrease of the quasi-\nuniform mode amplitude at this drive frequency. If the\ndrive frequency is shifted away from1\n2!nto either higher\nor lower values, the damping decreases, which can re-\nsult in an increase of the quasi-uniform mode amplitude\nj\u0016aj. Therefore, the amplitude of the quasi-uniform mode\nj\u0016aj(!) can exhibit a local minimum at !=1\n2!n. Due to\nits nonlinear origin, the tendency to form a local min-\nimum inj\u0016aj(!) at1\n2!nis enhanced with increasing j\u0016aj.\nSincej\u0016ajis large near the resonance frequency !0, tun-\ning!0to be equal to1\n2!ngreatly ampli\fes the e\u000bect of\nlocal minimum formation in j\u0016aj(!). This qualitative ar-\ngument based on Equation (5) explains the data in Fig. 2\n{ the split-peak nonlinear resonance of the quasi-uniform\nmode is only observed when external magnetic \feld tunes\nthe spin wave eigenmode frequencies to the three-magnon\nresonance condition !0=1\n2!n.\nEquation (6) reveals that the nonlinear frequency shift\nof the quasi-uniform mode is also a resonant function of\nthe drive frequency. In contrast to the nonlinear damping\nresonance described by Equation (5), the frequency shift\nresonance is an antisymmetric function of !\u00001\n2!n. The\nnonlinear shift is negative for ! <1\n2!nand thus causes\na fold-over towards lower frequencies while it is positive\nfor!>1\n2!ncausing fold-over towards higher frequencies.\nAt the center of the resonance pro\fle, the three-magnon\nprocess induces no frequency shift. This double-sided\nfold-over also contributes to the formation of the split-\npeak lineshape of the resonance shown in Figs. 2(c) and\n2(d) and to the linewidth broadening. As with the non-\nlinear damping resonance, the antisymmetric nonlinear\nfrequency shift and the double-sided fold-over become\ngreatly ampli\fed when the spin wave mode frequencies\nare tuned near the three-magnon resonance !0=1\n2!n.\nEquations (5) and (6) also shed light on the origin\nof the quasi-uniform mode line broadening by the an-\ntidamping spin torque. The antidamping spin torque in-\ncreases the quasi-uniform mode amplitude j\u0016ajvia transfer\nof angular momentum from spin current to the mode [57].\nSince the nonlinear damping and the nonlinear frequency\nshift are both proportional to j\u0016aj2and both contribute to\nthe line broadening, the antidamping spin torque can in-\ndeed give rise to the line broadening. Equation (5) reveals\ntwo competing e\u000bects of the antidamping spin torque on\nthe quasi-uniform mode damping parameter \u000be\u000b\n0: spin6\ntorque from Idcdecreases the linear component of the\ndamping parameter \u000b0!\u000b0(1\u0000Idc=Ij0i\nc) and increases\nthe nonlinear component via increased j\u0016aj2. Whether the\nantidamping spin torque decreases or increases the spec-\ntral linewidth of the mode depends on the system param-\neters. Our numerical solution of Eqs. (3) and (4) shown\nin Fig. 3 clearly demonstrates that the antidamping spin\ntorque can strongly increase the linewidth of the quasi-\nuniform mode when the three-magnon resonance condi-\ntion!0=1\n2!nis satis\fed. Furthermore, we \fnd that\nthe three-magnon process exhibits no threshold behav-\nior upon increasing amplitude (Supplemental Material)\nor decreasing intrinsic damping.\nThe key requirement for observation of the resonant\nnonlinear damping is the discreteness of the magnon\nspectrum imposed by geometric con\fnement in the\nnanoscale ferromagnet. The split-peak nonlinear reso-\nnance discovered in this work cannot be realized in bulk\nferromagnets because the three-magnon resonance con-\ndition in bulk is not only valid at the uniform mode\nfrequency!0=1\n2!nbut instead in a broad frequency\nrange. Owing to the magnon spectrum continuity in\nbulk, shifting the excitation frequency away from !0does\nnot suppress the three-magnon scattering of the uniform\nmode { it simply shifts it from one group of magnons to\nanother [29, 40]. Therefore, the amplitude of the uni-\nform mode does not increase when the drive frequency is\nshifted away from !0and the split-peak resonance is not\nrealized.\nWe expect that the resonant nonlinear damping dis-\ncovered in this work will have strong impact on the\nperformance of spin torque devices such as spin torque\nmagnetic memory, spin torque nanooscillators and spin\ntorque microwave detectors. Since all these devices rely\non large-amplitude oscillations of magnetization driven\nby spin torque, the amplitude limiting resulting from the\nresonant nonlinear damping is expected to have detri-\nmental e\u000bect on the device performance.V. CONCLUSIONS\nIn conclusion, our measurements demonstrate that\nmagnetic damping of spin wave modes in a nanoscale\nferromagnet has a strong nonlinear component of reso-\nnant character that appears at a discrete set of magnetic\n\felds corresponding to resonant three-magnon scattering.\nThis strong resonant nonlinearity can give rise to unusual\nspin wave resonance pro\fle with a local minimum at the\nresonance frequency in sharp contrast to the properties\nof the linear and nonlinear spin wave resonances in bulk\nferromagnets. The resonant nonlinearity has a profound\ne\u000bect on the response of the nanomagnet to spin torque.\nAntidamping spin torque, that reduces the quasi-uniform\nspin wave mode damping at magnetic \felds far from the\nresonant three-magnon regime, can strongly enhance the\ndamping in the resonant regime. This inversion of the\ne\u000bect of spin torque on magnetization dynamics by the\nresonant nonlinearity is expected to have signi\fcant im-\npact on the performance of nanoscale spin torque devices\nsuch as magnetic memory and spin torque oscillators.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFoundation through Grants No. DMR-1610146, No.\nEFMA-1641989 and No. ECCS-1708885. We also ac-\nknowledge support by the Army Research O\u000ece through\nGrant No. W911NF-16-1-0472 and Defense Threat Re-\nduction Agency through Grant No. HDTRA1-16-1-0025.\nA. M. G. thanks CAPES Foundation, Ministry of Educa-\ntion of Brazil for \fnancial support. R.E.A acknowledges\nFinanciamiento Basal para Centros Cienti\fcos y Tec-\nnologicos de Excelencia under project FB 0807 (Chile),\nand Grant ICM P10-061-F by Fondo de Innovacion para\nla Competitividad-MINECON. 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Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F´ ısica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nI. METHODS\nA. Linewidth evaluation\nAll measurements presented were carried out with magnetic field applied along the easy axis of the MTJ devices so\nthat the magnetic moments of the free and pinned layers are collinear to each other. In this geometry, the ST-FMR sig-\nnals are dominated by photo-resistance contribution and are proportional to the square of the transverse component of\nthe dynamic magnetization magnetization [1], which allows us to directly compare calculated |a|2(ω) resonance curves\nto measured ST-FMR resonance curves ˜Vmix(f) and toVmix(f) approximated by numerical integration/integraltext˜Vmix(f)df.\nWhenVmix(f) and|a|2(ω) are single-peak curves, they are fit to a sum of symmetric and antisymmetric Lorentzian\ncurves with identical central frequencies and linewidth parameters as described in Ref. [2], and the spectral linewidth\nis determined as half-width at the half-maximum of the symmetric Lorentzian curve.\nIn order to quantify the linewidth of the split-peak resonance profile, we introduce a fitting function that is a sum\nof two Lorentzian curves with different central frequencies separated by δf. The half width of the resonance profile\n∆f0is then defined as the average of the half widths of the two Lorentzians plus δf/2.\nSupplemental Figure 1. Spatial profiles of spin wave eigenmodes. Normalized amplitude and phase of the three lowest frequency\nspin wave eigenmodes of the MTJ free layer, given by micromagnetic simulations.\nB. Micromagnetic simulations\nMicromagnetic simulations were performed using OOMMF software [3, 4]. To account for all magnetic interactions\nin the MTJ, a three dimensional model was employed with three ferromagnetic layers: free, SAF top and SAF bottom.\nWe use material parameters obtained from the measurements and/or their accepted literature values (see Ref. [2] for2\nthe MTJ structure and fabrication details). Magnetization dynamics is excited by a combined pulse of spin torque\nand Oersted field, resulting from a sinc-shaped spatially uniform current pulse. The spatial profile of the Oersted\nfield corresponds to that of a long wire with elliptical cross section. The direction of the spin torque vector acting\non the free layer is determined by the magnetization orientation of the SAF top layer. The spectrum of spin wave\neigenmodes is obtained via fast Fourier transform (FFT) of the time dependent components of the layers’ magnetic\nmoment. Spatial mapping of the resulting Fourier amplitude and phase at a given frequency provides the mode\nprofiles (Supplmental Fig. 1). The observed excitations are confirmed to be spin wave modes localized to the free\nlayer. SAF modes are found at much higher frequencies than the free layer modes, and their frequencies are found to\nbe incommensurable to the free layer quasi-uniform mode frequency [5].\nII. SOLUTION OF THE EQUATIONS OF MOTION\nThe Hamiltonian equations of motion describing the coupled dissipative dynamics of the quasi-uniform ( a) and the\nhigher-order ( b) spin wave modes are:\nida\ndt=∂H\n∂a†+∂Q\n∂(da†/dt)(1)\nidb\ndt=∂H\n∂b†+∂Q\n∂(db†/dt)(2)\nwhereHis the Hamiltonian of the system and Qis the dissipation function, given by:\nH=ω0a†a+ωnb†b+1\n2Ψ0a†a†aa+1\n2Ψnb†b†bb+ (ψ∗\nnaab†+ψna†a†b) +ζ{exp(−iωt)a†+ exp(iωt)a} (3)\nQ=da†\ndtda\ndt(α0+η0a†a) +db†\ndtdb\ndt(αn+ηnb†b) (4)\nBy using Eq. (3) and Eq. (4) in Eq. (1) and Eq. (2), the Hamiltonian equations can be written as:\nida\ndt−(α0+η0a†a)da\ndt=ω0a+ 2ψna†b+ Ψ 0a†aa+ζexp(−iωt) (5)\nidb\ndt−(αn+ηnb†b)db\ndt=ωnb+ψ∗\nnaa+ Ψnb†bb (6)\nUsing a periodic ansatz a= ¯aexp(−iωt) andb=¯bexp(−2iωt) in Eq. (5) and Eq. (6), where ¯ aand¯bare complex\namplitudes, reduces the Hamiltonian equations to a set of two algebraic equation for the complex amplitudes:\n/parenleftbig\nω−ω0−Ψ0|¯a|2+i(α0+η0|¯a|2)ω/parenrightbig\n¯a−2ψn¯a∗¯b=ζ (7)\n/parenleftbig\n2ω−ωn−Ψn|¯b|2+ 2i(αn+ηn|¯b|2)ω/parenrightbig¯b=ψ∗\nn¯a2(8)\nWe solve Eq. (8) for ¯band multiply the numerator and denominator of this expression by the complex conjugate of\nthe denominator:\n¯b=ψ∗\nn¯a2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n−i2(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(9)\nthen we multiply Eq. (9) by2ψn¯a∗\n¯aand evaluate the real and imaginary parts.\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|22/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(10)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|2−4(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(11)\nBy taking the modulus of Eq. (8), we obtain:\n|¯a|2=|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2 (12)3\nUsing Eq. (12) in Eqs. (10-11), we derive:\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(13)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(14)\nTaking the modulus squared of Eq. (7):\n/braceleftBigg/parenleftbigg\nω−ω0−Ψ0|¯a|2−/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2\n+/parenleftbigg\n(α0+η0|¯a|2)ω−/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2/bracerightBigg\n|¯a|2=ζ2(15)\nand using Equations (12)–(14) in Eq. (15) gives us an algebraic equation for the absolute value of the higher order\nmode amplitude|¯b|:\n\n\n\nω−ω0−Ψ0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2−2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n+\n\n/parenleftbigg\nα0+η0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2/parenrightbigg\nω−−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n\n×\n|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2=ζ2\n(16)\nAfter numerically solving Eq. (16) for |¯b|, and using it in Eq. (12), we can calculate the amplitude of the quasi-uniform\nmode|¯a|.\nIII. EFFECTS OF THE DRIVE AMPLITUDE AND INTRINSIC NONLINEARITITES\nTo understand the impact of the intrinsic nonlinearity parameters (Ψ 0, Ψn,η0,ηn) on the quasi-uniform spin wave\nmode resonance, we plot the numerical solution of Eq. (16) in Supplemental Figure 2. Each panel of this figure shows\na reference lineshape of the resonance calculated with all intrinsic nonlinearity parameters set to zero (red curve) and\na lineshape calculated with one of the intrinsic nonlinearity parameter different from zero (blue curve). This figure\nreveals that increasing η0decreases the mode amplitude and slightly increases the linewidth. Increasing ηndecreases\nthe degree of the double-peak lineshape splitting. Increasing Ψ nincreases the lineshape asymmetry. Increasing Ψ 0\nincreases lineshape asymmetry and induces fold-over.\nSupplemental Figure 3 shows the linewidth as a function of the drive amplitude for three scenarios, where the\nintrinsic nonlinearities Ψ 0, Ψn,η0,ηnare set to zero for simplicity. If the coupling parameter is zero, ψn= 0, the\nlinewidth does not depend on the drive amplitude, as expected for a single-mode linear oscillator. The second case\ndemonstrates that the linewidth remains constant when the product ψn·ζis constant. For a constant non-zero\ncoupling parameter, the linewidth shows an increase with the drive amplitude. This observation allows us to employ a\nsingle fitting parameter ( ψn·ζ) to fit the data in Fig. 1b. This conjecture can be confirmed analytically by introducing\na normalized spin wave amplitude ˆ a=ψn¯a, which allows us to rewrite Eq. (16) omitting all intrinsic nonlinearities\ninto the following form:\nω/bracketleftbigg\n1 +iα0+i4αn|ˆa|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\nˆa−ω0ˆa−2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2|ˆa|2ˆa=ψnζ (17)\nThis equation describes an effective single-mode nonlinear oscillator with renormalized excitation amplitude ψnζ.4\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nη0\n2 2.5 3 3.500.010.02\nFrequency (GHz)|a|20\nηn\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨ0\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨn|a|2|a|2|a|2a b\nc d\nSupplemental Figure 2. Effect of intrinsic nonlinearities on the quasi-uniform spin wave resonance lineshape. Spectral lineshape\nof the quasi-uniform spin wave mode resonance |¯a|2(ω) at the three-magnon resonance condition 2 ω0=ωncalculated by\nnumerically solving Eq. (16). The red curve is a reference lineshape calculated with all intrinsic nonlinearity parameters\n(η0,ηn,Ψ0,Ψn) set to zero. The blue lineshape in each panel is calculated with one of the intrinsic nonlinearity parameters set\nto a non-zero value: (a) η0= 1.325·10−24J, (b)ηn= 3.313·10−24J, (c) Ψ 0= 1.325·10−24J, (d) Ψ n= 1.325·10−23J. Other\nparameters employed in the calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662,αn= 0.03042 atIdc= 0;\nψn·ζ=h2·0.006 GHz2, wherehis the Planck constant.\n0 0.05 0.1 0.15 0.20.150.20.25\nζ h-1 (GHz)∆f0 (GHz)\n(i)(ii)(iii)\nSupplemental Figure 3. Effect of the drive amplitude on linewidth in the resonant three-magnon regime. Calculated linewidth\nof the quasi-uniform spin wave mode as a function of the drive amplitude ζfor different values of the mode coupling parameter\nψn. (i) Green: ψn= 0, (ii) red: variable ψnwith a constraint ψn·ζ=h2·0.006 GHz2, and (iii) blue: ψn=h·0.1 GHz. All\nintrinsic nonlinearity parameters: Ψ 0, Ψn,η0andηnare set to zero. his the Planck constant. Other parameters employed in\nthe calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662 andαn= 0.03042 atIdc= 0.5\nIV. EFFECTIVE SINGLE-MODE NONLINEAR OSCILLATOR APPROXIMATION\nIf we neglect intrinsic nonlinearities Ψ nandηnof the higher order spin wave mode, Eq. (16) can be reduced to a\ncubic equation for ¯ aand solved analytically. This approximation allows us to obtain several important qualitative\ninsights into the properties of the resonant nonlinear damping of the quasi-uniform mode. By setting Ψ n= 0 and\nηn= 0 in Eq. (8), we obtain an exact solution for ¯b:\n¯b=ψ∗\nn¯a2\n2ω(1 +iαn)−ωn(18)\nUsing this result, we reduce Eq. (16) to a cubic algebraic equation for ¯ a:\nω/bracketleftbigg\n1 +i(α0+η0|¯a|2) +i4|ψn|2αn|¯a|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n¯a−ω0¯a−/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2¯a=ζ (19)\nThis equation describes the amplitude ¯ aof an effective single-mode nonlinear oscillator.\nIt is evident from Eq. (19) that the frequency of the quasi-uniform mode experiences a nonlinear shift:\nωeff\n0=ω0+/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(20)\nThe nonlinear frequency shift has a well-pronounced antisymmetric resonant character near the resonance frequency\nωn/2, that arises from the resonant three-magnon scattering.\nFurther, it is clear from Eq. (19) that the effective damping of the quasi-uniform mode also acquires a term arising\nfrom the three-magnon interaction:\nαeff\n0=α0+/bracketleftbigg\nη0+4|ψn|2αn\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(21)\nThe last term describes a resonant enhancement of the nonlinear damping by three-magnon scattering near the\nresonance frequency ωn/2. Strikingly, the magnitude of the resonant damping enhancement at ωn/2 increases when\nthe intrinsic damping of the higher order mode αndecreases. In the limit αn→0, the effective damping becomes\nαeff\n0→α0+/bracketleftbigg\nη0+2π|ψn|2\nωδ(2ω−ωn)/bracketrightbigg\n|¯a|2(22)\nwhereδis Dirac’s delta function. Equation (21) suggests that the effective damping of the quasi-uniform mode αeff\n0\ncan increase with increasing antidamping spin torque applied to the nanomagnet. Indeed, the antidamping spin torque\ntends to increase the amplitude [6] of the quasi-uniform mode |¯a|and decrease the intrinsic damping parameter of the\nhigher order mode αn→αn(1−Idc/I|n/angbracketright\nc), both enhancing the nonlinear damping term in Eq. (19). For a sufficiently\nlarge mode coupling parameter ψn, the enhancement of the nonlinear damping term by the antidamping spin torque\ncan exceed the reduction of the linear damping parameter α0→α0(1−Idc/I|0/angbracketright\nc) by the torque, leading to an increase\nofαeff\n0byIdc>0 and broadening of the quasi-uniform mode resonance by the antidamping spin torque. This scenario\nis indeed realized in the MTJ devices studied here as demonstrated by the data and calculations in Fig. 3.\nV. MODE COUPLING PARAMETER\nIn this Supplementary Note, we discuss how the coupling parameter between the spin wave modes, ψnin Eq. (3),\ncan be calculated. We consider a very thin, magnetically soft ferromagnetic disk with elliptical cross section, that\nis magnetized in-plane. Within a classical micromagnetic model, we include Zeeman, dipolar and exchange terms in\nthe free energy. An applied field Halong thexdirection (long axis of the ellipse) magnetizes the sample to a nearly\nuniform state. Through a classical Holstein-Primakoff transformation [7] we introduce variables c(/vector x,t) andc∗(/vector x,t) to\ndescribe the magnetization such that the magnetization magnitude is conserved:\nmx= 1−cc∗, m +=c√2−cc∗, m−=c∗√2−cc∗, (23)\nwhere/vector m=/vectorM/Ms, andm±≡mz±imy. Approximating the exchange energy to the fourth order in candc∗, the\nnormalized free energy of the disk, U≡E/4πM2\ns, is given by\nU/similarequal−hx/integraldisplay\n(1−cc∗) dV+ (lex)2/integraldisplay/bracketleftbigg\n/vector∇c·/vector∇c∗+1\n4c2(/vector∇c∗)2+1\n4c∗2(/vector∇c)2/bracketrightbigg\ndV−1\n2/integraldisplay\ndV/vectorhD(/vector m)·/vector m, (24)6\nwithhx≡H/4πMs,lex≡/radicalbig\nA/2πM2sis the exchange length, and /vectorhD(/vector m) =/vectorHD(/vector m)/4πMsis the normalized\ndemagnetizing field. The Landau-Lifshitz equations of motion in the new variables are: i˙c=δU/δc∗,i˙c∗=−δU/δc\nwitht/prime= 4πMs|γ|t.\nAssuming the normal modes involved in three magnon scattering dominate the magnetization dynamics, the free\nenergy in Eq. (24) can be written in terms of amplitudes of these modes, by expressing cin terms of aandb:\nc(/vector x,t)/similarequala(t)f(/vector x) +a∗(t)g(/vector x) +b(t)p(/vector x) +b∗(t)q(/vector x) (25)\nThe functions f,g,p,q can be determined from calculating the linear modes of oscillation of the sample. The terms of\nthe free energy proportional to aab∗anda∗a∗bdescribe the three-magnon process and the magnitude of these terms\ngives the coupling parameter ψn.\nIf the magnetization state is approximated as exactly uniform, the dipolar energy for a very thin film may be\napproximated as UD=m2\nz/2 = (c+c∗)2(1−cc∗/2), and in this case all three-magnon terms are zero. However, when\nthe effects due to the sample edges (such as spatial inhomogeneity of the demagnetization field and edge roughness)\nare taken into account, the equilibrium magnetization configuration is generally nonuniform. In this case, there are\nnon-zero three-magnon terms in the free energy expression. An explicit calculation of the corresponding overlap\nintegrals is necessary for a quantitative prediction of ψn. Refs. [8, 9] show such extensive calculations for circular disks\nand include explicit expressions for the exchange and dipolar energies.\n[1] Michael Harder, Yongsheng Gui, and Can-Ming Hu, “Electrical detection of magnetization dynamics via spin rectification\neffects,” Phys. Rep. 661, 1–59 (2016).\n[2] A. M. Gon¸ calves, I. Barsukov, Y.-J. Chen, L. Yang, J. A. Katine, and I. N. Krivorotov, “Spin torque ferromagnetic\nresonance with magnetic field modulation,” Appl. Phys. Lett. 103, 172406 (2013).\n[3] M. J. Donahue and D. G. Porter, OOMMF User’s Guide (National Institute of Standards and Technology, Gaithersburg,\nMD, 1999).\n[4] Robert D. McMichael and Mark D. Stiles, “Magnetic normal modes of nanoelements,” J. Appl. Phys. 97, 10J901 (2005).\n[5] P. S. Keatley, V. V. Kruglyak, A. Neudert, R. J. Hicken, V. D. Poimanov, J. R. Childress, and J. A. Katine, “Resonant\nenhancement of damping within the free layer of a microscale magnetic tunnel valve,” J. Appl. Phys. 117, 17B301 (2015).\n[6] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, “Spin-transfer-driven\nferromagnetic resonance of individual nanomagnets,” Phys. Rev. Lett. 96, 227601 (2006).\n[7] T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58,\n1098–1113 (1940).\n[8] D. Mancilla-Almonacid and R. E. Arias, “Instabilities of spin torque driven auto-oscillations of a ferromagnetic disk mag-\nnetized in plane,” Phys. Rev. B 93, 224416 (2016).\n[9] D. Mancilla-Almonacid and R. E. Arias, “Spin-wave modes in ferromagnetic nanodisks, their excitation via alternating\ncurrents and fields, and auto-oscillations,” Phys. Rev. B 95, 214424 (2017)."    },    {        "title": "1903.10135v1.Distributed_Inter_Area_Oscillation_Damping_Control_for_Power_Systems_by_Using_Wind_Generators_and_Load_Aggregators.pdf",        "content": "arXiv:1903.10135v1  [math.OC]  25 Mar 20191\nDistributed Inter-Area Oscillation Damping Control\nfor Power Systems by Using Wind Generators and\nLoad Aggregators\nZhiyuan Tang, Yue Song, Tao Liu, and David J. Hill, Life Fellow, IEEE\nAbstract —This paper investigates the potential of wind turbine\ngenerators (WTGs) and load aggregators (LAs) to provide sup -\nplementary damping control services for low frequency inte r-area\noscillations (LFOs) through the additional distributed da mping\ncontrol units (DCUs) proposed in their controllers. In orde r\nto provide a scalable methodology for the increasing number\nof WTGs and LAs, a novel distributed control framework is\nproposed to coordinate damping controllers. Firstly, a dis tributed\nalgorithm is designed to reconstruct the system Jacobian ma trix\nfor each damping bus (buses with damping controllers). Thus ,\nthe critical LFO can be identified locally at each damping bus by\napplying eigen-analysis to the obtained system Jacobian ma trix.\nThen, if the damping ratio of the critical LFO is less than a\npreset threshold, the control parameters of DCUs will be tun ed\nin a distributed and coordinated manner to improve the dampi ng\nratio and minimize the total control cost at the same time. Th e\nproposed control framework is tested in a modified IEEE 39-bu s\ntest system. The simulation results with and without the pro posed\ncontrol framework are compared to demonstrate the effectiv eness\nof the proposed framework.\nIndex Terms —Low frequency oscillation, load-side control,\nwind generator, distributed control\nI. I NTRODUCTION\nLow frequency inter-area oscillations (LFOs) have always\nbeen a matter of concern to power system operators due to\ntheir potential threats to the power system stability [1]. W ith\nthe development of the electricity market and growing power\ndemand, future power systems will become more stressed\nand operate closer to their stability limits, which highlig hts\nthe need to improve the damping ratio of LFOs and prevent\nsustained oscillations that can result in serious conseque nces\nsuch as system separations or even large-area blackouts [1] .\nThe power system stabilizers (PSSs) installed on conven-\ntional synchronous machines are the most important compon-\nents to improve system damping against LFOs. However, the\nincreasing penetration of wind power limits the availabili ty\nof PSSs to provide sufficient damping against LFOs. For one\nthing, the displacement of conventional synchronous gener -\nators with wind turbine generators (WTGs) may reduce the\nThis work was fully supported by the Research Grants Council of the\nHong Kong Special Administrative Region under the Theme-ba sed Research\nScheme through Project No. T23-701/14-N.\nZ. Tang, Y . Song, and T. Liu are with the Department of Electri cal and\nElectronic Engineering, The University of Hong Kong, Hong K ong (email:\nzytan@eee.hku.hk; yuesong@eee.hku.hk; taoliu@eee.hku. hk).\nD. J. Hill is with the Department of Electrical and Electroni c Engineering,\nThe University of Hong Kong, Hong Kong. He is also with the Sch ool of\nElectrical and Information Engineering, The University of Sydney, NSW 2006,\nAustralia (email: dhill@eee.hku.hk; david.hill@sydney. edu.au).damping ratio of inter-area modes by the reconfiguration of\nline power flows, reduction of system inertia, and interacti on\nof converter controls with power system dynamics [2]. For\nanother thing, once the conventional synchronous machines\nare replaced by WTGs, the associated PSSs are also removed\nfrom the system with no replacement controllers for WTGs to\nprovide damping control services. Thus, if no new alternati ve\ncontrollers are developed to provide supplementary dampin g\ncontrol services, insufficient system controls may jeopard ize\nthe system security and stability. To solve this issue, in th is\npaper, we are looking for solutions from both the generation\nand load sides.\nFor the generation side, we utilize the converter interface d\nWTGs which can provide damping torques for LFOs by\nquickly adjusting their active power outputs though a prope r\ncontrol of electronic devices that interface them with the\ngrid [3], [4]. For the load side, the option of using highly\ndistributed controllable loads (demand control) is appeal ing.\nDue to properties such as instantaneous responses and spati al\ndistributions, demand control has gained a lot of attention\n[5]–[7]. In particular, demand control has been utilized to\naccomplish important system support tasks such as frequenc y\ncontrol [5], voltage control [6], and small-disturbance an gle\nstability enhancement [7]. However, the ability of demand\ncontrol to provide supplementary damping control services\nagaist LFOs has not been thoroughly investigated yet. In thi s\npaper, the load aggregators (LAs) will be coordinated with\nWTGs to provide damping torques against LFOs through the\nadditional distributed damping control units (DCUs) devel oped\nin their controllers.\nIn the literature, numerous methods have been proposed\nto coordinate traditional damping controllers (e.g. PSS) [ 8]–\n[10] and new damping controllers (e.g. FACTS and HVDC)\n[11]–[13]. Approaches based on robust control theories and\nlinear matrix inequalities have been utilized to deal with t he\nuncertainties of operating conditions [9]–[11]. For examp le,\nin [10], the synthesis of the controller is formulated as a\nmixedH2/H∞output feedback control problem with regional\npole placement that is resolved through a linear matrix in-\nequality approach. However, such a robust controller desig n\nmethod is too conservative and unable to incorporate all\nsystem constraints (e.g. hard limits on the control signals ).\nApproaches based on model predictive control have been\nutilized to incorporate all system constraints. For exampl e,\nthe authors of [12] propose a model predictive control based\nHVDC supplementary controller which can incorporate plant2\nconstraints explicitly. Unfortunately, the model used in s uch a\nmethod is developed at a pre-given operating point, and henc e,\nthe obtained controller cannot directly guarantee robustn ess\naround the other operating points. Approaches based on fuzz y\nlogic have been utilized to handle the variations of operati ng\npoints [13]. For example, a fuzzy logic adaptive control uni t is\nproposed in [13] to adjust control gains for different opera ting\npoints. However, this fuzzy logic based method becomes very\ncomplicated when the number of damping controllers becomes\nlarge. Moreover, all the methods mentioned above are carrie d\nout in a centralized manner that lacks scalability and flexib ility,\ni.e., a new damping controller is added into the original con trol\nsystem, the whole control law need to be redesigned.\nTo overcome the drawbacks of the abovementioned meth-\nods, in this paper, a novel distributed control framework is\nproposed to coordinate damping controllers, which can be\nimplemented by local measurements and limited communic-\nations between neighboring buses. The proposed distribute d\ncontrol framework consists of two modules: a critical LFO\nidentification module and a controller parameters tuning mo d-\nule where the communication network used in each module\nis different. The critical LFO identification module aims at\nreconstructing the system Jacobian matrix for each damping\nbus (a bus with damping controller) in a distributed manner\nwhere the communication network used covers all buses in\nthe system. Thus, the critical LFO (the LFO with the least\ndamping ratio) can be identified locally at each damping bus\nby applying eigen-analysis to the obtained system Jacobian\nmatrix. Further, if the damping ratio of the critical LFO is\nless than a preset threshold, the parameters of DCUs will be\ntuned in a distributed manner to improve the damping ratio\nof the critical LFO and minimize the total control cost at the\nsame time where the communication network used only covers\nthose damping buses. The contributions of this paper are lis ted\nbelow:\n•A novel two-step communication based distributed con-\ntrol framework is proposed to coordinate LAs and WTGs.\nThe proposed control method can survive one-point fail-\nure in the communication network and is suitable in\npractice for its scalability.\n•In the critical LFO identification module, based on struc-\ntural properties of the original power grid, a distributed\ncalculation algorithm is developed to recover the Jacobian\nmatrices for each damping bus.\n•In the controller parameters tuning module, based on the\neigenvalue sensitivities, a controller tuning problem is\nformulated and solved in a distributed manner.\nThe rest of the paper is organized as follows. Section II\nintroduces the DCU and the power system model to be stud-\nied. The proposed distributed control framework is explici tly\npresented in Section III. Section IV presents a case study by\nusing a modified IEEE 39-bus test system. Conclusions are\ngiven in Section V .\nNotations\nDenoteRandCas the set of real numbers and complex\nnumbers, respectively. An m-dimensional vector is denoted\nFigure 1. The control block diagram of the proposed DCU.\nasx= [xi]∈Rm. The transpose of a vector or a matrix\nis defined as (·)T. The notation Im∈Rm×mdenotes the\nidentity matrix, 0is a zero vector or matrix with an appropriate\ndimention, and ei∈Rpdenotes the vector with the ithentry\nbeing one and others being zeros. The notation |x|(∠x) takes\nthe modulus (angle) of a complex number x∈C. The notation\nV(A)means converting the matrix A= [a1,...,ap]∈Rm×p\nwithai∈Rmwithi= 1,...,p to a vector, i.e., V(A) =\n[aT\n1,...,aT\np]T∈Rmp. The symbols /bardbl · /bardbl and/bardbl· /bardbl∞denote\nthel2andl∞norms for a vector, respectively.\nII. N ETWORK DESCRIPTION\nIn this section, we firstly introduce the DCU proposed for\neach damping controller. Then, the power system network to\nbe studied is introduced, which will be used to design the\ncontrol framework in Section III.\nA. Distributed Damping Control Unit\nFig. 1 shows the block diagram of the proposed DCU\nwhich mimics the structure of PSS. The input is the local bus\nvoltage angle θi, and the output is posciwhich is added to the\nreference active power demand of the WTG or LA to provide\nsupplementary damping control services. The mathematical\nmodel of the ithDCU is given by\n˙x1i=−1\nTwi(Kiθi+x1i)\n˙x2i=1\nT2i/parenleftbigg\n(1−T1i\nT2i)(Kiθi+x1i)−x2i/parenrightbigg\n˙x3i=1\nT4i/parenleftbigg\n(1−T3i\nT4i)/parenleftbigg\nx2i+/parenleftbiggT1i\nT2i(Kiθi+x1i)/parenrightbigg/parenrightbigg\n−x3i/parenrightbigg\nposci=x3i+T3i\nT4i/parenleftbigg\nx2i+T1i\nT2i(Kiθi+x1i)/parenrightbigg\n.\n(1)\nThe dynamics can be written in a compact form as ˙xCi=\nfCi(xCi,θi)wherexCi= [x1i,x2i,x3i]Tis the supplement-\nary state variables, Kiis the gain, Twiis the wash-out time\nconstant,T1i,T2i,T3i, andT4iare time constants for lead-lag\ncompensation. In the proposed control framework, Ki,T1i,\nT2i,T3i, andT4iwill be tuned to improve the damping ratio\nof the critical LFO.\nB. Power system network\nConsider a connected power system consisting of Nbuses\nwithNGsynchronous generators (SGs), NWWTGs,NL\nloads, andNTtransfer buses where N=NG+NW+NL+NT.\nThe SG (WTG or load) bus refers to a bus that connects a\nSG (WTG or load) only. The transfer bus is a bus with no3\ngeneration or demand. We number the SG buses as VG=\n{1,...,N G}, WTG buses as VW={NG+1,...,N G+NW},\nload buses as VL={NG+NW+1,...,N G+NW+NL},\nand transfer buses as VT={NG+NW+NL+1,...,N}.\n1) SG model: To highlight the effectiveness of the proposed\ndamping controllers, PSSs are not included in the SG models.\nWith the 4th-order two-axis synchronous machine model and\nIEEE standard exciter model (IEEET1), the mathematical\nmodel of the ithSG is written as:\n˙xGi=fGi(xGi,θi,vi)\npGi=gpGi(xGi,θi,vi)\nqGi=gqGi(xGi,θi,vi), i∈ VG(2)\nwhere the state variable xGiis defined as xGi=\n[e′\nqi,e′\ndi,δi,ωi,xmi,xr1i,xr2i,xfi]T;e′\nqiande′\ndiare transient\nd-axis and q-axis voltages, respectively; δiandωiare the\nrotor angle and speed, respectively; xmi,xr1i,xr2iandxfi\nare the state variables corresponding to the IEEET1 exciter .\nThe algebraic variables are the local bus voltage angle θiand\nmagnitudevi. The active and reactive power injections of the\nithSG bus are denoted as pGiandqGi, respectively. The\ndetailed descriptions of nonlinear functions fGi,gpGi,gqGi\ncan be found in [14], which is given in the Appendix A for\nself-completeness.\n2) WTG model: Fully rated converter WTGs are adopted,\nwhich employ the configuration of a synchronous machine\nwith a permanent magnet rotor [15]. Normally, the controlle r\nof WTG gives a reference active power demand to optimize\nthe wind energy capture based on the measured rotor speed\n(see the lower branch in Fig. 2). In this paper, two additiona l\ncontrol units are added into the original WTG’s controller t o\nadapt the active power reference set point, i.e., the primar y\nfrequency support unit proposed in [16] (see the upper branc h\nin Fig. 2) and the DCU (see the middle branch in Fig. 2). The\nmathematical model of the ithWTG is written as:\n˙θi=ωi\n˙xWi=fWi(xWi,ωi,θi,vi)\npWi=gpWi(xWi,ωi,θi,vi)\nqWi=gqWi(xWi,ωi,θi,vi), i∈ VW(3)\nwhereωiis the local bus frequency. The state variable\nxWi= [ωmi,θpi,isqi,icdi,xT\nCi]Twhereωmiis the rotor\nspeed;θpiis the pitch angle used for maximum power control;\niqsiis the generator stator quadrature current used for active\npower/speed control; and icdiis the converter direct current\nused for reactive power/voltage control; xCi= [x1i,x2i,x3i]T\nare state variables corresponding to the DCU. The active and\nreactive power injections of the ithWTG bus are denoted\naspWiandqWi, respectively. The detailed descriptions of\nnonlinear functions fWi,gpWi,gqWiare given in the Appendix\nB.\n3) Load model: The active power of each load pLiis\ndivided into two parts: the controllable part di=posci(xLi,θi)\n(referred to as LA in this paper) and static voltage frequenc y\ndependent part, whereas the reactive power of each load qLi\nis assumed to be static voltage frequency dependent. With th e\nFigure 2. Control block diagram of the controller for WTG.\nadditional DCU, the mathematical model of the ithload bus\nis given as follows:\n˙θi=ωi\n˙xLi=fLi(xLi,θi)\npLi=poi(vi)αi(1+kpfi(ωi−ω0))+di(xLi,θi)\n:=gpLi(xLi,ωi,θi,vi)\nqLi=qoi(vi)βi(1+kqfi(ωi−ω0))\n:=gqLi(ωi,vi), i∈ VL(4)\nwhere the state variable xLi= [x1i,x2i,x3i]Tcorresponds to\nthe DCU;poiandqoiare the nominal values; αiandβiare\nvoltage coefficients; kpfiandkqfiare frequency coefficients.\n4) Transfer bus: As transfer buses have no generations or\nloads, theithtransfer bus is simply modeled as:\npTi= 0, qTi= 0, i∈ VT (5)\nwherepTiandqTiare the active and reactive power injections,\nrespectively.\n5) Network power flows: The network power flows are\nrepresented by the usual set of algebraic power flow equation s,\nwhich are used to couple all buses power injection equations\nmentioned above. For the ithbus in the system, the power\nflow equations are given as:\n0 =−pinj\ni+viN/summationdisplay\nj=1vj(Gijcosθij+Bijsinθij)\n0 =−qinj\ni+viN/summationdisplay\nj=1vj(Gijsinθij−Bijcosθij), i∈ V(6)\nwhereGijandBijare the real and imaginary parts of Yij\nwhich is the (i,j)entry of the admittance matrix Y; the nota-\ntionθijis the short for θi−θj; the setV=VG∪VW∪VL∪VT;\npinj\niandqinj\niare injected active and reactive power of the ith\nbus, respectively. In particular, for SG buses, pinj\ni=pGiand\nqinj\ni=qGi; for WTG buses, pinj\ni=pWiandqinj\ni=qWi; for\nload buses, pinj\ni=−pLiandqinj\ni=−qLi; and for transfer\nbuses,pinj\ni=pTiandqinj\ni=qTi.\n6) Overall system: Combining (2)-(6), the overall system\ncan be expressed as differential-algebraic equations:\n˙x=f(x,y)\n0=h(x,y)(7)\nwhere the vector x= [xT\nG,θT\nW,xT\nW,θT\nL,xT\nL]Tand the\nvectory= [θT\nG,ωT\nW,ωT\nL,θT\nT,vT\nG,vT\nW,vT\nL,vT\nT]T;xk=\n[xT\nk1,...,xT\nki,...,xT\nkNk]T, i∈ Vk,k∈ {G,W,L};θk=4\n[θi]∈RNk, i∈ Vk,k∈ {G,W,L,T };vk= [vi]∈\nRNk, i∈ Vk,k∈ {G,W,L,T };ωk= [ωi]∈RNk, i∈ Vk,\nk∈ {W,L}. The nonlinear functions fandhrepresent\nthe system dynamics and network power flow equations,\nrespectively.\nLinearizing system (7) gives the following linear model:\n/bracketleftbigg∆˙x\n0/bracketrightbigg\n=/bracketleftbiggAsBs\nCsDs/bracketrightbigg/bracketleftbigg∆x\n∆y/bracketrightbigg\n(8)\nwhere the detailed expressions of the matrices As,Bs,Cs,\nandDsare given in the Appendix C. Assuming Dsis nonsin-\ngular (it is a common assumption adopted in the literature [1 7])\nand eliminating ∆yin (8) gives:\n∆˙x=Ar∆x (9)\nwhereAr=As−BsD−1\nsCs∈RNA×NAwithNA= 8NG+\n8NW+4NL.\nIII. D ISTRIBUTED CONTROL FRAMEWORK\nIn this section, the critical LFO identification module and\ncontroller parameters tuning module that form the proposed\ndistributed control framework will be introduced in detail s.\nA. Critical LFO identification module\nAs mentioned earlier, this module aims at identifying the\ncritical LFO for each damping bus in a distributed manner.\nIt is known that the critical LFO can be investigated by\napplying eigenvalue analysis based on the global informati on,\ni.e., the system Jacobian matrix Arin (9) which is usually\nobtained in a centralized manner [1]. However, in this paper ,\nwe will reconstruct the matrix Arfor each damping bus in a\ndistributed manner by revealing the structure properties o f the\npower grid contained in the matrices As,Bs,Cs, andDs.\nBy performing an elementary column operation, the\nmatricesAs,Bs,Cs, andDscan be reformulated as:\n/bracketleftbiggAsBs\nCsDs/bracketrightbigg\n=/bracketleftbiggK1K2\nK3Jpf/bracketrightbigg\nT (10)\nwhere the matrix Tis the elementary column operator and Jpf\nis the power flow Jacobian matrix. The detailed expressions\nof the matrices T,K1,K2,K3, andJpfare given in the\nAppendix C.\nThrough the matrix transform (10), we can see that the\nmatricesAs,Bs,Cs, andDscan be reconstructed by all\nJacobian matrices K∧\n∨(refer to (52) in the Appendix C for\ndetails),Jpf, identity matrices, and T. For identity matrices\nandT, since they are constant, they can be broadcasted or\nstored at each damping bus in advance. For all Jacobian\nmatricesK∧\n∨(all are block diagonal matrices) and Jpf, we\nadopt the distributed algorithm proposed in [17] that has to tal\n2Nsteps to calculate their elements, where the communication\nnetwork used covers all buses in the system and has the same\ntopology as the physical grid. The communication network ca n\nbe described by the undirected graph G1={V,E}, whereV\nis the set of nodes (buses) and E ⊆ V×V represents the set of\nedges (branches). The set of neighbors of node iis represent\nbyNi={j∈ V: (j,i)∈ E} with cardinality |Ni|=Di.We assume that 1) each bus knows the parameters of its local\nmachine (or load) and lines connecting it; 2) each damping\nbus knows the model structure of SG, WTG, and load; 3)\neach bus knows its own bus number and total number of\nbusesN; 4) each bus in the system has the capability of local\nmeasurement, storing data, processing data, communicatin g\nwith its neighbors, and calculation; and 5) communication\ndelays are negligible.\nAt each step, bus i, i∈ V has four columns of data for\ncommunication, denoted as γa\ni,̟a\ni,γb\ni,̟b\ni∈R2N. The\ndata update process is designed as follows:\n[Xa(τ+1),Xb(τ+1)] =Jpf[Xa(τ),Xb(τ)] (11)\nwhereXa(τ),Xb(τ)∈R2N×2Nare the data matrices at the\nτthstep iteration with the definitions as follows:\nXa(τ) = [γa\n1(τ),...,γa\nN(τ),̟a\n1(τ),...,̟a\nN(τ)]T\nXb(τ) = [γb\n1(τ),...,γb\nN(τ),̟b\n1(τ),...,̟b\nN(τ)]T(12)\nwhich are initialized by Xa(0) =I2NandXb(0) =\n[γb\n1(0),...,γb\nN(0),̟b\n1(0),...,̟b\nN(0)]T. The vectors γb\ni(0)\nand̟b\ni(0)assigned to the ithbus satisfies:\n[γb\ni(0);̟b\ni(0)] = [V(KfGixGi);V(KfGi\nθi);V(KfGivi);V(KhpGixGi);\nV(KhqGixGi);0], i∈ VG;\n[γb\ni(0);̟b\ni(0)] = [V(KfWi\nθi);V(KfWixWi);V(KfWiωi);V(KfWivi);\nV(KhpWixWi);V(KhpWiωi);V(KhqWixWi);\nV(KhqWiωi);0], i∈ VW;\n[γb\ni(0);̟b\ni(0)] = [V(KfLi\nθi);V(KfLixLi);V(KhpLixLi);V(KhpLiωi);\nV(KhqLiωi);0], i∈ VL;\n[γb\ni(0);̟b\ni(0)] = [0], i∈ VT.\n(13)\nThe designed update process (11) can be realized in a dis-\ntributed manner via the communication network G1={V,E}\nmentioned earlier since\n1) the initial values of vectors γa\ni(0),̟a\ni(0),γb\ni(0), and\n̟b\ni(0)can be assigned locally for each bus i, because\ni) the vectors γa\ni(0),̟a\ni(0)can be assigned locally as\neach bus knows its own bus number and ii) the elements\nof vectorsγb\ni(0),̟b\ni(0)can be calculated based on local\nmeasurements θi,vi,pinj\niandqinj\ni[14], [15];\n2) for each sub-matrix Jhp\nθ,Jhp\nv,Jhq\nθ,Jhq\nvofJpf(see\n(52) and (53) in Appendix C for details), the nonzero\nelements of the ithrow are functions of measurements\nof busiand its neighboring bus j∈ Ni[17], [18].\nDuring the update process, at each step τ,0< τ≤2N,\neach damping bus i, i∈ VW∪ VLstores its own data and\ndata from its neighboring buses (which can be realized via\ncommunication links between neighboring buses). Thus, the\nwhole distributed algorithm is expressed as:\n[Xa(τ+1),Xb(τ+1)] =Jpf[Xa(τ),Xb(τ)]\n[ξa\ni(τ),ξb\ni(τ)] =Si[Xa(τ),Xb(τ)], i∈ VW∪VL(14)\nwhere the matrix Si= [ei,ej,eN+i,eN+j]T∈R2(Di+1)×2N,\nj∈ Niselects the rows with respect to the damping bus iand5\nits neighboring buses j, j∈ Ni;ξa\ni(τ),ξb\ni(τ)∈R2(Di+1)×2N\ndenote the data collected by the damping bus i. We assume\nthe discrete-time system (14) is observable, which usually\nholds in practice [17], i.e., rank(Oi) = 2NwhereOi∈\nR4(Di+1)N×2Nis defined as\nOi= [ST\ni,(SiJpf)T,...,(SiJ2N−1\npf)T]T. (15)\nAfter the update process (14), each damping bus i, i∈\nVW∪VLcan recoverJpfandXb(0)via the data it collected\nξa\ni(τ)andξb\ni(τ), τ= 0,1,...,2N. For simplicity, we define\nthe following data matrices:\nΞa\ni1= [ξa\ni(0)T,...,ξa\ni(2N−1)T]T∈R4(Di+1)N×2N\nΞa\ni2= [ξa\ni(1)T,...,ξa\ni(2N)T]T∈R4(Di+1)N×2N\nΞa\ni= [Ξa\ni1T,Ξa\ni2T]T∈R8(Di+1)N×2N\nΞb\ni1= [ξb\ni(0)T,...,ξb\ni(2N−1)T]T∈R4(Di+1)N×2N.(16)\nThe singular value decomposition of Ξa\niis also needed, which\nis given as:\nΞa\ni= [˜Uξi,˜U0\nξi]/bracketleftbigg\nΣξi\n0/bracketrightbigg\n˜VT\nξi=˜UξiΣξi˜VT\nξi(17)\nwhereΣξi,˜Vξi∈R2N×2N,˜Uξi∈R8(Di+1)N×2N,˜U0\nξi∈\nR8(Di+1)N×(8(Di+1)N−2N). Based on the matrices given in\n(16) and (17), each damping bus i, i∈ VW∪VLcan recover\nJpfandXb(0)by the following equations:\nJpf= (˜UT\nξi1Ξa\ni1)−1Θi˜UT\nξi1Ξa\ni1 (18a)\nXb(0) = (Ξa\ni1)†Ξb\ni1 (18b)\nwhere˜Uξi1,˜Uξi2∈R4(Di+1)N×2Nare sub-matrices of ˜Uξi\nwith˜Uξi= [˜UT\nξi1,˜UT\nξi2]T,Θi= (˜UT\nξi1˜Uξi2)(˜UT\nξi1˜Uξi1)−1∈\nR2N×2N, and the superscript †denotes the Moore-Penrose\ninverse. The mathematical proof of (18a)-(18b) can be found\nin [17].\nAs mentioned earlier, each damping bus is assumed to\nknow the model structures of SG, WTG, and load. Thus, each\ndamping bus can identify the type of bus i(i.e, SG, WTG,\nload, or transfer bus) based on the γb\ni(0)and̟b\ni(0)ofXb(0)\nobtained, and hence can recover all K∧\n∨Jacobian matrices\ninK1,K2, andK3of (10) fromXb(0)obtained based on\n(13). Combined with Jpfobtained, each damping bus can\nreconstructArby (9) and (10). Therefore, the critical LFO\ncan be calculated by applying eigenvalue analysis to Arat\neach damping bus.\nRemark 1: In the proposed update process (14), we assume\nthat the sum of the length of all vectorized K∧\n∨matrices related\nto each type of bus (i.e., SG, WTG, load, or transfer bus) is le ss\nthan the length of the data vectors [γb\ni;̟b\ni], i∈ V assigned\nfor each type of bus that is 4N(refer to (13) for details).\nIf there exist one type of bus whose sum of the length of\nall vectorized K∧\n∨matrices is more than 4N, additional data\nvectorsγc\ni,̟c\ni∈R2Nare assigned for each bus to form the\nadditional data matrix Xc∈R2N×2N. For the type of bus\nwhose sum of the length of all the vectorized K∧\n∨matrices is\nmore than 4N,[γc\ni(0);̟c\ni(0)] is initialized by the remaining\nelements. For the other types of buses whose sum of the length\nof the vectorized K∧\n∨matrices is less than 4N,[γc\ni(0);̟c\ni(0)]\nFigure 3. The closed-loop representation of the system.\nis initialized by zeros. The additional data matrix Xc(0)can\nbe recovered by each damping bus via the same way as the\ndata matrixXb(0)is recovered.\nB. Controller parameters tuning module\nIn order to guarantee an adequate stability margin, the\ndamping ratio ςcof the critical LFO λc=σc+jωcshould\nsatisfyςc≥ς⋆whereςc=−σc/|λc|andς⋆>0is the preset\nthreshold. Once the damping ratio of the critical LFO is less\nthanς⋆, the parameters of each DCU (i.e., Ki,T1i,T2i,T3i,\nandT4i) of each damping bus will be tuned coordinately to\nimprove the damping ratio of the critical LFO.\nWithout loss of generality, we firstly study the impact of\nthe parameter changes of the ith, i∈ VW∪ VLDCU (i.e.,\nthe DCU of the bus NG+i) onλc. For analysis purposes,\nthe system model (8) is rewritten in the following form by\nreordering the variables of xin (8):\n/bracketleftbigg\n∆˙˜xi\n0/bracketrightbigg\n=/bracketleftbigg˜Asi˜Bsi\n˜Csi˜Dsi/bracketrightbigg/bracketleftbigg∆˜xi\n∆y/bracketrightbigg\n(19)\nwhere∆˜xi= [∆xT\ni,∆xT\nCi]T,xi∈RNA−3includes all state\nvariables inxexceptxCi∈R3that is the corresponding state\nof theithDCU;˜Asi=T−1\niAsTi(hereTi∈RNA×NAis\ninvertable which is the corresponding elementary row opera tor\nsuch thatx=Ti˜xi);˜Bsi=T−1\niBs;˜Csi=CsTi; and\n˜Dsi=Ds. Then the system model (19) can be written in the\nclosed-loop form [19]. In the closed-loop form, the system\nmodel is partitioned into two subsystems. For subsystem 1,\nwhich does not depend on parameters of the ithDCU, we\nhave the following state space description:\n/bracketleftbigg∆˙xi\n0/bracketrightbigg\n=/bracketleftbiggAiBi\nCiDi/bracketrightbigg/bracketleftbigg∆xi\n∆y/bracketrightbigg\n+/bracketleftbiggEi\nFi/bracketrightbigg\n∆ui.(20)\nwhereui=posciis the output of the ithDCU. Assuming Di\nis nonsingular and eliminating ∆yin (20) gives:\n∆˙xi=Asi∆xi+Bsi∆ui; ∆θi=Csi∆xi (21)\nwhereAsi=Ai−BiD−1\niCi∈R(NA−3)×(NA−3),Bsi=\nEi−BiD−1\niFi∈RNA−3andCT\nsi∈RNA−3. For subsystem\n2, which only depends on the parameters of the ithDCU, we\nhave the following state space description:\n/bracketleftbigg∆˙xCi\n∆ui/bracketrightbigg\n=/bracketleftbiggACiBCi\nCCiDCi/bracketrightbigg/bracketleftbigg∆xCi\n∆θi/bracketrightbigg\n. (22)\nwhereACi,BCi,CCi, andDCican be easily obtained from\n(1). A transfer function description for (22) is given as:\nFi(s,Ki) =CCi(sI−ACi)−1BCi+DCi (23)6\nwhereKi∈Ris the gain factor in the ithDCU model. Based\non (21) and (23), the schematic diagram of the closed-loop\nform is given in Fig. 3.\nThen the sensitivity of λcwith respect to Kiof the transfer\nfunctionFi(s,Ki)is given by [19]:\n∂λc\n∂Ki=Ri∂Fi(s,Ki)\n∂Ki/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=λc(24)\nwhereRi=CsiφsiψT\nsiBsi∈Cis the residue with respect to\nthe critical eigenvalue λc;φsi∈RNA−3andψsi∈RNA−3\nare the right and left eigenvectors of λc, respectively. Here,\nφsi(ψsi) consists of the first NA−3elements ofφi∈RNA\n(ψi∈RNA) which is the right (left) eigenvector of λcwith\nrespect to ˜Arithat is obtained by eliminating ∆yin (19), i.e.,\n˜Ari=˜Asi−˜Bsi˜D−1\nsi˜Csi=T−1\niArTi. (25)\nCombining (24) and the transfer function of DCU given in\nFig. 1, the sensitivity of λcwith respect to Kibecomes:\nsi=∂λc\n∂Ki=Ri·10λc\n1+10λc·1+T1iλc\n1+T2iλc·1+T3iλc\n1+T4iλc.(26)\nHere, the wash-out time constant of each DCU is assumed to\nbe 10, i.e.,Twi= 10 .\nIt follows from (26) that the tuning process of DCUs\ncan be split into two parts: 1) tuning parameters of lead-lag\ncompensation of the ithDCU such that ∠si= 180◦; and then\n2) tuning gain factors Kiof all DCUs such that λcmoves\nto the desired location, i.e.,/summationtextNW+NL\ni=1|si|∆Ki≥∆ℜ(λc)⋆\nwhere∆ℜ(λc)⋆=ς⋆|ωc|//radicalbig\n1−(ς⋆)2+σcis the expected\nreal part change of λc. For part 1), the parameters of T1i,T2i,\nT3i, andT4ican be calculated by [20]\n\n\nαi= (1+sin( ∠Ki)/2)/(1−sin(∠Ki)/2)\nT1i=T3i= (√αi)/ωc\nT2i=T4i= 1/(√αiωc)(27)\nwhere∠Ki= 180◦−∠Ri. For part 2), the gain factor change\n∆Kiis calculated by solving the following optimization\nproblem:\nminNW+NL/summationdisplay\ni=1ci (28)\ns.t.NW+NL/summationdisplay\ni=1|si|∆Ki≥∆ℜ(λc)⋆(29)\n∆Kmin\ni≤∆Ki≤∆Kmax\ni,i= 1,...,N W+NL\n(30)\nwhere∆Kmin\ni and∆Kmax\ni are the lower and upper bounds\non the gain factor of the ithDCU, respectively. To account for\nthe damping controller adjustments, in this work, we introd uce\na simple quadratic cost function for the ithdamping bus\nwhich has been widely used in the literature (e.g., [12]), i. e.,\nci=πi∆K2\niandπi>0is the cost parameter assigned for\ntheithdamping bus. The objective (28) is to minimize the\ntotal control cost. For convenience, the convex optimizati on\nproblem (28)-(30) is rewritten in a compact form as:\nmin\n∆KNW+NL/summationdisplay\ni=1ci(∆Ki)s.t. g(∆K)≤0,∆Ki∈∆ˆKi(31)where∆K= [∆K1,...,∆KNW+NL]Tdenotes the gain\nfactor changes of NW+NLDCUs;g(∆K)≤0represents the\nglobal constraint in (29); ∆ˆKirepresents the local constraint\nin (30).\nAs mentioned earlier, in this module, the proposed two-part\ntuning process will be realized in a distributed manner. For\nthe first part tuning process, it is realized locally as the Ri\nrequired of the ithdamping bus can be obtained locally. It\nfollows from (24) that Rican be calculated by ˜Ari,Bsi, and\nCsi. ForCsi, based on (21), it can be easily obtained as the\nithdamping bus knows the order of variables in xi. For˜Ari,\nit can be calculated by (25) as Tiis known locally and Ar\nhas been obtained in the critical LFO identification module\nfor each damping bus. For Bsi, based on (19)-(22), we have\n˜Asi=/bracketleftbiggAi+EiDCiCsiEiCCi\nBCiCsiACi/bracketrightbigg\n,˜Bsi=/bracketleftbiggBi\n0/bracketrightbigg\n˜Csi=/bracketleftbig\nCi+FiDCiCsiFiCCi/bracketrightbig\n,˜Dsi=Di.\n(32)\nTheBsican be obtained by (21) locally as: 1) matrices ACi,\nBCi,CCi, andDCiis known locally, 2) according to (19),\nmatrices ˜Asi,˜Bsi,˜Csi, and˜Dsican be calculated based\nonAs,Bs,Cs, andDswhich have been obtained by each\ndamping bus in the critical LFO identification module, and 3)\nCsican be obtained locally, then based on the matrix relations\nin (32), matrices Ai,Bi,Ci,Di,Ei, andFican be calculated.\nFor the second part tuning process, in order to solve the\nconvex optimization problem (28)-(30) in a distributed man ner,\nwe decompose the Lagrange function of (31) into a sum of\nNW+NLlocal Lagrange functions where each of them is\nassigned to a damping bus:\nL(∆K,µ) =NW+NL/summationdisplay\ni=1Li(∆K,µ) (33)\nwhereLi(∆K,µ) =ci(∆Ki) +µg(∆K), scalarµis the\nLagrange multiplier for g(∆K)≤0in (31).\nInspired by (33), based on the distributed Lagrangian\nprimal-dual sub-gradient algorithm proposed in [21], a dis -\ntributed algorithm is designed to update the decision varia bles\n∆Kand Lagrangian multiplier µvia communication between\nneighboring damping buses. The communication network used\nonly covers damping buses and is allowed to have a different\ntopology from the physical grid, which can be described by\nthe undirected graph G2={V2,E2,W}, whereV2=VW∪VL,\nE2⊆ V2× V2, andW={wij} ∈R(NW+NL)×(NW+NL). If\n(i,j)∈ E2,i/ne}ationslash=j, thenwij=wji>0and/summationtextNW+NL\nj=1,i/negationslash=jwij<1;\notherwise,wij=wji= 0. We define the diagonal entry wii\nof the matrix Waswii= 1−/summationtextNW+NL\nj=1,i/negationslash=jwij. In the proposed\ndistributed algorithm, the following assumptions are adop ted:\n1) The function gin (31) is known to all damping buses.\n2) The topology of the communication network G2is\nundirected and connected, and communication delays\nare negligible.\nFor assumption 1), since As,Bs,Cs, andDshave been ob-\ntained by each damping bus via the critical LFO identificatio n\nmodule, then all sensitivities siin functiongcan be calculated7\nlocally for each damping bus via the same method used for\ncalculatingRiin the first part tuning process.\nBased on the abovementioned assumptions, the update pro-\ncess of decision variables ∆Kand Lagrangian multiplier µis\nexpressed as follows:\n∆Ki(τ+1) =P∆ˆKi[∆¯Ki(τ)−ς(τ)DLi,∆¯Ki(τ)]\nµi(τ+1) =PˆUi[¯µi(τ)+ς(τ)DLi,¯µi(τ)](34)\nwhere∆Ki∈RNW+NLandµi∈Rare the information\ndata assigned for the ithdamping bus. We use ∆¯Ki(τ) =/summationtextNW+NL\nj=1wij∆Kj(r)and¯µi(r) =/summationtextNW+NL\nj=1wijµj(r)for\nshort. At each time τ+ 1, theithdamping bus calculates\nvectorsDLi,∆¯Ki=∂Li/∂(∆¯Ki)andDLi,¯µi=∂Li/∂¯µi\nin the gradient direction of its local Li. Combined with\ninformation received from its neighboring buses ∆¯Ki(τ)and\n¯µi(τ), theithdamping bus updates its own decision variables\n∆Ki(τ+ 1) andµi(τ+ 1) by taking a projection onto its\nlocal constraint ∆ˆKiandˆUi={µi≥0}, respectively. Here,\nthe projection operator P∆ˆKiis defined by the definition\nofP∆ˆKi[¯x] = argminx∈∆ˆKi/bardbl¯x−x/bardbl, where¯xis a given\nvector. The projection operator PˆUiis defined in the same way\nasP∆ˆKi. The diminishing step size is ς(r)which satisfies\nlimr→+∞ς(r) = 0 ,/summationtext+∞\nr=0ς(r) = +∞, and/summationtext+∞\nr=0ς(r)2<\n+∞. It has been proven in [21] that for a convex optimization\nproblem, the proposed distributed algorithm will asymptot ic-\nally converge to a pair of primal-dual optimal solutions (i. e.,\nlimτ→∞∆Ki(τ) = ∆K∗, i= 1,...,N W+NLwhere\n∆K∗= [∆K∗\n1,···,∆K∗\nNW+NL]Tis the optimal solution)\nunder the Slater’s condition, assumptions 1) and 2) mention ed\nabove. In our case, the optimization problem (28)-(30) is a\nconvex optimization program whose global optimal solution s\ncan be solved in a distributed way via the algorithm (34).\nIt is worth mentioning that, different damping buses have\ndifferent geometric controllability/obserbility measur es (COs)\nof the critical LFO λcunder different operating conditions\n[10]. The definition of the CO of the ithdamping bus is given\nasCOi=|ψT\nsiBsi|\n||ψsi||||Bsi||·|Csiφsi|\n||Csi||||φsi||which can be calculated\nlocally. In the proposed two-part tuning process, only the\ndamping buses with high COs participate the tuning process.\nIn other words, if the CO of the ithdamping bus satisfies\nCOi< CO⋆whereCO⋆is a threshold, then this damping\nbus does not participate the first part tuning process and the\nsecond part tuning process by setting ∆Kmin\ni= ∆Kmax\ni= 0\nin (30) locally.\nIV. C ASE STUDY\nIn this section, the modified IEEE 39-bus test system used\nfor simulation is introduced firstly. Then, the simulation r esults\nand explanations will be presented.\nA. Test system\nFig. 4 shows the modified IEEE 39-bus test system that is\nused to demonstrate the proposed distributed control frame -\nwork. In the modified 39-bus system, the SG at bus 10 is\nreplaced by a FRC-WTG with the same size of maximum\npower generation. All buses are renumbered according to the\nFigure 4. The modified IEEE 39-bus system\nrules described in Section II-B, i.e., NG= 9 ,NW= 1 ,\nNL= 17 , andNT= 12 . The damping buses considered\nare 1 WTG bus and 17 load buses. The model and system\nparameters are taken from [22]. For model parameters that\nare not provided in [22], we use the default values of models\ngiven in the library developed in PSAT/MATLAB [23].\nFor the communication network G2used in the control\nparameters tuning module, each edge is assigned a weight\nwhich can be calculated by a simplified computational method :\nwij=1\n1+max{˜Di,˜Dj}, i∈ V2, j∈˜Ni (35)\nwhere˜Nidefines the set of adjacent damping buses of the ith\ndamping bus with the definition of ˜Ni={j∈ V2: (j,i)∈ E2}\nand cardinality |˜Ni|=˜Di.\nB. Simulation results\nFollowing the procedure described in Section III-A, the cri t-\nical LFO identified by each damping bus is −0.0476±j1.7311\nwith the damping ratio ςc= 0.0275 (the preset threshold\nς⋆= 0.1) and oscillation frequency equal to 0.275Hz, where\nG2-G9 oscillate against G1 (see Fig. 6(a)). Then, the contro ller\nparameter tuning module is activated to tune the correspond ing\nparameters as described in Section III-B. The price paramet ers\nneeded for the optimization problem (28)-(30) are given in\nTable I. In this case study, for simplicity, we assume the\ngain limits for DCUs are the same (i.e. ∆Kmin=−10\nand∆Kmax= 60 ). As mentioned in Section III-B, only the\ndamping buses with high COs participate the tuning process.\nFig. 4 shows the COs of all 18 damping buses, and the\nthresholdCO⋆= 10−4. It follows from Fig. 4 that buses\n10, 17, 20, and 21 participate in the parameter tuning proces s.\nThe obtained optimal gain factor changes are also given in\nTable I. The Table II compares the original λc, expectedλc,\nand the new λc. It can be seen from Table II that the critical\nLFO is stabilized as desired.\nTo illustrate the effectiveness of the proposed distribute d\ncontrol framework, we investigate the variation of rotor an gle\nof G1 after a three-phase fault before and after the proposed\ntuning process. The three-phase fault happens at 1 s for 0.1\nseconds on bus 25. From Fig. 6(b) we can see that the system8\n10111213141516171819202122232425262700.511.5x 10−4\nBus numberControllability/obserbility (CO) measures\nFigure 5. CO measures of all damping buses\nTable I\nTHE PARAMETERS FOR DAMPING CONTROLLERS\nBus no. π∆K⋆Bus no. π∆K⋆\n10 0.8692 37 19 0.8524 0\n11 0.9566 0 20 0.9367 60\n12 0.7578 0 21 0.7306 33\n13 1.2769 0 22 0.9391 0\n14 0.8650 0 23 0.7993 0\n15 1.3035 0 24 1.1363 0\n16 1.3578 0 25 1.0443 0\n17 0.9937 42 26 0.7898 0\n18 1.0715 0 27 0.9862 0\nTable II\nORIGINAL ,EXPECTED ,AND NEW λc\nOriginalλc Expected λc Newλc\n−0.0476±j1.7311−0.1749±j1.7311−0.1785±j1.7340\n(a) Mode shape (b) Rotor angle response\n0 10 20 30-0.148-0.146-0.144-0.142-0.14-0.138-0.136-0.134-0.132\nTime (s)Rotor angle of G1 (p.u.)Before tuning\nAfter tuning\n0.51\n30\n21060\n24090\n270120\n300150\n330180 0G1G2-G9\nFigure 6. (a) Compass plot of relative mode shape (ref. G6) an d (b) rotor\nangle of G1 responses to a three-phase fault on bus 25\nperformance is improved with the presence of the proposed\ndistributed control framework.\nV. C ONCLUSION\nIn this paper, WTGs and LAs have been coordinated\nto provide damping torques for the critical low frequency\noscillation by adapting their active power generations and\nconsumptions, respectively. In order to provide a scalable\ncontrol framework for the increasing number of WTGs andLAs, a novel distributed control framework has been propose d\nwhich consists of a critical LFO identification module and a\ncontroller parameters tuning module. The simulation resul ts\nhave shown that the proposed distributed control framework\nis feasible and effective.\nREFERENCES\n[1] P. Kundur, Power System Stability and Control . New York: McGraw-\nHill, 1994.\n[2] M. Garmroodi, D. J. Hill, G. Verbic, and J. Ma, “Impact of t ie-line\npower on inter-area modes with increased penetration of win d power,”\nIEEE Trans. Power Syst., vol. 31, no. 4, pp. 3051-3059, Jul. 2016.\n[3] M. Singh, A. J. Allen, E. Muljadi, V . Gevorgian, Y . Zhang, and S.\nSantoso, “Interarea oscillation damping controls for wind power plants,”\nIEEE Trans. Sustain. Energy, vol. 6, no. 3, pp. 967-975, Jul. 2015.\n[4] A. E. Leon and J. A. Solsona, “Power oscillation damping i mprovement\nby adding multiple wind farms to wide-area coordinating con trols,” IEEE\nTrans. Power Syst., vol. 29, no. 3, pp. 1356-1364, May 2014.\n[5] T. Liu, D. J. Hill, and C. Zhang, “Non-disruptive load-si de control for\nfrequency regulation in power systems,” IEEE Trans. Smart Grid, vol. 7,\nno. 4, pp. 2142-2153, Jul. 2016.\n[6] Z. Tang, D. J. Hill, T. Liu, and H. Ma, “Hierarchical volta ge control of\nweak subtransmission networks with high penetration of win d power,”\nIEEE Trans. Power Syst., vol. 33, no. 1, pp. 187-197, Jan. 2018.\n[7] B. Ramanathan and V . Vittal, “Small-disturbance angle s tability en-\nhancement through direct load control part I-framework dev elopment,”\nIEEE Trans. Power Syst., vol. 21, no. 2, pp. 773-781, May 2006.\n[8] X. Zhang, C. Lu, S. Liu, and X. Wang, “A review on wide-area\ndamping control to restrain inter-area low frequency oscil lation for large-\nscale power systems with increasing renewable generation, ”Renew. and\nSustain. Energy Rev., vol. 57, pp. 45-58, May. 2016.\n[9] R. A. Ramos, A. C. Martins, and N. G. Bretas, “An improved m ethodo-\nlogy for the design of power system damping controllers,” IEEE Trans.\nPower Syst., vol. 20, no. 4, pp. 1938-1945, Nov. 2005.\n[10] Y . Zhang and A. Bose, “Design of wide-area damping contr ollers for\ninterarea oscillations,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1136-\n1143, Aug. 2008.\n[11] J. Deng, C. Li, and X. P. Zhang, “Coordinated design of mu ltiple robust\nFACTS damping controllers: A BMI-based sequential approac h with\nmulti-model systems,” IEEE Trans. Power Syst., vol. 30, no. 6, pp. 3150-\n3159, Nov. 2015.\n[12] S. P. Azad, R. Iravani, and J. E. Tate, “Damping inter-ar ea oscillations\nbased on a model predictive control (MPC) HVDC supplementar y\ncontroller,” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 3174-3183, Aug.\n2013.\n[13] D. Z. Fang, X. Yang, T. S. Chung, and K. P. Wong, “Adaptive fuzzy-logic\nSVC damping controller using strategy of oscillation energ y descent,”\nIEEE Trans. Power Syst., vol. 19, no. 3, pp. 1414-1421, Aug. 2004.\n[14] P. W. Sauer and M. A. Pai, Power System Dynamics and Stability . Upper\nSaddle River, New Jersey: Prentice Hall, 1998.\n[15] A. Perdana, “Dynamic models of wind turbines,” Ph.D. di ssertation,\nChalmers Univ. Technol., Gotteborg, Sweden, 2008.\n[16] J. Morren, S. W. De Haan, W. L. Kling, and J. A. Ferreira, “ Wind\nturbines emulating inertia and supporting primary frequen cy control,”\nIEEE Trans. Power Syst., vol. 21, no. 1, pp. 433-434, Feb. 2006.\n[17] Y . Song, D. J. Hill, T. Liu, and Y . Zheng, “A distributed f ramework\nfor stability evaluation and enhancement of inverter-base d microgrids,”\nIEEE Trans. Smart Grid, vol. 8, no. 6, pp. 3020-3034, Nov. 2017.\n[18] Z. Tang, D. J. Hill, and T. Liu, “Fully distributed volta ge control in\nsubtransmission networks via virtual power plants,” in Proc. IEEE Int.\nConf. on Smart Grid Commun., pp. 193-198, 2016.\n[19] F. L. Pagola, I. J. Perez-Arriaga, and G. C. Verghese, “O n sensitivities,\nresidues and participations: applications to oscillatory stability analysis\nand control,” IEEE Trans. Power Syst., vol. 4, no. 1, pp. 278-285, Feb.\n1989.\n[20] L. Rouco and F. L. Pagola, “An eigenvalue sensitivity ap proach to\nlocation and controller design of controllable series capa citors for\ndamping power system oscillations,” IEEE Trans. Power Syst., vol. 12,\nno. 4, pp. 1660-1666, Nov. 1997.\n[21] M. H. Zhu and S. Mart´ ınez, “On distributed convex optim ization under\ninequality and equality constraints,” IEEE Trans. Autom. Control, vol.\n57, no. 1, pp. 151-164, Jan. 2012.9\n[22] “IEEE PES Task Force on Benchmark Systems for Stability Controls,”\nI. A. Hisken, 2013 [online]. Available: http://eioc.pnnl. gov/benchmark/\nieeess/index.htm\n[23] F. Milano, “An open source power system analysis toolbo x,”IEEE Trans.\nPower Syst., vol. 20, no. 3, pp. 1199-1206, Aug. 2005.\nAPPENDIX\nA. SG model\nWith4th-order two-axis synchronous machine model\nand IEEE standard exciter model (IEEET1), the resulting\ndifferential-algebraic equations for the ithSG bus are given\nas:\n1) Differential equations:\n˙e′\nqi=1\nT′\ndoi/parenleftbig\n−e′\nqi−(xdi−x′\ndi)idi+vfi/parenrightbig\n˙e′\ndi=1\nT′\nqoi/parenleftbig\n−e′\ndi−(xqi−x′\nqi)iqi/parenrightbig\n˙δi=ω0(ωi−1)\n˙ωi=1\nMi/parenleftbig\npmi−e′\ndiidi−e′\nqiiqi−(x′\nqi−x′\ndi)idiiqi\n−Di(ωi−1))\n˙vmi=1\nTri(vi−vmi)\n˙vr1i=1\nTai/parenleftbigg\nKai(vrefi−vmi−vr2i−Kfi\nTfivfi)−vr1i/parenrightbigg\n˙vr2i=−1\nTfi(Kfi\nTfivfi+vr2i)\n˙vfi=−1\nTei(vfi(Kei+Sei(vfi)−vri))\n(36)\nwhereω0is the base frequency, T′\ndoiandT′\nqoi;xdiandxqi;\nx′\ndiandx′\nqi;idiandiqiare the d-axis and q-axis transient\ntime constant; reactance; transient reactance; current, r espect-\nively;pmi,Di, andMiare the mechanical power, damping\ncoefficient, and moment of inertia, respectively; vfiandvrefi\nare the field and reference voltages, respectively; Tri,Tai,Tfi,\nandTeiare measurement, amplifier, stabilizer, and field circuit\ntime constants, respectively; Kai,Kfi, andKeiare amplifier,\nstabilizer, and field circuit gains, respectively; Seiis the ceiling\nfunction.\n2) Algebraic equations: The stator algebraic equations are\ngiven as:\npGi=idivisin(δi−θi)+iqivicos(δi−θi)\nqGi=idivicos(δi−θi)−iqivisin(δi−θi).(37)\nIn order to express network voltages in the polar form, idiand\niqiin (36) and (37) are expressed in terms of state variables\nxGiand algebraic variables vi,θi:\n/bracketleftbiggidi\niqi/bracketrightbigg\n=/bracketleftbiggrsi−x′\nqi\nx′\ndirsi/bracketrightbigg−1/bracketleftbigge′\ndi−visin(δi−θi)\ne′\nqi−vicos(δi−θi)/bracketrightbigg\n(38)\nwherersiis the stator resistance. Substitution of (38) into (36)\nand (37) gives\n˙xGi=fGi(xGi,θi,vi)\npGi=gpGi(xGi,θi,vi)\nqGi=gqGi(xGi,θi,vi), i∈ VG(39)B. WTG model\nThe model of a WTG includes models of the direct drive\nsynchronous generator (DDSG), controller, and converter.\n1) DDSG model: As the stator and rotor flux dynamics\nare fast in comparison with grid dynamics and the converter\ncontrols decoupled the generator from the grid, the steady- state\nelectrical equations of DDSG are assumed. The differential\nand algebraic equations for DDSG of the ithWTG are given\nas:\n˙ωmi=1\n2Hmi(τmi−τei)\npsi=vsdiisdi+vsqiisqi\nqsi=vsqiisdi−vsdiisqi(40)\nwith\nτmi=pwi(θpi)\nωmi\nτei=ψsdiisqi−ψsqiisdi\nvsdi=−rsiisdi−ωmiψsqi\nvsqi=−rsiisqi+ωmiψsdi\nψsdi=−xsdiisdi+ψpmi\nψsqi=−xsqiisqi(41)\nwhereHmiis the rotor inertia; pwi(θpi)is the mechanical\npower which is the function of pitch angle θpi;τmiandτei\nare the mechanical and electrical torques, respectively; vsdi\nandvsqi;isdiandisqi;xsdiandxsqi;ψsdiandψsqiare stator\nd-axis and q-axis voltages; currents; reactances; and fluxe s,\nrespectively; psiandqsiare produced active and reactive\npower, respectively; rsiis the stator resistance; ψpmi is the\npermanent magnet flux of rotor. Assuming the power factor\nequal to 1 (permanent magnet rotor), the reactive power outp ut\nof the DDSG equals zero, i.e., qsi= 0. Substituting (41) into\n(40) and expressing isqiwithisdibased onqsi= 0 in (40)\ngives:\n˙ωmi=fmi(ωmi,θpi,isqi)\npsi=gspi(isqi).(42)\n2) Controller: The model of the controller includes models\nof the pitch angle control unit, primary frequency control u nit,\nand the DDCU. For pitch angle control unit, its dynamic is\ndescribed by the differential equation:\n˙θpi=1\nTpi(Kpiφi(ωmi−ωmrefi)−θpi) (43)\nwhereKpi,ωmrefi , andTpiare pitch control gain, reference\nrotor speed, and pitch control time constant, respectively ;φiis\na function which allows varying the pitch angle set point onl y\nwhen the difference ωmi−ωmrefi exceeds a predefined value\n±∆ωmi. For the primary frequency control unit, its control is\ngiven as:\npfi=Kfi(ωi−ω0) (44)10\nwithωi=˙θiwhereω0is the nominal frequency and Kfiis\nthe control gain. For the DDCU, its model is already given in\nSection II-A and repeated here for completeness:\n˙x1i=−1\nTwi(Kiθi+x1i)\n˙x2i=1\nT2i/parenleftbigg\n(1−T1i\nT2i)(Kiθi+x1i)−x2i/parenrightbigg\n˙x3i=1\nT4i/parenleftbigg\n(1−T3i\nT4i)/parenleftbigg\nx2i+/parenleftbiggT1i\nT2i(Kiθi+x1i)/parenrightbigg/parenrightbigg\n−x3i/parenrightbigg\nposci=x3i+T3i\nT4i/parenleftbigg\nx2i+T1i\nT2i(Kiθi+x1i)/parenrightbigg\n.\n(45)\n3) Converter model: Converter dynamics are highly simpli-\nfied as they are fast in comparison with the electromechanica l\ntransients. Thus, the converter are modeled as an ideal curr ent\nsource where isqiandidciare state variables and are used for\nthe active power/speed control and the reactive power/volt age\ncontrol, respectively. The differential equations for the con-\nverter of the ithWTG are given as:\n˙isqi=1\nTpri(isqrefi−isqi)\n˙idci=1\nTVi((vrefi−vi)−icdi)(46)\nwith\nisqrefi=prefi(ωmi)+pfi+posci\nωmi(ψpmi−xsdiisdi)(47)\nwhereisqrefi is the reference current, prefi(ωmi)is the power-\nspeed characteristic which roughly optimizes the wind ener gy\ncapture and is calculated by based on current rotor speed ωmi.\nThe active and reactive power injected into the grid from the\nconverter are given as:\npci=vcdiicdi+vcqiicqi\nqci=vcqiicdi−vcdiicqi(48)\nwherevcdi=−visinθiandvcqi=vicosθi.\nAssuming a lossless converter, the outputs of the WTG\nbecome\npWi=pci=psi\nqWi=vi/parenleftbigg\nicdicosθi+sinθi(psi+viicdisinθi)\nvicosθi/parenrightbigg\n.(49)\nSubstituting pfiin (44) andposciin (45) into (47), combining\n(42), (43), (45), (46), and (49) gives\n˙θi=ωi\n˙xWi=fWi(xWi,ωi,θi,vi)\npWi=gpWi(xWi,ωi,θi,vi)\nqWi=gqWi(xWi,ωi,θi,vi), i∈ VW.(50)\nC. Matrices\n1) MatricesAs,Bs,Cs, andDs:Refer to (51) on next\npage for the detailed definition, where the notation K∧\n∨(J∧\n∨)\nexpresses Jacobian matrix of the ∨in the subscript with respect\nto the∧in the superscript. It should be noted that all the\nJacobian matrices K∧\n∨are block diagonal matrices.2) MatricesK1,K2,K3, andJpf:Refer to (52) on next\npage for the detailed definition. All the Jacobian matrices J∧\n∨\nin (52) form the power flow Jacobian matrix Jpf∈R2N×2N\nwhere\nJpf=/bracketleftBigg\nJhp\nθJhp\nv\nJhq\nθJhq\nv/bracketrightBigg\n. (53)\n3) Elementary column operator matrix T:\n\nI8NG0 0 0 0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 INW0 0 0 0 0 0\n0 0I7NW0 0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 INL0 0 0 0 0\n0 0 0 0 I3NL0 0 0 0 0 0 0 0\n0 0 0 0 0 ING0 0 0 0 0 0 0\n0INW0 0 0 0 0 0 0 0 0 0 0\n0 0 0 INL0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 INT0 0 0 0\n0 0 0 0 0 0 0 0 0 ING0 0 0\n0 0 0 0 0 0 0 0 0 0 INW0 0\n0 0 0 0 0 0 0 0 0 0 0 INL0\n0 0 0 0 0 0 0 0 0 0 0 0 INT\n\n(54)11\n/bracketleftbiggAsBs\nCsDs/bracketrightbigg\n=\nKfGxG0 0 0 0 KfG\nθG0 0 0 KfGvG0 0 0\n0 0 0 0 0 0 INW0 0 0 0 0 0\n0KfW\nθWKfWxW0 0 0 KfWωW0 0 0 KfWvW0 0\n0 0 0 0 0 0 0 INL0 0 0 0 0\n0 0 0 KfL\nθLKfLxL0 0 0 0 0 0 0 0\nKhpGxGJhpG\nθW0JhpG\nθL0JhpG\nθG0 0JhpG\nθTJhpGvGJhpGvWJhpGvLJhpGvT\n0JhpW\nθWKhpWxWJhpW\nθL0JhpW\nθGKhpWωW0JhpW\nθTJhpWvGJhpWvWJhpWvLJhpWvT\n0JhpL\nθW0JhpL\nθLKhpLxLJhpL\nθG0KhpLωLJhpL\nθTJhpLvGJhpLvWJhpLvLJhpLvT\nKhqGxGJhqG\nθW0JhqG\nθL0JhqG\nθG0 0JhqG\nθTJhqGvGJhqGvWJhqGvLJhqGvT\n0JhqW\nθWKhqWxWJhqW\nθL0JhqW\nθGKhqWωW0JhqW\nθTJhqWvGJhqWvWJhqWvLJhqWvT\n0JhqW\nθW0JhqL\nθL0JhqL\nθG0KhqLωLJhqL\nθTJhqLvGJhqLvWJhqLvLJhqLvT\n(51)\n/bracketleftbiggK1K2\nK3Jpf/bracketrightbigg\n=\nKfGxG0 0 0 0 KfG\nθG0 0 0 KfGvG0 0 0\n0INW0 0 0 0 0 0 0 0 0 0 0\n0KfWωWKfWxW0 0 0 KfW\nθW0 0 0 KfWvW0 0\n0 0 0 INL0 0 0 0 0 0 0 0 0\n0 0 0 0 KfLxL0 0KfL\nθL0 0 0 0 0\nKhpGxG0 0 0 0 JhpG\nθGJhpG\nθWJhpG\nθLJhpG\nθTJhpGvGJhpGvWJhpGvLJhpGvT\n0KhpWωWKhpWxW0 0JhpW\nθGJhpW\nθWJhpW\nθLJhpW\nθTJhpWvGJhpWvWJhpWvLJhpWvT\n0 0 0 KhpLωLKhpLxLJhpL\nθGJhpL\nθWJhpL\nθLJhpL\nθTJhpLvGJhpLvWJhpLvLJhpLvL\nKhqGxG0 0 0 0 JhqG\nθGJhqG\nθWJhqG\nθLJhqG\nθTJhqGvGJhqGvWJhqGvLJhqGvT\n0KhqWωWKhqWxW0 0JhqW\nθGJhqW\nθWJhqW\nθLJhqW\nθTJhqWvGJhqWvWJhqWvLJhqWvT\n0 0 0 KhqLωL0JhqL\nθGJhqL\nθWJhqL\nθLJhqL\nθTJhqLvGJhqLvWJhqLvLJhqLvT\n(52)"    },    {        "title": "2108.07542v3.Application_of_Herglotz_s_Variational_Principle_to_Electromagnetic_Systems_with_Dissipation.pdf",        "content": "Application of Herglotz’s Variational Principle to\nElectromagnetic Systems with Dissipation\naJordi Gaset∗,bAdrià Marín-Salvador†\naEscuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Spain.\nbMathematical Institute, University of Oxford, United Kingdom.\nDecember 15, 2021\nAbstract\nThis work applies the contact formalism of classical mechanics and classical field theory, introduced\nby Herglotz and later developed in the context of contact geometry, to describe electromagnetic systems\nwith dissipation. In particular, we study an electron in a non-perfect conductor and a variation of the\ncyclotron radiation. In order to apply the contact formalism to a system governed by the Lorentz force,\nit is necessary to generalize the classical electromagnetic gauge and add a new term in the Lagrangian.\nWe also apply the k-contact theory for classical fields to model the behaviour of electromagnetic fields\nthemselves under external damping. In particular, we show how the theory describes the evolution\nof electromagnetic fields in media under some circumstances. The corresponding Poynting theorem is\nderived. We discuss its applicability to the Lorentz dipole model and to a highly resistive dielectric.\nKeywords: Contact geometry, Lagrangian systems, dissipation, field theories, Maxwell equations, electro-\nmagnetic gauge.\nMSC 2020 codes: 37K58, 37L05, 53D10, 35Q53\nContents\n1 Introduction 2\n2 Contact and k-contact Lagrangian systems 4\n2.1 k-contact Lagrangian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n∗jordi.gaset@unir.net( ORCID:0000-0001-8796-3149).\n†adria.marin@st-hughs.ox.ac.uk (ORCID: 0000-0001-8054-1576).\n1arXiv:2108.07542v3  [physics.class-ph]  14 Dec 20213 Study of Particles under a Lorentz Force 8\n3.1 Symplectic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.1.1 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.1.2 Classical Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.2 Contact Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.2.1 A First Attempt at the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 10\n3.2.2 Generalized Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n3.2.3 The Equations of Motion Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n3.2.4 Equivalent Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n3.2.5 Example. Electron in a non-Perfect Conductor . . . . . . . . . . . . . . . . . . . . . 14\n3.2.6 Example. Particle in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n4 Electromagnetic Fields 18\n4.1 Covariant Formulation of Classical Electromagnetism . . . . . . . . . . . . . . . . . . . . . . 18\n4.2k-contact Formulation for Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . 20\n4.3 Generalized Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n4.4k-contact Maxwell’s equations as Maxwell’s equations in matter . . . . . . . . . . . . . . . . 22\n4.5 Example. The Lorentz Dipole Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n4.6 Example. Highly Resistive Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n4.7 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n5 Conclusions 24\n1 Introduction\nAround 1930, G. Herglotz introduced a variational method to model mechanical systems with dissipation\n[24,25]. Herglotz allowed the Lagrangian of the system to depend on the action itself, and obtained a\nset of generalized Euler-Lagrange equations that happened to be useful when considering some dissipative\nphenomena. The framework of such description was later discovered to be contact geometry [13,19].\nRecently, there has been a renewed interest in contact geometry due to its success in modelling several\nsystems. Some of these applications include thermodynamics [2,21,31,33,34], statistical mechanics [3],\ngeometric optics [1], hydrodynamics [14], circuit theory [20] and control theory [8,35].\nIn parallel, contact geometry has been expanded with new structures and frameworks. The Lagrangian\nformulation and its symmetries have been studied [9–11,15,17], and several generalizations have been\ndeveloped: the higher-order case [5], the Hamilton-Jacobi theory [7] and the unified formalism [6].\nRecently, a higher-dimensional analogue of contact geometry has been proposed, called k-contact ge-\nometry, which can be applied to model field theories with dissipation [16,18]. This description generalizes\nprevious attempts to define an action principle for action-dependant Lagrangian field densities [27].\nThe present work applies the contact formalism of classical mechanics and classical field theory to\nelectromagnetism. This has, to the best of our knowledge, not been done explicitly in literature, apart from\n2a small example in [27]. Electromagnetism provides an interesting context in which to apply the theory. On\nthe one hand, the study of a particle under a Lorentz force allows for the use of the contact formalism of\nclassical mechanics. On the other, one can focus on the study of the evolution of the electromagnetic vector\nfields themselves applying the k-contact geometry theory. In addition, one can think of multiple examples\nin which a particle moves under the influence of a Lorentz force in the presence of external dissipation\n[32, Ch. 6]. It is also known how electromagnetic fields can be damped when in a medium [22, Sec. 11.5].\nIt should be clear that contact geometry does not provide a description of all dissipative phenomena, but\nrather simply produces a set of equations of motion from a Lagrangian function that is action-dependant.\nThus, not all dissipative systems can be explained via this theory and its use is not constrained to such\nmodels. This work explores to what extent dissipation in electromagnetic systems can be modelled using\ncontact formalism.\nDuring this work, some objects will be referred to as dissipative although they actually represent a break\nin energy conservation, which might be an increase or a decrease of the energy. It will we specified when\nthis is the case and when we actually refer to energy loss.\nDriven by previous successful applications of contact geometry in dissipation [10,16–18,27], we have\ndecided to produce the equations of motion from a Lagrangian function which is the classical symplectic\nLagrangian plus a linear term in the action. We allow the linear term to be tuned a posteriori in order to\nfit the experimental results in every particular example.\nThe currently developed contact techniques for classical mechanics and classical field theory only apply\nto autonomous Lagrangians. That is, in the classical mechanical case, the Lagrangians cannot depend\nexplicitly on time, and in the k-contact framework, the Lagrangian densities cannot depend explicitly on the\ncomponents of the spacetime. This is the reason for which during this work it will be asked that the objects\nappearing in the Lagrangian do not have explicit terms in such components. However, using variational\nprinciples, one can show that the produced equations of motion are still valid in the non-autonomous case.\nFuture research is needed in order to develop the mathematical tools to describe such Lagrangians using\nthe contact formalism.\nThisworkisstructuredasfollows. Section2presentsamathematicalintroductiontoLagrangiancontact\nsystems and Lagrangian k-contact systems. In Section 3, we apply the contact formalism of classical\nmechanics to study particles under a Lorentz force. When producing the equations of motion from the\nclassical Lagrangian of a Lorentz force, one finds that they are not gauge invariant. This can be solved\nby slightly changing the Lagrangian and introducing a new characterization of the electromagnetic gauge,\nwhich reduces to the classical one for non-dissipative systems. The theory is applied to two particular\nexamples, an electron in a non-perfect conductor and a particle in a magnetic field with damping. Finally,\nin section 4 we apply the k-contact theory to electromagnetic fields. Apart of the dissipative term, we\nconsider a particular metric to model lineal materials. In some cases, the classical Lagrangian density in\nvacuum can model electromagnetic fields in media when adding a linear term in the action. We derive\nsufficient and necessary conditions that a system must satisfy so that the theory applies. Finally, the\napplicability of the produced equations of motion is discussed in two real-life examples.\n32 Contact and k-contact Lagrangian systems\nLagrangian contact systems are an special case of contact systems, which we will introduce briefly. The\nreader can find a more detailed exposition in [10,11,17].\nConsider the manifold TQ\u0002R, whereQis anndimensional manifold which represents the configuration\nspace of the system, with local coordinates (qi;vi;s). The canonical endomorphism Sand the Liouville\nvector field \u0001ofTQextend toTQ\u0002Rin the usual way. Their local expressions are\nS=@\n@vi\ndqi;\n\u0001 =vi@\n@vi:\nA Lagrangian is a function L:TQ\u0002R!R. The associated contact form is \u0011L=ds\u0000S\u0003(dL), with local\nexpression\n\u0011L=ds\u0000@L\n@vidqi:\nThe Reeb vector field of \u0011Lis\nRL=@\n@s\u0000Wij@2L\n@vj@s@\n@vi\nwhere (Wij)is the inverse of the Hessian of L. The associated Lagrangian energy is EL= \u0001(L)\u0000L. Its\nlocal expression is\nEL=vi@L\n@vi\u0000L:\nThen, wehavethecontactsystem (\u0011L;EL). Itssolutionsareintegralcurvesofavectorfield X2X(TQ\u0002R),\nwhich is a SODE, and satisfies the generalized Euler-Lagrange equations\n\u0013Xd\u0011L=dEL\u0000\u0000\nLRLEL\u0001\n\u0011L;\n\u0013X\u0011L=\u0000EL:\nFor a holonomic curve \u001b(t) = (qi(t);vi(t);s(t)), these equations take the local expression\n_vj@2L\n@vi@vj+vj@2L\n@vi@qj+ _s@2L\n@vi@s\u0000@L\n@qi=@L\n@vi@L\n@s;\n_s=L: (1)\nHence,scan be interpreted as the action of the system. We are therefore modelling systems in which\nthe Lagrangian depends on the action itself.\nThe generalized Euler-Lagrange equations can also be obtained via a variational method, as seen in\n[10, Section 5], which was introduced by G. Herglotz in [24]. Let L2C1(TQ\u0002R)be a Lagrangian\nfunction and consider two points x;y2Q. Consider [0;1]\u0012Rand let us denote by Sthe space of all\nsmooth curves \u000b:[0;1]!Qsuch that\u000b(0) =xand\u000b(1) =y.\nLet us denote by C1([0;1]!X)the set of all smooth mappings from [0;1]to a manifold X. One can\ndefine the functional\nZ:C1([0;1]!Q)!C1([0;1]!R)\n\u00187! Z (\u0018);\n4whereZ(\u0018)is the unique solution to\n8\n><\n>:dZ(\u0018)(t)\ndt=L(\u0018(t);_\u0018(t);Z(\u0018)(t))\nZ(\u0018)(0) = 0:(2)\nAfter the previous discussion on the problem and particularly Equation (1), it is clear why one would\ndefine such an operator Z: given a curve \u0018, its imageZ(\u0018)is the action of the system associated to the\npath\u0018. Hence, the physically realisable path between xandyis the curve \u00182Sthat minimizes\nZ(\u0018)(1);\nthe action at the endpoint, see [10, Thm. 2]. Note that, from Equation (2), we find\nZ(\u0018)(1) =Z1\n0L(\u0018(t);_\u0018(t);Z(\u0018)(t))dt: (3)\nLet\u0011(t) =\u0000\n\u00111(t);\u00112(t);\u00113(t)\u0001\n2C1([0;1]!Q)such that\u0011(0) =\u0011(1) = 0and considerZ(\u0018+\"\u0011)(1).\nThe variational problem (3) implies that\n0 =dZ(\u0018+\"\u0011)(1)\nd\"\f\f\f\n\"=0=Z1\n0\u0010@L\n@qi\u0011i+@L\n@vi_\u0011i\u0011\ndt+Z1\n0@L\n@sdZ(\u0018+\"\u0011)(t)\nd\"\f\f\f\n\"=0dt: (4)\nLet us now define \u0010(r) =dZ(\u0018+\"\u0011)(r)\nd\"j\"=0andA(t) =@L\n@qi\u0011i+@L\n@vi_\u0011i. Note that, for r2(0;1],\n\u0010(r) =Zr\n0A(t)dt+Zr\n0@L\n@s\u0010(t)dt; (5)\nwhich implies\nd\u0010\ndr(r) =A(r) +@L\n@s\u0010(r):\nHence, solving the ODE with the initial condition \u0010(0) = 0,\n\u0010(r) = exp\u0010Zr\n0@L\n@s(\u0012)d\u0012\u0011\n\u0001Zr\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011\nA(t)dt:\nLetusimposethat \u0010(1) = 0, whichisequivalenttoimposingthattheactionreachesarelativeextremum.\nSince the first factor cannot vanish, this reads\n0 =Z1\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011\nA(t)dt=Z1\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011\u0010@L\n@qi\u0011i+@L\n@vi_\u0011i\u0011\ndt;\nand integrating by parts the second term\n0 =Z1\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011@L\n@qi\u0011idt+Z1\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011@L\n@vi_\u0011idt=\n=Z1\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011@L\n@qi\u0011idt\u0000Z1\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011hd\ndt\u0010@L\n@vi\u0011\n\u0000@L\n@s@L\n@vii\n\u0011idt=\n=Z1\n0exp\u0010\n\u0000Zt\n0@L\n@s(\u0012)d\u0012\u0011h@L\n@qi\u0000d\ndt\u0010@L\n@vi\u0011\n+@L\n@s@L\n@vii\n\u0011idt;\nwhich, by the fundamental lemma of calculus of variations [26, Lemm. 1.1.1], implies the generalized\nEuler-Lagrange equations\n@L\n@qi\u0000d\ndt\u0010@L\n@vi\u0011\n+@L\n@s@L\n@vi= 0:\n52.1 k-contact Lagrangian systems\nThe k-contact structure was introduced in [16] in order to generalize contact mechanics to field theories.\nThe Lagrangian formalism, which we use in Section 4, was developed in [18]. In this section the principle\nelements of this formalism are stated. Moreover, we will give a variational formulation of the k-contact\nEuler-Lagrange equations.\nThe Lagrangian k-contact formalism of a system with an k-dimensional configuration space Qover ank-\ndimensional space-time takes place in \bkTQ\u0002Rk. The bundle\bkTQoverQis the Whitney sum of kcopies\nof the tangent bundle, each one representing the derivative of the coordinate over the different directions\nof space-time. Moreover, kdissipative variables are considered. Natural coordinates on \bkTQ\u0002Rkare\n(qi;qi\n\u0016;s\u0016), where 1\u0014i\u0014nand1\u0014\u0016\u0014k.\nA Lagrangian is a function L:\bkTQ\u0002Rk!R. On this work we only consider a particular class of\nLagrangians, those with linear dependence on s\u0016with constant coefficients.\nThe k-contact structure is formed by the k1-forms\n\u0012\u0016\nL=ds\u0016\u0000@L\n@qi\u0016dqi:\nThe Reeb vector fields are a set of kvector fields such that\n\u0013R\u000b\u0012\f=\u000e\f\n\u000b;\u0013R\u000bd\u0012\f= 0: (6)\nFor the particular class of Lagrangians we will consider in this work, they can be chosen to be R\u0016=@\n@s\u0016.\nFinally, the Lagrangian energy is given by\nEL=@L\n@qi\u0016qi\n\u0016\u0000L:\nThe solutions are holonomic functions \u001b:Rk!\bkTQ\u0002Rk, which are integrable sections of a k-\ndimensional distribution. This distribution can be described by kvector fields (X\u0016)and a section \u001bis\nintegral of (X\u0016)if\nT\u001b\u000e@\n@x\u0016=X\u0016\u000e\u001b:\nThe k-contact Euler-Lagrange equations are\n\u0013X\u0016d\u0012\u0016\nL=dEL\u0000\u0000\nLR\u0016EL\u0001\n\u0012\u0016\nL;\n\u0013X\u0016\u0012\u0016\nL=\u0000EL:(7)\nFor a holonomic function \u001b= (qi(x\u0016);qi\n\u0016(x\u0016);s\u0016(x\u0016), these equations take the local expression\n@qj\n\u0017\n@x\u0016@2L\n@qi\u0016@qj\n\u0017+qj\n\u0016@2L\n@qi\u0016@qj+@s\u0017\n@x\u0016@2L\n@qi\u0016@s\u0017\u0000@L\n@qi=@L\n@qi\u0016@L\n@s\u0016;\n@s\u0016\n@x\u0016=L:\nThe Lagrangian k-contact formalism presented in [18] is developed for regular Lagrangians. Unfortunately,\nthe Lagrangian used in section 4 is singular. Nevertheless, one can try to follow all the steps described\nabove, but there is one problem: Equations (6) do not have a unique solution for singular Lagrangians. To\n6circumvent this issue, in section 4 we will use an idea introduced in [10] for the mechanical case: to proof\nthat Equations (7) are independent of the solution of (6) chosen.\nJust like in the classical contact formulation, it is possible to derive the k-contact Euler-Lagrange\nequations for fields as a result of a variational principle. Let L(qi;qi\n\u0016;s\u0016)be a Lagrangian function on\n\bkTQ\u0002Rk. A field on Qis a smooth map\n\t:\n\u001aRk! Q\u0002Rk\nx\u0017!(\t\u0016(x\u0017);\ts\u0016(x\u0017)\u0011s\u0016(x\u0017))\nfrom an open subset \n, and the action related to such field is defined as\nS(\t) =Z\n\nL(\t\u0016;@\u0017\t\u0016;s\u0016)d4x:\nThen, the equations of motion can be obtained by minimizing the action S(\t)with respect to \tunder the\nconstraint\n@\u0016s\u0016=L(\t\u0016;@\u0017\t\u0016;s\u0016):\nBy the Lagrange multiplier Theorem for Banach spaces [29, Thm. 9.3.1], extremizing Swith the above\nconstraint is equivalent to extremizing the following function with respect to \t,\nf(\t\u0016;@\u0017\t\u0016;s\u0016;@\u0017s\u0016;\u0015) =Z\n\nh\nL(\t\u0016;@\u0017\t\u0016;s\u0016)\u0000\u0015(x\u000b)\u0000\n@\u0016s\u0016\u0000L(\t\u0016;@\u0017\t\u0016;s\u0016)\u0001i\nd4x\n=Z\n\nh\n@\u0016s\u0016\u0000\u0015(x\u000b)\u0000\n@\u0016s\u0016\u0000L(\t\u0016;@\u0017\t\u0016;s\u0016)\u0001i\nd4x;\nwhere\u0015:\n!Ris a smooth function which we call Lagrange multiplier function. The Euler-Lagrange\nequations for fread8\n>>>>>><\n>>>>>>:@\u0017\u0010\n\u0015(x\u000b)@L\n@(@\u0017\t\u0016)\u0011\n=\u0015(x\u000b)@L\n@\t\u0016\n\u0000@\u0016\u0015(x\u000b) =\u0015(x\u000b)@L\n@s\u0016\n0 =@\u0016s\u0016\u0000L(\t\u0016;@\u0017\t\u0016;s\u0016);\nwhere the last equation comes from imposing that the Euler-Lagrange equations are also satisfied for \u0015.\nExpanding the first equation,\n\u0015(x\u000b)@L\n@\t\u0016=@L\n@(@\u0017\t\u0016)@\u0017\u0015(x\u000b) +\u0015(x\u000b)@\u0017\u0010@L\n@(@\u0017\t\u0016)\u0011\n=\u0000\u0015(x\u000b)@L\n@(@\u0017\t\u0016)@L\n@s\u0017+\u0015(x\u000b)@\u0017\u0010@L\n@(@\u0017\t\u0016)\u0011\n;\nwhich, dividing by \u0015(x\u000b), implies\n@\u0017\u0010@L\n@(@\u0017\t\u0016)\u0011\n\u0000@L\n@\t\u0016=@L\n@(@\u0017\t\u0016)@L\n@s\u0017;\nthek-contact Euler-Lagrange equations for fields.\n73 Study of Particles under a Lorentz Force\n3.1 Symplectic Formulation\n3.1.1 The Equations of Motion\nLetQ\u0012R3be an open subset of R3on which a particle of mass mand charge kmoves under the influence\nof the electromagnetic force\nF=k(E+v\u0002B)\ninduced by an electric field Eand a magnetic field Bwhich are time independent. Let q= (q1;q2;q3)be\ncoordinates defined on Q. Assume, in addition, that the electromagnetic potentials\n\u001e:Q!R\nq7!\u001e(q)A:Q!R3\nq7!A(q)\nthat define EandBare also time independent. Recall that Aand\u001eare such that\nE=\u0000r\u001eand B=r\u0002A:\nThe Lagrangian associated with the Lorentz force is\nL=m\n2v\u0001v+kA\u0001v\u0000k\u001e:\nThe Lagrangian energy function and the Lagrangian symplectic form read\nEL=@L\n@v\u0001v\u0000L=\u0000\nmv+kA\u0001\n\u0001v\u0000L=m\n2v\u0001v+k\u001e\nand\n!L=\u00003X\ni=1d\u0010@L\n@vi\u0011\n^dqi=\u00003X\ni=1d(mvi+kAi)^dqi=m3X\ni=1dqi^dvi+k3X\ni=1\u0010X\nj6=i@Ai\n@qjdqi^dqj\u0011\n:\nThe dynamics of the Lagrangian dynamical system (TQ;!L;EL)are encoded in the vector field XL2\nX(TQ)solution to\n\u0013XL!L=dEL: (8)\nOne can compute\ndEL=k@\u001e\n@q\u0001dq+mv\u0001dv:\nIf we write\nXL=3X\ni=1Qi@\n@qi+3X\ni=1Vi@\n@vi;\nthen Equation (8) reads\nm3X\ni=1Qidvi+3X\ni=1 \n\u0000mVi+kX\nj6=i\u0010@Aj\n@qi\u0000@Ai\n@qj\u0011\nQj!\ndqi=k3X\ni=1@\u001e\n@qidqi+m3X\ni=1vidvi:(9)\n8Since all the differentials are linearly independent, the previous equation implies\n8\n>><\n>>:Qi=vi\nmVi=kX\nj6=i\u0010@Aj\n@qi\u0000@Ai\n@qj\u0011\nQj\u0000k@\u001e\n@qi:(10)\n(11)\nHence, the time evolution of the particle in configuration space is given by (q(t);v(t))with _q(t) =v(t),\nand\nm_v(t) =kv(t)\u0002(r\u0002A)\u0000kr\u001e; (12)\nwhich are, of course, the equations of motion obtained by using the Euler-Lagrange equations. Note that\nwe findm_v=kE+kv\u0002B, which are the equations of motion obtained by applying Newton’s second law\nto the Lorentz force.\n3.1.2 Classical Gauge\nThe choice of vector and scalar potentials at the beginning of Section 3.1.1 is not unique. Indeed, let\nf2C1(Q\u0002R)be a smooth function on Q\u0002R, where the Rcoordinate depicts time, and let\n\u001e0=\u001e\u0000@f\n@tand A0=A+rf:\nAssume that the primed potentials are also time independent, which is equivalent to imposing\n@2f\n@t2=@2f\n@t@qi= 0:\nThen,\nr\u001e0=r\u001e\u0000r@f\n@t=r\u001e\u00003X\ni=1@2f\n@qi@t=r\u001e\nand\nr\u0002A0=r\u0002A+r\u0002(rf) =r\u0002A;\nand hence the induced electric and magnetic fields remain unchanged. This freedom in choosing the vector\nand scalar potentials is known as gauge freedom, and the choice of a particular pair (\u001e;A)is called a gauge\nfixing or a choice of gauge. It is believed that the gauge is not measurable since, as deduced in Section\n3.1.1, the equations of motion only depend on the observable fields EandB.\nAlthough the equations of motion remain unchanged when the gauge changes, the Lagrangian does not.\nIndeed, let us compute\nL0=m\n2v\u0001v+kA0\u0001v\u0000k\u001e0=L+k(rf)\u0001v+k@f\n@t=L+d(kf)\ndt:\nIt is known that the addition of a full time derivative to the Lagrangian does not change the extrema\nof the action and hence it does not change the equations of motion. However, it is important to note that\nthe Lagrangian and the action themselves do depend on the choice of gauge.\n93.2 Contact Formulation\n3.2.1 A First Attempt at the Equations of Motion\nConsider again a particle of mass mand charge kmoving in an open subset Q\u0012R3under the influence\nof a Lorentz force F=k(E+v\u0002B). Assume EandBare time independent and let (\u001e;A)be a choice\nof gauge for the electric and magnetic fields such that the potentials are also time independent. Driven\nby previous successful applications of contact formalism to mechanical systems with dissipation [10,17], we\npropose the following contact Lagrangian on TQ\u0002R:\nL=m\n2v\u0001v+kA\u0001v\u0000k\u001e\u0000\rs\nfor some\r2R.\nThe Lagrangian energy associated to this system is\nEL=@L\n@v\u0001v\u0000L= (mv+kA)\u0001v\u0000L=m\n2v\u0001v+k\u001e+\rs;\nand the canonical contact form equals\n\u0011L=ds\u0000@L\n@v\u0001dq=ds\u0000mv\u0001dq\u0000kA\u0001dq:\nSince all the second derivatives@2L\n@s@vivanish, the Reeb vector field is simply R=@\n@s. It is clear that the\ndirectional derivative of ELin the direction of@\n@sis the constant \r. Thus, the generalized Euler-Lagrange\nequations simplify to\n\u0013XLd\u0011L=dEL\u0000\r\u0011L\n\u0013XL\u0011L=\u0000EL(13)\n(14)\nfor an unknown vector field XL2X(TQ\u0002R). Let us compute the differentials\ndEL=kr\u001e\u0001dq+mv\u0001dv+\rds\nand\nd\u0011L=m3X\ni=1dqi^dvi+k3X\ni;j=1@Aj\n@qidqj^dqi:\nIf we write the unknown vector field XLin coordinates as\nXL=3X\ni=1Qi@\n@qi+3X\ni=1Vi@\n@vi+S@\n@s;\nthen Equation (14) reads\nS\u0000@L\n@v\u0001Q=\u0000@L\n@v\u0001v+L\nand if we impose that the system is holonomic, that is Qi= _qi=vi, then the solution curve (q(t);v(t);s(t))\nwill satisfy\n_s=L;\nand the Lagrangian is interpreted to depend on the actionR\nLdtitself.\n10If we now let _vi=Vi, Equation (13) implies\n\u0000m_v\u0001dq+k3X\ni=1X\nj6=i\u0010@Aj\n@qi\u0000@Ai\n@qj\u0011\nQjdqi=kr\u001e\u0001dq+\r(mv+kA)\u0001dq;\nand given the linear independence of the 1-forms dqi,\nm_v(t) =\u0000k(r\u0002A)\u0002v(t)\u0000kr\u001e\u0000\r(mv(t) +kA): (15)\nNote that we find the same two terms as in Equation (12), together with a dissipative term in velocities\n\u0000\rmv(t)and an interaction term between the dissipation and the vector potential, \u0000\rkA.\nThe obtained equations of motion are not invariant under the classical gauge defined in Section 3.1.2.\nIndeed, if we make a change of gauge\n\u001e0=\u001e\u0000@f\n@tand A0=A+rf\nbetween time independent potentials, the first two terms remain invariant, but the interaction term does\nnot, and hence then the equations of motion become\nm_v(t) =\u0000k(r\u0002A)\u0002v(t)\u0000kr\u001e\u0000\r(mv(t) +kA)\u0000\rkrf:\nNotethatachangeof gauge withasmoothfunction fintroducesafulltimederivativeintheLagrangian,\nas discussed in Section 3.1.2. Hence, we observe that, in the contact framework, adding a total time\nderivative does not, in general, preserve the equations of motion. The Lorentz force provides an example\nof how producing equivalent Lagrangians in the contact framework differs from the symplectic case.\nSince we want the equations of motion to be gauge invariant, and the gauge to be non-observable, we will\nintroduce a new characterization of the gauge and propose a new contact Lagrangian in the next section.\n3.2.2 Generalized Gauge\nIn this section we introduce a new characterization of the classical gauge that generalizes the one defined\nin Section 3.1.2. Instead of a pair (\u001e;A), a choice of gauge will now be a triplet\n(\u001e;A;f);\nwhere (\u001e;A)is a classical gauge and f2C1(Q)is a smooth function on Q. As discussed in previous\nsections, only time independent scalar and vector potentials are considered. Note that, since fis a function\nonQ, it is time independent.\nWe will define two choices of gauge (\u001e;A;f);(\u001e0;A0;f0)to be related by the gauge if there exists a\nsmooth function g2C1(Q\u0002R)such that\n8\n>>>><\n>>>>:\u001e0=\u001e\u0000@g\n@t\nA0=A+rg\nf0=f\u0000g:\nFor both gauges to be time independent, it will be necessary and sufficient that@g\n@t= 0. Note that this\nis equivalent to g2C1(Q)and that it implies that the scalar potential remains unchanged. Note also that\n11the relation defined on the choices of gauge is an equivalence relation, and hence it produces equivalence\nclasses of choices of gauge. Let [(\u001e;A;f)]denote the equivalence class of the choice of gauge (\u001e;A;f).\nWe claim that, in each class [(\u001e;A;f)], there exists a unique choice of gauge of the type (\u001e0;A0;0).\nIndeed,\n[(\u001e;A;f)] = [(\u001e\u0000@f\n@t;A+rf;f\u0000f)] = [(\u001e\u0000@f\n@t;A+rf;0)];\nand if [(\u001e;A;0)] = [(\u001e0;A0;0)]the function g2C1(Q)that relates them must satisfy 0 = 0\u0000g, and hence\nmust beg= 0. Then,\u001e=\u001e0andA=A0,\nWe propose, for a choice of gauge (\u001e;A;f), the contact Lagrangian\nL=m\n2v\u0001v+kA\u0001v+k@f\n@q\u0001v\u0000k\u001e\u0000\rs; (16)\nwhere we explicitly make use of the function f2C1(Q)of the gauge.\n3.2.3 The Equations of Motion Revisited\nAssume a particle of mass mand charge kis moving in an open subset Q\u0012R3under the influence of an\nelectromagnetic force. Let (\u001e;A;f)be a choice of gauge as defined in Section 3.2.2, and let the contact\nLagrangian of the system be the one defined in Equation (16),\nL=m\n2v\u0001v+kA\u0001v+k@f\n@q\u0001v\u0000k\u001e\u0000\rs: (17)\nThe Lagrangian energy density associated with this Lagrangian is\nEL=@L\n@v\u0001v\u0000L=m\n2v\u0001v+k\u001e+\rs;\nand the contact 1-form is\n\u0011L=ds\u0000@L\n@v\u0001dq=ds\u0000mv\u0001dq\u0000kA\u0001dq\u0000k@f\n@q\u0001dq:\nJustlikeinSection3.2, theReebvectorfieldis RL=@\n@s, andthedirectionalderivativeoftheenergywith\nrespect to the Reeb vector field is LRLEL=@EL\n@s=\r. Hence, the generalized Euler-Lagrange equations\nread\n\u0013XLd\u0011L=dEL\u0000\r\u0011L\n\u0013XL\u0011L=\u0000EL:(18)\n(19)\nA straightforward computation gives\ndEL=mv\u0001dv+kr\u001e\u0001dq+\rds\nand\nd\u0011L=m3X\ni=1dqi^dvi+k3X\ni;j=1\u0010@Aj\n@qi\u0011\ndqj^dqi:\nSince the only difference with respect to Equations (13) and (14) is the extra term \u0000krf\u0001dqin the\ncontact form, the equations of motion are\n_s=L\n12and\nm_v(t) =\u0000k(r\u0002A)\u0002v(t)\u0000kr\u001e\u0000\r\u0000\nmv(t) +kA\u0001\n\u0000\rkrf; (20)\nonce we have imposed that the system is holonomic, i.e. _q=v.\nWe claim that Equation (20) is now independent of the choice of gauge. Indeed, if we take g2C1(Q)\nand let 8\n>>><\n>>>:\u001e0=\u001e\nA0=A+rg\nf0=f\u0000g;\nEquation (20) becomes\nm_v(t) =\u0000k(r\u0002A)\u0002v(t)\u0000kr\u001e\u0000\r\u0000\nmv(t) +kA\u0001\n\u0000\rkrg\u0000\rkrf+\rkrg=\n=\u0000k(r\u0002A)\u0002v(t)\u0000kr\u001e\u0000\r\u0000\nmv(t) +kA\u0001\n\u0000\rkrf;\nand thus, it remains unchanged. Actually, note that the Lagrangian (16) is itself invariant under a change\nof gauge, unlike in the symplectic case.\nIn this new proposed framework, the observable fields are\n8\n>>><\n>>>:E=\u0000r\u001e\nR=A+rf\nB=r\u0002A=r\u0002R(21)\nfor a choice of gauge (\u001e;A;f), and the equations of motion are\nm_v=kv(t)\u0002B+kE\u0000\rmv(t)\u0000\rkR;\nwhere all three fields are invariant under the gauge. Note that knowing Rallows us to know Aup to a\ngradient, which is exactly the same freedom for Awhen knowing B. In addition, the generalized moment\nof a particle under the proposed Lagrangian is\n@L\n@v=mv+kA+krf=mv+kR;\nwhich is also an observable, whilst in the symplectic case the generalized moment is not an observable.\n3.2.4 Equivalent Lagrangians\nIn order to have a more complete understanding of the gauge freedom of the Lagrangian, let us analyze its\nequivalent Lagrangians.\nTwo Lagrangians are equivalent if they lead to the same solutions. In symplectic mechanics, two La-\ngrangians that differ by a total derivative are equivalent. In contact mechanics, one can construct equivalent\nLagrangians by considering transformations for the svariable, which can be thought of as constructing new\nactions with the same critical points. One can generalize the symplectic result to the contact setting by\nconsidering transformations of the form s!\u0010(qi;s) =s+h(qi).\nFor a Lagrangian Land a transformation \u0010, the corresponding equivalent Lagrangian is given by:\n\u0016L(x;q;v;\u0010 ) =L(x;q;v;s ) +rh\u0001v:\n13Notice that the action variable for the new Lagrangian is \u0010. Considering the Lagrangian (17) we are\ninterested in, for any function h(qi)we have an equivalent Lagrangian\n\u0016L(x;q;v;\u0010 ) =m\n2v\u0001v+kA\u0001v+krf\u0001v\u0000k\u001e\u0000\r(\u0010\u0000h) +rh\u0001v:\nSettingh=kfwe have a particularly interesting equivalent Lagrangian:\n\u0016L(x;q;v;\u0010 ) =m\n2v\u0001v+kA\u0001v\u0000k\u001e\u0000\r(\u0010+kf); (22)\nwhich is another gauge invariant realization for a particle moving with dissipation under a Lorentz force\ngiven by (\u001e;A;f). Indeed, performing a gauge transformation given by a function g, we obtain\n\u0016\u0016L(x;q;v;\u0010 ) =m\n2v\u0001v+kA\u0001v+krg\u0001v\u0000k\u001e\u0000\r(\u0010\u0000kg+kf);\nwhich is an equivalent Lagrangian to (22) given by the transformation s!s+kg.\nWhen performing a gauge transformation on the Lagrangian (22) we obtain an equivalent Lagrangian,\nas in the symplectic case. On the other hand, a gauge transformation leaves (17) invariant, with no need\nto invoke equivalence theorems. Notice that this is also the case when we recover the symplectic case\nsetting\r= 0. The triplet description of the electromagnetic gauge (\u001e;A;f), together with the term\nkA\u0001v+krf\u0001v\u0000k\u001ein the Lagrangian gives us a variational description of the Lorentz force where the\nLagrangian is invariant under gauge transformations (in both the classical and the contact settings).\n3.2.5 Example. Electron in a non-Perfect Conductor\nWe shall now apply the formalism discussed in Section 3.2.3 to describe the motion of an electron in a\nnon-perfect conductor.\nConsider a sufficiently large but finite cylindrical non-perfect conductor of length Land cross-section\nA. Let\u001bdenote the conductivity of the material. Assume a voltage \u0001Vis applied between the ends of the\nconductor, which is known to generate an electric field of constant magnitude\nE=\u0000\u0001V\nL\nin the longitudinal direction of the conductor. We take this direction to correspond to be the xcoordinate.\nAssume no magnetic fields intervene in the problem. Hence, one can take the vector potential to be A= 0\nand the gauge to be\n(\u001e;A;f) =\u0010\u0001V\nLx;0;0\u0011\nto describe the problem. Indeed, the observables for this choice as described in Equation (21) are\nE=\u0000\u0001V\nLexand B=R= 0;\nwhere exdenotes the unit vector in the xdirection. The equations of motion (20) read\n_v(t) =\u0000\u0016\u0001V\nL\u0000\rv(t) (23)\n14in thexdirection, where \u0016:=ke\nme<0, forkeandmethe charge and mass of the electron respectively. If\nwe assume the electron starts at rest at x= 0, the previous ODE can be solved for the velocity to find\nv(t) =\u0016\u0001V\n\rL\u0010\ne\u0000\rt\u00001\u0011\n; (24)\nand Equation (24) can be integrated to obtain\nx(t) =\u0000\u0016\u0001V\n\r2L(\rt+e\u0000\rt\u00001):\nNote that the electron reaches a limit velocity\nvlim= lim\nt!1v(t) =\u0000\u0016\u0001V\n\rL>0: (25)\nGiven that the conductor is ohmic, the limit velocity (or drift velocity) of an electron in the conductor can\nbe shown to be [12, p. 187]\nvlim=\u0000mmol\u001b\u0001V\n\u001akenL;\nwheremmolis the molecular mass of the conductor, \u001ais the density of the conductor and nthe number\nof free electrons per molecule of conductor. Hence, one can impose the limit velocity in (25) is the drift\nvelocity of the electron to find\n\r=\u0016ke\u001an\nmmol\u001b:\nIf we take copper as the material the conductor is made of, one finds that \r\u00194\u00011013s\u00001and hence\nthe limit velocity is achieved at the order of t\u001910\u000013s. This means that the electron has travelled\napproximately 3\u000110\u000016m. Thus, the assumption that Lis large enough so that the limit velocity is\nachieved is physically realizable.\nThe energy of the electron in the described electric field is known to be,\nE=me\n2v2+ke\u0001V\nLx\nand hence it dissipates at a rate\ndE\ndt=mev(t) _v(t) +ke\u0001V\nLv(t)\n=\u0000ke\u0016(\u0001V)2\n\rL2\u0010\ne\u0000\rt\u00001\u0011\ne\u0000\rt+ke\u0016(\u0001V)2\n\rL2\u0010\ne\u0000\rt\u00001\u0011\n=\n=\u0000ke\u0016(\u0001V)2\n\rL2\u0010\ne\u0000\rt\u00001\u00112\n;\nwhich, ast!1, when the drift velocity is achieved, becomes\ndE\ndt=\u0000ke\u0016(\u0001V)2\n\rL2=\u0000mmol\u001b(\u0001V)2\nn\u001aL2: (26)\nAssume there is no interaction between the electrons in the conductor, which implies that all of them\ndissipate energy at the same rate when they reach the drift velocity. The number of electrons in the\nconductor is given in the variables of the problem by\nne=nMT\nmmol=n\u001aAL\nmmol;\n15whereMTdenotes the total mass of the conductor. Hence, the total energy dissipation rate within the\nconductor when all electrons have achieved the drift velocity is\ndET\ndt=nedE\ndt=\u0000A\u001b\nL(\u0001V)2;\nwhich is precisely Joule’s heating law.\n3.2.6 Example. Particle in a Magnetic Field\nConsider a particle of charge kand massm, in a magnetic field B= (0;0;B). A choice of vector potential\nforBis\nR=A=B\n2(\u0000y;x;0);\ntakingf= 0, and if we assume no electric fields are present, the equations of motion (20) read\n8\n><\n>:_vx=\u0016Bvy\u0000\rvx+\r\u0016\n2By\n_vy=\u0000\u0016Bvx\u0000\rvy\u0000\r\u0016\n2Bx;\nwhere\u0016:=k\nm. This system of equations is not easy to discuss for arbitrary values of \r. However, if we\ndefine the energy of the particle as its kinetic energy, we see that\ndE\ndt=m\n2d\ndt(v2\nx+v2\ny) =m(vx_vx+vy_vy) =m\u0010\nvx(\u0016Bvy\u0000\rvx+\r\u0016\n2By) +vy(\u0000\u0016Bvx\u0000\rvy\u0000\r\u0016\n2Bx)\u0011\n=\u0000\rm(v2\nx+v2\ny) +m\r\u0016B\n2(vxy\u0000xvy) =\u0000\rmv2\u0000\r\u0016\n2B\u0001L:\nIt is known that a charged particle spinning in a magnetic field which is perpendicular to its velocity\nexperiences a dissipation of its energy due to the emission of radiation [30]. This effect is known as cyclotron\nradiation. The frequency of a particle of mass memitting cyclotron radiation in the classical limit is\nf=kB\n2\u0019m;\nand the energy dissipated due to the cyclotron radiation satisfies [28, Eq. 18.8]\ndE\ndt=\u0000\u001btB2v2\nc\u00160;\nwhere\u001bt=8\u0019\n3\u0010\nk2\n4\u0019\u000f0mc2\u00112\nis the Thomson total cross-section.\nIn order to argue about the nature of the motion of the particle, let us assume that the cross terms\n\r\u0016\n2Bxand\r\u0016\n2Bycan be neglected. Then, the system reads\n(_vx=\u0016Bvy\u0000\rvx\n_vy=\u0000\u0016Bvx\u0000\rvy;\nand can be solved by 8\n<\n:vx(t) =e\u0000\rt\u0000\nvx0cos(\u0016Bt) +vy0sin(\u0016Bt)\u0001\nvy(t) =e\u0000\rt\u0000\nvy0cos(\u0016Bt)\u0000vx0sin(\u0016Bt)\u0001\n;\nfrom which we deduce that the particle describes a decreasing spiral in the plane if \r >0. Note that the\nfrequency of the motion is exactly the frequency of a particle emitting cyclotron radiation.\n16Let the energy of the particle be\nE(t) =mv2(t)\n2=mv2\nx+v2\ny\n2=me\u00002\rt\n2v2\n0;\nwherev2\n0=v2\nx0+v2\ny0is the square of its initial velocity. Then, the energy is dissipated at a rate\ndE\ndt=\u0000m\re\u00002\rtv2\n0=\u0000m\rv2;\nand we only obtain the term in the cyclotron dissipation. Let us impose that the dissipated energy of the\nmodel is precisely the dissipation in the cyclotron radiation. Then,\n\r=\u001btB2\nc\u00160m>0: (27)\nIfwetaketheparticletobeanelectron, then \u0016\u0019\u00001:76\u00011011C=kg. IfweletB= 1T, then\r\u00190:1938s\u00001.\nLet usreturn now to thegeneral case, consideringthe cross-terms. Letus assume we canfix \ras inEquation\n(27) in order to estimate the solutions of the system. Assume that 0<\r\u001cj\u0016Bj. If we let\u0011=x+iy, then\n\u0011=\u0000_\u0011\u0000\ni\u0016B+\r\u0001\n\u0000i\r\u0016B\n2;\nand the eigenvalues of the characteristic polynomial are\n8\n>>><\n>>>:\u0000\r\u0000i\u0016B\n2+i\n2p\n\u00162B2\u0000\r2\u0019\u0000\r\n2\u0000i\r2\n4\u0016B\n\u0000\r\u0000i\u0016B\n2\u0000i\n2p\n\u00162B2\u0000\r2\u0019\u0000\r\n2\u0000i\u0016B+i\r2\n4\u0016B:\nThe frequency of the motion is altered in the order O(\r2\n\u0016B). In addition, if 0< \r\u001cj\u0016Bj, the particle\nwill describe a slightly perturbed decreasing spiral motion for small times. The term\n\u0000\r\u0016\n2B\u0001L\nis always positive, but it can be seen that, when 0<\r\u001cj\u0016Bj, it is smaller in norm than the dissipative\nterm due to the cyclotron radiation. Hence, the particle does indeed lose energy, but at a lower rate than in\nthe cyclotron. We are modelling the small-time behaviour of a particle inside a magnetic field that dissipates\nenergy due to the emission of a cyclotron radiation that has been altered by the interaction between the\nexternal magnetic field and the angular momentum of the particle itself.\n17Figure 1: Position (left) and energy (right) of an electron starting at the origin with velocity \u00001:76\u0001\n1011m=seybetweent= 0sandt= 10s. Darker color represents larger times. Note that the energy of the\nelectron follows an exponential pattern plus an oscillating fashion of period \u00185\u000110\u000011sand amplitude of\nthe order of 10\u000018J.\n4 Electromagnetic Fields\n4.1 Covariant Formulation of Classical Electromagnetism\nIn Section 4.1, we introduce the main objects and tools of the covariant formulation of classical elec-\ntromagnetism, so that our work is self-contained. Throughout Section 4, Einstein’s summation conven-\ntion will be used unless specifically mentioned, and the considered metric in Minkowski space will be\n\u0011\u000b\u0016=\u0011\u000b\u0016=diag(1;\u00001;\u00001;\u00001).\nRecall that the four-displacement tensor is defined as x\u0016= (ct;x;y;z ), wherecis the speed of light in\nvacuum, and the covariant four-gradient is @\u0016=@\n@\u0016=\u0010\n1\nc@\n@t;r\u0011\n. IfC\u000bis a four-tensor, we will use the\nnotation@\u0016C\u000b=C\u000b;\u0016.\nThe main object of this formulation of electromagnetism is the covariant antisymmetric tensor\nF\u000b\u0016=0\nBBBBBB@0\u0000Ex=c\u0000Ey=c\u0000Ez=c\nEx=c 0\u0000BzBy\nEy=c Bz 0\u0000Bx\nEz=c\u0000ByBx 01\nCCCCCCA(28)\nfor a pair of electric and magnetic vector fields E= (Ex;Ey;Ez);B= (Bx;By;Bz). The tensor F\u000b\u0016is\nknown as the electromagnetic tensor. If \u001eandAare a choice of scalar and vector potentials of Eand\nBunder the classical gauge defined in Section 3.1.2, the electromagnetic four-potential is defined to be\nA\u000b=\u0010\n\u001e\nc;A\u0011\n, which satisfies F\u000b\u0016=@\u000bA\u0016\u0000@\u0016A\u000b=A\u0016;\u000b\u0000A\u000b;\u0016.\nFinally, an electric charge density \u001aand an electric current density jdefine the tensor J\u000b= (c\u001a;j),\nwhich is known as the four-current.\nThe four Maxwell’s equations in vacuum in vector notation reduce to two tensor equations. The first,\n18known as the Gauss-Faraday law, reads\n@\u0016\u00101\n2\u000f\u0016\u000b\f\u001cF\f\u001c\u0011\n= 0; (29)\nwhere\u000f\u0016\u000b\f\u001cis the Levi-Civita tensor, and it comes from the fact that @\u0016F\u0017\u0015+@\u0017F\u0015\u0016+@\u0015F\u0016\u0017= 0.\nThe Gauss-Faraday law is the same in vacuum and in media, and thus it will not be central in our\ndiscussion. The other Maxwell’s equation is known as de Gauss-Ampère law and reads\n@\u0016F\u0016\u000b=\u00160J\u000b; (30)\nwhere\u00160is the magnetic permeability of vacuum.\nThe Lagrangian density for classical electromagnetism is defined to be\nL=\u00001\n4\u00160F\u000b\u0016F\u000b\u0016\u0000A\u000bJ\u000b; (31)\nand it derives the Gauss-Ampère law via the Euler-Lagrange equation for fields [4, Ch. 1.10]. The electro-\nmagnetic energy density is defined as u=\u000f0\n2E\u0001E+1\n2\u00160B\u0001B, and the energy flux density is given by the\nPoynting’s vector field S=E\u0002B\n\u00160.\nWhen considering electromagnetic fields in matter, the polarization density and magnetization density\nvector fields PandMencode the response of the medium to the incoming electric and magnetic vector\nfields, see [12]. The electric displacement vector is then defined as D=\u000f0E+P, where\u000f0is the electric\npermittivity of vacuum, and the magnetic intensity is H=1\n\u00160B\u0000M.\nThese quantities can be absorbed into the antisymmetric magnetisation-polarisation tensor\nM\u000b\u0016=0\nBBBBBB@0Pxc Pyc Pzc\n\u0000Pxc 0\u0000MzMy\n\u0000Pyc Mz 0\u0000Mx\n\u0000Pzc\u0000MyMx 01\nCCCCCCA(32)\nand the antisymmetric electromagnetic displacement tensor\nD\u000b\u0016=0\nBBBBBB@0\u0000Dxc\u0000Dyc\u0000Dzc\nDxc 0\u0000HzHy\nDyc Hz 0\u0000Hx\nDzc\u0000HyHx 01\nCCCCCCA; (33)\nwhich are related to the electromagnetic tensor via\nD\u000b\u0016=1\n\u00160F\u000b\u0016\u0000M\u000b\u0016: (34)\nLet us also recall that the bound current in a material is defined as\nJ\u000b\nbd:= (c\u001abd;jbd):=@\u000bM\u000b\u0016=\u0000\n\u0000cr\u0001P;@P\n@t+r\u0002M\u0001\n: (35)\nAs discussed above, the Gauss-Faraday law (29) does not change when considering fields in matter.\nHowever, the Gauss-Ampère law (30) becomes\n@\u0016D\u0016\u000b=J\u000b; (36)\n19which is known as the Gauss-Ampère law in matter. The electromagnetic energy density in a material\nbecomesumatter =1\n2\u0000\nE\u0001D+B\u0001H\u0001\nand the Poynting’s vector is Smatter =E\u0002H.\n4.2 k-contact Formulation for Electromagnetic Fields\nIn the current section, we develop a theory that allows us to model linear and dissipative electromagnetic\nsystems, as well as systems with contributions from both. The linear features are obtained by replacing the\nMinkowski metric of the spacetime with a diagonal metric that encodes information of the material, whilst\nthe dissipative facet is modelled by introducing a linear term in the action to the Lagrangian density. Our\ndiscussion generalises the results derived in [27].\nRecall that a linear material is such that there exist constants \u001feand\u001fmfor which\n8\n><\n>:P=\u000f0\u001feE\nM=1\n\u00160\u001fm\n1 +\u001fm:=\u001fm\n\u0016B:(37)\nThe Gauss-Ampère law for a linear material can be obtained from the electromagnetic Lagrangian\ndensity in vacuum where the Minkowski metric has been replaced. For a particular four-current J\u000b:U\u001a\nM4!Q\u001aR4, define\nL=\u00001\n4\u00160g\u000b\u0016g\f\u0017F\u0016\u0017F\u000b\f\u0000A\u000bJ\u000b2C1\u00104M\ni=1TQ\u0011\n;\nwith the symmetric bilinear form\ng=g\u0016\u0017=1p1 +\u001fmdiag\u0010\n(1 +\u001fe)(1 +\u001fm);\u00001;\u00001;\u00001\u0011\nonU.\nThe Euler-Lagrange equations for fields imply g\u0016\u001bg\u000b\u001c@\u0016F\u001b\u001c=\u00160J\u000b, which reads\n8\n>><\n>>:\u00160c\u001a=\u00160J0=g\u0016\u001bg0\u001c@\u0016F\u001b\u001c=g00gii@iFi0=1 +\u001fe\ncr\u0001E() r\u0001 D=\u001a\n\u00160ji=\u00160Ji=giig\u0016\u001b@\u0016F\u001bi=giig00\nc2@Ei\n@t+ (gii)2\u0000\nr\u0002B\u0001\ni()j=\u0000@D\n@t+r\u0002H;\nthe non-geometric Maxwell’s equations for a material satisfying (37).\nWe now add a linear term in the action to this Lagrangian density in order to model a larger subset of\nmaterials. Let\nL=\u00001\n4\u00160g\u000b\u0016g\f\u0017F\u0016\u0017F\u000b\f\u0000A\u000bJ\u000b\u0000\r\u000bs\u000b2C1\u00104M\ni=1TQ\u0002R\u0011\n:\nFollowing the discussion in [18], the Lagrangian energy density defined by Lis\nEL=A\u0016;\u000b@L\n@A\u0016;\u000b\u0000L=1\n\u00160g\u0016\u0017g\u000b\fA\u0016;\u000bF\u0017\f+1\n4\u00160g\u0016\u0017g\u000b\fF\f\u0017F\u000b\u0016+A\u000bJ\u000b+\r\u000bs\u000b\nand gives rise to four contact 1-forms\u0012\u000b\nL2\n\u00104L\ni=1TQ\u0002R4\u0011\ndefined by\n\u0012\u000b\nL=ds\u000b\u0000@L\n@A\u0016;\u000bdA\u0016=ds\u000b+1\n\u00160g\u000b\fg\u0016\u0017F\f\u0017dA\u0016:\n20The usual Reeb vector fields of the form R\u000b=@\n@s\u000bfullfill the conditions\n\u0013R\u000b\u0012\f=\u000e\f\n\u000b;\u0013R\u000bd\u0012\f= 0;\nbut they are not unique. We can construct all the solutions of the previous equations by adding a general\nterm of the form R\u000b=R\u000b+F\u0016\u0017;\u000b@\n@A\u0016;\u0017withF\u0016\u0017;\u000b\u0000F\u0017\u0016;\u000b = 0. Fortunately, \u0013(R\u000b)dEL=\r\u000bfor any\npossible (antisymetric) F\u0016\u0017;\u000b, therefore the choice of Reeb vector fields does not change the equations. From\nnow on we will use R\u000b=@\n@s\u000b.\nThe equations of motion for a k-vector field X\u000b\n\u0013X\u000bd\u0012\u000b\nL=dEL\u0000\r\u000b\u0012\u000b\nL\n\u0013X\u000b\u0012\u000b\nL=\u0000EL\nhence imply\n@\u000bs\u000b=L\nF\u001c\f=A\f;\u001c\u0000A\u001c;\f\n\u00160J\u0016=g\u0017\u001bg\u0016\u001c\u0010\n@\u0017F\u001b\u001c+\r\u0017F\u001b\u001c\u0011\n:(38)\n(39)\n(40)\nLetting\r\u0016= (\r\nc;\r\r\r), Equation (40) reads in vector notation as\n(1 +\u001fe)\u0000\nr\u0001E+\r\r\r\u0001E\u0001\n=\u001a\n\u000f0\n\u0000(1 +\u001fe)\u000f0\u0000@E\n@t+\rE\u0001\n+1\n1 +\u001fm\u0000\nr\u0002B\n\u00160+\r\r\r\u0002B\n\u00160\u0001\n=j;(41)\n(42)\nwhich we refer to as the k\u0000contact Maxwell’s equations. These reduce to the Gauss-Ampère law for linear\nmaterials (or vacuum) when \r\u0017= 0.\n4.3 Generalized Poynting’s Theorem\nWe next derive a generalized Poynting’s Theorem for the obtained k\u0000contact Maxwell equations in order\nto discuss when the modelled systems are dissipative. Dot multiplying Equation (42) by the vector field E\nand using Faraday’s law of induction ( r\u0002E=\u0000@B\n@t), we find\nE\u0001j=\u0000(1 +\u001fe)\u000f0\u00101\n2@E\u0001E\n@t+\rE\u0001E\u0011\n\u00001\n1 +\u001fm\u00101\n2\u00160@B\u0001B\n@t+r\u0001S+\r\r\r\u0001S\u0011\n;\nand integrating over a volume Vwe obtain\nZ\nV@u\n@tdV+Z\nS=@VS\u0001^ndS\n=\u0000Z\nVE\u0001jdV\u0000Z\nVE\u0001\u0010\n\u001fe\u000f0@E\n@t+ (1 +\u001fe)\u000f0\rE+\u001fm\n1 +\u001fmr\u0002B\n\u00160\u00001\n1 +\u001fm\r\r\r\u0002B\n\u00160\u0011\ndV;\nwhich we refer to as the generalised Poynting’s theorem. In order to shed some light on the utility of this\nresult, let us consider the case in which \u001fe=\u001fm= 0. Then, the generalised Poynting’s theorem reads\nZ\nV@u\n@tdV+Z\nS=@VS\u0001^ndS=\u0000Z\nVE\u0001jdV\u0000Z\nVE\u0001\u0010\n\u000f0\rE\u0000\r\r\r\u0002B\n\u00160\u0011\ndV;\nand the energy of the electromagnetic fields is being dissipated by both the real current jand by a virtual\ncurrent j\r:=\u000f0\rE\u0000\r\r\r\u0002B\n\u00160. Note that E\u0001j\r=\u000f0\rE\u0001E+\r\r\r\u0001S. Whenever the vector \r\r\rvanishes and \r >0,\nthen the volume integralZ\nVE\u0001j\rdV=\u000f0\rZ\nVE\u0001EdV\n21is positive, and thus the modelled system is indeed dissipative. This discussion agrees with that made in\n[27]. In general, the virtual current is j\r=\u001fe\u000f0@E\n@t+ (1 +\u001fe)\u000f0\rE+\u001fm\n1+\u001fmr\u0002B\n\u00160\u00001\n1+\u001fm\r\r\r\u0002B\n\u00160, and\nE\u0001j\r=1\n2@\n@t\u0010\n\u001fe\u000f0E\u0001E\u0000\u001fm\n1 +\u001fm1\n\u00160B\u0001B\u0011\n+ (1 +\u001fe)E\u0001E+\u001fm\n1 +\u001fmr\u0001S+1\n1 +\u001fm\r\r\r\u0001S:\nOne can also argue as follows. Following the constitutive relations for DandHin a linear material, let\n~D:= (1+\u001fe)\u000f0Eand~H:=1\n1+\u001fm1\n\u00160B. Then, define ~u:=1\n2\u0010\nE\u0001~D+B\u0001~H\u0011\n=1\n2\u0010\n(1+\u001fe)\u000f0E\u0001E+1\n1+\u001fm1\n\u00160B\u0001B\u0011\nand~S:=E\u0002~H=1\n1+\u001fmS. Then, the generalized Poynting’s theorem implies\nZ\nV@~u\n@tdV+Z\nS=@V~S\u0001^ndS=\u0000Z\nVE\u0001jdV\u0000Z\nVE\u0001\u0010\n\r~D\u0000\r\r\r\u0002~H\u0011\ndV;\nand the virtual current dissipating energy from the system becomes ~j\r=\r~D\u0000\r\r\r\u0002~H.\n4.4 k-contact Maxwell’s equations as Maxwell’s equations in matter\nAs discussed at the end of Section 4.2, the k-contact Maxwell’s equations are precisely Maxwell’s equations\nfor a linear material or vacuum when \r\u0017= 0. In general, imposing that the obtained equations of motion\nare precisely the Gauss-Ampère law is equivalent to the existence of constants \u001feand\u001fmand a four tensor\n\r\u0017for which the identity g\u0017\u001bg\u0016\u001c\u0010\n@\u0017F\u001b\u001c+\r\u0017F\u001b\u001c\u0011\n=\u00160\u0011\u0017\u000b\u0011\u0016\f@\u0017D\u000b\fis satisfied. Looking at the k-contact\nMaxwell’s equations, it is sufficient that the material satisfies\nr\u0001P=\u001fe\u000f0r\u0001E+ (1 +\u001fe)\u000f0\r\r\r\u0001E\n@P\n@t=\u001fe\u000f0@E\n@t+ (1 +\u001fe)\u000f0\rE\nr\u0002M=\u001fm\n1 +\u001fmr\u0002B\n\u00160\u00001\n1 +\u001fm\r\r\r\u0002B\n\u00160(43)\nfor certain constants \u001fe;\u001fm;\r2Rand\r\r\r2R3. Note that it is necessary that the incoming electric field\nfulfils\r\r\r\u0001@E\n@t=\r0r\u0001E.\nNote that whenever these relations are satisfied, and hence our framework models electromagnetism\nin matter, the bound current jbdinduced by the material is precisely the virtual current appearing in the\ngeneralized Poynting’s theorem in Section 4.3, that is\njbd=@P\n@t+r\u0002M=\u001fe\u000f0@E\n@t+ (1 +\u001fe)\u000f0\rE+\u001fm\n1 +\u001fmr\u0002B\n\u00160\u00001\n1 +\u001fm\r\r\r\u0002B\n\u00160=j\r:\nHence, our framework models dissipative electromagnetic systems whenever the bound current jbd=j\r\nis a dissipative term, as seen in the generalized Poynting’s Theorem.\n4.5 Example. The Lorentz Dipole Model\nThe Lorentz dipole models the interaction between an oscillating electric field and an electron [23, Ch. 3].\nThe model assumes the electron behaves like a harmonic oscillator attached to the nucleus by a hypothetical\nspring of constant C, and that the oscillations are driven by the electric field. The source of damping is not\nspecified but comes from a drag force ~F=\u0000\u0016~ veon the electron. Let mbe the mass of the electron and\nqe>0the absolute value of it’s charge.\n22The solution to the equations of motion can be found assuming the incoming field to be of the form\nE(r;t) =E0eik\u0001r\u0000i!t, where kis called the wave vector and !is the frequency. One can derive the\nconstitutive relation of the polarization P=\u000f0(\u001f0\u0000i\u001f00)Eof a material formed of Nsuch atoms that do\nnot interact with each other, where the electric susceptibility becomes a complex quantity with\n\u001f0=!2\np(!2\n0\u0000!2)\n(!2\n0\u0000!2)2+ (!=\u001c)2\u001f00=!2\np!=\u001c\n(!2\n0\u0000!2)2+ (!=\u001c)2;\nwhere!0=q\nC\nmis the natural frequency of the electron-nucleus system and !2\np=Nqe\n\u000f0m. We have also\ndefined\u001c=m\n\u0016. The complex part of the susceptibility models the absorption of light by the material, and\nhence the dissipation of electromagnetic energy of the incoming field.\nNote that@P\n@t=\u000f0(\u001f0\u0000i\u001f00)@E\n@t=\u000f0\u001f0@E\n@t\u0000\u000f0\u001f00!E\nr\u0001P=\u000f0(\u001f0\u0000i\u001f00)r\u0001E=\u000f0\u001f0r\u0001E+\u000f0\u001f00k\u0001E;\nand since no magnetic effects are considered, it is sufficient to define \u001fe=\u001f0,\u001fm= 0and\r=\u0000\u001f00\n1+\u001f0!,\n\r\r\r=\u001f00\n1+\u001f0kfor Equations (43) to be satisfied.\n4.6 Example. Highly Resistive Dielectric\nConsider a parallel capacitor filled with a dielectric. The induced vector field is E=\u0001V\ndez, where \u0001V\nis the constant applied voltage difference, dis the distance between the plates and ezis the unit vector\northogonal to the plates. Assume no external electric or magnetic fields intervene and that the dielectric is\nnon-magnetic.\nAssume that the dielectric is composed of spherical molecules with a high resistivity. These are of radius\nRand separated a distance s\u001dR. The electric field inside the dielectric away from the centre of the spheres\nis essentially constant and equal to\u0001V\ndez.\nDue to the high resistivity of the molecules, the accumulation of charge at the poles as a response\nto the applied electric field is described as a rate, rather than a magnitude, which is proportional to the\nexternal electric field, see [22, Sec. 11.5]. Thus, we expect@P\n@tto be proportional to E, contrary to when\nthe spheres are perfect conductors, which makes the dielectric linear and hence Pproportional to E. Thus,\nthe polarization constitutive law for a short initial period of time reads\n@P\n@t=\u000bE;\nwhere\u000bis a positive real constant dependant only on the properties and geometry of the spheres.\nHence, one can take \u001fe=\u001fm= 0and\r=\u000b\n\u000f0c. Also, if no interaction between the spheres is considered,\nr\u0001D= 0andr\u0001P= 0. Also, since the magnetization of the molecules is zero, r\u0002M= 0and if we let\n\r\r\r= 0,\n0 =\r\r\r\u0001E=r\u0001P= 0\nand\n0 =\u0000\r\r\r\u0002B=\u00160r\u0002M= 0\n23and hence Equations (43) are satisfied. Since \r\r\r= 0and\r0>0, the bound current is a source of dissipation,\nas discussed in Section 4.3. This coincides with the conclusions drawn in [22, Sec. 11.5] by means of physical\narguments.\n4.7 Gauge Invariance\nIn the current section we show how the developed k-contact theory for electromagnetic fields is invariant\nunder the classical gauge theory for fields.\nLetQ=R4be the space in which the four tensor A\u0016takes its values, and consider a general function\nf:Q!R. The change of gauge given by fis\nA0\n\u0016=A\u0016\u0000@\u0016f:\nLet us consider the case in which the external current J\u000bvanishes. Let the primed variables denote the\nobjects in the new gauge, and the unprimed variables denote the objects in the old gauge. Note first that\nF0\n\u000b\u0016=@\u000bA0\n\u0016\u0000@\u0016A0\n\u000b=F\u000b\u0016\u0000@\u000b\u0016f+@\u000b\u0016f=F\u000b\u0016;\nand henceF\u000b\u0016is invariant under the gauge. Thus, the Lagrangian\nL=\u00001\n4\u00160g\u000b\u0016g\f\u0017F\u0016\u0017F\u000b\f\u0000\r\u000bs\u000b\nis invariant under the gauge. Also, the energy satisfies\nE0\nL=1\n\u00160g\u0016\u0017g\u000b\fA0\n\u0016;\u000bF0\n\u0017\f+1\n4\u00160g\u0016\u0017g\u000b\fF0\n\f\u0017F0\n\u000b\u0016+\r\u000bs\u000b=EL\u00001\n\u00160g\u0016\u0017g\u000b\fF\u0017\f@\u000b\u0016f\nwhere the last equality follows from the fact that the tensor @\u000b\u0016fis symmetric while F\u000b\u0016is antisymmetric.\nIt is also clear that the equations of motion\n0 =g\u0017\u001bg\u0016\u001c\u0010\n@\u0017F\u001b\u001c+\r\u0017F\u001b\u001c\u0011\nare invariant under the gauge.\nThus, ourk-contact theory and all of the objects involved are invariant under the classical gauge theory\nfor electromagnetic fields.\n5 Conclusions\nThe work presented here contributes to the understanding of how contact geometry can be applied in classi-\ncal mechanics and classical field theory to describe systems with dissipation and damping. A lot of research\nhas been done lately to build a general theory, discovering conservation and dissipation theorems and ana-\nlogues to classical results such as Noether’s theorem [10,16–18], but no extensive studies on applications to\ncurrent theories and real-life examples have been made apart from small examples in [10,16–18,27].\nThe present work focuses on how contact formalism can be applied to electromagnetic systems with\ndissipation. It has been studied how the techniques introduced by G. Herglotz in 1930, and recently\nformalized using the language of contact geometry, can be applied to particles under the influence of a\n24Lorentz force and external damping. Moreover, a recent theory developed in [16,18], which generalizes\ncontact formalism to study classical field theory, has been used to discuss electromagnetic fields themselves\nwhen being dissipated by external phenomena.\nThe contact formalism of classical mechanics deals with systems whose Lagrangian function or density\ndepends on the action itself. Driven by previous successful applications of the theory, in the present work\nthe Lagrangians under study have been taken to be the classical Lagrangians of electromagnetism (the\nLagrangian of the Lorentz force and the Lagrangian density of electromagnetic fields) plus a linear term in\nthe action.\nWhen applying the theory to particles under the Lorentz force, the produced equations of motion\nhave turned out to be not invariant under the current gauge theory. This has driven us to propose a\nnew gauge approach for electromagnetism which generalizes the classical one and reduces to the currently\naccepted theory for non-dissipative symplectic systems. It has also been necessary to slightly change the\nLagrangian of the Lorentz force, adding an extra term which produces the same equations of motion in\nthe non-dissipative case but that is key in the contact formalism. All of these generalizations make a new\nmathematical observable vector field appear, which can be seen to vanish from the equations of motion\nin the symplectic case. Moreover, in this new paradigm, the generalized momentum of a particle under a\nLorentz force becomes an observable, whilst in the symplectic theory of electromagnetism the generalized\nmomenta are not gauge-invariant.\nThe developed approach for particles under a Lorentz force in the presence of external damping has been\napplied to two real-life scenarios. Firstly, our theory is able to model the behaviour of an electron inside\na non-perfect conductor. When imposing that the limit velocity of the particle agrees with the current\ntheory, Joules’ heating law is recovered. It has also been discussed how our description can be applied to a\nparticle in a magnetic field with dissipation.\nWhen considering vector fields, we have produced a new set of Gauss-Ampère equations which reduce\nto the classical Gauss-Ampère equation when omitting the dissipation parameters. In the general case,\nwe have showed how our theory models electromagnetic fields in matter for a certain type of systems. It\nhas been argued how these systems might have in common that the bound four-current generated by the\nmaterial is precisely a dissipative term. Further research is needed in order to better understand such\nsystems and characterize them further. Our theory has been seen to be able to describe a particular regime\nof the Lorentz dipole model.\nThis work pretends to be a first approach towards the use of contact geometry techniques to model\ndissipation in electromagnetism. It should be further studied whether the generalizations needed to make\nthe theory gauge invariant for the Lorentz force can actually be seen as generalizations of the current\ntheory of electromagnetism. For fields, more research is needed to better understand how the modelled\nsystems behave and find whether other variations of the Lagrangian can describe different systems. It is\nalso necessary to produce more and more relevant examples of how our equations can be applied to real\nproblems in Physics and reproduce current data and predict new phenomena. 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Vassiliou, Contact geometry and its application to control , Advances in the theory of control, signals and systems\nwith physical modeling, 2010, pp. 225–237.\n27"    },    {        "title": "1601.00069v2.Atmospheric_Circulation_of_Hot_Jupiters__Dayside_Nightside_Temperature_Differences.pdf",        "content": "Submitted to The Astrophysical Journal\nPreprint typeset using L ATEX style emulateapj v. 5/2/11\nATMOSPHERIC CIRCULATION OF HOT JUPITERS: DAYSIDE-NIGHTSIDE TEMPERATURE\nDIFFERENCES\nThaddeus D. Komacek1and Adam P. Showman1\n1Department of Planetary Sciences, University of Arizona, Tucson, AZ, 85721\ntkomacek@lpl.arizona.edu\nSubmitted to The Astrophysical Journal\nABSTRACT\nThe full-phase infrared light curves of low-eccentricity hot Jupiters show a trend of increasing\ndayside-to-nightside brightness temperature di\u000berence with increasing equilibrium temperature. Here\nwe present a three-dimensional model that explains this relationship, in order to shed insight on\nthe processes that control heat redistribution in tidally-locked planetary atmospheres. This three-\ndimensional model combines predictive analytic theory for the atmospheric circulation and dayside-\nnightside temperature di\u000berences over a range of equilibrium temperature, atmospheric composition,\nand potential frictional drag strengths with numerical solutions of the circulation that verify this\nanalytic theory. This analytic theory shows that the longitudinal propagation of waves mediates\ndayside-nightside temperature di\u000berences in hot Jupiter atmospheres, analogous to the wave adjust-\nment mechanism that regulates the thermal structure in Earth's tropics. These waves can be damped\nin hot Jupiter atmospheres by either radiative cooling or potential frictional drag. This frictional drag\nwould likely be caused by Lorentz forces in a partially ionized atmosphere threaded by a background\nmagnetic \feld, and would increase in strength with increasing temperature. Additionally, the ampli-\ntude of radiative heating and cooling increases with increasing temperature, and hence both radiative\nheating/cooling and frictional drag damp waves more e\u000eciently with increasing equilibrium tempera-\nture. Radiative heating and cooling play the largest role in controlling dayside-nightside temperature\ntemperature di\u000berences in both our analytic theory and numerical simulations, with frictional drag\nonly being important if it is stronger than the Coriolis force. As a result, dayside-nightside temper-\nature di\u000berences in hot Jupiter atmospheres increase with increasing stellar irradiation and decrease\nwith increasing pressure.\nSubject headings: hydrodynamics - methods: numerical - methods: analytical - planets and satel-\nlites: gaseous planets - planets and satellites: atmospheres - planets and satel-\nlites: individual (HD 189733b, HD 209458b, WASP-43b, HD 149026b, WASP-14b,\nWASP-19b, HAT-P-7b, WASP-18b, WASP-12b)\n1.INTRODUCTION\nHot Jupiters, gas giant exoplanets with small semi-\nmajor axes and equilibrium temperatures exceeding\n1000 K, are the best characterized class of exoplanets to\ndate. Since the \frst transit observations of HD 209458b\n(Henry et al. 2000; Charbonneau et al. 2000), infrared\n(IR) phase curves have been obtained for a variety of\nobjects (e.g. Knutson et al. 2007; Cowan et al. 2007;\nBorucki et al. 2009; Knutson et al. 2009a,b; Cowan et al.\n2012; Knutson et al. 2012; Demory et al. 2013; Maxted\net al. 2013; Stevenson et al. 2014; Zellem et al. 2014;\nWong et al. 2015a). Such phase curves allow for the\nconstruction of longitudinally resolved maps of surface\nbrightness. These maps exhibit a wide diversity, showing\nthat|across the class of hot Jupiters|the fractional dif-\nference between dayside and nightside \rux varies drasti-\ncally from planet to planet. Figure 1 shows the fractional\ndi\u000berence between dayside and nightside brightness tem-\nperatures as a function of equilibrium temperature for\nthe nine low-eccentricity transiting hot Jupiters with\nfull-phase IR light curve observations. This fractional\ndi\u000berence in dayside-nightside brightness temperature,\nAobs, has a value of zero when hot Jupiters are longitu-\ndinally isothermal and unity when the nightside has ef-\nfectively no emitted \rux relative to the dayside. As seenin Figure 1, the fractional dayside-nightside tempera-\nture di\u000berence increases with increasing equilibrium tem-\nperature. The correlation between fractional dayside-\nnightside temperature di\u000berences and stellar irradiation\nshown in Figure 1 has also been found by Cowan & Agol\n(2011), Perez-Becker & Showman (2013), and Schwartz\n& Cowan (2015).\nMotivated by these observations, a variety of groups\nhave performed three-dimensional (3D) numerical sim-\nulations of the atmospheric circulation of hot Jupiters\n(e.g. Showman & Guillot 2002; Cooper & Showman 2005;\nMenou & Rauscher 2009; Showman et al. 2009; Thrastar-\nson & Cho 2010; Heng et al. 2011b,a; Perna et al. 2012;\nRauscher & Menou 2012a; Dobbs-Dixon & Agol 2013;\nMayne et al. 2014; Showman et al. 2015). These general\ncirculation models (GCMs) generally exhibit day-night\ntemperature di\u000berences ranging from \u0018200{1000 K (de-\npending on model details) and fast winds that can ex-\nceed several km s\u00001. When such models include realis-\ntic non-grey radiative transfer, they allow estimates of\nday-night temperature and IR \rux di\u000berences that can\nbe quantitatively compared to phase curve observations.\nSuch comparisons are currently the most detailed for HD\n189733b, HD 209458b, and WASP-43b because of the ex-\ntensive datasets available for these \\benchmark\" planets\n(Showman et al. 2009; Zellem et al. 2014; Kataria et al.arXiv:1601.00069v2  [astro-ph.EP]  18 Feb 20162 T.D. Komacek & A.P. Showman\n1000 1500 2000 2500\nEquilibrium Temperature (Kelvin)0.00.20.40.60.81.0Aobs=(Tb,day−Tb,night)/Tb,dayHD 189733b\nWASP-43b\nHD 209458b\nHD 149026b\nWASP-14b\nWASP-19b\nHAT-P-7b\nWASP-18b\nWASP-12b\nFig. 1.| Fractional dayside to nightside brightness temper-\nature di\u000berences Aobsvs. global-average equilibrium tempera-\nture from observations of transiting, low-eccentricity hot Jupiters.\nHere we de\fne the global-average equilibrium temperature, Teq=\n[F?=(4\u001b)]1=4, whereF?is the incoming stellar \rux to the planet\nand\u001bis the Stefan-Boltzmann constant. Solid points are from\nthe full-phase observations of Knutson et al. (2007, 2009b, 2012)\nfor HD 189733b, Cross\feld et al. (2012); Zellem et al. (2014) for\nHD 209458b, Knutson et al. (2009a) for HD 149026b, Wong et al.\n(2015a) for WASP-14b, Wong et al. (2015b) for WASP-19b and\nHAT-P-7b, and Cowan et al. (2012) for WASP-12b. The error bars\nfor WASP-43b (Stevenson et al. 2014) and WASP-18b (Nymeyer\net al. 2011; Maxted et al. 2013) show the lower limit on Aobsfrom\nthe nightside \rux upper limits (and hence fractional temperature\ndi\u000berence lower limits). See Appendix A for the data and method\nutilized to make this \fgure. There is a clear trend of increasing\nAobswith increasing equilibrium temperature, and hence dayside-\nnightside temperature di\u000berences at the photosphere are greater\nfor planets that receive more incident \rux.\n2015).\nDespite the proliferation of GCM investigations, our\nunderstanding of the underlying dynamical mechanisms\ncontrolling the day-night temperature di\u000berences of hot\nJupiters is still in its infancy. It is crucial to empha-\nsize that, in and of themselves, GCM simulations do\nnot automatically imply an understanding: the under-\nlying dynamics is often su\u000eciently complex that careful\ndiagnostics and a hierarchy of simpli\fed models are of-\nten necessary (e.g., Held 2005; Showman et al. 2010).\nThe ultimate goal is not simply matching observations\nbut also understanding physical mechanisms and con-\nstructing a predictive theory that can quantitatively ex-\nplain the day-night temperature di\u000berences, horizontal\nand vertical wind speeds, and other aspects of the circu-\nlation under speci\fed external forcing conditions. Taking\na step toward such a predictive theory is the primary goal\nof this paper.\nThe question of what controls the day-night temper-\nature di\u000berence in hot Jupiter atmospheres has been a\nsubject of intense interest for many years. Most stud-\nies have postulated that the day-night temperature dif-\nferences are controlled by a competition between radi-\nation and atmospheric dynamics|speci\fcally, the ten-\ndency of the strong dayside heating and nightside cool-\ning to create horizontal temperature di\u000berences, and the\ntendency of the atmospheric circulation to regulate those\ntemperature di\u000berences by transporting thermal energy\nfrom day to night. Describing this competition using a\ntimescale comparison, Showman & Guillot (2002) \frst\nsuggested that hot Jupiters would exhibit small frac-\ntional dayside-nightside temperature di\u000berences when\n\u001cadv\u001c\u001cradand large fractional dayside-nightside tem-perature di\u000berences when \u001cadv\u001d\u001crad. Here,\u001cadvis\nthe characteristic timescale for the circulation to advect\nair parcels horizontally over a hemisphere, and \u001cradis\nthe timescale over which radiation modi\fes the thermal\nstructure (e.g., the timescale to relax toward the local ra-\ndiative equilibrium temperature). Since then, numerous\nauthors have invoked this timescale comparison to de-\nscribe how the day-night temperature di\u000berences should\ndepend on pressure, atmospheric opacity, stellar irradia-\ntion, and other factors (e.g., Cooper & Showman 2005;\nShowman et al. 2008b, 2009; Fortney et al. 2008; Lewis\net al. 2010; Rauscher & Menou 2010; Cowan & Agol 2011;\nMenou 2012; Perna et al. 2012; Ginzburg & Sari 2015).\nOne would expect that hot Jupiters have short advec-\ntive timescales due to their fast zonal winds. This is\nobservationally evident from phase curves from tidally-\nlocked planets with \u001cadv\u0018\u001crad, which consistently show\na peak in brightness just before secondary eclipse. This\nindicates that the point of highest emitted \rux (\\hot\nspot\") is eastward of the substellar point (the point of\npeak absorbed \rux), due to downwind advection from a\nsuperrotating1equatorial jet (Showman & Polvani 2011).\nThe full-phase observations of HD 189733b (Knutson\net al. 2007), HD 209458b (Zellem et al. 2014), and\nWASP-43b (Stevenson et al. 2014) show hot spot o\u000bsets.\nThese o\u000bsets agree with those predicted from correspond-\ning circulation models (Showman et al. 2009; Kataria\net al. 2015). Hence, we have observational con\frmation\nthat hot Jupiters have fast ( \u0018kilometers/second) zonal\nwinds.\nAs discussed above, fast zonal winds are a robust fea-\nture of hot Jupiter general circulation models (GCMs),\nand notably also exist when the model hot Jupiter has\na large eccentricity (Lewis et al. 2010; Kataria et al.\n2013). However, these circulation models show a range\nof dayside-nightside temperature di\u000berences, showing no\nclear trend with wind speeds. Perna et al. (2012) ex-\namined how heat redistribution is a\u000bected by incident\nstellar \rux, showing that the nightside/dayside \rux ra-\ntio decreases with increasing incident stellar \rux. This\ntrend is akin to the observational trend shown in Fig-\nure 1. This trend was explained in Perna et al. (2012) by\ncalculating the ratio of \u001cadv=\u001crad, which increases with\nincident stellar \rux in their models.\nAs pointed out by Perez-Becker & Showman (2013),\nthere exist several issues with the idea that a timescale\ncomparison between \u001cadvand\u001cradgoverns the amplitude\nof the day-night temperature di\u000berences. First, although\nit is physically motivated, this timescale comparison has\nnever been derived rigorously from the equations of mo-\ntion; as such, it has always constituted an ad-hoc albeit\nplausible hypothesis, as opposed to a theoretical result.\nSecond, the comparison between \u001cadvand\u001craddoes not\ninclude any obvious role for other timescales that are\nimportant, including those for planetary rotation, hori-\nzontal and vertical wave propagation, and frictional drag\n(if any). These processes in\ruence the circulation and\nthus one might expect the timescale comparison to de-\npend on them. Third, the comparison is not predictive|\n\u001cadvdepends on the horizontal wind speeds, which are\n1Superrotation occurs where the zonal-mean atmospheric circu-\nlation of a planet has a greater angular momentum per unit mass\nthan the planet itself at the equator.Dayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 3\nonly known a posteriori . Hence, it is only possible to\nevaluate the comparison between advective and radia-\ntive timescales if one already has a numerical model (or\ntheory) for the atmospheric circulation, which is neces-\nsarily related to other relevant timescales governing the\ncirculation.\nTo show how other timescales play a large role in de-\ntermining the atmospheric circulation, consider the equa-\ntorial regions of Earth. On Earth, horizontal tempera-\nture gradients in the tropics are weak, and the radia-\ntive cooling to space that occurs in Earth's troposphere\nis balanced primarily by vertical advection rather than\nhorizontal advection|a balance known as the weak tem-\nperature gradient (WTG) regime (Polvani & Sobel 2001;\nSobel et al. 2001; Sobel 2002). This vertical advection\ntimescale is related to the e\u000ecacy of lateral wave prop-\nagation. When gravity, Rossby, or Kelvin waves propa-\ngate, they induce vertical motion, which locally advects\nthe air parcels upward or downward. If these waves are\nable to propagate away, and rotation plays only a mod-\nest role, then this wave-adjustment tends to leave behind\na state with \rat isentropes|which is equivalent to eras-\ning the horizontal temperature di\u000berences (Bretherton\n& Smolarkiewicz 1989; Showman et al. 2013b). Given\nthat adjustment of isentropes due to propagating waves\nis known to occur on planetary scales on both Earth\n(Matsuno 1966; Gill 1980) and exoplanets (Showman &\nPolvani 2010, 2011; Tsai et al. 2014), Perez-Becker &\nShowman (2013) suggested that wave adjustment like-\nwise acts to lessen horizontal temperature di\u000berences in\nhot Jupiter atmospheres.\nIn hot Jupiter atmospheres, planetary-scale Kelvin and\nRossby waves are generated by the large gradient in\nradiative heating from dayside to nightside (Showman\n& Polvani 2011). Unlike in Earth's atmosphere, these\nwaves exhibit a meridional half-width stretching nearly\nfrom equator to pole, as the Rossby deformation radius\nis approximately equal to the planetary radius (Show-\nman & Guillot 2002). These waves cause horizontal con-\nvergence/divergence that forces vertical motion through\nmass continuity. This vertical motion moves isentropes\nvertically, and if the waves are not damped this leads to a\n\fnal state with \rat isentropes. However, if these Kelvin\nand Rossby waves cannot propagate (i.e. are damped),\nthen this process cannot occur. Hence, the ability of\nwave adjustment processes to lessen horizontal temper-\nature gradients can be weakened by damping of propa-\ngating waves. As shown in Figure 1, damping processes\nthat increase day-night temperature di\u000berences seem to\nincrease in e\u000ecacy with increasing equilibrium temper-\nature. The most natural damping process is radiative\ncooling, which should increase in e\u000eciency with the cube\nof equilibrium temperature (Showman & Guillot 2002).\nAdditionally, frictional drag on the atmosphere can re-\nduce the ability of wave adjustment to reduce longitudi-\nnal temperature gradients. This drag could either be due\nto turbulence (Li & Goodman 2010; Youdin & Mitchell\n2010) or the Lorentz force in a partially ionized atmo-\nsphere threaded by a dipole magnetic \feld (Batygin &\nStevenson 2010; Perna et al. 2010; Menou 2012; Batygin\net al. 2013; Rauscher & Menou 2013; Rogers & Showman\n2014; Rogers & Komacek 2014). Both of these processes\nshould increase fractional day-night temperature di\u000ber-\nences with increasing equilibrium temperature, helpingexplain Figure 1. However, it is not obvious a priori\nwhether radiative e\u000bects or drag should more e\u000eciently\ndamp wave adjustment in hot Jupiter atmospheres.\nTo understand the mechanisms controlling day-night\ntemperature di\u000berences|including their dependence on\nradiative and frictional e\u000bects|Perez-Becker & Show-\nman (2013) introduced a shallow-water model with a\nsingle active layer representing the atmosphere, which\noverlies a deeper layer, representing the interior, whose\ndynamics are \fxed and assumed to be quiescent. Perez-\nBecker & Showman (2013) performed numerical simula-\ntions over a broad range of drag and radiative timescales,\nwhich generally showed that strong radiation and fric-\ntional drag tend to promote larger day-night temperature\ndi\u000berences. They then compared their model results to\nderived analytic theory and found good agreement be-\ntween the expected fractional dayside-nightside temper-\nature di\u000berences and model results, albeit with minor ef-\nfects not captured in the theory. This theory showed rig-\norously that wave adjustment allows for reduced horizon-\ntal temperature di\u000berences in hot Jupiter atmospheres.\nThey also showed that horizontal wave propagation is\nmainly damped by radiative e\u000bects, with potential drag\nplaying a secondary, but crucial, role. As a result, they\nfound that the strength of radiative heating/cooling is\nthe main governor of dayside-nightside temperature dif-\nferences in hot Jupiter atmospheres. Nevertheless, be-\ncause the model is essentially two-dimensional (in longi-\ntude and latitude) and lacks a vertical coordinate, ques-\ntions exist about how the results would carry over to a\nfully three-dimensional atmosphere.\nHere we extend the work of Perez-Becker & Show-\nman (2013) to fully three-dimensional atmospheres. The\nuse of the full three-dimensional primitive equations en-\nables us to present a predictive analytic understanding\nof dayside-nightside temperature di\u000berences and wind\nspeeds that can be directly compared to observable quan-\ntities. Our analytic theory is accompanied by numerical\nmodels which span a greater range of radiative forcing\nand drag parameter space than Perez-Becker & Showman\n(2013), enabling quantitative validation of these analytic\nresults. We keep the radiative forcing simple in order to\npromote a physical understanding. Despite this simpli\f-\ncation, we emphasize that this is the \frst fully predictive\nanalytic theory for the day-night temperature di\u000berences\nof hot Jupiter atmospheres in three dimensions.\nThis paper is organized as follows. In Section 2, we de-\nscribe our methods, model setup, and parameter space\nexplored. We discuss the results of our numerical param-\neter study of dayside-nightside temperature di\u000berences in\nSection 3. In Section 4, we develop our theory in order\nto facilitate a comparison to numerical results in Sec-\ntion 5. In Section 6, we explore the implications of our\nmodel results in the context of previous observations and\ntheoretical work, and express conclusions in Section 7.\n2.MODEL\nWe adopt the same physical model for both our nu-\nmerical and analytic solutions. GCMs with accurate ra-\ndiative transfer have proven essential for detailed com-\nparison with observations (Showman et al. 2009, 2013a;\nKataria et al. 2015); however, our goal here is to promote\nanalytic tractability and a clean environment in which to\nunderstand dynamical mechanisms, and so we drive the4 T.D. Komacek & A.P. Showman\ncirculation using a simpli\fed Newtonian heating/cooling\nscheme (e.g. Showman & Guillot 2002; Cooper & Show-\nman 2005). This enables us to systematically vary the\ndayside-nightside thermal forcing and control the rate at\nwhich temperature relaxes to a \fxed radiative equilib-\nrium pro\fle. We also incorporate a drag term in the\nequations to investigate how day-night temperature dif-\nferences are modi\fed by the combined e\u000bects of atmo-\nspheric friction and di\u000berential stellar irradiation. Our\nmodel setup is nearly identical to that in Liu & Showman\n(2013).\n2.1. Dynamical equations\nWe solve the horizontal momentum, vertical momen-\ntum, continuity, energy equation, and ideal gas equation\nof state (i.e. the hydrostatic primitive equations), which,\nin pressure coordinates, are:\ndv\ndt+f^k\u0002v+r\b =Fdrag+DS; (1)\n@\b\n@p+1\n\u001a= 0; (2)\nr\u0001v+@!\n@p= 0; (3)\nTd(ln\u0012)\ndt=dT\ndt\u0000!\n\u001acp=q\ncp+ES; (4)\np=\u001aRT: (5)\nWe use the following symbols: pressure p, density\u001a, tem-\nperatureT, speci\fc heat at constant pressure cp, spe-\nci\fc gas constant R, potential temperature2\u0012, horizon-\ntal velocity (on isobars) v, horizontal gradient on isobars\nr, vertical velocity in pressure coordinates !=dp=dt ,\ngeopotential \b = gz, Coriolis parameter f= 2\nsin\u001e\n(with \n planetary rotation rate, here equivalent to or-\nbital angular frequency, and \u001elatitude), and speci\fc\nheating rate q. In this coordinate system, the total\n(material) derivative is d=dt =@=@t +v\u0001r+!@=@p .\nFdragrepresents a drag term that we use to represent\nmissing physics (Rauscher & Menou 2012b), for exam-\nple drag due to turbulent mixing (Li & Goodman 2010;\nYoudin & Mitchell 2010), or the Lorentz force (Perna\net al. 2010; Rauscher & Menou 2013). The terms DS\nandESrepresent a standard fourth-order Shapiro \flter,\nwhich smooths grid-scale variations while minimally af-\nfecting the \row at larger scales, and thereby helps to\nmaintain numerical stability in our numerical integra-\ntions. Because the Shapiro \flter terms do not a\u000bect the\nglobal structure of our equilibrated numerical solutions,\nthey are negligible in comparison to the other terms in\nthe equations at the near-global scales captured in our\nanalytic theory. As a result, we neglect them from our\nanalytic solutions.\n2Potential temperature is de\fned as \u0012=T(p=p0)\u0014, where\n\u0014=R=cp, which is here assumed constant. The potential temper-\nature is the temperature that an air parcel would have if brought\nadiabatically to an atmospheric reference pressure p0. We choose\na reference pressure of p0= 1 bar, but the solution is independent\nof the value of p0chosen.2.2. Thermal forcing and frictional drag\nWe represent the radiative heating and cooling using\na Newtonian heating/cooling scheme, which relaxes the\ntemperature toward a prescribed radiative equilibrium\ntemperature, Teq, over a speci\fed radiative timescale\n\u001crad:\nq\ncp=Teq(\u0015;\u001e;p )\u0000T(\u0015;\u001e;p;t )\n\u001crad(p): (6)\nIn this scheme, Teqdepends on longitude \u0015, latitude\u001e,\nand pressure, while, for simplicity, \u001cradvaries only with\npressure. The radiative equilibrium pro\fle is set to be\nhot on the dayside and cold on the nightside:\nTeq(\u0015;\u001e;p ) =\u001aTnight;eq(p) + \u0001Teq(p)cos\u0015cos\u001edayside;\nTnight;eq(p) nightside :\n(7)\nHere\u0015is longitude, Tnight;eq(p) is the radiative\nequilibrium temperature pro\fle on the nightside and\nTnight;eq(p) + \u0001Teq(p) is that at the substellar point. To\nacquire the nightside heating pro\fle, Tnight;eq, we take\nthe temperature pro\fle of HD 209458b from Iro et al.\n(2005) and subtract our chosen \u0001 Teq(p)=2. We specify\n\u0001Teqas in Liu & Showman (2013), setting it to be a\nconstant \u0001 Teq;topat pressures less than peq;top, zero at\npressures greater than pbot, and varying linearly with log\npressure in between:\n\u0001Teq(p) =8\n>><\n>>:\u0001Teq;top p<p eq;top\n\u0001Teq;topln(p=pbot)\nln(peq;top=pbot)peq;top<p<p bot\n0 Kelvin p>p bot:\n(8)\nAs in Liu & Showman (2013), we assume peq;top= 10\u00003\nbars andpbot= 10 bars. However, we vary \u0001 Teq;top\nfrom 1000\u00000:001 Kelvin, ranging from highly nonlinear\nto linear numerical solutions.\nRadiative transfer calculations show that the radiative\ntime constant is long at depth and short aloft (Iro et al.\n2005; Showman et al. 2008a). To capture this behav-\nior, we adopt the same functional form for \u001cradas Liu &\nShowman (2013): \u001cradis set to a large constant \u001crad;bot\nat pressures greater than pbot, a (generally smaller) con-\nstant\u001crad;topat pressures less than prad;top, and varies\ncontinuously in between:\n\u001crad(p) =8\n>><\n>>:\u001crad;top p<p rad;top\n\u001crad;bot\u0012p\npbot\u0013\u000b\nprad;top<p<p bot\n\u001crad;bot p>p bot;\n(9)\nwith\n\u000b=ln(\u001crad;top=\u001crad;bot)\nln(prad;top=pbot): (10)\nHere, as in Liu & Showman (2013), we set prad;top=\n10\u00002bars andprad;bot= 10 bars. Note that the pres-\nsures above which \u0001 Teqand\u001cradare \fxed to a constant\nvalue at the top of the domain are di\u000berent, in order\nto be fully consistent with the model setup of Liu &\nShowman (2013). The model is set up such that the cir-\nculation forced by Newtonian heating/cooling has three-\ndimensional temperature and wind distributions that areDayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 5\n103\n104\n105\n106\n107\nτrad (sec)10-4\n10-3\n10-2\n10-1\n100\n101\n102Pressure (bar)\n103104105106107108\nτdrag (sec)10-810-710-610-510-410-3kv (sec−1)\nFig. 2.| Radiative forcing and drag pro\fles used in the numerical model. Left: Radiative timescale vs. pressure, for all assumed\n\u001crad;top= 103\u0000107sec. Right: Drag timescale as a function of pressure, for each of assumed spatially constant \u001cdrag= 103\u00001 sec. The\ndrag constant kv=\u001c\u00001\ndragis shown on the upper x-axis.\nsimilar to results from simulations driven by radiative\ntransfer. This motivated the choices of prad;topand\npeq;top, along with the values of other \fxed parameters\nin the model.\nThe various \u001crad-pressure pro\fles used in our models\n(for di\u000berent assumed \u001crad;top) are shown on the left hand\nside of Figure 2. We choose \u001crad;bot= 107sec, which\nis long compared to relevant dynamical and rotational\ntimescales but short enough to allow us to readily inte-\ngrate to equilibrium. For the purposes of our study, we\nvary\u001crad;topfrom 103\u0000107sec, corresponding to a range\nof radiative forcing.\nWe introduce a linear drag in the horizontal momen-\ntum equation, given by\nFdrag=\u0000kv(p)v; (11)\nwherekv(p) is a pressure-dependent drag coe\u000ecient.\nThis drag has two components:\n\u000fFirst, we wish to examine how forces that crudely\nparameterize Lorentz forces a\u000bect the day-night\ntemperature di\u000berences. This could be represented\nwith a drag coe\u000ecient that depends on longitude,\nlatitude, and pressure and, moreover, di\u000bers in all\nthree dimensions (e.g. Perna et al. 2010; Rauscher\n& Menou 2013). However, the Lorentz force should\ndepend strongly on the ionization fraction and\ntherefore the local temperature, requiring full nu-\nmerical magnetohydrodynamic solutions to exam-\nine the e\u000bects of magnetic \\drag\" in detail, e.g.\nBatygin et al. (2013); Rogers & Komacek (2014);\nRogers & Showman (2014). For simplicity and\nanalytic tractability, we represent this component\nwith a spatially constant drag timescale \u001cdrag, cor-responding to a drag coe\u000ecient \u001c\u00001\ndrag. We system-\natically explore \u001cdrag values (in sec) of 103, 104,\n105, 106, 107, and1. The latter corresponds to\nthe drag-free limit. Such a scheme was already ex-\nplored by Showman et al. (2013a).\n\u000fSecond, following Liu & Showman (2013), we in-\ntroduce a \\basal\" drag at the bottom of the do-\nmain, which crudely parameterizes interactions be-\ntween the vigorous atmospheric circulation and\na relatively quiescent planetary interior. For\nthis component, the drag coe\u000ecient is zero at\npressures less than pdrag;topand is\u001c\u00001\ndrag;bot(p\u0000\npdrag;top)=(pdrag;bot\u0000pdrag;top) at pressures greater\nthanpdrag;top, wherepdrag;botis the mean pres-\nsure at the bottom of the domain (200 bars) and\npdrag;topis the lowest pressure where this basal\ndrag component is applied. Thus, the drag coef-\n\fcient varies from \u001c\u00001\ndrag;botat the bottom of the\ndomain to zero at a pressure pdrag;top; this scheme\nis similar to that in Held & Suarez (1994). We\ntakepdrag;top= 10 bars, and set \u001cdrag;bot= 10 days.\nThus, basal drag acts only at pressures greater than\n10 bars and has a minimum characteristic timescale\nof 10 days at the bottom of the domain, increas-\ning to in\fnity (meaning zero drag) at pressures less\nthan 10 bars. We emphasize that the precise value\nis not critical. Changing the drag time constant at\nthe base to 100 days, for example (corresponding\nto weaker drag) would lead to slightly faster wind\nspeeds at the base of the model, and would require\nlonger integration times to reach equilibrium, but\nwould not qualitatively change our results.6 T.D. Komacek & A.P. Showman\nTo combine the two drag schemes, we simply set the drag\ncoe\u000ecient to be the smaller of the two individual drag\ncoe\u000ecients at each individual pressure, leading to a \fnal\nfunctional form for the drag coe\u000ecient:\nkv(p) = max\u0014\n\u001c\u00001\ndrag;\u001c\u00001\ndrag;bot(p\u0000pdrag;top)\n(pdrag;bot\u0000pdrag;top)\u0015\n(12)\nThe righthand side of Figure 2 shows the various \u001cdrag(p)\npro\fles used in our models. Corresponding values for\nkv(p) are given along the top axis.\nWe adopt planetary parameters ( cp,R, \n,g,R)\nrelevant for HD 209458b. This includes speci\fc heat\ncp= 1:3\u0002104J kg\u00001K\u00001, speci\fc gas constant R=\n3700 J kg\u00001K\u00001, rotation rate \n = 2 :078\u000210\u00005s\u00001,\ngravityg= 9:36 m s\u00002, and planetary radius a= 9:437\u0002\n107m. Though we use parameters relevant for a given\nhot Jupiter, our results are not sensitive to the precise\nvalues used. Moreover, we emphasize that the qual-\nitative model behavior should not be overly sensitive\nto numerical parameters such as the precise values of\nprad;top,peq;top,pbot, and so on. Modifying the values of\nthese parameters over some reasonable range will change\nthe precise details of the height-dependence of the day-\nnight temperature di\u000berence and wind speeds but will\nnot change the qualitative behavior or the dynamical\nmechanisms we seek to uncover. We would \fnd simi-\nlar behavior regardless of the speci\fc parameters used,\nas long as they are appropriate for a typical hot Jupiter.\n2.3. Numerical details\nOur numerical integrations are performed using the\nMITgcm (Adcroft et al. 2004) to solve the equations\ndescribed above on a cubed-sphere grid. The horizon-\ntal resolution is C32, which is roughly equal to a global\nresolution of 128 \u000264 in longitude and latitude. There\nare 40 vertical levels, with the bottom 39 levels evenly\nspaced in log-pressure between 0.2 mbars and 200 bars,\nand a top layer that extends from 0.2 mbars to zero pres-\nsure. Models performed at resolutions as high as C128\n(corresponding to a global resolution of 512 \u0002256) by\nLiu & Showman (2013) behave very similar to their C32\ncounterparts, indicating that C32 is su\u000ecient for current\npurposes. All models are integrated to statistical equi-\nlibrium. We integrate the model from a state of rest with\nthe temperatures set to the Iro et al. (2005) temperature-\npressure pro\fle. Note that this system does not exhibit\nsensitivity to initial conditions (Liu & Showman 2013).\nFor the most weakly nonlinear runs described in Sec-\ntion 3.1, reaching equilibration required 25 ;000 Earth\ndays of model integration time. However, our full grid of\nsimulations varying radiative and drag timescales with\na \fxed equilibrium day-night temperature di\u000berence re-\nquired .5;000 days of integration.\n3.NUMERICAL RESULTS\n3.1. Parameter space exploration\nGiven the forcing and drag prescriptions speci\fed in\nSection 2 and the planetary parameters for a typical hot\nJupiter, the problem we investigate is one governed by\nthree parameters|\u0001 Teq;top,\u001crad;top, and\u001cdrag. Our goal\nis to thoroughly explore a broad, two-dimensional grid of\nGCM simulations varying \u001crad;topand\u001cdragover a wide\n10−310−210−110010110210310−410−310−210−1100101102103104\n∆ Teq,top (K)Urms (m/s)\n  \nτdrag = 105 secτdrag = basalFig. 3.| Root-mean-square (RMS) horizontal wind speed at a\npressure of 80 mbars plotted against equilibrium dayside-nightside\ntemperature di\u000berences \u0001 Teq;top. These day-night temperature\ndi\u000berences set the forcing amplitude of the circulation. When\n\u0001Teq;topis small, the RMS velocities respond linearly to forcing,\nand when \u0001 Teq;topis large, the RMS velocities respond nonlin-\nearly. The transition from nonlinear to linear response occurs at\ndi\u000berent \u0001Teq;topdepending on whether or not spatially constant\ndrag is applied. When there is not spatially constant drag (blue\nopen circles) and only basal drag is applied, the transition occurs at\n\u0001Teq;top\u00180:1 Kelvin. When \u001cdrag = 105sec (red \flled circles),\nthe transition occurs at \u0001 Teq;top\u001810 Kelvin. For comparison,\na linear relationship between Urmsand \u0001Teq;topis shown by the\nblack line.\nrange. Here we \frst explore the role of \u0001 Teqso that we\nmay make appropriate choices about the values of \u0001 Teq\nto use in the full grid.\nThe day-night radiative-equilibrium temperature con-\ntrast \u0001Teqrepresents the amplitude of the imposed ra-\ndiative forcing, and controls the amplitude of the result-\ning \row. In the low-amplitude limit (\u0001 Teq!0), the\nwind speeds and temperature perturbations are weak,\nand the nonlinear terms in the dynamical equations\nshould become small compared to the linear terms. Thus,\nin this limit, the solutions should behave in a mathemat-\nically linear3manner: the spatial structure of the circu-\nlation should become independent of forcing amplitude,\nand the amplitude of the circulation|that is, the wind\nspeeds and day-night temperature di\u000berences|should\nvary linearly with forcing amplitude. On the other hand,\nat very high forcing amplitudes (large \u0001 Teq), the wind\nspeeds and temperature di\u000berences are large, and the so-\nlutions behave nonlinearly.\nTherefore, we \frst performed a parameter sweep of\n\u0001Teqto understand the transition between linear and\nnonlinear forcing regimes, and to determine the value of\n\u0001Teqat which this transition occurs. For this param-\neter sweep, we performed a sequence of models varying\n\u0001Teq;topfrom 1000 to 0 :001 Kelvin4. We did one such\nsweep using \u001crad;top= 104s and\u001cdrag =1(meaning\nbasal-drag only), and another such sweep using \u001crad;top=\n3By this we mean that as \u0001 Teq!0, the solutions of the full\nnonlinear problem should converge toward the mathematical solu-\ntions to versions of Equations (1){(4) that are linearized around a\nstate with zero wind and the background T(p) pro\fle.\n4Speci\fcally, we tested values of \u0001 Teq;top =\n1000;500;200;100;10;1;0:1;0:01;and 0:001K.Dayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 7\n⌧rad,top(sec)\u0000Teq= 1000KBasal103104105106107\n  \nTemperature (K)9001000110012001300140015001600107106105104103\nlongitude (degrees)\nlatitude (degrees)\n−150−100−50050100150−80−60−40−20020406080Basal103104105106107107106105104103longitude (degrees)\nlatitude (degrees)−90090−45045\nFig. 4.| Maps of temperature (colors) and wind (vectors) for suite of 30 GCM simulations varying \u001crad;topand\u001cdragwith \u0001Teq;top= 1000\nKelvin. All maps are taken from the 80-mbar statistical steady-state end point of an individual model run. All plots share a color scheme\nfor temperature but have independent overplotting of horizontal wind vectors. The substellar point is located at 0\u000e;0\u000ein each plot, with\nthe lower right plot displaying latitude & longitude axes.\n104s and\u001cdrag= 105s. These sweeps verify that we are\nindeed in the linear limit (where variables such as wind\nspeed and temperature respond linearly to forcing) at\n\u0001Teq;top.0:1 Kelvin for any \u001cdrag. Figure 3 shows\nhow the root-mean-square (RMS) horizontal wind speed\nvaries with \u0001 Teq;topfor these two parameter sweeps.\nHere, the RMS horizontal wind speed, Urms, is de\fned\nat a given pressure as:\nUrms(p) =rR\n(u2+v2)dA\nA: (13)\nHere the integral is taken over the globe, with Athe hori-\nzontal area of the globe and u;vthe zonal and meridional\nvelocities at a given pressure level, respectively. It is no-table that the linear limit is reached at di\u000berent \u0001 Teq;top\nvalues depending on the strength of the drag applied.\nWith a spatially constant \u001cdrag= 105sec, the models\nrespond linearly to forcing at \u0001 Teq;top.10 Kelvin, and\nare very nearly linear throughout the range of \u0001 Teq;top\nconsidered. This causes the Urms-\u0001Teq;toprelationship\nfor\u001cdrag= 105sec to be visually indistinguishable from\na linear slope in Figure 3. However, without a spa-\ntially constant drag, the linear limit is not reached until\n\u0001Teq;top.0:1 Kelvin, and the dynamics are nonlinear\nfor \u0001Teq;top&1K.\nThe main grid presented involves a parameter study\nmimicking that of Perez-Becker & Showman (2013), but\nusing the 3D primitive equations. We do so in order\nto understand mechanisms behind hot Jupiter dayside-8 T.D. Komacek & A.P. Showman\n\u0000Teq=0.001KBasal103104105106107\n  \nTemperature (K)1285.63541285.63551285.63561285.63571285.63581285.63591285.636107106105104103\nlongitude (degrees)\nlatitude (degrees)\n−150−100−50050100150−80−60−40−20020406080\nBasal103104105106107107106105104103longitude (degrees)\nlatitude (degrees)−90090−45045⌧rad,top(sec)\nFig. 5.| Same as Figure 4, except with \u0001 Teq;top= 0:001 Kelvin.\nnightside temperature di\u000berences with the full system of\nnonlinear primitive equations. We varied \u001crad;topfrom\n103\u0000107sec and\u001cdrag from 103\u00001 sec (the range\nof timescales displayed in Figure 2), extending an or-\nder of magnitude lower in \u001cdrag than Perez-Becker &\nShowman (2013). We ran this suite of models for both\n\u0001Teq;top= 1000 Kelvin (nonlinear regime, see Figure\n4) and \u0001Teq;top= 0:001 Kelvin (linear regime, see Fig-\nure 5) to better understand the mechanisms controlling\ndayside-nightside temperature di\u000berences.\n3.2. Description of atmospheric circulation over a wide\nrange of radiative and frictional timescales\nFigure 4 shows latitude-longitude maps of tempera-\nture (with overplotted wind vectors) at a pressure of\n80 mbars for the entire suite of models performed at\n\u0001Teq;top= 1000 Kelvin. These are the statisticallysteady-state end points of 30 separate model simulations,\nwith\u001crad;topvarying from 103\u0000107sec and\u001cdragranging\nfrom 103\u00001 sec. All plots have the same temperature\ncolorscale for inter-comparison.\nOne can identify distinct regimes in this \u001crad;topand\n\u001cdragspace. First, there is the nominal hot Jupiter regime\nwith\u001cdrag=1and\u001crad;top.105sec, which has been\nstudied extensively in previous work. This regime has a\nstrong zonal jet which manifests as equatorial superrota-\ntion. However, there is no zonal jet when \u001cdrag.105sec.\nHence, there is a regime transition in the atmospheric\ncirculation at \u001cdrag\u0018106sec between a strong coher-\nent zonal jet and weak or absent zonal jets. Addition-\nally, there are two separate regimes of dayside-nightside\ntemperature di\u000berences as \u001crad;topvaries. When \u001crad;top\nis short (103\u0000104sec), the dayside-nightside temper-Dayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 9\nature di\u000berences are large. When \u001crad;top&106sec,\ndayside-nightside temperature di\u000berences are small on\nlatitude circles. However, \u001crad;topis not the only control\non dayside-nightside temperature di\u000berences. If atmo-\nspheric friction is strong, with \u001cdrag.104sec, dayside-\nnightside temperature di\u000berences are large unless \u001crad;top\nis extremely long.\nEquivalent model inter-comparison to Figure 4 for the\n\u0001Teq;top= 0:001 Kelvin case is presented in Figure 5.\nThe same major trends apparent in the \u0001 Teq;top=\n1000 Kelvin results are seen here in the linear limit. That\nis,\u001crad;topis still the key parameter control on dayside-\nnightside temperature di\u000berences. If \u001crad;topis small,\ndayside-nightside temperature di\u000berences are large, and\nif\u001crad;topis large, dayside-nightside di\u000berences are small.\nThis general trend is modi\fed slightly by atmospheric\nfriction|if \u001cdrag.104sec, drag plays a role in deter-\nmining the dayside-nightside temperature di\u000berences.\nA key di\u000berence between simulations at high and low\n\u0001Teqis that, under conditions of short \u001cradand weak\ndrag (upper-left quadrant of Figures 4{5), the maximum\ntemperatures occur on the equator in the nonlinear limit,\nbut they occur in midlatitudes in the linear limit. This\ncan be understood by considering the force balances.\nNamely, because the Coriolis force goes to zero at the\nequator, the only force that can balance the pressure\ngradient at the equator is advection. Advection is a non-\nlinear term that scales with the square of wind speed,\nand hence this force balance is inherently nonlinear. As\na result, the advection term and pressure gradient force\nboth weaken drastically at the equator in the linear limit,\nand the nonlinear balance cannot hold. This causes the\nequator to be nearly longitudinally isothermal in the lin-\near limit, rather than having large day-night temperature\ndi\u000berences as in the nonlinear case. This phenomenon\nwas already noted by Showman & Polvani (2011) and\nPerez-Becker & Showman (2013) in one-layer shallow wa-\nter models, and Figure 5 represents an extension of it to\nthe 3D system.\nAnother di\u000berence in the temperature maps between\nthe nonlinear and linear limit is the orientation of the\nphase tilts in the long \u001cdrag, short\u001cradupper left quad-\nrant of Figure 5. These phase tilts are the exact op-\nposite of those that are needed to drive superrotation.\nThe linear dynamics in the weak-drag limit causing these\ntilts has been examined in detail by Showman & Polvani\n(2011), see their Appendix C. They showed analytically\nthat in the limit of long \u001cdrag, the standing Rossby waves\ndevelop phase tilts at low latitudes that are northeast-\nto-southwest in the northern hemisphere and southeast-\nto-northwest in the southern hemisphere. This is the\nopposite of the orientation needed to transport eastward\nmomentum to equatorial regions and drive equatorial su-\nperrotation. Hence, the upper left quadrant of Figure 5\nshows key distinctions from the same quadrant in Fig-\nure 4. In the linear limit, there is no superrotation, and\nthe ratio of characteristic dayside and nightside temper-\natures at the equator is much smaller than in the case\nwith large forcing amplitude.\nOur model grids in Figures 4 and 5 exhibit a striking re-\nsemblance to the equivalent grids from the shallow-water\nmodels of Perez-Becker & Showman (2013, see their Fig-\nures 3 and 4). This gives us con\fdence that the same\nmechanisms determining the day-night temperature dif-ferences in their one-layer models are at work in the full\n3D system. Additionally, by extending \u001cdragone order\nof magnitude shorter, we have reached the parameter\nregime where drag can cause increased day-night tem-\nperature di\u000berences even when \u001cradis extremely long, al-\nlowing a more robust comparison to theory in this limit.\nDespite the distinctions between the nonlinear and lin-\near limits discussed above, both grids show similar over-\nall parameter dependences on radiative forcing and fric-\ntional drag. As radiative forcing becomes stronger (i.e.,\n\u001crad;topbecomes shorter), day-night temperature di\u000ber-\nences increase, with drag only playing a role if it is ex-\ntremely strong. Additionally, drag is the key factor to\nquell the zonal jet. The fact that the same general trends\nin day-night temperature di\u000berences occur in both the\nnonlinear and linear limit suggests that the same quali-\ntative mechanisms are controlling day-night temperature\ndi\u000berences in both cases, although nonlinearities will of\ncourse introduce quantitative di\u000berences at su\u000eciently\nlarge \u0001Teq. This makes it likely that a simple analytic\ntheory can explain the trends seen in Figures 4 and 5. We\ndevelop such a theory in Section 4, and continue in Sec-\ntion 5 to compare our results to the numerical solutions\npresented in this section.\n4.THEORY\n4.1. Pressure-dependent theory\nWe seek approximate analytic solutions to the problem\nposed in Sections 2{3. Speci\fcally, here we present solu-\ntions for the pressure-dependent day-night temperature\ndi\u000berence and the characteristic horizontal and vertical\nwind speeds as a function of the external control param-\neters (\u0001Teq;top,\u001crad;top,\u001cdrag, and the planetary param-\neters). In this theory, we do not distinguish variations\nin longitude from variations in latitude. As a result, we\nassume that the day-to-night and equator-to-pole tem-\nperature di\u000berences are comparable. Most GCM studies\nproduce relatively steady hemispheric-mean circulation\npatterns (e.g., Showman et al. 2009; Liu & Showman\n2013), including our own simulations shown in Section\n3, and so we seek steady solutions to the primitive equa-\ntions (1){(5). For convenience, we here cast these in\nlog-pressure coordinates (Andrews et al. 1987; Holton &\nHakim 2013):\nv\u0001rv+w?@v\n@z?+fk\u0002v=\u0000r\b\u0000v\n\u001cdrag; (14)\n@\b\n@z?=RT; (15)\nr\u0001v+ez?@(e\u0000z?w?)\n@z?= 0; (16)\nv\u0001rT+w?N2H2\nR=Teq\u0000T\n\u001crad: (17)\nEquation (14) is the horizontal momentum equation,\nEquation (15) the vertical momentum equation (hydro-\nstatic balance), Equation (16) the continuity equation,\nand Equation (17) the thermodynamic energy equation.\nIn this coordinate system, z?is de\fned as\nz?\u0011\u0000lnp\np00; (18)10 T.D. Komacek & A.P. Showman\nwithp00a reference pressure, and the vertical velocity\nw?\u0011dz?=dt, which has units of scale heights per sec-\nond (such that 1 =w?is the time needed for air to \row\nvertically over a scale height). Nis the Brunt-Vaisala fre-\nquency and H=RT=g is the scale height. This equation\nset is equivalent to the steady-state version of Equations\n(1-5). Note that drag is explicitly set on the horizontal\ncomponents of velocity. We use the steady-state system\nhere in order to facilitate comparison with our models,\nwhich themselves are run to steady-state with kinetic en-\nergy equilibration.\nGiven the set of Equations (14){(17) above, we can now\nutilize scaling to give approximate solutions for compar-\nison to both our fully nonlinear (high \u0001 Teq) and linear\n(low \u0001Teq) numerical solutions. Here, we step systemat-\nically through the equations, starting with the continuity\nequation, then the thermodynamic energy equation, and\n\fnally the momentum equations.\n4.1.1. Continuity equation\nFirst, consider the continuity equation (16). The scal-\ning for the \frst term on the left hand side is subtle.\nWhen the Rossby number Ro &1, we expect that\nr\u0001v\u0018U=L, whereUis a characteristic horizontal ve-\nlocity andLa characteristic lengthscale of the circula-\ntion. However, when Ro .1, geostrophy holds and in\nprinciple we could have r\u0001v\u001c U=L. For a purely\ngeostrophic \row, r\u0001v\u0011\u0000\fv=f , withvmeridional ve-\nlocity (see Showman et al. 2010, Eq. 32). On a sphere,\n\f= 2\ncos\u001e=a, and hence \f=f = cot\u001e=a. As a result,\nr\u0001v\u0019Ucot\u001e=a. For hot Jupiters, L\u0018a, and hence it\nturns out that for geostrophic \row not too close to the\npole thatr\u0001v\u0018U=L. Hence, the scaling r\u0001v\u0018U=L\nholds throughout the circulation regimes considered here.\nNow consider the second term on the left side of (16).\nExpanding out the derivative yields \u0000w?+@w?=@z?. The\nterm@w?=@z?scales asw?=\u0001z?, where \u0001z?is the ver-\ntical distance (in scale heights) over which w?varies by\nits own magnitude. Thus, one could write\nU\nL\u0018max\u0014\nw?;w?\n\u0001z?\u0015\n: (19)\nPrevious GCM studies suggest that w?maintains co-\nherency of values over several scale heights (e.g. Par-\nmentier et al. 2013), suggesting that \u0001 z?is several (i.e.,\ngreater than one), in which case the \frst term on the\nright dominates. De\fning an alternate characteristic ver-\ntical velocityW=Hw?, which gives the approximate\nvertical velocity in m s\u00001, the continuity equation be-\ncomes simply\nU\nL\u0018W\nH: (20)\n4.1.2. Thermodynamic energy equation\nWe next consider the thermodynamic energy equation,\nwhich contains the sole term that drives the circulation\n(i.e., radiative heating and cooling). The quantity Teq\u0000T\non the rightmost side of Equation (17) represents the\nlocal di\u000berence between the radiative-equilibrium and\nactual temperature. This di\u000berence varies spatially in\nvalue and sign, as it is typically positive on the dayside\nand negative on the nightside. Here we seek an expres-\nsion for its characteristic magnitude. We note that if\nLongitude \nTemperature Day Night Night \n∆T \n∆Teq \n|Teq-T|night |Teq-T|day \nActual temperature Radiative equilibrium temperature Fig. 6.| Simpli\fed diagram of the model, schematically display-\ning longitudinal pro\fles of actual and radiative equilibrium temper-\nature. We show this schematic to help explain Equation (22). The\ndi\u000berence between the characteristic actual and radiative equilib-\nrium temperature di\u000berences from dayside to nightside, \u0001 Teq\u0000\u0001T,\nis approximately equal to the sum of the characteristic di\u000berences\non each hemisphere, jTeq\u0000Tjday+jTeq\u0000Tjnight.\njTeq\u0000Tjglobalis de\fned as\njTeq\u0000Tjglobal\u0011jTeq\u0000Tjday+jTeq\u0000Tjnight;(21)\nwhere the di\u000berences on the right hand side are charac-\nteristic di\u000berences for the appropriate hemisphere, one\ncan write\n\u0001Teq\u0000\u0001T\u0018jTeq\u0000Tjglobal: (22)\nIn Equation (22), \u0001 Tand \u0001Teqare de\fned to be the\ncharacteristic di\u000berence between the dayside and night-\nside temperature and equilibrium temperature pro\fles,\nrespectively. Figure 6 shows visually this approximate\nequality between \u0001 Teq\u0000\u0001TandjTeq\u0000Tjglobal.\nWith this formalism for characteristic dayside-\nnightside temperature di\u000berences, we can write an ap-\nproximate version of Equation (17) as:\n\u0001Teq\u0000\u0001T\n\u001crad\u0018max\u0014U\u0001T\nL;WN2H\nR\u0015\n: (23)\nThe quantitiesU,W, \u0001T,H, andNare implicitly func-\ntions of pressure. For the analyses that follow, we use a\nvalue ofLapproximately equal to the planetary radius.\nWhat is the relative importance of the two terms on\nthe right-hand side of (23)? Using Equation (20) and the\nde\fnition of Brunt-Vaisala frequency, the second term\ncan be expressed as U=Ltimes\u000eTstrat, where\u000eTstrat is\nthe di\u000berence between the actual and adiabatic tempera-\nture gradients integrated vertically over a scale height|\nor, equivalently, can approximately be thought of as\nthe change in potential temperature over a scale height.\nThus, the \frst term (horizontal entropy advection) dom-\ninates over the second term (vertical entropy advection)\nonly if the day-night temperature (or potential temper-\nature) di\u000berence exceeds the vertical change in potential\ntemperature over a scale height. For a highly strati\fed\ntemperature pro\fle like those expected in the observable\natmospheres of hot Jupiters, the vertical change in po-\ntential temperature over a scale height is a signi\fcant\nfraction of the temperature itself.5\n5For a vertically isothermal temperature pro\fle, \u000eTstrat =Dayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 11\nThus, one would expect that vertical entropy advec-\ntion dominates over horizontal entropy advection unless\nthe fractional day-night temperature di\u000berence is close to\nunity. This is just the weak temperature gradient regime\nmentioned in the Introduction. Considering this WTG\nbalance, we have6\n\u0001Teq\u0000\u0001T\n\u001crad\u0018WN2H\nR: (24)\nGiven our full solutions, we will show in Section 5.4.2 that\nhorizontal entropy advection is indeed smaller than ver-\ntical entropy advection in a hemospheric-averaged sense,\ndemonstrating the validity of (24).\n4.1.3. Hydrostatic balance\nHydrostatic balance relates the geopotential to the\ntemperature, and thus we can use it to relate the\nday-night temperature di\u000berence, \u0001 T, to the day-night\ngeopotential di\u000berence (alternatively pressure gradient)\non isobars. Hydrostatic balance implies that \u000e\b = R\nRTdlnp, where here \u000e\b is a vertical geopotential di\u000ber-\nence. Then, consider evaluating \u000e\b on the dayside and\nnightside, for two vertical air columns sharing the same\nvalue of \b at their base, at points separated by a hori-\nzontal distanceL. Given that these two points have the\nsame \b at the bottom isobar, we can di\u000berence \u000e\b at\nthese locations to solve for the geopotential change from\ndayside to nightside. Doing so, we \fnd the horizontal\ngeopotential di\u000berence7:\n\u0001\b\u0019RZpbot\np\u0001Tdlnp0: (25)\nIn Equation (25), both \u0001\b and \u0001 Tare functions of pres-\nsure. Here we only have to integrate to the pressure level\nat which our prescribed equilibrium dayside-nightside\ntemperature di\u000berence \u0001 Teqgoes to zero, which is la-\nbeledpbot.\nGiven that di\u000berences in scalar quantities from day-\nside to nightside are a function of pressure in our model,\nwe can use our Newtonian cooling parameterizations as\na guide to the form this pressure-dependence will take.\nHence, we take a form of \u0001 Tsimilar to that of \u0001 Teq,\nfocusing only on the region with pressure dependence:\n\u0001T= \u0001Ttopln(p=pbot)\nln(peq;top=pbot): (26)\nNow, integrating Equation (25) and dropping a factor of\ntwo, we \fnd\n\u0001\b\u0019R\u0001Tln\u0012pbot\np\u0013\n: (27)\n4.1.4. Momentum equation\nNext we analyze the approximate horizontal momen-\ntum equation, where the pressure gradient force driving\ngH=cp=RT=cp, and ifR=cp= 2=7 as appropriate for an H 2\natmosphere, then \u000eTstrat\u0019400 K for a typical hot Jupiter with a\ntemperature of 1500 K.\n6This balance was \frst considered for hot Jupiters by Showman\n& Guillot (2002, Eq. 22).\n7Note that this is essentially the hypsometric equation (e.g.\nHolton & Hakim 2013 pp. 19 or Wallace & Hobbs 2004 pp. 69-72).the circulation can be balanced by horizontal advection\n(in regions where the Rossby number Ro \u0011U=fL\u001d 1),\nvertical advection, the Coriolis force (Ro \u001c1), or drag:\nr\b\u0018max\u0014U2\nL;UW\nH;fU;U\n\u001cdrag\u0015\n: (28)\nNote that, given our continuity equation (20), the hor-\nizontal and vertical momentum advection terms U2=L\nandUW=Hare identical. As a result, we expect that\nthe \fnal solutions for \u0001 T(p),U(p), andW(p) should be\nthe same if horizontal momentum advection balances the\npressure gradient force as for the case when vertical mo-\nmentum advection balances it.\nExpressingr\b as \u0001\b=L, and relating \u0001\b to \u0001 Tus-\ning (27), we evaluate Equation (28) for every possible\ndominant term on the right hand side, now with explicit\npressure-dependence. This leads to approximate hori-\nzontal velocities that scale as\nU(p)\u00188\n>>><\n>>>:R\u0001T(p)\u0001lnp \u001cdrag(p)\nLDrag\nR\u0001T(p)\u0001lnp\nLfCoriolis\np\nR\u0001T(p)\u0001lnp Advection;\n(29)\nwhere we de\fne \u0001 ln p= ln(pbot=p) as the di\u000berence in\nlog pressure from the deep pressure pbot= 10 bars, where\nthe day-night forcing goes to zero, to some lower pressure\nof interest. As foreshadowed above, the same solution,\ngiven by the \fnal expression in (29), is attained when\neither horizontal or vertical momentum advection bal-\nances the pressure gradient force. The expressions for\nUin Equation (29) for advection and Coriolis force bal-\nancing pressure gradient match those of Showman et al.\n(2010) (see their Equations 48 and 49). Hence, we recap-\nture within our idealized model the previous expectations\nfor characteristic horizontal wind speeds on hot Jupiters\nbased on simple force balance. Note that these are ex-\npressions for characteristic horizontal wind speed, which\nrepresents average day-night \row and hence is not the\nzonal-mean zonal wind associated with superrotation.\nInvoking the relationship between UandWfrom Equa-\ntion (20), we can now write expressions for the vertical\nwind speed as a function of \u0001 T:\nW(p)\u00188\n>>>>><\n>>>>>:R\u0001T(p)H(p)\u0001lnp \u001cdrag(p)\nL2Drag\nR\u0001T(p)H(p)\u0001lnp\nL2fCoriolis\nH(p)\nLp\nR\u0001T(p)\u0001lnp Advection:\n(30)\n4.2. Full Solution for fractional dayside-nightside\ntemperature di\u000berences and wind speeds\nTo obtain our \fnal solution, we simply combine Equa-\ntion (24) and (30) to yield expressions for \u0001 T(p) and12 T.D. Komacek & A.P. Showman\nW(p) as a function of control parameters:\n\u0001T(p)\n\u0001Teq(p)\u00188\n>>>>>>><\n>>>>>>>:\u0012\n1 +\u001crad(p)\u001cdrag(p)\n\u001c2wave(p)\u0001lnp\u0013\u00001\nDrag\n\u0012\n1 +\u001crad(p)\nf\u001c2wave(p)\u0001lnp\u0013\u00001\nCoriolis\np\n\r(p) + 4\u0001Teq(p)\u0000p\n\r(p)p\n\r(p) + 4\u0001Teq(p) +p\n\r(p)Advection;\n(31)\nand\nW\u00188\n>>>>>>>>>>>>>>>>>>>>>>>>><\n>>>>>>>>>>>>>>>>>>>>>>>>>:RH(p)\nL2\u001c2\nwave(p)\n\u001crad(p)\u0001Teq(p)\n\u0002\"\n1\u0000\u0012\n1 +\u001crad(p)\u001cdrag(p)\n\u001c2wave(p)\u0001lnp\u0013\u00001#\nDrag\nRH(p)\nL2\u001c2\nwave(p)\n\u001crad(p)\u0001Teq(p)\n\u0002\"\n1\u0000\u0012\n1 +\u001crad(p)\nf\u001c2wave(p)\u0001lnp\u0013\u00001#\nCoriolis\nRH(p)\nL2\u001c2\nwave(p)\n\u001crad(p)\u0001Teq(p)\n\u0002\"\n1\u0000p\n\r(p) + 4\u0001Teq(p)\u0000p\n\r(p)p\n\r(p) + 4\u0001Teq(p) +p\n\r(p)#\nAdvection:\n(32)\nThe solution forU(p) is simply that for W(p) timesL=H.\nIn these solutions, we have de\fned \r(p) =\n\u001c2\nrad(p)L2\u0001lnp=(\u001c4\nwave(p)R). Moreover, we have substi-\ntuted a Kelvin wave propagation timescale \u001cwave(p) =\nL=(N(p)H(p)), as in Showman et al. (2013a). This wave\npropagation timescale is derived by taking the long ver-\ntical wavelength limit of the Kelvin wave dispersion rela-\ntionship, giving the fastest phase and group propagation\nspeedsc= 2NH. Hence, the wave propagation time\nacross a hemisphere is approximately L=(NH).\nOur drag and Coriolis-dominated equations are the\nsame expressions as in Perez-Becker & Showman (2013),\nbut the advection-dominated cases di\u000ber. Note that as in\nPerez-Becker & Showman (2013), the dayside-nightside\ntemperature (thickness in their case) di\u000berences for the\ndrag and Coriolis-dominated regimes are equal if \u001cdrag\u0018\n1=f. The solutions for \u0001 T=\u0001Teqin the drag and Corio-\nlis cases are independent of the forcing amplitude \u0001 Teq,\nwhereas the solution in the advection case depends on\n\u0001Teq, as one would expect given the nonlinear nature of\nthe momentum balance.\nWith these \fnal solutions for the dayside-nightside\ntemperature di\u000berence and characteristic wind speeds,\nwe can compare our theory to the GCM results in Sec-\ntion 5. However, we \frst must diagnose which solution\nis appropriate for any given combination of parameters.\n4.3. Momentum equation regime\nTo determine which regime is appropriate in our solu-\ntions from Section 4.2 for given choices of control pa-\nrameters (\u001crad;top,\u001cdrag, and \u0001Teq;top), we perform a\ncomparison of the magnitudes of the drag, Coriolis, and\nmomentum advection terms. We do this by comparingthe relative amplitues of their characteristic timescales:\n\u001cdrag, 1=f, and\u001cadv. Using Equations (20) and (32), we\ncan write\u001cadvas\n\u001cadv(p) =L2\u001crad(p)\nR\u001c2wave(p)\u0001Teq(p)\n\u0002\"\n1\u0000p\n\r(p) + 4\u0001Teq(p)\u0000p\n\r(p)p\n\r(p) + 4\u0001Teq(p) +p\n\r(p)#\u00001\n:\n(33)\nNow, we can write the condition that determines which\nregime the solution is in. If \u001cdrag<min(1=f;\u001c adv),\nthe solution is in the drag-dominated regime. If 1 =f <\nmin(\u001cdrag;\u001cadv), it is in the Coriolis regime. If \u001cadv<\nmin(1=f;\u001c drag), it is in the advection-dominated regime.\nTo evaluate these conditions, we proceed as follows.\nOne can start by evaluating the Rossby number Ro =\nU=fLusing Eqs. (20) and (32) for given external pa-\nrameters, and this determines the relative magnitudes\nof Coriolis and advective forces8. First, we consider the\nregime where Ro <1, which corresponds to the regions\nwhere the Coriolis force has a magnitude larger than\nthat of advection. This is appropriate in the mid-to-\nhigh latitudes of a planet, with the latitudinal extent\nof this regime increasing with increasing rotation rate.\nThough the Coriolis force dominates over advection, if\n\u001cdrag is very short, drag can have a larger magnitude\nthan the Coriolis force. Hence, we consider a three-term\nforce balance in this regime, with drag balancing pres-\nsure gradient if \u001cdrag<1=fand Coriolis balancing the\npressure gradient if 1 =f <\u001c drag. Given this, we can write\na solution for the fractional day-night temperature dif-\nferenceA(p)\u0018\u0001T(p)=\u0001Teq(p) (labeled such for future\ncomparison with Aobsin Section 5) in the Ro <1 regime:\nA(p)\u00188\n>><\n>>:\u0012\n1 +\u001crad(p)\u001cdrag(p)\n\u001c2wave(p)\u0001lnp\u0013\u00001\n\u001cdrag<f\u00001\n\u0012\n1 +\u001crad(p)\nf\u001c2wave(p)\u0001lnp\u0013\u00001\n\u001cdrag>f\u00001:\n(34)\nIn the regime where Ro >1, the advection term has a\ngreater magnitude than the Coriolis force. This is rele-\nvant to the equatorial regions of any given planet, where\nf!0 due to the latitudinal dependence of the Coriolis\nforce. Additionally, if advection is very strong (or rota-\ntion very slow), this regime could extend to the mid-to-\nhigh latitudes. The three-term force balance in regions\nwhere Ro>1 is hence between drag, advection, and the\nday-night pressure gradient force. Using the same rea-\nsoning as above, we can write the fractional day-night\n8Alternatively, one can calculate the Rossby number from the\nadvective timescale, as we take fas an external parameter and\nU=L=\u001c\u00001\nadv. Hence, one can calculate the Rossby number (as a\nfunction of latitude) as Ro( \u001e) = [f(\u001e)\u001cadv]\u00001, where\u001cadvis given\nby Equation (33).Dayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 13\ntemperature di\u000berence at the equator Aeq(p):\nAeq(p)\u00188\n>>>>>>><\n>>>>>>>:\u0012\n1 +\u001crad(p)\u001cdrag(p)\n\u001c2wave(p)\u0001lnp\u0013\u00001\n\u001cdrag(p)<\u001cadv(p)\np\n\r(p) + 4\u0001Teq(p)\u0000p\n\r(p)p\n\r(p) + 4\u0001Teq(p) +p\n\r(p)\u001cdrag(p)>\u001cadv(p);\n(35)\nwhere\u001cadvis evaluated from Eq. (33).\n4.4. Transition from low to high day-night temperature\ndi\u000berences\nFrom the expressions above for the fractional day-night\ntemperature di\u000berence, it is clear that a comparison be-\ntween wave propagation timescales and other relevant\ntimescales (radiative, drag, Coriolis) controls the ampli-\ntude of the day-night temperature di\u000berence. Here, we\nuse our theory to obtain timescale comparisons for the\ntransition from small to large day-night temperature dif-\nference (relative to that in radiative equilibrium).\nUsing the expression for overall Afrom Equation (34),\nthe transition between low fractional dayside-nightside\ntemperature di\u000berences ( A!0) and high fractional\ndayside-nightside temperature di\u000berences ( A!1) oc-\ncurs when9:\n\u001cwave(p)\u0018(q\n\u001cdrag(p)\u001crad(p)\u0001lnp \u001c drag(p)<f\u00001\np\nf\u00001\u001crad(p)\u0001lnp \u001c drag(p)>f\u00001:\n(36)\nThis timescale comparison is valid in regions where Ro <\n1, which occurs at nearly all latitudes (except those bor-\ndering the equator, where Ais given by Equation 35).\nWhen\u001cwave is smaller than the expression on the right-\nhand side of Equation (36), the day-night temperature\ndi\u000berences are small, while if it is larger the day-night\ntemperature di\u000berences are necessarily large. It is no-\ntable that the transition between low and high day-night\ntemperature di\u000berence always involves a comparison be-\ntween wave and radiative timescales, no matter what\nregime the system is in. Meanwhile, drag is only im-\nportant if\u001cdrag<min(1=f;\u001c adv), which is why drag only\nplays a role in determining the day-night temperature\ndi\u000berences if \u001cdrag<1=f\u0018105sec for our choice of \n.\nWe can motivate the timescale comparison in Equa-\ntion (36) by considering the time it takes for adiabatic\nvertical motions to smooth out a day-night temperature\ndi\u000berence equal to \u0001 T. The contribution of vertical mo-\ntion to the time variation of temperature is WN2H=R\n(see Equation 23), which implies that we can express the\ntimescale\u001cvertfor day-night temperature di\u000berences to\nbe erased by vertical entropy advection as:\n\u001cvert\u0018\u0001TR\nWN2H: (37)\nFor concreteness, consider the drag-dominated regime\nwhere\u001cdrag<min(1=f;\u001c adv). Substituting our solutions\n9Note that the two expressions in Equation (36) are equivalent\nto a pressure-dependent versions of Equations (24) in Perez-Becker\n& Showman (2013)for \u0001TandWfrom Equations (31) and (32), respec-\ntively, we can write:\n\u001cvert\u0018\u001c2\nwave\n\u001cdrag\u0001lnp: (38)\nIf\u001cradis on the order of \u001cvert, the approximate equality\n\u001crad\u0018\u001c2\nwave\n\u001cdrag\u0001lnp(39)\nholds, which is precisely the same timescale comparison\nas in Equation (36) for the strong-drag regime. Per-\nforming the same exercise for the regime where 1 =f <\nmin(\u001cdrag;\u001cadv),\n\u001cvert\u0018\u001crad\u0018f\u001c2\nwave\n\u0001lnp: (40)\nThis is precisely the same timescale comparison as in\nEquation (36) for the weak-drag regime. Note that if\n\u001cvert\u001c\u001crad, the solution approaches the limit of small\nA, with\u001cwave\u001cminhp\nf\u00001\u001crad\u0001lnp;p\n\u001cdrag\u001crad\u0001lnpi\n.\nIn the other limit of \u001cvert\u001d\u001crad, the solution approaches\nthat in radiative equilibrium, with A!1. This link\nbetween\u001cvertand\u001cwave is a natural consequence of the\nWTG limit, where vertical motion that moves isentropes\nvertically (parameterized here by \u001cvert) is forced by hor-\nizontal convergence/divergence caused by propagating\nwaves (parameterized by \u001cwave).\n5.TEST OF THE THEORY WITH 3D SIMULATIONS\n5.1. Computing fractional dayside-nightside temperature\ndi\u000berences from numerical results\nThough we have expressions for the dayside-nightside\ntemperature di\u000berence relative to the equilibrium\ndayside-nightside temperature di\u000berence, we must relate\nthis to a metric we can compute using our numerical so-\nlutions. To compute this metric for the overall dayside-\nnightside temperature contrast, we follow a method sim-\nilar to Perez-Becker & Showman (2013). First, we eval-\nuate the longitudinal RMS variations in temperature, to\n\fndA(\u001e;p), the overall fractional dayside-nightside tem-\nperature di\u000berence as a function of latitude and pressure:\nA(\u001e;p) =(R2\u0019\n0\u0002\nT(\u0015;\u001e;p )\u0000\u0016T(\u001e;p)\u00032d\u0015\nR2\u0019\n0\u0002\nTeq(\u0015;\u001e;p )\u0000\u0016T(\u001e;p)\u00032d\u0015)1=2\n:(41)\nHere \u0016Tis the zonal-mean temperature at a given latitude\nand pressure. Then, we average A(\u001e;p) with latitude\nover a 120\u000eband, centered around the equator10to \fnd\nA, a numerical representation of \u0001 T=\u0001Teq:\nA(p) =3\n2\u0019Z\u0019=3\n\u0000\u0019=3A(\u001e;p)d\u001e: (42)\nAfter this averaging, we choose a nominal pressure level,\n\u001880 mbars, to calculate near-photospheric dayside-\nnightside temperature di\u000berences. This Ais relevant to\n10We do not include the poles in this average, as the dayside-\nnightside forcing is strongest at low latitudes in the Newtonian\ncooling scheme used due to the cos( \u001e) dependence of radiative-\nequilibrium temperature. Additionally, their larger projected area\nimplies that low latitudes dominate the emitted \rux seen in phase\ncurves.14 T.D. Komacek & A.P. Showman\nthe Ro<1 limit, where Equation (34) gives our theoreti-\ncal prediction for A. To compare with the predictions for\nday-night temperature di\u000berences at the equator, given\nby Equation (35), we simply evaluate A(\u001e;p) from Equa-\ntion (41) at the equator itself, where \u001e= 0\u000e. We also\ntake the pressure-dependence of Ainto account, which\ninforms us about how dayside-nightside temperature dif-\nferences vary with depth. Now we can directly compare\nour theoretical expressions for A(p) with numerical solu-\ntions.\n5.2. Comparison of theory to numerical results\nWe now use our expectations for Afrom Section 4.3\nto directly compare model results and theoretical ex-\npectations. First, we compare the model overall A\nfrom the theoretical predictions in Equation (34) for the\n\u0001Teq;top= 1000 Kelvin case at two pressure levels, 80\nmbars and 1 bar, shown in Figure 7. We show the\nsame comparison for the \u0001 Teq;top= 0:001 Kelvin case\nin Figure 8. These are predictions that use a momentum\nbalance between either the Coriolis force and pressure\ngradient or drag and pressure gradient. In this and all\nthe comparisons that follow, we calculate \u001cwavefrom the\nlocal scale height and vertically-isothermal limit of the\nBrunt-Vaisala frequency. The black line on the theory\nplots demarcates the transition between these balances.\nAbove the black line, the Coriolis force balances pres-\nsure gradient, while below the black line drag balances\nthe pressure-gradient force.\nFrom Figures 7 and 8, the similarity between the GCM\nresults and theory show that our analysis quantitatively\nreproduces the overall dayside-nightside temperature dif-\nferences well. In both the numerical results and theory,\nfractional dayside-nightside temperature di\u000berences de-\ncrease as\u001cradincreases. Additionally, if drag is stronger\nthan the Coriolis force, the dayside-nightside tempera-\nture di\u000berences are larger. That the transition between\nhigh and low dayside-nightside temperature di\u000berences\ndepends on both \u001cdragand\u001cradin the region where drag is\nstronger than the Coriolis force can be understood from\nEquation (36). Here, A(p) is inversely correlated with\nboth\u001cdragand\u001crad, while above the black line (where\nthe Coriolis force is stronger than drag) A(p) only de-\npends on\u001crad.\nThe main discrepancy between the theoretical prediction\nand GCM results for overall dayside-nightside tempera-\nture di\u000berences occurs at high pressures ( \u00181 bar) when\nradiative forcing is strong. This can be seen in the upper-\nleft hand quadrant of the 1 bar comparison in Figures 7\nand 8. This is due to the circulation (Gill) pattern, which\nallows for a signi\fcant horizontal temperature contrast\nin these cases. In this region of \u001crad;\u001cdrag parameter\nspace, winds do not necessarily blow straight from day to\nnight across isotherms as assumed in our theory, but cir-\ncle around hot/cold regions in large-scale vortices. This\ncan be seen at lower pressures in Figures 4 and 5, and\npersists to\u00181 bar pressure. Similar circulation patterns\nhave been seen previously, for example in the simulations\nof Showman et al. (2008a) using Newtonian cooling and\nby Showman et al. (2009) when including non-grey radia-\ntive transfer. Hence, at pressures of \u00181 bar, our theory\n(Equation 31) does not predict the day-night temper-\nature di\u000berence well in the weak-drag, strong-radiation\ncorner of the parameter space. However, at even greaterpressures this pattern does not exist and A(p) again be-\ncomes a proper metric for the dayside-nightside temper-\nature contrast.\nThough the overall comparison between theoretical ex-\npectations and model results for dayside-nightside tem-\nperature di\u000berences in Figures 7 and 8 is useful, the\nobservable dayside-nightside temperature di\u000berences are\nexpected to be dominated by equatorial \row. Hence,\nwe examine a comparison between model Aeqand the\ntheoretical Aeqcalculated from Equation (35). We do\nso for both the nonlinear case (see Figure 9) and linear\ncase (Figure 10). A similar general trend is seen as for\nthe overall A, with the theory capturing well the change\ninAeqwith\u001cradand\u001cdragand with the same mismatch\nin the weak-drag, strong-radiative cooling regime in the\nnonlinear limit.\nThe quantitative comparison between theory and re-\nsults is especially good in the \u0001 Teq;top= 0:001 Kelvin\ncase (Figure 10). This increased compatibility is ex-\nplained as in Perez-Becker & Showman (2013) by the low-\nered strength of horizontal advection as day-night forcing\ngoes to zero. This causes the timescale comparison at the\nequator to be almost exactly vertical momentum advec-\ntion against drag. Hence, in the linear limit the equa-\ntorial zonal-mean momentum equation becomes almost\nexactly pressure gradient force balancing either advec-\ntion or drag. This enables the three-term analytic the-\nory to predict dayside-nightside temperature di\u000berences\nmost e\u000bectively, as we do not include the superrotating\nequatorial jet in our theory.\nOur theory also matches, to \frst order, the depth-\ndependence of dayside-nightside temperature di\u000ber-\nences over the range of radiative forcing and fric-\ntional drag strengths studied. Figure 11 compares\nthe pressure-dependent theoretical predictions for the\ndayside-nightside temperature di\u000berence with GCM re-\nsults throughout the \u001crad,\u001cdragparameter space consid-\nered. This \u0001 T=TRMS;day\u0000TRMS;night is a di\u000berence\nbetween the RMS dayside and RMS nightside temper-\natures, and hence is a representative dayside-nightside\ntemperature di\u000berence, not the peak to trough di\u000ber-\nence.\nThe same general trends are seen in both the theo-\nretical predictions and numerical results, with \u0001 Tbeing\nsmallest at long \u001crad. As here the dominant comparison\nin the momentum equation is between pressure-gradient\nforces and either drag and Coriolis forces, the theoretical\nlines in a given column when \u001cdrag&105sec are equiv-\nalent. Referring back to Equation (34), this means that\ndrag does not have an e\u000bect on wave timescales unless\nit is stronger than the Coriolis force. That drag has a\nsmall e\u000bect on wave propagation timescales can also be\nseen in our numerical solutions by the similarity between\nthe model curves for \u0001 Tfor\u001cdrag&105sec. Hence, this\nis a re-iteration of the trend from Figure 4 that \u001cradis\nthe main damping of wave adjustment processes in our\nmodel. Drag is a secondary e\u000bect that only becomes\nimportant at \u001cdrag.104sec. Crucially, the dayside-\nnightside temperature di\u000berence decreases with increas-\ning pressure in both our theoretical predictions and nu-\nmerical simulations. This is a key result which is due\nto the fact that \u001cradincreases with depth. Though this\nincrease of the radiative timescale with depth is model-\nprescribed, it is expected from previous analytic (Show-Dayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 15\n\u0000Teq= 1000K80 mbar\n1 bartheory\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7log10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7log10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9GCM\nFig. 7.| Overall fractional dayside-nightside temperature temperature di\u000berence, A, plotted as a function of \u001crad;top(abscissa) and\n\u001cdrag (ordinate) for both GCM results (left) and theoretical predictions (right), and two di\u000berent pressure levels: 80 mbars (top row), 1\nbar (bottom row). The model and theory plots for a given pressure level have the same color scale. The theoretical predictions are based\non the competition between pressure-gradient forces and either drag or Coriolis forces, see Equation (34). The black line on the righthand\nplots demarcates the regions where drag or Coriolis forces dominate: above the black line, Coriolis force is balanced with pressure gradient,\nand below drag is the dominant balancing term. The colormap is scaled such that red corresponds to high temperature di\u000berences and\nblue to small temperature di\u000berences.\n80 mbar\n1 bar\u0000Teq=0.001KtheoryGCM\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7\nFig. 8.| As in Figure 7, but for the case with \u0001 Teq;top= 0:001 Kelvin.16 T.D. Komacek & A.P. Showman\n\u0000Teq= 1000K80 mbar\n1 bartheory\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7log10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7log10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9GCM\nFig. 9.| As in Figure 7, but with Aonly evaluated at the planetary equator. In this \fgure, the theoretical predictions are based on the\ncompetition between drag and advection, see Equation (35). The black line demarcates this transition: above the black line, advection is\nthe dominant balance with the pressure gradient force, while below the black line drag is the controlling term.\n80 mbar\n1 bar\u0000Teq=0.001K\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7log10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.7log10τrad (sec)log10τdrag (sec)\n  \n34567345678\nA0.10.20.30.40.50.60.70.80.9theoryGCM\nFig. 10.| As in Figure 9, but for the case with \u0001 Teq;top= 0:001 Kelvin.Dayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 17\n⌧rad,top(sec)\n\u0000Teq= 1000KBasal10310410510650100150200250107∆ T (Kelvin)10750100150200250∆ T (Kelvin)10650100150200250∆ T (Kelvin)10550100150200250∆ T (Kelvin)10450100150200250∆ T (Kelvin)\n103104105106103Pressure (Pa)104105106Pressure (Pa)104105106Pressure (Pa)104105106Pressure (Pa)104105106050100150200250∆ T (Kelvin)Pressure (Pa)\nFig. 11.| Comparison of theoretical prediction (dashed lines) and GCM results (solid lines) for characteristic overall dayside-nightside\ntemperature di\u000berence \u0001 T=TRMS;day\u0000TRMS;night as a function of pressure for the case where \u0001 Teq;top= 1000 Kelvin. Since our\ntheory predicts that the characteristic Rossby number is .1 over most of the parameter space, the theory curves utilize the three-way\nforce balance between Coriolis, drag, and pressure-gradient forces in the momentum equation. \\Basal\" here means \u001cdrag =1, and only\nthe Held-Suarez drag applies. Note that the theory curves in a given column when \u001cdrag&105sec are similar, which is a statement that\nthe wave propagation timescales are approximately equivalent. There is good agreement between the numerical solutions and theoretical\npredictions for \u0001 Tover much of the parameter space. The dayside-nightside temperature di\u000berence decreases with increasing pressure in\nboth the theoretical predictions and numerical simulations.\nman & Guillot 2002; Ginzburg & Sari 2015) and numer-\nical (Iro et al. 2005; Showman et al. 2008a) work which\nmotivated this prescription.\n5.3. Vertical and horizontal velocities\nUsing our predictive scaling for horizontal and verti-\ncal velocities, we compare GCM results with theoretical\npredictions for wind speeds in Figure 12. Here, model\nand theory contours of velocities at 80 mbars pressure\nare plotted as a function of \u001cradand\u001cdrag, as in the A\nplots in Figures 7 - 10. The model results are RMS windspeeds, with Urmsde\fned in Equation (13) and\nWrms(p) =rR\nw2dA\nA; (43)\nwherewis the vertical velocity at a given pressure\nlevel, calculated from our GCM results as w=\u0000!H=p\n(Lewis et al. 2010). The theoretical predictions require\nonly knowledge of the model input parameters (radia-\ntive equilibrium temperature pro\fle, radiative and drag\ntimescales). The same general trend of wind speeds in-\ncreasing at shorter \u001cradand longer \u001cdragis seen in both18 T.D. Komacek & A.P. Showman\n\u0000Teq= 1000KtheoryGCM\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nUrms (m/s)20040060080010001200\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nUrms (m/s)\n020040060080010001200\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nWrms (m/s)12345678910\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nWrms (m/s)1234567891080 mbar\n80 mbar\nFig. 12.| Root-mean-square velocities at 80 mbars, for model results (left) and theoretical prediction (right). The top row shows\nhorizontal wind speeds, and the bottom row shows vertical wind speeds. The wind speeds are calculated using Equations (20) and (32),\nwith an overall three-way balance between Coriolis force, drag, and pressure gradient relevant to Ro <1. The contours of wind speed are\nplotted as a function of \u001crad;top(abscissa) and \u001cdrag(ordinate), with the same contour scale for model and theory plots at a given pressure\nlevel. The black line divides the regions where Coriolis force dominates (above the line) and drag dominates (below the line). There is\nagreement within a factor of 2 for both horizontal and vertical wind speeds throughout almost all of parameter space.\nthe model and theory plots. The theoretical wind speeds\nagree everywhere to within a factor of \u00182.\nThe largest discrepancy between theoretical and nu-\nmerical horizontal wind speeds lies in the high \u001crad, high\n\u001cdragregime. This discrepancy is due to the fact that, in\nthis corner of the parameter space, the GCM simulations\nexhibit longitudinal variations of temperature between\nday and night that are much smaller than the equator-\nto-pole temperature di\u000berences (see Figures 4-5). In con-\ntrast, our theory only has one horizontal temperature\ndi\u000berence| we are implicitly assuming that the day-\nnight temperature di\u000berence (as measured on latitude\ncircles) is comparable to the equator-to-pole tempera-\nture di\u000berence. This assumption starts to become prob-\nlematic in the high \u001crad, high\u001cdragportion of parameter\nspace. In this region of parameter space, the equator-to-\npole temperature di\u000berence is larger than the day-night\ntemperature di\u000berence. This equator-to-pole tempera-\nture di\u000berence then drives strong horizontal winds, which\nare not predicted from our theory. This same discrepancy\ncan be seen in the theory of Perez-Becker & Showman\n(2013), see their Figure 6.\nGiven the simplicity of our theory, the agreement be-\ntween predicted characteristic velocities and simulation\nRMS velocities is very good. This agreement is especially\ngood for vertical velocities, because they are derived di-\nrectly from the thermodynamic weak temperature gradi-\nent balance. The horizontal velocities also agree within\na factor of 2 over almost all of parameter space consid-\nered, except when drag and radiative forcing are bothextremely weak. As a result, our velocity scaling can\nbe used to predict characteristic velocities using dayside-\nnightside \rux di\u000berences, which are attainable through\nphase curve observations. This is a powerful \frst deter-\nminant of the circulation of a given tidally locked ex-\noplanet where the dayside-nightside temperature di\u000ber-\nences are larger than equator-to-pole temperature di\u000ber-\nences.\n5.4. Checks of regime determinations\n5.4.1. Momentum equation\nTo con\frm that the the dynamical regimes our model\nlies in agrees with theoretical predictions from Sec-\ntion 4.3, we examine the approximate momentum equa-\ntion (Equation 28) at the equator and mid-latitudes. To\ncalculate each term, we use the RMS values of each vari-\nable, settingU=Urms,W=Wrms, and calculating H\nusing the RMS value of temperature at a given pres-\nsure level. Figure 13 shows this comparison as a func-\ntion of\u001cdrag. This shows the dominant balancing term\nwith the pressure gradient force for the low and high\ndrag regimes at both the equator and mid-latitudes. We\nchoose\u001e= 34\u000eas a representative mid-latitude, as this\nis well northward of the region where the Coriolis forces\nand horizontal momentum advection terms are approx-\nimately equal, i.e. where Ro \u00191. This latitude is\n\u00190\u000025\u000eat 80 mbar pressure, depending on the values of\n\u001cdragand\u001crad, as the values of these timescales strongly\na\u000bect the wind speeds (see Figure 12) and therefore the\nmagnitude of the momentum advection term. Note thatDayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 19\n34567810−510−410−310−2Equatoracceleration (m/s2)\nlog10τdrag (sec)34567810−510−410−310−2Mid−Latitudesacceleration (m/s2)\nlog10τdrag (sec)34567810−510−410−310−2Equator\nlog10τdrag (sec)acceleration (m/s2)\n  \npressure gradientdraghorizontal advectionvertical advectionCoriolis34567810−510−410−310−2Equator\nlog10τdrag (sec)acceleration (m/s2)\n  \npressure gradientdraghorizontal advectionvertical advectionCoriolis\nFig. 13.| Comparison of absolute magnitudes of all terms in the approximate momentum equation, Equation (28), as a function of\n\u001cdragfor the \u0001Teq;top= 1000 Kelvin case with \u001crad;top= 104sec at 80 mbars. The comparison is done both at the planetary equator (0\u000e\nlatitude, left) and mid-latitudes (34\u000elatitude, right).\n\u0000Teq= 1000K80 mbar\n1 bartheoryGCM\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nlog10(Ev/Eh)\n012345log10τrad (sec)log10τdrag (sec)\n  \n34567345678\nlog10(Ev/Eh)\n01234\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nlog10(Ev/Eh)\n00.511.522.533.54\nlog10τrad (sec)log10τdrag (sec)\n  \n34567345678\nlog10(Ev/Eh)\n012345\nFig. 14.| Ratio of vertical and horizontal entropy advection terms, plotted at pressures of 80 mbar (top) and 1 bar (bottom) for both\nGCM results (left) and theoretical prediction (right) as a function of \u001crad;top(abscissa) and \u001cdrag(ordinate). Model and theory plots for a\ngiven pressure level share a contour scale. The vertical entropy advection term is substantially larger than or comparable to the horizontal\nentropy advection term throughout all of the parameter space considered, in both the numerical results and theoretical predictions. This\nrelative dominance of vertical entropy advection continues up to \u001810 mbar pressure. Our choice of the weak temperature gradient regime\nfor our analyses is therefore consistent with our numerical results.20 T.D. Komacek & A.P. Showman\nthis is a comparison performed using our approximate\nmomentum equation, and hence is a test both of our nu-\nmerical results and theoretical approximations valid to\nthe order-of-magnitude level.\nIt is evident from Figure 13 that drag balances the\npressure gradient force for all \u001cdrag.105sec. At larger\n\u001cdrag, di\u000berent forces balance pressure gradient at the\nequator and mid-latitudes.\nAt the equator, horizontal and vertical momentum ad-\nvection are the dominant balancing terms when \u001cdragis\nlarge. These approximate vertical and horizontal mo-\nmentum advection terms are comparable over all \u001crad,\n\u001cdragparameter space, showing that our determination\nof the approximate continuity equation in Equation (20),\nwhich directly relates horizontal and vertical momentum\nadvection, is appropriate.\nAt mid-latitudes, the Coriolis force is the dominant\nbalancing term with pressure gradient when drag is weak,\nas it is orders of magnitude larger than the momentum\nadvection terms. Hence, it was valid to consider two\nseparate three-term force balances, at both the equator\nand mid-latitudes. At the equator, a three-way balance\nbetween advection, drag, and pressure gradient forces ap-\nplies. At higher latitudes, the balance is between Cori-\nolis, drag, and pressure gradient, as expected from our\ntheory.\n5.4.2. Energy equation\nEarth's tropics are in a weak temperature gradient\nregime, where vertical entropy advection dominates over\nhorizontal entropy advection (Sobel 2002). In Section\n4.1.2, we determined using our analytical theory that the\natmospheres of hot Jupiters lie in a similar regime. How-\never, one might naively expect that horizontal entropy\nadvection is more important in hot Jupiter atmospheres\ndue to the large day to night forcing.\nTo determine whether horizontal or vertical entropy\nadvection is the controlling term in Equation (23) over\nthe\u001crad,\u001cdrag parameter space studied, we can ana-\nlyze our numerical solutions. Additionally, we exam-\nine the ratio of vertical and horizontal entropy advec-\ntion terms from our analyses to show that the theory is\nself-consistent. To do so, we consider the approximate\noverall momentum balance between the Coriolis or drag\nforces and pressure gradient. This results in values of A\nand hence \u0001 Tgiven by Equation (34). Figure 14 shows\nthe approximate ratio of vertical and horizontal entropy\nadvection terms, Ev=Eh:\nEv\nEh\u0019WrmsN2H\nR\u0012UrmsA\u0001Teq\nL\u0013\u00001\n: (44)\nThis ratio is plotted as a function of \u001cradand\u001cdragat two\nseparate pressure levels (80 mbars and 1 bar) for both our\nnumerical results and theoretical predictions. Note that\nwe approximate \u0001 T\u0019A\u0001Teq, in order to calculate it\nconsistently from our numerical solutions.\nVertical entropy advection is comparable to or greater\nthan horizontal entropy advection over the entire\n\u001crad;\u001cdragparameter space, up to \u001810 mbars pressure.\nThe vertical entropy advection term is only comparable\nto horizontal entropy advection in our numerical results\nwhen\u001cdragis very large and \u001cradshort. Otherwise, the\nvertical entropy advection term is orders of magnitudelarger than the horizontal entropy advection term. Ad-\nditionally, our theory is self-consistent, as the predicted\nratio of vertical to horizontal advection is always larger\nthan unity. Hence, hot Jupiters, like the equatorial re-\ngions of Earth, lie largely in the weak temperature gra-\ndient regime|at least in a hemisphere-integrated sense\n(nevertheless, there will likely be local regions, particu-\nlarly near the terminators, in which horizontal advection\ndominates).. This validates our use of the weak temper-\nature gradient limit in our theory. This is the same limit\nfound in Perez-Becker & Showman (2013) and similar to\nthat used by Wordsworth (2015) to examine day-night\ndi\u000berences on tidally-locked terrestrial exoplanets.\n6.DISCUSSION\nAs discussed in Section 5.4.2, hot Jupiters lie in a weak\ntemperature gradient regime, similar to the tropics of\nEarth (Sobel et al. 2001; Sobel 2002; Showman et al.\n2013b). In Earth's tropics, the propagation of gravity\nwaves plays a key role in the adjustment of the thermal\nstructure. Similarly, we interpret our theory to identify\nthe mechanism that regulates dayside to nightside tem-\nperature di\u000berences on hot Jupiters as wave adjustment,\nanalagous to the process that mediates temperature dif-\nferences in Earth's tropics. This wave propagation can\nbe damped, causing higher dayside-nightside tempera-\nture di\u000berences. This damping is largely due to radiative\ncooling, with a second order e\u000bect from potential atmo-\nspheric drag. Moreover, a simple theory (as presented in\nSection 4) can explain the bulk of our model results for\nfractional dayside-nightside temperature di\u000berences over\n\u00154 orders of magnitude variation in radiative and drag\ntimescales.\nAdditionally, these dayside-nightside temperature dif-\nferences are a strong function of pressure. Namely, we\nexpect that at depth the atmospheric circulation is more\ne\u000ecient at reducing day-night temperature di\u000berences.\nThis depth-dependence is due in part to the vertical\ncoupling between atmospheric levels but especially to\nthe depth-dependence of radiative cooling, as \u001cradin-\ncreases with increasing pressure in our model, as justi\fed\nby previous calculations of the radiative time constant\n(Iro et al. 2005; Showman et al. 2008a). At pressures\n&1 bar, fractional dayside-nightside temperature dif-\nferences are very small for the entire parameter regime\nstudied, except in the case of extremely strong drag\n(\u001cdrag.104sec), or extremely short radiative timescales\n(\u001crad;top.104sec).\nThe result that a competition between equatorial wave\npropagation and radiative cooling is the key balance\nmediating dayside-nightside temperature di\u000berences was\npreviously found by Perez-Becker & Showman (2013),\nand our results con\frm theirs in 3D. More recently, Koll\n& Abbot (2015) applied a similar understanding to ter-\nrestrial exoplanets. They found through numerical sim-\nulations of tidally-locked terrestrial planets that the ra-\ntio\u001cwave=\u001cradis a key determinant of dayside-nightside\ntemperature di\u000berences. This is evident in our theory\nsimply from how the fraction \u001cwave=\u001cradappears in ev-\nery expression for A(p) in Equation (31). We expect that\nthis comparison between wave and radiative timescales\nhas consequences for the dayside-nightside temperature\ndi\u000berences of a variety of tidally-locked gaseous exoplan-\nets. Therefore, this comparison is applicable not onlyDayside-Nightside Temperature Di\u000berences in Hot Jupiter Atmospheres 21\nfor the hot Jupiter regime but extends to terrestrial ex-\noplanets.\nHowever, one might note that unlike wave adjust-\nment in response to cumulus convection in Earth's\ntropics (Bretherton & Smolarkiewicz 1989), which is a\nhighly time-dependent process, we are here considering\na steady forced/damped system, rather than examin-\ning the impact of freely propagating gravity waves. In\nthe freely propagating case, these waves cause horizon-\ntal convergence/divergence, raising and lowering isen-\ntropes and causing lateral propagation with a (Kelvin)\nwave timescale \u001cwave\u0018 L=NH . Even through we do\nnot consider freely propagating waves in our system,\nour solutions from Section 4 \fnd the same natural wave\ntimescale. This is unsurprising, as the strong heating\non the dayside lowers isentropes relative to that on the\nnightside, leading to a system with isentropes that slope\nsigni\fcantly from day to night. This sets up a horizontal\npressure gradient force, and the dynamical response to\nthis force causes \row that reduces the entropy di\u000berence\nbetween dayside and nightside at a given pressure level.\nThis process is essentially the same physical mechanism\nas gravity wave adjustment, and likewise acts on a wave\ntimescale (Perez-Becker & Showman 2013).\nThe situation on a rotating planet has been examined\nby Perez-Becker & Showman (2013) (see their Section\n5.3), which shows that there is a tight coupling between\nour numerical solutions and wave dynamics. The spatial\nstructure of temperature in Figures 4 and 5 was shown\nto be due to standing, planetary-scale Rossby and Kelvin\nwaves by Showman & Polvani (2011). In the short \u001crad,\nlong\u001cdragregime, one can identify the mid-latitude vor-\nticial winds and temperature maximum as dynamically\nsimilar to a Rossby wave. Similarly, the equatorial diver-\ngence near the substellar point is analogous to a Kelvin\nwave. The standing solution in the linear limit represents\nbehavior that is dynamically similar to the corresponding\npropagating modes, and is normally interpreted in terms\nof wave dynamics (e.g., Matsuno 1966; Gill 1980). Hence,\nwe can re-iterate that the same mechanisms which cause\npropagation of these planetary-scale Kelvin and Rossby\nwaves in the time-dependent case are the same mech-\nanisms that regulate the steady-state dayside-nightside\ntemperature di\u000berences in our hot Jupiter models. As\nsuch, the wave propagation timescale of a Kelvin wave\nis the fundamental wave timescale determining the tran-\nsition from low to high day-to-night temperature di\u000ber-\nences.\nThe importance of this Kelvin wave timescale for pre-\ndicting the dayside-nightside temperature di\u000berences in\nhot Jupiter atmospheres was examined in Section 4.4.\nThe transition from low-to-high dayside-nightside tem-\nperature di\u000berences necessarily depends on whether\ndrag, Coriolis, or advection balances pressure-gradient\nforces in the momentum equation. However, we expect\nthat over most latitudes Coriolis dominates over advec-\ntion, and at temperatures &1;500 K where magnetic\ne\u000bects become important (Perna et al. 2010; Rogers &\nKomacek 2014; Ginzburg & Sari 2015) the day-night\ntemperature di\u000berences will necessarily be large due to\nthe very short \u001crad/T\u00003(Showman & Guillot 2002).\nHence, referring back to Equation (36), a comparison be-\ntween\u001cwave andp\nf\u00001\u001crad(p)\u0001lnpdetermines whetherday-night temperature di\u000berences will be low or high. If\n\u001cwaveis shorter thanp\nf\u00001\u001crad(p)\u0001lnp, the propagating\ntropical waves can extend longitudinally enough to re-\nduce day-night temperature di\u000berences, leading to values\nofAnear zero. However, ifp\nf\u00001\u001crad(p)\u0001lnpis shorter\nthan\u001cwave, these equatorial Kelvin and Rossby waves\nare damped before they can zonally propagate, leading\nto high day-night temperature di\u000berences, with A\u00181.\nOur theory also enables estimation of horizontal and\nvertical velocities to within a factor of \u00182 given dayside-\nnightside temperature di\u000berences. Notably, the veloc-\nity scaling forUin Equation (29) allows estimation of\nhorizontal velocities in the nominal hot Jupiter regime\ngiven only \u0001 Tand \n, both of which can be calculated\nwith phase curve information, assuming planet mass and\nradius are known. Additionally, our scaling for Win\nEquation (30) enables estimation of vertical velocities\ngiven \u0001T, \n, and an estimate for Kelvin wave propaga-\ntion timescales \u001cwaveand radiative timescale \u001crad. These\ntimescales can be computed with knowledge of the atmo-\nspheric composition and temperature. Hence, our theory\nenables \frst-order calculation of horizontal and vertical\nvelocities directly from observable information, without\nthe need for numerical simulations. This theory is a\npowerful tool for \frst insights on atmospheric circulation\ngiven only basic information about planet properties and\ndayside-nightside \rux di\u000berences.\n7.CONCLUSIONS\nTo summarize the main conclusions of our work from\nthe discussion above:\n1. Wave adjustment processes, analogous to those\nthat mediate temperature di\u000berences in Earth's\ntropics, regulate dayside-to-nightside temperature\ndi\u000berences in hot Jupiter atmospheres. The damp-\ning of wave adjustment processes in hot Jupiter\natmospheres is mainly due to radiative cooling,\nwith frictional drag playing a secondary role. Hot\nJupiters have small dayside-nightside temperature\ndi\u000berences at long \u001cradand large dayside-nightside\ndi\u000berences at short \u001crad, in accord with the observa-\ntional trend of increasing dayside-nightside bright-\nness temperature di\u000berences with increasing inci-\ndent stellar \rux.\n2. When drag occurs on characteristic timescales\nfaster than the rotation rate, dayside-nightside\ntemperature di\u000berences are nominally large. Drag\nonly has a second-order e\u000bect on dayside-nightside\ntemperature di\u000berences in other cases, but does af-\nfect zonal wind speeds strongly, being a dominant\nmehanism to quell zonal winds.\n3. We presented a simple analytic theory that explains\nreasonably well the day-night temperature di\u000ber-\nences and their dependence on radiative time con-\nstant, drag time constant, and pressure in three-\ndimensional models. Importantly, our analytic the-\nory has zero free parameters, no tuning, and does\nnot require any information from the GCM simu-\nlations to evaluate. This is the \frst fully predictive\ntheory for the day-night temperature di\u000berences on\nhot Jupiters in three dimensions. Notably, a three-\nway force balance in conjunction with the weak22 T.D. Komacek & A.P. Showman\ntemperature gradient limit can be used to describe\ndayside-nightside temperature di\u000berences at both\nthe equator and higher latitudes. At the equator,\nthe main forces balancing the day-to-night pressure\ngradient are momentum advection and drag, and at\nmid-latitudes drag and Coriolis forces balance the\npressure gradient force.\n4. This analytic theory predicts that the transition\nbetween small and large day-night temperature dif-\nferences is governed by a comparison between \u001cwave\nandp\nf\u00001\u001crad(p)\u0001lnp, with low day-night temper-\nature di\u000berences if \u001cwave<p\nf\u00001\u001crad(p)\u0001lnpand\nhigh day-night temperature di\u000berences if \u001cwave>p\nf\u00001\u001crad(p)\u0001lnp. If frictional drag has a charac-\nteristic timescale shorter than 1 =f, the transition is\ninstead governed by a similar comparison between\n\u001cwave andp\n\u001cdrag(p)\u001crad(p)\u0001lnp. The theory also\ncovers the situation where both drag and Corio-\nlis forces are weak compared to advective forces, a\nsituation which should be relevant to planets that\nrotate especially slowly.\n5. The same theory used to predict dayside-nightside\ntemperature di\u000berences can also be inverted to cal-\nculate characteristic horizontal and vertical veloc-\nities from phase curve information. Horizontal\nvelocities may be estimated given only the rota-\ntion rate and dayside-nightside \rux di\u000berences on\na given tidally locked planet. Similarly, vertical\nvelocities may be calculated with the additional\nknowledge of atmospheric composition and tem-\nperature at a given pressure. This velocity scaling\nis a powerful tool to gain information on atmo-\nspheric circulation from data attainable through\nphase curves and may be extended to investigate\nproperties of the circulation for the suite of tidally\nlocked exoplanets without a boundary layer.\nThis research was supported by NASA Origins grant\nNNX12AI79G to APS. TDK acknowledges support from\nNASA headquarters under the NASA Earth and Space\nScience Fellowship Program Grant PLANET14F-0038.\nWe thank the anonymous referee for useful comments.We also thank Josh Lothringer, Xianyu Tan, and Xi\nZhang for helpful comments on the manuscript. Re-\nsources supporting this work were provided by the\nNASA High-End Computing (HEC) Program through\nthe NASA Advanced Supercomputing (NAS) Division at\nAmes Research Center.\nAPPENDIX\nA.OBSERVED FRACTIONAL DAYSIDE-NIGHTSIDE\nBRIGHTNESS TEMPERATURE DIFFERENCES\nTable 1 displays the data utilized to calculate the\nAobs\u0000Teqpoints shown in Figure 1, along with appropri-\nate references. To compile this data, we collected either\nthe error-weighted or photometric dayside brightness\ntemperature and dayside-nightside brightness tempera-\nture di\u000berences. If no error-weighted values were given,\nwe calculated each brightness temperature as an arith-\nmetic mean of that provided at each wavelength. Specif-\nically, we computed arithmetic means for HD 189733b,\nWASP-43b, and WASP-18b, with the other transiting\nplanets either having an error-weighted value provided\nor only one photometric wavelength with available data.\nFor both WASP-19b and HAT-P-7b, we utilized the\n4:5\u0016m day-night brightness temperature di\u000berence, as\nonly upper limits on the nightside brightness tempera-\ntures at 3:6\u0016m were available. Note that though HAT-\nP-7b seems to have a day-night brightness temperature\ndi\u000berence somewhat below that expected, the lower limit\nonAobsat 3:6\u0016m is 0:483 (Wong et al. 2015b), 33 :4%\nlarger than the value of Aobsat 4:5\u0016m.\nTo compute a lower limit on Aobsfor WASP-43b, we\nutilized the 1 \u001bupper limit on the nightside bolometric\n\rux from Stevenson et al. (2014). We used a similar\nmethod for WASP-18b, averaging the lower limits on\nAobsfrom the 3:6\u0016m and 4:5\u0016m nightside 1 \u001b\rux up-\nper limits from Maxted et al. (2013). 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We consider the transmission problem for a coupled system of\nundamped and structurally damped plate equations in two suffi ciently smooth\nand bounded subdomains. It is shown that, independently of t he size of the\ndamped part, the damping is strong enough to produce uniform exponential\ndecay of the energy of the coupled system.\n1.Introduction\nIn this paper, we investigate a coupled system of linear plate equatio ns where an\nundamped plate and a structurally damped plate are coupled throug h transmission\nconditions. From the point of view of applications, there is a connect ion to the\nsuppression of vibration of elastic structures which is a main topic in m aterial\nscience. The undamped plate equation can be seen as a linear model f or vibrating\nstiff objects where the potential energy is related to curvature- like terms, resulting\nin the bi-Laplacianoperatoras the main elastic operator, see, e.g., [ 12], Chapter 12.\nFor the purely undamped plate, we have no energy dissipation, and t he governing\nsemigroup is unitary. The model of structural damping is widely used to describe\nsmoothing effects and loss of energy (cf. [ 20] for a discussion of the model). Here,\nwe consider the damping term which has order two in the spatial varia bles, so it is\nof half order of the leading elastic term, see also [ 8] and [9] for the analysis of the\nstructurally damped plate equation.\nFrom a theoretical point of view, the resulting system can be seen a s a trans-\nmission problem of mixed type: While the structurally damped plate equ ation is\nof parabolic nature, the undamped part is of dissipative nature. Be low we will see\nthat the damping is strong enough (independent of the size of the d amped part) to\nobtain exponential stability for the semigroup of the coupled syste m. The analog\nresult for a coupled system of thermoelastic / elastic plates was obt ained in [ 16].\nThe question of analyticity of the semigroup for a coupled thermoela stic plate /\nplate system is discussed in [ 10]. In [14], a plate / plate transmission problem with\ndamping only on a part of the boundary with resulting polynomial deca y was stud-\nied, see also [ 4] for the proof of exponential stability for a boundary stabilized pla te\n/ plate transmission problem. Transmission problems of plate / plate t ype can also\nbe seen as an equation with coefficients having jumps, cf. [ 13].\nIn the system we consider the damping effect acting only through th e transmis-\nsion interface. Closely related is the question of boundary damping, see, e.g., [ 17]\nor [22]. In the literature, there are many results on coupled systems of p late / wave\ntype (cf. [ 3] and the references therein). In particular, in [ 6] and [7], the exponen-\ntial stability for an abstract wave equation coupled with a plate-like e quation on\nthe boundary is studied. To our knowledge, the undamped / struct urally damped\nplate system has not yet been studied in literature.\nDate: February 2, 2017.\n2010Mathematics Subject Classification. 74K20; 74H40; 35B40; 35Q74.\nKey words and phrases. Plate equation, transmission problem, exponential stabil ity.\n12 ROBERT DENK AND FELIX KAMMERLANDER\nLet Ω⊂Rnbe a bounded domain with boundary Γ 1:=∂Ω, and let Ω 2⊂Ω be a\nnon-empty bounded domain satisfying Ω2⊂Ω. We set Γ := ∂Ω2and Ω 1:= Ω\\Ω2.\nThen, Γ is the common interface (transmission interface) between Ω1and Ω 2, and\n∂Ω1=∂Ω∪Γ (see Figure 1for the geometrical situation). All domains are assumed\nto be of class C4. For technical reasons, we assume n≤4, including the physically\nmost relevant cases n= 1 and n= 2. Let νdenote the outer unit normal on Γ 1.\nOn Γ, we choose νto be the outer unit normal with respect to Ω 2. Thus,νis the\ninner unit normal vector on Γ with respect to Ω 1, see Figure 1. Note that, apart\nfrom the smoothness, we do not impose a geometrical condition on t he domains.\nν\nν\nΩ2Ω1\nΓ1\nΓ\nFigure 1. The set Ω = Ω 1∪Γ∪Ω2.\nWe consider a transmission problem for thin plates where the plate in Ω 2is\nundamped and the material in Ω 1is structurally damped. More precisely, we are\nlooking for solutions ui: Ωi→Cof the system\n∂2\ntu1+∆2u1−ρ∆∂tu1= 0 in (0 ,∞)×Ω1, (1-1)\n∂2\ntu2+∆2u2= 0 in (0 ,∞)×Ω2 (1-2)\nwithclamped boundary conditions\nu1=∂νu1= 0 on Γ 1 (1-3)\nHere,ρ∈(0,∞) is the damping factor. The transmission conditions on Γ are given\nby\nu1=u2, (1-4)\n∂νu1=∂νu2, (1-5)\n∆u1= ∆u2, (1-6)\n−ρ∂ν∂tu1+∂ν∆u1=∂ν∆u2. (1-7)\nThe problem is completed by the initial conditions\nu1(0,·) =u0\n1, ∂tu1(0,·) =u1\n1in Ω1, (1-8)\nu2(0,·) =u0\n2, ∂tu2(0,·) =u1\n2in Ω2. (1-9)\nThe energy of the system ( 1-1)-(1-9) is defined as\nE(t) :=1\n2/integraldisplay\nΩ1|∂tu1|2+|∆u1|2dx+1\n2/integraldisplay\nΩ2|∂tu2|2+|∆u2|2dx. (1-10)\nIf (u1,u2) is a solution, integration by parts yields the estimate\nd\ndtE(t) =−ρ/ba∇dbl∇∂tu1/ba∇dbl2\nL2(Ω1)≤0. (1-11)\nNote that ui=∂νui= 0 on Γ iimplies∂tui=∂ν∂tui= 0 on Γ ifori= 1,2.The\nestimate shows that the energy of the transmission problem is decr easing in time\nand the dissipation is caused by the damped part u1.EXPONENTIAL STABILITY FOR A COUPLED SYSTEM 3\nOur main result, Theorem 4.5below, states that the damping in Ω 1is strong\nenough to achieve exponential decrease of the energy, i.e. there exist constants\nC,κ >0 such that\nE(t)≤CE(0)e−κt\nholds for all t≥0.To prove this, we first study the resolvent and the spectrum of\nthe first-ordersystem related to( 1-1)–(1-7) in Section 2. In the proofofexponential\nstability, we also need an a priori estimate on the damped part which is obtained in\nSection 3 with the help of the interpolation-extrapolation scales of B anach spaces.\nFinally, the results from Section 2 and 3 are used to prove the main re sult on\nexponential stability in Section 4.\n2.The spectrum of the first-order system\nSettingU:= (u1,u2,v1,v2)⊤withvj:=∂tuj, we rewrite the transmission prob-\nlem (1-1)-(1-9) as\n∂tU(t)−AU(t) = 0 (t >0), U(0) =U0 (2-1)\nwhere the operator Aacts in form of the matrix\nA(D) :=\n0 0 1 0\n0 0 0 1\n−∆20ρ∆ 0\n0−∆20 0\n.\nAs the basic space for the first two components ( u1,u2), we will choose\nX(Ω) :=/braceleftbig\n(u1,u2)∈H2(Ω1)×H2(Ω2) :u1=∂νu1= 0 on Γ 1,\nu1=u2on Γ, ∂νu1=∂νu2on Γ/bracerightbig\n.\nRemark 2.1. a) Let (u1,u2)∈H2(Ω1)×H2(Ω2). Then the conditions u1=u2,\n∂νu1=∂νu2on Γ are equivalent to u:=χΩ1u1+χΩ2u2∈H2(Ω), where χΩjstands\nfor the characteristic function of Ω j, i.e.χΩj(x) = 1 for x∈ΩjandχΩj(x) = 0\nelse. Therefore, we have\nX(Ω) ={(u|Ω1,u|Ω2) :u∈H2\n0(Ω)}.\nIn the following, we will several times use the identification of ( u1,u2) andu.\nb) The norm in X(Ω) is defined as\n/ba∇dbl(u1,u2)/ba∇dblX(Ω):=/parenleftig\n/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)+/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)/parenrightig1/2\n.\nNotethatthisnormisequivalenttothestandardnorm( /ba∇dblu1/ba∇dbl2\nH2(Ω1)+/ba∇dblu2/ba∇dbl2\nH2(Ω2))1/2.\nIn fact, due to the invertibilityofthe Dirichlet Laplacianin Ω, the norm s/ba∇dbl∆u/ba∇dblL2(Ω)\nand/ba∇dblu/ba∇dblH2(Ω)are equivalent on the space H2(Ω)∩H1\n0(Ω). As H2\n0(Ω) is a closed\nsubspace of H2(Ω)∩H1\n0(Ω), these norms are also equivalent on H2\n0(Ω), and now\nthe assertion follows from part a) (see also [ 11], Proposition 2.1 and Proposition\n2.2).\nWe say that the transmission conditions ( 1-6) and (1-7) are weakly satisfied if\n/a\\}b∇acketle{t∆2u1−ρ∆v1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆2u2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)\n=/a\\}b∇acketle{t∆u1,∆ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆u2,∆ϕ2/a\\}b∇acket∇i}htL2(Ω2)+ρ/a\\}b∇acketle{t∇v1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1)(2-2)\nholds for all ( ϕ1,ϕ2)∈X(Ω). Let\nH:=X(Ω)×L2(Ω1)×L2(Ω2).\nThen we define the operator A:H ⊃D(A)→ Hby4 ROBERT DENK AND FELIX KAMMERLANDER\nD(A) :=/braceleftbig\n(u1,u2,v1,v2)∈X(Ω)×X(Ω) : ∆2u1∈L2(Ω1),∆2u2∈L2(Ω2),\n(1-6) and (1-7) are weakly satisfied/bracerightbig\nandAU:=A(D)U(U∈D(A)).\nWe will see in Lemma 2.4below that the functions in D(A) are sufficiently\nsmooth and the transmission conditions hold in the sense of traces.\nTheorem 2.2. The operator Ais the generator of a C0-semigroup of contractions\non the Hilbert space H.Therefore, for all U0∈D(A)the Cauchy problem (2-1)has\na unique classical solution U∈C1([0,∞),H)withU(t)∈D(A)for allt≥0.\nProof.By the definition of D(A) and the weak transmission conditions ( 2-2), it is\nimmediately seen that\nRe/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−ρ/ba∇dbl∇v1/ba∇dbl2\nL2(Ω1)(U∈D(A)).\nHence,Ais dissipative. We want to show that 1 −Ais surjective. For this, let\nF= (f1,f2,g1,g2)⊤∈ H.We have to find U= (u1,u2,v1,v2)⊤∈D(A) satisfying\nu1−v1=f1,\nu2−v2=f2,\nv1+∆2u1−ρ∆v1=g1,\nv2+∆2u2=g2.\nPlugging in vi=ui−fifori= 1,2 in the third and fourth equation yields that we\nhave to solve\nu1+∆2u1−ρ∆u1=g1+f1−ρ∆f1, (2-3)\nu2+∆2u2=g2+f2 (2-4)\nas equalities in L2(Ω1) andL2(Ω2),respectively.\nWe define the continuous sesquilinear form B:X(Ω)×X(Ω)→Cby\nB(u,ϕ) =/a\\}b∇acketle{tu1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆u1,∆ϕ1/a\\}b∇acket∇i}htL2(Ω2)+ρ/a\\}b∇acketle{t∇u1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1)\n+/a\\}b∇acketle{tu2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)+/a\\}b∇acketle{t∆u2,∆ϕ2/a\\}b∇acket∇i}htL2(Ω2)\nforu= (u1,u2),ϕ= (ϕ1,ϕ2)∈X(Ω). Since\nReB(u,u)≥ /ba∇dbl(u1,u2)/ba∇dbl2\nX(Ω)(u∈X(Ω)),\nBis coercive. Obviously, the mapping Λ: X(Ω)→Cdefined by\nΛ(ϕ) :=/a\\}b∇acketle{tg1+f1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+ρ/a\\}b∇acketle{t∇f1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{tg2+f2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)\nforϕ= (ϕ1,ϕ2)∈X(Ω) is linear and continuous. By the theorem of Lax-Milgram,\nthere exists a unique u= (u1,u2)∈X(Ω) such that B(u,ϕ) = Λ(ϕ) holds for all\nϕ∈X(Ω). In particular, choosing ( ϕ1,ϕ2)∈C∞\n0(Ω1)×C∞\n0(Ω2)⊂X(Ω), we see\nthat (2-3) and (2-4) hold in the sense of distributions in Ω 1and Ω 2, respectively.\nAs the right-hand side of ( 2-3) belongs to L2(Ω1), the same holds for the left-hand\nside. Due to u1∈H2(Ω1), this yields ∆2u1∈L2(Ω1). In the same way, we see\nthat (2-4) holds as equality in L2(Ω2) and that ∆2u2∈L2(Ω2).\nSetv1:=u1−f1andv2:=u2−f2. By (2-3)–(2-4), we have\n∆2u1−ρ∆v1=−u1+g1+f1,\n∆2u2=−u2+g2+f2.(2-5)\nLetϕ= (ϕ1,ϕ2)∈X(Ω). Then, because of ( 2-5) andB(u,ϕ) = Λ(ϕ), we get\n/a\\}b∇acketle{t∆2u1−ρ∆v1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆2u2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)EXPONENTIAL STABILITY FOR A COUPLED SYSTEM 5\n=/a\\}b∇acketle{t−u1+g1+f1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t−u2+g2+f2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)\n=/a\\}b∇acketle{t∆u1,∆ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{t∆u2,∆ϕ2/a\\}b∇acket∇i}htL2(Ω2)+ρ/a\\}b∇acketle{t∇v1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1).\nTherefore, the weak transmissionconditions ( 2-2) aresatisfied. Altogether, we have\nseen that U:= (u1,u2,v1,v2)⊤belongs to D(A). Because of ( 2-3)–(2-4) and the\ndefinition of v1,v2, we also have (1 −A)U=F. Therefore, 1 −Ais surjective which\nimplies that Ais densely defined (see [ 18], Theorem 4.6). An application of the\nLumer-Phillips theorem now yields the statement of the theorem. /square\nRemark 2.3. In the same way as in the previous proof, one can show that the\noperator Ais continuously invertible, i.e. 0 belongs to the resolvent set ρ(A). To\nshow this, we now have to consider\n∆2u1=g1−ρ∆f1, (2-6)\n∆2u2=g2 (2-7)\ninstead of ( 2-3)–(2-4). The sesquilinear form Band the functional Λ are now\ndefined by B(u,ϕ) =/a\\}b∇acketle{tu,ϕ/a\\}b∇acket∇i}htX(Ω)and\nΛ(ϕ) :=/a\\}b∇acketle{tg1,ϕ1/a\\}b∇acket∇i}htL2(Ω1)+ρ/a\\}b∇acketle{t∇f1,∇ϕ1/a\\}b∇acket∇i}htL2(Ω1)+/a\\}b∇acketle{tg2,ϕ2/a\\}b∇acket∇i}htL2(Ω2)\nforu= (u1,u2),ϕ= (ϕ1,ϕ2)∈X(Ω).\nAs before, we see that there exists a unique solution u= (u1,u2)∈X(Ω)\nsatisfying B(u,ϕ) = Λ(ϕ) for all ϕ∈X(Ω). Moreover, setting vj:=−fj, the\nvectorU:= (u1,u2,v1,v2)⊤belongs to D(A) and satisfies −AU=F.\nOn the other hand, if /tildewideU∈D(A) solves−A/tildewideU=F, thenB(/tildewideu,ϕ) = Λ(ϕ) holds for\nallϕ∈X(Ω) due to the definition of D(A) and the weak transmission conditions.\nTherefore, U=/tildewideU, andA:D(A)→ His a bijection. Since Ais the generator of a\nC0-semigroup, Ais closed and the continuity of A−1:H → Hfollows. Therefore,\n0∈ρ(A).\nLemma 2.4. a) The domain of Ais given by\nD(A) =/braceleftbig\n(u1,u2,v1,v2)∈/parenleftbig\nH4(Ω1)×H4(Ω2)/parenrightbig\n∩X(Ω)×X(Ω) :\n∆u1= ∆u2onΓ,−ρ∂νv1+∂ν∆u1=∂ν∆u2onΓ/bracerightbig\n.(2-8)\nHere, the equalities on Γcan be understood as equalities in the trace spaces H3/2(Γ)\nandH1/2(Γ), respectively.\nb) The operator Ahas compact resolvent and, consequently, discrete spectru m.\nProof.a) Let/tildewideU∈D(A) andF= (f1,f2,g1,g2)⊤:=−A/tildewideU∈ H. To show the\nstatement, we construct a strong solution Uof−AU=Fbelonging to the right-\nhand side of ( 2-8) and show that U=/tildewideU. So we consider\n∆2u1=g1−ρ∆f1, (2-9)\n∆2u2=g2 (2-10)\ninL2(Ω1)×L2(Ω2) with boundary conditions\nu1=∂νu1= 0 on Γ 1 (2-11)\nand transmission conditions\nu1−u2= 0, (2-12)\n∂νu1−∂νu2= 0, (2-13)\n∂2\nνu1−∂2\nνu2= 0, (2-14)\n∂3\nνu1−∂3\nνu2=−ρ∂νf1. (2-15)6 ROBERT DENK AND FELIX KAMMERLANDER\nConcerningthehigher-ordertransmissionconditions( 2-14)and(2-15), notethatfor\nall(u1,u2)∈H4(Ω1)×H4(Ω2)satisfying ( 2-12) and(2-13) alltangentialderivatives\nofu1−u2and∂νu1−∂νu2alongΓdisappear. Therefore, forsuch uthe transmission\nconditions ( 2-14)–(2-15) are equivalent to the conditions\n∆u1−∆u2= 0,\n∂ν∆u1−∂ν∆u2=−ρ∂νf1.\nDefine the operator B:L2(Ω)⊃D(B)→L2(Ω) byD(B) :=H4(Ω)∩H2\n0(Ω)\nandBw:= ∆2w. Then, Bis a selfadjoint operator with 0 ∈ρ(B). To construct\na strong solution of the transmission problem ( 2-9)–(2-15), we first eliminate the\ninhomogeneity on the right-hand side of ( 2-15). By [21], Section 4.7.1, p. 330, the\nmapping\nRh:=/parenleftbig\nh|∂Ω1,∂νh|∂Ω1,∂2\nνh|∂Ω1,∂3\nνh|∂Ω1/parenrightbig⊤\nis a retraction from H4(Ω1) onto/producttext3\nj=0H4−1/2−j(∂Ω1).Therefore, there exists a\nfunction h∈H4(Ω1) such that\nRh= (0,0,0,−χΓρ∂νf1)⊤.\nHere again χΓstands for the characteristic function of Γ. We define w:=B−1G∈\nH4(Ω)∩H2\n0(Ω),where\nG=χΩ1(g1−ρ∆f1−∆2h)+χΩ2g2∈L2(Ω).\nFinally, we set u1:=w|Ω1+handu2:=w|Ω2.Then,u= (u1,u2)∈H4(Ω1)×\nH4(Ω2) satisfies the strong transmission problem ( 2-9)–(2-15). Therefore, U:=\n(u1,u2,v1,v2)⊤withvj:=−fjbelongs to the right-hand side of ( 2-8) and solves\n−A(D)U=F.\nOn the other hand, using integration by parts and the fact that usolves the\nstrong transmission problem, we see that Usatisfies the weak transmission condi-\ntions (2-2). Therefore, Ubelongs to D(A) and solves −AU=F. By Remark 2.3,\nthis solution is unique which implies U=/tildewideU.\nb) Due to a), we have\nD(A)⊂/parenleftbig\nH4(Ω1)×H4(Ω2)/parenrightbig\n∩X(Ω)×X(Ω).\nBy the Rellich-Kondrachov theorem, the space on the right-hand s ide is compactly\nembedded into H. Therefore, A−1is compact, and the spectrum of Ais discrete.\n/square\nWe already know that the spectrum of Ais discrete and that 0 is no eigenvalue.\nIn fact, there are no purely imaginary eigenvalues of A, as the next result shows.\nTheorem 2.5. The imaginary axis is a subset of the resolvent set of A,i.e.iR⊂\nρ(A).\nProof.Assume that U= (u1,u2,v1,v2)⊤∈D(A) satisfies ( −iλ+A)U= 0 with\nλ∈R\\{0}. Thenvj= iλujforj= 1,2, and (u1,u2) satisfies\n−∆2u1+iλρ∆u1+λ2u1= 0 in Ω 1, (2-16)\n−∆2u2+λ2u2= 0 in Ω 2 (2-17)\nwith boundary conditions u1=∂νu1= 0 on Γ 1and transmission conditions\nu1=u2,\n∂νu1=∂νu2,\n∆u1= ∆u2,\n−iλρ∂νu1+∂ν∆u1=∂ν∆u2EXPONENTIAL STABILITY FOR A COUPLED SYSTEM 7\non the common interface Γ .\nWe will show that ( u1,u2) = 0.We multiply ( 2-16) and (2-17) withu1andu2,\nrespectively. Summing up and performing an integration by parts yie lds\n−/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)−iλρ/ba∇dbl∇u1/ba∇dbl2\nL2(Ω1)+λ2/ba∇dblu1/ba∇dbl2\nL2(Ω1)\n−/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)+λ2/ba∇dblu2/ba∇dbl2\nL2(Ω2)= 0.\nHere we have used the boundary conditions as well as the transmiss ion conditions\non Γ.Considering only the imaginary part we get /ba∇dbl∇u1/ba∇dblL2(Ω1)= 0.Together with\nu1|Γ1= 0 we obtain u1= 0.Therefore, u2satisfies the boundary value problem\n−∆2u2+λ2u2= 0 in Ω 2, (2-18)\nu2=∂νu2= ∆u2=∂ν∆u2= 0 on Γ = ∂Ω2. (2-19)\nBecause of ( 2-19), the trivial extension /tildewideu2by zero to Rnbelongs to H4(Rn) and\nsatisfies ∆2/tildewideu2=λ2/tildewideu2inRn. As ∆2inL2(Rn) has no eigenvalues, this implies\n/tildewideu2= 0 and therefore u2= 0. Altogether we have seen U= 0. /square\nThe last results already implies strong stability of the semigroup ( T(t))t≥0gen-\nerated by A, i.e., for any U0∈ Hwe have /ba∇dblT(t)U0/ba∇dblH→0 (t→ ∞) (see [5],\nTheorem 2.4). We will see in Section 4 that Tis even exponentially stable.\n3.A priori estimates for the damped plate equation\nFor the proof of exponential stability of the coupled damped–unda mped plate\nequation, we need some a priori estimates for the damped part. Fo r this, we\nwill apply the theory of interpolation-extrapolation scales due to Am ann (see [ 2],\nChapter V).\nThroughout this section, let U⊂Rnbe a bounded C4-domain. We define the\noperator Ain the space H2\n0(U)×L2(U) by\nD(A) := (H4(U)∩H2\n0(U))×H2\n0(U),\nA:=/parenleftbigg0 1\n−∆2ρ∆/parenrightbigg\n.(3-1)\nIt was shown in [ 8], Proposition 3.1 (see also [ 9], Theorem 5.1) that Agenerates an\nanalytic exponentially stable C0-semigroup in H2\n0(U)×L2(U). To extrapolate this\nresult to spaces of negative regularity, we need to determine the a djoint operator A′\nconsidered in the dual spaces. In the following, /a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htX′×Xdenotes the dual pairing\nin a Banach space X. We begin with a small observation on the bi-Laplacian\noperator.\nRemark 3.1. Under the above assumptions on U, the operator ∆2:H2\n0(U)→\nH−2(U) is an isomorphism. In fact, we have the coercive estimate\n/a\\}b∇acketle{t∆2u,u/a\\}b∇acket∇i}htH−2(U)×H2\n0(U)=/ba∇dbl∆u/ba∇dbl2\nL2(U)≥C/ba∇dblu/ba∇dbl2\nH2(U)(u∈H2\n0(U)).\nHere the last inequality holds by elliptic regularity and invertibility of the Dirichlet\nLaplacian ∆ D:H2(U)∩H1\n0(U)→L2(U). Now an application of the Lax-Milgram\ntheorem yields the invertibility of ∆2:H2\n0(U)→H−2(U).\nLemma 3.2. The adjoint operator A′ofAis given by\nA′:H−2(U)×L2(U)⊃D(A′) :=L2(U)×H2\n0(U)→H−2(U)×L2(U),\nA′:=/parenleftbigg0−∆2\n1ρ∆/parenrightbigg\n.8 ROBERT DENK AND FELIX KAMMERLANDER\nProof.We define E:=H2\n0(U)×L2(U) and/tildewideD:=L2(U)×H2\n0(U)⊂E′,where\nE′=H−2(U)×L2(U).Then, for all v= (v1,v2)∈/tildewideDand\nu= (u1,u2)∈D(A) =/parenleftbig\nH4(U)∩H2\n0(U)/parenrightbig\n×H2\n0(U),\nintegration by parts and the definition of distributional derivatives yield\nv(Au) =/a\\}b∇acketle{tv1,u2/a\\}b∇acket∇i}htL2(U)+/a\\}b∇acketle{tv2,−∆2u1/a\\}b∇acket∇i}htL2(U)+/a\\}b∇acketle{tv2,ρ∆u2/a\\}b∇acket∇i}htL2(U)\n=/a\\}b∇acketle{tv1,u2/a\\}b∇acket∇i}htL2(U)+/a\\}b∇acketle{t−∆v2,∆u1/a\\}b∇acket∇i}htL2(U)+/a\\}b∇acketle{tρ∆v2,u2/a\\}b∇acket∇i}htL2(U)\n=/a\\}b∇acketle{t−∆2v2,u1/a\\}b∇acket∇i}htH−2(U)×H2\n0(U)+/a\\}b∇acketle{tv1,u2/a\\}b∇acket∇i}htL2(U)+/a\\}b∇acketle{tρ∆v2,u2/a\\}b∇acket∇i}htL2(U)\n=w1(u1)+w2(u2),\nwithw1:=−∆2v2∈H−2(U) andw2:=v1+ρ∆v2∈L2(U).Therefore, we set\n/tildewideA:=/parenleftbigg\n0−∆2\n1ρ∆/parenrightbigg\nwithD(/tildewideA) :=/tildewideD.With this definition, we have v(Au) = (/tildewideAv)(u) for allu∈D(A)\nand allv∈/tildewideD.Moreover, for all v∈/tildewideDthe mapping [ u/mapsto→v(Au)]:D(A)→Cis\ncontinuous with respect to /ba∇dbl·/ba∇dblE.Hence, we have /tildewideA⊂A′.\nLetu∈D(A) andv∈E′.Then\nv(Au) =/a\\}b∇acketle{tv1,u2/a\\}b∇acket∇i}htH−2(U)×H2\n0(U)+/a\\}b∇acketle{tv2,−∆2u1/a\\}b∇acket∇i}htL2(U)+/a\\}b∇acketle{tv2,ρ∆u2/a\\}b∇acket∇i}htL2(U).(3-2)\nNow, let v∈D(A′).Then, the mapping [ u/mapsto→v(Au)]:D(A)→Ccan be extended\nto a linear, continuous mapping from EtoC.In particular, considering\n|/a\\}b∇acketle{tv2,∆2u1/a\\}b∇acket∇i}htL2(U)|=|v(A(u1,0))| ≤C/ba∇dbl(u1,0)/ba∇dblE=C/ba∇dblu1/ba∇dblH2(U)\nforu1∈H4(U)∩H2\n0(U),it holds that\nϕ:H4(U)∩H2\n0(U)→C, u1/mapsto→ϕ(u1) :=/a\\}b∇acketle{tv2,∆2u1/a\\}b∇acket∇i}htL2(U)(3-3)\nis continuous with respect to /ba∇dbl·/ba∇dblH2(U).By Remark 3.1,\n∆2:H2\n0(U)→H−2(U) (3-4)\nis an isomorphism. Therefore, ( 3-3) and (3-4) imply that\n/bracketleftbig\n/tildewideu1/mapsto→ϕ/parenleftbig\n(∆2)−1/tildewideu1/parenrightbig\n=/a\\}b∇acketle{tv2,/tildewideu1/a\\}b∇acket∇i}htL2(U)/bracketrightbig\n:L2(U)→C\nis continuousconsideredasamapping from( L2(U),/ba∇dbl·/ba∇dblH−2(U)) toC.By the density\nofL2(U)⊂H−2(U), there exists a unique continuous extension\n/tildewideϕ∈/parenleftbig\nH−2(U)/parenrightbig′=H2\n0(U)\nof this mapping. Together with\n/a\\}b∇acketle{t/tildewideϕ,/tildewideu1/a\\}b∇acket∇i}htH2\n0(U)×H−2(U)=/a\\}b∇acketle{tv2,/tildewideu1/a\\}b∇acket∇i}htH2\n0(U)×H−2(U)\nfor/tildewideu1∈L2(U),we deduce v2=/tildewideϕ∈H2\n0(U).\nThe fact that v2∈H2\n0(U) implies that the last term in ( 3-2),\n/bracketleftig\nu2/mapsto→ /a\\}b∇acketle{tv2,ρ∆u2/a\\}b∇acket∇i}htL2(U)=/a\\}b∇acketle{tv2,ρ∆u2/a\\}b∇acket∇i}htH2\n0(U)×H−2(U)/bracketrightig\n:H2\n0(U)→C,\nis continuous on L2(U). Since ( 3-2) needs to be continuous, by setting u1= 0 it\nfollows that also the first term/bracketleftig\nu2/mapsto→ /a\\}b∇acketle{tv1,u2/a\\}b∇acket∇i}htH−2(U)×H2\n0(U)/bracketrightig\n:H2\n0(U)→C\ncan be extended continuously to L2(U),which means v1∈L2(U).\nWe have shown that v∈D(A′) implies v2∈H2\n0(U) andv1∈L2(U),i.e.v∈/tildewideD.\nHence, we obtain /tildewideD=D(A′) and therefore /tildewideA=A′. /squareEXPONENTIAL STABILITY FOR A COUPLED SYSTEM 9\nIn the following,\nΣϕ={z∈C\\{0}:|arg(z)|< ϕ}\ndenotes the open sector in C.\nTheorem 3.3. There exists a constant C0>0such that for any λ∈ρ(A)⊃\nΣπ/2\\{0}and any F∈H2\n0(U)×L2(U)the unique solution u= (u1,v1)∈D(A)of\n(λ−A)u=F∈H2\n0(U)×L2(U) (3-5)\nsatisfies the estimate\n/ba∇dblu/ba∇dblH2+θ(U)×Hθ(U)≤C0/ba∇dblF/ba∇dblHθ(U)×H−2+θ(U)(θ∈[0,2]). (3-6)\nIn particular, for u= (u1,v1)∈D(A)solving\n(λ−A)u=/parenleftbigg0\nf/parenrightbigg\nwithf∈L2(U)we obtain the estimate\n/ba∇dblu1/ba∇dblH2+θ(U)≤C0/ba∇dblf/ba∇dblH−2+θ(U)(θ∈[0,2]). (3-7)\nProof.By [8], Proposition 3.1, Ais the generator of an analytic, exponentially sta-\nble, strongly continuous semigroup on H2\n0(U)×L2(U).Therefore, ( 3-5) is uniquely\nsolvable, and we have the uniform resolvent estimate\n/ba∇dblu/ba∇dblH4(U)×H2(U)≤C1/ba∇dblF/ba∇dblH2(U)×L2(U) (3-8)\nwith some constant C1>0 independent of Fandλ.\nLetA♯:=A′be the adjoint operator of Aand set\nE0:=H2\n0(U)×L2(U),\nE1:=D(A) =/parenleftbig\nH4(U)∩H2\n0(U)/parenrightbig\n×H2\n0(U),\nE♯\n0:=E′\n0=H−2(U)×L2(U),\nE♯\n1:=D(A♯).\nObviously, E0is reflexive and E1is dense in E0.SinceAis the generator of an\nanalytic C0-semigroup on E0with domain E1, in symbols A∈ H(E1,E0),by [2],\np. 13, Proposition 1.2.3, the same holds true for A♯onE♯\n0with domain E♯\n1,i.e.\nA♯∈ H(E♯\n1,E♯\n0).\nHence, we can define the interpolation-extrapolation scales {(Aα,Eα) :α∈R}and\nits dual scale {(A♯\nα,E♯\nα) :α∈R}.Then, Theorem 1.5.12 in [ 2] states that Eαis\nreflexive and we have\n(Eα)′=E♯\n−αand (Aα)′=A♯\n−α\nfor allα∈R.Moreover, by [ 1], Theorem 6.1 and [ 2], Theorem 2.1.3it holds that Aα\nandA♯\nαare generators of analytic C0-semigroups in Eαwith domain Eα+1andE♯\nα\nwith domain E♯\nα+1for allα∈R,respectively. Again, we write Aα∈ H(Eα+1,Eα)\nandA♯\nα∈ H(E♯\nα+1,E♯\nα).\nInparticular, A−1isthegeneratorofananalytic C0-semigroupon E−1with domain\nE0.By [2], Theorem 2.1.3, λ−A−1is an isomorphism from E0toE−1and we have\n/ba∇dbl(µ−A−1)−1/ba∇dblL(E−1,E0)≤C/ba∇dbl(µ−A)−1/ba∇dblL(E0,E1)≤C′\nfor allµ∈ρ(A) with a constant C′independent of µ.By Lemma 3.2, the space\nE−1equals\nE−1= (E−1)′′=/parenleftig\nE♯\n1/parenrightig′\n= (D(A′))′=/parenleftbig\nL2(U)×H2\n0(U)/parenrightbig′=L2(U)×H−2(U).10 ROBERT DENK AND FELIX KAMMERLANDER\nTherefore, there exists a constant C2>0 such that\n/ba∇dblu/ba∇dblH2(U)×L2(U)≤C2/ba∇dblF/ba∇dblL2(U)×H−2(U). (3-9)\nNow the inequality ( 3-6) follows by (real) interpolation between ( 3-8) and (3-9)\nwithC0:= max{C1,C2}.\nConsidering the particular case F=/parenleftbig0\nf/parenrightbig\nand only the first component of u, we\nobtain (3-7). /square\n4.Exponential stability\nIn this section, we continue the analysis of the coupled system ( 1-1)–(1-2). We\nwill estimates the resolvent ( A−iλ)−1of the corresponding first-order system on\nthe imaginaryaxisfor λ∈Rwith|λ|large. Bya result due to Pr¨ uss([ 19], Corollary\n4), uniform boundedness of the resolvent on the imaginary axis implie s exponential\nstability of the semigrroup.\nWe start with some identities which will be useful for our estimates. I n the\nfollowing, we will shortly write xfor the identity function x/mapsto→x. For vectors\ny,z∈Cnwe sety·z:=/summationtextn\nj=1yjzj(note that this is not the scalar product in Cn).\nLemma 4.1. LetU⊂Rnbe aC4-domain, let w∈H4(U), and let ν:∂U→Rn\nbe the outer unit normal vector. Then,\n2Re/integraldisplay\nU(x·∇w)∆2wdx= (4−n)/ba∇dbl∆w/ba∇dbl2\nL2(U)+/integraldisplay\n∂U(x·ν)|∆w|2dS\n+2Re/integraldisplay\n∂U/bracketleftig\n(x·∇w)∂ν∆w−∆w∂ν(x·∇w)/bracketrightig\ndS.\nProof.This follows by straightforward calculation from the divergence the orem,\napplied to the vector field\nV:=|∆w|2x+2(x·∇w)∇∆w−2∆w∇(x·∇w).\nNote that\ndivV=n|∆w|2+x·∇|∆w|2+2(x·∇w)∆2w−2∆w∆(x·∇w)\nand Re(div V) = 2Re( x·∇w)∆2w+(n−4)|∆w|2. A more general variant of the\nstatement is also known as Rellich’s identity, see, e.g., [ 15], Proposition 2.2, or [ 14],\np. 238. /square\nLemma 4.2. LetU⊂Rnbe aC4-domain, and let w∈H4(U)be a solution of\n−∆2w+λ2w=zwithλ∈Randz∈L2(U). Then we have\nnλ2/ba∇dblw/ba∇dbl2\nL2(U)+(4−n)/ba∇dbl∆w/ba∇dbl2\nL2(U)+/integraldisplay\n∂U(x·ν)|∆w|2dS\n=−2Re/integraldisplay\nU(x·∇w)zdx+λ2/integraldisplay\n∂U(x·ν)|w|2dS\n−2Re/integraldisplay\n∂U/bracketleftig\n(x·∇w)∂ν∆w−∆w∂ν(x·∇w)/bracketrightig\ndS.\nProof.Applying the divergence theorem to the vector field |w|2xand taking the\nreal part, we obtain\n2Re/integraldisplay\nU(x·∇w)wdx=−n/ba∇dblw/ba∇dbl2\nL2(U)+/integraldisplay\n∂U(x·ν)|w|2dS.\nFrom this and ∆2w=λ2w−zwe get\n2Re/integraldisplay\nU(x·∇w)∆2wdx=−2Re/integraldisplay\nU(x·∇w)zdx−nλ2/ba∇dblw/ba∇dbl2\nL2(U)EXPONENTIAL STABILITY FOR A COUPLED SYSTEM 11\n+λ2/integraldisplay\n∂U(x·ν)|w|2dS.\nPlugging this into the statement of Lemma 4.1, the assertion follows. /square\nThe following result can be found, e.g., in [ 14], Proof of Theorem 2.2.\nLemma 4.3. LetU⊂Rnbe aC3-domain, and let S⊂∂Ube a nontrivial part\nof the boundary. Then, for every w∈H3(U)withw=∂νw= 0onSwe have\n∂ν(x·∇w) = (x·ν)∆wonS.\nIn the next step, we considerthe resolventequation ( −iλ+A)U=Ffor a partic-\nular right-hand side F= (0,0,0,g2)⊤with inhomogeneous transmission conditions.\nMore precisely, we consider\n−iλu1+v1= 0 in Ω 1, (4-1)\n−∆2u1+ρ∆v1−iλv1= 0 in Ω 1, (4-2)\n−iλu2+v2= 0 in Ω 2, (4-3)\n−∆2u2−iλv2=g2in Ω2 (4-4)\nwith transmission conditions\n∆u1= ∆u2,\n−iλρ∂νu1+∂ν∆u1=∂ν∆u2+iλρ∂νw1/bracerightbigg\non Γ. (4-5)\nThefollowingaprioriestimatewillbethecrucialstepfortheproofo fexponential\nstability.\nProposition 4.4. Letw1∈H4(Ω1)andg2∈L2(Ω2)be given. Then, there exists\nλ0>0and a constant C >0(only depending on n,ρ,δ0andλ0) such that for\nany solution U= (u1,u2,v1,v2)⊤∈X(Ω)×X(Ω)withui∈H4(Ωi)fori= 1,2of\n(4-1)–(4-5)the estimate\n/ba∇dblU/ba∇dblH≤C/parenleftbig\n/ba∇dblg2/ba∇dblL2(Ω2)+|λ|/ba∇dbl∂νw1/ba∇dblL2(Γ)/parenrightbig\n(λ∈R,|λ|> λ0)\nholds.\nIn the following proof, we will use a generic constant Cindependent of λ,U,and\nF. Moreover, an estimate of the form /ba∇dbl·/ba∇dbl ≤ε/ba∇dbl·/ba∇dbl1+Cε/ba∇dbl·/ba∇dbl2has to be understood\nin the sense that for every small ε >0 there exists a constant Cε>0 such that\nthe inequality holds. Again Cεdenotes a generic constant. Note that all constants\nmay depend on ρ.\nProof.We have to estimate\n/ba∇dblU/ba∇dblH=/parenleftig\n/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)+/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)+/ba∇dblv1/ba∇dbl2\nL2(Ω1)+/ba∇dblv2/ba∇dbl2\nL2(Ω2)/parenrightig1/2\n.\nThe proof is done in several steps.\n(i)Estimate of v1.Letλ∈Rwith|λ| ≫1 andU= (u1,u2,v1,v2)∈X(Ω)×\nX(Ω) be a solution of ( 4-1)–(4-5). Hence, ( u1,u2) is a solution of\n−∆2u1+iλρ∆u1+λ2u1= 0, (4-6)\n−∆2u2+λ2u2=g2 (4-7)\nin Ω1×Ω2satisfying the transmission conditions ( 4-5). By the definition of X(Ω),\nwe have u1=∂νu1= 0 on Γ 1. In order to show the assertion of the theorem, we\nneed to establish an estimate of the form\n/ba∇dblU/ba∇dbl2\nH≤ε/ba∇dblU/ba∇dbl2\nH+Cε/parenleftig\n/ba∇dblg2/ba∇dbl2\nL2(Ω2)+|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightig12 ROBERT DENK AND FELIX KAMMERLANDER\nSimilar to the proof of the dissipativity of Ain Theorem 2.2, we obtain\nRe/a\\}b∇acketle{tF,U/a\\}b∇acket∇i}htH= Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−ρ/ba∇dbl∇v1/ba∇dbl2\nL2(Ω1)−Re/integraldisplay\nΓiλρv1∂νw1dS.\nTherefore, Poincar´ e and Young’s inequality yield\n/ba∇dblv1/ba∇dbl2\nH1(Ω1)≤C/parenleftbig\n/ba∇dblg2/ba∇dblL2(Ω2)/ba∇dblU/ba∇dblH+|λ|/ba∇dbl∂νw1/ba∇dblL2(Γ)/ba∇dblv1/ba∇dblH1(Ω1)/parenrightbig\n≤ε/parenleftig\n/ba∇dblU/ba∇dbl2\nH+/ba∇dblv1/ba∇dbl2\nH1(Ω1)/parenrightig\n+Cε/parenleftig\n/ba∇dblg2/ba∇dbl2\nL2(Ω2)+|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightig\n,\nthat is\n/ba∇dblv1/ba∇dbl2\nH1(Ω1)≤ε/ba∇dblU/ba∇dbl2\nH+Cε/parenleftig\n/ba∇dblg2/ba∇dbl2\nL2(Ω2)+|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightig\n.(4-8)\nTogether with ( 4-1) this implies\n/ba∇dblu1/ba∇dbl2\nH1(Ω1)≤ε|λ|−2/ba∇dblU/ba∇dbl2\nH+Cε/parenleftig\n|λ|−2/ba∇dblg2/ba∇dbl2\nL2(Ω2)+/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightig\n.(4-9)\n(ii)Estimate of ∆u1and∆u2.We multiply ( 4-6) by−u1and (4-7) by−u2.\nIntegration by parts and summing up yields\n/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)+/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)+iλρ/ba∇dbl∇u1/ba∇dbl2\nL2(Ω2)\n=λ2/parenleftig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightig\n−/a\\}b∇acketle{tg2,u2/a\\}b∇acket∇i}htL2(Ω2)+iλρ/integraldisplay\nΓu1∂νw1dS,\nwhere we have used the transmission conditions ( 4-5) andu1=∂νu1= 0 on Γ 1.\nTaking the real part and observing vj=iλuj, we see that\n/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)+/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)≤ |λ|2/parenleftbig\n/ba∇dblu1/ba∇dbl2\nL2(Ω1)+/ba∇dblu2/ba∇dbl2\nL2(Ω2)/parenrightbig\n+/ba∇dblg2/ba∇dblL2(Ω2)/ba∇dblu2/ba∇dblL2(Ω2)+|λ|ρ/ba∇dblu1/ba∇dblL2(Γ)/ba∇dbl∂νw1/ba∇dblL2(Γ)\n=/parenleftbig\n/ba∇dblv1/ba∇dbl2\nL2(Ω1)+/ba∇dblv2/ba∇dbl2\nL2(Ω2)/parenrightbig\n+/ba∇dblg2/ba∇dblL2(Ω2)/ba∇dblu2/ba∇dblL2(Ω2)+ρ/ba∇dblv1/ba∇dblL2(Γ)/ba∇dbl∂νw1/ba∇dblL2(Γ).(4-10)\nAssuming |λ| ≥1, we get with the trace theorem and Young’s inequality\n/ba∇dblg2/ba∇dblL2(Ω2)/ba∇dblu2/ba∇dblL2(Ω2)≤1\n2/ba∇dblg2/ba∇dbl2\nL2(Ω1)+1\n2/ba∇dblv2/ba∇dbl2\nL2(Ω2),\nρ/ba∇dblv1/ba∇dblL2(Γ)/ba∇dbl∂νw1/ba∇dblL2(Γ)≤ρ\n2/ba∇dblv1/ba∇dbl2\nH1(Ω1)+ρ\n2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ).\nInserting this into ( 4-10) yields\n/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)+/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)≤(1+ρ\n2)/ba∇dblv1/ba∇dbl2\nH1(Ω1)+/ba∇dblv2/ba∇dbl2\nL2(Ω2)\n+/ba∇dblg2/ba∇dbl2\nL2(Ω1)+ρ\n2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)\n≤ε/ba∇dblU/ba∇dbl2\nH+C/ba∇dblv2/ba∇dbl2\nL2(Ω2)\n+Cε/parenleftbig\n/ba∇dblg2/ba∇dbl2\nL2(Ω2)+|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightbig\n. (4-11)\nHere, in the last step we estimated /ba∇dblv1/ba∇dblH1(Ω1)due to (4-8).\n(iii)Estimate of v2.We apply Lemma 4.2inU= Ω2withw=u2andz=g2\nand obtain, noting iλu2=v2,\nn/ba∇dblv2/ba∇dbl2\nL2(Ω2)=−(4−n)/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)−/integraldisplay\nΓ(x·ν)|∆u2|2dS\n−2Re/integraldisplay\nΩ2(x·∇u2)g2dx+λ2/integraldisplay\nΓ(x·ν)|u2|2dS\n−2Re/integraldisplay\nΓ/bracketleftig\n(x·∇u2)∂ν∆u2−∆u2∂ν(x·∇u2)/bracketrightig\ndS. (4-12)\nIn the same way, we apply Lemma 4.2inU= Ω1withw=u1andz=−ρ∆v1.\nHere we remark that −∆2u1+λ2u1=−ρ∆v1by (4-1) and (4-2). Moreover, theEXPONENTIAL STABILITY FOR A COUPLED SYSTEM 13\nnormal vector νis the outer normal at the part Γ 1of the boundary ∂Ω1, butνis\nthe inner normal at the part Γ of ∂Ω1. We obtain\nn/ba∇dblv1/ba∇dbl2\nL2(Ω1)=−(4−n)/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)−/integraldisplay\nΓ1(x·ν)|∆u1|2dx\n+/integraldisplay\nΓ(x·ν)|∆u1|2dx+2Re/integraldisplay\nΩ1(x·∇u1)ρ∆v1dx\n+λ2/integraldisplay\nΓ1(x·ν)|u1|2dS−λ2/integraldisplay\nΓ(x·ν)|u1|2dS\n−2Re/integraldisplay\nΓ1/bracketleftig\n(x·∇u1)∂ν∆u1−∆u1∂ν(x·∇u1)/bracketrightig\ndS\n+2Re/integraldisplay\nΓ/bracketleftig\n(x·∇u1)∂ν∆u1−∆u1∂ν(x·∇u1)/bracketrightig\ndS. (4-13)\nDue to the condition ( u1,u2)∈X(Ω) and the transmission conditions ( 4-5), we\nhave\nu1=u2,∇u1=∇u2,∆u1= ∆u2, ∂ν∆u2=∂ν∆u1−iλρ∂ν(u1+w1) on Γ.\n(4-14)\nLet/tildewideu2∈H4(Ω) be a regular extension of u2to Ω, and define ϕ:=u1−/tildewideu2|Ω1∈\nH4(Ω1). Thenϕ=∂νϕ= 0 on Γ, and an application of Lemma 4.3yields\n∂ν(x·∇ϕ) = (x·ν)∆ϕ= (x·ν)(∆u1−∆u2) = 0 on Γ\nwhich gives\n∂ν(x·∇u1) =∂ν(x·∇u2) on Γ. (4-15)\nMoreover, with Lemma 4.3again we get\nu1= 0,∇u1= 0, ∂ν(x·∇u1) = (x·ν)∆u1on Γ1. (4-16)\nAdding ( 4-12) and (4-13) and taking into account ( 4-14)–(4-16), we obtain\nn/parenleftbig\n/ba∇dblv1/ba∇dbl2\nL2(Ω1)+/ba∇dblv2/ba∇dbl2\nL2(Ω2)/parenrightbig\n=−(4−n)/parenleftbig\n/ba∇dbl∆u1/ba∇dbl2\nL2(Ω1)+/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)/parenrightbig\n+2Re/bracketleftig\niλρ/integraldisplay\nΩ1(x·∇u1)∆u1dx/bracketrightig\n−2Re/bracketleftig/integraldisplay\nΩ2(x·∇u2)g2dx/bracketrightig\n+2Re/bracketleftbig\niλρ/integraldisplay\nΓ(x·∇u1)∂ν(u1+w1)dS/bracketrightig\n+/integraldisplay\nΓ1(x·ν)|∆u1|2dS.\nTherefore,\n/ba∇dblv2/ba∇dbl2\nL2(Ω2)≤C/bracketleftig\n|λ|/ba∇dblu1/ba∇dblH1(Ω1)/ba∇dbl∆u1/ba∇dblL2(Ω1)+/ba∇dbl∇u2/ba∇dblL2(Ω2)/ba∇dblg2/ba∇dblL2(Ω2)\n+|λ|/ba∇dblu1/ba∇dbl2\nH1(Γ)+|λ|/ba∇dblu1/ba∇dblH1(Γ)/ba∇dbl∂νw1/ba∇dblL2(Γ)+/ba∇dblu1/ba∇dbl2\nH2(Γ1)/bracketrightig\n.(4-17)\nWe estimate the first four terms on the right-hand side of ( 4-17) while the last term\nwill be treated in part (iv) of this proof.\nThe first term in ( 4-17) can be estimated by ( 4-8),/ba∇dbl∆u1/ba∇dblL2(Ω1)≤ /ba∇dblU/ba∇dblHand\nYoung’s inequality. We obtain\n|λ|/ba∇dblu1/ba∇dblH1(Ω1)/ba∇dbl∆u1/ba∇dblL2(Ω1)≤/parenleftig\nε/ba∇dblU/ba∇dblH+Cε/parenleftbig\n/ba∇dblg2/ba∇dblL2(Ω2)+|λ|/ba∇dbl∂νw1/ba∇dblL2(Γ)/parenrightbig/parenrightig\n/ba∇dblU/ba∇dblH\n≤ε/ba∇dblU/ba∇dbl2\nH+Cε/parenleftbig\n/ba∇dblg2/ba∇dblL2(Ω2)+|λ|/ba∇dbl∂νw1/ba∇dblL2(Γ)/parenrightbig2.(4-18)\nFor the second term in ( 4-17), we apply Green’s formula, using u1=u2and\n∂νu1=∂νu2on Γ to see that\n/ba∇dbl∇u2/ba∇dbl2\nL2(Ω2)≤ /ba∇dblu2/ba∇dblL2(Ω2)/ba∇dbl∆u2/ba∇dblL2(Ω2)+/ba∇dblu1/ba∇dblL2(Γ)/ba∇dbl∇u1/ba∇dblL2(Γ)\n≤1\n2/ba∇dblu2/ba∇dbl2\nL2(Ω2)+1\n2/ba∇dbl∆u2/ba∇dbl2\nL2(Ω2)+C/ba∇dblu1/ba∇dbl2\nH2(Ω1)14 ROBERT DENK AND FELIX KAMMERLANDER\n≤C/parenleftbig\n/ba∇dblu1/ba∇dbl2\nH2(Ω1)+/ba∇dblu2/ba∇dbl2\nH2(Ω2)/parenrightbig\n≤C/ba∇dblU/ba∇dbl2\nH.\nFor the last inequality, we have applied Remark 2.1b). Therefore, the second term\nin (4-17) can be estimated by\n/ba∇dbl∇u2/ba∇dblL2(Ω2)/ba∇dblg2/ba∇dblL2(Ω2)≤ε/ba∇dblU/ba∇dbl2\nH+Cε/ba∇dblg2/ba∇dbl2\nL2(Ω2).\nFor the third term in ( 4-17) we use interpolation to see that\n|λ|/ba∇dblu1/ba∇dbl2\nH1(Γ)≤C|λ|/ba∇dblu1/ba∇dbl2\nH3/2(Ω1)≤C|λ|/ba∇dblu1/ba∇dblH1(Ω1)/ba∇dblu1/ba∇dblH2(Ω1)\n≤ε/ba∇dblU/ba∇dbl2\nH+Cε/parenleftbig\n/ba∇dblg2/ba∇dbl2\nL2(Ω2)+|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightbig\n.(4-19)\nSimilarly, for the fourth term in ( 4-17) we write\n|λ|/ba∇dblu1/ba∇dblH1(Γ)/ba∇dbl∂νw1/ba∇dblL2(Γ)≤1\n2/ba∇dblu1/ba∇dbl2\nH1(Γ)+1\n2|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ).\nAs we have |λ| ≥1, this again can be estimated by the right-hand side of ( 4-19).\nAltogether, we obtain\n/ba∇dblv2/ba∇dbl2\nL2(Ω2)≤ε/ba∇dblU/ba∇dbl2\nH+Cε/parenleftbig\n/ba∇dblg2/ba∇dbl2\nL2(Ω2)+|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightbig\n+C/ba∇dblu1/ba∇dbl2\nH2(Γ1).(4-20)\n(iv)Estimate of u1onΓ1.The only term still left is /ba∇dblu1/ba∇dblH2(Γ1). We introduce\na cut-off function χ∈C∞(Ω1),0≤χ≤1,satisfying χ= 1 in a neighbourhood of\nΓ1andχ= 0 in a neighbourhood of the transmission interface Γ .Now, set\nz1:=χu1, z2:= iλz1, z:= (z1,z2)⊤.\nThen, since u1is a solution of ( 4-6),zsatisfies\n(−iλ+A)z=/parenleftbigg0\n/tildewidef/parenrightbigg\n,\nwhereAis defined as in ( 3-1) and\n/tildewidef= (−∆2χ)u1−2(∇∆χ)·∇u1−∆χ∆u1−2∆(∇χ·∇u1)\n−∆χ∆u1−2∇χ·∇∆u1+iλρ((∆χ)u1+2∇χ·∇u1)∈L2(Ω1)\n=B3(D,χ)u1+iλρ((∆χ)u1+2∇χ·∇u1)\nwith aλ-independent differential operator B3(D,χ) of order3 with coefficients only\nconsisting of derivatives of the C∞-function χ.Hence,\nB3(D,χ)∈L(H3/2(Ω1),H−3/2(Ω1)).\nFrom Theorem 3.3withθ=1\n2, we obtain\n/ba∇dblu1/ba∇dblH2(Γ1)=/ba∇dblz1/ba∇dblH2(Γ1)≤C/ba∇dblz1/ba∇dblH5/2(Ω1)≤C/ba∇dbl/tildewidef/ba∇dblH−3/2(Ω1)\n≤C/parenleftbig\n/ba∇dblB3(D,χ)u1/ba∇dblH−3/2(Ω1)+|λ|/ba∇dblu1/ba∇dblH1(Ω1)/parenrightbig\n≤C/parenleftbig\n/ba∇dblu1/ba∇dblH3/2(Ω1)+|λ|/ba∇dblu1/ba∇dblH1(Ω1)/parenrightbig\n≤C/parenleftbig\n|λ|1/2/ba∇dblu1/ba∇dblH3/2(Ω1)+/ba∇dblv1/ba∇dblH1(Ω1)/parenrightbig\n.\nNow, (4-8) and (4-19) yield\n/ba∇dblu1/ba∇dbl2\nH2(Γ1)≤ε/ba∇dblU/ba∇dbl2\nH+Cε/parenleftbig\n/ba∇dblg2/ba∇dbl2\nL2(Ω2)+|λ|2/ba∇dbl∂νw1/ba∇dbl2\nL2(Γ)/parenrightbig\n.(4-21)\nThe assertion of the Proposition now follows from ( 4-8), (4-11), (4-20), and\n(4-21). /squareEXPONENTIAL STABILITY FOR A COUPLED SYSTEM 15\nTheorem 4.5. There exists a constant C=C(ρ)>0such that\n/ba∇dbl(−iλ+A)−1/ba∇dblL(H)≤C(λ∈R\\{0},|λ|> λ0)\nfor some λ0>0.Consequently, the C0-semigroup (T(t))t≥0generated by Ais\nexponentially stable, i.e. there exist constants M >0andκ >0such that\n/ba∇dblT(t)U0/ba∇dblH≤Me−κt/ba∇dblU0/ba∇dblH(t≥0)\nholds for all U0∈ H.\nProof.Letλ∈Rwith|λ| ≫1 and let F= (f1,f2,g1,g2)∈ H.Furthermore, let\nU= (u1,u2,v1,v2)∈D(A) be the unique solution of\n(−iλ+A)U=F,\ni.e.Usatisfies\n−iλu1+v1=f1in Ω1,\n−∆2u1+ρ∆v1−iλv1=f2in Ω1,\n−iλu2+v2=g1in Ω2,\n−∆2u2−iλv2=g2in Ω2.\nIn order to show the assertion, we will subtract the solution Wof a structurally\ndamped plate equation with clamped boundary conditions on the whole domain\nΩ fromU.For this difference we will be able to use the a-priori estimate from\nProposition 4.4, whereas for Wan appropriate estimate is known.\nRecall the definition of the operator Afrom (3-1) and define\n/tildewiderW= (w,z)∈D(A) =/parenleftbig\nH4(Ω)∩H2\n0(Ω)/parenrightbig\n×H2\n0(Ω)\nby\n/tildewiderW:= (−iλ+A)−1/parenleftbiggχ1f1+χ2f2\nχ1g1+χ2g2/parenrightbigg\n,\nwhereχiis the characteristic function on Ω ifori= 1,2.Sinceχ1f1+χ2f2∈H2\n0(Ω)\nandχ1g1+χ2g2∈L2(Ω) due to the definition of H,by Theorem 3.3,/tildewiderWis well-\ndefined. In the following, we denote the restrictions of the compon ents of/tildewiderWby\nwi:=w|Ωiandzi:=z|Ωifori= 1,2.Finally, we set\nW:= (w1,w2,z1,z2)∈X(Ω)×X(Ω).\nNote that ui∈H4(Ωi) fori= 1,2.With this definitions, we obtain that the\ndifference U−Wsatisfies\n(−iλ+A)(U−W) =F−(−iλ+A)W= (0,0,0,/tildewideg2)⊤(4-22)\nwith/tildewideg2=ρ∆z2∈L2(Ω2),subject to the transmission conditions\n/braceleftigg\n∆(u1−w1) = ∆(u2−w2),\n−iλρ∂ν(u1−w1)+∂ν∆(u1−w1) =∂ν∆(u2−w2)+iλρ∂νw1.\nThanks to Proposition 4.4, we have\n/ba∇dblU−W/ba∇dblH≤C/parenleftbig\n/ba∇dbl/tildewideg2/ba∇dblL2(Ω2)+|λ|/ba∇dbl∂νw1/ba∇dblL2(Γ)/parenrightbig\n. (4-23)\nAn application of Theorem 3.3withU= Ω and θ= 2 gives\n/ba∇dbl/tildewideg2/ba∇dblL2(Ω2)=ρ/ba∇dbl∆z2/ba∇dblL2(Ω2)≤C/ba∇dblz2/ba∇dblH2(Ω2)≤C/ba∇dbl/tildewiderW/ba∇dblH4(Ω)×H2(Ω)\n≤C/vextenddouble/vextenddouble/vextenddouble/parenleftbiggχ1f1+χ2f2\nχ1g1+χ2g2/parenrightbigg/vextenddouble/vextenddouble/vextenddouble\nH2(Ω)×L2(Ω)≤C/ba∇dblF/ba∇dblH.16 ROBERT DENK AND FELIX KAMMERLANDER\nSinceAis the generator of a bounded, analytic C0-semigroup on H2\n0(Ω)×L2(Ω)\nby Theorem 3.3, we see that\n|λ|/ba∇dbl∂νw1/ba∇dblL2(Γ)≤C|λ|/ba∇dblw1/ba∇dblH2(Ω1)≤C/ba∇dblF/ba∇dblH.\nTherefore, ( 4-23) yields\n/ba∇dblU−W/ba∇dblH≤C/ba∇dblF/ba∇dblH.\nInvoking Theorem 3.3again, we deduce\n/ba∇dblU/ba∇dblH≤ /ba∇dblU−W/ba∇dblH+/ba∇dblW/ba∇dblH≤C/ba∇dblF/ba∇dblH\nwith a constant C=C(ρ)>0.This proves the theorem. /square\nReferences\n[1] H. Amann. Parabolic evolution equations in interpolati on and extrapolation spaces. J. Funct.\nAnal., 78(2):233–270, 1988.\n[2] H. Amann. Linear and quasilinear parabolic problems. Vol. I , volume 89 of Monographs in\nMathematics . Birkh¨ auser Boston, Inc., Boston, MA, 1995.\n[3] K. Ammariand S. Nicaise. Stabilization of a transmissio nwave/plate equation. J. Differential\nEquations , 249(3):707–727, 2010.\n[4] K. Ammari and G. Vodev. Boundary stabilization of the tra nsmission problem for the\nBernoulli-Euler plate equation. Cubo, 11(5):39–49, 2009.\n[5] W.Arendtand C.J.K.Batty. Tauberian theorems and stabi lity ofone-parameter semigroups.\nTrans. Amer. Math. Soc. , 306(2):837–852, 1988.\n[6] G. Avalos. The exponential stability of a coupled hyperb olic/parabolic system arising in\nstructural acoustics. Abstr. Appl. Anal. , 1(2):203–217, 1996.\n[7] G. Avalos and I. Lasiecka. The strong stability of a semig roup arising from a coupled hyper-\nbolic/parabolic system. Semigroup Forum , 57(2):278–292, 1998.\n[8] S. P. Chen and R. Triggiani. Proof of extensions of two con jectures on structural damping\nfor elastic systems. Pacific J. Math. , 136(1):15–55, 1989.\n[9] R. Denk and R. Schnaubelt. A structurally damped plate eq uation with Dirichlet–Neumann\nboundary conditions. Journal of Differential Equations , 259(4):1323–1353, 2015.\n[10] H. D. Fern´ andez Sare and J. E. Mu˜ noz Rivera. Analytici ty of transmission problem to ther-\nmoelastic plates. Quart. Appl. Math. , 69(1):1–13, 2011.\n[11] F. Hassine. Logarithmic stabilization of the Euler-Be rnoulli transmission plate equation with\nlocally distributed Kelvin-Voigt damping. arXiv preprint arXiv:1403.0356 , 2014.\n[12] R. Leis. Initial-boundary value problems in mathematical physics . B. G. Teubner, Stuttgart;\nJohn Wiley & Sons, Ltd., Chichester, 1986.\n[13] W. Liu and G. H. Williams. Exact controllability for pro blems of transmission of the plate\nequation with lower-order terms. Quart. Appl. Math. , 58(1):37–68, 2000.\n[14] S. Mansouri. Boundary stabilization of coupled plate e quations. Palest. J. Math. , 2(2):233–\n242, 2013.\n[15] E. Mitidieri. A Rellich type identity and applications .Comm. Partial Differential Equations ,\n18(1-2):125–151, 1993.\n[16] J. E. Mu˜ noz Rivera and H. Portillo Oquendo. A transmiss ion problem for thermoelastic\nplates.Quart. Appl. Math. , 62(2):273–293, 2004.\n[17] M. I. Mustafa and G. A. Abusharkh. Plate equations with v iscoelastic boundary damping.\nIndag. Math. (N.S.) , 26(2):307–323, 2015.\n[18] A. Pazy. Semigroups of linear operators and applications to partial differential equations ,\nvolume 44. Springer Science & Business Media, 2012.\n[19] J. Pr¨ uss. On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. , 284(2):847–857, 1984.\n[20] D. L. Russell. Mathematical models for the elastic beam and their control-theoretic implica-\ntions. In Semigroups, theory and applications, Vol. II (Trieste, 198 4), volume 152 of Pitman\nRes. Notes Math. Ser. , pages 177–216. Longman Sci. Tech., Harlow, 1986.\n[21] H. Triebel. Interpolation Theory, Function Spaces, Differential Opera tors. North Holland,\n1978.\n[22] J. C. Vila Bravo and J. E. Mu˜ noz Rivera. The transmissio n problem to thermoelastic plate\nof hyperbolic type. IMA J. Appl. Math. , 74(6):950–962, 2009.EXPONENTIAL STABILITY FOR A COUPLED SYSTEM 17\nUniversit ¨at Konstanz, Fachbereich f ¨ur Mathematik und Statistik, 78457 Konstanz,\nGermany\nE-mail address :robert.denk@uni-konstanz.de\nUniversit ¨at Konstanz, Fachbereich f ¨ur Mathematik und Statistik, 78457 Konstanz,\nGermany\nE-mail address :felix.kammerlander@uni-konstanz.de"    },    {        "title": "2010.05614v2.Decays_rates_for_Kelvin_Voigt_damped_wave_equations_II__the_geometric_control_condition.pdf",        "content": "DECAY RATES FOR KELVIN-VOIGT DAMPED WAVE EQUATIONS II: THE\nGEOMETRIC CONTROL CONDITION\nNICOLAS BURQ AND CHENMIN SUN\nAbstract. We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric\ncontrol condition. When the damping coe\u000ecient is su\u000eciently smooth ( C1vanishing nicely, see (1.3)) we show\nthat exponential decay follows from geometric control conditions (see [5, 12] for similar results under stronger\nassumptions on the damping function).\n1.Introduction\nIn this paper we investigate decay rates for Kelvin-Voigt damped wave equations under geometric control\nconditions. We work in a smooth bounded domain \n \u001aRdand consider the following equation\n(1.1)8\n><\n>:(@2\nt\u0000\u0001)u\u0000div(a(x)rx@tu) = 0\nujt=0=u02H1\n0(\n); @tujt=0=u12L2(\n)\nuj@\n= 0\nwith a non negative damping term a(x). The solution can be written as\n(1.2) U(t) =\u0012u\n@tu\u0013\n=eAt\u0012u0\nu1\u0013\n;\nwhere the generator Aof the semi-group is given by\nA=\u0012\n0 1\n\u0001 divar\u0013\u0012\nu0\nu1\u0013\n;\nwith domain\nD(A) =f(u0;u1)2H1\n0\u0002L2; \u0001u0+ divaru12L2;u12H1\n0g:\nThe energy of solutions\nE(u)(t) =Z\n\n(jrxuj2+j@tuj2)dx\nsatis\fes\nE((u0;u1))(t)\u0000E((u0;u1))(0) =\u0000Zt\n0Z\n\na(x)jrx@tuj2(s;x)ds:\nOur purpose here is to show that if the damping ais su\u000eciently smooth, the exponential decay rate holds,\ndropping some unnecessary assumptions on the behaviour of the damping term where it becomes positive in\nprevious works [5]. Namely we shall assume a(x)>0 isC1(\n) and satisfy the regularity hypothesis\njraj6Ca1\n2: (1.3)\nOur main result is\nTheorem 1. Assume that \nis a compact Riemannian manifold with smooth boundary. Let a2C1(\n)be a\nnonnegative function satisfying (1.3) , such that the following geometric control condition is satis\fed:\n\u000fThere exists \u000e > 0such that all rays of geometric optics (straight lines) re\recting on the boundary\naccording to the laws of geometric optics eventually reach the set !\u000e=fx2\n :a(x)>\u000egin \fnite time.\n1arXiv:2010.05614v2  [math.AP]  19 Mar 20212 N. BURQ AND C-M. SUN\nThen there exists \u000b>0, such that for all t>0and every (u0;u1)2H1\n0(\n)\u0002L2(\n), the energy of solution u(t)\nof(1.1) with initial data (u0;u1)satis\fes\nE[u](t)6e\u0000\u000btE[u](0):\nTo prove this result, we \frst reduce it very classicaly in Section 2 to resolvent estimates. Since the low\nfrequency estimates are true, we are reduced to the high frequency regime. The proof relies on resolvent\nestimates which are proved through a contradiction argument that we establish in Section 2. In Section 3 we\nprove a priori estimates for our sequences. The main task then is to prove a propagation invariance for these\nmeasures. A main di\u000eculty to overcome is that it is not possible to put the damping term in the r.h.s. of the\nequation (1.1) and treat it as a perturbation . Instead we have to keep it on the left hand side and revisit the\nproof of the propagation property from [7]. In Section 4, we introduce the geometric tools necessary to tackle the\nboundary value problem and de\fne semi-classical measures associated to our sequences. In Section 5 we prove\nthe interior propagation result for our measures. Finally, in Section 6, we \fnish the proof of the contradiction\nargument by establishing the invariance of the semi-classical measures we de\fned up to the boundary. Here the\nproof uses crucially the main result in [7, Th\u0013 eor\u0012 eme 1].\nRemark 1.1. Throughout this note, we shall prove that some operators of the type P\u0000\u0015Id,\u00152R(resp.\n\u00152iR) are invertible with estimates on the inverse. All these operators share the feature that they have\ncompact resolvent, i.e. 9z02C; (P\u0000z0)\u00001exists and is compact (or it will be possible to reduce the question\nto this situation). As a consequence, since\n(P\u0000\u0015) = (P\u0000z0)\u00001(Id + (z0\u0000\u0015))\u00001);\nand (Id + (z0\u0000\u0015)\u00001) is Fredholm with index 0, to show that ( P\u0000\u0015) is invertible with inverse bounded in norm\nbyA, it is enough to bound the solutions of ( P\u0000\u0015)u=fand prove\n(P\u0000\u0015)u=f)kukL26AkfkL2:\nRemark 1.2. Assume that ais the restriction to \n of a nonnegative C2(Rd) function. Then the hypothesis (1.3)\nis satis\fed.\nProof. It is enough to prove (1.3) for \n = Rd,a2C2(\n). Letx02Rdand denote by z0=ra(x0) From\nTaylor's formula, we have for any s2R, there exists \u00122(0;1), such that\na(x0+sz0) =a(x0) +sjz0j2+s2\n2a00(x0+\u0012sz0)(z0;z0)>0\nSince this polynomial in sis non negative, we deduce tat its discriminant is non positive\njz0j4\u00002ka00k1jz0j2a(z0)60)jrxa(x0)jj262ka00k1a(z0):\nNotice that in the above lemma, the condition cannot be relaxed to a2C2(\n);a>0. Indeed, consider the\nfollowing example: \n = B(0;1) anda(x) = 1\u0000jxj2forjxj61. Then obviously a2C2(\n),a>0 , but on the\nboundary,rxa6= 0, whilea= 0. \u0003\nAcknowledgment. The \frst author is supported by Institut Universitaire de France and ANR grant ISDEEC,\nANR-16-CE40-0013. The second author is supported by the postdoc programe: \\Initiative d'Excellence Paris\nSeine\" of CY Cergy-Paris Universit\u0013 e and ANR grant ODA (ANR-18-CE40- 0020-01).\n2.Contradiction argument\nIt is well known that decay estimates for the evolution semi-group follow from resolvent estimates [1, 2, 4].\nHere we shall need the classical (see e.g. [6, Proposition A.1])\nTheorem 2. The exponential decay of the Kelvin Voigt semi-group is equivalent to the following resolvent\nestimate: There exists Csuch that for all \u00152R, the operator (A\u0000i\u0015)is invertible from D(A)toHand its\ninverse satis\fes\n(2.1) k(A\u0000i\u0015)\u00001kL(H)6CKELVIN-VOIGT DAMPING 3\nLet us \frst recall that\n(2.2) ( A\u0000i\u0015)\u0012u\nv\u0013\n=\u0012f\ng\u0013\n,\u001a\u0000i\u0015u+v=f\n\u0001u+ divarxv\u0000i\u0015v=g\nFrom [8, Section 2], we have the following low frequencies estimates of the resolvent of the operator A:\nProposition 2.1. Assume that a2L1is non negative a>0and non trivialR\n\na(x)dx > 0. Then for any\nM > 0, there exists C > 0such that for all \u00152R;j\u0015j6M, the operatorA\u0000i\u0015is invertible from D(A)toH\nwith estimate\n(2.3) k(A\u0000i\u0015)\u00001kL(H)6C:\nAs a consequence, to prove Theorem 1 it is enough to study the high frequency regime \u0015!+1and prove\nProposition 2.2. Assume that a2C1(\n)is a nonnegative function satisfying (1.3) . Then under the geometric\ncontrol condition, there exists \u00030>0such that for any j\u0015j>\u00030we have\nk(A\u0000i\u0015)\u00001kL(H)6C:\nBy standard argument, we can reduce the proof of Proposition 2.2 to a semi-classical estimate. We denote\nby 0<h=j\u0015j\u00001\u001c1 and\nPh=\u0000h2\u0001\u00001\u0000ihdiva(x)r:\nProposition 2.3. There exists C > 0, such that for all 0<h\u001c1,\nkP\u00001\nhkL(L2)6Ch\u00001: (2.4)\nFor the proof of Proposition 2.3, we argue by contradiction. Assume that there exist sequences ( un)\u001a\nH2\\H1\n0;(fn)\u001aL2andhn!0, such that Phnun=fn,kunkL2= 1 andkfnkL2=o(hn). We will use a\nsemi-classical notation and denote by ( uh;fh) the sequences with the properties\nkuhkL2= 1;kfhkL2=o(h); Phuh=fh: (2.5)\nSometimes we even omit the subindex for uh;fh. In the following subsections, we will prove propagation\nestimates for such sequences.\n3.A priori estimates\nIn this section we establish a series of a priori estimates for the sequence de\fned in (2.5).\nLemma 3.1. Assume that a2L1(\n)is a non-negative function, then\n(1)Z\n\n\u0000\njuhj2\u0000jhruhj2\u0001\n= ReZ\n\nfhuh=o(h); (3.1)\n(2)Z\n\na(x)jhruhj2=hImZ\n\nfhuh=o(h2):; (3.2)\n(3)h2kuhkH2=O(1): (3.3)\nProof. We get (1) and (2) by multiplying the equation Phu=fbyu, integrating by part and taking the real\nand imaginary parts repectively. For (3), from the equation, we have\nh2\u0001u+u+ihra\u0001ru+iha\u0001u=\u0000f;i.e.h2\u0001u=\u0000u+f+ira\u0001hru\n1 +ih\u00001a(x):\nFrom the global estimate of the Poisson equation\nkwkH26Ck\u0001wkL2;8w2H2\\H1\n0;\nwe obtain thatkh2ukH2=O(1). \u0003\nCorollary 3.2. Assume that a2C1(\n)is a non-negative function satisfying (1.3) , then\nka1\n2uhkL2+ka1\n2hruhk+ka1\n2h2\u0001uhkL2=o(h): (3.4)4 N. BURQ AND C-M. SUN\nProof. We only need to estimateR\n\na(x)juj2, sinceR\n\na(x)jhruj2=o(h2) is just (3.2). Multiplying Phu=fby\nauand taking the real part, we haveZ\n\na(x)juj2= ReZ\n\nhru\u0001hr(au)\u0000ImhZ\n\na(x)ru\u0001r(a(x)u) + ImZ\n\na(x)fu:\nSincejraj6Ca1\n2, the \frst term on the r.h.s. can be bounded by\nka1\n2hruk2\nL2+hkrahrukL2kukL2=o(h2):\nThe third term of r.h.s is bounded by o(h)ka1\n2ukL2, and the second term can be bounded by\nh\f\f\fZ\n\nara\u0001uru\f\f\f6hka1\n2ukL2ka1\n2rarukL26Cka1\n2hrukka1\n2ukL2=o(h)ka1\n2ukL2:\nFor the second derivative, we observe that\na(x)1\n2h2\u0001u=\u0000a1\n2u+a1\n2f+iha1\n2ra\u0001hru\n1 +ih\u00001a(x);\nthuska1\n2h2\u0001ukL2=o(h). This completes the proof of Corollary 3.2. \u0003\nLet\u0017be the out-normal vector \feld on @\n. We denote by L2(@) =L2(@\n). The following hidden regularity\nholds:\nLemma 3.3. Assume that a2C1(\n)is a nonnegative function satisfying (1.3) , then\nkh@\u0017ukL2(@)=O(1);ka1\n2h@\u0017ukL2(@)=O(h1\n2):\nProof. We use the standard multiplier method. Let L=bj(x)@jbe anC2extension of the out-normal vector\n\feld\u0017, wherebj's are supported in a neighborhood of @\n. WritePh=Ph;0+iMh, where\nPh;0=\u0000h2\u0001\u00001; Mh=\u0000hdiva(x)r\nare self-adjoint operators. Consider the commutator [ Ph;L] = [Ph;0;L] +i[Mh;L]. Note that [ Ph;0;L] =\n1\nh[Ph;0;hL] belongs to h2Op (S2)\u0003, we deduce that\u0000\n[Ph;0;L]u;u\u0001\nL2=O(1). By direct computation, we have\n\u0000\u0000\n[Mh;L]u;u\u0001\nL2=\u0000\nh@k[a@k;bj@j]u;u\u0001\nL2+h\u0000\n[@k;bj@j]a@ku;u\u0001\nL2\n=\u0000\nh@k(a(@kbj)@ju\u0000bj(@ja)@ku);u\u0001\nL2+\u0000\n(@kbj)h@j(a@ku);u\u0001\nL2\n=\u0000\u0000\n(a@kbj)@ju\u0000bj(@ja)@ku;h@ku\u0001\nL2\u0000\u0000\na@ku;h@j((@kbj)u):\u0001\nL2\nFrom Corollary 3.2, the absolute value of the r.h.s. can be bounded by constant times\nkarukL2+krarukL2=o(1):\nTherefore,\u0000\n[Ph;L]u;u\u0001\nL2=O(1). On the other hand, by developing the commutator and exploiting the\nequation, we have\u0000\n[Ph;L]u;u\u0001\nL2=\u0000\nPhLu;u\u0001\nL2\u0000\u0000\nLf;u\u0001\nL2\n=\u0000\nLu;P\u0003\nhu\u0001\nL2\u0000\u0000\nf;L\u0003u\u0001\nL2+h2k@\u0017uk2\nL2(@)+ihka1\n2@\u0017uk2\nL2(@):\nObserve that\n\u0000\nLu;P\u0003\nhu\u0001\nL2=\u0000\nLu;f\u00002Mhu\u0001\nL2=o(1)\u00002\u0000\nLu;Mhu\u0001\nL2=o(1)\u00002\u0000\nLu;hra\u0001ru+ha\u0001u\u0001\nL2:\nSinceLis a \frst order di\u000berential operator and from Corollary 3.2 that ka1\n2h\u0001ukL2=o(1), we have\nj\u0000\nLu;hra\u0001ru+ha\u0001u\u0001\nL2j6khrukL2krarukL2+ka1\n2rukL2ka1\n2h\u0001ukL2=o(1):\nTherefore,\nkh@\u0017uk2\nL2(@)+ihka1\n2@\u0017uk2\nL2(@)=O(1):\n\u0003Strictly speaking, the symbol of Lis not C1, but here we only need1\nh[Ph;0; hL] =O(h2) onL2.KELVIN-VOIGT DAMPING 5\nThe proof of Lemma 3.3 is then completed by taking real and imaginary parts. \u0003\nLet\u001f2C1\nc(R) such that \u001f(z)\u00111 forjzj61 and\u001f(z)\u00110 forjzj>2. We decompose\n(3.5) uh=vh+wh; vh=\u001f\u0000\nah\u00001\u0001\nuh; wh=\u0000\n1\u0000\u001f\u0000\nah\u00001\u0001\u0001\nuh:\nIn the rest of this note, we always assume that a2C1(\n) is a nonnegative function satisfying (1.3).\nLemma 3.4. We have\nkwhkL2+khrwhkL2=o(h1\n2);ka1\n2vhkL2+ka1\n2hrvhkL2=o(h);\nand\nkuhkH1\nh(a>ch)+kvhkH1\nh(a>ch)=o(h1\n2):\nProof. By de\fnition,Z\n\n\u0000\njwj2+jhrwj2\u0001\n6Z\na>h\u0000\njuj2+jhruj2+jraj2juj2\u0001\n:\nThe conclusion then follows from Corollary 3.2 and the fact that jraj26Ca. Similarly, for any other cuto\u000b\nto the region a>ch, we deduce that kuhkH1\nh(a>ch)=o(h1\n2):For the estimate of v, note that a1\n2hrv=\na1\n2\u001fhru+a1\n2ra\u001f0v, from Corollary 3.2, we have\nka1\n2hrvkL26ka1\n2hrukL2+k\u001f0a1\n2(a1\n2v)kL2=o(h):\nThis completes the proof of Lemma 3.4. \u0003\n4.Geometry, semi-classical measures\nHaving the a priori estimates of the previous section at hand, we can now study vh. For some subsequence\nofvh, we will associate it a semi-classical measure and then prove the invariance of the measure under the\ngeneralized geodesic \row. First recall some geometric preliminaries from [7].\n4.1.Geometry. Denote bybT\n the bundle of rank dwhose sections are the vector \felds tangent to @\n,bT\u0003\nthe dual bundle (Melrose's compressed cotangent bundle) and j:T\u0003\n!bT\u0003\n the canonical map. In any\ncoordinate system where \n = fx= (xd>0;x0)g), the bundlebT\n is generated by the \felds@\n@x0,xd@\n@xdandj\nis de\fned by\n(4.1) j(xd;x0;\u0018d;\u00180) = (xd;x0;v=xd\u0018d;\u00180):\nDenote by Car P0the semi-classical characteristic manifold of P0=\u0000h2\u0001\u00001 andZits projection\n(4.2) Car P0=\b\n(x;\u0018) = (x0;xd;\u00180;\u0018d)2T\u0003Rdj\n;p(x;\u0018) = 0\t\n; Z =j(CarP0):\nThe setZis a locally compact metric space.\nConsider, near a point x02@\n a geodesic system of coordinates for which x0= (0;0), \n =f(xd;x0)2\nR+\u0002Rd\u00001gand the operator P0has the form (near x0)\n(4.3) Ph;0=\u0000h2\u0001\u00001 =h2D2\nxd\u0000R(xd;x0;hDx0) +hQ(x;hDx);\nwithRa second order tangential operator and Qa \frst order operator.\nWe recall now the usual decomposition of T\u0003@\n (in this coordinate system). Denote by r(x0;xd;\u00180) the\nsemi-classical principal symbol of Randr0=rjxd=0. ThenT\u0003@\n is the disjoint union of E[G[H with\n(4.4) E=fr0<0g;G=fr0= 0g;H=fr0>0g:\nRemark that jgives a natural identi\fcation between Zj@MandH[G\u001aT\u0003@M. InGwe distinguish between the\ndi\u000bractive pointsG2;+=fr0= 0;r1=@xdrjxd=0>0gand the gliding pointsG\u0000=fr0= 0;r1=@xdrjxd=060g.\nWe will make the assumption (\n has no in\fnite order contact with its tangents) that for any %02T\u0003@M, there\nexistsN2Nsuch that\nHN\nr0(r1)6= 06 N. BURQ AND C-M. SUN\nThe de\fnition of the generalized bicharacteristic \row, 'sassociated to the operator P0is essentially the\nde\fnition given in [11]:\nDe\fnition 4.1. A generalized bicharacteristic curve \r(s) is a continuous curve from an interval I\u001aRtoZ\nsuch that\n(1) ifs02Iand\r(s0)2T\u0003\n then close to s0,\ris an integral curve of the Hamiltonian vector \feld Hep\n(2) Ifs02Iand\r(s0)2H[G2;+then there exists \">0 such that for 0 <js\u0000s0j<\",xd(\r(s))>0\n(3) Ifs02Iand\r(s0)2G\u0000then for any function f2C1(T\u0003Rdj\n) satisfying the symmetry condition\n(4.5) 8%02Z;8b%0;e%02j\u00001(%0)\\Car(eP);f(b%0) =f(e%0)\nthen\nd\ndsf(j(\r(s))js=s0=Hepjj\u00001(\r(s0))f(j\u00001(\r(s0)))\nIt is proved in [11] that under the assumption of no in\fnite order contact, through every point %o2bT\u0003Mnf0g\nthere exists a unique generalized bicharacteristic (which is furthermore a limit of bicharacteristics having only\nhyperbolic contacts with the boundary). This de\fnes the \row \b.\n4.2.Wigner measures. Consider functions a=ai+a@withai2C1\n0(T\u0003M), anda@2C1\n0(R2d\u00001). Such\nsymbols are quantized in the following way: take 'i2C1\n0(M) (resp'@2C1\n0(Rd)) equal to 1 near the\nx-projection of supp( ai) (resp the x-projection of supp( a@)) and de\fne\n(4.6) Op'i;'@\nh(a)(x;hDx)f=1\n(2\u0019h)dZ\nei(x\u0000y)\u0001\u0018=hai(x;\u0018)'i(y)f(y)dyd\u0018\n+1\n(2\u0019h)d\u00001Z\nei(x0\u0000y0)\u0001\u00180=ha\u000e(xd;x0;\u0018)'\u000e(xd;y0)f(xd;y0)dy0d\u00180:\nRemark that according to the symbolic semi-classical calculus, the operator Op'i;'@\nh(a) does not depend on the\nchoice of functions 'i;'@, modulo operators on L2of norms bounded by O(h1). For conciseness we shall in\nthe sequel drop the index 'i;'@.\nDenote byAhthe space of the operators which are a \fnite sum of operators obtained as above in suitable\ncoordinate systems near the boundary and for B2A, byb=\u001b(B) the semiclassical symbol of the operator A.\nFor such functions bwe can de\fne \u0014(b)2C0(Z) by\n(4.7) \u0014(b)(\u001a) =b(j\u00001(\u001a))\n(the value is independent of the choice of j\u00001(\u001a) since the operator is tangential).\nThe set\n(4.8) f\u0014(b);b=\u001b(B);B2Ahg\nis a locally dense subset of C0\nc(Z).\n4.2.1. Elliptic regularity. The sequence vhsatis\fes (with \u001fh=\u001f(a=h))\nPhvh=\u001fhPhuh\u0000h2div(r\u001fhuh)\u0000h2r\u001fhruh\u0000ihdiv(ar\u001fhuh)\u0000ihar\u001fh\u0001ruh:\nSincer\u001fh=h\u00001\u001f0(a=h)raandjraj.a1\n2, by Corollary 3.2, we have\nPhvh=oL2(h) +oH\u00001(h2):\nThus\n(h2\u0001 + 1)vh=\u0000ihdiv(ar(\u001fhu)) +oL2(h) +oH\u00001(h2):\nUsing Corollary 3.2 again and the fact that a.hon the support of \u001fh, we deduce that har(\u001fhu) =oL2(h3\n2),\nhence\n(h2\u0001 + 1)vh=o(h3\n2)H\u00001(\n)+oL2(h):\nWe deduce, by standard elliptic regularity resultsKELVIN-VOIGT DAMPING 7\nProposition 4.2. Ifaiis equal to 0near Car(P0)then\n(4.9) lim\nk!+1(Ophk(ai)vhk;vhk)L2= 0;\nwhile near the boundary (see e.g. [7, Appendice A.1] in a slightly di\u000berent context) we get\nProposition 4.3. Ifa@is equal to 0nearZ(i.e.aiis supported in the elliptic region) then\n(4.10) lim\nk!+1(a@(x0;xd;hkDx0)vhk;vhk)L2= 0:\nRemark 4.4. Note that if we regard the damping term hdiv(ar(\u001fhu)) =oH\u00001(h3\n2) as a source term, we are\nnot able to use the classical propagation theorem for h2\u0001 + 1 as a black box, as such a strategy would require\nsmaller r.h.s., namely oH\u00001(h2) +oL2(h). On the other hand, an integration by parts shows from Lemma 3.4\n\u0010\ndiv(ar(\u001fhu));\u001fhu\u0011\nL2=\u0000ka1=2rx\u001fhuk2\nL2=o(1);\nand this will ensure that in the propagation estimates such terms are invisible. The key of our analysis in the\nsequel will be to systematically uses this procedure: testing the damping term on expressions like Qh\u001fhu, doing\nthe integration by part and then balancing a1\n2to the other side. It is to perform this analysis that we need the\nconditionjraj6Ca1\n2to ensure the gain O(h) from the commutator [ a1\n2;Qh]. More precisely, we shall need the\nfollowing lemma:\nLemma 4.5. Assume that Qh;B0;h;B1;hare tangential h-pseudodi\u000berential operators of order 0andBh=\nB0;h+B1;hhDxd, then\nh\u00001\u0000\nQhMhu;Bhu\u0001\nL2=o(1);\nwhereMh=\u0000hdivar.\nProof. SinceMhu=\u0000rahru\u0000ah\u0001u, from Corollary 3.2, we have\nh\u00001\f\f\u0000\nQhrahru;Bhu\u0001\nL2\f\f6CkrarukL2kukH1\nh=o(1):\nTo estimate\u0000\nQha\u0001u;Bhu\u0001\nL2, we writeQha\u0001u=a1\n2Qha1\n2\u0001u+ [Qh;a1\n2]a1\n2\u0001u. From Corollary 3.2, a1\n2\u0001u=\noL2(h\u00001). By Corollary A.2, [ Qh;a1\n2];[Bh;a1\n2] =OL(L2)(h). Therefore,\n\u0000\nQha\u0001u;Bhu\u0001\nL2=\u0000\nQha1\n2\u0001u;Bha1\n2u\u0001\nL2+o(1):\nAgain from Corollary 3.2, we have Bha1\n2u=OL2(h), hence (Qha\u0001u;Bhu)L2=o(1). The proof of Lemma 4.5\nis complete. \u0003\n4.2.2. De\fnition of the measure. The following results gives the existence of semi-classical measures.\nProposition 4.6. Let(vhkp)be a sequence bounded in L2(\n). There exists a subsequence (kp)and a Radon\npositive measure \u0016onZsuch that\n(4.11) 8Q2Ahkplim\np!1(Qvhkp;vhkp)L2=h\u0016;\u0014(\u001b(Q))i:\nThe proof of this result relies on the G\u0017 arding inequality for tangential operators (see G. Lebeau [10] for a\nproof in the classical context and [3, 9] for the semi-classical construction). As before, we drop the indexes kp\nand denote by ( vh) the extracted sequence.\nProposition 4.7 (First properties of the measure \u0016).We have\n(4.12) \u0016(H) = 0:\nMoreover, for any tangential symbol b,\n(4.13) lim sup\nk!+1j(Oph(b)hkDxdvhk;vhk)L2j6C sup\n%2supp(b)jrj1=2jbj:8 N. BURQ AND C-M. SUN\nProof. The \frst property (4.12) follows from the fact that the trajectories near a hyperbolic point is transversal\nto the boundary. It follows from [7], with additional attention to the damping term ihdivarvh. We factorize\nPh;0=\u0000h2\u0001\u00001 as (hDxd\u0000L+\nh)(hDxd\u0000L\u0000\nh)+OH1(h1) (see [7, Lemme 6.1]) near \u001a02Hand choose L\u0006\nhwith\nprincipal symbols \u0006l(x0;xd;\u00180) =\u0006p\nr(x0;xd;\u00180). we denote by q0(x0;\u00180)2C1\nc(H) andq\u0006(y;x0;\u00180) solutions\nof\n@xdq\u0006\u0007fl;q\u0006g= 0; q\u0006jxd=0=q0:\nDenote by\nu\u0006:= (xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u;\nwhere (xd)\u00111 if 06xd6\u000f0. We have\n(hDxd\u0000L\u0006\nh)u\u0006= (xd)[hDxd\u0000L\u0006\nh;Q\u0006\nh](hDxd\u0000L\u0007\nh)u+Q\u0006\nhfh+h\ni 0(xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u\n+iQ\u0006\nhMhu+OL2(h1)\nwhereMh=hdivarandfh=oL2(h). By de\fnition of q\u0006, the \frst term of r.h.s. is O(h2), hence\n(hDxd\u0000L\u0006\nh)u\u0006=g\u0006\nh\u0000ih 0(xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u+iQ\u0006\nhMhu; g\u0006\nh=oL2(h): (4.14)\nWe have\nhd\ndxd(u\u0006;u\u0006)L2(@)=\u00002 Im\u0000\ng\u0006\nh\u0000ih 0(xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u+iQ\u0006\nhMhu;u\u0006\u0001\nL2(@)+i\u0000\n(L\u0006\nh\u0000L\u0006;\u0003\nh)u\u0006;u\u0006\u0001\nL2(@):\nFory06\u000f0, we have\nku\u0006(y0)k2\nL2(@)6ku\u0006(0)k2\nL2(xd=0)+Ch\u00001kg\u0006\nhkL2(xd6y0)ku\u0006kL2(xd6y0)+Cku\u0006k2\nL2(xd6y0)\n+Ch\u00001\f\f(Q\u0006\nhMhu;u\u0006)L2(xd6y0)\f\f\nThe second line of r.h.s. is o(1), due to Lemma 4.5, and the \frst line of r.h.s. can be bounded by\nku\u0006(0)k2\nL2(xd=0)+Cku\u0006k2\nL2(xd6y0)+o(1):\nIntegrating both sides over y06\u000f0, lettingh!0 and then \u000f0!0, we deduce that h\u00161y=0;q0i= 0. This proves\n(4.12).\nFor (4.13), it su\u000eces to prove the inequality for uinstead ofv. By Cauchy-Schwarz,\n\f\f\u0000\nOph(b)h@xdu;u\u0001\nL2\f\f6\f\f\u0000\nOph(b)\u0003Oph(b)h@xdu;h@xdu\u0001\nL2\f\f1\n2kukL2:\nDoing integration by part, we have\n\u0000\nOph(b)\u0003Oph(b)h@xdu;h@xdu\u0001\nL2=\u0000\u0000\nOph(b)\u0003Oph(b)h2@2\nxdu;u\u0001\nL2+O(h):\nReplacingh2@2\nxduby equation h2@2\nxdu=\u0000Rhu\u0000iMhu+OL2(h);we deduce that\n\f\f\u0000\nOph(b)\u0003Oph(b)h2@2\nxdu;u\u0001\nL2\f\f6\f\f\u0000\nOph(jbj2)Rhu;u\u0001\nL2\f\f+O(h) +\f\f\u0000\nOph(b)\u0003Oph(b)Mhu;u\u0001\nL2\f\f:\nFrom Lemma 4.5, the third term of r.h.s. is o(h). Passingh!0, we complete the proof of Proposition 4.7. \u0003\n4.3.Invariance of the measure. The key to prove the invariance of the measure will be to apply the propa-\ngation theorem in [7, Th\u0013 eor\u0012 eme 1].\nTheorem. The two following properties are equivalent\n(1)The measure \u0016is invariant along the generalised \row.\n(2)The measure \u0016satis\fes _\u0016= 0and\u0016(G+\n2) = 0 in the sense thath\u0016;fp;qgi= 0holds for any even symbol\nq2C1\nc(Car(P0)), i.e.q(x0;xd= 0;\u00180;\u0018d) =q(x0;xd= 0;\u00180;\u0000\u0018d).KELVIN-VOIGT DAMPING 9\nRemark 4.8. Technically, Theorem 4.3 is proved in [7] for time dependent measures, i.e. measures depending\nin addition on two additional variables ( t;\u001c)2T\u0003R, andpis replaced by p\u0000\u001c2. However, it is easy to apply\nthe results from [7] by considering the measure\n(4.15) e\u0016=\u0016x;\u0018\ndt\n\u000e\u001c=1;\nwhich is supported in Car( \u0000\u0001 +@2\nt) and satis\fes _e\u0016= 0 in the sense that h\u0016;fp\u0000\u001c2;qgi= 0 holds for any even\nsymbolq2C1\nc(Car(P0\u0000\u001c2)), i.e.q(x0;xd= 0;t;\u00180;\u0018d;\u001c) =q(x0;xd= 0;t;\u00180;\u0000\u0018d;\u001c). Remark that though we\nshall not use it, the measure e\u0016is, in the sense of [7, Section 2], the microlocal defect measure on the sequence\nvn(t;x) =hneith\u00001\nnun(x) (the pre-factor hncomes from the H1normalisation of the sequence vnin [7]). Now,\nthe generalised bicharacteristic \row for p0\u0000\u001c2, \tsis given in terms of the generalised bicharacteristic \row for\np0, sby\n\ts(t;x;\u001c = 1;\u0018) = (t\u00002s;\u001c= 1; s(x;\u0018));\nThe set of di\u000bractive points eG2;+in the time dependent frame-work is given by\neG2;+=G2;+\u0002R\u0002f\u001c=\u00061g\nand consequently,\n\u0016(G2;+) = 0,e\u0016(eG2;+) = 0;\nand in view of the particular form (4.15), the invariance of e\u0016by \tsis equivalent to the invariance of \u0016by s.\nLet us now brie\ry explain the procedure we are going to follow.\n\u000fFirst from Proposition 4.6 and the elllipticity (Proposition 4.9, Proposition 4.10), the measure \u0016is\nde\fned on Z=j(Car(P0)) by testing on symbols of the form q=qi+q@;qi2C1\nc(T\u0003\n) andq@\ntangential (which is dense in C0(Z)).\n\u000fUsing the fact \u0016(H) = 0 (Proposition 4.7), the measure \u0016can be extended to test on functions of\nCar(P0) which admits a representation (thanks to Malgrange's theorem)\nq(x0;xd;\u00180;\u0018d) =q0(x0;xd;\u00180) +\u0018dq1(x0;xd;\u00180);on\u00182\nd=r(x0;xd;\u00180):\nThen, we will show in Proposition 4.9 that for tangential h-pseudodi\u000berential operators B0;h;B1;h, the\nquadratic form\n((B0;hk+B1;hk1\nihk@xd))vhk;vhk)\nconverges toh\u0016;b0+b1\u0018d1\u001a=2Hi, by a suitable limit procedure for symbols in Ah. Consequently, for any\nq2C1\nc(Car(P0)), we can make sense of the expression\nh\u0016;fp;qgi\n\u0016-a.e., by viewing fp;qg= 2\u0018d@xdq1\u001a=2H\u0000fr;qg:We remark that to calculate fp;qg, it is enough to\nchoose one representation q=q0+q1\u0018don Car(P0), sincefp;pg= 0 andp= 0 on supp( \u0016).\n\u000fFinally, to prove that the measure \u0016is invariant along the Melrose-Sj ostrand \row, we apply Theorem 4.3,\nfor which we need to verify the following conditions:\n(a)\u0016(G2;+) = 0\n(b) _\u0016= 0, in the sense that h\u0016;fp;qgi= 0 holds for any even symbol q2C1\nc(Car(P0)), i.e.q(x0;xd=\n0;\u00180;\u0018d) =q(x0;xd= 0;\u00180;\u0000\u0018d).\nThe veri\fcation of (a),(b) in our context is based on the propagation formula: Proposition 5.1 and Proposi-\ntion 6.1. Especially, starting from Proposition 6.1, by choosing suitable test symbols of the form q0+q1\u0018d, we\nare able to verify the conditions (a) and (b).\nProposition 4.9. IfB0;h;B1;hare two tangential h-pseudodi\u000berential operators of with principal symbols b0;b1\nof order 0, then we have\nlim\nk!1((B0;hk+B1;hk1\nihk@xd)vhk;vhk)L2=h\u0016;b0+b1\u0018d1\u001a=2Hi:10 N. BURQ AND C-M. SUN\nProof. SinceB0;handB1;hare all tangential, by the de\fnition of the measure, the \frst term ( B0;hkvhk;vhk)L2\nconverges toh\u0016;b0i. It remains to prove the convergence of the second term ( B1;hk1\nihk@xdvhk;vhk)L2. For this,\nwe pick\u000f>0;\u000e> 0 and de\fne\nB1;hk;\u000f=\u0000\n1\u0000 \u0000xd\n\u000f\u0001\u0001\nB1;hk\u0000\n1\u0000 \u0000xd\n2\u000f\u0001\u0001\n; B\u000f\n1;hk=B1;hk\u0000B1;hk;\u000f;\nB\u000f;\u000e\n1;hk= Ophk\u0000\n \u0000r\n\u000e\u0001\u0001\nB\u000f\n1;hk; B\u000f\n1;hk;\u000e=B\u000f\n1;hk\u0000B\u000f;\u000e\n1;hk;\nwhere is a cuto\u000b function which is 1 near 0. Now by the de\fnition of \u0016and the dominating convergence,\nlim\n\u000f!0lim\nk!1(B1;hk;\u000f1\nihk@xdvhk;vhk)L2=h\u0016;b1\u0018d1xd>0i=h\u0016;b1\u0018d1\u001a=2Hi;\nsince\u0016(E) =\u0016(H) = 0. Now from Proposition 4.7, the contribution of\nlim\n\u000f!0lim sup\nk!1j(B\u000f;\u000e\n1;hkhk@xdvhk;vhk)L2j6C\u000e1\n2;\nwhich converges to 0 if we let \u000e!0. Finally, by Cauchy-Schwarz,\nj(B\u000f\n1;hk;\u000ehk@xdvhk;vhk)L2j6khk@xdvhkkL2k(B\u000f;\u000e\n1;hk)\u0003vhkkL2:\nNotice that\nlim\nk!1k(B\u000f;\u000e\n1;hk)\u0003vhkk2\nL2= lim\nk!1(B\u000f;\u000e\n1;hk(B\u000f;\u000e\n1;hk)\u0003vhk;vhk)L2\n=h\u0016;\u0000\n1\u0000 \u0000r\n\u000e\u0001\u0001\u0002\n \u0000xd\n\u000f\u0001\u0000\n1\u0000 \u0000xd\n2\u000f\u0001\u0001\nb1+\u0000\n1\u0000 \u0000xd\n\u000f\u0001\n \u0000xd\n2\u000f\u0001\u0001\nb1\u0003\ni;\ntaking the double limit lim sup\u000e!0lim sup\u000f!0, we obtain that\nlim sup\n\u000e!0lim sup\n\u000f!0lim\nk!1k(B\u000f;\u000e\n1;hk)\u0003vhkk2\nL26h\u0016;b2\n11xd=01r6=0i= 0;\nsince\u00161E[H= 0. This completes the proof of Proposition 4.9. \u0003\n5.Interior propagation estimate\nProposition 5.1 (Interior propagation) .LetQh=e\u001fQhe\u001fbe ah-pseudodi\u000berential operator of order 0, where\ne\u001f2C1\nc(\n), then we have\n1\nih\u0000\n[h2\u0001 + 1;Qh]vh;vh\u0001\nL2=o(1):\nProof. Denote byPh=Ph;0+iMhwithMh=\u0000hdivarandPh;0=\u0000h2\u0001\u00001, we have\n1\nih\u0000\n[Ph;0;Qh]v;v\u0001\nL2=1\nih\u0000\nQhv;Ph;0v\u0001\nL2\u00001\nih\u0000\nPh;0v;Q\u0003\nhv\u0001\nL2\n=1\nih\u0000\nQhv;\u001fPh;0u\u0001\nL2\u00001\nih\u0000\n\u001fPh;0u;Q\u0003\nhv\u0001\nL2+R1\nwithR1=1\nih\u0000\nQhv;[Ph;0;\u001f]u\u0001\nL2\u00001\nih\u0000\n[Ph;0;\u001f]u;Q\u0003\nhv\u0001\nL2:\nBy using the equation Ph;0u=f\u0000iMhu, we have\n1\nih\u0000\n[Ph;0;Qh]v;v\u0001\nL2=R1+R2+o(1); (5.1)\nwhere\nR2=1\nh\u0000\nQhv;\u001fMhu\u0001\nL2+1\nh(\u001fMhu;Q\u0003\nhv)L2:\nNote that\n\u0000[Ph;0;\u001f] =hr\u0000\nra\u001f0(a=h)\u0001\n+ 2ra\u001f0(a=h)hr;\nsincehr(\u001f(a=h)) =ra\u001f0(a=h).\n\u000fClaim 1:R1=o(1)KELVIN-VOIGT DAMPING 11\nIt su\u000eces to show that ih\u00001\u0000\nBhv;[Ph;0;\u001f]u\u0001\nL2=o(1) for any compact supported h-pseudoBhof degree 0. By\nintegration by part,\n\u00001\nih\u0000\nBhv;[Ph;0;\u001f]u\u0001\nL2=\u00001\nih\u0000\nBhv;hdiv (\u001f0(a\nh)rau) +\u001f0(a\nh)rahru\u0001\nL2\nand we simply apply Corollary 3.2, to get for each term o(1).\n\u000fClaim 2:R2=o(1)\nIt su\u000eces to prove that ( Qhv;\u001fdiv (aru))L2=o(1). We write\n\u0000\nQhv;\u001fdivaru\u0001\nL2=\u0000\u0000\n(r\u001f)Qhv;aru\u0001\nL2\u0000\u0000\n\u001f[r;Qh]v;aru\u0001\nL2\u0000\u0000\n\u001fQhrv;aru\u0001\nL2:\nSinceja1\n2r\u001fj=h\u00001ja1\n2\u001f0raj6C, from Corollary 3.2, the \frst term of r.h.s. can be bounded by\nkQhvkL2ka1\n2rukL2=o(1):\nThe second term of r.h.s. can be bounded by o(h). Observe that r(a1\n2) =1\n2a\u00001\n2rais bounded, thus from\nCorollary A.2,\n[a1\n2;Qh] =OL(L2)(h):\nTherefore,\f\f\u0000\n\u001fQhrv;aru\u0001\nL2\f\f6\f\f\u0000\n\u001fQha1\n2rv;a1\n2ru\u0001\nL2\f\f+\f\f\u0000\n\u001f[a1\n2;Qh]rv;a1\n2ru\u0001\nL2\f\f:\nThe second term is bounded by ChkrvkL2ka1\n2rukL2=o(1), and the \frst term can be bounded by o(1), due\nto Lemma 3.4. This completes the proof of Proposition 5.1. \u0003\n6.Propagation near the boundary\nRecall that vh=\u001f(a=h)uh. Consider the operator\nBh=B0;h+B1;hh\ni@xd\nwhereBj;h=e\u001f1Oph(bj)e\u001f1,j= 0;1 are two tangential operators and e\u001f1has compact support near a point\nz02@\n. Note that in the local coordinate system,\nPh;0=\u0000h2\u0001\u00001 =\u00001p\njgjh@xdp\njgjh@xd\u0000Rh;\nwhereRhis a self-adjoint tangential operator of order 2. The operator involving the damping can be written as\nMh=\u0000hp\njgj@xdp\njgja@xd\u0000hp\njgj@x0\nkp\njgjagjk@x0\nj\nProposition 6.1 (Boundary propagation) .\n1\nih\u0000\n[Ph;0;Bh]v;v\u0001\nL2=\u0000\nB1;hjxd=0(h@xdv)jxd=0;(h@xdv)jxd=0\u0001\nL2(@)+o(1):\nProof. We give the proof in the case B0;h= 0. TheB0;hterms are handled by slightly simpler versions of the\nsame computations. By developing the commutator, we have\n1\nih\u0000\n[Ph;0;Bh]v;v\u0001\nL2=1\nih\u0000\nBhv;Ph;0v\u0001\nL2\u00001\nih\u0000\nBhPh;0v;v\u0001\nL2+\u0000\nB1;hjxd=0(h@xdv)jy=0;h@xdvjxd=0\u0001\nL2(@);\nwhere the boundary term (the third) comes from the integration by part of the term\n1\nih\u00001p\njgjh@xdp\njgjh@xdv;v\u0001\nL2;\nsinceRhis self-adjoint tangential operator. It su\u000eces to show that\nIh:=1\nih\u0000\nB1;hh@xdv;Ph;0v\u0001\nL2\u00001\nih\u0000\nB1;hh@xdPh;0v;v\u0001\nL2=o(1): (6.1)12 N. BURQ AND C-M. SUN\nSincev=\u001fuandPh;0u=Phu\u0000iMhu=fh\u0000iMhu, we have\nPh;0v=\u001fPh;0u+ [Ph;0;\u001f]u=\u001ffh\u0000i\u001fMhu+ [Ph;0;\u001f]u:\nTherefore,\nIh=o(1) + Ih;1+ Ih;2;\nwhere\nIh;1=1\nih\u0000\nB1;hh@xdv;[Ph;0;\u001f]u\u0001\nL2\u00001\nih\u0000\nB1;hh@xd[Ph;0;\u001f]u;v\u0001\nL2\nand\nIh;2=1\nh\u0000\nB1;hh@xdv;\u001fMhu\u0001\nL2\u00001\nh\u0000\nB1;hh@xd\u001fMhu;v\u0001\nL2\n\u000fClaim 1:Ih;1=o(1).\nIndeed, from integration by part, the second term\nih\u00001(B1;hh@xd[Ph;0;\u001f]u;v)L2=ih\u00001([Ph;0;\u001f]u;h@xdAhv)L2\nfor some tangential operator Ah, hence it has the same structure as the \frst term. It su\u000eces to show that\nh\u00001\u0000\nB1;hh@xdv;[Ph;0;\u001f]u\u0001\nL2=o(1):\nSince\n\u0000[Ph;0;\u001f]u=hr\u0001(ra\u001f0(a=h))u+ 2ra\u001f0(a=h)\u0001hru;\ndoing integration by part, we obtain that\n\u0000h\u00001\u0000\nB1;hh@xdv;[Ph;0;\u001f]u\u0001\nL2=\u0000\u0000\nr(uB1;hh@xdv);ra\u001f0\u0001\nL2+ 2h\u00001\u0000\nB1;hh@xdv;ra\u001f0hru\u0001\nL2\n=\u0000\u0000\nrB1;hh@xdv;ra\u001f0u\u0001\nL2+h\u00001\u0000\nB1;hh@xdv;ra\u001f0hru\u0001\nL2:\nNote thatv=\u001fu, if one of the derivatives h@xd,hrfall on\u001f(a=h) we can bound them from Corollary 3.2 by\no(h). If all the derivatives fall on uin anyone of the two terms, by Lemma 3.1 and Corollary 3.2, these terms\ncan be bounded by\nkhr@xdukL2kraukL2+h\u00001kh@xdukL2krahrukL2=o(1):\n\u000fClaim 2: Ih;2=o(1).\nIt su\u000eces to prove that\nh\u00001\u0000\nB1;hh@xd(\u001fu);\u001fMhu\u0001\nL2=o(1):\nNote that\u0000Mhu=rahru+ah\u0001u=oL2(1) andh@xd(\u001fu) =@xda\u001f0u+\u001fh@xdu. We have\nh\u00001j\u0000\nB1;h@xda\u001f0u;\u001fMhu\u0001\nL2j6h\u00001kraukL2k\u001fMhukL2=o(1);\nsincekraukL2=o(h) from Corollary 3.2. It remains to show that\nh\u00001\u0000\nB1;h\u001fh@xdu;\u001f(ra\u0001hru+ah\u0001u)\u0001\nL2=o(1):\nSincekrahrukL2=o(h), we haveh\u00001\u0000\nB1;h\u001fh@xdu;\u001f(ra\u0001hru)\u0001\nL2=o(1). Finally, we show that\nh\u00001\u0000\nB1;h\u001fh@xdu;\u001fah \u0001u\u0001\nL2=o(1):\nRecall that from jr(a1\n2)j6Cand Corollary A.2,\n[B1;h;a1\n2] =OL(L2)(h);\nwe have\nh\u00001j\u0000\nB1;h\u001fh@xdu;\u001fah \u0001u\u0001\nL2j6h\u00001j\u0000\nB1;ha1\n2\u001fh@xdu;\u001fa1\n2h\u0001u\u0001\nL2j+h\u00001j\u0000\n[B1;h;a1\n2]\u001fh@xdu;\u001fa1\n2h\u0001u\u0001\nL2j\n6Ch\u00001ka1\n2hrukL2ka1\n2h\u0001ukL2+CkhrukL2ka1\n2h\u0001ukL2=o(1):\nThis completes the proof of Proposition 6.1. \u0003KELVIN-VOIGT DAMPING 13\nTo show that the semi-classical measure \u0016of (vhk) is invariant along the Melrose-Sj ostrand \row (to complete\nthe proof of Proposition 2.3), we need to verify the condition (2) in Theorem 4.3. We will make use of the\npropagation formula, i.e. Proposition 6.1. Formally, for Bh=B0;h+B1;h1\nih@xd, the principal symbol of\ni\nh[Ph;0;Bh] is given by\nf\u00112\u0000r;b0+b1\u0018dg=a0+a1\u0018d+a2\u00182\nd;\nwhere\na0=b1@xdr\u0000fr;b0g0; a 1= 2@xdb0\u0000fr;b1g0; a 2= 2@xdb1; (6.2)\nandf\u0001;\u0001g0is the Poisson bracket for ( x0;\u00180) variables. On the other hand, by calculating the commutator, we\n\fnd\ni\nh[Ph;0;Bh] =A0+A1hDxd+A2h2D2\nxd+hOph(S0\n@+S0\n@\u0018d); (6.3)\nwhereA0;A1;A2are tangential operators with symbols a0;a1;a2, with respectively. We will prove the following\npropagation formula:\nCorollary 6.2. Assume that Bh=Bh;0+Bh;1hDxd, whereBh;0;Bh;1are tangential operators of order 0 with\nsymbolsb0;b1, with respectively. Assume that b=b0+b1\u0018d. De\fne the formal Poisson bracket\nfp;bg= (a0+a2r) +a1\u0018d1\u001a=2H;\nwherea0;a1;a2are given by (6.2) . Then the defect measure \u0016satis\fes the equation\nh\u0016;fp;bgi=\u0000h\u0017@;b1i;\nwhere\u0017@is the semiclassical measure of (h@xdvhjxd=0):Moreover, if bis an even symbol (i.e. b(x0;xd=\n0;\u00180;\u0018d) =b(x0;xd= 0;\u00180;\u0000\u0018d)), then we have\nh\u0016;fp;bgi= 0:\nIn particular, by combining Proposition 5.1, we have _\u0016= 0.\nProof. From Proposition 6.1 and the decomposition (6.3), we have\n\u0000\n(A0+A1hkDxd+A2h2\nkD2\nxd)vhk;vhk\u0001\nL2=\u0000h\u0017@;b1i+o(1): (6.4)\nFrom Lemma 3.4, we can also replace the function vhkon the l.h.s. by uhk. Using the equation of uhk:\n(h2D2\nxd\u0000Rhk)uhk=iMhkuhk\u0000fhk+OL2(hk);\nwe deduce that\n(A2h2\nkD2\nxduhk;uhk) = (A2Rhkuhk;uhk)L2+o(1);\nthanks to Lemma 4.5. Therefore, from Proposition 4.9,\nlim\nkh!1((A0+A1hkDxd+A2Rhk)uhk;uhk)L2=h\u0016;a0+a1\u0018d1\u001a=2H+a2ri=h\u0016;fp;bgi:\nNow ifb=b0+b1\u0018dis an even symbol, we must have b1jxd=0= 0, therefore,h\u0016;fp;bgi=\u0000h\u0017@;b1i= 0. The\nproof of Lemma 6.2 is complete. \u0003\nCorollary 6.3. We have\u0016(G2;+) = 0 .\nProof. We will make use of the formula h\u0016;fp;bgi=\u0000h\u0017@;b1iby choosing b=b1;\u000f\u0011with\nb1;\u000f(x0;xd;\u00180) = \u0000xd\n\u000f1\n2\u0001\n \u0000r(xd;x0;\u00180)\n\u000f\u0001\n\u0014(xd;x0;\u00180);\nwhere 2C1\nc(R) equals to 1 near the origin and \u0014(y;x0;\u00180)>0 near a point \u001a02G2;+. Sincefp;b\u000fg=\n(a0+a2r) +a1\u0018d1\u001a=2H, anda0;a1;a2are given by the relation (6.2). In particular for our choice, by direct\ncalculation we have\na0=b1;\u000f@xdr; a 1=\u0000fr;\u0014g0 \u0000xd\n\u000f1\n2\u0001\n \u0000r\n\u000f\u0001\n;14 N. BURQ AND C-M. SUN\nand\na2= 2@xdb1;\u000f= 2\u000f\u00001\n2 0\u0000xd\n\u000f1\n2\u0001\n \u0000r\n\u000f\u0001\n\u0014+ 2@xdr\n\u000f \u0000xd\n\u000f1\n2\u0001\n 0\u0000r\n\u000f\u0001\n\u0014+ 2 \u0000xd\n\u000f1\n2\u0001\n \u0000r\n\u000f\u0001\n@xd\u0014:\nNote thata2is uniformly bounded in \u000fand for any \fxed ( y;x0;\u00180),ra2!0 as\u000f!0. Thus by dominating\nconvergence, we have\nlim\n\u000f!0h\u0016;fp;b\u000fgi=h\u0016;\u0014jxd=0@xdr1r=0i>0\nsince@xdr>0 onG2;+. However,\u0000h\u0017@;b\u000fi60, we must have \u00161G2;+= 0. This completes the proof of Lemma\n6.3. \u0003\nFrom Lemma 6.2 and Lemma 6.3, we have veri\fed that _ \u0016= 0 and\u0016(G2;+) = 0, thus from Theorem 4.3,\nthe semi-classical \u0016is invariant along the Melrose-Sj ostrand \row. Thanks to the geometric control condition\nand the fact that a1\n2vhk=oL2(1), we deduce that \u0016= 0. This contradicts to the assumption that kvhkkL2=\nkuhkkL2= 1 +o(1), ask!1 . The proof of Proposition 2.3 is now complete.\nAppendix A.Some commutator estimates\nLemma A.1. Assume that b(x;y;\u0018 )2L1(R3d\nx;y;\u0018)such that\nj@\u000b\n\u0018b(x;y;\u0018 )j.\u000bh\u0018i\u0000(j\u000bj+1)\nfor all multi-index \u000b2Nd;j\u000bj6d+ 1. Then the operator Thassociated with the Schwartz kernel\nKh(x;y) :=1\n(2\u0019h)dZ\nRdb(x;y;\u0018 )ei(x\u0000y)\u0001\u0018\nhd\u0018\nis bounded on L2(Rd), uniformly in 0<h61.\nProof. Using the Littlewood-Paley decomposition, we can decompose the operator Th=P\nj>0Th;jwhere each\nTh;jhas the Schwartz kernel\nKh;j(x;y) =1\n(2\u0019h)dZ\nRdbj(x;y;\u0018 )ei(x\u0000y)\u0001\u0018\nhd\u0018;\nwithbj(x;y;\u0018 ) =b(x;y;\u0018 ) j(\u0018) and j(\u0018) = (2\u0000j\u0018) for some  2C1\nc(1\n26j\u0018j62), ifj>1 and 0(\u0018) is\nsupported onj\u0018j61. Note that\n(x\u0000y)\u000bKh;j(x;y) =i\u0000j\u000bjhj\u000bj\n(2\u0019h)dZ\nRdD\u000b\n\u0018bj(x;y;\u0018 )\u0001ei(x\u0000y)\u0001\u0018\nhd\u0018;\nwe have\njKh;j(x;y)j.\u000bhj\u000bj\u0000d\njx\u0000yjj\u000bj\u00012\u0000j(j\u000bj+1\u0000d):\nWe have another trivial bound jKh;j(x;y)j.\u000b2jdh\u0000d:Therefore, for \fxed x2Rd, by choosingj\u000bj=d+ 1, we\nhaveZ\nRdjKh;j(x;y)jdy.Z\nRdmin\b\n2\u0000j2\u0000jh\njx\u0000yjd+1;2jdh\u0000d\t\ndy\n6Z\njzj62\u0000j\nd+1\u00012\u0000jh2jdh\u0000ddz+Z\njzj>2\u0000j\nd+1\u00012\u0000jh2\u0000j2\u0000jh\njzjd+1dz.2\u0000jd\nd+1:\nSimilarly, for \fxed y2Rd,Z\nRdjKh;j(x;y)jdx.2\u0000jd\nd+1:\nBy Schur's test, we have kTh;jkL(L2).2\u0000jd\nd+1. Using the triangle inequality, we obtain that This bounded on\nL2(Rd), uniformly in 0 <h61. The proof of Lemma A.1 is now complete. \u0003KELVIN-VOIGT DAMPING 15\nCorollary A.2. Assume that \u00142W1;1(Rd)andb2S0(R2d)is a symbol of order zero, then we have\nk[Oph(b);\u0014]kL(L2)=O(h):\nProof. The kernel of [Oph(b);\u0014] is given by\nK(x;y) =1\n(2\u0019h)dZ\nRdb(x;\u0018)(\u0014(y)\u0000\u0014(x))ei(x\u0000y)\u0001\u0018\nhd\u0018:\nSince\u00142W1;1, there exists \t2L1(Rd;Rd) such that\n\u0014(y)\u0000\u0014(x) = (y\u0000x)\u0001\t(x;y):\nThus\nK(x;y) =\u0000dX\nj=1h\ni(2\u0019h)dZ\nRd@\u0018jb(x;\u0018)\tj(x;y))ei(x\u0000y)\u0018\nhd\u0018\nApplying Lemma A.1 to each @\u0018jb(x;\u0018)\tj(x;y), the proof of Corollary A.2 is complete. \u0003\nReferences\n[1] Charles J. K. Batty and Thomas Duyckaerts. Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. ,\n8(4):765{780, 2008.\n[2] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann. , 347(2):455{478,\n2010.\n[3] N. Burq. Mesures semi-classiques et mesures de d\u0013 efaut. S\u0013 eminaire Bourbaki , Mars 1997.\n[4] N. Burq. D\u0013 ecroissance de l'\u0013 energie locale de l'\u0013 equation des ondes pour le probl\u0012 eme ext\u0013 erieur et absence de r\u0013 esonance au\nvoisinage du r\u0013 eel. Acta Mathematica , 180:1{29, 1998.\n[5] N. Burq and H. Christianson. Imperfect geometric control and overdamping for the damped wave equation. Comm. Math.\nPhys. , 336(1):101{130, 2015.\n[6] N. Burq and P. G\u0013 erard. Stabilization of wave equations on the torus with rough dampings. To appear Pure and Applied\nAnalysis. preprint arxiv https://arxiv.org/abs/1801.00983 , 2020.\n[7] N. Burq and G. Lebeau. Mesures de d\u0013 efaut de compacit\u0013 e, application au syst\u0012 eme de Lam\u0013 e. Ann. Sci. \u0013Ecole Norm. Sup. (4) ,\n34(6):817{870, 2001.\n[8] Nicolas Burq. Decays for Kelvin-Voigt damped wave equations I: the black box perturbative method. SIAM J. Control Optim. ,\n58(4):1893{1905, 2020.\n[9] P. G\u0013 erard and E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Mathematical Journal ,\n71:559{607, 1993.\n[10] G. Lebeau. Equation des ondes amorties. In A. Boutet de Monvel and V. Marchenko, editors, Algebraic and Geometric Methods\nin Mathematical Physics , pages 73{109. Kluwer Academic, The Netherlands, 1996.\n[11] R.B. Melrose and J. Sj ostrand. Singularities of boundary value problems II. Communications in Pure Applied Mathematics ,\n35:129{168, 1982.\n[12] Louis Tebou. A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping. C. R.\nMath. Acad. Sci. Paris , 350(11-12):603{608, 2012.\nUniversit \u0013e Paris-Saclay, Laboratoire de math \u0013ematiques d'Orsay, UMR 8628 du CNRS, B ^atiment 307, 91405 Orsay\nCedex, France and Institut Universitaire de France\nEmail address :nicolas.burq@math.u-psud.fr\nUniversit \u0013e de Cergy-Pontoise, Laboratoire de Math \u0013ematiques AGM, UMR 8088 du CNRS, 2 av. Adolphe Chauvin\n95302 Cergy-Pontoise Cedex, France\nEmail address :chenmin.sun@u-cergy.fr"    },    {        "title": "1403.7377v1.The_Confrontation_between_General_Relativity_and_Experiment.pdf",        "content": "arXiv:1403.7377v1  [gr-qc]  28 Mar 2014The Confrontation between General Relativity\nand Experiment\nClifford M. Will\nDepartment of Physics\nUniversity of Florida\nGainesville FL 32611, U.S.A.\nemail: cmw@physics.ufl.edu\nhttp://www.phys.ufl.edu/ ~cmw/\nAbstract\nThe status of experimental tests of general relativity and o f theoretical frameworks for\nanalyzing them are reviewed and updated. Einstein’s equiva lence principle (EEP) is well\nsupported by experiments such as the E¨ otv¨ os experiment, t ests of local Lorentz invariance\nand clock experiments. Ongoing tests of EEP and of the invers e square law are searching\nfor new interactions arising from unification or quantum gra vity. Tests of general relativity\nat the post-Newtonian level have reached high precision, in cluding the light deflection, the\nShapiro time delay, the perihelion advance of Mercury, the N ordtvedt effect in lunar motion,\nand frame-dragging. Gravitational wave damping has been de tected in an amount that agrees\nwith general relativity to better than half a percent using t he Hulse–Taylor binary pulsar, and\na growing family of other binary pulsar systems is yielding n ew tests, especially of strong-field\neffects. Current and future tests of relativity will center o n strong gravity and gravitational\nwaves.\n1Contents\n1 Introduction 3\n2 Tests of the Foundations of Gravitation Theory 6\n2.1 The Einstein equivalence principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.1.1 Tests of the weak equivalence principle . . . . . . . . . . . . . . . . . . . . . 7\n2.1.2 Tests of local Lorentz invariance . . . . . . . . . . . . . . . . . . . . . . . . 9\n2.1.3 Tests of local position invariance . . . . . . . . . . . . . . . . . . . . . . . . 12\n2.2 Theoretical frameworks for analyzing EEP . . . . . . . . . . . . . . . . . . . . . . . 16\n2.2.1 Schiff’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6\n2.2.2 The THǫµformalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n2.2.3 The c2formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n2.2.4 The Standard Model Extension (SME) . . . . . . . . . . . . . . . . . . . . . 21\n2.3 EEP, particle physics, and the search for new interactions . . . . . . . . . . . . . . 22\n2.3.1 The “fifth” force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3\n2.3.2 Short-range modifications of Newtonian gravity . . . . . . . . . . . . . . . . 24\n2.3.3 The Pioneer anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n3 Metric Theories of Gravity and the PPN Formalism 26\n3.1 Metric theories of gravity and the strong equivalence principle . . . . . . . . . . . . 26\n3.1.1 Universal coupling and the metric postulates . . . . . . . . . . . . . . . . . 26\n3.1.2 The strong equivalence principle . . . . . . . . . . . . . . . . . . . . . . . . 26\n3.2 The parametrized post-Newtonian formalism . . . . . . . . . . . . . . . . . . . . . 28\n3.3 Competing theories of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n3.3.1 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3\n3.3.2 Scalar-tensor theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34\n3.3.3f(R) theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36\n3.3.4 Vector-tensor theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36\n3.3.5 Tensor–vector–scalar (TeVeS) theories . . . . . . . . . . . . . . . . . . . . . 37\n3.3.6 Quadratic gravity and Chern-Simons theories . . . . . . . . . . . . . . . . . 39\n3.3.7 Massive gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n4 Tests of Post-Newtonian Gravity 40\n4.1 Tests of the parameter γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\n4.1.1 The deflection of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\n4.1.2 The time delay of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\n4.1.3 Shapiro time delay and the speed of gravity . . . . . . . . . . . . . . . . . . 44\n4.2 The perihelion shift of Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45\n4.3 Tests of the strong equivalence principle . . . . . . . . . . . . . . . . . . . . . . . . 46\n4.3.1 The Nordtvedt effect and the lunar E¨ otv¨ os experiment . . . . . . . . . . . . 46\n4.3.2 Preferred-frame and preferred-location effects . . . . . . . . . . . . . . . . . 47\n4.3.3 Constancy of the Newtonian gravitational constant . . . . . . . . . . . . . . 48\n4.4 Other tests of post-Newtonian gravity . . . . . . . . . . . . . . . . . . . . . . . . . 49\n4.4.1 Search for gravitomagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n4.4.2 Geodetic precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1\n4.4.3 Tests of post-Newtonian conservation laws . . . . . . . . . . . . . . . . . . . 51\n4.5 Prospects for improved PPN parameter values . . . . . . . . . . . . . . . . . . . . 52\n25 Strong Gravity and Gravitational Waves: Tests for the 21st Century 54\n5.1 Strong-field systems in general relativity . . . . . . . . . . . . . . . . . . . . . . . . 54\n5.1.1 Defining weak and strong gravity . . . . . . . . . . . . . . . . . . . . . . . . 54\n5.1.2 Compact bodies and the strong equivalence principle . . . . . . . . . . . . . 55\n5.2 Motion and gravitational radiation in general relativity: A history . . . . . . . . . 56\n5.3 Compact binary systems in general relativity . . . . . . . . . . . . . . . . . . . . . 58\n5.3.1 Einstein’s equations in “relaxed” form . . . . . . . . . . . . . . . . . . . . . 58\n5.3.2 Equations of motion and gravitational waveform . . . . . . . . . . . . . . . 60\n5.4 Compact binary systems in scalar-tensor theories . . . . . . . . . . . . . . . . . . . 61\n5.4.1 Scalar-tensor equations in “relaxed” form . . . . . . . . . . . . . . . . . . . 62\n5.4.2 Equations of motion and gravitational waveform . . . . . . . . . . . . . . . 63\n5.4.3 Binary systems containing black holes . . . . . . . . . . . . . . . . . . . . . 65\n6 Stellar System Tests of Gravitational Theory 67\n6.1 The binary pulsar and general relativity . . . . . . . . . . . . . . . . . . . . . . . . 67\n6.2 A zoo of binary pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71\n6.3 Binary pulsars and alternative theories . . . . . . . . . . . . . . . . . . . . . . . . . 73\n6.4 Binary pulsars and scalar-tensor gravity . . . . . . . . . . . . . . . . . . . . . . . . 74\n7 Gravitational-Wave Tests of Gravitational Theory 77\n7.1 Gravitational-wave observatories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77\n7.2 Gravitational-wave amplitude and polarization . . . . . . . . . . . . . . . . . . . . 77\n7.2.1 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7\n7.2.2 Alternative theories of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 78\n7.3 Gravitational-wave phase evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 81\n7.3.1 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1\n7.3.2 Alternative theories of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 82\n7.4 Speed of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82\n8 Astrophysical and cosmological tests 84\n9 Conclusions 85\n10 Acknowledgments 86\n1 Introduction\nWhen general relativity was born 100yearsago, experimental con firmation was almost a side issue.\nAdmittedly, Einstein did calculate observable effects of general rela tivity, such as the perihelion\nadvance of Mercury, which he knew to be an unsolved problem, and t he deflection of light, which\nwas subsequently verified. But compared to the inner consistency and elegance of the theory, he\nregarded such empirical questions as almost secondary. He famou sly stated that if the measure-\nments of light deflection disagreed with the theory he would “feel so rry for the dear Lord, for the\ntheoryiscorrect!”.\nBy contrast, today experimental gravitation is a major componen t of the field, characterizedby\ncontinuing efforts to test the theory’s predictions, both in the sola r system and in the astronomical\nworld, to detect gravitational waves from astronomical sources , and to search for possible gravita-\ntional imprints of phenomena originating in the quantum, high-energ y or cosmological realms.\nThemodernhistoryofexperimentalrelativitycanbedividedroughly intofourperiods: Genesis,\nHibernation, a Golden Era, and the Quest for Strong Gravity. The G enesis (1887–1919)comprises\n3the period of the two great experiments which were the foundation of relativistic physics – the\nMichelson–Morley experiment and the E¨ otv¨ os experiment – and th e two immediate confirmations\nof general relativity – the deflection of light and the perihelion advan ce of Mercury. Following this\nwas a period of Hibernation (1920–1960) during which theoretical w ork temporarily outstripped\ntechnology and experimental possibilities, and, as a consequence, the field stagnated and was\nrelegated to the backwaters of physics and astronomy.\nBut beginning around 1960, astronomical discoveries (quasars, p ulsars, cosmic background\nradiation)andnewexperimentspushedgeneralrelativitytothefo refront. Experimentalgravitation\nexperienced a Golden Era (1960–1980) during which a systematic, w orld-wide effort took place\nto understand the observable predictions of general relativity, t o compare and contrast them with\nthe predictions of alternative theories of gravity, and to perform new experiments to test them.\nNew technologies – atomic clocks, radar and laser ranging, space pr obes, cryogenic capabilities, to\nmention only a few – played a central role in this golden era. The period began with an experiment\nto confirm the gravitationalfrequency shift of light (1960) and en ded with the reported decrease in\nthe orbitalperiod ofthe Hulse-Taylorbinary pulsarat a rate consis tent with the generalrelativistic\nprediction of gravitational-wave energy loss (1979). The results a ll supported general relativity,\nand most alternative theories of gravity fell by the wayside (for a p opular review, see [389]).\nSince that time, the field has entered what might be termed a Quest f or Strong Gravity. Much\nlike modern art, the term “strong” means different things to differe nt people. To one steeped in\ngeneral relativity, the principal figure of merit that distinguishes s trong from weak gravity is the\nquantityǫ∼GM/Rc2, whereGis the Newtonian gravitational constant, Mis the characteristic\nmass scale of the phenomenon, Ris the characteristic distance scale, and cis the speed of light.\nNear the event horizon of a non-rotating black hole, or for the exp anding observable universe,\nǫ∼1; for neutron stars, ǫ∼0.2. These are the regimes of strong gravity. For the solar system,\nǫ<10−5; this is the regime of weak gravity.\nAn alternative view of “strong” gravity comes from the world of par ticle physics. Here the\nfigure of merit is GM/R3c2∼ℓ−2, where the Riemann curvature of spacetime associated with the\nphenomenon, represented by the left-hand-side, is comparable t o the inverse square of a favorite\nlength scale ℓ. Ifℓis the Planck length, this would correspond to the regime where one e xpects\nconventional quantum gravity effects to come into play. Another p ossible scale for ℓis the TeV\nscale associated with many models for unification of the forces, or m odels with extra spacetime\ndimensions. Fromthis viewpoint, stronggravityiswherethe curvat ureis comparableto the inverse\nlength squared. Weak gravity is where the curvature is much smaller than this. The universe at\nthe Planck time is strong gravity. Just outside the event horizon of an astrophysical black hole is\nweak gravity.\nConsiderations of the possibilities for new physics from either point o f view have led to a wide\nrange of questions that will motivate new tests of general relativit y as we move into its second\ncentury:\n•Are the black holes that are in evidence throughout the universe tr uly the black holes of\ngeneral relativity?\n•Do gravitational waves propagate with the speed of light and do the y contain more than the\ntwo basic polarization states of general relativity?\n•Does general relativity hold on cosmological distance scales?\n•Is Lorentz invariance strictly valid, or could it be violated at some det ectable level?\n•Does the principle of equivalence break down at some level?\n•Are there testable effects arising from the quantization of gravity ?\n4In this update of our Living Review , we will summarize the current status of experiments,\nand attempt to chart the future of the subject. We will not provid e complete references to early\nwork done in this field but instead will refer the reader to selected re cent papers and to the\nappropriatereviewarticlesandmonographs,specificallyto Theory and Experiment in Gravitational\nPhysics[388], hereafter referred to as TEGP; references to TEGP will be b y chapter or section,\ne.g., “TEGP 8.9”. Additional reviews in this subject are [34, 333, 364 ]. The “Resource Letter” by\nthe author [396], contains an annotated list of 100 key references for experimental gravity.\n52 Tests of the Foundations of Gravitation Theory\n2.1 The Einstein equivalence principle\nThe principle of equivalence has historically played an important role in t he development of gravi-\ntation theory. Newton regarded this principle as such a cornersto ne of mechanics that he devoted\nthe openingparagraphofthe Principia toit. In 1907, Einsteinused the principle asabasicelement\nin his development of general relativity (GR). We now regard the prin ciple of equivalence as the\nfoundation, not of Newtonian gravity or of GR, but of the broader idea that spacetime is curved.\nMuch of this viewpoint can be traced back to Robert Dicke, who cont ributed crucial ideas about\nthe foundations of gravitation theory between 1960 and 1965. Th ese ideas were summarized in\nhis influential Les Houches lectures of 1964 [117], and resulted in wh at has come to be called the\nEinstein equivalence principle (EEP).\nOne elementary equivalence principle is the kind Newton had in mind when he stated that\nthe property of a body called “mass” is proportional to the “weight ”, and is known as the weak\nequivalence principle (WEP). An alternative statement of WEP is that the trajectory of a freely\nfalling “test” body (one not acted upon by such forces as electrom agnetism and too small to be\naffected by tidal gravitational forces) is independent of its intern al structure and composition. In\nthe simplest case of dropping two different bodies in a gravitational fi eld, WEP states that the\nbodies fall with the same acceleration (this is often termed the Unive rsality of Free Fall, or UFF).\nThe Einstein equivalence principle (EEP) is a more powerful and far-r eaching concept; it states\nthat:\n1. WEP is valid.\n2. The outcome of any local non-gravitational experiment is indepe ndent of the velocity of the\nfreely-falling reference frame in which it is performed.\n3. The outcome of any local non-gravitational experiment is indepe ndent of where and when in\nthe universe it is performed.\nThe second piece of EEP is called local Lorentz invariance (LLI), and the third piece is called local\nposition invariance (LPI).\nFor example, a measurement of the electric force between two cha rged bodies is a local non-\ngravitational experiment; a measurement of the gravitational fo rce between two bodies (Cavendish\nexperiment) is not.\nThe Einstein equivalence principle is the heart and soul of gravitation al theory, for it is pos-\nsible to argue convincingly that if EEP is valid, then gravitation must be a “curved spacetime”\nphenomenon, in other words, the effects of gravity must be equiva lent to the effects of living in a\ncurved spacetime. As a consequence of this argument, the only th eories of gravity that can fully\nembody EEP are those that satisfy the postulates of “metric theo ries of gravity”, which are:\n1. Spacetime is endowed with a symmetric metric.\n2. The trajectories of freely falling test bodies are geodesics of th at metric.\n3. In local freely falling reference frames, the non-gravitational laws of physics are those written\nin the language of special relativity.\nThe argument that leads to this conclusion simply notes that, if EEP is valid, then in local\nfreely falling frames, the laws governing experiments must be indepe ndent of the velocity of the\nframe (local Lorentz invariance), with constant values for the va rious atomic constants (in order\nto be independent of location). The only laws we know of that fulfill th is are those that are\ncompatible with special relativity, such as Maxwell’s equations of elect romagnetism. Furthermore,\n6in local freely falling frames, test bodies appear to be unaccelerate d, in other words they move\non straight lines; but such “locally straight” lines simply correspond t o “geodesics” in a curved\nspacetime (TEGP 2.3 [388]).\nGeneral relativity is a metric theory of gravity, but then so are man y others, including the\nBrans–Dicke theory and its generalizations. Theories in which varyin g non-gravitational constants\nareassociatedwith dynamicalfields that couple to matter directly a renot metric theories. Neither,\nin this narrow sense, is superstring theory (see Section 2.3), which , while based fundamentally on a\nspacetime metric, introduces additional fields (dilatons, moduli) tha t can couple to material stress-\nenergyin a way that can lead to violations, say, ofWEP. It is importan t to point out, however, that\nthere is some ambiguity in whether one treats such fields as EEP-viola ting gravitational fields, or\nsimply as additional matter fields, like those that carry electromagn etism or the weak interactions.\nStill, the notion of curved spacetime is a very general and fundamen tal one, and therefore it\nis important to test the various aspects of the Einstein equivalence principle thoroughly. We\nfirst survey the experimental tests, and describe some of the th eoretical formalisms that have\nbeen developed to interpret them. For other reviews of EEP and its experimental and theoretical\nsignificance,see[168,220]; forapedagogicalreviewofthevariety ofequivalenceprinciples,see[115].\n2.1.1 Tests of the weak equivalence principle\nAdirecttestofWEPisthecomparisonoftheaccelerationoftwolabo ratory-sizedbodiesofdifferent\ncomposition in an external gravitational field. If the principle were v iolated, then the accelerations\nof different bodies would differ. The simplest way to quantify such pos sible violations of WEP in\na form suitable for comparison with experiment is to suppose that fo r a body with inertial mass\nmI, the passive gravitational mass mPis no longer equal to mI, so that in a gravitational field\ng, the acceleration is given by mIa=mPg. Now the inertial mass of a typical laboratory body\nis made up of several types of mass-energy: rest energy, electr omagnetic energy, weak-interaction\nenergy, and so on. If one of these forms of energy contributes t omPdifferently than it does to mI,\na violation of WEP would result. One could then write\nmP=mI+/summationdisplay\nAηAEA\nc2, (1)\nwhereEAis the internal energy of the body generated by interaction A,ηAis a dimensionless\nparameter that measures the strength of the violation of WEP indu ced by that interaction, and c\nis the speed of light. A measurement or limit on the fractional differen ce in acceleration between\ntwo bodies then yields a quantity called the “E¨ otv¨ os ratio” given by\nη≡2|a1−a2|\n|a1+a2|=/summationdisplay\nAηA/parenleftbiggEA\n1\nm1c2−EA\n2\nm2c2/parenrightbigg\n, (2)\nwhere we drop the subscript “I” from the inertial masses. Thus, e xperimental limits on ηplace\nlimits on the WEP-violation parameters ηA.\nMany high-precision E¨ otv¨ os-type experiments have been perfo rmed, from the pendulum exper-\niments of Newton, Bessel, and Potter to the classic torsion-balanc e measurements of E¨ otv¨ os [133],\nDicke [118], Braginsky [57], and their collaborators. In the modern to rsion-balance experiments,\ntwo objects of different composition are connected by a rod or plac ed on a tray and suspended\nin a horizontal orientation by a fine wire. If the gravitational accele ration of the bodies differs,\nand this difference has a component perpendicular to the suspensio n wire, there will be a torque\ninduced on the wire, related to the angle between the wire and the dir ection of the gravitational\nacceleration g. If the entire apparatus is rotated about some direction with angu lar velocity ω, the\ntorque will be modulated with period 2 π/ω. In the experiments of E¨ otv¨ os and his collaborators,\n71900 1920 1940 1960 1970 1980 1990 2000 10 -8 \n10 -9 \n10 -10 \n10 -11 \n10 -12 \n10 -13 \n10 -14 \nYEAR OF EXPERIMENT dE¨ otv¨ os \nRenner \nPrinceton \nMoscow Boulder \nE¨ ot-Wash \nE¨ ot-Wash Free-fall \nFifth-force \n  searches \nLLR               TESTS OF THE \nWEAK EQUIVALENCE PRINCIPLE \nd\u001b    a1-a 2\n (a 1+a 2)/2 \n2010 Matter waves\nFigure 1: Selected tests of the weak equivalence principle, showing b ounds onη, which measures\nfractional difference in acceleration of different materials or bodies . The free-fall and E¨ ot-Wash\nexperiments were originally performed to search for a fifth force ( green region, representing many\nexperiments). The blue band shows evolving bounds on ηfor gravitating bodies from lunar laser\nranging (LLR).\nthe wire and gwere not quite parallel because of the centripetal acceleration on the apparatus due\nto the Earth’s rotation; the apparatus was rotated about the dir ection of the wire. In the Dicke\nand Braginsky experiments, gwas that of the Sun, and the rotation of the Earth provided the\nmodulation of the torque at a period of 24 hr (TEGP 2.4 (a) [388]). Be ginning in the late 1980s,\nnumerous experiments were carried out primarily to search for a “fi fth force” (see Section 2.3.1),\nbut their null results alsoconstituted tests ofWEP. In the “free- fall Galileo experiment” performed\nat the University of Colorado, the relative free-fall acceleration o f two bodies made of uranium and\ncopper was measured using a laser interferometric technique. The “E¨ ot-Wash” experiments car-\nried out at the University of Washington used a sophisticated torsio n balance tray to compare the\naccelerations of various materials toward local topography on Ear th, movable laboratory masses,\nthe Sun and the galaxy [350, 25], and have reached levels of 2 ×10−13[1, 326, 372].\nThe recent development of atom interferometry has yielded tests of WEP, albeit to modest\naccuracy,comparabletothatoftheoriginalE¨ otv¨ osexperimen t. Intheseexperiments, onemeasures\nthelocalaccelerationofthetwoseparatedwavefunctionsofana tomsuchasCesiumbystudyingthe\n8interferencepatternwhenthewavefunctionsarecombined, and comparesthatwiththeacceleration\nof a nearby macroscopic object of different composition [254, 269 ]. A claim that these experiments\ntest the gravitational redshift [269] was subsequently shown to b e incorrect [408].\nTheresultingupperlimits on ηaresummarizedinFigure1(TEGP14.1[388]; forabibliography\nof experiments up to 1991, see [140]).\nA number of projects are in the development or planning stage to pu sh the bounds on ηeven\nlower. The project MICROSCOPE is designed to test WEP to 10−15. It is being developed\nby the French space agency CNES for launch in late 2015, for a one- year mission [256]. The\ndrag-compensated satellite will be in a Sun-synchronous polar orbit at 700 km altitude, with a\npayload consisting of two differential accelerometers, one with elem ents made of the same material\n(platinum), and another with elements made of different materials (p latinum and titanium). Other\nconcepts for future improvements include advanced space exper iments (Galileo-Galilei, STEP),\nexperiments on sub-orbital rockets, lunar laser ranging (see Sec . 4.3.1), binary pulsar observations,\nand experiments with anti-hydrogen. For a recent focus issue on p ast and future tests of WEP,\nsee Vol. 29, Number 18 of Classical and Quantum Gravity [343].\n2.1.2 Tests of local Lorentz invariance\nAlthough special relativity itself never benefited from the kind of “c rucial” experiments, such as\nthe perihelion advance of Mercury and the deflection of light, that c ontributed so much to the\ninitial acceptance of GR and to the fame of Einstein, the steady acc umulation of experimental\nsupport, together with the successful merger of special relativ ity with quantum mechanics, led\nto its acceptance by mainstream physicists by the late 1920s, ultima tely to become part of the\nstandard toolkit of every working physicist. This accumulation includ ed\n•the classic Michelson–Morley experiment and its descendents [255, 329, 190, 61],\n•the Ives–Stillwell, Rossi–Hall, and other tests of time-dilation [184, 31 6, 136],\n•tests of the independence of the speed of light of the velocity of th e source, using both binary\nX-ray stellar sources and high-energy pions [59, 7],\n•tests of the isotropy of the speed of light [66, 313, 216].\nIn addition to these direct experiments, there was the Dirac equat ion of quantum mechanics\nand its prediction of anti-particles and spin; later would come the stu nningly successful relativistic\ntheoryofquantumelectrodynamics. Forapedagogicalreviewont heoccasionofthe2005centenary\nof special relativity, see [394].\nIn 2015, on the 110th anniversary of the introduction of special r elativity, one might ask “what\nis there to test?” Special relativity has been so thoroughly integra ted into the fabric of modern\nphysics that its validity is rarely challenged, except by cranks and cr ackpots. It is ironic then, that\nduring the past several years, a vigorous theoretical and exper imental effort has been launched, on\nan international scale, to find violations of special relativity. The mo tivation for this effort is not\na desire to repudiate Einstein, but to look for evidence of new physic s “beyond” Einstein, such as\napparent, or “effective” violations of Lorentz invariance that migh t result from certain models of\nquantum gravity. Quantum gravity asserts that there is a fundam ental length scale given by the\nPlanck length, ℓPl= (/planckover2pi1G/c3)1/2= 1.6×10−33cm, but since length is not an invariant quantity\n(Lorentz–FitzGerald contraction), then there could be a violation of Lorentz invariance at some\nlevel in quantum gravity. In brane-world scenarios, while physics ma y be locally Lorentz invariant\nin the higher dimensional world, the confinement of the interactions of normal physics to our four-\ndimensional “brane” could induce apparent Lorentz violating effect s. And in models such as string\ntheory, the presence of additional scalar, vector, and tensor lo ng-range fields that couple to matter\n9ofthe standard model could induce effective violationsof Lorentzs ymmetry. These and other ideas\nhave motivated a serious reconsideration of how to test Lorentz in variance with better precision\nand in new ways.\nA simple and useful way of interpreting some of these modern exper iments, called the c2-\nformalism, is to suppose that the electromagnetic interactions suff er a slight violation of Lorentz\ninvariance, through a change in the speed of electromagnetic radia tioncrelative to the limiting\nspeed of material test particles ( c0, made to take the value unity via a choice of units), in other\nwords,c∝ne}ationslash= 1 (see Section 2.2.3). Such a violation necessarily selects a preferr ed universal rest\nframe, presumably that of the cosmic background radiation, thro ugh which we are moving at\nabout 370 km s−1[233]. Such a Lorentz-non-invariant electromagnetic interaction w ould cause\nshifts in the energy levels of atoms and nuclei that depend on the or ientation of the quantization\naxis of the state relative to our universal velocity vector, and on t he quantum numbers of the state.\nThe presence or absence of such energy shifts can be examined by measuringthe energyof one such\nstate relative to another state that is either unaffected or is affec ted differently by the supposed\nviolation. One way is to look for a shifting of the energy levels of state s that are ordinarily equally\nspaced, such as the Zeeman-split 2 J+1 ground states of a nucleus of total spin Jin a magnetic\nfield; another is to compare the levels of a complex nucleus with the at omic hyperfine levels of a\nhydrogen maser clock. The magnitude of these “clock anisotropies ” turns out to be proportional\ntoδ≡ |c−2−1|.\nThe earliest clock anisotropy experiments were the Hughes–Dreve r experiments, performed in\nthe period 1959–60 independently by Hughes and collaborators at Y ale University, and by Drever\nat Glasgow University, although their original motivation was somewh at different [178, 123]. The\nHughes–Drever experiments yielded extremely accurate results, quoted as limits on the parameter\nδ≡c−2−1 in Figure 2. Dramatic improvementswere made in the 1980susing lase r-cooledtrapped\natoms and ions [298, 221, 71]. This technique made it possible to reduc e the broading of resonance\nlines caused by collisions, leading to improved bounds on δshown in Figure 2 (experiments labelled\nNIST, U. Washington and Harvard, respectively).\nAlso included for comparison is the corresponding limit obtained from M ichelson–Morley type\nexperiments (for a review, see [169]). In those experiments, when viewed from the preferred frame,\nthe speed of light down the two arms of the moving interferometer is c, while it can be shown using\nthe electrodynamics of the c2formalism, that the compensating Lorentz–FitzGerald contractio n of\nthe parallel arm is governed by the speed c0= 1. Thus the Michelson–Morley experiment and its\ndescendants also measure the coefficient c−2−1. One of these is the Brillet–Hall experiment [61],\nwhich used a Fabry–Perot laser interferometer. In a recent serie s of experiments, the frequencies\nof electromagnetic cavity oscillators in various orientations were co mpared with each other or with\natomic clocks as a function of the orientation of the laboratory [407 , 234, 268, 17, 347]. These\nplaced bounds on c−2−1 at the level of better than a part in 109. Haugan and L¨ ammerzahl [167]\nhave considered the bounds that Michelson–Morley type experimen ts could place on a modified\nelectrodynamics involving a “vector-valued” effective photon mass .\nThec2framework focusses exclusively on classical electrodynamics. It h as recently been ex-\ntended to the entire standard model of particle physics by Kostele ck´ y and colleagues [82, 83, 210].\nThe “Standard Model Extension” (SME) has a large number of Lore ntz-violating parameters,\nopening up many new opportunities for experimental tests (see Se ction 2.2.4). A variety of clock\nanisotropy experiments have been carried out to bound the electr omagnetic parameters of the\nSME framework [209]. For example, the cavity experiments describe d above [407, 234, 268] placed\nbounds on the coefficients of the tensors ˜ κe−and ˜κo+(see Section 2.2.4 for definitions) at the lev-\nels of 10−14and 10−10, respectively. Direct comparisons between atomic clocks based on different\nnuclear species place bounds on SME parameters in the neutron and proton sectors, depending\non the nature of the transitions involved. The bounds achieved ran ge from 10−27to 10−32GeV.\nRecent examples include [409, 340].\n101900 1920 1940 1960 1970 1980 1990 2000 10 -2 \n10 -6 \n10 -10\n10 -14\n10 -18\n10 -22\n10 -26\nYEAR OF EXPERIMENTδMichelson-Morley\nJoos\nHughes DreverBrillet-HallJPL              TESTS OF  \nLOCAL LORENTZ INVARIANCE\nδ = 1/c2 - 1TPA\nCentrifuge\nNIST\nHarvard\nU. Washington\n2010 Cavities\nFigure 2: Selected tests of local Lorentz invariance showing the bo unds on the parameter δ, which\nmeasuresthe degreeofviolationofLorentzinvariancein electroma gnetism. TheMichelson–Morley,\nJoos, Brillet–Hall and cavity experiments test the isotropy of the r ound-trip speed of light. The\ncentrifuge, two-photon absorption (TPA) and JPL experiments t est the isotropy of light speed\nusing one-way propagation. The most precise experiments test iso tropy of atomic energy levels.\nThe limits assume a speed of Earth of 370 km s−1relative to the mean rest frame of the universe.\nAstrophysical observations have also been used to bound Lorent z violations. For example, if\nphotons satisfy the Lorentz violating dispersion relation\nE2=p2c2+EPlf(1)|p|c+f(2)p2c2+f(3)\nEPl|p|3c3+..., (3)\nwhereEPl= (/planckover2pi1c5/G)1/2is the Planck energy, then the speed of light vγ=∂E/∂pwould be given,\nto linear order in the f(n)by\nvγ\nc≈1+/summationdisplay\nn≥1(n−1)f(n)\nγEn−2\n2En−2\nPl. (4)\nSuch a Lorentz-violating dispersion relation could be a relic of quantu m gravity, for instance. By\nbounding the difference in arrival time of high-energy photons from a burst source at large dis-\ntances, one could bound contributions to the dispersion for n >2. One limit, |f(3)|<128 comes\n11from observations of 1 and 2 TeV gamma rays from the blazar Marka rian 421 [42]. Another limit\ncomes from birefringence in photon propagation: In many Lorentz violating models, different pho-\nton polarizations may propagate with different speeds, causing the plane of polarization of a wave\nto rotate. If the frequency dependence of this rotation has a dis persion relation similar to Eq. (3),\nthen by studying “polarization diffusion” of light from a polarized sour ce in a given bandwidth,\none can effectively place a bound |f(3)|<10−4[158]. Measurements of the spectrum of ultra-high-\nenergy cosmic rays using data from the HiRes and Pierre Auger obse rvatories show no evidence\nfor violations of Lorentz invariance [349, 41]. Other testable effect s of Lorentz invariance viola-\ntion include threshold effects in particle reactions, gravitational Ce renkov radiation, and neutrino\noscillations.\nFor thorough and up-to-date surveys of both the theoretical f rameworks and the experimental\nresults for tests of LLI see the reviews by Mattingly [250], Liberati [231] and Kosteleck´ y and\nRussell [211]. The last article gives “data tables” showing experiment al bounds on all the various\nparameters of the SME.\n2.1.3 Tests of local position invariance\nThe principle of local position invariance, the third part of EEP, can b e tested by the gravitational\nredshift experiment, the first experimental test of gravitation p roposed by Einstein. Despite the\nfact that Einstein regarded this as a crucial test of GR, we now rea lize that it does not distin-\nguish between GR and any other metric theory of gravity, but is only a test of EEP. The iconic\ngravitational redshift experiment measures the frequency or wa velength shift Z≡∆ν/ν=−∆λ/λ\nbetween two identical frequency standards (clocks) placed at re st at different heights in a static\ngravitational field. If the frequency of a given type of atomic clock is the same when measured in\na local, momentarily comoving freely falling frame (Lorentz frame), in dependent of the location or\nvelocity ofthat frame, then the comparisonof frequencies of two clocks at rest at different locations\nboils down to a comparison of the velocities of two local Lorentz fram es, one at rest with respect\nto one clock at the moment of emission of its signal, the other at rest with respect to the other\nclock at the moment of reception of the signal. The frequency shift is then a consequence of the\nfirst-order Doppler shift between the frames. The structure of the clock plays no role whatsoever.\nThe result is a shift\nZ=∆U\nc2, (5)\nwhere ∆Uis the difference in the Newtonian gravitational potential between t he receiver and the\nemitter. If LPI is not valid, then it turns out that the shift can be wr itten\nZ= (1+α)∆U\nc2, (6)\nwhere the parameter αmay depend upon the nature of the clock whose shift is being measur ed\n(see TEGP 2.4 (c) [388] for details).\nThe first successful, high-precisionredshift measurement was th e seriesof Pound–Rebka–Snider\nexperiments of 1960–1965 that measured the frequency shift of gamma-ray photons from57Fe as\nthey ascended or descended the Jefferson Physical Laboratory tower at Harvard University. The\nhigh accuracy achieved – one percent – was obtained by making use o f the M¨ ossbauer effect to\nproduce a narrow resonance line whose shift could be accurately de termined. Other experiments\nsince 1960 measured the shift of spectral lines in the Sun’s gravitat ional field and the change in\nrate of atomic clocks transported aloft on aircraft, rockets and satellites. Figure 3 summarizes the\nimportant redshift experiments that have been performed since 1 960 (TEGP 2.4 (c) [388]).\nAfter almost 50 years of inconclusive or contradictory measureme nts, the gravitational redshift\nof solar spectral lines was finally measured reliably. During the early y ears of GR, the failure to\n121960 1970 1980 1990 2000 10 /uni0000-1\n10 /uni0000-2\n10 /uni0000-3\n10 /uni0000-4\n10 /uni0000-5\nYEAR OF EXPERIMENTαPound-/uni0000Rebka Millisecond Pulsar              TESTS OF  \nLOCAL POSITION INVARIANCE\nH /uni0000maser GPAPound\n/uni0000Snider  \nSaturn\nSolar spectra\nClocks in rockets \nspacecraft & planes\nΔν/ν = (1+α)Δ U/c210 /uni0000-6\n2010 10 /uni0000-7Null experimentsSAO-\nStanford\nFountain\nClocks\nFigure 3: Selected tests of local position invariance via gravitationa l redshift experiments, showing\nbounds onα, which measures degree of deviation of redshift from the formula ∆ ν/ν= ∆U/c2. In\nnull redshift experiments, the bound is on the difference in αbetween different kinds of clocks.\nmeasure this effect in solar lines was siezed upon by some as reason to doubt the theory (see [85]\nfor an engaging history of this period). Unfortunately, the measu rement is not simple. Solar\nspectral lines are subject to the “limb effect”, a variation of spect ral line wavelengths between\nthe center of the solar disk and its edge or “limb”; this effect is actua lly a Doppler shift caused\nby complex convective and turbulent motions in the photosphere an d lower chromosphere, and is\nexpected to be minimized by observing at the solar limb, where the mot ions are predominantly\ntransverse. The secret is to use strong, symmetrical lines, leadin g to unambiguous wavelength\nmeasurements. Successful measurements were finally made in 196 2 and 1972 (TEGP 2.4 (c) [388]).\nIn 1991, LoPresto et al. [238] measured the solar shift in agreemen t with LPI to about 2 percent\nby observing the oxygen triplet lines both in absorption in the limb and in emission just off the\nlimb.\nThe most precisestandardredshift test to date wasthe Vessot– Levinerocketexperiment known\nas Gravity Probe-A (GPA) that took place in June 1976 [370]. A hydr ogen-maser clock was flown\n13onarockettoanaltitude ofabout10,000kmanditsfrequencycomp aredtoahydrogen-maserclock\non the ground. The experiment took advantage of the masers’ fr equency stability by monitoring\nthe frequency shift as a function of altitude. A sophisticated data acquisition scheme accurately\neliminated all effects of the first-order Doppler shift due to the roc ket’s motion, while tracking data\nwere used to determine the payload’s location and the velocity (to ev aluate the potential difference\n∆U, and the special relativistic time dilation). Analysis of the data yielded a limit|α|<2×10−4.\nA “null” redshift experiment performed in 1978 tested whether the relativerates of two dif-\nferent clocks depended upon position. Two hydrogen maser clocks and an ensemble of three\nsuperconducting-cavity stabilized oscillator (SCSO) clocks were co mpared over a 10-day period.\nDuring the period of the experiment, the solar potential U/c2within the laboratory was known\nto change sinusoidally with a 24-hour period by 3 ×10−13because of the Earth’s rotation, and to\nchange linearly at 3 ×10−12per day because the Earth is 90 degrees from perihelion in April. How-\never, analysis of the data revealed no variations of either type with in experimental errors, leading\nto a limit on the LPI violation parameter |αH−αSCSO|<2×10−2[361]. This bound has been\nimproved using more stable frequency standards, such as atomic f ountain clocks [159, 299, 29, 55].\nThe best current bounds, from comparing a Rubidium atomic founta in with a Cesium-133fountain\nor with a hydrogen maser [164, 292], and from comparing transitions of two different isotopes of\nDysprosium [227], hover around the one part per million mark.\nThe Atomic Clock Ensemblein Space (ACES) project will place both a co ld trapped atom clock\nbased on Cesium called PHARAO (Projet d’Horloge Atomique par Refro idissement d’Atomes en\nOrbite), and an advanced hydrogen maser clock on the Internatio nal Space Station to measure\nthe gravitational redshift to parts in 106, as well as to carry out a number of fundamental physics\nexperimentsandtoenableimprovementsinglobaltimekeeping[308]. L aunchiscurrentlyscheduled\nfor May 2016.\nThevaryinggravitationalredshiftofEarth-boundclocksrelative tothehighlystablemillisecond\npulsar PSR 1937+21, caused by the Earth’s motion in the solar gravit ational field around the\nEarth-Moon center of mass (amplitude 4000 km), was measured to about 10 percent [353]. Two\nmeasurements of the redshift using stable oscillator clocks on spac ecraft were made at the one\npercent level: One used the Voyager spacecraft in Saturn’s gravit ational field [215], while another\nused the Galileo spacecraft in the Sun’s field [217].\nThe gravitational redshift could be improved to the 10−10level using an array of laser cooled\natomic clocks on board a spacecraft which would travel to within fou r solar radii of the Sun [247].\nSadly, the Solar Probe Plus mission, scheduled for launch in 2018, has been formulated as an\nexclusivelyheliophysicsmission, andthuswillnotbeabletotestfunda mentalgravitationalphysics.\nModern advances in navigation using Earth-orbiting atomic clocks an d accurate time-transfer\nmust routinely take gravitational redshift and time-dilation effects into account. For example, the\nGlobal Positioning System (GPS) provides absolute positional accur acies of around 15 m (even\nbetter in its military mode), and 50 nanoseconds in time transfer acc uracy, anywhere on Earth.\nYet the difference in rate between satellite and ground clocks as a re sult of relativistic effects is\na whopping 39 microseconds per day (46 µs from the gravitational redshift, and −7µs from time\ndilation). If these effects were not accurately accounted for, GP S would fail to function at its\nstated accuracy. This represents a welcome practical application of GR! (For the role of GR in\nGPS, see [21, 22]; for a popular essay, see [392].)\nA final example of the almost “everyday” implications of the gravitat ional redshift is a remark-\nable measurement using optical clocks based on trapped aluminum ion s of the frequency shift over\na height of 1/3 of a meter [70].\nLocal position invariance also refers to position in time. If LPI is satis fied, the fundamental\nconstants of non-gravitational physics should be constants in tim e. Table 1 shows current bounds\non cosmological variations in selected dimensionless constants. For discussion and references to\nearlywork, see TEGP 2.4 (c) [388] or[124]. For a comprehensive rec ent review both of experiments\n14and of theoretical ideas that underly proposals for varying const ants, see [367].\nExperimental bounds on varying constants come in two types: bou nds on the present rate of\nvariation, and bounds on the difference between today’s value and a value in the distant past.\nThe main example of the former type is the clock comparison test, in w hich highly stable atomic\nclocks of different fundamental type are intercompared over per iods ranging from months to years\n(variants of the null redshift experiment). If the frequencies of the clocks depend differently on\nthe electromagnetic fine structure constant αEM, the electron-proton mass ratio me/mp, or the\ngyromagnetic ratio of the proton gp, for example, then a limit on a drift of the fractional frequency\ndifference translates into a limit on a drift of the constant(s). The d ependence of the frequencies on\nthe constants may be quite complex, depending on the atomic specie s involved. Experiments have\nexploited the techniques of laser cooling and trapping, and of atom f ountains, in order to achieve\nextreme clockstability, and comparedthe Rubidium-87 hyperfine tr ansition [248], the Mercury-199\nion electric quadrupole transition [43], the atomic Hydrogen 1 S–2Stransition [144], or an optical\ntransition in Ytterbium-171 [291], against the ground-state hyper fine transition in Cesium-133.\nMore recent experiments have used Strontium-87 atoms trapped in optical lattices [55] compared\nwith Cesium to obtain ˙ αEM/αEM<6×10−16yr−1, compared Rubidium-87 and Cesium-133 foun-\ntains [164] to obtain ˙ αEM/αEM<2.3×10−16yr−1, or compared two isotopes of Dysprosium [227]\nto obtain ˙αEM/αEM<1.3×10−16yr−1,.\nThe secondtype ofbound involvesmeasuringthe relics oforsignal f rom a processthat occurred\nin the distant past and comparing the inferred value of the constan t with the value measured in\nthe laboratory today. One sub-type uses astronomical measure ments of spectral lines at large\nredshift, while the other uses fossils of nuclear processes on Eart h to infer values of constants early\nin geological history.\nTable 1: Bounds on cosmological variation of fundamental constan ts of non-gravitational physics.\nFor an in-depth review, see [367].\nConstantk Limit on ˙k/k Redshift Method\n(yr−1)\nFine structure constant\n(αEM=e2//planckover2pi1c)<1.3×10−160 Clock comparisons\n[55, 164, 227]\n<0.5×10−160.15 Oklo Natural Reactor\n[91, 151, 293]\n<3.4×10−160.45187Re decay in meteorites\n[287]\n(6.4±1.4)×10−160.2–3.7 Spectra in distant quasars\n[376, 271, 199]\n<1.2×10−160.4–2.3 Spectra in distant quasars\n[344, 67, 302, 192, 229]\nWeak interaction constant\n(αW=Gfm2\npc//planckover2pi13)<1×10−110.15 Oklo Natural Reactor\n[91]\n<5×10−12109Big Bang nucleosynthesis\n[246, 307]\ne-p mass ratio <3.3×10−150 Clock comparisons\n[55]\n<3×10−152.6–3.0 Spectra in distant quasars\n[183]\n15Earlier comparisons of spectral lines of different atoms or transitio ns in distant galaxies and\nquasars produced bounds αEMorgp(me/mp) on the order of a part in 10 per Hubble time [410].\nDramatic improvementsin the precisionof astronomicaland laborat oryspectroscopy, in the ability\ntomodelthecomplexastronomicalenvironmentswhereemissionan dabsorptionlinesareproduced,\nand in the ability to reach large redshift have made it possible to improv e the bounds significantly.\nIn fact, in 1999, Webb et al. [376, 271] announced that measureme nts of absorption lines in Mg,\nAl, Si, Cr, Fe, Ni, and Zn in quasars in the redshift range 0 .5< Z <3.5 indicated a smaller\nvalue ofαEMin earlier epochs, namely ∆ αEM/αEM= (−0.72±0.18)×10−5, corresponding to\n˙αEM/αEM= (6.4±1.4)×10−16yr−1(assuming a linear drift with time). The Webb group\ncontinues to report changes in αover large redshifts [199]. Measurements by other groups have so\nfar failed to confirm this non-zero effect [344, 67, 302]; An analysis o f Mg absorption systems in\nquasars at 0 .4<Z <2.3 gave ˙αEM/αEM= (−0.6±0.6)×10−16yr−1[344]. Recent studies have\nalso yielded no evidence for a variation in αEM[192, 229]\nAnother important set of bounds arises from studies of the “Oklo” phenomenon, a group of\nnatural, sustained235U fission reactors that occurred in the Oklo region of Gabon, Africa , around\n1.8 billion years ago. Measurements of ore samples yielded an abnorma lly low value for the ratio of\ntwo isotopes of Samarium,149Sm/147Sm. Neither of these isotopes is a fission product, but149Sm\ncan be depleted by a flux of neutrons. Estimates of the neutron flu ence (integrated dose) during\nthe reactors’ “on” phase, combined with the measured abundanc e anomaly, yield a value for the\nneutron cross-section for149Sm 1.8 billion years ago that agrees with the modern value. However,\nthe capture cross-section is extremely sensitive to the energy of a low-lying level ( E∼0.1 eV),\nso that a variation in the energy of this level of only 20 meV over a billion years would change\nthe capture cross-section from its present value by more than th e observed amount. This was first\nanalyzed in 1976 by Shlyakter [336]. Recent reanalyses of the Oklo da ta [91, 151, 293] lead to a\nbound on ˙αEMat the level of around 5 ×10−17yr−1.\nIn a similar manner, recent reanalyses of decay rates of187Re in ancient meteorites (4.5 billion\nyears old) gave the bound ˙ αEM/αEM<3.4×10−16yr−1[287].\n2.2 Theoretical frameworks for analyzing EEP\n2.2.1 Schiff’s conjecture\nBecausethe three partsofthe Einstein equivalenceprinciple discus sedabovearesoverydifferent in\ntheir empirical consequences, it is tempting to regard them as indep endent theoretical principles.\nOn the other hand, any complete and self-consistent gravitation t heory must possess sufficient\nmathematical machinery to make predictions for the outcomes of e xperiments that test each prin-\nciple, and because there are limits to the number of ways that gravit ation can be meshed with the\nspecial relativistic laws of physics, one might not be surprised if ther e were theoretical connections\nbetween the three sub-principles. For instance, the same mathem atical formalism that produces\nequations describing the free fall of a hydrogen atom must also pro duce equations that determine\nthe energy levels of hydrogen in a gravitational field, and thereby t he ticking rate of a hydrogen\nmaser clock. Hence a violation of EEP in the fundamental machinery o f a theory that manifests\nitself as a violation of WEP might also be expected to show up as a violatio n of local position\ninvariance. Around 1960, Leonard Schiff conjectured that this kin d of connection was a necessary\nfeature of any self-consistent theory of gravity. More precisely , Schiff’s conjecture states that any\ncomplete, self-consistent theory of gravity that embodies WEP necessarily embodies EEP . In other\nwords, the validity of WEP alone guarantees the validity of local Lore ntz and position invariance,\nand thereby of EEP.\nIf Schiff’s conjecture is correct, then E¨ otv¨ os experiments may be seen as the direct empirical\nfoundation for EEP, hence for the interpretation of gravity as a c urved-spacetime phenomenon. Of\n16course, a rigorous proof of such a conjecture is impossible (indeed , some special counter-examples\nare known [286, 275, 81]), yet a number of powerful “plausibility” ar guments can be formulated.\nThe most general and elegant of these arguments is based upon th e assumption of energy\nconservation. This assumption allows one to perform very simple cyc lic gedanken experiments\nin which the energy at the end of the cycle must equal that at the be ginning of the cycle. This\napproachwaspioneered by Dicke, Nordtvedt, and Haugan (see, e .g., [166]). A system in a quantum\nstateAdecays to state B, emitting a quantum of frequency ν. The quantum falls a height Hin\nan external gravitational field and is shifted to frequency ν′, while the system in state Bfalls with\nacceleration gB. At the bottom, state Ais rebuilt out of state B, the quantum of frequency ν′,\nand the kinetic energy mBgBHthat stateBhas gained during its fall. The energy left over must\nbe exactly enough, mAgAH, to raise state Ato its original location. (Here an assumption of local\nLorentz invariance permits the inertial masses mAandmBto be identified with the total energies\nof the bodies.) If gAandgBdepend on that portion of the internal energy of the states that was\ninvolved in the quantum transition from AtoBaccording to\ngA=g/parenleftbigg\n1+αEA\nmAc2/parenrightbigg\n, g B=g/parenleftbigg\n1+αEB\nmBc2/parenrightbigg\n, E A−EB≡hν (7)\n(violation of WEP), then by conservation of energy, there must be a corresponding violation of\nLPI in the frequency shift of the form (to lowest order in hν/mc2)\nZ=ν′−ν\nν′= (1+α)gH\nc2= (1+α)∆U\nc2. (8)\nHaugan generalized this approach to include violations of LLI [166] ( TEGP 2.5 [388]).\nBox 1. The THǫµformalism\nCoordinate system and conventions:\nx0=t: time coordinate associated with the static nature of the static sp herically symmetric\n(SSS) gravitational field; x= (x,y,z): isotropic quasi-Cartesian spatial coordinates; spatial\nvector and gradient operations as in Cartesian space.\nMatter and field variables:\n•m0a: rest mass of particle a.\n•ea: charge of particle a.\n•xµ\na(t): world line of particle a.\n•vµ\na=dxµ\na/dt: coordinate velocity of particle a.\n•Aµ=: electromagnetic vector potential; E=∇A0−∂A/∂t,B=∇×A.\nGravitational potential:\nU(x).\nArbitrary functions:\nT(U),H(U),ǫ(U),µ(U); EEP is satisfied if ǫ=µ= (H/T)1/2for allU.\n17Action:\nI=−/summationdisplay\nam0a/integraldisplay\n(T−Hv2\na)1/2dt+/summationdisplay\naea/integraldisplay\nAµ(xν\na)vµ\nadt+(8π)−1/integraldisplay\n(ǫE2−µ−1B2)d4x.\nNon-metric parameters:\nΓ0=−c2\n0∂\n∂Uln[ǫ(T/H)1/2]0,Λ0=−c2\n0∂\n∂Uln[µ(T/H)1/2]0,Υ0= 1−(TH−1ǫµ)0,\nwherec0= (T0/H0)1/2and subscript “0”refersto achosenpoint in space. IfEEPis satisfi ed,\nΓ0≡Λ0≡Υ0≡0.\n2.2.2 The THǫµformalism\nThe first successful attempt to prove Schiff’s conjecture more f ormally was made by Lightman and\nLee [232]. They developed a framework called the THǫµformalism that encompasses all metric\ntheories of gravity and many non-metric theories (see Box 1). It r estricts attention to the behavior\nof charged particles (electromagnetic interactions only) in an exte rnal static spherically symmetric\n(SSS) gravitational field, described by a potential U. It characterizes the motion of the charged\nparticles in the external potential by two arbitrary functions T(U) andH(U), and characterizes\nthe response of electromagnetic fields to the external potential (gravitationally modified Maxwell\nequations) by two functions ǫ(U) andµ(U). The forms of T,H,ǫ, andµvary from theory to\ntheory, but every metric theory satisfies\nǫ=µ=/parenleftbiggH\nT/parenrightbigg1/2\n, (9)\nfor allU. This consequence follows from the action of electrodynamics with a “minimal” or metric\ncoupling:\nI=−/summationdisplay\nam0a/integraldisplay\n(−gµνvµ\navν\na)1/2dt+/summationdisplay\naea/integraldisplay\nAµ(xν\na)vµ\nadt−1\n16π/integraldisplay√−ggµαgνβFµνFαβd4x,(10)\nwhere the variables are defined in Box 1, and where Fµν≡Aν,µ−Aµ,ν. By identifying g00=T\nandgij=Hδijin a SSS field, Fi0=EiandFij=ǫijkBk, one obtains Eq. (9). Conversely, every\ntheory within this class that satisfies Eq. (9) can have its electrody namic equations cast into\n“metric” form. In a given non-metric theory, the functions T,H,ǫ, andµwill depend in general\non the full gravitational environment, including the potential of th e Earth, Sun, and Galaxy, as\nwell as on cosmological boundary conditions. Which of these factor s has the most influence on a\ngiven experiment will depend on the nature of the experiment.\nLightman and Lee then calculated explicitly the rate of fall of a “test ” body made up of inter-\nacting charged particles, and found that the rate was independen t of the internal electromagnetic\nstructure of the body (WEP) if and only if Eq. (9) was satisfied. In o ther words, WEP ⇒EEP\nand Schiff’s conjecture was verified, at least within the restrictions built into the formalism.\nCertain combinations of the functions T,H,ǫ, andµreflect different aspects of EEP. For\ninstance, position or U-dependence of either of the combinations ǫ(T/H)1/2andµ(T/H)1/2signals\nviolations of LPI, the first combination playing the role of the locally me asured electric charge or\nfine structure constant. The “non-metric parameters” Γ 0and Λ0(see Box 1) are measures of such\nviolations of EEP. Similarly, if the parameter Υ 0≡1−(TH−1ǫµ)0is non-zero anywhere, then\n18violations of LLI will occur. This parameter is related to the differenc e between the speed of light\nc, and the limiting speed of material test particles c0, given by\nc= (ǫ0µ0)−1/2, c 0=/parenleftbiggT0\nH0/parenrightbigg1/2\n. (11)\nIn many applications, by suitable definition of units, c0can be set equal to unity. If EEP is valid,\nΓ0≡Λ0≡Υ0= 0 everywhere.\nThe rate of fall of a composite spherical test body of electromagn etically interacting particles\nthen has the form\na=mP\nm∇U, (12)\nmP\nm= 1+EES\nB\nMc2\n0/bracketleftbigg\n2Γ0−8\n3Υ0/bracketrightbigg\n+EMS\nB\nMc2\n0/bracketleftbigg\n2Λ0−4\n3Υ0/bracketrightbigg\n+..., (13)\nwhereEES\nBandEMS\nBare the electrostatic and magnetostatic binding energies of the bo dy, given\nby\nEES\nB=−1\n4T1/2\n0H−1\n0ǫ−1\n0/angbracketleftBigg/summationdisplay\nabeaeb\nrab/angbracketrightBigg\n, (14)\nEMS\nB=−1\n8T1/2\n0H−1\n0µ0/angbracketleftBigg/summationdisplay\nabeaeb\nrab[va·vb+(va·nab)(vb·nab)]/angbracketrightBigg\n, (15)\nwhererab=|xa−xb|,nab= (xa−xb)/rab, and the angle brackets denote an expectation value\nof the enclosed operator for the system’s internal state. E¨ otv ¨ os experiments place limits on the\nWEP-violating terms in Eq. (13), and ultimately place limits on the non-m etric parameters\n|Γ0|<2×10−10and|Λ0|<3×10−6. (We set Υ 0= 0 because of very tight constraints on it from\ntests of LLI; see Figure 2, where δ=−Υ0.) These limits are sufficiently tight to rule out a number\nof non-metric theories of gravity thought previously to be viable (T EGP 2.6 (f) [388]).\nTheTHǫµformalism also yields a gravitationally modified Dirac equation that can b e used\nto determine the gravitational redshift experienced by a variety o f atomic clocks. For the redshift\nparameterα(see Eq. (6)), the results are (TEGP 2.6 (c) [388]):\nα=\n\n−3Γ0+Λ0hydrogen hyperfine transition, H-Maser clock,\n−1\n2(3Γ0+Λ0) electromagnetic mode in cavity, SCSO clock,\n−2Γ0 phonon mode in solid, principal transition in hydrogen.(16)\nThe redshift is the standard one ( α= 0), independently of the nature of the clock if and only\nif Γ0≡Λ0≡0. Thus the Vessot–Levine rocket redshift experiment sets a limit o n the parameter\ncombination |3Γ0−Λ0|(see Figure 3); the null-redshift experiment comparing hydrogen- maser\nand SCSO clocks sets a limit on |αH−αSCSO|=3\n2|Γ0−Λ0|. Alvarez and Mann [8, 9, 10, 11, 12]\nextended the THǫµformalism to permit analysis of such effects as the Lamb shift, anoma lous\nmagnetic moments and non-baryonic effects, and placed interestin g bounds on EEP violations.\n2.2.3 The c2formalism\nTheTHǫµformalism can also be applied to tests of local Lorentz invariance, bu t in this context\nit can be simplified. Since most such tests do not concern themselves with the spatial variation of\nthe functions T,H,ǫ, andµ, but rather with observations made in moving frames, we can treat\n19them as spatial constants. Then by rescaling the time and space co ordinates, the charges and the\nelectromagnetic fields, we can put the action in Box 1 into the form (T EGP 2.6 (a) [388])\nI=−/summationdisplay\nam0a/integraldisplay\n(1−v2\na)1/2dt+/summationdisplay\naea/integraldisplay\nAµ(xν\na)vµ\nadt+(8π)−1/integraldisplay\n(E2−c2B2)d4x,(17)\nwherec2≡H0/(T0ǫ0µ0) = (1−Υ0)−1. This amounts to using units in which the limiting speed c0\nof massive test particles is unity, and the speed of light is c. Ifc∝ne}ationslash= 1, LLI is violated; furthermore,\nthe form of the action above must be assumed to be valid only in some p referred universal rest\nframe. The natural candidate for such a frame is the rest frame o f the microwave background.\nThe electrodynamical equations which follow from Eq. (17) yield the b ehavior of rods and\nclocks, just as in the full THǫµformalism. For example, the length of a rod which moves with\nvelocity Vrelative to the rest frame in a direction parallel to its length will be obs erved by a\nrest observer to be contracted relative to an identical rod perpe ndicular to the motion by a factor\n1−V2/2 +O(V4). Notice that cdoes not appear in this expression, because only electrostatic\ninteractions are involved, and cappears only in the magnetic sector of the action (17). The energy\nand momentum of an electromagnetically bound body moving with veloc ityVrelative to the rest\nframe are given by\nE=MR+1\n2MRV2+1\n2δMij\nIViVj+O(MV4),\nPi=MRVi+δMij\nIVj+O(MV3),(18)\nwhereMR=M0−EES\nB,M0is the sum of the particle rest masses, EES\nBis the electrostatic binding\nenergy of the system (see Eq. (14) with T1/2\n0H0ǫ−1\n0= 1), and\nδMij\nI=−2/parenleftbigg1\nc2−1/parenrightbigg/bracketleftbigg4\n3EES\nBδij+˜EESij\nB/bracketrightbigg\n, (19)\nwhere\n˜EESij\nB=−1\n4/angbracketleftBigg/summationdisplay\nabeaeb\nrab/parenleftbigg\nni\nabnj\nab−1\n3δij/parenrightbigg/angbracketrightBigg\n. (20)\nNote that ( c−2−1) corresponds to the parameter δplotted in Figure 2.\nThe electrodynamics given by Eq. (17) can also be quantized, so tha t we may treat the\ninteraction of photons with atoms via perturbation theory. The en ergy of a photon is /planckover2pi1times its\nfrequencyω, while its momentum is /planckover2pi1ω/c. Using this approach, one finds that the difference in\nround trip travel times of light along the two arms of the interferom eter in the Michelson–Morley\nexperiment is given by L0(v2/c)(c−2−1). The experimental null result then leads to the bound\non (c−2−1) shown on Figure 2. Similarly the anisotropy in energy levels is clearly illu strated by\nthe tensorial terms in Eqs. (18, 20); by evaluating ˜EESij\nBfor each nucleus in the various Hughes–\nDrever-type experiments and comparing with the experimental limit s on energy differences, one\nobtains the extremely tight bounds also shown on Figure 2.\nThe behavior of moving atomic clocks can also be analyzed in detail, and bounds on ( c−2−1)\ncan be placed using results from tests of time dilation and of the prop agation of light. In some\ncases, it is advantageous to combine the c2frameworkwith a “kinematical” viewpoint that treats a\ngeneral class of boost transformations between moving frames. Such kinematical approaches have\nbeen discussed by Robertson, Mansouri and Sexl, and Will (see [38 6]).\nFor example, in the “JPL” experiment, in which the phases of two hyd rogen masers connected\nby a fiberoptic link were compared as a function of the Earth’s orient ation, the predicted phase\ndifference as a function of direction is, to first order in V, the velocity of the Earth through the\n20cosmic background,\n∆φ\n˜φ≈ −4\n3(1−c2)(V·n−V·n0), (21)\nwhere˜φ= 2πνL,νis the maser frequency, L= 21 km is the baseline, and where nandn0are\nunit vectors along the direction of propagation of the light at a given time and at the initial time\nof the experiment, respectively. The observed limit on a diurnal var iation in the relative phase\nresulted in the bound |c−2−1|<3×10−4. Tighter bounds were obtained from a “two-photon\nabsorption” (TPA) experiment, and a 1960s series of “M¨ ossbaue r-rotor”experiments, which tested\nthe isotropy of time dilation between a gamma ray emitter on the rim of a rotating disk and an\nabsorber placed at the center [386].\n2.2.4 The Standard Model Extension (SME)\nKosteleck´ y and collaborators developed a useful and elegant fra mework for discussing violations of\nLorentz symmetry in the context of the Standard Model of partic le physics [82, 83, 210]. Called\nthe Standard Model Extension (SME), it takes the standard SU(3 )×SU(2)×U(1) field theory of\nparticle physics, and modifies the terms in the action by inserting a va riety of tensorial quantities\nin the quark, lepton, Higgs, and gaugeboson sectorsthat could ex plicitly violate LLI. SME extends\nthe earlier classical THǫµandc2frameworks, and the χ−gframework of Ni [275] to quantum\nfield theory and particle physics. The modified terms split naturally int o those that are odd under\nCPT (i.e. that violate CPT) and terms that are even under CPT. The r esult is a rich and complex\nframework, with many parameters to be analyzed and tested by ex periment. Such details are\nbeyond the scope of this review; for a review of SME and other fram eworks, the reader is referred\nto the Living Review by Mattingly [250] or the review by Liberati [231]. The review of the SME\nby Kosteleck´ y and Russell [211] provides “data tables” showing ex perimental bounds on all the\nvarious parameters of the SME.\nHere we confine our attention to the electromagnetic sector, in or der to link the SME with the\nc2framework discussed above. In the SME, the Lagrangian for a sca lar particle φwith charge e\ninteracting with electrodynamics takes the form\nL= [ηµν+(kφ)µν](Dµφ)†Dνφ−m2φ†φ−1\n4/bracketleftbig\nηµαηνβ+(kF)µναβ/bracketrightbig\nFµνFαβ,(22)\nwhereDµφ=∂µφ+ieAµφ, where (kφ)µνis a real symmetric trace-free tensor, and where ( kF)µναβ\nis a tensor with the symmetries of the Riemann tensor, and with vanis hing double trace. It has 19\nindependent components. Therecould alsobe aCPT-oddterm in Lofthe form( kA)µǫµναβAνFαβ,\nbut because of a variety of pre-existing theoretical and experime ntal constraints, it is generally set\nto zero.\nThe tensor ( kF)µανβcan be decomposed into “electric”, “magnetic”, and “odd-parity” compo-\nnents, by defining\n(κDE)jk=−2(kF)0j0k,\n(κHB)jk=1\n2ǫjpqǫkrs(kF)pqrs,\n(κDB)kj=−(kHE)jk=ǫjpq(kF)0kpq.(23)\n21In many applications it is useful to use the further decomposition\n˜κtr=1\n3(κDE)jj,\n(˜κe+)jk=1\n2(κDE+κHB)jk,\n(˜κe−)jk=1\n2(κDE−κHB)jk−1\n3δjk(κDE)ii,\n(˜κo+)jk=1\n2(κDB+κHE)jk,\n(˜κo−)jk=1\n2(κDB−κHE)jk.(24)\nThe first expression is a single number, the next three are symmetr ic trace-free matrices, and the\nfinal is an antisymmetric matrix, accounting thereby for the 19 com ponents of the original tensor\n(kF)µανβ.\nIn the rest frame of the universe, these tensors have some form that is established by the global\nnature of the solutions of the overarching theory being used. In a frame that is moving relative to\nthe universe, the tensors will have components that depend on th e velocity of the frame, and on\nthe orientation of the frame relative to that velocity.\nIn the case wherethe theory is rotationallysymmetric in the prefer red frame, the tensors ( kφ)µν\nand (kF)µναβcan be expressed in the form\n(kφ)µν= ˜κφ/parenleftbigg\nuµuν+1\n4ηµν/parenrightbigg\n, (25)\n(kF)µναβ= ˜κtr/parenleftBig\n4u[µην][αuβ]−ηµ[αηβ]ν/parenrightBig\n, (26)\nwhere[ ]aroundindicesdenoteantisymmetrization, andwhere uµisthe four-velocityofanobserver\nat rest in the preferred frame. With this assumption, all the tenso rial quantities in Eq. (24) vanish\nin the preferred frame, and, after suitable rescalings of coordina tes and fields, the action (22) can\nbe put into the form of the c2framework, with\nc=/parenleftbigg1−3\n4˜κφ\n1+1\n4˜κφ/parenrightbigg1/2/parenleftbigg1−˜κtr\n1+ ˜κtr/parenrightbigg1/2\n. (27)\n2.3 EEP, particle physics, and the search for new interactio ns\nThus far, we have discussed EEP as a principle that strictly divides th e world into metric and non-\nmetric theories, and have implied that a failure of EEP might invalidate m etric theories (and thus\ngeneral relativity). On the other hand, there is mounting theoret ical evidence to suggest that EEP\nislikelyto be violated at some level, whether by quantum gravity effects, by effects arising from\nstring theory, or by hitherto undetected interactions. Roughly s peaking, in addition to the pure\nEinsteinian gravitational interaction, which respects EEP, theorie s such as string theory predict\nother interactions which do not. In string theory, for example, th e existence of such EEP-violating\nfields is assured, but the theory is not yet mature enough to enable a robust calculation of their\nstrength relative to gravity, or a determination of whether they a re long range, like gravity, or\nshort range, like the nuclear and weak interactions, and thus too s hort-range to be detectable.\nIn one simple example [116], one can write the Lagrangian for the low-e nergy limit of a string-\ninspired theory in the so-called “Einstein frame”, in which the gravita tional Lagrangian is purely\n22general relativistic:\n˜L=/radicalbig\n−˜g/parenleftbigg\n˜gµν/bracketleftbigg1\n2κ˜Rµν−1\n2˜G(ϕ)∂µϕ∂νϕ/bracketrightbigg\n−U(ϕ)˜gµν˜gαβFµαFνβ\n+˜ψ/bracketleftBig\ni˜eµ\naγa/parenleftBig\n∂µ+˜Ωµ+qAµ/parenrightBig\n−˜M(ϕ)/bracketrightBig\n˜ψ/parenrightbigg\n, (28)\nwhere ˜gµνisthenon-physicalmetric, ˜RµνistheRicci tensorderivedfromit, ϕisadilatonfield, and\n˜G,Uand˜Marefunctionsof ϕ. TheLagrangianincludesthatfortheelectromagneticfield Fµν, and\nthat for particles, written in terms of Dirac spinors ˜ψ. This is not a metric representation because\nof the coupling of ϕto matter via ˜M(ϕ) andU(ϕ). A conformal transformation ˜ gµν=F(ϕ)gµν,\n˜ψ=F(ϕ)−3/4ψ, puts the Lagrangian in the form (“Jordan” frame)\nL=√−g/parenleftbigg\ngµν/bracketleftbigg1\n2κF(ϕ)Rµν−1\n2F(ϕ)˜G(ϕ)∂µϕ∂νϕ+3\n4κF(ϕ)∂µF∂νF/bracketrightbigg\n−U(ϕ)gµνgαβFµαFνβ+ψ/bracketleftBig\nieµ\naγa(∂µ+Ωµ+qAµ)−˜M(ϕ)F1/2/bracketrightBig\nψ/parenrightbigg\n.(29)\nOne may choose F(ϕ) = const./˜M(ϕ)2so that the particle Lagrangian takes the metric form (no\nexplicit coupling to ϕ), but the electromagneticLagrangianwill still couple non-metrically toU(ϕ).\nThe gravitational Lagrangian here takes the form of a scalar-ten sor theory (see Section 3.3.2). But\nthe non-metric electromagnetic term will, in general, produce violatio ns of EEP. For examples of\nspecific models, see [354, 105]. Another class of non-metric theorie s are included in the “varying\nspeed of light (VSL)” theories; for a detailed review, see [245].\nOntheotherhand, whetheroneviewssucheffectsasaviolationofE EPoraseffectsarisingfrom\nadditional “matter” fields whose interactions, like those of the elec tromagnetic field, do not fully\nembody EEP, is to some degree a matter of semantics. Unlike the field s of the standard model of\nelectromagnetic, weak and strong interactions, which couple to pr operties other than mass-energy\nand are either short range or are strongly screened, the fields ins pired by string theory couldbe\nlong range (if they remain massless by virtue of a symmetry, or at be st, acquire a very small mass),\nandcancouple to mass-energy, and thus can mimic gravitational fields. Still, there appears to be\nno way to make this precise.\nAs a result, EEP and related tests are now viewed as ways to discove r or place constraints\non new physical interactions, or as a branch of “non-accelerator particle physics”, searching for\nthe possible imprints of high-energy particle effects in the low-energ y realm of gravity. Whether\ncurrent or proposed experiments can actually probe these pheno mena meaningfully is an open\nquestion at the moment, largely because of a dearth of firm theore tical predictions.\n2.3.1 The “fifth” force\nOn the phenomenological side, the idea of using EEP tests in this way m ay have originated in the\nmiddle 1980s, with the search for a “fifth” force. In 1986, as a res ult of a detailed reanalysis of\nE¨ otv¨ os’ original data, Fischbach et al. [141] suggested the exis tence of a fifth force of nature, with\na strength of about a percent that of gravity, but with a range (a s defined by the range λof a\nYukawa potential, e−r/λ/r) of a few hundred meters. This proposal dovetailed with earlier hint s\nof a deviation from the inverse-square law of Newtonian gravitation derived from measurements\nof the gravity profile down deep mines in Australia, and with emerging id eas from particle physics\nsuggesting the possible presence of very low-mass particles with gr avitational-strength couplings.\nDuring the next four years numerous experiments looked for evide nce of the fifth force by searching\nfor composition-dependent differences in acceleration, with varian ts of the E¨ otv¨ os experiment or\n23withfree-fallGalileo-typeexperiments. Althoughtwoearlyexperim entsreportedpositiveevidence,\nthe others allyielded null results. Overthe rangebetween one and 104meters, the null experiments\nproduced upper limits on the strength of a postulated fifth force b etween 10−3and 10−6of the\nstrengthofgravity. InterpretedastestsofWEP(correspond ingtothelimitofinfinite-rangeforces),\nthe results of two representative experiments from this period, t he free-fall Galileo experiment and\ntheearlyE¨ ot-Washexperiment, areshowninFigure1. Atthesame time, testsoftheinverse-square\nlaw of gravity were carried out by comparing variations in gravity mea surements up tall towers or\ndown mines or boreholes with gravity variations predicted using the in verse square law together\nwith Earth models and surface gravity data mathematically “continu ed” up the tower or down the\nhole. Despite early reports of anomalies, independent tower, bore hole, and seawater measurements\nultimately showed no evidence of a deviation. Analyses of orbital dat a from planetary range\nmeasurements, lunar laser ranging (LLR), and laser tracking of th e LAGEOS satellite verified\nthe inverse-square law to parts in 108over scales of 103to 105km, and to parts in 109over\nplanetaryscalesofseveralastronomicalunits[352]. Aconsensus emergedthattherewasnocredible\nexperimental evidence for a fifth force of nature, of a type and r ange proposed by Fischbach et al.\nFor reviews and bibliographies of this episode, see [140, 142, 143, 3, 385].\n2.3.2 Short-range modifications of Newtonian gravity\nAlthough the idea of an intermediate-range violation of Newton’s gra vitational law was dropped,\nnew ideas emerged to suggest the possibility that the inverse-squa re law could be violated at very\nshort ranges, below the centimeter range of existing laboratory v erifications of the 1 /r2behavior.\nOne set of ideas [16, 18, 304, 303] posited that some of the extra s patial dimensions that come with\nstring theory could extend over macroscopic scales, rather than being rolled up at the Planck scale\nof 10−33cm, which was then the conventional viewpoint. On laboratory dista nces large compared\nto the relevant scale of the extra dimension, gravity would fall off as the inverse square, whereas\non short scales, gravity would fall off as 1 /R2+n, wherenis the number of large extra dimensions.\nMany models favored n= 1 orn= 2. Other possibilities for effective modifications of gravity at\nshort range involved the exchange of light scalar particles.\nFollowing these proposals, many of the high-precision, low-noise met hods that were developed\nfor tests of WEP were adapted to carry out laboratory tests of t he inverse square law of Newto-\nnian gravitation at millimeter scales and below. The challenge of these e xperiments has been to\ndistinguish gravitation-like interactions from electromagnetic and q uantum mechanical (Casimir)\neffects. No deviations from the inverse square law have been found to date at distances between\ntens of nanometers and 10 mm [237, 177, 176, 69, 236, 193, 4, 360 , 157, 351, 39, 411, 200]. For a\ncomprehensive review of both the theory and the experiments circ a 2002, see [2].\n2.3.3 The Pioneer anomaly\nIn 1998, Anderson et al. [14] reported the presence of an anoma lous deceleration in the motion of\nthe Pioneer 10 and 11 spacecraft at distances between 20 and 70 a stronomical units from the Sun.\nAlthough the anomaly was the result of a rigorous analysis of Doppler data taken over many years,\nit might havebeen dismissed as havingno real significancefor new phy sics, where it not for the fact\nthat the acceleration, of order 10−9m/s2, when divided by the speed of light, was strangely close\nto the inverse of the Hubble time. The Pioneer anomaly prompted an o utpouring of hundreds of\npapers, most attempting to explain it via modifications of gravity or v ia new physical interactions,\nwith a small subset trying to explain it by conventional means.\nSoon after the publication of the initial Pioneer anomaly paper [14], Ka tz pointed out that\nthe anomaly could be accounted for as the result of the anisotropic emission of radiation from\nthe radioactive thermal generators (RTG) that continued to pow er the spacecraft decades after\ntheir launch [194]. At the time, there was insufficient data on the per formance of the RTG over\n24time or on the thermal characteristics of the spacecraft to just ify more than an order-of-magnitude\nestimate. However, the recoveryof an extended set of Doppler d ata covering a longer stretch of the\norbits of both spacecraft, together with the fortuitous discove ry of project documentation and of\ntelemetrydatagivingon-boardtemperatureinformation,madeitp ossiblebothtoimprovetheorbit\nanalysis and to develop detailed thermal models of the spacecraft in order to quantify the effect\nof thermal emission anisotropies. Several independent analyses n ow confirm that the anomaly is\nalmost entirely due to the recoil of the spacecraft from the anisot ropic emission of residual thermal\nradiation [312, 366, 267]. For a thorough review of the Pioneer anom aly published just as the new\nanalyses were underway, see the Living Review by Turyshev and Toth [365].\n253 Metric Theories of Gravity and the PPN Formalism\n3.1 Metric theories of gravity and the strong equivalence pr inciple\n3.1.1 Universal coupling and the metric postulates\nThe empirical evidence supporting the Einstein equivalence principle, discussed in Section 2, sup-\nports the conclusion that the only theories of gravity that have a h ope of being viable are metric\ntheories, or possibly theories that are metric apart from very wea k or short-range non-metric cou-\nplings (as in string theory). Therefore for the remainder of this re view, we shall turn our attention\nexclusively to metric theories of gravity, which assume that\n1. there exists a symmetric metric,\n2. test bodies follow geodesics of the metric, and\n3. in local Lorentz frames, the non-gravitational laws of physics a re those of special relativity.\nThe property that all non-gravitational fields should couple in the s ame manner to a single\ngravitational field is sometimes called “universal coupling”. Because of it, one can discuss the\nmetric as a property of spacetime itself rather than as a field over s pacetime. This is because its\npropertiesmaybemeasuredandstudied usingavarietyofdifferent experimentaldevices, composed\nof different non-gravitational fields and particles, and, because o f universal coupling, the results\nwill be independent of the device. Thus, for instance, the proper t ime between two events is a\ncharacteristic of spacetime and of the location of the events, not of the clocks used to measure it.\nConsequently, ifEEPisvalid, thenon-gravitationallawsofphysicsm aybeformulatedbytaking\ntheir special relativistic forms in terms of the Minkowski metric ηand simply “going over” to new\nforms in terms of the curved spacetime metric g, using the mathematics of differential geometry.\nThe details of this “going over” can be found in standard textbooks (see [265, 377, 327, 297],\nTEGP 3.2. [388]).\n3.1.2 The strong equivalence principle\nIn any metric theory of gravity, matter and non-gravitational fie lds respond only to the spacetime\nmetricg. In principle, however, there could exist other gravitational fields besides the metric, such\nas scalar fields, vector fields, and so on. If, by our strict definition of metric theory, matter does\nnot couple to these fields, what can their role in gravitation theory b e? Their role must be that\nof mediating the manner in which matter and non-gravitational fields generate gravitational fields\nand produce the metric; once determined, however, the metric alo ne acts back on the matter in\nthe manner prescribed by EEP.\nWhat distinguishes one metric theory from another, therefore, is the number and kind of\ngravitational fields it contains in addition to the metric, and the equa tions that determine the\nstructure and evolution of these fields. From this viewpoint, one ca n divide all metric theories of\ngravity into two fundamental classes: “purely dynamical” and “prio r-geometric”.\nBy “purely dynamical metric theory” we mean any metric theory who se gravitational fields\nhave their structure and evolution determined by coupled partial d ifferential field equations. In\nother words, the behavior of each field is influenced to some extent by a coupling to at least one\nof the other fields in the theory. By “prior geometric” theory, we m ean any metric theory that\ncontains “absolute elements”, fields or equations whose structur e and evolution are given a priori,\nandareindependent ofthestructureandevolutionofthe otherfi eldsofthe theory. These“absolute\nelements” typically include flat background metrics ηor cosmic time coordinates t.\nGeneral relativity is a purely dynamical theory since it contains only o ne gravitational field,\nthe metric itself, and its structure and evolution are governed by p artial differential equations\n26(Einstein’s equations). Brans–Dicke theory and its generalizations are purely dynamical theories;\nthe field equation for the metric involves the scalar field (as well as th e matter as source), and\nthat for the scalar field involves the metric. Visser’s bimetric massive gravity theory [371] is a\nprior-geometric theory: It has a non-dynamical, Riemann-flat bac kground metric η, and the field\nequations for the physical metric ginvolveη.\nBydiscussingmetrictheoriesofgravityfromthisbroadpointofview , itispossibletodrawsome\ngeneral conclusions about the nature of gravity in different metric theories, conclusions that are\nreminiscent of the Einstein equivalence principle, but that are subsu med under the name “strong\nequivalence principle”.\nConsider a local, freely falling frame in any metric theory of gravity. L et this frame be small\nenough that inhomogeneities in the external gravitational fields ca n be neglected throughout its\nvolume. On the other hand, let the frame be large enough to encomp ass a system of gravitating\nmatter and its associated gravitational fields. The system could be a star, a black hole, the solar\nsystem, or a Cavendish experiment. Call this frame a “quasi-local L orentz frame”. To determine\nthe behavior of the system we must calculate the metric. The compu tation proceeds in two stages.\nFirst we determine the external behaviorofthe metric and gravita tionalfields, thereby establishing\nboundary values for the fields generated by the local system, at a boundary of the quasi-local frame\n“far” from the local system. Second, we solve for the fields gener ated by the local system. But\nbecause the metric is coupled directly or indirectly to the other fields of the theory, its structure\nand evolution will be influenced by those fields, and in particular by the boundary values taken on\nby those fields far from the local system. This will be true even if we w orkin a coordinate system in\nwhich the asymptotic form of gµνin the boundary regionbetween the local system and the external\nworld is that of the Minkowski metric. Thus the gravitational enviro nment in which the local\ngravitating system resides can influence the metric generated by t he local system via the boundary\nvalues of the auxiliary fields. Consequently, the results of local gra vitational experiments may\ndepend on the location and velocity of the frame relative to the exte rnal environment. Of course,\nlocalnon-gravitational experiments are unaffected since the gravitationa l fields they generate are\nassumed to be negligible, and since those experiments couple only to t he metric, whose form can\nalways be made locally Minkowskian at a given spacetime event. Local g ravitational experiments\nmight include Cavendish experiments, measurement of the accelera tion of massive self-gravitating\nbodies, studies of the structure of stars and planets, or analyse s of the periods of “gravitational\nclocks”. We can now make several statements about different kind s of metric theories.\n•A theory which contains only the metric gyields local gravitational physics which is inde-\npendent of the location and velocity of the local system. This follows from the fact that\nthe only field coupling the local system to the environment is g, and it is always possible\nto find a coordinate system in which gtakes the Minkowski form at the boundary between\nthe local system and the external environment (neglecting inhomo geneities in the external\ngravitational field). Thus the asymptotic values of gµνare constants independent of location,\nand are asymptotically Lorentz invariant, thus independent of velo city. GR is an example of\nsuch a theory.\n•A theorywhich containsthe metric gand dynamicalscalarfields ϕAyields local gravitational\nphysics which may depend on the location of the frame but which is inde pendent of the\nvelocity of the frame. This follows from the asymptotic Lorentz inva riance of the Minkowski\nmetric and of the scalar fields, but now the asymptotic values of the scalar fields may depend\non the location of the frame. An example is Brans–Dicke theory, whe re the asymptotic scalar\nfield determines the effective value of the gravitational constant, which can thus vary as ϕ\nvaries. On the other hand, a form of velocity dependence in local ph ysics can enter indirectly\nif the asymptotic values of the scalar field vary with time cosmologically . Then the rateof\nvariation of the gravitational constant could depend on the velocit y of the frame.\n27•A theory which contains the metric gand additional dynamical vector or tensor fields or\nprior-geometric fields yields local gravitational physics which may ha ve both location and\nvelocity-dependent effects.\nThese ideas can be summarized in the strong equivalence principle (SE P), which states that:\n1. WEP is valid for self-gravitating bodies as well as for test bodies.\n2. The outcome of any local test experiment is independent of the v elocity of the (freely falling)\napparatus.\n3. The outcome of any local test experiment is independent of wher e and when in the universe\nit is performed.\nThe distinction between SEP and EEPis the inclusion ofbodies with self- gravitationalinteractions\n(planets, stars) and ofexperiments involvinggravitationalforce s (Cavendishexperiments, gravime-\nter measurements). Note that SEP contains EEP as the special ca se in which local gravitational\nforces are ignored. For further discussion of SEP and EEP, see [11 5].\nThe above discussion of the coupling of auxiliary fields to local gravita ting systems indicates\nthat if SEP is strictly valid, there must be one and only one gravitation al field in the universe, the\nmetricg. These arguments are only suggestive however, and no rigorous p roof of this statement is\navailable at present. Empirically it has been found that almost every m etric theory other than GR\nintroduces auxiliary gravitational fields, either dynamical or prior g eometric, and thus predicts\nviolations of SEP at some level (here we ignore quantum-theory insp ired modifications to GR\ninvolving “R2” terms). The one exception is Nordstr¨ om’s 1913conformally-flat scalar theory [278],\nwhich can be written purely in terms of the metric; the theory satisfi es SEP, but unfortunately\nviolates experiment by predicting no deflection of light. General rela tivity seems to be the only\nviable metric theory that embodies SEP completely. In Section 4.3, we shall discuss experimental\nevidence for the validity of SEP.\n3.2 The parametrized post-Newtonian formalism\nDespite the possible existence of long-range gravitational fields in a ddition to the metric in var-\nious metric theories of gravity, the postulates of those theories d emand that matter and non-\ngravitational fields be completely oblivious to them. The only gravitat ional field that enters the\nequations of motion is the metric g. The role of the other fields that a theory may contain can\nonly be that of helping to generate the spacetime curvature assoc iated with the metric. Matter\nmay create these fields, and they plus the matter may generate th e metric, but they cannot act\nback directly on the matter. Matter responds only to the metric.\nThus the metric and the equations of motion for matter become the primary entities for calcu-\nlating observable effects, and all that distinguishes one metric theo ry from another is the particular\nway in which matter and possibly other gravitational fields generate the metric.\nThe comparison of metric theories of gravity with each other and wit h experiment becomes\nparticularly simple when one takes the slow-motion, weak-field limit. Th is approximation, known\nas the post-Newtonian limit, is sufficiently accurate to encompass mo st solar-system tests that\ncan be performed in the foreseeable future. It turns out that, in this limit, the spacetime metric\ngpredicted by nearly every metric theory of gravity has the same st ructure. It can be written\nas an expansion about the Minkowski metric ( ηµν= diag(−1,1,1,1)) in terms of dimensionless\ngravitational potentials of varying degrees of smallness. These po tentials are constructed from the\nmatter variables (see Box 2) in imitation of the Newtonian gravitation al potential\nU(x,t)≡/integraldisplay\nρ(x′,t)|x−x′|−1d3x′. (30)\n28The “order of smallness” is determined according to the rules U∼v2∼Π∼p/ρ∼ǫ,vi∼\n|d/dt|/|d/dx| ∼ǫ1/2, and so on (we use units in which G=c= 1; see Box 2 for definitions and\nconventions).\nA consistent post-Newtonian limit requires determination of g00correct through O(ǫ2),g0i\nthrough O(ǫ3/2), andgijthrough O(ǫ) (for details see TEGP 4.1 [388]). The only way that one\nmetric theory differs from another is in the numerical values of the c oefficients that appear in front\nof the metric potentials. The parametrized post-Newtonian (PPN) formalism inserts parameters\nin place of these coefficients, parameters whose values depend on t he theory under study. In the\ncurrent version of the PPN formalism, summarized in Box 2, ten para meters are used, chosen\nin such a manner that they measure or indicate general properties of metric theories of gravity\n(see Table 2). Under reasonable assumptions about the kinds of po tentials that can be present at\npost-Newtonian order (basically only Poisson-like potentials), one fi nds that ten PPN parameters\nexhaust the possibilities.\nTable2: The PPNParametersand theirsignificance(note that α3hasbeen showntwice toindicate\nthat it is a measure of two effects).\nParameter What it measures relative\nto GRValue\nin GRValue in semi-\nconservative\ntheoriesValue in fully\nconservative\ntheories\nγ How much space-curva-\nture produced by unit rest\nmass?1 γ γ\nβ How much “nonlinearity”\nin the superposition law\nfor gravity?1 β β\nξ Preferred-location effects? 0 ξ ξ\nα1 Preferred-frame effects? 0 α1 0\nα2 0 α2 0\nα3 0 0 0\nα3 Violation of conservation 0 0 0\nζ1 of total momentum? 0 0 0\nζ2 0 0 0\nζ3 0 0 0\nζ4 0 0 0\nThe parameters γandβare the usual Eddington–Robertson–Schiffparametersused to d escribe\nthe “classical” tests of GR, and are in some sense the most importan t; they are the only non-\nzero parameters in GR and scalar-tensor gravity. The parameter ξis non-zero in any theory of\ngravity that predicts preferred-location effects such as a galaxy -induced anisotropy in the local\ngravitational constant GL(also called “Whitehead” effects); α1,α2,α3measure whether or not\nthe theory predicts post-Newtonian preferred-frame effects; α3,ζ1,ζ2,ζ3,ζ4measure whether or\nnot the theory predicts violations of global conservation laws for t otal momentum. In Table 2 we\nshow the values these parameters take\n1. in GR,\n2. in any theory of gravity that possesses conservation laws for t otal momentum, called “semi-\nconservative” (any theory that is based on an invariant action prin ciple is semi-conservative),\n29and\n3. in any theory that in addition possesses six global conservation la ws for angular momentum,\ncalled “fully conservative” (such theories automatically predict no p ost-Newtonian preferred-\nframe effects).\nSemi-conservative theories have five free PPN parameters ( γ,β,ξ,α1,α2) while fully conservative\ntheories have three ( γ,β,ξ).\nThePPNformalismwaspioneeredbyKennethNordtvedt[280], whost udiedthepost-Newtonian\nmetric of a system of gravitating point masses, extending earlier wo rk by Eddington, Robertson\nand Schiff (TEGP 4.2 [388]). Will [382] generalized the framework to per fect fluids. A general and\nunified version of the PPN formalism was developed by Will and Nordtve dt [397]. The canonical\nversion, with conventions altered to be more in accord with standar d textbooks such as [265], is\ndiscussed in detail in TEGP 4 [388]. Other versions of the PPN formalism have been developed\nto deal with point masses with charge, fluid with anisotropic stresse s, bodies with strong internal\ngravity, and post-post-Newtonian effects (TEGP 4.2, 14.2 [388]).\nBox 2. The Parametrized Post-Newtonian formalism\nCoordinate system:\nThe frameworkuses a nearly globally Lorentz coordinate system in w hich the coordinates are\n(t,x1,x2,x3). Three-dimensional, Euclidean vector notation is used throughou t. All coordi-\nnate arbitrariness (“gauge freedom”) has been removed by spec ialization of the coordinates\nto the standard PPN gauge (TEGP 4.2 [388]). Units are chosen so th atG=c= 1, where G\nis the physically measured Newtonian constant far from the solar sy stem.\nMatter variables:\n•ρ: density of rest mass as measured in a local freely falling frame mome ntarily comoving\nwith the gravitating matter.\n•vi= (dxi/dt): coordinate velocity of the matter.\n•wi: coordinate velocity of the PPN coordinate system relative to the m ean rest-frame\nof the universe.\n•p: pressure as measured in a local freely falling frame momentarily com oving with the\nmatter.\n•Π: internal energy per unit rest mass (it includes all forms of non-r est-mass, non-\ngravitational energy, e.g., energy of compression and thermal en ergy).\nPPN parameters:\nγ,β,ξ,α1,α2,α3,ζ1,ζ2,ζ3,ζ4.\nMetric:\n30g00=−1+2U−2βU2−2ξΦW+(2γ+2+α3+ζ1−2ξ)Φ1+2(3γ−2β+1+ζ2+ξ)Φ2\n+2(1+ζ3)Φ3+2(3γ+3ζ4−2ξ)Φ4−(ζ1−2ξ)A−(α1−α2−α3)w2U−α2wiwjUij\n+(2α3−α1)wiVi+O(ǫ3),\ng0i=−1\n2(4γ+3+α1−α2+ζ1−2ξ)Vi−1\n2(1+α2−ζ1+2ξ)Wi−1\n2(α1−2α2)wiU\n−α2wjUij+O(ǫ5/2),\ngij= (1+2γU)δij+O(ǫ2).\nMetric potentials:\nU=/integraldisplayρ′\n|x−x′|d3x′,\nUij=/integraldisplayρ′(x−x′)i(x−x′)j\n|x−x′|3d3x′,\nΦW=/integraldisplayρ′ρ′′(x−x′)\n|x−x′|3·/parenleftbiggx′−x′′\n|x−x′′|−x−x′′\n|x′−x′′|/parenrightbigg\nd3x′d3x′′,\nA=/integraldisplayρ′[v′·(x−x′)]2\n|x−x′|3d3x′,\nΦ1=/integraldisplayρ′v′2\n|x−x′|d3x′,\nΦ2=/integraldisplayρ′U′\n|x−x′|d3x′,\nΦ3=/integraldisplayρ′Π′\n|x−x′|d3x′,\nΦ4=/integraldisplayp′\n|x−x′|d3x′,\nVi=/integraldisplayρ′v′\ni\n|x−x′|d3x′,\nWi=/integraldisplayρ′[v′·(x−x′)](x−x′)i\n|x−x′|3d3x′.\nStress–energy tensor (perfect fluid):\nT00=ρ(1+Π+v2+2U),\nT0i=ρvi/parenleftbigg\n1+Π+v2+2U+p\nρ/parenrightbigg\n,\nTij=ρvivj/parenleftbigg\n1+Π+v2+2U+p\nρ/parenrightbigg\n+pδij(1−2γU).\nEquations of motion:\n31•Stressed matter: Tµν;ν= 0.\n•Test bodies:d2xµ\ndλ2+Γµ\nνλdxν\ndλdxλ\ndλ= 0.\n•Maxwell’s equations: Fµν;ν= 4πJµ, F µν=Aν;µ−Aµ;ν.\n323.3 Competing theories of gravity\nOne of the important applications of the PPN formalism is the comparis on and classification of\nalternative metric theories of gravity. The population of viable theo ries has fluctuated over the\nyears as new effects and tests have been discovered, largely thro ugh the use of the PPN framework,\nwhich eliminated many theories thought previously to be viable. The th eory population has also\nfluctuated as new, potentially viable theories have been invented.\nIn this review, we shall focus on GR, the general class of scalar-te nsor modifications of it, of\nwhich the Jordan–Fierz–Brans–Dicke theory (Brans–Dicke, for s hort) is the classic example, and\nvector-tensor theories. The reasons are several-fold:\n•A full compendium of alternative theories circa 1981 is given in TEGP 5 [3 88].\n•Many alternative metric theories developed during the 1970s and 19 80s could be viewed as\n“straw-man” theories, invented to prove that such theories exis t or to illustrate particular\nproperties. Few of these could be regarded as well-motivated theo ries from the point of view,\nsay, of field theory or particle physics.\n•A number of theories fall into the class of “prior-geometric” theor ies, with absolute elements\nsuch as a flat background metric in addition to the physical metric. M ost of these theories\npredict “preferred-frame” effects, that have been tightly cons trained by observations (see\nSection 4.3.2). An example is Rosen’s bimetric theory.\n•A large number of alternative theories of gravity predict gravitatio nal wave emission sub-\nstantially different from that of GR, in strong disagreement with obs ervations of the binary\npulsar (see Section 9).\n•Scalar-tensor modifications of GR have become very popular in unific ation schemes such as\nstring theory, and in cosmologicalmodel building. Because the scala r fields could be massive,\nthe potentials in the post-Newtonian limit could be modified by Yukawa- like terms.\n•Theoriesthat alsoincorporatevectorfields haveattractedrece ntattention, in the spiritofthe\nSME (see Section 2.2.4), as models for violations of Lorentz invarianc e in the gravitational\nsector, and as potential candidates to account for phenomena s uch as galaxy rotation curves\nwithout resorting to dark matter.\n3.3.1 General relativity\nThe metric gis the sole dynamical field, and the theory contains no arbitrary fun ctions or parame-\nters, apart from the value of the Newtonian coupling constant G, which is measurablein laboratory\nexperiments. Throughout this article, we ignore the cosmological c onstant Λ C. We do this despite\nrecent evidence, from supernova data, of an accelerating univer se, which would indicate either a\nnon-zero cosmological constant or a dynamical “dark energy” co ntributing about 70 percent of\nthe critical density. Although Λ Chas significance for quantum field theory, quantum gravity, and\ncosmology, on the scale of the solar-system or of stellar systems it s effects are negligible, for the\nvalues of Λ Cinferred from supernova observations.\nThe field equations of GR are derivable from an invariant action princip leδI= 0, where\nI= (16πG)−1/integraldisplay\nR(−g)1/2d4x+Im(ψm,gµν), (31)\nwhereRis the Ricci scalar, and Imis the matter action, which depends on matter fields ψm\nuniversally coupled to the metric g. By varying the action with respect to gµν, we obtain the field\n33Table 3: Metric theories and their PPN parameter values ( α3=ζi= 0 for all cases). The\nparameters γ′,β′,α′\n1, andα′\n2denotecomplicatedfunctions ofthearbitraryconstantsand mat ching\nparameters.\nTheory Arbitrary Cosmic PPN parameters\nfunctions matching\nor constants parameters γ β ξ α 1α2\nGeneral relativity none none 1 1 0 0 0\nScalar-tensor\nBrans–Dicke ωBD φ01+ωBD\n2+ωBD1 0 0 0\nGeneral,f(R)A(ϕ),V(ϕ)ϕ01+ω\n2+ω1+λ\n4+2ω0 0 0\nVector-tensor\nUnconstrained ω,c1,c2,c3,c4u γ′β′0α′\n1α′\n2\nEinstein-Æther c1,c2,c3,c4 none 1 1 0 α′\n1α′\n2\nTensor-Vector-Scalar k,c1,c2,c3,c4φ0 1 1 0 α′\n1α′\n2\nequations\nGµν≡Rµν−1\n2gµνR= 8πGTµν, (32)\nwhereTµνis the matter energy-momentum tensor. General covariance of t he matter action implies\nthe equations of motion Tµν;ν= 0; varying Imwith respect to ψmyields the matter field equations\nof the Standard Model. By virtue of the absence of prior-geometric elements, the equations of\nmotionarealsoaconsequenceofthe field equationsviathe Bianchiid entitiesGµν;ν= 0. According\nto our choice of units, we set G= 1.\nThe general procedure for deriving the post-Newtonian limit of met ric theories is spelled out\nin TEGP 5.1 [388], and is described in detail for GR in TEGP 5.2 [388]. (see a lso Chapters 6 – 8\nof [297]). The PPN parameter values are listed in Table 3.\n3.3.2 Scalar-tensor theories\nThese theories contain the metric g, a scalar field ϕ, a potential function V(ϕ), and a coupling\nfunctionA(ϕ) (generalizations to more than one scalar field have also been carrie d out [92]).\nFor some purposes, the action is conveniently written in a non-metr ic representation, sometimes\ndenoted the “Einstein frame”, in which the gravitational action look s exactly like that of GR and\nthe scalar action looks like a minimally coupled scalar field with a potential:\n˜I= (16πG)−1/integraldisplay/bracketleftBig\n˜R−2˜gµν∂µϕ∂νϕ−V(ϕ)/bracketrightBig\n(−˜g)1/2d4x+Im/parenleftbig\nψm,A2(ϕ)˜gµν/parenrightbig\n,(33)\nwhere˜R≡˜gµν˜Rµνis the Ricci scalarofthe “Einstein”metric ˜ gµν. (Apart from the scalarpotential\ntermV(ϕ), this corresponds to Eq. (28) with ˜G(ϕ)≡(4πG)−1,U(ϕ)≡1, and˜M(ϕ)∝A(ϕ).)\nThis representation is a “non-metric” one because the matter field sψmcouple to a combination\n34ofϕand ˜gµν. Despite appearances, however, it is a metric theory, because it c an be put into a\nmetric representation by identifying the “physical metric”\ngµν≡A2(ϕ)˜gµν. (34)\nThe action can then be rewritten in the metric form\nI= (16πG)−1/integraldisplay/bracketleftbig\nφR−φ−1ω(φ)gµν∂µφ∂νφ−φ2V/bracketrightbig\n(−g)1/2d4x+Im(ψm,gµν),(35)\nwhere\nφ≡A(ϕ)−2,\n3+2ω(φ)≡α(ϕ)−2,\nα(ϕ)≡d(lnA(ϕ))\ndϕ.(36)\nThe Einstein frame is useful for discussing general characteristic s of such theories, and for some\ncosmological applications, while the metric representation is most us eful for calculating observ-\nable effects. The field equations, post-Newtonian limit and PPN param eters are discussed in\nTEGP 5.3 [388], and the values of the PPN parameters are listed in Table 3.\nThe parameters that enter the post-Newtonian limit are\nω≡ω(φ0), λ≡/bracketleftbiggφdω/dφ\n(3+2ω)(4+2ω)/bracketrightbigg\nφ0, (37)\nwhereφ0is the value of φtoday far from the system being studied, as determined by approp riate\ncosmological boundary conditions. The Newtonian gravitational co nstantGN, which is set equal\nto unity by our choice of units, is related to the coupling constant G,φ0andωby\nGN≡1 =G\nφ0/parenleftbigg4+2ω\n3+2ω/parenrightbigg\n0. (38)\nIn Brans–Dicke theory ( ω(φ)≡ωBD= const.), the larger the value of ωBD, the smaller the effects\nof the scalar field, and in the limit ωBD→ ∞(α0→0), the theory becomes indistinguishable from\nGR in all its predictions. In more general theories, the function ω(φ) could have the property that,\nat the present epoch, and in weak-field situations, the value of the scalar field φ0is such that ωis\nvery large and λis very small (theory almost identical to GR today), but that for pa st or future\nvalues ofφ, or in strong-field regions such as the interiors of neutron stars, ωandλcould take on\nvalues that would lead to significant differences from GR. It is useful to point out that all versions\nof scalar-tensor gravity predict that γ≤1 (see Table 3).\nDamour and Esposito-Far` ese [92] have adopted an alternative pa rametrization of scalar-tensor\ntheories, in which one expands ln A(ϕ) about a cosmological background field value ϕ0:\nlnA(ϕ) =α0(ϕ−ϕ0)+1\n2β0(ϕ−ϕ0)2+... (39)\nA precisely linear coupling function produces Brans–Dicke theory, w ithα2\n0= 1/(2ωBD+ 3), or\n1/(2+ωBD) = 2α2\n0/(1+α2\n0). The function ln A(ϕ) acts as a potential for the scalar field ϕwithin\nmatter, and, if β0>0, then during cosmologicalevolution, the scalar field naturally evolv es toward\nthe minimum of the potential, i.e. toward α≈0,ω→ ∞, or toward a theory close to, though not\nprecisely GR [100, 101]. Estimates of the expected relic deviations fr om GR today in such theories\ndepend on the cosmological model, but range from 10−5to a few times 10−7for|γ−1|.\nNegative values of β0correspond to a “locally unstable” scalar potential (the overall th eory is\nstill stable in the sense of having no tachyons or ghosts). In this ca se, objects such as neutron stars\n35can experience a “spontaneous scalarization”, whereby the inter ior values of ϕcan take on values\nvery different from the exterior values, through non-linear intera ctions between strong gravity and\nthe scalar field, dramatically affecting the stars’ internal structu re and leading to strong violations\nof SEP. On the other hand, in the case β0<0, one must confront that fact that, with an unstable\nϕpotential, cosmological evolution would presumably drive the system away from the peak where\nα≈0, toward parameter values that could be excluded by solar system experiments.\nScalar fields coupled to gravityor matter are also ubiquitous in partic le-physics-inspiredmodels\nofunification,suchasstringtheory[354,243,105,102,103]. Ins omemodels, thecouplingtomatter\nmay lead to violations of EEP, which could be tested or bounded by the experiments described in\nSection 2.1. In many models the scalar field could be massive; if the Com pton wavelength is of\nmacroscopic scale, its effects are those of a “fifth force”. Only if t he theory can be cast as a metric\ntheory with a scalar field of infinite range or of range long compared t o the scale of the system in\nquestion (solar system) can the PPN framework be strictly applied. If the mass of the scalar field\nis sufficiently large that its range is microscopic, then, on solar-syst em scales, the scalar field is\nsuppressed, and the theory is essentially equivalent to general re lativity.\nFor a detailed review of scalar-tensor theories see [152].\n3.3.3f(R)theories\nThese are theories whose action has the form\nI=c3\n16πG/integraldisplay\nf(R)(−g)1/2d4x+Im(ψm,gµν), (40)\nwherefis a function chosen so that at cosmological scales, the universe will experience accelerated\nexpansionwithout resortingto either a cosmologicalconstant ord ark energy. However, it turns out\nthat such theories are equivalent to scalar-tensor theories: rep lacef(R) byf(χ)−f,χ(χ)(R−χ),\nwhereχis a dynamical field. Varying the action with respect to χyieldsf,χχ(R−χ) = 0, which\nimplies that χ=Ras long asf,χχ∝ne}ationslash= 0. Then defining a scalar field φ≡f,χ(χ) one puts the action\nintotheformofascalar-tensortheorygivenbyEq.(35), with ω(φ) = 0andφ2V=φχ(φ)−f(χ(φ)).\nAs we will see, this value of ωwould ordinarily strongly violate solar-system experiments, but it\nturns out that in many models, the potential V(φ) has the effect of giving the scalar field a large\neffective mass in the presence of matter (the so-called “chameleon mechanism” [198]), so that the\nscalar field is suppressed at distances that extend outside bodies lik e the Sun and Earth. In this\nway, with only modest fine tuning, f(R) theories can claim to obey standard tests, while providing\ninteresting, non general-relativistic behavior on cosmic scales. For detailed reviews of this class of\ntheories, see [341] and [110].\n3.3.4 Vector-tensor theories\nThese theories contain the metric gand a dynamical, typically timelike, four-vector field uµ. In\nsome models, the four-vector is unconstrained, while in others, ca lled Einstein-Æther theories it\nis constrained to be timelike with unit norm. The most general action f or such theories that is\nquadratic in derivatives of the vector is given by\nI= (16πG)−1/integraldisplay/bracketleftBig\n(1+ωuµuµ)R−Kµν\nαβ∇µuα∇νuβ+λ(uµuµ+1)/bracketrightBig\n(−g)1/2d4x+Im(ψm,gµν),\n(41)\nwhere\nKµν\nαβ=c1gµνgαβ+c2δµ\nαδν\nβ+c3δµ\nβδν\nα−c4uµuνgαβ. (42)\nThe coefficients ciare arbitrary. In the unconstrained theories, λ≡0 andωis arbitrary. In the\nconstrained theories, λis a Lagrange multiplier, and by virtue of the constraint uµuµ=−1, the\n36factorωuµuµin front of the Ricci scalar can be absorbed into a rescaling of G; equivalently, in the\nconstrained theories, we can set ω= 0. Note that the possible term uµuνRµνcan be shown under\nintegration by parts to be equivalent to a linear combination of the te rms involving c2andc3.\nUnconstrained theories were studied during the 1970s as “straw- man” alternatives to GR. In\naddition to having up to four arbitrary parameters, they also left t he magnitude of the vector field\narbitrary, since it satisfies a linear homogenous vacuum field equatio n of the form Luµ= 0 (c4= 0\nin all such cases studied). Indeed, this latter fact was one of most serious defects of these theories.\nEach unconstrained theory studied corresponds to a special cas e of the action (41), all with λ≡0:\nGeneral vector-tensor theory; ω,τ,ǫ,η(see TEGP 5.4 [388])\nThegravitationalLagrangianforthisclassoftheorieshadthefor mR+ωuµuµR+ηuµuνRµν−\nǫFµνFµν+τ∇µuν∇µuν, whereFµν=∇µuν−∇νuµ, corresponding to the values c1= 2ǫ−τ,\nc2=−η,c1+c2+c3=−τ,c4= 0. In these theories γ,β,α1, andα2are complicated\nfunctions of the parameters and of u2=−uµuµ, while the rest vanish.\nWill–Nordtvedt theory (see [397])\nThis is the special case c1=−1,c2=c3=c4= 0. In this theory, the PPN parameters are\ngiven byγ=β= 1,α2=u2/(1+u2/2), and zero for the rest.\nHellings–Nordtvedt theory; ω(see [172])\nThis is the special case c1= 2,c2= 2ω,c1+c2+c3= 0 =c4. Hereγ,β,α1andα2are\ncomplicated functions of the parameters and of u2, while the rest vanish.\nEinstein-Æther theory; c1, c2, c3, c4\nThe Einstein-Æther theories were motivated in part by a desire to ex plore possibilities for\nviolations of Lorentz invariance in gravity, in parallel with similar studie s in matter interac-\ntions, such as the SME. The general class of theories was analyzed by Jacobson and collab-\norators [186, 251, 187, 132, 148], motivated in part by [212]. Analyz ing the post-Newtonian\nlimit, they were able to infer values of the PPN parameters γandβas follows [148]:\nγ= 1, (43)\nβ= 1, (44)\nξ=α3=ζ1=ζ2=ζ3=ζ4= 0, (45)\nα1=−8(c2\n3+c1c4)\n2c1−c2\n1+c2\n3, (46)\nα2=−4(c2\n3+c1c4)\n2c1−c2\n1+c2\n3−(2c13−c14)(c13+c14+3c2)\nc123(2−c14), (47)\nwherec123=c1+c2+c3,c13=c1+c3,c14=c1+c4, subject to the constraints c123∝ne}ationslash= 0,\nc14∝ne}ationslash= 2, 2c1−c2\n1+c2\n3∝ne}ationslash= 0. By requiring that gravitational wave modes have real (as\nopposed to imaginary) frequencies, one can impose the bounds c1/c14≥0 andc123/c14≥0.\nConsiderations of positivity of energy impose the constraints c1>0,c14>0 andc123>0.\n3.3.5 Tensor–vector–scalar (TeVeS) theories\nThis class of theories was invented to provide a fully relativistic theor y of gravity that could mimic\nthe phenomenological behavior of so-called Modified Newtonian Dyna mics (MOND), whereby in\na weak-field regime, Newton’s laws hold, namely a=Gm/r2wheremis the mass of a central\nobject, as long as ais large compared to some fundamental scale a0, but in a regime where\na<a0, the equations of motion would take the form a2/a0=Gm/r2[259]. With such a behavior,\nthe rotational velocity of a particle far from a central mass would h ave the form v∼√ar∼\n37(Gma0)1/4, thus reproducing the flat rotation curves observed for spiral g alaxies, without invoking\na distribution of dark matter.\nDevising such a theory turned out to be no simple matter, and the fin al result, TeVeS was\nrather complicated [31]. Furthermore, it was shown to have unexpe cted singular behavior that\nwas most simply cured by incorporating features of the Einstein-Æt her theory [337]. The extended\ntheory is based on an “Einstein” metric ˜ gµν, related to the physical metric gµνby\ngµν≡e−2φ˜gµν−2uµuνsinh(2φ), (48)\nwhereuµis a vector field, and φis a scalar field. The action for gravity is the standard GR\naction of Eq. (31), but defined using the Einstein metric ˜ gµν, while the matter action is that of a\nstandard metric theory, using gµν. These are supplemented by the vector action, given by that of\nEinstein-Æther theory, Eq. (41), and a scalar action, given by\nIS=−c3\n2k2ℓ2G/integraldisplay\nF(kℓ2hµνφ,µφ,ν)(−g)1/2d4x, (49)\nwherekis a constant, ℓis a distance, and hµν≡˜gµν−uµuν, indices being raised and lowered\nusing the Einstein metric. The function F(y) is chosen so that µ(y)≡dF/dyis unity in the high-\nacceleration, or normal Newtonian and post-Newtonian regimes, a nd nearly zero in the MOND\nregime.\nThe PPN parameters of the theory [319] have the values γ=β= 1 andξ=α3=ζi= 0, while\nthe parameters α1andα2are given by\nα1= (α1)Æ−16Gκc1(2−c14)−c3sinh4φ0+2(1−c1)sinh22φ0\n2c1−c2\n1+c2\n3, (50)\nα2= (α2)Æ−2G/parenleftbig\nA1κ−2A2sinh4φ0−A3sinh22φ0/parenrightbig\n, (51)\nwhere (α1)Æand (α2)Æare given by Eqs. (46) and (47), where\nA1≡(2c13−c14)2\nc123(2−c14)+4c1(2−c14)\n2c1−c2\n1+c2\n3−6(1+c13−c14)\n2−c14, (52)\nA2≡(2c13−c14)2\nc123(2−c14)2−4(1−c1)\n2c1−c2\n1+c2\n3+2(1−c13)\n2−c14/parenleftbigg2\nc123+3\n2−c14/parenrightbigg\n, (53)\nA3≡(2c13−c14)2\nc123(2−c14)2+4c3\n2c1−c2\n1+c2\n3+2\n(2−c14)/parenleftbigg3(1−c13)\nc123−2c13−c14\n2−c14/parenrightbigg\n,(54)\nwhereκ≡k/8π,\nG≡1\n2/parenleftbigg2−c14\n1+κ(2−c14)/parenrightbigg\n, (55)\nandφ0is the asymptotic value of the scalar field. In the limit κ→0 andφ0→0,α1andα2reduce\nto their Einstein-Æther forms.\nHowever, these PPN parameter values are computed in the limit wher e the function F(y) is\na linear function of its argument y=kℓ2hµνφ,µφ,ν. When one takes into account the fact that\nthe function µ(y) =dF/dymust interpolate between unity and zero to reach the MOND regime,\nit has been found that the dynamics of local systems is more strong ly affected by the fields of\nsurrounding matter than was anticipated. This “external field effe ct” (EFE) [260, 51, 52] produces\na quadrupolar contribution to the local Newtonian gravitational po tential that depends on the\nexternal distribution of matter and on the shape of the function µ(y), and that can be significantly\nlarger than the galactic tidal contribution. Although the calculation s of EFE have been carried\n38out using phenomenological MOND equations, it should be a generic ph enomenon, applicable to\nTeVeS as well. Analysis of the orbit of Jupiter using Cassini data has p laced interesting constraints\non the MOND interpolating function µ(y) [171].\nFor thorough reviews of MOND and TeVeS, and their confrontation with the dark-matter\nparadigm, see [338, 135].\n3.3.6 Quadratic gravity and Chern-Simons theories\nQuadratic gravity is a recent incarnation of an old idea of adding to th e action of GR terms\nquadratic in the Riemann and Ricci tensors or the Ricci scalar, as “e ffective field theory” models\nfor more fundamental string or quantum gravity theories. The ge neral action for such theories can\nbe written as\nI=/integraldisplay/bracketleftbigg\nκR+α1f1(φ)R2+α2f2(φ)RαβRαβ+α3f3(φ)RαβγδRαβγδ+α4f4(φ)∗RR\n−β\n2/parenleftbigg\ngµν∂µφ∂νφ+2V(φ)/parenrightbigg/bracketrightbigg\n(−g)1/2d4x+Im(ψm,gµν), (56)\nwhereκ= (16πG)−1,φis a scalar field, αiare dimensionless coupling constants (if the functions\nfi(φ) are dimensionless), and βis a constant whose dimension depends on that of φ, and where\n∗RR≡∗RαβγδRβαγδ, where∗Rαβγδ≡1\n2ǫγδρσRαβρσis the dual Riemann tensor.\nOne challenge inherent in these theories is to find an argument or a me chanism that evades\nmaking the natural choice for each of the αparameters to be of order unity. Such a choice makes\nthe effects of the additional terms essentially unobservable in most laboratory or astrophysical\nsituations because of the enormous scale of κ∝1/ℓ2\nPlanckin the leading term. This class of\ntheories is too vast and diffuse to cover in this review, and no good re view is available, to our\nknowledge.\nChern-Simons gravity is the special case of this class of theories in w hich only the parity-\nviolating term∗RRis present ( α1=α2=α3= 0). It can arise in various anomaly cancellation\nschemes in the standard model of particle physics, or in cancelling th e Green-Schwarz anomaly in\nstring theory. It can also arise in loop quantum gravity. The action in this case is given by\nI=/integraldisplay/bracketleftbigg\nκR+α\n4φ∗RR−β\n2/parenleftbigg\ngµν∂µφ∂νφ+2V(φ)/parenrightbigg/bracketrightbigg\n(−g)1/2d4x+Im(ψm,gµν),(57)\nwhereαandβare coupling constants with dimensions ℓA, andℓ2A−2, assuming that the scalar\nfield has dimensions ℓ−A.\nThere are two different versions of Chern-Simons theory, a non-d ynamical version in which\nβ= 0, so that φ, givena priori as some specified function of spacetime, plays the role of a\nLagrange multiplier enforcing the constraint∗RR= 0, and a dynamical version, in which β∝ne}ationslash= 0.\nThe PPN parameters for a non-dynamical version of the theory wit hα=κandβ= 0 are\nidenticalto those ofGR; however,there isan additional, parity-ev enpotential in the g0icomponent\nof the metric that does not appear in the standard PPN framework , given by\nδg0i= 2dφ\ndt(∇×V)i. (58)\nAlexander and Yunes [5] give a thorough review of Chern-Simons gr avity.\n3.3.7 Massive gravity\nMassive gravity theories attempt to give the putative “graviton” a mass. The simplest attempt\nto implement this in a ghost-free manner suffers from the so-called v an Dam–Veltman–Zakharov\n39(vDVZ) discontinuity [368, 414]. Because of the 3 additional helicity states available to the massive\nspin-2 graviton, the limit of small gravitonmass does not coincide with pure GR, and the predicted\nperihelion advance, for example, violates experiment. A model theo ry by Visser [371] attempts\nto circumvent the vDVZ problem by introducing a non-dynamical flat -background metric. This\ntheory is truly continuous with GR in the limit of vanishing graviton mass ; on the other hand, its\nobservational implications have been only partially explored. Branew orld scenarios predict a tower\nor a continuum of massive gravitons, and may avoid the vDVZ discont inuity, although the full\ndetails are still a work in progress [113, 86]. Attempts to avert the v DVZ problem involve treating\nnon-linearaspects of the theory at the fundamental level; many m odels incorporatea second tensor\nfield in addition to the metric. For recent reviews, see [173, 111], and a focus issue in Vol. 30,\nNumber 18 of Classical and Quantum Gravity .\n4 Tests of Post-Newtonian Gravity\n4.1 Tests of the parameter γ\nWith the PPN formalism in hand, we are now ready to confront gravita tion theories with the\nresults of solar-system experiments. In this section we focus on t ests of the parameter γ, consisting\nof the deflection of light and the time delay of light.\n4.1.1 The deflection of light\nA light ray (or photon) which passes the Sun at a distance dis deflected by an angle\nδθ=1\n2(1+γ)4M⊙\nd1+cosΦ\n2(59)\n(TEGP 7.1 [388]), where M⊙is the mass of the Sun and Φ is the angle between the Earth-Sun line\nand the incoming direction of the photon (see Figure 4). For a grazin g ray,d≈d⊙, Φ≈0, and\nδθ≈1\n2(1+γ)1.′′7505, (60)\nindependent of the frequency of light. Another, more useful exp ression gives the change in the\nrelative angular separation between an observed source of light an d a nearby reference source as\nboth rays pass near the Sun:\nδθ=1\n2(1+γ)/bracketleftbigg\n−4M⊙\ndcosχ+4M⊙\ndr/parenleftbigg1+cosΦ r\n2/parenrightbigg/bracketrightbigg\n, (61)\nwheredanddrare the distances of closest approach of the source and referen ce rays respectively,\nΦris the angular separation between the Sun and the reference sour ce, andχis the angle between\nthe Sun-source and the Sun-reference directions, projected o n the plane of the sky (see Figure 4).\nThus, for example, the relative angular separation between the tw o sources may vary if the line of\nsight of one of them passes near the Sun ( d∼R⊙,dr≫d,χvarying with time).\nIt is interesting to note that the classic derivations of the deflectio n of light that use only the\ncorpuscular theory of light (Cavendish 1784, von Soldner 1803 [38 4]), or the principle of equiv-\nalence (Einstein 1911), yield only the “1/2” part of the coefficient in f ront of the expression in\nEq. (59). But the result of these calculations is the deflection of ligh t relative to local straight\nlines, asestablished for example by rigid rods; however,because of space curvaturearound the Sun,\ndetermined by the PPN parameter γ, local straight lines are bent relative to asymptotic straight\nlines far from the Sun by just enough to yield the remaining factor “ γ/2”. The first factor “1/2”\n40xr\nxex⊕\nnnr\nddrΦΦrχReference Source\nSourceEarth\nSun\nFigure 4: Geometry of light deflection measurements.\nholds in any metric theory, the second “ γ/2” varies from theory to theory. Thus, calculations that\npurport to derive the full deflection using the equivalence principle a lone are incorrect.\nThe prediction ofthe full bending oflight by the Sun was oneof the gr eatsuccessesofEinstein’s\nGR. Eddington’s confirmation of the bending of optical starlight obs erved during a solar eclipse\nin the first days following World War I helped make Einstein famous. How ever, the experiments\nof Eddington and his co-workers had only 30 percent accuracy (fo r a recent re-evaluation of Ed-\ndington’s conclusions, see [197]). Succeeding experiments were no t much better: the results were\nscattered between one half and twice the Einstein value (see Figure 5), and the accuracies were\nlow. For a history of this period see [85].\nHowever,thedevelopmentofradiointerferometery,andlaterof very-long-baselineradiointerfer-\nometry (VLBI), produced greatly improved determinations of the deflection of light. These tech-\nniquesnowhavethecapabilityofmeasuringangularseparationsand changesinanglestoaccuracies\nbetter than 100 microarcseconds. Early measurements took adv antage of a series of heavenly coin-\ncidences: Each year, groups of strong quasistellar radio sources pass very close to the Sun (as seen\nfrom the Earth), including the group 3C273, 3C279, and 3C48, and the group 0111+02, 0119+11,\nand 0116+08. As the Earth moves in its orbit, changing the lines of sig ht of the quasars relative\nto the Sun, the angular separation δθbetween pairs of quasars varies (see Eq. (61)). The time\nvariation in the quantities d,dr,χ, and Φ rin Eq. (61) is determined using an accurate ephemeris\nfor the Earth and initial directions for the quasars, and the result ing prediction for δθas a func-\ntion of time is used as a basis for a least-squares fit of the measured δθ, with one of the fitted\nparameters being the coefficient1\n2(1+γ). A number of measurements of this kind over the period\n1969–1975 yielded an accurate determination of the coefficient1\n2(1+γ), or equivalently γ−1. A\n1995 VLBI measurement using 3C273 and 3C279 yielded γ−1 = (−8±34)×10−4[224], while a\n2009 measurement using the VLBA targeting the same two quasars plus two other nearby radio\nsources yielded γ−1 = (−2±3)×10−4[147].\nIn recent years, transcontinental and intercontinental VLBI o bservations of quasars and radio\ngalaxies have been made primarily to monitor the Earth’s rotation (“V LBI” in Figure 5). These\nmeasurements are sensitive to the deflection of light over almost th e entire celestial sphere (at 90◦\nfrom the Sun, the deflection is still 4 milliarcseconds). A 2004 analysis of almost 2 million VLBI\nobservations of 541 radio sources, made by 87 VLBI sites yielded (1 +γ)/2 = 0.99992±0.00023,\nor equivalently, γ−1 = (−1.7±4.5)×10−4[335]. A 2009 analysis that incorporated data through\n2008 yielded γ−1 = (−1.6±1.5)×10−4[219].\n41OpticalRadio\nVLBI\nHipparcosDEFLECTION\n  OF LIGHT\nPSR 1937+21\nVoyager\nVikingSHAPIRO\nTIME\nDELAY\n1920 1940 1960 1970 1980 1990 20001.10\n1.05\n1.00\n0.95\n1.05\n1.00\n0.95 (1+γ)/2 \nYEAR OF EXPERIMENTTHE PARAMETER  (1+γ)/2\nCassini\n(1X10-5 )2X10-4 \nGalactic\nLensing\n201010 -4 \nFigure 5: Measurements of the coefficient (1+ γ)/2 from light deflection and time delay measure-\nments. Its GR value is unity. The arrows at the top denote anomalou sly large values from early\neclipse expeditions. The Shapiro time-delay measurements using the Cassini spacecraft yielded\nan agreement with GR to 10−3percent, and VLBI light deflection measurements have reached\n0.02 percent. Hipparcos denotes the optical astrometry satellite , which reached 0.1 percent.\nAnalysis ofobservationsmade by the Hipparcosoptical astrometr ysatellite yielded a test at the\nlevel of 0.3 percent [150]. A VLBI measurement of the deflection of lig ht by Jupiter was reported\nin 1991; the predicted deflection of about 300 microarcseconds wa s seen with about 50 percent\naccuracy [359].\nFinally, a remarkable measurement of γon galactic scales was reported in 2006 [56]. It used\ndata on gravitational lensing by 15 elliptical galaxies, collected by the Sloan Digital Sky Survey.\nThe Newtonian potential Uof each lensing galaxy (including the contribution from dark matter)\nwas derived from the observed velocity dispersion of stars in the ga laxy. Comparing the observed\nlensing with the lensing predicted by the models provided a 10 percent bound onγ, in agreement\nwith general relativity. Unlike the much tighter bounds described pr eviously, which were obtained\non the scale of the solar system, this bound was obtained on a galact ic scale.\nThe results of light-deflection measurements are summarized in Figu re 5.\n424.1.2 The time delay of light\nA radar signal sent across the solar system past the Sun to a plane t or satellite and returned to\nthe Earth suffers an additional non-Newtonian delay in its round-tr ip travel time, given by (see\nFigure 4)\nδt= 2(1+γ)M⊙ln/parenleftbigg(r⊕+x⊕·n)(re−xe·n)\nd2/parenrightbigg\n, (62)\nwherexe(x⊕) are the vectors, and re(r⊕) are the distances from the Sun to the source (Earth),\nrespectively (TEGP 7.2 [388]). For a ray which passes close to the Sun ,\nδt≈1\n2(1+γ)/bracketleftbigg\n240−20ln/parenleftbiggd2\nr/parenrightbigg/bracketrightbigg\nµs, (63)\nwheredis the distance of closest approach of the ray in solar radii, and ris the distance of the\nplanet or satellite from the Sun, in astronomical units.\nIn the two decades following Irwin Shapiro’s 1964 discovery of this eff ect as a theoretical con-\nsequence of GR, several high-precision measurements were made using radar ranging to targets\npassing through superior conjunction. Since one does not have ac cess to a “Newtonian” signal\nagainst which to compare the round-trip travel time of the observ ed signal, it is necessary to do a\ndifferential measurement of the variations in round-trip travel tim es as the target passes through\nsuperior conjunction, and to look for the logarithmic behavior of Eq . (63). In order to do this\naccurately however, one must take into account the variations in r ound-trip travel time due to\nthe orbital motion of the target relative to the Earth. This is done b y using radar-ranging (and\npossiblyother)dataonthetargettakenwhenitisfarfromsuperio rconjunction(i.e.whenthetime-\ndelay term is negligible) to determine an accurate ephemeris for the t arget, using the ephemeris\nto predict the PPN coordinate trajectory xe(t) near superior conjunction, then combining that\ntrajectory with the trajectory of the Earth x⊕(t) to determine the Newtonian round-trip time and\nthe logarithmic term in Eq. (63). The resulting predicted round-trip travel times in terms of\nthe unknown coefficient1\n2(1 +γ) are then fit to the measured travel times using the method of\nleast-squares, and an estimate obtained for1\n2(1+γ).\nThe targets employed included planets, such as Mercury or Venus, used as passive reflectors\nof the radar signals (“passive radar”), and artificial satellites, su ch as Mariners 6 and 7, Voy-\nager 2, the Viking Mars landers and orbiters, and the Cassini space craft to Saturn, used as active\nretransmitters of the radar signals (“active radar”).\nThe results for the coefficient1\n2(1+γ) of all radar time-delay measurements performed to date\n(including a measurement of the one-way time delay of signals from th e millisecond pulsar PSR\n1937+21) are shown in Figure 5 (see TEGP 7.2 [388] for discussion and references). The 1976\nViking experiment resulted in a 0.1 percent measurement [306].\nA significant improvement wasreported in 2003from Doppler trackin g of the Cassini spacecraft\nwhile it was on its way to Saturn [38], with a result γ−1 = (2.1±2.3)×10−5. This was made\npossible by the ability to do Doppler measurements using both X-band (7175 MHz) and Ka-band\n(34316 MHz) radar, thereby significantly reducing the dispersive e ffects of the solar corona. In\naddition, the 2002 superior conjunction of Cassini was particularly favorable: with the spacecraft\nat 8.43 astronomical units from the Sun, the distance of closest ap proach of the radar signals to\nthe Sun was only 1 .6R⊙.\nFrom the results of the Cassini experiment, we can conclude that t he coefficient1\n2(1+γ) must\nbe within at most 0.0012 percent of unity. Scalar-tensor theories m ust haveω >40,000 to be\ncompatible with this constraint.\n434.1.3 Shapiro time delay and the speed of gravity\nIn 2001, Kopeikin [203] suggested that a measurement of the time d elay of light from a quasar\nas the light passed by the planet Jupiter could be used to measure th e speed of the gravitational\ninteraction. He argued that, since Jupiter is moving relative to the s olar system, and since gravity\npropagateswithafinitespeed, the gravitationalfieldexperienced bythelightrayshouldbeaffected\nby gravity’s speed, since the field experienced at one time depends o n the location of the source a\nshort time earlier, depending on how fast gravity propagates. Acc ording to his calculations, there\nshould be a post1/2-Newtonian correction to the normal Shapiro time-delay formula (6 2) which\ndepends on the velocity of Jupiter and on the velocity of gravity. On September 8, 2002, Jupiter\npassed almost in front of a quasar, and Kopeikin and Fomalont made p recise measurements of the\nShapiro delay with picosecond timing accuracy, and claimed to have me asured the correction term\nto about 20 percent [146, 207, 204, 205].\nHowever, several authors pointed out that this 1.5PN effect does notdepend on the speed\nof propagation of gravity, but rather only depends on the speed o f light [20, 393, 320, 64, 321].\nIntuitively, if one is working to only first order in v/c, then all that counts is the uniform motion of\nthe planet, Jupiter (its acceleration about the Sun contributes a h igher-order, unmeasurably small\neffect). But if that is the case, then the principle of relativity says t hat one can view things from\nthe rest frame of Jupiter. In this frame, Jupiter’s gravitational fi eld is static, and the speed of\npropagation of gravity is irrelevant. A detailed post-Newtonian calc ulation of the effect was done\nusing a variant of the PPN framework, in a class of theories in which th e speed of gravity could be\ndifferent from that of light [393], and found explicitly that, at first o rder inv/c, the effect depends\non the speed of light, not the speed of gravity, in line with intuition. Eff ects dependent upon the\nspeed of gravity show up only at higher order in v/c. Kopeikin gave a number of arguments in\nopposition to this interpretation [205, 208, 206]. On the other han d, thev/ccorrection term does\nshow a dependence on the PPN parameter α1, which could be non-zero in theories of gravity with\na differing speed cgof gravity (see Eq. (7) of [393]). But existing tight bounds on α1from other\nexperiments (see Table 4) already far exceed the capability of the J upiter VLBI experiment.\nTable 4: Current limits on the PPN parameters.\nParameter Effect Limit Remarks\nγ−1 time delay 2 .3×10−5Cassini tracking\nlight deflection 2 ×10−4VLBI\nβ−1 perihelion shift 8 ×10−5J2⊙= (2.2±0.1)×10−7\nNordtvedt effect 2 .3×10−4ηN= 4β−γ−3 assumed\nξ spin precession 4 ×10−9millisecond pulsars\nα1 orbital polarization 10−4Lunar laser ranging\n7×10−5PSR J1738+0333\nα2 spin precession 2 ×10−9millisecond pulsars\nα3 pulsar acceleration 4 ×10−20pulsar˙Pstatistics\nζ1 — 2 ×10−2combined PPN bounds\nζ2 binary acceleration 4 ×10−5¨Ppfor PSR 1913+16\nζ3 Newton’s 3rd law 10−8lunar acceleration\nζ4 — — not independent (see Eq. (71))\n444.2 The perihelion shift of Mercury\nThe explanation of the anomalous perihelion shift of Mercury’s orbit w as another of the triumphs\nof GR. This had been an unsolved problem in celestial mechanics for ov er half a century, since\nthe announcement by Le Verrier in 1859 that, after the perturbin g effects of the planets on Mer-\ncury’s orbit had been accounted for, and after the effect of the p recession of the equinoxes on the\nastronomical coordinate system had been subtracted, there re mained in the data an unexplained\nadvance in the perihelion of Mercury. The modern value for this discr epancy is 43 arcseconds\nper century. A number of ad hocproposals were made in an attempt to account for this excess,\nincluding, among others, the existence of a new planet Vulcan near t he Sun, a ring of planetoids,\na solar quadrupole moment and a deviation from the inverse-square law of gravitation, but none\nwas successful. General relativity accounted for the anomalous s hift in a natural way without\ndisturbing the agreement with other planetary observations.\nThe predicted advance per orbit ∆˜ ω, including both relativistic PPN contributions and the\nNewtonian contribution resulting from a possible solar quadrupole mo ment, is given by\n∆˜ω=6πm\np/parenleftbigg1\n3(2+2γ−β)+1\n6(2α1−α2+α3+2ζ2)η+J2R2\n2mp/parenrightbigg\n, (64)\nwherem≡m1+m2andη≡m1m2/m2are the total mass and dimensionless reduced mass\nof the two-body system respectively; p≡a(1−e2) is the semi-latus rectum of the orbit, with\nthe semi-major axis aand the eccentricity e;Ris the mean radius of the oblate body; and J2\nis a dimensionless measure of its quadrupole moment, given by J2= (C−A)/m1R2, whereC\nandAare the moments of inertia about the body’s rotation and equatoria l axes, respectively (for\ndetails of the derivation see TEGP 7.3 [388]). We have ignored prefer red-frame and galaxy-induced\ncontributions to ∆˜ ω; these are discussed in TEGP 8.3 [388].\nThe first term in Eq. (64) is the classical relativistic perihelion shift, w hich depends upon\nthe PPN parameters γandβ. The second term depends upon the ratio of the masses of the two\nbodies; it is zero in any fully conservative theory of gravity ( α1≡α2≡α3≡ζ2≡0); it is also\nnegligible for Mercury, since η≈mMerc/M⊙≈2×10−7. We shall drop this term henceforth.\nThe third term depends upon the solar quadrupole moment J2. For a Sun that rotates uni-\nformly with its observed surface angular velocity, so that the quad rupole moment is produced by\ncentrifugal flattening, one may estimate J2to be∼1×10−7. This actually agrees reasonably well\nwith values inferred from rotating solar models that are in accord wit h observations of the nor-\nmal modes of solar oscillations (helioseismology); the latest inversion s of helioseismology data give\nJ2= (2.2±0.1)×10−7[252, 15]; for a review of measurements of the solar quadrupole mom ent,\nsee [317]. Substituting standard orbital elements and physical con stants for Mercury and the Sun\nwe obtain the rate of perihelion shift ˙˜ω, in seconds of arc per century,\n˙˜ω= 42.′′98/parenleftbigg1\n3(2+2γ−β)+3×10−4J2\n10−7/parenrightbigg\n. (65)\nThe most recent fits to planetary data include data from the Messe nger spacecraft that orbited\nMercury, thereby significantly improving knowledge of its orbit. Ado pting the Cassini bound on γ\na priori, these analyses yield a bound on βgiven byβ−1 = (−4.1±7.8)×10−5. Further analysis\ncould push this bound even lower [137, 369], although knowledge of J2would have to improve\nsimultaneously. A slightly weaker bound β−1 = (0.4±2.4)×10−4from the perihelion advance\nof Mars (again adopting the Cassini bound on γ) was obtained by exploiting data from the Mars\nReconnaissance Orbiter [202]\nLasertrackingofthe Earth-orbitingsatellite LAGEOS II led to a mea surementofits relativistic\nperigee precession (3 .4 arcseconds per year) in agreement with GR to 0 .2 percent [241].\n454.3 Tests of the strong equivalence principle\nThe next class of solar-system experiments that test relativistic g ravitational effects may be called\ntests of the strong equivalence principle (SEP). In Section 3.1.2 we p ointed out that many metric\ntheories of gravity (perhaps all except GR) can be expected to vio late one or more aspects of\nSEP. Among the testable violations of SEP are a violation of the weak e quivalence principle for\ngravitating bodies that leads to perturbations in the Earth-Moon o rbit, preferred-location and\npreferred-frame effects in the locally measured gravitational con stant that could produce observ-\nable geophysical effects, and possible variations in the gravitationa l constant over cosmological\ntimescales.\n4.3.1 The Nordtvedt effect and the lunar E¨ otv¨ os experiment\nIn a pioneering calculation using his early form ofthe PPNformalism, No rdtvedt [279] showed that\nmany metric theories of gravity predict that massive bodies violate t he weak equivalence principle\n– that is, fall with different accelerations depending on their gravita tional self-energy. Dicke [315]\nargued that such an effect would occur in theories with a spatially var ying gravitational constant,\nsuch as scalar-tensor gravity. For a spherically symmetric body, t he acceleration from rest in an\nexternal gravitational potential Uhas the form\na=mp\nm∇U,\nmp\nm= 1−ηNEg\nm,\nηN= 4β−γ−3−10\n3ξ−α1+2\n3α2−2\n3ζ1−1\n3ζ2,(66)\nwhereEgis the negative of the gravitationalself-energyof the body ( Eg>0). This violation of the\nmassive-body equivalence principle is known as the “Nordtvedt effec t”. The effect is absent in GR\n(ηN= 0) but present in scalar-tensor theory ( ηN= 1/(2+ω)+4λ). The existence of the Nordtvedt\neffect does not violate the results of laboratory E¨ otv¨ os experim ents, since for laboratory-sized\nobjectsEg/m≤10−27, far below the sensitivity of current or future experiments. Howe ver, for\nastronomicalbodies, Eg/mmay be significant (3 .6×10−6for the Sun, 10−8for Jupiter, 4 .6×10−10\nfor the Earth, 0 .2×10−10for the Moon). If the Nordtvedt effect is present ( ηN∝ne}ationslash= 0) then the Earth\nshould fall toward the Sun with a slightly different acceleration than t he Moon. This perturbation\nin the Earth-Moon orbit leads to a polarization of the orbit that is dire cted toward the Sun as\nit moves around the Earth-Moon system, as seen from Earth. This polarization represents a\nperturbation in the Earth-Moon distance of the form\nδr= 13.1ηNcos(ω0−ωs)t[m], (67)\nwhereω0andωsaretheangularfrequenciesoftheorbitsoftheMoonandSunarou ndtheEarth(see\nTEGP 8.1 [388] for detailed derivations and references; for improve d calculations of the numerical\ncoefficient, see [284, 108]).\nSince August 1969,whenthe firstsuccessful acquisitionwasmade ofalasersignalreflectedfrom\nthe Apollo 11 retroreflector on the Moon, the LLR experiment has m ade regular measurements\nof the round-trip travel times of laser pulses between a network o f observatories and the lunar\nretroreflectors, with accuracies that are approaching the 5 ps ( 1 mm) level. These measurements\nare fit using the method of least-squares to a theoretical model f or the lunar motion that takes into\naccountperturbationsdue to the Sun and the otherplanets, tida l interactions, and post-Newtonian\ngravitational effects. The predicted round-trip travel times bet ween retroreflector and telescope\nalso take into account the librations of the Moon, the orientation of the Earth, the location of the\n46observatories, and atmospheric effects on the signal propagatio n. The “Nordtvedt” parameter ηN\nalong with several other important parameters of the model are t hen estimated in the least-squares\nmethod. For a review of lunar laser ranging, see [253].\nNumerous ongoing analyses of the data find no evidence, within expe rimental uncertainty, for\nthe Nordtvedt effect [405, 406] (for earlier results see [119, 40 4, 270]). These results represent\na limit on a possible violation of WEP for massive bodies of about 1.4 parts in 1013(compare\nFigure 1).\nHowever, at this level of precision, one cannot regard the results of LLR as a “clean” test of\nSEP until one eliminates the possibility of a compensating violation of WE P for the two bodies,\nbecause the chemical compositions of the Earth and Moon differ in th e relative fractions of iron\nand silicates. To this end, the E¨ ot-Wash group carried out an impro ved test of WEP for laboratory\nbodies whose chemical compositions mimic that of the Earth and Moon . The resulting bound of\n1.4 parts in 1013[25, 1] from composition effects reduces the ambiguity in the LLR bou nd, and\nestablishes the firm SEP test at the level of about 2 parts in 1013. These results can be summarized\nby the Nordtvedt parameter bound |ηN|= (4.4±4.5)×10−4.\nAPOLLO, the Apache Point Observatory for Lunar Laser ranging O peration, a joint effort by\nresearchers from the Universities of Washington, Seattle, and Ca lifornia, San Diego, has achieved\nmm ranging precision using enhanced laser and telescope technology , together with a good, high-\naltitude site in New Mexico. However models of the lunar orbit must be im proved in parallel in\norder to achieve an order-of-magnitude improvement in the test o f the Nordtvedt effect [273]. This\neffort will be aided by the fortuitous 2010 discovery by the Lunar Re connaissance Orbiter of the\nprecise landing site of the Soviet Lunokhod I rover, which deployed a retroreflector in 1970. Its\nuncertain location made it effectively “lost” to lunar laser ranging for almost 40 years. Its location\non the lunar surface will make it useful in improving models of the lunar libration [272].\nIn GR, the Nordtvedt effect vanishes; at the level of several cen timeters and below, a number\nof non-null general relativistic effects should be present [284].\nTests of the Nordtvedt effect for neutron stars have also been c arried out using a class of\nsystems known as wide-orbit binary millisecond pulsars (WBMSP), whic h are pulsar–white-dwarf\nbinary systems with small orbital eccentricities. In the gravitation al field of the galaxy, a non-\nzero Nordtvedt effect can induce an apparent anomalous eccentr icity pointed toward the galactic\ncenter [106], which can be bounded using statistical methods, given enough WBMSPs (see [345]\nfor a review and references). Using data from 21 WBMSPs, including recently discovered highly\ncircular systems, Stairs et al. [346] obtained the bound ∆ <5.6×10−3, where ∆ = ηN(Eg/M)NS.\nBecause(Eg/M)NS∼0.1for typical neutron stars, this bound does not compete with the bound on\nηNfrom LLR; on the other hand, it does test SEP in the strong-field re gimebecause of the presence\nof the neutron stars. The 2013 discovery of a millisecond pulsar in or bit with twowhite dwarfs\nin very circular, coplanar orbits [305] may lead to a test of the Nord vedt effect in the strong-field\nregime that surpasses the precision of lunar laser ranging by a subs tantial factor (see Sec. 6.2).\n4.3.2 Preferred-frame and preferred-location effects\nSome theories of gravity violate SEP by predicting that the outcome s of local gravitational experi-\nments may depend on the velocity of the laboratory relative to the m ean rest frame of the universe\n(preferred-frame effects) or on the location of the laboratory r elative to a nearby gravitating body\n(preferred-location effects). In the post-Newtonian limit, prefe rred-frame effects are governed by\nthe values of the PPN parameters α1,α2, andα3, and some preferred-location effects are governed\nbyξ(see Table 2).\nThe most important such effects are variations and anisotropies in t he locally-measured value\nof the gravitational constant which lead to anomalous Earth tides a nd variations in the Earth’s\nrotation rate, anomalous contributions to the orbital dynamics of planets and the Moon, self-\n47accelerations of pulsars, and anomalous torques on the Sun that w ould cause its spin axis to be\nrandomly oriented relative to the ecliptic (see TEGP 8.2, 8.3, 9.3, and 1 4.3 (c) [388]).\nA tight bound on α3of 4×10−20was obtained from the period derivatives of 21 millisecond\npulsars [32, 346]. The best bound on α1, comes from the orbit of the pulsar–white-dwarf system\nJ1738+0333 [331]. Early bounds on on α2andξcame from searches for variations induced by\nan anisotropy in Gon the acceleration of gravity on Earth using gravimeters, and (in t he case\nofα2) from limiting the effects of any anomalous torque on the spinning Sun over the age of the\nsolar system. Today the best bounds on α2andξcome from bounding torques on the solitary\nmillisecond pulsars B1937+21 and J1744–1134 [331, 330, 332]. Becau se these later bounds involve\nsystems with strong internal gravity of the neutron stars, they should strictly speaking be regarded\nas bounds on “strong field” analogues of the PPN parameters. Her e we will treat them as bounds\non the standard PPN parameters, as shown in Table 4.\n4.3.3 Constancy of the Newtonian gravitational constant\nMosttheoriesofgravitythatviolateSEPpredictthatthelocallymea suredNewtoniangravitational\nconstantmayvarywithtime astheuniverseevolves. Forthe scalar -tensortheorieslistedin Table3,\nthe predictions for ˙G/Gcan be written in terms of time derivatives of the asymptotic scalar fi eld.\nWhereGdoes change with cosmic evolution, its rate of variation should be of t he order of the\nexpansion rate of the universe, i.e. ˙G/G∼H0, whereH0is the Hubble expansion parameter, given\nbyH0= 73±3 km s−1Mpc−1= 7.4×10−11yr−1.\nSeveral observational constraints can be placed on ˙G/G, one kind coming from bounding the\npresent rate of variation, another from bounding a difference bet ween the present value and a past\nvalue. The first type of bound typically comes from LLR measuremen ts, planetary radar-ranging\nmeasurements, and pulsar timing data. The second type comes fro m studies of the evolution of the\nSun, stars and the Earth, big-bang nucleosynthesis, and analyse s of ancient eclipse data. Recent\nresults are shown in Table 5.\nTable5: Constancyofthegravitationalconstant. Forbinarypuls ardata, theboundsaredependent\nupon the theory of gravity in the strong-field regime and on neutro n star equation of state. Big-\nbang nucleosynthesis bounds assume specific form for time depend ence ofG.\nMethod ˙G/G Reference\n(10−13yr−1)\nMars ephemeris 0.1±1.6 [202]\nLunar laser ranging 4±9 [405]\nBinary & millisecond pulsars −7±33 [114, 223]\nHelioseismology 0±16 [165]\nBig Bang nucleosynthesis 0±4 [84, 27]\nThe best limits on a current ˙G/Gcome from improvements in the ephemeris of Mars using\nrange and Doppler data from the Mars Global Surveyor (1998–200 6), Mars Odyssey (2002–2008),\nand Mars Reconnaissance Orbiter (2006–2008), together with imp roved data and modeling of the\neffects of the asteroid belt [294, 202]. Since the bound is actually on v ariations of GM⊙, any future\nimprovements in ˙G/Gbeyond a part in 1013will have to take into account models of the actual\nmass loss from the Sun, due to radiation of light and neutrinos ( ∼0.7×10−13yr−1) and due to\nthe solar wind ( ∼0.2×10−13yr−1). Another bound comes from LLR measurements ([405]; for\nearlier results see [119, 404, 270]).\n48Although bounds on ˙G/Gfrom solar-system measurements can be correctly obtained in a\nphenomenological manner through the simple expedient of replacing GbyG0+˙G0(t−t0) in\nNewton’s equations of motion, the same does not hold true for pulsa r and binary pulsar timing\nmeasurements. The reason is that, in theories of gravity that viola te SEP, such as scalar-tensor\ntheories, the “mass” and moment of inertia of a gravitationally boun d body may vary with G.\nBecause neutron stars are highly relativistic, the fractional varia tion in these quantities can be\ncomparable to ∆ G/G, the precise variation depending both on the equation of state of n eutron\nstar matter and on the theory of gravity in the strong-field regime . The variation in the moment\nof inertia affects the spin rate of the pulsar, while the variation in the mass can affect the orbital\nperiod in a manner that can subtract from the direct effect of a var iation inG, given by ˙Pb/Pb=\n−2˙G/G[283]. Thus, the bounds quoted in Table 5 for binary and millisecond puls ars are theory-\ndependent and must be treated as merely suggestive.\nIn a similar manner, bounds from helioseismology and big-bang nucleos ynthesis (BBN) assume\na model for the evolution of Gover the multi-billion year time spans involved. For example, the\nconcordance of predictions for light elements produced around 3 m inutes after the big bang with\nthe abundances observed indicate that Gthen was within 20 percent of Gtoday. Assuming a\npower-law variation of G∼t−αthen yields a bound on ˙G/Gtoday shown in Table 5.\n4.4 Other tests of post-Newtonian gravity\n4.4.1 Search for gravitomagnetism\nAccording to GR, moving or rotating matter should produce a contr ibution to the gravitational\nfield that is the analogue of the magnetic field of a moving charge or a m agnetic dipole. In\nparticular, one can view the g0ipart of the PPN metric (see Box 2) as an analogue of the vector\npotential of electrodynamics. In a suitable gauge (not the standa rd PPN gauge), and dropping the\npreferred-frame terms, it can be written\ng0i=−1\n2(4γ+4+α1)Vi. (68)\nAt PN order, this contributes a Lorentz-type acceleration v×Bgto the equation of motion, where\nthe gravitomagnetic field Bgis given by Bg=∇×(g0iei).\nGravitomagnetism plays a role in a variety of measured relativistic effe cts involving moving\nmaterial sources, such as the Earth-Moon system and binary puls ar systems. Nordtvedt [282,\n281] has argued that, if the gravitomagnetic potential (68) were turned off, then there would be\nanomalous orbital effects in LLR and binary pulsar data.\nRotation also produces a gravitomagnetic effect, since for a rotat ing body, V=−1\n2x×J/r3,\nwhereJis the angularmomentum of the body. The result is a “draggingof iner tial frames” around\nthe body, also called the Lense–Thirring effect. A consequence is a p recession of a gyroscope’s spin\nSaccording to\ndS\ndτ=ΩLT×S,ΩLT=−1\n2/parenleftbigg\n1+γ+1\n4α1/parenrightbiggJ−3n(n·J)\nr3, (69)\nwherenis a unit radial vector, and ris the distance from the center of the body (TEGP 9.1 [388]).\nIn2011theRelativityGyroscopeExperiment(GravityProbeBorGP B)carriedoutbyStanford\nUniversity, NASA and Lockheed–Martin Corporation [162], finally com pleted a space mission to\ndetect this frame-dragging or Lense–Thirring precession, along w ith the “geodetic” precession (see\nSection 4.4.2). Gravity Probe B will very likely go down in the history of s cience as one of the\nmost ambitious, difficult, expensive, and controversial relativity ex periments ever performed.1It\n1Full disclosure: The author served as Chair of an external NA SA Science Advisory Committee for Gravity\nProbe B from 1998 to 2011.\n49was almost 50 years from inception to completion, although only abou t half of that time was spent\nas a full-fledged, approved space program.\nThe GPBspacecraftwaslaunchedonApril 20, 2004into analmostpe rfectly circularpolarorbit\nat an altitude of 642 km, with the orbital plane parallel to the directio n of a guide star known as\nIM Pegasi (HR 8703). The spacecraft contained four spheres made of fuze d quartz, all spinning\nabout the same axis (two were spun in the opposite direction), which was oriented to be in the\norbital plane, pointing toward the guide star. An onboard telescop e pointed continuously at the\nguide star, and the direction of each spin was compared with the dire ction to the star, which was at\na declination of 16orelative to the Earth’s equatorial plane. With these conditions, the precessions\npredicted by GR were 6630 milliarcsecond per year for the geodetic e ffect, and 38 milliarcsecond\nper year for frame dragging, the former in the orbital plane (in the north-south direction) and the\nlatter perpendicular to it (in the east-west direction).\nIn order to reduce the non-relativistic torques on the rotors to a n acceptable level, the rotors\nwere fabricated to be both spherical and homogenous to better t han a few parts in 10 million.\nEach rotor was coated with a thin film of niobium, and the experiment w as conducted at cryogenic\ntemperatures inside a dewar containing 2200 litres of superfluid liquid helium. As the niobium\nfilm becomes a superconductor, each rotor develops a magnetic mo ment parallel to its spin axis.\nVariations in the direction of the magnetic moment relative to the spa cecraft were then measured\nusing superconducting current loops surrounding each rotor. As the spacecraft orbits the Earth,\nthe aberration of light from the guide star causes an artificial but p redictable change in direction\nbetween the rotors and the on-board telescope; this was an esse ntial tool for calibrating the con-\nversion between the voltages read by the current loops and the ac tual angle between the rotors and\nthe guide star. The motion ofthe guide starrelativeto distant inert ial frameswasmeasuredbefore,\nduring and after the mission separately by radio astronomersat Ha rvard/SAOand elsewhere using\nVLBI (IM Pegasi is a radio star) [334].\nThe mission ended in September 2005, as scheduled, when the last of the liquid helium boiled\noff. Although all subsystems of the spacecraft and the apparatu s performed extremely well, they\nwere not perfect. Calibration measurements carried out during th e mission, both before and after\nthe science phase, revealed unexpectedly large torques on the ro tors. Numerous diagnostic tests\nworthy of a detective novel showed that these were caused by ele ctrostatic interactions between\nsurface imperfections (“patch effect”) on the niobium films and the spherical housings surrounding\neach rotor. These effects and other anomalies greatly contaminat ed the data and complicated its\nanalysis, but finally, in October 2010, the Gravity Probe B team anno unced that the experiment\nhad successfully measured both the geodetic and frame-dragging precessions. The outcome was in\nagreement with general relativity, with a precision of 0 .3 percent for the geodetic precession, and\n20 percent for the frame-dragging effect [134]. For a commentary on the GPB result, see [402].\nThe full technical and data analysis details of GPB are expected to b e published as a special issue\nofClassical and Quantum Gravity in 2015.\nAnother way to look for frame-dragging is to measure the precess ion of orbital planes of bodies\ncircling a rotating body. One implementation of this idea is to measure t he relative precession,\nat about 31 milliarcseconds per year, of the line of nodes of a pair of la ser-ranged geodynam-\nics satellites (LAGEOS), ideally with supplementary inclination angles; t he inclinations must be\nsupplementary in order to cancel the dominant (126 degrees per y ear) nodal precession caused\nby the Earth’s Newtonian gravitational multipole moments. Unfortu nately, the two existing LA-\nGEOS satellites are not in appropriately inclined orbits. Nevertheless , Ciufolini and collabora-\ntors [76, 78, 75] combined nodal precession data from LAGEOS I an d II with improved models\nfor the Earth’s multipole moments provided by two orbiting geodesy s atellites, Europe’s CHAMP\n(Challenging Minisatellite Payload) and NASA’s GRACE (Gravity Recover y and Climate Exper-\niment), and reported a 10 percent confirmation of GR [75]. In earlier reports, Ciufolini et al.\nhad reported tests at the the 20–30 percent level, without the be nefit of the GRACE/CHAMP\n50data [73, 77, 72]. Some authors stressed the importance of adequ ately assessing systematic errors\nin the LAGEOS data [311, 181].\nOn February 13, 2012, a third laser-ranged satellite, known as LAR ES (Laser Relativity Satel-\nlite) was launched by the Italian Space Agency [288]. Its inclination was very close to the required\nsupplementary angle relative to LAGEOS I, and its eccentricity was v ery nearly zero. However,\nbecause its semimajor axis is only 2 /3 that of either LAGEOS I or II, and because the Newtonian\nprecession rate is proportional to a−3/2, LARES does not provide a cancellation of the Newtonian\nprecession. Nevertheless, combining data from all three satellites with continually improving Earth\ndata from GRACE, the LARES team hopes to achieve a test of frame -dragging at the one percent\nlevel [74].\n4.4.2 Geodetic precession\nA gyroscope moving through curved spacetime suffers a precessio n of its spin axis given by\ndS\ndτ=ΩG×S,ΩG=/parenleftbigg\nγ+1\n2/parenrightbigg\nv×∇U, (70)\nwherevis the velocity of the gyroscope, and Uis the Newtonian gravitational potential of the\nsource (TEGP 9.1 [388]). The Earth-Moon system can be considered as a “gyroscope”, with its\naxisperpendicular to the orbitalplane. The predicted precessionis about 2 arcsecondsper century,\nan effect first calculated by de Sitter. This effect has been measure d to about 0.6 percent using\nLLR data [119, 404, 405].\nFor the GPB gyroscopes orbiting the Earth, the precession is 6.63 a rcseconds per year. GPB\nmeasured this effect to 3 ×10−3; the resulting bound on the parameter γis not competitive with\nthe Cassini bound.\n4.4.3 Tests of post-Newtonian conservation laws\nOfthefive“conservationlaw”PPNparameters ζ1,ζ2,ζ3,ζ4, andα3, onlythree, ζ2,ζ3, andα3, have\nbeen constrained directly with any precision; ζ1is constrained indirectly through its appearance\nin the Nordtvedt effect parameter ηN, Eq. (66). There is strong theoretical evidence that ζ4,\nwhich is related to the gravity generated by fluid pressure, is not re ally an independent parameter\n– in any reasonable theory of gravity there should be a connection b etween the gravity produced\nby kinetic energy ( ρv2), internal energy ( ρΠ), and pressure ( p). From such considerations, there\nfollows [383] the additional theoretical constraint\n6ζ4= 3α3+2ζ1−3ζ3. (71)\nA non-zero value for any of these parameters would result in a violat ion of conservation of\nmomentum, or of Newton’s third law in gravitating systems. An altern ative statement of Newton’s\nthird law for gravitating systems is that the “active gravitational m ass”, that is the mass that\ndetermines the gravitational potential exhibited by a body, should equal the “passive gravitational\nmass”, the mass that determines the force on a body in a gravitatio nal field. Such an equality\nguarantees the equality of action and reaction and of conservatio n of momentum, at least in the\nNewtonian limit.\nA classic test of Newton’s third law for gravitating systems was carr ied out in 1968 by Kreuzer,\nin which the gravitational attraction of fluorine and bromine were co mpared to a precision of 5\nparts in 105.\nA remarkable planetary test was reported by Bartlett and van Bur en [28]. They noted that\ncurrent understanding of the structure of the Moon involves an ir on-rich, aluminum-poor mantle\n51whose center of mass is offset about 10 km from the center of mass of an aluminum-rich, iron-\npoor crust. The direction of offset is toward the Earth, about 14◦to the east of the Earth-Moon\nline. Such a model accounts for the basaltic maria which face the Ear th, and the aluminum-rich\nhighlands on the Moon’s far side, and for a 2 km offset between the ob served center of mass\nand center of figure for the Moon. Because of this asymmetry, a v iolation of Newton’s third\nlaw for aluminum and iron would result in a momentum non-conserving se lf-force on the Moon,\nwhose component along the orbital direction would contribute to th e secular acceleration of the\nlunar orbit. Improved knowledge of the lunar orbit through LLR, an d a better understanding of\ntidal effects in the Earth-Moon system (which also contribute to th e secular acceleration) through\nsatellite data, severely limit any anomalous secular acceleration, with the resulting limit\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(mA/mP)Al−(mA/mP)Fe\n(mA/mP)Fe/vextendsingle/vextendsingle/vextendsingle/vextendsingle<4×10−12. (72)\nAccording to the PPN formalism, in a theory of gravity that violates c onservation of momentum,\nbut that obeys the constraint of Eq. (71), the electrostatic bind ing energyEeof an atomic nucleus\ncould make a contribution to the ratio of active to passive mass of th e form\nmA=mP+1\n2ζ3Ee. (73)\nThe resulting limit on ζ3from the lunar experiment is ζ3<1×10−8(TEGP 9.2, 14.3 (d) [388]).\nNordtvedt[285]hasexaminedwhetherthisboundcouldbeimproved byconsideringtheasymmetric\ndistribution of ocean water on Earth.\nAnother consequence of a violation of conservation of momentum is a self-acceleration of the\ncenter of mass of a binary stellar system, given by\naCM=−1\n2(ζ2+α3)m\na2µ\naδm\nme\n(1−e2)3/2nP, (74)\nwhereδm=m1−m2,ais the semi-major axis, and nPis a unit vector directed from the center of\nmasstothepointofperiastronof m1(TEGP9.3[388]). Aconsequenceofthisaccelerationwouldbe\nnon-vanishingvalues for d2P/dt2, wherePdenotes the period ofany intrinsic processin the system\n(orbit, spectra, pulsar periods). The observed upper limit on d2Pp/dt2of the binary pulsar PSR\n1913+16 places a strong constraint on such an effect, resulting in t he bound |α3+ζ2|<4×10−5.\nSinceα3has already been constrained to be much less than this (see Table 4) , we obtain a strong\nsolitary bound on ζ2<4×10−5[387].\n4.5 Prospects for improved PPN parameter values\nAnumber ofadvancedexperiments orspacemissionsareunder dev elopmentorhavebeen proposed\nwhich could lead to significant improvements in values of the PPN param eters, ofJ2of the Sun,\nand of˙G/G.\nLLR at the Apache Point Observatory (APOLLO project) could impr ove bounds on the\nNordvedt parameter to the level 3 ×10−5and on˙G/Gto better than 10−13yr−1[406].\nThe BepiColumbo Mercury orbiter is a joint project of the European and Japanese space\nagencies, scheduled for launch in 2015 [33]. In a two-year experime nt, with 6 cm range capability,\nit could yield improvements in γto 3×10−5, inβto 3×10−4, inα1to 10−5, in˙G/Gto 10−13yr−1,\nand inJ2to 3×10−8. An eight-year mission could yield further improvements by factors of 2–5\ninβ,α1, andJ2, and a further factor 15 in ˙G/G[258, 23].\nGAIA isahigh-precisionastrometricorbitingtelescopelaunchedby E SAin 2013(asuccessorto\nHipparcos) [154]. With astrometric capability ranging from 10 to a few hundred microsarcseconds,\n52plus the ability measure the locations of a billion stars down to 20th mag nitude, it could measure\nlight-deflection and γto the 10−6level [257].\nLATOR (Laser Astrometric Test of Relativity) is a concept for a NAS A mission in which two\nmicrosatellites orbit the Sun on Earth-like orbits near superior conj unction, so that their lines of\nsight are close to the Sun. Using optical trackingand an optical inte rferometeron the International\nSpaceStation, itmaybepossibletomeasurethedeflectionoflightwit hsufficientaccuracytobound\nγto a part in 108andJ2to a part in 108, and to measure the solar frame-dragging effect to one\npercent [362, 363].\nAnother concept, proposed for a European Space Agency medium -class mission, is ASTROD\nI (Astrodynamical Space Test of Relativity using Optical Devices), a variant of LATOR involving\na single satellite parked on the far side of the Sun [58]. Its goal is to me asureγto a few parts in\n108,βto six parts in 106andJ2to a part in 109. A possible follow-on mission, ASTROD-GW,\ninvolving three spacecraft, would improve on measurements of tho se parameters and would also\nmeasure the solar frame-dragging effect, as well as look for gravit ational waves.\n535 Strong Gravity and Gravitational Waves: Tests for the\n21st Century\n5.1 Strong-field systems in general relativity\n5.1.1 Defining weak and strong gravity\nIn the solar system, gravity is weak, in the sense that the Newtonia n gravitational potential and\nrelated variables ( U(x,t)∼v2∼p/ρ∼ǫ) are much smaller than unity everywhere. This is the\nbasis for the post-Newtonian expansion and for the “parametrize d post-Newtonian” framework\ndescribed in Section 3.2. “Strong-field” systems are those for whic h the simple 1PN approximation\nof the PPN framework is no longer appropriate. This can occur in a nu mber of situations:\n•Thesystemmaycontainstronglyrelativisticobjects,suchasneut ronstarsorblackholes,near\nand inside which ǫ∼1, and the post-Newtonian approximation breaks down. Neverthe less,\nunder some circumstances, the orbital motion may be such that th e interbody potential and\norbital velocities still satisfy ǫ≪1 so that a kind of post-Newtonian approximation for the\norbital motion might work; however, the strong-field internal gra vity of the bodies could\n(especially in alternative theories of gravity) leave imprints on the or bital motion.\n•The evolution of the system may be affected by the emission of gravit ational radiation. The\n1PN approximation does not contain the effects of gravitational ra diation back-reaction. In\nthe expression for the metric given in Box 2, radiation back-reactio n effects in GR do not\noccur until O(ǫ7/2) ing00,O(ǫ3) ing0i, andO(ǫ5/2) ingij. Consequently, in order to describe\nsuch systems, one must carry out a solution of the equations subs tantially beyond 1PNorder,\nsufficient to incorporate the leading radiation damping terms at 2.5PN order. In addition,\nthe PPN metric described in Section 3.2 is valid in the near zone of the system, i.e. within\none gravitational wavelength of the system’s center of mass. As s uch it cannot describe the\ngravitational waves seen by a detector.\n•The system may be highly relativistic in its orbital motion, so that U∼v2∼1 even\nfor the interbody field and orbital velocity. Systems like this include t he late stage of the\ninspiral of binary systems of neutron stars or black holes, driven b y gravitational radiation\ndamping, prior to a merger and collapse to a final stationary state. Binary inspiral is one\nof the leading candidate sources for detection by the existing LIGO -Virgo network of laser\ninterferometric gravitational-wave observatories and by a futur e space-based interferometer.\nA proper description of such systems requires not only equations f or the motion of the binary\ncarried to extraordinarily high PN orders (at least 3.5PN), but also r equires equations for\nthe far-zone gravitational waveform measured at the detector , that are equally accurate to\nhigh PN orders beyond the leading “quadrupole” approximation.\nOf course, some systems cannot be properly described by any pos t-Newtonian approximation\nbecause their behavior is fundamentally controlled by strong gravit y. These include the imploding\ncores of supernovae, the final merger of two compact objects, the quasinormal-mode vibrations of\nneutron stars and black holes, the structure of rapidly rotating n eutron stars, and so on. Phe-\nnomena such as these must be analyzed using different techniques. Chief among these is the full\nsolution of Einstein’s equations via numerical methods. This field of “n umerical relativity” has be-\ncome a mature branchof gravitationalphysics, whose description is beyond the scope ofthis review\n(see [228, 161, 30] for reviews). Another is black-hole perturbat ion theory (see [261, 201, 323, 37]\nfor reviews).\n545.1.2 Compact bodies and the strong equivalence principle\nWhen dealing with the motion and gravitational wave generation by or biting bodies, one finds\na remarkable simplification within GR. As long as the bodies are sufficient ly well-separated that\none can ignore tidal interactions and other effects that depend up on the finite extent of the bodies\n(such as their quadrupole and higher multipole moments), then all as pects of their orbital behavior\nand gravitational wave generation can be characterized by just t wo parameters: mass and angular\nmomentum. Whether their internal structure is highly relativistic, a s in black holes or neutron\nstars,ornon-relativisticasinthe EarthandSun, onlythe massand angularmomentumareneeded.\nFurthermore, both quantities are measurable in principle by examinin g the external gravitational\nfield of the bodies, and make no reference whatsoever to their inte riors.\nDamour [90] calls this the “effacement” of the bodies’ internal stru cture. It is a consequence of\nthe Strong Equivalence Principle (SEP), described in Section 3.1.2.\nGeneral relativity satisfies SEP because it contains one and only one gravitational field, the\nspacetime metric gµν. Consider the motion of a body in a binary system, whose size is small\ncompared to the binary separation. Surround the body by a region that is large compared to\nthe size of the body, yet small compared to the separation. Becau se of the general covariance of\nthe theory, one can choose a freely-falling coordinate system whic h comoves with the body, whose\nspacetime metric takes the Minkowski form at its outer boundary ( ignoring tidal effects generated\nby the companion). There is thus no evidence of the presence of th e companion body, and the\nstructure of the chosen body can be obtained using the field equat ions of GR in this coordinate\nsystem. Far from the chosen body, the metric is characterized by the mass and angular momentum\n(assuming that one ignores quadrupole and higher multipole moments of the body) as measured\nfar from the body using orbiting test particles and gyroscopes. Th ese asymptotically measured\nquantities are oblivious to the body’s internal structure. A black ho le of massmand a planet of\nmassmwould produce identical spacetimes in this outer region.\nThe geometry of this region surrounding the one body must be matc hed to the geometry\nprovided by the companion body. Einstein’s equations provide consis tency conditions for this\nmatching that yield constraints on the motion of the bodies. These a re the equations of motion.\nAs a result, the motion of two planets of mass and angular momentum m1,m2,J1, andJ2is\nidentical to that of two black holes of the same mass and angular mom entum (again, ignoring tidal\neffects).\nThis effacement does not occur in an alternative gravitional theory like scalar-tensor gravity.\nThere, in addition to the spacetime metric, a scalarfield φis generatedby the masses of the bodies,\nand controls the local value of the gravitational coupling constant (i.e.GLocalis a function of φ).\nNow, in the local frame surrounding one of the bodies in our binary sy stem, while the metric can\nstill be made Minkowskian far away, the scalar field will take on a value φ0determined by the\ncompanion body. This can affect the value of GLocalinside the chosen body, alter its internal\nstructure (specifically its gravitational binding energy) and hence alter its mass. Effectively, each\nbody can be characterized by several mass functions mA(φ), which depend on the value of the\nscalar field at its location, and several distinct masses come into play , such as inertial mass,\ngravitational mass, “radiation” mass, etc. The precise nature of the functions will depend on the\nbody, specifically on its gravitational binding energy, and as a result , the motion and gravitational\nradiation may depend on the internal structure of each body. For compact bodies such as neutron\nstars and black holes these internal structure effects could be lar ge; for example, the gravitational\nbinding energy of a neutron star can be 10–20 percent of its total mass. At 1PN order, the leading\nmanifestation of this phenomenon is the Nordtvedt effect.\nThis is how the study of orbiting systems containing compact object s provides strong-field\ntests of GR. Even though the strong-field nature of the bodies is e ffaced in GR, it is not in other\ntheories, thus any result in agreement with the predictions of GR co nstitutes a kind of “null” test\n55of strong-field gravity.\n5.2 Motion and gravitational radiation in general relativi ty: A history\nAt the most primitive level, the problem of motion in GR is relatively straig htforward, and was\nan integral part of the theory as proposed by Einstein2. The first attempts to treat the motion of\nmultiple bodies, each with a finite mass, were made in the period 1916–1 917by Lorentz and Droste\nand by de Sitter [239, 112]. They derived the metric and equations of motion for a system of N\nbodies, in what today would be called the first post-Newtonian appro ximation of GR (de Sitter’s\nequations turned out to contain some important errors). In 1916 , Einstein took the first crack at\na study of gravitational radiation, deriving the energy emitted by a body such as a rotating rod or\ndumbbell, heldtogetherbynon-gravitationalforces[129]. Hemade someunjustifiedassumptionsas\nwell as a trivial numerical error(later corrected by Eddington [127 ]), but the underlying conclusion\nthat dynamical systems would radiate gravitational waves was cor rect.\nThe next significant advance in the problem of motion came 20 years la ter. In 1938, Einstein,\nInfeld and Hoffman published the now legendary “EIH” paper, a calcu lation of the N-body equa-\ntions of motion using only the vacuum field equations of GR [130]. They t reated each body in the\nsystem as a spherically symmetric object whose nearby vacuum ext erior geometry approximated\nthat of the Schwarzschildmetric of a static spherical star. They t hen solved the vacuum field equa-\ntions for the metric between each body in the system in a weak field, s low-motion approximation.\nThen, using a primitive version of what today would be called “matched asymptotic expansions”\nthey showed that, in order for the nearby metric of each body to m atch smoothly to the interbody\nmetric at each orderin the expansion, certainconditions on the mot ion ofeach body had to be met.\nTogether, these conditions turned out to be equivalent to the Dro ste-Lorentz N-body equations\nof motion. The internal structure of each body was irrelevant, ap art from the requirement that\nits nearby field be approximately spherically symmetric, a clear illustra tion of the “effacement”\nprinciple.\nAround the same time, there occurred an unusual detour in the pr oblem of motion. Using\nequations of motion based on de Sitter’s paper, specialized to two bo dies, Levi-Civita [230] showed\nthat the center of mass of a binary star system would suffer an acc eleration in the direction of the\npericenter of the orbit, in an amount proportional to the differenc e between the two masses, and to\nthe eccentricity of the orbit. Such an effect would be a violation of th e conservation of momentum\nfor isolated systems caused by relativistic gravitational effects. L evi-Civita even went so far as to\nsuggestlookingforthis effect in selected nearbyclosebinarystars ystems. However,Eddingtonand\nClark [126] quickly pointed out that Levi-Civita had based his calculatio ns on de Sitter’s flawed\nwork; when correct two-body equations of motion were used, the effect vanished, and momentum\nconservation was upheld. Robertson confirmed this using the EIH e quations of motion [314]. Such\nan effect can only occur in theories of gravity that lack the appropr iate conservation laws (Sec.\n4.4.3).\nTherewasongoingconfusionoverwhether gravitationalwavesar erealorareartifactsofgeneral\ncovariance. Although Eddington was credited with making the unfor tunate remark that gravita-\ntional waves propagate “with the speed of thought”, he did clearly elucidate the difference between\nthe physical, coordinate independent modes and modes that were p urely coordinate artifacts [127].\nBut in 1936, in a paper submitted to the Physical Review , Einstein and Rosen claimed to prove\nthat gravitational waves could not exist; the anonymous referee of their paper found that they had\nmade an error. Upset that the journal had sent his paper to a ref eree (a newly instituted practice),\nEinstein refused to publish there again. A corrected paper by Einst ein and Rosen showing that\n2This history is adapted from Ref. [403]. For a detailed techn ical and historical review of the problem of motion,\nsee Damour [90]\n56gravitational waves didexist – cylindrical waves in this case – was published elsewhere [131]. Fif ty\nyears later it was revealed that the anonymous referee was H. P. R obertson [195].\nRoughly 20 more years would pass before another major attack on the problem of motion.\nFock in the USSR and Chandrasekhar in the US independently develop ed and systematized the\npost-Newtonian approximation in a form that laid the foundation for modern post-Newtonian\ntheory [145, 68]. They developed a full post-Newtonian hydrodyna mics, with the ability to treat\nrealistic, self-gravitating bodies of fluid, such as stars and planets . In the suitable limit of “point”\nparticles, or bodies whose size is small enough compared to the inter body separations that finite-\nsize effects such as spin and tidal interactions can be ignored, their equations of motion could be\nshown to be equivalent to the EIH and the Droste-Lorentz equatio ns of motion.\nThe next important period in the history of the problem of motion was 1974 –1979, initiated\nby the 1974 discovery of the binary pulsar PSR 1913+16 by Hulse and Taylor [180]. Around the\nsame time there occurred the first serious attempt to calculate th e head-on collision of two black\nholes using purely numerical solutions of Einstein’s equations, by Sma rr and collaborators [339].\nThe binary pulsar consists of two neutron stars, one an active puls ar detectable by radio\ntelescopes, the other very likely an old, inactive pulsar (Sec. 6.1). E ach neutron star has a mass\nof around 1.4 solar masses. The orbit of the system was seen immedia tely to be quite relativistic,\nwith an orbital period of only eight hours, and a mean orbital speed o f 200 km/s, some four times\nfaster than Mercury in its orbit. Within weeks of its discovery, nume rous authors pointed out that\nPSR 1913+16 would be an important new testing ground for GR. In pa rticular, it could provide\nfor the first time a test of the effects of the emission of gravitation al radiation on the orbit of the\nsystem.\nHowever, the discovery revealed an ugly truth about the “problem of motion”. As Ehlers\net al.pointed out in an influential 1976 paper [128], the general relativistic problem of motion\nand radiation was full of holes large enough to drive trucks through . They pointed out that most\ntreatments of the problem used “delta functions” as a way to appr oximate the bodies in the system\nas point masses. As a consequence, the “self-field”, the gravitat ional field of the body evaluated at\nits own location, becomes infinite. While this is not a major issue in Newto nian gravity or classical\nelectrodynamics, the non-linear nature of GR requires that this infi nite self-field contribute to\ngravity. In the past, such infinities had been simply swept under the rug. Similarly, because\ngravitational energy itself produces gravity it thus acts as a sour ce throughout spacetime. This\nmeans that, when calculating radiative fields, integrals for the multip ole moments of the source\nthat are so useful in treating radiation begin to diverge. These dive rgent integrals had also been\nroutinely swept under the rug. Ehlers et al.further pointed out that the true boundary condition\nfor any problem involvingradiationby an isolated system should be one of “no incoming radiation”\nfrom the past. Connecting this boundary condition with the routine use of retarded solutions of\nwaveequations was not a trivial matter in GR. Finally they pointed out that there was no evidence\nthat the post-Newtonian approximation, so central to the proble m of motion, was a convergent or\neven asymptotic sequence. Nor had the approximation been carrie d out to high enough order to\nmake credible error estimates.\nDuring this time, some authorsevenarguedthat the “quadrupolef ormula”for the gravitational\nenergy emitted by a system (see below), while correct for a rotatin g dumbell as calculated by\nEinstein, was actually wrongfor a binary system moving under its own gravity. The discovery in\n1979 that the rate of decay of the orbit of the binary pulsar was in a greement with the standard\nquadrupole formula made some of these arguments moot. Yet the q uestion raised by Ehlers et al.\nwas still relevant: is the quadrupole formula for binary systems an a ctual prediction of GR?\nMotivated by the Ehlers et al.critique, numerous workers began to address the holes in the\nproblem of motion, and by the late 1990s most of the criticisms had be en answered, particularly\nthose related to divergences. For a detailed history of the ups and downs of the subject of motion\nand gravitational waves, see [196].\n57The problem of motion and radiation in GR has received renewed intere st since 1990, with pro-\nposals for construction of large-scale laser interferometric grav itational wave observatories. These\nproposals culminated in the construction and operation of LIGO in th e US, VIRGO and GEO600\nin Europe, and TAMA300 in Japan, the construction of an undergro und observatory KAGRA in\nJapan, and the possible construction of a version of LIGO in India. A dvanced versions of LIGO\nand VIRGO are expected to be online and detecting gravitational wa ves around 2016. An inter-\nferometer in space has recently been selected by the European Sp ace Agency for a launch in the\n2034 time frame.\nA leading candidate source of detectable waves is the inspiral, driven by gravitational radiation\ndamping, of a binary system of compact objects (neutron stars o r black holes) (for a review of\nsources of gravitational waves, see [324]). The analysis of signals f rom such systems will require\ntheoretical predictions from GR that are extremely accurate, we ll beyond the leading-order predic-\ntion of Newtonian or even post-Newtonian gravityfor the orbits, a nd well beyond the leading-order\nformulae for gravitational waves.\nThis presented a major theoretical challenge: to calculate the mot ion and radiation of systems\nof compact objects to very high PN order, a formidable algebraic ta sk, while addressing the issues\nof principle raised by Ehlers et al., sufficiently well to ensure that the results were physically\nmeaningful. Thischallengehasbeen largelymet, sothatwemaysoons eearemarkableconvergence\nbetweenobservationaldataandaccuratepredictionsofgravita tionaltheorythatcouldprovidenew,\nstrong-field tests of GR.\n5.3 Compact binary systems in general relativity\n5.3.1 Einstein’s equations in “relaxed” form\nHere we give a brief overview of the modern approach to the problem of motion and gravitational\nradiation in GR. For a full pedagogical treatment, see [297].\nThe Einstein equations Gµν= 8πGTµνare elegant and deceptively simple, showing geometry\n(in the form ofthe Einstein tensor Gµν, which is afunction ofspacetime curvature)being generated\nby matter (in the form of the material stress-energy tensor Tµν). However, this is not the most\nuseful form for actual calculations. For post-Newtonian calculat ions, a far more useful form is\nthe so-called “relaxed” Einstein equations, which form the basis of t he program of approximating\nsolutions of Einstein’s equations known as post-Minkowskian theory and post-Newtonian theory.\nThe starting point is the so-called “gothic inverse metric”, defined b ygαβ≡√−ggαβ, whereg\nis the determinant of gαβ. One then defines the gravitational potential hαβ≡ηαβ−gαβ. After\nimposing the the de Donder or harmonic gauge condition ∂hαβ/∂xβ= 0 (summation on repeated\nindices is assumed), one can recast the exact Einstein field equation s into the form\n/squarehαβ=−16πGταβ, (75)\nwhere/square≡ −∂2/∂t2+∇2is the flat-spacetime wave operator. This form of Einstein’s equatio ns\nbears a striking similarity to Maxwell’s equations for the vector poten tialAαin Lorentz gauge:\n/squareAα=−4πJα,∂Aα/∂xα= 0. There is a key difference, however: The source on the right han d\nside of Eq. (75) is given by the “effective” stress-energy pseudot ensor\nταβ= (−g)/parenleftBig\nTαβ+tαβ\nLL+tαβ\nH/parenrightBig\n, (76)\nwheretαβ\nLLandtαβ\nHare the Landau-Lifshitz pseudotensor and a harmonic pseudoten sor, given by\nterms quadratic (and higher) in hαβand its derivatives (see [297], Eqs. (6.5, 6.52, 6.53) for explicit\nformulae). In GR, the gravitational field itself generates gravity, a reflection of the nonlinearity of\nEinstein’s equations, and in contrast to the linearity of Maxwell’s equa tions.\n58Eq. (75) is exact, and depends only on the assumption that the rele vant parts of spacetime\ncan be covered by harmonic coordinates. It is called “relaxed” beca use it can be solved formally\nas a functional of source variables without specifying the motion of the source, in the form (with\nG= 1)\nhαβ(t,x) = 4/integraldisplay\nCταβ(t−|x−x′|,x′)\n|x−x′|d3x′, (77)\nwherethe integrationis overthe past flat-spacetimenull cone Cofthe field point ( t,x). The motion\nof the source is then determined either by the equation ∂ταβ/∂xβ= 0 (which follows from the\nharmonic gauge condition), or from the usual covariant equation o f motionTαβ;β= 0, where the\nsubscript ;βdenotes a covariant divergence. This formal solution can then be it erated in a slow\nmotion (v <1) weak-field ( ||hαβ||<1) approximation. One begins by substituting hαβ\n0= 0 into\nthe sourceταβin Eq. (77), and solving for the first iterate hαβ\n1, and then repeating the procedure\nsufficiently many times to achieve a solution of the desired accuracy. For example, to obtain the\n1PN equations of motion, twoiterations are needed (i.e. hαβ\n2must be calculated); likewise, to\nobtain the leading gravitational waveform for a binary system, two iterations are needed.\nAt the same time, just as in electromagnetism, the formal integral (77) must be handled dif-\nferently, depending on whether the field point is in the far zone or th e near zone. For field points\nin the far zone or radiation zone, |x|>R, whereRis a distance of the order of a gravitational\nwavelength, the field can be expanded in inverse powers of R=|x|in a multipole expansion,\nevaluated at the “retarded time” t−R. The leading term in 1 /Ris the gravitational waveform.\nFor field points in the near zone or induction zone, |x| ∼ |x′|<R, the field is expanded in powers\nof|x−x′|about the local time t, yielding instantaneous potentials that go into the equations of\nmotion.\nHowever, because the source ταβcontainshαβitself, it is not confined to a compact region,\nbut extends over all spacetime. As a result, there is a danger that the integrals involved in the\nvarious expansions will diverge or be ill-defined. This consequence of the non-linearity of Einstein’s\nequations has bedeviled the subject of gravitational radiation for decades. Numerous approaches\nhave been developed to try to handle this difficulty. The post-Minkow skian method of Blanchet,\nDamour, and Iyer[46, 47, 48, 96, 49, 44] solvesEinstein’s equation sby two different techniques, one\nin the near zone and one in the far zone, and uses the method of sing ular asymptotic matching to\njoin the solutions in an overlap region. The method provides a natura l “regularization” technique\nto control potentially divergent integrals (see [45] for a thorough review). The “Direct Integration\nof the Relaxed Einstein Equations” (DIRE) approach of Will, Wiseman, and Pati [398, 289, 290]\nretains Eq. (77) as the global solution, but splits the integration int o one over the near zone and\nanother over the far zone, and uses different integration variable s to carry out the explicit integrals\noverthe two zones. In the DIRE method, all integralsarefinite and convergent. Itoh and Futamase\nused an extension of the Einstein–Infeld–Hoffman matching approa ch combined with a specific\nmethod for taking a point-particle limit [182], while Damour, Jaranowsk i, and Sch¨ afer pioneered\nan ADM Hamiltonian approach that focuses on the equations of motio n [188, 189, 97, 98, 99].\nThese methods assume from the outset that gravity is sufficiently w eak that ||hαβ||<1 and\nharmonic coordinates exists everywhere, including inside the bodies . Thus, in order to apply the\nresults to cases where the bodies may be neutron stars or black ho les, one relies upon the SEP\nto argue that, if tidal forces are ignored, and equations are expr essed in terms of masses and\nspins, one can simply extrapolate the results unchanged to the situ ation where the bodies are\nultrarelativistic. While no general proof of this exists, it has been sh own to be valid in specific\ncircumstances, such as through 2PN order in the equations of mot ion [163, 266], and for black\nholes moving in a Newtonian background field [90].\nMethods such as these have resolved most of the issues that led to criticism of the foundations\nof gravitational radiation theory during the 1970s.\n595.3.2 Equations of motion and gravitational waveform\nAmong the results of these approaches are formulae for the equa tions of motion and gravitational\nwaveform of binary systems of compact objects, carried out to h igh orders in a PN expansion. For\na review of the latest results of high-order PN calculations, see [45 ]. Here we shall only state the\nkey formulae that will be needed for this review. For example, the re lative two-body equation of\nmotion has the form\na=dv\ndt=m\nr2{−ˆ n+A1PN+A2PN+A2.5PN+A3PN+A3.5PN+...}, (78)\nwherem=m1+m2is the total mass, r=|x1−x2|,v=v1−v2, andˆ n= (x1−x2)/r. The\nnotation AnPNindicates that the term is O(ǫn) relative to the Newtonian term −ˆ n. Explicit\nand unambiguous formulae for non-spinning bodies through 3.5PN or der have been calculated by\nvarious authors [45]. Here we quote only the 1PN corrections and th e leading radiation-reaction\nterms at 2.5PN order:\nA1PN=/braceleftbigg\n(4+2η)m\nr−(1+3η)v2+3\n2η˙r2/bracerightbigg\nˆ n+(4−2η)˙rv, (79)\nA2.5PN=8\n5ηm\nr/braceleftbigg/parenleftbigg\n3v2+17\n3m\nr/parenrightbigg\n˙rˆ n−/parenleftBig\nv2+3m\nr/parenrightBig\nv/bracerightbigg\n, (80)\nwhereη=m1m2/(m1+m2)2. The radiation-reaction acceleration is expressed in the so-called\nDamour-Deruelle gauge. These terms are sufficient to analyze the o rbit and evolution of the binary\npulsar (see Section 6.1). For example, the 1PN terms are responsib le for the periastron advance of\nan eccentric orbit, given by\n˙ω=6παmfb\na(1−e2). (81)\nwhereaandeare the semi-major axis and eccentricity of the orbit, respectively , andfbis the\norbital frequency, given to the needed order by Kepler’s third law 2 πfb= (m/a3)1/2.\nAnother product is a formula for the gravitational field far from th e system, written schemati-\ncally in the form\nhij=2m\nR/braceleftBig\nQij+Qij\n0.5PN+Qij\n1PN+Qij\n1.5PN+Qij\n2PN+Qij\n2.5PN+.../bracerightBig\n, (82)\nwhereRis the distance from the source, and the variables are to be evaluat ed at retarded time\nt−R. The leading term is the so-called quadrupole formula\nhij(t,x) =2\nR¨Iij(t−R), (83)\nwhereIijis the quadrupole moment of the source, and overdots denote time derivatives. For a\nbinary system this leads to\nQij= 2η/parenleftbigg\nvivj−mˆniˆnj\nr/parenrightbigg\n. (84)\nFor binary systems, explicit formulae for the waveform through 3.5 PN order have been derived\n(see [50] for a ready-to-use presentation of the waveform to 2P N order for circular orbits; see [45]\nfor a full review).\nGiventhe gravitationalwaveform,onecancomputethe rateat wh ichenergyiscarriedoffbythe\nradiation (schematically/integraltext˙h˙hdΩ, the gravitational analog of the Poynting flux). The lowest-orde r\nquadrupole formula leads to the gravitational wave energy flux\n˙E=8\n15µη\nr/parenleftBigm\nr/parenrightBig3/parenleftbig\n12v2−11˙r2/parenrightbig\n. (85)\n60This has been extended to 3.5PN order beyond the quadrupole form ula [45]. Formulae for fluxes\nof angular and linear momentum can also be derived. The 2.5PN radiatio n-reaction terms in the\nequation of motion (78) result in a damping of the orbital energytha t precisely balances the energy\nflux (85) determined from the waveform. Averaged over one orbit , this results in a rate of increase\nof the binary’s orbital frequency given by\n˙fb=192π\n5f2\nb(2πMfb)5/3F(e),\nF(e) = (1−e2)−7/2/parenleftbigg\n1+73\n24e2+37\n96e4/parenrightbigg\n,(86)\nwhereMis the so-called “chirp” mass, given by M=η3/5m. Notice that by making precise\nmeasurements of the phase Φ( t) = 2π/integraltexttf(t′)dt′of either the orbit or the gravitational waves (for\nwhichf= 2fbfor the dominant component) as a function of the frequency, one in effect measures\nthe “chirp” mass of the system.\nThese formalisms have also been generalized to include the leading effe cts of spin-orbit and\nspin-spin coupling between the bodies as well as many next-to-leadin g-order corrections [45].\nAnother approach to gravitational radiation is applicable to the spe cial limit in which one\nmass is much smaller than the other. This is the method of black hole pe rturbation theory. One\nbegins with an exact background spacetime of a black hole, either th e non-rotating Schwarzschild\nor the rotating Kerr solution, and perturbs it according to gµν=g(0)\nµν+hµν. The particle moves\non a geodesic of the background spacetime, and a suitably defined s ource stress-energy tensor for\nthe particle acts as a source for the gravitational perturbation a nd wave field hµν. This method\nprovides numerical results that are exact in v, as well as analytical results expressed as series\nin powers of v, both for non-rotating and for rotating black holes. For non-rot ating holes, the\nanalytical expansions have been carried to the impressive level of 2 2PN order, or ǫ22beyond the\nquadrupole approximation [153], and for rotating Kerr black holes, t o 20PN order [328]. All results\nofblackholeperturbationagreepreciselywith the m1→0limit ofthe PNresults, up tothe highest\nPN order where they can be compared (for reviews of earlier work s ee [261, 201, 323]).\n5.4 Compact binary systems in scalar-tensor theories\nBecause of the recent resurgence of interest in scalar-tensor t heories of gravity, motivated in part\nby string theory and f(R) theories, considerable work has been done to analyze the motion a nd\ngravitational radiation from systems of compact objects in this cla ss of theories. In earlier work,\nEardley [125] was the first to point out the existence of dipole gravit ational radiation from self-\ngravitating bodies in Brans-Dicke theory, and Will [401] worked out the lowest-order monopole,\ndipole and quadrupole radiation flux in general scalar-tensor theor ies (as well as in a number of\nalternative theories) for bodies with weak self-gravity. Using the a pproach pioneered by Eard-\nley [125] for incorporating strongly self-gravitating bodies into sca lar-tensor calculations, Will and\nZaglauer [400] calculated the 1PN equations of motion along with the m onopole-quadrupole and\ndipole energy flux for compact binary systems; Alsing et al.[6] extended these results to the case\nof Brans-Dicke theory with a massive scalar field. However, the exp ressions for the energy flux\nin those works were incomplete, because they failed to include some im portant post-Newtonian\ncorrections in the scalar part of the radiation that actually contrib ute at the same order as the\nquadrupole contributions from the tensor part. Damour and Espo sito-Far` ese[93] obtained the cor-\nrect monopole-quadrupole and dipole energy flux, working in the Eins tein-frame representation of\nscalar-tensortheories, andgavepartialresultsforthe equatio nsofmotionto 2PNorder. Mirshekari\nand Will [262] obtained the complete compact-binary equations of mo tion in general scalar-tensor\ntheories through 2.5PN order, and obtained the energy loss rate in complete agreement with the\n61flux result from Damour and Esposito-Far` ese. Lang [222] obtaine d the gravitational-wave signal\nto 2PN order.\nNotwithstanding the very tight bound on the scalar-tensor couplin g parameter ωfrom Cassini\nmeasurements in the solar system, this effort is motivated by a desir e to test this theory in strong-\nfield situations, whether by binary pulsar observations, or by meas urements of gravitational radi-\nation from compact binary inspiral. Here we summarize the key result s in a manner that parallels\nthe results for GR.\n5.4.1 Scalar-tensor equations in “relaxed” form\nThe field equations of scalar-tensor theory can be cast in a form sim ilar to the “relaxed” equations\nof GR. Here one works in terms of an auxiliary metric ˜ gαβ≡ϕgαβ, whereϕ≡(φ/φ0) andφ0is the\nasymptotic value of the scalar field, and defines the auxiliary gothic in verse metric ˜gαβ≡√−˜g˜gαβ,\nand the auxiliary tensor gravitational potential ˜hαβ≡ηαβ−˜gαβ, along with the harmonic gauge\ncondition∂˜hαβ/∂xβ= 0. The field equations then take the form\n/square˜hαβ=−16πG˜ταβ, (87)\nwhere/square≡ −∂2/∂t2+∇2is again the flat-spacetime wave operator, and where\n˜ταβ= (−˜g)/parenleftbiggϕ\nφ0Tαβ+˜tαβ\nφ+˜tαβ\nLL+˜tαβ\nH/parenrightbigg\n, (88)\nwhere˜tαβ\nφ≡(3+2ω)ϕ−2ϕ,µϕ,ν(˜gµα˜gνβ−1\n2˜gµν˜gαβ) is a scalar stress-energy tensor, and where ˜tαβ\nLL\nand˜tαβ\nHhave exactly the same forms, when written in terms of ˜hαβ, as their counterparts in GR do\nin terms of hαβ. Note that this is equivalent to formulating the relaxed equations of scalar-tensor\ntheory in the Einstein conformal frame. The field equation for the s calar field can be written in\nthe form /squareϕ=−8πG˜τs, where ˜τsis a source consisting of a matter term, a scalar energy density\nterm and a term that mixes ˜hαβandϕ(see [262] for details).\nIn order to incorporate the internal gravity of compact, self-gr avitating bodies, it is common to\nadopt an approach pioneered by Eardley [125], based in part on gene ral arguments dating back to\nDicke, in which one treats the matter energy-momentum tensor as a sum of delta functions located\nat the position of each body, but assumes that the mass of each bo dy is a function MA(φ) of the\nscalar field. This reflects the fact that the gravitational binding en ergy of the body is controlled\nby the value of the gravitational constant, which is directly related to the value of the background\nscalar field in which the body finds itself. The underlying assumption is t hat the timescale for\norbital motion is long compared to the internal dynamical timescale o f the body, so that the\nbody’s structure evolves adiabatically in response to the changing s calar field. Consequently, the\nmatter action will havean effective dependence on φ, and as a result the field equations will depend\non the “sensitivity” of the mass of each body to variations in the sca lar field, holding the total\nnumber of baryons fixed. The sensitivity of body Ais defined by\nsA≡/parenleftbiggdlnMA(φ)\ndlnφ/parenrightbigg\n, (89)\nevaluated at a value of the scalar field far from the body. For neutr on stars, the sensitivity depends\non the mass and equation of state of the star and is typically of orde r 0.2; in the weak-field limit,\nsAis proportional to the Newtonian self-gravitational energy per un it mass of the body. From a\ntheorem of Hawking [170], for stationary black holes, it is known that sBH= 1/2. This means,\namong other things, that the source ˜ τsfor the scalar field will contain an explicit term dependent\nupon∂T/∂φ, because of the dependence on MA(φ).\n62Table 6: Parameters used in the equations of motion\nParameter Definition\nScalar-tensor parameters\nζ 1/(4+2ω0)\nλ (dω/dϕ)0ζ2/(1−ζ)2\nSensitivities\nsA [dlnMA(φ)/dlnφ]0\ns′\nA [d2lnMA(φ)/dlnφ2]0\nEquation of motion parameters\nα 1−ζ+ζ(1−2s1)(1−2s2)\n¯γ −2α−1ζ(1−2s1)(1−2s2)\n¯β1α−2ζ(1−2s2)2(λ(1−2s1)+2ζs′\n1)\n¯β2α−2ζ(1−2s1)2(λ(1−2s2)+2ζs′\n2)\n5.4.2 Equations of motion and gravitational waveform\nBy following the methods of post-Minkowskian theory adapted to sc alar-tensor theory, it has\nbeen possible to derive the equations of motion for binary systems o f compact bodies to 2.5PN\norder [262] and the gravitational-wavesignal and energy flux to 1P N order beyond the quadrupole\napproximation. Here we shall quote selected results in parallel with t hose quoted in Sec. 5.3.2.\nThe relative two-body equation of motion has the form\na=dv\ndt=αm\nr2{−ˆ n+A1PN+A1.5PN+A2PN+A2.5PN+A3PN+A3.5PN+...}.(90)\nThe key difference between this PN series and that in GR is the presen ce of a radiation-reaction\nterm at 1.5PN order, caused by the emission of dipole gravitational r adiation. The key parameters\nthat appear in the two-body equations of motion are given in Table 6. Notice that αplays the role\nof a two-body gravitationalinteraction parameter; ¯ γand¯βAare the two-body versions of γ−1 and\nβ−1 respectively. In the limit of weakly self-gravitating bodies ( sA→0),α→1, ¯γ→γ−1 =−2ζ\nand¯βA→β−1 =ζλ(compare with Table 3).\nHere we quote only the 1PN corrections and the leading radiation-re action terms at 1.5PN and\n2.5PN order:\nA1PN=/braceleftbigg\n(4+2η+2¯γ+2¯β+−2ψ¯β−)αm\nr−(1+3η+ ¯γ)v2+3\n2η˙r2/bracerightbigg\nˆ n\n+(4−2η+2¯γ)˙rv, (91)\nA1.5PN=4\n3ηαm\nrζS2\n−(3˙rˆ n−v), (92)\nA2.5PN=8\n5ηαm\nr/braceleftBig/parenleftBig\na1v2+a2αm\nr+a3˙r2/parenrightBig\n˙rˆ n−/parenleftBig\nb1v2+b2m\nr+b3˙r2/parenrightBig\nv/bracerightBig\n,(93)\n63where\na1= 3−5\n2¯γ+15\n2¯β++5\n8ζS2\n−(9+4¯γ−2η)+15\n8ζψS−S+,\na2=17\n3+35\n6¯γ−95\n6¯β+−5\n24ζS2\n−/bracketleftbig\n135+56¯γ+8η+32¯β+/bracketrightbig\n+30ζS−/parenleftbiggS−¯β++S+¯β−\n¯γ/parenrightbigg\n−5\n8ζψS−/parenleftbigg\nS+−32\n3S−¯β−+16S+¯β++S−¯β−\n¯γ/parenrightbigg\n−40ζ/parenleftbiggS+¯β++S−¯β−\n¯γ/parenrightbigg2\n,\na3=25\n8/bracketleftbig\n2¯γ−ζS2\n−(1−2η)−4¯β+−ζψS−S+/bracketrightbig\n,\nb1= 1−5\n6¯γ+5\n2¯β+−5\n24ζS2\n−(7+4¯γ−2η)+5\n8ζψS−S+,\nb2= 3+5\n2¯γ−5\n2¯β+−5\n24ζS2\n−/bracketleftbig\n23+8¯γ−8η+8¯β+/bracketrightbig\n+10\n3ζS−/parenleftbiggS−¯β++S+¯β−\n¯γ/parenrightbigg\n−5\n8ζψS−/parenleftbigg\nS+−8\n3S−¯β−+16\n3S+¯β++S−¯β−\n¯γ/parenrightbigg\n,\nb3=5\n8/bracketleftbig\n6¯γ+ζS2\n−(13+8¯γ+2η)−12¯β+−3ζψS−S+/bracketrightbig\n. (94)\nwhere\n¯β±≡1\n2(¯β1±¯β2),\nψ≡(m1−m2)/m,\nS−≡ −α−1/2(s1−s2),\nS+≡α−1/2(1−s1−s2). (95)\nThe periastron advance that results from these equations is given by\n˙ω=6παmfb\na(1−e2)/bracketleftbigg\n1+2¯γ−¯β+−ψ¯β−\n3/bracketrightbigg\n. (96)\nwhere 2πfb= (αm/a3)1/2.\nThe tensor part of the gravitational waveform has the schematic form\n˜hij=2(1−ζ)m\nR/braceleftBig\nQij+Qij\n0.5PN+Qij\n1PN+Qij\n1.5PN+Qij\n2PN+.../bracerightBig\n, (97)\nwhere\nQij= 2η/parenleftbigg\nvivj−αmˆniˆnj\nr/parenrightbigg\n. (98)\nContributions to the tensor waveform through 2PN order have be en derived by Lang [222]. The\nscalar waveform is given by φ=φ0(1+Ψ), where,\nΨ =ζηα1/2m\nR{Ψ−0.5PN+Ψ0PN+Ψ0.5PN+Ψ1PN+...}, (99)\n64where, ignoring terms that are constant in time,\nΨ−0.5PN= 4S−(ˆN·v),\nΨ0PN= 2(S+−ψS−)/bracketleftBig\n(ˆN·v)2−αm\nr(ˆN·x)2/bracketrightBig\n−2αm\nr/bracketleftbigg\n3S+−ψS−−8/parenleftbiggS+¯β++S−¯β−\n¯γ/parenrightbigg/bracketrightbigg\n,\nΨ0.5PN=−∂\n∂t/braceleftbigg\n(ˆN·x)/bracketleftbigg\n(3−4η)S−−ψS++8ψ/parenleftbiggS+¯β++S−¯β−\n¯γ/parenrightbigg\n−8/parenleftbiggS−¯β++S+¯β−\n¯γ/parenrightbigg/bracketrightbigg/bracerightbigg\n+1\n3[(1−2η)S−−ψS+]∂3\n∂t3(ˆN·x)3, (100)\nwhereˆNis a unit vector directed toward the observer.\nThe energy flux is given by\ndE/dt=−4\n3ζµη\nr/parenleftBigαm\nr/parenrightBig3\nS2\n−−8\n15µη\nr/parenleftBigαm\nr/parenrightBig3/parenleftbig\nκ1v2−κ2˙r2/parenrightbig\n, (101)\nwhere the first term is the contribution of dipole radiation (formally o f -1PNorder), and the second\nterm (formally of 0PN order, according to the conventional rules o f counting) is a combination of\nquadrupole radiation, PN corrections to monopole and dipole radiatio n, and even a cross-term\nbetween dipole and octupole radiation. The coefficients κ1andκ2are given by [262]\nκ1= 12+5¯γ−5ζS2\n−(3+ ¯γ+2¯β+)+10ζS−/parenleftbiggS−¯β++S+¯β−\n¯γ/parenrightbigg\n+10ζψS2\n−¯β−−10ζψS−/parenleftbiggS+¯β++S−¯β−\n¯γ/parenrightbigg\n,\nκ2= 11+45\n4¯γ−40¯β+−5ζS2\n−/bracketleftbig\n17+6¯γ+η+8¯β+/bracketrightbig\n+90ζS−/parenleftbiggS−¯β++S+¯β−\n¯γ/parenrightbigg\n+40ζψS2\n−¯β−−30ζψS−/parenleftbiggS+¯β++S−¯β−\n¯γ/parenrightbigg\n−120ζ/parenleftbiggS+¯β++S−¯β−\n¯γ/parenrightbigg2\n.(102)\nThese results are in complete agreement with the total energy flux to−1PNand 0PNorders,\nas calculated by Damour and Esposito-Far` ese [92]. They disagree w ith the flux formula of Will\nand Zaglauer [400], as repeated in earlier versions of this Living Review as well as in [6]. Will\nand Zaglauer [400] failed to take into account PN corrections to the dipole term induced by PN\ncorrections in the equations of motion, and a dipole-octupole cross term in the scalar energy\nflux, all of which contribute to the flux at the same 0PN order as the quadrupole and monopole\ncontributions.\nIn the limit of weakly self-gravitating bodies the equations of motion a nd energy flux (including\nthe dipole term) reduce to the standard results quoted in TEGP [388 ].\n5.4.3 Binary systems containing black holes\nRoger Penrose was probably the first to conjecture, in a talk at th e 1970 Fifth Texas Symposium,\nthat black holes in Brans-Dicke theory are identical to their GR coun terparts [357]. Motivated by\nthis remark, Thorne and Dykla showed that during gravitational co llapse to form a black hole,\nthe Brans-Dicke scalar field is radiated away, in accord with Price’s th eorem, leaving only its\nconstant asymptotic value, and a GR black hole [357]. Hawking [170] p roved on general grounds\nthat stationary, asymptotically flat black holes in vacuum in BD are th e black holes of GR. The\n65basic idea is that black holes in vacuum with non-singular event horizon s cannot support scalar\n“hair”. Hawking’s theorem was extended to the class of f(R) theories that can be transformed\ninto generalized scalar-tensor theories by Sotiriou and Faraoni [3 42].\nA consequence of these theorems is that, for a stationary black h ole,s= 1/2. Another way to\nsee this is to note that, because all information about the matter t hat formed the black hole has\nvanished behind the event horizon, the only scale on which the mass o f the hole can depend is the\nPlanck scale, and thus M∝MPlanck∝G−1/2∝φ1/2. Hences= 1/2.\nIf both bodies in the binary system are black holes, then setting sA= 1/2 for each body,\nall the parameters ¯ γ,¯βAandS±vanish identically, and α= 1−ζ. But since αappears only\nin the combination with αm, a simple rescaling of each mass puts all equations into complete\nagreement with those of GR. This is also true for the 2PN terms in the equations of motion [262].\nThus, in the class of scalar-tensor theories discussed here, binar y black holes are observationally\nindistinguishable from their GR counterparts, at least to high order s in a PN approximation.\nIf one of the members of the binary system, say body 2, is a black ho le, withs2= 1/2,\nthenα= 1−ζ, ¯γ=¯βA= 0, and hence, through 1PN order, the motion is again identical to\nthat in GR. At 1 .5PN order, dipole radiation reaction kicks in, since s1<1/2. In this case,\nS−=S+=α−1/2(1−2s1)/2, and thus the 1 .5PN coefficients in the relative equation of motion\n(92) take the form\nA1.5PN=5\n8Q,\nB1.5PN=5\n24Q, (103)\nwhere\nQ≡ζ\n1−ζ(1−2s1)2=1\n3+2ω0(1−2s1)2, (104)\nwhile the coefficients in the energy loss rate simplify to\nκ1= 12−15\n4Q,\nκ2= 11−5\n4Q(17+η). (105)\nThe result is that the motion of a mixed compact binary system throu gh 2.5PN order differs from\nits general relativistic counterpart only by terms that depend on a single parameter Q, as defined\nby Eq. (104).\nIt should be pointed out that there areways to induce scalar hair on a black hole. One is\nto introduce a potential V(φ), which, depending on its form, can help to support a non-trivial\nscalar field outside a black hole. Another is to introduce matter. A co mpanion neutron star is\nan obvious choice, and such a binary system in scalar-tensor theor y is clearly different from its\ngeneral relativistic counterpart. Another possibility is a distributio n of cosmological matter that\ncan support a time-varying scalar field at infinity. This possibility has b een called “Jacobson’s\nmiracle hair-growth formula” for black holes, based on work by Jaco bson [185, 175].\n666 Stellar System Tests of Gravitational Theory\n6.1 The binary pulsar and general relativity\nThe 1974 discovery of the binary pulsar B1913+16 by Joseph Taylor and Russell Hulse during a\nroutine search for new pulsars provided the first possibility of prob ing new aspects of gravitational\ntheory: the effects of strong relativistic internal gravitational fi elds on orbital dynamics, and the\neffects of gravitational radiation reaction. For reviews of the disc overy, see the published Nobel\nPrize lectures by Hulse and Taylor [179, 355]. For reviews of the curr ent status of testing general\nrelativity with pulsars, including binary and millisecond pulsars, see [24 0, 345, 381]; specific details\non every pulsar discovered to date, along with orbit elements of puls ars in binary systems, can be\nfound at the Australia Telescope National Facility (ATNF) online pulsa r catalogue [24]. Table 7\nlists the current values of the key orbital and relativistic paramete rs for B1913+16, from analysis\nof data through 2006 [379].\nTable 7: Parameters of the binary pulsar B1913+16. The numbers in parentheses denote errors in\nthe last digit. Data taken from [379] and defined as of 11 December 2003 (MJD 52984.0). Note\nthatγ′is not the same as the PPN parameter γ(see Eqs. (106)).\nParameter Symbol Value\n(units)\n(i) “Physical” parameters:\nRight Ascension α 19h15m27.s99999(2)\nDeclination δ 16◦06′27.′′4034(4)\nPulsar period Pp(ms) 59 .0299983444181(5)\nDerivative of period ˙Pp 8.62713(8) ×10−18\n(ii) “Keplerian” parameters:\nProjected semimajor axis apsini(s) 2.341782(3)\nEccentricity e 0.6171334(5)\nOrbital period Pb(day) 0 .322997448911(4)\nLongitude of periastron ω0(◦) 292 .54472(4)\nJulian date of periastron T0(MJD) 52144 .90097841(4)\n(iii) “Post-Keplerian” parameters:\nMean rate of periastron advance ∝an}b∇acketle{t˙ω∝an}b∇acket∇i}ht(◦yr−1) 4.226598(5)\nRedshift/time dilation γ′(ms) 4 .2992(8)\nOrbital period derivative ˙Pb(10−12)−2.423(1)\nThe system consists of a pulsar of nominal period 59 ms in a close binar y orbit with an unseen\ncompanion. The orbital period is about 7.75 hours, and the eccentr icity is 0.617. From detailed\nanalyses of the arrival times of pulses (which amounts to an integra ted version of the Doppler-shift\nmethods used in spectroscopicbinarysystems), extremelyaccur ateorbitaland physicalparameters\nfor the system have been obtained (see Table 7). Because the orb it is so close ( ≈1R⊙) and because\nthere is no evidence of an eclipse of the pulsar signal or of mass tran sfer from the companion, it\nis generally agreed that the companion is compact. Evolutionary arg uments suggest that it is\nmost likely a dead pulsar, while B1913+16 is a “recycled” pulsar. Thus t he orbital motion is\n67very clean, free from tidal or other complicating effects. Further more, the data acquisition is\n“clean” in the sense that by exploiting the intrinsic stability of the puls ar clock combined with the\nability to maintain and transfer atomic time accurately using GPS, the observers can keep track of\npulse time-of-arrival with an accuracy of 13 µs, despite extended gaps between observing sessions\n(including a several-year gap in the middle 1990s for an upgrade of th e Arecibo radio telescope).\nThe pulsar has shown no evidence of “glitches” in its pulse period.\nThree factors made this system an arena where relativistic celestia l mechanics must be used:\nthe relatively large size of relativistic effects [ vorbit≈(m/r)1/2≈10−3], a factor of 10 larger than\nthe corresponding values for solar-system orbits; the short orb ital period, allowing secular effects\nto build up rapidly; and the cleanliness of the system, allowing accurat e determinations of small\neffects. Because the orbital separation is large compared to the n eutron stars’ compact size, tidal\neffects can be ignored. Just as Newtonian gravity is used as a tool f or measuring astrophysi-\ncal parameters of ordinary binary systems, so GR is used as a tool for measuring astrophysical\nparameters in the binary pulsar.\nThe observationalparametersthat are obtained from a least-sq uaressolution ofthe arrival-time\ndata fall into three groups:\n1. non-orbital parameters, such as the pulsar period and its rate of change (defined at a given\nepoch), and the position of the pulsar on the sky;\n2. five “Keplerian” parameters, most closely related to those appr opriate for standard Newto-\nnian binary systems, such as the eccentricity e, the orbital period Pb, and the semi-major\naxis of the pulsar projected along the line of sight, apsini; and\n3. five “post-Keplerian” parameters.\nThe five post-Keplerian parameters are: ∝an}b∇acketle{t˙ω∝an}b∇acket∇i}ht, the average rate of periastron advance; γ′, the am-\nplitude of delays in arrival of pulses caused by the varying effects of the gravitational redshift and\ntime dilation as the pulsar moves in its elliptical orbit at varying distance s from the companion\nand with varying speeds; ˙Pb, the rate of change of orbital period, caused predominantly by gr av-\nitational radiation damping; and rands= sini, respectively the “range” and “shape” of the\nShapiro time delay of the pulsar signal as it propagates through the curved spacetime region near\nthe companion, where iis the angle of inclination of the orbit relative to the plane of the sky. A n\nadditional 14 relativistic parameters are measurable in principle [107 ].\nIn GR, the five post-Keplerian parameters can be related to the ma sses of the two bodies and\nto measured Keplerian parameters by the equations (TEGP 12.1, 14 .6 (a) [388])\n∝an}b∇acketle{t˙ω∝an}b∇acket∇i}ht= 6πfb(2πmfb)2/3(1−e2)−1,\nγ′=e(2πfb)−1(2πmfb)2/3m2\nm/parenleftBig\n1+m2\nm/parenrightBig\n,\n˙Pb=−192π\n5(2πMfb)5/3F(e),\nr=m2,\ns= sini,(106)\nwherem1andm2denote the pulsar and companion masses, respectively. The formu la for∝an}b∇acketle{t˙ω∝an}b∇acket∇i}ht\nignores possible non-relativistic contributions to the periastron sh ift, such as tidally or rotationally\ninduced effects caused by the companion (for discussion of these e ffects, see TEGP 12.1 (c) [388]).\nThe formula for ˙Pbincludes only quadrupole gravitational radiation; it ignores other so urces of\nenergy loss, such as tidal dissipation (TEGP 12.1 (f) [388]). Notice th at, by virtue of Kepler’s third\nlaw, (2πfb)2=m/a3, (2πmfb)2/3=m/a∼ǫ, thus the first two post-Keplerian parameters can\nbe seen as O(ǫ), or 1PN corrections to the underlying variable, while the third is an O(ǫ5/2), or\n681.41 \n1.40 \n1.39 \n1.38 \n1.37 \n1.42 1.43 1.44 1.45 1.46 \nMASS OF PULSAR (sol ar masses)·MASS OF COMPANION ( solar masses) \na'(0.02%) dt/dt  \n(0.0001 %) dP \nb/dt (0.2 %) a\n0          1        2         3 3\n2\n1\nFigure6: Constraintsonmassesofthe pulsarand its companionfro mdata onB1913+16,assuming\nGR to be valid. The width of each strip in the plane reflects observatio nal accuracy, shown as a\npercentage. An inset shows the three constraints on the full mas s plane; the intersection region\n(a) has been magnified 400 times for the full figure.\n2.5PN correction. The current observed values for the Keplerian a nd post-Keplerian parameters\nare shown in Table 7. The parameters randsare not separately measurable with interesting\naccuracy for B1913+16 because the orbit’s 47◦inclination does not lead to a substantial Shapiro\ndelay, however they are measurable in the double pulsar, for examp le.\nBecausefbandeare separately measured parameters, the measurement of the t hree post-\nKeplerian parameters provides three constraints on the two unkn own masses. The periastron shift\nmeasures the total mass of the system, ˙Pbmeasures the chirp mass, and γ′measures a complicated\nfunction of the masses. GR passes the test if it provides a consiste nt solution to these constraints,\nwithin the measurement errors.\nFromtheintersectionofthe ∝an}b∇acketle{t˙ω∝an}b∇acket∇i}htandγ′constraintsweobtainthevalues m1= 1.4398±0.0002M⊙\nandm2= 1.3886±0.0002M⊙. The third of Eqs. (106) then predicts the value ˙Pb=−2.402531±\n0.000014×10−12. In ordertocomparethe predictedvalue for ˙Pbwith the observedvalueofTable 7,\nit is necessary to take into account the small kinematic effect of a re lative acceleration between the\nbinary pulsar system and the solar system caused by the differentia l rotation of the galaxy. Using\ndata on the location and proper motion of the pulsar, combined with t he best information available\non galactic rotation; the current value of this effect is ˙Pgal\nb≃ −(0.027±0.005)×10−12. Subtracting\nthis from the observed ˙Pb(see Table 7) gives the corrected ˙Pcorr\nb=−(2.396±0.005)×10−12, which\n69agrees with the prediction within the errors. In other words,\n˙Pcorr\nb\n˙PGR\nb= 0.997±0.002. (107)\nThe consistency among the measurements is displayed in Figure 6, in w hich the regions allowed by\nthe three most precise constraints have a single common overlap. U ncertainties in the parameters\nthat go into the galactic correction are now the limiting factor in the a ccuracy of the test of\ngravitational damping.\nFigure 7: Plot of the cumulative shift of the periastron time from 197 5–2005. The points are\ndata, the curve is the GR prediction. The gap during the middle 1990s was caused by a closure of\nArecibo for upgrading [379].\nA third way to display the agreement with GR is by comparing the obser ved phase of the orbit\nwithatheoreticaltemplatephaseasafunctionoftime. If fbvariesslowlyintime, thentofirstorder\nin a Taylorexpansion, the orbital phaseis givenby Φ b(t) = 2πfb0t+π˙fb0t2. The time ofperiastron\npassagetPis given by Φ( tP) = 2πN, whereNis an integer, and consequently, the periastron\ntime will not grow linearly with N. Thus the cumulative difference between periastron time tP\nandN/fb0, the quantities actually measured in practice, should vary accordin g totP−N/fb0=\n−˙fb0N2/2f3\nb0≈ −(˙fb0/2fb0)t2. Figure 7 shows the results: The dots are the data points, while\nthe curve is the predicted difference using the measured masses an d the quadrupole formula for\n˙fb0[379].\nThe consistency among the constraints provides a test of the ass umption that the two bodies\nbehave as “point” masses, without complicated tidal effects, obey ing the general relativistic equa-\ntions of motion including gravitational radiation. It is also a test of st rong gravity, in that the\n70highly relativistic internal structure of the neutron stars does no t influence their orbital motion,\nas predicted by the SEP of GR.\nObservations [213, 378] indicate that the pulse profile is varying with time, which suggests\nthat the pulsar is undergoing geodetic precession on a 300-year tim escale as it moves through the\ncurved spacetime generated by its companion (see Section 4.4.2). T he amount is consistent with\nGR, assuming that the pulsar’s spin is suitably misaligned with the orbita l angular momentum.\nUnfortunately, the evidence suggests that the pulsar beam may p recess out of our line of sight by\n2025.\n6.2 A zoo of binary pulsars\nMore than 70 binary neutron star systems with orbital periods less than a day are now known.\nWhile somearelessinterestingfortestingrelativity, somehaveyielde d interestingtests, andothers,\nnotably the recently discovered “double pulsar” are likely to continu e to produce significant results\nwell into the future. Here we describe some of the more interesting or best studied cases;\nThe “double” pulsar: J0737-3039A, B. This binary pulsar system, discovered in 2003 [63],\nwas already remarkable for its extraordinarily short orbital period (0.1 days) and large periastron\nadvance (16 .88◦yr−1), but then the companion was also discovered to be a pulsar [242]. Be cause\ntwo projected semi-major axes could be measured, the mass ratio was obtained directly from the\nratio of the two values of apsini, and thereby the two masses could be obtained by combining\nthat ratio with the periastron advance, assuming GR. The results a remA= 1.337±0.005M⊙\nandmB= 1.250±0.005M⊙, whereAdenotes the primary (first) pulsar. From these values, one\nfinds that the orbit is nearly edge-on, with sin i= 0.9991, a value which is completely consistent\nwith that inferred from the Shapiro delay parameter. In fact, the five measured post-Keplerian\nparameters plus the ratio of the projected semi-major axes give s ix constraints on the masses\n(assuming GR): as seen in Fig. 8, all six overlap within their measureme nt errors [214]. Because\nof the location of the system, galactic proper-motion effects play a significantly smaller role in\nthe interpretation of ˙Pbmeasurements than they did in B1913+16; this and the reduced effe ct of\ninterstellar dispersion means that the accuracy of measuring the g ravitational-wave damping may\nsoon beat that from the Hulse-Taylor system. The geodetic prece ssion of pulsar B’s spin axis has\nalso been measured by monitoring changes in the patterns of eclipse s of the signal from pulsar A,\nwith a result in agreement with GR to about 13 percent [60]; the const raint on the masses from\nthat effect (assuming GR to be correct) is also shown in Fig. 8.\nJ1738+0333: A white-dwarf companion. This is a low-eccentricity, 8 .5-hour period system\nin which the white-dwarf companion is bright enough to permit detailed spectroscopy, allowing\nthe companion mass to be determined directly to be 0 .181M⊙. The mass ratio is determined\nfrom Doppler shifts of the spectral lines of the companion and of th e pulsar period, giving the\npulsar mass 1 .46M⊙. Ten years of observation of the system yielded both a measureme nt of the\napparent orbital period decay, and enough information about its p arallax and proper motion to\naccount for the substantial galactic kinematic effect to give a value of the intrinsic period decay\nof˙Pb= (−25.9±3.2)×10−15ss−1in agreement with the predicted effect [149]. But because of\nthe asymmetry of the system, the result also places a significant bo und on the existence of dipole\nradiation, predicted by many alternative theories of gravity (see S ec. 6.3 below for discussion).\nData from this system were also used to place the tight bound on the PPN parameter α1shown\nin Table 4.\nJ1141-6545: A white-dwarf companion. This system is similar in some ways to the Hulse-\nTaylor binary: short orbital period (0 .20 days), significant orbital eccentricity (0 .172), rapid pe-\nriastron advance (5 .3 degrees per year) and massive components ( Mp= 1.27±0.01M⊙,Mc=\n1.02±0.01M⊙). The key difference is that the companion is again a white dwarf. The intrinsic\n71Figure 8: Constraints on masses of the pulsar and its companion fro m data on J0737-3039A,B,\nassuming GR to be valid. The inset shows the intersection region magn ified by a factor of 80.\n(Figure courtesy of M. Kramer).\norbit period decay has been measured in agreement with GR to about six percent, again placing\nlimits on dipole gravitational radiation [40].\nJ0348+0432: The most massive neutron star. Discovered in 2011, this is another neutron-\nstar white-dwarf system, in a very short period (0 .1 day), low eccentricity (2 ×10−6) orbit. Timing\nof the neutron star and spectroscopy of the white dwarf have led to mass values of 0 .172M⊙for the\nwhite dwarf and 2 .01±0.04M⊙for the pulsar, making it the most massive accurately measured\nneutron star yet. This result ruled out a number of heretofore via ble soft equations of state for\nnuclear matter. The orbit period decay agrees with the GR predictio n within 20 percent and is\nexpected to improve steadily with time.\nJ0337+1715: Two white-dwarf companions. This remarkable system was reported in 2014\n[305]. It consists of a 2.73 millisecond pulsar ( M= 1.44M⊙) with extremely good timing pre-\ncision, accompanied by twowhite dwarfs in coplanar circular orbits. The inner white dwarf\n(M= 0.1975M⊙) has an orbital period of 1 .629 days, with e= 6.918×10−4, and the outer\nwhite dwarf ( M= 0.41M⊙) has a period of 327 .26 days, with e= 3.536×10−2. This is an ideal\nsystem for testing the Nordtvedt effect in the strong-field regime . Here the inner system is the\nanalogue of the Earth-Moon system, and the outer white dwarf pla ys the role of the Sun. Because\nthe outer semi-major axis is about 1/3 of an astronomical unit, the basic driving perturbation is\ncomparable to that provided by the Sun. However, the self-gravit ational binding energy per unit\nmass of the neutron star is almost a billion times larger than that of th e Earth, greatly amplifying\n72the size of the Nordtvedt effect. Depending on the details, this sys tem could exceed lunar laser\nranging in testing the Nordtvedt effect by several orders of magn itude.\nOther binary pulsars. Twoofthe earliestbinarypulsars, B1534+12andB2127+11C,disco vered\nin 1990, failed to live up to their early promise despite being similar to the Hulse-Taylor system in\nmost respects (both were believed to be double neutron-star sys tems). The main reason was the\nsignificant uncertainty in the kinematic effect on ˙Pbof local accelerations, galactic in the case of\nB1534+12, and those arising from the globular cluster that was hom e to B2127+11C.\nTable 8: Parameters of other binary pulsars. References may be f ound in the text. Values for orbit\nperiod derivatives include corrections for galactic kinematic effects\nParameter J0737–3039(A, B) J1738+0333 J1141–6545\n(i) Keplerian:\napsini(s) 1.41504(2)/1.513(3) 0 .34342913(2) 1 .858922(6)\ne 0.087779(5) (3 .4±1.1)×10−70.171884(2)\nPb(day) 0.102251563(1) 0 .354790739872(1) 0 .1976509593(1)\n(ii) Post-Keplerian:\n∝an}b∇acketle{t˙ω∝an}b∇acket∇i}ht(◦yr−1) 16.90(1) 5.3096(4)\nγ′(ms) 0.382(5) 0.77(1)\n˙Pb(10−12) −1.21(6) −0.026(3) −0.401(25)\nr(µs) 6.2(5)\ns= sini 0.9995(4)\n6.3 Binary pulsars and alternative theories\nSoon after the discovery of the binary pulsar it was widely hailed as a n ew testing ground for\nrelativistic gravitational effects. As we have seen in the case of GR, in most respects, the system\nhas lived up to, indeed exceeded, the early expectations.\nIn another respect, however, the system has only partially lived up to its promise, namely\nas a direct testing ground for alternative theories of gravity. The origin of this promise was the\ndiscovery [125, 401] that alternative theories of gravity generica lly predict the emission of dipole\ngravitational radiation from binary star systems. In GR, there is n o dipole radiation because\nthe “dipole moment” (center of mass) of isolated systems is uniform in time (conservation of\nmomentum), and because the “inertial mass” that determines the dipole moment is the same as\nthe mass that generates gravitational waves (SEP). In other th eories, while the inertial dipole\nmoment may remain uniform, the “gravity wave” dipole moment need n ot, because the mass that\ngenerates gravitational waves depends differently on the interna l gravitational binding energy of\neach body than does the inertial mass (violation of SEP). Schematic ally, in a coordinate system\nin which the center of inertial mass is at the origin, so that mI,1x1+mI,2x2= 0, the dipole part\nof the retarded gravitational field would be given by\nh∼1\nRd\ndt(mGW,1x1+mGW,2x2)∼ηm\nRv/parenleftbiggmGW,1\nmI,1−mGW,2\nmI,2/parenrightbigg\n, (108)\nwherev=v1−v2andηandmare defined using inertial masses. In theories that violate\nSEP, the difference between gravitational wave mass and inertial m ass is a function of the internal\n73gravitationalbinding energy ofthe bodies. This additional form ofg ravitationalradiationdamping\ncould, at least in principle, be significantly stronger than the usual q uadrupole damping, because\nit depends on fewer powers of the orbital velocity v, and it depends on the gravitational binding\nenergy per unit mass of the bodies, which, for neutron stars, cou ld be as large as 20 percent (see\nTEGP 10 [388] for further details). As one fulfillment of this promise, Will and Eardley worked\nout in detail the effects of dipole gravitational radiation in the bimetr ic theory of Rosen, and,\nwhen the first observation of the decrease of the orbital period w as announced in 1979, the Rosen\ntheory suffered a terminal blow. A wide class of alternative theories also fails the binary pulsar\ntest because of dipole gravitational radiation (TEGP 12.3 [388]).\nOn the other hand, the early observations of PSR 1913+16 already indicated that, in GR, the\nmasses of the two bodies were nearly equal, so that, in theories of g ravity that are in some sense\n“close” to GR, dipole gravitational radiation would not be a strong eff ect, because of the apparent\nsymmetryofthe system. TheRosentheory, andotherslikeit, are not“close”toGR, exceptintheir\npredictions for the weak-field, slow-motion regime of the solar syst em. When relativistic neutron\nstars are present, theories like these can predict strong effects on the motion of the bodies resulting\nfrom their internal highly relativistic gravitationalstructure (viola tions of SEP). As a consequence,\nthe masses inferred from observations of the periastron shift an dγ′may be significantly different\nfrom those inferred using GR, and may be different from each other , leading to strong dipole\ngravitational radiation damping. By contrast, the Brans–Dicke th eory is “close” to GR, roughly\nspeaking within 1 /ωBDof the predictions of the latter, for large values of the coupling con stant\nωBD. Thus, despite the presence of dipole gravitational radiation, the Hulse-Taylor binary pulsar\nprovides at present only a weak test of pure Brans–Dicketheory, not competitive with solar-system\ntests.\nHowever, the discovery of binary pulsar systems with a white dwarf companion, such as\nJ1738+0333, J1141-6545 and J0348+0432 has made it possible to p erform strong tests of the\nexistence of dipole radiation. This is because such systems are nece ssarily asymmetrical, since\nthe gravitational binding energy per unit mass of white dwarfs is of o rder 10−4, much less than\nthat of the neutron star. Already, significant bounds have been p laced on dipole radiation using\nJ1738+0333 and J1141-6545 [149, 40].\nBecause the gravitational-radiationand strong-field properties o f alternative theories of gravity\ncan be dramatically different from those of GR and each other, it is diffi cult to parametrize these\naspects of the theories in the manner of the PPN framework. In ad dition, because of the generic\nviolation of the Strong Equivalence Principle in these theories, the re sults can be very sensitive\nto the equation of state and mass of the neutron star(s) in the sy stem. In the end, there is no\nway around having to analyze every theory in turn. On the other ha nd, because of their relative\nsimplicity, scalar-tensor theories provide an illustration of the esse ntial effects, and so we shall\ndiscuss binary pulsars within this class of theories.\n6.4 Binary pulsars and scalar-tensor gravity\nMaking the usual assumption that both members of the system are neutron stars, and using\nthe methods summarized in TEGP 10–12 [388] (see also [262]) one can obtain formulas for the\nperiastronshift, the gravitationalredshift/second-orderDop pler shift parameter, the Shapiro delay\ncoefficients, and the rate of change of orbital period, analogous t o Eqs. (106). These formulas\ndepend on the masses of the two neutron stars, on their sensitivit iessA, and on the scalar-tensor\nparameters, as defined in Table 6 (and on a new sensitivity κ∗, defined below). First, there is a\nmodification of Kepler’s third law, given by\n2πfb=/parenleftBigαm\na3/parenrightBig1/2\n. (109)\n74Then the predictions for ∝an}b∇acketle{t˙ω∝an}b∇acket∇i}ht,γ′,˙Pb,randsare\n∝an}b∇acketle{t˙ω∝an}b∇acket∇i}ht= 6πfb(2παmfb)2/3(1−e2)−1P, (110)\nγ′=e(2πfb)−1(2παmfb)2/3m2\nmα−1/bracketleftBig\n1−2ζs2+αm2\nm+ζκ∗\n1(1−2s2)/bracketrightBig\n,(111)\n˙Pb=−192π\n5(2παMfb)5/3F′(e)−8πζ(2πµfb)S2G(e), (112)\nr=m2(1−ζ), (113)\ns= sini, (114)\nwhere\nP= 1+1\n3/parenleftbig\n2¯γ−¯β++ψ¯β−/parenrightbig\n, (115)\nF′(e) =1\n12(1−e2)−7/2/bracketleftbigg\nκ1/parenleftbigg\n1+7\n2e2+1\n2e4/parenrightbigg\n−1\n2κ2e2/parenleftbigg\n1+1\n2e2/parenrightbigg/bracketrightbigg\n, (116)\nG(e) = (1−e2)−5/2/parenleftbigg\n1+1\n2e2/parenrightbigg\n, (117)\nwhereκ1andκ2are defined in Eq. (102). The quantity κ∗\nAis defined by\nκ∗\nA=/parenleftbigg∂(lnIA)\n∂(lnφ)/parenrightbigg\n, (118)\nand measures the “sensitivity” of the moment of inertia IAof each body to changes in the scalar\nfield for a fixed baryonnumber N(see TEGP 11, 12 and 14.6(c) [388] for further details). The sen-\nsitivitiessAandκ∗\nAwill depend on the neutron-star equation of state. Notice how the violation of\nSEP in scalar-tensor theory introduces complex structure-depe ndent effects in everything from the\nNewtonian limit (modification of the effective coupling constant in Keple r’s third law) to gravita-\ntional radiation. In the limit ζ→0, we recover GR, and all structure dependence disappears. The\nfirst term in ˙Pb(see Eq. (112)) is the combined effect of quadrupole and monopole g ravitational\nradiation, post-Newtonian corrections to dipole radiation, and a dip ole-octupole coupling term, all\ncontributing at 0PN order, while the second term is the effect of dipo le radiation, contributing at\nthe dominant −1PN order.\nUnfortunately, becauseofthenearequalityofneutronstarmas sesintypicaldoubleneutronstar\nbinary pulsars, dipole radiation is somewhat suppressed, and the bo unds obtained are typically not\ncompetitive with the Cassini bound on γ, except for those generalized scalar-tensor theories, with\nβ0<0 where the strong gravity of the neutron stars induces spontan eous scalarization effects [94].\nFigure 9 illustrates this: the bounds on α0andβ0from the three binary neutron star systems\nB1913+16, J0737-3039, and B1534+12 are not close to being comp etitive with the Cassini bound\nonα0, except for very negative values of β0(recall that α0= (3+2ω0)−1/2).\nOntheotherhand, abinarypulsarsystemwithdissimilarobjects, su chasawhitedwarforblack\nhole companion, provides potentially more promising tests of dipole ra diation. As a result, the\nneutron-star–white-dwarfsystems J1141-6545and J1738+03 33yield much more stringent bounds.\nIndeed, the latter system surpasses the Cassini bound for β0>1 andβ0<−2, and is close to that\nbound for the pure Brans-Dicke case β0= 0 [149].\nIt is worth pointing out that the bounds displayed in Fig. 9 havebeen c alculated using a specific\nchoice of scalar-tensor theory, in which the function A(ϕ) is given by\nA(ϕ) = exp/bracketleftbigg\nα0(ϕ−ϕ0)+1\n2β0(ϕ−ϕ0)2/bracketrightbigg\n, (119)\n75whereα0, andβ0, are arbitrary parameters, and ϕ0is the asymptotic value of the scalar field. In\nthe notation for scalar-tensor theories used here, this theory c orresponds to the choice\nω(φ) =−1\n2/parenleftbigg\n3−1\nα2\n0−β0lnφ/parenrightbigg\n, (120)\nwhereφ0=A(ϕ0)−2= 1. The parameters ζandλare given by\nζ=α2\n0\n1+α2\n0,\nλ=1\n2β0\n1+α2\n0. (121)\nThe bounds were also calculated using a polytropic equation of state , which tends to give lower\nmaximum masses for neutron stars than do more realistic equations of state.\nLLR LLR \nSEP\nJ1141–6545 B1534+12 \nB1913+16 J0737–3039 \nJ1738+0333 \n−6−4−2 2 4 6 00\n00|\n10 \n10 \n10 10 \n10 \nCassini\nFigure 9: Bounds on the scalar-tensor parameters α0andβ0from solar-system and binary pulsar\nmeasurements. Bounds from tests of the Nordtvedt effect using lunar laser ranging and circular\npulsar–white-dwarf binary systems are denoted LLR and SEP, res pectively. Credit Ref. [149]\nBounds on various versions of TeVeS theories have also been estab lished, with the tightest\nconstraints again coming from neutron-star–white-dwarf binarie s [149]; in the case of TeVeS, the\ntheory naturally predicts γ= 1 in the post-Newtonian limit, so the Cassini measurements are\nirrelevant here.\n767 Gravitational-Wave Tests of Gravitational Theory\n7.1 Gravitational-wave observatories\nSoon after the publication of this update, a new method of testing r elativistic gravity will be real-\nized, when a worldwide network of upgraded laser interferometric g ravitational wave observatories\nin the U.S. (LIGO Hanford and LIGO Livingston) and Europe (VIRGO a nd GEO600) begins\nregular detection and analysis of gravitational wave signals from as trophysical sources. Within a\nfew years, they may be joined by an underground cryogenic interf erometer (KAGRA) in Japan,\nand around 2022, by a LIGO-type interferometer in India. These b road-band antennas will have\nthe capability of detecting and measuring the gravitational wavefo rms from astronomical sources\nin a frequency band between about 10 Hz (the seismic noise cutoff) a nd 500 Hz (the photon count-\ning noise cutoff), with a maximum sensitivity to strain at around 100 Hz ofh∼∆l/l∼10−22\n(rms), for the kilometer-scale LIGO/VIRGO projects. The most p romising source for detection\nand study of the gravitational wave signal is the “inspiralling compac t binary” – a binary system\nof neutron stars or black holes (or one of each) in the final minutes of a death spiral leading to\na violent merger. Such is the fate, for example, of the Hulse–Taylor binary pulsar B1913+16 in\nabout 300 Myr, or the “double pulsar” J0737-3039 in about 85 Myr. Given the expected sensitiv-\nity of the advanced LIGO-Virgo detectors, which could see such so urces out to many hundreds of\nmegaparsecs, it has been estimated that from 40 to several hund red annual inspiral events could be\ndetectable. Other sources, such as supernova core collapse eve nts, instabilities in rapidly rotating\nnewborn neutron stars, signals from non-axisymmetric pulsars, a nd a stochastic background of\nwaves, may be detectable (see [324] for a review).\nIn addition, plans are being developed for orbiting laser interferome ter space antennae, such\nas DECIGO in Japan and eLISA in Europe. The eLISA system would con sist of three spacecraft\norbiting the sun in a triangular formation separated from each othe r by a million kilometers, and\nwould be sensitive primarily in the very low-frequency band between 1 0−4and 10−1Hz, with peak\nstrain sensitivity of order h∼10−23.\nA third approach that focuses on the ultra low-frequency band (n anohertz) is that of Pulsar\nTiming Arrays (PTA), whereby a network of highly stable millisecond pu lsars is monitored in a\ncoherent way using radio telescopes, in hopes of detecting the fluc tuations in arrival times induced\nby passing gravitational waves.\nFor recent reviews of the status of all these approaches to grav itational-wave detection, see the\nProceedings of the 8th Edoardo Amaldi Conference on Gravitation al Waves [249].\nIn addition to opening a new astronomical window, the detailed obser vation of gravitational\nwaves by such observatories may provide the means to test gener al relativistic predictions for the\npolarizationandspeedofthewaves,forgravitationalradiationda mpingandforstrong-fieldgravity.\nThese topics have been thoroughly covered in two recent Living Reviews by Gairet al.[155] and\nby Yunes and Siemens [413]. Here we present a brief overview.\n7.2 Gravitational-wave amplitude and polarization\n7.2.1 General relativity\nA generic gravitational wave detector can be modelled as a body of m assMat a distance Lfrom a\nfiducial laboratory point, connected to the point by a spring of res onant frequency ω0and quality\nfactorQ. From the equation of geodesic deviation, the infinitesimal displacem entξof the mass\nalong the line of separation from its equilibrium position satisfies the eq uation of motion\n¨ξ+2ω0\nQ˙ξ+ω2\n0ξ=L\n2/parenleftBig\nF+(θ,φ,ψ)¨h+(t)+F×(θ,φ,ψ)¨h×(t)/parenrightBig\n, (122)\n77whereF+(θ,φ,ψ) andF×(θ,φ,ψ) are “beam-pattern” factors that depend on the direction of th e\nsource (θ,φ) and on a polarization angle ψ, andh+(t) andh×(t) are gravitational waveforms\ncorresponding to the two polarizations of the gravitational wave ( for pedagogical reviews, see [356,\n297]). In a source coordinate system in which the x–yplane is the plane of the sky and the\nz-direction points toward the detector, these two modes are given by\nh+(t) =1\n2(hxx\nTT(t)−hyy\nTT(t)), h ×(t) =hxy\nTT(t), (123)\nwherehij\nTTrepresent transverse-traceless(TT) projections of the calcu lated waveform of Eq. (82),\ngiven by\nhij\nTT=hkl/bracketleftbigg/parenleftBig\nδik−ˆNiˆNk/parenrightBig/parenleftBig\nδjl−ˆNjˆNl/parenrightBig\n−1\n2/parenleftBig\nδij−ˆNiˆNj/parenrightBig/parenleftBig\nδkl−ˆNkˆNl/parenrightBig/bracketrightbigg\n,(124)\nwhereˆNjis a unit vector pointing toward the detector. The beam pattern fa ctors depend on the\norientation and nature of the detector. For a wave approaching a long the laboratory z-direction,\nandforamasswhoselocationonthe x–yplanemakesanangle φwith thex-axis, the beampattern\nfactors are given by F+= cos2φandF×= sin2φ. For a laser interferometer with one arm along\nthe laboratory x-axis, the other along the y-axis, with ξdefined as the differential displacement\nalong the two arms, the beam pattern functions are\nF+=1\n2(1+cos2θ)cos2φcos2ψ−cosθsin2φsin2ψ,\nF×=1\n2(1+cos2θ)cos2φsin2ψ+cosθsin2φcos2ψ. (125)\nHere, we assume that ω0≈0 in Eq. (122), corresponding to the essentially free motion of the\nsuspended mirrors in the horizontal direction. The waveforms h+(t) andh×(t) depend on the\nnature and evolution of the source. For example, for a binary syst em in a circular orbit, with an\ninclination irelative to the plane of the sky, and the x-axis oriented along the major axis of the\nprojected orbit, the quadrupole approximation of Eq. (84) gives\nh+(t) =−2M\nR(2πMfb)2/3(1+cos2i) cos2Φ b(t), (126)\nh×(t) =−2M\nR(2πMfb)2/3(2cosi) cos2Φ b(t), (127)\nwhere Φ b(t) = 2π/integraltexttfb(t′)dt′is the orbital phase.\n7.2.2 Alternative theories of gravity\nAgenericgravitationalwavedetectorwhosescaleissmallcompare dtothegravitationalwavelength\nmeasures the local components of a symmetric 3 ×3 tensor which is composed of the “electric”\ncomponents of the Riemann curvature tensor, R0i0j, via the equation of geodesic deviation, given,\nfor a pair of freely falling particles by ¨ xi=−R0i0jxj, wherexidenotes the spatial separation. In\ngeneral there are six independent components, which can be expr essed in terms of polarizations\n(modes with specific transformationproperties under rotationsa nd boosts); fora wavepropagating\nin thez-direction, they can be displayed by the matrix\nSjk=\nAS+A+A×AV1\nA×AS−A+AV2\nAV1AV2AL\n. (128)\n78Three modes ( A+,A×, andAS) are transverse to the direction of propagation, with two repres ent-\ning quadrupolar deformations and one representing a monopolar “b reathing” deformation. Three\nmodes are longitudinal, with one ( AL) an axially symmetric stretching mode in the propagation\ndirection, and one quadrupolar mode in each of the two orthogonal planes containing the propa-\ngation direction ( AV1andAV2). Figure 10 shows the displacements induced on a ring of freely\nfallingtest particlesbyeachofthesemodes. Generalrelativitypre dictsonlythe first twotransverse\nquadrupolar modes (a) and (b) independently of the source; thes e correspond to the waveforms\nh+andh×discussed earlier (note the cos2 φand sin2φdependences of the displacements).\nMassless scalar-tensor gravitational waves can in addition contain the transverse breathing\nmode (c). This can be obtained from the physical waveform hαβ, which is related to ˜hαβandϕto\nthe required order by\nhαβ=˜hαβ+Ψηαβ, (129)\nwhere Ψ = ϕ−1. In this case, A+(−)∝˜h+(−), whileAS∝Ψ (see Eqs. (97), (98), (99) and (100)\nfor the leading contributions to these fields). In massive scalar-te nsor theories, the longitudinal\nmode (d) can also be present, but is suppressed relative to (c) by a factor (λ/λC)2, whereλis the\nwavelength of the radiation, and λCis the Compton wavelength of the massive scalar.\nMore general metric theories predict additional longitudinal modes , up to the full complement\nof six (TEGP 10.2 [388]).\nA suitable array of gravitational antennas could delineate or limit the number of modes present\nin a given wave. The strategy depends on whether or not the sourc e direction is known. In general\nthere are eight unknowns (six polarizations and two direction cosine s), but only six measurables\n(R0i0j). If the direction can be established by either association of the wa ves with optical or other\nobservations, or by time-of-flight measurements between separ ated detectors, then six suitably\noriented detectors suffice to determine all six components. If the direction cannot be established,\nthen the system is underdetermined, and no unique solution can be f ound. However, if one assumes\nthat only transverse waves are present, then there are only thr ee unknowns if the source direction\nis known, or five unknowns otherwise. Then the corresponding num ber (three or five) of detectors\ncan determine the polarization. If distinct evidence were found of a ny mode other than the two\ntransverse quadrupolar modes of GR, the result would be disastro us for GR. On the other hand,\nthe absence of a breathing mode would not necessarily rule out scala r-tensor gravity, because the\nstrength of that mode depends on the nature of the source.\nFor laser interferometers, the signal controlling the laser phase o utput can be written in the\nform\nS(t) =1\n2/parenleftbig\nej\n1ek\n1−ej\n2ek\n2/parenrightbig\nSjk, (130)\nwheree1ande2are unit vectors directed along the two arms of the interferomete r. The final\nresult is\nS(t) =FSAS+FLAL+FV1AV1+FV2AV2+F+A++F×A×, (131)\nwhere the angular pattern functions FA(θ,φ,ψ) are given by\nFS=−1\n2sin2θcos2φ,\nFL=1\n2sin2θcos2φ,\nFV1=−sinθ(cosθcos2φcosψ−sin2φsinψ),\nFV2=−sinθ(cosθcos2φsinψ+sin2φcosψ),\nF+=1\n2(1+cos2θ)cos2φcos2ψ−cosθsin2φsin2ψ,\nF×=1\n2(1+cos2θ)cos2φsin2ψ+cosθsin2φcos2ψ, (132)\n79xy\nxy\nzx\nzyxy\nzy(b)\n(d)\n(f) (e)(c)(a)Gravitational−Wave Polarization\nFigure 10: The six polarization modes for gravitational waves permit ted in any metric theory of\ngravity. Shown is the displacement that each mode induces on a ring o f test particles. The wave\npropagates in the + zdirection. There is no displacement out of the plane of the picture. I n (a),\n(b), and (c), the wave propagates out of the plane; in (d), (e), a nd (f), the wave propagates in\nthe plane. In GR, only (a) and (b) are present; in massless scalar-t ensor gravity, (c) may also be\npresent.\n(see [297] for detailed derivations and definitions). Note that the s calar and longitudinal pattern\nfunctions are degenerate and thus no array of laser interferome ters can measure these two modes\nseparately.\nSome of the details of implementing such polarization observations ha ve been worked out\nfor arrays of resonant cylindrical, disk-shaped, spherical, and tr uncated icosahedral detectors\n(TEGP10.2[388], forrecentreviewssee[235,373]). Earlyworktoas sesswhether theground-based\n80or space-based laser interferometers (or combinations of the tw o types) could perform interesting\npolarization measurements was carried out in [374, 62, 244, 156, 38 0]; for a recent detailed anal-\nysis see [276]. Unfortunately for this purpose, the two LIGO obser vatories (in Washington and\nLouisiana states, respectively) have been constructed to have t heir respective arms as parallel as\npossible, apart from the curvature of the Earth; while this maximize s the joint sensitivity of the\ntwo detectors to gravitational waves, it minimizes their ability to det ect two modes of polarization.\nIn this regard the addition of Virgo, and the future KAGRA and LIGO -India systems will be cru-\ncial to polarization measurements. The capability of space-based in terferometers to measure the\npolarization modes was assessed in detail in [358, 277]. For pulsar timin g arrays, see [226, 13, 65].\n7.3 Gravitational-wave phase evolution\n7.3.1 General relativity\nIn the binary pulsar, a test of GR was made possible by measuring at le ast three relativistic\neffects that depended upon only two unknown masses. The evolutio n of the orbital phase under\nthe damping effect of gravitational radiation played a crucial role. A nother situation in which\nmeasurement of orbital phase can lead to tests of GR is that of the inspiralling compact binary\nsystem. The key differences are that here gravitationalradiation itself is the detected signal, rather\nthan radio pulses, and the phase evolution alone carries all the infor mation. In the binary pulsar,\nthe first derivative of the binary frequency ˙fbwas measured; here the full nonlinear variation of fb\nas a function of time is measured.\nBroad-band laser interferometers are especially sensitive to the p hase evolution of the gravi-\ntational waves, which carry the information about the orbital pha se evolution. The analysis of\ngravitational wave data from such sources will involve some form of matched filtering of the noisy\ndetector output against an ensemble of theoretical “template” w aveforms which depend on the\nintrinsic parameters of the inspiralling binary, such as the componen t masses, spins, and so on,\nand on its inspiral evolution. How accurate must a template be in orde r to “match” the waveform\nfrom a given source (where by a match we mean maximizing the cross- correlation or the signal-to-\nnoise ratio)? In the total accumulated phase of the wave detecte d in the sensitive bandwidth, the\ntemplate must match the signal to a fraction of a cycle. For two insp iralling neutron stars, around\n16,000 cycles should be detected during the final few minutes of insp iral; this implies a phasing\naccuracy of 10−5or better. Since v∼1/10 during the late inspiral, this means that correction\nterms in the phasing at the level of v5or higher are needed. More formal analyses confirm this\nintuition [89, 138, 87, 296].\nBecause it is a slow-motion system ( v∼10−3), the binary pulsar is sensitive only to the lowest-\norder effects of gravitational radiation as predicted by the quadr upole formula. Nevertheless, the\nfirst correction terms of order vandv2to the quadrupole formula were calculated as early as\n1976 [375] (see TEGP 10.3 [388]).\nButforlaserinterferometricobservationsofgravitationalwave s,thebottomlineisthat, inorder\ntomeasurethe astrophysicalparametersofthe sourceandto t est the propertiesofthe gravitational\nwaves,it is necessaryto derivethe gravitationalwaveformandth e resultingradiationback-reaction\non the orbit phasing at least to 3PN order beyond the quadrupole ap proximation.\nFor the special case of non-spinning bodies moving on quasi-circular orbits (i.e. circular apart\nfrom a slow inspiral), the evolution of the gravitationalwave freque ncyf= 2fbthrough 2PN order\nhas the form\n˙f=96π\n5f2(πMf)5/3/bracketleftbigg\n1−/parenleftbigg743\n336+11\n4η/parenrightbigg\n(πmf)2/3+4π(πmf)\n+/parenleftbigg34103\n18144+13661\n2016η+59\n18η2/parenrightbigg\n(πmf)4/3+O[(πmf)5/3]/bracketrightbigg\n,(133)\n81whereη=m1m2/m2. The first term is the quadrupole contribution (compare Eq. (86)) , the\nsecond term is the 1PN contribution, the third term, with the coeffic ient 4π, is the “tail” contribu-\ntion, and the fourth term is the 2PN contribution. Two decades of in tensive work by many groups\nhave led to the development of waveforms in GR that are accurate t o 3.5PN order, and for some\nspecific effects, such as those related to spin, to 4.5PN order (see [45] for a thorough review).\nSimilar expressions can be derived for the loss of angular momentum a nd linear momentum.\nExpressions for non-circular orbits have also been derived [160, 95 ]. These losses react back on the\norbit to circularizeit and causeit to inspiral. The result is that the orb ital phase (and consequently\nthe gravitational wave phase) evolves non-linearly with time. It is th e sensitivity of the broad-\nband laser interferometric detectors to phase that makes the hig her-order contributions to df/dt\nso observationally relevant.\nIf the coefficients of each of the powers of fin Eq. (133) can be measured, then one again\nobtains more than two constraints on the two unknowns m1andm2, leading to the possibility\nto test GR. For example, Blanchet and Sathyaprakash [53, 54] ha ve shown that, by observing a\nsource with a sufficiently strong signal, an interesting test of the 4 πcoefficient of the “tail” term\ncould be performed.\nAnother possibility involvesgravitationalwavesfrom a small mass or biting and inspirallinginto\na (possibly supermassive) spinning black hole. A general non-circula r, non-equatorial orbit will\nprecess around the hole, both in periastron and in orbital plane, lea ding to a complex gravitational\nwaveform that carries information about the non-spherical, stro ng-field spacetime around the hole.\nAccording to GR, this spacetime must be the Kerr spacetime of a rot ating black hole, uniquely\nspecified by its mass and angular momentum, and consequently, obs ervation of the waves could\ntest this fundamental hypothesis of GR [318, 295].\n7.3.2 Alternative theories of gravity\nIn general, alternative theories of gravity will predict rather differ ent phase evolution from that of\nGR,notablyviatheadditionofdipolegravitationalradiation. Forexa mple,thedipolegravitational\nradiationpredicted byscalar-tensortheoriesmodifies thegravita tionalradiationback-reaction,and\nthereby the phase evolution. Including only the leading 0PN and −1PN (dipole) contributions,\none obtains,\n˙f=96π\n5f2(παMf)5/3κ1\n12/bracketleftBig\n1+b(πmf)−2/3/bracketrightBig\n, (134)\nwhereM=η3/5m, andbis the coefficient of the dipole term, given to first order in ζbyb=\n(5/24)ζα−5/3S2, whereκ1is given by Eq. (102), S=α−1/2(s1−s2) andζ= 1/(4 + 2ω0).\nDouble neutron star systems are not promising because the small r ange of masses available near\n1.4M⊙results in suppression of dipole radiation by symmetry. For black hole s,s= 0.5 identically,\nconsequently double black hole systems turn out to be observation allyidentical in the two theories.\nThus mixed systems involving a neutron star and a black hole are pref erred. However, a number of\nanalysesof the capabilities of both ground-basedand space-base d (LISA) observatorieshave shown\nthat observing waves from neutron-star–black-hole inspirals is no t likely to bound scalar-tensor\ngravity at a level competitive with the Cassini bound, with future so lar-system improvements, or\nwith binary pulsar observations [390, 218, 94, 325, 399, 35, 36].\nThese considerations suggest that it might be fruitful to attempt to parametrize the phasing\nformulae in a manner reminiscent of the PPN framework for post-Ne wtonian gravity. A number\nof approaches along this line have been developed [412, 264].\n7.4 Speed of gravitational waves\nAccording to GR, in the limit in which the wavelength of gravitational wa ves is small compared\nto the radius of curvature of the background spacetime, the wav es propagate along null geodesics\n82of the background spacetime, i.e. they have the same speed cas light (in this section, we do not\nsetc= 1). In other theories, the speed could differ from cbecause of coupling of gravitation to\n“background” gravitational fields. For example, in the Rosen bimet ric theory with a flat back-\nground metric η, gravitational waves follow null geodesics of η, while light follows null geodesics\nofg(TEGP 10.1 [388]).\nAnother way in which the speed of gravitational waves could differ fr omcis if gravitation were\npropagated by a massive field (a massive graviton), in which case vgwould be given by, in a local\ninertial frame,\nv2\ng\nc2= 1−m2\ngc4\nE2, (135)\nwheremgandEare the graviton rest mass and energy, respectively.\nThe most obvious way to test this is to compare the arrival times of a gravitational wave and\nan electromagnetic wave from the same event, e.g., a supernova or a prompt gamma-ray burst.\nFor a source at a distance D, the resulting value of the difference 1 −vg/cis\n1−vg\nc= 5×10−17/parenleftbigg200 Mpc\nD/parenrightbigg/parenleftbigg∆t\n1 s/parenrightbigg\n, (136)\nwhere ∆t≡∆ta−(1+Z)∆teis the “time difference”, where ∆ taand ∆teare the differences in\narrival time and emission time of the two signals, respectively, and Zis the redshift of the source.\nIn many cases, ∆ teis unknown, so that the best one can do is employ an upper bound on ∆ te\nbased on observation or modelling. The result will then be a bound on 1 −vg/c.\nFor a massive graviton, if the frequency of the gravitational wave s is such that hf≫mgc2,\nwherehis Planck’s constant, then vg/c≈1−1\n2(c/λgf)2, whereλg=h/mgcis the graviton\nCompton wavelength, and the bound on 1 −vg/ccan be converted to a bound on λg, given by\nλg>3×1012km/parenleftbiggD\n200 Mpc100 Hz\nf/parenrightbigg1/2/parenleftbigg1\nf∆t/parenrightbigg1/2\n. (137)\nThe foregoing discussion assumes that the source emits bothgravitational and electromagnetic\nradiation in detectable amounts, and that the relative time of emissio n can be established to\nsufficient accuracy, or can be shown to be sufficiently small.\nHowever, there is a situation in which a bound on the graviton mass ca n be set using gravi-\ntational radiation alone [391]. That is the case of the inspiralling comp act binary. Because the\nfrequency of the gravitational radiation sweeps from low frequen cy at the initial moment of obser-\nvation to higher frequency at the final moment, the speed of the g ravitons emitted will vary, from\nlower speeds initially to higher speeds (closer to c) at the end. This will cause a distortion of the\nobserved phasing of the waves and result in a shorter than expect ed overall time ∆ taof passage of\na given number of cycles. Furthermore, through the technique of matched filtering, the parameters\nof the compact binary can be measured accurately (assuming that GR is a good approximation to\nthe orbital evolution, even in the presence of a massive graviton), and thereby the emission time\n∆tecan be determined accurately. Roughly speaking, the “phase inter val”f∆tin Eq. (137) can\nbe measured to an accuracy 1 /ρ, whereρis the signal-to-noise ratio.\nThus one can estimate the bounds on λgachievable for various compact inspiral systems, and\nfor various detectors. For stellar-mass inspiral (neutron stars or black holes) observed by the\nLIGO/VIRGO class of ground-based interferometers, D≈200 Mpc,f≈100 Hz, and f∆t∼\nρ−1≈1/10. The result is λg>1013km. For supermassive binary black holes (104to 107M⊙)\nobserved by the proposed laser interferometer space antenna ( LISA),D≈3 Gpc,f≈10−3Hz,\nandf∆t∼ρ−1≈1/1000. The result is λg>1017km.\nA full noise analysis using proposed noise curves for the advanced L IGO and for LISA weakens\nthese crude bounds by factors between two and 10 [391, 399, 35 , 36, 19, 348]. For example, for the\n83inspiral of two 106M⊙black holes with aligned spins at a distance of 3 Gpc observed by LISA, a\nbound of 2 ×1016km could be placed [35]. Other possibilities include using binary pulsar dat a to\nbound modifications of gravitational radiation damping by a massive g raviton [139], using LISA\nobservationsof the phasing ofwavesfrom compact white-dwarfb inaries, eccentric galacticbinaries,\nand eccentric inspiral binaries [88, 191], and using pulsar timing arra ys [225].\nBoundsobtainablefromgravitationalradiationeffects shouldbeco mparedwiththe solidbound\nλg>2.8×1012km [352] derived from solar system dynamics, which limit the presence of a Yukawa\nmodification of Newtonian gravity of the form\nV(r) =GM\nrexp(−r/λg), (138)\nand with the model-dependent bound λg>6×1019km from consideration of galactic and cluster\ndynamics [371].\nMirshekari et al.[263] studied bounds that could be placed on more general gravito n dispersion\nrelations that could emerge from alternative theories with Lorentz violation, in which the effective\npropagation speed is given by\nv2\ng\nc2= 1−m2\ngc4\nE2−AEα−2, (139)\nwhereAandαare parameters that depend on the theory.\n8 Astrophysical and cosmological tests\nOne of the central difficulties of testing GR in the strong-field regime is the possibility of contam-\nination by uncertain or complex physics. In the solar system, weak- field gravitational effects can\nin most cases be measured cleanly and separately from non-gravita tional effects. The remarkable\ncleanliness of many binary pulsars permits precise measurements of gravitational phenomena in a\nstrong-field context.\nUnfortunately, nature is rarely so kind. Still, under suitable conditio ns, qualitative and even\nquantitative strong-field tests of GR could be carried out.\nOne example is the exploration of the spacetime near black holes and n eutron stars. Studies of\ncertain kinds of accretion known as advection-dominated accretio n flow (ADAF) in low-luminosity\nbinary X-ray sources may yield the signature of the black hole event horizon [274]. The spectrum\nof frequencies of quasi-periodic oscillations (QPO) from galactic blac k hole binaries may permit\nmeasurement of the spins of the black holes [300]. Aspects of stro ng-field gravity and frame-\ndragging may be revealed in spectral shapes of iron fluorescence lin es from the inner regions of\naccretiondisks[309,310]. UsingsubmmVLBI,acollaborationdubbed theEventHorizonTelescope\ncould image our galactic center black hole SgrA* and the black hole in M8 7 with horizon-scale\nangular resolution; observation of accretion phenomena at these angular resolutions could provide\ntests of the spacetime geometry very close to the black hole [120]. Tracking of hypothetical stars\nwhose orbits are within a fraction of a milliparsec of SgrA* could test t he black hole “ho-hair”\ntheorem, via a direct measurement of both the angular momentum Jand quadrupole moment Q\nof the black hole, and a test of the requirement that Q=−J2/M[395].\nBecause of uncertainties in the detailed models, the results to date of studies like these are sug-\ngestive at best, but the combination of future higher-resolution o bservations and better modelling\ncould lead to striking tests of strong-field predictions of GR.\nFor a detailed review of strong-field tests of GR using electromagne tic observations, see [301].\nAnother example is in cosmology. From a few seconds after the big ba ng until the present, the\nunderlying physics of the universe is well understood, in terms of a S tandard Model of a nearly spa-\ntially flat universe, 13.6 Gyr old, dominated by cold dark matter and da rk energy (ΛCDM). Some\n84alternative theories of gravity that are qualitatively different from GR fail to produce cosmologies\nthat meet even the minimum requirements of agreeing qualitatively wit h big-bang nucleosynthesis\n(BBN) or the properties of the cosmic microwave background (TEG P 13.2 [388]). Others, such\nas Brans–Dicke theory, are sufficiently close to GR (for large enoug hωBD) that they conform to\nall cosmological observations, given the underlying uncertainties. The generalized scalar-tensor\ntheories and f(R) theories, however, could have small effective ωat early times, while evolving\nthrough the attractor mechanism to large ωtoday.\nOne way to test such theories is through big-bang nucleosynthesis , since the abundances of the\nlight elements produced when the temperature of the universe was about 1 MeV are sensitive to\nthe rate of expansion at that epoch, which in turn depends on the s trength of interaction between\ngeometry and the scalar field. Because the universe is radiation-do minated at that epoch, uncer-\ntainties in the amount of cold dark matter or of the cosmological con stant are unimportant. The\nnuclear reaction rates are reasonably well understood from labor atoryexperiments and theory, and\nthe number of light neutrino families (3) conforms to evidence from p article accelerators. Thus,\nwithin modest uncertainties, one can assess the quantitative differ ence between the BBN predic-\ntions of GR and scalar-tensorgravity under strong-field condition s and compare with observations.\nFor recent analyses, see [322, 104, 79, 80].\nIn addition, many alternative theories, such as f(R) theories have been developed in order\nto provide an alternative to the dark energy of the standard ΛCDM model, in particular by\nmodifying gravity on large, cosmological scales, while preserving the conventional solar and stellar\nsystem phenomenology of GR. Since we are now in a period of what may be called “precision\ncosmology”, one can begin to envision trying to test alternative the ories using the accumulation of\ndata on many fronts, including the growth of large scale structure , cosmic backgroundfluctuations,\ngalactic rotation curves, BBN, weak lensing, baryon acoustic oscilla tions, etc. The “parametrized\npost-Friedmann” frameworkis one initial foray into this arena [26]. O ther approachescan be found\nin [109, 122, 121, 415, 174].\n9 Conclusions\nGeneral relativity has held up under extensive experimental scrut iny. The question then arises,\nwhybother to continue totest it? One reasonis that gravityis a fun damental interactionofnature,\nand as such requires the most solid empirical underpinning we can pro vide. Another is that all\nattempts to quantize gravityand to unify it with the other forces s uggest that the standard general\nrelativity of Einstein may not be the last word. Furthermore, the pr edictions of general relativity\nare fixed; the pure theory contains no adjustable constants so n othing can be changed. Thus every\ntest of the theory is either a potentially deadly test or a possible pro be for new physics. Although\nit is remarkable that this theory, born 100 years ago out of almost p ure thought, has managed to\nsurvive every test, the possibility of finding a discrepancy will contin ue to drive experiments for\nyears to come. These experiments will search for new physics beyo nd Einstein at many different\nscales: the large distance scales of the astrophysical, galactic, an d cosmological realms; scales of\nvery short distances or high energy; and scales related to strong or dynamical gravity.\n8510 Acknowledgments\nThis work has been supported since the initial version in part by the N ational Science Foundation,\nGrantNumbersPHY96-00049,00-96522,03-53180,06-52448, 0965133,12-60995and13-06069,and\nby the National Aeronautics and Space Administration, Grant Numb ers NAG5-10186 and NNG-\n06GI60G.We alsogratefullyacknowledgethecontinuinghospitalityo fthe Institut d’Astrophysique\nde Paris, where portions of this update were completed. We thank L uc Blanchet for helpful\ncomments, and Michael Kramer and Norbert Wex for providing usef ul figures.\n86References\n[1] Adelberger, E.G., “New tests of Einstein’s equivalence principle and Newton’s inverse-square\nlaw”,Class. Quantum Grav. ,18, 2397–2405 (2001). 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[DOI],[ADS],[arXiv:1110.3830\n[astro-ph.CO]] .\n113"    },    {        "title": "1703.07859v1.Bergman_Lorentz_spaces_on_tube_domains_over_symmetric_cones.pdf",        "content": "arXiv:1703.07859v1  [math.CA]  22 Mar 2017BERGMAN-LORENTZ SPACES ON TUBE DOMAINS\nOVER SYMMETRIC CONES\nDAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nAbstract. We study Bergman-Lorentz spaces on tube domains\nover symmetric cones, i.e. spaces of holomorphic functions which\nbelong to Lorentz spaces L(p,q).We establish boundedness and\nsurjectivity of Bergman projectors from Lorentz spaces to the cor-\nrespondingBergman-Lorentzspacesandrealinterpolationbetw een\nBergman-Lorentz spaces. Finally we ask a question whose positive\nanswer would enlarge the interval of parameters p∈(1,∞) such\nthat the relevant Bergman projector is bounded on Lpfor cones of\nrankr≥3.\n1.Introduction\nThe notations and definitions are those of [9]. We denote by Ω an\nirreducible symmetric cone in Rnwith rankrand determinant ∆ .We\ndenoteTΩ=Rn+iΩ the tube domain in Cnover Ω.Forν∈R,we\ndefine theweighted measure µonTΩbydµ(x+iy) = ∆ν−n\nr(y)dxdy.We\nconsiderLebesguespaces Lp\nνandLorentzspaces Lν(p,q)onthemeasure\nspace (TΩ,dµ).The Bergman space Ap\nν(resp. the Bergman-Lorentz\nspaceAν(p,q)) is the subspace of Lp\nν(resp. ofLν(p,q)) consisting of\nholomorphic functions.\nOur first result is the following.\nTheorem 1.1. Let1<p≤ ∞and1≤q≤ ∞.\n(1) Forν≤n\nr−1,the Bergman-Lorentz space Aν(p,q)is trivial,\ni.eAν(p,q) ={0}.\n(2) Suppose ν >n\nr−1.Equipped with the norm induced by the\nLorentz space Lν(p,q),the Bergman-Lorentz space Aν(p,q)is a\nBanach space.\nForp= 2,the Bergman space A2\nνis a closed subspace of the Hilbert\nspaceL2\nνand the Bergman projector Pνis the orthogonal projector\n12 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nfromL2\nνtoA2\nν.We adopt the notation\nQν= 1+ν\nn\nr−1.\nOur boundedness theorem for Bergman projectors on Lorentz s paces\nis the following.\nTheorem 1.2. Letν >n\nr−1and1≤q≤ ∞.The weighted Bergman\nprojectorPγ, γ≥ν+ν\nr−1(resp. the Bergman projector Pν)extends\nto a bounded operator from Lν(p,q)toAν(p,q)for all1< p < Q ν\n(resp. for all 1+Q−1\nν<p<1+Qν). In this case, under the restriction\n1< q <∞(resp.1≤q <∞), thenPγforγsufficiently large (resp.\nPν)is the identity on Aν(p,q).\nFor the Bergman projector Pν,following recent developments, this\ntheorem is extended below to a larger interval of exponents pon tube\ndomains over Lorentz cones ( r= 2) (see section 6). Finally our real in-\nterpolation theorem between Bergman-Lorentz spaces is the follo wing.\nTheorem 1.3. Letν >n\nr−1.\n(1) For all 1< p0< p1< Qν(resp.1 +Q−1\nν< p0< p1<\n1+Qν),1≤q0,q1≤ ∞and0< θ <1,the real interpolation\nspace\n[Aν(p0,q0),Aν(p1,q1)]θ,q\nidentifies with Aν(p,q),1\np=1−θ\np0+θ\np1,1< q <∞(resp.\n1≤q<∞) with equivalence of norms.\n(2) For1<p0<1+Q−1\nν, Qν<p1<1+Qν,1≤q0,q1≤ ∞,the\nBergman-Lorentz spaces Aν(p,q),1+Q−1\nν<p<Q ν,1≤q <\n∞are real interpolation spaces between Aν(p0,q0)andAν(p1,q1)\nwith equivalence of norms.\nThese results were first presented in the PhD dissertation of the s ec-\nond author [10].\nThe plan of the paper is as follows. In section 2, we overview defi-\nnitions and properties of Lorentz spaces on a non-atomic σ-finite mea-\nsure space. This section encloses results on real interpolation bet ween\nLorentz spaces. In section 3, we define Bergman-Lorentz space s on aBERGMAN-LORENTZ SPACES 3\ntube domain TΩover a symmetric cone Ω and we establish Theorem\n1.1. In section 4, we study the density of the subspace Aν(p,q)∩At\nγin\nthe Banach space Aν(p,q) forν,γ >n\nr−1,1<p,t≤ ∞,1≤q <∞.\nRelying on boundedness results for the Bergman projectors Pγ, γ≥\nν+n\nr−1 andPνon Lebesgue spaces Lp\nν[2, 3], we then provide a\nproof of Thorem 1.2. In section 5, we prove Theorem 1.3. In section 6,\nwe ask a question whose positive answer would enlarge the interval o f\nparamaersp∈(1,∞) such that the Bergman projector Pνis bounded\nonLp\nνfor upper rank cones ( r≥3). A second question addresses the\ndensity of the subspace Aν(p,q)∩At\nγin the Banach space Aν(p,q).The\nthird question concerns a possible extension of Theorem 1.3.\n2.Lorentz spaces on measure spaces\n2.1.Definitions and preliminary topological properties. Through-\nout this section, the notation ( E,µ) is fixed for a non-atomic σ-finite\nmeasure space. We refer to [14], [11], [6] and [5]. Also cf. [10].\nDefinition 2.1. Letfbe a measurable function on ( E,µ).The distri-\nbution function µfoffis defined on [0 ,∞) by\nµf(λ) =µ({x∈E:|f(x)|>λ}).\nThe non-increasing rearrangement function f⋆offis defined on [0 ,∞)\nby\nf⋆(t) = inf{λ≥0 :µf(λ)≤t}.\nTheorem 2.2 (Hardy-Littlewood Theorem) .Letfandgbe two mea-\nsurable functions on (E,µ).Then\n/integraldisplay\nE|f(x)g(x)|dµ(x)≤/integraldisplay∞\n0f⋆(s)g⋆(s)ds.\nIn particular, let gbe a positive measurable function on (E,µ)and let\nFbe a measurable subset of Eof bounded measure µ(F).Then\n/integraldisplay\nFg(x)dµ(x)≤/integraldisplayµ(F)\n0g⋆(s)ds.4 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nDefinition 2.3. Let 1≤p,q≤ ∞.The Lorentz space L(p,q) is the\nspace of measurable functions on ( E,µ) such that\n||f||⋆\np,q=/parenleftbigg/integraldisplay∞\n0/parenleftBig\nt1\npf⋆(t)/parenrightBigqdt\nt/parenrightbigg1\nq\n<∞if 1≤p<∞and 1≤q<∞\n(resp.\n||f||⋆\np,∞= sup\nt>0t1\npf⋆(t)<∞if 1≤p≤ ∞).\nWe recall that for p=∞andq <∞,this definition gives way to\nthe space of vanishing almost everywhere functions on ( E,µ).It is also\nwell known that for p=q,the Lorentz space L(p,p) coincides with the\nLebesgue space Lp(E,dµ).More precisely, we have the equality\n(2.1) ||f||p=/parenleftbigg/integraldisplay∞\n0f⋆(t)pdt/parenrightbigg1\np\n.\nIn the sequel we shall adopt the following notation:\nLp=Lp(E,dµ).\nWe shall need the following two results.\nProposition 2.4. The functional ||·||⋆\np,qis a quasi-norm (it satisfies\nall properties of a norm except the triangle inequality) onL(p,q). This\nfunctional is a norm if 1≤q≤p<∞orp=q=∞.\nLemma 2.5. Let1≤p <∞.Then for every measurable function f\non(E,µ),we have\n||f||⋆\np,∞= sup\nλ>0λµf(λ)1\np.\nWe next define a norm on L(p,q) which is equivalent to the quasi-\nnorm|| · ||⋆\np,q.For a measurable function fon (E,µ),we define the\nfunctionf⋆⋆on [0,∞) by\nf⋆⋆(t) =1\nt/integraldisplayt\n0f⋆(u)du.\nFor 1≤p,q≤ ∞,we define the functional ||·||p,qby\n||f||p,q=/parenleftbigg/integraldisplay∞\n0/parenleftBig\nt1\npf⋆⋆(t)/parenrightBigqdt\nt/parenrightbigg1\nq\nif 1≤p<∞and 1≤q<∞BERGMAN-LORENTZ SPACES 5\n(resp.\n||f||p,∞= sup\nt>0t1\npf⋆⋆(t) if 1 ≤p≤ ∞).\nProposition 2.6. Let1< p≤ ∞and1≤q≤ ∞.The functional\n||·||p,qis a norm on L(p,q).\nLemma 2.7. Let1< p≤ ∞and1≤q≤r≤ ∞.Then for every\nmeasurable function fon(E,µ),we have\n||f||p,r≤ ||f||p,q\nand\n||f||⋆\np,r≤ ||f||⋆\np,q.\nThe following lemma asserts that the norm ||·||p,qis equivalent to the\nquasi-norm ||·||⋆\np,qwhen 1<p≤ ∞and 1≤q≤ ∞.\nLemma 2.8. For all1< p <∞,1≤q≤ ∞and for every f∈\nLν(p,q),we have\n||f||⋆\np,q≤ ||f||p,q≤p\np−1||f||⋆\np,q.\nWe now state the following theorem.\nTheorem 2.9. Let1< p <∞,1≤q≤ ∞.Equipped with the norm\n||·||p,q,the Lorentz space L(p,q)is a Banach space.\nWe shall also need the following proposition.\nProposition 2.10. Let1< p <∞and1≤q≤ ∞.Every Cauchy\nsequence in the Lorentz space (L(p,q),||·||p,q)contains a subsequence\nwhich converges a.e. to its limit in L(p,q).\nWe finally record the following duality theorem [11].\nTheorem 2.11. (1)Let1≤p <∞.The topological dual space\n(L(p,1))′of the Lorentz space L(p,1)identifies with the Lorentz space\nL(p′,∞)with respect to the duality pairing\n(⋆) (f,g) =/integraldisplay\nTΩf(z)g(z)dµ(z).\n(2)Let1< p <∞and1< q <∞.The topological dual space\n(L(p,q))′of the Lorentz space L(p,q)identifies with the Lorentz space\nL(p′,q′)with respect to the same duality pairing (⋆).6 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\n2.2.Interpolation via the real method between Lorentz spaces.\nWe begin with an overview of the theory of real interpolation betwee n\nBanach spaces (cf. e.g. [6]).\nDefinition 2.12. A pair (X0,X1) of Banach spaces is called a compat-\nible couple if there is some Hausdorff topological vector space in which\nX0andX1are continuously embedded.\nDefinition2.13. Let(X0,X1)beacompatiblecoupleofBanachspaces.\nDenoteX=X0+X1.Lett>0 anda∈X.We define the functional\nK(t,a,X) by\nK(t,a,X) = inf{||a0||X0+t||a1||X1:a=a0+a1,a0∈X0,a1∈X1}.\nFor 0<θ<1,1≤q <∞or 0≤θ≤1, q=∞,the real interpolation\nspace between X0andX1is the space\n[X0,X1]θ,q:={a∈X:||a||θ,q,X<∞}\nwith\n||a||θ,q,X:=/parenleftbigg/integraldisplay∞\n0/parenleftbig\nt−θK(t,a,X)/parenrightbigqdt\nt/parenrightbigg1\nq\nif 1≤q<∞(resp.\n||a||θ,∞,X:= sup\nt>0t−θK(t,a,X)\nifq=∞).\nProposition 2.14. For0< θ <1,1≤q <∞or0≤θ≤1, q=\n∞,the functional || · ||θ,q,Xis a norm on the real interpolation space\n[X0,X1]θ,q.Endowed with this norm, [X0,X1]θ,qis a Banach space.\nDefinition 2.15. Let (X0,X1) and (Y0,Y1) be two compatible couples\nofBanachspacesandlet Tbealinearoperatordefinedon X:=X0+X1\nand taking values in Y:=Y0+Y1.ThenTis said to be admissible with\nrespect to the couples XandYif fori= 0,1,the restriction of Tis a\nbounded operator from XitoYi.\nTheorem 2.16. Let(X0,X1)and(Y0,Y1)be two compatible couples of\nBanach spaces and let 0<θ <1,1≤q <∞or0≤θ≤1, q=∞.BERGMAN-LORENTZ SPACES 7\nLetTbe an admissible linear operator with respect to the couples X\nandYsuch that\n||Tfi||Yi≤Mi||fi||Xi(fi∈Xi, i= 0,1).\nThenTis a bounded operator from [X0,X1]θ,qto[Y0,Y1]θ,q.More pre-\ncisely, we have\n||Tf||[Y0,Y1]θ,q≤M1−θ\n0Mθ\n1||f||[X0,X1]θ,q\nfor allf∈[X0,X1]θ,q.\nThe following theorem gives the real interpolation spaces between\nLebesgue spaces and Lorentz spaces on the measure space ( E,µ).\nTheorem 2.17. Let0< θ <1,1≤q≤ ∞.Let1≤p0< p1≤ ∞\nand define the exponent pby1\np=1−θ\np0+θ\np1.We have the identifications\nwith equivalence of norms:\na)[Lp0ν,Lp1ν]θ,q=Lν(p,q);\nb)[Lν(p0,q0),Lν(p1,q1)]θ,q=Lν(p,q)for1< p0< p1<∞,1≤\nq0,q1≤ ∞.\nDefinition 2.18. Let(R,µ)and(S,ν)betwonon-atomic σ-finitemea-\nsure spaces. Suppose 1 ≤p<∞,1≤q≤ ∞.LetTbe a linear opera-\ntor defined on the simple functions on ( R,µ) and taking values on the\nmeasurable functions on ( S,ν).ThenTis said to be of restricted weak\ntype (p,q) if there is a positive constant Msuch that\nt1\nq(TχF)⋆(t)≤Mµ(F)1\np(t>0)\nfor all measurable subsets FofR.This estimate can also be written in\nthe form\n||TχF||q,∞≤M||χF||p,1\nor equivalently in view of Lemma 2.5,\nsup\nλ>0λµχF(λ)1\nq≤Mµ(F)1\np.\nIn the next two statements, LR(p,1) andLS(q,∞) denote the corre-\nsponding Lorentz spaces on the respective measure spaces ( R,µ) and\n(S,ν).8 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nProposition 2.19. Let(R,µ)and(S,ν)be two non-atomic σ-finite\nmeasure spaces. Suppose 1≤p <∞,1≤q≤ ∞.LetTbe a linear\noperator defined on the simple functions on (R,µ)and taking values on\nthe measurable functions on (S,ν).We suppose that Tis of restricted\nweak type (p,q).ThenTuniquely extends to a bounded operator from\nLR(p,1)toLS(q,∞).\nTheorem 2.20 (Stein-Weiss) .Let(R,µ)and(S,ν)be two measure\nspaces. Suppose 1≤p0< p1<∞and1≤q0,q1≤ ∞withq0/ne}ationslash=\nq1.Suppose further that Tis a linear operator defined on the simple\nfunctions on (R,µ)and taking values on the measurable functions on\n(S,ν)and suppose that Tis of restricted weak types (p0,q0)and(p1,q1).\nIf1≤r≤ ∞,thenThas a unique extension to a linear operator, again\ndenoted by T,which is bounded from LR(p,r)intoLS(q,r)where\n1\np=1−θ\np0+θ\np1,1\nq=1−θ\nq0+θ\nq1,0<θ<1.\nIf in addition, the inequalities pj≤qj(j= 0,1)hold, then Tis of\nstrong type (p,q),i.e. there exists a positive constant Csuch that\n||Tf||Lq(S,ν)≤C||f||Lp(R,µ)(f∈Lp(R,µ)).\nWe finish this section with the Wolff reiteration theorem. Let ( X,||·\n||X) and (Y,||·||Y) be two normed vector spaces over the field C.We\nsay thatXcontinuously embeds in Yand we write X ֒→YifX⊂Y\nand there exists a positive constant Csuch that\n||x||Y≤C||x||X∀x∈X.\nIt is easy to check that Xidentifies with Yif and only if X ֒→Yand\nY ֒→X.\nDefinition 2.21. If (X0,X1) is a compatible couple of Banach spaces,\nthen a Banach space Xis said to be an intermediate space between X0\nandX1ifXis continuously embedded between X0∩X1andX0+X1,\ni.e.\nX0∩X1֒→X ֒→X0+X1.BERGMAN-LORENTZ SPACES 9\nWeremindthereaderthattherealinterpolationspace[ X0,X1]θ,q,0<\nθ <1,1≤q≤ ∞is an intermediate space between X0andX1.In\nthis direction, we recall the following density theorem.\nTheorem 2.22. [6, Theorem 2.9] Let(X0,X1)be a compatible couple\nof Banach spaces and suppose 0< θ <1,1≤q <∞.Then the\nsubspaceX0∩X1is dense in [X0,X1]θ,q.\nWe next state the Wolff reiteration theorem.\nTheorem 2.23 ( [15, 12]) .LetX2andX3be intermediate Banach\nspaces of a compatible couple (X1,X4).Let0< ϕ,ψ < 1and1≤\nq,r≤ ∞and suppose that\nX2= [X1,X3]ϕ,q, X3= [X2,X4]ψ,r.\nThen (up to equivalence of norms)\nX2= [X1,X4]ρ,q, X3= [X1,X4]θ,r\nwhere\nρ=ϕψ\n1−ϕ+ϕψ, θ=ψ\n1−ϕ+ϕψ.\n3.The Bergman-Lorentz spaces on tube domains over\nsymmetric cones\n3.1.Symmetric cones: definitions and preliminary notions.\nMaterials of this section are essentially from [9]. We give some def-\ninitions and useful results.\nLet Ω be an irreducible open cone of rank r inside a vector space\nVof dimension n, endowed with an inner product ( .|.) for which Ω is\nself-dual. Such a cone is called a symmetric cone in V.LetG(Ω) be the\ngroupof transformations of Ω, and Gits identity component. It is well-\nknown that there exists a subgroup HofGacting simply transitively\non Ω, that is every y∈Ω can be written uniquely as y=gefor some\ng∈Hand a fixed e∈Ω. The notation ∆ is for the determinant of Ω .\nWe first recall the following lemma.\nLemma 3.1. [2]The following inequality is valid.\n∆(y)≤∆(y+v) (y,v∈Ω).10 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nWedenoteby dΩtheH-invariantdistanceonΩ .Thefollowinglemma\nwill be useful.\nLemma 3.2. [2]Letδ >0.There exists a positive constant γ=γ(δ,Ω)\nsuch that for ξ,ξ′∈ΩsatisfyingdΩ(ξ,ξ′)≤δ,we have\n1\nγ≤∆(ξ)\n∆(ξ′)≤γ.\nIn the sequel, we write as usual V=Rn.\n3.2.Bergman-Lorentz spaces on tube domains over symmetric\ncones. Proof of Theorem 1.1. Let Ω be an irreducible symmetric\ncone inRnwith rankr,determinant ∆ and fixed point e.We denote\nTΩ=Rn+iΩ the tube domain in Cnover Ω.Forν∈R,we define\nthe weighted measure µonTΩbydµ(x+iy) = ∆ν−n\nr(y)dxdy.For a\nmeasurablesubset AofTΩ,wedenoteby |A|the(unweighted) Lebesgue\nmeasure of A,i.e.|A|=/integraltext\nAdxdy.\nDefinition 3.3. Since the determinant ∆ is a polynomial in Rn,it\ncan be extended in a natural way to Cnas a holomorphic polynomial\nwe shall denote ∆/parenleftbigx+iy\ni/parenrightbig\n.It is known that this extension is zero free\non the simply connected region TΩinCn.So for each real number α,\nthe power function ∆αcan also be extended as a holomorphic function\n∆α/parenleftbigx+iy\ni/parenrightbig\nonTΩ.\nThe following lemma will be useful.\nLemma 3.4. [2]Letα>0.\n(1) We have/vextendsingle/vextendsingle/vextendsingle∆−α/parenleftBigz\ni/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤∆−α(ℑmz) (z∈TΩ).\n(2) We suppose ν >n\nr−1.The following estimate\n/integraldisplay\nΩ/parenleftbigg/integraldisplay\nRn/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆−α/parenleftbiggx+i(y+e)\ni/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\ndx/parenrightbigg\n∆ν−n\nr(y)dy<∞\nholds if and only if α>ν+2n\nr−1\np.\nWe denote by dthe Bergman distance on TΩ.We remind the reader\nthat the group Rn×Hacts simply transitively on TΩ.The following\nlemma will also be useful.BERGMAN-LORENTZ SPACES 11\nLemma 3.5. The measure ∆−2n\nr(y)dxdyisRn×H-invariant on TΩ.\nThe following corollary is an easy consequence of Lemma 3.2.\nCorollary 3.6. Letδ >0.There exists a positive constant C=C(δ)\nsuch that for z,z′∈Ωsatisfyingd(z,z′)≤δ,we have\n1\nC≤∆(ℑmz)\n∆(ℑmz′)≤C.\nThe next proposition will lead us to the definition of Bergman-\nLorentz spaces.\nProposition 3.7. The measure space (TΩ,µ)is a non-atomic σ-finite\nmeasure space.\nIn view of this proposition, all the results of the previous section ar e\nvalid on the measure space ( TΩ,µ).We shall denote by Lν(p,q) the\ncorresponding Lorentz space on ( TΩ,µ).Moreover, we write Lp\nνfor the\nweighted Lebesgue space Lp(TΩ,dµ) on (TΩ,µ).\nThe following corollary is an immediate consequence of Theorem 2.22\nand assertion a) of Theorem 2.17.\nCorollary 3.8. The subspace C∞\nc(TΩ)consisting of C∞functions with\ncompact support on TΩis dense in the Lorentz space Lν(p,q)for all\n1<p<∞,1≤q <∞.\nDefinition 3.9. The Bergman-Lorentz space Aν(p,q),1≤p,q≤ ∞\nis the subspace of the Lorentz space Lν(p,q) consisting of holomorphic\nfunctions. In particular Aν(p,p) =Ap\nν,whereAp\nν=Hol(TΩ)∩Lp\nνis\nthe usual weighted Bergman space on TΩ.In fact,Ap\nν,1≤p≤ ∞\nis a closed subspace of the Banach space Lp\nν.The Bergman projector\nPνis the orthogonal projector from the Hilbert space L2\nνto its closed\nsubspaceA2\nν.\nExample 3.10. Letν >n\nr−1,1< p <∞,1≤q≤ ∞.The\nfunctionF(z) = ∆−α(z+ie\ni) belongs to the Bergman-Lorentz Aν(p,q) if\nα>ν+2n\nr−1\np.Indeed we can find positive numbers p0andp1such that\n1≤p0< p < p 1≤ ∞andα >ν+2n\nr−1\npi(i= 0,1).By assertion 2)\nof Lemma 3.4, the holomorphic function Fbelongs toLpiν(i= 0,1).12 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nThe conclusion follows by assertion a) of Theorem 2.17 and Theorem\n2.22.\nLemma 3.11. Letν∈R,1< p≤ ∞,1≤q≤ ∞and letf∈\nAν(p,q).For every compact set KofCncontained in TΩ,there is a\npositive constant CKsuch that\n|f(z)| ≤CK||f||p,q(z∈K).\nProof.Suppose first p=∞.The interesting case is q=∞.In this case,\nLν(p,q) =L∞\nνandAν(p,q) =A∞\nνis the space of bounded holomorphic\nfunctions on ( TΩ,µ).The relevant result is straightforward.\nWe next suppose 1 < p <∞,1< q <∞andf∈Aν(p,q).Since\nLν(p,q) continuously embeds in Lν(p,∞) (Lemma 2.7) it suffices to\nshow that for each f∈Aν(p,∞),we have\n|f(z)| ≤CK||f||p,∞(z∈K).\nFor a compact set KinCncontained in TΩ,we callρthe Euclidean dis-\ntance from Kto the boundary of TΩ.We denoteB(z,ρ\n2) the Euclidean\nball centered at z,with radiusρ\n2.We apply successively\n- the mean-value property,\n- the fact that the function\nu+iv∈TΩ/mapsto→∆ν−n\nr(v)\nis uniformly bounded below on every Euclidean ball B(z,ρ\n2)\nwhenzlies onKand\n- the second part of Theorem 2.2,\nto obtain that\n|f(z)|=1\n|B(z,ρ\n2)|/vextendsingle/vextendsingle/vextendsingle/integraltext\nB(z,ρ\n2)f(u+iv)dudv/vextendsingle/vextendsingle/vextendsingle\n≤C\n|B(z,ρ\n2)|/integraltext\nB(z,ρ\n2)|f(u+iv)|∆ν−n\nr(v)dudv\n≤C\n|B(z,ρ\n2)|/integraltextµ(B(z,ρ\n2))\n0f⋆(t)dt\n≤C\n|B(z,ρ\n2)|/integraltextµ(Kρ)\n0t1\npf⋆(t)t1\np′dt\nt\n≤C||f||p,∞\n|B(z,ρ\n2|/integraltextµ(Kρ)\n0t1\np′dt\nt≤CK||f||p,∞\nforeachz∈KwithKρ=/uniontext\nz∈KB(z,ρ\n2)andCK=Cp′1\n|B(z,ρ\n2)|(µ(Kρ))1\np′.\nWe recall that there is a positive constant Cnsuch that for all z∈CnBERGMAN-LORENTZ SPACES 13\nandρ>0,we have|B(z,ρ)|=Cnρ2nand we check easily that µ(Kρ)<\n∞.\n/square\nProof of Theorem 1.1 .(1) We suppose that ν≤n\nr−1.It suffices\nto show that Aν(p,∞) ={0}for all 1<p<∞.GivenF∈Aν(p,∞),\nwe first prove the following lemma.\nLemma 3.12. For general ν∈R,the following estimate holds.\n(3.1)|F(x+i(y+e))|∆ν+n\nr\np(y+e)≤Cp||F||p,∞(x+iy∈TΩ).\nProof of the Lemma. We recall the following inequality [2]:\n|F(x+i(y+e))| ≤C/integraldisplay\nd(x+i(y+e), u+iv)<1|F(u+iv)|dudv\n∆2n\nr(v)\n(3.2)≤C′∆−ν−n\nr(y+e)/integraldisplay\nd(x+i(y+e), u+iv)<1|F(u+iv)|dµ(u+iv).\nThe latter inequality follows by Corollary 3.6. Now by Theorem 2.2,\nwe have\n/integraldisplay\nd(x+i(y+e),u+iv)<1|F(u+iv)|dµ(u+iv)≤/integraldisplayµ(Bberg(x+i(y+e),1))\n0t1\npF⋆(t)t1\np′dt\nt\n(3.3) ≤p′||F||p,∞(µ(Bberg(x+i(y+e),1))1\np′,\nwhereBberg(·,·) denotes the Bergman ball in TΩ.By Lemma 3.5 and\nCorollary 3.6, we obtain that\n(3.4) µ(Bberg(x+i(y+e),1))≃∆ν+n\nr(y+e).\nThen combining (3.2), (3.3) and (3.4) gives the announced estimate\n(3.1). /square\nWe next deduce that the function\nz∈TΩ/mapsto→F(z+ie)∆−α(z+ie)\nbelongs to the Bergman space A1\nνwhenαis sufficiently large. We\ndistinguish two cases: 1) ν≤ −n\nr; 2)−n\nr<ν≤n\nr−1.14 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nCase1). We suppose that ν≤ −n\nr.We takeα >−ν−n\nr\npand we apply\nassertion (1) of Lemma 3.4 to get\n/vextendsingle/vextendsingleF(x+i(y+e))∆−α(x+i(y+e))/vextendsingle/vextendsingle\n=|F(x+i(y+e))|/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆−α−ν+n\nr\np(x+i(y+e))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆ν+n\nr\np(x+i(y+e))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ |F(x+i(y+e))|/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆−α−ν+n\nr\np(x+i(y+e))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆ν+n\nr\np(y+e)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Cp||F||p,∞|∆−α−ν+n\nr\np(x+i(y+e))|.\nForthe latter inequality, we applied estimate (3.1) of Lemma 3.12. The\nconclusion follows because by assertion (2) of Lemma 3.4, the funct ion\n∆−α−ν+n\nr\np(x+i(y+e)) is integrable on TΩwhenαis sufficiently large.\nCase2). We suppose that −n\nr< ν≤n\nr−1.Sinceν+n\nr>0 and\n∆(y+e)≥∆(y) by Lemma 3.1, it follows from (3.1) that the function\nz∈TΩ/mapsto→F(z+ie) is bounded on TΩ.The conclusion easily follows.\nFinally we remind that A1\nν={0}ifν≤n\nr−1 (cf. e.g. [2]). We\nconclude that the function F(·+ie) vanishes identically on TΩ.An ap-\nplicationoftheanalyticcontinuationprinciple thenimplies theidentity\nF≡0 onTΩ.\n(2) We suppose that ν >n\nr−1.It suffices to show that Aν(p,q) is a\nclosed subspace of the Banach space ( Lν(p,q),||·||p,q).Forp=∞,the\ninteresting case is q=∞and thenAν(p,q) =A∞\nν; the relevant result\nis easy to obtain.\nWe next suppose that 1 <p<∞and 1≤q≤ ∞.In view of Lemma\n3.11, every Cauchy sequence {fm}∞\nm=1in (Aν(p,q),||·||p,q) converges to\na holomorphic function f:TΩ→Con compact sets in Cncontained\ninTΩ.On the other hand, since the sequence {fm}∞\nm=1is a Cauchy se-\nquenceintheBanachspace( Lν(p,q),||·||p,q),itconverges withrespect\nto theLν(p,q)-norm to a function g∈Lν(p,q).Now by Proposition\n2.10, this sequence contains a subsequence {fmk}∞\nk=1which converges\nµ-a.e. tog.The uniqueness ofthe limit implies that f=ga.e. Wehave\nproved that the Cauchy sequence {fm}in (Aν(p,q),||·||p,q) converges\nin (Aν(p,q),||·||p,q) to the function f. /squareBERGMAN-LORENTZ SPACES 15\n4.Density in Bergman-Lorentz spaces. Proof of Theorem\n1.2\n4.1.Density in Bergman-Lorentz spaces. We adopt the following\nnotation given in the introduction:\nQν= 1+ν\nn\nr−1.\nWe shall refer to the following result. For its proof, consult [13] and\n[2].\nTheorem 4.1. Letν >n\nr−1.The weighted Bergman projector Pγ, γ≥\nν+ν\nr−1(resp. the Bergman projector Pν)extends to a bounded operator\nfromLp\nνtoAp\nνfor all1≤p<Qν(resp. for all 1+Q−1\nν<p<1+Qν).\nThe following corollary follows from a combination of Theorem 4.1,\nTheorem 2.16 and Theorem 2.17.\nCorollary 4.2. Letν >n\nr−1.TheweightedBergmanprojector Pγ, γ≥\nν+ν\nr−1(resp. the Bergman projector Pν)extends to a bounded operator\nfromLν(p,q)toAν(p,q)for all1< p < Q ν(resp. for all 1+Q−1\nν<\np<1+Qν)and1≤q≤ ∞.\nThe following proposition was proved in [2].\nProposition 4.3. We suppose that ν,γ >n\nr−1.Let1≤p<∞.The\nsubspaceAp\nν∩At\nγis dense in the Banach space Ap\nν.Moreover, if the\nweighted Bergman projector Pγextends to a bounded operator on Lp\nν\nand ifPγis the identity on At\nγ,thenPγis the identity on Ap\nν.\nThe next corollary is a consequence of Theorem 4.1 and Proposition\n4.3.\nCorollary 4.4. Letγ >n\nr−1.For allt∈(1 +Q−1\nγ,1 +Qγ),the\nBergman projector Pγis the identity on At\nγ.\nWe shall prove the following density result for Bergman-Lorentz\nspaces.\nProposition 4.5. We suppose that γ≥ν >n\nr−1,1<p,t<∞and\n1≤q <∞.The subspace Aν(p,q)∩At\nγis dense in the Banach space\nAν(p,q)in the following three cases.16 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\n(1)p=q;\n(2)γ=ν >n\nr−1,1+Q−1\nν<p,t<1+Qνand1≤q <p;\n(3)p,q∈(t,∞).\nProof. 1) Forp=q, Aν(p,p) =Ap\nν,the result is known, cf. e.g. [2].\n2) We suppose now that γ=ν >n\nr−1,1 +Q−1\nν< p,t < 1+Qν\nand 1≤q < p.GivenF∈Aν(p,q),by Corollary 3.8, there exists a\nsequence {fm}∞\nm=1ofC∞functions with compact support on TΩsuch\nthat{fm}∞\nm=1→F(m→ ∞) inLν(p,q).Eachfmbelongs to Lt\nν∩\nLν(p,q).By Corollary 4.2 and Theorem 4.1, the Bergman projector Pν\nextends to a bounded operator on Lν(p,q) and onLt\nνrespectively. So\nPνfm∈Aν(p,q)∩At\nνand{Pνfm}∞\nm=1→PνF(m→ ∞) inAν(p,q).\nNotice that Aν(p,q)⊂Ap\nνbecauseq <p.By Corollary 4.4, we obtain\nthatPνF=F.This finishes the proof of assertion 2).\n3)Letlbeaboundedlinearfunctionalon Aν(p,q)suchthatl(F) = 0\nfor allF∈Aν(p,q)∩At\nγ.We must show that l≡0 onAν(p,q).\nWe first prove that the holomorphic function Fm,αdefined onTΩby\nFm,α(z) = ∆−α/parenleftbiggz\nm+ie\ni/parenrightbigg\nF(z)\nbelongs to Aν(p,q)∩At\nγwhenα >0 is sufficiently large. By Lemma\n3.4 (assertion (1)) and Lemma 3.1, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆−α/parenleftbiggz\nm+ie\ni/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤∆−α(e) = 1.\nSo|Fm,α| ≤ |F|; this implies that Fm,α∈Aν(p,q).\nWe next show that Fm,α∈At\nγwhenαis large. We obtain that\nI:=/integraltext\nTΩ/vextendsingle/vextendsingle/vextendsingle∆−α/parenleftBigx+iy\nm+ie\ni/parenrightBig\nF(x+iy)/vextendsingle/vextendsingle/vextendsinglet\n∆γ−n\nr(y)dxdy\n=/integraltext\nTΩ/vextendsingle/vextendsingle/vextendsingle∆−α/parenleftBigx+iy\nm+ie\ni/parenrightBig\nF(x+iy)/vextendsingle/vextendsingle/vextendsinglet\n∆γ−ν(y)dµ(x+iy).\nWe takeαt>γ−ν.By Lemma 3.4 (assertion (1)) and Lemma 3.1, we\nobtain that\nI≤/integraldisplay\nTΩ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆−αt+γ−ν/parenleftBiggx+iy\nm+ie\ni/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|F(x+iy)|tdµ(x+iy).BERGMAN-LORENTZ SPACES 17\nObserving that( |f|t)⋆= (f⋆)t,wenoticethat |F|t∈Lν(p\nt,q\nt).ByTheo-\nrem2.11,itsufficestoshowthatthefunction z∈TΩ/mapsto→∆−αt+γ−ν/parenleftBigx+iy\nm+ie\ni/parenrightBig\nbelongs toLν((p\nt)′,(q\nt)′) whenαis large. The desired conclusion follows\nby Example 3.10.\nSo our assumption implies that\n(4.1) l(Fm,α) = 0.\nBy the Hahn-Banach theorem, there exists a bounded linear funct ional\n/tildewidelonLν(p,q) such that/tildewidel|Lν(p,q)=land||/tildewidel||=||l||.Furthermore, by\nTheorem 2.11, there exists a function ϕ∈Lν(p′,q′) such that\n/tildewidel(f) =/integraldisplay\nTΩf(z)ϕ(z)dµ(z)∀f∈Lν(p,q).\nWe must show that\n(4.2)/integraldisplay\nTΩF(z)ϕ(z)dµ(z) = 0 ∀F∈Aν(p,q).\nThe equation (4.1) can be expressed in the form\n/integraldisplay\nTΩFm,α(z)ϕ(z)dµ(z) =/integraldisplay\nTΩ∆−α/parenleftbiggz\nm+ie\ni/parenrightbigg\nF(z)ϕ(z)dµ(z) = 0.\nAgain by Theorem 2.11, the function Fϕis integrable on TΩsince\nF∈Lν(p,q) andϕ∈Lν(p′,q′).We also have/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆−α/parenleftbiggz\nm+ie\ni/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1∀z∈TΩ, m= 1,2,···\nAn application of the Lebesgue dominated theorem next gives the an -\nnounced conclusion (4.2). /square\nWe next deduce the following corollary.\nCorollary 4.6. Letν >n\nr−1.\n(1) For all 1< p < Q νand1< q <∞,there exists a real index\nγ≥ν+n\nr−1such that the Bergman projector Pγis the identity\nonAν(p,q).\n(2) For all p∈(1 +Q−1\nν,1 +Qν)and1≤q <∞,the Bergman\nprojectorPνis the identity on Aν(p,q).\nProof. (1) By Corollary 4.4, for all t∈(1 +Q−1\nγ,1 +Qγ), Pγis the\nidentity on At\nγ.Next let 1 < p < Q νand 1< q <∞.Let the real18 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nindexγbe such that γ≥ν+n\nr−1 and 1+Q−1\nγ<min(p,q).We taket\nsuch that 1+ Q−1\nγ<t<min(p,q).It follows from the assertion ( 3)of\nProposition 4.5 that the subspace Aν(p,q)∩At\nγis dense in the Banach\nspaceAν(p,q).But by Corollary 4.2, Pγextends to a bounded operator\nonLν(p,q).We conclude then that Pγis the identity on Aν(p,q).\n(2)By Corollary 4.4, for all t∈(1+Q−1\nν,1+Qν),Pνis the identity\nonAt\nν.Next let 1+ Q−1\nν< p <1+Qν.By Corollary 4.2, Pνextends\nto a bounded operator on Lν(p,q).It then suffices to show that the\nsubspaceAν(p,q)∩At\nνis dense in the Banach space Aν(p,q) for some\nt∈(1 +Q−1\nν,1 +Qν).If 1 +Q−1\nν< q <∞,we taketsuch that\n1 +Q−1\nν< t <min(p,q) : the conclusion follows by assertion (3) of\nProposition 4.5. Otherwise, if 1 ≤q≤1+Q−1\nν,the conclusion follows\nby assertion (2) of the same proposition.\n/square\n4.2.Proof of Theorem 1.2. Combine Corollary 4.2 with Corollary\n4.6.\n5.Real interpolation between Bergman-Lorentz spaces.\nProof of Theorem 1.3.\nFor all 1< p0< p1<∞and 1≤q0, q1≤ ∞and 0< θ <1,by\nDefintion 2.13 (the definition of real interpolation spaces), we have at\nonce\n[Aν(p0,q0),Aν(p1,q1)]θ,q֒→[Lν(p0,q0),Lν(p1,q1)]θ,q=Lν(p,q),\nwith1\np=1−θ\np0+θ\np1.Weconcludethat[ Aν(p0,q0),Aν(p1,q1)]θ,q֒→Hol(TΩ)∩\nLν(p,q) =Aν(p,q).\nWe next show the converse, i.e. Aν(p,q)֒→[Aν(p0,q0),Aν(p1,q1)]θ,q.\nIf we denote A0=Aν(p0,q0), A1=Aν(p1,q1),A=A0+A1,we must\nshow that there exists a positive constant Csuch that\n(5.1) ||F||[A0,A1]θ,q≤C||F||Lν(p,q)∀F∈Aν(p,q).\nWe recall that\n(5.2) ||F||[A0,A1]θ,q=/parenleftbigg/integraldisplay∞\n0/parenleftbig\nt−θK(t,F,A)/parenrightbigqdt\nt/parenrightbigg1\nqBERGMAN-LORENTZ SPACES 19\nwith\nK(t,F,A) = inf{||a0||A0+t||a1||A1:F=a0+a1,a0∈A0,a1∈A1}.\nWedenoteL0=Lν(p0,q0), L1=Lν(p1,q1),L=L0+L1.SinceLν(p,q)\nidentifies with [ Lν(p0,q0),Lν(p1,q1)]θ,qwith equivalence of norms (The-\norem 2.17), we have\n(5.3) ||F||Lν(p,q)=/parenleftbigg/integraldisplay∞\n0/parenleftbig\nt−θK(t,F,L)/parenrightbigqdt\nt/parenrightbigg1\nq\n.\nComparing(5.2)and(5.3), theestimate(5.1)will followifwe canprove\nthat\nK(t,F,A)≤CK(t,F,L)∀F∈Aν(p,q).\nWe first suppose that 1 < p0< p1< Qν,1≤q0,q1≤ ∞and\n0< θ <1.Let1\np=1−θ\np0+θ\np1,1< q <∞and letF∈Aν(p,q).By\nTheorem1.2, for γsufficiently largewehave F=PγFandbyCorollary\n4.2,Pγextends to bounded operator from LitoAi, i= 0,1.So the\ninclusion\n{(a0,a1)∈A0×A1:F=a0+a1}\n⊃ {(Pγa0,Pγa1) : (a0,a1)∈L0×L1,F=a0+a1}\nimplies that\nK(t,F,A)\n≤inf{||Pγa0||L0+t||Pγa1||L1:F=a0+a1,a0∈L0,a1∈L1}\n≤inf{||Pγ||0||a0||L0+t||Pγ||1||a1||L1:F=a0+a1,a0∈L0,a1∈L1}\nwhere|| · ||idenotes the operator norm on Li(i= 0,1).We have\nshown that\nK(t,F,A)≤/parenleftbigg\nmax\ni=0,1||Pγ||i/parenrightbigg\nK(t,F,L).\nRemind that max\ni=0,1||Pγ||i<∞.This completes the proof in this first\ncase.\nWe next suppose that 1+ Q−1\nν<p0<p1<1+Qν,1≤q0,q1≤ ∞\nand 0< θ <1.Let1\np=1−θ\np0+θ\np1,1≤q <∞and letF∈Aν(p,q).\nAgain by Theorem 1.2, we have F=PνFand by Corollary 4.2, Pν\nextends to bounded operator from LitoAi, i= 0,1.So the inclusion\n{(a0,a1)∈A0×A1:F=a0+a1} ⊃ {(Pνa0,Pνa1) : (a0,a1)∈L0×L1,F=a0+a1}20 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nimplies that\nK(t,F,A)≤inf{||Pνa0||L0+t||Pνa1||L1:F=a0+a1,a0∈L0,a1∈L1}\n≤inf{||Pν||0||a0||L0+t||Pν||1||a1||L1:F=a0+a1,a0∈L0,a1∈L1}\n≤/parenleftbigg\nmax\ni=0,1||Pν||i/parenrightbigg\nK(t,F,L).\nThis finishes the proof in the second case.\nWe now prove the second assertion of Theorem 1.3. We combine the\nWolff reiteration theorem (Theorem 2.23) with the first assertion of the\ntheorem. Here X1=Aν(p0,q0) andX4=Aν(p1,q1).We takeX2=\nAν(p2,r) andX3=Aν(p3,q) with 1+Q−1\nν<p2<p3<Qν,1≤q<∞\nand 1<r<∞.By assertion (1) of the theorem, we have\nX2= [X1,X3]ϕ,qwith1\np2=1−ϕ\np0+ϕ\np3\nand\nX3= [X2,X4]ψ,rwith1\np3=1−ψ\np2+ψ\np1.\nBy Theorem 2.23, we conclude that\nX2= [X1,X4]ρ,q, X3= [X1,X4]θ,r\nwith\nρ=ϕψ\n1−ϕ+ϕψ, θ=ψ\n1−ϕ+ϕψ.\nIn other words, in the present context, we have\nAν(p2,q) = [Aν(p0,q0),Aν(p1,q1)]ρ,q, Aν(p3,r) = [Aν(p0,q0),Aν(p1,q1)]θ,r\nwith\nρ=ϕψ\n1−ϕ+ϕψ, θ=ψ\n1−ϕ+ϕψ.\nTo conclude, given 1+ Q−1\nν<p<Q ν,take for instance p3=p.\n6.Open questions.\n6.1.Statement of Question 1. In [3], the following conjecture was\nstated.BERGMAN-LORENTZ SPACES 21\nConjecture. Letν >n\nr−1.Then the Bergman projector Pνadmits\na bounded extension to Lp\nνif and only if\np′\nν<p<pν:=ν+2n\nr−1\nn\nr−1−(1−ν)+\nn\nr−1.\nThis conjecture was completely settled recently for the case of tu be\ndomains over Lorentz cones ( r= 2). The proof of this result is a\ncombinationofresultsof[1]and[8](cf. [4]; cf. also[7]forthepart icular\ncase where ν=n\n2). In this case, Theorem 1.3 can be extended as\nfollows.\nTheorem 6.1. We restrict to the particular case of tube domains over\nLorentz cones (r=2). Letν >n\n2−1.\n(1) For all 1< p0< p1< Qν(resp.p′\nν< p0< p1< pν) and\n0<θ <1,the real interpolation space [Aν(p0,q0),Aν(p1,q1)]θ,q\nidentifies with Aν(p,q),1\np=1−θ\np0+θ\np1,1< q <∞(resp.1≤\nq <∞)with equivalence of norms.\n(2) For1< p0< p′\nν< p1< pνand1≤q0,q1≤ ∞,the Bergman-\nLorentz spaces Aν(p,q), p′\nν< p < Q ν,1< q <∞are real\ninterpolation spaces between Aν(p0,q0)andAν(p1,q1).\nIn the upper rank case ( r≥3), we always have (1 −ν)+= 0 and the\nconjecture has the form\np′\nν<p<pν:=ν+2n\nr−1\nn\nr−1.\nThe best result so far is Theorem 1.2. If we could show that Pνis\nof restricted weak type ( p1,p1) for some 1 + Qν< p1< pν(resp.\np′\nν< p1<1 +Q−1\nν),then by the Stein-Weiss interpolation theorem\n(Theorem 2.20), we would obtain that Pνadmits a bounded extension\ntoLp\nνfor all 1+Qν≤p < p1(resp.p′\nν< p1<1+Q−1\nν).This would\nimprove Theorem 1.2.\nQuestion 1. Provetheexistenceofanexponent p1∈(1+Qν,pν)(resp.\np1∈(p′\nν,1+Q−1\nν)) such that Pνis of restricted weak type ( p1,p1).That\nis, there exists a positive constant Cp1such that for each measurable22 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA\nsubsetEofTΩ,we have\nsup\nλ>0λp1µχE(λ)≤Cp1µ(E).\nFor Lorentz cones ( r= 2), the latter property holds for all p1∈\n(p′\nν,1+Q−1\nν)∪(1+Qν,pν).\n6.2.Question 2. The second question is twofold; it is induced by\nProposition 4.5 and Corollary 4.6. Let ν >n\nr−1.\n(1) Here 1 + Q−1\nν≤q <∞.IsAν(p,q)∩At\nνdense inAν(p,q) for\nsome (all) 1+ Q−1\nν<p≤t<1+Qνandp∈(q,∞)?\n(2) Hereq= 1.Suppose that p∈(1,1+Q−1\nν].IsAν(p,1)∩At\nγdense\ninAν(p,1) for some (all) t∈(1,∞) andγ >n\nr−1?\n6.3.Question 3. Can the second assertion of Theorem 1.3 be ex-\ntended to some values p∈(1,1+Q−1\nν] orp∈[Qν,1+Qν)?\nAcknowledgements. The authors wish to express their gratitude to\nAline Bonami, Jacques Faraut, Gustavo Garrig´ os and Chokri Yaco ub\nfor valuable discussions.\nReferences\n[1]D. B´ekoll´e, A. Bonami, G. Garrig ´os, F. Ricci ,Littlewood-Paley decom-\npositions related to symmetric cones and Bergman projectio ns in tube domains ,\nProc. London Math. Soc. (3) 89(2004), 317-360.\n[2]D. B´ekoll´e, A. Bonami, G. Garrig ´os, C. Nana, M. M. Peloso, F.\nRicci,Lecture Notes on Bergman projectors in tube domains over con es: an an-\nalytic and geometric viewpoint , IMHOTEP-Afr. J. Pure Appl. 5(2004), Expos´ e\nI, Proceedings of the International Workshop in Classical Analysis , Yaound´ e\n2001.\n[3]D. B´ekoll´e, A. Bonami, G. Garrig ´os, F. Ricci, B. Sehba ,Hardy-type\ninequalities and analytic Besov spaces in tube domains over symmetric cones ,\nJ. Reine Angew. Math. 647(2010), 25-56.\n[4]D. B´ekoll´e, J. Gonessa and C. Nana , Lebesgue mixed norm estimates for\nBergman projectors: from tube domains over homogeneous cone s to homoge-\nneous Siegel domains of type II, preprint.\n[5]J. Bergh and J. L ¨ofstr¨om,Interpolation Spaces An Introduction , Springer-\nVerlag Berlin Heidelberg New York (1976).\n[6]C. Bennett and R. Sharpley ,Interpolation of Operators , Pure and Applied\nMath., Vol. 129, Academic Press, Inc. (1988).\n[7]A. Bonami and C. Nana ,Some questions related to the Bergman projection\nin Symmetric domains . Adv. Pure Appl. Math. 6, No. 4, 191-197 (2015).BERGMAN-LORENTZ SPACES 23\n[8]J. Bourgain and C. Demeter ,The proof of the l2-decoupling conjecture ,\narXiv:1403.5335v3[math.CA] 26 Jul 2015.\n[9]J. Faraut, A. Kor ´anyi,Analysis on symmetric cones , Clarendon Press, Ox-\nford, (1994).\n[10]J. Gonessa ,Espaces de type Bergman dans les domaines homog` enes de Sieg el\nde type II: dcomposition atomique et interpolation . Th` ese de Doctorat/Ph.D,\nUniversit´ e de Yaound´ e I (2006).\n[11]R.A. Hunt ,OnL(p,q)spaces.L’Ens. Math. 12(1966), 249-275.\n[12]S. Janson, P. Nilsson, J. Peetre (with Appendix by M. Zafran) ,\nNotes on Wolff’s note on interpolation spaces , Proc. London Math. Soc. 48(3)\n(1984), 283-299.\n[13]B. F. Sehba ,Bergman type operators in tubular domains over symmetric\ncones, Proc. Edin. Math. Soc. 52(2) (2009), 529–544.\n[14]E. M. Stein, G. Weiss ,Introduction to Fourier Analysis on Euclidean\nSpaces, Princeton University Press, Princeton (1971).\n[15]T. Wolff ,A note on interpolation spaces , Harmonic Analysis (1981 Proceed-\nings), Lecture Notes in Math. 908, Springer Verlag (1982), 199-204.\nUniversity of Ngaound ´er´e, Faculty of Science, Department of Math-\nematics, P. O. Box 454, Ngaound ´er´e, Cameroon\nE-mail address :dbekolle@univ-ndere.cm\nUniversit ´e de Bangui, Facult ´e des Sciences, D ´epartement de math ´ematiques\net Informatique, BP. 908 Bangui, R ´epublique Centrafricaine\nE-mail address :gonessa.jocelyn@gmail.com\nFaculty ofScience, Department of Mathematics, University of Buea,\nP.O. Box 63, Buea, Cameroon\nE-mail address :nana.cyrille@ubuea.cm"    },    {        "title": "1801.06735v1.Long_time_dynamics_for_weakly_damped_nonlinear_Klein_Gordon_equations.pdf",        "content": "arXiv:1801.06735v1  [math.AP]  20 Jan 2018LONG TIME DYNAMICS FOR WEAKLY DAMPED NONLINEAR\nKLEIN-GORDON EQUATIONS\nN. BURQ, G. RAUGEL, W. SCHLAG\nAbstract. We continue our study of damped nonlinear Klein-Gordon equa tions.\nIn [4] we considered fixed positive damping and proved a form of the soliton res-\nolution conjecture for radial solutions. In contrast, here we consider damping\nwhich decreases in time to 0. In the class of radial data we again establish soliton\nresolution provided the damping goes to 0sufficiently slowly. While [ 4] relied on\ninvariant manifold theory, here we use the Łojasiewicz-Sim on inequality applied\nto a suitable Lyapunov functional.\nKeywords : Klein-Gordon equation with dissipation, subcri tical focusing non-\nlinearity, convergence to an equilibrium, soliton resolut ion, Łojasiewicz-Simon in-\nequality, Ambrosetti-Rabinowitz condition, observation inequalities, Strichartz es-\ntimates.\n1.Introduction\nA central question in the theory of nonlinear dispersive evo lution equations con-\ncerns the long-term behavior of solutions. For completely i ntegrable evolution equa-\ntions the inverse scattering transform yields explicit mul ti-soliton solutions and the\nasymptotic shape of solutions (but under quite restrictive conditions on the data) is\nknown. In absence of complete integrability the theory is mu ch less developed, and\nlargely remains in its infancy.\nFor the class of semi-linear equations the last five years hav e witnessed the influx\nof new ideas. Around 2012 Duyckaerts, Kenig, and Merle [ 11] obtained the complete\ndescription of radial energy solutions to the three-dimens ional critical wave equation\nutt´∆u´u5“0.\n1991Mathematics Subject Classification. 35BXX, 35B40, 35L05, 35L71, 37L10, 37L50, 37L45.\nThe first author was partially supported by the ANR through AN R-13-BS01-0010-03 (ANAÉ)\nand ANR-16-CE40-0013 (ISDEEC), the second author was parti ally supported by the ANR\nthrough ANR-16-CE40-0013 (ISDEEC) and the third author was partially supported by the NSF\nthrough DMS-1500696. The third author thanks the Institute for Advanced Study, Princeton, for\nits hospitality during the 2017-18 academic year.\n12 N. BURQ, G. RAUGEL, W. SCHLAG\nThe data are radial and belong to 9H1ˆL2pR3q, which is the natural space from the\nperspective of wellposedness. The main result in [ 11] states that global solutions\nasymptotically decouple into radiation (possibly of large energy) and finitely many\nrescaled solitons WλjptqpxqwhereWpxq “ p1` |x|2{3q´1\n2andWλ:“λ1\n2Wpλ¨q. The\npositive parameters λjptq ą0depend continuously on time and their ratios either\ntend to0or8astÑ 8. In other words, the energy decouples into the radiation plu s\nthe energy of Wcounted with the multiplicity of the number of solitons. A pa rticular\nconsequence of this result is that global solutions are unif ormly bounded in time in\nthe norm of 9H1ˆL2pR3q. For subcritical equations Cazenave [ 7] obtained such a\nbound many years ago, but under a restrictive condition on th e nonlinearity. For\nexample, in three dimensions, powers between 3and5are not covered by Cazenave’s\nmethod. It is unknown at this point if global solutions to foc using nonlinear Klein-\nGordon equations remain bounded in H1ˆL2pR3qfor powers in that regime.\nFor solutions which are not global, [ 11] establishes a corresponding representation\ninto a fixed pair of functions instead of the radiation, and a s um of solitons. Thus,\nonly type-II blowup can occur. For nonradial solutions, a pa rtial characterization\nof an analogous nature was obtained in [ 12] along a sequence of times.\nThe channel of energy method, which was pioneered in [ 11], has found a number of\napplications over the past few years that we will not describ e here in detail. It does\nnot apply in the subcritical regime due to the fact that the fr ee radial Klein-Gordon\nequation does not exhibit nonzero asymptotic energy outsid e a backward or forward\nlight cone; the energy at times ˘8in the region |x| ě |t|is vacuous. This is due to\nthe fact that for Klein-Gordon the group velocity is ă1whereas for the free wave\nequation it is exactly 1, leading to a fixed percentage of the aforementioned exterio r\nenergy either in forward or backward times. The transition f rom wave to Klein-\nGordon is due to passing from critical to subcritical equati ons: indeed, in contrast\nto the critical equation, subcritical wave equations do not exhibit stationary soliton\nsolutions due to the Pohozaev identity. It is therefore nece ssary to add the mass\nterm for a natural formulation of the subcritical soliton resolution problem .\nIn view of a lack of any techniques currently known to attack t he subcritical\nHamiltonian problem, the authors of this paper set out in [ 4] to study the dissipative\ncase. The idea underlying the addition of a damping term is th e availability of\nmethods originating in dynamical systems. In [ 4] we relied on results from invariant\nmanifolds and center dynamics, whereas here we employ Lyapu nov functionals and\nthe Łojasiewicz-Simon inequality. We now describe the cont ents of this paper in\nmore detail.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 3\nWe consider the damped Klein-Gordon equation\nutt`2αptqut´∆u`u´fpuq “0,\npup0q,utp0qq “ pϕ0,ϕ1q PHrad,pKGqα\nwhere\nH“H1pRdq ˆL2pRdq,\nand\nHrad“H1\nradpRdq ˆL2\nradpRdq,\nwhere1ďdď6. We shall denote /vector uptq “ puptq,utptqqthe solution. In [ 4], we\nassumed that αis a positive constant. Here we assume that αptq ą0is a positive\nfunction of class C1intě0, which converges to 0astgoes to 8, see below for\nmore details. The nonlinearity f:yPRÞÑfpyq PRis an oddC1-function which\nsatisfiesf1p0q “0. Moreover, it satisfies a condition of Ambrosetti-Rabinowi tz type,\ni.e., there exists γą0such thatż\nRd`\n2p1`γqFpϕpxqq ´ϕpxqfpϕpxqq˘\ndxď0,@ϕPH1pRdq, pH.1qf\nwhereFpyq “şy\n0fpsqds. In dimensions dě2we impose the following growth\ncondition on f\n|f1pyq| ďCmax`\n|y|β,|y|θ´1˘\n,@yPR,\n|f1py1q ´f1py2q| ďC|y1´y2|β`\n1` |y1|θ´1´β` |y2|θ´1´β˘\n,@y1,y2PR,pH.2qf\nwhere1ăθăθ˚,0ăβăθ´1,βď1,θ˚“2˚´1and where 2˚“ 8ifd“1,2\nand2˚“2d\nd´2ifdě3. We notice that, when dě3,θ˚“d`2\nd´2. In other words,\nthe growth of fis energy subcritical for large y“0, and we also assume that f1\nisβ-Hölder continuous. For sake of simplicity in the proofs bel ow, we may assume,\nwithout loss of generality, that 0ăβăminp1,θ´1,2\ndq.\nClassical examples of a function fsatisfying hypotheses pH.1qfand pH.2qfare as\nfollows\nfpuq “m1ÿ\ni“1ai|u|pi´1u´m2ÿ\nj“1bj|u|qj´1u,with#\n1ăqjăpiăd`2\nd´2,@i,j\nai,bjě0,am1ą0.(1.1)\nIfαptqdecays to 0too quickly as tgoes to 8, then the equation becomes a\nperturbation of the conservative case. See [ 38] and [ 39] for a discussion in the\ncontext of linear equations. Minimal assumptions on the dis sipation rate are\nαptq ą0,lim\ntÑ8αptq “0,ż8\n0αpsqds“ 8, α ptqnon-increasing . pH.1qα4 N. BURQ, G. RAUGEL, W. SCHLAG\nWe will assume more, namely\nαptq “1\np1`tqa,0ďaă1\n3. pH.2qα\nThe main results of the paper are as follows.\nTheorem 1.1. Under the conditions pH.1qf,pH.2qf, and pH.2qα, any solution /vector uptq\nofpKGqα\n(1)either blows-up in finite time,\n(2)or exist globally and converges strongly to an equilibrium p oint pQ,0qof\npKGqα, astÑ 8. More precisely, Dc0,c1;@tě0,\n}/vector uptq ´ pQ,0q}Hďc0exp`\n´c1p1`tq1´a˘\n. (1.2)\nTheorem 1.2. Under the additional assumption that every equilibrium poi ntpϕ,0q P\nHradofpKGqα(with energy smaller than ℓ) is isolated, the same conclusions as in\nTheorem 1.1hold (for initial data with energy smaller than ℓ) with the assump-\ntion pH.2qαrelaxed to\nαptq “1\np1`tqa,0ďaă1\n2. pH.3qα\nWe remark that our arguments below do not depend on the existe nce or uniqueness\nof a ground state solution, which in any case is notguaranteed by Hypothesis pH.1qf\nalone. We further note that Hypothesis pH.1qfmay actually be replaced by the\nfollowing weaker oneż\nRd`\n2p1`γqFpϕpxqq ´ϕpxqfpϕpxqq˘\ndxď0,for}ϕ}H1large enough ,pH.1bisqf\nbut, for sake of simplicity, we assume pH.1qfthroughout.\nWe denote by Sαpt,sq,αě0, the local non-autonomous system generated by the\nequation pKGqαonHas well as on Hrad, when the initial data are considered at\ntimes. We introduce the energy functional (also called Lyapunov f unctional in the\ncase of positive damping αptq ą0)\nEpϕ0,ϕ1q “ż\nRdˆ1\n2|∇ϕ0|2`1\n2ϕ2\n0`1\n2ϕ2\n1´Fpϕ0q˙\ndx. (1.3)\nWe recall that, as long as /vector upsq “ pupsq,utpsqqexists, fortět0ě0, we have,\nEp/vector uptqq ´Ep/vector upt0qq “ ´2żt\nt0αpsq}utpsq}2\nL2pRdqds. (1.4)\nIn the case of constant positive damping, the relation easil y implies that every\nelement in the ω-limit set of a global trajectory is necessarily an equilibr ium. UnderLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 5\nthe aforementioned properties on the damping, it clearly fo llows from Theorem 1.1.\nUnder weaker damping assumptions, we do not apriori know that global trajectories\nare bounded in H. Hence, it is completely unclear whether or not the ω-limit set of a\nglobal trajectory is nonempty and whether it contains only e quilibria. The following\nresult which requires only the much weaker pH.1qαis thus perhaps surprising.\nTheorem 1.3. Supposeαptq ą0is continuous and satisfies Hypothesis pH.1qα. Let\n/vector ube a forward global solution of pKGqα. Then, for any rą0, there exist a sequence\nof timestnÑ 8asngoes to infinity, and an equilibrium point pQ,0qsuch that\nlim\nnÑ`8}/vector uptnq ´ pQ,0q}Hrad“0,lim\nnÑ`8żtn`r\ntn´r}utpsq}2\nL2pRdqds“0. (1.5)\nThe plan of the paper is as follows. In Section 2.1, we state th e basic properties of\nthe non-autonomous damped Klein-Gordon equation pKGqα, in particular the local\nexistence and uniqueness of mild solutions. We also recall S trichartz inequalities\nsatisfied by the corresponding linear damped Klein-Gordon e quation. In Section 2.2,\nwe recall spectral properties of the linearized elliptic eq uation around an equilibrium\npoint of pKGqα. In Section 3, we introduce the functional K0, which plays a central\nrole in the proofs of Theorems 1.1 to 1.3. We also prove, by gen eralizing arguments\nof [7], that the L2-norm of global solutions is bounded. In Lemma 3.3, we give\na sufficient condition on K0for blow-up in finite time of the solutions of pKGqα.\nSection 4 is devoted to the construction of special time sequ ences, which are useful in\nthe proofs of our main results. In particular, these time seq uences allow to construct\na time sequence t˚\nnso thatK0pupt˚\nnqqtends to0asngoes to infinity (see Section 4.2).\nIn Section 4.3, we use this time sequence t˚\nnin order to prove Theorem 1.3, namely\nthat, if a solution /vector uptqdoes not blow up in finite positive time, then the ω-limit set\nωp/vector up0qqcontains at least one equilibrium point pQ,0qunder the weak hypothesis\npH.1qα. Sections 5 and 6 are the core of this paper. In Section 5, we pr ove Theorem\n1.1when1ďdď6andd‰3or whend“3and the nonlinearity fsatisfies the\ncondition 1ăθď4, that is, that, under the more restrictive hypothesis pH.2qα,\nany solution /vector uptqofpH.2qαeither blows-up in finite time or strongly converges to\nthe above mentioned equilibrium point pQ,0qastgoes to infinity. In Section 6,\nwe prove Theorem 1.1in the more delicate case, where d“3and the nonlinearity\nfsatisfies the condition 4ăθă5. In this case, we need to work with averaged\nquantities in order to exploit (integral) Strichartz estim ates. In this case, we also\nneed to use an observability inequality proved in [ 5]. The short Section 7 is devoted\nto the proof of Theorem 1.2. Finally in the Appendix A, we indicate the proof of\nthe Łojasiewicz-Simon inequality in the framework of this p aper. Our results are\ntrue in dimensions dď6. It might be possible to extend them to higher dimensions6 N. BURQ, G. RAUGEL, W. SCHLAG\nmodulo technical complications. For concision, most of the proofs in the paper will\nbe written only for 3ďdď6(the casesd“1,2being a priori simpler).\n2.Basic properties\n2.1.Local existence results. Let us first consider the following linear perturbed\nKlein-Gordon equation, written as a first order system\nBtˆu\nut˙\n“ˆ\n0 1\n∆´1 0˙ˆu\nut˙\n`ˆ0\n´2αptqut˙\n”A/vector u`ˆ0\n´2αptqut˙\n,\n/vector upt0q “ˆ\nϕ0\nϕ1˙\nPH,(2.1)\nwheret0ě0and/vector u:“`u\nut˘\n. It is well-known that Agenerates a linear C0-group on\nHand onHrad. Sinceαptqis a continuous function of tě0and that ´2αptqutis\na uniformly Lipschitz continuous function on L2pRdq, it follows from [ 33, Theorem\n6.1.2] that, for every /vector u0” pϕ0,ϕ1q PH, the system ( 2.1) has a unique mild solution\n/vector uPC0prt0,`8q,Hq. Moreover, the mapping /vector u0Ñ/vector uis Lipschitz continuous from\nHintoC0prt0,`8q,Hq(resp. from HradintoC0prt0,`8q,Hradq). In addition,\nsinceαptqis aC1function of tě0, then the solution /vector uptqof (2.1) with/vector u0P\nH2pRdq ˆH1pRdqis a classical solution of ( 2.1), that is, the solution /vector uptqbelongs to\nC0prt0,`8q,H2pRdq ˆH1pRdqq XC1prt0,`8q;H1pRdq ˆL2pRdqqand the equation\n(2.1) is satisfied for any tP rt0,`8q.\nLetΣ0p.q:HÑHbe the linear C0-group generated by the linear conservative\nKlein-Gordon equation. By [ 33, Corollary 4.2.2], the quantity\nżt\nt0Σ0pt´sqp0,2αpsqutpsqqds\nis a continuous function from rt0,`8q toHand the solution /vector uptqof the linear\nequation ( 2.1) can be written as\n/vector uptq “Σ0pt´t0q/vector upt0q `żt\nt0Σ0pt´sqp0,´2αpsqutpsqqds. (2.2)\nArguing exactly as Martinez (see [ 29] for example) and introducing the following\nenergy functional on H,\nEp/vector vq “1\n2ż\nRdpv2\n2` |∇v1|2`v2\n1qdx, @/vector v“ pv1,v2q PH, (2.3)\nwe may prove the following subexponential decay of the H-norm of the solution /vector uptq\nof (2.1).LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 7\nTheorem 2.1. Assume that the function αptqsatisfies the hypothesis pH.1qα. Then,\nthere exist positive constants ωandKě1such that, if /vector uptq PHis a solution of the\nequation (2.1), then we have, for any sě0, for anytěs,\nEp/vector uqptq ďEp/vector uqpsqexpp1´ωżt\ns2αpτqdτq ďEp/vector uqpsqKexpp´2ωżt\nsαpτqdτq.(2.4)\nProof. We first prove the above inequalities for s“0. We do not repeat the proof\nhere since it is the same, mutatis mutandis, as in [29, pages 301–304] . Afterwards,\nwe prove it for są0by doing a simple change of time variable, replacing /vector uptqand\nαptqby/vector ups`t˚qandαps`t˚q. Notice that the constants ωandKdo not depend\nof the initial time s. /square\nThe above considerations, Theorem 2.1together with, for example, [ 33, Sections\n5.2 and 5.3] (see in particular [33, Theorem 5.3.1] ) imply that the equation ( 2.1) de-\nfines a unique evolution process, Σαpt,sq,0ďsďtď `8 , and/vector uptq “Σαpt,t0q/vector upt0q.\nWe now want to show that the solution of ( 2.1) also satisfies the classical Strichartz\ninequalities for the wave equation. First we recall these in equalities in the case of\nthe affine conservative Klein-Gordon equation for 0ďtďT,\nvttptq ´∆vptq `vptq “hptq, /vector vp0q “ pv0,v1q PH, (2.5)\nwherehptq PL1pp0,8q,L2pRdqq.\nThe following proposition is well-known\nProposition 2.2. In all dimensions dě1, the solution vof(2.5)satisfies the\nfollowing energy bounds,\nsup\ntě0}pv,Btvqptq}H1ˆL2ďC0“\n}pv0,v1q}H`ż8\n0}hpsq}L2ds‰\n, (2.6)\nas well as the Strichartz estimates, in dimensions dě2,\n}v}Lq\ntLppRdqďC˚\n0“\n}pv0,v1q}H` }h}L˜q1\ntL˜p1\nx‰\n, (2.7)\nwhere1\nq`d\np“d\n2´1“1\n˜q1`d\n˜p1´2,2ďp,˜pă 8,2ďq,˜q, and1\nq`d´1\n2pďd´1\n4,\n1\n˜q`d´1\n2˜pďd´1\n4and whereC˚\n0“C˚\n0pp,q,p1,q1qis a positive constant.\nWe next apply the previous proposition to the solution /vector uptqof the linear equation\n(2.1). Let pp,qqbe a pair satisfying the conditions of Proposition 2.2. We denote\nbyP1:/vector vPHÑvPH1pRdqthe projection onto the first component. Since /vector uptq,\ntět0, satisfies the integral equality ( 2.2), we deduce from the Strichartz estimate\n(2.7) withh“ ´2αptqutand p˜p1,˜q1q “ p2,1qthat,\n}u}Lqppt0,`8q,LppRdqqďC˚\n0`\n}/vector upt0q}H`ż`8\nt0}2αpsqutpsq}L2ds˘\n. (2.8)8 N. BURQ, G. RAUGEL, W. SCHLAG\nFinally, we apply Theorem 2.1to the inequality ( 2.8) to get,\n}u}Lqppt0,`8q,LppRdqqďC˚\n0}/vector upt0q}H`\n1`K1{2ż`8\nt0αpsqexpp´ωżs\nt0αpσqdσqds˘\nďC˚\n0}/vector upt0q}Hp1`K1{2ω´1q.(2.9)\nThe decay of the norm of /vector uptqinHis subexponential, whereas the upper bound of\ntheLq\ntLppRdq-norm ofuptq, obtained in the above inequality, is a simple constant.\nBy the above properties and [ 33, Theorems 5.2.2, Theorem 5.2.3 and 5.3.1], the\nsystem ( 2.1) generates a unique evolution process (or evolution system )Σαpt,sq,0ď\nsďtinH, and ( 2.1) has a unique solution /vector uptq “Σαpt,t0q/vector upt0qinH, satisfying the\nproperties of Theorem 5.3.1 of [ 33]. And we can state the following result..\nProposition 2.3. Lett0ą0and/vector upt0q PH. System (2.1)has a unique solution\n/vector uptq “Σαpt,t0q/vector upt0qinC0prt0,`8q,Hq, satisfying the properties of Theorem 5.3.1\nof[33]. And, for any pair pp,qqsatisfying the conditions of Proposition 2.2, there\nexists a positive constant C0“C0pp,qqso that,\n}u}Lqppt0,8q,LppRdqqďC0}/vector upt0q}H. (2.10)\nWe finally turn to the affine damped Klein-Gordon equation, for t0ďtďT,\nuttptq `2αptqutptq ´∆uptq `uptq “Gptq, /vector upt0q “ pϕ0,ϕ1q PH, (2.11)\nwhereGptq PL1pp0,Tq,L2pRdqq, forTą0(or evenT“ `8 ). Again, by [ 33,\nSection 4.2.], the quantity\ngptq ”Σ0pt´t0q/vector upt0q `żt\nt0Σ0pt´sqp0,Gpsqqds\nis a continuous function from rt0,TsintoH. By [ 33, Corollary 6.1.3], the integral\nequation\n/vector uptq “Σ0pt´t0q/vector upt0q `żt\nt0Σ0pt´sqp0,Gpsqqds`żt\nt0Σ0pt´sqp0,´2αpsqutpsqqds,\n(2.12)\nhas a unique solution /vector uptq PC0prt0,Ts,Hq. This integral solution coincides actually\nwith the mild solution (in the non-autonomous sense) of the e quation ( 2.11), that\nis,\n/vector uptq “Σαpt,t0q/vector upt0q `żt\nt0Σαpt,sqp0,Gpsqqds. (2.13)\nUsing the above integral formula for the solution /vector uptqof Equation ( 2.11) and apply-\ning Theorem 2.1and Proposition 2.3, we obtain the following result.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 9\nProposition 2.4. There exists c0such that, for any t0ą0and any/vector upt0q PH,\nSystem (2.11)has a unique mild solution /vector uptqinC0prt0,Ts,Hqand the following\nestimate holds,\n}/vector uptq}Hďc0“\n}/vector upt0q}Hexpp´ωżt\nt0αpsqdsq `żt\nt0expp´ωżt\nsαpτqdτq}Gpsq}L2ds‰\n,\n(2.14)\nwhereωą0has been defined in Theorem 2.1.\nMoreover, for any pair pp,qqsatisfying the conditions of Proposition 2.2, there exists\nCą0such that for any t0ăT,\n}u}Lqppt0,Tq,LppRdqqďCr}/vector upt0q}H`żT\nt0}Gpsq}L2pRdqdss. (2.15)\nProof. 1) Due to the integral form of the solution /vector uptqof (2.11), the inequality is a\ndirect consequence of Theorem 2.1.\n2) Using the integral form ( 2.13) of the solution /vector upτq,t0ďτďt, of (2.11) and\nProposition 2.3, we obtain, by applying the Minkowski inequality, that\n}u}Lqppt0,Tq,LppRdqqďC}/vector upt0q}H` }żT\nt0|P1Σαpτ,sq1τěsp0,Gpsqq|ds}Lqppt0,Tq,LppRdqq,\nďC}/vector upt0q}H`żT\nt0}P1Σαpτ,sq1τěsp0,Gpsqq}Lqppt0,Tq,LppRdqqds,\nďC}/vector upt0q}H`CżT\nt0}Gpsq}L2pRdqds.\nand we are done. /square\nWe are now able to state the theorem of local existence of solu tions of the equation\npKGqα. This local existence theorem is analogous to the local exis tence theorem,\ngiven in [ 4] (see Theorem 2.3 there) in the case of a constant damping αą0. The\nproof of the following theorem follows the lines of the proof of [4, Theorem 2.3] if one\nreplaces the Strichartz estimates given in [ 4, Lemma 2.2] by the Strichartz estimates\ngiven in the above Theorem 2.4.\nThe theorem below does not only give the local existence of so lutions to Equation\npKGqα, but also to the slightly more general non-autonomous dampe d Klein-Gordon\nequation for t0ě0,\nutt`2αptqut´∆u`u´fpuq “0,\npupt0q,utpt0qq “/vector u0PHrad.pKGqα,t010 N. BURQ, G. RAUGEL, W. SCHLAG\nTheorem 2.5. Letdď6. Under assumptions pH.2qfand pH.1qα, for anyrą0\nthere exists TěC{rδ,δą0, such that for any t0and any/vector u0PH“H1pRdq ˆ\nL2pRdq,}/vector u0}Hďrthe equation pKGqα,t0has a unique solution (which is radial if\nthe initial data are radial)\n/vector uptq P t/vector vptq ” pvptq,Btvptqq PC0prt0,t0`Ts,Hq,vptq PLqppt0,t0`Tq;LppRdqqu,\nwith pq,pqas in Proposition 2.3suitably chosen. This solution is bounded by 2}/vector u0}H\nin\nXt0,T“/visualspace\n/vector u“ pu0,u1qptq PCprt0,t0`Ts,H1pRdqq XC1prt0,t0`Ts,L2pRdqq,\nuptq PLqppt0,t0`Tq;LppRdqq(\n.\nIn particular, if 3ďdď6, we can take q“θ˚,p“2θ˚. Furthermore, the\nfollowing properties hold.\n1)If the above solution /vector uptq ”Sαpt,t0q/vector u0with initial data /vector u0PHexists for\ntP rt0,t0`˜Ts, then there exists a neighborhood VinHsuch that, for every /vector v0PV,\nthe equation pKGqα,t0has a unique solution Sαpt,t0q/vector v0”/vector vptq ” pvptq,Btvptqqwith\nvPXt0,˜T. And the solution\npt,/vector v0q P rt0,˜Ts ˆVÞÑSαpt,t0q/vector v0PH\nis jointly continuous.\n2)For anyt0ďτďt0`˜T, the map/vector v0PVÞÑSαpt0,τq/vector v0PHis Lipschitz\ncontinuous on the bounded sets of V.\n3)The map/vector v0PVÞÑvptq PXt0,˜TXLθ˚ppt0,t0`˜Tq,L2θ˚pRdqqis aC1-map.\n4)LetT˚`t0ąT`t0ąt0be the maximal time of existence. If T˚ă 8, then\nlimsup\ntÑT˚}/vector uptq}H“ `8.\n5)If/vector u0PH2pRdq ˆH1pRdq, then\nuPCprt0,t0`Tq,H2pRdqq XC1prt0,t0`Tq,H1pRdqq.\n6)The energy (1.3)decreases: for any t2ět1ět0, we have,\nEp/vector upt2qq ´Ep/vector upt1qq “ ´2żt2\nt1αpsq}Btupsq}2\nL2pRdqds, (2.16)\nand in particular,\nEp/vector upt2qq ďEp/vector upt0qq. (2.17)\n7)If}/vector upt0q}H!1, then the solution exists globally, and if furthermore\nαptq “1\np1`tqa,0ďaă1. (2.18)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 11\n}/vector uptq}Hconverges sub-exponentially to 0astÑ 8,\nDc0,ωą0;}/vector u0}Hďr0ñ @tąt0,}/vector uptq}Hďc0expp´ωżt0`t\nt0αpsqdsq}/vector u0}H.(2.19)\nProof. The proofs of this result follows the lines of the proof of the similar result [ 4,\nTheorem 2.3]. The points 1) to 6) are proved by using Duhamel f ormulation and\nperforming a fixed point in a ball of the space\nY”Yt0,T” t/vector uPL8ppt0,t0`Tq,Hqu,\nin dimension 1;2or\nY”Yt0,T” t/vector u“ pu0,u1q PL8ppt0,t0`Tq,Hq,u0PLθ˚ppt0,t0`Tq,L2θ˚pRdqqu,\n(2.20)\nin dimensions 3ďdď6, which is possible if 0ăTis small enough depending on\nthe size of the initial data, due to the subcriticality of the problem. We turn to the\nproperty (7), that is, we consider the case of small data. We p roceed in two steps.\nWe first recall that small initial data generate global solut ions\nLemma 2.6 (Small data a priori estimates) .There exists r0ą0such that for any\nt0ě0and any initial data satisfying\n}/vector u0}Hďr0, (2.21)\nthe solution of (pKGqα)exists on rt0,`8q. and satisfies for any pq,pqsatisfying\nthe conditions of Proposition 2.2there exists C,K ą0such that for any initial data\nsatisfying (2.21), the solution satisfies\n}/vector u}L8ppt0,Tq;HqďC}/vector u0}H, (2.22)\n}u}Lqppt0,Tq;LppRdqqďCp1`TqK}/vector u0}H. (2.23)\nProof. According to ( pH.2qf), and Sobolev embedding, for the potential part of the\nenergy ( 1.3) we have\nż\nRd|Fpu0|dxďC`\n}/vector u0}β`2\nH` }/vector u0}θ`1\nH˘\n.\nWe deduce, since 2ăβ`2ăθ`1, that forǫą0small enough, on H0\nǫ, the\nconnected component of p0,0qin the set\nHǫ“ t/vector uPH,Ep/vector u0q ďǫu,\nthe energy and Hnorm are equivalent\n@p/vector u0q PH0\nǫ,}/vector u0}2\nHď2Ep/vector u0q ď3}/vector u0}2\nH.12 N. BURQ, G. RAUGEL, W. SCHLAG\nThe global existence with the bound ( 2.22) follows for small initial data from the\nfact that the energy decays (and hence solutions starting in H0\nǫremain in H0\nǫ). To\nprove ( 2.23), we use that from the local existence theory, for t1ět0, theLqpt1,t1`\nCr´δ\n0;Lp-norm is bounded by 2}pupt1,utpt1qq}Hnorm which in turn is bounded by\n6}pupt0,utpt0qq}H, and theTKestimate in ( 2.23) follows from gluying these estimates\ntogether. /square\nIn a second step we prove the subexponential decay. From Prop osition 2.14, we\nhave for any tąsět0,\n}p/vector uptq}Hďc0“\nexpp´ωżt\nsαpsqdsq}/vector upsq}H.` }fpuq}L1ps,tq;L2pRdqq‰\n(2.24)\nFrom pH.2qf(2.23) (starting from t1) and Sobolev embeddings to control the L1`β\nx-\nnorm by the H1-norm, we get\n}fpuq}L1pps,tq;L2pRdqqďC`\n}u}1`β\nL1`βpps,tq;H1q` }u}θ\nLθpps,tq;L2θq˘\nďCp1` |t´s|qK`\n}/vector upsq}1`β\nH` }/vector upsq}θ\nH˘\n.(2.25)\nWe are now going to apply ( 2.24) and ( 2.25) on sequences of intervals\ntn“κnβ,β“2´a\n1´aą1,ně1,\nwithκsufficiently large so that\nc0expp´ωżtn`1\ntnapsqds„nÑ`8c0expp´ω\n1´aβκq ď1\n8.\nLetn0such that for any něn0,\nc0expp´ωżtn`1\ntnapsqdsq ď1\n4.\nWe get for něn0,\n}p/vector uptnq}Hď“1\n4`Cp1`npβ´1qKq`\n}/vector uptn´1q}β\nH` }/vector uptn´1q}θ´1\nH˘‰\n}/vector uptn´1q}H.\nFixNąn0so that for all pěN,\nCp1`pβKq2´βpp´n0qď1\n8. (2.26)\nUsing ( 2.22), we can choose the size of the initial data r0ą0small enough so that\n@n0ďnďN, C p1`nβKq`\n}/vector uptn´1q}β\nH`}/vector uptn´1q}θ´1\nHďC1p1`NβKq`\nrβ\n0`rθ´1\n0˘\nď1\n4.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 13\nThis will guarantee that for all n0ďnďN\n}p/vector uptnq}Hď1\n2}/vector uptn´1q}Hď2´pn´n0qr0.\nThen ifNis large enough and r0ď1, using ( 2.26) and a straightforward induction\nargument shows that for any něN\nCp1`nβKq`\n}/vector uptn´1q}β` }/vector uptn´1q}θ´1ď1\n4,\nand\n}p/vector uptnq}Hď1\n2/vector uptn´1q}Hñ }p/vector uptnq}HďC12´βpn´n0qr0.\nSincetn„nβ, this shows the sub-exponential decay along the sequence of timestn.\nTo get the full decay, we just use ( 2.22) betweentnandtn`1. /square\nRemark 2.7. Since here we are mainly interested in the behavior of the sol utions\n/vector uptqofpKGqα,t0when the time tąt0goes to `8, we stated and proved the local\nexistence and continuity properties on time intervals rt0`0,t0`Ts, withTą0. Of\ncourse, one shows in the same way that there exists 0ăTďt0so that the properties\n(1) to (6) of the above Theorem 2.5also hold on time intervals r´T`t0,t0s.\n2.2.Spectral properties. Let us recall first that the kernel of the linearized elliptic\noperator around an equilibrium point is at most of dimension one in the radial setting\n(see [4] for a proof). Suppose now that we have a stationary solution ϕ0PH1\nradpRdq\ntopKGqα, namely,\n´∆ϕ0`ϕ0´fpϕ0q “0.\nBy elliptic theory, these solutions are exponentially deca ying, and lie in C3,β˚for\nsomeβ˚ą0. Solving pKGqαforu“ϕ0`vyields\nvtt`2αptqvt´∆v`v´f1pϕ0qv“Npϕ0,vq, (2.27)\nwhereNpϕ0,vq “fpϕ0`vq ´fpϕ0q ´f1pϕ0qv. SetL“ ´∆`I´f1pϕ0q.\nThis leads us to consider the linearized equation below\nvtt`2αptqvt´∆v`v´f1pϕ0qv“0, t ěs, /vector v psq “/vector vs. (2.28)\nWe denote Σ˚\nαpt,sqthe evolution operator associated with ( 2.28), that is,/vector vptq “\nΣ˚\nαpt,sq/vector vs.\nWe next recall the spectral properties of L(see [4, Section 2.3]).\nProposition 2.8. The operator Lis self-adjoint with domain H2pRdq. The spec-\ntrumσpLqconsists of an essential part r1,8q, which is absolutely continuous, and\nfinitely many eigenvalues of finite multiplicity all of which fall into p´8,1s. The14 N. BURQ, G. RAUGEL, W. SCHLAG\neigenfunctions are C2,β˚withβ˚ą0and the ones associated with eigenvalues below\n1are exponentially decaying. Over the radial functions, all eigenvalues are simple.\n3.Boundedness in L2of global solutions and the functional K0.\n3.1.Definition of K0.We consider the functional K0:ϕPH1pRdq ÞÑK0pϕq PR,\ndefined by\nK0pϕq “ż\nRdp|∇ϕ|2`ϕ2´ϕfpϕqqdx.\nAs in [ 4],K0plays an important role in this paper. The “Ambrosetti-Rabi nowitz”\nhypothesis pH.1qfallows one to prove the following lemmas, which will be used\nthroughout this paper.\nLemma 3.1. For any pϕ,ψq PH1pRdq ˆL2pRdq, we have\nγp}ϕ}2\nH1` }ψ}2\nL2q ď2p1`γqEppϕ,ψqq ´K0pϕq. (3.1)\nProof. We simply write\nγp}ϕ}2\nH1` }ψ}2\nL2q\n“2p1`γqEppϕ,ψqq ´K0pϕq ´ }ψ}2\nL2`ż\nRd`\n2p1`γqFpϕq ´ϕpxqfpϕpxqq˘\ndx\nď2p1`γqEppϕ,ψqq ´K0pϕq,(3.2)\nwhere the integral is nonpositive by pH.1qf. /square\nCorollary 3.2. Suppose/vector uptq “ puptq,Btuptqqis a strong solution of pKGqαdefined\non the maximal interval 0ďtăT˚. Assume\ninf\n0ďtăT˚K0puptqq ą ´8.\nThenT˚“ 8, i.e., the solution is global.\nThe proof of the next lemma uses a convexity argument. In the c ase ofα“0, it\nhas been proved in [ 32] and [ 30, Corollary 2.13]. In the case where αis a positive\nconstant, it has been proved in [ 4, Lemma 2.7]. For the following two results it is\nsufficient to impose condition pH.1qα.\nLemma 3.3. Assume that /vector uptq ” puptq,Btuptqqis a solution of pKGqαdefined on\nr0,T˚qwhereT˚P p0,8sis maximal. If K0puptqq ď ´δ(whereδą0), fort0ďtă\nT˚, thenT˚ă 8, i.e., the solution blows up in finite time.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 15\nCorollary 3.4. Suppose that the initial energy Ep/vector u0qis non positive (and that the\nsolutionis not identically zero). Then the solution blows- up in finite time T˚ă\n8. In particular, there does not exist a non-trivial equilibr ium point/vector u0satisfying\nEp/vector u0q ď0.\nProof. The case of negative energy is clear from ( 3.1) and Lemma 3.3. If the initial\nenergy is zero, and the solution is not stationary, then the e nergy becomes negative\nin finite time and we are done. If the solution is stationary an d equal to φ, then\nK0pφq “0. However, this contradicts ( 3.1). /square\nProof of Lemma 3.3.We assume without loss of generality that t0“0, and towards\na contradiction that T˚“ 8. In order to show that /vector uptqblows up in finite time, we\nuse a standard convexity argument see [ 32] or [4]. We set\nyptq “1\n2}uptq}2\nL2.\nWe have\n9yptq “ puptq,9uptqq,\n:yptq “ } 9uptq}2\nL2` puptq,:uptqq\n“ }9uptq}2\nL2` puptq,p∆u´u`fpuqqptqq ´2αptqpuptq,9uptqq\n“ }9uptq}2\nL2´K0puptqq ´2αptqpuptq,9uptqq\n“ }9uptq}2\nL2´K0puptqq ´2αptq9yptq.(3.3)\nHence, for all tět0,\n:yptq `2αptq9yptq ěδ,\nor withAptq “2şt\nt0αpτqdτ,\nd\ndt`\neAptq9yptq˘\něδeAptq, 9yptq ěe´Aptq9ypt0q `δżt\nt0eApsq´Aptqds.\nBy our assumptions on αptq,Aptq Ñ 8 astÑ 8 and for any Lě1and all\nsP rt´L,tswe haveAptq ´Apsq ď1providedtis sufficiently large. By the\npreceding remark,\n9yptq ěe´Aptq9ypt0q `δLe´1,16 N. BURQ, G. RAUGEL, W. SCHLAG\nfor all large t. In particular, 9yptq Ñ 8 astÑ 8 and soyptq Ñ 8 astÑ 8. Next,\nwe note that\n:yptq `2αptq9yptq “ } 9uptq}2\nL2´K0puptqq (3.4)\n“ p2`γq}9uptq}2\nL2`γ}uptq}2\nH1´2p1`γqEptq (3.5)\n´ż\nRd`\n2p1`γqFpuptqq ´uptqfpuptqq˘\ndx, (3.6)\nwhere we have set for simplicity Eptq “Eppuptq,9uptqqq. Using pH.1qf, we can also\nwrite, fortět0sufficiently large,\n:yptq `2αptq9yptq ě p2`γq}9uptq}2\nL2`γ}uptq}2\nH1´2p1`γqEp0q (3.7)\ně p2`γq}9uptq}2\nL2`γ\n2}uptq}2\nH1 (3.8)\ně2`γ\n29y2ptq\nyptq`γyptq. (3.9)\nUsing the Young inequality, one easily shows that, for any εą0, and all sufficiently\nlarget,\nε9y2ptq\nyptq`γyptq ě2αptq9yptq.\nIn conclusion, there exist κą0andTlarge enough so that\n:yptq ě p1`κq9y2ptq\nyptq,@těT.\nThis means that yptq´κis concave which contradicts that yptq Ñ 8 (for details of\nthis last step, see [ 32] or [4]). /square\n3.2.Convexity argument and L2bound. Our aim here is to check that global\n(forward) solutions /vector uptq “ puptq,utptqqhave the property that }uptq}L2remains uni-\nformly bounded with respect to t. We will closely follow the work of Cazenave [ 7,\nProposition 3.1, Page 42] and [ 7, Lemma 2.4, Page 39]).\nLet/vector uptqbe a global solution of pKGqα, not identically 0. By Corollary 3.4,\nEp/vector uptqq ą0for anytě0. To simplify the notation in this section, we set\nhptq “ }uptq}2\nL2“2yptq,\nwhereyptqhas been defined in the proof of Lemma 3.3. It has been proved there\nthatyptq, and thushptq, is of class C2.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 17\nUsing the condition pH.1qf, we deduce from the equality ( 3.4) that\nh2ptq ě p4`2γq}ut}2\nL2`2γ}∇u}2\nL2`2γ}u}2\nL2´4ż\nαptqupx,tqutpx,tqdx\n´2p2`2γqEp/vector uptqq.(3.10)\nThis inequality together with the estimate ( 1.4) imply that, for tět1ě0,\nh2ptq ě p4`2γq}ut}2\nL2`2γ}∇u}2\nL2`2γ}u}2\nL2´4ż\nαptqupx,tqutpx,tqdx\n´2p2`2γqEp/vector upt1qq.(3.11)\nWe next choose t1ą0so that, for tět1,\n2αptq ďγ. (3.12)\nThus, the inequalities ( 3.11) and ( 3.12) imply that, for tět1,\nh2ptq ě˜Φp/vector uptqq ´2p2`2γqEp/vector upt1qq, (3.13)\nwhere\n˜Φpu,vq ” p4`γq}v}2\nL2`2γ}∇u}2\nL2`γ}u}2\nL2. (3.14)\nNext we remark that, for any ηą0and anyBą0, we can write\nB|h1ptq| ďBη}uptq}2\nL2`B\nη}utptq}2\nL2ďγ}uptq}2\nL2` p4`γq}utptq}2\nL2. (3.15)\nTo achieve this inequality, we set\nη“a\nγp4`γq´1, B “a\nγp4`γq.\nThe inequalities ( 3.15) and ( 3.13) imply the second inequality in the lemma below.\nWe thus have proved and analog of [ 7, Lemma 2.4].\nLemma 3.5. Let/vector uptqbe a nonzero global solution of pKGqα. Then there exists\nt1ą0, such that for tět1,\nh2ptq ě˜Φp/vector uptqq ´4p1`γqEp/vector upt1qq, (3.16)\nand\nh2ptq ěa\nγp4`γq`\n|h1ptq| ´4p1`γqa\nγp4`γqEp/vector upt1qq˘\n. (3.17)\nWith this lemma, we can now show the boundedness of the L2-norm ofuptqby\nfollowing [ 7, Section 3]. In this step, we only require αptq Ñ0in infinite time.18 N. BURQ, G. RAUGEL, W. SCHLAG\nProposition 3.6. Let/vector uptqbe a solution of pKGqα, which is global in positive time.\nLett1ą0be as in Lemma 3.5. We have the estimates\nd\ndtp|2γhpuptqq ´4p2`2γqEp/vector upt1qq|`q ď0,@t1ďtă 8, (3.18)\nhpuptqq ďsup`\nhpupt1qq,4p1`γq\nγEp/vector upt1qq˘\n,@tět1, (3.19)\nand there exists τ1ět1such that, for těτ1,\nhpuptqq ď8p1`γq\nγEp/vector upt1qq. (3.20)\nProof. The estimates hold trivially if the solution vanishes ident ically. Thus we may\nassume that Ep/vector uptqq ą0for alltě0and we prove ( 3.18) by contradiction, as in [ 7,\nPage 43]. One sets\ngptq “hptq ´4p1`γq\nγEp/vector upt0qq.\nIf (3.18) does not hold, there exists t2ět1so that\ng1pt2q ą0, g pt2q ą0.\nMoreover, from ( 3.16), we infer that\ng2ptq ěγgptq,@tP rt1,8q.\nHencegis a convex increasing function on rt2,8qwithlimtÑ8gptq “ 8 . Since\ngptq ě0fortět2, (3.16) implies that, for tět2,\nh2ptq ě p4`γq}utptq}2\nL2.\nMultiplying by hptq, we get\nhptqh2ptq ě4`γ\n4ph1ptqq2.\nHence phptqq´γ{4is concave on rt2,8q. Since phptqq´γ{4Ñ0astÑ 8, this is\nimpossible and ( 3.18) holds. Now ( 3.19) is a direct consequence of ( 3.18), and to\nprove ( 3.20) we also proceed by contradiction. If ( 3.20) is not true, there exists\nt2ąτ1, where,\nτ1”t1` p2`2γq´1Ep/vector upt1qq´1|h1pt1q{2|,\nsuch that\nhpt2q ą8p1`γq\nγEp/vector upt1qq. (3.21)\nThe property ( 3.18) or (3.19) implies that\nhptq ěhpt2q,@t1ătăt2. (3.22)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 19\nIf we apply the inequality ( 3.16), we deduce from the inequalities ( 3.21) and ( 3.22)\nthat, for any t1ďtďt2,\nh2ptq ě8p1`γqEp/vector upt1qq ´4p1`γqEp/vector upt1qq ě4p1`γqEp/vector upt1qq ą0.\nTherefore,\nh1pt2q ě4p1`γqEp/vector upt1qqpt2´t1q `h1pt1q ą |h1pt1q| `h1pt1q,\nand hence\nh1pt2q ą0.\nThis contradicts the property ( 3.18) fort“t2. Thus ( 3.20) is true. /square\n4.Construction of special time sequences\n4.1.Preliminary lemmata. We begin with a few elementary lemmata, in which\nwe use the decay property of the energy Ep/vector uptqqwhen/vector uis a global solution of\npKGqα. Indeed, in this case, Ep/vector uptqq ě0and thus,\n2ż8\n0αpsq}utpsq}2\nL2dsďEp/vector up0qq. (4.1)\nLemma 4.1. Assume pH.1qα. Then, for any global solution /vector uptqofpKGqαand for\nanyrą0, there exists a sequence tnÑ 8, asngoes to infinity, so that\nlim\nnÑ8żtn`r\ntn}utpsq}2\nL2ds“0. (4.2)\nMoreover, there exist three sequences ˜tj\nn,j“0,1,2so that˜tj\nnP rtn`jr{3,tn` pj`\n1qr{3qand\nlim\nnÑ8}utp˜tj\nnq}L2“0. (4.3)\nProof. Assume that the property ( 4.2) does not hold. Then there exist εą0and\nt1ą0so that, for tět1,żt`r\nt}utpsq}2\nL2dsěε,\nwhich implies, due to the hypothesis pH.1qα, that\nż8\nt1αpsq}utpsq}2\nL2dsě8ÿ\nn“0żt1`pn`1qr\nt1`nrαpsq}utpsq}2\nL2ds\něε8ÿ\nn“0αpt1` pn`1qrq “ 8.(4.4)20 N. BURQ, G. RAUGEL, W. SCHLAG\nThis leads to a contradiction since by ( 4.1) the left-hand side is bounded, and\ntherefore, for any nlarge enough, there exists tněnso that,\nżtn`r\ntn}utpsq}2\nL2dsď1\nn, (4.5)\nand thus ( 4.2) holds. The property ( 4.3) is an obvious consequence of ( 4.2)./square\nWe will need a stronger quantitative decay.\nLemma 4.2. Assume that the hypothesis pH.2qαholds. Let/vector uptqbe a global solution\nofpKGqα. There exists a sequence nm,mPN, so that the following properties hold:\nlim\nmÑ8żpnm`1q3{2\nn3{2\nms1\n3}utpsq}2\nL2ds”lim\nmÑ8εmprq “0, (4.6)\nandżpnm`1q3{2\nn3{2\nm}utpsq}L2dsďCε1{2\nm. (4.7)\nProof. We also prove ( 4.6) by contradiction. Assume that ( 4.6) does not hold. Then,\nthere exist δą0andn0, such that, for něn0\nżpn`1q3{2\nn3{2s1\n3}utpsq}2\nL2dsěδ,\nwhich implies that\nż8\nn3{2\n0αpsq}utpsq}2\nL2dsě8ÿ\nn“n0żpn`1q3{2\nn3{2αpsq}utpsq}2\nL2ds\něδ8ÿ\nn“n01\np1` pn`1q3\n2aqpn`1q1\n2“ 8,(4.8)\nif\n3\n2a`1\n2ď1, (4.9)\nwhich leads to a contradiction. Now, using Cauchy-Schwarz i nequality, we get\nżpnm`1q3{2\nn3{2\nm}utpsq}L2dsďppnm`1q3{2´n3{2\nmq1{2\nn1{4\nm`żpnm`1q3{2\nn3{2\nms1\n3}utpsq}2\nL2ds˘1{2\nďCε1{2\nmppnm`1q3{2´n3{2\nmq1{2\nn1{4\nmďCε1{2\nm.\n(4.10)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 21\nWe notice that in what follows, we will use\npnm`1q3{2´n3{2\nměn3a{2\nm. (4.11)\n/square\nUnder the weaker hypothesis pH.3qα, the same proof gives the following result.\nLemma 4.3. Assume that the hypothesis pH.3qαholds. Let/vector uptqbe a global solution\nofpKGqα. There exists a sequence nm,mPN, so that the following property holds:\nlim\nmÑ8żpnm`1q2\nn2m}utpsq}2\nL2ds”lim\nmÑ8εm“0, (4.12)\n4.2.The vanishing of K0pupt˚\nnqqin the limit nÑ 8.We have the following\ntrichotomy for the forward evolution of pKGqα\n(FTB)/vector uptqblows up in finite positive time.\n(GEB)/vector uptqexists globally and the forward trajectory /vector uptqis bounded in Hraduni-\nformly intě0.\n(GEU)/vector uptqexists globally and is unbounded in forward time.\nWe shall later exclude the third possibility.\nLemma 4.4. Under the hypothesis pH.1qα, let/vector uptqbe a global trajectory of pKGqα.\nAnd letrą0be fixed. There exist a sequence of times t˚\nnand a sequence of numbers\nδn, such that t˚\nnÑ 8asnÑ 8and\nlim\nnÑ8żt˚\nn`r\nt˚n´r}utpsq}2\nL2ds“0 lim\nnÑ8K0pupt˚\nnqq ”lim\nnÑ8δn“0. (4.13)\nProof. We will argue by contradiction. We assume for simplicity r“1.\n1) By Lemma 4.1, there exist sequences tnÑ 8,εnÑ0(nÑ 8), and three\nsequences ˜tj\nnP rtn`j{3,tn` pj`1q{3q,j“0,1,2, such that\nżtn`2\ntn´1}utpsq}2\nL2ds`3ÿ\nj“0}utp˜tj\nnq}L2ďεn. (4.14)\n2) We argue by contradiction and assume first that there exist δą0and an\ninfinite number of intervals In“ rtn,tn`1sso that\n@tPIn,K0puptqq ěδ. (4.15)\nBy Proposition 3.6, the function ypsq “ }upsq}2\nL2{2is bounded by a positive constant\nM2. Using ( 3.3) we get\n:yptq “ } 9uptq}2\nL2´K0puptqq ´2αptqpuptq,9uptqq ď3\n2}9uptq}2\nL2´δ`2M2α2ptq,22 N. BURQ, G. RAUGEL, W. SCHLAG\nwhich implies by integration between ˜t0\nnand˜t2\nn\n9yp˜t2\nnq ´9yp˜t0\nnq ď3\n2εn´δ{3`2M2max\ntnďtďtn`1α2ptq ď ´δ{4. (4.16)\nOn the other hand,\n´δ{4ě9yp˜t2\nnq ´9yp˜t0\nnq “ pup˜t2\nnq,9up˜t2\nnqq ´ pup˜t0\nnq,9up˜t0\nnqq ě ´2Mεn. (4.17)\nBut, fornlarge enough 2M˜εnăδ{4, which lead to a contradiction and shows\nthat ( 4.15) cannot be true.\n3) Assume next that there exist δą0and an infinite number of intervals Inso\nthat\n@tPIn,K0puptqq ď ´δ. (4.18)\nIn this case, there exists n0such that, for něn0, for anytPIn,\n:yptq “ } 9uptq}2\nL2´K0puptqq ´2αptqpuptq,9uptqq (4.19)\ně1\n2}9uptq}2\nL2`δ´2M2max\ntnďtďtn`1α2ptq ě3\n4δ, (4.20)\nwhich implies by integration between ˜t0\nnand˜t2\nnthat, forněn0,\n2M˜εně9yp˜t2\nnq ´9yp˜t0\nnq ě1\n4δ. (4.21)\nChoosingnlarge enough again leads to a contradiction. Hence, the prop erty (4.18)\ncannot be true on an infinite number of intervals In.\n4) The properties 2) and 3) imply that there exist a subsequen cenp,pÑ 8, two\ntimesˇtnpPInp,pandˆtnpPInp,pso that\nK0pupˇtnpqq ě ´1\np, K0pupˆtnpqq ď1\np. (4.22)\nIf\n´1\npďK0pupˆtnpqq ď1\np,\nwe may set t˚\nn“ˆtnpand the properties ( 4.13) are proved. If K0pupˆtnpqq ă ´1\np,\nthen, by ( 4.22) and the intermediate value theorem, there exists t˚\nnPInp,pso that\nK0pupt˚\nnqq “ ´1\npand again the properties ( 4.13) hold. /square\nRemark 4.5. We point out that the results of Sections 4.1and4.2also hold in the\nnon-radial case, whereas the convergence property of the ne xt section uses the radial\nassumption.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 23\n4.3.Proof of Theorem 1.3.This is similar to the proof of [ 4, Theorem 3.3]. For\nthe reader’s convenience we present the argument. We recall that the sequence t˚\nn,\nně0, is given in Lemma 4.4together with the constant rą0. >From Lemma 3.1,\nwe conclude that\nsup\nně0}pupt˚\nnq,Btupt˚\nnqq}Hă 8.\nWe consider the equations\n#\nBttunptq `2αpt˚\nn`tqBtunptq ´∆unptq `unptq ´fpunptqq “0,\npunp0q,Btunp0qq “ pupt˚\nnq,Btupt˚\nnqq.pKGqn\nα\nBy wellposedness (see Theorem 2.5), there exist Tą0andCą0such that, for any\nn, the solution punptq,Btunptqqexists on r´T,Tsand, for ´TďtďT,\n}punptq,Btunptqq}HďC. (4.23)\nWithout loss of generality, we may choose 0ăTďr.\nIn dimensions d“1ord“2, (4.23) implies that }un}L8pp´T,Tq,LppRdqqďC, for all\n2ďpă 8. For dimensions 3ďdď6, the Strichartz estimates give\n}un}Lθ˚pp0,Tq,L2θ˚pRdqqďC, (4.24)\nwhereθ˚“d`2\nd´2. By uniqueness, unptq “upt˚\nn`tq. For anys,tP r´T,Ts, we have\nthe following time equicontinuity\nż\nRd|unptq ´unpsq|2dx“ż\nRdˇˇˇˇżt\nsBtunpσqdσˇˇˇˇ2\ndx\nď |t´s|ż\nRdżt\ns|Btunpσq|2dσdx ď |t´s|żt`t˚\nn\ns`t˚n}Btupσq}2\nL2dσ,\nand thus, by Lemma 4.4,\n}unptq ´unpsq}2\nL2ď |t´s|żt`t˚\nn\ns`t˚n}Btupσq}2\nL2dσ (4.25)\nď2rżtn`r\ntn´r}Btupσq}2\nL2dσÝ Ñ0,asnÑ `8. (4.26)\nFors,tP r´T,Ts, and any fixed pP p2,2˚q, we deduce from the above inequality,\nby using an interpolation argument, that there exist b”bppq P p0,1qand a uniform24 N. BURQ, G. RAUGEL, W. SCHLAG\nconstant ˜Cą0so that,\n}unptq ´unpsq}Lpď }unptq ´unpsq}b\nL2˚}unptq ´unpsq}1´b\nL2\nď˜C|t´s|1´b\n2`żt˚\nn`r\nt˚n´r}Btupσq}2\nL2dσ˘1´b\n2Ñ0asnÑ `8.(4.27)\nWe choose 2ăp0ăp1ă2˚(to be fixed later) and define X“Xp0,p1”Lp0pRdq X\nLp1pRdq. We consider the family of functions punptqqninC0pr´T,Ts;Xq. By the\nproperty ( 4.23), we know that punqis bounded in C0r´T,Ts;H1XC1r´T,Ts;L2,\nhence also in Cαr´T,Ts;H1´αand for any 2ăpă2˚inCαppqp´T,Ts;Lp. Due\nto the compact embedding of H1\nradpRdqintoLq,2ăqă2˚, we deduce that the\nsequence punqis equicontinuous in C0pr´T,Ts;Kq, where pKqis a compact subset\nofX. Thus, by the Ascoli’s theorem, (after possibly extracting a subsequence) the\nsequence punqconverges in C0pr´T,Ts;Xqto a function u˚which according to ( 4.25)\nis constant on r´T,Ts. We have proved\n‚unptq Ñu˚asnÑ `8 inC0pr´T,Ts;Xqandu˚:“u˚ptq\n‚ Btunptq Ñ0asnÑ `8 inL2pp´T,Tq;L2pRdqq\n‚ punptq,Btunptqqnis uniformly bounded in L8pp´T,Tq;Hqand, in particular\ninL2pp´T,Tq;Hq.\nTo pass from these weak convergences to strong convergences in\nCpr´T,Ts;H1q XC1pr´T,Ts;L2pRdqq,\nwe are first going to show that u˚is an equilibrium point of the (undamped) Klein\nGordon equation. To pass to the limit in the equation ( pKGqα) satisfied by unit\nis enough to show that we can pass to the limit in the non linear ity (all the linear\nterms passing easily to the limit in the distribution sense) . The argument here is\nthat we can combine the slack from, sub criticality of the non linearity with the\ncompact embedding H1\nradial ÑLp,2ăpă2˚.\nLemma 4.6. We can choose p0,p1sufficiently close to 2,2˚, respectively (depending\nonpH.2qf) so that\nlim\nnÑ`8sup\ntPr´T,Tsż\nRd|fpunptqqunptq ´fpu˚qu˚|dx“0. (4.28)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 25\nProof. Using Hypothesis pH.2qfand the Hölder inequality, we see that for tP\nr´T,Ts,\n|ż\nRdpfpu˚qu˚´fpunptqqunqptqdx|\nďcż\nRd|u˚´unptq|p|unptq|θ` |unptq|β`1` |u˚|θ` |u˚|β`1qdx,(4.29)\nand the result follows from the choice\n2ăp0ďβ`2ăθ`2ďp1ă2˚,\nand the convergence of untou˚inCpr´T,Ts;Lp0XLp1q.\n/square\nSinceK0punqp0q Ñ0,nÑ `8 , we get\nlim\nnÑ`8}unp0q}2\nH1“lim\nnÑ`8ż\nRdfpunp0qqunp0q\n“ż\nRdfpu˚qu˚dx“ż\nRdp|∇u˚|2` pu˚q2qdx“ }u˚}2\nH1,(4.30)\nwhere in the third equality we used that u˚is an equilibrium of pKGqα.\nOn the other hand, the sequence punp0qqconverges to u˚inX, hence weakly in\nH1. From ( 4.30) we deduce that the H1convergence is actually strong. We also\nhave\nuptq “up0q `żs\n0Bsupsqds\nñ }unptq´u˚}L8p´T,Tq;L2pRdqď }unp0q´u˚}L2pRdq`}Bsu}L1pp´T,Tq;L2pRdqqÑnÑ`80.\n(4.31)\nTo finish the proof of the theorem it remains to show that\nBtunp0q Ñ0inL2pRdq. (4.32)\nThe difference ˜un:“un´u˚satisfies\n$\n’&\n’%Btt˜un´∆˜un`˜un“fpunq ´fpu˚q ´2αpt`t˚\nnqBt˜un\n˜unp0q “unp0q ´u˚Ñ0asnÑ `8 inH1pRdq\nBt˜unp0q “ Btunp0q.(4.33)26 N. BURQ, G. RAUGEL, W. SCHLAG\nOne hasun´u˚“wn`vnwherewnandvnare solutions of the equations\n#\nBttwn´∆wn`wn“fpunq ´fpu˚q ´2αpt`t˚\nnqBtun\nwnp0q “unp0q ´u˚, Btwnp0q “0,(4.34)\nand#\nBttvn´∆vn`vn“0\nvnp0q “0, Btvnp0q “ Btunp0q.(4.35)\nBy energy estimates for all ´TďtďT,\n}pwn,Btwnqptq}HďC“\n}unp0q ´u˚}H1`C?\n2TˆżT\n´T}Btunpsq}2\nL2ds˙1\n2\n`żT\n´T}fpunqpsq ´fpu˚q}L2ds‰\n,(4.36)\nand since the two first terms in the r.h.s. of ( 4.36) tend to 0, the following lemma\nallows to conclude that\nlimnÑ`8 }pwn,Btwnqptq}L8pp´T,Tq;Hq“0. (4.37)\nLemma 4.7. We have\nlim\nnÑ`8}fpunqpsq ´fpu˚}L1pp´T,Tq;L2pRdqq“0.\nProof. In view of Hypothesis pH.2qfand the fact that 0ăβăθ´1, one has,\nżT\n´T}fpunqpsq ´fpu˚q}L2ds\nďCżT\n´T}|unpsq ´u˚|p|un|β` |u˚|β` |un|θ´1` |u˚|θ´1q}L2ds\nď2CżT\n´T}|unpsq ´u˚|p1` |un|θ´1` |u˚|θ´1q}L2ds\nď2C“\n}un´u˚}L8pp´T,Tq,L2pRdqq`żT\n´T}|unpsq ´u˚|p|un|θ´1` |u˚|θ´1q}L2pRdqds.\n(4.38)\nSinceunÑu˚inC0pI,L2pRdqq, the first term on the right-hand side of ( 4.38)\nvanishes in the limit nÑ 8. Since, by elliptic regularity, u˚is uniformly boundedLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 27\ninL8pRdq, using ( 4.31), we have\nżT\n´T}pun´u˚q|u˚|θ´1q}L2dsď2T}un´u˚}L8pL2pRdqq}u˚}θ´1\nL8pL8q\nďC}un´u˚}L8pL2pRdqqÑ0,pnÑ `8q.(4.39)\nTo bound the remaining term in ( 4.38), we argue as in the proof of Theorem 2.5, by\nusing the Hölder and Strichartz inequalities (see the estim ates (2.7)), and we obtain\nżT\n´T}punpsq ´u˚q|unpsq|θ´1}L2dsď }unpsq}pθ´1qη1\nθ\nL8pI,L2pRdqq}unpsq ´u˚}η1\nθ\nL8pI,L2pRdqq\nˆ p2Tqη1`żT\n´T}unpsq ´u˚}θ˚\nL2θ˚ds˘1´η1\nθˆ`żT\n´T}unpsq}θ˚\nL2θ˚ds˘pθ´1qp1´η1q\nθ,(4.40)\nwhereη1“d`2´θpd´2q\n4. The right-hand side of the inequality ( 4.40) tends to 0asn\ngoes to infinity. /square\nBy construction, vn“ pun´u˚q´wnand, in particular, Btvn“ Btun´Btwn. From\n(4.2) and ( 4.37), we infer that\nlim\nnÑ`8}Btvn}L2pp´T,Tq;L2pRdqq“0. (4.41)\nTo conclude the proof of Theorem 1.3, we turn this L2\ntaveraged estimate into a\nuniform estimate with\nLemma 4.8 ( [4, Lemma 3.4]) .For anyT0ą0, there exists a positive constant\ncpT0q ą0, independent of n, such that for any solution of the linear Klein Gordon\nequation (4.35),\n}Btvnp0q}2\nL2pRdqďcpT0qżT0\n´T0ż\nRd|Btvn|2dxds. (4.42)\n4.4.Trapping the trajectory near an equilibrium for a long time. The\nfollowing lemma is the key mechanism by which a global trajec tory will be trapped\nnear an equilibrium. This lemma requires nothing on the diss ipation other than\nαě0. So it also holds in the conservative case.\nLemma 4.9. Let/vector ube a global trajectory and suppose that for some equilibrium Q\nand for some time t0\n}/vector upt0q ´ pQ,0q}Hăρ,\nwhere0ăρă1is arbitrary but fixed. There exists a small δ0“δ0pρ,d,Q,f q\n(independent of α) with the following property if }uptq ´Q}L2ďδ0for alltP rt0,t1s28 N. BURQ, G. RAUGEL, W. SCHLAG\nwheret1ět0is arbitrary, then\n}/vector uptq ´ pQ,0q}Hď2ρ, (4.43)\nfor alltP rt0,t1s.\nProof. Define\nµptq:“1\n2}uptq ´Q}2\nH1`1\n2}ut}2\nL2.\nIn view of xu,vyH1“ xp´∆`1qu,vyL2one has\n9µptq “ xuptq,utyH1´ xQ,utyH1` x∆u´u,utyL2´2αptq}utptq}2\nL2` xfpuptqq,utyL2\nďd\ndt“\n´ xp´∆`1qQ,uyL2`ż\nRdFpuptqqdx‰\n,\nwhence for all t0ďtďt1and with constants depending on Qand the nonlinearity\nfvia the constants in pH.2qf,\nµptq ďµpt0q ´ xfpQq,uptq ´upt0qyL2`ż\nRdFpuptqq ´Fpupt0qqdx\nď1\n2ρ2`Cδ0`Cż\nRdp|uptq| ` |upt0q| ` |uptq|θ` |upt0q|θq|uptq ´upt0q|dx\nď1\n2ρ2`Cδ0`Cp}uptq}θ\nθ`1` }upt0q}θ\nθ`1q}uptq ´upt0q}θ`1.\nUsing a classical Sobolev embedding and an interpolation in equality, we may write\nthat, for some η“ηpθ,dq P p0,1q,\n}uptq ´upt0q}θ`1ďC}uptq ´upt0q}η\n2}uptq ´upt0q}1´η\nH1,\nwhich further implies that for all t0ďtďt1,\nµptq ď1\n2ρ2`Cδ0`Cδη\n0p1`µptqqp1´ηqpθ`1q. (4.44)\nBy the method of continuity, it follows that for δ0sufficiently small (independently\noft0) one hasµptq ď2ρ2for allt0ďtďt1. /square\nFor future reference we remark that the condition on δ0is\nCδη\n0ďρ2, (4.45)\nwith constant C“CpQ,d,f qandη“ηpd,θq “ηpd,fq, as can be seen from ( 4.44).\nThe following immediate corollary of Lemma 4.9is the most useful form of the\ntrapping property for our purposes.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 29\nCorollary 4.10. Let/vector ube a global trajectory and suppose that for some equilibrium\nQand for some times 0ďt0ďt1\n}/vector upt0q ´ pQ,0q}Hăρ, }upt0q ´Q}L2`żt1\nt0}utptq}L2dtďδ0,\nwhere0ăρă1is arbitrary, and δ0“δ0pρ,d,Q,f qis small. Then\n}/vector uptq ´ pQ,0q}Hď2ρ, (4.46)\nfor alltP rt0,t1s.\nProof. Lemma 4.9applies since\nmax\nt0ďtďt1}uptq ´Q}L2ď }upt0q ´Q}2`żt1\nt0}utpsq}L2dsďδ0,\nand we are done. /square\nCombining this corollary with Lemma 4.2yields the following result which guar-\nantees trapping along increasingly long intervals Im.\nProposition 4.11. Under the hypothesis pH.2qα, let/vector uptqbe a global solution of\npKGqα. Letnm,mPNbe the sequence obtained in Lemma 4.2. Then there exists\nan equilibrium pQ,0qofpKGqαand, for any 0ăρă1there exists m0such that\n}/vector upsq ´ pQ,0q}Hďρ, @sPIm:“ rn3{2\nm`1,pnm`1q3{2s, (4.47)\nfor allměm0.\nProof. The proof of Theorem 1.3applies to any sequence JmĂImof intervals of\nfixed length, say |Jm| “1. In particular, we may take Jm“ rn3{2\nm,n3{2\nm`1s. This\nyields a sequence of times t˚,1\nnmfor which the properties ( 4.13) hold, and such that\n/vector upt˚,1\nnmqconverges strongly to an equilibrium point pQ,0q.\nIn view of ( 4.7) we may apply the previous corollary to the interval rt˚,1\nnm,pnm`\n1q3{2s. /square\n4.5.Trapping the trajectory near an equilibrium in the case of is olated\nequilibria.\nProposition 4.12. Assume that the hypothesis pH.3qαholds. Let/vector uptqbe a global\nsolution of pKGqα. Suppose moreover that all the equilibria pQ,0qwith energy\nEpQ,0q ďEp/vector up0qqare isolated. Let nm,mPN, be the sequence obtained in\nLemma 4.3. Then there exist an equilibrium pQ,0qofpKGqαandρ0ą0, so that\nfor any0ăρăρ0, there exists m0ą0such that\n}/vector upsq ´ pQ,0q}Hďρ, @sPI2\nm:“ rn2\nm`1,pnm`1q2s, (4.48)30 N. BURQ, G. RAUGEL, W. SCHLAG\nfor allměm0.\nProof. The proposition is proved in two steps.\nStep 1 As in the previous proposition, we remark that the proof of Th eorem 1.3\napplies to any sequence JmĂI2\nmof intervals of fixed length, say |Jm| “1. In\nparticular, we may take Jm“ rn2\nm,n2\nm`1s. This yields a sequence of times t˚,1\nnm\nfor which the properties ( 4.13) hold, and such that /vector upt˚,1\nnmqconverges strongly to an\nequilibrium point pQ,0q. Since pQ,0qis isolated, there exists δą0such that the ball\nBHppQ,0q,δqdoes not contain another equilibrium. We next choose 0ăρ0ăδ{2.\nBy the uniform continuity of Sαpt,sqatpQ,0q, for any 0ăρăρ0, there exist ηą0,\nso that, if\n}pQ,0q ´/vector w}Hďη, (4.49)\nthen\n}pQ,0q ´Sαpt,t˚,1\nnmq/vector w}Hăρ, n2\nmďtďn2\nm`1. (4.50)\nThus, since /vector upt˚,1\nnmqconverges to pQ˚,0q, there exists m1, so that, for měm1, we\nhave\n}pQ,0q ´/vector uptq}Hăρ, n2\nmďtďn2\nm`1. (4.51)\nStep 2 If the property ( 4.51) does not hold for every tPI2\nm, then there exists\nsnmP pn2\nm`1,pnm`1q2ssuch that/vector upsnmqbelongs to the sphere SppQ,0q,ρqof\ncenter pQ,0qand radius ρ.\nWe next choose 0ără1smaller than the local time Tą0of existence of\nsolutions of pKGqαwith initial data in the ball BppQ,0q,2δqof center pQ,0qand\nradius2δ.\nWe now consider the sequence of times snm. We first apply Lemma 4.4to the\nsequence of intervals rsnm´r,snm`rs, obtaining then a sequence of times s˚\nnmfor\nwhich the properties ( 4.13) hold. Afterwards, we notice that the proof of Theo-\nrem1.3also applies to the sequence of intervals rsnm´r,snm`rsand thus that\n/vector ups˚\nnmqconverges to an equilibrium denoted pQ2,0q. Arguing as in Step 1, we show\nthat there exists m2ěm1such that, for měm2, we have\n}pQ2,0q ´/vector uptq}Hăρ, snm´rďtďsnm`r, (4.52)\nand in particular,\n}pQ2,0q ´/vector upsnmq}Hăρ. (4.53)\nThe property ( 4.53) also implies that\n}pQ2,0q ´ pQ,0qq}Hă2ρăδ, (4.54)\nwhich leads to a contradiction, unless Q2“Q. And Proposition 4.12is proved.\n/squareLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 31\n5.Proof of Theorem 1.1ifdą3ord“3andθď4\n5.1.The main functional Hν.For technical reasons we first present these cases.\nThe remaining one, i.e., d“3and4ăθă5, is more complicated and we turn to\nit later. The proof proceeds by considering suitable functi onals of Lyapunov type.\nHowever, the analysis is somewhat delicate as the functiona ls only give limited in-\nformation (for example, only locally in time). We will rely o n the Łojasiewicz-Simon\ninequality, which is a well-known method for gradient syste ms (see for example,[ 36,\n19,20,21]) and used first by Simon [ 36]. Throughout this section /vector uis a global solu-\ntion, andImis the interval from Proposition 4.11. In particular, pQ,0qwill always\ndenote the equilibrium of that proposition. Let ρ0be given by the Łojasiewicz-Simon\ntheorem of Appendix A, and we set ρ“ρ0in Proposition 4.11.\nDefinition 5.1. Assume hypothesis pH.2qα. Let/vector ube a global trajectory, and let\npQ,0qbe the equilibrium from Proposition 4.11. Fix a large positive number R1so\nthatR1ě8R0, withR0ěmaxp4R,2ρ0qandR“ }Q}H1. For anytě0let\nHνptq:“Ep/vector uptqq ´EpQ,0q `ε0\np1`tqaνx´∆uptq `uptq ´fpuptqq,utptqyH´1,(5.1)\nwhere1ěε0ą0will be specified later and where νą1is chosen so that aνă1\n3.\nLetTmąn3{2\nmbe the first time where }/vector uptq}H“R1, orTm“ 8if this never\noccurs, and define ˜Im:“ rn3{2\nm,Tms.\nNote that by Proposition 4.11and our choice of parameters, we have Tmě pnm`\n1q3\n2. Thus,˜ImextendsImstrictly to the right. See Lemma 5.3for basic properties\nofHνrelating to differentiability and monotonicity. The condit ions to be satisfied\nbyε0will appear in the proof below. We notice that the condition aνă1{3imposes\nan additional condition on aonly ifνą1is large. In fact, it suffices to take νą1\nclose to1. The functional Hνwitha“0has been used by Haraux and Jendoubi\nin [20] on bounded domains and for globally bounded trajectories i n the case where\n1ăθďd{pd´2qand provided the damping is a positive constant. A functiona l\nof this type with ν“1was used by Haraux and Jendoubi in [ 21] for bounded\ntrajectories for finite-dimensional gradient systems, and dampingαptq ą0under a\ncondition similar to pH.1qα. Amongst several others, our challenge is that we are\ndealing with potentially unbounded trajectories. However , we will use in a crucial\nway that the trajectories are bounded on the time intervals Im, withmlarge enough.\nIf we knew a priori that the trajectory is bounded, then the pr oof of Theorem 1.1\nwould be much simpler.\nWe begin with an estimate on the nonlinear expression arisin g inH1\nν.32 N. BURQ, G. RAUGEL, W. SCHLAG\nLemma 5.2. Assume that Hypothesis pH.2qfholds, with the restriction 1ăθď4\nin the case d“3. Then, the following inequality holds\n|xf1puqut,utyH´1| ďc}ut}2\nL2}u}θ´1\nH1. (5.2)\nProof. a) The case d“3: Assume that 1ăθď4, then we can for example write\n|xf1puqut,utyH´1| ď }f1puqut}H´2}ut}L2ďsup\n}ϕ}H2“1|ż\nR3f1puqutϕdx|}ut}L2.(5.3)\nUsing condition pH.2qfon the nonlinearity with }ϕ}H2“1fixed we have\n|ż\nR3f1puqutϕdx| ďsup\n|v|ď1|f1pvq|}ut}L2}ϕ}L2`C}|u|θ´1ut}1}ϕ}8 (5.4)\nďCp1` }|u|θ´1}L2q}ut}L2, (5.5)\nwhich implies, since 2pθ´1q ď6, that\n|xf1puqut,utyH´1| ďCp1` }u}θ´1\nL2pθ´1qq}ut}2\nL2ďCp1` }u}θ´1\nH1q}ut}2\nL2, (5.6)\nand the estimate ( 5.2) is proved in this case.\nb) The case 5ďdď6: In this case, H2pRdqis embedded in L2d{pd´4qpRdq.\nApplying the estimate ( 5.3) and the Hölder inequality, we get\n|xf1puqut,utyH´1| ďC}ut}2\nL2sup\n}ϕ}H2“1`\n}ϕ}L2d{pd´4q}u}θ˚´1\nL2d{pd´2q` }ϕ}L2˘\n(5.7)\nďC}ut}2\nL2p1` }u}θ˚´1\nH1q, (5.8)\nand the estimate ( 5.2) is again proved in this case.\nc) The case d“4: Ifd“4,H2pRdq ĂLppRdq, for any 1ďpă 8. This case is\neven simpler and is left to the reader. /square\nWe can now derive basic properties of Hν.\nLemma 5.3. Assume that /vector u‰ pQ,0q. ThenHνis a decreasing C1-function as long\nas}/vector uptq}HăR1. In particular, Hνis decreasing on ˜Imwithměm0.\nProof. The functional Hνis well-defined and continuous. Indeed, it suffices to use\nthe continuity properties of /vector uand the following estimates,\n|x´∆u`u,utyH´1| ď }u}H1}ut}H´1ďC}u}H1}ut}L2, (5.9)\nand, due to Hypothesis pH.2qf,\n|xfpuq,utyH´1| ď }fpuq}H´1}ut}H´1ďCp}u}H1` }u}θ˚\nH1q}ut}L2. (5.10)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 33\nTo obtain the final estimate here, we note that\n}fpuq}H´1ďC`\n}u}L2` }|u|θ˚}H´1˘\nďC`\n}u}L2` }|u|θ˚}L2d{pd`2q˘\nďC`\n}u}L2` }u}θ˚\nH1˘\n,(5.11)\nsinceL2d{pd`2qpRdqãÑH´1pRdqandθ˚¨2d\nd`2“2d\nd´2“2˚. Next, one needs to\nshow that Hνis differentiable and that the derivative is continuous. To s how\nthatHνis differentiable and C1int, we first assume that the initial data /vector up0q\nbelong toH2pRdq ˆH1pRdq. Then the solution /vector uptqis classical and belongs to\nC1pr0,8q,H1pRdqq XC0pr0,8q,H2pRdqq XC2pr0,8q,L2pRdqq. In this case, one can\nalso compute the derivative of Hνin a classical way and one finds that, for tě0,\nH1\nνptq “ ´2αptq}utptq}2\nL2´aνε0\np1`tqaν`1x´∆u`u´fpuq,utyH´1\n`ε0\np1`tqaνx´∆u`u´fpuq,uttyH´1\n`ε0\np1`tqaνx´∆ut`ut´f1puqut,utyH´1,\nwhich implies that, for tě0,\nH1\nνptq “ ´2αptq}utptq}2\nL2´ε0νa\np1`tqaν`1x´∆u`u´fpuq,utyH´1\n´ε0\np1`tqaν} ´∆u`u´fpuq}2\nH´1\n´2ε0\np1`tqapν`1qx´∆u`u´fpuq,utyH´1\n`ε0\np1`tqaνx´∆ut`ut´f1puqut,utyH´1.(5.12)\nIn the above expression of H1\nνptq, the first to the fourth terms are always well-defined\n(see (5.9) and ( 5.10)) and one has good bounds even if the initial data are only in\nH. It remains to prove that it is also the case for the last term. In the previous\nlemma we established the bound\n|xf1puqut,utyH´1| ďC}ut}2\nL2p1` }u}θ´1\nH1q,\nsee (5.2). In addition, one has the following estimate\n|x´∆ut`ut,utyH´1| ďC}ut}2\nL2. (5.13)34 N. BURQ, G. RAUGEL, W. SCHLAG\nWe next apply Lemma 5.2and the Cauchy-Schwarz inequality to ( 5.12). Thus there\nexistε0ą0small enough, and t0ą0large enough, so that for all tět0,\nH1\nνptq ď ´p2´ε0CpR1qq\np1`tqa}ut}2\nL2´ε0\n4p1`tqaν} ´∆u`u´fpuq}2\nH´1ď0,(5.14)\nprovided that }/vector uptq}HďR1, and, say,\n2´ε0CpR1q ě3{2. (5.15)\nIfH1\nνp˜tq “0, thenut“0and ´∆u`u´fpuq “0at timet“˜t. Therefore, uis\nan equilibrium p˜Q,0q. By Proposition 4.11we then must have Q“˜Q. Thus in fact\nH1\nνptq ă0. These estimates are still valid when the initial data belon g toHonly.\nThus, arguing by density, we can show that Hνis still of class C1even if the initial\ndata only belong to H. /square\nWe derive the following conclusions from the previous lemma : Assume that the\ntrajectory/vector uptqis bounded in Hfortě0by say the constant R1. In Theorem 1.3,\nwe have seen that there exists a sequence of times tnmsuch that/vector uptnmqconverges\ntopQ,0qastnmgoes to infinity, which means that Hνptnmqgoes to0whentnmgoes\nto infinity. On the other hand, since /vector uptqis bounded in Hby the constant R1, the\nfunctionHνptqis decreasing for tět0. Since it is bounded from below, the function\nHνptqconverges as tgoes to infinity and thus we may conclude that\nlim\ntÑ`8Hνptq “0. (5.16)\nThus, without loss of generality, we may assume that, by choo sing the above time\nt0ą0large enough, Hνptq ą0fortět0.\nIf we do not know a priori that the trajectory /vector uptqremains bounded in H, then\nwe cannot conclude that Hνptqstays positive for tět0. The above arguments\n(see (5.14)) only show that, for tP rt0,T1s, where rt0,T1sis defined as the maximal\ntime interval (beginning at t0) on which }uptq}HďR1, the function Hνis non\nincreasing. It could be that on this time interval the functi onHνbecomes negative.\nIndeed, the function Hνcould become again positive later, when }uptq}HąR1. This\npossibility makes the proof below more delicate, as several different cases will need\nto be considered.\n5.2.Hνbecomes negative at some time. The main technical result in this case\nis the following one.\nProposition 5.4. Assume that there exists t1PIm0so thatHνpt1q ď0, withm0\nsufficiently large. Denote the right end-point of ˜Im0byT1. ThenHνptqis decreasingLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 35\nand thusHνptq ă0onpt1,T1s. Furthermore, if νis sufficiently close to 1, then\nżT1\nt1}utpsq}L2dsďCt´apν´1q\n1, (5.17)\nwhereCis a positive absolute constant (independent of m0).\nWe remark that T1always lies to the right of Im0. The proof of Proposition 5.4\nis quite involved, so we first show how to deduce T1“ 8from it. This is the key\nstep in the proof of Theorem 1.1. To this end, fix a small positive constant δ0ďρ0\n8\n(the precise condition will be specified later). By Proposit ion4.11, there exists m0,\nso that, for měm0, we have,\n}pQ,0q ´/vector upsq}Hďδ0\n2ďρ0\n16,@sPIm. (5.18)\nWe also choose m0large enough so that, for tět0”n3{2\nm0, the non-increasing\nproperty ( 5.14) of Lemma 5.3holds, as long as tět0and provided that }/vector uptq}Hď\nR1.\nCorollary 5.5. Supposeδ0ą0is sufficiently small and m0is large. Then under\nthe conditions of Proposition 5.4,\n}/vector uptq ´ pQ,0q}Hďρ0\n2, (5.19)\nfor all times tět1. In particular, T1“ 8.\nProof. We choose m0large so that\nżT1\nt1}utpsq}L2dsďCt´apν´1q\n1 ďδ0\n2, (5.20)\n}upt1q ´Q}L2ďδ0\n2. (5.21)\nBy Corollary 4.10for alltP rt1,T1s,\n}/vector uptq ´ pQ,0q}Hďρ0\n2, (5.22)\nifδ0ą0is chosen small enough.\nRecall that rt1,T1sis the maximal interval on which }/vector uptq}HďR1. The property\n(5.22) says that on this same time interval, cf. Definition 5.1,\n}/vector uptq}Hďρ0\n2`R0{4ăR0{2ăR1{16,36 N. BURQ, G. RAUGEL, W. SCHLAG\nwhich leads to a contradiction if T1ă 8. Thus we have shown that, if there exists\nt1PIm0so thatHνpt1q ă0, then/vector uptq,tět1never leaves the ball BHppQ,0q,R1{16q\nand thus that, for tět1, the estimate ( 5.22) holds for all times. /square\nProof of Proposition 5.4.IfHνptq ď0, then\n0ăEp/vector uptqq ´EppQ,0qq ď ´ε0\np1`tqaνx´∆uptq `uptq ´fpuptqq,utptqyH´1.(5.23)\nTherefore on this time interval, we have the following two ki nds of inequalities\n0ăEp/vector uptqq ´EppQ,0qq ď2ε0\np1`tqapν`1q}utptq}2\nH´1`ε0\n2p1`tqaνB\nBt}utptq}2\nH´1,\n0ăEp/vector uptqq ´EppQ,0qq ďε0C2\np1`tqaν}utptq}H´1ďε0C2\np1`tqaν}utptq}L2,\n(5.24)\nfor someC2“C2pR1q ą0. The estimates in the first line follows from ( 5.23) and\nthe PDE, whereas the second line follows from ( 5.23) and the estimates ( 5.9) and\n(5.10). To bound the time integral in Proposition 5.4, we introduce the positive\nfunctional\nGptq “Ep/vector uptqq ´EppQ,0qq, (5.25)\nthe derivative of which is\nG1ptq “ ´2αptq}utptq}2\nL2. (5.26)\nPositivity of Gis guaranteed since /vector uis not stationary by assumption. The second\nline of ( 5.24) and the equality ( 5.26) imply that Gsatisfies the differential inequality\nG1ptq `C3p1`tqap2ν´1qG2ptq ď0 (5.27)\nonrt1,T1s, whereC3“2pε0C2q´2. Integrating this inequality, we find that, for\ntP rt1,T1s,\nGptq ď´\nG´1pt1q `C3\nap2ν´1q `1rp1`tqap2ν´1q`1´ p1`t1qap2ν´1q`1s¯´1\n,(5.28)\nand also, by using the inequalities ( 5.24) and ( 5.18),\nGptq ď´8p1`t1qaν\nC4ρ0`C3\nap2ν´1q `1rp1`tqap2ν´1q`1´ p1`t1qap2ν´1q`1s¯´1\nď´p1`t1qaν\nC4R1`C3\nap2ν´1q `1rp1`tqap2ν´1q`1´ p1`t1qap2ν´1q`1s¯´1\n,(5.29)\nwhereC4“ε0C2.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 37\nWe next estimate the quantityşt`τ\nt}utpsq}L2ds, for anytě0,τě0. From ( 5.26)\nand the Cauchy-Schwarz inequality,\nżt`τ\nt}utpsq}L2dsď`żt`τ\ntp1`sqads˘1{2`\n2żt`τ\ntαpsq}utpsq}2\nL2ds˘1{2\nďτ1{2p1`t`τqa{2pGptq ´Gpt`τqq1{2\nďτ1{2p1`t`τqa{2Gptq1{2.(5.30)\nIn a first step, we shall assume that t1ďtďT1and use ( 5.29) to control Gptq. Note\nthat we do not require that t`τďT1. Indeed, the value of Gpt`τqplays no role\nin the previous estimate, so ( 5.29) only enters in the estimation of Gptq.\n5.2.1. The case 2t1ďtďT1.We deduce from ( 5.29) and ( 5.30) that,\nż2t\nt}utpsq}L2dsď2a{2pap2ν´1q `1q1{2t1{2p1`tqa{2\nC1{2\n3p1`tqpap2ν´1q`1q{2r1´ p1`t1\n1`2t1qap2ν´1q`1s1{2\nďC5pap2ν´1q `1q1{2p1`tq´apν´1q\nr1´ p1`t1\n1`2t1qap2ν´1q`1s1{2.(5.31)\nWithout loss of generality, we may assume that we have chosen m0large enough so\nthatt0”nγ\nm0,γ“3\n2, satisfies\n1`t1\n1`2t1ď1`t0\n1`2t0ď3\n4.\nThus, we deduce from the estimate ( 5.31) that\nż2t\nt}utpsq}L2dsďC6p1`tq´apν´1q, (5.32)\nwhereC6“2C5pap2ν´1q `1q1{2. LetC7“maxpC5,C6q. Sinceνą1andaą0,\nthe estimates ( 5.31) and ( 5.32) imply that\nżT1\n2t1}utpsq}L2dsď8ÿ\nn“1C7\n2anpν´1qtapν´1q\n1ďC7\ntapν´1q\n1 p1´2´apν´1qqďC8t´apν´1q\n1.(5.33)\n5.2.2. The caset1ďtďminp2t1,T1q.We setT2:“minp2t1,T1q. Going back to the\ninequality ( 5.24), we now exploit the inequality on the first line. To do so, we n eed\nto take the sign ofd\ndt}utptq}2\nH´1into account.\nCase 1 }utptq}H´1is nondecreasing on some interval rt1,t1`τs Ă rt1,T2s, withτą0.\nIn particular,d\ndt}utptq}H´1ě0for allt1ďtďt1`τ. We now maximize τwith\nthis property. I.e., t˚“t1`τďT2is maximal with the property that }utptq}H´138 N. BURQ, G. RAUGEL, W. SCHLAG\nis nondecreasing on rt1,t˚s. Note that if t˚ăT2, then there exists a decreasing\nsequence tsku8\nk“1inrt˚,T2swith\nd\ndt}utpskq}H´1ă0“d\ndt}utpt˚q}H´1, s kÑt˚. (5.34)\nWe remark that we cannot claim that }utpt˚q}H´1is a maximum, since }utptq}H´1\nmight oscillate to the right of t“t˚. Let us recall that\n}v}H´1ď }v}L2,@vPL2,\nas can be seen by the Fourier transform and Plancherel. Indee d,\n}v}H´1“ }p1`4π2|ξ|2q´1\n2ˆvpξq}L2ď }ˆvpξq}L2“ }v}L2.\nFrom the property ( 5.24), we deduce that\n2żt˚\nt1αpsq}utpsq}2\nL2dsďε0C1\np1`t1qaν}utpt1q}H´1. (5.35)\nThus, there exists 0ďτ1ďt˚´t1so that\n2αpt1`τ1q}utpt1`τ1q}2\nL2ďε0C1\npt˚´t1qp1`t1qaν}utpt1q}H´1,\nand hence\n}utpt1`τ1q}2\nL2ďε0C1\npt˚´t1qp1`t1qapν´1q}utpt1q}H´1. (5.36)\nSince by our nondecreasing assumption on t1,\n}utpt1q}H´1ď }utpt1`τ1q}H´1, (5.37)\nit follows from ( 5.36) and ( 5.37) that\n}utpt1q}2\nH´1ď }utpt1`τ1q}2\nL2ďε0C1\npt˚´t1qp1`t1qapν´1q}utpt1q}H´1,\nand\n}utpt1q}H´1ďε0C1\npt˚´t1qp1`t1qapν´1q. (5.38)\nUsing again the inequality ( 5.24), we deduce from ( 5.38) and ( 5.35) (replacing t˚on\nthe left-hand side by 8) that\n2ż8\nt1αpsq}utpsq}2\nL2dsďε2\n0C2\n1\npt˚´t1qp1`t1qap2ν´1q. (5.39)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 39\nLetℓbe a fixed positive number, which will made more precise later . It follows from\n(5.39) that\nżt˚`ℓ\nt1}utpsq}L2dsď2a{2Cpℓqpt˚´t1q1{2p1`t1qa{2´ż8\nt1αpsq}utpsq}2\nL2ds¯1{2\nďε0C1Cpℓq\np1`t1qapν´1q.(5.40)\nSince we may take t1larger than any fixed time, it follows from ( 5.40) and Corol-\nlary4.10thatT1, and therefore T2, cannot be arbitrarily close to t1. More precisely,\nfor any fixed Lą0we can assume that T1ěT2ět1`L.\nIt remains to estimate the termşT2\nt˚}utpsq}L2dswhent˚`ℓďT2. We set\nℓ““1`2apν´1q\naν‰\n`1, (5.41)\nwhere“\nx‰\ndenotes the integral part of x. We will show by recursion that there exists\na finite sequence of numbers 0ď˜τjď1,j“1,2,...,ℓ ´1, such that\nż8\nt˚`řj“ℓ´1\nj“1˜τj1\np1`sqa}utpsq}2\nL2dsďεℓ\n022pℓ´1qa\np1`t˚qapν`1q`pℓ´1qaν}utpt˚q}2\nH´1. (5.42)\nFrom the first line of ( 5.24) and the vanishing in ( 5.34), we deduce that\nż8\nt˚1\np1`sqa}utpsq}2\nL2dsďε0\np1`t˚qapν`1q}utpt˚q}2\nH´1. (5.43)\nHence, there exists τ1with0ďτ1ď1such that\n}utpt˚`τ1q}2\nL2ďε0p1`t˚`τ1qa\np1`t˚qapν`1q}utpt˚q}2\nH´1ďε02a\np1`t˚qaν}utpt˚q}2\nH´1.(5.44)\nIfd\ndt}utpt˚`τ1q}2\nH´1ď0, we set ˜τ1“τ1. If this is not the case, we may choose\n0ă˜τ1ăτ1so thatt˚`˜τ1is the closest time on the left of t˚`τ1so thatd\ndt}utpt˚`\n˜τ1q}2\nH´1ď0(in fact, one then has “0here). The existence of ˜τ1follows from the\nintermediate value theorem in view of the sequence skin (5.34). Consequently,\n}utpt˚`˜τ1q}H´1ă }utpt˚`τ1q}H´1ď }utpt˚`τ1q}L2. (5.45)40 N. BURQ, G. RAUGEL, W. SCHLAG\nApplying again the property ( 5.24), we obtain,\nż8\nt˚`˜τ11\np1`sqa}utpsq}2\nL2dsďε0\np1`t˚`˜τ1qapν`1q}utpt˚`˜τ1q}2\nH´1\nďε0\np1`t˚`˜τ1qapν`1q}utpt˚`τ1q}2\nL2\nďε2\n02a\np1`t˚`˜τ1qapν`1qp1`t˚qaν}utpt˚q}2\nH´1,(5.46)\nwhere the final estimate uses ( 5.44). Therefore, there exists τ2with0ďτ2ď1such\nthat\n}utpt˚`˜τ1`τ2q}2\nL2ďε2\n022a\np1`t˚`˜τ1qaνp1`t˚qaν}utpt˚q}2\nH´1. (5.47)\nWe proceed as before. Ifd\ndt}utpt˚`˜τ1`τ2q}2\nH´1ď0, we set˜τ2“τ2. If this is not the\ncase, we may choose 0ď˜τ2ăτ2ď1so thatt˚`˜τ1`˜τ2is the closest time on the\nleft oft˚`˜τ1`τ2so thatd\ndt}utpt˚`˜τ1`˜τ2q}2\nH´1ď0. It is important to note that\nwe allow ˜τ2“0at this step. Thus, using in view of ( 5.24), and ( 5.47) we obtain,\nż8\nt˚`˜τ1`˜τ21\np1`sqa}utpsq}2\nL2ds\nďε0\np1`t˚`˜τ1`˜τ2qapν`1q}utpt˚`˜τ1`˜τ2q}2\nH´1\nďε0\np1`t˚`˜τ1`˜τ2qapν`1q}utpt˚`˜τ1`τ2q}2\nL2\nďε3\n022a\np1`t˚`˜τ1`˜τ2qapν`1qp1`t˚`˜τ1qaνp1`t˚qaν}utpt˚q}2\nH´1.(5.48)\nArguing by recursion, we finally prove that there exists a seq uence of numbers\n0ď˜τjď1,j“1,2,...,ℓ ´1, such that\nż8\nt˚`ℓ´11\np1`sqa}utpsq}2\nL2dsďż8\nt˚`řj“ℓ´1\nj“1˜τj1\np1`sqa}utpsq}2\nL2ds\nďεℓ\n022pℓ´1qa\np1`t˚qapν`1q`pℓ´1qaν}utpt˚q}2\nH´1\nďC˚pℓqR2\n1\np1`t˚qℓaν`a,(5.49)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 41\nwhereC˚pℓqis a positive constant depending on ℓonly. It follows from ( 5.49) that\nżT2\nt˚`ℓ}utpsq}L2dsď p2t1q1{2p1`2t1qa{2´ż8\nt˚`ℓ´11\np1`sqa}utpsq}2\nL2ds¯1{2\nďC˚˚pℓqR1\np1`t1qapν´1q,(5.50)\nwhereC˚˚pℓqis a positive constant depending on ℓonly. The final estimate here\nuses the definition ( 5.41).\nThe inequalities ( 5.33), (5.40), and ( 5.50) imply that\nżT1\nt1}utpsq}L2dsďC˚\n8t´apν´1q\n1, (5.51)\nwhereC˚\n8ą0is a positive constant (independent of m0).\nCase 2 }utptq}H´1fails to be nondecreasing on any interval rt1,t1`τs Ă rt1,T2s,\nwithτą0.\nThis means that either (i)d\ndt}utpt1q}H´1ă0or (ii)d\ndt}utpt1q}H´1“0. In the latter\ncase, there exists a decreasing sequence skÑt1as in ( 5.34). Clearly, this sequence\nalso exists if we have negativity as in (i). Setting t˚“t1we may therefore assert\nthat ( 5.50) holds with t˚“t1. Indeed, the recursive procedure that lead to this\nbound did not require vanishing in ( 5.34) att˚, but only nonpositivity. So the same\nargument goes through here, too. It remains to establish the analogue of ( 5.40) for\nthe integral over rt1,t1`ℓs, i.e.,\nżt1`ℓ\nt1}utpsq}L2dsďε0C1Cpℓq\np1`t1qapν´1q. (5.52)\nSinced\ndt}utpt1q}H´1ď0, we will apply the first inequality in ( 5.24), which implies\nthatż`8\nt1αpsq}utpsq}2\nL2dsď2ε0CpR1q\np1`t1qapν`1q,\nand thus, by using a Hölder inequality, we obtain,\nżt1`ℓ\nt1}utpsq}L2dsď´żt1`ℓ\nt1p1`sqads¯1{2´żt1`ℓ\nt1αpsq}utpsq}2\nL2ds¯1{2\nďp2ℓε0CpR1qq1{2p1`t1`ℓqa{2\np1`t1qapν`1q{2ďCpℓ,R1qp1`t1q´aν{2\nďCpℓ,R1qp1`t1q´apν´1q,(5.53)\nsince we could choose νą1close to1(choose1ăνď2so thataνă1{3)./square42 N. BURQ, G. RAUGEL, W. SCHLAG\n5.3.Hνě0and the Łojasiewicz-Simon inequality. In this section we assume\nthatHνpt0q ą0, wheret0“nγ\nm0`1, withγ“3\n2, and also that Hνptq ą0on the\nmaximal time interval rt0,T1son which }/vector uptq}HďR1. We recall that the functional\nHνcan be written as\nHνptq “Jpuptqq ´JpQq `1\n2}utptq}2\nL2`ε0\np1`tqaνx´∆uptq `uptq ´fpuptqq,utptqyH´1.\n(5.54)\nTheorem A.1of Appendix A implies that, as long as }uptq ´Q}H1ďρ0and in\nparticular, as long as }/vector uptq ´ pQ,0q}Hďρ0, we have\nHνptq ďC0} ´∆uptq `uptq ´fpuptqq}2\nH´1`1\n2}utptq}2\nL2\n`ε0\n2p1`tqaν`\n} ´∆uptq `uptq ´fpuptqq}2\nH´1` }utptq}2\nL2˘\nďC1`\n} ´∆uptq `uptq ´fpuptqq}2\nH´1` }utptq}2\nL2˘\n.(5.55)\nThis implies in particular that Hνptq ďCpR1qfor alltP rt0,T1s. From the estimates\n(5.14) and ( 5.55), we deduce that, for tět0, as long as }/vector uptq ´ pQ,0q}Hďρ0, we\nhave\n´H1\nνptq ěC2ε0\np1`tqaνHνptq, (5.56)\nor also\nH1\nνptq `C2ε0\np1`tqaνHνptq ď0. (5.57)\nWe now set\nt˚\n0“supttět0|}/vector upsq ´ pQ,0q}Hďρ0,@sP rt0,tsu. (5.58)\nBy the choice of parameters, t˚\n0ăT1. We recall that we have chosen t0“nγ\nm0`1,\nsee (5.18). Hence\n}pQ,0q ´/vector upsq}Hďδ0\n2ďρ0\n16,@sPIm0.\nThis implies that t0ď pnm0`1qγăt˚\n0. Applying Lemma 4.2, we can also choose\nm0so thatżpnm0`1qγ\nnγ\nm0}utpsq}L2dsďδ0\n2ďρ0\n16. (5.59)\nNote that\npnm0`1qγ´nγ\nm0„n1\n2m0„t1\n3\n0. (5.60)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 43\nNote that the differential inequality ( 5.57) is only useful provided Hνptq ą0. Inte-\ngrating ( 5.57) yields, for t0ďtďt˚\n0,\nHνptq ďHνpt0qexp´\n´żt\nt0C2ε0\np1`sqaνds¯\nďHνpt0qexp´\n´C2ε0\n1´aνrp1`tq1´aν´ p1`t0q1´aνs¯\n.(5.61)\nIn view of ( 5.14), (5.15)\nżp1`βqt\nt3\n2p1`sqa}utpsq}2\nL2dsďHνptq ´Hνpp1`βqtq, (5.62)\nfort0ďtď p1`βqtďt˚\n0, where0ăβď1. Thus, since Hνptq ě0ont0ďtďT1,\nwe infer that\nżp1`βqt\nt1\np1`sqa}utpsq}2\nL2dsď2Hνpt0q\n3exp´\n´C2ε0\n1´aνrp1`tq1´aν´ p1`t0q1´aνs¯\n.\n(5.63)\nBy Cauchy-Schwarz, for all t0ďtď p1`βqtďt˚\n0,\nżp1`βqt\nt}utpsq}L2dsď´żp1`βqt\ntp1`sqads¯1{2´żp1`βqt\nt1\np1`sqa}utpsq}2\nL2ds¯1{2\nďβ1{2t1{2p1`tqa{2Hνpt0q1{2exp´\n´C2ε0\n2p1´aνqrp1`tq1´aν´ p1`t0q1´aνs¯\n.(5.64)\nIft“t0`τwhere0ăτďt0, we see that\nexp´\n´C2ε0\n2p1´aνqrp1`tq1´aν´ p1`t0q1´aνs¯\n“exp´\n´τC2ε0\n2p1`t1\n0qaν¯\n,\nfor somet0ďt1\n0ďt. For the right-hand side to be small we need τětaν`η\n0where\nηą0. Thus, since aνă1\n3by our assumptions, we fix aν`η“1\n3. In view of ( 5.59)\nand (5.60) the interval t´t0ďt1\n3\n0has already been dealt with.44 N. BURQ, G. RAUGEL, W. SCHLAG\nLet us now take tďt˚\n0so thatt0`t1\n3\n0ďtď2t0and sett“t0`t1\n3\n0`τ. Then the\nestimate ( 5.64) implies that\nżt0`t1\n3\n0`τ\nt0`t1\n3\n0}utpsq}L2ds\nď2t1{2\n0p1`t0qa{2Hνpt0q1{2exp´\n´C2ε0\n2p1´aνqrp1`t0`t1\n3\n0q1´aν´ p1`t0q1´aνs¯\nď2t1{2\n0p1`t0qa{2Hνpt0q1{2exp´\n´C2ε0t1\n3\n0\n2p1`t0qaν¯\n.(5.65)\nSinceHνis uniformly bounded on rt0,T1sbyCpR1q, the right-hand side of the\nestimate ( 5.65) tends to 0ast0goes to infinity. Therefore, we choose m0(and thus\nt0) large enough so that\nżt0`t1\n3\n0`τ\nt0`t1\n3\n0}utpsq}L2dsďC9t´µ\n0ďδ0\n2ďρ0\n16, (5.66)\nwhereµą0. If2t0ďtďt˚\n0, we deduce from ( 5.64) that\nż2t\nt}utpsq}L2ds\nď2t1{2p1`tqa{2Hνpt0q1{2exp´\n´C2ε0\n2p1´aνqp1`tq1´aν`\n1´ p1`t0\n1`2t0q1´aν˘¯\n.\n(5.67)\nAs in the analysis of negative Hνabove, we may choose m0large enough so that\n1`t0\n1`2t0ď3\n4.\nThen, the inequality ( 5.67) implies that\nż2t\nt}utpsq}L2dsď2t1{2p1`tqa{2Hνpt0q1{2exp´\n´C2ε0\n8p1´aνqp1`tq1´aν¯\n.(5.68)\nChoosingm0(that ist0) large enough, which in turn implies that tis large enough,\nwe deduce from the estimate ( 5.68) that\nż2t\nt}utpsq}L2dsďC10p1`tq´µ. (5.69)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 45\nAs in ( 5.33), the estimates ( 5.66) and ( 5.69) imply that, for m0large enough, we\nhaveżt˚\n0\nt0`t1\n3\n0}utpsq}L2dsď8ÿ\nn“0C11\n2nµtµ\n0ďC11\ntµ\n0p1´2´µqďC12t´µ\n0, (5.70)\nwhereC11“maxpC9,C10q. Choosing m0larger if necessary, we can ensure that\nżt˚\n0\nt0`t1\n3\n0}utpsq}L2dsďC12t´µ\n0ďδ0\n2. (5.71)\nBy Corollary 4.10, we conclude that for all tP rt0`t1\n3\n0,t˚\n0s,\n}/vector uptq ´ pQ,0q}Hďρ0\n2, (5.72)\nifδ0ą0is chosen small enough. The property ( 5.72) leads to a contradiction if\nt˚\n0ă 8. Thust˚\n0“T1“ 8.\n5.3.1. The case where Hνptqchanges sign on the interval rpnm0`1qγ,T1s.The only\nremaining scenario left to deal with is as follows: Hνptqis never negative on Im0, but\ndoes not remain positive on all of ˜Im0(the extension of Im0for which/vector ustays in the\nR1ball inH). Recall that Hνdecreases on that larger interval. Say Hνpt2q “0with\nt2P˜Im0zIm0. If so, then ( 5.72) holds up to time t2. After that time, Proposition 5.4\napplies with t2taking the role of t1. The only reason that t1PIm0in that proposition\nlies with the fact that it guarantees the initial closeness ( 5.18) to pQ,0q. But this\nis now provided by ( 5.72). Hence, the argument following Proposition 5.4applies\nunchanged and /vector uremains close to the equilibrium for all times, cf. ( 5.22).\n5.4.Conclusion of the proof for dą3ord“3andθď4.Up to this point\nwe have only proved that the trajectory /vector uptqis bounded in Hand stays forever in\nthe ballBHp0,R1qfortět0, witht0large enough. But, taking into account that\nHνptnq Ñ0for some sequence tnÑ 8, we also have shown that, for t0large enough\nHνdoes not become negative. Moreover, we claim that proof in th e case of positive\nHνyields that /vector uptq ´ pQ,0qconverges to 0at a sub-exponential rate. Indeed, since\nthe trajectory /vector uptqstays forever in the ball BHp0,R1qfortě2t0, witht0large\nenough, we may choose ν“1and we obtain by arguing as in ( 5.62) that, for any\nτąt,żτ\nt3\n2p1`sqa}utpsq}2\nL2dsďH1ptq, (5.73)\nand thus, for tlarge enough,\nż`8\nt1\np1`sqa}utpsq}2\nL2dsďKpt0qexp´\n´C2ε0\n1´ap1`tq1´a¯\n, (5.74)46 N. BURQ, G. RAUGEL, W. SCHLAG\nwhereKpt0qis a positive constant. From ( 5.74), we deduce that, for tě2t0,\nEp/vector uptqq ´EpQ,0q ďCKpt0qexp´\n´C2ε0\n1´ap1`tq1´a¯\n. (5.75)\nNext, arguing as in ( 5.67) to (5.70), we obtain that, for tě2t0,\nż`8\nt}utpsq}L2dsďK˚pt0qt´µexp`\n´kp1`tq1´a˘\n. (5.76)\nThe inequality ( 5.74) implies that, for sątą2t0, we have\n}upsq ´uptq}L2ďK˚pt0qt´µexp`\n´kp1`tq1´a˘\n. (5.77)\nSince we alreay know that the sequence uptnqconverges to Q, whenngoes to infinity\n(see Theorem 1.3), we may pass to the limit in s“tnto wit\n}uptq ´Q}L2ďK˚pt0qt´µexp`\n´kp1`tq1´a˘\n. (5.78)\nFinally, from Lemma 4.9and (4.45) we conclude that, for tě2t0,\n}/vector uptq ´ pQ,0q}HďK1pt0qexp`\n´ηkp1`tq1´a˘\n, (5.79)\nwhereηą0depends only on the power of the non-linearity. We have there fore\nestablished sub-exponential convergence of the entire tra jectory to an equilibrium.\n6.Proof of Theorem 1.1ifd“3and4ăθă5\n6.1.Applying observation inequalities. Throughout this section, we let d“3\nand assume that the Hypothesis pH.2qfholds, with 4ăθă5.\nThe difficulty with d“3and these values of θlies with the fact that Lemma 5.2\nfails in that case. It is not possible to make an estimate such as (5.2) pointwise in\ntime. Rather, we will bound an averaged expression of the lef t-hand side of ( 5.2)\nby means of Strichartz estimates and the following observat ion inequality from a\ncompanion paper [ 5].\nProposition 6.1. Letube a solution of pKGqαon the interval I“ rt0,t1swith\n}/vector uptq}HďMfortPI. There exists C“CpM,|I|qso that\n}Btu}L8pI,L2qďCpM,|I|q}Btu}L2pI,L2q. (6.1)\nBy inspection, CpM,|I|qis decreasing in |I|. This allows us to establish the\nfollowing space-time bound analogous to Lemma 5.2.\nLemma 6.2. Letube a solution of pKGqαon the interval rt,t`τswhere0ăτď1\nwith }/vector upsq}HďMfor alltďsďt`τ. Then\nżt`τ\nt|xf1puqut,utyH´1|dsďCpM,τ qżt`τ\nt}ut}2\nL2ds, (6.2)LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 47\nwhere the constant CpM;τq ą0depends on M,τand the nonlinearity f.\nProof. By the embedding H2pR3qãÑL8pR3qand its dual L1pR3qãÑH´2pR3qwe\nmay estimate\nżt`τ\nt|xf1pupsqqutpsq,utpsqyH´1|ds\nďCżt`τ\nt}f1pupsqqutpsq}H´2}utpsq}2ds\nďCżt`τ\nt}p|upsq|β` |upsq|θ´1qutpsq}L1}utpsq}2ds\nďC}ut}L8pI,L2qżt`τ\nt´\n}upsq}β\nL2β` }upsq}θ´1\n2pθ´1q¯\n}utpsq}L2ds,(6.3)\nwhereI“ rt,t`τs. Due to the hypotheses made on β, we may boundşt`τ\nt´\n}upsq}β\nL2β`\n}upsq}θ´1\n2pθ´1q¯\n}utpsq}L2dsbyşt`τ\nt´\n1` }upsq}L2` }upsq}θ´1\n2pθ´1q¯\n}utpsq}L2ds. Bounding\nthe first term by Proposition 6.1we may continue bounding the above expression\nby\nďCż\nI}utpsq}2\n2dsżt`τ\nt´\n1` }upsq}L2` }upsq}θ´1\n2pθ´1q¯\nds\nďCpM,τ qż\nI}utpsq}2\n2ds.(6.4)\nTo pass to the final inequality we bound }upsq}2ďMand we use Strichartz estimates\nto controlżt`τ\nt}upsq}θ´1\n2pθ´1qdsďCpM,τ q, (6.5)\nwhich follows by local wellposedness, via the assumption }/vector upsq}HďM. Indeed,\nfor the endpoint θ“5one has an L4\ntprt,t`δs,L8\nxpR3qqStrichartz estimate with\nδ“δpMq ą0and forθ“4we control the L3prt,t`δs,L6\nxpR3qqnorm of the\nsolution. We can then cover the interval rt,t`τswithδ-intervals. It now follows\nthat we can similarly handle the entire range 4ăθă5. /square\nIn the previous proof we applied the observation inequality twice, the second time\nto pass from ( 6.3) to (6.4). This was strictly speaking not necessary, as we could\nhave bounded ( 6.3) by means of Cauchy-Schwarz. This would then lead to an L8\nt,x\nStrichartz estimate for Klein-Gordon, which is precisely t he optimal one on the scale\nofL8\nxpR3qfor the wave equation with energy data.48 N. BURQ, G. RAUGEL, W. SCHLAG\nIn analogy with Lemma 5.3we can use the previous lemma to obtain decay of\nHν, but we can only compare times which are not closer than τ0. Recall that\nHνptq:“Ep/vector uptqq ´EpQ,0q `ε0\np1`tqaνx´∆uptq `uptq ´fpuptqq,utptqyH´1.\nFor the remainder of this section, we fix some τ0P p0,1s, sayτ0“1\n2(the constants\nare then uniform in1\n2ďτ0ď1). We remark that ( 6.2) holds for any τ0ě1\n2with\nthe same constant, provided that we remain in the interval of existence (but we are\ninterested in global solutions only here). This follows by a dding up the individual\nestimates ( 6.2) over intervals of size1\n2. As in the previous section, we take 0ăaă1\n3\nandνą1fixed.\nCorollary 6.3. Letuptqbe a solution of pKGqαfort0ďtďt1wheret1ět0`1.\nAssume }/vector uptq}HďMfor allt0ďtďt1. Then for t0ět˚\n0“t˚\n0pMqone has\nHνpt1q ´Hνptq ď0 @t0ďtďt`1ďt1ďt1 (6.6)\nżt1\nt3\n2p1`sqa}ut}2\nL2dsďHνptq ´Hνpt1q. (6.7)\nThe inequality in (6.6)is strict if/vector u‰ pQ,0q.\nProof. Arguing by density, we first assume that the data /vector up0q PH2pR3q ˆH1pR3q.\nThen the solution /vector uptqis classical, i.e.,\nuPC1pr0,`8q,H1pR3qq XC0pr0,`8q,H2pR3qq XC2pr0,`8qq,L2pR3qq\nandH1\nνsatisfies the equality ( 5.12). Hence, for t,t1PJ:“ rt0,t1sas above,\nHνpt1q ´Hνptq “żt1\ntH1\nνpsqds\n“żt1\nt”\n´2αpsq}utpsq}2\nL2´νaε0\np1`sqaν`1x´∆upsq `upsq ´fpupsqq,utpsqyH´1\n´ε0\np1`sqaν} ´∆upsq `upsq ´fpupsqq}2\nH´1\n´2ε0\np1`sqapν`1qx´∆upsq `upsq ´fpupsqq,utpsqyH´1\n`ε0\np1`sqaνx´∆utpsq `utpsq ´f1pupsqqutpsq,utpsqyH´1ı\nds.LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 49\nApplying Lemma 6.2and the Cauchy-Schwarz inequality (in space, not time, cf. L emma 5.3),\none infers that there exists some constant C1“C1pMqso that\nHνpt1q ´Hνptq ďżt1\nt”\n´p2´ε0C1p1`sq´apν´1qq\np1`sqa}ut}2\nL2\n´ε0\n2p1`sqaν} ´∆u`u´fpuq}2\nH´1ı\ndsď0,(6.8)\nand forsět0large,\n2´ε0C1p1`sq´apν´1qě3{2. (6.9)\nUsing a density argument, one shows that the above estimates (6.8) still hold when\nthe initial data only belong to H. /square\nWe remark that we may also take ε0small rather than taking t0large. For\nexample, if the power of the nonlinearity is at most 3, then we can set ν“1which\nthen requires making ε0small. Next, we combine Lemma 6.2with the Łojasiewicz-\nSimon inequality. The small parameter ρ0ą0is from Theorem A.1.\nCorollary 6.4. Assumeuis a solution of pKGqαon the interval rt0,t1ssatisfying\n}/vector upsq ´ pQ,0q}Hďρ0, Hνpsq ą0, @t0ďsďt1.\nThen\nHνpt1q ďHνptqexp´\n´żt1\ntC2ε0\np1`sqaνds¯\n, (6.10)\nfor allt0ďtďt`1ďt1ďt1, whereC2is a constant that depends on Q.\nProof. Using Theorem A.1we obtain the inequality ( 5.55), i.e.,\nHνpsq ďC1`\n} ´∆upsq `upsq ´fpupsqq}2\nH´1` }utpsq}2\nL2˘\n.\nWe next choose a constant C2,0ăC2ă1\n8C1so that\nC2ε0Hνpsq ďε0\n8`\n} ´∆upsq `upsq ´fpupsqq}2\nH´1` }utpsq}2\nL2˘\n. (6.11)\nTherefore, with Ωpsq “Ωpt,sq “şs\ntC2ε0\np1`uqaνdu,\neΩpt`τ0qHνpt`τ0q ´Hνptq “żt`τ0\ntd\nds”\neΩpsqHνpsqı\nds\n“żt`τ0\nteΩpsq“C2ε0\np1`sqaνHνpsq `H1\nνpsq‰\ndsď0.(6.12)50 N. BURQ, G. RAUGEL, W. SCHLAG\nThe final inequality holds due to\nżt`τ0\nteΩpsqH1\nνpsqdsďżt`τ0\nteΩpsq”\n´3\n2p1`sqa}utpsq}2\nL2\n´ε0\n2p1`sqaν} ´∆upsq `upsq ´fpupsqq}2\nH´1ı\ndsď0,(6.13)\nwhich follows by basically the same proof as that of Corollar y6.3(note thateΩpsqď\nCpτ0qfor alltďsďt`τ0). We infer from ( 6.12) that\nHνpt`τ0q ďHνptqexp´\n´żt`τ0\ntC2ε0\np1`sqaνds¯\n. (6.14)\nIterating this estimate along an arithmetic sequence t`jτ0, with a suitably chosen\n1\n2ďτ0ď1, concludes the proof. /square\n6.2.The main argument. We now indicate how to adapt the proof technique of\nTheorem 1.1from Section 5to fit this section. First, we deal with the case in which\nHνcan assume negative values in the interval Im0, cf. Proposition 5.4. Reflecting\nthe fact that monotonicity of Hνnow holds in a slightly weaker sense, we need to\nmodify that proposition accordingly.\nProposition 6.5. Assume that there exists t1PIm0withHνpt1q ď0. Let rt1,T1s\nbe the maximal time interval on which }/vector uptq}HďR1. ThenHνď0onrt1`1,T1s\nprovidedm0is large enough. In addition, if νis sufficiently close to 1, then\nżT1\nt1`1}utpsq}L2dsďCt´apν´1q\n1, (6.15)\nwhereCis a positive absolute constant (independent of m0).\nProof. By (6.6) we have that Hνptq ď0for alltP rt1`1,T1s. Then the exact same\nproof as that of Proposition 5.4gives the desired conclusion. /square\nTheorem 1.1follows from this result in the same fashion as in the previou s section.\nAs already noted there, T1lies arbitrarily far to the right of Im0ifm0is large.\nMoreover,/vector upt1`1qwill still satisfy ( 4.47) whence Corollary 5.5applies in this\ncontext.\nIf, on the other hand, t1as in the previous proposition does not exist, then due\nto Corollary 6.3the proof of Section 5.3applies verbatim, provided the assumption\nwhich we made there holds, i.e., that Hνą0on˜Im0. Finally, if this is not the\ncase thenHνpt1q “0for somet1P˜Im0zIm0. Then by the monotonicity properties\nderived in this section, Hνptq ă0for alltP˜Im0withtět1`1. Up to time t1,\nSection 5.3applies, and by well-posedness the closeness ( 5.72) remains correct forLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 51\nallt1ďtďt1`1(m0large). Hence, we can then apply the argument for negative\nHνto conclude as before.\n7.Proof of Theorem 1.2\nMutatis mutandis, we argue as in the proofs of Theorem 1.1in Sections 5 and\n6. The only changes are that we apply Lemma 4.3and Proposition 4.12instead of\nLemma 4.2and Proposition 4.11and that we choose νą1so thataνă1{2.\nAppendix A.The Łojasiewicz-Simon inequality\nIn this section we prove the following estimate which plays a n instrumental role\nin our main argument.\nTheorem A.1. Assume that the hypotheses pH.1qfand pH.2qfhold. Let pQ,0qbe\nan equilibrium point of pKGqαinHrad. Then there exist 0ăρ0ă1andC0ą0\nsuch that, for any uPBH1\nradpQ,ρ0q, we have\n|Jpuq ´JpQq| ďC0} ´∆u`u´fpuq}2\nH´1pRdq, (A.1)\nwhereJpuq “Epu,0q.\nResults of this type go back to Simon [ 37, Section 3.12, pages 74]. We will\ninvoke a somewhat more abstract formulation of Simon’s theo rem by Haraux and\nJendoubi [ 20]. To state it, let VandHbe two Hilbert spaces, with Vdense inH.\nIn addition, [ 20] requires the embedding VãÑHto be compact. But this is not\nnecessary as we will explain below. We identify Hwith its dual H˚, whenceHãÑV˚\nis bounded. Let GPC1pVqbe real-valued, and set M:“∇G. This means that\nxp∇Gqpvq,wy “d\ndtˇˇˇ\nt“0Gpv`twq,@v,w PV.\nThe pairing on the left-hand side is the duality pairing betw eenVandV˚. Thus,\nM:VÑV˚as a continuous map.\nTheorem A.2 (Simon, Haraux-Jendoubi) .AssumeMPC1pV,V˚q. LetφPVbe\nsuch that Mpφq “0. SupposeL“M1pφq, which is a bounded linear transformation\nVÑV˚, has the form\nL“Λ`B,\nwhereΛ :VÑV˚is an isomorphism, and B:VÑV˚is compact. Denote\nN“KerpLq, which is of dimension dă 8. Ifdą0we assume that M´1p0qnear\nφis a smooth d-dimensional manifold, which then has Nas tangent space at φ.\nUnder these assumptions, there exist δą0and a constant Csuch that\n@}u´φ}Hăδ,|Gpuq ´Gpφq| ďC}Mpuq}2\nV˚.52 N. BURQ, G. RAUGEL, W. SCHLAG\nProof. We present the beginning of the proof here in some detail sinc e we do not\nassume that VãÑHis compact. The case d“dimpNq “0is easy since we\nmay then invert Λby Fredholm’s theorem. If dą0, we letΦbe the orthogonal\nprojection in HontoN. Now define L:VÑV˚,L“Π`L. SinceΠhas finite\nrank, it is compact as a map VÑV˚. We do not need compactness of VãÑH\nfor this step. We claim that KerpLq “ t0u. If so, Fredholm’s theorem implies that\nL“Λ`Π`B:VÑV˚is an isomorphism.\nTo prove the claim we need to invoke the symmetry of L. This means that\n@v,w PV, xLv,w y “ xLw,v y.\nThe pairing on the left is again the duality pairing between VandV˚. This identity\nreflects the symmetry of the Hessian in calculus and follows f rom the fact that\nxLv,w y “ Btˇˇˇ\nt“0Bsˇˇˇ\ns“0Gpφ`sv`twq (A.2)\n“ Bsˇˇˇ\ns“0Btˇˇˇ\nt“0Gpφ`sv`twq “ xLw,v y. (A.3)\nNow suppose Lv“0whenceLv“ ´ΠvPHãÑV˚. Then\nxLv,Πvy “ ´xΠv,Πvy “ ´}Πv}2\nH.\nThe final equality here follows from Riesz representation an d density of VinH. On\nthe other hand,\nxLv,Πvy “ xLΠv,vy “0,\nsinceΠvPN“KerL. ThusΠv“0and soLv“0which means that vPN. But\nthenv“Πv“0and the claim holds.\nThe remainder of the proof is identical to Haraux-Jendoubi a nd we refer the reader\nto pages 452–55 in [ 20]. /square\nThe rest of this appendix is devoted to the proof of Theorem A.1. We first recall\nsome well-known facts. Let us consider the solutions QPH1\nradpRdqof the elliptic\nequation\n´∆Q`Q´fpQq “0. (A.4)\nBy elliptic theory, see for example [ 2], the solutions of ( A.4) are exponentially de-\ncaying, and lie in C3,bfor somebą0. We set\nL” ´∆`I´f1pQq.\nIn [4, Lemma 2.9], we have stated the following properties. The op eratorL:\nH2pRdq ĂL2pRdq ÑL2pRdqis self-adjoint with domain H2pRdq. The spectrum\nσpLqconsists of an essential part r1,8q, which is absolutely continuous, and finitelyLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 53\nmany eigenvalues of finite multiplicity all of which fall int op´8,1s. The eigenfunc-\ntions areC2,b˚withb˚ą0and the ones associated with eigenvalues below 1are\nexponentially decaying. Over the radial functions, all eig envalues are simple. Notice\nthatLcan be extended to a continuous operator from H1pRdqintoH´1pRdq(as well\nas fromH1\nradpRdqintoH´1\nradpRdq).\nIn the case where QPH1\nradpRdqis a hyperbolic solution in H1\nradpRdqof the ellip-\ntic equation ( A.4), the proof of Theorem A.1is elementary and we recall it now.\nNote that this is precisely the case dimN“0which we skipped in the proof of\nTheorem A.2above.\nProof of (A.1)whenQis an hyperbolic equilbrium (in H1\nradpRdq).Letu“Q`v.\nBy Taylor’s formula,\nJpuq ´JpQq “ż\nRdr1\n2p|∇pv`Qq|2´ |∇Q|2` |v`Q|2´ |Q|2q\n´Fpv`Qq `FpQqsdx\n“1\n2}v}2\nH1´ż\nRdpFpv`Qq ´FpQq ´fpQqvqdx\n“1\n2}v}2\nH1´ż\nRdż1\n0p1´sqf1pQ`svqv2dsdx.(A.5)\nUsing the hypothesis pH.2qfand the fact that }v}H1ď1, we deduce from ( A.5) that\n|Jpuq ´JpQq| ďC}v}2\nH1. (A.6)\nNext, we write\n´∆u`u´fpuq “ ´∆v`v´ pfpv`Qq ´fpQqq (A.7)\n“Lv´ż1\n0pf1pQ`svq ´f1pQqqvds. (A.8)\nSinceLis an isomorphism from H1\nradpRdqintoH´1\nradpRdq, we infer from ( A.7) that\nv“L´1p´∆u`u´fpuqq ´L´1pż1\n0pf1pQ`svq ´f1pQqqvdsq. (A.9)\nUsing the continuous Sobolev embedding of Lq1pRdqintoH´1pRdq, whereq1“\n2d{pd`2q, together with Hypothesis pH.2qf, we obtain that\n›››ż1\n0pf1pQ`svq ´f1pQqqvds›››\nH´1\nďC`\n}v}β`1\nL2dpβ`1q{pd`2q` }v}θ\nL2dθ{pd`2q˘\nď˜C}v}1`β\nH1,(A.10)54 N. BURQ, G. RAUGEL, W. SCHLAG\nwhere0ăβăθ´1. Ifρ0ą0is chosen so that 1´˜Cρβ\n0ą0, we deduce from\nthe inequalities ( A.6) to (A.10), that there exists a positive constant C0so that the\ninequality ( A.1) holds.\n/square\nProof of (A.1)in the general case. We will apply Theorem A.2. To fix notation, we\nhaveH1\nradpRdq “V,H“L2\nradpRdqandV1“H´1\nradpRdq. The functional Gpuqis the\nenergy functional Jpuq. Clearly,\n∇Gpuq “Mpuq “ ´∆u`u´fpuq.\nIn our case, Mis inC1pH1\nradpRdq,H´1\nradpRdqqandM1pQq “L. We notice that\nLv“ ´∆v`v´f1pQqvand that ´∆`Iis an isomorphism from H1\nradpRdqinto\nH´1\nradpRdqand also from H2\nradpRdqintoL2\nradpRdq. To fulfill the hypothesis (ii) of\nTheorem 1.1 of [ 20], we need to prove that the operator of multiplication by ´f1pQq\nis a compact operator from H1pRdqintoH´1pRdqand even from H1pRdqintoL2pRdq.\nLetRą1. Using Hypothesis pH.2qf, we can write, for any vPH1pRdq,\nż\n|x|ąR|f1pQqv|2dxďC1\nR}v}2\nL2}|x||Qpxq|2pθ´1}L8ď˜C1\nR}v}2\nL2. (A.11)\nLet nextvn,ně1, be a bounded sequence in H1pRdq(bounded by C0). Without loss\nof generality, we may assume that vnconverges weakly in H1pRdqtov˚PH1pRdq.\nFor anyεm“1{m,mPN, we can choose Rmą1large enough so that, by ( A.11),\nż\n|x|ąRm|f1pQqv|2dxď˜CC2\n01\nRmďε2\nm\n4.\nSince the embedding of H1pBH1p0,RmqqintoL2pBH1p0,Rmqqis compact, there exists\na subsequence vnmofvnsuch thatvnmconverges to v˚inL2ppBH1p0,Rmqq. In\nparticular, there exists nm0such that for mąm0,\n}vnm´v˚}L2pBH1p0,Rmqqďεm\n2.\nWe deduce from the above two inequalities that, for nmąnm0,\n}vnm´v˚}L2pRdqďεm.\nRepeating this argument, we can construct a subsequence vnjofvnsuch thatvnj\nconverges to v˚inL2pRdqasnjÑ `8 . Thus the multiplication operator by ´f1pQq\nis a compact operator from H1pRdqintoL2pRdq. This implies that Lis a Fredholm\noperator of index 0fromH2\nradpRdqintoL2\nradpRdqor fromH1\nradpRdqintoH´1\nradpRdq./squareLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 55\nReferences\n[1] B. Aulbach, Continuous and Discrete Time Dynamics Near Manifolds of Equ ilibria , Springer\nVerlag, Berlin-Heidelberg, 1984.\n[2] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely m any\nsolutions , Arch. Rational Mech. Anal. 82(1983), pp. 347–375.\n[3] P. Brunovsky and P. Poláčik, On the local structure of ω-limit sets of maps , Z. Angew. Math.\nPhys.48(1997), pp. 976–986.\n[4] N. Burq, G. 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Pazy, Semigroups of Linear Operators and Applications to Partial Di fferential Equations ,\nAppl. Math. Sci. 44, Springer-Verlag, New-York, 1983.\n[34] G. Raugel, Dynamics of Partial Differential Equations on Thin Domains, C IME Course,\nMontecatini Terme , Lecture Notes in Mathematics 1609, Springer Verlag, (1995), pp. 208–\n315.\n[35] W. Schlag, Spectral theory and nonlinear partial differential equations : a survey , Discrete\nContin. Dyn. Syst. 15(2006), pp. 703–723.\n[36] L. Simon Asymptotics for a class of nonlinear evolution equations, w ith applications to geo-\nmetric problems , Ann. of Math. (2) 118(1983), pages 525–571.\n[37] L. Simon, Theorems on regularity and singularity of energy minimizing m aps, Lect. in Math.,\nETH Zürich, Birkhäuser, 1996.\n[38] J. Wirth, Wave equations with time-dependent dissipation I. Non-effec tive dissipation , J. Dif-\nferential Equations (2006)\n[39] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation , J. Differ-\nential Equations (2007)\n[40] T. J. Zelenyak, Stabilization of solutions of boundary value problems for a s econd order para-\nbolic equation with one space variable , Diff. Equations 4(1968), pp. 17–22.\nNicolas Burq: Univ Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay\nCedex, F-91405; CNRS, Orsay cedex, F-91405, FranceLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 57\nGeneviève Raugel: CNRS, Laboratoire de Mathématiques d’Or say, Orsay Cedex,\nF-91405; Univ Paris-Sud, Orsay cedex, F-91405, France\nWilhelm Schlag: University of Chicago, Department of Mathem atics, 5734 South\nUniversity Avenue, Chicago, IL 60636, U.S.A."    },    {        "title": "1608.02655v2.Damping_Functions_correct_over_dissipation_of_the_Smagorinsky_Model.pdf",        "content": "DAMPING FUNCTIONS CORRECT OVER-DISSIPATION OF THE SMAGORINSKY\nMODEL\nALI PAKZAD\nAbstract. This paper studies the time-averaged energy dissipation rate h\"SMD (u)ifor the combination\nof the Smagorinsky model and damping function. The Smagorinsky model is well known to over-damp.\nOne common correction is to include damping functions that reduce the e\u000bects of model viscosity near\nwalls. Mathematical analysis is given here that allows evaluation of h\"SMD (u)ifor any damping function.\nMoreover, the analysis motivates a modi\fed van Driest damping. It is proven that the combination of\nthe Smagorinsky with this modi\fed damping function does not over dissipate and is also consistent with\nKolmogorov phenomenology.\n1.introduction\nExperience with the Smagorinsky model (SM) indicates it over dissipates (p.247 of Sagaut [30]). This\nextra dissipation can laminarize the numerical approximation of a turbulent \row and prevent the transition\nto turbulence (p.192 of [21]). Model re\fnements aim at reducing model dissipation occur as early as 1975 [31]\nand continues with dynamic parameter selection (Germano, Piomelli, Moin and Cabot [7] and Swierczewska\n[34]), structural sensors (Hughes, Oberai and Mazzei [12]) and near wall models (e.g., Piomelli and Balaras\n[28], John, Layton and Sahin [15] and John and Liakos [16]). The classical approach is to multiply the\nturbulent viscosity with damping function \f(x) (such as van Driest damping [35]) with \f(x)!0 asx!\nwalls. There has been many numerical tests but little analytic support of this combination.\nThis paper analyzes this combination of the SM with damping function \f(x) in the \row domain \n = (0 ;L)3,\n(1.1) ut+u\u0001ru\u0000\u0017\u0001u+rp\u0000r\u0001 (\f(x)(Cs\u000e)2jrujru) = 0 andr\u0001u= 0 in \n:\nIn (1.1)uis the velocity, pis the pressure, \u0017is the kinematic viscosity, \u000e << 1 is a model length scale and\nCs'0:1 is the standard model parameter (Lilly [23]). To evaluate the e\u000bect of damping function in the near\nwall region, we study the time-averaged energy dissipation rate of (1.1) for shear \row. L-periodic boundary\nconditions in xandydirections are imposed. z= 0 is a \fxed wall and the wall z=Lmoves with velocity U\n(Figure 1),\n(1.2) u(x;y;0;t) = (0;0;0)>andu(x;y;L;t ) = (U;0;0)>:\nThe Reynolds number is Re=UL\n\u0017. The time-averaged energy dissipation rate for model (1.1) includes\ndissipation due to the viscous forces and turbulent di\u000busion reduced by the damping function \f(x). It is\ngiven by\n(1.3) h\"SMD (u)i= lim sup\nT!11\nTZT\n0(1\nj\njZ\n\n\u0017jruj2+ (cs\u000e)2\f(x)jruj3dx)dt:\n1arXiv:1608.02655v2  [math.AP]  30 Jul 20172 ALI PAKZAD\nFigure 1. Shear \row.\nThis paper estimates h\"SMD (u)i(Theorem 3.1) for a damping function \f(x) in terms of its integral on an\n\r=O(Re\u00001) strip along the moving wall as\nh\"SMD (u)i\u0014\u0002\nC1+C2(Cs\u000e\nL)2Re31\nLZL\nL\u0000\rL\f(z)dz\u0003U3\nL:\nFor an algebraic approximation of van Driest damping (Section 2.2), Corollary 3.2 shows\nh\"SMD (u)i\u0014\u0002\nC2+C2(Cs\u000e\nL)2Re2\u0003U3\nL:\nThe above estimate goes toU3\nLfor \fxedReas\u000e!0, but blows up as Re!1 for \fxed\u000e, suggesting\nover-dissipation. On the other hand, damping with a classical mixing length formula (3.19) of Prandtl, we\nobtain in Corollary 3.4\nh\"SMD (u)i\u0014CU3\nL;\nwhich is consistent with Kolmogorov phenomenology.\n1.1.Related work. In the theory of turbulence, the time-averaged energy dissipation rate is a fundamental\nquantity (Sreenivasan [33], Pope [29], Lesieur [22], and Frisch [6]) determines the smallest persistent length\nscales and the dimension of any global attractor [20]. Moreover, the smallest length scale of turbulent \row\nsimulation can be estimated by using upper bounds of the energy dissipation rate. In turbulent \rows, the\nenergy dissipation is often observed approach a limit independent of the viscosity [18]. No rigorous proof of\nthis fact has been given. For shear \row between parallel plates, Busse [3] and Howard [11] estimated h\"(u)i\nunder the assumptions that the \row is statistically stationary. Doering and Constantin [4] proved an upper\nbound on the time-averaged energy dissipation rate for general weak solutions of the Navier-Stokes equations\n(NSE),h\"(u)i\u0014CU3\nL. Similar estimations have been proven by Marchiano [24], Wang [36] and Kerswell [17]\nin more generality.\nThe Smagorinsky model [32], \f(x)\u00111 in (1.1), is a common turbulence model used in Large Eddy\nSimulation (e.g., [1], [30], [14], [26], [27] and [8]). The extra term with respect to the NSE can be generally\njusti\fed as follows. In turbulence, dissipation occurs non-negligibly only at very small scales, smaller than\ntypical mesh. The balance between energy input at the largest scales and energy dissipation at the smallest\nis a critical selection mechanism for determining statistics of turbulent \rows. To get an accurate simulation,DAMPING FUNCTIONS CORRECT OVER-DISSIPATION OF THE SMAGORINSKY MODEL 3\nonce a mesh is selected, an extra dissipative term must be introduced to model the e\u000bect of the unresolved\n\ructuations, which are smaller than the mesh width, upon the resolved velocity.\nThe energy dissipation rate of the Smagorinsky model for shear \row with boundary layers was estimated\nin [20] as\n(1.4) h\"s(u)i'[1 +C2\ns(\u000e\nL)2(1 +Re)2]U3\nL:\nThis estimate blows up for \u000e\fxed asRe!1 , which is consistent with the numerical evidence (e.g., Iliescu\nand Fischer [13] and Moin and Kim [25]). Surprisingly, it was shown in [19] that the energy dissipation rate\nof the Smagorinsky model in the absence of boundary layers satis\fes\n(1.5) h\"s(u)i'U3\nL:\nComparing these two results (1.4) and (1.5) suggests that the model over dissipation is due to the action\nof the model viscosity in boundary layers rather than in interior small scales generated by the turbulent\ncascade. To reduce the e\u000bect of model viscosity in the boundary layers damping functions \f(x), which\ngo to zero at the walls, are often used (Pope [29]). In this case most of the tools of analysis, such as\nK orn's inequality, the Poincar\u0013 e-Friederichs inequality, and Sobolev's inequality, no longer hold. Thus, the\nmathematical development of the SM under no-slip boundary conditions with damping function is cited in\n[1] p.78 as an important open problem.\nAcknowledgments. The author would like to thank Professor William Layton for suggesting this problem\nand for many fruitful discussions. A.P. was Partially supported by NSF grants DMS 1522267 and CBET\n1609120.\n2.Mathematical preliminaries\nWe use the standard notations Lp(\n);Wk;p(\n);Hk(\n) =Wk;2(\n) for the Lebesgue and Sobolev spaces\nrespectively. The inner product in the space L2(\n) will be denoted by ( \u0001;\u0001) and its norm by jj\u0001jj for scalar,\nvector and tensor quantities. Norms in Sobolev spaces Hk(\n);k > 0, are denoted by jj\u0001jjkand the usual\nLpnorm is denoted by jj\u0001jjLp. The symbols CandCifori= 1;2;3 stand for generic positive constant\nindependent of the \u0017,LandU.ruis the gradient tensor ( ru)ij=@uj\n@xifori;j= 1;2;3.\nDe\fnition 2.1. The velocity at a given time tis sought in the space\nX(\n) :=fu2H1(\n) :u(x;y;0) = (0;0;0)>; u(x;y;L ) = (U;0;0)>; uisL-periodic in xandydirectiong:\nThe test function space is\nX0(\n) :=fu2H1(\n) :u(x;y;0) = (0;0;0)>; u(x;y;L ) = (0;0;0)>; uisL-periodic in xandydirectiong:\nThe pressure at time tis sought in\nQ(\n) :=L2\n0(\n) =fq2L2(\n) :R\n\nqdx= 0g:\nAnd the space of divergence-free functions is denoted by\nV(\n) :=fu2X(\n) : (r\u0001u;q) = 08q2Qg:\nDe\fnition 2.2. (Trilinear from) De\fneb:X\u0002X\u0002X!Rasb(u;v;w ) := (u\u0001rv;w).\nLemma 2.1. The nonlinear term b(\u0001;\u0001;\u0001)is continuous on X\u0002X\u0002X(and thus on V\u0002V\u0002Vas well).\nMoreover, we have the following skew-symmetry property for b\nb(u;v;v ) = 08u2V;v2X:4 ALI PAKZAD\nFigure 2. The background \row.\nProof. The proof is standard and the one with zero boundary conditions can be found in p.114 of Girault\nand Raviart [9]. \u0003\n2.1.Construction of background \row. One key step to the upper bound on h\"SMD (u)iis to construct\nan appropriate background \row, \b 2X(\n), following Hopf [10] and Doering and Constantin [4]. This is a\ndivergence-free function extending the boundary condition (1.2) to the interior of \n. Moreover, u\u0000\b2X0(\n)\nand will be used as a test function in the weak form (3.1). The choice of \r2(0;1) will be determined by the\nneeds of the estimates in (3.16) and it will be chosen to be \r=1\n5:1(Re)\u00001.\nDe\fnition 2.3. (The background \row) De\fne \b(x;y;z ) := (\u001e(z);0;0)>, where\n\u001e(z) =(\n0 if z2[0;L\u0000\rL]\nU\n\rL(z\u0000(L\u0000\rL)) ifz2[L\u0000\rL;L ]:\n\u001e(z) is sketched in Figure 2. We collect two properties for \b in Lemmas 2.2 and 2.3.\nLemma 2.2. \bsatis\fes\nk\bk1\u0014U; a) kr\bk1\u0014U\n\rL; b) k\bk2\u0014U2\rL3\n3; c) kr\bk2\u0014U2L\n\r: d)\nProof. They all are the immediate consequence of the De\fnition 2.3. We show ( c) here as an example.\nk\bk2=L2RL\n0j\u001e(z)j2dz=L2RL\nL\u0000\rLU2\n(\rL)2(z\u0000(L\u0000\rL))2dz=U2\rL3\n3:\n\u0003\nWe will need the well-known dependence of the Poincar\u0013 e -Friedrichs inequality constant on the domain.\nA straightforward argument in the thin domain O\rLimplies Lemma 2.3.\nLemma 2.3. LetO\rL=f(x;y;z )2\n :L\u0000\rL\u0014z\u0014Lgbe the region close to the upper boundary. Then\nwe have\n(2.1) ku\u0000\bkL2(O\rL)\u0014\rLkr(u\u0000\b)kL2(O\rL):\nProof. First letvbe aC1function onO\rLthat vanishes for z=L. Then component-wise ( i= 1;2;3), we\nhave\nvi(x;y;z ) =vi(x;y;L )\u0000ZL\nzdvi\nd\u0018(x;y;\u0018 )d\u0018:DAMPING FUNCTIONS CORRECT OVER-DISSIPATION OF THE SMAGORINSKY MODEL 5\nObserving that vi(x;y;L ) = 0, squaring both sides, and using the Cauchy-Schwarz inequality, we get\nv2\ni(x;y;z )\u0014\rLZL\nL\u0000\rL(dvi\nd\u0018(x;y;\u0018 ))2d\u0018:\nIntegrating both sides with respect to zgives\nZL\nL\u0000\rLv2\ni(x;y;z )dz\u0014(\rL)2ZL\nL\u0000\rL(dvi\nd\u0018(x;y;\u0018 ))2d\u0018:\nThen integrating with respect to xandyand summing from i= 1 to 3, we obtain\nkvk2\nL2(O\rL)\u0014(\rL)2krvk2\nL2(O\rL):\nThis proves the lemma for v2C1. Finally use a density argument and take v=u\u0000\b. \u0003\n2.2.The kinetic energy. Before proving the main theorem, we prove boundedness of the kinetic energy,\n1\n2jjujj2, and thath\"SMD (u)iis well-de\fned. The proof of the model (1.1) is similar to the NSE case \frst\npresented in Hopf [10]. We need the following proposition \frst.\nProposition 2.4. Letrv2Lp(O\rL)and0<p<1. Ifv(x;y;L;t ) = 0 then\njjv(x;y;z )\nL\u0000zjjLp(O\rL)\u0014p\np\u00001jj@v\n@z(x;y;z )jjLp(O\rL):\nProof. Using B. Hardy's inequality (P.313 of Brezis [2]) when z2[L\u0000\rL;L ] for \fxedxandygives\njjv(x;y;z )\nL\u0000zjjLp([L\u0000\rL;L ])\u0014p\np\u00001jj@v\n@z(x;y;z )jjLp([L\u0000\rL;L ]):\nRaising both sides to power p, then taking a double integral with respect to xandyforx;y2[0;L] implies\nthe result,\njjv(x;y;z )\nL\u0000zjjp\nLp(O\rL)\u0014(p\np\u00001)pjj@v\n@z(x;y;z )jjp\nLp(O\rL)\u0014(p\np\u00001)pjjrvjjp\nLp(O\rL):\n\u0003\nLemma 2.5. The kinetic energy and the time averages of the energy dissipation of the solution to (1.1) with\nthe boundary conditions (1.2) are uniformly bounded in time, i.e.\nsup\nt2(0;1)jju(t)jj\u0014C <1 and sup\nt2(0;1)1\nTZT\n0(1\nj\njZ\n\n\u0017jruj2+ (cs\u000e)2\f(x)jruj3dx)dt\u0014C <1:\nProof. The strategy is to subtract o\u000b the inhomogeneous boundary conditions (1.2). Consider v=u\u0000\b,\nthenvsatis\fes homogeneous boundary conditions. Substituting u=v+ \b in the equation (1.1) yields\n(2.2)vt+v\u0001rv\u0000\u0017\u0001v+rp+\u001e(z)@v\n@x+v3\u001e0(z)(1;0;0)\u0000r\u0001 (\f(x)(Cs\u000e)2jr(v+ \b)jr(v+ \b)) = 0;\nr\u0001v= 0:\nwith boundary conditions\n(2.3) v(x;y;0;t) = (0;0;0)>andv(x;y;L;t ) = (0;0;0)>;\nand\n(2.4) v(x+L;y;z;t ) =v(x;y;z;t ) and v(x;y+L;z;t ) =v(x;y;z;t ):6 ALI PAKZAD\nTaking inner product with v= (v1;v2;v3) and integrating over \n give\n(2.5)1\n2@\n@tjjvjj2+\u0017jjrvjj2+Z\n\n\u0000\n\u001e(z)@v\n@x\u0001v+v1v3\u001e0(z)\u0001\ndx+ (\f(x)(Cs\u000e)2jr(v+ \b)jr(v+ \b);rv) = 0:\nSince any integral containing \u001eand\u001e0will be zero outside the strip O\rL, by integrating by part we have\nZ\n\n\u001e(z)@v\n@x\u0001vdx=1\n2Z\nO\rL\u001e(z)@\n@xjvj2dx=1\n2Z\nO\rL@\n@x(\u001e(z)jvj2)dx\u00001\n2Z\nO\rL\u001e0(z)jvj2dx=\u00001\n2U\n\rLZ\nO\rLjvj2dx:\nInserting this identity in (2.5) and using the triangle inequality on the last term give\n@\n@tjjvjj2+ 2\u0017jjrvjj2+U\n\rLZ\nO\rL\u0000\n2v1v3\u0000jvj2\u0001\ndx+ 2Z\n\n\f(x)(Cs\u000e)2jrvj3dx\n\u00144Z\n\n(\f(x)(Cs\u000e)2jrvj2jr\bjdx+ 2Z\n\n(\f(x)(Cs\u000e)2jrvjjr\bj2dx:(2.6)\nThe rest of analysis requires to approximate various term in the above. Let p= 2 in Proposition 2.4 and\nthe two terms on the LHS can be bounded above as\nU\n\rLZ\nO\rLjvj2dx=U\n\rLZ\nO\rLd(z)2jv\nd(z)j2dx\u0014U\n\rL(\rL)2Z\nO\rLjv\nd(z)j2dx\n\u00142U\rLZ\nO\rLjrvj2dx\u00142U\rLjjrvjj2:(2.7)\nSimilarly\n(2.8)U\n\rLZ\nO\rL2jv1v3jdx\u00144U\rLjjrvjj2:\nTo bound the two terms on the RHS of (2.6), use H older's inequality and Young inequality jjfgjjL1\u0014\njjfjjLpjjgjjLq\u0014\u000f\npjfjjp\nLp+\u000f\u0000q\np\nqjjgjjq\nLq. Consider the \frst term, for p=3\n2,q= 3 and\u000f= 0:6 we have\n4Z\n\n(\f(x)(Cs\u000e)2jrvj2jr\bjdx\u00144(Cs\u000e)2\u0000Z\n\n\f(x)jrvj3dx\u00012\n3\u0000Z\n\n\f(x)jr\bj3dx\u00011\n3\n\u00141:6Z\n\n\f(x)(Cs\u000e)2jrvj3dx+ 4Z\n\n\f(x)(Cs\u000e)2jr\bj3dx:(2.9)\nThe second term is estimated exactly like the last term for p= 3,q=3\n2and\u000f= 0:6 as\n2Z\n\n(\f(x)(Cs\u000e)2jrvjjr\bj2dx\u00142(Cs\u000e)2\u0000Z\n\n\f(x)jrvj3dx\u00011\n3\u0000Z\n\n\f(x)jr\bj3dx\u00012\n3\n\u00140:4Z\n\n\f(x)(Cs\u000e)2jrvj3dx+ 2Z\n\n\f(x)(Cs\u000e)2jr\bj3dx:(2.10)\nInserting these last four estimates into the energy inequality (2.6) for vgives\n@\n@tjjvjj2+ (2\u0017\u00006U\rL)jjrvjj2\u00146Z\n\n\f(x)(Cs\u000e)2jr\bj3dx:\nThus, if\ris chosen small enough that\n\r <1\n3(Re)\u00001;\nthen (2\u0017\u00006U\rL) becomes positive. Applying the Poincar\u0013 e-Friedrichs inequality jjvjj\u0014CjjrvjjgivesDAMPING FUNCTIONS CORRECT OVER-DISSIPATION OF THE SMAGORINSKY MODEL 7\n@\n@tjjvjj2+Cjjvjj2\u00146Z\n\n\f(x)(Cs\u000e)2jr\bj3dx:\nSince RHS is uniformly bounded in time, a standard Gr onwall's inequality shows that\nsup\nt2(0;1)jjv(t)jj\u0014C <1;\nWhich proves the boundedness of the kinetic energy,1\n2jjujj2. From this and standard arguments it follows\nthat\n1\nTZT\n0(1\nj\njZ\n\n\u0017jruj2+ (cs\u000e)2\f(x)jruj3dx)dt\u0014C <1;\nwhich meansh\"SMD (u)iis well-de\fned. \u0003\n2.3.van Driest damping. To modify the mixing-length model van Driest proposed [35], with some the-\noretical support but mainly as a good \ft to data (p.77 of Wilcox [37]), that the mixing length `should be\nmultiplied by the damping function so that `(x)!0 asx!wall. The van Driest damping function is\n(2.11) fw(z) = 1\u0000e\u0000z+\nA+;\nwhereA+= 26 is the van Driest constant and z+is the non-dimensional distance from the wall (p.76 of\nWilcox [37])\n(2.12) z+=u\u001c(L\u0000z)\n\u0017;\nwhich determines the relative importance of viscous and turbulent phenomena. u\u001cis the wall shear velocity\ngiven by\n(2.13) u\u001c=s\n\u0017@u\n@z\f\f\f\fwall:\nu\u001cis still unknown, the analysis herein will require a speci\fc value for u\u001c. To this end, it can be estimated\nas follows. Near the wall ru'@u\n@z, then\n(2.14) u\u001c=s\n\u0017@u\n@z\f\f\f\fwall'q\n\u0017rujwall=4q\n\u00172(rujwall)2'4p\n\u0017h\u0016\u000fwi;\nwhere \u0016\u000fwis a spatial-average energy dissipation rate near the wall. After assuming a non-zero fraction occurs\nin near-wall region and therefore neglecting the e\u000bects of viscosity far from the boundary layer, dissipation\noccurs mainly in the boundary layers near the bottom and top walls which both have a volume of L3\r. Hence\nh\u000fi= 21\nL3(L3\r)h\u0016\u000fwi= 2\rh\u0016\u000fwi;\nOn the other hand, based on the statistical equilibrium h\u000fi=U3\nL, therefore\nh\u000fi=U3\nL= 2\rh\u0016\u000fwi;\nUsing\r=O(Re\u00001) gives\n(2.15) h\u0016\u000fwi'1\n2U4\n\u0017:8 ALI PAKZAD\nFigure 3. van Driest.\n Figure 4. Algebraic approximation.\nThenu\u001cis estimated by inserting (2.15) in (2.14) to be\n(2.16) u\u001c'U\n4p\n2:\nHence van Driest damping function is approximated as (Figure 3)\n(2.17) fw(z)'1\u0000exp(\u0000U(L\u0000z)\n264p\n2\u0017):\nUsing Taylor series to approximate (2.17) in the boundary layer O\rLgives\n(2.18) fw(z)'kX\nn=1[1\n264p\n2n!Re(1\u0000z\nL)]n+O(Rek+1(1\u0000z\nL)k+1):\nNote that the above approximation (2.18) is valid when the reminder Re(1\u0000z\nL) is less than 1, and this\noccurs when L\u0000\rL\u0014z\u0014L:Moreover, approximation (2.18) suggests ( Re)\u000b(1\u0000z\nL)\u000bfor\u000b\u00151 as a damping\nfunction only on the top layer. Thus (2.19) is an algebraic approximation to the van Driest damping on the\nwhole domain (Figure 4, for \u000b= 2).\n(2.19) \fw(z) =8\n><\n>:(Re)\u000b(z\nL)\u000bifz2[0;\rL]\n1 if z2[\rL;L\u0000\rL]\n(Re)\u000b(1\u0000z\nL)\u000bifz2[L\u0000\rL;L ]:\nRemark 2.6.\fwplays the role of \fin (1.1).\n3.Analysis of the Smagorinsky with Damping Function\nTheorem 3.1. Supposeu02L2(\n) and let1\r=1\n5:1Re\u00001, then for any positive damping function\n\f(z)2L1(\n),h\"SMD (u)isatis\fes\nh\"SMD (u)i\u0014\u0002\nC1+C2(Cs\u000e\nL)2Re31\nLZL\nL\u0000\rL\f(z)dz\u0003U3\nL:\nProof. The weak form of (1.1) is obtained by taking the scalar product v2X0andq2L2\n0with (1.1) and\nintegrating over the space \n.\n1In fact\rcan be\u0014(1\n5Re\u00001) for any\u00142(0;1). Without loss of generality, \ris taken to be1\n5:1Re\u00001for simplicity in calculations.DAMPING FUNCTIONS CORRECT OVER-DISSIPATION OF THE SMAGORINSKY MODEL 9\n(3.1)(ut;v) +\u0017(ru;rv) +b(u;u;v )\u0000(p;r\u0001v) + (\f(x)(Cs\u000e)2jrujru;rv) = 08v2X0;\n(r\u0001u;q) = 08q2L2\n0;\n(u(x;0)\u0000u0(x);v) = 08v2X0:\nTakev=u\u0000\b in (3.1). Using the skew-symmetry of b(\u0001;\u0001;\u0001) (Lemma 2.1) and r\u0001\b = 0 gives\n(ut;u\u0000\b) +\u0017(ru;ru\u0000r\b) +b(u;u;u\u0000\b) + (\f(x)(Cs\u000e)2jrujru;ru\u0000r\b) = 0;\nwhich is equivalent to the following\n1\n2d\ndtkuk2+\u0017kruk2+ (Cs\u000e)2(\f(x)jrujru;ru) =d\ndt(u;\b) +b(u;u;\b) +\u0017(ru;r\b)\n+ (Cs\u000e)2(\f(x)jrujru;r\b):(3.2)\nIntegrating with respect to time from above equation gives\n1\n2ku(T)k2\u00001\n2ku(0)k2+\u0017ZT\n0kruk2dt+ZT\n0(Z\n\n(Cs\u000e)2\f(x)jruj3dx)dt= (u(T);\b)\u0000(u(0);\b)\n+ZT\n0b(u;u;\b)dt+\u0017ZT\n0(ru;r\b)dt+ (Cs\u000e)2ZT\n0(\f(x)jrujru;r\b)dt:(3.3)\nThe proof continues by bounding, term by term, each term on the right-hand side of the energy equality\n(3.3). Using the Cauchy-Schwarz Young inequality and Lemma 2.2, the \frst three terms are estimated as\nfollows.\n(3.4) ( u(T);\b)\u00141\n2ku(T)k2+1\n2k\bk2=1\n2ku(T)k2+U2\rL3\n6:\n(3.5) ( u(0);\b)\u0014ku(0)kk\bk=r\r\n3UL3\n2ku(0)k:\n(3.6) \u0017ZT\n0(ru;r\b)dt\u0014\u0017\n2ZT\n0kruk2+kr\bk2dt=\u0017\n2ZT\n0kruk2dt+\u0017\n2U2L\n\rT:\nFor the nonlinear term b(\u0001;\u0001;\u0001) in (3.3) add and subtract terms and then use skew-symmetry. This gives\nb(u;u;\b) =b(u\u0000\b;u\u0000\b;\b) +b(\b;u\u0000\b;\b)\n=1\n2b(u\u0000\b;u\u0000\b;\b)\u00001\n2b(u\u0000\b;\b;u\u0000\b)\n+1\n2b(\b;u\u0000\b;\b)\u00001\n2b(\b;\b;u\u0000\b):(3.7)\nTo estimate the four terms in (3.7), use Lemma 2.2, Lemma 2.3, the Cauchy-Schwarz Young inequality.\nMoreover, apply the fact that b(u;u;\b) is an integration on O\rLsince supp(\b) =O\rL. For the \frst term in\n(3.7) we have\n(3.8)b(u\u0000\b;u\u0000\b;\b)\u0014k\bkL1(O\rL)ku\u0000\bkL2(O\rL)kr(u\u0000\b)kL2(O\rL)\u0014\rLUkr(u\u0000\b)k2\nL2(O\rL)\n\u0014\rLUkru\u0000r\bk2\nL2\u0014UL\r(kruk+kr\bk)2\u0014UL\r(2kruk2+ 2kr\bk2)\n\u0014UL\r(2kruk2+ 2U2L\n\r) = 2UL\rkruk2+ 2U3L2:10 ALI PAKZAD\nFor the second term we have\n(3.9)b(u\u0000\b;\b;u\u0000\b)\u0014kr \bkL1(O\rL)ku\u0000\bk2\nL2(O\rL)\u0014U\n\rL\r2L2kr(u\u0000\b)k2\nL2(O\rL)\n\u0014\r2L2U\n\rL(2kruk2+ 2U2L\n\r) = 2\rLUkruk2+ 2U3L2:\nThe third one is estimated as\nb(\b;u\u0000\b;\b)\u0014k\bkL1(O\rL)kr(u\u0000\b)kL2(O\rL)k\bkL2(O\rL)\u0014Ur\nU2\rL3\n3(kruk+kr\bk)\n\u0014Ur\nU2\rL3\n3(kruk+s\nU2L\n\r)\u0014U2\r1\n2L3\n2p\n3kruk+U3L2\np\n3\n= [U3\n2Lp\n3] [(U\rL)1\n2kruk] +U3L2\np\n3\u0014(U3L2\n6) +1\n2UL\rkruk2+U3L2\np\n3\n=1\n2UL\rkruk2+ (p\n3\n3+1\n6)U3L2:(3.10)\nAnd \fnally the last one satis\fes\nb(\b;\b;u\u0000\b)\u0014k\bkL1(O\rL)kr\bkL2(O\rL)ku\u0000\bkL2(O\rL)\u0014Us\nU2L\n\r\rLkr(u\u0000\b)kL2(O\rL)\n\u0014U2\r1\n2L3\n2(kruk+kr\bk)\u0014U2\r1\n2L3\n2(kruk+ (U2L\n\r)1\n2)\n=U2\r1\n2L3\n2kruk+U3L2= [U3\n2L] [(UL\r)1\n2kruk] +U3L2\n\u00141\n2(U3L2) +1\n2UL\rkruk2+U3L2=1\n2UL\rkruk2+3\n2U3L2:(3.11)\nUsing (3.8), (3.9), (3.10) and (3.11) in (3.7) gives the \fnal estimation for the non-linear term as below.\n(3.12) jb(u;u;\b)j\u00145\n2UL\rkruk2+19\n6U3L2:\nFinally the last term on the RHS of (3.3) can be estimated as the follows. Using H older's inequality and\nYoung inequality for p=3\n2andq= 3 gives\nj(\f(x)jrujru;r\b)j\u0014Z\n\nj\f(x)jjruj2jr\bjdx\n=Z\n\n(\f2\n3jruj2) (j\fj1\n3jr\bj)dx\n\u0014[Z\n\n\fjruj3dx]2\n3[Z\n\n\fjr\bj3dx]1\n3\u00142\n3Z\n\n\fjruj3dx+1\n3Z\n\n\fjr\bj3dx:(3.13)\nInserting (3.4), (3.5), (3.6), (3.12) and (3.13) in (3.3) implies\n1\n2ku(T)k2\u00001\n2ku(0)k2+\u0017ZT\n0kruk2dt+ (Cs\u000e)2ZT\n0(Z\n\n\f(x)jruj3dx)dt\u00141\n2ku(T)k2+U2\rL3\n6\n+r\r\n3UL3\n2ku(0)k+5\n2UL\rZT\n0kruk2dt+19\n6U3L2T+\u0017\n2ZT\n0kruk2dt+\u0017\n2U2L\n\rT\n+2\n3(Cs\u000e)2ZT\n0(Z\n\n\f(x)jruj3dx)dt+1\n3(Cs\u000e)2ZT\n0(Z\n\n\f(x)jr\bj3dx)dt:(3.14)DAMPING FUNCTIONS CORRECT OVER-DISSIPATION OF THE SMAGORINSKY MODEL 11\nSince the kinetic energy is bounded (Lemma 2.5), the above inequality becomes\n(1\n2\u00005\n2\u0017\rLU)ZT\n0\u0017kruk2dt+1\n3(Cs\u000e)2ZT\n0(Z\n\n\f(x)jruj3dx)dt\u00141\n2ku(0)k2+1\n6U2\rL3\n+r\r\n3UL3\n2ku(0)k+19\n6U3L2T+\u0017\n2\rLU2T+1\n3(Cs\u000e)2ZT\n0(Z\n\n\f(x)jr\bj3dx)dt:(3.15)\nFinally dividing both sides of (3.15) by Tandj\nj=L3and taking the limit superior leads to\n(3.16) minf1\n2\u00005\n2\rLU\n\u0017;1\n3gh\"SMD (u)i\u001419\n6U3\nL+\u0017\n2\rU2\nL2+1\n31\nL3(Cs\u000e)2Z\n\n\f(x)jr\bj3dx:\nThe above inequality leads to the last step when C1andC2are positive and independent of viscosity, diam(\n)\nand lid velocity. Take \r=1\n5:1Re\u00001, then (1\n2\u00005\n2\u0017\rLU) becomes positive and therefore\n(3.17) h\"SMD (u)i\u0014C1U3\nL+C21\nL3(Cs\u000e)2Z\n\n\f(x)jr\bj3dx:\nBecause the background \row \b vanishes on (\n nO\rL), we have\n(3.18)Z\n\n\f(x)jr\bj3dx= (U\n\rL)3ZL\n0ZL\n0ZL\nL\u0000\rL\f(x;y;z )dxdydz =U3\n\r3LZL\nL\u0000\rL\f(z)dz:\nInserting (3.18) in (3.17) proves Theorem 3.1. \u0003\n3.1.Evaluation of Damping Functions. Theorem 3.1 is the starting point for the evaluation of damping\nfunctions. It is next applied to two damping functions in Corollaries 3.2 and 3.4 and the result compared.\nCorollary 3.2. For the algebraic approximation of the van Driest damping function, \fw(z)in (2.10), we\nhave\nh\"SMD (u)i\u0014\u0002\nC1+C21\n\u000b+ 1(Cs\u000e\nL)2Re2\u0003U3\nL:\nProof. The result is a calculation by applying \fw(z) in (2.10) to the Theorem 3.1. \u0003\nThe upper bound in Corollary 3.2 is a function of the global velocity U, domain diameter L, the eddy\nsize\u000e, and surprisingly, the Reynolds number. Moreover, it blows up as Re!1 . Due to this fact one can\npropose the following modi\fcation to \fw(z). Consider \fd(z)2C1(\n) in (3.19) which is based on a connection\nof the algebraic damping near the wall smoothly to the no damping in the interior by hermite interpolation.\nIt is given by\n(3.19) \fd(z) =8\n>>>>>><\n>>>>>>:(z\nL)\u000b(1\u0000z\nL)\u000bifz2[0;\rL]\na1(z\u0000\rL)3+b1(z\u0000\rL)2+c1(z\u0000\rL) +d1ifz2[\rL;2\rL]\n1 if z2[2\rL;L\u00002\rL]\na2(z+ 2\rL\u0000L)3+b2(z+ 2\rL\u0000L)2+ 1 if z2[L\u00002\rL;L\u0000\rL]\n(z\nL)\u000b(1\u0000z\nL)\u000bifz2[L\u0000\rL;L ];\nwhere\u000b\u00150 anda1;a2;b1;b2;c1andd1are constant such that\n\u000fa1=\u00002\n\r3L3+1\nL3\u000b\r\u000b\u00003(1\u0000\r)\u000b\u00001(1\u00002\r) +2\nL3\r\u000b\u00003(1\u0000\r)\u000b,\n\u000fb1=3\n\r2L2\u00002\nL2\u000b\r\u000b\u00002(1\u0000\r)\u000b\u00001(1\u00002\r)\u00003\nL2\r\u000b\u00002(1\u0000\r)\u000b,\n\u000fc1=1\nL\u000b\r\u000b\u00001(1\u0000\r)\u000b\u00001(1\u00002\r),\n\u000fd1=\r\u000b(1\u0000\r)\u000b,12 ALI PAKZAD\nFigure 5. Proposed damping function (3.19).\n\u000fa2=\u0000a1,\n\u000fb2=\u00003\n\r2L2+1\nL2\u000b\r\u000b\u00002(1\u0000\r)\u000b\u00001(1\u00002\r) +3\nL2\r\u000b\u00002(1\u0000\r)\u000b.\nRemark 3.3.All the constants above are calculated such that \fd(z)2C1.\fd, which plays the role of \fin\n(1.1), is sketched in Figure 5 and compared to \fw. They both are symmetric, bounded and vanish at z= 0\nandz=L. Moreover, they are almost 1 on the whole domain except on the thin boundary layers.\nCorollary 3.4. Supposeu02L2(\n)and\fdgiven by (3.19) with 2\u0014\u000b2N. Then, for anyRe\u00151we have\nh\"SMD (u)i\u0014\u0002\nC1+C2(Cs\u000e\nL)2\u0003U3\nL:\nProof. Considering \fdis symmetric on the whole domain implies\nZL\nL\u0000\rL\fd(z)dz=Z\rL\n0(z\nL)\u000b(1\u0000z\nL)\u000bdz:\nNow applying the Binomial Theorem on (1 \u0000z\nL)\u000band then taking integral gives\nZL\nL\u0000\rL\fd(z)dz=L\r\u000b+1\u00001\n\u000b+ 1\u00001\n\u000b+ 2\u000b\r+1\n\u000b+ 3\u000b(\u000b\u00001)\n2\r2\n+:::+ (\u00001)\u000b1\n2\u000b\u000b\r\u000b\u00001+ (\u00001)\u000b1\n2\u000b+ 1\r\u000b\u0001\n:(3.20)\nAfter dropping negative terms, since \r=1\n5:1(Re)\u00001\u001c1 the RHS of (3.20) can be bounded above by a\nconstant,C\u000b, which depends on \u000b. Therefore\n(3.21)ZL\nL\u0000\rL\fd(z)dz\u0014C\u000bL\r\u000b+1;\nUsing\r=1\n5:1(Re)\u00001and inserting the above in the Theorem 3.1 imply\nh\"SMD (u)i\u0014\u0002\nC1+C2(Cs\u000e\nL)2(1\n5:1)3C\u000b\r\u000b\u00002\u0003U3\nL:\nUse the assumptions \u000b\u00152 and\r=1\n5:1(Re)\u00001\u001c1 imply\r\u000b\u00002\u00141 and now the corollary is proved.\n\u0003\nCorollary 3.4 is in accordance with the Kolmogorov theory of turbulence. It establishes that the combina-\ntion of SM with damping function \fd(z) given by (3.19) does not over dissipate, and the energy input rateDAMPING FUNCTIONS CORRECT OVER-DISSIPATION OF THE SMAGORINSKY MODEL 13\nU3\nLis balanced byh\"SMD (u)i. This estimate is consistent with the rate proven for the NSE in [4] and [5]; it\nis also dimensionally consistent.\nThe assumption \u000b\u00152 is a signi\fcant one in the analysis. When \u000b= 1 we obtain the following corollary.\nCorollary 3.5. Suppose\u000b= 1in Corollary 3.4, then\nh\"SMD (u)i\u0014\u0002\nC1+C2(Cs\u000e\nL)2Re\u0003U3\nL:\nProof. The proof follows that of \u000b\u00152 in the Corollary 3.4 except the inequality (3.21) is modi\fed to\nZL\nL\u0000\rL\fd(z)dz\u0014CL(Re)\u00002;\nfor\u000b= 1: \u0003\n4.Conclusion\nThe key parameter is \u000b= the order of contact of the damping function at the wall. Comparing h\"SMD (u)i'\nU3\nLfor\u000b\u00152 in Corollary 3.4 with h\"SMD (u)i'\u0002\nC1+C2(Cs\u000e\nL)2Re\u0003U3\nLfor\u000b= 1 in Corollary 3.5 suggests\nthat the model over dissipates \rows for \u000b= 1. If the upper bounds are sharp (an open problem), the accurate\nsimulation would need \u000b\u00152. The next logical step is to study h\"SMD (uh)iafter discretization by \fxed mesh\nhand extend the results in this paper, specially when the mesh does not resolve the boundary layers.\nReferences\n[1] L.C. Berselli, T. Iliescu, and W.J. 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Wilcox, Turbulence modeling for CFD , DCW Industries, Inc., 2006.\nDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA\nE-mail address :alp145@pitt.edu"    },    {        "title": "1107.2978v1.Aspects_of_General_Relativity__Pseudo_Finsler_extensions__Quasi_normal_frequencies_and_Multiplication_of_tensorial_distributions.pdf",        "content": "arXiv:1107.2978v1  [gr-qc]  15 Jul 2011Aspects of General Relativity:\nPseudo-Finsler extensions,\nQuasi-normal frequencies and\nMultiplication of tensorial\ndistributions\nby\nJozef Sk´ akala\nA thesis\nsubmitted to the Victoria University of Wellington\nin fulfilment of the\nrequirements for the degree of\nDoctor of Philosophy\nin Mathematics.\nVictoria University of Wellington\n2018Abstract\nThis thesis is based on three different projects, all of them are directly linked to\ntheclassical general theory of relativity, but they might have consequen ces for\nquantum gravity as well.\nThe first chapter deals with pseudo-Finsler geometric exten sions of the classi-\ncal theory, these being ways of naturally representing high -energy Lorentz sym-\nmetry violations. In this chapter we prove a certain type of “ no-go” result for\nsignificant number of theories. This seems to have important consequences for\nthe question of whether some weaker formulation of Einstein ’s equivalence prin-\nciple is sustainable, if (at least) certain types of Lorentz violations occur.\nThe second chapter deals with the problem of highly damped qu asi-normal\nmodes related to different types of black hole spacetimes. F irst, we apply to this\nproblem the technique of approximation by analytically sol vable potentials. We\nuse the Schwarzschild black hole as a consistency check for o ur method and de-\nrive many new and interesting results for the Schwarzschild -de Sitter (S-dS) black\nhole. One of the most important results is the equivalence be tween having a ratio-\nnal ratio of horizon surface gravities and periodicity of qu asi-normal modes. By\nanalysing the complementary set of analytic results derive d by the use of mon-\nodromy techniques we prove that all our theorems almost comp letely generalize\nto all the known analytic results. This relates to all the typ es of black holes for\nwhich quasi-normal mode results are currently known.\nThe third chapter is related to the topic of multiplication o f tensorial distri-\nbutions. We focus on an alternative approach to the ones pres ently known. The\nnew approach is fully based on the Colombeau equivalence rel ation, but techni-\ncally avoids the Colombeau algebra construction. The advan tage of this approach\nis that it naturally generalizes the covariant derivative o perator into the general-ized tensor algebra. It also operates with much more general concept of piecewise\nsmooth manifold, which is in our opinion natural to the langu age of distributions.Acknowledgments\nI’m deeply thankful to my supervisor Prof. Matt Visser for ma ny priceless dis-\ncussions and a lot of support throughout my PhD years. It was m y great pleasure\nto spend more than three years in a friendly environment crea ted by him.\nI’m also deeply thankful to the Victoria University of Welli ngton for support-\ning me with a Vice-Chancellor’s Strategic Research Scholar ship and also with\ntravel grants. Part of my expenses were covered by the Marsde n grant, the credit\nfor this goes again to my supervisor.\nFurthermore I would like to thank to the School of Mathematic s, Statistics and\nOperations Research for providing me with all the facilitie s and service, to the\ndepartment staff for being friendly and helpful whenever ne eded.\nI would like to thank to the students from the applied mathema tics group,\nparticularly to C´ eline Catto¨ en, Gabriel Abreu, Petarpa B oonserm, Bethan Cropp,\nNicole Walters, Jonathan Crook, Valentina Baccetti and Kyl e Tate, for strongly con-\ntributing to a friendly and inspiring environment. It was my pleasure to exchange\nideas with them and also to get to know them personally throug hout the years.\nI would also like to thank to many other physicists and mathem aticians with\nwhich I had opportunity to exchange ideas, especially to som e of the people from\nmy old Comenius University in Bratislava, Slovakia.\nFurthermore, I would like to thank to the thesis examiners fo r valuable sug-\ngestions that certainly improved the content of the thesis.\nLast, but not least I would like to thank to all the people that are close to me,\nin particular to my family, to my girlfriend and to all my frie nds. Life is extremely\neasy with their love and support.\niContents\nIntroduction 1\n1 Pseudo-Finsler extensions to gravity 4\n1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.2 Basics of (pseudo-)Finsler geometry . . . . . . . . . . . . . . . . . . 9\n1.3 Analogue model: Birefringent crystal . . . . . . . . . . . . . . . . . . 11\n1.3.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n1.3.2 Space versus space-time: Interpretations of the Fins ler and\nco-Finsler structures . . . . . . . . . . . . . . . . . . . . . . . 12\n1.3.3 Technical issues and problems . . . . . . . . . . . . . . . . . 17\n1.4 General bi-metric situations . . . . . . . . . . . . . . . . . . . . . . . 28\n1.5 General construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n1.5.1 General constraints . . . . . . . . . . . . . . . . . . . . . . . . 29\n1.5.2 How one proceeds for general bi-metric situations . . . . . . 30\n1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\n2 Analytic results for highly damped QNMs 34\n2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34\n2.2 Approximation by analytically solvable potentials . . . . . . . . . . 38\n2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38\n2.2.2 Schwarzschild black hole . . . . . . . . . . . . . . . . . . . . 41\n2.2.3 Schwarzschild - de Sitter black hole . . . . . . . . . . . . . . 4 6\n2.4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68\niiCONTENTS iii\n2.5 Monodromy results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69\n2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69\n2.5.2 Particular results . . . . . . . . . . . . . . . . . . . . . . . . . 70\n2.5.3 Rational ratios of surface gravities . . . . . . . . . . . . . . . 79\n2.6.1 Irrational ratios of surface gravities . . . . . . . . . . . . . . 82\n2.7.1 Analysis of particular cases . . . . . . . . . . . . . . . . . . . 87\n2.7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99\n3 Multiplication of tensorial distributions 100\n3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100\n3.2 Overview of the present state of the theory . . . . . . . . . . . . . . 104\n3.2.1 The theory of Schwartz distributions . . . . . . . . . . . . . . 104\n3.2.2 The standard Rntheory of Colombeau algebras . . . . . . . 108\n3.2.3 Distributions in the geometric approach . . . . . . . . . . . . 113\n3.2.4 Colombeau algebra in the geometric approach . . . . . . . . 114\n3.2.5 Practical application of the standard results . . . . . . . . . . 120\n3.3 How does our approach relate to current theory? . . . . . . . . . . . 122\n3.4 New approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125\n3.4.1 The basic concepts/definitions . . . . . . . . . . . . . . . . . 12 5\n3.4.2 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129\n3.4.3 Generalized tensor fields . . . . . . . . . . . . . . . . . . . . 131\n3.4.4 The relation of equivalence ( ≈) . . . . . . . . . . . . . . . . . 137\n3.4.5 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . 149\n3.4.6 Basic discussion of previous results and open questio ns . . . 159\n3.4.7 Some notes on the initial value problem within the part ial\ndifferential equivalence relations ( ≈) onD′m\nn(M). . . . . . . 161\n3.5 What previous results can be recovered, and how? . . . . . . . . . . 164\n3.5.1 Generalization of some particular statements . . . . . . . . . 164\n3.5.2 Relation to practical computations . . . . . . . . . . . . . . . 166\n3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167CONTENTS iv\nConclusions 168\nA Bi-refringent crystals 170\nA.1 Basic characteristics of the crystal media . . . . . . . . . . . . . . . . 170\nA.2 Group velocity and ray equation . . . . . . . . . . . . . . . . . . . . 1 71\nA.3 Phase velocity and Fresnel equation . . . . . . . . . . . . . . . . . . 173\nA.4 Connecting the ray and the wave vectors . . . . . . . . . . . . . . . 175\nA.5 Optical axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175\nB Special functions: some important formulas 178\nB.1 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . 178\nB.2 Gamma function identities and approximations . . . . . . . . . . . 178\nB.3 Hypergeometric function identities . . . . . . . . . . . . . . . . . . . 179\nB.4 Bessel function identities and expansions . . . . . . . . . . . . . . . 179\nTable of symbols and terms for Chapter 3 182\nPublications and papers 186\nBibliography 188Introduction\nThis thesis is based on three different projects. All of thes e three projects lie some-\nwhere on the boundary between the classical general theory of relativity and the\nso far unknown quantum theory of gravity. All of them deal wit h the phenom-\nena that can be fully understood by classical (non-quantum) language, but at the\nsame time describe, or at least strongly indicate, high ener gy modifications of the\nclassical theory, mostly due to quantum gravity.\nEver since Einstein’s general theory of relativity was esta blished as a highly\nsuccessful theory of gravity, physicists have also realize d its unavoidable limits.\nThe first set of problems is due to the theory itself. As is prov en by rigorous math-\nematical theorems, physically reasonable situations lead in general relativity to\nsolutions containing singularities. The other set of probl ems is due to the fact that\nthe theory is classical . To be able to consistently describe the fullinteraction be-\ntween quantum fields and gravity, one naturally needs to go be yond the classical\nlanguage and somehow quantize gravity. These two sets of pro blems are, how-\never, closely related. Singularities appear in the situati ons where we do notexpect\nthe classical description to be relevant.\nThere are presently many ideas, suggestions and conjecture s about the quan-\ntum theory of gravity. They are related to the many different approaches to the\nproblem that have appeared over the last decades, some of the m having quite\ndifferent backgrounds. (These are approaches such as strin g theory, loop quan-\ntum gravity, non-commutative geometries, etc.) Despite th e fact that the basic ap-\nproaches are very different, there is a wide agreement about the results obtained\nwithin the semi-classical regime. (For example, the effect of Hawking radiation.)\n1Surprisingly, the results of semi-classical approach alre ady seem to give us con-\nsiderable information about some of the objects of fullquantum gravity, such as\nblack holes. These results (such as the relation between hor izon area and black\nhole entropy) must be recovered by every quantum theory of gr avity, that has\nambitions to be correct.\nFirst two chapters in my thesis present results obtained in c lose collaboration\nwith my supervisor Prof. Matt Visser. They were published in various journals\n(see the list of publications at the end of the thesis). The la st chapter is related to\nmy own work and its significantly shortened version will be pr epared for journal\nsubmission in the immediate future.\nThe first chapter of the thesis deals with some possible classical geometric ex-\ntensions of the general theory of relativity. Such geometri c extensions are related\nto possible high-energy Lorentz symmetry violations, by ma ny physicists being\nassumed to be one of the possible effects of the future quantu m gravity theory. To\nsearch for a link between some generalized geometry and poss ible high-energy\nLorentz violations seems to be, with respect to the spirit of Einstein’s equivalence\nprinciple, a very natural approach.\nThe second chapter deals with a semi-analytic approach to th e highly damp-\ned quasi-normal modes of various classical black hole space -times. It is in fact\na topic from classical general relativity, but it is also of p articular interest of the\nquantum gravity community. This is because the asymptotic q uasi-normal modes\nbehaviour is suspected [77, 110] to be connected to the area s pectrum of the quan-\ntum black holes.\nThe third chapter represents a topic from the field of mathema tical physics\nclosely related to the general theory of relativity. It offe rs new ideas about how\ntofully generalize the language of differential geometry into the d istributional\nframework. It is again a conceptual extension which has dire ct relevance for the\nclassical theory, but it might not be unreasonable to assume that it can have im-\nportant consequences for quantizing gravity as well.\nAt the end of the thesis we summarize our results. This is foll owed by ap-\npendices containing some of the most common physics and math ematics results\n2relevant for the calculations in this thesis.\n3Chapter 1\nPseudo-Finsler extensions to gravity\n1.1 Introduction\nPossible low-energy manifold-like limits of quantum gravi ty One of the most\nsignificant problems in the recent history of theoretical ph ysics is the problem\nof quantization of gravity. Most of the candidates for a quan tum gravity theory\nsuggest that the description of spacetime is at the fundamen tal level far from the\ntraditional concept of manifold.\nDespite this fact, one might be still interested in whether t hese theories have a\ndeeper manifold-like low-energy limit; one more subtle tha n the ordinary pseudo-\nRiemannian geometry. If they have such limit, it must be defin itely an extension\nof pseudo-Riemannian geometry, but in the same time it must b e “close to” the\nhighly successful concept of pseudo-Riemannian geometry.\nUltra-high energy violations of Lorentz symmetry The theoretical physics com-\nmunity has recently exhibited increasing interest in the po ssibility of ultra-high-\nenergy violations of Lorentz invariance [84, 85, 86, 87, 108 , 154, 155, 175]. Specif-\nically, recent speculations regarding Lorentz symmetry br eaking and/or funda-\nmental anisotropies and/or multi-refringence arise separ ately in the many and\nvarious approaches to quantum gravity.\n4CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 5\nSuch phenomena arise in loop quantum gravity [25, 59], strin g models [96,\n112, 113], and causal dynamical triangulations [1, 2], and a re also part and parcel\nof the “analogue spacetime” programme [15], and of many atte mpts at devel-\noping “emergent gravity” [16, 24, 97]. Recently, the ultra- high energy breaking\nof Lorentz invariance has been central to the Horava-Lifshi tz models [78, 79, 80,\n154, 155, 175]. Of course not all models of quantum gravity le ad to high-energy\nLorentz symmetry breaking, and the comments below should be viewed as ex-\nploring one particular class of interesting models.\nThe connection between Lorentz violations and geometry The extensions of\npseudo-Riemannian geometry can typically modify dispersi on relations, so one\ncan be interested in seeing if such modified dispersions rela tions can be natu-\nrally embedded in some extension of pseudo-Riemannian geom etry. These exten-\nsions could be then naturally viewed as a low energy manifold -like non-pseudo-\nRiemannian limit of a given quantum gravity approach.\nIn the other direction, if we wish to follow the spirit of Eins tein’s equivalence\nprinciple and develop a geometric spacetime framework for r epresenting Lorentz\nsymmetry breaking, either due to spacetime anisotropies or multi-refringence,\nthen it certainly cannot be standard pseudo-Riemannian geo metry.\nThis strongly suggests that carefully thought out extensio ns and modifications\nof pseudo-Riemannian geometry might be of real interest to b oth the general rel-\nativity and high-energy communities.\nWhy focus on the light cone structure? In particular, when attempting to gen-\neralize pseudo-Riemannian geometry, the interplay betwee n the “signal cones” of\na multi-refringent theory and the generalized spacetime ge ometry is an issue of\nconsiderable interest:\n•In multi-refringent situations it is quite easy to unify all the signal cones in\none single Fresnel equation that simultaneously describes all polarization\nmodes on an equal footing.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 6\n•In a standard manifold setting, where we retain the usual com mutative co-\nordinates, we shall see that it is natural to demand that each polarization\nmode can be assigned a specific geometric object. This object is in fact a\nLorentzian analogue of what mathematicians know as Finsler norm (see the\nnext section for details).\n•In standard general relativity the (single, unique) signal cone almost com-\npletely specifies the spacetime geometry — one needs only sup plement the\nsignal cone structure with one extra degree of freedom at eac h point in\nspacetime, an overall conformal factor, in order to complet ely specify the\nspacetime metric, and thereby completely specify the geome try. This is ul-\ntimately due to the fact that in standard pseudo-Riemannian geometry the\nscalar product is a simple bi-linear operation. Unfortunat ely in the more\ngeneral pseudo-Finsler geometry1life is more difficult, but one might still\nhave a hope that following the guideline of simplicity the li ght-cone behav-\nior could hold a crucial piece of information about the overa ll geometry.\nBi-refringent crystal and beyond Considerable insight into such Finsler-like\nmodels can be provided by considering the “analogue spaceti me” programme,\nwhere analogue models of curved spacetime emerge at some lev el from well un-\nderstood physical systems [15]. In particular, the physics of bi-axial bi-refringent\ncrystals [29] provides a particularly simple physical anal ogue model for the math-\nematical object introduced some 155 years ago by Bernhard Ri emann [132], and\nnow known as Finsler distance2(again see the next section for more details). We\nshall soon see that this mathematical object can reasonably easily be extended to\na Lorentzian signature pseudo-Finsler spacetime, with an a ppropriate pseudo-\nFinsler norm.\nNote that we are not particularly interested in the properti es of bi-refringent\n1For the details of what we mean by “pseudo-Finsler geometry” see the next section.\n2It must be emphasized that, despite many misapprehensions t o the contrary, uni-axial bire-\nfringent crystals are relatively uninteresting in this reg ard; they do not lead to Finsler 3-spaces,\nbut “merely” yield bi-metric Riemannian 3-geometries.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 7\ncrystals per se, we use them only as an exemplar of Finsler 3-s pace and Finsler\nspace-time, as a guidepost to more complicated things that m ay happen in Fins-\nlerian extensions to general relativity. As a result of this fact in the next step we\nshow that all our observations from the particular bi-refri ngent crystal case hold\nin general bi-metric situations3. This is because our ultimate goal is to be able\nto say something about the (presumed) low-energy manifold- like limit of what-\never quantum theory (or class of quantum theories) is leadin g to a bi-metric /\nbi-refringent theory approximately reproducing Einstein gravity.\nThe no-go result While the use of Finsler 3-spaces to describe crystal optics is\nreasonably common knowledge within the community of mathem aticians and\nphysicists studying Finsler spaces, it is very difficult to g et a clear and concise\nexplanation of exactly what is going on when one generalizes to Lorentzian sig-\nnature space-time. In particular the fact that any relativi stic formulation of Finsler\nspace needs to work in Lorentzian signature (- +++), instead of the Euclidean\nsignature (+ +++) more typically used by the mathematical co mmunity, leads to\nmany technical subtleties (and can sometimes completely in validate naive con-\nclusions). Moreover unlike the spacetime Finsler norm, defining a spacetime Finsler\nmetric is fraught with technical problems. These problems s eem to be fundamen-\ntal and hold in arbitrary bi-metric situations.\nSo the basic things we assert are:\n•What is exceedingly difficult, and we shall argue is in fact ou tright impossi-\nble within this framework, is to construct a unified and still simple formal-\nism that moves “off-shell” (off the signal cones).\n•This is a negative result, a “no-go theorem”, which we hope wi ll focus at-\ntention on what can and cannot be accomplished in any natural way when\ndealing with multi-refringent anisotropic Finsler-like e xtensions to general\nrelativistic spacetime.\n3For the difference between bi-refringence and bi-metricit y see for example [176].CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 8\nTo this end, our “no-go” result indicates that the popular as sumption that\nanisotropies and multi-refingence are likely to occur in “qu antum gravity” leads\nto significant difficulties for the Einstein equivalence pri nciple — since even the\nloosest interpretation of the Einstein equivalence princi ple would imply the neces-\nsity of a coherent formalism for dealing with all signal come s, and the spacetime\ngeometry, in some unified manner. We conclude that, despite t he fact that space-\ntime anisotropies and multi-refringence are very popularl y assumed to be natural\nfeatures of “quantum gravity”, and while these features hav e a straightforward\n“on-shell” implementation in terms of a suitably defined Fre snel equation, there\nis no natural way of extending them “off-shell” and embeddin g them into a single\nover-arching spacetime geometry.\nBut to remind the reader: if one steps outside of the usual man ifold picture,\neither by adopting non-commutative coordinates, or even mo re abstract choices\nsuch as spin foams, causal dynamical triangulations, or str ing-inspired models,\nthen the issues addressed in this chapter are moot — our consi derations are rele-\nvant only insofar as one is interested in the first nontrivial deviations from exact\nlow-energy Lorentz invariance, and only relevant insofar a s these first nontrivial\neffects can be placed in a Finsler-like setting.\nThe structure of this chapter This chapter begins with introducing the concept\nof Finsler geometry. This is followed by proving our “no-go” result for the partic-\nular example of the bi-refringent crystal analogue model. A fter this we show that\nthe result holds in arbitrary bi-metric situation. At the en d of this chapter we add\na section where we explore the general conditions, which any pseudo-Finslerian\ngeometry must fulfil in order to give bi-refringence. But as a consequence of our\n“no-go” result, such constructs represent only a “complica ted” and non-intuitive\nroute for how to recover bi-refringence by pseudo-Finsleri an geometry.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 9\n1.2 Basics of (pseudo-)Finsler geometry\nMathematically, we define a Finsler function (Finsler norm, Finsler distance func-\ntion) [21, 58] to be a C-valued function F(x,v)on the tangent bundle to a manifold,\nsuch that it is homogeneous of degree 1:\nF(x,κv) =κF(x,v), κ>0, x∈M, v∈TxM. (1.1)\nThis then allows one to define a notion of distance on the manif old, as the minimal\nvalue of the functional\nS(x(ti),x(tf)) =/integraldisplaytf\nti/vextendsingle/vextendsingle/vextendsingle/vextendsingleF/parenleftbigg\nx(t),dx(t)\ndt/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledt, (1.2)\nwhich is now guaranteed to be independent of the specific para meterization t.\nBy “pseudo-Finsler geometry” we mean Finsler geometry with Lorentzian sig-\nnature. Now by Lorenzian signature of the general Finsler me tric (see the equation\n1.4) we mean, that for any arbitrary vector taken as an argume nt of the metric we\nobtain matrix with (−+ ++) signature. A pseudo-Riemannian norm is only a\nspecial case of a pseudo-Finsler norm. For a pseudo-Riemann ian manifold with\nmetricgab(x)one would take\nF(x,v) =/radicalbig\ngab(x)vavb, (1.3)\nbut for a general pseudo–Finslerian manifold the function F(x,v)is arbitrary ex-\ncept for the 1-homogeneity constraint in vand the metric signature constraint.\nNote that in Euclidean signature (where gab(x,v)is taken to be for any arbitrary v\na positive definite matrix), the general Finsler function F(x,v)is typically smooth\nexcept atv= 0. In Lorentzian signature however, F(x,v)is typically non-smooth\nfor all null vectors — so that non-smoothness issues have gro wn to affect (and in-\nfect) the entire null cone (signal cone). As we shall subsequ ently see below, some-\ntimes a suitable higher algebraic power ,F2n(x,v), of the pseudo-Finsler norm is\nsmooth.\nTo ensure smoothness of the (pseudo-)Finsler metric , defined below, it is enou-\ngh to weaken the condition that F(x,v)shall be smooth and to demand only thatCHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 10\nthe square F2(x,v)be smooth, except possibly at v= 0. It is standard to define the\n(pseudo-)Finsler metric as\ngab(x,v)≡1\n2∂2[F2(x,v)]\n∂va∂vb(1.4)\nwhich then satisfies the constraint that it is homogeneous of order zero\ngab(x,κv) =gab(x,v), κ> 0. (1.5)\nThis can be viewed as a “direction-dependent metric”, and is clearly a significant\ngeneralization of the usual (pseudo-)Riemannian case. One can immediately see\nthat pseudo-Riemannian metric fulfills this definition with respect to the pseudo-\nRiemannian norm (1.3).\nAlmost all of the relevant mathematical literature has been developed for the\nEuclidean signature case. Because of this assumption, any m athematical result\nthat depends critically on the assumed positive definite nat ure of the matrix of\nmetric coefficients cannot be carried over into the physically interesting pseudo-\nFinsler regime, at least not without an independent proof th at avoids the posi-\ntive definite assumption. (Unfortunately it is not uncommon to find significant\nmathematical errors in the pseudo-Finsler physics literat ure due to neglect of this\nelementary point.) Basic references within the mathematic al literature include\n[21, 58].\nThe Legendre transformation between a vector tangent space at the point x\nand its dual: Vx→V∗\nx, is defined as\nlb(v)≡gab(x,v)va(1.6)\nThen the dual (pseudo-)Finsler norm F∗can be defined by the condition:\nF∗(l(v))≡F(v). (1.7)\nThe dual metric is again naturally obtained as:\ngab(x,v)≡1\n2∂2[F∗2(v,x)]\n∂va∂vb. (1.8)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 11\nAll this is a natural generalization from the (pseudo-)Riem annian case. The\nconstruction of a full (pseudo-)Finsler geometry is in gene ral significantly more\ncomplicated than in the (pseudo-)Riemannian subcase. But s ince the definition\nof objects like non-linear connection, Finsler connection , (etc.), is not needed for\nthe purpose of this chapter, it will be omitted here and left f or the specialized\nliterature (see for example [133]).\n1.3 Analogue model: Birefringent crystal\n1.3.1 Outline\nPurely for the purposes of developing a useful analogy, whic h we shall use as a\nguide to the mathematics we wish to develop, we will focus on t he optical physics\nof bi-axial bi-refringent crystals. After the basic definit ions are presented, we will\nshow how various purely spatial 3-space Finsler structures arise. (Many purely\ntechnical details, when not directly involved in the logic fl ow, will be relegated to\nthe appendix A.) We again emphasize that uni-axial bi-refri ngent crystals, which\nare what much of the technical literature and textbook prese ntations typically fo-\ncus on, are for our purposes rather uninteresting — uni-axia l bi-refringent crystals\n“merely” lead to bi-metric Riemannian space-times and are f rom a Finslerian per-\nspective “trivial”. Such crystals are only one particular e xample demonstrating\ngeneral difficulties with Finsler representation of bi-met ric theories. The gener-\nalization from this example to any bi-metric case is present ed in the following\nsection.\nWe shall soon see that even in three-dimensional space there are at least four\nlogically distinct Finsler structures of interest: On the t angent space each of the\ntwo photon polarizations leads, via study of the group veloc ity, to two quite dis-\ntinct Finsler spacetimes. On the co-tangent space each of th e two photon polariza-\ntions leads, via study of the phase velocity, to two quite dis tinct co-Finsler space-\ntimes. The inter-relations between these four structures i s considerably more sub-\ntle than one might naively expect.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 12\nAdditionally, (apart from some purely technical difficulti es along the optical\naxes in bi-axial crystals), each of these four 3-dimensiona l spatial Finsler struc-\ntures has a natural 4-dimensional extension to a spacetime p seudo-Finsler struc-\nture. Beyond that, there are reasonably natural ways of merg ing the two photon\npolarizations into “unified” Finsler and co-Finsler norms, closely related to the\nappropriate Fresnel equation, though the associated Finsl er metrics are consider-\nably more problematic — all these mathematical constructio ns do come with a\nprice — and we shall be careful to point out exactly where the t echnical difficul-\nties lie. Finally, using this well-understood physical sys tem as a template, we shall\n(in the spirit of analogue spacetime programme) then ask wha t this might tell us\nabout possible Finslerian extensions to general relativit y, and in particular to the\nsubtle relationship between bi-refringence and bi-metric ity, (or more generally,\nmulti-refringence and multi-metricity).\nSpecifically, we have investigated the possibility of wheth er one can usefully\nand cleanly deal with both Finsler structure (anisotropy) a nd multi-refringence\nsimultaneously. That is, given two (or more) “signal cones” : Is it possible to natu-\nrally and intuitively construct a “unified” pseudo-Finsler spacetime such that the\npseudo-Finsler metric specifies null vectors on these “sign al cones”, but has no\nother zeros or singularities? Our results are much less enco uraging than we had\noriginally hoped, and lead to a “no-go” result.\n1.3.2 Space versus space-time: Interpretations of the Fins ler and\nco-Finsler structures\nThe key physics point in bi-axial bi-refringent crystal opt ics is that the group ve-\nlocities, and the phase velocities, are both anisotropic an d depend on direction in\na rather complicated way [29]. Technical details that would detract from the flow\nof the text are relegated to appendix A.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 13\nFrom group velocity to pseudo-Finsler norms\nWe can summarize the situation by pointing out that the group velocity is given\nby\nv2\ng(n) =¯q2(n,n)±/radicalbig\n¯q2(n,n)2−¯q0(n,n) (n·n)\n¯q0(n,n), (1.9)\nwhere¯q2(n,n)and¯q0(n,n)are known quadratic functions of the direction nand\nare given as\n˜q0(n,n) =n2\nxv−2\nyv−2\nz+n2\nyv−2\nxv−2\nz+n2\nzv−2\nxv−2\ny; (1.10)\n˜q2(n,n) =1\n2(n2\nx(v−2\ny+v−2\nz)+n2\ny(v−2\nx+v−2\nz)+n2\nz(v−2\nx+v−2\ny)). (1.11)\nThe coefficients in these quadratic forms are explicit funct ions of the compo-\nnents of the 3×3permittivity tensor. (See appendix A.) The function vg(n)so\ndefined is homogeneous of degree zero in the components of n:\nvg(κn) =vg(n) =vg(ˆn). (1.12)\nThe homogeneous degree zero property should remind one of th e relevant feature\nexhibited by the Finsler metric. There is a natural connecti on between the concept\nof group velocity and the geometric objects on a tangent spac e (rather than a co-\ntangent space). This is given by the fact that group velocity describes how energy\npropagates.\nLet us now first define the quantities\nF3±(n) =||n||\nvg(n)=/radicalBig\n¯q2(n,n)∓/radicalbig\n¯q2(n,n)2−¯q0(n,n) (n·n), (1.13)\nor adopt the perhaps more transparent notation\nF3±(dx) =||dx||\nvg(dx)=/radicalBig\n¯q2(dx,dx)∓/radicalbig\n¯q2(dx,dx)2−¯q0(dx,dx) (dx·dx).(1.14)\nThe quantity 1/vg(n)appearing in (1.14) is in the literature often called “slow-\nness”. (1.14) is by inspection a 3-dimensional (Riemannian ) Finsler distance de-\nfined on space , having all the correct homogeneity properties, F3±(κdx) =|κ|F3±(dx).CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 14\nPhysically, the Finsler distance is in this situation the ti me taken for the wavepacket\nto travel a distance dx.\nTo now extend the construction given above to full (3+1) dime nsional space-\ntime, we first define a generic 4-vector\ndX= (dt;dx), (1.15)\nand then formally construct\nF4±(dX) =/radicalbig\n−(dt)2+F3±(dx)2. (1.16)\nThat is\nF4±(dX) =/radicalBigg\n−(dt)2+dx·dx\nvg(dx)2. (1.17)\nEven more explicitly, one may write\nF4±(dX)\n=/bracketleftBig\n−(dt)2+ ¯q2(dx,dx)∓/radicalbig\n¯q2(dx,dx)2−¯q0(dx,dx) (dx·dx)/bracketrightBig1/2\n. (1.18)\nThe null cones (signal cones) of F4±(dX)are defined by\nF4±(dX) = 0 ⇔ || dx||=vg(dx) dt. (1.19)\nSo far this has given us a very natural pair of (3+1)-dimensional pseudo–Finsler\nstructures in terms of the ray velocities corresponding to t he two photon polariza-\ntions.\nFor future use, let us now formally define the quantity\nds4={F4(dX)}4(1.20)\n={F4+(dX)F4−(dX)}2\n= (dt)4−2(dt)2¯q2(dx,dx)+ ¯q0(dx,dx) (dx·dx).\nThis certainly provides an example of a specific and simple 4th-root Finsler norm\nthat can naturally and symmetrically be constructed from th e two polarizationCHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 15\nmodes, and its properties (and defects) are certainly worth investigating. Phys-\nically the condition ds= 0 defines a double-sheeted conoid (a double-sheeted\ntopological cone) that is the union of the propagation cone o f the individual pho-\nton polarizations. This Finsler norm defines the Finsler geo metry naturally uni-\nfying the two original geometries. It is also very close to qu artic extension of the\nnotion of distance that Bernhard Riemann speculated about i n his inaugural lec-\nture (see [132]).\nHowever we shall soon see that when it comes to defining a Finsl er space-\ntime metric this construction nevertheless leads to a number of severe t echnical\ndifficulties; difficulties that can be tracked back to the fac t that we are working in\nnon-Euclidean signature.\nFrom phase velocity to pseudo-co-Finsler norms\nIn counterpoint, as a function of wave-vector the phase velo city is\nv2\np(k) =q2(k,k)±/radicalbig\nq2(k,k)2−q0(k,k) (k·k)\n(k·k), (1.21)\nwhere the quadratics q2(k,k)andq0(k,k)are now given by equations\nq0(k,k) =k2\nxv2\nyv2\nz+k2\nyv2\nxv2\nz+k2\nzv2\nxv2\ny; (1.22)\nand\nq2(k,k) =1\n2/parenleftbig\nk2\nx(v2\ny+v2\nz)+k2\ny(v2\nx+v2\nz)+k2\nz(v2\nx+v2\ny)/parenrightbig\n. (1.23)\nThis expression is homogeneous of order zero in k, so that\nvp(κk) =vp(k) =vp(ˆk). (1.24)\nAgain, we begin to see a hint of Finsler structure emerging. B ecausekis a wave-\nvector it transforms in the same way as the gradient of the pha se; thuskis most\nnaturally thought of as living in the 3-dimensional space of co-tangents to physical\n3-space. Let us now define a co-Finsler structure on that co-tangent space by\nG3±(k) =vp(k)||k||=/radicalBig\nq2(k,k)±/radicalbig\nq2(k,k)2−q0(k,k) (k·k).(1.25)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 16\nWe use the symbol Grather than Fto emphasize that this is a co-Finsler structure,\nand note that this object satisfies the required homogeneity property\nG3±(κk) =|κ|G3±(k). (1.26)\nNow let us go for a (3+1) dimensional spacetime interpretati on: Consider the 4-\nco-vector\nk= (ω;k), (1.27)\nand define\nG4±(k) =/radicalbig\n−ω2+G3±(k)2. (1.28)\nThat is\nG4±(k) =/radicalBig\n−ω2+vp(k)2(k·k). (1.29)\nMore explicitly\nG4±(k) =/bracketleftBig\n−ω2+q2(k,k)±/radicalbig\nq2(k,k)2−q0(k,k) (k·k)/bracketrightBig1/2\n. (1.30)\nWe again see that this object satisfies the required homogene ity property\nG4±(κk) =|κ|G4±(k), (1.31)\nso that this object is indeed suitable for interpretation as a co-Finsler structure.\nFurthermore the null co-vectors ofG4are defined by\nG4±(k) = 0 ⇔ω=vp(k)||k||, (1.32)\nwhich is exactly the notion of dispersion relation for allowed “on mass shell” wave-\n4-vectors that we are trying to capture. Thus G4±lives naturally on the co-tangent\nspace to physical spacetime, and we can interpret it as a pseu do-co-Finsler struc-\nture.\nWe can again define a “unified” quantity\nG4(k)4={G4+(k)G4−(k)}2\n=ω4−2ω2q2(k,k)+q0(k,k) (k·k). (1.33)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 17\nPhysically, the condition G4(k) = 0 simultaneously encodes both dispersion re-\nlations for the two photon polarizations. It defines a double -sheeted conoid (a\ndouble-sheeted topological cone) that is the union of the di spersion relations of\nthe individual photon polarizations. The vanishing of G4(k)can be viewed as a\nFresnel equation, and can indeed be directly related to Fres nel’s condition for the\npropagation of a mode of 4-wavenumber k= (ω;k). As is the case for F4(dX),\nwe shall soon see that this construction (once one tries to ex tract a spacetime co-\nFinsler metric ) nevertheless leads to a number of severe technical difficul ties; dif-\nficulties that can again be tracked back to the fact that we are now working in\nnon-Euclidean signature.\n1.3.3 Technical issues and problems\nThe situation as presented so far looks very pleasant and com pletely under control\n— and if what we had seen so far were all there was to the matter, then the study\nof pseudo-Finsler space-times would be very straightforwa rd indeed — but now\nlet us indicate where potential problems are hiding.\n•Note that up to this stage we have not established any direct c onnection\nbetween the Finsler functions F3±(n)and the co-Finsler functions G3±(k).\nPhysically it is clear that they must be very closely related , but (as we shall\nsoon see) establishing the precise connection is tricky.\n•Furthermore, the transition from Finsler distance to Finsler metric requires at\nleast two derivatives. Even in Euclidean signature this pla ces some smooth-\nness constraints on the Finsler distance, smoothness const raints that are non-\ntrivial and not always satisfied.\n•Especially, there are problematic technical issues involv ing the 4-dimension-\nalspacetime Finsler and co-Finsler metrics — certain components of the m et-\nric are infinite, and this time the potential pathology is wid espread. (In\nLorentzian-like signature situations potential problems tend to infect the en-\ntire null cone.)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 18\nThe Finsler and co-Finsler 3-metrics\nThe standard definition used to generate a Finsler metric from a Finsler distance is\nto set:\ngij(n) =1\n2∂2[F3±(n)2]\n∂ni∂nj, (1.34)\nwhich in this particular case implies\ngij(n) =1\n2∂2[¯q2(n,n)∓/radicalbig\n¯q2(n,n)2−¯q0(n,n) (n·n)]\n∂ni∂nj. (1.35)\nIt is convenient to rewrite the quadratics as\n¯q2(n,n) = [¯q2]ijninj; (1.36)\n¯q0(n,n) = [¯q0]ijninj; (1.37)\nsince then we see\ngij(n) = [¯q2]ij∓(discriminant contributions) . (1.38)\nUnfortunately we shall soon see that the contributions comi ng from the discrimi-\nnant are both messy, and in certain directions, ill-defined. This is obvious from the\nfact that squares of both Finsler functions are not even ever ywhere differentiable.\nSimilarly we can construct a Finsler co-metric:\nhij(k) =1\n2∂2[G3±(k)2]\n∂ki∂kj, (1.39)\nwhich specializes to\nhij(k) =1\n2∂2[q2(k,k)∓/radicalbig\nq2(k,k)2−¯q0(k,k) (k·k)]\n∂ki∂kj. (1.40)\nIt is again convenient to rewrite the quadratics as\nq2(k,k) = [q2]ijkikj; (1.41)\nq0(k,k) = [q0]ijkikj; (1.42)\nsince then we see\nhij(k) = [q2]ij∓(discriminant contributions) . (1.43)\nAgain we shall soon see that the contributions coming from th e discriminant are,\nin certain directions, problematic.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 19\nTechnical problems with the Finsler 3-metric\nConsider the (ray) discriminant\n¯D= ¯q2(n,n)2−¯q0(n,n) (n·n). (1.44)\nThere are three cases of immediate (mathematical) interest :\nIsotropic: Ifvx=vy=vzthen¯D= 0; in this case the two Finsler functions\nF±are equal to ech other. F3(dx)then describes an ordinary Riemannian geome-\ntry, andF4(dX)an ordinary pseudo–Riemannian geometry. This is the standa rd\nsituation, and is for our current purposes physically uninteresting.\nUni-axial: If one of the principal velocities is distinct from the other two, then\nwe can without loss of generality set vx=vy=voandvz=ve. The discriminant\nthen factorizes into a perfect square\n¯D=/braceleftbigg(v2\no−v2\ne)(n2\nx+n2\ny)\n2v2\nov2\ne/bracerightbigg2\n. (1.45)\nIn this case it is immediately clear that both F3±(dx)reduce to simple quadratics,\nand so describe two ordinary Riemannian geometries. Indeed\nF3+(n) =n·n\nv2o; (1.46)\nand\nF3−(n) =n2\nx+n2\ny\nv2\ne+n2\nz\nv2\no. (1.47)\nIn the language of crystal optics voandveare the “ordinary” and “extraordinary”\nray velocities of a uni-axial birefringent crystal. In geom etrical language the two\nphoton polarizations “see” distinct Riemannian 3-geometr iesF3±(dx)and distinct\npseudo-Riemannian 4-geometries F4±(dX)— this situation is referred to as “bi-\nmetric”. This situation is for our current purposes physically uninteresting.\nBi-axial: The full power of the Finsler approach is only needed for the bi-axial\nsituation where the three principal velocities are distinc t. This is the only situation\nof real physical interest for us, as it is the only situation t hat leads to a non-trivial\nFinsler metric. In this case we can without loss of generalit y orient the axes so thatCHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 20\nvx> vy> vz. There are now two distinct directions in the x–zplane where the\ndiscriminant vanishes — these are the called the (ray) optic al axes. After some ma-\nnipulations that we relegate to Appendix A.5, the discrimin ant can be factorized\nas\n¯D=(v2\nx−v2\nz)2\n4v4\nxv4\nz×/bracketleftbig\n(n·n)−(¯e1·n)2/bracketrightbig/bracketleftbig\n(n·n)−(¯e2·n)2/bracketrightbig\n, (1.48)\nwhere the two distinct (ray) optical axes are\n¯e1,2=/parenleftBigg\n±vy\nvx/radicalBigg\nv2x−v2y\nv2\nx−v2\nz; 0 ;vy\nvz/radicalBigg\nv2y−v2z\nv2\nx−v2\ny/parenrightBigg\n. (1.49)\nNote that ¯e1,2are unit vectors (in the ordinary Euclidean norm) so that the dis-\ncriminant ¯Dvanishes for any n∝¯e1,2, and does not vanish anywhere else. We\ncan thus introduce projection operators ¯P1and¯P2and write\n¯P1(n,n) = (n·n)−(¯e1·n)2; (1.50)\n¯P2(n,n) = (n·n)−(¯e2·n)2. (1.51)\nCombining this with our previous results:\n{F3±(n)}2= ¯q2(n,n)∓(v2\nx−v2\nz)\n2v2\nxv2\nz/radicalBig\n¯P1(n,n)¯P2(n,n). (1.52)\nIf we now calculate the Finsler metric [g3±(n)]ijwe shall rapidly encounter tech-\nnical difficulties due to the discriminant term. To make this a little clearer, let us\ndefine\n[¯P3(n)]ij=∂2/radicalBig\n¯P1(n,n)¯P2(n,n)\n∂ni∂nj, (1.53)\nsince then\n[g3(n)]ij= [¯q2(n)]ij∓(v2\nx−v2\nz)\n2v2xv2z[¯P3(n)]ij. (1.54)\nTemporarily suppressing the argument n, we have\n[¯P3]ij=1\n2∂i\n∂j¯P1/radicalBigg\n¯P2\n¯P1+∂j¯P2/radicalBigg\n¯P1\n¯P2\n. (1.55)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 21\nA brief computation now yields the rather formidable result\n[¯P3]ij=1\n2\n∂i∂j¯P1/radicalBigg\n¯P2\n¯P1+∂i∂j¯P2/radicalBigg\n¯P1\n¯P2\n\n+1\n4/bracketleftBigg\n∂i¯P1∂j¯P2+∂i¯P2∂j¯P1/radicalbig¯P1¯P2−∂i¯P1∂j¯P1¯P1/2\n2\n¯P3/2\n1−∂i¯P2∂j¯P2¯P1/2\n1\n¯P3/2\n2/bracketrightBigg\n=1\n2/radicalbig¯P1¯P2/bracketleftbig\n∂i∂j¯P1¯P2+∂i∂j¯P2¯P1/bracketrightbig\n+1\n4/radicalbig¯P1¯P2/bracketleftbigg\n∂i¯P1∂j¯P2+∂i¯P2∂j¯P1−∂i¯P1∂j¯P1¯P2\n¯P1−∂i¯P2∂j¯P2¯P1\n¯P2/bracketrightbigg\n.(1.56)\nFrom this expression it is clear that along either optical ax is, (as long as the op-\ntical axes are distinct, which is automatic in any bi-axial s ituation), some of the\ncomponents of [¯P3]ij, and therefore some of the components of the Finsler metric\n[g3±]ij= [¯q2]ij±(constant) ×[¯P3]ij, will be infinite .\nTo see this in an invariant way, let uandwbe two 3-vectors and consider\n[¯P3(n)](u,w) = [¯P3]ijuiwj. (1.57)\nAfter a brief computation:\n[¯P3(n)](u,w) =1\n2/radicalbig¯P1(n,n)¯P2(n,n)/bracketleftbig¯P1(u,w)¯P2(n,n)+¯P2(u,w)¯P1(n,n)/bracketrightbig\n+1\n4/radicalbig¯P1(n,n)¯P2(n,n)/bracketleftBigg\n¯P1(n,u)¯P2(n,w)+¯P2(n,u)¯P1(n,w)\n−¯P1(n,u)¯P1(n,w)¯P2(n,n)\n¯P1(n,n)−¯P2(n,u)¯P2(n,w)¯P1(n,n)\n¯P2(n,n)/bracketrightBigg\n.(1.58)\nThis quantity will tend to infinity as ntends to either optical axis provided:\n•The optical axes are distinct.\n(If the optical axes are coincident then ¯P1=¯P2and so¯P3degenerates to\n[¯P3(n)](u,w)→¯P1(u,w) =¯P2(u,w). (1.59)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 22\nOne recovers the [ for our purposes physically uninteresting] result for a uni-\naxial crystal.)\n•One is not considering the special case u=w=n.\n(In this particular special case ¯P3degenerates to\n[¯P3(n)](n,n)→/radicalBig\n¯P1(n,n)¯P2(n,n), (1.60)\nwhich is well-behaved on either optical axis.)\nIn summary:\n•The spatial Finsler 3-metric is [g3±]ijgenerically ill-behaved on either optical axis .\n•This feature will also afflict the spacetime pseudo-Finsler 4-metric [g4±]abde-\nfined by suitable derivatives of the Finsler 4-norm F4±.\n•This particular feature is annoying, but seems only to be a te chnical problem\nto do with the specifics of crystal optics, it does not seem to u s to be a critical\nobstruction the developing a space-time version of Finsler geometry. Ulti-\nmately it arises from the fact that the “null conoid” is given by a quartic; this\nleads to two topological cones that for topological reasons always intersect,\nthis intersection defining the optical axes.\n•It is the technical problems associated with the (3+1) spacetime Finsler metric,\nto be discussed below, which much more deeply concern us.\nTechnical problems with the co-Finsler 3-metric\nThe phase discriminant\nD=q2(k,k)2−q0(k,k) (k·k), (1.61)\narising from the Fresnel equation (and considerations of th e phase velocity) ex-\nhibits features similar to those arising for the ray discrim inant. There are three\ncases:CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 23\nIsotropic: If the crystal is isotropic, then D= 0. (This again is for our purposes\nphysically uninteresting.)\nUni-axial: If the crystal is uni-axial, then Dis a perfect square\nD=/braceleftbigg(v2\no−v2\ne)(k2\nx+k2\ny)\n2/bracerightbigg2\n, (1.62)\nand so the co-Finsler structures G3±are both Riemannian:\nG3+(k) =v2\nok·n; (1.63)\nG3−(k) =v2\ne(k2\nx+k2\ny)+v2\nok2\nz. (1.64)\n(This situation again is for our purposes physically uninteresting.)\nBi-axial: Only in the bi-axial case are the co-Finsler structures G3±“truly” Fins-\nlerian. There are now two distinct (phase) optical axes (wav e-normal optical axes)\nalong which the discriminant is zero, these optical axes bei ng given by\nˆe1,2=/parenleftBigg\n±/radicalBigg\nv2x−v2y\nv2x−v2z; 0 ;/radicalBigg\nv2y−v2z\nv2x−v2z/parenrightBigg\n, (1.65)\nin terms of which the phase discriminant also factorizes\nD=(v2\nx−v2\nz)2\n4/bracketleftbig\n(k·k)−(k·e1)2/bracketrightbig/bracketleftbig\n(k·k)−(k·e2)2/bracketrightbig\n. (1.66)\nThe co-Finsler norm is then (now using projection operators P1andP2based on\nthe phase optical axes e1,2)\n{G3±(k)}2=q2(k,k)∓(v2\nx−v2\nz)\n2/radicalbig\nP1(k,k)P2(k,k). (1.67)\nThe co-Finsler metric is defined in the usual way\n[h3±]ij(k) =1\n2∂2[G3±(k)2]\n∂ki∂kj. (1.68)\nThis now has the interesting “feature” that some of its compo nents are infinite\nwhen evaluated on the (phase) optical axes. That is: The co-F insler 3-metric is\n[h3±]ijgenerically ill-behaved on either optical axis. This featu re will also afflict\nthe pseudo-co-Finsler 4-metric [h4±]abdefined by suitable derivatives of the Finsler\n4-normG4±.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 24\nTechnical problems with the (3+1) spacetime interpretatio n\nThe (3+1)-dimensional spacetime objects that give the best way how to merge two\n(3+1) pseudo-Finsler norms and pseudo-co-Finsler norms in to one geometry are\nthe quantities\nF4(dX)4={F4+(dX)F4−(dX)}2; (1.69)\nand\nG4(k)4={G4+(k)G4−(k)}2. (1.70)\nas defined in equation (1.21) and (1.33). This is tantamount t o taking\nF4(dX) =/radicalbig\nF4+(dX)F4−(dX); (1.71)\nand\nG4(k) =/radicalbig\nG4+(k)G4−(k). (1.72)\nNowF4(dX)andG4(k)are by construction perfectly well behaved Finsler and\nco-Finsler norms , with the correct homogeneity properties — and with the nice\nand concise physical interpretation that the vanishing of F4(dX)defines a double-\nsheeted “signal cone” that includes both polarizations, wh ile the vanishing of\nG4(k)defines a double-sheeted “dispersion relation” (“mass shel l”) that includes\nboth polarizations. (Thus F4(dX)andG4(k)successfully unify the “on-shell” be-\nhaviour of the signal cones in a Fresnel-like manner.)\nWhile this is not directly a “problem” as such, the norms F4(dX)andG4(k)do\nhave the interesting “feature” that they pick up non-trivia l complex phases: Since\nF4±(dX)2is always real, (positive inside the propagation cone, nega tive outside),\nit follows that F4±(dX)is either pure real or pure imaginary. But then, thanks to\nthe additional square root in defining F4(dX), one has:\n•F4(dX)is pure real inside both propagation cones.\n•F4(dX)is proportional to√\ni=(1+i)√\n2between the two propagation cones.\n•F4(dX)is pure imaginary outside both propagation cones.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 25\nSimilar comments apply to the co-Finsler norm G4(k).\nA considerably more problematic point is this: In the usual E uclidean signa-\nture situation the Finsler norm is taken to be smooth everywh ere except for the\nzero vector — this is usually phrased mathematically as “smo oth on the slit tan-\ngent bundle”. What we see here is that in a Lorentzian-like si gnature situation\nthe Finsler norm cannot be smooth as one crosses the propagat ion cones — what\nwas in Euclidean signature a feature that only arose at the ze ro vector of each tan-\ngent space has in Lorentzian-like signature situation grow n to affect (and infect)\nall null vectors. The Finsler norm is here at best “smooth on t he tangent bundle\nexcluding the null cones”. (In a mono-refringent case the sq uared norm, F4(dX)2,\nis smooth across the propagation cones, but in the bi-refrin gent case one has to go\nto the fourth power of the norm, F4(dX)4, to get a smooth function.)\nA “no go” result: Unfortunately, when attempting to bootstrap these two rea-\nsonably well-behaved norms to Finsler and co-Finsler metrics one encounters ad-\nditional and more significant complications. We have alread y seen that there are\nproblems with the spatial 3-metrics [g3±]ij(n)and[h3±]ij(k)on the optical axes,\nproblems which are inherited by the single-polarization sp acetime (3+1)-metrics\n[g4±]ab(n)and[h4±]ab(k), again on the optical axes.\nBut now the spacetime (3+1)-metrics\n[g4]ab(n) =1\n2∂2[F4(n)2]\n∂na∂nb, (1.73)\nand\n[h4]ab(k) =1\n2∂2[G4(k)2]\n∂ka∂kb, (1.74)\nboth have (at least some) infinite components — [g4]ab(n)has infinities on the en-\ntire signal cone, and [h4]ab(k)has infinities on the entire mass shell. Since the ar-\ngument is essentially the same for both cases, let us perform a single calculation:\ngab=1\n2∂a∂b/radicalBig\n[F2\n+F2\n−] (1.75)\n=1\n4∂a/bracketleftbigg\n∂b[F2\n+]F−\nF++∂b[F2\n−]F+\nF−/bracketrightbigg\n, (1.76)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 26\nso that\ngab=1\n4/bracketleftbigg\n∂a∂b[F2\n+]F−\nF++∂b∂b[F2\n−]F+\nF−/bracketrightbigg\n+1\n2/bracketleftbigg\n∂aF+∂bF−+∂aF−∂bF+−∂aF+∂bF+F−\nF+−∂aF−∂bF−F+\nF−/bracketrightbigg\n.(1.77)\nThat is, tidying up:\ngab=1\n2/bracketleftbigg\n(g+)abF2\nF1+(g−)abF1\nF2/bracketrightbigg\n+1\n2/bracketleftbigg\n∂aF+∂bF−+∂aF−∂bF+−∂aF+∂bF+F−\nF+−∂aF−∂bF−F+\nF−/bracketrightbigg\n.(1.78)\nThe problem is that this “unified” metric gab(n)has singularities on both of the\nsignal cones. The (relatively) good news is that the quantit ygab(n)nanb=F2(n),\nand so on either propagation cone F→0, soF(n)itself has a well defined limit.\nBut now let nabe the vector the Finsler metric depends on, and let wabe some\nother vector. Then\ngab(n)nawb=1\n2nawb∂a∂b[F2] (1.79)\n=1\n2wb∂b[F2] (1.80)\n=1\n2wb∂b/radicalbig\nF+F− (1.81)\n=1\n4wb/bracketleftbigg\n∂b[F2\n+]F−\nF++∂b[F2\n−]F+\nF−/bracketrightbigg\n(1.82)\n=1\n2/braceleftbigg\n([g+]abnawb)F−\nF++([g−]abnawb])F+\nF−/bracerightbigg\n. (1.83)\nThe problem now is this: g+andg−have been carefully constructed to be individ-\nually well defined and finite (except at worst on the optical ax es). But now as we\ngo to propagation cone “ +” we have\ngab(n)nawb→1\n2([g+]abnawb)F−\n0=∞, (1.84)\nand as we go to the other propagation cone “ −” we have\ngab(n)nawb→1\n2([g−]abnawb)F+\n0=∞. (1.85)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 27\nSo at least some components of this “unified” Finsler metric gab(n)are unavoid-\nably singular on the propagation cones. Related (singular) phenomena have pre-\nviously been encountered in multi-component BECs, where mu ltiple phonon mo-\ndes can interact to produce Finslerian propagation cones [1 83].\nThings are just as bad if we pick uandwto be twovectors distinct from “the\ndirection we are looking in”, n. In that situation\ngab(n)uawb=1\n2/bracketleftBigg\ng+(u,w)F−\nF++g−(u,w)F+\nF−\n+g+(u,n)g−(w,n)+g+(w,n)g−(u,n)\nF+F−\n−g+(u,n)g+(w,n)F−\nF3\n+−g−(u,n)g−(w,n)F+\nF3\n−/bracketrightBigg\n. (1.86)\nAgain, despite the fact that both g+andg−have been very carefully set up to be\nregular on the propagation cones (except for the known, isol ated, and tractable\nproblems on the optical axes), the “unified” metric gab(n)is unavoidably singular\nthere — unless, that is, you only choose to look in the nndirection.\nIf we give up the condition of mathematical simplicity in the construction of\nthe “unified” space-time (co-)Finsler norms, then F4andG4do not have to fac-\ntorize into a product of powers of the Finsler functions for i ndividual polariza-\ntions. We could then look for more complicated ways of buildi ng “unified” Finsler\nand co-Finsler structures (constrained mainly by giving th e correct light propaga-\ntion cones in the birefringent crystal), and we might be able to find an appropri-\nate pseudo-Finsler geometry (which might also fulfill some a dditional reasonable\nphysical conditions). The necessary conditions for such a g eometry are relatively\neasy to formulate (and this is done in the last section), but b ecause of the lack of\nany intuitive interpretation, and the corresponding lack o f direct physical motiva-\ntion, the physical meaning of such an approach is highly doub tful [145].CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 28\n1.4 General bi-metric situations\nIn the previous part one could observe that the arguments (gi ven the way they\nwere constructed) might apply for a large set of different si tuations than the bi-\nrefringent crystal. In fact they generally hold within the c lass of bi-metric theories.\nTake arbitrary bi-metric theory. Such theories contain two distinct pseudo-\nRiemannian metrics g±\nab, so we can define two distinct “elementary” pseudo-Rie-\nmannian norms\nF±(x,v) =/radicalBig\ng±\nab(x)vavb. (1.87)\nSuppose one now wants a combined Finsler norm that simultane ously encodes\nboth signal cones — then the natural thing is always to take\nF(x,v) =/radicalbig\nF+(x,v)F−(x,v);gabcd=g+\n(abg−\ncd). (1.88)\nThis construction for F(x,v)is automatically 1-homogeneous in v. Generally\nthe vanishing of F(x,v)correctly encodes the two signal cones. So this definition\nofF(x,v)provides a perfectly good Finsler norm.\nHowever only from the fact that the individual F±are not positive definite\nautomatically follows that:\n•F(x,v)is proportional to√\ni=1+i√\n2between the two propagation cones,\nhence picks up a non-trivial phase.\n•As a result this Finsler norm is at best “smooth on the tangent bundle ex-\ncluding the null cones”.\nFor the “unified” metric, from the fundamental definition we s ee\ngab(x,v) =\n1\n2/bracketleftbigg\ng+\nabF−\nF++g−\nabF+\nF−/bracketrightbigg\n+1\n2/bracketleftbigg\n∂aF+∂bF+−∂aF+∂bF−F−\nF+−∂aF−∂bF−F+\nF−/bracketrightbigg\n.(1.89)\nThis “unified” and “natural” Finsler metric gab(x,v)has necessarily singularities\non both of the signal cones.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 29\nThe quantity gab(x,v)vavb=F2(x,v)is still being well defined, but if vais the\nvector the Finsler metric depends on, and wasome other (arbitrary) vector, then\ngab(x,v)vawb\n=1\n2vawb∂a∂b/radicalBig\nF2\n+F2\n−=1\n2wb∂b/radicalbig\nF+F−\n=1\n4wb/bracketleftbigg\n∂b[F2\n+]F−\nF++∂b[F2\n−]F+\nF−/bracketrightbigg\n=1\n2/bracketleftbigg\n(g+\nabvawb)F−\nF++(g−\nabvawb)F+\nF−/bracketrightbigg\n. (1.90)\nAs we go to propagation cone “+”, necessarily\ngab(x,v)vawb→1\n2(g+\nabvawb)F−\n0=∞, (1.91)\nand as we go to the other propagation cone “-”, necessarily\ngab(x,v)vawb→1\n2(g−\nabvawb)F+\n0=∞. (1.92)\nSo at least some components of this “unified” Finsler metric gab(x,v)are (com-\npletely generally) unavoidably singular on the propagatio n cones. And again,\nthings are just as bad if we pick uandwto be two vectors distinct from v.\nIn other words we see that all the problems which arose in the p articular case\nof bi-refringent bi-axial crystal are in fact also features of arbitrary bi-metric situ-\nation.\n1.5 General construction\n1.5.1 General constraints\nIn the previous sections we observed that an “intuitive” way of encoding bi-\nmetricity into pseudo-Finsler geometry leads to significan t problems which seem\ninevitable and unavoidable. The other problematic part whi ch was omitted is\nwhether such intuitive construction leads to appropriate p seudo-Riemannian limitCHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 30\nfor, (in some frame), slowly moving objects. Here we discuss all the general con-\nstrains on a pseudo-Finsler geometry recovering arbitrary bi-refringence.\nWhat are the specific physical constraints given by bi-refri ngent theories on\nour geometry? We consider it meaningful to impose the follow ing constraints:\na) Locally there must exist a coordinate frame in which holds the following:\nAt any arbitrary point from the domain of these coordinates t ake within this\nframe the purely time-oriented, (v,0)vector. Then on some neighborhood\nof this vector the pseudo-Finsler norm approaches the pseud o-Riemannian\nnorm. Also each “constant proper time” hypersurface, speci fied byF2(v) =\n−τ2\n0, is connected, and contains vectors (v,0).\nb) The pseudo-norm must break Lorentz invariance as encoded in the “Fres-\nnel” equation giving the bi-refringence.\nLet us discuss these conditions a little bit: The first is just requirement that we\nwant to recover, in some frame, low-energy physics as we know it (in a pseudo-\nRiemann form). It also means that if we have a massive particl e, it should not\nbe able to have two distinct sets of four-velocities between which it is not able\nto undergo a smooth transition. The last constraint it gives is that any massive\nparticle should be able to move arbitrarily slowly with resp ect to the used “pre-\nferred” frame. The second condition is trivial: it is the spe cific contribution of the\ngiven bi-refringent model (unlike the first condition, whic h can be considered as\ngeneric).\n1.5.2 How one proceeds for general bi-metric situations\nCo-metric structure construction\nTake the general bi-metric situation and define\nG(x,k)4=G+(x,k)2G−(x,k)2(1.93)\nwhere\nG±(x,k)2=gab\n±(x)kakb. (1.94)CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 31\nThis suggests a generic candidate for the pseudo-co-Finsle r norm, which can be\nwritten as ˜G(x,k)2∝G(x,k)4or more precisely ˜G(x,k)2=M1(x,k)G(x,k)4,\nwhereM1(x,k)is an otherwise arbitrary R-valued function fulfilling following\nconstraints:\na)∂[M1(x,k)G(x,k)4]\n∂kiis a bijection from V∗→V;\nb)M1(x,k)is inka homogeneous map of degree −2;\nc)M1(x,k)is smooth on V∗/{0};\nd)M1(x,k)is inside both signal cones negative, outside both positive , between\nthem nonzero;\ne)M1(x,k)G(x,k)4must approximate the pseudo-Riemann norm (squared) lo-\ncally in some coordinates for kclose to(v,0)and such vectors must lie on\nsome hypersurface (mass-shell) given by: M1(x,k)G(x,k)4=−m2, which\nmust be always connected.\nThese conditions follow trivially from the conditions we im posed on any physi-\ncally meaningful Finslerian geometry.\nMetric structure construction\nTake the given bi-metric situation and define\nF(x,v)4=F+(x,v)2F−(x,v)2, (1.95)\nwith\nF±(x,v)2=g±\nab(x)vavb. (1.96)\nThe generic Finsler pseudo-norm is in this case expressed ex actly in the same way\nas was the co-Finsler pseudo-norm in the previous case, by th e function ˜F(x,v)2=\nM2(x,v)F(x,v)4, whereM2(x,v)is again an arbitrary function fulfilling exactly\nthe same conditions as the M1(x,k)function, we just have to exchange V∗forV.CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 32\nInterconnecting the metric structure with the co-metric st ructure\nNow the last step in putting constraints on geometric constr uction is to relate the\nFinsler and co-Finsler structures. The items which are as ye t undetermined are the\nfunctionsM1(x,v), andM2(x,k). So they have to fulfill the last condition, which is\nthe condition of forming the full united geometry. This cond ition is given by the\nLegendre transform relation:\n/bracketleftBigg\n˜G/parenleftBigg\nx,∂[˜F(x,v)2]\n∂vi/parenrightBigg/bracketrightBigg2\n=˜F(x,v)2. (1.97)\nIn our language, the function M2(x,v)must be connected to M1(x,k)by:\nM1/parenleftbigg\nx,∂[M2(x,v)F(x,v)4]\n∂vi/parenrightbigg\nG/parenleftbigg\nx,∂[M2(x,v)F(x,v)4]\n∂vi/parenrightbigg4\n=M2(x,v)F(x,v)4. (1.98)\nThus we see that if we have somehow found an appropriate M2(x,v), thenM1(x,k)\nwill be uniquely determined4.\nTo find solutions of such an abstract (although precisely for mulated) mathe-\nmatical exercise is not an easy task. At this early stage it is quite doubtful whether\ngeneral solutions exist.\n1.6 Conclusions\nIn conclusion: Our “no-go” theorem suggests that while pseu do-Finsler space-\ntimes are certainly useful constructs, in bi-refringent si tuations it does not appear\npossible to naturally and intuitively construct a “unified” pseudo-Finsler space-\ntime such that the pseudo-Finsler metric is null on both “sig nal cones”, but has\nno other zeros or singularities — it seems physically more ap propriate to think of\nphysics as taking place in a single topological manifold tha t carries two distinct\npseudo-Finsler metrics, one for each polarization mode.\n4It holds both ways, also M2(x,k)is uniquely determined by M1(x,k).CHAPTER 1. PSEUDO-FINSLER EXTENSIONS TO GRAVITY 33\nThis means that in the case of high-energy Lorentz violation s it is highly doubt-\nful whether one has to follow the idea to try to formulate the t heory geometrically.\nThere is no analogy between this case and the very successful Minkowski geomet-\nric formulation of special theory relativity. This leads to a considerable concern\nabout how the equivalence principle might be sustainable if we admit the high-\nenergy Lorentz violations.\nPhysically this might suggest that high energy Lorentz viol ations – if they ac-\ntually occur in nature – should be “universal”. That is, all p articles should see\nthesame Lorentz violation. In this case one could at least have a sing le “signal\ncone”, avoid all the problems that were demonstrated in this chapter and so have\na reasonable chance of satisfying the Einstein equivalence principle.Chapter 2\nAnalytic results for highly damped\nquasi-normal modes\n2.1 Introduction\nBrief introduction into the topic Black hole quasi-normal modes are physically\nintuitive gravitational perturbations of various types of black hole spacetimes.\nTake black hole spacetimes with spherical symmetry, where t he metric is of the\nform:\ngµν=−f(r)dt2+f(r)−1dr2+r2dΩ2. (2.1)\nOne decomposes the general perturbation into a tensorial ge neralization of spher-\nical harmonics (for a detailed introduction into the topic s ee [123]):\nhµν=∞/summationdisplay\nl=0l/summationdisplay\nm=−l10/summationdisplay\nn=1Ψn\nlm(t,r)/braceleftBig\n(Yn\nlm)µν(θ,φ)/bracerightBig\n. (2.2)\nThey split into scalar, vector and tensor perturbations. Mo reover we can split\nthem with respect to parity into two sets: axial and polar perturbations. Both fol-\n34CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 35\nlow the one-dimensional equation:1\n∂2Ψlm(t,x)\n∂t2−∂2Ψlm(t,x)\n∂x2+V(x)Ψlm(t,x) = 0. (2.3)\nHere we transformed the coordinate rto the tortoise coordinate x≡/integraltextdr\nf(r). The\nV(x)function is called the Regge-Wheeler potential (axial pert urbations) or Zerilli\npotential (polar perturbations). Both Regge-Wheeler and Z erilli potentials (in gen-\neral different) depend on the background geometry, the spin of the perturbation,\nand the wave-mode number of the perturbation.\nIf we are interested only in perturbations with the “harmoni c” time depen-\ndenceΨlm(t,x) =eiωtψlm(x), we obtain:\n∂2ψlm(x)\n∂x2−(V(x)−ω2)ψlm(x) = 0. (2.4)\nDespite the fact that this equation formally reminds us of the Schr¨ odinger equa-\ntion forL2Hilbert space self-adjoint operator eigenfunctions, it sh ould be kept in\nmind that here the whole situation is very different. As a res ult of this fact the ω2\n“eigenvalues” are in general non-real numbers.\nThe quasi-normal modes (QNMs) are solutions of (2.4) with th e boundary con-\nditions giving purely outgoing radiation:\nψlm(x)→C±e±iωxx→ ∓∞. (2.5)\nThese particular perturbations are of a general interest be cause:\n•The boundary conditions are the physically intuitive ones.\n•It can be proven [123] that after some time scale these pertur bations become\ndominant within arbitrary black hole perturbation.\n•In the field of quantum gravity, there exists Hod’s conjectur e [77], and more\nrecently Maggiore’s [110] conjecture, concerning the conn ection between the\nhighly damped QNMs and the black hole area spectrum.\n1This equation reminds us of the Klein-Gordon equation with a potential, describing a relativis-\ntic scalar particle scattering.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 36\n•The QNMs related to asymptotically anti-de Sitter black hol es are interesting\nfor people working with AdS/CFT correspondence.\nThe second point suggests QNMs describe the characteristic “sound” of black\nholes. They are in principle observable in black hole oscill ations and ring-down\nphenomena. As we see the modes are characterized by the QNM fr equenciesω\n(QNFs - quasinormal frequencies). It can be generally prove n that for the QNM\nfrequencies we have:\n•the quasinormal frequencies ωform an infinite but countable set,\n•theω-s are complex with Im(ω)>0, and hence describe stable perturba-\ntions,\n•they have real parts symmetrically spaced with respect to th e imaginary\naxis.\nThe literature lists many techniques that have been used to c alculate the QNM\nfrequencies. The basic ones are:\n•WKB inspired approximations [57, 74, 82, 90, 185];\n•phase-amplitude methods [3, 4];\n•continued fraction approximations [105, 106, 107];\n•monodromy techniques [51, 65, 118, 119];\n•Born approximations [42, 114, 115, 125];\n•approximation by analytically solvable potentials.\nSome of these techniques were used only to calculate the fund amental (least damp-\ned) QNM frequencies (like the approximation by the analytic ally solvable P¨ oschl-\nTeller potential [57]), other techniques were used to estim ate the asymptotic be-\nhavior of highly damped frequencies as well.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 37\nThe basic focus and results of our work The main focus of this work is the an-\nalytic results for highly damped QNM frequencies. These can be obtained by two\ndifferent methods: approximation by analytically solvabl e potentials, and mon-\nodromy techniques. Unfortunately the first method was previ ously used only\nto estimate the fundamental frequencies, and one of the main contributions of\nthis work is using this method to also explore the highly damp ed QNMs. Unlike\nthe method of approximation by analytically solvable poten tials, the monodromy\ntechnique was used many times to understand the asymptotic Q NM behavior.\nThere are striking similarities between our results obtain ed by the analytically\nsolvable potentials and the known monodromy results. These allow us to analyze\nthe monodromy results in a new way bringing much deeper under standing about\nthe behaviour of the asymptotic QNM frequencies encoded in t he monodromy\nformulae.\nA problem of special interest is the following: for the highl y damped frequen-\ncies the asymptotic behavior\nωn=ω0+in·gap+O(n−m), m> 0, (2.6)\n(where “gap” denotes some real constant), was often observe d [118, 119]. On the\nother hand situations have been observed where this behavio r seems to fail. The\nanalysis of the equations derived by both the analytically s olvable potentials and\nmonodromy techniques gives us an indication to when such beh avior is to be\nexpected. It also tells us how the gap spacing is given.\nThe structure of this chapter In the first part of this chapter we analyze the\nhighly damped QNMs by using the idea of approximation by anal ytically solv-\nable potentials. We verify our method by applying it to the Sc hwarzschild black\nhole, where the results are known and widely accepted. After that we explore the\nmuch less known Schwarzschild-de Sitter (S-dS) case. As a re sult of our approach\nwe will prove interesting theorems about the highly damped Q NM behavior for\nthe S-dS black hole. We will also discuss our results in the co ntext of black hole\nthermodynamics. In the second part we explore the complemen tary set of ana-CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 38\nlytic results obtained by monodromy calculations (related to many different types\nof black holes) and prove that all of the results follow the pa tterns discovered by\nour approximation. This means we are able to generalize our t heorems for almost\nevery analytic result presently known. This suggests the be havior discovered is\nvery generic also between different black hole spacetimes.\n2.2 Approximation by analytically solvable potentials\n2.2.1 Introduction\nAs previously mentioned, one of the ways to derive analytic a pproximate expres-\nsions for the highly damped QNMs is to approximate the real Re gge-Wheeler-\n/Zerilli potential by analytically solvable potentials. I n the past this method has\nbeen used to give a formula for the fundamental QNM frequenci es. Ferrari and\nMashoon [57] used the P¨ oschl-Teller (Eckart) potential\nV(x) =V0/cosh2(αx) (2.7)\nto approximate the Regge-Wheeler potential at the peak (by fi tting theV0andα\nparameters by the peak height and peak curvature). The QNM fr equencies of the\nP¨ oschl-Teller potential are given by the formula\nωn=±/radicalbigg\nV0−α2\n4−iα/parenleftbigg\nn+1\n2/parenrightbigg\n. (2.8)\nThe widely accepted result (see for example [119]) for highl y damped QNM\nfrequencies of Schwarzschild black hole is\nωn=±ln3\n2πκ+iκ/parenleftbigg\nn+1\n2/parenrightbigg\n+O/parenleftbig\nn−1/2/parenrightbig\n, (2.9)\nwhereκis the surface gravity at the black hole horizon. The interes ting observa-\ntion is the following:\n•the asymptotic formula (2.9) can be obtained by the appropri ate fitting of\nP¨ oschl-Teller potential, by setting α=κ,CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 39\n•such P¨ oschl-Teller potential qualitatively recovers the behavior of one tail2\nof the Regge-Wheeler potential.\nThis fact seems not to be a coincidence as one can also turn the logic around. The\nwavepacket formed of highly damped QNMs close to the peak wil l quickly spread\nout from the peak region to the region of the tails of the poten tial. That suggests\nthe tails are the most important factor in determining the hi ghly damped QNMs.\nThere is also a general observation that the wavelength (giv en by theRe(ω)) is\nhigher for highly damped modes, so this supports the view tha t these modes must\nbe more sensitive to the asymptotic behavior of the potentia l. So the general ex-\npectation is that a good approximation to the tails of the Regge-Wheeler/Zeri lli potentials\nshould give a good qualitative estimate for the behavior of t he highly damped quasinormal\nmodes .\nTo give this statement an exact meaning it is appropriate to e xactly define what\nwe mean by a good qualitative vs.quantitative asymptotic estimate. We say that\nsequence of QNFs ω1nquantitatively matches with the sequence ω2n(for the asymp-\ntotic QNFs), if\nlim\nn→∞(ω1n−ω2n) = 0. (2.10)\nWe say that sequence ω1nmatches qualitatively with the sequence ω2nif\nlim\nn→∞|ω1n−ω2n|\n|ω1n|= lim\nn→∞|ω1n−ω2n|\n|ω2n|= 0. (2.11)\nSo in the first case the “error” goes to zero, in the second case only the “relative\nerror” (error compared to the result) goes to zero. One can ob serve that if ω1nand\nω2nare both equispaced, then if they match quantitatively they must be equal. On\nthe other hand for qualitative matching only the same gap spacing is required (they\ncan have different ω0modes). The second statement can be proven immediately\n2By the “tail of the potential” we mean the potential in one of t he asymptotic regions, given\nas|x|>>0. By the “peak of the potential” we mean the potential in the re gion near its global\nmaximum, typically near x= 0.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 40\nfrom the definition\nlim\nn→∞|ω1n−ω2n|\n|ω1,2n|=|gap(ω1n)−gap(ω2n)|\n|gap(ω1,2n)|, (2.12)\nwheregap(ω1n),gap(ω2n)denote the “gap” constants from (2.6) related to sequences\nω1n,ω2n. Then the condition of qualitative matching implies\ngap(ω1n)−gap(ω2n) = 0, (2.13)\nand nothing else.\nAll this means the question whether the tails of the real pote ntials can be ap-\nproximated by piecewise smooth analytically tractable pot entials is of high inter-\nest. As we will see this is the case of 1 and 2 horizon situation s.\nOne horizon situations\nThis is the situation in the Schwarzschild geometry. Since i t is asymptotically flat,\nonly one side of the potential is approaching the (black hole ) horizon.\nFor the specific case of a Schwarzschild black hole the tortoi se coordinate is\ngiven by\ndr\ndx= 1−2m\nr;x(r) =r+2mln/bracketleftbiggr−2m\n2m/bracketrightbigg\n; (2.14)\nand the Regge–Wheeler potential is\nV(x(r)) =/parenleftbigg\n1−2m\nr/parenrightbigg/bracketleftbiggℓ(ℓ+1)\nr2+2m(1−s2)\nr3/bracketrightbigg\n. (2.15)\nHeresis the spin of the particle and ℓis the angular momentum of the specific\nwave mode under consideration, with ℓ≥s. Asx→ −∞ we haver→2mand\nV(x)→exp/parenleftbiggx−2m\n2m/parenrightbiggℓ(ℓ+1)+(1 −s2)\n(2m)2=V0exp(2κx), (2.16)\nwhereκis the black hole surface gravity. The Zerilli potential is m ore complicated,\nbut leads to the same asymptotic behavior. This specific beha viour in terms ofCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 41\nthe surface gravity generalizes beyond the Schwarzschild b lack hole and for an\narbitrary black hole in an asymptotically flat spacetime one has\nV(x)→\n\nV0−exp(−2κ|x|), x→ −∞;\nV0+(2m)2/x2, x→+∞.(2.17)\nFor the highly damped modes this behavior suggests we shall fi t the real poten-\ntials by a proper combination of P¨ oschl-Teller and 1/x2(inverted harmonic oscila-\ntor) potential.\nTwo horizon situations\nIf one turns to asymptotically de Sitter black holes (or more generally any two\nhorizon system) the situation is different, but following t he same patterns — the\nRegge–Wheeler potential and the Zerilli potential have the asymptotic behaviour\nV(x)→\n\nV0−exp(−2κ−|x|), x → −∞;\nV0+exp(−2κ+|x|), x →+∞;(2.18)\nwhere the two surface gravities are now (in general) distinc t. It is this situation\nthat we will model using a piecewise Eckart potential (P¨ oschl–Teller potential). A\nconsiderable amount of analytic information can be extract ed from this model,\ninformation which in the concluding discussion we shall att empt to relate back to\n“realistic” black hole physics.\n3 (and more) horizon situations\nThere is no simple or practicable way of dealing with three-h orizon situations\nusing semi-analytic techniques.\n2.2.2 Schwarzschild black hole\nPotential\nWe work with the Regge-Wheeler/Zerilli equationCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 42\n−ψ′′(x)+(V(x)−ω2)ψ(x) = 0. (2.19)\nThe potential recovering the tails behavior of the given Reg ge-Wheeler/Zerilli\npotential is\nV(x) =\n\nV0−sech2(x/b−)forx<0;\nV0+/(x+a)2forx>0.(2.20)\nHereb−is defined by b−≡κ−1. The potential can be discontinuous at the origin\nand the infinite peak can be placed anywhere in the negative pa rt of the real line,\nsoacan be taken to be an arbitrary positive real number.\nWavefunction\nWe are interested in the solutions of the Regge-Wheeler/Zer illi equation (2.4) with\nthe boundary constraints\nψ+(x→+∞)→C+e−iωx;ψ−(x→ −∞)→C−e+iωx, (2.21)\ndefining the quasi-normal modes. The solutions are (for V0+≥ −1/4)\nψ+(x) =C+/radicalbigg\nx+a\nω/bracketleftbig\nJα+(ω(x+a))−e−iα+πJ−α+(ω(x+a))/bracketrightbig\n, (2.22)\nwhereJα+(x), J−α+(x)are Bessel functions with\nα+≡/radicalbig\n1+4V0+ (2.23)\nand\nψ−=C−eiωx\n2F1/parenleftbigg1\n2+α−,1\n2−α−,1+ib−ω,1\n1+e−2x/b−/parenrightbigg\n, (2.24)\nwhere 2F1(...)is the hypergeometric function withCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 43\nα−≡\n\n/radicalBig\n1\n4−V0b2\n− forV0b2\n−<1/4;\ni/radicalBig\nV0b2\n−−1\n4forV0b2\n−>1/4.(2.25)\nJunction condition\nWe need to match the functions ψ±(x)and their derivatives at the origin. We can\ncombine the two matching conditions in such a way, that one of them will give\nthe equation for quasinormal modes, while the other will onl y relate the normal-\nization constants C±, hence will not be of pressing interest. The first equation\nwe obtain by equating the ψ′\n±(0)/ψ±(0)ratios. (In special situations where ψ±(0)\nmight accidentally equal zero one might need to perform a spe cial case analysis.\nThe generic situation is ψ±(0)∝\\e}atio\\slash= 0, and will prove sufficient for almost everything\nwe need to calculate.)\nFor theψ′\n+(0)/ψ+(0)ratio it holds:\nψ′\n+(0)\nψ+(0)=−1\n2a+\n+ω\n2·J(α+−1)(aω)−J(α++1)(aω)−e−iα+π/bracketleftbig\nJ(−α+−1)(aω)−J(−α++1)(aω)/bracketrightbig\nJα+(aω)−e−iα+πJ−α+(aω).(2.26)\nThis equation is obtained after using known differential id entities for Bessel\nfunctions (see identity B.9 in appendix B.4).\nThe key step in obtaining the ratio ψ′\n−(0)/ψ−(0)is to calculate the logarithmic\nderivative. By choosing the variable z= 1/(1+e−2x/b−), note thatx= 0maps into\nz= 1/2. Then using the Leibnitz rule and the chain rule one has:\nψ′\n−(0)\nψ−(0)=iω+1\n2b−d ln/braceleftbig\n2F1/parenleftbig1\n2+α−,1\n2−α−,1+ib−ω,z/parenrightbig/bracerightbig\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=1/2.(2.27)\nInvoking the differential identity (B.8) in appendix B.3, w e see\nψ′\n−(0)\nψ−(0)=iω2F1/parenleftbig1\n2+α−,1\n2−α−,ib−ω,z/parenrightbig\n2F1/parenleftbig1\n2+α−,1\n2−α−,1+ib−ω,z/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=1/2.(2.28)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 44\nNow using Bailey’s theorem (B.7) to evaluate the hypergeome tric functions at\nz→1\n2we have the exact result\nψ′\n−(0)\nψ−(0)=2\nb−Γ(α−+iωb−\n2+3\n4) Γ(−α−+iωb−\n2+3\n4)\nΓ(α−+iωb−\n2+1\n4) Γ(−α−+iωb−\n2+1\n4). (2.29)\nThe asymptotic approximations\nThe exact junction condition we wish to apply at x= 0is\nψ′\n+(0)\nψ+(0)=ψ′\n−(0)\nψ−(0). (2.30)\nWe see that equating the right sides of (2.29) and (2.26) give s a messy equation and\nthe presence of the Gamma and Bessel functions above makes th is exact junction\ncondition intractable. Fortunately, from the beginning we are interested only in\nthe asymptotic quasi-normal modes. This means Im(ω)≫0andIm(ω)≫Re(ω).\nIn such a case we can use asymptotic formulas for both the Bess el and Gamma\nfunctions.\nFor Bessel functions one takes the asymptotic expansion (B. 10) from the ap-\npendix B.4. In the dominant (zero-th) order, for Re(x)bounded and Im(x)→\n∞, the functions P(α,x)andQ(α,x), (see B.11 and B.12 in appendix B.4), be-\nhave asP(α,x)→1, whileQ(α,x)→0. Hence for x≈Im(x)≫0the\nexpansion (B.10) gives:\nJα(x)≈/radicalbigg\n2\nπxcos/parenleftBig\nx−πα\n2−π\n4/parenrightBig\n. (2.31)\nBy substituting this into (2.26), after some additional cal culations using trivial\ntrigonometric relations we obtain a very simple asymptotic formula:\nψ′\n+(0)\nψ+(0)≈ −1\n2a+Im(ω)≈Im(ω). (2.32)\nThe case (2.29) is slightly more difficult. In this case consi der the following: if\nωhas a large positive imaginary part, then the Gamma function arguments above\ntend towards the negative real axis, a region where the Gamma function has manyCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 45\npoles. This is computationally inconvenient, and to obtain a more tractable result\nit is extremely useful to use the reflection formula (B.4) of a ppendix B.2 to derive\nΓ(α−+iωb−\n2+3\n4)\nΓ(α−+iωb−\n2+1\n4)=Γ(1−α−+iωb−\n2−1\n4) sin(π[α−+iωb−\n2+1\n4])\nΓ(1−α−+iωb−\n2−3\n4) sin(π[α−+iωb−\n2+3\n4])\n=Γ(−α−+iωb−\n2+3\n4) sin(π[α−+iωb−\n2+1\n4])\nΓ(−α−+iωb−\n2+1\n4) sin(π[α−+iωb−\n2+1\n4]+π\n2)\n=Γ(−α−+iωb−\n2+3\n4)\nΓ(−α−+iωb−\n2+1\n4)×tan/parenleftbigg\nπ/bracketleftbiggα−+iωb−\n2+1\n4/bracketrightbigg/parenrightbigg\n.(2.33)\nThis leads to the exact result\nψ′\n−(0)\nψ−(0)=2\nb−Γ(−α−−iωb−\n2+3\n4) Γ(α−−iωb−\n2+3\n4)\nΓ(−α−−iωb−\n2+1\n4) Γ(α−−iωb−\n2+1\n4)\n×tan/parenleftbigg\nπ/bracketleftbiggα−+iωb−\n2+1\n4/bracketrightbigg/parenrightbigg\ntan/parenleftbigg\nπ/bracketleftbigg−α−+iωb−\n2+1\n4/bracketrightbigg/parenrightbigg\n.(2.34)\nIfωhas a large positive imaginary part, then the Gamma function arguments\nabove now tend towards the positive real axis, a region where the Gamma func-\ntion is smoothly behaved — all potential poles in the logarit hmic derivative have\nbeen isolated in the trigonometric functions. We can also us e one of the trigono-\nmetric identities (B.1) of appendix B.1 to rewrite this as\nψ′\n−(0)\nψ−(0)=2\nb−Γ(−α−−iωb−\n2+3\n4)Γ(α−−iωb−\n2+3\n4)\nΓ(−α−−iωb−\n2+1\n4)Γ(α−−iωb−\n2+1\n4)×cos(πα−)−cos(π[iωb−+1/2])\ncos(πα−)+cos(π[iωb−+1/2]),\nwhich we can rewrite (still an exact result) as\nψ′\n−(0)\nψ−(0)=2\nb−Γ(−α−−iωb−\n2+3\n4) Γ(α−−iωb−\n2+3\n4)\nΓ(−α−−iωb−\n2+1\n4) Γ(α−−iωb−\n2+1\n4)×cos(πα−)+sin(iπωb−)\ncos(πα−)−sin(iπωb−).(2.35)\nWe have already seen how to eliminate the Bessel functions in the approxima-\ntion thatIm(ω)is very large and Re(ω)bounded. Now we have to do the same\nwith Gamma functions. Fortunately this is possible as well. As long as we are\nprimarily focussed on the highly damped QNFs ( Im(ω)→ ∞ ,Re(ω)bounded)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 46\nwe can employ the Stirling approximation in the form (B.6) in dicated in appendix\nB.2 to deduce\nΓ(−α−−iωb−\n2+3\n4)\nΓ(−α−−iωb−\n2+1\n4)=/radicalbigg\n−α−−iωb−\n2+1\n4×/bracketleftbigg\n1+O/parenleftbigg1\nIm(ωb−)/parenrightbigg/bracketrightbigg\n=/radicalbigg\nIm(ω)b−\n2×/bracketleftbigg\n1+O/parenleftbigg1\nIm(ωb−)/parenrightbigg/bracketrightbigg\n. (2.36)\nThis allows us to approximate the junction condition by\n1 =cos(πα−)+sin(iπωb−)\ncos(πα−)−sin(iπωb−)(2.37)\nleading directly to\nsin(iπωb−) = 0 (2.38)\nand hence\nω=in\nb−=inκ. (2.39)\nDiscussion\nThis result means that the QNMs are purely imaginary and equi spaced with the\ngap being given by the surface gravity at the black hole horiz on. This result can be\nconsidered to qualitatively (but not fully quantitatively ) match with the well known\nresult (2.9).\n2.2.3 Schwarzschild - de Sitter black hole\nPotential\nThe model we are interested in investigating is\n−ψ′′(x)+(V(x)−ω2)ψ(x) = 0, (2.40)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 47\nwith\nV(x) =\n\nV0−sech2(x/b−)forx<0;\nV0+sech2(x/b+)forx>0.(2.41)\nHere we again take b±≡κ−1\n±. We will allow a discontinuity in the potential at\nx= 0. The standard case that is usually dealt with is for\nV0−=V0+=V0;b−=b+=b;V(x) =V0\ncosh2(x/b). (2.42)\nA related model where V0−=V0+=V0butb+∝\\e}atio\\slash=b−has been explored by\nSuneeta [159], but our current model is more general, and we w ill take the analysis\nmuch further.\nWavefunction\nWe start again by imposing quasi-normal boundary condition s (outgoing radia-\ntion boundary conditions)\nψ+(x→+∞)→C+e−iωx;ψ−(x→ −∞)→C−e+iωx. (2.43)\nOn each half line ( x <0, andx >0) the exact wavefunction (see especially page\n405 of the article by Beyer [23]) is now:\nψ±(x) =C±e∓iωx\n2F1/parenleftbigg1\n2+α±,1\n2−α±,1+ib±ω,1\n1+e±2x/b±/parenrightbigg\n, (2.44)\nwhere\nα±≡\n\n/radicalBig\n1\n4−V0b2\n± forV0b2\n±<1/4;\ni/radicalBig\nV0b2\n±−1\n4forV0b2\n±>1/4.(2.45)\nThe same as in previous section, “all” we need to do is to appro priately match\nthese wavefunctions at the origin.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 48\nJunction condition\nThe junction condition is again\nψ′\n+(0)\nψ+(0)=ψ′\n−(0)\nψ−(0)(2.46)\nand the same remarks related to zero values of ψ±(0)as in the previous section\nhold. The ratio ψ′\n±(0)/ψ±(0)can be rewritten through Gamma functions as:\nψ′\n±(0)\nψ±(0)=∓2\nb±Γ(α±+iωb±\n2+3\n4)Γ(−α±+iωb±\n2+3\n4)\nΓ(α±+iωb±\n2+1\n4)Γ(−α±+iωb±\n2+1\n4). (2.47)\nThe asymptotic approximations\nHere both sides of the junction condition contain Gamma func tions in the same\nway as the right side of the junction condition from the previ ous section. Hence\nall the analysis can be repeated (rewriting some of the Gamma functions through\nthe basic Gamma functions relations and using the Stirling a pproximation for the\nasymptotic modes). Following exactly the same arguments on e obtains the fol-\nlowing approximation to the junction condition:\ncos(πα+)+sin(iπωb+)\ncos(πα+)−sin(iπωb+)=−cos(πα−)+sin(iπωb−)\ncos(πα−)−sin(iπωb−), (2.48)\nwhich is accurate up to fractional corrections of order O(1/Im(ωb±)). This is now\nanapproximate “quantization condition” for calculating the QNFs being as ymp-\ntotically increasingly accurate for the highly-damped mod es.\nQNF condition\nThe asymptotic QNF condition above can, by cross multiplica tion and the use of\ntrigonometric identities, be rewritten in any one of the equ ivalent forms:\nsin(−iπωb+)sin(−iπωb−) = cos(πα+)cos(πα−); (2.49)\nsinh(πωb+)sinh(πωb−) =−cos(πα+)cos(πα−); (2.50)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 49\ncos(−iπω[b+−b−])−cos(−iπω[b++b−]) = 2 cos(πα+)cos(πα−); (2.51)\ncosh(πω[b+−b−])−cosh(πω[b++b−]) = 2 cos(πα+)cos(πα−). (2.52)\nWhich particular form one chooses to use is a matter of taste t hat depends on ex-\nactly what one is trying to establish. It is sometimes useful to split the asymptotic\nQNF condition into real and imaginary parts. To do so note\ncos(A+iB) = cos(A)cos(iB)−sin(A)sin(iB)\n= cos(A)cosh(B)−isin(A)sinh(B), (2.53)\nso that\ncos(−iπω[b+−b−]) = cos( Im(ω)π[b+−b−])cosh(Re(ω)π[b+−b−])\n−isin(Im(ω)π|b+−b−|)sinh(Re(ω)π|b+−b−|).(2.54)\nTherefore the asymptotic QNF condition implies both\ncos(Im(ω)π[b+−b−])cosh(Re(ω)π[b+−b−])\n= cos(Im(ω)π[b++b−])cosh(Re(ω)π[b++b−])+2 cos(πα+)cos(πα−),(2.55)\nand\nsin(Im(ω)π|b+−b−|)sinh(Re(ω)π|b+−b−|)\n= sin(Im(ω)π[b++b−])sinh(Re(ω)π[b++b−]). (2.56)\nWe shall now seek to apply this QNF condition, in its many equi valent forms, to\nextract as much information as possible regarding the distr ibution of the QNFs.\nSome general observations\nWe shall start with some general observations regarding the QNFs.\n1. Note that the α±are either pure real or pure imaginary.\n2. Consequently cos(πα+)cos(πα−)is always pure real ∈[−1,+∞).CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 50\n3. Ifcos(πα+)cos(πα−)>0, then there are no pure real QNFs.\nProof: Consider equation (2.50) and note that under this condition the LHS\nis positive while the RHS is negative.\n4. Ifcos(πα+)cos(πα−)<0, then there is a pure real QNF.\nProof: Consider equation (2.50) and note that under this condition the RHS\nis positive. The LHS is positive and by continuity there will be a a real root\nω∈(0,∞).\n5. Ifcos(πα+)cos(πα−)>1, then there are no pure imaginary QNFs.\nProof: Consider equation (2.49) and note that under this condition the LHS\n≤1while the RHS >1.\n6. There are infinitely many pure imaginary solutions to thes e asymptotic QNF\nconditions provided cos(πα+)cos(πα−)≤Q(b+,b−)≤1; that is, whenever\ncos(πα+)cos(πα−)is “sufficiently far” below 1.\nProof: Define\nQ(b+,b−) = max\nω/braceleftbiggcos(|ω|π[b+−b−])−cos(|ω|π[b++b−])\n2/bracerightbigg\n≤1. (2.57)\nThen by inspection equation (2.51) will have an infinite numb er of pure\nimaginary solutions as long as\ncos(πα+)cos(πα−)≤Q(b+,b−). (2.58)\n7. For any purely imaginary ωthere will be some choice ofb±,α±that makes\nthis a solution of the asymptotic QNF condition.\nProof: Consider the specific case\nω=i(α++1\n2)\nb+=i(α−+1\n2)\nb−, (2.59)\nand note this satisfies the QNF condition but enforces only tw o constraints\namong the four unknowns b±,V0±.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 51\nRational ratios for the falloff\nSupposeb+/b−is rational, that is\nb+\nb−=p+\np−∈Q, (2.60)\nand suppose we now define b∗by\nb+=p+b∗;b−=p−b∗;b∗= hcf(b+,b−), (2.61)\nwhere “hcf” means “highest common factor”. ( p+,p−are relatively prime.3) Then\nthe asymptotic QNF condition is given by\nsin(−iωπp+b∗)sin(−iωπp−b∗) = cos(πα+)cos(πα−). (2.62)\nIfω∗isanyspecific solution of this equation, then\nωn=ω∗+i2n\nb∗=ω∗+i2nlcm/parenleftbigg1\nb+,1\nb−/parenrightbigg\n(2.63)\nwill also be a solution. (Here “lcm” stands for “least common multiplier”.) But are\nthese the only solutions? Most definitely not. For instance, consider (for rational\nb+/b−) the set of all QNFs for which\nIm(ω)<1\nb∗, (2.64)\nand label them as\nω0,aa∈ {1,2,3...N}. (2.65)\nThen the set of all QNFs decomposes into a set of families\nωn,a=ω0,a+in\nb∗;a∈ {1,2,3...N};n∈ {0,1,2,3...}; (2.66)\nwhereNis yet to be determined. But for rational b+/b−we can rewrite the QNF\ncondition as\ncos(−iωπb∗|p+−p−|)−cos(−iωπb∗[p++p−]) = 2cos(πα+)cos(πα−).(2.67)\n3Later in this chapter will the symbols pi/pjautomatically mean a rational number given by\nthe relatively prime integers pi,pj.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 52\nNow define z= exp(ωπb∗), then the QNF condition can be rewritten as\nz|p+−p−|+z−|p+−p−|−z[p++p−]−z−[p++p−]= 4cos(πα+)cos(πα−), (2.68)\nor equivalently\nz2[p++p−]−z|p+−p−|+[p++p−]\n−4cos(πα+)cos(πα−)z[p++p−]−z−|p+−p−|+[p++p−]+1 = 0,(2.69)\nthat is\nz2[p++p−]−z2p+−z2p−−4cos(πα+)cos(πα−)z[p++p−]+1 = 0. (2.70)\nThis is a polynomial of degree N= 2(p++p−), so it has exactly Nrootsza. In\nterms of equi-spaced families of QNM the situation is a littl e bit more tricky. If\np+·p−is odd then the gap spacing is only1\nb∗and the generic number of different\nfamilies is only p++p−, while ifp+·p−is even, then the gap is given by2\nb∗and the\ngeneric number of families is 2(p++p−). This can be summarized as:\nTheorem 2.3. Takeb+/b−=p+/p−∈Qandb∗= hcf(b+,b−). The QNFs are, with the\nimaginary part of the logarithm lying in [0,2π)given by\nωn,a=ln(za)\nπb∗+i2n\ngb∗a∈/braceleftbigg\n1,2,3...N≤2(p++p−)\ng/bracerightbigg\nn∈ {0,1,2,3,...},\n(2.71)\nwithg= 2 forp+·p−odd andg= 1 forp+·p−even.\nSo for rational b+/b−all modes form equi-spaced families, all families have\nthe same gap spacing and are characterized by distinct offse tsln(za)/(πb∗). That\nis:Arbitrary rational ratios of b+/b−automatically imply the ωn= offset + in·gap\nbehaviour.\nIrrational ratios for the falloff\nNow suppose b+/b−is irrational, that is\nb∗= hcf(b+,b−) = 0. (2.72)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 53\nThen all of the “families” considered above only have one ele ment\nω0,aa∈ {1,2,3...∞}. (2.73)\nThat is, there will be no “pattern” in the QNFs, and they will n ot be regularly\nspaced. (Conversely, if there is a “pattern” then b+/b−is rational.) Stated more\nformally, it is possible to derive a theorem as below.\nTheorem 2.4. Suppose we have at least one family of equi-spaced QNFs such t hat\nωn=ω0+in·gap, (2.74)\nthenb+/b−is rational.\nProof. If we have a family of QNFs of the form given in equation (2.74) then we\nknow that ∀n≥0\ncos(−iω0π|b+−b−|+nKπ|b+−b−|)−cos(−iω0π[b++b−]+nKπ|b++b−|)\n= cos(−iω0π|b+−b−|)−cos(−iω0π[b++b−]). (2.75)\nLet us write this in the form ∀n≥0\ncos(A+nJ)−cos(B+nL) = cos(A)−cos(B), (2.76)\nand realize that this also implies\ncos(A+[n+1]J)−cos(B+[n+1]L) = cos(A)−cos(B), (2.77)\nand\ncos(A+[n+2]J)−cos(B+[n+2]L) = cos(A)−cos(B). (2.78)\nNow appeal to the trigonometric identity (based on equation (B.3))\ncos(A+[n+2]J)+cos(A+nJ) = 2cos(J)cos(A+[n+1]J), (2.79)\nto deduce\ncos(J)cos(A+[n+1]J)−cos(L)cos(B+[n+1]L) = cos(A)−cos(B).(2.80)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 54\nThat is,∀n≥0we have both\ncos(A+[n+1]J)−cos(B+[n+1]L) = cos(A)−cos(B), (2.81)\nand\ncos(J)cos(A+[n+1]J)−cos(L)cos(B+[n+1]L) = cos(A)−cos(B).(2.82)\nThe first of these equations asserts that all the points\n/parenleftBig\ncos(A+[n+1]J),cos(B+[n+1]L)/parenrightBig\n(2.83)\nlie on the straight line of slope 1 that passes through the poi nt(0,cosB−cosA).\nThe second of these equations asserts that all the points\n/parenleftBig\ncos(A+[n+1]J),cos(B+[n+1]L)/parenrightBig\n(2.84)\nalso lie on the straight line of slope cos(J)/cos(L)that passes through the point\n(0,[cosB−cosA]/cosL). We then argue as follows:\n•IfcosJ∝\\e}atio\\slash= cosLthen these two lines are not parallel and so meet only at a\nsingle point, let’s call it (cosA∗,cosB∗), whence we deduce\ncos(A+[n+1]J) = cosA∗; cos(B+[n+1]L) = cosB∗. (2.85)\nBut then both JandLmust be multiples of 2π, and so cosJ= 1 = cos L\ncontrary to hypothesis.\n•IfcosJ= cosL∝\\e}atio\\slash= 1then we have both\ncos(A+[n+1]J)−cos(B+[n+1]L) = cos(A)−cos(B), (2.86)\nand\ncos(J)[cos(A+[n+1]J)−cos(B+[n+1]L)] = cos(A)−cos(B).(2.87)\nbut these are two parallel lines, both of slope 1, that never i ntersect unless\ncos(J) = 1 . ThuscosJ= 1 = cosLcontrary to hypothesis.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 55\n•We therefore conclude that both JandLmust be multiples of 2πso that\ncosJ= 1 = cosL(in which case the QNF condition is certainly satisfied).\nBut now\n|b+−b−|\nb++b−=J\nL∈Q, (2.88)\nand therefore\nb+\nb−∈Q. (2.89)\nThat is: Rational ratios of b+/b−are implied by the ωn= offset+ingapbehaviour .\nExplicit examples\n•Ifb+=b−=b∗, but we do not necessarily demand α+=α−, then the\nasymptotic QNFs are exactly calculable and are given by\nωn=icos−1{1−2 cos(πα+)cos(πα−)}\n2πb∗+in\nb∗. (2.90)\nProof: The asymptotic QNF condition reduces to\n1−cos(−i2πωb∗) = 2 cos(πα+)cos(πα−), (2.91)\nwhence\ncos(−i2πωb∗) = 1−2 cos(πα+)cos(πα−). (2.92)\nThis is easily solved to yield\n−i2πωnb∗= cos−1{1−2 cos(πα+)cos(πα−)}+2nπ, (2.93)\nwhence\nωn=icos−1{1−2 cos(πα+)cos(πα−)}\n2πb∗+in\nb∗. (2.94)\nComment: These QNFs are pure imaginary for cos(πα+)cos(πα−)≤1, and\noff-axis complex for cos(πα+)cos(πα−)>1. We can always, for convenience,CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 56\nchoose to define Re(cos−1(x))∈[0,2π); then because cos−1(·)is double val-\nued we see\nθ∈[0,π]andcosθ=ximpliescos−1(x) ={θ,π−θ}, (2.95)\nθ∈[π,2π)andcosθ=ximpliescos−1(x) ={θ,3π−θ}. (2.96)\nWith this notation\nωn=ω0+in\nb∗;ω0=icos−1{1−2 cos(πα+)cos(πα−)}\n2πb∗; (2.97)\nwith0≤Im(ω0)<1/b∗andn∈ {0,1,2,3,...}. Note that because of the\ndouble valued nature of cos−1(·)there are actually two branches of QNFs\nhiding in this notation — which we will need if we wish to regai n the known\nstandard result when we specialize to α−=α+. (We shall subsequently\ngeneralize this specific result, but it is explicit enough an d compact enough\nto make it worthwhile presenting it in full. Furthermore we s hall need this\nas input to our perturbative analysis.)\n•Ifb+= 3b−, that isb+=3\n2b∗andb−=1\n2b∗, but we do not necessarily demand\nα+=α−, then the asymptotic QNFs are calculable and are given by\nωn=i\nπb∗cos−1/parenleftBigg\n1±/radicalbig\n9−16 cos(πα+)cos(πα−))\n4/parenrightBigg\n+2in\nb∗. (2.98)\nProof: To see this note that in this situation |b+−b−|=b∗= (b++b−)/2.\nTherefore the QNF condition reduces to\ncos(−iπωb∗)−cos(−i2πωb∗) = 2 cos(πα+)cos(πα−), (2.99)\nimplying\ncos(−iπωb∗)−2cos2(−iπωb∗)+1 = 2 cos( πα+)cos(πα−). (2.100)\nThat is\n2cos2(−iπωb∗)−cos(−iπωb∗)−1+2 cos(πα+)cos(πα−) = 0, (2.101)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 57\nwhence\ncos(−iπωb∗) =1±/radicalbig\n1+8(1−2 cos(πα+)cos(πα−))\n4, (2.102)\nso that\ncos(−iπωb∗) =1±/radicalbig\n9−16 cos(πα+)cos(πα−))\n4, (2.103)\nimplying\n−iπωnb∗= cos−1/parenleftBigg\n1±/radicalbig\n9−16 cos(πα+)cos(πα−))\n4/parenrightBigg\n+n2π. (2.104)\nFinally\nωn=i\nπb∗cos−1/parenleftBigg\n1±/radicalbig\n9−16 cos(πα+)cos(πα−))\n4/parenrightBigg\n+2in\nb∗. (2.105)\nComment: This gives us another specific example of asymptotic off-axi s com-\nplex QNF’s — now with b+∝\\e}atio\\slash=b−. Note that because of the ±and the double-\nvalued nature of cos−1(·)there are actually 4 branches of QNFs hiding in this\nnotation.\n•This particular trick can certainly be extended to the cubic and quartic poly-\nnomials, for which general solutions exist.\n–The quadratic corresponds to\nb+−b−\nb++b−= 2;b+= 3b−. (2.106)\n–The cubic corresponds to\nb+−b−\nb++b−= 3;b+= 2b−. (2.107)\n–The quartic corresponds to\nb+−b−\nb++b−= 4;b+=5\n3b−. (2.108)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 58\nSome approximate results\nA number of approximate results can be extracted by looking a t special regions of\nparameter space.\nCaseb−≈b+Supposeb−≈b+. Then the quantity −i(b+−b−)ωis slowly vary-\ning over the range where −i(b++b−)ωchanges by 2π. Letω∗be any solution of\nthe approximate QNF condition, and define b∗= (b++b−)/2. Then for nearby\nfrequencies we are trying to (approximately) solve\ncos(−iω∗π[b+−b−])−cos(−iωπ[b++b−]) = 2cos(πα+)cos(πα−), (2.109)\nthat is\ncos(−iω∗π[b+−b−])−cos(−iω2πb∗) = 2cos(πα+)cos(πα−), (2.110)\nand the solutions of this are approximately\nωn≈ω∗+in\nb∗valid for |n| ≪b++b−\n|b+−b−|. (2.111)\nThus approximate result will subsequently be incorporated into a more general\nperturbative result to be discussed below.\nCaseb−≪b+Now suppose b−≪b+. Then the quantity −ib−ωis slowly varying\nover the range where −ib+ωchanges by 2π. Letω∗be any solution of the approxi-\nmate QNF condition, then for nearby frequencies we are tryin g to (approximately)\nsolve\nsin(−iωπb+)sin(−iω∗πb−) = cos(πα+)cos(πα−), (2.112)\nand the solutions of this are approximately\nωn≈ω∗+2in\nb+valid for |n| ≪b+\nb−. (2.113)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 59\nCaseα−≈1/2This corresponds to\nV0−b2\n−≈0, (2.114)\nin which case the QNF condition becomes\nsin(−iωπb+)sin(−iωπb−)≈0. (2.115)\nTherefore one obtains either (the physically relevant cond ition)\n−iωb+=n⇒ω=in\nb+, (2.116)\nor (the physically uninteresting situation)\n−iωb−=n⇒ω=in\nb−. (2.117)\nNote: If you go to the limit α−= 1/2by setting V0−= 0, one sees on physical\ngrounds that b−is irrelevant, so it cannot contribute to the physical QNF. A lterna-\ntively if you hold V0−∝\\e}atio\\slash= 0but driveb−→0, then these QNF’s are driven to infinity\n— and so decouple from the physics. Either way, the only physi cally interesting\nQNFs areω=in/b+. We explore these limits more fully below.\nSome special cases\nA number of special cases can now be analyzed in detail to give us an overall feel\nfor the general situation.\nCaseα−= 0 =α+This corresponds to\nV0−b2\n−=1\n4=V0+b2\n+, (2.118)\nin which case the QNF condition becomes\nsin(iωπb+)sin(iωπb−) = 1. (2.119)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 60\nLet us look for pure imaginary QNFs. (We do not claim that thes e are the only\nQNFs.) This implies that we must simultaneously satisfy both\nsin(−iωπb+) = sin(−iωπb−) = 1, (2.120)\norboth\nsin(−iωπb+) = sin(−iωπb−) =−1. (2.121)\nThat is both\n−iωb+= 2n++1\n2;−iωb−= 2n−+1\n2, (2.122)\norboth\n−iωb+= 2n+−1\n2;−iωb−= 2n−−1\n2. (2.123)\nTherefore either\nb+\nb−=2n++1\n2\n2n−+1\n2=4n++1\n4n−+1, orb+\nb−=2n+−1\n2\n2n−−1\n2=4n+−1\n4n−−1. (2.124)\nIn either case we need b+/b−to be rational, so that b+=p+b∗andb−=p−b∗.\nThis special case is thus evidence that there is something ve ry special about the\nsituation where b+/b−is rational, as we know from the theorem (2.3). Then either\n−iω=2n++1\n2\np+b∗;−iω=2n−+1\n2\np−b∗; (2.125)\nor\n−iω=2n+−1\n2\np+b∗;−iω=2n−−1\n2\np−b∗. (2.126)\nNow write\nn+=m++np+;n−=m−+np−; (2.127)\nwithm+<p+andm−<p−. (Whilen∈ {0,1,2,3,...}.) Then in the first case\nω=i/braceleftbigg2m++1\n2\np+b∗+n\nb∗/bracerightbigg\n=i/braceleftbigg2m−+1\n2\np−b∗+n\nb∗/bracerightbigg\n=ω∗+in\nb∗, (2.128)\nwhile in the second case\nω=i/braceleftbigg2m+−1\n2\np+b∗+n\nb∗/bracerightbigg\n=i/braceleftbigg2m−−1\n2\np−b∗+n\nb∗/bracerightbigg\n=ω∗+in\nb∗. (2.129)\nAs we already know all this is only a consequence of our genera l result (2.3).CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 61\nCaseV0−= 0 We can best analyze this situation by working directly with t he\nexact wavefunction. If V0−= 0thenα−= 1/2and\nψ−(0) = 1; ψ′\n−(0) = +iω;ψ′\n−(0)\nψ−(0)= +iω. (2.130)\nThe exact QNF boundary condition is then\niω=−2\nb+Γ(α++iωb+\n2+3\n4) Γ(−α++iωb+\n2+3\n4)\nΓ(α++iωb+\n2+1\n4) Γ(−α++iωb+\n2+1\n4). (2.131)\nBut this we can rewrite as\nω=2i\nb+Γ(−α+−iωb+\n2+3\n4) Γ(α+−iωb+\n2+3\n4)\nΓ(−α+−iωb+\n2+1\n4) Γ(α+−iωb+\n2+1\n4)×cos(πα+)−sin(−iπωb+)\ncos(πα+)+sin(−iπωb+). (2.132)\nThis certainly has pure imaginary roots. If we write ω=i|ω|then asymptotically\n(|ω| → ∞ ) this becomes\n1 =cos(πα+)−sin(π|ω|b+)\ncos(πα+)+sin(π|ω|b+), (2.133)\nimplying\nsin(π|ω|b+) = 0; ⇒π|ω|b+=nπ;⇒ω=in\nb+. (2.134)\nThis agrees with our previous calculation for α−≈1/2and as expected gives the\nSchwarzschild result (2.39).\nCaseb−→0This is best dealt with by using a Taylor expansion to show tha t\nψ′\n−(0)\nψ−(0)= +iω+V0−b−+O(b2\n−). (2.135)\nThat is\nlim\nb−→0ψ′\n−(0)\nψ−(0)= +iω. (2.136)\nThe analysis then follows that for the case V0−= 0above, and furthermore agrees\nwith our previous calculation for α−≈1/2.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 62\nCaseb−→ ∞ This is best dealt with by using the Stirling approximation t o-\ngether with a Taylor expansion to show that\nψ′\n−(0)\nψ−(0)=i/radicalbig\nω2−V0−+O(1/b2\n−). (2.137)\nThat is\nlim\nb−→∞ψ′\n−(0)\nψ−(0)=i/radicalbig\nω2−V0−. (2.138)\nThe exact QNF boundary condition is then\ni/radicalbig\nω2−V0−=−2\nb+Γ(α++iωb+\n2+3\n4) Γ(−α++iωb+\n2+3\n4)\nΓ(α++iωb+\n2+1\n4) Γ(−α++iωb+\n2+1\n4). (2.139)\nBut this we can rewrite as\n/radicalbig\nω2−V0−\n=2i\nb+Γ(−α+−iωb+\n2+3\n4) Γ(α+−iωb+\n2+3\n4)\nΓ(−α+−iωb+\n2+1\n4) Γ(α+−iωb+\n2+1\n4)×cos(πα+)−sin(−iπωb+)\ncos(πα+)+sin(−iπωb+).(2.140)\nIf we write ω=i|ω|then asymptotically, ( |ω| → ∞ , withV0−held fixed, implying\nthatV0−effectively decouples from the calculation), this becomes\n1 =cos(πα+)−sin(π|ω|b+)\ncos(πα+)+sin(π|ω|b+), (2.141)\nimplying\nsin(π|ω|b+) = 0; ⇒π|ω|b+=nπ;⇒ω=in\nb+. (2.142)\nThe importance of this observation is that it indicates that for “one sided” poten-\ntials it is only the side for which the potential has exponent ial falloff that con-\ntributes to the “gap”. This again confirms (2.39), since this is the case when cos-\nmological horizon surface gravity goes to 0 and we get the res ult appropriate to a\nSchwarzschild black hole.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 63\nSystematic first-order perturbation theory\nSometimes it is worthwhile to adopt a perturbative approach and to estimate\nshifts in the QNFs from some idealized pattern. Define\nb=b++b−\n2; ∆ = |b+−b−|; (2.143)\nand rewrite the asymptotic QNF condition as\ncos(−iπω∆)−cos(−i2πωb) = 2 cos(πα+)cos(πα−), (2.144)\nwhere we are implicitly holding α±fixed. When ∆ = 0 we have previously seen\nthat the QNF are explicitly calculable with\nˆωn=icos−1{1−2 cos(πα+)cos(πα−)}\n2πb+in\nb. (2.145)\nCan we now obtain an approximate formula for the the QNF’s whe n∆∝\\e}atio\\slash= 0? It is\na good strategy to define the dimensionless parameter ǫby\n∆ = 2ǫb, (2.146)\nand to set\nω= ˆω+δω;δω=O(ǫ); (2.147)\nso that the asymptotic QNF condition becomes\ncos(−i2π[ˆω+δω]ǫb)−cos(−i2π[ˆω+δω]b) = 2 cos(πα+)cos(πα−). (2.148)\nThen to first order in ǫ\ncos(−i2πˆωǫb)−cos(−i2π[ˆω+δω]b) = 2 cos(πα+)cos(πα−), (2.149)\nwhere implicitly this approximation requires ǫ|δω|b≪1. Subject to this condition\nwe have\ncos(−i2π[ˆω+δω]b) = cos(−i2ˆπωǫb)−2 cos(πα+)cos(πα−), (2.150)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 64\nwhence\n−i2π[ˆωn+δωn]b= cos−1{cos(−i2πˆωnǫb)−2 cos(πα+)cos(πα−)}+2πn, (2.151)\nso that\nˆωn+δωn=icos−1{cos(−i2πˆωnǫb)−2 cos(πα+)cos(πα−)}\n2πb+in\nb. (2.152)\nBut we know that the unperturbed QNFs satisfy ˆωn= ˆω0+in/b, so we can also\nwrite this as\nδωn=icos−1{cos(−i2πˆωnǫb)−2 cos(πα+)cos(πα−)}\n2πb−ˆω0. (2.153)\nUsing the definition of ˆωnthis can now be cast in the form\nδωn=icos−1{cos(−i2πˆωnǫb)+cos(−i2πˆωnb)−1}\n2πb−ˆω0, (2.154)\nor the slightly more suggestive\nδωn=icos−1{cos(−i2πˆωnb)+cos(−i2πˆωnǫb)−1}\n2πb−ˆω0, (2.155)\nwhich can even be simplified to\nδωn=icos−1{cos(−i2πˆω0b)+cos(−i2πˆωnǫb)−1}\n2πb−ˆω0. (2.156)\nNote that this manifestly has the correct limit as ǫ→0. Note that we have not\nasserted or required that ˆωnǫb≪1, in fact when n≫1/ǫthis is typically nottrue.\n(Consequently cos(−i2πˆωnǫb)is relatively unconstrained.) Note furthermore that\nIm(δωn)≤1/b.\nDiscussion\nThe key lesson to be learned from our semi-analytic model for the QNFs is that the\ncommonly occurring ωn= offset+igap·nbehaviour is common but not universal .\nSpecifically, in our semi-analytic model the key point is whe ther or not the ratio\nb+/b−is a rational number.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 65\nRelation to the real physical Regge-Wheeler / Zerilli poten tials. The very in-\nteresting question is the following: If the physically relevant results are by this approx-\nimate potentials recovered only qualitatively (so only the correct gap structure is recov-\nered), does also the multi-family splitting transfer to the physically relevant case? This is\nbecause all the different families have the same gap structu re, so the multi-family\nsplitting might be only uninteresting artefact of our appro ximate model. The an-\nswer to this question comes with the following theorem:\nTheorem 2.4.1. Take forb+/b−=p+/p−∈Qthe gap structure as gap = 1/b∗, where\nb∗=b±/p±and assume that the number of equi-spaced families is bounde d with respect\ntob+,b−. Then the function ω(b+,b−)is discontinuous at every ˜b+,˜b−;˜b+/˜b−∈Qfor\ninfinite number of QNM frequencies.\nProof. Take any ˜b+/˜b−=p+/p−and arbitrary monotonically growing sequence of\nprimesPl. Take a sequence\n(bl+,bl−)≡/parenleftbiggPl\nPl−1˜b+,Pl−2\nPl−1˜b−/parenrightbigg\n. (2.157)\nIt is obvious that\nlim\nl→∞(bl+,bl−) = (˜b+,˜b−). (2.158)\nBut then\nbl∗=˜b∗gcd(Pl−2, p+)\nPl−1. (2.159)\nHere “gcd” means “greatest common divisor” and that means gcd(Pl−2,p+)is\nfrom the definition integer. Also it clearly holds that 1≤gcd(Pl−2,p+)≤p+,\nhence it is upper and lower bounded and ˜b∗=˜b±/p±. But then\nlim\nl→∞gap(bl+,bl−) = lim\nl→∞Pl−1\n˜b∗gcd(Pl−2, p+)=∞. (2.160)\nBut if the number of families is bounded with respect to b+,b−then only finite\nnumber of modes can be obtained at ˜b+,˜b−as a limitCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 66\nlim\nl→∞ωN(l)(bl+,bl−) =ωN(˜b+,˜b−). (2.161)\nThis proves the theorem.\nBut this just means that if we do not want to end up with extreme ly strongly\ndiscontinuous function ω(b+,b−), we have to accept the fact that the given 1/b∗gap\ndependence automatically implies unbounded number of QNM f amilies (with respect to\nb+,b−), as it was in the case of our analytically solvable potentia l.\nWe also suspect that it might be possible to generalize the mo del potential\neven further — the “art” would lie in picking a piecewise pote ntial that is still\nanalytically solvable (at least for the highly damped modes ) but which might be\ncloser in spirit to the Regge–Wheeler (Zerilli) potential t hat is the key physical\nmotivation for the current work. (Of course if we temporaril y forget the black hole\nmotivation, it may already be of some mathematical and physi cal interest that we\nhave a nontrivial extension of the Eckart potential that is a symptotically exactly\nsolvable — one could in principle loop back to Eckart’s origi nal article and start\nasking questions about tunnelling probabilities for elect rons encountering such\npiecewise Eckart barriers.)\nRelation to the black hole thermodynamics. The last point we would like to\nbriefly discuss is the relation of our results to the conjectu red connection between\nthe highly damped QNMs and the black hole thermodynamics. Th ere are two ba-\nsic conjectures: The first conjecture, due to Hod [77], gives some strong arguments\nsupporting the idea that there is a connection between the re al part of asymptotic\nQNMs and the quantum black hole area spacing. The basic princ iple underlying\nthis conjecture is Bohr’s correspondence principle. The co njecture was also used\nin the context of Loop Quantum Gravity by Dreyer [53]. The sec ond conjecture,\nsuch that it modifies Hod’s original proposal, is due to Maggi ore [110]. It solves\nsome controversies of Hod’s conjecture, (see [91]), and giv es the relation betweenCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 67\nblack hole mass spectra and highly damped QNMs as:\n∆M≈/planckover2pi1·∆/radicalbig\nRe(ωn)2+Im(ωn)2≈/planckover2pi1·∆Im(ωn), n>> 0. (2.162)\nHere we naturally assume that Re(ωn)is bounded and Im(ωn)is growing to in-\nfinity. But it means, that if we are interested in mass quantum ∆minM:\n∆minM≈/planckover2pi1·(Im(ωn)−Im(ωn−1)), n>> 0. (2.163)\nThe area quantum for the Schwarzschild black hole one obtain s from the formula:\n∆minA= 32πM∆minM= 8πl2\np=const., (2.164)\n(wherelpdenotes the Planck length). In our case, if we naively extrap olate Mag-\ngiore’s conjecture to S-dS spacetime and the ratio of surfac e gravities is rational,\nwe obtain black hole mass quanta as:4\n∆minM≈/planckover2pi1·gap(ωn) =/planckover2pi1·2\nglcm(κ+,κ−), n>> 0. (2.165)\nHere all the symbols are defined as in the proof of the theorem 2 .3 and in the equa-\ntion (2.12). Now there is a fascinating result by Choudhury a nd Padmanabhan\n[43], that clearly “fits” very well into all these ideas. It sa ys that in the S-dS space-\ntime there exists a coordinate system with globally defined t emperature, (hence\nsome kind of thermodynamic equilibrium), if and only if the r atio of surface grav-\nities is rational. (For a very nice discussion of what might b e the physical meaning\nof rational ratios of surface gravities and the role of highl y damped QNMs as a\npotential source of information see again [43].) The global temperature is given as\n[43]:\nTsds=hcf(κ+,κ−)\n2π. (2.166)\nThen the gap spacing is, in our case:\ngap(ωn) =2\nglcm(κ+,κ−) =2\ngp+p−hcf(κ+,κ−) =4πp+p−Tsds\ng. (2.167)\n4As the reader might have noted: we claim that the expression f or the gap in the spacing of the\nQNMs in the asymptotic formula, (the one derived in this sect ion), is an exact result, “unharmed”\nby the fact that we used only approximate potential.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 68\nThis is a different result from the Schwarzschild spacetime , where it is\ngap(ωn) = 2πTs. (2.168)\nBut the fact that something special happens with both the S-d S spacetime ther-\nmodynamics and highly damped QNMs when the surface gravitie s have rational\nratio is very interesting. Especially because it allows us t o relate the constant gap\nin the QNM spacing and global spacetime temperature, and str ikingly they both\nexist under the same condition. Moreover, in terms of Maggio re’s conjecture, the\nexistence of equispaced families is very interesting, beca use the gap in the QNM\nspacing multiplied by Planck constant is simply the quantum of black hole mass.\nThese considerations suggest that something very interest ing is happening, but\nthe topic needs clearly further exploration. What is writte n here has much more a\ncharacter of ambiguous indications than of a well founded id eas, but it gives very\nexciting suggestions for future work. Even more because, as we will show in the\nnext section, the link between rational ratio of surface gra vities and equispaced\nfamilies of highly damped QNMs is very generic for the multi- horizon black hole\nspacetimes.\n2.4.1 Conclusions\nThe arguments that highly damped modes qualitatively depen d on the tails of the\npotential were confirmed for Schwarzschild black hole by der iving the behavior\nof (2.9). That was the only straight computational test that could have been done.\nAfter this the method was used to obtain new results for Schwa rzschild-de Sitter\nblack holes. One of the nice features of the semi-analytic mo del related to S-dS\nblack holes is that a quite surprising amount of semi-analyt ic information can be\nextracted, in terms of general qualitative results, approx imate results, perturba-\ntive results, and reasonably explicit computations. Our re sults also might have\nimportant implications for the black hole thermodynamics.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 69\n2.5 Monodromy results\n2.5.1 Introduction\nIn monodromy approaches one works with an analytic continua tion into the com-\nplex radial plane. They are part of a wider approach, the phas e integral method\n(for details see [5]). One has to choose branch cuts (as a part of analytic continua-\ntion), identify the singular points of the solutions in the c omplex plane, locate the\nStokes and Anti-Stokes lines and calculate the monodromy ar ound the singulari-\nties [4, 5, 22, 33, 42, 49, 51, 88, 109, 118, 119, 120, 121, 141] .\nWhile many technical details differ, both between the semi- analytic and mon-\nodromy approaches, and often among various authors seeking to apply the mon-\nodromy technique, there is widespread agreement that not on ly the semi-analytic\napproximation, but also the monodromy approaches lead to QN F master equa-\ntions of the general form:\nN/summationdisplay\nA=1CAexp/parenleftBiggH/summationdisplay\ni=1ZAiπω\nκi/parenrightBigg\n= 0. (2.169)\nHereκiis the surface gravity of the i-th horizon, His the number of horizons,\nthe matrix ZAialways has rational entries (and quite often is integer-val ued). The\nphysics contained in the master equation is invariant under substitutions of the\nformZAi→ZAi+(1,...,1)T\nAhi, where the hiare arbitrary rational numbers. Either/summationtext\niZAi= 0, or it can without loss of generality be made zero. Furthermo reNis\nsome reasonably small positive integer. (In fact N≤2H+ 1in all situations we\nhave encountered, and typically N > H .) TheCAare a collection of coefficients\nthat are often but not always integers, though in all known ca ses they are at least\nreal. Finally in almost all known cases the rectangular N×HmatrixZAihas\nrankH, and the QNF master equation is almost always irreducible (t hat is, non-\nfactorizable). We shall first demonstrate that all known mas ter equations (whether\nbased on semi-analytic or monodromy techniques) can be cast into this form. Then\nwe will generalize the results for rational/irrational sur face gravities ratios fromCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 70\nthe special cases obtained within the approximations by the analytically solvable\npotentials to this more general case (2.169).\n2.5.2 Particular results\nSurvey of the monodromy results\nOne horizon: In the one-horizon situation there is general agreement tha t the\nrelevant master equation is\nexp/parenleftBigπω\nκ/parenrightBig\n+1+2cos( πj) = 0. (2.170)\nUnfortunately there is distressingly little agreement ove r the precise status of the\nparameter j. References [4, 22, 118, 119, 120, 121] assert that this is th e spin of\nthe perturbation under consideration, but with some disagr eement as to whether\nthis applies to all spins and all dimensions. In contrast in r eference [51] a partic-\nular model for the spacetime metric is adopted, and in terms o f the parameters\ndescribing this model these authors take\nj=qd\n2−1. (2.171)\nHere, (and also later in this section), the symbol ddenotes spacetime dimension,\nandqis a parameter related to the power with which the general d-dimensional\nblack hole metric coefficients\n−f(r)dt2+dr2\ng(r)+ρ2(r)dΩ2\nd (2.172)\nbehave close to the singularity, (see [51]). Reference [109 ] asserts that for spin 1\nperturbations\nj=2(d−3)\nd−2. (2.173)\nBe this as it may, there is universal agreement on the form of the QNF master\ncondition, and it is automatically of the the form of equatio n (2.169), with H= 1\nandN= 2terms. The vector CAand the matrix ZAiare\nCA=/bracketleftBigg\n+1\n1+2cos(πj)/bracketrightBigg\n;ZA1=/bracketleftBigg\n+1\n0/bracketrightBigg\n. (2.174)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 71\nBy multiplying through by exp(−πω/(2κ))we can re-cast the QNF condition as\nexp/parenleftBigπω\n2κ/parenrightBig\n+{1+2cos(πj)}exp/parenleftBig\n−πω\n2κ/parenrightBig\n= 0. (2.175)\nThis now corresponds to\nCA=/bracketleftBigg\n+1\n1+2cos(πj)/bracketrightBigg\n;ZA1=/bracketleftBigg\n+1/2\n−1/2/bracketrightBigg\n, (2.176)\nand in this form we have/summationtext\nAZAi= 0. (This is one of rather few cases where it is\nconvenient to take the ZAito be rational-valued rather than integer-valued.)\nTwo horizons: For two-horizon situations the analysis is slightly differ ent for\nSchwarzschild–de Sitter spacetimes (Kottler spacetimes) versus Reissner–Nord-\nstr¨ om spacetimes.\n•ForSchwarzschild–de Sitter spacetimes there is general agreement that the\nrelevant master equation for the QNFs is\n{1+2cos(πj)}cosh/parenleftbiggπω\nκ++πω\nκ−/parenrightbigg\n+cosh/parenleftbiggπω\nκ+−πω\nκ−/parenrightbigg\n= 0. (2.177)\nWe shall again adopt conventions such that κ±are both positive. Again,\nthere is unfortunately distressingly little agreement ove r the precise status\nof the parameter j. References [4, 22, 118, 119, 120, 121] assert that this is th e\nspin of the perturbation under consideration, but with some disagreement\nas to whether this applies to all spins and all dimensions. In contrast in\nreference [65] a particular model for the spacetime metric i s again adopted,\nand in terms of the parameters describing this model they tak e\nj=qd\n2−1. (2.178)\nOne still has to perform a number of trigonometric transform ations to turn\nthe quoted result of reference [65] for d∝\\e}atio\\slash= 5\ntanh/parenleftbiggπω\nκ+/parenrightbigg\ntanh/parenleftbiggπω\nκ−/parenrightbigg\n=2\ntan2(πj/2)−1, (2.179)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 72\ninto the equivalent form (2.177) above. For d= 5the authors of [141] assert\nthe equivalent of\n{1+2cos(πj)}sinh/parenleftbiggπω\nκ++πω\nκ−/parenrightbigg\n+sinh/parenleftbiggπω\nκ+−πω\nκ−/parenrightbigg\n= 0. (2.180)\nReference [109] again asserts that for spin 1 perturbations\nj=2(d−3)\nd−2. (2.181)\nBe this as it may, there is again universal agreement on the form of the QNF\nmaster condition, and converting hyperbolic functions int o exponentials, it\ncan be transformed into the form of equation (2.169), with H= 2andN= 4\nterms. The vector CAand matrix ZAiare\nCA=\n1+2cos(πj)\n+1\n+1\n1+2cos(πj)\n;ZAi=\n+1 +1\n+1−1\n−1 +1\n−1−1\n. (2.182)\nNote that we explicitly have/summationtext\nAZAi= 0. There are two exceptional cases:\n–Ifj= 2m+1withm∈Zthen\nCA=\n−1\n+1\n+1\n−1\n;ZAi=\n+1 +1\n+1−1\n−1 +1\n−1−1\n. (2.183)\nIn this situation the QNF master equation factorizes\nsinh/parenleftbiggπω\nκ+/parenrightbigg\nsinh/parenleftbiggπω\nκ−/parenrightbigg\n= 0. (2.184)\nThis appears to be the physically relevant case for spin 1 par ticles. The\nrelevant QNF spectrum is that of equation (2.115).CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 73\n–Ifcos(πj) =−1\n2, which does not appear to be a physically relevant situ-\nation but serves to illustrate potential mathematical path ologies, then\nCA=\n0\n+1\n+1\n0\n;ZAi=\n+1 +1\n+1−1\n−1 +1\n−1−1\n. (2.185)\nBut in this situation the top row and bottom row do not contrib ute to\nthe QNF master equation and one might as well delete them. Tha t is,\none might as well write\nCA=/bracketleftBigg\n+1\n+1/bracketrightBigg\n;ZAi=/bracketleftBigg\n+1−1\n−1 +1/bracketrightBigg\n. (2.186)\nThis is a situation (albeit unphysical) where the matrix ZAidoes not\nhave maximal rank. The QNF master equation degenerates to\nsinh/parenleftbiggπω\nκ+−πω\nκ−/parenrightbigg\n= 0. (2.187)\nIn this situation the QNF spectrum is\nωn=inκ+κ−\n|κ+−κ−|, (2.188)\nwith no restriction on the relative values of κ±. This situation is how-\never clearly non-generic (and outright unphysical).\n•ForReissner–Nordstr¨ om spacetime one has [119]\nexp/parenleftbigg2πω\nκ+/parenrightbigg\n+2{1+cos(πj)}exp/parenleftbigg\n−2πω\nκ−/parenrightbigg\n+{1+2cos(πj)}= 0,(2.189)\nwhereκ+is the surface gravity of the outer horizon and κ−is the surface\ngravity of the inner horizon. There is again some disagreeme nt on the status\nof the parameter j. Reference [119] now takes j=1\n3for spin 0, and j=5\n3CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 74\nfor spins 1 and 2 (in any dimension). Reference [121] asserts that for general\ndimension\nj=d−3\n2d−5for spin 0, 2, and j=3d−7\n2d−5for spin 1.(2.190)\nBe this as it may, there is universal agreement on the form of the QNF master\ncondition, and it is automatically of the form of equation (2 .169), with H= 2\nandN= 3terms. The vector CAand matrix ZAiare\nCA=\n+1\n2{1+cos(πj)}\n1+2cos(πj)\n;ZAi=\n+2 0\n0−2\n0 0\n. (2.191)\nIf we multiply through by a suitable factor then we can write t he QNF con-\ndition in the equivalent form\nexp/parenleftbigg2πω\n3κ++πω\n3κ−/parenrightbigg\n+2{1+cos(πj)}exp/parenleftbigg\n−πω\n3κ+−2πω\n3κ−/parenrightbigg\n+{1+2cos(πj)}exp/parenleftbigg\n−πω\n3κ++πω\n3κ−/parenrightbigg\n= 0, (2.192)\nThis corresponds to\nCA=\n+1\n2{1+cos(πj)}\n1+2cos(πj)\n;ZAi=\n+2/3 1/3\n−1/3−2/3\n−1/3 1/3\n. (2.193)\nIn this form we now explicitly have/summationtext\nAZAi= 0. (This is one of rather few\ncases where it is convenient to take the ZAito be rational-valued rather than\ninteger-valued.) Returning to the original form in equatio n (2.189), there are\ntwo exceptional cases:\n–Ifj= 2m+ 1 withm∈Z, (this does not appear to be a physically\nrelevant situation but again this serves to illustrate the p ossible mathe-\nmatical pathologies one might encounter), then\nCA=\n+1\n0\n−1\n;ZAi=\n+2 0\n0−2\n0 0\n. (2.194)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 75\nBut then (without loss of information) one might as well elim inate the\nsecond row, to obtain\nCA=/bracketleftBigg\n+1\n−1/bracketrightBigg\n;ZAi=/bracketleftBigg\n+2 0\n0 0/bracketrightBigg\n. (2.195)\nFurthermore, since κ−now decouples, we might as well eliminate the\nsecond column, to obtain\nCA=/bracketleftBigg\n+1\n−1/bracketrightBigg\n;ZAi=/bracketleftBigg\n2\n0/bracketrightBigg\n. (2.196)\nThe QNF master equation then specializes to\nexp/parenleftbigg2πω\nκ+/parenrightbigg\n−1 = 0. (2.197)\n–Ifcos(πj) =−1\n2, which does not appear to be a physically relevant situ-\nation but serves to illustrate potential mathematical path ologies, then\nCA=\n+1\n+1\n0\n;ZAi=\n+2 0\n0−2\n0 0\n. (2.198)\nBut in this situation the bottom row does not contribute to th e QNF\nmaster equation and one might as well delete it. That is, one m ight as\nwell write\nCA=/bracketleftBigg\n+1\n+1/bracketrightBigg\n;ZAi=/bracketleftBigg\n+2 0\n0−2/bracketrightBigg\n. (2.199)\nWe can rearrange the terms in the master equation to have the Q NF\nmaster equation specialize to\nexp/parenleftbigg2πω\nκ++2πω\nκ−/parenrightbigg\n+1 = 0. (2.200)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 76\nThis corresponds to\nCA=/bracketleftBigg\n+1\n+1/bracketrightBigg\n;ZAi=/bracketleftBigg\n+2 +2\n0 0/bracketrightBigg\n. (2.201)\nNote that in this exceptional case ZAiis not of maximal rank. In this\nsituation the QNF spectrum is\nωn=(2n+1)iκ+κ−\nκ++κ−, (2.202)\nwith no restriction on the relative values of κ±. This situation is how-\never clearly non-generic (and outright unphysical).\nThree horizons: For three horizons the natural example to consider is that of\nReissner–Nordstr¨ om–de Sitter spacetime. References [12 1, 141] agree that (for d∝\\e}atio\\slash=\n5)\ncosh/parenleftbiggπω\nκ+−πω\nκC/parenrightbigg\n+{1+cos(πj)}cosh/parenleftbiggπω\nκ++πω\nκC/parenrightbigg\n+2{1+cos(πj)}cosh/parenleftbigg2πω\nκ−+πω\nκ++πω\nκC/parenrightbigg\n= 0. (2.203)\nHereκ±refer to the inner and outer horizons of the central Riessner –Nordstr¨ om\nblack hole, while κCis now the surface gravity of the cosmological horizon. All\nthese surface gravities are taken positive. In contrast for d= 5one has\nsinh/parenleftbiggπω\nκ+−πω\nκC/parenrightbigg\n+{1+cos(πj)}sinh/parenleftbiggπω\nκ++πω\nκC/parenrightbigg\n+2{1+cos(πj)}sinh/parenleftbigg2πω\nκ−+πω\nκ++πω\nκC/parenrightbigg\n= 0, (2.204)\nAgain\nj=d−3\n2d−5for spin 0, 2, and j=3d−7\n2d−5for spin 1. (2.205)\nThere is universal agreement on the form of the QNF master condition, and con-\nverting hyperbolic functions into exponentials, it can be t ransformed into the formCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 77\nof equation (2.169), with H= 3 andN= 6 terms. The vector CAand matrix ZAi\nare\nCA=\n+1\n1+cos(πj)\n2{1+cos(πj)}\n±2{1+cos(πj)}\n±{1+cos(πj)}\n±1\n;ZAi=\n+1−1 0\n+1 +1 0\n+1 +1 +2\n−1−1−2\n−1−1 0\n−1 +1 0\n. (2.206)\nGenerically, ZAihas maximal rank H= 3. Note that we explicitly have/summationtext\nAZAi=\n0.\nThe only exceptional case is cos(πj) =−1in which case\nCA=\n+1\n0\n0\n0\n0\n±1\n;ZAi=\n+1−1 0\n+1 +1 0\n+1 +1 +2\n−1−1−2\n−1−1 0\n−1 +1 0\n. (2.207)\nBut then the 2ndto5throws decouple and may as well be removed, yielding\nCA=/bracketleftBigg\n+1\n±1/bracketrightBigg\n;ZAi=/bracketleftBigg\n+1−1 0\n−1 +1 0/bracketrightBigg\n. (2.208)\nThe3rdcolumn, corresponding to κ−, now decouples and may as well be removed,\nyielding\nCA=/bracketleftBigg\n+1\n±1/bracketrightBigg\n;ZAi=/bracketleftBigg\n+1−1\n−1 +1/bracketrightBigg\n. (2.209)\nNote that in this exceptional case ZAiis not of maximal rank, and the QNF master\nequation degenerates to\ncosh/parenleftbiggπω\nκ+−πω\nκC/parenrightbigg\n= 0, orsinh/parenleftbiggπω\nκ+−πω\nκC/parenrightbigg\n= 0, (2.210)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 78\nrespectively. In this situation the QNF spectrum is\nωn=(2n+1)iκ+κC\n2|κ+−κC|,orωn=inκ+κC\n|κ+−κC|, (2.211)\nrespectively, with no restriction on the relative values of κ±. This situation is how-\never clearly non-generic (and outright unphysical).\nRewriting the analytically solvable potentials results in to our general form\nOne horizon For highly damped QNFs the master equation is derived in prev i-\nous section and also in references [150, 153], in a form equiv alent to\nsinh/parenleftBigπω\nκ/parenrightBig\n= 0. (2.212)\nThis means\nCA=/bracketleftBigg\n1\n−1/bracketrightBigg\n, (2.213)\nand\nZAi=/bracketleftBigg\n1\n−1/bracketrightBigg\n. (2.214)\nTwo horizons For highly damped QNFs the master equation is derived in pre-\nvious section and also in references [150, 151, 153] in a form equivalent to\ncosh/parenleftbiggπω\nκ++πω\nκ−/parenrightbigg\n−cosh/parenleftbiggπω\nκ+−πω\nκ−/parenrightbigg\n+2 cos(πα+)cos(πα−) = 0. (2.215)\nThis means\nCA=\n1\n1\n−1\n−1\n2cos(πα+)cos(πα−)\n, (2.216)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 79\nand\nZAi=\n1 1\n−1−1\n1−1\n−1 1\n0 0\n. (2.217)\nNote thatα±=/radicalBig\n1\n4−V0±\nκ2\n±and henceα±=1\n2is a physically degenerate case corre-\nsponding either to V0±= 0(in which case the corresponding κ±is physically and\nmathematically meaningless), or κ±=∞, (in which case the QNF master equa-\ntion is vacuous). Either of these situations is unphysical s o one must have α±∝\\e}atio\\slash=1\n2.\nThis QNF condition above is irreducible (non-factorizable ) unlessα±=m+1\n2with\nm∈Z. This occurs when\nV0±=−m(m+1)κ2\n±, (2.218)\nand in this exceptional situation the QNF master equation fa ctorizes to\nsinh/parenleftbiggπω\nκ+/parenrightbigg\nsinh/parenleftbiggπω\nκ−/parenrightbigg\n= 0, (2.219)\nand hence takes the same shape as (2.184).\n2.5.3 Rational ratios of surface gravities\nFrom master equation to polynomial\nFirst let us suppose that the ratios Rij=κi/κjare all rational numbers. This is\nnot as significant a constraint as one might initially think. In particular, since the\nrationals are dense in the reals one can always with arbitrar ily high accuracy make\nan approximation to this effect. Furthermore since floating point numbers are\nessentially a subset of the rationals, all numerical invest igations implicitly make\nsuch an assumption, and all numerical experiments should be interpreted with\nthis point kept firmly in mind.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 80\nProvided that the ratios Rij=κi/κjare all rational numbers, it follows that\nthere is a constant κ∗and a collection of relatively prime integers misuch that\nκi=κ∗\nmi. (2.220)\nThe QNF master equation then becomes\nN/summationdisplay\nA=1CAexp/parenleftBiggH/summationdisplay\ni=1ZAimiπω\nκ∗/parenrightBigg\n= 0, (2.221)\nNow define z= exp(πω/κ∗), and define a new set of integers ˜mA=/summationtextH\ni=1ZAimi.\n(There is no guarantee or requirement that the ˜mAbe relatively prime, and some of\nthe special cases we had to consider in the previous section a nd in reference [151]\nultimately depend on this observation.) Then\nN/summationdisplay\nA=1CAz˜mA= 0. (2.222)\nThis is (at present) a Laurent polynomial, as some exponents may be (and typically\nare) negative. Multiplying through by z−˜mminconverts this to a regular polynomial\nwith a nonzero constant z0term and with degree\nD= ˜mmax−˜mmin. (2.223)\nIf we write ¯mA= ˜mA−˜mminthen the relevant regular polynomial is\nN/summationdisplay\nA=1CAz¯mA= 0. (2.224)\nNote that the polynomial is typically “sparse” — the number o f termsNis small\n(typicallyN≤2H+1) but the degree Dcan easily be arbitrarily large. There are\nat mostDdistinct roots for the polynomial za, and the general solution of the QNF\ncondition is\nωa,n=κ∗ln(za)\nπ+2inκ∗;a∈ {1,...,D};n∈ {0,1,2,3,...}. (2.225)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 81\nIf the¯mAare not relatively prime, define a degeneracy factor g= hcf{¯mA}. Then\nthe roots will fall into D/g classes where the gdegenerate members of each class\ndiffer only by the various g-th roots of unity. In this situation we can somewhat\nsimplify the above QNF spectrum to yield\nωa,n=κ∗ln(za)\nπ+2inκ∗\ng;a∈ {1,...,D/g};n∈ {0,1,2,3,...}.(2.226)\nWe again emphasize that behaviour of this sort certainly doe s occur in practice.\nThere is no guarantee or requirement that the ¯mAbe relatively prime, and some of\nthe special cases we had to consider in the previous section a nd in reference [151]\nultimately depend on this observation.\nSo let us summarize what we just proved in a theorem:\nTheorem 2.6. Take quasi-normal frequencies to be given by the equation (2.221) and,\ntake the ratio of arbitrary pairs of surface gravities to be r ational. Then the quasi-normal\nfrequencies are given by the formula (2.226) , wherezaare, (in general), D/g solutions of\nthe equation (2.224) .\nThere is a (slightly) weaker condition that also leads to pol ynomial master\nequations and the associated families of QNFs. Suppose that we know that the\nratios\nRAB=/summationtextH\ni=1ZAi/κi/summationtextH\ni=1ZBi/κi∈Q (2.227)\nare always rational numbers. Then it follows that there is a s et of integers ˆmAsuch\nthat\nH/summationdisplay\ni=1ZAi\nκi=ˆmA\n¯κ∗, (2.228)\nwhere the ˆmAare all relatively prime. (Note ¯κ∗does not have to equal κ∗). This\nis actually a (slightly) weaker condition than Rij=κi/κjbeing rational, since it\nis only ifZAiis of rankHthat one can derive Rij∈QfromRAB∈Q. Assuming\nRAB∈Qthe QNF master equation becomes\nN/summationdisplay\nA=1CAexp/parenleftbigg\nˆmAπω\n¯κ∗/parenrightbigg\n= 0. (2.229)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 82\nThis can now be converted into a polynomial in exactly the sam e manner as pre-\nviously, leading to families of QNFs as above. Provided both RABandRijare\nrational we can identify ¯κ∗=κ∗/g.\nFactorizability\nNow it is mathematically conceivable that in certain circum stances the master\nequation might factorize into a product over two disjoint se ts of horizons\n/bracketleftBiggN1/summationdisplay\nA=1C1Aexp/parenleftBiggH1/summationdisplay\ni=1Z1Aiπω\nκ1i/parenrightBigg/bracketrightBigg /bracketleftBiggN2/summationdisplay\nA=1C2Aexp/parenleftBiggH2/summationdisplay\ni=1Z2Aiπω\nκ2i/parenrightBigg/bracketrightBigg\n= 0. (2.230)\nPhysically one might in fact expect this if the horizons inde xed byi∈ {1,...,H 1}\nare very remote (in physical distance) from the other horizo ns indexed by i∈\n{1,...,H 2}. If such a factorization were to occur then the QNFs would fal l into\ntwo completely disjoint classes, being independently and d isjointly determined\nby these two classes of horizon.\n2.6.1 Irrational ratios of surface gravities\nWe now wish to work “backwards” to see if the existence of a fam ily of equi-\nspaced QNFs can lead to constraints on the ratios Rij=κi/κj. Such an analysis\nhas already been performed for the specific class of QNF maste r equations arising\nfrom semi-analytic techniques, and we now intend to general ize the argument to\nthe generic class of QNF master equations presented in equat ion (2.169). Let us\ntherefore assume the existence of at least one “family” of QN Fs of the form:\nωn=ω0+ingap;n∈ {0,1,2,3,...}. (2.231)\nThen we are asserting\nN/summationdisplay\nA=1CAexp/parenleftBiggH/summationdisplay\ni=1ZAiπ(ω0+ingap)\nκi/parenrightBigg\n= 0;n∈ {0,1,2,3,...}. (2.232)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 83\nThat is\nN/summationdisplay\nA=1/braceleftBigg\nCAexp/parenleftBiggH/summationdisplay\ni=1ZAiπω0\nκi/parenrightBigg/bracerightBigg\nexp/parenleftBigg\ninπgapH/summationdisplay\ni=1ZAi\nκi/parenrightBigg\n= 0; (2.233)\nn∈ {0,1,2,3,...}.\nWe can rewrite this as\nN/summationdisplay\nA=1DAexp(2πinJA) = 0;n∈ {0,1,2,3,...}. (2.234)\nA priori, there is no particular reason to expect either the DAor theJAto be real.\nCase 1\nOne specific solution to the above collection of constraints is\nN/summationdisplay\nA=1DA= 0; exp(2 πiJA) =r. (2.235)\nFurthermore, as long as no proper subset of the DA’s sums to zero , we assert that this\nis the only solution. To see this let us define\nλA= exp(2πiJA);MAB= (λA)B−1;A,B∈ {1,2,3,...,N}. (2.236)\nThenMABis a square N×NVandermonde matrix, and then equation (2.234)\nimplies\nN/summationdisplay\nA=1DAMAB= 0, (2.237)\nwhencedet(MAB) = 0 . But from the known form of the Vandermonde determi-\nnant we have\ndet(MAB) =/productdisplay\nA>B(λA−λB) = 0, (2.238)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 84\nimplying that at least two of the λAare equal. Without loss of generality we can\nshuffle the λA’s so that the two which are guaranteed to be equal are λ1andλ2.\nThen equation (2.234) implies\n(D1+D2)λB−1\n1+N/summationdisplay\nA=3DA(λA)B−1= 0;B∈ {1,2,3,...,N−1}. (2.239)\nBut by hypothesis D1+D2∝\\e}atio\\slash= 0, so this equation can be rewritten in terms of a\nnon-trivial reduced (N−1)×(N−1)Vandermonde matrix, whose determinant\nmust again be zero, so that two more of the λA’s must be equal. Proceeding in this\nway one reduces the size of the Vandermonde matrix by unity at each step and\nfinally has\nλA=r, (2.240)\nas asserted. We then see\nJA=−iln(r)\n2π+mA;mA∈Z. (2.241)\nExpressed directly in terms of the surface gravities this yi elds\nH/summationdisplay\ni=1ZAigap\nκi=−iln(r)\nπ+2mA;mA∈Z. (2.242)\nBy assumption, we have asserted the existence of at least one solution to these\nconstraint equations. (Otherwise the family we used to star t this discussion would\nnot exist.) We have seen that we can choose to present the mast er equation in such\na manner that/summationtextN\nA=1ZAi= 0. But then\n0 =−iln(r)\nπN+2N/summationdisplay\nA=1mA;mA∈Z. (2.243)\nThis implies that\niln(r)\nπ= 2/summationtextN\nA=1mA\nN=q∈Q. (2.244)\nThat is, there is a rational number qsuch that\nH/summationdisplay\ni=1ZAigap\nκi=q+2mA;q∈Q;mA∈Z. (2.245)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 85\nThis is already enough to imply that the ratios RABare rational. If in addition\nZAiis of rankHthen, (either using standard row-echelon reduction of the a ug-\nmented matrix, or invoking the Moore–Penrose pseudo-inver se and noting that\nthe Moore-Penrose pseudo-inverse of an integer valued matr ix has rational ele-\nments), we see that for each horizon the ratio (gap)/κimust be a rational number,\nand consequently the ratios Rij=κi/κjmust all be rational numbers.\nThat is:\nTheorem 2.7. Take quasi-normal frequencies to be given by the equation (2.221) , and\nsuppose that we have a family of QNFs as described by equation (2.231) . If no proper\nsubset of the DA’s from (2.221) sums to 0, and the matrix ZAifrom the same equation is\nof rankH, then the ratio of arbitrary pair of surface gravities must b e rational.\nCase 2\nMore generally, if some proper subset of the DA’s sums to zero , subdivide the Nterms\nA∈ {1,2,3,...,N}into a cover of disjoint irreducible proper subsets Basuch that\nN/summationdisplay\nA∈BaDA= 0. (2.246)\nThen the solutions of equation (2.234) are uniquely of the fr om\nexp(2πiJA∈Ba) =λA∈Ba=ra. (2.247)\nIt is trivial to see that under the stated conditions this is a solution of equation (2.234),\nthe only technically difficult step is to verify that these ar e the only solutions.\nOne again proceeds by iteratively using the Vandermonde mat rixMAB= (λA)B−1\nand considering its determinant. Instead of showing that allof theλA’s equal\neach other, we now at various stages of the reduction process use the condition/summationtext\nA∈BaDA= 0 to completely decouple the corresponding λA∈Ba=rafrom the\nremaining λA/negationslash∈Ba. Proceeding in this way we finally obtain equation (2.247) as\nclaimed.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 86\nWe then see\nJA∈Ba=−iln(ra)\n2π+mA∈Ba;mA∈Ba∈Z. (2.248)\nExpressed directly in terms of the surface gravities this yi elds\nH/summationdisplay\ni=1ZA∈Bagap\nκi=−iln(ra)\nπ+2mA∈Ba;mA∈Ba∈Z. (2.249)\nWith the obvious notation of a(A)denoting the index of the particular disjoint set\nBathatAbelongs to, we can write this as\nH/summationdisplay\ni=1ZAigap\nκi=−iln{ra(A)}\nπ+2mA;mA∈Z. (2.250)\nThis result now is somewhat more subtle to analyze. Let AandBboth belong to\na particular set Ba. Then\nH/summationdisplay\ni=1{ZAi−ZBi}gap\nκi= +2{mA−mB};mA,mB∈Z;A,B∈ Ba.(2.251)\nThat is\ngap =2{mA−mB}\nH/summationdisplay\ni=1{ZAi−ZBi}\nκi; gap ∈R;A,B∈ Ba; (2.252)\nso we see that the gap is real. (Furthermore, the gap is seen to be a sort of “integer-\nweighted harmonic average” of the κi.) But reality then implies that ra=eiφaso\nthat\nH/summationdisplay\ni=1ZAigap\nκi=φa(A)\nπ+2mA;mA∈Z. (2.253)\nBy using/summationtext\nAZAi= 0we see that\n¯φ\nπ=/summationdisplay\naφa|Ba|\nNπ∈Q, (2.254)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 87\nso that\nH/summationdisplay\ni=1ZAigap\nκi=φa(A)−¯φ\nπ+2(mA−¯m);mA∈Z. (2.255)\nUnfortunately in the general case there is little more than c an be said and one has\nto resort to special case-by-case analyses. One last point w e can make is that even\nthough in this situation the Rij=κi/κjare sometimes irrational we can make the\nweaker statement that\nH/summationdisplay\ni=1{ZAi−ZBi}\nκi\nH/summationdisplay\ni=1{ZCi−ZDi}\nκi∈Q;A,B,C,D ∈ Ba; (2.256)\nThat is, certain weighted averages of the surface gravities are guaranteed to be\nrational. If we wish to analyze whether rational ratios of Rij=κi/κjare implied\nin each of the particular cases of interest, we need to:\na) Check if there exists some ω0giving non-trivial subsets Ba, leading to (2.246).\nb) Analyze the sets of equations (2.251) implied by such an ω0.\nBy proceeding in this way we are able to prove that periodicit y of the QNFs im-\nplies rational ratios for the surface gravities in the follo wing physically interesting\ncases:\na) Forj= 2min equation (2.177).\nb) For equation (2.189) when j∝\\e}atio\\slash= 2m+1andcos(πj)∝\\e}atio\\slash=−1\n2.\nc) For equation (2.203) when j= 2m.\n2.7.1 Analysis of particular cases\nNow explore the familiar cases, which can serve also as parti cular examples\ndescribed by this theorem. In the first part of each particula r example we ex-\nplore whether periodicity implies rational ratios of the su rface gravities (withinCHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 88\nthis particular case). If the rational ratios are implied, w e can use the results of\nsection 2.5.3 to determine the families and the gap structur e. Despite having the\nanalysis from section 2.5.3, we will derive (for each case) t he gap structure also by\nanalysing the equations (2.251). This serves as:\n•a consistency check,\n•to bring more understanding in how the ideas used in the secti on 2.6.1 work.\nThis derivation is made in the second part of each case analys is (to be exact, it is\nmade for illustrative reasons for one set splitting only).\nSome common notation\nWe will use the following notation:\nfA≡H/summationdisplay\ni=1ZAi\nκi, (2.257)\nbut if there is some index ˜A, such that f˜A=−fA, we rename it to ˜A=−A. So\nalwaysf−A=−fA. We will also use mA,B≡mA−mBand for the equation (2.251)\nof the form\ngap(ωn)(fA−fB) = 2mA,B (2.258)\nwe use the symbol EA,B. Furthermore let us add one conceptual explanation: We\nsay that equations EA,BandE˜A,˜Bare linearly independent if there do not exist\nsuch integers mA,B,m˜A,˜B, that the equations EA,BandE˜A,˜Bwill be linearly depen-\ndent in the usual sense of the word. If EA,BandE˜A,˜Bare not linearly independent,\nwe say they are linearly dependent.\n2 Horizons case, S-dS black hole by monodromy calculations:\n(1) spin 1 perturbations, (2) spin 0 and 2 perturbations\nTake the first case, which is formula derived by the use of mono dromy techniques\nfor the S-dS black hole:\ncosh/parenleftbiggπω\nκ−−πω\nκ+/parenrightbigg\n+[1+2cos( πj)]cosh/parenleftbiggπω\nκ−+πω\nκ+/parenrightbigg\n= 0. (2.259)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 89\nHerej= 0 for spin 0, 2 and j= 1 for spin 1.\nThis becomes for spin 1 perturbation:\ne(πω\nκ−−πω\nκ+)+e−(πω\nκ−−πω\nκ+)−e(πω\nκ−+πω\nκ+)−e−(πω\nκ−+πω\nκ+)= 0, (2.260)\nand for spin 0 and 2 perturbations:\ne(πω\nκ−−πω\nκ+)+e−(πω\nκ−−πω\nκ+)+3e(πω\nκ−+πω\nκ+)+3e−(πω\nκ−+πω\nκ+)= 0. (2.261)\nIn the case of spin 1 perturbation we already know that ratios of surface gravities\nmight be completely arbitrary and we still get periodic solu tions.\nThe question of surface gravities rational ratios The matrix ZAihas rank 2 ( =\nH) hence if some subset of DAdoes not sum to 0, the rational ratio of surface\ngravities is implied.\nLet us have a look what happens here: The functions fiare given as\nf1=1\nκ−−1\nκ+,\nf2=1\nκ−+1\nκ+,\nand we have also f−1andf−2. If the coefficients DAsplit into two sets each\nhaving two elements (which is the only way how they can be non- trivially split in\nthis case), there are two equations and three possibilities how one can split them:\n• {D1,D−1}and{D2,D−2},\n• {D1,D2}and{D−1,D−2},\n• {D1,D−2}and{D2,D−1}.\nIn terms of equations this leads to the following combinatio ns:\n•E1,−1, E2,−2gives 2 linearly independent equations,CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 90\n•E1,2, E−1,−2gives only 1 linearly independent equation,\n•E1,−2, E2,−1gives only 1 linearly independent equation.\nBecause we have two surface gravities and two linearly indep endent equations\nthe first combination of equations leads to the condition tha t the ratio of surface\ngravities must be rational. That means one needs to explore o nly the second and\nthe third combination of equations. Here we have to check the stepa)from the end\nof the previous section. If there exists ω0giving us Basets leading to the second,\nor the third combination of equations, it must fulfill the fol lowing conditions:\n•In the case of the second combination ( E1,2, E−1,−2) it must fulfil the equa-\ntions\nπω0\nκ+=i2πm1,2+ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC1\nC2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (2.262)\n−πω0\nκ+=i2πm−1,−2+ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC−1\nC−2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.263)\n•In the case of the third combination ( E1,−2, E−1,2) it must fulfil the equations\nπω0\nκ−=i2πm1,−2+ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC1\nC−2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (2.264)\n−πω0\nκ−=i2πm−1,2+ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC−1\nC2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.265)\nBut for the second combination this means:\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC1\nC2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=−ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC−1\nC−2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (2.266)\nand for the third combination this means:\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC1\nC−2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=−ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleC−1\nC2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.267)\nFrom (2.260) we see that for spin 1 perturbation it holds\n|C1|=|C2|=|C−1|=|C−2|= 1. (2.268)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 91\nAs a result of (2.268) the conditions (2.266) and (2.267) are trivially fulfilled, since\nall the logarithms are 0. But for spin 0 and 2 |C1|=|C−1|= 1 and|C2|=|C−2|=\n3, which means that in each case we get ln(1\n3). This means there is no way how to\nfulfil (2.266), or (2.267), hence split the coefficients in su ch way that we do notget\nthe surface gravities rational ratio condition. This means that for spin 0 and spin 2\nperiodicity implies the rational ratio of surface gravitie s.\nFor the spin 1 perturbation we already showed we can find expli citω0-s lead-\ning to two different families of QNMs, each family related to different horizon\nsurface gravity ( inκ±). This means that in the case of spin 1 perturbation rational\nratios are notimplied. Note also that each of the two families is just a resu lt of\ndifferent splitting of coefficients DA.\n“Gap” derivation by using our approach (spin 1 perturbation ) As previously\nnoted, the general solutions split into two families, one re lated to one surface grav-\nity (ω=inκ−), the other to another ( ω=inκ+). Now take one solution from the\nset{inκ−}(for example ω=iκ−) and substitute it in (2.260) to get the coefficients\nDA. We see that the two set splitting of DAis:\n/braceleftbigg\nD1=−e−iπκ−\nκ+, D−2=e−iπκ−\nκ+/bracerightbigg\n,/braceleftbigg\nD−1=−eiπκ−\nκ+, D2=eiπκ−\nκ+/bracerightbigg\n.(2.269)\n(We can easily observe that the sum of those couples of coeffic ients within each\nset is 0). Now, since E1,−2andE−1,2are linearly dependent, the second equation\ncan be taken only as a definition of m−1,2. The only independent equation is then:\ngap(ωn) =m1,−2κ−.\nTo obtain from this equation the basic gap structure, we have to choose the integer\nm1,−2to be such, that we obtain the “narrowest” gap. This is obtain ed bym1,−2=\n1, confirming the known result.\nIf we start with the second set of solutions related to the cos mological horizon\nsurface gravity, we can proceed completely analogically an d verify also the second\nresult. The gap will be in such case given by the cosmological horizon surface\ngravity.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 92\n“Gap” derivation by using our approach (spin 0 and 2 perturba tion) Here we\nproved we have 2 linearly independent equations (the rest of the equations are\nonly definitions of mA,Bterms):\n•gap(ωn) =−m1,2κ+,\n•gap(ωn) =−m1,−2κ−.\nNow, they immediately lead to the conditionκ−\nκ+=m1,2\nm1,−2, hence surface gravities\nratio being rational.\nTo explore the gap structure one has to explore all the other i ndependent con-\nditions as well (here there are three independent condition s obtained by equating\n4 terms). This is because definitions of mA,Bmight put some additional constrains\non the gap function. (The constraints come from the fact that themA,Bhave to\nalways be integers. This will be seen in the analysis of the fo llowing case.) But in\nthis case the third independent condition can be given as\ngap(ωn)/parenleftbigg1\nκ−−1\nκ+/parenrightbigg\n=m1,−1 (2.270)\nand is consistent with taking the “narrowest” gap as\ngap(ωn) =κ∗=κ−m1,−2=κ−p−=κ+m1,2=κ+p+=p−\nb−=p+\nb+=1\nb∗.\n(Hereb∗is the highest common divisor of b−,b+with respect to integers.) This\nmeans:−m1,2=p+and−m1,−2=p−.\n2 Horizons, S-dS black hole by analytically solvable potent ials\nThe equation we obtained is the following:\ncosh/parenleftbiggπω\nκ−−πω\nκ+/parenrightbigg\n−cosh/parenleftbiggπω\nκ−+πω\nκ+/parenrightbigg\n= 2cos(πα+)cos(πα−). (2.271)CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 93\nThe question of surface gravities rational ratios In the previous section we\nalready proved that in this case the rational ratios are impl ied by the periodicity.\nWe can prove the same by analyzing all the possibilities how t o split coefficients\ninto different sets {DA}. It is an alternative proof to the one in the previous\nsection. It is definitely a more complicated proof, but its ad vantage is that it is\npart of more general approach, which generates proofs for al l the cases where the\nimplication holds.\nLet us show the “ugly” way of proving it by the new general meth od: The\nmatrixZAiis of rank 2 ( =H) and hence if the DAcoefficients do not split into\nnontrivial subsets, the rational ratios of surface graviti es is proven. Now let us\nexplore what ways of splitting the coefficients one can obtai n and what they gen-\nerally mean. We can rewrite the equation (2.271) to get the fo llowing functions:\nf1=3\nκ−+1\nκ+,f2=1\nκ−+3\nκ+,f3= 3(1\nκ−+1\nκ+),\nf4=1\nκ−+1\nκ+,f5= 2/parenleftBig\n1\nκ−−1\nκ+/parenrightBig\n.\nLet us explore what happens when we split the DAcoefficients into two sets,\none having three and the other two elements (these are the onl y possible nontriv-\nial ways of splitting the coefficients). This gives for each s plitting 3 equations.\nOne can analyze all the combinations of equations in the foll owing way: Pick f˜A\nbelonging to a coefficient in the three element set and take al l combinations of\nsuch equations EA,B(related to all the possible ways of splitting the coefficien ts\nin whichD˜Ais in the set of three elements), that they do notinvolve the given f˜A.\nIf these combinations give 2 independent equations we are fin ished, and do not\nneed to explore the 3-rd equation involving f˜A. If there is only one independent\nequation from the given couple, we need to explore if there is a way how to add an\nequation involving f˜A(E˜A,B), by keeping the number of independent equations\nto be still 1. The I/D letters in the table mean linearly independent/dependent:CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 94\nf˜Acombination 1 I/D combination 2 I/D combination 3 I/D\nf1E2,3, E4,5 IE2,4, E3,5 IE2,5, E3,4 I\nf2E1,3, E4,5 IE1,4, E3,5 IE1,5, E3,4 I\nf3E1,2, E4,5 IE1,4, E2,5 IE1,5, E2,4 I\nf4E1,2, E3,5 IE1,3, E2,5 IE1,5, E2,3 I\nf5E1,2, E3,4 IE1,3, E2,4 DE1,4, E2,3 D\nWe see that only the last two cases give linearly dependent eq uations, but it\nis very easy to check that in both of these cases it holds, that if we add arbitrary\nthird equation of the type E5,B(hence involving f5), the number of linearly inde-\npendent equations grows to 2. This is a result of the fact that inE1,3, E2,4there is\nonlyκ+present and in E1,4, E2,3there is only κ−present. This proves that here\nthe rational ratios are implied by the periodicity.\nThe “gap” derivation by using our approach If there are no non-trivial ways\nof splitting the coefficients, then we get from our result (2. 251) 4 independent\nconditions (although, in the sense defined, only 2 linearly i ndependent equations)\nand they are:\n1)gap(ωn)/parenleftBig\n1\nκ−−1\nκ+/parenrightBig\n=m1,2,\n2)gap(ωn) =−m1,3κ+,\n3)gap(ωn) =m1,4κ−,\n4)gap(ωn)/parenleftBig\n1\nκ−−1\nκ+/parenrightBig\n= 2m1,5.\nNow any two from the first three equations5give surface gravity rational ratio\ncondition. Choose for this purpose, for example, equations 2) and 3). (Notice here\nthat in order to get the surface gravities ratio positive, m1,3,m1,4must have oppo-\nsite signs.) Equation 1) is then uninteresting since it is on ly defining m1,2, without\nconstraining the possible gap function. Now the whole probl em is encoded in\n5The same holds for any two of the last three equations.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 95\nthe equation 4) and particularly in the fact that there is 2 on the right side of the\nequation. In fact it leads to m1,4+m1,3= 2m1,5, and that means one can fulfil the\nequation 4) only if both m1,4,m1,3are odd or both are even. This means that the\ndefinition of m1,5constrains the gap function. Now realize that only the ratio of\nm1,3,m1,4is determined by the ratio of the surface gravities p−/p+, so there is still\nfreedom to multiply both, p+andp−, by the same arbitrary scaling integer. We\nhave to determine the integer to fulfill the equation 4) and si multaneously to give\nthe “narrowest” gap. But then, if the ratio\nκ−\nκ+=b+\nb−=p+\np−\nis in its most reduced form given by one odd and one even number (p+·p−is\neven), we have to take both m1,4,m1,3even by:m1,4= 2p−and−m1,3= 2p+. Then\nthe gap will be gap(ωn) = 2κ∗=2\nb∗/parenleftBig\nκ∗=κ−p−=κ+p+=1\nb∗/parenrightBig\n. In the case both\np−,p+are odd numbers (hence p−·p+is odd), the 4)-th equation is automatically\nfulfilled and to get the “narrowest” gap one chooses the scali ng integer to be ±1,\nhencem1,4=p−andm1,3=−p+. Then the gap is obtained as gap(ωn) =κ∗=1\nb∗.\n2 Horizons, R-N black hole\nTake as another example the equation (2.189) with j= 0. We can rewrite it as:\neπω(1\nκ++1\nκ−)+3e−πω(1\nκ++1\nκ−)+3eπω(1\nκ−−1\nκ+)= 0. (2.272)\nThere cannot be any non-trivial subset of coefficients DAsumming up to 0. (This is\nbecause there are three non-zero coefficients, so it cannot h appen that two of them\nwill give 0 by the summation). That means we have always two in dependent\nconditions:\n1)gap(ωn) =m1,3κ+,\n2)gap(ωn) =−m2,3κ−.\nBut1)and2)are also linearly independent equations, so this immediate ly implies\nthe same results, as in the previous case. These are:CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 96\n•the rational ratio of surface gravities given as −m1,3/m2,3(note again that to\nget the ratio positive, m1,3m2,3have to be chosen with the opposite signs),\n•the “narrowest” gap given by m1,3=p+andm2,3=−p−, hence\ngap(ωn) =κ∗=κ−p−=κ+p+=1\nb∗.\nSo in this case the rational ratios are implied by the periodicity.\n3 Horizons, Monodromy results: R-N-dS black hole\nTake the equation (2.203) with any j, for which the coefficients are non-zero (trivial\nexample is standard j= 0).\nThe question of surface gravities rational ratios Now this is a case with six\nfunctionsfA:\nf1=1\nκ+−1\nκC,f2=1\nκ++1\nκC,f3=2\nκ−+1\nκ+−1\nκCandf−1, f−2, f−3.\nHere we have 3 surface gravities and the rank of ZAimatrix is 3, so if there is no\nnontrivial subset of DAcoefficients summing to 0, the surface gravities rational\nratios are implied.\nLet us explore the nontrivial ways of splitting the coefficie nts. The splitting of\ncoefficients giving the lowest number of independent condit ions is the splitting\ninto three sets, each having 2 elements. It gives 3 equations (and we need three\nlinearly dependent). There are three basic ways how to split the coefficients:\n•The first splitting is giving E1,−1, E2,−2, E3,−3, but these are necessarily 3\nlinearly independent equations as f1, f2, f3are linearly independent set of\nfunctions.\n•The other type of splitting is EA1,A2, E−A2,A3, E−A3,−A1,Ai=±1,±2,±3,\n|Ai| ∝\\e}atio\\slash=|Aj|ifi∝\\e}atio\\slash=j. But all these ways of splitting the coefficients give 3\nlinearly independent equations as well.CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 97\n•The third basic way how to split coefficients into three two el ement sets gives\nEA1,−A1, EA2,A3, E−A2,−A3,Aibeing defined as before. This splitting gives\nonly 2 linearly independent equations.\nThat means in the last case we have to take the second step. For tunately here we\ncan use for different CAcoefficients multiplying different cosh(·)terms the same\nargument as in the case of S-dS and spin 0 and 2 perturbations. This argument\nexcludes the possibility of such ways of splitting the coeffi cients.\nAny splitting into two sets, one having 2 and the other 4 eleme nts is just more\nconstrained version of some “three set each having two eleme nts” splitting. This\nmeans the arguments provided in the previous case transfer a utomatically to this\ncase.\nThe last, deeply nontrivial splitting is when DAcoefficients split into two sets,\neach having three elements. In such case we have 4 independen t conditions. There\nare two possible basic ways of splitting the equations:\n•EA1,−A1, EA1,A2, E−A2,A3, EA3,−A3\n•EA1,A2, EA2,A3, E−A1,−A2, E−A2,−A3\nNow the first splitting leads to three independent equations , but the second split-\nting only to two independent equations. So in the second spli tting case one has\nto proceed to the next step: One can observe that if there exis ts anω0giving the\nsecond splitting, then the following two equations have to b e fulfilled. Take:\n•z≡eπω0,\n•α≡1\nκ+−1\nκC,\n•β≡ ±/parenleftBig\n1\nκ++1\nκC/parenrightBig\n,\n•δ≡ ±/parenleftBig\n2\nκ−+1\nκ++1\nκC/parenrightBig\nand\n•K≡1+cos(πj).CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 98\nThen the following holds:\nzα+Kzβ+2Kzδ= 0, (2.273)\nz−α+Kz−β+2Kz−δ= 0. (2.274)\nBut then by substituting to (2.274) z−δ=−K\nzα+Kzβwe will obtain\n1−3K2\nK+zα−β+z−(α−β)= 0,\nand then we get the result:\nzα−β=−1−3K2\n2K±1\n2/radicalBigg/parenleftbigg1−3K2\nK/parenrightbigg2\n−4.\nForK >1√\n3we havezα−β>0. We see that if we take j=n\n2, the only nontrivial\ncases (K∝\\e}atio\\slash= 0) areK= 1,2and fulfil this condition. But the equation (2.273) is\nin fact:\nzα−β+K+2Kz(α−β)δ−β\nα−β= 0. (2.275)\nDefine byu≡zα−β. Then by solving (2.275) we obtain\nδ−β\nα−β= logu/parenleftbigg\n−u+K\n2K/parenrightbigg\n.\nBut since the logarithm argument is for K >1√\n3negative (and uis in such case\npositive), it means thatδ−β\nα−βmust be nonreal. But considering how α,β,δ are de-\nfined, this cannot happen for surface gravities being real nu mbers. Hence it is\nproven that there is no ω0giving the splitting considered. This means we proved\nthat for (2.203) the periodicity implies rational ratios of surface graviti es.\nThe “gap” analysis by using our approach If we consider again only such ω0-s,\nfor which there is no nontrivial splitting of DAcoefficients, we obtain the follow-\ning 5 independent conditions:\n•gap(ωn)1\nκ−=m2,3,CHAPTER 2. ANALYTIC RESULTS FOR HIGHLY DAMPED QNMS 99\n•gap(ωn)1\nκ+=m1,−2,\n•gap(ωn)1\nκC=−m1,2,\n•gap(ωn)(1\nκ+−1\nκC) =m1,−1,\n•gap(ωn)1\nκ−=m−2,−3.\nNow note: the last two equations are uninteresting, since th ey are only definitions\nofm1,−1andm−2,−3without putting any constraint on the gap function. On the\nother hand the first three equations tell us that the ratio bet ween arbitrary two\nsurface gravities is a rational number. Then to get the “narr owest” gap, consider\nthe following:\nκ−\nκ+=p+\np1−=b+\nb−,κ−\nκC=pC\np2−=bC\nb−.\nFurtherκ∗=1\nb∗=κ−p1−=κ+p+andκ′\n∗=1\nb′∗=κ−p2−=κCpC. Then the gap\nmust be given by gap(ωn) = lcm{κ∗,κ′\n∗}=l1κ∗=l2κ′\n∗, (l1,l2∈Z). That means the\nmintegers must be taken as: m2,3=l1·p1−,m1,−2=l1·p+and−m1,2=l2·pC.\n2.7.2 Conclusions\nIt seems to be very hard (if not impossible) to find a more const rained form of the\nmonodromy results than the formulation (2.169). Since in (2 .169) it is not a general\nresult (but seems to be generic enough) that the rational rat ios of surface gravities\nare implied by periodicity (see S-dS spin 1 perturbations as counter-example), we\ndeveloped a general method how to prove the implication for e very case in which\nit holds. The necessary step was to prove the theorem at the be ginning of the sec-\nond section. By using our method we proved that the implicati on (periodicity →\nrational ratios) holds in every monodromy case (from the cas es described before),\napart of the one given counter-example.Chapter 3\nMultiplication of tensorial\ndistributions\n3.1 Introduction\nThis chapter is devoted to a topic from the field of mathematic al physics which is\nclosely related to the general theory of relativity. It offe rs possible significant con-\nceptual extensions of general relativity at short distance s/high energies, and gives\nanother arena in which one can conceptually/physically mod ify the classical the-\nory. But the possible meaning of these ideas is much wider tha n just the general\ntheory of relativity. It is related to the general questions of how one defines the\ntheory of distributions for any geometrically formulated physics describing interac-\ntions .\nThe main reasons why we “bother” Let us start by giving the basic reasons\nwhy one should work with the language of distributions rathe r than with the old\nlanguage of functions:\n•First there are deep physical reasons for working with distr ibutions rather\nthan with smooth tensor fields. We think distributions are mo re than just a\nconvenient tool for doing computations in those cases in whi ch one cannot\n100CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 101\nuse standard differential geometry. We consider them to be m athematical\nobjects which much more accurately express what one actuall y measures in\nphysics experiments, more so than when we compare them to the old lan-\nguage of smooth functions. The reason is that the question: “ What is the\n‘amount’ of physical quantity contained in an open set?” is i n our view\na much more reasonable physical question, (reasonable from the point of\nview of what we can ask the experimentalists to measure), tha n the ques-\ntion: “What is the value of quantity at a given point?”. But “p oint values”\nas “recovered” by delta distributions do seem to give a preci se and reason-\nable meaning to the last question. There is also a strong intu ition that the\n“amount” of a physical quantity in the open set Ω1∪Ω2, whereΩ1,Ω2are\ndisjoint is the sum of the “amounts” of that quantity in Ω1andΩ2. That\nmeans it is more appropriate to speak about distributions ra ther than gen-\neral smooth mappings from functions to the real numbers (the mappings\nshould also be linear).\n•The second reason is that many physical applications sugges t the need for\na much richer language than the language of smooth tensor fiel ds. Actually\nwhen we look for physically interesting solutions it might b e always a matter\nof importance to have a much larger class of objects availabl e than the class\nof smooth tensor fields.\n•The third reason (which is a bit more speculative) is the rela tion of the lan-\nguage defining the multiplication of distributions to quant um field theories\n(but specifically to quantum gravity). Note that the problem s requireing\ndistributions, that means problems going beyond the langua ge of classical\ndifferential geometry, might be related to physics on small scales. At the\nsame time understanding some operations with distribution s, specifically\ntheir product has a large formal impact on the foundations of quantum field\ntheory, particularly on the problem with interacting fields . (See, for example,\n[73].) As a result of this it can have significant consequence s for quantum\ngravity as well.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 102\nThe intuition behind our ideas Considerations about language intuitiveness\nlead us to an interesting conclusion: the language of distri butions (being con-\nnected with our intuition) strongly suggests that the prope rties of classical tensor\nfields should not depend on the sets having (in every chart) Le besgue measure\n0. If we follow the idea that these measure zero sets do not hav e any impact on\nphysics, we should naturally expect that we will be able to ge neralize our lan-\nguage from a smooth manifold into a piecewise smooth manifol d (which will\nbring higher symmetry to our conceptual network). The first t races of piecewise\nsmooth coordinate transformations are already seen, for ex ample, in [100].\nThe current situation is “strange” Now it is worth noting how strange the cur-\nrent situation of the theory of distributions is: we have a us eful and meaningful\nlanguage of distributions, which can be geometrically gene ralized, but this lan-\nguage works only for linear physics. But linear physics is on ly a starting point (or\nat best a rough approximation) for describing real physical interactions, and hence\nnonlinear physics. So one naturally expects that the “physi cal” language of dis-\ntributions will be a result of some mathematical language de fining their product.\nMoreover, at the same time we want this language to contain th e old language of\ndifferential geometry (as a special case), as in the case of l inear theories. It is quite\nobvious that the nonlinear generalization of the geometric distributional theory\nand the construction of generalized differential geometry are just two routes to\nthe same mathematical theory. The natural feeling that such theory might exist is\nthe main motivation for this work. The practical need of this language is obvious\nas well, as we see in the numerous applications [11, 61, 73, 76 , 104, 156, 158, 173].\n(Although this is not the main motivation of our work.) But is it necessary that\nsuch more general mathematical language exists as a full wel l defined theory?\nNo, not at all. The potentially successful uses of distribut ions that go behind the\nSchwarz original theory might be only “ad hoc” from the funda mental reasons.\nTake the classical physics. The success of such uses of distr ibutions here might\nnot be a result of some more general language than smooth tens or calculus being\na classical limit of the more fundamental physics. There mig ht be only hiddenCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 103\nspecific reasons in the more fundamental physics why such “ad hoc” calculations\nin some of the particular cases work. But it is certainly very interesting and im-\nportant to explore and answer the question whether: (i) such a full mathematical\nlangauge exists, and (ii) to which extent it is useful to the p hysics community.\nFor the first question this work suggests a positive answer, t o answer the second\nquestion much more work has to be done.\nSo what did we achieve? The main motivation for this work is the develop-\nment of a language of distributional tensors, strongly conn ected with physical in-\ntuition. (This also means it has to be based on the concept of a piecewise smooth\nmanifold.) This language must contain all the basic tensori al operations in a gen-\neralized way, enabling us to understand the results the comm unity has already\nachieved, and also the problems attached to them. It is worth to stress that some\nof this motivation results from a shift in view regarding the foundations of clas-\nsical physics (so it is given by “deeper” philosophical reas ons), but it can have\nalso an impact on practical physical questions. The scale of this impact has still\nto be explored. We claim that the goals defined by our motivati on (as described\nat the beginning) are achieved in this work. Particularly, w e have generalized all\nthe basic concepts from smooth tensor field calculus (includ ing the fundamental\nconcept of the covariant derivative) in two basic direction s:\n•The first generalization goes in the direction of the class of objects that, (in\nevery chart on the piecewise smooth manifold), are indirect ly related to sets\nof piecewise continuous functions. This class we call D′m\nnEA(M). For a de-\ntailed understanding see section 3.4.3.\n•The second generalization is a generalization to the class o f objects naturally\nconnected with a smooth manifold belonging to our piecewise smooth man-\nifold (in the sense that the smooth atlas of the smooth manifo ld is a subatlas\nof our piecewise smooth atlas). This is a good analogy to the g eneralization\nknown from the classical distribution theory. The class of s uch objects we\ncallD′m\nn(So)(M). For detailed understanding see again the section 3.4.3.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 104\nWe view our calculus as the most natural and straightforward construction achiev-\ning these two particular goals. The fact that such a natural c onstruction seems to\nexist supports our faith in the practical meaning of the math ematical language\nhere developed.\nThe last goal of this chapter is to suggest much more ambitiou s, natural gen-\neralizations, which are unfortunately at present only in th e form of conjectures.\nLater in the text we provide the reader with such conjectures .\nThe structure of this chapter The structure of this chapter is the following: In\nthe first part we want to present the current state of the Colom beau algebra the-\nory and its geometric formulations. We want to indicate wher e its weaknesses are.\nThis part is followed by several technical sections, in whic h we define our theory\nand prove the basic theorems. First we define the basic concep ts underlying our\ntheory. After that we define the concept of generalized tenso r fields, their impor-\ntant subclasses and basic operations on the generalized ten sor fields (like tensor\nproduct). This is followed by the definition of the basic conc ept of our theory, the\nconcept of equivalence between two generalized tensor field s. The last technical\npart deals with the definition of the covariant derivative op erator and formulation\nof the initial value problem in our theory. All these technic al parts are followed\nby explanatory sections, where we discuss our results and sh ow how our theory\nrelates to the practical results already achieved (as descr ibed in the first part of the\nchapter).\n3.2 Overview of the present state of the theory\n3.2.1 The theory of Schwartz distributions\nAround the middle of the 20-th century Laurent Schwartz foun d a mathemati-\ncally rigorous way for extending the language of physics fro m the language of\nsmooth functions into the language of distributions. Physi cists such as Heavi-\nside and Dirac had already given good physics reasons for bel ieving that such aCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 105\nmathematical structure might exist.\nThe classical formulations of the distribution theory were directly connected\nwithRnand were non-geometric. Distributions in such a formulatio n are typically\nunderstood as continuous, linear maps from compactly suppo rted smooth func-\ntions (on Rn) to real numbers. (The class of such functions is typically d enoted\nbyD(Rn). The dual to such space, which is what distributions are, is t ypically\ndenoted by D′(Rn).) Here the word “continuous” refers to the following topol-\nogy on the given space of compactly supported smooth functio ns:The sequence\nof smooth compactly supported functions fl(xi)converges for l→ ∞ to a smooth com-\npactly supported function f(xi)iff an arbitrary degree derivative with respect to arbitrar y\nvariables∂nfl(xi)\n∂xm1\n1...∂xmk\nk/parenleftBig/summationtextk\ni=1mk=n/parenrightBig\nconverges uniformly on each compact Rnsubset to\n∂nf(xi)\n∂xm1\n1...∂xmk\nk.\nThe alternative classical way of formulating the theory of d istributions is to\nextend the space of test objects to be such that they still fol low appropriate fall-off\nproperties. The properties can be summarized as:\nf(xi)belongs to such space if it is smooth and\n(∀α,∀β) lim\nx1,...xk→∞/parenleftbigg\nxn1\n1...xnk\nk∂αf(xi)\n∂xm1\n1...∂xmk\nk/parenrightbigg\n= 0, (3.1)\nk/summationdisplay\ni=1ni=β,k/summationdisplay\nj=1mj=α.\nThis is a topological space with topology given by a set of sem i-normsPα,βde-\nfined1as\nPα,β= sup\nxi/vextendsingle/vextendsingle/vextendsingle/vextendsinglexn1\n1...xnk\nk∂αf(xi)\n∂xm1\n1...∂xmk\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (3.2)\nk/summationdisplay\ni=1ni=β,k/summationdisplay\nj=1mj=α.\nNaturally, the space of continuous and linear maps on such a s pace is more re-\nstricted as in the first, more common formulation. Objects be longing to such duals\n1We admit that the notation Pα,βmight be somewhat misleading, since the α,β values do not\nspecify the semi-norm in a unique way.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 106\nare called “tempered distributions”. The advantage of this more restricted version\nis that Fourier transform is a well defined mapping on the spac e of tempered dis-\ntributions (see [122]).\nIf we refer to the more common, first formulation, the space of distributions\naccommodates the linear space of smooth functions by the map ping:\nf(xi)→/integraldisplay\nRnf(xi)···dnx. (3.3)\nHeref(xi)is a smooth function on Rnand/integraltext\nRnf(x)···dxis a distribution de-\nfined as the mapping:\nΨ(xi)→/integraldisplay\nRnf(xi)Ψ(xi)dnx,Ψ(xi)∈D(Rn). (3.4)\nMoreover, by use of this mapping one can map into the space of d istributions\nany Lebesgue integrable function (injectively up to a funct ion the absolute value\nof which has Lebesgue integral 0) . The distributional objec ts defined by the im-\nages of the map (3.3) are called regular distributions. The m ap (3.3) always pre-\nserves the linear structure, hence the space of Lebesgue int egrable functions is a\nlinear subspace of the space of distributions. But the space of distributions is a\nlarger space than the space of regular distributions. This c an be easily demon-\nstrated by defining the delta distribution\nδ(Ψ)≡Ψ(0) (3.5)\nand showing that such mapping cannot be obtained by a regular distribution.\nNote particularly that delta distribution had an immediate use in physics in the\ndescription of point-like sources. (Its intuitive use in th e work of Paul A.M. Dirac\nbefore the theory of Schwartz distributions was found, was o ne of the main physics\nreasons why people searched for such language extension.)\nThese considerations show that the space of distributions i ssignificantly larger\nthan the space of smooth functions. The space of distributio ns is classically taken\nto be a topological space with the weak ( σ−) topology. In this topology the space\nof regular distributions given by smooth functions is a dens e set.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 107\nMoreover one can continuously extend the derivative operat or from the space\nofC1(Rn)functions to the space of distributions by using the definiti on:\nT, xi(Ψ)≡ −T(Ψ, xi), (3.6)\n(whereTdenotes a distribution). This means that C∞(Rn)functions form not\nonly a linear subspace in the space of distributions, but als o a differential linear\nsubspace. It looks like there stands “almost” nothing in the way of fully and satis-\nfactorily extending the language of C1(Rn)functions to the language of Schwartz\ndistributions. Unfortunately there is still one remaining trivial operation and this\nis the operation of multiplication. That means we need to obt ain some distribu-\ntional algebra having as subalgebra the algebra of Lebesgue integrable functions\nfactorized by functions the absolute value of which has Lebe sgue interal 0. Unfor-\ntunately, shortly after Laurent Schwartz formulated the th eory of distributions he\nproved the following “no-go” result [138]:\nThe requirement of constructing an algebra that fulfills the following three con-\nditions is inconsistent:\na) the space of distributions is linearly embedded into the a lgebra,\nb) there exists a linear derivative operator, which fulfills the Leibniz rule and\nreduces on the space of distributions to the distributional derivative,\nc) there exists a natural number ksuch, that our algebra has as subalgebra the\nalgebra ofCk(Rn)functions.\nThis is called the Schwartz impossibility theorem. There is a nice example show-\ning where the problem is hidden: Take the Heaviside distribu tionH. Suppose\nthata)andb)hold and we multiply the functions/distributions in the usu al way.\nThen since Hm=Hthe following must hold:\nH′= (Hm)′=δ=mHm−1δ. (3.7)\nBut this actually implies that δ= 0, which is nonsense.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 108\nThe closest one can get to fulfill the conditions a)−c)from the Schwarz impos-\nsibility theorem is the Colombeau algebra (as defined in the n ext section) where\nconditions a)−c)are fulfilled with the exception that in the c)conditionkis taken\nto be infinite. This means that only the smooth functions form a subalgebra of\nthe Colombeau algebra. This is obviously not satisfactory, since we know (and\nneed to recover) rules for multiplying multiply much larger classes of functions\nthan only smooth functions. In the case of Colombeau algebra s this problem is\n“resolved” by the equivalence relation, as we will see in the following section.\n3.2.2 The standard Rntheory of Colombeau algebras\nThe special Colombeau algebra and the embedding of distributions\nThe so called special Colombeau algebra is on Rndefined as:\nG(Rn) =EM(Rn)/N(Rn). (3.8)\nHereEM(Rn)(moderate functions) is defined as the algebra of functions:\nRn×(0,1]→R (3.9)\nthat are smooth on Rn(this is usually called E(Rn)), and for any compact subset\nKofRn(for which we will henceforth use the notation K⊂⊂Rn) it holds that:\n∀α∈Nn\n0,∃p∈Nsuch that2sup\nx∈K|Dαfǫ(xi)| ≤O(ǫ−p)asǫ→0. (3.10)\nTheN(Rn)(negligible functions) are functions from E(Rn)where for any K⊂⊂\nRnit holds that:\n∀α∈Nn\n0,∀p∈Nwe have sup\nxi∈K|Dαfǫ(xi)| ≤O(ǫp)asǫ→0. (3.11)\nThe first definition tells us that moderate functions are thos e whose partial deriva-\ntives of arbitrary degree (with respect to variables xi) do not diverge faster then\n2In this definition, (and also later in the text), the symbol Nn\n0denotes a sequence of nmembers\nformed of natural numbers (with 0 included).CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 109\nany arbitrary negative power of ǫ, asǫ→0. Negligible functions are those mod-\nerate functions whose partial derivatives of arbitrary deg ree go to zero faster than\nany positive power of ǫ, asǫ→0. This simple formulation can be straightfor-\nwardly generalized into general manifolds just by substitu ting the concept of Lie\nderivative for the “naive” derivative used before.\nIt can be shown, by using convolution with an arbitrary smoot hing kernel (or\nmollifier), that we can embed a distribution into the Colombe au algebra. By a\nsmoothing kernel we mean, in the widest sense a compactly sup ported, smooth\nfunctionρǫ(xi), withǫ∈(0,1], such that:\n•supp(ρǫ)→ {0}for (ǫ→0),\n•/integraltext\nRnρǫ(xi)dnx→1for (ǫ→0),\n• ∀η >0∃C,∀ǫ∈(0,η) supxi|ρǫ(xi)|<C.\nThis most generic embedding approach is mentioned for examp le in [100] (in\nsome sense also in [104]).\nMore “restricted” embeddings to G(Rn)are also commonly used. We can\nchoose for instance a subclass of mollifiers called A0(Rn), which are smooth func-\ntions from D(Rn)(smooth, compactly supported) and (i.e. [72]) such that\n∀ǫholds/integraldisplay\nRnρǫ(xi)dnx= 1. (3.12)\nTheir dependence on ǫis given3as\nρǫ(xi)≡1\nǫnρ/parenleftBigxi\nǫ/parenrightBig\n. (3.13)\nSometimes the class is even more restricted. To obtain such a formulation, we\nshall define classes Am(Rn)as classes of smooth, compactly supported functions,\nsuch that /integraldisplay\nRnxi\n1···xj\nlφ(xi)dnx=δ0kfori+···+j=k≤m. (3.14)\n3Later in the text will the notation ρǫ(xi),ψǫ(xi)(etc.) automatically mean dependence on the\nvariableǫas in (3.13).CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 110\nClearlyAm+1(Rn)⊂Am(Rn). Then the most restricted class of mollifiers is taken\nto be the class A∞(Rn). This approach is taken in the references [76, 124, 156, 158,\n173].\nEven in the case of the more restricted class of mollifiers the embeddings are\ngenerally non-canonical [72, 173]. The exception are smoot h distributions, where\nthe difference between two embeddings related to two differ ent mollifiers is al-\nways a negligible function.\nThe fullColombeau algebra and the embedding of distributions\nWhat is usually considered to be the canonical formulation o f Colombeau algebras\ninRnis the following: The theory is formulated in terms of functi ons\nRn×A0(Rn)→R(call themF). (3.15)\nThe Colombeau algebra is defined in such way that it is a factor algebra of mod-\nerate functions over negligible functions, where:\n•Moderate functions are functions from Fthat satisfy:\n∀m∈Nn\n0,∀K⊂⊂Rn∃N∈Nsuch that if φ∈AN(Rn),there areα,ρ>0,\nsuch that sup\nxi∈K|DmF(φǫ,xi)| ≤αǫ−Nif0<ǫ<ρ. (3.16)\n•Negligible functions are functions from Fthat satisfy:\n∀m∈N,∀K⊂⊂Rn,∀p∈N∃q∈Nsuch that if φ∈Aq(Rn),∃α,ρ>0,\nwe have sup\nxi∈K|DmF(φǫ,xi)| ≤αǫpif0<ǫ<ρ. (3.17)\nThen ordinary distributions automatically define such func tions by the convo-\nlution ([46, 50, 104, 173] etc.):4\nB→Bx/bracketleftbigg1\nǫnφ/parenleftbiggyi−xi\nǫ/parenrightbigg/bracketrightbigg\n, φ∈A0(Rn). (3.18)\n4Here the “Bx” notation means that xis the variable removed by applying the distribution.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 111\nImportant common feature of both formulations\nAll of these formulations have two important consequences. Given that Cdenotes\nthe embedding mapping:\n•Smooth functions ( C∞(Rn)) form a subalgebra of the Colombeau algebra\n(C(f)C(g) =C(f·g)forf,gbeing smooth distributions).\n•Distributions form a differential linear subspace of Colom beau algebra (this\nmeans for instance that C(f′) =C′(f)).\nThe relation of equivalence in the special Colombeau algebra\nWe can formulate a relation of equivalence between an elemen t of the special\nColombeau algebra fǫ(xi)and a distribution T. We call them equivalent, if for\nanyφ∈D(Rn), we have\nlim\nǫ→0/integraldisplay\nRnfǫ(xi)φ(xi)dnx=T(φ). (3.19)\nThen two elements of Colombeau algebra fǫ(xi),gǫ(xi)are equivalent, if for any\nφ(xi)∈D(Rn)\nlim\nǫ→0/integraldisplay\nRn(fǫ(xi)−gǫ(xi))φ(xi)dnx= 0. (3.20)\nFor the choice of A∞(Rn)mollifiers the following relations are respected by the\nequivalence:\n•It respects multiplication of distribution by a smooth dist ribution [173] in\nthe sense that: C(f·g)≈C(f)·C(g), wherefis a smooth distribution and\ng∈D′(Rn).\n•It respects (in the same sense) multiplication of piecewise continuous func-\ntions (we mean here regular distributions given by piecewis e continuous\nfunctions) [46].\n•Ifgis a distribution and f≈g, then for arbitrary natural number nit holds\nDnf≈Dng[50].CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 112\n•Iffis equivalent to distribution g, and ifhis a smooth distribution, then f·h\nis equivalent to g·h[50].\nThe relation of equivalence in the fullColombeau algebra\nIn the canonical formulation the equivalence relation is ag ain formulated either\nbetween an element of the Colombeau algebra and a distributi on, or analogously\nbetween two elements of the Colombeau algebra: If there ∃m, such that for any\nφ∈Am(Rn), and for any Ψ(xi)∈D(Rn), it holds that\nlim\nǫ→0/integraldisplay\nRn(f(φǫ,xi)−g(φǫ,xi))Ψ(xi)dnx= 0, (3.21)\nthen we say that fandgare equivalent ( f≈g). For the canonical embedding and\ndifferentiation we have the same commutation relations as i n the non-canonical\ncase. It can be also proven that for f1...fnbeing regular distributions given by\npiecewise continuous functions it follows that\nC(f1)...C(fn)≈C(f1...fn), (3.22)\nand forfbeing arbitrary distribution and gsmooth distribution it holds that\nC(f)·C(g)≈C(f·g). (3.23)\nHow this relates to some older Colombeau papers\nIn older Colombeau papers [46, 47] all these concepts are for mulated (equiva-\nlently) as the relations between elements of a Colombeau alg ebra taken as a sub-\nalgebra of the C∞(D(Rn))algebra. The definitions of moderate and negligible\nelements are almost exactly the same as in the canonical form ulation, the only\ndifference is that their domain is taken here to be the class D(Rn)×Rn(being\na larger domain than Ao(Rn)×Rn). The canonical formulation is related to the\nelements of the class C∞(D(Rn))through their convolution with the objects from\nthe classD(Rn). It is easy to see that you can formulate all the previous rela tions\nas relations between elements of the C∞(D(Rn))subalgebra (with pointwise mul-\ntiplication), containing also distributions.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 113\n3.2.3 Distributions in the geometric approach\nThis part is devoted to review the distributional theory in t he geometric frame-\nwork. How to define arbitrary rank tensorial distribution on arbitrary manifolds\nby avoiding reference to preferred charts? Usually we mean b y a distribution rep-\nresenting an (m,n)tensor field an element from the dual to the space of objects\ngiven by the tensor product of (m,n)tensor fields and smooth compactly sup-\nportedk-form fields (on kdimensional space). That means for example a regular\ndistributional (m,n)tensor field Bµ...\nν...is introduced as a map\nT⊗ω→/integraldisplay\nBν...β\nµ...αTµ...α\nν...βω. (3.24)\nThis is very much the same as to say that the test space are smoo th compactly\nsupported tensor densities Tµ....α\nν...β [60, 61]. The topology taken on this space is the\nusual topology of uniform convergence for arbitrary deriva tives related to arbi-\ntrary charts (so the convergence from Rntheory should be valid in all charts). The\nderivative operator acting on this space is typically Lie de rivative. (Lie derivative\nalong a smooth vector field ξwe denote Lξ.) This does make sense, since:\n•To use derivatives of distributions we automatically need d erivatives along\nvector fields.\n•Lie derivative preserves p-forms.\n•In case of Lie derivatives, we do not need to apply any additio nal geometric\nstructure (such as connection in the case of covariant deriv ative).\nThere is an equivalent formulation to [61], given by [173], w hich takes the\nspace of tensorial distributions to be D′(M)⊗Tm\nn(M). HereD′(M)is the dual\nto the space of smooth, compactly supported k-form fields ( kdimensional space).\nOr in other words, it is the space of sections with distributi onal coefficients. In\n[101] the authors generalize the whole construction by taki ng the tensorial distri-\nbutions to be the dual to the space of compactly supported sec tions of the bundle\nE∗⊗Vol1−q. HereVol1−qis a space of ( 1−q)-densities and E∗is a dual to aCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 114\ntensor bundle E(hence to the dual belong objects given as E⊗Volq). In all these\nformulations Lie derivative along a smooth vector field repr esents the differential\noperator5.\nLet us mention here that there is one unsatisfactory feature of these construc-\ntions, namely that for physical purposes we need much more to incorporate the\nconcept of the covariant derivative rather than the Lie deri vative. There was some\nwork done in this direction [75, 111, 174], but it is a very bas ic sketch, rather than\na full and satisfactory theory. It is unclear (in the papers c ited) how one can obtain\nfor the covariant derivative operator the expected and mean ingful results outside\nthe class of smooth tensor fields.\n3.2.4 Colombeau algebra in the geometric approach\nScalar special Colombeau algebra\nFor arbitrary general manifold it is easy to find a covariant f ormulation of the spe-\ncialColombeau algebra. At the end of the day you obtain a space of ǫ-sequences\nof functions on the general manifold M. For the non-canonical case the defini-\ntions are similar to the non-geometric formulations, the ba sic difference is that Lie\nderivative plays here the role of the Rnpartial derivative. Thus the definition of\nthe Colombeau algebra will be again:\nG(Ω) =EM(Ω)/N(Ω). (3.25)\nEM(Ω)(moderate functions) are defined as algebra of functions Ω×(0,1]→R,\nsuch that are smooth on Ω(this is usually called E(Ω)) and for any K⊂⊂Ωwe\ninsist that\n∀k∈Nn\n0,∃p∈Nsuch that ∀ξ1...ξkwhich are smooth vector fields,\nsup\nx∈K|Lξ1...Lξkfǫ(x)| ≤O(ǫ−p)asǫ→0. (3.26)\n5We can mention also another classical formulation of distri butional form fields, which comes\nfrom the old book of deRham [52] (it uses the expression “curr ent”). It naturally defines the space\nof distributions to be a dual space to space of all compactly s upported form fields.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 115\nN(Ω)(negligible functions) we define exactly in the same analogy to the non-\ngeometric formulation6:\n∀K⊂⊂Ω,∀k∈Nn\n0,∀p∈N,∀ξ1...ξkwhich are smooth vector fields,\nsup\nx∈K|Lξ1...Lξkfǫ(x)| ≤O(ǫp)asǫ→0. (3.27)\nTensor special Colombeau algebra and the embedding of distributions\nAfter one defines the scalar special Colombeau algebra, it is easy to define the\ngeneralized Colombeau tensor algebra as the tensor product of sections of a tensor\nbundle and Colombeau algebra. This can be formulated more ge nerally [101] in\nterms of maps from Mto arbitrary manifold. One can define them by changing\nthe absolute value in the definition (3.27) to the expression “any Riemann measure\non the target space”. Then you get the algebra [101, 103, 173] :\nΓC(X,Y) =ΓM(X,Y)/N(X,Y). (3.28)\nThe tensor fields are represented when the target space is tak en to be the TM\nmanifold7. It is clear that any embedding of distributions into such al gebra will\nbe non-canonical from various reasons. First, it is non-can onical even on Rn. An-\nother, second reason is that this embedding will necessaril y depend on some pre-\nferred class of charts on M. The embedding one defines as [101]:\nWe pick an atlas, and take a smooth partition of unity subordi nate toVαi, (Θj,\nsupp(Θ)⊆Vαjj∈N) and we choose for every j,ξj∈D(Vαj), such that ξj=\n1on supp (ξj). Then we can choose in fixed charts an A∞(Rn)elementρ, and the\nembedding is given by\n∞/summationdisplay\nj=1/parenleftBig\n((ξj(Θj◦ψ−1\nαj)uαj)∗ρǫ)◦ψαj/parenrightBig\nǫ, (3.29)\n6This version is due to [76]. There are also different definiti ons: in [101] the authors use instead\nof “for every number of Lie derivatives along all the possibl e smooth vector fields” the expression\n“for every linear differential operator”, but they prove th at these definitions are equivalent. This\nis also equivalent to the statement that in any chart holds: Φ∈ EM(Rn)(see [101]).\n7This is equivalent to G(X)⊗Γ(X,E)tensor valued Colombeau generalized functions [103].CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 116\nwhereψαjis a coordinate mapping and ∗is a convolution.\nThe equivalence relation\nNow let us define the equivalence relation in analogy to the Rncase. Since in [101]\nthe strongest constraint on the mollifier is taken, one would expect that strong re-\nsults will be obtained, but the definition is more complicate d. And in fact, stan-\ndard results (such as embedding of smooth function multiply ing distribution is\nequivalent to product of their embeddings) are not valid her e [101]. That is why\nthe stronger concept of k-association is formulated. It states that U∈ΓCisk\nassociated to function f, if\nlim\nǫ→0Lξ1...Lξl(Uǫ−f)→0 (3.30)\nuniformly on compact sets for all l≤k. The cited paper does not contain a precise\ndefinition of kequivalence between two generalized functions, but it can b e easily\nderived.\nThe older formulation of scalar fullColombeau algebra\nIf we want to get a canonical formulation, we certainly canno t generalize it straight\nfrom the Rncase (the reason is that the definition of the classes An(Rn)is not\ndiffeomorphism invariant). However, there is an approach p roviding us with a\ncanonical formulation of generalized scalar fields [68]. Th e authors define the\nspaceE(M)as a space of C∞(M×A0(M)), whereA0(M)is the space of n-forms\n(n-dimensional space), such that/integraltext\nω= 1. Now the authors define a smoothing\nkernel asC∞map from\nM×I→A0(M), (3.31)\nsuch that it satisfies:\n(i)∀K⊂⊂M∃ǫ0, C >0∀p∈K∀ǫ≤ǫ0,suppφ(ǫ,p)⊆BǫC(p),CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 117\n(ii)∀K⊂⊂M,∀k,l∈N0,∀X1,···Xk,Y1···Ylsmooth vector fields,\nsupp∈K,q∈M/vextenddouble/vextenddoubleLY1···LYl(L′\nX1+LXk)···(L′\nXk···LXk)Φ(ǫ,p)(q)/vextenddouble/vextenddouble=\n=O(ǫ−(n+1)).\nHereL′is defined as:\nL′\nXf(p,q) =LX(p→f(p,q)) =d\ndtf((Flx\nt)(p),q)|0. (3.32)\nBǫCis a ball centered at Chaving radius ǫmeasured relatively to arbitrary Rie-\nmannian metric. Let us call the class of such smoothing kerne lsA0(M). Then in\n[68] classes Am(M)are defined as the set of all Φ∈A0(M)such that ∀f∈C∞(M)\nand∀K⊂⊂M(compact subset) it holds:\nsup/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(p)−/integraldisplay\nMf(q)Φ(ǫ,p)(q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=O(ǫm+1). (3.33)\nModerate and the negligible functions are defined in the foll owing way:\nR∈ E(M)is moderate if ∀K⊂⊂M∀k∈N0∃N∈N∀X1,....Xk(X1,....Xkare\nsmooth vector fields) and ∀Φ∈A0(M)one has:\nsup\np∈K∝bardblLX1.....LXk(R(Φ(ǫ,p),p))∝bardbl=O(ǫ−N). (3.34)\nR∈ E(M)is negligible if ∀K⊂⊂M,∀k,l∈N0∃m∈N∀X1,...Xk(X1,...Xkare\nagain smooth vector fields) and ∀Φ∈Am(M)one has:\nsup\np∈K∝bardblLX1...LXk(R(Φ(ǫ,p),p)∝bardbl=O(ǫl). (3.35)\nNow we can define the Colombeau algebra in the usual way as a fac tor algebra\nof moderate functions over negligible functions. Scalar di stributions, defined as\ndual ton-forms, can be embedded into such algebra in a complete analo gy to the\ncanonical Rnformulation. Also association is in this case defined in the “ usual”\nway (integral with compactly supported smooth n-form field) and has for mul-\ntiplication the usual properties. However any attempt to ge t a straightforward\ngeneralization from scalars to tensors brings immediate pr oblems, since the em-\nbedding does not commute with the action of diffeomorphisms . This problem\nwas finally resolved in [69].CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 118\nTensor fullColombeau algebra and the embedding of distributions\nThe authors of reference [69] realized that diffeomorphism invariance can be achiev-\ned by adding some background structure defining how tensors t ransport from\npoint to point, hence a transport operator. Colombeau (m,n)rank tensors are\nthen taken from the class of smooth maps C∞(ω,q,B)having values in (Tm\nn)qM,\nwhereω∈A0(M),q∈MandBis from the class of compactly supported trans-\nport operators. After defining how Lie derivative acts on suc h objects and the\nconcept of the “core” of a transport operator, the authors of reference [69] define\n(in a slightly complicated analogy to the previous case) the moderate and the neg-\nligible tensor fields. Then by usual factorization they obta in the canonical version\nof the generalized tensor fields (for more details see [69]). The canonical embed-\nding of tensorial distributions is the following: The smoot h tensorial objects are\nembedded as\n˜t(p,ω,B) =/integraldisplay\nt(q)B(p,q)ω(q)dq (3.36)\nwhere as expected ω∈A0(M),tis the smooth tensor field and Bis the transport\noperator. Then the arbitrary tensorial distribution sis embedded (to ˜s) by the\ncondition\n˜s(ω,p,B)·t(p) = (s,B(p,.)·t(p)⊗ω(.)), (3.37)\nwhere on the left side we are contracting the embedded object with a smooth\ntensor field t, and on the right side we are applying the given tensorial dis tribution\nsin the variable assigned by the dot. It is shown that this embe dding fulfills all\nthe important properties, such as commuting with the Lie der ivative operator [69].\nAll the other results related to equivalence relation (etc. ) are obtained in complete\nanalogy to the previous cases.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 119\nThe generalized geometry (in special Colombeau algebras)\nIn [101, 102, 103] the authors generalized all the basic geom etric structures, like\nconnection, covariant derivative, curvature, or geodesic s into the geometric for-\nmulation of the special Colombeau algebra. That means they d efined the whole\ngeneralized geometry.\nWhy is this somewhat unsatisfying?\nHowever in our view, the crucial part is missing. What we woul d like to see is an\nintuitive and clear definition of the covariant derivative o perator acting on the dis-\ntributional objects in the canonical Colombeau algebra for mulation, on one hand\nreproducing all the classical results, and on the other hand extending them in the\nsame natural way as in the classical distributional theory w ith the classical deriva-\ntive operator. Whether there is any way to achieve this goal b y the concept of gen-\neralized covariant derivative acting on the generalized te nsor fields, as defined in\n[103], is unclear. Particularly it is not clear whether such generalized geometry\ncan be formulated also within the canonical Colombeau algeb ra approach. There\nexist definitions of the covariant derivative operator with in the distributional ten-\nsorial framework [75, 111]. (The reference [75] gives parti cularly nice application\nof such distributional tensor theory to signature changing spacetimes.) But these\napproaches are still “classical”, in the sense, that they do not fully involve the op-\neration of the tensor product of distributional tensors. Th ere cannot be any hope\nof finding a more appropriate, generalized formulation of cl assical physics with-\nout finding such a clear and intuitive definition of both, the c ovariant derivative\nand the tensor product. All that can be in this situation achi eved is to use these\nconstructions to solve some specific problems within the are a of physics. But as\nwe see, the more ambitious goal can be very naturally achieve d by our own con-\nstruction, which follows after this overview.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 120\n3.2.5 Practical application of the standard results\nNow we will briefly review various applications of the Coulom beau theory pre-\nsented before.\nClassical shock waves\nThe first application we will mention is the non-general-rel ativistic one. In [48] the\nauthors provide us with weak solutions of nonlinear partial differential equations\n(using Colombeau algebra) representing shock waves. They u se a special version\nof the Colombeau algebra, and specifically the relation\nHnH′≈1\nn+1H′, (3.38)\n(which is related to mollifiers from A0(Rn)class). The more general analysis re-\nlated to the existence and uniqueness of weak solutions of no nlinear partial dif-\nferential equations can be found in [124].\nBlack hole “distributional” spacetimes\nIn general relativity there are many results obtained by the use of Colombeau al-\ngebras. First, we will focus on the distributional Schwarzs child geometry, which\nis analysed for example in [76]. The authors of [76] start to w ork in Schwarzschild\ncoordinates using the special Colombeau algebra and A∞(Rn)classes of mollifiers.\nThey obtain the delta-functional results (as expected) for the Einstein tensor, and\nhence also for the stress-energy tensor. But in Schwarzschi ld coordinates there\nare serious problems with the embedding of the distribution al tensors, since these\ncoordinates do not contain the 0 point. As a result, if one loo ks for smooth em-\nbeddings, one does not obtain an inverse element in the Colom beau algebra in the\nneighbourhood of 0 for values of ǫclose to 0. (Although there is no problem, if we\nrequire that the inverse relation should apply only in the se nse of equivalence.)\nProgress can be made by turning to Eddington-Finkelstein co ordinates [76]. The\nmetric is obtained in Kerr-Schild form, in which one is able t o compute Rab,GabCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 121\nand hence Tabas delta-functional objects (which is expected). (The (1,1 ) form of\nthe field equations is used since the metric dependence has a r elatively simple\nform in Eddington-Finkelstein coordinates.) This result d oes not depend on the\nmollifier (see also [173]), but one misses the analysis of the relation between dif-\nferent embeddings given by the different coordinate system s8. Even in the case\nof Kerr geometry there is a computation of/radicalbig\n(g′)Rµν(wheregab=g′ab+fkakb)\ngiven by Balasin, but this is mollifier dependent [11, 173]. H ere the coordinate\ndependence of the results is even more unclear.\nAichelburg metric\nThere exists an ultrarelativistic weak limit of the Schwarz schild metric. It is taken\nin Eddington-Finkelstein coordinates u=t+r∗,v=t−r∗, (wherer∗is tortoise\ncoordinate) by taking the boost in the weak v→climit. We obtain the “delta func-\ntional” Aichelburg metric. Reference [104] provides a comp utation of geodesics\nin such a geometry. The authors of [104] take the special Colo mbeau algebra (and\ntakeA0(Rn)as their class of mollifiers), and they prove that geodesics a re given\nby the refracted lines. The results are mollifier independen t. This is again ex-\npected. Moreover, what seems to be really interesting is tha t there is a continuous\nmetric which is connected with the Aichelburg metric by a gen eralized coordinate\ntransformation [100, 173].\nConical spacetimes\nThe other case we want to mention are conical spacetimes. One of the papers\nwhere the conical spacetimes are analysed is an old paper of G eroch and Traschen\n[61]. In [61] it is shown that conical spacetimes can not be an alysed through the\nconcept of gt-metric. These are metrics which provide us with a distribut ional\n8It seems that the authors use relations between Rµνand components of gµνobtained in\nEddington-Finkelstein coordinates by using algebraic ten sor computations. Then it is not obvi-\nous, whether these results can be obtained by computation in the Colombeau algebra using the ≈\nrelation, since in such case some simple tricks (such as subs titution) cannot in general be used.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 122\nRicci tensor in a very naive sense. The multiplication is giv en just by a simple\nproduct of functions defining the regular distributions. A c alculation of the stress\nenergy tensor was given by Clarke, Vickers and Wilson [44], b ut this is mollifier\ndependent (although it is coordinate independent [173]).\n3.3 How does our approach relate to current theory?\nGeneral summary How does our own approach (to be described in detail in\nthe next section) relate to all what has presently been achie ved in the Colombeau\ntheory? We can summarize what we will do in three following po ints:\n•We will define tensorial distributional objects, and the bas ic related opera-\ntions (especially the covariant derivative ). The definition directly follows our\nphysical intuition (there is a unique way of constructing it ). This generalizes\nthe Schwartz Rndistribution theory.\n•We will formulate the Colombeau equivalence relation in our approach and\nobtain all the usual equivalence results from the Colombeau theory.\n•We will prove that the classical results for the covariant de rivative opera-\ntor (as known within differential geometry) significantly g eneralize in our\napproach.\nThe most important point is that our approach is fully based o n the Colombeau\nequivalence relation translated to our language. That mean s we take and use\nonly this particular feature of the Colombeau theory and com pletely avoid the\nColombeau algebra construction.\nThe advantages compared to usual Colombeau theory What are the advan-\ntages of this approach comparing to Colombeau theory?\n•First, by avoiding the algebra factorization (which is how C olombeau al-\ngebra is constructed) we fulfill the physical intuitiveness condition of the\nlanguage used.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 123\n•Second, we can naturally and easily generalize the concept o f the covari-\nant derivative in our formalism, which has not been complete ly satisfacto-\nrily achieved by the Colombeau algebra approach9. This must be taken as\nan absolutely necessary condition that any generalization of a fundamental\nphysical language must fulfill.\n•It is specifically worth discussing the third advantage: Why is the classi-\ncal approach so focused on Colombeau algebras? The answer is simple:\nWe want to get an algebra of C∞(M)functions as a subalgebra of our al-\ngebra. (This is why we need to factorize by negligible functi ons, and we\nneed to get the largest space where they form an ideal, which i s the space\nof moderate functions.) But there is one strange thing: all o ur efforts are\naimed at reaching the goal of getting a more rich space than is provided by\nthe space of smooth functions. But there is no way of getting a larger dif-\nferential subalgebra than the algebra of smooth functions, as is shown by\nSchwartz impossibility result. That is why we use only the eq uivalence re-\nlation instead of straight equality. But then the question r emains: Why one\nshould still prefer smooth functions? Is not the key part of a ll the theory\nthe equivalence relation? So why is it that we are not satisfie d with the way\nthe equivalence relation recovers multiplication of the sm ooth objects (we\nrequire something stronger), but we are satisfied with the wa y it recovers\nmultiplication within the larger class? Unlike the Colombe au algebra based\napproach, we are simply taking seriously the idea that one sh ould treat all\nthe objects in an equal way. This means we do not see any reason to try\nto achieve “something more” with smooth objects than we do wi th objects\noutside this class. And the fact that we treat all the objects in the same way\nprovides the third advantage of our theory; it makes the theo ry much more\nnatural than the Colombeau algebra approach.\n•The fourth advantage is that it naturally works with the much more general\n9As previously mentioned, the current literature lists some ambiguous attempts to incorporate\nthe covariant derivative, but no satisfactory theory.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 124\n(and for the language of distributions natural) concept of p iecewise smooth\nmanifolds, so the generalization of the physics language in to such concep-\ntual framework will give it much higher symmetry.\n•The fifth and last, “small” advantage comes from the fact that by avoiding\nthe Colombeau algebra construction we automatically remov e the problem\nof how to canonically embed arbitrary tensorial distributi ons into the alge-\nbra. But one has to acknowledge that this problem was already solved also\nwithin the Colombeau theory approach [69].\nThe disadvantages compared to usual Colombeau theory What are the disad-\nvantages of this approach compared to Colombeau theory? A co nservative person\nmight be not satisfied with the fact that we do not have a smooth tensor algebra as\na subalgebra of our algebra. This means that the classical sm ooth tensorial fields\nhave to be considered to be solutions of equivalence relatio ns only (as opposed\nto the classical, “stronger” view to take them as solutions o f the equations). But\nin my view, this is not a problem at all. The equivalence relat ions contain all the\nclassical smooth tensorial solutions (equivalently smoot h tensor fields), for the\nsmooth initial value problem. So for the smooth initial valu e problem the equiv-\nalence relations reduce to classical equations. We will sug gest how to extend the\ninitial value problem for larger classes of distributions a nd show that it has unique\nsolutions. It is true that the equivalence relations might h ave also many other\n(generally non-linear) solutions in the space constructed ,10but the situation that\nthere exist many physically meaningless solutions is for ph ysicists certainly noth-\ning new. Our previous considerations suggest that we shall l ook only for distribu-\ntional solutions (they are the ones having physical relevan ce) where the solution\nis provided to be unique (up to initial values obviously), if it exists. (This will also\nrecover the common, but also many “less” common physical res ults [173].)\n10By “solution” we mean here any object fulfilling the particul ar equivalence relations.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 125\n3.4 New approach\n3.4.1 The basic concepts/definitions\nBefore saying anything about distributional tensor fields, we have to define the\nbasic concepts which will be used in all the following mathem atical constructions.\nThis task is dealt with in this section, so this is the part cru cially important for un-\nderstanding all the subsequent theory. An attentive reader , having read through\nthe introduction and abstract will understand why we are par ticularly interested\nin defining these concepts.\nDefinition of (M, A)\nDefinition 3.4.1. By piecewise smooth function we mean a function from an open\nsetΩ1(⊆R4)→Rmsuch, that there exists an open set Ω2(in the usual “open ball”\ntopology on R4) on which this function is smooth and Ω2= Ω1\\Ω′(whereΩ′has\na Lebesgue measure 0).\nTakeMas a 4D paracompact11, Hausdorff locally Euclidean space, on which\nthere exists a smooth atlas S. Hence (M,S)is a smooth manifold. Now take\na ordered couple (M,A), whereAis the maximal atlas, where all the maps are\nconnected by piecewise smooth transformations such that:\n•the transformations and their inverses have on every compac t subset of R4\nall the first derivatives (on the domains where they exist) bo unded\n(hence Jacobians, inverse Jacobians are on every compact se t bounded),\n•it contains at least one maximal smooth subatlas S ⊆ A ,\n(coordinate transformations between maps are smooth there ).\n11We will use specifically 4-dimensional manifolds, but one ca n immediately generalize all the\nfollowing constructions for n-dimensional manifolds.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 126\nNotation. The following notation will be used:\n•By the letter Swe will always mean some maximal smooth subatlas of A.\n•Every subset of Mon which there exists a chart from our atlas A, we call\nΩCh. An arbitrary chart on ΩChfrom our atlas Ais denoted Ch(ΩCh).\nNotation. Take some set ΩCh. Take some open subset of that set Ω′⊂ΩCh. Then\nCh(ΩCh)|Ω′is defined simply as Ch(Ω′), which is obtained from Ch(ΩCh)by limit-\ning the domain to Ω′.\nDefinition of “continuous to the maximal possible degree”\nDefinition 3.4.2. We call a function on Mcontinuous to the maximal possible\ndegree, if on arbitrary Ωof Lebesgue measure 0 it is only in such cases:12\na) either undefined,\nb) or defined and discontinuous,\nin which there does not exist a way of turning it into a functio n continuous on Ω\nby\n•in the case of a)extending its domain by the Ωset,\n•in the case of b)re-defining it on Ω.\nJacobians and algebraic operations with Jacobians\nNow it is obvious that since transformations between maps do not have to be\neverywhere once differentiable, the Jacobian and inverse J acobian may always be\n12The expression “Lebesgue measure 0 set” will have in this cha pter extended meaning. It refers\nto such subsets of a general manifold Mthat they have in arbitrary chart Lebesgue measure 0.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 127\nundefined on a set having Lebesgue measure 0. Now if we underst and product in\nthe sense of a limit, then the relation\nJµ\nα(J−1)α\nν=δµ\nν, (3.39)\nfor example, might hold even at the points where both Jacobia n and inverse Jaco-\nbian are undefined. This generally means the following: any a lgebraic operation\nwith tensor fields is understood in such way, that in every cha rt it gives sets of\nfunctions continuous to a maximal possible degree. From thi s follows that the\nmatrix product (3.39) must be, for µ=ν, equal to 1 and, for µ∝\\e}atio\\slash=ν, 0.\nTensor fields on M\nWe understand the tensor field on Mto be an object which is:\n•Defined relative to the 1-differentiable subatlas of Aeverywhere except for\na set having Lebesgue measure 0 (this set is a function of the g iven 1-dif-\nferentiable subatlas).\n•In every chart from Ait is given by functions continuous to a maximal pos-\nsible degree.\n•It transforms ∀ΩChbetween charts Ch1(ΩCh), Ch2(ΩCh)∈ A in the tensorial\nway\nTµ...\nν... Ch 2=Jµ\nα(J−1)β\nν···Tα...\nβ... Ch 1a.e.,13(3.40)\nwhereJµ\nαis the Jacobian of the coordinate transformation from Ch1(ΩCh)\ntoCh2(ΩCh), andTµ...\nν... Ch 1,Tµ...\nν... Ch 2are tensor field components in charts Ch1,\nCh2. As we already mentioned: If Tµ...\nν... Ch 1is at some given point undefined,\nso in some chart Ch1(ΩCh)the tensor components do not have a defined\n13This expression means “almost everywhere”, that is, “every where apart from a set having\nLebesgue measure 0”.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 128\nlimit, this limit can still exist in Ch2(ΩCh), since Jacobians and inverse Jaco-\nbians of the transformation from Ch1(ΩCh)toCh2(ΩCh)might be undefined\nat that point as well. This limit then defines Tµ...\nν... Ch 2at the given point.\nImportant classes of test objects\nNotation. The following notation will be used:\n•We denote by CP(M)the class of 4-form fields on M, such that they are\ncompactly supported and their support lies within some ΩCh. For such 4-\nform fields we will generally use the symbol ω.\n•For the scalar density related to ω∈CP(M)we use always the symbol ω′.\n•ByCP(ΩCh)we mean a subclass of CP(M), given by 4-form fields hav-\ning support inside ΩCh. Note that only the CP(ΩCh)subclasses form linear\nspaces.\n•Take such maximal atlas ˜S, (∃S ⊂˜S ⊂ A ), that there exist 4-forms from\nCP(M), such that they are given in this atlas by everywhere smooth s calar\ndensityω′. (“Maximal” here means that these 4-forms have in every char t\noutside this atlas non-smooth scalar densities.) By CP\nS(˜S)(M)we mean a class\nof all such elements from CP(M), that they have everywhere smooth scalar\ndensity in ˜S.\n•The letter ˜Swill from now on be reserved for maximal atlases defining\nCP\nS(˜S)(M)classes.\n•CP\nS(M)is defined as: CP\nS(M)≡ ∪˜SCP\nS(˜S)(M).\n•CP\nS(˜S)(ΩCh)(orCP\nS(ΩCh)) meansCP\nS(˜S)(M)(orCP\nS(M)) element having sup-\nport inside the given ΩCh.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 129\nTopology on CP\nS(˜S)(ΩCh)\nConsider the following topology on each CP\nS(˜S)(ΩCh): A sequence from ωn∈\nCP\nS(˜S)(ΩCh)converges to an element ωfrom that set if all the supports of ωnlie in\na single compact set, and in any chart Ch(ΩCh)∈˜S, for arbitrary kit is true that\n∂kω′\nn(xi)\n∂xl1..∂xlkconverges uniformly to∂kω′(xi)\n∂xl1..∂xlk.\n3.4.2 Scalars\nThis section deals with the definition of scalar distributio ns as the easiest par-\nticular example of a generalized tensor field. The explanato ry reasons are the\nmain ones why we deal with scalars separately, instead of tak ing more “logical”,\nstraightforward way to tensor fields of arbitrary rank.\nDefinition of D′(M), and hence of linear generalized scalar fields\nDefinition 3.4.1. We say that B, being a function that maps some subclass of\nCP(M)toRn, is linear, if the following holds: Take such ω1andω2from the do-\nmain ofB, that they belong to the same class CP(ΩCh). Whenever the domain\nofBcontains also their linear combination λ1ω1+λ2ω2, whereλ1,λ2∈R, then\nB(λ1ω1+λ2ω2) =λ1B(ω1)+λ2B(ω2).\nDefinition 3.4.2. Now take the space of linear maps F→R, whereFis such set\nthat∃˜S, such that CP\nS(˜S)(M)⊆F⊆CP(M). These linear maps are also required\nto be for every ΩChonCP\nS(˜S)(ΩCh)continuous, (relative to the topology taken in\nsection 3.4.1). Call set of such maps D′(M), or in words, the set of linear generalized\nscalar fields .\nImportant subclasses of D′(M)\nNotation. The following notation will be used:\n•Now take a subset of D′(M)given by regular distributions defined as in-\ntegrals of piecewise continuous functions (everywhere on CP(M), where itCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 130\nconverges). We denote it by D′\nE(M).14\n•Take such subset of D′\nE(M)that there ∃Sin which the function under the\nintegral is smooth. Call this class D′\nS(M).\n•Now take subsets of D′(M)such that they have some common set\n∪nCP\nS(˜Sn)(M)belonging to their domains and are ∀ΩCh,∀ncontinuous on\nCP\nS(˜Sn)(ΩCh). Denote such subsets by D′\n(∪n˜Sn)(M). Obviously T∈D′\n(∪n˜Sn)(M)\nmeansT∈ ∩nD′\n(˜Sn)(M).\n•ByD′\n(∪n˜Sno)(M)we mean objects such that they belong to D′\n(∪n˜Sn)(M)and\ntheir full domain is given as ∪nCP\nS(˜Sn)(M).\n•If we use the notation D′\nE(∪n˜Sno)(M), we mean objects defined by integrals of\npiecewise continuous functions, with their domain being th e class∪nCP\nS(˜Sn)(M).\n•By usingD′\nS(∪n˜Sno)(M)we automatically mean subclass of D′\nE(∪n˜Sno)(M),\nsuch that it is given by an integral of a smooth function in som e smooth\nsubatlas S ⊆ ∪ n˜Sn.\nD′\nA(M), hence generalized scalar fields\nNotation. Let us for any arbitrary set D′\n(˜S)(M)15construct, by the use of point-\nwise multiplication of its elements, an algebra. Another wa y to describe the alge-\nbra is that it is a set of multivariable arbitrary degree poly nomials, where different\nvariables represent different elements of D′\n(˜S)(M). Call itD′\n(˜S)A(M).\nBy pointwise multiplication of linear generalized scalar fi eldsB1, B2∈\nD′\n(˜S)(M)we mean a mapping from ωinto product of the images (real numbers)\nof theB1,B2mappings: (B1·B2)(ω)≡B1(ω)·B2(ω). The domain on which\nthe product (and the linear combination as well) is defined is an intersection of\n14Actually, it holds that if and only if the function is integra ble in every chart on every compact\nset inRn, then this function defines a regular D′(M)distribution and is defined at least on the\nwholeCP\nS(M)class.\n15The union is here trivial, it just means one element ˜S.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 131\ndomains of B1andB2(trivially always nonempty, containing CP\nS(˜S)(M)at least).\nNote also that the resulting arbitrary element of D′\n(˜S)A(M)has in general all the\nproperties defining D′(M)objects, except that of being necessarily linear.\nDefinition 3.4.3. The set of objects obtained by the union ∪˜SD′\n(˜S)A(M)we denote\nD′\nA(M), and call them generalized scalar fields (GSF).\nTopology on D′\n(˜So)(M)\nIf we take objects from D′\n(˜So)(M)and we take a weak ( σ−) topology on that set,\nwe know that any object is a limit of some sequence from D′\nS(˜So)(M)objects from\nthat set (they form a dense subset of that set). That is known f rom the classical\ntheory. Such a space is complete.\n3.4.3 Generalized tensor fields\nThis section is of crucial importance. It provides us with definitions of all the basic\nobjects we are interested in, the generalized tensor fields a nd all their subclasses\nof special importance as well.\nThe classD′m\nn(M)oflinear generalized tensor fields\nFirst let us clearly state how to interpret the Jµ\nνJacobian in all the following def-\ninitions. It is a matrix of piecewise smooth functions Ω1\\Ω2→R,Ω1being a\nopen subset of R4andΩ2having Lebesgue measure 0. Let it represent transfor-\nmations from Ch1(ΩCh)toCh2(ΩCh). We can map the Jacobian by the inverse of\ntheCh1(ΩCh)coordinate mapping to ΩChand it will become a matrix of functions\nΩCh\\Ω′→R,Ω′having Lebesgue measure 0. The object Jµ\nν·ωis then under-\nstood as a matrix of 4-forms from CP(ΩCh), which also means that outside ΩChwe\ntrivially define them to be 0.\nDefinition 3.4.1. Take some set F,F1⊆F⊆F2, whereF1us such set that ∃˜SandCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 132\n∃S ⊆˜S(Sis some maximal smooth atlas) defining it as:\nF1≡ ∪ΩCh/braceleftBig/parenleftbig\nω,Ch(ΩCh)/parenrightbig\n:ω∈CP\nS(˜S)(ΩCh), Ch(ΩCh)∈ S/bracerightBig\n.\nF2is defined as:\nF2≡ ∪ΩCh/braceleftbig/parenleftbig\nω,Ch(ΩCh)/parenrightbig\n:ω∈CP(ΩCh), Ch(ΩCh)∈ A)/bracerightbig\n.\nBy aD′m\nn(M)object, the linear generalized tensor field , we mean a linear mapping\nfromF→R4m+nfor which the following holds:16\n• ∀ΩChit is∀Chk(ΩCh)∈ S ⊂˜Scontinuous on the class CP\nS(˜S)(ΩCh). (BothS,\n˜Sare from the definition of F1.)\n•This map also ∀ΩChtransforms between two charts from its domain,\nCh1(ΩCh),Ch2(ΩCh), as:\nTµ...γ\nν...δ(Ch1,ω) =Tα...λ\nβ...ρ/parenleftbig\nCh2,Jµ\nα...Jγ\nλ(J−1)β\nν....(J−1)ρ\nδ·ω/parenrightbig\n.\n•The following consistency condition holds: If Ω′\nCh⊂ΩCh, thenTµ...α\nν...δ gives\nonω×Ch(ΩCh)|Ω′\nCh,ω∈CP(Ω′\nCh)the same results17as onω×Ch(ΩCh).\nWe can formally extend this notation also for the case m=n= 0. This means\nscalars, exactly as defined before. So from now on m,ntake also the value 0,\nwhich means the theory in the following sections holds also f or the scalar objects.\n16We could also choose for our basic objects maps taking ordere d couples from CP(ΩCh)×\nCh(Ω′\nCh), (ΩCh∝\\e}atio\\slash= Ω′\nCh). The linearity condition then automatically determines t heir values, since\nforω∈CP(ΩCh), whenever it holds that Ω′\nCh∩supp(ω) ={0}, they must automatically give\n0 for any chart argument. Hence these two definitions are triv ially connected and choice between\nthem is just purely formal (only a matter of “taste”).\n17By the “same results” we mean that they are defined on the same d omains, and by the same\nvalues.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 133\nImportant subclasses of D′m\nn(M)\nNotation. The following notation will be used:\n•By a complete analogy to scalars we define classes D′m\nnE(M): On arbitrary\nΩCh, being fixed in arbitrary chart Ch1(ΩCh)∈ A we can express it in an-\nother arbitrary chart Ch2(ΩCh)∈ A, as an integral from a multi-index matrix\nof piecewise continuous functions on such subset of CP(M), on which the\nintegral is convergent18.\n•Analogously the class D′m\nnS(M)⊂D′m\nnE(M)is defined by objects which can,\nfor some maximal smooth atlas S, in arbitrary charts19Ch1(ΩCh),Ch2(ΩCh)∈\nS, be expressed by an integral from a multi-index matrix of smo oth func-\ntions.\n•D′m\nn(∪lAt(˜Sl))(M)means a class of objects being for every ΩChin everyCh(ΩCh)∈\nAt(˜Sl)continuous on CP\nS(˜Sl)(ΩCh). (HereAt(˜S)stands for a map from atlases\n˜Sto some subatlases of A.)\n•D′m\nn(∪lAt(˜Sl)o)(M)means a class of objects from D′m\nn(∪lAt(˜Sl))(M)having as their\ndomain the union\n∪ΩCh∪l/braceleftBig\n(ω,Ch(ΩCh)) :ω∈CP\nS(˜Sl)(ΩCh), Ch(ΩCh)∈At(˜Sl)/bracerightBig\n.\n•If we have classes D′m\nn(∪lAt(˜Sl))(M)andD′m\nn(∪lAt(˜Sl)o)(M)whereAt(˜Sl) =Sl⊂\n˜Sl, we use the simple notation D′m\nn(∪lSl)(M),D′m\nn(∪lSlo)(M).20\n18Actually we will use the expression “multi-index matrix” al so later in the text and it just means\nspecifically ordered set of functions.\n19The first chart is an argument of this generalized tensor field and the second chart is the one\nin which we express the given integral.\n20We have to realize that the subatlas Snspecifies completely the atlas ˜Sn, since taking forms\nsmooth in Sndetermines automatically the whole set of charts in which th ey are still smooth. This\nfact contributes to the simplicity of this notation.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 134\nDefinition of D′m\nnA(M), hence generalized tensor fields\nDefinition 3.4.2. Now define D′m\nn(S)A(M)to be the algebra constructed from the ob-\njectsD′m\nn(S)(M)by the tensor product, exactly in analogy to the case of scala rs (this\nreduces for scalars to the product already defined). The obje ct, being a result of\nthe tensor product, is again a mapping V→R4m+n, defined in every chart by com-\nponentwise multiplication. Now denote by D′m\nnA(M)a set given as ∪SD′m\nn(S)A(M),\nmeaning a union of all possible D′m\nn(S)A(M). Call the objects belonging to this set\nthe generalized tensor fields (GTF) .\nNotation. Furthermore let us use the same procedure as in the previous d efini-\ntion, just instead of constructing the algebras from the cla ssesD′m\nn(S)(M), we now\nconstruct them only from the classes D′m\nn(∪lAt(˜Sl))(M)∩D′m\nn(S)(M), (again by tensor\nproduct). For the union of such algebras we use the notation D′m\nn(∪lAt(˜Sl))A(M).\nDefinition of Γ−objects, their classes and algebras\nNotation. Now let us define the generalized space of objects Γl(M). (For ex-\nample the Christoffel symbol would fall into this class.) Th ese objects are defined\nexactly in the same way as D′m\nn(M)(m+n=l) objects, we just do not require that\nthey transform between charts in the tensorial way, (second point in the definition\nof generalized tensor fields).\nNote the following:\n•The definition of Γl(M)includes also the case m= 0. Now we see, that the\nscalars can be taken as subclass of Γ0(M), given by objects that are constants\nwith respect to the chart argument.\n•Note also that for a general Γm(M)object there is no meaningful differenti-\nation between “upper” and “lower” indices, but we will still use formally\ntheTµ...\nν...notation (for all cases).CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 135\nNotation. In the same way, (by just not putting requirements on the tran sforma-\ntion properties), we can generalize the classes\n•D′m\nn(∪lAt(˜Sl))(M)toΓm+n\n(∪lAt(˜Sl))(M),\n•D′m\nn(∪lAt(˜Sl)o)(M)toΓm+n\n(∪lAt(˜Sl)o)(M),\n•D′m\nn(∪lSl)(M)toΓm+n\n(∪lSl)(M),\n•D′m\nn(∪lSlo)(M)toΓm+n\n(∪lSlo)(M),\n•D′m\nnE(M)toΓm+n\nE(M),\n•D′m\nn(∪lAt(˜Sl))A(M)toΓm+n\n(∪lAt(˜Sl))A(M), and\n•D′m\nnA(M)toΓm+n\nA(M).\nIt is obvious that all the latter classes contain all the form er classes as their\nsubclasses, (this is a result of what we called a “generaliza tion”).\nNote that when we fix Γm\nE(M)objects in arbitrary chart from A, they must be\nexpressed by integrals from multi-index matrix of function s integrable on every\ncompact set. In the case of the D′m\nnE(M)subclass it can be required in only one\nchart, since the transformation properties together with b oundedness of Jacobians\nand inverse Jacobians, provide that it must hold in any other chart from A. The\nspecific subclass of Γm\nE(M)isΓm\nS(M), which is a subclass of distributions given\nin any chart from A, (being an argument of the given Γ−object), by integrals\nfrom multi-index matrix of smooth functions (when we expres s the integrals in\nthe same chart, as the one taken as the argument). Γm\nS(∪nSno)(M)stands again for\nΓm\nS(M)objects with domain limited to\n∪ΩCh∪n/braceleftBig\n(ω,Ch(ΩCh)) :ω∈CP\nS(˜Sn)(ΩCh), Ch(ΩCh)∈ Sn/bracerightBig\n,\nwhere˜Snis given by the condition Sn⊂˜Sn.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 136\nNotation. Take some arbitrary elements Tµ...\nν...∈Γm\nE(M),ω∈CP(ΩCh), and\nChk(ΩCh)∈ A. TheTµ...\nν...(Chk,ω)can be always expressed as/integraltext\nΩChTµ...\nν...(Chk)·ω.\nHereTµ...\nν...(Chk)appearing under the integral denotes some multi-index matr ix of\nfunctions continuous to a maximal possible degree on ΩCh. ForTµ...\nν...∈D′m\nnE(M)\ntheTµ...\nν...(Chk)multi-index matrix components can be obtained from a tensor field\nby:\n•expressing the tensor field components in Chk(ΩCh)on some subset of R4,\n•mapping the tensor field components to ΩChby the inverse of Chk(ΩCh).\nFurthermore ω′(Chk)will denote the 4-form scalar density in the chart Chk(ΩCh).\nTopology on Γm\n(∪nSno)(M)\nIf we take the class of Γm\n(∪nSno)(M), and we impose on this class the weak (point or\nσ−) topology, then the subclass of Γm\n(∪nSno)(M)defined as Γm\nS(∪nSno)(M)is dense\ninΓm\n(∪nSno)(M). The same holds for D′m\nn(∪lSlo)(M)andD′m\nnS(∪lSlo)(M).\nDefinition of contraction\nDefinition 3.4.3. We define the contraction of a Γm\nA(M)object in the expected\nway: It is a map that transforms the object T...µ...\n...ν...∈Γm\nA(M)to the object T...µ...\n...µ...∈\nΓm−2\nA(M).\nNow contraction is a mapping D′m\nn(M)→D′m−1\nn−1(M)andΓm(M)→Γm−2(M),\nbut it is not in general the mapping D′m\nnA(M)→D′m−1\nn−1A(M), onlyD′m\nnA(M)→\nΓm+n−2\nA(M).\nInterpretation of physical quantities\nThe interpretation of physical observables as “amounts” of quantities on the open\nsets is dependent on our notion of volume. So how shall we get t he notion of\nvolume in the context of our language? First, by volume we mea n a volume ofCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 137\nan open set. But we will consider only open sets belonging to s omeΩCh. So take\nsomeΩChand some arbitrary Ω′⊂ΩCh. Let us now assume that we have a\nmetric tensor from D′m\nnE(M). This induces a (volume) 4-form. Multiply this 4-\nform by a noncontinuous function χΩ′defined to be 1 inside Ω′and everywhere\nelse 0. Call it ωΩ′. Then by volume of an open set Ω′we understand:/integraltext\nωΩ′. Also\nωΩ′is object from CP(M)(particularly from CP(ΩCh)). The “amounts” of physical\nquantities on Ω′we obtain, when the D′m\nnA(M)objects act on ωΩ′.\n3.4.4 The relation of equivalence ( ≈)\nThis section now provides us with the fundamental concept of the theory, the con-\ncept of equivalence of generalized tensor fields. Most of the first part is devoted\nto fundamental definitions, the beginning of the second part deals with the basic,\nimportant theorems, which just generalize some of the basic Colombeau theory\nresults to the tensor product of generalized tensor fields. I t adds several impor-\ntant conjectures as well. The first part ends with the subsect ion “some additional\ndefinitions” and the second part with the subsection “some ad ditional theory”.\nThey both deal with much less central theoretical results, b ut they serve very well\nto put light on what equivalence of generalized tensor fields means “physically”.\nThe necessary concepts to define the equivalence relation\nNotation. Take some subatlas of our atlas, this will be a maximal subatl as of\ncharts, which are maps to the whole of R4. Such maps exist on each set ΩChand\nthey will be denoted as Ch′(ΩCh). We say that a chart Ch′(ΩCh)is centered at the\npointq∈ΩCh, if this point is mapped by this chart to 0 (in R4). We will use the\nnotationCh′(q,ΩCh).\nNotation. Take some ΩCh,q∈ΩChandCh′(q,ΩCh)∈˜S. The set of 4-forms\nωǫ∈An(˜S,Ch′(q,ΩCh))is defined in such way that ωǫ∈CP\nS(˜S)(ΩCh)belongs to\nthis class if:CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 138\na) in the given Ch′(q,Ω′\nCh),∀ǫit holds that:\n/integraltext\nCh′(Ω′\nCh)(/producttext\nixki\ni)ω′\nǫ(x)d4x=δk0,/summationtext\niki=k,k≤n,n∈N,\nb) the dependence on ǫis inCh′(q,Ω′\nCh)given asǫ−4ω′(x\nǫ).\nNotation. Take an arbitrary q,ΩCh(q∈ΩCh),Ch′(q,ΩCh)∈˜Sand some nat-\nural number n. For anyωǫ∈An(Ch′(q,ΩCh),˜S)we can, relatively to Ch′(ΩCh),\ndefine a continuous set of maps (depending on the parameter y)\nAn(Ch′(q,ΩCh),˜S)→CP\nS(˜S)(ΩCh), (3.41)\nsuch that they are, on ΩChand inCh′(ΩCh), given as ω′(x\nǫ)ǫ−4→ω′(y−x\nǫ)ǫ−4.\n(To remind the reader ω′is the density expressing ωin this chart.) This gives us\n(depending on the parameter y∈R4) variousCP\nS(˜S)(ΩCh)objects, such that they\nare in the fixed Ch′(ΩCh)expressed by ω′(x−y\nǫ)ǫ−4dx1/logicalandtext.../logicalandtextdx4. Denote these\n4-form fields by ˜ωǫ(y).\nNotation. Now, take any Tµ...\nν...∈Γm\n(At(˜S))A(M). By applying it in an arbitrary\nfixed chart Chk(ΩCh)∈At(˜S,ΩCh)on the 4-form field ˜ωǫ(y), obtained\nfromωǫ∈An(Ch′(q,ΩCh),˜S)through the map (3.41), we get a function R4→\nR4m+n. As a consequence, the resulting function depends on the fol lowing objects:\nTµ...\nν...(∈Γm\nA(M)), ωǫ(∈An(q,ΩCh,Ch′(ΩCh))) andChk(ΩCh). We denote it by:\nF′µ...\nν.../parenleftbig\nTµ...\nν...,˜S,ΩCh,Ch′(q,ΩCh),n,˜ωǫ(y),Chk(ΩCh)/parenrightbig\n.\nDefinition of the equivalence relation\nDefinition 3.4.1. Bµ...\nν...,Tµ...\nν...∈Γm\nA(M)are called equivalent ( Bµ...\nν...≈Tµ...\nν...), if:\n•they belong to the same classes Γm\n(At(˜S))A(M),\n• ∀ΩCh,∀q(q∈ΩCh),∀Ch′(q,ΩCh)∈˜S ( such that Bµ...\nν...,Tµ...\nν...∈\nΓm\n(At(˜S))A(M)),∀Ch(ΩCh)∈At(˜S,ΩCh)∃n, such that ∀ωǫ∈CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 139\nAn(Ch′(q,ΩCh),˜S)and for any compactly supported, smooth function\nR4→R,φ, it holds:\nlim\nǫ→0/integraldisplay\nR4/braceleftbigg\nF′µ...\nν.../parenleftbig\nBµ...\nν...,q,Ω′\nCh,Ch′(Ω′\nCh),n,˜ωǫ(y),Ch(ΩCh)/parenrightbig\n−F′µ...\nν.../parenleftbig\nTµ...\nν...,q,Ω′\nCh,Ch′(Ω′\nCh),n,˜ωǫ(y),Ch(ΩCh)/parenrightbig/bracerightbigg\n·φ(y)d4y= 0.(3.42)\nNote that for Bµ...\nν...,Cµ...\nν...,Dµ...\nν...,Tµ...\nν...having the same domains and being from the\nsameΓn\n(At(˜S))(M)classes, it trivially follows that: Tµ...\nν...≈Bµ...\nν...,Cµ...\nν...≈Dµ...\nν...implies\nλ1Tµ...\nν...+λ2Cµ...\nν...≈λ1Bµ...\nν...+λ2Dµ...\nν...forλ1,λ2∈R.\nNotation. Now since we have defined an equivalence relation, it divides the ob-\njectsΓm\nA(M)naturally into equivalence classes. The set of such equival ence classes\nwill be denoted as ˜Γm\nA(M). Later we may also use sets of more limited classes of\nequivalence ˜D′m\nnA(M),˜D′m\nnEA(M)(etc.), which contains equivalence classes (only)\nof the objects belonging to D′m\nnA(M),D′m\nnEA(M)(etc.).\nNotation. In some of the following theorems, (also for example in the de finition\nof the covariant derivative), we will use some convenient no tation: Take some\nobjectBµ...\nν...∈Γm\nE(M). The expression Tµ...\nν...(Bα...\nβ...ω)will be understood in the fol-\nlowing way: Take Chk(ΩCh)×ω(ω∈CP(ΩCh)) from the domain of Tµ...\nν.... Then\nBα...\nβ...(Chk)·ωis a multi-index matrix of CP(ΩCh)objects. This means that outside\nΩChset they are defined to be trivially 0. We substitute this mult i-index matrix of\nCP(ΩCh)objects toTµ...\nν..., with the chart Chk(ΩCh)taken as the argument.\nRelation to Colombeau equivalence\nA careful reader now understands the relation between our co ncept of equiva-\nlence and the Colombeau equivalence relation. It is simple: The previous defi-\nnition just translates the Colombeau equivalence relation (see [47]) into our lan-\nguage and the equivalence classes will naturally preserve a ll the features of the\nColombeau equivalence classes (this will be proven in the fo llowing theorems).CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 140\nSome additional definitions (concepts of associated field an dΛclass)\nWe define the concept (of association) to bring some insight t o what our concepts\nmean in the most simple (but most important and useful) cases . It enables us to\nsee better the relation between the calculus we defined (conc erning equivalence)\nand the classical tensor calculus. It brings us also better u nderstanding of what\nequivalence means in terms of physics (at least in the simple cases). It just means\nthat the quantities might differ on the large scales, but tak e the same small scale\nlimit (for the small scales they approach each other).\nDefinition 3.4.2. TakeTµ...\nν...∈Γm\nA(M). Assume that:\na)∀S, such that Tµ...\nν...∈Γm\n(S)A(M),∀ΩCh and∀Chk(ΩCh)∈ S\n∃ΩCh\\Ω′(Chk), (the set Ω′(Chk)being 0 in any Lebesgue measure),\nsuch that ∀q∈ΩCh\\Ω′(Chk),∀Ch′(q,ΩCh)∈ S ⊂ ˜S ∃n,\nsuch that ∀ωǫ∈An(Ch′(q,ΩCh),˜S)\n∃lim\nǫ→0Tµ...\nν...(Chk,ωǫ). (3.43)\nb) The limit (3.43) is ∀Ch′(q,ΩCh)∈ S ⊂ ˜S,∀ωǫ∈An(Ch′(q,ΩCh),˜S)the\nsame.\nIf botha)andb)hold, then the object defined by the limit (3.43) is a mapping:\nCh(ΩCh)(∈ S))×ΩCh\\Ω′(Ch)→R4m+n. (3.44)\nWe call this map the field associated to Tµ...\nν...∈Γm\nA(M), and we use the expression\nAs(Tµ...\nν...). (It necessarily fulfills the same consistency conditions f orΩ1\nCh⊂Ω2\nChas\ntheΓm\nA(M)objects.)\nDefinition 3.4.3. Denote by Λ⊂Γm\nE(∪nAt(˜Sn)o)(M)a class of objects, such that each\nTµ...\nν...∈Λcan be ∀ΩCh,∀n,∀ω∈CP\nS(˜Sn)(ΩCh),∀S ⊂˜Sn∩At(˜Sn),\n∀Chk(ΩCh)∈ S expressed as a map\n(ω,Chk)→/integraldisplay\nΩChTµ...\nν...(Chk)·ω, (3.45)CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 141\nwhere forTµ...\nν...(Chk)holds the following: In each chart from Sfor every point z0,\nwhereTµ...\nν...(Chk)is continuous ∃δ >0,∃Kµ\nν>0, such that ∀ǫ(0≤ǫ≤δ))and\nfor arbitrary unit vector n(in the Euclidean metric on R4)\nTµ...\nν...(Chk,z0)−Kµ...\nν....ǫ≤Tµ...\nν...(Chk,z0+nǫ)≤Tµ...\nν...(Chk,z0)+Kµ...\nν...ǫ. (3.46)\nNotation. Take from (3.4.3) arbitrary, fixed Tµ...\nν...,ΩChandChk(ΩCh). By the\nnotation ˜Ω(Chk)⊂ΩChwe denote a set (having Lebesgue measure 0) on which is\nTµ...\nν...(Chk)discontinuous.\nReproduction of the basic results by the equivalence relati on\nTheorem 3.4.1. Any class ˜Γm\n(∪nAt(˜Sn)o)A(M)contains maximally one linear element.\nProof. We need to prove that there do not exist such two elements of Γm(M),\nwhich are equivalent. Take two elements BandTfrom the class Γm\n(∪nAt(˜Sn)o)A(M)\n(both with the given domains and continuity). Take arbitrar yΩCh, arbitrary ˜S\nfrom their domains, and arbitrary Ch′(ΩCh)∈˜S. Map all the CP\nS(˜S)(ΩCh)objects\nto smooth, compact supported functions on R4through this fixed chart mapping.\nNow bothBandTgive, in fixed but arbitrary Ch(ΩCh)∈At(˜S)linear, continuous\nmaps on the compactly supported smooth functions. (The only difference from\nColombeau distributions is that it is in general a map to Rm, so the difference is\nonly “cosmetic”.)\nNow after applying this construction, our concept of equiva lence reduces for\neveryCh(ΩCh)∈At(˜S)to Colombeau equivalence from [47]. The same results\nmust hold. One of the results says that there are no two distri butions being equiv-\nalent. All the parameters are fixed but arbitrary and all the 4 -forms from domains\nofBandTcan be mapped to the R4functions for some proper fixing of ΩCh\nand˜S. Furthermore, the CP\nS(M)4-forms are arguments of BandTonly in the\ncharts, in which BandTwere compared as maps on the spaces of R4functions.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 142\nSo this “arbitrary chart fixing” covers all their domain. As a resultBandTmust\nbe identical and that is what needed to be proven.\nTheorem 3.4.2. Any class of equivalence ˜Γm\nEA(M)contains maximally one linear\nelement.\nProof. First notice that the elements of Γm\nEA(M)are continuous and defined on\neveryCP\nS(˜S)(M)in every chart from A, so they are required to be compared in\nany arbitrary chart from A. By taking this into account, we can repeat the pre-\nvious proof. There is one additional trivial fact one has to n otice: the CP\nS(M)\ndomain also uniquely determines how the Γm\nE(M)element acts outside CP\nS(M).\nSo ifB, T∈Γm\nEA(M)give the same map on CP\nS(M), they give the same map\neverywhere.\nTheorem 3.4.3. The following statements hold:\na) TakeTµ...α\nν...β∈Γa\nEA(M)such that ∀ΩCh,∀Chk(ΩCh)∈ A and∀ω∈CP(ΩCh)\nTµ...α\nν...β is defined as a map\n(Chk,ω)→/integraldisplay\nΩChTµ...\n1ν...(Chk)ω.../integraldisplay\nΩChTα...\nNβ...(Chk)ω. (3.47)\nThen the class of equivalence ˜Γa\nEA(M), to which Tµ...\nν...belongs, contains a\nlinear element defined (on arbitrary ΩCh) as the map: ∀ω∈CP(ΩCh),\n∀Chk(ΩCh)∈ A ,\n(Chk,ω)→/integraldisplay\nΩChTµ...\n1ν...(Chk)...Tα...\nNβ...(Chk)ω, (3.48)\nif and only if ∀ΩCh,∀Chk(ΩCh)∈ A ∃Chl(ΩCh)∈ A , such that\n/integraldisplay\nChl(ΩCh)|Ω′Tµ...\n1ν...(Chk)...T...α\nN...β(Chk)d4x (3.49)\nconverges on every compact set Ω′⊂ΩCh.\nThe same statement holds, if we take instead of Γm\nEA(M)its subclass D′a\nbEA(M)\nand instead of the equivalence class ˜Γm\nEA(M), the equivalence class ˜D′a\nbEA(M).CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 143\nThe same statement also holds if we take instead of Γm\nEA(M)andD′m\nnEA(M)\nclasses, the classes Γm\nEA(∪lAt(˜Sl)o)(M)andD′m\nnEA(∪lAt(˜Sl)o)(M), (with the excep-\ntion that the given convergence property shall be considere d only for charts\nfrom∪lAt(˜Sl)).\nb) For any distribution Aα...\nβ...∈Γa\nS(∪nSno)(M), and an element Tµ...\nν...∈\nΓm\n(∪nSn)(M), we have that Aα...\nβ...Tµ...\nν... is equivalent to an element of\nΓm+a\n(∪nSno)(M), (and for subclasses D′a\nb(∪nSn)(M)⊂Γa+b\n(∪nSn)(M)and\nD′k\nlS(∪nSno)(M)∈Γk+l\nS(∪nSno)(M)it is equivalent to an element of\nD′k+a\nl+b(So)(M)). The element is on its domain given as the mapping\n(ω,Chk)→Tµ...\nν...(Aα...\nβ...ω). (3.50)\nc) For any tensor distribution Aα...\nβ...∈Γa\nS(M)and an element Tµ...\nν...∈\nΓm\n(∪nSno)(M), we have that Aα...\nβ...Tµ...\nν... is equivalent to an element of\nΓm+a\n(∪nSno)(M). The element is on its domain given as mapping\n(ω,Chk)→Tµ...\nν...(Aα...\nβ...ω). (3.51)\nProof. a) Use exactly the same construction as in the previous proof . For ar-\nbitraryΩChand arbitrary Chk(ΩCh)∈ A , we see that Tµ...\nν...is for every\nω∈CP\nS(˜S)(ΩCh)given by (3.47) (it is continuous in arbitrary chart on every\nCP\nS(˜S)(ΩCh)). We can express the map (3.47) in some chart Chl(ΩCh)as\n(Chk,ω′)\n→/integraldisplay\nChl(ΩCh)Tµ...\n1ν...(Chk)ω′d4x.../integraldisplay\nChl(ΩCh)Tα...\nNβ...(Chk)ω′d4x. (3.52)\nThen it is a result of Colombeau theory that if\n(Chk,ω′)→/integraldisplay\nChl(ΩCh)Tµ...\n1ν...(Chk)...T...α\nN...β(Chk)ω′d4x (3.53)\nis defined as a linear mapping on compactly supported, smooth R4func-\ntionsω′(in our case they are related by Chl(ΩCh)to givenCP\nS(M)objects),CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 144\nit is equivalent to (3.52). Now everything was fixed, but arbi trary, so the\nresult is proven. From this proof we also see that the simple t ransforma-\ntion properties of the D′m\nn(M)objects21are fulfilled by the map (3.53) if the\nobjects multiplied are from D′m\nnA(M). So the second result can be proven im-\nmediately. The last two results concerning the classes with limited domains\ntrivially follow from the previous proof.\nb) is proven completely in the same way, we just have to unders tand that be-\ncause of the “limited” domain of the D′a\nbS(∪nSno)(M)objects, we can effec-\ntively use the concept of smoothness in this case.\nc) is just the same as b), the only difference is that the domai n of the product is\nlimited because of the “second” term in the product.\nNote that this means that tensor product gives, on appropria te subclasses of\nD′m\nnEA(M), the mapping ˜D′a\nbEA(M)טD′m\nnEA→˜D′a+m\nb+nEA(M). It also means that this\nprocedure gives, on appropriate subclasses of Γm\nEA(M), the mapping ˜Γa\nEA(M)×\n˜Γm\nEA→˜Γa+m\nEA(M). The disappointing fact is that this cannot be extended to D′m\nnA(M).\nTheorem 3.4.4. TakeTµ...\nν...∈Γm\n(∪nSno)A(M),Bµ...\nν...∈Γm\n(∪nSno)(M)andLα...\nβ...∈\nΓn\nS(∪lSlo)(M). ThenTµ...\nν...≈Bµ...\nν...implies22(L⊗T)α...µ...\nβ...ν...≈(L⊗B)α...µ...\nβ...ν... .\nProof. Use the same method as previously. It trivially follows from the results\nof Colombeau theory (especially from the theorem saying tha t if a Colombeau\nalgebra object is equivalent to a distribution, then after m ultiplying each of them\nby a smooth distribution, they remain equivalent).\nTheorem 3.4.5. Contraction (of µandνindex) is always, for such objects T...µ...\n...ν...∈\nD′m\nnEA(M)that they are equivalent to some linear element, a map to some element\nof the equivalence class from ˜Γm+n−2\nEA(M). The equivalence class from\n21We include also the scalar objects here.\n22It is obvious that we can extend the definition domains either ofTµ...\nν...andBµ...\nν..., or ofLµ...\nν....CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 145\n˜Γm+n−2\nEA(M)is such, that it contains (exactly) one element from D′m−1\nn−1E(M)and this\nelement is defined as the map: ∀ΩCh,∀Chk(ΩCh)∈ A ,∀ω∈CP(ΩCh),\n(ω,Chk)→/integraldisplay\nΩChT...α...\n...α...(Chk)ω. (3.54)\nProof. The proof trivially follows from the fact that contraction c ommutes with\nthe relation of equivalence (this trivially follows from ou r previous note about\naddition and equivalence).\nSome interesting conjectures\nConjecture 3.4.1. Tensor product gives these two maps:\n•˜D′a\nbEA(M)טD′m\nnEA(M)→˜D′a+m\nb+nEA(M),\n•˜Γa\nEA(M)טΓb\nEA(M)→˜Γa+b\nEA(M).\nConjecture 3.4.2. Take some Bµ...\nν...∈Γa\n(∪nAt(˜Sn)o)(M). Take an element Tµ...\nν...∈\nΓb\nE(M), such that ∀˜S ⊆ ∪ n˜Sn,∀ΩCh,∀Chk(ΩCh)∈At(˜S)it holds that ∀ω∈\nCP\nS(˜S)(M)the elements of the multi-index matrix Tµ...\nν...(Chk)·ωremain to be from\nthe classCP\nS(˜S)(M)⊂ ∪nCP\nS(˜Sn)(M). Then it holds that Bα...\nβ...Tµ...\nν...is equivalent to\nan element of Γa+b\n(∪nAt(˜Sn)o)(M). (For subclasses D′m\nn(∪lAt(˜Sl)o)(M)andD′a\nbE(M)it is\nequivalent to an element D′m+a\nn+b(∪lAt(˜Sl)o)(M).) The element is on its domain given\nas mapping\n(ω,Chk)→Bµ...\nν...(Tα...\nβ...ω). (3.55)\nSome additional theory\nTheorem 3.4.1. Any arbitrary Tµ...\nν...∈Λ(as defined by 3.4.3) defines an As(Tµ...\nν...)\nobject onM. Take any arbitrary ˜Sfrom the domain of Tµ...\nν..., any arbitrary S ⊂\nAt(˜S)∩˜S, any arbitrary ΩChand any arbitrary Chk(ΩCh)∈ S. Then for ΩCh\\\n˜Ω(Chk)it holds that multi-index matrix of functions Tµ...\nν...(Chk)can be obtained\nfrom the tensor components of As(Tµ...\nν...)inChk(ΩCh)by the inverse mapping to\nChk(ΩCh).CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 146\nProof. For∀ΩCh, take fixed but arbitrary Chk(ΩCh)∈ S and takeTµ...\nν...(Chk).\nThen∀q∈ΩCh,∀Ch′(ΩCh,q)∈ S, and∀ωǫ∈An(˜Sl,Ch′(q,ΩCh)), we see that\nω′(Ch′)is a delta-sequence. That means we just have to show, that on t he set\nwhereTµ...\nν...(Chk)is continuous in the 3.4.3 sense, the delta-sequencies give the\nvalue of this multi-index matrix. So write the integral:\n/integraldisplay\nCh′(ΩCh)Tµ...\nν...[(Chk)(x)]1\nǫ4ω′/parenleftBigx\nǫ/parenrightBig\nd4x.\nBy substitution x=ǫ.zwe obtain:\n/integraldisplay\nCh′(ΩCh)Tµ...\nν...[(Chk) (ǫ.z)]ω′(z)d4z.\nBut from the properties of Tµ...\nν...[(Chk)(x)]it follows that\n/integraldisplay\nCh′(ΩCh)(Tµ...\nν...[(Chk) (z0)−Kµ...\nν...ǫ)]ω′(z)d4z\n≤/integraldisplay\nCh′(ΩCh)Tµ...\nν...[(Chk)(z0+nǫ)]ω′(z)d4z\n≤/integraldisplay\nCh′(ΩCh)(Tµ...\nν...[(Chk)(z0)]+Kµ...\nν...ǫ)ω′(z)d4z,\nfor someǫsmall enough.\nBut we are taking the limit ǫ→0which, considering the fact that ω(x)are\nnormed to 1, means that the integral must give Tµ...\nν...[(Chk)(z0)]. The set, where it\nis not continuous in the sense of 3.4.3, has Lebesgue measure 0. That means the\nΩChpart, which is mapped to this set has Lebesgue measure 0. But t hen the values\nof the multi-index matrix in the given chart at this arbitrar y, but fixed point give\nus an associated field (and are independent on delta sequence obviously).\nTheorem 3.4.2. The field associated to a Tµ...\nν...∈Λ∩D′m\nnE(M), transforms for\neachΩCh, for every pair of charts from its domain, Ch1(ΩCh), Ch2(ΩCh)on\nsomeM/(˜Ω(Ch1)∪˜Ω(Ch2)), as an ordinary tensor field with piecewise smooth\ntransformations23.\n23Of course, some transformations in a generalized sense migh t be defined also on the ˜Ω(Ch1)∪\n˜Ω(Ch2)set.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 147\nProof. All this immediately follows from what was done in the previo us proof,\nand from the fact that union of sets with Lebesgue measure 0 ha s Lebesgue mea-\nsure 0.\nNote, that if there exists such point that for the object Γm\nA(M)we have\nlim\nǫ→0Tµ...\nν...(ωǫ) =±∞\nat that point, then the field associated to this object, can be associated to another\nobject, which is nonequivalent to this object. This means th at the same field can\nbe associated to mutually non-equivalent elements of Γm\nA(M). This is explicitly\nshown and proven by the next example.\nTheorem 3.4.3. Takeδ(q,Chk(ΩCh))∈D′\n(So)(M)being defined as mapping from\neach 4-form CP\nS(˜S)(M) (S ⊆˜S)to the value of this form’s density at the point qin\nthe chartChk(ΩCh)∈ S, (q∈ΩCh). Then any power n∈N+ofδ(q,Chk(ΩCh))is\nassociated to the function being defined on the domain M\\{q}and everywhere 0.\nNote that this function is associated to any power ( n∈N+) (being a nonzero nat-\nural number) of δ(q), but different powers of δ(q)are mutually nonequivalent24.\nProof. Contracting powers of δ(q,Chk(ΩCh))with a sequence of 4-forms from ar-\nbitraryAn(Ch′(q′,Ω′\nCh),˜S),(q′∝\\e}atio\\slash=q)(they have a support converging to another\npoint than q) will give 0. For q=q′(3.43) gives\nlim\nǫ→0ǫ−4ω′[(Chk)(0)] =±∞.\nNow explore the equivalence between different powers of δ(Chk,q).\nδn(q,Chk(ΩCh))applied to ωǫ(x)∈An(Ch′\nk(q,ΩCh),˜S)will lead to the expression\nǫ−4nωn(x\nǫ). Then if we want to compute\n24It is hard to find in our theory a more “natural” definition gene ralizing the concept of delta\nfunction from Rn. But there is still another natural generalization: it is an object from Γ0\n(∪nSno)(M),\ndefined as: δ(Chk(ΩCh),q,ω) =ω′[(Chk)(˜q)],Chk(ΩCh)∈ Sn,ω∈CP\nS(˜Sn)(ΩCh)(Sn⊂˜Sn),˜qis\nimage ofqgiven by the chart mapping Chk(ΩCh). So it gives value of the density ω′in the chart\nChk(ΩCh), at the chart image of the point q.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 148\nlim\nǫ→0/integraldisplay\nCh′(ΩCh)/parenleftbigg1\nǫ4nωn/parenleftBigx\nǫ/parenrightBig\n−1\nǫ4mωm/parenleftBigx\nǫ/parenrightBig/parenrightbigg\nΦ(x)d4x\nit leads to\nlim\nǫ→01\nǫ4m−4Φ(0)/integraldisplay\nCh′(ΩCh)/parenleftbigg\nωn(x)−1\nǫ4(n−m)ωm(x)/parenrightbigg\nd4x\nwhich is for n∝\\e}atio\\slash=m, n,m ∈N+clearly divergent, hence nonzero.\nNote that despite of the fact that within our algebras we, nat urally, have all the\nn∈N+powers of the delta distribution, they are for n >1, unfortunately, not\nequivalent to any distribution.\nTheorem 3.4.4. We see that the map Asis linear (in the sense analogous to 3.4.1),\nand for arbitrary number of gµ...\nν...,...,hµ...\nν...∈Λ∩Γm\nE(∪nAt(˜Sn)o)(M)one has: Take\n∀ΩCh,∀n,∀S ⊂˜Sn∩At(˜S),∀Chk(ΩCh)∈ S,∪i˜Ωi(Chk)to be the union of all\n˜Ωi(Chk)related to the objects gµ...\nν...,...,hµ...\nν.... Then\nAs(gα...\nβ...⊗...⊗hµ...\nν...) =As(gα...\nβ...)⊗...⊗As(hµ...\nν...)\nonΩCh\\ ∪i˜Ωi(Chk). Here the first term is a product between Λobjects and the\nsecond is the classical tensor product.\nProof. It is trivially connected with previous proofs: Note that fr om the definition\n(3.4.2) for appropriate 4-form fields ωǫwe have\nAs(gα...\nβ...⊗...⊗hµ...\nν...)(Chk) = lim\nǫ→0gα...\nβ...(Chk,ωǫ)...hµ...\nν...(Chk,ωǫ). (3.56)\nBut for the objects gα...\nβ...,...,hα...\nβ...∈Λ, with respect to the theorem (3.4.1) neces-\nsarily\nlim\nǫ→0gα...\nβ...(Chk,ωǫ)...hµ...\nν...(Chk,ωǫ) =As(gα...\nβ...)⊗...⊗As(hµ...\nν...). (3.57)\nThis proves the theorem.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 149\nThis means that the result of tensor multiplication of eleme nts from Λ(it has\nproduct of two scalars as a subcase) is always equivalent to s ome element from Λ.\nThis is a result closely related to the theorem (3.4.3). It te lls us that multiplication\nis a mapping between equivalence classes formed of more cons trained classes as\nthose mentioned in the part a)of the theorem (3.4.3).\nConjecture 3.4.1. IfTµ...\nν...∈D′m\nn(So)(M)has an associated field, then it transforms\non its domains as a tensor field.\nConjecture 3.4.2. IfTµ...\nν...∈Γm\n(So)(M)has an associated field and Lα...\nβ...∈Γn\nS(M),\nthenAs(Tµ...\nν...⊗Lα...\nβ...) =As(Tµ...\nν...)⊗As(Lα...\nβ...). (The⊗sign has again slightly different\nmeaning on the different sides of the equation).\nConjecture 3.4.3. TakeCµ...\nν...,Dµ...\nν...,Fα...\nβ...,Bα...\nβ...∈Γm\nA(M), such that they belong to\nthe same classes Γm\n(At(˜S))A(M). Also assume that ∀˜S, such that Cµ...\nν...,Dµ...\nν...,Fα...\nβ...,\nBα...\nβ...∈Γm\n(At(˜S))A(M), and∀S ⊆˜S, each of the elements Cµ...\nν...,Dµ...\nν...,Fα...\nβ...,Bα...\nβ...has\nfor everyAt(˜S)associated fields defined on the whole M. ThenFα...\nβ...≈Bα...\nβ...and\nCµ...\nν...≈Dµ...\nν...implies (C⊗F)µ...α...\nν...β...≈(D⊗B)µ...α...\nν...β... .\n3.4.5 Covariant derivative\nThe last missing fundamental concept is the covariant deriv ative operator on\ngeneralized tensor fields (GTF). This operator is necessary to formulate an appro-\npriate language for physics and generalize physical laws. S uch an operator must\nobviously reproduce our concept of the covariant derivativ e on the smooth tensor\nfields (through the given association relation to the smooth manifold). This is pro-\nvided in the following section. The beginning of the first par t is again devoted to\nfundamental definitions. The beginning of the second part gi ves us fundamental\ntheorems, again just generalizing Colombeau results for ou r case. After these the-\norems we, (similarly to previous section), formulate conje ctures representing the\nvery important and natural extensions of our results (bring ing a lot of new sig-\nnificance to our results). The last subsection in the second p art being again calledCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 150\n“some additional theory” brings (analogously to previous s ection) just physical\ninsight to our abstract calculus and is of lower mathematica l importance.\nDefinition of ∂-derivative and connection coefficients\nDefinition 3.4.1. We define a map, called the ∂-derivative, given by smooth vector\nfieldUi(smooth in the atlas S) as a mapping Γm\n(S)(M)→Γm\n(S)(M), given on its\ndomain ∀ΩChandChk(ΩCh)as:\nTµ...\nν... ,(U)(Chk,ω)≡ −Tµ...\nν...(Chk,(Uαω),α)ω∈CP(ΩCh).\nHere(Uαω),αis understood in the following way: We express Uαin the chart\nChk(ΩCh)and take the derivatives (Uα(Chk)ω′(Chk)),αin the same chart25\nChk(ΩCh). They give us some function in Chk(ΩCh), which can be (in this chart)\ntaken as expression for density of some object from CP(ΩCh). This means we\ntrivially extend it to Mby taking it to be 0 everywhere outside ΩCh. This is the\nobject used as an argument in Tµ...\nν....\nTo make a consistency check: This Tµ...\nν...,(U)is an object which is defined at\nleast on the domain CP\nS(˜S)(M)forS ⊆˜S(Srelated toUiin the sense that Uiis\nsmooth in S), and is continuous on the same domain. This means it belongs to\nthe class Γn(M). To show this take some arbitrary ΩChand some arbitrary chart\nChk(ΩCh)∈ S. We see that within Chk(ΩCh)the expression (Uαω′),αis smooth\nand describes 4-forms, which are compactly supported, with their support being\nsubset of ΩCh. Hence they are from the domain of Tµ...\nν...in every chart from S.\nIn any arbitrary chart from Swe trivially observe, (from the theory of distribu-\ntions), that if ωn→ω, thanTµ...\nν...,(U)(ωn)→Tµ...\nν...,(U)(ω). It means that Tµ...\nν...,(U)(ω)is\ncontinuous.\nDefinition 3.4.2. Now, by generalized connection we denote an object from Γ3(M)\nsuch that:\n25We will further express that the derivative is taken in Chk(ΩCh)by using the notation\n(Uαω′(Chk)),[(Chk)α].CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 151\n•The set\n∪ΩCh∪˜S/braceleftbigg/parenleftbig\nω,Ch(ΩCh)/parenrightbig\n:ω∈CP\nS(˜S)(ΩCh), Ch(ΩCh)∈ A/bracerightbigg\nbelongs to its domain.\n•It is∀˜S,∀ΩCh,∀Chk(ΩCh)∈ A continuous on CP\nS(˜S)(ΩCh)withChk(ΩCh)\ntaken as its argument.\n•It transforms as:\nΓα\nβγ(Ch2,ω) = Γµ\nνδ(Ch1,((J−1)ν\nβ(J−1)δ\nγJα\nµ−Jα\nm(J−1)m\nβ,γ)ω). (3.58)\nDefinition of covariant derivative\nDefinition 3.4.3. By a covariant derivative (on ΩCh) in the direction of a vector\nfieldUi(M), smooth with respect to atlas S, of an object Tµ...\nν...∈Γm\n(S)(ΩCh), we\nmean:\nDC(U)Tµ...\nν...(ω)≡Tµ...\nν... ,(U)(ω)+Γµ\nαρ(ω)Tρ...\nν...(Uαω)−Γα\nνρ(ω)Tµ...\nα...(Uρω). (3.59)\nThis definition (3.59) automatically defines covariant deri vative everywhere\nonΓm\n(S)(M). This can be easily observed: The ∂-derivative still gives us an object\nfromΓm\n(S)(M), and the second term containing generalized connection is f rom\nΓm\n(S)(M)trivially too.\nDefinition 3.4.4. Furthermore, let us extend the definition of covariant deriv ative\nto the class Γm\n(S)A(M)just by stating that on every nonlinear object (note that eve ry\nsuch object is constructed by tensor product of linear objec ts) it is defined by the\nLeibniz rule. (This means it is a standard derivative operat or, since it is trivially\nlinear as well.)CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 152\nTheS′\nnclass\nNotation. Take as Dnsomen-times continuously differentiable subatlas of A.\nThen the class S′\nnrelated to the atlas Dnis formed by objects Tµ...\nν...∈Γa\nE(∪mAt(˜Sm)o)A(M),\nsuch that:\n• ∪m˜Sm=Dnand∀m, At(˜Sm) =Dn,\n•it is given ∀ΩCh,∀Chk(ΩCh)∈ Dn,∀m,∀ω∈CP\nS(˜Sm)(ΩCh)as a map\n(ω,Chk)→/integraldisplay\nΩChTµ...\nν...(Chk)ω.\nHereTµ...\nν...(Chk)is a multi-index matrix of n-times continuously differen-\ntiable functions (if it is being expressed in any arbitrary c hart from Dn).\nBasic equivalence relations related to differentiation an d some of the interest-\ning conjectures\nTheorem 3.4.1. The following statements hold:\na) Take a vector field Ui, which is smooth at S ⊂ D n+1⊂ Dnforn≥1, (for-\nmally including also n=n+1 =∞, henceS), plus a generalized connection\nΓµ\nνα∈Γ3\nE(M), andTµ...\nν...∈D′a\nb(M)∩S′\nn.S′\nnis here related to the given Dn.\nThen it follows that: DC(U)Tµ...\nν...is an object from S′\nn+1(being related to the\ngivenDn+1). Moreover, the equivalence class ˜S′\nn+1of the image contains\nexactly one linear element given for every chart from its dom ain as integral\nfrom some multi-index matrix of piecewise continuous funct ions. Particu-\nlarly this element is for ∀ΩChand arbitrary such ω∈CP(ΩCh),Chk(ΩCh)\nthat are from its domain a map:\n(Chk,ω)→/integraldisplay\nUαTµ...\nν...;α(Chk)ω. (3.60)\n(Here “;” means the classical covariant derivative related to the “c lassical”\nconnection, components of which are in Chkgiven by Γα\nβδ(Chk), and the\ntensor field, components of which are in Chkgiven byTµ...\nν...(Chk).)CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 153\nb) TakeUibeing smooth in SandΓµ\nνα∈Γ3\nS(M). Then the following holds:\nThe covariant derivative is a map D′m\nn(So)(M)→˜Γm+n\n(So)A(M)and the classes\n˜Γm+n\n(So)A(M)of the image contain (exactly) one element of D′m\nn(So)(M).\nProof. a) Covariant derivative is, on its domain, given as\n(ω,Chk)→Tµ...\nν... ,(U)(ω)+Γµ\nαρ(ω)Tρ...\nν...(Uαω)−Γα\nνρ(ω)Tµ...\nα...(Uρω).\nTake the first term Tµ...\nν...,(U). Express it on ΩChinChk(ΩCh)∈ Dnas the map:\nω→/integraldisplay\nChk(ΩCh)Tµ....\nν...(Chk)ω′(Chk)d4x.\nSo analogously:\nTµ...\nν...,(U)(Chk,ω) =−/integraldisplay\nChk(ΩCh)Tµ...\nν...(Chk)(Uα(Chk)ω′(Chk)),[(Chk)α]d4x.\nHereω′is in arbitrary chart from Dn, beingn-times continuously differen-\ntiable, (the domain is limited to such objects by the second c ovariant deriva-\ntive term, which is added to the ∂-derivative), and such that the expression\n(Ui(Chk)ω′(Chk)),[(Chk)i]is in anyChk(ΩCh)∈ Dnn-times continuously dif-\nferentiable.\nNow by using integration by parts, (since all the objects und er the integral\nare at least continuously differentiable ( n≥1), it can safely be used), and\nconsidering the compactness of support we obtain:\nTµ...\nν...,(U)(Chk,ω)\n=−/integraldisplay\nChk(ΩCh)Tµ...\nν...(Chk)(Uα(Chk)ω′(Chk)),[(Chk)α]d4x\n=/integraldisplay\nChk(ΩCh)Tµ...\nν...(Chk),[(Chk)α]Uα(Chk)ω′(Chk)d4x. (3.61)CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 154\nThen it holds that:\nTµ...\nν...,(U)(Chm(ΩCh),ω)\n=/integraldisplay\nChm(ΩCh)Tµ...\nν...(Chm),[(Chm)α](Uα(Chm)ω′(Chm)d4x\n=/integraldisplay\nChk(ΩCh)(Jµ\nβ(J−1)δ\nν....Tδ...\nβ...(Chk)),[(Chk)α]Uα(Chk)ω′(Chk)d4x. (3.62)\nWe see that this is defined and continuous in any arbitrary cha rt fromDn+1\nfor everyCP\nS(˜S)(M), such that ∃S ⊂˜SandS ⊂ D n+1.\nNow we see that the second term in the covariant derivative ex pression is\nequivalent to the map (with the same domain):\n(Chk,ω)→/integraldisplay\n(Γµ\nαρUαTρ...\nν...−Γα\nνρUρTµ...\nα...)(Chk)ω,\n(see theorem 3.4.3), and between charts the objects appeari ng inside the in-\ntegral transform exactly as their classical analogues. Thi s must hold, since\nTµ...\nν...is everywhere continuous (in every chart considered), henc e on every\ncompact set bounded, so the given object is well defined. This means that\nwhen we fix this object in chart Chm(ΩCh), and express it through the chart\nChk(ΩCh)and Jacobians (with the integral expressed at chart Chk(ΩCh)), as\nin the previous case, we discover (exactly as in the classica l case), that the\nresulting object under the integral transforms as some obje ctD′m\nnE(M), with\nthe classical expression for the covariant derivative of a t ensor field appear-\ning under the integral.\nb) The resulting object is defined particularly only on CP\nS(˜S)(M)(S ⊂˜S). We\nhave to realize that Tµ...\nν...can be written as a ( N→ ∞ ) weak limit (in every\nchart from S) ofTµ...\nν...N∈D′m\nnS(So)(M). It is an immediate result of previous\nconstructions and Colombeau theory, that\n−Γα\nνρ(.)Tµ...\nα...(Uρ.)≈ −Tµ...\nα...(Γα\nνρUρ.) (3.63)CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 155\nNow take ∀ΩChboth,Tµ...\nν...,(U)and (3.63) fixed in arbitrary Chk(ΩCh)∈\nS. Write both of those objects as limits of integrals of some se quence of\n“smooth” objects in Chk(ΩCh). But now we can again use for Tµ...\nν...,(U)an\nintegration per parts and from the “old” tensorial relation s; we get the “ten-\nsorial” transformation properties under the limit. This me ans that the re-\nsulting object, which is a limit of those objects transforms in the way the\nD′m\nn(M)objects transform.\nTheorem 3.4.2. Parta)of the theorem (3.4.1) can be also formulated through a\ngeneralized concept of covariant derivative, where we do no t require the Uivector\nfield to be smooth at some S, but it is enough if it is n+1differentiable in Dn+1.\nProof. We just have to follow our proof and realize that the only reas on why we\nused smoothness of UiinSwas that it is required by our definition of covariant\nderivative (for another good reasons related to different c ases).\nThis statement has a crucial importance, since it shows that not only all the\nclassical calculus of smooth tensor fields with all the basic operations is contained\nin our language (if we take the equivalence instead of equali ty being the crucial\npart of our theory), but it can be even extended to arbitrary o bjects from S′\nn. (If\nthe covariant derivative is obtained through connection fr om the class Γ3\nE(M).) In\nother words it is more general than the classical tensor calc ulus.\nWe can now think about conjectures extending our results in a very important\nway:\nConjecture 3.4.1. Take an arbitrary piecewise smooth, and on every compact set\nbounded26vector field Ui. Take also Γµ\nνα∈Γ3\nE(M)and suchTµ...\nν...∈D′m\nnE(M),\n26To be exact, the expression “covariant derivative” is used i n this and the following conjecture\nin a more general way, since we do not put on Uithe condition of being smooth in some subatlas\nS.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 156\nthat∀ΩCh,∀Chk(ΩCh)∈ A ∃Chl(ΩCh)∈ A, in which27\n/integraldisplay\nChl|Ω′(ΩCh)Tµ...\nν...(Chk)Γα\nβδ(Chk)d4x\nconverges on every compact set Ω′⊂ΩCh. Then the following holds: The covari-\nant derivative (along Ui) maps this object to an element of some equivalence class\nfrom˜Γm+n\nA(M). This class contains (exactly) one element from D′m\nn(M).\nConjecture 3.4.2. TakeUibeing a piecewise smooth vector field, and Γµ\nνα∈\nΓ3\nS(M). Then the following holds: The covariant derivative along t his vector field\nis a map:D′m\nn(∪lSlo)(M)→˜Γm+n\n(∪lSlo)A(M), and the classes ˜Γm+n\n(∪lSlo)A(M)of the image\ncontain (exactly) one element of D′m\nn(M).\nTheorem 3.4.3. ForUibeing a smooth tensor field in Swith the connection taken\nfromΓ3\nS(M),Tµ...\nν...∈Γm\n(So)A(M)andBµ...\nν...∈Γm\n(So)(M), it holds that Tµ...\nν...≈Bµ...\nν...im-\npliesDn\nC(U)Tµ...\nν...≈Dn\nC(U)Bµ...\nν...for arbitrary natural number n.\nProof. Pick an arbitrary ΩChand an arbitrary fixed chart Ch′(ΩCh)∈ S. Such a\nchart maps all the 4-forms from the domain of Γm\n(So)A(M)objects to smooth com-\npact supported functions (given by densities expressed in t hat chart). The objects\nTµ...\nν...((Uαω),α)andBµ...\nν...((Uαω),α)are taken as objects of the Colombeau algebra\n(the connection, fixed in that chart, is also an object of the C olombeau algebra)\nand are equivalent to Uα(ω)Tµ...\nν...,α(ω)andUα(ω)Bµ...\nν...,α(ω). Here the derivative\nmeans the ”distributional derivative” as used in the Colomb eau theory (fulfilling\nthe Leibniz rule) and Uα(ω)is simply a D′m\nnS(So)(M)object with the given vec-\ntor field appearing under the integral. But in the Colombeau t heory one knows\nthat if some object is equivalent to a distributional object , then their derivatives\nof arbitrary degree are also equivalent. It also holds that i f any arbitrary object is\nequivalent to a distributional object, then they remain equ ivalent after being mul-\ntiplied by arbitrary smooth distribution. In the fixed chart we have (still in the\nColombeau theory sense),\n27This means we are trivially integrating Tµ...\nν...Γα\nβδon compact sets within subset of R4given as\nimage of the given chart mapping.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 157\nTµ...\nν...≈Bµ...\nν....\nBut since their ∂-derivatives were, in the fixed chart, obtained only by the\ndistributional derivatives and multiplication by a smooth function, also their ∂-\nderivatives must remain equivalent. The same holds about th e second covariant\nderivative term (containing connection). So the objects fr om classical Colombeau\ntheory, (classical theory just trivially extended to what w e call multi-index matri-\nces of functions), obtained by the chart mapping of the covar iant derivatives of\nTµ...\nν...andBµ...\nν..., are equivalent in the sense of the Colombeau theory. But the ΩCh\nset was arbitrary and also the chart was an arbitrary chart fr om the domain of\nTµ...\nν..., Bµ...\nν.... SoTµ...\nν...andBµ...\nν...are equivalent with respect to our definition.\nWe can try to extend this statement to a conjecture:\nConjecture 3.4.3. TakeUibeing piecewise smooth tensor field, take connection\nfrom the class Γ3\nS(M),Tµ...\nν...∈Γm\n(∪lSlo)A(M), andBµ...\nν...∈Γm\n(∪lSlo)(M). Then it holds\nthatTµ...\nν...≈Bµ...\nν...impliesDn\nC(U)Tµ...\nν...≈Dn\nC(U)Bµ...\nν...for arbitrary natural number n,\nif such covariant derivative exists.\nThis conjecture in fact means that if we have connection from the class Γ3\nS(M),\nthen the covariant derivative is a map from such element of th e class˜Γm\n(∪lSlo)A(M),\nthat it contains some linear element, to ˜Γm\nA(M). Note that we can also try to prove\nan extended version of the conjecture, taking the same state ment and just extend-\ning the classes Γm\n(∪lSlo)A(M),Γm\n(∪lSlo)(M)to the classes Γm\n(∪lSl)A(M),Γm\n(∪lSl)(M).\nSome additional theory\nTheorem 3.4.4. TakeUito be vector field smooth in some S ⊂ D n, withΓµ\nνα∈\nΛandTµ...\nν...∈S′\nn∩D′a\nb(M)(n≥1,S′\nnis related to Dn). ThenDC(U)Tµ...\nν...has\nan associated field which is on M\\˜Ω(Ch)the classical covariant derivative of\nAs(Tµ...\nν...). (˜Ω(Ch)is a set on which is Γµ\nνα(Ch)continuous and is of 0 Lebesgue\nmeasure.) It is defined on the whole Dn+1(S ⊂ D n+1). That means association\nand covariant differentiation in this case commute.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 158\nProof. Just take the definition of the classical covariant derivati ve, and define the\nlinear mapping given ∀ΩChand arbitrary Chk(ΩCh)∈ Dnas\n(ω,Chk)→/integraldisplay\nΩChUν(Chk)[As(Tµ...\nν...);µ](Chk)ω. (3.64)\nThis is an internally consistent definition, since [As(Tµ...\nν...);µ](Chk)is defined\neverywhere apart of a set having L measure 0. Now from our prev ious results\nfollows that everywhere outside ˜Ω(Ch)\n[As(Tµ...\nν...);µ](Chk) =Tµ...\nν...;µ(Chk),\nand so the linear mapping (3.64) is equivalent to the object DC(U)Tµ...\nν.... Then\nUνAs(Tµ...\nν...);µ=As(DC(U)Tµ...\nν...)everywhere outside the set ˜Ω(Ch).\nTheorem 3.4.5. An extended analogy of theorem (3.4.4), can be proven, if we u se\ngeneralized concept of covariant derivative, without assu ming that vector field is\nsmooth in some S, but onlyn+1continuously differentiable within Dn+1.\nProof. Exactly the same as before.\nThis means that the aim to define a concept of covariant deriva tive, “lifted”\nfrom the smooth manifold and smooth tensor algebra to GTF in s ense of associa-\ntion, has been achieved. It completes the required connecti on with the old tensor\ncalculus.\nConjecture 3.4.4. TakeUismooth in S,Tµ...\nν...∈D′m\nn(So)(M), such that ∃As(Tµ...\nν...),\nΓµ\nνα∈Γ3\nS(M). Then∃As(DC(U)Tµ...\nν...)and holds that\nAs(DC(U)Tµ...\nν...) =UαAs(Tµ...\nν...);α.\n(Hence, similarly to ⊗, the covariant derivative operator commutes with associa-\ntion for some significant number of objects.)\nNote that for every class Γm\nAt(S)(M)we can easily define the operator of Lie\nderivative along arbitrary in Ssmooth vector field V(not along the generalizedCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 159\nvector field, but even in the case of covariant derivative we d id not prove anything\nabout larger classes of vector fields than smooth vector field s). Lie derivative can\nbe defined as (LVT,ω)≡(T,LVω). This is because the Lie derivative preserves\nn-forms and also preserves the properties of such CP\n(˜S)(M)classes, for which it\nholds that S ⊂˜S.\n3.4.6 Basic discussion of previous results and open questio ns\nWe have constructed the algebra of GTFs, being able to incorp orate the concept of\ncovariant derivative, (with the given conditions on vector fields and connection),\nfor a set of algebras constructed from specific distribution al objects. The use of\nthese ideas in physics is meaningful where the operations of tensor product and\ncovariant derivative give a map from appropriate subclass o fD′m\nn(M)class to the\nelements of ˜Γm(M)containing a D′m\nn(M)element. This is always guaranteed\nto work between appropriate subclasses of piecewise contin uous distributional\nobjects, but a given physical equivalence might specify a la rger set of objects for\nwhich these operations provide such mapping. Note that the w hole problem lies\nin the multiplication of distributions outside the D′m\nnE(M)class. (For instance it\ncan be easily seen that square of δ(Chk,q)as introduced before is not equivalent\nto any distribution.) This is because the product does not ha ve to be necessarily\nequivalent to a distributional object. Even worse, in case i t is not equivalent to\na distributional object, the product is not necessarily a ma pping between equiv-\nalence classes of the given algebra elements ( ˜D′m\nnA(M)). The same holds about\ncontraction.\nBut even in such cases there can be a further hope. For example we can aban-\ndon the requirement that certain quantities must be linear, (for example the con-\nnection), and only some results of their multiplication are really physical (mean-\ning linear). Then it is a question whether they should be cons tructed (constructed\nfrom the linear objects as for example metric connection fro m the metric tensor)\nthrough the exact equality or only through the equivalence. If we take only the\nweaker (equivalence) condition, then there is a vast number of objects we canCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 160\nchoose, and many other important questions can be posed. Eve n in the case that\nthe mathematical operations do depend on particular repres entatives of the equiv-\nalence classes, there is no necessity to give up; in such situ ation it might be an\ninteresting question if there are any specific “paths” which can be used to solve\nthe physical equivalence relations. The other point is that if these operations do\ndepend on the class members, then we can reverse this process . It means that\nfor example in the case of multiplication of two delta functi ons we can find their\nnonlinear equivalents first and then take their square, thus obtaining possibly an\nobject belonging to an equivalence class of a distribution.\nAs I mentioned in the introduction, these are not attempts to deal with phys-\nical problems in a random, ad hoc way. Rather I want to give the following in-\nterpretation to what is happening: The differential equati ons in physics should\nbe changed into equivalence relations. For that reason they have plentiful solu-\ntions in the given algebra28. (By a “solution” one here means any object fulfilling\nthe given relations.) One obtains much “more” solutions tha n in the case of clas-\nsical partial differential equations (but all the smooth di stributions representing\n“classical” solutions of the “classical” initial value pro blem are there), but what is\nunder question is the possibility to formulate the initial v alue problem for larger\nclasses of objects than D′m\nnE(M)andD′m\nn(So)(M)(see the next section). Moreover, if\nthis is possible then there remains another question about t he physical meaning\nof those solutions. It means that even in the case we get nonli near objects as so-\nlutions of some general initial value problem formulation, this does not have to\nbe necessarily something surprising; the case where physic al laws are solved also\nby physically meaningless solutions is nothing new. The set of objects where we\ncan typically search for physically meaningful solutions i s defined by most of the\ndistributional mappings (that is why classical calculus is so successful), but they\ndo not have to be necessarily the only ones.\n28After one for example proves that covariant derivative is a w ell defined operator on all GTF\nelements, then it is the whole GTF algebra.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 161\n3.4.7 Some notes on the initial value problem within the part ial\ndifferential equivalence relations ( ≈) onD′m\nn(M)\nIn this section we will suggest how to complete the mathemati cal structure de-\nveloped and will get some idea how a physical problem can be fo rmulated in our\nlanguage. It is again divided into what we call “basic ideas” and “some additional\nideas”. The first part is of a considerable importance, the se cond part is less im-\nportant, it just gives a suggestion how to recover the classi cal geometric concept\nof geodesics in our theory.\nThe basic ideas\nThe approach giving the definition of the initial value probl em Take a hyper-\nsurface (this can be obviously generalized to any submanifo ld of lower dimen-\nsion)N⊂M, which is such that it gives in some subatlas AN⊂ A a piecewise\nsmooth submanifold. In the same time ANis such atlas that ∃S S ⊆ A N.\nIf we consider space of 3-form fields living on N(we give up on the idea of\nrelating them to 4-forms on M), we get two types of important maps:\n•Take such D′m\nnE(M)objects, that they have in every chart from ANasso-\nciated (tensor) fields defined everywhere on N, apart from a set having 3\ndimensional Lebesgue measure equal to 0. Such objects can be , in every\nsmooth subatlas S ⊂ A N, mapped to the class D′m\nnE(N)by embedding their\nassociated tensor fields29intoN. This defines a tensor field Tµ...\nν...living on a\npiecewise smooth manifold N. Furthermore if A3Dis some largest piece-\nwise smooth atlas on N, the tensor field Tµ...\nν...defines on its domain a map\n∀ΩCh⊂N,Chk(ΩCh)∈ A3Dandω∈CP(ΩCh)\n(ω,Chk)→/integraldisplay\nΩChTµ...\nν...(Chk)ω. (3.65)\n29As previously noted, the given Tµ\nν∈D′m\nnE(M)we can define in every chart by (Chk,ω)→/integraltext\nΩCh[As(Tµ\nν)](Chk)ω.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 162\nThis is object from the class D′m\nnE(N). What remains to be proven is that in\nany smooth subatlas we map the same D′m\nnE(M)object toD′m\nn(N), otherwise\nthis formulation is meaningless.\n•The other case is a map D′m\nn(So)(M)→D′m\nn(So)(N)(S ⊂ A N), defined in a\nsimple way: The objects from D′m\nnS(So)(M)are mapped as associated smooth\ntensor fields (in the previous sense). After this step is take n, one maps the\nrest of distributional objects from D′m\nn(So)(M)by using the fact that they are\nweak limits of smooth distributions D′m\nnS(So)(M). (This is coordinate inde-\npendent for arbitrary tensor distributions.) So in the case of objects outside\nthe classD′m\nnS(So)(M)we embed the smooth distributions first, and take the\nlimit afterwards (exchanging the order of operations). The basic conjecture\nis that if this limit exists on M, it will exist on N(in the weak topology), by\nusing the embedded smooth distributions.\nNow we can say that initial value conditions of, (for example ), second-order\npartial differential equations are given by two distributi onal objects from\nD′m\nn(So)(N1),D′m\nn(So)(N2)(D′m\nnE(N1),D′m\nnE(N2)) on two hypersurfaces N1,N2(not\nintersecting each other). The solution is a distributional object from the same class,\nwhich fulfills the ≈equation and is mapped (by the maps introduced in this\nsection) to these two distributional objects.\nUseful conjecture related to our approach Note, that we can possibly (if the\nlimit commutes) extend this “initial value” approach throu gh theD′m\nnS(M)class\nto all the weak topology limits of the sequences formed by the objects from this\nclass. This means extension to the class of objects belongin g toD′m\nn(M), such that\nfor any chart from Athey have the full CP(M)domain and D′m\nnS(M)is dense in\nthis class.\nSome additional ideas\n“Null geodesic solution” conjecture Let us conjecture the following:CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 163\nConjecture 3.4.1. Pick some atlas S. Pick some gµν∈D0\n2S(So)(M), such that it has\nAs(gµν), being a Lorentzian signature metric tensor field. Take some ΩChand two\nspacelike hypersurfaces H1, H2,H1∩H2={0},H1∩ΩCh∝\\e}atio\\slash={0},H2∩ΩCh∝\\e}atio\\slash=\n{0}. Furthermore H1,H2are such that there exist two points q1∈H1∩ΩCh,\nq2∈H2∩ΩChseparated by a null curve geodesics (relatively to As(gµν)), and\nthe geodesics lies within ΩCh. Construct such chart Chk(ΩCh)∈ S, that both of\nthe hypersurfaces are hypersurfaces (they are smooth manif olds relatively to S)\ngiven byu=const. condition (uis one of the coordinates) and the given geodesics\nis representing u-coordinate curve.\nTake classical free field equation with equivalence: /squaregΦ≈0(withgµνas\npreviously defined). Then look for the distributional solut ion of this equation with\nthe initial value conditions being δ/parenleftbig\nCh1k(ΩCh∩H1),q1/parenrightbig\n∈D′m\nn(So)(H1)on the first\nhypersurface and δ/parenleftbig\nCh2k(ΩCh∩H2),q2/parenrightbig\n∈D′m\nn(So)(H2)on the second hypersurface.\n(HereCh1k(ΩCh∩H1), Ch2k(ΩCh∩H2)are coordinate charts, which are the same\non the intersection of the given hypersurface and ΩChas the original coordinates\nwithoutu.) Then the solution of this initial value problem is a mappin gΦ, which\nis such, that it can be in chart Chk(ΩCh)expressed as\nω→/integraldisplay\ndu/integraldisplay/productdisplay\nidxi/parenleftbig\nδ(xi(q1))ω′(Chk)(u,xj)/parenrightbig\n(wherexi(q1)is image of q1in chart mapping Chk(ΩCh)).\nWe can formulate similar conjectures for timelike and space like geodesics, we\njust have to:\n•instead of point separation by null curve, consider the sepa ration by timelike\nor spacelike curve,\n•instead of the “massless” equation we have to solve the (/squareg±m2)Φ≈\n0equation (mbeing arbitrary nonzero real number). Here ±depends on the\nsignature we use and on whether we look for timelike or spacel ike geodesics.\nThe rest of the conditions are unchanged (see 3.4.1). Some in sight to our con-\njectures can be brought by calculating the massless case for flat Minkowski space,CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 164\nusing modified cartesian coordinates ( u=x−ct,x,y,z ). We get the expected re-\nsults.\n3.5 What previous results can be recovered, and how?\nAs was already mentioned, our approach is in some sense a gene ralization of\nColombeau approach from [46], which is equivalent to canoni calRnapproach.\nSo forΩCh, after we pick some Ch′(ΩCh)∈ S, (which determines the classes\nAn(M)related to this chart), and by considering only the objects CP\nS(˜S)(ΩCh)(S ⊂\n˜S), (hence considering the D′m\nn(So)(M)class only), we obtain from our construction\nthe mathematical language used in [46]. But all the basic equ ivalence relations\nfrom Colombeau approach have been generalized first to the cl assD′m\nn(So)A(M)\nand also to appropriate subclasses of the D′m\nnEA(M)class.\n3.5.1 Generalization of some particular statements\nNow there are certain statements in Rn, where one has to check whether they are\njust a result of this specific reduction, or not. A good exampl e is a statement\nHnδ≈1\nn+1δ. (3.66)\n(His Heaviside distribution.) What we have to do is to interpre t the symbols\ninside this equation geometrically. This is a R1relation.His understood as a\nD′\n(So)(M)element and defined on the manifold (one dimensional, so the g eom-\netry would be quite trivial) by integral given by a function ( onM) obtained by\nthe inverse coordinate mapping substituted to H. Now take some fixed chart\nChk(R1). The derivative is a covariant derivative along the smooth v ector field\nU, which is constant and unit in the fixed coordinates Chk(R1). Thenδcan be\nreinterpreted as δ(Chk,q), whereqis the 0 point in the chart Chk(R1). Then the\nrelation can be generalized, since it is obvious that (see th e covariant derivativeCHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 165\nsection)DC(U)H=δ(q,Chk)and so\nDC(U)H=DC(U)L(Hf)\n=DC(U)L(Hn+1\nf)≈DC(U)(Hn+1) = (n+1)HnDC(U)H. (3.67)\nByLwe mean here a regular distribution defined by the function in the brackets,\nhence an object from D′\nE(M). (To be precise and to avoid confusion in the nota-\ntion, we used for the Heaviside function the symbol Hf, while for the Heaviside\ndistribution the usual symbol H.) This is nice, but rather trivial illustration.\nThis can be generalized to more nontrivial cases. Take the fla tRntopological\nmanifold. Fix such chart Chk(Rn)covering the whole manifold, that we can\nexpress Heaviside distribution in this chart through Hf(x1)∈D′\n(So)(M). This\nmeans the hypersurface where Hfis discontinuous is given in Chk(Rn)asx1= 0.\nNow the derivative will be a covariant derivative taken alon g a smooth vector\nfield being perpendicular (relatively to the flat space metri c30) to the hypersurface\non which is Hfdiscontinuous. We easily see that the covariant derivative ofH\nalong such vector field gives a distribution (call it ˜δ∈D′\n(So)(M)), which is in the\nchartChk(Rn)expressed as\n˜δ(φ) =δx1/parenleftbigg/integraldisplay\nφ(x1,x2...xn)dx2...dxn/parenrightbigg\n. (3.68)\nThis distribution reminds us in some sense the “geodesic” di stribution from the\nprevious part. Then the following holds:\nHn˜δ≈1\nn+1˜δ. (3.69)\nThis generalized form of our previous statement can be used f or computations\nwith Heaviside functional metrics (computation of connect ion in fixed coordi-\nnates).\n30Note that since it is flat space it makes sense to speak about a p erpendicular vector field, since\nwe can uniquely transport vectors to the hypersurface.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 166\n3.5.2 Relation to practical computations\nThese considerations (for example) imply that the result fr om canonical Colom-\nbeauRntheory derived by [46] can be derived in our formalism as well . This is\nalso true for the geodesic computation in curved space geome try from [104]. The\nresults derived in special Colombeau algebras (in geometri cally nontrivial cases)\nare more complicated, since in such cases the strongest, A∞(Rn), version of the\ntheory is used. This version is not contained in our chart rep resentations. (This is\nbecause we are using only An(Rn)with finiten.) It is clear that all the equivalence\nrelations from our theory must hold in such stronger formula tion (since obviously\nA∞(M)⊂An(M)∀n∈N) and the uniqueness of distribution solution must hold\nas well. This means that at this stage there seems to be no obst acle to reformulate\nour theory by using A∞(Chk,q,ΩCh)classes (and taking elements from D′\n(˜So)(M)\nat least), if necessary. But it is unclear whether one can tra nsfer all the calculations\nusingA∞(Chk,q,ΩCh)classes to our weaker formulation.\nThe strong formulation was used also in the Schwarzschild ca se [76], but there\nis a problem. The fact that the authors of [76] regularize var ious functions piece\nby piece does not have to be necessarily a problem in Colombea u theory31. But,\nas already mentioned, the problem lies in the use of formula f orRν\nµ, origi-\nnally derived within smooth tensor field algebra. If we want t o derive in the\nSchwarzschild case Ricci tensor straight from its definitio n, we cannot avoid mul-\ntiplications of delta function by a non-smooth function. Th is is in Colombeau\ntheory deeply non-trivial.\nIn the cases of Kerr’s geometry and conical spacetimes theor y this problem\nappears as well. As a consequence of this fact, calculations are mollifier depen-\ndent, not being (in the strict sense) results of our theory an ymore. On the other\nhand there is no reasonable mathematical theory in which the se calculations make\nsense. This means that a better understanding of these resul ts will be necessary.\nBy better understanding of these results provided by our the ory we mean their\n31Although the authors use (in the first part, not being necessa rily connected with the results)\nquite problematic embeddings.CHAPTER 3. MULTIPLICATION OF TENSORIAL DISTRIBUTIONS 167\nderivation by a net of equivalence relations, by taking some intermediate quanti-\nties to be nonlinear. So the results should follow from the pr inciple that the equiv-\nalence relations are the fundamental part of all the mathema tical formulation of\nphysics.\n3.6 Conclusions\nThe main objectives of this work were to build foundations of a mathematical\nlanguage reproducing the old language of smooth tensor calc ulus and extending\nit at the same time. The reasons for these objectives were giv en at the beginning\nof this chapter. This work is a first step to such theory, but it already achieves its\nbasic goals. That means we consider these results as useful i ndependently of how\nsuccessful future work on the topic will turn out to be. On the other hand, the\nterritory it opens for further exploration is in my opinion l arge and significant. It\noffers a large area of possibilities for future work.\nJust to summarize: the result of our work is a theory based pur ely on equiv-\nalence relations instead of equalities, using a well defined concept of generalized\ntensor field and the covariant derivative operator. This ope rator is well defined at\nleast on the proper subclass of generalized tensor fields. We also defined (using\nsome conjectures) the initial value problem for partial dif ferential equivalence re-\nlations. Our theory naturally relates to many results beyon d the classical smooth\ntensor calculus, already derived.Conclusions\nAs we mentioned in the introduction, this thesis is based on t hree separate re-\nsearch projects.\nThe first project was related to pseudo-Finsler extensions o f the general theory\nof relativity, and represented an attempt to find a natural ge ometric framework\nfor possible high energy Lorentz violations. The reason why one was interested\nin such construction is the question of whether it is possibl e to find a weaker in-\nterpretation of Einstein’s equivalence principle consist ent with Lorentz symmetry\nviolations. The result obtained was, when mathematical sim plicity was taken as\nthe guidance principle, unfortunately a “no-go” theorem, a t least for significant\nnumber of cases. The cases particularly affected were the bi -metric theories, but\nthe analogue model based on bi-refringent crystal optics in dicated that the prob-\nlem might affect much larger class of theories. The problem l ies in the fact that,\nunlike what one would naturally expect, introducing Lorent zian signature puts\nvery tight constraints on Finsler geometry, at least if one w ants to keep some of\nthe basic geometric concepts well defined and meaningful. Th is “no-go” result\nwe consider to be disappointing, (which is the case of most “n o-go” results), but\ncertainly very useful.\nThe second project focused on the highly damped quasi-norma l modes of dif-\nferent black hole spacetimes. The method of approximation b y analytically solv-\nable potentials was used to estimate the highly damped modes for the Schwarz-\nschild and the Schwarzschild-de Sitter (S-dS) black holes. The first served more as\na consistency check (since the asymptotic formula for the hi ghly damped QNMs\nof the Schwarzschild black hole is well known). But for the Sc hwarzschild-de Sit-\n168ter black hole a lot of new information was extracted from tho se models, especially\nthe link between rational ratios of horizon surface graviti es to the periodic be-\nhaviour of the QNMs. Also, when periodic, in general the high ly damped modes\ndo not form only one equi-spaced family as in the case of Schwa rzschild black\nhole, but split into multiple families. Strikingly, as we di scovered, the same pat-\nterns can be observed in a complementary set of analytic esti mations for the highly\ndamped modes, those approximations being obtained by monod romy techniques.\nThis holds for all types of black hole spacetime so far analys ed by those tech-\nniques. That means we were able to significantly generalize o ur theorems about\nthe highly damped mode behaviour to all the presently known a nalytic results.\nOur results might be interesting also from the viewpoint of t he black hole ther-\nmodynamics, as the asymptotic QNM behaviour is suspected to be linked to the\nblack hole area spectrum [77, 110].\nThe third project dealt with the problem of multiplication o f tensorial distri-\nbutions. Despite the fact that lot has been done in the field in the past [71, 173],\nthe full generalization of the covariant derivative operat or was, for example, not\nyet achieved. On the other hand practical results confirm a ne ed for such gen-\neralization (see [173]). We built an alternative construct ion, which fully operates\nwith the Colombeau equivalence relation, but technically a voids Colombeau al-\ngebra construction. It generalizes the concept of covarian t derivative to tensorial\ndistributions and operates on the much more general, but for the language of\ndistributions natural, piecewise smooth manifolds. We are convinced that such\nlanguage might offer conceptual extension of general relat ivity and might have\npossibly interesting consequences for quantum gravity as w ell.\n169Appendix A\nBi-refringent crystals\nA.1 Basic characteristics of the crystal media\nThe basic optics reference we shall use is Born and Wolf, Principles of Optics [29]. In\nparticular we shall focus on Chapter XV , “Optics of crystals ”, pages 790–818. See\nespecially pages 796–798 and pages 808–811. Specific page, c hapter, and section\nreferences below are to the 7th (expanded) edition, 1999/20 03.\nThe theory of bi-refringent crystal optics is formulated in the preferred inertial\nsystem, the inertial system of the crystal. The optical medi um of the crystal is\ncharacterized by permeability and permittivity. Permeabi lityµis taken to be a\nscalar, permittivity ǫija “spatial”, (relative to the inertial system of the crystal ),\n3×3tensor. (This is an excellent approximation for all known optically active\nmedia.) By going to the principal axes we can, without loss of generality, take ǫij\nto be diagonal\nǫij=\nǫx0 0\n0ǫy0\n0 0ǫz\n. (A.1)\nThis fixes the relativistic inertial coordinate system in a u nique way.\nWe furthermore define “principal velocities”\nvx=c√µǫx;vy=c√µǫy;vz=c√µǫz. (A.2)\n170APPENDIX A. BI-REFRINGENT CRYSTALS 171\nNote (this is a tricky point that has the potential to cause co nfusion) that vxisnot\nthe velocity of light in the xdirection — since ǫx(and sovx) is related to the prop-\nerties of the electric field in the xdirection, the principal velocity vxis instead the\nvelocity of a light wave whose electric field is pointing in th exdirection. That is,\nfor light waves propagating in the y-zplane, oneof the polarizations will propa-\ngate with speed vx.\nA.2 Group velocity and ray equation\nThe group velocity, vg, in the framework used by Born and Wolf, is identical to\nthe “ray velocity”, and is controlled by the so-called “ray e quation”. See (15.2.29),\npage 797. To set some conventions, ˆnwill always denote a unit vector in physical\nspace — a unit with respect to the usual Euclidean norm, while nis a generic\nposition in physical 3-space. In contrast, ˆkwill be reserved for a unit wave-vector\nin the dual “wave-vector space”.\nBorn and Wolf exhibit the ray equation in a form equivalent (B orn and Wolf\nusetwhere we use n) to:\nˆn2\nx\n1/v2g−1/v2x+ˆn2\ny\n1/v2g−1/v2y+ˆn2\nz\n1/v2g−1/v2z= 0. (A.3)\nHere the group velocity (ray velocity) is defined by looking a t the energy flux and\nvg=vgˆn. (A.4)\nWe can rewrite this as\nˆn2\nxv2\nx\nv2g−v2x+ˆn2\nyv2\ny\nv2g−v2y+ˆn2\nzv2\nz\nv2g−v2z= 0. (A.5)\nThis form of the ray equation encounters awkward “division b y zero” problems\nwhen one looks along the principal axes, so it is advisable to eliminate the denom-\ninators by multiplying through by the common factor (v2\ng−v2\nx)(v2\ng−v2\ny)(v2\ng−v2\nz),APPENDIX A. BI-REFRINGENT CRYSTALS 172\nthereby obtaining:\nˆn2\nxv2\nx(v2\ng−v2\ny)(v2\ng−v2\nz)+ ˆn2\nyv2\ny(v2\ng−v2\nz)(v2\ng−v2\nx)\n+ˆn2\nzv2\nz(v2\ng−v2\nx)(v2\ng−v2\ny) = 0. (A.6)\nIt is this form of the ray equation that, (because it is much be tter behaved), we shall\nuse as our starting point. Now this is clearly a quartic in vg, and by regrouping it\nwe can write\nv4\ng/bracketleftbig\nˆn2\nxv2\nx+ ˆn2\nyv2\ny+ ˆn2\nxv2\nz/bracketrightbig\n−v2\ng/bracketleftbig\nˆn2\nxv2\nx(v2\ny+v2\nz)+ ˆn2\nyv2\ny(v2\nz+v2\nx)+ ˆn2\nzv2\nz(v2\nx+v2\ny)/bracketrightbig\n+/bracketleftbig\nv2\nxv2\nyv2\nz/bracketrightbig\n= 0. (A.7)\nEquivalently\nv4\ng/bracketleftbig\nˆn2\nxv−2\nyv−2\nz+ ˆn2\nyv−2\nzv−2\nx+ ˆn2\nzv−2\nxv−2\ny/bracketrightbig\n−v2\ng/bracketleftbig\nˆn2\nx(v−2\ny+v−2\nz)+ ˆn2\ny(v−2\nz+v−2\nx)+ ˆn2\nz(v−2\nx+v−2\ny)/bracketrightbig\n+1 = 0. (A.8)\nNow define two quadratics (in terms of the three direction cos inesˆni)\n¯q0(ˆn,ˆn) =/bracketleftbig\nˆn2\nxv−2\nyv−2\nz+ ˆn2\nyv−2\nzv−2\nx+ ˆn2\nzv−2\nxv−2\ny/bracketrightbig\n, (A.9)\n¯q2(ˆn,ˆn) =1\n2/bracketleftbig\nˆn2\nx(v−2\ny+v−2\nz)+ ˆn2\ny(v−2\nz+v−2\nx)+ ˆn2\nz(v−2\nx+v−2\ny)/bracketrightbig\n,(A.10)\nthen\nv2\ng(ˆn) =¯q2(ˆn,ˆn)±/radicalbig\n¯q2(ˆn,ˆn)2−¯q0(ˆn,ˆn)\n¯q0(ˆn,ˆn). (A.11)\nBut, since ˆnis a unit vector, we could equally well rewrite this as\nv2\ng(ˆn) =¯q2(ˆn,ˆn)±/radicalbig\n¯q2(ˆn,ˆn)2−¯q0(ˆn,ˆn) (ˆn·ˆn)\n¯q0(ˆn,ˆn). (A.12)APPENDIX A. BI-REFRINGENT CRYSTALS 173\nIn this form both numerator and denominator are manifestly h omogeneous and\nquadratic in the components of ˆn, so for any 3-vector n(now notnecessarily of\nunit norm) we can take the further step of writing\nv2\ng(n) =¯q2(n,n)±/radicalbig\n¯q2(n,n)2−¯q0(n,n) (n·n)\n¯q0(n,n). (A.13)\nThe function vg(n)so defined is homogeneous of degree zero in the components\nofn:\nvg(κn) =vg(n) =vg(ˆn). (A.14)\nThe homogeneous degree zero property should remind one of th e relevant feature\nexhibited by the Finsler metric. It is also useful to note tha t\n1\nvg(n)2=¯q2(n,n)∓/radicalbig\n¯q2(n,n)2−¯q0(n,n) (n·n)\n(n·n). (A.15)\nA.3 Phase velocity and Fresnel equation\nIn contrast, the phase velocity, in the framework used by Bor n and Wolf, is con-\ntrolled by the so-called “equation of wave normals”, also kn own as the “Fresnel\nequation”. See equation (15.2.24), page 796. The relevant c omputations are similar\nto, but not quite identical to, those for the group velocity.\nLet us consider a plane wave exp(i[k·x−ωt])and define the phase velocity by\nvp=vpˆk=ω\nkˆk, (A.16)\nthen the Fresnel equation is equivalent (Born and Wolf use swhere we use ˆk) to\nˆk2\nx\nv2p−v2x+ˆk2\ny\nv2p−v2y+ˆk2\nz\nv2p−v2z= 0. (A.17)\nThis form of the equation exhibits “division by zero” issues if you try to look\nalong the principal axes, so it is for many purposes better to multiply through byAPPENDIX A. BI-REFRINGENT CRYSTALS 174\nthe common factor (v2\np−v2\nx)(v2\np−v2\ny)(v2\np−v2\nz)thereby obtaining the equivalent of\ntheir equation (15.3.1) on page 806:\nˆk2\nx(v2\np−v2\ny)(v2\np−v2\nz)+ˆk2\ny(v2\np−v2\nz)(v2\np−v2\nx)\n+ˆk2\nz(v2\np−v2\nx)(v2\np−v2\ny) = 0. (A.18)\nThis is clearly a quartic in vpand by regrouping it, and using ˆk·ˆk= 1, we can\nwrite\nv4\np−v2\np/bracketleftBig\nˆk2\nx(v2\ny+v2\nz)+ˆk2\ny(v2\nz+v2\nx)+ˆk2\nz(v2\nx+v2\ny)/bracketrightBig\n+/bracketleftBig\nˆk2\nxv2\nyv2\nz+ˆk2\nyv2\nzv2\nx+ˆk2\nzv2\nxv2\ny/bracketrightBig\n= 0. (A.19)\nLet us now define two quadratics (in terms of the direction cos inesˆki)\nq2(ˆk,ˆk) =1\n2/bracketleftBig\nˆk2\nx(v2\ny+v2\nz)+ˆk2\ny(v2\nz+v2\nx)+ˆk2\nz(v2\nx+v2\ny)/bracketrightBig\n, (A.20)\nand\nq0(ˆk,ˆk) =/bracketleftBig\nˆk2\nxv2\nyv2\nz+ˆk2\nyv2\nzv2\nx+ˆk2\nzv2\nxv2\ny/bracketrightBig\n, (A.21)\nso as a function of direction the phase velocity is\nv2\np(ˆk) =q2(ˆk,ˆk)±/radicalBig\nq2(ˆk,ˆk)2−q0(ˆk,ˆk). (A.22)\nThis is very similar to the equations obtained for the ray vel ocity. In fact, we can\nnaturally extend this formula to arbitrary wave-vector kby writing\nv2\np(k) =q2(k,k)±/radicalbig\nq2(k,k)2−q0(k,k) (k·k)\n(k·k). (A.23)\nThis expression is now homogeneous of order zero in k, so that\nvp(κk) =vp(k) =vp(ˆk). (A.24)\nAgain, we begin to see a hint of Finsler structure emerging.APPENDIX A. BI-REFRINGENT CRYSTALS 175\nA.4 Connecting the ray and the wave vectors\nConnecting the ray-vector ˆnand the wave-vector ˆkin birefringent optics is rather\ntricky — for instance, Born and Wolf provide a rather turgid d iscussion on page\n798 — see section 15.2.2, equations (34)–(39). The key resul t is\nvg(n) ˆni\nvg(n)2−v2\ni=vp(k)ˆki\nvp(k)2−v2\ni, (A.25)\nwhich ultimately can be manipulated to calculate ˆnas a rather complicated “ex-\nplicit” function of ˆk— albeit an expression that is so complicated that even Born\nand Wolf do not explicitly write it down. Unfortunately if it comes to pseudo-\nFinsler geometry, all the extra technical machinery provid ed by Finsler notions of\nnorm and distance do not serve to simplify the situation. (Th e fact that phase and\ngroup velocities can be used to define quite distinct, and in s ome situations com-\npletely unrelated, effective metrics has also been noted in the context of acoustics\n[177, 178].)\nA.5 Optical axes\nTo find the ray optical axes we (without loss of generality) ta kevz>vy>vz, and\ndefine quantities ¯∆±(this is of course the result of considerable hindsight) by:\nv2\nx=v2\ny\n1−v2y¯∆2\n+;v2\nz=v2\ny\n1+v2y¯∆2\n−. (A.26)\nFurthermore eliminate ˆnyby using\nˆn2\ny= 1−ˆn2\nx−ˆn2\nz, (A.27)\nthen\n¯D= ¯q2\n2−¯q0\n=1\n4/bracketleftbig\n(ˆnx¯∆++ ˆnz¯∆−)2−(¯∆2\n++¯∆2\n−)/bracketrightbig\n×/bracketleftbig\n(ˆnx¯∆+−ˆnz¯∆−)2−(¯∆2\n++¯∆2\n−)/bracketrightbig\n. (A.28)APPENDIX A. BI-REFRINGENT CRYSTALS 176\nThus the (ray) optical axes are defined by\n(ˆnx¯∆+±ˆnz¯∆−)2= (¯∆2\n++¯∆2\n−). (A.29)\nBut thanks to the Cauchy–Schwartz inequality\n(ˆnx¯∆+±ˆnz¯∆−)2≤(ˆn2\nx+ ˆn2\nz)(¯∆2\n++¯∆2\n−)≤(¯∆2\n++¯∆2\n−). (A.30)\nTherefore on the (ray) optical axis we must have ˆny= 0, and(ˆn2\nx+ˆn2\nz) = 1 . So (up\nto irrelevant overall signs)\n¯e1,2=/parenleftBigg\n±¯∆+/radicalbig¯∆2\n++¯∆2\n−; 0 ;¯∆−/radicalbig¯∆2\n++¯∆2\n−/parenrightBigg\n, (A.31)\nwhich we can recast in terms of the principal velocities as\n¯e1,2=\n±/radicalBigg\n1/v2y−1/v2x\n1/v2z−1/v2x; 0 ;/radicalBigg\n1/v2z−1/v2y\n1/v2z−1/v2x\n, (A.32)\nor\n¯e1,2=/parenleftBigg\n±vz\nvy/radicalBigg\nv2x−v2y\nv2x−v2z; 0 ;vx\nvy/radicalBigg\nv2y−v2z\nv2x−v2z/parenrightBigg\n. (A.33)\nThese are the two ray optical axes. (Compare with equation (1 5.3.21) on p. 811 of\nBorn and Wolf.)\nA similar computation can be carried through for the phase op tical axes. We\nagain take vz>vy>vz, and now define\nv2\nx=v2\ny+∆2\n+;v2\nz=v2\ny−∆2\n−. (A.34)\nEliminate ˆkyby using\nˆk2\ny= 1−ˆk2\nx−ˆk2\nz. (A.35)\nThen\nD=q2\n2−q4 (A.36)\n=1\n4/bracketleftBig\n(ˆkx∆++ˆkz∆−)2−(∆2\n++∆2\n−)/bracketrightBig\n×/bracketleftBig\n(ˆkx∆+−ˆkz∆−)2−(∆2\n++∆2\n−)/bracketrightBig\n. (A.37)APPENDIX A. BI-REFRINGENT CRYSTALS 177\nThis tells us that the discriminant factorizes, always . The discriminant vanishes if\n(ˆkx∆+±ˆkz∆−)2= ∆2\n++∆2\n−. (A.38)\nBut by the Cauchy–Schwartz inequality\n(ˆkx∆+±ˆkz∆−)2≤(ˆk2\nx+ˆk2\nz)(∆2\n++∆2\n−)≤(∆2\n++∆2\n−). (A.39)\nThus on the (phase) optical axis we must have ˆky= 0and(ˆk2\nx+ˆk2\nz) = 1 . The two\nunique directions (up to irrelevant overall sign flips) that make the discriminant\nvanish are thus\ne1,2=/parenleftBigg\n±∆+/radicalbig\n∆2\n++∆2\n−; 0 ;∆−/radicalbig\n∆2\n++∆2\n−/parenrightBigg\n, (A.40)\nwhich can be rewritten as\ne1,2=/parenleftBigg\n±/radicalBigg\nv2\nx−v2\ny\nv2x−v2z; 0 ;/radicalBigg\nv2\ny−v2\nz\nv2x−v2z/parenrightBigg\n. (A.41)\nThese are the two phase optical axes. (Compare with equation (15.3.11) on p. 810\nof Born and Wolf.)Appendix B\nSpecial functions: some important\nformulas\nB.1 Trigonometric identities\nIn the body of the thesis we needed to use some slightly unusua l trigonometric\nidentities. They can be derived from standard ones without t oo much difficulty\nbut are sufficiently unusual to be worth mentioning explicit ly:\ntanAtanB=cos(A−B)−cos(A+B)\ncos(A−B)+cos(A+B); (B.1)\ntan/parenleftbiggA+B\n2/parenrightbigg\ntan/parenleftbiggA−B\n2/parenrightbigg\n=cosB−cosA\ncosB+cosA; (B.2)\nand\ncos(A+2B)+cosA= 2cosBcos(A+B). (B.3)\nB.2 Gamma function identities and approximations\nThe key Gamma function identity we need is\nΓ(z) Γ(1−z) =π\nsin(πz). (B.4)\n178APPENDIX B. SPECIAL FUNCTIONS: SOME IMPORTANT FORMULAS 179\nThe Stirling approximation for Gamma function gives\nΓ(x) =/radicalbigg\n2π\nx/parenleftBigx\ne/parenrightBigx/parenleftbigg\n1+O/parenleftbigg1\nx/parenrightbigg/parenrightbigg\n;Re(x)>0,|x| → ∞. (B.5)\nWe also need the following asymptotic estimate based on the S tirling approxima-\ntion\nΓ(x+1\n2)\nΓ(x)=√x/bracketleftbigg\n1+O/parenleftbigg1\nx/parenrightbigg/bracketrightbigg\n;Re(x)>0,|x| → ∞. (B.6)\nB.3 Hypergeometric function identities\nThe key hypergeometric function identities we need are Bail ey’s theorem\n2F1/parenleftbigg\na,1−a,c,1\n2/parenrightbigg\n=Γ(c\n2)Γ(c+1\n2)\nΓ(c+a\n2)Γ(c−a+1\n2), (B.7)\nwhich is easily found in many standard references (for examp le [166]), and the\nparticular differential identity\nd{2F1(a,b,c,z)}\ndz=c−1\nz[2F1(a,b,c−1,z)−2F1(a,b,c,z)], (B.8)\nwhich is easy enough to verify once it has been presented.\nB.4 Bessel function identities and expansions\nThe important differential identity which holds for Bessel functions is the follow-\ning:\ndJα(x)\ndx=1\n2/bracketleftbig\nJα−1(x)−Jα+1(x)/bracketrightbig\n. (B.9)\nWhat was also needed was the asymptotic expansion (see for ex ample [166]):\nJα(x) =/radicalbigg\n2\nπx/bracketleftbigg\nP(α,x)cos/parenleftbigg\nx−πα\n2−π\n4/parenrightbigg\n−Q(α,x)sin/parenleftbigg\nx−πα\n2−π\n4/parenrightbigg/bracketrightbigg\n, (B.10)APPENDIX B. SPECIAL FUNCTIONS: SOME IMPORTANT FORMULAS 180\nwhere\nP(α,x)≡∞/summationdisplay\nn=0(−1)nΓ/parenleftbig1\n2+α+2n/parenrightbig\n(2x)2n(2n)!Γ/parenleftbig1\n2+α−2n/parenrightbig (B.11)\nand\nQ(α,x)≡∞/summationdisplay\nn=0(−1)nΓ/parenleftbig1\n2+α+2n+1/parenrightbig\n(2x)2n+1(2n+1)!Γ/parenleftbig1\n2+α−2n−1/parenrightbig. (B.12)181Table of symbols and terms for\nChapter 3\nSymbol: Definition/Explanation: Page:\nD(Rn) compactly supported, smooth functions\n(C∞(Rn))101\nD′(Rn) space of distributions on the space of com-\npactly supported, smooth functions102\nfǫ(xi),xi∈Rn 1\nǫnf/parenleftbigxi\nǫ/parenrightbig\n106\nEM(Rn) Moderate functions 105,107,111\nN(Rn) Negligible functions 105,107,111\nmollifier, (usually)ǫ−sequence of smooth, compactly 105\nsmoothing kernel supported functions with integral normed to\n1 and with support “stretching” to 0 as ǫ→0\nC(f) embedding of distributions into Colombeau\nalgebra by a convolution with a mollifier107\nM (in the section 3.4) manifold, on which one can\ndefine a smooth atlas121\nA maximal piecewise smooth atlas, where the\ntransformation Jacobians are bounded on ev-\nery compact set121\nS maximal smooth subatlas of A 122\nΩCh subset of a manifold, such that it can be\nmapped to Rnby a chart mapping122\n182Symbol: Definition/Explanation: Page:\nCh(ΩCh) chart from the atlas Aon the set ΩCh 122\nCP(ΩCh) class of compactly supported piecewise\nsmooth 4-forms, (we work from the be-\nginning with a 4D manifold), having their\nsupport inside the set ΩCh124\n˜S maximal subatlas of A, such that there exist el-\nements of the class CP(ΩCh), that have in this\nsubatlas smooth scalar densities124\nCP\nS(˜S)(ΩCh) set of all 4-forms from CP(ΩCh), that are in ˜S\ngiven by smooth scalar densities124\nCP\nS(ΩCh) ∪˜SCP\nS(˜S)(ΩCh) 124\nCP(M) ∪ΩChCP(ΩCh) 124\nCP\nS(˜S)(M) ∪ΩChCP\nS(˜S)(ΩCh)\nCP\nS(M) ∪ΩChCP\nS(ΩCh)\nD′(M) linear generalized scalar fields 125\nD′\n(˜S)(M) linear generalized scalar fields defined (at\nleast) on the class CP\nS(˜S)(M)126\nD′\n(˜So)(M) linear generalized scalar fields defined exclu-\nsively on the class CP\nS(˜S)(M)126\nD′\nE(M),\nD′\nE(∪n˜Sno)(M)distributional analogue of piecewise smooth\nscalar fields125, 126\nD′\nS(M),\nD′\nS(∪n˜Sno)(M)objects as close as possible to a distributional\nanalogue of smooth scalar fields125, 126\nD′\nA(M) generalized scalar fields 126\nD′\nEA(M),\nD′\nSA(M)(etc.)generalized scalar fields constructed from the\nelements of the classes D′\nE(M),D′\nS(M)(etc.)129\nD′m\nn(M) linear generalized tensor fields of rank (m,n)127-128\nAt(˜S) specific function, which maps atlases to at-\nlases129\n183Symbol: Definition/Explanation: Page:\nD′m\nn(∪lAt(˜Sl))(M) linear generalized tensor fields of rank (m,n),\ndefined (at least) on all the classes CP\nS(˜Sl)(M),\nlabeled by l, in the atlases At(˜Sl)129\nD′m\nn(∪lAt(˜Sl)o)(M) linear generalized tensor fields of rank (m,n),\ndefined exclusively on the classes CP\nS(Sl)(M),\nlabeled by l, in the atlases At(˜Sl)128\nD′m\nnE(M) distributional analogue of rank (m,n)piece-\nwise smooth tensor fields128\nD′m\nnS(M) objects as close as possible to a distributional\nanalogue of rank (m,n)smooth tensor fields128\nD′m\nn(∪l˜Sl)(M) subclass of D′m\nn(∪lAt(˜Sl))(M), such that At(˜Sl)\nhas as image of ˜Slthe atlas Sl⊆˜Sl129\nD′m\nnA(M) generalized tensor fields of rank (m,n) 129\nD′m\nnEA(M),D′m\nnSA(M)(etc.) generalized tensor fields of rank (m,n)con-\nstructed from the elements of the classes\nD′m\nnE(M),D′m\nnS(M)129\nΓm(M),Γm\nE(M),Γm\nS(M),\nΓm\n(∪nAt(˜Sn))(M),\nΓm\n(∪nAt(˜Sn)o)(M),\nΓm\n(∪n˜Sn)(M)Gamma objects (generalized from linear gen-\neralized tensor fields of rank (a,b), by impos-\ning no condition on the transformation prop-\nerties)129-130\nΓm\nA(M) sets of algebras constructed from the elements\nof the class Γm(M)(generalization of Dm\nnA(M))130\nΓm\nEA(M),Γm\nSA(M)(etc.) sets of algebras constructed from the elements\nof the classes Γm\nE(M),Γm\nS(M)(etc.) (general-\nization from D′m\nnEA(M),D′m\nnSA(M))130\nTµ...\nν...(Chk) multi-index matrix obtained from the tensor\ncomponents of Tµ...\nν...in the chart Chk131\nCh′(q,ΩCh) chart mapping from ΩChto the whole R4, such\nthat it maps the point qto 0133\nAn(˜S,Ch′(q,ΩCh)) specific subclass of the class CP\nS(˜S)(ΩCh) 133\n184Symbol: Definition/Explanation: Page:\nωǫ(y) specific one parameter class of elements\nof the class CP\nS(˜S)(M)133\n≈ the equivalence relation 133-134\n˜Γm(M),˜Γm\nE(M),\n˜D′m\nnA(M),˜D′m\nnEA(M)\n(etc.)sets of equivalence classes constructed\nfrom the elements of the classes Γm(M),\nΓm\nE(M),D′m\nnA(M),D′m\nnEA(M)(etc.)134\nΛ specific subclass of the class\nΓm\nE(∪nAt(˜Sn)o)(M)135-136\nAs(Tµ...\nν...) tensor field associated to Tµ...\nν...∈Γm\nA(M) 135\n˜Ω(Chk) set of Lebesgue measure 0, on which a\nparticular multi-index matrix Tµ...\nν...(Chk)\nis discontinous136\n∂-derivative specific operator acting on the elements\nof the class Γm\nA(M)and generalizing the\noperator of the partial derivative (as a\npart of the covariant derivative defini-\ntion)144-145\nDC(U) covariant derivative along the vector\nfieldUi145-146\nDn n-times differentiable subatlas of A 146\nS′\nnrelated to the\natlasDnspecific subclass of Γm\nE(∪nAt(˜Sn)o)A(M) 146\n185Publications and papers\nJournal\n•J. Skakala and M. Visser, “Generic master equations for quasi-normal frequen-\ncies” , Journal of High Energy Physics (JHEP) 1007 :070, (2010), [arXiv:\ngr-qc/1009.0080]\n•J. Skakala and M. Visser, “Semi-analytic results for quasi-normal frequen-\ncies” , Journal of High Energy Physics (JHEP) 1008 :061, (2010), [arXiv:\ngr-qc/1004.2539]\n•J. Skakala and M. Visser, “Highly-damped quasi-normal frequencies for piece-\nwise Eckart potentials” , Physical Review D, 81:125023, (2010), [arXiv:\ngr-qc/1007.4039]\n•J. Skakala and M. Visser, “The causal structure of spacetime is a parametrized\nRanders geometry” , Classical and Quantum Gravity, 28:065007, (2011),\n[arXiv: gr-qc/1012.4467]\n•J. Skakala and M. Visser, “Bi-metric pseudo-Finslerian spacetimes” , Journal\nof Geometry and Physics, 61:1396-1400, (2011), [arXiv: gr-qc/1008.0689]\n•J. Skakala and M. Visser, “Pseudo-Finslerian spacetimes and multi-refringence” ,\nInternational Journal for Modern Physics D, 19:1119-1146, (2010), [arXiv:\ngr-qc/0806.0950]\n186Conference proceedings\n•J. Skakala and M. Visser, “Birefringence in pseudo-Finsler spacetimes” , NEBXIII\nconference ( Recent Developments in Gravity), Thessalonik a (Greece), June\n2008, J.Phys.Conf.Ser. 189:012037, (2009), [arXiv: gr-qc/0806.0950]\n•J. Skakala and M. Visser, Quasi-normal frequencies: Semi-analytic results for\nhighly damped modes” , The Spanish relativity meeting - ERE 2010, Granada\n(Spain), September 2010, will be published in J.Phys.Conf. Ser. [arXiv:\ngr-qc/1011.4634]\n•J. Skakala, “New ideas about multiplication of tensorial distribution s”, 12-th\nMarcel Grossman meeting (MG12), Paris (France), July 2009, soon to be\npublished\nE-print only\n•J. Skakala, “New ideas about multiplication of tensorial distribution s”, [arXiv:\ngr-qc/0908.0379]\n•J. Skakala and M. 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D 63\n(2001) 124004 [arXiv: hep-th/0101133].\n205Some work that did not appear in the\nthesis\nJozef Skakala and Matt Visser, The causal structure of space time\nis a parametrized Randers geometry\nAbstract:\nThere is a by now well-established isomorphism between stat ionary 4-dimensional\nspacetimes and 3-dimensional purely spatial Randers geome tries - these Randers\ngeometries being a particular case of the more general class of 3-dimensional\nFinsler geometries. We point out that in stably causal space times, by using the\n(time-dependent) ADM decomposition, this result can be ext ended to general\nnon-stationary spacetimes - the causal structure (conform al structure) of the full\nspacetime is completely encoded in a parameterized (time-d ependent) class of\nRanders spaces, which can then be used to define a Fermat princ iple, and also to\nreconstruct the null cones and causal structure.\npublished in Classical and Quantum Gravity 28, (2011), 0650 07,\n[arXiv: gr-qc/1012.4467].\n206Jozef Skakala and Matt Visser, Birkhoff-like theorem for ro tating\nstars in (2+1) dimensions\nAbstract:\nConsider a rotating and possibly pulsating ”star” in (2+1) d imensions. If the star\nis axially symmetric, then in the vacuum region surrounding the star, (a region\nthat we assume at most contains a cosmological constant), th e Einstein equations\nimply that under physically plausible conditions the geome try is in fact stationary.\nFurthermore, the geometry external to the star is then uniqu ely guaranteed to be\nthe (2+1) dimensional analogue of the Kerr-de Sitter spacet ime, the BTZ geometry.\nThis Birkhoff-like theorem is very special to (2+1) dimensi ons, and fails in (3+1)\ndimensions. Effectively, this is a ”no hair” theorem for (2+ 1) dimensional axially\nsymmetric stars: the exterior geometry is completely speci fied by the mass, angu-\nlar momentum, and cosmological constant.\narXiv: gr-qc/0903.2128.\n207"    },    {        "title": "1903.00540v2.Comprehensive_Study_of_Neutrino_Dark_Matter_Mixed_Damping.pdf",        "content": "Prepared for submission to JCAP\nComprehensive Study of\nNeutrino-Dark Matter Mixed Damping\nJulia Stadlera;1C\u0013 eline B\u001bhmb;c;d; 2Olga Menae\naInstitute for Particle Physics Phenomenology, Durham University, South Road, Durham,\nDH1 3LE, United Kingdom\nbSchool of Physics, Physics Road, The University of Sydney, NSW 2006 Camperdown, Syd-\nney, Australia\ncLAPTH, U. de Savoie, CNRS, BP 110, 74941 Annecy-Le-Vieux, France\ndPerimeter institute, 31 Caroline Street N., Waterloo, Ontario, N2L 2Y5, Canada\neInstituto de F\u0013 \u0010sica Corpuscular (IFIC), CSIC-Universitat de Val\u0012 encia,\nApartado de Correos 22085, E-46071, Spain\nE-mail: julia.j.stadler@durham.ac.uk, celine.boehm@sydney.edu.au,\nomena@i\fc.uv.es\nAbstract. Mixed damping is a physical e\u000bect that occurs when a heavy species is cou-\npled to a relativistic \ruid which is itself free streaming. As a cross-case between collisional\ndamping and free-streaming, it is crucial in the context of neutrino-dark matter interac-\ntions. In this work, we establish the parameter space relevant for mixed damping, and we\nderive an analytical approximation for the evolution of dark matter perturbations in the\nmixed damping regime to illustrate the physical processes responsible for the suppression\nof cosmological perturbations. Although extended Boltzmann codes implementing neutrino-\ndark matter scattering terms automatically include mixed damping, this e\u000bect has not been\nsystematically studied. In order to obtain reliable numerical results, it is mandatory to re-\nconsider several aspects of neutrino-dark matter interactions, such as the initial conditions,\nthe ultra-relativistic \ruid approximation and high order multiple moments in the neutrino\ndistribution. Such a precise treatment ensures the correct assessment of the relevance of\nmixed damping in neutrino-dark matter interactions.\n1https://orcid.org/0000-0001-5888-023X\n2https://orcid.org/0000-0002-5074-9998\n3https://orcid.org/0000-0001-5225-975XarXiv:1903.00540v2  [astro-ph.CO]  26 Jul 2019Contents\n1 Introduction 1\n2 Mixed damping regime and parameter space 3\n2.1 Neutrino decoupling by weak interactions 3\n2.2 Neutrino decoupling set by dark matter interactions 4\n3 Numerical evolution of cosmological perturbations 5\n3.1 Boltzmann Equations 5\n3.2 The Ultra-relativistic Fluid Approximation 7\n3.3 Initial conditions 8\n4 The physics of mixed damping 8\n4.1 Comparison of decoupling histories 9\n4.2 Analytical mixed damping evolution 10\n4.3 Evolution of density perturbations and the matter power spectrum 12\n5 Conclusions 14\nA Interaction formalism 19\nB Evaluation of the e\u000bect of the ultra-relativistic \ruid approximation 19\nC Initial conditions for coupled neutrinos 21\n1 Introduction\nDark matter (DM) is a required ingredient in our universe to explain e.g. the galactic rota-\ntion curves, gravitational lensing, Cosmic Microwave Background (CMB) measurements and\nthe growth of matter perturbations. Large-scale-structures observations by galaxy surveys\nand other measurements seem to be perfectly consistent with a Cold Dark Matter (CDM)\ncomponent, that is, with a pressureless, non-interacting \ruid. However, small-scale discrep-\nancies [1] indicate that the properties of dark matter could be more complex than the CDM\nhypothesis.\nIn this regard, dark matter interactions with Standard Model particles can change\nobservations at small scales [2{28]. Mixed damping [2, 5, 6] refers to the physical damping\nphenomenon in which dark matter is kinetically coupled to another species which itself is\nfree streaming. In the context of interacting dark matter, mixed damping is particularly\nrelevant for dark matter-neutrino interactions, the main focus of this study, but it might also\napply to other dark matter interactions, e.g. with a dark radiation component [29]. Note\nthat a similar e\u000bect, Silk damping [30], exists in the evolution of the baryon-photon plasma,\nwhere photons decouple brie\ry before the end of the baryon drag epoch [31, 32]. However,\nin the interacting dark matter scenarios considered in this study the dark matter decoupling\nhistory is very distinct from the baryonic one. Dark matter perturbations evolve in the\nmixed damping regime for a much longer time, while simultaneously radiation perturbations\n{ 1 {are largely una\u000bected. Hence, the e\u000bect of mixed damping on dark matter and neutrino\nperturbations considerably di\u000bers from what is observed in the baryon-photon sector.\nThe mixed damping e\u000bect is a cross between collisional damping and free-streaming;\ndark matter perturbations are damped because the dark matter follows the free-streaming\nneutrinos. The condition for mixed damping to occur is thus\n\u0000DM\u0000\u0017>H > (\u0000\u0017\u0011\u0000\u0017\u0000e+ \u0000 \u0017\u0000DM); (1.1)\nwhere the collision rates between dark matter and neutrinos are given by\n\u0000\u0017\u0000DM=nDM\u001b\u0017DM; (1.2a)\n\u0000DM\u0000\u0017=4\u001a\u0017\n3\u001aDM\u0000\u0017\u0000DM: (1.2b)\nThe condition in Eq. (1.1) assumes that dark matter scattering is the only non-standard\ninteraction of the neutrino species beyond the usual weak interactions with electrons occurring\nat a rate \u0000 \u0017\u0000e=\u001b\u0017\u0000ene.\nTo progress further, one needs to specify the dark matter number density nDM. The\nobserved dark matter relic density suggests that, if the dark matter was produced thermally,\nit has either annihilated or decayed since then1. In a thermal annihilating dark matter\nscenario, where the dark matter and anti dark matter number densities are supposed to be\nexactly the same, the observed relic density imposes a condition on the ratio of the dark\nmatter mass to the freeze-out temperature (for a given annihilation cross section), which\ncan be used to determine whether one needs to account for the evolution of the dark matter\nnumber density while neutrinos are decoupling [6]. In the following, we will disregard such\na scenario and assume that dark matter has already annihilated when the neutrino \ruid\ndecouples. As a result, the bounds that we quote in the following should be used with\nprecaution when dark matter is lighter than mDM.O(MeVs).\nIn the past, several constraints on dark matter-neutrino interactions have been derived\nfrom a plethora of cosmological observations. Forecasts for next-generation of CMB and large\nscale structure surveys are also available in the literature, see e.g. [13]. Constraints are given\nin terms of a single parameter, namely\nu\u0017DM=\u001b\u0017DM\n\u001bTh\u0010mDM\n100 GeV\u0011\u00001\n; (1.3)\nwhere the dark matter-neutrino scattering cross section is normalised to the Thomson scat-\ntering rate \u001bTh. The dark matter-neutrino scattering cross-section may be a power-law of\nthe neutrino temperature, i.e. \u001b\u0017DM/Tn\u0017DM\u0017 . It is convenient to further de\fne\nu\u0017DM=u\u0017DM;0\u0002a\u0000n\u0017DM; (1.4)\nwhereu\u0017DM;0is a constant. The most recent analyses provide u\u0017DM\u0014(4:5\u00009:0)\u000210\u00005for\nn\u0017DM= 0 andu\u0017DM;0\u0014(3:0\u00005:4)\u000210\u000014forn\u0017DM= 2 [13, 19, 33]. Mixed damping is\nan important e\u000bect for these values, as we discuss in Sec. 2.2. Although the e\u000bects of mixed\ndamping are automatically accounted for in the Boltzmann hierarchy for collisional neutrinos,\na systematic description of the mixed damping regime is lacking to date. Furthermore, there\n1There may be other mechanism in the thermal scenario that could explain why the dark matter number\ndensity nowadays is so small. Alternatively dark matter may have been produced non-thermally.\n{ 2 {is a number of subtleties which have to be taken into account in the numerical evolution of\nperturbations if dark matter interacts with neutrinos. In the present manuscript we focus on\nall these aspects.\nThe structure of this paper is as follows. We start by examining the parameter space\nfor which mixed damping is important in Sec. 2, describing the conditions which Eq. (1.1)\nimposes on the scale factor at decoupling, the scales which can be a\u000bected by mixed damping\nand the values of the interactions strength parameter u\u0017DM. We then turn to the numer-\nical evolution of perturbations in Sec. 3, where we introduce the Boltzmann equations for\nneutrino-dark matter scattering. We adopt a self consistent approach to account for the\nangular dependence of the matrix elements for the scattering process in the evolution of\nhigher-order multipoles. We shall also examine the validity of the ultra-relativistic \ruid ap-\nproximation in the presence of dark matter-neutrino interactions and present the required\nmodi\fcations to the initial conditions in the presence of a coupled \ruid. In order to investi-\ngate the mixed damping physics we derive an analytical approximation for the evolution of\ndark matter perturbations in this scenario in Sec. 4. We conclude in Sec. 5.\n2 Mixed damping regime and parameter space\nAs mentioned in the introduction, mixed damping occurs when neutrinos decouple from dark\nmatter before the reverse happens. In general, the two interaction rates \u0000 DM\u0000\u0017and \u0000 \u0017\u0000DM\nare not expected to be equal because of the di\u000berences between the neutrino and dark matter\nnumber densities (c.f. Eq. (1.2)). In the Standard Model, neutrinos kinetically decouple from\nthe thermal bath when they decouple from the electrons. However, in models where neutrinos\ninteract with dark matter, the neutrino interaction rate is given by \u0000 \u0017\u0011\u0000\u0017\u0000DM+\u0000 \u0017\u0000e. The\nneutrino kinetic decoupling is thus determined by their last interactions, which might be\nwith either the electrons (if \u0000 \u0017\u0018\u0000\u0017\u0000e'H) or dark matter (if \u0000 \u0017\u0018\u0000\u0017\u0000DM'H). Each\noption corresponds to a di\u000berent dark matter decoupling epoch and to a di\u000berent maximum\ndamping scale, which we further examine in Sec. 2.1 and Sec. 2.2, respectively.\nIn any case, however, mixed damping can only occur if there is an epoch where \u0000 DM\u0000\u0017>\nH > \u0000\u0017. This condition provides a lower bound on the scales which can be a\u000bected by\nmixed damping and a corresponding minimum scattering rate. It is set by the simultane-\nous decoupling of dark matter from neutrinos and of neutrinos from electrons and leads to\numin\n\u0017DM;0>2:4\u000210\u000014, 1:4\u000210\u000033and 7:4\u000210\u000053forn\u0017DM= 0, 2 and 4 respectively. The scale\nfactor at decoupling in this con\fguration is amin=a(T\r= 1 MeV) = 2 :35\u000210\u000010and the\ncorresponding \"mixed damping scale\" is rather small, lying beyond the current experimental\nreach:rmin= (aH)\u00001\nT\r=1 MeV = 0:11 kpc, orMmin= 0:2M\f.\n2.1 Neutrino decoupling by weak interactions\nThe mixed damping e\u000bect is the sole process responsible for erasing dark matter perturbations\nlarger than rminwhen the neutrino decoupling is set by its Standard Model interactions,\ni.e. \u0000 DM\u0000\u0017>\u0000\u0017\u0000e>\u0000\u0017\u0000DM(assuming a monotonous thermal evolution of both dark\nmatter and neutrinos in the early Universe). In this case, the magnitude of the e\u000bect is\ndetermined by the dark matter decoupling only (\u0000 DM\u0000\u0017'H); the later the decoupling,\nthe bigger the e\u000bect. Still, \u0000 DM\u0000\u0017can not become arbitrarily large. Indeed the condition\n\u0000\u0017\u0000DM<\u0000\u0017\u0000ein combination with Eq. (1.2b) implies that the condition for dark matter\n{ 3 {decoupling, \u0000 DM\u0000\u0017'Hcan be recasted as\n4\u001a\u0017\n3\u001aDM\u0000\u0017\u0000e'H; (2.1)\nfor the maximum neutrino-dark matter interaction rate value allowed. The corresponding\nmaximum scale on which mixed damping can occur, rwdec, is 79 kpc, 2 :6 kpc and 0 :8 kpc for\nn\u0017DM= 0;2 and 4 respectively (or, in terms of enclosed mass, Mwdec = 8\u0002107M\f;3\u0002\n103M\f;1\u0002102M\ffor the respective power laws).\nNeutrino decoupling by weak interactions can only occur if the dark matter interaction\nrate lies below a maximum value. Combined with the previous section's discussion, the mixed\ndamping condition on the interaction strength parameter in this regime is umin\n\u0017DM;0<u \u0017DM;0<\nuwdec\n\u0017DM;0. In numbers, the upper bound is uwdec\n\u0017DM;0= 1:3\u000210\u00008;7:4\u000210\u000028;4:1\u000210\u000047for\nn\u0017DM= 0;2 and 4 respectively. For these limiting parameters we depict the evolution of the\ninteraction rates in Fig. 1, where the left edge of the green shaded region indicates the time\nof neutrino decoupling in the Standard Model.\n2.2 Neutrino decoupling set by dark matter interactions\nMixed damping also plays an important role if the neutrino kinetic decoupling is determined\nby the neutrino-dark matter interactions. This is relevant for larger modes, which are easier\nto access observationally. To illustrate the situation, \frst consider a small mode, which enters\nthe horizon during a period where \u0000 DM\u0000\u0017>\u0000\u0017\u0000DM>H. This mode will be initially subject\nto collisional damping and, upon neutrino decoupling, will experience a transition to the\nmixed damping regime. A larger mode, on the other hand, enters the horizon later and thus\ncan be subject to mixed damping only. However, the transition from the collisional to the\nmixed damping regime can only occur if the ratio of densities in Eq. (1.2b) is larger than\nunity by the time dark matter decouples from the neutrino \ruid. The corresponding upper\nlimit on the decoupling time is\n4\u001a\u0017\n3\u001aDM\f\f\f\f\namax= 1,amax= 1:9\u000210\u00004\u0012\nDMh2\n0:1186\u0013\u00001\n: (2.2)\nThe largest scales that will be a\u000bected by mixed damping have therefore to enter the\nhorizon before amax. The comoving Hubble radius at amaxis\nrmax=71:0 Mpc\u0010\n0:1186\n\nDMh2\u0011\nr\n1:0 + 0:069\u0010\n\nbh2\n0:0223\u0011\u0010\n0:1186\n\nDMh2\u0011; (2.3)\nwhich roughly corresponds to a mass of Mmax'6\u00021016M\f.\nFinally, the criterion of neutrinos decoupling from the dark matter before amaxtranslates\ninto a maximum value of the scattering rate\numax\n\u0017DM;0= 1:97\u000210\u00002\u0002an\u0017DMmax\u0002\u00120:1186\n\nDMh2\u00132s\n1:0 + 0:066\u0012\nbh2\n0:0223\u0013\u00120:1186\n\nDMh2\u0013\n:(2.4)\nWe \fnducrit\n\u0017DM;0'1:98\u000210\u00002, 7:12\u000210\u000010and 2:56\u000210\u000017, forn\u0017DM= 0, 2 and 4\nrespectively. Note that these values fall within the current observational limits [13, 17],\n{ 4 {10−910−710−5100102104106108101010121014\nnνDM= 0uνDM= 2×10−2\nuνDM= 1×10−8\n10−910−710−5nνDM= 2uνDM= 7×10−10\nuνDM= 7×10−28\n10−910−710−5nνDM= 4uνDM= 3×10−17\nuνDM= 4×10−47\n0.0 0.2 0.4 0.6 0.8 1.0\nscale factor a0.00.20.40.60.81.0scatttering rate [Mpc−1]Figure 1 . The evolution of \u0000 \u0017\u0000DM(solid lines), \u0000 DM\u0000\u0017(dashed lines) and the Hubble rate H(black\nsolid line) as a function of the scale factor afor three possible temperature dependent scenarios and\ntwo possible values of the normalised interacting cross section. The green region indicates the dark\nmatter decoupling times at which mixed damping is possible.\nindicating the importance of mixed damping while deriving bounds on scenarios with dark\nmatter-neutrino interactions.\nThe evolution of scattering rates in this limiting case is depicted in Fig. 1 along with\nthe Hubble rate. The right edge of the green region indicates amax. Irrespective of the value\nchosen foru\u0017DM;0andn\u0017DMthe dark matter and the neutrino scattering rate are always the\nsame atamax. Mixed damping occurs whenever umin\n\u0017DM;0< u \u0017DM;0< umax\n\u0017DM;0, or, in terms of\nFig. 1, whenever the neutrino and dark matter decoupling times fall within the green region.\nDepending on the size of a mode it might either be subject to mixed damping solely or to a\nperiod of collisional damping followed by mixed damping.\n3 Numerical evolution of cosmological perturbations\nUp to now we have discussed the mixed damping on the level of the homogeneous background\nevolution. In order to investigate in detail how mixed damping a\u000bects cosmological perturba-\ntions, we now turn to the linearised set of Einstein and Boltzmann equations. After showing\nhow the relevant equations are altered by the presence of a scattering term between dark\nmatter and neutrinos, we shall address several numerical subtleties absolutely mandatory to\nensure the correctness of the formalism.\n3.1 Boltzmann Equations\nIn the Newtonian gauge,\nds2=a2(\u001c)\u0002\n\u0000(1 + 2 )d\u001c2+ (1\u00002\u001e)dxidxi\u0003\n; (3.1)\n{ 5 {where\u001eand are the metric perturbations, the scattering processes between dark matter\nand massless neutrinos are governed, for the dark matter \ruid, by its evolution equations,\n_\u000eDM=\u0000\u0012DM+ 3_\u001e; (3.2a)\n_\u0012DM=k2 \u0000H\u0012DM\u0000R_\u0014\u0017DM(\u0012DM\u0000\u0012\u0017): (3.2b)\nHereR= 4\u001a\u0017=(3\u001aDM), the scattering rate is de\fned as _ \u0014\u0017DM=anDM\u001b\u0017DMand the reduced\nHubble rate given by H=aH. For massless neutrinos the modi\fed Boltzmann hierarchy\nis [34, 35]\n_\u000e\u0017=\u00004\n3\u0012\u0017+ 4_\u001e; (3.3a)\n_\u0012\u0017=k2\u0012\u000e\u0017\n4\u0000\u001b\u0017\u0013\n+k2 \u0000_\u0014\u0017DM(\u0012\u0017\u0000\u0012DM); (3.3b)\n2 _\u001b\u0017=8\n15\u0012\u0017\u00003\n5kF\u0017;3\u0000\u000b2_\u0014\u0017DM\u001b\u0017; (3.3c)\n_F\u0017;l=k\n2l+ 1[lF\u0017;l\u00001\u0000(l+ 1)F\u0017;l+1]\u0000\u000bl_\u0014\u0017DMF\u0017;l; (3.3d)\n_F\u0017;lmax=k\u0014\nF\u0017;lmax\u00001\u0000lmax+ 1\nk\u001cF\u0017;lmax\u0015\n\u0000\u000bl_\u0014\u0017DMF\u0017;lmax: (3.3e)\nThe angular coe\u000ecients \u000bl, which appear in the interaction terms of the higher-order multi-\npoles, are generally of O(1). Nevertheless, their precise numerical value is set by the depen-\ndence of the matrix element for the scattering process jM\u0017DMj2on the cosine of the angle\nbetween the incoming and the outgoing neutrino \u0016. Previous works on dark matter-neutrino\nscattering have adopted di\u000berent choices for \u000bl, concretely \u000b2= 2 and\u000bl= 1 forl\u00153 [13], or\n\u000b2= 9=5 and\u000bl= 1 forl\u00153 [19]. Here, we follow Ref. [35], which provides a self-consistent\nformalism to compute the higher-order multipole coe\u000ecients,\n\u000bl=R\ndp p4\u0010\n@f\u0017\n@p\u0011\n[A0(p)\u0000Al(p)]\nR\ndp p4\u0010\n@f\u0017\n@p\u0011\n[A0(p)\u0000A1(p)]: (3.4)\nNeutrinos follow a Fermi-Dirac statistic, denoted by f\u0017, and the matrix element a\u000bects the\ncoe\u000ecients via its projection on the Legendre polynomials Pl\nAl(p) =1\n2Z1\n\u00001d\u0016 Pl(\u0016)\u00121\n\u0011DM\u0011\u0017jM\u0017DMj2\u0013\f\f\f\ft=2p2(\u0016\u00001)\ns=m2\nDM+2mDMp: (3.5)\nIn the low energy limit, the momentum pof the neutrino does not to change in the scattering\nprocess,tandsare Mandelstam variables, and \u0011DMand\u0011\u0017denote the internal degrees of\nfreedom of the dark matter particle and neutrinos respectively. Within this formalism, the\nscattering rate can be expressed as [35]\n_\u0014\u0017DM=anDM\n128\u00193m2\nDM\u0011\u0017\n\u001a\u0017Z1\n0dp\u0012@f\u0017\n@p\u0013\np4[A0(p)\u0000A1(p)]: (3.6)\n{ 6 {We make use of the classi\fcation of neutrino-dark matter interaction scenarios introduced\nin [20] to give an overview of the derived values for _ \u0014\u0017DMand\u000blin Tab. 1 in Appendix A.\nTo evolve cosmological perturbations numerically, we have modi\fed the publicly available\nBoltzmann solver code CLASS2(version v2.7), introducing three new input parameters.\nNamely,u\u0017DMrepresents the coupling strength parameter (see Eq. (1.3)), n\u0017DMgoverns the\ntemperature dependence of the cross-section (see Eq. (1.4)) and \u000blrefers to the higher-order\nmultipole coe\u000ecients appearing in Eq. (3.3).\n3.2 The Ultra-relativistic Fluid Approximation\nThe evolution of neutrino perturbations is described by an in\fnite hierarchy of moment\nequations (c.f. Eqs. (3.3)), that must be truncated at some maximum multipole lmax. At\nearly times lmaxis typically ofO(10). Once neutrino perturbations are well inside the horizon,\nneutrino multipoles over the range 2 <l\u001ck\u001care suppressed, and the evolution of the highest\nand the lowest multipoles decouples [36]. Note that, during the radiation dominated era, the\nsize of the comoving Hubble radius is given by the conformal time, H\u00001'\u001c. In this regime,\nCLASS employs the ultra-relativistic \ruid approximation (UFA) [36], which truncates the\nmultipole hierarchy after l= 2 once that k\u001c > (k\u001c)UFA. The advantage is twofold. First, the\ncomputational costs to describe the late time evolution of neutrino perturbations are reduced.\nSecond, to avoid unphysical re\rections, which are caused by the inevitable truncation of the\na priori in\fnite Boltzmann hierarchy, for a mode of wavenumber kat some time \u001cone has to\nchoose the maximum multipole moment lmax\u0015k\u001c. Truncating the Boltzmann hierarchy in\na consistent way as earlier as possible then allows to choose a smaller value for lmaxduring\nthe early evolution and hence also bene\fts the computational costs prior to the UFA3.\nIn neutrino-dark matter interacting schemes the UFA method can also be applied, but the\nconditions must be carefully revised. This is the aim of this section.\nThe UFA truncation scheme in general di\u000bers from the ordinary truncation scheme for\nBoltzmann equations proposed in Ref. [34] and assumes free streaming neutrinos. In the\n\u0003CDM scenario this assumption is always valid on all scales larger than rmin, which enter\nthe horizon after neutrino-electron decoupling. But, as we discussion in Sec. 2.2, the coupling\nbetween dark matter and neutrinos can delay the epoch of neutrino free streaming. Previous\nstudies dealing with neutrino-dark matter interactions have tried to generalise the existing\nUFA expression, just by including an interaction term. Since the origin of such a term is\nnot fully understood, we follow here a di\u000berent avenue. We choose ( k\u001c)UFAlarge enough to\nmake sure that neutrinos have decoupled from dark matter when the UFA starts. Then, we\nevolve the neutrino perturbations accordingly to Eqs. (3.3) while the coupling to dark matter\nis active. Once neutrinos have decoupled from dark matter, we switch to the standard UFA\ntruncation formula which does not include interactions, c.f. Eq. (B.1). In parallel, we ensure\nthatlmaxis large enough to avoid unphysical re\rections while the Boltzmann hierarchy is\nevolved.\nThe conditions on ( k\u001c)UFAare detailed in Appendix B, where we also analyse how the\npremature application of the ultra-relativistic \ruid approximation a\u000bects the cosmological\nobservables. Since the e\u000bect of the UFA treatment on CMB temperature and polarisation\nspectra is well below the experimental sensitivity for u\u0017DM= 4:5\u000210\u00005(n\u0017DM= 0) and for\n2Our modi\fed CLASS version is publicly available and can be downloaded from\nhttps://gitlab.dur.scotgrid.ac.uk/dm-interactions/class v2.7 ndm.git\n3The default values for these two parameters in CLASS are ( k\u001c)UFA= 30 andlmax= 17.\n{ 7 {u\u0017DM;0= 5:4\u000210\u000014(n\u0017DM= 2), the constraints obtained in Refs. [19, 33] remain valid,\nregardless the UFA was the default one.\nDiscrepancies are however evident in the matter power spectrum, c.f. Fig. 5. For those\ninteraction strengths interesting to mixed damping and to cosmological constraints, the dif-\nferences only arise at small scales on which the matter power spectrum is already suppressed\nby several orders of magnitude with respect to the \u0003CDM prediction. Di\u000berences at this level\nwill be completely erased by non-linear e\u000bects and, more importantly, are negligible when\ncompared to the expected systematic e\u000bects in galaxy surveys. In Appendix B we further\nillustrate this by estimating the e\u000bect of the UFA treatment on the predicted number of\nMilky Way satellites. Discrepancies arising from the treatment of the UFA regime are well\nbelow Poisson scatter, which the counting is subject to.\n3.3 Initial conditions\nIf neutrinos are coupled to dark matter this a\u000bects not only the evolution of perturbations\nbut also their initial conditions. The reason for this is obvious from the traceless space-space\ncomponent of the linearised Einstein equations\nk2(\u001e\u0000 ) = 12 \u0019Ga2(\u001a+p)\u001b: (3.7)\nIn the radiation dominated era the anisotropic stress is dominated by neutrino perturbations,\nand\u001b\u0017will be suppressed if neutrinos scatter o\u000b dark matter. Even though we aim to study\nthe mixed damping regime, where neutrinos are free streaming, we have shown in Sec. 2 that\nthe transition from the collisional to the mixed damping regime is continuous and smooth.\nA perturbation of \fxed wavelength can start its evolution in the collisional regime and then\nevolve towards the mixed damping one.\nIn Appendix C we give the initial conditions in the tight neutrino-dark matter coupling\nlimit. Obviously, these expressions are only applicable to the smallest modes, which enter\nthe horizon earliest. The largest modes, on the other hand, enter the horizon when neutrinos\nhave completely decoupled and hence are described by the default initial conditions. The\nratio \u0000 \u0017\u0000DM=Hfor each mode prior at horizon entry determines which case should one follow.\nNamely, if \u0000 \u0017\u0000DM=H\u001c1, we proceed with the default initial conditions. In the opposite\ncase, we start the integration su\u000eciently early such that \u0000 \u0017\u0000DM=H\u001d1 and use instead\nthe tightly coupled initial conditions. Nevertheless, by comparing the CMB and the matter\npower spectra obtained for the default initial conditions to those merging from our devoted\nanalyses, we \fnd that the di\u000berences in the CMB spectra are below the \u0016K2level and that\nthe relative di\u000berence in the matter power spectrum reaches 10% only in that range where\nthis observable is already suppressed by several orders of magnitude. Hence the constraints\nand conclusions obtained in previous works on the phenomenology of neutrino-dark matter\nscattering (c.f. Refs [13, 19, 33]) are stable against the initial conditions.\n4 The physics of mixed damping\nHaving discussed all the relevant technical aspects required to obtain accurate predictions for\nthe evolution of perturbations in the mixed damping scenario, we now turn to the evolution\nof individual modes and discuss how these are a\u000bected by mixed damping. We start with a\nbrief discussion of the evolution of perturbations in the canonical \u0003CDM case and compare\nhow the decoupling history of the baryon photon plasma di\u000bers from neutrino-dark matter\n{ 8 {decoupling in the mixed damping regime. These di\u000berences are important for the derivation\nof an analytical approximation to the evolution of dark matter perturbations presented in the\nfollowing. Finally, we compare our analytical results to the full numerical results to elucidate\nthe physical mechanism behind the mixed damping e\u000bect.\nThe scale factors at which dark matter and neutrinos decouple from the respective other\nspecies relate as\na\u0017;dec=\u0012aDM;dec\namax\u0013 1\nn\u0017DM+1\naDM;dec: (4.1)\nMixed damping requires that aDM;dec< a max. For a \fxed dark matter decoupling time,\nneutrinos decouple earlier the smaller the power law index n\u0017DMis, and the mixed damping\nregime lasts longer, see Fig. 1. Correspondingly, a larger range of modes is subject to mixed\ndamping only, making this regime more accessible. In addition, the numerical solution is\nmore stable for small values of n\u0017DM, because for a given decoupling time the scattering\nrates in Eqs. (3.3) and (3.2) are smaller when the numerical integration starts, and hence\nthe system of di\u000berential equations is less sti\u000b. We therefore focus the discussion here on the\nn\u0017DM= 0 case. The generalisation to other temperature dependencies of the scattering cross\nsection is however straightforward.\n4.1 Comparison of decoupling histories\nAmong all species, the evolution of dark matter perturbations in the \u0003CDM scenario is\nprobably the simplest: they are constant while the Hubble radius is smaller than the mode's\nsize and receive a boost upon horizon crossing which sets them in a growing mode. The\ngrowth is proportional to log awhile the universe is radiation dominated and proceeds as\n\u000eDM/aonce the universe has transitioned to matter domination [31, 32]. The evolution of\nfour modes in the radiation dominated epoch is indicated in Fig. 2 by pink, dashed lines. Upon\nhorizon crossing, neutrino perturbations receive the same boost from gravitational infall as\ndark matter. Being relativistic, neutrinos di\u000buse out of overdense regions and the competition\nbetween gravitational forces and the neutrino pressure leads to damped oscillations in \u000e\u0017that\ncan be clearly noticed in Fig. 2.\nPhotons and baryons are initially tightly coupled by Thomson scattering and oscillate\nas a single \ruid. Upon recombination, the phase of the oscillation becomes imprinted on\nthe CMB angular spectrum, causing the characteristic peak-though structure. Silk damping\nof the photon perturbations causes a decrease in height of the acoustic peaks at higher\nmultipoles. The end of the baryon drag epoch is delayed with respect to photon decoupling\n[31, 32], and photon di\u000busion damps baryon \ructuations due to Compton drag. At the end of\nthe Compton drag epoch, baryons fall into the dark matter potential wells, and \ructuations in\nthe baryons become imprinted on the matter power spectrum as baryon acoustic oscillations\n(BAO).\nWhile the introduction of neutrino-dark matter interactions could make the dynamics\nof this coupled sector to be equivalent to that of the baryon-photon plasma, an important\ndi\u000berence arises due to the very di\u000berent decoupling histories. The formation of neutral\nhydrogen at the epoch of recombination implies a steep decrease in the free electron fraction\nand hence in the photon baryon scattering rate. Still, decoupling of baryons and photons does\nnot occur simultaneously but very closely. The Planck collaboration obtained z\u0003= 1090\u00060:41\nfor the redshift of photon decoupling and zdrag= 1059:39\u00060:46 for the end of the baryon\ndrag epoch [37]. In contrast, the dark matter and the neutrino scattering rate evolve as\n{ 9 {power laws of the scale factor for the entire cosmological history relevant in this context, and\nthe di\u000berence between the respective decoupling times can be considerable. Indeed, this is\nexactly the characteristic shaping mixed damping - while neutrinos have already decoupled,\ndark matter is dragged along their free streaming evolution, and the growth of dark matter\nperturbations is inhibited.\n4.2 Analytical mixed damping evolution\nThe basic premises of mixed damping allow for some simplifying assumptions which we use\nhere to derive an analytical approximation for the evolution of dark matter perturbations.\nFirstly, mixed damping can only occur if dark matter decouples from neutrinos before matter\nradiation equality (c.f. Sec. 2). At these times, the metric potentials are dominated by\nradiation perturbations and are insensitive to alterations in the dark matter evolution to good\napproximation. Secondly, neutrinos evolve in the free-streaming regime and are coupled to\nall other components by gravitational interactions only. Hence, in the radiation dominated\nlimit, neutrino perturbations are not a\u000bected by the modi\fed dynamics of the dark matter\nsector. With these two assumptions, the dark matter evolution Eq. (3.2) can be reformulated\nin terms of an external source function S(k;\u001c)\n\u000eDM+\u00121\n\u001c+C\u0014\n\u001c3\u0013\n_\u000eDM=S(k;\u001c): (4.2)\nDuring radiation domination, the reduced Hubble rate is H=\u001c\u00001and forn\u0017DM= 0 we\ndecompose the scattering rate as R_\u0014\u0017DM=C\u0014\u001c\u00003, whereC\u0014is a constant. The source\nfunction depends on the metric potentials and on the neutrino velocity divergence as\nS(k;\u001c) = 3 \u001e+3\n\u001c_\u001e\u0000k2 +C\u0014\n\u001c3\u0010\n3_\u001e\u0000\u0012\u0017\u0011\n: (4.3)\nNeglecting the neutrino anisotropic stress implies \u001e= (see Eq. (3.7)), and the metric\nperturbations deep in the radiation dominated epoch are approximated by [38]\n\u001e(k;\u001c) = 3\u001ep(k) \nsin\u0000\nk\u001c=p\n3\u0001\n\u0000\u0000\nk\u001c=p\n3\u0001\ncos\u0000\nk\u001c=p\n3\u0001\n\u0000\nk\u001c=p\n3\u00013!\n; (4.4)\nwhere\u001epis the primordial magnitude of the \ructuation, i.e. \u001e(k;\u001c!0) =\u001ep(k). For the\nneutrino free-streaming evolution we take [36]\n\u0012\u0017(k;\u001c) =3k\n4(\u000e\u0017+ 4 )j\u001c=0j1(k\u001c) + 6kZ\u001c\n0d\u001c0_\u001ej1\u0002\nk\u0000\n\u001c\u0000\u001c0\u0001\u0003\n: (4.5)\nThe full solution to the time evolution of dark matter perturbations (4.2) is a linear com-\nbination of the homogeneous solutions and a particular solution, constructed from Green's\nmethod\n\u000e(k;\u001c) =C1+C2Ei\u0012C\u0014\n2\u001c2\u0013\n+Z\u001c\n0d\u001c0S\u0000\nk;\u001c0\u0001\u001c02\n2e\u0000C\u0014\n2\u001c0\u0014\nEi\u0012C\u0014\n2\u001c02\u0013\n\u0000Ei\u0012C\u0014\n2\u001c2\u0013\u0015\n:(4.6)\nFor small values of \u001c, when the mode is well outside the Hubble radius, \u000e(k;\u001c) is constant.\nHence the initial conditions dictate C2= 0 andC1=\u000e(k;0). We evaluate the integral\nnumerically for several modes at consecutive conformal times. The results are indicated as\n{ 10 {−8−6−4−20δ(k= 10h/Mpc)\nν\nDM\nDM, analytically\n−202δ(k= 20h/Mpc)\n−202δ(k= 40h/Mpc)\n10−210−1100101\nτ[Mpc]−3−2−10δ(k= 60h/Mpc)\nFigure 2 . Evolution of density perturbations for individual modes in the \u0003CDM case (dashed, pastel\nlines) and for a neutrino-dark matter interaction strength of u\u0017DM= 10\u00006andn\u0017DM= 0 (dark, solid\nlines). The analytic prediction for the dark matter evolution obtained from the integration of Eq. (3.2)\nis depicted by the black points.\n{ 11 {black crosses in Fig. 2. Notice that our \fnal results capture all relevant qualitative features,\neven if our approximations are not perfect4. Because our main intention in deriving the\nanalytical solution is to shed light on the underlying physics, we keep the simple analytic\nforms given above with no attempt to improve the accuracy using e.g. \ftting formulas [31, 32].\n4.3 Evolution of density perturbations and the matter power spectrum\nThe matter power spectrum for several neutrino-dark matter interacting scenarios is shown in\nFig. 3. For the largest values of u\u0017DM, perturbations on small scales are subject to a mixture\nof collisional and mixed damping, while for larger wavenumbers and smaller scattering rates\nall power suppression with respect to \u0003CDM is caused by mixed damping only. We indicate\nthe transition between these regimes in Fig. 3 by coloured arrows. In general, the matter\npower spectra show a common set of features, namely:\n\u000fAt the largest scales, or correspondingly for the smallest wavenumbers, the matter\npower spectrum does not di\u000ber from the \u0003CDM case.\n\u000fContinuing to smaller wavenumbers, a steep decrease in power is followed by a bump\nafter which the matter power spectrum continues to decrease again.\n\u000fA small plateau is encountered in the subsequent damping tail, after which the pertur-\nbations continue to decrease further.\nThe \frst of these features, i.e. the \u0003CDM-like behaviour on large scales, can be easily\nunderstood. These scales enter the Hubble radius after dark matter has decoupled from the\nneutrinos and hence are not a\u000bected by the interaction. An estimate of the scale which enters\nthe Hubble radius when \u0000 DM\u0000\u0017=Hyields\nkDM;dec= 2\u0019H0p\n\nr'6:67\u0002103\npu\u0017DMh=Mpc; (4.7)\nforn\u0017DM= 0, reproducing the onset of the damping rather well. In terms of the analytic\nsolution of Eq. (4.6), it is reassuring to note that it recovers the \u0003CDM behaviour in the\nlimitC\u0014\u001c\u001c2. In this limit the exponential integral can be expanded as\nEi\u0012C\u0014\n2\u001c2\u0013\n=\rE+ ln (C\u0014=2)\u00002 ln (\u001c) +O\u0012C\u0014\n2\u001c2\u0013\n; (4.8)\nand the \frst two terms can be absorbed in the de\fnition of C1. The source function S(k;\u001c),\non the other hand, reduces to the \u0003CDM case if the last term of Eq. (4.3) can be neglected,\nsuch that one recovers the \u0003CDM result (see e.g. Ref. [38]).\nWe show several examples for the time evolution of neutrino and dark matter density\nperturbations at smaller wavenumbers for u\u0017DM= 10\u00006in Fig. 2 and compare them to the\n\u0003CDM evolution. The largest of the modes lies at a position in the matter power spectrum\nin which there is already a suppression of power but before the \frst bump. As previously\ndiscussed, gravitational infall upon horizon crossing causes a bump in both the neutrino and\ndark matter perturbations, and after this initial kick, the neutrino perturbation undergoes\n4While the analytic form for the metric evolution follows the numerical results rather well, its \frst and\nsecond derivatives are less accurate. We also \fnd that our expression for the neutrino velocity gets damped\nslightly slower than in the numerical case.\n{ 12 {10−1100101102\nwavenumber k[h/Mpc]10−910−610−3100103P(k)/bracketleftbig\n(Mpc/h)3/bracketrightbig\nΛCDM\nuνDM= 10−6\nuνDM= 10−5\nuνDM= 10−4\nuνDM= 10−3Figure 3 . The matter power spectrum in presence of dark matter-neutrino interactions. Depending\non the size of a mode and the particular value of u\u0017DM, the suppression with respect to \u0003CDM is\ncaused by mixed damping only or a by combination of collisional and mixed damping. Arrows in the\nsame colours as the graphs indicate the wavenumbers for which mixed damping is the sole mechanism\nresponsible for the suppression of power. For the smallest scattering rates shown here these bounds\nlie outside the plotted range.\ndamped oscillations around the zero point. After horizon crossing in the radiation dominated\nera the gravitational potentials decay, hence the source function (4.3) becomes zero at late\ntimes and the dark matter evolution at \u001c >\u001c latecan be approximated as\n\u000e(k;\u001c) =\u000e(k;\u001clate) + 2\u0012Z\u001clate\n0d\u001c0S(k;\u001c0)\u001c0\n2e\u0000C\u0014\n2\u001c02\u0013\nln (\u001c); (4.9)\nwhere we have used the decoupled limit, Eq. (4.8). The result indicates that, after dark\nmatter decoupling in the radiation dominated era, the perturbation still grows /lna. The\ngrowth at late times is clearly noticeable in the top panel of Fig. 2 for both the analytical\nand the numerical result. However, the proportionality constant in front of the logarithm is\nmodi\fed by the presence of the neutrino velocity in the source function, and also the value\nof\u000e(k;\u001clate) is altered in comparison to the \u0003CDM result.\nIn the \u0003CDM case, \u000e(k;\u001clate) and the integral in front of the logarithmic term in Eq. (4.9)\nhave always negative values, therefore the dark matter density perturbations grow in the\nnegative direction. This is evident for all modes in Fig. 3. The neutrino velocity modi\fes\nthis behaviour, and its e\u000bect on the source function is larger as kincreases. Figure 4 compares\nthe source function for \u0003CDM to that of an interacting model with u\u0017DM= 10\u00006for three\nvalues ofkillustrated previously in Fig. 2 and clearly con\frms this relation. The source\nfunction is modi\fed more severely for larger values of kdue to the larger value of \u0012\u0017, see\nEq. (4.5). Notice from Fig. 4 that a very steep decrease in the source function at early times\nappears in interacting scenarios. However, this gets largely suppressed by the smallness of\nthe exponential function in the integral of Eq. (4.6). Instead, the relevant feature is the\ndevelopment of a positive peak which grows with kand can make the value of \u000e(k\u001clate)\npositive. This is precisely what occurs for the second mode in Fig. 2, resulting from the fact\nthat the dark matter \ruid follows the neutrinos for long enough.\n{ 13 {10−1100\nτ[Mpc]−1000100S(k,τ)/bracketleftbig\nMpc−2/bracketrightbig\nk= 10h/Mpc\nk= 20h/Mpc\nk= 40h/MpcFigure 4 . External source function driving the evolution of dark matter perturbations at early times\naccording to Eqs. (4.2) and (4.3). The two cases shown are \u0003CDM (pastel, dashed lines) and an\ninteracting scenario with u\u0017DM= 10\u00006(dark, solid lines).\nFinally, for the smallest mode of Fig. 2, the onset of logarithmic growth is delayed\nbeyond the depicted time interval. Well after horizon crossing the gravitational potentials\nhave decayed and oscillations in the neutrino density average to zero in the integral over\nthe source function, i.e. the perturbation remains constant. An o\u000bset between \u000e\u0017and\u000eDM\ncan be noticed in Fig. 2 for the smallest mode both in the analytical and the numerical\nevolution. This, however, does not indicate a decoupling between the two species but the\nfact that interactions are e\u000bective at the level of velocity perturbations, in such a way that\n\u0012DMclosely tracks \u0012\u0017.\n5 Conclusions\nNon-standard dark matter scenarios, which predict a lower number of structures at small-\nscales with respect to \u0003CDM, have become popular in the last decade, as they have the\npotential to alleviate the various problems that \u0003CDM may face. This includes interacting\ndark matter scenarios with massless or very light Standard Model particles, such as photons\nor neutrinos. In these scenarios, dark matter \ructuations are usually erased by collisional\ndamping.\nHowever, there exists an additional (and equally relevant) damping regime, dubbed\nmixed damping . Mixed damping is of particular relevance for dark mater-neutrino interaction\nscenarios. It takes place when dark matter remains coupled to neutrinos after they started\nto free-stream. Extended Boltzmann codes, which describe interactions between dark matter\nand neutrinos, automatically account for the mixed damping e\u000bect. Nevertheless, the physical\nmechanism which leads to the suppression of structures in the mixed damping regime is\ndistinct from the one operating in the collisional damping case. Our exploration of the\nparameter space shows that current constraints already exclude the parameter range for\nwhich only collisional damping is possible.\nFuture cosmological observations will be sensitive to very small scales and therefore will\ntest even smaller dark mater-neutrino interaction rates. If these observations reveal that\n{ 14 {the matter power spectrum departs from \u0003CDM due to neutrino-dark matter interactions,\nmixed damping is very important for the suppression of structure. In particular, larger\nmodes, where the suppression sets in, are a\u000bected by mixed damping only, and this is where\ndiscrepancies are expected to be observed \frst. In contrast, smaller modes are subject to\na mixture of collisional and mixed damping. Here, we provide a comprehensive study of\nthe mixed damping e\u000bect. Comparing our analytic approximation for the evolution of dark\nmatter density perturbations to the full numerical results, we can explain all features which\nappear in the damped matter power spectrum. Thus we are able to provide a clear physical\npicture of the mixed damping e\u000bect.\nBeyond that, we also re-examine the numerical treatment of neutrino-dark matter inter-\nactions in the linear regime carefully. Our results reveal several discrepancies with previous\nstudies. Higher-order terms in the Boltzmann hierarchy are not entirely independent from the\nspeci\fc form of the matrix element for the neutrino-dark matter interactions. We use a sim-\npli\fed model approach to characterise all possibilities and \fnd that the angular dependence\nof the matrix elements does not in\ruence the \fnal spectrum of cosmological perturbations\non a detectable level. We also revise the ultra-relativistic \ruid approximation (UFA) in the\npresence of dark matter neutrino interactions and point out that it can only be consistently\napplied after neutrinos have decoupled from dark matter. If the UFA is not considered care-\nfully, it causes unphysical artefacts at small scales in the matter power spectrum. Finally, we\npoint out that the initial conditions for the numerical integration have to be revised in the\npresence of neutrino-dark matter interactions. We derive the appropriate expressions and test\ntheir e\u000bect on the CMB and matter power spectra. We \fnd that neither the modi\fcations\nto the UFA approach, nor the revised treatment of the initial conditions a\u000bects the theory\npredictions at a signi\fcant level. Hence, previous cosmological constraints on the strength of\nneutrino-dark matter interactions remain robust.\nLast but not least, the interest and the reach of our results are not limited to the mixed\ndamping regime. Rather, the correct description of the higher-order multipole coe\u000ecients,\ntogether with a suitable treatment of the UFA regime, are basic and indispensable pieces\nto analyse dark matter interactions with any light or massless degrees of freedom. The\ncalculations carried out in this paper should be of broad interest for a large number of non-\nstandard cosmological perturbation theory scenarios.\nAcknowledgments\nWe thank Andr\u0013 es Olivares-Del Campo for useful discussion and for sharing the matrix el-\nements for neutrino-dark matter scattering in the simpli\fed model approach. JS receives\nfunding/support from the European Union's Horizon 2020 research and innovation pro-\ngramme under the Marie Sklodowska-Curie grant agreement No 674896.OM is supported by\nthe Spanish MINECO Grants FPA2017-85985-P and SEV-2014-0398. JS thanks IFIC and\nthe Fermilab Theoretical Department for hospitality. This work is supported by the European\nUnion's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie\ngrant agreements No. 690575 and 674896.This research was supported in part by Perimeter\nInstitute for Theoretical Physics. Research at Perimeter Instituteis supported by the Govern-\nment of Canada through the Department of Innovation, Science, and EconomicDevelopment,\nand by the Province of Ontario through the Ministry of Research and Innovation.\n{ 15 {References\n[1] J. S. Bullock and M. Boylan-Kolchin, Small-Scale Challenges to the \u0003CDM Paradigm ,Ann.\nRev. Astron. Astrophys. 55(2017) 343 [ 1707.04256 ].\n[2] C. Boehm, P. Fayet and R. 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D83 (2011) 103521 [ 1012.0569 ].\n{ 17 {scenario0\n@1\n\u0011\u0017\u0011DMX\nspinsjM\u0017DMj21\nAs=m2\nDM+ 2mDMp1\nt= 2p2\n1(\u0016\u00001)\np1\u001cmM; mDMscattering rate _ \u0014\u0017DMhigher order\ncoe\u000ecients\n\u000bl; l\u00152\ncomplex DM\nDirac mediator NRg4m4\nDM\u00002sm2\nDM+s2+ts\n(t+s\u0000mN)22g4p2\n1m2\nDM(\u0016+ 1)\n(m2\nN\u0000m2\nDM)2155\n588anDM\u0019g4\n(m2\nN\u0000m2\nDM)2T2\n\u0017\u000bl=3\n2\nreal DM\nDirac mediator NRg4\u0000\n2s+t\u00002m2\nDM\u00012\u0002\nm4\nDM\u00002smDM+s(s+t)\u0003\n\u0000\ns\u0000m2\nN\u00012\u0000\ns+t+m2\nN\u00002m2\nDM\u0001232g4m4\nDMp4\n1(1 +\u0016)\u0000\nm2\nN\u0000m2\nDM\u00014508\n21g4\u00193m2\nDManDM\u0000\nm2\nN\u0000m2\nDM\u00014T4\n\u0017\u000bl=3\n2\ncomplex DM\nMajorana mediator\nNLg4m4\nDM\u00002sm2\nDM+s2+ts\n(t+s\u0000mN)22g4p2\n1m2\nDM(\u0016+ 1)\n(m2\nN\u0000m2\nDM)2155\n588anDM\u0019g4\n(m2\nN\u0000m2\nDM)2T2\n\u0017\u000bl=3\n2\nreal DM\nMajorana mediator\nNLg4\u0000\n2s+t\u00002m2\nDM\u00012\u0002\nm4\nDM\u00002smDM+s(s+t)\u0003\n\u0000\ns\u0000m2\nN\u00012\u0000\ns+t+m2\nN\u00002m2\nDM\u0001232g4m4\nDMp4\n1(1 +\u0016)\u0000\nm2\nN\u0000m2\nDM\u00014508\n21g4\u00193m2\nDManDM\u0000\nm2\nN\u0000m2\nDM\u00014T4\n\u0017\u000bl=3\n2\nDirac DM\nscalar mediator \u001eg4\u0000\ns+t\u0000m2\nDM\u00012\n2\u0010\ns+t+m2\n\u001e\u00002m2\nDM\u00112g4m2\nDMp2\n1\u0010\nm2\nDM\u0000m2\n\u001e\u00112155\n392\u0019g4anDM\u0010\nm2\n\u001e\u0000m2\nDM\u00112T2\n\u0017\u000bl= 1\nMajorana DM\nscalar mediator \u001eg4\n22\n64\u0000\ns\u0000m2\nDM\u00012\n\u0010\ns\u0000m2\n\u001e\u00112+\u0000\ns+t\u0000m2\nDM\u00012\n\u0010\ns+t+m2\n\u001e\u00002m2\nDM\u001123\n754g4m2\nDMp2\n1\u0010\nm2\n\u001e\u0000m2\nDM\u00112155\n196\u0019g4anDM\u0010\nm2\n\u001e\u0000m2\nDM\u00112T2\n\u0017\u000bl= 1\nvector DM\nfermionic mediator\nNL- - -16g4m2\nDMp2\n1(3\u0000\u0016)\n3\u0000\nm2\nDM\u0000m2\nN\u000121240\n441\u0019g4anDM\u0000\nm2\nDM\u0000m2\nN\u00012T2\n\u0017\u000bl=9\n8\nscalar DM\nvector mediator Z\u0016g44\u0002\nm2\nDM\u00002m2\nDMs+s(s+t)\u0003\n\u0000\nm2\nZ\u0000t\u000128g4mDMp2\n1(\u0016+ 1)\nm4\nZ3\n2310\n441\u0019g4anDM\nm4\nZT2\n\u0017\u000bl=3\n2\nDirac DM\nvector mediator Z\u0016g44\u0000\nm2\nDM\u0000s\u00012+ 4st+ 2t2\n\u0000\nm2\nZ\u0000t\u000128g4mdm2p2\n1(1 +\u0016)\nm4\nZ155\n147g4\u0019an DM\nm4\nZT2\n\u0017 \u000bl=3\n2\nMajorana DM\nvector mediator Z\u0016g42m4\nDM+ 2s2+ 2st+t2\u00004m2\nDM(s+t)\n(t\u0000m2z)22g4mdm2p2\n1(3\u0000\u0016)\nm4\nZ155\n441\u0019g4anDM\nm4zT2\n\u0017 \u000bl=9\n8\nvector DM\nvector mediator Z\u0016- - - 48g4m2\nDMp2\n1(1 +\u0016)\nm4\nZ310\n49\u0019g4anDM\nm4\nZT2\n\u0017 \u000bl=9\n8\nTable 1 . Possible scenarios for dark matter interactions and the corresponding matrix element, scattering cross section and coe\u000ecients for the\nhigher multipoles in the Boltzmann equations.\n{ 18 {A Interaction formalism\nWe present in Tab. 1 the values of _ \u0014\u0017DMand\u000blwhich allow for a proper calculation of high-\norder multipoles in the neutrino-dark matter interacting sector. We exploit the classi\fcation\nof dark matter-neutrino interaction scenarios provided in Ref. [20].\nB Evaluation of the e\u000bect of the ultra-relativistic \ruid approximation\nWith the ultra-relativistic \ruid approximation the Boltzmann hierarchy for neutrino pertur-\nbations (Eqs. (3.3)) is cut after the anisotropic shear once perturbations are well inside the\nhorizon, i.e. when k\u001c > (k\u001c)UFA. The UFA truncation scheme in general di\u000bers from the or-\ndinary truncation scheme for Boltzmann equations proposed in Ref. [34]. Its implementation\nin CLASS reads [36]\n_\u001bur=\u00003\n\u001c\u001bur+2\n3\u0010\n\u0012ur\u00006_\u001e\u0011\n: (B.1)\nwhere the subscript \\ur\" refers to any ultra-relativistic species. Previous studies dealing with\ndark matter-neutrino interactions have tried to generalise the expression above as\n_\u001bur=\u00003\n\u001c\u001bur+2\n3\u0010\n\u0012ur\u00006_\u001e\u0011\n\u0000_\u0014\u0017DM\u001bur: (B.2)\nHowever, we could not \fnd a consistent derivation for the latter expression and adopt an\nalternative approach, in which we delay the onset of the UFA until neutrinos have decou-\npled. In the mixed damping scenario neutrinos have to decouple before matter radiation\nequality (note that amax< a eq). The scale factor at decoupling is de\fned by the condition\n\u0000\u0017\u0000DM(a\u0017;dec)\u0011H(a\u0017;dec) and approximately given by\nan\u0017DM+1\n\u0017;dec=3MP2\n8\u0019\nDMp\nru\u0017DM\u001bThH0\n100 GeV= 1:19\u000210\u00002\u0002u\u0017DM;0\u0002\u0012\nDMh2\n0:1186\u0013\n: (B.3)\nCorrespondingly, the ultra-relativistic \ruid approximation should not be employed before\n\u001cUFA=a\u0017;dec\nH0p\nr=8\n>>>><\n>>>>:5:53\u0002103Mpc\u0002u\u0017DM\u0002\u0010\n\nDM\n0:1186\u0011\nifn\u0017DM= 0\n10:6\u0002104Mpc\u0002u\u0017DM1\n3\u0002\u0010\n\nDM\n0:1186\u00111\n3ifn\u0017DM= 2\n19:2\u0002104Mpc\u0002u\u0017DM1\n5\u0002\u0010\n\nDM\n0:1186\u00111\n5ifn\u0017DM= 4; (B.4)\nThe trigger value ( k\u001c)UFAfor the onset of the UFA is determined by the wavenumber kmax\nof the smallest mode we are interested in, and we require\nkmax\u001cUFA\u0015(k\u001c)UFA: (B.5)\nBefore the onset of the UFA, the Boltzmann hierarchy is cut at some large multipole lmax.\nTo avoid unphysical re\rections in this regime we make sure to choose lmax\u0015(k\u001c)UFA.\nWe consider two benchmark scenarios to investigate the impact of the UFA approach\non the CMB temperature auto-correlation, E-mode polarisation auto-correlation and the\ntemperature-E-mode cross-correlation spectra. Namely, the parameters we consider are\nn\u0017DM= 0 andu\u0017DM= 4:5\u000210\u00005(upper limit from Ref. [19]) and n\u0017DM= 2 andu\u0017DM;0=\n5:4\u000210\u000014(upper limit derived from Ref. [33]). The remaining six \u0003CDM parameters are\n{ 19 {set to the mean values obtained in the Planck 2018 data release [39], and we consider three\nmassless neutrinos with identical interactions to dark matter. For each scenario, we delay the\nUFA regime by increasing the value of ( k\u001c)UFA, both varying and keeping \fxed the value of\nlmax. We \fnd that the e\u000bect of delaying the UFA regime on the CMB spectra is completely\nnegligible for both scenarios.\nHowever, the impact on the matter power spectrum is clearly noticeable, as we shall\nnow illustrate, focusing exclusively on the n\u0017DM= 0 case. Figure 5 shows that, up to\nthe \frst oscillation peak, the power spectra computed with CLASS UFA default settings,\n(Eq. (B.1), see dashed lines) and with a delayed UFA regime approach (solid lines) typically\nagree. However, at smaller scales, non-negligible discrepancies show up. We also compare the\nresults obtained with a truncation according to Eq. (B.1) and Eq. (B.2) for the default and\nthe delayed UFA settings. Note that Ref. [33] uses the truncation scheme of Eq. (B.2) but\nwith default CLASS parameters for the onset of the UFA. Either truncation scheme results\nin a very similar power spectrum if the UFA regime is delayed su\u000eciently. This behaviour\nis expected since, in this case, the scattering term in Eq. (B.2) is small and should have no\nimpact. On the other hand, increasing lmaxwhile not delaying the onset of the UFA method\nleaves the power spectrum unchanged. Most importantly, however, the modi\fed truncation\nscheme of Ref. [33] is not able to reproduce the result to which both codes converge for\na delayed UFA regime for default UFA parameters on the smallest scales. We therefore\nconclude that, to obtain precise predictions for the matter power spectrum on small scales, it\nis indeed necessary to delay the onset of the UFA regime. The more economical approach of\nincluding the interaction term to the truncation equation (B.2) can not reproduce the correct\nsmall scale behaviour.\nAs the power law of the dark matter-neutrino interaction cross section steepens, i.e.\nn\u0017DMin Eq. (1.4) increases, the duration of the mixed damping regime is shortened. This\ntrend is clearly visible in Fig. 1 (see also Eq. (4.1) and the discussion there). Hence, the\ntime interval during which default UFA setting would erroneously treat neutrinos as free\nstreaming particles tends to be shorter for larger values of n\u0017DM. Nevertheless, we advocate\nto conduct a careful convergence study, as the accuracy of the results will depend on the\nprecise combination of u\u0017DM;0andn\u0017DMconsidered.\nEven though the default UFA settings fail to correctly predict the matter power spec-\ntrum on small scales, we do not expect that these discrepancies a\u000bect current or future\nconstraints from observations of the power spectrum. Di\u000berences at this level will likely be\nerased by non-linear e\u000bects during structure formation and, in addition, they are well below\nthe expected statistical and systematic uncertainties. To further illustrate this point, we use\nthe matter power spectrum with both the default and the delayed UFA settings to estimate\nthe number of Milky Way satellites following Ref. [40]5. Table 2 summarises the results\nin then\u0017DM= 0 case for di\u000berent values of the neutrino-dark matter interaction parameter\nu\u0017DM. Di\u000berences at this level are negligible compared to the expected Poisson scatter, sys-\ntematic uncertainties from baryonic feedback and/or the modelling of non-linear structure\nformation.\n5See also the works of Refs. [7, 12, 14] for devoted simulations within di\u000berent possible interacting dark\nmatter scenarios.\n{ 20 {10−1100101102\nwavenumber k[h/Mpc]10−1110−810−510−2101104P(k)/bracketleftbig\n(Mpc/h)3/bracketrightbignνDM= 0\nΛCDM\nuνDM= 10−3\nuνDM= 10−4\nuνDM= 10−5\nuνDM= 10−6\nuνDM= 10−7Figure 5 . The matter power spectrum computed with the default CLASS UFA settings (dashed lines)\nand with a delayed onset of the UF approximation ( k\u001c=lmax= 200, solid lines). Non-negligible\ndi\u000berences appear beyond the \frst oscillation peak.\nscenario default UFA settings delayed UFA regime\n\u0003CDM 160 160\nu\u0017DM= 1\u000210\u0000730.8 30.7\nu\u0017DM= 5\u000210\u000078.5 8.5\nu\u0017DM= 1\u000210\u000065.8 5.9\nu\u0017DM= 5\u000210\u000060.72 0.74\nu\u0017DM= 5\u000210\u000050.05 0.06\nTable 2 . Number of Milky Way satellite galaxies Nsatcomputed from the linear matter power spectra\nobtained with default UFA settings and with a delayed UFA regime ( k\u001c=lmax= 200). We consider\nthen\u0017DM= 0 case for di\u000berent dark matter-neutrino cross sections.\nC Initial conditions for coupled neutrinos\nThe initial conditions for cosmological perturbations in CLASS are given to second order in\nk\u001cand zeroth order in the inverse Thomson scattering rate [41]. These expressions need to\nbe modi\fed when neutrinos interact with dark matter, because the scattering suppresses the\nneutrino anistotropic stress. To derive the modi\fed expressions we follow the steps of [34]\nand obtain\n\u001eini= ini=4\n3Cini; (C.1a)\n\u0012\r;ini=\u000e\u0017;ini=4\n3\u000eb;ini=4\n3\u000eDM;ini=\u00008\n3Cini; ; (C.1b)\n\u0012\u0017;ini=\u0012\r;ini=\u0012DM;ini=\u0012b;ini=\u0000Cini\n18k4\u001c3; (C.1c)\n\u001b\r;ini=\u001b\u0017;ini= 0; (C.1d)\nwhere the subscript \\b\" refers to baryons and Ciniis an integration constant.\n{ 21 {10−810−710−6\nscale factor a0.00.20.40.6metric perturbationsΛCDM\ndefault initial conditions\ninteracting- νinitial conditionsuνDM= 10−5\nk= 20h/Mpc\nφ\nψFigure 6 . Comparison of the evolution of metric perturbation with the default and the revised\ninitial conditions. For the case of revised initial conditions the integration starts at an earlier time to\nensure that the approximation of neutrino-dark matter tight coupling is still valid.\nTo further illustrate how dark matter-neutrino scattering a\u000bects the initial conditions\nof the metric perturbations we show their evolution for a single mode in Fig. 6. The mode\nhas a wavenumber of k= 20h=Mpc and enters the horizon during radiation domination,\nclose to the time when neutrinos decouple. The numerical solution starts earlier if the initial\nconditions take into account neutrino interactions because Eq. (C.1) is only applicable in the\nlimit of tight coupling between dark matter and neutrinos. Notice from Fig. 6 that the two\ntreatments of the initial conditions converge to a common evolution quickly and therefore in\nour \fnal result we observe that the choice of initial conditions has a negligible impact.\n{ 22 {"    },    {        "title": "1701.05092v1.Two_types_of_spurious_damping_forces_potentially_modeled_in_numerical_seismic_nonlinear_response_history_analysis.pdf",        "content": "16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017 Paper N° 3187 Registration Code: S-I1464709076\t\r TWO TYPES OF SPURIOUS DAMPING FORCES POTENTIALLY MODELED IN NUMERICAL SEISMIC NONLINEAR RESPONSE HISTORY ANALYSIS  P. Jehel(1), P. Léger(2)  (1) Assistant Professor, Laboratoire MSSMat, CNRS-UMR8579, CentraleSupélec, pierre.jehel@centralesupelec.fr (2) Full Professor, Department of Civil, Geological and Mining Engineering, Polytechnique Montréal, pierre.leger@polymtl.ca  Abstract The purpose of this paper is to provide practitioners with further insight into spurious damping forces that can be generated in nonlinear seismic response history analyses (RHA). The term ‘spurious’ is used to refer to damping forces that are not present in an elastic system and appear as nonlinearities develop: such damping forces are not necessarily intended and appear as a result of modifications in the structural properties as it yields or damages due to the seismic action. In this paper, two types of spurious damping forces are characterized. Each type has often been treated separately in the literature, but each has been qualified as ‘spurious’, somehow blurring their differences. Consequently, in an effort to clarify the consequences of choosing a particular viscous damping model for nonlinear RHA, this paper shows that damping models that avoid spurious damping forces of one type do not necessarily avoid damping forces of the other type.  Keywords: Spurious damping forces; Rayleigh damping; Caughey damping; Wilson-Penzien damping; structural damage  16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n2\t\r 1. Introduction During severe ground motion caused by an earthquake, the seismic energy imparted to the structures has to be dissipated. A structure collapses if it does not have the capacity to dissipate it. Otherwise, it eventually gets back to rest usually in a damaged state. Damping can be defined as the overall phenomenon that causes energy dissipation in response history analysis (RHA) of a structure subjected to dynamic loading. Numerous mechanisms contribute to damping and need to be modeled to predict the level of structural damage in a post-earthquake state. This damage can then be communicated in terms of structural performance for post-disaster planning. In computational earthquake engineering, energy dissipation results on the one hand from modeling the nonlinear hysteretic response of the structural system. On the other hand, it results from additional damping that accounts for energy dissipative mechanisms not otherwise represented in the hysteretic model for the structural system. In seismic structural RHA, practitioners have widely been using Rayleigh damping as the model for additional damping. The consequences of this choice have been thoroughly investigated in the past and it is now well known that it can lead to unrealistic (spurious) damping forces. The equations of motions of a multi degrees-of-freedom (MDOF) structural system with viscous damping and DOFs without associated mass can be written as \n (1) where the mass and damping matrices (M and C) along with the resisting and external forces vectors (f r and f e) and the displacements vector (u) have been split into parts pertaining to the P DOFs with mass and the S massless DOFs. An increase in the resisting forces at time t can be written as Δf r(t) = K(t)Δu(t) where K(t) is the tangent stiffness matrix. As the structure is elastic we have K(t) = K0. A dot over a quantity indicates derivative with respect to time. Damping forces f d = Cdu/dt are added to account for energy dissipation sources not otherwise already represented in the structural model. There are many reasons why damping forces are added in seismic nonlinear response history analysis. From a numerical point of view, damping models can be designed so that they damp out non-realistic phenomena coming from high frequencies generated by a small time step in the algorithm [1]. From a physical point of view, structural models rarely explicitly account for all potential energy dissipation mechanism: there are for instance non-structural elements that are generally absent in the model. Potentially, there are numerous energy dissipation mechanisms that can be activated at different scales during the seismic motion history, as illustrated in Fig. 1 for reinforced concrete (RC) buildings. This makes it challenging to develop reliable structural models with explicit representation of all these mechanisms. Accordingly, these numerous mechanisms are all considered together in hysteretic force-displacement laws at the structural element section level, or at the material level in fiber elements.  In elastic analysis, an efficient method for solving Eq. (1) consists in using the modal basis of the classically damped system yielding M=P+S equations of the form \n (2) that can be solved readily. Assuming classical damping – which means that the damping matrix is diagonal in the modal basis – modal analysis is performed with the undamped system because the modal basis of the undamped system coincides with those of the damped system [2]. Accordingly, the eigenvalues ωm2 and eigenvectors ϕm of the system are computed solving \n (3) In Eq. (2) and (3), we used the orthogonality properties of the modal basis and the following notations: \n (4) 16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n3\t\r \n Fig. 1 – Some of the numerous energy dissipation mechanisms potentially activated in RC buildings in seismic motion. The hysteresis in the force-displacement curves used in nonlinear structural engineering results from a series of nonlinear phenomena from material to structural scales. There are several options available to practitioners to build a viscous damping matrix C for seismic inelastic RHA. The consequences of using damping models initially developed for elastic systems in nonlinear RHA has been attracting the attention of researchers and practitioners for several decades (see [3] for a recent thorough review), and still is a topic of intense discussions (see the recent papers and discussion [3, 4, 5, 6]). The purpose of this paper is to provide practitioners with further insight into spurious damping forces that can be generated in nonlinear RHA. The term ‘spurious’ is used to refer to damping forces that are not present in an elastic system and develop as nonlinearities develop: such damping forces are not necessarily intended and appear as dependent on the structural history. In the following, two types of spurious damping forces are pointed out. Each type has often been treated separately in the literature, but each has been qualified as ‘spurious’, somehow blurring their differences. Then, in an effort to clarify the consequences of choosing a particular viscous damping model for nonlinear RHA, this paper shows that damping models that avoid spurious damping forces of one type do not necessarily avoid damping forces of the other type. Next section reviews popular damping models that have been proposed for seismic nonlinear response history analysis: Caughey and Rayleigh damping along with Wilson-Penzien damping. In Section 3, two different types of spurious damping forces are presented and some strategies available for avoiding them – or controlling them – are discussed. Applications in Section 4 illustrate how viscous damping models that avoid one type of spurious forces still have the potential to generate spurious forces of another type. Finally, the issue of whether spurious viscous damping forces should really be avoided is raised and some conclusions with future directions toward a rational modeling of damping in inelastic RHA are formulated. 2. Models of viscous damping forces Several damping models have been developed over the past decades for earthquake engineering. Rayleigh [7], Caughey [2, 8], and Wilson-Penzien [9] damping have been widely discussed in the literature. As Rayleigh damping is a particular Caughey series, we consider both together hereafter. All these three categories of damping models have been developed as nonlinear structural seismic analysis was very rare or 16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n4\t\r even inexistent. In the context of the elastic analysis of multi-DOF structures, the main motivation was deriving damping matrices that would satisfy the hypothesis of classical – or proportional – damping. 2.1. Caughey and Rayleigh damping The classical viscous damping model presented in Caughey’s work for elastic structures [2, 8] is based on the following damping matrix: \n (5) Any J significant structural eigenmodes, out of the total number of modes M, are considered in this relation. Stiffness matrix K is either defined as the initial stiffness matrix of the undamaged structure, or as the tangent stiffness. Besides, reduced forms of stiffness matrix can be used where for instance, as it will be discussed later, the components that pertain to massless DOFs can be set to zero. As J0 = 0 and J1 = 2 in Eq. (1), Rayleigh damping matrix is retrieved: \n (6) Viscous damping models of the Rayleigh category are pervasive in the Earthquake Engineering community. This category of damping models is rooted in the work of Lord Rayleigh [7] for elastic structural systems. It has been introduced as a mathematically convenient ad hoc procedure for representing damping.  Several kinds of Caughey or Rayleigh damping matrices have been implemented for nonlinear RHA depending on which stiffness matrix K is used (initial, tangent, reduced) and on whether the coefficients a0 and a1 are calculated once for all from initial structural properties or are updated throughout RHA. Theoretically, these damping coefficients can be computed for any J of the M structural eignemodes and at any time t in the structure history. 2.2. Wilson-Penzien damping Wilson and Penzien [9] have proposed the following damping matrix that yields classical damping in linear systems: \n (7) This damping matrix does not benefit from the diagonal pattern of mass and stiffness matrices with the finite element procedure and has therefore not been popular in practice for computational reasons. However, it has been shown in [4] that, in the context of nonlinear RHA, the computational demand of this model is comparable with those of tangent stiffness-based Rayleigh damping. Besides, it has been shown in [3] that this model can be efficiently used when the structural damping matrix is built from the assembly of elemental damping matrices. 3. Spurious damping forces The damping models presented above have been developed in the context of elastic structural analysis. As seismic nonlinear RHA emerged, the same damping models have been used in the simulations. However in nonlinear RHA, it has been observed on numerous occasions for several decades that these damping models can generate unintended spurious damping forces, in the sense that they can generate damping forces in the nonlinear regime that are not generated while the system remains elastic. 3.1. Spurious damping forces of type S As Caughey or Rayleigh viscous damping model is used, there is no damping forces generated at the S massless DOFs in elastic structures. However, as the structure yields, damping forces appear at these S DOFs, and such forces have been qualified as ‘spurious’ because they develop in consequence of the 16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n5\t\r yielding of structural elements, which seems anomalous [4]. This has first been pointed out in [10] and it has then been given a rational explanation in [11], which we briefly recall here. Starting from Eq. (2), damping forces read \n (8) And inertia forces read  \n (9) Also, from Eq. (3), the following relation holds \n (10) which, introduced in Eq. (5) and using Eqs. (8) and (9), yields \n (11) This means that modal damping and modal inertia forces have the same shape. Accordingly, because f im,s = 0 for all modes m at massless DOFs, there is also f dm,s = 0 and eventually no damping forces generated at any of the S massless DOFs. In the same paper [11], it is also shown that \n (12) which means that there is no spurious damping forces of type S for any damping matrix that satisfies Css(t) = 0 at any time t throughout the analysis. Keeping this in mind, and remembering that Mps = Msp = Mss = 0, it is clear that Caughey damping with J1 < 2 (see Eq. (5)) as well as Wilson-Penzien damping (see Eq. (7)) do not produce spurious forces of type S. According to Eq. (12), another straightforward approach for avoiding damping forces of type S consists in assigning zeros to the components of the damping matrix that pertain to a massless DOF, thus building a reduced stiffness matrix that satisfies the condition Css(t) = 0 at any time t throughout the analysis. This can be done either with static condensation (with zero loading at S massless DOFs) [11] or without [12]. Still, of course, if stiffness is null after yielding, using the tangent stiffness for building the damping matrix also guarantees there is no such spurious damping forces [12]. In [13] spurious damping forces of type S are avoided using elastic beam elements with zero-length semi-rigid rotational plastic hinges at their ends and assigning zero stiffness to the S massless DOFs when building Rayleigh damping matrix. 3.2. Spurious damping forces of type P The work presented in [14] has pointed out that another kind of spurious damping forces can appear in seismic nonlinear RHA as the natural frequencies of the structure drop in consequence of yielding or damage. This shift of the natural dynamic properties of the structure can result in a shift of the effective viscous damping ratios and consequently of the damping forces in the system. Following the assumption that viscous damping ratios should remain the same throughout inelastic RHA (as discussed in [4]), that is in the elastic range as well as after yielding or damaging, this shift of damping forces can be qualified as spurious. This second kind of spurious damping forces can affect both mass and massless DOFs depending on the damping model that is selected. We refer to them as spurious damping forces of type P. For both Caughey and Wilson-Penzien damping models, spurious damping forces of type P are generated in nonlinear RHA if the damping coefficients are set once for all, say at time t0, while the structural modal properties change as the structure becomes nonlinear. The time history of the effective viscous damping ratios has been analytically derived in [15] for Rayleigh damping. This has also been investigated in [16]. A basic assumption in the formula derived in [15] is that the off-diagonal terms in the 16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n6\t\r modal damping matrix can be neglected. Indeed, as the structure yields at time ty, the matrix ϕT(ty)C(t0)ϕ(ty) is not necessarily diagonal because C(t0) is built from modal properties that are not computed at the same time ty. We also use this assumption in what follows and therefore express effective modal viscous damping ratios time history from the relation: \n (13) We focus on Caughey series with negative powers and on Wilson-Penzien damping. It has been shown that those two models avoid spurious damping forces of type S [4, 11], but it is shown now that they do not necessarily avoid spurious damping forces of type P: 1) Introducing Eq. (5) with negative powers into Eq. (13), it comes (J ≥1) \n (14) Therefore, if the coefficients aj are calculated once for all, say at time t0, the modal viscous damping ratios change according to the evolution of the modal properties. 2) Introducing now Eq. (7) into Eq. (13), we have \n (15) Here again, it is clear that the modal damping ratios change with the modal properties as nonlinearity appears. For both models, as shown in Section 3.1, there is no damping forces generated at S DOFs. However, spurious damping forces are introduced at P DOFs if we adopt the assumption that modal viscous damping ratios should remain unchanged in the elastic and nonlinear regimes. With this assumption, any damping force that would appear in the nonlinear regime while absent in the elastic one is qualified as spurious. Solutions to avoid spurious damping forces of type P have been proposed in the past. In [14], Rayleigh damping is introduced in nonlinear RHA with coefficients updated at each time step, which leads to constant viscous damping ratios for the two modes that are considered for building Rayleigh damping model. In [17], a capped viscous damping model is presented based on the introduction of bounds for the effective viscous damping ratios throughout the nonlinear RHA. This notion of bounds has also been considered later in [15] with emphasis put on the consequences of using initial or tangent stiffness for Rayleigh damping: it is shown that there is no choice that is intrinsically better than the other but that it is easier to control these bounds when tangent stiffness is adopted. In [12], the author provides recommendations to manage the issues raised by spurious damping forces of both types P and S, and ultimately comes up with the conclusion that viscous damping model should ideally be eliminated and replaced by hysteretic laws that would represent the actual sources of the overall structural damping. Very recently, the approach that has been presented in [3] combines the benefits of Wilson-Penzien damping [4, 8] (control over a large range of modes) and updated damping coefficients [14] (control over the viscous damping ratios history), and solves the computational issue raised by the need to re-compute the modal basis at each time step. This approach is based on building a viscous damping matrix from the finite element assembly of elemental damping matrices (as for stiffness or mass matrices) instead of building a global (structural) damping matrix. This allows for using analytical expression for solving the eigenvalue problem at the element level. 4. Applications The applications below show that damping models that avoid spurious forces of type S do not necessarily avoid spurious damping forces of type P.   16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n7\t\r  4.1. Inelastic structure and yielding scenarios The structure used for this application is the same as the one introduced in [12], which we briefly present here. The structure is a five-story building modeled as a system of five DOFs – the horizontal displacements – connected by inelastic columns, which all have the same elastic properties. At each DOF the same mass is lumped. Two different yielding scenarios are considered: 1. The entire stiffness matrix is assumed to uniformly drop to 50% of its original value; 2. The structural elements non-uniformly yield along the building height: Nth-story stiffness is reduced to    10% + (N – 1) × 20% of its original value, that is 10% for the 1st story, 30% for the 2nd… 90% for the 5th. To investigate the history of the damping ratios while the structure is yielding, we arbitrarily set the total duration of the inelastic RHA to T = 1 s and divide the analysis into 5 steps (0, 0.2 s, . . . 1 s). At each time step the structure suddenly damages. Here is the damaging history for scenario 1 (uniform damaging): at t = 0 structural stiffness matrix K is equal to the initial stiffness matrix K0, then at t = 0.2 s it suddenly drops to K = 90% × K0, at t = 0.4 s to K = 80% × K0, and so on until t = 1 s where K = 50% × K0. The structure has 5 eigenmodes, with corresponding angular eigenfrequencies in the initial (undamaged) state: ω1(t0), …, ω5(t0) = 5.56, 16.23, 25.58, 32.87, 37.49 rad/s.  4.2. Caughey and Rayleigh damping The performance of initial and tangent stiffness-based Rayleigh damping with either frozen or updated coefficients, along with Rayleigh damping with reduced stiffness (with for instance zeros assigned to the components pertaining to the S DOFs) has been investigated in [15]. Here we focus on the Caughey damping model proposed in [11] to solve the problem of spurious damping forces of type S: \n (16) This model corresponds to Caughey series (Eq. (5)) with J0 = J1 = 0 (negative power) and is based on the initial stiffness matrix K0. Viscous damping ratios time histories are computed from Eq. (14) with  J = 2. The coefficients a0 and a1 are computed from the initial structural properties (t = t0) so that the viscous damping ratios for modes 1 and 3 are ξ1(t0) = ξ3(t0) = ξ0 = 2%; these coefficients are set once for all and kept frozen throughout the inelastic RHA. Angular eigenfrequencies ωm(t) and factors h m(t) are computed at every time step of the nonlinear RHA (t = 0.2 s, …, t = 1 s). The evolution of the effective modal viscous damping ratios ξm(t) is plotted in Fig. 2. In case of uniform stiffness degradation (top row in Fig. 2), effective viscous damping ratio for mode 1 increases constantly from the initially set ξ1 = 2% (at t = 0) to almost ξ1 = 6% at the end of the RHA at t = 1 s. In case of non-uniform stiffness degradation (bottom row in Fig. 2), this increase is even larger, with an effective viscous damping ratio for mode 1 getting larger than 12% at the end of the RHA. These large increases are due to the shift of the modal frequencies because of structural yielding. It is therefore obvious here that, although the damping model described in Eq. (16) does not generate spurious damping forces of type S as discussed earlier, it can generate spurious damping forces of type P. 4.3. Wilson-Penzien damping We now focus on Wilson-Penzien damping model. The damping matrix CWP (see Eq. (7)) is built from the initial structural properties (t = t0) with J = 5. The viscous damping ratios are set to ξj(t0) = ξ0 = 2% for all 5 modes j = 1, …, 5. The modal structural properties are computed at each of the 5 time steps considered in the inelastic RHA and the corresponding time history of the effective viscous damping ratios is computed using Eq. (15). The evolution of the effective modal viscous damping ratios is plotted in Fig. 3. In the case of uniform stiffness degradation (top row in Fig. 3), the damping ratios constantly and uniformly increase from ξ1(t=0) = 16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n8\t\r ξ0 = 2% to more than ξ(t=1s) = 2.8% as the structure yields. In the case of non-uniform stiffness degradation (bottom row in Fig. 3), the effective viscous damping ratio for mode 1 constantly increases from ξ1(t=0) = ξ0 = 2% to ξ1(t=1s) = 5.2%. As for Caughey or Rayleigh damping, it is obvious that, although the damping model described in Eq. (7) does not generate spurious damping forces of type S as discussed above, it can generate spurious damping forces of type P.   \n Fig. 2 – Viscous damping ratios time history for modes 1 (×), 2 (o), 3 (☐), 4 (★) and 5 (◊) with damping matrix as in Eq. (16) in case of uniform [top] and non-uniform [bottom] structural yielding scenarios. Damping coefficients a0 and a1 are identified once for all according to the initial elastic structural properties (at time t = 0) so that initial damping ratios ξ1 = ξ3 = ξ0 = 2%. Dashed lines represent the curve ξ(ω; t = 0). 5. Should we really avoid ‘spurious’ damping forces? As mentioned on several occasions in this paper, the term ‘spurious’ has been used to qualify damping forces that are triggered by the yielding of the structure and that are not present as the structure remains in its initial elastic domain. However, as the structure yields, it is fair to recognize that the numerous energy dissipative mechanisms that can be activated in yielding or damaging structural systems (see Fig. 1) are not all well understood, let alone well modeled. Consequently, additional damping forces generated in inelastic incursions could be seen as physical, for instance in the case where nonstructural elements – which are generally not modeled in the hysteretic structural response – participate to the overall damping in the structure. On another hand, there is no evidence that the energy dissipative mechanisms that generate overall damping in the elastic regime remain unchanged in the nonlinear regime, although this is a tacit assumption in nonlinear RHA [4]. Nevertheless, as far as spurious damping forces of type P are concerned, this is because of this assumption that damping forces are referred to as ‘spurious’ as an increase of the effective modal viscous damping ratios is observed.   16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n9\t\r \n Fig. 3 – Viscous damping ratios time history for modes 1 (×), 2 (o), 3 (), 4 (★) and 5 (◊) with damping matrix as in Eq. (7) in case of uniform [top] and non-uniform [bottom] structural yielding scenarios. Initial damping ratios (at time t = 0) are set to ξj = ξ0 = 2% for all modes j = 1,…, 5.  6. Conclusions In this paper, two types of ‘spurious’ damping forces have been identified in nonlinear seismic response history analysis (RHA). ‘Spurious’ damping forces are forces that arise – possibly unintended – during inclusions in the nonlinear regime and that are absent in the elastic domain. ‘Spurious’ damping forces of type S develop at massless DOFs as the very pervasive Rayleigh damping model is used, as well as for any other Caughey series. ‘Spurious’ damping forces of type P develop as the structural modal frequencies drop in consequence of yielding or damaging, whether Rayleigh or Caughey or Wilson-Penzien damping is used. According to the state of the practice in the field of seismic inelastic RHA, it is recommended to choose viscous damping models that avoid both types of spurious damping forces. However, as pointed out in Section 5, whether so-called spurious viscous damping forces should be avoided or not is not that straightforward because the fact is that the actual damping forces in nonlinear structural systems are not always well identified nor modeled. Nevertheless, it is of utter importance that what is actually modeled in nonlinear RHA be controlled and it is therefore worth gaining better knowledge on the damping forces that are actually generated when using this or that viscous damping model. This is what this paper intends to provide. Toward the purpose of designing reliable well-controlled damping models, next step to proceed to is, on the one hand, the quantification of the viscous damping forces that are actually modeled and, on the other hand, the identification of the damping forces that are actually needed. The concept of discrepancy forces introduced in [18] mingles structural simulation and experimental data, which allows quantifying the actual damping forces needed to satisfy equilibrium. From these discrepancy forces, damping models could be identified on a rational basis: this is what our ongoing investigations on this topic are oriented toward. 6. Acknowledgements This research has been partly supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme (proposal No. 275928). 16th World Conference on Earthquake Engineering, 16WCEE 2017 Santiago Chile, January 9th to 13th 2017   \n10\t\r 7. References [1] Filiatrault A, Léger P, Tinawi R (1994): On the computation of seismic energy in inelastic structures. Engineering Structures, 16 (6), 425-436. [2] Caughey TK (1960): Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics, 27 (2), 269-271. [3] Puthanpurayil AM, Lavan O, Carr AJ, Dhakal RP (2016): Elemental damping formulation: an alternative modelling of inherent damping in nonlinear dynamic analysis. Bulletin of Earthquake Engineering, 14 (8), 2405-2434. [4] Chopra AK, McKenna F (2016): Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation. Earthquake Engineering & Structural Dynamics, 45 (2), 193-211. [5] Hall JF (2016): Discussion of ‘Modelling viscous damping in nonlinear response history analysis of buildings for earthquake excitation’ by Anil K. Chopra and Frank McKenna. Earthquake Engineering & Structural Dynamics, 45 (13), 2229-2233. [6] Chopra AK, McKenna F (2016): Response to John Hall’s Discussion (EQE-16-0008) to Chopra and McKenna’s paper, ‘Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation’. Earthquake Engineering & Structural Dynamics, 45 (13), 2235-2238. [7] Strutt JW (Lord Rayleigh) (1894): The Theory of Sound (vol. 1). McMillan and Co., 2nd edition. [8] Caughey TK, O’Kelly MEJ (1965): Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics, 32 (3), 583-588. [9] Wilson EL, Penzien J (1972): Evaluation of orthogonal damping matrices. International Journal for Numerical Methods in Engineering, 14, 5-10. [10] Chrisp D (1980): Damping models for inelastic structures. Master’s Thesis, University of Canterbury, Christchurch, New Zealand. [11] Bernal D (1994): Viscous damping in inelastic structural response. Journal of Structural Engineering, 120 (4), 1240-1254. [12] Charney FA (2008): Unintended consequences of modeling damping in structures. Journal of Structural Engineering, 134 (4), 581-592. [13] Zareian F, Medina RA (2010): A practical method for proper modeling of structural damping in inelastic plane structural systems. Computers and Structures, 88, 45–53. [14] Léger P, Dussault S (1992): Seismic-energy dissipation in MDOF structures. Journal of Structural Engineering, 118 (6), 1251-1267. [15] Jehel P, Léger P, Ibrahimbegovic A (2014): Initial versus tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses. Earthquake Engineering & Structural Dynamics, 43 (3), 467-484. [16] Chang S-Y (2013): Nonlinear performance of classical damping. Earthquake Engineering and Engineering Vibration, 12, 279-296. [17] Hall JF (2006): Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake Engineering & Structural Dynamics, 35, 525-545. [18] Jehel P (2014): A critical look into Rayleigh damping forces for seismic performance assessment of inelastic structures. Engineering Structures, 78, 28-40.  "    },    {        "title": "1908.00384v1.The_Temperature_dependent_Damping_of_Propagating_Slow_Magnetoacoustic_Waves.pdf",        "content": "DRAFT VERSION AUGUST 2, 2019\nTypeset using L ATEXpreprint2 style in AASTeX62\nThe Temperature-dependent Damping of Propagating Slow Magnetoacoustic Waves\nS. K RISHNA PRASAD ,1D. B. J ESS,1, 2AND T. V ANDOORSSELAERE3\n1Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK\n2Department of Physics and Astronomy, California State University Northridge, Northridge, CA 91330, U.S.A.\n3Centre for mathematical Plasma Astrophysics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium\nABSTRACT\nThe rapid damping of slow magnetoacoustic waves in the solar corona has been extensively\nstudied in previous years. Most studies suggest that thermal conduction is a dominant con-\ntributor to this damping, albeit with a few exceptions. Employing extreme-ultraviolet (EUV)\nimaging data from SDO/AIA, we measure the damping lengths of propagating slow magne-\ntoacoustic waves observed in several fan-like loop structures using two independent methods.\nThe dependence of the damping length on temperature has been studied for the first time. The\nresults do not indicate any apparent decrease in damping length with temperature, which is\nin contrast to the existing viewpoint. Comparing with the corresponding theoretical values\ncalculated from damping due to thermal conduction, it is inferred that thermal conduction is\nsuppressed in hotter loops. An alternative interpretation that suggests thermal conduction is\nnot the dominant damping mechanism, even for short period waves in warm active region\nloops, is also presented.\nKeywords: magnetohydrodynamics (MHD) — methods: observational — Sun: atmosphere\n— Sun:corona — Sun: oscillations\n1.INTRODUCTION\nPropagating waves along fan-like active region\nloops have been a common observational feature\nsince their initial discovery in the solar corona\n(Ofman et al. 1997; Deforest and Gurman 1998;\nBerghmans and Clette 1999; De Moortel et al.\n2000). Recent multi-wavelength observations have\nestablished the origin of these waves in the pho-\ntosphere (Jess et al. 2012; Krishna Prasad et al.\n2015), from where they could be channelled by\nmagnetic fields into the corona (De Pontieu et al.\n2005; Erd ´elyi 2006; Khomenko et al. 2008). It\nCorresponding author: S. Krishna Prasad\nkrishna.prasad@qub.ac.ukis believed that these waves are a manifestation of\npropagating slow magnetoacoustic oscillations that\nare generated via mode conversion (Spruit 1991;\nCally et al. 1994) in the lower atmospheric lay-\ners. Their physical properties found in a variety\nof coronal structures have been extensively stud-\nied both from theoretical and observational van-\ntage points (De Moortel 2009; Wang 2011; Kr-\nishna Prasad et al. 2012b; Banerjee and Krishna\nPrasad 2016). In the solar corona, the slow mag-\nnetoacoustic waves undergo rapid damping and are\nconsequently visible only over a small fraction of\nthe loop length. A number of physical mechanisms\nsuch as thermal conduction, compressive viscos-\nity, optically thin radiation, gravitational stratifi-\ncation, divergence of the magnetic field, etc., arearXiv:1908.00384v1  [astro-ph.SR]  1 Aug 20192 K RISHNA PRASAD ET AL .\nknown to affect the amplitude of slow magnetoa-\ncoustic waves. Thermal conduction, however, has\nbeen put forward as the dominant contributor for\ntheir damping (De Moortel and Hood 2003, 2004).\nIt must be noted that slow magnetoacoustic waves\nalso exhibit frequency-dependent damping, with\nstronger dissipation at higher frequencies (Krishna\nPrasad et al. 2014, 2017) which, as well, is shown\nto be consistent with generalized damping via ther-\nmal conduction (Mandal et al. 2016).\nMarsh et al. (2011) studied slow magnetoa-\ncoustic waves propagating within a coronal loop\nusing stereoscopic images from STEREO/EUVI\n(Wuelser et al. 2004) and simultaneous spectro-\nscopic data from Hinode/EIS (Culhane et al. 2007).\nIt was found that thermal conduction was insuffi-\ncient to explain the observed damping, and instead\nmagnetic field divergence appeared to be the dom-\ninant factor. Marsh et al. (2011) explained that\nthe discrepancy was due to the relatively longer\noscillation periods ( \u001810minutes) and colder tem-\nperatures (\u00180:84MK) observed within the loop.\nFollowing a method developed by Van Doorsse-\nlaere et al. (2011), Wang et al. (2015) estimated\nthe polytropic index from the temperature and\ndensity perturbations corresponding to a stand-\ning slow magnetoacoustic wave observed in a hot\nflare loop. Based upon the value of the polytropic\nindex (\r= 1:64\u00060:08) they obtained, Wang\net al. (2015) inferred that thermal conduction is\nsuppressed and that the observed damping could\nbe explained by a slightly enhanced compressive\nviscosity term, which was later validated through\nmagnetohydrodynamic (MHD) simulations (Wang\net al. 2018). Krishna Prasad et al. (2018) investi-\ngated propagating slow magnetoacoustic waves in\na number of active region fan-like loops and found\na temperature dependency of the polytropic in-\ndices, whereby hotter loops corresponded to larger\npolytropic index values. However, the authors\nconcluded that the polytropic index could be, in\nfact, affected by a range of physical processes, in-\ncluding an unknown heating mechanism, radiativelosses, plasma flows, turbulence, etc., suggesting\nthat a direct association between the polytropic\nindex and thermal conduction cannot be unequivo-\ncally deduced. Indeed, Zavershinskii et al. (2019,\nprivate communication) found that an imbalance\nin the embedded plasma heating and cooling pro-\ncesses can actually cause temperature-dependent\nvariations in the polytropic index. Here, we study\nthe damping of short period ( \u00183 min) oscillations\nin quiescent active region fan-like loop structures,\nwhich was previously suggested to be the result of\nthermal conduction. The temperature dependency\nof the damping length is also investigated to find\nwhether there are signatures of thermal conduction\nbeing suppressed in hotter loops. Details on the\nobservational data used, the analysis methods em-\nployed, and the results obtained are presented in\nthe subsequent sections, followed by a discussion\nof the obtained results and their implications for\nfuture studies of the solar corona.\n2.OBSERV ATIONS\nExtreme-ultraviolet (EUV) imaging observations\nof solar coronal fan loops taken by the Atmo-\nspheric imaging assembly (AIA; Lemen et al.\n2012) on-board the Solar Dynamics Observa-\ntory (SDO; Pesnell et al. 2012) are utilized for\nthe present study. AIA captures the entire Sun\nin 10 different wavelength channels, from which\n6 are mainly dedicated to coronal observations.\nUsing online data browsing tools, such as He-\nlioviewer1, we selected 30 different active regions\n(ARs) with fan-like loop structures, where propa-\ngating oscillation signatures are clearly observed.\nThe observations of these ARs span from 2011\nto 2016, although a majority of them were taken\nbetween 2012 and 2014 (i.e., during the last so-\nlar maximum). For each active region, a one-\nhour-long image sequence, comprising of a small\nsubfield (\u001918000\u000218000) surrounding the desired\nloop structures, is extracted for all 6 dedicated\n1https://helioviewer.org/TEMPERATURE -DEPENDENT DAMPING OF SLOW WAVES 3\nSDO/AIA coronal channels (94 ˚A, 131 ˚A, 171 ˚A,\n193˚A, 211 ˚A, and 335 ˚A). The spatial sampling\nand the cadence of the data are 0.600per pixel and\n12 s, respectively. All of the data were processed\nusing the aiaprep routine, which is available\nwithin the Solar SoftWare (SSW) environment,\nto perform the roll angle and plate scale correc-\ntions required for subsequent scientific analysis.\nTo achieve accurate alignment between the data\nfrom multiple channels, and to successfully im-\nplement some of the above processing steps, we\nemployed the robust pipeline developed by Rob\nRutten2. This dataset was previously used in the\nstudy by Krishna Prasad et al. (2018), where com-\nplete observational details, including the locations,\nstart times, active region numbers, etc., of the indi-\nvidual image sequences are listed.\n3.ANALYSIS & RESULTS\nThe fan-like loop structures within a sample ac-\ntive region (NOAA AR 12553) from the selected\ndataset are displayed in Fig. 1a. Compressive os-\ncillations, with a periodicity of \u0019180 s, are found\npropagating outwards along these loop structures.\nIn order to identify the oscillations and understand\ntheir propagation behavior, a time-distance map\n(De Moortel et al. 2000) is constructed from one\nof the loop segments bounded by the two solid\nblue lines marked in Fig. 1a. The specific details\nof the method employed here are described in Kr-\nishna Prasad et al. (2012a), but in general, the in-\ntensities corresponding to the pixels across a se-\nlected loop segment are averaged to build a one-\ndimensional intensity profile along the loop, with\nsimilar profiles from successive images stacked to-\ngether to generate a time-distance map. The final\nmap obtained is shown in Fig. 1b, with the x-axis\ndisplaying time in minutes and the y-axis display-\ning distance along the loop in megameters (Mm).\nSlanted ridges of alternating brightness, visible in\nthis map, reveal the propagating waves along the\n2http://www.staff.science.uu.nl/ rutte101/rridl/sdolib/selected loop. Previous studies of such oscilla-\ntions, especially those propagating along similar\nfan-like loop structures that are usually rooted in\nsunspots, confirm their nature as propagating slow\nmagnetoacoustic waves (e.g., Kiddie et al. 2012;\nKrishna Prasad et al. 2012b). To enhance the vis-\nibility of the ridges, the time series at each spa-\ntial position was filtered to allow only a narrow\nband of frequencies around the dominant oscilla-\ntion period to remain. The filtered time-distance\nmap is displayed in Fig. 1c. It is clear that the am-\nplitude of the oscillations is not constant, but in-\nstead varies with time and, in particular, decreases\nwith distance along the loop from the correspond-\ning foot point. The temporal modulation of the\noscillation amplitude has been linked to the char-\nacteristics of the wave driver, with closely-spaced\nfrequencies causing a beat-like phenomenon (e.g.,\nKrishna Prasad et al. 2015), whereas the spatial\ndamping is mainly due to physical wave dissipa-\ntion and some geometrical factors. As discussed in\nSection 1, thermal conduction is believed to play a\nkey role in the observed spatial damping.\nWe identify a total of 35 loop structures from\nthe 30 active regions, where signatures of prop-\nagating slow magnetoacoustic waves are promi-\nnent. The prominence of oscillations is determined\nthrough a visual inspection of time-distance maps\nconstructed from multiple loop structures within\neach active region. It may be noted that these loop\nstructures are the same as those studied by Kr-\nishna Prasad et al. (2018), where the periodicity\nof the oscillations observed, the temperature and\ndensity of the plasma within the loop structures,\nthe polytropic index, among other parameters, are\ndiscussed. The temperature, in particular, was de-\nrived from the peak location in the corresponding\ndifferential emission measure (DEM) curve, which\nwas extracted by employing a regularized inver-\nsion method (Hannah and Kontar 2012) on the\nnear-simultaneously observed intensities across all\n6 SDO/AIA coronal channels. However, the main\nfocus in the present study is on the damping char-4 K RISHNA PRASAD ET AL .\nFigure 1. (a) A snapshot of the fan-like loop structures from NOAA AR 12553 captured in the SDO/AIA 171 ˚A\nchannel. The blue solid lines mark the boundaries of a chosen loop segment. (b) Time-distance map depicting\nthe evolution of the loop segment shown in (a). The alternating slanted bands of brightness apparent in this map\nindicate the presence of propagating compressive oscillations due to slow magnetoacoustic waves. (c) Same as (b),\nbut processed to enhance the visibility of the brightness ridges by filtering the time series at each spatial position to\nallow only a narrow band of frequencies around the dominant oscillation period. The white dashed line marks the\ntemporal location chosen to study the spatial damping characteristics shown in Fig. 2.\nacteristics of the oscillations. In order to study\nthe damping properties of slow magnetoacous-\ntic waves across the different loop structures se-\nlected, we estimate a characteristic damping length\nemploying two independent methods, namely, a\nphase tracking method and an amplitude tracking\nmethod, as described in the following sections.\n3.1. Phase tracking method\nA temporal location (marked by a white-dashed\nline in Fig. 1c), where the oscillation amplitude is\nrelatively strong, is initially chosen to investigate\nthe spatial variation of the oscillation phase. Note\nthat the selection of this location is purely based on\nthe strength of the oscillations as may be seen from\nFig. 1c. Same criterion is applied to all the other\nloop structures studied. The filtered intensities\nfrom three consecutive frames (i.e., \u000612s) around\nthe selected temporal location are averaged to im-\nprove the signal-to-noise, then normalized by the\ncorresponding background to construct a represen-\ntative spatial intensity profile such as that shown\nin Fig. 2a. The background is constructed from theintensities obtained by smoothing the original ob-\nserved values to remove any oscillations with pe-\nriodicities below 10 minutes. The spatial profile\nclearly demonstrates a rapid decrease in the oscil-\nlation amplitude with distance along the loop. The\nvertical bars indicate respective uncertainties in the\nimaging intensities that are estimated from noise\ncontributions in the SDO/AIA 171 ˚A channel (fol-\nlowing the methodology of Yuan and Nakariakov\n2012), which includes noise from various sources\nbesides the dominant photon and readout compo-\nnents (Jess et al. 2019). An exponentially decaying\nsine wave function of the form,\nI(x) =A0e\u0010\n\u0000x\nLd\u0011\nsin\u00122\u0019x\n\u0015+\u001e\u0013\n+B0+B1x;\n(1)\nis fitted to the spatial profile. Here, Iis the nor-\nmalized pixel intensity, xis the distance along\nthe loop,B0andB1are appropriate constants,\nandA0,Ld,\u0015and\u001eare the amplitude, damping\nlength, wavelength and phase of the oscillation,\nrespectively. Applying the Levenberg-Marquardt\nleast-squares minimization technique (MarkwardtTEMPERATURE -DEPENDENT DAMPING OF SLOW WAVES 5\n2009), the best fitment to the data is shown as a\nblack solid curve in Fig. 2a. The damping length of\nthe oscillation, as estimated from the fitted curve,\nis3:7\u00060:4Mm. The orange diamond symbols\nin Fig. 2b display the damping lengths obtained\nfrom all 35 selected loop structures, plotted as a\nfunction of the corresponding localized tempera-\nture on a log-log scale. The vertical bars denote\nthe respective uncertainties on damping length de-\nrived from the fit whereas the horizontal bars high-\nlight the associated uncertainties on temperature\npropagated from the respective errors given by the\nregularized inversion method (Hannah and Kontar\n2012). Since the temperature is determined from\na double-Gaussian fit to the individual DEMs (Kr-\nishna Prasad et al. 2018), the uncertainty on peak\nlocation is estimated by scaling the corresponding\nerror on the nearest point by a factor of 1=p\nN,\nwhere,Nis the number of data points involved\nin the fit. Subsequently, to get a representative\ntemperature value for each loop, a weighted mean\nacross all spatio-temporal locations near the foot\npoint is considered. The associated uncertainty\nis then estimated from the weighted standard de-\nviation of values across the same locations. It\nmay be noted that the uncertainties on loop tem-\nperature reported by Krishna Prasad et al. (2018)\nare fairly small compared to those shown here\n(Fig. 2b) which is because the authors did not in-\ncorporate the temperature errors given by the DEM\ninversion method but simply quoted the errors ob-\ntained from the Gaussian fit alone. The actual tem-\nperature values might also marginally differ be-\ncause of the weighted averages employed here.\nAnother important aspect to note here is that in\nabout 5 cases, the damping lengths are measured\nfrom pairs of loops from the same active region\nsome of which exhibit distinct values. The differ-\nences in values obtained in such cases reflect the\ndifferent physical conditions of the loop structures\ndespite belonging to the same active region.\n3.2. Amplitude tracking methodSince the phase tracking method involves man-\nually choosing a specific temporal location from\neach of the time-distance maps, it is possible that\nsuch human intervention naturally biases the ob-\ntained results. Also, it is not trivial to apply this\ntechnique to all temporal locations since the signal-\nto-noise at a large number of locations is low due\nto aspects of amplitude modulation. To circum-\nvent this problem and verify the reliability of our\nresults, we estimate the damping lengths using the\nalternative technique of amplitude tracking. In this\nmethod, the amplitude of the oscillation, A, at each\nspatial position is directly measured in relation to\nthe standard deviation, \u001b, of the respective filtered\ntime series using A=p\n2\u001b. This formula assumes\nthat the observed oscillations can be represented\nby a pure sinusoidal signal. The time-averaged\nintensities from the original time series (i.e., col-\nlapsing the time domain in Fig. 1b) are used as\nthe background for normalization to obtain relative\namplitudes as a function of distance along the loop.\nSince the amplitude at each spatial position is de-\nrived from the full time series, the median error\non respective pixel intensities is used to estimate\nthe corresponding uncertainty. The diamond sym-\nbols in Fig. 3a show the spatial dependence of am-\nplitude values thus obtained for the fan-like loop\nstructure highlighted in Fig. 1a. The vertical bars\nrepresent the associated uncertainties. These data\nwere then fit with a decaying exponential model\nsatisfying the functional form,\nA(x) =A0e\u0010\n\u0000x\nLd\u0011\n+C ; (2)\nwherexis the distance along the loop, A0andC\nare appropriate constants, and Ldis the damping\nlength.\nThe initial few locations where the wave ampli-\ntude is found to increase are ignored in order to\nisolate the purely decaying phase of the oscillation\nfor fitment. The black solid line in Fig. 3a repre-\nsents the best exponential fit obtained. The corre-\nsponding damping length is 4:8\u00061:5Mm, which is\non the same order as that obtained using the phase6 K RISHNA PRASAD ET AL .\nFigure 2. (a) Spatial variation of the relative intensity at the temporal location marked by the white dashed line in\nFig. 1c. The vertical bars denote the respective uncertainties. The solid curve represents the best fit to the data for an\nexponentially decaying sine wave model following Eq. 1. The obtained damping length value from the fitted curve\nis listed in the upper-right corner of the plot. (b) Damping lengths extracted from all of the selected loop structures,\nplotted as a function of the localized temperature on a log-log scale. The orange diamonds represent the values\nobtained following the phase tracking method shown in panel (a), whereas the green circles represent the theoretical\nvalues estimated from damping due to thermal conduction. The open and filled circles, respectively, highlight the\nvalues computed from a constant \r(= 5=3) and those computed using \rvalues extracted from observations (Krishna\nPrasad et al. 2018). The vertical and horizontal bars on the observed values denote the corresponding propagated\nuncertainties.\ntracking method outlined in Section 3.1. Following\nthe same procedure, the damping lengths for the\noscillations observed in all 35 selected loop struc-\ntures have been estimated. For a handful of loop\nstructures, it is found that the model does not con-\nverge properly, producing damping lengths either\nfar greater than the loop length itself ( >1000 Mm)\nor less than one pixel ( <0:1Mm). Upon in-\nspection of the time-distance maps corresponding\nto these individual cases, we found that there are\nunusual brightenings, perhaps in the form of tran-\nsient events manifesting in the loop background,\nat certain spatio-temporal locations, which natu-\nrally contaminate the amplitude extraction process\nand thereby prevent a robust fitment of the data.\nWhile this could be avoided by manually restrict-\ning the time series for each particular case, themain strength of this method was in the allevia-\ntion of human intervention. As such, we chose to\nignore specific loop structures where the model fit-\nting did not converge to commonly expected val-\nues. The orange diamond symbols in Fig. 3b rep-\nresent the damping lengths obtained from the re-\nmaining 31 cases, plotted as a function of the loop\ntemperature on a log-log scale. The vertical bars\nhighlight the corresponding uncertainties on damp-\ning length whereas the horizontal bars denote the\nrespective uncertainties on temperature. The loop\ntemperature and the associated uncertainties are es-\ntimated in the same way as that described in Sec-\ntion 3.1.\n3.3. Theoretical calculationsTEMPERATURE -DEPENDENT DAMPING OF SLOW WAVES 7\nFigure 3. (a) Relative amplitudes of the oscillations as a function of distance along the loop segment marked by the\nsolid blue lines in Fig. 1a. The vertical bars denote the respective uncertainties. The black solid line represents an\nexponential fit to the decaying phase of the data following Eq. 2. The obtained damping length value from the fitted\ncurve is listed in the plot. (b) Damping lengths extracted from all the selected loop structures plotted as a function\nof the local temperature on a log-log scale. The orange diamonds represent the observational values following the\namplitude tracking method shown in panel (a), whereas the green circles represent the theoretical values estimated\nfrom the damping due to thermal conduction. The open and filled circles, respectively, highlight the values computed\nfrom a constant \r(= 5=3) and those computed using \rvalues extracted from observations (Krishna Prasad et al.\n2018). The vertical and horizontal bars on the observed values denote the corresponding uncertainties.\nTheoretical and numerical calculations in the\npast have suggested that thermal conduction is the\ndominant physical mechanism responsible for the\ndamping of slow magnetoacoustic waves in the so-\nlar corona (De Moortel and Hood 2003; Klimchuk\net al. 2004). Considering one-dimensional linear\nwave theory for slow magnetoacoustic waves with\nthermal conduction as the damping mechanism\n(e.g., De Moortel and Hood 2003; Krishna Prasad\net al. 2012b), the dispersion relation between the\nwave number, k, and the angular frequency, !, can\nbe shown to be,\ndc4\nsk4+i!c2\nsk2\u0000\rd!2c2\nsk2\u0000i!3= 0:(3)\nHere,csis the sound speed, d=(\r\u00001)\u0014kT0\n\rc2sp0is the\nthermal conduction parameter, \u0014k=\u00140T5=2\n0is the\nparallel thermal conduction, and p0= 2n0kBT0is the gas pressure, where \u00140is the thermal con-\nduction coefficient, T0is the equilibrium tempera-\nture, andn0is the number density. For propagating\nwaves, the frequency, !, is constant and in the limit\nof weak thermal conduction (i.e., when d!\u001c1),\nthe solutions for wave number, k, may be found as\nk=!\ncs\u0000i1\nLd, whereLd=2cs\nd!2(\r\u00001)is the damping\nlength. The interested reader is referred to Mandal\net al. (2016) for a detailed derivation.\nAs described in Section 3, Krishna Prasad et al.\n(2018) studied the same set of loop structures\nthat are presented here. They applied a regular-\nized inversion method (Hannah and Kontar 2012)\non observed intensities in 6 coronal channels of\nSDO/AIA to compute corresponding DEM. Sub-\nsequently, by employing a double-Gaussian fit to\nthe DEM curve, the temperature and the density\nof the plasma are calculated from the peak loca-8 K RISHNA PRASAD ET AL .\ntion and the area under the curve, respectively.\nThe density is estimated by assuming the appar-\nent width of the loop as equivalent to the emission\ndepth along the line of sight effectively ignoring\nany background/foreground emission although a\ncontribution to the latter from hot plasma is care-\nfully discarded from the double-Gaussian fit. Us-\ning the theoretical relation between the relative\noscillation amplitudes in temperature and density\nassociated with a slow wave, the polytropic in-\ndex of the plasma is determined after eliminating\nthe corresponding phase shifts. The periodicity of\nthe oscillations is also calculated through a sim-\nple Fourier analysis on the intensity fluctuations.\nUtilizing the respective values of these parameters\ncomputed by Krishna Prasad et al. (2018) for each\nloop, we estimate the expected damping lengths\nfrom the above theory.\nThe filled green circle symbols shown in Figs. 2b\n& 3b represent the theoretically computed values.\nIt may be noted that the respective d!values were\nfound to reside in the interval 0:01\u00000:16, so\nthe assumption of weak thermal conduction (i.e.,\nd!\u001c1) is inherently valid across our range\nof coronal datasets. The classical Spitzer values\nfor thermal conduction, following \u0014k= 7:8\u0002\n10\u00007T5=2\n0ergs cm\u00001s\u00001K\u00001, are employed in our\ncalculations. As can be seen from the figures, the\ndamping lengths are expected to be considerably\nshorter for hotter loops. Since it is very common to\nassume the polytropic index, \r, is equal to 5=3in\nthe solar corona, we additionally compute damp-\ning lengths arising from a constant ( 5=3) value for\n\r. The open green circles shown in Figs. 2b &\n3b represent these values which suggest a simi-\nlar but much shallower dependence on tempera-\nture. Moreover, the damping lengths in this case\nare shorter by up to an order of magnitude or more\nwhich clearly divulges the effect of polytropic in-\ndex on the damping length. It is worth noting here\nthat the scatter in the theoretically computed damp-\ning lengths is mainly due to the different physical\nconditions of the loop structures studied.4.DISCUSSION & CONCLUDING REMARKS\nThe spatial damping characteristics of propagat-\ning slow magnetoacoustic waves, observed in 35\nfan-like loop structures selected from 30 different\nactive regions, have been studied. The damping\nlength, in particular, is measured using two inde-\npendent methods: a phase tracking method and an\namplitude tracking method. Employing the tem-\nperature information acquired from DEM analysis,\nthe temperature dependence of the damping length\nhas been investigated for the first time (Figs. 2b\n& 3b). These results do not indicate any appar-\nent decrease in damping length with temperature\nas would be expected by the stronger thermal con-\nduction in that case. It may be noted that the re-\nsults from previous studies (e.g., Krishna Prasad\net al. 2012b), who based their conclusions on the\nmeasurement of damping lengths for a single loop\nstructure observed in multiple temperature chan-\nnels, are inconsistent with the current findings.\nHowever, those studies are purely qualitative and\nthe results are often based on just two tempera-\nture channels. Furthermore, the sensitivities of the\nmeasured damping lengths from the intensity per-\nturbations to the filter/instrument used (e.g., Klim-\nchuk et al. 2004) are also not taken into consider-\nation in previous studies. In the present case, we\nemploy damping length measurements from mul-\ntiple loop structures observed in the same filter\n(SDO/AIA 171 ˚A). Hence, we naturally consider\nthe current results more reliable due to the conser-\nvation of instrument characteristics across all inde-\npendent measurements.\nUtilizing the temperatures, densities, polytropic\nindices, and oscillation periods that have previ-\nously been derived for the same set of loop struc-\ntures (Krishna Prasad et al. 2018), we calculated\nthe theoretical damping lengths expected from the\ndissipation due to thermal conduction. In contrast\nto the observations, the theoretical calculations\nshow a steep decrease in the damping length with\ntemperature. Damping lengths were also computed\nassuming a fixed value, 5=3, for the polytropic in-TEMPERATURE -DEPENDENT DAMPING OF SLOW WAVES 9\ndex, in line with the previous studies. These values\ndisplay a similar but shallower dependence. The\ndiscrepancy between the observational and theo-\nretical dependences perhaps indicates that thermal\nconduction is suppressed in hotter loop structures.\nIn fact, the increase in the polytropic indices of\nthese loops with temperature, as reported by Kr-\nishna Prasad et al. (2018), also implies the suppres-\nsion of thermal conduction in hotter loops (e.g.,\nin accordance with Wang et al. 2015), although\na direct conclusion could not be drawn from these\nresults alone since the polytropic index of the coro-\nnal plasma is dependent on several other physical\nprocesses besides thermal conduction. The current\nresults, on the other hand, appear to show direct ev-\nidence for the suppression of thermal conduction\nwith increasing temperature.\nAlternatively, one could argue that thermal con-\nduction is perhaps not the dominant damping\nmechanism for slow magnetoacoustic waves, as\npreviously reported by Marsh et al. (2011) and\nWang et al. (2015). Indeed, as can be seen from\nFigs. 2b & 3b, the theoretical damping lengths are\n2\u00003orders of magnitude higher than those ob-\ntained from the observations. One may also note\nthat a simple visual inspection of oscillation am-\nplitudes, in Figs. 1b & 1c for example, reveals sig-\nnificant damping within 10 Mm scales whereas the\nexpected damping lengths due to thermal conduc-\ntion are at least 100 Mm or more which clearly\ndemonstrates the extent of mismatch between the\nobservations and the theory. The differences in\nthe temperature dependence would further add to\nthis discrepancy. We note, however, that the in-\ncongruity between the theory and observations is\nless if we consider the calculations for \r= 5=3.\nAlso, the distances measured along the observed\nloop structures are projected onto the image plane,\nmeaning the obtained damping lengths are only\nlower limits. Nevertheless, the difference between\nthe theoretical and the observed scales is too large\nto ignore, and would not likely be accounted for\neven if a fractional contribution from the otherdamping mechanisms (e.g., compressive viscosity\nand optically thin radiation) is included.\nLastly, we would like to bring out some of the\nmajor caveats of our results. The temperature\nrange of the loops investigated is limited especially\nconsidering the large uncertainties on temperature.\nWhile the magnitude of change in the expected\ndamping lengths over the same temperature range\nand the extent of mismatch between the observed\nand theoretical values still make our results valid,\nit is imperative to state that a larger temperature\nrange would make the results more reliable. Ad-\nditionally, it should be noted that different DEM\ninversion methods can result in different peak tem-\nperatures although the difference can be marginal\ndepending on the temperature range investigated.\nAlso, it can be argued whether the peak emis-\nsion in a DEM sufficiently represents the plasma\nwithin the loop. Keeping these limitations in mind,\nwe believe further investigations, both theoretical\nand observational, are necessary to understand the\ndamping of slow magnetoacoustic waves in the so-\nlar corona. In particular, the impetus is on increas-\ning the temperature range studied to include hotter\nloop structures to examine whether these traits are\nconsistent across the full spectrum of coronal mag-\nnetism.\nCONFLICT OF INTEREST STATEMENT\nThe authors declare that the research was con-\nducted in the absence of any commercial or finan-\ncial relationships that could be construed as a po-\ntential conflict of interest.\nAUTHOR CONTRIBUTIONS\nSKP and DBJ planned and designed the study;\nSKP processed the data and performed the anal-\nysis; SKP wrote the first draft of the manuscript;\nTVD assisted in theoretical calculations; DBJ and\nTVD contributed to the interpretation of the re-\nsults; All authors took part in the manuscript re-\nvision, read and approved the submitted version.\nFUNDING10 K RISHNA PRASAD ET AL .\nThe authors gratefully acknowledge the financial\nsupport from the following research grants:\n\u000fSTFC — ST/K004220/1; ST/L002744/1;\nST/R000891/1\n\u000fRandox Laboratories Ltd. — 059RDEN-1\n\u000fEuropean Research Council (ERC) — 724326\n\u000fKU Leuven — GOA-2015-014\nACKNOWLEDGMENTS\nThe authors thank all the referees for their valu-\nable comments/suggestions. SKP would like to\nthank the UK Science and Technology Facilities\nCouncil (STFC) for support. DBJ gratefully ac-\nknowledges STFC for the award of an Ernest\nRutherford Fellowship, in addition to Invest NI\nand Randox Laboratories Ltd. for the award ofa Research & Development Grant (059RDEN-1).\nThis project has received funding from the Eu-\nropean Research Council (ERC) under the Euro-\npean Union’s Horizon 2020 research and innova-\ntion programme and KU Leuven. The SDO/AIA\nimaging data employed in this work are courtesy\nof NASA/SDO and the AIA science team. We ac-\nknowledge the use of pipeline developed by Rob\nRutten to extract, process, and co-align AIA cutout\ndata.\nDATA A V AILABILITY STATEMENT\nThe data analyzed in this study was obtained\nby the Atmospheric Imaging Assembly onboard\nNASA’s Solar Dynamics Observatory. 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Standing Slow-Mode Waves in Hot\nCoronal Loops: Observations, Modeling, and\nCoronal Seismology. SSRv 158, 397–419.\nhttps://doi.org/10.1007/s11214-010-9716-1\nWang, T., Ofman, L., Sun, X., Provornikova, E., and\nDavila, J. M. (2015). Evidence of Thermal\nConduction Suppression in a Solar Flaring Loop by\nCoronal Seismology of Slow-mode Waves. ApJL\n811, L13.\nhttps://doi.org/10.1088/2041-8205/811/1/L13\nWang, T., Ofman, L., Sun, X., Solanki, S. K., and\nDavila, J. M. (2018). Effect of Transport\nCoefficients on Excitation of Flare-induced Standing\nSlow-mode Waves in Coronal Loops. ApJ860, 107.\nhttps://doi.org/10.3847/1538-4357/aac38aWuelser, J.-P., Lemen, J. R., Tarbell, T. D., Wolfson,\nC. J., Cannon, J. C., Carpenter, B. A., et al. (2004).\nEUVI: the STEREO-SECCHI extreme ultraviolet\nimager. In Telescopes and Instrumentation for Solar\nAstrophysics , eds. S. Fineschi and M. A. Gummin.\nvol. 5171 of Proc. SPIE , 111–122.\nhttps://doi.org/10.1117/12.506877\nYuan, D. and Nakariakov, V . M. (2012). Measuring\nthe apparent phase speed of propagating EUV\ndisturbances. A&A 543, A9.\nhttps://doi.org/10.1051/0004-6361/201218848\nZavershinskii, D. I., Kolotkov, D. Y ., Nakariakov,\nV . M., Molevich, N. E., and Ryaschikov, D. S.\n(2019). Quasi-periodicity of magnetoacoustic waves\nin a plasma with heating/cooling misbalance. PPCF\nUnder review"    },    {        "title": "1111.6330v2.Dispersion_Relations_Explaining_OPERA_Data_From_Deformed_Lorentz_Transformation.pdf",        "content": "arXiv:1111.6330v2  [hep-ph]  2 Dec 2011Dispersion Relations Explaining OPERA Data\nFrom Deformed Lorentz Transformation\nGang Guo1, Xiao-Gang He1,2\n1INPAC, Department of Physics, Shanghai Jiao Tong Universit y, Shanghai, China\n2Department of Physics and Center for Theoretical Sciences,\nNational Taiwan University, Taipei, Taiwan\n(Dated: November 5, 2018)\nAbstract\nOPERA collaboration has reported evidence of superluminal phenomenon for neutrinos. One of the\npossible ways to explain the superluminality is to have Lore ntz symmetry violated. It has been shown\nthat dispersion relations put forwards has the problem of ph ysics laws vary in different inertial frames\nleading to conflicting results on Cherenkov-like radiation , pion decay and high energy neutrino cosmic\nray. For theories with deformed Lorentz symmetry, by modify ing conservation laws corresponding to\nenergy and momentum in the usual Lorentz invariant theory, i t is possible to have superluminal effect\nand at the same time avoid to have conflicts encountered in Lor entz violating theories. We study\ndispersion relations from deformed Lorentz symmetry. We fin d that it is possible to have dispersion\nrelations which can be consistent with data on neutrinos. We show that once the superluminality δvas\na function of energy is known the corresponding dispersion r elation in the deformed Lorentz symmetry\ncan be determined.\n1The OPERA collaboration has reported evidence of superluminal beh avior for muon neutrino\nνµwith energies of a few tens of GeV [1]. More recently, they have rep orted new data with\na shorter beam bunch allowing the measurement of the neutrino fligh t time more accurately.\nThey confirmed their earlier results [1]. The mean neutrino energy is 1 7 GeV and neutrinos\narrived earlier by δt= (57.8±7.8(stat.)+8.3\n−5.9(sys.)) ns than expected if neutrinos are traveling\nwith the speed of light c. This implies a superluminal velocity vof neutrinos by relative amount\nof superluminality δv= (v−c)/c= (2.37±0.32(stat.)+0.34\n−0.24)×10−5. This has generated exten-\nsive discussions. Existence of superluminal phenomena as large as O PERA observed challenges\nthe completeness of relativity theory based on Lorentz symmetry [2]. It is important that the\nOPERA result be tested by other experiments. Many of the studies in the literature aimto study\ntheoretical explanations and implications of the data. Different mec hanisms have been proposed\nto explain the data. Lorentz violation is a popular scenario[3–5].\nIt has been shown that explanation of the OPERA data based on cer tain Lorentz violating\ntheory is inconsistent with other known results from the same expe riment [6, 7]. Cohen and\nGlashow argued that in the energy range of neutrino, the beam wou ld loose much of their\nenergy, via Chereov-like processes, such as νµ→νµ+e++e−, on their way from CERN to the\nOPERA detector at Gran Sasso, and OPERA would not detect neutr inos with energy exceeding\n12 GeV in certain Lorentz violating theories. Bi et al and Gonzalez-M estres showed[7] that with\nthe large superluminal effect of δv= 2.37×10−5, neutrinos cannot have energies exceeding 5\nGeV if they are from π+→µ+νµwhich is believed to provide the substantial amount of the\nmuon neutrino flux detected at Gran Sasso. It is interesting to not e that physics laws are altered\nin different inertial frames. In the νµ→νµ+e++e−case, in an inertial frame in which the\nmuon neutrino has less than two times of the electro rest mass, the processes cannot happen,\nbut in another inertial frame the initial neutrino travels with a larger velocity and therefore a\nhigher energy than two rest electron mass, this process can happ en. In the π+→µ+νµcase, in\nan inertial frame pion has low energy, the process can happen, but with higher energy, it may\nnot happen. This, by itself, is an very interesting result. This is in con trary to our common\nbelieves and needs an explanation. This conclusion applies to many Lor entz violating theories.\nSeveral solutions to these conflicts have been proposed. If neut rinos are Tachyons, they always\nhave a velocity larger than the speed of light and is consistent with Lo rentz symmetry. However,\nithasbeenshownthatthistypeoftheorieshasproblemswhentakin gintoaccount otheravailable\ndata[5], such as possible dependence of neutrino maximal speed with energy. Another interesting\npossibility of explaining the data by keeping the principle of relativity of inertial frames is based\non deformed Lorentz symmetry[8–11]. This type of theory allows su perluminal effect but can\navoid the problems above by keeping physical laws not changed in diffe rent inertial frames, if the\nconservation lawscorresponding totheenergy andmomentum inth eusual Lorentzsymmetry are\nsuitably modified. In this work, we will work in the framework of defor med Lorentz symmetry\nto find dispersion relations which can have superluminal effect to exp lain the OPERA data and\nalso other related neutrino data including those from neutrino oscilla tion.\nA general dispersion relation between energy Eand momentum pcan be written as[8],\nE2f2(E,p)−p2g2(E,p) =m2, which can also be re-written without loss of generality as\nF2(E,p)E2−p2=m2. (1)\n2F2(E,p) =f2(E,p)−(g2(E,p)−1)p2/E2. One can also solve for Eas a function of pfirst and\nthen insert the expression for EinF(E,p) to transform Fas a function of pandm. We will\nwriteFasF(p,m).\nIf the generator of Lorentz boost Niis modified in such a way that\n[Ni,F2(p,m)E2−p2] = 0, (2)\nthen the relativity of the inertial frames are preserved. In the limit of usual Lorentz symmetry\nF(p,m) = 1. If one also wants to keep the meaning of mass as rest energy, thenF(0,m) = 1.\nThe above commutation relation can be achieved by[10]\n[Ni,pj] =F(p,m)Eδij,[Ni,E] =pi/parenleftbigg1\nF(p,m)−2E2∂F(p,m)\n∂p2/parenrightbigg\n. (3)\nThe above defines a deformed Lorentz symmetry. With this deform ed Lorentz symmetry,\nthe conserved quantities corresponding to the energy momentum conservation laws in the usual\nLorentz symmetry is modified. They are\nF(p,m)E , /vector p . (4)\nIt has been shown that with the deformed Lorentz symmetry, the previously mentioned prob-\nlems will be naturally avoided. The theory is self consistent even supe rluminal effects exist.\nHere we demonstrate this by showing how an example problem discuss ed in Ref.[7] can be re-\nsolved with deformed Lorentz symmetry, the problem of π+→µ+νµ. The quantity Fν(pν,mν)Eν\nplays the role of energy in the usual Lorentz symmetry theory. Th e conservations of energy and\nmomentum in this case are given by\n/vector pπ=/vector pµ+/vector pν,Eπ=Eµ+F(p,m)Eν. (5)\nThe above conservation law for energy looks dramatically different t han the usual one. However,\nif one replaces F(pν,mν)Eν=/radicalbig\n/vector p2ν+m2νusing eq.(1), the energy and momentum conservation\nequations become\n/vector pπ=/vector pµ+/vector pν,/radicalbig\n/vector p2π+m2π=/radicalBig\n/vector p2µ+m2µ+/radicalbig\n/vector p2ν+m2ν. (6)\nThe above equations are the usual energy and momentum conserv ation laws expressed in terms\nof momenta. π+→µ+νµis not forbidden to decay in any inertial frame in this theory.\nAt present stage we do not have a underlying theory to provide the deformed Lorentz sym-\nmetry to dictate what are the conserved quantities, one can ther efore, without loss of gener-\nality, translate any given dispersion relation into the above form and choose conserved quan-\ntities accordingly. One can also choose to re-write the general disp ersion relation in the form\nE2−G2(p,m)p2=m2. F and G are related by G2= 1+(1 −F2)E2/p2. In this case, the con-\nserved quantities are EandG(E,m)/vector p. In the following we will work with the form of dispersion\nrelation in eq.(1).\n3A straightforward calculation for the velocity vof a particle satisfying the dispersion relation\nabove gives\nv=dE\ndp=p\nE/parenleftbigg1\nF2(p,m)−2E2\nF(p,m)dF(p,m)\ndp2/parenrightbigg\n=p/radicalbig\np2+m21\nF(p,m)+/radicalbig\np2+m2d\ndp1\nF(p,m). (7)\nSince the neutrino mass mis small which can be neglected for practical discussions. In this cas e\nthe relation between superluminality δvandF=F(p,0) is simple with\nδv=1\nF(p,m)+pd\ndp1\nF−1. (8)\nWhether the theory allows superluminal velocity depends on the spe cific form of F(p,m). For\nexample, the dispersion relation[8]\nE2−p2=m2+2E2p2/M2, (9)\ngivesF2= 1−2p2/M2, and a superluminal velocity vwith\nv=p/radicalbig\n(p2+m2)1+m2/M2\n(1−2p2/M2)3/2≈1−m2\n2p2+3p2\nM2. (10)\nwherem << M has been assumed for the approximation in the last step above.\nIf neutrinos satisfy deformed Lorentz symmetry, there is a chan ce that they have superluminal\nvelocity without the problems given in Refs.[6, 7]. What form of F(p,m) the neutrinos take is\nnot known unless an underlying theory is known. Unfortunately at t his stage, the study of\ndeformed Lorentz symmetry is not yet ready to provide such insigh t. We therefore will consider,\nat phenomenological level, dispersion relations which can provide larg e superluminal velocity\nindicated by the OPERA data and at the same time satisfy the constr aints from other neutrino\ndata.\nIn general F(p,m) can depend on neutrino spices, that is Fi=νe,νµ,ντ(p,mi) may be different for\ndifferent neutrinos. However, with a small difference in Fibetween different “i”, the coherence\nof neutrino mixing will be destroyed leading to no oscillation between ne utrinos. For this reason,\nwe consider that the form of Fis universal for all neutrinos. With this assumption, one then\nis required to explain the energy dependence of the superlumnality f or electron neutrino νeand\nmuon neutrino νµ. In the energy range of the OPERA data, no obvious energy depen dence\nof the superluminality is observed. At energies 13.8 GeV and 40.7 GeV, the associated early\narrival times are δt1= (54.7±18.4(stat.)+7.3\n−6.9(sys.))nsandδt2= (68.1±19.1(stat.)+7.3\n−6.9(sys.))ns\nwhich implies a superluminality δvof order a few times of 10−5and the averaged δv= (2.37±\n0.32(stat.)+0.34\n−0.24)×10−5. However, observationaldatafromSN1987aconstrainthesupe rluminality\nδvof electron neutrino to be less than 2 ×10−9with an energy of 10 MeV or so. This can be\ntaken as an indication of energy dependence since we are assuming u niversal dispersion relation.\nThere arealso constraintsonneutrino superluminality athigher ene rgies upto200GeV. Between\nMINOS[12] energy of a few GeV and the FERMILAB 1979[13] energy o f up to 200 GeV, the\n4/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s69/s45/s49/s51/s49/s69/s45/s49/s50/s49/s69/s45/s49/s49/s49/s69/s45/s49/s48/s49/s69/s45/s57/s49/s69/s45/s56/s49/s69/s45/s55/s49/s69/s45/s54/s49/s69/s45/s53/s49/s69/s45/s52\n/s32/s100/s97/s116/s97\n/s32/s40/s49/s45/s50/s112/s50\n/s47/s77/s50\n/s41/s45/s51/s47/s50\n/s45/s49\n/s32/s98/s112/s50\n/s47/s40/s112/s50\n/s43/s97/s50\n/s41\n/s32/s98/s112/s51\n/s47/s40/s112/s51\n/s43/s97/s51\n/s41\n/s32/s98/s112/s52\n/s47/s40/s112/s52\n/s43/s97/s52\n/s41\n/s70/s101/s114/s109/s105/s108/s97/s98/s32/s49/s57/s55/s57/s77/s73/s78/s79/s83\n/s79/s80/s69/s82/s65/s79/s80/s69/s82/s65\n/s32/s32\n/s118\n/s110/s101/s117/s116/s114/s105/s110/s111/s32/s109/s111/s109/s101/s110/s116/s117/s109/s32/s105/s110/s32/s71/s101/s86\nFIG. 1: Constraints on neutrino superluminalities from dat a and predictions from dispersion relations\ndiscussed in the text for M= 6×103GeV,a= 10 GeV and b= 2.37×10−5.\nenergy dependence is not obvious. The experimental data are dep icted in Fig. 1. A viable\ndispersion relation should be able to fit the OPERA, SN1987a and othe r data simultaneously.\nThe dispersion relation in eq.(9) gives δv= 1/(1−2p2/M2)3/2. We now check if this can fit\nthe data. This dispersion relation has only one unknown parameter M. To have δv= 2.37×10−5\nwith energy equal to 17 GeV, the mass parameter Mis determined to be about 6 ×103GeV.\nThe predictions for superluminality at other energies of this dispers ion relation is plotted in Fig.\n1. It can be seen that the SN1987a limit on neutrino superluminality is s atisfied. However, at\nhigher energies, this dispersion related produces a too large super luminality than that allowed\nfrom FERMILAB 1979 data. This dispersion relation cannot be a viable possible candidate.\nOther dispersion relation is needed to fit the data. To find a realistic s uperluminality profile,\nwe note that there is no obvious energy dependence between the e nergy (around 17 GeV) of\nthe OPERA data and higher energies (up to 200 GeV) from FERMILAB 1979 data with a\nsuperluminality of order 10−5or lower, but at the energies of the SN1987a neutrino, a few MeV,\nthe profile drops down to a upper limit less than 10−9. This suggest a form for δv\nδv=bpn\nan+pn, (11)\nwhere a and b are constants. For large enough p,δvapproaches a constant b, and for small\nenoughp,δv≈bpn/anwhich goes to zero quickly for a positive n.\nIn Fig. 1, we plot the superluminaluty δvfor n =2, 3 and 4. We agree with the observation\nin ref.[4] that n= 3 can fit data well. We use a= 10 GeV and b= 2.37×10−5. In fact the class\nof solutions in eq.(11) can all be consistent with present data. With a larger n, it is much easier\nto have a small superluminality for the SN1987a data after fitting th e superluminality for the\nOPERA data and FERMILAB 1979 data.\nIf the energy profile of the superlominality δv=P(p) is known, one can then use eq.(8) to\n5solve for Fand therefore the dispersion relation for neutrinos with negligibly sm all massm≈0.\nWe have\n1\nF+pd\ndp1\nF= 1+P(p). (12)\nThe solution for 1 /Fis given by\n1\nF=1\np/parenleftbigg/integraldisplayp\n(1+P(p′))dp′+c/parenrightbigg\n, (13)\nwherecis an integration constant set by the condition F(0) = 1.\nFor a known value of n for δvin eq.(11), one can obtain the dispersion relation factor Fusing\nthe above equation. We write down the general solution for Fwith integer n, Fn(p),\n1\nFn(p)= 1+b\n−ab\np\n\n2\nn/summationtextn/2−1\nk=0[Qksin((2k+1)π\nn)−Pkcos((2k+1)π\nn)],n = even\n1\nnln((1+p\na)+2\nn/summationtext(n−3)/2\nk=0[Qksin((2k+1)π\nn)−Pkcos((2k+1)π\nn)]−C ,n = odd .,(14)\nwhere\nPk=1\n2ln[p2\na2−2p\nacos((2k+1)π\nn)+1],\nQk= arctan/parenleftBiggp\na−cos((2k+1)π\nn)\nsin((2k+1)π\nn)/parenrightBigg\n,\nC=(n−3)/2/summationdisplay\nk=0((2k+1)π\nn−π\n2)(sin(2k+1)π\nn). (15)\nForn= 2 and n= 3, we have\nF2(p) =p/a\n(1+b)p/a−darctan(p/a), (16)\nF3(p) =6√\n3p/a\n6√\n3(1+b)p/a−b(π+6arctan[(2 p−a)/√\n3a]+√\n3ln[(a+p)2/(a2−ap+p2)]).\nWe therefore have found a class of deformed Lorentz symmetry w hich can fit the current data\non neutrino superluminality δv. As have been shown earlier, with the function F(p,m) known\none also knows how the energy- momentum conservation laws are mo dified. The conserved\nquantities are F(p,m)Eand/vector p. The momentum conservation law is not changed compared to\nthe usual Lorentz symmetry. The energy conservation law is modifi ed. In principle one can test\ntheF(p,m)Elaw against the usual Econservation law in the usual Lorentz symmetry. However,\nthe deviation of F(p,m) from 1 in the dispersion relation is very small, at most of order a few\ntimes of 10−5as shown in Fig.2 for n =2, 3 and 4 cases. It is, therefore, difficult to t est these\n6/s49 /s49/s48 /s49/s48/s48/s48/s46/s57/s57/s57/s57/s55/s53/s48/s46/s57/s57/s57/s57/s56/s48/s48/s46/s57/s57/s57/s57/s56/s53/s48/s46/s57/s57/s57/s57/s57/s48/s48/s46/s57/s57/s57/s57/s57/s53/s49/s46/s48/s48/s48/s48/s48/s48\n/s32/s32\n/s70\n/s110/s101/s117/s116/s114/s105/s110/s111/s32/s109/s111/s109/s101/s110/s116/s117/s109/s32/s105/s110/s32/s71/s101/s86\nFIG. 2:F(p) as functions for δv= 1/(1−2p2/M2)3/2andδv=bpn/(pn+an) withn= 2,3,4.\nmodifiedenergy conservationlaws atpresent. The deviationathigh energies for F2= 1−2p2/M2\ncase can be larger. Unfortunately, this case also produce too larg e a superluminality at energy\nlarger than OPERA neutrino energy and therefore it cannot be a via ble candidate.\nWe have studied dispersion relations which can be interpreted as fro m deformed Lorentz\nsymmetry. We find that it is possible to have dispersion relations which can explain data on\nneutrino superluminality data. We have shown that once the superlu minality δvas a function of\nmomentum is known, one can determine the corresponding dispersio n relation and therefore the\nconservation law corresponding to the energy conservation in a th eory with the usual Lorentz\nsymmetry. The energy conservation law is modified which in principle ca n be used to test\nthe models. The modifications are, however, small and cannot be te sted at the present. Future\nexperiments which can measure neutrino energy to high precision ma y provide useful information\nabout deformed Lorentz symmetry.\nAcknowledgments\nThis work was supported in part by NSC and NCTS of ROC, and SJTU 98 5 grant and NNSF\ngrant of PRC.\n[1] T. Adam et al., [OPERA Collaboration], arXiv:1109.4897 .\n[2] B. Alles, arXiv:1111.0805;\n[3] F. R. Klinkhamer, arXiv:1109.5671 [hep-ph]; Z. Lingli a nd B. -Q. Ma, arXiv:1109.6097 [hep-ph];\nJ. Alexandre, J. Ellis and N. E. Mavromatos, arXiv:1109.629 6 [hep-ph]; F. R. Klinkhamer and\nG. E. Volovik, Pisma Zh. Eksp. Teor. Fiz. 94, 731 (2011) [arXiv:1109.6624 [hep-ph]]; E. Ciuffoli,\n7J. Evslin, J. Liu and X. Zhang, arXiv:1109.6641 [hep-ph]; L. Maccione, S. Liberati and D. M. Mat-\ntingly, arXiv:1110.0783 [hep-ph]; N. Qin and B. -Q. Ma, arXi v:1110.4443 [hep-ph]; S. Nojiri and S.\nOdintsov, arXiv:1110.0889[hep-ph]; Z. Lingli and B. -Q. Ma , arXiv:1111.1574 [hep-ph]; A. Coutant,\nS. Finazzi, S. Liberati and R. Parentani, arXiv:1111.4356 [ gr-qc].\n[4] G. F. Guidice, S. Sibiryakov, A. Strumia, arXiv:1109.56 82[hep-ph].\n[5] M. Li and T. Wang, arXiv:1109.5924 [hep-ph].\n[6] A. G. Cohen and S. Glashow, Phys. Rev. Lett. 107, 181803(2011)[arXiv:1109.6630[HEP-PH]].\n[7] L.Gonzalez-Mestres. arXiv:1109.6630; X.-J.Bi, P.-F. Yin, Z.-H.Yu, Q.Yuan, arXiv:1109.6667[hep-\nph]; R. Cowsik, S. Nussinov, U. Sarkar, arXiv:1110.0241[he p-ph].\n[8] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L . Smolin, arXiv:1110.0521[hep-ph]\n[9] G. Amelino-Camelia et al., arXiv:1111.0993[hep-ph]\n[10] Yi Ling, arXiv:1111.3716[hep-ph]\n[11] Yunjie Hou et al., arXiv:1111.4994[hep-ph]\n[12] P. Adamson et al., [MINOS Collaboration], Phys. Rev. D76, 072005(2007).\n[13] G. R. Kalbfleisch, N. Baggett, E.C. Fowler, J. Alspector , Phys. Rev. Lett. 43, 1361(1979).\n8"    },    {        "title": "1506.05585v1.Damping_of_MHD_turbulence_in_partially_ionized_plasma__implications_for_cosmic_ray_propagation.pdf",        "content": "arXiv:1506.05585v1  [astro-ph.HE]  18 Jun 2015DRAFT VERSION SEPTEMBER 25, 2018\nPreprint typesetusingL ATEX styleemulateapjv. 5/2/11\nDAMPINGOF MHDTURBULENCE INPARTIALLYIONIZEDPLASMA:IMPL ICATIONSFORCOSMICRAY\nPROPAGATION\nSIYAOXU1, HUIRONGYAN2ANDA. LAZARIAN3\nDraftversion September 25,2018\nABSTRACT\nWe study the dampingfrom neutral-ioncollisions of both inc ompressible and compressible magnetohydro-\ndynamic (MHD) turbulence in partially ionized medium. We st art from the linear analysis of MHD waves\napplying both single-fluid and two-fluid treatments. The dam ping rates derived from the linear analysis are\nthenusedindeterminingthedampingscalesofMHD turbulenc e. Thephysicalconnectionbetweenthedamp-\ningscale ofMHD turbulenceandcutoffboundaryof linearMHD wavesis investigated. Our analyticalresults\nare shown to be applicable in a variety of partially ionized i nterstellar medium (ISM) phases and solar chro-\nmosphere. As a significant astrophysical utility, we introd uce damping effects to propagation of cosmic rays\nin partially ionizedISM. The important role of turbulenced ampingin both transit-time dampingand gyrores-\nonanceisidentified.\nSubjectheadings: magnetohydrodynamics(MHD)-turbulence-ISM:cosmicrays\n1.INTRODUCTION\nPartially ionized plasmas are universal in both space and\nastrophysical environments. They fill the most volume\nthroughout the Galaxy and build up molecular clouds from\nwhich stars form. The presence of neutrals affects the\nplasmadynamicsindirectly,mediatedbythefrictionbetwe en\nions and neutrals (see studies by e.g. Piddington 1956;\nKulsrud&Pearce1969).\nOn the other hand, astrophysical plasmas are generally\nturbulent and magnetized (see e.g. Armstrongetal. 1995;\nChepurnov& Lazarian2010). Bothincompressibleandcom-\npressible MHD turbulence in fully ionized medium have\nbeen ealier studied (Goldreich&Sridhar 1995, hereafter\nGS95, Lithwick&Goldreich 2001; Cho &Lazarian 2002,\n2003). The GS95 model of turbulence and its extensions\nin compressible fluids successfully reproduce the proper-\nties of MHD turbulence and are supported by numerical\ntests(Maron&Goldreich2001; Cho&Lazarian2002,2003;\nKowal&Lazarian2010).\nThe turbulence theory for a partially ionized medium\nshould be constructed under the consideration of the damp-\ning by neutrals. There are extensive studies in existing\nliterature regarding neutral-ion collisional damping, us ing\neither single-fluid (e.g. Braginskii 1965; Balsara 1996;\nKhodachenkoet al.2004;Fortezaet al.2007)ortwo-fluidde-\nscription(e.g. Pudritz 1990; Balsara 1996; Kumar& Roberts\n2003; Zaqarashvilietal. 2011; Mouschoviasetal. 2011;\nSoleret al. 2013b,a). The single-fluid approach has simpler\nformalism and hence physical transparency, but is restrict ed\non the limited wave frequencies lower than the neutral-ion\ncollisional frequency. While the two-fluid approach treats\nion-electron and neutral fluids separately and is valid over a\nbroadrangeofwavefrequencies.\nBut all these studies are restricted to linear MHD waves.\nThe modern advances of MHD turbulence reveal it is a\n1Department ofAstronomy, Schoolof Physics,Peking Univers ity, Bei-\njing 100871, China\n2Kavli Institute for Astronomy and Astrophysics, Peking Uni versity,\nBeijing 100871, China; hryan@pku.edu.cn\n3Department of Astronomy, University of Wisconsin, 475 Nort h Char-\nter Street, Madison, WI53706, USAhighly non-linear phenomenon, with the energy injected at\na large scale cascading successively downwards at the tur-\nbulence eddy turnover rate (see e.g. Cho &Lazarian 2003;\nChoetal. 2003). The effect of neutral-ion collisions on the\ncascade of compressible MHD turbulence was investigated\nby Lithwick&Goldreich (2001), but they mainly focused\non a highly-ionized medium with gas pressure overwhelm-\ning magnetic pressure. The role of neutral viscous damping\nin a mostly neutral gas in turbulent motions was thoroughly\nstudied in Lazarianet al. (2004). Combining the two damp-\ning effects, a recent work by Xuet al. (2014) (hereafter Pa-\nper I) carried out a detailed study on damping of the Alfv´ en\ncomponent of MHD turbulence in a variety of turbulence\nregimes and demonstrated its significance in explaining the\nobservedlinewidthdifferencesbetweenneutralsandions, and\nthe potential value in measuring magnetic field in molecu-\nlar clouds. In the current work, we extend the theory de-\nvelopedin Paper I to include compressible MHD turbulence.\nTheweakcouplingofAlfv´ entocompressiblemodeswassug-\ngestedbyGS95. ThedecompositionofMHD turbulenceinto\nAlfv´ en,fast,andslowmodeshasbeenconfirmedandrealized\nthroughastatisticalprocedureinbothFourierandrealspa ces\n(Cho& Lazarian 2002, 2003; Kowal&Lazarian 2010). On\nthisbasis,wecandealwiththethreecomponentsofMHDtur-\nbulenceseparately. Wetakeadvantageofthewell-establis hed\nlineartheoryonwavepropagationanddissipationinstudyi ng\nthedampingprocessesofMHDturbulence.\nThe output of this investigation has a wide range of astro-\nphysicalapplications,forinstance,cosmicray(CR) propa ga-\ntion. The propagation of CRs in ISM is usually described\nin terms of diffusive transport of CRs in a turbulent mag-\nnetic field (see e.g. Schlickeiser 2002). The well establish ed\nstatistics on the scattering of CRs by magnetic irregulari-\nties can successfully explain the high-degree isotropy of t he\ncomic radiation. But the conventionally used, simple model\nof CR diffusion assuming a universal diffusion coefficient\nfaces big challenges in interpreting more and more observa-\ntional findings (e.g. Hunteret al. 1997; Guillianet al. 2007 ;\nAdrianiet al. 2011; Ackermannet al. 2012). With the de-\nvelopment and acceptance of the modern concept of MHD\nturbulence (see recent reviews by Brandenburg&Lazarian2\n2014; Beresnyak& Lazarian 2015), significant refinements\nandmodificationshavebeenprogressivelymadeinthepictur e\nof CR propagation, especially in the aspect of quantitative ly\nrelating the CR diffusion and the properties of the turbulen t\nmediumthroughwhichtheypass.\nTheconsiderationofturbulencedampingisnecessaryfora\nproperdescriptionoftheinteractionbetweenCRparticles and\nMHD turbulence. The study in Yan&Lazarian (2002, 2004,\n2008) showed the important influence of turbulence proper-\ntiesanditsdampingonCRscatteringandpropagationinfull y\nionized plasma. The ionization fraction possessed by ISM\nvaries over a vast expanse, ranging from fully ionized to al-\nmost entirely neutral media (Spitzer 1978; Draine 2011). In\nthepresenceofneutrals,thedampingprocessinpartiallyi on-\nizedmediumiscompletelydifferentfromthatinfullyioniz ed\nmedium. Accordingly, the propagation of CRs in partially\nionized medium is likely to present distinctive features an d\ndeservesaspecialattention. Withthismotivation,weproc eed\nthe earlier research in partially ionized medium and aim to\nachievearealistic modelofCR transportinISM.\nThe organization of the paper is as follows. In Section 2,\nwe introduce the concept of neutral-ion and ion-neutral de-\ncoupling. We then provide a linear analysis on MHD waves\nusingbothtwo-fluidandsingle-fluidapproachesinSection3 .\nWith the damping rates obtained from linear analysis, damp-\ningofMHDturbulenceispresentedinSection4. Thenumer-\nical tests of our analytical results in diverse partially io nized\nISM conditions and solar chromosphere are given in Section\n5. Section6containsanapplicationonCRpropagationinpar -\ntiallyionizedmedium. Section7presentsfurtherdiscussi ons.\nThesummaryis givenin Section8.\n2.COUPLING BETWEENNEUTRALS ANDIONS\nThecouplingofneutralsandionsdependsonthewavefre-\nquencyofinterestincomparisonwiththeneutral-ioncolli sion\nfrequency,\nνni=γdρi. (1)\nHereρiis ion density and γdis the drag coefficient defined\nin Shu (1992). νniis used when we consider the drag force\nexerted on the neutrals by the ions. It is related with ion-\nneutralcollisionfrequency νinby\nνniρn=νinρi, (2)\nwhereρnis neutral density. The condition ω∼νniand\nω∼νincorrespond to neutral-ion decoupling scale k−1\ndec,ni\nandion-neutraldecouplingscale k−1\ndec,inrespectively. Ina pre-\ndominantlyneutralmedium, k−1\ndec,niismuchlargerthan k−1\ndec,in.\nAt greater scales than k−1\ndec,ni, neutrals and ions are perfectly\ncoupledand oscillate together. In this regime the single-fl uid\napproximation is adequate to study the behavior and damp-\ning of MHD waves. Below k−1\ndec,ni, neutrals start to decouple\nfromions. Perturbationsofmagneticfieldcannotbeinstant ly\ntransmitted to neutrals, thus Alfv´ en waves are suppressed in\nneutralfluid.\nThe expressions of kdecare different for different wave\nmodes, depending on the form of their respective wave fre-\nquencies. In the limit case of low- β, whereβ= 2c2\ns/V2\nAis defined as the ratio of gas and magnetic pressure, both\nAlfv´ en and fast modes propagate with Alfv´ en speed. Their\nneutral-ion decoupling scales are determined by the condi-\ntionsVAkcosθ=νniandVAk=νnirespectively, where θis the wave propagationangle relative to magnetic field. The\ncosθappears for Alfv´ en modes because their waves travel\nalongmagneticfield,whilefastwavespropagateisotropica lly\nandtheir decouplingscales do not dependon propagationdi-\nrection. The intrinsic nature of neutrals allows them to sup -\nport their own sound waves after they decouple from ions.\nOnce satisfying the condition csnk=νni, neutrals are able\nto develop their sound waves and become independent from\nions’ modulation. Thus the k−1\ndec,niof slow modes depends on\nωof sound waves. So we can summarize in low- βplasma,\nneutral-ion decoupling scale sets the largest scale for the ab-\nsence of Alfv´ en waves in neutral fluid for Alfv´ en and fast\nmodes,butthepresenceofsoundwavesinneutralsinthecase\nofslow modes.\nIn the meantime, ions are still subject frequent collisions\nwith surrounding neutrals above k−1\ndec,in. Therefore the re-\nmaining wave motions in ions can be strongly influenced\nby the collisional friction. On the other hand, the single-\nfluid approximation can still be used to describe the MHD\nwaves since ions maintain the coupling state with neutrals.\nAt shorter scales than k−1\ndec,in, ions also become ineffectively\ncoupledwith neutrals. Thescale k−1\ndec,inis determinedby ωof\nMHD waves in ions alone. For the weakly coupled ions and\nneutrals, the assumption of strong coupling breaks down, so\nthetwo-fluiddescriptionwithnorestrictionimposedonwav e\nfrequencyrangesneedstobeadopted. Inthefollowingofthe\npaper, we use ”strongly” and ”weakly” coupled regimes by\nmeaningk < kdec,niandk > kdec,inrespectively.\nBasedonaboveanalysis,Table1liststhedecouplingscales\nfordifferentwavemodesandregimes.\nTABLE1\nDECOUPLING SCALES OF MHDWAVES.\nkdec,ni kdec,in\nAlfv´ enνni\nVAcosθνin\nVAicosθ\nFastνni\nVAνin\nVAi\nSlowνni\ncsnνin\ncsicosθ\n3.MHD WAVESIN PARTIALLYIONIZEDPLASMA\nThe study of damping process is based on the linear anal-\nysis of MHD waves in partially ionized plasma. We present\nthewavefrequenciesobtainedfrombothtwo-fluidandsingle -\nfluid approaches. And we mainly focus on the solutions in a\nlow-βlimit.\n3.1.Two-fluidapproach\nAlfv´en waves In a neutral-dominated medium, the dissi-\npation of Alfv´ en waves can be enhanced by both netral-ion\ncollisional dampingand the viscosity effects in neutrals. The\nrelativeimportanceofthetwodampingmechanismscanvary\nin different phases of ISM. When the neutral-ion collisiona l\ndamping is much more efficient and viscous damping can be\nsafelyneglected,Alfv´ enwaveshaveawell-studieddisper sion\nrelation (see e.g., Piddington 1956; Kulsrud&Pearce 1969;\nSoleretal. 2013b),\nω3+i(1+χ)νniω2−k2cos2θV2\nAiω−iνnik2cos2θV2\nAi= 0,\n(3)3\nwhereχ=ρn/ρi. Under the assumption of weak damping\n|ωI| ≪ |ωR|, theapproximatesolutionsare\nω2\nR=k2cos2θV2\nAi((1+χ)ν2\nni+k2cos2θV2\nAi)\n(1+χ)2ν2\nni+k2cos2θV2\nAi,(4a)\nωI=−νniχk2cos2θV2\nAi\n2((1+χ)2ν2\nni+k2cos2θV2\nAi). (4b)\nItisreducedto\nω2\nR=V2\nAk2cos2θ, (5a)\nωI=−ξnV2\nAk2cos2θ\n2νni. (5b)\ninstronglycoupledregime,i.e. ω≪νin, and\nω2\nR=V2\nAik2cos2θ, (6a)\nωI=−νin\n2(6b)\ninweaklycoupledregime,i.e. ω≫νin.\nThe assumption of weak damping holds in both strongly\nand weakly coupled regimes, but fails at intermediate wave\nfrequencies. The strongly damped region can be confined by\nthe condition |ωI|=|ωR|. FromEq. (5) and(6), it yieldsthe\nboundarywavenumbers,\nk+\nc=2νni\nVAξncosθ, (7a)\nk−\nc=νin\n2VAicosθ. (7b)\n|ωI|convergeswith |ωR|atk±\nc. Within[k+\nc,k−\nc], the Alfv´ en\nwaves become nonpropagating and have purely imaginary\nwavefrequencies. Althoughtheprecisecalculationofthec ut-\noff interval of Alfv´ en waves should involve the discrimina nt\nof Eq. (3) (e.g. Soleret al. 2013b), numericalresults confir m\nthe boundaries [k+\nc,k−\nc]obtained from the equality between\n|ωI|and|ωR|canverywellconfinethenonpropagatinginter-\nval withωR= 0. Thereforewe choosethis simple derivation\nanduse theresulting k±\ncasthelimit wavenumbersofthecut-\noffregion.\nBy comparing with Table 1, we find the cutoff boundaries\nareconnectedwith thedecouplingscalesby\nk+\nc,/bardbl=2\nξnkdec,ni,/bardbl, (8a)\nk−\nc,/bardbl=1\n2kdec,in,/bardbl, (8b)\nshowing the cutoff region is slightly smaller than the\n[kdec,ni,kdec,in]range.\nWhen neutral viscosity dominatesover the neutral-ioncol-\nlisional damping, the approximate damping rate in strongly\ncoupledregimeis (Lazarianet al. 2004;PaperI)\n|ωI|=τ−1\nυ=k2νn, (9)\nwhereνn=csn/(nnσnn)is the kinematic viscosity in neu-\ntrals. By equalingthe above |ωI|and|ωR|fromEq. (5a), the\ncorrespondingcutoffwavenumberis\nk+\nc=VAcosθ\nνn. (10)The wave frequencies and k−\ncin weakly coupled regime are\nthe same as those in the case of neutral-ioncollisional damp -\ning.\nFast and slow waves We then turn to the magnetoacous-\ntic waves. We consider neutral-ion collisional friction as the\ndominantdampingmechanism. We adoptthedispersionrela-\ntion given by Soleretal. (2013a) (see also Zaqarashviliet a l.\n2011). Taking advantage of the weak damping assumption,\nanalyticsolutionscanbeobtainedinthelimitcaseofstron gly\ncoupledfluids,\nω2\nR=1\n2/bracketleftbigg\n(c2\ns+V2\nA)±/radicalBig\n(c2s+V2\nA)2−4c2sV2\nAcos2θ/bracketrightbigg\nk2,\n(11a)\nωI=−k2[ξnV2\nA(c2\nsk2−ω2\nR)+ξic2\nsω2\nR]\n2νni[k2(c2s+V2\nA)−2ω2\nR]. (11b)\nHerecs=/radicalbig\nc2\nsiξi+c2snξnis the sound speed in strongly\ncoupledionsandneutrals. Theabovesolutionsare consiste nt\nwith those in earlier work, e.g. Ferriereetal. (1988). Dif-\nferent from Alfv´ en waves, the compressibility introduces the\nimportant parameter β. Its definition was given in Section 2.\nIn a low-βplasma, the solutions in strongly coupled regime\nhavesimpleexpressions,\nω2\nR=V2\nAk2, (12a)\nωI=−ξnV2\nAk2\n2νni, (12b)\nforfast modes,and\nω2\nR=c2\nsk2cos2θ, (13a)\nωI=−ξnc2\nsk2\n⊥\n2νni, (13b)\nforslow modes.\nIn weakly coupled regime, under the low- βcondition, the\npropagatingcomponentofwavefrequencybecomes\nω2\nR=V2\nAik2(14)\nforfast modes,and\nω2\nR=c2\nsik2cos2θ (15)\nfor slow modes. And they have the same damping rate as\nAlfv´ enwavesat highwave frequencies(Eq. (6b)).\nIf we compare the expressions of ωIof the three modes\nin strongly coupled regime, we find Alfv´ en and fast modes\nboth have the damping rate proportional to their quadratic\nwave frequencies (Eq. (5b), (12b)), as they can always in-\nduce oscillations of ions orthogonal to magnetic field. The\nadditional magnetic restoring force exerted on ions result s in\nrelative drift between ions and neutrals and damping to the\nwavemotions. Itindicatesfasterpropagatingwavesaremor e\nstronglydamped. But forslowmodes,wesee fromEq. (13b)\nthat there is no damping to purely parallel propagation with\nθ= 0, since in low- βplasma, slow modespropagatingalong\nmagneticfieldareanalogoustosoundwavesintheabsenceof\nmagneticfield,andhenceundamped.\nWe then compare the expressions of ωIin both strongly\nand weakly coupled regimes. We find the neutral-ion colli-\nsionsatlowwavefrequenciesareresponsibleforcouplingt he\ntwofluidstogether,andmorefrequentcollisionsbringweak er\ndampingto MHD waves. Differently,the dampingratein the4\nweak coupling regime ((6b)) purely depends on ion-neutral\ncollisional frequency. The collisional friction plays the role\nofdissipatingwavesinions.\nAnalogically,thecutoffboundariescanbedeterminedfrom\n|ωI|=|ωR|. Fast modeshave\nk+\nc=2νni\nVAξn, (16a)\nk−\nc=νin\n2VAi, (16b)\nwhich are the same as Eq. (7) for Alfv´ en waves except k±\nc,/bardbl\narereplacedby k±\nc. Slowmodeshave\nk+\nc=2νnicosθ\ncsξnsin2θ, (17a)\nk−\nc=νin\n2csicosθ. (17b)\nThe relations between k±\ncand the decoupling scales of fast\nwaves (see Table 1) are similar to those of Alfv´ en waves in\nparalleldirectionto magneticfield,\nk+\nc=2\nξnkdec,ni, (18a)\nk−\nc=1\n2kdec,in. (18b)\nSlow waves have the same k−\nc,/bardbl=1\n2kdec,in,/bardblas Alfv´ en waves,\nbuttherelationbetween k+\ncandkdec,niislessstraightforward.\nWe will discussthischaracterofslow modesin Section4.\nIn fact, in the context of anisotropic MHD turbulence, the\nwave propagation angle θappearing in k±\ncand decoupling\nscales also depends on scales. In Section 4 we will present\ntheirexactexpressions.\n3.2.Single-fluidapproach\nAlthough the single-fluid approach has a restriction on the\nrange of wave frequencies, i.e. ω < νin, it has the advantage\nof a simpler mathematical treatment. We take the dispersion\nrelations from Balsara (1996) under the single-fluid approx i-\nmation. We also follow the approximation of negligible ion\ndensityappliedintheirworkforthissubsection.\nThedispersionrelationofAlfv´ enwavesis\nλ2+i˜kVAncosθλ−V2\nAncos2θ= 0.(19)\nThesamenotationsareusedasdefinedin theirpaper,\nλ=ω\nk,˜k=k˜L,˜L=VAncosθ\nνni. (20)\nNotice that Eq. (19) can be directly obtainedfromEq. (3) by\nomitting the first term ω3in Eq. (3), which is unimportantat\nlowwavefrequencies.\nTheexactsolutionstoEq. (19)arestraightforward,\nλ=−i˜k±/radicalbig\n4−˜k2\n2VAncosθ. (21)\nAs is shown by Kulsrud&Pearce (1969), the solutions be-\ncome purely imaginary when ˜k >2, which is equivalent to\nk/bardbl>2νni/VAn. From Eq. (7a), we see 2νni/VAn≈k+\nc,/bardblin\nmostlyneutralplasma. Itshowsthesingle-fluidapproachal so\ncancapturethelowercutoffboundary.At˜k <2, we express λas(ωR+iωI)/k. Together with\nEq. (20),we get\nω2\nR=4−˜k2\n4V2\nAnk2cos2θ, (22a)\nωI=−1\n2˜kkcosθVAn=−V2\nAnk2cos2θ\n2νni.(22b)\nIn comparison with the solutions to the two-fluid dispersion\nrelation in strongly coupled regime (Eq. (5)), with ˜k2≪4\nat low wave frequencies, Eq. (22) coincides with Eq. (5) in\nweaklyionizedplasma.\nThe dispersion relation of compressible modes is (Balsara\n1996),\nλ4+i˜kVAnsecθλ3−(c2\nn+V2\nAn)λ2\n−i˜kVAnsecθc2\nnλ+V2\nAnc2\nncos2θ= 0.(23)\nRecallthatwederivedtheanalyticalsolutionstothetwo-fl uid\ndispersion relation under the weak damping approximation\nin Section 3.1. To better capture the wave behavior beyond\ntheweakdampinglimit,herewefollowa differentprocedure\ncalled Newton’s method to obtain the approximatesolutions .\nTosimplifytheproblem,weconsideraparadigmaticcaseofa\nlow-βplasma. Theapproximateexpressionsofwavefrequen-\nciesinsometractablelimitcasesareprovidedinAppendixB .\nWe treat the left-hand side of Eq. (23) as a function fand\nadopt the approximate roots of f= 0at the extreme β→0\nasthe startingpoint,\nλ1,2\n0=−i˜ksecθ±/radicalbig\n4−˜k2sec2θ\n2VAn,(24a)\nλ3,4\n0=±cncosθ. (24b)\nλ1,2\n0andλ3,4\n0correspondto fast and slow waves respectively.\nBasedon λ0,a betterapproximationisgivenby\nλ1=λ0−f(λ0)\nf′(λ0), (25)\nwheref′(λ0)is thederivativeof fatλ0. Thisone-stepitera-\ntioncanalreadyconvergetotheactualsolutionswell. Butt he\nexpressions of λ1can be too complicated to be illustrative.\nTherefore,instead of applying λ1(not shown for simplicity),\nwe findλ1can be furtherreducedat low- βconditionand be-\ncome\nλ1,2\n1≈−i˜ksecθ±/radicalbig\n4−˜k2sec2θ\n2VAn,(26a)\nλ3,4\n1≈ ±cncosθ−c2\nn˜ktan2θcosθ\n2VAni. (26b)\nWe keepλ1,2\n1the same as λ1,2\n0since the modification is neg-\nligible. From Eq. (26a), we see fast waves become nonprop-\nagating when ˜k >2cosθ. That is, k >2νni/VAn, in accor-\ndancewith k+\ncgivenbyEq. (16a).\nWhen˜k <2cosθ, we rewrite Eq. (26) in terms of ωRand\nωI, andget\nω2\nR=4−˜k2sec2θ\n4V2\nAnk2, (27a)\nωI=−˜kksecθ\n2VAn=−V2\nAnk2\n2νni(27b)5\nforfast waves,and\nω2\nR=c2\nsnk2cos2θ, (28a)\nωI=−c2\nn˜kktan2θcosθ\n2VAn=−c2\nnk2\n⊥\n2νni(28b)\nfor slow waves. In weakly ionized medium, they are in good\nagreementwith Eq. (12) and(13) derivedusingtwo-fluidap-\nproach.\nZaqarashvilietal. (2012) arguedthat the appearanceof the\ncutoff wavenumber in Alfv´ en waves is not a physical phe-\nnomenon under the single-fluid description. Nevertheless,\nabove analysis demonstrates in strongly coupled regime, th e\nsingle-fluid approach provides the same analytical wave fre -\nquenciesasthoseobtainedusingtwo-fluidapproachinneutr al\ndominatedmedium. Furthermore,thedifferentanalyticalp ro-\ncedure we applied for single-fluid MHD can inform us with\nthe lowercutoffboundarydirectlyfromthe expressionof ωR\nin the cases of Alfv´ en and fast waves. It indicates the singl e-\nfluid approach is adequate to deal with turbulence damping\nwhendampingoccursin stronglycoupledregime.\nIn addition, we also need to point out, similar to the sim-\nplifiedexpressionof thetwo-fluid ωIat low-βcondition(Eq.\n(13b)), we also see from the single-fluid ωI(Eq. (28b)) that\ndampingisabsentinthecaseofpurelyparallelpropagation of\nslowmodes. Thisconclusionwasearlierreachedthroughnor -\nmal mode analysis in single-fluid approach by Fortezaetal.\n(2007), whereasthe two-fluid approachby Zaqarashvilietal .\n(2011) and the energy equations given by Braginskii (1965)\nbothshownon-zerodampingrateforpurelyparallelpropaga -\ntion of slow modes. Indeed, accordingto the general expres-\nsion ofωIin Eq. (11b), ωIis not zero at θ= 0. But it does\nnot mean the simplified expression of ωIat low-βmedium\n(Eq. (13b)and(28b))isinvalid. Becausefirst,fromEq. (11b )\nwesee thetermwith k/bardblissubdominantandcanbesafelyne-\nglectedatlow- βcondition;andsecond, k/bardblbecomesmoreand\nmore insignificant compared with k⊥towards smaller scales\nduetotheturbulenceanisotropy(seenextsection). Theabo ve\nanalyticaldampingrateswillbenumericallytestedinSect ion\n5.\n4.DAMPINGOF MHD TURBULENCE\nMHD turbulence is a highly non-linear phenomenon, as a\ncascade of turbulent energy towards the undamped smallest\nscale. The remarkable feature of the GS95 model of tur-\nbulence is the hydrodynamic-like mixing motions of mag-\nnetic field lines in perpendicular direction have the same\ntime scale as that of the wave-like motions along field lines.\nThat is, Alfv´ enic turbulence cascades over one wave period ,\nk/bardblVA∼k⊥v⊥, whichis calledthecritical balancecondition.\nCombining the Kolmogorov scaling for the hydrodynamic-\nlike motions v⊥∝k−1/3\n⊥, the critical balance condition nat-\nurally leads to the scale-dependent anisotropy, k/bardbl∼k2/3\n⊥. It\ncannot be overemphasized that the GS95 laws are true only\nin local reference system in respect with the local config-\nuration of magnetic field lines. The importance of the lo-\ncalnotion was not pointed out by GS95, but demonstrated\nin Lazarian&Vishniac (1999) and later numerically testi-\nfied in Cho&Vishniac (2000); Maron& Goldreich (2001);\nChoet al. (2002). This specification on reference system\nmakesthe originalGS95 picturemoreself-consistentanden -\ntails the dependence on scales of turbulence anisotropy. Fo r\nthe sake of tradition and uniformity, we use the notations k/bardblandk⊥whenapplyingthescalingrelations,butkeepinmind\nthat instead of ordinaryFourier componentsof wave vectors ,\nthey should be treated as inverse of scales measured paralle l\nandperpendicularto thelocalmeanmagneticfield.\nThe scaling relations were first introduced for trans-\nAlfv´ enicturbulenceinGS95,andlatergeneralizedfordif fer-\nentregimesofMHDturbulenceincludingsub-Alfv´ enicturb u-\nlence (Lazarian&Vishniac 1999). With the scale-dependent\nanisotropy of MHD turbulence taken into account, the above\nlinearanalysisof wavedampingcanbe usedto set the damp-\ning scales of MHD turbulence. We next briefly describe\nthe cascading of MHD turbulence and list the corresponding\ndampingscalesderivedin PaperI.\n4.1.Alfv´enmodes\nAlfv´ en modes cascade independently from compressible\nmodesandhavecascadingrate (GS95,Lazarian2006)\nτ−1\ncas=/braceleftBigg\nk2/3L−1/3VL, lA<1/k < L, (29a)\nk2/3\n⊥L−1/3VL,1/k < l A,(29b)\nfor super-Alfv´ enicturbulence,i.e., MA=VL/VA>1. Here\nlAis the scale where turbulence anisotropy starts to develop\nand increases towards smaller scales. At scales smaller tha n\nlA, the parallel and perpendicular components of the wave\nvectorwithrespecttothelocaldirectionofmagneticfielda re\nrelatedby\nk/bardbl∼l−1\nA(k⊥lA)2/3. (30)\nSub-Alfv´ enicturbulencehas MA<1,andacascadingrate\n(Lazarian& Vishniac1999)\nτ−1\ncas=\n\n(kvl)2\nωA=k2V2\nL\nk⊥VA, l tr<1/k < L,(31a)\nvl/l⊥=k2\n3\n⊥L−1\n3VLM1\n3\nA,1/k < l tr,(31b)\nwhereltris the transition scale from weak to strong turbu-\nlence. Weak turbulence only evolves in perpendicular direc -\ntion with k⊥increasing during cascade, while strong turbu-\nlencehasthescalingrelation\nk/bardbl∼L−1(k⊥L)2/3M4/3\nA. (32)\nThe dampingscale of turbulencecan be approximatelyde-\ntermined by setting |ωI| ∼τ−1\ncas. With both damping effects\nincluded,thegeneralexpressionofdampingscale is\nkdam,/bardbl=−(νn+V2\nAi\nνin)+/radicalBig\n(νn+V2\nAi\nνin)2+8VAνnlA\nξn\n2νnlA,\n(33a)\nkdam=kdam,/bardbl/radicalBig\n1+lAkdam,/bardbl,1/kdam< lA.(33b)\nforsuper-Alfv´ enicturbulence,and\nkdam,/bardbl=−(νn+V2\nAi\nνin)+/radicalBig\n(νn+V2\nAi\nνin)2+8VAνnLM−4\nA\nξn\n2νnLM−4\nA,\nkdam=kdam,/bardbl/radicalBig\n1+LM−4\nAkdam,/bardbl,1/kdam< ltr.\n(34)\nforsub-Alfv´ enicturbulence. Inmostsituations,onlythe dom-\ninant damping effect needs to be considered. In the case6\nwhen damping is dominated by neutral-ion collisions, from\nEq. (5b),(29b), (30), (31b)and(32), weobtain\nkdam=/parenleftbigg2νni\nξn/parenrightbigg3\n2\nL1\n2V−3\n2\nL (35)\nforsuper-Alfv´ enicturbulencewhen k−1\ndam< lA,and\nkdam=/parenleftbigg2νni\nξn/parenrightbigg3\n2\nL1\n2V−3\n2\nLM−1\n2\nA (36)\nforsub-Alfv´ enicturbulencewhen k−1\ndam< ltr.\nBy applying the scaling relations Eq. (30) and (32) to the\ndecouplingscalesofAlfv´ enmodesinTable1,wefind kdamin\naboveequationsaredirectlyrelatedwith kdec,niby\nkdam= (2\nξn)3\n2kdec,ni (37)\nforbothsuper-andsub-Alfv´ enicturbulence. And kdec,intakes\ntheform\nkdec,in= (νin\nVAi)3\n2L1\n2M−3\n2\nA(38)\ninsuper-Alfv´ enicturbulence,and\nkdec,in= (νin\nVAi)3\n2L1\n2M−2\nA (39)\nin sub-Alfv´ enic turbulence. Here we use the perpendicular\ncomponent k⊥to represent total k, since we assume the de-\ncouplinganddampingscalesaresufficientlysmall wheretur -\nbulence anisotropy is prominent, which is a common occur-\nrencein astrophysicalenvironments.\nWhenneutralviscousdampingplaysamoreimportantrole,\nacombinationofEq. (9),(29b), (30),(31b)and(32) yields\nkdam=ν−3\n4nL−1\n4V3\n4\nL,1/kdam< lA,(40)\nforsuper-Alfv´ enicturbulence,and\nkdam=ν−3/4\nnL−1/4V3/4\nLM1/4\nA,1/kdam< ltr,(41)\nfor sub-Alfv´ enic turbulence. The assumption made here is\nthemeanfreepathofneutral-neutralcollisions lnisrelatively\nsmall compared with lAin super-Alfv´ enic turbulence and L\nin sub-Alfv´ enicturbulence, which is reasonable in most IS M\nconditions.\nItisinstructivetocomparethecutoffwavenumbersderived\nfrom linear MHD waves and the damping scales of MHD\nturbulence. Due to the critical balance, |ωR|=τ−1\ncasholds\nfor Alfv´ enic turbulence. Consequently, the cutoff condit ion\n|ωR|=|ωI|ofMHDwavesisequivalenttothedampingcon-\nditionτ−1\ncas=|ωI|of MHD turbulence. Notice that the wave\npropagation direction, cosθ=k/bardbl/k, follows the scaling re-\nlations given by Eq. (30) and (32), and is scale-dependent.\nBy takingthisintoaccount,the k+\nccorrespondingtoEq. (7a)\nand(10)arefullyconsistentwith kdamexpressedbyEq. (35),\n(36),(40),and(41). InTable2,wesummarizetheexpression s\nforthecutoffboundaries k±\ncofthethreeturbulencemodesby\ntaking the scaling relations of MHD turbulence into account .\n”NI” and ”NV” represent neutral-ion collisional and neutra l\nviscous damping. The asterisks indicate k+\nccoincides with\nkdamincorrespondingsituations.Regarding the relation between the cutoff boundaries and\ndecouplingscales, fromEq. (37), (38) and(39),we deduce\nk+\nc= (2\nξn)3\n2kdec,ni, (42a)\nk−\nc= 2−3\n2kdec,in. (42b)\ninthe caseof neutral-ioncollisionaldamping. Thediffere nce\nbetweenthemisindependentofturbulenceproperties.\nTABLE2\nCUTOFF BOUNDARIES OF MHDMODES\nk+\nc k−\nc\nAlfv´ enSuperNIEq. (35)∗/parenleftbigνin\n2VAi/parenrightbig3\n2L1\n2M−3\n2\nANV Eq. (40)∗\nSubNIEq. (36)∗/parenleftbigνin\n2VAi/parenrightbig3\n2L1\n2M−2\nANV Eq. (41)∗\nFast Eq. (16a) Eq. (16b)\nSlowSuper/parenleftbig2νni\ncsξn/parenrightbig3\n4L−1\n4M3\n4\nA∗/parenleftbigνin\n2csi/parenrightbig3\n2L1\n2M−3\n2\nA\nSub/parenleftbig2νni\ncsξn/parenrightbig3\n4L−1\n4MA∗/parenleftbigνin\n2csi/parenrightbig3\n2L1\n2M−2\nA\n4.2.Compressiblemodes\nCho&Lazarian(2003)foundfastmodesforminbothhigh-\nand low-βmedia and follow a cascade similar to the one\nin acoustic turbulence. The cascading rate of fast modes is\n(Cho& Lazarian2003; Yan&Lazarian2004)\nτ−1\ncas=\n\n(k\nlA)1\n2V2\nA\nVf, M A>1,(43a)\n(k\nL)1\n2V2\nL\nVf, M A<1,(43b)\nwhereVfis thephasespeedoffast modes. It takesthe form\nVf=/radicalbigg\n1\n2(c2s+V2\nA)+1\n2/radicalBig\n(c2s+V2\nA)2−4c2sV2\nAcos2θ\n(44)\ninstronglycoupledregime,and\nVf=/radicalbigg\n1\n2(c2\nsi+V2\nAi)+1\n2/radicalBig\n(c2\nsi+V2\nAi)2−4c2\nsiV2\nAicos2θ\n(45)\ninweaklycoupledregime. Thetransitionof Vftakesplaceat\nkdec,iniftheturbulencesurvivesdamping. Fastmodescascade\nradially,so the wave propagationdirectioncan be consider ed\nfixedduringthe cascade. Theintersectionscale between τ−1\ncas\nwithVffromEq. (44)and |ωI|(Eq. (11b))correspondstothe\ndampingscale\nkdam=\n\nl−1\n3\nA\n2νniV2\nA(c2\ns+V2\nA−2V2\nf)\nVf/bracketleftBig\nξnV2\nA(c2s−V2\nf)+ξic2sV2\nf/bracketrightBig\n2\n3\n,(46a)\nL−1\n3\n2νniV2\nL(c2\ns+V2\nA−2V2\nf)\nVf/bracketleftBig\nξnV2\nA(c2s−V2\nf)+ξic2sV2\nf/bracketrightBig\n2\n3\n,(46b)7\nforsuper-andsub-Alfv´ enicturbulencerespectively. Inl ow-β\nplasma,theaboveexpressionscanbewrittenas\nkdam=\n\n/parenleftbigg2νni\nξn/parenrightbigg2/3\nl−1/3\nAV−2/3\nA, M A>1,(47a)\n/parenleftbigg2νni\nξn/parenrightbigg2/3\nV4/3\nLL−1/3V−2\nA, MA<1.(47b)\nBy comparing the damping scale with the neutral-iondecou-\nplingscale (Table1),we find\nkdam\nkdec,ni=\n\n/parenleftbigg2\nξn/parenrightbigg2\n3\n(kdec,nilA)−1\n3, M A>1,(48a)\n/parenleftbigg2\nξn/parenrightbigg2\n3\n(kdec,niL)−1\n3M4\n3\nA, MA<1.(48b)\nSince the terms (kdec,nilA)−1and(kdec,niL)−1are usually\nmuch smaller than 1,kdam< kdec,niis a commonoccurrence.\nItmeansthedampingoffastmodestakesplacebeforeneutral -\nion decoupling. It is worthwhile to mention that since the\ncutoffscale isalwayssmaller thanthe neutral-iondecoupl ing\nscale for fast modes (Eq. (18a)), the cutoff does not play a\nroleinthedampingoffast modes.\nSlow modes cascade passively and have the same τ−1\ncasas\nAlfv´ en modes (GS95). According to the critical balance, th e\ndampingcondition τ−1\ncas=|ωI|is equivalentto k/bardblVA=|ωI|,\nwhile the cutoff condition |ωR|=|ωI|givesk/bardblcs=|ωI|.\nIn a low-βmedium, τ−1\ncasis always larger than |ωR|on a cer-\ntain scale. It means the cutoff of slow waves happens before\nthe equilibrium between τ−1\ncasand|ωI|is reached. Therefore\nthe dampingscale of slow modesdependson the cutoffscale\n1/k+\nc(see Table 2 for the expressions). On the other hand,\nslow modes dissipate subsequently after Alfv´ en modes are\ndampedout,sotheirdampingscaleshouldnotbesmallerthan\nthat of Alfv´ en modes. Accordingly, in combination with Eq.\n(35)and(36),we have kdamofslowmodesas\nkdam=min/bracketleftBigg/parenleftbigg2νni\ncsξn/parenrightbigg3\n4\nL−1\n4M3\n4\nA,/parenleftbigg2νni\nξn/parenrightbigg3\n2\nL1\n2V−3\n2\nL/bracketrightBigg\n.\n(49)\nwhen1/kdam< lAin super-Alfv´ enicturbulence,and\nkdam=min/bracketleftBigg/parenleftbigg2νni\ncsξn/parenrightbigg3\n4\nL−1\n4MA,/parenleftbigg2νni\nξn/parenrightbigg3\n2\nL1\n2V−3\n2\nLM−1\n2\nA/bracketrightBigg\n.\n(50)\nwhen1/kdam< ltrin sub-Alfv´ enic turbulence. In fact, the\nratio between the two terms, k+\ncof slow modes and kdamof\nAlfv´ enmodesinEq. (49) and(50) reveals\nk+\nc,s\nkdam,A=\n\n/bracketleftbigg/parenleftbiggξn\n2/parenrightbigg/parenleftbiggVA\ncs/parenrightbigg\n(kdec,ni,/bardbllA)−1/bracketrightbigg3\n4\n, MA>1,\n(51a)\n/bracketleftbigg/parenleftbiggξn\n2/parenrightbigg/parenleftbiggVA\ncs/parenrightbigg\n(kdec,ni,/bardblL)−1M4\nA/bracketrightbigg3\n4\n, MA<1,\n(51b)\nwherekdec,ni,/bardbl=νni/VA. The terms (kdec,ni,/bardbllA)−1and\n(kdec,ni,/bardblL)−1are much smaller than 1. If theβvalue of the\nmediumis not extremelysmall, k+\nc,s< kdam,Ausually stands.Thus we can safely take kdam=k+\ncfor slow modes in most\ncases.\nSlow modes have the same relation between k−\ncandkdec,in\nas Alfv´ en modes (see Eq. (42b)). We then examine the rela-\ntion between k+\ncandkdec,ni. The neutral-iondecouplingcon-\nditioncsnk=νnicanbewrittenas\nξnc2\nsnk2\n2νni=ξncsnk\n2, (52)\nwhilethecutoffconditionis(see Eq. (13))\nξnc2\nsk2\n⊥\n2νni=csk/bardbl. (53)\nUnderthe conditionof strong turbulenceanisotropy,i.e. k∼\nk⊥andk⊥≫k/bardbl, we see the cutoffconditioncan be reached\non a larger scale than the neutral-ion decouplingscale. Thu s\nslowmodesfallintoadifferentsituationfromfastandAlfv ´ en\nmodes (see Eq. (18a) and (42a)) and can have k+\nc< kdec,ni.\nBut in terms of damping, due to kdam=k+\nc, slow modes are\nsimilar to fast modesand also dampedout when neutralsand\nionsarestronglycoupledtogether.\nComparingthissectionwithSection3,wecanseethemain\ndifference between the damping processes of linear MHD\nwaves and MHD turbulence is the consideration of scale-\ndependent anisotropy, which depends on turbulence proper-\nties, e.g. super- or sub-Alfv´ enic turbulence, and turbule nce\nregimes, e.g. weak or strong turbulence. For our study on\nturbulence damping of Alfv´ en and slow modes, a fixed wave\npropagation direction, e.g. purely parallel or perpendicu lar\npropagation, has no physical meaning. Furthermore, as re-\ngards scaling estimations, e.g. evaluation of the damping\nscales, turbulence anisotropy is the essential physical in gre-\ndient. Ignoranceof it can lead to severely wrongconclusion s\ninrelatedastrophysicalapplications.\n5.NUMERICAL TESTSINDIFFERENTPHASES OF\nPARTIALLYIONIZEDISMAND SOLAR\nCHROMOSPHERE\nThe generality of the two-fluid approximationallows us to\napplyourresultstoa widevarietyofastrophysicalsituati ons.\nWenextapplytheanalysisonturbulencedampingindifferen t\nphasesofpartiallyionizedISM.Toavoidconfusion,westre ss\nthat bynumericaltests, we meannumericallysolvingthe ful l\ndispersion relations with given physical parameters, but n ot\nperforminganyMHD simulations.\nTABLE3\nPARAMETERS USED FOR DIFFERENT PHASES OF PARTIALLY\nIONIZED ISMAND SOLAR CHROMOSPHERE .\nWNM CNM MC DC SC\nnH[cm−3]0.4 30 300 1044.2×1012\nne/nH0.1 10−310−410−61.78×10−2\nT[K] 6000 100 20 10 6220\nB[µG]8.66 8.66 8.66 86 .6 6.96×107\nβ 0.22 0.23 0.20 0.03 0 .03\nMA 0.4 2.9 9.2 5.3 0 .4\nTable 3 lists the typical physicalparametersfor warm neu-\ntralmedium(WNM),coldneutralmedium(CNM),molecular\ncloud (MC) and dense core in a molecular cloud (DC). The8\nparameters nH,ne/nH, andTare taken from Lazarianetal.\n(2004). We assume temperatures are equal for all species.\nMagnetic field strengths are√\n3×the half maximum values\nof the Zeeman measurements from figure 1 in Crutcheretal.\n(2010) as the total strength of the three-dimensional mag-\nnetic field. In addition, we adopt the drag coefficient as\nγd= 3.5×1013cm3g−1s−1from Draineet al. (1983)4, and\nthedrivingconditionofturbulence\nL= 30pc, VL= 10kms−1. (54)\nThe masses of ions and neutrals are mi=mn=mHfor\nWNM and CNM, and mi= 29mH,mn= 2.3mHfor the\notherenvironments(Shu1992).\nNot only ISM, observations confirm MHD waves are also\nubiquitous in the solar atmosphere (e.g. DePontieuetal.\n2007,2012). ThedampingprocessofMHDwavesinthepar-\ntiallyionizedsolaratmosphere,i.e. photosphereandchro mo-\nsphere,hasbeenwidelysuggestedasaheatingmechanismof\ntheexternalsolaratmosphere(e.g. Osterbrock1961;Hollw eg\n1986; Narain&Ulmschneider 1996; Goodman 2000, 2001).\nThus we also extend our analysis to the solar chromosphere\n(SC) environment. In reality, the physical parameters vary\nwithaltitudefromthe photospherelevel. Forthesake ofsim -\nplicity, we neglect the vertical variation, but consider a t oy\nmodelinahomogeneousmediumwithallparameterstakenat\namedianheight. Thissimplifiedanalysisservesasaparadig -\nmatic case of dampingin chromosphere-likeenvironment. A\nmoresophisticatedmodelmayberequiredwhenmakingcom-\nparisonswith observationaldata.\nWe adopt model C for quiet sun in Vernazzaet al. (1981)\nat a height 1280km and only consider hydrogen. Magnetic\nfield strength is obtained by assuming B=Bph(ρ/ρph)0.3\n(Leake& Arber 2006). With Bph= 1.5×103G and\nρph= 2.73×10−7g cm−3at the photospheric level\nused, we get B= 69.6G. For the neutral-ion collisional\ncross-section, we use the value ( ≈1×10−14cm2) pro-\nposed by Vranjes&Krstic (2013). Their approach with en-\nergy dependenceand quantumeffects containedyields a col-\nlisional cross-section two orders of magnitude larger than\nthat obtained from the hard sphere model (Braginskii 1965).\nThen drag coefficient can be calculated with the expression\n(Braginskii1965; Soleret al.2013a),\nγd=1\n2mH/radicalbigg\n16kBT\nπmHσni. (55)\nBesides, we use the height of chromosphere as the injection\nscaleL= 2500km, and assume turbulent velocity at Las\nVL= 30km s−1based on the velocity amplitude measure-\nment of the chromosphericwaves by Okamoto& DePontieu\n(2011). Theparametersare alsolisted inTable3.\nAccording to the values of βlisted in Table 3, in the fol-\nlowing comparisons with numerical results, we will use the\nsimplified analytical results introduced in previous secti ons\nfordealingwithlow- βmedia.\n5.1.Alfv´enmodes\nWe assume the energy is equally distributed into Alfv´ en,\nfast, and slow modes at the scale of turbulence driving.\n4This constant γdapplies in conditions like molecular clouds where the\nrelative velocities between neutrals and ions are relative ly low and the ef-\nfective cross-section of neutral-ion collisions is much la rger than geometric\ncross section (seeShu 1992).Fig. 1(a) illustrates the damping of the sub-Alfv´ enic turb u-\nlenceinWNM.ThecascadingrateisgivenbyEq. (31b). No-\nticethescalesshownarewithinthestrongturbulenceregim e.\nWe show the damping rates obtained by numerically solving\nthe two-fluid dispersion relations with only neutral-ion co l-\nlisional damping (Eq. (3)) and both neutral-ion collisiona l\nand viscous damping (see Paper I). In this case, neutral-ion\ncollisionsandneutralviscosityhavecomparabledampinge f-\nficiencies in the strongly coupled regime, so the general ex-\npression of kdam(Eq. (34)) applies. The analytical damping\nrate is from Eq. (4b), which coincideswith the numerical re-\nsultatbothsmallandlargewavenumbers,butunderestimate s\nthe numerical result at intermediate wavenumbers when |ωI|\nand|ωR|areofthesameorder. Thesingle-fluiddampingrate\nis givenby Eq. (22b), which tracesthe numericalone well at\nsmallwavenumbers. Thenumericalsolutionsdonotexhibita\nnonpropagatingintervalwith ωR= 0.\nFig. 1(b) and 1(c) display the damping of super-Alfv´ enic\nturbulence in CNM and DC. The solutions with both damp-\ning effects are in full coincidence with those considering\nonlyneutral-ioncollisionaldampingduetonegligiblevis cous\ndamping. Accordingly, kdamforneutral-ioncollisionaldamp-\ning (Eq. (35)) applies, which coincides with the lower cutof f\nboundary k+\nc. By comparing with numerical results, we are\nconvinced that the single-fluid approach can provide a good\napproximationof the actual damping rate down to the damp-\ning scale (or cutoff boundary) of Alfv´ en modes. The pure\nimaginary solutions are omitted from the numerical results ,\ncorrespondingto thecutoffregions.\nFig. 1(d) shows the damping of Alfv´ en modes in sub-\nAlfv´ enicSC. Thefilledcirclesrepresenttheanalyticalda mp-\ningrate includingbothdampingeffects(seePaperI),\n|ωI|=/bracketleftbig\nτ−1\nυ(τ−1\nυ+(1+χ)νni)+k2cos2θV2\nAi/bracketrightbig\nχνni\n2[k2cos2θV2\nAi+χτ−1υνni+(τ−1υ+(1+χ)νni)2].\n(56)\nObviously neutral viscosity dominates over neutral-ions c ol-\nlisions in damping Alfv´ en modes. It meets the prediction in\nPaper I that neutral viscosity can play a more important role\nofdampinginsub-Alfv´ enicturbulencethaninsuper-Alfv´ enic\nturbulence. Theexactcriteriafordeterminingtherelativ eim-\nportance between neutral viscosity and neutral-ion collis ions\nin damping super- and sub-Alfv´ enic turbulence can be found\ninPaperI.Asinthecase ofWNM, thegeneralexpressionby\nEq. (34) including both damping effects can give a good ap-\nproximationof kdam. But herewe present kdamwitha simpler\nform (Eq. (41)) derived from viscous damping alone as an\napproximate kdam. We see in SC condition, owing to the ex-\nistenceofneutralviscosity,thedampingscaleisconsider ably\nlarger than that predicted by neutral-ioncollisional damp ing,\nand is also larger than the neutral-ion decoupling scale. It\nshowsnotonlyneutral-ioncollisions,neutralviscositys hould\nalso be considered as an important mechanism in damping\nAlfv´ enmodes.\nBesides dampingrate, the propagatingcomponentof wave\nfrequency |ωR|is also present in Fig. 2 as an example in the\ncase of Alfv´ en modes in MC. The same symbols are used as\ninFig. 1,exceptthatthedashedlineandopensquaresarethe\nnumericalandanalytical(Eq. (5a)and(6a)) |ωR|respectively.\nNoticethat aspointedout earlierinSection 4, dueto the cri t-\nical balance, |ωR|in strongly coupled regime coincides with\nthe cascadingrate. We find |ωR|starts to decay at kdec,ni, and\niscutoffat k+\nc. Then|ωR|arisesagainat k−\nc,butisonlyfully9\nresumed after reaching kdec,in. Thus the region [kdec,ni,kdec,in]\ncontainsthecutoffzone [k+\nc,k−\nc]. Differentfromwavedamp-\ning, which has dampingrate insensitive to decouplingscale s,\nthe propagating behavior of waves depends on properties of\nfluid coupling,andcan only be studiedoutside [kdec,ni,kdec,in]\nregion.\n5.2.Fastmodes\nFig. 3(a)-3(d) display the comparison between the analyt-\nical and numerical damping rates of fast modes in different\nISM conditionsand SC. The cascading rate is from Eq. (43),\nwith the wave propagationangle fixed at 45◦. The analytical\ntwo-fluid (Eq. (12b) and (6b)) and single-fluid (Eq. (27b))\ndamping rates are in excellent agreement with the numerical\nresultonscaleslargerthanthe cutoffboundary 1/k+\nc.\nBoth|ωR|and|ωI|are shown in Fig. 4 for fast modes in\nMC. The numerical |ωR|is obtained by numerically solving\nthe dispersion relation (equation (51) in Soleretal. 2013a ).\nAnalytical |ωR|is from Eq. (12a) and (14). Similar to the\ncase of Alfv´ en modes, the cutoff interval is contained with in\n[kdec,ni,kdec,in], and only outside [kdec,ni,kdec,in], fast waves\nhavephasespeedequaltotheAlfv´ enspeed.\nIn all the conditions present, fast modes are damped out\nwhen neutrals and ions are strongly coupled, i.e. kdam<\nkdec,ni. Inparticular,fastmodesinWNMarethemostseverely\ndamped among the ISM phases. WNM has much lower to-\ntal density than other ISM conditions, which leads to a faste r\nwavephasespeedandlowerneutral-ioncollisionalfrequen cy.\nAccordingly, fast modes in WNM have a lower cascading\nrate (Eq. (43)) but a higher damping rate (see discussions\nin Section 3.1). It means the intersection between the two\nrates can take place at a large scale. In SC, high wave phase\nspeedcomesfromstrongmagneticfield strengthand leadsto\nalowcascadingrate. Althoughlargeiondensityyieldsahig h\nneutral-ion collisional frequency and low damping rate (Eq .\n(12b)), it can still exceed the substantially slow cascade o n\na large scale. Therefore fast modes in SC are also severely\ndamped. Theverticaldashedlinesindicatethedampingscal e\ngiven by Eq. (47). It shows the approximate kdamat low-β\nlimit is consistent with numerical results. In fact, accord ing\nto Eq. (46), kdamhas a dependence on the wave propaga-\ntiondirection. Fig.5presentsthedampingscale givenbyEq .\n(46b) as a function of θin WNM as an example. The damp-\ning scale in other cases show similar results. We find despite\ntheexistenceofanisotropy,thedependenceof kdamonθisso\nweakthataconstantdampingscalecanbeapplied. Itleadsto\na conclusion that the energy distribution of fast modes keep s\nisotropicoverall existentscales.\n5.3.Slowmodes\nFig. 6(a)-6(d) show the results of slow modes. The ana-\nlytical damping rate for two-fluid description is given by Eq .\n(13b) for k < k+\ncand Eq. (6b) for k > k−\nc, and the single-\nfluid damping rate is from Eq. (28b). The analytical |ωR|\nfrom Eq. (13a) for k < k+\ncand Eq. (15) for k > k−\ncis also\nshownin Fig. 7 in thecase of MC. FromFig. 7, we findsim-\nilar to other wave modes at high wave frequencies, although\nωRreemergesatthecutoffboundary k−\nc,thephasespeedcan\nonly reach csiat the slightly smaller ion-neutral decoupling\nscale.\nSlow modes exhibit a distinctive feature on wavenumbers\nabovekdec,ni. Besides the slow modes in ions, a new sort of\nslow modes emerges in neutrals, which is the ”neutral slowmode” reported in Zaqarashviliet al. (2011). It is also the\n”neutral acoustic mode” coupled with the magnetoacoustic\nwaves discussed in Soleretal. (2013a). It is generated be-\nlowtheneutral-iondecouplingscalewhentheslowmodeson\nlarger scales impose compression and produce perturbation s\nin neutral fluid. In the rest of the paper,we will term the two\nbranchesofslowmodesbelowthescale 1/kdec,nias”neutral”\nand ”ion” slow modesrespectively, and term the slow modes\nin stronglycoupledregimeand ”ion”slow modestogetheras\n”usual”slowmodes. Wenextputourfocusontheneutraland\nionslow modeswithinthe wavenumberrange [kdec,ni,kdec,in].\nIf we extend the wave frequencies of slow modes in\nstronglycoupledregime(Eq. (13))tothescale kdec,ni,wefind\ntheyreachthevalues\nω2\nR(kdec,ni)≈ν2\nnicos2θ, (57a)\nωI(kdec,ni)≈ −νni\n2sin2θ, (57b)\nwhereweusetheapproximations ξn∼1andcsn∼cs. Start-\ningfromkdec,ni,the slowmodesarisinginneutralshavewave\nfrequenciesas\nω2\nR=c2\nsnk2, (58a)\nωI=−νni\n2sin2θ. (58b)\nThe propagating component ωRhas the expression corre-\nsponding to pure acoustic waves in neutrals. The imagi-\nnary part ωIhas the same value as ωI(kdec,ni)in Eq. (57b).\nBy setting |ωR|=|ωI|, we can get its cutoff wavenumber\nνni/(2csn), which is equal to kdec,ni/2. Below kdec,in, ions\nstart to separate from neutrals. ωIof the neutral slow modes\nbecomes\nωI=−νni\n2. (59)\nThe above analytical |ωI|(Eq. (58b) and (59)) are shown by\nfilled circles in Fig. 6 and 7. Due to the strong anisotropy at\nrelatively small scales, sinθ∼1and the difference between\nEq. (58b) and (59)) is indistinguishable on the plots. The\nfilled squares in Fig. 7 specifically show the analytical |ωR|\n(Eq. (58a)). Although the slow modes in neutrals have the\nproperties of pure acoustic waves, it is induced by the slow\nmodesin strongly coupledregime and is one of the solutions\ntothedispersionrelationofthemagnetoacousticwavesinp ar-\ntially ionized plasma. We treat it as an additional branch of\nslowmodesinouranalysis.\nTherelativedampingefficiencyof theionandneutralslow\nmodes at high wave frequencies depends on the ionization\nfraction of the plasma. The ratio of their damping rates at\nk > k dec,inisχ(=ρn/ρi). The higher degree of ionization\nin WNM and SC results in a smaller ratio between the two\ndampingratescomparedwithotherphases.\nAnotherdifferencebetweenWNM andotherenvironments\nis the absence of cutoff region in WNM. Also, we see from\nFig. 6(a) that the cascading rate of slow modes is above the\ndamping rate of ion slow modes over all the scales. Thus\nthe usual slow modes in the case of WNM survive neutral-\nioncollisional damping. For the rest conditions,the dampi ng\nscales all coincide with the cutoff boundary k+\nc(see Table 2\nfortheexpression).\nWe next turn to the ion slow modes within the interval\n[kdec,ni,kdec,in]. In the vicinity of kdec,ni, ions stay tightly cou-\npledwith neutrals,andtheirmotionsareoverwhelmedbythe\nacoustic perturbations in neutrals. That leads to inefficie nt10\n10010110210310410510−210−1100101\nkVA\nνni|ωI|/ ν nikdec,nikdamkdec,in\n(a) WNM10−110010110210310410510−310−210−1100101102103104\nkVA\nνni|ωI|/ ν ni\n  \nkc−\n(kc+)kdec,nikdamkdec,in\n(b) CNM\n10−110010110210310410510610710−310−210−1100101102103104105106\nkVA\nνni|ωI|/ ν ni\n  \nkdec,in\n(kc+)kdec,nikdamkc−\n(c) DC10−210−110010110210310410510610710810−710−610−510−410−310−210−1100101102103\nkVA\nνni|ωI|/ ν ni\n(kc+)kdec,inkdamkdec,nikc−\n(d) SC\nFIG. 1.—|ωI|/νnivs.kVA/νniofAlfv´ enmodesinvariousenvironments. Solidlinesarenu mericaldampingratesofneutral-ion collisionaldamping. Dashed\nlines are numerical damping rates including both neutral-i on collisional and viscous damping effects. Open circles ar e analytical damping rates corresponding\nto neutral-ion collisional damping. Triangles show the sam e as open circles, but for single-fluid approach. The filled ci rcles in (d) show the analytical damping\nrate considering both damping effects. Dash-dotted lines a re the cascading rates of Alfv´ en modes. They intersect with the damping rates at kdam. Theanalytical\nwavenumbers kdec,ni,kdec,in,kdam,andk±\ncare denoted by vertical dashed lines.\n10−110010110210310410510−310−210−1100101102103104\nkVA\nνni|ω|/ ν ni(kc+)kdamkdec,nikc−kdec,in\nFIG. 2.—|ωR|/νniand|ωI|/νnivs.kVA/νniof Alfv´ en modes in MC.\nThesamesymbols asin Fig. 1are used,except that thedashed l ine and open\nsquares arethe numerical and analytical |ωR|.collisional friction and substantially reduceddampingra te of\nthe ion slow modes. The damping rate is determined by the\ndifferencebetween νni/2and|ωI|ofneutralslowmodes(Eq.\n(58b)),\n|ωI|=νni\n2−νni\n2sin2θ=νni\n2cos2θ.(60)\nOn the other hand, ions are constrained to magnetic field\nlines. Driven by the magnetic perturbation, the slow modes\nwithaneffectivesoundspeed\ncs,eff=/radicalbig\nξicsi=/radicalbig\n2γkBT/mr (61)\nas the phase speed are generated in ions, where the reduced\nmass ismr=ρ/ni. It originates from the coupling state\nions remain with neutralsin this range of wavenumbers. No-\ntice thatcs,effis differentfrom cs=/radicalbig\nγkBT/mrof the slow\nmodesin stronglycoupledregime. The reducedmass in csis\nmr=ρ/n, wheren= 2ni+nnis the total numberdensity.\nIt does not distinct the neutral and ionized componentssinc e\nneutrals and ions are frozen together and move as one fluid.\nBut in the range [kdec,ni,kdec,in], ions experiencefrequent col-11\n10−310−210−110010110210−610−510−410−310−210−1100101102\nkVA\nνni|ωI|/νnikdec,nikdamkdec,in\n(a) WNM10−210−110010110210310410−410−310−210−1100101102103104\nkVA\nνni|ωI|/νnikdamkdec,inkc−kc+\nkdec,ni\n(b) CNM\n10−210−110010110210310410−510−410−310−210−1100101102103104105\nkVA\nνni|ωI|/νnikdamkdec,in\nkc−kc+kdec,ni\n(c) DC10−310−210−110010110210310−610−510−410−310−210−1100101102\nkVA\nνni|ωI|/νnikdamkdec,inkc+kc−kdec,ni\n(d) SC\nFIG. 3.— Sameas Fig. 1 butfor fast modes. Hereweset θ= 45◦as an example.\n10−210−110010110210310410−410−310−210−1100101102103104\nkVA\nνni|ω|/νni\nkdamkc+kc−\nkdec,nikdec,in\nFIG. 4.— Sameas Fig. 2 butfor fast modes.\nlisionswithneutralsandhencecannotoscillatefreely. Me an-\ntime, the occasional collisions acting on neutrals are inad e-\nquate to transfer inertia instantly to neutrals. As a result , we\nsee fromthe expressionof cs,effthat ions alone carrythe total01020304050607080908.158.28.258.38.358.48.458.5x 10−20\nθkdam\nFIG. 5.—kdamof fast modes as afunction of θin WNM.\nmassofthe mediumto move.\nThe ion slow modes can only become propagating when\nthe real wave frequency |ωR|=cs,effkcosθbecomes larger\nthanthe dampingrate |ωI|(Eq. (60)). By applyingthe cutoff12\ncondition |ωR|=|ωI|, weobtainthe cutoffscale\nkc,t1=νnicosθ\n2cs,eff, (62)\nas the lower wavenumber boundary of the ion slow modes\nwithin[kdec,ni,kdec,in]. Theionslowmodesemergeat kc,t1,but\nthephasespeed cs,effcanonlybereachedabovethewavenum-\nber\nkdec,t1=νnicosθ\ncs,eff, (63)\nsatisfying |ωR|=νnicos2θ. It actually signifies ions no\nlonger follow the acoustic motions in neutrals, but start to\ndevelop their own wave modes propagating along magnetic\nfield.\nTheappearanceofthepropagatingionslowmodesinduces\na new component of the damping rate, associated with the\nwave motions. Since the ion slow modes are driven by mag-\nnetic perturbationand the propagationis guided by magneti c\nfield, no transverse compression can be produced. Therefore\ndifferent from the case for slow modes in strongly coupled\nregime, there is only damping to the parallel propagation.\nCorrespondingly,thenewcomponentofdampingrateis\n|ωI|=ω2\nR\n2νni=c2\ns,effk2cos2θ\n2νni. (64)\nWenowgetthecompleteexpressionsofthewavefrequencies\noftheionslowmodeswithin [kdec,ni,kdec,in]\nω2\nR=c2\ns,effk2cos2θ, (65a)\nωI=−/parenleftbiggνni\n2cos2θ+c2\ns,effk2cos2θ\n2νni/parenrightbigg\n.(65b)\nTherelativeimportanceofthetwotermsinEq. (65b)changes\nwithk. By equaling the two terms of the damping rate, we\nobtainthetransitionscale\nktran=νni\ncs,eff. (66)\nWhenk < k tran, the first term dominates the damping rate.\nIt is independent of ωRand corresponds to the decreasing\n|ωI|withkshown in Fig. 6 and Fig. 7 under the condi-\ntion of scale-dependent anisotropy. While on wavenumbers\nbeyondktran, the second terms becomes dominant, which is\nproportional to ω2\nRand increases towards larger k. Notice at\nktran,|ωR|=νnicosθcan be fulfilled, which coincides with\nthe|ωR|value atkdec,ni(Eq. (57a)) and is equivalent to the\ncondition cs,effk=νni. We see ktrannot only represents the\ntransitionindampingrate,butalsocorrespondstothe crit ical\nwavelength within which the effective sound crossing time\nis equal to the neutral-ion collisional time. That is, below\nthe scale1/ktran, the disturbanceassociatedwith the ionslow\nmodes cannot be effectively transmitted to neutrals throug h\nneutral-ion collisions. Neutrals further drift apart from ions,\nandionsarelesseffectivelycoupledtoneutrals. Thediffe ren-\ntial motions between ions and neutrals become more signifi-\ncant, which consequently causes stronger damping to the ion\nslowmodes.\nThepropagatingcomponent |ωR|startstodecaywhen |ωR|\nreachesthe valueof νni,correspondingtothe scale\nkdec,t2=νni\ncs,effcosθ. (67)Neutrals become essentially unaffected by the ion slow\nmodes. Subsequently |ωR|and|ωI|are in equality and the\nionslow modesare cutoffatthescale\nkc,t2=2νni\ncs,effcosθ, (68)\nwhichisalsotheupperwavenumberboundaryoftheionslow\nmodeswithin [kdec,ni,kdec,in].\nThe analytical |ωI|(Eq. (65b)) is shown by open circles\nwithin[kdec,ni,kdec,in]forall the environmentsin Fig. 6 and7.\nAnd|ωR|givenbyEq. (65a)isalsodisplayedbyopensquares\nin Fig. 7. The scales kc,t1andkc,t2are indicated by vertical\ndashedlinesinFig. 6(a)-6(d). Thethreeshortverticaldas hed\nlines within [kc,t1,kc,t2]in Fig. 7 indicate kdec,t1,ktran,kdec,t2\nfromlarge to small scales respectively. Their expressions us-\ningscale-dependentanisotropyare givenin AppendixC.\nTable 4 is a summary diagram of the critical scales and\nwave frequencies of ion slow modes, as well as those of the\nslowmodesinbothstronglyandweaklycoupledregimes. All\nthe expressions apply in low- βenvironment. The scale ex-\npressionspresentherecontainthepropagationdirectiona ngle\nθ. If we disregard the variance of cosθwith scales, we find\ntheyaresymmetricallylinkedby\nktran=kdec,t1\ncosθ=2kc,t1\ncosθ≈ξnk+\nc\n2√ξicosθ(69a)\n=kdec,t2cosθ=kc,t2cosθ\n2=2√ξicosθk−\nc\nξn,(69b)\nwhere the last term in Eq. (69a) is obtained by assuming\nsin2θ∼1andcs∼csi. If we only focus on the relation\namongthe criticalscalessmallerthan k−1\ntran, we find\n2kdec,t2=kc,t2=4√ξi\nξnk−\nc=2√ξi\nξnkdec,in.(70)\nThe same relation also holdsfor kdec,ni,k+\nc,k−\nc, andkdec,inof\nAlfv´ en and fast modes. That is (see expressions in Table 1\nandEq. (7) and(16))\n2kdec,ni=k+\nc=4√ξi\nξnk−\nc=2√ξi\nξnkdec,in (71)\nwhenξn∼1. And again we do not consider the change of\ncosθwithkforAlfv´ enwaveshere. Infact,thewavebehavior\nof the ion slow modes below the scale k−1\ntranfully resembles\nAlfv´ enwavesoverthewholerangeofscaleswepresent,with\nVAireplacedby csi. Noticesimilarto cs,eff=√ξicsi, thereis\nVA=√ξiVAi.\nAboveanalysissuggests that with ktranacting as a dividing\nline, the wave spectrum can be divided into two zones, i.e.\nk < k tranandk > k tran. The two zones have similarities in\nthesensethatthescales k+\nc,kc,t1,kdec,t1inonezoneplaysim-\nilar roles as kc,t2,k−\nc,kdec,inin the other. But there also exist\ndifferences,resultingfromthevaryingcouplingstatebet ween\nneutralsandionswithscales. Asalreadydiscussed,oneoft he\nmain differences is for the slow modes in the strongly cou-\npledregime,thereisnodampingforparallelpropagation(E q.\n(13b)), but for the ion slow modes within [ktran,kc,t2], damp-\ning only appears for purely parallel propagation (Eq. (65b) ).\nIn addition, we observe |ωI|within[kc,t1,ktrans](Eq. (65b))\ncanbeapproximatelywrittenas\n|ωI| ∼ξicos2θ\nξnνin\n2. (72)13\nThefactor ξicos2θ/ξnreflectsmuchweakerfrictionaldamp-\ning of the ion slow waves compared with |ωI|(Eq. (6b)) of\ntheslow modesinweaklycoupledregime.\nTo reinforce our understanding on the wave behavior, we\nalsoprovidetheforceanalysisforbothfastandslowwavesi n\nAppendixD.\nThe neutral slow modes originate from the compression\nproducedby magnetoacoustic waves. It can only develop af-\nter neutrals decouple from ions and propagate with a sound\nspeed in neutrals5. In terms of turbulence properties, be-\nlow the scale 1/kdec,ni, for Alfv´ en and fast modes in low- β\nmedium,hydrodynamicturbulencestartstoevolveinneutra ls\nwhere no sound waves arise. But in the case of slow modes,\nneutralsbegintocarryacousticturbulencecausedbyinter act-\ningsoundwaves(Zakharov1967;Zakharov& Sagdeev1970;\nL’vovetal. 2000) at scales smaller than 1/kdec,ni. The wave\nmotionsofneutralfluidalsoleadtoanotherdistinctivefea ture\nof slow modes from other modes. Instead of having a purely\nnonpropagating interval, propagating ion slow waves appea r\nwithin[k+\nc,k−\nc]. Itsuggeststhatincomparisonwithhydrody-\nnamicmotions,thewavemotionsofneutralslowmodescause\nweaker friction against magnetic pressure force on ions, an d\nthus the ion slow modes can be driven by the persistent per-\nturbations of magnetic field and sustained in ions even at the\n”cutoff”wavenumbers.\nNotethatinTable4andtheforceanalysisofslowwavesin\nAppendixD,weadoptafixedwavepropagationdirectionand\ndisregardits dependenceon length scales for simplicity. I t is\nnecessary to point out, this treatment using fixed propagati on\nangle can indeed provide simpler and more intuitively obvi-\nousdescriptionof wave behaviorthan that consideringscal e-\ndependent anisotropy, but it is only applicable in analysis of\nlinearMHDwaves.\nThe above comparisons with numerical results show that\nthesingle-fluidapproachisabletodepictthedampingbehav -\niorofMHDmodescorrectlyat largescalesuntilreachingthe\ncutoff boundary k+\nc. We see for Alfv´ en and slow modes, the\ncutoff boundary k+\ncof linear waves is also the kdamof turbu-\nlence in CNM, MC, DC and SC. However, WNM does not\nexhibit a cutoff region for all the three wave modes. The pa-\nrameter space required for the existence of wave cutoffs can\nbeconfinedbyequaling k+\ncandk−\nc. Duetothehigherioniza-\ntiondegreethanotherISMphases,MHD wavesin WNM are\nless affected by collisions with neutrals and can avoid bein g\ncutoff.\nAnother remark needs to be made is about the relation be-\ntween wave cutoff and fluid decoupling. We found kdec,ni\n(kdec,in) andk+\nc(k−\nc) are of the same orderofmagnitude,and\ntheintervalof [kdec,ni,kdec,in]isrelativelylargerthan [k+\nc,k−\nc].\nTheirphysicalconnectionisobvious. Onlyafterneutralss ep-\narate themselves from the MHD wave motions of ions, can\ntheydeveloptheirownmotionsandexertsignificantinfluenc e\non ions, i.e. collisional friction strong enough for cuttin g off\nthe MHD waves. On the other hand, although propagating\nwaves reemerge at k−\nc, they can only be fully resumed after\nionsgetfreefromcouplingwith neutralsat kdec,in.\n6.APPLICATIONSON CRPROPAGATIONIN\nPARTIALLYIONIZEDMEDIUM\n5Inhigh-βplasma,sincethefastmodeswillbehavelikesoundwaves,th e\nsound waves in neutrals will become the neutral ”fast” modes .It is necessaryto introduceturbulencedampingfor achiev-\ningacorrectunderstandingontheinteractionofparticles and\nMHD turbulence. Yan&Lazarian (2002, 2004, 2008) de-\nscribed CR transport in fully ionized medium and clarified\nfastmodesarethemosteffectivescattererofCRsdespiteth eir\ndamping. It is instructive to extend their study to cover par -\ntially ionized mediumand build up a complete picture of CR\npropagationintheISM.\n6.1.Pitch-anglescatteringofCRs\nWe investigate pitch-angle scattering of CRs based on the\nformalism of diffusion coefficients obtained from Fokker-\nPlanck theory (Jokipii 1966; Schlickeiser& Miller 1998;\nYan& Lazarian 2002, 2004). The energy spectra of dif-\nferent modes we adopt are obtained from three-dimensional\nMHD simulations (Choetal. 2002; Yan& Lazarian 2003).\nBesides, we follow the nonlinear theory (NLT) employed in\nYan& Lazarian(2008) for ourcalculationongyroresonance.\nThe fluctuationsof Bin MHD turbulence result in variations\nofparticlevelocitiesin bothperpendicularandparalleld irec-\ntions with regards to magnetic field lines. That substantial ly\nbroadens the δ-function resonance predicted in quasi-linear\nscatteringtheory,especiallyforlarge-amplitudeMHDwav es.\nThefullsetofexpressionsforFokker-Planckdiffusioncoe ffi-\ncientDµµin low-βmediumandnotationsusedarepresented\nin Appendix E (see Yan&Lazarian 2004, 2008 for more de-\ntailsonthederivation).\nWe choose the phases WNM and MC as the representative\nexamplesofsub-andsuper-Alfv´ enicturbulence. Fig. 8and 9\nshow the calculated Dµµ(normalized by Ω) as a function of\nµfor100PeV CRs in WNM and 30TeV CRs in MC, where\nµis cosine of particle pitch angle. CRs’ energy is chosen\nwith their Larmor radius rLlarger than the maximum damp-\ningscale amongthethreeturbulencemodes. Here\nrL=v⊥\nΩ,Ω =eB\nΓmpc. (73)\nWe separately calculate the transit-time damping (TTD,\nsolid line) for fast and slow modes, and gyroresonance in-\nteractions(dashedline) for all turbulentmodes. The gyror es-\nonance result calculated using quasi-linear theory (QLT; s ee\nJokipii 1966; Schlickeiser 2002) is also displayed for com-\nparison (open circles). Crosses show the approximate resul t\nusingQLT forgyroresonancewith fastmodes,givenby\nDG\nµµ(QLT) =vπ√µ(1−µ2)\n4L√\nR/parenleftbigg2\n7−2/radicalbig\n1−µ2\n21µ2/parenrightbigg\n.(74)\nThis simplified formula is obtained under the condition\nkdam,⊥rL<1. We see good consistency between it and the\nnumericalintegralat small pitchangles.\nThe diffusion coefficients in WNM and MC exhibit simi-\nlar behavior. Amongthe three modes, fast turbulencehas the\nlargest damping scale. Despite severe damping, TTD from\nfast modes dominates CR scattering at large pitch angles.\nOn the contrary,gyroresonanceis moreefficient in scatteri ng\nat small pitch angles. The overall contribution from Alfv´ e n\nmodesis smaller comparedwith that from fast modes. Espe-\ncially in WNM, due to the more prominent anisotropy over\nall scalesin sub-Alfv´ enturbulence,Alfv´ enmodesare lar gely\nsuppressed in CR scattering. Comparing Fig. 8(b) and 8(d),\nwefindfastmodescontributealoneinthetotaldiffusioncoe f-\nficient. Theseresultsareinaccordancewiththeearlyfindin gs\ninYan& Lazarian(2002,2004,2008). Thescatteringbyslow14\nTABLE4\nCRITICAL SCALES AND WAVE FREQUENCIES OF THE SLOW MODES\nRegimes strongly coupled ions coupled with neutrals weakly coupled\nScalesk+\nc kc,t1 kdec,t1 ktran kdec,t2 kc,t2 k−\nc kdec,in\n2νnicosθ\ncsξnsin2θνnicosθ\n2cs,effνnicosθ\ncs,effνni\ncs,effνni\ncs,effcosθ2νni\ncs,effcosθνin\n2csicosθνin\ncsicosθ\n|ωR|< cskcosθ< cs,effkcosθ cs,effkcosθ < cs,effkcosθ< csikcosθcsikcosθ\n|ωI|ξnc2\nsk2sin2θ\n2νniνnicos2θ\n2+c2\ns,effk2cos2θ\n2νniνin\n2\n10−210−110010110210310410510610−510−410−310−210−1100101102103\nkVA\nνni|ωI|/ ν nikdec,inltr−1kdec,ni\n(a) WNM10−210−110010110210310410510610710810−610−510−410−310−210−1100101102103104105\nkVA\nνni|ωI|/ ν nikdam\n(kc+)lA−1\nkc,t1kc,t2kdec,ni\nkc−kdec,in\n(b) CNM\n10−210−110010110210310410510610710810910101011101210−610−510−410−310−210−1100101102103104105106\nkVA\nνni|ωI|/ ν nilA−1kdamkdec,ni \n(kc+)\nkc,t2kdec,inkc,t1kc−\n(c) DC10−210−110010110210310410510610710810−710−610−510−410−310−210−1100101102103104\nkVA\nνni|ωI|/ ν nikdam\n(kc+)\nkc,t1kc,t2kdec,ni\nkc−kdec,ni\n(d) SC\nFIG. 6.— Same as Fig. 1 but for slow modes. Only one branch of slow m odes is present in strongly coupled regime ( k < kdec,ni). On wavenumbers beyond\nkdec,ni, there are both neutral and ion slow modes. Their analytical damping rates are represented by filled and open circles resp ectively. Four cutoff boundaries\nk±\nc,kc,t1,kc,t2areindicated in thefigures.\nmodes depends on βof the medium (see Eq. (E16)-(E18)).\nLowβvalueinISMconditions(seeTable3)makestheeffect\nofslow modesnegligible. Inaddition,bycomparingwith the\nresults of gyroresonance from QLT, we find nonlinear effect\nresultsinamuchwiderrangeofpitchangles,whileQLTgives\nDµµ= 0at large pitch angles due to its discrete resonances,\nk/bardbl,res= Ω/(vµ). Thisresonanceconditioncannotbesatisfied\natsmallµastheturbulenceisdampedbeforecascadingdown\nto small scales. Only at the large µend, gyroresonancefrom\nQLTcomesintocoincidencewith thatfromNLT.To examinethe dampingeffect on CR scattering, we again\ncalculatethediffusioncoefficientsbutforCRswithrelati vely\nlowenergies. CRs’ energyischosenwiththeirLarmorradius\nrLsmallerthantheminimumdampingscale amongthethree\nturbulence modes. But since in WNM, slow modes survive\nneutral-ioncollisional damping(see Section 5.3) and can e x-\ntend to the gyroscale of ions, we choose 10 TeV CRs with\nrLsmallerthanthedampingscale ofAlfv´ enmodes,whichis\nsmaller than that of fast modes. For MC, Alfv´ en modeshave\nthe smallest damping scale. Accordingly, we adopt 10 GeV15\n10−210−110010110210310410510610710−510−410−310−210−1100101102103104105\nkVA\nνni|ω|/ ν ni(kc+)lA−1kdamkdec,ni\nkc,t1kc,t2kc−kdec,in\nFIG. 7.—SameasFig. 2butforslowmodes. Thefilledsymbolscorre spond\nto the neutral slow modes. Theshort vertical dashed lines wi thin[kc,t1,kc,t2]\nrepresent kdec,t1,ktran,kdec,t2from left to right respectively.\nastheCRenergy.\nFig. 10 and Fig. 11 show the results in WNM and MC\nrespectively. Theasterisksrepresentthesimplifiedexpre ssion\nforTTD withfast modes,\nDT\nµµ(NLT) =v√π(1−µ2)2\n8L∆µexp/parenleftbigg\n−µ2\n(∆µ)2/parenrightbigg\n(/radicalbig\nkdamL−1),\n(75)\nwhich is obtained under the condition kdam,⊥rL<1and\nagrees well with integral result for low-rigidity CRs over a ll\npitch angles. Obviously, TTD with fast modes becomes the\nonly significant scattering effect. The remaining gyroreso -\nnance of slow modes in WNM has negligible effect. Com-\nparedwiththeresultsforhigh-energyCRswith rLlargerthan\nthe dampingscales of turbulencemodesin Fig. 8 and 9, here\nwecanclearlyseetheresonancegapatsmallpitchanglesdue\ntotheabsenceofgyroresonance. TTDalonecannotfulfillthe\nscatteringofCRs withsmall pitchangles.\n6.2.ScatteringfrequencyofTTD andgyroresonance\nTo gain a deeperunderstandingon the dependenceof TTD\nand gyroresonanceon CR energy and the influence of turbu-\nlence damping, we illustrate the scattering frequency ν=\n2Dµµ/(1−µ2)of TTD and gyroresonance separately in\nWNM(Fig.12)andMC(Fig. 13). Sincethemajorscattering\nagentdiffersatsmallandlargepitchangles,wepresentres ults\nofTTDat µ= 0.1inFig.12(a)and13(a),andgyroresonance\natµ= 0.8in Fig. 12(b)and 13(b). The vertical dashed lines\nindicatethe CR energieswith rLequalto the dampingscales\nof Alfv´ en ( Edam,A) and fast ( Edam,f) modes. Slow modes are\nnotconsideredduetotheirnegligibleeffect.\nIn WNM, as shown in Fig. 12(a), when dampingis absent,\nν(dashed line) throughTTD with fast modes decreases with\nCR energy. Otherwise ν(solidline)keepssteadyuntil reach-\ningEdam, f(vertical dashed line), and then coincides with the\nresult without damping. The analytical scattering frequen cy\n(asterisks) is derived by using Eq. (75), and can apply to\nthe energy range below Edam, f. As illustrated by Eq. (75),\nfor CRs with rL< k−1\ndam,DT\nµµis determined by the damping\nscale. Since CR velocity is approximately equal to the light\nspeedand its changewith energyis marginal,Eq. (75) yields\na constant DT\nµµandνat a fixed µ. It implies only the turbu-lenceonscaleslargerthan rLcontributesto TTDinteraction.\nAsa result,in thecase ofnoturbulencedamping, νoflower-\nenergyCRs hasa highervaluebecauseturbulenceona larger\nrange of scales is involved in TTD scattering. While in the\ncase with damping, νof TTD is independent of CR energy\nfor CRs with rL<1/kdam, and decreases with energy when\nrL>1/kdam.\nFig. 12(b) shows the total νof gyroresonance by both\nAlfv´ enand fast modesin WNM. The results freeof damping\nand in the presence of damping come into coincidence when\nCR energyis over Edam,f. Therespectiverolesof Alfv´ enand\nfast modes are explicitly displayed. When there is no damp-\ning, Alfv´ en modes have negligible effect over the whole en-\nergy range due to strong turbulence anisotropy in WNM. So\nνfor gyroresonance with fast modes (open triangles) coin-\ncides with the total ν. The crosses are analytical νby em-\nploying Eq. (74). Although Eq. (74) is the simplified DG\nµµ\nusing QLT for kdam,⊥rL<1, the gyroresonance condition\nk/bardbl= Ω/(vµ) =/radicalbig\n1−µ2/(rLµ)can still be satisfied at\nlargeµ, so that it can reach a good agreement with the nu-\nmerical result using NLT.6Similar to TTD, gyroresonance\nwith fast modes has decreasing νwith increasing CR ener-\ngies, but their physical origins are different. For TTD, the re\nisnota specific resonantscale. Turbulenceoverall the scal es\naboverLcontributestoparticlescattering. However,gyrores-\nonance can only become effective on specified scales which\nare determined by CR energy. We observe in Eq. (74) that\nDG\nµµ∝1/√\nR, whereRis CR rigidity. It demonstrates\nthathigher-energyCRsarescatteredbylarger-sizeturbul ence\neddies, and consequently have larger mean free paths and\nsmallerDG\nµµandν.\nWhen damping is taken into account, since gyroresonance\nhas a preferential resonant scale, νcorresponding to Alfv´ en\n(filled circles) and fast (filled triangles) modes only becom e\nefficient after rLexceeds their damping scales. The analyt-\nicalDG\nµµgiven by Eq. (74) can only apply to CRs with en-\nergies larger than Edam,fin this situation. Therefore, TTD is\ntheonlyeffectiveinteractionandgovernsparticlescatte ringat\nlower energies. Since Alfv´ en modes have a smaller damping\nscale than fast modes, their contribution can be seen within\nthe energy range [ Edam,A,Edam,f]. But fast modes dominate\nthescatteringofCR withenergieshigherthan Edam,f.\nIn the case of super-Alfv´ enic MC, we see much higher ν\nvaluesforbothTTDandgyroresonance.The νofTTDinFig.\n13(a) shows a similar trend as that in Fig. 12(a) for WNM.\nBut forνof gyroresonance (Fig. 13(b)), the Alfv´ en modes\nbecome more important in MC compared with WNM due to\nless prominent turbulence anisotropy. Fig. 13(c) and 13(d)\ndisplay the components of νfrom Alfv´ en and fast contribu-\ntions for cases without and with damping. When damping is\nabsent,νof gyroresonancewith Alfv´ en modes(opencircles)\nincreases towards higher energies. The opposite trend com-\nparedwithgyroresonancewithfastmodesoriginatesfromth e\nscale-dependentturbulenceanisotropy. Lower-energyCRs in-\nteract with many elongated eddies with k⊥≫k/bardbl∼1/rL\nwithinonegyroperiodandthisrandomwalkcausesinefficien t\nscattering. While higher-energyCRs are scattered by large r-\nsizeeddieswhicharelessanisotropicandhencethescatter ing\nefficiency increases (Yan& Lazarian 2004). We see in Fig.\n13(c), approaching lA, weak turbulence anisotropy makes\n6In fact, Eq. (74) is only valid for µ >0.5314to ensure a positive value\nofDµµ.16\n00.10.20.30.40.50.60.70.80.9100.511.52x 10−4\nµDµµ/Ω\n(a) Alfv´ en00.10.20.30.40.50.60.70.80.9100.0050.010.0150.020.025\nµDµµ/Ω\n(b) Fast\n00.10.20.30.40.50.60.70.80.9110−1010−910−810−710−610−510−410−3\nµDµµ/Ω\n(c) Slow00.10.20.30.40.50.60.70.80.9100.0050.010.0150.020.025\nµDµµ/Ω\n(d) Total\nFIG. 8.—Pitch-angle diffusion coefficients (normalized by Ω)of100PeVCRsin(a)Alfv´ en,(b)fast,and(c)slowmodesinW NM.Solidanddashedlinesrefer\ntoTTDandgyroresonance byapplying NLT.Open circles areQL Tresults forgyroresonance. Crosses in(b)showtheapproxi mate QLTresultforgyroresonance\ngiven by Eq. (74). (d) Total diffusion coefficients fromthre e modes. Thedash-dotted line is the sumof the solid and dashe d lines.\nAlfv´ en modes have comparable efficiency as fast modes in\ngyroresonancescattering. Their joint effect flattens νat high\nenergies.\nIn brief, we find both TTD and gyroresonance with fast\nmodes decrease with CR energy, while gyroresonance with\nAlfv´ enmodesincreaseswith energy. Thedampingeffectcan\nonly make a difference in CR scattering when rL<1/kdam.\nThen TTD becomes the only scattering agent and has a scat-\ntering efficiency independent of particle energy. At higher\nenergieswith rL>1/kdam, the importanceof gyroresonance\nwith Alfv´ en modes depends on the anisotropy degree of the\nturbulence. In super-Alfv´ enic turbulence, Alfv´ en modes can\nbecomemoreefficientthanfastmodesingyroresonancescat-\nteringat largescalescomparableto lA.\n6.3.ResultsonparallelmeanfreepathofCRs\nGiventhe results of Dµµ, the parallel meanfree path λ/bardblof\nCRs canbeevaluatedbyintegratingover µ,\nλ/bardbl=3v\n4/integraldisplay1\n0dµ(1−µ2)2\nDµµ(µ), (76)\nwhereDµµis the total contributions from both TTD and\ngyroresonance with the three modes calculated using NLT.\nFig. 14(a)-14(d) present λ/bardblas a function of CR energy Ek.The upper limit of the energy range is restrained by rL< L\ninWNM and rL< lAinotherenvironments.\nWe can clearly see the impact of turbulence damping on\nλ/bardbl. In the case of WNM (Fig. 14(a)), even in the undamped\nrangeofturbulence,neitherAlfv´ ennorslowmodescaneffe c-\ntivelyscatter CRs. Only whenCR energyapproaches Edam, f,\ncanCRsbeconfinedbygyroresonancewithfastmodes. Nev-\nertheless, the resulting λ/bardblis larger than ltrover the whole\nenergyrange. It shows in WNM, direct interactionswith tur-\nbulence modes are invalid in scattering CRs. Instead, other\nmechanisms, e.g. field line wandering, can play a more im-\nportantrolein determining λ/bardbl.\nIn the other ISM phases (Fig. 14(b)-14(d)), we observe\na nonmonotonic U-shaped dependence of λ/bardblon CR energy.\nThe dramatical decrease of λ/bardblatEdam,Aand the further de-\ncreaseofλ/bardblatEdam,fcomefromthecontributionofgyroreso-\nnancewith Alfv´ enandfast modesrespectively,whilethe ri se\nofλ/bardblatEk> Edam, fis in accordance with the decreasing\nνof TTD with energy. The marginal change of λ/bardblatEdam,s\nreflectstheinsignificantroleofslow modesin CRscattering .\nOn the other hand, although TTD contributes over all en-\nergy range and is more efficient in CR scattering, it is inca-\npabletoscatterCRswithsmallpitchangles. ThusTTDalone\nis not enough to confine CRs and leads to infinite λ/bardblin the17\n00.10.20.30.40.50.60.70.80.9100.0050.010.0150.020.0250.030.035\nµDµµ/Ω\n(a) Alfv´ en00.10.20.30.40.50.60.70.80.9100.010.020.030.040.050.060.07\nµDµµ/Ω\n(b) Fast\n00.10.20.30.40.50.60.70.80.912468101214x 10−5\nµDµµ/Ω\n(c) Slow00.10.20.30.40.50.60.70.80.9100.010.020.030.040.050.060.070.080.090.1\nµDµµ/Ω\n(d) Total\nFIG. 9.— Sameas Fig. 8 butfor 30 TeV CRs in MC.\n00.10.20.30.40.50.60.70.80.9100.511.522.533.5x 10−6\nµDµµ/Ω\n(a) Fast00.10.20.30.40.50.60.70.80.911e−301e−201e−101\nµDµµ/Ω\n(b) Slow\nFIG. 10.— Sameas Fig. 8but for 10 TeV CRs in (a) fast and (b) slow mo des in WNM.Theasterisks in (a) are analytical result given b y Eq. (75).18\n00.10.20.30.40.50.60.70.80.9100.511.522.533.5x 10−5\nµDµµ/Ω\n(a) Fast00.10.20.30.40.50.60.70.80.9100.20.40.60.81x 10−7\nµDµµ/Ω\n(b) Slow\nFIG. 11.— Sameas Fig. 8but for 10 GeV CRs in (a) fast and (b) slow mo des in MC.\n10310410510610710810−1210−1110−1010−910−810−710−6\nEk [GeV]ν [s−1]Edam,f\n(a) TTD1031041051061071081e−201e−181e−161e−141e−121e−101e−81e−6\nEk [GeV]ν [s−1]Edam,AEdam,f\n(b) Gyroresonance\nFIG. 12.— Scattering frequency vs energy of CRs in WNM from (a) TT D atµ= 0.1and (b) gyroresonance at µ= 0.8. Dashed and solid lines correspond\nto cases in the absence of damping and in the presence of dampi ng. Circles and triangles in (b) show the separate contribut ions from Alfv´ en and fast modes.\nFilled and open symbols correspond to situations with and wi thout damping. Asterisks and crosses represent the analyti cal results by using Eq. (75) and (74)\nrespectively.19\n10010110210310410510−710−610−510−4\nEk [GeV]ν [s−1]Edam, f\n(a) TTD10010110210310410510−1010−910−810−710−610−510−4\nEk [GeV]ν [s−1]Edam, AEdam, f\n(b) Gyroresonance\n10010110210310410510−1010−910−810−710−610−510−4\nEk [GeV]ν [s−1]\n(c) Gyroresonance, without damping10010110210310410510−1010−910−810−710−610−510−4\nEk [GeV]ν [s−1]Edam, AEdam, f\n(d) Gyroresonance, with damping\nFIG. 13.— Sameas Fig. 12 butfor MC.(b) isreplotted in (c) and (d) , showing the separate contributions fromAlfv´ en (circles ) and fast (triangles) modes.\nenergy range where gyroresonance is absent. Therefore, the\nlower energy limit of effective CR scattering and finite λ/bardblis\ndeterminedbythedampingscale ofAlfv´ enmode.\nTable 5 summarizes the lowest energies of CRs with their\nscattering unaffectedby turbulence dampingin different p ar-\ntially ionized ISM, which are determined by the damping\nscales of fast modes. The scattering and acceleration of\nCRs with lower energies are subject to turbulence damp-\ning. Notice that different from the situation discussed her e,\nYan&Lazarian 2004showed in fully ionized plasma, damp-\ning of fast modesstrongly dependson the angle θbetweenk\nandB, which results in anisotropic distribution of fast mode\nenergy at small scales. In their case, the lowest energy for\nCR scattering unaffected by turbulence damping should be a\nfunctionof θ.\n6.4.OthermechanismsonconfiningCRs\nOur results show that in partially ionized medium, MHD\nturbulence, especially fast modes, is severely damped by\nneutral-ion collisions. Consequently, only high-energy C Rs\ncan be efficiently scattered through interactions with turb u-\nlence modes. Besides scattering, the diffusion of CRs can\nalso arise from the spatial wandering of magnetic field lines\n(Jokipii 1966). In three dimensional turbulence, the wande r-TABLE5\nTHE MINIMUM ENERGY OF CRS UNAFFECTED BY\nTURBULENCE DAMPING IN DIFFERENT ISMPHASES.\nISMphasesk−1\ndamEk,min\nAlfv´ en fast slow\nWNM 0.003pc4.0pc — 45.3PeV\nCNM 0.005pc0.1pc0.04pc1.2PeV\nMC 6.7AU0.002pc98.2AU18.9TeV\nDC 35.0AU0.009pc261.7AU0.99PeV\ning of field lines is induced by turbulent motionsand charac-\nterized by turbulenceproperties. Lazarian&Vishniac (199 9)\nquantitively described the field line wandering according t o\nthe scaling laws of GS95-typeMHD turbulence, and demon-\nstrated its key role in determining the rate that magnetic\nreconnection proceeds. Based on the Lazarian& Vishniac\n(1999) prescription for magnetic field wandering and the re-\nlated field line diffusion, significant theoretical reformu la-\ntionsone.g. thermalproduction(Narayan&Medvedev2001;20\n105106107108102103104105\nEk [GeV]λ|| [pc]EtrEdam,f\n(a) WNM1051061070.511.52\nEk [GeV]λ|| [pc]Edam,fEdam,sEdam,A\nlA\n(b) CNM\n1021031041050.020.040.060.080.1\nEk [GeV]λ|| [pc]\nlAEdam,sEdam,fEdam,A\n(c) MC1041051061070.10.20.30.40.50.6\nEk [GeV]λ|| [pc]Edam,fEdam,sEdam,A\nlA\n(d) DC\nFIG. 14.— Parallel mean free path of CRs as a function of their ene rgies in (a) WNM, (b) CNM, (c) MC and (d) DC. Vertical dashed li nes represent the CR\nenergies with their rLequal to the damping scales of Alfv´ en ( Edam, A), slow (Edam, s), and fast ( Edam, f) modes. Notice that Edam,Ais below the energy range\nshown in (a). Etris the energy corresponding to ltr. Thehorizontal dashed lines in (b),(c) and (d) refer to the l ength scales of lA.\nLazarian 2006) and CR propagation (Yan&Lazarian 2008,\n2012; Lazarian&Yan2014),havebeenachieved.\nOthermechanismswhichcanenhancescatteringofCRsin-\nclude streaming instability (Wentzel 1974; Cesarsky 1980)\nand gyroresonance instability (Lazarian&Beresnyak 2006;\nYan&Lazarian 2011). The additional MHD waves excited\nby CRs can in turn efficiently scatter CRs within a range of\nenergies(e.g., /lessorsimilar100GeV).Yanet al.(2012)investigatedthe\nshock acceleration of CRs in the presence of streaming in-\nstability. They found the CR flux near supernova remnants\nis strongly enhanced compared with typical Galactic values ,\nso the growthrates of streaminginstability can overcometh e\nbackgroundturbulencedampingandboost scatteringofCRs.\nThisfindingenablesthemto reproducethe observedgamma-\nrayemissionfromthesupernovaremnantW28.\nThegrowthrateoftheresonantwavesexcitedbystreaming\nCRs is\nΓg,CR∼Ω0nCR(>Γ)\nni/parenleftbiggvstream\nVA−1/parenrightbigg\n,(77)\nwhereΩ0=eB/m pcis the non-relativistic CR gyrofre-\nquency.nCR(>Γ)is the number density of CRs with their\nenergies larger than ΓGeV, and the rLvalue correspond-\ning toΓgives the wavevector of the generated waves, i.e.\nrL∼1/k/bardbl∼1/k, which propagate closely parallel to mag-neticfield. And vstreamisthestreamingvelocityofCRs,which\nshouldbelargerthan VAsoasto amplifywaves.\nThe instability can only set in when Γg,CRexceeds the to-\ntal damping rate, which includes the neutral-ion collision al\nand neutral viscous damping discussed in this paper, and\nalso the nonlinear damping by background MHD turbulence\n(Yan&Lazarian 2002, 2004; Farmer& Goldreich 2004;\nBeresnyak&Lazarian 2008; Yan& Lazarian 2011). The\nnon-linear damping rate takes the form as (Yan&Lazarian\n2011)\nΓturb=r−1\n2\nLl−1\n2\nAVA (78)\nforrL< lAinsuper-Alfv´ enicturbulence,and\nΓturb=r−1\n2\nLL−1\n2VL (79)\nforrL< ltrin sub-Alfv´ enicturbulence.\nAs an example, Fig. 15 compares the growth and damp-\ning rates at different CR energies in the WNM environment.\nThe neutral-ion collisional damping rate for parallel Alfv ´ en\nwaves (Eq. (4b) with θ= 0) applies, which has the value\nof|ωI|=νin/2at large wavenumbers correspondingto low\nCR energies. Notice neutral viscous damping is unimportant\nforthe generatedwavesin thisparticularcase. The nonline ar\ndampingonlyexistsforCRswiththeir rLlargerthanthepar-21\nallel damping scale of the background Alfv´ enic turbulence .\nFor the growth rate represented by the solid line, we assume\nvstreamhas the same order of magnitude as VAand adopt the\nCR density near the sun, nCR(>Γ) = 2×10−10Γ−1.6cm−3\n(Wentzel 1974). It shows the streaming instability can only\novercome the collisional damping and effectively contribu te\nto particle scattering for CRs with energies lower than ∼10\nGeV. Crosses show a higher growth rate with the CR density\nincreasedbythreeordersofmagnitude. Asaresult,CRswith\nupto∼1TeVcanbesignificantlyscatteredandconfined.\n10010110210310410510610710810−1110−1010−910−810−710−610−510−410−310−210−1100101102103104105\nEk [GeV]Γ/νni\n  \n|ωI| \nΓturb \nΓg,CR\nΓg,CREdam,||,A\nFIG. 15.—Growthanddampingrates(normalized by νni)ofstreaming in-\nstability as afunction of CR energy in WNM. Thedashed line is the neutral-\nion collisional damping rate. The dash-dotted line is the no nlinear damping\nrate. The growth rates with different CR number densities, nCR(>Γ) =\n2×10−10Γ−1.6cm−3(solidline),and nCR(>Γ) = 2×10−7Γ−1.6cm−3\n(crosses) are shown. The vertical dashed line indicates the energy corre-\nsponding to the parallel damping scale of the background Alf v´ en modes.\nFrom above example we see to establish a comprehen-\nsive picture of CR propagation, other scattering mechanism s\nshould also be incorporated in addition to direct interac-\ntions with turbulence modes. For relatively low-energyCRs ,\nstreaming instability can be a promising mechanism of CR\nconfinementbyscattering. Forhigh-energyCRswiththeir rL\nexceeding the damping scale of MHD turbulence, TTD and\ngyroresonancescatteringoperate. RegardingtheCRswithi n-\ntermediate energies, their confinement is mainly attribute d to\nfieldlinewanderingasmentionedearlier.\nWe againstressthat inanalyzingthesescatteringprocesse s\ninapartiallyionizedmedium,neutral-ioncollisionaldam ping\nisanessentialphysicalingredient,whichdeterminesthed issi-\npationscaleofturbulencecascade,andisthemajorconstra int\nforthegrowthof instabilities. Takingthedampingeffecti nto\naccountisnecessarytoattainrealisticcalculationofdif fusion\ncoefficients and proper understanding of CR propagation in\npartiallyionizedmedium.\n7.DISCUSSION\nIn partially ionized medium, the wave behavior strongly\ndepends on the coupling state between neutrals and ions.\nWe consider low- βmedium and specifically distinguish the\nneutral-ion and ion-neutral decoupling scales of three mod es\nand the regimeswith differentcoupling degreesseparated b y\nthem. The cutoff boundaries of waves are closely correlatedwiththedecouplingscales. Infact,thecutoffintervalapp roxi-\nmatelysharesthesamedomainasthatbetweentheneutral-io n\nandion-neutraldecouplingscales. Thiscoincidenceindic ates\nfluiddecouplingcanbethephysicaloriginofthewavecutoff .\nUnlike Alfv´ en and fast waves, propagating waves can arise\nwithintheoriginalcutoffregionofslow waves,whichresul ts\nin two cutoff intervals sitting among three propagating wav e\nbranches. We forthe first time providefull analytical expre s-\nsions of all the cutoff boundaries and the wave frequencies\nof both ”neutral” and ”ion” slow waves in different coupling\nregimes. Slow waves on scales within [kdec,ni,kdec,in]exhibit\nmore complicated properties in both propagating and damp-\ning components of the wave frequencies. Accordingly, the\nwavespectrumonintermediatescalesisalsodividedintodi f-\nferent coupling regimes, confined by multiple critical scal es\nwhich are linked together in a symmetric patten (see Section\n5.3).\nInordertotest the validityofthe single-fluidapproach,we\nadopted Newtons iteration method to solve the single-fluid\ndispersion relation. By comparing the solutions with those\nto the two-fluid dispersion relation under weak damping as-\nsumption, we see consistent results at large scales down to\nthe lower cutoff boundary k+\nc. We further numerically con-\nfirmedthevalidityofsingle-fluidapproximationindescrib ing\nMHDwavesinpartiallyionizedISMandSCenvironments. It\nmeans for practical purposes, single-fluid treatment can al so\nbe used to determinethe dampingscales of bothincompress-\nible and compressible turbulence modes in strongly coupled\nregimeinsimilarconditions.\nTo obtain the damping scales of MHD turbulence, we fol-\nlowthepresent-dayunderstandingofturbulenceandtreatt he\nturbulence cascades of Alfv´ en, fast and slow modes sepa-\nrately. It turns out the scale-dependent anisotropy of turb u-\nlenceplaysacriticalroleinregulatingwavebehaviorandd e-\nriving turbulence damping. We see a close relation between\nthe cutoffs appearing in linear MHD waves and the damping\nofMHDturbulence. Particularly,becauseoftheintrinsicc rit-\nical balance of the turbulent motions (GS95), the cascade of\nAlfv´ enmodesistruncatedatthelowercutoffboundarywher e\nthe nonpropagatingwaves arise. As a result, the wave cutoff\nshould also be taken into account when deriving turbulence\ndampingscales.\nIn addition, we apply the analytical results on turbulence\ndampingtoavarietyofpartiallyionizedISMphasesandsola r\nchromosphere.\n- We find neutral viscosity plays a significant role in\ndampingAlfv ´en modesin WNM andSC , while neutral-\nion collisions act as the dominant damping effect in\notherenvironments.\n- Fastmodesinallconditionsaredampedoutinstrongly\ncoupled regime. And the damping is especially severe\nin WNM and SC because the cascading rates in both\nenvironmentsare substantially low and in addition,the\ndamping rate in WNM is relatively high due to its low\ndensity. Besides,differentfromthecaseinfullyionized\nmedium, the damping of fast modes only has a weak\ndependenceonwavepropagationdirection.\n- In the case of slow modes, since cutoff appears earlier\nthan turbulence damping, the damping scales of slow\nmodesaregivenbythelargestcutoffscalesinallcondi-\ntionsexceptforWNM.Duetotheabsenceofcutoffin-\ntervals, slow modesin WNM survive neutral-ioncolli-22\nsionaldamping. Infact,allthethreemodesinWNMdo\nnothavecutoffs. Thephysicalexplanationisduetothe\nrelatively high ionization fraction in WNM, coupling\nof neutralswith ionsonlargerscalesis easier since ev-\nery neutral has more chances to collide with ions, and\ndecoupling of ions from neutrals on smaller scales is\nalso easier since everyion hasfewerchancesto collide\nwithneutrals. Consequently,theseparationbetweenthe\nneutral-ion and ion-neutral decoupling scales is much\nshortenedandcutoffcanbe avoided.\nRegardingtheturbulencecascadeofslow modesbelowthe\nneutral-ion decoupling scale in WNM, although the MHD\nturbulence in ions survives neutral-ion collisional dampi ng,\nsinceslowmodesareslavedtoAlfv´ en,itmaybequenchedat\na smaller scale with the dampingof Alfv´ en modes. Provided\nthe cascade of ”ion” slow modes can extend to the regime\nwhere collisionless dampingis dominant,we refer the reade r\nto Lithwick& Goldreich (2001) on the slow mode damping\nbelow the proton mean free path. For the acoustic turbu-\nlenceof”neutral”slowmodes,itissubjecttobothneutral- ion\ncollisional and neutral viscous damping. The damping scale\ncan be determined by comparing the cascading rate and both\ndamping rates, which falls beyond the scope of the current\nworkandwill bediscussedinfuturework.\nAs one important application of the present study, we ex-\nploredthedampingeffectonCRpropagationinpartiallyion -\nized medium. The local reference system where GS95-type\nscalings stand is also very important for studying CR propa-\ngation, since the scattering of a CR particle is determined b y\nits interaction with the local magnetic field perturbation i n-\nstead of background field. That ensures realistic statistic s of\nmagnetic fluctuations which we use for calculating CR scat-\ntering.\nAlfv´ en modes are found to be an ineffective scatterer of\nCRs (Chandran 2000; Yan& Lazarian 2002, 2004). During\ntheenergycascadeofAlfv´ enmodes,turbulenteddiesbecom e\nmoreandmoreelongatedalongmagneticfieldlines,withen-\nergy concentrating in the direction perpendicularto the lo cal\nmean magnetic field. Thus a CR particle with rLcompara-\nble to the parallel scale of eddies interacts with many uncor -\nrelated eddies in perpendicular direction within one gyrop e-\nriod. Therandomwalkleadstoveryinefficientgyroresonanc e\nscattering of CRs by Alfv´ en modes. But in the presence of\nneutral-ion collisional damping, turbulence cascade is he av-\nily damped and truncated at a large scale, where turbulence\nanisotropy is relatively weak. Especially in super-Alfv´ e nic\nturbulence, turbulent eddies are nearly isotropic when rLap-\nproaches lA. As a result, high-energyCRs can be effectively\nscattered by gyroresonance with Alfv´ en modes, as shown\nabove.\n8.SUMMARY\nIn this paper we continued the work in Paper I on damp-\ning of MHD turbulence in partially ionized medium. We put\nmoreemphasisoncompressiblemodesandapplytheanalyti-\ncalresultstoavarietyoflow- βISMphasesandsolarchromo-\nsphere. Asanimportantapplication,westudiedtheCRprop-\nagation in partially ionized medium considering the neutra l-ion collisional damping, as well as neutral viscous damping ,\nofMHD turbulence. Herewe summarizethemainresults.\n1. We present explicit correlation between the cutoff\nboundariesof MHD waves and decouplingscales. Under the\nconsiderationofscale-dependentanisotropy,theyare\nk+\nc= (2\nξn)3\n2kdec,ni, (80a)\nk−\nc= 2−3\n2kdec,in. (80b)\nforAlfv´ enmodes,and\nk+\nc=2\nξnkdec,ni, (81a)\nk−\nc=1\n2kdec,in. (81b)\nforfast modes. Slow modeshavethesame\nk−\nc= 2−3\n2kdec,in (82)\nas Alfv´ en modes, but the relation between k+\ncandkdec,nide-\npendsonturbulenceproperties.\n2. Single-fluid approach is capable of correctly describ-\ningwavebehavioranddampingpropertiesinstrongcoupling\nregime.\n3. CutoffofMHD wavesshouldalso betakeninto account\nwhenderivingdampingscalesofturbulencecascade.\n4. We showedthe importanceofneutralviscosityin damp-\ningAlfv´ enmodesinWNMandSC.EspeciallyfortheAlfv´ en\nmodesinSC,neutralviscosityisthedominantdampingeffec t\ninsteadofneutral-ioncollisions.\n5. We thoroughly investigated the behavior of slow waves\nandprovidedfullexpressionsofthemultiplecutoffsandwa ve\nfrequenciesindifferentcouplingregimes.\n6. Thescale-dependentanisotropyofGS95-typeturbulence\nisimportantforunderstandingwavebehavior,calculating tur-\nbulence damping scales, and studying interactions between\nCRsandmagneticperturbations.\n7. We evaluated the scattering efficiencies of TTD and gy-\nroresonance with the three turbulence modes, and found due\ntotheseveredampinginpartiallyionizedmedium,onlyhigh -\nenergy CRs can be effectively scattered through direct inte r-\nactionswithturbulencemodes. Comparedwithotherenviron -\nments,CRsinWNMarepoorlyconfinedduetoitsprominent\nturbulenceanisotropy,aswell as the largestdampingscale of\nfastmodes. AsforCRswithlowerenergies,othereffectssuc h\nasfieldlinewanderingandstreaminginstabilitycansetina nd\ncontributein confiningCRs.\nACKNOWLEDGEMENT\nS. X. acknowledges the support from China Scholarship\nCouncilduringherstay in Universityof Wisconsin-Madison .\nS. X. and H. Y. are supported by the NSFC grant AST-\n11473006. S. X. also thanks Heshou Zhang for the stimu-\nlating discussions. A. L. acknowledges the NSF grant AST\n1212096, a distinguished visitor PVE/CAPES appointment\nat the Physics Graduate Program of the Federal University\nof Rio Grande do Norte and thanks the INCT INEspao and\nPhysicsGraduateProgram/UFRNforhospitality.\nAPPENDIX\nA SUMMARY OF THE NOTATIONS USED IN THIS PAPER23\ndrag coefficient γd\nion density ρi\nneutral density ρn\ntotal density ρ\nproton mass mp\nmass of hydrogen atom mH\nion mass mi\nneutral mass mn\nreduced mass mr\nion number density ni\nneutral number density nn\ntotal number density n\nneutral-ion collisional cross-section σni\nelectronic charge e\nneutral-ion collision frequency νni\nion-neutral collision frequency νin\nion fraction ξi\nneutral fraction ξn\nratio of neutral and ion densities χ\nwave frequency ω\nreal partof wave frequency ωR\nimaginary part of wave frequency ωI\nwave number k\nx,y,zcomponent of k kx,ky,kz\nkcomponent parallel to local magnetic field k/bardbl\nkcomponent perpendicular to local magnetic field k⊥\nwave propagation angle with regard of magnetic field θ\nAlfv´en speed VA\nAlfv´en speed ofion-electron gas VAi\nmagnetic field B\nsound speed cs\nsound speed ofneutral gas csn\nsound speed ofion-electron gas csi\nion gas pressure Pi\nneutral gas pressure Pn\ntemperature T\nBoltzmann constant kB\nadiabatic constant γ\nratio of gaspressure to magnetic pressure β\ninjection scale of turbulence L\ninjection scale of MHDturbulence in super-Alfv ´enic turbulence lA\ntransition scale from weak to strong MHDturbulence in sub-A lfv´enic turbulence ltr\nturbulent velocity at L VL\nturbulent velocity at l vl\nturbulent velocity at ltr vtr\nAlfv´enic Mach number MA\ndamping scale ld\nion velocity vi\nneutral velocity vn\nmean free path for aneutral particle ln\ncross section for aneutral-neutral collision σnn\nspeed of light c\nLorentz factor Γ\nCR particle’s velocity v\nCR particle’s gyrofrequency Ω\nCR particle’s non-relativistic gyrofrequency Ω0\nCR particle’s pitch angle cosine µ\npitch angle diffusion coefficient Dµµ\nscattering frequency ν\ngrowth rate of streaming instability Γg,CR\nstreaming speed of CRs vstream\nnumber density of CRs nCR\nnonlinear damping rate Γturb\nAPPROXIMATE SOLUTIONS TO SINGLE-FLUID DISPERSION RELATIO N\nHere we list the analytical solutions to the single-fluid dis persion relation at some tractable limits following the met hod de-\nscribedin Section3.2.\n1.θ→0◦\nλ1,2=±cn,λ3,4=VA\n2(−i˜k±/radicalBig\n4−˜k2). (B1)\n2.θ→90◦\nλ1,2=±cn. (B2)\n3.β→0\nλ1,2=±cncosθ\nλ3,4=VA\n2(−i˜ksecθ±/radicalbig\n4−˜k2sec2θ).(B3)24\n4.β→ ∞\nλ1,2=±cn,λ3,4=−iVAcosθ(˜ksec2θ±/radicalbig˜k2sec4θ−4)\n2. (B4)\n5.˜k→0(idealMHDcase)\nλ1,2=±/radicalbig\nc2n+V2\nA+∆√\n2,λ3,4=±/radicalbig\nc2n+V2\nA−∆√\n2. (B5)\nwhere\n∆ =/radicalBig\n(c2n+V2\nA)2−4c2nV2\nAcos2θ. (B6)\n6. Smallθandβ→0\nλ1,2=±1+cos2θ\n2cn\nλ3=−VAζ\n2i−4ζ−2˜ksecθζ2+ζ3\n4(2ζ2+3˜ksecθζ−4)i\nλ4=−VAκ\n2i−4κ−2˜ksecθκ2+κ3\n4(2κ2−3˜ksecθκ−4)i(B7)\nwhere\nζ=˜k−i/radicalBig\n4−˜k2,κ=˜k+i/radicalBig\n4−˜k2. (B8)\n7. Smallθandβ→ ∞\nλ1,2=±cn\nλ3=−VAζ\n2i−4cos2θ−2˜ksecθζ+ζ2\n4(˜ksecθ−ζ)i\nλ4=−VAκ\n2i−4cos2θ−2˜ksecθκ+κ2\n4(˜ksecθ−κ)i(B9)\n8.θ≤90◦\nλ1=cn+cnsin2θV2\nA\n2(c2n+icn˜ksecθVA−V2\nA),\nλ2=−cn−cnsin2θV2\nA\n2(c2\nn−icn˜ksecθVA−V2\nA),\nλ3=−iVA\n˜ksec3θ.(B10)\n9. Lowβandθ→0\nλ1,2=±cn,λ3=−VAζ\n2i,λ4=−VAκ\n2i. (B11)\n10. Lowβandθ→90◦\nλ1,2=VA\n2(−i˜ktanθ±/radicalBig\n4−˜k2tanθ2). (B12)\n11. High βandθ→0\nλ1,2=±cn,λ3,4=−VA(˜k±/radicalbig˜k2−4)\n2i. (B13)\n12. High βandθ→90◦\nλ1,2=±cn,λ3=−iVA˜ksecθ. (B14)\n13. Small ˜kandβ→0\nλ1,2=±cn√\n2,\nλ3=VA−i˜kVAsecθ\n3i˜ksecθ+2,λ4=−VA+˜kVAsecθ\n3˜ksecθ+2i.(B15)\n14. Small ˜kandβ→ ∞\nλ1,2=±cn,\nλ3=VA√\n2−i˜kVAsecθ√\n2(i˜ksecθ+√\n2),λ4=−VA√\n2−i˜kVAsecθ√\n2(−i˜ksecθ+√\n2).(B16)\n15. Small ˜kandθ→0\nλ1,2=±cn,\nλ3=VA+i˜kV2\nA(V2\nA−c2\nn)\ni˜kVA(c2n−3V2\nA)+2VA(c2n−V2\nA),λ4=−VA+i˜kV2\nA(V2\nA−c2\nn)\n−i˜kVA(c2n−3V2\nA)+2VA(c2n−V2\nA).(B17)\n16. Small ˜kandθ→90◦\nλ1,2=±2(V2\nA+c2\nn)3\n2\n3V2\nA+2c2n. (B18)25\nCRITICAL SCALES OF ION SLOW MODES CONSIDERING SCALE-DEPEND ENT ANISOTROPY\nBysubstitutingthescalingrelationsdescribedinEq. (30) and(32)intoEq. (62), (63), (67),and(68), we getexpressio ns\nkc,t1=/parenleftbiggνni\n2√ξicsi/parenrightbigg3\n4\nL−1\n4M3\n4\nA, kc,t2=/parenleftbigg2νni√ξicsi/parenrightbigg3\n2\nL1\n2M−3\n2\nA, (C1)\nkdec,t1=/parenleftbiggνni√ξicsi/parenrightbigg3\n4\nL−1\n4M3\n4\nA, kdec,t2=/parenleftbiggνni√ξicsi/parenrightbigg3\n2\nL1\n2M−3\n2\nA, (C2)\nfor1/k < l Ainsuper-Alfv´ enicturbulence,and\nkc,t1=/parenleftbiggνni\n2√ξicsi/parenrightbigg3\n4\nL−1\n4MA, kc,t2=/parenleftbigg2νni√ξicsi/parenrightbigg3\n2\nL1\n2M−2\nA, (C3)\nkdec,t1=/parenleftbiggνni√ξicsi/parenrightbigg3\n4\nL−1\n4MA, kdec,t2=/parenleftbiggνni√ξicsi/parenrightbigg3\n2\nL1\n2M−2\nA, (C4)\nfor1/k < l trin sub-Alfv´ enicturbulence,undertheconsiderationofst rongturbulenceanisotropy.\nFORCE ANALYSIS OF MAGNETOACOUSTIC WAVES\nThe wave behavior can be better understoodwith the aid of for ce analysis. Here we follow the methodin Soleret al. (2013b)\nfor Alfv´ en waves to perform an analysis on the forces for mag netoacoustic waves. We start from the momentum equations of\nionsandneutrals(see e.g. Zaqarashviliet al.2011),\n−iωvix=−ic2\nsik2\nx\nω(vix+viz)−i2V2\nAik2\nx\nωvix\n−νin(vix−vnx), (D1a)\n−iωviz=−ic2\nsik2\nx\nω(vix+viz)−νin(viz−vnz), (D1b)\n−iωvnx=−ic2\nsnk2\nx\nω(vnx+vnz)+νni(vix−vnx), (D1c)\n−iωvnz=−ic2\nsnk2\nx\nω(vnx+vnz)+νni(viz−vnz). (D1d)\nHereviandvnare velocityperturbationsin ionsand neutrals. We conside r the constant magnetic field is along z-direction,and\nwavespropagateinx-zplane. Foraqualitativestudyonwave behavior,wefixthewavepropagationdirectionas 45◦withregards\ntomagneticfield,i.e. kx=kz,tosimplifythealgebraicprocedures. Toconducttheforce analysisforionandneutralparticlesin\ndifferentdirections,fromaboveequationswe get\nfi,x=|Pi|+|M|−|Fx|, (D2a)\nfi,z=|Pi|−|Fz|, (D2b)\nfn,x=|Pn|+|Fx|, (D2c)\nfn,z=|Pn|+|Fz|. (D2d)\nHere|Pi|and|Pn|are gas pressure forces acting on ions and neutrals. |M|is magnetic pressure force and only acts on ions in\nperpendiculardirection to magnetic field. |Fx|and|Fz|are friction forces between neutrals and ions in x and z direc tions. The\nexpressionsof forcesare not displayedfor simplicity. We n exttake the parametersof MC phase and use the numericallyso lved\nωof fast and slow waves to calculate the correspondingforces . To further simply the problem, we also set mi= 2mnto have\nequalsoundspeedsinneutralsandions.\nInfigure16,we displaythewavefrequenciesanddifferentfo rces|f|normalizedby |M|formagnetoacousticwavesusingthe\nparametersofMC.Theanalyticalexpressionsof |ωR|and|ωI|ofsomebranchesarepresentspecifically.\n(a)Fastwaves We seefromFig. 16(b), |Fx| ∼ |M|(solidline)atlargescalesmakestwofluidsstronglycouple dandoscillate\ntogether. Theforcesin Eq. (D2) satisfy\nfi,x\nρi≈fn,x\nρn>fi,z\nρi≈fn,z\nρn(D3)\nto make neutrals move together with ions. The perpendicular component of force is larger due to the involvement of magnet ic\npressure. |Fz|/|M|(dashedline)increasestowards kdec,niandexceeds |Pi|/|M|(opencircles)at kdec,ni. Withinthecutoffinterval,\n|Fx|/|M|also increasesto be largerthan 1. Thestrongfrictionforceactingonionsefficientlydissip atestheoscillatorymotions\nofions. Inthemeantime,duetotheweakcouplingwithions, |Pn|/|M|(filledcircles)inneutralssubstantiallydropstothesame\nlevelas|Pi|/|M|afterthe cutoffinterval,andfurtherdecreasestogetherw ith|Fz|/|M|towardssmaller scales. Accordingly,no\nwavemotionscanbedevelopedinneutralsaftertheydecoupl efromionsat kdec,ni.26\nAt scalessmallerthan 1/kdec,in,the forceshaverelations\nfi,x\nρi≫fn,x\nρn,fi,z\nρi≫fn,z\nρn, (D4a)\nfi,x≫fi,z, fn,x≫fn,z. (D4b)\nWith decreasing frictions in both directions, ions oscilla te under the pressure forces, and marginally affected by neu trals. The\ndampedfast wavesreemerge,butonlyinions,with aphasespe edVAiinsteadof VA(showninFig. 16(a)).\n(b)Usual slow waves Owning to the constant propagationdirection ( θ= 45◦) adopted, we observe different wave behavior\nof the slow waves here (Fig. 16(c)) from that in Fig. 7 despite the same environment parameters used. First, since the rela tion\nk/bardbl≪k⊥∼kbreaks down, kdec,niis smaller than k+\nc(see Eq. (52) and (53)). Thus kdec,nibecomes the cutoff scale of the\nusual slow waves in this case. Second, according to the expre ssion in Table 4, the ratio kc,t1/kc,t2substantially increases due\nto the relatively large value of cosθ. Meanwhile, the ratio |ωI|/|ωR|of the ion slow waves within [kc,t1,kc,t2]also increases,\nwhichleadstomoreefficientdampingandashorterwavenumbe rintervalofthiswavebranch. Inaddition,withaconstant cosθ,\nwe cannot observe a decreasing |ωI|withk. From above comparison we clearly see that the wave propagat ion direction can\nsignificantlyinfluencetheslow wavebehavior.\nSimilar analysistofast modesappliestothe usual slow mode son scaleslargerthan 1/kdec,ni. Within theinterval [kdec,ni,kc,t1],\nions are still strongly coupled with neutrals and undergoth e same forces as neutrals, which we will discuss later for the neutral\nslow waves. The propagating ion slow waves arise at kc,t1after|Pi|/|M|substantially increases and |M|dominates over |Fx|.\nBut due to the rise of |Fz|/|M|, the wave is strongly damped with |ωI|>|ωR|(see Fig. 16(c)) and cut off shortly at kc,t2. The\nionslowmodescanonlybefullyresumedbeyond kdec,inwhen|Fz|/|M|decreasesto 1. Thenfrictionforcesfurtherdecreaseand\nbecomenegligiblecomparedwith |M|towardssmallerscales. Theforceshaverelationsas\nfi,x\nρi>fi,z\nρi≫fn,z\nρn>fn,x\nρn. (D5)\n(c)Neutral slow waves As shown in Fig. 16(e), with θ= 45◦used, we can clearly see the change in the damping rate |ωI|\nof the neutral slow modes (see Eq. (58b) and (59)). Fig. 16(f) illustrates the forces corresponding to the neutral slow mo des.\n|Pn|/|M|significantlyincreasesandresultsina nearlyisotropicfo rceonneutrals. Thecompressionproducedbytheusualslow\nmodesatlargerscalesdrivesoscillationsinneutrals. Wit houtmuchinterferencefromions,isotropicallypropagati ngsoundwaves\nare able to arise and maintain in the neutral fluid. Before rea chingkdec,in, ions are still coupled with neutrals. As a results, the\nforcesare\nfn,x\nρn≈fn,z\nρn>fi,x\nρi≫fi,z\nρi. (D6)\nAt scalessmallerthan k−1\ndec,in,|Fx|/|M|dropsandconvergeswith |Fz|/|M|. Accordingly fi,x/ρifurtherdecreasesandbecomes\nsignificantlysmallercomparedwith theforcesonneutrals.\nTHE FOKKER-PLANCK DIFFUSION COEFFICIENTS\n(a)Super-Alfv ´enic turbulence Since the hydrodynamicturbulence over [L,lA]does not contribute to particle scattering, the\nintegral interval of Dµµis from the injection scale of GS95 turbulence lAto the dampingscale. In the followingexpressionsof\nDµµforthecase ofsuper-Alfv´ enicturbulence,weuse lAasthe injectionscale LandMA= 1appliescorrespondingly.\nInAlfv´enicturbulence,byapplyingNLT,we canget\nDG\nµµ(NLT) =4v√π(1−µ2)\n3LR2/integraldisplaykdamL\n1xx−10/3\n⊥dx/integraldisplay1\n0(1+µVA\nv)2dη\nη∆µJ2\n1(w)\nw2exp/bracketleftBigg\n−x/bardbl\nx2/3\n⊥−(µ−1\nηxR)2\n∆µ2/bracketrightBigg\n,(E1)\nwhere ’G’ refers to gyroresonance. Here the dimensionless q uantities are: R=v/(ΩL)(CR rigidity), x=kL,η= cosθ.\nAnd∆µ= ∆v/bardbl/vrepresents the dispersion in particle pitch angles due to th e nonlinear effect. Jn(w)is Bessel function with\nw=k⊥v⊥/Ω. InQLT,withthe δ-functionresonanceconditionused, DG\nµµbecomes\nDG\nµµ(QLT) =4vπ(1−µ2)\n3LR2/integraldisplaykdamL\n1x2x−10/3\n⊥dx/integraldisplay1\n0(1+µVA\nv)2dη\nηµJ2\n1(w)\nw2exp/parenleftBigg\n−x/bardbl\nx2/3\n⊥/parenrightBigg\nδ/parenleftbigg\nx−1\nηµR/parenrightbigg\n.(E2)\nForfastmodes,weobtain\nDG\nµµ(NLT) =v√π(1−µ2)\nLR2/integraldisplaykdamL\n1x−5/2dx/integraldisplay1\n0(1+µVA\nvη)2ηdη\n∆µ(J′\n1(w))2exp/bracketleftBigg\n−(µ−1\nηxR)2\n∆µ2/bracketrightBigg\n, (E3)\nDT\nµµ(NLT) =v√π(1−µ2)\nLR2/integraldisplaykdamL\n1x−5/2dx/integraldisplay1\n0(1+µVA\nvη)2ηdη\n∆µJ2\n1(w)exp/bracketleftBigg\n−(µ−VA\nηv)2\n∆µ2/bracketrightBigg\n, (E4)27\n10−110010110210310−210−1100101102103104105\nkVA\nνni|ω|/ ν niVAik\nVAkkdec,inkc−kdec,nikc+\n(a) Wavefrequencies of fastmodes10−110010110210310410510610−1010−910−810−710−610−510−410−310−210−1100101\nkVA\nνni|f|/|M|\nkc−kdec,nikc+kdec,in\n(b) Forces of fast modes\n10−110010110210310410510610−410−310−210−1100101102103104105106\nkVA\nνni|ω|/ ν ni\ncsk||csik||\nξi1/2csik||kc,t1kc,t2kdec,nikc−kdec,in\n(c) Wavefrequencies of usual slow modes10−110010110210310410510610−710−610−510−410−310−210−1100101102\nkVA\nνni|f|/|M|\nkc,t1kc,t2kc−kdec,inkdec,ni\n(d) Forces of usual slow modes\n10−110010110210310410510610−210−1100101102103104105106\nkVA\nνni|ω|/ ν nikdec,ni\ncsnk\nνnisin2θ/2νni/2kc−kdec,in\n(e) Wavefrequencies of neutral slow modes10−110010110210310410510610−410−310−210−1100101102103104105106\nkVA\nνni|f|/|M|kdec,nikc−kdec,in\n(f) Forces of neutral slow modes\nFIG. 16.— Wavefrequencies andforceratios vs. normalized kofmagnetoacoustic wavesusingMCparameters. In(a),(c),a nd(e),thepropagating ( |ωR|)and\ndamping( |ωI|)componentsofwavefrequenciesarerepresentedbysolidan ddashedlinesrespectively. In(b),(d),and(f),thedispla yedforceratiosare |Fx|/|M|\n(solid line), |Fz|/|M|(dashed line), |Pi|/|M|(open circles), and |Pn|/|M|(filled circles). Thehorizontal dashed lines indicate the p osition|f|/|M|= 1.\ncorresponding to gyroresonance and TTD interaction. For lo w-rigidity CRs with their rLshorter than the damping scale of\nturbulence, Jn(w)cantake theasymptoticform (w/2)n/n!atsmall argument. Then DT\nµµ(NLT)approximatelybecomes\nDT\nµµ(NLT) =v√π(1−µ2)2\n8L∆µexp/parenleftbigg\n−µ2\n(∆µ)2/parenrightbigg\n(/radicalbig\nkdamL−1). (E5)28\nTheDG\nµµinQLT takesthe form\nDG\nµµ(QLT) =vπ(1−µ2)\nLR2/integraldisplaykdamL\n1x−3/2dx/integraldisplay1\n0(1+µVA\nvη)2ηdη\nµ(J′\n1(w))2δ/parenleftbigg\nx−1\nηµR/parenrightbigg\n. (E6)\nItssimplifiedformforlow-rigidityCRsis\nDG\nµµ(QLT) =vπ√µ(1−µ2)\n4L√\nR/parenleftbigg2\n7−2/radicalbig\n1−µ2\n21µ2/parenrightbigg\n. (E7)\nDiffusioncoefficientsinthe caseofslow modesare\nDG\nµµ(NLT) =v√π(1−µ2)β2\n3LR2/integraldisplaykdamL\n1xx−10/3\n⊥dx/integraldisplay1\n0(1+µcs\nv)2(1−η2)η3dη\n∆µ(J′\n1(w))2exp/bracketleftBigg\n−x/bardbl\nx2/3\n⊥−(µ−1\nηxR)2\n∆µ2/bracketrightBigg\n,(E8)\nDT\nµµ(NLT) =v√π(1−µ2)β2\n3LR2/integraldisplaykdamL\n1xx−10/3\n⊥dx/integraldisplay1\n0(1+µcs\nv)2(1−η2)η3dη\n∆µJ2\n1(w)exp/bracketleftBigg\n−x/bardbl\nx2/3\n⊥−(µ−cs\nv)2\n∆µ2/bracketrightBigg\n,(E9)\nDG\nµµ(QLT) =vπ(1−µ2)β2\n3LR2/integraldisplaykdamL\n1x2x−10/3\n⊥dx/integraldisplay1\n0(1+µcs\nv)2(1−η2)η3dη\nµ(J′\n1(w))2exp/parenleftBigg\n−x/bardbl\nx2/3\n⊥/parenrightBigg\nδ/parenleftbigg\nx−1\nηµR/parenrightbigg\n.(E10)\n(b)Sub-Alfv´enic turbulence Sub-Alfv´ enic turbulence have different energy spectra fo r weak and strong turbulence regimes.\nAccordingly,thediffusioncoefficientisa sumofthecompon entsinbothweakandstrongturbulence.\nForthegyroresonancewithAlfv´ enmodes,wehave DG\nµµ=DG\nµµ,w+DG\nµµ,s. Here\nDG\nµµ,w(NLT) =2v√π(1−µ2)M2\nA\nLR2/integraldisplayM−2\nA\n1x−2\n⊥dx⊥(1+µVA\nv)21\n∆µJ2\n1(w)\nw2exp/bracketleftbigg\n−(µ−1\nR)2\n∆µ2/bracketrightbigg\n, (E11)\nisDG\nµµin weakturbulence,and\nDG\nµµ,s(NLT) =4v√π(1−µ2)M4/3\nA\n3LR2/integraldisplaykdamL\nM−2\nAxx−10/3\n⊥dx/integraldisplay1\n0(1+µVA\nv)2dη\nη∆µJ2\n1(w)\nw2exp/bracketleftBigg\n−x/bardbl\nx2/3\n⊥M4/3\nA−(µ−1\nηxR)2\n∆µ2/bracketrightBigg\n,\n(E12)\nisDG\nµµin strong turbulence. We can see DG\nµµ,s(NLT)is similar to the DG\nµµ(NLT)in super-Alfv´ eniccase (Eq. (E1)), but with a\nfactorM4/3\nAadded(MA<1).\nIn QLT, since weak turbulence only has smaller structures de veloped in perpendicular direction, i.e. k/bardbl=L−1over[L,ltr],\nweak turbulence does not contribute to gyroresonance scatt ering unless CRs have rLcomparable to L. Thus we only consider\nthecontributionofstrongturbulenceandget\nDG\nµµ,s(QLT) =4vπ(1−µ2)M4/3\nA\n3LR2/integraldisplaykdamL\nM−2\nAx2x−10/3\n⊥dx/integraldisplay1\n0(1+µVA\nv)2dη\nηµJ2\n1(w)\nw2exp/parenleftBigg\n−x/bardbl\nx2/3\n⊥M4/3\nA/parenrightBigg\nδ/parenleftbigg\nx−1\nηµR/parenrightbigg\n.\n(E13)\nTheenergyspectrumoffast modesarenotdependentonturbul enceregimes. Fast modeshavesimilardiffusioncoefficient sin\nsub-Alfv´ enicturbulenceasthoseinsuper-Alfv´ eniccase . TheexpressionsEq. (E3)-(74)stillapply,buthere Listhedrivingscale\nofturbulenceand M2\nAshouldbe multipliedaccountingforthe ratiobetweenkinet icandmagneticenergiesofturbulence.\nSlowmodeshave\nDG\nµµ,w(NLT) =v√π(1−µ2)β2M2\nA\n2LR2/integraldisplayM−2\nA\n1x−2\n⊥dx⊥(1+µcs\nv)2(1−η2)η4\n∆µ(J′\n1(w))2exp/bracketleftbigg\n−(µ−1\nR)2\n∆µ2/bracketrightbigg\n,(E14)\nDT\nµµ,w(NLT) =v√π(1−µ2)β2M2\nA\n2LR2/integraldisplayM−2\nA\n1x−2\n⊥dx⊥(1+µcs\nv)2(1−η2)η4\n∆µJ2\n1(w)exp/bracketleftbigg\n−(µ−cs\nv)2\n∆µ2/bracketrightbigg\n.(E15)\ninweakturbulence,and\nDG\nµµ,s(NLT) =v√π(1−µ2)β2M4/3\nA\n3LR2/integraldisplaykdamL\nM−2\nAxx−10/3\n⊥dx/integraldisplay1\n0(1+µcs\nv)2(1−η2)η3dη\n∆µ(J′\n1(w))2exp/bracketleftBigg\n−x/bardbl\nx2/3\n⊥M4/3\nA−(µ−1\nηxR)2\n∆µ2/bracketrightBigg\n,\n(E16)\nDT\nµµ,s(NLT) =v√π(1−µ2)β2M4/3\nA\n3LR2/integraldisplaykdamL\nM−2\nAxx−10/3\n⊥dx/integraldisplay1\n0(1+µcs\nv)2(1−η2)η3dη\n∆µJ2\n1(w)exp/bracketleftBigg\n−x/bardbl\nx2/3\n⊥M4/3\nA−(µ−cs\nv)2\n∆µ2/bracketrightBigg\n,\n(E17)29\ninstrongturbulence.\nQLTin strongturbulencegives\nDG\nµµ(QLT) =vπ(1−µ2)β2M4/3\nA\n3LR2/integraldisplaykdamL\nM−2\nAx2x−10/3\n⊥dx/integraldisplay1\n0(1+µcs\nv)2(1−η2)η3dη\nµ(J′\n1(w))2exp/parenleftBigg\n−x/bardbl\nx2/3\n⊥M4/3\nA/parenrightBigg\nδ/parenleftbigg\nx−1\nηµR/parenrightbigg\n.\n(E18)\nREFERENCES\nAckermann, M.,et al. 2012, ApJ,750, 3\nAdriani, O.,etal. 2011, Science, 332, 69\nArmstrong, J.W.,Rickett, B.J.,&Spangler, S. 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Z.1970, Soviet Physics Doklad y, 15,439\nZaqarashvili, T.V.,Carbonell, M.,Ballester, J.L.,&Khod achenko, M. L.\n2012, A&A,544, A143\nZaqarashvili, T.V.,Khodachenko, M. L.,&Rucker, H.O.2011 , A&A,529,\nA82"    },    {        "title": "2007.12994v2.Decay_for_the_Kelvin_Voigt_damped_wave_equation__Piecewise_smooth_damping.pdf",        "content": "arXiv:2007.12994v2  [math.AP]  20 Dec 2021DECAY FOR THE KELVIN-VOIGT DAMPED WAVE EQUATION:\nPIECEWISE SMOOTH DAMPING\nNICOLAS BURQ AND CHENMIN SUN\nAbstract. We study the energy decay rate of the Kelvin-Voigt damped wave eq uation with\npiecewise smooth damping on the multi-dimensional domain. Under suit able geometric as-\nsumptions on the support of the damping, we obtain the optimal poly nomial decay rate which\nturns out to be different from the one-dimensional case studied in [ LR05]. This optimal de-\ncay rate is saturated by high energy quasi-modes localised on geome tric optics rays which hit\nthe interface along non orthogonal neither tangential directions . The proof uses semi-classical\nanalysis of boundary value problems.\n1.Introduction\n1.1.Kelvin-Voigt damped wave equation. In this article, we study the decay rate of the\nKelvin-Voigt damped wave equation on the multi-dimensional bounded domain Ω ⊂Rd,d≥2:\n\n\n/parenleftbig\n∂2\nt−∆−diva(x)∇∂t/parenrightbig\nu= 0,(t,x)∈Rt×Ω,\nu(t,·)|∂Ω= 0,\n(u,∂tu)|t=0= (u0,u1)(1.1)\nThe damping a(x)≥0 is assumed to be piecewise smooth. Denote by H1=H1\n0×L2. The\nsolution of ( 1.1) can be written as\nU(t) =/parenleftbiggu(t)\n∂tu(t)/parenrightbigg\n=etA/parenleftbiggu0\nu1/parenrightbigg\n,\nwhere the generator\nA=/parenleftbigg0 1\n∆ diva∇/parenrightbigg\n(1.2)\nwith domain\nD(A) ={(u0,u1)∈H1\n0×L2: ∆u0+diva∇u1∈L2,u1∈H1\n0}.\nNote that the energy\nE[u](t) =1\n2/ba∇dbletA(u0,u1)/ba∇dbl2\nH1=1\n2/integraldisplay\nΩ/parenleftbig\n|∂tu|2+|∇u|2/parenrightbig\ndx\nsatisfies\nE[u](t)−E[u](0) =−/integraldisplayt\n0/integraldisplay\nΩa(x)|∇x∂tu|2(s,x)ds\n12 N. BURQ AND C-M. SUN\nIt was proved in [ BC15] and [BS20] (see also [ LR06],[Te16] for related results) that if ais\nsmooth, vanishing nicely and the region {x∈Ω :a(x)>0}controls geometrically Ω, then the\nrate of decay of the energy is exponential:\nE[u](t)≤Ce−ctE[u](0).\nIn this article, we investigate the different case where the damping a(x) is piecewise smooth\nand has a jump across some hypersurface Σ ⊂Ω. Unlike the smooth damping vanishing nicely,\nthe problem with piecewise damping can be seen as an elliptic-hyperbolic transmission system\non the two sides of the interface Σ connected by some transmission condition. The interface be-\ncomesawalltoreducetheenergytransmissionfromthehyperbolic regiontothedampedregion.\nThis phenomenon is known as overdamping . It turns out that this discontinuous Kelvin-Voigt\ndamping ∇ ·(a(x)∇∂tu) does not follow the principle that the “geometric control condition ”\nimplies the exponential stabilization, which holds for the wave equatio n with localized viscous\ndampinga(x)∂tu(see [LR05,Zh18] for results using the multiplier methods)\n1.2.The main result. To state our main result, we first make some geometric assumptions .\nLet Ω⊂Rdwithd≥2. We consider the piecewise smooth damping a∈C∞(Ω1),a|Ω\\Ω1= 0,\nsuch that there exists α0>0,\ninf\nx∈∂Ω1a(x)≥α0,\nwhere Ω 1⊂Ω.We assume that ∂Ω1consists of ∂Ω and Σ = ∂Ω1\\∂Ω where Σ ⊂Ω. Denote\nby Ω2= Ω\\(Ω1∪Σ), then∂Ω2= Σ is the interface. We will fix this geometry in this article\nand assume that Ω 1,Ω2and Σ are smooth ( C∞, though this assumption could be relaxed to a\nfinite number of derivatives).\nΩ1:a(x)≥α0\nΩ2:a(x) = 0\nν\nΣ\nGeometry of the damped region\nDefinition 1.1 (Geometric control condition) .We say that Ω1satisfies the geometric control\ncondition, if all generalized rays (geometric optics reflec ting on the boundary ∂Ωaccording to\nthe laws of geometric optics) of Ωeventually reach the set Ω1in finite time.\nAn alternative (equivalent in this context) property is the followingKELVIN-VOIGT DAMPED WAVE EQUATION 3\n(H) All the bicharacteristics of Ω 2will reach a non-diffractive point (with respect to the\ndomain Ω 2) at the boundary Σ.\nTheorem 1. Assume that Ω,Ω1,Ω2anda(x)satisfy the above geometric conditions. Then\nunder the hypothesis (H), there exists a uniform constant C >0, such that for every (u0,u1)∈\nD(A)andt≥0,\n/ba∇dbletA(u0,u1)/ba∇dbl ≤C\n1+t/ba∇dbl(u0,u1)/ba∇dblD(A). (1.3)\nMoreover, the decay rate is optimal in the following sense: w henΩ⊂Rd,d≥2andΩ2=D⊂Ω\nis a unit ball, Ω1= Ω\\Ω2, the semi-group etAassociated with the damping a(x) =1Ω1(x)satisfies\nsup\n0/\\egatio\\slash=(u0,u1)∈D(A)/ba∇dbletA(u0,u1)/ba∇dblH1\n/ba∇dbl(u0,u1)/ba∇dblD(A)≥C′\n1+t, (1.4)\nfor allt≥0, whereC′>0is a uniform constant.\nRemark 1.2. In [Bu19], under the geometric control condition, a weaker decay rate, na mely\n1√1+twas achieved with a simpler and very robust general proof requiring much less rigidity on\nthe geometric setting. Notice also that in dimension 1, a stronger de cay rate, namely1\n(1+t)2is\nknown to hold [ LR05, Section 3, Example 1]. It is hence remarquable that in higher dimensio ns\nwe can construct examples of geometries where the1\n(1+t)decay rate is saturated. This phenom-\nenon is linked to the fact that in higher dimensions there exists seque nces of eigenfunctions of\nthe Laplace operator in Ω 2with Dirichlet boundary condidtions (or at least high order quasi-\nmodes), with mass concentrated along rays which do not encounte r the boundary at normal\nincidence (a fact which is clearly false in dimension 1, seeing that in this c ase the incidence is\nalways normal).\nΩ1\nΩ2\nΣ\nAngle of incidence is acute\nRemark 1.3. Let us mention that the non-exponential stability for ( 1.1) and a more general\n(theromo)viscoelastic system were studied in [ MRa], where the authors obtained a rougher\npolynomial decay rate O(t−1\n3). Moreover, in our result, the damped region (Ω 1) only needs to\nsatisfy the geometric control condition, so the geometric configu ration in Munoz Rivera-Racke\nis contained in our assumption.4 N. BURQ AND C-M. SUN\nRemark 1.4. The choice of Dirichlet boundary conditions on ∂Ω plays no particular role, and\nwe could have taken any type of boundary conditions for which the s ystem is well posed and\nwe have propagation of singularities (e.g. Neumann boundary condit ions)\nRemark 1.5. The picture for Kelvin Voigt damping is now quite complete for smooth ( essen-\ntiallyC2) dampings [ BC15] and [BS20] (and also [ LR06],[Te16]), or discontinuous dampings,\nsee in dimension 1 [ LR05, Section 3, Example 1], and the present paper. It would be interest ing\nto understand the intermediate situation ( Cα,α∈(0,2), dampings). We refer to [ HZZ] for\nresuts in this direction in dimension 1.\nRemark 1.6. In this article, we do not treat the case where Σ ∩∂Ω/\\e}atio\\slash=∅. In that case, ∂Ω2can\nbe only Lipchitz, and more technical treatments for the propagat ion of singularities are needed\nnear the points Σ ∩∂Ω.\nTheorem 2. We have Spec (A)∩iR=∅. Moreover, there exists C >0, such that for all\nλ∈R,|λ| ≥1,/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble\nL(H)≤C|λ|. (1.5)\nMoreover, when Ω⊂Rd,d≥2andΩ2=D⊂Ωis a unit ball, Ω1= Ω\\Ω2, we actually have\na lower bound:\nlimsup\nλ→+∞λ−1/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble\nL(H)=c>0.\nIn other words, there exist sequences (Un)⊂ H1andλn→+∞such that\n/ba∇dblUn/ba∇dblH= 1,/ba∇dbl(iλn−A)Un/ba∇dblH=O(λ−1\nn). (1.6)\n•Theorem 1and Theorem 2are essentially equivalent. Indeed, the equivalence between the\nresolvent estimate ( 1.5) and the decay rate ( 1.3) is covered by Theorem 2.4 of [ BoT10]. It\nis very likelly that ( 1.4) and (1.5) are also equivalent. However, we prove here only the fact\nthat (1.6) implies ( 1.4). We argue as follows: Let Unbe a sequence of quasi-modes associated\nwithλn(λn→+∞) that saturates ( 1.5). Denote by Fn= (iλn−A)Un. We have\n/ba∇dblUn/ba∇dblH= 1,/ba∇dblFn/ba∇dblH=O(λ−1\nn),/ba∇dblUn/ba∇dblD(A)∼λn.\nDefineUn(t) = etAUnand we write\nUn(t) = eiλntUn+Rn(t),\nthen\n(∂t−A)Rn=−(iλn−A)eitλnUn=OH(λ−1\nn), Rn(0) = 0.\nSince\nRn(t) =−/integraldisplayt\n0e(t−s)A(iλn−A)eisλnUnds,\nwe deduce that /ba∇dblRn(t)/ba∇dblH=O(λ−1\nnt) fort >0. Assume that κ(t) is the optimal decay rate of\nthe energy, then by E[Un(t)]1\n2=/ba∇dblUn(t)/ba∇dblH≤κ(t)1\n2/ba∇dblUn/ba∇dblD(A)we have\nC1κ(t)1\n2λn≥1−/ba∇dblRn(t)/ba∇dblH= 1−C2λ−1\nnt.KELVIN-VOIGT DAMPED WAVE EQUATION 5\nFor fixedt>0, we choose nlarge enough such that C2λ−1\nnt=1\n2, thus we obtain that\nκ(t)1\n2≥1\n2C1λn=1\nC1C2t.\nThis proves ( 1.4). As a consequence, we shall in the sequel reduce the analysis to t he proof of\nTheorem 2.\nThis article is organized as follows. We present the proof of ( 1.5) of Theorem 2in Section\n2, Section 3 and Section 4. The proof follows from a contradiction ar gument which reduces\nthe matter to study the associated high energy quasi-modes. In S ection 2, we reduce the\nequation of quasi-modes to a transmission problem, consisting of an elliptic system in Ω 1and a\nhyperbolic system in Ω 2, coupled at the interface Σ. Both systems are semi-classical but w ith\ndifferent scales h,/planckover2pi1=h1/2. Next in Section 3, we study the elliptic system and obtain the\ninformation of the quasi-modes restricted to the interface by tra nsmission conditions. Then in\nSection 4, we prove the propagation theorem for the hyperbolic pr oblem in Ω 2which will lead\nto a contradiction. We need to analyze two semi-classical scales cor responding to the elliptic\nand hyperbolic region, connected by the transmission condition on t he interface. Finally in\nSection 5, we construct a sequence of quasi-modes saturating th e inequality ( 1.5) in a simple\ngeometry. In particular this proves the optimality of the resolvent estimate. We collect various\ntoolboxes in the final section of the appendix.\nThroughout this article, we adopt the standard notations in semi-c lassical analysis (see for\nexample [ Zw12]). We will use the standard quantization for classical and semi-class ical pseudo-\ndifferential operators Op ,Oph,Op/planckover2pi1. We will also adopt the usual asymptotic notations, such\nasO(hα),O(/planckover2pi1α) ando(hα),o(/planckover2pi1α), ash→0. Moreover, for a Banach space Xandh-dependent\nfamilies of functions fh,gh, we meanfh=OX(hα),gh=oX(hα), if\n/ba∇dblfh/ba∇dblX=O(hα),/ba∇dblgh/ba∇dblX=o(hα),\nash→0.\nAcknowledgment. The first author is supported by Institut Universitaire de France a nd\nANR grant ISDEEC, ANR-16-CE40-0013. The second author is sup ported by the postdoc\nprograme: “Initiative d’Excellence Paris Seine” of CY Cergy-Paris Un iversit´ e and ANR grant\nODA (ANR-18-CE40- 0020-01).\n2.Reduction to a transmission problem\nIt was proved by the first author in [ Bu19] that\n/ba∇dbl(A−iλ)−1/ba∇dblL(H)≤Cec|λ|\nundermoregeneralconditionsforthedamping. Therefore, thep roofofthefirstpartofTheorem\n2(i.e. (1.5)) is reduced to the high energy regime |λ| →+∞. For this, we argue by contradic-\ntion. Assume that ( 1.5) is not true, then there exist h-dependent functions U=/parenleftbigu\nv/parenrightbig\n,F=/parenleftbigf\ng/parenrightbig\n,6 N. BURQ AND C-M. SUN\nsuch that\n/ba∇dblUj/ba∇dblH1×L2=O(1),/ba∇dblFj/ba∇dblH1×L2=o(h) (2.1)\n(A−ih−1)U=F. (2.2)\nLetνbe the unit normal vector pointing to the undamped region Ω. Denot e bya1(x) =\na(x)1Ω1LetU=/parenleftbigu\nv/parenrightbig\nandF=/parenleftbigf\ng/parenrightbig\n. Then forU∈D(A) andF∈ H, the equation\n(A−iλ)U=F\nis equivalent to ( h=λ−1) the following system for uj=u1Ωj,fj=f1Ωj,andgj=g1Ωj,\nj= 1,2:\n\n\nu1=ih(f1−v1),in Ω1\nh∆u1+h∇x·(a1(x)∇v1)−iv1=hg1,in Ω1\nu2=ih(f2−v2),in Ω2\nh∆u2−iv2=hg2,in Ω2(2.3)\nwith boundary condition on the interface\nu1|Σ=u2|Σ, ∂νu2|Σ= (∂νu1+a1∂νv1)|Σ, (2.4)\nIndeed, the equations inside Ω 1,Ω2can be verified directly. The first boundary condition is just\nthe fact that the function uequal toujin Ωjmust have no jump at the interface to enssure\ntaht ity belongs to H1(Ω). To check the second boundary condition, we take an arbitrar y test\nfunctionϕ∈C∞\nc(Ω) and multiply the equation h∆u−iv+hdiva∇v= 0 byϕ. We obtain that\n0 =−h/integraldisplay\nΩ∇u·∇ϕ−h/integraldisplay\nΩa∇v·∇ϕ−i/integraldisplay\nΩvϕ\n=−2/summationdisplay\nj=1/integraldisplay\nΩj/parenleftbig\nh∇uj·∇ϕ−ivjϕ/parenrightbig\n−h/integraldisplay\nΩ1a1(x)∇v1·∇ϕ\n=2/summationdisplay\nj=1/integraldisplay\nΩj/parenleftbig\nh∆ujϕ−ivjϕ/parenrightbig\n+/integraldisplay\nΩ1h∇x·(a1(x)∇v1)·ϕ+h/integraldisplay\nΣ(∂νu2−∂νu1−a1∂νv1)|Σ·ϕ.\nUsing the differential equations in Ω 1,Ω2, the last term on the right side is equal to\nh/integraldisplay\nΣ(∂νu2−∂νu1−a1∂νv1)|Σ·ϕ|Σ,\nhence it must vanish for all ϕ. This verifies ( 2.4).\nFirst we prove an a priori estimate for these functions:\nLemma 2.1 (A priori estimate) .Denote byUj=/parenleftbiguj\nvj/parenrightbig\n,Fj=/parenleftbigfj\ngj/parenrightbig\n, forj= 1,2. Assume that\n/ba∇dblUj/ba∇dblH1×L2=O(1)and/ba∇dblFj/ba∇dblH1×L2=o(h), then we have\n/ba∇dbl∇v1/ba∇dblL2=o(h1\n2),/ba∇dblv1/ba∇dblL2=o(h)KELVIN-VOIGT DAMPED WAVE EQUATION 7\nand\n/ba∇dbl∇u1/ba∇dblL2=o(h3\n2),/ba∇dblu1/ba∇dblL2=o(h2).\nConsequently, by the trace theorem, we have\n/ba∇dblu1/ba∇dblH1\n2(Σ)=o(h3\n2),/ba∇dblv/ba∇dblH1\n2(Σ)=o(h1\n2).\nProof.First we observe that, from the relation between uandv, we deduce that ∇v∈L2(Ω)\nand\n/ba∇dbl∇vj/ba∇dblL2(Ωj)=O(h−1), j= 1,2. (2.5)\nMoreover, by the trace theorem, v1|Σ=v2|Σas functions in H1\n2(Σ). From the system ( 2.3), we\nhave\n(∇u1,∇v1)L2(Ω1)\n=ih(∇f1,∇v1)L2(Ω1)−ih/ba∇dbl∇v1/ba∇dbl2\nL2(Ω1)−(∇u1,∇v1)L2(Ω1)\n−/ba∇dbla1/2\n1∇v1/ba∇dbl2\nL2(Ω1)+(∂νu1+a1∂νv1,v1)L2(Σ) (2.6)\n=ih−1/ba∇dblv1/ba∇dbl2\nL2(Ω1)+(g1,v1)L2(Ω1)−(∇u2,∇v2)L2(Ω2) (2.7)\n=ih(∇f2,∇v2)L2(Ω2)−ih/ba∇dbl∇v2/ba∇dbl2\nL2(Ω2)−(∇u2,∇v2)L2(Ω2)−(∂νu2,v2)L2(Σ)(2.8)\n=ih−1/ba∇dblv2/ba∇dbl2\nL2(Ω2)+(g2,v2)L2(Ω2). (2.9)\nTaking the real part of ( 2.6)+(2.7)-(2.8)+(2.9), we deduce that /ba∇dbl∇v1/ba∇dbl2\nL2(Ω1)=o(h), thanks to\nthe boundary condition ( 2.4) andv1|Σ=v2|Σ. Therefore, from the first equation of ( 2.3), we\nhave/ba∇dbl∇u1/ba∇dbl2\nL2(Ω1)=o(h3). Then, using this fact and the second equation of ( 2.3), we deduce\nthat\niv1=h∆u1+h∇·(a1∇v1)−hg1=OH−1(Ω1)(h3\n2).\nBy interpolation, we have v1=oL2(Ω1)(h), and from u1=ih(f1−v1),u1=oL2(Ω1)(h2). This\ncompletes the proof of Lemma 2.1. /square\n3.Estimates of the elliptic system\n3.1.Standard theory. Webrieflyrecallthesemiclassical ellipticboundaryvalueproblemnear\nthe interface Σ. In what follows, we will sketch the parametrix cons truction for ( 3.1), following\n[BL03]. Near a point x0∈Σ, we use the coordinate system ( y,x′) where Ω 1={(y,x′) :y>0}\nnearx0.\nL/planckover2pi1w=κ=oL2(/planckover2pi12), w|Ω1=oH1(/planckover2pi1), w|Σ=oH1\n2(/planckover2pi1) (3.1)\nwhere in the local coordinate chart,\nL/planckover2pi1:=/planckover2pi12D2\ny−R(y,x′,/planckover2pi1Dx′)+d−1/summationdisplay\nj=1/planckover2pi1Mj(y,x′)/planckover2pi1∂x′\nj+/planckover2pi1H(y,x′)/planckover2pi1∂y.8 N. BURQ AND C-M. SUN\nHereR(y,x′,/planckover2pi1Dx′) is a second order semiclassical differential operator in x′with the principal\nsymbolr(y,x′,ξ′). The principal symbol of L/planckover2pi1is\nl(y,x′,η,ξ′) =η2−r(y,x′,ξ′),\nand we denote by\nm(y,x′,η,ξ′) =d−1/summationdisplay\nj=1Mj(y,x′)ξ′\nj+H(y,x′)η.\nThe set of elliptic points in T∗∂Ω is given by\nE:={(y= 0,x′,ξ′) :r(0,x′,ξ′)<0}\nBy homogeneity, near a point ρ0∈ E\n−r(y,x′,ξ′)≥c(ρ0)|ξ′|2. (3.2)\nDenote by w:=w1y≥0the extension by zero of w, and the same for κ,etc. Thenwsatisfies\nthe equation\nL/planckover2pi1w=−/planckover2pi1(/planckover2pi1∂yw)|y=0⊗δy=0+/planckover2pi12w|y=0⊗δ′\ny=0+/planckover2pi12H(0,x′)w|y=0⊗δy=0+κ.(3.3)\nLetϕ(y,x′) be a cut-off to the local chart. Let ψ∈C∞(Rd−1), be a Fourier multiplier in S0\nξ′\nsuch that on the support of ϕ(y,x′)ψ(ξ′), (3.2) holds and ϕ(y,x′)ψ(ξ′) = 1 nearρ0. We define\ne0(y,x′,η,ξ′) :=ϕ(y,x′)ψ(ξ′)\nl(y,x′,η,ξ′)(3.4)\nandej,j≥1 inductively by\ne1·l=−/summationdisplay\n|α|=11\ni∂α\nξ′,ηe0·∂α\nx′,yl−e0·m,\nej·l=−/summationdisplay\n|α|+k=n,k/\\egatio\\slash=n1\ni|α|∂α\nξ′,ηek·∂α\nx′,yl−/summationdisplay\n|α|+k=n−11\ni|α|∂α\nξ′,ηek·∂α\nx′,ym.\nFor anyN∈N, we define\neN=N/summationdisplay\nj=0/planckover2pi1jej, EN= Op/planckover2pi1(eN),\nand then\nENL/planckover2pi1=ϕ(y,x′)ψ(ξ′)Id+RN,\nwhere\nRN=O(/planckover2pi1N+1) :L2\nx′,y→L2\nx′,y, RN=O(/planckover2pi1N+1−2M) :Hs\nx′,y→Hs+2M\nx′,y,\nand\nEN=O(1) :L2\nx′,y→L2\nx′,y, EN=O(/planckover2pi1−2) :Hs\nx′,y→Hs+2\nx′,y,KELVIN-VOIGT DAMPED WAVE EQUATION 9\nthanks to Lemma 6.2. Applying ENto the equation ( 3.3), we obtain that\nϕ(y,x′)ψ(/planckover2pi1Dx′)w=−/planckover2pi12EN((∂yw)|y=0⊗δy=0)+/planckover2pi12EN(w|y=0⊗δ′\ny=0)+/planckover2pi12EN(Hw|y=0⊗δy=0)\n+ENκ−RNw.\nNote thateN(y,x′,η,ξ′) is meromorphic in ηwith poles η±=±i/radicalbig\n−r(y,x′,ξ′). Denote by\nG(x′) =∂yw(0,x′)+H(0,x′)w(0,x′), we calculate for y>0,x′∈Rd−1that\n/planckover2pi12EN((∂yw+Hw)|y=0⊗δy=0)(y,x′)\n=/planckover2pi12\n(2π/planckover2pi1)d/integraldisplay\nG(/tildewidex′)ei(x′−/tildewidex′)·ξ′\n/planckover2pi1d/tildewidex′dξ′/integraldisplay\neN(y,x′,η,ξ′)eiyη\n/planckover2pi1dη\n=i/planckover2pi1\n(2π/planckover2pi1)d−1/integraldisplay\neiyη+\n/planckover2pi1nN(y,x′,ξ′)ei(x′−/tildewiderx′)·ξ′\n/planckover2pi1G(/tildewidex′)d/tildewidex′dξ′,\nwherenN(y,x′,ξ′) = Res(eN(y,x′,η,ξ′);η=η+). Similarly, for y>0,x′∈Rd−1,\n/planckover2pi12EN(w|y=0⊗δ′\ny=0)(y,x′) =i/planckover2pi1\n(2π/planckover2pi1)d/integraldisplay\nw(0,/tildewidex′)ei(x′−/tildewidex′)·ξ′\n/planckover2pi1d/tildewidex′dξ′/integraldisplay\nηeiyη\n/planckover2pi1eN(y,x′,η,ξ′)dη\n=−1\n(2π/planckover2pi1)d−1/integraldisplay\neiyη+\n/planckover2pi1dN(y,x′,ξ′)ei(x′−/tildewidex′)·ξ′\n/planckover2pi1w(0,/tildewidex′)d/tildewidex′dξ′,\nwheredN=η+nN. Therefore,\nϕ(y,x′)ψ(/planckover2pi1Dx′)w\n=iOp/planckover2pi1(eiyη+//planckover2pi1nN(y,·))/parenleftbig\n−(/planckover2pi1∂yw)|y=0+/planckover2pi1(Hw)|y=0/parenrightbig\n−Op/planckover2pi1(eiyη+//planckover2pi1dN(y,·))(w|y=0)\n+ENκ−RNw,(3.5)\nwhere the two operators in the expression above are tangential. N ote that by Lemma 6.2\nRNw, ENκ=oL2\nx′,y(/planckover2pi12) =oH2\nx′,y(1),\nhence from the interpolation and the trace theorem, we have\n(RNw)|y=0=oH1/2\nx′(/planckover2pi1),(ENκ)|y=0=oH1/2\nx′(/planckover2pi1).\nTaking the trace y= 0 for (3.5), we obtain that\nOp/planckover2pi1(ϕ(0,x′)ψ(ξ′)+dN(0))(w|y=0) =−Op/planckover2pi1(inN(0))((/planckover2pi1∂yw)|y=0+/planckover2pi1(Hw)|y=0)+oH1/2\nx′(/planckover2pi1).(3.6)\nNote that the principal symbols of nN(0),dN(0) are\nσ(inN(0)) =ϕ(0,x′)ψ(ξ′)\n2/radicalbig\n−r(0,x′,ξ′), σ(dN(0)) =ϕ(0,x′)ψ(ξ′)\n2.\nIn summary, there exists (near ρ0) a/planckover2pi1-P.d.ON/planckover2pi1, elliptic and of order 1 classic and of order 0\nsemi-classic, in the sense that\nN/planckover2pi1=O(/planckover2pi1) :Hs\nx′→Hs−1\nx′,\nsuch that\n(/planckover2pi1∂yw)|y=0=N/planckover2pi1(w|y=0+OH1/2(/planckover2pi1)).10 N. BURQ AND C-M. SUN\n3.2.Control of the semi-classical wave-front set of the trace. For the further need, we\nshould also control the wave front set of the precise elliptic equatio n (with/planckover2pi1=h1\n2)\n/planckover2pi12∆w−i\na1w+/planckover2pi1∇a1\na1·/planckover2pi1∇w=κ,\nwhere theh-semiclassical wave front set of the Neumann data WF h(∂νw|Σ).Here we need to\npay attention to two different semi-classical scales.\nProposition 3.1. Assume that wsatisfies the /planckover2pi1-semiclassical elliptic equation (with /planckover2pi1=h1\n2)\n/planckover2pi12∆w−i\na1w+/planckover2pi1∇a1\na1·/planckover2pi1∇w=κ\nwith Neumann trace ∂νw|ΣandWFh(∂νw|Σ)is contained in a compact subset of T∗Σ\\ {0}.\nAssume moreover that w=OH1(h1\n2)andκ=OL2(h), then we have\nWFh(w|Σ)⊂WFh(∂νw|Σ)∪π/parenleftbig\nWFh(κ)/parenrightbig\n,\nwhereπ:T∗Ω1→T∗Σis the projection defined for points near T∗Σ, and\nπ/parenleftbig\nWFh(κ)/parenrightbig\n=/braceleftbig\nρ0∈T∗Σ :∃ρ∈T∗Ω1,nearT∗Σ,such thatρ∈WFh(κ)andπ(ρ) =ρ0/bracerightbig\n.\nProof.Let (x0,ξ0)/∈WFh(∂νw|Σ)∪π/parenleftbig\nWFh(κ)/parenrightbig\n. Locally near x0∈Σ, we can choose local\ncoordinate system as in the previous subsection. Here the cutoff ψ(ξ′) can be chosen as 1, since\nthe operator /planckover2pi12∆−iis always elliptic. Consider the tangential h-P.d.OAhwhich is elliptic\nnear (x0,ξ0) and its principal symbol is supported away from WF h(∂νw|Σ)∪π/parenleftbig\nWFh(κ)/parenrightbig\n. We\nneed to show that ( Ahw)|y=0=OL2(Σ)(h∞).\nFrom (3.5) we have\nϕ(y,x′)w=iOp/planckover2pi1/parenleftbig\neiyη+\n/planckover2pi1nN(y)/parenrightbig/parenleftbig\n−(/planckover2pi1∂yw)|y=0+/planckover2pi1(Hw)|y=0/parenrightbig\n−Op/planckover2pi1/parenleftbig\neiyη+\n/planckover2pi1dN(y)/parenrightbig\n(w|y=0)\n+ENκ+OH1(hN\n2),\nwhere we gain /planckover2pi1NforRNw. Bytaking thetrace y= 0andusing thefactthat dN(0) =1\n2ϕ(0,x′),\nwe obtain that\n(Ahϕ(y,x′)w)|y=0+/parenleftbig\nAhOph(dN(0))w/parenrightbig\n|y=0−i/planckover2pi1(AhOp/planckover2pi1(nN)(Hw)|y=0\n=−iAhOp/planckover2pi1(nN(0))(/planckover2pi1∂yw)|y=0+(AhENκ)|y=0+O\nH1\n2\ny=0(hN/2).\nWe claim that it suffices to show that\nAhOp/planckover2pi1(nN(0))(/planckover2pi1∂yw)|y=0=OL2\ny=0(h∞) and (AhENκ)|y=0=OL2\ny=0(h∞).(3.7)\nIndeed, once this is done, we obtain that, at least ( Ahw)|y=0=OL2(/planckover2pi1)1. Now we can replace Ah\nby another tangential operator /tildewideAhwith principal symbol /tildewideasuch that /tildewideais supported in a slightly\n1The operator Ahis of the form χAhχfor some cutoff χsuch that χ≡1 on supp( ϕ).KELVIN-VOIGT DAMPED WAVE EQUATION 11\nlarger region containing supp( a) and/tildewidea= 1 on supp( a). We still have ( /tildewideAhw)|y=0=OL2(/planckover2pi1). Now\nwe write\n/planckover2pi1AhOp/planckover2pi1(nN)Hw=/planckover2pi1AhOp/planckover2pi1(nN)H/tildewideAhw+/planckover2pi1AhOp/planckover2pi1(nN)H(1−/tildewideAh)w.\nFrom Lemma 3.2, the trace of the second term on the right side is OL2(h∞). Therefore, the\ntrace of the first term on the right side is OL2(/planckover2pi12), hence (Ahw)|y=0=OL2(/planckover2pi12). Then we can\ncontinuously apply this argument to conclude.\nIt remains to prove ( 3.7). For this, we just need to interchange the operator AhwithENand\nOp/planckover2pi1(nN(0)). Here additional attentions are needed, since Op/planckover2pi1(nN(0)),ENare/planckover2pi1-P.d.O. This\ncan be verified from the following lemma:\nLemma 3.2. Assume that a,b,q∈S0(Rn\nx×Rn\nξ), compactly supported in the xvariable such\nthat\ndist/parenleftbig\nsupp(a),supp(b)/parenrightbig\n≥c0>0.\nThen for any s∈R,N∈N,N≥2n, we have\na(x,hDx)q(x,h1\n2Dx)b(x,hDx) =OL2→L2(hN).\nProof.Denote by\nA(x,y,ξ,η) =a(x,h(ξ+η))q(x+y,h1\n2ξ).\nThen from Lemma 6.3,\na(x,hDx)q(x,h1\n2Dx) =/summationdisplay\n|β|≤NOp/parenleftbigh|β|\ni|β|β!(∂β\nξa)(x,hξ)(∂β\nxq)(x,h1\n2ξ)/parenrightbig\n+OL(L2)(hN+1−n),\nsince for any β∈N2n,\nsup\n|α|=N+1sup\n(x,ξ)/integraldisplay/integraldisplay\nR2n|∂β\nx,ξ∂α\nz∂α\nζ(a(x,h(ξ+ζ))q(x+z,h1\n2ξ))|dzdζ=O(hN+1−n).\nUsing the fact thath|β|\ni|β|β!∂β\nξa·∂β\nxq·b= 0, thanks to the support property, we have, using again\nLemma6.3,\na(x,hDx)q(x,h1\n2Dx)b(x,hDx) =OL(L2)(hN)\nfor anyNlarge enough. This completes the proof. /square\nTherefore the proof of Proposition 3.1is now complete. /square12 N. BURQ AND C-M. SUN\n3.3.Estimate of the traces. Letu1,v1be solutions of the first two equations of ( 2.3). Con-\nsiderw=u1+a1v1, then under the assumption of Lemma 2.1,\nw=oH1(h1\n2), w=oL2(h), w|Σ=oH1\n2(h1\n2).\nNote thatwsatisfies the elliptic equation (with /planckover2pi1=h1\n2)\n/planckover2pi12∆w+/planckover2pi1∇a1\na1·/planckover2pi1∇w−i\na1w=/planckover2pi12g1−/planckover2pi12∆a1·v1+/planckover2pi12|∇a1|2\na1v1−/planckover2pi1∇a1\na1·/planckover2pi1∇u1+i\na1u1(3.8)\nIn particular,\n/planckover2pi12∆w−i\na1w+/planckover2pi1∇a1\na1·/planckover2pi1∇w=oL2(/planckover2pi12).\nIn this case, N/planckover2pi1defined in the last subsection is the usual /planckover2pi1-semiclassical Dirichlet-Neumann\noperator:\nN/planckover2pi1(w|Σ+oH1/2(/planckover2pi12)) := (/planckover2pi1∂νw)|Σ.\nWe can apply the standard theory (to h−1w)with the particular choice ψ(ξ′)≡1 in (3.4) and\nobtain the following:\nProposition 3.3. Letχ∈C∞\nc(R). Then under the hypothesis of Lemma 2.1and in the local\nchart near Σ, we haveϕχ(hDx′)ϕ(∂yw)|y=0=oL2\nx′(1), whereh=/planckover2pi12. Consequently,\nu2|Σ=oH1/2(h3\n2), ϕχ(hDx′)ϕh∂yu2|y=0=oL2(Σ)(h).\nProof.Assume that ϕ,ϕ1are supported in a local chart and satisfy ϕ1|supp(ϕ)= 1. In apri-\nori, we have ϕ∂y(ϕ1w)|y=0=ϕ/planckover2pi1−1N/planckover2pi1((ϕ1w)|y=0) =oH−1/2\nx′(/planckover2pi1). Thus by Lemma 6.2we have\nϕ/planckover2pi1−1χ(/planckover2pi12Dx′)ϕN/planckover2pi1((ϕ1w)|y=0) =oL2\nx′(1).\n/square\n4.Propagation estimate\nIn this section, we will deal with the propagation estimate for u2inH1, satisfying\n(h2∆+1)u2=ihf2+h2g2=oH1(h2)+oL2(h3),in Ω2,\n/ba∇dblu2/ba∇dblH1(Ω2)=O(1),/ba∇dblu2|Σ/ba∇dblH1/2(Σ)=o(h3/2),\n/ba∇dblh∂νu2|Σ/ba∇dblH−1/2(Σ)=o(h3/2),/ba∇dblϕψ(hDx′)ϕh∂νu2|Σ/ba∇dblL2(Σ)=o(h).(4.1)\nSetw2=h−1u2. From (4.1),\n−/ba∇dbl∇u2/ba∇dbl2\nL2(Ω2)+/ba∇dblh−1u2/ba∇dbl2\nL2(Ω2)=/angbracketleftbig\n(∂νu2)|Σ·u2|Σ/angbracketrightbigH1/2(Σ)\nH−1/2(Σ)+o(1) =o(1).\nHence we could equivalently deal with the propagation estimate for w2inL2, satisfying\n(h2∆+1)w2=if2+hg2=oH1(h)+oL2(h2),in Ω2,\n/ba∇dblw2/ba∇dblH1(Ω2)=O(h−1),/ba∇dblw2/ba∇dblL2(Ω2)=O(1),/ba∇dblw2|Σ/ba∇dblH1/2(Σ)=o(h1/2),\n/ba∇dblh∂νw2|Σ/ba∇dblH−1/2(Σ)=o(h1/2),/ba∇dblϕψ(hDx′)ϕh∂νw2|Σ/ba∇dblL2(Σ)=o(1).(4.2)KELVIN-VOIGT DAMPED WAVE EQUATION 13\nThe goal of this section is to prove the invariance of the semiclassica l measureµassociated\nwith (a subsequence of) w2and finally prove that µ= 0 from the boundary conditions in ( 4.2)\non the interface Σ. This will end the contradiction argument.\n4.1.Propagation away from Σ.The defect measure in the interior of Ω 2foru2is defined\nvia the following quadratic form:\nφ(Qh,w2) = (Qhw2,w2)L2(Ω2):=/integraldisplay\nΩ2Qhw2·w2dx.\nProposition 4.1 (Interior propagation) .LetQh=/tildewideχQh/tildewideχbe ah-pseudodifferential operator of\norder 0, where /tildewideχ∈C∞\nc(Ω2), then we have\n1\nih/parenleftbig\n[h2∆+1,Qh]w2,w2/parenrightbig\nL2=o(1).\nProof.By developing the commutator and using the equation ( 4.1), we have\n/parenleftbig1\nih[h2∆+1,Qh]w2,w2/parenrightbig\n=1\nih/parenleftbig\n(h2∆+1)Qhw2,w2/parenrightbig\n−1\nih/parenleftbig\nQh(if2+hg2),w2/parenrightbig\n=1\nih/parenleftbig\nQhw2,if2+hg2/parenrightbig\n+1\nih/parenleftbig\nQh(if2+hg2),w2/parenrightbig\n=o(1),\nwhere we used the integration by part without boundary terms, sin ce the kernel of Qhis\nsupported away from the boundary Σ = ∂Ω2. This completes the proof of Proposition 4.1./square\n4.2.Geometry near the interface. Near Σ =∂Ω2, we adopt the local coordinate system\nx= (y,x′) inU:= (−ǫ0,ǫ0)y×Xx′for the tubular neighborhood of Σ, similar as in the previous\nsection but with the new convention Ω 2∩U= (0,ǫ0)y×Xx′. In this coordinate system, the\nEuclidean metric dx2is identified as the matrix\ng=/parenleftbigg1 0\n0g(y,x′)/parenrightbigg\n,org−1:=/parenleftbigg1 0\n0g−1(y,x′)/parenrightbigg\n.\nNear Σ, the defect measure µforw2is defined via the quadratic form for tangential operators:\nφ(Qh,w2) :=/integraldisplay\nUQhw2·w2/radicalbig\n|g|dydx′,\nwhere|g|:= det(g). The principal symbol of the operator Ph,0=−(h2∆+1) is\np(y,x′,η,ξ′) =η2+|ξ′|2\ng−1 :=η2+/a\\}b∇acketle{tξ′,g−1ξ′/a\\}b∇acket∇i}htRd−1−1.\nBy Char(P) we denote the characteristic variety of p:\nChar(P) :={(x,ξ)∈T∗Rd|Ω:p(x,ξ) = 0}.14 N. BURQ AND C-M. SUN\nDenote bybTΩ2the vector bundle whose sections are the vector fields X(p) onΩ2withX(p)∈\nTp∂Ω2ifp∈∂Ω2. Moreover, denote bybT∗Ω2the Melrose’s compressed cotangent bundle\nwhich is the dual bundle ofbTΩ2. Let\nj:T∗Ω2→bT∗Ω2\nbe the canonical map. In our geodesic coordinate system near ∂Ω2,bTΩ2is generated by the\nvector fields∂\n∂x′\n1,···,∂\n∂x′\nd−1,y∂\n∂yand thusjis defined by\nj(y,x′;η,ξ′) = (y,x′;v=yη,ξ′).\nLetZ:=j(Char(P)). By writing in another way\np=η2−r(y,x′,ξ′), r(y,x′,ξ′) = 1−|ξ′|2\ng,\nwe have the standard decomposition\nT∗∂Ω2=E ∪H∪G,\naccording to the value of r0:=r|y=0where\nE={r0<0},H={r0>0},G={r0= 0}.\nThe sets E,H,Gare called elliptic, hyperbolic and glancing, with respectively. We define also\nthe set\nHδ:={δ <r0<1−δ}\nwith 0< δ <1\n2for the non-tangential and non incident points. Note that here th e elliptic\npointsEis different from those defined in Section 3.\nTo classify different situations as a ray approaching the boundary, we need more accurate\ndecomposition of the glancing set G. Letr1=∂yr|y=0and define\nGk+3={(x′,ξ′) :r0(x′,ξ′) = 0,Hj\nr0(r1) = 0,∀j≤k;Hk+1\nr0(r1)/\\e}atio\\slash= 0},k≥0}\nG2,±:={(x′,ξ′) :r0(x′,ξ′) = 0,±r1(x′,ξ′)>0},G2:=G2,+∪G2,−.\nNext we recall the definition of the generalized bicharacteristic:\nDefinition 4.2. A generalized bicharacteristic of Ω2is a piecewise continuous map from Rto\nbT∗Ω2such that at any discontinuity point s0, the left and right limits γ(s0∓)exist and are\nthe two points above the same hyperbolic point on the boundar y (this property translates the\nspecular reflection of geometric optics) and except at these isolated points the curve is C1and\nsatisfies\n•dγ\nds(s) =Hp(γ(s))ifγ(s)∈T∗Ω2orγ(s)∈ G2,+\n•dγ\nds(s) =Hp(γ(s))−H2\npy\nH2ypHyifγ(s)∈ G \\G2,+whereyis the boundary defining function.KELVIN-VOIGT DAMPED WAVE EQUATION 15\nRemark 4.3. The first property in the definition above is the fact that the curve is a geodesic\nin the interior or passing though a non diffractive point. The second o ne is that passing through\na non diffractive gliding point it is curved to be forced to remain in the int erior ofT∗∂Ω2for a\nwhile. When the domainis smoothanddoes not have infinite order of co ntact with its tangents,\nthen (see [ MS]) through each point passes a unique generalized bicharacteristic. In general only\nexistence is known.\nRemark 4.4. In the statement of the geometric control condition 1.1, the generalized rays are\nthe projection of the generalized bicharacteristics of Ω onto Ω.\n4.3.Elliptic regularity.\nLemma 4.5. Denote byλ(y,x′,ξ′) =/radicalBig\n|ξ′|2\ng−1. Letψ∈C∞(Rd−1),ϕ1,ϕ2∈C∞\nc(Rd), such\nthat on the support of ψ(ξ′)ϕ1(y,x′)andψ(ξ′)ϕ2(y,x′),|ξ′|g>1+δfor someδ >0. Then we\nhave\nOph/parenleftbig\n1y≥0ϕ2e−yλ\nhψ(ξ′)/parenrightbig\nϕ1=O(1) :H−1\n2(Rd−1\nx′)→L2(Rd\n+)\nProof.Denote by Ty:= Oph/parenleftbig\n1y≥0ϕ2e−yλ\nhψ(ξ′)/parenrightbig\n. By definition, we have for f0∈H−1/2\nx′and\ny>0 that\n(Tyf0)(x′) :=1\n(2πh)d−1/integraldisplay/integraldisplay\ne−yλ(y,x′,ξ′)\nhψ(ξ′)ϕ2(y,x′)ei(x′−z′)·ξ′\nhf0(z′)dz′dξ′.\nDenote byF0:=/a\\}b∇acketle{tD′\nx/a\\}b∇acket∇i}ht−1/2f0, then this term can be written as\nOp/parenleftbig\ne−yλ(y,x′,hξ′)\nhψ(hξ′)/a\\}b∇acketle{tξ′/a\\}b∇acket∇i}ht1\n2ϕ2(y,x′)/parenrightbig\nF0.\nFor fixedy >0, from the Calder´ on-Vaillancourt theorem and the support prop erty ofψ, we\nhave, for any M >0 that\n/vextenddouble/vextenddoubleOp/parenleftbig\ne−yλ(y,x′,hξ′)\nhψ(hξ′)/a\\}b∇acketle{tξ′/a\\}b∇acket∇i}ht1\n2ϕ2(y,x′)/parenrightbig\nF0/vextenddouble/vextenddouble\nL2\nx′≤CMh−1\n2e−cy\nh/parenleftbig\n1+yM\nhM/parenrightbig\n/ba∇dblF0/ba∇dblL2\nx′.\nand the constants CM,care independent of y. Squaring the inequality above and integrating in\nyyields the bound O(1)/ba∇dblF0/ba∇dbl2\nL2\nx′=O(1)/ba∇dblf0/ba∇dbl2\nH−1\n2\nx′.This completes the proof of Lemma 4.5./square\nProposition 4.6.\nµ1E= 0.\nProof.Applying ( 3.5) toκ=oH1(h)+oL2(h2) and/planckover2pi1=h, we obtain that\nϕ(y,x′)ψ(hDx′)w2=−Oph/parenleftbig\neiyη+\nhinN(y,·)/parenrightbig\n(h∂yw2|y=0)−Oph/parenleftbig\neiyη+\nhdN(y,·)/parenrightbig\n(w2|y=0)\n+Oph/parenleftbig\neiyη+\nhinN(y,·)/parenrightbig\n(hH(0,x′)w2|y=0)+oL2\ny,x′(h2).(4.3)16 N. BURQ AND C-M. SUN\nApplying Lemma 4.5, we have Oph/parenleftbig\neiyη+\nhinN(y,·)/parenrightbig\n(h∂yw2|y=0) =oL2\ny,x′(h1\n2).By the same way,\nthe other terms on the right side of ( 4.3) are at most oL2\ny,x′(h1\n2). Henceϕ(y,x′)ψ(hDx′)w2=\noL2\ny,x′(h1\n2), and this completes the proof of Proposition 4.6. /square\n4.4.Propagation formula near the interface. Consider the operator\nBh=B0,h+B1,hh∂y\nwhereBj,h=/tildewideχ1Oph(bj)/tildewideχ1,j= 0,1 are two tangential operators and /tildewideχ1has compact support\nnear a point z0∈Σ. The symbols bjare compactly supported in ( x′,ξ′) variables. Note that\nin the local coordinate system,\nPh,0=−h2∆−1 =−1/radicalbig\n|g|h∂y/radicalbig\n|g|h∂y−Rh,\nwhereRhis a self-adjoint tangential differential operator of order 2 classic and of order 0\nsemiclassic.\nLemma 4.7 (Boundary propagation) .Let(/tildewidewh)be ah-dependent family of functions satisfying\n/tildewidewh=OL2(Ω2)=O(1)and/tildewidewh=OH1(Ω2)(h−1). Assume moreover that /tildewidewhsatisfies the equation\nPh,0/tildewidewh=oH1(Ω2)(h)+oL2(Ω2)(h2)\nand the boundary condition: /tildewidewh|Σ=oH1\n2(h1\n2)andh∂ν/tildewidewh=OH−1\n2(h1\n2). Then we have\n1\nih/parenleftbig\n[Ph,0,Bh]/tildewidewh,/tildewidewh/parenrightbig\nL2(Ω2)=i/parenleftbig\nB1,h|y=0(h∂y/tildewidewh)|y=0,(h∂y/tildewidewh)|y=0/parenrightbig\nL2(Σ)+o(1).(4.4)\nProof.First we remark that the right hand side of ( 4.4) makes sense, since B1,h|y=0is a classical\nsmoothing operator (but of semi-classical order 0). We denote by /tildewidew=/tildewidewhfor simplicity.\nWithout loss of generality, we may assume that B0,h= 0, since the treatment for the term\n1\nih/parenleftbig\n[Ph,0,B0,h]/tildewidew,/tildewidew/parenrightbig\nL2is the same as in the proof of Proposition 4.1, which contributes only o(1)\nterms. By expanding the commutator, we have\n1\nih/parenleftbig\n[P0,h,Bh]/tildewidew,/tildewidew/parenrightbig\nL2\n=1\nih/parenleftbig\nP0,hB1,hh∂y/tildewidew,/tildewidew/parenrightbig\n−1\nih/parenleftbig\nB1,hh∂yP0,h/tildewidew,/tildewidew/parenrightbig\nL2\n=1\nih/parenleftbig\nB1,hh∂y/tildewidew,P0,h/tildewidew/parenrightbig\nL2−1\nih/parenleftbig\nB1,hh∂yP0,h/tildewidew,/tildewidew/parenrightbig\nL2\n+i/parenleftbig\nB1,h|y=0(h∂y/tildewidew)|y=0,(h∂y/tildewidew)|y=0/parenrightbig\nL2(Σ)−i/parenleftbig\n(h∂yB1,hh∂y/tildewidew)|y=0,/tildewidew|y=0/parenrightbig\nL2(Σ)\nObserve that B1,hh∂y/tildewidew=OL2(Ω2)(1),P0,h/tildewidew=oH1(Ω2)(h) +oL2(Ω2)(h2), andB1,hh∂yP0,h/tildewidew=\noL2(Ω2)(h)+oH−1(Ω2(h2), thus\n1\nih/parenleftbig\nB1,hh∂yw2,P0,h/tildewidew/parenrightbig\nL2−1\nih/parenleftbig\nB1,hh∂yP0,h/tildewidew,/tildewidew/parenrightbig\nL2=o(1)KELVIN-VOIGT DAMPED WAVE EQUATION 17\nash→0. Writeh∂yB1,hh∂y/tildewidew=h(∂yB1,h)h∂y/tildewidew+B1,hh2∂2\ny/tildewidewand using the equation satisfied\nby/tildewidew,we obtain that\nh∂yB1,hh∂y/tildewidew=Ahh∂y/tildewidew−B1,hRh/tildewidew−B1,hPh,0/tildewidew,\nwhereAhis a tangential operator of order 0 semi-classic. Thanks to Lemma 6.2,B1,h=\nOL2→H1(h−1), thusB1,hPh,0/tildewidew=oH1(Ω2)(h) andby thetracetheorem( B1,hPh,0/tildewidew)|Σ=oH1\n2(Σ)(h).\nNext since Rh/tildewidew|Σ=oH1\n2(Σ)(h1\n2) +oH−3\n2(Σ)(h5\n2), we have B1,hRh/tildewidew|Σ=oH1\n2(Σ)(h1\n2). We then\ndeduce that ( h∂yB1,hh∂y/tildewidew)|y=0=OH−1\n2(Σ)(h1\n2), which implies that\n/parenleftbig\n(h∂yB1,hh∂y/tildewidew)|y=0,/tildewidew|y=0/parenrightbig\nL2(Σ)=o(1).\nThis completes the proof of Lemma 4.7. /square\nTo derive the propagation formula for the semiclassical measure, w e consider a family of\nfunctions ( /tildewidewh) satisfying the equation\nPh,0/tildewidewh=oH1(Ω2)(h)+oL2(Ω2)(h2)\nwith a weaker boundary conditions, compared with ( 4.2).\n/ba∇dbl/tildewidewh/ba∇dblL2(Ω2)=O(1),/ba∇dblh∇/tildewidewh/ba∇dblL2(Ω)=O(1),/ba∇dbl/tildewidewh|Σ/ba∇dblH1\n2(Σ)=o(h1\n2),/ba∇dbl(h∂ν/tildewidewh)|Σ/ba∇dblH−1\n2(Σ)=O(h1\n2).\nDenote byµis the semiclassical measure associated with ( /tildewidewh).\nProposition 4.8.\n(1)µ1H= 0; (4.5)\n(2) limsup\nh→0/vextendsingle/vextendsingle/parenleftbig\nOph(b0)h∂y/tildewidewh,/tildewidewh/parenrightbig\nL2/vextendsingle/vextendsingle≤sup\nρ∈supp(b0)|r(ρ)|1\n2|b0(ρ)|, (4.6)\nfor any tangential symbol b0(y,x′,ξ′)of order 0.\nProof.(1) follows from the transversality of the rays reaching H, and the proof is the same as\nin [BL03] (see also the proof of Proposition 2.14 in [ BS20] by taking Mh= 0 there). The proof\nof (2) is also similar as in [ BS20], with an additional attention when doing the integration by\npart. Indeed, by Cauchy-Schwartz,\n/vextendsingle/vextendsingle/parenleftbig\nOph(b0)h∂y/tildewidewh,/tildewidewh/parenrightbig\nL2/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/parenleftbig\nOph(b0)h∂y/tildewidewh,Oph(b0)h∂y/tildewidewh/parenrightbig/vextendsingle/vextendsingle1\n2/ba∇dbl/tildewidewh/ba∇dblL2.\nDoing the integration by part,\n/parenleftbig\nOph(b0)h∂y/tildewidewh,Oph(b0)h∂y/tildewidewh/parenrightbig\nL2=O(h)−/parenleftbig\nOph(b0)h2∂2\ny/tildewidewh,Oph(b0)/tildewidewh/parenrightbig\nL2,\nwhereO(h) comes from the commutators and the boundary term, since by th e assumption\non/tildewidewh,/parenleftbig\nOph(b0)(h∂y/tildewidewh)|y=0,hOph(b0)(/tildewidewh|y=0)/parenrightbig\nL2\nx′=o(h2). For the rest argument, we just\nreplace−h2∂2\ny/tildewidewhby−Rh/tildewidewhplus errors in OL2(h). Fromthesymbolic calculus, the contribution\nsupρ|r|1\n2|b0(ρ)|comes from the principal term/vextendsingle/vextendsingle/parenleftbig\nOph(b0)Rh/tildewidewh,Oph(b0)/tildewidewh/parenrightbig\nL2/vextendsingle/vextendsingle1\n2,after taking\nlimsup inh. This completes the proof of Proposition 4.8. /square18 N. BURQ AND C-M. SUN\nLemma 4.9. LetB0,h,B1,hare tangential semiclassical operators of order 0, with principal\nsymbolsb0,b1with respectively, supported near a point ρ0ofT∗Σ. Then\n/parenleftbig/parenleftbig\nB0,h+B1,hh\ni∂y/parenrightbig\n/tildewidewh,/tildewidewh/parenrightbig\n=/a\\}b∇acketle{tµ,b0+b1η/a\\}b∇acket∇i}ht+o(1), (4.7)\nash→0.\nProof.First we remark that the expression /a\\}b∇acketle{tµ,b0+b1η/a\\}b∇acket∇i}htis well-defined, since µbelongs to\nthe dual of C0(Z) andµ(H) = 0, and in particular, by elliptic regularity, µ1|η|>1= 0. The\nconvergenceofthequadraticform( B0,h/tildewidewh,/tildewidewh)to/a\\}b∇acketle{tµ,b0/a\\}b∇acket∇i}htisjustthedefinitionofthesemiclassical\nmeasureµ. Ifρ0∈ E, the contributions of both sides of ( 4.7) iso(1), thanks to the elliptic\nregularity (see the proof of Proposition 4.6). Next we assume that ρ0∈ H ∪ G . Takeϕ∈\nC∞\nc(−1,1),ϕis equal to 1 in a neighborhood of ( −1/2,1/2). Forǫ>0, we write\nB1,h,ǫ:=/parenleftbig\n1−ϕ/parenleftbigy\nǫ/parenrightbig/parenrightbig\nB1,h, Bǫ\n1,h:=B1,h−B1,h,ǫ.\nTakingh→0 first we obtain that\n/parenleftbig\nB1,h,ǫh\ni∂y/tildewidewh,/tildewidewh/parenrightbig\nL2(Ω2)→ /a\\}b∇acketle{tµ,/parenleftbig\n1−ϕ/parenleftbigy\nǫ/parenrightbig/parenrightbig\nb1η/a\\}b∇acket∇i}ht.\nIfρ0∈ H, then taking ǫ→0, we obtain that\nlim\nǫ→0/a\\}b∇acketle{tµ,/parenleftbig\n1−ϕ/parenleftbigy\nǫ/parenrightbig/parenrightbig\nb1η/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tµ,1y>0b1η/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tµ,b1η/a\\}b∇acket∇i}ht,\nsinceµ1H∪E= 0. It remains to estimate the contribution of/parenleftbig\nBǫ\n1,hh\ni∂y/tildewidewh,/tildewidewh/parenrightbig\n. For fixed ǫ>0,\nwe have/vextendsingle/vextendsingle/vextendsinglelimsup\nh→0/parenleftbig/parenleftbig\nBǫ\n1,hh∂y/tildewidewh,/tildewidewh/parenrightbig/parenrightbig\nL2(Ω2)/vextendsingle/vextendsingle/vextendsingle≤limsup\nh→0/parenleftBig\n/ba∇dblϕ(y/ǫ)B∗\n1,h/tildewidewh/ba∇dblL2(Ω2)/ba∇dblh∂y/tildewidewh/ba∇dblL2(Ω2)/parenrightBig\n.\nSince on supp( µ1y>0),|η| ≤1, together with the fact that ρ0∈ H∩E, we deduce that the right\nside converges to 0 as ǫ→0.\nNow suppose that ρ0∈ G. For anyǫ >0,δ >0, we decompose B1,h=Bǫ\n1,h+Bǫ,δ\n1,h+Bǫ\n1,h,δ,\nwith\nB1,h,ǫ=/parenleftBig\n1−ϕ/parenleftbigy\nǫ/parenrightbig/parenrightBig\nB1,h,\nBǫ,δ\n1,h= Oph/parenleftBig\nϕ/parenleftbigy\nǫ/parenrightbig\nϕ/parenleftbigr\nδ/parenrightbig/parenrightBig\nB1,h,\nBǫ\n1,h,δ= Oph/parenleftBig\nϕ/parenleftbigy\nǫ/parenrightbig/parenleftBig\n1−ϕ/parenleftbigr\nδ/parenrightbig/parenrightBig/parenrightBig\nB1,h.\nBy the same argument, we have\nlim\nǫ→0lim\nh→0/parenleftbig\nB1,h,ǫhDy/tildewidewh,/tildewidewh/parenrightbig\nL2(Ω2)=/a\\}b∇acketle{tµ,b1η1y>0/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tµ,b1η1ρ/∈H/a\\}b∇acket∇i}ht,\nsinceµ1H∪E= 0. Next, from (2) of Proposition 4.8, we have\nlimsup\nǫ→0limsup\nh→0/parenleftbig\nBǫ,δ\n1,hhDy/tildewidewh,/tildewidewh/parenrightbig\nL2(Ω2)≤Cδ,KELVIN-VOIGT DAMPED WAVE EQUATION 19\nwhich converges to 0 if we let δ→0. Finally, by Cauchy-Schwartz,\n/vextendsingle/vextendsingle/parenleftbig\nBǫ\n1,h,δhDy/tildewidewh,/tildewidewh/parenrightbig\nL2(Ω2)/vextendsingle/vextendsingle≤ /ba∇dblhDy/tildewidewh/ba∇dblL2(Ω2)/vextenddouble/vextenddouble/vextenddoubleB∗\n1,hOph/parenleftBig\nϕ/parenleftbigy\nǫ/parenrightbig/parenleftBig\n1−ϕ/parenleftbigr\nδ/parenrightbig/parenrightBig/parenrightBig∗\n/tildewidewh/vextenddouble/vextenddouble/vextenddouble\nL2.\nTaking the triple limit, we have\nlimsup\nδ→0limsup\nǫ→0limsup\nh→0/vextendsingle/vextendsingle/parenleftbig\nBǫ\n1,h,δhDy/tildewidewh,/tildewidewh/parenrightbig\nL2(Ω2)/vextendsingle/vextendsingle≤ /a\\}b∇acketle{tµ,|b1|21y=01r/\\egatio\\slash=0/a\\}b∇acket∇i}ht= 0,\nsinceµ1E∪H= 0. This completes the proof of Lemma 4.9. /square\nAs in [BL03], we define the function\nθ(y,x′;η,ξ′) =η\n|ξ′|ify>0;θ(y,x′,η,ξ′) =i/radicalbig\n−r0(x′,ξ′)\n|ξ′|onE.\nSinceµ1H= 0,θisµalmost everywhere defined as a function on Z. Formally,\nσ/parenleftbigi\nh[Ph,0,Bh]/parenrightbig\n={η2−r,b0+b1η}=a0+a1η+a2η2,\nwhere\na0=b1∂yr−{r,b0}′, a1= 2∂yb0−{r,b1}′, a2= 2∂yb1, (4.8)\nand{·,·}′is the Poisson bracket for ( x′,ξ′) variables. By expanding the commutator, we find\ni\nh[Ph,0,Bh] =A0+A1hDy+A2h2D2\ny+hOph(S0\n∂+S0\n∂η), (4.9)\nwhereA0,A1,A2are tangential operators with symbols a0,a1,a2, with respectively, and S0\n∂\nstands for the tangential symbol class of order 0. We now have all the ingredients to present\nthe propagation formula for the defect measure in the spirit of [ BL03]:\nProposition 4.10. Assume that Bh=Bh,0+Bh,1hDy, whereBh,0,Bh,1are tangential operators\nof order 0 with symbols b0,b1, with respectively. Assume that b=b0+b1η. Define the formal\nPoisson bracket\n{p,b}= (a0+a2r)+a1θ|ξ′|1ρ/∈H,\nwherea0,a1,a2are given by (4.8). Then any defect measures µ,ν0of(/tildewidewh),(h∂ν/tildewidewh)|Σsatisfy\nthe relation\n/a\\}b∇acketle{tµ,{p,b}/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{tν0,b1/a\\}b∇acket∇i}ht.\nMoreover, if b∈C0(Z), we have\n/a\\}b∇acketle{tµ,{p,b}/a\\}b∇acket∇i}ht= 0. (4.10)\nProof.See [BL03]. /square\nMoreover, we have\nProposition 4.11. µ/parenleftbig\nG2,+/parenrightbig\n= 0.20 N. BURQ AND C-M. SUN\nAs showed in [ BL03], we obtain that the measure µis invariant by the flow of Melrose-\nSj¨ ostrand. More precisely, we have\nTheorem 3 ([BL03]).Assume that µis a semi-classical measure onbT∗Ωassociated with the\nsequence (/tildewidewh)satisfying (4.10)and Proposition 4.11. Thenµis invariant under the Melrose-\nSj¨ ostrand flow φs.\nRemark 4.12. This is a consequence of Theorem 1 in [ BL03] which asserts the equivalence\nbetweenthemeasureinvarianceandthepropagationformula Hp(µ) = 0togetherwith µ(G2,+) =\n0. Though Theorem 1 in [ BL03] is stated and proved in the context of micro-local defect\nmeasure, it also holds true in the context of the semiclassical measu re from the word-by-word\ntranslation.\n4.5.The last step to the proof of the resolvent estimate in Theore m2.In this subsec-\ntion, we take /tildewidewh=w2. To finish the contradiction argument in the proof of ( 1.5), it suffices to\nshow thatµ= 0. Letµbe the corresponding semiclassical measure and ν0be the semiclassical\nmeasure of h∂νu2|Σ. Sinceh∂νu2|Σ=oH−1\n2(Σ(h1\n2), we have that /a\\}b∇acketle{tν0,b1/a\\}b∇acket∇i}ht= 0 for any compactly\nsupported symbol b1(x′,ξ′). Thanks to (H), along the Melrose-Sj¨ ostrand flow ofbT∗Ω2issued\nfrom points inbT∗Ω2, there must be some points reaching H(Σ)∪G2,−(Σ). By the property of\nthe Melrose-Sj¨ ostrand flow onbT∗Ω2, to show that µ= 0, we need to verify that\nµ/parenleftbig\nG2,−(Σ)/parenrightbig\n= 0\nandµ= 0 near a neighborhood of ρ0∈ H(Σ).\nProposition 4.13. µ/parenleftbig\nG2,−(Σ)/parenrightbig\n= 0.\nProof.The proof is exactly the same as the proof of Proposition 4.11. We will make use of the\nformula\n/a\\}b∇acketle{tµ,{p,b}/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tν0,b1/a\\}b∇acket∇i}ht\nby choosing b=b1,ǫηwith\nb1,ǫ(y,x′,ξ′) =ψ/parenleftbigy\nǫ1\n2/parenrightbig\nψ/parenleftbigr(y,x′,ξ′)\nǫ/parenrightbig\nκ(y,x′,ξ′),\nwhereψ∈C∞\nc(R) equals to 1 near the origin and κ(y,x′,ξ′)≥0 near a point ρ0∈ G2,−. Note\nthat{p,bǫ}= (a0+a2r)+a1η1ρ/∈H, anda0,a1,a2are given by the relation ( 4.8). In particular\nfor our choice, by direct calculation we have\na0=b1,ǫ∂yr, a1=−{r,κ}′ψ/parenleftbigy\nǫ1\n2/parenrightbig\nψ/parenleftbigr\nǫ/parenrightbig\n,\nand\na2= 2∂yb1,ǫ= 2ǫ−1\n2ψ′/parenleftbigy\nǫ1\n2/parenrightbig\nψ/parenleftbigr\nǫ/parenrightbig\nκ+2∂yr\nǫψ/parenleftbigy\nǫ1\n2/parenrightbig\nψ′/parenleftbigr\nǫ/parenrightbig\nκ+2ψ/parenleftbigy\nǫ1\n2/parenrightbig\nψ/parenleftbigr\nǫ/parenrightbig\n∂yκ.KELVIN-VOIGT DAMPED WAVE EQUATION 21\nObserve that a2is uniformly bounded in ǫand for any fixed ( y,x′,ξ′),ra2→0 asǫ→0. Thus\nfrom the dominating convergence, we have\nlim\nǫ→0/a\\}b∇acketle{tµ,{p,bǫ}/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tµ,κ|y=0∂yr1r=0/a\\}b∇acket∇i}ht.\nSince∂yr <0 onG2,−, while−/a\\}b∇acketle{tν0,bǫ/a\\}b∇acket∇i}ht= 0, we deduce that µ1G2,−= 0. This completes the\nproof of Proposition 4.13. /square\nProposition 4.14. Letρ0∈ H(Σ). Letb(y,x′,ξ′)be a tangential symbol, supported near ρ0.\nThen\n/ba∇dblOph(b)w2/ba∇dblL2(Ω2)+/ba∇dblh∂yOph(b)w2/ba∇dblL2(Ω2)=o(1),\nash→0.\nProof.Since the Melrose-Sj¨ ostrand flow is transverse to H, by localizing the symbol b, it suffices\nto prove the same estimate by replacing btoq±, whereq±is the solutions of\n∂yq±∓H√r(y,x′,ξ′)q±= 0, q±|y=0=q0,\nandq0is supported in a sufficiently small neighborhood of ρ0. Nearρ0, it follows from [ BL03]\nthat we can factorize Ph,0as/parenleftbig\nhDy−Λ+\nh(y,x′,hDx′)/parenrightbig/parenleftbig\nhDy+Λ−\nh(y,x′,hDx′)/parenrightbig\n+OH∞(h∞), and\nalso/parenleftbig\nhDy−/tildewideΛ+\nh(y,x′,hDx′)/parenrightbig/parenleftbig\nhDy+/tildewideΛ−\nh(y,x′,hDx′)/parenrightbig\n+OH∞(h∞),whereΛ±\nhand/tildewideΛ±\nhhaveprincipal\nsymbols±/radicalbig\nr(y,x′,ξ′). Denote by Q±\nh= Oph(q±) and set\nw+\n2:=ϕ(y)Q+\nh(hDy−Λ−\nh)w2, w−\n2:=ϕ(y)Q−\nh(hDy−/tildewideΛ+\nh)w2,\nwhere the cutoff ϕ(y) is supported on 0 ≤y≤ǫ0≪1 and is equal to 1 for 0 ≤y≤ǫ0/2. From\nthe equation of w2, we have\n(hDy−Λ+\nh)w+\n2=ϕ(y)[hDy−Λ+\nh,Q+\nh](hDy−Λ−\nh)w2−ihϕ′(y)Q+\nh(hDy−Λ−\nh)w2+oL2\ny,x′(h)\n=−ihϕ′(y)Q+\nh(hDy−Λ−\nh)w2+oL2\ny,x′(h),\nsince the principal symbol of1\nih[hDy−Λ+\nh,Q+\nh] is zero, thanks to the choice of symbols q±.\nMultiplying by w+\n2to both sides and integrating, we have for y≤ǫ0/2 (thusϕ′(y) = 0) that\nh/ba∇dblw+\n2(y,·)/ba∇dbl2\nL2\nx′≤h/ba∇dblw+\n2(0,·)/ba∇dbl2\nL2\nx′+o(h). (4.11)\nSince Oph(q0)(h∂yw2)|y=0=oL2\nx′(1), we deduce by definition that w+\n2(0) =oL2\nx′(1). This\ntogetherwith( 4.11)yieldsw+\n2(y) =oL2\nx′(1),uniformlyforall0 ≤y≤ǫ0/2. Thusw+\n2=oL2\ny,x′(1).\nSimilar argument for w−\n2yieldsw−\n2=oL2\ny,x′(1). Note that hDy−Λ−\nhis elliptic on the support of\nq+, we deduce that Q+\nhw2=oL2\ny,x′(1). This means that µis zero near the support of q+, hence\nthe proof of Proposition 4.14is complete. /square22 N. BURQ AND C-M. SUN\nConsequently, we have shown that the measure µis invariant along the bicharacteristic flow\non Ω2, it vanishes near every hyperbolic point of Σ, and µ(G2,−) = 0. Thus µis supported on\nbicharacteristics which encounter Σ only at points of\nG2<=∪k≥3Gk.\nThese bicharacteristics are consequently near Σ integral curves ofHp(because in Definition 4.2,\nthetwo vector fields HpandHp−H2\np(y\nH2ypHycoincide on G2<). However, according to thegeometric\nconditionassumption, allsuchbicharacteristics mustleaveΩ 2. Asaconsequence, µissupported\non the emptyset, and hence µ= 0. This gives a contradiction. The proof of ( 1.5) in Theorem 2\nis now complete.\n5.Optimality of the resolvent estimate\nIn this section we prove the second part of Theorem 2. For simplicity, we consider the model\ncase Ω 2=D:={x∈R2:|x|<1}anda(x) =1Ω1and Σ = S1. To prove the second part in\nTheorem 2we need to construct functions u1,v1,u2,v2, such that\n/ba∇dbl(uj,vj)/ba∇dblH1×L2(Ωj)∼1,/ba∇dbl(fj,gj)/ba∇dblH1×L2(Ωj)=O(h), j= 1,2\n\n\nu1=ih(f1−v1),in Ω1\nh∆u1+h∆v1−iv1=hg1,in Ω1\nu2=ih(f2−v2),in Ω2\nh∆u2−iv2=hg2,in Ω2\ntogether with the boundary condition on the interface\nu1|Σ=u2|Σ, ∂νu2|Σ= (∂νu1+∂νv1)|Σ,\nThe key point in the construction is that in Ω 1, we construct quasi-modes concentrated at the\nscale|Dx| ∼/planckover2pi1−1=h−1\n2while in Ω 2, the quasi-modes are concentrated at the scale |Dx′| ∼\n|Dy| ∼ |Dx| ∼h−1near the interface Σ, where x′is the tangential variable near Σ and yis the\nnormal variable. Now we describe the construction.\n•Step 1: Construction at the zero order: We first choose u(0)\n2, such that\nh2∆u(0)\n2+u(0)\n2= 0, u(0)\n2|Σ= 0;/ba∇dbl∇u(0)\n2/ba∇dblL2(Ω2)∼h−1/ba∇dblu(0)\n2/ba∇dblL2(Ω2)∼1.\nMoreover, we require that u(0)\n2such that they are hyperbolically localized, in the sense that\n/ba∇dbl∂νu(0)\n2|Σ/ba∇dblHs(Σ)∼h−s,WFh(∂νu(0)\n2|Σ)⊂ Hδ(Σ) :={(x′,ξ′) :δ <r0(x′,ξ′)<1−δ}(5.1)\nfor some 0 < δ <1\n2. The existence of such sequence of eigenfunctions is not difficult to prove\nin the case of a disc or an ellipse, we postpone this fact in Lemma 6.5of the Appendix. This\nwill actually be the only point where in Theorem 1we use the particular choice Ω 2=D.KELVIN-VOIGT DAMPED WAVE EQUATION 23\nNext we define\nv(0)\n2=ih−1u(0)\n2, f(0)\n2=g(0)\n2= 0.\nFrom (5.1), we have\n∂νu(0)\n2|Σ=\n\nOL2(Σ)(1)\nOH−1\n2(Σ)(h1\n2)\nOH1\n2(Σ)(h−1\n2).(5.2)\nWe remark that here we use the fact that the dimension d≥2.\nNext we solve the elliptic equation with the mixed Dirichlet-Neumann dat a (with/planckover2pi1=h1\n2):\n(/planckover2pi12∆−i)w(0)= 0, ∂νw(0)|Σ=∂νu(0)\n2|Σ, w(0)|∂Ω1\\Σ= 0.\nFrom Proposition 6.1, there exists a unique solution w(0)of this system, which satisfies\nw(0)=\n\nOH2(Ω1)(/planckover2pi1−1)\nOH1(Ω1)(/planckover2pi1)\nOL2(Ω1)(/planckover2pi12),(5.3)\nand hence by interpolation\nw(0)=OH3\n2(Ω1)(1)\nand by trace theorems\nw(0)|Σ=/braceleftBigg\nOH1\n2(Σ)(/planckover2pi1)\nOH1(Σ)(1)(5.4)\nMoreover, from the information of WF h(∂νu(0)\n2|Σ) and Proposition 3.1, we have\nWFh(w(0)|Σ)⊂WFh(∂νw0|Σ)⊂ Hδ(Σ).\nHence\n/ba∇dblw(0)|Σ/ba∇dblH1(Σ)∼h−1\n2/ba∇dblw(0)|Σ/ba∇dblH1\n2(Σ)=O(1)\nNext we define u(0)\n1,v(0)\n1such that\nv(0)\n1=ih−1u(0)\n1, w(0)=u(0)\n1+v(0)\n1= (1+ih−1)u(0)\n1;f(0)\n1= 0, g(0)\n1=ih−1u(0)\n1=v(0)\n1.\nThis implies\nu(0)\n1=/braceleftBigg\nOH1(Ω1)(h3\n2)\nOL2(Ω1)(h2),(5.5)\nand consequently g(0)\n1=OL2(Ω1)(h).24 N. BURQ AND C-M. SUN\nIn summary, as the first step, we have constructed quasi-modes (u(0)\n1,v(0)\n1;u(0)\n2,v(0)\n2) and\n(f(0)\n1= 0,g(0)\n1;f(0)\n2= 0,g(0)\n2= 0) such that\n\n\nh∆(u(0)\n1+v(0)\n1)−iv(0)\n1=hg(0)\n1, g 1=OL2(Ω1)(h)\nu(0)\n1=−ihv(0)\n1\nh∆u(0)\n2−iv(0)\n2= 0\nu(0)\n2=−ihv(0)\n2\n∂νu(0)\n2|Σ=/parenleftbig\n∂νu(0)\n1+∂νv(0)\n1/parenrightbig\n|Σ,/parenleftbig\nu(0)\n2−u(0)\n1/parenrightbig\n|Σ=OH1\n2(Σ)(h3\n2),/parenleftbig\nv(0)\n2−v(0)\n1/parenrightbig\n|Σ=OH1\n2(Σ)(h1\n2),(5.6)\nand to conclude the proof of Theorem 2, it remains to eliminate the error term in the last\nboundary condition in ( 5.6). An important point is that both u(0)\n2andu(0)\n1(and hence also\nuv(0)2andv(0)\n1) have their wave front included in Hδ(Σ).\n•Step 2: Construction at the first order: We now introduce correction terms to eliminate\nthe error term in the last boundary condition of ( 5.6). We are looking for a correction term\ne(1)\n2,\nu(1)\n2=u(0)\n2+e(1)\n2, v(1)\n2=ih−1u(1)\n2=v(0)\n2+ih−1e(1)\n2,\nwhile keeping all other terms identical\nu(1)\n1=u(0)\n1, v(1)\n1=v(0)\n1,\nFirst, using the geometric optics construction (see Appendix), we construct /tildewidee(1)\n2, solving near\nΣ, solving for Nlarge enough to be fixed later\n(h2∆+1)/tildewidee(1)\n2=OL2(hN)\nnear Σ, and the boundary conditions\n/tildewidee(1)\n2|Σ= (u(0)\n1−u(0)\n2)|Σ+OL2(Σ)(hN), ∂ ν/tildewidee(1)\n2|Σ=OL2(Σ)(hN). (5.7)\nwithh-semiclassical wave front sets of all the functions are localized nea rHδ(Σ).\n(h2∆+1)/tildewidee(1)\n2=OL2(Ω2)(hN),/tildewidee(1)\n2= (u(0)\n1−u(0)\n2)|Σ+OHN(Σ)(hN), h∂ν/tildewidee(1)\n2|Σ=OHN(Σ)(hN)\n(5.8)\nlocallynear x0∈Σ. Wethentakeacutoff χ, suchthatχ≡1onΣ, andwithsupportsufficiently\nclose to Σ so that /tildewideeis defined on the support of χ(i.e.χvanishes along the bicharacteristics,\nbefore the formation of the caustics). Let e(1)\n2:=χ/tildewidee(1)\n2.Hence\n(h2∆+1)e(1)\n2= [h2∆,χ]/tildewidee(1)\n2+OL2(Ω2)(h4) =OL2(Ω2)(h3),\ne(1)\n2|Σ=/tildewidee(1)\n2|Σ, h∂νe(1)\n2|Σ=h∂ν/tildewidee(1)\n2|Σ.(5.9)KELVIN-VOIGT DAMPED WAVE EQUATION 25\nAgain, all the functions and the errors are microlocalized near ( x0,ξ0)∈ Hδ(Σ). More-\nover, from the boundary conditions ( 5.7) which determine the values of the symbols b±in the\ngeometric optics construction, we have\n/ba∇dble(1)\n2/ba∇dblH1(Ω2)=O(h),/ba∇dble(1)\n2/ba∇dblL2(Ω2)=O(h2), ∂νe(1)\n2|Σ=OL2(Σ)(hN−1).\nThe geometric optics constructions in the appendix are local, but us ing a partition of unity of\nΣ, we choose a finite cutoff functions ( χj)M\nj=1to replaceχand modify the function e(1)\n2by\ne(1)\n2:=M/summationdisplay\nj=1χj/tildewidee(1)\n2,j,\nwhere/tildewidee(1)\n2,jis the corresponding geometric optics near supp( χj).\nNext we define g(1)\n2=h−2·(h2∆+1)e(1)\n2. We now have\n\n\nh∆(u(1)\n1+v(1)\n1)−iv(1)\n1=hg(1)\n1, g 1=OL2(Ω1)(h)\nu(1)\n1+ihv(1)\n1= 0\nh∆u(1)\n2−iv(1)\n2=hg(1)\n2, g(1)\n2=OL2(Ω1)(h)\nu(1)\n2=−ihv(1)\n2\n∂νu(1)\n2|Σ=/parenleftbig\n∂νu(1)\n1+∂νv(1)\n1/parenrightbig\n|Σ+OHN(Σ)(hN)/parenleftbig\nu(1)\n2−u(1)\n1/parenrightbig\n|Σ=OHN(Σ)(hN),/parenleftbig\nv(1)\n2−v(1)\n1/parenrightbig\n|Σ=OHN(Σ)(hN−1),(5.10)\nIt now remains to eliminate completely the errors in the last boundary condition in ( 5.10). For\nthis we just use the trace operators. Recall that if s>3\n2, the map\nΓ :u∈Hs(Ω1)/ma√sto→(u|Σ,∂νu|Σ)∈Hs−1/2(Σ)×Hs−3/2(Σ)\nis continuous surjective and admits a bounded right inverse. As a co nsequence, if Nis large\nenough, there exists e(2)\n2∈HN−3\n2(supported near Σ) such that\n/ba∇dble(2)\n2/ba∇dblHN−3\n2(Ω2)=O(hN), e(2)\n2|Σ= (u(1)\n1−u(1)\n2)|Σ, ∂νe(2)\n2|Σ=/parenleftbig\n∂νu(1)\n1−∂νv(1)\n1/parenrightbig\n|Σ−∂νu(1)\n2|Σ\nChoosing now\nu(2)\n2=u(1)\n2+e(2)\n2, v(2)\n2=v(1)\n2+ih−1e(2)\n2, g(2)\n2=g(1)\n2+h−1(h2∆+1)e(2)\n2\nand keeping the other terms identical\nu(2)\n1=u(0)\n1, v(2)\n1=v(0)\n1, g(2)\n1=g(1)\n1,26 N. BURQ AND C-M. SUN\nwe get (ifNis large enough)\n\n\nh∆(u(2)\n1+v(2)\n1)−iv(2)\n1=hg(2)\n1, g 1=OL2(Ω1)(h)\nu(2)\n1+ihv(2)\n1= 0\nh∆u(2)\n2−iv(2)\n2=hg(2)\n2, g(2)\n2=OL2(Ω1)(h)\nu(2)\n2=−ihv(2)\n2\n∂νu(2)\n2|Σ=/parenleftbig\n∂νu(2)\n1+∂νv(2)\n1/parenrightbig\n|Σ/parenleftbig\nu(2)\n2−u(2)\n1/parenrightbig\n|Σ= 0,/parenleftbig\nv(2)\n2−v(2)\n1/parenrightbig\n|Σ= 0(5.11)\nThis ends the proof of the construction of quasi-modes in Theorem 2. /square\n6.Appendix: Technical ingredients\n6.1.Elliptic problem with mixed Dirichlet Neumann data. LetU⊂Rdbe a bounded\ndomain with smooth boundary. For F∈C∞(U), we denote by\nγ0(F) =F|∂U, γ1(F) = (∂νF)|∂U\nthe Dirichlet and Neumann trace, with respectively. From the trace theorem, we know that\nγ0:Hs(U)→Hs−1\n2(U)\nis bounded and surjective. Let\nH1\n0(Ω1) ={v∈H1(Ω);v|∂Ω1\\Σ= 0},\nWe prove the following existence result of the mixed Dirichlet-Neuman n boundary value\nproblem:\nProposition 6.1. For anyF∈H−1\n2(Σ), the boundary value problem (note that ∂Ω1= Σ∪∂Ω\nandΣ,∂Ωare separated)\n(/planckover2pi12∆−i)w= 0, (6.1)\n∂νw|Σ=F, w |∂Ω1\\Σ= 0 (6.2)\nadmits a unique solution w∈ H1\n0(Ω1)satisfying\n/parenleftBig\n/planckover2pi1/ba∇dbl∇xw/ba∇dblL2(Ω1)+/ba∇dblw/ba∇dblL2(Ω1)/parenrightBig\n≤C/planckover2pi1/ba∇dblF/ba∇dblH−1\n2(Σ).\nFurthermore, if F∈H1\n2(Σ), thenw∈H2(Ω1)and\n/ba∇dbl∇2\nxw/ba∇dblL2(Ω1)≤C/parenleftBig\n/ba∇dblF/ba∇dblH1\n2(Σ)+/planckover2pi1−1/ba∇dblF/ba∇dblH−1\n2(Ω1)/parenrightBig\n.KELVIN-VOIGT DAMPED WAVE EQUATION 27\nProof.We just sketch the proof which is a variation around very classical id eas. Multiply-\ning (6.1) byϕvanishing on ∂Ω1\\Σ and integrating by parts using Greens formula, we get\n0 =/integraldisplay\nΩ1(/planckover2pi12∆−i)wϕ(x)dx=/integraldisplay\nΩ1−/planckover2pi12∇xw∇xϕ−iwϕ(x)dx+/integraldisplay\nΣ/planckover2pi12∂νwϕ(x)dσ\nAs a consequence, if the function wsatisifes ( 6.1) (6.2) if an only if\n∀v∈ H1\n0(Ω1), Q(w,v) :=/integraldisplay\nΩ1/planckover2pi12∇xw∇xv+iwv(x)dx=TF(v) :=/integraldisplay\nΣ/planckover2pi12Fv(x)dσ.(6.3)\nFrom the trace theorem, the map\nv∈ H1\n0(Ω1)/ma√sto→v|Σ∈H1\n2(Σ)\nis continuous and hence for any F∈H−1\n2(Σ), the map\nv/ma√sto→TF(v)∈C\nis a continuous antilinear form on H1\n0(Ω1).\nThe existence of a unique solution to ( 6.3) (and consequently the solution to ( 6.1), (6.2))\nnow follows from Lax-Milgram Theorem. Applying ( 6.3) tov=w, we get\n/ba∇dblh∇xw/ba∇dbl2\nL2(Ω1)+/ba∇dblw/ba∇dbl2\nL2(Ω1)≤2|TF(w)| ≤C/planckover2pi12/ba∇dblF/ba∇dblH−1\n2(Σ)/ba∇dblw/ba∇dblH1(Ω1),\nwhich implies\n/ba∇dblw/ba∇dblH1(Ω1)≤C/ba∇dblF/ba∇dblH−1\n2(Σ),\nand using again ( 6.3) withv=w,\n/ba∇dblw/ba∇dbl2\nL2(Ω1)≤/planckover2pi12/ba∇dbl∇xw/ba∇dblL2(Ω1)+/planckover2pi12|TF(w)| ≤C/planckover2pi12/ba∇dblF/ba∇dbl2\nH−1\n2(Σ).\nThis proves the first part in Proposition 6.1. The proof of the second part is standard elliptic\nregularity results. Indeed, we have\n∆w=i/planckover2pi1−2w, ∂ νw|Σ=F∈H1\n2(Σ), w|∂Ω1\\Σ= 0,\nand we deduce by standard elliptic regularity results,\n/ba∇dblw/ba∇dblH2(Ω1)≤C/parenleftBig\n/planckover2pi1−2/ba∇dblw/ba∇dblL2(Ω1)+/ba∇dblF/ba∇dblH1\n2(Σ)/parenrightBig\n≤C/parenleftBig\n/planckover2pi1−1/ba∇dblF/ba∇dblH−1\n2(Σ)+/ba∇dblF/ba∇dblH1\n2(Σ)/parenrightBig\nThis completes the proof of Proposition 6.1. /square28 N. BURQ AND C-M. SUN\n6.2.Estimates for some operators.\nLemma 6.2. Ifb(x,ξ)∈S−m(m≥0) is compactly supported in x∈Rn, then for any s∈R,\nOph(b) =O(h−θ) :Hs(Rn)→Hs+θ(Rn),∀θ∈[0,m].\nProof.First we show that Oph(b) is bounded from HstoHs. It is equivalent to show that the\noperatorTh:=/a\\}b∇acketle{tDx/a\\}b∇acket∇i}htsOph(b)/a\\}b∇acketle{tDx/a\\}b∇acket∇i}ht−sis bounded (independent of h) fromL2toL2. By definition,\nwe have\n/hatwider(Thf)(ξ) =1\n(2π)d/integraldisplay\nRn/a\\}b∇acketle{tξ/a\\}b∇acket∇i}hts/hatwideb(ξ−η,hη)/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht−s/hatwidef(η)dη,\nwhere/hatwideb(ζ,η) = (Fx→ζa)(ζ,η)is a well-defined function. Thus /hatwidestThfcan be viewed as anoperator\nacting on /hatwidef∈L2(Rd\nξ) with Schwartz kernel\nKh(ξ,η) :=1\n(2π)d/a\\}b∇acketle{tξ/a\\}b∇acket∇i}hts/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht−s/hatwideb(ξ−η,hη).\nBy Schur’s test, to check the boundedness of this operator, it su ffices to check that\nsup\nξ,h/integraldisplay\nRn|Kh(ξ,η)|dη<∞,sup\nη,h/integraldisplay\nRn|Kh(ξ,η)|dξ<∞.\nSinceKh(ξ,η) is rapidly decaying in /a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht, these conditions can be simply verified by the\nelementary convolution inequalities:/integraldisplay\nRn1\n/a\\}b∇acketle{tη/a\\}b∇acket∇i}hts/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}htMdη≤CM/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−s,∀M >d,s ≥0, (6.4)\nand/integraldisplay\nRn/a\\}b∇acketle{tη/a\\}b∇acket∇i}htσ\n/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}htMdη≤CM,σ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htσ,∀M >d+σ,σ≥0. (6.5)\nBy interpolation, to finish the proof, it suffices to estimate the oper ator bound of Oph(b) from\nHstoHs+m. Similarly, we need to check that the kernel\nGh(ξ,η) =hm/a\\}b∇acketle{tξ/a\\}b∇acket∇i}hts+m/hatwideb(ξ−η,hη)/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht−s\nsatisfies the conditions for Schur’s test. First note that for any α∈Nn,\n(i(ξ−η))α/hatwideb(ξ−η,η) =1\n(2π)d/integraldisplay\nRn(∂α\nxb)(x,η)e−ix·(ξ−η)dx,\nthus/hatwideb(ξ−η,hη) =O/parenleftbig\n/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht−M/a\\}b∇acketle{thη/a\\}b∇acket∇i}ht−m/parenrightbig\nfor anyM∈N. Note that\n/a\\}b∇acketle{thm/a\\}b∇acket∇i}ht−m∼(1+h|η|)−m≤h−m/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht−m.\nThis implies that\n|Gh(ξ,η)| ≤CM/a\\}b∇acketle{tξ/a\\}b∇acket∇i}hts+m/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht−(s+m)/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht−M.\nNow the boundeness of the integration/integraltext\nGh(ξ,η)dηor/integraltext\nGh(ξ,η)dξfollows from the same\nconvolution inequalities ( 6.4) and (6.5). This completes the proof of Lemma 6.2. /squareKELVIN-VOIGT DAMPED WAVE EQUATION 29\nLemma 6.3. Leta∈S0(R2n),b∈S0(R2n)be two symbols with compact support in the x\nvariable. Then for any N∈N,N≥2n,\n/vextenddouble/vextenddouble/vextenddoubleOp(a)Op(b)−/summationdisplay\n|α|≤N1\ni|α|α!Op/parenleftbig\n∂α\nξa∂α\nxb/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nL(Hs→Hs)\n≤CN/summationdisplay\n|β|≤K(n)sup\n|α|=N+1sup\n(x,ξ)∈R2n/integraldisplay/integraldisplay\nR2n/vextendsingle/vextendsingle∂β\nx,ξ∂α\nz∂α\nζA(x,z,ξ,ζ)/vextendsingle/vextendsingledzdζ,\nwhere\nA(x,x,ξ,ζ) =a(x,ξ+ζ)b(x+z,ξ).\nProof.The symbol of the operator\nOp(a)Op(b)−/summationdisplay\n|α|≤N1\ni|α|α!Op/parenleftbig\n∂α\nξa∂α\nxb/parenrightbig\nis given by\nrN(x,ξ) :=1\nN!/integraldisplay/integraldisplay\nR2n/integraldisplay1\n0(1−t)N/summationdisplay\n|α1|+|α2|=N+1(∂α1\ny∂α2\nηA)(x,tz,ξ,tζ )zα1ζα2e−iz·ζdzdζdt,\nwith\nA(x,z,ξ,ζ) =a(x,ξ+ζ)b(x+z,ξ).\nUsing the identity\nzα1ζα2e−iz·ζ=iN+1∂α2\nz∂α1\nζ(e−iz·ζ)\nand doing the integration by part, we have\nrN(x,ξ) =/summationdisplay\n|α|=N+1iN+1\nN!/integraldisplay1\n0(1−t)NtN+1dt/integraldisplay/integraldisplay\nR2n(∂α\nz∂α\nζA)(x,tz,ξ,tζ )e−iz·ζdzdζ\n=/summationdisplay\n|α|=N+1iN+1\nN!/integraldisplay1\n0(1−t)NtN+1−2ndt/integraldisplay/integraldisplay\nR2n(∂α\nz∂α\nζA)(x,z,ξ,ζ)e−it−2z·ζdzdζ\nHence the integral converges absolutely. Viewing rN(x,ξ) as a symbol of order 0, we obtain\nthe desired bound, thanks to the Caldr´ on-Vaillancourt theorem. /square\n6.3.Special sequence of eigenfunctions of a disc. First we recall that\nJm(z) =/parenleftBigz\n2/parenrightBigm∞/summationdisplay\nk=0(−1)k/parenleftbigz\n2/parenrightbig2k\nk!(m+k)!\nare the Bessel functions satisfying the Bessel differential equat ion:\nz2J′′\nm(z)+zJ′\nm(z)+(z2−m2)Jm(z) = 0.30 N. BURQ AND C-M. SUN\nBy definition, one has\nJm+1(z)+Jm−1(z) =2m\nzJm(z), Jm−1(z)−Jm+1(z) = 2J′\nm(z). (6.6)\nDenote byλm,nthen-th zero ofJm(z). It is well known that\nλm,1<λm,2<···<λm,n<···\nand the functions\nϕm,n(r,θ) =Jm(λm,nr)eimθ\nform an orthogonal sequence of eigenfunctions of ∆ D, associated with eigenvalues {λ2\nm,n:m∈\nZ,n∈N}. We will chose a special sequence\nJαn(λαn,nr)eiαnθ\nfor someα∈N, to be fixed later. Let us recall some facts about the zeros of Bes sel functions:\nProposition 6.4 ([E84]).There exists a continuous function ι: [−1,∞), such that\nλαn,n<nι(α),andlim\nn→∞λαn,n\nn=ι(α).\nMoreover, there exists 0<β1<β2, such that for all α≥1,\n1+β1α−2\n3<ι(α)\nα≤1+β2α−2\n3.\nThanks to this proposition, we have:\nLemma 6.5. Fixα∈N, large enough and let\nwn:=ϕαn,n\nλαn,n/ba∇dblϕαn,n/ba∇dblL2(D).\nThen we have\n/ba∇dbl(∂νwn)|∂D/ba∇dblL2(∂D)=O(1),WFh(∂νwh|Σ)⊂ Hδ(∂D) :={δ<r0<1−δ}\nwhereh= (hn)n∈N,hn=λ−1\nαn,n∼(ι(α)n)−1and the semiclassical wave-front set is taken for the\nsequence (wn)n∈N, with a little abuse of the notation.\nProof.To simplify the notation, we write m=αnandι:=ι(α).From Proposition 6.4, we have\n1+β1α−2\n3−o(1)<ι\nα−o(1) =λm,n\nm<ι\nα≤1+β2α−2\n3,asn→ ∞.\nNote that at r= 1,∂ν=∂rand|∇w|2=|∂rw|2+1\nr2|∂θw|2. The hyperbolicity at the boundary\nis essentially due to the fact that\n∂θwn=imwn\nand\n|m|\nλm,n=α\nι+o(1)≤1−δ(α) (6.7)KELVIN-VOIGT DAMPED WAVE EQUATION 31\nforn≫1. Let 0< ǫ0< δ(α),χ∈C∞(R) such that χ(s)≡0 if|s|>1−ǫ0. From ( 6.7)\nwe havewn=χ(hn∂θ)wn.Since∂2\nθis just the Laplace operator on L2(∂D), we have, near\n∂D, WFh(wn) is contained in r > ǫ0>0, thuswnis microlocalized near H(Σ). The estimate\n/ba∇dbl∂rwn|r=1/ba∇dblL2(∂D)=O(1) then follows from the hyperbolicity and the fact that /ba∇dbl∇wn/ba∇dblH1(D)= 1.\nThis completes the proof of Lemma 6.5. /square\nConcentration of the eigenfunctions ϕαn,nasn→ ∞\n6.4.Geometric optics construction. Inthispartwerecallthegeometricopticsconstruction\nfor the hyperbolic boundary value problem. In the tubular neighbor hood of the interface Σ, we\nuse the geodesic normal coordinate x= (y,x′), such\n∆ =1\nκ∂y(κ∂y)+1\nκ∂i(gij\n0κ∂j),\nwhereκ=/radicalbig\ndet(g0) and∂j=∂x′\nj. The semiclassical operator\nPh=h2∆g0+1 =h2∂2\ny+h2gij\n0∂i∂j+1+h\nκ(∂yκ)h∂y+h\nκ∂i(gij\n0κ)h∂j.\nLetf±\n0∈L2(Rd−1\nx′) such that WF h(f±\n0) lies in a neighborhood of ( y= 0,x′\n0;η= 0,ξ′\n0), such\nthat\nr0(0,x′\n0,ξ′\n0)≥c0>0.\nDenote byθ±(ξ) =Fh(χf±\n0)(ξ), whereχ∈C∞\nc(Rd−1\nx′), supported near x′\n0.\nConsider the semi-classical Fourier integral operators U±, represented by\nU±(χf±\n0)(y,x′) =1\n(2πh)d−1/integraldisplay\nRd−1ei\nhϕ±(y,x′,ξ′)b±(y,x′,ξ′)θ±(ξ′)dξ′.\nWe have\nPh(U±(χf±\n0)) =1\n(2πh)d−1/integraldisplay\nRd−1(h2∆g+1)(eiϕ±\nhb±)θ±(ξ′)dξ′.\nObserving that\n(h2∆g0+1)(eiϕ±\nhb±) =(1−|∇g0ϕ±|2)b±eiϕ±\nh+ih(2∇g0ϕ±·∇g0b±+∆g0ϕ±b±)eiϕ±\nh\n+h2(∆g0b±)eiϕ±\nh.32 N. BURQ AND C-M. SUN\nNear WF h(f0) and for small y, we can solve the eikonal equation\n1−|∇g0ϕ±|2= 0, ϕ±|y=0(x′,ξ′) =x′·ξ′. (6.8)\nNote that |∇g0ϕ±|2=|∂yϕ±|2+gjk\n0∂jϕ±∂kϕ±.Near (x′\n0,ξ′\n0)∈ H(Σ), for each fixed ξ′, we find\na Lagrangian submanifold of T∗Σ, locally of the form\nL0,ξ′:={(x′,ξ=∂x′ϕ0(x,ξ′)) :ϕ0(x′,ξ′) =x′·ξ′}.\nAt each point ( x′,ξ=ξ′)∈ L0,ξ′, there are two distinct roots η±of the equation\nη2+gjk\n0ξ′\njξ′\nk= 1,\nand each root determines a flow Φ±\nyof the bicharacteristics p=η2−r(y,x′,ξ′) on{p= 0}.\nThen we can define the Lagrangian L±\ny,ξ′:= exp(Φ±\ny)(L0,ξ′) locally, which is again a Lagrangian\nofT∗Σ (viewing yas a parameter) and can be written locally as L±\ny,ξ′={(x′,∂x′ϕ±)}. Then\nϕ±is the desired solutions of ( 6.8) with the property\n∂yϕ++∂yϕ−= 0,aty= 0.\nNext we set\nb±(y,x′,ξ′) =N/summationdisplay\nj=0hjb±\nj(y,x′,ξ′),\nwith coefficients bjsolving the transport equations\n2∂yϕ±∂yb±\n0+gjk\n0∂jϕ±∂kb±\n0+(∆g0ϕ±)b±\n0= 0,\n2i∂yϕ±∂yb±\nj+igjk\n0∂jϕ±∂kb±\nj+i(∆g0ϕ±)b±\nj+∆g0b±\nj−1= 0,1≤j≤N.(6.9)\nThen\nPh(χf±\n0) =hN+2\n(2πh)d−1/integraldisplay\nRd−1eiϕ±\nh∆g0b±\nN(y,x′,ξ′)θ±(ξ′)dξ′=OL2(hN+2).\nTo determine the datum b±\nj|y=0, we need the boundary conditions. Note that the approximate\nquasi-mode is given by\nuh=/summationdisplay\n±U±(χf±\n0)\nand we want to determine f±\n0.\nDenote byB±\nh= Oph(b±) andB0,±\nh= Oph(b±|y=0), then the Dirichlet trace is given by\nB0,+\nh(χf+\n0)+B0,−\nh(χf−\n0),\nand the Neumann trace is given by\n/summationdisplay\n±±Oph(√r0b±|y=0)(χf±\n0)+h/summationdisplay\n±Oph(∂yb±|y=0)|y=0(χf±\n0).KELVIN-VOIGT DAMPED WAVE EQUATION 33\nNow we choose b+\n0|y=0=b−\n0|y=0=χ(x′)ψ(ξ′) localized near ( x′\n0,ξ′\n0) andb±\nj|y=0= 0 for all\n1≤j≤N. Then the symbol (matrix-valued)\nΘ = Θ 0+h/parenleftbigg0 0\n∂yb+\n0∂yb−\n0/parenrightbigg\n|y=0\nwith\nΘ0:=/parenleftbiggb+\n0b−\n0√r0b+\n0−√r0b−\n0/parenrightbigg\n|y=0\nis invertible. For such an elliptic system, we can construct a symbol ( matrix-valued) Υ, such\nthat\nOph(Θ)Oph(Υ) = Id+ OHs→Hs+m(hN+1−m).\nIn particular, for a given Dirichlet trace σDirand Neumann trace σNeuwith wave front sets\nlocated near ( x′\n0,ξ′\n0), we find\n/parenleftbiggχf+\n0\nχf−\n0/parenrightbigg\n=χOph(Υ)χ/parenleftbiggσDir\nσNeu/parenrightbigg\n.\nThen microlocally near ( x′\n0,ξ′\n0)∈ H(Σ),uhsatisfies\n(h2∆g0+1)uh=OL2(hN), u|h=0=σDir+OH1\n2(hN), h∂yuh|y=0=σNeu+OH−1\n2(hN),\nand microlocally near ( x′\n0,ξ′\n0),uh=OL2(1),WF h(uh) lies in a small neighborhood of ( x′\n0,ξ′\n0).\nFinally, due to the microlocalisation in the hyperbolic region, we can exc hange in the error\nterms powers of hagainst derivatives, leading to\n(h2∆g0+1)uh=OHk(hN−k), u|h=0=σDir+OH1\n2+k(hN−k), h∂yuh|y=0=σNeu+OH−1\n2+k(hN−k).\nReferences\n[AHR18] K. Ammari, F. Hassine, L. Robbiano, Stabilization for the wave equation with singular kelvin-v oigt\ndamping , preprint, arXiv:1805.10430.\n[BLR92] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, an d\nstabilization of waves from the boundary , SIAM J. Control Optim., 30 (1992), 1024-1065.\n[BoT10] A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigro ups, Math. Ann.,\n347(2):455–478, 2010.\n[Bu04] N. Burq, Smoothing effect for Schr¨ odinger boundary value problems. Duke Math. Jour., 123 (2004),\n403-427.\n[Bu19] N. Burq. Decays for Kelvin-Voigt damped wave equation I , preprint, arXiv:1904.08318.\n[BC15] N. Burq, H. Christianson. Imperfect geometric control and overdamping for the damped wave equation ,\nComm. Math. Phys., 336 (1): 101-130, 2015.\n[BL03] N. Burq, G. Lebeau, Mesures de d´ efaut de compacit´ e, application au syst` eme d e Lam´ e, Ann. Scient. Ec.\nNorm. Sup., (4) 34 (2001), 817-870.\n[BG96] N. Burq, P. G´ erard, Condition n´ ecessaire et suffisante pour la contrˆ olabilit´ e exacte des ondes , Comptes\nRendus de L’Acad´ emie des Sciences, 325 (1997), 749-752.\n[BS20] N. Burq, C-M. Sun, Decay rates for the Kelvin-Voigt damped wave equation II: Th e geometric control\ncondition , to appear in Proceeding AMS.34 N. BURQ AND C-M. SUN\n[E84] A. Elbert, An asymptotic relation for the zeros of Bessel functions , J. Math. Anal. Appl., 98 (1984),\n502-511.\n[Ge91] P. G´ erard, Microlocal defect measures , Comm. Partial Differential Equations, 16.11 (1991), 1761-1794.\n[HZZ] Z.-J. Han, Z. Liu and Q. Zhang Sharp Stability of a String with Local Degenerate Kelvin-Vo igt Damping\nPreprint https://arxiv.org/abs/2111.09500. 2021\n[LR06] K. Liu, B. Rao. Exponential stability for the wave equation with local kelv in-voigt damping , Z. Angew.\nMath. Phys., 57 (3): 419-432, 2006.\n[LR05] Z. Liu, B. Rao. Characterization of polynomial decay rate for the solution of linear evolution equation ,\nZ. Angew. Math. Phys., 56 (4): 630-644, 2005.\n[MS] R.B. Melrose, J. Sj¨ ostrand, Singularities of boundary value problems I . Communications in Pure Applied\nMathematics, 31:593-617, 1978.\n[MRa] J.E. Munoz Rivera, R. Racke, Transmission problems in (thermo)viscoelasticity with Ke lvin-Voigt damp-\ning: non-exponential, strong and polynomial stability , SIAM J. Math. Anal. Vol. 49 (2017), No. 5, pp.\n3741–3765.\n[RZh18] L. Robbiano, Q. Zhang, Logarithmic decay of a wave equation with kelvin-voigt damp ing, Mathematics\n2020, 8(5), 715, p.2–19.\n[Tay] M. E. Taylor. Partial Differential Equations I , Second edition (2011), Springer.\n[Te16] L. Tebou. Stabilization of some elastodynamic systems with localize d Kelvin-Voigt damping , Discrete\nContin. Dyn. Syst., 36 (12): 7117-7136, 2016.\n[Zh18] Q. Zhang. Polynomial decay of an elastic/viscoelastic waves interac tion system . Z. Angew. Math. Phys.,\n69 (4): Ast. 88, 10, 2018.\n[Zw12] M. Zworski, Semiclassical analysis , Graduate Studies in Mathematics, (2012) AMS.\nUniversit ´e Paris-Saclay, Laboratoire de math ´ematiques d’Orsay, UMR 8628 du CNRS, B ˆat.\n307, 91405 Orsay Cedex, France and Institut Universitaire d e France\nEmail address :nicolas.burq@universite-paris-saclay.fr\nUniversit ´e de Cergy-Pontoise, Laboratoire de Math ´ematiques AGM, UMR 8088 du CNRS, 2\nav. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France\nEmail address :chenmin.sun@u-cergy.fr"    },    {        "title": "1710.03585v1.A_four_field_gyrofluid_model_with_neoclassical_effects_for_the_study_of_the_rotation_velocity_of_magnetic_islands_in_tokamaks.pdf",        "content": "A four-\feld gyro\ruid model with neoclassical e\u000bects for the study of\nthe rotation velocity of magnetic islands in tokamaks\nA. Casolari\nJune 11, 2022\n1 Introduction\nAt equilibrium, in a tokamak, magnetic \feld lines lie on surfaces forming a family of nested tori, named\nmagnetic surfaces. This structure of nested magnetic surfaces can be a\u000bected by instabilities. One of the most\nimportant ones is the so-called tearing mode, which is an instability \"tearing\" and reconnecting magnetic \feld\nlines. Magnetic reconnection locally breaks the topology of magnetic surfaces leading to a more energetically-\nfavorable con\fguration. Magnetic islands result from the nonlinear evolution of tearing modes and represent a\nserious obstacle for obtaining nuclear fusion in magnetic con\fnement devices. In fact, the breaking of magnetic\nsurfaces causes an increase in the heat and particle \ruxes. The uncontrolled growth of magnetic islands can\nalso lead to major disruptions, causing serious damage to the device.\nMany e\u000borts have been made in the past decades to develop a theory of magnetic island dynamics in tokamaks.\nThe interest in this kind of studies is to understand the conditions for the onset of the islands in the tokamak\nexperiments and to control them to prevent their growth to large amplitudes and the consequent negative e\u000bects\non con\fnement. Magnetic islands arise from the nonlinear evolution of tearing modes [1]. In the presence\nof an equilibrium density and temperature gradient, the tearing mode acquires a propagation frequency and\nthe instability is said a drift-tearing mode [2]. According to the linear drift-tearing dispersion relation, the\npropagation frequency of the instability should be close to the electron diamagnetic frequency, !\u0000!E\u0019!\u0003e\n[3, 2], where the frequency is related to the velocity through the wave vector k,!=k\u0001v. The tearing\nmode is an instability characterized by a long wavelength, which corresponds to a small wavevector. !Eis\ntheE^B-drift frequency, due to the equilibrium electric \feld. In fact, the plasma as a whole rotates with\ntheE^Bvelocity, so that this contribution must be subtracted from the island rotation velocity (Doppler\nshift). Experimental observations of magnetic islands in tokamaks, under speci\fc conditions, show a rotation\nfrequency closer to the ion diamagnetic frequency, !\u0000!E\u0019!\u0003i[4, 5, 6]. This disagreement between the\npredictions of the linear theory and the experimental observations raises doubts on the validity of the most\ncredited theoretical models describing magnetic island dynamics. According to recently developed models, in\nthe presence of signi\fcant electron temperature gradients, the introduction of the so-called \"mode inductivity\"\n[7] in the Ohm's law permits the existence of modes propagating with the ion diamagnetic frequency. This\ne\u000bect arises naturally in the linear regime, but the experimental observations of the island rotation concern\nnonlinear islands, thus a direct check of the validity of this model is not currently possible. Another widely\naccepted interpretation of the observed rotation velocity is that, when the island width becomes larger than\nthe ion-acoustic radius, the ion \ruid cannot cross the island separatrix, thus the island is forced to propagate\nwith the velocity of the ion \row [8]. This explanation works for islands which are large enough, but it cannot\naccount for the transition from one diamagnetic velocity to the other. Nonlinear island dynamics is still not\nfully understood, and the processes that determine the island rotation velocity are under investigation.\nAttempts to study the stationary rotation of magnetic islands have been made by Fitzpatrick & Waelbroeck\nin a series of papers on the subject [8, 9, 10, 11] by solving an improved version of the four-\feld model,\npreviously deduced by Hazeltine, Kotschenreuther and Morrison [12], which is a reduction of the two-\ruid\nplasma description. The result of their studies is that both the island width and the neoclassical e\u000bects in\ruence\nthe island rotation. In particular, the critical parameters which determine the island dynamics are the ratio\nbetween the island width wand the ion-acoustic radius \u001ason one side, and the ratio between the collision\nfrequency\u0017iand the bounce frequency !bon the other side. The \frst parameter determines if the island is\nin the sonic or hypersonic regime, which is related to the relative role of ion-acoustic waves on the \rattening\nof the density pro\fle inside the separatrix. The second parameter determines if the plasma is in the weak or\nin the intermediate damping regime, which is related to the relative strength of the neoclassical e\u000bects. The\n1arXiv:1710.03585v1  [physics.plasm-ph]  10 Oct 2017simultaneous presence of both the e\u000bects in a tokamak plasma makes it particularly di\u000ecult to determine the\nislands rotation velocity.\nThe purpose of this work is to investigate the magnetic island dynamics in tokamaks, in particular as regards\nisland rotation. The attempts to study the island rotation by Fitzpatrick & Waelbroeck rested on the inclusion\nof the neoclassical e\u000bects in their equations by using simpli\fed expressions for the neoclassical terms, together\nwith the possibility to keep the island-size e\u000bects by using an appropriate normalization for the \felds. Although\ntheir work shows results consistent with the experimental observations, their results come from a system of\n\ruid equations which did not include the Finite Larmor Radius (FLR) corrections from the start. In this work\nwe attempt to improve their results by starting from a set of gyro\ruid equations, which result from taking\nthe moments of the gyrokinetic equation [13, 14], and then reducing them to a four-\feld model analogous\nto that used by Fitzpatrick & Waelbroeck. The neoclassical e\u000bects are included in the model by using the\nsame simpli\fed expressions, with an important di\u000berence. To be consistent with the gyro\ruid equations, we\ncompute the lowest order FLR corrections to the poloidal \row damping by solving the gyrokinetic equation in an\nappropriate limit and then computing the poloidal \row damping by following the same approach adopted in the\nbook \"Collisional transport of impurities in plasmas\" by Helander & Sigmar [15]. The equations thus obtained\nhave been solved by adopting a series of perturbative expansions introduced by Fitzpatrick & Waelbroeck in\ntheir works and based on the multiple-scale approach [16]. The \fnal equations have been solved, in two di\u000berent\nregimes of collisionality, together with the torque balance condition, imposing that the total electromagnetic\nforce acting on the freely-rotating islands is zero. The solution of this system of equations provides the \feld\npro\fles and the self consistent phase velocity of the islands. Attempts to study both analitically and numerically\nthe FLR e\u000bects on magnetic island evolution have been done [17, 18]. In these works the focus of the authors\nwas mainly on the analysis of the island dynamics and related phenomena, such as the emission of drift waves\nand the \rattening of the density pro\fle, given the island phase velocity, which was just a parameter of their\nmodels. The approach we choose, which is the same used by Fitzpatrick & Waelbroeck, is to deduce the island\nrotation frequency consistently with the \feld pro\fles in a stationary regime.\nThe paper is organized as follows. In Section 2 an analytical solution of the gyrokinetic equation is deduced and\nthe lowest-order FLR corrections to the poloidal \row damping are calculated. In Section 3 a four-\feld gyro\ruid\nmodel with neoclassical e\u000bects is deduced and a series of simpli\fcations is performed on these equations to apply\nthem to the study of the stationary rotation of a chain of magnetic islands. In Section 4 the torque balance\ncondition is introduced and its explicit form is deduced for the case being considered. In Section 5 the model\nis applied to the study of the weak-damping regime. In Section 6 the model is applied to the study of the\nintermediate-damping regime, where a new term appears which contains the lowest-order FLR corrections to\nthe poloidal \row damping. In Section 7 the results of the numerical integration of the system of equations in\nthe weak and the intermediate regime are displayed. Conclusions are drawn in Section 8.\n2 A particular solution of the gyrokinetic equation\nThe \frst step in our calculation is to deduce a particular solution for the gyrokinetic equation\n@f\n@t+ (b0vk+vd)\u0001rf\u0000\u001a1\nm\u0012\n1 +\u0016B1?\nB0\u0013\u0000\nZerk\u0016\u001e1+\u0016rk(B0+\u0016B1k)\u0001\n+\n+1\nmvkvd\u0001\u0000\nZer\u0016\u001e1+\u0016r(B0+\u0016B1k)\u0001\u001b@f\n@vk=C(f)(1)\nunder speci\fc simplifying hypotheses. This solution will be then used to \fnd the FLR corrections to the\nneoclassical e\u000bects which occur in a tokamak. The resolution will follow the method outlined in [15, 19].\n2.1 FLR expansion of the gyrokinetic equation\nBy starting from Eq.1, the distribution function fis expanded in an equilibrum Maxwellian part plus a small\nperturbation, ordered with \u000e=\u001ai=L\u001c1:f=FM+f1.\u001aiis the ion Larmor radius, while Lis a macroscopic\nlength scale. The equilibrium solution is assumed a stationary \rux function, that is: @FM=@t= 0,rkFM= 0.\nThe following orderings are used\n@\n@t=O(\u000e2vth=L);Ze\u0016\u001e1\nT=O(\u000e);vd\nvth=O(\u000e) (2)\n2With these orderings, the gyrokinetic equation Eq.1 to order \u000ebecomes\nvkrkf1+vd\u0001r(f1+FM)\u0000\u0016\nmrkB0@f1\n@vk\u00001\nm\u0014\nZerk\u0016\u001e1+\u0016\u0012\nrk\u0016B1k+\u0016B1?\nBrkB0\u0013\u0015@FM\n@vk\u0000\n\u00001\nmvkvd\u0001[Zer\u0016\u001e1+\u0016r(B0+\u0016B1k)]@FM\n@vk+Ze\nmE(A)\nk@FM\n@vk= 0(3)\nThe term proportional to the parallel induced electric \feld E(A)\nkhas been introduced to include the e\u000bect of the\nmagnetic \rux variation in a tokamak. We can meake the further assumption that rB0\u001cr\u001e1;rB1k, meaning\nthat the equilibrium magnetic \feld is almost uniform. Using the following identity to express vdin terms of vk:\nvd\u0001rf=Ivkrk\u0010vk\n\n\u0011@f\n@ (4)\nwhereI=RB'is a \rux function, the gyrokinetic equation becomes:\nvkrk\u0014\nf1+Ivk\n\n@\n@ (f1+FM)\u0015\n=\u0000FM\nTvkn\n[Zerk\u0016\u001e1\u0000ZeE(A)\nk+\u0016rk\u0016B1k]+\n+rk\u0014Ivk\n\n\u0012\nZe@\u0016\u001e1\n@ +\u0016@\u0016Bk1\n@ \u0013\u0015\u001b (5)\nGyrokinetic theory is usually used to study turbulent transport, which is typically much larger than the col-\nlisional one. For this reason the gyrokinetic equation we started from didn't have the collisional term on the\nright-hand side. To deal with neoclassical e\u000bects, we need to include the e\u000bect of collisions by using an appro-\npriate collision operator. The contribution from E(A)\nkcan be absorbed in a Spitzer function fs, as customary\nin the drift-kinetic case. This one is neglected in respect to f1because, for the ions, fs\u001cf1. For the \felds\nperturbations caused by the onset of a magnetic island, the leading term is \u0016B?1, so that we can neglect \u0016Bk1.\nWith these simpli\fcations:\nvkrk\u0014\nf1+Ivk\n\n@\n@ (f1+FM)\u0015\n=\u0000FMvk\nT\u001a\nZerk\u0016\u001e1+rk\u0014Ivk\n\n\u0012\nZe@\u0016\u001e1\n@ \u0013\u0015\u001b\n+C(f1) (6)\nIn the low-collisional regime, we can expand f1in a power series of the collisionality \u0017\u0003, so that we can write\nf1=f(0)\n1+f(1)\n1+\u0001\u0001\u0001[15, 19]. To the two lowest orders\nrk\u0014\nf(0)\n1+Ivk\n\n@\n@ (f(0)\n1+FM)\u0015\n+FMZe\nTrk\u0014\n\u0016\u001e1+Ivk\n\n@\u0016\u001e1\n@ \u0015\n= 0\nvkrk\"\nf(1)\n1+Ivk\n\n@f(1)\n1\n@ #\n=C(f(0)\n1)(7)\nUsing the fact that rkFM=rkT= 0 (neglecting signi\fcant perturbations to the temperature), the lowest\norder equation becomes\nrk\u0014\nf(0)\n1+Ivk\n\n@\n@ (f(0)\n1+FM) +FMZe\nT\u0012\n\u0016\u001e1+Ivk\n\n@\u0016\u001e1\n@ \u0013\u0015\n= 0 (8)\nBy integrating once, we \fnd the following equation for f(0)\n1:\n\u0012\n1 +Ivk\n\n@\n@ \u0013\nf(0)\n1=g\u0000Ivk\n\n@FM\n@ \u0000FMZe\nT\u0012\n1 +Ivk\n\n@\n@ \u0013\n\u0016\u001e1 (9)\nwithgan unknown function such that rkg= 0. Eq.9 can be solved formally, by writing the solution f(0)\n1in an\nintegral form. Every time we deal with an equation of this form\n\u0012\n1 +ad\ndx\u0013\nf(x) =K(x) (10)\nthe particular solution takes the form [20]:\nf(x) =e\u0000x=a\naZx\nx0ey=aK(y)dy (11)\n3Eq.9 is in the form Eq.10, so the solution for f(0)\n1becomes\nf(0)\n1=c1(v)e\u0000 = s+e\u0000 = s\n sZ \n 0e\u001f= s\u0014\ng(\u001f)\u0000 s@FM\n@\u001f\u0000ZeFM\nT\u0012\n1 + s@\n@\u001f\u0013\n\u0016\u001e1\u0015\nd\u001f (12)\nwhere s=Ivk=\n has the dimensions of a magnetic \rux. c1(v)e\u0000 = sis the solution of the homogeneus\nequation. Once Eq.12 has been solved, we can multiply both members of the second equation of Eq.7 by B=vk\nand take the \rux surface average, so that we are left with the equation\n\u001cB\nvkC(f(0)\n1)\u001d\n= 0 (13)\n2.2 Analytical solution\nTthe collision operator Ccan be chosen in the following form [15, 19]:\nCii(fi) =\u0017ii\nD(v)\u0012\nL(fi1) +mivkui\nTifMi\u0013\n(14)\nBy introducing the following de\fnition of the Lorentz operator\nL=2hvk\nv2@\n@\u0015\u0015vk@\n@\u0015(15)\nwhereh\u0011B0=Bis the toroidal metric coe\u000ecient and \u0015is related to the particles pitch angle by \u0015\u0011hsin2\u000b,\nEq.13 becomes:\n*\nB(\n2h\nv2@\n@\u0015\u0015vk@\n@\u0015 \ng\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s@\u001fg+e\u0000 = sc1!\n+I\n\n[@ FM+\n+ZeFM\nT@ \u0016\u001e1\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+miui\nTFM)+\n= 0(16)\nWe de\fne the auxiliary function J:\nJ=g\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s@\u001fg+e\u0000 = sc1 (17)\nThe equation for Jis\n@\n@\u0015\u0015\nvk\u000b@\n@\u0015J=\u0000v2\n2\u001aI\nh\n\u0014\n@ FM+ZeFM\nT@ \u0016\u001e1\u0000\n\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+Dui\nhEmi\nTFM) (18)\nFrom the form of Eq.18, we can deduce that Jplays the role of the function which, in the drift-kinetic equation,\nvanishes in the trapped particle space, so that the solution of Eq.18 is\nJ=H(\u0015c\u0000\u0015)v2\n2Z\u0015c\n\u0015d\u00150\n\nvk(\u00150)\u000b\u001aI\nh\n\u0014\n@ FM+ZeFM\nT@ \u0016\u001e1\u0000\n\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+Dui\nhEmi\nTFM) (19)\n4whereHis the Heaviside function. By using a few results from drift-kinetik theory, we \fnd the following\nsolution:\nf(0)\n1=\u0000Ivk\n\n\"\n@ FM+ZeFM\nT@ \u0016\u001e1\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+\n+IHVk\nh\n\u0012mv2\n2T\u00001:33\u0013dlogT\nd FM\u0000Ze\u0016\u001e1\nTFM+e\u0000 = sZ \n 0d\u001fe\u001f= sZe@\u001f\u0012FM\nT\u0016\u001e1\u0013\n\u0000\n\u0000IHVk\nh\ne\u0000 = sZ \n 0d\u001fe\u001f= s\u0014\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0015\n\u0000\n\u0000IHVk\nh\nFM\b\n\u0017ii\nD\t(\n\u0017ii\nDe\u0000 = sZ \n 0d\u001fe\u001f= s\"\n@2\n\u001fFM\nFM+Ze\nFM@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013#)(20)\nThe curly braces in the last line of Eq.20 represent the velocity-space average, which is de\fned as:\nfFg=Z\nd3vFmv2\nnTFM (21)\nThe solution Eq.20 still contains terms in an integral form. However, it contains the FLR corrections which\nprovide, after velocity-space integration, the modi\fed transport coe\u000ecients in the di\u000berent collisionality regimes.\n2.3 Poloidal \row damping\nEq.20 can be used to compute the neoclassical e\u000bects, in particular the poloidal \row damping, which comes\nfrom the toroidal geometry, through the equation [15, 19]\nhB\u0001r\u0001\u0019i=D\nB(Fk+nZeE(A)\nk)E\n(22)\nwhere\u0019is the stress tensor and Fkis the parallel component of the friction force, which is de\fned in terms of\nthe distribution function\nFk\u0011Z\nmvkC(f1)d3v (23)\nWe can now use the Spitzer function to eliminate the term proportional to the inductive electric \feld and remind\nthat, for the ions, the Spitzer function is negligible in respect to the function f1. Using the particular form for\nthe collision operator Eq.14, together with Eq.23, Eq.22 becomes\nhB\u0001r\u0001\u0019i=\u001c\nBZ\nmvk\u0017ii\nD\u0012\nL(f(0)\n1) +mvkUki\nTFM\u0013\nd3v\u001d\n(24)\nUkiis the parallel \row velocity of the ions, which is de\fned as\nUki=Z\nvkf(0)\n1d3v (25)\nEq.20 contains the FLR e\u000bects in terms of integral expressions. Such quantities can be expanded in a power\nseries in respect to  sperforming an integration by parts\ne\u0000 = sZ\nd\u001fe\u001f= sF= sF\u0000 2\ns@ F+O( 3\ns@2\n F) (26)\nThis expansion is made possible by the smallness of the ion Larmor radius: in fact, after velocity integration,\n s= =\u001ai=L\u001c1. From this result we notice that, when taking the velocity moments of the distribution\nfunction, only the terms which have the correct parity will remain and the others will be zero. This is particularly\nimportant because, from the lowest order expansion Eq.26, only terms proportional to \u001a2\niwill remain. When\napplying this expansion to Eq.20 and applying it to Eq.25, we \fnd:\nUk=\u0000I\n\nZ\nd3vv2\nk\u0012\n@ FM+ZeFM\nT@ \u0016\u001e1\u0013\n+I\n\nZ\nd3vv2\nk\u0012mv2\n2T\u00001:33\u0013dlogT\nd FM+\n+I\n\nZ\nd3vv2\nkZe@ \u0012FM\nT\u0016\u001e1\u0013\n+I3\n\n3Z\nd3vv2\nkFM\nf\u0017ii\nDg(\n\u0017ii\nDv2\nk\"\n@3\n FM\nFM+Ze\nFM@2\n \u0012FM\nT@ \u0016\u001e1\u0013#) (27)\n5Since the radial derivatives are steep, we only keep the highest order derivatives in Eq.27;\n@3\n FM+Ze@2\n \u0012FM\nT@ \u0016\u001e1\u0013\n=\u00141\nPd3P\nd 3+ 2Ze\nTd3\u0016\u001e1\nd 3+\u0012mv2\n2T\u00005\n2\u00131\nTd3T\nd 3\u0015\nFM (28)\nInserting the solution Eq.20 expanded according to Eq.26 into Eq.24, and by using Eq.28:\nhB\u0001r\u0001\u0019i\u0019B\u001601minft\u0017iiITi\nmi\ni\u001a\n1:17dlogTi\nd +\n+I2Ti\nmi\n2\ni\u0014\n2:70\u00121\nPd3P\nd 3+ 2Ze\nTd3\u0016\u001e1\nd 3\u0013\n\u00000:701\nTd3T\nd 3\u0015\u001b (29)\nwhere\u00160i=f\u0017ii\nDg,ftis the fraction of trapped particles and the following properties have been used [15]:\nf\u0017ii\nDg\u00190:53f\u0017ii\nDx2g\u00190:71f\u0017ii\nDx4g\u00191:59 (30)\nwherex2=v2=v2\nth. The \frst term in Eq.29 is the result from drift-kinetic theory. The additional terms are the\n\frst order FLR corrections, which are proportional to \u001a2\ni.\n3 Four-\feld gyro\ruid model\nThe system of gyro\ruid equations originally developed by P. B. Snider [21] consists of six equations, evolving\nthe density, the parallel velocity, the parallel and perpendicular pressure and the parallel and perpendicular\nheat \rux for each particle species. Here we just need the \frst two of them, together with the vorticity equation,\nwhich can be deduced from the quasi-neutrality condition\nne= \u00001=2\n0ni+n0Ze\nTi(\u00000\u00001)\u001e (31)\nwhere \u0000 0=\nJ2\n0\u000b\nis the velocity average of the zero-order Bessel function, with argument k2\n?\u001a2\ni, which comes\nfrom the gyroaverage involved in the gyrocenter transformation. The second term on the right-hand side of\nEq.31 is the so-called polarization density. The momentum equation for the two species electrons and ions can\nbe written as:\n@\u0016ni\n@t+v\b\u0001r\u0016ni+n0\u0016rk\u0016uki= 0\n@ne\n@t+v\u001e\u0001rne+n0rkuke= 0\nmin0\u0012@\u0016uki\n@t+v\b\u0001r\u0016uki\u0013\n=\u0000T0i\u0016rk\u0016ni+en0\u0012@\t\n@t\u0000rk\b\u0013\n+Fie\nmen0\u0012@uke\n@t+v\u001e\u0001ruke\u0013\n=\u0000T0erkne\u0000en0\u0012@ \n@t\u0000rk\u001e\u0013\n+Fei(32)\nwhere \b = \u00001=2\n0\u001e, \t = \u00001=2\n0 are the gyroaveraged \felds, FieandFeiare the collisional friction forces, whose\nparallel component is de\fned in Eq.23. \u0016rkis the parallel gradient performed along the gyroaveraged magnetic\n\feld. \u0016niand \u0016ukiare the ion density and parallel velocity expressed in the gyrocenter coordinates. Owing to\nmomentum conservation in Coulomb collisions, the property Fie=\u0000Feiholds. We used P=nT0, withT0\nuniform and constant. The common equilibrium density n0multiplies the electric force term in the momentum\nequations because the electric \feld is perturbative. By proceeding similarly to the quasi-neutrality calculation,\nwe can \fnd the gyrokinetic de\fnition of the current:\nJk=\u0000en0(uke\u0000\u00001=2\n0\u0016uki) (33)\nIntroducing the Debye length \u0015Di=p\nTi=(nie2), Eq.31 becomes:\n1\n\u00152\nDi(\u00000\u00001)\u001e=e(ne\u0000\u00001=2\n0\u0016ni) (34)\n6Taking the time derivative of Eq.34 and using the equations above, we obtain the following vorticity equation:\n\u0012@\n@t+v\u001e\u0001r\u0013\u00121\n\u00152\nDi(\u00000\u00001)\u001e+e(\u00001=2\n0\u00001)\u0016ni\u0013\n=rkJk+e(\u00001=2\n0\u00001)v\u001e\u0001r\u0016ni\u0000erk(\u00001=2\n0\u00001)\u0016uki\u0000e(\u0016rk\u0000rk)\u0016uki\n(35)\nWe can further simplify this system of equations by neglecting the electron inertia in the electron momentum\nequation, which becomes the generalized Ohm law. Then we use the quasi-neutrality condition Eq.34 to express\nnein terms of \u0016 niand we sum the momentum equations of the ions and the electrons.\nThe system of equations we get is:\n@\u0016ni\n@t+v\b\u0001r\u0016ni+n0\u0016rk\u0016uki= 0\n@ \n@t\u0000rk\u001e+T0e\nen0rk\u0014\n\u00001=2\n0\u0016ni+n0(\u00000\u00001)e\u001e\nT0i\u0015\n=\u0011Jk\nmi\u0012@\n@t+v\b\u0001r\u0013\n\u0016uki=\u0000T0e\nn0\u0016rk\u0014\n(\u001c+ \u00001=2\n0)\u0016ni+ (\u0000 0\u00001)e\u001e\nT0i\u0015\n\u0012@\n@t+v\u001e\u0001r\u0013\u00121\n\u00152\nDi(\u00000\u00001)\u001e+e(\u00001=2\n0\u00001)\u0016ni\u0013\n=rkJk+e(\u00001=2\n0\u00001)v\u001e\u0001r\u0016ni(36)\nwhere\u001c=T0i=T0e. We neglected in the parallel momentum equation the electric force coming from the di\u000berence\nbetween the \felds  ,\u001eand their gyroaverage. v\band \u0016rkare theE^Bvelocity and the parallel gradient on\nthe \felds calculated with the gyroaveraged \felds: B0v\b\u0001r= (^ez^r\u00001=2\n0\u001e)\u0001r,B0\u0016rk= (^ez^r\u00001=2\n0 )\u0001r.\nThe quantity 1 =\u00152\nDi(\u00000\u00001)\u001e+e(\u00001=2\n0\u00001)\u0016niis the gyrokinetic vorticity. In the limit of large wavelengths, this\nquantity reduces to \u001a2\nie(noe=Tir2\n?\u001e+ 1=2r2\n?ni). The \frst term corresponds to the E^Bdrift velocity, while\nthe second one represents the contribution from the diamagnetic velocity. The factor 1 =2 appearing in front of\nthis term comes from the expansion of the gyroverage operator [22]. Attempts to study an Hamiltonian version\nof these equations, both analytically and numerically, has been done by di\u000berent authors [23, 24].\nGyro\ruid equations surpass the \ruid equations because they include the FLR e\u000bects which come from the\ngyrokinetic theory. However, these FLR e\u000bects are present in the form of nonlinear di\u000berential operators, quite\ndi\u000ecult to deal with both analytically and numerically. Several attempts have been made by di\u000berent authors to\ndeal with these operators by approximating them with power expansions and elementary functions. An overview\nof these attempts is provided in [14]. The operators \u0000 0and \u00001=2\n0involve all the even powers of b=k2\n?\u001a2\ni:\n\u00000=\nJ2\n0\u000b\n=I0(b)e\u0000b= 1\u0000b+O(b2)\n\u00001=2\n0\u0019\nJ2\n0\u000b1=2=I1=2\n0(b)e\u0000b=2= 1\u0000b=2 +O(b2)(37)\nwhereI0is the modi\fed Bessel function. The Taylor expansion of these operators provides the FLR corrections\nto all orders in b. In the limit of large wavelengths k2\n?\u001a2\ni\u001c1 the power expansion can be truncated to a low\norder (usually the second order is already a good approximation). However, in the limit of small wavelengths\nk2\n?\u001a2\ni\u001d1 (or more realistically, k2\n?\u001a2\ni=O(1)), the power expansion isn't a good approximation any longer.\nIf we introduce the following normalization for the \felds [11]:\n^ =Ls\nB0w2 ; ^n=\u0000Ln\nw\u0016ni\nn0;^\u001e=\u0000\u001e\nwV\u0003eB0+ ^xVp\n^V=\u000f\nqVki\nV\u0003e; ^J=LS\u00160\nB0\u000eeJk;^\u0011=\u0011\n\u00160k\u0012V\u0003ew2(38)\nwherewis the island width, \u000ee=\f=\u000b2,\u000b2=w2=\u001a2\nsL2\nn=L2\nsand\u001as=cs=\niis the ion-acoustic radius. The\nx-derivatives are normalized to w, they-derivatives are normalized to Ln= 1=k\u0012and the time derivatives are\nnormalized to k\u0012V\u0003e.Vpis the unknown island rotation velocity. The additional term ^ xVpin the normalized\nelectrostatic potential represents the contribution from the island-induced electric \feld. With this choice for\nthe normalization, the gradients length-scale of the \felds in the radial direction is comparable with the island\nwidthw.\n3.1 Neoclassical e\u000bects\nThe neoclassical e\u000bects come from the inhomogeneity of the magnetic \feld and the low collisionality of the\nplasma. Neoclassical theory and the poloidal \row damping have been thoroughly described in [19, 15]. In\n7addition to the poloidal damping caused by the toroidal shape of the tokamak, there is a similar phenomenon\ncaused by non-axisymmetric e\u000bects, such as the magnetic islands. The broken poloidal symmetry of the torus\ncauses the travelling particles to experience a magnetic-mirror e\u000bect, which leads to the phenomenon of banana\norbits and the consequent poloidal \row damping. Analogously, the broken axisymmetry caused by the presence\nof magnetic islands leads to a situation of \"helically-trapped particles\", causing a braking e\u000bect on the plasma\nrotation called \"island-induce \row damping\" [25, 26]. The procedure to obtain the non-axisymmetric e\u000bects\non the plasma rotation is analogous to that used in the axisymmetric case but the calculations are much more\ninvolved because of the complex shape of the \rux surfaces. The island-induced \row damping is proportional\nto the island width wsquared [25, 27], so that its e\u000bect becomes signi\fcative only when the FLR e\u000bects are\nnegligible. For this reason we chose not to compute the FLR corrections to this term by solving the gyrokinetic\nequation. The only correction we are going to keep is the usual lowest-order expansion of the gyroaveraged\nelectrostatic potential. That said, let us consider the divergence of the ion stress tensor we deduced in the\nsection above: according to [27], the \row damping can be included in the system of equations we are using by\nimposing that, under the e\u000bect of this damping, the poloidal rotation velocity tends to relax to its neoclassical\nvalue. This amounts to introducing the following damping term:\nmini\u0017\u0012(Vi\u0001e\u0012\u0000Vnc\n\u0012) (39)\nAfter switching from  tor, simplifying a few terms and introducing the notation T0=dT=dr , Eq.39 becomes:\nmini\u0017\u0012\u001a\nVki^b\u0001e\u0012+1\nB0\u0016\u001e0\n1+Ti\nen0B0\u0016n0\ni(1\u0000c\u0012)\u0000Ti\nen0B0\u0014\nc1n000\ni\nn0+c2Ze\nTi\u0016\u001e000\n1\u0015\u001b\n(40)\nwherec\u0012= 1:17\u0011i,c1= 2:70 + 2\u0011i,c2= 5:40 and\u0011i=Ln=LTis the ratio between the length scales of the\ndensity and temperature gradients. If we use the normalization for the \felds introduced in Eq.38, we \fnd the\nfollowing adimensional expression:\n\u0000^\u0017\u0012n\n^V+Vp\u0000@^x[\u00001=2\n0^\u001e+\u001c^n(1\u0000c\u0012)] +\u001c\u001a2@^x[c1@2\n^x^n+c2@2\n^x^\u001e]o\n(41)\nwhere ^\u0017\u0012=\u0017\u0012=(k\u0012V\u0003e) is the poloidal damping coe\u000ecient, which is determined by the kinetic theory. An analo-\ngous expression exists for the island-induced perpendicular \row damping [25, 27]. When using the normalization\nEq.38, it takes the following form:\n^\u0017?@^x[\u00001=2\n0^\u001e+\u001c^n(1\u0000c?)] (42)\n^\u0017?=\u0017?=(k\u0012V\u0003e) is the perpendicular damping coe\u000ecient, which is proportional to w2, andc?= 2:37. As\nemphasized in [27], the island-induced \row damping acts in the perpendicular direction, so that it doesn't\ncontribute to the parallel momentum equation. When including these e\u000bects in Eq.36, the normalized system\nof equations becomes:\n@^n\n@^t= [^\u001e;^n] + [^V;^ ] +\u001a2\n2f[@2\n^x^\u001e;^n] + [^V;@2\n^x^ ]g\n@^ \n@^t= [^\u001e\u0000^n;^ ] + ^\u0011\u000ee^J\u0000\u001a2[@2\n^x^\u001e=\u001c+@2\n^x^n=2;^ ]\n@^V\n@^t= [^\u001e;^V] +\u000b2(1 +\u001c)[^n;^ ] +\u001a2\u001a1\n2[@2\n^x^\u001e;^V] +\u000b2\n2\u0010\n(1 +\u001c)[^n;@2\n^x^ ] + [@2\n^x^n;^ ]\u0011\n+1\n\u001c[^ ;@2\n^x^\u001e]\u001b\n\u0000\n\u0000^\u0017\u0012\u0012\u000f\nq\u00132\u001a\n^V+Vp\u0000@^x\u0014\u0012\n1 +\u001a2\n2@2\n^x\u0013\n^\u001e+\u001c^n(1\u0000c\u0012)\u0000\u001cc1\u001a2@2\n^x^n\u0000\u001cc2\u001a2@2\n^x^\u001e\u0015\u001b\n@\n@^t@2\n^x\u0010\n^\u001e+\u001c\n2^n\u0011\n=h\n^\u001e;@2\n^x\u0010\n^\u001e+\u001c\n2^n\u0011i\n+\u001c\n2[^n;@2\n^x^\u001e] + [^J;^ ] +\u001a2nh\n^\u001e;@4\n^x\u0010\n^\u001e+\u001c\n4^n\u0011i\n+\u001c\n4[^n;@4\n^x^\u001e]o\n\u0000\n\u0000\u0017?@2\n^x\u0014\u0012\n1 +\u001a2\n2@2\n^x\u0013\n^\u001e+\u001c^n(1\u0000c?)\u0015\n+\u0017\u0012@^x\u001a\n^V\u0000@^x\u0014\u0012\n1 +\u001a2\n2@2\n^x\u0013\n^\u001e+\u001c^n(1\u0000c\u0012)\u0000\u001cc1\u001a2@2\n^x^n\u0000\u001cc2\u001a2@2\n^x^\u001e\u0015\u001b\n(43)\nNote that the FLR corrections coming from the analytical resolution of the gyrokinetic equation are consistent\nwith those coming from the the small-Larmor-radius expansion of the gyro\ruid equations. In fact, although the\nperturbation to the distribution function f1was assumed ordered with \u000e=\u001ai=L\u001c1, the distinction between\nthe parallel and the perpendicular length scales was appropriately addressed, so that the FLR corrections have\nnaturally emerged from our calculations.\n83.2 Simpli\fcation of the system of equations\nEq.43 provides a system of equations describing a plasma in the presence of an island whose width wis larger\nthan the ion Larmor radius \u001ai, so that the FLR corrections enter only to order \u001a2. In this approximation, we can\nreasonably assume \u000b2=O(1) [11, 27]. If the ordering \u001a2\u001c1 holds, we can expand the \felds in the following\nway:\n\u001e=\u001e0+\u001a2\u001e1+O(\u001a4) (44)\nIn the following calculations we are going to use the constant-  approximation, which holds as long as j\u00010wj;\u000ee\u001c\n1, where \u00010is the tearing mode stability parameter and \u000eewas de\fned previously. If this condition holds, the\nmagnetic \rux function takes the form:\n (x;y) =x2\n2+ cosy (45)\nEq.45 describes a magnetic island centered in x= 0, with the O-point in y= 0. The region inside the separatrix\ncorresponds to\u00001<  < 1 and the region outside the separatrix corresponds to  \u00151. From now on, the\nmagnetic \rux function  will no longer be an unknown and, wherever possible, we will express the \felds as\nfunctions of  or its derivatives. For consistency with the results of Fitzpatrick [11, 27], we assume the zero-order\n\felds to be \rux functions, so that the \frst-order \felds are going to be the lowest order FLR corrections. We\nde\fne the following functions:\nM\u0011d\u001e0\nd ; L\u0011dn0\nd ; V0\n0\u0011dV0\nd (46)\nThe dissipative terms represented by the resistivity and the neoclassical viscosity are generally small, so that\nwe can neglect them in \frst instance. We introduce the \rux-surface average operation, which is de\fned as [11]:\nhf(\u001b; ;y )i\u00118\n<\n:1\n2\u0019H\ndyf(\u001b; ;y )p\n2( \u0000cosy)( \u00151)\n1\n2\u0019P\n\u001bRy0\n\u0000y0dyf(\u001b; ;y )p\n2( \u0000cosy)(\u00001< < 1)(47)\nwhere\u001b= sign(x) andy0= cos\u00001 . The \rux surface average is the annihilator of the parallel gradient, so that\nevery term in the form [ A; ] in the equations is deleted by this operator. By using the small-Larmor-radius\nexpansion, we can \fnd explicit expressions of the \frst-order \felds in terms of the zero-order quantities M,Land\nV0\n0. To \fnd the zero-order \felds we need to introduce a second ordering which involves the transport coe\u000ecients.\nThis new ordering assumes that the \frst order \felds are as small as the transport coe\u000ecients, which are in turn\nmuch smaller than the FLR parameter \u001a2.\nTo recover the correct form of the equations [11], we introduce a phenomenological perpendicular viscosity \u0016\nand a di\u000busion coe\u000ecient D(which is related to resistivity through the parallel compressibility [8]). The \felds\nappearing in the \fnal equations obey the following boundary conditions for x! 1 [11]:\nn!x\n\u001e!xVp\nV!V1(48)\nThe \frst condition means that the density gradient becomes constant far from the island. The gradient of the\nelectrostatic potential tends to a constant value which is the electric \feld induced by the island rotation. The\nasymptotic velocity V1is determined by the neoclassical theory.\n4 Torque balance\nThe linear stability index \u00010comes from the equilibrium current which causes the mode to be unstable, but\nevery other contributions to the current a\u000bect the mode growth. It can be easily shown [28] that, with the\nchoice Eq.45 for the magnetic \rux function, the contributions to the mode growth can be parametrized by this\nquantity :\nJc= 4Z+1\n\u00001hJcosyid (49)\n9where the angular brackets represent the \rux-surface average operation. Eq.49 means that the only currents\nthat contribute to the mode growth are those which have the cos y-symmetry. Analogously, there is a similar\nexpression parametrizing the contributions to the torque which is exerted on the island by external currents:\nJs= 4Z+1\n\u00001hJsinyid (50)\nEq.50 means that the only currents that contribute to the torque on the magnetic island are those which have\nthe siny-symmetry. By solving the lowest-order vorticity equation:\n[J(0)\n0; ]\u0000[\u001e(0)\n0;@2\nx(\u001e(0)\n0+\u001c=2n(0)\n0)]\u0000\u001c=2[n(0)\n0;@2\nx\u001e(0)\n0] = 0 (51)\nwe \fnd out that the solution for the current is:\nJ(0)\n0=\u0010\nM0\u0010\nM+\u001c\n2L\u0011\n+\u001c\n2ML0\u0011\nfx2=1\n2[M(M+\u001cL)]0fx2 (52)\nSincefx2=x2\u0000\nx2\u000b\n, this term doesn't contribute to the torque. To \fnd the lowest-order contribution to the\ntorque, we have to consider the following \frst-order vorticity equation:\n[J(1)\n0; ] + [\u001e(1)\n0;@2\nx(\u001e(0)\n0+\u001c=2n(0)\n0)] +\u001c=2[n(1)\n0;@2\nx\u001e(0)\n0] + [\u001e(0)\n0;@2\nx(\u001e(1)\n0+\u001c=2n(1)\n0)] +\u001c=2[n(0)\n0;@2\nx\u001e(1)\n0]+\n+\u0016@4\nx(\u001e(0)\n0+\u001c=2n(0)\n0)\u0000\u0017?@2\nx[\u001e(0)\n0+\u001cn(0)\n0(1\u0000c?)] +\u0017\u0012@xfV(0)\n0\u0000@x[\u001e(0)\n0+\u001cn(0)\n0(1\u0000c\u0012)]g= 0(53)\nIt follows by just performing the calculations and applying the boundary conditions, that the following identity\nholds: Z+1\n\u00001h[J; ]xid =\u0000Z+1\n\u00001hJsinyid (54)\nEq.54 enables us to compute the lowest order contribution to the torque by just multiplying Eq.53 by x, solving\nit for [J(1)\n0; ]xand operating on it with the \rux-surface average and the  -integration. For an isolated island,\nwhich is not interacting with an external electromagnetic \feld, the total torque is zero. By doing this and\napplying again the boundary conditions, the torque-balance condition becomes:\nZ+1\n\u00001d f\u0017\u0012[(V0\u0000V1+Vp+\u001c(1\u0000c\u0012))h1i+ (M+\u001cL(1\u0000c\u0012))] +\u0017?[(Vp+\u001c(1\u0000c?))h1i+ (M+\u001cL(1\u0000c?))]g= 0\n(55)\nBy using the system of equations for the \felds we have deduced above, with their boundary conditions, together\nwith the torque balance condition, we can \fnd the phase velocity of the island in the following way: we \frst\nchoose a value for the phase velocity Vpand we solve the di\u000berential equations for the \felds by the shooting\nmethod, we substitute these solutions in the torque balance condition and we \fnd a new value for Vp, we use\nthis new value in the equations again and we iterate until convergence is reached.\nV1represents the velocity of the plasma far from the island, which depends on the damping e\u000bects. We will\nsee in the following sections that a solution can be found in two di\u000berent collisionality regime, namely the weak\ndamping and the intermediate damping regimes.\n5 Weak damping regime\nIn the weak-damping regime, the following ordering holds:\n1\u001dD;\u0016;\u0011;\u0017\u0012\u001d\u0017? (56)\nIn this case, we can disregard the terms where the products between the FLR parameter \u001a2and the transport\ncoe\u000ecients appear, as well as the perpendicular damping coe\u000ecient \u0017?. Also the product ( \u000f=q)2\u0017\u0012is small and\n10can be neglected. In the weak-damping regime, the equations become:\nDD\n@2\nxn(0)\n0E\n+\u001a2=2D\n[x2M0;n(1)\n0]E\n= 0\n\u0016D\n@2\nxV(0)\n0E\n+\u001a2nD\n[\u001e(0)\n1;V(1)\n0]E\n+D\n[\u001e(1)\n0;V(0)\n1]E\n+ 1=2D\n[x2M0;V(1)\n0]Eo\n= 0\n\u0016D\n@4\nx(\u001e(0)\n0+\u001cn(0)\n0)E\n+D\n[\u001e(1)\n0;x2(M0+\u001c=2L0)]E\n\u0000\u0017\u0012D\n@xfV(0)\n0\u0000@x[\u001e(0)\n0+\u001cn(0)\n0(1\u0000c\u0012)]gE\n+\n+\u001c=2D\n[n(1)\n0;x2M0]E\n+\u001a2nD\n[\u001e(1)\n0;@2\nx(\u001e(0)\n1+\u001c=2n(0)\n1)]E\n+D\n[\u001e(1)\n0;x4(M000+\u001c=2L000) + 6x2(M00+\u001c=2L00)]E\n+\n+\u001c=2D\n[n(1)\n0;@2\nx\u001e(0)\n1]E\n+\u001c=4D\n[n(1)\n0;x4L000+ 6x2L00]Eo\n= 0\n(57)\nEqs.57 neglect the FLR corrections to the neoclassical \row damping. After many mathematical steps, the\nsystem of equations reduces to\nd\nd \u0000\nL\nx2\u000b\u0001\n= 0;d\nd \u0012\nx2\u000bdV0\nd \u0013\n= 0\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\u0012D\n\u0016ML0+V00\n0\u0013D\nfx2fx2E\n\u0000\n\u0000\u0017\u0012\n\u0016\b\n(V0\u0000V1+Vp+\u001c(1\u0000c\u0012)) + [M+\u001cL(1\u0000c\u0012)]\nx2\u000b\t\n= 0(58)\nwhere we used V0=V(0)\n0. Note that the solution of the \frst equation, compatible with the boundary condition\nV!V1, is [11]:\nL=\u00001=\nx2\u000b\n(59)\nThe solution for V0, compatibly with the boundary condition V0!V1, isV0=V1.\nWe can neglect the island-induced \row damping in the torque-balance condition Eq.55, which becomes:\nZ+1\n\u00001d f(V0\u0000V1+Vp+\u001c(1\u0000c\u0012))h1i+ (M+\u001cL(1\u0000c\u0012))g= 0 (60)\nNow we consider the fact that \u001e0andn0are \rux functions, so that they must have the same x-symmetry of\nthe \rux function  , which is an even function. However, for the tearing symmetry, both \u001e0andn0are even in\nrespect tox. The only way to solve this contradiction is by imposing that they must be zero inside the separatrix\n[11], that is for\u00001< < 1. By using this result in the parallel momentum equation, we \fnd out that V0=\u0000Vp\ninside the separatrix. Furthermore, the quantity c\u0012which drives the intrinsic poloidal rotation, depends on the\ntemperature gradient and it is thus zero inside the separatrix. Outside the separatrix, instead, the system of\nequations Eq.58 hold. The \frst two equations bring to the solutions we have already seen L=\u00001=\nx2\u000b\nand\nV0=V1. SinceV0is a constant, the relation V0=V1holds in all the region 1 <  < +1. By imposing\nthatV0is continuous across the separatrix, we also \fnd that Vp=\u0000V1. By putting these results in the torque\nbalance condition Eq.60, we \fnd an equation for Vp, whose solution is\nVp=\u0000\u001c\u0014\n1\u0000c\u0012\nI1(I2\u0000I3)\u0015\n\u00001\nI1Z+1\n1d (M+\u001cL) (61)\nwhere we introduced the quantities I1=R+1\n\u00001d h1i,I2=R+1\n1d h1iandI3=\u0000R+1\n1d L. The remaining\nunknown function Mmust be determined by solving the following equation:\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000D\n\u0016\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\nML0D\nfx2fx2E\n\u0000\u0017\u0012\n\u0016(Vp+M\nx2\u000b\n) = 0 (62)\nwhereH=M(L\u0000M)+\u000b2(1+\u001c). By solving Eq.62 with the boundary condition M!Vp=p2 and computing\nVpby using Eq.61 iteratively, we can obtain the radial pro\fle of Mand the phase velocity Vp.\n6 Intermediate damping regime\nIn the intermediate-damping regime, the following ordering holds:\n1\u001d\u0017\u0012\u001dD;\u0016;\u0011;\u0017? (63)\n11In this case, we have to keep the terms where the product between \u001a2and the poloidal damping coe\u000ecient \u0017\u0012\nappear, together with the non-axisymmetric contributions \u0017?. By neglecting again the products \u001a2D,\u001a2\u0016and\n\u001a2\u0017\u0012(\u000f=q)2, the set of surface-averaged equations becomes:\nd\nd \u0012\nx2\u000bdV0\nd \u0013\n\u0000\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\nf(V0+Vp)h1i+ [M+\u001cL(1\u0000c\u0012)]g= 0\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000\u0017\u0012\n\u0016(V0\u0000V1+Vp+M\nx2\u000b\n)\u0000\u0017?\n\u0016(Vp+M\nx2\u000b\n)\u0000\n\u0000\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\"\u0012D\n\u0016ML0+V00\n0\u0013D\nfx2fx2E\n+\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\n[M+\u001cL(1\u0000c\u0012)]D\nfx2~xE#\n+\n+\u001a2\u0017\u0012\n\u0016\u001c\nxd\nd \u0014A\n\u001cHfx2\u0000\u00121\n2\u0000\u001cc2\u0013\nM0x2\u0000B\nHfx2(1\u0000c\u0012) +\u001cc1L0x2\u0015\u001d\n= 0(64)\nwhereH\u0011V0\n0+M(L\u0000M) +\u000b2(1 +\u001c),A\u0011M2(M0+\u001c=2L0) +\u001c=2MLM0\u0000\u000b2(1 +\u001c) (M0+\u001c=2L0) +\n\u001c=2(V0\n0M0+\u000b2L0)\u0000M0andB\u0011A+H(M0+\u001c=2L0). In the torque balance, we have to include the island-\ninduced damping terms proportional to \u0017?. In the internal region, the same considerations hold as before. In\nthe external region, however, we can impose the further condition that the plasma velocity tends to the intrinsic\npoloidal velocity far from the island. This condition is equivalent to imposing that the poloidal \row damping\ntends to zero far from the island, that is:\n\u0017\u0012fV+Vp\u0000@x[\u001e+\u001cn(1\u0000c\u0012)]g ! 0 (65)\nBy imposing the boundary conditions, Eq.65 becomes V1=\u001c(1\u0000c\u0012). By using this result, the torque-balance\ncondition Eq.55 becomes an equation for Vp, whose solution is\nVp=\u0000\u001c\u0014\n1\u0000\u0017\u0012c\u0012+\u0017?c?\n(\u0017\u0012+\u0017?)I1(I2\u0000I3)\u0015\n\u0000\u0017\u0012\n(\u0017\u0012+\u0017?)I1Z+1\n1d h1i(\u001c(1\u0000c\u0012)\u0000V0)\u00001\nI1Z+1\n1d (M+\u001cL) (66)\nIn the limit \u0017?!0 andV0=\u001c(1\u0000c\u0012), which is the case of the weak-damping regime, Eq.66 reduces to Eq.61.\nThe remaining unknown functions V0andMmust be determined by solving the system Eqs.64. By taking a\nfew more steps, the term containing the FLR corrections to the poloidal \row damping can be written more\nexplicitly, so that Eqs.64 become:\nd\nd \u0012\nx2\u000bdV0\nd \u0013\n\u0000\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\nf(V0+Vp)h1i+ [M+\u001cL(1\u0000c\u0012)]g= 0\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000\u0017\u0012\n\u0016(V0+Vp\u0000\u001c(1\u0000c\u0012) +M\nx2\u000b\n)\u0000\u0017?\n\u0016(Vp+M\nx2\u000b\n)\u0000\n\u0000\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\"\u0012D\n\u0016ML0+V00\n0\u0013D\nfx2fx2E\n+\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\n[M+\u001cL(1\u0000c\u0012)]D\nfx2~xE#\n+\n+\u001a2\u0017\u0012\n\u0016d\nd \u0014\u00121\n\u001cH[A+\u001cB(1\u0000c\u0012)] (1\u0000h1i)\u0000\u00121\n2\u0000\u001cc2\u0013\nM0+\u001cc1L0\u0013\nx2\u000b\u0015\n= 0(67)\nwhereH=V0\n0+M(L\u0000M) +\u000b2(1 +\u001c),AandBhave been de\fned above. Just as in the weak-damping case,\nby solving Eq.67 with the boundary conditions M!Vp=p2 ,V0!\u001c(1\u0000c\u0012) and computing Vpby using\nEq.66 iteratively, we can obtain the radial pro\fles of MandV0and the phase velocity Vp.\n6.1 Further simpli\fcation\nEqs.67 can be further simpli\fed by considering the limit of small-Larmor-radius, \u001a2!0, and the ordering\nEq.63. By using these simpli\fcations, V0outside the separatrix can be deduced by solving the second equation\nof Eqs.67:\nV0=\u001c(1\u0000c\u0012)\u0000(Vp+M\nx2\u000b\n)\u0017\u0012+\u0017?\n\u0017\u0012(68)\nFor the reasons explained above, V0=\u0000Vpinside the separatrix. Substituing Eq.68 in Eq.55\nVp=\u0000\u001c\u0014\n1 +(\u0017\u0012+\u0017?)I2\n\u0017?(I1\u0000I2)\u0000(\u0017\u0012c\u0012+\u0017?c?)\n\u0017?(I1\u0000I2)(I2\u0000I3)\u0015\n\u0000(\u0017\u0012+\u0017?)\n\u0017?(I1\u0000I2)Z+1\n1d [M(1\u0000\nx2\u000b\nh1i) +\u001cL] (69)\n12By substituing Eq.68 in Eqs.67, we get:\n\u0017\u0012+\u0017?\n\u0017\u0012d\nd \u0014\nx2\u000bd\nd \u0000\nM\nx2\u000b\u0001\u0015\n\u0000\u0017?\n\u0016\u0012\u000f\nq\u00132\n[Vp+M\nx2\u000b\n]h1i+\n+\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\b\nM(1\u0000\nx2\u000b\nh1i) +\u001c(1\u0000c\u0012)(h1i+L)\t\n= 0(70)\nBy solving simultaneously Eq.69 and Eq.70, with the appropriate boundary conditions, we can obtain the radial\npro\fle ofMand the phase velocity Vp. The systems of equations Eq.61+Eq.62 and Eq.69+Eq.70 represent\nlimit cases which can be easily solved numerically, and they can be both deduced from Eq.66+Eq.67 under\nappropriate limits. To further simplify the calculations, we introduce the variable k=p\n(1\u0000 )=2. Then we\nde\fne the following quantities, E=\nx2\u000b\n=(2k),F= 2kh1iand we de\fne the variable Qso as to absorb the\nfactorVp+\u001c(1\u0000c\u0012):\nQ=\u001c(1\u0000c\u0012)\u00002kEM\n\u001c(1\u0000c\u0012) +Vp(71)\nBy using these quantities, Eq.69 and Eq.70 become:\nVp=\u0000\u001c\u0014\n1 +I6\nI4+I5(c\u0012\u0000c?)\u0000c\u0012I5\nI4+I5\u0015\n(72)\n\u0017\u0012+\u0017?\n4\u0017\u0012d\ndk\u0014\nEdQ\ndk\u0015\n\u0000\u0017?\n\u0016\u0012\u000f\nq\u00132\nF(Q\u00001)\u0000\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\nQ\u0012\nF\u00001\nE\u0013\n= 0 (73)\nwhereI4=R1\n02kh1idk,I5= (\u0017\u0012+\u0017?)=\u0017?R+1\n1Q(F\u00001=E)dkandI6=R+1\n1(F\u00001=E)dk.\n7 Numerical results\nIt is possible to determine the magnetic islands rotation velocity by numerically integrating the system of\nequations consisting in Eqs.61,62 for the weak damping regime and Eqs.72,73 for the intermediate damping\nregime, in the limit of small Larmor radius. Unfortunately, the numerical integration of Eqs.66,67 still represents\na challenge too di\u000ecult to solve. We leave the analysis of the more general case to a future work.\nThe weak damping regime is characterized by the parameters \u000b, which enters the denominator H, and\u0017\u0012=\u0016.\n\u000bis proportional to the ratio between the island width wand the ion-acoustic radius \u001as, and it is a measure\nof the importance of the ion-acoustic waves on the \rattening of the density pro\fle inside the separatrix. \u0017\u0012=\u0016\ndepends on the plasma collisionality and measures the importance of the poloidal \row damping. The result of\nthe numerical integration of Eqs.61,62 for di\u000berent values of \u000band\u0017\u0012=\u0016is displayed in Fig.1. The presence of\nFigure 1: Island phase velocity Vpversus the ion-acoustic parameter \u000bfor di\u000berent choices of the poloidal \row\ndamping parameter \u0017\u0012=\u0016\nthe resonant denominator Hin Eq.62 prevents the solution from converging for the smaller values of \u000b, which\n13correspond to the small island width limit. The interesting feature of Fig.1 is the transition of the value of Vp\nfrom positive values to negative values as the parameter \u0017\u0012=\u0016is increased from values much smaller than one to\nvalues close to one. A positive phase velocity corresponds to an island rotating in the direction of the electron\n\ruid, while a negative value corresponds to the direction of the ion \ruid. The hypotheses of zero equilibrium\nelectric \feld means that the E^Bdrift has been subtracted from the plasma velocity.\nThe intermediate damping regime, in the limit of small Larmor radius, is characterized by the parameters\nw=\u001as, which enters the perpendicular damping coe\u000ecient, and \u0017i, which enters both the poloidal and the\nperpendicular damping coe\u000ecients. From Eq.72, it is evident that the island phase velocity Vpis determined\nby the neoclassical velocities c\u0012andc?, which are proportional to the radial temperature gradient through the\nparameter\u0011i=Ln=LT. The result of the numerical integration of Eqs.61,62 for di\u000berent values of w=\u001asand\u0017i\nis displayed in Figs.2,3,4 for the choices of \u0011i= 0:5,\u0011i= 1 and\u0011i= 2.\nFigure 2: Island phase velocity Vpversus the normalized island width w=\u001asfor di\u000berent choices of the collision\nfrequency\u0017iwith\u0011i= 0:5\nFigure 3: Island phase velocity Vpversus the normalized island width w=\u001asfor di\u000berent choices of the collision\nfrequency\u0017iwith\u0011i= 1\nBecause of the absence of a resonant denominator, the solution converges even in the limit of small island\nwidth. However, the equations we integrate are valid only in the limit of small Larmor radius, so that the results\nlose validity for w=\u001as<1. The interesting feature of these pictures is the transition of the value of Vpfrom\nnegative values to positive values as the parameter \u0011iis increased from values less than one to values larger than\none. Note that LnandLTare of the same order in realistic tokamak plasmas, so that very large values and\nvery small values of \u0011are unrealistic. The di\u000berent slopes of the curves corresponding to the di\u000berent values of\n14Figure 4: Island phase velocity Vpversus the normalized island width w=\u001asfor di\u000berent choices of the collision\nfrequency\u0017iwith\u0011i= 2\n\u0017ishow that, the smaller the collisionality, the more e\u000bective the neoclassical \row damping is in relaxing the\nisland velocity towards the neoclassical value, which is determined by the parameters c\u0012andc?. Note that the\nnumerical integration of Eqs.61,62 for the weak damping regime was performed with the choice \u0011i= 1, but the\nresults are not signi\fcantly a\u000bected by the choice of \u0011i. In all our integrations we chose \u001c= 1.\n8 Conclusions\nIn this paper we addressed the issue of determining the phase velocity of a chain of freely rotating magnetic\nislands by using a four \feld gyro\ruid system of equations which includes the neoclassical \row damping e\u000bects\nand the lowest order FLR corrections. To do that, we \frst solved the gyrokinetic equation under some simplifying\nhypotheses and we computed the FLR corrections to the poloidal \row damping. Then we deduced a four \feld\ngyro\ruid model by starting from a set of gyro\ruid equations and we closed it by using a simpli\fed form for the\ndivergence of the stress tensor, which provides the neoclassical \row damping e\u000bects. By following the method\ndescribed by Fitzpatrick & Waelbroek [8, 9, 10, 11], we managed to obtain a system of equations whose solution\nprovides the islands rotation velocity consistently with the \felds radial pro\fles close to the resonant surface.\nWe applied this system of equations to the investigation of two collisionality regimes, namely the weak damping\nregime and the intermediate damping regime. In the second case, which corresponds to the low collisionality\nregime, an additional term, containing the lowest order FLR corrections to the poloidal \row damping, appeared\nin the equations. The numerical integration of Eqs.61,62 in the weak damping regime shows that the island\nphase velocity moves from positive values to negative values as the poloidal damping parameter \u0017\u0012=\u0016is increased\nfrom values much smaller than one to values close to one. A positive phase velocity is associated with a magnetic\nisland rotating in the direction of the electron \ruid, while negative values means that the island rotates in the\ndirection of the ions. The numerical integration of Eqs.72,73 in the intermediate damping regime shows that\nthe phase velocity Vpmoves from negative values to positive values as the parameter \u0011iis increased from values\nless than one to values larger than one. These results are in agreement with what was already known about\nthe subject, but they are valid only within the limitations of their hypotheses. Unfortunately, the numerical\nintegration of Eqs.66,67 still represents a challenge too di\u000ecult to solve. We leave the analysis of the more\ngeneral case to a future work. In the case of large, saturated islands, we expect \u001a2to be small, so that the\nFLR corrections we found should be small as well. 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Cambridge University\nPress, 2005.\n[16] J. Kevorkian and J. D. Cole, Multiple scale and singular perturbation methods , vol. 114. Springer Science\n& Business Media, 2012.\n[17] F. Waelbroeck, J. Connor, and H. Wilson, \\Finite larmor-radius theory of magnetic island evolution,\"\nPhysical review letters , vol. 87, no. 21, p. 215003, 2001.\n[18] M. Siccinio, E. Poli, W. Hornsby, and A. Peeters, \\Gyrokinetic investigation of magnetic islands in toka-\nmaks,\" in Journal of Physics: Conference Series , vol. 260, p. 012019, IOP Publishing, 2010.\n[19] S. Hirshman and D. Sigmar, \\Neoclassical transport of impurities in tokamak plasmas,\" Nuclear Fusion ,\nvol. 21, no. 9, p. 1079, 1981.\n[20] K. Zhukovsky, \\Inverse derivative and solutions of some ordinary di\u000berential equations,\" Applied Mathe-\nmatics , vol. 2, no. 2, pp. 34{39, 2012.\n[21] P. B. Snyder, Gyro\ruid theory and simulation of electromagnetic turbulence and transport in tokamak\nplasmas . PhD thesis, Citeseer, 1999.\n16[22] X. Xu, P. Xi, A. Dimits, I. Joseph, M. Umansky, T. Xia, B. Gui, S. Kim, G. Park, T. Rhee, et al. , \\Gyro-\n\ruid and two-\ruid theory and simulations of edge-localized-modesa),\" Physics of Plasmas (1994-present) ,\nvol. 20, no. 5, p. 056113, 2013.\n[23] L. Comisso, D. Grasso, E. Tassi, and F. Waelbroeck, \\Numerical investigation of a compressible gyro\ruid\nmodel for collisionless magnetic reconnection,\" Physics of Plasmas (1994-present) , vol. 19, no. 4, p. 042103,\n2012.\n[24] F. L. Waelbroeck and E. Tassi, \\A compressible hamiltonian electromagnetic gyro\ruid model,\" Communi-\ncations in Nonlinear Science and Numerical Simulation , vol. 17, no. 5, pp. 2171{2178, 2012.\n[25] K.-C. Shaing, \\Plasma \row and con\fnement in the vicinity of a rotating island in collisional tokamak\nplasmas,\" Physics of Plasmas (1994-present) , vol. 11, no. 2, pp. 625{632, 2004.\n[26] K. Shaing, T. Tsai, M. Chu, and S. Sabbagh, \\Theory for island induced neoclassical toroidal plasma\nviscosity in tokamaks,\" Nuclear Fusion , vol. 51, no. 4, p. 043013, 2011.\n[27] R. Fitzpatrick and F. Waelbroeck, \\Spontaneous healing and growth of locked magnetic island chains in\ntoroidal plasmas,\" Physics of Plasmas (1994-present) , vol. 19, no. 11, p. 112501, 2012.\n[28] R. Fitzpatrick and F. Waelbroeck, \\E\u000bect of drift-acoustic waves on magnetic island stability in slab\ngeometry,\" Physics of Plasmas (1994-present) , vol. 12, no. 12, p. 122511, 2005.\n17"    },    {        "title": "1111.3913v2.Three_player_quantum_Kolkata_restaurant_problem_under_decoherence.pdf",        "content": "arXiv:1111.3913v2  [quant-ph]  17 Nov 2011Three-player quantum Kolkata restaurant problem under dec oherence\nM. Ramzan\nDepartment of Physics Quaid-i-Azam University\nIslamabad 45320, Pakistan\n(Dated: June 13, 2018)\nEffect of quantum decoherence in a three-player quantum Kolkata restaurant problem\nis investigated using tripartite entangled qutrit states. Amplitude d amping, depolarizing,\nphase damping, trit-phase flip and phase flip channels are considere d to analyze the\nbehaviour of players payoffs. It is seen that Alice’s payoff is heavily infl uenced by the\namplitude damping channel ascomparedto the depolarizingand flippin g channels. However,\nfor higher level of decoherence, Alice’s payoff is strongly affected b y depolarizing noise.\nWhereas the behaviour of phase damping channel is symmetrical ar ound 50 % decoherence.\nIt is also seen that for maximum decoherence ( p= 1),the influence of amplitude damping\nchannel dominates over depolarizing and flipping channels. Whereas , phase damping\nchannel has no effect on the Alice’s payoff. Therefore, the problem becomes noiseless\none at maximum decoherence in case of phase damping channel. Furt hermore, the Nash\nequilibrium of the problem does not change under decoherence.\nKeywords: Decoherence; qutrit channels; Kolkata restaura nt problem.\nI. INTRODUCTION\nGame theory a branch of applied mathematics, was initially d eveloped for use in economics\nby von Neumann and Morgenstern [1] and important contributi ons were given by John Nash [2].\nIt attempts to capture mathematical behavior in strategic s ituations, in which an individual’s\nsuccess of making a choice depends on the choice of the other p layers. It is usually used to model\nthe behavior of biological, economical and computer system s. During last few years, a number of\nclassical games has been converted into the realm of quantum mechanics [3-18]. Recently, Miszczak\net al. [19] has studied a qubit flip game on a Heisenberg spin ch ain. They have shown that being2\nwell aware of the dimensionality of the system, a player can a chieve a mean payoff equal to almost\n1. More recently, Sharif et al. [20] has proposed the quantum solution to a three-player Kolkata\nRestaurant problem. The Kolkata Paise Restaurant (KPR) [21 ] is a repeated game similar to the\nMinority games, played between a large number of agents havi ng no interaction among themselves.\nSincequantumminority games [22-25] hasattracted much att ention in recent years. Theyhave also\nbeen analyzed under the influence of decoherence by Flitney a nd Hollenberg [26]. It is therefore,\nimportant to study the behaviour of quantum restaurant prob lem in the presence of environmental\ninfluences.\nSince, itis notpossibletocompletely isolate aquantumsys temfromits environment. Therefore,\nit is important to study the system-environment dynamics in the presence of environmental effects.\nQuantum games may provide a feasible platform for implement ing quantum information processing\nin physical systems [27] and can be used to probe the influence of decoherence in such systems\n[6, 28-31]. In this connection, quantum channels provide a n atural theoretical framework for the\nstudy of decoherence in noisy quantum communication system s. Quantum error correction [32-33]\nand entanglement purifications [34] can be employed to avoid the problem of decoherence.\nIn this paper, the effect of quantum decoherence in a three-pla yer quantum Kolkata restaurant\nproblem is studied using entangled qutrit states. By consid ering different noisy qutrit channels\nparameterized by decoherence parameter psuch that p∈[0,1]. The lower and upper limits of\ndecoherence parameter represent the fully coherent and ful ly decohered system, respectively. It is\nseen that for lower level of decoherence, amplitude damping channel heavily influences the payoffs\nas compared to the depolarizing and flipping channels. Howev er, for higher level of decoherence,\nthe payoff is strongly affected by depolarizing noise. On the ot her hand, the behaviour of phase\ndamping channel is symmetrical around 50 % decoherence. Fur thermore, the Nash equilibrium of\nthe problem does not change under decoherence.\nII. DECOHERENCE AND QUANTUM KOLKATA RESTAURANT PROBLEM\nIn the Kolkata Paise Restaurant (KPR) problem, Nnon-communicating agents have to choose\nbetween nchoices. The agents receive a gain in their utility if their c hoice is not too crowded,\ni.e. the number of agents that made the same choice is under so me threshold limit. The choices\ncan also have different values of utility associated with them , accounting for a preference profile\nover the set of choices. Therefore, in KPR, Nprospective customers choose from Nrestaurants\neach evening in a parallel decision mode. Each restaurant ha ve identical price but different rank k3\n(agreed by the all the Nagents) and can serve only one customer. If more than one agen ts arrive\nat any restaurant on any evening, one of them is randomly chos en and is served and the rest do\nnot get dinner that evening.\nFor the sake of simplicity, let the three agents, Alice, Bob a nd Charlie have three possible\nchoices: security 0, security 1 and security 2. They receive a payoff $ = 1 if their choice is unique,\notherwise they receive $ = 0. Therefore, the game is a one shoo t game, that is, it is a non-iterative,\nand the agents have no information from previous rounds. Sin ce the agents cannot communicate,\ntherefore, there is nothing left to do other than randomizin g between the choices. Randomization\ngives the agent ian expected payoff of Ec($) = 4/9, where the superscript crepresents the classical\nstrategy.\nIn this problem, let Alice, Bob and Charlie share a general tr ipartite entangled qutrit state of\nthe form\nρin=f|Ψin/an}bracketri}ht/an}bracketle{tΨin|+(1−f)\n27I27 (1)\nwhere the parameter fcontrols the degree of entanglement and\n|Ψin/an}bracketri}ht=1√\n3(|000/an}bracketri}ht+|111/an}bracketri}ht+|222/an}bracketri}ht) (2)\nIn order to analyze the effect of entanglement, another genera l initial state is also considered as\ngiven below\n|Ψin/an}bracketri}ht= sinθcosφ|000/an}bracketri}ht+sinθsinφ|111/an}bracketri}ht+cosθ|222/an}bracketri}ht (3)\nwhere 0 ≤θ≤πand 0≤θ≤2π.If we set θ=π/4,3π/4 andφ=±cos−1(1/√\n3) in the above\nequation, the three-qutrit state becomes the maximally ent angled state. The strategies of the\nplayers can be defined by the unitary operator Uacting on the initial qutrit state of the problem\ngiven as [35]\nU=\nz1¯ω1¯z2ω3−¯z3ω2\nz2¯ω2¯z3ω1−¯z1ω3\nz3¯ω3¯z1ω2−¯z2ω1\n(4)\nwhere\n− →z=\nsinθcosφeiα1\nsinθsinφeiα2\ncosθeiα3\n(5)4\nand\n− →ω=\ncosχcosθcosφei(β1−α1)+sinχsinφei(β2−α1)\ncosχcosθsinφei(β1−α2)−sinχcosφei(β2−α2)\n−cosχsinθei(β1−α3)\n(6)\nwhere 0≤χ≤π/2 and 0≤β1,β2≤2π.After the action of players unitary operators the state of\nthe game transform to\nρ´f= (U†\nA⊗U†\nB⊗U†\nC)(|Ψin/an}bracketri}ht/an}bracketle{tΨin|)(UA⊗UB⊗UC) (7)\nThe evolution of the state of a quantum system in a noisy envir onment can be described by the\nsuper-operator Φ in the Kraus operator representation as [1 ]\n˜ρf= Φρf=/summationdisplay\nkEkρfE†\nk(8)\nwhere the Kraus operators Eisatisfy the following completeness relation\n/summationdisplay\nkE†\nkEk=I (9)\nWe have constructed the Kraus operators for the game from the single qutrit Kraus operators (as\ngiven in equations (9-11) below) by taking their tensor prod uct over all n2combination of π(i)\nindices\nEk=⊗\nπeπ(i) (10)\nwherenis the number of Kraus operators for a single qutrit channel. The single qutrit Kraus\noperators for the amplitude damping channel are given by [36 ]\nE0=\n1 0 0\n0√1−p0\n0 0√1−p\n, E1=\n0√p0\n0 0 0\n0 0 0\n, E2=\n0 0√p\n0 0 0\n0 0 0\n(11)\nSimilarly, the single qutrit Kraus operators for the phase d amping channel are given as [36]5\nE0=/radicalbig\n1−p\n1 0 0\n0 1 0\n0 0 1\n, E1=√p\n1 0 0\n0ω0\n0 0ω2\n, (12)\nwhereω=e2πi\n3.The single qutrit Kraus operators for the depolarizing chan nel are given by [37]\nE0=/radicalbig\n1−pI, E1=/radicalbiggp\n8Y, E2=/radicalbiggp\n8Z, E3=/radicalbiggp\n8Y2, E4=/radicalbiggp\n8YZ (13)\nE5=/radicalbiggp\n8Y2Z, E6=/radicalbiggp\n8YZ2, E7=/radicalbiggp\n8Y2Z2, E8=/radicalbiggp\n8Z2(14)\nwhere\nY=\n0 1 0\n0 0 1\n1 0 0\n, Z=\n1 0 0\n0ω0\n0 0ω2\n(15)\nThe single qutrit Kraus operators for the phase flip channel a re given by\nE0=\n1 0 0\n0√1−p0\n0 0√1−p\n, E1=\n0√p0\n0 0 0\n0 0 0\n, E2=\n0 0√p\n0 0 0\n0 0 0\n(16)\nand the single qutrit Kraus operators for the trit-phase flip channel are given by\nE0=/radicalbigg\n1−2p\n3\n1 0 0\n0 1 0\n0 0 1\n, E1=/radicalbiggp\n3\n0 0 e2πi\n3\n1 0 0\n0e−2πi\n30\n,\nE2=/radicalbiggp\n3\n0e−2πi\n30\n0 0 e2πi\n3\n1 0 0\n, E3=/radicalbiggp\n3\n0e2πi\n30\n0 0e−2πi\n3\n1 0 0\n(17)\nwherep= 1−e−Γtrepresents the quantum noise parameter usually termed as de coherence param-\neter. Here the bounds [0 ,1] ofpcorrespond to t= 0,∞respectively. The final state of the game6\nafter the action of the channel can be written as\nρf= Φp(ρf) (18)\nwhere Φ αis the super-operator realizing the quantum channel parame trized by the real number p\n(decoherence parameter). The payoff operator for ithplayer (say Alice) can be written as\nPA=2/summationdisplay\nx3,x2,x1=0|x3x2x1/an}bracketri}ht/an}bracketle{tx3x2x1|,x3/ne}ationslash=x2/ne}ationslash=x1\n+2/summationdisplay\nx3,x2,x1=0|x3x2x1/an}bracketri}ht/an}bracketle{tx3x2x1|,x3=x2/ne}ationslash=x1\n(19)\nThe payoff of ithplayer can be calculated as\nEi($) = Tr{PA˜ρf} (20)\nwhere Tr represents the trace of the matrix. The optimal stra tegy for players is found to be Uopt\ngiven by\nUopt(θ,φ,χ,α 1,α2,α3,β1,β2) = (π\n4,cos−1(1/√\n3),π\n4,5π\n18,5π\n18,5π\n18,π\n3,11π\n6)\n(21)\nIt is seen that the results are consistent with Ref. [20] for p= 0.\nIn order to interpret the effect of decoherence on the three-pl ayer quantum Kolkata restaurant\nproblem, different graphs has been plotted as a function of dec oherence parameter. In figure (1),\nAlice’s payoff is plotted as a function of decoherence parame terpfor (a)f= 0.2,(b)f= 0.5,\n(c)f= 1 and θ=π\n4, φ= cos−1(1/√\n3) for amplitude damping, depolarizing, phase damping,\ntrit-phase flip and phase flip channels, where AD, Dep, PD, TPF and PF represent the amplitude\ndamping, depolarizing, phase damping, trit-phase flip and p hase flip channels respectively. It is\nseen that Alice’s payoff is heavily influenced by the amplitud e damping channel as compared to\nthe depolarizing and flipping channels. In figures (2 and 3), A lice’s payoff is plotted as a function\nofθandφforp= 0.3 andp= 0.7 (a) amplitude damping, (b) phase damping, (c) depolarizin g\nand (d) trit-phase flip channels, respectively, for the stat e of equation (3). It is seen that for\nhigher level of decoherence (see figure 3), Alice’s payoff is s trongly affected by depolarizing noise.\nWhereas the behaviour of phase damping channel remains symm etrical around 50 % decoherence.\nIn figures (4), Alice’s payoff is plotted as a function of θandφforp= 1 (a) amplitude damping, (b)7\nphase damping, (c) depolarizing and (d) trit-phase flip chan nels, respectively. It is shown that for\nmaximum decoherence i.e. p= 1,amplitude damping channel dominates over the depolarizing and\nflippingchannels having considerable reduction in the payo ff. Whereas, phase dampingchannel has\nno effect on the Alice’s payoff. In case of phase damping channel, the problem becomes noiseless\nat maximum decoherence. However, maximal entanglement giv es the maximum payoff (6 /9 at\np= 0) and it reduces as one changes the degree of entanglement f rom its maxima or introduces\nthe value of decoherence parameter p >1. Furthermore, the Nash equilibrium of the problem does\nnot change under decoherence.\nIII. CONCLUSIONS\nQuantumthree-player Kolkata restaurant problem is invest igated in thepresenceof decoherence\nusingtripartiteentangledqutritstatesusingamplituded amping,depolarizing, phasedamping, trit-\nphase flip and phase flip channels. It is seen that for lower lev el of decoherence, amplitude damping\nchannel heavily influences the payoffs as compared to the depol arizing and flipping channels. How-\never, for higher level of decoherence, the payoff is strongly affected by depolarizing noise. It is\nalso seen that for maximum level of decoherence, amplitude d amping channel dominates over the\ndepolarizing and flipping channels. Whereas, phase damping channel has no effect on the Alice’s\npayoff at p= 1. Therefore, the problem becomes noiseless at maximum dec oherence for phase\ndamping channel only. Furthermore, the Nash equilibrium of the problem does not change under\ndecoherence.\n[1] vonNeumann JandMorgensteinO,Thetheoryofgamesandecon omicbehaviour,PrincetonUniversity\nPress, Princeton, NJ (1944).\n[2] Nash J, Equilibrium points in n-person games, Proc. National Acad emy of Science 36 p. 48 (1950).\n[3] Meyer D A 1999 Phys. Rev. Lett. 821052\n[4] Eisert J, Wilkens M and Lewenstein M 1999 Phy. Rev. Lett. 833077\n[5] Marinatto L and Weber T 2000 Phys. Lett. A 272291\n[6] Flitney A P and Abbott D 2005 J. Phys. A 38449\n[7] Cheon T and Iqbal A 2008 J. Phys. Soc. Japan 77024801\n[8] Iqbal A and Abbott D 2009 J. Phy. Soc. Japan 78014803\n[9] Iqbal A, Cheon T and Abbott D 2008 Phys. Lett. A 3726564\n[10] Iqbal A and Toor A H 2001 Phys. Lett. A 2802498\n[11] Flitney A P and Abbott D 2002 Phys. Rev. A 65062318\n[12] Iqbal A and Toor A H 2002 Phys. Rev. A 65022036\n[13] Eisert J and Wilkens M 2000 J. Mod. Opt. 472543\n[14] Ramzan M and Khan M K 2008 J. Phys. A: Math. Theor. 41435302\n[15] Flitney A P, Ng J and Abbott D 2002 Physica A 31435\n[16] Iqbal A and Toor A H 2002 Phys. Lett. A 293103\n[17] Johnson N F 2001 Phys. Rev. A 63020302(R)\n[18] D’Ariano G M, Gill R D, Keyl M, Kuemmerer B, Maassen H and Werner R F 2002 Quant. Inf. Comp.\n2355\n[19] Miszczak J A, Gawron P and Puchala Z, (arXiv:1108.0642 [quant-p h], 2011).\n[20] Puya Sharif and Hoshang Heydari, (arXiv:1111.1962 [quant-ph], 2011).\n[21] Chakrabarti A S, Chakrabarti B K, Chatterjee A, Mitra M, Ph ysica A 388 (2009) 2420-2426.\n[22] Benjamin S C and Hayden P M 2001, Phys. Rev. A, 64, 030301(R).\n[23] Chen Q, Wang Y, 2004 Physics Letters A, A 327, 98,102\n[24] A. P. Flitney, A. Greentree, (Elsiever Science, feb 2008).\n[25] C. Schmid, A.P. Flitney, (arXiv:0901.0063v1 [quant-ph], 2008).\n[26] Flitney A P and Hollenberg L C L 2007 Quantum Inf. Comput. 7111\n[27] Pakula I, 2007 Phys. Stat. Sol. B 244(7), 2513\n[28] Ramzan M, Nawaz A, Toor A H and Khan M K 2008 J. Phys. A: Math. Theor.41055307\n[29] Gawron P and Sladkowski J 2008 Int. J. Quant. Info. 6667\n[30] Chen L K, Ang H, Kiang D Kwek L C and Lo C F 2003 Phys. Lett. A 316317\n[31] Ramzan M and Khan M K 2010 Quant. Inform. Process. 9, 667\n[32] Steane A 1996 Phys. Rev. Lett. 77793\n[33] Preskill J 1998 Proc. R. Soc. Lond. A 454385\n[34] Deutsch D, Ekert A, Josza R, MacchiavelloC, Popescu S and San pera A 1996 Phys. Rev. Lett. 772818\n[35] Mathur M and Sen D 2001 J.Math.Phys. 424181-4196.\n[36] Gregg jeager 2007 Journal of modern optics 541\n[37] Berg A 2002 IEEE Transaction on Information Theory 4830969\nFigures captions\nFigure 1 . (Color online). Alice’s payoff is plotted as a function of de coherence parameter pfor (a)\nf= 0.2,(b)f= 0.5,(c)f= 1 and θ=π\n4, φ= cos−1(1/√\n3) for amplitude damping, depolarizing,\nphase damping, trit-phase flip and phase flip channels.\nFigure 2 . (Color online). Alice’s payoff is plotted as a function of θandφforp= 0.3 (a)\namplitude damping, (b) phase damping, (c) depolarizing and (d) trit-phase flip channels for the\nstate of equation.\nFigure 3 . (Color online). Alice’s payoff is plotted as a function of θandφforp= 0.7 (a)\namplitude damping, (b) phase damping, (c) depolarizing and (d) trit-phase flip channels for the\nstate of equation.\nFigure 4 . (Color online). Alice’s payoff is plotted as a function of θandφforp= 1 (a) amplitude\ndamping, (b) phase damping, (c) depolarizing and (d) trit-p hase flip channels for the state of\nequation.10\n00.20.40.60.8 10.450.50.55\npPayoff(a) f=0.2\n  \nAD\nDep\nPD\nTPF\nPF\n00.20.40.60.8 10.450.50.550.6\npPayoff(b) f=0.5\n  \nAD\nDep\nPD\nTPF\nPF\n00.20.40.60.8 10.450.50.550.60.650.7\npPayoff(c) f=1\n  \nAD\nDep\nPD\nTPF\nPF\n00.20.40.60.8 10.450.50.550.60.650.7\npPayoff(d) θ=π/4 & φ=cos(1/sqrt(3))−1\n  \nAD\nDep\nPD\nTPF\nPF\nFIG. 1: (Color online). Alice’s payoff is plotted as a function of decohe rence parameter pfor (a)f= 0.2,\n(b)f= 0.5,(c)f= 1 and θ=π\n4, φ= cos−1(1/√\n3) for amplitude damping, depolarizing, phase damping,\ntrit-phase flip and phase flip channels.\n02\n050.40.50.6\nθ(a) Amplitude Damping \nφPayoff\n02\n050.460.480.50.52\nθ(b) Phase damping\nφPayoff\n02\n050.450.50.55\nθ(c) Depolarizing\nφPayoff\n02\n050.450.50.55\nθ(d) Trit−phase flip\nφPayoff\nFIG. 2: (Color online). Alice’s payoff is plotted as a function of θandφforp= 0.3 (a) amplitude damping,\n(b) phase damping, (c) depolarizing and (d) trit-phase flip channels for the state of equation.11\n02\n050.460.480.50.52\nθ(a) Amplitude Damping \nφPayoff\n02\n050.460.480.50.52\nθ(b) Phase damping\nφPayoff\n02\n050.50.510.52\nθ(c) Depolarizing\nφPayoff\n02\n050.50.510.52\nθ(d) Trit−phase flip\nφPayoff\nFIG. 3: (Color online). Alice’s payoff is plotted as a function of θandφforp= 0.7 (a) amplitude damping,\n(b) phase damping, (c) depolarizing and (d) trit-phase flip channels for the state of equation.\n0123\n050.460.4650.47\nθ(a) Amplitude Damping \nφPayoff\n0123\n050.40.50.6\nθ(b) Phase damping\nφPayoff\n0123\n050.50.5050.510.5150.52\nθ(c) Depolarizing\nφPayoff\n0123\n050.50.5050.510.5150.52\nθ(d) Trit−phase flip\nφPayoff\nFIG. 4: (Color online). Alice’s payoff is plotted as a function of θandφforp= 1 (a) amplitude damping,\n(b) phase damping, (c) depolarizing and (d) trit-phase flip channels for the state of equation."    },    {        "title": "2308.10412v1.Quasinormal_modes_of_the_bumblebee_black_holes_with_a_global_monopole.pdf",        "content": "arXiv:2308.10412v1  [gr-qc]  21 Aug 2023Quasinormal modes of the bumblebee black holes with a global\nmonopole\nZainab Malik1\n1Institute of Applied Sciences and Intelligent Systems, H-1 5, Pakistan\nAbstract\nWe engage in the computation of quasinormal modes associate d with scalar and Dirac fields\naround spherically symmetric black holes with a global mono pole in bumblebee gravity. The fre-\nquency of oscillation and the damping rate exhibit significa nt decreasing as the global monopole\nparameter is increased. An intriguing observation arises i n the extreme limit, where the quasi-\nnormal modes manifest a form of universal behavior: the actu al oscillation frequency remains\nunaltered despite variations in the Lorentz symmetry break ing (LSB) parameter. Our calculations\nare conducted through two distinct methods, both of which yi eld results that align remarkably\nwell. Furthermore, we derive an analytical formula for quas inormal modes within the eikonal\napproximation.\n1I. INTRODUCTION\nBumblebee Gravity is an unconventional theory proposing deviation s from Einstein’s\nGeneral Relativity, involving the violation of Lorentz invariance and in troducing additional\nfields, within the framework of effective field theory. This concept s uggests that gravity\nmight exhibit altered behavior under extreme conditions or at small s cales, connecting it to\nthe quest for a theory of quantum gravity. The theory’s implication s span cosmology, poten-\ntially explaining challenging observations, and it raises questions abou t dark matter, strong\ngravitational fields, and unresolved cosmological puzzles. The pro spect of experimental or\nobservational tests is a prominent feature of discussions surrou nding ”Bumblebee Gravity,”\nreflecting the dynamic and exploratory nature of theoretical phy sics. [1–10].\nIn the Bumblebee Gravity deviations from General Relativity sugges t that black holes\nstructure and surrounding accretion processes may differ from c onventional models. The\nintroduction of extra fields and violations of Lorentz invariance cou ld lead to modified dy-\nnamics near black holes, impacting phenomena like accretion disk form ation and radiation\nemission. Therefore, special attention was devoted to proper os cillation frequencies of black\nholes in this and other theories with broken Lorentz invariance - qua sinormal modes [ 11–26].\nQuasinormal modes (QNMs) of black holes are a fundamental aspec t of their behavior,\nunveiling unique insights into the vibrational properties of black holes [27–29]. In the realm\nof general relativity, QNMs represent the characteristic oscillatio ns that arise when a pertur-\nbation disturbs the equilibrium of a black hole, leading to damped, expo nentially decaying\nwaveforms. These modes serve as a signature of black hole spacet ime, revealing (together\nwith the electromagnetic spectra [ 30–32]) information about its mass, angular momentum,\nand charge. QNMs are pivotal in studying the stability and dynamics o f black holes, as\nwell as probing the nature of gravity and matter in extreme conditio ns. Analyzing QNMs\nfacilitates the detection and interpretation of gravitational wave s from black hole mergers,\noffering a powerful tool for testing theories of gravity and explor ing the properties of black\nhole spacetime in various astrophysical scenarios.\nOne of the aspects we are interested here is black hole solution with a global monopole,\nas proposed by Vilenkin [ 33], introduces a scenario where the presence of a topological\ndefect affects the geometry of spacetime near a black hole. This so lution, stemming from\nthe interplay between gravitational and scalar fields, sheds light on the intriguing coupling\n2between geometry and fundamental forces in certain theoretica l frameworks. While the\nquasinormal modes and grey-body factors of black holes in the Bum blebee Gravity has been\nstudied in a number of works, no such studies was proposed for the case of the additional\nfactor of the global monopole, except a recent study in [ 34] where the case of the scalar field\nwas considered.\nIn this analysis, we will thoroughly examine the scenario characteriz ed by asymptotic\nflatness. We accomplish this by investigating the propagation of sca lar and neutrino fields\nacross a range of parameter values, encompassing the quasi-ext remal regime. By doing so,\nwe corroborate and augment the findings outlined in a recent publica tion [34]. We ascertain\nquasinormal modes utilizing both the WKB method and time-domain inte gration, achieving\na notable alignment with results from the Leaver method.\nA notable observation is the pronounced decrease in the frequenc y of oscillation and\ndamping rate as the global monopole parameter is augmented. At th e point of extremal\nconditions, the quasinormal modes exhibit a type of uniform behavio r: the actual oscilla-\ntion frequency Re(ω) is unaffected by variations in the Lorentz symmetry breaking (LSB )\nparameter. Furthermore, we will derive an analytical formula for q uasinormal modes within\nthe eikonal approximation.\nThe structure of this paper is outlined as follows. In Section II, we d educe the wave-like\nequation and present the fundamental aspects of the methodolo gies utilized for identifying\nquasinormal modes. In Section III, we delve into a discussion regar ding the quasinormal\nmodes, and finally, we consolidate our findings within the Conclusions s ection.\nII. THE WAVE-LIKE EQUATION AND METHODS FOR STUDY QUASINOR-\nMAL MODES\nA. The metric\nThe spherically symmetric black hole has the line element\nds2=f(r)dt2−dr2\nh(r)−r2dθ2−r2sin2θdφ2(1)\nwheref(r) andh(r) are functions of the radial coordinate r. Here it is assumed that the\nbumblebee field is purely radial. Under this supposition the metric func tion describing a\n3black hole has the form [ 35]\nf(r) =1−κη2−2M\nr,\nh(r) =L+1\nf(r),(2)\nwhereL≡ξb2is the LSB parameter, and Mis the Arnowitt-Deser-Misner mass. The\nevent horizon is located at rh= 2M/(1−κη2), independently of the value of the bumblebee\nparameter.\nB. Wave-like equation\nWe will consider the test scalar field Φ which, thereby, is described by the covariant\nKlein-Gordon equation,\n1√−g∂µ/parenleftbig\ngµν√−g∂νΦ/parenrightbig\n= 0. (3)\nUsing the separation of variable,\nΦ =ψl(r)\nrYlm(θ,φ)e−iωt, (4)\nthedynamicalequationforthescalarfieldcanbereducedtothefo llowingwave-like equation:\nd2ψl\ndx2+/bracketleftbig\nω2−U(x)/bracketrightbig\nψl= 0. (5)\nHere, the effective potential is (see, for example, [ 36,37] for the deduction of the effective\npotential for the case of two metric functions)\nU(r) =f(r)l(l+1)\nr2+(f(r)h(r))′\n2r. (6)\nThe tortoise coordinate xis\ndx\ndr=√\n1+L\nf(r), (7)\nSimilarly, for the general covariant Dirac field we have\n[γaeµ\na(∂µ+Γµ)]Ψ = 0, (8)\nwhereγaare the Dirac matrices, eµ\nais the inverse of the tetrad ea\nµ(gµν=ηabea\nµeb\nν),ηabis the\nMinkowski metric. The spin connections Γ µare defined as follows:\nΓµ=1\n8[γa,γb]eν\naebν;µ. (9)\n4The above equation can be reduced to a couple of wave-like equation s with the following\neffective potentials\nV±1/2=/radicalBig\n˜f|l|\nr2/parenleftBig\n|l|/radicalBig\n˜f±r\n2d˜f\ndr∓˜f/parenrightBig\n, (10)\nwhere\n˜f(r) =f(r)/√\nL+1.\nC. Higher order WKB method\nThe main method we use here is the semi-analytic WKB method, applied f or the first\ntime for finding quasinormal modes by B. Schutz and C. Will in [ 38]. The Schutz-Will\nformula was then extended to higher orders [ 39–41] and made even more accurate when\nusing the Pad´ e approximants [ 41]. The general WKB formula has the form [ 42],\nω2=V0+A2(K2)+A4(K2)+A6(K2)+...−iK/radicalbig\n−2V2/parenleftbig\n1+A3(K2)+A5(K2)+A7(K2)+.../parenrightbig\n,\n(11)\nwhereK=n+ 1/2 is half-integer. The corrections Ak(K2) of the order kto the eikonal\nformulaarepolynomialsof K2withrationalcoefficients. Theydependonthevaluesofhigher\nderivatives of the effective potential V(r) in its peak. With the procedure of Matyjasek and\nOpala [41] using the Pad´ e approximants the whole procedure is even more ac curate. Here\nwe will use the sixth order WKB method with ˜ m= 4 and 5, where the definition of ˜ mand\nall details on this method can be found in [ 41,42]. Higher order WKB method has been\nused for finding quasinormal modes and grey-body factors in grea t number of publications\n( see for example [ 43–52] and references therein) and comparison with accurate, conver gent\nLeaver method proved that the WKB formula is sufficiently accurate in the majority of cases\n[40].\nD. Time-domain integration\nThe integration in the time domain that we employ is founded on the wav elike equation\nexpressed using the light-cone variables u=t−r∗andv=t+r∗. We implement the\ndiscretization approach introduced by Gundlach-Price-Pullin [ 53] as follows:\nΨ(N) = Ψ(W)+Ψ(E)−Ψ(S)−∆2V(S)Ψ(W)+Ψ(E)\n4+O/parenleftbig\n∆4/parenrightbig\n,(12)\n5κη26WKB 6WKB(Pade-5) 6WKB(Pade-6)\n0.0.479592-0.078865 i 0.479593-0.078864 i 0.479592-0.078864 i\n0.10.408788-0.063857 i 0.408789-0.063856 i 0.408788-0.063856 i\n0.20.342003-0.050437 i 0.342003-0.050436 i 0.342003-0.050436 i\n0.30.279447-0.038601 i 0.279447-0.038601 i 0.279447-0.038601 i\n0.40.221379-0.028349 i 0.221379-0.028349 i 0.221379-0.028349 i\n0.50.168119-0.019680 i 0.168119-0.019680 i 0.168119-0.019680 i\n0.60.1200890-0.0125902 i 0.1200890-0.0125902 i 0.1200890-0.0125902 i\n0.70.0778652-0.0070793 i 0.0778652-0.0070793 i 0.0778652-0.0070793 i\n0.80.0423110-0.0031451 i 0.0423110-0.0031451 i 0.0423110-0.0031451 i\n0.90.0149332-0.0007860 i 0.0149332-0.0007860 i 0.0149332-0.0007860 i\n0.940.00693547-0.00028291 i 0.00693547-0.00028291 i 0.00693547-0.00028291 i\n0.980.001333800-0.000031429 i 0.001333800-0.000031429 i 0.001333800-0.000031429 i\nTABLE I. Fundamental quasinormal mode for the scalar field fo rl= 2,L= 0.5 and various values\nofκη2.\nHere, the notation for the points is given by: N≡(u+∆,v+∆),W≡(u+∆,v),E≡\n(u,v+∆), andS≡(u,v). The Gaussian initial data are imposed on the two null surfaces,\nspecifically u=u0andv=v0. The quasinormal frequencies can be extracted from the\ntime-domain profiles reasonably accurately using the Prony method , as detailed in, for\ninstance, [ 29]. Although the time-domain integration method is indeed an accurate and\nconvergent technique for generating time-domain profiles, the ex traction of frequencies with\nhigh precision can be challenging. Generally, achieving a convergence with the exact results\nfrom the Leaver method, with a discrepancy of less than one perce nt, is attainable for the\nfirst and higher multipoles [ 54,55].\n6κη26WKB 6WKB(Pade-5) 6WKB(Pade-6)\n0.0.477556-0.068236 i 0.477556-0.068236 i 0.477556-0.068236 i\n0.10.407222-0.055256 i 0.407222-0.055255 i 0.407222-0.055255 i\n0.20.340835-0.043647 i 0.340835-0.043646 i 0.340835-0.043646 i\n0.30.278610-0.033408 i 0.278610-0.033407 i 0.278610-0.033407 i\n0.40.220808-0.024537 i 0.220808-0.024537 i 0.220808-0.024537 i\n0.50.167757-0.017035 i 0.167757-0.017035 i 0.167757-0.017035 i\n0.60.1198814-0.0108993 i 0.1198814-0.0108993 i 0.1198814-0.0108993 i\n0.70.0777640-0.0061291 i 0.0777640-0.0061291 i 0.0777640-0.0061291 i\n0.80.0422742-0.0027232 i 0.0422742-0.0027232 i 0.0422742-0.0027232 i\n0.90.0149267-0.0006806 i 0.0149267-0.0006806 i 0.0149267-0.0006806 i\n0.940.00693365-0.00024499 i 0.00693365-0.00024499 i 0.00693365-0.00024499 i\n0.980.001333683-0.000027218 i 0.001333683-0.000027218 i 0.001333683-0.000027218 i\nTABLE II. Fundamental quasinormal mode for the scalar field f orl= 2,L= 1 and various values\nofκη2.\nIII. QUASINORMAL MODES\nFirst and foremost, our primary objective is to establish a compara tive analysis between\nthe outcomes of our research and those recently published in [ 34]. Regrettably, [ 34] does not\npresent numerical data in tabular form, with the exception of a sam ple illustration involving\nspecific parameter values: L= 0.5,l= 2,M= 1, andκη2= 0.2. The quasinormal modes\nwere computed in [ 34] utilizing both the time-domain integration and the continued fractio n\nmethod (which seemingly corresponds to the Leaver method [ 56]). The following outcomes\nwere obtained for the aforementioned parameters:\nω= 0.342031−0.0504228i(time−domain LJZ ),\n7κη26WKB 6WKB(Pade-5) 6WKB(Pade-6)\n0.0.475513-0.055662 i 0.475513-0.055662 i 0.475513-0.055662 i\n0.10.405650-0.045078 i 0.405650-0.045078 i 0.405650-0.045078 i\n0.20.339663-0.035610 i 0.339663-0.035610 i 0.339663-0.035610 i\n0.30.277770-0.027259 i 0.277770-0.027259 i 0.277770-0.027259 i\n0.40.220236-0.020023 i 0.220236-0.020023 i 0.220236-0.020023 i\n0.50.167394-0.013902 i 0.167394-0.013902 i 0.167394-0.013902 i\n0.60.1196735-0.0088958 i 0.1196735-0.0088958 i 0.1196735-0.0088958 i\n0.70.0776627-0.0050029 i 0.0776627-0.0050029 i 0.0776627-0.0050029 i\n0.80.0422374-0.0022231 i 0.0422374-0.0022231 i 0.0422374-0.0022231 i\n0.90.0149202-0.0005557 i 0.0149202-0.0005557 i 0.0149202-0.0005557 i\n0.940.00693184-0.00020002 i 0.00693184-0.00020002 i 0.00693184-0.00020002 i\n0.980.001333567-0.000022223 i 0.001333567-0.000022223 i 0.001333567-0.000022223 i\nTABLE III. Fundamental quasinormal mode for the scalar field forl= 2,L= 2 and various values\nofκη2.\nω= 0.342003−0.0504359i(continued fraction LJZ ),\nwhere ”LJZ” signifies the Lin-Jiang-Zhai outcomes reported in [ 34].\nThis value of ωaligns with the depiction in figure 3 of [ 34], hence we will utilize it for\nthe purpose of comparison. Ordinarily, the precision of the time-do main integration method\nis limited due to the inherent conditionality of the period of quasinorma l oscillations. The\nexact determination of this period becomes challenging due to the int erplay between the\ninitial outburst and the asymptotic tails. Therefore, it was rather surprising to observe such\na remarkable concurrence between the time-domain integration ap proach and the Leaver\nmethod, which is renowned for its accuracy.\nForthesame valuesoftheparameters weobtainedthefollowing valu esofthequasinormal\nmodes:\nω= 0.342468−0.0504003i(time−domain),\nω= 0.342003−0.050437i(WKB6th order),\n8κη26WKB 6WKB(Pade-5) 6WKB(Pade-6)\n0.0.282706-0.068607 i 0.282711-0.068593 i 0.282710-0.068592 i\n0.10.240492-0.055528 i 0.240495-0.055519 i 0.240494-0.055518 i\n0.20.200799-0.043839 i 0.200801-0.043834 i 0.200800-0.043833 i\n0.30.163739-0.033537 i 0.163740-0.033534 i 0.163740-0.033534 i\n0.40.1294496-0.0246193 i 0.1294500-0.0246180 i 0.1294498-0.0246179 i\n0.50.0981035-0.0170827 i 0.0981037-0.0170822 i 0.0981036-0.0170821 i\n0.60.0699301-0.0109239 i 0.0699301-0.0109237 i 0.0699301-0.0109237 i\n0.70.0452468-0.0061395 i 0.0452469-0.0061395 i 0.0452469-0.0061395 i\n0.80.0245342-0.0027264 i 0.0245342-0.0027264 i 0.0245342-0.0027264 i\n0.90.00864045-0.00068101 i 0.00864045-0.00068101 i 0.00864045-0.00068101 i\n0.940.00400944-0.00024508 i 0.00400944-0.00024508 i 0.00400944-0.00024508 i\n0.980.000770406-0.000027221 i 0.000770406-0.000027221 i 0.000770406-0.000027221 i\nTABLE IV. Fundamental quasinormal mode for the scalar field l= 1,L= 1 and various values of\nκη2.\nω= 0.342003−0.050436i(WKB6th order Pade ).\nAs anticipated, the time-domain method demonstrates a remarkab ly close agreement\nwith the 6th order WKB formula when employing Pade approximants. T he relative error\nremainsaminute fraction, well withintheconfinesofonepercent. T heaccuracy oftheWKB\napproach diminishes when Pade approximants are excluded. Remark ably, our WKB data,\ncomplemented byPadeapproximants, exhibit animpressive alignment withthemeticulously\naccurate data obtained through the continued fraction method in [34].\nUpon examining tables I-VI, it becomes evident that the foremost p arameter influencing\nthe quasinormal frequencies is the global monopole parameter κη2. This parameter sig-\nnificantly suppresses both the real and imaginary components of t he modes. The Lorentz\nsymmetry breaking (LSB) parameter exerts a milder influence but a lso results in a decrease\nin the quasinormal modes. Of particular interest is the observation that, for a substantial\nimpact of the global monopole (leading to near-extreme states, e.g .,κη2= 0.98), the real\n9κη26WKB 6WKB(Pade-5) 6WKB(Pade-6)\n0.0.186091-0.079016 i 0.186060-0.078971 i 0.186043-0.078977 i\n0.10.159413-0.063965 i 0.159395-0.063933 i 0.159385-0.063936 i\n0.20.134041-0.050510 i 0.134031-0.050488 i 0.134025-0.050490 i\n0.30.1100775-0.0386487 i 0.1100726-0.0386348 i 0.1100694-0.0386360 i\n0.40.0876462-0.0283780 i 0.0876441-0.0283700 i 0.0876426-0.0283707 i\n0.50.0668992-0.0196952 i 0.0668985-0.0196913 i 0.0668979-0.0196916 i\n0.60.0480309-0.0125976 i 0.0480307-0.0125960 i 0.0480305-0.0125961 i\n0.70.0313027-0.0070822 i 0.0313026-0.0070817 i 0.0313026-0.0070817 i\n0.80.0170968-0.0031459 i 0.0170968-0.0031458 i 0.0170968-0.0031458 i\n0.90.00606518-0.00078607 i 0.00606518-0.00078606 i 0.00606518-0.00078606 i\n0.940.00282267-0.00028293 i 0.00282267-0.00028293 i 0.00282267-0.00028293 i\n0.980.000543962-0.000031430 i 0.000543962-0.000031430 i 0.000543962-0.000031430 i\nTABLE V. Fundamental quasinormal mode for the Dirac field for l= 1,L= 1/2 and various\nvalues of κη2.\ncomponent of the quasinormal modes becomes independent of the other parameter L.\nQuasinormal modes can be found in the analytical approximate form in the regime of\nlarge values of the multipole moment l, which corresponds to high frequencies of oscillations.\nIndeed, using the expression for the peak of the effective potent ial in the eikonal regime,\nrmax=3M\n1−κη2,\nand substituting it into the first order WKB formula\ni(ω2−Vmax)/radicalbig\n−2V′′max= 0, (13)\nwe find an expression for the quasinormal mode at high l:\nω=−/parenleftbig\nl+1\n2/parenrightbig√\nL+1(η2κ−1)3+i(L+1)/parenleftbig\nn+1\n2/parenrightbig/radicalBig\n−(η2κ−1)7\n(L+1)3/2\n3√\n3(L+1)M/radicalBig\n−(η2κ−1)3\n√L+1(14)\n10κη26WKB 6WKB(Pade-5) 6WKB(Pade-6)\n0.0.187640-0.068328 i 0.187629-0.068301 i 0.187623-0.068303 i\n0.10.160612-0.055321 i 0.160606-0.055302 i 0.160602-0.055304 i\n0.20.134940-0.043691 i 0.134937-0.043678 i 0.134935-0.043679 i\n0.30.1107255-0.0334358 i 0.1107240-0.0334284 i 0.1107228-0.0334290 i\n0.40.0880896-0.0245543 i 0.0880890-0.0245502 i 0.0880885-0.0245505 i\n0.50.0671819-0.0170442 i 0.0671818-0.0170423 i 0.0671816-0.0170424 i\n0.60.0481936-0.0109037 i 0.0481936-0.0109029 i 0.0481935-0.0109030 i\n0.70.0313823-0.0061308 i 0.0313823-0.0061306 i 0.0313823-0.0061306 i\n0.80.0171259-0.0027237 i 0.0171259-0.0027237 i 0.0171259-0.0027237 i\n0.90.00607033-0.00068067 i 0.00607033-0.00068067 i 0.00607033-0.00068067 i\n0.940.00282411-0.00024500 i 0.00282411-0.00024500 i 0.00282411-0.00024500 i\n0.980.000544054-0.000027219 i 0.000544054-0.000027219 i 0.000544054-0.000027219 i\nTABLE VI. Fundamental quasinormal mode for the Dirac field fo rl= 1,L= 1 and various values\nofκη2.\nWhenL=η2κ= 0 we reproduce the Schwarzschild limit,\nω=l−i/parenleftbig\nn+1\n2/parenrightbig\n+1\n2\n3√\n3M.\nThe aforementioned eikonal formula holds particular interest due t o its alignment with the\ncharacteristicsofnullgeodesicspertinenttotestfields, encomp assingtheLyapunovexponent\nand frequency at the unstable circular orbit [ 57]. Consequently, this formula also maintains\na connection with the diameter of the shadow cast by a black hole [ 58,59]. While [ 60,61]\ndemonstrates thatthecorrespondence istypically upheld, there areinstances where itcanbe\ndisrupted, such asforcertainnon-minimally coupledfields, or itmight remainincomplete for\nthe asymptotically de Sitter scenario. However, one can readily ver ify through calculations\nof the pertinent geodesic parameters that the correspondence indeed holds true within our\nspecific case.\n11IV. CONCLUSIONS\nGlobal monopoles hold significance in the realm of physics as they stem from the sponta-\nneous breakdown of symmetries within specific field theories. Their e xistence offers insights\ninto the structure of the early universe and the evolution of cosmic defects. Delving into\nthe study of global monopoles aids in unraveling the intricate relation ship between particle\nphysics and cosmology. This exploration contributes to comprehen ding the distribution of\nmatter and energy on a large scale, shedding light on the fundament al forces that shaped\nthe cosmos during its formative stages.\nIn this current work, we embarked on the calculation of quasinorma l modes associated\nwith scalar and Dirac fields around asymptotically flat bumblebee black holes coupled with\na global monopole. We have demonstrated that the quasinormal mo des experience signif-\nicant suppression due to the global monopole parameter, while the in fluence of the LSB\nparameter is more gentle. Interestingly, a form of universal beha vior emerges in the vicinity\nof the extreme limit. Exploring the Leaver method could facilitate the determination of\nhigher overtones within the spectrum, which characterize the eve nt horizon properties [ 62].\nSimilarly, our analysis can be extended to encompass cases with a non -zero cosmological\nconstant, which holds intrigue due to the emergence of a novel bra nch of (purely de Sitter)\nmodes. These modes are relevant to the Strong Cosmic Censorship conjecture [ 63–70]. Fur-\nthermore, the investigation of grey-body factors and Hawking ra diation associated with the\naforementioned black holes remains a pivotal avenue that we leave f or future exploration.\n[1] Q. G. Bailey and V. A. Kostelecky, “Signals for Lorentz vi olation in post-Newtonian gravity,”\nPhys. Rev. D 74, 045001 (2006) ,arXiv:gr-qc/0603030 .\n[2] R. Bluhm and V. A. Kostelecky, “Spontaneous Lorentz viol ation, Nambu-Goldstone modes,\nand gravity,” Phys. Rev. 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It follows that these small dampings a re spectrally undetectable.\n1 Introduction\nConsider a classical or a quantum system described by the damped wave equation\nutt+a(x)ut−∆xu= 0 (1)\ninthespace-timevariables( x,t)∈Rd×(0,∞), wherethe‘damping’ aisacomplex-valuedfunction.\nThe positive (respectively, negative) part of the real part of acorresponds to the dissipation\n(respectively, excitation) of mechanical or electromagnetic wave s, while the imaginary part of a\ncan be interpreted as a conservative perturbation of a Dirac (qua si-)particle.\nWritingU:= (u,ut)T, it is customary to replace the scalar second-order equation (1) b y the\nfirst-order evolution system Ut=AaUwith the matrix-valued damped wave operator\nAa:=/parenleftbigg\n0 1\n∆−a/parenrightbigg\n,domAa:=H2(Rd)×˙H1(Rd), (2)\nacting in the Hilbert space H:=˙H1(Rd)×L2(Rd). A careful analysis of the stationary problem\nAaΨ =µΨ inH, whereµ∈Cis a spectral parameter, provides information on the behaviour of\nthe time-dependent solutions uof (1). It is easy to see that the spectral problem is related to the\noperator pencil (see Lemma 1 below)\nSµaψ:=−∆ψ+µaψ=−µ2ψinL2(Rd). (3)\nAs an example of usefulness of the spectral data, let us recall the collaboration of the first\nauthor with P. Freitas [19], where it is shown that the damped waveeq uation (1) becomes unstable\nwhenever the real-valued damping aachieves a sufficiently negative minimum. The strategy of [19]\nis based on establishing spectral asymptotics of the family of Schr¨ odinger operators Sµaasµ→\n+∞. Consequently, Aapossesses a real positive point µin the spectrum, which is responsible for\na global instability of (1).\nThe present paper is partially motivated by a remark of T. Weidl [30] t hat Lieb–Thirring\ninequalities known for Sµacould provide more quantitative information on the location of the real\nspectrum of Aa. Moreover, in view of the unprecedented interest in non-self-adj oint Schr¨ odinger\n1operators in the near past, new complex extensions of the Lieb–Th irring inequalities have been\nderived in recent years. Consequently, the T. Weidl’s observation h as become interesting for the\ncomplex spectrum of Aaas well. In this paper we go beyond the usual setting by considering e ven\ncomplex-valued dampings.\nThe literature on damped wave systems is ratherextensive and we r estrict ourselveson quoting\nthe following recent works and refer the interested reader for fu rther references therein. The\nrelationship between the damped wave and Dirac equations is discuss ed on an abstract level\nin [22] and related one-dimensional Lieb-Thirring-type inequalities ca n be found in [8, 9]. An\nextensive spectral analysis of the wave operator with possibly unb ounded damping is performed\nin [20]. Basic eigenvalue bounds for weakly damped wave systems can b e deduced from [18]. The\nperturbation of eigenvalues of abstract damped wave operators has been recently studied in the\nframework of Krein spaces in [27]. Resolvent estimates for an abstr act dissipative operator are\nderived in [4, 29].\nThe structure of this paper is as follows. In Section 2 we comment on basic properties of\nthe damped wave operator Aaand state a crucial correspondence between its eigenvalue proble m\nand the operator pencil (3). Self-adjoint and non-self-adjoint L ieb–Thirring-type inequalities are\napplied in Sections 3 and 4, respectively. In particular, in Theorem 4 w e prove that the point\nspectrum of the one-dimensional damped wave operator Aais empty provided that /ba∇dbla/ba∇dblL1(R)<2.\nWe interpret this result as that small dampings cannot be detected by measuring eigenfrequencies\nof the wave system. In Section 5 we strengthen this observation t o a complete lack of spectral\ndetectability by showing that the non-self-adjoint operator Aais actually similar to the skew-\nadjoint undamped operator A0under the same smallness condition.\n2 Preliminaries\nAlthough the damped wave equation can be made meaningful for cer tain unbounded dampings\n(cf.[20]), our standing hypothesis is that the damping a:Rd→Cis bounded, i.e.,\na∈L∞(Rd). (4)\nThenitiseasytoseethatthereexistsaconstant csuchthatAa+cisageneratorofa C0-semigroup\nof contractions ( cf.[19, App. B]), so (1) is well posed. Equivalently, Aais a quasi-m-accretive\noperator. In particular, the operator Aais closed, so its spectrum is well defined.\nNotice that Aais skew-adjoint ( i.e.,iAais self-adjoint) if, and only if, ais purely imaginary.\nIn particular, the undamped operator A0is skew-adjoint. To have this symmetry result, it is\nimportant that we consider the homogeneous Sobolev space ˙H1(Rd) (defined as the completion of\nC∞\n0(Rd) with respect to the norm ψ/ma√sto→ /ba∇dbl∇ψ/ba∇dbl) in the definition of the Hilbert space H, see (2).\nOn the other hand, it is important to keep in mind that ˙H1(Rd) is not a subset of L2(Rd).\nWith an abuse of notation, we denote by −∆ the distributional Laplacian as well as its self-\nadjoint realisation in L2(Rd) with domain dom( −∆) :=H2(Rd). For any bounded potential\nV:Rd→C, the Schr¨ odinger operator SV=−∆ +Vwith domSV=H2(Rd) is a well defined\nm-sectorial operator.\nWe say that Visvanishing at infinity and writeV∞− →0, provided that\nlim\nR→+∞/ba∇dblV/ba∇dblL∞(Rd\\BR(0))= 0,\nwhereBR(0) denotes the ball of radius R>0 centred at the origin. Then Vis a relatively compact\nperturbation of −∆. Consequently, σe(SV) =σe(S0) = [0,+∞) for any choice of the essential\nspectrum (namely, σe1,...,σ e5in the notation of [11, Chapt. IX]). More specifically, it is clear\nfor the components σe1,...,σ e4, which are preserved by relatively compact perturbations, and th e\nresult extends to the widest choice σe5becauseSVis m-sectorial ( cf.[24, Prop. 5.4.4]).\nIt is easy to see that the spectrum of the undamped operator A0is purely continuous and equal\nto the imaginary axis; in particular, σ(A0) =σe(A0) =iR. Ifais vanishing at infinity, then the\ndamping represents a relatively compact perturbation of A0. Consequently, σek(Aa) =σek(A0) =\n2iRfork= 1,...,4. To see that it is true also for σe5, it is enough to notice that each connected\ncomponent of C\\iRintersects the resolvent set of Aadue to (4). In summary,\na∞− →0 =⇒σe(Aa) =σe(A0) =iR.\nSince the essential spectrum is independent of a, the present paper focuses on the point spec-\ntrum ofAa. The following lemma specifies the equivalence between the spectral problem for (2)\nand the operator pencil (3) in the case of eigenvalues (not necess arily discrete).\nLemma 1. Assume (4). For every µ∈C,\n−µ2∈σp(Sµa)⇐⇒µ∈σp(Aa).\nProof.Assuming −µ2∈σp(Sµa), there exists a non-trivial function ψ1∈H2(Rd) such that\nSµaψ1=−µ2ψ1. Then Ψ := ( ψ1,µψ1)T∈domAaand\n(AaΨ)T= (µψ1,∆ψ1−aµψ1) = (µψ1,−Sµaψ1) =µ(ψ1,µψ1) =µΨT.\nConversely, assuming µ∈σp(Aa), there exists a non-trivial Ψ = ( ψ1,ψ2)T∈domAasuch that\nAaΨ =µΨ. In other words, ψ1∈H2(Rd),ψ2∈˙H1(Rd) andψ2=µψ1, ∆ψ1−aψ2=µψ2.\nCombining the latter two equations, we get Sµaψ1=−µ2ψ1withψ1/\\e}atio\\slash= 0.\n3 Real spectrum and real-valued damping\nIn this section, we assume that the damping asatisfying (4) is real-valued and focus on real\neigenvalues µ∈R. Recallingthe correspondenceofLemma 1, it is enough to consider t he auxiliary\nSchr¨ odinger operators SVwith real-valued potentials V. ThenSVis self-adjoint, so its spectrum\nis purely real. Let us denote by {λn(V)}N\nn=1the non-decreasing sequence of negative discrete\neigenvalues of SV, where each eigenvalue is repeated according to its multiplicity. The s et can be\neither empty ( N= 0) or finite (1 ≤N <∞) or infinite ( N=∞).\nOur starting point are the self-adjoint Lieb–Thirring inequalities ( cf.[26, 25])\nN/summationdisplay\nn=1|λn(V)|γ≤Lγ,d/integraldisplay\nRdVγ+d\n2\n−, (5)\nwhereLγ,dis a positive constant independent of VandV±:=1\n2(|V|±V). More specifically, it is\nknown that such a bound holds with a finiteconstantLγ,dif, and only if,\nγ≥1\n2ifd= 1,\nγ >0 ifd= 2,\nγ≥0 ifd≥3.(6)\nThe sharp values of the constants Lγ,dare not known explicitly for all the admissible values of γ.\nIn special cases, however, one knows that L1\n2,1=1\n2and\nLγ,d=Lcl\nγ,d:=Γ(γ+1)\n2dπd\n2Γ(γ+d\n2+1)ford≥1, γ≥3\n2\nare the best possible values.\nA direct combination of the Lieb–Thirring inequalities with Lemma 1 yields the following basic\nresults.\nTheorem 1. Suppose that ais real-valued and assume (4). Letγbe any number satisfying (6).\nIfµis a positive (respectively, negative) eigenvalue of Aaanda−∈Lγ+d\n2(Rd)(respectively,\na+∈Lγ+d\n2(Rd)), then\nµγ−d\n2≤Lγ,d/integraldisplay\nRdaγ+d\n2\n−/parenleftbigg\nrespectively, (−µ)γ−d\n2≤Lγ,d/integraldisplay\nRdaγ+d\n2\n+/parenrightbigg\n.\n3On the other hand, if a−∈Ld(Rd)(respectively, a+∈Ld(Rd)) and\n/integraldisplay\nRdad\n−<1\nLd\n2,d/parenleftBigg\nrespectively,/integraldisplay\nRdad\n+<1\nLd\n2,d/parenrightBigg\n,\nthenAahas no positive (respectively, negative) eigenvalues.\nProof.From Lemma 1 we get that real µ∈σp(Aa) if, and only if, there exists n∈Nsuch that\nλn(µa) =−µ2. Hence, (5) implies\n\n\nµ2γ=|λn(µa)|γ≤Lγ,dµγ+d\n2/integraldisplay\nRdaγ+d\n2\n− forµ∈σp(Aa)∩(0,+∞),\n|µ|2γ=|λn(µa)|γ≤Lγ,d|µ|γ+d\n2/integraldisplay\nRdaγ+d\n2\n+forµ∈σp(Aa)∩(−∞,0).(7)\nThis proves the first part of the theorem. The case 0 ∈σp(Aa) cannot occur because the spectrum\nofS0is purely continuous. Dividing (7) by |µ|γ+d\n2, and eventually putting γ=d\n2, which can be\ndone for all d≥1, we conclude with the second part of the theorem.\nTo continue, we restrict ourselves to d= 1. In this case the Buslaev–Faddeev–Zakharov trace\nformulae ( cf.[12]) provides us with a lower bound for the sum of square roots of th e eigenvalues\nofSµa, namely\nN/summationdisplay\nn=1|λn(µa)|1\n2≥ −µ\n4/integraldisplay\nRa. (8)\nOf course, the inequality is non-trivial only if µ/integraltext\nRa <0. The latter is known to be a sufficient\ncondition which guarantees that inf σ(Sµa)<0. Assuming in addition a∞− →0, it follows that Sµa\npossesses at least one negative eigenvalue. The number of eigenva luesNofSµais controlled from\nabove by the Bargmann bound ( cf.[28, Problem 22])\nN≤1+|µ|/integraldisplay\nR|a(x)||x|x..\nConsequently, for µsatisfying the inequality\n|µ|</parenleftbigg/integraldisplay\nR|a(x)||x|x./parenrightbigg−1\n(9)\nthe operator Sµahas exactly one negative eigenvalue λ1(µa) and we get from (8) the estimate\n|µ|=|λ1(µa)|1\n2≥ −µ\n4/integraldisplay\nRa. (10)\nThis implies/integraltext\nRa≥ −4 forµ >0 and/integraltext\nRa≤4 forµ <0. In summary, we have established the\nfollowing result.\nTheorem 2. Letd= 1. Suppose that ais real-valued and in addition to (4)assumea∈L1(R,|x|x.)\nanda∞− →0. Letµbe a real eigenvalue of Aa. Ifµ>0and/integraltext\nRa<−4(orµ<0and/integraltext\nRa>4),\nthen\n|µ| ≥/parenleftbigg/integraldisplay\nR|a(x)||x|x./parenrightbigg−1\n.\nFinally, combining the Lieb–Thirring inequalities with the Buslaev–Fadde ev–Zakharov trace\nformulae, we obtain the following quantitative bounds on the location of real eigenvalues of the\none-dimensional damped wave operator. The presence of the cou pling parameter αis useful for\nstudying the stability of solutions of (1) in the spirit of [17, 21, 19].\n4Theorem 3. Letd= 1. Suppose that ais real-valued and in addition to (4)assumea∈L1(R,|x|x.)\nanda∞− →0. Let/integraltext\nRa <0(respectively,/integraltext\nRa >0). For any µ >0(respectively, µ <0) satisfy-\ning(9), there exists exactly one α>0satisfying\n2/parenleftbigg/integraldisplay\nRa−/parenrightbigg−1\n≤α≤ −4/parenleftbigg/integraldisplay\nRa/parenrightbigg−1/parenleftBigg\nrespectively, 2/parenleftbigg/integraldisplay\nRa+/parenrightbigg−1\n≤α≤4/parenleftbigg/integraldisplay\nRa/parenrightbigg−1/parenrightBigg\nsuch thatµ\nαis an eigenvalue of Aαa.\nProof.Assumingµ >0 and/integraltext\nRa <0 together with a∞− →0 and (9), the operator Sµapossesses\nexactly one negative eigenvalue. Applying (7) with γ=1\n2and (10), we get\n−µ\n4/integraldisplay\nRa≤ |λ1(µa)|1\n2≤µ\n2/integraldisplay\nRa−.\nSimilarly, for µ<0 and/integraltext\nRa>0, we get\n−µ\n4/integraldisplay\nRa≤ |λ1(µa)|1\n2≤|µ|\n2/integraldisplay\nRa+.\nThese estimates together with Lemma 1 prove the theorem.\n4 Complex spectrum and complex-valued damping\nIn this section, we use the recent progress in spectral theory of non-self-adjoint Schr¨ odinger op-\nerators and state results about complex eigenvalues of the damped wave operator. There is no\nobstacle to consider complex-valued dampings at the same time.\nLet us start with dimension d= 1. The celebrated result of E. B. Davies et al.(see [1, Thm. 4]\nand [10, Corol. 2.16]) states that every eigenvalue λ(V) ofSVwithV∈L1(R) satisfies the bound\n|λ(V)|1\n2≤1\n2/integraldisplay\nR|V(x)|x.. (11)\nMoreover, the bound is known to be optimal for step-like potentials approximating the Dirac\npotential. Now, assuminginadditionto(4)that a∈L1(R), ifµisanyeigenvalueof Aa(necessarily\nit must be non-zero, cf.the proof of Theorem 1), Lemma 1 ensures that there exists λ(µa)∈\nσp(Sµa) such that\n|µ|=|λj(µa)|1\n2≤1\n2|µ|/integraldisplay\nR|a(x)|x.,\nwhere the inequality is due to (11). Dividing by |µ|, it follows that the point spectrum of Aa\nis empty provided that the L1-norm ofais small, namely /ba∇dbla/ba∇dblL1(R)<2. Let us summarise the\nobservation into the following theorem.\nTheorem 4. Letd= 1. In addition to (4)assumea∈L1(R). If/ba∇dbla/ba∇dblL1(R)<2, thenσp(Aa) =∅.\nMoreover, the constant 2is optimal.\nProof.It remains to argue about the optimality. Our strategy is to show th at for any number\nslightly greater than 2 there exists a damping Wwith theL1-norm equal to this number such\nthatAWhas an eigenvalue. For this we choose the analytically computable cas e, where the\ndamping is a step-like potential\nW(x) :=\n\n0 ifx<−b,\naif−b<x<b,\n0 ifx>b,witha<0<b.\n5The eigenvalue equation AWΨ =µΨ reduces to ∆ ψ−µWψ−µ2ψ= 0, where ψis the first\ncomponent of Ψ. It is enough to analyse the situation of real µ. Forµ= 0 orµ=−awe\nget just a trivial solution to the eigenvalue equation. Also, using the spectral correspondence of\nLemma 1, we know that all the real eigenvalues of AWmust be positive, provided that W≤0\nandµ≤ /ba∇dbla/ba∇dblL∞(R)=−a. Thus the only interval where we can find a real eigenvalue is (0 ,−a).\nNow, forψto lie inH2(R)⊂C1(R) and be non-trivial, it can be easily verified that the secular\nequation\nF(µ) := 2/radicalbig\n−(µa+µ2) cos/parenleftbig\n2b/radicalbig\n−(µa+µ2)/parenrightbig\n+(a+2µ) sin/parenleftbig\n2b/radicalbig\n−(µa+µ2)/parenrightbig\n= 0\nmust be satisfied. For µ∈(0,−a) this is equivalent to G(µ) :=/radicalbig\n−(µa+µ2)F(µ) = 0. We\ncompute\nlim\nµ→0+G(µ) = 0 = lim\nµ→−a−G(µ),\nlim\nµ→0+G′(µ) = 2a(−1+c),\nlim\nµ→−a−G′(µ) = 2a(1+c),\nwherec:=−ba=1\n2/ba∇dblW/ba∇dblL1(R). We observe that for c>1 both the derivatives have the same sign\nwhich, together with the fact that the limit points of this continuous functionG(µ) are the same,\nimplies that there exists µ∗∈(0,−a) such that G(µ∗) = 0. Hence, µ∗∈σp(AW) which proves the\ndesired optimality.\nRemark 1. Takingb:= (2|a|)−1andα∈C, the complexified step-like potential αWconverges\nin the sense of distributions to αδasa→ −∞, whereδis the Dirac delta function. Replacing\n(formally)abyαδin (3), one arrives at the pencil problem\n\n\n−ψ′′=−µ2ψinR\\{0},\nψ(0+)−ψ(0−) = 0,\nψ′(0+)−ψ′(0−) =µαψ(0).(12)\nThere exists no admissible solution ψ∈H2(R\\ {0}), unlessα=−2 (respectively, α= 2) in\nwhich case every µ∈Cwithℜµ >0 (respectively, with ℜµ <0) is an ‘eigenvalue’! Since\n/ba∇dblαW/ba∇dblL2(R)=|α|, this is another support for the optimality of the constant 2 in Theo rem 4.\nHowever, we are not aware of any result about a spectral approx imation of the operator pencil (3)\nby bounded potentials.\nThe damped wave equation on a finiteinterval with the damping being a Dirac delta function\nwas previously studied in [3], [2, Sec. 4.1.1] and [7].\nIn Section 5, we argue that the absence of eigenvalues follows as a c onsequence of the similarity\nofAato the undamped wave operator A0, provided that /ba∇dbla/ba∇dblL1(R)<2.\nIn higher dimensions d≥2, we use the robust result of R. Frank et al.(see [15, Thm. 1] and\n[16, Thm. 3.2]) stating that every eigenvalue λ(V) ofSVwithV∈Lγ+d\n2(Rd), 0<γ≤1\n2, satisfies\nthe bound\n|λ(V)|γ≤Dγ,d/integraldisplay\nRd|V(x)|γ+d\n2x., (13)\nwhereDγ,dis a constant independent of V. Using Lemma 1, it follows that any eigenvalue µofAa\nsatisfies\n|µ|2γ=|λj(µa)|γ≤Dγ,d|µ|γ+d\n2/integraldisplay\nRd|a(x)|γ+d\n2x.. (14)\nWe therefore conclude with the following theorem.\n6Theorem 5. Letd≥2. In addition to (4)assumea∈Lγ+d\n2(Rd)with0<γ≤1\n2. There exists a\nconstantDγ,dsuch that, for any eigenvalue µ∈σp(Aa), one has\n|µ|γ−d\n2≤Dγ,d/integraldisplay\nRd|a|γ+d\n2.\nThe formal analogue of (13) for γ= 0 andd≥3 states that there exists a dimensional positive\nconstantD0,dsuch that if\nD0,d/integraldisplay\nRd|V(x)|d\n2x.<1, (15)\nthenSVhas no eigenvalues. The case of discrete eigenvalues is due to R. Fra nk [15, Thm. 2], while\npossibly embedded eigenvalues were covered by [16, Thm. 3.2]. Inde pendently, the total absence\nof eigenvalues under a weaker hypothesis for d= 3 and alternative conditions in higher dimensions\nwas obtained by L. Fanelli, D. Krejˇ ciˇ r´ ık and L. Vega in [13] (see als o [14] and [6, 5]). In fact, the\nspectral stability of SVfor potentials Vwhich are small in some sense goes back to the abstract\nresult of T. Kato’s [23] that we shall recall for other purposes in th e following section. Here we\njust mention that a straightforward combination of Lemma 1 and (1 5) implies that (14) holds also\nforγ= 0 andd≥3. Consequently, we get the following formal analogue of Theorem 5 .\nTheorem 6. Letd≥3. In addition to (4)assumea∈Ld\n2(Rd). There exists a positive con-\nstantD0,dsuch that, for any eigenvalue µ∈σp(Aa), one has\n|µ|−d\n2≤D0,d/integraldisplay\nRd|a|d\n2.\n5 Similarity\nIn this section, we come back to the one-dimensional setting of The orem 4. We find the result\nappealing because it implies that small dampings cannot be distinguishe d from the undamped\nsystem just by measuring eigenfrequencies. In fact, the following result shows a much stronger\nresult that small dampings are indeed spectrally undetectable .\nTheorem 7. Letd= 1. In addition to (4)assumea∈L1(R). If/ba∇dbla/ba∇dblL1(R)<2, then the\noperatorsAaandA0are similar to each other. Moreover, the constant 2is optimal.\nHere the similarity means that there exists an operator W∈B(H) such that W−1∈B(H)\nandAa=WA0W−1. In otherwords, iAaisquasi-self-adjoint (cf.[24]), because iA0is self-adjoint.\nThen the spectra of AaandA0coincide. In particular, Aamust have the same eigenvalues as A0.\nSince the spectrum of A0is purely continuous, Theorem 4 follows as a direct consequence of\nTheorem 7.\nProof.The optimality of the constant 2 follows from the proof of Theorem 4 . Indeed, for any\nnumber strictly larger than 2, there exists a damping awhoseL1-norm equals that number,\nwhileAapossesses eigenvalues. This would violate the similarity.\nTo prove the similarity, we use the abstract result of T. Kato [23, T heorem 1.5]. Writing\niAa=iA0+(−isgnaB)∗BwithB:=/parenleftbigg0 0\n0|a|1\n2/parenrightbigg\n,\nwhere sgn is the complex sign (defined by sgn f:=f/|f|iff/\\e}atio\\slash= 0 and sgn f:= 0 iff= 0), it is\nenough to show that the bounded operators Band−isgnaBarerelatively smooth with respect\ntoiA0(cf.[23, Def. 1.2]) and\nsup\nξ∈C\\R/ba∇dblKξ/ba∇dbl<1,whereKξ:=BR(ξ,iA0)Bisgna\n7is the Birman–Schwingeroperator. Here we use the notation R(ξ,T) := (T−ξ)−1for the resolvent\nof an operator Tat pointξ∈C\\σ(T). The relative smoothness is a rather complicated condition\nin general, but it reduces to reasonable criteria when iA0is self-adjoint ( cf.[23, Thm. 5.1]).\nUsing in addition the simple structure of the intertwining operators Band−isgnaBin our case,\neverything reduces to verifying the unique condition\nsup\nξ∈C\\R/ba∇dbl˜Kξ/ba∇dbl<1,where ˜Kξ:=BR(ξ,iA0)B. (16)\nIt can be easily verified that\nR(ξ,iA0) =R(ξ2,−∆)/parenleftbigg\nξ i\ni∆ξ/parenrightbigg\n.\nConsequently,\n˜Kξ=|a|1\n2ξR(ξ2,−∆)|a|1\n2/parenleftbigg\n0 0\n0 1/parenrightbigg\n,\nand therefore\n/ba∇dbl˜Kξ/ba∇dbl=|ξ|/vextenddouble/vextenddouble/vextenddouble|a|1\n2R(ξ2,−∆)|a|1\n2/vextenddouble/vextenddouble/vextenddouble,\nwhere/ba∇dbl · /ba∇dbldenotes the operator norm both in HandL2(R) on the left- and right-hand side,\nrespectively. The latter will be estimated by the Hilbert–Schmidt nor m/ba∇dbl·/ba∇dblHS. To this purpose,\nwe recall the explicit formula for the integral kernel of R(z,−∆):\nGz(x,y) =e−√−z|x−y|\n2√−z,\nwherex,y∈R,z∈C\\[0,+∞) and the principal branch of the square root is used. Using the\nelementary estimate\n|Gz(x,y)| ≤1\n2/radicalbig\n|z|,\nwe therefore get\n/ba∇dbl˜Kξ/ba∇dbl2≤ |ξ|2/vextenddouble/vextenddouble/vextenddouble|a|1\n2R(ξ2,−∆)|a|1\n2/vextenddouble/vextenddouble/vextenddouble2\nHS\n=|ξ|2/integraldisplay\nR×R|a(x)||Gξ2(x,y)|2|a(y)|2x.y.\n≤/ba∇dbla/ba∇dbl2\nL1(R)\n4,\nwhere the last bound is independent of ξ. Recalling (16), T. Kato’s similarity condition [23,\nTheorem 1.5] holds provided that /ba∇dbla/ba∇dblL1(R)<2.\nAcknowledgment\nThe research was partially supported by the GACR grants No. 18-0 8835S and 20-17749X.\nReferences\n[1] A. A. Abramov, A. Aslanyan, and E. B. Davies, Bounds on complex eigenvalues and reso-\nnances, Journal of Physics A: Mathematical and General 34(2001), no. 1, 57.\n[2] K Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback , Springer,\n2015.\n[3] A. Bamberger, J. Rauch, and M. Taylor, A model for harmonics on stringed instruments ,\nArch. Ration. Mech. Anal. 79(1982), 267–290.\n8[4] J.-M. 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Kato, Wave operators and similarity for some non-selfadjoint ope rators, Math. Ann. 162\n(1966), 258–279.\n9[24] D. Krejˇ ciˇ r´ ık and P. Siegl, Elements of spectral theory without the spectral theorem , InNon-\nselfadjoint operators in quantum physics: Mathematical as pects(432 pages), F. Bagarello,\nJ.-P. Gazeau, F. H. Szafraniec, and M. Znojil, Eds., Wiley-Interscie nce, 2015.\n[25] A. Laptev, Spectral inequalities for Partial Differential Equations a nd their applications. ,\nAMS/IP Stud. Adv. Math 51(2012), 629–643.\n[26] A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities , Journ´ ees ´Equations aux\nd´ eriv´ ees partielles (2000), 1–14.\n[27] I. Naki´ cand K. Veseli´ c, Perturbation of eigenvalues of the Klein–Gordon operators , Rev. Mat.\nComplut. 33(2020), 557–581.\n[28] M. Reed and B. Simon, Methods in mathematical physics, vol. IV: Analysis of opera tors,\nAcademic Press, 1978.\n[29] J. Royer, Local decay for the damped wave equation in the energy space , J. Inst. Math. Jussieu\n(2016), 1–32.\n[30] T. Weidl, private communication in Stuttgart (April 2005).\n10"    },    {        "title": "1805.06535v3.Stabilization_rates_for_the_damped_wave_equation_with_Hölder_regular_damping.pdf",        "content": "arXiv:1805.06535v3  [math.AP]  28 Nov 2018Stabilization rates for the damped wave equation with\nH¨ older-regular damping\nPerry Kleinhenz\nAbstract\nWe study the decay rate of the energy of solutions to the damped w ave equation\nin a setup where the geometric control condition is violated. We cons ider damping\ncoefficients which are 0 on a strip and vanish like polynomials, xβ. We prove that the\nsemigroup cannot be stable at rate faster than 1 /t(β+2)/(β+3)by producing quasimodes\nof the associated stationary damped wave equation. We also prove that the semigroup\nis stable at rate at least as fast as 1 /t(β+2)/(β+4). These tworesults establish an explicit\nrelation between the rate of vanishing of the damping and rate of de cay of solutions.\nOur result partially generalizes a decay result of Nonnemacher in whic h the damping\nis an indicator function on a strip.\n1 Introduction\nLetM= (M,g) be a Riemannian manifold. Fix some W∈L∞(M),W≥0. We study the\nasymptotic behavior as t→ ∞of solutions to the damped wave equation\n\n\n∂2\ntu−∆u+W(x)∂tu= 0 inM×R+\n(u,∂tu)|t=0= (u0,u1) inM\nu|∂M= 0 if ∂M/ne}ationslash=∅.(1)\nThe quantity of particular interest is the energy\nE(u,t) =1\n2/parenleftBig\n||∇u(·,t)||2\nL2+||∂tu(·,t)||2\nL2/parenrightBig\n. (2)\nIn this paper we will work on the square M= [−b,b]×[−b,b] or torusM=T2, which we\nparametrize by ( x,y). We will give detailed proofs in the case of the square then s how how\nthose results can be applied to the torus.\nFor some fixed β≥0 anda,σ>0, such that a+σ<b, we study damping W∈L∞(M)\nof the form\nW(x,y) =\n\n0 |x|<a\n(|x|−a)βa<|x|<a+σ\nc(|x|)>0a+σ<|x|<b,(3)\n1In particular note that the damping is invariant in the ydirection.\nRemark. The particular form of c(x) does not affect our result so long as c(x)∈\nL∞(a+σ,b) and it is uniformly bounded away from 0. However choosing a csuch thatW\nis smooth for |x|>aandW=C >0 for|x|>a+2σis perhaps the most interesting case\nin the context of existing results.\nDefinition 1. Letf(t) be a function such that f(t)→0 ast→ ∞. We say that (1) is\nstable at rate f(t) if there exists a constant C >0 such that for all ( u0,u1)∈(H2(M)∩\nH1\n0(M))×H1\n0(M) (orH2(M)×H1(M) if∂M=∅) ifusolves (1) with ( u0,u1) as Cauchy\ndata then\nE(u,t)≤Cf(t)2/parenleftBig\n||u0||2\nH2(M)+||u1||2\nH1(M)/parenrightBig\nfor allt>0.\nOur main result is\nTheorem 1.1. For allε>0, withWas in(3)the equation (1)cannot be stable at rate\nt−β+2\nβ+3−ε.\nMore precisely if we use m1(t) to denote the best possible ffor which definition 1 holds\nthen this result along with Lemma 4.6 of [AL+14] and Proposition 3 of [BD08] show that\nm1(t)≥C/(1+t)β+2\nβ+3for someC >0, where we use the notation of [BD08].\nWe also show that\nTheorem 1.2. ForWas in(3)withβ >0the equation (1)is stable at rate\nt−β+2\nβ+4.\nAgain using Lemma 4.6 of [AL+14] and Proposition 3 of [BD08] this shows m1(t)≤\nC/(1+t)β+2\nβ+4for someC >0.\nThe decay of energy of the damped wave equation is a well studi ed question. The\nstrongestpossibledecay isuniformstabilization, whichi sdefinedastheexistence of F(t)→\n0 ast→ ∞such that, for all usolving (1) with initial data in H1(M)×L2(M)\nE(u,t)≤F(t)E(u,0), t≥0.\nIt was established by [RT75] that uniform stabilization occ urs withF=Ce−κtfor\nsomeκ,C >0, when∂M=∅,W∈C0(M) and supp Wsatisfies the geometric control\ncondition (GCC). We recall that a set Usatisfies the GCC if there exists T >0 such that\nfor every geodesic γonMof lengthT,γ∩U/ne}ationslash=∅. The reverse implication, that uniform\nstabilization with any Fimplies supp Wsatisfies the GCC, was shown in [Ral69]. These\nresults were extended to the case M/ne}ationslash=∅by [CBR92] and [BG97]. This in turn guarantees\nthat when uniform stabilization occurs one can always let F=Ce−κtfor someκ,C >0.\nFor a finer discussion of when uniform stabilization occurs f orL∞damping see [BG18].\n2A natural next question to ask is what occurs when the GCC does not hold for supp W.\nBecause of the necessity of the GCC for uniform stabilizatio n, as soon as it does not hold\nwe must adjustthe kindof decay wehopefor. Thenextbest thin gis thestability definedin\nDefinition 1, which comes from [Leb96] and requires initial d ata with an additional spatial\nderivative.\nIn [Bur98], the author showed that the energy of a solution to (1) decays at least\nlogarithmically ( f(t) = 1/log(2 +t)) as soon as the damping W(x)≥c >0 on some\nopen, nonempty set. In [Leb96] the author gave explicit exam ples of domains on which\nf(t) = 1/log(2+t) is the exact decay rate, in particular when M=S2and{W >0}does\nnot intersect a neighborhood of the equator. For related wor k see also [LR97].\nIn the case of the square when the damped region contains a ver tical strip, [LR05]\nestablished a decay rate of f(t) = (log(t)/t)1/2. This was expanded to the case of partially\nrectangular domains when {W >0}contains a neighborhood of the nonrectangular part\nin [BH07]. Additionally in [BH07] a relation between vanish ing rate of the damping and\ndecay rate for the damped wave equation was established.\nThese results were refined by [AL+14]. The authors established a decay rate of f(t) =\n1/t1/2for the damped wave equation in a more general setting, so lon g as the associated\nSchr¨ odinger equation is controllable. This includes the c ase of not identically vanishing\ndamping on the 2 dimensional square (or torus) as a consequen ce of [Jaf90] (resp. [Mac10],\n[BZ12]).\nContinuing in the case of the 2 dimensional square [AL+14] also show that the system\ncan not be stable at rate f(t) = 1/t1+εfor anyε>0, when{W >0}does not satisfy GCC,\n(this condition is referred to as the GCC being strongly viol ated). They further show the\nexistence of a smooth damping coefficient which strongly viol ates the GCC for which the\nenergy decays at rate f(t) = 1/t1−εfor anyε>0.\nIn an appendix to [AL+14], Nonnenmacher shows that when the damping is the in-\ndicator function on a strip on the square or torus the system c annot be stable at rate\nf(t) = 1/t2/3+ε, for anyε>0. The complementary result was shown in [Sta17] to estab-\nlishf(t) = 1/t2/3as the exact rate of decay when damping is a strip on the square or torus.\nThe difference in behavior between the smooth and discontinuo us damping led the authors\nof [AL+14] to pose the problem of establishing an explicit relation between the vanishing\nrate of the damping and the decay rate.\nAnexplicitrelation wasestablishedby[LL17]inaslightly differentsetting. Theauthors\nstudy the case of a general manifold in which the damping is su pported everywhere but a\nflat subtorus. In the 2 dimensional case this is an example of t he GCC not holding but also\nnot being strongly violated. The damping is required to be in variant along this subtorus\nand must satisfy W(x)≤C|x|βnear where it vanishes. When this is the case the authors\nshow that the system cannot be stable at rate f(t) = 1/tβ+2\nβ+ε, for anyε >0. They also\nshow that if the vanishing behavior of the damping is further limited toC−1\n1|x|β≤W(x)≤\nC1|x|βthe system is stable at exactly the rate f(t) = 1/tβ+2\nβ(See also [BZ15]).\n3Note that in [LL17] decreasing βcorrespondsto faster vanishing(i.e. less regular damp-\ning) which produces faster decay, which is counter to the beh avior exhibited in [AL+14],\n[BH07] and our own result, namely that faster vanishing (i.e . less regular damping) pro-\nduces slower decay. However the dynamics in 2 dimensions in t he two situations are\ndifferent, with only one undamped orbit in the former as oppose d to a whole family in the\nlatter.\nOur paper provides a partial answer to the problem posed by [A L+14]. We establish an\nexplicit relation between the rate of vanishing of the dampi ng and the stability rate of the\nsystem, in a case where the GCC is strongly violated on the squ are orT2. Our work also\npartially extends that of Nonnenmacher in the appendix to [A L+14], which agrees with\nour first theorem when β= 0, although that result follows from the existence of modes\nof the stationary equation where we produce quasimodes. Our two results provide further\nevidence for the fact, discussed in [AL+14], [LL17], [BH07], that once the support of the\ndamping is fixed the rate of vanishing of the damping is the mos t significant feature when\ndetermining the decay rate.\nWe note that our second result improves that of Theorem 1.2 of [BH07], which gives a\ndecay rate of f(t) = 1/tβ/(β+4)(see also [AL+14] Theorem 2.6). We also note that there\nis a gap between our two results. Closing this gap would be an i nteresting area for further\nwork.\nIn the next section we outline the proof of Theorem 1.1. Secti ons 3, 4 and 5 contain\nthe details of the proof. Section 6 contains the proof of Theo rem 1.2.\nAcknowledgements The author would like to thank Jared Wunsch for his advice\nand guidance. The author would also like to thank Matthieu L´ eautaud for comments on\nan early manuscript. The author would also like to thank the r eferees for their prompt,\ndetailed and constructive comments. This research was supp orted in part by the National\nScience Foundation grant ”RTG: Analysis on manifolds” at No rthwestern University.\n2 Outline of Proof of Theorem 1.1\nTo prove Theorem 1.1 we rely on the following result from [AL+14] (Proposition 2.4) and\n[BT10] which relates energy decay of the damped wave equatio n to resolvent estimates of\nthestationary dampedwave equation. Withthisresultitiss ufficienttoproducesufficiently\ngood quasimodes, to do so we reduce the problem to one dimensi on and then the interval\n[0,b]. We will then show that we can use solutions to a related prob lem, the complex\nabsorbing potential, on the half line to produce the desired quasimodes. We finally show\nthat suchsolutionsof thecomplex absorbingpotential prob lemexistbyproducingsolutions\non (0,a) and (a,∞) separately, the latter following from a rescaling argumen t, we are able\nto glue these solutions together via a compatibility condit ion which we satisfy via the\nimplicit function theorem.\n4Proposition 2.1. Fixα, if there exist sequences {qj} ∈C,{uj} ∈H2(M)∩H1\n0(M),(or\nH2(M)if∂M=∅) and some j0∈N, such that for all j >|j0|\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∆uj+iqjW(x)uj−q2\njuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(M)≤C\n|Re(qj)|2/α/parenleftBig\n||uj||2\nH1(M)+|qj|2||uj||2\nL2(M)/parenrightBig\n,(4)\nand\n|qj| → ∞,|Im(qj)| ≤C\n|Re(qj)|1/α, (5)\nthen for all ε>0the system (1)is not stable at rate\n1/tα+ε.\nRemark. Although Proposition 2.1 has ||uj||2\nH1on the right hand side of (4) the\nquasimodes ujwe will apply it to satisfy a stronger inequality, namely\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∆uj+iqjW(x)uj−q2\njuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(M)≤C\n|Re(qj)|2||uj||2\nL2(M). (6)\nThestrengthoftheconclusionweobtainfromapplyingPropo sition2.1tothesequasimodes\nis instead limited by the qjfor which we have such an estimate due to (5).\nNotethatproducingsequences qjandujwhichsatisfythehypothesesofthisproposition\nwithα=β+2\nβ+3proves Theorem 1.1.\nWe will make two simplifications before proceeding. First we will reducethe problem to\nobtaining quasimodes of an ordinary differential equation on [−b,b]. We will then further\nrestrict our attention to the same equation on [0 ,b]. After making these simplifications\nwe will introduce three key parameters and the complex absor bing potential problem on\n(0,∞), solutions of which we will use to produce our desired quasi modes.\nFor the first simplification note that for any sequence of inte gersmjif/tildewideujis a sequence\nof functions on [ −b,b] which satisfy\n\n\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nx/tildewideuj+iqjW/tildewideuj+/parenleftbigg\n4π2m2\nj\nb2−q2\nj/parenrightbigg\n/tildewideuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(−b,b)≤C\n|Re(qj)|2||/tildewideuj||2\nL2(−b,b)asj→ ∞\n/tildewideuj(x) = 0|x|=b,\n(7)\nthenuj(x,y) =/tildewideuj(x)sin/parenleftBig2πmjy\nb/parenrightBig\nsatisfy (6). Therefore it is enough for us to find functions\nwhich satisfy (7) with qjwhich satisfy (5) with α=β+2\nβ+3.\nThe second simplification we make is to limit our attention to [0,b] from [−b,b]. Since\nour damping is even, if we find integers mjand functions /tildewideujon [0,b] which satisfy\n\n\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nx/tildewideuj+iqjW/tildewideuj+/parenleftbigg\n4π2m2\nj\nb2−q2\nj/parenrightbigg\n/tildewideuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤C\n|Re(qj)|2||/tildewideuj||2\nL2(0,b)asj→ ∞\n/tildewideuj(x) = 0x=b\n/tildewideuj(0) = 0 or/tildewideu′\nj(0) = 0(8)\n5we can extend the /tildewideujto−b≤x<0 by setting/tildewideuj(−x) =−/tildewideuj(x) (or/tildewideuj(−x) =/tildewideuj(x) resp.)\nand the resulting functions satisfy (7). Therefore it is eno ugh for us to find functions which\nsatisfy (8) with qjwhich satisfy (5) with α=β+2\nβ+3.\nBefore we introduce the complex absorbing potential we intr oduce three new parame-\nters. Leth∈[0,1) be a small parameter which will be sent to 0 and haveb\n2πh2∈N. Let\nlbe a bounded parameter, when working in the case u(0) = 0 we impose l∈Zand when\nu′(0) = 0 we impose l+1\n2∈Zbut otherwise leave lfree in relation to h. We define\nλh=πlh\na+Chh(β+4)/(β+2)Ch=Oh(1)∈C. (9)\nWe will eventually specify Chmore completely in Section 4. When appropriate we will\nrefer to sequences of these parameters as hj,λhj,ljandChj, where we emphasize that λhj\nandChjdepend onhj.\nNow we introduce the complex absorbing potential problem on (0,∞)\n/braceleftBigg\n0 =−h2∂2\nxv+i(x−a)β\n+v−λ2\nhv\nv(0) = 0 orv′(0) = 0.(10)\nIn order to relate this to (8) we make an ansatz for relations b etween the parameters.\nIfvjare a sequence of solutions of (10) for some hj,lj,λhj,Chj, we define qjandmjas\nfollows\nmj=b\n2πh2\nj∈N (11)\nqj=1\nh2\nj+λ2\nhj\n2=1\nh2\nj+π2l2\njh2\nj\na2+2Chjπlj\nah(2β+6)/(β+2)\nj +C2\nhjh(2β+8)/(β+2)\nj .\nNote that in this regime\nRe(qj) =1\nh2\nj+O(h2\nj) Im(qj) =2Im(Chj)πlj\nah(2β+6)/(β+2)\nj +O(h(2β+8)/(β+2)\nj ),\nso\nqj→ ∞and|Im(qj)| ≤C\n|Re(qj)|(β+3)/(β+2)asj→ ∞.\nAs we will see shortly, solutions of (10) in this regime satis fy the inequality in (8) but\nnot necessarily the boundary condition at x=b. In order to ensure they do we multiply\nthese solutions by a cutoff function which is 0 in a neighborho od ofb. We will see the\nresulting functions still satisfy the inequality in (8) as t he solutions of (10) in this regime\nhave rapid decay on the support of the potential (see Lemma 3. 1), which is exactly where\nerrors introduced by the cutoff function appear.\n6Remark We note also that it is because we are working with solutions t o (10) on the\nhalf-line that we will only produce quasimodes rather than r eal modes. It is necessary for\nus to work on the half-line in order to perform a rescaling tha t allows us to set h= 0, if we\nwere working on a finite interval the rescaling would make the interval depend on hwhich\nour approach is not well adapted to address.\nFixδ>0 such that a+σ<b−2δ; we define φ∈C∞(0,∞) to satisfy\nφ(x) =/braceleftBigg\n1x<b−2δ\n0x>b−δ(12)\nProposition 2.2. FixM >0, let{vj} ∈H2(0,∞)be a sequence of solutions of (10)with\neigenvalues\nλhj=πljhj\na+Chjh(β+4)/(β+2)\nj, Chj=O(1)∈C,\nwhere|lj| ≤Mandhj→0asj→ ∞andb\n2πh2\nj∈N. Set\nuj(x) =φ(x)vj(x).\nThen forjlarge enough so that hj<σβ/2the functions ujwithqj,mjas defined in (11)\nsatisfy(8)and(5).\nIt remains to be seen that we can indeed find solutions to the co mplex absorbing\npotential problem with eigenvalues of this form.\nTheorem 2.3. For alll∈Z,(orl+1\n2∈Z), there exists h0>0andK >0such that for\nallh∈(0,h0), there exists a Chwith|Ch|<Kandv∈H2(0,∞)∩H1\n0(0,∞),v/ne}ationslash= 0(resp.\nH2(0,∞)) satisfying (10)withv(0) = 0(resp.v′(0) = 0) withλgiven by (9).\nUsing Theorem 2.3 we obtain a sequence {vj}of solutions of (10) which satisfy the hy-\npotheses of Proposition 2.2 which in turn produces sequence s which satisfy the hypotheses\nof Propositions 2.1 which in turn proves Theorem 1.1.\nRemark. It is straightforward to extend these results to the case M=T2. We\nparametrize T2by [−b,b]×[−b,b] with parallel edges identified. Thus it is enough to show\nthat the quasimodes we produced on the square satisfy period ic boundary conditions and\nare thus functions on the torus. Our quasimodes are of the for m\nu(x,y) =vj(x)φ(x)sin/parenleftbigg2πmjy\nb/parenrightbigg\n,\nso it is straightforward to see they satisfy periodic bounda ry conditions in yandx(as\nu(x,y)≡0 for|x−b|<δand|b+x|<δ).\nWe will prove Proposition 2.2 in Section 3, we will then prove Theorem 2.3 in Section\n4. We prove a necessary estimate in Section 5. We finally prove Theorem 1.2 in Section 6.\n73 Proof of Proposition 2.2\nWe begin by stating an estimate necessary for the proof.\nLemma 3.1. Letv∈H1(0,∞)be a solution of (10)with eigenvalue λ=O(h)and letφ\nbe as in (12). Fixs∈Rthen forh<σβ/2for allNthere exists CN,s>0such that\n||φv||2\nHs\nh(a+σ,b)≤CN,shN||φv||2\nL2(0,b). (13)\nThis will be proved in Section 5 using the semiclassical elli pticity of −h2∂2\nx+i(x−a)β\n+−\nλ2\nhon (a+σ/4,b).\nProof of Proposition 2.2. We have a sequence vjof solutions of\n0 =−h2\nj∂2\nxvj+i(x−a)β\n+vj−λ2\nhjvj, x∈(0,∞),\nwith\nλhj=πlhj\na+O/parenleftBig\nh(β+4)/(β+2)\nj/parenrightBig\n, hj→0 asj→ ∞.\nIt is clear that uj=φvjhasuj(b) =φ(b)vj(b) = 0. Recalling (11) and the subsequent\ndiscussionqj,mjsatisfy (5). It remains to be seen that ujsatisfies the inequality in (8).\nBy (11) and (10) ujsatisfies\n−∂2\nxuj+iqjW(x)uj+/parenleftBigg\n4π2m2\nj\nb2−q2\nj/parenrightBigg\nuj\n=φ/parenleftBigg\niλ2\nhj\n2(x−a)β\n+vj−λ4\nhj\n4vj/parenrightBigg\n−φ′′vj−2φ′v′\nj+iqj/parenleftBig\nW(x)−(x−a)β\n+/parenrightBig\nφvj.\nThus\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nxuj+iqjW(x)uj+/parenleftBigg\n4π2m2\nj\na2−q2\nj/parenrightBigg\nuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤λ4\nhj\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x−a)β\n+uj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)+λ8\nhj\n16||uj||2\nL2(0,b)+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′vj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)\n+4/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′v′\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)+|qj|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(W−(x−a)β\n+)φvj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b).\nSinceb−δ>a+σ, by Lemma 3.1 for any N >0\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′vj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′v′\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤C\nh2||φvj||H2\nh(b−δ,b)≤CNhN||φvj||2\nL2(0,b).\nFurthermore by Lemma 3.1 for any N >0\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(W−(x−a)β\n+)φvj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤C||φvj||2\nL2(a+σ,b)dx≤CNhN||φvj||2\nL2(0,b).\n8Therefore\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nxuj+iqj(x−a)β\n+uj+/parenleftBigg\n4π2m2\nj\nb2−q2\nj/parenrightBigg\nuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤Cλ4\nhj\n4||uj||2\nL2(0,b)+λ8\nhj\n16||uj||2\nL2(0,b)+CNhN||uj||2\nL2(0,b)\n≤C\n|Re(qj)|2||uj||2\nL2(0,b).\nWhere we used that |Re(qj)|= 1/h2\nj+O(h2\nj) andλhj=O(hj).\nWe now show that there are solutions of (10) with the desired e igenvalues.\n4 Proof of Theorem 2.3\nFrom this point on we focus on solutions of (10) on (0 ,∞). We begin by considering the\ncases when v(0) = 0 orv′(0) = 0, but focus on the case v(0) = 0 for the bulk of the section.\nWe then explain how the proof changes for v′(0) = 0.\nIn order to produce solutions to (10) with the desired eigenv alues we will solve it on\n(0,a) and (a,∞) separately. That is given solutions vl,vr∈H2of (10) on (0 ,a) and (a,∞)\nrespectively, with the same values of λhandh, if there exists B∈Csuch that\n/braceleftBigg\nvl(a) =Bvr(a)\nv′\nl(a) =Bv′\nr(a),(14)\nthen\nv=/braceleftBigg\nvl(x)x<a\nBvr(x)a<x,\nsolves (10) on (0 ,∞) with the same λhandhandv∈H2(0,∞)∩H1\n0(0,∞) (orH2(0,∞)\nifv′\nl(0) = 0) . We will refer to equations (14) as the compatibility condition.\nWe will explicitly solve (10) on (0 ,a). We will then use a rescaling of the equation on\n(a,∞) and the implicit function theorem to show that the compatib ility condition can be\nsatisfied when λhis of the form (9) and his small enough.\nOn (0,a) (10) is solved by\nvl(x) =eiλh(x−a)/h+Ref(λh,h)e−iλh(x−a)/h,\nwhere we choose Ref( λh,h) to ensure the boundary condition at 0 is satisfied. That is\nRef(λh,h) =/braceleftBigg\n−e−2iλha/hv(0) = 0,\ne−2iλha/hv′(0) = 0.\nWe will work now specify to the case where v(0) = 0 and work through it in detail and\nthen summarize how the proof changes for v′(0) = 0.\n9We now rescale the equation on ( a,∞). IfFsolves\n/braceleftBigg\n0 =−F′′(x)+/parenleftBig\nixβ−λ2\nh\nh2β/(β+2)/parenrightBig\nF(x)x∈(0,∞)\nF′(0) = 1,(15)\nthenvr(x) =F(h−2/(β+2)(x−a)) solves (10) on ( a,∞) withv′\nr(a) =h−2/(β+2). This follows\nimmediately from the definition of Fand (10).\nRemark. This rescaling is necessary; in order to show the compatibil ity condition can\nbe satisfied for all hin a neighborhood of 0 we will apply the implicit function the orem and\nso must be able to set h= 0. This can not be done in a satisfactory way with solutions o f\n(10); however for λhof the form (9), (15) is well defined at h= 0 as\nλ2\nh\nh2β/(β+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nh=0=1\nh2β/(β+2)Ch2+O(h(2β+6)/(β+2))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nh=0= 0.\nBefore we apply the implicit function theorem we first establ ish an inequality for u∈\nH1(0,∞) using some facts about the Neumann spectrum of −∂2\nx+xβon (0,∞). We then\nuse this inequality to establish the existence and uniquene ss ofH2(0,∞) solutions to (15)\nand to show the boundary value F(0) is bounded away from 0 uniformly in the spectral\nparameter. We then explain how we can use the implicit functi on theorem to satisfy the\ncompatibility condition and make use of these properties of Fto do so.\nLetσN(−∂2\nx+xβ) be the spectrum of −∂2\nx+xβon (0,∞) with Neumann boundary\nconditions, and let\n/tildewiderλ1= infσN(−∂2\nx+xβ).\nLemma 4.1.\n/tildewiderλ1>0\nand\n/tildewiderλ1||u||2\nL2(0,∞)≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleu′/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞), (16)\nfor allu∈H1(0,∞).\nProof.The Neumann spectrum is discrete (this follows for instance from [HS12] Theorems\n5.10 and 10.7) so we know that /tildewiderλ1is the lowest eigenvalue of the operator. We also know\nthat the spectrum doesn’t contain 0, since\n||∂xu||2\nL2(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞)>0,\nfor nontrivial u∈H1. Thus/tildewiderλ1>0.\nThe inequality follows immediately from the variational pr inciple for the spectrum of\nself adjoint operators (see [HS12] Corollary 12.2).\n10Lemma 4.2. For any |η|</tildewiderλ1, there exists a unique H2(0,∞)solution of\n/braceleftBigg\n0 =−F′′(x)+ixβF(x)−ηF(x)\nF′(0) = 1.(17)\nFurthermore if we let F(0,η)be the value of this function at x= 0there exists C >0such\nthat for all |η| ≤/tildewiderλ1\n2\n1/C≤ |F(0,η)| ≤C.\nandF(0,η)is holomorphic in ηon that same neighborhood.\nProof.We first show the existence of a solution. Let ψ∈C∞\n0(0,∞) withψ′(0) = 1,ψ(0) =\n0. DefineQ(η,ψ) as\nQ(η,ψ) :=ψ′′−ixβψ+ηψ.\nNow letJsolve /braceleftBigg\n−J′′+ixβJ−ηJ=Q\nJ′(0) = 0,(18)\nand note that F=ψ+Jsolves (17). We will apply the Lax-Milgram theorem to show th e\nexistence of solutions to (18).\nLet\nH=H1(0,∞)∩x−β/2L2(0,∞),\nand define the norm\n||u||2\nH=||u||2\nH1(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞),\nnoting that His a Hilbert space with this norm.\nWe define the sesquilinear form B:H×H→C\nB[u,v] =ˆ∞\n0u′¯v′+ixβu¯v−ηu¯v.\nFor anyu,v∈H\n|B[u,v]| ≤ˆ\n|u′||v′|+xβ|u||v|+|η||u||v| ≤C||u||H||v||H.\nFurthermore for u∈H⊂H1using (16)\n|B[u,u]| ≥ˆ\n|u′|2+xβ|u|2−|η||u|2dx≥/parenleftbigg\n1−|η|\n/tildewiderλ1/parenrightbiggˆ\n|u′|2+xβ|u|2dx≥C||u||2\nH.\nTherefore by Lax-Milgram for any Q∈Hthere exists a unique J∈Hsuch that\nB[J,v] =ˆ\nQ¯vdx,\n11for allv∈H.\nTherefore there exists an F∈Hsolving (17) given by F=J+ψ.\nNow to show that F(0,η) is holomorphic in ηwe restate the result of our application\nof Lax-Milgram. We have shown that when |η|</tildewiderλ1, for allQ∈Hthere exists a unique\nJ∈Hsuch that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(−∂2\nx+ixβ−η)−1Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=||J||H≤C||Q||H.\nTherefore( −∂2\nx+ixβ−η)isbijective withaboundedinversefor |η|</tildewiderλ1. Thustheresolvent\n(−∂2\nx+ixβ−η)−1exists for |η|</tildewiderλ1and by [HS12] Theorem 1.2 it is holomorphic in η\nthere as well.\nNow recall that the trace operator T:H1(0,∞)→Ris linear and continuous on H.\nThusTJ=J(0,η) is also holomorphic in η. Recall that F(0,η) =J(0,η)+ψ(0) =J(0,η),\nsoF(0,η) is holomorphic in η.\nTo see that Fis unique assume otherwise, so there exists F1∈H1(0,∞) which also\nsolves (17) with F1(0)/ne}ationslash=F(0). SetF2=F1−F, thenF′\n2(0) = 0,F2∈H1(0,∞) and\n0 =−∂2\nxF2+ixβF2−ηF2.\nMultiply both sides by ¯F2and integrate then integrate by parts\n0 =ˆ∞\n0|∂xF2|2+ixβ|F2|2−η|F2|2dx.\nThen using (16)\n0≥ˆ\n|∂xF2|2+xβ|F2|2−|η||F2|2dx≥/parenleftbigg\n1−|η|\n/tildewiderλ1/parenrightbiggˆ\n|∂xF2|2+xβ|F2|2dx>0,\nsinceF2∈H1(0,∞). The inequality is strict since |η|</tildewiderλ1, which is a contradiction.\nTo establish the stated regularity of Fnote that the equation (17) is elliptic and the\nleft hand side is 0 so in fact F∈H∞(0,∞) and thusF∈H2(0,∞) in particular.\nNow we show that F(0,η) is bounded away from 0 and ∞. The upper bound follows\nimmediately by the holomorphy of F(0,η) and the boundedness of η.\nTo see|F(0,η)|>1/Cit is enough to show that F(0,η)/ne}ationslash= 0 sinceFis continuous\nand we are considering ηin a compact set. Assume otherwise, so F(0,η0) = 0 for some\nη0∈C,|η0| ≤/tildewiderλ1/2. Multiply both sides of (17) by ¯Fthen integrate and integrate by parts\n0 =ˆ∞\n0|∂xF|2+ixβ|F|2−η0|F|2dx.\nThen using (16) (noting that here F∈H1\n0⊂H1)\n0≥ˆ\n|∂xF|2+xβ|F|2−|η0||F|2dx≥/parenleftbigg\n1−|η0|\n/tildewiderλ1/parenrightbiggˆ\n|∂xF|2+xβ|F|2dx>0,\nWe note that the inequality is strict since |η0|</tildewiderλ1, which is a contradiction.\n12Now we introduce µh∈C. We use it to clarify the dependence of λhonhand will\nimplicitly solve for it in terms of hin order to show that the compatibility condition can\nbe satisfied. If F0is the Dirichlet data of the unique H2solution of (15) for h=λh= 0,\nwe setC1=A1+µhso our definition of λhin (9) becomes\nλh=πlh\na+A1h(β+4)/(β+2)+µhh(β+4)/(β+2),\nwherel∈Zand\nA1=πlF0\na2.\nNow takevr(x) =F(h−2/(β+2)(x−a)),whereFis the unique H2solution of (15) with\nthe aboveλh. Recalling the explicit form of vl(x), the compatibility condition becomes\niλh\nh(1−Ref(µh,h)) =Bh−2/(β+2)\n1+Ref(µh,h) =BF(0,µh,h),\nwhereF(0,µh,h) denotes the Dirichlet data of Fand Ref(µh,h) = Ref(λh,h) and both are\nwritten this way to emphasize their dependence on µhandh. Divide the top equation by\nthe bottom\n/parenleftbiggπli\na+A1ih2/(β+2)+iµh2/(β+2)/parenrightbigg/parenleftBig\n1+exp(−2πil−2A1iah2/(β+2)−2iaµhh2/(β+2))/parenrightBig\nF(0,µh,h)\n=h−2/(β+2)/parenleftBig\n1−exp(−2πil−2A1iah2/(β+2)−2iaµhh2/(β+2))/parenrightBig\n.\nNow Taylor expand the exponentials for small h\n/parenleftbiggπli\na+A1ih2/(β+2)+iµhh2/(β+2)/parenrightbigg/parenleftBig\n2−2A1iah2/(β+2)−2iaµhh2/(β+2)+g(h)/parenrightBig\nF(0,µh,h)\n=h−2/(β+2)/parenleftBig\n2A1iah2/(β+2)+2iaµhh2/(β+2)−g(h)/parenrightBig\n,\nwhereg(h) is the remainder term from the Taylor expansion with g(h) =O(h4/(β+2)).\nSo in order to prove Theorem 2.3 it is enough to establish that for allhnear 0∈[0,∞)\nthere exists µh∈Csuch that the following function has a zero at ( µh,h);\nG(µ,h) =/parenleftbiggπli\na+A1ih2/(β+2)+iµh2/(β+2)/parenrightbigg/parenleftBig\n2−2A1iah2/(β+2)−2iaµh2/(β+2))+g(h)/parenrightBig\nF(0,µ,h)\n−2A1ia−2iaµ+g(h)h−2/(β+2). (19)\nTo do so we apply the implicit function theorem with weak regu larity hypotheses to solve\nforµh. We recall the implicit function theorem as stated in Theore m 11.1 of [LS90] (pp.\n166) can be applied if there exists some h0such that\n131.G(0,0) = 0\n2.Gis continuous on [0 ,1]×[0,h0)\n3.DµGexists and is continuous on [0 ,1]×[0,h0)\n4.DµG(0,0) is invertible.\nTo begin we see immediately that\nG(0,0) =2πli\naF0−2A1ia=2πli\naF0−2πli\naF0= 0.\nIn the following two lemmas we show that points 2 and 3, and 4 ar e satisfied.\nLemma 4.3. There exists h0>0such thatGas defined in (19)is continuous and∂\n∂µG\nexists and is continuous on {µ∈C;|µ|<1}×[0,h0).\nProof.To see that Gis continuous it will be enough to show that F(0,µ,h) is continuous\nas the other terms in G(µ,h) are clearly continuous in µandh. Similarly to see that∂\n∂µG\nexists and is continuous it is enough to see that∂\n∂µF(0,µ,h) exists and is continuous in µ\nandh.\nWe recall that F(0,µh,h) is the Dirichlet data for the L2solution of (17) with spectral\nparameter\nη=λ2\nh\nh2β/(β+2)=π2l2h4/(β+2)+(2A1πl+2πlµ)h6/(β+2)\na+(A2\n1+µ2+A1µ)h8/(β+2).\nBy Lemma 4.2 the Dirichlet data for the L2solution is holomorphic in |η|</tildewiderλ1. Thisηis\na sum of functions which are jointly continuous in µandhand continuously differentiable\ninµ, thereforeF(0,µ,h) is as well.\nFurthermore since |µ|<1 and there is a positive power of hin each term of ηthere\nexists some h0such that |η|</tildewiderλ1/2 for|µ|<1 andh∈[0,h0).\nLemma 4.4. WithGdefined as in (19)\n∂\n∂µG(µ,h)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nh=0,µ=0/ne}ationslash= 0,\nand thus is invertible.\nProof.Note\nG(µ,0) =2πli\naF(0,µ,0)−2A1ai−2aiµ\nso\n∂\n∂µG(µ,0) =2πli\na/parenleftbigg∂\n∂µF(0,µ,0)/parenrightbigg\n−2ai.\n14Whenh= 0 the equation Fsolves is 0 = −F′′(x)+ixβF(x).Therefore when h= 0 there\nis no dependence on µso∂\n∂µF(0,µ,h= 0) = 0. Thus\n∂\n∂µG(µ= 0,h= 0) =−2ai/ne}ationslash= 0.\nNow that we have shown we can apply the implicit function theo rem we conclude the\nproof of Theorem 2.3. That is given some lwe leth0be as in the proof of Lemma 4.3 and\nletK=π|F0|\na2+1. Then we choose an h∈(0,h0) and use the implicit function theorem to\nproduce aµh∈[0,1] such that G(µh,h) = 0 and |µh|<1. We setCh=πF0\na2+µhwe are\nthen able to solve (10) on (0 ,a) and (a,∞) forλh=πlh\na+A1h(β+4)/(β+2)+µhh(β+4)/(β+2)\nsuch that the compatibility conditions are satisfied giving us a solution v∈H2(0,∞)∩\nH1\n0(0,∞),v/ne}ationslash= 0 with |Ch|<Kandλof the appropriate form.\n4.1 Case u′(0) = 0\nWe now discuss how these proofs change when u′(0) = 0. When this is the case\nRef(λh,h) =e−2iλha/h.\nBecause of this we take l+ 1/2∈Zrather than l∈Z. This changes the specific steps\ntaken when going from the compatibility condition to the defi nition ofGin (19) but the\neventual definition of Gis the same. The specific form of lis otherwise not used so the\nremaining proofs in this section hold unchanged.\n5 Proof of Lemma 3.1\nProof.To show this result we consider our operator Ph=−h2∂2\nx+i(x−a)β\n+−λ2\nhon\n(a+σ/4,b). We note that the coefficients are smooth away from aand so it makes sense\nto look at the pricipal symbol\n|ξ|2+i(x−a)β\n+.\nIt is straightforward to see from this that Phis semiclassically elliptic on ( a+σ/4,b).\nRecall in our definition of φthat we required a+σ<b−2δand in (12) that φsatisfies\nφ(x) =/braceleftBigg\n1x<b−2δ\n0b−δ<x.\nWe now define ψ∈C∞(0,b) satisfying\nψ=/braceleftBigg\n0x<a+σ/2\n1a+σ<x<b,\n15so thatφψ∈Ψ0\nh(0,b) withWFh(φψ)⊂ellh(Ph).\nIfvis asolution of Phv= 0, by standardsemiclassical elliptic estimates (see fori nstance\n[Zwo12] Theorem 7.1) there exists χ∈C∞\n0(a+σ/4,b) such that for all s∈R\n||φψv||Hs\nh(a+σ/2,b)≤O(h∞)||χv||L2(a+σ/4,b)\nIn particular\n||φv||Hs\nh(a+σ,b)≤O(h∞)||v||L2(0,b).\nIt remains to show that\n||v||L2(0,b)/lessorsimilar||φv||L2(0,b).\nTo proceed we multiply both sides of (10) by ¯ vthen integrate and integrate by parts\n0 =h2ˆ∞\n0|∂xv|2dx+iˆ∞\n0(x−a)β\n+|v|2dx−λ2\nhˆ∞\n0|v|2dx.\nTake the imaginary part of both sides and rearrange\nˆ∞\n0(x−a)β\n+|v|2dx= Im(λ2\nh)ˆ∞\n0|v|2dx≤O(h2)ˆ∞\n0|v|2dx.\nFurthermore\nˆ∞\na+σσβ|v|2dx≤ˆ∞\na+σ(x−a)β\n+|v|2dx≤ˆ∞\n0(x−a)β\n+|v|2dx.\nSo ˆ∞\na+σ|v|2dx≤h2\nσβˆ∞\n0|v|2dx.\nAdd´∞\n0|v|2dxto both sides and rearrange\n/parenleftbigg\n1−h2\nσβ/parenrightbiggˆ∞\n0|v|2dx≤ˆa+σ\n0|v|2dx.\nNotice that ( b−2δ,b)⊂(0,∞) and (0,a+σ)⊂(0,b−2δ) so\n/parenleftbigg\n1−h2\nσβ/parenrightbiggˆb\nb−2δ|v|2dx≤/parenleftbigg\n1−h2\nσβ/parenrightbiggˆ∞\n0|v|2dx≤ˆa+σ\n0|v|2dx≤ˆb−2δ\n0|v|2dx.\nTherefore for h<σβ/2\nˆb\n0|v|2dx=ˆb−2δ\n0|v|dx+ˆb\nb−2δ|v|2dx≤/parenleftbigg\n1+σβ\nσβ−h2/parenrightbiggˆb−2δ\n0|v|2dx≤Cˆb\n0|φv|2dx.\n166 Proof of Theorem 1.2\nIn this section we establish a rate of decay of the energy by ad apting and improving\nSection 3 of [BH07] for our particular setup. That result and our result rely heavily on an\nobservability result by Burq and Zworski (Proposition 6.1) in [BZ04].\nTo prove Theorem 1.2 we again rely on Proposition 2.4 of [AL+14] (see also [BT10])\nwhich we state a variant of.\nProposition 6.1. If there exists C >0andq0≥0such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(−∆+iqW(x)−q2)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2→L2≤C|q|1\nα−1(20)\nfor allq∈R,|q| ≥q0then(1)is stable at rate 1/tα.\nProof of Theorem 1.2. Consideru∈H2(M) solving\n(−∆+iqW−q2)u=f∈L2(M), u|∂M= 0, q≫1. (21)\nLet 0≤χ∈C∞\n0(R) be a cutoff function with χ= 0 for|x| ≥2 andχ= 1 for|x| ≤1.\nNow letχq=χ(qγW(x)) withγ=β\nβ+2. Note that χqis identically 0 for |x|>a+σ/4 for\nqlarge enough since W >0 on [a+σ,b]. Because of this χqand it’s derivatives are only\nsupported where Whas the form ( |x|−a)β\n+. Furthermore on the support of χ′\nq(x)\nW(x)∼1\nqγ, q≫1. (22)\nRemark. The proof in [BH07] uses an analogous setup with γ= 1. The key change we\nmake is to set γ=β\nβ+2. The rest of our argument is similar to the proof in [BH07] but we\ndetail it for the convenience of the reader and to explain why this value of γis ideal.\nThe function χqustill vanishes on ∂M(or if∂M=∅it still satisfies the periodicity\ncondition) and satisfies on M\n(−∆−q2)χqu=χqf+χ′′\nqu−2∂x(χ′\nqu)−iqW(x)χqu. (23)\nWe apply Proposition 6.1 of [BZ04] to this equation choosing the control region ωx=\n[a+σ/4,a+σ/2] and setting ω:=wx×[−b,b] to obtain\n||χqu||2\nL2≤C/parenleftBig/vextendsingle/vextendsingle/vextendsingle/vextendsingleχqf+χ′′\nqu−2∂x(χ′\nqu)−iqW(x)χqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH−1\nxL2y(M)/parenrightBig\n+||χqu|ω||2\nL2(ω)\n≤C/parenleftBig\n||χqf||2\nL2+/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+||qWχqu||2\nL2/parenrightBig\n. (24)\nWe emphasize that χqvanishes on ω, which allowed us to drop that term. We now estimate\nthe remaining terms on the right hand side. Using that Wis exactly ( |x| −a)β\n+on the\nsupport ofχqand its derivatives we obtain the following bound on the deri vative ofχq,\n|χ′\nq|=|qγχ′\nq(qγW(x))W′(x)| ≤Cqγ/β, (25)\n17and similarly\n|χ′′\nq| ≤Cq2γ/β.\nNote that on the support of χ′\nqandχ′′\nqthe damping Wis smooth, so this computation is\nvalid for all β >0.\nNow write\nχ′\nqu=χ′\nqW1/2u\nW1/2,\nthen using (25) and (22)/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′\nq\nW1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤qγ(1/2+1/β),\nand consequently/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Cqγ(1+2/β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2. (26)\nWe estimate the L2norm ofχ′′\nquin a similar way\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤O(1)qγ(1+4/β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2. (27)\nFinally a similar argument shows\n||qWχqu||2\nL2≤O(1)q2−γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2. (28)\nThe smaller the terms on the right of (26), (27) and (28) are th e stronger the resolvent\nestimate is. Because of this we would like to minimize\nmax{2−γ,γ(1+4/β),γ(1+2/β)}.\nThis is attained when\n2−γ=γ(1+4/β),\ni..e.γ=β/(β+2). Therefore (26), (27), (28) along with (24) give\n||χqu||2\nL2≤O(1)/parenleftbigg\n||f||2\nL2+q(β+4)/(β+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2/parenrightbigg\n.\nNow note that pairing (21) with ¯ u\n|q|ˆ\nW|u|2dx≤ ||f||L2||u||L2. (29)\nTherefore\n||χqu||2\nL2≤O(1)/parenleftBig\n||f||2\nL2+q2/(β+2)||f||L2||u||L2/parenrightBig\n.\n18It remains to control the L2norm of (1 −χq)u. To do so we remark that 1 −χqis supported\nin the set where W≥1/qγ. Using (29) again\n||(1−χq)u||2\nL2≤qγˆ\n(1−χq)W|u|2dx≤qγ−1||f||L2||u||L2.\nTherefore\n||u||2\nL2≤O(1)/parenleftBig\n||f||2\nL2+q1/(β+2)||f||L2||u||L2/parenrightBig\nand thus we obtain\n||u||L2≤O(1)q2/(β+2)||f||L2, q∈R,|q| ≫1,\nwhich along with Proposition 6.1 gives stability at the stat ed rate.\nReferences\n[AL+14] N. Anantharaman, M. L´ eautaud, et al. Sharp polynomial d ecay rates for the\ndamped wave equation on the torus. Anal. PDE , 7(1):159–214, 2014.\n[BD08] C. J. K. Batty and T. Duyckaerts. Non-uniform stabili ty for boundedsemi-groups\non banach spaces. Journal of Evolution Equations , 8(4):765–780, Nov 2008.\n[BG97] N. Burq and P. G´ erard. Condition n´ ecessaire et suffis ante pour la contrˆ olabilit´ e\nexacte des ondes. Comptes Rendus de l’Acadˆ emie des Sciences - Series I - Math-\nematics, 325(7):749 – 752, 1997.\n[BG18] N. Burq and P. G´ erard. Stabilisation of wave equatio ns on the torus with rough\ndampings. arXiv preprint arXiv:1801.00983 , 2018.\n[BH07] N. Burq and M. Hitrik. Energy decay for damped wave equ ations on partially\nrectangular domains. Mathematical Research Letters , 14(1):35–47, 2007.\n[BT10] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator\nsemigroups. Mathematische Annalen , 347(2):455–478, 2010.\n[Bur98] N. Burq. D´ ecroissance de l’´ energie locale de l’´ e quation des ondes pour le probl` eme\next´ erieur et absence de r´ esonance au voisinage du r´ eel. Acta Math. , 180(1):1–29,\n1998.\n[BZ04] N.BurqandM.Zworski. Geometriccontrol intheprese nceofablack box. Journal\nof the American Mathematical Society , 17(2):443–471, 2004.\n[BZ12] N. Burq and M. Zworski. Control for Schr¨ odinger oper ators on tori. Mathematical\nResearch Letters , 19, 06 2012.\n19[BZ15] N. Burq and C. Zuily. Laplace eigenfunctions and damp ed wave equation on prod-\nuct manifolds. Applied Mathematics Research eXpress , 2015(2):296–310, 2015.\n[CBR92] G. Lebeau C. Bardos and J. Rauch. Sharp sufficient cond itions for the obser-\nvation, control, and stabilization of waves from the bounda ry.SIAM Journal on\nControl and Optimization , 30(5):1024–1065, 1992.\n[HS12] P. Hislop and I. Sigal. Introduction to spectral theory: With applications to\nSchr¨ odinger operators , volume 113. Springer Science & Business Media, 2012.\n[Jaf90] S. Jaffard. Contrˆ ole interne exact des vibrations d’ une plaque rectangulaire.\n(Internal exact control for the vibrations of a rectangular plate).Port. Math. ,\n47(4):423–429, 1990.\n[Leb96] G. Lebeau. Equation des ondes amorties. In Algebraic and Geometric Methods\nin Mathematical Physics: Proceedings of the Kaciveli Summe r School, Crimea,\nUkraine, 1993 , pages 73–109. Springer Netherlands, Dordrecht, 1996.\n[LL17] M. L´ eautaud and N. Lerner. Energy decay for a locally undamped wave equation.\nAnnales de la facult des sciences de Toulouse Sr.6 , 26(1):157–205, 2017.\n[LR97] G. Lebeau and L. Robbiano. Stabilisation de l´ equati on des ondes par le bord.\nDuke Math. J. , 86(3):465–491, 1997.\n[LR05] Z. Liu and B. Rao. Characterization of polynomial dec ay rate for the solution\nof linear evolution equation. Zeitschrift f¨ ur angewandte Mathematik und Physik\nZAMP, 56(4):630–644, 2005.\n[LS90] L.H. Loomis and S. Sternberg. Advanced Calculus . Jones and Bartlett Publishers,\nInc., 1990.\n[Mac10] F. Maci` a. High-frequency propagation for the Schr ¨ odinger equation on the torus.\nJ. Funct. Anal. , 258(3):933–955, 2010.\n[Ral69] J. Ralston. Solutions of the wave equation with loca lized energy. Communications\non Pure and Applied Mathematics , 22(6):807–823, 1969.\n[RT75] J. Rauch and M. Taylor. Exponential decay of solution s to hyperbolic equations\nin bounded domains. Indiana Univ. Math. J. , 24:7986, 1975.\n[Sta17] R. Stahn. Optimal decay rate for the wave equation on a square with constant\ndamping on a strip. Zeitschrift f¨ ur angewandte Mathematik und Physik , 68(2):36,\n2017.\n[Zwo12] M. Zworski. Semiclassical analysis , volume 138. American Mathematical Soc.,\n2012.\n20"    },    {        "title": "1607.03749v1.The_Impact_of_Lorentz_Violation_on_the_Klein_Tunneling_Effect.pdf",        "content": "arXiv:1607.03749v1  [hep-ph]  12 Jul 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nThe Impact of Lorentz Violation on the Klein Tunneling Effect\nZhi Xiao\nIndiana University Center for Spacetime Symmetries\nBloomington, IN 47405, USA\nDepartment of Mathematics and Physics, North China Electri c Power University\nBeijing 102206, China\nWe discuss the impact of a tiny Lorentz-violating bµterm on the one dimen-\nsional motion of aDiracparticle scattering on a rectangula r barrier. We assume\nthe experiment is performed in a particular inertial frame, where the compo-\nnents of bµare assumed constants. The results show that Lorentz-viola tion\nmodification to the transmission rate depends on the observe r Lorentz nature\nofbµ. For a spacelike or lightlike bµthe induced resonant frequency shift\ndepends on the polarization, while for timelike bµthere is essentially no mod-\nification.\n1. Basic assumptions\nIn this report, we try to use the Standard-Model Extension1to explore the\nimpact of a tiny Lorentz symmetry violation(LV) bµon the Klein tunneling\neffect.2The complete analysis is presented in Ref. 3. For simplicity, we\nassume that we are working in a particular inertial reference frame , where\nall the components of bµremain constant. The hamiltonian of the LV\nmodified Dirac equation is\nˆHb=/vector α·ˆ/vectorP+γ0m−b0γ5+/vectorb·/vectorΣ+U(Z), (1)\nwhereU(Z) =V0[Θ(Z)−Θ(Z−L)] and Σi≡1\n2ǫijkσjk, withǫijkbeing a\ntotally antisymmetric tensor with ǫ123= 1. Next, we will use the spacelike\nbµterm asan example toshow the LVimpact on the Klein tunneling, which\nlets us get a glimpse of how the LV impact depends on the observerLo rentz\nnature of bµ.\n2. The impact of the LV bµterm on Dirac tunneling\nFor spacelike bµ, we can use observer Lorentz symmetry to transform it\nintobµ= (0,/vectorb), and further assume /vectorb(θ,φ) =b(sinθcosφ,sinθsinφ,cosθ)Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\nfor convenience. The static wave function can take the form\nΨR(Z,T) =e−iET/braceleftBig\n[φi(p)eipZ+φr(p′)e−ip′Z]Θ(−Z)\n+[φf(q)eiqZ+φg(q′)e−iq′Z][Θ(Z)−Θ(Z−L)]\n+φt(p)eipZΘ(Z−L)/bracerightbig\n, (2)\nwhereφn=/parenleftbiggξn\nηn/parenrightbigg\nwith subscript n=i,r,f,g,t respectively. According to\nthe Dirac equation i˙ΨR(Z,T) =ˆHbΨR(Z,T), eachφnmust satisfy\n/parenleftBigg\nm−[E−U(Z)]+/vectorb·/vector σ /vector σ ·ˆ/vectorP\n/vector σ·ˆ/vectorP [U(Z)−E]−m+/vectorb·/vector σ/parenrightBigg\nφn= 0.(3)\nFrom the lower equation of (3), we determine ηnand substitute it back into\nthe upper equation of (3) to get the dispersion relation\n(p2+m2−E2)2−2b2(p2cos2θ+E2+m2)+b4= 0. (4)\nA similar relationapplies to qwith the replacements p→q, E→E−V0in\nEq. (4). Assume further that ξn≡ Nξi, whereN=R,F,G,Trespectively.\nBy using the continuity equation at points Z= 0, L, in principle we can\nsolve for all the proportionality constants R,F,G,T. For example, using\nthe method above, we can determine the transmission amplitude as\nT=e−ipL\ncos[qL]−i\n2(Kp\nq+q\nKp)sin[qL], (5)\nwhereK ≡1−V0\nE+m−bs, s=±1, and we have used p=p′,q=q′since\nEq. (4) is an even function of p. Next, supposing /vectorb=bˆeZ, we get from\nEq. (5) the resonant energy\nERes(n,s) =V0+bs+/radicalbig\nm2+(nπ/L)2 (6)\nwhenE > V 0+m+bs. This is the relativistic counterpart of ordinary\nquantum-mechanical resonant transmission.4For a specific resonance num-\nbern, the resonant energy difference between states with opposite he lic-\nity isδERes(n)≡ERes(n,+1)−ERes(n,−1) = 2b. For a specific barrier\nheightV0and energy Eof the incoming electron, the resonant barrier\nlength is L(n,s) =nπ√\n(E−V0−bs)2−m2. The resonant-length difference be-\ntween opposite-helicity states with the same nis\nδL(n) =L(n,+1)−L(n,−1)≃2bE−V0\n(E−V0)2−m2LLI(n),(7)Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\nwhereLLI(n) =nπ√\n(E−V0)2−m2is the Lorentz-invariant resonant-barrier\nlength. So the resonant-length difference δL(n) increases with the bar-\nrier length L. In principle, we may use this and a precisely controllable\nlong barrier to amplify the tiny LV bµeffect. See Fig. 1, where we plot\nwith an impractically large b∝0.001meto make the LV helicity-dependent\nshift of the resonant spectrum distinguishable. A complete discuss ion using\nother types of bµis similar; see Ref. 3.\nLVs/Equal1\nLVs/Equal/Minus1\nLICase\n2L0 3L0 4L0L0.20.40.60.81.0Trate\nFig. 1. Klein tunneling rate as a function of barrier length. The solid black curve\ncorresponds to the Lorentz-invariant (LI) tunneling rate, while the dashed gray and\nblack curves correspond to the Lorentz-violating tunnelin g rate with helicity s= +1,−1\nrespectively. The length unit is L0=π//radicalbig\n(E−V0)2−m2.\nAcknowledgments\nThe author appreciates valuable discussions with Alan Kosteleck´ y, Hai\nHuang, Ralf Lehnert, and Herb Fertig, and is grateful for IUCSS h ospi-\ntality and for study-abroad funding from the Chinese Scholarship C ouncil.\nReferences\n1. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998); V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n2. O. Klein, Z. Phys. 53, 157 (1929).\n3. Z. Xiao, Phys. Rev. D 93, 125022 (2016).\n4. Z. Xiao, H. Huang, and X.X. Lu, Int. J. Mod. Phys. B 29, 1550052 (2015)."    },    {        "title": "1201.6047v1.The_exponential_of_the_spin_representation_of_the_Lorentz_algebra.pdf",        "content": "arXiv:1201.6047v1  [math-ph]  29 Jan 2012The exponential of the spin representation of\nthe Lorentz algebra\njason hanson∗\nAugust 14, 2018\nAbstract\nAs discussed in a previous article, any (real) Lorentz algeb ra ele-\nment possess a unique orthogonal decomposition as a sum of tw o mu-\ntually annihilating decomposable Lorentz algebra element s. Inthis ar-\nticle, this concept is extended to thespin representation o f the Lorentz\nalgebra. As an application, a formula for the exponential of the spin\nrepresentation is obtained, as well as a formula for the spin represen-\ntation of a proper orthochronous Lorentz transformation.\n1 Orthogonal decomposition\nLetgbe a Lorentz metric on R4. That is, gis a symmetric nondegenerate\ninner product with determinant −1. For our purposes, we need not specify\na signature for g. TheLorentz group O(g) is the Lie group of linear transfor-\nmations Λ on R4such that ΛTgΛ =g, and the Lorentz algebra so(g) is the\nLie algebra of transformations Lfor which LTg+gL= 0.\n1.1 Decomposition of bivectors\nElements of so(g) are called Lorentz bivectors. A special type of Lorentz\nbivector is the simple bivector , which takes the form u∧gvfor four–vectors\nu,vinR4, where\n(u∧gv)(w).=g(v,w)u−g(u,w)v (1)\n∗jhanson@digipen.edu\n1whenappliedtothefour–vector w. Inindexnotation,( u∧gv)α\nβ=uαvβ−vαuβ.\nSimple bivectors arealso called decomposable bivectors, and arecharacterized\nby the condition det( u∧gv) = 0.\nWhile not every Lorentz bivector Lis simple, it is the sum of simple\nbivectors. In fact, it can be shown that any nonsimple Lorentz bive ctorL\nadmits an orthogonal decomposition :L=L++L−, withL±simple and\nL+L−= 0 =L−L+. This decomposition is unique. Indeed, the summands\nof the decomposition of Lare given by\nL±=±L3−µ∓L\nµ+−µ−(2)\nwhereµ±are the positive and negative roots of the equation x2+(tr2L)x+\ndetL= 0, or equivalently, the solutions of the simultaneous equations\nµ++µ−=−tr2Landµ+µ−= detL. (3)\nHere tr 2Lis the second order trace of L, which can be computed by the\nformula tr 2L=−1\n2trL2(this identity holds for any traceless matrix). In\nparticular, tr 2L±=−µ±. See [3] for details.\n1.2 Spin representation of the Lorentz algebra\nRepresentations of so(g) may be constructed from representations of the\nClifford algebra Cl(g) onR4. Recall that Cl(g) is the quotient of the tensor\nalgebraT∗(R4)bythesubalgebrageneratedbytherelation uv+vu= 2g(u,v)\nforu,v∈R4. Letρbea Cliffordalgebra representation; i.e., a(possibly com-\nplex) vector space Vand a linear map ρ:Cl(g)→Hom(V,V) that respects\nClifford multiplication: ρ(uv) =ρ(u)ρ(v). We obtain the spin representation\nσ:so(g)→Hom(V,V) by setting\nσ(u∧gv).=1\n4ρ(uv−vu) (4)\nfor simple Lorentz bivectors, and extending linearly to all of so(g). One\nshows that σis a Lie algebra homomorphism: σ([L1,L2]) =σ(L1)σ(L2)−\nσ(L2)σ(L1) for all Lorentz bivectors L1,L2(see [2], for example).\nA natural choice for the representation ρwould be gamma matrices; i.e.,\nρ(u).=uαγα. However, arepresentationmaybeconstructeddirectlyfromth e\nClifford algebra itself: view V=Cl(g) as a sixteen–dimensional real vector\n2space, andtake ρtobetheidentity. Unlikethegammamatrixrepresentation,\nthis is not an irreducible Clifford algebra representation. In the follow ing,\nwe will not have the need to make a specific choice for ρ, and we simply\nrefer toσas “the” spin representation of so(g). We remark that all formulas,\nwith the exception of those that appear in section 1.4, are actually v alid\nfor summands of the spin representation. In particular, they are valid for\nhalf–spin representations.\nThey key property of the spin representation σthat we will make use\nof is the following, which makes apparent the usefulness of decompo sing a\nbivector into a sum of simple bivectors. Here we write I=ρ(1).\nTheorem 1. SupposeL=u∧gvis a simple Lorentz bivector. Then tr2L=\ng(u,u)g(v,v)−g(u,v)2andσ(L)2=−1\n4(tr2L)I.\nProof.From equation (1), one computes that tr 2L=−1\n2trL2is given by the\nstated expression (see also [3]). Now compute using equation (4) an d the\nClifford algebra relation:\nσ(u∧gv)2=1\n16ρ(uvuv−uv2u−vu2v+vuvu)\n=1\n16ρ/parenleftbig\nu[−uv+2g(v,u)]v−g(v,v)g(u,u)\n−g(u,u)g(v,v)+v[−vu+2g(u,v)]u/parenrightbig\n=1\n16ρ/parenleftbig\n2g(u,v)(uv+vu)−4g(u,u)g(v,v)/parenrightbig\nwhich implies the stated expression for σ(L)2.\n1.3 Decomposition of a spin representation\nIfLis a Lorentz bivector, then L3is as well. So we may apply the spin\nrepresentation directly to each summand in equation (2) to obtain σ(L±) in\nterms ofσ(L) andσ(L3). However, we would like an expression that involves\nonly powers of σ(L).\nTheorem 2. IfL=L++L−is the orthogonal decomposition of a nonsimple\nLorentz bivector, then\nσ(L±) =±2\nµ+−µ−/braceleftbig1\n4(µ∓+3µ±)σ(L)−σ(L)3/bracerightbig\nwithµ±as in equation (3).\n3Proof.Since (∗)σ(L) =σ(L+) +σ(L−), we have that σ(L)3=σ(L+)3+\n3σ(L+)2σ(L−)+3σ(L+)σ(L−)+σ(L−)3. Using theorem 1 to reduce powers,\nwe may rewrite this as ( ∗∗)σ(L)3=1\n4(µ++3µ−)σ(L+)+1\n4(µ−+3µ+)σ(L−).\nThe determinant of the linear system ( ∗) and (∗∗) is1\n2(µ+−µ−), which is\nnonzero if Lis nonsimple, and the system may be solved to yield the stated\nexpressions for σ(L±).\nThe summands of the orthogonal decomposition of a nonsimple Lore ntz\nbivector are mutually annihilating. Although their images under the sp in\nrepresentation do not share this property, they do commute.\nTheorem 3. IfL=L++L−is the orthogonal decomposition of the nonsim-\nple Lorentz bivector L, thenσ(L+)σ(L−) =σ(L−)σ(L+) =1\n8(tr2L)I+1\n2σ(L)2.\nProof.AsL+,L−trivially commute, [ σ(L+),σ(L−)] =σ([L+,L−]) = 0. By\ntheorem 1 and equation (3), we then have σ(L)2=σ(L+)2+2σ(L+)σ(L−)+\nσ(L−)2= 2σ(L+)σ(L−)−1\n4(tr2L)I.\n1.4 A computational digression\nTo use the formula in theorem 2, we need to know the values of µ±, which\nare obtained from the invariants tr 2Land detLofL. However, we would\nlike to deduce these values in the event we only have knowledge of σ(L).\nLemma 1. For any four–vectors a,b,u,v\ntrρ(abuv) = trI{g(a,b)g(u,v)−g(a,u)g(b,v)+g(a,v)g(b,v)}.\nThis generalizes the well–known identity for gamma matrices, so we ne ed\nnot repeat the computation here. We do note however, that tr I= trρ(1) is\nthe dimension of the representation: tr I= 16 for the Clifford algebra Cl(g)\nitself, and tr I= 4 for the gamma matrix representation.\nLemma 2. IfL=L++L−is the orthogonal decomposition of a Lorentz\nbivector with L+=a∧gbandL−=u∧gv, then\ng(a,v)g(b,u)−g(a,u)g(b,v) = 0.\nProof.From equation (1), one computes ( a∧gb)(u∧gv) =g(b,u)avTg−\ng(b,v)auTg−g(a,u)bvTg+g(a,v)buTg. Taking the trace of both sides, we\nget tr{(a∧gb)(u∧gv)}= 2g(b,u)g(v,a)−2g(a,u)g(v,b), which is necessarily\nzero, since ( a∧gb)(u∧gv) =L+L−= 0.\n4Theorem 4. IfL=L++L−is the orthogonal decomposition of a Lorentz\nbivector, then tr{σ(L+)σ(L−)}= 0.\nProof.WriteL+=a∧gbandL−=u∧gv. Thenσ(L+)σ(L−) =1\n16ρ(ab−\nba)ρ(uv−vu) =1\n16/parenleftbig\nρ(abuv)−ρ(abvu)−ρ(bauv)+ρ(bavu)/parenrightbig\n. Take the trace\nand apply the previous two lemmas.\nTheorem 5. For any Lorentz bivector L,tr2L=−4trσ(L)2/trIanddetL=\n4trσ(L)4/trI−4tr2σ(L)2/tr2I.\nProof.For the first formula, take the trace of the formula in theorem 3. F or\nthe second, use the orthogonal decomposition L=L++L−and theorem 1 to\ncompute σ(L)4={σ(L+)+σ(L−)}4=1\n16µ2\n+I−µ+σ(L+)σ(L−)+3\n8µ+µ−I−\nµ−σ(L+)σ(L−) +1\n16µ2\n−I. Taking traces, we get tr σ(L)4=1\n16(trI)(µ2\n++\n6µ+µ−+µ2\n−) =1\n16(trI){(µ++µ−)2+ 4µ+µ−}=1\n16(trI)(tr2\n2L+ 4detL),\ncourtesy of equation (3). Solving for det Land using the first formula yields\nthe desired expression.\n2 Exponential of the spin representation\nBy general principles, the exponential operation exp( L).=/summationtext\nn≥0Ln/n! is a\nmap exp : so(g)→O(g), whose image is the connected component SO+(g)\nofO(g) containing the identity transformation; i.e., the set of all proper\northochronous Lorentz transformations. A closed formula for e xp(L) was\nobtained in [1]. Here we give a closed expression for exp( σ(L)) for both\nsimple and nonsimple Lorentz bivectors.\nTheorem 6. IfLis a simple Lorentz bivector, then exp(σ(L)) = ¯c+¯sσ(L),\nwhere\niftr2L >0,¯c.= cos1\n2/radicalbig\ntr2Land¯s.=2√tr2Lsin1\n2/radicalbig\ntr2L\niftr2L <0,¯c.= cosh1\n2/radicalbig\n−tr2Land¯s.=2√−tr2Lsinh1\n2/radicalbig\n−tr2L\nand iftr2L= 0, then¯c= 1 = ¯s.\nProof.In the case tr 2L >0, theorem 1 implies σ(L)2p= (−θ2)p= (−1)pθ2p,\nwhereθ.=1\n2√tr2L. Consequently, we may write σ(L)2p+1=σ(L)2pσ(L) =\n(−1)pθ2p+1σ(L)/θ. Thus the series/summationtext\nn≥0σ(L)n/n! =/summationtext\np≥0σ(L)2p/(2p)! +\n5/summationtext\np≥0σ(L)2p+1/(2p+1)! is summed using the usual Taylor series expansion\nfor sine and cosine. The case when tr 2L <0 is similar, and the case tr 2L= 0\nis trivial.\nRecall that exp( A+B) = exp( A)exp(B) whenever the matrices A,B\ncommute. Thus, exp( σ(L)) = exp( σ(L+))exp(σ(L−)). Theorems 3 and 6\nthen lead to the following.\nTheorem 7. Suppose L=L++L−is the orthogonal decomposition of a\nnonsimple Lorentz bivector. Define θ±.=1\n2√∓tr2L±,¯c+.= coshθ+,¯c−.=\ncosθ−,¯s+.= sinhθ+/θ+, and¯s−.= sinθ−/θ−. Then\nexp(σ(L)) = ¯c+¯c−+ ¯s+¯c−σ(L+)+¯c+¯s−σ(L−)+ ¯s+¯s−σ(L+)σ(L−)\nWe derive an alternative formula for exp( σ(L)) as a polynomial in σ(L).\nBy combining theorems 2 and 3 with theorem 7, we obtain the following.\nTheorem 8. IfLis a nonsimple Lorentz bivector, then\nexp(σ(L)) =α0+α1σ(L)+α2σ(L)2+α3σ(L)3\nα0.= ¯c+¯c−−1\n8(µ++µ−)¯s+¯s− α2.=1\n2¯s+¯s−\nα1.=1\n4N/braceleftbig\n(µ−+3µ+)¯s+¯c−−(µ++3µ−)¯c+¯s−/bracerightbig\nα3.=N(¯c+¯s−−¯s+¯c−)\nwithµ±as in equation (3),¯c±,¯s±as in theorem 7, and N.= 2/(µ+−µ−).\n3 Spin representation of a Lorentz transfor-\nmation\nThe spin representation σ:so(g)→Hom(V,V) on the Lie algebra level\ninduces a projective representation Σ : SO+(g)→SL(V)/±on the Lie\ngroup level ([4]). We would like to deduce an explicit formula for Σ.\nWe define a Lorentz transformation Λ ∈SO+(g) to besimpleif it is the\nimage of a simple Lorentz bivector under the exponential map. A crit erion\nfor simplicity is that tr 2Λ = 2(trΛ −1). Here, the second order trace may\nbe computed from the general formula tr 2Λ =1\n2(tr2Λ−trΛ2). Moreover, it\nshould be noted that for any proper orthochronous Lorentz tra nsformation\n(simple or not), trΛ ≥0. See [3] for more details.\n6We will need the following fact for computing the logarithm of a simple\nLorentz transformation, as given in [3]. The special case when trΛ = 0 will\nbe handled later.\nProposition 1. IfΛ∈SO+(g)is a simple Lorentz transformation with\ntrΛ>0, thenΛ = exp( L)andtr2L=−µ, whereL=1\n2k(Λ−Λ−1)and\n1. if0<trΛ<4, thenk=√−µ\nsin√−µand√−µ= cos−1(1\n2trΛ−1),\n2. iftrΛ>4, thenk=√µ\nsinh√µand√µ= cosh−1(1\n2trΛ−1),\n3. iftrΛ = 4, thenk= 0andµ= 0.\nTheorem 9. Suppose Λ∈SO+(g)is simple. If trΛ>0, then up to an\noverall sign,\nΣ(Λ) =1\n2√\ntrΛ/braceleftbig\ntrΛ+2σ(Λ−Λ−1)/bracerightbig\n.\nProof.LetLbe a simple Lorentz bivector such that Λ = exp( L). By the gen-\neral properties of the exponential map on a Lie algebra, Σ(Λ) = exp (σ(L)).\nWritingL=1\n2k(Λ−Λ−1) as in proposition 1, we have exp( σ(L)) = ¯c+\n1\n2k¯sσ(Λ−Λ−1),accordingtotheorem6. Thevaluesof ¯ c,¯sdependonthevalue\noftr2L. Weconsiderthecasetr 2L >0,sothat µ <0inthenotationofpropo-\nsition 1, which occurs when 0 <trΛ<4. The other two cases are similar. In\nthis case, we have cos√−µ=1\n2trΛ−1. Now, ¯ c= cos1\n2√tr2L= cos1\n2√−µ.\nSimilarly,\n1\n2k¯s=1\n2√−µ\nsin√−µ2√−µsin1\n2√−µ=sin1\n2√−µ\nsin√−µ=1\n2cos1\n2√−µ\nOn the other hand, we have cos21\n2√−µ=1\n2(1−cos√−µ) =1\n4trΛ, so that\ncos1\n2√−µ=±1\n2√\ntrΛ. Although the choice of Ldetermines the sign here,\nthe choice of Lsuch that exp( L) = Λ is not unique. Indeed, if we take\nL′=αL, withαchosen such that√tr2L′=√tr2L+2π, then exp( L′) = Λ.\nHowever,1\n2√−µ′=1\n2√−µ+π, so that exp( σ(L′)) =−exp(σ(L)).\nTo obtain an analogous formula for a nonsimple Lorentz transforma tion,\nwe will make use of the fact that such a transformation is a product of\n7commuting simple transformations. Indeed, since SO+(g) is exponential, we\nmay write Λ = exp( L) for some Lorentz bivector L. Using the orthogonal\ndecomposition L=L++L−, we have exp( L) = exp( L+)exp(L−). Taking\nΛ±.= exp(L±), we may write Λ = Λ +Λ−, with Λ +,Λ−commuting simple\nLorentz transformations. In [3], the following explicit formula is obta ined.\nProposition 2. IfΛ∈SO+(g)is nonsimple, then Λ = Λ +Λ−, whereΛ±\nare the commuting simple Lorentz transformations\nΛ±=±1\n2(c+−c−)/braceleftbig\n(1+2c±)I−Λ−1−(1+2c∓)Λ+Λ2/bracerightbig\nwithc±=1\n4(trΛ±√\n∆)and∆.= tr2Λ−4tr2Λ+8,c+>1, and1≤c−<1.\nUsing this decomposition, we use the fact that Σ is a group homomor-\nphismtowriteΣ(Λ) = Σ(Λ +)Σ(Λ−), whereeachfactorontherighthandside\nmay be computed using theorem 9. Note that Σ(Λ ±) necessarily commute.\nTheorem 10. IfΛis a nonsimple proper orthochronous Lorentz transfor-\nmation with 2+2trΛ+tr 2Λ/ne}ationslash= 0, then up to sign\nΣ(Λ) =1\n2√2+2trΛ+tr 2Λ/braceleftbig\n(2+trΛ+tr 2Λ−1\n4tr2Λ)\n+(trΛ+2) σ(Λ−Λ−1)−σ(Λ2−Λ−2)+σ(Λ−Λ−1)2/bracerightbig\nProof.For brevity, we set τ±\nk.= trkΛ±. From theorem 9, we compute Σ(Λ) =\nΣ(Λ+)Σ(Λ−) to be:\nΣ(Λ) =1\n4/radicalbig\nτ+\n1τ−\n1/braceleftbig\nτ+\n1τ−\n1+2τ+\n1σ(Λ−−Λ−1\n−)+2τ−\n1σ(Λ+−Λ−1\n+)\n+4σ(Λ+−Λ−1\n+)σ(Λ−−Λ−1\n−)/bracerightbig\n(5)\nNow from proposition 2, one shows that ( ⋆)τ1.= trΛ = 2( c++c−) and\nτ2.= tr2Λ = 4c+c−+ 2, and that ( ⋆⋆)τ±\n1= 2(1 +c±). Moreover, since for\nany Lorentz transformation Λ−1=g−1ΛTg, we find that\nΛ±−Λ−1\n±=∓1\n2(c+−c−)/braceleftbig\n2c∓(Λ−Λ−1)−(Λ2−Λ−2)/bracerightbig\n(6)\nWe now rewrite the summands of equation (5) in terms of Λ and its firs t and\nsecond order traces. For the first summand, using ( ⋆⋆) and (⋆) we obtain\nτ+\n1τ−\n1= 4(1+c+)(1+c−) = 2+2 τ1+τ2 (7)\n8For the second and third summands in (5), using ( ⋆⋆) and (6) one computes\nτ+\n1σ(Λ−−Λ−1\n−)+τ−\n1σ(Λ+−Λ−1\n+) = 2(1+ c++c−)σ(Λ−Λ−1)−σ(Λ2−Λ−2).\nThus by ( ⋆),\nτ+\n1σ(Λ−−Λ−1\n−)+τ−\n1σ(Λ+−Λ−1\n+) = (τ1+2)σ(Λ−Λ−1)−σ(Λ2−Λ−2) (8)\nFor the fourth summand in (5), we similarly compute (although we also need\nto use the fact that σ(Λ−Λ−1) andσ(Λ2−Λ−2) commute, as σis a Lie\nalgebra homomorphism)\n4σ(Λ+−Λ−1\n+)σ(Λ−−Λ−1\n−) =1\n(c+−c−)2/braceleftbig\n(2−τ2)σ(Λ−Λ−1)2\n+τ1σ(Λ−Λ−1)σ(Λ2−Λ−2)−σ(Λ2−Λ−2)2/bracerightbig\n(9)\nHowever, the terms in this expression are not algebraically independ ent. To\nfind a relation, we make use of the fact that Λ ±are simple: from theorem\n1,σ(L±)2=−1\n4tr2L±, whereL±are such that exp( L±) = Λ±, and may be\ncomputed using proposition 1. Indeed, L+=1\n2k+(Λ+−Λ−1\n+), with tr 2L+=\n−µ+,c+= cosh√µ+, andk+=√µ+/sinh√µ+. The equation σ(L+)2=\n−1\n4tr2L+and (6) then lead to\n4c2\n−σ(Λ−Λ−1)2−4c−σ(Λ−Λ−1)σ(Λ2−Λ−2)+σ(Λ2−Λ−2)2\n= 4(c+−c−)2(c2\n+−1)\n(note that sinh2√µ+=c2\n+−1). Similarly, σ(L−)2=−1\n4tr2L−, proposition\n1, and (6) imply\n4c2\n+σ(Λ−Λ−1)2−4c+σ(Λ−Λ−1)σ(Λ2−Λ−2)+σ(Λ2−Λ−2)2\n= 4(c+−c−)2(c2\n−−1)\nAdding these two equations together and using ( ⋆) then yields the relation\n(τ2\n1−2τ2+4)σ(Λ−Λ−1)2−2τ1σ(Λ−Λ−1)σ(Λ2−Λ−2)\n+2σ(Λ2−Λ−2)2= (c+−c−)2(τ2\n1−2τ2−4) (10)\nCombining (9) and (10) together, we see that we may write the four th sum-\nmand in (5) as\n4σ(Λ+−Λ−1\n+)σ(Λ−−Λ−1\n−) = 2σ(Λ−Λ−1)2−1\n2(τ2\n1−2τ2−4) (11)\nCombining (5), (7), (8), and (11) give the desired formula for Σ(Λ) .\n93.1 Special case\nWe consider the case when a Lorentz transformation Λ ∈SO+(g) satisfies\nthe identity\n2+2trΛ+tr 2Λ = 0 (12)\nIn the case when Λ is simple, so that tr 2Λ = 2(trΛ −1), this condition\nreduces to trΛ = 0. It should be noted that a simple Lorentz transf ormation\nis traceless only if Λ = exp( L) for some simple Lorentz bivector with tr 2L=\nπ2. A nonsimple transformation satisfies (12) only if its decomposition in\nproposition 2, Λ = Λ +Λ−, is such that the simple factor Λ −is traceless.\nWe will not be able to obtain an explicit formula for the spin represen-\ntation Σ(Λ) of such a Lorentz transformation. However, we can g ive an\nalgorithm. The key lies in the following two facts from [3].\nProposition 3. Letu,vbe four–vectors, and set L.=u∧gv. The two–\nplanePspanned by u,vis nondegenerate (that is, gis nondegenerate when\nrestricted to P) if and only if tr2L/ne}ationslash= 0, in which case PL.=−L2/tr2Lis\ng–orthogonal projection onto P. Conversely, if Pisg–orthogonal projection\nonto a two–plane, then the two–plane is nondegenerate and P=PL, where\nL=u∧gvandu,vare any linearly independent four–vectors in the image\nofP.\nProposition 4. IfΛis a simple Lorentz transformation with trΛ = 0, then\nΛ2=I,PΛ.=1\n2(I−Λ)isg–orthogonal projection onto a nondegenerate\ntwo–plane, and −π2PΛis the square of a simple Lorentz bivector LΛwith\nexp(LΛ) = Λandtr2LΛ=π2.\nTheorem 11. SupposeΛ∈SO+(g)is simple with trΛ = 0. Letu,vbe any\ntwo linearly independent vectors in the image of PΛ=1\n2(I−Λ). Then up to\nsign,Σ(Λ) = (2 //radicalbig\ntr2(u∧gv))σ(u∧gv).\nProof.SetL.=u∧gv. ThenPLisg–orthogonal projection onto the (nec-\nessarily nondegenerate) two–plane Pspanned by u,v. By uniqueness of\ng–orthogonal projection, we must have PL=PΛ. NowLΛ=u′∧gv′,\nwhereu′,v′lie inP. Writing u′=au+bvandv′=cu+dv, we com-\npute that u′∧gv′= (ad−bc)u∧gv; i.e.,LΛ=αLfor some scalar α.\nTaking 2–traces, we get π2= tr2LΛ=α2tr2L, so that α=±π/√tr2L.\nSince exp( LΛ) = Λ, Σ(Λ) = exp σ(LΛ). On the other hand, by theorem 6,\nexpσ(LΛ) = (2/π)σ(LΛ) = (2α/π)σ(L).\n10References\n[1] Bortolom´ e Coll and Fernando San Jos´ e, On the exponential of the 2–\nforms in relativity, General Relativity and Gravitation, 22 (7), 811–826,\n1990.\n[2] WilliamFultonandJoeHarris, Representation Theory, Springer–Verlag,\n1991.\n[3] jason hanson, Orthogonal decomposition of Lorentz transformations,\narXiv:1103.1072v1 [gr-qc].\n[4] Eugene Wigner, Unitary representations of the inhomogeneous Lorentz\ngroup,Annals of Mathematics, 40, 149–204, 1939.\n11"    },    {        "title": "1512.03610v1.Ultra_low_magnetic_damping_of_a_metallic_ferromagnet.pdf",        "content": "Ultra -low magnetic damping of a metallic ferromagnet  \nMartin A. W. Schoen,1, 2 Danny Thonig,3 Michael L. Schneider,1 T. J. Silva,1 Hans T. Nembach,1 Olle \nEriksson,3 Olof Karis,3 and Justin M. Shaw1* \n \n1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO, 80305 , USA  \n2Institute of Experimental and Applied Physics, University of Regens burg, 93053 Regensburg, Germany   \n3Department of Physics and Astronomy, University Uppsala S -75120 Uppsala, Sweden  \nDated: 12/11/2015   \nPACS numbers: 71.70.Ej,76.50.+g, 75.78.n,71.20.Be  \n*Corresponding author: justin.shaw@nist.gov  \n \n \nThe phenomenology of magnetic damping is of critical importance for devices that seek to exploit \nthe electronic spin degree of freedom since damping strong ly affects the energy required and speed \nat which a device can operate. However, theory has struggled to quantitatively predict the \ndamping, even in common ferromagnetic materials1–3. This presents a challenge for a broa d range \nof applications in spin tronics4 and spin -orbitron ics that depend on materials and structures with \nultra -low damping. Such systems enable many experimental investigations that further our \ntheoretical understanding of  numerous magnetic phenomena such  as damping and spin -\ntransport5 mediated by chirality6 and the Rashba effect. Despite this requirement, it is believed  \nthat achieving ultra -low damping in metallic ferromagnets is limited  due to the scattering of \nmagnons by the conduction electrons. However, we report on a binary alloy of Co and Fe that \novercomes this obstacle and exhibits a damping parameter approaching 10-4, which is comparable \nto values reported  only for ferrimagnetic insulators7,8. We explain this phenomenon by a unique \nfeature of the bandstructure  in this system: T he density of states exhibits a sharp minimum at the \nFermi level at the same alloy concentration at whic h the minimum in the magnetic damping is \nfound. This discovery provides both a significant fundamental understanding of damping \nmechanisms as well as a test of the theoretical predictions put forth by Mankovsky et al.3.  \n \nIn recent  decades, several theoretical appr oaches have  attempt ed to quantitatively  predict \nmagnetic damping  in metallic systems.  One of the  early promising theories was that of Kambersky , \nwho introduced the so -called breathing Fermi surface  model .9–11 More recently,  Gilmore  and Stiles2 as \nwell as Thonig  et al.12 demonstra ted a generalized torque  correlation model that include  both intraband  \n(conductivity -like) and  interband (resistivity -like) transitions.  The use of scattering theory  to describe \ndamping was later applied by Brataas et al.13 and Liu et al.14 to describe  damping  in transition metals . \nA numerical realization of a linear response damping model  was implemented  by Mankovsky3 for Ni -\nCo, Ni -Fe, Fe -V and Co -Fe alloys. For the Co -Fe alloy , these calculations predict a minimum  intrinsic \ndamping  of αint ≈ 0.000 5 at a Co -concentration of 1 0 % to 20 %, but was not experimentally \nobserved.15  \nUnderlying this theoretical work is the goal of achieving new systems with ultra -low damping \nthat are required in many magnonic and spin-orbitronics applications .7,8  Ferrimagnetic insulators such \nas yttrium -iron-garnett (YIG) have long been the workhorse for these investigations since  YIG films \nas thin as 25 nm have experimental damping parameters  as low as 0.9×10−4.16 Such ultra-low damping \ncan be achieved in insulating ferrimagnets in part due to the absence of conduction electrons and \ntherefore the suppression of magnon -electron scattering. H owever, insulators cannot  be used in most  \nspintronic and spin -orbitronic applications where a charge current through the magnetic material is \nrequired  nor is the requirement of growth on gadolinium g allium g arnet  templates compatible with \nspintronics and c omplementary metal -oxide semiconductor (CMOS) fabrication processes.  One \nproposed alternative class of materials are Heusler alloys, some  of which are theoretically predicted to \nhave damping parameters as low as 10-4.17 While such values have yet to be realized, damping \nparameters as low as 0.001 have been reported for Co 2FeAl18 and NiMnSb19. However, Heusler alloys have non -trivial fabrication constraints, such as high -temperature annealing , which  are also \nincompatible with spintronic and CMOS device fabrication constraints . \nIn contrast, metallic ferromagnets such a s 3d transition metals are ideal  candidate  materials for \nthese applications since high quality ma terials can be produced at room temperature (RT)  without the \nrequirement of annea ling. However, ultra -low damping is thought to be unachievable in metallic \nsystems since damping in conductors is dominated by magnon -electron scattering in the conduction \nband  resulting in a damping parameter  over an order  of magnitude higher than those found in high -\nquality YIG.   \n Inspired  by Mankovsky’s theoretical prediction of ultra -low damping in the CoxFe1−x alloy system3, \nwe systematically studied the compositional dependence of the  damping parameter in Co xFe1−x alloy s, \ninclu ding careful evaluation of spin -pumping and radiative damping contributions.  Polycrystalline \nCoxFe1−x alloy films , 10 nm thick,  were sputter -deposited at RT with  Cu/Ta seed and capping layers . \nX-ray diffraction ( XRD) reveals that  the CoFe  alloys display a body -centered -cubic (bcc) phase over a \nCo concentration of 0 % to 60 %, a face -centered -cubic (fcc) phase above 80 % Co, and a mixed phase \nbetween 60 % and 80 % Co, in good agreement with the bulk phase diagram of this system. The \ndamping  parameter is determined from broadband ferromagnetic resonance (FMR) spectroscopy \nwhich measures the susceptibility over frequencies spanning 5  GHz to 40 GHz. An example of 𝑆21(𝐻) \nvector -network -analyzer transmission data is shown in Figure 1a  and b, toge ther with fits to the \ncomplex susceptibility for the real and imaginary parts, respectively. The total damping parameter αtot \nis determined from  the frequency depen dence of the linewidth obtained from these su sceptibility fits, \naccording to equation (1),  \n  \n∆𝐻=2ℎ𝛼tot\n𝑔μ0μB 𝑓+∆𝐻0     (1) \n \nwhere µ0 is the vacuum permeability, µB is the Bohr magneton, h is Planck’s constant, g is the Landé \ng-factor , and ΔH0 is the inhomogeneous linewidth . \n \n   \nFigure 1: Ferromagnetic resonance spectra, measured via FMR and the resulting linewidth as a \nfunction of frequency . a and b, respectively , show the real and imaginary part s of the  S21(H) \ntransmission parameter (black circles) with the complex susceptibility fit (red li nes).  c, the line -widths \n(symbols) are plotted versus the frequency for Co, Fe, Co 20Fe80 and Co 25Fe75. The uncertainties in the \nline-widths were ob tained by means of the standard method for the determination of  confidence limits \non estimated parameters for nonlinea r models under the assu mption of Gaussian white noise . The lines \nare error -weighted fits to equation (1), which are used to determine  both the total damping αtot and the \ninhomogeneous linewidth broadening for each alloy.  \n \nThe measured total damping αtot versus alloy composition for 10 nm films is plotted in Figure  2. αtot  \nshows a clear minimum of  (2.1 ±  0.15)  × 10−3 at a Co concentration of 25  %. However, as a result of the \nutilized measurement  geometry and the structure of the sample, there are several extrinsi c contributions \nto αtot that are independent of αint.  \nThe first contribution —the result of the inductive coupling of the precessing magnetization and the \ncoplanar waveguide  (CPW)—is radiative damping αrad.20 The FMR system is designed and optimized \nto couple microwaves into the ferromagnet, and therefore, by virtue of reciprocity, the system is efficient \nat coupling mi crowaves out of the ferromagnet.  For very thin films or films with low saturation \nmagnetization , αrad is typically not a significant contribution to the total damping and can be ignored. \nHowever, in the present case, αrad must be accounted for in the anal ysis due to the combination of a very \nhigh saturation magnetization and the exceptionally small value of αint. As described in the supplemental \ninformation section  (SI), we calculate and experimentally validate the contribution of αrad to the total \ndamping, which is  plotted in Figure  2. \n \n \n \n \n \nFigure 2: The total measured damping with radiative and interfacial contributions. The total \ndamping αtot (red crosses with lines), spin - pumping αsp (gray line)  and radiative αrad (green line) \ncontributions to the damping, and the intrinsic damping αint are plotted against the respective Co \nconcentration. The errors are propagated from the line -width fits.  The crystal structure of the alloys, \nobtained from XRD, signified by the color  regions in the plot . \n  \nThe second non -negligible contribution to the total damping is the damping enhancement due to \nspin-pumping  into the adjacent Cu/Ta layers.  The spin-pumping contribution αsp can be determined from \nthe thickness dependence of ( αtot - αrad) since it behaves as an interfacial damping term21. Indeed, we \nmeasured the thickness dependence of ( αtot - αrad) for many alloy samples in order to quantify and account \nfor αsp (see SI), which is displayed  in Figure 2.   \nContributions from  eddy -current damping22 are estimat ed to be smaller than 5% and are neglected. \nFinally, two -magnon scattering is minimized in the perpendicular geometry used in this investigation \nand its contribution is disregarded23. \n \n  \n \nFigure 3: Calculated electron  density of states  (DOS ) and it s comparison to the intrinsic damping.  \na Electronic structure of bulk Co xFe1-x. The DOS  is shown for different Co concentrations, as indicated. \nEF is the Fermi energy . b The extracted intrinsic damping (black squares, left axis) is compared to the \ntheory in Mankovsky et al.3 , for a temperature of 0 K (blue line)  and for a temperature of 300 K for \npure Fe (blue cross).  The calculated DOS  at the Fermi energy  n(EF) is plotted on the right axis  (red line) . \nThe y-offset of n(EF) is chosen deliberately in order to demonstrate that the concentration -dependent \nfeatures o f the damping directly correlate to features of n(EF). We cannot exclude concentration -\nindependent contributions to the damping, which are accounted for by the 0.4 eV-1 y-offset.  \nThe total measured damping becomes , αtot =  αint +  (αrad /2) +  αsp, allowing the intrinsic damping αint \nto be determined, which is presented in Figure  2. For many values of  αint, the contributions of αsp and \nαrad are of similar  magnitude, showing the importance of accounting for these  contributions . For 25 % \nCo αint now di splays a sharp minimum in damping of (5 ± 1.5) × 10−4, which  is astonishing for a \nconductor.  Indeed, αint < 0.001 have been measured only  in ferrimagnetic insulators .24  \nThese results raise the question why αint can be  so low in the presence of conduction electrons. To \ngain a deeper understanding, we performed electronic structure calculations for Co xFe1-x within a full -\nrelativistic , multiple -scattering approach (Korringa -Kohn -Rostoker method25, KKR) using the coherent \npotential approximation (CPA)26,27 over the entire  range of compositions  (see SI).  Several  representative  \nexamples are given  in Fig ure 3a.  \nThe d−states  (peak in the DOS below EF) for pure Fe are not fully occupied . Consistent with  the rigid \nband model28, the d−states shift to lower energies when the concentration of Co increases , and become \nfully oc cupied at 25 %  Co, coinciding with the minimum in n(EF) displayed in Figure 3a , which \noriginates from the hybridization between majority Fe eg and minority Co t2g states.  \nEbert et al.1 suggested that αint is proportional to n(EF) in the breathing Fermi -surface model  (i.e., \nintraband transitions)  in the  cases of a minimally  varying spin -orbit coupling  (SOC ) (as is the case for \nthe Co xFe1-x system)  and small electron -phonon  coupling2,29. Alternatively, i nterband transitions become \nsignificant only if bands have a finite overlap due to band broadening, caused for example  by coupling \nto the phonon s.  As a result, interband transitions suppressed at low -temperature and energy dissipatio n \nbecomes dominated by intraband transitions. Our RT measurements of Co xFe1-x satisfy this “low -\ntemperature” condition since the electron -phonon coupling is < 20 meV for pure bcc Fe, and < 30 meV \nfor pure hcp Co . Band broadening due to disorder is about 15  meV for the , at EF dominating,  eg states  \n(50 meV for the t 2g states)  in  Co25Fe75 and varies up to 55 meV for the e g states  (150 meV for the t 2g \nstates) over the whole range of composition. T hese calculations show that the band broadening  effect at \nRT is too small to provide significant interband damping, consistent with the  almost perfect \nproportionality between n(EF) for all alloy compositions in the bcc phase (0 % to 60 % Co). Such a \nproportionality requires an offset of 0.4 eV-1, which originates f rom the fact the n(EF) is a superposition \nof all states, some of which do not contribute significantly to the damping.   \nThe calculations of αint by Mankovsky3 (included in Fig 3b) show a minimum value of αint ≈ 0.0005 \nbetween 10 % and 20 % Co, which differs from the sharp minimum we find at 25 % Co in both the \nexperimental data and the calculated values of n(EF). Remarkably, with the exception of pure Fe, the \ncalculated values of αint (at 0 K) agree with our r esults within a factor of ≈ 2. Furthermore, the agreement \nis greatly improved for pure Fe, when a finite temperature of 300 K is included. While not perfect, the \nagreement between those calculations and our results is compelling and provides the critical f eedback \nneeded for further refinement of theory.  \nWe therefore demonstrate and conclude that αint is largely determined by n(EF) in the limit of \nintraband scattering. Secondly, our work shows that a theoretical understanding of damping requires an \naccurate account of all contributions to the damping parameter.  Furthermore, if the theoretical \nexplanation put forth here to explain low damping holds in general, it is natural to utilize data -mining \nalgorithms to screen larger groups of materials in o rder to iden tify additional low -damping systems. \nExamples of such studies to identify new materials for use , for example,  in scintillators have been \npublished30, and the generalization to applications in mag netization dynamics is straight forward . \n \nAuthor responsibility  \nMAWS and DT wrote the manuscript, JS and HN conceived of the experiment, MAWS deposited the \nsamples, and performed the SQUID measurements and analysis, MAWS, MLS and HN performed the \nFMR measurements and analysis, JS performed the XRD measurements and anal ysis, DT and OE \nperformed the first -principle s DFT calculations. All authors contributed to the interpretation of the \nresults.  \n \nAcknowledgements  \n \nThe authors are grateful to Mark Stiles and Eric Edwards for valuable discussions.  \n \nReferences.  \n1. Ebert, H., Mankovsky, S., Ködderitzsch, D. & Kelly, P. J. Ab Initio Calculation of the Gilbert Damping \nParameter via the Linear Response Formalism. Phys. Rev. Lett.  107, 66603 –66607 (2011).  \n2. Gilmore, K., Idzerda, Y. U. & Stiles, M. D. Identification of the Dominant Precession -Damping Mechanism \nin Fe, Co, and Ni by First -Principles Calculations. Phys. Rev. Lett.  99, 027204 (2007).  \n3. Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. First -principles calculation o f the Gilbert \ndamping parameter via the linear response formalism with application to magnetic transition metals and \nalloys. Phys. Rev. B  87, 014430 (2013).  \n4. Žutic, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys . 76, 323 –\n410 (2004).  \n5. Emori, S., Bauer, U., Ahn, S. -M., Martinez, E. & Beach, G. S. D. Current -driven dynamics of chiral \nferromagnetic domain walls. Nat Mater  12, 611 –616 (2013).  \n6. Jué, E. et al.  Chiral damping of magnetic domain walls. accepted in Nat. Materials  (2015).  \n7. Allivy Kelly, O. et al.  Inverse spin Hall effect in nanometer -thick yttrium iron garnet/Pt system. Applied \nPhysics Letters  103, (2013).  \n8. Onbasli, M. C. et al.  Pulsed laser deposition of epitaxial yttrium iron garnet films with l ow Gilbert damping \nand bulk -like magnetization. APL Materials  2, (2014).  \n9. Kamberský, V. FMR linewidth and disorder in metals. Czechoslovak Journal of Physics B  34, 1111 –1124 \n(1984).  \n10. Kamberský, V. On ferromagnetic resonance damping in metals. Czechosl ovak Journal of Physics B  26, \n1366 –1383 (1976).  \n11. Kambersky, V. & Patton, C. E. Spin -wave relaxation and phenomenological damping in ferromagnetic \nresonance. Phys. Rev. B  11, 2668 –2672 (1975).  \n12. Thonig, D. & Henk, J. Gilbert damping tensor within the b reathing Fermi surface model: anisotropy and non -\nlocality. New J. Phys.  16, 013032 (2014).  \n13. Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Scattering Theory of Gilbert Damping. Phys. Rev. Lett.  101, \n037207 (2008).  14. Liu, Y., Starikov, A. A., Yuan, Z. & Kelly, P. J. First -principles calculations of magnetization relaxation in \npure Fe, Co, and Ni with frozen thermal lattice disorder. Phys. Rev. B  84, 014412 (2011).  \n15. Oogane, M. et al.  Magnetic Damping in Ferromagnetic Thin Films. Japanese Journal of Applied Physics  45, \n3889 –3891 (2006).  \n16. Chang, H. et al.  Nanometer -Thick Yttrium Iron Garnet Films With Extremely Low Damping. Magnetics \nLetters, IEEE  5, 1–4 (2014).  \n17. Liu, C., Mewes, C. K. A., Chshiev, M., Mewes, T. & Butler, W. H. Origin of low Gilbe rt damping in half \nmetals. Applied Physics Letters  95, (2009).  \n18. Mizukami, S. et al.  Low damping constant for Co2FeAl Heusler alloy films and its correlation with density \nof states. Journal of Applied Physics  105, (2009).  \n19. Dürrenfeld, P. et al.  Tunabl e damping, saturation magnetization, and exchange stiffness of half -Heusler \nNiMnSb thin films. ArXiv e -prints  (2015).  \n20. Schoen, M. A. W., Shaw, J. M., Nembach, H. T., Weiler, M. & Silva, T. J. Radiative damping in waveguide -\nbased ferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered \npermalloy films. Phys. Rev. B  92, 184417 (2015).  \n21. Tserkovnyak, Y., Brataas, A. & Bauer, G. E. W. Enhanced Gilbert Damping in Thin Ferromagnetic Films. \nPhys. Rev. Lett.  88, 117601 (2 002).  \n22. Lock, J. M. Eddy current damping in thin metallic ferromagnetic films. British Journal of Applied Physics  \n17, 1645 (1966).  \n23. Hurben, M. J. &  Patton, C. E. Theory of two magnon scattering microwave relaxation and ferromagnetic \nresonance linewidth in magnetic thin films. Journal of Applied Physics  83, 4344 –4365 (1998).  \n24. Sun, Y. et al.  Damping in Yttrium Iron Garnet Nanoscale Films Capped by P latinum. Phys. Rev. Lett.  111, \n106601 (2013).  \n25. Zabloudil, J., Hammerling, R., Szunyogh, L. & Weinberger, P. Electron Scattering in Solid Matter . \n(Springer: Berlin, 2005).  \n26. Durham, P. J., Gyorffy, B. L. &  Pindor, A. J. On the fundamental equations of the Korringa -Kohn -Rostoker \n(KKR) version of the coherent potential approximation (CPA). J. Phys. F  10, 661 (1980).  \n27. Faulkner, J. S. & Stocks, G. M. Calculating properties with the coherent -potential approxi mation. PRB 21, \n3222 (1980).  \n28. Stern, E. A. Rigid -Band Model of Alloys. Phys. Rev.  157, 544 (1967).  \n29. Kamberský, V. On the Landau -Lifshitz relaxation in ferromagnetic metals. Can. J. Phys.  48, 2906 (1970).  \n30. Ortiz, C., Eriksson, O. & Klintenberg, M. Data mining and accelerated electronic structure theory as a tool \nin the search for new functional materials. Computational Materials Science  44, 1042 –1049 (2009).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Methods: Ultra -low magnetic damping of a metallic \nferromagnet  \nMartin A. W. Schoen,1, 2 Danny Thonig,3 Michael Schneider,1 Thomas J. Silva,1 Hans T. Nembach,1 Olle \nEriksson,3 Olof Karis ,3 and Justin M. Shaw1* \n \n1Quantum  Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO, 80305  \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany  \n3Department of Physics and Astronomy, University Uppsala S -75120 Uppsala, Sweden  \n \nDated: 12/11/2015  \n PACS numbers: 71.70.Ej,76.50.+g, 75.78.n,71.20.Be  \n*Corresponding author:  justin.shaw@nist.gov  \n \n \nSample preparation  \nThe sample s were deposited  by DC magnetron sputtering at an Ar pressure of approximately 0.67 Pa ( 5 \n× 10-3 Torr) in a chamber with a base -pressure of less than 5 × 10-6 Pa (4 × 10-8 Torr).  The alloys were \ndeposited by co -sputtering from two targets with the deposition rates calibrated by X -ray reflectometry \n(XRR). The repeatability of the deposition rates was found to be better than 3% variation over the course \nof this study. For all deposited alloys, the combined deposition rate was kept at approximately 0.25 \nnm/s, to ensure similar growth conditions. Furthermore the Co 20Fe80 and the Co 25Fe75 samples were \nreplicated by depositing from single stoichiometric targets, to prove  the reproducibility of the results. \nSamples with a thickness of 10 nm were fabricated over the full alloy composition range and additional \nthickness series (7 nm, 4nm, 3nm and 2 nm) were fabricated for the pure elements and select \nintermediate alloy concentr ations (20% Co, 25% Co, 50% Co and 85% Co).  \n \nX-ray diffraction measurement.  \nThe crystal structure  of the alloys  was determined by X -ray diffraction XRD using an in -plane geometry \nwith parallel beam optics and a Cu K α X-ray source. The in -plane geometry all ows the signal from the \nCo-Fe alloys to be isolated from the high -intensity signal coming from the silicon substrate. These \nmeasurements yield both the in -plane lattice constant s and the crystal structure, as shown in the \nsupplemental material, section 1.  The deposition rates were calibrated using XRR using the same system \nconfigured for the out -of-plane geometry.  \n \nSQUID measurement.  \nWe measured the in -plane hysteresis curves at 300 K to determine the magnetic moment of the sample.  \nSample were first diced  with a precision diamond saw such that an accurate value of the volume of the \nsample could be calculated. The s aturation magnetization MS for all alloy samples is then determined \nby normalizing the measured moment to the volume of the CoFe in the sample . These values are  shown \nin the supplemental material.  \n \n \nVNA -FMR measurement  \nThe FMR measurement utilized a room -temperature -bore superconducting magnet, capable of applying \nstatic magnetic fields as high as 3 T. An approximately 150 nm p oly(methyl methacryl ate) (PMMA) \ncoat was first applied  to the samples to prevent electrical shorting of the co -planar waveguide (CPW) \nand to protect the sample surface. Sample were placed face down on the CPW and m icrowave fields \nwere applied in  the plane of the sample, at a frequenc ies that ranged from 10 GHz to 40 GHz. A vector -network -analyzer ( VNA ) was connected to both sides of the CPW and the complex S21(H) transmission \nparameter was determined. The iterative susceptibility fit of S21(H) was done with the method describe d \nby Nembach et al.1. All fits were constrained to a field window that was 3 times the linewidth around \nthe resonance field in order to minimize the fit residual. We verified that this does not change the results , \nbut reduces the influence of measurement noise on the error bars of the fitted values.  \n \nCalculation of the DOS  \nElectronic ground state calc ulations have been performed by a full -relativistic multiple -scattering \nGreen’s function method ( Korringa -Kohn -Rostoker method2, KKR ) that relies on the local s pin-\ndensity approximation (LDA) to density functional theory. We utilize d Perdew –Wang exc hange \ncorrelation functionals3–6.  \n \nIn our multiple -scattering theory , the electronic structure is described by scattering path operators τij \n(i,j lattice site indices)  2, where , in the spin -angular -mome ntum representation , we consider angular \nmomenta up to lmax = 3 and  up to 60 x 60 x 60 points in reciprocal space. The substitutional alloys are \ntreated within the coherent potential approximation (CPA)7,8. Co impurities in the Fe host lattice are \ncreated in the effective CPA medium by defect matrices. The CPA medium is described by scattering \npath operators. The site -dependent potentials are con sidered in the atomic sphere approximation \n(potentials are spherically symmetric within muffin -tin spheres and constant in the interstitials).  \nThe DOS is obtained from the integrated spectral density2 \n \n𝑛(𝐸)= 1\n𝜋∫ 𝐼𝑚[tr(𝐺(𝐸+i𝜂,𝒌))]d𝒌\nΩBZ \n  \nwith the small positive energy η. The integration in reciprocal space k runs over the first Brillouin \nzone ΩBZ.  \n \n \n \nReferences  \n \n1. Nembach, H. T. et al.  Perpendicular ferromagnetic resonance measurements of damping and Lande g -factor \nin sputtered (Co_2Mn)_1 -xGe_x films. Phys. Rev. B  84, 054424 (2011).  \n2. Zabloudil, J., Hammerling, R., Szunyogh, L. & Weinberger, P. Electron Scattering in Solid Matter . (Springer: \nBerlin, 2005).  \n3. Becke, A. D. Density -functional exchange -energy approximation with correct asymptotic behavior. Phys. Rev. \nA 38, 3098 –3100 (1988).  \n4. Langreth, D. C. & Mehl, M. J. Beyond the local -density approximation in calculations of gro und-state \nelectronic properties. Phys. Rev. B  28, 1809 –1834 (1983).  \n5. Perdew, J. P. et al.  Erratum: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient \napproximation for exchange and correlation. Phys. Rev. B  48, 4978 –4978 (1993).  \n6. Perdew, J. P. & Wang, Y. Accurate and simple analytic representation of the electron -gas correlation energy. \nPhys. Rev. B  45, 13244 –13249 (1992).  \n7. Asada, T. & Terakura, K. Generalized -gradient -approximation study of the magnetic and cohesive p roperties \nof bcc, fcc, and hcp Mn. Phys. Rev. B  47, 15992 –15995 (1993).  \n8. Paxton, A. T., Methfessel, M. & Polatoglou, H. M. Structural energy -volume relations in first -row transition \nmetals. Phys. Rev. B  41, 8127 –8138 (1990).  \n \n \n \n \n \n \n Supplemental Inform ation: Ultra -low magnetic damping of \na metallic ferromagnet  \nMartin A. W. Schoen,1, 2 Danny Thonig,3 Michael Schneider,1 Thomas J. Silva,1 Hans T. Nembach,1 Olle \nEriksson,3 Olof Karis,3 and Justin M. Shaw1 \n \n1Quantum  Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO, 80305  \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany*  \n3Department of Physics and Astronomy , University Uppsala S -75120 Uppsala, Sweden  \n \nDated: 12/11/2015   \n \nX-ray diffraction measurement  \n \nFigure S1 shows several in -plane geometry X -ray diffraction (XRD) spectra for a series of select alloys. \nFrom 100 % to 80 % Co, there is no evidence of a body -centered cubic peak and we can conclude that \nthe phase of the CoFe  is purely  face-centered cubic (fcc). At a concentration of 75% (not shown), a bcc \npeak emerges in the spectrum. An fcc (220) peak is always visible in the spectra due to the signal from \nthe Cu seed and capping layers. The presence of the signal from the C u layers complicates the analysis \nsince the point at which the crystalline phase transitions from a mixed fcc and bcc to a pure bcc phase  \nfor the Co xFe1-x alloys cannot be exactly determined. However , there is strong evidence that the phase \ntransition occurs at a Co concentration in the vicinity of 60 % to 70 % Co.  This can be observed in Fig \nS2b which shows a plot of the lattice constants as a function of the concentration. For samples with a \nCo conce ntration smaller than 70  %, the lattice constant for the fcc phase becomes constant at a value \nthat is expected for pure Cu.  This is in contrast to the rapidly changing value of the lattice constant \nobserved in alloys with a concentration of Co exceeding 7 0 %. To put these results in context , the bulk \nphase diagram for this system predicts a fcc to bcc transition at  approximately 75  % Co1. Thus, our \nresults shows that our thin films initiate a bcc phase at a slightly higher value of Co concentration.  \n \nFigure S1:  XRD spectra.  The spectra are shown for select alloy concentrations. The Cu fcc peak is \nalways visible and is overlaid with the fcc Co pe ak. The Co bcc peak is  visible only for Co concentrations \nbelow 70%.  \n \n \nIt is important to point out that we do not measure the expected hexagonal close -packed (hcp) crystal \nstructure for the pure Co  samples .1 This suppression of the hcp structure was previously found in growth \nof Co on Cu and explained by the strained growth of the Co on Cu suppressing the hcp phase.2–4 Indeed, \nwe confirm this result by comparing pure Co films grown with a Ta/Cu seed  and capping  layer and one \ngrown with only a  Ta seed and capping layer . Figure S2a shows an XRD plot in the vicinity of the \nhcp(010) peak for the 10 nm pure Co samples with a Ta/Cu seed and a Ta seed. The sample with Ta \nseed layer exhibits a clear hcp(010) peak, indicating an hcp structure. In contrast, the sample that \ninclude s Cu in the seed layer shows no evidence of the hcp(010) peak. .  \n \n \n  \nFigure S2:  XRD characterization.  In a the XRD spectrum of pure Co samples with a Ta and a Ta/Cu \nseed layer. The via XRD determined lattice constant is pl otted against the respective Co concentration \nin b. The crystalline structure is denoted and coded in color.  \n \nSaturation and effective magnetization.  \n \nThe magnetization of all of our samples were measured and verified using two approaches. The \nfirst is the direct measurement of saturation magnetization Ms by superconducting quantum interference \ndevice (SQUID) magnetometry. These measured values of Ms are displayed in Figure S3 as a function \nof the concentration of Co in the alloy. For comparison we also include the Slater -Pauling curve1,5,6 in \nthe plot. In the bcc phase , MS displays almost perf ect agreement with the Slater -Pauling curve indicative \nof the high quality of the samples used in this study. Howe ver, below 80 % Co, the Slater -Pauling curve \nunderestimates Ms slightly when the crystalline phase of the CoFe alloy becomes exclusively fcc.  \n Ferromagnetic resonance (FMR) also provides a measurement of the effective magnetization \nMeff, which is equal to Ms−Hk, where Hk is the perpendicular anisotropy of the material. As a result of \nthe very small thickness of our samples, we expect a non -negligible value of Hk to result from the \ninterfaces of the material with the seed and capping layers7,8. Indeed, Meff in Figure S3 are smaller than \nMs, indicative of the presence of a perpendicular interface anisotropy. If the difference between Meff and \nMs is interfacial in origin, a thickness dependence of the Meff should produce the condition of Ms=Meff \nas the thickness t→ ∞. For a select subset of alloys, we varied the thickness of the alloy and measured \nMeff. From an extrapolation of thicknes s t→ ∞ we determined Ms. These values are included in Figure \nS3 and show reasonable agreement with the values of Ms determined from SQUID magnetometry, \ndemonstrating the equivalency of both measurement methods.  \nThe minimum of the DOS curve at EF, shown in Figure 3, is coupled to the Slater -Pauling maximum of \nthe magnetization curve, which appears in the present work at Co concentration of approximately 25 %. \nFor this concentration the large peak in the DOS, immediately above the Fermi level, is entirely of spin-\ndown character, and alloys with higher Co concentration unavoidably populate this peak, which reduces \nthe magnetization. In the present work, and in several other theoretical9–11 and experimental12,13 works \nthe maximum of the magnetization (and the minimum in the density of states) appears between 20 % \nand 30 % Co, while in the work of Ebert14,15 and, thus, also in the calculations of Mankovsky16 it appears \nat lower Co concentrations of approximately 15 %. This may explain why in the work of Mankovsky16 \nthe theory predicts a minimum of the damping at this concentration.  \n \n  \nFigure S3:  The variation of the saturation magnetization and effective magnetization with \nconcentration.  Meff (black squares) determined from FMR and Ms (blue triangles) determined from \nSQUID magnetometry are plotted versus alloy concentration. For comparison , Ms is also determined \nby extrapolating  the linear regression of Meff  vs. 1/ t, as described in the text. The extrapolated values \nfor Ms are included in the plot (red crosses) and they match Ms determined by SQUID at those alloy  \nconcentrations reasonably well . This shows that SQUID and FMR meas urement are consistent with \neach other. For comparison , the Slater -Pauling curve1 is also shown (gray dotted line). The crystalline \nphase of th e alloys are also indicated in the figure.  \n \n \nCalculation and direct measurement of the radiative damping contribution . \n \nRadiative damping in the perpendicular FMR geometry, used in our study, arises from the inductive \ncoupling of the dynamic in -plane components of the magnetization to the wave -guide. By Faraday’s \nlaw, only the magnetization component perpendicular to the direction of the wave -guide can couple \neffectively, therefore the radiative damping is anisotropic and has to be calculated according ly. Thus, \nassuming a uniform magnetization profile and excitation field in the sample, the radiative damping αrad \nin our measurement is calculated via17  \n  \n𝛼rad= 𝛾 𝜇02 𝑀𝑠𝑡𝑙\n8 𝑍0𝑤𝑐𝑐,   (S1) \n \n \n \nwhere γ=gμB/ħ is the gyromagnetic ratio, Z0 =50 Ω the impedance,  wcc is the width of the wave guide, \nand t and l are the thickness and length of the sample on the waveguide. This calculated value of αrad can \nalso be measured directly,  by placing a spacer between sample and waveguide. Following the argument \nin Schoen et al.17, a 100 µm glass spa cer decreases αrad by approximately a factor of ten, making it , \nwithin errors , undetectable in our measurement. Indeed, when the spacer is inserted , the damping for \nboth the Co 20Fe80 and Co 25Fe75 decreases significantly, shown in Table  S1, and the damping decrease \nmatches the calculated αrad = 0.00065 for both samples  reasonably well . \n \n Sample  α without spacer  α with spacer  αrad \nCo20Fe80 0.00230±0.00003  0.0016±0.00015  0.0007±0.00015  \nCo25Fe75 0.0021±0.00015  0.0018±0.0002  0.0003±0.00025  \n \nTable S1:  Direct measurement of the radiative damping contribution . The damping of the \nCo20Fe80 and the Co 25Fe75 sample measured with and without a 100 µm spacer between sample \nand waveguide. The radiative damp ing αrad is determined from the difference in damping with \nand without spacer.  \n \nDetermination of the interfacial damping enhancement  \n \nThe flow of spin-angular momentum into adjacent layers (in our sample geometry the Cu/Ta cap and \nseed layers) further enhances the measured damping . This non -local damping or spin -pumping \ncontribution αsp is purely interfacial and can therefore  be determined from the thickness dependence of \nthe damping for a given  alloy concentration18. We measured the damping of thickness series of the pure \nelements and select alloy concentrations (20 % Co, 25 % Co, 50 % Co and 85 % Co). In order to correctly \naccount for the inte rfacial contribution αsp  we first remove the radiative contribution αrad from αtot and in \nFigure S4a plot the dependence of (αtot - αrad ) on the inverse thickness 1 /t for select alloy concentrations. \nαsp is described by \n 𝛼sp= 2𝑔𝑒𝑓𝑓↑↓𝜇𝐵𝑔\n4𝜋𝑀𝑆𝑡.   (S2) \nThe factor of 2 accounts for the two nominally equal interfaces of the alloy to the seed and cap layers. \nThe effective spin mixing conductance 𝑔eff↑↓ then is determined from the slope of the linear fits to the \n(αtot - αrad)  vs. 1/ t data and plott ed in Fig ure S4b. All values of 𝑔eff↑↓  are in the range of expected values \nbased on previous reports19,20. Using interpola ted values of 𝑔eff↑↓ for all alloy concentrations (gray line in \nFigure S4b), the non -local interface effects for all alloy compositions are straightforwardly determined \nwith equation S2.  \n  \nFigure S4:  Determination of the interfacial damping contribution . The (αtot - αrad) vs. 1/t dependence \n(symbols) for the pure elements and select alloy concentrations (20% Co, 25% Co, 50% Co and 85% \nCo) with fits to Equation 2 (lines) is plotted in a. From these fits the effective spin mixing conductance \ngeff↑↓ is calculated and plotted in b. The gray line is an interpolation of the calculated values.  \n \n \nComparison of αint to other theories  \n \nGilmore et al.21 calculated the damping as a funct ion of the electron -phonon self -energy Γ. They found \na minimum in damping for the pure elements bcc  Fe (𝛼int=0.0013 ), hcp  Co (𝛼int=0.0011 ) and \nfcc Ni (𝛼int=0.017). A comparison of  these values to the extracted 𝛼𝑖𝑛𝑡 for bcc  Fe 𝛼int=0.0024 ) and \nfcc Co (𝛼int=0.0026 ) shows that these calculated values are well below what we estimate to be the \nintrinsic damping. However, an adjustment of  the electron -phonon  self-energies Γ to 3 meV  for Fe and \n2 meV  for Co covers our findings. Nevertheless, we acknowledge that our sputtered films contain some \ndisorder and strain. Thus, it is conceivable that epitaxially grown, single crystal CoFe alloys may exhibit \neven lower  values of damping.  \n  \n \n \n \n1. Bozorth, R. M. Ferromagnetism . (IEEE Press: 2003).  \n2. Kief, M. T. & Egelhoff, W. F. Growth and structure of Fe and Co thin films on Cu(111), Cu(100), \nand Cu(110): A comprehensive study of metastable  film growth. Phys. Rev. B  47, 10785 –10814 \n(1993).  \n3. Pelzl, J. et al.  Spin-orbit -coupling effects on g -value and damping factor of the ferromagnetic \nresonance in Co and Fe films. Journal of Physics: Condensed Matter  15, S451 (2003).  \n4. Tischer, M. et al.  Enhancement of Orbital Magnetism at Surfaces: Co on Cu(100). Phys. Rev. Lett.  \n75, 1602 –1605 (1995).  \n5. Pauling, L. The Nature of the Interatomic Forces in Metals. Phys. Rev.  54, 899 –904 (1938).  \n6. Slater, J. C. Electronic Structure of Alloys. Journal of A pplied Physics  8, 385 –390 (1937).  \n7. Farle, M., Mirwald -Schulz, B., Anisimov, A. N., Platow, W. & Baberschke, K. Higher -order \nmagnetic anisotropies and the nature of the spin -reorientation transition in face -centered -tetragonal \nNi(001)/Cu(001). Phys. Rev. B 55, 3708 –3715 (1997).  \n8. Shaw, J. M., Nembach, H. T. & Silva, T. J. Measurement of orbital asymmetry and strain in \nCo_90Fe_10/Ni multilayers and alloys: Origins of perpendicular anisotropy. Phys. Rev. B  87, \n054416 (2013).  \n9. James, P., Eriksson, O., Johansson, B. & Abrikosov, I. A. Calculated magnetic properties of binary \nalloys between Fe, Co, Ni, and Cu. Phys. Rev. B  59, 419 –430 (1999).  \n10. Richter, R. & Eschrig, H. Spin -polarised electronic structure of disordered BCC FeCo alloys from \nLCAO -CPA. Journal of Physics F: Metal Physics  18, 1813 (1988).  \n11. Wu, D., Zhang, Q., Liu, J. P., Yuan, D. & Wu, R. First -principles prediction of enhanced magnetic \nanisotropy in FeCo alloys. Applied Physics Letters  92, (2008).  \n12. Paduani, C. & Krause, J. C. Electronic structure and magnetization of Fe -Co alloys and multilayers. \nJournal of Applied Physics  86, 578 –583 (1999).  \n13. Stearns, M. B. Magnetic Properties of 3d, 4d and 5d Elements, Alloys and Compounds . (Springer \nBerlin: 1984).  \n14. Chadov, S. et al.  Orbital magnetism in transition metal systems: The role of local correlation \neffects. EPL 82, 37001 (2008).  \n15. Ebert, H. & Battocletti, M. Spin and orbital polarized relativistic multiple scattering theory - With \napplications to Fe, Co, Ni and Fe(x)Co(1 -x). Solid State Communications  98, 785 –789 (1996).  \n16. Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. First -principles calculation of the \nGilbert damping parameter via the linear response formalism with application to magneti c \ntransition metals and alloys. Phys. Rev. B  87, 014430 (2013).  \n17. Schoen, M. A. W., Shaw, J. M., Nembach, H. T., Weiler, M. & Silva, T. J. Radiative damping in \nwaveguide -based ferromagnetic resonance measured via analysis of perpendicular standing spin \nwaves in sputtered permalloy films. Phys. Rev. B  92, 184417 (2015).  \n18. Tserkovnyak, Y., Brataas, A. & Bauer, G. E. W. Enhanced Gilbert Damping in Thin Ferromagnetic \nFilms. Phys. Rev. Lett.  88, 117601 (2002).  \n19. Czeschka, F. D. et al.  Scaling Behavior of t he Spin Pumping Effect in Ferromagnet -Platinum \nBilayers. Phys. Rev. Lett.  107, 046601 (2011).  \n20. Weiler, M., Shaw, J. M., Nembach, H. T. & Silva, T. J. Detection of the DC Inverse Spin Hall \nEffect Due to Spin Pumping in a Novel Meander -Stripline Geometry.  Magnetics Letters, IEEE  5, \n1–4 (2014).  \n21. Gilmore, K., Idzerda, Y. U. & Stiles, M. D. Identification of the Dominant Precession -Damping \nMechanism in Fe, Co, and Ni by First -Principles Calculations. Phys. Rev. Lett.  99, 027204 (2007).  \n \n "    },    {        "title": "0705.1990v1.Identification_of_the_dominant_precession_damping_mechanism_in_Fe__Co__and_Ni_by_first_principles_calculations.pdf",        "content": "arXiv:0705.1990v1  [cond-mat.str-el]  14 May 2007Identification of the dominant precession damping mechanis m in Fe, Co, and Ni by\nfirst-principles calculations\nK. Gilmore1,2, Y.U. Idzerda2, and M.D. Stiles1\n1National Institute of Standards and Technology, Gaithersb urg, MD 20899-8412\n2Physics Department, Montana State University, Bozeman, MT 59717\n(Dated: October 23, 2018)\nThe Landau-Lifshitz equation reliably describes magnetiz ation dynamics using a phenomenolog-\nical treatment of damping. This paper presents first-princi ples calculations of the damping param-\neters for Fe, Co, and Ni that quantitatively agree with exist ing ferromagnetic resonance measure-\nments. This agreement establishes the dominant damping mec hanism for these systems and takes\na significant step toward predicting and tailoring the dampi ng constants of new materials.\nMagnetic damping determines the performance of\nmagnetic devices including hard drives, magnetic ran-\ndom access memories, magnetic logic devices, and mag-\nnetic field sensors. The behavior of these devices can be\nmodeled using the Landau-Lifshitz (LL) equation [1]\n˙m=−|γ|m×Heff−λ\nm2m×(m×Heff),(1)\nor the essentially equivalent Gilbert (LLG) form [2, 3].\nThe first term describes precession of the magnetization\nmabouttheeffectivefield Heffwhereγ=gµ0µB/¯histhe\ngyromagnetic ratio. The second term is a phenomeno-\nlogical treatment of damping with the adjustable rate λ.\nTheLL(G) equationadequatelydescribesdynamicsmea-\nsured by techniques as varied as ferromagnetic resonance\n(FMR) [4], magneto-optical Kerr effect [5], x-ray absorp-\ntion spectroscopy [6], and spin-current driven rotation\nwith the addition of a spin-torque term [7, 8].\nAccess to a range of damping rates in metallic mate-\nrials is desirable when constructing devices for different\napplications. Ideally, one would like the ability to de-\nsign materials with any desired damping rate. Empiri-\ncally, dopingNiFe alloyswith transitionmetals[9]orrare\nearths [10] has produced compounds with damping rates\nin the range of α= 0.01 to 0.8. A recent investigation\nof adding vanadium to iron resulted in an alloy with a\ndamping rate slightly lower than that for pure iron [11],\nthe systemwiththe lowestpreviouslyknownvalue. How-\never, the damping rate of a new material cannot be pre-\ndicted because there has not yet been a first-principles\ncalculation of damping that quantitatively agrees with\nexperiment. The challenging pursuit of new materials\nwith specific or lowered damping rates is further com-\nplicated by the expectation that, as device size contin-\nues to be scaled down, material parameters, such as λ,\nshould change [12]. A detailed understanding of the im-\nportant damping mechanisms in metallic ferromagnets\nand the ability to predictively calculate damping rates\nwould greatly facilitate the design of new materials ap-\npropriate for a variety of applications.\nThe temperature dependence of damping in the tran-\nsition metals has been carefully characterized through\nmeasurement of small angle dynamics by FMR [13].While one might na¨ ıvely expect damping to increase\nmonotonically with temperature, as it does for Fe, both\nCo and Ni also exhibit a dramatic rise in damping at low\ntemperature as the temperature decreases. These ob-\nservations indicate that two primary mechanisms are in-\nvolved. Subsequent experiments [14, 15] partition these\nnon-monotonic damping curves into a conductivity-like\nterm that decreases with temperature and a resistivity-\nliketermthatincreaseswithtemperature. Thetwoterms\nwere found to give nearly equal weight to the damping\ncurve of Ni and have temperature dependencies similar\nto those of the conductivity and resistivity, suggesting\ntwo distinct roles for electron-lattice scattering.\nThe torque-correlation model of Kambersky [16] ap-\npears to qualitatively match the data. However, like\nmost of the various models presented by Kambersky\n[16, 17, 18, 19] and others [20], it has not been quan-\ntitatively evaluated in a rigorous fashion. This has left\nthe community to speculate, based on rough estimates or\nless, astowhichdampingmechanismsareimportant. We\nresolve this matter in the present work by reporting first-\nprinciples calculations of the Landau-Lifshitz damping\nconstant according to Kambersky’s torque-correlation\nexpression. Quantitative comparison of the present cal-\nculations to the measured FMR values [13] positively\nidentifies this damping pathway as the dominant effect\nin the transition metal systems. In addition to present-\ning these primary conclusions, we also describe the re-\nlationship between the torque-correlation model and the\nmore widely understood breathing Fermi surface model\n[18, 21], showing that the results of both models agree\nquantitatively in the low scattering rate limit.\nThe breathing Fermi surface model of Kambersky pre-\ndicts\nλ=g2µ2\nB\n¯h/summationdisplay\nn/integraldisplaydk3\n(2π)3η(ǫn,k)/parenleftbigg∂ǫn,k\n∂θ/parenrightbigg2τ\n¯h.(2)\nThis model offers a qualitative explanation for the low\ntemperature conductivity-like contribution to the mea-\nsureddamping. The modeldescribesdamping ofuniform\nprecession as due to variations ∂ǫn,k/∂θin the energies\nǫn,kof the single-particle states with respect to the spin2\ndirection θ. The states are labeled with a wavevector\nkand band index n. As the magnetization precesses,\nthespin-orbitinteractionchangestheenergyofelectronic\nstatespushingsomeoccupiedstatesabovetheFermilevel\nand some unoccupied states below the Fermi level. Thus,\nelectron-hole pairs are generated near the Fermi level\neven in the absence of changes in the electronic popula-\ntions. The ηfunction in Eq. (2) is the negative derivative\nof the Fermi function and picks out only states near the\nFermi level to contribute to the damping. gis the Land´ e\ng-factor and µBis the Bohr magneton. The electron-hole\npairs created by the precession exist for some lifetime τ\nbeforerelaxingthroughlattice scattering. The amountof\nenergy and angular momentum dissipated to the lattice\ndepends on how far from equilibrium the system gets,\nthus damping by this mechanism increases linearly with\nthe electron lifetime as seen in Eq.2. Since the electron\nlifetime is expected to decrease as the temperature in-\ncreases, this model predicts that damping diminishes as\nthe temperature is raised.\nBecause the predicted damping rate is linear in the\nscattering time the damping rate cannot be calculated\nmore accurately than the scattering time is known. For\nthis reason it is not possible to make quantitative com-\nparisonsbetween calculationsof the breathing Fermi sur-\nface and measurements. Further, while the breathing\nFermi surface model can explain the dramatic temper-\nature dependence observed in the conductivity-like por-\ntion of the data it fails to capture the physics driving the\nresistivity-like term. This is a significant limitation from\na practical perspective because the resistivity-like term\ndominates damping at room temperature and above and\nis the only contribution observed in iron [13] and NiFe\nalloys [22]. For these reasons it is necessary to turn to\nmore complete models of damping.\nKambersky’s torque-correlation model predicts\nλ=g2µ2\nB\n¯h/summationdisplay\nn,m/integraldisplaydk3\n(2π)3/vextendsingle/vextendsingleΓ−\nnm(k)/vextendsingle/vextendsingle2Wnm(k) (3)\nand we will show that it both incorporates the physics of\nthe breathing Fermi surface model and also accounts for\ntheresistivity-like terms. The matrix elements Γ−\nnm(k) =\n/angbracketleftn,k|[σ−, Hso]|m,k/angbracketrightmeasure transitions between states\nin bands nandminduced by the spin-orbit torque.\nThese transitionsconservewavevector kbecausethey de-\nscribe the annihilation of a uniform precession magnon,\nwhich carries no linear momentum. The nature of these\nscattering events, which are weighted by the spectral\noverlapWnm(k) = (1/π)/integraltext\ndω1η(ω1)Ank(ω1)Amk(ω1),\nwill be discussedin moredetail below. The electronspec-\ntral functions Ankare Lorentzians centered around the\nband energies ǫnkand broadenedbyinteractionswith the\nlattice. The width ofthe spectralfunction ¯ h/τprovidesa\nphenomenological account for the role of electron-lattice\nscattering in the damping process. The ηfunction is theh/τ(eV)\n108109λ (1/s)\n0.001α0.001 0.01 0.1 1\nFe\n1081091010λ (1/s)\n101310141015\n 1/τ (1/s)0.0010.010.1αNi108109λ (1/s)\n0.0010.01αCo\nFIG. 1: Calculated Landau-Lifshitz damping constant for Fe ,\nCo, and Ni. Thick solid curves give the total damping param-\neter while dotted curves give the intraband and dashed lines\nthe interband contributions. The top axis is the full-width -\nhalf-maximum of the electron spectral functions.\nsame as in Eq. (2) and enforces the requirement of spec-\ntral overlap at the Fermi level.\nEquation (3) captures two different types of scatter-\ning events: scattering within a single band, m=n, for\nwhich the initial and final states are the same, and scat-\ntering between two different bands, m/negationslash=n. As explained\nin [16] the overlap of the spectral functions is propor-\ntional (inverse) to the electron scattering time for intra-\nband (interband) scattering. From this observation the\nqualitative conclusion is made that the intraband contri-\nbutions matchthe conductivity-like terms while the inter-\nband contributions give the resistivity-like terms. While\nthis seems promising, evaluation of Eq. (3) is more com-\nputationally intensive than that of the breathing Fermi\nsurface model and until now only a few estimates for Ni\nand Fe have been made [19].3\nTABLE I: Calculated and measured [13] damping parameters. V alues for λ, the Landau-Lifshitz form, are reported in 109s−1,\nvalues of α, the Gilbert form, are dimensionless. The last two columns l ist calculated damping due to the intraband contribution\nfrom Eq. (3) and from the breathing Fermi surface model [12], respectively. Values for λ/τare given in 1022s−2. Published\nnumbers from [13] and [12] have been multiplied by 4 πto convert from the cgs unit system to SI.\nαcalcλcalcλmeasλcalc/λmeas(λ/τ)intra(λ/τ)BFS\nbcc Fe/angbracketleft001/angbracketright0.0013 0.54 0.88 0.61 1.01 0.968\nbcc Fe/angbracketleft111/angbracketright0.0013 0.54 – – 1.35 1.29\nhcp Co/angbracketleft0001/angbracketright0.0011 0.37 0.9 0.41 0.786 0.704\nfcc Ni/angbracketleft111/angbracketright0.017 2.1 2.9 0.72 6.67 6.66\nfcc Ni/angbracketleft001/angbracketright0.018 2.2 – – 8.61 8.42\nWe have performed first-principles calculations of the\ntorque-correlation model Eq. (3) with realistic band\nstructures for Fe, Co, and Ni. Prior to evaluating Eq. (3)\nthe eigenstates and energies of each metal were found us-\ning the linear augmented plane wave method [23] in the\nlocal spin density approximation (LSDA) [24, 25, 26].\nDetails of the calculations for these materials are de-\nscribed in [27]. The exchange field was fixed in the cho-\nsen equilibrium magnetization direction. Calculations of\nEq. (3) presented in this paper are converged to within\na standard deviation of 3 %, which required sampling\n(160)3k-points for Fe, (120)3for Ni, and (100)2k-points\ninthe basalplaneby57alongthe c-axisforCo. Electron-\nlattice interactions were treated phenomenologically as a\nbroadeningofthe spectralfunctions. The Fermi distribu-\ntion was smeared with an artificial temperature. Results\ndid not vary significantly with reasonable choices of this\ntemperature since the broadening of the Fermi distribu-\ntion was considerably less than that of the bands. The\ndamping rate was calculated for a range of scattering\nrates (spectral widths) just as damping has been mea-\nsured over a range of temperatures.\nThe results ofthese calculations arepresented in Fig. 1\nand are decomposed into the intraband and interband\nterms. The downward sloping line in Fig. 1 represents\nthe intraband contribution to damping. Damping con-\nstants were recently calculated using the breathing Fermi\nsurface model [12, 21] by evaluating the derivative of\nthe electronic energy with respect to the spin direc-\ntion according to Eq. (2). The results of the breathing\nFermi surface prediction are indistinguishable from the\nintraband terms of the present calculation even though\nthe computational approaches differed significantly; the\nagreement is quantified in Table I.\nThe breathing Fermi surface model could not be quan-\ntitatively compared to the experimental results because\nthe temperature dependence of the scattering rate has\nnot been determined sufficiently accurately. While the\npresent calculations also require knowledge of the scat-\ntering rate to determine the damping rate the non-monotonic dependence of damping on the scattering rate\nproduces a unique minimum damping rate. In the same\nmanner that the calculated curves of Fig. 1 have a mini-\nmumwithrespecttoscatteringrate,themeasureddamp-\ning curves exhibit minima with respect to temperature.\nWhatever the relation between temperature and scatter-\ning rate, the calculated minima may be compared di-\nrectly and quantitatively to the measured minima. Ta-\nble I makes this comparison. The agreement between\nmeasured and calculated values shows that the torque-\ncorrelationmodelaccountsforthedominantcontribution\nto damping in these systems.\nOur calculated values are smaller than the measured\nvalues. Using measured gvalues instead of setting g=\n2 would increase our results by a factor of ( g/2)2, or\nabout 10 % for Fe and 20 % for Co and Ni. Other pos-\nsible reasons for the difference include a simplified treat-\nment ofelectron-latticescatteringin which the scattering\nrates for all states were assumed equal, the mean-field\napproximation for the exchange interaction, errors asso-\nciatedwith thelocalspindensityapproximation(LSDA),\nand numerical convergence (discussed below). Other\ndamping mechanisms may also make small contributions\n[28, 29, 30].\nSince the manipulations involved with the equation of\nmotion techniques employed in deriving Eq. (3) obscure\nthe underlying physics we now discuss the two scatter-\ning processes and connect the intraband terms to the\nbreathing Fermi surface model. The intraband terms in\nEq. (3) describe scattering from one state to itself by\nthe torque operator, which is similar to a spin-flip oper-\nator. A spin-flip operation between some state and itself\nis only non-zero because the spin-orbit interaction mixes\nsmall amounts of the opposite spin direction into each\nstate. Since the initial and final states are the same, the\noperation is naturally spin conserving. The matrix ele-\nments do not describe a real transition, but rather pro-\nvide a measure of the energy of the electron-hole pairs\nthat are generated as the spin direction changes. The\nelectron-hole pairs are subsequently annihilated by a real4\nelectron-lattice scattering event.\nTo connect the derivatives ∂ǫ/∂θin Eq. (2) and the\ntorque matrix elements in Eq. (3) we imagine first point-\ning the magnetization in some direction ˆ z. The only\nenergy that changes with the magnetization direction is\nthe spin-orbit energy Hso. As the spin of a single parti-\ncle state |/angbracketrightrotates along ˆθabout ˆxits spin-orbit energy\nis given by ǫ(θ) =/angbracketleft|eiσxθHsoe−iσxθ|/angbracketright. The derivative\nwith respect to θis∂ǫ(θ)/∂θ=i/angbracketleft|eiσxθ[σx, Hso]e−iσxθ|/angbracketright.\nEvaluating this derivative at the pole ( θ= 0) gives\n∂ǫ/∂θ=i/angbracketleft|[σx, Hso]|/angbracketright. Similarly, rotating the spin along\nˆθabout ˆyleads to ∂ǫ/∂θ=i/angbracketleft|[σy, Hso]|/angbracketright. The torque\nmatrix elements in Eq. (3) are Γ−=/angbracketleft|[σ−, Hso]|/angbracketright=\n/angbracketleft|[σx, Hso]|/angbracketright−i/angbracketleft|[σy, Hso]|/angbracketright. Using the relations between\nthe commutators and derivatives just found the torque is\nΓ−=−i(∂ǫ/∂θ)x−(∂ǫ/∂θ)ywhere the subscripts in-\ndicate the rotation axis. Squaring the torque matrix\nelements gives |Γ−|2= (∂ǫ/∂θ)2\nx+ (∂ǫ/∂θ)2\ny. For high\nsymmetry directions ( ∂ǫ/∂θ)x= (∂ǫ/∂θ)yand we de-\nduce|Γ−|2= 2(∂ǫ/∂θ)2demonstrating that the intra-\nband terms of the torque-correlation model describe the\nsame physics as the breathing Fermi surface.\nThe monotonically increasing curves in Fig. 1 indi-\ncate the interband contribution to damping. Uniform\nmode magnons, which have negligible energy, may in-\nduce quasi-elastic transitions between states with differ-\nent energies. This occurs when lattice scattering broad-\nens bands sufficiently so that they overlap at the Fermi\nlevel. Thesewavevectorconservingtransitions, whichare\ndriven by the precessing exchange field, occur primarily\nbetween states with significantly different spin character.\nThe process may roughly be thought of as the decay of\na uniform precession magnon into a single electron spin-\nflip excitation. These events occur more frequently as\nthe band overlaps increase. For this reason the interband\nterms, which qualitatively match the resistivity-like con-\ntributions in the experimental data, dominate damping\nat room temperature and above.\nWe have calculated the Landau-Lifshitz damping pa-\nrameterfortheitinerantferromagnetsFe, Co, andNiasa\nfunction ofthe electron-latticescatteringrate. Theintra-\nband and interband components match qualitatively to\nconductivity- andresistivity-like terms observed in FMR\nmeasurements. A quantitative comparison was made be-\ntweentheminimaldampingratescalculatedasafunction\nof scattering rate and measured with respect to temper-\nature. This comparison demonstrates that our calcula-\ntions account for the dominant contribution to damping\nin these systemsand identify the primarydamping mech-\nanism. At room temperature and above damping occurs\noverwhelmingly through the interband transitions. The\ncontribution of these terms depends in part on the band\ngap spectrum around the Fermi level, which could be\nadjusted through doping.\nK.G. and Y.U.I. acknowledge the support of the Officeof Naval Research through grant N00014-03-1-0692 and\nthroughgrantN00014-06-1-1016. We wouldlike tothank\nR.D. McMichael and T.J. Silva for valuable discussions.\n[1] L. LandauandE. Lifshitz, Phys. Z.Sowjet. 8, 153(1935).\n[2] T. L. Gilbert, Armour research foundation project No.\nA059, supplementary report, unpublished (1956).\n[3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[4] D. Twisselmann and R. McMichael, J. Appl. Phys. 93,\n6903 (2003).\n[5] T. Gerrits, J. Hohlfeld, O. Gielkens, K. Veenstra, K. Bal ,\nT. Rasing, and H. van den Berg, J. Appl. Phys. 89, 7648\n(2001).\n[6] W. Bailey, L. Cheng, D. Keavney, C. Kao, E. Vescovo,\nand D. Arena, Phys. Rev. B 70, 172403 (2004).\n[7] I. Krivorotov, D. Berkov, N. Gorn, N. Emley, J. Sankey,\nD. Ralph, and R. Buhrman, Phys. Rev. B (2007).\n[8] M. Stiles and J. Miltat, Spin dynamics in confined mag-\nnetic structures III (Springer, Berlin, 2006).\n[9] J. Rantschler, R. McMichael, A. Castiello, A. Shapiro,\nJ. W.F. Egelhoff, B. Maranville, D.Pulugurtha, A. Chen,\nand L. Conners, J. Appl. Phys. 101, 033911 (2007).\n[10] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE\nTrans. Mag. 37, 1749 (2001).\n[11] C. Scheck,L.Cheng, I.Barsukov, Z.Frait, andW.Bailey ,\nPhys. Rev. Lett. 98, 117601 (2007).\n[12] D. Steiauf and M. Faehnle, Phys. Rev. B 72, 064450\n(2005).\n[13] S. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n[14] B. Heinrich, D. Meredith, and J. Cochran, J. Appl. Phys.\n50, 7726 (1979).\n[15] J.F.Cochran and B. Heinrich, IEEE Trans. Magn. 16,\n660 (1980).\n[16] V. Kambersky, Czech. J. Phys. B 26, 1366 (1976).\n[17] B. Heinrich, D. Fraitova, and V. Kambersky,\nPhys. Stat. Sol. 23, 501 (1967).\n[18] V. Kambersky, Can. J. Phys. 48, 2906 (1970).\n[19] V. Kambersky, Czech. J. Phys. B 34, 1111 (1984).\n[20] V. Korenman and R. Prange, Phys. Rev. B 6, 2769\n(1972).\n[21] J. Kunes and V. Kambersky, Phys. Rev. B 65, 212411\n(2002).\n[22] S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczews ki,\nP. Trouilloud, and R. Koch, Phys. Rev. B 66, 214416\n(2002).\n[23] L. Mattheiss and D. Hamann, Phys. Rev. B 33, 823\n(1986).\n[24] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864\n(1964).\n[25] W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965).\n[26] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).\n[27] M. Stiles, S. Halilov, R. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[28] R. McMichael and A. Kunz, J. Appl. Phys. 91, 8650\n(2002).\n[29] E. Rossi, O. G. Heinonen, and A. H. MacDonald,\nPhys. Rev. B 72, 174412 (2005).\n[30] Y. Tserkovnyak, G. Fiete, and B. Halperin,\nAppl. Phys. Lett. 84, 5234 (2004)."    },    {        "title": "1908.01359v3.Efficient_spin_excitation_via_ultrafast_damping_like_torques_in_antiferromagnets.pdf",        "content": "Efficient spin excitation via ultrafast damping-like torques in antiferromagnets\nChristian Tzschaschel,1,\u0003Takuya Satoh,2and Manfred Fiebig1\n1Department of Materials, ETH Z ¨urich, 8093 Zurich, Switzerland\n2Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan\n(Dated: November 25, 2021)\nABSTRACT\nDamping effects form the core of many emerging concepts for high-speed spintronic applications. Impor-\ntant characteristics such as device switching times and magnetic domain-wall velocities depend critically on\nthe damping rate. While the implications of spin damping for relaxation processes are intensively studied,\ndamping effects during impulsive spin excitations are assumed to be negligible because of the shortness of\nthe excitation process. Herein, we show that, unlike in ferromagnets, ultrafast damping plays a crucial role\nin antiferromagnets because of their strongly elliptical spin precession. In time-resolved measurements, we\nfind that ultrafast damping results in an immediate spin canting along the short precession axis. The inter-\nplay between antiferromagnetic exchange and magnetic anisotropy amplifies this canting by several orders\nof magnitude towards large-amplitude modulations of the antiferromagnetic order parameter. This leverage\neffect discloses a highly efficient route towards the ultrafast manipulation of magnetism in antiferromagnetic\nspintronics.\nINTRODUCTION\nAntiferromagnets (AFMs) are a promising material class for spintronic applications1,2. Their robustness against\nexternal magnetic fields, along with the possibility of picosecond antiferromagnetic switching, could help to substitute\nsemiconductor-based information technologies with antiferromagnetic spintronic devices3,4. In fact, an electrically\nswitched antiferromagnetic bit has been presented recently5and synthetic AFMs have substantially improved the\nspeed and data density of racetrack memories6. Efficient manipulation of AFMs relies on spin currents acting as\neffective magnetic fields generating staggered (anti-) damping-like torques7. Such electrical control, however, has\ntwo disadvantages. On the one hand, the required current densities are in the range of \u0018106\u0000108A cm\u00002, which\ninvolves a considerable amount of Joule heating. On the other hand, timescales accessible with electrical currents\nare typically too slow to explore the aforementioned picosecond dynamics. In contrast, optical laser pulses provide\nthe time-resolution required to track THz-spin dynamics. The inverse Faraday effect (IFE), i.e. the optical generation\nof effective sub-picosecond magnetic field pulses with circularly polarised light, allows to impulsively create a spin\ncanting in AFMs8–15.\nFor ferromagnetic excitations, the magnetic anisotropy typically leads to a circular or only slightly elliptical spin\nprecession16. In AFMs, however, the competition between magnetic anisotropy and the &100 times stronger ex-\nchange interaction results in strongly elliptical trajectories. An oscillating net magnetisation occurs along the minorarXiv:1908.01359v3  [cond-mat.mtrl-sci]  5 Dec 20202\naxis of the ellipse, whereas the antiferromagnetic order parameter is modulated due to a collective in-phase spin\ncanting along the major axis15,17,18. Therefore, the efficiency of impulsive spin excitations in antiferromagnets de-\npends strongly on the direction of the initial spin canting. In particular, an impulsively generated net magnetisation in\nan AFM would be highly efficient in triggering large-amplitude changes of the antiferromagnetic order.\nAntiferromagnetic spin dynamics are typically described by the Landau-Lifshitz-Gilbert equation19\ndMi\ndt=\u0000\r\u00160Mi\u0002Hi\u0000\r\u00160\u000b\nM0Mi\u0002(Mi\u0002Hi); (1)\nwhere Midenotes the magnetisation of the i-th antiferromagnetic sublattice with magnitude M0;Hiis the field act-\ning on Mi, and\u000bis the Gilbert damping parameter. Further, \rand\u00160are the gyromagnetic ratio and magnetic\npermeability, respectively. The first term of Equation (1) is commonly referred to as the field-like torque TFL, whereas\nthe second term corresponds to the damping-like torque TDL7. For fully compensated AFMs excited by the IFE, TFL\nmodulates the antiferromagnetic order parameter, but does not induce a net magnetisation at t= 0(see Supple-\nmentary Movie 1). Therefore, the impulsive generation of a net magnetisation by the effective magnetic field of the\nIFE can only occur via an ultrafast damping-like torque TDL(see Supplementary Movie 2). So far, however, damping\nhas been considered to describe only the spin-relaxation after the excitation20. The ultrafast damping-like torque of\nthe IFE was neglected for the excitation process itself based on the shortness of the interaction and the suspected\nsmallness of the torque.\nHere, we highlight the crucial role of the ultrafast damping-like torque during the excitation process for efficient\nimpulsive spin excitations in AFMs. We detect such damping in the initial phase of a coherent spin precession\nof hexagonal YMnO 3and HoMnO 3. We tune the spin damping strength via temperature variations and study the\ninfluence of magnetic order–order and order–disorder transitions on the damping. Our phenomenological theory\nconsiders both damping and magnetic anisotropy effects during the excitation, and thus develops and quantifies\na consistent picture of impulsively induced coherent spin dynamics. Strikingly, we find that the hitherto neglected\nultrafast spin damping during the impulsive spin excitation can even provide the dominant spin excitation mechanism.\nRESULTS\nImpulsive spin excitations in damped h- RMnO 3\nYMnO 3orders antiferromagnetically at the N ´eel temperature TN\u001873 K with a transition from P63cm10(T >T N) to\nP60\n3cm0(T <T N) symmetry21(Fig. 1a). HoMnO 3undergoes a P63cm10!P60\n3c0mtransition at TN\u001875 K followed\nby a first-order spin-reorientation transition (SRT) to P60\n3cm0atTSR\u001835 K22,23. In a mean-field approach, the free\nenergy density for both YMnO 3and HoMnO 3reads\nF=\u0015X\nhi;jiMi\u0001Mj+DzX\niM2\ni;z+DyX\niM2\ni;y+ \u0001X\niM2\ni;xM2\ni;y; (2)3\nx1y1a b\nWP\nBPDx\nzprobe\nHIFEpump\nh-RMnO3zx2\nx3\nFLDLMi HIFE cP���\ny\nFIG. 1. Experimental geometry. a Projection of the YMnO 3crystal structure onto the basal plane. Mn3+ions (violet) in grey and\nwhite areas are located in planes at z= 0andz=c=2, respectively. The three sublattice magnetisations Mipoint along equivalent\nxaxesxias indicated in the coordinate system. Note that spins in HoMnO 3point along equivalent yaxes forTSR< T < T N\n(Supplementary Fig. 1). bSchematic of the setup. The pump and probe pulses are circularly and linearly polarised, respectively.\n(P - polariser, \u0015=4- quarter-wave plate, WP - Wollaston prism, BPD - balanced photodiode). cVisualisation of optical Z-mode\nexcitation with field-like and damping-like torques TFLandTDLexerted by the effective field HIFEof the IFE on the magnetisation\nMik^ xi(^ xidenotes the unit vector in the direction xi). Black lines illustrate the ensuing strongly elliptical spin precession.\nDashed ellipses show the expected trajectory without spin excitation via TDL. Precession amplitudes are significantly reduced.\nwhere Midenotes the magnetisation at site i(i= 1;2;3). Furthermore \u0015>0andDz>0parametrise the antiferro-\nmagnetic intra-plane exchange interaction and the easy-plane anisotropy, respectively. The inter-plane exchange is\ntwo orders of magnitude weaker than the intra-plane exchange; thus, we neglect it24.Dy<0stabilises the ground\nstate asP60\n3c0m(Mithen points along equivalent crystallographic yaxes), whereas Dy>0stabilisesP60\n3cm0(with\nMialong equivalent crystallographic xaxes). \u0001\u00150is a weak fourth-order in-plane anisotropy, which becomes\nrelevant at the SRT, where Dycrosses zero. The three-sublattice antiferromagnet exhibits three optically excitable\nspin precessions called X,Y, andZmode18,25, which relate to an oscillating net magnetisation along the x,y, orz\naxes, respectively. We investigate the damping-like torque in relation to spin excitations via the IFE, which leaves us\nwith theZ-mode excitation illustrated in Fig. 1c.\nFigure 2a shows an exemplary measurement of the time-resolved Faraday rotation in YMnO 3after optical excita-\ntion with a circularly polarised pump pulse at 30 K. Two regimes can be distinguished: (i) a pronounced peak around\n0 psreflecting the direct interaction between pump and probe pulse and (ii) a damped sinusoidal modulation of the\nFaraday rotation that marks the ensuing spin precession and relaxation.\nFrom a fit of the latter, we extract the relaxation time \u001c, frequency !and initial phase \u001e0of the precession (see\nMethods). Figure 2b shows a magnified view of the region of the temporal overlap along with an extrapolation of\nthe sinusoidal fit. Most strikingly, we find a finite magnetisation at 0 ps. This is unexpected because the conventional\nfield-like torque of the IFE only perturbs the antiferromagnetic order, but does not induce a net magnetisation at the\nmoment of the excitation14,26–28. We will show that the observed immediate onset of a finite net magnetisation is a4\n05 01 001 502 00Faraday rotation (arb. units)P\nump-probe delay (ps)T = 30 K05 0100150200cD\nelay (ps)/s116/s119-\n10 1 bD\nelay (ps)initial phase /s1020a\nFIG. 2.Z-mode precession in YMnO 3. atime-resolved Faraday rotation following the impulsive optical excitation at time t= 0.\nbMagnified view of the region around t= 0.cOffset-corrected oscillation (see Methods). The red line is an exponentially\ndamped sine-fit with the dashed line as its envelope. The extrapolation of the fit towards t= 0is shown in band reveals a finite\ndeflection at t= 0. This initial deflection yields a temporal shift of the spin precession by approximately 280 fs , which amounts\nto an initial phase of 0.1 for a precession with a period of 18:2 ps.\nconsequence of spin damping during the impulsive spin excitation, which has been neglected so far.\nOriginally, YMnO 3is in the magnetic ground state with Mi(t <0) =M0^ xi, whereM0is the sublattice saturation\nmagnetisation. The optomagnetic field pulse of the IFE can be modelled as HIFE(t) =H\u0012\u000e(t)^ z, whereHis the\neffective magnetic field strength, \u0012is the laser pulse duration, and \u000e(t)denotes the Dirac delta function29. Analytically\nintegrating Equation (1) from t=\u00001to+0withHi=HIFEyields:\nMi(+0) =M00\nBBB@1\n\r\u00160H\u0012\n\u000b\r\u0016 0H\u00121\nCCCA: (3)\nThe canting along the zaxis is typically significantly smaller than the ycanting because \u000b\u001c1. However, any\nspin canting along the magnetic hard axis ( z) will be enhanced during the spin precession as a combined effect of\nantiferromagnetic exchange and magneto-crystalline anisotropy. This so-called exchange enhancement1,14,17,29,30is\nan effect that is intrinsic to AFMs, but is absent in ferromagnets. Thus, whereas damping effects during the spin\nexcitation may be neglected for ferromagnets, the neglecting is not generally justified in AFMs. In fact, we will show\nthat despite \u000b\u001c1, the damping-like contribution to the total spin excitation may even dominate over the field-like\ncontribution.\nThe threefold rotational symmetry, which is preserved during the Z-mode precession, allows us to describe the an-5\ntiferromagnetic dynamics by considering only one spin precessing in effective out-of-plane and in-plane anisotropies\nJ=3\n2\u0015+Dz+jDyjandD= \u0001M2\n0+jDyj, respectively (Supplementary Note 1). For the zcomponent of the\noscillating net magnetisation M, this yields a damped sinusoidal oscillation with\n!=!0p\n1\u0000A2\u000b2=4; (4)\n\u001c=2\nA\u000b! 0; (5)\ntan\u001e0=A\u000bp\n1\u0000A2\u000b2=4\n1\u0000A2\u000b2=2: (6)\nHere, ~!0= 2g\u0016BM0pJDis the precession frequency for the undamped case with the Land ´e factorg= 2and the\nBohr magneton \u0016B.A=p\nJD\u00001\u001d1is the exchange enhancement factor1,29. Geometrically, Acorresponds to the\naspect ratio of the elliptical spin precession14,17. Note that the initial phase \u001e0depends on the damping during the\nimpulsive spin excitation in direct relation to the initial phase in Fig. 2 and the ultrafast damping-like torque TDL.\nUltrafast spin damping in HoMnO 3\nWe further investigate the role of the damping-like torque by performing temperature-dependent measurements of\nthe optically induced spin dynamics. Temperature affects both the Gilbert-damping parameter \u000b(and therefore the\nmagnitude ofTDL) and the magneto-crystalline anisotropy (and therefore the exchange enhancement A). In partic-\nular, we trace the Zmode through the first-order SRT in HoMnO 3and towards the second-order antiferromagnetic-\nparamagnetic phase transition in YMnO 3(see Supplementary Fig. 2 for exemplary time-domain data). Conceptually,\nwe thus investigate an order–order and order–disorder phase transition, thereby showcasing the general importance\nof spin damping during ultrafast antiferromagnetic spin excitations.\nWe first consider the case of HoMnO 3. Based on measurements as in Fig. 2, we extract !0,\u001cand\u001e0as depicted in\nFigs. 3a–c, respectively. The SRT causes a dip of !0around 33 K. The fact that the frequency does not drop to zero\nat the SRT highlights the importance of the fourth-order anisotropy term in Equation (2). All anisotropy parameters\ncan be extracted from a fit of the temperature dependence of !0with a minimal model (see Methods). (The deviations\nbelow 25 K are caused by the incipient ordering of the Ho(4b) moments24,31,32, see Supplementary Fig. 3).\nMoreover, using Eqs. (4) and (5), we determine the exchange-enhanced damping A\u000band, in combination with\nthe extracted anisotropy constants, its constituents Aand\u000b(Figs. 3d–f). We find a significant increase in A\u000b\nbetween 30 K and45 K. Together with Equation (6) we can then calculate the temperature dependence of the initial\nphase and compare it with the direct measurement (Fig. 3c). The agreement around the SRT and towards higher\ntemperatures is impressive and demonstrates the importance of the previously unrecognised damping-like torque\nTDL. More strikingly, combining Eqs. (2) and (3) yields the energy density transferred into the magnetic lattice during\nthe excitation as6\nRelaxation timea\nHo(4b)\norderingTSR\nb\ncd\ne\nf\nFIG. 3. Temperature-dependent Z-mode characterisation in HoMnO 3. aFrequency, brelaxation time and ctangent of the\ninitial phase of the optically induced spin precession as function of temperature. Red: model calculations (see Methods). In\nthe blue-shaded region .25 K , incipient ordering of the Ho(4b) moments leads to discrepancies between data and the model\n(see Supplementary Fig. 3). Grey line: exponential decay as guide to the eye. dExchange-enhanced damping parameter A\u000b\nextracted from Z-mode frequency and relaxation time (Eqs. (4) and (5)). eTemperature-dependent aspect ratio Aof the spin-\nprecession ellipses. fResulting Gilbert damping parameter \u000b. Yellow line: function proportional to Bose-Einstein distribution\nwith quasiparticle energy of 4:3 meV .\n\u0001F=F(+0)\u0000F(\u00001)\u00193(A2\u000b2+ 1)DM2\ny; (7)\nwhere the first term originates from the ultrafast damping-like torque along the zaxis, whereas the second term is re-\nlated to the field-like in-plane torque of the IFE. Thus, as A\u000bexceeds one, spin excitation via the damping-like torque\nof the IFE even becomes the dominant excitation mechanism. The enhancement of Aaround the SRT is based on\nchanges of the magneto-crystalline anisotropy, while the anomalous increase of \u000bmatches the observations of a\ntemporary small-domain state in passing the SRT23. Such a small-domain state exhibits a high density of domain\nwalls which will locally affect the spin precession frequency. Thus, we attribute the enhanced Gilbert damping to\ndephasing due to magnon scattering at the walls33.7\ntan �0 Frequency       (meV) Lifetime    (ns)TN\nCorrected model\nUncorrected model\nFIG. 4. Temperature-dependent Z-mode characterisation in YMnO 3. aFrequency, brelaxation time, and ctangent of the initial\nphase of the optically induced spin precession as function of temperature. Green and red: model calculations with and without\ncorrection for non-damping-based excitation mechanisms, respectively. Grey line: fitted exponential decay.\nUltrafast spin damping in YMnO 3\nThe general trend of \u000bis captured well by an offset Bose-Einstein distribution for a quasiparticle with an energy of\n\u00184:3 meV , suggesting that scattering with the crystal-field excitations of the Ho 4felectrons (\u00193:5 meV24) contributes\nsignificantly to the spin damping. This is confirmed by measurements on YMnO 3(Fig. 4), yielding relaxation times\nthat are\u001820 times longer than those for HoMnO 3in line with the unoccupied 4forbital. Accordingly, the extracted\nvalues for the temperature-dependent Gilbert-damping parameter \u000bin YMnO 3(Fig. 5) are an order of magnitude\nlower than those in HoMnO 3. At low temperature, \u000bcan be approximated by a Bose-Einstein distribution with a\nquasiparticle energy of 2:6 meV , which corresponds to a hybrid spin-lattice mode in YMnO 334. Assuming that the\nfrequency of this mode is, just like the Zmode frequency, proportional to the magnitude of the antiferromagnetic\norder parameter, we can consistently describe the temperature dependence up to the N ´eel temperature.\nNote, however, that despite good qualitative agreement, Fig. 4c reveals a difference between the measured and\nmodelled value of the initial phase \u001e0. This indicates secondary, non-damping-based excitation mechanisms, such\nas an optical spin transfer torque35or optical orientation36, which could become relevant because of the smallness\nof\u000bin YMnO 3as compared with that in HoMnO 3. These mechanisms involve an angular momentum transfer from8\n0123G\nilbert damping parameter /s97x10-3T N1\n0203040506070800.000.050.100.150.200.25T\nemperature (K)Exchange-enhanced damping A/s97\nFIG. 5. Temperature-dependent damping of the Zmode in YMnO 3.Gilbert-damping parameter \u000band exchange-enhanced\ndampingA\u000bare shown in the same plot with A= 69:3(see Methods, Table 1). Full line: offset Bose-Einstein distribution with\nconstant quasiparticle energy E=2:6 meV . Dashed line: offset Bose-Einstein distribution with E/!0andE(0 K) =2:6 meV .\nthe circularly polarised pump pulse to the material. As the angular momentum of the circular laser pulse is directed\nalong thezaxis, we heuristically include such a contribution in our model by setting Mi;z(+0) =\u000b\r\u0016 0H\u0012M 0+\u000ein\nEquation (3), which changes the initial phase to\ntan\u001e0\u0019A\u0012\n\u000b+\u000e\n\r\u00160H\u0012M 0\u0013\n: (8)\nWith this, we restore the quantitative agreement between the measured and modelled initial phase in Fig. 4c.\nDISCUSSION\nSummarising, we thus demonstrated the indispensable role of the previously neglected ultrafast damping-like\ntorque during the optical excitation of AFMs, both experimentally and theoretically. The torque impulsively generates\na magnetisation along the minor axis of the intrinsically highly elliptical spin precession in AFMs. The extreme ellip-\nticity converts the small damping-related magnetisation into a large precessional canting. Because of this leverage\neffect, this damping-based mechanism can even facilitate the dominant spin excitation mechanism in striking contrast\nto its complete neglect thus far. In addition to studying the spin dynamics deep in the antiferromagnetic phase, we\nalso traced the associated Zmode in the vicinity of both first- and second-order phase transitions. This enabled us\nto verify the role of the ultrafast damping-like torque in distinctly different scenarios. We find that the Gilbert-damping\nparameter\u000bremains almost constant throughout the SRT in HoMnO 3, while the changes of the magneto-crystalline\nanisotropy driving the SRT lead to a two-fold increase in the exchange enhancement factor A. In contrast, as we\napproach the N ´eel temperature TN(here in YMnO 3), the magneto-crystalline anisotropy remains unchanged, but \u000b\ndiverges. Although fundamentally different, we find an enhanced ultrafast damping-like torque in both cases, which\nagain highlights the general importance of this hitherto neglected excitation mechanism.9\nMore specifically, in terms of damping mechanisms we identified scattering with the 4fcrystal-field excitations\nof HoMnO 3as the leading contribution to the spin damping. Therefore, the choice of the A-site ion enables us\nto tune the strength of the damping across several orders of magnitude. This is, on one hand, an useful degree\nof freedom for tuning material properties to specific spintronics applications. On the other hand, it allowed us to\nidentify the presence of additional non-damping-based mechanisms inducing a net magnetisation. With respect to\nthis, we note that any ultrafast mechanism that induces a net magnetisation in an AFM will benefit from the observed\nexchange enhancement. While field-like excitations via the IFE or using THz pulses typically induce spin-cantings of\nthe order of 1\u000e15,37, there are various possibilities beyond increasing the pump fluence that may allow us to exceed\na90\u000espin canting and thus induce antiferromagnetic switching. Exploiting different excitation mechanisms such as\nthermally and optically induced ultrafast spin-transfer torques35,38,39may drastically increase the impulsively induced\nnet magnetisation. Increasing the exchange enhancement, e.g. by tuning the system close to a SRT, will increase\nthe ellipticity of the spin precession. Therefore, the impulsively induced net magnetisation translates into a larger\nmodulation of the antiferromagnetic order parameter thereby reducing the switching threshold. Ultimately, our results\nshow that both field-like and damping-like torques — a combination that has proved to be powerful for the electrical\ncontrol of magnetic order7,40— are available optically, too, and can be utilized to act on the magnetic order. Thus,\noptically generated torques might provide the long sought-after tool enabling the efficient realisation of ultrafast\ncoherent precessional switching of AFMs.\nMETHODS\nSetup\nBoth samples are flux-grown and polished, (0001)-oriented platelets of approximately 55µmthickness. Our setup\nis schematically shown in Fig. 1b. The fundamental light source is a regeneratively amplified laser system providing\n130 fs pulses at 1:55 eV photon energy and 1 kHz repetition rate. We use the output of an optical parametric amplifier\ncombined with a quarter-wave plate to generate circularly polarised pump pulses at 0:95 eV photon energy, which\ncreate an effective magnetic field in the sample via the IFE. The pump fluence on the sample is approximately\n60 mJ cm\u00002. By pumping the material far below the nearest absorption resonance at \u00191:6 eV41, we minimise parasitic\nheating effects42. We then probe the ensuing dynamics by measuring the time-resolved Faraday rotation \b(t)of a\nlinearly polarised probe pulse at 1:55 eV with balanced photodetectors. The probe fluence is <0:1 mJ cm\u00002. Both\nthe pump and probe beams are arranged in a quasi-collinear geometry with the beam propagation direction along\nthe hexagonal zaxis, which is the optic axis. Thus, the incident light experiences optical isotropy and complications\narising from birefringence or other anisotropic optical effects are avoided. The sample is mounted in a cryostat for\nvarying the temperature between \u00195 Kand300 K .\nThe oscillatory part of the signal (Fig. 2a) is fitted by\n\b(t) =A0+A1e\u0000t=t1+A2e\u0000t=\u001csin (!t+\u001e0): (9)10\nWhile the first two terms model parasitic contributions to the signal arising from thermal electron dynamics, we extract\nthe frequency !(T), relaxation time \u001c(T)and initial phase \u001e0(T)from the oscillatory third term. Error bars in Figs. 3,\n4 and 5 reflect the standard error of that fit or its propagation through Eqs. (4)-(6).\nTemperature dependence of !0\nAs stated in the main text, the analytic solution of the Landau-Lifshitz-Gilbert equation yields (Supplementary\nNote 1): ~!0= 2g\u0016BM0pJD. According to mean-field theory, the temperature-dependent sublattice magnetisation\nM0(T)can be described as a paramagnet aligned by the exchange field of the neighbouring sublattices, i.e. M0(T) =\nNg\u0016 BSBS=2(M0(T)TNT\u00001), whereBS=2is the Brillouin function of an S= 2 Mn spin16,43,44andN= 6=Vwith\nV=0:374 nm3, being the unit cell volume, is the number density of Mn atoms. We model the SRT in HoMnO 3\nby introducing a linear dependence of the anisotropy parameter Dyon temperature. The SRT temperature TSR\nis defined as the point where Dyvanishes24, i.e.Dy(T) =\u0014(T\u0000TSR). For YMnO 3, assuming a temperature-\nindependent value for Dyyields a reasonable fit. This model is fitted to the measured temperature dependence of\n!0. The results are summarised in Table I.\nParameter Unit HoMnO 3 YMnO 3\nTN [K] 69.2(7) 74.7(1)\nTSR [K] 33.5(2) —\n\u0015 [T2cm3J\u00001] 70.84571.125\nDz [T2cm3J\u00001] 11.04513.925\nDy(T) [T2cm3J\u00001]\u0014(TSR\u0000T) 0.0251(1)\n\u0014 [T2cm3J\u00001K\u00001] 0.0093(6) 0\n\u0001\u0001(g\u0016B)2[T2cm3J\u00001] 0.0354(9) 0\nJ [T2cm3J\u00001] 117.2 +jDyj 120.6\nD [T2cm3J\u00001] \u0001M2\n0+jDyj 0.0251(1)\nA=p\nJ=D — Fig. 3e 69.3\nTABLE I. Fit parameters. Values for\u0015=J\nN(g\u0016B)2andDz=Kz\nN(g\u0016B)2were calculated from the exchange interaction Jand the\nout-of-plane anisotropy constant Kzgiven in the respective references.\nDATA AVAILABILITY\nThe data that support the findings of this study are available from the corresponding author on reasonable request.\nSource data are provided with this paper.\n\u0003ctzschaschel@fas.harvard.edu\nCurrent address : Department of Chemistry and Chemical Biology, Harvard University, Cambridge 02138, USA11\n[1] O. Gomonay, V. Baltz, A. 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Kiryukhin, P . Jain, and M. R. Fitzsimmons, Magnetic structures and dynamics of multiferroic systems\nobtained with neutron scattering, npj quantum mater. 1, 16003 (2016).\nACKNOWLEDGMENT\nThe authors thank B. A. Ivanov and A. V. Kimel for valuable discussions and R. Gm ¨under for experimental assis-\ntance. T.S. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Nos. JP19H01828,13\nJP19H05618, JP19K21854 and JP26103004) and the JSPS Core-to-Core Program (A. Advanced Research Net-\nworks). C.T. acknowledges support by the SNSF under fellowship P2EZP2-191801. C.T. and M.F . acknowledge\nsupport from the SNSF project 200021/147080 and by FAST, a division of the SNSF NCCR MUST.\nAUTHOR CONTRIBUTIONS\nC.T. and T.S. conceived the project. C.T. designed and evaluated the experiments. T.S. and M.F . supervised the\nproject. All authors discussed the results and contributed to writing the manuscript.\nCORRESPONDING AUTHOR\nCorrespondence to Christian Tzschaschel.\nCOMPETING INTERESTS\nThe authors declare no competing interests."    },    {        "title": "1807.00881v1.Thermoplasmonic_behavior_of_semiconductor_nanoparticles__A_comparison_with_metals.pdf",        "content": "Thermoplasmonic response of semiconductor\nnanoparticles: A comparison with metals\nVaibhav Thakore1,*, Janika Tang2,z, Kevin Conley2,y, Tapio Ala-Nissila2,3,4,§, and Mikko\nKarttunen1,5,$\n1Department of Applied Mathematics, Western University, 1151 Richmond Street, London, Ontario N6A 5B7,\nCanada\n2QTF Center of Excellence, Department of Applied Physics, Aalto University School of Science, FIN-00076, Aalto,\nEspoo, Finland\n3Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USA\n4Interdisciplinary Centre for Mathematical Modelling, Department of Mathematical Sciences, Loughborough\nUniversity, Loughborough LE11 3TU, UK\n5Department of Chemistry, Western University, 1151 Richmond Street, London, Ontario N6A 5B7, Canada\n*Email: vthakore@knights.ucf.edu, Phone: +1 226 239 4731\n‡janika.tang@aalto.fi\n†kevin.conley@aalto.fi\n§tapio.ala-nissila@aalto.fi\n$mkarttu@uwo.ca\nABSTRACT\nA number of applications in nanoplasmonics utilize noble metals, gold (Au) and silver (Ag), as the materials of choice. However,\nthese materials suffer from problems of poor thermal and chemical stability accompanied by significant dissipative losses\nunder high-temperature conditions. In this regard, semiconductor nanoparticles have attracted attention with their promising\ncharacteristics of highly tunable plasmonic resonances, low ohmic losses and greater thermochemical stability. Here, we\ninvestigate the size-dependent thermoplasmonic properties of semiconducting silicon and gallium arsenide nanoparticles to\ncompare them with metallic Au nanoparticles using Mie theory. To this end, we employ experimentally estimated models of\ndielectric permittivity in our computations. Among the various permittivity models for Au, we further compare the Drude-Lorentz\n(DL) and the Drude and critical points (DCP) models. Results show a redshift in the scattering and absorption resonances\nfor the DL model while the DCP model presents a blueshift. The dissipative damping in the semiconductor nanoparticles\nis strongest for the sharp electric octupole resonances followed by the quadrupole and dipole modes. However, a reverse\norder with strongest values for the broad dipole resonance is observed for the Au nanoparticles. A massive Drude broadening\ncontributes strongly to the damping of resonances in Au nanoparticles at elevated temperatures. In contrast, the semiconductor\nnanoparticles do not exhibit any significant deterioration in their scattering and absorption resonances at high temperatures.\nIn combination with low dissipative damping, this makes the semiconductor nanoparticles better suited for high-temperature\napplications in nanoplasmonics wherein the noble metals suffer from excessive heating.\nINTRODUCTION\nPlasmonically enhanced absorption or scattering of radiation at the mesoscale forms the basis of promising applications in\na wide variety of fields. Applications as diverse as: biosensing for lab-on-chip devices[1]; photothermal therapy in cancer\ntreatment[2,3]; energy-efficient spatiotemporal control of photocatalysis[4]; solvothermal chemistry for radiation-controlled\nsynthesis of materials[5]; energy harvesting using solar/thermophotovoltaics and thermoelectrics[6]; heat-assisted magnetic\nrecording for high density data storage[7]; and, control of radiative heat transfer in insulators[8]- all exploit the plasmonic\nbehavior of nano- or microstructured materials. Also, nearly all of these applications either lead to generation of heat due to\nlosses in the plasmonic materials/devices or require their operation at elevated temperatures. For many applications based\non plasmonics the noble metals - gold (Au) and silver (Ag) - have been the materials of choice. However, it is also common\nknowledge that these materials are characterized by problems of poor thermal and chemical stability due to large ohmic\ndissipation at high temperatures[9,10]. Furthermore, the tunability of the plasmonic resonances in the noble metals is limited\n1arXiv:1807.00881v1  [cond-mat.mes-hall]  2 Jul 2018Thermoplasmonic response of semiconductor nanoparticles\nto the ultraviolet and visible frequency regimes of electromagnetic radiation[11]. These issues have prompted a quest for\nmaterials with better thermoplasmonic properties for applications that require operating in harsh environments[12]. The use of\nthermochemically stable semiconducting materials has thus assumed importance since they present low dissipative damping\nand highly tunable plasmonic resonances through bandgap engineering and control over dopant concentration[11,13]. The\nwidespread use of noble metals in applications based on plasmonics has also meant that their thermoplasmonic properties have\nbeen extensively studied and are well-characterized[14,15,16]. However, despite their obvious technological relevance, there have\nnot been any studies reported, to our knowledge, on the thermoplasmonic properties of semiconducting nanoparticles.\nA large proportion of the energy absorbed by a plasmonic particle from the incident radiation is eventually either dissipated\nas heat or scattered away. The particle size and material dependent strength of the electron-electron ( e-e), electron-phonon\n(e-f) interactions and the magnitude of the surface scattering of the charge carriers determine the contribution of the dissipative\ndamping to the portion of the absorbed energy that goes into generating heat. This dissipation of the absorbed energy as\nheat thus leads to a rise in the temperature of the nanoparticle. Depending on the material and the size of the nanoparticle,\nthe resulting rise in temperature may cause a further increase in the absorption of the incident radiation and consequently its\ntemperature through a nonlinear feedback effect[17,18]. Additionally, there is a material, particle size and temperature dependent\nemission of photons by the plasmonic excitations referred to as radiation damping that leads to a decay of the absorbed energy\nthrough scattering. Such plasmonically enhanced scattering serves to minimize the temperature increase due to dissipative\ndamping and thus helps cool the nanoparticle[19]. It is therefore important to understand the strength and the nature of these\ntwo competing damping mechanisms, both for making an informed choice of a material for a target application and for the\ntailoring of its desired thermoplasmonic behavior.\nHere, we report results from a computational study based on the Mie theory for size-dependent thermoplasmonic properties of\nundoped silicon (Si) and gallium arsenide (GaAs) with indirect and direct bandgaps, respectively. We further compare and\ncontrast our results from spherical Si and GaAs nanoparticles with those obtained from temperature dependent plasmonic\nbehavior of Au nanoparticles. Our results indicate that there is no significant deterioration in the scattering and absorption\nresonances at elevated temperatures for the semiconductor nanoparticles as compared to the metallic Au nanoparticles.\nFurthermore, the Au nanoparticles exhibit increased Drude damping of plasmonic resonances and enhanced absorption at\nlonger wavelengths that is responsible for excessive heating at high temperatures. The theory and methods section presents the\ndetails of the dielectric permittivity models employed for computing the temperature dependent complex refractive indices of\nSi, GaAs and Au followed by the relevant details from the Mie theory.\nTHEORY AND METHODS\nThe thermoplasmonic response of a material is a function of its frequency and temperature dependent complex dielectric\npermittivity. The dielectric permittivity, in turn, is modulated by the changes in the scattering of the charge carriers and their\nconcentration based on the energy of the incident photons. The scattering of the charge carriers has contributions from collisions\namong themselves due to Coulombic interactions and from collisions with phonons that typically increase in density with a rise\nin the system temperature[15,20]. In confined systems, such as a nanoparticle, another contribution to scattering comes from the\nboundaries or surfaces[21]that becomes significant for metallic nanoparticles with radii less than \u001810nm. In reality, however,\nthe system boundaries are not always smooth and one needs to additionally account for the effect of the surface roughness as\nwell[22]. In even smaller nanoclusters (radii \u00181-5nm), the number of charge carriers for both metallic and semiconducting\nsystems becomes really small and the discreteness of the band-structure needs to be taken into account to model dielectric\npermittivity and plasmonic response[23,24,25,26]. However, to understand and isolate the differences in the thermoplasmonic\nresponse of semiconducting and metallic nanoparticles, we limit ourselves to the temperature dependent response of ideal\nspherically smooth particles whose behavior can be modeled using the Mie theory[8,27]. We further include nanoparticles that\ncan be treated as large enough to exhibit a continuous band structure. In addition, we employ experimentally estimated realistic\nmodels of the dielectric permittivity of Au, Si and GaAs to compute their thermoplasmonic response[15,28,29].\nDielectric permittivity\nThe temperature dependent refractive indices of Au and the intrinsic semiconductors (Si and GaAs) required for the computation\nof the particle size-dependent thermoplasmonic response are computed in the wavelength ranges l=400-1450 nm and\n500-1450 nm respectively using the following methodologies.\n2/11Thermoplasmonic response of semiconductor nanoparticles\nGold\nFor modeling the dielectric permittivity of metals either a Drude-Lorentz (DL) or a Drude and critical points (DCP) model is\nemployed. The Drude-Lorentz model can in principle include an arbitrary number of oscillator terms to approximate the spectral\nline shapes due to interband transitions with possibly no direct physical relevance to the optical response of a material[15,17].\nThe Drude and critical points model on the other hand is causal and physically motivated as it takes into account the asymmetric\nline shapes due to interband transitions at the critical points[15]. However, the use of the simple DCP model may also present\nproblems[14]. Therefore, we employ both the DL and DCP models to understand (a) the important differences between them,\nand, (b) their predictions of the thermoplasmonic behavior of Au particles.\nBriefly, for the DL model, we compute the dielectric permittivity for Au based on the model estimated by Rakic et al. and given\nby[20]\ne(w) =e¥\u0000W2\np\nw2+iGDw+5\nå\nj=1Cjw2\np\n(w2\nj\u0000w2)+iwgj: (1)\nHere, Wp=pf0wpis the plasma frequency associated with the intraband transitions of oscillator strength f0and the Drude\ndamping factor GD;e¥,wandwpare the background dielectric constant, frequency of the incident radiation and the plasma\nfrequency, respectively. The parameters Cj,wj, and gjrepresent the oscillator strength, characteristic frequency and damping\nrespectively for the five Lorentz oscillators considered in the DL model[20]. The temperature dependence of the permittivity in\nthe DL model comes from the second term in equation (1)through the plasma frequency Wpand the Drude damping factor GD.\nHowever, the Lorentz oscillator parameters Cj,wjandgjin the DL model, as estimated by Rakic et al. from experimental\nellipsometric data, are assumed to be independent of the temperature[20]. The temperature dependence of the plasma frequency\nwp[=Ne2=m\u0003e0] is related to the changes in the carrier concentration Nunder the influence of the lattice thermal expansion\nand is given by[17]\nwp(T) =wp(T0)p\n1+3aL(T\u0000T0): (2)\nHere, aLis the coefficient of linear thermal expansion for Au, Tis the temperature, T0=293:15K is the reference temperature,\nm\u0003is the effective mass of the electrons, eis the electric charge and e0is the permittivity of the free space. The Drude\ndamping GD(=Gee+Gef) has temperature dependent contributions from both electron-electron ( e-e) and electron-phonon\n(e-f) interactions through the corresponding damping factors GeeandGefthat are given by[17,30,31]\nGee=p3GD\n12EF\u0014\n(kBT)2+\u0012}w\n2p\u00132\u0015\n(3)\nand\nGef=Go\u00142\n5+4\u0012T\nQD\u00135ZQD=T\n0z4\nez\u00001dz\u0015\n: (4)\nHere, G=0:55is a constant determining the average scattering probability over the Fermi surface; D=0:77is the fractional\numklapp scattering; EF=5:51eV is the Fermi energy of the free electrons in Au; QD=177K is the Debye temperature for\nAu; and, Go=0:07eV is a material dependent constant obtained by fitting the bulk permittivity of Au for energies of incident\nradiation less than those corresponding to the onset of the interband transitions[17,30,31].\nFor the DCP model, we employ the experimentally estimated model by Reddy et al. with dielectric permittivity of the form[15]\ne(w) =e¥\u0000w2\np\nw2+iGDw+2\nå\nj=1Cjwj\u0014eifj\nwj\u0000w\u0000igj+e\u0000ifj\nwj+w+igj\u0015\n(5)\nwhere the parameter fjrepresents the oscillator phase for the two oscillators in the DCP model. The rest of the critical point\noscillator parameters Cj,wjandgjhave the same meaning as for the Lorentz oscillators in the DL model. The critical point\noscillators correspond to the two interband transitions observed in Au around \u0018470and\u0018330nm[32]. Reddy et al. estimate the\ntemperature dependent DCP model parameters for a number of crystalline and polycrystalline Au films of different thicknesses.\nThese films present different levels of thermophysical stability during the measurement of their optical spectra between 23-500\n\u000eC[15]. However, for our study, we employ the DCP model parameters estimated for the 200 nm thick single-crystalline Au film\nthat was observed to be stable at all temperatures under repeated cycles of measurement[15].\n3/11Thermoplasmonic response of semiconductor nanoparticles\nSilicon\nThe temperature dependence of the refractive index npfor Si is computed using an empirical power law\nz(T) =z(T0)(T=T0)b(6)\nproposed by Svantesson et al.[33]and employed by Green to model the dielectric dispersion in intrinsic Si[28]. Here, zis\nthe real ( h) or the imaginary ( k) part of the refractive index np,TandT0=300K are the temperature and the reference\ntemperature, respectively, and the exponent bis related to the normalized temperature coefficients Cz[\u0011(1=z)dz=dT] of the\noptical constants handkthrough the expression\nb=Cz(T)T=Cz(T0)T0: (7)\nThe optical constants handkfor Si are themselves estimated using the accurate spectroscopic ellipsometry data reported by\nHerzinger et al.[34]and consistently reconciled with the data from previous researchers by Green[28]. The calculations for the\noptical constants are based on a Kramers-Kronig analysis[35]that makes use of the reflectance Rto first compute its phase qin\nradians at the target energy Etas\nq(Et) =\u0000Et\np\u0014ZEt\u0000d\n0ln[R(E)]\nE2\u0000E2tdE+Z¥\nEt\u0000dln[R(E)]\nE2\u0000E2tdE\u0015\n(8)\nwhere d!0andEis a dimensionless quantity. The real ( h) and the imaginary ( k) parts of the refractive index are then\ncalculated as\nh(E) =1\u0000R(E)\n1+R(E)\u0000p\n2R(E)cosq(9)\nk(E) =2p\nR(E)sinq\n1+R(E)\u0000p\n2R(E)cosq: (10)\nThe temperature dependent dielectric permittivity for Si is then given by\ne(T) =n2\np(T) = [h(T)+ik(T)]2=h2\u0000k2+2ihk: (11)\nGaAs\nWe employ a heuristic model proposed and estimated by Reinhart for photon energies below, at and above the bandgap to model\nthe temperature dependence of the refractive index for the direct bandgap intrinsic GaAs[29]. This model realistically accounts\nfor the existence of the experimentally observed band tails and the non-parabolic shape of absorption above the bandgap\nin undoped GaAs. The model requires estimating the continuum ( ac) and the exciton ( aex) contributions to the absorption\ncoefficient a(=ac+aex)by fitting the following functions to the experimental absorption spectrum[29]\nac(E0) =Aexp[r(E0)]\u00141\n1+exp(\u0000E0=Es1)+aso\n1+exp((D\u0000E0)=Eso)\u0015\n; (12)\nand\naex(E0) =2\nå\nj=1\u0014axj\nexp[(E0+Exj)=Es1]+exp[\u0000(E0+Exj)=Es2]\u0015\n; (13)\nwhere\nr(E0) =r1E0+r2E02+r3E03andE0=E\u0000Eg(T):\nThe first term in equation (12) describes the increase in the absorption aabove the fundamental bandgap while the Fermi-\nfunction-type terms represent the sharp step-like absorption increase at the fundamental and the split-off gaps. The temperature\ndependent bandgap-shrinkage effect is modeled using[36]\nEg(T) =Eg(0)\u0000sQ\n2\u0014\nps\n1+\u00122T\nQ\u0013p\n\u00001\u0015\n: (14)\n4/11Thermoplasmonic response of semiconductor nanoparticles\nFor intrinsic GaAs: Eg(0) =1:5192 eV is the zero-temperature bandgap; s=0:475eV/K is the high-temperature limiting value\nof the forbidden-gap entropy; Q=222:4K is the material specific phonon temperature that represents the effective phonon\nenergy ( }weff\u0011kBQ,}is the reduced Planck’s constant); and, p=2:667is an empirical parameter that is a consequence of the\nglobally concave shape ( p>2) of the electron-phonon spectral distribution function in conjunction with the contributions from\nthe thermal expansion mechanism to the total gap-shrinkage effect[36].\nThe parameters Aandasoin equation (12) are the absorption amplitudes for the fundamental and the split-off valence band\ntransitions respectively whereas the coefficients rjdescribe the increase in the absorption for energies E>Eg. The energy\nslope parameters Es1andEs2in equation (13) along with Esoin equation (12) for the split-off valence band are a linear function\nof the temperature that account for the weak electron-phonon coupling and are given by[29]\nEs1=7(T+7)=304;Es2=4(T+7)=304 and ;Eso=40(T+7)=304 (in meV ): (15)\nThe temperature dependent discrete transition amplitudes axj(j=1;2)in equation (13) relate to the transitions of the excitonic\nground and first excited states. These are computed as[29]\nax1=ax0exp(\u0000T=230)[1\u0000exp(\u0000Ex1=kBT)] (16)\nand\nax2=ax0exp(\u0000T=230)[1\u0000exp(\u0000Ex2=kBT)]=8: (17)\nThe exciton binding energies of the fundamental (Ex1)and the first excited (Ex2)states are Ex1=4Ex2=4:22 meV .\nThe imaginary part kof the refractive index ncan thus be calculated from the absorption coefficient aas a function of the\nenergy ( E) using the Kramers-Kronig integral as[29,35]\nk(E) =c}\nqpZEu\n0a(E)dE\nE2\u0000E2: (18)\nHere, cis the speed of light and qis the magnitude of the electronic charge. The upper limit of the integral is taken to be\nEu=Eg+1:5eV . For the real part of the refractive index h, the dominant contribution from the high critical points E2=3and\nE3=5 eV is captured using[29]\nh(E) =s\n1+AK2\nE2\n2\u0000E2+AK3\nE2\n3\u0000E2+A4\nE2\n4\u0000E2: (19)\nThe coefficients AKirepresent the oscillator strengths (in eV2) related to the critical points and are quadratic functions of\nthe temperature T. The last term approximates the contribution of the reststrahl band with A4=0:0002382 (eV2) and\nE4=Eph=0:0336 eV the optical phonon energy. All these parameters for the real part of the refractive index are determined\nby fitting to the precise refractive index data obtained from experiments. The dielectric permittivity for GaAs can thus be\ncalculated using equation (11) in a manner similar to Si.\nMie scattering\nThe size and temperature dependent efficiencies for scattering ( Qsca) and absorption ( Qabs) are calculated in the temperature\nrange 200-650 K based on Mie theory using an algorithm by Wiscombe[37]. The expressions for the efficiencies are[8,27,37]\nQsca=2\nx2N\nå\nn=1(2n+1)(janj2+jbnj2); (20)\nQabs=2\nx2N\nå\nn=1(2n+1)(Re(an+bn)\u0000(janj2+jbnj2)): (21)\nThe variables anandbnrepresent the electric and magnetic Mie coefficients given by\nan=mmn2\nrjn(nrx)[x jn(x)]0\u0000mpjn(x)[nrx jn(nrx)]0\nmmn2rjn(nrx)[xhn(x)]0\u0000mphn(x)[nrx jn(nrx)]0; (22)\nbn=mpjn(nrx)[x jn(x)]0\u0000mmjn(x)[nrx jn(nrx)]0\nmpjn(nrx)[xhn(x)]0\u0000mmhn(x)[nrx jn(nrx)]0: (23)\n5/11Thermoplasmonic response of semiconductor nanoparticles\nHere, hn(x) =jn(x)+iyn(x)are the Hankel functions of order nwherein jnandynare the spherical Bessel functions of the first\nand second kind respectively; nr=np=nmis the relative refractive index; npis the complex refractive index of the particle; mp\nandmmare the permeabilities of the spherical particle and the host medium respectively; and, the primes indicate differentiation\nof the argument in the square parentheses with respect to the size parameter x(=2pr nm=l) of the particle. For our simulations,\nhowever, we assume both the host medium and the nanoparticles to be non-magnetic i.e.mm=mp=1. The surrounding\nmedium is also assumed to be isotropic and non-absorbing with a constant refractive index of nm=1:5. For more details on\nthe theory and computation of Mie efficiencies the reader is referred to our previous work[8]and the relevant text[27,37]. In all\nour computations here, we further assume that the modeled particles are in thermal equilibrium at a given temperature and no\nthermal transients occur. However, the radii r(T)of the Au, Si and GaAs nanoparticles are adjusted for thermal expansion in\nour calculations for the Mie efficiencies using\nr(T) =ro(1+aLT): (24)\nThe coefficient of linear thermal expansion aAu\nL(=1:42\u000210\u00005m/K) for Au nanoparticles is assumed to be constant with\ntemperature[17]. The linear coefficient of thermal expansion for the Si nanoparticles is computed using[38]\naSi\nL= [3:725[1\u0000exp(\u00005:88\u000210\u00003(T\u0000124))]+ 5:548\u000210\u00004T]10\u00006(25)\nwhile for the GaAs nanoparticles the thermal expansion data from Glazov et al.[39]is used after suitable interpolation and linear\nextrapolation to temperatures below 390 K. The effect of the surface scattering is taken into account for the Au nanoparticles by\nincluding a corresponding damping term in the computation of the Drude contribution to the dielectric permittivity of Au in\nequations (5) and (1)\nGD!Geff=GD+2pA}vF\nr; (26)\nwhere vF=1:4\u0002106m/s and A=0:33is a geometric form factor estimated using the best fitting value for absorption spectra\nfrom individual gold nanoparticles[17]. However, this term is neglected in our calculations for the intrinsic Si and GaAs\nnanoparticles because of the characteristic low density of charge carriers.\nRESULTS AND DISCUSSION\nComplex refractive indices\nWe first examine and compare the results for the temperature dependent complex refractive indices for the materials Au, Si and\nGaAs. Figure 1a-c, d-f shows the real ( h) and the imaginary ( k) parts of the refractive indices for Au, Si and GaAs respectively.\nThe results in Figure 1a,d for handkhave been computed using the DL model of temperature dependent dielectric permittivity\nfor Au while Figure S1a,b shows the results from the DCP model. For a temperature T, Figure 1a shows that hfor Au initially\ndecreases in the interval l=400-760nm (see inset, Figure 1a) and then increases continuously up to 1450 nm with an increase\nin the wavelength of the incident radiation. A similar behavior is seen for kwherein for a given temperature T, it decreases with\nan increase in wavelength until 450nm and monotonically increases thereafter. However, has a function of the temperature\nincreases monotonically with an increase in the temperature Tfrom 200to650K at any given wavelength in the 400-1450 nm\nrange. In contrast, kincreases for wavelengths in the range 400-510nm and then decreases until l=1450 nm with an increase\nin the temperature from 200 to 650 K (see inset, Figure 1b).\nThe results for the refractive index from the DCP model (Supplementary Information, Figure S1a,b ) show a variation that\nis very similar to the DL model as a function of the wavelength. Unlike the DL model, however, the results from the DCP\nmodel show a highly non-monotonic variation in handkwith an increase in temperature from 200-650K (Figure S1a,b). In\nthe wavelength interval 400-500nm,hdecreases with temperature until 560 K and thereafter increases slightly between 560\nand650K. Then there occurs a transition region in the interval between l=500-740nm (see inset: Figure S1a) wherein h\ninitially increases with an increase in Tfrom 200 to 470 K, then decreases between 470-560 K followed by an increase again\nforT>560K. Between l=740-1450 nm,hincreases monotonically with an increase in temperature. Also, notably the\nrelative increase in his greater at longer wavelengths (beyond 740nm) with nearly a doubling of the hvalue between 200and\n650K atl=1450 nm. As opposed to the decrease of kwith an increase in the temperature in the DL model, the kfor Au in\nthe DCP model increases monotonically at all wavelengths (except for a small region between 400-425nm) for temperatures\nbetween 200 and 560 K followed by a saturation between 560 and 650 K (inset: Figure S1b).\nThis temperature dependent behavior of the refractive index, at longer wavelengths and away from the interband transitions at\n\u0018470and\u0018330nm in the DCP model, can be attributed to the changes in the plasma frequency wpand the Drude damping\nGD[=}=tD,tD\u0011Drude relaxation time]. Reddy et al.[15]demonstrate through their experimental work that the temperature\n6/11Thermoplasmonic response of semiconductor nanoparticles\nFigure 1. Comparison of the temperature dependent (a-c) real ( h) and (d-f) imaginary ( k) parts of the refractive indices for\nAu (DL model), Si and GaAs respectively. In (a-c), his seen to increase monotonically with temperature, T, for all materials.\nThe inset in (d) shows the non-monotonic behavior of kfor Au: kincreases with temperature at shorter wavelengths followed\nby a decrease at the longer wavelengths. This is in contrast with the monotonic increase of kwith temperature for Si and GaAs\nin (e-f).\ndependence of wpandGDis a direct consequence of (a) the decrease in carrier density N[µ(1+3aLDT)\u00001] due to thermal\nexpansion, (b) decrease in the effective mass m\u0003, and, (c) an increase in the electron-phonon ( e-f) interaction with an increase\nin temperature. The counteracting effect of the decrease in Nandm\u0003is reflected in the initial increase of wpwith rising\ntemperatures up to \u0018473K wherein the decrease in m\u0003is dominant. At still higher temperatures ( T>473) K, a decrease in\nwpfollows due to a decrease in the charge carrier density as a result of thermal expansion[15]. In contrast, the temperature\ndependence of the plasma frequency in the DL model has contributions only from the changes in the carrier concentration Nas a\nresult of thermal expansion while the effective mass of the free electrons is assumed to be independent of the temperature[17,20].\nIn the DL model, the temperature dependence of the e-einteractions shown in equation (3) is quadratic in Tbut is masked by\nthe high frequencies ( w) of the incident radiation ( l=400-1450 nm)[17,30,31]. Reddy et al. show that in the DCP model the\ncontribution of the e-finteraction to GDfor temperatures Tabove the Debye temperature ( q=170K) is linear and directly\nproportional to Tthereby resulting in a steady increase in GDwith a rise in T[15]. In the DL model, however, no such assumption\nis made and the temperature dependence of the damping factor Gefin equation (4)is computed without any assumption of\nlinearity[17,30,31].\nConsidering next the semiconductors Si and GaAs, Figures 1b,e and 1c,f respectively show that the increase in handkwith\nincreasing temperatures is monotonic across the entire wavelength range of 500-1450 nm. The extinction coefficient for Si\n(k\u00180:1-0:001) and GaAs ( k\u00180:6-0:1) is also observed to stay almost constant with temperature for the shorter wavelengths\nwithin the absorption band. However, at the longest of wavelengths ( l\u00181450 nm), kcontinually increases with temperature\nfrom low values on the order of 10\u000017to about 10\u00006and10\u000020to10\u00007for Si and GaAs, respectively. Thus, with the values of\nkfor Au lying between 6-10at the longer wavelengths, the k-values for Si and GaAs are still about 7-8orders of magnitude\nsmaller compared to those for Au at the highest temperature ( T=650K). Such low values of kfor the intrinsic Si and\nGaAs point to the low number of charge carriers generated for longer wavelengths of the incident radiation away from the\nabsorption band. At the same time, an almost 13-14orders of magnitude increase in kpoints to the enhanced e-fcoupling\nat elevated temperatures. This leads to an increase in the intrinsic carrier concentrations ri[µexp(\u0000Eg(T)=2kBT)] due to\nbandgap shrinkage and elevated temperatures. Also, between Si and GaAs, the sharp band-edge representing a steep increase in\nkis clearly visible for the direct bandgap GaAs. This bandgap shrinks with an increase in the temperature as evidenced by the\nsignificant red-shift of the absorption band-edge from l\u0019900nm at 200K tol\u00191090 nm at 650K. In contrast, there exists\nno noticeable sharp increase in kfor the indirect bandgap Si and instead a gradual increase representing phonon-mediated\n7/11Thermoplasmonic response of semiconductor nanoparticles\nFigure 2. Mie scattering efficiency, Qsca, as a function of the wavelength, l, of the incident radiation and the particle radii rat\nthree different temperatures: T=200(a-c), 430(d-f) and 650K (g-i) for Au (DL model), Si and GaAs particles, respectively.\nThe strength of the resonances in Qscafor Au particles shows a decline with an increase in the temperature, T, whereas the Si\nand GaAs particles show no significant deterioration. (See Figure 4 also.)\ninterband transitions is observed. Although, similar to GaAs, an increase in temperature is accompanied by a shrinking of the\nbandgap in Si as well.\nMie scattering and absorption efficiencies\nFigures 2 and 3 show the variations in the Mie scattering ( Qsca) and absorption ( Qabs) efficiencies, respectively, at three different\ntemperatures T=200,430and650K as a function of the wavelength, l, of the incident radiation and the particle radii in the\n5-1500 nm range for Au (DL model; see Figures S2 and S3 for the DCP model), Si and GaAs particles. Here, two types of\nbroadening and a shift of spectral features are evident - one arising from the increase in the particle radii and the other due to\nan increase in the temperature. The second temperature dependent contribution to the broadening and shifting of the spectral\nfeatures at elevated temperatures is straight-forward to understand and interpret for the Au (DL model), Si and GaAs particles.\nHowever, this is rendered rather complicated in the case of the Au (DCP model) nanoparticles because of the non-monotonic\nbehavior of the dielectric permittivity ( e) or refractive index ( np) with rising temperatures in the DCP model (Figures S1\nand S5). Therefore, we first address the effects arising due to an increase in the particle radii and leave a discussion on the\ntemperature effects for later when we consider the thermoplasmonic response of individual nanoparticles with a given radii.\n8/11Thermoplasmonic response of semiconductor nanoparticles\nFigure 3. Mie absorption efficiency, Qabs, as a function of the wavelength, l, of the incident radiation and the particle radii r\nat three different temperatures T=200(a-c), 430(d-f) and 650K (g-i) for Au (DL model), Si and GaAs particles, respectively.\nA decrease in Qabsfor the smaller Au particles is observed with a rise in temperature at shorter wavelengths in contrast to an\nincrease at the longer wavelengths. The Si and GaAs particles exhibit no significant decline in Qabsat elevated temperatures\nand a clear redshift of the absorption resonances is seen. (See Figure 7 also.)\nA broadening and redshifting of the spectral features due to increasing particle radii are observed for Au, Si and GaAs particles\nalike as a result of the weakening of the restoring force that drives the plasmonic resonances (Figures 2, 3 and S2-S3). The\nincrease in the particle radii leads to an increased distance between the oscillating charges on the opposite sides of the particle\nand therefore lower plasmon energies corresponding to a redshift. This particle size-dependent redshift can be better understood\nwhen one considers the small particle limit such that the size parameter x\u001c1. The resonance condition for the nthorder electric\nmodes anin the case of a vanishingly small particle ( x!0), wherein the magnetic modes can be neglected, requires that the\ndenominator in equation (22) goes to zero and is given by[8,27]\ne(wn) =\u0000n+1\nnem: (27)\nHere, em=2:25is the dielectric permittivity for the non-absorbing host medium. The frequencies ( wn) obtained from the above\nresonance condition are complex and therefore the associated modes are referred to as virtual[8,27]. Thus, for real materials this\n9/11Thermoplasmonic response of semiconductor nanoparticles\ncondition can only be satisfied approximately. In such cases, the real frequency that approximates this resonance condition\nclosest is referred to as the Fr ¨ohlich frequency[27]. Furthermore, the imaginary part e00of the dielectric permittivity is required\nto vanish here because the left-hand side of equation (27) is complex while the right-hand side is real.\nFigure 4. Mie scattering efficiency ( Qsca) with dipole (blue), quadrupole (red) and octupole (green) contributions as a\nfunction of the wavelength, l, for (a, d, g, j) Au (DL model), (b, e, h, k) Si and (c, f, i, l) GaAs nanoparticles of radii\nr=20;80;140and200nm, respectively, at temperatures T=200(solid lines) and 650(dotted lines) K. The thick orange and\npink solid lines represent the total scattering efficiencies, Qsca, at temperatures T=200 and 650 K, respectively. The sharp\nhigher order resonances in Si and GaAs nanoparticles against a background of the broad dipole modes give rise to Fano\nresonances that do not decay significantly with an increase in the temperature. In contrast, the Mie resonances in the Au\nnanoparticles exhibit massive Drude broadening and a deterioration in strength with temperature increase due to dissipative\ndamping. Here, the Mie computations for the nanoparticles of different sizes take into account their thermal expansion,\nalthough the text labels indicate the values for nanoparticle radii at 200 K.\nFor a finite but small size parameter ( x\u001c1) (the case of interest here), the resonance condition for the Fr ¨ohlich mode is\nobtained by a power series expansion of the Hankel ( hn) and Bessel ( jn) functions in the denominator of equation (22) to second\norder in the size parameter x. The equation (27) then gets modified to[8,27,40]\ne0=\u0000(2+12x2=5)em: (28)\nNow, an examination of the dielectric permittivity in Figures S4a-c and S5a reveals that for all three materials, metallic Au (DL\nand DCP models) and the semiconducting Si and GaAs, the real part e0[=h2\u0000k2] of the dielectric permittivity increases with\ndecreasing wavelength in the range l=400-1450 nm and 500-1450 nm, respectively. Therefore, since the right-hand sides of\n10/11Thermoplasmonic response of semiconductor nanoparticles\nequations (27) and(28) are negative, an increase in the size parameter xdue to an increase in the particle radii redshifts the\nresonance frequency for the electric modes an.\nA further comparison of Qscain Figure 2 for the Au nanoparticles (DL model) with the Si and GaAs nanoparticles shows a\ngeneral decline in the scattering efficiency for the Au nanoparticles with an increase in the temperature as opposed to steady\nvalues of Qscafor the Si and GaAs nanoparticles. Also, it is observed that the results from the DCP model (Figure S2) show a\nstronger scattering efficiency than the Qscafor the DL model in Figure 2. Next, the distinct variations in the Mie absortpion\nefficiency Qabsare shown in Figure 3 as a function of the wavelength land the particle radii rvarying between 5-1500 nm at\nthree different temperatures ( T=200,430and650K) for Au (DL model), Si and GaAs particles. In Figure 3a,d,g, the Au\nnanoparticles exhibit increased absorption efficiencies at longer wavelengths with an increase in temperature for the larger\nparticles while for the smaller Au nanoparticles there occurs a decrease in Qabs. Figure 3b,e,h shows that for the Si particles, the\nabsorption appears diffuse characterized by a redshift to longer wavelengths with an increase in temperature. The diffuseness\nand the redshift of the Qabsspectra are a result of the phonon-mediated interband transitions due to an increase in the e-f\ninteraction and a shrinking of the indirect bandgap at high temperatures, respectively. With the notable exception of a sharp\nabsorption band-edge, a similar spectral redshift is also observed in Qabsfor the GaAs particles due to bandgap shrinkage\n(Figure 3c,f,i). The sharp band-edge is a consequence of the direct interband transitions in GaAs and becomes progressively\ndiffuse on account of increased e-fcoupling at elevated temperatures (compare subplots, Figure 3c,f,i).\nWe next examine the contributions of the different Mie modes to Qscato further understand the effect of the temperature and\nparticle size on its spectral behavior. Figure 4 presents the contributions from the dipole, quadrupole and octupole modes (order\nn\u00143) to the total scattering efficiencies for particles of radii 20;80;140and200nm at T=200and650K. Note, that for\ntemperatures greater than 200K we take into account the thermal expansion of the nanoparticles. Figure 4a-c and a comparison\nof Figures 5a-c and 6a-c show that for the smaller 20nm particles the strongest and the most significant contribution to Qsca\ncomes from the electric dipole mode for all three types of nanoparticles (Au (DL model), Si and GaAs). Upon increasing the\nparticle radii to 80nm (Figure 4d-f and Figures 5d-f, 6d-f), the dipole contribution to Qscabegins to exhibit a strong and distinct\nresonance that broadens and redshifts with an increase in temperature from 200to650K. Additionally, there appears a weak\nquadrupole contribution from both magnetic and electric modes at short wavelengths that is strongest for the Si nanoparticles\n(Figures 4e, 5e and 6e). For the 80nm Si and GaAs particles a large contribution to Qscacomes from the magnetic modes\nthat exhibit sharper resonances and occur at longer wavelengths than the electric dipole modes (Figures 5d-f and 6d-f). In all\ncases, the magnetic Mie resonances of the same order as the electric modes occur at longer wavelengths and hence have lower\nassociated energies. However, in the case of the 80nm Au nanoparticles the strongest contributor to the resonance in Qscais by\nfar still the electric dipole mode (Figures 4d, 5d and 6d). With an increase in the temperature there is also a weakening of the\ndipole contribution in the 80 nm nanoparticles for all materials.\nAs one transitions to the larger 140and200nm particles, a strengthening of the quadrupole and octupole modes is observed\nin all materials (Figures 4g-l, 5g-l and 6g-l). In Au nanoparticles, the quadrupole and octupole modes are also very broad\nand contribute to the already massive Drude broadening from the dipole modes (Figures 4g,j, 5g,j and 6g,j). The largest of\nthe Au nanoparticles with radii equal to 140and200nm have significant contributions to Qscafrom the magnetic modes bn.\nHowever, these are still weaker than contributions from the electric modes an(Figures 4g,j, 5g,j and 6g,j). In contrast, sharp\nmagnetic resonances bnoccur against a background of the broad electric modes anin the larger ( r=140,200nm) Si and\nGaAs nanoparticles. This gives rise to Fano resonances in the corresponding spectrum for Qsca(Figures 4h-i,k-l, 5h-i,k-l and\n6h-i,k-l). The Fano resonances from the octupole magnetic modes ( b3), in particular, are observed to be especially sharp for\nthe Si nanoparticles compared to those for the GaAs nanoparticles (Figures 4h-i,k-l, 5 and 6). For all radii and materials a\nredshifting, weakening and broadening of the Mie resonances of all orders is observed with an increase in temperature from\n200to650K. This is consistent with the increase in the Drude damping associated with free carrier absorption for Au and the\nincreased e-eande-finteractions in the semiconductors at elevated temperatures[15,17].\nFigure 7 shows the absorption efficiency Qabsas a function of lfor the Au (DL model), Si and GaAs nanoparticles of radii\nr=20;80;140and200nm at the temperatures 200and650K. Similar to the results for Qsca(Figure 4a-c), the strongest\ncontribution to Qabsfor the smaller 20nm particles comes from the electric dipole modes a1for the nanoparticles of all three\nmaterials (Figure 5a-c). However, a clear difference in Qabsbetween the metallic Au and the semiconductor nanoparticles\nis seen with an increase in the temperature from 200to650K. The plasmonic resonance peak in Qabsfor the 20nm Au\nnanoparticles broadens and decreases in height with an increase in the temperature. In contrast, the absorption efficiency Qabs\nfor the 20nm Si and GaAs particles exhibits an increase at higher temperatures. We note here that Yeshchenko and coworkers\nhave also reported similar experimental results for the temperature dependence of plasmonic resonances in Au nanoparticles\n(r=10nm) that are embedded in a silica matrix[16]. Here, the decrease in Qabsfor the 20nm Au nanoparticles occurs because\nof an increase in the Drude damping from the greater e-eande-fcoupling at elevated temperatures (equations (3)and(4)). On\n11/11Thermoplasmonic response of semiconductor nanoparticles\nFigure 5. Electric dipole (blue), quadrupole (red) and octupole (green) Mie modes ( an) as a function of the wavelength, l, for\n(a, d, g, j) Au (DL model), (b, e, h, k) Si and (c, f, i, l) GaAs nanoparticles of radii r=20;80;140and200nm, respectively, at\ntemperatures T=200 (solid lines) and 650 (dotted lines) K. A broadening and redshifting of the electric Mie resonances is\nobserved with increasing particle size and temperature for all materials. The larger Si and GaAs nanoparticles exhibit sharp\nhigher order electric resonances. These, alongwith the magnetic Mie modes ( bn) (Figure 6), are responsible for giving rise to\nFano resonances in QscaandQabsfor Si and GaAs nanoparticles (Figures 4 and 7). Here, the Mie computations for the\nnanoparticles of different sizes take into account their thermal expansion, although the text labels indicate the values for\nnanoparticle radii at 200 K.\nthe other hand, the increase in the Qabsfor the Si and GaAs nanoparticles is driven by the bandgap shrinkage effect and the\nincreased e-finteractions associated with the rise in temperature from 200to650K. This characteristic redshift and broadening\nof the absorption band at high temperatures is evident in the absorption efficiencies for the larger ( 80,140, and 200nm) Si\nand GaAs nanoparticles as well (Figures 7e-f,h-i,k-l, 5e-f,h-i,k-l, and 6e-f,h-i,k-l, respectively). The appearance of the broad\nhigher order electric modes ( n\u00152) at shorter wavelengths in the 140and200nm Au particles against a background of the\nmostly featureless and broad magnetic modes (except for the magnetic dipole mode b1for the 200nm Au particle) leads\nto small humps in Qabs. Unlike the smaller ( r=20nm) Au nanoparticles, these strong contributions from the higher order\nelectric modes to Qabsin the larger Au nanoparticles also counteract the decrease in the height of the plasmonic resonance\npeak due to Drude damping at elevated temperatures. The stronger absorption efficiency Qabsfor the large Au nanoparticles\n(r=80;140and200nm; Figure 7d,g,j) observed at the long wavelengths, away from the interband transitions, can be attributed\nto the accompanying free carrier absorption and the increased e-finteraction at higher temperatures. In contrast to the broad\nresonance absorption observed for the Au nanoparticles due to the electric modes, the Si and GaAs nanoparticles exhibit Fano\nresonances (Figure7e-f,h-i,k-l). These arise due to contributions from the higher order resonant Mie modes ( anandbn) that\n12/11Thermoplasmonic response of semiconductor nanoparticles\nFigure 6. Magnetic dipole (blue), quadrupole (red) and octupole (green) Mie modes ( bn) as a function of the wavelength, l,\nfor (a, d, g, j) Au (DL model), (b, e, h, k) Si and (c, f, i, l) GaAs nanoparticles of radii r=20;80;140and200nm, respectively,\nat temperatures T=200 (solid lines) and 650 (dotted lines) K. Similar to the electric Mie resonances an(Figure 5), the\nmagnetic resonances also broaden and redshift with an increase in particle size and temperature for all materials. The larger Si\nand GaAs nanoparticles exhibit sharp higher order magnetic Mie resonances. These, alongwith the electric Mie modes ( an)\n(Figure 5), are responsible for giving rise to the Fano resonances in QscaandQabsfor Si and GaAs nanoparticles (Figures 4 and\n7). Here, the Mie computations for the nanoparticles of different sizes take into account their thermal expansion, although the\ntext labels indicate the values for nanoparticle radii at 200 K.\noccur against a background of the broad quadrupole or dipole modes (Figures 5e-f,h-i,k-l and 6e-f,h-i,k-l). In particular, for the\nlarger Si nanoparticles ( r=140and200nm), the Fano resonances are sharper and their number increases with the size of the\nnanoparticle (Figures 5h-i,k-l and 6h-i,k-l, respectively)[8].\nThe electric ( an) or magnetic ( bn) modes that contribute the most to the resonant plasmonic absorption in the nanoparticles\nof these three materials are the higher order ( n\u00152) Mie modes that occur at wavelengths close to the interband transitions\n(Figures 5 and 6). This is not a coincidence as the maximal contribution Qn;max\nabsof the nthorder Mie mode to Qabsfor a small\nbut finite-sized particle is given by[41]\nQn;max\nabs=1\nx2\u0014\nn+1\n2\u0015\n: (29)\nThus, for small but finite-sized particles the maximal absorption efficiency Qn;max\nabsfor a Mie mode is directly proportional to\nits order n. Again, the resonance condition (equation (27)) for the higher order Mie modes also shows that these resonances\nalways occur at wavelengths less than those for the dipole modes. This is because for most materials, including Au, Si and\n13/11Thermoplasmonic response of semiconductor nanoparticles\nFigure 7. Mie absorption efficiency ( Qabs) as a function of the wavelength lfor (a, d, g, j) Au (DL model), (b, e, h, k) Si and\n(c, f, i, l) GaAs nanoparticles of radii r=20;80;140and200nm, respectively, at temperatures T=200(blue) and 650(red) K.\nFor the Au nanoparticles, the highest Qabsis observed for the smallest 20 nm particles. An increase in Qabswith temperature\nfor the larger Au nanoparticles is evident at the longer wavelengths due to uniform Drude broadening. The Si and GaAs\nnanoparticles exhibit higher Qabsfor the larger particles at short wavelengths and negligible values at longer wavelengths with\nincrease in temperature. A redshift in the resonances is also observed at elevated temperatures for the semiconductor\nnanoparticles. Here, the Mie computations for the nanoparticles of different sizes take into account their thermal expansion,\nalthough the text labels indicate the values for nanoparticle radii at 200 K.\nGaAs considered here, the real part e0of the dielectric permittivity is an increasing function of the decreasing wavelength of\nthe incident radiation (Figures S4a-c and S5a). The factor (n+1)=n2[2;1]forn2[1;¥]and the negative right hand side in\nequation (27) when combined with the preceding observation ensure that the higher order Mie resonances appear blueshifted\nrelative to the dipole modes a1andb1.\nDrude-Lorentz Vs Drude and critical points model\nThe Mie resonances in the DL model for Au nanoparticles discussed above exhibit a redshift with an increase in the temperature\n(Figures 4-7a,d,g,j). In contrast, the results from Mie computations based on the DCP model estimated by Reddy et al.[15]\npresent a blueshift (Figures S6-S9). The redshift of the resonances in QscaandQabs(Figures 4a,d,g,j and 7a,d,g,j respectively),\nobtained using the DL model here, is consistent with the experimental results reported by Yeshchenko et al.[16]on the\ntemperature dependent shift of the plasmon resonance for Au nanoparticles embedded in a silica matrix. The difference in the\nspectral shift of the Mie resonances between the results from the DL (Figures 4-7) and DCP (Figures S6-S9) models arises\nprimarily because the real part e0of the dielectric permittivity in the DCP model (Figure S5a) decreases with an increase in\n14/11Thermoplasmonic response of semiconductor nanoparticles\nthe temperature as opposed to an increase in the DL model (Figure S4a). The real part e0of the permittivity, we recall, is an\nincreasing function of the decreasing wavelength of the incident radiation (Figures S4a and S5a). Thus, the blueshift of the Mie\nresonances for the Au nanoparticles in the DCP model (Figures S6-S9) is a direct consequence of (a) the resonance condition in\nequation (27) for the electric modes an, and, (b) the decrease in e0as a function of the rising temperature (Figure S5a). Note here\nthat the DCP model for the temperature dependent dielectric permittivity of Au is estimated from the spectroscopic ellipsometry\nmeasurements of 200 nm thick Au films deposited on a mica substrate under ambient conditions[15]. Similarly, the DL model\nestimated by Rakic et al.[20]also makes use of the ellipsometric measurements of Au thin films for the estimation of the Drude\nand Lorentz oscillator parameters at room temperature. However, the key difference between the two models is that the Lorentz\noscillator parameters for the interband transitions in the DL model[20]are independent of temperature whereas the critical\npoint oscillator parameters in the DCP model[15]depend on the temperature. An additional difference between the two models\ncomes from the temperature dependence of the Drude contribution to the permittivity. In the DL model, the Drude contribution\nfrom the e-eande-finteractions (equations (3)and(4)respectively) is computed based on theoretical considerations[17,30,31]\nwhereas in the DCP model it is estimated directly through a fitting of the experimental data[15]. Furthermore, unlike the DL\nmodel, the contributions from the interband transitions to the imaginary part e00of the dielectric permittivity are negative and\nunphysical for the DCP model[14,15](Figures S10 and S11). Regardless, this should have no actual bearing on the final results\nfrom the DCP model because the computation of the Mie resonances here ultimately depends only on the dielectric permittivity\ne. The results from the DCP model are therefore independent of the way the interband transitions are computed if the model\nitself is estimated based on accurate experimental data and approximates it closely.\nRadiation Vs dissipative damping\nThe dissipative damping of the nthorder resonance in a plasmonic nanoparticle is characterized by the imaginary part\ne00\nr(ln) =e00(ln)=emof the relative dielectric permittivity of its material and is therefore straightforward to estimate as a\nfunction of the temperature. However, an estimation of the radiation damping for a resonant Mie mode associated with a\nplasmon resonance presents some challenges. Experimentally, the radiation damping for a plasmon resonance is estimated by\nfitting a Lorentzian characterized by a linewidth Geff(=GD+Grad) to the extinction data obtained from a suspension of ideally\nmonodisperse nanoparticles[42,43]. This estimated linewidth Geffrepresents the total damping associated with the plasmonic\nresonance. The radiation damping Grad(=2}xV)itself is taken to be proportional to the volume Vof the particle with xas its\nstrength. The radiation damping strength xis then estimated by a linear fitting of the total damping for the different volumes V\nof the plasmonic nanoparticle with GDandGradas free parameters in the linewidth, Geff[42,43]. The same procedure can also\nbe employed if the extinction curves have been obtained from the finite difference time domain simulations of the plasmonic\nresonance[43]. This approach to determine Gradhowever fails, if the plasmonic resonance occurs too close to an interband\ntransition as is the case in our results for the Si and GaAs nanoparticles[43].\nOn the theoretical side, one is again limited to using analytical results in the limit of a small but finite particle size. In such a\ncase, the radiation damping ( \u0011e00\nrad(ln)) for the nthorder Mie mode dominates over the dissipative damping characterized by\ne00\nr(ln)when the following condition is satisfied[44]\ne00\nr(ln)\u001cx2n+1\nn\nn[(2n\u00001)!!]2\u0011e00\nrad(ln): (30)\nFigure 8 shows the ratio Lof the measure of the radiative damping e00\nradto the imaginary part e00\nr(=e00=em)of the relative\ndielectric permittivity at the resonance wavelength ( ln) as a function of the temperature for the strongest of the electric modes\n(an). Additionally, Figure S12 shows the plots for e00\nrad(ln)ande00\nr(ln)separately while Figure S13 presents the actual electric\nresonances anat the temperatures 200and650K. The above expression in equation (30) for the radiation damping e00\nrad(ln)is\nstrictly valid only for x\u001c1. This condition is not satisfied by the resonance wavelengths lnfor the sizes ( r=95;120and145\nnm) of the nanoparticles considered here. Nevertheless, equation (30) does offer useful qualitative insights into the relative\nstrengths of the radiative and dissipative damping in the nanoparticles. For the above reasons and in the absence of a better\nalternative, we employ L(=e00\nrad=e00\nr)to understand the relative contributions of the radiative and dissipative dampings to the\nplasmonic resonances observed in the Au, Si and GaAs nanoparticles.\nIt is clearly seen in Figure 8 that Ldecreases with an increase in the temperature from 200to650K for all materials. This is a\ndirect consequence of an almost linear increase in dissipative damping, characterized by e00\nr(ln), with increasing temperatures\nfor all materials (Figure S12). Consistent with the experimental findings, a comparison of the plots in Figure 8 for the different\nradii shows that in general the radiation damping increases with an increase in the volume of the nanoparticles[42,43]. Among all\nthree materials, the metallic Au nanoparticles present the smallest values for Lpointing to the presence of a strong dissipative\nbehavior as a consequence of the Drude damping of free electrons in the conduction band (Figure 8a,d,g). This is also borne\n15/11Thermoplasmonic response of semiconductor nanoparticles\nFigure 8. The ratio, L, of radiative ( e00\nrad(ln)) to dissipative ( e00\nr(ln)) damping for the strongest of the electric dipole (blue),\nquadrupole (red) and octupole (green) modes as a function of the temperature, T, for (a, d, g) Au (DL model), (b, e, h) Si and\n(c, f, i) GaAs nanoparticles of radii r=95;120and145nm, respectively. A decrease in Lis observed with rising temperatures\nfor all materials. The Si and GaAs nanoparticles exhibit a Lthat is about 1-2 orders of magnitude higher compared to the Au\nnanoparticles primarily due to the strong dissipative damping in Au. See Figure S12 for separate plots of e00\nrad(ln)ande00\nr(ln).\nHere, the Mie computations for the nanoparticles of different sizes take into account their thermal expansion, although the text\nlabels indicate the values for nanoparticle radii at 200 K.\nby the stronger dissipative damping e00\nr(l1)in Au nanoparticles for the dipole mode a1that exhibits the broadest resonances\n(Figures S12-S13a,d,g). Figure S12 further shows that the dissipative damping in the Au nanoparticles is dominated by the\nelectric dipole mode followed by the quadrupole and octupole modes whereas for the semiconducting Si and GaAs nanoparticles\nthe order is completely inverted. In Si and GaAs nanoparticles (Figure 8b-c,e-f,h-i respectively), the lowest values for L\nare observed for the higher order quadrupole and octupole resonances that are strongly absorbing. These resonances occur\nat wavelengths wherein the dissipative damping ( µe00\nr) arising from the photogenerated free charge carriers is high (Figure\nS12b-c,e-f,h-i respectively). The same holds true for the Au nanoparticles as well with the exception of the 145nm particles\nthat exhibit a higher Lfor the octupole resonance compared to the quadrupole and dipole modes. This is a consequence of the\nbroad minima in e00\nr(Figure S4d) observed around the resonance wavelength l3=582nm for the electric octupole mode. In\ngeneral, however the rapidly increasing denominator in equation (30) with the order nof the resonant mode implies that the\nradiation damping e00\nraddecreases faster for the higher order Mie modes. Overall, Lis about one to two orders of magnitude\nstronger for the Si and GaAs nanoparticles compared to the Au nanoparticles (Figure 8).\n16/11Thermoplasmonic response of semiconductor nanoparticles\nSUMMARY AND CONCLUSIONS\nTypically, depending on the target application, there arises a need to either maximize or minimize the dissipation of the\nabsorbed energy through heat generation or scattering. Applications such as photothermal therapy[2,3], photocatalysis[45], high\ndensity magnetic memory devices[7], solar-thermophotovoltaics[6]seek to maximize and harness the dissipative damping that\ndrives the conversion of the incident radiation to heat. Conversely, thermal management through the development of high\ntemperature insulation[8], cool reflective coatings[46], spectrally selective filters[47]among others exploits the plasmonically\nenhanced scattering efficiencies resulting from a maximization of the radiative damping and a reduction of the dissipative\ndamping to reduce conductive heat transfer. On the other hand, nanoscale cavity resonators used in sensing applications[1]\nrequire a suppression of both dissipative and radiative damping mechanisms. It is therefore important to determine if a given\nmaterial is suitable for a target application by understanding the temperature dependent absorption and scattering behavior in\nplasmonic materials. To this end, we have employed the experimentally estimated models of temperature dependent dielectric\npermittivity with Mie theory to model the thermoplasmonic behavior of Au (metallic), and, Si (indirect bandgap) and GaAs\n(direct bandgap) semiconductor nanoparticles. In addition, we have investigated the differences in the thermoplasmonic\nbehavior of Au nanoparticles using the two widely employed Drude-Lorentz (DL) and the Drude and critical points (DCP)\nmodels of dielectric permittivity[15,17,20,30,31].\nBriefly, the real ( e0) and imaginary ( e00) parts of the dielectric permittivity for Au (DL model), Si and GaAs exhibit a monotonic\nincrease with temperature. In contrast, their temperature dependence obtained from the DCP model for Au is complex and\nnon-monotonic. Clear differences relating to the temperature dependence of the dielectric permittivity of Au exist between\nthe DL and DCP models. An increase in e0with temperature is observed for the DL model while the opposite occurs for the\nDCP model. As a result, this gives rise to a redshift in the scattering and absorption resonances computed with the DL model\nand a blueshift for the DCP model. The redshift of the Mie resonances with an increase in temperature is consistent with the\nexperimentally determined temperature dependence of the plasmonic resonance for Au nanoparticles embedded in a silica\nmatrix[16].\nA broadening and redshifting of the scattering and absorption resonances with an increase in the temperature and the particle\nradii is observed in nanoparticles of all three materials including Au (DL model), Si and GaAs. However, there is a massive\nDrude broadening of the Mie modes in the Au nanoparticles from free carrier absorption at longer wavelengths, away from the\ninterband transitions. This broadening is enhanced at elevated temperatures as a result of the increased e-eande-finteractions.\nThe smaller ( r=20nm) of the Au nanoparticles exhibit a decrease in the scattering and absorption efficiencies with temperature\nfor the dominant electric dipole modes due to an increase in the dissipative Drude damping. In the larger Au nanoparticles, the\ndecrease in the Mie efficiencies with temperature gets compensated with contributions from the higher order Mie modes.\nIn contrast, Fano resonances are seen in the scattering ( Qsca) and absorption ( Qabs) efficiencies for the Si and GaAs nanoparticles.\nThese can be ascribed to the occurrence of sharp higher order electric ( an) and magnetic ( bn) resonances against a background\nof the broad quadrupole or dipole modes. The strength of the scattering resonances with an increase in the temperature stays\nalmost the same for the semiconducting Si and GaAs nanoparticles unlike a decrease for the metallic Au particles. In general,\nthe Si and GaAs nanoparticles present higher values for QscaandQabscompared to those for Au with the exception of the\nsmaller 20 nm particles. However, the absorption efficiency farther away from the absorption band edge at longer wavelengths\ngoes to zero for the Si and GaAs nanoparticles in contrast to the Au nanoparticles. In addition, the absorption band edge\nbecomes progressively diffuse and redshifted for the Si and GaAs nanoparticles at higher temperatures.\nThe ratio of the radiative to dissipative damping, L, decreases with an increase in temperature for the nanoparticles of all\nthree materials. Furthermore, Lis observed to be one to two orders of magnitude higher for the Si and GaAs nanoparticles\nat all temperatures compared to those for Au. This points to the presence of strong dissipative damping in Au at elevated\ntemperatures. The ratio Lfor all materials is additionally observed to be strongest for the dipole, quadrupole and octupole\nmodes in that order. The only exception occurs for the larger 145nm Au particles wherein the octupole resonance presents\nthe highest Ldue to its occurrence at a broad minima in e00. In general, the dissipative damping in Si and GaAs particles is\nstrongest for the electric octupole resonance followed by the quadrupole and dipole modes while the opposite holds true for the\nAu nanoparticles.\nIn conclusion, we have investigated, analyzed and compared the thermoplasmonic behavior of Au, Si and GaAs nanoparticles\nto assess their feasibility for high-temperature applications. At elevated temperatures, the noble metals suffer from problems of\npoor thermophysical and chemical stability due to strong dissipative damping that results in excessive heating[9]. Our results\nshow that, unlike noble metal particles, semiconductor nanoparticles with their tunable plasmonic resonances, do not exhibit\nany significant deterioration in scattering and absorption efficiencies at high temperatures. Therefore, the use of semiconducting\nmaterials characterized by low dissipative damping is more suitable for applications that seek to exploit the plasmonic behavior\n17/11Thermoplasmonic response of semiconductor nanoparticles\nof materials on the nanoscale under high temperature conditions.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge funding and support from the Academy of Finland, COMP Center of Excellence Programs\n(2015-2017), Grant No. 284621; QTF Center of Excellence Program, Grant No. 312298; RADDESS Consortium Grant;\nthe Aalto Energy Efficiency Research Program EXPECTS; the Aalto Science-IT project; the Discovery Grants and Canada\nResearch Chairs Program of the Natural Sciences and Engineering Research Council (NSERC) of Canada; and Compute\nCanada (www.computecanada.ca).\nCONFLICT OF INTEREST\nThe authors declare no conflict of interest.\nSupplementary Information accompanies the manuscript on the Light: Science & Applications website (http:/www.nature.com/lsa/).\nReferences\n1.Wang, M. 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Sol Energy 81, 488–497 (2007).\n20/11Thermoplasmonic response of semiconductor nanoparticles\nSupplementary Information: Thermoplasmonic response of semiconductor nanoparticles: A\ncomparison with metals\nFigure S1. The temperature dependent (a) real ( h) and (d) imaginary ( k) parts of the refractive index of Au computed using\nthe Drude and critical points (DCP) model. Unlike the Drude-Lorentz (DL) model (Figure 1a), hin the DCP model behaves\nnon-monotonically with temperature, T:htransitions in the 500-740 nm wavelength region from decreasing at shorter\nwavelengths to increasing at longer wavelengths beyond l=740nm with rising temperature [see inset in (a)]. The values for\nkincrease monotonically at all wavelengths (except for a small region between l=400-425 nm) for temperatures between\n200 and 560 K and then saturate between 560 and 650 K [see inset in (b)].\n1/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S2. Mie scattering efficiency, Qsca, computed using the DCP model for Au as a function of the wavelength, l, of the\nincident radiation and the particle radii rat three different temperatures, T, equal to (a) 200, (b) 430and (c) 650K. Similar to\nthe DL model, the strength of the resonances in Qscafor Au particles show a decline with an increase in the temperature, T. See\nFigure S6 for the clear blueshift observed in the scattering resonances for the DCP model as opposed to a redshift in the DL\nmodel (Figures 2a,d,g and 4a,d,g,j in the main text).\n2/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S3. Mie absorption efficiency, Qabs, computed using the DCP model for Au as a function of the wavelength, l, of the\nincident radiation and the particle radii rat three different temperatures, T, equal to (a) 200, (b) 430and (c) 650K. See Figure\nS9 for the clear blueshift is observed in the absorption resonances for the DCP model as opposed to a redshift in the DL model\n(Figures 3a,d,g and 7a,d,g,j in the main text).\n3/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S4. Comparison of the temperature dependent (a-c) real ( e0) and (d-f) imaginary ( e00) parts of the dielectric\npermittivity for Au (DL model), Si and GaAs respectively. Both e0ande00are seen to increase monotonically with temperature,\nT, for all materials. Also, visible is the gradual rise in e00for the indirect bandgap Si in contrast to the sharp rise observed in\nGaAs at shorter wavelengths beyond the onset of the absorption band edge. In (d), e00for Au initially decreases with increasing\nwavelength and then increases monotonically for the longer wavelengths.\nFigure S5. The temperature dependent (a) real ( e0) and (b) imaginary ( e00) parts of the dielectric permittivity for Au computed\nusing the DCP model. e0displays a monotonic decrease with increasing temperatures across the entire wavelength range from\nl=400-1450 nm. On the other hand, the behavior of e00is highly non-monotonic: it decreases with increasing temperature\nuntil l=500nm followed by a transition region up to l=710nm and an increase with temperature thereafter until l=1450\nnm.\n4/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S6. Mie scattering efficiency ( Qsca) with dipole (blue), quadrupole (red) and octupole (green) contributions as a\nfunction of the wavelength, l, computed using the DCP model for Au nanoparticles of radii, r, equal to (a) 20, (b) 80, (c)140\nand (d) 200nm at temperatures T=200(solid lines) and 650(dotted lines) K. The thick orange and pink solid lines represent\nthe total scattering efficiencies, Qsca, at temperatures T=200 and 650 K, respectively. Just as in the DL model, the Mie\nresonances in the Au nanoparticles exhibit massive Drude broadening and a deterioration in strength with temperature increase\ndue to dissipative damping. However, unlike the DL model (Figure 4a,d,g,j in the main text), there is a clear blue shift in the\nresonances with an increase in temperature. This blueshift is more pronounced for the smaller particles but gradually becomes\nweaker with increase in particle size. This is because the blueshift due to the decrease in the real part e0of the dielectric\npermittivity at elevated temperatures is counteracted by the redshift due to plasmon retardation effect from an increase in the\nparticle size. Here, the Mie computations for the nanoparticles of different sizes take into account their thermal expansion\nwhile the values indicated above for the nanoparticle radii are at T=200 K.\n5/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S7. Electric dipole (blue), quadrupole (red) and octupole (green) Mie modes ( an) as a function of the wavelength, l,\ncomputed using the DCP model for Au nanoparticles of radii, r, equal to (a) 20, (b) 80, (c)140and (d) 200nm at temperatures\nT=200 (solid lines) and 650 (dotted lines) K. Unlike the DL model (Figure 5a,d,g,j in the main text), a blueshifting of the\nelectric Mie resonances is observed with increasing particle size and temperature. This blueshift is more pronounced for the\nsmaller particles but gradually becomes weaker with increase in particle size. This is because the blueshift, as a result of the\ndecrease in the real part e0of the dielectric permittivity at elevated temperatures, is counteracted by the redshift due to plasmon\nretardation effect from an increase in the particle size. Here, the Mie computations for the nanoparticles of different sizes take\ninto account their thermal expansion while the values indicated above for the nanoparticle radii are at T=200 K.\n6/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S8. Magnetic dipole (blue), quadrupole (red) and octupole (green) Mie modes ( bn) as a function of the wavelength, l,\ncomputed using the DCP model for Au nanoparticles of radii, r, equal to (a) 20, (b) 80, (c)140and (d) 200nm at temperatures\nT=200(solid lines) and 650(dotted lines) K. Similar to the electric Mie resonances an(Figure S7), the magnetic resonances\nalso broaden with an increase in particle size and temperature for all materials. However, unlike the electric modes an(Figure\nS7), a blueshift of the magnetic Mie resonances is not observed with increasing temperature. Here, the Mie computations for\nthe nanoparticles of different sizes take into account their thermal expansion while the values indicated above for the\nnanoparticle radii are at T=200 K.\n7/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S9. Mie absorption efficiency ( Qabs) as a function of the wavelength lcomputed using the DCP model for Au\nnanoparticles of radii, r, equal to (a) 20, (b) 80, (c) 140 and (d) 200 nm at temperatures T=200 (blue) and 650 (red) K.\nSimilar to the DL model (Figure 7a,d,g,j in the main text), the highest Qabsis observed for the smallest 20 nm particles. An\nincrease in Qabswith temperature for the larger Au nanoparticles is evident at the longer wavelengths due to uniform Drude\nbroadening but the increase is not as pronounced as seen in the DL model (Figure 7a,d,g,j in the main text). There is also a\nblueshift of the absorption resonances primarily due to contributions from the blue shifted electric resonances an(Figure S7).\nHere, the Mie computations for the nanoparticles of different sizes take into account their thermal expansion while the values\nindicated above for the nanoparticle radii are at T=200 K.\n8/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S10. The contribution of the temperature independent interband transitions, represented by the Lorentz oscillators, to\nthe imaginary part e00of the dielectric permittivity as a function of the wavelength, l, of the incident radiation for Au in the DL\nmodel. The contributions from all of the five Lorentz oscillators stay positive throughout the wavelength range from l=400to\n1450 nm.\nFigure S11. Temperature dependence of the contribution of interband transitions to the imaginary part, e00, of the dielectric\npermittivity as a function of the wavelength, l, of the incident radiation for Au in the DCP model. The contribution to\ndielectric permittivity from the interband transition occurring at l\u0019330nm in the DCP model becomes negative for l&600\nnm and hence is unphysical.\n9/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S12. The radiative ( e00\nrad(ln)) and dissipative ( e00\nr(ln)) damping for the strongest of the electric dipole (blue),\nquadrupole (red) and octupole (green) modes as a function of the temperature, T, for (a, d, g) Au (DL model), (b, e, h) Si and\n(c, f, i) GaAs nanoparticles of radii r=95;120and145nm, respectively. An almost linear increase in e00\nr(ln)is observed with\nrising temperatures for all materials. The values of e00\nr(ln)that govern dissipative damping are about 1-2 orders of magnitude\nhigher for Au nanoparticles compared to the Si and GaAs nanoparticles. This results in a L(=e00\nrad=e00\nr)that is about 1-2orders\nof magnitude higher for the Si and GaAs nanoparticles compared to the Au nanoparticles (see Figure 8 in the main text).\nAdditionally, consistent with experiments, the radiative damping is seen to increase with the particle size for all nanoparticles in\ngeneral. Here, the Mie computations for the nanoparticles of different sizes take into account their thermal expansion, although\nthe text labels indicate the values for nanoparticle radii at 200 K.\n10/11Thermoplasmonic response of semiconductor nanoparticles\nFigure S13. Electric dipole (blue), quadrupole (red) and octupole (green) Mie modes ( an) as a function of the wavelength, l,\nfor (a, d, g) Au (DL model), (b, e, h) Si and (c, f, i) GaAs nanoparticles of radii r=95;120 and 145 nm, respectively, at\ntemperatures T=200(solid lines) and 650(dotted lines) K. See Figure S12 and Figure 8 in the main text for the corresponding\nradiative and dissipative damping alongwith their ratio, L, respectively. Similar to the electric modes anin Figure 5 of the main\ntext, a broadening and redshifting of the electric Mie resonances is observed with increasing particle size and temperature for\nall materials.\n11/11"    },    {        "title": "1710.04766v1.Mode_Dependent_Damping_in_Metallic_Antiferromagnets_Due_to_Inter_Sublattice_Spin_Pumping.pdf",        "content": "Mode-Dependent Damping in Metallic Antiferromagnets Due to Inter-Sublattice Spin\nPumping\nQian Liu,1,\u0003H. Y. Yuan,2,\u0003Ke Xia,1, 3and Zhe Yuan1,y\n1The Center for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, 100875 Beijing, China\n2Department of Physics, South University of Science and Technology of China, Shenzhen, Guangdong, 518055, China\n3Synergetic Innovation Center for Quantum E\u000bects and Applications (SICQEA),\nHunan Normal University, Changsha 410081, China\n(Dated: October 16, 2017)\nDamping in magnetization dynamics characterizes the dissipation of magnetic energy and is essen-\ntial for improving the performance of spintronics-based devices. While the damping of ferromagnets\nhas been well studied and can be arti\fcially controlled in practice, the damping parameters of an-\ntiferromagnetic materials are nevertheless little known for their physical mechanisms or numerical\nvalues. Here we calculate the damping parameters in antiferromagnetic dynamics using the gen-\neralized scattering theory of magnetization dissipation combined with the \frst-principles transport\ncomputation. For the PtMn, IrMn, PdMn and FeMn metallic antiferromagnets, the damping coef-\n\fcient associated with the motion of magnetization ( \u000bm) is one to three orders of magnitude larger\nthan the other damping coe\u000ecient associated with the variation of the N\u0013 eel order ( \u000bn), in sharp\ncontrast to the assumptions made in the literature.\nDamping describes the process of energy dissipation in\ndynamics and determines the time scale for a nonequi-\nlibrium system relaxing back to its equilibrium state.\nFor magnetization dynamics of ferromagnets (FMs), the\ndamping is characterized by a phenomenological dissipa-\ntive torque exerted on the precessing magnetization [1].\nThe magnitude of this torque that depends on material,\ntemperature and magnetic con\fgurations, has been well\nstudied in experiment [2{10] and theory [11{16].\nRecently, magnetization dynamics of antiferromagnets\n(AFMs) [17{20], especially that controlled by an electric\nor spin current [21{32], has attracted lots of attention in\nthe process of searching the high-performance spintronic\ndevices. However, the understanding of AFM dynamics,\nin particular the damping mechanism and magnitude in\nreal materials, is quite limited. Magnetization dynam-\nics of a collinear AFM can be described by two coupled\nLandau-Lifshitz-Gilbert (LLG) equations corresponding\nto the precessional motion of the two sublattices, respec-\ntively [33], i.e. ( i= 1;2)\n_mi=\u0000\rmi\u0002hi+\u000bimi\u0002_mi; (1)\nwhere\ris the gyromagnetic ratio, miis the magnetiza-\ntion direction on the i-th sublattice and _mi=@tmi.hi\nis the e\u000bective magnetic \feld on mi, which contains the\nanisotropy \feld, the external \feld and the exchange \feld\narising from the magnetization on the both sublattices.\nThe last contribution to himakes the dynamic equation\nof one sublattice coupled to the equation of the other one.\nSpeci\fcally, if the free energy of the AFM is given by the\nfollowing formF[m1;m2]\u0011\u00160MsVE[m1;m2] with the\npermeability of vacuum \u00160, the magnetization on each\nsublatticeMsand the volume of the AFM V, one has\nhi=\u0000\u000eE=\u000emi.\u000biin Eq. (1) is the damping param-\neter representing the dissipation rate of the magnetiza-tionmi. Due to the sublattice permutation symmetry,\nthe damping magnitudes of the two sublattices should\nbe equal. This approach has been used to investigate the\nAFM resonance [33, 34], temperature gradient induced\ndomain wall (DW) motion [35] and spin-transfer torques\nin an AFMjFM bilayer [36].\nAn alternative way to deal with the AFM dynamics is\nintroducing the net magnetization m\u0011m1+m2and the\nN\u0013 eel order n\u0011m1\u0000m2so that the precessional motion\nofmandncan be derived from the Lagrangian equa-\ntion [26]. The damping e\u000bect is then included arti\fcially\nwith two parameters \u000bmand\u000bnthat characterize the\ndissipation rate of mandn, respectively. This approach\nis widely used to investigate spin super\ruid in an AFM\ninsulator [37, 38], AFM nano-oscillator [39], and DW mo-\ntion induced by an electrical current [26, 40], spin waves\n[41] and spin-orbit torques [42, 43]. Using the above def-\ninitions of mandn, one can reformulate Eq. (1) and\nderive the following dynamic equations\n_n= (\rhm\u0000\u000bm_m)\u0002n+ (\rhn\u0000\u000bn_n)\u0002m;(2)\n_m= (\rhm\u0000\u000bm_m)\u0002m+ (\rhn\u0000\u000bn_n)\u0002n;(3)\nwhere hnandhmare the e\u000bective magnetic \felds exerted\nonnandm, respectively. They can also be written as\nthe functional derivative of the free energy [26, 41], i.e.\nhn=\u0000\u000eE=\u000enandhm=\u0000\u000eE=\u000em. The damping pa-\nrameters in Eqs. (1{3) have the relation \u000bn=\u000bm=\n\u000b1=2 =\u000b2=2 [36]. Indeed, the assumption \u000bm=\u000bnis\ncommonly adopted in the theoretical study of AFM dy-\nnamics with only a few exceptions, where \u000bmis ignored in\nthe current-induced skyrmion motion in AFM materials\n[44] and the magnon-driven DW motion [45]. However,\nthe underlying damping mechanism of an AFM and the\nrelation between \u000bmand\u000bnhave not been fully justi\fed\nyet [46, 47].arXiv:1710.04766v1  [cond-mat.mtrl-sci]  13 Oct 20172\nIn this paper, we generalize the scattering theory of\nmagnetization dissipation in FMs [48] to AFMs and cal-\nculate the damping parameters from \frst-principles for\nmetallic AFMs PtMn, IrMn, PdMn and FeMn. The\ndamping coe\u000ecients in an AFM are found to be strongly\nmode-dependent with \u000bmup to three orders of magni-\ntude larger than \u000bn. By analyzing the dependence of\ndamping on the disorder and spin-orbit coupling (SOC),\nwe demonstrate that \u000bnarises from SOC in analog to the\nGilbert damping in FMs, while \u000bmis dominated by the\nspin pumping e\u000bect between sublattices.\nTheory.| In analogue to the scattering theory of mag-\nnetization dissipation in FMs [48], the damping parame-\nters in AFMs, \u000bnand\u000bm, can be expressed in terms of\nthe scattering matrix. Following the previous de\fnition\nof the free energy, the energy dissipation rate of an AFM\nreads\n_E=\u0000\u00160MsV_E=\u00160MsV\u0012\n\u0000\u000eE\n\u000em\u0001_m\u0000\u000eE\n\u000en\u0001_n\u0013\n=\u00160MsV(hm\u0001_m+hn\u0001_n): (4)\nBy replacing the e\u000bective \felds hmandhnby the time\nderivative of magnetization order and N\u0013 eel order using\nEq. (2) and (3), one arrives at [49]\n_E=\u00160MsV\n\r\u0000\n\u000bn_n2+\u000bm_m2\u0001\n: (5)\nIf we place an AFM between two semi-in\fnite nonmag-\nnetic metals, the propagating electronic states coming\nfrom the metallic leads are partly re\rected and trans-\nmitted. The probability amplitudes of the re\rection and\ntransmission form the so-called scattering matrix S[50].\nFor such a scattering structure with only the order pa-\nrameter nof the AFM varying in time (see the insets of\nFig. 1), the energy loss that is pumped into the reservoir\nis given by\n_E=~\n4\u0019Tr\u0010\n_S_Sy\u0011\n=~\n4\u0019Tr\u0012@S\n@n@Sy\n@n\u0013\n_n2\u0011Dn_n2:(6)\nHere we de\fne Dn\u0011(~=4\u0019)Tr[(@S=@n)(@Sy=@n)]. Com-\nparing Eqs. (S7) and (6), we obtain\nDn=\u00160MsA\n\r\u000bnL; (7)\nwhere we replace the volume Vby the product of the\ncross-sectional area Aand the length Lof the AFM. We\ncan express \u000bmin the same manner,\nDm=\u00160MsA\n\r\u000bmL (8)\nwithDm\u0011(~=4\u0019)Tr[(@S=@m)(@Sy=@m)]. Using\nEqs. (7) and (8), we calculate the energy dissipation as a\nfunction of the length Land extract the damping param-\neters\u000bn(m)via a linear least squares \ftting. Note thatthe above formalism can be generalized to include non-\ncollinear AFM, such as DWs in AFMs, by introducing\nthe position-dependent order parameters n(r) and m(r).\nIt can also be extended for the AFMs containing more\nthan two sublattices, which may not be collinear with\none another [51]. For the latter case, one has to rede\fne\nthe proper order parameters instead of nandm[52].\nFirst-principles calculations.| The above formalism is\nimplemented using the \frst-principles scattering calcu-\nlation and is applied here in studying the damping of\nmetallic AFMs including PtMn, IrMn, PdMn and FeMn.\nThe lattice constants and magnetic con\fgurations are the\nsame as in the reported \frst-principles calculations [53].\nHere we take tetragonal PtMn as an example to illustrate\nthe computational details. A \fnite thickness ( L) of PtMn\nis connected to two semi-in\fnite Au leads along (001) di-\nrection. The lattice constant of Au is made to match\nthat of the aaxis of PtMn. The electronic structures are\nobtained self-consistently within the density functional\ntheory implemented with a minimal basis of the tight-\nbinding linear mu\u000en-tin orbitals (TB LMTOs) [54]. The\nmagnetic moment of every Mn atom is 3.65 \u0016Band Pt\natoms are not magnetized.\nTo evaluate \u000bnand\u000bm, we \frst construct a lateral\n10\u000210 supercell including 100 atoms per atomic layer in\nthe scattering region, where the atoms are randomly dis-\nplaced from their equilibrium lattice sites using a Gaus-\nsian distribution with the root-mean-square (RMS) dis-\nplacement \u0001 [15, 55]. The value of \u0001 is chosen to repro-\nduce typical experimental resistivity of the corresponding\nbulk AFM. The scattering matrix Sare obtained using a\n\frst-principles \\wave-function matching\" scheme that is\nalso implemented with TB LMTOs [56] and its derivative\nis obtained by \fnite-di\u000berence method [49].\nFigure 1(a) shows the calculated energy pumping rate\nDnof PtMn as a function of Lfornalong thecaxis\nwith \u0001=a= 0:049. The total pumping rate (solid sym-\nbols) increases linearly with increasing the volume of\nthe AFM. A linear least squares \ftting yields \u000bn=\n(0:67\u00060:02)\u000210\u00003, as plotted by the solid line. The \fnite\nintercept of the solid line corresponding to the interface-\nenhanced energy dissipation, which is essentially the spin\npumping e\u000bect at the AFM jAu interface [57, 58]. The\nN\u0013 eel order induced damping \u000bncompletely results from\nspin-orbit coupling (SOC). If we arti\fcially turn SOC o\u000b,\nthe calculated pumping rate is independent of the volume\nof the AFM indicating \u000bn= 0. This is because the spin\nspace is decoupled from the real space without SOC and\nthe energy is then invariant with respect to the direction\nofn. The spin pumping e\u000bect is nearly unchanged by\nthe SOC.\nThe energy pumping rate Dmof PtMn with nalong\nthecaxis is plotted in Fig. 1(b), where we \fnd three\nimportant features. (1) The extracted value of \u000bm=\n0:59\u00060:02, which is nearly 1000 times larger than \u000bn.\n(2) Turning SOC o\u000b only slightly increases the calculated3\n0 5 10 15 20 25 30L (nm)0102030γDm/(µ0Ms A) (nm)0 0.5 1SOC Factor00.40.8αm103αn\n0.020.030.04γDn/(µ0Ms A) (nm)PtMn1Mn2\nm1 m2 \nm1 m2 \n ξSO=0 ξSO=0(a)\n(b) ξSO≠0\n ξSO≠0abc\nFIG. 1. Calculated energy dissipation rate as a function of\nthe length of PtMn due to variation of the order parameters\nn(a) and m(b).Ais the cross-sectional area of the lat-\neral supercell. Arrows in each panels illustrate the dynamical\nmodes of the order parameters. The empty symbols are cal-\nculated without spin-orbit interaction. The inset of panel (a)\nshows atomic structure of PtMn with collinear AFM order.\nThe inset in (b) shows calculated \u000bnand\u000bmas a function\nof the scaled SOC strength. The factor 1 corresponds to the\nreal SOC strength that is determined by the derivative of the\nself-consistent potentials.\n\u000bmindicating that SOC is not the main dissipative mech-\nanism of\u000bm. The di\u000berence between the solid and empty\ncircles in Fig. 1(b) can be attributed to the SOC-induced\nvariation of electronic structure near the Fermi level. To\nsee more clearly the di\u000berent in\ruence of SOC on \u000bmand\n\u000bn, we plot in the inset of Fig. 1(b) the calculated damp-\ning parameters as a function of SOC strength. Indeed,\nas the SOC strength \u0018SOis arti\fcially tuned from its real\nvalue to zero, \u000bndecreases dramatically and tends to\nvanish at\u0018SO= 0, while\u000bmis less sensitive to \u0018SOthan\n\u000bn. (3) The intercepts of the solid and dashed lines are\nboth vanishingly small indicating that this speci\fc mode\ndoes not pump spin current into the nonmagnetic leads.\nThe pumped spin current from an AFM generally reads\nIpump\ns/n\u0002_n+m\u0002_m[58]. For the mode depicted in\nFig. 1(b), one has _n= 0 and _mkmsuch that Ipump\ns = 0.\nTo explore the disorder dependence of the damping pa-\nrameters\u000bnand\u000bm, we further perform the calculation\nby varying the RMS of atomic displacements \u0001. Fig-\nure 2(a) shows that the calculated resistivity increases\nmonotonically with increasing \u0001. The resistivity \u001acwith\nnalongcaxis is lower than \u001aawithnalongaaxis.\n0.51.01.52.0αn (10-3)(a)80160240ρ (µΩ cm)n//a(b)n//c4.6 5.4 6.2∆/a (10-2)01020AMR (%)\n4.6 5.0 5.4 5.8 6.2∆/a (10-2)0.20.40.60.8αm4 6 8 10 12σ (105Ω-1 m-1)0.20.50.8αm(b)(c)FIG. 2. Calculated resistivity (a) and damping parameters\n\u000bn(b) and\u000bm(c) of PtMn as a function of the RMS of\natomic displacements. The red squares and black circles are\ncalculated with nalongaaxis andcaxis, respectively. The\ninset of (a) shows the calculated AMR. \u000bmis replotted as a\nfunction of conductivity in the inset of (c). The blue dashed\nline illustrates the linear dependence.\nThe anisotropic magnetoresistance (AMR) de\fned by\n(\u001aa\u0000\u001ac)=\u001acis about 10%, which slightly decreases with\nincreasing \u0001, as plotted in the inset of Fig. 2(a). The\nlarge AMR in PtMn is useful for experimental detection\nof the N\u0013 eel order. The calculated AMR seems to be an\norder of magnitude larger than the reported values in lit-\nerature [59{61]. We may attribute the di\u000berence to the\nsurface scattering in thin-\flm samples and other types of\ndisorder that have been found to decrease the AMR of\nferromagnetic metals and alloys [62].\n\u000bnof PtMn plotted in Fig. 2(b) is of the order of 10\u00003,\nwhich is comparable with the magnitude of the Gilbert\ndamping of ferromagnetic transition metals [2{4, 15]. For\nnalongaaxis,\u000bnshows a weak nonmonotonic depen-\ndence on disorder, while \u000bnfornalongcaxis increases\nmonotonically. With the relativistic SOC, the electronic\nstructure of an AFM depends on the orientation of n.\nWhen nvaries in time, the occupied energy bands may\nbe lifted above the Fermi level. Then a longer relax-\nation time (weaker disorder) gives rise to a larger energy\ndissipation, corresponding to the increase in \u000bnwith de-\ncreasing \u0001 at small \u0001. It is analogous to the intraband\ntransitions accounting for the conductivity-like behavior\nof Gilbert damping at low temperature in the torque-\ncorrelation model [11, 12]. Su\u000eciently strong disorder4\nrenders the system isotropic and the variation of ndoes\nnot lead to electronic excitation but scattering of conduc-\ntion electrons by disorder still dissipates energy into the\nlattice through SOC. The higher the scattering rate, the\nlarger is the energy dissipation rate corresponding to the\ncontribution of the interband transitions [11, 12]. There-\nfore,\u000bnshares the same physical origin as the Gilbert\ndamping of metallic FMs.\nThe value of \u000bmis about three orders of magnitude\nlarger than \u000bnand it decreases monotonically with in-\ncreasing the structural disorder, as shown in Fig. 2(c).\nThis remarkable di\u000berence can be attributed to the en-\nergy involved in the dynamical motion of mandn. While\nthe precession of nonly changes the magnetic anisotropy\nenergy in an AFM, the variation of mchanges the ex-\nchange energy that is in magnitude much larger than the\nmagnetic anisotropy energy.\nPhysically, \u000bmcan be understood in terms of spin\npumping [63, 64] between the two sublattices of an AFM.\nThe sublattice m2pumps a spin current that can be ab-\nsorbed by m1resulting in a damping torque exerted on\nm1as\u000b0m1\u0002[m1\u0002(m2\u0002_m2)]. Here\u000b0is a dimen-\nsionless parameter to describe the strength of the spin\npumping. This torque can be simpli\fed to be \u000b0m1\u0002_m2\nby neglecting the high-order terms of the total magne-\ntization m. In addition, the spin pumping by m1also\ncontributes to the damping of the sublattice m1that is\nequivalent to a torque \u000b0m1\u0002_m1exerted on m1. Tak-\ning the inter-sublattice spin pumping into account, we are\nable to derive Eqs. (2) and (3) and obtain the damping\nparameters \u000bn=\u000b0=2 and\u000bm= (\u000b0+ 2\u000b0)=2 [49]. Here\n\u000b0is the intrinsic damping due to SOC for each sublat-\ntice. It is worth noting that the spin pumping strength\nwithin a metal is proportional to its conductivity [65{67].\nWe replot\u000bmas a function of conductivity in the inset\nof Fig. 2(c), where a general linear dependence is seen for\nbothnalongaaxis andcaxis.\nWe list in Table I the calculated \u001a,\u000bnand\u000bmfor\ntypical metallic AFMs including PtMn, IrMn, PdMn and\nFeMn. For IrMn, \u000bmis only 10 times larger than \u000bn,\nwhile\u000bmof the other three materials are about three\norders of magnitude larger than their \u000bn.\nTABLE I. Calculated resistivity and damping parameters for\nthe N\u0013 eel order nalongaaxis andcaxis.\nAFM n\u001a(\u0016\n cm)\u000bn(10\u00003)\u000bm\nPtMnaaxis 119 \u00065 1.60 \u00060.02 0.49 \u00060.02\ncaxis 108 \u00064 0.67 \u00060.02 0.59 \u00060.02\nIrMnaaxis 116 \u00062 10.5 \u00060.2 0.10 \u00060.01\ncaxis 116 \u00062 10.2 \u00060.3 0.10 \u00060.01\nPdMnaaxis 120 \u00068 0.16 \u00060.02 1.1 \u00060.10\ncaxis 121 \u00068 1.30 \u00060.10 1.30 \u00060.10\nFeMnaaxis 90 \u00061 0.76 \u00060.04 0.38 \u00060.01\ncaxis 91 \u00061 0.82 \u00060.03 0.38 \u00060.01\n0 10 20 30 40Hext (kOe)00.040.080.12∆ω (THz)2.0 2.4ω (THz)-1.6 -1.2-Im χ (arb.units)Hext=20 kOe\nαm=αnαm=103αnm1 m2 \nm1 m2 \nHextHext ωL\n//// ωR ωL ωRFIG. 3. Linewidth of AFMR as a function of the external\nmagnetic \feld. The black dashed lines and red solid lines\nare calculated with \u000bm=\u000bnand\u000bm= 103\u000bn, respectively.\nInset: the imaginary part of susceptibility as a function of\nthe frequency for the external magnetic \feld Hext= 20 kOe\nand\u000bm= 103\u000bn. The cartoons illustrate the corresponding\ndynamical modes. Here we use HE= 103kOe,HA= 5 kOe\nand\u000bn= 0:001.\nAntiferromagnetic resonance.| Ke\u000ber and Kittel for-\nmulated antiferromagnetic resonance (AFMR) with-\nout damping [33] and determined the resonant fre-\nquencies that depend on the external \feld Hext, ex-\nchange \feld HEand anisotropy \feld HA,!res=\n\rh\nHext\u0006p\nHA(2HE+HA)i\n. Here we follow their ap-\nproach, in which Hextis applied along the easy axis and\nthe transverse components of m1andm2are supposed\nto be small. Taking both the intrinsic damping due to\nSOC and spin pumping between the two sublattices into\naccount, we solve the dynamical equations of AFMR and\n\fnd the frequency-dependent susceptibility \u001f(!) that is\nde\fned by n?(!) =\u001f(!)\u0001h?(!). Here n?andh?\nare the transverse components of the N\u0013 eel order and mi-\ncrowave \feld, respectively. The imaginary part of the\ndiagonal element of \u001f(!) withHext= 20 kOe is plotted\nin the inset of Fig. 3, where two resonance modes can be\nidenti\fed. The precessional modes for the positive ( !R)\nand negative frequency ( !L) are schematically depicted\nin Fig. 3. The linewidth of the AFMR \u0001 !can be deter-\nmined from the imaginary part of the (complex) eigen-\nfrequency [68] by solving det j\u001f\u00001(!)j= 0 and is plotted\nin Fig. 3 as a function of Hext. Without Hext, the two\nmodes have the same linewidth. A \fnite external \feld\nincreases the linewidth of !Rand decreases that of !L,\nboth linearly. By including the spin pumping between\ntwo sublattices, both the linewidth at Hext= 0 and the\nslope of \u0001!as a function of Hextincrease by a factor\nof about 3.5. It indicates that the spin pumping e\u000bect\nbetween the two sublattices plays an important role in\nthe magnetization dynamics of metallic AFMs.5\nConclusions.| We have generalized the scattering the-\nory of magnetization dissipation in FMs to be applicable\nfor AFMs. Using \frst-principles scattering calculation,\nwe \fnd the damping parameter accompanying the mo-\ntion of magnetization ( \u000bm) is generally much larger than\nthat associated with the motion of the N\u0013 eel order ( \u000bn)\nin metallic AFMs PtMn, IrMn, PdMn and FeMn. While\n\u000bnarises from the spin-orbit interaction, \u000bmis mainly\ncontributed by the spin pumping between the two sublat-\ntices in an AFM via exchange interaction. 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Saib, Modeling and design of microwave devices based on ferromagnetic nanowires\n(Presses Universitaires du Louvain, 2004).1\nSupplementary Material for \\Mode-Dependent Damping in Metallic\nAntiferromagnets Due to Inter-Sublattice Spin Pumping\"\nQian Liu,1;\u0003H. Y. Yuan,2;\u0003Ke Xia,1;3and Zhe Yuan1;y\n1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China\n2Department of Physics, Southern University of Science and Technology of China, Shenzhen, Guangdong, 518055, China\n3Synergetic Innovation Center for Quantum E\u000bects and Applications (SICQEA), Hunan Normal University, Changsha\n410081, China\nIn the Supplemental Material, we present the detailed derivation of the energy pumping arising from antiferromagnetic\ndynamics, the implementation of calculating the derivatives of scattering matrix and derivation of dynamic equations of nand\nmincluding the spin pumping between sublattices.\nDERIVATION OF ENERGY DISSIPATION IN ANTIFERROMAGNETIC DYNAMICS\nWe consider a collinear antiferromagnet (AFM) with two sublattices, both of which have the magnetization Ms.\nThe magnetization directions are denoted by the unit vectors m1andm2. Then we are able to de\fne the total\nmagnetization m=m1+m2and the N\u0013 eel order parameter n=m1\u0000m2. The dynamic equations of mandncan\nbe written as [S1, S2]\n_m=\u0000\r(m\u0002hm+n\u0002hn) +\u000bmm\u0002_m+\u000bnn\u0002_n; (S1)\n_n=\u0000\r(m\u0002hn+n\u0002hm) +\u000bmn\u0002_m+\u000bnm\u0002_n: (S2)\nHerehmandhnare the e\u000bective \felds acting on the total magnetization and the N\u0013 eel order. Speci\fcally, if the free\nenergy is written as F=\u00160MsVE, where\u00160is the vacuum permeability, Vis the volume of the AFM, and Eis a\nreduced free energy density, one has [S1]\nhm=\u0000\u000eE\n\u000em;andhn=\u0000\u000eE\n\u000en: (S3)\nIn Eqs. (S1) and (S2), \u000bmand\u000bnare used to characterize the damping due to the variation of the magnetization and\nthe N\u0013 eel order, respectively.\nIfmandnare the only time-varying parameters in the system, the energy dissipation can be represented by\n_E=\u0000_F=\u0000\u00160MsV_E\n=\u00160MsV\u0014\n_m\u0001\u0012\n\u0000\u000eE\n\u000em\u0013\n+_n\u0001\u0012\n\u0000\u000eE\n\u000en\u0013\u0015\n=\u00160MsV(_m\u0001hm+_n\u0001hn): (S4)\nWe then insert the dynamic Eqs. (S1) and (S2) into the above Eq. (S4) and obtain\n_E\n\u00160MsV= [\u0000\r(m\u0002hm+n\u0002hn) +\u000bmm\u0002_m+\u000bnn\u0002_n]\u0001hm\n+[\u0000\r(m\u0002hn+n\u0002hm) +\u000bmn\u0002_m+\u000bnm\u0002_n]\u0001hn\n=\u0000\rn\u0002hn\u0001hm+ (\u000bmm\u0002_m+\u000bnn\u0002_n)\u0001hm\u0000\rn\u0002hm\u0001hn+ (\u000bmn\u0002_m+\u000bnm\u0002_n)\u0001hn\n= (\u000bmm\u0002_m+\u000bnn\u0002_n)\u0001hm+ (\u000bmn\u0002_m+\u000bnm\u0002_n)\u0001hn\n=\u000bm(m\u0002_m\u0001hm+n\u0002_m\u0001hn) +\u000bn(n\u0002_n\u0001hm+m\u0002_n\u0001hn)\n=\u000bm(hm\u0002m\u0001_m+hn\u0002n\u0001_m) +\u000bn(hm\u0002n\u0001_n+hn\u0002m\u0001_n)\n=\u000bm(hm\u0002m+hn\u0002n)\u0001_m+\u000bn(hm\u0002n+hn\u0002m)\u0001_n\n=\u000bm1\n\r(_m\u0000\u000bmm\u0002_m\u0000\u000bnn\u0002_n)\u0001_m+\u000bn1\n\r(_n\u0000\u000bmn\u0002_m\u0000\u000bnm\u0002_n)\u0001_n\n=\u000bm\n\r_m2\u0000\u000bm\u000bn\n\rn\u0002_n\u0001_m+\u000bn\n\r_n2\u0000\u000bn\u000bm\n\rn\u0002_m\u0001_n\n=1\n\r\u0000\n\u000bm_m2+\u000bn_n2\u0001\n: (S5)2\nLeft Lead Right Lead Scattering Region \n!O!IOIm1 m2 \nFIG. S1. Schematic illustration of the scattering geometry that is used in the \frst-principles calculations. Since both the\nleft and right leads are semi-in\fnite with periodic crystalline structure, the propagating (incoming and outgoing) Bloch states\ncan be obtained by solving the Kohn-Sham equation self-consistently. Then the transmission and re\rection coe\u000ecients can be\nsolved using the numerical technique called \\wave function matching\" [S3].\nTherefore the energy dissipation during antiferromagnetic dynamics can be eventually obtained\n_E=\u00160MsV\n\r\u0000\n\u000bm_m2+\u000bn_n2\u0001\n: (S6)\nCALCULATING THE DERIVATIVE OF SCATTERING MATRIX\nNoting that the energy dissipation in a scattering geometry, i.e. the left lead{scattering region{the right lead (see\nFig. S1), can be written in terms of the parametric pumping [S4]\n_E=~\n4\u0019Tr\u0010\n_S_Sy\u0011\n: (S7)\nHereSis the scattering matrix. Supposing only the magnetic order \u0010(\u0010=mor\u0010=n) of the system is varying in\ntime, one can rewrite Eq. (S7) as\n_E=~\n4\u0019Tr\u0012@S\n@\u0010@Sy\n@\u0010\u0013\n_\u00102\u0011D\u0010_\u00102: (S8)\nThe quantity D\u0010is generally a positive-de\fnite and symmetric tensor [S5] with its elements de\fned by\nDij\n\u0010=~\n4\u0019Tr\u0012@S\n@\u0010i@Sy\n@\u0010j\u0013\n: (S9)\nNoting that @S=@\u0010i,@Sy=@\u0010jand their product are all matrices, so we rewrite Eq. (S9) in terms of the speci\fc matrix\nelements as\nDij\n\u0010=~\n4\u0019X\n\u0016\u0012@S\n@\u0010i@Sy\n@\u0010j\u0013\n\u0016\u0016=~\n4\u0019X\n\u0016X\n\u0017@S\u0016\u0017\n@\u0010i@\u0000\nSy\u0001\n\u0017\u0016\n@\u0010j=~\n4\u0019X\n\u0016X\n\u0017@S\u0016\u0017\n@\u0010i\u0012@S\u0016\u0017\n@\u0010j\u0013\u0003\n: (S10)\nIn particular, for i=j, we have the diagonal elements of D\u0010\nDii\n\u0010=~\n4\u0019X\n\u0016;\u0017\f\f\f\f@S\u0016\u0017\n@\u0010i\f\f\f\f2\n; (S11)\nwhich is a real number. All the remaining task is to numerically calculate the derivatives of the scattering matrix\nelements@S\u0016\u0017=@\u0010i.\nIn the following, we take \u0010=nas an example and illustrate the calculation of @S\u0016\u0017=@\u0010i. Considering the N\u0013 eel\norder along z-axis, i.e. m1=\u0000m2= ^zandn= 2^z, one can calculate the scattering matrix S(n). Then we add an\nin\fnitesimal transverse component \u0001 n=\u0011^xonto the N\u0013 eel order so that the new N\u0013 eel order becomes n0= 2^z+\u0011^x.\n(In practice, we \fnd that the calculated results are well converged with \u0011in the range of 10\u00003{10\u00005.) Under such a\nmagnetic con\fguration, we redo the scattering calculation to \fnd another scattering matrix S(n0). The derivatives of\nthe matrix element S\u0016\u0017can be obtained by\n@S\u0016\u0017\n@nx=S0\n\u0016\u0017\u0000S\u0016\u0017\n\u0011: (S12)3\nIn the same manner, we can \fnd another scattering matrix S00atn00= 2^z+\u0011^yand consequently we have\n@S\u0016\u0017\n@ny=S00\n\u0016\u0017\u0000S\u0016\u0017\n\u0011: (S13)\nFinally, we \fnd that the calculated o\u000b-diagonal elements Dxy\nn(m)andDyx\nn(m)are much smaller than the diagonal elements\nDxx\nn(m)andDyy\nn(m). The latter two are nearly the same. So we take their average in practice, i.e. Dn= (Dxx\nn+Dyy\nn)=2\nandDm= (Dxx\nm+Dyy\nm)=2.\nDYNAMICAL EQUATIONS WITH INTER-SUBLATTICE SPIN PUMPING\nWe start from the coupled dynamical equations of an AFM with the sublattice index i= 1;2,\n_mi=\u0000\rmi\u0002hi+\u000b0mi\u0002_mi: (S14)\nHerehiis the e\u000bective \feld exerted on mi, which can be calculated from the functional derivative of the free energy\nFas\nhi=\u00001\n\u00160MsV\u000eF\n\u000emi: (S15)\n\u000b0is the damping parameter, which must be equal for m1andm2because of the permutation symmetry. Now we\nconsider the spin pumping e\u000bect that discussed in the main text. The spin pumping by the sublattice m1contributes\na dissipative torque \u000b0m1\u0002_m1that is exerted on m1. Here\u000b0is a dimensionless parameter to quantify the magnitude\nof the inter-sublattice spin pumping. The pumped spin current by m1can be absorbed by m2resulting in a damping-\nlike torque m2\u0002[m2\u0002(\u000b0m1\u0002_m1)]\u0019\u000b0m2\u0002_m1, which is exerted on m2. In the same manner, we can identify\ntwo torques due to the spin pumping of m2:\u000b0m1\u0002_m2exerted on m1and\u000b0m2\u0002_m2exerted on m2. Eventually,\nwe obtain the coupled dynamical equations by including the inter-sublattice spin pumping as\n_m1=\u0000\rm1\u0002h1+ (\u000b0+\u000b0)m1\u0002_m1+\u000b0m1\u0002_m2;\n_m2=\u0000\rm2\u0002h2+ (\u000b0+\u000b0)m2\u0002_m2+\u000b0m2\u0002_m1: (S16)\nThe above form of the dynamical equations can be rigorously derived using the Rayleigh functional to describe the\ndissipation [S6].\nIn the following, we rewrite Eq. (S16) into the dynamical equations of the total magnetization m=m1+m2and\nthe N\u0013 eel order n=m1\u0000m2. The e\u000bective \feld hican be transformed as\nh1=\u00001\n\u00160MsV\u000eF\n\u000em1=\u00001\n\u00160MsV\u0012\u000eF\n\u000em@m\n@m1+\u000eF\n\u000en@n\n@m1\u0013\n=hm+hn;\nh2=\u00001\n\u00160MsV\u000eF\n\u000em2=\u00001\n\u00160MsV\u0012\u000eF\n\u000em@m\n@m2+\u000eF\n\u000en@n\n@m2\u0013\n=hm\u0000hn; (S17)\nwhere we have de\fned\nhm=\u00001\n\u00160MsV\u000eF\n\u000em;\nhn=\u00001\n\u00160MsV\u000eF\n\u000en: (S18)\nThen we \fnd\n_m=_m1+_m2=\u0000\r(m1\u0002h1+m2\u0002h2) + (\u000b0+\u000b0) (m1\u0002_m1+m2\u0002_m2) +\u000b0(m1\u0002_m2+m2\u0002_m1):(S19)\nUsing Eq. (S17), the \frst term in the right-hand side of Eq. (S19) can be simpli\fed as\n\u0000\r(m1\u0002h1+m2\u0002h2) =\u0000\r\u0014m+n\n2\u0002(hm+hn) +m\u0000n\n2\u0002(hm\u0000hn)\u0015\n=\u0000\r(m\u0002hm+n\u0002hn):(S20)4\nThe second and the third terms in the right-hand side of Eq. (S19) can be simpli\fed, respectively, as\n(\u000b0+\u000b0) (m1\u0002_m1+m2\u0002_m2) = (\u000b0+\u000b0)\u0012m+n\n2\u0002_m+_n\n2+m\u0000n\n2\u0002_m\u0000_n\n2\u0013\n=\u000b0+\u000b0\n2(m\u0002_m+n\u0002_n);\n(S21)\nand\n\u000b0(m1\u0002_m2+m2\u0002_m1) =\u000b0\u0012m+n\n2\u0002_m\u0000_n\n2+m\u0000n\n2\u0002_m+_n\n2\u0013\n=\u000b0(m\u0002_m\u0000n\u0002_n): (S22)\nFinally, Eq. (S19) is rewritten as\n_m=\u0000\r(m\u0002hm+n\u0002hn) +\u0010\u000b0\n2+\u000b0\u0011\nm\u0002_m+\u000b0\n2n\u0002_n: (S23)\nThe dynamical equation of the N\u0013 eel order ncan be obtained in the same way\n_n=\u0000\r(m\u0002hn+n\u0002hm) +\u0010\u000b0\n2+\u000b0\u0011\nn\u0002_m+\u000b0\n2m\u0002_n: (S24)\nComparing Eqs. (S23) and (S24) with Eqs. (S1) and (S2), we can identify the relations of the damping parameters,\ni.e.\n\u000bm=\u000b0\n2+\u000b0;and\u000bn=\u000b0\n2: (S25)\nThe above relations naturally show the spin pumping e\u000bect and is consistent with our \frst-principles calculations.\n\u0003These authors contributed equally to this work.\nyCorresponding author: zyuan@bnu.edu.cn\n[S1] K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, \\Phenomenology of current-induced dynamics in antiferromagnets,\"\nPhys. Rev. Lett. 106, 107206 (2011).\n[S2] E. V. Gomonay and V. M. Loktev, \\Spintronics of antiferromagnetic systems (review article),\" Low Temp. Phys. 40, 17\n(2014).\n[S3] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W. Bauer, \\First-principles scattering matrices for spin\ntransport,\" Phys. Rev. B 73, 064420 (2006).\n[S4] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, \\Optimal quantum pumps,\" Phys. Rev. Lett. 87, 236601 (2001).\n[S5] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, \\Magnetization dissipation in ferromagnets from scattering theory,\"\nPhys. Rev. B 84, 054416 (2011).\n[S6] H. Y. Yuan, Qian Liu, Ke Xia, Zhe Yuan, and X. R. Wang, \\Proper dissipative torques in antiferromagnetic dynamics,\"\nunpublished (2017)."    },    {        "title": "2302.06099v3.Thickness_and_temperature_dependent_damping_in_La___0_67__Sr___0_33__MnO___3___epitaxial_films.pdf",        "content": " \nThickness and temperature -dependent damping in La0.67Sr0.33MnO 3 \nepitaxial films  \n \nYifei Wang,1 Xinxin Fan,1 Xiaoyu Feng,1 Xiaohu Gao,1 Yunfei Ke,1 Jiguang Yao,1,2 Muhan \nGuo,1 Tao Wang,1 Lvkang Shen,3,4 Ming Liu,3,4 Desheng Xue,1 and Xiaolong Fan1 \n \nAFFILIATIONS  \n1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou \nUniversity,  Lanzhou 730000, People ’s Republic of China  \n2Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2  \n3School of Microelectronics, Xi ’an Jiaotong University, Xi ’an 710049, People ’s Republic of China  \n4State Key Laboratory for Mechanical Behavior of Materials,  Xi’an Jiaotong University, Xi ’an \n710049 , People ’s Republic of China  \n \nABSTRACT  \nThe damping of La 0.67Sr0.33MnO 3 (LSMO) epitaxial films as a function of thickness \nat different temperature s was studied . The competition between two scattering types \n(𝜌  like and 𝜎  like) with entirely distinct thickness  and temperature dependencies  \nresulted in complicated damping behavior. The behavior of 𝜎 like damping in LSMO \nfilms is consistent with the behavior in magnetic metal films. However, because 𝜌 like \ndamping is sensitive to the fine electron struc ture near the Fermi surface, the distortion \nof the oxygen octahedra controlled by the film thickness is an important factor in \ncontrolling the damping. Our study demonstrates that the complexity of damping in \nLSMO epitaxial films is a consequence of strong  correlation effects, which are \ncharacteristic of complex transition  metal oxides.  \n \nSince the rapid development of spintronics for next  generation memory and \nprocessor architectures ,1 the family of spintronic materials has extended from transition \nmetals to complex oxides with strongly correlated elec trical and magnetic properties .2,3 \nIn conventional metals, electron density and spin  orbit coupling are limite d owing to \ntheir robust electron structure. In contrast, transition  metal oxides with strongly coupled \ncharge, spin, and crystalline structure s are promising candidates for improved \nperformance and surprising spintronic effects .4–6 For example, a two  dimensional \nelectron system  with high mobility at oxide interfaces7 can achieve higher spin  charge \ninterconversion efficiency than conventional heavy me tals;8 current  induced \ndeterministic magnetic field  free switching has been found in all  oxide heterojunctions \nwith higher efficiency .9 \nAs a comple x transition metal oxide, LSMO film has attracted considerable attention \nin the field of spintronics because of its high Curie temperature (~360 K) among \nmanganese oxides10 and a spin polarization rate close to 100 %.11 The latter gives rise  \nnot only to an ultra  high tunnel magnetoresistance (1800% at 4 K)11 but also to an ultra  \nlow Gilbert damping (5.2 × 10 4, grown on NdGaO 3 substrate)12 because of the \nrestricted spin  dependent scattering around the Fermi surface with only one spin  \noriented state .13 In spintronics, Gilbert damping is not only an essential parameter in \nmagnetization dynamics14 but also limits the threshold current for spin  torque switching \nand auto oscillators15,16 and the decay length of diffusive spin waves .17 However, there \nis limited research on the damping mechanism of epitaxial LSMO films, and the \nspecifics of their damping tuning mechanism remain controversial. For instance, it is \ndisputed whether the temperature  dependent damping of LSMO films i s monotonic .18,19 \nTo address this discrepancy, our study focus ed on the influence of strong  correlation \neffects on the damping mechanism. Specifically, the correlation between the crystalline \nstructure and electronic momentum scattering  was systematically studied in epitaxial \nLSMO films.  \nLSMO films of different thicknesses ( 𝑡 77.1 38.4 nm) were epitaxially grown by \npulsed laser deposition  using a KrF laser with a wavelength  of 248 nm  on (001)  oriented \nSrTiO 3 (STO) substrates . The base pressure was greater than 3.6 × 10 8 Torr, and the \nfilms were grown at 620 °C with an oxygen pressure of 90 mTorr. The pulse energy \nwas 500 mJ with a frequency of 3 Hz. After deposition, the films were annealed in situ \nfor 1 h at 750 °C under an oxygen atmosphere of 300 To rr, and then cooled to 300 K at \na rate of 10 °C/min. The thickness es of the LSMO films were determined using X  ray \nreflectometry,  and their crystalline structure was characterized using high  resolution X  \nray diffraction (XRD) and  reflection high  energy electron diffraction (RHEED).  \nSurface topography was characterized using atomic force microscopy (AFM). The \nmagnetic hysteresis loop s were measured using a vibrating  sample magnetometer.  The \ntemperature  dependent ferromagnetic reson ance (FMR) and transport properties were \nmeasured using a self  built testing system with a Cryogenic  J4440 cryofree vector \nmagnet.  \nFigure 1(a) shows  the θ 2θ scan of XRD for LSMO (002) peaks, indicating the \nLSMO films are epitaxially grown on  STO substrates.  Local magnification s of the \nLSMO (002) peaks with thicknesses of 24.0 nm (green) and 38.4 nm (black) are shown \nin the insets. The clear Laue fringes demonstrate the flatness and uniformity of the \nepitaxial LSMO film.  Moreover, a shift in the L SMO (002) peak position to a high er \ndiffraction angle can be observed, indicating that the values of the out  of plane lattice \nconstant c decrease with increasing thickness.  \nTo quantify the variation between the lattice constant and thickness, the diffracti on \nangles of LSMO (002) were determined using Jade software, and the values of c were \nobtained as follows:  \n2𝑎𝑏𝑐 sin𝜃\n√ℎ2𝑏2𝑐2+𝑘2𝑎2𝑐2+𝑙2𝑎2𝑏2=𝜆                , (1) \nwhere 𝑎 and 𝑏 are the in  plane lattice constants, h, k, l are Miller indices, and 𝜆71.54 Å \nis the X  ray wavelength.  As the diffraction peaks are spread out, the calculated lattice  \nconstants are only meaningful as a statistical average.  It can be seen from Fig. 1( b) that \nc decreases with increasing thickness and tends to be saturated.  According to A. \nVailionis et al. ,20 the variation of the LSMO lattice constant with thickness is due to the  \nspontaneous presence of regions with different l attice constants in the thickness \ndirection of the films.  Here, we simplified this complicated situation into a bilayer \nmodel: an LSMO interfacial layer with N unit cells and a relatively large 𝑐i73.93 Å, \nand a bulk  like layer for the remaining part with 𝑐b73.84 Å ,20 where i and b denote \ninterface and bulk  like. Based on this model, the average lattice constant x (7a, b, c) as \na function of thickness 𝑡 is given by  \n𝑥̅=N𝑐i𝑥i+(𝑡−N𝑐i)𝑥b\n𝑡                         . (2)  \nAs shown in Fig. 1( b), the fitting curve is consistent with the data and gives N≈9u.c. \n \nFIG. 1. (a) XRD patterns around the (002) peaks of the films with different thicknesses; the \ninset is the local magnification of the data for thicknesses of 24.0 nm (green) and 38.4 nm \n(black) . Average (b) out  of plane, (c) in  plane lattice constants and (d) effective strain as a \nfunction of thickness, followed by a fitted curve of Eq. (2). (e) Schematic i llustration of the \nbilayer model, where the oxygen octahedra in the interface layer (bulk  like layer) are \ncompressive (tensile) strained.  \n \n \nTo comprehensively evaluate the structural variation of the films with different \nthicknesses, we also performed  XRD analysis of the LSMO (201) peak.  Similar to the \ncalculation of 𝑐, the average in plane lattice constants 𝑎 are shown in Fig. 1(c) as a \nfunction of thickness.  The parameter s 𝑎i=3.85 Å , 𝑎b=3.88 Å are obtained from the \nfitting of Eq. (2)  with fixing N=9u.c.. Compared with the data shown in Fig. 1( b), \nthe values of 𝑐 and 𝑎 are different, exhibiting opposite trends with the thickness. The \nvariation in the LSMO film structure is due to the distortion of the oxygen octahedron , \nand the degree of distorti on varies with thickness.  These deformations normally arise \nfrom two effects: the Jahn  Teller effect and the strain caused by lattice mismatch .21,22 \nAs both effects break the symmetry of the oxygen octahedron, we use a total average \neffective strain 𝜀eff as the quantitative parameter for describing the structural detail in \nLSMO films of different thicknesses, given by  𝜀eff=√1\n6(2𝜀𝑧𝑧−𝜀𝑥𝑥−𝜀𝑦𝑦) ,23,24 \nwhere 𝜀𝑧𝑧  (𝜀𝑥𝑥 , 𝜀𝑦𝑦 ) is the out of plane (in plane) strain . The calculations are \npresented in Fig. 1(d). Based on the bilayer model, the effective strain can be fitted \nsimilarly by replacing x with the effective strain in Eq. (2)  and the fitting gives 𝜀effi=\n1.79% (c>a, compressive) and 𝜀effb=−0.98% (c<a, tensile).    \n  As the strong  correlation effect is closely related to the deformation of the oxygen \noctahedron where Mn3+ is located, the details of the crystalline structure should be \nfurther investigated. It has been shown that the crystal field caused by the distorted \noctahedra degenerate s the eg levels of x2 y2 and 3 z2 r2 orbits.25,26 In LSMO films, tensile \nstrain favors x2 y2 occupancy, while compressive strain favors 3 z2 r2 occupancy .27–29 \nConsequently, the 3 z2 r2 (x2 y2) orbit in the interface (bulk  like) layer is preferentially \noccupied .29–31 Furthermor e, the splitting energy ∆Ei between the x2 y2 and 3 z2 r2 \norbitals in the interface layer is larger than that of the bulk  like layer ∆Eb, as shown in \nFig. 1(e)。 \nNext, w e investigated the dynamic magnetic response s of the films using frequency  \ndependent broadband FMR . Figure 2(a) shows representative FMR spectra obtained \nwith an in  plane [100] magnetic field at room temperature. The spectra were fitted with \nthe universal line  shape equation ,32 allowing us to extract the resonant field 𝐻0 and \nthe linewidth ∆𝐻, as shown in Figs. 2(b) and 2(c), respectively . The vertical shift in \nFig. 2(c) was performed to make the data clear.  In the absence of significant \nmagnetocrystalline anisotropy in the LSMO films studied here [Supplemental Material \nⅣ, Figs. S4(a)  (b)], the relation between 𝐻0 and the microwave angular frequency 𝜔 \nis given by  𝜔=𝛾√(𝐻0+4𝜋𝑀eff)𝐻0 , where 𝛾/2π 72.8 GHz/kOe  is the \ngyromagnetic ratio .33 We determined the effective Gilbert damping 𝛼eff by fitting ∆𝐻 \nas a function of 𝜔, which is given by : \n Δ𝐻=Δ𝐻0+𝛼eff𝜔\n𝛾                       .(3)  \nHere, 𝛥𝐻 0 represents the extrinsic linewidth, which can originate  from defects and \nmagnetic inhomogeneities . The effective Gilbert damping 𝛼eff  consists of intrinsic \nand extrinsic damping .34,35 In single  layer films, the latter is typically attributed to two  \nmagnon scattering (TMS), eddy current, radiative, and magnetoelastic damping. We  \nestimated the maximum contributions of eddy  current, radiative, and magnetoelastic \ndamping to be ≈1.1% (30.2 nm at 50 K), 1.8% (38.4 nm at 50 K), and 3.0% (7.1 nm \nat 50 K) of 𝛼eff, respectively (Supplemental Material  Ⅱ). Therefore, we can reasonably \nneglect these contributions in the following discussion.  \n \nFIG. 2. (a) Typical FMR spectra with applied in  plane magnetic field. (b) Microwave angular \nfrequency versus resonant field; solid lines are fitting of Kittel ’s formula. (c) Frequency  \ndependent resonance linewidth; the vertical shift was performed to make the data clear.  \n \nAccording to Xu et al., the disentanglement of various damping mechanisms in \nferromagnetic metal films can be achieved by measuring the thickness  dependent 𝛼eff, \nwhich is given by  \n𝛼eff=𝛼𝑏+𝛽𝑖\n𝑡+𝛽TMS\n𝑡2                        (4) \n,35 where 𝛼b  is the bulk damping, 𝛽i  and 𝛽TMS  are the coefficients of interface \ndamping (including spin pumping for multilayers) and TMS, respectively. Figure 3(a) \nshows the variation of 𝛼eff with thickness at room temperature, which exhibits a non \nmonotonic trend  and is inconsistent with Eq. (4). As the temperature decreases, the \nthickness  dependent damping gradually  becomes monotonic [Figs. 3(a)–(c)], and the \nmodel in Eq. (4) is adequate for fitting the data when the temperature is below 100 K \n[Supple mental Material Ⅲ , Fig. S2(e)].   \nAccording to previous studies on the damping of magnetic metal films ,35–37 the \nthickness  dependent damping is expected to be consistent with Eq. (4) at different \n \ntemperatures. Consequently, the data in Figs. 3(a) –(c) suggest a complicated damping \nmechanism for LSMO films.  Building upon prior discussions, the crystalline structure \nof the LSMO film exhibited variations in thickness, as indicated in Fig. 1(d). Despite \nthe relatively small deformation of the crystalline structure, it has a significant influence \non the electronic structure s of strongly correlated systems .4 As a result, significant \nchanges in the spin  related scattering in the LSMO film  induced by spin  orbit coupling \nare anticipated. Therefore, a comprehensi ve understanding of the coupling between the \nstructure and damping necessitates the separation of the damping contributed by \ndifferent types of scattering.   \n \nFIG. 3. Thickness  dependent 𝛼eff of LSMO films at (a) 300 K, (b) 200 K and (c) 50K. Blue dashed \nlines and solid lines represent the guided and fitted curves of Eq. (4), respectively. (d) Temperature  \ndependent normalized resistivity of LSMO films. All of the samples possess ferromagnetic order in \nthis work.  Symbols are temperature  dependent damping  in (e) 7.1 nm and (f) 38.4 nm LSMO films , \nsolid lines are fitting curves of Eq. (5), purple dotted lines and gray dashed lines represent the \ncontribution of 𝜌 like and 𝜎 like damping.  \n  \nKambersky's torque correlation model38,39 provides a framework for partitioning the \nGilbert damping in ferromagnetic transition metals into two scattering types: \n\"conductivity  like\" ( 𝜎  like) damping and \"resistivity  like\" (𝜌  like) damping, which \narise from intraband and interband scattering, respectively. The thickness  dependent \neffective damping parameter 𝛼eff  governed by Eq. (4) becomes increasingly \nsignificant as the temperature decreases [Figs. 3(a) –(c)], similar to the behavior of 𝜎 \nlike damping. Therefore, by referring to this method, we phenomenologically separated \nthe temperature  dependent 𝛼eff using different types of scattering:   \n𝛼eff=𝛼𝜌(𝑡)𝜌(𝑇)\n𝜌(300  K)+𝛼𝜎(𝑡)𝜎(𝑇)\n𝜎(300  K)                   ,(5) \n17,18,35 where 𝛼𝜌(𝑡)  and 𝛼𝜎(𝑡)  represent the interband and intraband scattering \nparameters, and 𝜌(300 K) and 𝜎(300 K) are the resistivity and conduct ivity at 300 K . \n \nFigure 3(d) displays the temperature  dependent normalized resistivity, which indicates \nthe promising half  metallic nature of LSMO films. Using the temperature  dependent \nresistivity, the effective damping is separated into two distinct contr ibutions by  Eq. (5) . \nThe fitting result is illustrated in Figs. 3(e) (f). The results show that the 𝜌 like ( 𝜎  like) \ndamping dominates 𝛼eff at high (low) temperature [ Figs. 3(e) (f)]. \nThe 𝜎  like damping [𝛼𝜎(𝑡,𝑇)≡𝛼𝜎(𝑡)𝜎(𝑇)\n𝜎(300  K) ] is shown in Figs. 4(a)–(c) as a \nfunction of 1𝑡⁄ at different temperatures. 𝛼𝜎(𝑡,𝑇) is linear with 1𝑡⁄ at 300 K [Fig. \n4(a)], in line with prior reports .12 The degree of nonlinearity in 𝛼𝜎(𝑡,𝑇) becomes more \nevident with decreasing temperature [Figs. 4(b) and (c)].  This suggests that the TMS \nbecomes more significant with decreasing temperature.  In contrast to 𝛼eff , the \nthickness  dependent 𝛼𝜎(𝑡,𝑇) shows agreement with Eq. (4) at different temperatures \n[blue line in Figs. 4(a)–(c)], which is consistent with previous studies  on ferromagnetic \nmetal films .35–37 The contributions of bulk 𝛼b, interface 𝛽i, and TMS 𝛽TMS extracted \nfrom 𝛼𝜎(𝑇,𝑡) is shown in Figs. 4(d) (f). \n \nFIG. 4.  Thickness  dependent 𝜎 like damping at (a) 300 K, (b) 200 K and (c) 50K. Blue lines are \nfitted curves of Eq. (4).  Temperature  dependent (d) bulk 𝛼b, (e) interface 𝛽i and (f) TMS 𝛽TMS \ncontribution to 𝜎 like damping.  \n \nAs shown in Fig. 4(d), 𝛼b  is quite small ( 10−4 ); hence, the enhancement of \n𝛼𝜎(𝑇,𝑡) at low temperatures can be attributed to the contributions of the interface (28% \nfor 7.1 nm at 50 K) and TMS (69%).  The temperature  dependent 𝛽i shown in Fig. 4(e) \nincreases with decreasing temperature. According to Haspot et al.  ,18 in LSMO ultrathin \nfilms, the precession of magnetization injects a pure spin current into the magnetically \nactive dead layer and contributes to damping, which increases with decreasing \ntemperature. Thus, we attribute  the interface contribution  𝛽i to spin pumping by the \nmagnetically active dead layer.  The linear proportionality of the temperature  dependent \n \n𝛼𝜎(𝑡,𝑇)  with 1𝑡⁄  at 300 K [Fig. 4(a)] and the trend of 𝛽TMS  with decreasing \ntemperature [Fig. 4(f)] indicate the transition from negligible to significant TMS \ncontribution as the temperature decreases. To confirm the negligible TMS, we \nconducted in  plane angle  dependent electron spin resonance with 9.46 GHz at 300  K \nand estimated the TMS contribution accounts for ≈ 3.7% of 𝛼𝜎 (300 K, 9.7 nm) \n[Supplemental Material Ⅳ, Fig. S4(e) ]. Furthermore, we attribute d the trend of  𝛽TMS \nincreasing with decreasing temperature to  the enhanced anisotropic field at the interface, \nas reported by Xu et al.  .35 \n \nFIG. 5. Thickness  dependent 𝜌 like damping (a) 300 K, (b) 200 K and (c) 50K. Blue lines are fitted \ncurves of Eq. (6). (d) The fitting parameter 𝐵(𝑇) depending on temperature. The inset shows a \nthickness  dependent av erage energy gap of eg level splitting calculated from ∆𝐸(𝑡)∝\n𝑁𝑐𝑖|𝜀eff𝑖|+(𝑡−𝑁𝑐𝑖)|𝜀𝑒ff𝑏|\n𝑡 based on the bilayer model in LSMO films.  \n \nWe next present a detailed discussion on thickness  dependent 𝜌  like damping  \n[𝛼𝜌(𝑡,𝑇)≡𝛼𝜌(𝑡)𝜌(𝑇)\n𝜌(300  K) ]. In Figs. 5(a) (c), 𝛼𝜌(𝑡,𝑇)  increases with increasing \nthickness, unlike 𝛼𝜎(𝑡,𝑇). For interband scattering, the magnetization dynamics excite \nelectron  hole pairs across different bands, which is related to the fine electronic \nstructure near the Fermi sur face.40,41 Therefore, the increase in 𝛼𝜌(𝑡,𝑇) with thickness \nmay be attributed to the differences in the fine electronic structure near the Fermi \nsurface resulting from thickness variation. Thus, we speculate that the damping \nresulting from interband scattering is influenced by the energy gap of the eg level \nsplitting due to octahedral deformation. Within the bilayer model, the average energy \ngap of the split orbital is proportional to the effective strain magnitude, i.e.,  ∆𝐸̅̅̅̅(𝑡)∝\n|𝜀̅eff|= 𝑁𝑐i|𝜀eff𝑖|+(𝑡−𝑁𝑐i)|𝜀eff𝑏|\n𝑡 [see insert of Fig. 5(d)].Therefore, the phenomenological \n \nexpression for 𝛼𝜌(𝑡,𝑇) as a function of thickness can be given by   \n𝛼𝜌(𝑡,𝑇)=A𝑒−∆𝐸̅̅̅̅(𝑡)/𝐵(𝑇)                   (6) \nwhere A is a normalized parameter and 𝐵(𝑇)  characterizes the energy required to \nexcite electron interband transitions  at different temperatures.  The results in Figs. 5(a) \n(c) show that this simple deduction fits well with the experimental data, indicating that \n𝛼𝜌(𝑡,𝑇) is related to the deformation of the oxygen octahedra. This is a manifestation \nof the strong  correlati on effect in tuning damping in LSMO films. Additionally, the \n𝐵(𝑇)  shown in Fig. 5(d), which saturates at low temperature s, decreases with \nincreasing temperature, and approaches zero near the Curie temperature  of LSMO (7.1 \nnm, ~340K). The dependence of 𝐵(𝑇)  on temperature is similar to that of \nmagnetization.  Note that, t he floor level energy of spin waves is proportional to ℏ𝜔=\nℏ𝛾√𝐻0(𝐻0+4𝜋𝑀eff), which is directly linked to the effective demagnetization field  \n4𝜋𝑀s . Hence, the energy required to excite electron interband transitions may be \nrelated to  magnons through spin  orbit coupling.  \nTherefore, the non  monotonic trend of 𝛼eff at high temperatures [Figs. 3(a) (b)] is \ncaused by the pronounced 𝛼𝜌(𝑡,𝑇) . Furthermore, both the interfac e and TMS \ncontributions explain why the thickness  dependent 𝛼eff  shown in Figs. 3(a)–(c) \ngradually agree with Eq. (4) , because 𝛼𝜎(𝑡,𝑇) is dominant at low temperature . Based \non the suppression of the contribution of 𝛼𝜌(𝑡,𝑇)  to 𝛼eff  with decreasing \ntemperature, we conclude that the different trends of thickness  dependent 𝛼eff  at \ndifferent temper atures is the result of competition between 𝛼𝜎(𝑡,𝑇)  and 𝛼𝜌(𝑡,𝑇) , \nwhich have entirely distinct thickness and temperature dependences.  \n In summary,  we studied the damping mechanism of LSMO  epitaxial films by \nthickness  dependent 𝜌  like and 𝜎  like damping at different temperatures. 𝛼𝜎(𝑡,𝑇) \nconsists of bulk, interface, and TMS contributions, which decrease with increasing \nthickness in line with the literature. However,  owing to the strong  correlation effects, \nvariations in the crystalline structure of LSMO can significantly influence the fin e \nelectronic structure near the Fermi surface to tune the 𝛼𝜌(𝑡,𝑇), which combined with \nthe competition with 𝛼𝜎(𝑡,𝑇) results in the complexity of the LSMO films damping. \nThrough spin  orbit coupling, the energy required to excite electron interband trans itions \nmay arise from the magnon  induced by magnetization dynamics. Our work \ndemonstrates that in spintronics it is necessary to understand the coupling between the \ncrystalline  and electronic structure s of complex transition metal oxides.  \n \nSee Supplemental  Material that includes information on Surface topography and \nstatic magnetic properties of LSMO films (Sect. Ⅰ), the calculated detail of eddy -current, \nradiative and magnetoelastic damping for eddy -current  (Sect. Ⅱ), the discussion of the \ntemperature -dependence of αeff (Sect. Ⅲ) and in -plane angle -dependent electeon spin \nresonance  (Sect. Ⅳ ).  \n \nThis project was supported by National Natural Science Foundation of China (Grant \nNo. 52171231), the Program for Changjiang Scholars and Innovative Re search Team \nin University (Grant No. IRT  16R35), and the 111 Project under Grant No. B20063. We \nwould  like to thank Editage (www.editage.cn) for English language  editing.  \n \nAUTHOR DECLARATIONS  \nConflict of Interest  \nThe authors have no conflicts to disclose . \n \nAuthor Contributions  \nYifei Wang : Data curation (lead); Investigation (lead); Methodology (lead); Writing    \noriginal draft  (lead). Xinxin Fan : Methodology  (equal) ; Writing – review & editing \n(supporting) . Xiaoyu Feng : Investigation (supporting); Writing – review & editing \n(supporting) . Xiaohu Gao : Writing – review & editing (supporting) . Yunfei Ke : \nWriting – review & editing (supporting) . Jiguang Yao : Investigation (supporting) . \nMuhan Guo : Writing – review & editing (supporting) . Tao Wang : Supervision (equal) ; \nWriting – review & editing (supporting) . Lvkang Shen : Supervision (equal) . Ming Liu : \nSupervision (equal) . Desheng Xue : Supervision (equal) . Xiaolong Fan : Project \nadministration (lead); Supervision (lead); Writing – review & editing (lead).  \n \nDATA A V AILABILITY  \nThe data that support the findings of this study are available from the  corresponding \nauthor upon reasonable request.  \n \nREFERENCES  \n1 J. Varignon, L. Vila, A. Barthé lé my, and M. Bibes, Nat. Phys. 14(4), 322 –325 (2018).  \n2 M. Bibes, and A. Barthelemy, IEEE Trans. Electron Devices 54(5), 1003 –1023 (2007).  \n3 S. Majumdar, and S. van Dijken, J. Phys. D. Appl. Phys. 47(3), 34010 (2014).  \n4 H.Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Nagaosa, and Y. Toku ra, Nat. Mater. 11(2), 103 –113 \n(2012).  \n5 M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. 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B 91(1), 014438 (2015).  \n \n "    },    {        "title": "1908.01710v2.Topics_in_Lorentz_Geometry.pdf",        "content": "OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nMAT6702 - Topics in Lorentz Geometry\nIvo Terek Couto\nAcknowledgement: This mini-course was supported in part by the departments of\nMathematics of The Ohio State University and of the University of São Paulo , and in part\nby aFAPESP-OSU 2015 Regular Research Award ( FAPESP grant: 2015/50265-6).\nPage 1arXiv:1908.01710v2  [math.DG]  2 Sep 2019OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nContents\nList of Figures 3\nForeword 4\n1 The spaces Rn\nn 5\n1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n1.2 Pseudo-orthogonal transformations . . . . . . . . . . . . . . . . . . . . . 10\n1.3 Relation with Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . 14\n1.4 Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\nProblems for the first section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n2 Curve theory in L322\n2.1 Admissible curves and the Frenet Trihedron . . . . . . . . . . . . . . . . 22\n2.2 Curves with lightlike osculating plane . . . . . . . . . . . . . . . . . . . . 24\n2.3 Lancret’s theorem and classification of helices . . . . . . . . . . . . . . . . 35\nProblems for the second section . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n3 Surface theory 40\n3.1 Causal characters (once more) and curvatures . . . . . . . . . . . . . . . . 40\n3.2 The Diagonalization Problem . . . . . . . . . . . . . . . . . . . . . . . . . 44\nProblems for the last section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\nExtra #1: Riemann’s classification of surfaces with constant K 50\nProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56\nExtra #2: Weierstrass’s representation of critical surfaces in L357\nAn introduction to split-complex algebra . . . . . . . . . . . . . . . . . . . . . 57\nWeierstrass-Enneper representation formulas . . . . . . . . . . . . . . . . . . . 62\nProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71\nReferences 73\nIndex 76\nPage 2OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nList of Figures\n1 Causal “regions” in L2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2 Causal types in L3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n3 Causal “regions” in L3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n4 Physical interpretation for causal characters. . . . . . . . . . . . . . . . . 15\n5 Understanding the cross products in R3\nn. . . . . . . . . . . . . . . . . . . . 18\n6 Orientations for a lightlike plane. . . . . . . . . . . . . . . . . . . . . . . . 26\n7 Cartan Trihedron for a lightlike helix. . . . . . . . . . . . . . . . . . . . . 30\n8 A “test” lightlike curve a. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\n9 Projections in the coordinate planes of the Cartan Trihedron. . . . . . . . 32\n10 The “spheres” in L3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\n11 Finding the causal character of graphs over the plane z=0. . . . . . . . 41\n12 Spanning a surface of revolution in L3. . . . . . . . . . . . . . . . . . . . . 41\n13 The totally umbilic surfaces in L3. . . . . . . . . . . . . . . . . . . . . . . . 45\n14 Construction of a Fermi chart. . . . . . . . . . . . . . . . . . . . . . . . . . 51\n15 Constant Gaussian curvature K=1. . . . . . . . . . . . . . . . . . . . . . 54\n16 Constant Gaussian curvature K=\u00001. . . . . . . . . . . . . . . . . . . . . 55\n17 Enneper surface in R3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66\n18 Spacelike Enneper surface in L3. . . . . . . . . . . . . . . . . . . . . . . . 67\n19 Catalan surface in R3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67\n20 Spacelike Lorentzian catenoid. . . . . . . . . . . . . . . . . . . . . . . . . 68\n21 Henneberg surface in R3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68\n22 Timelike Enneper surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70\n23 Timelike catenoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70\nPage 3OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nForeword\nThe present text was prepared for the mini-course MAT6702 - Topics in Lorentz Ge-\nometry , to be taught at the University of São Paulo, during the week from 03/11/19\nto 03/15/19. Due to time constraints, some very interesting topics (such as Lorentz\nboosts, the proof of the classification of matrices in O+\"\n1(3,R), and Bonnet rotations for\ntimelike surfaces) unfortunately had to be left out, but a list of references is provided\nin the end. As an attempt to engage the reader actively on what is happening here, a\nfew problems are suggested in the end of each section.\nIn general, the content of these notes is very introductory and meant to be a step-\nping stone for those interested in learning the subject without yet having advanced\nbackground, avoiding the “heavier” language of differentiable manifolds and assum-\ning only some knowledge of multivariable calculus, linear algebra, and differential\ngeometry of curves and surfaces in R3(on the level of [9] or [26] should be enough).\nFor this reason, instead of focusing on the similarities between Euclidean space R3\nand Lorentz-Minkowski space L3, we will devote our little time together in trying to\ngrasp some of the most striking differences between those ambients.\nI hope you enjoy reading this, and if you learn anything new at all here, it was\nworth the effort.\nColumbus, March of 2019\nIvo Terek Couto\nPage 4OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n1 The spaces Rn\nn\n1.1 Basic definitions\nDefinition 1.1. Letn>0 and 0\u0014n\u0014nbe non-negative integers. The pseudo-\nEuclidean space of index nis the pair Rn\nn.= (Rn,h\u0001,\u0001in), where the scalar product\nh\u0001,\u0001in:Rn\u0002Rn!Ris given by\nhx,yin.=x1y1+\u0001\u0001\u0001+xn\u0000nyn\u0000n\u0000xn\u0000n+1yn\u0000n+1\u0000\u0001\u0001\u0001\u0000 xnyn.\nParticular cases are the usual Euclidean space Rn\n0\u0011Rnand the Lorentz-Minkowski\nspaceLn\u0011Rn\n1, whose products are then denoted simply h\u0001,\u0001iEandh\u0001,\u0001iL, respectively.\nRegarding vectors in Rnas column-vectors, one may write hx,yin=x>Idn\u0000n,ny,\nwhere the identity matrix of index nis\nIdn\u0000n,n= (hn\nij)n\ni,j=1.=0\n@Idn\u0000n 0\n0\u0000Idn1\nA.\nNote that the product h\u0001,\u0001inis not positive-definite anymore, which is an obstacle for\ndefining a normk\u0001k n. We will insist on trying, and setting kxkn.=p\njhx,xinjanyway.\nThis “fake norm” k\u0001k nhas poor properties – we’ll see a couple of them soon. Despite\nthis perhaps-not-so-small issue, the product h\u0001,\u0001inhas the one property that allows us\nto develop the theory to some extent: non-degenerability. That is to say, if hx,yin=0\nfor every y2Rn\nn, we necessarily have x=0. Or in other words, the induced map\nRn\nn3x7!hx,\u0001in2(Rn\nn)\u0003is an isomorphism. Having lost the positivity of h\u0001,\u0001in, it is\nconvenient to sort vectors in Rn\nnin three classes:\nDefinition 1.2 (Causal character) .A non-zero vector x2Rn\nnis called:\n•spacelike ifhx,xin>0.\n•timelike ifhx,xin<0.\n•lightlike ifhx,xin=0.\nThe indicator ofxis 1,\u00001 or 0 according to the causal type of x, and it is denoted by ex.\nExample 1.3. If can = (e1, . . . ,en)is the standard basis of Rn\nn, then eiis spacelike for\n1\u0014i\u0014n\u0000nand timelike for n\u0000n+1<i\u0014n. If 1\u0014i\u0014n\u0000n<j\u0014n, then ei\u0006ej\nis lightlike. In L2andL3, we can actually make some sketches based in the equations\nx2\u0000y2=cand x2+y2\u0000z2=c(for positive, negative or zero c):\nFigure 1: Causal “regions” in L2.\nPage 5OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\ntimelike\nspacelikelightlike\nFigure 2: Causal types in L3\nFigure 3: Causal “regions” in L3.\nNot surprisingly, we call the collection of all lightlike vectors in the space its light\ncone.\nSoon we will generalize the notion of causal character for other objects than vectors,\nsuch as subspaces, curves and surfaces. One of the fundamental concepts in geometry\nis the one of orthogonality. So:\nDefinition 1.4. Two vectors x,y2Rn\nnaren-orthogonal ifhx,yin=0. A basis for Rn\nn\nis called n-orthogonal if all its vectors are pairwise n-orthogonal, and it is said to be\nn-orthonormal if all its vectors have scalar square 1 or \u00001. For n=1, one usually uses\nthe term “Lorentz-orthogonal” instead, and if there is no risk of confusion, we’ll do\naway with the n.\nDefinition 1.5. LetS\u0012Rn\nnbe any set. Let’s say that\nS?=fx2Rn\nnjhx,yin=0 for all y2Sg\nPage 6OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nis the subspace of Rn\nnorthogonal to S .\nRemark. S?is a vector subspace of Rn\nneven when Sis not.\nWe avoid the name “orthogonal complement” because when Sis a subspace of\nRn\nnwe might not have S\bS?=Rn\nn. For example, in L2, the line Sspanned by the\nlightlike vector (1, 1)satisfies S=S?=S+S?. So a natural question should be: when\ndo we have S\bS?=Rn\nn? We start with the recomforting result:\nProposition 1.6. Let S\u0012Rn\nnbe a subspace. Then dim S+dim S?=n and (S?)?=S.\nProof: The map Rn\nn3x7! hx,\u0001i\f\f\nS2S\u0003is linear, surjective (since h\u0001,\u0001inis non-\ndegenerate), and its kernel is S?. So the dimension formula follows from the rank-\nnullity theorem. Said formula applied twice also says that dim S=dim(S?)?, so\nS\u0012(S?)?implies S= (S?)?.\nWith this, we may also conclude the:\nCorollary 1.7. Let S\u0012Rn\nnbe a subspace. Then S \bS?=Rn\nnif and only if S is non-\ndegenerate (i.e.,h\u0001,\u0001in\f\f\nSis non-degenerate). It also follows that S is non-degenerate if and only\nif S?is also non-degenerate.\nProof: From dim (S+S?) = dim S+dim S?\u0000dim(S\\S?) = n\u0000dim(S\\S?)it\nfollows that S+S?=Rn\nnif and only if S\\S?=f0g, which in turn is equivalent to\nh\u0001,\u0001in\f\f\nSbeing non-degenerate.\nThis means that we may define orthogonal projections only onto non-degenerate sub-\nspaces. Back to the previous example, we may now see what went wrong there: the\nline spanned by (1, 1)inL2is degenerate, since h(1, 1),(l,l)iL=0 for all l2R. In\nLn, we may stick to the causal type terminology previously used:\nDefinition 1.8. LetS\u0012Lnbe a non-trivial vector subspace. We say that Sis:\n•spacelike ifh\u0001,\u0001iL\f\f\nSis positive-definite;\n•timelike ifh\u0001,\u0001iL\f\f\nSis negative-definite, or indefinite and non-degenerate;\n•lightlike ifh\u0001,\u0001iL\f\f\nSis degenerate;\nRemark. InRn\nn, one might also say that Sis timelike ifh\u0001,\u0001in\f\f\nSis negative definite, but\nifn>1 this does not have the same physical appeal (which we’ll get to in the next\nsection) as in Ln. Moreover, one can say that Sisnull ifh\u0001,\u0001in\f\f\nS=0, which in Lnis the\nsame as Sbeing lightlike and one-dimensional.\nHere’s the relation between causal characters of subspaces and orthogonality:\nTheorem 1.9. Let S\u0012Lnbe a subspace. Then S is spacelike if and only if S?is timelike; S is\nlightlike if and only S?is also lightlike.\nPage 7OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProof: The second part of the result is nothing more than a restatement of Corollary\n1.7, which will also be used to prove the first part. Assume that Sis spacelike. So\nSis non-degenerate and we write Ln=S\bS?. Then S?must necessarily contain\na timelike vector because Lndoes - more precisely, take v2Lntimelike and write\nv=x+ywith x2Sandy2S?, so thathx,xiL+hy,yiL=hv,viL<0 with\nhx,xiL\u00150 forces y2S?to be timelike. Conversely, assume now that Sis timelike,\nand take u2Stimelike. Since S?\u0012u?, it suffices now to show that u?is spacelike.\nWe know again from Corollary 1.7 that u?is not lightlike, and if we have v2u?\ntimelike, the plane spanned by uandvinLnhas dimension 2 while being negative-\ndefinite, which is impossible.\nHere’s another important result:\nTheorem 1.10. Let S\u0012Rn\nnbe a non-degenerate subspace. Then S has an orthogonal basis.\nProof: By induction. By hypothesis we may take u2Swithhu,uin6=0. Then the\northogonal complement of uinSis non-degenerate and has dimension one lower.\nTake an orthogonal basis for this complement and add uto this list. Fill any details\nyou may want.\nWe can conclude this section with some results about linear independence, in gen-\neral:\nTheorem 1.11. Letu1, . . . ,un+12Rn\nnbe pairwise lightlike orthogonal vectors. Then we have\nthat(u1, . . . ,un+1)is linearly dependent.\nProof: The space Rn\nnhas a natural decomposition as Rn\nn=Rn\u0000n\bRn\nn, so that for the\nstandard basis can = (e1, . . . ,en)ofRn\nn, we may decompose\nuj=xj+n\nå\ni=1aijen\u0000n+i, 1\u0014j\u0014n+1,\nfor some vectors xj2Rn\u0000n\bf0gand real coefficients aij, which actually define a\nlinear map A:Rn+1!Rn. The conditionhui,ujin=0 readily implies the equality\nhxi,xjin=ån\nk=1akiakj, for all 1\u0014i,j\u0014n+1. For dimensional reasons, we may also\nchoose a non-zero vector b= (bi)n+1\ni=12kerA. Putting all of this together, we see that\n*\nn+1\nå\ni=1bixi,n+1\nå\nj=1bjxj+\nn=n+1\nå\ni,j=1bibjn\nå\nk=1akiakj= (Ab)>(Ab) =0.\nHowever, the combination ån+1\ni=1bixilies in the spacelike subspace Rn\u0000n\bf0g, so the\nabove gives ån+1\ni=1bjxj=0. So,b2kerAnow gives us\nn+1\nå\nj=1bjuj=n+1\nå\nj=1bjxj+n\nå\ni=1 \nn+1\nå\nj=1bjaij!\nen\u0000n+1=0+0=0,\nas wanted.\nPage 8OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nAs a corollary, we obtain one of the most striking differences between Euclidean\nand Lorentzian geometry:\nCorollary 1.12. Two lightlike vectors in Lnare Lorentz-orthogonal if and only if they are\nparallel.\nThe previous proof might also hint that the matrix of coefficients of h\u0001,\u0001inwith\nrespect to a given basis (also called the Gram matrix ofh\u0001,\u0001inwith respect to said basis)\nwill play an important role in this whole theory.\nProposition 1.13. Letu1, . . . ,um2Rn\nnbe vectors such that the Gram matrix (hui,ujin)m\ni,j=1\nis invertible. Then (u1, . . . ,um)is linearly independent.\nProof: Write åm\ni=1aiui=0and applyh\u0001,ujinon both sides to get åm\ni=1aihui,ujin=0.\nThe hypothesis then implies that a1=\u0001\u0001\u0001=am=0 as wanted.\nWe know that for the usual Euclidean product in Rnthe converse to the above re-\nsult is true. It is not true, in general, in the pseudo-Euclidean spaces Rn\nn. As an extreme\ncounter-example, take any (non-zero) lightlike vector: it is linearly independent by it-\nself, but its 1\u00021 Gram matrix is just (0). As disappointing as this might be, this means\nthat we’ll have to add some extra conditions for this converse to hold. This leads us to\nwhat we may call the “non-degenerability chain conditions”. Here is an example:\nProposition 1.14. Let(u1, . . . ,um)be a m-uple of linearly independent vectors in Rn\nnsuch\nthat each intermediate subspace span(u1, . . . ,uk)is non-degenerate, for 1\u0014k\u0014m. Then the\nGram matrix (hui,ujin)m\ni,j=1is invertible.\nAnother example of this non-degenerability chain condition is related to the Gram-\nSchmidt orthogonalization process . Namely, if we start with linearly independent vectors\n(u1, . . . ,um)and try to produce from these vectors another set of orthogonal vectors\n(eu1, . . . ,fum)spanning the same subspace, at least in the Euclidean case we would pro-\nceed inductively, by setting\nguk+1=uk\u0000k\nå\ni=1huk+1,euii\nkeuik2eui.\nIn the pseudo-Euclidean case, not only we need to take into account the causal char-\nacter of each eui, but we need to ensure that none of those vectors are lightlike. The\ncondition that all the intermediate subspaces span (u1, . . . ,uk)are non-degenerate, for\n1\u0014k\u0014m, is again precisely what we need to safely do\nguk+1=uk\u0000k\nå\ni=1eeuihuk+1,euiin\nkeuik2neui\ninRn\nn. Usually it is a bad idea to insist on using the “fake norm” k\u0001k n: we’ll try to\navoid the absolute values the most we can. So we may alternatively write\nguk+1=uk\u0000k\nå\ni=1huk+1,euiin\nheui,euiineui\nPage 9OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\ninstead, which automatically takes into account the indicators of the eui. For the proof\nof Proposition 1.14 above and more details about the adapted Gram-Schmidt process,\nsee [27]. When we start discussing curve theory, we will see that we’ll have three\nclasses of curves: the admissible curves, the lightlike curves and the semi-lightlike\ncurves. The latter two require some special treatment precisely because they fail to\nrespect a certain non-degenerability chain condition (but you might have guessed this\nby now). We move on.\n1.2 Pseudo-orthogonal transformations\nWhen studying the geometry of any scalar product, it is essential to understand\nthe transformations of the ambient space which preserve said product:\nDefinition 1.15. A linear transformation L:Rn\nn!Rn\nnsuch thathLx,Lyin=hx,yin\nfor all x,y2Rn\nnis called a pseudo-orthogonal transformation . We denote the collection of\nthese transformations, maybe not surprisingly, by O n(n,R). When n=1,Lis called a\nLorentz transformation and O 1(n,R)is called the Lorentz group .\nLet’s get the following simple characterization out of the way:\nProposition 1.16. LetL:Rn\nn!Rn\nnbe a linear transformation. Then L2On(n,R)if and\nonly if L>Idn\u0000n,nL=Idn\u0000n,n. It follows that detL=\u00061, and so Lis an isomorphism.\nRemark. Another way to state the above is saying that the rows and columns of L\nform orthonormal bases of Rn\nn. This proposition also implies that O n(n,R)is a group\nclosed under matrix transposition (proof?).\nExample 1.17. Given j>0, the hyperbolic rotation Rh\nj:L2!L2given by\nRh\nj(x,y) = ( xcosh j+ysinhj,xsinhj+ycosh j)\nis a Lorentz transformation, whose inverse is naturally (Rh\nj)\u00001=Rh\u0000j(you should\ncheck this if you don’t immediately believe it, it is instructive). Up to a couple of\nsigns, this is actually the only Lorentz transformation in dimension 2. We’ll come back\nto that in Theorem 1.21.\nIn general, in the same way that a rigid motion of Rnis always the composition\nof an orthogonal map and a translation, the corresponding notion of “rigid motion”\ninRn\nnalso has this property. Rewriting the definition of a rigid motion in Rnwithout\nemployingk\u0001k leads to the:\nDefinition 1.18. Apseudo-Euclidean isometry inRn\nnis a map F:Rn\nn!Rn\nnsuch that\nhF(x)\u0000F(y),F(x)\u0000F(y)in=hx\u0000y,x\u0000yin,\nfor all x,y2Rn\nn. The collection of such maps is denoted by E n(n,R). When n=1,Fis\ncalled a Poincaré transformation and P (n,R) =E1(n,R)is called the Poincaré group .\nPage 10OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nTo justify the name “Poincaré group”, one has to check that pseudo-Euclidean\nisometries are indeed invertible, and that its inverse is also a pseudo-Euclidean isom-\netry. One possible way to do this is actually going over and beyond, and classifying\nthese maps. We can even say that the above definition was written precisely so that\nthe same strategy used in proving that every rigid motion in Rnis the composition of\na translation and an orthogonal map works. As such, we won’t provide a full proof,\nbut the main steps:\n(i) show that if F2En(n,R)is such that F(0) = 0, then F2On(n,R)(using a\npolarization formula for h\u0001,\u0001inand the result of Problem 4 ahead);\n(ii) apply (i) for L=F\u0000F(0), where F2En(n,R)is now any pseudo-Euclidean\nisometry, and conclude that F=TF(0)\u000eL, where TF(0)denotes translation by\nF(0);\n(iii) check that Ta1\u000eL1=Ta2\u000eL2implies a1=a2andL1=L2, for all a1,a22Rn\nn\nandL1,L22On(n,R), by simply evaluating both sides of the assumed equality\nat0.\nSee Problem 5 in the end of the chapter for another point of view about this.\nThe pseudo-Euclidean space has a natural decomposition as Rn\nn=Rn\u0000n\bRn\nn, as\nwe have explored before in the proof of Theorem 1.11. This allows us to understand\nthe structure of O n(n,R), by writing any Lin block-form as\nL= \nLSB\nCLT!\n,\nwhere LS2Mat(n\u0000n,R)eLT2Mat(n,R)are to be called the spatial and temporal\nparts of L. Since Lis an isomorphism and preserves causal types, we have that LSe\nLTare also non-singular. The blocks LSandLTare intimately related:\nTheorem 1.19. detLS=detLTdetL.\nProof: Let can = (ei)n\ni=1be the usual basis for Rn\nn, and also consider the orthonormal\nbasis of Rn\nnformed by the columns of L, namely, B=\u0000\nLe1, . . . ,Len\u0001\n. Write Lexplic-\nitly as L= (lij)1\u0014i,j\u0014n. Let’s “delete” the block B, defining a linear map T:Rn\nn!Rn\nn\nby\nT(Lej) =(\nLej, if 1\u0014j\u0014n\u0000ne\nån\ni=n\u0000n+1lijei, if n\u0000n<j\u0014n.\nWe immediately have [L]can,B=Idnand\n[T]B,can= \nLS0\nCLT!\n.\nCompute now the matrix [T]B. The expression T(Lej) =Lej, which holds for the\nindices 1\u0014j\u0014n\u0000n, tells us that the upper left and lower left blocks of [T]Bare,\nPage 11OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nrespectively, Id n\u0000nand 0. To compute the determinant of [T]Bby blocks, we need the\nlastncomponents of T(Lej)in the base B, for n\u0000n<j\u0014n. Using the shorthand\nek.=eek, we have:\nT(Lej) =n\nå\ni=n\u0000n+1lijei=n\nå\ni=n\u0000n+1lijn\nå\nk=1ekhei,LekinLek\n=n\nå\ni=n\u0000n+1n\nå\nk=1n\nå\n`=1eklijl`khei,e`inLek\n=n\nå\nk=1 \nn\nå\ni=n\u0000n+1n\nå\n`=1eklijl`khn\ni`!\nLek.\nThe desired last ncomponents correspond to n\u0000n<k\u0014n, and in these conditions,\nwe have that the entries of the lower right block of [T]Bare given by\nn\nå\ni=n\u0000n+1n\nå\n`=1\u0000lijl`k(\u0000di`) =n\nå\ni=n\u0000n+1lijlik,\nwhich we may recognize as the definition of the matrix product between L>\nTandLT.\nWe obtain:\n[T]B=0\n@Idn\u0000n\u0003\n0L>\nTLT1\nA.\nIn particular, it follows that det T= (detLT)2. Moreover:\n[TL]B= [T]B,can[L]can,B=0\n@LS0\nCLT1\nA.\nThus\n(detLT)2detL=detTdetL=det(TL) =detLTdetLS,\nand finally det LS=detLTdetL, as wanted.\nWith this result in our hands, we may label the elements in O n(n,R)by the signs of\nthe determinants of its spatial and temporal parts. This gives us a partition of O n(n,R):\nO+\"\nn(n,R).=fL2On(n,R)jdetLS>0 e det LT>0g\nO+#\nn(n,R).=fL2On(n,R)jdetLS>0 e det LT<0g\nO\u0000\"\nn(n,R).=fL2On(n,R)jdetLS<0 e det LT>0g\nO\u0000#\nn(n,R).=fL2On(n,R)jdetLS<0 e det LT<0g\nThen we may say that the elements of O+\u000f\nn(n,R)preserve the orientation of space ,\nwhile the elements of O\u000f\"\nn(n,R)preserve the orientation of time (i.e., they are orthochro-\nnous ). We know that det L>0 means that Lpreserves the algebraic orientation of\nthe vector space Rn\nn, but on the other hand, if L2On(n,R)and det LS>0 then L\nPage 12OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\npreserves the spatial orientation of the spacelike subspaces1ofRn\nn. Using convenient\ndiagonal matrices with only 1’s and \u00001’s, we conclude the:\nCorollary 1.20. O+#\nn(n,R),O\u0000#\nn(n,R)andO\u0000\"\nn(n,R)are cosets of O+\"\nn(n,R).\nThis means that we may focus our attention to the identity component O+\"\nn(n,R).\nIn low dimensions, we have the following classifications:\nTheorem 1.21.\nO+\"\n1(2,R) =\u001a\u0012cosh jsinhj\nsinhjcosh j\u0013\n2Mat(2,R)jj2R\u001b\n.\nProof: AnyL= (lij)2\ni,j=12O+\"\n1(2,R)satisfies\n8\n><\n>:l2\n11\u0000l2\n21=1\nl2\n12\u0000l2\n22=\u00001, and\nl11l12\u0000l21l22=0\nwith l11,l22\u00151. So we get unique t,s2R\u00150with l11=cosh tandl22=cosh s. The\nabove equations imply that jl21j=sinh tandjl12j=sinh s. The additional condition\ndetL=1 gives l12l21=cosh tcosh s\u00001\u00150, so l12andl21have the same sign. No\nmatter which sign, the the third equation above now says that\n0=cosh tsinh s\u0000sinh tcosh s=sinh(s\u0000t) =)s=t.\nThenLis one of the following matrices, for t>0:\n\u0012cosh tsinh t\nsinh tcosh t\u0013\nor\u0012cosh t\u0000sinh t\n\u0000sinh tcosh t\u0013\n.\nA somewhat similar strategy also gives us the classification in dimension 3:\nTheorem 1.22. AnyL2O+\"\n1(3,R)is conjugate to one of the following matrices:\n0\n@1 0 0\n0 cosh jsinhj\n0 sinh jcosh j1\nA,0\n@cosq\u0000sinq0\nsinqcosq0\n0 0 11\nA,or0\nB@1\u0000q q\nq1\u0000q2/2 q2/2\nq\u0000q2/21+q2/21\nCA,\nwhere j,q2R. The transformation Lis called hyperbolic, elliptic or parabolic, depending on\nits conjugacy class.\nRemark.\n• One can prove that any L2O+\"\n1(3,R)has at least one unit eigenvector, say v.\nThe causal character of vdecides what is the class of L. Namely, Lis hyperbolic\nifvis spacelike (so Lacts as an hyperbolic rotation in the timelike plane v?),\nelliptic if vis timelike (so Lacts as a Euclidean rotation in the spacelike plane\nv?), and parabolic if vis lightlike (so Lhas that shear-like action in the null line\ndefined by v).\n• This terminology is also useful in establishing the classification of helices in L3\n(Lancret’s theorem ), according to the causal type of the helix’s axis.\n1Now read this sentence again. Slowly.\nPage 13OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n1.3 Relation with Special Relativity\nHere we will motivate the names “spacelike”, “timelike” and “lightlike”, and try\nto give some relation between what we have done so far and the mathematics used\nin Special Relativity. We focus on Lorentz-Minkowski space L4, whose points are, in\nthis setting, called events . Fixing the inertial frame given by the standard basis of L4,\nwe write the coordinates in L4as(x,y,z,t). Assume that a particle with positive mass\nmoves in spacetime from event pto event q, through some time interval Dt6=0, and\nlet\nv=q\u0000p= (Dx,Dy,Dz,Dt)\nbe the spacetime displacement vector. The fact that the particle may not move at a\nspeed greater than the speed of light cmay be written as\n\u0012Dx\nDt\u00132\n+\u0012Dy\nDt\u00132\n+\u0012Dz\nDt\u00132\n<c2.\nSo, if we letev= (Dx/Dt,Dy/Dt,Dz/Dt)be the velocity vector of the worldline of the\nparticle, in R3\u0018=R3\bf0g\u0012L4, the above means that kevkE<c. We henceforth set\nthe so called geometric units , where c=1. With this in mind, computing\nhv,viL= (Dx)2+ (Dy)2+ (Dz)2\u0000(Dt)2\n= (Dt)2 \u0012Dx\nDt\u00132\n+\u0012Dy\nDt\u00132\n+\u0012Dz\nDt\u00132\n\u00001!\n= (Dt)2(kevk2\nE\u00001)\n= (Dt)2(kevkE+1)(kevkE\u00001)\nwe see that:\n(1) if vis timelike, thenkevkE<1, and so the event pmay influence event qifDt>0,\nand the other way around if Dt<0, e.g., via the propagation of a material wave.\n(2) if vis lightlike and Dt6=0, thenkevkE=1 and so the influence between the events\ncan only be given via the propagation of some eletromagnectic wave, or by the\nemission of some light signal sent by one of the events and reaching the other.\n(3) if vis spacelike with Dt6=0, there is no influence relation between the events,\nsincekevkE>1 means that the speed necessary for a particle starting at one event\nto reach the spatial location of the other must be greater than the speed of light,\nwhich is impossible: not even a photon or neutrino is fast enough to experience\nboth events. Both of them are not inside, or even in the boundary, of the other’s\nlightcone.\nPage 14OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nPq\nq\nq\nFigure 4: Physical interpretation for causal characters.\nLet’s try and make more precise this notion of causal influence. For this, we need\nto know what does it mean for a vector to point to the future (or past):\nDefinition 1.23. Leten= (0, . . . , 0, 1 )2Ln. A timelike or lightlike vector v2Lnis\nfuture-directed (resp. past-directed ) ifhv,eniL<0 (resp.hv,eniL>0).\nDefinition 1.24 (\u001cand4).Given p2Ln, we define the timecone and lightcone centered\natpby\nCT(p) =fq2Lnjq\u0000pis timelikegand CL(p) =fq2Lnjq\u0000pis lightikeg.\nNaturally, using the previous definition we may divide those in future cones C+\nT(p)and\nC+\nL(p), and past cones C\u0000\nT(p)and C\u0000\nL(p). We’ll say that pchronologically preceds q(resp.\ncausally preceds q) ifq2C+\nT(p)(resp. q2C+\nT(p)[C+\nL(p)). These relations will be\ndenoted by p\u001cqandp4q.\nLet’s list some properties of these relations:\nProposition 1.25. Given p,u,v2Ln, we have that:\n(i) if u,v2C+\nT(p), thenhu\u0000p,v\u0000piL<0;\n(ii) if u,v2CT(p)andhu\u0000p,v\u0000piL<0, then both uandvare in C+\nT(p)or C\u0000\nT(p);\n(iii)\u001cand4are transitive.\nGeometrically, they’re easy to understand, but their proofs rely on technicalities\nwith hyperbolic trigonometric functions. You are welcome to try and prove them, but\nyou can check the proofs on [22] or [27], and more general results in the contexts of\nspacetimes in General Relativity may be found on [6], [14] and [25].\nNow, we have previously mentioned that the “fake norm” k\u0001k Lhas poor proper-\nties, which is mainly due to the fact that it is not induced by a positive-definite inner\nproduct. In this context, here’s probably the best we can get:\nProposition 1.26 (Backwards Cauchy-Schwarz) .Letu,v2Lnbe timelike vectors. Then\njhu,viLj\u0015k ukLkvkL. Furthermore, equality holds if and only if uandvare proportional.\nPage 15OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProof: Write Ln=Ru\bu?and write v=lu+u0, with l2Randu0spacelike and\nLorentz-orthogonal to u. On one hand, we have hv,viL=l2hu,uiL+hu0,u0iL. On\nthe other, we compute:\nhu,vi2\nL=hu,lu+u0i2\nL\n=l2hu,ui2\nL\n=\u0000hv,viL\u0000hu0,u0iL\u0001hu,uiL\n\u0015hv,viLhu,uiL>0,\nusing that u0is spacelike and uis timelike. The result follow by taking roots. Note\nthat the equality holds if and only if u0=0, which is equivalent to uandvbeing\nproportional.\nWith this we may define the hyperbolic angle between timelike vectors, both future-\ndirected or past-directed, in the same fashion one defines the angle between vectors\nin a vector space with a positive-definite inner product. Since the image of cosh is\nthe interval [1,+¥[, there is a unique j\u00150 such thathu,viL=\u0000kukLkvkLcosh j.\nAnother consequence is the:\nCorollary 1.27 (Backwards triangle inequality) .Letu,v2Lntimelike vectors, both\nfuture-directed or past-directed. Then ku+vkL\u0015kukL+kvkL.\nAs a general strategy in Mathematics, once we have defined something (here, \u001c\nand4), it is natural to turn our attention to the mappings related to what we have\ndefined. So we write the:\nDefinition 1.28. A map F:Ln!Lnis called a causal automorphism if it is bijective and\nboth Fand F\u00001preserve4, that is:\nx4y() F(x)4F(y) and x4y() F\u00001(x)4F\u00001(y).\nRemark. It can be shown that preserving 4is the same as preserving \u001c.\nObvious examples of causal automorphisms are positive homotheties, translations,\nand orthochronous Lorentz transformations. Amazingly, that’s all of them:\nTheorem 1.29 (Alexandrov-Zeeman) .Let n\u00153and F :Ln!Lnbe a causal automor-\nphism. Then there is a positive constant c >0, an orthochronous Lorentz transformation L,\nand a vector a2Lnsuch that\nF(x) =cL(x) +a, for all x2Ln.\nMoreover, this decomposition is unique.\nThe proof of this theorem is actually difficult (except maybe for the uniqueness\npart2), employing a mix of results from the linear algebra we have seen so far, Dar-\nboux’s fundamental theorem of geometry (regarding certain doubly-ruled surfaces),\n2Proof: assume c1L1(x) +a1=c2L2(x) +a2for all x2Ln, according to the statement of the\ntheorem. Evaluate at 0to get a1=a2. Cancel the translation to get c1L1(x) = c2L2(x)for all x2Ln.\nTake the scalar square of both sides to get c2\n1=c2\n2. Since c1,c2>0, it follows that c1=c2. We conclude\nthatL1=L2.\nPage 16OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nand lifting properties of some maps. You can see more details in [22], for example.\nThe result is false for n=2, in view of some deeper results about the conformal struc-\nture of L2– a counter-example is discussed in [27]. Furthermore, this theorem has also\na topological flavor: the usual topology in Lndoes not properly capture the causal\nfeatures of this spacetime, in contrast with the so called Zeeman topology , whose home-\nomorphisms are precisely the causal automorphisms here discussed. For more about\nthese topologies, you may consult [21].\n1.4 Cross product\nWith a new scalar product h\u0001,\u0001in, comes together a new notion of cross product :\nDefinition 1.30. The index ncross product ofv1, . . . ,vn\u000012Rn\nnis the unique vector\nv2Rn\nnsuch thathv,xin=det(x,v1, . . . ,vn\u00001), for all x2Rn\nn. The existence and\nuniqueness of such vis ensured by the non-degenerability of h\u0001,\u0001in. We then denote v\nbyv1\u0002\u0001\u0001\u0001\u0002 vn\u00001, the index nbeing understood.\nRemark. Just like we denote the scalar products of R3andL3byh\u0001,\u0001iEandh\u0001,\u0001iL, we’ll\nfollows this convention for cross products, using \u0002Eand\u0002L, respectively.\nJust from the definition, we the cross product inherits some immediate properties\nfrom det, registered in the:\nProposition 1.31. The index ncross product in Rn\nnis(n\u00001)-multilinear, totally skew-\nsymmetric, and orthogonal to each of its arguments. If n =3, it additionaly satisfies the\nidentityhv1\u0002v2,v3in=hv1,v2\u0002v3in, for all v1,v2,v32R3\nn(comma commutes with \u0002).\nAs important as these properties are, they still do not tell us how to explicitly com-\npute cross products. Just like when you first learned about cross products in R3, we’ll\nkeep using formal determinants with a convenient Laplace expansion along the first\nrow:\nProposition 1.32. LetB= (ui)n\ni=1be a positive orthonormal basis for Rn\nnand let be given\nvectors vj=ån\ni=1vijui2Rn\nn, for 1\u0014j\u0014n\u00001. Using the shorthand ei.=euifor the\nindicators of the elements in B, we have:\nv1\u0002\u0001\u0001\u0001\u0002 vn\u00001=\f\f\f\f\f\f\f\f\fe1u1\u0001\u0001\u0001 enun\nv11\u0001\u0001\u0001 vn1\n.........\nv1,n\u00001\u0001\u0001\u0001 vn,n\u00001\f\f\f\f\f\f\f\f\f.\nProposition 1.33. Letu1, . . . ,un\u00001,v1, . . . ,vn\u000012Rn\nn. Then we have\nhu1\u0002\u0001\u0001\u0001\u0002 un\u00001,v1\u0002\u0001\u0001\u0001\u0002 vn\u00001in= (\u00001)ndet\u0000(hui,vjin)1\u0014i,j\u0014n\u00001\u0001\n.\nProof: If(ui)n\u00001\ni=1or(vj)n\u00001\nj=1is linearly dependent, there’s nothing to do. Assume then\nthat both are linearly independent. Since both sides of the proposed equality are linear\nin each of the 2 n\u00002 variables, and both the cross product and the determinant are\nPage 17OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\ntotally skew-symmetric, we may assume without loss of generality that uk=eikand\nv`=ej`, where (ei)n\ni=1is the standard basis for Rn\nnand\n1\u0014i1<\u0001\u0001\u0001<in\u00001\u0014ne 1\u0014j1<\u0001\u0001\u0001<jn\u00001\u0014n.\nWe will proceed with the analysis in cases, in terms of the indices i\u0003and j\u0003being\nomitted in each of the two (n\u00001)-uples of indices considered.\n• If i\u00036=j\u0003, both sides vanish. To wit, the left hand side equals hei\u0003,ej\u0003in=0, and\nthe determinant on the right hand side has the i\u0003-th row and the j\u0003-th column\nonly with zeros.\n• If 1\u0014i\u0003=j\u0003\u0014n\u0000n, the left hand side equals hei\u0003,ei\u0003in=1, and the right hand\nside equals (\u00001)ndet Id n\u00001,n= (\u00001)n(\u00001)n=1.\n• If n\u0000n<i\u0003=j\u0003\u0014n, the left hand side equals hei\u0003,ei\u0003in=\u00001, and the right\nhand side equals (\u00001)ndet Id n\u00001,n\u00001= (\u00001)n(\u00001)n\u00001=\u00001.\nCorollary 1.34 (Lagrange’s Identities) .Letu,v2R3\nn. Then:\nku\u0002Evk2\nE=kuk2\nEkvk2\nE\u0000hu,vi2\nE,\nhu\u0002Lv,u\u0002LviL=\u0000hu,uiLhv,viL+hu,vi2\nL.\nThe orientation of the bases chosen for Rn\nnwill be very important for defining con-\nvenient frames along lightlike and semi-lightlike curves in the next chapter. So we\nmight as well discuss this now in a bit greater generality. We follow the convention\nthat the standard basis for Rn\nnis, of course, positive.\nIfv1, . . . ,vn\u000012Rn\nnare linearly independent, do not span a lightlike hyperplane,\nand we denote v=v1\u0002\u0001\u0001\u0001\u0002 vn\u00001, thenB=\u0000\nv1, . . . ,vn\u00001,v\u0001\nis a basis for Rn\nn, and\nit would natural to ask ourselves when such basis is positive or negative. The answer\nis in the determinant of the matrix having these vectors in rows or columns. We have\ndet(v1, . . . ,vn\u00001,v) = (\u00001)n\u00001det(v,v1, . . . ,vn\u00001) = (\u00001)n\u00001hv,vin\nand, hence, positiveness of the basis Bdepends not only on the parity of n, but also\non the causal character of v. Explicitly: if vis spacelike, Bis positive if nis odd, and\nnegative if nis even; if vis timelike, Bis positive if nis even, and negative if nis odd.\nIn particular, for n=3 we may represent all the possible cross products between\nthe elements in the standard basis of R3\nnby the following diagrams:\n\u0002E\ne3e1\ne2\n(a) InR3.\ne1\ne3\u0002Le2\u0000e1\n\u0000e3\u0002L\n(b) In L3.\nFigure 5: Understanding the cross products in R3\nn.\nPage 18OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nThe cross products are obtained by following the arrows. For example, we have\ne2\u0002Ee3=e1ande1\u0002Le2=\u0000e3. InR3, following the arrows in the opposite direction,\nwe obtain the results with the swapped sign (since \u0002Eis skew), e.g., e1\u0002Ee3=\u0000e2.\nInL3this does not work anymore due to the presence of causal characters: note that\ne3\u0002Le2=\u0000e16=\u0000(\u0000e1) =e1. The cross products in L3which cannot be obtained di-\nrectly from the above diagram may be obtained by using that \u0002Lis skew, after finding\nthe up-to-sign correct product on the diagram.\nRemark. These diagrams remain valid using any positive orthonormal basis of the\nspace, provided that in L3the timelike vector (corresponding to e3) is the last one.\nWe’ll conclude the chapter stating two general facts from linear algebra, which will\nbe necessary for giving adequate definitions for the Gaussian and mean curvatures of\na surface later:\nLemma 1.35. Let B :Rn\nn\u0002Rn\nn!Z be a bilinear map, where Z is any vector space. If (vi)n\ni=1\nand(wi)n\ni=1are orthonormal bases for Rn\nn, then we have:\n(i)ån\ni=1eviB(vi,vi) =ån\ni=1ewiB(wi,wi);\n(ii)det\u0000(B(vi,vj))n\ni,j=1\u0001=det\u0000(B(wi,wj))n\ni,j=1\u0001\n, provided Z =R.\nThese quantities (which are then invariant under change of basis) are denoted by trh\u0001,\u0001inB and\ndeth\u0001,\u0001inB.\nPage 19OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProblems\nProblem 1 (Triangles of light) .Show that in Lnwe cannot have lightlike vectors u1,u2\nandu3with u1+u2+u3=0andfu1,u2,u3glinearly independent. Try to generalize.\nProblem 2. Show that if S\u0012Lnis lightlike then dim (S\\S?) =1, and conclude that if\nSis a lightlike hyperplane, then S?\u0012S. Give an example of a subspace S\u0012Rn\nnwith\nn>1 and dim (S\\S?)\u00152.\nProblem 3 (Sylvester’s Law of Inertia) .Show that every orthonormal basis for Rn\nnmust\nnecessarily have n\u0000nspacelike vectors, ntimelike vectors, and no lightlike vectors.\nHint. There’s a proof in [23], which you should try to at least understand if you cannot\ncome up with a solution on your own.\nProblem 4. Show that if a map L:Rn\nn!Rn\nnpreservesh\u0001,\u0001in, then it is automatically\nlinear (and hence in O n(n,R)).\nProblem 5. Consider the semi-direct product O n(n,R)nRn\nnwith operation\u0003given\nby\n(A,v)\u0003(B,w) = ( AB,Aw+v).\nProve that this operation is indeed associative with identity element (Idn,0), compute\n(A,v)\u00001for any (A,v)2On(n,R)nRn\nn, and show that F: On(n,R)nRn\nn!En(n,R)\ngiven by F(A,v) =Tv\u000eAis a group isomorphism.\nProblem 6. LetL2O1(n,R)be a Lorentz transformation.\n(a) Show that a non-lightlike eigenvector must have 1 or \u00001 as associated eigenvalue.\n(b) Show that the product of the eigenvalues associated with two linearly independent\nlightlike vectors is 1.\n(c) If W\u0012Lnis an eigenspace of Lcontaining a non-lightlike vector, show that every\nother eigenspace of Lis orthogonal to W.\n(d) If W\u0012Lnis a subspace, show that WisL-stable (i.e., L[W]\u0012W) if and only if\nW?isL-stable.\nProblem 7 (Margulis Invariant) .LetF2P(3,R)be a hyperbolic Poincaré transforma-\ntion, given by F(x) =Lx+w, with L2O+\"\n1(3,R)andw2L3.\n(a) Show that Lhas three positive eigenvalues 1/ l<1<l. The eigenspaces associ-\nated to land 1/ lare automatically null lines.\n(b) Let vlandv1/lbe future-directed eigenvectors associated to land 1/ l, and v1be\na unit eigenvector associated to 1 such that the base B= (vl,v1,v1/l)is positive.\nShow that Fleaves invariant a unique (affine) line parallel to v1, and acts on such\nline by translation. That is, show that there are p2L3andaF2Rsuch that\nF(p+tv1) =p+tv1+aFv1,\nfor all t2R. We say that aFis the Margulis invariant ofF.\nPage 20OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n(c) Show that aF=hw,v1iLand use this to show that if F1,F22P(3,R)are hyperbolic\nand conjugate by an element of O 1(3,R), then aF1=aF2(thus justifying the name\n“invariant”).\n(d) Show that for every non-zero integer n,aFn=naF. Be careful with the case n<0,\nand pay close attention to the orientation of the eigenbasis associated to L\u00001.\nProblem 8. Letx2L3be a spacelike vector. Show that T.=x\u0002_:L3!L3is\ndiagonalizable, and the directions of the null lines given by the intersection of the\ntimelike plane x?with the lightcone of L3are eigenvectors of T.\nProblem 9 (Lorentz gfactor) .Letv= (Dx1, . . . ,Dxn\u00001,Dt)2Lnbe the displacement\nvector of a particle, moving between two events in spacetime. Show that the hyper-\nbolic angle jbetween vandenis characterized by\ng.=cosh j=1q\n1\u0000kevk2\nE,\nwhereev= (Dx1/Dt, . . . ,Dxn\u00001/Dt)2Rn\u00001is the velocity vector of the particle’s\ntrajectory in Rn\u00001. Show also thatkevkE=tanh j.\nProblem 10 (Coordinate-free index raising) .LetB:Rn\nn\u0002Rn\nn!Rbe a bilinear map.\nThere is a unique linear operator T:Rn\nn!Rn\nnsuch that B(x,y) =hTx,yinfor all\nx,y2Rn\nn. Show that trh\u0001,\u0001inB=trTand deth\u0001,\u0001inB= (\u00001)ndetT.\nPage 21OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n2 Curve theory in L3\nRemark. All curves and functions will be assumed of class C¥(even though most of\nthe time C3orC4is enough), and Iwill always denote an open interval in R.\n2.1 Admissible curves and the Frenet Trihedron\nWe know from classical differential geometry in Euclidean space R3that:\n• any regular curve a:I!R3admits a reparametrization with unit speed, so we\nmay assume without loss of generality that ka0(s)kE=1;\n• we may define, for each s2I, a positive orthonormal frame (Ta(s),Na(s),Ba(s))\nforR3, pictured as attached to the point a(s)– these vectors are called the tangent ,\nnormal and binormal vectors to aats, and they form the so-called Frenet Trihedron\nofaats;\n• there are functions ka:I!R\u00150andta:I!R, called the curvature and torsion\nofa, such that\n0\n@T0\na(s)\nN0\na(s)\nB0\na(s)1\nA=0\n@0 ka(s) 0\n\u0000ka(s) 0 ta(s)\n0\u0000ta(s) 01\nA0\n@Ta(s)\nNa(s)\nBa(s)1\nA,\nfor all s2I.\nWith this data, one states and proves the Fundamental Theorem of Curves in R3,\nwhich basically says that up to rigid motions of R3,aitself is determined by the func-\ntions kaandta. More precisely:\nTheorem 2.1. Letk,t:I!Rbe given functions with k>0,p02R3, s02I and\n(T0,N0,B0)a positive orthonormal basis for R3. Then there exists a unique unit speed regular\ncurve a:I!R3such that:\n•a(s0) =p0;\n•(Ta(s0),Na(s0),Ba(s0)) = ( T0,N0,B0);\n•ka(s) =k(s)andta(s) =t(s)for all s2I.\nThe proof consists, briefly speaking, in solving the Frenet system for a. From\nthis point onwards, we focus our attention on three-dimensional Lorentz-Minkowski\nspace L3. Recall that the definition of a (parametrized) regular curve does not really\ndepend on the scalar product we have equipped the ambient space with. And in the\nsame way that a parametrized curve a:I!L3is regular if a0(t)6=0for all tinI\n(which is the same as saying that fa0(t)gis linearly independent for all t2I), we may\ntake one step further and say that aisbiregular iffa0(t),a00(t)gis linearly independent\nfor all t2I. You might be (correctly) guessing what a k-regular curve in Rn\nnis, by now.\nPage 22OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nSilly as this may seem, this together with a non-degenerability chain condition\n(such as the ones used to relate linear independence of a set of vectors with invertibil-\nity of the associated Gram matrix, or the one which allows us to perform the Gram-\nSchmidt orthogonalization process) is precisely what we need to adapt the classical\ncurve theory developed in R3forL3.\nDefinition 2.2. A curve a:I!L3is called admissible if it is biregular, and for each\nt2Iboth the tangent line spanned by a0(t)and the osculating plane span(a0(t),a00(t))\nare non-degenerate.\nWe might as well define the notion of causal character for curves now:\nDefinition 2.3. Leta:I!Rn\nnbe a regular curve and t02I. We say that ais:\n(i)spacelike at t 0ifa0(t0)is a spacelike vector;\n(ii)timelike at t 0ifa0(t0)is a timelike vector;\n(iii) lightlike at t 0ifa0(t0)is a lightlike vector.\nIf the causal type of a0(t)is the same for all t2Iaccording to the above, we attribute\nsaid causal type to aitself. If this is the case for curves in L3, we also define:\n(iv) the indicator eaofato be 1,\u00001 or 0 if ais spacelike, timelike or lightlike, respec-\ntively.\n(v) the coindicator haofato be 1,\u00001 or 0 if the osculating planes are spacelike, time-\nlike or lightlike, respectively.\nWith this out of the way, let’s analyze the recipe described for curves in R3. First,\nwe need a good parametrization for the curve. It turns out that for this first step,\nregularity is almost enough. Here’s a general statement:\nProposition 2.4. Leta:I!Rn\nnbe a regular curve, which is not lightlike (at any point).\nThen aadmits a reparametrization with unit speed.\nProof: Fixt02Iand define s:I!Rby\ns(t).=Zt\nt0ka0(u)kndu.\nBy the Fundamental Theorem of Calculus and the given hypotheses, we have that\ns0(t) =ka0(t)kn>0. So sis an increasing diffeomorphism from Iinto J.=s[I], with\ninverse h:J!I. Thenea.=a\u000ehhas unit speed.\nRemark. For timelike curves in Ln, we call such parameter the proper time ofaand\ndenote it by t. Physically, the condition ka0(t)kL=1 says that if arepresents the\ntrajectory of an observer carrying a clock, then t\u0000t0is the time lapse measured by\nsuch observer between the events a(t0)anda(t).\nNow, the admissibility condition allows us to apply Corollary 1.7 (p. 7) for the\nosculating planes to the curve and write the:\nPage 23OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nDefinition 2.5. Leta:I!L3be a unit speed admissible curve. The tangent vector to\naatsisTa(s).=a0(s). Since the conditions on aensure that the curvature ofaat s,\nka(s).=ka00(s)kL, never vanishes, we may define the normal vector toaatsto be the\nunique unit vector Na(s)such that T0\na(s) =ka(s)Ta(s). Then let the binormal vector\ntoaats,Ba(s), be the unique unit vector such that (Ta(s),Na(s),Ba(s))is a positive\northonormal basis for L3.\nRemark.\n• Note thatha0(s),a0(s)iL=eaimplies 2ha00(s),a0(s)iL=0, so indeed the vectors\nTa(s)andNa(s)are orthogonal.\n• It follows from our previous discussion regarding orientability of bases in Rn\nn\nthatBa(s) = (\u00001)neahaTa(s)\u0002Na(s)(of course, we’re interested in what will\nhappen for n=1 here) – this can be also checked by applying Lagrange’s identity\n(Corollary 1.34, p. 18) together with the definition of the index ncross product as\nthe vector representing the linear functional induced by det and its arguments.\nIn the same setting as the above definition, the torsion taofawill be the unique\nfunction such that\n0\n@T0\na(s)\nN0\na(s)\nB0\na(s)1\nA=0\n@0 ka(s) 0\n\u0000eahaka(s) 0 ta(s)\n0 (\u00001)n+1eata(s) 01\nA0\n@Ta(s)\nNa(s)\nBa(s)1\nA,\nfor all s2I. Setting n=0 and ea=ha=1, we recover the usual Frenet equations in\nR3. This means that the theory for admissible curves can be developed simultaneously\nin both ambients R3andL3. Here’s a more powerful version of Theorem 2.1:\nTheorem 2.6. Letk,t:I!Rbe given functions with k>0,p02R3\nn, s02I and\n(T0,N0,B0)a positive orthonormal basis for R3\nn. Then there exists a unique unit speed admis-\nsible curve a:I!R3\nnsuch that:\n•a(s0) =p0;\n•(Ta(s0),Na(s0),Ba(s0)) = ( T0,N0,B0);\n•ka(s) =k(s)andta(s) =t(s)for all s2I.\nA detailed proof of this version of the Fundamental Theorem of Curves, and also\nhow to adapt what was summarized here for admissible curves not necessarily having\nunit speed, see [27].\n2.2 Curves with lightlike osculating plane\nWe will continue to work with biregular curves (without further comments). In\nparticular, we are excluding null lines. Let’s say that a unit speed non-lightlike and\nnon-admissible curve is semi-lightlike (observe that such curves are automatically space-\nlike). That is to say, a non-admissible curve is either lightlike or semi-lightlike, accord-\ning to whether the tangent line or the osculating plane is degenerate. Or equivalently,\na lightlike curve has (ea,ha) = ( 0, 1)while a semi-lightlike curve has (ea,ha) = ( 1, 0).\nPage 24OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nIt is possible to treat both lightlike and semi-lightlike curves simultaneously. How-\never, there is an issue we must solve first: lightlike curves obviously do not admit\nreparametrizations with unit speed. One can also understand the reason for this bear-\ning in mind that lightlike curves may be seen as worldlines of photons or neutrinos –\nthe proper time measured by it is zero, and so it cannot be used as the curve parameter.\nIf we cannot have ka0(t)kL=1, we’ll move on to the next best thing: ka00(t)kL=1.\nMore precisely:\nLemma 2.7. Leta:I!Lnbe a lightlike curve with ka00(t)kL6=0for all t2I. Then a\nadmits an arc-photon reparametrization . Namely, there is an open interval J \u0012Rand a\ndiffeomorphism h :J!I such thatea=a\u000eh satisfieskea00(f)kL=1for all f2J.\nProof: Let’s check what such hmust satisfy, and see if said conditions are actually\nenough to define it. We should have ea(f) =a(h(f))for all f2J, and differentiating\neverything twice we get\nea00(f) =a00(h(f))h0(f)2+a0(h(f))h00(f).\nSince ais lightlike, a00(h(f))is orthogonal to a0(h(f)), and the given condition\nka00(h(f))kL6=0 says that a00(h(f))is spacelike. So, taking scalar squares on both\nsides yields\n1=ha00(h(f)),a00(h(f))iLh0(f)4,\nwhich readily implies that h0(f) =ka00(h(f))k\u00001/2\nL. This is a first order differential\nequation which depends continuously on h, and given f02Jand t02I, there is a\nunique solution hwith h(f0) = t0. For this h, defineea=a\u000eh. This is the desired\nreparametrization.\nExample 2.8. Consider the helix a:R!L3given by a(t) = ( rcost,rsint,rt), where\nr>0 is fixed. Since a0(t) = (\u0000rsint,rcost,r)is a lightlike vector for all t2R,aitself\nis lightlike. Moreover, a00(t) = (\u0000rcost,\u0000rsint, 0)satisfieska00(t)kL=r6=0 for all\nt2R. So, there is an arc-photon reparametrization. The differential equation to solve\nbecomes just h0(f) =1/pr. It follows that\nea(f) =\u0012\nrcos\u0012fpr\u0013\n,rsin\u0012fpr\u0013\n,p\nrf\u0013\n, f2R,\nis an arc-photon reparametrization of a.\nWhen treating both types of curves at the same time, we will omit the distinguished\nparameter sorf, to avoid notation clutter. The next step is, like before, to define an\nadapted frame for each point in the curve. But in this case, an orthonormal frame does\nnot carry geometric information about the curve’s acceleration vector. If we cannot\nnormalize the acceleration vector... we just don’t do it. We start with the:\nDefinition 2.9. Leta:I!L3be a lightlike or semi-lightlike curve. We define the\ntangent and normal vectors to the curve by\nTa.=a0and Na.=a00,\nrespectively.\nPage 25OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nWe have given up on orthonormality, but not on positiveness. To complete the\nframe, we need to find a third vector Ba, again to be called the binormal vector , such\nthat the basis (Ta,Na,Ba)is positive at each point of the curve.\nIn general, we may define the orientation of a basis (v,w)for a lightlike plane in\nterms of a choice of a euclidean-normal vector nto the plane. More precisely, we’ll say\nthat(v,w)ispositive if(v,w,n)is a positive basis for L3, with nfuture-directed. If vis\nlightlike and wis unit (and spacelike), we have that the cross product v\u0002Lwis also\nlightlike, and hence proportional to v. Writing v\u0002Lw=lvfor some l2R, we may\ngeometrically analyze the sign of las follows:\nvw\nnv\u0002\nEw\nv\u0002Lw\n(a)(v,w)positive ( l<0)\nnv\nw\nv\u0002\nEwv\u0002Lw\n(b)(v,w)negative ( l>0)\nFigure 6: Orientations for a lightlike plane.\nThis way, if (v,w)is positive then l<0 and, similarly, if (v,w)is negative we\nhave l>0.\nBack to defining (Ta,Na,Ba): we may assume (by reparametrizing aif necessary)\nthat the bases (Ta,Na)of the osculating planes are positive. In this case, to determine\nthe vector Ba, to be lightlike, we need to also prescribe the values of hTa,BaiLand\nhNa,BaiL. In view of the above, one of these values should be 0 (so that Bais Lorentz-\northogonal to the spacelike vector) and the other \u00001 (so that Bais not proportional to\nthe other lightlike vector, preserving linear independence). Which of these products\nwill be 0 and which will be \u00001 should naturally depend on the causal type of aitself.\nChoosing lightlike Basuch thathTa,BaiL=\u0000haandhNa,BaiL=\u0000ea, we can treat\nall the cases simultaneously. So:\nProposition 2.10. Leta:I!L3be a lightlike or semi-lightlike curve. The triple (Ta,Na,Ba)\nis a positive basis for L3, for each point in a.\nProof: Our goal is to show that det (Ta,Na,Ba)>0. Let’s do the case ea=0 and\nha=1. Writing Ba(f)as in terms of the basis\u0000\nTa(f),Na(f),Ta(f)\u0002ENa(f)\u0001\n, we\nsee that the only relevant component of Ba(f)for the determinant we’re going to\ncompute is the one in the direction of Ta(f)\u0002ENa(f)– call it m(f)Ta(f)\u0002ENa(f).\nPage 26OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nThen\ndet(Ta(f),Na(f),Ba(f)) = m(f)det(Ta(f),Na(f),Ta(f)\u0002ENa(f))| {z }\n>0,\nso that we only have to verify that m(f)>0. From Figure 6, we may write that\nTa(f)\u0002LNa(f) =l(f)Ta(f)for a certain coefficient l(f)<0 (since (Ta(f),Na(f))\nis positive). Applying Id 2,1on this equality, it follows that\nTa(f)\u0002ENa(f) =Id2,1\u0000\nTa(f)\u0002LNa(f)\u0001=l(f)Id2,1Ta(f) =)\n=) hTa(f)\u0002ENa(f), Id2,1Ta(f)iE<0.\nFinallly, since Ta(f)andNa(f)are Lorentz-orthogonal to Ta(f), we have that\n\u00001=hBa(f),Ta(f)iL=m(f)hTa(f)\u0002ENa(f), Id2,1Ta(f)iE,\nand so we conclude that m(f)>0.\nThe triple\u0000\nTa,Na,Ba\u0001\nis then called the Cartan Trihedron ofa.\nGeometrically, when the curve is lightlike, the situation is as follows: the vector\nNa(f)is spacelike, and so its orthogonal complement is a timelike plane which inter-\nsects the lightcone of L3in two null lines, with exactly one of them in the direction of\nTa(f). The binormal vector is then in the direction of the other null line in Na(f)?,\nbeing determined by the equation hBa(f),Ta(f)iL=\u00001. A similar interpretation can\nbe made for semi-lightlike curves.\nNow, recall that the Frenet equations arise when we write the derivatives of the\nvectors in the frame as a combination of the frame elements themselves. The equations\nwere then a quick consequence of the general formula for the orthonormal expansion\nof a given vector – formula that we no longer have in this setting. Here’s what we\nhave instead:\nLemma 2.11. Leta:I!L3andv2L3. So:\n(i) ifais lightlike, we have\nv=\u0000hv,Ba(f)iLTa(f) +hv,Na(f)iLNa(f)\u0000hv,Ta(f)iLBa(f),\nfor all f2I;\n(ii) if ais semi-lightlike, we have\nv=hv,Ta(s)iLTa(s)\u0000hv,Ba(s)iLNa(s)\u0000hv,Na(s)iLBa(s),\nfor all s2I.\nRemark. One possible mnemonic is: switch the position and sign only of the coeffi-\ncients corresponding to lightlike directions.\nPage 27OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProof: We will treat both cases at once, noting the relations en\na=ea,hn\na=hafor all\nn\u00151,eaha=0 and ea+ha=1, which follow from the fact that the only possibilities\nof pairs are (ea,ha) = ( 1, 0)and(ea,ha) = ( 0, 1). Moreover, recall that we are still\nassuming that (Ta,Na)is positive. That being said, write v=aTa+bNa+cBa.\nTaking all possible products with the elements of the Cartan Trihedron and organizing\nthe results in a matrix, we get\n0\n@hv,TaiL\nhv,NaiL\nhv,BaiL1\nA=0\n@ea 0\u0000ha\n0 ha\u0000ea\n\u0000ha\u0000ea 01\nA0\n@a\nb\nc1\nA.\nFrom the relations mentioned previously, the inverse of this coefficient matrix exists,\nand it is just the original matrix, so that:\n0\n@a\nb\nc1\nA=0\n@ea 0\u0000ha\n0 ha\u0000ea\n\u0000ha\u0000ea 01\nA0\n@hv,TaiL\nhv,NaiL\nhv,BaiL1\nA.\nWe are done.\nBefore this lemma comes into play, we have the:\nDefinition 2.12. Leta:I!L3be a lightlike or semi-lightlike curve. The pseudo-torsion\nofais the function da:I!Rgiven by da.=\u0000hN0\na,BaiL.\nRemark. The function dais also called the Cartan curvature ofa.\nTheorem 2.13. Leta:I!L3be a lightlike or semi-lightlike curve. Then we have that\n0\n@T0\na\nN0\na\nB0\na1\nA=0\n@0 1 0\nhadaeada ha\neahada\u0000eada1\nA0\n@Ta\nNa\nBa1\nA.\nRemark. Explicitly, the coefficient matrices when ais lightlike or semi-lightlike are,\nrespectively,0\n@0 1 0\nda(f) 0 1\n0da(f)01\nA e0\n@0 1 0\n0da(s) 0\n1 0\u0000da(s)1\nA.\nProof: The first equation is the very definition of the normal vector. For the second\none, we apply Lemma 2.11 regarding N0\naas a column vector to get\nN0\na=0\n@ea 0\u0000ha\n0 ha\u0000ea\n\u0000ha\u0000ea 01\nA0\n@hN0\na,TaiL\nhN0\na,NaiL\nhN0\na,BaiL1\nA\n=0\n@ea 0\u0000ha\n0 ha\u0000ea\n\u0000ha\u0000ea 01\nA0\n@\u0000ha\n0\n\u0000da1\nA=0\n@hada\neada\nha1\nA,\nPage 28OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nand so we have the second row of the sought coefficient matrix. Similarly for B0\na, we\nhave\nB0\na=0\n@ea 0\u0000ha\n0 ha\u0000ea\n\u0000ha\u0000ea 01\nA0\n@hB0\na,TaiL\nhB0\na,NaiL\nhB0\na,BaiL1\nA\n=0\n@ea 0\u0000ha\n0 ha\u0000ea\n\u0000ha\u0000ea 01\nA0\n@ea\nda\n01\nA=0\n@ea\nhada\n\u0000eada1\nA,\nand we obtain the last row.\nExample 2.14. Letr>0 and consider again the curve a:R!L3given by\na(f) =\u0012\nrcos\u0012fpr\u0013\n,rsin\u0012fpr\u0013\n,p\nrf\u0013\n,\nwhich is lightlike with arc-photon parameter. We readily have\nTa(f) =a0(f) =\u0012\n\u0000p\nrsin\u0012fpr\u0013\n,p\nrcos\u0012fpr\u0013\n,p\nr\u0013\nand\nNa(f) =a00(f) =\u0012\n\u0000cos\u0012fpr\u0013\n,\u0000sin\u0012fpr\u0013\n, 0\u0013\n.\nTo compute Ba(f), note that the cross product\nTa(f)\u0002ENa(f) =\u0012p\nrsin\u0012fpr\u0013\n,\u0000p\nrcos\u0012fpr\u0013\n,p\nr\u0013\n,\nseen in L3, is lightlike and future-directed, so that the basis (Ta(f),Na(f))of the\nosculating plane is always positive (so there is no need to further reparametrize a).\nFurthermore, in this case, we have one particularity: Ta(f)\u0002ENa(f)is also Lorentz-\northogonal to Na(f). This implies that Ba(f)must be a positive multiple of the\nTa(f)\u0002ENa(f). To obtainhBa(f),Ta(f)iL=\u00001, if suffices to take\nBa(f) =\u00121\n2prsin\u0012fpr\u0013\n,\u00001\n2prcos\u0012fpr\u0013\n,1\n2pr\u0013\n.\nFinally, we have:\nda(f) =\u0000hN0\na(f),Ba(f)iL=\u00001\n2rsin2\u0012fpr\u0013\n\u00001\n2rcos2\u0012fpr\u0013\n+0=\u00001\n2r.\nPage 29OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nFigure 7: Cartan Trihedron for awith r=1/4.\nFrom the names “pseudo-torsion” and “Cartan curvature”, you might be guessing\nthatdashould be halfway between kaandta. The next two results actually show how\nfardaactually is from ta:\nTheorem 2.15. The only plane lightlike curves in L3are null lines.\nProof: Clearly null lines are plane curves, and if ais not a null line, then it has an arc-\nphoton reparametrization. It then suffices to check that if a:I!L3is a lightlike curve\nwith arc-photon parameter and ha(f)\u0000p,viL=0 for all f2I, and certain p,v2L3,\nthen v=0. To wit, differentiating the given expression thrice we obtain:\nhTa(f),viL=hNa(f),viL=da(f)hTa(f),viL+hBa(f),viL=0.\nIf follows from Lemma 2.11 (p. 27) that v=0as wanted.\nExample 2.16. Let f:I!Rbe a smooth function with positive second derivative,\nand consider a:I!L3given by a(s) = (s,f(s),f(s)). We have that ais semi-\nlightlike with Ta(s) =a0(s) = ( 1,f0(s),f0(s))andNa(s) =a00(s) =(0,f00(s),f00(s)).\nAlso, Ta(s)\u0002ENa(s) = ( 0,\u0000f00(s),f00(s))is a future-directed lightlike vector, so that\n(Ta(s),Na(s))is positive. We look for a lightlike vector Ba(s) = ( a(s),b(s),c(s)),\nLorentz-orthogonal to Ta(s)and such thathBa(s),Na(s)iL=\u00001. Explicitly, we have\nthe system:8\n><\n>:a(s)2+b(s)2\u0000c(s)2=0\na(s) +f0(s)(b(s)\u0000c(s)) = 0\nf00(s)(b(s)\u0000c(s)) =\u00001\nBy substituting the third equation in the second one we obtain a(s) = f0(s)/f00(s).\nWith this, the first equation becomes\n(b(s)\u0000c(s))(b(s) +c(s)) = b(s)2\u0000c(s)2=\u0000f0(s)2\nf00(s)2=)b(s) +c(s) =f0(s)2\nf00(s),\nPage 30OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nafter using the third equation again. We then obtain\nBa(s) =1\n2f00(s)\u0010\n2f0(s),f0(s)2\u00001,f0(s)2+1\u0011\n.\nFinally, we compute\nda(s) =\u0000hN0\na(s),Ba(s)iL=f000(s)\nf00(s).\nIn particular, note that ais contained in the (lightlike) plane P:y\u0000z=0, but we may\nchoose functions ffor which the pseudo-torsion does not vanish.\nThe above example shows that, in general, the pseudo-torsion of a semi-lightlike\ncurve is not a measure of how much the curve deviates from being a plane curve.\nOne might wonder next whether the sign of dasays something about how the curve\ncrosses its own osculating planes (just like tadoes in R3). Again, the answer is a\nresounding no. Let a:I!L3be lightlike and assume that 0 2Ianda(0) =0. Taylor\nexpansion gives\na(f) =fa0(0) +f2\n2a00(0) +f3\n6a000(0) +R(f),\nwhere R(f)/f3!0asf!0. Organizing this in terms of the Cartan Trihedron\nF=\u0000\nTa(0),Na(0),Ba(0)\u0001\n, we see that the components of a(f)\u0000R(f)are\na(f)\u0000R(f) =\u0012\nf+da(0)f3\n6,f2\n2,f3\n6\u0013\nF.\nFigure 8: A “test” lightlike curve a.\nProjecting, independent of the sign of da(0), we get:\nPage 31OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nNaH0LBaH0L\n(a) Projection in the normal plane\nTaH0LBaH0L (b) Projection in the rectifying plane\nFigure 9: Projections in the coordinate planes of the Cartan Trihedron.\nIt might be worth noting here that even though the vectors of the Cartan Trihedron\nare not mutually orthogonal, we may still picture them as in the above figures, bear-\ning in mind that only their linear independence and the assumed positive orientation\nare relevant to concluding information about how acrosses the osculating plane. We\nconclude that no matter the sign of the pseudo-torsion, any lightlike curve crosses its\nosculating planes in the direction of the binormal vector.\nIfais semi-lightlike instead, a similar calculation gives\na(s)\u0000R(s) =\u0012\ns,s2\n2+da(0)s3\n6, 0\u0013\nF,\nwhich hints at a much more extreme situation:\nTheorem 2.17. Every semi-lightlike curve is plane and contained in a lightlike plane.\nProof: Ifa:I!L3is semi-lightlike, we seek p,v2L3, with lightlike v, such that\nha(s)\u0000p,viL=0 for all s2I. If this condition is satisfied, differentiating twice\ngiveshNa(s),viL=0, and we conclude that vshould be proportional to Na(s)(two\nLorentz-orthogonal lightlike vectors are parallel by Corollary 1.12, p. 9). Motivated by\nthis, we seek a smooth function l:I!Rsuch that v=l(s)Na(s)is constant. This\nwould lead us to\n0= (l0(s) +da(s)l(s))Na(s),\nfor all s2I. Define vin such a way, by taking\nl(s) =exp\u0012\n\u0000Zs\ns0da(x)dx\u0013\n,\nwhere s02Iis fixed. By construction, vis constant and then we just take p=a(s0).\nThis being understood, the justificative that such pevsatisfy everything we need is\nthe usual: consider f:I!Rgiven by f(s) =ha(s)\u0000a(s0),viL. Clearly f(s0) = 0\nand f0(s) =hTa(s),viL=0 for all s2I.\nPage 32OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nBack to the given Taylor expansion, we see that its only relevant projection is\ng:I!R2given by\ng(s) =\u0012\ns,s2\n2+da(0)s3\n6\u0013\n,\nand it would be natural to seek a relation between the curvature of gat 0 (as a plane\ncurve) and the pseudo-torsion da(0). There is a crucial detail here, however, which\nwill stop us from pursuing this question further: since the osculating plane is degen-\nerate, the “metric” to be used in this R2is noth\u0001,\u0001iEnorh\u0001,\u0001iL, but the ill-behaved\nproducthh(x1,x2),(y1,y2)ii.=x1y1. In view of this, the expression\ndet(g0(s),g00(s))\nkg0(s)k3=1+da(0)s\nmay no longer be seen as the curvature3ofg, sincehh\u0001,\u0001iiis degenerate. Even worse,\nthere is no reasonable notion of curvature here, since every curve of the form (s,f(s)),\nwhere fis a smooth function, can be mapped into the xaxis via F:R2!R2given\nbyF(x,y) = ( x,y\u0000f(x)). The derivative DF(x,y)is a linear map which preserves\nhh\u0001,\u0001ii, and so Fis a “rigid motion” of the degenerate plane. That is to say, all the\ngraphs of smooth functions are then congruent. Now, since every spacelike curve may\nbe parametrized as a graph over the xaxis and the lightlike curves are vertical lines,\nwe conclude that it is not possible to assign a geometric invariant which distinguishes\nthose curves.\nDespite all these technical issues, the pseudo-torsion is powerful enough by itself\nto classify all lightlike and semi-lightlike curves in L3up to Poincaré transformations.\nTheorem 2.18. Letd:I!Rbe a continuous function, p02L3, s0,f02I and (T0,N0,B0)\na positive basis for L3such that B0is a lightlike vector and (T0,N0)is a positive basis for a\nlightlike plane. Then:\n(i) if T0is lightlike, N0is unit spacelike and hT0,B0iL=\u00001, there is a unique lightlike\ncurve a:I!L3with arc-photon parameter such that\n•a(f0) =p0;\n•(Ta(f0),Na(f0),Ba(f0)) = ( T0,N0,B0);\n•da(f) =d(f)for all f2I.\n(ii) if T0is unit spacelike, N0is lightlike andhN0,B0iL=\u00001, there is a unique unit speed\nsemi-lightlike curve a:I!L3such that\n•a(s0) =p0;\n•(Ta(s0),Na(s0),Ba(s0)) = ( T0,N0,B0);\n•da(s) =d(s)for all s2I.\n3Recall here that if g:I!R2is a regular plane curve in the Euclidean plane, not necessarily with\nunit speed, then its curvature is given by kg(t) =det(g0(t),g00(t))/kg0(t)k3.\nPage 33OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProof: We will treat case (i). In a similar way done in the proof of the classical version\nof this result in R3, consider the following initial-value-problem in R9:\n8\n>>><\n>>>:0\nB@T0(f)\nN0(f)\nB0(f)1\nCA=0\nB@0 1 0\nd(f) 0 1\n0d(f)01\nCA0\nB@T(f)\nN(f)\nB(f)1\nCA\ne\u0000\nT(f0),N(f0),B(f0)\u0001=\u0000\nT0,N0,B0\u0001\n.\nSuch a system of linear ordinary differential equations has a unique globally defined\nsolution\u0000\nT(f),N(f),B(f)\u0001\n. We claim that this solution still satisfies, for all f2I, the\nsame conditions as in f0. Namely, we will have that T(f)andB(f)are lightlike, N(f)\nis unit spacelike and Lorentz-orthogonal to B(f), andhT(f),B(f)iL=\u00001. To wit, we\nnow consider the following initial-value-problem for a:I!R6:\n(\na0(f) = A(f)a(f),\na(f0) =\u0000\n0, 1, 0, 0,\u00001, 0\u0001\n,\nwhere\nA(f) =0\nBBBBBB@0 0 0 2 0 0\n0 0 0 2 d(f) 0 2\n0 0 0 0 0 2 d(f)\nd(f) 1 0 0 1 0\n0 0 0 d(f) 0 1\n0d(f)1 0 d(f) 01\nCCCCCCA.\nIf the components of a(f)are all the possible products between the frame vectors4\nT(f),N(f)andB(f), we conclude that the unique solution with the given initial\nvalues is the constant vector a0=\u0000\n0, 1, 0, 0,\u00001, 0\u0001\n, from where the claim follows. We\nmay then define\na(f).=p0+Zf\nf0T(x)dx.\nTo finish the proof, we must verify that this ais lightlike, has arc-photon parameter,\nandda=d. Clearly we have a(f0) =p0anda0(f) =T(f), whence ais lightlike.\nDifferentiating again, we obtain a00(f) =N(f), so that ahas an arc-photon parameter.\nThis way, Ta(f) =T(f)andNa(f) =N(f), and the positivity of these bases ensure\nthatBa(f) =B(f)too. Now, differentiating Na(f) =N(f)yields\nda(f)Ta(f) +Ba(f) =d(f)T(f) +B(f),\nand from all the equalities seen so far it follows that da(f) =d(f)for all f2I. The\nuniqueness of such ais verified in the same way as in the proof of the classical theorem:\nthe Cartan Trihedron for another curve bwill satisfy the same initial-value-problem,\nimplying that Ta=Tb, and so a(f0) =b(f0)gives a=b.\nCorollary 2.19. Two curves, both lightlike or semi-lightlike and with the same pseudo-torsion,\nwhose osculating planes are positively oriented, are congruent by a positive Poincaré transfor-\nmation of L3.\n4In order, a=\u0000hT,TiL,hN,NiL,hB,BiL,hT,NiL,hT,BiL,hN,BiL\u0001\n.\nPage 34OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n2.3 Lancret’s theorem and classification of helices\nHere, as an application of the fundamental theorems seen so far, we can classify\nhelices in R3\nn. InR3, you should remember that a helix is a curve admiting a direction\nwhich makes a constant angle with all the curve’s tangent lines. In L3, a priori we can\nonly speak of the hyperbolic angle between two timelike vectors pointing both to the\nfuture or to the past, defined just after Proposition 1.26 (p. 15). We would like to work\nwith a definition of helix that works on both ambients simultaneously. Here’s one:\nDefinition 2.20. Leta:I!R3\nnbe a regular curve. We will say that ais ahelix if there\nis a non-zero vector v2R3\nnsuch thathTa(t),viis constant. Furthermore, in L3, we\nwill say that the helix is\n(i)hyperbolic ifvis spacelike;\n(ii)elliptic ifvis timelike;\n(iii) parabolic ifvis lightlike.\nThe direction defined by vis called the helical axis ofa.\nFor admissible curves, we have the:\nTheorem 2.21 (Lancret) .Leta:I!R3\nnbe a unit speed admissible curve. Then ais a helix\nif and only if the ratio ta(s)/ka(s)is constant.\nProof: Assume that ais a helix whose helical axis is given by a vector v. If we define\nc.=hTa(s),vi, thenhka(s)Na(s),vi=0 readily implieshNa(s),vi=0, since we have\nka(s)6=0. Differentiating again, we get\n\u0000eahaka(s)c+ta(s)hBa(s),vi=0.\nTo see that the ratio ta(s)/kais constant, it suffices to verify that hBa(s),viis a non-\nzero constant. To wit:\nd\ndshBa(s),vi= (\u00001)n+1eata(s)hNa(s),vi=0.\nNow, ifhBa(s),vi=0 for all s, then c=0, and orthonormal expansion yields v=0,\ncontradicting the definition of helix. Hence ta(s)/ka(s)is a constant.\nConversely, assume that ta(s) = cka(s), for some c2R. Ifc=0 then ais a plane\ncurve and then Ba(s) =Bdefines the helical axis for a. Ifc6=0, we seek a constant\nvector\nv=v1(s)Ta(s) +v2(s)Na(s) +v3(s)Ba(s)\nsuch thathTa(s),viis also constant. This condition, in turn, is equivalent to v1(s) =v1\nbeing constant. Differentiating the expression for vgives us that\n0=\u0000eahaka(s)v2(s)Ta(s)\n+\u0010\nv1ka(s) +v0\n2(s) + (\u00001)n+1eacka(s)v3(s)\u0011\nNa(s)\n+\u0000\ncka(s)v2(s) +v0\n3(s)\u0001\nBa(s).\nPage 35OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nNow, linear independence implies that\n8\n><\n>:0=\u0000eahaka(s)v2(s)\n0=v1ka(s) +v0\n2(s) + (\u00001)n+1eacka(s)v3(s),\n0=cka(s)v2(s) +v0\n3(s).\nand hence\nv2(s) =0 e v3(s) =(\u00001)n\nceav1.\nEffectively, we have parametrized the helical axis for a, using v1as the real parameter.\nFor example, setting v1=1 we may see that\nv.=Ta(s) +(\u00001)n\nceaBa(s)\ndefines the helical axis for a.\nRemark.\n• In particular, this proof ensures the existence of precisely one helical axis for a\ngiven helix.\n• Ifais a parabolic helix, then ta(s) =\u0006ka(s). The converse holds provided that\nha=1.\nCorollary 2.22. A unit speed admissible helix a:I!R3\nnwith both constant curvature and\nconstant torsion is congruent, for a certain choice of a ,b2R, to a piece of precisely one of the\nfollowing standard helices :\n•b1(s) =\u0000\nacos(s/c),asin(s/c),bs/c\u0001\n;\n•b2(s) =\u0000\nacos(s/c),asin(s/c),bs/c\u0001\n;\n•b3(s) =\u0000\nbs/c,acosh(s/c),asinh(s/c)\u0001\n;\n•b4(s) =\u0000\nbs/c,asinh(s/c),acosh(s/c)\u0001\n;\n•b5(s) =\u0000\nas2/2,a2s3/6,s+a2s3/6\u0001\n;\n•b6(s) =\u0000\nas2/2,s\u0000a2s3/6,\u0000a2s3/6\u0001\n,\nwhere b1is seen in R3, the remaining ones in L3, c.=p\na2+b2forb1andb4, and\nc.=p\nja2\u0000b2jforb2andb3;\nProof: Let’s denote the curvature and torsion of a, respectively, by kandt. Ifais seen\ninR3, then it is congruent to b1. We focus then on what happens in L3. One vector\nspanning the helical axis is\nv=Ta(s)\u0000eak\ntBa(s),\nwhencehv,viL=ea\u0000\n1\u0000hak2/t2\u0001\n. In general, the causal type of all the curves given\nin the statement of the result is determined by the constants aand b. Thus, a timelike\nhelix is:\nPage 36OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n• hyperbolic if k>jtj, and hence congruent to b3;\n• elliptic if k<jtj, and hence congruent to b2, and;\n• parabolic if k=jtj, and hence congruent to b5.\nSimilarly, a spacelike helic with timelike normal is necessarily hyperbolic, and so it is\ncongruent to b4. Lastly, a spacelike helix with timelike binormal is:\n• hyperbolic if k<jtj, and hence congruent to b3;\n• elliptic if k>jtj, and hence congruent to b2, and;\n• parabolic if k=jtj, and hence congruent to b6.\nRemark. In each case above, it is possible to find out what aand bshould be in terms\nofkandt. Have fun (or not).\nNow, we move on to non-admissible curves. Since every semi-lightlike curve is\nplane, it is automatically a helix. For lightlike curves the situation becomes interesting\nagain, and we have the:\nTheorem 2.23 (Lancret, lightlike version) .Leta:I!L3be a lightlike curve with arc-\nphoton parameter. Then ais a helix if and only if its pseudo-torsion dais constant.\nProof: Assume that ais a helix and let v2L3define the helical axis. Namely, v\nis such thathTa(f),viL=c2Ris constant. Differentiating that relation twice we\ndirectly obtain\nhNa(f),viL=da(f)c+hBa(f),viL=0\nfor all f2I. We claim that c6=0 and thathBa(f),viLis constant, whence da(f)is\nalso constant. To wit, if c=0 then Lemma 2.11 (p. 27) says that v=0, contradicting\nthe definition of helix. Moreover, we have\nd\ndfhBa(f),viL=da(f)hNa(f),viL=0.\nConversely, assume that da(f) =dis a constant. If d=0, then v=Ba(f)defines\nthe helical axis for a. Ifd6=0, define\nv.=Ta(f)\u00001\ndBa(f).\nIndeed, we have that\ndv\ndf=Na(f)\u00001\nddNa(f) =0\nso that vis constant. It follows that hTa(f),viL=1/dis constant, as wanted.\nCorollary 2.24. A lightlike helix a:I!L3is congruent, for a certain choice of r >0, to a\npiece of precisely one of the following standard helices :\nPage 37OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n•g1(f) =\u0000prf,rcosh(f/pr),rsinh(f/pr)\u0001\n;\n•g2(f) =\u0000\nrcos(f/pr),rsin(f/pr),prf\u0001\n;\n•g3(f) =\u0012\n\u0000f3\n4+f\n3,f2\n2,\u0000f3\n4\u0000f\n3\u0013\n.\nProof: Let’s denote the constant pseudo-torsion of asimply by d. We know from the\nprevious proof that a vector defining the helical axis of aifd6=0 is\nv=Ta(f)\u00001\ndBa(f),\nwhencehv,viL=2/d, while we may take v=Ba(f)ifd=0 (and hencehv,viL=0).\nThus, we have that ais\n• hyperbolic if d>0, and hence congruent to g1;\n• elliptic if d<0, and hence congruent to g2, and;\n• parabolic if d=0, and hence congruent to g3.\nPage 38OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProblems\nProblem 11. Leta:I!Lnbe a timelike future-directed curve (i.e., each a0(t)is future-\ndirected), and a,b2Iwith a<b. Show that:\n(a) the difference a(b)\u0000a(a)is timelike and future-directed.\n(b)Zb\naka0(u)kLdu\u0014ka(b)\u0000a(a)kL, and equality holds if and only if the image of\nthe restriction aj]a,b[is the line segment joining a(a)anda(b). What does this\nmean physically?\nHint. In (a), write a= (b,xn), where b:I!Rn\u00001, and estimatekb(b)\u0000b(a)kE. For\n(b), use the backwards Cauchy-Schwarz inequality (Proposition 1.26, p. 15).\nProblem 12. Leta:I!L3be a unit speed admissible curve. Show that ais a plane\ncurve if and only if ta=0.\nProblem 13. Leta:I!Lnbe a lightlike curve, and suppose that ea1:J1!Lnand\nea2:J2!Lnare two arc-photon reparametrizations of a, so thatea1(f1(t)) =ea2(f2(t)).\nShow that f1(t) =f2(t) +afor some a2R. What is the meaning of the constant a?\nProblem 14. Check the remaining case ea=1 and ha=0 mentioned in the proof of\nProposition 2.10 (p. 26).\nProblem 15. Find the Cartan Trihedron and the pseudo-torsion of a:R!L3given\nby\na(f) =\u0012p\nrf,rcosh\u0012fpr\u0013\n,rsinh\u0012fpr\u0013\u0013\n,\nwhere r>0 is fixed.\nProblem 16. Work through the proof of case (ii) in Theorem 2.18 (p. 33).\nProblem 17. Show Corollary 2.19 (p. 34).\nProblem 18. Show that every semi-lightlike curve a:I!L3with non-zero constant\npseudo-torsion da(s) =d6=0, contained in the plane P:y=z, is of the form\na(s) =\u0012\n\u0006s+a,b\nd2eds+cs+d,b\nd2eds+cs+d\u0013\n,\nfor some constants a,b,c,d2R(perhaps up to reparametrization).\nPage 39OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n3 Surface theory\n3.1 Causal characters (once more) and curvatures\nThe usual definition of a regular surface in R3(embedded, with no self-intersections)\ndoes not depend whatsoever of the product h\u0001,\u0001iE, and so it still makes perfect sense\ninL3. This way, all the theory regarding the topology and calculus on surfaces is still\nvalid and applicable here. In particular, we assume known:\n• that inverse images of regular values of real-valued smooth functions on R3are\nregular surfaces;\n• what is the tangent plane to a surface at any given point;\n• what is the differential of a smooth function defined in a surface, as well as what\nare its partial derivatives computed with respect to a given coordinate chart.\nFor example, since 1 and \u00001 are both regular values of the scalar square function\nF:L3!Rgiven by F(p) =hp,piL, we conclude that the de Sitter space S2\n1=F\u00001(1)\nand the hyperbolic plane H2(the upper connected component of F\u00001(\u00001)) are regular\nsurfaces:\n(a)S2\n1\n(b)H2[H2\u0000\nFigure 10: The “spheres” in L3.\nThe producth\u0001,\u0001iLcomes into play when we want to generalize the notion of causal\ncharacter to surfaces:\nDefinition 3.1. LetM\u0012L3be a regular surface. We’ll say that Mis:\n(i)spacelike if, for all p2M,TpMis a spacelike plane;\n(ii)timelike if, for all p2M,TpMis a timelike plane;\n(iii) lightlike if, for all p2M,TpMis a lightlike plane.\nIn particular, we’ll say that Misnon-degenerate if no tangent plane TpMis lightlike\n(and degenerate otherwise). In this case, the indicator eMofMwill be\u00001 or 1 according\nto whether Mis spacelike or timelike.\nPage 40OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nExample 3.2.\n(1) Let U\u0012R2be open, and f:U!Rbe a smooth function. The graph gr (f)is:\n• spacelike, ifkrFk<1;\n• timelike, ifkrFk>1;\n• lightlike, ifkrFk=1.\nHere’s a picture:\n1timelikelightlike\nrfspacelike\nFigure 11: Finding the causal character of graphs over the plane z=0.\n(2) Let F:L3!Rbe a smooth function, a2Ra regular value for F, and M=F\u00001(a)\na level surface. Then it follows from Theorem 1.9 (p. 7) that Mis spacelike (resp.\ntimelike, lightlike) if and only if the usual gradient rFis always timelike (resp.\nspacelike, lightlike).\n(3) If a:I!L3is a smooth, regular and injective curve whose trace lies in the plane\ny=0 but does not touch the z-axis, then we obtain a regular surface Mby rotating\naaround the z-axis. The causal character of Mis the same one as a’s. One can\nunderstand this by noting that the parallels of revolution are always spacelike, so\nthe only way of obtaining a lightlike or timelike direction comes from a possible\ncontribution from a.\nFigure 12: Spanning a surface of revolution in L3.\nPage 41OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nWe have topological restrictions on the causal character of a surface:\nProposition 3.3. There is no compact regular surface with constant causal character in L3.\nProof: LetM\u0012L3be a compact regular surface. By compactness, both projections\nM3(x,y,z)7!x2Rand M3(x,y,z)7!z2R\nadmit critical points in M, say, pandq. Then TpMis timelike, while TqMis spacelike.\nJust like for regular surfaces in R3, we will say that the restriction of h\u0001,\u0001iLto the\ntangent planes of a regular surface M\u0012L3is its First Fundamental Form . If there’s\na first form, there should be at least a second one too. And as we might recall, for\nthis we needed some orientability condition. So we’ll say that a Gauss map for a non-\ndegenerate regular surface M\u0012L3is a smooth choice of unit normal vectors along\nM, that is, a smooth map N:M!L3such thatkN(p)kL=1 and N(p)?TpM,\nfor all p2M. For surfaces in R3, the codomain of a Gauss map is automatically the\nsphere S2, but in L3this depends on the causal character of M. Namely, the codomain\nofNis the de Sitter space S2\n1ifMis timelike, while it is the hyperbolic plane H2if\nMis spacelike with Nfuture-directed (or it’s reflection through the plane z=0 if\nNis past-directed). Moreover, we see that if Mhas a fixed causal character, then\neM=hN(p),N(p)iL. This is useful for keeping track of the correct signs for some\nformulas we’ll soon deduce.\nTo understand how a non-degenerate surface Mbends in space L3near a point\np2M, we may focus on the “linear approximation” to Matp: the tangent plane TpM.\nUnderstanding how the tangent planes change near pis the same as understanding\nhow their orthogonal complements N(p)change. The motto\n“rate of change = derivative”\nleads to the:\nDefinition 3.4. LetM\u0012L3be a non-degenerate regular surface, and Nbe a Gauss\nmap for M. The Weingarten operator forMatpis the differential\u0000dNp:TpM!TpM.\nThe Second Fundamental Form ofMatpis the bilinear map I I p:TpM\u0002TpM!(TpM)?\ncharacterized by the relation hI Ip(v,w),N(p)iL=h\u0000dNp(v),wiL, for all v,w2TpM.\nItsscalar versioneI Ipis just this common quantity, that is, eI Ip(v,w) =hI Ip(v,w),N(p)iL.\nRemark. Note that if Mis spacelike, then TpM\u0018=TN(p)(H2), since both planes are\nthe Lorentz-orthogonal complement of N(p). Similarly, is Mis timelike, for the same\nreason we have TpM\u0018=TN(p)(S2\n1), and this is why we may regard \u0000dNpas a linear\noperator in TpM. The negative sign, by the way, is meant to reduce signs in further\nformulas, is notrelated to the ambient L3, and appears naturally in the context of\nsubmanifold theory in pseudo-Riemannian geometry, in general.\nOne can prove, just like in R3, that d Npis a self-adjoint operator with respect\ntoh\u0001,\u0001iL, so that both I I pandeI Ipare symmetric. We will conclude this preliminary\ndiscussion by giving precise definitions of “curvature” and formulas for expressing\nthem in terms of a parametrization of the surface.\nPage 42OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nDefinition 3.5. LetM\u0012L3be a non-degenerate regular surface. The mean curvature\nvector and the Gaussian curvature ofMat a point p2Mare defined by\nH(p).=1\n2trh\u0001,\u0001iL(I Ip) =1\n2(ev1I Ip(v1,v1) +ev2I Ip(v2,v2))and\nK(p).=\u0000deth\u0001,\u0001iL(eI Ip) =\u0000det\u0000(I Ip(vi,vj))2\ni,j=1\u0001\n,\nwhere (v1,v2)is any orthonormal basis for TpM.\nRemark.\n• The negative sign in the definition of Kaccounts for the loss of information we\nhave when considering eI I instead of I I there.\n• If we write H(p) = H(p)N(p),H(p)is called the mean curvature ofMatp.\nChoosing the opposite Gauss map changes the sign of H, but not of H.\nStill assuming this whole setup, we recall the classical notation for the coefficients\nof the fundamental forms. If x:U!x[U]\u0012Mis a parametrization, then we set\nE.=\u001c¶x\n¶u,¶x\n¶u\u001d\nL,F.=\u001c¶x\n¶u,¶x\n¶v\u001d\nLand G.=\u001c¶x\n¶v,¶x\n¶v\u001d\nL,\nas well as\ne.=\u001c¶2x\n¶u2,N\u000ex\u001d\nL,f.=\u001c¶2x\n¶u¶v,N\u000ex\u001d\nLand g.=\u001c¶2x\n¶v2,N\u000ex\u001d\nL,\nso that (with a mild abuse of notation) we have\nI I\u0012¶x\n¶u\u0013\n=eMeN, I I\u0012¶x\n¶u,¶x\n¶v\u0013\n=eMfN, and I I\u0012¶x\n¶v\u0013\n=eMgN.\nTo produce orthonormal bases for the tangent planes to M, needed for computing\nHand Kvia the definitions, the Gram-Schmidt process comes to rescue. We obtain\nsimilar formulas for the ones in R3, which now take into account the causal character\nofMitself:\nProposition 3.6. Let M\u0012L3be a non-degenerate regular surface, and x:U!x[U]\u0012M\na parametrization for M. Then\nH\u000ex=eM\n2Eg+eG\u00002F f\nEG\u0000F2and K\u000ex=eMeg\u0000f2\nEG\u0000F2.\nDo note that setting eM=1 ifM\u0012R3, the above gives also correct results for the\nmean and Gaussian curvatures of M. The details of these maybe-not-so-short calcula-\ntions may be consulted, for example, in [27]. They also follow from the more general\ntheory developed in [23]. Here are some more examples:\nPage 43OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nExample 3.7.\n(1) Planes admit a constant Gauss map, so the Weingarten operator vanishes. Hence\nwe get H=K=0.\n(2) The position map N(p) =pis a Gauss map for both the de Sitter space S2\n1and the\nhyperbolic plane H2. Taking the causal characters into account, we obtain K=1\nand H=\u00001 forS2\n1, and K=\u00001 and H=1 for H2. If you studied anything\nabout hyperbolic geometry before, this serves both as a quick sanity check (hyper-\nbolic plane should have negative curvature) as well as another justification for the\npresence of the minus sign in the definition of K.\n(3) If f:U\u0012R2!Ris a smooth function for which the graph gr (f)\u0012L3is non-\ndegenerate, by applying the coordinate formulas given in Proposition 3.6 (p. 43),\nwe obtain\nK=f2\nuv\u0000fuufvv\n(\u00001+f2u+f2v)2and H=fuu(\u00001+f2\nv)\u00002fufvfuv+fvv(\u00001+f2\nu)\nj\u00001+f2u+f2vj3/2.\n3.2 The Diagonalization Problem\nWe know from linear algebra the Real Spectral Theorem : that if (V,h\u0001,\u0001i)is a finite-\ndimensional real vector space equipped with a positive-definite inner product, and\nT:V!Vis a linear operator which is self-adjoint with respect to h\u0001,\u0001i, then Vadmits\nan orthonormal basis of eigenvectors of T. This result is no longer true if h\u0001,\u0001iis not\npositive-definite, and non-degeneracy alone is not strong enough to ensure any good\nconclusions. There is one adaptation, though: if dim V\u00153 andhT(v),vi6=0 for all\nnon-zero v2Vwithhv,vi=0, then Vadmits an orthonormal basis of eigenvectors\nofT. A very surprising proof using integration and homotopy, due to Milnor, may be\nfound in [13].\nWe have seen that the Weingarten operator of any non-degenerate surface M\u0012L3\nis still self-adjoint with respect to the First Fundamental Form of M. So we conclude\nthat if Mis spacelike, then \u0000dNpis diagonalizable: the eigenvalues k1(p)andk2(p)\nare called the principal curvatures ofMatp, and the (orthogonal) eigenvectors are\ncalled the principal directions ofMatp. We cannot guarantee the existence of prin-\ncipal directions for timelike surfaces in M, even with the sharpened version of the\nSpectral Theorem mentioned above, since dim TpM=2<3.\nThat being said, our goal here is to understand precisely when do we have princi-\npal directions for timelike surfaces in L3.\nProposition 3.8. Let M\u0012L3be a non-degenerate regular surface with diagonalizable Wein-\ngarten operators. Then\nH(p) =eMk1(p) +k2(p)\n2and K (p) =eMk1(p)k2(p).\nRemark. Usually one defines Hand Kfor surfaces in R3by the above formulas (set-\ntingeM=1, of course). The reason why we went through the hassle of using metric\nPage 44OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\ntraces and determinants to define them in L3was just so we could have a unified\napproach that worked in all the cases simultaneously, even when we could not use\nprincipal curvatures. Also note that the expression for Hjustifies the name “mean”\ncurvature.\nWe might as well start understanding a class of surfaces which, in general, have\ndiagonalizable Weingarten operators.\nDefinition 3.9. LetM\u0012L3be a non-degenerate regular surface, and p2M. The\npoint pis called umbilic if there is l(p)2Rsuch that\neI Ip(v,w) =l(p)hv,wi,\nfor all v,w2TpM. We will also say that Mistotally umbilic if all its points are umbilic.\nInformally, a point is umbilic if there the two fundamental forms of Mare “linearly\ndependent”. In umbilical points, we have \u0000dNp=l(p)IdTpM. Indeed, for all vec-\ntorsv,w2TpMwe have thathl(p)v,wiL=eI Ip(v,w) =h\u0000dNp(v),wiL, and the\nconclusion follows from non-degeneracy of h\u0001,\u0001iLrestricted to TpM.\nYou might remember from the classical theory in R3that there, the only totally\numbilic surfaces are spheres and planes. Since the de Sitter space S2\n1and the hyperbolic\nplane H2(together with its reflection H2\u0000) play the role of spheres in L3, the following\nresult (with the same proof as in R3) should not be a surprise:\nTheorem 3.10 (Characterization of totally umbilic surfaces in L3).Let M\u0012L3be a\nnon-degenerate, regular, connected and totally umbilic surface. Then M is contained in some\nplane, or there is a center c2R3\nnand a radius r >0such that\n(i) if M is spacelike, then M \u0012H2(c,r)or M\u0012H2\u0000(c,r);\n(ii) if M is timelike, then M \u0012S2\n1(c,r).\nRemark. Here, we mean S2\n1(c,r) =fp2L3jhp\u0000c,p\u0000ciL=r2g, etc.. Moreover,\nin the timelike case, what decides between H2(c,r)orH2\u0000(c,r)is the direction of the\ntimelike vector p\u0000cfor some (hence all) p2M(due to connectedness).\nFigure 13: The totally umbilic surfaces in L3.\nPage 45OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nBack to the diagonalization problem. Let’s see necessary conditions for an affirma-\ntive answer to the problem.\nProposition 3.11. Let M\u0012L3be a non-degenerate regular surface, and p2M such that\nthe Weingarten operator at pis diagonalizable. Then H (p)2\u0000eMK(p)\u00150, with equality\nholding if and only if pis umbilic.\nProof: Directly, we have:\n0\u0014\u0012k1(p)\u0000k2(p)\n2\u00132\n=k1(p)2\u00002k1(p)k2(p) +k2(p)2\n4\n=k1(p)2+2k1(p)k2(p) +k2(p)2\n4\u0000k1(p)k2(p)\n=\u0012k1(p) +k2(p)\n2\u00132\n\u0000k1(p)k2(p)\n= (eMH(p))2\u0000eMK(p) =H(p)2\u0000eMK(p).\nEquality holds if and only if k1(p) =k2(p), that is to say, if pis umbilic.\nSo we have a necessary, but not sufficient condition for the diagonalizability of the\nWeingarten operators. What we can see, though, is that the quantity H(p)2\u0000eMK(p)\nwill play a big role in our analysis, which will be done in full detail in the proof of the\ndesired:\nTheorem 3.12 (Diagonalization in L3).Let M\u0012L3be a non-degenerate regular surface,\nNa Gauss map for M, and p2M. Then:\n(i) if H (p)2\u0000eMK(p)>0,\u0000dNpis diagonalizable;\n(ii) if H (p)2\u0000eMK(p)<0,\u0000dNpis not diagonalizable;\n(iii) if H (p)2\u0000eMK(p) = 0and M is spacelike, then pis umbilic, and hence \u0000dNpis\ndiagonalizable.\nRemark. IfH(p)2\u0000eMK(p) = 0 and Mis timelike, the criterion is inconclusive and\nthe Weingarten operator may or may not be diagonalizable.\nProof: Consider the characteristic polynomial c(t)of\u0000dNp, given by\nc(t) =t2\u0000tr(\u0000dNp)t+det(\u0000dNp) =t2\u00002eMH(p)t+eMK(p),\nwhose discriminant is:\n(\u00002eMH(p))2\u00004(eMK(p)) = 4(H(p)2\u0000eMK(p)).\n• If H(p)2\u0000eMK(p)>0, then c(t)has two distinct roots, which are the eigen-\nvalues of\u0000dNp, who then admits two linearly independent eigenvectors (hence\ndiagonalizable).\nPage 46OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n• If H(p)2\u0000eMK(p)<0,c(t)does not have any real roots. Thus \u0000dNphas no\nreal eigenvalues, and hence it is not diagonalizable.\n• Now assume that H(p)2\u0000eMK(p) = 0 and that Mis spacelike, that is, that\nK(p) =\u0000H(p)2. From the expression given for the discriminant of c(t), it fol-\nlows that\u0000H(p)is an eigenvalue of \u0000dNp. So, there is a unit (spacelike) vector\nu12TpMsuch that d Np(u1) = H(p)u1. Consider then an orthogonal basis\nB.= (u1,u2)ofTpM. Then:\n\u0002\ndNp\u0003\nB=\u0012H(p)a\n0 b\u0013\n, where d Np(u2) =au1+bu2.\nIt suffices to check that a=0 and b=H(p)to conclude the proof. Applying\nh\u0001,u1iL, we have:\na=hdNp(u2),u1iL=hu2, dNp(u1)iL=hu2,H(p)u1iL=H(p)hu2,u1iL=0.\nOn the other hand:\n\u0000H(p)2=K(p) =\u0000det(\u0000dNp) =\u0000det(dNp) =\u0000H(p)b,\nso that H(p)b=H(p)2. IfH(p) =0, then d Npis the zero map (hence diagonal-\nizable). If H(p)6=0, we obtain b=H(p), as wanted. Note that in this case pis\numbilic.\nObserve that in the above proof, we would not be able to control the causal type\nof the eigenvector u1in the last case discussed if Mwere timelike. If u1were lightlike,\nwe could not consider the basis Bto proceed with the argument. With this in mind,\nwe obtain the following extension of the theorem:\nCorollary 3.13. Let M\u0012L3be a timelike regular surface and p2M be a point with\nH(p)2\u0000K(p) =0. If\u0000dNphas no lightlike eigenvectors, then it is diagonalizable and pis\numbilic, with both principal curvatures equal to \u0000H(p).\nLet’s conclude the section exploring examples of timelike surfaces for which the\nequality H(p)2=K(p)holds and anything can happen with the Weingarten opera-\ntors.\nExample 3.14.\n(1) For the de Sitter space S2\n1, we had\u0000dNp=\u0000IdTp(S2\n1)(hence diagonalizable), with\nK=1 and H=\u00001, so that H2\u0000K=0.\n(2) Consider a lightlike curve a:I!L3with arc-photon parameter. Define the B-\nscroll associated to a,x:I\u0002R!L3given by x(f,t).=a(f) +tBa(f). Restricting\nenough the domain of x, we may assume that its image Mis a regular surface. Put,\nfor each f2I,D(f).=det\u0000\nTa(f),Na(f),Ba(f)\u0001>0. Computing the derivatives\nxf(f,t) =Ta(f) +tda(f)Na(f)and xt(f,t) =Ba(f),\nPage 47OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nwe immediately have that\n(gij(f,t))1\u0014i,j\u00142=\u0012\nt2da(f)2\u00001\n\u00001 0\u0013\n,\nwhence Mis timelike. Here, gijis shorthand for the coefficients of the First Funda-\nmental Form. Noting that jdet((gij(f,t))1\u0014i,j\u00142)j=1, we directly obtain that\nN(x(f,t)) = Ta(f)\u0002LBa(f) +tda(f)Na(f)\u0002LBa(f).\nComputing the second order derivatives\nxff(f,t) =tda(f)2Ta(f) + ( 1+td0\na(f))Na(f) +tda(f)Ba(f),\nxft(f,t) =da(f)Na(f)and\nxtt(f,t) =0,\nwe obtain the coefficients hijof the Second Fundamental Form:\n(hij(f,t))1\u0014i,j\u00142=\u0012(\u00001\u0000td0\na(f) +t2da(f)3)D(f)\u0000da(f)D(f)\n\u0000da(f)D(f) 0\u0013\n.\nIt follows that\nK(x(f,t)) =da(f)2D(f)2and H(x(f,t)) =da(f)D(f).\nWe then know that, in each point x(f,t),\u0000dNx(f,t)has only one eigenvalue (namely,\nda(f)D(f)). It suffices to check then that there are points in Mfor which the as-\nsociated eigenspace has dimension 1 – this shows that the Weingarten operators at\nthose points are not diagonalizable. To wit, we have\nh\n\u0000dNx(f,t)i\nBx=D(f)\u0012da(f) 0\n1+tda(f)da(f)\u0013\n,\nand the kernel of \u00120 0\n1+tda(f)0\u0013\nhas always dimension 1 when 1 +tda(f)6=0 (e.g., along aitself, setting t=0).\nPage 48OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProblems\nProblem 19. Work through Example 3.2 (p. 41).\nProblem 20 (Horocycles) .Letv2L3be a future-directed lightlike vector, and c<0.\nThe set Hv,c.=fx2H2j hx,viL=cgis called a horocycle of H2based on v. Let\na:I!Hv,cbe a unit speed curve (assume that 0 2I, reparametrizing if necessary).\n(a) Show that\na(s) =\u0000s2\n2cv+sw1+w2,\nfor some unit and orthogonal vectors w1andw2, with w1spacelike, w2timelike,\nhw1,viL=0 andhw2,viL=c.\n(b) Conclude that ais a semi-lightlike curve whose pseudo-torsion identically van-\nishes.\nProblem 21. Compute the Gaussian and mean curvatures for the surface of revolution\nspanned by a unit speed curve as in item (3) of Example 3.2 (p. 41).\nRemark. One can also study surfaces of revolution in L3generated by hyperbolic ro-\ntations about the x-axis instead of the timelike z-axis. See [27] for more about this.\nProblem 22. Leta:I!L3andb:J!L3be two smooth lightlike curves such that\nfa0(u),b0(v)gis linearly independent for all (u,v)2I\u0002J. Then, reducing Iand Jif\nnecessary, the image Mof the sum x:I\u0002J!L3given by x(u,v) =a(u) +b(v)is a\nregular surface. Show that Mis timelike with H=0.\nRemark. Actually, the “converse” holds: every timelike surface with H=0 admits\nparametrizations like this xabove. See [8] for more details.\nProblem 23. Prove Theorem 3.10 (p. 45).\nProblem 24. LetM\u0012L3be a non-degenerate regular surface, and N:M!L3a\nGauss map for M. Show that the Weingarten operator \u0000dNpis self-adjoint with re-\nspect toh\u0001,\u0001iL, for all p2M. Namely, show that given v,w2TpM, we have\nhdNp(v),wiL=hv, dNp(w)iL.\nHint. Use a parametrization of Mand do it locally.\nProblem 25. Make sure you understand how to obtain Corollary 3.13 (p. 47) by adapt-\ning the proof of Theorem 3.12 (p. 46).\nProblem 26. Consider the anti-de Sitter space H2\n1.=fp2R3\n2jhp,pi2=\u00001g. Try to\nunderstand how to translate the results discussed for the ambient L3for the ambient\nR3\n2and show that H2\n1has constant Gaussian curvature K=\u00001.\nPage 49OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nExtra #1: Riemann’s classification of surfaces with con-\nstant K\nUp to this moment, we know some surfaces with constant Gaussian curvature.\nNamely, we have met:\n• The planes R2andL2, with K=0;\n• The sphere S2and the de Sitter space S2\n1, with K=1;\n• The hyperbolic plane H2and the anti-de Sitter H2\n1, with K=\u00001.\nOur goal here is to show that, locally, every surface with constant Kis one of those\nsurfaces described above. More precisely, we want to prove the:\nTheorem 3.15 (Riemann) .Let(M,h\u0001,\u0001i)be a geometric surface with constant Gaussian\ncurvature K2f\u0000 1, 0, 1g. Then:\n(A) if the metric is Riemannian, every point in M has a neighborhood isometric to an open\nsubset of\n(i)R2, if K =0;\n(ii)S2, if K =1;\n(iii)H2, if K =\u00001,\n(B) while if the metric is Lorentzian, to an open subset of\n(i)L2, if K =0;\n(ii)S2\n1, if K =1;\n(iii)H2\n1, if K =\u00001.\nBygeometric surface , we mean an abstract surface (2-dimensional manifold) en-\ndowed with a metric tensor (called Riemannian if positive-definite, or Lorentzian if it\nhas index 1). The proof strategy consists in constructing parametrizations for which\nthe metric assumes a simple form. To actually do this, we will use geodesics , which are\nknow to be plentiful in any geometric surface.\nRecall here that given any regular parametrization x:U\u0012R2!x[U]\u0012Mof our\nsurface, we set gij=hxu,xvi, so that (gij)2\ni,j=1is the inverse matrix of (gij)2\ni,j=1, and the\nChristoffel symbols ofxare defined by\nGk\nij=2\nå\nr=1gkr\n2\u0012¶gir\n¶uj+¶gjr\n¶ui\u0000¶gij\n¶ur\u0013\n,\nwhere we identify u$u1,v$u2, and i,j,k2f1, 2g. Geodesics are curves g:I!M\nwith the property that given any parametrization xand writing g(t) =x(u(t),v(t)),\nwe have\n¨uk+k\nå\ni,j=1Gk\nij˙ui˙uj=0, k=1, 2.\nPage 50OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nFurther general facts about geodesics (which won’t be necessary here) can be con-\nsulted in pretty much any book (we list here [9], [23], [26] or [27], for concreteness). To\navoid singularities, we won’t consider lightlike geodesics.\nThus, we fix once and for all a geometric surface (M,h\u0001,\u0001i), with metric tensor of index\nn2f0, 1g, and a unit speed geodesic g:I!M. For each v2I, consider a unit speed\ngeodesic gv:Jv!M, which crosses gorthogonally at the point gv(0).=g(v). Setting\nU.=f(u,v)2R2jv2Ieu2Jvg,\ndefine x:U!x(U)\u0012Mbyx(u,v) =gv(u).\ngvg(v)g\nFigure 14: Construction of a Fermi chart x.\nDefinition 3.16. The chart xabove defined is called a Fermi chart forM, centered in g.\nWe’ll also fix until the end of the section this Fermi chart x:U!x(U)\u0012Mso\nconstructed.\nRemark.\n• Whenh\u0001,\u0001iis Lorentzian, we’ll have two types of Fermi charts, according to the\ncausal character of g. Moreover, recalling that geodesics have automatically con-\nstant causal character (hence determined by a single velocity vector), it follows\nthat if gis spacelike (resp. timelike), then all the gvare timelike (resp. spacelike),\nsincefg0(v),g0\nv(0)gis a orthonormal basis of Tg(v)M, for all v2I.\n• When necessary, if gis timelike, we might denote the coordinates by (t,J)in-\nstead of (u,v).\nProposition 3.17. The Fermi chart xis indeed regular in a neighborhood of f0g\u0002I (so that\nreducing U if necessary, we may assume that xitself is regular).\nProof: We’ll show that for all v2I, the vectors xu(0,v)andxv(0,v)are orthogonal.\nTo wit, we have by construction that\nhxu(0,v),xv(0,v)i=hg0\nv(0),g0(v)i=0.\nSince none of those vectors is lightlike, orthogonality implies linear independence. By\ncontinuity of x, the vectors xu(u,v)andxv(u,v)remain linearly independent for small\nenough values of u.\nPage 51OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProposition 3.18. The coordinate expression of h\u0001,\u0001iwith respect to the Fermi chart xis\nds2= (\u00001)negdu2+G(u,v)dv2.\nProof: All the gvare unit speed curves with the same indicator egv. We have that\nE(u,v) =hxu(u,v),xu(u,v)i=hg0\nv(u),g0\nv(u)i=egv.\nNow, egegv= (\u00001)nfor all v2I, whence E(u,v) = (\u00001)neg.\nProceeding, we see that by construction F(0,v) =0 for all v2I, so that it suffices to\ncheck that Fdoes not depend on the variable u. Fixed v02I, we have the expression\nx(u,v0) =gv0(u), and so the second geodesic equation for gv0yields G2\n11(u,v0) = 0.\nFrom the arbitrariety of v0it follows that G2\n11=0. On the other hand, by definition of\nG2\n11we have\nG2\n11(u,v) =(\u00001)neg\n(\u00001)negG(u,v)\u0000F(u,v)2Fu(u,v),\nso that Fu(u,v) =0, and we conclude that F(u,v) =0 for all (u,v)2U, as desired.\nRemark. Since G(0,v) =eg6=0, the continuity of Gallows us to assume, by reducing\nUagain if necessary, that G(u,v)has the same sign as egfor all (u,v)2U.\nCorollary 3.19. The Gaussian curvature of (M,h\u0001,\u0001i)is expressed in terms of the Fermi chart\nxby\nK\u000ex= (\u00001)n+1eg(p\njGj)uup\njGj.\nBefore starting the proof of Theorem 3.15 (p. 50), we only need to get one more\ntechnical lemma out of the way:\nLemma 3.20 (Boundary conditions) .The Fermi chart xsatisfies G u(0,v) =0, for all v2I.\nProof: Asg(v) =x(0,v), the first geodesic equation for gboils down to G1\n22(0,v) =0,\nfor all v2I. Since F(0,v) =0, it directly follows that\nG1\n22(0,v) =\u0000Gu(0,v)\n2eg,\nwhence Gu(0,v) =0, as desired.\nFinally:\nProof: [of Theorem 3.15] In all possible cases, the coefficient Gmust satisfy the follow-\ning differential equation:\n(q\njGj)uu+ (\u00001)negKq\njGj=0.\nNow, we solve this equation (in each case) forp\njGj, and use the boundary conditions\nG(0,v) =egand Gu(0,v) =0 to determine Gexplicitly.\nPage 52OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n(A) Assume thath\u0001,\u0001iis Riemannian.\n(i) For K=0, we have (p\nG)uu=0, and sop\nG(u,v) = A(v)u+B(v). The\nboundary conditions then give A(v) = 0 and B(v) = 1, so that G(u,v) = 1\nfor all (u,v)2U, and d s2=du2+dv2.\n(ii) When K=1, we have (p\nG)uu+p\njGj=0, whose solutions are of the\nformp\nG(u,v) = A(v)cosu+B(v)senu. Now, the boundary conditions\ngive A(v) = 1 and B(v) = 0, and so G(u,v) = cos2u, and it follows that\nds2=du2+cos2udv2: the metric in S2.\n(iii) If K=\u00001, the equation to be solved is (p\nG)uu\u0000p\njGj=0. We have\nthatp\nG(u,v) = A(v)eu+B(v)e\u0000u, and now the boundary conditions give\nA(v) = B(v) = 1/2, whence G(u,v) = cosh2uand we obtain the local ex-\npression d s2=du2+cosh2udv2. To recognize this in an easier way as the\nmetric in H2, we may let x=evtanh uand y=evsech u, so that\nds2=dx2+dy2\ny2,\nas desired.\n(B) Assume now that h\u0001,\u0001iis Lorentzian.\n(i) For K=0, just like above, we have d s2=\u0000du2+dv2=dt2\u0000dJ2.\n(ii) If K=1, we now have two cases to discuss. If gis spacelike, we again\nobtain (p\nG)uu\u0000p\nG=0, from where it follows that G(u,v) = cosh2uand\nwe get the S2\n1metric (expressed in the usual revolution parametrization):\nds2=\u0000du2+cosh2udv2.\nIfgis timelike instead, we have (p\n\u0000G)tt+p\n\u0000G=0, whose solution is\nG(t,J) =\u0000cos2t, and so d s2=dt2\u0000cos2tdJ2.\n(iii) If K=\u00001, the situation is dual to the previous one, switching “spacelike”\nand “timelike”, and also the signs of the metric expressions. Omitting re-\npeated calculations, we obtain\nds2=\u0000du2+cos2udv2=dt2\u0000cosh2tdJ2,\nwhich is the metric of H2\n1in suitable coordinates.\nWe will conclude the section by presenting surfaces in the ambients L3andR3\n2\nwhose metric’s coordinate expressions are the ones discovered in the proof above. For\nK=0 the situation is completely uninteresting. But for K6=0 we have the following:\nPage 53OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nExample 3.21.\n(1)K=1:\n• The metric d s2=\u0000du2+cos2udv2may be realized by the usual revolution\nparametrization x:R2!S2\n1\u0012L3given by\nx(u,v) = ( cosh ucosv, cosh usinv, sinh u),\nand also by y: cosh\u00001\u0000]1,p\n2[\u0001\u0002R!R3\n2given by\ny(u,v) =\u0012\ncosh ucosh v, cosh usinh v,Zu\n0q\n2\u0000cosh2tdt\u0013\n.\n• For d s2=dt2\u0000cos2tdJ2, consider x:]0, 2p[\u0002R!S2\n1\u0012L3given by\nx(t,J) = ( sint, costcoshJ, costsinhJ),\nand also by y:]\u0000p/2,p/2[\u0002R!R3\n2, given by\ny(t,J) =\u0012Zt\n0p\n1+sin2tdt, costcosJ, costsinJ\u0013\n.\nRemark. The periodicity condition y(t,J) = y(t+p,J)in the last given\nparametrization along with the fact that translations are isometries in R3\n2al-\nlow us to restrict everything to the given domains, which is maximal for non-\ndegenerability.\nTo summarize, when K=1 we have the following visualizations:\n(a) In L3\n(b) InR3\n2\nFigure 15: Constant Gaussian curvature K=1.\n(2)K=\u00001:\nPage 54OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n• The metric d s2=\u0000du2+cos2udv2may be realized by the parametrization\nx:]\u0000p/2,p/2[\u0002R!L3, given by\nx(u,v) =\u0012\ncosucosv, cos usinv,Zu\n0p\n1+sin2tdt\u0013\n,\nand also by y:]0, 2p[\u0002R!H2\n1\u0012R3\n2:\ny(u,v) = ( cosusinh v, cos ucosh v, sin u).\nIn this case, the same remark made for yin the case K=1 holds for xhere.\n• The metric d s2=dt2\u0000cosh2tdJ2may be realized by the parametrization\nx: cosh\u00001\u0000]1,p\n2[\u0001\u0002R!L3given by\nx(t,J) =\u0012Zt\n0q\n2\u0000cosh2tdt, cosh tcoshJ, cosh tsinhJ\u0013\nand by y:R\u0002]0, 2p[!H2\n1\u0012R3\n2,\ny(t,J) = ( sinht, cosh tcosJ, cosh tsinJ).\nSo in this case, we have:\n(a) InR3\n2\n(b) In L3\nFigure 16: Constant Gaussian curvature K=\u00001.\nLastly, we observe that the surfaces in the figures 15(a) and 16(a) are isometric when\nequipped by the metrics induced by R3, but on the pseudo-Riemannian ambients con-\nsidered, they have rotational symmetry along axes of distinct causal characters. The\nsame holds for the surfaces given in figures 15(b) and 16(b). Furthermore, note that S2\n1\nandH2\n1“fit better” in L3andR3\n2, respectively – switching the ambients require the use\nof parametrizations depending on certain elliptic integrals.\nPage 55OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProblems\nProblem 27. Prove Corollary 3.19 (p. 52).\nProblem 28. Show that if x=evtanh uand y=evsech u, then\ndu2+cosh2udv=dx2+dy2\ny2.\nProblem 29 (Riemann’s Formula) .Let(M,h\u0001,\u0001i)be a geometric surface equipped with\na Riemannian metric tensor, and x:U!x(U)\u0012Mbe a Fermi chart for M(on\nwhich the metric is expressed by d s2=du2+G(u,v)dv2). In some adequate domain,\nconsider the reparametrization x=ucosvand y=usenv. Show that\nds2=dx2+dy2+H(x,y)(xdy\u0000ydx)2,\nwhere H(x,y) = ( G(u,v)\u0000u2)/u4.\nRemark. The function Hmeasures, up to second order, how far is the metric from\nbeing Euclidean near the origin. The reason why is that one can show that if Hactually\nadmits a continuous extension to the origin, then the Gaussian curvature at the point\nwith coordinates (x,y) = ( 0, 0)is\u00003H(0, 0).\nProblem 30 (Revolution surfaces with constant K).Leta:I!R3\nnbe smooth, regular,\nnon-degenerate, injective and of the form a(u) = ( f(u), 0,g(u)), for certain functions\nfand gwith f(u)>0 for all u2I, and let Mbe the revolution surface spanned by a,\naround the z-axis. Assume that ahas unit speed, Mhas constant Gaussian curvature\nK, and consider the parametrization x:I\u0002]0, 2p[!I!x(U)\u0012Mgiven by\nx(u,v) = ( f(u)cosv,f(u)sinv,g(u)).\n(a) Show that, in general, fand gsatisfy\nf00(u) +eaK f(u) =0 and g(u) =Zq\n(\u00001)n(ea\u0000f0(u)2)du.\n(b) Verify that\nf(u) =8\n><\n>:Acos(peaKu) +Bsin(peaKu), se eaK>0\nAu+B, se K=0,\nAcosh(p\u0000eaKu) +Bsinh(p\u0000eaKu)seeaK<0,\nwhere in the case K=0 we necessarily have jAj\u0014 1 if the ambient is R3, while\njAj\u00151 if the curve is spacelike in L3(for timelike curves there are no restrictions).\n(c) Identify all the revolution surfaces with constant Gaussian curvature K2f\u0000 1, 0, 1g.\nPage 56OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nExtra #2: Weierstrass’s representation of critical surfaces\ninL3\nAn introduction to split-complex algebra\nWe start recalling a possible construction of the complex numbers: define in R2the\noperations\n(a,b) + ( c,d).= (a+c,b+d)and (a,b)(c,d).= (ac\u0000bd,ad+bc).\nSuch operations turn R2into a field, which is then denoted by C. Since we have the\nidentities (a,b) = ( a, 0) + ( b, 0)(0, 1)and(0, 1)2= (\u00001, 0), we may identify Rwith\nthe setf(a, 0)2R2ja2Rgand put i.= (0, 1), hence recovering the usual description\nC=fa+bija,b2Rand i2=\u00001g.\nGiven z=a+bi2C, the projections Re (z).=aand Im (z).=bare called the real and\nimaginary parts of z . The conjugate ofzis defined as z.=a\u0000bi, and the absolute value of\nzasjzj.=p\na2+b2=k(a,b)kE. Moreover, if z1=a1+b1iand z2=a2+b2iare two\ncomplex numbers, we have\nRe(z1z2) =h(a1,b1),(a2,b2)iE,\nwhich shows that Cencodes the geometry of the usual inner product in R2. One then\nproceeds to develop Calculus in a complex variable.\nOur goal here is to define a Lorentzian version of Cbased on the above review, and\nbriefly understand how calculus works in this new setting.\nDefinition 3.22 (Split-complex numbers) .The set C0of the split-complex numbers is the\nspace L2equipped with the operations\n(a,b) + ( c,d).= (a+c,b+d)and (a,b)(c,d).= (ac+bd,ad+bc).\nRemark. The split-complex numbers are also known as hyperbolic numbers . To justify\nthis terminology, work through Problem 31 in the end of the section.\nIt is easy to see that C0is a commutative ring with 1. Since this time we have the\nidentities (a,b) = ( a, 0) + ( b, 0)(0, 1)and(0, 1)2= (1, 0), we may again identify R\nwithf(a, 0)2L2ja2Rgand put h.= (0, 1)to obtain a similar description to the\nprevious one given for C:\nC0=fa+bhja,b2Rand h2=1g.\nDefinition 3.23. Letw=a+bh2C0.\n(i) The split-conjugate ofwis defined by w.=a\u0000bh.\n(ii) The split-complex absolute value ofwis given byjwj.=p\nja2\u0000b2j=k(a,b)kL.\nPage 57OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n(iii) The real part ofwis given by Re (w).=a, and its imaginary part is given by\nIm(w).=b.\nLet’s register some basic algebraic properties of C0in the following:\nProposition 3.24. Let w ,w1,w22C0.\n(i)w1+w2=w1+w2,w1w2=w1w2,w=w, and w=w if and only if w 2R. In\nfancier terms, conjugation in C0is still an involution preserving R;\n(ii) if 1/w exists, then 1/w=1/w;\n(iii)jwj=jwj,jwwj=jwj2;\n(iv)jw1w2j=jw1jjw2jand, if 1/w exists,j1/wj=1/jwj. In particular, if 1/w exists, we\nnecessarily havejwj6=0.\nTo justify that C0is indeed the Lorentzian version of Cthat we seek, note that if\nw1=a1+b1hand w2=a2+b2hare two split-complex numbers, then\nRe(w1w2) =h(a1,b1),(a2,b2)iL,\nwhich says that C0encodes the geometry of L2in the same way that Cdoes it for R2.\nThis also gives us a geometric interpretation for C0not being a field like C: the zero\ndivisors in C0correspond precisely to the lightlike directions in L2.\nWe proceed with some calculus. We endow C0with the usual topology of the plane.\nThat is to say, the open subsets of C0are the same ones as of C, and the overall notion\nof continuity is the same. In particular, if U\u0012C0is open and f:U0!C0is written in\nthe form\nf(x+hy) =f(x,y) +hy(x,y)\nfor some real-valued functions fandy, then fis continuous if and only if both fand\nyare.\nTo define holomorphicity in C0, we will again mimic the definition used in C, taking\ncare to not divide by “lightlike” directions:\nDefinition 3.25. LetU\u0012C0be an open set, w02Uand f:U!C0a function. We’ll\nsay that fisC0-differentiable at w0if the limit\nf0(w0).= lim\nDw!0\nDw62CL(0)f(w0+Dw)\u0000f(w0)\nDw\nexists. In this case, f0(w0)is called the derivative offatw0. And fis called split-\nholomorphic inw0if it isC0-differentiable in every point of some neighborhood of w0.\nThe usual rules hold:\nProposition 3.26. Let U\u0012C0be an open set, w 02U and f ,g:U!C0two\nC0-differentiable functions at w 0. Then\n(i) f+g isC0-differentiable at w 0and(f+g)0(w0) = f0(w0) +g0(w0).\nPage 58OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n(ii) f g is C0-differentiable at w 0and(f g)0(w0) =g(w0)f0(w0) +f(w0)g0(w0).\n(iii) if g does not assume any value in the lightlike directions of the plane, then f /g is\nC0-differentiable at w 0and(f/g)0(w0) = ( f0(w0)g(w0)\u0000f(w0)g0(w0))/g(w0)2.\nExample 3.27. Constant functions and the identity C0!C0are clearly split-holomor-\nphic. It follows that all polynomials are split-holomorphic, and its derivatives are\ngiven by the usual rules (e.g., the derivative of f(w) =w3+3w2isf0(w) =3w2+6w).\nThe same goes for rational functions, as long as the denominator does not take values\nin the lightlike directions of the plane.\nProposition 3.28 (Chain rule) .Let U 1,U2\u0012C0be open sets and f :U1!C0, g:U2!C0\nbe functions such that f (U1)\u0012U2. If f é C0-differentiable at w 0and g is C0-differentiable at\nf(w0), then g\u000ef isC0-differentiable at w 0and(g\u000ef)0(w0) =g0(f(w0))f0(w0)holds.\nIn the usual complex calculus, we know that the real and imaginary parts of a\nholomorphic function must satisfy the Cauchy-Riemann equations . InC0, we should\nexpect some sign change. Here’s what we get (with almost the same proof):\nProposition 3.29 (Revised Cauchy-Riemann) .Let U\u0012C0be an open set and fix w 02U.\nIf f:U!C0isC0-differentiable in w 0, and we write f (x+hy) =f(x,y) +hy(x,y), then\n¶f\n¶x(w0) =¶y\n¶y(w0)and¶f\n¶y(w0) =¶y\n¶x(w0).\nRemark. These revised equations may be expressed in a more concise way using split-\ncomplex versions of the so-called Wirtinger operators :\n¶\n¶w.=1\n2\u0012¶\n¶x+h¶\n¶y\u0013\nand¶\n¶w.=1\n2\u0012¶\n¶x\u0000h¶\n¶y\u0013\n.\nThe revised Cauchy-Riemann equations become only ¶f/¶w=0, in which case the\nformula f0(w) = ( ¶f/¶w)(w)holds.\nExample 3.30. Motivated by Euler’s formula ex+iy=ex(cosy+isiny)inC, we define\nexpC0:C0!C0by expC0(w) =ex(cosh y+hsinh y), where w=x+hy. We have that\nexpC0is split-holomorphic, with (expC0)0=expC0. When there is no risk of confusion,\none may simply write ew.\nAn important consequence of the revised Cauchy-Riemann equations is the ana-\nlogue in C0of the well-known fact that the real and imaginary parts of a holomorphic\nfunction are harmonic. We have the:\nCorollary 3.31. Let U\u0012C0be an open set and f :U!C0a split-holomorphic function.\nIf f=f+hy, then fandyare solutions of the wave equation :\u0003f=\u0003y=0. Here\n\u0003=¶2/¶x2\u0000¶2/¶y2is the wave operator (d’Alembertian ), and we say that fandyare\nLorentz-harmonic .\nRemark. Note that\u0003=4¶\n¶w¶\n¶w.\nPage 59OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nHere we see another stark contrast between CandC0. While it is difficult to solve\nexplicitly the heat equation 4F=0 (an elliptic partial differential equation), there\nare explicit solutions for the wave equation \u0003F=0 (a hyperbolic partial differen-\ntial equation). This can be used to completely classify all split-holomorphic functions\nwith a convex domain. In particular, this gives us a good source of split-holomorphic\nfunctions. It also follows from this classification that while being holomorphic and\ncomplex-analytic are the same thing, split-holomorphic functions are not necessarily\n“split-analytic” (or even of class C¥). We will not pursue this further here, but you\ncan see the details in [27].\nWe’ll conclude the discussion about differentiation stating the next two definitions,\nnecessary for what will come later.\nDefinition 3.32. LetU\u0012C0be an open set, w02Uand f:Unfw0g!C0. We’ll say\nthat w0is apole of order k\u00151 of fifkis the least integer for which (w\u0000w0)kf(w)is\nsplit-holomorphic.\nDefinition 3.33. LetU\u0012C0be an open set and P\u0012Ube discrete. We’ll say that a\nsplit-holomorphic function f:UnP!C0issplit-meromorphic in U ifPis precisely the\nset of poles of f.\nLet’s also register the bare minimum we need about integration in C0:\nDefinition 3.34. LetU\u0012C0be an open set, f:U!C0be a continuous function and\ng:I!Ua smooth curve. The integral of f along gis defined as\nZ\ngf(w)dw.=Z\nIf(g(t))g0(t)dt.\nRemark. This split-complex line integral can (obviously?) be expressed in terms of real\nline integrals. Moreover, this definition is naturally extended for piecewise smooth\ncurves in C0, and if gis closed we’ll just writeH\ngf(w)dwas usual.\nProbably the most important aspect of this integral is that we still have the:\nTheorem 3.35 (Fundamental Theorem of Calculus) .Let U\u0012C0be an open set,\nf:U!C0a continuous function, and g:[a,b]!U a piecewise smooth curve (actually\nC1is enough). If F :U!C0is aprimitive of f (i.e., F is split-holomorphic with F0=f ),\nthen Z\ngf(w)dw=F(g(b))\u0000F(g(a)).\nIn the next section, we will need some split-complex integrals to depend only on\nthe endpoints of the curve we’re integrating upon. In C, we had the Cauchy-Goursat\nTheorem. In C0we still have the same theorem, with the same proof (e.g., using\nGreen’s Theorem):\nTheorem 3.36 (Revised Cauchy-Goursat) .Let U\u0012C0be a simply-connected open set,\nf:U!C0be a split-holomorphic function with continuous derivative, and g:[a,b]!U a\nclosed and piecewise smooth curve (again, C1suffices), injective in ]a,b[. Then\nI\ngf(w)dw=0.\nPage 60OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nRemark. Note (again) that the assumption of f0being continuous, which is automati-\ncally satisfied in C, has to be explicitly stated here.\nThe two main corollaries are:\nCorollary 3.37. The integral of a split-holomorphic function (in the setting of the previous\ntheorem) along a given curve depends only on the endpoints of the curve. In this case, we\ndenote Z\ngf(w)dw=Zw\nw0f(w)dw,\nwhere gjoins w 0and w.\nAnd also:\nCorollary 3.38. Let U\u0012C0be a simply-connected open set, w 02U and f :U!C0be\ncontinuous. Then F :U!C0given by\nF(w) =Zw\nw0f(w)dw\nis split-holomorphic and satisfies F0=f .\nWe just need to get one more result out of our way:\nDefinition 3.39. LetU\u0012C0be an open set. Two functions f,y:U\u0012R2!Rare\ncalled Lorentz-conjugate iffu=yvandfv=yu. Such condition implies that both f\nandyare Lorentz-harmonic.\nTheorem 3.40. Let U\u0012R2\u0011C0be a simply-connected set, and f:U!Rbe a Lorentz-\nharmonic function. Then there exists a split-holomorphic function f :U!C0which has fas\nits real part. In particular, there is a Lorentz-conjugate function to f.\nProof: Define g.=fu+hfv. The condition\u0003f=0 ensures that gis split-holomorphic\nand, in particular, continuous, so that Ubeing simply-connected gives us the existence\nof a primitive G=y+hzforg. With this, G0=gtogether with the revised Cauchy-\nRiemann equations for Gyield\nfu+hfv=yu+hzu=yu+hyv.\nEquating real and imaginary parts, we obtain y=f+cfor some c2R. So f.=G\u0000c\nis the desired function.\nThe next step would be to look for a Cauchy-like formula in C0. Unfortunately,\nthere is not such a formula in this new setting. We are ready to move on and apply\nwhat we have seen here for surfaces in L3with H=0. For more about split-complex\nnumbers, see for example, [5], [7] and [27].\nPage 61OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nWeierstrass-Enneper representation formulas\nGiven a spacelike surface M\u0012R3\nnand a parametrization x:U!x[U]\u0012M, we\nmay identify R2withCin the usual way and use z=u+ivas a parameter. We may\nthen write\nx(z,z) = ( x1(z,z),x2(z,z),x3(z,z)),\nnoting the explicit dependence on the conjugate variable zis due to the fact that we\ndon’t know whether the components of xare holomorphic functions. For timelike\nsurfaces in L3, the parametrization domains will be identified with open subsets of C0\ninstead. Recall from calculus in a complex variable that if\n¶\n¶z=1\n2\u0012¶\n¶u\u0000i¶\n¶v\u0013\nand¶\n¶z=1\n2\u0012¶\n¶u+i¶\n¶v\u0013\n,\nthen the Laplacian operator can be expressed as\n4=¶2\n¶u2+¶\n¶v2=\u0012¶\n¶u+i¶\n¶v\u0013\u0012¶\n¶u\u0000i¶\n¶v\u0013\n=4¶\n¶z¶\n¶z.\nIt is sometimes convenient to study such parametrizations as the real part of curves\ninC3. We then consider an extension of the product in R3toC3, also to be denoted by\nh\u0001,\u0001iE, defined by\nh(z1,z2,z3),(w1,w2,w3)iE=z1w1+z2w2+z3w3.\nFor timelike surfaces, we’ll go instead to the complex Lorentzian space C3\n1, with the\nextended product h\u0001,\u0001iLdefined by\nh(z1,z2,z3),(w1,w2,w3)iL=z1w1+z2w2\u0000z3w3.\nWe’ll maintain the usual causal character terminology used so far.\nDefinition 3.41. LetUbe a open subset of CorC0, and x:U!R3\nnbe a regular and\nnon-degenerate parametrized surface.\n(i) The complex derivative ofxis\nf\u0011¶x\n¶z\u0011xz.=1\n2(xu\u0000ixv).\n(ii) The split-complex derivative ofxis\ny\u0011¶x\n¶w\u0011xw.=1\n2(xu+hxv).\nRemark. Note thathf,fiE=0 does not imply that f=0, since f(z,z)2C3for all z,\nand not necessarily in R3. Same holds a fortiori fory.\nProposition 3.42. If M\u0012R3\nnis a non-degenerate regular surface and x:U!x(U)\u0012M\nbe a parametrization of M, then xisisothermal (i.e.,jEj=jGj=l2for some smooth land\nF=0) if and only if:\nPage 62OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n(i)hf,fiE=0, when M\u0012R3;\n(ii)hf,fiL=0, when M\u0012L3is spacelike;\n(iii)hy,yiL=0, when M\u0012L3is timelike.\nProof: Let’s work through the proof when Mis spacelike. We have\n(xj\nz)2=\u00121\n2(xj\nu\u0000i xj\nv)\u00132\n=1\n4((xj\nu)2\u0000(xj\nv)2\u00002ixj\nuxj\nv),\nand summing over j=1, 2, 3 gives\nhf,fi=1\n4(E\u0000G\u00002iF),\nso that the conclusion follows from the fact that a complex number vanishes if and\nonly if its real and imaginary part vanish. For timelike M, one obtains\nhy,yiL=1\n4(E+G+2hF)\ninstead.\nLemma 3.43. Let M\u0012R3\nnbe a non-degenerate regular surface and x:U!x[U]\u0012M be\nan isothermal parametrization of M. Then:\n(i)hf,fiE=l2/26=0, when M\u0012R3;\n(ii)hf,fiL=l2/26=0, when M\u0012L3is spacelike;\n(iii)hy,yiL=eul2/26=0, when M\u0012L3is timelike.\nProof: Let’s work through the proof again assuming that Mis spacelike:\nxj\nzxj\nz=1\n4(xj\nu\u0000i xj\nv)(xj\nu+i xj\nv) =1\n4((xj\nu)2+ (xj\nv)2\u00002ixj\nuxj\nv),\nand summing over j=1, 2, 3 yields\nhf,fi=1\n4(l2+l2\u00002i\u00010) =l2\n2.\nProposition 3.44. Let M\u0012R3\nnbe a non-degenerate regular surface and x:U!x[U]\u0012M\nbe an isothermal parametrization of M. Then xiscritical (i.e., H =0) if and only if f\nis holomorphic or yis split-holomorphic, according to whether M is spacelike or timelike,\nrespectively.\nProof: It is a straightforward consequence of the formulas\n¶f\n¶z=¶2x\n¶z¶z=1\n44xand¶y\n¶w=¶2x\n¶w¶w=1\n4\u0003x.\nPage 63OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nIn view of this last result, we may conclude that at least locally any non-degenerate\ncritical surface may be represented by a triple:\n•f= (f1,f2,f3)of holomorphic functions satisfying\n(f1)2+ (f2)2+ (f3)2=0,\nifM\u0012R3;\n•f= (f1,f2,f3)of holomorphic functions satisfying\n(f1)2+ (f2)2\u0000(f3)2=0,\nifM\u0012L3is spacelike;\n•y= (y1,y2,y3)of split-holomorphic functions satisfying\n(y1)2+ (y2)2\u0000(y3)2=0,\nifM\u0012L3is timelike.\nA natural question at this point is: given such a triple, how to recover the starting\nsurface? The key to answering this lies in the next:\nProposition 3.45. Let M\u0012R3\nnbe a non-degenerate, regular and critical surface, U be a\nsimply-connected open set, and x:U!x[U]\u0012M be an isothermal parametrization of M.\nThen the components of xsatisfy:\n(i) xj(z,z) =cj+2 ReZz\nz0fj(x)dx, for some z 02U, if M is spacelike, and\n(ii) xj(w,w) =cj+2 ReZw\nw0yj(w)dw, for some w 02U, if M is timelike,\nwhere c j2Rare convenient constants.\nProof: Just for a change, let’s work this time the proof when Mis timelike. First note\nthat since Uis simply connected, xis isothermal and Mis critical, then yis split-\nholomorphic and the integrals in the statement of the proposition are path-independent.\nWith differentials, we have:\nyjdw=1\n2(xj\nu+h xj\nv)(du+hdv) =1\n2(xj\nudu+xj\nvdv+h(xj\nvdu+xj\nudv))\nyjdw=1\n2(xj\nu\u0000h xj\nv)(du\u0000hdv) =1\n2(xj\nudu+xj\nvdv\u0000h(xj\nvdu+xj\nudv))\nAdding both expressions, we obtain\ndxj=xj\nudu+xj\nvdv=yjdw+yjdw=2 Reyjdw,\nwhence\nxj(w,w) =cj+2 ReZw\nw0yj(w)dw,\nfor some cj2Rand w02U.\nPage 64OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nTheorem 3.46 (Enneper-Weierstrass I) .Let U\u0012Cbe a simply-connected open set, z 02U,\nand f ,g:U!Cfunctions with f holomorphic, g meromorphic, and f g2holomorphic. Then\nthe map x:U!R3\nndefined by x(z,z) = ( x1(z,z),x2(z,z),x3(z,z)), where\n(i) x1(z,z) =ReZz\nz0f(x)(1\u0000g(x)2)dx,\nx2(z,z) =ReZz\nz0i f(x)(1+g(x)2)dxand,\nx3(z,z) =2 ReZz\nz0f(x)g(x)dx, forxinR3or;\n(ii) x1(z,z) =ReZz\nz0f(x)(1+g(x)2)dx,\nx2(z,z) = ReZz\nz0i f(x)(1\u0000g(x)2)dxand,\nx3(z,z) =2 ReZz\nz0\u0000f(x)g(x)dx, forxinL3\nis a parametrized surface, regular in the points where the zeros of f have exactly twice the\norder than the order of the poles of g, and jgj6=1(this last condition necessary only in L3).\nFurthermore, its image is a spacelike critical surface.\nProof: The conditions over U,fand gensure that all integrals are path-indepdendent.\nMoreover, in R3, the complex derivative of xis precisely\nf=\u00121\n2f(1\u0000g2),i\n2f(1+g2),f g\u0013\n,\nwhich satisfies\nhf,fiE=\u00121\n2f(1\u0000g2)\u00132\n+\u0012i\n2f(1+g2)\u00132\n+ (f g)2=0,\nso that the expression given in Proposition 3.42 (p. 62) for hf,fiEgives us that E=G\nand F=0. A similar computations gives us the same conclusion in L3. This way, x\nis regular precisely when E=G6=0, which is equivalent to the condition given on\nzeros of fand poles of f. The necessity of jgj6=1 inL3follows from the fact that\nf=\u00121\n2f(1+g2),i\n2f(1\u0000g2),\u0000f g\u0013\n,\nthen\nl2\n2=hf,fiL=\f\f\f\f1\n2f(1+g2)\f\f\f\f2\n+\f\f\f\fi\n2f(1\u0000g2)\f\f\f\f2\n\u0000j\u0000f gj2=jfj2\n2(1\u0000jgj2)2.\nThis being the case, xis isothermal and spacelike. Furthermore, fis holomorphic and\nhence the image of xis critical.\nThe pair (f,g)is called the Weierstrass data for the surface. The geometry of the\nsurface can be described by this data. For more details in R3, see for example [24].\nWhen gis holomorphic and invertible, we may use it as a parameter itself and\nobtain an alternative representation:\nPage 65OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nTheorem 3.47 (Enneper-Weierstrass II) .Let U\u0012Cbe a simply-connected open set,\nz02U, and F :U!Ca holomorphic function. Then the map x:U!R3\nndefined by\nx(z,z) = ( x1(z,z),x2(z,z),x3(z,z)), where\n(i) x1(z,z) =ReZz\nz0(1\u0000x2)F(x)dx,\nx2(z,z) =ReZz\nz0i(1+x2)F(x)dxand,\nx3(z,z) =2 ReZz\nz0xF(x)dx, forxinR3, or;\n(ii) x1(z,z) =ReZz\nz0(1+x2)F(x)dx,\nx2(z,z) = ReZz\nz0i(1\u0000x2)F(x)dx, and\nx3(z,z) =\u00002ReZz\nz0xF(x)dx, forxandL3,\nis a parametrized surface, regular in the points where F (z)6=0(inR3) or F (z)6=0and\njzj6=1(inL3). Furthermore, its image is a spacelike critical surface.\nExample 3.48 (Critical spacelike surfaces in R3\nn).\n(1) Enneper surface in R3: consider the Weierstrass data f(z) = 1 and g(z) = z. We\nobtain the parametrization x:R2!R3given by\nx(u,v) =\u0012\nu\u0000u3\n3+uv2,\u0000v+v3\n3\u0000u2v,u2\u0000v2\u0013\n.\nFigure 17: Enneper surface in R3.\nThe same surface could be obtained by the single type II data F(z) =1.\n(2) Spacelike Enneper surface in L3: consider the same data as before, now in the\nLorentzian setting. We obtain the parametrization x:R2!R3given by\nx(u,v) =\u0012\nu+u3\n3\u0000uv2,\u0000v\u0000v3\n3+u2v,v2\u0000u2\u0013\n.\nPage 66OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nFigure 18: Spacelike Enneper surface in L3.\n(3) Catalan surface in R3: choose the type II data F(z) =i\u0010\n1\nz\u00001\nz3\u0011\n, which yields\nx(u,v) =\u0010\nu\u0000sinucosh v, 1\u0000cosucosh v,\u00004 sin\u0010u\n2\u0011\nsinh\u0010v\n2\u0011\u0011\nFigure 19: Catalan surface in R3.\n(4) Spacelike catenoid in L3: choose the type II data F(z) = 1/z2, which gives the\nparametrization\nx(u,v) =\u0012\nu\u0000u\nu2+v2,v\u0000v\nu2+v2,\u00002 log(u2+v2)\u0013\n.\nPage 67OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nFigure 20: Spacelike Lorentzian catenoid.\n(5) Henneberg surface in R3: the type II data is F(z) =1\u00001\nz4, producing the parametriza-\ntion\nx(u,v) =(2 sinh ucosv\u0000(2/3)sinh(3u)cos(3v),\n2 sinh (u)sin(v) + ( 2/3)sinh(3u)sin(3v), 2 cosh (2u)cos(2v)).\nFigure 21: Henneberg surface in R3.\nNow, let’s repeat this for timelike surfaces in L3:\nTheorem 3.49 (Enneper-Weierstrass I – timelike version) .Let U\u0012C0be a simply-\nconnected open set, w 02U, and f ,g:U!Cfunctions with f split-holomorphic, g split-\nmeromorphic, and f g2split-holomorphic. Then the map x:U\u0000\u0000\u0000! L3defined by\nPage 68OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nx(w,w) = ( x1(w,w),x2(w,w),x3(w,w)), where\nx1(w,w) =ReZw\nw0f(w)(1\u0000g(w)2)dw\nx2(w,w) =2 ReZw\nw0f(w)g(w)dw\nx3(w,w) = ReZw\nw0f(w)(1+g(w)2)dw,\nis a parametrized surface, regular in the points where the zeros of f have exactly twice the order\nthan the order of the poles of g, f (w)is not a zero-divisor and g (w)is not real. Furthermore,\nits image is a timelike critical surface.\nProof: The conditions over U,fand gagain ensure that all the above integrals are\npath-independent. In this case, the split-complex derivative of xis\ny=\u00121\n2f(1\u0000g2),f g,1\n2f(1+g2)\u0013\n.\nWe also have that\nhy,yiL=\u00121\n2f(1\u0000g2)\u00132\n+ (f g)2\u0000\u00121\n2f(1+g2)\u00132\n=0 and\nhy,yiL=ff\n4\u0010\n(1\u0000g2)(1\u0000g2) +4gg\u0000(1+g2)(1+g2)\u0011\n=\u0000ff\n2(g\u0000g)2,\nfrom where the conclusion follows.\nTheorem 3.50 (Enneper-Weierstrass II) .Let U\u0012C0be a simply-connected open set which\ndoes not touch the real axis, w 02U, and F :U!C0a split-holomorphic function. Then the\nmapx:U!L3defined by x(w,w) = ( x1(w,w),x2(w,w),x3(w,w)), where\nx1(w,w) =ReZw\nw0(1\u0000w2)F(w)dw\nx2(w,w) =2 ReZw\nw0wF(w)dw\nx3(w,w) =ReZw\nw0(1+w2)F(w)dw,\nis a parametrized surface, regular in the points where F (w)is not a zero divisor. Furthermore,\nits image is a timelike critical surface.\nExample 3.51 (Timelike critical surfaces in L3).\n(1) Timelike Enneper surface: for F(w) =1 we obtain\nx(u,v) =\u0012\nv\u0000u2v\u0000v3\n3, 2uv,v+u2v+v3\n3\u0013\n.\nPage 69OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nFigure 22: Timelike Enneper surface.\n(2) Timelike catenoid: for F(w) =1/w2we obtain\nx(u,v) =\u0012\n\u0000u\nu2\u0000v2\u0000u, log\u0010\n(u2\u0000v2)2\u0011\n,\u0000u\nu2\u0000v2+u\u0013\n,\nwhich is defined in all the plane L2, except for the two null lines, and regular\neverywhere minus on the real axis ( v=0).\nFigure 23: Timelike catenoid.\nFor more details about such representation formulas, you may consult [18] and [20].\nSuch techniques also have applications in the study of the so-called Bjorling problems –\nsee, for example, [2], [3], [10] and [11].\nPage 70OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProblems\nProblem 31 (Generalized complex numbers) .Given structure constants a,b2Rand a\nsymbol u, define\nCa,b.=fa+ubja,b2R,u2=a+bug,\nwhere the operations are defined in the obvious way.\n(a) Show that a+ub2Ca,bis invertible if and only if D.=a2+bab\u0000ab26=0.\n(b) If b6=0, then D/b2=0 may be regarded as a second degree equation in the\nvariable a/b, whose discriminant is D=b2+4a(check). The position of the point\n(a,b)in the plane relative to the parabola D=0 determines the possibility of\nrealizing divisions in Ca,b. Show that:\n• IfD<0,Ca,bis a field.\n• IfD\u00150, the zero divisors in Ca,bare precisely the elements a+ubsuch that\na+ (b+p\nD)b/2=0 or a+ (b\u0000p\nD)b/2=0, while all the other elements\nare invertible.\n(c) One says that Ca,bis an elliptic ,parabolic orhyperbolic system of numbers if D<0,\nD=0 orD>0, respectively. Justify this terminology by studying in terms of D\nthe conic x2+bxy\u0000ay2=0 in the plane.\nRemark. If one defines a+ub.=a+bb\u0000ub,Dis precisely the “squared norm” of the\nelement a+ub. The map D:Ca,b!Rthus defined has its behavior controlled by D.\nNamely, Dis positive-definite if D<0, degenerate for D=0 and indefinite for D>0.\nPolarizing D, we have that Ca,bis an algebraic model for the geometry of the bilinear\nform\nh(a,b),(c,d)ia,b.=ac+b\n2ad+b\n2bc\u0000abd\ninR2.\nProblem 32. LetU\u0012C0be a connected open set, and f:U!C0be a split-holomorphic\nfunction. Denote `= (1+h)/2. Show that given s,t2R, for all w2Usuch that\nw+t`,w+s`2Uwe have f(w) =`f(w+t`) +`f(w+s`).\nProblem 33. Show that f:C0!C0given by\nf(x+hy) =1+h\n1+e\u0000xe\u0000y\nis bounded and split-holomorphic (hence a counter-example for Liouville’s Theorem\ninC0).\nProblem 34. Letx,y:U!R3\nnbe two regular and conjugate (or Lorentz-conjugate)\nparametrized surfaces. Show that if xis isothermal, then so is y.\nPage 71OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nProblem 35. Letq2R. Show that the parametrized surface x:]0, 2p[\u0002R!R3\ngiven by\nx(u,v) = ( ucosq\u0006sinucosh v,v\u0006cosqcosusinh v,\u0006sinqcosucosh v)\nis isothermal and minimal.\nProblem 36. Prove Lemma 3.43 (p. 63) for timelike surfaces.\nProblem 37. LetM\u0012R3\nnbe a critical spacelike surface and x:U!M\u0012R3\nnbe a\ntype II Weierstrass parametrization defined by a holomorphic function F. Show that\nthe Gaussian curvature is given by\nK(x(u,v)) = (\u00001)n+1 4\njF(u+iv)j2((\u00001)n+u2+v2)4.\nProblem 38. LetM\u0012R3\nnbe a non-degenerate, regular and connected surface. Assume\nthatNis a Gauss map for M, which is a locally conformal map. Show that Mis critical\nor is contained in a piece of a sphere, de Sitter space or hyperbolic plane.\nPage 72OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nReferences\n[1] Alexandrov, D., A contribution to chronogeometry , Canadian J. Math. 19, pp. 1119-\n1128, 1967.\n[2] Alias, L.; Chaves, R. M. B.; Mira, P .; Bjorling roblem for maximal surfaces in Lorentz-\nMinkowski space , Math. Proc. Camb. Phil. Soc. (no. 134, pp. 289-316), 2003.\n[3] Chaves, R. M. B.; Dussan, M. P .; Magid, M.; Bjorling Problem for timelike surfaces in\nthe Lorentz-Minkowski space , Journal of Mathematical Analysis and Applciations\n(377, no. 2, pp; 481-494), 2011.\n[4] Anciaux, H., Minimal Submanifolds in Pseudo-Riemannian Geometry , World Scien-\ntific, 2011.\n[5] Antonuccio, F., Semi–Complex Analysis &Mathematical Physics (Corrected Version) ,\neprint arXiv:gr-qc/9311032, 1993, https://arxiv.org/pdf/gr-qc/9311032.\npdf.\n[6] Beem, J. K.; Ehrlich, P . E.; Easley, K. L., Global Lorentzian Geometry , CRC Press,\n1996.\n[7] Catoni et al., Geometry of Minkowski Spacetime , Springer-Verlag (Springer Briefs\nin Physics), 2011.\n[8] Chen, B. Y.; Pseudo-Riemannian Geometry, d-invariants and Applications , World Sci-\nentific, 2011.\n[9] do Carmo, M. P ., Geometria Diferencial de Curvas e Superfícies , SBM (Universitary\nTexts Collection, volume 04), 2014.\n[10] Dussan, M. P .; Franco Filho, A. P .; Magid, M.; The Bjorling Problem for timelike\nminimal surfaces in R4\n1, Annali di Matematica Pura ed Applicata (pp. 1-19), 2016.\n[11] Dussan, M. P .; Magid, M.; Bjorling Problem for timelike surfaces in R4\n2, Journal of\nGeometry and Physics (no. 73, pp. 187-199), 2013.\n[12] Goldman, W. M.; Margulis, G. A., Flat Lorentz 3-manifolds and cocompact Fuchsian\ngroups. Crystallographic groups and their generalizations (Kortrijk, 1999) Contempo-\nrary Mathematics 262, pp. 135—145, 2000.\n[13] Greub, W. H., Linear Algebra , Springer-Verlag, 1975.\n[14] Hawking, S., Ellis, G.; The Large Scale Structure of Spacetime , Cambridge Mono-\ngraphs on Mathematical Physics, 1973.\n[15] Hitzer, E., Non-constant bounded holomorphic functions of hyperbolic numbers – Can-\ndidates for hyperbolic activation functions , Proccedings of the First SICE Sympo-\nsium on Computational Intelligence, pp. 23–28, 2011.\nPage 73OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\n[16] Kosheleva, O.; Kreinovich, V ., Observable Causality Implies Lorentz Group:\nAlexandrov–Zeeman–Type Theorem for Space–Time Regions , Mathematical Struc-\ntures and Modeling 30, pp. 4–14, 2014.\n[17] Kobayashi, O., Maximal Surfaces in the 3-Dimensional Minkowski Space L3, Tokyo\nJ. of Math. Volume 06, pp. 297–309, 1983.\n[18] Konderak, J.; A Weierstrass Representation Theorem for Lorentz Surfaces , Complex\nVariables 50 (no. 5, pp.319-332), 2005.\n[19] López, R., Differential Geometry of Curves and Surfaces in Lorentz–Minkowski space ,\neprint arXiv:0810.3351, https://arxiv.org/pdf/0810.3351 , 2008.\n[20] Magid, M.; Minimal Timelike Surfaces via the Split-Complex Numbers , Proceedings\nof PADGE 2012, Shaker-Verlag, Aachen, 2013.\n[21] Naber, G. L., Spacetime and Singularities, An Introduction , Cambridge University\nPress, 1988.\n[22] Naber, G. L., The Geometry of Minkowski Spacetime: An Introduction to the Mathe-\nmatics of the Special Theory of Relativity , Springer–Verlag (Applied Mathematical\nSciences 92), 1992.\n[23] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity , Academic\nPress, 1983.\n[24] Osserman, R., A Survey of Minimal Surfaces , Dover, 1986.\n[25] Penrose, R., Techniques of Differential Topology in Relativity , AMS Coloquium Pub-\nlications (SIAM, Philadelphia), 1972.\n[26] Tenenblat, K., Introdução à Geometria Diferencial , Edgard Blücher, 2008.\n[27] Terek, I.; Lymberopoulos, A.; Introdução à Geometria Lorentziana: Curvas e Superfí-\ncies, SBM (Universitary Texts Collection, volume 21), 2018.\nPage 74OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nIndex\nn-orthogonality, 6\nAdmissible curve, 23\nAnti-de Sitter space, 49\nArc-photon reparametrization, 25\nBiregular curve, 22\nB-scroll (associated to a lightlike\ncurve), 47\nCartan\ncurvature, see also pseudo-torsion\nTrihedron, 27\nCatalan surface in R3, 67\nCausal\nautomorphism, 16\ncharacter, 5\nChain rule, 59\nChristoffel symbols, 50\nChronological and causal precedences,\n15\nCoindicator of a curve, 23\nComplex\nderivative, 62\nLorentzian space, 62\nCritical/minimal surface, 63\nCross product, 17\nd’Alembertian, 59\nde Sitter space, 40\nEnneper surface in R3, 66\nEnneper-Weierstrass representation\nformulas, 65, 69\nFermi chart, 51\nFirst Fundamental Form, 42\nFrenet Trihedron, 22\nFundamental Theorem\nfor lightlike and semi-lightlike\ncurves, 33\nof Calculus for split-complex\nintegrals, 60\nGauss map, 42\nGaussian curvature, 43Geodesic, 50\nGeometric surface, 50\nGeometric units, 14\nGram matrix, 9\nGram-Schmidt process, 9\nHelix, 35\nHenneberg surface in R3, 68\nHorocycle, 49\nHyperbolic\nnumbers, see also split-complex\nnumbers\nplane, 40\nrotation, 10\nIdentity matrix of index n, 5\nIndicator\nof a curve, 23\nof a surface, 40\nof a vector, 5\nIntegral along a curve, 60\nIsothermal parametrization, 62\nLagrange’s Identities, 18\nLancret’s theorem, 13\nLorentz-harmonic functions, 59\nLorentz-Minkowski space, 5\nMargulis invariant, 20\nMean curvature, 43\nvector, 43\nOrthochronous map, 12\nOrthogonal\nprojection, 7\nspace, 7\nOsculating plane, 23\nPole, 60\nPrincipal\ncurvatures, 44\ndirections, 44\nPseudo-Euclidean\nisometry, 10\nspace of index n, 5\nPage 75OSU/USP - MAT6702 Lecture Notes Ivo Terek Couto\nPseudo-orthogonal transformation, 10\nPseudo-torsion, 28\nReal Spectral Theorem, 44\nRevised\nCauchy-Goursat Theorem, 60\nCauchy-Riemann equations, 59\nRiemann’s\nclassification theorem, 50\nFormula, 56\nSecond Fundamental Form, 42\nSpacelike\ncatenoid in L3, 67\nEnneper surface in L3, 66\nSpatial and temporal parts of a\npseudo-orthogonal map, 11\nSplit-complexderivative, 62\nnumbers, 57\nSplit-holomorphic function, 58\nSplit-meromorphic function, 60\nStructure constants (for generalized\ncomplex numbers), 71\nTimelike\ncatenoid, 70\nEnneper surface, 69\nTotally umbilic surface, 45\nUmbilic point, 45\nWeingarten operator, 42\nWirtinger operators, 59\nZeeman topology, 17\nPage 76"    },    {        "title": "2108.06749v1.Exponential_stability_of_a_damped_beam_string_beam_transmission_problem.pdf",        "content": "arXiv:2108.06749v1  [math.AP]  15 Aug 2021EXPONENTIAL STABILITY OF A DAMPED\nBEAM-STRING-BEAM TRANSMISSION PROBLEM\nBIENVENIDO BARRAZA MART ´INEZ, JAIRO HERN ´ANDEZ MONZ ´ON,\nAND GUSTAVO VERGARA ROLONG\nAbstract. We consider a beam-string-beam transmission problem, wher e\ntwo structurally damped or undamped beams are coupled with a friction-\nally damped string by transmission conditions. We show that for this type of\nstructure, the dissipation produced by the frictional part is strong enough\nto produce exponential decay of the solution, no matter how s mall is its\nsize: for the exponential stability in the damped-damped-d amped situation\nwe use energy method and in the undamped-damped-undamped si tuation\nwe use a frequency domain method from the semigroups theory, which com-\nbines a contradiction argument with the multiplier techniq ue to carry out\na special analysis for the resolvent. Additionally, we show that the solution\nfirst defined by the weak formulation, in fact have higher Sobo lev space\nregularity.\n1.Introduction\nRecent advances in material science have provided new means of suppressing\nvibrations from elastic multi-link structures, for instan ces, by applying some\ntype of local or total dampings to them. These structures con sisting of\nconnected flexible elements such as strings, beams, plates a nd shells have\nmany applications in engineering areas such as in robot arms , frames, solar\npanels, aircrafts, satellite antennae, bridges and so on (s ee [6], [7], [15] and\nthe references therein). In this context we consider, in thi s paper, a coupled\nbeam-string-beam system, where we assume structural dampi ng/no damping\nfor the beams and frictional damping for the string. More pre cisely, we\nconsider a elastic structure composed by three parts. The fir t and the third\nare structurally damped or undamped beams in the open interv als (l0,l1) and\n(l2,l3), respectively, and the second is a frictionally damped str ing, occupying\nin equilibrium the open interval ( l1,l2), as shown in Figure 1.\nDate: August 17, 2021.\n2000Mathematics Subject Classification. 35M33, 35B35, 35B40, 93D23.\nKey words and phrases. Exponential stability, transmission problems, beams-str ings equa-\ntions, frictional damping.\nThe authors would like to thank COLCIENCIAS (Project 121571 250194) for the financial\nsupport.\n12 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nFigure 1\nDenoting by u=u(x,t),w=w(x,t) andv=v(x,t) the vertical displacements\nof the points on the beams and on the string with coordinates xat timet,\nrespectively, the mathematical model for the structure is g iven by the equations\nutt+uxxxx−ρ1utxx= 0,in (l0,l1)×(0,∞), (1.1)\nvtt−vxx+βvt= 0,in (l1,l2)×(0,∞), (1.2)\nwtt+wxxxx−ρ2wtxx= 0,in (l2,l3)×(0,∞), (1.3)\nwhereρi≥0,i= 1,2, andβ≥0 are fixed constants. The coefficients ρ1≥0\nandρ2≥0 describe the structural damping (or the absence of damping ) for the\nbeam equations ( 1.1) and (1.3), whereas β >0 in (1.2) describes a frictional\ndampingonthestring.Ontheendpoints l0,l3,ofthebeams,weimposeclamped\n(Dirichlet) boundary conditions\nu(l0,t) =ux(l0,t) =w(l3,t) =wx(l3,t) = 0, t∈(0,∞).(1.4)\nOn the interface{l1,l2}, we have transmission conditions\nu(l1,t) =v(l1,t) andv(l2,t) =w(l2,t), t∈(0,∞),(1.5)\nuxxx(l1,t)−ρ1utx(l1,t)+vx(l1,t) = 0, t∈(0,∞),(1.6)\nwxxx(l2,t)−ρ2wtx(l2,t)+vx(l2,t) = 0, t∈(0,∞),(1.7)\nuxx(l1,t) = 0, t∈(0,∞),(1.8)\nwxx(l2,t) = 0, t∈(0,∞).(1.9)\nCondition ( 1.5) is known as the continuity transmission condition, ( 1.6) and\n(1.7) mean that the two forces which are the shear force of the beam s and the\nstress of the string are such that one cancels the other, and ( 1.8) and (1.9)\ndescribe the fact that the the beams present possible inflect ion point on l1and\nl2(compare with [ 14], p. 1934).\nFinally, the boundary-transmission problem ( 1.1)–(1.9) is endowed with ini-\ntial conditions of the form\nu(x,0) =u0(x), ut(x,0) =u1(x), x∈(l0,l1), (1.10)\nv(x,0) =v0(x), vt(x,0) =v1(x), x∈(l1,l2), (1.11)\nw(x,0) =w0(x), wt(x,0) =w1(x), x∈(l2,l3). (1.12)EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 3\nThe aim of the present paper is to study well-posedness, regu larity, and\nexponential stability of the solution of ( 1.1)-(1.12).\nIn the recent years, the large-time behavior of structures c onsisting of elastic\nstrings and/or beams with different damping has been studied a lot. We\nrefer to [ 17], [19] and [24], where structures formed by beams were studied.\nFor instance, Shel in [ 24] showed, under certain conditions, the exponential\nstability of a network of elastic and thermoelastic Euler-B ernoulli beams. For\ntransmission problems between strings, see for example, [ 1], [9], [13], [18],\n[22] and [23]. For example, Alves, Mu˜ noz Rivera et al. considered in [ 1] a\ntransmission problem of a material composed of three compon ents; one of\nthem is a Kelvin–Voigt viscoelastic material, the second is an elastic material\n(no dissipation), and the third is an elastic material inser ted with a frictional\ndamping mechanism. They proved exponential stability of th e solution, if\nviscoelastic component is not in the middle of the material. Then, Rissel\nand Wang established in [ 23] exponential stability for a coupled system of\nelasticity and thermoelasticity with second sound confined by a purely elastic\none. Specifically, its structure was composed of three parts , where the first and\nthird were purely elastic and the second thermo-elastic. No w, regarding elastic\nstructures composed of string and beam we mention [ 2]-[5], [14], [16], [25], [26]\nand [27]. Ammari et al. in [ 2]-[5] considered the nodal feedback stabilization\nfor networks of strings and beams. They obtained that the dec ay rate of a\nclosed-loop system depends on the positions of the nodal fee dback controllers.\nHassine in [ 14] studied a elastic transmission wave/beam systems with a lo cal\nKelvin–Voigt damping. He showed that the energy of this coup led system\ndecays polynomially as the time variable goes to infinity, if the damping, which\nis locally distributed, acts through one part of the structu re. Li, Han and Xu\nin [16] obtained polynomial stability for a string-frictionally damped beam\nsystem and exponential stability for a frictionally damped string-beam system.\nShel in [25] considered transmission problems for a coupling of a strin g and a\nbeam with at least one of them being thermoelastic and establ ished that the\nassociated semigroup is exponentially stable when the stri ng is thermoelastic\nand polynomial stables when only the beam is thermoelastic a nd satisfies\ncertain additional condition. Wang in [ 26] obtained the strong stability of the\nsemigroup associated to a frictionally damped string-beam system. F. Wang\nand J. M. Wang in [ 27] established exponential stability for a beam-frictional ly\ndamped string system with some feedback at the interface poi nt.\nIn this paper, we study the well-posedness of the problem ( 1.1)-(1.12),\nthe higher regularity of the solution and the exponential st ability of the\nenergy of the system, depending on the dampings. By energy me thod, we\nprove the exponential stability of ( 1.1)-(1.12), if the beams and the string are4 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\ndamped (i.e. if ρ1,ρ2andβare positive). For the proof we do not need higher\nregularity of the solution. Moreover, by a frequency domain method from the\nsemigroups theory, we show the exponential stability of ( 1.1)-(1.12) in the\nundamped-damped-undamped situation, i.e. ρ1=ρ2= 0 andβ >0. For the\nproof we use the higher Sobolev space regularity of the solut ions, which implies\nthat the transmission conditions, first defined by the weak fo rmulation, hold in\nthe classical sense. For this we follow some analogous ideas from Section 4 of [ 8].\nThis paper is organized as follows: In Section 2, we define the basic spaces\nand operators. Then, in Section 3 we show the generation of a C0−semigroup\nof contractions (and therefore the well-posedness of ( 1.1)-(1.12)). Exponential\nstability for the cases damped-damped-damped and undamped -damped-\nundamped, as well as higher regularity of the solutions, are shown in Section 4.\nFinally, let us say some words about notation. Derivatives w ith respect to tof\na function will be eventually denoted by a “dot” over the name of the fuction.\nSo˙φwill denote the derivative of φwith respect to t. As usual we will also use\nthe notations ψ′,ψ′′,ψ′′′, or in general ψ(n)forn≥4, for the derivatives of a\nfunctionψ, which depends only on the one-dimensional spatial variabl ex.\n2.Base Spaces\nFor the existence and uniqueness of the solution of the probl em (1.1)-(1.12)\nwe will write this in an abstract form. For it, we will introdu ce some definitions\nand notations. For example, let us define the operator matrix A, which domain\nwill be explained later, by\nA:=\n0 0 0 1 0 0\n0 0 0 0 1 0\n0 0 0 0 0 1\n−∂4\n∂x40 0 ρ1∂2\n∂x20 0\n0∂2\n∂x20 0−β0\n0 0−∂4\n∂x40 0ρ2∂2\n∂x2\n.EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 5\nThen, ifzt:=∂z\n∂t,ztt:=∂2z\n∂t2andU:= (u,v,w,u t,vt,wt)⊤, we have that\nUt= (ut,vt,wt,utt,vtt,wtt)⊤and (1.1)-(1.3) can be written as\nUt=\nut\nvt\nwt\nutt\nvtt\nwtt\n=\nut\nvt\nwt\n−∂4u\n∂x4+ρ1∂2ut\n∂x2\n∂2v\n∂x2−βv2\n−∂4w\n∂x4+ρ2∂2wt\n∂x2\n=AU.\nSo, with the initial condition ( 1.12), we obtain\nUt(t) =AU(t) (t>0), U(0) =U0, (2.1)\nwhereU0:= (u0,v0,w0,u1,v1,w1)⊤.\nNow, we define the following spaces\nH2\nl0:={u∈H2(I1) :u(l0) =u′(l0) = 0},\nH2\nl3:={w∈H2(I3) :w(l3) =w′(l3) = 0},\nwith the inner products\n/an}bracketle{tu,˜u/an}bracketri}htH2\nl0:=/an}bracketle{tu′′,˜u′′/an}bracketri}htL2(I1), (2.2)\n/an}bracketle{tw,˜w/an}bracketri}htH2\nl3:=/an}bracketle{tw′′,˜w′′/an}bracketri}htL2(I3). (2.3)\nDue to the generalized Poincar´ e’s inequality the induced n orms/bardbl·/bardblH2\nl0and\n/bardbl·/bardblH2\nl3are equivalents to the standard norms /bardbl·/bardblH2(I1)and/bardbl·/bardblH2(I3)onH2\nl0\nandH2\nl3, respectively.\nNow, let define the spaces\nH:={(u,v,w)⊤∈H2\nl0×H1(I2)×H2\nl3:u(l1) =v(l1) yv(l2) =w(l2)},\nL:=L2(I1)×L2(I2)×L2(I3),\nequipped with the inner products\n/an}bracketle{t(u,v,w)⊤,(˜u,˜v,˜w)⊤/an}bracketri}htH:=/an}bracketle{tu,˜u/an}bracketri}htH2\nl0+/an}bracketle{tv′,˜v′/an}bracketri}htL2(I2)+/an}bracketle{tw,˜w/an}bracketri}htH2\nl3,(2.4)\n/an}bracketle{t(u,v,w)⊤,(˜u,˜v,˜w)⊤/an}bracketri}htL:=/an}bracketle{tu,˜u/an}bracketri}htL2(I1)+/an}bracketle{tv,˜v/an}bracketri}htL2(I2)+/an}bracketle{tw,˜w/an}bracketri}htL2(I3).(2.5)\nAgain, by the Poincar´ e’s inequality, the norm in H, induced by the inner\nproduct ( 2.4), is equivalent to the standard norm in the poduct space\nH2(I1)×H1(I2)×H2(I3). Due to the continuity of the trace operator, His6 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\na closed subspace of H2(I1)×H1(I2)×H2(I3) and therfore/parenleftbig\nH,/an}bracketle{t·,·/an}bracketri}htH/parenrightbig\nis a\nHilbert space.\nFinally, we define the Hilbert space\nH:=H×L,\nwith the inner product given by\n/an}bracketle{tU,˜U/an}bracketri}htH: =/an}bracketle{t(u1,v1,w1)⊤,(˜u1,˜v1,˜w1)⊤/an}bracketri}htH+/an}bracketle{t(u2,v2,w2)⊤,(˜u2,˜v2,˜w2)⊤/an}bracketri}htL\n=/an}bracketle{tu1,˜u1/an}bracketri}htH2\nl0+/an}bracketle{tv′\n1,˜v′\n1/an}bracketri}htL2(I2)+/an}bracketle{tw1,˜w1/an}bracketri}htH2\nl3+/an}bracketle{tu2,˜u2/an}bracketri}htL2(I1)\n+/an}bracketle{tv2,˜v2/an}bracketri}htL2(I2)+/an}bracketle{tw2,˜w2/an}bracketri}htL2(I3),\nfor allU= (u1,v1,w1,u2,v2,w2)⊤,˜U= (˜u1,˜v1,˜w1,˜u2,˜v2,˜w2)⊤∈H.\nSince the functions in the spaces defined above are not regula r enough to\nsatisfy the transmission conditions ( 1.6)-(1.9) in the classic sense, or even in\nthe trace sense, we interpretate this transmission conditi on first in a “weak”\nsense. For this, we consider U= (u1,v1,w1,u2,v2,w2)⊤∈Hand˜U=\n(˜u1,˜v1,˜w1,˜u2,˜v2,˜w2)⊤∈H×Hsufficiently smooth, such that the following\ncalculations make sense. Applying integration by parts we o btain\n/an}bracketle{tAU,˜U/an}bracketri}ht=a(U,˜U)+b(U,(˜u2,˜v2,˜w2)⊤),\nwhere\na(U,˜U) :=/an}bracketle{tu2,˜u1/an}bracketri}htH2\nl0+/an}bracketle{tv′\n2,˜v′\n1/an}bracketri}htL2(I2)+/an}bracketle{tw2,˜w1/an}bracketri}htH2\nl3−/an}bracketle{tu′′\n1˜u′′\n2/an}bracketri}htL2(I1)\n−ρ1/an}bracketle{tu′\n2,˜u′\n2/an}bracketri}htL2(I1)−β/an}bracketle{tv2,˜v2/an}bracketri}htL2(I2)−/an}bracketle{tv′\n1,˜v′\n2/an}bracketri}htL2(I2)−/an}bracketle{tw′′\n1,˜w′′\n2/an}bracketri}htL2(I3)\n−ρ2/an}bracketle{tw′\n2,˜w′\n2/an}bracketri}htL2(I3)\nand\nb(U,(˜u2,˜v2,˜w2)⊤)\n:=−u(3)\n1(l1)˜u2(l1)+u(3)\n1(l0)˜u2(l0)+u′′\n1(l1)˜u′\n2(l1)−u′′\n1(l0)˜u′\n2(l0)\n+ρ1u′\n2(l1)˜u2(l1)−ρ1u′\n2(l0)˜u2(l0)+v′\n1(l2)˜v2(l2)−v′\n1(l1)˜v2(l1)\n−w(3)\n1(l3)˜w2(l3)+w(3)\n1(l2)˜w2(l2)+w′′\n1(l3)˜w′\n2(l3)−w′′\n1(l2)˜w′\n2(l2)\n+ρ2w′\n2(l3)˜w2(l3)−ρ2w′\n2(l2)˜w2(l2).\nSince˜U∈H×H, we have that ˜ u2(l0) = ˜u′\n2(l0) = ˜w2(l3) = ˜w′\n2(l3) = 0, ˜v2(l1) =\n˜u(l1) y ˜v2(l2) = ˜w2(l2). Then, it follows that\nb(U,(˜u2,˜v2,˜w2)⊤)\n=/bracketleftbig\n−u(3)\n1(l1)+ρ1u′\n2(l1)−v′\n1(l1)/bracketrightbig\n˜u2(l1)+/bracketleftbig\nw(3)\n1(l2)−ρ2w′\n2(l2)+v′\n1(l2)/bracketrightbig\n˜w2(l2)\n+u′′\n1(l1)˜u′\n2(l1)−w′′\n1(l2)˜w′\n2(l2)\n= 0,EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 7\nif and only if\nu(3)\n1(l1)−ρ1u′\n2(l1)+v′\n1(l1) = 0,\nw(3)\n1(l2)−ρ2w′\n2(l2)+v′\n1(l2) = 0,\nu′′\n1(l1) = 0,\nw′′\n1(l2) = 0,\nwhich arethetransmissionconditions ( 1.6)-(1.9) that wehave in thedescription\nof the problem in Section 1, if there u2=∂tu1andw2=∂tw1. This motivates\nthe following definition.\nDefinition 2.1. U∈Hsatisfies the transmission conditions ( 1.6)-(1.9), in a\nweak sense, if and only if\n/an}bracketle{tAU,˜U/an}bracketri}ht=a(U,˜U),for all˜U∈H×H. (2.6)\nNow, we define the operator\nA:D(A)⊂H→H, U→AU:=AU, (2.7)\nwhere\nD(A) :=/braceleftig\nU∈H×H:u(4)\n1∈L2(I1),v′′\n1,∈L2(I2),w(4)\n1∈L2(I3),\nUsatisfies the ( 1.6)-(1.9) conditions in a weak sense/bracerightig\n.\nIn this way, the problem ( 1.1)-(1.12) can be written in abstract form by the\nCauchy problem\ndU\ndt(t) =AU(t) (t>0), U(0) =U0. (2.8)\n3.Well posedness\nNow, we will show that the problem ( 2.8) is well posed, which means, that\nfor everyU∈D(A), (2.8) has one and only classical solution which continu-\nously depends of U0. For it, we will proof that the operator Adefined on ( 2.7)\ngenerates a C0-semigroup of contractions on H. To achieve that, we will use\nthe Lumer-Phillips theorem.\nProposition 3.1. The following assertions hold:\na)Ais dissipative.\nb)I−Ais surjective.\nc)D(A)is dense in H.\nProof.LetU∈D(A). From ( 2.6) we have\n/an}bracketle{tAU,U/an}bracketri}htH=−ρ1||u′\n2||2\nL2(I1)−β||v2||2\nL2(I2)−ρ2||w′\n2||2\nL2(I3)≤0,(3.1)8 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nfromwhich a)follows. For thesurjectivity wewillusetheLax-Milgram t heorem.\nLetF= (f1,g1,h1,f2,g2,h2)⊤∈H. We need to show that there exists a\nU= (u1,v1,w1,u2,v2,w2)⊤∈D(A) such that ( I−A)U=F, i.e.\nu1−u2=f1∈H2\nl0, (3.2)\nv1−v2=g1∈H1(I2), (3.3)\nw1−w2=h1∈H2\nl3, (3.4)\nu(4)\n1+u2−ρ1u′′\n2=f2∈L2(I1), (3.5)\n−v′′\n1+(1+β)v2=g2∈L2(I2), (3.6)\nw(4)\n1+w2−ρ2w′′\n2=h2∈L2(I3). (3.7)\nPlugging ( 3.2)–(3.4) in (3.5)–(3.7), we have to solve\nu(4)\n1+u1−ρ1u′′\n1=f1+f2−ρ1f′′\n1, (3.8)\n−v′′\n1+(1+β)v1=g2+(1+β)g1, (3.9)\nw(4)\n1+w1−ρ2w′′\n1=h1+h2−ρ2h′′\n1. (3.10)\nWe define the sesquilinear form B:H×H→Cby\nB(Y,Φ) :=/an}bracketle{ty1,φ1/an}bracketri}htH2\nl0+ρ1/an}bracketle{ty′\n1,φ′\n1/an}bracketri}htL2(I1)+/an}bracketle{ty1,φ1/an}bracketri}htL2(I1)+/an}bracketle{ty′\n2,φ′\n2/an}bracketri}htL2(I2)\n+(1+β)/an}bracketle{ty2,φ2/an}bracketri}htL2(I2)+/an}bracketle{ty3,φ3/an}bracketri}htH2\nl3+ρ2/an}bracketle{ty′\n3,φ′\n3/an}bracketri}htL2(I3)+/an}bracketle{ty3,φ3/an}bracketri}htL2(I3),\nforY:= (y1,y2,y3)⊤,Φ := (φ1,φ2,φ3)⊤∈H. It is easy to see that\nB:H×H→Cis continuous and coercive.\nNow, for (f1,g1,h1,f2,g2,h2)⊤∈H, we define Λ : H→Cby\nΛ(Φ) :=/an}bracketle{tf1+f2,φ1/an}bracketri}htL2(I1)+ρ1/an}bracketle{tf′\n1,φ′\n1/an}bracketri}htL2(I1)+/an}bracketle{tg2+(1+β)g1,φ2/an}bracketri}htL2(I2)\n+/an}bracketle{th1+h2,φ3/an}bracketri}htL2(I3)+ρ2/an}bracketle{th′\n1,φ′\n3/an}bracketri}htL2(I3),\nfor all Φ∈H. It is also easy to see that Λ : H→Cis an antilinear continuous\nfunctional.\nThen, for the Lax-Milgram theorem exists one unique Y= (y1,y2,y3)⊤∈H\nsuch that\nB(Y,Φ) = Λ(Φ),for all Φ∈H. (3.11)\nIn particular, if φ1∈C∞\nc(I1) andφ2=φ3= 0, we have in ( 3.11) that\n/an}bracketle{ty1,φ1/an}bracketri}htH2\nl0+ρ1/an}bracketle{ty′\n1,φ′\n1/an}bracketri}htL2(I1)+/an}bracketle{ty1,φ1/an}bracketri}htL2(I1)\n=/an}bracketle{tf1+f2,φ1/an}bracketri}htL2(I1)+ρ1/an}bracketle{tf′\n1,φ′\n1/an}bracketri}htL2(I1),\nwhich can be written, in distibutional sense, as\n/an}bracketle{ty(4)\n1−ρ1y′′\n1+y1,φ1/an}bracketri}ht=/an}bracketle{tf1+f2−ρ1f′′\n1,φ1/an}bracketri}ht,EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 9\nfor allφ1∈C∞\nc(I1). This implies that\ny(4)\n1−ρ1y′′\n1+y1=f1+f2−ρ1f′′\n1\nin distributional sense. As y1,y′′\n1,f1,f2,f′′\n1∈L2(I1), we conclude that\ny(4)\n1∈L2(I1).\nAnalogously, we get y(4)\n3∈L2(I3) andy′′\n2∈L2(I2).\nNow,let(u1,v1,w1) := (y1,y2,y3)and(u2,v2,w2) := (u1−f1,v1−g1,w1−h1).\nThen, we have that U:= (u1,v1,w1,u2,v2,w2)⊤∈H×H,u(4)\n1∈L2(I1),v′′\n1∈\nL2(I2) yw(4)\n1∈L2(I3). For this Uand˜U= (˜u1,˜v1,˜w1,˜u2,˜v2,˜w2)⊤∈H×H\narbitrary, from ( 3.8)–(3.11) and (u2,v2,w2) = (u1−f1,v1−g1,w1−h1), we get\n/an}bracketle{t−u(4)\n1+ρ1u′′\n2,˜u2/an}bracketri}htL2(I1)+/an}bracketle{tv′′\n1−βv2,˜v2/an}bracketri}htL2(I2)+/an}bracketle{t−w(4)\n1+ρ2w′′\n2,˜w2/an}bracketri}htL2(I3)\n=/an}bracketle{tu1−(f1+f2),˜u2/an}bracketri}htL2(I1)+/an}bracketle{tv1−(g1+g2),˜v2/an}bracketri}htL2(I2)\n+/an}bracketle{tw1−(h1+h2),˜w2/an}bracketri}htL2(I3)\n=−/an}bracketle{tu1,˜u2/an}bracketri}htH2\nl0−ρ1/an}bracketle{t(u1−f1)′,˜u′\n2/an}bracketri}htL2(I1)−/an}bracketle{tv′\n1,˜v′\n2/an}bracketri}htL2(I2)−β/an}bracketle{t(v1−g1),˜v2/an}bracketri}htL2(I2)\n−/an}bracketle{tw1,˜w2/an}bracketri}htH2\nl3−ρ2/an}bracketle{t(w1−h1)′,˜w′\n2/an}bracketri}htL2(I3)\n=−/an}bracketle{tu1,˜u2/an}bracketri}htH2\nl0−ρ1/an}bracketle{tu′\n2,˜u′\n2/an}bracketri}htL2(I1)−/an}bracketle{tv′\n1,˜v′\n2/an}bracketri}htL2(I2)−β/an}bracketle{tv2,˜v2/an}bracketri}htL2(I2)\n−/an}bracketle{tw1,˜w2/an}bracketri}htH2\nl3−ρ2/an}bracketle{tw′\n2,˜w′\n2/an}bracketri}htL2(I3).\nThen,\n/an}bracketle{tAU,˜U/an}bracketri}htH=/an}bracketle{tu2,˜u1/an}bracketri}htH2\nl0+/an}bracketle{tv′\n2,˜v′\n1/an}bracketri}htL2(I2)+/an}bracketle{tw2,˜w1/an}bracketri}htH2\nl3+/an}bracketle{t−u(4)\n1+ρ1u′′\n2,˜u2/an}bracketri}htL2(I1)\n+/an}bracketle{tv′′\n1−βv2,˜v2/an}bracketri}htL2(I2)+/an}bracketle{t−w(4)\n1+ρ2w′′\n2,˜w2/an}bracketri}htL2(I3)\n=/an}bracketle{tu2,˜u1/an}bracketri}htH2\nl0+/an}bracketle{tv′\n2,˜v′\n1/an}bracketri}htL2(I2)+/an}bracketle{tw2,˜w1/an}bracketri}htH2\nl3−/an}bracketle{tu1,˜u2/an}bracketri}htH2\nl0\n−ρ1/an}bracketle{tu′\n2,˜u′\n2/an}bracketri}htL2(I1)−/an}bracketle{tv′\n1,˜v′\n2/an}bracketri}htL2(I2)−β/an}bracketle{tv2,˜v2/an}bracketri}htL2(I2)−/an}bracketle{tw1,˜w2/an}bracketri}htH2\nl3\n−ρ2/an}bracketle{tw′\n2,˜w′\n2/an}bracketri}htL2(I3).\nWhich means that Usatisfies the transmission conditions ( 1.6)-(1.9) in a weak\nsense. From this we conclude that U∈D(A), i.e.b) holds. Since His a Hilbert\nspace, we get from a) andb) thatD(A) is dense in Hdue to Theorem 4.6,\nChapter 1, in [ 21]. /square\nTheorem 3.2. the operator Ais the generator of a C0-semigroup (S(t))t≥0of\ncontractions on the Hilbert space H. In consequence, for each U0∈D(A)the\nCauchy problem ( 2.8) has a unique classical solution U∈C1([0,∞),H)which\ndepends continuously on the initial data.\nProof.Due to the Proposition 3.1and the Lumer-Phillips theorem we have that\nAis the generator of a contraction C0-semigroup over H. It follows that, for\neachU0∈D(A), the Cauchy problem ( 2.8) has a unique classical solution10 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nU∈C1([0,∞),H) which depends continuously on the initial data, i.e. the\nproblem is well posed. /square\n4.Exponential Stability\nIn this section, we will prove the main results of this paper. First we will\nprove the exponential stability of the semigroup ( S(t))t≥0generated by A, if\nwe have damping in the three subdomains, i.e., ρ1,ρ2andβare positive.\nForU0∈D(A), we have from Theorem 3.1 that U(t) :=S(t)U0(t≥0) is the\nclasical solution of ( 2.8). In this case the energy E(t) of the system is defined\nby\nE(t) : =1\n2||U(t)||2\nH\n=1\n2/parenleftig\n||u1(t)||2\nH2\nl0+||v1(t)′||2\nL2(I2)+||w1(t)||2\nH2\nl3+||u2(t)||2\nL2(I1)\n+||v2(t)||2\nL2(I2)+||w2(t)||2\nL2(I3)/parenrightig\n.\nNote that\nd\ndtE(t) = Re/an}bracketle{tAU(t),U(t)/an}bracketri}htH\n=−ρ1||u2(t)′||2\nL2(I1)−β||v2(t)||2\nL2(I2)−ρ2||w2(t)′||2\nL2(I3),(4.1)\nwhich shows that the system is dissipative if only one of the d amping term\nis active (ρ1+ρ2+β >0) and conservative if there is no damping at all\n(ρ1=ρ2=β= 0).\nNow, we will prove the first main result of this paper.\nTheorem 4.1. Letρ1>0,ρ2>0andβ >0. Then, the semigroup (S(t))t≥0\nis exponentially stable, i.e., for any U0∈D(A)andU(t) :=S(t)U0(t≥0)we\nhave\nE(t)≤Ce−αtE(0)\nwith positive constants Candα.\nProof.ForU0∈D(A) andt≥0, let\nU(t) := (u1(t),v1(t),w1(t),u2(t),v2(t),w2(t))⊤:=S(t)U0\nand\nF(t) :=/an}bracketle{tu1(t),u2(t)/an}bracketri}htL2(I1)+/an}bracketle{tv1(t),v2(t)/an}bracketri}htL2(I2)+/an}bracketle{tw1(t),w2(t)/an}bracketri}htL2(I3).\nThen,\n|F(t)|≤1\n2/parenleftig\n||u1(t)||2\nL2(I1)+||u2(t)||2\nL2(I1)+||v1(t)||2\nL2(I2)+||v2(t)||2\nL2(I2)\n+||w1(t)||2\nL2(I3)+||w2(t)||2\nL2(I3)/parenrightig\n≤1\n2||U(t)||2\nX,EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 11\nwithX:=H2(I1)×H1(I2)×H2(I3)×L. Because of the equivalence between\nthe standar norm in Xand the norm||·|| H, there exists c1>0 such that\n|F(t)|≤c1E(t). (4.2)\nDue tod\ndtU(t) =AU(t), i.e.,\n\n˙u1(t)\n˙v1(t)\n˙w1(t)\n˙u2(t)\n˙v2(t)\n˙w2(t)\n=\nu2(t)\nv2(t)\nw2(t)\n−u1(t)(4)+ρ1u2(t)′′\nv1(t)′′−βv2(t)′′\n−w1(t)(4)+ρ2w2(t)′′\n,\nit holds\nd\ndtF(t) =/an}bracketle{t˙u1(t),u2(t)/an}bracketri}htL2(I1)+/an}bracketle{tu1(t),˙u2(t)/an}bracketri}htL2(I1)+/an}bracketle{t˙v1(t),v2(t)/an}bracketri}htL2(I2)\n+/an}bracketle{tv1(t),˙v2(t)/an}bracketri}htL2(I2)+/an}bracketle{t˙w1(t),w2(t)/an}bracketri}htL2(I3)+/an}bracketle{tw1(t),˙w2(t)/an}bracketri}htL2(I3)\n=||u2(t)||2\nL2(I1)+||v2(t)||2\nL2(I2)+||w2(t)||2\nL2(I3)+/an}bracketle{tΦ(t),AU(t)/an}bracketri}htH\n=||u2(t)||2\nL2(I1)+||v2(t)||2\nL2(I2)+||w2(t)||2\nL2(I3)−||u1(t)||2\nHl0\n−ρ1/an}bracketle{tu1(t)′,u2(t)′/an}bracketri}htL2(I1)−||v1(t)′||2\nL2(I2)−β/an}bracketle{tv1(t),v2(t)/an}bracketri}htL2(I2)\n−||w1(t)||2\nH2\nl3−ρ2/an}bracketle{tw1(t)′,w2(t)′/an}bracketri}htL2(I3)\n=||(u2(t),v2(t),w2(t))||2\nL−||(u1(t),v1(t),w1(t))||2\nH\n−/an}bracketle{t(ρ1u1(t)′,βv1(t),ρ2w1(t)′),(u2(t)′,v2(t),w2(t)′)/an}bracketri}htL.\nThere, the weak transmission conditions were used with Φ( t) :=\n(0,0,0,u1(t),v1(t),w1(t))⊤.\nLetδ>0. By Young’s inequality and Poincar´ e’s inequality, there existsCδ>0\nandc2>0 such that\n−/an}bracketle{t(ρ1u1(t)′,βv1(t),ρ2w1(t)′),(u2(t)′,v2(t),w2(t)′)/an}bracketri}htL\n≤δ||(ρ1u1(t)′,βv1(t),ρ2w1(t)′)||2\nL+Cδ||(u2(t)′,v2(t),w2(t)′)||2\nL\n≤c2δ||(u1(t),v1(t),w1(t))||2\nH+Cδ||(u2(t)′,v2(t),w2(t)′)||2\nL.\nApplying Poincar´ e’s inequality to u2(t) andv2(t), and taking δsmall enough\nsuch thatc2δ≤1\n2, we obtain\nd\ndtF(t)≤||(u2(t),v2(t),w2(t))||2\nL−1\n2||(u1(t),v1(t),w1(t))||2\nH\n+Cδ||(u2(t)′,v2(t),w2(t)′)||2\nL\n≤c3||(u2(t)′,v2(t),w2(t)′)||2\nL−1\n2||(u1(t),v1(t),w1(t))||2\nH(4.3)12 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nfor a constant c3. Now, letL(t) :=c4E(t) +F(t) withc4a positive constant.\nThen, forc4large enough such that 2 c1≤c4and−c4max{ρ2\n1,β2,ρ2\n2}+c3≤−1\n2,\nit follows from ( 4), (4.3) and Poincar´ e’s inequality applyied to u2andw2that\nthere are constants c5andc6such that\nd\ndtL(t)≤−1\n2||(u2(t)′,v2(t),w2(t)′)||2\nL−1\n2||(u1(t),v1(t),w1(t))||2\nH\n≤−c5\n2||(u2(t),v2(t),w2(t))||2\nL−1\n2||(u1(t),v1(t),w1(t))||2\nH\n≤−min{c5,1}1\n2||U(t)||2\nH×L\n=−c6E(t).(4.4)\nAs|F(t)|≤c1E(t)≤c4\n2E(t), we have\nc4\n2E(t)≤L(t)≤3c4\n2E(t).\nTherefore, ( 4.4) yields thatd\ndtL(t)≤−αL(t) with some positive constan α. By\nGronwall’s lemma, L(t)≤Ce−αtL(0), which implies:\nE(t)≤2\nc4L(t)≤2\nc4e−αtL(0)≤e−αtE(0).\n/square\nNow, we will consider the case in which both beams are undampe d. For this\nwe will need to show some results about regularity.\nLemma 4.2. LetIan open interval in the real line. For each g∈L2(I)there\nexists a unique v∈H1\n0(I)such that\n/integraldisplay\nIv′φ′=−/integraldisplay\nIgφ(∀φ∈H1\n0(I)). (4.5)\nFurthermore, v∈H2(I).\nProof.It is easy to see that the bilinear form ( v,φ)/ma√sto→/integraltext\nIv′φ′is continuous\ninH1\n0(I)×H1\n0(I) and that, due to Poincar´ e’s inequality, it is also coerciv e in\nH1\n0(I). Then, Lax-Milgram theorem give the existence and uniquen ess of the\nsolutionvof (4.5), sinceφ/ma√sto→/integraltext\nIgφis a continuous linear functional on H1\n0(I),\nwheneverg∈L2(I). Now, ( 4.5) implies that ( v′)′=gin distributional sense.\nSinceg∈L2(I) we have that v′∈H1(I) and therefore v∈H2(I). /square\nWe will also need the resultin thefollowing lemma, the proof of which follows\nanalogously to the proof of Corollary 4.3 in [ 8].\nLemma 4.3. Leta<b,f∈L2((a,b))andz∈C. For sufficiently large λ>0\nthere exists a unique u∈H4((a,b))such that\nu(4)+λu=fin(a,b),\nu(a) =u′(a) = 0,EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 13\nu′′(b) = 0,\nu′′′(b) =z.\nNow, we can set the following result.\nTheorem 4.4. LetU= (u1,v1,w1,u2,v2,w2)⊤∈D(A). Thenu1∈H4(I1),\nv1∈H2(I2)andw1∈H4(I3). In particular, the transmission conditions ( 1.6)–\n(1.9) hold in the classical sense.\nProof.LetF= (f1,g1,h1,f2,g2,h2)⊤:=AU. Then,\nu2=f1∈H2\nl0, (4.6)\nv2=g1∈H1(I2), (4.7)\nw2=h1∈H2\nl3, (4.8)\n−u(4)\n1+ρ1u′′\n2=f2∈L2(I1), (4.9)\nv′′\n1−βv2=g2∈L2(I2), (4.10)\n−w(4)\n1+ρ2w′′\n2=h2∈L2(I3). (4.11)\nFrom (4.7) and (4.10) we have\nv′′\n1=βg1+g2. (4.12)\nNow, letv0be a smooth function on I2such thatv0(l1) =u1(l1) andv0(l2) =\nw1(l2) (we can take v0as an affine function for example) and set /hatwidev:=v1−v0.\nThen/hatwidev∈H1\n0(I2) and, due to ( 4.12), we have\n/hatwidev′′=βg1+g2−v′′\n0\nin distributional sense. Then /hatwidevsatisfies/integraldisplay\nI2/hatwidev′φ′=−/integraldisplay\nI2(βg1+g2−v′′\n0)φ(∀φ∈H1\n0(I2)).\nDue to Lemma 4.2it follows that /hatwidev∈H2(I2) sinceβg1+g2−v′′\n0∈L2(I2).\nTherefore,v1=v0+/hatwidev∈H2(I2) and, in particular, Sobolev embedding theorem\nimplies that v1∈C1(I2).\nOn the other side, we consider Φ := (0 ,0,0,φ,ψ,0)⊤withφ∈H2\nl0arbitrary\nandψ∈H1(I2) such that ψ(l1) =φ(l1) andψ(l2) = 0. Due to the definition of\nD(A) we obtain\n/angbracketleftbig\nAU,Φ/angbracketrightbig\nH=/an}bracketle{t−u(4)\n1+ρ1u′′\n2,φ/an}bracketri}htL2(I1)+/an}bracketle{tv′′\n1−βv2,ψ/an}bracketri}htL2(I2)\n=−/an}bracketle{tu′′\n1,φ′′/an}bracketri}htL2(I1)−ρ1/an}bracketle{tu′\n2,φ′/an}bracketri}htL2(I1)−β/an}bracketle{tv2,ψ/an}bracketri}htL2(I2)−/an}bracketle{tv′\n1,ψ′/an}bracketri}htL2(I2).\nClearing de term /an}bracketle{tu(4)\n1,φ/an}bracketri}htL2(I1)in the equality above, taking into account ( 4.6)\nand thatv1∈H2(I2), we see that integration by parts yields\n/an}bracketle{tu(4)\n1,φ/an}bracketri}htL2(I1)=/an}bracketle{tu′′\n1,φ′′/an}bracketri}htL2(I1)+/bracketleftbig\nρ1f′\n1(l1)−v′\n1(l1)/bracketrightbig\nφ(l1). (4.13)14 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nBy Lemma 4.3, for sufficiently large λ >0 there exists a unique ˜ u1∈H4(I1)\nsuch that\nλ˜u1+ ˜u(4)\n1=λu1+ρf′′\n1−f2inI1,\n˜u1(l0) = ˜u′\n1(l0) = 0,\n˜u′′\n1(l1) = 0,\n˜u′′′\n1(l1) =ρ1f′\n1(l1)−v′\n1(l1).\nThen, for all φ∈H2\nl0, we have by integration by parts twice and the boundary\nconditions above, that\n/an}bracketle{tλ˜u1+ ˜u(4)\n1,φ/an}bracketri}htL2(I1)=λ/an}bracketle{t˜u1,φ/an}bracketri}htL2(I1)+/an}bracketle{t˜u′′\n1,φ′′/an}bracketri}htL2(I1)\n+/bracketleftbig\nρ1f′\n1(l1)−v′\n1(l1)/bracketrightbig\nφ2(l1). (4.14)\nFor the same λ, adding up the therm /an}bracketle{tλu1,φ/an}bracketri}htL2(I1)in (4.13) we obtain\n/an}bracketle{tλu1+u(4)\n1,φ/an}bracketri}htL2(I1)=λ/an}bracketle{tu1,φ/an}bracketri}htL2(I1)+/an}bracketle{tu′′\n1,φ′′/an}bracketri}htL2(I1)\n+/bracketleftbig\nρ1f′\n1(l1)−v′\n1(l1)/bracketrightbig\nφ2(l1). (4.15)\nFrom (4.7) and (4.9) we have\nλu1+u(4)\n1=λu1+ρ1f′′\n1−f2=λ˜u1+ ˜u4\n1.\nSubtracting ( 4.15) from (4.14), with/hatwideu1:= ˜u1−u1, we get\n0 =λ/an}bracketle{t/hatwideu1,φ/an}bracketri}htL2(I1)+/an}bracketle{t/hatwideu′′\n1,φ′′/an}bracketri}htL2(I1)\nfor allφ∈H2\nl0. Since/hatwideu1∈H2\nl0we can set φ=/hatwideu1in the last equality and we\nget/hatwideu1= 0, which implies u1= ˜u1∈H4(I1). In particular, Sobolev embedding\ntheorem guarantees that u1∈C3(I1) and therefore the transmission conditions\nholdin theclassical sense. Theproofof w1∈H4(I3), andtherefore w1∈C3(I3),\nis similar. /square\nNow, we will use the following frequency domain result, whic h gives us a\nnecessaryandsufficientconditionfottheexponentialstabi lity ofaC0-semigroup\nof contractions. For the proof see [ 11], [12] and [20].\nProposition 4.5. Let(T(t))t≥0be aC0-semigroup of contractions in a Hilbert\nspaceH, generated by an operator A. Then the semigroup is exponentially stable\nif and only if\niR⊂ρ(A)and||(iλI−A)−1||L(H)≤C∀λ∈R. (4.16)\nThe second main result of this paper is the following.\nTheorem 4.6. Ifρ1=ρ2= 0andβ >0, then the semigroup (S(t))t≥0gener-\nated by Ais exponentially stable.EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 15\nProof.By the Proposition 4.5it is sufficient to show that Asatisfies ( 4.16).\nFirst, wewill showthat0 ∈ρ(A).LetF= (f1,g1,h1,f2,g2,h2)⊤∈H,thenwe\nhave to find a U= (u1,v1,w1,u2,v2,w2)⊤∈D(A) such that−AU=F, which\nis equivalent to equations ( 4.6)–(4.11) with−Finstead ofF. Then, plugging\nthe first three equations in the last three, we get\nu(4)\n1=f2−ρ1f′′\n1, (4.17)\n−v′′\n1=g2+βg1, (4.18)\nw(4)\n1=h2−ρ2h′′\n1. (4.19)\nNow, we define the sesquilinear form B0:H×H→Cby\nB0(Y,Φ) :=/an}bracketle{ty1,φ1/an}bracketri}htH2\nl0+/an}bracketle{ty′\n2,φ′\n2/an}bracketri}htL2(I2)+/an}bracketle{ty3,φ3/an}bracketri}htH2\nl3,\nforY:= (y1,y2,y3)⊤,Φ := (φ1,φ2,φ3)⊤∈H, and the antilinear functional\nΛ :H→Cby\nΛ(Φ) :=/an}bracketle{tf2,φ1/an}bracketri}htL2(I1)+ρ1/an}bracketle{tf′\n1,φ′\n1/an}bracketri}htL2(I1)+/an}bracketle{tg2+βg1,φ2/an}bracketri}htL2(I2)+/an}bracketle{th2,φ3/an}bracketri}htL2(I3)\n+ρ2/an}bracketle{th′\n1,φ′\n3/an}bracketri}htL2(I3),\nfor all Φ∈H. It is easy to see that B0:H×H→Cis continuous and coercive,\nand that Λ : H→Cis continuous. Then, by the Lax-Milgram theorem, there\nexists a unique Y:= (y1,y2,y3)⊤∈Hsuch that\nB0(Y,Φ) = Λ(Φ),for all Φ∈H. (4.20)\nIn the same way as in the proof of Proposition 3.1, we obtain that U:=\n(y1,y2,y3,−f1,−g1,−h1)⊤∈D(A) and satisfies−AU=F. On the other\nhand, if ˜U:= (˜u1,˜v1,˜w1,˜u2,˜v2,˜w2)⊤∈D(A) solves−A˜U=F, then\nB0((˜u1,˜v1,˜w1)⊤,Φ) = Λ(Φ) holds for all Φ ∈Hdue to definition of D(A)\nand the weak transmission conditions. Therefore U=˜UandAis a bijection.\nSince Ais the generator of C0-semigroup of contractions by Theorem 3.1, A\nis closed and hence 0 ∈ρ(A).\nBy the Sobolev’s embedding theorem, we obtain that A−1is a compact op-\nerator on H, and therefore, the spectrum of Aconsists of eigenvalues only.\nThus, we have to establish that there are not purely imaginar y eigenvalues. Let\nλ∈R,λ/ne}ationslash= 0, andU∈D(A) with AU=iλU, i.e.\niλu1=u2, (4.21)\niλv1=v2, (4.22)\niλw1=w2, (4.23)\niλu2+u(4)\n1= 0, (4.24)\niλv2−v′′\n1+βv2= 0, (4.25)\niλw2+w(4)\n1= 0. (4.26)16 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nDue to the dissipativity of A, it holds that\n0 = Re/an}bracketle{t(iλ−A)U,U/an}bracketri}ht= Re(iλ||U||H)−Re/an}bracketle{tAU,U/an}bracketri}ht=β||v2||2\nL2(I2).\nThen,v1=v2= 0, i.e.U= (u1,0,w1,u2,0,w2)⊤. Sincev1∈H1(I2) andn= 1,\nthe Sobolev’s inequality implies that v1(x) = 0 for all x∈I1= [l0,l1].\nMultiplying ( 4.21) byiλand substituting in ( 4.24), we get\nu(4)\n1−λu1= 0 in (l0,l1). (4.27)\nMoreoveru1satisfies the boundary conditions\nu1(l0) =u′\n1(l0) = 0 and u′′\n1(l1) =u′′′\n1(l1) = 0. (4.28)\nLetx:=l1+(l0−l1)η, η∈[0,1]. Then,\nu1(x) =u1(l1+(l0−l1)η) =:z(η) anddku1\ndxk=1\n(l0−l1)kdkz\ndηk.\nTherefore, the problem ( 4.27)–(4.28) can be transformed as follows:\nz(4)−a2z= 0 in (0 ,1),\nz(0) =z′′(0) =z′′′(0) = 0,\nz(1) =z′(1) = 0,(4.29)\nwherea:= (l0−l1)2|λ|/ne}ationslash= 0. The general solution of the ordinary differential\nequation in ( 4.29) is\nz(η) =c1cosh(√aη)+c2sinh(√aη)+c3cos(√aη)+c4sin(√aη).\nNow, we will see that the boundary conditions in ( 4.29) imply that c1=c2=\nc3=c4= 0 and therefore z≡0, which leads to iR⊂ρ(A). In fact, using the\nboundary condition in ( 4.29), we obtain\nc1+c3= 0, c1−c3= 0 and c2−c4= 0.\nThenc1=c3= 0 andc2=c4. Moreover, z′(1) = 0 implies c2(cosh(√a) +\ncos(√a)) = 0. Since cosh(√a) + cos(√a)/ne}ationslash= 0, we have c2= 0 and therefore\nz≡0, i.e.u1≡0. In similar way we obtain w1≡0. Thus,U= 0 and we\nconclude that iR⊂ρ(A).\nNow, we will show that\nsup\nλ∈R||(iλ−A)−1||L(H)<∞. (4.30)\nIf (4.30) is false, there are sequences ( λn)n∈N⊂Rand (Un)n∈N⊂D(A) such\nthat|λn|−−−→\nn→∞∞,||Un||H= 1 for alln∈Nand\n||(iλn−A)Un||H→0 (n→∞). (4.31)EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 17\nLetFn:= (iλn−A)Un. Due to the standard norm on H2(I1)×H1(I2)×\nH2(I3)×L2(I1)×L2(I2)×L2(I3) is equivalent to the norm ||·||HonH, (4.31)\nimplies\niλnu1,n−u2,n=f1,n→0 inH2(I1), (4.32)\niλnv1,n−v2,n=g1,n→0 inH1(I2), (4.33)\niλnw1,n−w2,n=h1,n→0 inH2(I3), (4.34)\niλnu2,n+u(4)\n1,n=f2,n→0 inL2(I1), (4.35)\niλnv2,n−v′′\n1,n+βv2,n=g2,n→0 inL2(I2), (4.36)\niλnw2,n+w(4)\n2,n=h2,n→0 inL2(I3), (4.37)\nwhereρ1=ρ2= 0. Due to the dissipativity of A, it follows\n0←Re(/an}bracketle{t(iλn−A)Un,Un/an}bracketri}ht) = Re[iλn||Un||2\nH−/an}bracketle{tAUn,Un/an}bracketri}htH] =β||v2,n||2\nL2(I2),\ni.e.\n||v2,n||L2(I2)→0. (4.38)\nNow, (4.33), (4.36) and (4.38) imply\n||v1,n||2\nL2(I2)→0,|λn|||v1,n||L2(I2)→0 and||λ−1\nnv′′\n1,n||L2(I2)→0.(4.39)\nFrom this and the Gagliardo-Nirenberg inequality, it follo ws that\n||v′\n1,n||L2(I2)≤||λnv1,n||1/2\nL2(I2)||λ−1\nnv′′\n1,n||1/2\nL2(I2)+||v1,n||L2(I2)→0.(4.40)\nSubstituting v2,n=iλnv1,n−g1,nin (4.36), we have\ng2,n=−λ2\nnv1,n−iλng1,n−v′′\n1,n+βv2,n. (4.41)\nNow, taking L2-product of ( 4.41) with (l2−x)v′\n1,n, we get\n/an}bracketle{tg2,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)\n=λn/an}bracketle{tv1,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)−||λnv1,n||2\nL2(I2)+(l2−l1)|λnv1,n(l1)|2\n−iλn/an}bracketle{tg1,n, v1,n/an}bracketri}htL2(I2)+/an}bracketle{ti(l2−x)g′\n1,n, λnv1,n/an}bracketri}htL2(I2)\n+i(l2−l1)λng1,n(l1)v1,n(l1)−||v′\n1,n||2\nL2(I2)+/an}bracketle{tv′′\n1,n,(l2−x)v1,n/an}bracketri}htL2(I2)\n+(l2−l1)|v′\n1,n(l1)|2+/an}bracketle{tβv1,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)\nor equivalently\n−||λnv1,n||2\nL2(I2)+(l2−l1)|λnv1,n(l1)|2−||v′\n1,n||2\nL2(I2)\n+(l2−l1)|v′\n1,n(l1)|2+2Re{/an}bracketle{tβv2,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)}\n=/an}bracketle{tg1,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)+/an}bracketle{tβv2,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)\n−λ2n/an}bracketle{tv1,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)+/an}bracketle{tig1,n, λnv1,n/an}bracketri}htL2(I2)\n−/an}bracketle{ti(l2−x)g′\n1.n, λnv1,n/an}bracketri}htL2(I2)−i(l2−l1)λng1,n(l1)v1,n(l1)\n−/an}bracketle{tv′′\n1,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2).(4.42)18 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nFrom (4.41) we have that βv2,n=g2,n+λ2\nnv1,n+iλng1,n+v′′\n1,nand therefore\n/an}bracketle{tβv2,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)\n=/an}bracketle{tg2,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)+λ2\nn/an}bracketle{tv1,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)\n+/an}bracketle{tig′\n1,n, λnv1,n/an}bracketri}htL2(I2)−/an}bracketle{ti(l2−x)g′\n1,n, λnv1,n/an}bracketri}htL2(I2)\n−i(l2−l1)λng1,n(l1)v1,n(l1)+/an}bracketle{tv′′\n1,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2).\nSubstituting this in ( 4.42), we obtain\n(l2−l1)|λnv1,n(l1)|2+(l2−l1)|v′\n1,n(l1)|2\n= 2Re/braceleftig\n/an}bracketle{tg2,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)−/an}bracketle{ti(l2−x)g′\n1,n, λnv1,n/an}bracketri}htL2(I2)\n−i(l2−l1)g1,n(l1)λnv1(l1)−/an}bracketle{tβv2,n,(l2−x)v′\n1,n/an}bracketri}htL2(I2)/bracerightig\n+/an}bracketle{tig′\n1,n, λnv1,n/an}bracketri}htL2(I2)+/an}bracketle{tig1,n, λnv1,n/an}bracketri}htL2(I2)+||λnv1,n||2\nL2(I2)\n+||v′\n1,n||2\nL2(I2).\nNow, by the Cauchy-Schwarz inequality and Young’s inequali ty, for each ǫ>0\nthere exists a Cǫ>0 such that\n(l2−l1)|λnv1,n(l1)|2+(l2−l1)|v′\n1,n(l1)|2\n≤2/bracketleftig\nl2||g2,n||L2(I2)||v1,n||L2(I2)+l2||g′\n1,n||L2(I2)||λnv1,n||L2(I2)+\n+(l2−l1)(ε|λnv1,n(l1)|2+Cε|g1,n(l1)|2)+βl2/bardblv2,n/bardblL2(I2)/bardblv′\n1,n/bardblL2(I2)/bracketrightig\n+2/bardblg1,n/bardblH1(I2)/bardblλnv1,n/bardblL2(I2)+||λnv1,n||2\nL2(I2)+||v′\n1,n||2\nL2(I2).\nFrom this, ( 4.33), (4.36), (4.38)–(4.40) and the trace theorem, it follows\n|λnv1,n(l1)|−−−→\nn→∞0 and|v′\n1,n(l1)|−−−→\nn→∞0. Thus,\n|λnu1,n(l1)|−−−→\nn→∞0 and|u′′′\n1,n(l1)|−−−→\nn→∞0 (4.43)\ndue to the transmission conditions ( 1.5) and (1.6) withρ1= 0.\nNow, substituting u2,n=iλnu1,n−f1,nin (4.35), we obtain\nf2,n=−λ2\nnu1,n−iλnf1,n+u(4)\n1,n. (4.44)\nTakingL2-product of ( 4.44) with (x−l0)u′\n1,n, we get with integration by parts\nthat\n/an}bracketle{tf2,n,(x−l0)u′\n1,n/an}bracketri}htL2(I1)\n=||λnu1,n||2\nL2(I1)+λ2\nn/an}bracketle{tu1,n,(x−l0)u′\n1,n/an}bracketri}htL2(I1)−(l1−l0)|λnu1,n(l1)|2\n+iλn/an}bracketle{tf1,n, u1,n/an}bracketri}htL2(I1)+/an}bracketle{ti(x−l0)f′\n1,n, λnu1,n/an}bracketri}htL2(I1)\n−i(l1−l0)f1,nλnu1,n(l1)+/bardblu′′\n1,n/bardbl2\nL2(I1)−/an}bracketle{tu′′\n1,n,(x−l0)u′′′\n1,n/an}bracketri}htL2(I1)\n+(l1−l0)u′′′\n1,n(l1)u′\n1,n(l1).\n(4.45)EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 19\nFrom (4.44) we haveλ2\nnu1,n=−f2,n−iλnf1,n+u(4)\n1,nand therefore\nλ2\nn/an}bracketle{tu1,n,(x−l0)u′\n1,n/an}bracketri}htL2(I1)\n=−/an}bracketle{tf2,n,(x−l0)u′\n1,n/an}bracketri}htL2(I1)+/an}bracketle{tif1,n, λnu1,n/an}bracketri}htL2(I1)\n+/an}bracketle{ti(x−l0)f′\n1,n, λnu1,n/an}bracketri}htL2(I1)−i(l1−l0)f1,n(l1)λnu1,n(l1)\n+2||u′′\n1,n||2\nL2(I1+/an}bracketle{tu′′\n1,n,(x−l0)u′′′\n1,n/an}bracketri}htL2(I1)+(l1−l0)u′′′\n1,n(l1)u′\n1,n(l1).(4.46)\nNow, plugging ( 4.46) in (4.45), we get\n||λnu1,n||2\nL2(I1)+3||u′′\n1,n||2\nL2(I1)\n= 2Re/braceleftbig\n/an}bracketle{tf2,n,(x−l0)u′\n1,n/an}bracketri}htL2(I1)−/an}bracketle{tif1,n, λnu1,n/an}bracketri}htL2(I1)\n−/an}bracketle{ti(x−l0)f′\n1,n, λnu1,n/an}bracketri}htL2(I1)+i(l1−l0)f1,n(l1)λnu1,n(l1)\n−(l1−l0)u′′′\n1,n(l1)u′\n1,n(l1)/bracerightbig\n+(l1−l0)|λnu1,n(l1)|2.(4.47)\nNote that the Gagliardo-Nirenberg inequality implies\n||u′\n1,n||L2(I1)≤||u1,n||1/2\nL2(I1)||u′′\n1,n||1/2\nL2(I1)+||u1,n||L2(I1)\nand thus\n||u′\n1,n||2\nL2(I1)≤3||u1,n||2\nL2(I1)+||u′′\n1,n||L2(I1). (4.48)\nMoreover, it follows from the trace theorem that there exist s a positive constant\nCsuch that\n|u′\n1,n(l1)|≤C||u1,n||H2(I1)≤C||Un||H=C. (4.49)\nLetε1,ε2andε3positive numbers. By Young’s inequality in ( 4.47), there are\npositive constants Cε1,Cε2andCε3such that\n||λnu1,n||2\nL2(I1)+3||u′′\n1,n||2\nL2(I1)\n≤ε1||u′\n1,n||2\nL2(I1)+Cε1||f2,n||2\nL2(I1)+ε2||λnu1,n||2\nL2(I1)+Cε2||f1,n||2\nL2(I1)\n+ε3||λnu1,n||2\nL2(I1)+Cε3||f′\n1,n||2\nL2(I1)+2(l1−l0)/braceleftbig\n|f1,n(l1)||λnu1,n(l1)|\n+|u′′′\n1,n(l1)||u′\n1,n(l1)|+|λnu1,n(l1)|2/bracerightbig\n≤3ε1||λnu1,n||2\nL2(I1)+ε1||u′′\n1,n||2\nL2(I1)+Cε1||f2,n||2\nL2(I1)\n+ε2||λnu1,n||2\nL2(I1)+Cε2||f1,n||2\nL2(I1)+ε3||λnu1,n||2\nL2(I1)+Cε3||f′\n1,n||2\nL2(I1)\n+2(l1−l0)/braceleftbig\n|f1,n(l1)||λnu1,n(l1)|+C|u′′′\n1,n(l1)|+|λnu1,n(l1)|2/bracerightbig\n.\n(4.50)\nThere we have used ( 4.48) and (4.49). Choosing ε1,ε2andε3small enough such\nthat 3ε1+ε2+ε3<1/2, we get from ( 4.32), (4.35), (4.43) and (4.50) that\n||λnu1,n||L2(I1)−−−→\nn→∞0 and||u′′\n1,n||L2(I1)−−−→\nn→∞0. (4.51)\nAnalogously we conclude that\n||λnw1,n||L2(I3)−−−→\nn→∞0 and||w′′\n1,n||L2(I3)−−−→\nn→∞0.(4.52)20 B. BARRAZA, J. HERN ´ANDEZ, AND G. VERGARA\nTherefore,||Un||H−−−→\nn→∞0 due to ( 4.51), (4.52), (4.32), (4.33), (4.38) and\n(4.40), which is a contradiction. Thus we have proved that the ( S(t))t≥0is\nexponentially stable. /square\nReferences\n[1] M. Alves,J. E.Mu˜ noz Rivera,M. Sep´ ulvedaandO.Vera.T helack ofexponential stability\nin certain transmission problems with localized Kelvin-Vo igt dissipation, SIAM J. Appl.\nMath. 74 (2) (2014): pp. 345–365.\n[2] K. Ammari, D. Jellouli, and M. Mehrenberger. Feedback st abilization of a coupled string-\nbeam system, Netw. Heterog. media 4 (1) (2009), 19-34.\n[3] K. Ammari and M. Mehrenberger. 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Energy decay estimates of elastic transmis sion wave/beam systems with a\nlocal kelvin voigt damping. International Journal of contr ol, 89(10) (2016) :1933–1950.\n[15] J. Lagnese, G. Leugering and E. J. P. G. Schmidt. Modelin g, Analysis of Dynamic Elastic\nMulti-link Structures, Birkh¨ auser, Boston-Basel-Berli n, 1994.\n[16] Y. F. Li, Z. J. Han and G. Q. Xu. Explicit decay rate for cou pled string-beam system\nwith localized frictional damping, Applied Mathematics Le tters, 78 (2018): 51–58.\n[17] K. Liu and Z. Liu. Exponential decay of energy of the eule r-bernoulli beam with locally\ndistributed kelvin voigt damping. SIAM J. Control Optim, 36 (3) (1998):1086–1098.\n[18] T. K. Maryati, J. Mu˜ noz Rivera, A. Rambaud and O. Vera. S tability of an n-component\nTimoshenko beam with localized Kelvin-Voigt and frictiona l dissipation, Electronic Jour-\nnal of Di erential Equations, Vol. 2018 (2018), No. 136, pp. 1 -18.\n[19] J. Mu˜ noz Rivera and H. Portillo. The transmission prob lem for thermoelastic beams.\nJournal of Thermal Stresses, 24 (12) (2001):1137–1158.\n[20] J. Pr¨ uss. On the spectrum of C0-semigroups, Trans. Am. Math. Soc. 284 (1984) 847–857.EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 21\n[21] A. Pazy. Semigroups of linear operators and applicatio ns to partial differential equations,\nvolume 44. Springer Science & Business Media, 2012.\n[22] C. Raposo,W. Bastos, andJ. ´Avila. Atransmission problemfor euler-bernoulli beamwit h\nkelvin-voigt damping. Applied Mathematics and informatio n sciences, 5 (1) (2011):17–28.\n[23] M. Rissel and Y. G. Wang. Remarks on exponential stabili ty for a coupled sys-\ntem of elasticity and thermoelasticity with second sound, J . Evol. Equ. (2020).\nhttps://doi.org/10.1007/s00028-020-00636-4.\n[24] F. Shel. Exponential Stability of a Network of Beams, J. Dyn. Control Syst. 21 (2015):\n443–460.\n[25] F. Shel. Thermoelastic stability of a composite materi al, J. Differential Equations, 269\n(2020): 9348–9383.\n[26] Ch. Wang. Spectral Analysis for a Wave/Plate Transmiss ion System, Advances in Math-\nematical Physics, vol. 2019, Article ID 7849561, 9 pages, 20 19.\n[27] F. Wang and J. M. Wang. Stability of an interconnected sy stem of Euler–Bernoulli beam\nand wave equation through boundary coupling, Systems & Cont rol Letters 138 (2020)\n104664.\nB. Barraza Mart ´ınez, Universidad del Norte, Departamento de Matem ´aticas y\nEstad´ıstica, Barranquilla, Colombia\nEmail address :bbarraza@uninorte.edu.co\nJ. Hern ´andez Monz ´on, Universidad del Norte, Departamento de Matem ´aticas\ny Estad ´ıstica, Barranquilla, Colombia\nEmail address :jahernan@uninorte.edu.co\nG. Vergara Rolong, Universidad de la Costa, Departamento de Ciencias Nat-\nurales y Exactas, Barranquilla, Colombia\nEmail address :gvergara@cuc.edu.co"    },    {        "title": "2101.09431v1.Signature_of_Lorentz_Violation_in_Continuous_Gravitational_Wave_Spectra_of_Ellipsoidal_Neutron_Stars.pdf",        "content": "Signature of Lorentz Violation in Continuous Gravitational-Wave Spectra of\nEllipsoidal Neutron Stars\u0003\nRui Xu,1,yYong Gao,2, 1and Lijing Shao1, 3,z\n1Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China\n2Department of Astronomy, School of Physics, Peking University, Beijing 100871, China\n3National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China\n(Dated: January 26, 2021)\nWe study e\u000bects of Lorentz-invariance violation on the rotation of neutron stars (NSs) in the\nminimal gravitational Standard-Model Extension framework, and calculate the quadrupole radi-\nation generated by them. Aiming at testing Lorentz invariance with observations of continuous\ngravitational waves (GWs) from rotating NSs in the future, we compare the GW spectra of a ro-\ntating ellipsoidal NS under Lorentz-violating gravity with those of a Lorentz-invariant one. The\nformer are found to possess frequency components higher than the second harmonic, which does\nnot happen for the latter, indicating those higher frequency components to be potential signatures\nof Lorentz violation in continuous GW spectra of rotating NSs.\nI. INTRODUCTION\nThe observation of gravitational waves (GWs) from the compact binary system GW170817 initiates the era of\nmultimessenger astronomy [1, 2]. Gravitational theories, including the renowned general relativity (GR), are exposed\nto unprecedented tests utilizing GW signals [3]. Lorentz invariance, incorporated locally in GR as well as many other\nalternative gravitational theories, is certainly one of the fundamental principles subjected to these tests [4{8]. By\nemploying the Standard-Model Extension (SME) framework [9{13], which is widely used to investigate consequences\nfrom possible violations of Lorentz invariance in terrestrial experiments and astrophysical observations [14], stringent\nbounds have been set for the coe\u000ecients for Lorentz violation in the gravitational sector of the SME framework after\nanalyzing the observed GW data [2, 5{7].\nBesides the coalescence of compact binary systems, another type of GW sources are deformed rotating neutron\nstars (NSs). Especially, when the angular velocity of a deformed NS is misaligned with its angular momentum, the\nstar precesses about the direction of the angular momentum, radiating out GWs continuously [15, 16]. The search for\nsuch continuous GW signals is ongoing [17{22]. Once detected, the continuous GW signals will tell us a substantial\npiece of information on NS structure and deformability. Furthermore, they will bring new tests for the laws of physics,\namong which lies Lorentz invariance as one of the fundamental principles (see e.g. Ref. [8]).\nTo test Lorentz invariance, an investigation of the scenario where it is violated is necessary. The e\u000bects of Lorentz\nviolation on rotating spheroidal stars are studied in detail in Ref. [23] under the minimal gravitational SME framework.\nThe modi\fcation to the free precession of a deformed star is depicted by the name twofold precession , as brie\ry\nspeaking, Lorentz violation causes the angular momentum to precess about a \fxed direction while at the same time\nthe star still precesses about the instantaneous direction of the angular momentum. The correction in the quadrupole\nradiation due to the modi\fcation of the rotation of the star is calculated in Ref. [23], and it is found that the\nquadrupole radiation from a spheroidal star a\u000bected by Lorentz violation has frequency components higher than\ntwice of the fundamental one.\nIn this work, we are going to extend the numerical results in Ref. [23] to ellipsoidal NSs. The characteristic higher\nharmonics due to Lorentz violation remain in the GW spectra as we expect. But more importantly, our numerical\ncalculation for the quadrupole radiation from an ellipsoidal NS in the absence of Lorentz violation indicates that\nthough the nonaxisymmetry of the star modulates the \frst and the second harmonics in the GW spectra as discussed\nin Refs. [16, 24, 25], it does not generate harmonics higher than the second for freely precessing NSs. Therefore,\nharmonics higher than the second are indeed possible signatures for Lorentz violation in the GW spectra of rotating\nsolitary NSs.\nWe organize the paper as follows. In Sec. II, we present the analytical equations to construct the quadrupole\nradiation from a rotating ellipsoid under Lorentz-violating gravity. Then in Sec. III, numerical solutions to the rotation\nequations for ellipsoids with uniform density are obtained and used to construct examples of the quadrupole radiation.\n\u0003Invited article to special issue \\Lorentz Violation in Astroparticles and Gravitational Waves\" in journal Galaxies .\nyCorresponding author: xuru@pku.edu.cn\nzCorresponding author: lshao@pku.edu.cnarXiv:2101.09431v1  [gr-qc]  23 Jan 20212\nFIG. 1. Euler angles transforming the X-Y-Zinertial frame to the x-y-zbody frame. First, rotate the X-Y-Zframe about\ntheZaxis with angle \u000bso that the X-axis aligns with the intersection line MN. Then, rotate the just obtained X-Y-Zframe\nabout the line MN with angle\fso that the Z-axis aligns with the z-axis. Last, rotate the new X-Y-Zframe about the z-axis\nwith angle\rso that it overlaps with the x-y-zframe.\nSubsequently, Fourier transformations are performed to extract the frequency components of the quadrupole GWs,\nand we will see that while the GW from an ellipsoid under the twofold precession contains harmonics higher than the\nsecond, the GW from an ellipsoid under free precession only has frequencies around the \frst and the second harmonics.\nIn the end, conclusions are summarized in Sec. IV. For simplicity in writing equations, we use the geometrized unit\nsystem where G=c= 1. However, standard units do appear when numerical estimations are desired for realistic\nNSs.\nII. THEORETICAL BASICS\nTo proceed with the calculation, we neglect relativistic corrections to NS structure and motion, and solve its\nmotion from the rotation equations for rigid bodies.1Assuming that in the body frame x-y-z, the surface of the star\nis described by\nx2\na2x+y2\na2y+z2\na2z= 1; (1)\nwith semi-axes ax; ayandaz, then the Lagrangian for the rotation of the star can be written as\nL=1\n2\u0000\nIxx(\nx)2+Iyy(\ny)2+Izz(\nz)2\u0001\n\u0000\u000eU; (2)\nwhereIxx; IyyandIzzare the eigenvalues of the moment of inertia tensor along the principal axes, and \nx;\nyand\n\nzare the components of the angular velocity of the star in the x-y-zframe. The orientation-dependent self-energy\n\u000eUis calculated from the anisotropic correction \u000e\b to the Newtonian potential \b in the minimal gravitational SME,\nnamely [13]\n\u000e\b =\u00001\n2\u0016sijZ(xi\u0000x0i)(xj\u0000x0j)\njx\u0000x0j3\u001a(x0)d3x0;\n\u000eU=\u00001\n4\u0016sijZ\u0000\nxi\u0000x0i\u0001\u0000\nxj\u0000x0j\u0001\njx\u0000x0j3\u001a(x)\u001a(x0)d3xd3x0; (3)\nwhere \u0016sij, withi;j=x;y;z , are the coe\u000ecients for Lorentz violation in the body frame [12, 13], and \u001ais the density\nof the star.\nIn the SME framework, the coe\u000ecients for Lorentz violation are assumed to be constant in inertial frames. Therefore,\nas the star rotates, the coe\u000ecients \u0016 sijdepend on the orientation of the star according to\n\u0016sij=RiIRjJ\u0016sIJ; (4)\n1Relativistic corrections are reasonably characterized by the compactness of the body, which is about 0 :1 for a NS.3\nwhereRiIrepresents the rotation matrix transforming an inertial frame X-Y-Zto the body frame x-y-z. The capital\nindices run over X; Y andZ, and \u0016sIJare constant coe\u000ecients for Lorentz violation. The orientation dependence of\n\u0016sij, originated from the rotation matrix, can be easily described by the Euler angles ( \u000b;\f;\r ) in Fig. 1, as the rotation\nmatrix in terms of the Euler angles is\nI\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000!\nRiI=\ni \u0000\u0000\u0000\u0000\u00000\n@cos\u000bcos\r\u0000sin\u000bcos\fsin\r sin\u000bcos\r+ cos\u000bcos\fsin\rsin\fsin\r\n\u0000cos\u000bsin\r\u0000sin\u000bcos\fcos\r\u0000sin\u000bsin\r+ cos\u000bcos\fcos\rsin\fcos\r\nsin\u000bsin\f \u0000cos\u000bsin\f cos\f1\nA:(5)\nTogether with the relations between the velocity components and the Euler angles [26],\n\nx= _\u000bsin\fsin\r+_\fcos\r;\n\ny= _\u000bsin\fcos\r\u0000_\fsin\r;\n\nz= _\u000bcos\f+ _\r; (6)\nwhere dots denote time derivatives, the Euler-Lagrange equations for the Euler angles can be obtained from the\nLagrangian (2). Given the shape and density of the star, the moment of inertia tensor and the integrals in \u000eUcan be\ncalculated, and then the Euler angles can be solved to describe the rotation of the star.\nOnce the rotation of the star is known, its gravitational quadrupole radiation can be calculated via the metric\nperturbation\nhIJ=\u00002\nrIIJ; (7)\nwhereris the distance from the distant star to the observer, and the double dots denote the second time derivative.\nIn Ref. [23], it is shown that IIJcan be written as\nIIJ=RiIRjJAij; (8)\nwith the body-frame quantities Aijbeing\nAxx= 2\u0010\n\u00012(\ny)2\u0000\u00013(\nz)2\u0011\n;\nAyy= 2\u0010\n\u00013(\nz)2\u0000\u00011(\nx)2\u0011\n;\nAzz= 2\u0010\n\u00011(\nx)2\u0000\u00012(\ny)2\u0011\n;\nAxy= \n(\u00013)2\nIzz+ \u0001 1\u0000\u00012!\n\nx\ny+\u00013\nIzz\u0000z;\nAxz= \n(\u00012)2\nIyy+ \u0001 3\u0000\u00011!\n\nx\nz+\u00012\nIyy\u0000y;\nAyz= \n(\u00011)2\nIxx+ \u0001 2\u0000\u00013!\n\ny\nz+\u00011\nIxx\u0000x; (9)\nfor any rigid body subjected to arbitrary rotations. The quantities \u0001 1;\u00012and \u0001 3are de\fned as\n\u00011=Iyy\u0000Izz;\u00012=Izz\u0000Ixx;\u00013=Ixx\u0000Iyy; (10)\nand the components of the torque, \u0000x;\u0000yand \u0000z, are calculated from the orientation-dependent self-energy \u000eUvia\n\u0000x=\u0000sin\r\nsin\f@\u000b\u000eU\u0000cos\r@\f\u000eU+ cot\fsin\r@\r\u000eU;\n\u0000y=\u0000cos\r\nsin\f@\u000b\u000eU+ sin\r@\f\u000eU+ cot\fcos\r@\r\u000eU;\n\u0000z=\u0000@\r\u000eU: (11)4\nFinally, the two physical degrees of freedom in the GW can be extracted from hIJby de\fning the plus and the\ncross modes for an observer whose colatitude and azimuth are \u0012oand\u001eoin theX-Y-Zframe [27],\nh+=1\n2\u0010\n^\u0012I\no^\u0012J\no\u0000^\u001eI\no^\u001eJ\no\u0011\nhIJ=\u00001\nr\u0010\n^\u0012I\no^\u0012J\no\u0000^\u001eI\no^\u001eJ\no\u0011\nIIJ;\nh\u0002=1\n2\u0010\n^\u0012I\no^\u001eJ\no+^\u001eI\no^\u0012J\no\u0011\nhIJ=\u00001\nr\u0010\n^\u0012I\no^\u001eJ\no+^\u001eI\no^\u0012J\no\u0011\nIIJ; (12)\nwhere ^\u0012I\noand^\u001eI\noare theXYZ -components of the transverse unit vectors\n^\u0012o= cos\u0012ocos\u001eo^eX+ cos\u0012osin\u001eo^eY\u0000sin\u0012o^eZ;\n^\u001eo=\u0000sin\u001eo^eX+ cos\u001eo^eY: (13)\nNote that ^eX;^eYand^eZare the unit vectors of the X-Y-Zframe, while ^ex;^eyand^ezwill be used as the unit\nvectors of the x-y-zframe.\nIII. NUMERICAL EXAMPLES\nNow we can use the above equations to numerically calculate the GW spectra of a rotating ellipsoidal NS a\u000bected\nby Lorentz violation. To simplify the calculation of \u000eU, we assume the density of the star to be constant. Extension\nto realistic nonuniform NSs is straightforward. Then the angular parts of the integrals in \u000eUcan be carried out\nanalytically. Speci\fcally speaking, de\fne\nUij:=1\n2Z\u0000\nxi\u0000x0i\u0001\u0000\nxj\u0000x0j\u0001\njx\u0000x0j3\u001a(x)\u001a(x0)d3xd3x0; (14)\nthen they are related to the Newtonian potential\n\b =\u0000Z\u001a(x0)\njx\u0000x0jd3x0; (15)\nvia\nUij=Z\n\u001a(x)xi@j\bd3x: (16)\nThe Newtonian potential of a uniform ellipsoid is known to be [27, 28]\n\b =\u0000\u0019\u001a\u0000\nA0\u0000Axx2\u0000Ayy2\u0000Azz2\u0001\n; (17)\nwhere\nA0=axayazZ1\n0duq\n(a2x+u)(a2y+u)(a2z+u); Ai=axayazZ1\n0du\n(a2\ni+u)q\n(a2x+u)(a2y+u)(a2z+u);(18)\nwithi=x;y;z . Consequently, the nonvanishing Uijare found to be\nUxx=8\u00192\n15\u001a2Axa3\nxayaz; Uyy=8\u00192\n15\u001a2Ayaxa3\nyaz; Uzz=8\u00192\n15\u001a2Azaxaya3\nz: (19)\nFor NSs, the density varies from the center to the surface. For our purpose, we will take a uniform density of\n1015g=cm3in numerical calculations. As for the semi-axes, because NSs are compact objects having tiny deformations\nif any, we can only say that they are all about 10 km, roughly the radius of a spherical NS predicted by GR. The often\nused parameters to characterize NS deformation are the oblateness \u000fand the nonaxisymmetry \u000e. They are de\fned as\n\u000f=Izz\u0000Ixx\nIxx; \u000e =Iyy\u0000Ixx\nIzz\u0000Ixx; (20)\nwith an assumption that Izzis the largest eigenvalue of the moment of inertia tensor. NS models have suggested that\n\u000fis less than 10\u00007[29], while the magnitude of \u000eis hardly known. For demonstration, we take 0.1 for both \u000fand\u000ein\nthe following numerical examples. In addition, we use 10 km for az, and then the values of axandayare determined\nby noticing\nIxx=4\u0019\n15\u001aaxayaz(a2\ny+a2\nz); Iyy=4\u0019\n15\u001aaxayaz(a2\nx+a2\nz); Izz=4\u0019\n15\u001aaxayaz(a2\nx+a2\ny); (21)5\nX−0.5\n0.0\n0.5Y−0.50.00.5Z\n0.20.40.60.8\nX−0.5\n0.0\n0.5Y−0.50.00.5Z\n0.20.40.60.8\nFIG. 2. Illustrations for an example of the Lorentz-violating twofold precession ( left) and an example of the Lorentz-invariant\nfree precession ( right). The green trajectories trace the tail of the body-frame unit vector ^ezin the inertial frame, while the\nblue trajectory in the left plot traces the tail of the angular momentum unit vector in the inertial frame. The red arrows are\nthe body-frame unit vectors ^ex;^eyand^ezatt= 0, while the blue arrows indicate the angular momentum unit vector at t= 0.\nThe angular momentum is conserved in free precessions so the blue arrow in the right plot remains unchanged with time. The\ninitial values for both solutions are \u000bjt=0= 0,\fjt=0\u00190:762,\rjt=0=\u0019=2, and _\u000bjt=0\u00191:26,_\fjt=0= 0:5, _\rjt=0\u00190:0449. Time\nand time derivatives are dimensionless under the time unit tcgiven by Eq. (23).\nfor uniform ellipsoids.\nThen to compute \u000eUas a function of the Euler angles, we take numerical values \u0016 sXX= 0:02;\u0016sYY= 0:01;\u0016sZZ=\n\u00000:04 and \u0016sXY= \u0016sXZ= \u0016sYZ= 0 for the coe\u000ecients for Lorentz violation in the inertial frame. This means that the\naxes of the inertial frame are the principal axes of the \u0016 sijtensor. Note that this is a theoretical inertial frame \fxed by\nthe coe\u000ecients for Lorentz violation. It generally does not coincide with the widely used experimental inertial frame,\nnamely the Sun-centered celestial-equatorial frame de\fned in Ref. [11].\nAll the parameters in the Lagrangian (2) have been set now. Numerical solutions for the Euler angles can be\nobtained once initial values are given. For numerical calculations, a dimensionless parametrization for the angular\nvelocities is helpful. This can be achieved by employing a time unit. To be consistent with the choice in Ref. [23], it\nis taken to be\ntc:=r\n2Ixx\nUyy\u0000Uzz: (22)\nFor a uniform ellipsoid, keeping only the leading contribution from \u000fand\u000e, it is\ntc=s\na2y+a2z\n\u0019\u001a\u0000\nAya2y\u0000Aza2z\u0001\u0019s\n15\n4\u0019\u001a\u000f(1\u0000\u000e)\u001810\u00003s; (23)\nwhere the magnitude estimation is made for \u001a= 1015g=cm3and\u000f=\u000e= 0:1. Therefore, a dimensionless angular\nvelocity at order unity in our numerical results corresponds to about 1000 rad =s.\nFigure 2 shows the trajectories of the tail of the unit vector ^ezin the inertial frame to intuitively illustrate the\nrotations of the star for a certain set of initial values. Our examples consist of two solutions: the plot on the left\nshows a twofold precession with \u0016 sIJtaking the above said values, and the plot on the right shows a free precession\nwithout Lorentz violation for comparison. The distinction is also re\rected by the trajectories of the tail of the angular\nmomentum unit vector: in the left plot, there is a nontrivial trajectory for the angular momentum unit vector, while\nin the right plot, the angular momentum unit vector does not change with time.\nWith the two solutions, we calculate the GWs according to Eq. (12) for an observer at \u0012o= 0:8 rad and\u001eo= 0. The\nresults are presented in Fig. 3. Their Fourier transformations are shown in Fig. 4; only the plus mode is shown as the\ncross mode has very much the same spectra. The spectra of the free precession shows a fundamental angular frequency\nat about 1:4=tc, and peaks around the second harmonic at about 2 :9=tc. We know that if the star is axisymmetric, free\nprecessions generate GWs having exactly two frequencies, with one being twice of the other. The nonaxisymmetry6\n−0.50.00.5rh+[tc]twofold precession\n0 50 100 150\nt[tc]−0.50.00.5rh×[tc]\n−0.50.00.5rh+[tc]free precession\n0 50 100 150\nt[tc]−0.50.00.5rh×[tc]\nFIG. 3. GWs from a rigid body undergoing the rotations in Fig. 2. The observer receiving the waves has colatitude \u0012o= 0:8 rad\nand azimuth \u001eo= 0 in theX-Y-Zframe. The geometrized unit of time and distance is the time unit tcgiven by Eq. (23).\n0 2 4 6 8\nAngular Frequency [1 /tc]0.000.020.040.060.080.100.12F(rh+) [tc]twofold precession\nfree precession\n4.35 4.40 4.45 4.50\nAngular Frequency [1 /tc]0.00000.00050.00100.00150.00200.00250.00300.0035twofold precession\nfree precession\nFIG. 4. Fourier transformations of the rh+waves in Fig. 3. The two noticeable peaks at about 1.4 and 2.9 in the left plot are\nthe \frst and the second harmonics for both twofold precession and free precession. The modulation due to nonaxisymmetry\nis clearly represented by the adjacent peak at about 2.7 close to the second harmonic for both kinds of motion. However, the\nbarely visible tiny peaks, re\recting modulations due to Lorentz violation, only exist for twofold precession. The right plot,\nwhich zooms in on the tiny peak between 4 and 5, demonstrates the point. Note that in the plots the geometrized unit of\nFourier amplitude is tc, while the geometrized unit of angular frequency is 1 =tc.7\nhere modulates both the fundamental frequency and the second harmonic. This has been discussed in Refs. [16, 24, 25].\nWhat we are showing in the left plot of Fig. 4 tells us that the twofold precession, namely the rotation of an otherwise\nfreely precessing NS under Lorentz-violating gravity, generates similar GW frequency components. However, more\ninterestingly, in the enlarged plot on the right, we clearly see the distinction that while the twofold precession generates\nfrequency components around the third harmonic, the free precession has no component of the third harmonic at all.\nHigher frequency components exist in the spectra of the Lorentz-violating twofold precession, but they can easily be\nmissed as they are too small.\nIV. CONCLUSION\nWe have presented the analytical formulae to calculate the rotation of NSs under Lorentz-violating gravity in the\nminimal gravitational SME framework, and to construct the quadrupole GWs emitted from these NSs. Numerical\nexamples are plotted to demonstrate our conclusion that while freely precessing NSs in the Lorentz-invariant gravity\ndo not emit quadrupole GWs at frequencies higher than the second harmonic, NSs undergoing the twofold precession\ndue to Lorentz violation do. Therefore, harmonics higher than the second in the spectra of continuous GWs are\nappealing signatures of Lorentz violation. Once continuous GWs from rotating NSs are detected, a potential test of\nLorentz invariance can be performed by examining harmonics higher than the second in the spectra. However, we\ndo notice a possible di\u000eculty in this test: there might be conventional torques, like the electromagnetic spin-down\ntorque [30{33], acting on the NS to cause similar twofold precession motions and to generate higher harmonics in the\nGW spectra. Although the questions whether the twofold precession caused by Lorentz violation can be distinguished\nfrom rotations of NSs under the electromagnetic spin-down torque and whether the GW spectra of the latter have\nfrequency components higher than the second harmonic lie beyond the scope of this work, they are certainly worth\nto be investigated further. Furthermore, a statistical study of continuous GWs from an ensemble of NSs might have\nthe potential to distinguish between the two scenarios, as the Lorentz violation is universal for all NSs while the\nastrophysical torques are di\u000berent for di\u000berent systems.\nACKNOWLEDGMENTS\nWe are grateful to Marco Schreck for the invitation to submit an article to this special issue. This work was\nsupported by the National Natural Science Foundation of China (11975027, 11991053, 11721303), the National SKA\nProgram of China (2020SKA0120300), the Young Elite Scientists Sponsorship Program by the China Association\nfor Science and Technology (2018QNRC001), the Max Planck Partner Group Program funded by the Max Planck\nSociety, and the High-Performance Computing Platform of Peking University. It was partially supported by the\nStrategic Priority Research Program of the Chinese Academy of Sciences through the Grant No. XDB23010200. 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J. Zhu,\nAstrophys. J. 897, 22 (2020), arXiv:2005.06544 [astro-ph.HE].\n[21] B. Steltner, M. Papa, H.-B. Eggenstein, B. Allen, V. Dergachev, R. Prix, B. Machenschalk, S. Walsh, S. Zhu, and\nS. Kwang, (2020), arXiv:2009.12260 [astro-ph.HE].\n[22] Y. Zhang, M. A. Papa, B. Krishnan, and A. L. Watts, Astrophys. J. Lett. 906, L14 (2021), arXiv:2011.04414 [astro-ph.HE].\n[23] R. Xu, Y. Gao, and L. Shao, (2020), arXiv:2012.01320 [gr-qc].\n[24] C. Van Den Broeck, Class. Quant. Grav. 22, 1825 (2005), arXiv:gr-qc/0411030.\n[25] Y. Gao, L. Shao, R. Xu, L. Sun, C. Liu, and R.-X. Xu, Mon. Not. Roy. Astron. Soc. 498, 1826 (2020), arXiv:2007.02528\n[astro-ph.HE].\n[26] L. D. Landau and E. M. Lifshitz, Mechanics (Oxford, 1960).\n[27] E. Poisson and C. M. Will, Gravity: Newtonian, Post-Newtonian, Relativistic (Cambridge University Press, Cambridge,\nEngland, 2014).\n[28] S. Chandrasekhar and N. R. Lebovitz, Astrophys. J. 136, 1037 (1962).\n[29] B. J. Owen, Phys. Rev. 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The ideal cancell ation of irradiative losses of these ionic\nexcitations in the chain is demonstrated by inclusion of app ropriately retarded near-, medium and\nfar-field components of dipole interaction between spheres in the chain. It is proved that the Lorentz\nfriction losses in each sphere are completely compensated b y the energy income from the rest of the\nchain for a wide sector of the plasmon-polariton wave vector domain. There is shown that the\ndamping of plasmon-polariton is reduced to only residual Oh mic losses much lower than irradiation\nlosses for the separate electrolyte sphere. The self-frequ encies and the group velocities of plasmon-\npolaritons for longitudinal and transversal (with respect to the chain orientation) polarizations are\ndetermined and assessed for various ion and electrolyte par ameters. It is proved that there exist\nweakly damped self-modes of plasmon-polaritons in the chai n for which the propagation range is\nlimited by relatively small Ohmic losses only. Completely u ndamped collective waves are also de-\nscribed in the case of the presence of persistent external ex citation of some fragment of the chain.\nThe possibility of application of the plasmon-polariton mo del to describe the so-called saltatory\nconduction in periodically myelinated nerve axons is preli minarily discussed.2\nI. INTRODUCTION\nIn the previous paper [1] we have developed the description of plasm on scillations in ionic spheres in analogy to\nplasmons in metallic nanospheres. We have demonstrated the existe nce of the surface and the volume modes of\nplasmons in electrolyte spheres and analyzed their attenuation inclu ding scattering and irradiation lesses, the later\nexpressed by the Lorentz friction of oscillating charge fluctuation s. Majority of details for ionic systems remain in\nanalogy to electrons in metals though with overall scale shifted 3 −4 orders in magnitudes toward lower frequencies,\nlarger system size, and larger wave lengths of resonances.\nIn the present paper we will utilize plasmonic properties of ionic electr olyte spheres in order to discuss kinetics of\neffective surface plasmon wave-type excitations in a chain constru cted of electrolyte spheres, in analogy to plasmon-\npolaritons propagating in metallic nano-chains. A question which we tr y to discuss in the present paper is the\npossibility of similar plasmon-polaritonproperties occurrence for ion ic carriersinstead of electrons and the verification\nof efficiency of electrolyte plasmon wave-guides. The electrolyte sp heres of the chain are assumed to be closed by\nsphere-shaped membranes similar as those frequent in biological st ructures. The utilizing of ions in cell signaling,\nmembrane transfer or nerve cell conductivity are the examples of functioning of such ionic structures and the question\non the soft plasmonics usefulness in there arises.\nFor the theoretical model we will consider an infinite equidistant cha in of spheres with radius a, though a general-\nization to oblate or prolate spheroids or even other shapes of the c hain elements is natural. The ionic systems in each\nelectrolyte sphere we will model upon the developed in the previous p aper [1] the simplified effective jellium approach\nfor ions. Taking in mind that metallic nanochains create very effective low-noise wave-guides for electromagnetic\nsignals in form of the wave-type plasmon-polaritons, we will model th e similar phenomenon in ionic sphere chains.\ntransversal polarization\n longitudinal polarization\nx\nyzchain of ionic spheres\nchain of ionic spheroids\n − +D\nlongituninal dipoleV\n− + − + − + − +\nFIG. 1. Schematic presentation of the chain of ionic spheres with propagating plasmon-polariton (upper); the dipole Dcreation\nin single spheroid and wave type dipole excitation in the cha in of spheroids (the longitudinal mode is depicted in the pro late\ngeometry) traversing the chain with the group velocity v(lower)\nIn the next section of the present paper we will derive the fundame ntal equation for dynamics of collective surface\nplasmons in the ionic sphere chain including the surface plasmon self-o scillations in each chain component, the\nJoule heat damping of plasmons due to irreversible energy dissipation (Ohmic losses) by microscopic scattering of\nions, the irradiation losses described by the Loretz friction hamper ing the charge movement and the energy mutual\nsupplementation in the chain via radiation of all elements of the chain. The balance of the energy will be analyzed\nin order to detect low-noise windows for ionic plasmon-polariton prop agation. The solution of dynamic equation\nin the homogeneous and inhomogeneous cases will be presented. All the detailed model calculations will be shifted\nto Appendices. In the last but one Appendix the precise analysis of s ome singularities occurring in the dynamical\nequation for plasmon-polariton in the chain and caused by the medium - and far-field components of dipole inter-\nsphere interaction in the chain will be presented in analogy to similar fe atures in metallic nanochains. In the last\nAppendix some links to electric signaling in nerve axons are preliminary d iscussed from point of view of the possible\napplication of the developed description of ionic plasmon-polaritons in ionic periodic systems. The explanation of\nso-called saltatory conduction in myleanated nerve axons is sugges ted by utilizing of plasmon-polaritons propagation\nin periodic linear ionic structure.3\nII. DYNAMIC EQUATION FOR COLLECTIVE SURFACE PLASMONS IN A CH AIN OF SPHERICAL\nIONIC SYSTEMS\nNow we will consider an infinite chain of equidistantly placed ionic spheric al systems with radius aseparated with\nthe distance dbetween the sphere centers aligned along the axis z. The question is to describe collective surface dipole\nplasmons in such system in analogy to propagation of plasmon-polarit ons in metallic nano-chains [2, 3]. Because of\nsimilarity of ionic plasmon excitations in finite electrolyte systems to pla smons in metallic nanosphere, one can expect\nalso a similarity in collective plasmon behavior in complex ionis systems to m etallic ordered nano-structures. Even\nthough in finite electrolyte systems plasmon oscillations have much low er energy in comparison to plasmons in metals,\ntheir irradiation properties with typical size-dependence may admit an effective plasmon energy transfer along the\nionic chain in analogy to metallic nano-chains. Such electrolyte chains w ould serve as the wave-guides for plasmon\nsignals along ionic structures with potential link to some biological ionic information channels.\nThe main property of metallic plasmonic wave-guides is their perfect t ransmittance, i.e., the absence of irradiation\nlosses and the reducing of signal damping to only Ohmic attenuation, which makes possible a long range almost\nundamped propagation. The second property is the reduction of s ignal group velocity (in metallic chains) to at least\noneorderlowervaluein comparisonto lightvelocityallowingsubdiffract ionarrangementofplasmon-polaritoncircuits.\nThoughwithshorterwave-lengthcollectiveplasmonexcitationsinth echainincomparisontothesameenergyphotons,\nthe former would efficiently transport information signals over large distances, with transfer parameters controlled\nin wide range by chain size and geometry. If these properties could b e repeated in ionic periodic structures (with\nthe appropriately shifted scale of corresponding quantities), the y would be of some importance for understanding of\nsignal transfer in biological systems.\nPeriodicity of the chain makes the system similar to a 1D crystal. The in teraction between chain elements can\nbe considered as of dipole-type coupling when surface dipole plasmon s are excited in them. For ionic sphere chain\none can adopt the result from metallic chain analysis, which justify th e dipole model of sphere interactions when the\ndistance between neighboring spheres is larger than the sphere ra dius, i.e., when d >3a(otherwise the multipole\nchannels may contribute inter-sphere interaction) [3–5]. One can n otice also, that the model of interacting dipoles\n[6, 7] was developed originally for description of stellar matter [8, 9] a nd next it has been adopted to metal particle\nsystems [10, 11].\nThe dipole interaction resolvesitself to electric and magnetic field cre ated by oscillating dipole D(r,t) in any distant\npoint. If this this point is marked by the vector r0(fixed to the end of r, where the dipole is placed), the electric field\nproduced by the dipole D(r,t) has the following form, including relativistic retardation [12, 13]:\nE(r,r0,t) =1\nε/parenleftBig\n−∂2\nv2∂t21\nr0−∂\nv∂t1\nr2\n0−1\nr3\n0/parenrightBig\nD(r,t−r0/v)\n+1\nε/parenleftBig\n∂2\nv2∂t21\nr0+∂\nv∂t3\nr2\n0+3\nr3\n0/parenrightBig\nn0(n0·D(r,t−r0/v)),(1)\nwhere,n0=r0\nr0andv=c√ε,εis the dielectric susceptibility of the medium surrounding the chain. Th e terms\nwith the denominators r3\n0,r2\n0andr0are usually called near-field, medium-field and far-field interaction co mponents,\nrespectively. The above formula allows for description of mutual int eraction of plasmon dipoles on each sphere in the\nchain. The spheres in the chain are numbered by integer land the equation for the surface plasmon oscillation of the\nlthsphere can be written as follows,\n/bracketleftBig\n∂2\n∂t2+2\nτ0∂\n∂t+ω2\n1/bracketrightBig\nDα(ld,t)\n=εω2\n1a3m=∞/summationtext\nm=−∞, m/negationslash=lEα/parenleftBig\nmd,t−|l−m|d\nc/parenrightBig\n+εω2\n1a3ELα(ld,t)+εω2\n1a3Eα(ld,t).(2)\nThefirsttermofther.h.s. inEq. (2)describesthedipolecouplingbet weenspheresandtheothertwotermscorrespond\nto plasmon attenuation due to the Lorentz friction (as described in the previous paper [1] addressed to the single ionic\nsphere) and the forcing field due to an external electric field, corr espondingly, ω1=omega p√\n3εis the Mie frequency of\ndipole surface plasmons [1]. The Ohmic losses are included by the term [1 4],\n1\nτ0=v\n2λB+Cv\n2a, (3)\nwhereλBis the mean free path of carriers in bulk electrolyte, vis the mean velocity of carries at the temperature\nT,v=/radicalBig\n3kT\nm,mis the mass of the effective ion, kis the Boltzmann constant, Cis the constant of unity order, a\nis the sphere radius. The first term in the expression for1\nτ0approximates ion scattering losses, the same as in bulk4\nelectrolyte, whereas the second term displays lossesdue to scatt ering of ions on boundary of the sphere with the radius\na. The index αindicates polarization, the longitudinal one, when α=zand the transversal one, when α=x(y) (the\nchain orientation is assumed in zdirection). According to Eq. (1) we get in Eq. (2),\nEz(md,t) =2\nεd3/parenleftBig\n1\n|m−l|3+d\nv|m−l|2∂\n∂t/parenrightBig\n×Dz(md,t−|m−l|d/v),\nEx(y)(md,t) =−1\nεd3/parenleftBig\n1\n|m−l|3+d\nv|m−l|2∂\n∂t+d2\nv2|l−d|∂2\n∂t2/parenrightBig\n×Dx(y)(md,t−|m−l|d/v).(4)\nDue to periodicity of the chain (in analogy 1D crystal) one can assume the wave type collective solution of the\ndynamical equation (2),\nDα(ld,t) =Dα(k,t)e−ikld,\n0≤k≤2π\nd.(5)\nThis corresponds to the form of a solution in the Fourier picture of E q. (2) (the discrete Fourier transform (DFT)\nwith respect to the positions and the continuous Fourier transfor m (CFT) with respect to time) similar to that one as\nused in the case of phonons in 1D crystal. Let us note that the DFT is defined for a finite set of numbers, therefore\nwe can consider the chain with 2 N+ 1 spheres, i.e., the chain of finite length L= 2Nd. Then, for any discrete\ncharacteristics f(l), l=−N,...,0,...,Nof the chain, like a selected polarization of dipole distribution, one dea ls with\nDFT picture f(k) =N/summationtext\nl=−Nf(l)eikld, wherek=2π\n2Ndn, n= 0,...,2N. This means that kd∈[0,2π) due to periodicity\nof the equidistant chain. On the whole system the Born-Karman bou ndary condition is imposed resulting in the form\nofkas given above. For the infinite length of the chain one can take finally the limit N→ ∞which causes that the\nvariable kis quasi-continuous, but still kd∈[0,2π).\nIn the manner which is described in details in Appendix A, we arrive at th e Fourier representation of Eq. (2) in\nthe following form,\n/parenleftBig\n−ω2−i2\nτ0ω+ω2\n1/parenrightBig\nDα(k,ω)\n=ω2\n1a3\nd3Fα(k,ω)Dα(k,ω)+εa3ω2\n1E0α(k,ω),(6)\nwith\nFz(k,ω) = 4∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3cos(mωd/v)\n+ωd/vcos(mkd)\nm2sin(mωd/v)/parenrightBig\n+2i/bracketleftbigg\n1\n3(ωd/v)3+2∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3sin(mωd/v)\n−ωd/vcos(mkd)\nm2cos(mωd/v)/parenrightBig/bracketrightBig\n,\nFx(y)(k,ω) =−2∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3cos(mωd/v)\n+ωd/vcos(mkd)\nm2sin(mωd/v)−(ωd/v)2cos(mkd)\nmcos(mωd/v)/parenrightBig\n−i/bracketleftbigg\n−2\n3(ωd/v)3+2∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3sin(mωd/v)\n+ωd/vcos(mkd)\nm2cos(mωd/v)−(ωd/v)2cos(mkd)\nmsin(mωd/v)/parenrightBig/bracketrightBig\n.(7)\nOne can analytically calculate the sums in functions ImFz(k,ω) andImFx(y)(k,ω) corresponding to the radiative\ndamping for longitudinal and transversal plasmon-polariton modes , when within the perturbative approach one takes\nω=ω1in these functions, as is done in Appendix A—Eqs (A6) and (A8). Both these functions perfectly vanish when\n0< kd±ωd/v <0 (the corresponding region is indicated in Fig. 3). Outside this region radiative damping is not\nzero, which for longitudinal and transversal modes is illustrated in F igs 4 and 5, respectively.\nIII. PLASMON-POLARITON SELF-MODES IN THE CHAIN AND PROPAGA TION OF E-M SIGNAL\nALONG PERIODIC IONIC STRUCTURE\nThe real part of the functions Fαrenormalizes the self-frequency of plasmon-polaritons in the chain , while its\nimaginary part renormalizes damping rate of these modes. ReFα(k,ω) andImFα(k,ω) are functions of kandω.5\nWithin the first order approximation one can put ω=ω1inReFαand also in residual nonzero ImFα, in the latter\nfunction, outside the region 0 < kd±ω1d/v <2π. Let us emphasize, however, that vanishing of ImFα(k,ω) inside\nthe region 0 < kd±ωd/v <2πholds for any value of ω[15] (thus also for exact solution for frequency and not only\nfor approximated ω=ω1).\nHence, within the perturbative approach one can rewrite the dyna mic equation (6) for plasmon-polariton modes in\nthe chain in the following form,\n/parenleftBig\n−ω2−i2\nτα(k)ω+ωα(k)2/parenrightBig\nDα(k,ω) =εa3ω2\n1E0α(k,ω), (8)\nwhere the renormalized attenuation rate,\n1\nτα(k)=/braceleftBigg1\nτ0, for0< kd±ω1d/v <2π,\n1\nτ0+a3ω1\n2d3ImFα(k,ω1), for kd −ω1d/v <0or kd+ω1d/a >2π,(9)\nand the renormalized self-frequency,\nω2\nα(k) =ω2\n1/parenleftbigg\n1−a3\nd3ReFα(k,ω1)/parenrightbigg\n. (10)\nEq. (8) can be easily solved both for inhomogeneous and homogeneo us (when E0α= 0) case. The general solution\nof Eq. (8) has a form of sum of the general solution of the homogen eous equation and of a single particular solution\nof the inhomogeneous equation. The first one includes initial conditio ns and describes damped self-oscillations with\nfrequency,\nω′\nα=/radicalBigg\nω2α(k)−1\nτ2α(k), (11)\ni.e., for each kandα,\nD′\nα(k,t) =Aα,kei(ω′\nαt+φα,k)e−t/τα(k), (12)\nwhere the constants Aα,kandφα,kare adjusted to the initial conditions.\nFor the inhomogeneous case the particular solution is as follows:\nD′′\nα(k,t) =εa3ω2\n1E0α(k)ei(γt+ηα,k) 1/radicalBig\n(ω2α(k)−γ2)2+4γ2\nτ2α(k), (13)\nsuitably to assumed single Fourier time-component of E0α(k,t) =E0α(k)eiγt, andtg(ηα,k) =2γ/τ(α,k)\nω(α,k)2−γ2as usual\nfor a forced oscillator. Let us emphasize that E0α(k) is the real function for E0α(ld)∗=E0α(ld) =E0α(−ld). An\nappropriate choice of the latter function, in practice a choice of th e number of externally excited spheres in the\nchain, e.g., by suitably focused external excitation, allows for mode ling of its Fourier picture E0α(k). This gives\nthe envelope of the wave packet if one inverts the Fourier transfo rm in the solution given by Eq. (13) back to\nthe position variables. For the case of external excitation of only s ingle sphere, the wave packet envelope includes\nhomogeneously all wave vectors kd∈[0,2π). The larger number of spheres is simultaneously excited the narro wer\ninkwave packet envelope can be selected. For E0α(ld)∗=E0α(ld) =E0α(−ld) the Fourier transform has the same\nproperties, i.e., E0α(k)∗=E0α(k) =E0α(−k) and the latter equality can be rewritten, due to the period2π\ndfork, as,\nE0α(−k) =E0α(2π\nd−k) =E0α(k). The inverse Fourier picture to Eq. (13) (its real part),\nD′′\nα(ld,t) =2π/d/integraldisplay\n0dkcos(kld−γt−ηα,k)εa3ω2\n1E0α(k)1/radicalBig\n(ω2α(k)−γ2)2+4γ2\nτ2\nα(k). (14)\nThis integral can be rewritten by virtue of the mean value theorem in the following form,\nD′′\nα(ld,t) =2π\ndcos(k∗ld−γt−ηα,k∗)εa3ω2\n1E0α(k∗)1/radicalBig\n(ωα(k∗)2−γ2)2+4γ2\nτ2α(k∗). (15)6\nThe above expression describes the undamped wave motion with the frequency γand the velocity, amplitude and\nphase shift determined by k∗. The amplitude attains its maximal value at the resonance, when\nγ=ωα(k∗)/radicalBigg\n1−2\n(τα(k∗)ωα(k∗))2. (16)\nIn the chain being the subject of persistent time dependent electr ic field excitation applied to some number (even\nsmall number) of spheres, one deals with undamped wave packed pr opagating along the whole chain. Such modes\ndepending of particular shaping of the wave packet by specific choic e of the chain excitation, may be responsible\nfor experimentally observed long range, practically undamped plasm on-polariton propagation in metallic nano-chains\n[14, 16–18]. The similar behavior is expected also in ionic micro-chains, d ue to the shift of all size parameters toward\nthe micro-scale for plasmon-polaritons in ionic systems.\nThe self-modes described by Eq. (12) are damped and their propag ation depends on appropriately prepared initial\nconditions admitting nonzero values of Aα,k. These initial conditions might be prepared by switching off the time\ndependent external electric field exciting initially some fragment of t he chain. The resulting wave packet may embrace\nthe wave-numbers kfrom some region of [0 ,2π). If only wave-numbers k, for which 0 < kd±ω1d/v <2πcontribute\nto the wave packet, its damping is only of Ohmic-type (as is shown in Ap pendix A). The value of corresponding1\nτ0\nlowers with growing a(cf. Eq. (3))—thus for longer range of these excitations in the ch ain the larger spheres are\nfavorable. The limiting value of Ohmic losses rate for large spheres is1\nτ0→v\n2λB1/s, which gives the maximal range of\npropagation for irradiation-free modes of plasmon-polariton. It d epends both of mean velocity of ions, vand of their\nmean free path length, λB; for exemplary v∼1100 m/s and λB∼50×10−8m, one obtains, for the group velocity of\nthe wave packed assumed as ∼0.01cm/s, the range of ionic plasmon-polariton ∼3×10−3m. The precisely estimated\ngroup velocities of plasmon-polaritons for both polarizations are pr esented in Appendix B.\nIV. CONCLUSIONS\nThe presented above analysis, including placed to Appendices quant itative estimations, demonstrates that in chains\nbuilt of electrolyte finite systems (for the model assumed in form of spheres with radius ain size-scale of micrometers)\nmay be excited collective wave types surface plasmon modes similar to plasmon-polariton in metallic nano-chains.\nThese ionic plasmon-polaritons can efficiently transfer information a nd energy over relatively large distances and\nionic system chains can act as low-losses plasmon wave-guides. Thes e features of the ionic chains, convenient for\nplasmon-polariton kinetics, are related to the following properties:\n•Ideal cancelation to zero of irradiation losses of collective surface plasmon in the chain for both polarizations\nof oscillations. This happens due to ideal compensation of irradiation energy losses caused by the Lorentz\nfriction of oscillating charges in each sphere of the chain by radiative income from all neighboring spheres. The\ncontributions of mutual radiation of all spheres in the chain, includin g near-, medium- and far-field components\nof dipole interaction, with precisely accounted for retardation effe cts, perfectly reduce all irradiation losses and\nsqueeze all the electro-magnetic field to interior of the chain result ing in ideal plasmon-polariton guidance along\nthe ionic chain. The distance of this plasmon-polariton propagation in the chain is limited only by residual\nOhmic type losses caused by microscopic scattering of ions, including collisions with other ions, with solvent\nparticles and admixtures and with boundaries of electrolyte sphere s. It must be emphasized that the ionic\nplasmon damping caused by these irreversible scattering processe s are much smaller than the irradiation losses\nin a single separate electrolyte sphere with large radius (of several micrometer order depending on electrolyte\nparameters). Note that the residual heat losses (Ohmic losses) o f plasmon-polaritons in the chain may be\ncompensated by not high outside energetical supporting in the for m of persistent excitation or coupling to an\nactive medium (for metal nanochain such an active medium may be the system of quantum dots coupled with\nmetallic nanospheres in the chain and working in regime of spaser [19]). The free kinetics of ionic plasmon-\npolaritons in the chain can reach milimeter scale for the range of prop agation, whereas in the case of external\ncompensation of heat dissipation may be practically unlimited.\n•Thegroupvelocityforionicplasmon-polaritonsmayvaryinawiderang edependingontheelectrolyteparameters\n(mostly on the ion concentration) and on the particularities of signa l wave packet shaping (selecting appropriate\nenvelope for the wave vectors included to the packet of plasmon-p olariton modes). Typically, the ionic plasmon-\npolariton mode velocities fall on the scale of at least two orders lower than light velocity for ion concentration\n∼10−2N0(N0is one-molar concentration) despite the very low mean value of veloc ity of ions in electrolytes\n(due to large their mass). The value of velocities of ionic plasmon-pola ritons in the chains is correlated also to\nits frequency, and for dense ion concentration is by 3 −4 orders lower than for plasmon frequency in metals (i.e.,7\n∼1011−121/s in comparison to 1015−161/s in metals). For small ion concentrations the group velocity can b e\nmore significantly lowered.\nThe main properties of ionic plasmon-polaritons as listed above, are a ssociated also with many specific particular-\nities, as for instance, with a local growth of the group velocities for both plarizations of oscillations in close vicinity\nof wave vectors satisfying interference condition, 0 = kd−ωd/v= 2π(kis a wave vector, dis spacing in the chain,\nωis the plasmon-polariton frequency, here v=c/√εis the light velocity in electrolyte medium) and occurring due\nto constructive interference of chain element radiation in far- and medium-field zones. Remarkably, the singular\ncontributions to dipole interaction are effectively quenched by nonlin earity of the dynamics and do not lead to the\ndivergences of the group velocities, though result in local maxima of truncated singularities, sharply cut on the light\nvelocity level (as presented in Appendix C). This local enhancement of the group velocity may lead to an occurrence\nof fast-moving but narrow wave packets of plasmon-polaritons in io nic chains taking advantage of higher velocity of\nplasmon-polariton modes close to singular points. It must be, howev er, emphasized that this property is efficiently\nquenched in finite length chains as for fully development of mentioned singularities the contribution of infinite num-\nber (in practice, of large number) of spheres in the chain is needed. The other properties of the plasmon-polariton\ndynamics in the chain turn out, however, much more robust against shortening of the chain, because the very quick\nconvergences of series representing mutual radiation influence o f chain elements, except for singular once (which needs\nmany elements to develop the divergence).\nIn the context of the rich physics of plasmons-polaritonsin ionic sys tems repeating properties of plasmon-polaritons\nin metallic nano structures, there arises the question on how to ver ify them experimentally. Ionic systems with\nsufficiently high ion concentrations are referred rather to liquid elec trolytes which do not allow for simple their\nfragmentation into spheres and chains. Nevertheless, the finite e lectrolyte systems confined by appropriately formed\nmembranes might serve as the practical realization of model ionic sy stems. Such matter organization is frequent in\nbiological systems and moreover, in dimension scale which well fits with size scale typical for ionic plasmon effects,\ni.e., the micrometer scale. Cells and internal their components are fr equently spheroid-shape structures and ionic\neffects are of primary significance for their functionality and cell sig naling. Looking for some linkage with plasmonic\ncollective effects of ions involved in these systems possess some nov elty aspect of approach. The very efficient low-\ndamped and relatively fast kinetics of plasmon-polaritonsin ionic syst em chains is an attractiveopportunity especially\nwith by several orders lower frequencies of plasmon oscillations of io ns in comparison to high frequency of plasmons\nin metals, moreover, tunable in wide range by ion concentration. Esp ecially interesting is the question of possibility\nfor modeling of ionic plasmon-polariton signaling in nerve cells, taking int o account the structure of the long axons\nassociated with prolate-spheroid shape Schwann cells attaching th e myelin sheath of axon and separated by Ranvier\nnodes (cf. Fig. 2). Though the core of the axon is a long continuous thin nerve cell insertion, its electric conductivity\nis relatively poor and there is commonly approved that the Schwann c ell structure (resembling ionic spheroid chain)\nis crucial in acceleration and maintenance of signals over long distanc es. The myelin sheath must be sufficiently thick\nto assure proper functionality of neuron linkage and a myelin deficien cy result in slowing down of the signals as it\nis encountered in Multiple Sclerosis disease. The latter may be caused not by a complete removing of the myelin\ncover but even only by diminishing its thickness. Note, however, tha t the of lower thickness myelin sheath still\nefficiently isolates electrically the core of axon from inter-cell surro undings but probably is insufficient and too thin\nfor dielectric separation needed for plasmon-polariton forming. Th is may be linked with a distinct role of myelin than\nonly electrical isolation, and its significant function in creation of app ropriately large dielectric/electrolyte periodic\nsegmentsrequiredto optimize plasmon-polaritonkinetics. In possib le plasmonicmechanismofsignalconductingalong\nthe axon it must contribute also electro-chemical mechanism of pola rization/depolarization of those fragments of the\nmembrane which are unmyelinated in Ranvier nodes. The signal depen dent reversible releasing of ions trough the\nlipid membrane sharply enhances and subsequently reduces its polar isation upon the cyclic scheme of neuron activity\nsignal, called as action potential. The incoming to the Ranvier node rela tively low signal triggers the opening of\nNa+andK+inter-membrane ion channels, which results in characteristic larger activation signal due to the transfer\nthrough opened gates of ions caused by different their concentra tions on both sides of the membrane. The whole\ncycle takes the time of few miliseconds, but initial rising of polarization due to quick Na+channel happens on single\nmilisecond scale. As the myelin layersurroundedbythe Schwanncell p rohibitsion across-membranetransfer, the local\npolarization/depolarizationofthe internal cytoplasmaofaxon tak esplace only in the Ranviernodes, which strengthen\nthe dipole of the axon fragmentwraped by the Schwanncell. If the p lasmon-polaritonfrequency is accomodated to the\ntime scale of activity of ion gate complex in the Ranvier nodes, then th e plasmon-polariton mode traversing the axon\nstructure with periodic polarized segments wrapped by Schwann ce lls can realize the one-by-one ignition of Ranvier\nnode sequence. The triggering role of plasmon-polariton might elucid ate thus how the action signal jump between\nneighboring active nodes in which the dipole oscillation amplitude is raised by functioning of nonlinear block of two\nion-channels for Na+andK+. Due to the nonlinearity (the mutually dependent feedbacks of bot h ion channels) of\nthis block the polarization amplitude saturates on the constant leve l (it is kept the same for all nodes). These active\nelements of the chain play the role of external energy supply which c ompensates Ohmic loses of ions and assure long8\ndistance transduction ofthe e-m signal with constantamplitude alo ngthe axon. The velocityofthe plasmon-polariton\nin this chain combined with blocks in Ranvier nodes must be accomodate d to plasmon frequency as is illustrated in\na simplified manner in Appendix D. The direct calculus of ion density part icipating in forming of plasmon-polariton\nwith frequency of sub-milisecond order, may give the velocity of plas mon-polariton of order of 100 m/s (cf. Appendix\nD), which well fits to observed signal velocity in long myelinated axons .\nAxon NucleusDendrite \nCell body \nMyelin sheath Node of Ranvier Axon terminal \nSchwann cell \nFIG. 2. The scheme of the long myelinated axon with chain of Sc hwann cells of ca 100 µm length periodically repeated sectors,\nseparated by unmyelinated Ranvier nodes, which gives of ord er of 10 000 segments per 1 m of the axon length\nThe plasmon-polariton scenario in the ionic chain, though simplified in co mparison to the real structure of neurons,\nmighthavesomethingincommonwiththeveryefficientandenergetica llyeconomicalelectricsignalinginnervesystems\ndespite rather poor ordinary conductivity of axons. To avoid slowin g down nerve signaling, the plasmon-polariton\nkinetics in periodic structure of axon is a convenient possibility which g uarantees high level of its performance: the\nfast signal propagation without any irradiation losses independent ly of very low ordinary conductivity, and with\npractically unlimited range when energy is persistently supplied. The la tter is produced by ATP/ADP mechanism in\ncells, energetically contributing to signal dependent opening and clo sing ofNa+andK+channels in Ranvier nodes\nand next in restoring of steady conditions, i.e., via an active transpo rt of ions across the membrane counter to the ion\nconcentration gradient. Moreover, the coincidence of micromete r scale of axon periodic structure of Schwann cells (of\nca. 100µm oflength) with typical for ionic chain size-requirements, support s the suggested plasmon-polaritonconcept\nfor explanation of action potential transduction along the axon, a s presented by the tentative fitting in Appendix D.\nIt must be also emphasized that plasmon-polariton does not interac t with electromagnetic-wave, or equivalently,\nwith photons (even with adjusted energy), which is a consequence of the large difference of group velocity of plasmon-\npolariton and of photon velocity ( c/√ε). The resulting large incommensurability of wave length of photon an d\nplasmon-polariton with the same energy, prohibits a mutual transf orming of both excitations owing to the momentum\nconservation constraints. Therefore plasmon-polariton signaling by dipole oscillations along the chain, cannot be\nneither detected nor perturbed by the external electromagnet ic radiation. Worth noting is also the independence\nof surface plasmon frequencies of the temperature in opposition t o the volume plasmon frequencies, as it has been\nshown in the previous our paper [1]. For plasmon-polaritons the temp erature influences, however, the mean velocity\nof ions,v=/radicalBig\n3kT\nm, which enhances Ohmic losses with the temperature rise (cf. Eq. (3 )) and therefore strengthens\nplasmon-polariton damping. Hence, at higher temperatures the hig her external energy supplementation is required to\nmaintain the same long range propagation of plasmon-polariton with t he same amplitude. This property is important\nfrom point of view of nerve signaling and seems to agree with the obse rvations.\nAcknowledgments Authors acknowledge the support of the present work upon the N CN project\nno. 2011/03/D/ST3/02643 and the NCN project no. 2011/02/A/ ST3/00116.\nAppendix A: Calculation of radiative damping of plasmon-po lariton in the chain\nBoth sides of Eq.(2) can be multiplied byei(kld−ωt)\n2π, and next one can perform the summation with respect to sphere\npositions and the integration over t. Taking into account that,\n1\n2π∞/integraltext\n∞N/summationtext\nl=−NDα/parenleftbig\n±md+ld;t−md\nv/parenrightbig\ne−i(kld−ωt)\n=ei(∓kmd+ωmd\nv)Dα(k,ω),(A1)9\none obtains thus the following equation in Fourier representation (t he discrete Fourier transform for sphere positions\nand the continuous Fourier transforms for time),\n/parenleftBig\n−ω2−i2\nτ0ω+ω2\n1/parenrightBig\nDα(k,ω)\n=ω2\n1a3\nd3Fα(k,ω)Dα(k,ω)+εa3ω2\n1E0α(k,ω),(A2)\nwherek=2πn\n2Nd, n= 0,1,...,2N, i.e.,kd∈[0,2π) due to periodicity of the chain with equidistant separation dof\nspheres (separation between sphere centers), and the form of kis due to Born-Karman boundary condition with the\nperiodL= 2Nd. ForN→ ∞(infinite chain limit) kis a quasi-continuous variable. In Eq. (A2),\nFz(k,ω) = 4∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3cos(mωd/v)+ωd/vcos(mkd)\nm2sin(mωd/v)/parenrightBig\n+2i/bracketleftbigg\n1\n3(ωd/v)3+2∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3sin(mωd/v)\n−ωd/vcos(mkd)\nm2cos(mωd/v)/parenrightBig/bracketrightBig\n,\nFx(y)(k,ω) =−2∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3cos(mωd/v)+ωd/vcos(mkd)\nm2sin(mωd/v)\n−(ωd/v)2cos(mkd)\nmcos(mωd/v)/parenrightBig\n−i/bracketleftbigg\n−2\n3(ωd/v)3+2∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3sin(mωd/v)ωd/vcos(mkd)\nm2cos(mωd/v)\n−(ωd/v)2cos(mkd)\nmsin(mωd/v)/parenrightBig/bracketrightBig\n.(A3)\nSome summations in the above equations can be performed analytica lly [20],\n\n\n∞/summationtext\nm=1sin(mz)\nm=π−z\n2, for0< z <2π,\n∞/summationtext\nm=1cos(mz)\nm=1\n2ln/parenleftBig\n1\n2−2cos(z)/parenrightBig\n,\n∞/summationtext\nm=1cos(mz)\nm2=π2\n6−π\n2z+1\n4z2, for0< z <2π,\n∞/summationtext\nm=1sin(mz)\nm3=π2\n6z−π\n4z2+1\n12z3, for0< z <2π.(A4)\nUsing the above formulae one can show that if 0 < kd±ωd/v <2π, then,\nImFz(k,ω) = 2∞/summationtext\nm=1/bracketleftBig\nsin(m(kd+ωd/v))−sin(m(kd−ωd/v))\nm3\n−(ωd/v)cos(m(kd+ωd/v))+cos(m(kd−ωd/v))\nm2/bracketrightBig\n+2\n3(ωd/v)3\n= 2/bracketleftBig\nπ2\n6(kd+ωd/v)−π\n4(kd+ωd/v)2+1\n12(kd+ωd/v)3/bracketrightBig\n−2/bracketleftBig\nπ2\n6(kd−ωd/v)−π\n4(kd−ωd/v)2+1\n12(kd−ωd/v)3/bracketrightBig\n−2(ωd/v)/bracketleftBig\nπ2\n6−π\n2(kd+ωd/v)+1\n4(kd+ωd/v)2/bracketrightBig\n−2(ωd/v)/bracketleftBig\nπ2\n6−π\n2(kd−ωd/v)+1\n4(kd−ωd/v)2/bracketrightBig\n+2\n3(ωd/v)3≡0.(A5)\nHowever, if kd−ωd/v <0 orkd+ωd/v >2πfor some values of the wave vector k, the more general formula must\nbe used (by application of Heaviside step function one can extend th e formulae (A4) to the second period of their left10\nsides). This extended form for ImFz(k,ω) is as follows (here we use dimensionless variables x=kd, y=d/a),\nImFz(k,ω) = Θ(2π−x−ωya/v)2/bracketleftBig\nπ2\n6(x+ωya/v)−π\n4(x+ωya/v)2+1\n12(x+ωya/v)3/bracketrightBig\nΘ(−2π+x+ωya/v)2/bracketleftBig\nπ2\n6(x+ωya/v−2π)−π\n4(x+ωya/v−2π)2+1\n12(x+ωya/v−2π)3/bracketrightBig\n−Θ(x−ωya/v)2/bracketleftBig\nπ2\n6(x−ωya/v)−π\n4(x−ωya/v)2+1\n12(x−ωya/v)3/bracketrightBig\n−Θ(−x+ωya/v)2/bracketleftBig\nπ2\n6(x−ωya/v+2π)−π\n4(x−ωya/v+2π)2+1\n12(x−ωya/v+2π)3/bracketrightBig\n−Θ(2π−x−ωya/v)2(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x+ωya/v)+1\n4(x+ωya/v)2/bracketrightBig\n−Θ(−2π+x+ωay/v)2(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x+ωya/v−2π)+1\n4(x+ωya/v−2π)2/bracketrightBig\n−Θ(−x+ωay/v)2(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x−ωya/v+2π)+1\n4(x−ωya/v+2π)2/bracketrightBig\n−Θ(x−ωay/v)2(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x−ωya/v)+1\n4(x−ωya/v)2/bracketrightBig\n+2\n3(ωay/v)3.(A6)\nThe function given by Eq. (A6) is depicted in Fig. 4. The expression (A 6) allows to account for the inconsistence of\nperiodic functions given by the sums of sines and cosines with non per iodic Lorentz friction term and disagreement of\narguments kd±ωd/vof trigonometric functions out of the first period. In Fig 3 we have p lotted the solution of the\nequation ( kd−ωd/v)(kd+ωd/v−2π) = 0,which determines the region for kd(denoted by x) versusd/a(denoted by\ny) inside which the exact cancellation of the Lorentz friction by radiat ive energy income from other nanospheres takes\nplace. In Fig. 4, the comparison of this cancellation for various nano sphere diameters is presented, for longitudinal\npolarization of plasmon collective excitations.\nFIG. 3. The region (gray, 0 < kd±ω1d/v <2π) in where the radiation losses vanish for infinite chains of e lectrolyte spheres\nwith various sphere radii and chain-separation d/a∈[3,6], forω=ω1= 1.2×1012(3.8×1013) 1/s for ion concentration\nn= 10−3(−2)N0,N0is one-molar concentration—left(right), v=c\nThe similar analysis can be done for transversal polarization, i.e., for ImFx(y)(k,ω). This function is exactly zero\nonly in the region for arguments 0 < kd−ωd/v <2πand 0< kd+ωd/v <2π, where one can write,\nImFx(y)(k,ω) =−∞/summationtext\nm=1/bracketleftBig\nsin(m(kd+ωd/v))−sin(m(kd−ωd/v))\nm3\n−(ωd/v)cos(m(kd+ωd/v))+cos(m(kd−ωd/v))\nm2\n−(ωd/v)2sin(m(kd+ωd/v))−sin(m(kd−ωd/v))\nm/bracketrightBig\n+2\n3(ωd/v)3\n=−/bracketleftBig\nπ2\n6(kd+ωd/v)−π\n4(kd+ωd/v)2+1\n12(kd+ωd/v)3/bracketrightBig\n+/bracketleftBig\nπ2\n6(kd−ωd/v)−π\n4(kd−ωd/v)2+1\n12(kd−ωd/v)3/bracketrightBig\n+(ωd/v)/bracketleftBig\nπ2\n6−π\n2(kd+ωd/v)+1\n4(kd+ωd/v)2/bracketrightBig\n+(ωd/v)/bracketleftBig\nπ2\n6−π\n2(kd−ωd/v)+1\n4(kd−ωd/v)2/bracketrightBig\n1\n2(ωd/v)2[π−kd−ωd/v]−1\n2[π−kd+ωd/v]+2\n3(ωd/v)3≡0.(A7)\nNevertheless, outside the region 0 < kd±ωd/v <2πthe value of ImFx(y)is not zero, as it is demonstrated in Fig.11\nd/c613a\nd/c614a\nd/c616a\n0 1 2 3 4 5 6/c451012345\nkdz d/c613a\nd/c614a\nd/c616a\n0 1 2 3 4 5 6024681012\nkdza3 iond/c613a\nd/c614a\nd/c616a\n0 1 2 3 4 5 602468101214\nkdza10mion\nd/c613a\nd/c614a\nd/c616a\n0 1 2 3 4 5 6/c45101234\nkdza 5/c109mion\nFIG. 4. The function ImFz(k;ω=ω1) for infinite chains of ionic spheres with radius aand chain-separations d= 3a,4a,5a\nfor effective ions mass m= 104me, effective charge q= 3e, dielectric susceptibility of surroundings ε= 2,T= 300 K, ion\nconcentration n= 10−3N0(upper), n= 10−2N0(lower), N0is one-molar electrolyte concentration\n5, and can be accounted for by the formula ( x=kd,y=d/a),\nImFx(y)(k,ω) =−Θ(2π−x−ωya/v)/bracketleftBig\nπ2\n6(x+ωya/v)−π\n4(x+ωya/v)2+1\n12(x+ωya/v)3/bracketrightBig\n−Θ(−2π+x+ωya/v)/bracketleftBig\nπ2\n6(x+ωya/v−2π)−π\n4(x+ωya/v−2π)2+1\n12(x+ωya/v−2π)3/bracketrightBig\n+Θ(x−ωya/v)/bracketleftBig\nπ2\n6(x−ωya/v)−π\n4(x−ωya/v)2+1\n12(x−ωya/v)3/bracketrightBig\n+Θ(−x+ωya/v)/bracketleftBig\nπ2\n6(x−ωya/v+2π)−π\n4(x−ωya/v+2π)2+1\n12(x−ωya/v+2π)3/bracketrightBig\n+Θ(2π−x−ωya/v)(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x+ωya/v)+1\n4(x+ωya/v)2/bracketrightBig\n+Θ(x−ωay/v)(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x−ωya/v)+1\n4(x−ωya/v)2/bracketrightBig\n+Θ(−2π+x+ωay/v)(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x+ωya/v−2π)+1\n4(x+ωya/v−2π)2/bracketrightBig\n+Θ(−x+ωay/v)(ωay/v)/bracketleftBig\nπ2\n6−π\n2(x−ωya/v+2π)+1\n4(x−ωya/v+2π)2/bracketrightBig\n+Θ(2π−x−ωya/v)1\n2(ωay/v)2[π−x−ωya/v]+Θ(−2π+x+ωya/v)1\n2(ωay/v)2[3π−x−ωya/v]\n−Θ(x−ωya/v)1\n2(ωay/v)2[π−x+ωya/v]−Θ(−x+ωay/v)2[−π−x+ωya/v]+2\n3(ωay/v)3.(A8)\nThis function is plotted in Fig. 5. The discontinuity jump on the border between the regions with vanishing\nradiative damping and with nonzero radiative attenuation is caused b y discontinuous function∞/summationtext\nn=1sin(nz)\nn(cf. Fig. 6)\nentering ImFx(y)but notImFz, cf. Eq.(A3).\nIn order to compare the magnitudes of various contributions to th e damping of collective plasmons in the chain\nlet us plot dimensionless values, for the longitudinal polarization,1\nω1τ=1\nω1τ0+a3\n2d3ImFz(k) (in red in Fig. 7) in\ncomparison to the Lorentz friction contribution1\n3/parenleftbigω1a\nv/parenrightbig3(blue line in Fig 7, cf. Eq. (9)). In this figure one can\nnote that for exemplary spheres the Lorentz term is much larger t han the Ohmic attenuation (the bottom of red line,\ncf. also Eq. (3)). For large spheres the Ohmic damping is governed b y the ratiov\nλBin Eq. (3), which is small due\nto extremely small mean velocity of carriers in the electrolyte despit e also small mean free path. The same can be\ndemonstrated for the transversal polarization, as it is shown in Fig . 7.\nAppendix B: Calculation of self-frequencies and group velo cities of plasmon-polaritons in ionic electrolyte\nchain\nAccording to Eqs (10) and (11) for self-frequencies of plasmon-p olariton in the chain one can write out their explicit\nforms using expressions for Fα(k,ω1) given by Eq (7). The real part of the functions Fα(k,ω1) renormalizes the12\nkd\nda 3\nda 4\nda 6\n0 1 2 3 4 5 6024681012\nkdxy\nda 3\nda 4\nda 6\n0 1 2 3 4 5 6/c4510123456\nkdxya 2 ionda 3\nda 4\nda 6\n0 1 2 3 4 5 6/c45101234\nkdxy da 3\nda 4\nda 6\n0 1 2 3 4 5 6051015\nkdxy\nFIG. 5. The function ImFx(y)(k;ω=ω1) for infinite chains of ionic spheres with radius aand chain-separations d= 3a,4a,5a\nfor effective ions mass m= 104me, effective charge q= 3e, dielectric susceptibility of surroundings ε= 2,T= 300 K, ion\nconcentration n= 10−3N0(upper), n= 10−2N0(lower), N0is one-molar electrolyte concentration\n5 10 15\n/c451.5/c451.0/c450.50.51.01.5\n-\n-\n-5 10\n/c450.50.51.01.5\n-5 10\n/c451.0/c450.50.5\n-\n-\nFIG. 6. Sums∞/summationtext\nn=1sin(nx)\nn(left),∞/summationtext\nn=1cos(nx)\nn2(center),∞/summationtext\nn=1sin(nx)\nn3(right), for x∈(−1,15)\ncorresponding frequencies for both polarization and they attain t he following forms (according to Eq (10)):\nω2\nz(k) =ω2\n1/bracketleftbigg\n1−a3\nd34∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3cos(mω1d/v)+ω1d/vcos(mkd)\nm2sin(mω1d/v)/parenrightBig/bracketrightbigg\n=ω2\n1/bracketleftbigg\n1−a3\nd32∞/summationtext\nm=1/parenleftBig\ncos(m(x+ω1ya/v))+cos(m(x−ω1ya/v))\nm3 +ω1ya/vsin(m(x+ω1ya/v))−sin(m(x−ω1ya/v))\nm2/parenrightBig/bracketrightbigg\n,(B1)\nω2\nx(y)(k) =ω2\n1/bracketleftbigg\n1+a3\nd32∞/summationtext\nm=1/parenleftBig\ncos(mkd)\nm3cos(mω1d/v)+ω1d/vcos(mkd)\nm2sin(mω1d/v)\n−(ω1d/v)2cos(mkd)\nmcos(mω1d/v)/parenrightBig/bracketrightBig\n=ω2\n1/bracketleftbigg\n1+a3\nd3∞/summationtext\nm=1/parenleftBig\ncos(m(x+ω1ya/v)+cos(m(x−ω1ya/v))\nm3 +ω1ay/vsin(m(x+ω1ya/v))−sin(m(x−ω1ya/v))\nm2\n−(ω1ay/v)2cos(m(x+ω1ya/v))+cos(m(x−ω1ya/v))\nm/parenrightBig/bracketrightBig\n,(B2)\nwherex=kdandy=d/a.\nThe shift of self-frequencies of plasmon-polaritons cased by their attenuation is accounted for by the formula (11)\nwhereωα(k) is given by Eqs (B1) and (B2) where τα(k) has a form as in Eq. (9). Taking into account the explicit\nform for ImFα(k,ω1), i.e., the expressions (A6) and (A8), one can easily calculate the se lf-frequencies ω′\nα(k) – these\nfunctions are shown in Figs 8 and 9 for longitudinal and transversal polarizations, respectively.\nNote that for the transversal polarization in Eq. (B2) the sum,\n∞/summationdisplay\nm=1cos(m(x+ω1ya/v))+cos(m(x−ω1ya/v))\nm, (B3)13\nlongitudinal mode\ntransversal mode\n0 1 2 3 4 5 60.000.020.040.060.080.10\nkd0 1 2 3 4 5 60.000.020.040.060.080.10\nkd0 1 2 3 4 5 60.000.020.040.060.080.10\nkd1\n1 10 1 2 3 4 5 60.000.020.040.060.080.10\nkd1\nFIG. 7. Comparison of various contributions to collective p lasmon damping in the chain—red line corresponds to total da mping\nratio (in dimensionless units)1\nω1τz=1\nω1τ0+a3\n2d3ImFz(k), the bottom of this line plot displays only Ohmic contribut ion at that\nsegment of the kperiod where all e-m losses vanish, the blue line indicates t he Lorentz friction value (in the same dimensionless\nunits)1\n3/parenleftbigω1a\nv/parenrightbig3; for effective ions mass m= 104me, effective charge q= 3e, dielectric susceptibility of surroundings ε= 2,\nT= 300 K, ion concentration, n= 10−3N0,N0is one-molar electrolyte concentration and ion mean free pa thλB= 5×10−9\nm (left) λB= 10−8m (right)\nd/c613a\nd/c614a\nd/c615a\n0 1 2 3 4 5 6/c456/c180106/c454/c180106/c452/c18010602/c1801064/c1801066/c180106\nkdz\n0 1 2 3 4 5 6/c454/c180106/c452/c18010602/c1801064/c180106\nkdza 20 ionic spheres d 3 a\nd/c613a\nd/c614a\nd/c615a\n0 1 2 3 4 5 60.900.951.001.05\nkdza 20 ionic spheres\nFIG. 8. The self-frequency ω′\nz(k) (inω1units) and the group velocity (in m/s) of plasmon polariton w ith longitudinal polar-\nization in infinite chain of electrolyte spheres with radius a= 20µm and chain-separation d= 3,4a,5a, ion concentration\nn= 10−3N0,T= 300 K, ω1= 1.2×10121/s; in the right panel effective removal of logarithmic sing ularity in the group\nvelocity due to the far-field contribution in the finite chain of 15 spheres; the logarithmic singularity for the infinite c hain in the\nperturbation formula for vz(though not in formula for ω′\nz) leads to local exceeding of c—this artifact of perturbation approach\nis removed by the exact solution, cf. Appendix C\ncan be performed analytically according to the formula (A4) resultin g in the contribution,\n−1\n2ln[(2−2cos(x+ω1ya/v))(2−2cos(x−ω1ya/v))] (B4)\n(the other sums in Eqs (B1) and (B2) have to be done numerically). T his logarithmic singularity in self-frequencies\nfor transversal plasmon-polaritons on the rim of the region 0 < x−ω1ya/v <2π(inside which radiative damping\nvanishes) is indicated in Fig. 9 left. This singularity causes hyperbolic d iscontinuity in transversal group velocity\n(cf. Fig. 9 right). Nevertheless, the logarithmic singularity in self-e nergy and the related hyperbolic discontinuity of\ntransversal braid group turn out to be an artifact of perturbat ion solution of Eq. (6). The exact numerical solution\nof this equation demonstrates an effective quenching of the logarit hmic singularity to small local minimum resulting\nin finite group velocity discontinuity, as it is shown in Appendix C. This pr operty of transversal propagation of\nplasmon-polaritons in the chain caused by constructive interferen ce of the far-field component of dipole interactions\nbetween spheres was analyzed for metallic chain numerically in [21] an d commented in [15, 22]. The numerical\nstudies of plasmon-polariton propagation in metallic nano-chain [21] indicated very narrow and weak but long range\nmode besides the wide spectrum of quickly damped modes. This ’faintin g’ long range mode can be associated with14\nd/c613a\nd/c614a\nd/c615a\n0 1 2 3 4 5 60.970.980.991.001.011.021.031.04\nkdxya 20/c109mionic spheres\nd/c613a\nd/c614a\nd/c615a\n0 1 2 3 4 5 6/c453/c180106/c452/c180106/c451/c18010601/c1801062/c1801063/c180106\nkdxya 20/c109mionic spheres\nFIG. 9. The self-frequency ω′\nx(y)(k) (inω1units) and the group velocity (in m/s) of plasmon polariton w ith transversal polar-\nization in infinite chains of electrolyte spheres with radiu sa= 20µm and chain-separation d= 3a,4a,5a, ion concentration\nn= 10−3N0,T= 300 K, ω1= 1.2×10121/s; the logarithmic singularity in ω′\nx(y)(k) due to constructive interference of far-field\ncontribution of dipole interaction, and the related hyperb olic singularity in transversal group velocity are, howeve r, an artifact\nof perturbation method of solution of Eq. (6)—in the exact so lution of this equation the singularities are quenched to sm all\nlocal minima only, cf. Appendix C\ninterference of far-field part of dipole-dipole interaction of nanos pheres in the chain, resulting in local enhancement\nof transversal group velocity in narrow vicinity of singularity points . The same can be addressed to electrolyte ionic\nchains.\nIn order to find the group velocities of particular self-modes of plas mon-polariton in the chain, the derivative\nofω′\nα(k) with respect to kmust be performed, which according to the expressions (B1), (B2 ), (11) and (9) is a\nstraightforward though an extended calculus. The sums in formula e forωα(k) still cannot be done analytically except\nforthesumwith lineardenominatorin(B2). Theresultinggroupveloc itycalculatednumericallyforbothpolarizations\nand forkd∈[0,2π) andd/a∈[3,10] are presented in Figs 8 and 9 (for exemplary chain parameters a nd for both\npolarizations).\nAppendix C: Exact solution of Eq. (6)—the problem of the loga rithmic divergence of the far-field\ncontribution to self-frequency for transversal polarizat ion of plasmon polariton and of the medium-field\ncontribution to group velocities for both polarizations\nThe solution of Eq. (6) in the homogeneous case gives the Fourier ex ponentωα(k), which is a complex function, in\ngeneral. Its imaginary part defines plasmon-polariton damping rate , whereas the real part of the ωfunction defines\nself-frequency of particular kmode with αpolarization. One can next find the group velocities for particular mo des\ntaking the derivative of the real part of ωα(k) with respect to the wave vector k. As we have noticed above, the\nfar-field contribution to the dipole interaction in the chain produces a logarithmically singular term in Eq. (6) of the\nform given by Eqs (B3)-(B4). Because of infinite divergence of this term one can not apply the perturbation method\nfor solution Eq. (6), in the vicinity to the singular points in kdomain. The perturbative method of solution of Eq.\n(6) transfers this logarithmic singularity onto solution for self-fre quency in the singular point k. The derivative of\nthis frequency, i.e., the group velocity of the singular mode k, acquires in this way the hyperbolic singularity. This\nhappens for perturbative formula for the transversal group ve locity. These infinite divergences are visible in Fig.9\n(left) close to the edges of the domain kd∈[0,2π).\nFor the longitudinal polarization upon perturbation approach one e ncounters also infinite singularity of the group\nvelocity in the same kas for the transversal polarization, though without any singularit y in the longitudinal self-\nfrequency. This astonishing singularity in the group velocity is of the logarithmic type and arises actually for both\npolarizations, when one takes the derivative with respect to kfrom Eqs (B1)-(B2). The contribution to self-energies\nfor both polarizations sourced by medium-field irradiation term with t he factor/summationtextcos(nx)\nn2, after taking the derivative\nwith respect to x(x=kd) is divergent. For the longitudinal polarization this divergence is visib le in the Fig. 8. For\nthe transversal polarization this logarithmic singularity interferen ces with above described hyperbolic one. These all\nsingularities occur at isolated points for which kd±ω1d/v=lπ(l- integer). These singularities are apparent artifact\nof the perturbation approach because lead to local exceeding of lig ht velocity.\nIn order to resolve and clarify this problem of unphysical divergenc e one has to solve Eq. (6) exactly not applying\nthe perturbationtechnique. This is clear, because ofdivergenceo fthe expression(B3) the correspondingcontributions\ncannot be treated as a small perturbation. The exact solution of E q. (6) can be found numerically. The result of\nnumerical solution (by the Newton type algorithm) for self-freque ncies with both polarization and for whole kdomain\nis plotted in Fig. 10.\nBy the comparison with the corresponding plots obtained within the p erturbation method and those in Fig 10, we\nnotice that the exact solutions for self-frequencies do not differ s ignificantly from those obtained in the perturbation15\n0 200 400 600 800 10000.9900.9951.0001.0051.0101.015\nkd2/c1121000xya 20 ion d 4 a\n0 200 400 600 800 1000/c454/c180106/c452/c18010602/c1801064/c180106\nkd2/c1121000xy0 200 400 600 800 10000.960.970.980.991.001.011.021.03\nkd2/c1121000za 20 ion d 4 a\n0 200 400 600 800 1000/c453/c180106/c452/c180106/c451/c18010601/c1801062/c1801063/c180106\nkd2/c1121000za 20 ion d 4 a\nx\nxxxxx\n---[m/s]\n/c112/c46\nx\nxxx\n--\n[m/s]\n/c112/c46./c112\n/c112\nFIG. 10. The exact solutions for self-frequencies for the lo ngitudinal and transversal modes of plasmon-polariton in t he ionic\nchain (ωinω1units) obtained by the exact solution of Eq. (6) in 1000 point s on the sector kd∈[0,2π)—left; and corresponding\ngroup velocities for both polarizations—right\nmanner. Nevertheless, the exact solutions for the group velocitie s do not have any singularities. As it follows from the\nthoroughanalysis[22]intheexactsolutionsforgroupvelocitiesalld ivergencesarecutattheleveloflightvelocity. This\nquenching of singularities in the way protecting not exceeding the ligh t velocity is the manifestation of the Lorentz\nrelativistic invariance of plasmon-polariton dynamics description. Th e retardation of electric signals prohibits the\ncollective excitation group velocity to exceed the light velocity. This q uenching concerns infinite singularities which\noccur in perturbation expressions for self-energy and next in per turbation formulae for group velocities. The exact\nsolution of the Lorentz invariant dynamical equation inherently pos ses also this property. Exact self-energies have\nsuitably regularized their dependence with respect to k, that their derivatives do not exceed v=c√ε. It is worth\nemphasizing for the sake of completeness of the description, that inclusion of magnetic field of dipoles does not\nmodify this scenario, because the magnetic field contribution to self -energies is a few orders lower in comparison to\nthe electric field contribution due to the velocity of ions being also simila rly lower in comparison to light velocity,\nwhich significantly reduces the Lorentz force caused by the magne tic field. Therefore, the magnetic field of the dipoles\n[12, 13],\nBω=ik(Dω×n)/parenleftbiggik\nr0−1\nr2\n0/parenrightbigg\neikr0, (C1)\nthough contributing to the far-field and medium-field parts of plasm on-polariton self-energies, does not change signif-\nicantly the appropriate terms caused by the electric field and cause s only corrections practically negligible.\nAppendix D: Fitting of plasmon-polariton features to param eters of an axon in the peripheral nervous system\nThebulkplasmonfrequencyislinkedtoionparametersviatheformula ,ωp=/radicalBig\nq2n\nε0m, (inGaussunits, ωp=/radicalBig\nq2n4π\nm),\nwhereqis the charge of the ion, nis the ion concentration, ε0is the dielectric susceptibility of the vacuum, mis the\nmass of the ion. For model we assume the charge of ions, q= 1.6×10−19C (the electron charge), and their mass,\nm= 104me, whereme= 9.1×10−31kg is the mass of the electron, n= 2.1×10141/m3, and thus we obtain the\nMie frequency for ionic dipole oscillations, ω1≃0.1ωp√\n3ε≃4×1061/s, where for the MHz frequency, the relative\npermittivity of water ε≃80 [23] (though for the visible light frequencies it drops, staring the fall at ca 10 GHz, to\nthe final value ca. 1.7, corresponding to the refractive index for w aterη≃√ε1= 1.33). For the strongly prolate and\nthin ionic inner cord we reduced the longitudinal Mie frequency by fac tor 0.1 [24, 25] in comparison to the isotropic\nspherical case, when it was ω1=ωp√\n3ε. Let us emphasize that in the axon we deal with the cord of small diam eter, 2r,\nand this thin cord is wrapped by the myelin sheath of the length 2 aper each segment, but for the model we consider\nelectrolyte sphere with the radius a. Thus the auxiliary concentration, n, of ions in the fictitious sphere corresponds\nin fact to the concentration of ions in the cord, n′=n4/3πa3\n2aπr2, which gives the typical cell concentration of ions n′∼10\nmM (i.e., ca. 6 ×10241/m3). This is due to the fact that in the dipole oscillation in the segment of le ngth of 2 a16\ntake part all ions despite as in the fictitious model sphere or, as in th e real system, comprised in fact to the much\nsmaller volume of the thin cord portion, whereas the insulating myelina ted sheath is a lipid substance without any\nions. This insulating, relatively thick myelin coverage creates the tun nel required for formation of plasmon-polariton\nand its propagationinside such a channel. To reduce a coupling with su rrounding inter-cellular electrolyte and protect\nagainst a leakage of the plasmon-polariton, the myelin sheath must b e sufficiently thick, much thicker than only for\nelectrical isolating role. Moreover, to fit better with conductivity p arameters we accounted for, that for highly prolate\ngeometry of actually oscillating ionic system, the Mie frequency of th e longitudinal oscillations is significantly lower\nthan that one for the sphere with diameter equal to the longer axis (for a rough estimation we assumed the factor\n0.1).\nFor this Mie frequency, ω1≃4×1061/s, one can determine plasmon polariton self-frequencies in the ch ain\nof spheres with radius 50 µm (for Schwann cell length of 2 a) and the small chain separation, d/a= 2.01,2.1,2.2\n(giving the Ranvier node length, 0 .5,10,20µm, respectively) using approach presented in the previous Append ices.\nThe derivative of the self-frequency with respect to the wave vec torkdetermines the group velocity of plasmon-\npolariton modes. The results are collected in Fig. 11. We notice that f or ion system parameters as listed above, the\ngroup velocity of the plasmon-polariton reaches 100 m/s, for longit udinal modes, and 40 m/s, for transversal modes.\nNeverheless,ifoneassumesalargertransversalplasmonoscillatio nfrequency[24,25]formorerealisticmodelofprolate\nspheroids (as for the inner Shwann cell cord with ions), one arrives also with the value of 100 m/s for transversalgroup\nvelocity of plasmon-polariton for only twice increase of the frequen cy (cf. Fig. 11). The transversal ion oscillations\nare, however, inconvenient in the considered structure and more over, the initial action potential post-synaptic or from\nthe synapse hillock, excites rather the longitudinal oscillations.\n2.01\n2.1\n2.2\n0 1 2 3 4 5 60.900.951.001.051.10\nkdx(y)2.01\n2.1\n2.2\n0 1 2 3 4 5 6/c45100/c4550050100\nkdx(y)a 50/c109mionic spheres\n2.01\n2.1\n2.2\n0 1 2 3 4 5 60.900.951.001.051.10\nkdx(y)a 50 ionic spheres\n2.01\n2.1\n2.2\n0 1 2 3 4 5 6/c4540/c452002040\nkdx(y)a 50 ionic spheres--\n--2.01\n2.1\n2.2\n0 1 2 3 4 5 60.70.80.91.01.11.2\nkdza 50 mionic spheres\n2.01\n2.1\n2.2\n0 1 2 3 4 5 6/c45100/c4550050100\nkdza 50 ionic spheres\n--\nFIG. 11. Solutions for the self-frequency and the group velo city of the longitudinal (upper) and transversal (middle, l ower)\nmodes of plasmon-polariton in the model ionic chain; ωinω1units, here ω1= 4×1061/s, for the chain of spheres with\nradiusa= 50µm and chain separation d/a= 1.01,1.1,1.2, the equivalent ion concentration in the inner ionic cord o f the\naxon,n′∼6×10241/m3, the group velocity amplitude (for the longitudinal mode) ≃100 m/s—in the upper panel; for the\ntransversal mode the similar velocity amplitude is reached for twice-larger frequency (2 ×ω1) assumed in the prolate geometry\n[24, 25]; in the lower panel the transversal mode characteri stics, but for not increased ω1\nNote, that for ω1= 4×1061/s and a= 50µm the interference condition kd−ω1d/c= 0 and kd+ω1d/c= 2π\nare fulfilled for extremely small values of kdand 2π−kd, respectively (of order of 10−6ford∈[2,2.5]), thus are\nnegligible for kinetics of plasmon-polariton at these conditions. Henc e, the singularities caused by far- and medium-\nfield contributions to dipole interaction are effectively removed by pu shing to borders of the kwave vector domain\nand moreover, practically whole the region kd∈[0,2π) corresponds to ideal cancellation of the radiative losses of the\nplasmon-polariton modes. Additionally, the mentioned singularities ch aracteristic for infinite chains, cannot be also\nfully developed as the nerve chains are of finite length.\nDespite that the model of ionic system chain for myelinated axon is an apparently crude approximation of the real17\nscheme of myelinated axon structure\nmyelinated sector\nSchwann cell\n2r diameter\nof an axon cord\nRanvier interval\nfictitious spheres for the model100 μm\nFIG.12. Schematically presentedperiodic fragment ofthem yelinatedlongaxon(myelinis amultilamellar lipidmembra newhich\nenwraps the axon cord in segments separated by nodes of Ranvi er; the myelin sheath is produced by special cells—Schwann\ncells in the peripheral nervous system, and oligodendrocyt es in the central nervous system), for the model the fictitiou s periodic\nspherical ionic system chain is proposed\naxon structure, it can serve for comparison of energy and time sc ale of plasmon-polariton propagation in the model\nwith kinetic parameters of nerve signals. In the model, the propaga ting in the axon chain plasmon-polariton, excited\nbythe initialactionpotentialonthe firstRanviernode(after thes ynapseorinthe synapsehillockforthe reversesignal\ndirection), ignitesone-by-onetheconsecutiveRanviernodeblock sofNa+,K+iongatesandtheresultingfireofaction\npotential traversesthe axon with velocity ofca. 100m/s actually o bservedin axons. Each new ignition releasesenergy\nsupplement at particular Ranvier node block (due to its nonlinearity t he signal growth saturates on the constant level\nand the overall timing of each action potential spike has stable shap e of local polarization/depolarization scheme of\nthe short fragment of cell membrane corresponding to the Ranvie r node). This permanent energy income contributes\nthe plasmon-polariton dynamics compensating Ohmic thermal losses and assures undamped its propagation over\nunlimited range. Though the whole signal cycle of action potential on a single Ranvier node block takes a time\nof several miliseconds (or even longer including the time for restorin g the steady conditions, which, on the other\nhand, blocks reversion of the signal), the subsequent nodes are ig nited earlier, accommodated to the velocity of the\ntriggering pasmon-polariton wave packet. The velocity direction of the wave packet of plasmon-polariton is adjusted\nto the semi-infinite chain geometry (actually the finite chain and excit ed at its beginning). The action potential firing\ntriggered by plasmon-polariton travels along the axon in one directio n also because the already fired nodes have Na+\ngates inactivated and need relatively long time to restore its original status (whole the block sodium/potassium needs\nsome time, of second order, and energy supply to bring their conce ntrations to the normal values by across-membrane\nactive ion pumps). The plasmon-polariton scheme of ignition of action potential spikes in the chain of Ranvier nodes\nalong the axon fits well with observed saltatory conduction of myelin ated axons.\nr=20 nm\n2.01\n2.1\n2.2\n0 1 2 3 4 5 6/c4540/c452002040\nkdza 50 ionic spheres\n2.01\n2.1\n2.2\n0 1 2 3 4 5 6/c45100/c4550050100\nkdza 50 ionic spheresr=50 nm\n2.01\n2.1\n2.2\n0 1 2 3 4 5 6/c45200/c451000100200\nkdza 50 ionic spheresr=100 nm\nFIG. 13. Comparison of the group velocity, in [m/s], for the l ongitudinal plasmon-polariton mode with respect to the wav e\nvectork∈[0,2π/d), in the axon model with the Schwann cells of the length 100 µm, Ranvier separation 0.5 µm, 5µm, 10µm\n(idicated by d= 2.01,2.1,2.2 in the figure) and radius r= 20,50,100 nm of the axon cord\nIn Fig. 13 the group velocity for action signal traversing in firing mye linated axon is compared for rising diameter\nof internal cord of the axon (cf. Fig. 13) (for concentration of io ns in the cord 8 mM (i.e., ∼3×10241/m3, typical\nfor cell ion concentration), 100 µm length of myelinated sectors wrapped with Schwann cells and Ranvie r interval of\nlength 0.5 µm, 5µm, 10µm. The dependence on the Ranvier interval length is weak (not impor tant) but the rise of\nthe velocity with the thickness of the internal cord is significant.\n[1] W. Jacak, “Plasmons in fnite spherical ionic systems,” s ubmitted.\n[2] F. J. G. de Abajo, “Optical excitations in electron micro scopy,” Rev. Mod. Phys. 82, p. 209, 2010.18\n[3] D. S. Citrin, “Plasmon polaritons in finite-length metal -nanoparticle chains: The role of chain length unravelled, ”Nano\nLetters5, p. 985, 2005.\n[4] L. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction s pectra of silver nanoparticle arrays: Influence of array str ucture\non plasmon resonance wavelength and width,” J. Phys. Chem. B 107, p. 7343, 2003.\n[5] S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle a rray structures that produce remarkably narrow plasmon lin e-\nshapes,” J. Chem. Phys. 120, p. 10871, 2004.\n[6] S. B. Singham and C. F. Bohren, “Light scattering by an arb itrary particle: a physical reformulation of the coupled di pole\nmethod,” Optics Letters 12, p. 10, 1987.\n[7] T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Elec trodynamics of noble metal nanoparticles and nanoparticle\nclusters,” Journal of Cluster Science 10, p. 295, 1999.\n[8] B. T. Draine, “The discrete-dipole approximation and it s application to interstellar graphite grains,” The Astrophysical\nJournal333, p. 848, 1988.\n[9] E. M. Purcell and C. R. Pennypacker, “Scattering and abso rption of light by nanospherical dielectric grains,” The Astro-\nphysical Journal 186, p. 705, 1973.\n[10] B. T. Draine and P. J. Flatau, “Discrete-dipole approxi mation for scattering calculations,” J. Opt. Soc. Am. A 11, p. 1491,\n1994.\n[11] V. A. Markel, “Coupled-dipole approach to scattering o f light from a onedimensional periodic dipole structure,” Journal\nof Modern Optics 40, p. 2281, 1993.\n[12] L. D. Landau and E. M. Lifshitz, Field Theory , Nauka, Moscow, 1973.\n[13] J. D. Jackson, Classical Electrodynamics , John Willey and Sons Inc., New York, 1998.\n[14] S. A. Maier, P. G. Kik, L. A. Sweatlock, H. A. Atwater, J. J . Penninkhof, A. Polman, S. Meltzer, E. Harel, A. Requicha,\nand B. E. Koel, “Energy transport in metal nanoparticle plas mon waveguides,” Mat. Res. Soc. Symp. Proc. 777, p. T7.1.1,\n2003.\n[15] W. A. Jacak, “On plasmon polariton propagation along me tallic nano-chain,” Plasmonics 8, p. 1317, 2013. DOI\n10.1007/s11468-013-9528-8.\n[16] J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lac route, J. P. Goudonnet, G. Schider, W. Gotschy, A. Leitner,\nF. R. Aussenegg, and C. Girard, “Squeezing the optical near- field zone by plasmon coupling of metallic nanoparticles,”\nPhys. Rev. Lett. 82, p. 2590, 1999.\n[17] S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse p ropagation in metal nanoparticle chain waveguides,” Phys.\nRev. B67, p. 205402, 2003.\n[18] S. A. Maier and H. A. Atwater, “Plasmonics: Localizatio n and guiding of electromagnetic energy in metal/dielectri c\nstructures,” J. Appl. Phys. 98, p. 011101, 2005.\n[19] E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vin ogradov, and A. A. Lisyansky, “Stationary behavior of a chai n\nof interacting spasers,” Phys. Rev. B 85, p. 165419, 2012.\n[20] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products , Academic Press, Inc., Boston, 1994.\n[21] V.A.Markel andA.K.Sarychev, “Propagation ofsurface plasmons inorderedanddisorderedchains ofmetal nanosphe res,”\nPhys. Rev. B 75, p. 085426, 2007.\n[22] W. Jacak, “Exact solution for velocity of plasmon-pola riton in metallic nano-chain,” Optics Express 22, p. 18958, 2014.\n[23] T. Meissner and F. Wentz, “The complex dielectric const ant of pure and sea water from microwave satellite observati ons,”\nIEEE THRS 42, p. 1836, 2004.\n[24] T. A. El-Brolossy, T. Abdallah, M. B. Mohamed, S. Abdall ah, K. Easawi, S. Negm, and H. Talaat, “Shape and size\ndependence of the surface plasmon resonance of gold nanopar ticles studied by photoacoustic technique,” Eur. Phys. J.\nSpecial Topics 153, p. 361, 2008.\n[25] L. M. Liz-Marzan, “Tuning nanorod surface plasmon reso nances,” SPIE Newsroom (10.1117/2.1200707.0798), 2007."    },    {        "title": "1408.2574v1.Applications_of_Lorentz_force_in_medical_acoustics__Lorentz_force_hydrophone__Lorentz_Force_Electrical_Impedance_Tomography__Imaging_of_shear_waves_induced_by_Lorentz_force.pdf",        "content": "THÈSE DE DOCTORAT\nSpécialité Ingénierie biologique et médicale\nÉcole Doctorale Interdisciplinaire Science-Santé\nPrésentée pour obtenir le grade de\nDOCTEUR DE L ’UNIVERSITÉ DE LYON\nPar\nPol G RASLAND -MONGRAIN\nApplications de la force de Lorentz\nen acoustique médicale\nN SChamp\nmagnétique\nCourant\nélectriqueOnde\nacoustique\nThèse soutenue le 12 décembre 2013 devant le jury composé de :\nM. Cyril L AFON Directeur de thèse\nM. Mickaël T ANTER Rapporteur\nMme Gail TER HAAR Rapporteur\nM. Stefan C ATHELINE\nM. Pierre C ELSIS\nM. Bruno G ILLES\nM. Eric M CADAMS\nM. Serge M ENSAH\nNuméro d’ordre : 280-2013arXiv:1408.2574v1  [physics.med-ph]  11 Aug 2014ENCADREMENT\nDirecteur de thèse : Cyril L AFON\nCo-directeur de thèse : Jean-Yves C HAPELON\nTITRE COMPLET DE LA THÈSE (FRANÇAIS )\nApplications de la force de Lorentz en acoustique médicale :\nHydrophone à force de Lorentz\nTomographie d’Impédance Electrique par Force de Lorentz\nImagerie d’ondes de cisaillement induites par force de Lorentz\nTITRE COMPLET DE LA THÈSE (ANGLAIS )\nApplications of Lorentz force in medical acoustics :\nLorentz force hydrophone\nLorentz Force Electrical Impedance Tomography\nImaging of shear waves induced by Lorentz force\nMOTS-CLÉS (FRANÇAIS )\nforce de Lorentz ; acoustique médicale ; hydrophone ; Ultrasons Focalisés à Haute Inten-\nsité ; Tomographie Magnéto-Acousto-Electrique ; Tomographie d’Impédance Electrique\npar Force de Lorentz ; impédance électrique ; élastographie\nMOTS-CLÉS (ANGLAIS )\nLorentz force ; medical acoustics ; hydrophone ; High Intensity Focused Ultrasound ;\nMagneto-Acoustic-Electrical Tomography ; Lorentz Force Electrical Impedance Tomo-\ngraphy ; electrical impedance ; elastography\nETABLISSEMENT DÉLIVRANT LE GRADE DE DOCTEUR\nUniversité Claude Bernard Lyon 1\nINTITULÉ ET ADRESSE DU LABORATOIRE D ’ACCUEIL\nLaboratoire de Thérapie Appliquée par Ultrasons,\nINSERM U1032,\n151 Cours Albert Thomas,\n69424 Lyon Cedex 03 France\nApplications de la force de Lorentz en acoustique médicale 3Remerciements\nEcrire des remerciements est un exercice délicat, surtout que cette partie sera pro-\nbablement la plus lue de ce manuscrit. Je souhaite donc adresser ma sincère gratitude\nà ceux qui m’ont aidé dans ma thèse et plus largement à tous ceux que mon travail a\nintéressés, en implorant par avance de l’indulgence pour ma mémoire imparfaite si j’en\noublie certains.\nJe commence donc par Cyril Lafon, mon directeur de thèse, qui m’a encadré tout au\nlong de mes trois années de thèse et m’a prodigué moult conseils pour ma future (j’es-\npère) carrière de chercheur. Cyril, merci pour tes encouragements, ta confiance et ton\nsoutien sans faille ! Je remercie également Jean-Yves Chapelon, directeur du LabTAU et\nco-directeur de thèse, pour m’avoir accueilli au sein du laboratoire et pour toutes nos\ndiscussions, scientifiques ou non.\nJe tiens à remercier Stefan Catheline, qui m’a encadré notamment sur la partie onde\nde cisaillement, pour son expertise, son intérêt, sa bonne humeur. Stefan, tu m’as mon-\ntré qu’on peut être à la fois un chercheur de grande renommée et simplement quelqu’un\nde cool ! Je voudrais aussi remercier Bruno Gilles, avec qui j’ai travaillé sur l’hydro-\nphone électromagnétique, qui m’a ouvert les yeux sur ce que je sais et surtout sur tout\nce que je ne sais pas en mécanique des fluides...\nJe remercie sincèrement les deux rapporteurs de ce manuscrit : Gail ter Haar, que\nj’ai pu rencontrer avec John Civale lors de mon séjour à l’ICR, et Mickaël Tanter, qui a\nété mon maître de stage officiel en master 2 et avec qui les échanges ont toujours été très\ncordiaux. Je suis flatté de pouvoir vous présenter ces travaux de recherche aujourd’hui !\nUn grand merci aux autres membres du jury : Eric McAdams, que j’ai rencontré lors\nd’un entretien très sympathique sur les problèmes d’impédance électrique, ainsi que\nSerge Mensah et Pierre Celsis, qui m’ont tous les deux été chaudement recommandés\npar mon directeur de thèse. J’espère que ce travail vous intéressera !\nApplications de la force de Lorentz en acoustique médicale 5J’adresse aussi mes remerciements les plus chaleureux à Jean-Martial Mari, sans qui\nje ne serai sans doute pas ici en ce moment à rédiger ces lignes. Jean-Martial, je loue\nici tes conseils sur mon travail, les fins de journée à essayer de faire fonctionner tant\nbien que mal une expérience, la dix-huitième relecture de cet article qu’on n’arrive plus\nà lire... Mais aussi les blagues, les gadgets, les sorties, le cat-sitting, le déménagement,\net bien d’autres... L’une de tes citations favorites1est également devenue mienne ! Je\nremercie aussi Rémi Souchon pour ses conseils précieux de méthodologie, pour son\nintérêt pour mon travail et sa bonne humeur quotidienne. Je voudrais aussi citer Amal-\nric Montalibet et Jacques Jossinet pour leurs travaux sur la Tomographie d’Impédance\nElectrique par Force de Lorentz au laboratoire.\nJe salue ici les stagiaires pas si anonymes que j’ai eu la chance d’encadrer directe-\nment ou non, qui ont tous contribué à une part de cette thèse : Sandra Montalescot,\nla première, qui a fait preuve d’un enthousiasme et d’un investissement qui forcent le\nrespect, Benjamin Roussel et Alexandre Petit, qui ont réalisé un travail très sérieux et\nce toujours dans une ambiance agréable, Florian Cartellier, qui a fait avancer le projet\nd’élastographie avec une grande maitrise tout en préparant un concours et en organi-\nsant son mariage, ainsi que Mickaël Parisi, qui a réussi à dépasser ses objectifs de stage.\nCes trois années de thèse se sont passées dans un contexte agréable au sein du la-\nboratoire, et je souhaite donc remercier tout particulièrement mes camarades de râle-\nrie, pardon, de bureau, à savoir Elodie et Apoutou. Je remercie aussi toutes ces per-\nsonnes avec qui j’ai eu des échanges, qu’ils soient professionnels ou amicaux, Adrien,\nAdrien et Adrien, Pauline, Charlène, Jacqueline, Cyril, Jérémy et Jérémy, Arash, An-\ndrew, Alain, Maxime, Giovanna, Ali, Mike, Françoise, Jean-Louis, Fabrice, Frédéric et\nFrédéric, Wilfried, Nicolas, David, Heldmuth El Colombiano, Au, Abbas, Thomas, Cé-\ncilia, Mathieu, Isabelle, Sandrine et Sandrine, Anthony, Lucie et Lucie, Irvin, Alexandre,\nJean-Christophe, Claude, Manuela, Yves, Bernard, Benjamin...\nJe remercie les personnes du Département de Mathématiques Appliquées à l’ENS\nde Paris qui m’ont toujours accueilli très chaleureusement, donc un grand merci à Ha-\nbib Ammari, Laurent Seppecher, Pierre Millien et Thomas Boulier, dont le manuscrit et\nla soutenance ont été une précieuse source d’inspiration.\nLors de mon monitorat, j’ai pu rencontrer des personnes formidables, notamment\nValérie Martinez qui fait preuve de qualités humaines exceptionnelles, mais aussi Julien\nDe Bonfils, Sylvain Mousset, Laurent Ducroux, ainsi que mes collègues de monitorat.\n1. “Hâtez-vous lentement, et sans perdre courage,\nVingt fois sur le métier remettez votre ouvrage,\nPolissez-le sans cesse, et le repolissez,\nAjoutez quelquefois, et souvent effacez.”\n6 Pol Grasland-MongrainJe pense aussi à Cédric Ray, le directeur de mon master 2, que j’apprécie énormément.\nEt j’adresse un remerciement tout particulier à Valérian Reithinger, quelqu’un ayant de\ngrandes qualités humaines et professionnelles. Valérian, dans un an, tu seras dans la\nmême situation que moi, bon courage !\nJe voudrais citer mes colocataires qui pour certains m’ont supporté pendant presque\ndeux années : Mathieu, un ami plus qu’un coloc, Mathilde, Lucie, et plus fugacement,\nSandra et Pauline. Cette coloc a été bien plus qu’une simple cohabitation, merci à vous !\nJ’ai également une pensée pour Aquarelle, qui au milieu de ma thèse a voulu vérifier\nexpérimentalement la loi de la chute des corps de Newton. Heureusement, elle ne s’est\npas intéressée aux travaux de Schrödinger. Et je remercie ici Florence pour avoir rendu\nla fin de ma thèse bien plus plaisante.\nJ’adresse un remerciement un peu spécial à Solène, qui m’a supporté tout au long\nde ma thèse, dans les moments faciles et surtout dans les moments difficiles, même si la\nfin de la thèse n’a pas été comme nous l’avions prévue. Solène, si mes heurs et malheurs\nne t’ont pas découragée d’entreprendre un doctorat, je te souhaite bonne chance !\nEnfin, last but not least comme le dirait un certain docteur, je souhaite dédier cette\nthèse à mon grand-père Eugène, qui n’a malheureusement pas pu en voir la fin, et plus\nlargement à ma famille. Sans elle, je n’en serai pas là aujourd’hui, et je remercie ici tout\nparticulièrement Loïc, Johanne, Erwan, Yiming ainsi que Tissia qui nous a rejoints il y\na plus de deux ans déjà !\nApplications de la force de Lorentz en acoustique médicale 7Le pessimiste se plaint du vent,\nl’optimiste espère qu’il va changer,\nle réaliste ajuste ses voiles.\nWilliam Arthur Ward,\nécrivain américainTable des matières\nTable des matières 11\n1 Introduction 15\n1.1 La force de Lorentz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n1.1.1 Historique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n1.1.2 Expressions de la force de Lorentz . . . . . . . . . . . . . . . . . . 16\n1.2 L’acoustique médicale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n1.2.1 La thérapie par ultrasons . . . . . . . . . . . . . . . . . . . . . . . 20\n1.2.2 L’imagerie acoustique . . . . . . . . . . . . . . . . . . . . . . . . . 21\n1.3 Structure de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n2 Hydrophone à force de Lorentz 25\n2.1 La mesure de champs ultrasonores . . . . . . . . . . . . . . . . . . . . . . 25\n2.1.1 Les ultrasons à haute puissance . . . . . . . . . . . . . . . . . . . . 25\n2.1.2 Les mesures de pression acoustique . . . . . . . . . . . . . . . . . 26\n2.1.2.1 Les hydrophones piézoélectriques . . . . . . . . . . . . . 26\n2.1.2.2 Les hydrophones à fibre optique . . . . . . . . . . . . . . 28\n2.1.2.3 L’imagerie Schlieren . . . . . . . . . . . . . . . . . . . . . 28\n2.1.3 Les réflecteurs ponctuels . . . . . . . . . . . . . . . . . . . . . . . . 29\n2.1.4 Les mesures d’intensité acoustique . . . . . . . . . . . . . . . . . . 30\n2.1.4.1 Les sondes thermiques . . . . . . . . . . . . . . . . . . . 30\n2.1.4.2 Les balances acoustiques . . . . . . . . . . . . . . . . . . 30\n2.1.5 Les mesures de déplacement ou de vitesse particulaire . . . . . . 30\n2.1.5.1 Les interféromètres . . . . . . . . . . . . . . . . . . . . . 30\n2.1.5.2 Les hydrophones à force de Lorentz . . . . . . . . . . . . 31\nApplications de la force de Lorentz en acoustique médicale 11Table des matières\n2.1.6 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34\n2.2 Principe de l’hydrophone à force de Lorentz . . . . . . . . . . . . . . . . 35\n2.2.1 Expression de la force de Lorentz . . . . . . . . . . . . . . . . . . . 35\n2.2.2 Lien entre vitesse du fluide et vitesse du fil . . . . . . . . . . . . . 36\n2.2.3 Reconstruction tomographique . . . . . . . . . . . . . . . . . . . . 38\n2.2.4 Résonance des fils . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n2.2.5 Dépendance angulaire . . . . . . . . . . . . . . . . . . . . . . . . . 40\n2.3 Dispositifs testés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41\n2.3.1 Preuve de concept avec un simple aimant et un fil . . . . . . . . . 41\n2.3.2 Deuxième prototype : hydrophone à force de Lorentz . . . . . . . 42\n2.4 Caractérisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\n2.4.1 Résolution spatiale . . . . . . . . . . . . . . . . . . . . . . . . . . . 44\n2.4.2 Réponse fréquentielle . . . . . . . . . . . . . . . . . . . . . . . . . 46\n2.4.3 Sensibilité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47\n2.4.4 Résistance à la cavitation . . . . . . . . . . . . . . . . . . . . . . . 49\n2.4.5 Réponse directionnelle . . . . . . . . . . . . . . . . . . . . . . . . . 52\n2.4.6 Influence de la tension du fil électrique . . . . . . . . . . . . . . . 52\n2.5 Bilan du chapitre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54\n2.5.1 Principaux résultats . . . . . . . . . . . . . . . . . . . . . . . . . . 54\n2.5.2 Avenir du dispositif . . . . . . . . . . . . . . . . . . . . . . . . . . 57\n3 Tomographie d’impédance électrique par force de Lorentz 59\n3.1 L’impédance électrique des tissus biologiques . . . . . . . . . . . . . . . . 59\n3.1.1 Les types de milieux biologiques . . . . . . . . . . . . . . . . . . . 59\n3.1.2 Les propriétés électriques des milieux biologiques . . . . . . . . . 60\n3.1.3 Les différents modèles de conductivité électrique des tissus bio-\nlogiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61\n3.1.4 Courbes représentatives (muscle, sang, graisse, os) . . . . . . . . 63\n3.1.5 Tissus sains et tissus tumoraux . . . . . . . . . . . . . . . . . . . . 63\n3.1.6 Les lésions thermiques dans les tissus . . . . . . . . . . . . . . . . 64\n3.2 Les méthodes d’imagerie d’impédance électrique . . . . . . . . . . . . . 65\n3.2.1 La tomographie d’impédance électrique . . . . . . . . . . . . . . . 65\n3.2.2 L’imagerie d’impédance électrique par résonance magnétique . . 65\n3.2.3 L’imagerie acousto-électrique . . . . . . . . . . . . . . . . . . . . . 66\n3.2.4 La tomographie d’impédance électrique par force de Lorentz . . 68\n3.2.5 La tomographie magnéto-acoustique . . . . . . . . . . . . . . . . 69\n12 Pol Grasland-MongrainTable des matières\n3.3 Principe de la tomographie d’impédance électrique par force de Lorentz 70\n3.3.1 Mouvement des ions sous l’action des ultrasons dans un champ\nmagnétique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70\n3.3.2 Densité de courant créée . . . . . . . . . . . . . . . . . . . . . . . . 73\n3.3.3 Détection du courant . . . . . . . . . . . . . . . . . . . . . . . . . . 73\n3.3.4 Existence d’un speckle acousto-électrique . . . . . . . . . . . . . . 74\n3.4 Dispositif expérimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76\n3.5 Expériences mises en œuvre . . . . . . . . . . . . . . . . . . . . . . . . . . 77\n3.5.1 Objectivation du signal . . . . . . . . . . . . . . . . . . . . . . . . 77\n3.5.1.1 Le signal détecté dépend-il de la présence d’ultrasons ? 78\n3.5.1.2 Le signal détecté dépend-il de la présence d’un champ\nmagnétique ? . . . . . . . . . . . . . . . . . . . . . . . . . 78\n3.5.1.3 Le signal détecté est-il décalé dans le temps si on dé-\nplace l’échantillon par rapport au transducteur ? . . . . 80\n3.5.1.4 Le signal détecté dépend-il de la conductivité électrique\nde l’échantillon ? . . . . . . . . . . . . . . . . . . . . . . . 81\n3.5.1.5 Le signal détecté dépend-il de l’émission/réception d’ul-\ntrasons par le transducteur ? . . . . . . . . . . . . . . . . 83\n3.5.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 84\n3.5.2 Image d’un échantillon de gélatine salée . . . . . . . . . . . . . . 85\n3.5.3 Image de tissus organiques . . . . . . . . . . . . . . . . . . . . . . 86\n3.5.4 Observation du speckle électro-acoustique . . . . . . . . . . . . . . 88\n3.5.5 Suivi de lésion thermique . . . . . . . . . . . . . . . . . . . . . . . 91\n3.6 Bilan du chapitre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94\n3.6.1 Principaux résultats . . . . . . . . . . . . . . . . . . . . . . . . . . 94\n3.6.2 Avenir de la méthode . . . . . . . . . . . . . . . . . . . . . . . . . 95\n4 Imagerie d’ondes de cisaillement induites par force de Lorentz 97\n4.1 L’élastographie par ondes de cisaillement . . . . . . . . . . . . . . . . . . 97\n4.1.1 L’élasticité des tissus biologiques . . . . . . . . . . . . . . . . . . . 97\n4.1.2 L’élastographie statique . . . . . . . . . . . . . . . . . . . . . . . . 100\n4.1.3 La vitesse des ondes mécaniques dans les tissus biologiques . . . 101\n4.1.4 L’élastographie par onde de cisaillement . . . . . . . . . . . . . . 103\n4.1.5 L’utilisation de la force de Lorentz . . . . . . . . . . . . . . . . . . 104\n4.2 Principe de la création d’ondes de cisaillement par force de Lorentz . . . 105\n4.2.1 La force volumique dans le milieu . . . . . . . . . . . . . . . . . . 105\nApplications de la force de Lorentz en acoustique médicale 13Table des matières\n4.2.2 La propagation d’une onde mécanique dans le milieu . . . . . . . 107\n4.2.3 Conversion de mode . . . . . . . . . . . . . . . . . . . . . . . . . . 108\n4.3 Dispositif expérimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108\n4.4 Expériences de création d’ondes de cisaillement par force de Lorentz . . 110\n4.4.1 Création d’onde de cisaillement par conversion de mode . . . . . 110\n4.4.2 Création d’onde de cisaillement par force de Lorentz . . . . . . . 111\n4.4.3 Influence éventuelle du mouvement des électrodes . . . . . . . . 113\n4.4.4 Influence éventuelle de l’effet Joule . . . . . . . . . . . . . . . . . 116\n4.4.5 Bilan partiel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n4.5 Expériences d’élastographie par force de Lorentz . . . . . . . . . . . . . . 118\n4.5.1 Elastographie sur un fantôme homogène . . . . . . . . . . . . . . 118\n4.5.2 Expérience sur un fantôme bicouche . . . . . . . . . . . . . . . . . 119\n4.5.3 Expérience sur un morceau de foie . . . . . . . . . . . . . . . . . . 123\n4.5.4 Bilan partiel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123\n4.6 Expériences d’imagerie de conductivité électrique . . . . . . . . . . . . . 124\n4.6.1 Expérience de localisation d’inhomogénéité de conductivité élec-\ntrique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125\n4.6.2 Expérience dans un fantôme avec deux conductivités électriques\ndifférentes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126\n4.6.3 Bilan partiel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128\n4.7 Bilan du chapitre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128\n4.7.1 Principaux résultats . . . . . . . . . . . . . . . . . . . . . . . . . . 128\n4.7.2 Avenir de la méthode . . . . . . . . . . . . . . . . . . . . . . . . . 130\n5 Conclusion 133\n5.1 Bilan général . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133\n5.2 Chronologie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134\n5.3 Communications scientifiques . . . . . . . . . . . . . . . . . . . . . . . . . 136\nAnnexe : Calcul du déplacement pour une force ponctuelle 141\nBibliographie 143\nCredits images 154\nAbstract 157\nRésumé 159\n14 Pol Grasland-MongrainCHAPITRE 1\nIntroduction\nLa force de Lorentz présente une propriété remarquable, celle de relier un cou-\nrant électrique à un mouvement mécanique. Ainsi, le déplacement d’un conducteur\ndans un champ magnétique induit par force de Lorentz un courant électrique, et vice-\nversa. La force de Lorentz, dans une acception large qui sera retenu dans cette thèse,\ninclut les phénomènes d’induction électromagnétique et de force électromotrice. Elle\nest d’ailleurs l’une des principales manifestations de l’interaction électromagnétique,\nqui fait partie des quatre interactions élémentaires de la physique.\nCette force a eu de nombreuses applications depuis le XIXe siècle, en étant la base\nd’objets aussi variés que les générateurs de courant électrique, les accélérateurs de par-\nticules ou les spectromètres de masse. Si elle a déjà été utilisée en acoustique, étant la\nbase de nombreux microphones, elle a été rarement employée en acoustique médicale.\nCe dernier est un domaine large d’utilisation des ondes acoustiques pour notamment\nl’imagerie et le diagnostic de pathologies mais aussi les traitements médicaux.\n1.1 La force de Lorentz\n1.1.1 Historique\nSi les premières manifestations du magnétisme ont été observées dès l’antiquité\ngrecque, il a fallu attendre le XVIIIe siècle pour voir les premières descriptions quanti-\ntatives d’une “force électromagnétique” [45] : une première relation a été proposée par\nJohann Tobias Mayer en 1760 pour des objets magnétiques et par Henry Cavendish en\nApplications de la force de Lorentz en acoustique médicale 15Chapitre 1. Introduction\n1762 pour les particules chargées électriquement, avec des vérifications expérimentales\npar Charles-Augustin de Coulomb en 1784.\nDe nombreuses manifestations de cette “force électromagnétique” ont été ensuite\nobservées au cours du XIXe siècle. Ainsi, Hans Oersted découvrit en 1820 qu’un fil\nparcouru par un courant électrique dévie l’aiguille d’une boussole à proximité – donc\nqu’un champ électrique induit un champ magnétique [120] ; Michel Faraday constate\nen 1831 qu’un champ magnétique variable induit un courant électrique. Les différentes\nthéories de l’électrostatique, de l’électrocinétique et du magnétisme ont finalement été\nsynthétisées par James Maxwell en 1865 par quatre équations [58], [24] :\ndivE=\u001a\n\u000f0;rot E =\u0000dB\ndt;\ndivB= 0; rot B =\u00160j+\u00160\u000f0dEdt(1.1)\nOn donne souvent les noms respectifs de Maxwell-Gauss, Maxwell-Faraday, Maxwell-\nflux et Maxwell-Ampère à ces quatre équations pour marquer la contribution de chaque\nchercheur.\nCes équations, bien qu’elles aient été données au milieu du XIXe siècle, ont “ré-\nsisté” à l’introduction de théories du XXe siècle comme la relativité restreinte et sont\ntoujours valables aujourd’hui. Ce domaine de la physique reste néanmoins l’objet de\nnombreuses recherches. Pour n’en citer qu’une, afin de rendre les quatre équations\nbeaucoup plus symétriques avec des expressions de EetBsimilaires, des physiciens\ncherchent des “monopoles magnétiques”, ou “charges magnétiques”, à l’instar des charges\nélectriques. Bien que Paul Dirac ait prédit une existence théorique possible dans le cadre\nde la physique quantique [29], l’absence de résultat malgré plus d’un siècle de recherche\net la description du champ magnétique comme une manifestation relativiste du champ\nélectrique laisse douter de l’existence de tels monopoles.\n1.1.2 Expressions de la force de Lorentz\nLa force de Lorentz peut s’exprimer de différentes manières selon que l’on s’inté-\nresse à une particule chargée simple ou une distribution de charges dans un volume.\nPour une particule simple Soit une particule de charge qse déplaçant à une vitesse v\ndans un champ électromagnétique (E;B)\nLa force de Lorentz que subit la particule vaut alors :\nF=qE+qv^B (1.2)\n16 Pol Grasland-Mongrain1.1. La force de Lorentz\navec^le produit vectoriel, qui permet de définir la direction et le sens du terme qv^B\ncomme rappelé dans le figure 1.1.\nv\nB\nv x B\nFIGURE 1.1:Règle de la main droite . En l’absence de champ électrique E, une particule\nde charge positive se déplaçant avec une vitesse vdans la direction et le sens du pouce\net un champ magnétique Bdans la direction et le sens de l’index subira une force dans\nla direction et le sens du majeur.\nConcrètement, cela signifie qu’une particule sera localement accélérée dans la di-\nrection du champ électrique E(dans un sens ou l’autre selon sa charge q) et qu’elle\nchangera de direction perpendiculairement à sa vitesse vet au champ magnétique B\n(dans un sens ou l’autre selon sa charge q).\nOn peut alors parler de force électrique pour le terme qEet de force magnétique pour\nle termeqv^B, la force de Lorentz étant la somme de ces deux termes. Cependant, cer-\ntains auteurs désignent improprement la force de Lorentz comme la force magnétique\nuniquement.\nPour un conducteur parcouru par un courant Lorsqu’un conducteur fin placé dans\nun champ magnétique Best parcouru par un courant I(pouvant être lui-même issu\nd’un champ électrique E), la force subie par ce conducteur est égale à :\nF=Z\nLIdl^B (1.3)\navecdlun petit élément du conducteur de longueur totale L.\nApplications de la force de Lorentz en acoustique médicale 17Chapitre 1. Introduction\nOn appelle parfois cette force la force de Laplace , plus fréquemment dans les ou-\nvrages francophones qu’anglophones1, mais on réserve généralement ce terme pour\nles conducteurs filiformes.\nOn peut réécrire cette équation sous une forme volumique, avec un petit élément\nde volume d\u001c:\ndF\nd\u001c=j^B (1.4)\navecjla densité de courant électrique dans l’élément de volume d\u001cdu conducteur.\nC’est sous cette forme que la force de Lorentz sera le plus souvent utilisée dans cette\nthèse : un courant électrique dans un champ magnétique induit un déplacement mé-\ncanique, et vice-versa. Cela peut s’expliquer simplement en observant le déplacement\nde particules conductrices, notamment les ions dans les tissus biologiques, qui est ré-\nsumé sur la figure 1.2. Sous l’influence d’un champ magnétique et d’un déplacement\nmécanique, les particules conductrices dérivent dans la même direction mais dans des\nsens opposés selon leur signe, ce qui induit un courant électrique. La force de Lorentz\nsera utilisée sous cette forme dans le chapitre 2 consacré à un hydrophone et dans le\nchapitre 3 qui étudie une méthode d’imagerie d’impédance électrique. Sous l’influence\nd’un champ magnétique et d’un courant électrique, toutes les particules conductrices\ndérivent dans la même direction et le même sens, ce qui induit un déplacement méca-\nnique global. La force de Lorentz sera utilisée ainsi dans le chapitre 4 consacré à une\ntechnique d’élastographie.\nInduction de Lorentz, induction de Neumann Une formulation particulière de la\nforce de Lorentz est donnée par la force électromotrice due à un mouvement d’un\nconducteur dans un champ magnétique [58]. Cette grandeur est égale à :\ne=Z\nL(v^B):dl (1.5)\noù l’intégration se fait sur la portion considérée du conducteur. Cette force électromo-\ntrice, malgré son nom, n’est pas une force : elle est homogène à une tension, et ce nom\nn’est encore utilisé que pour des raisons historiques. Ainsi, le mouvement d’un conduc-\nteur à une vitesse vdans un champ magnétique Bcréé une tension eà ses bornes. On\nparle alors d’ induction de Lorentz .\nPar ailleurs, si le champ magnétique est variable au cours du temps, il induit un\nchamp électrique, comme décrit par l’équation de Maxwell-Faraday. Ce champ élec-\n1. La nationalité française de Pierre-Simon de Laplace n’y est peut-être pas étrangère.\n18 Pol Grasland-Mongrain1.2. L’acoustique médicale\n(a)\n(b)Champ\nmagnétiqueCourant\nélectrique\nq > 0 q < 0Déplacement\nmécanique+\nDérive DériveChamp\nmagnétique\nCourant\nélectriqueDéplacement\nmécanique+\nDériveq > 0\nDériveq < 0\nFIGURE 1.2: (a) Sous l’influence d’un champ magnétique et d’un déplacement méca-\nnique, les particules conductrices dérivent dans la même direction mais dans des sens\nopposés selon leur signe : cela induit un courant électrique. (b) Sous l’influence d’un\nchamp magnétique et d’un courant électrique, toutes les particules conductrices dé-\nrivent dans la même direction et le même sens : cela induit un déplacement mécanique\nglobal.\ntrique peut alors produire une force électromotrice dans le conducteur même si celui-ci\nest immobile : on parle dans ce cas d’ induction de Neumann .\nCes deux approches ont été rassemblées en une seule à travers l’équation de Fara-\ndaye=\u0000d\b\ndtavec \ble flux de champ magnétique à travers la surface Sformée par le\ncircuit du conducteur (une boucle pour un circuit fermé). Cette équation remarquable,\nqui résume deux phénomènes a priori différents, a été l’une des motivations qui ont\nmené à la relativité restreinte [31]. Mais ceci dépasse le cadre de cette thèse, où l’on\nutilisera uniquement des champs magnétiques fixes.\n1.2 L’acoustique médicale\nL’acoustique médicale est un domaine utilisant les ondes acoustiques pour deux\ngrands types d’applications : pour le traitement des pathologies d’une part et pour\nl’imagerie et le diagnostic d’autre part.\nApplications de la force de Lorentz en acoustique médicale 19Chapitre 1. Introduction\n1.2.1 La thérapie par ultrasons\nLors de ses recherches sur le sonar pendant la première Guerre Mondiale, Paul Lan-\ngevin a constaté que les ondes acoustiques avaient un potentiel de destruction impor-\ntant [83]. Cela a conduit naturellement à l’utilisation des ondes acoustiques pour la\nthérapie. Ainsi, les frères Fry ont évalué les effets des ultrasons sur les tissus biolo-\ngiques dès les années 50 [38]. La thérapie par ultrasons utilise deux effets principaux :\nles effets mécaniques et les effets thermiques. Les effets mécaniques regroupent les phé-\nnomènes autres que thermiques dus au déplacement des tissus par l’onde acoustique.\nEn particulier, les fortes pressions acoustiques peuvent créer des bulles de cavitation\nd’énergie considérable. Les effets thermiques, eux, sont créés par l’absorption de l’onde\nultrasonore par les tissus.\nCes effets ont mené à l’élaboration de thérapies par ultrasons, et notamment les\ntraitements par HIFU (High Intensity Focused Ultrasound) qui, comme leur nom l’in-\ndique, utilisent une onde ultrasonore focalisée à haute puissance. Ainsi, on peut traiter\ndes tissus pathologiques en les brûlant, comme des tumeurs cancéreuses par exemple,\ncomme représenté sur la figure 1.3. Une zone entière peut alors être traitée précisément\npoint par point en déplaçant la zone focale des ultrasons, sans traiter les couches inter-\nmédiaires entre le transducteur et la zone focale. Cette technique a été appliquée entre\nautres pour le traitement du foie [65], [77], du sein [64], [44], de la prostate [99], [43]\n(notamment avec la commercialisation d’appareils cliniques comme l’Ablatherm et le\nSonablate [4]), etc.\nCependant, ce type de traitement est encore jeune et moins connu que la radiothéra-\npie ou la chimiothérapie, et les normes internationales pour assurer un traitement fiable\net précis évoluent encore aujourd’hui [112]. En particulier, la caractérisation précise du\nchamp ultrasonore de thérapie à pleine puissance reste aujourd’hui un challenge, par\nle développement de capteurs à la fois précis et résistants – dont cette thèse proposera\nun prototype.\nParmi les autres applications des ondes acoustiques à des fins de thérapie, on peut\nciter la lithotripsie qui cherche à détruire des calculs rénaux grâce à des “ondes de\nchoc” (des ondes acoustiques puissantes et brèves) [74] ; la sonochimiothérapie et la\nsonoporation, qui améliorent grâce à des ondes ultrasonores la délivrance de drogues,\nsoit en “activant” ces dernières, soit en améliorant leur pénétration dans les cellules\n[114] ; la régénération des os, qui est accélérée par l’application d’ondes acoustiques à\nfaible puissance [70].\n20 Pol Grasland-Mongrain1.2. L’acoustique médicale\nFIGURE 1.3: Principe d’une sonde de traitement du cancer de la prostate par onde ul-\ntrasonore focalisée à haute intensité. La sonde endorectale brûle au point focal le tissu\nprostatique sans endommager le rectum (source page 155).\n1.2.2 L’imagerie acoustique\nLa forme la plus répandue d’imagerie acoustique est l’échographie, technique que\nl’on retrouve dans le domaine médical, vétérinaire ou industriel, qui a été développée\ndès la fin des années 40 par Georges Ludwig puis John Wild [123].\nDans cette technique, une sonde émet une onde ultrasonore dans une direction.\nChaque interface acoustique va donner lieu à une onde réfléchie, ou écho, captée par la\nsonde initiale. En mesurant le temps d’aller-retour des différents échos, et en connais-\nsant la vitesse de propagation de l’onde, on peut en déduire la distance des interfaces.\nCela constitue l’échographie “mode A”. Si l’on convertit l’amplitude des échos en ni-\nveaux de gris et que l’on observe plusieurs directions successives grâce à l’onde ultra-\nsonore, on peut former ligne par ligne une image, ce qui est nommé échographie “mode\nB”, dont on peut voir un exemple sur la figure 1.4. Cette technique présente de nom-\nbreux avantages qui expliquent son succès : résolution spatiale de l’ordre du millimètre,\naffichage des images en temps réel, innocuité, facilité d’utilisation, coût faible de mise\nen œuvre. Cependant, la qualité d’une échographie dépend beaucoup de la position\nde la sonde qui n’est pas connue a priori : l’échographie est “opérateur-dépendant” où\nsouvent une même personne fait l’image et l’interprétation à la fois, contrairement à\ndes radiographies ou des images IRM.\nL’échographie Doppler, développée à partir des années 1960-70 notamment avec L.\nPourcelot [94], est issue de cette technique. Dans cette dernière, on observe les varia-\nApplications de la force de Lorentz en acoustique médicale 21Chapitre 1. Introduction\nFIGURE 1.4: Echographie “mode B” d’un fœtus de 9 semaines (source page 155).\ntions de fréquence de l’onde reçue par rapport à l’onde émise. Cela donne une infor-\nmation sur le déplacement dans le tissu biologique étudié, notamment les vaisseaux\nsanguins.\nMais on peut utiliser les ondes acoustiques d’autres manières pour l’imagerie mé-\ndicale. Par exemple, on peut utiliser le déplacement local du tissu dû aux ultrasons\npour en déduire des paramètres physiques – cela est utilisé entre autres dans la tomo-\ngraphie d’impédance électrique par force de Lorentz, présentée dans cette thèse. Une\nonde lumineuse ou un courant électrique peuvent également être perturbés par la pré-\nsence d’une onde acoustique, ce qui a été mis à profit dans les techniques d’imagerie\nacousto-optique [15] et acousto-électrique [7].\nEnfin, une voie prometteuse d’imagerie médicale a été ouverte avec l’élastographie\nultrasonore dynamique qui permet de réaliser des images d’élasticité des tissus [88].\nCette technique utilise des ondes acoustiques pour faire des images échographiques\net pour provoquer un déplacement dans un tissu, observable sur les images échogra-\nphiques. Cette technique sera décrite plus en détails dans le chapitre de cette thèse\nconsacré à une utilisation particulière de la force de Lorentz en élastographie.\n1.3 Structure de la thèse\nL’objectif de cette thèse a donc été d’étudier différentes façons d’utiliser la force de\nLorentz dans le domaine de l’acoustique médicale. La force de Lorentz présente en effet\nla propriété de relier, via un champ magnétique, un courant électrique à un mouvement\n22 Pol Grasland-Mongrain1.3. Structure de la thèse\nmécanique.\nLa première partie de cette thèse concerne le développement d’un hydrophone pour\neffectuer des champs de vitesse acoustique. Cet hydrophone était constitué d’un fil\nde cuivre vibrant dans un champ magnétique, ce qui induit par force de Lorentz un\ncourant électrique. Un modèle hydrodynamique a été élaboré pour trouver une relation\nentre le courant électrique mesuré et la pression ultrasonore. Un prototype a ensuite été\nconçu et sa résolution spatiale, réponse fréquentielle, sensibilité, résistance et réponse\ndirectionnelle ont été examinées. Cette partie est décrite dans le chapitre 2.\nUne méthode d’imagerie appelée Tomographie d’Impédance Electrique par Force\nde Lorentz a aussi été étudiée. Dans cette méthode, un tissu biologique conducteur est\ndéplacé par ultrasons dans un champ magnétique, ce qui induit un courant électrique\npar force de Lorentz. L’impédance électrique du tissu peut ensuite être déduite de la\nmesure du courant. Cette technique a été appliquée pour réaliser des images de fan-\ntômes de gélatine et de muscle de bœuf, qui ont donné un contraste supplémentaire\naux images ultrasonores. Nous avons aussi pu détecter une lésion thermique dans un\néchantillon de muscle de poulet avec cette méthode. Cela constitue le chapitre 3.\nEnfin, cette thèse a démontré la possibilité de générer des ondes de cisaillement\ndans des tissus mous en utilisant la force de Lorentz. Cela a été réalisé en appliquant\nun courant électrique par deux électrodes dans un solide mou placé dans un champ ma-\ngnétique. Des ondes de cisaillement ont été observées dans des échantillons de gélatine\net de foie. Les ondes de cisaillement ont été utilisées pour trouver l’élasticité des échan-\ntillons et ainsi que leur conductivité électrique. Ces techniques sont présentée dans le\nchapitre 4.\nEnfin, le chapitre 5 dresse un bilan des avancées scientifiques obtenues dans le cadre\nde cette thèse, avec une liste des communications scientifiques.\nApplications de la force de Lorentz en acoustique médicale 23CHAPITRE 2\nHydrophone à force de Lorentz\nAvec le développement des transducteurs ultrasonores destinés à l’imagerie ou la\nthérapie, la connaissance du champ acoustique émis est devenue cruciale [53]. En ef-\nfet, une incertitude sur le champ acoustique lors d’une thérapie par ultrasons focali-\nsés à haute intensité peut avoir des conséquences dramatiques [116], [111], [11] : les\nzones pathologiques peuvent être sous-traitées, des tissus vitaux endommagés. Il est\nd’ailleurs obligatoire aujourd’hui de préciser les paramètres acoustiques des équipe-\nments médicaux ultrasonores [2]. Ces techniques mesurent l’intensité acoustique, la\npression acoustique, le déplacement ou la vitesse particulaire [95]. Cependant, il y a\nencore peu de méthodes bien adaptées aux champs acoustiques à haute puissance [1].\nCe chapitre va donc présenter l’ensemble des techniques disponibles et introduira un\nconcept d’hydrophone basé sur la force de Lorentz, dont l’un des avantages serait de\npouvoir faire des mesures de champs acoustiques à haute puissance.\n2.1 La mesure de champs ultrasonores\n2.1.1 Les ultrasons à haute puissance\nLe terme de “haute puissance” en acoustique n’est pas trivial [55]. A haute puis-\nsance, deux phénomènes peuvent endommager les appareils de mesure : le chauffage\net les dommages créés par la cavitation. La cavitation, dépendant de plusieurs para-\nmètres, peut apparaitre dans des gammes de 100 kPa à 100 MPa, comme indiqué dans\nle tableau 2.1 et peut facilement endommager des capteurs fragiles. Pour un milieu\nApplications de la force de Lorentz en acoustique médicale 25Chapitre 2. Hydrophone à force de Lorentz\nApplication Fréquence\nd’utilisationNombre de\ncycles par\ntirPression\nacoustique\ntypiqueSeuil de\ncavitation\nImagerie\nultrasonore1 à 20 MHz 1 1 MPa 10 MPa\nLithotripsie 0,2 MHz 1 100 MPa 10 MPa\nImagerie\nDoppler1 à 30 MHz 1 à 100 1 à 10 MPa 1 à 10 MPa\nThérapie\nultrasonore0,5 à 10 MHz Continu 1 à 10 MPa 100 kPa\nTABLE 2.1: Revue des principales applications acoustiques médicales, avec les fré-\nquences typiques d’utilisation, le nombre de cycles acoustiques par tir, la pression\nacoustique généralement utilisée (plus exactement, l’amplitude du pic positif de la\npression) et la pression acoustique au-dessus de laquelle des nuages de cavitations\nrisquent d’apparaitre. Certaines applications évitent la cavitation, tandis que d’autres\nl’exploitent (source : [55]).\ndonné, le risque de cavitation augmente avec la pression et le nombre de cycles par\npulse et diminue avec la fréquence.\nD’autre part, une onde acoustique se modifie à cause d’effets non-linéaires lors de\nsa propagation. Cet effet augmente avec la puissance de l’onde. Ainsi, avec une onde\nultrasonore à haute puissance, on peut obtenir une “onde choquée”, qui présente une\nforme en dents de scie. Le spectre de cette onde possède de nombreuses harmoniques,\ndonc un capteur d’onde ultrasonore à haute puissance devra avoir une bande passante\nfréquentielle large pour correctement représenter le signal.\nUn paramètre majeur est constitué par l’intensité acoustique I, qui donne le taux\nd’énergie par unité de suface perpendiculaire à la direction de propagation :\nI=1\nTZ\npudt (2.1)\navecTla durée considérée, pla pression acoustique et ula vitesse particulaire.\n2.1.2 Les mesures de pression acoustique\n2.1.2.1 Les hydrophones piézoélectriques\nLes premiers hydrophones utilisaient, à l’instar des transducteurs ultrasonores clas-\nsiques, l’effet piézoélectrique pour mesurer des pressions ultrasonores. Par cet effet, la\npression acoustique sur une membrane piézoélectrique crée une tension aux bornes de\n26 Pol Grasland-Mongrain2.1. La mesure de champs ultrasonores\nce matériau, tension que l’on peut mesurer et donc relier à la pression. On distingue\nparfois les hydrophones piézoélectriques selon le matériau de la partie sensible, qui\npeut être piézocomposite ou pyroélectrique (comme le polyfluorure de vinylidène, plus\nconnu sous le nom de PVDF) [54].\nLes hydrophones à membrane sont composés d’une membrane piézoélectrique de\ndiamètre typique entre 0,2 mm et 75 mm [98] acoustiquement transparente. Ils pré-\nsentent de nombreuses caractéristiques intéressantes pour le mesure de pression acous-\ntique [27] : leur réponse est plate sur une large plage de fréquences [10], elle est linéaire\nsur une gamme importante de pressions, elle est stable au cours du temps et la mem-\nbrane est quasiment invisible aux ultrasons [113].\nLes hydrophones à aiguilles sont généralement plus fins que les hydrophones à\nmembrane, leur diamètre pouvant atteindre 0,04 mm [98]. Cela permet d’avoir une\nmeilleure résolution spatiale au prix d’une sensibilité moindre. La partie sensible est\nplacée à la pointe de l’aiguille, ce qui permet de peu perturber l’onde acoustique. Un\nhydrophone à aiguille de type “Lipstick” est représenté sur la figure 2.1.\nFIGURE 2.1: (a) Photographie d’un hydrophone piézoélectrique aiguille (PZT-Z44-0400-\nH816, Speciality Engineering Associates, Sunnyvale, CA, USA). (b) Photographie d’un\nhydrophone piézoélectrique de type “Lipstick” (HGL-200-1113, Onda Corporation,\nSunnyvale, CA, USA).\nApplications de la force de Lorentz en acoustique médicale 27Chapitre 2. Hydrophone à force de Lorentz\n2.1.2.2 Les hydrophones à fibre optique\nPlusieurs designs d’hydrophones utilisant des fibres optiques existent.\nUn premier type soumet une portion d’une fibre optique à l’onde acoustique. Celle-\nci modifie la propagation d’une onde lumineuse dans la fibre, comme l’indice de ré-\nfraction, un changement de polarisation... La résolution spatiale de cet hydrophone\nest cependant limitée, car les changements sont intégrés le long du fil soumis à l’onde\nacoustique [34].\nUn deuxième type d’hydrophone, plus fréquent, utilise une pointe de fibre optique\ncomme élément de mesure de la pression. En effet, l’indice de réfraction d’un fluide\ndépend de la pression locale, on peut donc observer des changements dans la réflexion\nde l’onde lumineuse à la pointe de la fibre, à cause du changement d’indice à l’inter-\nface fibre/milieu [66], [115]. On peut voir une vue éclatée de ce type d’hydrophone\nsur la figure 2.2 (FOPH 2000, RP acoustics, Leutenbach, Allemagne). Ces hydrophones\nsont réputés extrêmement résistants, jusqu’à plusieurs centaines de mégapascals, mais\nmanquent de sensibilité, avec une limite basse autour de 1 MPa [17]. Cela restreint\nl’utilisation de ces hydrophones à des signaux intenses comme celui produit par des\nlithotripteurs ou stables dans le temps pour effectuer des moyennes.\nFIGURE 2.2: Vue éclatée d’un hydrophone à fibre optique avec son support (source page\n155).\n2.1.2.3 L’imagerie Schlieren\nL’imagerie Schlieren est une méthode de visualisation des variations d’indice, dé-\npendant de la pression acoustique, dans un milieu transparent en utilisant un rayon\nlumineux, généralement un laser [52], [67], [82]. Les rayons lumineux émis traversent le\nmilieu étudié, puis sont focalisés par une lentille convergente sur une fine lame opaque.\nLes rayons lumineux n’ayant pas été déviés par les changements d’indice du milieu\n28 Pol Grasland-Mongrain2.1. La mesure de champs ultrasonores\nsont absorbés sur la lame opaque, tandis que les autres sont détectés par un capteur\noptique. Cette technique, qui donne des images telles que celle sur la figure 2.3, permet\nde visualiser des champs de pression rapidement, mais nécessite un milieu transpa-\nrent. Cette technique permet en théorie d’avoir des mesures quantitatives [107], mais\nne donne des résultats fiables que pour des basses pressions et des difficultés subsistent\ndans la fabrication du système [89], [92], [90].\nFIGURE 2.3: Exemple de champ de pression reconstruit par un système d’imagerie\nSchlieren. Les zones plus claires correspondent à des pressions plus élevées (source\npage 155).\n2.1.3 Les réflecteurs ponctuels\nIl est également possible de mesurer des propriétés acoustiques d’un champ ultra-\nsonore avec la diffusion des ultrasons par un obstacle. Cet obstacle est généralement\nune sphère métallique placée à la pointe d’un support [14]. Les ultrasons diffusés sont\nalors observés avec un transducteur externe, plus rarement le transducteur d’émission.\nLa mesure est cependant relative, car le transducteur ne capte qu’une faible partie de\nl’onde diffusée. L’un des facteurs limitants de cette technique est le support de la boule,\nqui ne doit pas perturber le champ acoustique, et la taille de la boule. Un fil long et fin\na donc été proposé comme obstacle [97], où ce fil est déplacé en translation et rotation\npour reconstruire le champ complet par tomographie.\nApplications de la force de Lorentz en acoustique médicale 29Chapitre 2. Hydrophone à force de Lorentz\n2.1.4 Les mesures d’intensité acoustique\n2.1.4.1 Les sondes thermiques\nL’un des premiers systèmes de mesure acoustique était simplement constitué d’un\nthermocouple, avec des premières évaluations autour de 1950 [40], [39]. La tempéra-\nture augmente en effet jusqu’à ce qu’il y ait équilibre entre les pertes de chaleur et\nl’absorption d’énergie par le thermocouple. Cet équilibre est généralement atteint en\nmoins d’une seconde [39]. Cette température d’équilibre permet alors d’obtenir l’in-\ntensité ultrasonore – les pertes de chaleur sont proportionnelles à la différence entre la\ntempérature de la sonde et celle du milieu [73]. Cette technique est utilisée à la fois pour\ndes mesures ponctuelles ou pour des mesures volumiques.\n2.1.4.2 Les balances acoustiques\nLorsqu’un obstacle est placé dans la direction de propagation des ultrasons, il va\nsubir une force de radiation acoustique. L’évaluation de cette force permet de calculer\nl’intensité acoustique. Ainsi, il existe un système nommé “balance acoustique” com-\nposé d’un absorbant ultrasonore suspendu à une balance ; lorsque des ultrasons sont\némis sur l’absorbant, ce dernier va être soumis à la fois à son poids et à la force de ra-\ndiation acoustique : la balance indiquera alors une masse différente. Cela permet d’en\ndéduire la force sur l’absorbant, et donc l’intensité ultrasonore [95], [25]. Cependant,\ncette technique ne permet d’obtenir qu’une intensité moyenne au cours du temps et\nsur la surface de la masselotte : la résolution spatiale et temporelle est donc en pratique\nfaible [55]. Dans le même esprit, un type de balance acoustique mesure le changement\nde poids d’un liquide absorbant, lié à son chauffage par ultrasons [111].\n2.1.5 Les mesures de déplacement ou de vitesse particulaire\n2.1.5.1 Les interféromètres\nUn interféromètre de Fabry-Pérot peut être utilisé comme hydrophone. En effet,\nl’onde acoustique peut modifier le chemin optique d’une cavité de Fabry-Pérot, ce qui\nmodule la phase des signaux. Ce changement de phase est détectable par interféromé-\ntrie, et on peut donc en déduire le déplacement particulaire à la surface de la cavité\n[22].\nIl est également possible d’utiliser un interféromètre de Michelson. Dans cet interfé-\nromètre, une membrane servant de miroir acoustique semi-réfléchissant est placée dans\n30 Pol Grasland-Mongrain2.1. La mesure de champs ultrasonores\nFIGURE 2.4: Photographie d’une balance acoustique. Un transducteur est placé dans le\ntube et émet des ultrasons sur l’absorbant placé en bas. La force de radiation acoustique\nest calculée grâce au changement de poids apparent de l’absorbant (RFB-2000 Radiation\nForce Balance, Onda Corporation, Sunnyvale, CA, USA)\nle champ acoustique. Une technique d’interféromètrie mesure alors les déplacements à\nla surface de la membrane, ce qui permet d’en déduire la vitesse des particules [95].\n2.1.5.2 Les hydrophones à force de Lorentz\nL’utilisation de la force de Lorentz pour détecter des ondes acoustiques par mesure\nde courant électrique date du début du XXe siècle, avec par exemple le microphone à\nruban présenté en 1929 par H. Olson [87] représenté sur la figure 2.5. Ce microphone\nest constitué d’un ruban conducteur placé dans l’entrefer d’un aimant. Lorsqu’un son\narrive sur le ruban, celui-ci vibre dans un champ magnétique, ce qui induit par force\nde Lorentz un courant électrique. Ce type de microphone est réputé pour la qualité du\nson rendu, grâce à une large bande passante, mais des bruits secs et intenses peuvent\ndéchirer la membrane. Cet appareil a donc laissé la place à d’autres microphones, même\nApplications de la force de Lorentz en acoustique médicale 31Chapitre 2. Hydrophone à force de Lorentz\nsi l’on constate un retour depuis quelques années notamment grâce à l’amélioration de\nla solidité de la membrane. Ce microphone n’est cependant pas adapté à la cartographie\nde champs de pression acoustiques, notamment car la résolution spatiale serait faible.\nFIGURE 2.5: (a) Principe d’un microphone à ruban : un ruban plissé est placé au milieu\nd’un aimant. Le son fait vibrer le ruban dans le champ magnétique, ce qui induit par\nforce de Lorentz un courant électrique mesurable (source page 155). (b) Photographie\nd’un microphone à ruban (source page 155).\nPlus tard, en 1969, L. Filipczynski a présenté le principe d’un hydrophone électro-\nmagnétique [33]. Celui-ci est constitué d’un absorbant ultrasonore sur lequel est fixé\nun fil électrique, le tout placé dans le champ magnétique d’un aimant. Lorsque des\nultrasons pénètrent dans l’absorbant, l’interface entre le milieu et l’absorbant vibre, et\nnotamment le fil fixé dessus. Comme le fil vibre dans un champ magnétique, cela in-\nduit un courant électrique. Contrairement au microphone à ruban, cet hydrophone est\n32 Pol Grasland-Mongrain2.1. La mesure de champs ultrasonores\ntrès résistant – des tests avec pressions de 30 MPa ne l’ont pas endommagé [32]. La\nrésolution spatiale est cependant limitée par la longueur du fil.\nEn 1997, Y. Sharf a amélioré le dispositif en proposant de placer un fil court sur l’ab-\nsorbant sans être relié à un appareil de mesure [110], comme représenté sur le schéma\n2.6. Le courant électrique est récupéré par induction magnétique, grâce à une bobine\nplacée autour du fil. L’intérêt est de pouvoir raccourcir la longueur du fil fixé à l’ab-\nsorbant, en améliorant par là la résolution spatiale. Mais cela se fait au détriment de la\nsensibilité de la mesure.\nFIGURE 2.6: Deux dispositifs d’hydrophone à force de Lorentz proposés par Sharf et al.\n(a) Dispositif avec un absorbant et un fil fixé autour. (b) Dispositif avec un morceau de\nfil fixé à un absorbant et une bobine réceptrice [110].\nL’hydrophone à force de Lorentz étudié dans cette thèse est un dispositif conver-\ntissant le mouvement du fluide dû aux ultrasons en courant électrique par force de\nLorentz. A la différence des hydrophones présentés ci-dessus, il utilise un fil long et\nn’a pas d’absorbant. L’hydrophone est composé d’un ou plusieurs aimants créant un\nchamp magnétique homogène dans lequel est placé un fil électrique. Lorsque les ultra-\nsons mettent en mouvement le milieu, cela déplace par frottement le fil – si ce dernier\nest suffisamment fin, il perturbe peu le champ acoustique. Le mouvement du milieu dé-\nplace par frottements le fil, ce qui induit un courant électrique dans celui-ci. Une bonne\nrésolution spatiale est alors atteinte en effectuant des rotations et des translations du fil,\nce qui permet de réaliser une tomographie du champ acoustique.\nApplications de la force de Lorentz en acoustique médicale 33Chapitre 2. Hydrophone à force de Lorentz\nType Paramètre\nmesuréRésolution\nspatialeRésistance\nà la\ncavitationCommentaires Réf.\nHydrophone\npiézoélectriquePression\nacoustique40 µm - [98],\n[55]\nHydrophone à\nfibre optiquePression\nacoustique10 µm + Pression mini-\nmale de quelque\ncentaines de kPa[55],\n[66],\n[115]\nImagerie\nSchlierenPression\nacoustiqueTaille du\nfaisceau\nlumineux+++ Utilise une tech-\nnique de recons-\ntruction tomogra-\nphique ; quantita-\ntif à basse pression\nseulement[107],\n[89],\n[92],\n[90]\nRéflecteurs\nponctuelsPression\nacoustique\nréfléchie500 µm 0 Mesure relative ;\nMeilleure résolu-\ntion spatiale avec\nune reconstruction\ntomographique[9],\n[14],\n[97]\nSonde\nthermiqueIntensité\nacoustique10 µm - Peu cher ef facile à\nutiliser[40],\n[39],\n[73]\nBalance\nacoustiqueIntensité\nacoustiqueTaille de\nl’absorbant0 Intensité moyenne\nau cours du temps[95],\n[25]\nInterféromètre Déplacement\nou vitesse\nparticulaire50 µm - [22],\n[95]\nHydrophone\nà force de\nLorentzDéplacement\nou vitesse\nparticulaire5 mm + Résolution spatiale\ndépendante de la\ndirection[33],\n[32],\n[110]\nTABLE 2.2: Caractéristiques des techniques de mesure des paramètres acoustiques pré-\nsenté dans la section. Les symboles -, 0 et + indiquent respectivement que la cavitation\nendommage facilement, modérément ou difficilement l’appareil de mesure. Le système\nd’imagerie Schlieren y est insensible.\n2.1.6 Résumé\nLes caractéristiques de l’ensemble des techniques vues ci-dessus sont résumées sur\nle tableau 2.2. On observe que le principal défaut des hydrophones à force de Lorentz\npar rapport aux autres techniques est sa mauvaise résolution spatiale.\n34 Pol Grasland-Mongrain2.2. Principe de l’hydrophone à force de Lorentz\n2.2 Principe de l’hydrophone à force de Lorentz\n2.2.1 Expression de la force de Lorentz\nOn suppose dans cette partie que le champ magnétique est dans la direction X, le fil\nélectrique dans la direction Y et l’axe ultrasonore dans la direction Z, comme représenté\nsur le schéma 2.7.\nN SChamp\nmagnétiqueZX\nY\nCourant\nélectriqueOnde\nacoustique\nFIGURE 2.7: Principe d’un hydrophone à force de Lorentz : un fil électrique plongé\ndans un champ magnétique vibre sous l’effet d’une onde ultrasonore, ce qui induit un\ncourant électrique.\nLe mouvement d’un conducteur dans un champ magnétique induit un courant élec-\ntrique perpendiculaire au mouvement et au champ magnétique. Ainsi, pour un conduc-\nteur de longueur L, la force électromotrice induite eest proportionnelle à la vitesse v\ndu conducteur multipliée par le champ magnétique B, intégrée le long du conducteur,\nce qui s’exprime :\ne=Z\nL(v^B):dl (2.2)\nSi le fil est isolé, ou si la conductivité électrique du milieu est suffisamment faible\ncomparée à celle du fil, les pertes de courant du fil au milieu environnant peuvent être\nnégligées.\nLe déplacement induisant un courant électrique provient de (a) la vibration interne\ndes charges dans le fil et (b) de la vibration du fil lui-même à cause des ultrasons. Pour\nun fil de diamètre D= 200 µm, à une longueur d’onde ultrasonore \u0015= 1,5 mm (cor-\nrespondant à une fréquence de 1 MHz) avec une vitesse du son égale à 1500 m.s\u00001, on\nconsidère que le diamètre du fil est négligeable face à la longueur d’onde. La vitesse\nApplications de la force de Lorentz en acoustique médicale 35Chapitre 2. Hydrophone à force de Lorentz\ndu son étant de 5000 m.s\u00001environ dans le fil, la variation de pression à l’intérieur du\nfil est par conséquence petite et toutes les charges peuvent être supposées se déplaçant\nensemble. Dans le cas présenté ici, le phénomène (a) est donc négligeable et le signal\nprovient quasi exclusivement du phénomène (b). La section suivante s’attachera donc\nà déterminer la vitesse du fil en fonction de la vitesse ultrasonore.\n2.2.2 Lien entre vitesse du fluide et vitesse du fil\nLa relation entre la vitesse du fluide u0et la celle du fil vpeut être évaluée théori-\nquement [69]. Dans la démonstration suivante, le fil est modélisé comme un cylindre\ninfiniment long. La démonstration se fait dans un plan 2D perpendiculaire au fil et\ntoutes les quantités physiques comme la force et l’énergie doivent être considérées par\nunité de longueur. Pour une onde plane, la pression pet la vitesse du fluide u0sont\nproportionnels :\nu0=p\n\u001ac(2.3)\navec\u001ala densité du fluide, cla vitesse du son.\nEn prenant p = 1 MPa (pression typique dans la zone focale des ultrasons en écho-\ngraphie),\u001a= 1000 kg.m\u00003(densité de l’eau) et c= 1480 m.s\u00001(vitesse du son dans de\nl’eau douce à 20oC), la vitesse du fluide u0vaut approximativement 0,7 m.s\u00001.\nPour évaluer l’importance des forces inertielles par rapport aux forces visqueuses,\non peut utiliser le nombre de Reynolds Requi est égal au rapport entre ces deux forces.\nDans le cas du cylindre infini, le nombre de Reynolds vaut\nRe=u0D\n\u0017(2.4)\navecDle diamètre du cylindre et \u0017la viscosité cinématique du milieu. En prenant D\n= 200 µm,\u0017=1:10\u00006m2.s\u00001(eau à 20oC) etu0= 0,7 m.s\u00001comme dans le cas précé-\ndent, le nombre de Reynolds vaut 180. Cette valeur indique que les forces visqueues\nsont négligeables, contrairerement à celles des forces de pression, et le fluide peut par\nconséquent être considéré comme parfait.\nD’autre part, si le cylindre est petit comparé à la longueur d’onde ultrasonore et\nque les turbulences sont faibles au cours du temps (compatible avec la valeur de Re\ninférieure à 1000), le fluide loin du cylindre est considéré irrotationnel avec une vitesse\nconstanteu0. Si les dimensions typiques des variations de pression (la longueur d’onde\nultrasonore \u0015) sont bien plus grandes que les dimensions de l’obstacle (le diamètre D),\n36 Pol Grasland-Mongrain2.2. Principe de l’hydrophone à force de Lorentz\nle fluide peut de plus être considéré comme incompressible. Sous ces conditions, dans\nun référentiel où le fluide loin du cylindre est au repos, la vitesse du fluide ~ uest égale\nau gradient d’un potentiel \u001esolution de l’équation de Laplace \u0001\u001e= 0.\nLes solutions pour \u001esatisfaisant les conditions aux limites sont les dérivées de ln( r),\noùrest le vecteur orthogonal à l’axe du cylindre, R0le rayon du cylindre et ~ vla vitesse\ndu cylindre dans le référentiel considéré. On peut donc écrire \u001esous la forme :\n\u001e=\u0000R2\n0\nr2~ v:r (2.5)\nLe gradient de potentiel ~ us’écrit alors :\n~ u=grad\u001e=R2\n0\nr2(2r(~ v:r)\nr2\u0000~ v) (2.6)\nUne relation plus pratique à utiliser entre ~ uet~ vpeut être trouvée en utilisant l’éner-\ngie cinétique totale du fluide Ec, simplement égale à :\nEc=\u001a\n2Z\nu2dV (2.7)\nLe domaine d’intégration de Ecest l’ensemble de l’espace considéré cylindrique de\nrayonRavecRtendant vers l’infini, moins le volume du cylindre, de rayon R0. On\npeut de plus réécrire ~u2en~v2+ ~u2\u0000~v2, ce qui donne pour l’énergie cinétique totale :\nEc=\u001a\n2Z\n~v2dV+\u001a\n2Z\n(~ u+~ v)(~ u\u0000~ v)dV (2.8)\nDans la seconde intégrale, on peut remplacer (~ u+~ v)pargrad (\u001e+~ v:r). En utilisant\nles équations de continuité div~ u= 0 etdiv~ v= 0, l’énergie cinétique totale peut être\nréécrite :\nEc=\u001a\n2Z\n~v2dV+\u001a\n2Z\ndiv((\u001e+~ v:r)(~ u\u0000~ v))dV (2.9)\nLa première intégrale est simplement égale à\u001a\n2~v2(\u0019R2\u0000\u0019R2\n0). En utilisant le théo-\nrème de Green-Ostrogradsky, la seconde intégrale peut être remplacée par l’intégrale\nde(\u001e+~ v:r)(~ u\u0000~ v):nsur la surface des cylindres de rayons RetR0. D’après les équa-\ntions de continuité, ~ v:n=~ u:nà la surface du cylindre de rayon R0, ce qui donne une\nintégrale nulle. Après avoir remplacé ~ uet\u001epar leurs expressions en 2.5 et 2.6 et en\nnégligeant les termes tendant vers 0 lorsque Rtend vers l’infini, on trouve finalement\nApplications de la force de Lorentz en acoustique médicale 37Chapitre 2. Hydrophone à force de Lorentz\npour l’énergie cinétique\nEc=\u001a\n2\u0019R2\n0~v2(2.10)\nEn appliquant le théorème de la puissance cinétique, on peut calculer une force de\ntrainée Fdéfinie à partir de l’énergie cinétique :\nEc=\u0000F:~ v=\u0000(\u001a\u0019R2\n0d~ v\ndt):~ v (2.11)\nDans le référentiel du laboratoire, on peut remplacer ~ vpar~ v\u0000~ u0avec~ vla vitesse\ndu cylindre et ~ u0la vitesse du fluide loin du cylindre. Fest alors égal à :\nF=\u001a\u0019R2\n0(d~ u0\ndt\u0000d~ v\ndt) (2.12)\nLa deuxième loi de Newton énonce que l’accélération du cylindre est égale à la\nsomme de la force de trainée et de la force reliée au déplacement du volume de fluide\négal à celui occupé par le cylindre, ce qui peut s’exprimer sous la forme :\n\u001a0\u0019R2\n0d~ v\ndt=\u001a\u0019R2\n0(d~ u0\ndt\u0000d~ v\ndt) +\u001a\u0019R2\n0d~ u0\ndt\n)(\u001a0+\u001a)\u0019R2\n0d~ v\ndt= 2\u001a\u0019R2\n0d~ u0\ndt(2.13)\nAprès intégration par rapport au temps, on trouve finalement la relation 2.14 entre\nla vitesse du fluide ~ u0et celle du cylindre ~ v:\n~ v=K(\u001a;\u001a0)~ u0 (2.14)\navecK(\u001a;\u001a0) =2\u001a\n\u001a0+\u001a. Cette relation, remarquablement simple, ne dépend que de la\ndensité du fluide et du cylindre. Remarquons que si le cylindre et le fluide ont la même\ndensité, ils se déplacent à la même vitesse.\n2.2.3 Reconstruction tomographique\nDans le principe de l’hydrophone à force de Lorentz, le signal induit par force de\nLorentz est créé tout le long du fil, donc la mesure n’est pas ponctuelle. Pour réaliser\ndes champs ultrasonores, on peut réaliser une tomographie, grâce à des rotations et\ntranslations successives du fil.\n38 Pol Grasland-Mongrain2.2. Principe de l’hydrophone à force de Lorentz\nLa transformée de Fourier de l’équation 2.2 s’écrit :\nE(k;\u0012) =Z\nvzB\u000ee\u0000j2\u0019\u000ekd\u000e (2.15)\navec\u000ela coordonnée le long de la direction kd’angle\u0012avec l’axe X, kétant défini par\nk=xcos\u0012+ysin\u0012:\nLa vitesse ultrasonore en chaque point est alors calculée en prenant la transformée\nde Fourier inverse à deux dimensions de l’équation 2.15, ce qui permet de donner la\nvitesse ultrasonore en chaque coordonnée (x;y:\nv(x;y) =1\nBZ\u0019\n0Z+1\n\u00001E(k;\u0012)jkjej2\u0019\u000ekd\u0012dk (2.16)\nLa mesure du courant le long de directions kpour différents angles \u0012donne donc une\nreconstruction du champ de vitesses : l’hydrophone à force de Lorentz mesure donc\nla vitesse (et non la pression) ultrasonore. Cet hydrophone pourrait mener à l’inten-\nsité ultrasonore s’il est combiné avec un hydrophone mesurant la pression ultrasonore\n[111].\nCependant, lorsque l’hypothèse d’onde plane peut être fait, la pression et la vi-\ntesse du fluide sont proportionnelles. Cela permet de calculer la pression ultrasonore\nen chaque coordonnée (x;y):\np(x;y) =\u001ac\nBZ\u0019\n0Z+1\n\u00001E(k;\u0012)jkjej2\u0019\u000ekd\u0012dk (2.17)\nEn pratique, le nombre de translations donnera la résolution spatiale de l’image\ntandis que la qualité de la reconstruction sera reliée au nombre de rotations.\n2.2.4 Résonance des fils\nLa partie sensible de l’hydrophone est donc un fil électrique maintenu à ses extré-\nmités. On peut chercher la fréquence de résonance mécanique de ce fil en le modélisant\ncomme une corde de guitare. Les fréquences de résonance fnsont alors définies par :\nfn=nc\n2L=n1\n2Lr\nT\nw(2.18)\nApplications de la force de Lorentz en acoustique médicale 39Chapitre 2. Hydrophone à force de Lorentz\naveccla vitesse du son dans le fil, Tla tension,wla masse par unité de longueur, Lla\nlongueur totale de fil et nl’harmonique considérée.\nEn prenant une contrainte de 220 MPa (contrainte maximale avant rupture pour\nle cuivre) pour une surface de S= 410\u00009m\u00002pour un fil de 70 µm de diamètre, ce\nqui donne une tension T= 0,9 N,w= 1,3 10\u00004kg.m\u00001etL= 15 cm, la fréquence de\nrésonance mécanique f0est égale à 300 Hz. Les harmoniques non négligeables seront\nprincipalement en-dessous de 1000 Hz : c’est deux à trois ordres de grandeurs sous\nles fréquences de travail, donc la résonance mécanique n’a quasiment pas d’effet sur le\nsignal.\n2.2.5 Dépendance angulaire\nLe modèle peut être affiné en regardant la réponse directionnelle de l’hydrophone.\nCette réponse directionnelle est différente si l’axe de rotation est parallèle ou perpendi-\nculaire au fil électrique. Nous supposerons donc dans la suite que le champ magnétique\nest toujours selon X, le fil selon Y, mais que les ultrasons se propagent dans le plan XZ\n(axe de rotation dans la direction Y) ou YZ (axe de rotation dans la direction X), comme\nreprésenté respectivement sur les figures 2.8-(a) et 2.8-(b).\nN SZX\nY\naxe de\nrotationα\nN SZX\nY\naxe de\nrotationβ\nFIGURE 2.8: (a) Réponse directionnelle avec un axe de rotation dans la direction Y. (b)\nRéponse directionnelle avec un axe de rotation dans la direction X.\nAvec un axe de rotation dans la direction Y, on peut s’attendre à une dépendance\nsinusoidale à cause du produit vectoriel dans l’expression de la force électromotrice\ninduite par force de Lorentz :\ne=Z\n(v^B):dl=Z\nvzBcos(\u000b)dl (2.19)\n40 Pol Grasland-Mongrain2.3. Dispositifs testés\navec\u000bl’angle entre l’axe de propagation des ultrasons et la direction du champ magné-\ntique.\nDe même, la réponse directionnelle avec un axe de rotation dans la direction X\ns’écrit :\ne=Z\n(v^B):dl=Z\nvzBcos(\f)dl (2.20)\navec\fl’angle entre l’axe de propagation des ultrasons et la direction du fil.\nDans ce deuxième cas, l’amplitude mesurée ne présentera pas forcément une dé-\npendance sinusoïdale, car le fil peut être exposé simultanément à des pics et creux\nde pression. En supposant que l’onde ultrasonore a une forme sinusoïdale, la vitesse\ndu fluideu(\u0018)étant égale à U0cos(2\u0019\n\u0015\u0018), avecU0la vitesse maximum du fluide et \u0018=\ndtan(\f),détant la largeur du faisceau ultrasonore. Le calcul de l’intégrale entre 0 et L\nde l’équation 2.20 donne finalement une force électromotrice :\ne=KU0B\ndsinc(2\u0019d\n\u0015tan(\f)) (2.21)\nIl faut cependant bien garder en tête que cette dernière relation dépend de la forme\ndu signal ultrasonore émis, et n’a donc pas un caractère général comme pour le cas où\nl’axe de rotation est dans la direction Y.\n2.3 Dispositifs testés\nL’hydrophone a été mis en œuvre en deux temps, avec d’abord un simple prototype\nservant de preuve de concept, puis un deuxième plus élaboré qui avait pour but d’être\ncaractérisé quantitativement.\n2.3.1 Preuve de concept avec un simple aimant et un fil\nLe premier prototype est composé d’un fil de cuivre verni de 200 µm de diamètre\nplacé dans l’entrefer d’un aimant permanent en forme de U. Cet aimant, visible sur la\nphotographie 2.9-(a) est composé de deux pôles, chacun formé par deux aimants NdFeB\nde 5x5x3 cm3(BLS Magnet, Villers la Montagne, France). L’espace entre les pôles était\nde 4,5 cm. Le champ magnétique était relativement inhomogène, comme le montre une\nsimulation avec le logiciel FEMM 4.2 (figure 2.9-(b)), de 200 à 350 mT le long du fil.\nLe fil électrique, visible sur la photographie 2.10 est soudé à un câble BNC puis relié à\nun amplificateur de courant de 1 MV .A\u00001(HCA-2M-1M, Laser Components, Olching,\nApplications de la force de Lorentz en acoustique médicale 41Chapitre 2. Hydrophone à force de Lorentz\nAllemagne), pour pouvoir observer le signal avec un oscilloscope. Le fil de cuivre était\ntenu de manière lâche de sorte qu’il puisse suivre en chaque point le mouvement du\nfluide tout en étant suffisamment tendu pour rester droit.\nCe prototype a servi de preuve de concept pour montrer qu’il est possible de faire\ndes mesures de champ acoustique à l’aide d’un dispositif simple composé d’un fil élec-\ntrique et d’un aimant.\n10 cm0 T\n0.2 T\n0.4 T\n0.6 T\n0.8 T\n>1 T\nFIGURE 2.9: (a) Photographie de l’aimant en U, composé d’une structure en acier re-\ncouvert d’un vernis protecteur, et de deux paires d’aimants NdFeB. (b) Simulation de\nl’intensité du champ magnétique créé par l’aimant en U avec les lignes de champ en\nnoir. Le champ magnétique atteint 350 mT au centre des deux pôles du bas.\n2.3.2 Deuxième prototype : hydrophone à force de Lorentz\nUn nouveau prototype a ensuite été conçu avec pour objectif, entre autres, d’obtenir\nun champ magnétique homogène le long du fil électrique, comme visible sur la figure\n2.11-(a). Nous avons opté pour un arrangement d’aimants dans une configuration de\nHalbach [49]. Cette configuration comporte huit aimants permanents créant un champ\nmagnétique de 200 mT le long du fil, comme le montre la simulation du logiciel FEMM\n4.2 (figure 2.11-(b)). Ces aimants étaient supportés par un cylindre de PVC. Le champ\nmagnétique a été vérifié à l’aide d’un teslamètre et s’est avéré être entre 160 et 200 mT\nle long du fil, le maximum étant au centre.\n42 Pol Grasland-Mongrain2.4. Caractérisation\nFIGURE 2.10: Photographie du fil électrique du premier prototype.\nUn fil a été introduit dans le creux du cylindre, ressortant à travers deux vis trouées.\nCes vis servaient notamment à tendre plus ou moins le fil électrique (la tension, comme\nil sera démontré lors d’une expérience, a une influence négligeable sur le signal). Ce\nfil était facilement interchangeable pour les besoins de l’expérience. Le fil avec un dia-\nmètre de 200 µm et était fait en cuivre verni. Il était soudé à un câble BNC puis relié à\nun amplificateur de courant de 1 MV .A\u00001(HCA-2M-1M, Laser Components, Olching,\nGermany), pour pouvoir observer le signal avec un oscilloscope.\n2.4 Caractérisation\nLa caractérisation de l’hydrophone à force de Lorentz s’est basée sur la norme Inter-\nnational Electrotechnical Commission 62127-3, afin d’avoir des éléments comparables\navec d’autres hydrophones. Certains éléments spécifiques ont été rajoutés, comme l’in-\nfluence de la tension du fil électrique, d’autre supprimés car non pertinents dans notre\ncas, comme la taille de la partie sensible.\nLe générateur de fonctions était un AFG 3022B (Tektronix, Beaverton, OR, Etats-\nUnis), l’amplificateur de signal avait une puissance de 200 W (LA200H, Kalmus Engi-\nneering, Rock Hill, SC, Etats-Unis). Le transducteur ultrasonore “principal” a été monté\nau laboratoire avec une membrane piézoélectrique (piezoelectric ceramic PZ-28, Kvist-\ngraad, Meggitt, Denmark) d’une fréquence centrale de 0,5 MHz. Il était sphérique avec\nun diamètre de 50 mm et un rayon de courbure de 50 mm. La largeur du faisceau à 0,5\nMHz était de 8 mm dans la zone focale.\nApplications de la force de Lorentz en acoustique médicale 43Chapitre 2. Hydrophone à force de Lorentz\nFIGURE 2.11: (a) Photographie du prototype d’hydrophone à force de Lorentz. Cet hy-\ndrophone est composé de huit aimants placés dans un support en PVC. Lors des ex-\npériences, le fil était plus tendu, grâce à deux vis non visibles sur la photographie. (b)\nSimulation du champ magnétique créé par le prototype d’hydrophone à force de Lo-\nrentz avec les lignes de champ en noir et la direction des aimants indiqués par une\nflèche. Le champ magnétique est égal à 0,17 \u00060,03 T au centre du cylindre.\nPour les expériences de résolution spatiale, de réponse fréquentielle et de sensibilité,\nles caractéristiques ont été comparées avec celles d’un hydrophone lipstick, avec une\npartie sensible de 200 µm de diamètre, calibré avec son propre préamplificateur à 20\ndB (HGL-200-1113 avec préamplificateur AH2010, Onda Corporation, Sunnyvale, CA,\nEtats-Unis).\n2.4.1 Résolution spatiale\nDans la théorie, nous avons vu que la pression ultrasonore ne peut être directement\nmesurée grâce à l’hydrophone à force de Lorentz : il faut réaliser une tomographie par\ndes translations et rotations successives du transducteur. La qualité des champs va être\nconfrontée à l’aide de deux références : une simulation par ordinateur et une mesure par\nun hydrophone piézoélectrique de dimensions comparables à celles de l’hydrophone à\nforce de Lorentz.\nMatériel et méthode Le générateur émettait un signal de 1 V pic à pic à 1,1 MHz avec\n2 sinusoïdes par salve toutes les 10 ms, amplifié, jusqu’à un transducteur de 1,1 MHz,\nde diamètre et de distance focale 50 mm. Pour réaliser un champ acoustique, il était né-\n44 Pol Grasland-Mongrain2.4. Caractérisation\ncessaire de réaliser des translations et des rotations de l’hydrophone. Pour des raisons\nd’encombrement et de poids, seul le transducteur (et non l’hydrophone) était déplacé.\nCe déplacement était réalisé grâce à deux platines motorisées dirigées par un contrô-\nleur (Newport Corporation MM4005, Irvine, CA, Etats-Unis). Le premier prototype a\nété utilisé dans cette expérience et le fil était placé à la zone focale du transducteur. Le\nchamp de pression était ensuite reconstruit à partir des amplitudes pic à pic enregis-\ntrées par l’oscilloscope à chaque position par une transformée de Radon, calculée avec\nla fonction iradon de Matlab.\nLa qualité du champ de pression donné par l’hydrophone à force de Lorentz a été\ncontrôlée de deux manières : en comparant avec une simulation numérique du champ\nde pression basée sur la résolution linéaire de l’intégrale de Rayleigh du champ acous-\ntique [19] et avec le champ de pression acquis par l’hydrophone piézoélectrique lipstick.\nRésultats et discussions Les figures 2.12-(a), -(b) et -(c) montrent respectivement le\nchamp de pression simulé par ordinateur, acquis par l’hydrophone piezoélectrique, et\npar l’hydrophone à force de Lorentz, dans un plan XY à 50 mm du transducteur (dis-\ntance focale). La figure 2.13 représente la pression normalisée le long d’une ligne pas-\nsant par le maximum central pour les trois champs de pression, afin de comparer la\nlargeur du pic central.\nX (mm)Y (mm)\n-4 -2 0 2 4-4-2024\nX (mm)Y (mm)\n-4 -2 0 2 4-4-2024\nX (mm)Y (mm)\n-4 -2 0 2 4-4-2\n024(a) (b) (c)\nFIGURE 2.12: (a) Champ de pression normalisé dans le plan focal du transducteur si-\nmulé par ordinateur. (b) Champ de pression normalisé dans le plan focal du trans-\nducteur acquis par l’hydrophone piezoélectrique. (c) Champ de pression normalisé\ndans le plan focal du transducteur reconstruit par l’hydrophone à force de Lorentz.\nLes contours extérieurs sont sensiblement identiques.\nOn peut remarquer une grande similarité entre les différents champs de pression.\nLa largeur de pic à mi-hauteur est de 1,62 mm pour la simulation par ordinateur, de\nApplications de la force de Lorentz en acoustique médicale 45Chapitre 2. Hydrophone à force de Lorentz\n-4 -2 0 2 400.51\nX (mm)Hydrophone à force de Lorentz\nSimulation par ordinateur\nHydrophone piézoélectriqueAmplitude normalisée\nFIGURE 2.13: Champ de pression normalisé dans le plan focal du transducteur le long\nd’une ligne passant par le maximum d’amplitude, selon la simulation par ordinateur,\nl’acquisition par l’hydrophone piezoélectrique et la reconstruction par l’hydrophone à\nforce de Lorentz. La largeur à mi-hauteur du pic central vaut 1,62 mm pour la simula-\ntion, 1,23 mm pour l’hydrophone piezoélectrique et 1,57 mm l’hydrophone à force de\nLorentz. Loin du pic central, l’hydrophone à force de Lorentz est moins bien résolu que\nles autres méthodes. Sa résolution spatiale est néanmoins millimétrique.\n1,23 mm pour la mesure par hydrophone piezoélectrique et de 1,57 mm pour la mesure\npar hydrophone à force de Lorentz. L’hydrophone à force de Lorentz donne donc une\nlargeur de pic central intermédiaire entre la simulation et l’hydrophone piezoélectrique,\nles deux références. A basse pression et loin du pic central, le champ est cependant de\nmoins bonne qualité, avec en particulier des valeur négatives par endroit.\nCette expérience permet cependant de montrer que la résolution de l’hydrophone à\nforce de Lorentz est au moins millimétrique.\n2.4.2 Réponse fréquentielle\nLe modèle hydrodynamique présenté dans la partie théorique suppose que le fil\nélectrique doit être fin comparé à la longueur d’onde ultrasonore. Un fil de 200 µm\nd’épaisseur est du même ordre de grandeur que la longueur d’onde d’un signal ultra-\n46 Pol Grasland-Mongrain2.4. Caractérisation\nsonore à 7,5 MHz. Nous avons donc évalué la réponse fréquentielle de l’hydrophone\navec plusieurs diamètres de fil électrique.\nMatériel et méthode Le générateur émettait un signal de 1 V pic à pic de fréquence\nentre 100 kHz à 5 MHz par pas de 20 kHz avec 4 sinusoïdes par salve toutes les 10 ms,\namplifié, jusqu’à un transducteur large bande centré à 2 MHz, de diamètre 50 mm et de\ndistance focale 210 mm. L’amplificateur de courant HCA-2M-1M avait une fréquence\nde coupure à 2 MHz, donc a été remplacé par un amplificateur large bande (HCA-10M-\n100k, Laser Components, Olching, Allemagne).\nLe deuxième prototype a été utilisé dans cette expérience et le fil électrique était\nplacé à la zone focale du transducteur. Quatre diamètres de fil ont été testés : 70 µm,\n100 µm, 200 µm et 400µm.\nD’autre part, afin de mieux comprendre la réponse fréquentielle, l’impédance élec-\ntrique de l’hydophone a été évaluée. Pour cela, l’hydrophone à force de Lorentz avec un\nfil à 200 µm de diamètre était plongé dans l’eau et branché directement (sans amplifica-\nteur) à un impédancemètre (ZVB-4 Vector Network Analyzer, Rohde Schwarz GmbH,\nMunich, Allemagne). Ce dernier évaluait l’impédance réelle et imaginaire de .15 à 100\nMHz.\nRésultats et discussions La réponse fréquentielle pour les quatre diamètres de fil est\nreprésentée sur la figure 2.14-(a). On remarque une bande passante plate de 150 kHz\nà 1 MHz environ, puis une diminution assez rapide. La taille du fil semble avoir une\nfaible influence sur la fréquence de coupure haute. La fréquence de coupure basse n’a\npas pu être mesurée.\nL’impédance électrique de l’hydrophone entre 0,15 et 10 MHz est représenté sur la\nfigure 2.14-(b). Le pic de résonance électrique le plus bas apparait vers 30 MHz. Aucune\nrésonance électrique n’influence donc la réponse fréquentielle de l’hydrophone.\nCette réponse fréquentielle ne s’explique pas bien par le modèle hydrodynamique\nretenu, où on suppose seulement que le fil électrique est de taille négligeable devant la\nlongueur d’onde des ultrasons, et devrait faire l’objet de recherches ultérieures.\n2.4.3 Sensibilité\nLa dépendance du signal induit en fonction de la pression a été étudiée afin de\ncomparer les résultats au modèle et surtout de voir à quelle gamme de pressions est\nadapté l’hydrophone.\nApplications de la force de Lorentz en acoustique médicale 47Chapitre 2. Hydrophone à force de Lorentz\n.1 .2 .5 1 2 5−40−200\nFréquence (MHz)Amplitude (dB)\n70 µm\n100 µm\n210 µm\n400 µm\nPartie réelle\nPartie imaginaire\nFréquence (Hz)Impédance (Ohm)\nFIGURE 2.14: (a) Réponse fréquentielle de l’hydrophone à force de Lorentz entre 0,1 et\n5 MHz pour quatre diamètres de fil : 70 µm (rouge), 100 µm (bleu), 210 µm (vert) et\n400 µm (noir). La réponse décroit avec la fréquence, avec une fréquence de coupure\nhaute entre 1 et 2 MHz. La fréquence de coupure varie faiblement avec la taille du\nfil. (b) Impédance réelle ( xbleus) et imaginaire ( +noirs) en fonction de la fréquence\nde l’hydrophone à force de Lorentz entre 0,15 et 100 MHz. Aucun pic de résonance\nn’apparait en-dessous de 30 MHz, donc cela n’explique pas la réponse fréquentielle\nobtenue.\nMatériel et méthode Le générateur émettait un signal de tension variable, de fré-\nquence 0,5 MHz avec 4 sinusoïdes par salve toutes les 10 ms, amplifié, jusqu’au trans-\nducteur principal. L’amplification pouvait être nulle, faite par l’amplificateur de puis-\nsance de 200 W, ou par un amplificateur de 500 W (1040L, Electronics and Innovation\nEngineering, Rochester, NY, Etats-Unis), afin d’obtenir une vaste gamme de pression\nmesurable.\nLe deuxième prototype a été utilisé dans cette expérience et le fil était placé à la\nzone focale du transducteur. Comme la tension de sortie maximale de l’amplificateur\nde signal HCA-2M-1M était égale à 4V , les plus hautes pressions ne pouvaient pas être\névaluées avec celui-ci. Deux expériences ont donc été réalisées, l’une avec amplificateur\nde signal et l’autre sans. Le modèle hydrodynamique a été utilisé pour comparer les\nrésultats avec la sensibilité théorique de l’hydrophone.\nRésultats et discussions Les figures 2.15-(a) et 2.15-(b) montrent le signal acquis par\nl’hydrophone respectivement sans et avec amplificateur de courant. Une régression li-\nnéaire donne une sensibilité de 1;4\u00060;1 10\u000011V/mm/Pa = -217 à -216 dB re µV/mm/Pa\navec un coefficient de régression linéaire R2égal à 0,997 sans l’amplificateur de signal\n48 Pol Grasland-Mongrain2.4. Caractérisation\net de 8;6\u00060;1 10\u00008V/mm/Pa = -141,4 à -141,1 dB re µV/mm/Pa avec un coefficient\nR2égal à 0,9981 avec l’amplificateur de signal. Les valeurs élevées de R2montrent une\nexcellente linéarité en fonction de la pression à la fréquence de 0,5 MHz de 10 kPa à 10\nMPa. Aux pressions faibles, les parasites électromagnétiques deviennent prédominants\npar rapport au signal induit.\nCes valeurs doivent néanmoins être comparées avec celles données par le modèle\nhydrodynamique. En prenant une densité de l’eau \u001a= 1000\u000610kg.m\u00003, la vitesse du\nson dans l’eau c= 1470\u000610m.s\u00001, le champ magnétique au niveau du fil électrique\nB= 0;17\u00060;03T, la largeur du faisceau ultrasonore l= 8\u00061mm, le facteur K(\u001a;\u001a0) =\n5;0\u00060;3avec\u001a0= 9000\u0006500kg.m\u00003, la sensibilité sans amplification est égale à 2;3\u0006\n0;7 10\u000011V .mm\u00001.Pa\u00001, soit -215 à -210 dB re µV/mm/Pa. Cela donne une différence\nde 40% avec la sensibilité expérimentale. Avec une amplification \u000b= 1106V .A\u00001et une\nrésistance du fil plus entrée de l’amplificateur R= 251\u00061 \n, la sensibilité est égale à\n9;4\u00062;7 10\u00008V .mm\u00001.Pa\u00001, soit -143 to -138 dB re µV .mm\u00001.Pa\u00001, ce qui donne une\ndifférence de 10% avec la sensibilité expérimentale. Notons que l’incertitude totale est\nmajoritairement déterminée par l’incertitude sur le champ magnétique, de l’ordre de\n20%.\nCes résultats montrent que le modèle hydrodynamique donne une compréhension\ncorrecte du phénomène, mais encore insuffisante : une calibration reste nécessaire pour\nfaire des mesures précises.\n2.4.4 Résistance à la cavitation\nComme vu dans l’introduction, à puissance ultrasonore élevée, deux phénomènes\npeuvent endommager les hydrophones : le chauffage et les dommages créés par la cavi-\ntation. L’augmentation de la température à cause de l’atténuation supérieure dans le fil\nsera faible car (1) le fil est petit par rapport à la longueur d’onde ultrasonore et (2) le fil\nest refroidi par le fluide environnant. Même si une augmentation globale de la tempé-\nrature peut changer la résistance du fil, celle-ci restera faible par rapport à la résistance\nd’entrée de l’amplificateur. Des nuages de cavitation peuvent cependant endommager\nle fil électrique et donc avoir une influence sur le signal induit. Une expérience a donc\nété mise en oeuvre en utilisant un matériel spécifique pour obtenir périodiquement un\nnuage de cavitation sur le fil pendant une longue période de temps.\nMatériel et méthode Un ordinateur avec un module d’émission / réception (module\nNI5781R, National Instruments, Austin, TX, Etats-Unis) produisait un signal électrique,\nApplications de la force de Lorentz en acoustique médicale 49Chapitre 2. Hydrophone à force de Lorentz\n52.1042.10\n41.10\nPression (MPa)Tension par unité de longueur (V/mm)\n2\nPression (MPa)Tension par unité de longueur (V/mm)\nFIGURE 2.15: (a) Tension acquise par l’hydrophone à force de Lorentz en fonction de\nla pression sans amplificateur de signal. Une régression linéaire donne une sensibilité\nde1;4\u00060;1 10\u000011V .mm\u00001.Pa\u00001, soit -217 à -216 dB re µV .mm\u00001.Pa\u00001avec un coef-\nficient de régression linéaire R2égal à 0,997. (b) Tension acquise par l’hydrophone à\nforce de Lorentz en fonction de la pression avec amplificateur de signal. Une régression\nlinéaire donne une sensibilité de 8;6\u00060;1 10\u00008V .mm\u00001.Pa\u00001, soit -141,4 à -141,1 dB re\nµV .mm\u00001.Pa\u00001avec un coefficient R2égal à 0,9981 avec l’amplificateur de signal.\namplifié par un amplificateur de 69 dB (Prana RD, Brive, France), pour un transducteur\nultrasonore de fréquence centrale égale à 550 kHz avec un diamètre et une distance\nfocale de 10 cm. Les mesures ont été réalisées en trois étapes :\n1. Excitation du transducteur avec un signal avant amplification de 0,05 V avec 28\nsinusoïdes par émissions toutes les 25 ms. La pression était enregistrée par le\ndeuxième prototype d’hydrophone à force de Lorentz placé à la distance focale.\n2. Excitation du transducteur avec un signal de tension régulé avant amplification\nd’environ 2 V avec un rapport cyclique de 0,1 toutes les 25 ms au transducteur.\nL’amplitude était régulée de telle sorte qu’un nuage de cavitation apparaissait\nchaque seconde pendant 4800 secondes dans la zone focale, avec une pression pic\nà pic de 15 MPa. L’index de cavitation, défini comme CI= 20 log(jEj), avecEla\ntransformée de Fourier du signal entre 0,1 et 3 MHz [28], est égal à 15 dB.\n3. Même excitation qu’en (1) avec un signal avant amplification de 0,05 V avec 28\nsinusoïdes par émissions toutes les 25 ms. La pression était enregistrée par le\ndeuxième prototype d’hydrophone à force de Lorentz placé à la distance focale.\n50 Pol Grasland-Mongrain2.4. Caractérisation\nLa modification du signal à cause de la cavitation était évaluée entre les deux si-\ngnaux grâce à leur coefficient de corrélation Rdéfini par :\nR=P\t1\t2pP\t2\n1P\t2\n2(2.22)\navec \t1et\t2les deux signaux considérés. Ce coefficient est égal à 1 quand deux si-\ngnaux \t1et\t2sont identiques, -1 quand exactement opposés et 0 quand totalement\nnon corrélés.\nRésultats et discussion L’amplitude des signaux avant et après 4800 secondes de ca-\nvitation est représentée respectivement en bleu continu et en pointillés noirs sur la fi-\ngure 2.16). Le coefficient de corrélation est égal à 0,9963, valeur qui montre que les\nsignaux sont quasiment identiques.\n0 20 40 60−1.5−1−0.500.511.5Avant cavitation\nAprès cavitationTension (V)\nTemps (µs)\nFIGURE 2.16: Signal avant (bleu continu) et après (pointillés noirs) 4800 s de cavitation\navec un nuage de cavitation par seconde et un index de cavitation égale à 15 dB. Les\nsignaux sont quasiment identiques, avec un coefficient de corrélation égal à 0,9963, ce\nqui montre que la cavitation a eu peu d’impact sur l’hydrophone.\nLa cavitation a donc peu d’impact sur le signal induit et l’hydrophone se révèle\nrobuste aux dommages à haute pression. L’hydrophone peut même faire des mesures\npendant la cavitation ; mais la longueur du fil rend la résolution spatiale médiocre,\ntandis qu’une tomographie ne donnera pas la valeur en temps réel, surtout que le nuage\nde cavitation a un caractère aléatoire qui ne sera pas rendu par la reconstruction.\nApplications de la force de Lorentz en acoustique médicale 51Chapitre 2. Hydrophone à force de Lorentz\n2.4.5 Réponse directionnelle\nLa réponse directionnelle présentée dans la partie théorique a été évaluée grâce à\nune expérience où le faisceau ultrasonore n’était pas orthogonal au champ magnétique\nou à la direction du fil électrique.\nMatériel et méthodes Le générateur émettait un signal de 1 V pic à pic, de fréquence\n0,5 MHz avec 4 sinusoïdes par salve toutes les 10 ms, amplifié, jusqu’au transducteur\nprincipal. L’hydrophone à force de Lorentz était placé au point focal avec différents\nangles par rapport au faisceau ultrasonore.\nL’angle entre le transducteur et l’hydrophone variait de -35oà +35oavec des pas\nde 0,5odans deux directions : avec l’axe de rotation selon la direction X ou Y. Des\nangles plus élevés n’ont pas pu être évalués à cause de l’espace disponible au centre du\nsupport de PVC de l’hydrophone.\nRésultats et discussions L’amplitude normalisée du signal en fonction de l’angle est\nreprésenté sur la figure 2.17-(a) pour un axe de rotation dans la direction Y et sur la\nfigure 2.17-(b) pour un axe de rotation dans la direction X. Les barres d’erreur repré-\nsentent la déviation standard obtenues par huit acquisitions différentes. Pour vérifier\nla théorie initiale, la fonction cosinus a été représentée sur la figure 2.17-(a) et celle don-\nnée en 2.21 est représentée sur la figure 2.17-(b). L’amplitude est supérieure à 80% du\nmaximum pour des angles de -35 à +35opour un axe de rotation dans la direction Y et\nsupérieure à 50% du maximum pour des angles de -25 à +25opour un axe de rotation\ndans la direction X.\nLa réponse directionnelle avec un axe de rotation dans la direction Y se vérifie plu-\ntôt bien tandis que les résultats avec l’axe de rotation dans la direction X sont moins\nproches, ce qui peut s’expliquer par les hypothèses d’onde sinusoïdale effectuées sur le\nsignal émis.\n2.4.6 Influence de la tension du fil électrique\nLe modèle hydrodynamique présenté suppose que le fil suit parfaitement le mou-\nvement du fluide en tout point, mais ceci peut être modifié par la tension du fil.\nMatériel et méthodes Le générateur émettait un signal de 1 V pic à pic, de fréquence\n0,5 MHz avec 4 sinusoïdes par salve toutes les 10 ms, amplifié, jusqu’au transducteur\nprincipal.\n52 Pol Grasland-Mongrain2.4. Caractérisation\n0 20 40 -20 -40Amplitude normalisée\n0 20 4000.20.40.60.81\nAngle(°)\n-20 -40Amplitude normalisée\nFIGURE 2.17: (a) Amplitude normalisée du signal en fonction d’un angle \u000bavec l’axe Z\npour un axe de rotation dans la direction Y et fonction cos(\u000b)en fonction de l’angle \u000b,\navec un écart-type sur huit expériences. La courbe théorique est bien adaptée aux don-\nnées expérimentales. L’amplitude est supérieure à 80% du maximum pour des angles\nde -35 à +35o. (b) Amplitude normalisée du signal en fonction d’un angle \favec l’axe\nZ pour un axe de rotation dans la direction X et fonction sinc (2\u0019\n\u0015tan(\f))en fonction de\nl’angle\f, avec un écart-type sur huit expérience. L’amplitude est supérieure à 50% du\nmaximum pour des angles de -25 à +25o.\nLe deuxième prototype a été utilisé dans cette expérience et le fil était placé à la\nzone focale du transducteur. Il était posé horizontalement, donc le fil électrique avait\nune forme de chainette à cause de son poids. Cette forme, décrite il y a plus de 300 ans\npar Bernoulli, Huyghens et Leibniz [103], a une hauteur yreliée à la position x:\ny(x) =acosh(x=a) +c (2.23)\naveccla hauteur aux extrémités du fil et a=H\nwgavecHla composante horizontale de\nla tension,wla masse par unité de longueur et gl’accélération de pesanteur égale à 9,81\nm.s\u00001.\nLes ultrasons étaient focalisés dans la zone la plus basse du fil et la force électro-\nmotrice induite mesurée. Ensuite, lorsque la tension mécanique était augmentée par\nla rotation de deux vis, le fil se tendait et le point le plus bas se rapprochait du trans-\nducteur. Le délai entre l’émission des ultrasons par le transducteur et sa réception par\nl’hydrophone, modifié, était mesuré. En prenant une vitesse du son de 1470 m/s, le\ndéplacement correspondant était calculé, ce qui permettait d’en déduire la tension. La\nApplications de la force de Lorentz en acoustique médicale 53Chapitre 2. Hydrophone à force de Lorentz\ntension a été ensuite divisée par la surface de la section du fil (un disque de 70 µm de\ndiamètre) pour obtenir une contrainte en pascals, afin d’étendre ce résultat plus faci-\nlement à d’autres fils. La contrainte initiale a été augmentée graduellement jusqu’à la\ndestruction estimée à 220 MPa [124].\nPour quantifier l’influence de la tension sur le signal, nous avons calculé le coeffi-\ncient de corrélation Rentre deux signaux comme défini précédemment par\nR=P\t1\t2pP\t2\n1P\t2\n2(2.24)\nDans cette expérience, le coefficient de corrélation a été calculé pour chaque signal (le\nretard éventuel de l’écho étant compensé) avec un signal de référence, le premier signal\navec une contrainte de 300 kPa.\nRésultats et discussions Le délai entre l’émission des ultrasons et leur réception par\nl’hydrophone à chaque augmentation de tension est représenté sur la figure 2.18. Le\ndélai diminue lorsque la tension augmente, mais les dix derniers déplacements étaient\nquasiment indétectables : ils ont été extrapolés à partir des premières mesures et de la\ncontrainte de rupture, avec un ajustement d’une exponentielle décroissante. Le dépla-\ncement maximal est égal à 1,5 mm (délai de 1 µs avec une vitesse de 1470 m/s), ce qui\nindique que la contrainte initiale était égale à 300 kPa.\nLe tableau 2.3 montre le coefficient de corrélation comme défini en 2.24 entre chaque\nsignal et le signal de référence (contrainte de 0,3 MPa) en fonction de la contrainte du\nfil estimée (en MPa). Le coefficient de corrélation est toujours supérieur à 0,97, avec des\nvariations indépendantes de l’augmentation de la tension, ce qui signifie que la tension\nn’a pas d’effet significatif sur le signal.\n2.5 Bilan du chapitre\n2.5.1 Principaux résultats\nL’objectif de ce travail était de concevoir un hydrophone basé sur la force de Lo-\nrentz. Un modèle hydrodynamique et électromagnétique de l’hydrophone a été pro-\nposé, à partir duquel un prototype a été fabriqué puis caractérisé en suivant la norme\nde l’International Electrotechnical Commission dédiée aux hydrophones.\nLe design est basé sur une approche tomographique avec une reconstruction du\nchamp acoustique par une transformée de Radon inverse. Utiliser la tomographie n’est\n54 Pol Grasland-Mongrain2.5. Bilan du chapitre\n0 5 10 15 20 2547.647.84848.248.448.648.849\nNombre de rotations des vis\nDélai (µs)\nFIGURE 2.18: Délai entre l’émission des ultrasons et leur réception par l’hydrophone à\nchaque augmentation de tension, plus ajustement des données par une exponentielle\ndécroissante. Lorsque la tension croît à cause des rotations des vis, le fil se tend et se\nrapproche du transducteur.\npas rare pour des mesures acoustiques, avec des utilisations en imagerie Schlieren [91]\nou l’utilisation de diffuseurs à fils pour la caractérisation de transducteurs [97]. Pour\ndiminuer le nombre total d’acquistions, on peut s’affranchir des rotations si l’on sait\nque le champ acoustique présente une symétrie de rotation. De plus, le nombre de fils\npeut être multiplié, ce qui permet de faire des acquistions simultanées. L’influence de\nchaque fil sur ses voisins devrait cependant être proprement évalué dans ce cas. A la fin\nde cette thèse, l’étude théorique, expérimentale et par simulation de la faisabilité d’un\nhydrophone multifils était encore en cours.\nLa première étape de la conception du prototype d’hydrophone a été de contrôler\nle champ magnétique au niveau du fil électrique. L’arrangement de Halbach a permis\nd’obtenir un champ magnétique homogène le long du fil, ce qui facilite l’approche to-\nmographique. D’autre part, il a fallu évaluer l’impact de la tension mécanique du fil\nélectrique sur le signal induit. Le coefficient de corrélation entre les signaux acquis à\ndifférentes tensions était supérieur à 0,97, ce qui indique que la tension a un impact\nquasiment négligeable. Le fil doit cependant être légèrement tendu pour éviter des er-\nreurs de localisation des signaux.\nComme suggéré par la norme IEC, la réponse fréquentielle a été étudiée. Les me-\nsures d’impédance électrique n’ont montré aucune résonance dans les fréquences de\nApplications de la force de Lorentz en acoustique médicale 55Chapitre 2. Hydrophone à force de Lorentz\nEstimation de la contrainte du fil (MPa) Coefficient de correlation R\n0,3 1\n0,4 0,9843\n0,7 0,9840\n1 0,9751\n2 0,9765\n3 0,9768\n5 0,9657\n7 0,9760\n10 0,9802\n20 0,9786\n40 0,9775\n50 0,9711\n80 0,9874\n100 0,9833\n200 0,9829\nTABLE 2.3: Coefficient de corrélation entre chaque signal et le signal de référence (ten-\nsion de 0,3 MPa) en fonction de l’estimation de la tension du fil (en MPa).\ntravail entre 0,15 et 10 MHz, avec un premier pic de résonance à 30 MHz. Les réso-\nnances électriques n’ont donc pas d’impact sur les mesures de l’hydrophone à force de\nLorentz. Une expérience a montré que la fréquence haute de coupure se trouve entre 1\net 2 MHz, valeur qui semble faiblement dépendre du diamètre du fil. Ceci est assez mal\nexpliqué par le modèle hydrodynamique que nous avons utilisé qui n’est valable que\npour un diamètre du fil petit comparé à la longueur d’onde ultrasonore. La fréquence\nbasse de coupure n’a pu être étudiée avec l’équipement disponible mais est inférieure\nà 100 kHz.\nLa réponse directionnelle a également été évaluée. Pour la plupart des hydrophones,\ncela a un impact sur la taille effective de l’élément sensible, mais la forme cylindrique\nde cet hydrophone à force de Lorentz rend ce paramètre peu pertinent. La réponse di-\nrectionnelle à -3 dB est supérieure à \u000635oavec un axe de rotation dans la direction Y,\nla limite étant l’espace intérieur du cylindre de PVC, et égale à \u000625oavec un axe de\nrotation dans la direction X. Une modèle de réponse directionnelle a été proposé et est\nen accord avec les résultats expérimentaux.\nLes sensibilités expérimentales et théoriques sont relativement proches, avec des\ndifférences inférieures à 40%. Le modèle hydrodynamique donne donc une bonne com-\npréhension du phénomène, même si une calibration reste nécessaire pour des mesures\nprécises. A basse pression, inférieure à 50 kPa, deux problèmes surviennent : (1) les pa-\n56 Pol Grasland-Mongrain2.5. Bilan du chapitre\nrasites électromagnétiques et le signal induit par force de Lorentz sont du même ordre\nde grandeur, donc un blindage serait nécessaire ; et (2) le nombre de Reynolds devient\ntrès proche de 1 (pour un fil avec un diamètre de 200 µm à 10 kPa, Re= 1;3) et la\nprincipale hypothèse du modèle hydrodynamique n’est pas satisfaite. Inversement, le\nsignal devrait être linéaire à des pressions supérieures à 12 MPa. De plus, l’hydrophone\ns’est révélé résistant à la cavitation. Pour des comparaisons avec d’autres hydrophones,\nil faut cependant garder à l’esprit que l’hydrophone à force de Lorentz mesure surtout\nla vitesse du fluide, et que la relation utilisée entre la vitesse et la pression n’est valide\nque dans certaines conditions, notamment d’onde plane.\n2.5.2 Avenir du dispositif\nLes prochaines études devraient améliorer le modèle, en particulier à l’aide de simu-\nlations informatiques. Cela pourrait aider notamment à trouver un moyen d’augmen-\nter la fréquence de coupure. En effet, la réponse fréquentielle de cet hydrophone est\nen-deçà des attentes pour des applications médicales. Cet hydrophone fonctionne bien\npour des fréquences inférieures au mégahertz, sans que le modèle hydrodynamique\npuisse expliquer sa forme.\nLe principal atout de l’hydrophone à force de Lorentz pour la caractérisation de\nchamps acoustiques est sa robustesse. Le fil électrique est très résistant aux hautes\npressions, quasiment insensible à la chaleur, et s’est révélé robuste à la cavitation. A\ntitre anecdotique, on peut remarquer que le principal défaut du microphone à ruban\nproposé par H. Olson, la fragilité de la partie sensible, se révèle paradoxalement être la\nprincipale force de l’hydrophone à force de Lorentz !\nUn autre avantage de cet hydrophone est son faible coût - si le fil est abimé, il peut\nêtre facilement remplacé à bas prix. La destruction du vernis sur le fil n’aurait qu’un\nimpact faible sur le signal, car la conductivité électrique de la plupart des électrolytes est\nfaible par rapport à la conductivité électrique du fil (conductivité d’une salée inférieure\nà 5 S.m\u00001, conductivité du cuivre environ égale à 60 MS.m\u00001, et les pertes de courant\ndans le milieu environnant seraient être négligeables.\nLa mesure de vitesse plutôt que de pression peut être intéressante pour des mesures\nprécises acoustique. Cet hydrophone pourrait en particulier être combiné avec un hy-\ndrophone mesurant la pression ultrasonore afin calculer une intensité acoustique sans\napproximation d’onde plane, voire pour étudier les non-linéarités acoustiques [111].\nEn l’état, une application possible de cet hydrophone est donc la caractérisation de\ndispositifs médicaux à haute puissance et faible fréquence, comme la lithotripsie.\nApplications de la force de Lorentz en acoustique médicale 57Chapitre 2. Hydrophone à force de Lorentz\nEnfin, une autre voie de développement pourrait être le domaine de l’acoustique\nsous-marine, qui utilise des fréquences inférieures au mégahertz et donc évite le prin-\ncipale défaut de ce dispositif.\n58 Pol Grasland-MongrainCHAPITRE 3\nTomographie d’impédance\nélectrique par force de Lorentz\n3.1 L’impédance électrique des tissus biologiques\n3.1.1 Les types de milieux biologiques\nDu point de vue électrique, les tissus biologiques sont des milieux complexes qui\nressemblent peu aux autres matériaux et présentent une grande variabilité de proprié-\ntés au sein du corps humain [75]. Pour étudier les propriétés électriques des tissus, on\npeut néanmoins les classer en trois grands groupes : les liquides, les tissus conjonctifs\net l’épithélium, ces tissus composant les organes tels que le cerveau, le cœur, le foie...\nLes liquides sont composés essentiellement d’eau, de minéraux et d’éléments or-\nganiques. Cela rassemble des liquides aussi variés que le liquide céphalo-rachidien,\nl’urine, l’humeur aqueuse...\nLes tissus conjonctifs sont composés de cellules séparées par un milieu intercel-\nlulaire. Il peut s’agir par exemple du sang, dont le milieu intercellulaire est liquide,\nd’os, où le milieu intercellulaire calcifié, de tissus adipeux majoritairement composés\nde poches de graisse, etc.\nEnfin, l’épithélium est composé de cellules jointes. C’est le cas de l’épiderme à la\nsurface de la peau, mais on en trouve également dans le corps humain, comme l’épi-\nthélium intestinal ou l’épithélium gastrique.\nApplications de la force de Lorentz en acoustique médicale 59Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\n3.1.2 Les propriétés électriques des milieux biologiques\nD’un point de vue électromagnétique, les tissus biologiques peuvent être caractéri-\nsés comme amagnétiques, conducteurs et diélectriques à pertes [13].\nMises à part certaines protéines comme la méthémoglobine qui sont paramagné-\ntiques, l’ensemble des tissus biologiques est en effet diamagnétique. Le corps humain\nest donc globalement considéré comme insensible au champ magnétique, et l’on prend\nquasi systématiquement une perméabilité magnétique égale à celle du vide.\nLes propriétés électriques d’un milieu sont décrites grâce à sa conductivité élec-\ntrique complexe \u001b(!), définie comme :\nj=\u001b(!)E (3.1)\navecjla densité de courant en un point, Ele champ électrique en ce point et !la\npulsation du champ électrique. Cette conductivité électrique complexe peut être écrite\ncomme la somme d’une partie réelle et imaginaire selon la relation : \u001b(!) =\u001b0(!) +\ni\u001b00(!), avec\u001b0(!)la conductivité électrique réelle, ile nombre complexe dont le carré\nvaut -1 et\u001b00(!)la conductivité électrique imaginaire. Cette grandeur représente la\nconductance d’un volume de matière et a pour unité les Siemens par mètre (S.m\u00001).\nLa résistivité \u001aest définie comme l’inverse de \u001b(!), avec pour unité des \n.m.\nLa conductivité électrique réelle \u001b0(!)représente l’ensemble des pertes dans le mi-\nlieu dû au déplacement des charges libres et liées. Cette grandeur peut être séparée en\nla somme de deux composantes, \u001bSqui représente les pertes ohmiques dans le milieu\n(résistance au mouvement des charges libres) et \u001bd(!)qui représente les pertes diélec-\ntriques, dépendant de la fréquence (résistance au mouvement des charges liées).\nLa conductivité électrique imaginaire \u001b00(!)traduit la polarisabilité du matériau,\ndonc sa capacité à stocker de l’énergie sous forme d’un champ électrique. Plusieurs\nmécanismes de polarisation existent, comme la polarisation d’orientation de molécules\ndipolaires rigides, la polarisation ionique, la déformation du nuage électronique des\natomes... Chaque type de polarisation présente une réponse maximale à une fréquence\ndu champ électrique définie, ce qui correspond à une résonance conduisant à une ab-\nsorption d’énergie par le milieu – c’est ce que l’on appelle le phénomène de relaxation.\nOn peut définir la permittivité relative \u000f0\nrcomme la conductivité électrique imaginaire\ndivisée par la pulsation : \u000f0=\u001b00\n\u000f0!. Cette grandeur est plus souvent utilisée pour les\ntissus biologiques car elle présente une amplitude de variation moins importante que\n\u001b00(!).\n60 Pol Grasland-Mongrain3.1. L’impédance électrique des tissus biologiques\nComme représenté sur la figure 3.1 issue d’un modèle donné par Schwan et al. [108],\non observe expérimentalement quatre grands phénomènes de relaxation dans les tissus\nbiologiques à des fréquences inférieures au GHz, que l’on nomme respectivement \u000b,\f,\n\u000eet\r[47].\n103106109\nFréquence (Hz)100Permittivité relative εrConductivité σ (S/m)\n103\n100106\n10\n1εr\nσα\nβ\nγδ\nFIGURE 3.1: Variation de la conductivité électrique et de la permittivité relative typique\nd’un tissu en fonction de la fréquence. On observe plusieurs sauts, correspondants à\ndes phénomènes de relaxation, que l’on nomme \u000b,\f,\u000eet\r[108]\nLa relaxation \u000bapparait entre quelques Hz et quelques kHz. Elle correspond à une\ndiffusion des ions au niveau de la membrane des cellules : la formation d’une double\ncouche d’ions à la surface des membranes fait apparaitre des dipôles électriques locaux.\nLa conductivité électrique réelle varie peu tandis que la permittivité relative \u000frdiminue\nfortement.\nLa relaxation \fse situe entre 500 kHz et 20 MHz. A ces fréquences, la membrane\nn’est plus isolante, et la conductivité électrique réelle devient représentative de la conduc-\ntivité intracellulaire et extracellulaire. D’autre part, la membrane se polarise moins bien,\ndonc la permittivité relative diminue.\nLa relaxation \u000e, autour de 50 MHz, est plutôt faible et est due à la relaxation de\nmolécules d’eau au voisinage de macromolécules.\nEnfin, la relaxation \rn’intervient qu’à très haute fréquence, autour de 17 GHz, et a\npour origine l’orientation dipolaire des molécules d’eau.\n3.1.3 Les différents modèles de conductivité électrique des tissus\nbiologiques\nLes liquides biologiques sont bien modélisés par des électrolytes. Différents mo-\ndèles empiriques ont été proposés pour modéliser la dépendance fréquentielle des pro-\nApplications de la force de Lorentz en acoustique médicale 61Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\npriétés électriques des autres tissus, en particulier les tissus conjonctifs [30].\nLe modèle de Fricke Pour des milieux biologiques peu complexes, on peut se conten-\nter d’un modèle de conductivité électrique simple comme le modèle de Fricke [37].\nCe modèle suppose que l’on a un tissu conjonctif isotrope, sans inhomogénéité ni mo-\nlécules polaires, et essentiellement composé de cellules baignant dans un liquide. On\nconsidère que les cellules sont composées d’une membrane de quelques micromètres\nd’épaisseur, de capacité électrique Cm, renfermant un noyau et diverses structures que\nl’on désignera sous le terme de milieu intracellulaire, avec une résistance Ri. Cette\nmembrane, partiellement perméable, isole le milieu intracellulaire de l’extérieur, le mi-\nlieu extracellulaire, de résistance Re. On peut alors donner un circuit électrique équi-\nvalent avec en parallèle une résistance Riet un dipôle condensateur Cmet résistance\nRe, comme représenté sur la figure 3.2.\nRe\nRiCm\nFIGURE 3.2: Modélisation par un circuit électrique d’un tissu biologique simple. Ce mi-\nlieu est composé de cellules dont la membrane a une capacité Ciet un milieu intracel-\nlulaire de résistance Ri, et baignent dans un milieu extracellulaire de résistance Re\nLe modèle de Debye La plupart des tissus ne peuvent cependant pas être modéli-\nsés aussi simplement. En particulier, les tissus biologiques présentent plusieurs phéno-\nmènes de relaxation qui ne sont pas prédits par le modèle ci-dessus. Si on considère\nplusieurs seuils de relaxation k, on peut adopter le modèle de Debye [26]. Dans ce mo-\ndèle, la conductivité électrique \u001b(!)peut être écrite sous la forme :\n\u001b(!) =\u001b1+i!\u000f0\u000f1+X\nk\u0001\u001bk\n1\u0000i!\u001ck(3.2)\n62 Pol Grasland-Mongrain3.1. L’impédance électrique des tissus biologiques\navec\u001b1la conductivité électrique du tissu limite à haute fréquence, \u000f1la permittivité\ndiélectrique à très haute fréquence, \u0001\u001bkla variation de conductivité électrique pour la\nrelaxationket\u001ckla constante de temps du phénomène de relaxation kconsidéré.\nLe modèle de Cole-Cole Le modèle de Debye peut être amélioré en prenant en compte\nune distribution de constantes de temps plutôt qu’une série discrète, ce qui donne le\nmodèle de Cole-Cole [21] défini selon l’équation :\n\u001b(!) =\u001b1+i!\u000f0\u000f1+X\nk\u0001\u001bk\n1 + (\u0000i!\u001ck)1\u0000\u000bk(3.3)\navec\u000bkun paramètre empirique qui caractérise le temps caractéristique du phénomène\nde relaxation k. A noter que ce modèle est essentiellement empirique mais n’a pas spé-\ncialement de justification physique [76].\n3.1.4 Courbes représentatives (muscle, sang, graisse, os)\nLa conductivité électrique de plusieurs milieux biologiques est représentée sur la\nfigure 3.3, à partir d’une revue de Gabriel et al. faisant référence en la matière [41], [42].\nDans les expériences de ce chapitre, on se placera principalement à une fréquence de\n500 kHz. A cette fréquence, on observe que la conductivité électrique d’un morceau de\ntissu adipeux, égale à 0,08 S/m, est environ dix fois inférieure à celle d’un morceau de\nmuscle, égale à 0,5 S/m (on ne tient pas compte des effets d’anisotropie du muscle, qui\nsont faibles à une fréquence supérieure à une centaine de kilohertz). Des valeurs issues\nd’un modèle de Cole-Cole sont également indiquées en annexe.\n3.1.5 Tissus sains et tissus tumoraux\nPlusieurs études montrent que les tissus tumoraux présentent des propriétés élec-\ntriques différentes des tissus sains. Par exemple, Haemmerich et al. [48] montre qu’à 1\nMHz, les tumeurs hépatiques présentent une conductivité électrique de (2,69 \u00060,91)\n10\u00001S.m\u00001tandis que les tissus sains avaient une conductivité électrique de (4,61 \u0006\n0,42) 10\u00001S.m\u00001: cela montre que les deux types de tissus peuvent être différenciés par\nleur conductivité électrique. La mesure à différentes fréquences (spectroscopie d’impé-\ndance électrique) permet par ailleurs de bien distinguer les tissus normaux des tissus\ncancéreux, par exemple dans le sein [62], pour la peau [3] ou pour la prostate [51].\nApplications de la force de Lorentz en acoustique médicale 63Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\n103106109103106graisse\nmuscle\nsang\nos\n10310610910-1graisse\nmuscle\nsang\nosConductivité σ (S/m)100101\n10-2\n(a) (b)Fréquence (Hz) Fréquence (Hz)Permittivité relative εr\nFIGURE 3.3: Variation de la conductivité électrique (a) et de la permittivité relative (b)\nde la graisse, du sang, d’un muscle et d’un os en fonction de la fréquence. A 500 kHz,\nfréquence principale étudiée dans cette étude, on observe une conductivité électrique\ndix fois inférieure pour la graisse que pour le muscle [41].\n3.1.6 Les lésions thermiques dans les tissus\nLa température affecte notablement la conductivité électrique des tissus. Duck [30]\nmontre par exemple qu’un morceau de muscle chauffé entre 20 et 40°C présente à une\nfréquence de 100 kHz une augmentation de conductivité électrique de 2,1 % .oC\u00001(va-\nleur qui se retrouve dans de nombreux tissus). On peut donc faire des suivis de tempé-\nrature par imagerie de conductivité électrique. Cela serait particulièrement utile pour\nsuivre la formation de lésions thermiques [63], notamment par ultrasons focalisés à\nhaute intensité.\nUn modèle particulièrement intéressant a été proposé par Pop et al. [93], où le chan-\ngement de conductivité électrique dans les tissus chauffés est séparé en deux parties :\nune partie réversible, simplement liée à l’augmentation de la température, et une par-\ntie irréversible, liée à des modifications structurelles du tissu. De cette étude ressort\nque les tissus peuvent emmagasiner une certaine quantité de chaleur sans dommage.\nLorsque la quantité de chaleur dépasse une certaine valeur, des dommages irréversibles\napparaissent. Ainsi, lorsque le tissu est refroidi à sa température initiale, les valeurs de\nconductivité et de permittivité ne reviennent pas à leur valeur initiale si la quantité de\nchaleur absorbée a été suffisante.\n64 Pol Grasland-Mongrain3.2. Les méthodes d’imagerie d’impédance électrique\n3.2 Les méthodes d’imagerie d’impédance électrique\nBien que les propriétés électriques des tissus biologiques aient été intensément étu-\ndiées comme montré dans la section précédente, peu de méthodes existent pour faire\ndes mesures non invasives de conductivité électrique. En imagerie médicale, la mé-\nthode la plus ancienne et la plus développée est la Tomographie d’Impédance Elec-\ntrique, mais ne reste que rarement utilisée en clinique. Plusieurs autres méthodes tentent\nde produire des images de l’impédance électrique des tissus, mais elles n’en sont à\nl’heure actuelle qu’au stade de recherche et développement.\n3.2.1 La tomographie d’impédance électrique\nLa technique la plus avancée aujourd’hui se nomme la Tomographie d’Impédance\nElectrique [20] (EIT pour “Electrical Impedance Tomography”). Dans cette technique,\nschématisée sur la figure 3.4, des dizaines d’électrodes sont placées sur le patient. Un\ncourant électrique suffisamment faible pour être inoffensif est injecté entre deux élec-\ntrodes et le potentiel dans le tissu est mesuré par toutes les autres. Puis on change\nsuccessivement d’électrodes d’injection afin d’avoir un ensemble de mesures. A partir\nde l’ensemble des mesures, une image tridimensionnelle de la conductivité électrique\npeut être reconstruite. Il s’agit d’un problème inverse dit “mal posé” à cause du faible\nnombre de mesures comparé au nombre de points à reconstruire. Ainsi, malgré des\nprogrès mathématiques importants et l’utilisation d’hypothèses de reconstruction, la\nrésolution spatiale reste faible par rapport aux autres méthodes d’imagerie médicale\n[16]. Cette technique est cependant utilisée cliniquement pour certaines pathologies du\npoumon [36], en complément d’une technique d’imagerie classique. D’autres appareils\nsont en cours de développement, comme pour l’imagerie de la prostate avec une sonde\nendoscopique [61]. On peut également citer un appareil utilisé cliniquement nommé\nT-scan basé sur ce principe, même s’il ne produit pas d’image à proprement parler. Cet\nappareil permet de détecter des cancers du sein par l’injection dans le corps d’une pa-\ntiente d’un courant électrique récupéré au niveau du sein : un courant anormal peut\nindiquer la présence de nodules cancéreux [8].\n3.2.2 L’imagerie d’impédance électrique par résonance magnétique\nLa Tomographie d’Impédance Electrique par Résonance Magnétique (MREIT, pour\n“Magnetic Resonance Electrical Impedance Tomography”) est une technique dévelop-\npée à partir de 2002 d’imagerie d’impédance électrique utilisant, comme son nom l’in-\nApplications de la force de Lorentz en acoustique médicale 65Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nVVI\nLignes de courantEquipotentielles\nFIGURE 3.4: (a) Principe de la tomographie d’impédance électrique. Des électrodes sont\nplacées tout autour d’un tissu à étudier. Un courant électrique est injecté dans le tissu\npar deux électrodes et le potentiel électrique est capté par toutes les autres. Le schéma\nreprésente des lignes de courant en rouge et des équipotentielles en beige. (b) Résultat\nd’une tomographie d’impédance électrique au niveau de la poitrine d’un patient, avec\nun niveau de couleur relié à la conductivité électrique locale. La résolution spatiale est\nmédiocre, mais permet pour un œil expérimenté de savoir si les poumons sont sains ou\nnon [72].\ndique, un appareil d’imagerie par résonance magnétique [68], [125]. Le principe, repré-\nsenté sur le figure 3.5, est d’acquérir des images IRM du patient. Ensuite, un courant\nélectrique faible est envoyé dans un organe à l’aide par des électrodes. Le courant se\npropage dans les tissus, avec une densité de courant supérieure dans les zones d’im-\npédance électrique faible. La propagation de ce courant modifie le champ magnétique\nprincipal B z, ce qui perturbe donc le signal de résonance magnétique. Ces signaux per-\nturbés permettent alors de calculer l’impédance électrique locale. La résolution spatiale\nest identique à celle de l’IRM [6]. L’une des difficultés actuelle est la détermination la\nquantité de courant à injecter pour avoir un rapport signal sur bruit suffisant tout en ne\nprésentant pas de risque pour le patient [85].\n3.2.3 L’imagerie acousto-électrique\nSi un courant électrique est injecté dans un organe par des électrodes, il se propage\ndans toutes les directions, avec une intensité qui dépend de la conductivité électrique\nlocale. Mais il est difficile de savoir exactement par où le courant électrique passe :\nc’est l’une des principales difficultés de la tomographie de conductivité électrique. Cet\ninconvénient est contourné dans la technique d’imagerie acousto-électrique[7] (aussi\nappelée tomographie acousto-électrique [128]), proposée au début des années 2000. Le\n66 Pol Grasland-Mongrain3.2. Les méthodes d’imagerie d’impédance électrique\nFIGURE 3.5: (a) Image IRM d’un fantôme d’agar cylindrique contenant trois échantillons\nde tissu biologique. (b) Image de conductivité électrique du même échantillon en uti-\nlisant la méthode de tomographie d’impédance électrique par résonance magnétique.\n(c) Image des lignes de courant et de l’amplitude de la densité de courant J en chaque\npoint. Le courant était injecté de l’électrode de gauche vers celle de droite. (d) Densité\nde flux de champ magnétique B modifié par la densité de courant montré en (c) [109].\nprincipe de cette technique, schématisée sur la figure 3.6, est de faire passer un cou-\nrant électrique dans un tissu, puis de modifier la conductivité électrique en un point en\ny focalisant une onde acoustique. Cela modifie alors la propagation locale du courant\nélectrique. Et plus la conductivité électrique dans la zone focale de l’onde acoustique\nApplications de la force de Lorentz en acoustique médicale 67Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nest forte, plus la modification du courant électrique sera importante, ce qui est mesu-\nrable avec des électrodes [71], [60]. Ainsi, en focalisant l’onde acoustique en différents\nendroits du tissu, on peut obtenir une image de conductivité électrique des tissus avec\nune résolution spatiale de l’ordre du millimètre [86].\nImpedance\nMeasurementImpedance\nMeasurement(a) (b)\nFIGURE 3.6: Principe de l’imagerie acousto-électrique. Un courant électrique est injecté\ndans un tissu via deux électrodes. Une onde ultrasonore est focalisée en un point du\ntissu, et les changements mesurés au niveau des électrodes permettent de déduire la\nconductivité électrique dans la zone focale des ultrasons.\n3.2.4 La tomographie d’impédance électrique par force de Lorentz\nUne autre technique d’imagerie d’impédance électrique a été proposée sous le nom\nd’Imagerie par Effet Hall (HEI pour “Hall Effect Imaging”) en 1998 [122]. Ce nom\na ensuite été critiqué par certains auteurs [102], car le phénomène utilisé n’est selon\neux pas l’effet Hall, si bien que d’autres appellations ont été proposées, comme la\nTomographie Magnéto-Acousto-Electrique (MAET, pour “Magneto-Acousto-Electrical\nTomography”) [126] ou la Tomographie d’Impédance Electrique par Force de Lorentz\n(LFEIT pour “Lorentz Force Electrical Impedance Tomography”) [46]. Ce dernier nom,\nqui a été proposé dans le cadre de cette thèse, permet à la fois de caractériser le para-\nmètre étudié (l’impédance électrique) et la méthode pour y parvenir (la force de Lo-\nrentz). Dans cette méthode, représentée sur la figure 3.7 et qui sera décrite plus am-\n68 Pol Grasland-Mongrain3.2. Les méthodes d’imagerie d’impédance électrique\nplement dans la suite de ce chapitre, un transducteur émet des ultrasons dans un tissu\nbiologique soumis à un champ magnétique. Le tissu, conducteur, se déplace donc dans\nun champ magnétique, ce qui induit un courant électrique par force de Lorentz. Ce\ncourant peut être mesuré par des électrodes placées autour de l’échantillon. Il a été\nmontré que ce courant est relié à l’impédance électrique, ou plus précisément aux va-\nriations d’impédance électrique, dans la zone focale des ultrasons. En focalisant dans\ndifférentes zones du tissu, une image des interfaces d’impédance électrique peut donc\nêtre reconstruite, avec une résolution spatiale proche de celle des ultrasons [100].\nB B\nMesure du\ncourantCourant I(a) (b)\nFIGURE 3.7: Principe de la tomographie d’impédance électrique par force de Lorentz.\nDes ultrasons sont focalisés dans un tissu (a), et celui-ci est mis en mouvement dans la\nzone focale (ellipse marron). Le déplacement du tissu dans un champ magnétique crée\nun courant électrique mesurable par des électrodes (b).\n3.2.5 La tomographie magnéto-acoustique\nOn peut utiliser le principe de la tomographie d’impédance électrique par force\nde Lorentz en sens “inverse” [119], [122], [78]. Cette technique combine un courant\nélectrique et un champ magnétique pour créer des ondes ultrasonores.\nApplications de la force de Lorentz en acoustique médicale 69Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nDans cette technique, schématisée sur les figures 3.8-(a) et -(b), on émet dans un tissu\nbiologique un courant électrique à l’aide de deux électrodes. Le tissu étant placé dans\nun champ magnétique, des mouvements seront induits par force de Lorentz. Le mouve-\nment sera d’amplitude différente de part et d’autre de chaque interface de conductivité\nélectrique, ce qui va donner lieu à la formation d’une onde de compression – aux fré-\nquences utilisées, supérieures à la centaine de kilohertz, on peut négliger l’apparition\ndes ondes de cisaillement car celles-ci sont très rapidement atténuées. Les ondes de\ncompression peuvent être détectées par des transducteurs ultrasonores placés tout au-\ntour de l’échantillon, ce qui permet grâce à leurs temps de propagation de retrouver les\nsources des ondes, donc de réaliser des images des interfaces de conductivité électrique\n[79], [80].\nUne amélioration du dispositif a été proposée où le courant électrique est injecté\ndans l’échantillon non pas à l’aide d’une paire d’électrodes, mais par induction avec\nun champ magnétique variable créé une bobine : cela a donné lieu à la tomographie\nmagnéto-acoustique avec induction magnétique (MAT-MI pour “Magneto-Acoustic To-\nmography with Magnetic Induction”) [127]. Cette méthode présente l’avantage d’être\nsans contact et donc d’éviter les problèmes de contact avec les électrodes. Les derniers\nrésultats sont plutôt encourageants, avec des essais sur des tissus biologiques ex-vivo\n[56], [57], dont un résultat est montré sur les figures 3.8-(A) et -(B).\n3.3 Principe de la tomographie d’impédance électrique par\nforce de Lorentz\nOn supposera dans cette partie que le champ magnétique est selon l’axe X, la direc-\ntion de propagation des ultrasons selon l’axe Z et le courant électrique principalement\nselon l’axe Y. Les calculs suivants se basent sur ceux effectués par A. Montalibet [78].\n3.3.1 Mouvement des ions sous l’action des ultrasons dans un champ\nmagnétique\nLorsqu’un champ ultrasonore est appliqué dans un tissu biologique, modélisé comme\nun électrolyte, le tissu est mis en mouvement. Les ions présents dans la solution sont\nsupposés solvatés ; on peut alors décrire leur mouvement comme s’il était créé par une\nforce de friction avec le milieu déplacé par les ultrasons. Ce mouvement suit générale-\nment le mouvement global du milieu. Cependant, en présence d’un champ magnétique\nB, les ions sont en plus déviés par force de Lorentz à cause de leur charge. Cela permet\n70 Pol Grasland-Mongrain3.3. Principe de la tomographie d’impédance électrique par force de Lorentz\nB(a) (b)\nB\nCourant I\nFIGURE 3.8: Principe de la tomographie magnéto-acoustique. Un courant électrique est\nappliqué dans un tissu conducteur soumis à un champ magnétique (a), ce qui induit\naux interfaces de conductivité électrique (ellipse beige) une onde ultrasonore détec-\ntable par des transducteurs placés autour du tissu (b). Dans le cas de la tomographie\nmagnéto-acoustique avec induction magnétique, le courant électrique est induit grâce\nà un champ magnétique transitoire. (A) Photographie d’un échantillon comprenant un\nmorceau de muscle de chèvre et un morceau de graisse de porc dont une image de to-\nmographie magnéto-acoustique avec induction magnétique. (B) Image de conductivité\nélectrique prise avec la technique MAT-MI de l’échantillon présenté en (A) [56].\nd’écrire l’équation du mouvement pour un ion :\nmsdu\ndt=\u0000k(u\u0000v) +qu^B (3.4)\navecmsmasse de l’ion solvaté, uvitesse de l’ion, v=vezvitesse du fluide mise en\nmouvement par les ultrasons, kcoefficient de frottement, Bchamp magnétique et q\ncharge de l’ion.\nLe coefficient kpeut simplement être évalué avec une expérience où un ion est dé-\nplacé par un champ électrique Edans un fluide où il a atteint sa vitesse limite vlim,\nApplications de la force de Lorentz en acoustique médicale 71Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\ncomme décrit par l’équation 3.5 :\nkvlim =qE\n)k\nq=E\nvlim=1\n\u0016(3.5)\n\u0016étant défini comme la mobilité de l’ion. Ce coefficient vaut typiquement 5.108T\u00001\npour un ion solvaté.\nEn remplaçant kparq=\u0016dans l’équation (3.4), on obtient donc simplement\ndu\ndt+q\n\u0016ms(u\u0000vez) =q\nmsu^B (3.6)\nIl n’y a aucun mouvement dans la direction X, donc on peut étudier le mouvement\ndans le plan YZ seulement. L’équation 3.6 se décompose selon les directions Y et Z en\ndeux équations différentielles couplées.\nOn suppose que vzest de la forme Vsin(!t)à une fréquence f= 500 kHz, soit\n!\u00193:106rad.s\u00001, et on prend comme ordre de grandeur q= 1;6:10\u000019C,ms=200\n6;02:1023\nkg,B= 1T et\u0016= 5:108T\u00001.\nLa résolution de ces équations donnent pour les expressions de uzetuy:\n(\nuz=V!\n\u000b(cos!t\u0000e\u0000\u000btcos\ft) +V!2\f\n\u000b2(\u0000e\u0000\u000btsin\ft\u0000cos!t) +Vsin(!t)\nuy=V!\n\u000b(\u0000e\u0000\u000btsin\ft ) +V!2\f\n\u000b2(e\u0000\u000btcos\ft\u0000cos!t)\u0000V!2\f\n\u000b2cos(!t)\n(3.7)\navec\u000b=q\n\u0016ms\u00199:109s\u00001et\f=qB\nms\u0019500s\u00001\nEn gardant les termes dominants à t\u00191=!\u00192µs, on obtient alors :\n(\nuz=v0sin!t\nuy=v0\u0016Bsin!t(3.8)\nOn retrouve ainsi l’équation donnée par Montalibet [78] : uy=\u0016Buz. Tout se passe\ncomme si le mouvement selon Z était uniquement dû aux ultrasons et le mouvement\nselon Y à la déviation par le champ magnétique. Dans toute la suite, on supposera donc\nque la vitesse uzdes ions sera égale à vzet que la vitesse uysera égale à \u0016Buz. Cela\ndonne, pour un déplacement de l’ordre de un micromètre dans la direction Z, une dé-\nviation d’environ dix femtomètres dans la direction Y, soit quelques noyaux atomiques !\n72 Pol Grasland-Mongrain3.3. Principe de la tomographie d’impédance électrique par force de Lorentz\n3.3.2 Densité de courant créée\nLe déplacement différent des charges positives et négatives entraine l’apparition\nd’une densité de courant électrique j, définie parP\nk\u001akuk, avec\u001akla densité de charge\nketukla vitesse moyenne des charges k.\nEn décomposant selon les axes Y et Z, on peut donc écrire\nj=X\nk\u001akuk;zez+X\nk\u001akuk;yey (3.9)\nAprès un temps de transition négligeable, la vitesse des ions uk;zest égale à la vi-\ntesse du milieu vz, pour les ions positifs comme pour les ions négatifs. On obtient alorsP\u001akuk;zez=vzezP\u001ak=0, car le milieu est globablement neutre. La composante\nselon Z de la densité de courant est donc nulle.\nRemplaçons uk;ypar\u0016Buk;z=\u0016Bvz: cela donneP\u001akuk;yey=vzBeyP\u001ak\u0016k.\nCette grandeur n’est pas nulle car la mobilité dépend de la charge. Cette grandeur est\ndéfinie comme égale à \u001b, la conductivité électrique du milieu.\nLa densité de courant locale peut donc être définie par :\nj=\u001bvzBey (3.10)\n3.3.3 Détection du courant\nEn mesurant la densité de courant jsur une surface S(par exemple, la surface de\nl’électrode de mesure, avec une normale selon ez), on obtient un courant électrique\nglobalI:\nI=Z Z\nj:dS=Z Z\n\u001bvzBdxdz (3.11)\nLa vitesse particulaire et la pression d’un fluide sont reliées par la relation \u001adv\ndt=\n\u0000rp, que l’on peut réécrire sous la forme vz=\u00001\n\u001aRdp\ndzdt. On peut donc écrire :\nI=\u0000Z Z\n(\u001b\n\u001a(Zdp\ndzdt)Bdxdy ) (3.12)\nUne intégration par parties de l’équation (3.12) donne alors :\nI= [\u0000\u001b\n\u001aBZ\npdt]z2z1+Zd(\u001bB\n\u001a)\ndz(Z\npdt)dz) (3.13)\nApplications de la force de Lorentz en acoustique médicale 73Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nLe premier terme est nul si l’onde ultrasonore n’atteint pas les électrodes [78].\nSupposons que l’onde soit progressive et non atténuée dans la zone de mesure : on\na donc un lien entre le temps et la distance etR\np(t;z)dt=f(t\u0000z=c). En supposant\nque le champ magnétique Bet la densité du milieu \u001asont homogènes dans le milieu\n(ce dernier terme présentant des variations inférieures à 10 % dans les tissus mous), on\nobtient :\nI(t) =B\u001a\ncZ1\n0H(\u001c)f(t\u0000\u001c)d\u001c=B\u001a\nc(H?P )(t) (3.14)\navec\u001c=z=c,H(\u001c) =d\u001b\ndzet?le symbole du produit de convolution. Cette équation\nest donc le produit de convolution entre la variation de conductivité électrique Het\nla forme du signal ultrasonore P. En d’autre mots, le courant électrique détecté par\nles électrodes représente temporellement le signal ultrasonore à chaque interface de\nconductivité électrique, ce qui est représenté sur la figure 3.9\n3.3.4 Existence d’un speckle acousto-électrique\nComparons le signal donné par l’équation 3.14 à un signal d’échographie. Suppo-\nsons que l’onde acoustique, se propageant toujours selon l’axe Z, ne soit que très peu\natténuée par les hétérogénéités acoustiques : celles-ci ne réfléchissent donc qu’une faible\npartie des ultrasons, et on reste dans un régime de diffusion simple (non multiple).\nLa pression mesurée au temps t sera égale à 3.15 :\np(t) =X\nkR(zk)f(t\u00002zk=c) (3.15)\naveczkla position de chaque hétérogénéité acoustique k,R(zk)le coefficient de ré-\nflexion de chaque hétérogénéité (petit devant 1), fla forme du signal ultrasonore.\nDans un modèle continu, cela donne avec \u001c=z=c:\np(t) =Z1\n0R(\u001c)f(t\u00002\u001c)d\u001c (3.16)\nCe qui correspond à un facteur 2 près au produit de convolution entre la distribu-\ntion de coefficients de réflexion Ret la forme du signal ultrasonore f. Cela signifie que\nsi les hétérogénéités sont suffisamment proches les unes des autres et que le signal ul-\ntrasonore est suffisamment long, on peut observer des interférences entre les ondes ré-\nfléchies. Cela donne des zones brillantes ou sombres, selon que les interférences soient\n74 Pol Grasland-Mongrain3.3. Principe de la tomographie d’impédance électrique par force de Lorentz\nz\nt=z/cConductivité\nélectrique\nPression\nultrasonore p\n0\nSignal électrique\ninduit it=z/c\n0σ \nFIGURE 3.9: Supposons que la conductivité électrique présente trois variations dans la\ndirection Z, direction selon laquelle se propage une impulsion ultrasonore composée\nd’une seul sinusoïde. Le courant électrique mesuré par les électrodes, proportionnel\nau produit de convolution entre le gradient de conductivité électrique avec la pres-\nsion ultrasonore, aura temporellement la forme d’une sinusoïde à chaque variation de\nconductivité électrique.\nrespectivement constructives ou destructives : on nomme ce phénomène les tavelures\nultrasonores , ou plus fréquemment speckle ultrasonore [18] par analogie avec le speckle\noptique .\nLes équations (3.16) et (3.14) sont finalement tout à fait similaires, en remplaçant\nle coefficient de réflexion R(relié à la variation d’impédance acoustique) par la varia-\ntion de conductivité électriqued\u001b\ndz. On peut donc définir un speckle électro-acoustique :\nil est de nature électrique, car il s’agit d’inhomogénéités de conductivité électrique, et\nacoustique, car la taille des inhomogénéités est reliée à la longueur d’onde acoustique.\nApplications de la force de Lorentz en acoustique médicale 75Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\n3.4 Dispositif expérimental\nL’expérience effectuée est schématisée sur la figure 3.12. Un courant électrique de\n1 V à 0,5 MHz de trois sinusoïdes par salve à une fréquence de répétition de 100 Hz\nétait émis par un générateur (HP33120A, Agilent, Santa Clara, CA, Etats-Unis) relié à\nun amplificateur de puissance de 200 W (LA200H, Kalmus Engineering, Rock Hill, SC,\nEtats-Unis). Ce courant était transmis à un transducteur à 0,5 MHz, de diamètre 50 mm\net focalisé à 210 mm placé dans une cuve d’eau dégazée de 100x50x50 cm3. Les champs\nde pression simulés montrent que le faisceau ultrasonore a une largeur à mi-hauteur\nd’environ 2 cm, avec une pression efficace au point focal de 3 MPa, comme montré par\nles figures 3.10 et 3.11.\nFIGURE 3.10: Simulation du champ de pression du transducteur dans le plan YZ, pour\nX = 0 mm, Y = -40 à 40 mm, Z = 0 à 400 mm, avec pas de 0,2 mm. Le faisceau est à peu\nprès cylindrique de Z = 100 à 400 mm, avec un diamètre un peu inférieur à 20 mm. La\npression diminue de 3,2 MPa à Z = 150 mm à 1 MPa à Z = 400 mm.\nUne cuve d’huile de 20x4x15 cm3était placé de 15 à 35 cm du transducteur le long\nde l’axe ultrasonore. Cette cuve d’huile était placée au milieu d’un aimant permanent\nen forme de U décrit dans le chapitre 2, avec photographie et simulation de champ ma-\ngnétique sur la figure 2.9. Cette cuve servait d’isolant pour éviter tout courant parasite\nentre les électrodes et le transducteur.\nLes différents échantillons étaient placés dans cette cuve de 22 à 28 cm du trans-\nducteur. Trois types d’échantillons ont été testés : des petites cuves d’eau salée à 30%\nen sel (soit 300 g/L) avec le long de l’axe acoustique des membranes perméables aux\nultrasons ; des fantômes de gélatine à 10% en sel et 5% en gélatine ; ou des morceaux de\nmuscle animal, souvent avec une interface de graisse visible (bœuf, porc, poulet).\nUne paire d’électrodes en cuivre de taille 10x3x0,1 cm3était placée au contact de\nl’échantillon, respectivement dessus et dessous. Les électrodes étaient reliées via un\n76 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\nPosition Y (mm)Position X (mm)\n−40 −20 0 20 40−40\n−30\n−20\n−10\n0\n10\n20\n30\n40\nMPa\n0.511.52\nFIGURE 3.11: Simulation du champ de pression du transducteur dans le plan XY, pour\nX = -40 à 40 mm, Y = -40 à 40 mm, Z = 210 mm, avec pas de 0,2 mm. On observe\nune décroissance du faisceau autour de l’axe, avec quelques oscillations. La largeur à\nmi-hauteur du faisceau vaut 18 mm.\namplificateur de courant de 1 MV/A (HCA-2M-1M, Laser Components, Olching, Ger-\nmany) et un filtre de 0,1 à 2 MHz (NF Corporation FV-628B, Yokohama, Japon) à un os-\ncilloscope avec une entrée à 50 \n(WaveSurfer 422, LeCroy, Chestnut Ridge, NY, Etats-\nUnis) qui effectuait des moyennes sur 2000 acquisitions. Les signaux étaient ensuite\nenregistrés et analysés par le logiciel Matlab (The MathWorks, Natick, MA, Etats-Unis).\nCe logiciel filtrait les signaux entre 0,1 et 1 MHz et calculait la valeur absolue de la trans-\nformée de Hilbert du signal pour trouver l’enveloppe de ce signal. Cette enveloppe était\nensuite convertie en niveaux de couleur.\nComme vu dans la partie théorique, le signal donne le gradient de conductivité élec-\ntrique le long d’une ligne seulement. Afin de réaliser une image complète des gradients\nde conductivité électrique, le transducteur pouvait être déplacé pas à pas verticalement\npour former une image ligne par ligne.\n3.5 Expériences mises en œuvre\n3.5.1 Objectivation du signal\nCette partie avait pour but de créer un courant par force de Lorentz, de détecter\ncelui-ci et de s’assurer que ce courant est bien dû à l’interaction entre les ultrasons et\nApplications de la force de Lorentz en acoustique médicale 77Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nFIGURE 3.12: Dans le dispositif de base, un transducteur à 500 kHz émet des ultrasons\nsur un échantillon placé entre deux électrodes dans une cuve d’huile au milieu d’un\nchamp magnétique d’environ 300 mT créé par un aimant en U. Le courant électrique\ninduit est reçu par les deux électrodes puis mesuré. Le tout est placé dans une cuve\nd’eau dégazée.\nle champ magnétique. Dans ces expériences, l’échantillon était une petite cuve d’eau\nsalée isolée du milieu externe par une membrane perméable aux ultrasons. La salve\nultrasonore était composée de dix sinusoïdes, afin que le signal soit bien visible.\n3.5.1.1 Le signal détecté dépend-il de la présence d’ultrasons ?\nNous avons d’abord voulu vérifier que la présence d’ultrasons dans le fantôme est\nbien nécessaire à l’obtention du signal.\nMatériel et méthodes Pour cela, une expérience a été faite en plaçant un absorbant\nultrasonore épais de 2 cm entre le transducteur ultrasonore et l’échantillon.\nRésultats et discussions On observe sur la figure 3.13 que le premier ensemble de\npics reste quasiment inchangé entre les deux expériences, alors que le second disparait.\nLes ultrasons sont donc nécessaires au deuxième ensemble de pics mais pas au premier.\n3.5.1.2 Le signal détecté dépend-il de la présence d’un champ magnétique ?\nD’après le modèle théorique, le champ magnétique est nécessaire à l’obtention d’un\ncourant électrique par force de Lorentz.\n78 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\n0 5 10 15\nx 10−5−5−4−3−2−1012345x 10−4\nsans absorbant ultrasonore\navec absorbant ultrasonore\nAmplitude (V)\nTemps (s)\nFIGURE 3.13: Le signal issu des électrodes est mesuré sans absorbant entre le trans-\nducteur et le fantôme (signal bleu) puis avec un absorbant entre le transducteur et le\nfantôme (signal rouge). On observe que le premier ensemble de pics est quasiment in-\nchangé tandis que le deuxième disparait après ajout de l’absorbant.\nMatériel et méthodes Cette hypothèse a été testée en retirant l’aimant sans autre mo-\ndification dans le dispositif.\nRésultats et discussion Les résultats, visibles sur la figure 3.14, montrent que le pre-\nmier ensemble de pics reste quasiment inchangé entre les deux expériences, mais que le\ndeuxième disparait. Le champ magnétique est donc nécessaire au deuxième ensemble\nde pics mais pas au premier.\nApplications de la force de Lorentz en acoustique médicale 79Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\n0 5 10 15\nx 10−5−6−4−20246x 10−4\nAvec champ B\nSans champ B\nAmplitude (V)\nTemps (s)\nFIGURE 3.14: Le signal issu des électrodes est mesuré avec un aimant autour du fan-\ntôme (signal bleu) puis sans aimant (signal rouge). Le premier ensemble de pics est\nquasiment inchangé tandis que le deuxième disparait après retrait de l’aimant.\n3.5.1.3 Le signal détecté est-il décalé dans le temps si on déplace l’échantillon par\nrapport au transducteur ?\nNous avons regardé si la distance entre le transducteur et l’échantillon a une in-\nfluence sur le temps de réception du signal.\nMatériel et méthodes Une expérience a été réalisée avec l’échantillon placé à 16 puis\nà 10 cm du transducteur.\nRésultats et discussions On observe sur la figure 3.15 que le deuxième ensemble de\npics commence vers 70 µs après l’émission avec le fantôme à 10 cm du transducteur, et\n80 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\n0 5 10 15\nx 10−5−5−4−3−2−1012345x 10−4\nAmplitude (V)\nTemps (s)Echantillon à 16 cm du transducteur\nEchantillon à 10 cm du transducteur\nFIGURE 3.15: Le signal issu des électrodes est mesuré avec le fantôme à 16 cm du trans-\nducteur (signal bleu) puis à 10 cm du transducteur (signal rouge). On observe que le\nsignal arrive plus tôt, avec un temps compatible avec la durée de propagation des ul-\ntrasons.\nvers 110 µs avec le fantôme à 16 cm du transducteur. Avec une vitesse de propagation\nde 1500 m.s\u00001, cela correspond à des distances entre l’émission et la réception de 10,5\net 15,5 cm respectivement ce qui correspond approximativement à la distance entre le\ntransducteur et le fantôme. Cependant, le premier ensemble de pics n’a pas été modifié\net n’est donc pas été influencé par la position du fantôme.\n3.5.1.4 Le signal détecté dépend-il de la conductivité électrique de l’échantillon ?\nLe signal électrique est supposé dépendre au gradient de conductivité électrique.\nLa relation étant plutôt complexe (produit de convolution), nous avons simplement\nvérifié qu’un gradient de conductivité électrique plus faible donnait effectivement un\nApplications de la force de Lorentz en acoustique médicale 81Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nsignal plus faible.\nMatériel et méthodes Une expérience a donc été mise en oeuvre avec un échantillon\nà 30 g.L\u00001en sel puis un deuxième de taille identique au précédent sans ajout de sel,\ndonc avec une conductivité électrique égale à celle de l’eau du robinet.\n0 5 10 15\nx 10−5−2−1.5−1−0.500.511.52x 10−3\nAmplitude (V)\nTemps (s)Echantillon saturé en sel\nEchantillon non salé\nFIGURE 3.16: Le signal issu des électrodes est mesuré avec échantillon salé (signal bleu)\npuis un échantillon sans sel (signal rouge). Le premier ensemble de pics est quasiment\ninchangé, et le deuxième diminue notablement lorsque le gradient de conductivité élec-\ntrique est plus faible.\nRésultats et discussions On constate sur la figure 3.16 que l’amplitude de l’ensemble\nde pics apparaissant vers 100 µs diminue drastiquement entre l’échantillon salé, qui\ndonne un gradient de conductivité électrique important, qu’avec l’échantillon non salé,\navec un gradient de conductivité électrique plus faible. Cependant, le premier ensemble\nde pics n’est pas modifié notablement.\n82 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\n3.5.1.5 Le signal détecté dépend-il de l’émission/réception d’ultrasons par le\ntransducteur ?\nLes expériences précédentes montrent un signal au moment de l’émission qui res-\nsemble fortement au signal électrique émis par le générateur. Ce signal pourrait être\nexpliqué par la conversion du courant électrique en ultrasons qui émet des parasites\nélectromagnétiques détectés par les électrodes.\nMatériel et méthodes Pour vérifier cela, un réflecteur de PVC de 3 cm de large a\nété placé entre le fantôme et le transducteur, à 2 cm de ce dernier. Il était orthogonal\nà l’axe ultrasonore de façon à ce que les échos ultrasonores reviennent au niveau du\ntransducteur.\n0 5 10 15\nx 10−5−2−1.5−1−0.500.511.52x 10−3\nAmplitude (V)\nTemps (s)\nFIGURE 3.17: Le signal issu des électrodes est mesuré avec un réflecteur de PVC face au\ntransducteur, à 2 cm de celui-ci. On observe l’apparition d’un nouvel ensemble de pics\nà environ 25 µs de l’émission.\nApplications de la force de Lorentz en acoustique médicale 83Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nRésultats et discussions On remarque sur la figure 3.17 qu’un signal est apparu à\npartir de 25 µs environ. Cela correspond à une distance d’environ 4 cm, soit 2 cm aller-\nretour. On constate également dans les expériences ultérieures la présence systématique\nde ce signal à une distance deux fois supérieure aux surfaces réfléchissantes (toujours\navec une vitesse de propagation de 1500 m.s\u00001. On peut donc en conclure que le méca-\nnisme de conversion électrique/mécanique produit un signal électrique au niveau des\nélectrodes, que ce soit à l’émission ou la réception.\n3.5.1.6 Conclusion\nD’après les éléments ci-dessus, le signal détecté est composé de deux parties :\n• des parasites électromagnétiques émis lors de l’émission ou de la réception d’ul-\ntrasons par le transducteur. Ces parasites se propagent du transducteur aux élec-\ntrodes qui font office d’antenne, et ce à une vitesse très élevée (proche de la vitesse\nde la lumière) face à la vitesse moyenne des ultrasons (1500 m/s). Ces parasites\npeuvent être réduits grâce à un blindage et une mise à la masse des différents\néléments du dispositif.\n• un signal qui dépend des ultrasons, du champ magnétique, de la conductivité\nélectrique et qui apparait aux interfaces de conductivité électrique. Il s’agit du\nsignal acousto-magnétique que l’on souhaite détecter.\nAfin de s’affranchir des parasites électromagnétiques qui ont une amplitude sou-\nvent proche voire supérieure au signal recherché, on peut blinder les éléments du dis-\npositif, mais cela s’est avéré insuffisant dans notre dispositif. Il est également possible\nenregistrer le signal acoustique réfléchi, comme en échographie mode A, et soustraire\ncelui-ci au signal acousto-magnétique. Cette idée s’avère néanmoins très sensible au\nbruit et donc n’a pas pu être utilisée dans nos expériences. Enfin, on peut laisser une\ndistance suffisante entre le transducteur et le fantôme pour que les signaux acousto-\nmagnétiques recherchés apparaissent avant que les échos ultrasonores n’arrivent au\nniveau du transducteur. Cette méthode, facile à mettre en oeuvre, a été retenue pour la\nsuite.\nEnfin, nous avons remarqué que les signaux sont tout à fait reproductibles au cours\ndu temps, aucune variation n’étant détectée d’une expérience à l’autre, toute choses\nétant égales par ailleurs.\n84 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\n3.5.2 Image d’un échantillon de gélatine salée\nMatériel et méthodes La technique d’imagerie a été appliquée sur un fantôme que\nl’on peut voir sur la figure 3.18, composé de 10% en gélatine et 10 g.L\u00001de sel, donnant\nune conductivité électrique égale à 2 S/m. Ce fantôme avait une forme particulière, avec\ndeux interfaces d’entrée et deux interfaces de sortie. Des acquisitions ont été faites ligne\npar ligne, avec 140 lignes séparées chacune de 0,25 mm. Avec 2000 acquisitions par ligne\nà 2 minutes par ligne, l’acquisition a pris approximativement 5 heures. Simultanément,\nune image ultrasonore a été acquise et reconstruite ligne par ligne à partir des échos\nultrasonores.\nFIGURE 3.18: Photographie du fantôme de gélatine utilisé dans l’expérience. On ob-\nserve deux interfaces d’entrée et deux interfaces de sortie, comme indiqué par les quatre\nflèches violettes.\nRésultats et discussions L’image ultrasonore du fantôme est représentée sur la figure\n3.19-(a) et l’image d’impédance électrique par force de Lorentz (que l’on nommera sim-\nplement “image d’impédance électrique” par la suite) est visible sur la figure 3.19-(b).\nOn peut reconnaitre les interfaces d’entrée et de sortie (flèches violettes) à la fois sur\nl’image ultrasonore et l’image d’impédance électrique. Le faisceau ultrasonore ayant\nApplications de la force de Lorentz en acoustique médicale 85Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nune largeur supérieure à un centimètre, on voit des interfaces différentes sur une même\nligne.\nCette expérience montre que le dispositif est adapté pour observer des interfaces de\nconductivité électriques.\n20 24 282\n1\n0\nz (cm)y (cm)\n(a)\n20 24 282\n1\n0\nz (cm)y (cm)\n(b)\nFIGURE 3.19: (a) Image ultrasonore du fantôme de gélatine. (b) Image d’impédance\nélectrique correspondante du fantôme de gélatine. Le faisceau ultrasonore ayant une\nlargeur de presque deux centimètres, on voit plusieurs interfaces sur une même ligne.\nOn distingue néanmoins sur l’image d’impédance électrique comme sur l’image ultra-\nsonore les deux interfaces d’entrée et les deux de sortie comme indiqué par les flèches\nviolettes.\n3.5.3 Image de tissus organiques\nMatériel et méthodes La technique d’imagerie a été appliquée sur un morceau de côte\nde bœuf issu d’une boucherie, que l’on peut voir sur la figure 3.20. Cet échantillon pré-\n86 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\nsente deux interfaces d’entrée et une couche de graisse marquée ; le muscle lui-même\nest peu homogène, avec de nombreuses inclusions de graisse. Une image a été faite\nligne par ligne, avec 96 lignes séparées chacune de 0,5 mm. Avec 2000 acquisitions par\nligne à 2 minutes par ligne, l’acquisition a pris approximativement 3,5 heures.\nFIGURE 3.20: Photographie de la côte de boeuf utilisée dans l’expérience. Cet échan-\ntillon présente deux interfaces d’entrée, une couche de graisse marquée et des couches\nde muscle peu homogène.\nRésultats et discussions L’image ultrasonore du fantôme est représentée sur la figure\n3.21-(a) tandis que l’image d’impédance électrique est visible sur la figure 3.21-(b).\nOn peut reconnaitre les interfaces d’entrée (flèches violettes) à la fois sur l’image\nultrasonore et l’image d’impédance électrique, bien que l’interface du bas soit diffici-\nlement visible sur l’image d’impédance électrique. L’interface de graisse semble être\nvisible sur l’image d’impédance électrique sans l’être sur l’image ultrasonore – ce qui\nsemblerait normal vu la grande différence et de conductivité électrique et la faible dif-\nférence d’impédance acoustique entre la graisse et le muscle. La résolution spatiale est\ncomparable entre l’image ultrasonore et l’image d’impédance électrique, maise reste\ncependant médiocre avec notamment un ensemble de taches qui seront discutées dans\nle paragraphe suivant. La résolution spatiale pourrait être améliorée dans l’axe ultra-\nApplications de la force de Lorentz en acoustique médicale 87Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\n3\n2\n145\nz (cm)y (cm)\n20 22 24 26(a)\n3\n2\n145\n20 22 24\nz (cm)26y (cm)\n(b)\nFIGURE 3.21: (a) Image ultrasonore de la côte de boeuf ; (b) Image d’impédance élec-\ntrique de la côte de boeuf.\nsonore en augmentant la fréquence ultrasonore pour diminuer la longueur d’onde et\ndans l’axe transverse en utilisant des faisceaux ultrasonores plus fins.\n3.5.4 Observation du speckle électro-acoustique\nD’après la partie théorique, les images d’impédance électrique de tissus présen-\ntant des hétérogénéités devraient présenter une texture granuleuse similaire de spe-\nckle acousto-électrique similaire au speckle ultrasonore. Cependant, le speckle électro-\nacoustique n’est pas forcément identique au speckle ultrasonore, car il peut y avoir des\nhétérogénéités électriques et non acoustiques et vice-versa : par exemple une petite in-\nclusion de graisse aura une conductivité électrique différente du muscle alentours, mais\nquasiment la même impédance acoustique.\nMatériel et méthodes Afin de caractériser ce speckle , une expérience a été faite en mo-\ndifiant l’angle d’imagerie. Deux images faites à partir d’un angle différent présenteront\nen effet un speckle différent si l’angle est suffisamment important : il s’agit typiquement\nd’un angle donnant dans l’image un déplacement d’une longueur d’onde [121]. L’ima-\ngerie composée (ou compound imaging ) en ultrasons est d’ailleurs basée sur ce principe :\non réalise des images à partir de plusieurs angles proches, la moyenne des images di-\nminue le speckle des images tout en conservant les interfaces d’impédance acoustique\n[59]. Pour cela, des images d’un morceau de côte de porc visible sur la figure 3.22 ont\nété faites avec cinq angles différents. Les images ultrasonores et d’impédance électrique\n88 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\nont été acquises simultanément. Les angles choisis étaient de -2oà 2oavec un pas de 1\no, l’axe Z étant pris comme angle égal à 0oet le centre de rotation étant au milieu du\ntransducteur. La distance entre le transducteur et le fantôme étant de 22 cm, cela donne\nun déplacement approximatif vertical de -8 à +8 mm, la longueur d’onde étant de 3 mm\nà 500 kHz.\nFIGURE 3.22: Photographie de la côte de boeuf utilisée dans l’expérience. On observe\nune couche de graisse, puis un morceau de muscle avec une forme en “L”\nRésultats et discussions La figure 3.23 représente les images ultrasonores et d’impé-\ndance électrique à 5 angles différents : -2o, -1o, 0o, +1oet +2o.\nOn peut observer sur les images que les interfaces principales sont nettement vi-\nsibles. On voit néanmoins des changements de texture entre ces interfaces, d’autant\nplus importants que l’angle est différent, bien que cela reste difficile à quantifier. Ces\nchangements ne sont pas dus à des variations temporelles aléatoires des signaux, étant\ndonné la reproductibilité des expériences au cours du temps.\nOn peut donc concevoir une image composée en réalisant des moyennes sur les ac-\nquisitions à cinq angles différents, comme représenté sur l’image 3.24. Cela permet de\ndiminuer la présence du speckle sur les images, et donc améliorer légèrement la qualité\ndes images - même si cela n’est pas évident sur l’image présentement obtenue. Ce-\npendant, l’application la plus intéressante serait non pas de réduire ce speckle , mais de\nl’utiliser pour caractériser les tissus et leur état, comme le font les médecins en échogra-\nphie.\nApplications de la force de Lorentz en acoustique médicale 89Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\nFIGURE 3.23: (a) Images ultrasonores de la côte de porc, aux angles -2, -1, 0, 1 et 2o. (b)\nImage d’impédance électrique correspondantes de la côte de porc, aux angles -2, -1, 0,\n1 et 2o. Les interfaces principales sont nettement marquées sur l’ensemble des images.\nLes variations plus faibles, interprétées comme du speckle , varient entre les différents\nangles, surtout entre les angles “extrêmes”.\n90 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\nFIGURE 3.24: (a) Image ultrasonore composée à partir d’une moyenne des acquisitions\ncinq angles différents. (b) Image d’impédance électrique dcomposée à partir d’une\nmoyenne des acquisitions cinq angles différents.\n3.5.5 Suivi de lésion thermique\nDans la section 3.1.6 consacré à l’influence de la température sur l’impédance élec-\ntrique des tissus, nous avons vu qu’une lésion thermique modifiait la conductivité élec-\ntrique des tissus. Le but de cette expérience était la méthode était suffisamment efficace\npour observer un changement suite à une lésion thermique.\nMatériel et méthodes Pour cela, une image d’un échantillon de blanc de poulet de\n4x3x5 cm3était observé par le transducteur avec 60 lignes séparées de 2 mm. Afin d’ob-\ntenir un signal plus important, chaque salve était composée de cinq sinusoïdes au lieu\nde trois. Cet échantillon était placé de 22 à 27 cm du transducteur. Puis, le transducteur\nétait retiré à l’aide d’un banc motorisé pour laisser place à un transducteur de théra-\npie ultrasonore. Ce deuxième transducteur, placé à 2 cm de la face avant de l’échan-\ntillon, était focalisé à 50 mm et avait une fréquence centrale de 0,5 MHz. Un générateur\namplifié par un amplificateur 500 W (1040L, Electronics and Innovation Engineering,\nRochester, NY, Etats-Unis) émettait durant 60 secondes avec un cycle de 50 % un si-\ngnal électrique de puissance 16W comme mesuré par un wattmètre (392.4017.02 NAP ,\nRhode & Schwartz GmbH, Munich, Allemagne). Le transducteur a effectué 30 lésions\nsuccessives séparées de 1 mm chacune, formant une bande verticale de lésions sur 3\ncm de hauteur et une zone non insonifiée de 1 cm dessous. Puis, le transducteur était\nretiré, et après 10 minutes d’attente pour que le tissu refroidisse, le transducteur initial\nd’imagerie était replacé à sa position initiale pour acquérir une nouvelle image. Enfin,\nl’échantillon a été retiré et coupé en deux afin d’évaluer l’étendue de la lésion HIFU.\nRésultats et discussions La figure 3.25-(a) montre la face arrière de l’échantillon de\npoulet après lésion thermique. La lésion s’est propagée jusqu’aux bords de l’échan-\nApplications de la force de Lorentz en acoustique médicale 91Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\ntillon, sauf celui du bas. La figure 3.25-(b) montre une coupe centrale du blanc de poulet\naprès lésion thermique. On observe une bande de tissu lésé de 3 cm de haut, de 1 cm de\nla face avant et à la face arrière. Une bande 1 cm de haut de tissu non lésé est présente\nsous la lésion thermique.\nFIGURE 3.25: (a) Photographie de la face arrière de l’échantillon de blanc de poulet\naprès lésion thermique. (b) Photographie d’une coupe centrale l’échantillon de blanc\nde poulet après lésion thermique.\nLes images ultrasonore et d’impédance électrique avant et après lésion thermique\nsont représentées sur les figures 3.26-(a) et (b) et 3.27-(a) et (b) respectivement. On ob-\nserve sur toutes les images l’interface d’entrée et une texture granuleuse à l’intérieur\ndu tissu. Les taches blanches en haut à gauche et l’autre en bas à gauche de l’image\nultrasonore, indiquées par des flèches violettes, correspondent à des réflexions sur les\nentretoises du support ; ces entretoises, plongées dans le bain d’huile non conducteur,\nsont elles-mêmes non conductrices et donc ne sont pas visibles sur l’image d’impédance\nélectrique. Les interfaces ont une épaisseur assez importante et ont un aspect de bande-\nlettes verticales, surtout sur l’image d’impédance électrique, ce qui est dû à l’utilisation\nde cinq sinusoïdes lors des séquences d’imagerie.\nOn voit sur les deux images après lésion thermique les mêmes interfaces d’entrée\nque sur les images avant lésion thermique. Cependant, plus en profondeur et sur la\npartie haute de l’image, dans la zone encadrée par un pointillé violet, on remarque une\nmodification de texture. Cette zone se superpose à la zone lésée. En-dehors de cette\nzone, on observe quelques changements sur l’image ultrasonore, mais quasiment pas\nsur l’image d’impédance électrique. Ces observations sont plus claires si l’on soustrait\nles signaux acquis avant à ceux acquis après après lésion thermique, ce qui permet de\nreconstruire une image des différences comme sur la figure 3.28.\nCette expérience a montré que la tomographie d’impédance électrique par force de\nLorentz peut être potentiellement utilisée pour suivre des lésions thermiques. Les dif-\n92 Pol Grasland-Mongrain3.5. Expériences mises en œuvre\n(a)2\n1\n034y (cm)\nz (cm)22 24 26\n(b)2\n1\n034y (cm)\nz (cm)22 24 26\nFIGURE 3.26: (a) Image ultrasonore de l’échantillon avant lésion thermique. (b) Image\nd’impédance électrique de l’échantillon avant lésion thermique.\n(a)2\n1\n034y (cm)\nz (cm)22 24 26\n(b)2\n1\n034\nz (cm)y (cm)\n22 24 26\nFIGURE 3.27: (a) Image ultrasonore de l’échantillon après lésion thermique, avec la zone\ncorrespondant à la lésion thermique encadrée en pointillés violets. (b) Image d’impé-\ndance électrique de l’échantillon avant lésion thermique, avec la zone correspondant à\nla lésion thermique encadrée en pointillés violets.\nférences sur les images d’impédance électrique avant et après lésion sont certainement\ndues aux modifications de conductivité électrique dus à la lésion thermique. Cepen-\ndant, il n’est pas suffisamment clair sur les images s’il s’agit de modifications d’hété-\nrogénéités changeant le speckle acousto-électrique, de changements globaux de conduc-\ntivité électrique, voire éventuellement d’effets de la température si le tissu n’était pas\nparfaitement refroidi. Pour investiguer cela, il est nécessaire d’améliorer la qualité des\nApplications de la force de Lorentz en acoustique médicale 93Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\n2\n1\n034\nz (cm)y (cm)\n22 24 26\nFIGURE 3.28: Différence entre l’image d’impédance électrique avant etaprès lésion ther-\nmique. Les principales modifications sont observées dans la zone lésée, comme indi-\nquée par l’encadré violet.\nimages, et tout particulièrement la résolution spatiale.\n3.6 Bilan du chapitre\n3.6.1 Principaux résultats\nL’objectif de ce travail était d’étudier une méthode d’imagerie de la conductivité\nélectrique que nous avons nommée Tomographie d’Impédance Electrique par Force de\nLorentz. Dans cette méthode, on met en mouvement grâce à des ultrasons focalisés un\nmorceau de tissu biologique placé dans un champ magnétique. Cela induit un courant\nélectrique par force de Lorentz, mesurable par des électrodes. Ce courant est relié aux\nvariations d’impédance électrique le long de l’axe ultrasonore. Le principal intérêt de\ncette méthode réside dans sa résolution spatiale : celle-ci est en théorie proche de la\nlongueur d’onde ultrasonore.\nLa première partie de ce travail a consisté à objectiver le signal afin s’assurer que\nle courant détecté est effectivement celui induit par force de Lorentz. Il s’avère que\nl’émission et la réception d’ultrasons par le transducteur s’ajoute au signal induit, ce\nqui a posé plusieurs difficultés expérimentales.\nLa technique a ensuite été appliquée sur des fantôme de gélatine et de morceaux\nde tissus biologiques. Les images d’impédance électrique et les images ultrasonores\n94 Pol Grasland-Mongrain3.6. Bilan du chapitre\nsont cohérentes entre elles, lorsque les interfaces d’impédance électrique et acoustique\nétaient présentes aux mêmes emplacements. L’interface de graisse d’un tissu biologique\na notamment été observée sur une image d’impédance électrique sans être visible sur\nl’image ultrasonore associée, ce qui a été expliqué par la forte variation d’impédance\nélectrique et la faible variation d’impédance acoustique de cette couche par rapport au\nreste du muscle.\nA l’instar des échographies, les images issues de cette méthode sont censées pré-\nsenter un speckle électroacoustique. Celui-ci est en effet dû à des hétérogénéités de na-\nture électrique mais reliées à la longueur d’onde acoustique. La résolution spatiale des\nimages étant cependant médiocre et le signal détecté faible, ce speckle est difficilement\nobservable. Néanmoins, des variations faibles d’angle d’imagerie montrent des inter-\nfaces quasiment identiques, mais des motifs à l’intérieur du tissu légèrement différents,\nce qui est un indice de la présence du speckle recherché. Le principal intérêt ne serait\ncependant pas de réduire ce speckle avec des techniques d’imagerie composée, comme\nprésentée dans ce chapitre, mais de l’utiliser pour caractériser les tissus.\nEnfin, nous avons tenté de faire des images de lésion thermique avec cette méthode.\nEn effet, comme les tissus lésés thermiquement présentent une conductivité électrique\ndifférente des tissus normaux, la méthode peut potentiellement localiser ces zones. Mal-\ngré la faible résolution spatiale de la méthode, on observe quand même une modifica-\ntion dans l’image d’impédance électrique, sans que cette modification ne soit clairement\ndélimitée.\n3.6.2 Avenir de la méthode\nLa tomographie d’impédance électrique par force de Lorentz induite par ultrasons\nprésente de nombreuses similarités avec l’échographie : images des interfaces d’im-\npédance, présence de speckle , résolution millimétrique. Plusieurs des méthodes déri-\nvées de l’échographie standard sont donc théoriquement envisageables dans cette tech-\nnique, comme l’imagerie composée. Afin de caractériser le speckle électroacoustique\ncomme présenté dans cette thèse, il serait intéressant de faire des images à différentes\nfréquences ultrasonores afin de’éavluer l’évolution de la taille typique des grains – cela\nn’a pu être fait avec le matériel disponible au laboratoire.\nLa position des électrodes n’étant pas critique [78], on peut envisager une sonde\nspéciale comprenant un transducteur ultrasonore et deux électrodes placées de part et\nd’autre pour récupérer le courant.\nLa principale difficulté reste cependant la faiblesse du courant électrique créé. Il est\nApplications de la force de Lorentz en acoustique médicale 95Chapitre 3. Tomographie d’impédance électrique par force de Lorentz\npossible d’augmenter celui-ci d’un facteur 10 environ à l’aide d’un champ magnétique\nplus puissant. En utilisant un appareil IRM, même si l’on perdrait en même temps en\nlégèreté et en coût on gagnerait en puissance et en homogénéité du champ magné-\ntique. L’utilisation de larges électrodes permet de recueillir plus de courant, mais il est\nprobable que le gain potentiel soit inférieur à un ordre de grandeur par rapport aux\nconfigurations actuellement utilisées. Le capteur de courant peut par contre être lar-\ngement optimisé : comme on connait exactement le signal recherché, les méthodes de\nfiltrage peuvent extraire le signal utile sans être obligé d’effectuer des moyennes sur de\nnombreuses acquisitions.\nUn autre axe de développement serait de développer une méthode d’imagerie avec\ndes ondes ultrasonores planes acquises à différents angles, comme en échographie ul-\ntrarapide. On gagnerait en vitesse d’acquisition et peut-être en qualité de reconstruc-\ntion. Ceci se rapproche notamment de techniques d’imagerie acousto-électrique ultra-\nrapide actuellement en développement [96].\nPlusieurs applications sont envisageables. Comme évoqué précédemment, cette mé-\nthode présente un contraste différent de l’IRM, de l’échographie ou des radiographies :\non peut donc observer des choses invisibles avec d’autres méthodes, notamment grâce\nauspeckle des images. De plus, comme cela a été fait durant cette thèse, le matériel peut\nréaliser simultanément une échographie, donnant par là une image multimodale en\ntemps réel.\nUne application particulièrement intéressante serait d’utiliser l’influence de la tem-\npérature sur la conductivité électrique des tissus, afin de réaliser suivi en temps réel des\ntraitements par ultrasons focalisés.\n96 Pol Grasland-MongrainCHAPITRE 4\nImagerie d’ondes de cisaillement\ninduites par force de Lorentz\n4.1 L’élastographie par ondes de cisaillement\n4.1.1 L’élasticité des tissus biologiques\nComme montré sur le tableau 4.1, certains tissus pathologiques comme les tumeurs\ndu sein ou de la prostate présentent une élasticité différente des tissus sains, élasticité\nque l’on peut définir par le module d’Young Edu tissu.\nType de tissu biologique module d’Young (kPa)\nSein Normal adipeux 18-24\nNormal glandulaire 28-66\nTissu fibreux 96-244\nCarcinome 22-560\nProstate Normal antérieur 55-63\nNormal postérieur 62-71\nHypertrophie bénigne 36-41\nCarcinome 96-241\nFoie Normal 0,4-6\nCirrhose 15-100\nTABLE 4.1: Quelques valeurs du module d’Young de tissus biologiques sains et patho-\nlogiques.\nApplications de la force de Lorentz en acoustique médicale 97Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nLe module d’Young d’un tissu est défini à partir de la loi de Hooke. Cette loi, valable\npour des faibles déformations en traction ou compression, énonce que l’allongement\n\u0001lest proportionnel à la force appliquée F, avec un coefficient de proportionnalité k:\nF=k\u0001l.\nEn divisant cette équation par la surface d’application de la force et par la longueur\ntotale du matériau, on peut relier la contrainte \u001ben pascals et la déformation relative \u000f\nen pourcent, avec un coefficient de proportionnalité E(voir schéma 4.1) :\n\u001b=E\u000f (4.1)\nLe coefficient Eest le module d’Young, homogène à une pression.\nσ\nFIGURE 4.1: Le module d’Young est le coefficient de proportionnalité entre la contrainte\nappliquée\u001bet l’allongement relatif \u000fdu matériau.\nDe la même façon, pour de faibles déformations en cisaillement, la contrainte de\ncisaillement (ou cission) \u001cen pascals est reliée à l’angle de déformation relatif \u000ben\nradian (voir schéma 4.2) :\n\u001c=\u0016tan(\u000b) (4.2)\navec\u0016appelé module de cisaillement (aussi appelé deuxième coefficient de Lamé),\nhomogène à une pression.\nEnfin, la déformation relative totale \u000ftoten pourcent est proportionnelle à la contrainte\nisostatique \u001btoten pascals (voir schéma 4.3) :\n\u001btot=K\u000ftot (4.3)\n98 Pol Grasland-Mongrain4.1. L’élastographie par ondes de cisaillement\nατ\nFIGURE 4.2: Le module de cisaillement est le coefficient de proportionnalité entre la\ncontrainte appliquée \u001bet la tangente de l’angle de déformation \u000bdu matériau.\navecKle module d’élasticité isostatique en Pa.\nσ\nσ\nσ\nσσσ\nFIGURE 4.3: Le module d’Young est le coefficient de proportionnalité entre la contrainte\nappliquée isostatique \u001btotet l’allongement relatif total \u000ftotdu matériau.\nCes trois grandeurs sont liées entre elles selon l’équation :\nE=9K\u0016\n3K+\u0016(4.4)\nAlors que le module d’élasticité isostatique Kvarie peu selon les tissus, le module\nde cisaillement montre une grande variabilité au sein du corps humain, comme repré-\nsenté sur la figure 4.4\nApplications de la force de Lorentz en acoustique médicale 99Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\n10210310410510610710810910101011Module de compression\nisostatique (Pa)\nModule de cisaillement (Pa)Tissus\nmous OsLiquides\nOs Epiderme,\nCartilageDerme,\nMuscle\ncontractéFoie,\nGraisse,\nMuscle\ndécontracté\nFIGURE 4.4: Le module d’élasticité isostatique Kvaut environ 10 GPa pour la plupart\ndes tissus biologiques. Selon les tissus, le module de cisaillement varie de plusieurs\nordres de grandeurs, de 100 Pa à 10 GPa [106].\n4.1.2 L’élastographie statique\nLa palpation manuelle d’un organe, principalement sensible au module de cisaille-\nment du tissu, permet de détecter des nodules durs. Cette technique assez simple est\nmalheureusement non quantitative, dépend de l’opérateur et nécessite d’avoir des no-\ndules gros et durs pour être détectables [50]. Pour améliorer les diagnostics des méde-\ncins, des méthodes d’imagerie de l’élasticité des tissus ont été développées à partir des\nannées 1990.\nUne première méthode a été introduite par Ophir et al. en 1991 [88], que l’on nomme\naujourd’hui élastographie statique. Une première image ultrasonore est réalisée, puis\nune contrainte appliquée grâce à une tige extérieure, suivie d’une deuxième image ul-\ntrasonore. Cette technique obtient le déplacement à partir de corrélations locales entre\nles deux images, technique appelée interférométrie des tavelures ultrasonores, ou plus\nsouvent speckle-tracking . Le principal problème vient de la loi de Hooke, qui a la formu-\nlation d’une matrice 6x6 lorsqu’on étudie des volumes à trois dimensions :\n\u001bij=Cijkl\u000fkl (4.5)\navec la convention de sommation d’Einstein pour i,j,k,l, chacune prenant une va-\nleur parmi X, Y et Z. Ainsi, lors d’une déformation appliquée manuellement, de nom-\nbreuses valeurs de contraintes sont inconnues, donc il est difficile d’en déduire les Cijkl\net le module d’Young du matériau. Pour éviter ce problème, des méthodes dynamiques\nutilisant la propagation d’ondes mécaniques dans les tissus ont été proposées.\n100 Pol Grasland-Mongrain4.1. L’élastographie par ondes de cisaillement\n4.1.3 La vitesse des ondes mécaniques dans les tissus biologiques\nComme dans tous les solides, deux types d’ondes mécaniques existent dans les tis-\nsus biologiques : les ondes de compression et les ondes de cisaillement, dont la propa-\ngation dans un milieu infini est illustrée sur la figure 4.5.\n(a) Onde de compression\n(b) Onde de cisaillementλ\nλSens de propagation\nSens de propagation\nFIGURE 4.5: (a) Onde de compression plane se propageant dans un milieu infini. La\npropagation se fait dans la même direction que le déplacement local. (b) Onde de ci-\nsaillement plane se propageant dans un milieu infini. La propagation se fait dans un\nsens orthogonal au déplacement local.\nRappelons que l’onde de compression n’est longitudinale (propagation dans la di-\nrection du déplacement) et celle de cisaillement transversale (propagation dans une\ndirection orthogonale au déplacement) que pour des ondes planes. Pour une onde non\nplane, il est donc tout à fait possible d’avoir des ondes de cisaillement longitudinales.\nLa démonstration de l’existence et la description de ces deux ondes peut se faire à\nApplications de la force de Lorentz en acoustique médicale 101Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\npartir de l’équation de Navier, aussi appelée équation de l’élastodynamique :\n\u001ad2u\ndt2= (K+4\n3\u0016)grad divu+\u0016rot rot u +dF\nd\u001c(4.6)\navec\u001ala densité du milieu, ule déplacement local du tissu, Kle module de compres-\nsion isostatique, \u0016le module de cisaillement etdF\nd\u001cune force volumique extérieure.\nSupposons qu’aucune force extérieure n’est appliquée sur le tissu. On peut prendre\nla divergence de l’équation 4.6, ce qui donne en définissant p= div u:\ndp2\ndt2=v2\np\u0001p (4.7)\navecvp=q\n(K+4\n3\u0016)=\u001a. Cette équation correspond donc à la propagation de la diver-\ngence du déplacement local, ce qui permet de définir vpcomme la vitesse de l’onde de\ncompression.\nDe même, on peut prendre le rotationnel de l’équation 4.6, ce qui donne en définis-\nsants=rot u :\nds2\ndt2=v2\ns\u0001s (4.8)\navecvs=p\n\u0016=\u001a. Cette équation correspond à la propagation du rotationnel du dépla-\ncement local, ce qui permet de définir vscomme la vitesse des ondes de cisaillement.\nDans les tissus biologiques, on observe que vpvaut typiquement environ 1500 m/s\netvsde 1 à 10 m/s, ce qui donne une valeur de Ksupérieur de quatre à cinq ordres de\ngrandeur à \u0016. L’expression du module d’Young donnée dans l’équation 4.4 peut dans\nce cas être simplifiée en E\u00193\u0016= 3\u001av2\ns.\nD’après l’expression de vsdans l’équation 4.8 et la relation approchée entre le mo-\ndule d’Young et le module de cisaillement, la mesure de la vitesse d’une onde de ci-\nsaillement en un point d’un milieu permet de calculer son module d’Young en ce point :\nc’est le principe de l’élastographie par onde de cisaillement. Cette technique est com-\nposée de deux parties : un procédé pour créer un déplacement mécanique source d’une\nonde de cisaillement et un dispositif d’imagerie des déplacements dus à la propagation\nde l’onde de cisaillement.\nCependant, les ondes de cisaillement sont rapidement atténuées dans les tissus bio-\nlogiques, avec des coefficients d’atténuation de l’ordre de 105dB/cm pour une fré-\nquence de 10 MHz, alors que les ondes de compression ont un coefficient d’atténuation\nde l’ordre de 9 dB/cm à cette fréquence. Les ondes de cisaillement ne peuvent donc être\n102 Pol Grasland-Mongrain4.1. L’élastographie par ondes de cisaillement\nobservées qu’à basse fréquence, de quelques dizaines à quelques centaines de Hertz.\n4.1.4 L’élastographie par onde de cisaillement\nLes premières mesures par ondes de cisaillement se sont faites grâce à des ondes\nde cisaillement monochromatiques, donc avec une excitation continue. Sur ce principe,\nGreenleaf et al. [81] ont développé une méthode d’élastographie visualisée par IRM. Le\nprincipal défaut de cette méthode est de mélanger plusieurs types d’ondes, notamment\nles ondes de compression et celles de cisaillement. La vitesse mesurée en chaque point\nest donc une combinaison a priori non connue de vpet devs, ce qui induit des erreurs\ndans les images d’élasticité des tissus. Pour pallier ce problème, Sarvazyan et al. [106]\net Catheline et al. [18] ont proposé d’utiliser des impulsions plutôt que des signaux\ncontinus, le premier avec une visualisation par IRM et le deuxième par ultrasons. Ainsi,\ngrâce à la différence de vitesse de propagation, il est possible de séparer la vitesse de\nl’onde de compression de celle de l’onde de cisaillement, et donc d’obtenir des images\nquantitatives d’élasticité. Mais l’une des principales difficultés réside dans le contrôle\nde la source de ces ondes.\nLa première méthode employée pour générer une onde de cisaillement dans un\ntissu a été de placer un vibreur externe à la surface du corps d’un patient. Le déplace-\nment créé à la surface de la peau d’un patient se transmet effectivement à la fois sous\nla forme d’une onde de compression et d’une onde de cisaillement en profondeur, et\nle tout peut être observé par ultrasons ou IRM. Ce principe a notamment été utilisé\navec succès pour mesurer la dureté du foie et en déduire son état de fibrose [105] – un\nappareil industriel a d’ailleurs été commercialisé sous le nom de Fibroscan [35].\nLa pression de radiation ultrasonore peut également induire un déplacement dans\nun tissu [118]. Ce déplacement, qui peut être simplement utilisé pour en déduire la\ndureté du tissu dans la zone focale (méthode ARFI pour Acoustic Radiation Force Ima-\nging [84]), crée également une onde de cisaillement in-situ . Ainsi, la technique peut être\nappliquée avec un seul transducteur sans nécessiter d’appareil supplémentaire, ce qui\nfacilite la tâche aux médecins habitués à réaliser des échographies. De plus, si l’on ef-\nfectue des images ultrasonores à une cadence dite ultrarapide avec plusieurs milliers\nd’images par seconde, on peut suivre la propagation des ondes de cisaillement dans\nle tissu, ce qui permet de reconstruire directement une image d’élasticité [12]. Cepen-\ndant, si des obstacles comme des couches de graisses ou de l’os sont présents entre\nle transducteur ultrasonore et la zone focale des ultrasons, l’image peut présenter des\naberrations voire être impossible à former.\nApplications de la force de Lorentz en acoustique médicale 103Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\n4.1.5 L’utilisation de la force de Lorentz\nLa force de Lorentz peut aussi être utilisée pour créer des ondes de cisaillement.\nEn effet, lorsqu’un courant traverse un solide conducteur placé dans un champ magné-\ntique, la force de Lorentz provoque un déplacement local du milieu. Selon le courant,\nce déplacement peut se propager comme une onde de compression, de cisaillement\nou une combinaison des deux. Ceci est utilisé dans les transducteurs électromagné-\ntiques acoustiques (EMAT pour ElectroMagnetic Acoustic Transducer) [117]. Ces trans-\nducteurs, dont l’un des designs est représenté figure 4.6, sont utilisés dans l’industrie\nsur des matériaux ferromagnétiques. Ils sont composés d’un petit circuit électrique en\nforme de serpent ou de spirale et d’un aimant. Le circuit électrique induit un courant\nélectrique dans le matériau. Le courant électrique combiné au champ magnétique de\nl’aimant est une source d’onde de compression et de cisaillement. Ces ondes sont alors\ndétectées pour mesurer d’éventuels défauts dans le matériau.\nNS\nCisaillement\nen surfaceCourants induits\nOndes\nmécaniquesAimant permanent\nCircuit électrique\nFIGURE 4.6: Un circuit électrique crée par induction un courant électrique dans la sur-\nface d’un matériau ferromagnétique soumis à un champ magnétique. Des mouvements\napparaissent par force de Lorentz, ce qui crée une combinaison d’ondes de cisaillement\net de compression.\nL’application de ce principe aux tissus biologiques n’est pas évidente, principale-\nment à cause de la faible conductivité électrique de ceux-ci – typiquement un million\nde fois inférieure à celle d’un métal. Une différence importante concerne l’épaisseur de\npeau : à une fréquence donnée, l’épaisseur de peau des tissus biologiques est bien plus\nfaible que celle des matériaux ferromagnétiques. Ainsi, à la différence de ces derniers,\n104 Pol Grasland-Mongrain4.2. Principe de la création d’ondes de cisaillement par force de Lorentz\nla source des ondes dans un tissu biologique ne sera pas localisée vers la surface et\nn’est pas ponctuelle. Cela peut être vu comme un avantage ou un inconvénient selon\nles applications.\nBien que la force de Lorentz ait déjà été utilisée pour produire des ondes mécaniques\ndans les tissus biologiques, avec la tomographie magnéto-acoustique par induction ma-\ngnétique présentée dans la section 3.2.5, aucune méthode à l’heure actuelle ne s’est in-\ntéressée à l’utilisation de la force de Lorentz pour générer des ondes de cisaillement\ndans un but d’élastographie.\n4.2 Principe de la création d’ondes de cisaillement par force de\nLorentz\nDans l’élastographie par force de Lorentz, telle qu’étudiée dans cette thèse, un cou-\nrant électrique est injecté dans un échantillon conducteur. La présence d’un champ\nmagnétique perpendiculaire au courant électrique induit un déplacement par force de\nLorentz. Ce déplacement est la source d’une onde de compression et d’une onde de ci-\nsaillement. Une barrette d’échographie observe alors la propagation des déplacements\ndu milieu grâce à une technique de speckle-tracking comme présenté par Ophir et al. [88].\nCette méthode compare des images échographiques à chaque instant et essaie d’en dé-\nduire les déplacements locaux à partir de corrélations entre les images. Comme pour\nles autres méthodes d’élastographie dynamique, la vitesse des ondes de cisaillement\npermet de retrouver l’élasticité des tissus.\nDans les expériences d’élastographie par force de Lorentz dans un échantillon de\ntaille finie, les ondes de cisaillement peut être créé de différentes manières : soit par la\nforce de Lorentz elle-même ; soit par conversion de mode à partir d’une onde de com-\npression, aux bords d’un échantillon. Les deux derniers facteurs sont d’ailleurs com-\nmuns aux autres techniques d’élastographie.\n4.2.1 La force volumique dans le milieu\nComme précédemment, on suppose dans cette partie que le champ magnétique est\ndans la direction X et l’axe ultrasonore dans la direction Z. En élastographie par force\nde Lorentz, on applique une différence de potentiel \u0001Ventre deux électrodes. Cette\ndifférence de potentiel induit un champ électrique E=\u0000gradVdans tout l’espace. A\nl’intérieur du tissu, que l’on suppose isotrope de conductivité électrique \u001b, la loi d’Ohm\ns’écrit j=\u001bEen suivant les notations introduites au chapitre 3.\nApplications de la force de Lorentz en acoustique médicale 105Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nLa force de Lorentz induite Fpeut par ailleurs s’écrire :\ndF\nd\u001c=j^B (4.9)\navecd\u001cun petit élémént de volume et Ble champ magnétique.\nEn présence de cette force, l’équation de Navier s’écrit alors :\n\u001ad2u\ndt2= (K+4\n3\u0016)grad divu+\u0016rot rot u +j^B (4.10)\nOn supposera par la suite que la densité \u001adu milieu est homogène. Ce paramètre\nvarie de moins de 10 % dans les tissus mous du corps humain, donc cette hypothèse est\ngénéralement assez bien vérifiée.\nEn calculant la divergence de l’équation 4.10, on obtient avec p= div u:\ndp2\ndt2=v2\np\u0001p+1\n\u001adiv(j^B) (4.11)\nLa source de l’onde de compression dépend donc simplement de la divergence du\nproduit vectoriel de la densité de courant et du champ magnétique [101].\nEn calculant le rotationnel de l’équation 4.10, on obtient avec s=rot u :\nds2\ndt2=v2\nsrs+1\n\u001arot(j^B) (4.12)\nOn peut utiliser l’identité vectorielle suivante : rot(j^B) = (div B)j\u0000(divj)B+\n(B:grad )j\u0000(j:grad )B. En utilisant la deuxième équation de Maxwell divB= 0et en\nremarquant que divj= 0 en l’absence d’accumulation de charges dans le milieu, on\nobtient l’équation :\nds2\ndt2=v2\ns\u0001s\u00001\n\u001a(B:grad )j\u00001\n\u001a(j:grad )B (4.13)\nOn peut alors distinguer deux cas notables. Si Best uniforme dans le milieu, que\nl’on suppose selon l’axe X, on a alors, en remplaçant jpar\u001bE, alors on a la relation :\nds2\ndt2=v2\ns\u0001s\u0000\u001bBx\n\u001adE\ndx+Bx\n\u001ad\u001b\ndxE (4.14)\n106 Pol Grasland-Mongrain4.2. Principe de la création d’ondes de cisaillement par force de Lorentz\nSiEest uniforme selon l’axe du champ magnétique, la source de l’onde de cisaillement\ndépend alors du gradient de \u001bselon cet axe, et si \u001best uniforme selon l’axe du champ\nmagnétique, la source de l’onde de cisaillement dépend alors du gradient de Eselon\ncet axe.\nDans le deuxième cas, seul jest uniforme, que l’on supposera selon l’axe Y (le\nchamp Bn’étant plus uniforme), ce qui donne :\nds2\ndt2=v2\ns\u0001s\u0000jy\n\u001adB\ndy(4.15)\nLa source de l’onde de cisaillement est alors due au gradient de Bselon l’axe Y.\n4.2.2 La propagation d’une onde mécanique dans le milieu\nIl faut cependant bien comprendre que l’équation précédente indique la création\nlocale d’une onde de cisaillement. Si l’on reprend l’équation de Navier avec une force\nvolumique f(t)dépendant du temps, localisée en un point de coordonnées ret orientée\ndans une direction i, on obtient :\n\u001ad2u\ndt2\u0000(K+1\n3\u0016)grad divu\u0000\u0016\u0001u=f(t)\u000e(r) (4.16)\nLa solution de cette équation est donnée par une fonction de Green, que l’on nom-\nmeraG(r;t). Connaissant la solution G(r;t)pour une source ponctuelle, la solution\npour une source étendue spatialement f(t)S(r)est donnée par la convolution de la\nfonction de Green avec la fonction spatiale de la source soit G(r;t)\u0003rS(r), où\u0003rindique\nla convolution sur la variable d’espace. La création de l’onde de cisaillement peut donc\nêtre calculée grâce à cette dernière équation.\nPour un volume isotrope, homogène et infini (sans conditions aux limites), le calcul\ndont le détail est donné en annexe donne :\nui(r;t) =1\n4\u0019\u001ar3\u00123rirj\nr2\u0000\u000eij\u0013Zr=vs\nr=vp\u001cf(t\u0000\u001c) d\u001c)\nchamp proche\n+1\n4\u0019\u001av2prrirj\nr2f\u0012\nt\u0000r\nvp\u0013 \u001b\nonde de compression\n+1\n4\u0019\u001av2sr\u0010rirj\nr2\u0000\u000eij\u0011\nf\u0012\nt\u0000r\nvs\u0013 \u001b\nonde de cisaillement(4.17)\nApplications de la force de Lorentz en acoustique médicale 107Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\navecr=krketi;jles directions X, Y et Z. Ainsi, on voit que la solution est la somme de\ntrois termes : deux termes de champs lointain, décroissant en1\nr, et un terme de champ\nproche, avec une décroissance plus rapide. Le champ lointain se décompose lui-même\nen deux termes, celui correspondant à des ondes de compression polarisées longitu-\ndinalement se propageant à la vitesse vp, et un deuxième correspondant à des ondes\nde cisaillement polarisées latéralement se propageant à la vitesse vs. Ces deux termes\ncorrespondent d’ailleurs à la propagation d’ondes planes dans un milieu isotrope, ho-\nmogène et infini comme vu au début du chapitre. Cependant, le terme de champ proche\nne peut pas être décomposé de la même façon en deux termes distincts. On peut donc,\nproche de la source, observer des ondes de cisaillement polarisées longitudinalement.\nCes dernières ont d’ailleurs été mises en évidence expérimentalement par Catheline et\nal.[18].\nLe déplacement total du tissu par force de Lorentz nécessite donc la résolution des\nfonctions de Green.\n4.2.3 Conversion de mode\nEnfin, des ondes de cisaillement peuvent être produites par conversion de mode.\nCeci arrive lorsqu’une onde mécanique rencontre un obstacle lors de sa propagation\n(paroi rigide par exemple). Dans ce cas, une onde de compression peut donner lieu à\nla création d’une onde de cisaillement. Les conditions aux bords d’un échantillon vont\ndonc influer sur l’apparition d’ondes de cisaillement.\n4.3 Dispositif expérimental\nLe dispositif est représenté sur la figure 4.7.\nLe principe de l’expérience est de faire circuler une impulsion électrique brève dans\nun échantillon placé dans un champ magnétique, et d’observer les mouvements induits\navec une sonde ultrasonore.\nLe courant électrique était composé d’une sinusoide de 10 ms à 80 V pic à pic et\nétait émis par un générateur (AFG 3022B, Tektronix, Beaverton, OR, USA) couplé à\nun amplificateur (600W A500, Behringer, Willich, Germany). Une résistance de 1 Ohm\nplacée en série permettait de connaitre l’intensité du courant, comprise en 0,2 et 1 A\npic à pic selon les expériences. Ce courant était injecté par deux électrodes en cuivre\n10x3x0,1 cm3dans l’échantillon. L’échantillon est un fantôme d’alcool polyvinylique\n(PVA) de taille 6x4x4 cm3composé de PVA à 50 g.L\u00001[23], de 10 g/L de sel pour lui\n108 Pol Grasland-Mongrain4.3. Dispositif expérimental\nGénérateur\nd'impulsionLignes de courant\nélectrique\nDéplacementEchographe\nultrarapide\nChamp magnétique uniformeX\nYZEchantillon\nentre deux\nélectrodes\nFIGURE 4.7: Un courant électrique est injecté par deux électrodes dans un échantillon\nplacé dans un champ magnétique. La propagation du déplacement induit est observée\npar une sonde ultrasonore.\ndonner une conductivité électrique de 2 S.m\u00001, un peu supérieure à celle des tissus\nbiologiques, et de 1 g.L\u00001de poudre de graphite.\nCet échantillon était placé dans l’entrefer de l’aimant en U utilisé dans les parties\nprécédentes (photographie figure 2.9 et simulation de champ magnétique figure 2.11)\npour avoir un champ magnétique global de 350 mT.\nLe déplacement était visualisé avec une barrette multiéléments à 10 MHz fonction-\nnant en mode ultrarapide relié à un échographe Verasonics (Verasonics V-1, Redmond,\nWA, USA). Dans le mode ultrarapide, on émet une onde plane ultrasonore qui va être\nréfléchie par les interfaces d’impédance acoustique. On peut alors localiser celles-ci\navec les temps d’arrivée des ondes réfléchies sur les différents éléments du transduc-\nteur. Avec une vitesse ultrasonore d’environ 1500 m.s\u00001, on arrive donc théoriquement\npour une profondeur d’exploration de 10 cm à une cadence d’image de 7500 images\npar seconde. La qualité de l’image peut alors être améliorée en faisant de l’imagerie\ncomposée, en utilisant plusieurs angles d’incidence de visualisation [104]). Dans notre\ndispositif, nous avions une cadence de 1000 images par seconde avec un seul angle\npour la barrette multiéléments.\nLa poudre de graphite présente dans l’échantillon faisait office de diffuseurs acous-\ntiques. Ainsi, les échantillons présentaient un speckle sur les images ultrasonores. Lors\nde la propagation de l’onde, le tissu était légèrement déformé, ce qui déplace les grains\ndespeckle de quelques micromètres. En cherchant un maximum de corrélation entre\nune ligne et la même ligne sur l’image suivante, il est possible de déduire le mouve-\nment probable du tissu en cet endroit le long de la ligne ultrasonore, comme développé\ndans la thèse de S. Catheline [18]. Concrètement, il est possible d’observer une ampli-\ntude des déplacements inférieure au micromètre, alors que la longueur d’onde des ul-\nApplications de la force de Lorentz en acoustique médicale 109Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\ntrasons est approximativement de 150 µm (longueur d’onde à 10 MHz) : ceci est rendu\npossible par l’efficacité des algorithmes de speckle-tracking qui, comme les méthodes in-\nterférométriques en optique, atteignent des résolution spatiales bien en-dessous de la\nlongueur d’onde.\nLes signaux représentant les déplacements en chaque point au cours du temps\nétaient enfin filtrés par le logiciel Matlab. Pour cela, la transformée de Fourier temporel\ndu signal était calculée, la valeur nulle était donnée pour les fréquences supérieures à\n200 Hz, et la transformée de Fourier inverse était calculée. Ceci permettait de diminuer\nle bruit présent sur les images de déplacements.\n4.4 Expériences de création d’ondes de cisaillement par force\nde Lorentz\nLa première étape de ce travail a été d’obtenir une première observation expérimen-\ntale d’onde S par force de Lorentz dans de la matière molle. Un soin particulier a été\nporté à l’objectivation du phénomène, pour d’une part s’assurer que l’on crée effecti-\nvement des ondes de cisaillement, et d’autre part de vérifier que ces ondes ne sont pas\ninduites par un effet indésirable comme le mouvement des électrodes ou une dilation\npar effet Joule.\n4.4.1 Création d’onde de cisaillement par conversion de mode\nLa première expérience a consisté simplement à voir si l’injection d’un courant élec-\ntrique dans un tissu plongé dans un champ magnétique donnait un déplacement du\ntissu suffisamment important pour être visible. De plus, le dispositif était fait de sorte\nque l’on puisse observer la création d’une onde de cisaillement par conversion de mode\nsur les bords de l’échantillon, grâce au contact avec des parois rigides.\nMatériel et méthodes Une expérience a été faite avec deux électrodes en contact res-\npectivement avec la partie supérieure et inférieure de l’échantillon, le long de la tranche,\ncomme représenté sur le schéma 4.8-(a). L’échographie du milieu est représentée figure\n4.8-(b). L’apparence granuleuse de l’image est due à l’ajout de graphite.\nRésultats et discussions Les cartes des déplacements à 0, 5, 10, 15, 20, 30, 40 et 50 ms\nsont représentés sur la figure 4.9.\n110 Pol Grasland-Mongrain4.4. Expériences de création d’ondes de cisaillement par force de Lorentz\nGénérateur\nd'impulsionLignes de courant\nélectrique\nDéplacementEchographe\nultrarapide\nChamp magnétique uniformeX\nYZ\n10 20 30 40 50 60−10\n0\n10\nPosition Z (mm)Position Y (mm)Echographie\nFIGURE 4.8: (a) Schéma de l’expérience de création d’onde de cisaillement par conver-\nsion de mode – Un courant électrique est injecté par deux électrodes planes dans un\nfantôme soumis à un champ magnétique. Cela entraine des déplacements observés par\nune sonde ultrasonore. (b) Echographie obtenue par la sonde ultrasonore – Le fan-\ntôme a un aspect granulaire dû à l’ajout de graphite. L’interface du fond du fantôme est\nvisible sur la droite.\nAu moment de déclencher le courant, aucun déplacement n’est visible. Lorsqu’on\ninjecte un courant électrique dans un sens (partie positive de la sinusoïde émise), à 5 ms,\nun premier déplacement du fantôme apparait, globablement dans le même sens. Puis, à\n10 ms, correspondant à l’injection d’un courant dans le sens opposé (partie négative de\nla sinusoïde émise), on observe un déplacement globalement dans le sens contraire. A\n15, 20, 30, 40 et 50 ms, on observe la propagation d’une onde avec une longueur d’onde\ncentimétrique. Cette onde correspond à une onde de cisaillement dont la polarisation\nn’est pas transversale. Comme les champs EetBsont quasiment uniformes et que\nla conductivité électrique du fantôme est homogène, il ne s’agit pas de la force qui\ncrée intrinsèquement l’onde de cisaillement. Cependant, une déformation importante\na lieu au voisinage des plaques, ce qui donne lieu par conversion de mode à la création\nd’ondes de cisaillement.\n4.4.2 Création d’onde de cisaillement par force de Lorentz\nSi la force a une extension spatiale limitée, elle produit un déplacement qui se pro-\npage selon l’équation 4.17. Ainsi, même si la force ne produit pas de cisaillement in-\ntrinsèque, la taille de la source peut en produire. Les ondes de cisaillement ont alors\nune polarisation à la fois transverse et longitudinale, qui n’est transverse qu’en champ\nlointain. Nous avons réalisé ici une expérience avec une source ayant un rayonnement\nde type dipolaire pour induire une onde de cisaillement.\nMatériel et méthodes Une expérience a été réalisée où deux électrodes filaires sont\nplacées au milieu du fantôme, comme représenté sur la figure 4.10-(a). Le courant élec-\ntrique émis restait localisé entre celles-ci. L’échographie du milieu est représenté sur la\nApplications de la force de Lorentz en acoustique médicale 111Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012t = 0 ms\nPosition Z (mm)\nt = 20 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012t = 5 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012t = 30 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012t = 10 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012t = 40 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012t = 15 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012t = 50 ms\nPosition Z (mm)\nFIGURE 4.9:Résultats de l’expérience de création d’ondes de cisaillement par conver-\nsion de mode – Champ des déplacements dus à la traversée d’un courant électrique par\ndeux électrodes situées au-dessus et en-dessous d’un fantôme placé dans un champ ma-\ngnétique, à 0, 5, 10, 15, 20, 30, 40, 50 ms après le début du passage du courant (plan YZ).\nOn observe une onde de cisaillement provenant d’un bord de l’échantillon, produite\npar conversion de mode.\nfigure 4.10-(b). On distingue deux points blancs correspondants à la position des deux\nélectrodes filaires au milieu du fantôme.\nRésultats et discussions La figure 4.11 représente les champs des déplacements de\nl’échantillon à 0, 3, 6, 9, 12, 15, 18, 21, 24 et 27 millisecondes ms après le début du pas-\nsage du courant. On observe, notamment sur les champs à 10, 13, 15 et 18 ms des dé-\nplacements correspondant à une onde de cisaillement. De plus, les déplacements sont\n112 Pol Grasland-Mongrain4.4. Expériences de création d’ondes de cisaillement par force de Lorentz\nGénérateur\nd'impulsionLignes de courant\nélectriqueDéplacement\nEchographe\nultrarapide\nChamp magnétique uniformeX\nYZ\nPosition Z (mm)10 20 30 40 50 60 70−10\n0\n10EchographiePosition Y (mm)\nFIGURE 4.10: (a) Schéma de l’expérience sur la création d’onde de cisaillement par\nforce de Lorentz – Deux électrodes filaires créent un courant électrique dans un fan-\ntôme soumis à un champ magnétique, et les déplacements sont observés par une sonde\nultrasonore en mode ultrarapide. (b) Echographie obtenue par la sonde ultrasonore –\nOn distingue au milieu deux points blancs correspondant à la zone où se trouvent les\ndeux électrodes filaires.\nd’amplitude plus importante au-dessus et en-dessous de la paire d’électrodes que sur\nles côtés, même si du même ordre de grandeur : ceci est caractéristique de la directivité\nd’une onde de cisaillement en champ proche, avec la présence d’ondes de cisaillement\npolarisées longitudinalement.\n4.4.3 Influence éventuelle du mouvement des électrodes\nAvec un champ magnétique global autour du fantôme et des électrodes, deux phé-\nnomènes peuvent se produire, sans être exclusifs : l’onde de cisaillement peut être créée\npar un déplacement dû (1) au courant électrique traversant le fantôme plongé dans le\nchamp magnétique, ou bien (2) au mouvement des électrodes elles-mêmes. Même si\nles électrodes sont bien fixées, des mouvements de quelques micromètres d’amplitude\nrestent possibles et peuvent se transmettre au reste du fantôme. Une expérience avec\nun champ magnétique localisé permet donc d’observer l’origine du mouvement.\nMatériel et méthodes Une expérience a été réalisée où deux électrodes rectangulaires\nen contact avec deux côtés du fantôme émettaient un courant électrique, tandis qu’un\npetit aimant de 35x20x20 mm3produisait un champ magnétique localisé sur le fantôme,\ncomme représenté sur la figure 4.13-(a). Cet aimant donnait un champ magnétique lo-\ncalisé de 150\u000650 mT à une distance de 2 cm, comme représenté sur la simulation de\nchamp magnétique 4.12 réalisée dans le plan XY. Une attention particulière a été por-\ntée pour que cet aimant ne soit pas en contact avec le fantôme et qu’il soit tenu par\nun support indépendant du fantôme – mis à part la table de l’expérience – pour éviter\ntout contact électrique ou mécanique. Tout mouvement ne peut donc provenir que de\nl’intérieur du fantôme et non pas par contact avec les électrodes ou l’aimant.\nApplications de la force de Lorentz en acoustique médicale 113Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 12 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 3 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 15 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 6 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 18 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 9 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 21 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 24 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nt = 27 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−2−1012\nFIGURE 4.11: Schéma de l’expérience sur la création d’onde de cisaillement par force\nde Lorentz – Champ des déplacements dus à la traversée d’un courant électrique dans\nun fantôme par deux électrodes filaires au centre, le tout placé dans un champ magné-\ntique, à 0, 3, 6, 9, 12, 15, 18, 21, 24 et 27 ms après le début du passage du courant (plan\nYZ). On observe une onde se propageant dans toutes les directions, avec une amplitude\nplus élevée dans la direction Y. Ceci est caractéristique de la directivité d’une onde de\ncisaillement en champ proche.\n114 Pol Grasland-Mongrain4.4. Expériences de création d’ondes de cisaillement par force de Lorentz\n0 T\n0.2 T\n0.4 T\n0.6 T\n0.8 T\n>1 T 2 cmEmplacement\nde l'échantillon\nFIGURE 4.12: Simulation du champ magnétique créé par un aimant de 35x20x20 mm3\navec les lignes de champ en noir et la direction de l’aimant indiqué par une flèche (plan\nXY). Le champ magnétique est égal à 0,15 \u00060,05 T à 2 cm de l’aimant.\nL’échographie du milieu est représenté sur la figure 4.13-(b). On observe l’interface\ndu fond du fantôme sur la droite.\nGénérateur\nd'impulsionLignes de courant\nélectrique Déplacement\nEchographe\nultrarapide\nChamp magnétiqueX\nYZ\n10 20 30 40 50 60−10\n−5\n0\n5\n10\nPosition Z (mm)Position Y (mm)Echographie\nFIGURE 4.13: (a) Schéma de l’expérience sur l’influence éventuelle du mouvement des\nélectrodes – Deux électrodes longues créent un courant électrique homogène dans un\nfantôme dont seulement une partie est soumise à un champ magnétique, et les déplace-\nments sont observés par une sonde ultrasonore en mode ultrarapide. (b) Echographie\nobtenue par la sonde ultrasonore – L’interface du fond du fantôme est visible sur la\ndroite.\nRésultats et discussions La figure 4.11 représente six champs de déplacements au\ncours du temps, à 0, 2, 5, 7, 10 et 12 millisecondes après le début du passage du courant\nélectrique. On peut voir que les ondes sont issues de la zone à fort gradient de champ\nmagnétique. Ici, le mouvement est créé au milieu du fantôme, donc il n’y a pas d’effets\nde bords ou de conversion de mode. Cela signifie que le mouvement a bien été produit\nau sein de l’échantillon par la force de Lorentz et non pas par contact avec les électrodes.\nL’amplitude des déplacements est environ dix fois plus faible que dans les expériences\nApplications de la force de Lorentz en acoustique médicale 115Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nprécédentes, mais le champ magnétique est également moins intense. Le phénomène\n(1) n’est donc pas négligeable devant le phénomène (2) lors des autres expériences.\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−1−0.500.51\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−1−0.500.51t = 7 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−1−0.500.51t = 2 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−1−0.500.51t = 10 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−1−0.500.51t = 5 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60−10\n−5\n0\n5\n10µm\n−1−0.500.51t = 12 ms\nPosition Z (mm)\nFIGURE 4.14: Résultats de l’expérience sur l’influence éventuelle du mouvement des\nélectrodes – Champ des déplacements dus à la traversée d’un courant électrique dans\nle fantôme avec un champ magnétique localisé, à 0, 2, 5, 7, 10 et 12 ms après le début\ndu passage du courant (plan YZ). On observe que le mouvement initial apparait au\ncentre du fantôme, là où le champ magnétique est présent, et non pas des bords, où se\ntrouvent les électrodes.\n4.4.4 Influence éventuelle de l’effet Joule\nTout passage d’un courant dans un conducteur échauffe celui-ci par effet Joule. Cet\néchauffement provoque une dilatation locale, ce qui pourrait être la source du déplace-\nment observé, à l’instar de la technique d’imagerie photoacoustique.\nMatériel et Méthodes Une expérience a été mise en oeuvre sans champ magnétique,\nen retirant l’aimant, comme représenté sur la figure 4.15-(a). Si l’effet Joule a un effet\nimportant, on devrait alors voir un déplacement lors du passage du courant. L’écho-\n116 Pol Grasland-Mongrain4.4. Expériences de création d’ondes de cisaillement par force de Lorentz\ngraphie du milieu est représentée figure 4.15-(b), où l’on voit l’interface du fond du\nfantôme sur la droite.\nGénérateur\nd'impulsionLignes de courant\nélectrique\nEchographe\nultrarapideX\nYZ\nPosition Z (mm)10 20 30 40 50 60 70−10\n0\n10EchographiePosition Y (mm)\nFIGURE 4.15: (a) Schéma de l’expérience sur l’influence de l’effet Joule – Deux élec-\ntrodes longues créent un courant électrique homogène dans un fantôme en l’absence\nde champ magnétique, et les déplacements sont observés par une sonde ultrasonore en\nmode ultrarapide. (b) Echographie obtenue par la sonde ultrasonore – L’interface du\nfond du fantôme est visible sur la droite.\nRésultats et discussions La figure 4.16 représente quatre champs de déplacements\nau cours du temps à 0, 2, 5 et 10 millisecondes après le début du passage du courant.\nLa barre de couleur va de -0,5 µm à 0,5 µm, donc d’une amplitude quatre fois plus\nfaible que l’expérience initiale. Le courant mesuré est de 0,5 A pour une résistance du\nfantôme de 250 Ohm, ce qui est similaire aux expériences précédentes. On n’observe\naucune propagation de mouvement dans le milieu, bien que l’interface du fond semble\nlégèrement déplacée. Ce dernier phénomène est très probablement dû à des artefacts\nde l’algorithme de speckle-tracking , qui essaie de trouver un déplacement même en l’ab-\nsence de speckle . La dilatation par effet Joule n’explique donc pas les phénomènes ob-\nservés dans les parties précédentes.\n4.4.5 Bilan partiel\nCes expériences ont montré que la force de Lorentz peuvent être utilisée pour in-\nduire des ondes de cisaillement dans les tissus mous. Les ondes de cisaillement peut\napparaitre à cause de la force elle-même ou par conversion de mode lors de la propa-\ngation d’une onde mécanique.\nIl a par ailleurs été montré que les déplacements observés n’étaient pas des artefacts\ndû au mouvement des électrodes elles-mêmes, qui peuvent se déplacer à cause de la\nforce de Lorentz, ou à la dilatation par effet Joule.\nApplications de la force de Lorentz en acoustique médicale 117Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n10 20 30 40 50 60 70−10\n0\n10µm\n−0.500.5\nPosition Y (mm)\n10 20 30 40 50 60 70−10\n0\n10µm\n−0.500.5t = 5 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60 70−10\n0\n10µm\n−0.500.5t = 2 ms\nPosition Z (mm)\nPosition Y (mm)\n10 20 30 40 50 60 70−10\n0\n10µm\n−0.500.5t = 10 ms\nPosition Z (mm)\nFIGURE 4.16: Résultats de l’expérience sur l’influence de l’effet Joule – Champ des\ndéplacements dus à un courant électrique injecté par deux électrodes placé au-dessus\net en-dessous du fantôme, sans champ magnétique, à 0, 2, 5 et 10 ms (plan YZ). On\nn’observe aucune propagation d’onde et quasiment aucun mouvement, ce qui indique\nque la dilatation due au chauffage par effet Joule n’explique pas les ondes observées.\n4.5 Expériences d’élastographie par force de Lorentz\nL’objectif de cette section est de montrer qu’il est possible de réaliser des expé-\nriences d’élastographie avec des ondes de cisaillement induites par force de Lorentz.\nNous avons mesuré une vitesse d’onde de cisaillement dans un fantôme homogène,\npuis dans un fantôme composé de deux couches. Enfin, la méthode a été appliquée sur\nun échantillon ex-vivo de foie.\n4.5.1 Elastographie sur un fantôme homogène\nMatériel et méthodes Une expérience a été faite avec deux électrodes en contact avec\nle fond du fantôme, le long de la tranche. La barrette ultrasonore était placée de l’autre\ncôté, comme représenté sur le schéma 4.17-(a). L’échographie du milieu est représentée\nfigure 4.17-(b), où l’on voit au fond deux points blancs correspondant à la zone de\ncontact avec les deux électrodes.\nRésultats et discussions La figure 4.18 représente des champs de déplacement au\ncours du temps à 0, 5, 10, 15, 20, 25, 30, 35, 40 et 45 millisecondes après le début du\npassage du courant. On observe un mouvement entre 0 et 10 ms qui correspond au\ncourant électrique émis, puis à la propagation de ce mouvement au cours du temps.\nUne mesure le long de la ligne centrale en fonction du temps, représentée sur la figure\n118 Pol Grasland-Mongrain4.5. Expériences d’élastographie par force de Lorentz\nGénérateur\nd'impulsionLignes de courant\nélectrique\nDéplacementEchographe\nultrarapide\nChamp magnétique uniformeX\nYZ\nPosition Z (mm)Position Y (mm)\n30 40 50 60 70 80−10\n0\n10Echographie\nFIGURE 4.17: (a) Schéma de l’expérience d’élastographie sur un fantôme homogène\n– Deux électrodes proches créent un courant électrique dans un fantôme soumis à un\nchamp magnétique, et les déplacements sont observés par une sonde ultrasonore en\nmode ultrarapide. (b) Echographie obtenue par la sonde ultrasonore – On distingue au\nmilieu deux points blancs correspondant à la zone de contact avec les deux électrodes.\n4.19, donne une vitesse de propagation des ondes de 1,3 \u00060,1 m/s. Cette vitesse est\ncompatible avec la vitesse des ondes de cisaillement, entre 1 et 10 m.s\u00001pour un tissu\nmou. Avec la formule vs=p\n\u0016=\u001a, cela correspond pour une densité \u001ade 1000 kg.m\u00003\nà une élasticité Ede 5,0\u00060,8 kPa.\nL’onde de cisaillement créée est similaire au rayonnement d’une source ponctuelle\nde surface en champ proche, d’où la présence d’ondes de cisaillement polarisées longi-\ntudinalement.\n4.5.2 Expérience sur un fantôme bicouche\nMatériel et méthodes Une expérience a été faite avec deux électrodes en contact avec\nle fond du fantôme, le long de la tranche. La barrette ultrasonore était placée de l’autre\ncôté, comme représenté sur le schéma 4.20-(a). Cette fois-ci, le fantôme était composé\nde deux parties d’élasticité différente. La couche supérieure, plus dure, était faite en\nPVA, tandis que la couche inférieure, plus molle, était faite en gélatine. Le fantôme ne\ntouchait pas les parois, car comme vu lors de la toute première expérience, le contact\ncrée des ondes de cisaillement. La cadence d’acquisition était quatre fois plus élevée\n(4000 Hz au lieu de 1000 Hz) afin d’avoir une meilleure résolution temporelle, mais en\ncontrepartie, on observe des déplacements quatre fois plus faibles entre chaque image.\nL’échographie du milieu est représentée figure 4.20-(b), où l’on distingue l’interface\nentre les deux couches par la ligne plus claire au centre. L’interface du fond n’est pas\nvisible.\nRésultats et discussions La figure 4.21 représente huit champs de déplacements au\ncours du temps à 0, 3, 7, 9, 11, 13, 15 et 17 millisecondes après le début du passage du\ncourant. L’onde initiale visible à t = 3 ms est une onde de compression. A 7, 9 et 11\nApplications de la force de Lorentz en acoustique médicale 119Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nt = 0.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 25.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 5.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 30.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 10.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 35.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 15.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 40.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 20.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 45.00 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nFIGURE 4.18: Résultats de l’expérience d’élastographie sur un fantôme homogène –\nChamp des déplacements dus à un courant électrique injecté par deux électrodes au\nfond du fantôme placé dans un champ magnétique, à 0, 5, 10, 15, 20, 25, 30, 35, 40 et 45\nms (plan YZ). On observe une propagation des ondes à une vitesse de 1,3 \u00060,1 m/s.\nms, une onde de cisaillement en bleu apparait et se propage à deux vitesses différentes\ndans la partie supérieure et dans la partie inférieure. De même, l’onde de cisaillement\nen rouge visible à 13, 15 et 17 ms se propage à des vitesses différentes entre les deux\nzones. La vitesse des ondes de cisaillement dans les parties supérieure et inférieure vaut\nrespectivement 5,1 \u00060,1 et 2,5\u00060,1 m.s\u00001, ce qui correspond à des élasticités E1= 78\n120 Pol Grasland-Mongrain4.5. Expériences d’élastographie par force de Lorentz\nTemps (ms)Position Z (mm)Déplacements ligne y = 0 mm\n20 40 60 80 100 120 140 160 18050\n100\n150\n200\n250\nµm\n−2−1012Compression Cisaillement\nFIGURE 4.19: Résultats de l’expérience sur la vitesse des ondes – Déplacements locaux\nselon une ligne ultrasonore au cours du temps. On observe d’abord une vitesse de pro-\npagation très élevée (pointillés noirs), correspondant à la propagation d’une onde de\ncompression. Puis une autre onde se propageant à 1,3 \u00060,1 m/s (ligne pleine noire),\ncorrespondant à une onde de cisaillement.\nPVA (mou)\nGélatine (dur)Générateur\nd'impulsionLignes de courant\nélectrique\nDéplacementEchographe\nultrarapide\nChamp magnétique uniformeX\nYZ\n10 20 30−10\n0\n10\nPosition Z (mm)Position Y (mm)Echographie\nFIGURE 4.20: (a) Schéma de l’expérience d’élastographie sur un fantôme bicouche –\nDeux électrodes proches créent un courant électrique dans un fantôme soumis à un\nchamp magnétique, et les déplacements sont observés par une sonde ultrasonore en\nmode ultrarapide. Le fantôme présente deux couches d’élasticité différentes, une plus\ndure sur le dessus et une plus molle en-dessous. (b) Echographie obtenue par la sonde\nultrasonore – La ligne plus claire horizontale au centre de l’échographie correspondant\nà la séparation entre les deux couches, mais est à peine visible.\n\u00063 kPa etE2= 19\u00062 kPa.\nApplications de la force de Lorentz en acoustique médicale 121Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20 25 30−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 11 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20 25 30−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 3 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20 25 30−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 13 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20 25 30−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 7 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20 25 30−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 15 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20 25 30−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 9 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 25 30−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\n20\nt = 17 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20 25 30−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nFIGURE 4.21: Résultats de l’expérience d’élastographie sur un fantôme bicouche –\nChamp des déplacements dus à un courant électrique injecté par deux électrodes au\nfond du fantôme placé dans un champ magnétique, 0, 3, 7, 9, 11, 13, 15 et 17 ms (plan\nYZ). Une onde de compression se propageant rapidement et sans différence entre les\ndeux parties apparait à t = 3 ms. A 7, 9 et 11 ms, on voit l’onde en bleu se propager à\nune vitesse plus élevée dans la partie supérieure que dans la partie inférieure, de même\navec l’onde rouge à 13, 15 et 17 ms. Les ondes de cisaillement se propagent donc à des\nvitesses différentes dans la partie supérieure et dans la partie inférieure, respectivement\nà 5,1\u00060,1 et 2,5\u00060,1 m.s\u00001.\n122 Pol Grasland-Mongrain4.5. Expériences d’élastographie par force de Lorentz\n4.5.3 Expérience sur un morceau de foie\nMatériel et méthodes Un foie congelé de porc abattu dans un bloc opératoire a été ré-\ncupéré. Ce foie a été dégelé dans une solution physiologique, composée d’eau distillée\net de chlorure de sodium à 9% pour avoir une conductivité électrique proche de celle\ndes tissus mous. Un morceau de foie de 3x3x3 cm3a ensuite été découpé au scalpel\net posé au-dessus de deux électrodes, avec la barrette ultrasonore sur le côté, comme\nreprésenté sur le schéma 4.22-(a). Les configurations présentées dans les expériences\nprécédentes ne pouvaient en effet pas être utilisées, car l’échantillon de foie avait une\natténuation beaucoup plus forte, donc on ne pouvait pas vraiment faire d’images de\nprofondeur supérieure à 2 cm, comme on peut le voir sur l’échographie figure 4.22-(b).\nCette configuration permettait donc d’observer la propagation d’une onde de cisaille-\nment dans une direction Y.\nGénérateur\nd'impulsionLignes de courant\nélectriqueDéplacement\nEchographe\nultrarapide\nChamp magnétique uniformeX\nYZ\nPosition Y (mm)\n5 10 15 20−10\n0\n10\nPosition Z (mm)Echographie\nFIGURE 4.22: (a) Schéma de l’expérience d’élastographie sur un échantillon de foie –\nDeux électrodes proches créent un courant électrique dans un échantillon de foie sou-\nmis à un champ magnétique, et les déplacements sont observés par une sonde ultra-\nsonore en mode ultrarapide placé sur le côté. (b) Echographie obtenue par la sonde\nultrasonore – On observe de nombreux points clairs et une atténuation importante.\nRésultats et discussions La figure 4.23 représente des champs de déplacements à 0,\n10, 20, 30, 40 et 50 millisecondes après injection du courant. Un bruit plus important\nest visible dans cet échantillon par rapport aux fantômes. On observe néanmoins la\npropagation d’ondes de cisaillement vers le haut, à une vitesse de 0,5 \u00060,1 m.s\u00001. Cela\ncorrespont à un module d’Young E= 800\u0006300 Pa, ce qui confirme que l’échantillon\nétait plutôt mou.\n4.5.4 Bilan partiel\nCes expériences ont montré que les ondes de cisaillement induites par effet Joule\npeuvent être réalisées pour des expériences d’élastographie. Le courant injecté devra\nApplications de la force de Lorentz en acoustique médicale 123Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 20 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 40 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 10 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 30 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nt = 50 ms\nPosition Z (mm)Position Y (mm)\n5 10 15 20−10\n−5\n0\n5\n10µm\n−0.1−0.0500.050.1\nFIGURE 4.23: Résultats de l’expérience d’élastographie sur un échantillon de foie –\nChamp des déplacements dus à un courant électrique injecté par deux électrodes sous\nun échantillon de foie placé dans un champ magnétique à 0, 10, 20, 30, 40 et 50 ms\n(plan YZ). Le bruit est plus important que dans les fantômes. On observe une onde de\ncisaillement se propageant à 0,5 \u00060,1 m.s\u00001, ce qui correspond à un module d’Young\nEde 800\u0006300 kPa.\ncependant être diminué d’un facteur cent environ avant de pouvoir tester l’expérience\nsur des animaux.\n4.6 Expériences d’imagerie de conductivité électrique\nSi l’on a un champ magnétique et un champ électrique uniformes, seules les inho-\nmogénéités de conductivité électrique peuvent donner lieu à des ondes de cisaillement,\nsi l’on ne prend pas en compte les conversions de modes par effets de bord. En s’inté-\nressant à la source des déplacements plutôt qu’à leur propagation, on peut obtenir des\ncartes de conductivité électrique des tissus. Des expériences ont donc été réalisées pour\nvoir comment l’on pourrait détecter des variations de conductivité électrique avec ce\nprincipe.\n124 Pol Grasland-Mongrain4.6. Expériences d’imagerie de conductivité électrique\n4.6.1 Expérience de localisation d’inhomogénéité de conductivité\nélectrique\nNous avons d’abord voulu détecter une inhomogénéité de conductivité électrique\nforte, en plaçant un fil électrique au milieu du fantôme.\nMatériel et méthodes Une expérience, représentée figure 4.24-(a), a été faite avec\ndeux électrodes en contact respectivement avec la partie supérieure et inférieure de\nl’échantillon. L’échantillon était un fantôme composé de 5% de PVA, 0,1 % de graphite\net de 5% de sel, ainsi qu’un fil électrique de 100 µm de diamètre dans la direction Y.\nCe fil a été placé pendant la solidification du fantôme afin d’éviter de percer ce dernier\npar introduction du fil. Ce fil est environ cent mille fois plus conducteur que le reste\ndu fantôme, afin d’avoir une inhomogénéité marquée. L’image acquise par la barrette\nultrasonore était cette fois-ci dans le plan XZ, de façon à être perpendiculaire au fil\nélectrique. L’échographie du milieu est représentée figure 4.24-(b).\nGénérateur\nd'impulsionLignes de courant\nélectrique\nDéplacementEchographe\nultrarapide\nChamp magnétique uniformeX\nYZ\nFil électrique\nconducteur\n10 20 30 40 50 60 70−10\n0\n10Position Y (mm)\nPosition Z (mm)Echographie\nFIGURE 4.24: (a) Schéma de l’expérience sur la localisation d’inhomogénéité de\nconductivité électrique – Deux électrodes proches créent un courant électrique dans\nun fantôme soumis à un champ magnétique et contenant un fil électrique, et les dépla-\ncements sont observés par une sonde ultrasonore en mode ultrarapide. (b) Echographie\nobtenue par la sonde ultrasonore – L’échographie a été prise dans le plan XZ. On dis-\ntingue au milieu un point blanc correspondant au fil électrique et l’interface du fond\nsur la droite.\nRésultats et discussions La figure 4.25 représente les champs de déplacements à 0,\n2, 5, 7, 10, 12, 15 et 17 ms. Les champs de déplacements dans un plan perpendicu-\nlaire (plan XZ) aux mêmes temps ont également été représentés, sur la figure 4.26. On\npeut voir dans les deux cas que des ondes de cisaillement sont créées au niveau du fil,\ndonc que l’on peut trouver grâce à des ondes de cisaillement des inhomogénéités de\nconductivité électrique.Les inhomogénéités dans un tissu biologique seraient bien sûr\nmoins contrastées que dans cette expérience, et des développements en terme de qualité\nd’image sont à prévoir. Nous voyons cependant que l’inhomogénéité est particulière-\nApplications de la force de Lorentz en acoustique médicale 125Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nment détectable au début de l’injection du courant, avec un mouvement d’amplitude\nplus forte.\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 10 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 2 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 12 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 5 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 15 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 7 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 17 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nFIGURE 4.25: Résultats d’expérience sur la localisation d’inhomogénéité de conduc-\ntivité électrique – Champ des déplacements due à la traversée d’un courant électrique\ndans un fantôme placé dans un champ magnétique à 0, 2, 5, 7, 10, 12, 15 et 17 ms après\nle début du passage du courant. On observe une onde de cisaillement créée au niveau\ndu fil électrique se propageant dans toutes les directions.\n4.6.2 Expérience dans un fantôme avec deux conductivités électriques\ndifférentes\nLa force de Lorentz dépend de la densité de courant, donc le déplacement résultant\naussi. La densité de courant électrique dépend, d’après la loi d’Ohm, de la conducti-\nvité électrique et du champ électrique. Une expérience a donc été menée pour évaluer\nl’influence de la conductivité électrique sur l’amplitude du déplacement.\nMatériel et méthodes Une expérience, représentée figure 4.27-(a) a été faite avec deux\nélectrodes en contact respectivement avec la partie supérieure et inférieure de l’échan-\ntillon, le long de la tranche. L’échantillon est composé d’une couche de deux morceaux\n126 Pol Grasland-Mongrain4.6. Expériences d’imagerie de conductivité électrique\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 10 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 2 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 12 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 5 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 15 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 7 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nt = 17 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−2−1012\nFIGURE 4.26: Résultats d’expérience sur la localisation d’inhomogénéité de conduc-\ntivité électrique – Champ des déplacements due à la traversée d’un courant électrique\ndans un fantôme placé dans un champ magnétique à 0, 2, 5, 7, 10, 12, 15 et 17 ms après\nle début du passage du courant (plan XZ). On observe une onde de cisaillement créée\nau niveau du fil électrique se propageant dans toutes les directions.\nde gélatine de 3 cm d’épaisseur, composée de 5% de PVA, 0,1% de graphite et l’un\nà 0,05% de sel (salinité de l’eau du robinet), l’autre à 5% de sel. Cela donne pour la\ncouche supérieure une conductivité électrique de 0,1 S/m et celle inférieure de 10 S/m.\nLe fantôme a été fait en une fois, en commençant par couler la partie inférieure, et en\nattendant la gélification de celle-ci pour couler la partie supérieure. Cela permet d’évi-\nter une interface d’air entre les deux, tout en ayant une élasticité identique des deux\ncôtés. L’échographie du milieu est représentée figure 4.27-(b). La séparation entre les\ndeux couches est à peine visible (ligne horizontale au centre).\nApplications de la force de Lorentz en acoustique médicale 127Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\nGénérateur\nd'impulsionLignes de courant\nélectriqueDéplacement\nEchographe\nultrarapide\nChamp magnétique uniformeX\nYZ\nσ = 10 S/mσ = 0,1 S/m\n10 20 30 40 50 60 70−10\n0\n10Position Y (mm)\nPosition Z (mm)Echographie\nFIGURE 4.27: (a) Schéma de l’expérience dans un fantôme avec deux conductivités\nélectriques différentes – Deux électrodes proches créent un courant électrique dans un\nfantôme soumis à un champ magnétique, et les déplacements sont observés par une\nsonde ultrasonore en mode ultrarapide. Le fantôme présente deux couches de conduc-\ntivité électrique différentes, une plus isolante sur le dessus et une plus conductrice\nen-dessous. (b) Echographie obtenue par la sonde ultrasonore – La ligne plus claire\nhorizontale au centre de l’échographie correspondant à la séparation entre les deux\ncouches, mais est à peine visible.\nRésultats et discussions Les champs de déplacement à 0, 2, 4, 6, 8 et 10 millisecondes\naprès injection du courant sont représentées sur la figure 4.28. La première image n’in-\ndique aucun déplacement. Celles à 2 et 4 ms montrent clairement une amplitude de\ndéplacement plus importante dans la partie inférieure que dans la partie supérieure.\nLes images à 8 et 10 ms, prises au moment où le courant passe dans un sens inverse\npar rapport aux deux images précédentes, montrent également une amplitude de dé-\nplacement plus importante dans la partie inférieure. Les images suivantes, dont celle à\n22 ms, sont faites de mouvements plus complexes à interpréter et ne montrent aucune\ndifférence d’amplitude entre les deux parties. D’autre part, on n’observe aucun dépla-\ncement dans la zone proche de la sonde ultrasonore, probablement à cause du contact\nentre celle-ci et le fantôme qui gêne tout déplacement.\n4.6.3 Bilan partiel\nA travers deux expériences, nous avons vu que la conductivité électrique avait une\ninfluence sur les déplacements induits force de Lorentz. On peut ainsi envisager des\nmoyens de faire des images de conductivité électrique grâce à cette méthode.\n4.7 Bilan du chapitre\n4.7.1 Principaux résultats\nLe but de ce chapitre était d’étudier la génération d’ondes de cisaillement en appli-\nquant un courant électrique dans un tissu placé dans un champ magnétique.\n128 Pol Grasland-Mongrain4.7. Bilan du chapitre\nt = 0 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nt = 8 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nt = 2 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nt = 10 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nt = 4 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nt = 12 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nt = 6 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nt = 22 ms\nPosition Z (mm)Position Y (mm)\n  \n10 20 30 40 50 60 70−10\n−5\n0\n5\n10µm\n−1−0.500.51\nFIGURE 4.28: Expérience dans un fantôme avec deux conductivités électriques diffé-\nrentes – Champ des déplacements dus à la traversée d’un courant électrique dans un\nfantôme contenant un fil électrique, le tout placé dans un champ magnétique, à 0, 2, 4, 6,\n8, 10, 12 et 22 ms après le début du passage du courant (plan YZ). On observe une am-\nplitude de déplacement plus élevée dans la zone inférieure, de plus forte conductivité\nélectrique.\nLa partie théorique a permis de voir que la force de Lorentz peut générer des ondes\nde compression et de cisaillement dans un milieu homogène. Pour un champ magné-\ntique uniforme, il faut avoir une variation de la conductivité électrique ou du champ\nélectrique dans la direction du champ magnétique pour générer des ondes. Il est cepen-\ndant difficile de vérifier expérimentalement ces résultats à cause des effets de bords qui\ndonnent lieu aussi à des ondes de cisaillement.\nLes premières expériences se sont d’abord attachées à élaborer un dispositif de dé-\ntection des ondes de cisaillement et de s’assurer que les ondes créées sont bien dues\nà la force de Lorentz. Après plusieurs dispositifs non décrits ici, nous avons choisi un\nApplications de la force de Lorentz en acoustique médicale 129Chapitre 4. Imagerie d’ondes de cisaillement induites par force de Lorentz\néchographe Verasonics fonctionnant en mode ultrarapide. La génération des ondes de\ncisaillement se sont faites par l’injection d’un courant électrique basse fréquence (50-100\nHz) d’amplitude de quelques centaines de milliampères dans un échantillon placé dans\nun champ magnétique créé par un aimant permanent. Cela a permis d’observer des\ndéplacements qui se propagent sous forme d’ondes de cisaillement. Puis, nous avons\nvérifié que le déplacement était bien créé au sein de l’échantillon par la force de Lo-\nrentz et non pas par contact avec les électrodes ou le support, en utilisant un champ\nmagnétique localisé. Enfin, nous avons retiré le champ magnétique pour vérifier que le\nphénomène observé n’est pas dû à l’effet Joule.\nPar la suite, nous avons voulu voir si on peut utiliser ces ondes de cisaillement pour\nréaliser des expériences d’élastographie dynamique, technique basée sur la vitesse de\npropagation d’ondes de cisaillement. Nous avons pu calculer une vitesse de propaga-\ntion dans un fantôme homogène qui donnait une élasticité crédible. Puis, nous avons\npu voir dans un autre fantôme une vitesse de propagation nettement differente entre les\ndeux couches d’élasticités différentes. Enfin, des manipulations sur un organe ex-vivo\nont été réalisées, avec toutefois une profondeur d’exploration moindre.\nD’autre part, la théorie indique que les inhomogénéités de conductivité électrique\ndonnent lieu à des ondes de cisaillement. Nous avons donc regardé s’il était possible\nde donner des informations sur la conductivité électrique des tissus à partir des dépla-\ncements générés par la force de Lorentz. Nous avons pu en particulier localiser de cette\nfaçon une inhomogénéité de conductivité électrique, à savoir un fil électrique conduc-\nteur dans un fantôme. D’autre part, nous avons observé deux amplitudes différentes\ndans un fantôme composé de deux couches de conductivités électriques différentes.\n4.7.2 Avenir de la méthode\nPour l’instant, les deux techniques présentées ne sont pas directement applicable au\ncorps humain. En particulier, l’intensité du courant électrique, de quelques centaines\nde milliampères, est trop élevée par rapport au maximum acceptable dans le corps hu-\nmain, de quelques milliampères. Cependant, la puissance du champ magnétique peut\nêtre facilement accrue d’un facteur dix en utilisant le champ magnétique d’un IRM.\nUne voie à explorer est donc de tester la méthode par IRM. En effet, un champ ma-\ngnétique intense y est déjà présent ; et comme un champ magnétique variable induit un\nchamp électrique, il serait possible d’utiliser les bobines de champs pour induire le cou-\nrant électrique. De plus, de nombreuses techniques d’élastographie par IRM existent,\nce qui permet d’avoir des algorithmes de visualisation de la propagation des ondes de\n130 Pol Grasland-Mongrain4.7. Bilan du chapitre\ncisaillement.\nUne autre piste de développement serait d’induire un courant électrique grâce à\nun champ magnétique variable, comme cela est fait dans la technique de Tomographie\nMagnéto-Acoustique par Induction Magnétique. Le courant électrique risque cepen-\ndant de ne pas être induit en un endroit précis mais dans tout un volume. Par exemple,\non pourrait induire des ondes de cisaillement dans le cerveau de cette façon, d’autant\nplus si le courant électrique est créé par induction magnétique.\nIl est difficile de prévoir aujourd’hui quel est l’aspect le plus intéressant de l’image-\nrie d’ondes de cisaillement générées par force de Lorentz. Nous avons proposé d’uti-\nliser le phénomène pour des techniques d’élastographie dynamique, et ainsi observer\ndes ondes de cisaillement en profondeur – mais les variations de conductivité électrique\ndans les tissus peuvent perturber la localisation exacte de la source.\nL’autre aspect est la détection d’inhomogénéités de conductivité électrique. Les der-\nnières manipulations a montré que l’on peut détecter des inhomogénéités. Plusieurs\npistes se présentent : essayer de localiser la source des ondes de cisaillement par retour-\nnement temporel ; ne regarder que les premières images de déplacements, juste après\ninjection de courant ; augmenter la fréquence de l’impulsion ultrasonore, afin de gar-\nder un déplacement par force de Lorentz mais empêcher toute propagation par onde\nde cisaillement ; envoyer un signal blanc et à partir des mouvements d’amplitude sta-\ntistiquement plus élevées, en déduire les zones les plus conductrices. Cette dernière ap-\nproche profitera certainement des développements de l’élastographie passive au sein\ndu laboratoire qui utilise des techniques de corrélation puissantes.\nApplications de la force de Lorentz en acoustique médicale 131CHAPITRE 5\nConclusion\n5.1 Bilan général\nCette thèse s’est intéressée à plusieurs applications de la force de Lorentz pour\nl’acoustique médicale. L’intérêt de la force de Lorentz est de relier un mouvement mé-\ncanique à un courant électrique via un champ magnétique. Cette force a été appliquée\nen acoustique médicale, qui regroupe l’utilisation d’ondes acoustiques pour l’image-\nrie médicale et la thérapie. Les trois principales applications étudiées sont liées par la\nmême physique, mais ont chacune leurs spécificités propres : la métrologie et la mé-\ncanique des fluides pour la première partie, l’impédance électrique des tissus et les\nméthodes de reconstruction ultrasonore pour la deuxième, ou encore l’élastographie\npour la troisième.\nLa première application envisagée dans cette thèse a été l’étude d’un hydrophone\nultrasonore basé sur la force de Lorentz. Une étude théorique a donné une relation\nentre la vitesse d’un fluide due à une onde acoustique et la vitesse d’un fil placé dans ce\nfluide. A partir de cette relation, une preuve de concept expérimentale a été envisagée.\nPuis, un prototype d’hydrophone a été fabriqué et caractérisé en suivant notamment\nune norme internationale, sur des aspects de résolution spatiale, de réponse fréquen-\ntielle, de directivité, de sensibilité, de résistance à haute pression et d’influence de la\ntaille des fils. Le principal avantage de l’hydrophone réside dans sa résistance à haute\npression, tandis que le point principal à améliorer concerne la réponse fréquentielle\nhaute, limitée pour l’instant à environ 1 MHz.\nLa deuxième application étudiée a été une technique d’imagerie médicale introduite\nApplications de la force de Lorentz en acoustique médicale 133Chapitre 5. Conclusion\nsous le nom d’ “Imagerie par Effet Hall” en 1998, renommée “Tomographie d’Impé-\ndance Electrique par Force de Lorentz” dans cette thèse car le phénomène étudié ne\nrelève pas strictement de l’effet Hall. Cette technique permet d’obtenir une image de\nl’impédance électrique des tissus, qui offre potentiellement un bon contraste entre les\ndifférents types de tissus, les tissus sains ou pathologiques, les tissus chauffés ou non,\netc. Elle utilise des ultrasons focalisés dans un tissu biologique lui-même placé dans\nun champ magnétique. La vibration due aux ultrasons dans un champ magnétique in-\nduit par force de Lorentz un courant électrique. La mesure de ce courant permet d’en\ndéduire les variations de conductivité électrique le long d’une ligne ultrasonore. Dans\ncette thèse, un dispositif expérimental a été élaboré. Après avoir objectivé le signal,\ndes images de conductivité électrique de morceau de gélatine et de muscle animal ont\nété réalisées. Par la suite, nous avons essayé d’observer la formation d’une lésion ther-\nmique grâce au dispositif. Les images obtenues sont de faible résolution spatiale à cause\nde la taille importante du faisceau ultrasonore et de la fréquence utilisée mais montrent\nnéanmoins le potentiel offert par la méthode.\nEnfin, une troisième application a été de générer des ondes de cisaillement par force\nde Lorentz. Dans cette technique, on injecte un courant électrique dans un tissu biolo-\ngique placé dans un champ magnétique. La force de Lorentz induit alors un déplace-\nment, qui se propage à basse fréquence sous la forme d’une onde de cisaillement. Cette\nonde de cisaillement peut alors être utilisée pour réaliser des expériences d’élastogra-\nphie. Le travail ici a consisté à concevoir la technique, réaliser une étude théorique et\nmettre en place un dispositif expérimental. Nous nous sommes tout d’abord assuré que\nle signal que nous détections était bien créé par force de Lorentz. Puis, nous avons fait\nde l’élastographie ultrasonore par le principe, d’abord sur des échantillons de gélatine,\npuis sur des tissus biologiques animaux. Par ailleurs, nous avons remarqué lors des\nexpériences la possibilité d’obtenir des images de conductivité électrique par la mé-\nthode. Des expériences sur des échantillons de gélatine avec inclusion métallique ont\nété réalisé comme preuves de concept.\n5.2 Chronologie\nLe début de cette thèse s’est penché sur la tomographie d’impédance électrique par\nforce de Lorentz, en se basant sur les précédents travaux d’Amalric Montalibet et de\nJacques Jossinet au sein du laboratoire. L’élaboration d’un dispositif fonctionnel a prati-\nquement pris une année, notamment avec l’aide précieuse de Jean-Martial Mari : le cou-\nrant électrique induit est en effet très faible et facilement perturbé par des parasites exté-\n134 Pol Grasland-Mongrain5.2. Chronologie\nrieurs. Les premières mesures étaient faites sur des fantômes de gélatine salée. A cause\nde la faiblesse du courant électrique, ces fantômes étaient souvent remplacés par un fil\nélectrique afin de vérifier les différents éléments de l’expérience (ultrasons, champ ma-\ngnétique, amplification) grâce à sa conductivité électriqué élevée. Nous nous sommes\nalors aperçus que cela constituait un hydrophone rudimentaire. J’ai donc continué, en\nparallèle à la technique d’imagerie, à étudier ce dispositif. Un modèle hydrodynamique\na alors été élaboré avec Bruno Gilles pour relier le courant mesuré à la pression ultraso-\nnore. Nous avons finalement opté pour un design à fil long couplé à une technique de\ntomographie pour assurer une bonne résolution spatiale. Par la suite, nous avons voulu\nréaliser un prototype fonctionnel qui a été caractérisé en suivant les recommandations\nd’une norme IEC, avec l’aide d’un stagiaire, Benjamin Roussel. Enfin, deux stagiaires,\nFayssel Hamimi puis Mickaël Parisi, ont participé à l’élaboration de simulations avec\nle logiciel Comsol pour mieux comprendre les phénomènes se produisant autour du\nfil électrique – certaines expériences donnant des résultats non prévus par notre mo-\ndèle hydrodynamique. Cette partie consacrée à l’hydrophone a représenté la majorité\nde mes communications scientifiques, avec des présentations à cinq congrès interna-\ntionaux dont deux proceedings et deux publications, dans les revues “Applied Physics\nLetters” et “IEEE Transactions in Ultrasonics, Ferroelectrics and Frequency Controls”\n(actuellement en cours de révision).\nLes résultats sur l’imagerie d’impédance électrique ne donnaient cependant pas des\nrésultats aussi bons qu’espérés, et il m’a été difficile de présenter des résultats particu-\nlièrement meilleurs que ceux déjà publiés. Les images résultent d’un compromis entre\nplusieurs facteurs : une hausse de la fréquence ultrasonore améliore la résolution spa-\ntiale mais diminue l’amplitude des signaux, et il a donc fallu se contenter d’une ré-\nsolution spatiale de l’ordre du centimètre. Ces expériences ont néanmoins mené à un\narticle publié dans “Innovation and Research in BioMedical engineering”. Nous avons\nun peu étudié la partie théorique avec un stagiaire, Alexandre Petit, sans que les résul-\ntats soient confirmés par les expériences. En parallèle, un chercheur et deux doctorants\ndu département de Mathématiques Appliqués de l’ENS, Habib Ammari, Laurent Sep-\npecher et Pierre Millien, se sont intéressés à une modélisation mathématique du phéno-\nmène. Cela a permis de confronter un modèle théorique de la méthode sur des données\nexpérimentales “réelles”.\nCes résultats présentaient des débouchés moins prometteurs que la méthode en sens\n“inverse”, où l’on crée un mouvement à partir d’un courant électrique et d’un champ\nmagnétique. Cette dernière partie de ma thèse a été initiée avec l’arrivée de Stefan Ca-\ntheline au laboratoire grâce à son expertise en élastographie. Ainsi, au lieu d’utiliser la\nApplications de la force de Lorentz en acoustique médicale 135Chapitre 5. Conclusion\nforce de Lorentz en sens inverse à des fréquences ultrasonores, comme le fait l’équipe de\nBin He depuis plusieurs années, nous avons choisi de se placer à basse fréquence pour\ncréer des ondes de cisaillement. Avec l’aide de Rémi Souchon et d’une stagiaire, Sandra\nMontalescot, nous avons réussi après quelques temps à détecter l’onde de cisaillement\nrecherchée – mais des effets stroboscopiques rendaient les résultats difficilement pré-\nsentables tels quels. Nous avons donc poursuivi le travail avec un transducteur mo-\nnoélément, notamment suite à l’arrivée de Florian Cartellier en stage. Nous pouvions\nalors obtenir des images en déplaçant le transducteur ligne par ligne. Cela prenait du\ntemps, mais on obtenait des films de propagation d’ondes convaincants. L’objectiva-\ntion du signal a été un obstacle, car nous n’étions pas sûrs que les mouvements étaient\nvraiment dus à un déplacement du fantôme et non pas des électrodes. Puis, avec l’ar-\nrivée de l’échographe ultrarapide Verasonics, sur lequel Ali Zorgani et Rémi Souchon\nont beaucoup travaillé, nous avons pu faire des images en mode ultrarapide. Le dispo-\nsitif était alors prêt pour faire une caractérisation avancée. Cela a convaincu l’INSERM\nde déposer un brevet sur ce sujet, et nous avons ensuite communiqué extensivement\nsur ce sujet à la fin de ma thèse : présentation à trois congrès internationaux dont un\nproceeding et une publication (actuellement en cours d’écriture).\n5.3 Communications scientifiques\nArticles dans des journaux à comité de lecture\n•Imaging of Shear Waves Induced by Lorentz Force in Soft Tissues ,\nP Grasland-Mongrain, S Catheline, R Souchon, F Cartellier, A Zorgani, JY Chape-\nlon, C Lafon,\nPhysical Review Letters (soumis)\n•Lorentz Force Hydrophone Characterisation ,\nP Grasland-Mongrain, JM Mari, B Gilles, B Roussel, A Poizat, JY Chapelon, C\nLafon,\nIEEE Ultrasonics, Ferroelectrics and Frequency Control, 2014 (in press)\n•Lorentz Force Electrical Impedance Tomography ,\nP Grasland-Mongrain, JM Mari, JY Chapelon, C Lafon,\nInnovation and Research in BioMedical engineering, 34 (4), pp 357-360, 2013\n•Electromagnetic hydrophone with tomographic system for absolute velocity field map-\nping,\nP Grasland-Mongrain, JM Mari, B Gilles, JY Chapelon, C Lafon,\nApplied Physics Letters, 100 (24), 2012\n136 Pol Grasland-Mongrain5.3. Communications scientifiques\n•Ordered packings of bubbles in columns of square cross-section ,\nS Hutzler, J Barry, P Grasland-Mongrain, D Smyth, D Weaire,\nColloids and Surfaces A : Physicochemical and Engineering Aspects, 344 (1), 2009\n(travaux précédant la thèse)\nBrevets1\n•Shear Wave Imaging Method and Installation for Collecting Information on a Soft Solid ,\nS Catheline, JY Chapelon, P Grasland-Mongrain, C Lafon, R Souchon,\nEurope n° 13305987.3, 11 juillet 2013\nPrésentations à des congrès internationaux2\n•Ultrasound velocity mapping with Lorentz Force Hydrophone ,\nP Grasland-Mongrain , JM Mari, B Gilles, B Roussel, A Poizat, JY Chapelon, C\nLafon,\nAcoustical Society of America Symposium, San Francisco, 1-4 décembre 2013\n•Thermal lesion imaging using Lorentz force : Proofs of concept ,\nP Grasland-Mongrain , S Catheline, R Souchon, F Cartellier, A Zorgani, JY Cha-\npelon, C Lafon,\nAcoustical Society of America Symposium, San Francisco, 1-4 décembre 2013\n•Imaging of Shear Waves Induced by Lorentz Force in Soft Tissues ,\nP Grasland-Mongrain , S Catheline, R Souchon, F Cartellier, A Zorgani, JY Cha-\npelon, C Lafon,\nInternational Tissue Elasticity Conference, Lingfield (UK), 1-4 octobre 2013\n•Imaging of Shear Waves Induced by Lorentz Force - avec proceeding,\nP Grasland-Mongrain , S Catheline, R Souchon, F Cartellier, A Zorgani, S Monta-\nlescot, JY Chapelon, C Lafon,\nIEEE Ultrasonics, Ferroelectrics and Frequency Control, Prague, 21-25 juillet 2013\n•Lorentz Force Hydrophone Prototype Characterization ,\nP Grasland-Mongrain , JM Mari, B Gilles, B Roussel, A Poizat, JY Chapelon, C\nLafon,\nIEEE Ultrasonics, Ferroelectrics and Frequency Control, Prague, 21-25 juillet 2013\n•Electromagnetic hydrophone for high-intensity focused ultrasound (HIFU) measurement\n- avec proceeding,\n1. Inventeurs par ordre alphabétique\n2. En gras : personne ayant fait la présentation\nApplications de la force de Lorentz en acoustique médicale 137Chapitre 5. Conclusion\nP Grasland-Mongrain , JM Mari, B Gilles, JY Chapelon, C Lafon,\nAcoustical Society of America Symposium, Montréal, 2-7 juin 2013\n•Electromagnetic Hydrophone for High Intensity Focused Ultrasound Measurement ,\nP Grasland-Mongrain, JM Mari, B Gilles, JY Chapelon, C Lafon ,\nInternational Symposium on Thérapeutic Ultrasound, Shanghai, 12-15 mai 2013\n(poster)\n•Electromagnetic Hydrophone Characterization ,\nP Grasland-Mongrain, JM Mari, B Gilles, JY Chapelon, C Lafon ,\nInternational Congress on Ultrasonics, Singapour, 2-5 mai 2013\n•Electromagnetic tomographic ultrasonic sensor - avec proceeding,\nP Grasland-Mongrain , JM Mari, B Gilles, JY Chapelon, C Lafon,\nAcoustical Society of America Symposium, Hong-Kong, 13-18 mai 2012\n•Detection of fat tissues with ultrasonically-induced Lorentz force ,\nP Grasland-Mongrain , JM Mari, C Lafon,\nAcoustics, Nantes, 23-27 avril 2012\n•Combination of shape-constrained and inflation deformable models with application to\nthe segmentation of the left atrial appendage - avec proceeding,\nP Grasland-Mongrain , J Peters, O Ecabert,\nIEEE International Symposium on Biomedical Imaging, Rotterdam, 14-17 avril\n2010 (poster) (travaux précédant la thèse)\nPrésentations à des congrès nationaux3\n•Lorentz force prototype to characterize HIFU pressure fields ,\nP Grasland-Mongrain , JM Mari, JY Chapelon, C Lafon,\nColloque Recherche en Imagerie et Technologies pour la Santé, Bordeaux, 8-11\navril 2013 (poster)\n•Détection de gradients de conductivité électrique par ultrasons ,\nP Grasland-Mongrain , JM Mari, JY Chapelon, C Lafon,\nColloque Recherche en Imagerie et Technologies pour la Santé, Bordeaux, 8-11\navril 2013 (poster)\n•Détection de gradients de conductivité électrique par ultrasons ,\nP Grasland-Mongrain , JM Mari, C Lafon,\nJournées Imag’in vivo, Lyon, 11-13 décembre 2012 (poster)\n•Détection de gradients de conductivité électrique par ultrasons ,\nP Grasland-Mongrain , JM Mari, C Lafon,\n3. En gras : personne ayant fait la présentation\n138 Pol Grasland-Mongrain5.3. Communications scientifiques\nJournées Campus Santé, Lyon, 25-26 juin 2012 (poster)\n•Détection de tissus adipeux par force de Lorentz induite par ultrasons ,\nP Grasland-Mongrain , JM Mari, JY Chapelon, C Lafon,\nJournées Signal et Image en Acoustique Médicale, Paris, 1-2 décembre 2011\n•Mesure d’anisotropie de conductivité électrique par interaction acousto-magnétique ,\nP Grasland-Mongrain , JM Mari, C Lafon,\nJournées d’Acoustique Physique, Sous-Marine et Ultrasonore, Lille, 8-10 juin 2011\nSéminaires grand public\n•L’imagerie médicale aujourd’hui... et demain ? ,\nEcole Normale Supérieure de Cachan, 2 décembre 2013\nEcole Normale Supérieure de Lyon, 30 janvier 2013\n•La recherche en imagerie médicale ,\nEcole Normale Supérieure de Cachan, 10 mars 2011, 16 février 2012, 14 février\n2013 (3 séminaires)\nPrix et distinctions\n•2e prix meilleure présentation étudiante , Acoustical Society of America Symposium,\nMontréal, 2-7 juin 2013\n•Prix jeune chercheur , Colloque Recherche en Imagerie et Technologies pour la Santé,\nBordeaux, 8-11 avril 2013\n•Prix du meilleur poster , Journées Imag’in vivo, Lyon, 11-13 décembre 2012\n•Prix du meilleur poster , Journées Campus Santé, Lyon, 25-26 juin 2012\nApplications de la force de Lorentz en acoustique médicale 139Annexe : Calcul du déplacement\npour une force ponctuelle\nIl est intéressant de connaître la solution de l’équation de Navier pour une force\nponctuelle, avec f(t)une fonction temporelle quelconque. On suppose pour le moment\nqu’elle est orientée une direction X.\n\u001ad2u\ndt2\u0000(K+4\n3\u0016)grad divu\u0000\u0016rot rot u =f(t)\u000e(r) (1)\nLe calcul de cette équation est donné dans un livre de Aki et Richards [5]. Le prin-\ncipe consiste décomposer le vecteur déplacement comme u=grad (\u001e) +rot( ), et on\ncherche les potentiels \u001eet associés. On commence par effectuer la même décompo-\nsition pour le terme source, afin d’obtenir les potentiels dits de Helmholtz correspon-\ndants. On peut montrer que l’on a la décomposition suivante où l’on note r=krkla\nnorme du vecteur r:\nf(t)\u000e(r) =grad (\bf) +rot(\tf) (2)\navec :\n8\n<\n:\bf(r;t) =\u0000kf(t)k\n4\u0019@\n@x\u00001\nr\u0001\n\tf(r;t) =kf(t)k\n4\u0019\u0010\n0;@\n@z\u00001\nr\u0001\n;\u0000@\n@y\u00001\nr\u0001\u0011 (3)\nOn peut alors réécrire (1) comme deux équations portant sur les potentiels \u001eet .\nApplications de la force de Lorentz en acoustique médicale 141Chapitre 5. Conclusion\n8\n<\n:\u001ad2\u001e\ndt2\u0000(K+4\n3\u0016) \u0001\u001e=\u0000kf(t)k\n4\u0019@\n@x\u00001\nr\u0001\n\u001ad2 \ndt2\u0000\u0016\u0001 =kf(t)k\n4\u0019\u0010\n0;@\n@z\u00001\nr\u0001\n;\u0000@\n@y\u00001\nr\u0001\u0011 (4)\nL’étape suivante consiste à résoudre (4). On obtient alors les potentiels suivants, où\nl’on a posé vp=r\nK+4\n3\u0016\n\u001aetvs=q\n\u0016\n\u001a.\n8\n>>><\n>>>:\u001e(r;t) =\u00001\n4\u0019\u001a@\n@x\u00121\nr\u0013Zr\n\u000b\n0\u001cf(t\u0000\u001c) d\u001c\n (r;t) =\u00001\n4\u0019\u001a\u0012\n0;@\n@z\u00121\nr\u0013\n;\u0000@\n@y\u00121\nr\u0013\u0013Zr\n\f\n0\u001cF(t\u0000\u001c) d\u001c(5)\nOn passe ensuite à des directions quelconques i;jà la place de X, Y et Z. 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International Society for Optics and Photonics, 2004.\nApplications de la force de Lorentz en acoustique médicale 153Credits images\n• Image 1.3 : Sonde Ablatherm www.edap-tms.com - reproduction autorisée par\nles auteurs\n• Image 1.4 : commons.wikimedia.org/wiki/File :Medecine_Echographie.jpg - li-\ncence libre\n• Image 2.2 : manuel Fiber Optic Probe Hydrophone 2000, RP Acoustics - repro-\nduction autorisée par l’auteur\n• Image 2.3 : en.wikipedia.org/wiki/File :Schlieren_HIFU_1MHz.JPG - domaine\npublic\n• Image 2.5-(a) : runeberg.org/nfcr/0059.html - domaine public\n• Image 2.5-(b) : en.wikipedia.org/wiki/File :Rca44.png, LuckyLouie 23 :22, 4 No-\nvember 2007 (UTC) - licence Creative Commons (CC-BY-SA)\n• Image 2.6 : reproduction autorisée par les auteurs\n• Image 3.1 : reproduction personnelle\n• Image 3.3 : reproduction personnelle\n• Image 3.4-(b) : eprints.ma.man.ac.uk/1053/01/trieste.pdf - reproduction autori-\nsée par les auteurs\n• Image 3.5 : reproduction autorisée par les auteurs\n• Image 3.8 : reproduction autorisée par les auteurs\n• Image 4.4 : reproduction personnelleAbstract\nThe ability of the Lorentz force to link a mechanical displacement to an electrical\ncurrent presents a strong interest for medical acoustics, and three applications were\nstudied in this thesis.\nIn the first part of this work, a hydrophone was developed for mapping the particle\nvelocity of an acoustic field. This hydrophone was constructed using a thin copper wire\nand an external magnetic field. A model was elaborated to determine the relationship\nbetween the acoustic pressure and the measured electrical current, which is induced by\nLorentz force when the wire vibrates in the acoustic field of an ultrasound transducer.\nThe built prototype was characterized and its spatial resolution, frequency response,\nsensitivity, robustness and directivity response were investigated.\nAn imaging method called Lorentz Force Electrical Impedance Tomography was\nalso studied. In this method, a biological tissue is vibrated by ultrasound in a magnetic\nfield, which induces an electrical current by Lorentz force. The electrical impedance of\nthe tissue can be deduced from the measurement of the current. This technique was\napplied for imaging a gelatin phantom, a beef muscle sample, and a thermal lesion in a\nchicken breast sample. This showed the method may be useful for providing additional\ncontrast to conventional ultrasound imaging.\nFinally, this thesis demonstrated that shear waves can be generated in soft tissues\nusing Lorentz force. This work was performed by applying an electrical current with\ntwo electrodes in a soft solid placed in a magnetic field. Shear waves were observed in\ngelatin phantom and liver sample. The speed of the shear waves were used to compute\nelasticity and their source to map the electrical conductivity of the samples.Résumé\nLa capacité de la force de Lorentz à relier un déplacement mécanique à un courant\nélectrique présente un intérêt certain pour l’acoustique médicale, et trois applications\nont été étudiées dans cette thèse.\nDans la première partie de ce travail, un hydrophone a été développé pour effectuer\ndes champs de vitesse acoustique. Cet hydrophone était constitué d’un fil de cuivre\nvibrant dans un champ magnétique. Un modèle a été élaboré pour déterminer une\nrelation entre la pression acoustique et le courant électrique mesuré, qui est induit par\nforce de Lorentz lorsque le fil vibre dans un champ acoustique. Un prototype a ensuite\nété conçu et sa résolution spatiale, sa réponse fréquentielle, sa sensibilité, sa résistance\net sa réponse directionnelle ont été examinées.\nUne méthode d’imagerie appelée Tomographie d’Impédance Electrique par Force\nde Lorentz a aussi été étudiée. Dans cette méthode, un tissu biologique est déplacé par\nultrasons dans un champ magnétique, ce qui induit un courant électrique par force de\nLorentz. L’impédance électrique du tissu peut ensuite être déduite de la mesure du cou-\nrant. Cette technique a été appliquée pour réaliser des images d’un fantôme de gélatine,\nd’un muscle de bœuf, et d’une lésion thermique dans un échantillon de poulet. Cela a\nmontré que la méthode est potentiellement utile pour fournir un contraste supplémen-\ntaire à des images ultrasonores classiques.\nEnfin, cette thèse a démontré que des ondes de cisaillement peuvent être générées\ndans des tissus mous par force de Lorentz. Cela a été réalisé en appliquant un courant\nélectrique par deux électrodes dans un solide mou placé dans un champ magnétique.\nDes ondes de cisaillement ont été observées dans des échantillons de gélatine et de foie.\nLa vitesse des ondes de cisaillement a été utilisée pour calculer l’élasticité et leur source\npour cartographier la conductivité électrique des échantillons."    },    {        "title": "1307.3969v1.Classification_of_minimal_Lorentz_surfaces_in_indefinite_space_forms_with_arbitrary_codimension_and_arbitrary_index.pdf",        "content": "arXiv:1307.3969v1  [math.DG]  15 Jul 2013CLASSIFICATION OF MINIMAL LORENTZ SURFACES IN\nINDEFINITE SPACE FORMS WITH ARBITRARY\nCODIMENSION AND ARBITRARY INDEX\nBANG-YEN CHEN\nAbstract. Since J. L. Lagrange initiated in [18] the study of minimal\nsurfaces of Euclidean 3-space in 1760, minimal surfaces in r eal space\nforms have been studied extensively by many mathematicians during\nthe last two and half centuries. In contrast, so far very few r esults on\nminimal Lorentz surfaces in indefinite space forms are known . Hence, in\nthis paperwe investigate minimal Lorentz surfaces inarbit rary indefinite\nspace forms. As a consequence, we obtain several classificat ion results\nfor minimal Lorentz surfaces in indefinite space forms. In pa rticular, we\ncompletely classify all minimal Lorentz surfaces in a pseud o-Euclidean\nspaceEm\nswith arbitrary dimension mand arbitrary index s.\n1.Introduction.\nLetEm\nsdenote the pseudo-Euclidean m-space with the canonical metric\nof indexsgiven by\ng0=−/summationdisplays\ni=1dx2\ni+/summationdisplaym\nj=s+1dx2\nj, (1.1)\nwhere (x1,...,x m) is a rectangular coordinate system of Em\ns. Thelight cone\nLCofEm+1\nsis defined by LC={x∈Em+1\ns:/an}b∇acketle{tx,x/an}b∇acket∇i}ht= 0}.\nWe put\nSk\ns(c) ={x∈Ek+1\ns|/an}b∇acketle{tx,x/an}b∇acket∇i}ht=c−1>0}, (1.2)\nHk\ns(−c) ={x∈Ek+1\ns+1|/an}b∇acketle{tx,x/an}b∇acket∇i}ht=−c−1<0}, (1.3)\nwhere/an}b∇acketle{t,/an}b∇acket∇i}htdenotes the indefinite inner product on Ek+1\nt. TheSk\ns(c) and\nHk\ns(−c) are complete pseudo-Riemannian manifolds with index sand of\nconstant curvature cand−c, respectively.\n2000Mathematics Subject Classification. Primary: 53C40; Secondary 53C50.\nKey words and phrases. Minimal surface, Lorentz surface, pseudo-Euclidean space ,\npseudo sphere, pseudo-hyperbolic space.\n12 B.-Y. CHEN\nTheSk\ns(c) andHk\ns(−c) are called pseudok-sphereandpseudo hyperbolic\nk-space, respectively. The pseudo-Riemannian manifolds Ek\ns,Sk\nsandHk\ns\nare known as the indefinite space forms . In particular, Ek\n1,Sk\n1andHk\n1are\ncalled Minkowski, de Sitter and anti-de Sitter spacetimes, which play very\nimportant roles in relativity theory.\nThe history of minimal surfaces goes back to J. L. Lagrange (1 736-1813)\nwho initiated in 1760 the study of minimal surfaces in Euclid ean 3-space\n(see [18]). Since then minimal surfaces have attracted many mathematician.\nIn particular, minimal surfaces in real space forms have bee n studied very\nextensively during the last two and half centuries (see, [4, pages 207–249]\nand [21, 23] for details).\nIn [24, 25], L. Verstraelen and M. Pieters studied some famil ies of Lorentz\nsurfaces in 4-dimensional indefinite space forms with index 2. Recently,\nparallelLorentzsurfacesinindefinitespaceformswitharb itrarycodimension\nand arbitrary index were completely classified in a series of articles [8]-[14]\n(see also [1, 15, 16, 19]). Moreover, Lorentz surfaces with p arallel mean\ncurvature vector in an arbitrary pseudo-Euclidean space we re classified in\n[7] (see also [17]). Further, minimal Lorentz surfaces in Lo rentzian complex\nspace forms ˜M2\n1(c) with complex index one were investigated in [5, 6].\nIn this paper, we study minimal Lorentz surfaces in indefinit e space forms\nRm\ns(c) with arbitrary codimension and arbitrary index s. In parti cular,\nwe completely classify minimal Lorentz surfaces in an arbit rary pseudo-\nEuclidean space in section 4. In section 5, we classify minim al Lorentz\nsurfaces of constant curvature one in an arbitrary pseudo m-sphereSm\ns(1).\nThe classification of minimal Lorentz surfaces of constant c urvature −1 in\na pseudo-hyperbolic m-spaceHm\ns(−1) are obtained in section 6. In the last\ntwo sections, we provide many explicit examples of minimal L orentz surfaces\ninSm\ns(1) and inHm\ns(−1).\n2.Basics formulas, equations and definitions.\nLetRm\ns(c) beanm-dimensionalindefinitespaceformofconstant sectional\ncurvaturecand with index s. The curvature tensor of Rm\ns(c) is given by\n˜R(X,Y)Z=c{/an}b∇acketle{tY,Z/an}b∇acket∇i}htX−/an}b∇acketle{tX,Z/an}b∇acket∇i}htY}. (2.1)\nLetψ:M2\n1→Rm\ns(c) be an isometric immersion of a Lorentz surface M2\n1\nintoRm\ns(c). Denote by ∇and˜∇the Levi-Civita connections on M2\n1andMINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 3\n˜Rm\ns(c), respectively. Let X,Ybe vector fields tangent to M2\n1andξnormal\ntoM2\n1inRm\ns(c). The formulas of Gauss and Weingarten are given by (cf.\n[2, 3, 22]):\n˜∇XY=∇XY+h(X,Y), (2.2)\n˜∇Xξ=−AξX+DXξ. (2.3)\nThese formulas define h,AandD, which are called the second fundamental\nform, the shape operator and the normal connection, respect ively.\nFor each normal vector ξ∈T⊥\nxM2\n1, the shape operator Aξatξis a\nsymmetric endomorphism of the tangent space TxM2\n1,x∈M2\n1. The shape\noperator and the second fundamental form are related by\n/an}b∇acketle{th(X,Y),ξ/an}b∇acket∇i}ht=/an}b∇acketle{tAξX,Y/an}b∇acket∇i}ht. (2.4)\nThe mean curvature vector HofM2\n1inRm\ns(c) is defined by\nH=1\n2traceh. (2.5)\nA Lorentz surface in an indefinite space form is called totally geodesic if its\nsecond fundamental form vanishes identically. It is called minimal if its\nmean curvature vector vanishes identically.\nThe equations of Gauss, Codazzi and Ricci are given respecti vely by\nR(X,Y)Z=c{/an}b∇acketle{tY,Z/an}b∇acket∇i}htX−/an}b∇acketle{tX,Z/an}b∇acket∇i}htY}+Ah(Y,Z)X−Ah(X,Z)Y, (2.6)\n(∇Xh)(Y,Z) = (∇Yh)(X,Z), (2.7)\n/angbracketleftbig\nRD(X,Y)ξ,η/angbracketrightbig\n=/an}b∇acketle{t[Aξ,Aη]X,Y/an}b∇acket∇i}ht, (2.8)\nfor vector fields X,Y,Ztangent toM2\n1,ξnormal toM2\n1, where∇his defined\nby\n(2.9) ( ∇Xh)(Y,Z) =DXh(Y,Z)−h(∇XY,Z)−h(Y,∇XZ),\nandRDis the curvature tensor associated to the normal connection D, i.e.,\nRD(X,Y)ξ=DXDYξ−DYDXξ−D[X,Y]ξ. (2.10)\nA vectorvinRm\ns(c) is called spacelike (respectively, timelike, orlight-\nlike) if/an}b∇acketle{tv,v/an}b∇acket∇i}ht>0 (respectively, /an}b∇acketle{tv,v/an}b∇acket∇i}ht<0, or/an}b∇acketle{tv,v/an}b∇acket∇i}ht= 0 andv/ne}ationslash= 0). A curve\nz(x) inRm\ns(c) is called spacelike (respectively, timelike or null) if it s velocity\nvectorz′(x) is spacelike (respectively, timelike or lightlike) at eac h point.4 B.-Y. CHEN\n3.A special coordinate system on a Lorentz surface.\nLetM2\n1be a Lorentz surface. We may choose a local coordinate system\n{x,y}onM2\n1such that the metric tensor is given by\ng=−E2(x,y)(dx⊗dy+dy⊗dx) (3.1)\nfor some positive function E.\nThe Levi-Civita connection of gsatisfies\n∇∂\n∂x∂\n∂x=2Ex\nE∂\n∂x,∇∂\n∂x∂\n∂y= 0,∇∂\n∂y∂\n∂y=2Ey\nE∂\n∂y(3.2)\nand the Gaussian curvature Kis given by\nK=2EExy−2ExEy\nE4. (3.3)\nIf we put\ne1=1\nE∂\n∂x, e2=1\nE∂\n∂y, (3.4)\nthen{e1,e2}forms a pseudo-orthonormal frame satisfying\n/an}b∇acketle{te1,e1/an}b∇acket∇i}ht=/an}b∇acketle{te2,e2/an}b∇acket∇i}ht= 0,/an}b∇acketle{te1,e2/an}b∇acket∇i}ht=−1. (3.5)\nWe define the connection 1-form ωby the following equations:\n∇Xe1=ω(X)e1,∇Xe2=−ω(X)e2. (3.6)\nFrom (3.2) and (3.4) we find\n(3.7)∇e1e1=Ex\nE2e1,∇e2e1=−Ey\nE2e1,\n∇e1e2=−Ex\nE2e2,∇e2e2=Ey\nE2e2.\nBy comparing (3.6) and (3.7), we get\nω(e1) =Ex\nE2, ω(e2) =−Ey\nE2. (3.8)\nLetψ:M2\n1→Rm\ns(c) be an isometric immersion of M2\n1intoRm\ns(c). Then\nit follows from (2.5) and (3.5) that the mean curvature vecto r ofM2\n1is given\nby\nH=−h(e1,e2). (3.9)\nTherefore, M2\n1is a minimal surface of Rm\ns(c) if and only if h(e1,e2) = 0\nholds identically.MINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 5\n4.Minimal Lorentz surfaces in Em\ns.\nIn this section, we completely classify minimal Lorentz sur face in an arbi-\ntrary pseudo-Euclidean m-spaceEm\ns. More precisely, we prove the following.\nTheorem 4.1. A Lorentz surface in a pseudo-Euclidean m-space Em\nsis\nminimal if and only if locally the immersion takes the form L(x,y) =z(x)+\nw(y), wherezandware null curves satisfying /an}b∇acketle{tz′(x),w′(y)/an}b∇acket∇i}ht /ne}ationslash= 0.\nProof.LetL:M2\n1→Em\nsbe an isometric immersion of a Lorentz surface\nM2\n1into a pseudo-Euclidean m-space Em\nswith index s≥1. We choose a\nlocal coordinate system {x,y}onM2\n1satisfying\ng=−E2(x,y)(dx⊗dy+dy⊗dx) (4.1)\nThen we have (3.2)-(3.9).\nIfM2\n1is a minimal surface, it follows from (3.9) that h(e1,e2) = 0 holds.\nHence, we may put\nh(e1,e1) =ξ, h(e1,e2) = 0, h(e2,e2) =η (4.2)\nfor some normal vector fields ξ,η. After applying (2.2), (3.2), (3.4), and\n(4.2), we obtain\n(4.3)Lxx=2Ex\nELx+E2ξ, Lxy= 0, Lyy=2Ey\nELy+E2η.\nAfter solving the second equation in (4.3), we find\nL(x,y) =z(x)+w(y) (4.4)\nfor some vector-valued functions z(x),w(y). Thus, by applying (3.1) and\n(4.4), we obtain /an}b∇acketle{tz′,z′/an}b∇acket∇i}ht=/an}b∇acketle{tw′,w′/an}b∇acket∇i}ht= 0,and/an}b∇acketle{tz′,w′/an}b∇acket∇i}ht=−E2.Therefore,zand\nware null curves satisfying /an}b∇acketle{tz′,w′/an}b∇acket∇i}ht /ne}ationslash= 0.\nConversely, if L:M2\n1→Em\nsis an immersion of a Lorentz surface M2\n1into\nE4\nssuch thatL=z(x)+w(y) for somenull curves z,wsatisfying /an}b∇acketle{tz′,w′/an}b∇acket∇i}ht /ne}ationslash= 0,\nthen we obtain /an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0,/an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht /ne}ationslash= 0,andLxy= 0.Thus,\nM2\n1is surface with induced metric given by g=F(x,y)(dx⊗dy+dy⊗dx)\nfor some nonzero function F. Moreover, it follows from (3.9) and Lxy= 0\nthatLis a minimal immersion. /square\nRemark 4.1.Flat minimal Lorentz surfaces in the Lorentzian complex pla ne\nC2\n1have been completely classified in [5]. Moreover, if m= 3, this theorem\nis due to [20, Theorem 3.5].6 B.-Y. CHEN\nIn particular, if M2\n1is a flat Lorentz surface, we have the following.\nCorollary 4.1. A flat Lorentz surface in a pseudo-Euclidean m-space Em\ns\nis minimal if and only if locally the immersion takes the form\nL(x,y) =z(x)+w(y), (4.5)\nwherezandware null curves satisfying /an}b∇acketle{tz′,w′/an}b∇acket∇i}ht=constant /ne}ationslash= 0.\nProof.LetL:M2\n1→Em\nsbe an isometric immersion of a flat Lorentz surface\nM2\n1into a pseudo-Euclidean m-space Em\ns. Then we may choose a local\ncoordinate system {x,y}onM2\n1satisfying\ng=−(dx⊗dy+dy⊗dx) (4.6)\nThen we find from (4.3) that the immersion Lsatisfies\nLxx=ξ, Lxy= 0, Lyy=η (4.7)\nfor some normal vector fields ξ,η. After solving the second equation in (4.7)\nwe find\nL(x,y) =z(x)+w(y) (4.8)\nfor some vector functions z,w. Thus, by applying (4.6), we find /an}b∇acketle{tz′,z′/an}b∇acket∇i}ht=\n/an}b∇acketle{tw′,w′/an}b∇acket∇i}ht= 0 and /an}b∇acketle{tz′,w′/an}b∇acket∇i}ht=−1. Consequently, zandware null curves\nsatisfying /an}b∇acketle{tz′,w′/an}b∇acket∇i}ht=constant /ne}ationslash= 0.\nConversely, consider a map Ldefined byL(x,y) =z(x)+w(y), wherez\nandware null curves satisfying /an}b∇acketle{tz′,w′/an}b∇acket∇i}ht=constant /ne}ationslash= 0 . Then we have\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0,/an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=constant /ne}ationslash= 0.\nThus, with respect to the induced metric, (4.5) defines an iso metric im-\nmersion of a flat Lorentz surface M2\n1intoEm\ns. The remaining follows from\nTheorem 4.1. /square\n5.Minimal Lorentz surfaces in Sm\ns(1).\nLetψ:M2\n1→Sm\ns(1) be an isometric immersion of a Lorentz surface into\nSm\ns(1). Denote by L=ι◦ψ:M2\n1→Em+1\nsthe composition of ψand the\ninclusionι:Sm\ns(1)⊂Em+1\nsvia (1.2).\nObviously, every totally geodesic Lorentz surface in an ind efinite space\nformRm\ns(c) is of constant curvature c. A natural question is the following:MINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 7\nQuestion. Besides totally geodesic ones how many minimal Lorentz sur-\nfaces of constant curvature cinRm\ns(c)are there?\nTheorem 5.1 of [5] provides the answer to this basic question forc= 0.\nIn this section, we give an answer to this question for c >0. More\nprecisely, we classify all minimal Lorentz surfaces of cons tant curvature one\nin the pseudo-sphere Sm\ns(1) with arbitrary mand arbitrary index s.\nTheorem 5.1. LetM2\n1be a Lorentz surface of constant curvature one. Then\nan isometric immersion ψ:M2\n1→Sm\ns(1)is minimal if and only if one of\nthe following three cases occurs:\n(a)M2\n1is an open portion of a totally geodesic S2\n1(1)⊂Sm\ns(1).\n(b)The immersion L=ι◦ψ:M2\n1→Sm\ns(1)⊂Em+1\nsis locally given by\nL(x,y) =z(x)\nx+y−z′(x)\n2, (5.1)\nwherez(x)is a spacelike curve with constant speed 2lying in the light cone\nLCsatisfying /an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 0andz′′′/ne}ationslash= 0.\n(c)The immersion L=ι◦ψ:M2\n1→Sm\ns(1)⊂Em+1\nsis locally given by\nL(x,y) =z(x)+w(y)\nx+y−z′(x)+w′(y)\n2, (5.2)\nwherezandware curves in Em+1\nssatisfying\n/angbracketleftBigz(x)+w(y)\nx+y−z′(x)+w′(y)\n2,z(x)+w(y)\nx+y−z′(x)+w′(y)\n2/angbracketrightBig\n= 1, (c.1)\n2/angbracketleftbig\nz+w,z′′′/angbracketrightbig\n= (x+y)/angbracketleftbig\nz′+w′,z′′′/angbracketrightbig\n, (c.2)\n2/angbracketleftbig\nz+w,w′′′/angbracketrightbig\n= (x+y)/angbracketleftbig\nz′+w′,w′′′/angbracketrightbig\n. (c.3)\nProof.Assume that ψ:M2\n1→Sm\ns(1) is an isometric immersion of a Lorentz\nsurfaceM2\n1of constant curvature one into Sm\ns(1). IfM2\n1is totally geodesic\ninSm\ns(1), we obtain case (a). Hence, let us assume that M2\n1is non-totally\ngeodesic in Sm\ns(1).\nSinceM2\n1is of constant curvature one, we may choose local coordinate s\n{x,y}such that the metric tensor is given by\ng=−2\n(x+y)2(dx⊗dy+dy⊗dx). (5.3)\nHence the Levi-Civita connection satisfies\n∇∂\n∂x∂\n∂x=−2\nx+y∂\n∂x,∇∂\n∂x∂\n∂y= 0,∇∂\n∂y∂\n∂y=−2\nx+y∂\n∂y. (5.4)8 B.-Y. CHEN\nLet us put\n∂\n∂x=√\n2e1\nx+y,∂\n∂y=√\n2e2\nx+y. (5.5)\nThen we get\n/an}b∇acketle{te1,e1/an}b∇acket∇i}ht=/an}b∇acketle{te2,e2/an}b∇acket∇i}ht= 0,/an}b∇acketle{te1,e2/an}b∇acket∇i}ht=−1. (5.6)\nBecauseM2\n1isminimalin Sm\ns(1), itfollowsfrom(3.9)and(5.3)that h(e1,e2) =\n0. Hence, we may put\nh(e1,e1) =ξ, h(e1,e2) = 0, h(e2,e2) =η (5.7)\nforsomenormal vector fields ξ,η. Without loss of generality, wemay assume\nξ/ne}ationslash= 0. Since K= 1, it follows from the equation (2.6) of Gauss and (5.3)\nthat/an}b∇acketle{tξ,η/an}b∇acket∇i}ht= 0.\nCase(i):η= 0. From formula (2.2) of Gauss, (5.3)-(5.5), and (5.7), we g et\n(5.8)Lxx=2ξ\n(x+y)2−2Lx\nx+y, Lxy=2L\n(x+y)2, Lyy=−2Ly\nx+y.\nAfter solving the last two equations in (5.8) we obtain\nL(x,y) =z(x)\nx+y−z′(x)\n2(5.9)\nfor some Em+1\ns-valued function z(x). Since the metric tensor is given by\n(5.3), one finds\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0 and /an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=−2\n(x+y)2.\nThus, it follows from (5.8), (5.9) and /an}b∇acketle{tL,L/an}b∇acket∇i}ht= 1 thatz(x) satisfies\n/an}b∇acketle{tz,z/an}b∇acket∇i}ht=/angbracketleftbig\nz′′,z′′/angbracketrightbig\n= 0 and/angbracketleftbig\nz′,z′/angbracketrightbig\n= 4.\nMoreover, by substituting (5.9) into the first equation (5.8 ) we find\nξ=−(x+y)2z′′′(x)\n4. (5.10)\nCombining this with ξ/ne}ationslash= 0 givesz′′′(x)/ne}ationslash= 0. Consequently, we obtain case\n(b).\nConversely, suppose that Lis given by (5.1), where z(x) is a spacelike\ncurve with constant speed 2 lying in the light cone LC⊂Em+1\nssatisfying\n/an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 0 andz′′′/ne}ationslash= 0. ThenLsatisfies (5.9) with ξgiven by (5.10). From\nthe assumption, we have\n/an}b∇acketle{tz,z/an}b∇acket∇i}ht=/angbracketleftbig\nz,z′/angbracketrightbig\n=/angbracketleftbig\nz′′,z′′/angbracketrightbig\n= 0,/angbracketleftbig\nz′,z′/angbracketrightbig\n=−/angbracketleftbig\nz,z′′/angbracketrightbig\n= 4. (5.11)MINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 9\nBy using (5.9) and (5.11) we know that /an}b∇acketle{tL,L/an}b∇acket∇i}ht= 1 and the induced metric\ntensor is given by (5.3). Moreover, the second equation in (5 .8) shows that\nthe second fundamental form of ψsatisfiesh(∂\n∂x,∂\n∂y) = 0. Consequently, the\nimmersionψis a minimal immersion.\nCase(ii):η/ne}ationslash= 0. After applying formula (2.2) of Gauss, (5.3)-(5.5), and\n(5.7), we obtain\n(5.12)\nLxx=2ξ\n(x+y)2−2Lx\nx+y, Lxy=2L\n(x+y)2, Lyy=2η\n(x+y)2−2Ly\nx+y.\nThe compatibility conditions of (5.12) are given by\n˜∇∂\n∂yξ=2ξ\nx+y,˜∇∂\n∂xη=2η\nx+y. (5.13)\nSolving (5.13) gives\nξ= (x+y)2A(x), η= (x+y)2B(y) (5.14)\nfor some Em+1\ns-valued functions A(x),B(y). Substituting (5.14) into (5.12)\nyields\n(5.15)Lxx=A(x)−2Lx\nx+y, Lxy=2L\n(x+y)2, Lyy=B(y)−2Ly\nx+y.\nAfter solving system (5.15), we obtain\nL(x,y) =z(x)+w(y)\nx+y−z′(x)+w′(y)\n2, (5.16)\nwherez(x),w(y) areEm+1\ns-valued functions satisfying\nz′′′(x) =−4A(x), w′′′(y) =−4B(y). (5.17)\nFrom/an}b∇acketle{tL,L/an}b∇acket∇i}ht= 1 and (5.16), we obtain condition (c.1) in Theorem 5.1.\nBy combining (5.15) and (5.17), we obtain\n(5.18)Lxx=−z′′′\n4−2Lx\nx+y, Lxy=2L\n(x+y)2, Lyy=−w′′′(y)\n4−2Ly\nx+y.\nSince the metric tensor of M2\n1is given by (5.3), we find\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0,/an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=−2\n(x+y)2. (5.19)\nBecause /an}b∇acketle{tL,L/an}b∇acket∇i}ht= 1, we have /an}b∇acketle{tLxx,L/an}b∇acket∇i}ht=−/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht= 0. Thus, we obtain\ncondition (c.2) from (5.16), (5.18) and (5.19)\nSimilarly, due to /an}b∇acketle{tLyy,L/an}b∇acket∇i}ht=−/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0, we may also derive condition\n(c.3) from (5.16) and (5.18).10 B.-Y. CHEN\nConversely, assume that Lis defined by\nL(x,y) =z(x)+w(y)\nx+y−z′(x)+w′(y)\n2, (5.20)\nwherez(x),w(y) are curves satisfying conditions (c1), (c.2) and (c.3). Th en\nit follows from (5.20) that Lsatisfies system (5.18). Also, it follows from\n(5.20) and condition (c.1) that /an}b∇acketle{tL,L/an}b∇acket∇i}ht= 1. Thus, we have\n/an}b∇acketle{tL,Lx/an}b∇acket∇i}ht=/an}b∇acketle{tL,Ly/an}b∇acket∇i}ht= 0, (5.21)\nwhich implies that\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lxx/an}b∇acket∇i}ht,/an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lxy/an}b∇acket∇i}ht,/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lyy/an}b∇acket∇i}ht.(5.22)\nBy applying (5.22), (c.1) and the first equation in (5.22), we obtain\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lxx/an}b∇acket∇i}ht=2\n(x+y)2/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht, (5.23)\nwhich shows that /an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht= 0.\nSimilarly, from (5.22) and (c.3) we find /an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0. Also, after applying\n(c.1), (5.22) and the second equation in (5.18), we find /an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=−2/(x+\ny)2. Consequently, theinducedmetric tensor via Lis given by (5.3). Finally,\nit follows from (3.9) and the second equation in (5.18) that ψ:M2\n1→Sm\ns(1)\nis a minimal immersion. /square\n6.Minimal Lorentz surfaces in Hm\ns(−1).\nLetψ:M2\n1→Hm\ns(−1) be an isometric immersion of a Lorentz surface\nintoHm\ns(−1). Denote by L=ι◦ψ:M2\n1→Em+1\ns+1the composition of ψand\nthe inclusion ι:Hm\ns(−1)⊂Em+1\ns+1via (1.2).\nIn this section, we provide the following answer to the basic question\nproposed in section 5 for c<0.\nTheorem 6.1. LetM2\n1be a Lorentz surface of constant Gauss curvature −1.\nThen an isometric immersion ψ:M2\n1→Hm\ns(−1)is a minimal immersion\nif and only if one of the following three cases occurs:\n(i)M2\n1is an open portion of a totally geodesic H2\n1(−1)⊂Hm\ns(−1).\n(ii)The immersion L=ι◦ψ:M2\n1→Hm\ns(−1)⊂Em+1\ns+1is locally given by\nL(x,y) =z(x)tanh/parenleftBigx+y√\n2/parenrightBig\n−z′(x)√\n2, (6.1)MINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 11\nwherez(x)is a timelike curve with constant speed√\n2lying in the light cone\nLC⊂Em+1\ns+1satisfying /an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 4andz′′′/ne}ationslash= 2z′.\n(iii)The immersion L=ι◦ψ:M2\n1→Hm\ns(−1)⊂Em+1\ns+1is locally given by\nL(x,y) = (z(x)+w(y))tanh/parenleftBigx+y√\n2/parenrightBig\n−z′(x)+w′(y)√\n2, (6.2)\nwherezandware curves satisfying\n/angbracketleftBig\n(z+w)tanh/parenleftBigx+y√\n2/parenrightBig\n−z′+w′\n√\n2,(z+w)tanh/parenleftBigx+y√\n2/parenrightBig\n−z′+w′\n√\n2/angbracketrightBig\n=−1,(iii.1)\n√\n2/angbracketleftbig\nz+w,2z′−z′′′/angbracketrightbig\ntanh/parenleftBigx+y√\n2/parenrightBig\n=/angbracketleftbig\nz′+w′,2z′−z′′′/angbracketrightbig\n,(iii.2)\n√\n2/angbracketleftbig\nz+w,2w′−w′′′/angbracketrightbig\ntanh/parenleftBigx+y√\n2/parenrightBig\n=/angbracketleftbig\nz′+w′,2w′−w′′′/angbracketrightbig\n.(iii.3)\nProof.Assume that ψ:M2\n1→Hm\ns(−1) is an isometric immersion of a\nLorentz surface M2\n1of constant curvature −1 intoHm\ns(−1). IfMis totally\ngeodesic in Hm\ns(−1), we obtain (i). Hence, let us assume that M2\n1is non-\ntotally geodesic.\nSinceM2\n1is assumed to be of constant curvature −1, we may choose local\ncoordinates {x,y}such that the metric tensor is given by\ng=−sech2/parenleftBigx+y√\n2/parenrightBig\n(dx⊗dy+dy⊗dx). (6.3)\nHence, the Levi-Civita connection satisfies\n(6.4)∇∂\n∂x∂\n∂x=−√\n2tanh/parenleftBigx+y√\n2/parenrightBig∂\n∂x,\n∇∂\n∂x∂\n∂y= 0,\n∇∂\n∂y∂\n∂y=−√\n2tanh/parenleftBigx+y√\n2/parenrightBig∂\n∂y.\nLet us put\n∂\n∂x= sech/parenleftBigx+y√\n2/parenrightBig\ne1,∂\n∂y= sech/parenleftBigx+y√\n2/parenrightBig\ne2. (6.5)\nThen we get\n/an}b∇acketle{te1,e1/an}b∇acket∇i}ht=/an}b∇acketle{te2,e2/an}b∇acket∇i}ht= 0,/an}b∇acketle{te1,e2/an}b∇acket∇i}ht=−1. (6.6)12 B.-Y. CHEN\nBecauseM2\n1is minimal, it follows from (3.9) and (6.3) that h(e1,e2) = 0\nholds. Hence, we may put\nh(e1,e1) =ξ, h(e1,e2) = 0, h(e2,e2) =η (6.7)\nforsomenormal vector fields ξ,η. Without loss of generality, wemay assume\nξ/ne}ationslash= 0. Since M2\n1is of curvature −1, the equation of Gauss and (6.7) imply\nthat/an}b∇acketle{tξ,η/an}b∇acket∇i}ht= 0.\nCase(i):η= 0. By applying formula (2.2) of Gauss, (6.3)-(6.5), and (6. 7),\nwe obtain\n(6.8)Lxx= sech2/parenleftBigx+y√\n2/parenrightBig\nξ−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLx,\nLxy=−sech2/parenleftBigx+y√\n2/parenrightBig\nL,\nLyy=−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLy.\nAfter solving the last two equations in (6.8) we have\nL(x,y) =z(x)tanh/parenleftBigx+y√\n2/parenrightBig\n−z′(x)√\n2(6.9)\nfor some Em+1\ns+1-valued function z. It follows from (6.3), (6.8), (6.9) and\n/an}b∇acketle{tL,L/an}b∇acket∇i}ht=−1 thatz(x) satisfies /an}b∇acketle{tz,z/an}b∇acket∇i}ht= 0,/an}b∇acketle{tz′,z′/an}b∇acket∇i}ht=−2, and/an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 4.\nMoreover, substituting (6.9) into the first equation (6.8) y ields\nξ=/parenleftbigg√\n2z′(x)−z′′′(x)√\n2/parenrightbigg\ncosh/parenleftBigx+y√\n2/parenrightBig\n. (6.10)\nCombining this with ξ/ne}ationslash= 0 givesz′′′(x)/ne}ationslash= 2z′(x). Consequently, we obtain\n(ii).\nConversely, supposethat Lisgiven by (6.2), where z(x) isatimelike curve\nwith constant speed√\n2 lying in the light cone LCsatisfying /an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 4\nandz′′′/ne}ationslash= 2z′. Then,Lsatisfies (6.8) with ξgiven by (6.10). Moreover, from\nthe assumption, we have\n/an}b∇acketle{tz,z/an}b∇acket∇i}ht=/angbracketleftbig\nz,z′/angbracketrightbig\n= 0,/angbracketleftbig\nz,z′′/angbracketrightbig\n=−/angbracketleftbig\nz′,z′/angbracketrightbig\n= 2,/angbracketleftbig\nz′′,z′′/angbracketrightbig\n= 4. (6.11)\nHence, weknow from (6.9) and(6.11) that theinducedmetrict ensor is given\nby (6.3). Consequently, we see from (6.8) that the second fun damental form\nofψsatisfiesh(∂\n∂x,∂\n∂y) = 0. Therefore, the immersion ψis minimal.MINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 13\nCase(ii):η/ne}ationslash= 0. By applying formula (2.2) of Gauss, (6.3)-(6.5) and (6.7 ),\nwe obtain\n(6.12)Lxx= sech2/parenleftBigx+y√\n2/parenrightBig\nξ−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLx,\nLxy=−sech2/parenleftBigx+y√\n2/parenrightBig\nL,\nLyy= sech2/parenleftBigx+y√\n2/parenrightBig\nη−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLy.\nThe compatibility conditions of (6.12) are given by\n˜∇∂\n∂yξ=√\n2ξtanh/parenleftBigx+y√\n2/parenrightBig\n,˜∇∂\n∂xη=√\n2ηtanh/parenleftBigx+y√\n2/parenrightBig\n. (6.13)\nSolving (6.13) gives\nξ=A(x)cosh2/parenleftBigx+y√\n2/parenrightBig\n, η=B(y)cosh2/parenleftBigx+y√\n2/parenrightBig\nfor some Em+1\ns-valued functions A(x),B(y) satisfying /an}b∇acketle{tA,B/an}b∇acket∇i}ht= 0. Substi-\ntuting these into (6.12) yields\n(6.14)Lxx=A(x)−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLx,\nLxy=−sech2/parenleftBigx+y√\n2/parenrightBig\nL,\nLyy=B(y)−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLy.\nAfter solving system (6.14) we obtain\nL(x,y) = (z(x)+w(y))tanh/parenleftBigx+y√\n2/parenrightBig\n−z′(x)+w′(y)√\n2, (6.15)\nA(x) =√\n2z′(x)−z′′′(x)√\n2, B(y) =√\n2w′(y)−w′′′(y)√\n2(6.16)\nfor some Em+1\ns-valued functions z,w. From (6.15) and /an}b∇acketle{tL,L/an}b∇acket∇i}ht=−1, we\nobtain condition (iii.1) of the theorem.\nAfter differentiating (6.15) we get\n(6.17)Lx=z′(x)tanh/parenleftBigx+y√\n2/parenrightBig\n+z(x)+w(y)√\n2sech2/parenleftBigx+y√\n2/parenrightBig\n−z′′(x)√\n2,\nLy=w′(y)tanh/parenleftBigx+y√\n2/parenrightBig\n+z(x)+w(y)√\n2sech2/parenleftBigx+y√\n2/parenrightBig\n−w′′(y)√\n2.\nSince the metric tensor of M2\n1is given by (6.3), we find\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0,/an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=−sech2/parenleftBigx+y√\n2/parenrightBig\n. (6.18)14 B.-Y. CHEN\nFrom (6.14) and (6.16), we obtain\n(6.19)Lxx=√\n2z′(x)−z′′′(x)√\n2−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLx,\nLxy=−sech2/parenleftBigx+y√\n2/parenrightBig\nL,\nLyy=√\n2w′(y)−w′′′(y)√\n2−√\n2tanh/parenleftBigx+y√\n2/parenrightBig\nLy.\nBecause /an}b∇acketle{tL,L/an}b∇acket∇i}ht=−1, we have /an}b∇acketle{tLxx,L/an}b∇acket∇i}ht=−/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht= 0. Thus, we derive\nfrom (6.18) and (6.19) that\n√\n2/angbracketleftbig\nz+w,2z′−z′′′/angbracketrightbig\ntanh/parenleftBigx+y√\n2/parenrightBig\n=/angbracketleftbig\nz′+w′,2z′−z′′′/angbracketrightbig\n. (6.20)\nSimilarly, from /an}b∇acketle{tLyy,L/an}b∇acket∇i}ht=−/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht= 0, we have\n√\n2/angbracketleftbig\nz+w,2w′−w′′′/angbracketrightbig\ntanh/parenleftBigx+y√\n2/parenrightBig\n=/angbracketleftbig\nz′+w′,2w′−w′′′/angbracketrightbig\n. (6.21)\nThese give conditions (iii.2) and (iii.3). Therefore, we ob tain case (iii).\nConversely, if Lis given by (6.2) such that z(x),w(y) satisfy conditions\n(iii.1), (iii.2) and (iii.3), then we know from (6.2) that Lsatisfies (6.19).\nAlso, it follows from (6.2) and (iii.1) that /an}b∇acketle{tL,L/an}b∇acket∇i}ht=−1. Thus, we have\n/an}b∇acketle{tL,Lx/an}b∇acket∇i}ht=/an}b∇acketle{tL,Ly/an}b∇acket∇i}ht= 0, (6.22)\nwhich implies that\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lxx/an}b∇acket∇i}ht,/an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lxy/an}b∇acket∇i}ht,/an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lyy/an}b∇acket∇i}ht.(6.23)\nBy applying (6.19), (iii.1) and the first equation in (6.23), we obtain\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht=−/an}b∇acketle{tL,Lxx/an}b∇acket∇i}ht=√\n2tanh/parenleftBigx+y√\n2/parenrightBig\n/an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht, (6.24)\nwhichyields /an}b∇acketle{tLx,Lx/an}b∇acket∇i}ht= 0. Similarly, from(6.19)and(iii.3)wefind /an}b∇acketle{tLy,Ly/an}b∇acket∇i}ht=\n0. Also, after applying (iii.1), (6.19) and the second equat ion in (6.23), we\nobtain/an}b∇acketle{tLx,Ly/an}b∇acket∇i}ht=−sech2(x+y√\n2). Consequently, the induced metric tensor\nviaLis given by (6.3). Therefore, it follows from (3.9) and the se cond\nequation in (6.19) that the immersion ψ:M2\n1→Hm\ns(−1) is minimal. /squareMINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 15\n7.Explicit examples of minimal Lorentz surfaces in Sm\ns(1).\nThere exist infinitely many spacelike curves with constant s peed 2 lying\nin the light cone LC⊂Em+1\nssatisfying /an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 0 andz′′′/ne}ationslash= 0.\nExample 7.1. Consider the curve z=z(x) inE7\n3defined by\nz(x) =/parenleftbigg\nacoshpx,/radicalbig\n4r2+a2p2(p2−r2)\nq/radicalbig\nr2−q2coshqx,/radicalbig\n4q2+a2p2(p2−q2)\nr/radicalbig\nr2−q2sinhrx,\nasinhpx,/radicalbig\n4r2+a2p2(p2−r2)\nq/radicalbig\nr2−q2sinhqx,/radicalbig\n4q2+a2p2(p2−q2)\nr/radicalbig\nr2−q2coshrx,\n/radicalbig\n4(q2+r2)+a2(p2−r2)(p2−q2)\nqr/parenrightbigg\n,\nwherea,p,q,r are real numbers satisfying p > r > q > 0. It is easy to\nverify that zis a spacelike curve of constant speed 2 lying in LCsatisfying\n/an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 0,z′′′/ne}ationslash= 0.\nIt is direct to verify that the immersion defined by\nL(x,y) =z(x)\nx+y−z′(x)\n2\ngives rise to minimal Lorentz surfaces of constant curvatur e one inS6\n3(1).\nThus, there exist infinitely many minimal Lorentz surfaces o f type (b) of\nTheorem 5.1.\nThere exist infinitely many pairs ( z,w) of curves satisfying conditions\n(c.1), (c.2) and (c.3) of Theorem 5.1. Here we provide some ex amples of\nsuch pairs.\nExample 7.2. Letp,q,rbe positive numbers satisfying\n315\n4p2>80+189r2−64q2>35p2.16 B.-Y. CHEN\nConsider curves z(x) andw(y) inE14\n6defined by\nz(x) =/parenleftbigg/radicalbig\n256q2+369r2\n4√\n15cosh2x,/radicalbig\n16q2+609r2\n4√\n15sinh4x,rcosh5x,0,0,0,\n/radicalbig\n256q2+369r2\n4√\n15sinh2x,/radicalbig\n16q2+609r2\n4√\n15cosh4x,rsinh5x,0,0,0,q,0/parenrightbigg\n,\nw(y) =/parenleftbigg\n0,0,0,pcosh/parenleftBig3y\n2/parenrightBig\n,/radicalbig\n320+225 p2+756r2−256q2\n8√\n15sinh2y,\n/radicalbig\n315p2+1024q2−3024r2−1280\n4√\n15sinhy,0,0,0,psinh/parenleftBig3y\n2/parenrightBig\n,\n/radicalbig\n320+225 p2+756r2−256q2\n8√\n15cosh2y,/radicalbig\n315p2+1024q2−3024r2−1280\n4√\n15coshy,\n0,/radicalbig\n320+756 r2−35p2−256q2\n8/parenrightbigg\n.\nThenz,ware constant speed curves lying in the light cone LC⊂E14\n6satis-\nfying\n/angbracketleftbig\nz′,z′/angbracketrightbig\n=64q2−189r2\n20,/angbracketleftbig\nw′,w′/angbracketrightbig\n=80+189 r2−64q2\n20,\n/an}b∇acketle{tz,w/an}b∇acket∇i}ht=/angbracketleftbig\nz,z′′′/angbracketrightbig\n=/angbracketleftbig\nz′,z′′′/angbracketrightbig\n=/angbracketleftbig\nz′′,z′′/angbracketrightbig\n=/angbracketleftbig\nw,w′′′/angbracketrightbig\n=/angbracketleftbig\nw′,w′′′/angbracketrightbig\n=/angbracketleftbig\nw′′,w′′/angbracketrightbig\n= 0.\nMoreover, it is easy to see that conditions (c.1), (c.2) and ( c.3) are satisfied.\nIt is straightforward to verify that the immersion:\nL(x,y) =z(x)+w(y)\nx+y−z′(x)+w′(y)\n2\nvia (z,w) defines a minimal Lorentz surface of constant curvature one in\nS13\n6(1).\n8.Explicit examples of minimal Lorentz surfaces in Hm\ns(−1).\nThere exist infinitely many timelike curves with constant sp eed√\n2 lying\nin the light cone LC⊂Em+1\ns+1satisfying /an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht= 4 andz′′′/ne}ationslash= 2z′.\nExample 8.1. Leta,b,p,qbe positive numbers satisfying\np2<4+a2\n2+a2<q2andb2>a2(q2−1)(1−p2)+2(p2+q2−2)\np2q2.MINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 17\nConsider the curve z=z(x) inE8\n4defined by\nz(x) =/parenleftbigg\nb,acoshx,/radicalbig\nq2(2+a2)−(4+a2)\np/radicalbig\nq2−p2sinhpx,/radicalbig\n4+a2−p2(2+a2)\nq/radicalbig\nq2−p2sinhqx,\nasinhpx,/radicalbig\nq2(2+a2)−(4+a2)\np/radicalbig\nq2−p2coshpx,/radicalbig\n4+a2−p2(2+a2)\nq/radicalbig\nq2−p2coshqx\n/radicalbig\nb2p2q2−a2(q2−1)(1−p2)−2(p2+q2−2)\npq/parenrightbigg\n.\nIt is easy to see that zis curve lying in LCsatisfying /an}b∇acketle{tz′,z′/an}b∇acket∇i}ht=−2,/an}b∇acketle{tz′′,z′′/an}b∇acket∇i}ht=\n4 andz′′′/ne}ationslash= 2z′. A direct computation shows that\nL(x,y) =z(x)tanh/parenleftBigx+y√\n2/parenrightBig\n−z′(x)√\n2,\ndefines a minimal Lorentz surfaces of constant curvature −1 inH7\n3(−1).\nHence, there exist infinitely many minimal Lorentz surfaces of type (ii) of\nTheorem 6.1.\nThere are many pairs ( z,w) of curves satisfying conditions (iii.1)-(iii.3)\nof Theorem 6.1. Here we provide infinitely many examples of su ch pair of\ncurves.\nExample 8.2. Leta,b,p,q,r,s be positive numbers satisfying\n(8.1)a,b<1, p,q,r,s> 1, p2<1\n1−b2<q2, r2<1\n1−a2<s2,\nb2<p2+q2−2\np2q2, a2<r2+s2−2\nr2s2.\nConsider curves z(x) andw(y) inE14\n8defined by\nz(x) =/parenleftBigg\nb,/radicalbig\np2+q2−b2p2q2−2/radicalbig\n(p2−1)(q2−1)coshx,/radicalbig\n1−p2(1−b2)/radicalbig\n(q2−p2)(q2−1)sinhqx,\n/radicalbig\nq2(1−b2)−1/radicalbig\n(q2−p2)(p2−1)sinhpx,0,0,0,0,/radicalbig\np2+q2−b2p2q2−2/radicalbig\n(p2−1)(q2−1)sinhx,\n/radicalbig\n1−p2(1−b2)/radicalbig\n(q2−p2)(q2−1)coshqx,/radicalbig\nq2(1−b2)−1/radicalbig\n(q2−p2)(p2−1)coshpx,0,0,0/parenrightBigg\n,18 B.-Y. CHEN\nw(y) =/parenleftBigg\n0,0,0,0,a,√\nr2+s2−a2r2s2−2/radicalbig\n(r2−1)(s2−1)coshy,/radicalbig\n1−r2(1−a2)/radicalbig\n(s2−r2)(s2−1)sinhsy,\n/radicalbig\ns2(1−a2)−1/radicalbig\n(s2−r2)(r2−1)sinhry,0,0,0,√\nr2+s2−a2r2s2−2/radicalbig\n(r2−1)(s2−1)sinhy,\n/radicalbig\n1−r2(1−a2)/radicalbig\n(s2−r2)(s2−1)coshqy,/radicalbig\ns2(1−a2)−1/radicalbig\n(s2−r2)(r2−1)coshpy/parenrightBigg\n.\nIt is easy to verify that zandwsatisfy conditions (iii.1), (iii.2) and (iii.3).\nThe associated map\nL(x,y) = (z(x)+w(y))tanh/parenleftBigx+y√\n2/parenrightBig\n−z′(x)+w′(y)√\n2\ndefines a minimal Lorentz surface of constant curvature −1 inH13\n7(−1).\nRemark 8.1.There exist many positive numbers a,b,p,q,r,s satisfying the\nconditions given in (8.1). For instance, a=b= 1/√\n2,p=r= 1.1 and\nq=s= 1.5 satisfy all conditions given in (8.1).\nReferences\n[1] C. Blomstrom, Symmetric immersions in pseudo-Riemanni an space forms, Global\nDifferential Geometry and Global Analysis , 30–45, Lecture Notes in Math. 1156,\nSpringer, Berlin, 1985.\n[2] B.-Y. Chen, Geometry of Submanifolds, Mercer Dekker, New York , 1973.\n[3] B.-Y. Chen, Total Mean Curvature and Submanifolds of Fin ite Type, World Scien-\ntific, New Jersey , 1984.\n[4] B.-Y. Chen, Riemannian submanifolds, Handbook of Differential Geometry , Vol. I,\n187–418, North-Holland, Amsterdam, 2000 (eds. F. Dillen an d L. Verstraelen).\n[5] B.-Y. Chen, Minimal flat Lorentzian surfaces in Lorentzi an complex space forms,\nPubl. Math. Debrecen 73(2008), 233–248.\n[6] B.-Y. Chen, Nonlinear Klein-Gordon equations and Loren tzian minimal surfaces in\nLorentzian complex space forms, Taiwanese J. Math. 13(2009), 1–24.\n[7] B.-Y. Chen, Complete classification of Lorentz surfaces with parallel mean curvature\nvector in arbitrary pseudo-Euclidean space, Kyushu J. Math. 64(2010), 261–279.\n[8] B.-Y. Chen, Complete classification of parallel spatial surfaces in pseudo-Riemannian\nspace forms with arbitrary index and dimension, J. Geom. Phys. 60(2010) 260–280.\n[9] B.-Y. Chen, Complete classification of parallel Lorentz surfaces in 4D neutral pseudo-\nsphere,J. Math. Phys. 51(2010), no. 8, 083518, 22 pages.\n[10] B.-Y. Chen, Complete classification of parallel Lorent z surfaces in neutral pseudo\nhyperbolic 4-space, Cent. Eur. J. Math. 8(2010), 706–734.\n[11] B.-Y. Chen, Explicit classification of parallel Lorent z surfaces in 4D indefinite space\nforms with index 3, Bull. Inst. Math. Acad. Sinica (N.S.) 5(2010), 311–345.MINIMAL LORENTZ SURFACES IN INDEFINITE SPACE FORMS 19\n[12] B.-Y. Chen, Complete explicit classification of parall el Lorentz surfaces in arbitrary\npseudo-Euclidean space, J. Geom. Phys. 60(2010), 1333–1351.\n[13] B.-Y. Chen, F. Dillen and J. Van der Veken, Complete clas sification of parallel\nLorentzian surfaces in Lorentzian complex space forms, Intern. J. Math. 21(2010),\n665–686.\n[14] B.-Y. Chen and J. Van der Veken, Complete classification of parallel surfaces in\n4-dimensional Lorentzian space forms, Tohoku Math. J. 61(2009), 1–40.\n[15] L. K. Graves, On codimension one isometric immersions b etween indefinite space\nforms,Tsukuba J. Math. 3(1979), 17–29.\n[16] L. K. Graves, Codimension one isometric immersions bet ween Lorentz spaces, Trans.\nAmer. Math. Soc. 252(1979), 367–392.\n[17] Y. Fu and Z. H. Hou, Classification of Lorentzian surface s with parallel mean curva-\nture vector in pseudo-Euclidean spaces, J. Math. Anal. Appl. 371(2010), 25–40.\n[18] J. L. Lagrange, Essai d’une nouvelle m´ ethode pour d´ et erminer les maxima et les\nminima des formules int´ egrales ind´ efinies, Miscellanea Taurinensia ,2(1760), 173–\n195.\n[19] M. A. Magid, Isometric immersions of Lorentzspace with parallel second fundamental\nforms,Tsukuba J. Math. 8(1984), 31–54.\n[20] L. V. McNertney, One-parameter families of surfaces wi th constant curvature in\nLorentz 3-space, Ph.D. Thesis, Brown Univ. , 1980.\n[21] J. Nitsche, Lectures on Minimal Surfaces, Cambridge Univ. Press, Cambridge , 1989.\n[22] B. O’Neill, Semi-Riemannian Geometry with Applicatio ns to Relativity, Academic\nPress, New York , 1982.\n[23] R. Osserman, A Survey of Minimal Surfaces, Van Nostrand, New York , 1969.\n[24] L. Verstraelen and M. Pieters, Some immersions of Loren tz surfaces into a pseudo-\nRiemannian space of constant curvature and of signature (2 ,2),Rev. Roumaine Math.\nPures Appl. 19(1974), 107–115.\n[25] L. Verstraelen and M. Pieters, Some immersions of Loren tz surfaces into a pseudo-\nRiemannian space of constant curvature and of signature (2 ,2),Rev. Roumaine Math.\nPures Appl. 21(1976), 593–600.\nDepartment of Mathematics, Michigan State University, Eas t Lansing, MI\n48824, U.S.A.\nE-mail address :bychen@math.msu.edu"    },    {        "title": "2012.02932v1.On_Periodical_Damping_Ratio_of_a_Controlled_Dynamical_System_with_Parametric_Resonances.pdf",        "content": "NOVEMBER 2020 1\nOn Periodical Damping Ratio of a Controlled\nDynamical System with Parametric Resonances\nXin Xu, Student Member, IEEE, Kai Sun, Senior Member, IEEE\nAbstract\nThis report provides an interpretation on the periodically varying damping ratio of a dynamical system with direct control\nof oscillation or vibration damping. The principal parametric resonance of the system and a new type of parametric resonance,\nnamed ”zero-th order” parametric resonance, are investigated by using the method of multiple scales to find approximate, analytical\nsolutions of the system, which provide an interpretation on such damping variations.\nKeywords - Method of multiple scales, damping, parametric resonance.\nI. I NTRODUCTION\nTHIS study is motivated by the direct control of oscillation or vibration damping with a weakly-damped dynamical system.\nThe dynamics regarding a specific mode can be approximated by a one-degree-of-freedom (1-DOF) system having a\ndirect damping improvement as modeled by (1):\nx+ (\u00002\u001b+u) _x+ (\u001b2+!2)x= 0 (1)\nwherexis a state. Parameter uis the increased damping by, e.g., a feedback controller that measures real-time damping ratio\nand eliminates its error from a reference damping ratio. If u= 0, the eigenvalue \u0015is calculated by (2), where \u0010andwnare\nthe damping ratio and natural angular frequency, respectively.\n\u00151;2=\u001b\u0006j!=\u0000\u0010!n\u0006j!np\n1\u0000\u00102 (2)\nConsidering the dynamics of uover time, approximate uby a periodic function Kcos (\nt), whereKand\nare its amplitude\nand angular frequency, respectively. Eq. (1) becomes (3). By using the method of multiple scales (MMS), it will be shown\nthata)the principal parametric resonance can be excited when \n\u00192!andb), the “zero-th order” parametric resonance will\nbe excited when \n\u00190, which is why it is called “zero-th order”.\nx+ (\u00002\u001b+Kcos (\nt)) _x+ (\u001b2+!2)x= 0 (3)\nIn the rest of the report, principal and ”zero-th order” parametric resonances of the system are respectively investigated in\nsections II and III, and conclusions are drawn in section IV .\nII. P RINCIPAL PARAMETRIC RESONANCE\nWhen \n\u00192!, the principal parametric resonance can be observed in the response of x. For instance, let \u0010= 0.0098,!n=\n3.8072 rad/s, K= 0.5, and consequently, \u001b= 0.0373 and != 3.8070 rad/s. Then, if let \n= 6.9115 rad/s and let the initial value\nbex= 1;_x= 0, the response of xis shown in Fig. 1 with its envelop being marked and the damping ratio estimated, e.g.,\nfrom Prony’s method is given in Fig. 2. The periodic variation in the measured damping ratio shows the parametric resonance\ndue to \n\u00192!.\nThe parametric resonance can be investigated by finding an asymptotic solution via the MMS for (3), through which some\nproperties could be revealed. The basic idea of MMS is to find an asymptotic solution of a perturbed system considering\ndifferent time scales. The use of MMS follows the methodology in [1], [2], and [3]. First, the periodic parameter Kcos (\nt)\nis treated as a perturbation by inserting a small dimensionless parameter \">0like in (4), where K\"=K=\".\nx+ (\u00002\u001b+\"K\"cos(\nt)) _x+ (\u001b2+!2)x= 0 (4)\nThen, a first order uniform solution of (4) is sought in the form of (5):\nx(\";T0;T1)\u0019x0(T0;T1) +\"x1(T0;T1)\nT0=t;T1=\"t(5)\nX. Xu and K. Sun are with the Department of Electrical and Computer Engineering, the University of Tennessee, Knoxville, TN, 37919.\nE-mail: xxu30@vols.utk.edu, kaisun@utk.eduarXiv:2012.02932v1  [math.DS]  5 Dec 2020NOVEMBER 2020 2\nFig. 1. Principal parametric resonance: response of x.\nFig. 2. Principal parametric resonance: measured damping ratio.\nwhereT1is introduced as a slow-scale time variable, such that (5) could consider the evolution of the solution over long time-\nscales of the order \"\u00001. Note thatx0is exactly the solution of (4) when \"= 0, andx1is the part caused by the perturbation.\nSubstitute (5) into (4) and equate the coefficients of powers of \":\nD2\n0x0\u00002\u001bD0x0+ (\u001b2+!2)x0= 0 (6)\nD2\n0x1\u00002\u001bD0x1+ (\u001b2+!2)x1=\u00002D1D0x0\u0000K\"cos(\nT0)D0x0+ 2\u001bD1x0 (7)\nwhereDn=@=@Tn.\nThe solution of (6) can be expressed in a complex form:\nx0(T0;T1) =A(T1)e(\u001b+j!)T0+\u0016A(T1)e(\u001b\u0000j!)T0(8)\nwhere the bar denotes the complex conjugate. Substitute (8) into (7):\nD2\n0x1\u00002\u001bD0x1+ (\u001b2+!2)x1=\u00002(\u001b+j!)e(\u001b+j!)T0D1A\u00002\u0010!ne(\u001b+j!)T0D1A\n\u0000K\"\n2h\n(\u001b+j!)e(\u001b+j(\n+!))T0A+ (\u001b\u0000j!)e(\u001b+j(\n\u0000!))T0\u0016Ai\n+C:C:(9)\nwhereC:C: denotes the complex conjugates of the former two terms. Introduce a detuning parameter \u0018such that \n = 2!+\"\u0018,\n(9) can be converted to:\nD2\n0x1\u00002\u001bD0x1+ (\u001b2+!2)x1=\u0000\u0014\n2(\u001b+j!)D1A\u00002\u001bD1A+K\"\n2(\u001b\u0000j!)ej\"\u0018T 0\u0016A\u0015\ne(\u001b+j!)T0\n\u0000K\"\n2(\u001b+j!)e(\u001b+j(\n+!))T0A+C:C:(10)\nIt can be verified that the condition (11) should be met to avoid generating secular terms in the solution of x. The explanation\nof secular term can be found in [3]. In this problem it will be a term grows linearly in t, such that the identified solution will\nbe unbounded, while in fact the true solution is bounded.\n2(\u001b+j!)D1A\u00002\u001bD1A+K\"\n2(\u001b\u0000j!)ej\"\u0018T 0\u0016A= 0 (11)\nWith the condition (11), the solution of (10) is given below.NOVEMBER 2020 3\nx1(T0;T1) =K\"(\u001b+j!)A\n2\u0012e(\u001b+j(\n+!))T0\n\n(\n + 2!)+e(\u001b\u0000j!)T0\n2!(\n + 2!)\u0000e(\u001b+j!)T0\n2!\n\u0013\n+C:C: (12)\nx1could be ignored compared to x0, since usually it is numerically small. Hence, assume x(t)\u0019x0(t).A(T1)can be\ndetermined by solving (11). For a weakly-damped system, \u001bcan be assumed to be zero and (11) can be converted to:\ndA\ndT1=K\"\n4ej\u0018T1\u0016A (13)\nThe analytical solutions of A(T1)andx(t)can be found for three cases, 1)(\n\u00002!)2\u0000K2=4>0,2)(\n\u00002!)2\u0000K2=4<0,\nand3)(\n\u00002!)2\u0000K2=4 = 0 . For the sake of convenience, define !K=p\nj\u00182\u0000K2\"=4j=p\nj(\n\u00002!)2\u0000K2=4j=\", and\nletA0=Are0+jAim0be the given initial value of A(T1).\nA. Case 1: (\n\u00002!)2\u0000K2=4>0\nThe solution of A(T1)is:\nA(T1) = (C1ej(r1\u0000!KT1)+C2ejr2)ej\u0018+!K\n2T1(14)\n8\n>>>>>>>>>>>>><\n>>>>>>>>>>>>>:C1=s\n\u0018+!K\n2!2\nKq\n\u0018(A2\nre0+A2\nim0) +K\"Are0Aim0\nC2=s\n\u0018\u0000!K\n2!2\nKq\n\u0018(A2\nre0+A2\nim0) +K\"Are0Aim0\nr1=arctan\u00122Aim0(\u0018+!K) +K\"Are0\n2Are0(\u0018+!K) +K\"Aim0\u0013\nr2=arctan\u00122Aim0(\u0018\u0000!K) +K\"Are0\n2Are0(\u0018\u0000!K) +K\"Aim0\u0013(15)\nThe solution of x(t)is obtained by substituting (14) into (5) and ignoring x1:\nx(t) = 2C1e\u001btcos(!Ct+r1\u0000\"!Kt) + 2C2e\u001btcos(!Ct+r2) (16)\n!C=!+\u0018+!K\n2\" (17)\nAt the first glance, x(t)seems to depend on \"and\u0018. Actually, such dependence can be immediately removed by substituting\n\u0018= (\n\u00002!)=\",!K=p\nj(\n\u00002!)2\u0000K2=4j=\", (15) and (17) into (16). The final expression is not given here for the sake\nof brevity.\nThe resulting x(t)consists of two components. The magnitude of the first component is 2C1and the frequency is !C\u0000\"!K.\nThe magnitude of the second component is 2C2and the frequency is !C. The validity of the approximated solution can be\nvisualized by the case when \n= 6.9115 rad/s. The comparison of the true response of x(t)and the approximated x(t)from\n(16) is shown in Fig. 3. The approximated x(t)is almost the same as the true response.\nThe principal parametric resonance in this case can be interpreted as follows. Without loss of generality, only consider the case\nwhenC2is larger than C1. The second component can be more dominant than the first component. Since \"!Kis much smaller\nthan!C, the term\"!Ktcan be viewed as a slow change of phase of the first component. Then, the change of damping of x(t)\ncan be interpreted as the periodically phase shift between the two components. When r1\u0000\"!Kt\u0019r2+2m\u0019;m = 0;\u00061;\u00062;:::,\nthe two components are in-phase and the magnitude of x(t)is amplified. When r1\u0000\"!Kt\u0019r2+2m\u0019+\u0019;m = 0;\u00061;\u00062;:::,\nthe two components are out-of-phase and the magnitude of x(t)is reduced. This change is periodical at a frequency equal to\n\"!K=p\n(\n\u00002!)2\u0000K2=4, and it makes the response of x(t)to exhibit a periodically varying damping.\nB. Case 2: (\n\u00002!)2\u0000K2=4<0\nThe solution of A(T1)is:\nA(T1) = (C3ejr3+C4ej(r4\u0000!KT1))ej\u0018+!K\n2T1(18)NOVEMBER 2020 4\nFig. 3. Comparison of true response and approximated solution: (\n\u00002!)2\u0000K2=4>0.\n8\n>>>>>>>>>>>>>>>><\n>>>>>>>>>>>>>>>>:C3=s\nK\"(2K\"+!K)\n!2\nK\f\f\f\fAre0+Aim02\u0018\nK\"+ 2!K\f\f\f\f\nC4=s\nK\"(2K\"\u0000!K)\n!2\nK\f\f\f\fAre0+Aim02\u0018\nK\"\u00002!K\f\f\f\f\nr3=arctan\u0012\u0000Aim0(K\"\u00002!K)\u00002\u0018Are0\nAre0(K\"+ 2!K) + 2\u0018Aim0\u0013\nr4=arctan\u0012Aim0(K\"+ 2!K) + 2\u0018Are0\n\u0000Are0(K\"\u00002!K)\u00002\u0018Aim0\u0013(19)\nThe solution of x(t)is obtained by substituting (18) into (5) and ignoring x1. Similarly to (16), the solution (20) also does\nnot depend on \"and\u0018.\nx(t) = 2C3e(\u001b+\"!K\n2)tcos\u0012\n(!+\n2)t+r3\u0013\n+ 2C4e(\u001b\u0000\"!K\n2)tcos\u0012\n(!+\n2)t+r4\u0013\n(20)\nThe resulting x(t)consists of two components. The magnitude of the first component is 2C3and the magnitude of the second\ncomponent is 2C4. The frequency of both components is !+ \n=2. The two components of x(t)have damping coefficient\ndifferent from \u001b. This validity of the approximated solution can be verified by the case when \nis changed to 7.5524 rad/s.\nThe comparison of the true response of x(t)and the approximated x(t)from (20) is shown in Fig. 4. The approximated x(t)\nis almost the same as the true response. Note that the response diverges since the damping part of the first component becomes\npositive.\nIn this case, the response of x(t)will not exhibit a periodically varying damping, but might exhibit a time-variant damping.\nIfC3is much larger than C4, the first component will be dominant at the early stage and the response of x(t)will be damped\nin a fast pace. Then, the first component will take the dominance after the second component is damped out.\nC. Case 3: (\n\u00002!)2\u0000K2=4<0\nThe solution of A(T1)is:NOVEMBER 2020 5\nFig. 4. Comparison of true response and approximated solution: (\n\u00002!)2\u0000K2=4<0.\nA(T1) =A0ejK\"\n4T1+K\"\n4(\u0016A0\u0000jA0)T1ejK\"\n4T1; if \n\u00002!=K\n2\nA(T1) =A0e\u0000jK\"\n4T1\u0000K\"\n4(\u0016A0\u0000jA0)T1e\u0000jK\"\n4T1; if \n\u00002!=\u0000K\n2(21)\nThe solution of x(t)is obtained by substituting (21) into (5) and ignoring x1:\nx(t) = 2e\u001bt\u0002\u0000\nAre0+K\n4(Are0+Aim0)t\u0001\ncos\u0000\n2t\u0001\n\u0000\u0000\nAim0\u0000K\n4(Are0+Aim0)t\u0001\nsin\u0000\n2t\u0001\u0003\n; if \n\u00002!=K\n2\nx(t) = 2e\u001bt\u0002\u0000\nAre0\u0000K\n4(Are0+Aim0)t\u0001\ncos\u0000\n2t\u0001\n\u0000\u0000\nAim0+K\n4(Are0+Aim0)t\u0001\nsin\u0000\n2t\u0001\u0003\n; if \n\u00002!=\u0000K\n2(22)\nEach of the resulting x(t)consists of two components at the frequency \n=2. Note part of the results depends on t, which\nmay cause a time-variant damping.\nThis validity of the approximated solution can be verified by the case when \nis changed to 7.3639 rad/s. The comparison\nof the true response of x(t)and the approximated x(t)from (22) is shown in Fig. 5. The approximated x(t)is almost the\nsame as the true response.\nSince the condition (\n\u00002!)2\u0000K2=4 = 0 can hardly be met, this case is usually ignored in industrial applications.\nIII. “Z ERO-THORDER ” PARAMETRIC RESONANCE\nWhen \n\u00190, the parametric resonance can also be observed in the response of x, which is named as “zero-th order”\nparametric resonance here. For instance, let \u0010= 0.0098,!n= 3.8072 rad/s, K= 0.5, and consequently, \u001b= 0.0373 and !=\n3.8070 rad/s. Then, if let \n= 0.6283 rad/s and let the initial value be x= 1;_x= 0, the response is shown in Fig. 6 with the\nenvelop being marked and the measured damping ratio from the Prony’s method is given in Fig. 7. The periodic variation in\nthe measured damping ratio shows the parametric resonance due to \n\u00190.\nThrough MMS, some properties of such a resonance is revealed. Again, consider a small dimensionless parameter \"as in\n(4), and take the same derivation as from (4) to (9). By introducing a detuning parameter \u0018such that \n =\"\u0018, (9) can be\nconverted to (23).\nD2\n0x1\u00002\u001bD0x1+ (\u001b2+!2)x1=\u0000\u0014\n2(\u001b+j!)D1A\u00002\u001bD1A+K\"\n2(\u001b+j!)ej\"\u0018T 0A\u0015\ne(\u001b+j!)T0\n\u0000K\"\n2(\u001b\u0000j!)e(\u001b+j(\"\u0018+!))T0\u0016A+C:C:(23)\nIt can be verified that the condition (24) should be met to avoid generating secular terms in the solution of x(t).NOVEMBER 2020 6\nFig. 5. Comparison of true response and approximated solution: (\n\u00002!)2\u0000K2=4 = 0 .\nFig. 6. “Zero-th order” parametric resonance: response of x.\nFig. 7. “Zero-th order” parametric resonance: measured damping ratio.\n2(\u001b+j!)D1A\u00002\u001bD1A+K\"\n2(\u001b+j!)ej\"\u0018T 0A= 0 (24)\nThen, the solution of (23) is given below.\nx1(T0;T1) =K\"(\u001b\u0000j!)\u0016A\n2\u0012e(\u001b+j(\n\u0000!))T0\n\n(\n\u00002!)+e(\u001b\u0000j!)T0\n2!\n\u0000e(\u001b+j!)T0\n2!(2!\u0000\n)\u0013\n+C:C: (25)NOVEMBER 2020 7\nx1could be ignored compared to x0, since usually it is numerically small. Hence, assume x(t)\u0019x0(t).A(T1)can\nbe determined by solving (24). First convert (24) to (26). Then, the analytical solution of A(T1)is shown in (27), where\nA0=Are0+jAim0is the initial value of A(T1).\ndA\ndT1=K\"(\u0000!+j\u001b)\n4!ej\u0018T1A (26)\nA(T1) =eK\"(\u001b+j!)\n4\u0018!ej\u0018T1+A0\u0000eK\"(\u001b+j!)\n4\u0018! (27)\nSubstitute (27) into (5) and ignore x1:\nx(t) = 2eKp\n\u001b2+!2\n4\n!cos(\nt+\u0012)+\u001btcos \nKp\n\u001b2+!2\n4\n!sin(\nt+\u0012) +!t!\n+ 2e\u001bt\u0012\nAre0cos(!t)\u0000Aim0sin(!t)\u0000eK\u001b\n4!\ncos(!t+K\n4\n)\u0013 (28)\n\u0012=arctan\u0010!\n\u001b\u0011\n(29)\nThe solution includes two components. Note that the first component has a periodically varying parameter cos(\nt+\u0012)\nbeing added to the original damping part \u001bt, which could lead to a periodically varying damping in the response of x(t). The\nsin(\nt+\u0012)term in the first component can be viewed as a periodically varying phase of frequency \nthat leads to a periodic\nphase shift relative to the second component. Hence, the solution x(t)will exhibit periodically varying damping ratio, and the\nfrequency is close to \n.\nThe validity of the approximated solution can be verified by the case when \nis changed to 0.6283 rad/s. The comparison\nof the true response of x(t)and the approximated x(t)from (28) is shown in Fig. 8. The approximated x(t)is almost the\nsame as the true response.\nFig. 8. Comparison of true response and approximated solution: “zero-th order” parametric resonance.\nIV. C ONCLUSION\nThe response of a weakly-damped 1-DOF system with direct control of its damping ratio can exhibit parametric resonances\nunder principal parametric excitation and “zero-th order” parametric excitation. It is shown that the principal parametric\nresonance can be classified into three cases depending on (\n\u00002!)2\u0000K2=4>0;<0, or= 0. Specifically, when (\n\u00002!)2\u0000NOVEMBER 2020 8\nK2=4>0, the magnitude of xperiodically variation in time at a frequency close top\n(\n\u00002!)2\u0000K2=4, which manifests\nperiodical changes of its damping ratio. When “zero-th order” parametric resonance is excited, the magnitude of x(t)can\nperiodically vary in time at a frequency close to \n.\nREFERENCES\n[1] L. D. Zavodney, A. H. Nayfeh,“The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a fundamental parametric\nresonance,” Journal of Sound and Vibration , vol. 120, no. 1, pp. 63-93, 1988.\n[2] L. D. Zavodney, A. H. Nayfeh, N. E. Sanchez,“The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a principal\nparametric resonance,” Journal of Sound and Vibration , vol. 129, no. 3, pp. 417-442, 1989.\n[3] J. K. Kevorkian, J. D. Cole, Multiple scale and singular perturbation methods . Springer Science & Business Media, 2012."    },    {        "title": "1103.1072v2.Orthogonal_decomposition_of_Lorentz_transformations.pdf",        "content": "arXiv:1103.1072v2  [gr-qc]  26 Nov 2012Orthogonal decomposition of Lorentz\ntransformations\njason hanson∗\nDigiPen Institute of Technology\nAbstract\nThe canonical decomposition of a Lorentz algebra element in to a\nsum of orthogonal simple (decomposable) Lorentz bivectors is defined\nand discussed. This decomposition on the Lie algebra level l eads to\na natural decomposition of a proper orthochronous Lorentz t ransfor-\nmation into a product of commuting Lorentz transformations , each\nof which is the exponential of a simple bivector. While this l ater re-\nsult is known, we present novel formulas that are independen t of the\nform of the Lorentz metric chosen. As an application of our me th-\nods, we obtain an alternative method of deriving the formula s for the\nexponential and logarithm for Lorentz transformations.\n1 Introduction\nIt is common to define a three–dimensional rotation by specifying its axis\nand rotation angle. However, since the axis of rotation is fixed, we m ay also\ndefine the rotation in terms of the two–plane orthogonal to the ax is. This\npoint of view is useful in higher dimensions, where one no longer has a s ingle\nrotation axis. Indeed we may define a rotation in n–dimensional space by\nchoosing a two–plane and a rotation angle in that plane; vectors ort hogonal\nto the plane are fixed by the rotation. The most general rotation, however,\nis a composition of such planar, or simple, rotations. Moreover, the planes\n∗jhanson@digipen.edu\n1of rotation may be taken to be mutually orthogonal to each other, so that\nthe simple factors commute with one another.\nOn the Lie algebra level, where rotations correspond to antisymmet ric\nmatrices, a simple rotation corresponds to a decomposable matrix; i.e., a\nmatrix of the form u∧v=uvT−vuT, whereu,vare (column) vectors.\nGeometrically, the vectors u,vspan the plane of rotation. A general rotation\ncorresponds to the sum of decomposable matrices: A=/summationtext\niui∧vi, and\nthe mutual orthogonality of the two–planes of rotation correspo nds to the\nmutual annihilation of summands: ( ui∧vi)(uj∧vj) = 0 fori/negationslash=j. Note\nthat mutual annihilation trivially implies commutation, so that exp(/summationtext\niui∧\nvi) =/producttext\niexp(ui∧vi), and we recover algebraically the commutation of simple\nfactors in a rotation.\nIn the particular case of four–dimensional Euclidean space, a gene ral ro-\ntation is the composition of two simple commuting rotations in orthogo nal\ntwo–planes. Correspondingly, a 4 ×4 antisymmetric matrix can be written\nas the sum of two mutually annihilating decomposable matrices. It is po ssi-\nble (although apparently not of sufficient interest to be publishable) to write\ndownexplicit formulasforthisdecomposition—bothontheLiealgebra level\nand on the level of rotations. It is natural to ask if such an orthog onal de-\ncomposition is possible for Lorentz transformations. This article an swers this\nquestion in the affirmative. If one assumes a Minkowski metric and allo ws\nthe use of complex numbers, a Wick rotation may be applied to the for mulas\nfor four–dimensional Euclidean rotations. However our discussion will not\nproceed along this line; instead, we will limit ourselves to algebra over t he\nreals (see below) and develop the formulas from first principles, with out any\nassumptions on the specific form of the Lorentz metric.\nIt should be noted that the orthogonal decomposition of a Lorent z trans-\nformation Λ into commuting factors is distinct from the usual polar d ecom-\nposition Λ = BR, whereBis a boost and Ris a rotation. While BandRare\nsimple in the sense that they are exponentials of decomposable Lie alg ebra\nelements:B= exp(b) andR= exp(r), they do not necessarily commute,\nnor do the Lie algebra elements b,rannihilate each other. In particular,\nexp(b+r)/negationslash= exp(b)exp(r) in general.\nFrom the point of view of the geometry of Lorentz transformation s, there\nis nothing new presented in this work. It is known that any proper or -\nthochronous Lorentz transformation is the product of commutin g factors\nwhich act only on orthogonal two–planes ([5]). That the correspo nding\nLorentz algebra element can be written as the sum of mutually annihila t-\n2ing simple elements is a natural consequence. However, the formula s given\nhere, which actually allow one to compute these decompositions using only\nbasic arithmetic operations, appear to be original. Moreover they a re not\nmetric–specific, and so are applicable to any space–time manifold in a n atu-\nral way, without the need to use Riemann normal coordinates.\nSince it may seem oddly unnecessary to restrict ourselves only to alg e-\nbra over the reals, we briefly explain our motivation for this. The aut hor\nis interested in efficient numerical interpolation of Lorentz transfo rmations,\nwith a view towards a real–time relativistic simulation. That is, we are giv en\n(proper) Lorentz transformations Λ 0and Λ 1, which we think of as giving co-\nordinate transformations, measured at times t= 0 andt= 1, between an ob-\nserver andanobjectinnonuniformmotion. Foranytime twith0≤t≤1, we\nwould like tofind aLorentz transformationΛ( t) that is“between” Λ 0andΛ 1.\nAnalogous to linear interpolation, we may use geodesic interpolation in the\nmanifold of all Lorentz transformations: Λ( t) = Λ 0exp(tlogΛ−1\n0Λ1). Thus\nwe need to efficiently compute the logarithm and exponential for Lor entz\ntransformations. Although many computer languages offer arithm etic over\nthe complex numbers, not all do, and in any case they are typically no t\nsupported at the hardware level, so their usage introduces additio nal com-\nputational overhead.\n2 Preliminaries\n2.1 Generalized trace\nWe will make use of the k–th order trace trkMof an×nmatrixM, which\nis defined by the identity\ndet(I+tM) =n/summationdisplay\nk=0(trkM)tk. (1)\nEquivalently, the ( n−k)–th degree term of the characteristic polynomial\ndet(λI−M) ofMhas coefficient ( −1)ktrkM. In particular, the zeroth order\ntrace is unity, the first order trace is the usual trace, and the n–th order trace\nis the determinant. For k/negationslash= 1, tr kMis not linear. However, the identities\n(i) trkMT= trkM, (ii) tr kMN= trkNM, and (iii) tr kαM=αktrkMfor\nany scalar α, are always satisfied for any k. These properties follow from\n3equation (1) and the basic properties of determinants. Of primary interest\nis the second order trace, for which we have the explicit formula\ntr2M=1\n2(tr2M−trM2) (2)\nThis follows by computing the second derivative of equation (1) and e valu-\nating att= 0. In engineering, tr 2Mis often denoted by II M.\n2.2 Antisymmetry and decomposability\nIn preparation for the next section, let us derive a few facts that we will need\nconcerning 4 ×4 antisymmetric matrices. Such a matrix can be written in\nthe form\nAxy=/parenleftbigg\n0xT\n−xWy/parenrightbigg\nwhereWy=\n0−y3y2\ny30−y1\n−y2y10\n (3)\nfor some x,y∈R3. HereWyis chosen so that it has the defining property\nWyv=y×v.\nBy computing the determinant of (3) directly, we obtain the following ,\nwhich is a special case of the general fact that the determinant of an anti-\nsymmetric matrix (in any dimension) is the square of its Pfaffian.\nLemma 2.1. detAxy= (x·y)2\nWe may identify the wedge (exterior) product of four–vectors u,vwith the\nmatrixu∧v.=uvT−vuT. This matrix is evidently antisymmetric, and if we\nwriteu= (u0,u),v= (v0,v), we find that u∧v=Axy, wherex=u0v−v0u\nandy=v×u. In particular, lemma 2.1 implies that det u∧v= 0. This is\nagain a special case of a general fact: the determinant of a wedge is zero in\ndimensions greater than two, since we can always find a vector orth ogonal to\nthe vectors u,v. In dimension four, the converse is also true.\nLemma 2.2. Axyis a wedge–product if and only if detAxy= 0.\nProof.For the sufficiency, first assume that y/negationslash=0. Choose vectors u,vsuch\nthatv×u=y. Sincex·y= 0 (lemma 2.1), we may write x=u0v−v0u\nfor some scalars u0,v0. Ify=0, then take u= (1,0) andv= (0,x).\nIngeneral, thevector spaceoftwo–formsin ndimensions canbeidentified\nwith the space of n×nantisymmetric matrices. In dimensions n≥4, not\nevery two–form is decomposable; i.e., the wedge of two one–forms. Lemma\n2.2 gives a simple test for decomposability in dimension n= 4.\n43 Lorentz algebra element decomposition\nLetgbe a Lorentz inner product on R4; i.e.,gis symmetric, nondegenerate,\nand detg <0. Recall that the Lorentz algebra so(g) is the collection of\nall linear transformations LonR4for whichLTg+gL= 0; or equivalently,\nLT=−gLg−1. We will refer to an element of so(g) as abivector .\n3.1 Algebra invariants and properties\nLemma 3.1. Ag∈so(g)if and only if Ais antisymmetric.\nProof.Sincegis symmetric, ( Ag)Tg+g(Ag) =g(AT+A)g, which is zero if\nand only if AT=−A.\nThus any Lorentz bivector Lcan be written in the form L=Agfor\nsome antisymmetric matrix A. Although this identification gives us a vector\nspace isomorphism between so(g) and the space of all antisymmetric 4 ×4\nmatrices, theidentificationisnotcompatiblewiththeir respective Lie algebra\nstructures; i.e., does not give a Lie algebra isomorphism.\nLemma 3.2. For anyL∈so(g),trL= 0 = tr 3L.\nProof.trkL= trkLT= trk(−gLg−1) = trk(−L) = (−1)ktrkL.\nLemma 3.3. detL≤0for allL∈so(g).\nProof.WriteL=Ag. Then det L= detAdetg≤0, by lemma 2.1.\nBy the comments in section 2.1, the characteristic equation for a Lo rentz\nbivectorLis\nλ4+(tr2L)λ2+detL= 0. (4)\nThe squared–roots (that is, the roots of x2+(tr2L)x+detL= 0) are\nµ±=1\n2/parenleftbig\n−tr2L±/radicalBig\ntr2\n2L−4detL/parenrightbig\n(5)\nSince detL≤0,µ±are real numbers. Note that they are solutions to the\nequationsµ++µ−=−tr2Landµ+µ−= detL, and thatµ+≥0 andµ−≤0.\nMoreover, det L/negationslash= 0 if and only if µ+>0 andµ−<0.\nLemma 3.4. L4+(tr2L)L2+(detL)I= 0.\n5Proof.Thisfollowsfromequation(4)andtheCayley–Hamiltontheorem.\nThe determinant and second order trace of a Lorentz bivector ma y be\ncomputed from the traces of its powers.\nLemma 3.5. tr2L=−1\n2trL2anddetL=1\n8(tr2L2−2trL4).\nProof.Lemma 3.2 and equation (2) imply the first identity. For the second,\nlemma 3.4 implies tr L4+tr2LtrL2+4detL= 0.\n3.2 Simple bivectors\nGiventwofour–vectors u,v,wemayconstructthe simple Lorentz bivector\nu∧gv, which is the 4 ×4 matrix\nu∧gv.=uvTg−vuTg (6)\nIn covariant coordinate notation, ( u∧gv)β\nα=uβvα−vβuα. Observe that\n(u∧gv) = (u∧v)g, whence lemma 3.1 implies that u∧gv∈so(g).\nLemma 3.6. L∈so(g)is simple if and only if detL= 0.\nProof.WriteL=Ag, withAantisymmetric. Since det L= detAdetg, the\nlemma follows from lemma 2.2.\nLemma 3.7. L∈so(g)is simple if and only if L3+(tr2L)L= 0. Moreover\nifL=u∧gv, thentr2L= (uTgu)(vTgv)−(uTgv)2.\nProof.ForL=u∧gv, one computes directly from (6) that ( ∗) trL2= 2γ\nandL3=γL, whereγis the scalar ( uTgv)2−(uTgu)(vTgv). Lemma 3.5 and\n(∗) imply that tr 2L=−γ. For the converse, L3+(tr2L)L= 0 implies that\nL4+(tr2L)L2= 0. Lemma 3.5 then yields det L=−1\n4(trL4+tr2LtrL2) = 0,\nand lemma 3.6 applies.\nThe four–vectors u,vare parallel if and only if u∧gv= 0. However, even\nifu,vare linearly independent (so that u∧gv/negationslash= 0), it is possible that the\nmetric is no longer nondegenerate when restricted to the plane spa nned by\nu,v. We will call such a two–plane degenerate . The next lemma (combined\nwith the previous lemma) shows that degenerate two–planes are sy nonymous\nwithnull two–planes (two–flats) in the sense of [6].\nLemma 3.8. u,vspan a nondegenerate two–plane if and only if tr2u∧gv/negationslash= 0.\n6Proof.Forw=αu+βv, we havewTgu=α(uTgu)+β(vTgu) andwTgv=\nα(uTgv)+β(vTgv). This linear system has a unique solution if and only if\nits determinant ( uTgu)(vTgv)−(uTgv)2= tr2Lis nonzero.\nThe following lemma implies that almost all simple Lorentz bivectors are\ndetermined up to constant multiple by their squares.\nLemma 3.9. SupposeL=u∧gvis such that tr2L/negationslash= 0. ThenPL.=−L2/tr2L\nis orthogonal projection (with respect to g) onto the nondegenerate two–plane\nspanned by u,v; that is,P2\nL=PL,trPL= 2, and(gPL)T=gPL. Conversely,\nifPis orthogonal projection onto a two–plane, the two–plane is necessarily\nnondegenerate, and we have P=PL, whereL=u∧gvandu,vare any\nlinearly independent four–vectors in the image of P.\nProof.The first statement follows fromlemmas 3.7, 3.5, and3.8. Forthe sec -\nond statement, let Pdenote the two–plane forming the image of P. Suppose\nthatw∈ Pis such that wTgy= 0 for all y∈ P. Thus for any four–vector\nx,wTgx= (Pw)Tgx=wTgPx= 0; hence w= 0, sincegis nondegenerate.\nThusPis nondegenerate. The remainder of the lemma follows from the\nuniqueness of orthogonal projection. Indeed, given a four–vec torx, for any\ny∈ Pwe have (Px)Tgy=xTgPy=xTgy. Sincegis nondegenerate on P,\nthe value of Pxis thus uniquely determined by xandg.\n3.3 Orthogonal sum of simple bivectors\nAlthough an element L∈so(g) will not be a simple bivector in general, we\nmay write it as a sum of simple bivectors. Such a sum is not unique; howe ver,\nwe will do so in a canonical way using the operators\nP±.=±L2−µ∓I\nµ+−µ−(7)\nHereµ±are defined as in equation (5). For P±to be well–defined, Lcannot\nbe a simple bivector; that is, det L/negationslash= 0, since by the comments following\nequation (5), this guarantees that µ+−µ−/negationslash= 0.\nTheorem 1. IfL∈so(g)is such that detL/negationslash= 0, then (i)P++P−=I,\n(ii)P+P−= 0 =P−P+, (iii)P2\n±=P±, and (iv)P±L=LP±. Moreover,\nP±Lare simple Lorentz bivectors with L=P+L+P−L,(P+L)(P−L) = 0 =\n(P−L)(P+L), andtr2P±L=−µ±.\n7Proof.SetN.=µ+−µ−. ThenL4+(tr2L)L2+(detL)I= (L2−µ+I)(L2−\nµ−I) =−N2P+P−. ThusP+P−= 0 by lemma 3.4. Properties (i), (iii),\n(iv) also follow from the definitions of P±, coupled with lemma 3.4 and the\nremarks after equation (5). The remaining two algebraic statemen ts follow\nfrom (i), (ii), and (iv), so it suffices to establish that P±Lare simple Lorentz\nbivectors with the second order trace as stated. First, note tha t (L3)T=\n(LT)3= (−gLg−1)3=−gL3g−1. It then follows from equation (7) that\n(P±L)T=−g(P±L)g−1; henceP±L∈so(g). Second, ( P±L)3=P±L3=\n±(L5−µ∓L3)/N. Using lemma 3.4, we have L5=−(tr2L)L3−(detL)L.\nTherefore, ( P±L)3=±µ±(L3−µ∓L)/N=µ±P±L. From lemma 3.7, P±L\nis a simple bivector with tr 2P±L=−µ±.\nCorollary 3.1. Any nonsimple Lorentz bivector Lmay be written as the\nsum of two simple Lorentz bivectors: L=L++L−withL+L−= 0 =L−L+,\nwhere\nL±=±L3−µ∓L\nµ+−µ−\nandtr2L±=−µ±, withµ±are defined as in equation (5).\nWe shall refer to the decomposition L=L++L−in the corollary as\ntheorthogonal decomposition ofL. We remark that since µ+>0 and\nµ−<0 for a nonsimple Lorentz bivector, lemma 3.7 implies that the two–\nplane associated to L+(as in lemma 3.9) is time–like (intersects the null–\ncone) while that of L−isspace–like (does not intersect the null–cone) in\nSynge’s classification of two–planes given in [6].\n3.4 Special case: Minkowski metric\nIn the case when g=η.= diag(−1,1,1,1) is the Minkowski metric, we may\nmake use of the explicit parametrization of antisymmetric matrices g iven in\nequation (3). That is, any element of so(η) can be written in the form\nLxy=/parenleftbigg\n0xT\nxWy/parenrightbigg\n(8)\nfor some x,y∈R3; i.e.,Lxy=Axyη. Note that det Lxy=−(x·y)2, so that\nLxyis simple if and only if x·y= 0. One computes, using lemma 3.5, that\ntr2Lxy=|y|2−|x|2.\n8Theorem 2. Ifx·y/negationslash= 0, thenLxy=La+b++La−b−is the orthogonal\ndecomposition of Lxy:La+b+La−b−= 0 =La−b−La+b+, where\na±=±µ±x+(x·y)y\nµ+−µ−andb±=±µ±y−(x·y)x\nµ+−µ−\nandµ±=1\n2/parenleftbig\n|x|2−|y|2±/radicalbig\n(|x|2−|y|2)2+4(x·y)2/parenrightbig\n.\nProof.Using equation (8), compute L3\nxy=Lx′y′, wherex′= (|x|2−|y|2)x+\n(x·y)y, andy′=−(x·y)x+(|x|2−|y|2)y. Now use corollary 3.1.\nOnemayshowthat a+·a−= 0,b+·b−= 0, and a±×b∓= 0. Necessarily,\na±·b±= 0, sinceLa+b+andLa−b−are simple.\n3.4.1 Relation with the Hodge dual\nWe may also interpret orthogonal decomposition in terms of Hodge d uals.\nWrite the Lorentz bivector Las the two–form L=1\n2Lα\nβeβ∧eα, whereeα\n(α= 0,1,2,3) is a basis of R4andeβ=ηαβeαis the dual basis (the sum-\nmation convention is used). Observe that we may do this as Lη−1=Lη\nis antisymmetric. The Hodge dual (with respect to η) ofAxyisA(−y)x, so\nthat∗Lxy=L(−y)x. Moreover, one computes that ( x·y)a±=µ±b∓and\n(x·y)b±=−µ±a∓. It then follows that\n∗La+b+=µ+\nx·yLa−b−and∗La−b−=−x·y\nµ+La+b+.\n3.4.2 Relation with the spin group\nIn addition, we describe the orthogonal decomposition in terms of t he double\ncovering group homomorphism Ψ : SL2C→SO+(η). Since we are using the\nmathematicians choice of metric signature, we use ρ(u).=/parenleftbigu1+iu2u0+u3\nu0−u3u1−iu2/parenrightbig\nas\nour linear embedding of R4into the algebra of 2 ×2 complex matrices, along\nwith the involution/parenleftbigA B\nC D/parenrightbig⋆.=/parenleftbig¯D¯B¯C¯A/parenrightbig\n, so thatρ(u)⋆=ρ(u) and detρ(u) =\nuTηu. The map Ψ sends the 2 ×2 complex matrix Mwith unit determinant\nto the Lorentz transformation u/ma√sto→ρ−1(Mρ(u)M⋆).\nOn the Lie algebra level, the map ψ:sl2C→so(η) corresponding to\nΨ sends the traceless complex matrix m=/parenleftbiga b\nc−a/parenrightbig\nto the Lorentz bivector\nu/ma√sto→ρ−1(mρ(u) +ρ(u)m⋆). One computes that ψ(m) =Lxy, wherex=/parenleftbig\nRe(b+c),Im(b−c),2Re(a)/parenrightbig\nandy=/parenleftbig\nIm(b+c),−Re(b−c),2Im(a)/parenrightbig\n.\n9Thusx·y= 2Im(a2+bc) and|y|2− |x|2=−4Re(a2+bc). In particular,\nψ(m) is simple if and only if a2+bcis real. While it is possible to work\noutψ−1(La+b+), the resulting expression is not particularly nice. However,\none computes that ψ(im) =∗Lxy, so that by the previous discussion, if\nψ(m+) =La+b+, thenψ(iθm+) =La−b−for some real θ.\nTo get back to the Lie group level, we only need to exponentiate: if\nψ(m) =L, then Ψ(exp( m)) = exp(L). We will return to this briefly at the\nend of section 5.4.\n4 Exponential on so(g)\nRecall that the exponential of a square matrix Mis defined as exp( M) =/summationtext∞\nn=01\nn!Mn. The series always converges, and the exponential of a Lorentz\nbivector lies in the identity component the Lie group of all Lorentz tr ansfor-\nmations.\nTheorem 3. IfLis a simple Lorentz bivector, then exp(L) =I+d1L+d2L2,\nwhere\nd1=sinh√−tr2L√−tr2Landd2=1−cosh√−tr2L\ntr2Liftr2L<0,\nd1=sin√tr2L√tr2Landd2=1−cos√tr2L\ntr2Liftr2L>0,\nand iftr2L= 0, thend1= 1andd2=1\n2.\nProof.Lemma 3.7 implies that L2k+1= (−1)k(tr2L)kLfor allk/negationslash= 0, which\nis used to sum the exponential series.\nThisformula appearsin[2]with −tr2Lreplaced bythesymbol α; however\nin that reference, αis not identified with a matrix invariant, the coefficient of\nL2is incorrect, and the limiting case when tr 2L= 0 is not handled explicitly.\nFor nonsimple Lorentz bivectors, one may sum the exponential ser ies\nusing the identity Lk=µk\n+L++µk\n−L−for allk≥0, which one verifies by\ninduction in conjunction with lemma 3.4. The resulting formula, which is\ncubic inLappears in [2] and in [1], although the formula in the latter takes a\ndifferent formthanthatintheformer; bothreferences usesome what different\nmethods than ours, and in particular, make use of algebra over the complex\nnumbers.\n10Alternatively, we may make use of corollary 3.1 to obtain the exponen tial\nof a nonsimple Lorentz algebra element. Since L+,L−trivially commute,\nwe have exp( L) = exp(L+)exp(L−), and each factor can be computed from\ntheorem 3 to obtain the following.\nTheorem 4. IfL∈so(g)is nonsimple and L=L++L−is the orthogonal\ndecomposition of Las in corollary 3.1, then\nexp(L) =I+d+\n1L++d−\n1L−+d+\n2L2\n++d−\n2L2\n−\nd+\n1= sinh√µ+/√µ+ d−\n1= sin√−µ−/√−µ−\nd+\n2= (cosh√µ+−1)/µ+d−\n2= (cos√−µ−−1)/µ−\nNote that if we write L±in terms of L(according to corollary 3.1), we\nobtain a polynomial of degree six in L. However, the degree can be reduced\nto three by lemma 3.4; the resulting expression is necessarily the sam e as\nthat obtained in [1] and [2].\n5 Lorentz transformation factorization\nTheLorentz group O(g) is set of all linear transformations Λ on R4that\npreserve the metric: ΛTgΛ =g. Necessarily detΛ = ±1, and Λ is said to be\nproperif detΛ = 1. It is well–known that the group of all proper Lorentz\ntransformations has two connected components; the Lorentz t ransformations\nwithin the identity component are said to be orthochronous . In the case\nwheng=ηis the Minkowski metric, the orthochronous transformations are\ncharacterized by the component Λ0\n0being positive. The set of all proper\northochronous Lorentz group elements will be denoted by SO+(g).\nA note on proofs. In the remainder, all proofs will use the following\nabbreviations without warning: τk.= trkΛ,lk.= trkL,x.=√−l2when\nl2<0,y.=√l2whenl2>0,c+.= coshx,c−.= cosy,s+.= sinhx/x, and\ns−.= siny/y. Note that since Λ−1=g−1ΛTg, we have tr kΛ−1=τk= trkΛ.\nWe remind the reader that tr 2Λ =1\n2(tr2Λ−trΛ2), from equation (2).\n5.1 Some Lorentz transformation relations\nLemma 5.1. For allΛ∈SO+(g),tr3Λ = trΛ. Consequently, Λsatisfies the\nrelationΛ4−(trΛ)Λ3+(tr2Λ)Λ2−(trΛ)Λ+I= 0.\n11Proof.The characteristic polynomials of Λ and Λ−1are the same, namely\nλ4−τ1λ3+τ2λ2−τ3λ+1. By the Cayley–Hamilton theorem, ( ∗) Λ4−τ1Λ3+\nτ2Λ2−τ3Λ+I= 0 and ( ∗∗) Λ−4−τ1Λ−3+τ2Λ−2−τ3Λ−1+I= 0. Multiplying\n(∗∗) by Λ4and comparing with ( ∗), we getτ3=τ1.\nLemma 5.2. IfΛ∈SO+(g), thenΛk+ Λ−k=AkI+Bk(Λ +Λ−1), where\nA2=−tr2Λ,B2= trΛ,A3= (2−tr2Λ)trΛ,B3= tr2Λ−tr2Λ−1,A4=\n(2−tr2Λ)tr2Λ+tr2\n2Λ−2, andB4= (tr2Λ−2tr2Λ)trΛ\nProof.From lemma 5.1, we have Λ2−τ1Λ+τ2I−τ1Λ−1+Λ−2= 0; so that\nΛ2+Λ−2=−τ2I+τ1(Λ+Λ−1). Moreover, Λ3−τ1Λ2+τ2Λ−τ1I+Λ−1= 0\nand Λ−3−τ1Λ−2+τ2Λ−1−τ1I+ Λ = 0; thus Λ3+ Λ−3=τ1(Λ2+ Λ−2)−\n(τ2+1)(Λ+Λ−1)+2τ1I=τ1(2−τ2)I+(τ1B2−τ2−1)(Λ+Λ−1). Similarly,\nwe have Λ4+ Λ−4=τ1(Λ3+ Λ−3)−τ2(Λ2+ Λ−2) +τ1(Λ + Λ−1)−2I=\n(τ1A3−τ2A2−2)I+(τ1B3−τ2B2+τ1)(Λ+Λ−1).\n5.2 Simple Lorentz transformations\nWe will say that a Lorentz transformation is simpleif it is the exponential\nof a simple Lorentz bivector. Our first goal is to give an algebraic crit erion\nfor simplicity. We will make use of the fact that the orthochronous L orentz\ngroup is exponential (see [3]); that is, exp : so(g)→SO+(g) is surjective.\nLemma 5.3. For nonsimple Λ∈SO+(g), there is a nonsimple Lorentz\nbivectorLwith orthogonal decomposition L=L++L−such that Λ = exp(L)\nandtr2L±=−µ±, where√µ+= cosh−11\n4(trΛ+√\n∆),√−µ−= cos−11\n4(trΛ−√\n∆), and∆ = tr2Λ−4tr2Λ+8. Moreover, we have trΛ = 2(c++c−)and\ntr2Λ = 4c+c−+2, wherec+.= cosh√µ+,c−.= cos√−µ−.\nProof.SinceΛisnotsimple, Λ = exp( L)forsomenonsimpleLorentzbivector\nL, and theorem 4 applies. Moreover, using lemma 3.7 one computes\nΛ2=I+2d+\n1(1+µ+d+\n2)L++2d−\n1(1+µ−d−\n2)L−+A+L2\n++A−L2\n−(9)\nwhereA±.= (2d±\n2+d±2\n1+µ±d±2\n2) = 2(c2\n±−1)/µ±. Computing the traces\nin theorem 4 and equation (9), we obtain trΛ = 4 + d+\n2trL2\n++d−\n2trL2\n−and\ntrΛ2= 4 +A+trL2\n++A−trL2\n−. Using trL2\n±=−2tr2L±= 2µ±, one then\ncomputes that ( ⋆) trΛ = 2(c++c−) and trΛ2= 4(c2\n++c2\n−−1). Consequently,\n(⋆⋆) tr2Λ =1\n2(tr2Λ−trΛ2) = 4c+c−+2. Equations ( ⋆) and (⋆⋆) can then be\nsolved forc±to obtain the formulas in the statement of the lemma.\n12Lemma 5.4. Λis simple if and only if tr2Λ = 2trΛ −2andtrΛ≥0.\nProof.If Λ is simple, then Λ = exp( L) for some simple Lorentz bivector\nL. From theorem 3 and lemma 3.5, ( ∗)τ1= 4−2d2l2. Moreover, we\nhave Λ2=I+ 2d1L+ (2d2+d2\n1)L2+ 2d1d2L3+d2L4, and one computes\nτ2=1\n2(τ2\n1−trΛ2) = 6+(d2\n1−6d2)l2+d2\n2l2\n2using lemma 3.5 and det L= 0.\nThus (∗∗)τ2−2τ1+ 2 = (d2\n1−2d2)l2+d2\n2l2\n2. By theorem 3, if l2<0,\nthend2l2= 1−coshxandd2\n1l2=−sinh2x= 1−cosh2x. Thus (∗) yields\nτ1= 2(1+cosh x)≥4, and (∗∗) yieldsτ2−2τ1+2 = 0. Similarly, if l2>0,\nthenτ1= 2(1+ cos y)≥0 andτ2−2τ1+2 = 0; and if l2= 0, thenτ1= 4\nandτ2−2τ1+2 = 0.\nNow suppose that Λ is not simple, and write Λ = exp( L) as in lemma\n5.3. Thusτ2−2τ1+2 =−4(c+−1)(1−c−), which is only zero when c+= 1\norc−= 1. As Λ is nonsimple, µ+>0 andµ−<0; whencec+>1, and\nc−= 1 only if√−µ−is a nonzero multiple of 2 π. However in the latter case,\ntheorem 4 implies that exp( L) = exp(L+), which cannot be the case, since\nΛ is assumed to be nonsimple.\nWe note that the proof of lemma 5.4 implies that if Λ is nonsimple, then\nc+>1 and−1≤c−<1. In particular, trΛ >0.\n5.3 Simple logarithm\nTheorem 5. Suppose Λ∈SO+(g)is simple. For trΛ>0, let us define\nLΛ=1\n2k(Λ−Λ−1), where\n(i)k=√µ\nsinh√µand√µ= cosh−1(1\n2trΛ−1), iftrΛ>4\n(ii)k=√−µ\nsin√−µand√−µ= cos−1(1\n2trΛ−1), if0<trΛ<4\n(iii)k= 1andµ= 0, iftrΛ = 4\nThenLΛis a simple Lorentz bivector with exp(LΛ) = Λandtr2LΛ=−µ.\nFortrΛ = 0,PΛ=1\n2(I−Λ)is orthogonal projection onto a nondegenerate\ntwo–plane such that −π2PΛis the square of a simple Lorentz bivector Lwith\nexpL= Λandtr2L=π2.\n13Thus if trΛ /negationslash= 0, thenLΛmay be taken as the logarithm of Λ: LΛ= logΛ.\nIn the case trΛ = 0, we may reconstruct logΛ from the two independ ent\nfour–vectors in the image of PΛ; it should be noted that in this case, Λ is\nnecessarily an involution: Λ2=I. In general, the logarithm is not unique,\nsee [4]. In particular if Lis simple with tr 2L= 2nπ, for any integer n, then\nΛ.= exp(L) =Iby theorem 3. In this case, the above theorem gives LΛ= 0.\nA version of the formula for logΛ in the case trΛ /negationslash= 0 that involves algebra\nover the complex numbers appears in [1].\nProof.Write Λ = exp( L) for some simple L. By theorem 3, we have ( ⋆)\nd1L=1\n2(Λ−Λ−1). As in the proof of lemma 5.4, ( ⋆⋆)τ1= 4−2d2l2.\nFrom theorem 3, we see that d1= 0 only if l2>0 and√l2is a nonzero\nmultiple of π; by the periodicity of the sine and cosine, we may assume that\nl2=π2. In this case, d2= 2/π2and by (⋆⋆),τ1= 0. Conversely, if τ1= 0,\nthen (⋆⋆) impliesd2l2= 2, which only happens if l2>0 and√l2is a nonzero\nmultiple of π. Thus ifτ1= 0, then Λ = I+(2/π2)L2=I−2PL, withPLas\nin lemma 3.9. Thus, PL=PΛ.\nIfτ1/negationslash= 0, thend1/negationslash= 0; and from ( ⋆) and theorem 3, we only need to\ndeduce the value of l2in terms of Λ in order to obtain the formulas (i)—(iii).\nIf we assume that l2<0, then theorem 3 states that d2l2= 1−coshx; and\nso (⋆⋆) can be solved to yield cosh x=1\n2τ1−1. Note that this equation has\na solution for x>0 if and only if τ1>4. The cases when l2>0 andl2= 0\nare handled similarly.\n5.4 Decomposition into simple factors\nIn the case when Λ is nonsimple, then L= logΛ is a nonsimple bivector,\nand we have the orthogonal decomposition L=L++L−into simple mu-\ntually annihilating summands. Thus Λ = exp( L+)exp(L−) is a product of\ncommuting simple factors. We give explicit formulas.\nTheorem 6. IfΛ∈SO+(g)is nonsimple, then Λ = exp(L)whereLis a\nnonsimple Lorentz bivector with projection operators\nP±=±1\n2(Λ+Λ−1)−c∓I\nc+−c−\nwherec±are as in lemma 5.3.\n14Proof.From theorem 4, we have ( ∗) Λ−1= exp(−L) =I−d+\n1L+−d−\n1L−+\nd+\n2L2\n++d−\n2L2\n−, and we obtain the equation1\n2(Λ+Λ−1) =I+d+\n2L2\n++d−\n2L2\n−,\nwhich we rewrite as ( ◦)d+\n2L2\n++d−\n2L2\n−=−I+1\n2(Λ + Λ−1). In addition,\nfrom equation (9), we have1\n2(Λ2+ Λ−2) =I+A+L2\n++A−L2\n−. However,\nby lemma 5.2, this equation may be rewritten as ( ◦◦)A+L2\n++A−L2\n−=\n−1\n2(τ2+2)I+1\n2τ1(Λ+Λ−1). Equations ( ◦) and (◦◦) form a linear system in\nL2\n±, whose determinant is computed to be 2( c+−1)(1−c−)(c+−c−)/µ+µ−,\nwhich is never zero. After inverting the system ( ◦), (◦◦) , one computes\nL2=L2\n++L2\n−=1\nc+−c−(µ−c+−µ+c−)I+1\n2(µ+−µ−)(Λ+Λ−1)\nEquation (7) then yields the desired formulas for P±.\nTheorem 7. SupposeΛ∈SO+(g)is nonsimple. Let c±, andµ±be defined\nas in lemma 5.3. If µ−/negationslash=−π2, thenΛ = exp(L)withLa nonsimple Lorentz\nbivector whose orthogonal decomposition L=L++L−is given by\nL±=∓1\n(c+−c−)s±/braceleftbig1\n2c∓(Λ−Λ−1)−1\n4(Λ2−Λ−2)/bracerightbig\nwheres+.= sinh√µ+/√µ+ands−.= sin√−µ−/√−µ−. Moreover, tr2L±=\n−µ±.\nProof.From equation (9) and equation ( ∗) in the proof of the previous theo-\nrem, we obtain the equations1\n2(Λ−Λ−1) =d+\n1L++d−\n1L−and1\n4(Λ2−Λ−2) =\nd+\n1(1 +µ+d+\n2)L++d−\n1(1 +µ−d−\n2)L−. These define a linear system for L±.\nThe determinant is −(c+−c−)s+s−, which is zero only when√−µ−is a\nnonzero multiple of π. Inverting the linear system yields the formulas in the\nstatement of the theorem.\nTheorem 8. For nonsimple Λ∈SO+(g),Λ = Λ +Λ−, whereΛ±are the\ncommuting simple Lorentz transformations given by\nΛ±=±1\n2(c+−c−)/braceleftbig\n(1+2c±)I−Λ−1−(1+2c∓)Λ+Λ2/bracerightbig\nwherec±are as in lemma 5.3.\nProof.LetLbe a nonsimple Lorentz bivector such that Λ = exp( L), andP±\nas in theorem 6. Observe that exp( P±L) =P∓+P±Λ. Indeed, ( P±L)n=\nPn\n±Ln=P±Lnfor alln >0, sinceP±is a projection that commutes with\nL. Thus, exp( P±L) =I+/summationtext\nn>01\nn!(P±L)n=I+P±(/summationtext\nn>01\nn!Ln) =I+\nP±(exp(L)−I). The formulas for Λ ±then follow from theorem 6.\n15As we noted in section 3.4.2, in the special case when g=ηis the\nMinkowski metric, we may write L±in terms of the Lie algebra homomor-\nphismψ:sl2C→so(η). Namely, ψ(m+) =L+andψ(iθm+) =L−for\nsome traceless 2 ×2 complex matrix m+=/parenleftbiga b\nc−a/parenrightbig\nsuch thata2+bcis real.\nIt follows that M+.= exp(m+) andM−.= exp(iθm+) give (commuting)\nmatrices in SL2Cwith Ψ(M±) = Λ ±; i.e.,M±gives the decomposition of\nΨ(M+M−) = Λ into simple factors. Observe that exp( m+) is readily com-\nputed, since m2\n+= (a2+bc)I.\nWe may view the decomposition of a nonsimple Lorentz transformatio n\ninto commuting simple factors as a generalization of Synge’s 4–screw: a\nproduct of a boost and a rotation in orthogonal two–planes. While it is\nknown that every proper nonsimple Lorentz transformation can b e expressed\nastheproductoftwocommutingsimpleLorentztransformations( see[5]),the\nformulas presented here are ostensibly original, and in any case inde pendent\nof the specific form of the Lorentz metric.\nReferences\n[1] Bortolom´ e Coll and Fernando San Jos´ e, On the exponential of the 2-\nforms in relativity, GeneralRelativityandGravitation,22(7),811–826,\n1990.\n[2] Christopher Geyer, Catadioptric Projective Geometry: Theory and Ap-\nplications, Ph.D. thesis, University of Pennsylvania, December 2002.\n[3] Mitsuru Nishikawa, On the Exponential Map of the Group O(p,q)0,\nMemoirs of the Faculty of Science, Kyushu University, Series A, Mat h-\nematics, Vol. 37, 63–69, 1983.\n[4] Ronald Shaw and GrahamBowtell, The bivector logarithm of a Lorentz\ntransformation, Quarterly Journal of Mathematics, 20 (1), 497–503,\n1969.\n[5] E. J. Schremp, Simply transitive subgroups G4of the Poincar´ e group:\na new finding in the special theory of relativity, technical report of the\nInternational Centre for Theoretical Physics, Trieste, IC–68– 92, 1968.\n[6] J. L. Synge, Relativity: The Special Theory, second edition, North–\nHolland Pub. Co., Amsterdam, 1965.\n16"    },    {        "title": "2106.12134v1.Regularization_of_central_forces_with_damping_in_two_and_three_dimensions.pdf",        "content": "arXiv:2106.12134v1  [math-ph]  23 Jun 2021Regularization of central forces with damping in two and\nthree-dimensions\nE. Harikumar∗and Suman Kumar Panja†\nSchool of Physics, University of Hyderabad,\nCentral University P.O, Hyderabad-500046, Telangana, India\nPartha Guha‡\nDepartment of Mathematics\nKhalifa University of Science and Technology P.O. Box 127788, Abu Dh abi, UAE\nJune 24, 2021\nAbstract\nRegularization of damped motion under central forces in two and th ree-dimensions are investi-\ngated and equivalent, undamped systems are obtained. The dynam ics of a particle moving in1\nr\npotential and subjected to a damping force is shown to be regulariz ed a la Levi-Civita. We then\ngeneralize this regularization mapping to the case of damped motion in the potential r−2N\nN+1. Fur-\nther equation of motion of a damped Kepler motion in 3-dimensions is ma pped to an oscillator with\ninverted sextic potential and couplings, in 4-dimensions using Kusta anheimo-Stiefel regularization\nmethod. It is shown that the strength of the sextic potential is giv en by the damping co-efficient of\nthe Kepler motion. Using homogeneous Hamiltonian formalism, we esta blish the mapping between\nthe Hamiltonian of these two models. Both in 2 and 3-dimensions, we sh ow that the regularized\nequation is non-linear, in contrast to undamped cases. Mapping of a particle moving in a harmonic\npotential subjected to damping to an undamped system with shifte d frequency is then derived using\nBohlin-Sudman transformation.\n1 Introduction\nDissipation being unavoidable in natural systems, is of int rinsic interest. The investigation of such\nsystems has a long history and continues to be an active area o f research [1–3]. Damped harmonic\noscillator is one of the systems that has been studied vigoro usly as a prototype of dissipative systems.\nVarious approaches such as (i) coupling the system to a heat b ath, (ii) use of Bateman-Caldirola-\nKanai(BCK) [4–6] model which uses a time dependent Lagrangi an/Hamiltonian have been developed\nand adopted for studying different aspects of dissipative sys tems. Each of these methods though having\nunique advantages, has some unsatisfactory features [7–10 ].\nBCK model has been shown to be plagued by difficulties in interp retation even at the classical level,\nas the time dependent BCK-Lagrangian/Hamiltonian leading to correct damped equation of motion\nis shown to describe a variable mass system rather than a trul y damped oscillator [7,9]. It has been\nargued that the equivalence between BCK model and damped har monic oscillator is not valid globally(\ni.e., not for all times) and they are equivalent only for finit e time scales [10].\n∗harisp.uoh@nic.in\n†sumanpanja19@gmail.com\n‡partha.guha@ku.ac.ae\n1Generalization of action principle which allows to include non-conservative systems in its ambit\nwas presented long ago [11] and this method can be used to stud y dissipative systems. Use of contact\nmanifolds to study dissipative systems is a new upsurge of in terests among mathematicians [12–14].\nRegularization is considered to be a tool for converting sin gularities of a system of differential equations\ninto regular ones. This is carried out by changing the depend ent and/or independent variables of a\nsystem of (singular) differential equations so that the new sy stem is free of singularities [15]. A well\ndeveloped theory of regularization in celestial mechanics might be attributed to Bohlin, Sundman and\nLevi-Civita [16,17]. The notion of Levi-Civita regulariza tion is now used for the regularization of the\nbinary collisions in the planar (2-dimensional) Kepler pro blem [17]. Generalization of these approach\nto 3-dimensions was obtained by Kustaanheimo and Stiefel [1 8]. These regularization schemes also lead\nto linear differential equations. There are other schemes of r egularization such as those developed by\nMoser [19] and Ligon-Schaaf [20]. Moser showed that the flow o f then-dimensional Kepler problem on\nthe surface of constant negative energy is conjugate to the g eodesic flow on the unit tangent bundle of\nSn. The Ligon-Schaaf regularization procedure tackles toget her all the surfaces of negative energies.\nKepler problem is the most studied system which has singular ity. Keplerian orbit where the sep-\naration of particle from the central mass goes to zero is call ed a collision orbit. Detailed analysis of\nsuch orbits are necessary for artificial satellite missions , at the starting point as well as at the final\nlanding point in moon/planet/exoplanet, the separation di stance vanishes. Another situation that re-\nquires regularization is the low energy orbits where there i s no collision, but the probe passes very\nclose to moon/planet to achieve boost (known as slingshot effe ct) where the separation is negligibly\nsmall compared to other length scales involved [21]. Regula rization also becomes handy in dealing with\nperturbations to Kepler problem due to effects such as presenc e of a third body, change in the shape of\nbodies from spherical one assumed.\nMotion under the influence of central force in presence of fri ction arise in systems such as Rydberg\natom and binary stars and has been of interest. Effect of fricti on on the shape of orbits has been\nstudied [22–24]. Kepler problem with drag term, which is lin ear in velocity and inversely proportional\nto the square of the separation distance was studied and anal ytical solutions were obtained [25–27]. In\nthese studies, existence of a conserved quantity which is re lated to angular momentum was exploited in\nderiving the solutions. Further, for a specific choice of coe fficient of the drag term, this equation was\nshown to be related to that of harmonic oscillator [26]. Kepl er equations augmented with drag term\nwas used to model the orbits of artificial satellites landing on other planets/moon, for orbits of re-entry\nin to earth’s atmosphere as well as for analyzing motion of lo w orbit satellites. In this paper, we study\nregularization of collision orbits of particle moving unde r the influence of inverse power law potentials\nand also subjected to velocity dependent damping, in 2 and 3 d imensions.\nIn this paper, we use the approaches of Levi-Civita and Kusta anheimo and Stiefel to regularize\nproblem of a particle moving in central potentials and subje cted to damping in 2 and 3-dimensions. In\nparticular, we investigate the construction of equivalent model corresponding to systems with damping,\nviz: (i) particle in 2and3-dimensions, movingundertheact ion ofthe1\nrpotential subjectedto damping,\nand (ii) particle in 2-dimensions moving under the influence of a potential of the form r−2N/N+1where\nN∈Z, subjected to damping. We then study regularization of coll ision orbits of the particle in these\nsituations. The equations of motion describing the damped m otion, as well as the corresponding La-\ngrangian/Hamiltonian in the above cases have explicit time -dependence. Thus for these systems, energy\nis not a conserved quantity and this hinders the direct appli cation of Levi-Civita/K-S-transformations\nto these systems with damping. Also it is of interest to see ho w these equations transform under the\nre-parametrization of time, which is an integral part of the se regularizations. The re-parametrization\nof time which makes the motion near the collision point “slow ” do affect the form of the equations of\nmotion in terms of the new co-ordinates, through velocity an d acceleration. Since in the case of damped\nsystems we study, not only Lagrangian and Hamiltonian, but t he equations of motion also dependent\nexplicitly on the time,it is of interest to see how this expli cit time dependence affect the regularization.\nIn our analysis, we start with a Lagrangian (which can be rela ted to a BCK type, time dependent\nLagrangian) whose equation of motion describes a particle s ubjected to Kepler potential, in addition to\n2a velocity dependent damping. We first map these equations us ing a time dependent point transforma-\ntion such that the transformed equations follow as Euler-La grange equations from a time independent\nLagrangian. This allows us to construct conserved energy, w hich in turn allow the implementation of\nthe mapping of dynamics on a constant energy surface. We appl y Levi-Civita map /K-S transformation\nto these equations, after re-expressing them in terms of com plex co-ordinates/quaternions. The equa-\ntion in terms of the new complex co-ordinates/quaternions a nd time parameter is shown to describe a\nharmonic oscillator augmented by an inverted sextic potent ial in 2-dimensions/4-dimensions.\nThe regularization scheme known as Levi-Civita transforma tion for 2-dimensional Kepler problem\nandits generalization to3-dimensions knownasKustaanhei mo-Stiefel (K-S) transformation, dolinearize\nthe differential equations describing the motion of a particl e under the influence of gravitational force\nexcreted by a central body and also removes the singularity o f the equations when the separation of\nthese two bodies vanishes. The procedure consists of three steps ;\n1. In the first step , one implements a re-parametrization of time so that the vel ocity defined in terms\nof “new” time variable, do not diverge as the separation dist ance approaches zero. One now\nuses chain rule and re-express the time derivatives of the po sition co-ordinates appearing in the\ndifferential equation in terms of the “new” time variable.\n2. In the second step , one applies Levi-Civita conformal squaring (of the comple x co-ordinates) for\n2-dimensional case and K-S transformation which relates co -ordinates of 3-dimensional space to\nsquare of quaternions. Thus the equations of motion are now r e-expressed in terms of new co-\nordinates and their derivatives with respect to “new” time p arameter. Thus obtained equation is\nregular but non-linear.\n3. In the third or final step , one starts with the conserved energy associated with the in itial system,\nwritten in terms of kinetic and potential energies. One re-e xpresses the velocities appearing in\nkinetic part in terms of the derivative of new co-ordinates w ith respect to “new” time parameter\nand co-ordinates in potential energy are re-written in term s of new co-ordinates. This conserved\nquantity is then used to re-express the equations of motion o btained in the second step above.\nStraight forward calculations then converts the equations of motion to be regular and linear one.\nThis equation is then solved and by fixing the numerical value of the conserved quantity, one maps\norbits of fixed energy to regular solutions of the linear equa tion.\nIn this paper, we apply these regularization methods to Kepl er problem in 2 and 3 dimensions as\nwell as to a generic inverse power law potential, all subject ed to velocity dependent damping. We show\nthat the mapping leads to regular, but coupled equations. Fu rther, the coupling terms all have the\ndamping parameter as co-efficient. We also study the applicat ion of Bohlin-Sudman transformation to\nthe equation describing damped harmonic oscillator in 2-di mensions. After applying a time dependent\nco-ordinate transformation, we map the damped harmonic mot ion to that of a shifted harmonic oscil-\nlator. As earlier, this allows us to define conserved energy w hich facilitate the mapping of dynamics\nfrom a constant energy surface. This equation, after re-exp ressing in terms of complex co-ordinates, are\nmapped to that of 2-dim Kepler problem by Bohlin-Sudman tran sformation [16].\nThis paper is organized as follows. In the next section, we sh ow that the Levi-Civita transformation\nmaps the Kepler’s equations with damping in 2-dimensions to the equations of motion of a harmonic\noscillator with an additional, inverted, sextic potential . This is done by first mapping the damped\nKepler equations to an equivalent set of equations which do n ot have explicit time dependence. These\nequations comes from a time independent Lagrangian and thus the corresponding energy is a conserved\nquantity. These equations are mapped by Levi-Civita regula rization map to equations describing a\nharmonic oscillator, augmented with inverted, sextic pote ntial. In section 3, we use Levi-Civita map\nto obtain a regularized equation corresponding to equation describing a particle moving, in presence of\nfriction, under the generic inverse power law potential of t he form r−2N/N+1, 0≤2N/N+1<2 . In\nobtaining this mapping we follow the same steps as we have use d in section 2, in deriving equivalence\n3between 2-dimensional Kepler motion with drag to that of har monic oscillator in presence of inverted\nsextic potential. In both these cases, we express the confor mal squaring of the co-ordinates (discussed\nabove) as a matrix equation and further show that this equati on can beexpressed usinga generic matrix\nconnecting old and new co-ordinate. This matrix is written i n terms of two permutation operators which\ndocommuteamongthemselves. Insection 4, westudytheKeple r problemin3-dimensionsin presenceof\ndamping. After a brief summary of quaternions, in subsectio n 4.2, we show that the K-S transformation\nmaps the equations of motion of 3-dimensional Kepler proble m with damping to that of a 4-dimensional\nharmonic oscillator with the inverted sextic potential. In subsection 4.3, we show the mapping of these\ntwo systems at the level of Hamiltonians, using homogeneous Hamiltonian formalism. Our concluding\nremarks are given in section 5. In appendix A, we show the mapi ng of damped harmonic motion to\nthat of Kepler problem, using Bohlin-Sudman transformatio n.\n2 Kepler problem in presence of damping in 2-dimensions: Lev i-\nCivita map\nIn this section, we apply Levi-Civita map to damped Kepler pr oblem in 2-dimensions and obtain an\nequivalent but regularized formulation. We begin with a BCK type, time dependent Lagrangian whose\nequation of motion describes a particle moving in Kepler pot ential and also subjected to velocity depen-\ndent damping. We start by mapping these equations using a tim e dependent point transformation, so\nthat the modified equations are Euler-Lagrange equations fr om a time independent Lagrangian. This\npermits us to build conserved energy, which allows us to appl y the mapping of dynamics on a constant\nenergy surface. After re-expressing these equations in ter ms of complex co-ordinates, we use the Levi-\nCivita map to get equations for a harmonic oscillator with in verted sextic potential and interactions.\nWe note that the equation of motion following from a BCK type L agrangian\nL=eλt/bracketleftigg\nm\n2(˙x2\n1+ ˙x2\n2)+ke−3λt\n2\n˜r/bracketrightigg\n(2.1)\nwhere ˜r=/radicalbig\nx2\n1+x2\n2,m=m1+m2\nm1m2andk=Gm1m2is\nm¨xi+λm˙xi+ke−3λt\n2\n˜r3xi= 0, i= 1,2, (2.2)\nwhere ˙xi=dxi\ndtand ¨xi=d2xi\ndt2. These equations describe motion of a particle of mass m1under the influ-\nence of a gravitational potential of another body of mass m2as well as a damping force, in 2-dimensions.\nHere when λ−→0, above Eqn.2.2 becomes equation of motion for well known Ke pler problem in 2-\ndimensions. Note that this equation has explicit time depen dence and the re-parametrization of time\napplied in the first step of regularization, will not remove t his explicit dependence on the “old” time.\nAlso, the energy of this system is not conserved and thus impl ementing the third step of regularization is\nnot possible. In order to avoid these complications, we now r e-write these equation of motion in terms\nof a new set of co-ordinates ( X1,X2) which are related to old co-ordinates through time depende nt\ntransformation given by\nX1=x1eλt\n2,\nX2=x2eλt\n2. (2.3)\nIn terms of these co-ordinates, Eqn.(2.2) become\nm¨X1−mλ2\n4X1+kX1\n(X2\n1+X2\n2)3\n2= 0, (2.4)\nm¨X2−mλ2\n4X2+kX2\n(X2\n1+X2\n2)3\n2= 0. (2.5)\n4These equations do not have explicit time dependence and firs t difficulty in implementing the regular-\nization is now circumvented.\nIt is easy to see that these are Euler-Lagrange equations fol lowing from the Lagrangian\nL=m\n2(˙X2\n1+˙X2\n2)+mλ2\n8(X2\n1+X2\n2)−mλ\n2(X1˙X1+X2˙X2)+k\nr(2.6)\nNote that the third term can be re-written as −mλ\n4d\ndt(X2\n1+X2\n2), a total derivative and thus will\nnot contribute to equations of motion. Following the standa rd procedure, we calculate the canonical\nconjugate momenta corresponding to X1andX2as\nPX1=m˙X1−λ\n2mX1, (2.7)\nPX2=m˙X2−λ\n2mX2 (2.8)\nand using these derive the Hamiltonian as\nH=(P2\nX1+P2\nX2)\n2m+λ\n2(X1PX1+X2PX2)−k\nr, (2.9)\nwhere we now use r=/radicalbig\nX2\n1+X2\n2. This Hamiltonian does not have any explicit time dependenc e and is\ninvariant under time translations and hence total energy of this system is conserved. Thus, the second\ndifficulty mentioned above in implementing regularization i s also avoided.\nWe note that the angular momentum L=X1PX2−X2PX1is also conserved, as in the usual Kepler\nproblem. For later purposes, we express (negative of) this H amiltonian, which is a conserved quantity,\nin terms of velocities and co-ordinates, viz:\n−E=m\n2(˙X2\n1+˙X2\n2)−λ2\n8m(X2\n1+X2\n2)−k\nr. (2.10)\nWe re-express the Eqn.(2.4) and Eqn.(2.5) in terms of comple x variable Zgiven by Z=X1+iX2\nas\nm¨Z+k\n|Z|3Z=mλ2\n4Z, (2.11)\nwhere|Z|=/radicalbig\nX2\n1+X2\n2=r. Note that the derivative with respect to the time variable tis represented\nby ‘overdot’, in the above equations.\nWe now apply a re-parametrization of time and demand1\nd\ndt=c\nrd\ndτ. (2.12)\nNote here that the cappearing in the above equation is a proportionality consta nt(and not the velocity\nof light). Applying this re-parametrization, we find\ndZ\ndt=c\nrdZ\ndτ(2.13)\nd2Z\ndt2=c2\nr2d2Z\ndτ2−c2\nr3dr\ndτdZ\ndτ. (2.14)\nUsing these, we re-express Eqn.(2.11) as\nc2(rd2Z\ndτ2−dr\ndτdZ\ndτ)+µZ=λ2\n4r3Z, (2.15)\nwhereµ=k/m.\n1Conservedangular momentumin dampedKeplerproblem andper turbedharmonic oscillator (seeEqn.(2.6),Eqn.(2.30))\nallows us to equate these numbers and this leads to the relati on in Eqn.(2.12).\n5Next we apply a co-ordinate transformation(Levi-Civita re gularization) and re-write above equation\nin terms new complex co-ordinate U=U1+iU2which is related to Zas\nZ=γU2(2.16)\nwhich we can rewrite in matrix representation as following,\n/parenleftbiggX1\nX2/parenrightbigg\n=γA(U) =γˆU1/parenleftbiggU1\nU2/parenrightbigg\n, (2.17)\nwhere\nA(U) =ˆU1=/parenleftbiggU1−U2\nU2U1/parenrightbigg\n. (2.18)\nWhich we can rewrite as follows,\nA(U) =/parenleftig\nP(1)U,P(2)U/parenrightig\n, (2.19)\nwhere we have considered following set of permutations,\nP(1)U= (U1,U2)T,P(2)U= (−U2,U1)T. (2.20)\nWhich have properties like,\nP(i)P(j)=P(j)P(i)and(Pi)TPj=δij. (2.21)\nIn Eqn.(2.16), γis constant having dimensions of inverse length(without lo se of generality, we set this\nγto be one from now onwards). This sets r=|Z|=U¯U=|U|2,r2=¯ZZ. We also finddZ\ndτ= 2UdU\ndτ,\nd2Z\ndτ2= 2(dU\ndτ)2+2Ud2U\ndτ2anddr\ndτ=¯UdU\ndτ+d¯U\ndτU, using which, the Eqn.(2.15) becomes\n2c2U¯U/parenleftigg\nUd2U\ndτ2+/parenleftbiggdU\ndτ/parenrightbigg2/parenrightigg\n−c2/parenleftbigg\n¯UdU\ndτ+d¯U\ndτU/parenrightbigg\n2UdU\ndτ+µU2=λ2\n4U2(¯UU)3.(2.22)\nThis equation, after straight forward algebra gives\n2rd2U\ndτ2−2/vextendsingle/vextendsingle/vextendsingle/vextendsingledU\ndτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nU−λ2\n4c2r3U+µ\nc2U= 0. (2.23)\nNext,we express the conserved quantity −Ein terms ofdU\ndτ. For this, we first re-express −Ein terms of\nZanddZ\ndtas\n−E=m\n2d¯Z\ndtdZ\ndt−mλ2\n8¯ZZ−k\n|Z|(2.24)\nand apply re-parametrization to replace derivative with re spect to twith derivative with respect to τ\nand finally, re-express these terms using the derivative of U. This gives\n−E=m\n2c2\nr2d¯Z\ndτdZ\ndτ−mλ2\n8r2−k\nr(2.25)\n≡2mc2\nr/vextendsingle/vextendsingle/vextendsingle/vextendsingledU\ndτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−mλ2\n8r2−k\nr. (2.26)\nUsing this, we re-write/vextendsingle/vextendsingledU\ndτ/vextendsingle/vextendsingle2in Eqn.(2.23) and find\nd2U\ndτ2+/parenleftbiggE\n2mc2−3λ2\n16c2r2/parenrightbigg\nU= 0, (2.27)\n6wherer2=|U|4. This equation describes a perturbed harmonic oscillator f orλ <<1. To see this\nclearly, we re-write the above equation in U1andU2as\nU′′\n1+1\n2c2/parenleftbiggE\nm−3\n8λ2(U2\n1+U2\n2)2/parenrightbigg\nU1= 0 (2.28)\nU′′\n2+1\n2c2/parenleftbiggE\nm−3\n8λ2(U2\n1+U2\n2)2/parenrightbigg\nU2= 0 (2.29)\nwhere a ‘prime’ over Ustands for derivative with respect to the new time variable, τ. It is easy to see\nthat these equations follow from the Lagrangian\nL=m\n2(U′2\n1+U′2\n2)−E\n4c2(U2\n1+U2\n2)+1\n32mλ2\nc2/parenleftbig\nU2\n1+U2\n2/parenrightbig3. (2.30)\nWenowseethatthesystemdescribedbytheaboveLagrangian i soscillator in2-dimensionswithinverted\nsextic potential(i.e., with coefficients of U2\n1andU2\n2are negative) and couplings. Note that, for small λ,\nwe can treat this system as uncoupled harmonic oscillators w ith perturbations involving couplings and\ninverted sextic potential2.\nThe Hamiltonian following from the above Lagrangian is give n by\nH=P2\nU1\n2m+P2\nU1\n2m+E\n4c2(U2\n1+U2\n2)−1\n32mλ2\nc2/parenleftbig\nU2\n1+U2\n2/parenrightbig3. (2.31)\nNote that the λdependent term is with negative coefficient.\nThus we have shown that the equations of motion following fro m the Lagrangian in Eqn.(2.6)\nare mapped by Levi-Civita map to equations following from th e Lagrangian in Eqn. (2.30). This\ngeneralizes the equivalence of the Kepler problem to the har monic oscillator in 2-dimensions to the\nequivalence of the Kepler problem in presence of a damping fo rce, to a perturbed harmonic oscillator.\nThis is shown by first mapping the Eqn.(2.2) describing the da mped Kepler equations in 2-dimensions\nto the Eqns.(2.4,2.5) (which follow from the Lagrangian giv en in Eqn.(2.6)), which are then mapped to\nthat of harmonic oscillator in 2-dimensions with specific co uplings and inverted sextic potential. This\nmapping is obtained for the surface defined by the constant va lue ofE. From Eqn.(2.24), we note that\nthis conserved quantity reduces to energy of the Kepler syst em in the limit λ→0 and thus, in the limit\nλ→0, we get back the well known equivalence between Kepler moti on and harmonic oscillations in\n2-dimensions, on the constant energy surface.\n3 Mapping of motion in1\nr2N\nN+1,0≤2N/N+1<2potential with damping\nin 2-dimensions\nIn this section, we show that the Levi-Civita mapping regula rizes the damped motion in generic inverse\npower law potential given by1\nr2N\nN+1,0≤2N/N+1<2. In the undamped case, such a generalization\nwas shown in [28]. After expressing the transformations bet ween old and new co-ordinates as a matrix\nequation, in [28], this matrix was written using a pair of per mutation operators. The commuting nature\nof these permutation operators was crucial in proving vario us identities which in turn were used for this\ngeneralization of Levi-Civita transformation. Here, we ap ply this generalization to the system moving\nunder the influence of the potential1\nr2N\nN+1,0≤2N/N+1<2 and subjected to damping.\nWe start from a BCK type Lagrangian given by\nL=eλt/bracketleftigg\nm\n2(˙x2\n1+ ˙x2\n2)+ke−2N+1\nN+1λt\n˜r2N\nN+1/bracketrightigg\n(3.1)\n2Inserting γback into above equation, we find the coefficient of the last ter m in the Lagrangian ismλ2γ2\n32c2.\n7where ˜r=/radicalbig\nx2\n1+x2\n2,m=m1+m2\nm1m2andk=Gm1m2. The equations of motion following from this\nLagrangian are\nm¨xi+λm˙xi+/parenleftbigg2N\nN+1/parenrightbiggke−2N+1\nN+1λt\n˜r4N+2\nN+1xi= 0, i= 1,2, (3.2)\nwhere ˙xi=dxi\ndtand ¨xi=d2xi\ndt2. Here when N= 1, we get back equation of motions for perturbed Kepler\nproblem given in Eqn.2.2 and additionally when λ−→0 , above Eqn.3.2 becomes equation of motion\nfor well known Kepler problem in 2-dimensions\nWe now re-write these equation of motion in terms a new set of c o-ordinates ( X1,X2) which are\nrelated to old co-ordinates through time dependent transfo rmation given by\nX1=x1eλt\n2,\nX2=x2eλt\n2. (3.3)\nIn terms of these co-ordinates, Eqn.(3.2) become\nm¨Xi−mλ2\n4Xi+/parenleftbigg2N\nN+1/parenrightbiggkXi\nr4N+2\nN+1= 0, i= 1,2 (3.4)\nwherer=/radicalbig\nX2\n1+X2\n2. Here we can see that these equations following from the Lagr angian\nL=m\n2(˙X2\n1+˙X2\n2)+mλ2\n8(X2\n1+X2\n2)−mλ\n2(X1˙X1+X2˙X2)+k\nr2N\nN+1, (3.5)\nwhich has no explicit time dependence. Note that here N= 1 leads to Lagrangian of perturbed\nKepler problem given in Eqn.(2.6) and third term as a total de rivative term will not contribute to\nequation of motions. Following the standard procedure, we c alculate the canonical conjugate momenta\ncorresponding to X1andX2as\nPX1=m˙X1−λ\n2mX1 (3.6)\nPX2=m˙X2−λ\n2mX2 (3.7)\nand using these we derive the Hamiltonian as\nH=(P2\nX1+P2\nX2)\n2m+λ\n2(X1PX1+X2PX2)−k\nr2N\nN+1, (3.8)\nwherer=/radicalbig\nX2\n1+X2\n2and atN= 1 above Hamiltonian leads to Eqn.(2.9). This Hamiltonian d oes\nnot have any explicit time dependence and is invariant under time translations and hence total energy\nof this system is conserved. For later purposes, we express ( negative of) this Hamiltonian, which is a\nconserved quantity, in terms of velocities and co-ordinate s, viz:\n−E=−E\nm=1\n2(˙X2\n1+˙X2\n2)−λ2\n8(X2\n1+X2\n2)−µ\nr2N\nN+1, (3.9)\nwherek\nm=µ\nWe re-express the Eqn.(3.4) in terms of complex variable Zgiven by Z=X1+iX2as\nm¨Z+/parenleftbigg2N\nN+1/parenrightbiggk\n|Z|4N+2\nN+1Z=mλ2\n4Z, (3.10)\nwhere|Z|=/radicalbig\nX2\n1+X2\n2=r. Note that the derivative with respect to the time variable tis represented\nby ‘overdot’, in the above equations.\n8We now apply a re-parametrization of time to avoid presence o f singularity from the equations of\nmotion and demand\nd\ndt=1\ng(r)d\ndτ, where g (r) = (N+1)2r2N\nN+1 (3.11)\nNote here that the N= 1 leads to Eqn.(2.12), time re-parametrization used for pe rturbed Kepler\nproblem. Applying this re-parametrization, we find\ndZ\ndt=1\n(N+1)2r2N\nN+1dZ\ndτ(3.12)\nd2Z\ndt2=1\n(N+1)4/parenleftig\nr2N\nN+1/parenrightig2d2Z\ndτ2−/parenleftbigg2N\n(N+1)5/parenrightbigg1\nr5N+1\nN+1dr\ndτdZ\ndτ. (3.13)\nUsing these, we re-express Eqn.(3.10) as\n1\n(N+1)4r4N\nN+1d2Z\ndτ2−2N\n(N+1)51\nr5N+1\nN+1dr\ndτdZ\ndτ+/parenleftbigg2N\nN+1/parenrightbiggµ\nr4N+2\nN+1Z=λ2\n4Z, (3.14)\nwhereµ=k/m.\nNext we apply a co-ordinate transformation(Levi-Civita re gularization) and consider following rela-\ntions ( [28]),\nZ=ˆUNU (3.15)\nwhere,\nU(N)=ˆUN−1UandˆUN=/parenleftig\nP(1)U(N),P(2)U(N)/parenrightig\n, N≥1 (3.16)\nwithˆU0=IandˆU1=A(U) as explainedin Eqn.(2.18). Hereweconsidersamedefinitio nof permutation\nmatrices as discussed previously in Eqn.(2.20) i.e,\nP(1)U= (U1,U2)T,P(2)U= (−U2,U1)T, (3.17)\nwhich have following properties like,\nP(i)P(j)=P(j)P(i)and(Pi)TPj=δij. (3.18)\nWe also consider following relations ( [28]),\nˆUT\nNˆUN=ˆUNˆUT\nN=R2NI2×2and(U(N+1))′= (N+1)ˆUNU′, (3.19)\nwhere (′) denotes time derivative with respect to fictious time τ.\nThese set r=RN+1,r′= (N+1)RNR′. We also find (in matrix representation) Z′= (N+1)ˆUNU′,\nZ′′= (N+1)ˆU′\nNU′+(N+1)ˆUNU′′, using which, we can rewrite Eqn.(3.14) in matrix represent ation\nas,\n(N+1)ˆU′\nNU′+(N+1)ˆUNU′′−1\nR(2N)(N+1)R′(ˆUNU′)−λ2\n4(N+1)4R4NˆUNU+µ(2N)(N+1)3\nR2ˆUNU= 0\n(3.20)\nThis equation, after straight forward algebra gives\nU′′−λ2\n4(N+1)3R4NU+/braceleftbiggµ(2N)(N+1)2\nR2−N(/summationtext(U′\ni)2)\nR2/bracerightbigg\nU= 0. (3.21)\nHere for N= 1, we get back Eqn.(2.22) with c being1\n4. Next,we express the conserved quantity −Ein\nterms of new-coordinates. For this, we first re-express −Ein terms of ZanddZ\ndtas\n−E=1\n2d¯Z\ndtdZ\ndt−λ2\n8Z¯Z−µ\n|Z|2N\nN+1(3.22)\n9and apply re-parametrization of time, Eqn.(3.11) and finall y, re-express these terms of matrix represen-\ntation using the derivative of U. This gives\n−E=1\n21\n(N+1)2R2N(/summationdisplay\n(U′\ni)2)−λ2\n8R2N+2−µ\nR2N(3.23)\nUsing this, in Eqn.(3.21) we get,\nU′′−λ2\n4(2N+1)(N+1)2R4NU+ER2N−2(2N)(N+1)2U= 0, (3.24)\nwhereR2=U2\n1+U2\n2. we re-write the above equation in U1andU2as\nU′′\ni−λ2\n4(2N+1)(N+1)2(U2\n1+U2\n2)2NUi+E(U2\n1+U2\n2)N−1(2N)(N+1)2Ui= 0, i= 1,2 (3.25)\nwhere a ‘prime’ over Ustands for derivative with respect to the new time variable, τ. These equations\nare obvious extensions of the Lagrangian\nL=1\n2m(U′\n12+U′\n22)−Em(N+1)2(U2\n1+U2\n2)N+λ2\n8m(N+1)2(U2\n1+U2\n2)2N+1(3.26)\nNote that for N= 1, these equations of motions lead to Eqn.(2.28) and Eqn.(2 .29) i.e, equation\nof motions for perturbed harmonic oscillator problem and in addition λ→0, these describe motion of\nusual harmonic oscillator. Note that the λ2dependent terms are non-linear and these terms come due\nto the damping present in the original system.\n4 Kepler problem in 3-dimensions in presenceof damping: Kus taanheimo-\nStiefel transformation\nThe Kustaanheimo-Stiefel (KS) transformation, which is a t hree-dimensional extension of the Levi-\nCivita transformation, maps the singular equations of moti on of the three-dimensional Kepler problem\nto the linear and regular equations of a four-dimensional ha rmonic oscillator. Here we investigate\nthe regularization of damped Kepler problem in 3-dimension s. We first give a brief summary of the\ndefinitions and identities involving quaternions, which ar e defined as\nU=U0+iU1+jU2+kU3,withi2=j2=k2=−1, (4.1)\ni·j=−j·i=k;j·k=−k·j=i;k·i=−i·k=j (4.2)\nWe define the complex conjugate as\n¯U=U0−iU1−jU2−kU3 (4.3)\nand a∗-operation as\nU∗=U0+iU1+jU2−kU3. (4.4)\nWe note\n¯UU=|U|=3/summationdisplay\ni=0U2\ni=¯U∗U∗=|U∗|2. (4.5)\nWe also define a three vector\nX=X0+iX1+jX2 (4.6)\nwhich is given in terms of the quaternions using the KS transf ormation as\nX=UU∗. (4.7)\n10Since¯X=X, we find (¯UU∗) =X. One may write the above equation in the matrix form (with\nX3= 0) as\n\nX0\nX1\nX2\nX3\n=A(U)\nU0\nU1\nU2\nU3\n, (4.8)\nwhere\nA(U) =\nU0−U1−U2U3\nU1U0−U3−U2\nU2U3U0U1\nU3−U2U1−U0\n. (4.9)\nThis we can rewrite as follows,\nA(U) =/parenleftig\nP(0)U,P(1)U,P(2)U,P(3)U/parenrightig\n, (4.10)\nwhere we have considered following set of permutations,\nP(0)U= (U0,U1,U2,U3)T,P(1)U= (−U1,U0,U3,−U2)T\nP(2)U= (−U2,−U3,U0,U1)T,P(3)U= (U3,−U2,U1,−U0)T. (4.11)\nHere above used permutation matrices has some properties li ke,\nP(i)P(j)/ne}ationslash=P(j)P(i)and(Pi)TPj=δij. (4.12)\nWe note that X3= 0, which naturally comes from the requirement UU∗= (UU∗)∗. We also note that\nA(U) satisfy\nA(U)TA(U) =rI. (4.13)\nDefining r=/radicalig/summationtext3\ni=1X2\ni, we find\nr=/radicalbig\n|X|2=|U|2=|U∗|2. (4.14)\nDefining derivative as∂X\n∂τ=X′, we find\nUdU∗=dUU∗(4.15)\nX′=UU′∗+U′U∗= 2UU′∗(4.16)\nX′′= 2U′U′∗(4.17)\nr′=U′¯U+U¯U′(4.18)\nWe also find\ndX= 2A(U)dU. (4.19)\nDefining the momenta as\ndXi\ndt=Pi,dUi\ndτ= ˜pi,i= 0,1,2,3. (4.20)\nshows that the compatibility with Eqn.(4.19) sets\ndτ\ndt=1\n4r. (4.21)\nIt can be verified easily that the anzats\nP=1\n2rA(U)˜p (4.22)\nguarantee that the transformation ( Xi,Pi)→(Ui,˜pi) is a canonical transformation.\n11This is manifesting the generalization of Levi-Civita’s co nformal squaring. Hence the KS transfor-\nmation maps U= (U0,U1,U2,U3)∈R4toX= (X0,X1,X2)∈R3. Since the mapping does not preserve\nthe dimension, its inverse in the usual sense does not exist.\nNow, instead of considering the squaring of complex numbers as in the case of Levi-Civita regular-\nization, we define mapping in (4.7). The image U/mapsto→X:=UU∗is the set of quaternions with vanishing\nkcomponents, which can be identified with R3. By direct computation (4.7) yields\nX0=U2\n0−U2\n1−U2\n2+U2\n3, X1= 2(U0U1−U2U3), X2= 2(U0U2+U1U3), (4.23)\nfollows directly from X:=UU∗, thusX=X∗. This is exactly the K-S transformation in its clas-\nsical form. Denoting the space coordinates by ( X0,X1,X2) with conjugate momenta ( P0,P1,P2), the\nsymplectic form becomes\nω=dX0∧dP0+dX1∧dP1+dX2∧dP2\nhencewecanwrite {Xl,Pn}=δlnforl,n= 0,1,2. Wemustrecall that ωgenerates thefoldedsymplectic\nstructure [30], if we choose U3= 0, then\nω= 2(U0dU0−U1dU1−U2dU2)∧dP0+(U0dU1+U1dU0)∧dP1+(U0dU2+U2dU0)∧dP2,\nwhich yields ω∧ω∧ω=U0(U2\n0−U2\n1−U2\n2)dU0∧dP0∧dU1∧dP1∧dU2∧dP2, hence this is a hyperboliclike\nm-folded symplectic structure. A folded symplectic structu re is a closed 2-form which is nondegenerate\nexcept on a hypersurface.\nBy direct computation one can check\ndX=d(UU∗) =dUU∗+U∗dU= 2dUU∗= 2U∗dU, (4.24)\nprovided the ‘ k’ component vanishes, which yields a bilinear relation [15, 18,29],\nU0dU3−U3dU0+U2dU1−U1dU2= 0. (4.25)\nThis condition appears since KS transformation is a mapping fromR4toR3, it therefore, leaves one\ndegree of freedom in the parametric space undetermined. By i mposing this bilinear relation (4.25), the\ntangential map in (4.23) change into a linear map and also thi s yields bilinear constraint [15,18,29],\nU3˜p0−U2˜p1+U1˜p2−U0˜p3= 0, which makes P3= 0.\nAgain by taking poisson bracket operation between coordina tes, we get,\n{Ui,˜pj}=δiji,j= 0,1,2,3. (4.26)\nThese relations implies that the mapping in (4.7), is canoni cal in nature.\n4.1 Damped Kepler Problem\nIn thissubsection, we studytheregularization of collisio n orbits of aparticle’s motion in1\nrpotential that\nis also exposed to a dampingforce in 3-dimensions, which fol low from a time dependent Lagrangian. We\nfirst use a time dependent point transformation to turn these equations into Euler-Lagrange equations\nderived from this time independentLagrangian. This helps u sto construct conserved energy, which then\nallows us pursuance of the mapping of dynamics on the constan t energy surface. After re-expressing\nthese equations in terms of quaternions, we apply the K-S map to them. The equations are shown to\nrepresent a harmonic oscillator with inverted sextic poten tial and interactions.\nWe start with the Lagrangian describing Damped Kepler probl em in 3-dimensions\nL=eλt/parenleftbiggm\n2˙x2\ni+k\n˜re−3λt\n2/parenrightbigg\n, i= 0,1,2 (4.27)\nwhere ˜r=/radicalig\nx2\niand ˙xi=dxi\ndt. Euler-Lagrange equation following from this Lagrangian i s\nm¨xi+λm˙xi+k\n˜r3e−3λt\n2= 0, i= 0,1,2 (4.28)\n12Applying the transformations\nxi→Xi=xieλt\n2, i= 0,1,2 (4.29)\nwe re-write the above Lagrangian in terms of new co-ordinate sXias\nL=m\n2/parenleftbigg\n˙X2\ni+λ2\n4X2\ni−λXi˙Xi/parenrightbigg\n+k\nr, (4.30)\nherer=/radicalig\nX2\niand we have used summation convention. Note that the damping parameter λin\nEqn(4.27),Eqn(4.28) is now appearing a the co-efficient of X2\niandXi˙Xiterms.\nEquations of motion following from this Lagrangian are\nm¨Xi−mλ2\n4Xi+k\nr3Xi= 0, i= 0,1,2 (4.31)\nWe next derive the corresponding Hamiltonian correspondin g to this Lagrangian. Calculating the\nconjugate momentum using\nPi=∂L\n∂˙Xi=m˙Xi−mλ\n2Xi, i= 0,1,2 (4.32)\nand using this we obtain\nH=P2\ni\n2m+λ\n2XiPi−k\nr, (4.33)\n(Here and below we have used summation convention) which is a constant of motion. For later purposes,\nwe express the total Hamiltonian in terms of the velocities a s\nE=m\n2˙X2\ni−mλ2\n8X2\ni−k\nr(4.34)\nand define\n−E=−E\nm=˙X2\ni\n2−λ2\n8X2\ni−µ\nr(4.35)\nwhich is a constant of motion.\nNext, we define a re-parametrization of time through the rela tion\nd\ndt=1\n4rd\ndτ(4.36)\nand thus we find\n˙Xi=1\n4rX′\ni;¨Xi=1\n16r2X′′\ni−1\n16r3r′X′\ni (4.37)\nUsing the above re-parametrization, we re-express the equa tions of motion (4.31) as\nX′′\nr2−1\nr2r′X′−4λ2X+16µ\nr3X= 0, (4.38)\nwhereµ=k\nm. Here, we relabel X1→X0,X2→X1andX3→X2for notational compatibility ( see\neqn.(4.6)).\nNow, using Eqns.(4.16,4.17,4.18), we re-express eqn.(4.3 8) as\n2rUU′′∗−2|U′|2UU∗+16µUU∗−4r3λ2UU∗= 0 (4.39)\nNext, we apply re-parametrization to the conserved quantit y,−Ein eqn.(4.35) and re-express it as\n−E=1\n32r2|X′|2−λ2\n8r2−µ\nr(4.40)\nwhich in terms of quaternions become −E=1\n8r|U′∗|2−λ2\n8r2−µ\nrand we find\n−16Er= 2|U′∗|2−2λ2r3−16µ. (4.41)\n13Using this, we obtain from eqn(4.39)\nU′′∗+/parenleftbig\n8E −3λ2r2/parenrightbig\nU∗= 0. (4.42)\nSincer=|U|2, we have r2=|U|4and using this, we re-express the above equation, explicitl y, in\ncomponents, as\nU′′\ni+/parenleftbig\n8E −3λ2|U|4/parenrightbig\nUi= 0, i= 0,1,2,3. (4.43)\nThese equations are the Euler-Lagrange equations followin g from the Lagrangian\nLSO=1\n2m(U′\ni)2−4mEU2\ni+mλ2\n2(U2\ni)3, i= 0,1,2,3. (4.44)\nThis Lagrangian, describesa harmonicoscillator withsext ic potential andcouplings in 4-dimensions.\nThe Hamiltonian following from this is given by\nH=3/summationdisplay\ni=0˜Pi2\n2m+4mE(3/summationdisplay\ni=0U2\ni)−mλ2\n2(3/summationdisplay\ni=0U2\ni)3. (4.45)\nNote that the equations of perturbed Kepler problem in 3-dim ensions given in Eqn.(4.31) is mapped\nunder K-S transformation and re-parametrization of time va riable, to equations of Harmonic oscillator\nwith inverted sextic potential and couplings. Here too, we s ee that the regularized equation is non-linear\nand all the non-linear terms have λas the co-efficient, indicating that their origin is due to the damping\nin the original system.\n4.2 Homogeneous Hamiltonian\nWe now establish the equivalence between the damped Kepler p roblem in 3-dimensions to harmonic\noscillator with inverted sextic potential and interaction s in 4-dimensions at the level of Hamiltonians.\nThis is shown using homogeneous Hamiltonian formalism.\nHomogeneous Hamiltonian formalism [15,31–33] treats both space co-ordinates and time on an equal\nfooting. This is achieved (i) by treating the time parameter tas a function of another parameter, say\ns( then,x(t)≡x(s)), and (ii) by introducing a conjugate momentum for the time t(s). This approach\nhas been shown to be suitable to derive relativistic wave equ ations, using the usual replacement of\nmomentum operators with the derivatives with respect to con jugate variables [33]. This approach\nprovide a natural way to study the quantization of systems wi th time dependent constraints [32]. This\napproach naturally fits into the canonical treatment of regu larization transformations. In both Levi-\nCivitatransformationandK-Stransformation, timeistrea tedasafunctionofnewparameter. Usingthis\nframework, one can easily show that the regularization tran sformations are canonical transformations\n[15]. Homogeneous formalism of non-relativistic H-atom wa s used to show its equivalence to harmonic\noscillator in 4-dimensions [34].\nWe have seen that the damped Kepler problem in 3-dimensions i s equivalent to the system described\nby the Lagrangian in Eqn.(4.30) and here we show that homogen eous Hamiltonian correspondingto this\nmodel is equivalent to the Hamiltonian for 4-dimensional os cillator with inverted sextic potential and\ninteractions. Note that the Lagrangian obtained by time-de pendent point transformation of damped\nKepler problem in 3-dimensions given in Eqn.(4.30) is equiv alent to\nL=m\n2/parenleftbigg\n˙X2\ni+λ2\n4X2\ni/parenrightbigg\n+k\nr(4.46)\nas they differ only by a total derivative term and thus both lead to same equations of motion. The\nHamiltonian derived from the above Lagrangian is\nH=1\n2mP2\ni−λ2m2\n8X2\ni−k\nr. (4.47)\n14Starting from this Hamiltonian, we derive the Hamiltonian d escribing harmonic oscillator with inverted\nsextic potential and interactions. For this, we first define t he corresponding homogeneous Hamiltonian\nas\nHH= 4r(H+Pt) =2r\nmP2\ni−rλ2m2\n2X2\ni−4k+4Ptr. (4.48)\nWe now re-express this Hamiltonian in terms of ˜ piandUi, where we also use r=/summationtext3\ni=0U2\niand/summationtext2\ni=0X2\ni= (/summationtext3\ni=0U2\ni)2, to obtain\n¯HH=3/summationdisplay\ni=0˜p2\ni\n8m+4E(3/summationdisplay\ni=0U2\ni)−λ2m2\n8(3/summationdisplay\ni=0U2\ni)3+ps. (4.49)\nIn the above, we have identified Ptwith 4Eand 4kwith−pswhich is the conjugate of the new time\nparameter τ. Note that the above homogeneous Hamiltonian ¯HH, is defined in 4-dimension. The\ncorresponding non-homogeneous Hamiltonian is\n¯H=3/summationdisplay\ni=0˜p2\ni\n2m+4E(3/summationdisplay\ni=0U2\ni)−λ2m2\n2(3/summationdisplay\ni=0U2\ni)3(4.50)\nand it describes inverted sextic oscillator with couplings in 4-dimensions.We see that this Hamiltonian\nin Eqn.(4.50) is exactly same as one obtained in Eqn.(4.45) w ith identification,\n˜pi=˜Pi, E=mE (4.51)\nNote that U2\nidependent term in the Hamiltonian in Eqn.(4.45) and in Eqn.( 4.50), is in the form of\nharmonic oscillator potential with identification of relat ion between energy of perturbedKepler problem\nand strength (angular frequency) of harmonic oscillator i. e, 4E=1\n2mω2\n0. Here in Eqn.(4.45) and\nEqn.(4.50), damping parameter λappear as the co-efficient of inverted sextic potential and co uplings.\n5 Conclusion\nIn this paper, we have studied the regularization of central force systems with damping, both in 2\nand 3 dimensions. The systems we have analyzed are (i) Kepler problem in 2-dimensions with friction\nincluded, (ii) particle moving under the influence of generi c power law potential of the form r−2N/N+1,\n0≤2N/N+ 1<2 subjected also to a force linear in velocities, and (iii) da mped Kepler problem\nin 3-dimensions. We have shown that the Levi-Civita transfo rmation do regularize the equations of\nmotion of first two systems while K-S transformations does th e same for third system. In the first and\nthird cases, the regularized equations describe harmonic o scillator with inverted sextic potential and\ninteractions, in 2 and 4 dimensions, respectively. In the se cond case, regularized equation is that of a\nharmonic oscillator with potential and interaction that de pend on the actual value of N, 0≤N <2.\nIn all the three cases, we note that the non-linear terms in th e regularized equations have λas the\nco-efficient where λis the damping parameter in the original models.\nThe explicit time-dependence of damped systems studied her e is reflected in the fact that the cor-\nresponding Lagrangians/Hamiltonians also have explicit t ime dependence and thus breaks time trans-\nlation invariance that the undamped systems enjoy. Thus, in these undamped systems, we do not have\nconserved energy as expected for dissipative systems. Exis tence of conserved quantity is essential for\nimplementing Levi-Civita and/or K-S transformations. To o vercome this obstacle, we have mapped\nthe time-dependent equations (equivalently, Lagrangians ) describing the damped systems to equivalent\nsystems without explicit time dependence(see Eqns. (2.6, 3 .5, 4.30). These systems do have conserved\nenergies and we use them in applying regularization transfo rmations, in three steps as discussed in the\nintroduction. The regularized equations obtained here are non-linear, unlike their undamped counter-\nparts. Our results reduced to the well known, undamped cases in the limit λ→0, as expected.\nWe note that in the case of generic potential r−2N/N+1,0≤2N/N+ 1<2 with damping in\n2-dimensions, the permutation operators(see Eqn.(3.18)) were still commuting as in the undamped\n15case. But in the case of 3-dimensions, we note that the permut ation operators(see Eqn.(4.12)) are not\ncommuting. This feature of permutation operators mutually commuting is not due to damping and\neven when there is no damping, these operators in the 3-dimen sions would be non-commuting. Thus\nthe identities satisfied by the transformation of co-ordina tes in 2-dimensions shown in [28] will not be\nvalid for 3-dimensional Kepler problem. Thus generalizati on of K-S transformation to the case generic\npotential r−2N/(N+1)as this was done in 2-dimensions fails.\nOnecouldactuallyconstructconservedquantitiesfromthe timedependentLagrangians/Hamiltonians\nof the damped systems considered here by adapting/generali sing the method used in [35]) for deriving\nsuch a conserved quantity in 1-dimensions. These conserved quantities will reduce to the familiar con-\nserved energy whenthedampingparameter λvanishes. But usingtheseconserved quantities inapplying\nregularization transformations will lead to equations tha t depend on both new and old time parameters.\nIn an interesting paper, Andrade et al [36] studied Levi-Civ ita regularization problem of the Kepler\nproblem on surfaces of constant curvature. We would also lik e to study the regularization problem of\nthe Kepler equation with a drag force on surfaces of constant curvature., both positive and negative, S2\nandH2respectively.\nAcknowledgements\nSKP thank UGC, India for support through JRF scheme(id.1916 20059604). Work by the author PG\nwas supported by the Khalifa University of Science and Techn ology under grant number FSU-2021-014.\nAppendix : A Motion in r2potential with damping in 2-dimensions:\nBohlin-Sudman Map\nIn this appendix, we study the application of Bohlin-Sudman mapping to the equations of motion\ndescribing a damped harmonic oscillator. After mapping the se equations to that of a shifted harmonic\noscillator by a time dependent point transformation, we re- express the equations in terms of complex\nco-ordinates. Then by applying a re-parametrization of tim e followed by a co-ordinate change, we map\nthese equations describing a dynamics on a constant energy s urface to that of a particle moving in1\nr\npotential.\nWe start from the equations of motions\nq′′\ni+λq′\ni+Ω2qi, i= 1,2 (A.1)\ndescribing damped harmonic motion in 2-dimensions3. Hereq′\ni=dqi\ndτandq′′\ni=d2qi\ndτ2. We now apply the\ntime-dependent co-ordinate transformation\nxi=qieλτ\n2,i= 1,2 (A.2)\nand re-write the above equations of motion as\nx′′\n1+˜Ω2x1= 0, (A.3)\nx′′\n2+˜Ω2x2= 0, (A.4)\nwhere˜Ω2= Ω2−λ2\n4. These are Euler-Lagrange equations following from the Lag rangian\nL=m\n2/parenleftbig\nx′2\n1+x′2\n2/parenrightbig\n−m\n2˜Ω2(x2\n1+x2\n2)−mλ\n2(x1x′\n1+x2x′\n2). (A.5)\nWe re-express these equations using complex co-ordinate\nω=x1+ix2 (A.6)\n3These equations follow from the BCK Lagrangian L=1\n2m/parenleftbig\n˙q2\ni−Ω2q2\ni/parenrightbig\neλτ,i= 1,2\n16as\nω′′+˜Ω2ω= 0. (A.7)\nWe now apply Bohlin-Sudman transformation\nω→Z=ω2. (A.8)\nand also implement re-parametrization of time4using\n¯ZdZ\ndt=¯ω\n2dω\ndτ(A.9)\nUsing Eqn.(A.8) in Eqn.(A.9), we get\nd\ndt=1\n4¯ωωd\ndτ(A.10)\nand using this we find\n˙Z=dZ\ndt=1\n2¯ωdω\ndτ(A.11)\n¨Z=d2Z\ndt2=1\n8¯ωω/bracketleftbigg1\n¯ωd2ω\ndτ2−1\n¯ω2/parenleftbiggd¯ω\ndτ/parenrightbigg/parenleftbiggdω\ndτ/parenrightbigg/bracketrightbigg\n(A.12)\nFrom Eqn.(A.7), we have ω′′=−˜Ω2ωand using this, we re-write the second equation in the above a s\n¨Z=−1\n8¯ωω/bracketleftbigg1\n¯ω˜Ω2ω+1\n¯ω2/parenleftbiggd¯ω\ndτ/parenrightbigg/parenleftbiggdω\ndτ/parenrightbigg/bracketrightbigg\n(A.13)\n=−Z\n8|Z|3/bracketleftbigg/parenleftbiggd¯ω\ndτ/parenrightbigg/parenleftbiggdω\ndτ/parenrightbigg\n+˜Ω2¯ωω/bracketrightbigg\n(A.14)\nWe now derive the conserved “energy” associated with the Lag rangian in Eqn.(A.5) and re-express\nthe terms in the [ ] appearing in the above equation. To this en d, we first obtain the conjugate momenta\ncorresponding to xias\npi=mx′\ni−mλ\n2xi,i= 1,2 (A.15)\nand construct the Hamiltonian in terms of velocities as\nH=m\n2/parenleftbig\nx′2\n1+x′2\n2/parenrightbig\n+m˜Ω2\n2/parenleftbig\nx2\n1+x2\n2/parenrightbig\n(A.16)\n=m\n2/bracketleftig\n¯ω′ω′+˜Ω2¯ωω/bracketrightig\n≡E. (A.17)\nSinceEabove is a constant, we use it to re-express Eqn.(A.14) as\n¨Z=−E\n4mZ\n|Z|3. (A.18)\nWe note that with the identification of conserved Ewith 4k=m˜Ω2, the strength of Kepler potential,\nE\n4=k, (A.19)\ntheEqn.(A.18)istheKepler’sequationin2-dimension, wri tteninthecomplexco-ordinate Z=X1+iX2,\nThus we have mapped the equation of ‘damped’ harmonic oscill ator in 2-dimension, to the equation\nof motion corresponding to the (undamped) Kepler problem in 2-dimension.\n4Bohlin-Sudman transformation has been applied to derive th e mapping between 2-dimension harmonic oscillator to\n2-dimension Kepler problem. Here, angular momentum is cons erved in both systems and demanding these constants\nof motion are proportional to each other, results in the rela tion between time parameters of these two systems. In the\npresent case too, angular momentum is conserved for damped h armonic oscillator as well as (damped) Kepler problem in\n2-dimension.\n17We now start with the expression for energy of 2-dim Kepler sy stem, described in terms of ¯Zand\nZand re-express it in terms of ¯ ωandω, i.e.,\nEKepler=m\n2d¯Z\ndtdZ\ndt−k\n|Z|(A.20)\n=m\n8/bracketleftbigg1\n¯ωωd¯ω\ndτdω\ndτ/bracketrightbigg\n−k\n¯ωω(A.21)\nUsing above equation and Eqn.(A.19), we get\n−EKepler=m\n8/parenleftbigg\nΩ2−λ2\n4/parenrightbigg\n. (A.22)\nThus we find\n1. The equations of motion of 2-dimensional damped Harmonic oscillator in Eqn.(A.1) are first\nmapped to equations of a shifted harmonic oscillator given i n Eqn.(A.4) which are then mapped\nto that of (undamped) Kepler problem in 2-dimensions.\n2. The strength of the Kepler potential is related to the cons erved energy of the shifted harmonic\noscillator.\nReferences\n[1] H. Dekker, Phys. Rep. 80, 1 (1981).\n[2] R. W. Hasse, J. Math. Phys. 16, 2005 (1975).\n[3] U. Weiss, Quantum Dissipative Systems, 4th ed.2008 (Wor ld Scientific, 2008).\n[4] H. Bateman, Phys. Rev. 38, 815 (1931).\n[5] P. Caldirola, Nuovo Cimento 18, 393 (1941).\n[6] E. Kanai, Prog. Theor. Phys. 3, 440 (1950).\n[7] D. M. Greenberger, J. Math. Phys. 20, 762 (1979).\n[8] I. K. Edwards, Am. J. Phys. 47, 153 (1979).\n[9] J. R. Ray, Am. J. Phys. 47, 626 (1979).\n[10] M. C. Baldiotti, R. Fresneda and D. M. Gitman, Phys. Lett .A375, 1630 (2011).\n[11] G. Herglotz, Lectures at the University of G¨ ottingen, G¨ ottingen (1930); R. B. Guenther, C. M.\nGuenther and J. A. Gottsch, The Herglotz Lectures on Contact Transformations and Hamiltonian\nSystems, Lecture Notes in Non-linear Analysis, Vol.1, J. Sc hauder Center for non-linear Studies,\nNicholas Copernicus university, Torun (1996).\n[12] A. Bravetti, Entropy, 19(12), 535 (2017).\n[13] J. F. Carinena and P. Guha, Int. J. Geo. Meth. Mod. Phys. 16, 1940001 (2019).\n[14] M. de Le´ on and M. L. Valcazar, J. Math. Phys. 60, 102902 (2019).\n[15] E. L. Stiefel and G. Scheifele, Linear and Regular Celes tial Mechanics (Springer- Verlag, 1971).\n[16] K. Bohlin, Bull. Astro. 28, 144 (1911).\n[17] T. Levi-Civita, Acta Math. 42, 99 (1920).\n18[18] P. Kustaansheimo and E. Stiefel, J. Reine Angew. Math. 218204 (1965).\n[19] J. Moser, Commun. Pure Appl. Math. 23609 (1970).\n[20] T. Ligon and M, Schaaf, Rep. Math. Phys. 9281 (1976).\n[21] W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Ross, Dynamica l Systems, the three-body problem\nand space mission Design, Proceedings of the International Conference on Differential Equations,\nEquadiff 99 (2000) 1167. Ed:B. Fiedler, K. Groger and J. Sprek els.\n[22] B. Hamilton and M. Crescimanno, J. Phys. A: Math. Theor. 41235205 (2008).\n[23] S. Casertano, E. S. Phimmey and J. V. Villumsen, “Struct ure and Dynamics of elliptical galaxies”,\nProceedings of IAU symposium, Ed. P. T. Timotheus, Volume 12 7, page 475.\n[24] H. Aceves and M. Colosimo, Am. J. Phys. 75139 (2007).\n[25] J. M. A. Danby, Fundamentals of Celestial Mechanics, (M acMillian Co. New York, 1962).\n[26] D. Mittleman and D. Jezewski, Cel. Mech. 28401 (1982).\n[27] V. M. Gorringe and P. G. L. Leach, Cel. Mech. 41125 (1988).\n[28] S. Blanes and C. J. Budd, Celest. Mech. and Dyn. Astron. 89, 383-405 (2004).\n[29] Waldvogel, J. Quaternions and the perturbed Kepler pro blem. Celestial Mechanics and Dynamical\nAstronomy 95, 201-212 (2006b), Waldvogel, J. Quaternions f or regularizing Celestial Mechanics:\nthe right way. Celest Mech Dyn Astr 102, 149–162 (2008).\n[30] A. Delshams, A. Kiesenhofer and E. Miranda, J. Geo. Phys .11589-97 (2017).\n[31] A. Mercier, Analytical and Canonical Formalism in Phys ics, (North-Holland Publishing Company,\nAmsterdam, 1959).\n[32] D. M. Gitman and I. V. Tutin, Quantisation of Fields with Constraints, (Springer-Verlag, 1991).\n[33] A. A. Jackson, ‘Homogeneous Canonical Formalism and Re lativistic Wave Equations’, Thesis(\nMA), (North Texas State University , 1967).\n[34] H. Grinberg, J. Maranon and H. Vucetich, J. Math. Phys. 252648 (1984).\n[35] N. A. Lemos, Am. J. Phys. 47857 (1979).\n[36] J. Andrade, N. D´ avila, E Perez-Chavela and C. Vidal, Dy namics and regularization of the Kepler\nproblem on surfaces of constant curvature, Canad. J. Math. 69961-991 (2017).\n19"    },    {        "title": "1008.1344v2.Fermi_acceleration_and_suppression_of_Fermi_acceleration_in_a_time_dependent_Lorentz_Gas.pdf",        "content": "arXiv:1008.1344v2  [nlin.CD]  17 Aug 2010Fermi acceleration and suppressionof Fermiacceleration i n a\ntime-dependentLorentz Gas.\nDiegoF. M.Oliveira\nMaxPlanck Institute for Dynamics and Self-Organization – B unsenstrasse10 - D-37073 - G¨ ottingen - Germany\nCAMTP- Center ForApplied Mathematics and Theoretical Phys ics – University ofMaribor - Krekova 2 - SI-2000 -\nMaribor - Slovenia.\nJ¨ urgenVollmer\nMax PlanckInstitute for Dynamics and Self-Organization – B unsenstrasse 10 - D-37073 - G¨ ottingen -Germany.\nEdsonD. Leonel\nDepartamento deEstat´ ıstica, Matem´ atica Aplicada eComp uta¸ c˜ ao – Instituto deGeociˆ encias eCiˆ encias Exatas –\nUniversidade Estadual Paulista –\nAv. 24A,1515 – Bela Vista – CEP:13506-900 – Rio Claro –SP – Bra zil.\nAbstract\nWe study some dynamical properties of a Lorentz gas. We have c onsidered both the static and\ntime dependentboundary. For the static case we have shown th at the system hasa chaotic com-\nponent characterized with a positive Lyapunov Exponent. Fo r the time-dependent perturbation\nwe describe the model using a four-dimensional nonlinear ma p. The behaviour of the average\nvelocity is considered in two situations (i) non-dissipati veand (ii) dissipative. Our results show\nthat the unlimited energy growth is observed for the non-dis sipative case. However, when dis-\nsipation, via damping coe fficients, is introduced the senary changes and the unlimited e ngergy\ngrowthissuppressed. Thebehaviouroftheaveragevelocity isdescribedusingscalingapproach.\nKeywords: Billiard,LorentzGas, Lyapunovexponents,FermiAccelera tion,Scaling.\n1. Introduction\nThe process in which a classical particle acquies unlimited energy growth from collisions\nwith heavy boundaries,also called as phenomenonof Fermi ac celeration, were first reportedby\nEnricoFermi[1]asanattempttoexplaintheaccelerationof cosmicrays. Heproposedthatsuch\nbehaviourwas dueto interactionbetweenchargedparticles andtime-dependentmagneticfields.\nSoon after [1] some alternative models have been proposedus ing different approacheswith ap-\nplicationin differentfieldsofscienceincludingmolecularphysics[2],opt ics[3],nanostructures\n[4], quantumdots[5].\nEmailaddresses: diegofregolente@gmail.com (Diego F.M.Oliveira), juergen.vollmer@ds.mpg.de (\nJ¨ urgen Vollmer), edleonel@rc.unesp.br ( Edson D.Leonel)\nPreprint submitted to Elsevier November 4, 2018One of the most studied versions of the problem is the well kno wn one-dimensionalFermi-\nUlam model (FUM) [6, 7, 8, 9, 10, 11, 12]. The model consists of a classical particle confined\nand bouncing between two rigid walls in which one of them is fix ed and the other one moves\naccording to a periodic function. It is well known that the ph ase space, in the absence of dis-\nsipation, shows a mixed structure in the sense that dependin g on the combinations of control\nparameters and initial conditions, both invariant spannin g curves (also called as invariant tori),\nchaotic seas and Kolmogorov- Arnold-Moser (KAM) islands ar e all observed. An alternative\nmodel was latter proposed by Pustylnikov [13, 14]. Such syst em consists of a classical parti-\ncle bouncingin a vertical movingplatformunderthe e ffect of an externalconstant gravitational\nfield [15, 16, 17, 18, 19, 20, 21]. Both models seems to be quite similar. However there are\nmany differences between them. The main di fference is, that in the FUM framework the Fermi\naccelerationisnotobserved. Ontheotherhand,forspecific combinationsofbothcontrolparam-\neters and initial conditions the phenomenonof unlimited en ergy growth can be observed in the\nbouncer model. This apparent contradictoryresult was latt er discussed and explainedby Licht-\nenbergandLieberman[22, 23] andcan beeasily understoodby lookingat the spacephase. The\nFUM has a set of invariant spanning curves limiting the size o f the chaotic sea (as well as the\nparticle’s velocity), but such invariant tori, which could be interpreted as a barrier, they are not\nobservedin the bouncermodel and the energygrowsunbounded . However,in two-dimensional\nsystems it is not so simple to say if the phenomenon of Fermi ac celeration will be observed or\nnot. In this sense a conjecture was proposed by Loskutov-Rya bov-Akinshin(LRA) [24]. Such\nconjecture, known as LRA-conjecture, states that a chaotic componet in the phase space with\nstatic boundaryisa su fficientconditionto observeFermi accelerationwhena pertur bationis in-\ntroduced. Resultsthatcorroboratethevalidityofthiscon jectureincludethetimedependentoval\nbilliard[25],stadiumbilliard[26].\nWhen dissipationis introduce,it causesa drasticchangeon the phase space. Invariantspan-\nningcurvesaredestroyed. theellipticfixedpointsturnsin tosinksandthechaoticseaisreplaced\nby a chaotic attractor [27]. However, the influence of dissip ation on the averave velocity is still\nnotfullyunderstood. ItwasdiscussedbyLeonel[30]somere sultsforaone-dimensionalFermi-\nUlammodel. Itiswellknownthatsuchsysteminitsoriginalf ormulationtheparticledonothave\nunlimited energy growth. If one consider the motion of the mo ving wall to be random the phe-\nnomenon of Fermi acceleration is observed. However, the int roduction of inelastic collision is\nenoughtosuppressFermiacceleration. Resultsconsiderin galsotheonedimensionFermi-Ulam\nmodel underan externalforceof type sawtooth[31] undere ffects of dissipation generatedfrom\na sliding of a body against a rough surface has also been consi dered. The external perturbation\nofsawtoothtypewaschosenbecausetheoscillatingwallalw aysgivesenergytotheparticleafter\ncollisions. The main question is: Would it be possibel to sup press Fermi acceleration under the\nefectofdissipationgeneratedfromaslidingofabodyagain staroughsurface? Theanswerisnot\nsosimpleanditdependesonboth,initialconditionandthec ombinationofcontrolparameters. It\nwas observedthat for a certainrangeof parametersFermi acc elerationindeedhappen,however,\nit issuppressesunderspecificconditions.\nIn this paper, we study the problem of a Lorentz Gas consideri ng both the static and the\ntime-dependent boundary. In the first part of our work we stud y the Lorentz Gas with static\nboundary. We derived a two dimensional nonlinear mapping th at describes the dynamics of\nthe model. We obtain the phase space and we show that it is chao tic with positive Lyapunov\nExponent. Then,in the second part of our paperwe introducea time-dependentperturbationon\nthe boundary. There are many di fferent ways to introduce time dependent perturbation and the\nmostcommonmethodsare: stochasticcase,wheretheboundar ychangesaccordingtoa random\n2function[28]; regularcase, wherethesize of theboundaryv ariesaccordingto anharmoniclaw\n[29]. However, in both situations the center of mass it is ass umed to be fixed. Now, for the\nfirst time, we introduce a di fferent kind of time dependent perturbation for a Lorentz gas. We\nassume that the radius of the scatters are fixed and the center of mass changes according to an\nharmonic function. Our main goal in this part of our work is to verify the validity of the LRA\nconjecture, which is confirmed when we studied the behaviour of the average velocity for an\nensembleofparticles. Since thephenomenonofFermiAccele rationispresentinthismodelour\nnext approach is to introduce dissipation into the model via damping coefficients and trying to\nunderstandwhatistheinfluenceofdissipationonthepartic lesbehaviour. Ourresultsallowusto\nconfirm that when inelastic collisions is introduced into th e model it is a sufficient condition to\nbreak down the phenomenonof Fermi acceleration. In both, co nservative as well as dissipative\ncase, wedescribesthebehaviourofaveragevelocityusings calingformalism.\nThepaperisorganizedasfollows. Insection2wedescribeho wtoobtainthetwo-dimensional\nmappingthatdescribesthedynamicsofthestaticsystem. Se ction3isdevotedtodiscussthetime\ndependentmodelaswellasournumericalresults. Finally,c onclusionandacknowledgmentsare\ndrawnin section4.\n2. A staticLorentzGasand themapping\nInthis sectionwe discussall thedetails neededforthe cons tructionofa non-linearmapping\nthatdescribesthedynamicsoftheproblem. Themodelconsis tsofaclassical particleofmass m\nsufferingelastic collisionswithcircularscatters(seeFig1) . Wechooseatriangulararrangement\nofthescatters(whichalsoseemstobetheStarofDavidifone connectcircles1,5and9and11,7\nand 3) in order to avoid particles traveling infinitely far be tween collision and the fixed lattice\nspacingato be twice the radius of the scatters, R=a/2. The system is described in terms of a\ntwo dimensional mapping Ξ(θn,bn)=(θn+1,bn+1) where the dynamical variable θndenotes the\ndirection of the trajectory while bnis the impact parameter. Given an initial condition ( θ0,b0),\nthe particle starts from the black circle (center in Fig. 1) a nd hit one of the 12 otherscircles. In\nthis sense, we specify the scattered hit in the collision n+1bysn=0,...,11 and we introduce\nl(sn) for the distance of this scatterer and the scatterer hit at t he collision n+1.l(sn) can assume\nthe values of 2 a/√\n3 and 2afor even and odd values of sn, respectively. Additionally,when the\nparticle hits the boundaryit is specularly reflected with th e same absolute velocity. The particle\ndoesnotsufferinfluencesofanyexternalfieldalongitslineartrajector y. Fromthegreentriangle\nin Fig2(a)onecaneasilyverify\nsin(θn−πsn\n6)=−(bn+1−bn)\nl(sn). (1)\nMoreover,fromFig. 2 (b)onecanfindthat\nα=arcsin(−bn+1/R), (2)\nβ=π−2α, (3)\nψ=2α−θn, (4)\nθn+1=π−2α+θn. (5)\nSucharesultallow usto obtain θn+1,\n3Figure 1: Illustration ofthe Lorentz gas with triangular configurati on.\nθn+1=π+θn+2arcsin(bn+1/R). (6)\nFromEq. 1it is easilytofindthat theimpactparameter, bn+1, isgivenby\nbn+1=bn−l(sn)sin(θn−πsn\n6). (7)\nThus,the mappingthatdescribethedynamicsofatwo dimensi onalLorentzgasis givenby\nΞ:/braceleftigg\nθn+1=π+θn+2arcsin(bn+1\nR)\nbn+1=bn−l(sn)sin(θn−πsn\n6), (8)\nby definition b∈[−R,R] andθ∈[0,2π] is a counterclockwise angle such that Ξis defined on\nthe fundamentaldomain [ −R,R]×[0,2π]. From the mapping Ξ, Eq. (8), one can easily obtain\nthe JacobianMatrix, J, whichisdefinedas\nJ=∂θn+1\n∂θn∂θn+1\n∂bn∂bn+1\n∂θn∂bn+1\n∂bn, (9)\nwith coefficientsgivenby\n∂θn+1\n∂θn=1−2/radicalig\nR2−b2\nn+1l(sn)cos(θn−πsn\n6), (10)\n∂θn+1\n∂bn=2/radicalig\nR2−b2\nn+1, (11)\n∂bn+1\n∂θn=−l(sn)cos(θn−πsn\n6), (12)\n∂bn+1\n∂bn=1. (13)\n4Figure 2: Dependence of(a) b n+1on bnandθn; (b)θn+1on bn+1andθn.\nAfter some easy calculation one can show that the mapping Ξpreserves the phase space\nmeasuresincedet( J)=1.\nIt is well knowthe Lyapunovexponentsare an importanttool t o identifywhether the model\nischaoticornot. Asdiscussedin [32], theLyapunovexponen tsaredefinedas\nλj=lim\nn→∞1\nnln|Λj|,j=1,2, (14)\nwhereΛjare the eigenvaluesof M=/producttextn\ni=1Ji(θi,bi) andJiis the Jacobianmatrixevaluatedover\nthe orbit(θi,bi). However,a directimplementationofa computationalalgo rithmto evaluateEq.\n(14) has a severe limitation to obtain M. Even in the limit of short n, the components of M\ncan assumeverydi fferentordersof magnitudeforchaoticorbitsandperiodicat tractorsyielding\nimpracticabletheimplementationofthealgorithm. Inorde rtoavoidsuchproblemwenotethat J\ncanbewrittenas J=ΘTwhereΘisanorthogonalmatrixand Tisarighttriangularmatrix. Thus\nwerewrite MasM=JnJn−1...J2Θ1Θ−1\n1J1,whereT1=Θ−1\n1J1. Aproductof J2Θ1definesanew\nJ′\n2. In a nextstep, it is easy to show that M=JnJn−1...J3Θ2Θ−1\n2J′\n2T1. The same procedurecan\nbeusedtoobtain T2=Θ−1\n2J′\n2andsoon. Usingthisproceduretheproblemisreducedtoeval uate\nthe diagonalelementsof Ti:Ti\n11,Ti\n22. Finally,theLyapunovexponentsarenowgivenby\nλj=lim\nn→∞1\nnn/summationdisplay\ni=1ln|Ti\njj|,j=1,2. (15)\nIfatleastoneofthe λjispositivethentheorbitisclassifiedaschaotic. Addition ally,inconserva-\ntivesystems λ1+λ2=0andindissipativesystems λ1+λ2<0. Thephasespaceforthemapping\n(8)isshowninFig. 3(a). Forthesamecontrolparametersuse dinFig. 3(a), a=2,wehavealso\nevaluated numericallythe positive Lyapunovexponentas on e can see in Fig. 3(b). The average\nofthepositiveLyapunovexponentfortheensembleofthe5ti meseriesgives ¯λ=0.371±0.001\nwherethevalue0 .001correspondstothe standarddeviationofthefivesamples .\n3. A Time DependentLorentzGas.\nInthissection we introducea newkindoftime dependentpert urbationin a Lorentzgas. We\nassume that the radius of the scatters are fixed and the center of mass changes according to an\nharmonicfunction f(t), inparticularwe assumethecase where\nfi(t)=ǫi[1+cos(t)],for i=x,y. (16)\n5Figure 3: (a) Phase space generated from iteration of mappin g (8); (b) Behaviour of the positive Lyapunov exponent of\nthe chaotic sea. Thecontrol parameter used in both figures we rea=2.\nwhereǫiis the amplitude of the time-dependent perturbation and tis the time. When we intro-\nducetimedependentperturbationintothemodel,twonewdyn amicalvariablesappears,namely,\nvelocity and time. Now, we have a two dimensionalsystem desc ribedin terms of a fourdimen-\nsion nonlinear mapping which relates the collision nthwith the ( n+1)th, i.e.,Ξ(θn,bn,Vn,tn)=\n(θn+1,bn+1,Vn+1,tn+1). The corresponding variables are: the direction of the tra jectory,θn; the\nimpact parameter, bn; the absolute velocity of the particle, Vnand the instant of the hit with the\nboundary, tn.\nAssumingthataninitialcondition( θ0,b0,V0,t0)isgiven,wecanobtaintheequationthatde-\nscribethedynamicsofthesystem. Thus,accordingtoourcon struction,thecartesiancomponents\nofRaregivenby\nX(δn,tn)=Rcos(δn)+ǫx[1+cos(tn)], (17)\nY(δn,tn)=Rsin(δn)+ǫy[1+cos(tn)]. (18)\nwhereδnistheangularpositionwhichisgivenby δn=π/2+θn−arcsin(bn/R). Sincewealready\nknow the angle that the particle’strajectory doeswith the h orizontal(θn+π/2)and the position\n6ofthehit at thecollision nth, we canobtainthevectorvelocityoftheparticlethatiswri ttenas\n− →Vn=|− →Vn|[cos(θn+π/2)/hatwidei+sin(θn+π/2)/hatwidej], (19)\nwhere/hatwideiand/hatwidejrepresenttheunitvectorswithrespecttotheXandYaxis,re spectively. Theabove\nexpressionsallowustoobtainthepositionoftheparticlea safunctionoftimefor t≥tn:\nXp(t)=X(δn,tn)+|− →Vn|cos(θn+π/2)(t−tn), (20)\nYp(t)=Y(δn,tn)+|− →Vn|sin(θn+π/2)(t−tn). (21)\nThe index pdenotes the corresponding coordinates of the particle. In o rder to know the\npositionoftheparticleat ( n+1)thcollisionweneedtosolvenumericallythefollowingequati on\nr=/radicalig\n[Xx(t)−Xp(t)]2+[Yy(t)−Yp(t)]2/simequalR, (22)\nwhereboth XxandYyaregivenby\nXx(t)=lx+ǫx[1+cos(t)], (23)\nYy(t)=ly+ǫy[1+cos(t)], (24)\nbeinglxandlytheX andY componentsof l(sn); thisdistanceis measuredfromtheoriginofthe\ncoordinatessystem to the center of the sn=0,...,11 scatters at ( n+1)thcollision. Since we al-\nreadyknowthepositionoftheparticleatthecollision( n+1)th,onecaneasilyfindthedistancebe-\ntweentwosuccessiveimpacts,whichisgivenby d=/radicalbig\n[Xp(t)−X(δn,tn)]2+[Yp(t)−Y(δn,tn)]2.\nThen,the timeat ( n+1)thcollisionisobtainedevaluatingtheexpression\ntn+1=tn+/radicalbig\n[Xp(t)−X(δn,tn)]2+[Yp(t)−Y(δn,tn)]2\n|− →Vn|. (25)\nThenextstep istoobtainthe impactparameter, bn+1, whichis givenby\nbn+1=bn−l(sn)sin/parenleftbigg\nθn−ψ+π\n2/parenrightbigg\n, (26)\nwherel(sn)=/radicalbig\n(∆X)2+(∆Y)2andψ=arctan(∆X/∆Y) with\n∆X=lx+ǫx[cos(tn+1)−cos(tn)] (27)\n∆Y=ly+ǫy[cos(tn+1)−cos(tn)] (28)\nMoreover,thenewdirectionofthetrajectory, θn+1is\nθn+1=π+θn+2arcsin/bracketleftiggbn+1\nR/bracketrightigg\n. (29)\nWe already know ( θn+1,bn+1,tn+1), however we still have to find− →Vn+1. At the new angular\npositionδn+1, theunitarytangentandnormalvectorsare\n− →Tn+1=cos(δn+1)/hatwidei+sin(δn+1)/hatwidej, (30)\n− →Nn+1=−sin(δn+1)/hatwidei+cos(δn+1)/hatwidej. (31)\n7Since the referential frame of the boundary is moving, then, at the instant of the collision,\naccordingto ourconstruction,thefollowingconditionsmu stbematched\n− →V′n+1·− →Tn+1=γ− →V′n·− →Tn+1, (32)\n− →V′n+1·− →Nn+1=−δ− →V′n·− →Nn+1, (33)\nwhereγ∈[0,1] andδ∈[0,1] are damping coe fficients, which means that the particle can lose\nvelocity/energyuponcollision. Thecompleteinelasticcaseoccursw henγ=δ=0. Ontheother\nhand, when γ=δ=1 correspondsto the conservativecase. The upper prime indi cates that the\nvelocityoftheparticleismeasuredwith respectto themovi ngboundaryreferentialframe.\nHence,onecaneasilyfindthat\n− →Vn+1·− →Tn+1=γ− →Vn·− →Tn+1+(1−γ)− →Vb(tn+1)·− →Tn+1. (34)\n− →Vn+1·− →Nn+1=−δ− →Vn·− →Nn+1+(1+δ)− →Vb(tn+1)·− →Nn+1, (35)\nwhere− →Vb(tn+1) isthevelocityoftheboundarywhichiswrittenas\n− →Vb(tn+1)=−sin(tn+1)[ǫx/hatwidei+ǫy/hatwidej], (36)\nFinally,thevelocityat ( n+1)thcollisionisgivenby\n|− →Vn+1|=/radicalig\n(− →Vn+1·− →Tn+1)2+(− →Vn+1·− →Nn+1)2. (37)\n3.1. Numerical Results\nOur numerical results for the time-dependentLorentz Gas sh ows basically the behaviour of\nthe average velocity of the particle. Two di fferent procedures were applied in order to obtain\nthe average velocity. Firstly, we evaluate the average velo city over the orbit for a single initial\nconditionwhichisdefinedas\nVi=1\nn+1n/summationdisplay\nj=0Vi,j, (38)\nwhere the index icorresponds to a sample of an ensemble of initial conditions . Hence, the\naveragevelocityiswrittenas\nV=1\nMM/summationdisplay\ni=1Vi, (39)\nwhereMdenotes the number of di fferent initial conditions. We have considered M=1000 in\noursimulationsandfromnowon,we alsofixedthevalue a=2.\n8101102103104105106107\nn10-310-210-1100VV0=8x10-4\nV0=9x10-4\nV0=1x10-3\nV0=2x10-3\nV0=3x10-3\nV0=4x10-3\nV0=5x10-3\nV0=6x10-3\nV0=7x10-3\nV0=8x10-3\nV0=9x10-3\nV0=1x10-2\nV0=2x10-2\nV0=3x10-2\nV0=4x10-2\nV0=5x10-2\nV0=6x10-2\nV0=7x10-2\nV0=8x10-2\nV0=9x10-2\nV0=1x10-1\nFigure 4: Behaviour of ¯V×n for different initial velocities. The control parameters used were ǫx=10−4,ǫy=3×10−4\nand a=2\n3.2. Scalingresultsfortheconservativecase\nOur main goal in this section is describe a scaling present in the model for the conservative\ncase, where, in Eq. 32 and Eq. 33, we assume γ=δ=1, and also verify the validity of LRA\nconjecture.\nWebegindiscussingascalingobservedfortheaverageveloc ityoftheparticleasfunctionof\nV0andn. It is shown in Fig. 4 the behavior of the ¯V×nfor different initial velocities. Hence,\nthe control parameters used in Fig. 4 are ǫx=10−4,ǫy=3×10−4. Additionally, 21 di fferent\nvalues of V0were chosen and for each one we randomly chose t∈[0,2π],θ∈[0,2π] and\nb∈[1−(ǫx+ǫy),−1+(ǫx+ǫy)]. As one can see, all curves of the ¯Vbehave quite similarly\nin the sense that: for short n, the average velocity remainsconstant, then after a change over,all\nthe curves start growing with the same exponent. Such behavi our is typical in systems that can\nbe described using scaling approach. Based on the behavior s hown in Fig. 4, we propose the\nfollowinghypotheses:\n1. When n≪nx,¯Vbehavesaccordingto\n¯Vsat∝Vζ\n0, (40)\n92. Forn≫nx, theaveragevelocityisgivenby\n¯V∝nν, (41)\n3. The crossoveriteration numberthat marksthe changefrom constantvelocity to growthis\nwrittenas\nnx∝Vξ\n0, (42)\nwhereζandνarethecritical exponentsand ξisa dynamicexponent.\nAfter consider these three initial suppositions, we suppos e that the average velocity is de-\nscribedintermsofa generalizedhomogeneousfunctionofth etype\n¯V(V0,n)=l¯V(lpV0,lqn), (43)\nwherel isthe scalingfactor, pandqarescaling exponentsthat in principlemust be relatedto ζ,\nνandξ. Ifwe choselpV0=1,thenl=V−1/p\n0andEq. (43)is givenby\n¯V(V0,n)=V0−1/p¯V1(V−q/p\n0n), (44)\nwhere¯V1(V−q/p\n0n)=¯V(1,V−q/p\n0n)isassumedtobeconstantfor n≪nx. ComparingEq. (44)and\nEq. (40),weobtain ζ=−1/p.\nOntheotherhand,Choosingnowl =n−1/q,Eq. (43)isrewrittenas\n¯V(V0,n)=n−1/q¯V2(n−p/qV0), (45)\nwherethefunction ¯V2isdefinedas ¯V2(n−p/qV0)=¯V(n−p/qV0,1). Itisalsoassumedtobeconstant\nforn≫nx. Comparing Eq. (45) and Eq. (41) we find ν=−1/q. Given the two di fferent\nexpressions of the scaling factor l, we obtain a relation for the dynamic exponent ξ, which is\ngivenby\nξ=ζ\nν. (46)\nNotethat thescalingexponentsaredeterminedifthe critic alexponents ζandνwerenumer-\nically obtained. The exponent νis obtained from a power law fitting for the average velocity\nwhenn≫nx. Thus,an averageofthese valuesgives ν=0.49(1). Figure5 showsthe behaviour\nof (a),¯Vsat×V0and (b),nx×V0. Applying power law fittings we obtain ζ=1.00(1)/simequal1 and\nξ=2.01(3). Considering the previous values of both ζandνand usingξ=ζ/ν, we find that\nξ=2.04(2). Such result indeed agrees with our numerical data. In order to confirm the initial\nhypotheses and, since the values of the scaling exponents ζ,νandξare now known, we will\ncollapse all the curvesonto a single and universalplot, as d emonstratedin Fig. 6. Additionally,\nsuch result allow usto confirmthe validityof LRA conjecture since the ¯Vgrowsunboundedfor\nn>>nx.\n3.3. Scalingresultsforthedissipativecase.\nIn this section we will use the same scaling formalism as used in the previous section. We\nconcentratetocharacterizethebehaviouroftheaverageve locityintermsofthenumberofcolli-\nsionswiththeboundaryandasafunctionofthedampingcoe fficientalongthenormalcomponent\n100.001 0.004 0.02 0.06\nV00.0010.0040.020.06VsatNumerical Data\nBest Fit\n0.001 0.004 0.02 0.06\nV02×1013×1024×1037×104nx\nNumerical Data\nBest Fitζ=1.00(1)\nξ=2.01(3)(a)\n(b)\nFigure 5: (a) PlotofV sat×V0. (b) Behaviour ofn xasfunction ofV 0.\noftheparticle’svelocity, δ. WestudyadissipativeversionoftheLorentzGasclosetoth etransi-\ntionfromunlimitedtolimitedenergygrowth. Indeed,sucha transitionhappenswhenthecontrol\nparameterδ→1andit isbettercharacterizedifadoptthefollowingtrans formationδ→(1−δ).\nTo obtain the average velocity, each initial condition has a fixed initial velocity, V0=10−4and\nrandomlychose t∈[0,2π],θ∈[0,2π]andb∈[1−(ǫx+ǫy),−1+(ǫx+ǫy)]. Thecontrolparameter\nγwerefixedasbeen γ=1.\nItisshowninFig. 7 thebehaviourofaveragevelocityasfunc tionofthenumberofcollision\nfor different values of δ. Note that, for di fferent values of δ, the average velocity, for small\nn, starts to grow with the same slope and them they bend towards a regime of saturation for\nlong enough values of n. The changeover from growth to the saturation is marked by a t ypical\ncrossovernumber nx. Forsucha behaviour,we canproposethefollowingscalingh ypotheses:\n1. When n≪nxtheaveragevelocityis\nV∝nη, (47)\n2. For long time, n≫nx, the average velocity approaches a regime of saturation, th at is\n11101102103104105106107\nn4×10-32×10-26×10-22×10-11×100VV0=1x10-3\nV0=2x10-3\nV0=5x10-3\nV0=1x10-2\nV0=2x10-2\nV0=5x10-2\nV0=1x10-1\n102104106108101010121014\nn / V0ξ100101102103V / V0ζ(a)\n(b)\nFigure 6: (a)Behaviour ofaverage velocity for di fferent values ofV 0;(b)their collapse onto asingle and universal plot.\ndescribedas\nVsat∝(1−δ)σ, (48)\n3. The crossover numberthat marks the regime of growth to the constant velocity is written\nas\nnx∝(1−δ)z, (49)\nwhereσ,ηandzarecriticalexponents.\nThese scaling hypotheses allow us to describe the average ve locity in terms of a scaling\nfunctionofthetype\nV[n,(1−δ)]=lV[lpn,lq(1−δ)], (50)\nwherepeqare scaling exponents and l is a scaling factor. Since l is a sc aling factor, we can\nchoseit suchthatlpn=1,yielding\nV[n,(1−δ)]=n−1/pV1[(n)−q/p(1−δ)], (51)\n12Figure 7: Behaviour of V×n for different values of δ,as labeled in the figure.\nwhereV1[(n)−q/p(1−δ)]=V[1,(n)−q/p(1−δ)]isassumedtobeconstantfor n≪nx. Comparing\nEqs. (47) and (51), we obtain η=−1/p. A power law fitting gives us that η=0.471(2). Such\nvalue wasobtainedfromthe rangeof δ∈[0.99,0.99999]. Choosingnow lq(1−δ)=1, we have\nthat l=(1−δ)−1/qandEq. (50) isrewrittenas\nV[n,(1−δ)]=(1−δ)−1/qV2[(1−δ)−p/qn], (52)\nwhereV2[(1−δ)−p/qn]=V[(1−δ)−p/qn,1] is assumed to be constant for n≫nx. Comparing\nEqs. (48) and (52), we obtain −1/q=σ=−0.47(3)[see Fig. 8 (a)]. Using nowthe expressions\nobtainedforthescalingfactor l, wecaneasily showthat\nz=σ\nη=0.99(5), (53)\nwhichisquiteclosetothevalueobtainednumerically,assh ownin Fig. 8(b). Aconfirmationof\ntheinitialhypothesesismadebyacollapseallthecurvesof ¯V×nontoasingleanduniversalplot,\nas shown in Fig. 9, showing that the system is scaling invaria nt. We also shown that dissipation\ncauses a drastic change in the behavior of V. Note that when δ→1, implies that Eq. 48 and\nEq. 49 diverge, thus recovering the results for the conserva tive case, i.e., Fermi acceleration.\nHowever, when δis slightly less than 1, the average velocity grows and then r each a regime of\nsaturation for long enough time. Our results reinforce that dissipation introduced via damping\ncoefficientsisasufficientconditionto suppressthe phenomenonofFermiacceler ation.\n4. Conclusion\nIn this paper we consider the problem of a classical Lorentz G as considering both the static\nand time-dependent boundary. For the static case we obtain t he mapping that describes the dy-\nnamics of the system and we have shown that the modelhas a chao tic componentcharacterized\nwith positive Lyapunov Exponent. After that, we introduce a new type of the time-dependent\nperturbation on the boundary. Our results confirm the validi ty of LRA conjecture for the con-\nservative case since the phenomenon of Fermi acceleration i s observed. When dissipation, via\n132×10-52×10-41×10-31×10-21×10-1\n1−δ8×10-32×10-23×10-26×10-21×10-12×10-1VsatNumerical Data\nBest Fit\n2×10-52×10-41×10-31×10-21×10-1\n1−δ1×1002×1014×1028×1032×105nxNumerical Data\nBest Fitσ=−0.47(3)\nz=-1.04(7)(a)\n(b)\nFigure 8: (a) Behaviour of Vsat×(1−δ). (b) Behaviour of the crossover number n xagainst(1−δ). Apower law fitting\nin (a) furnishes σ=−0.47(3)while in (b) z=−1.04(7).\ndamping coefficients, is introduced into the model we observed that the ave rage velocity grows\nwith time and then reaches a constant value for large enough t ime confirmingthat Fermi accel-\neration is suppressed. Finally, the average quantities wer e described by scaling functions with\ncharacteristic exponentswhose validity was confirmed with the collapse of the curvesof ¯Vinto\nanuniversalplot.\nACKNOWLEDGMENTS\nD.F.M.O gratefully acknowledges Max Planck Institute for fi nancial support. E. D. L. is\ngratefultoFAPESP,CNPqandFUNDUNESP,Brazilianagencies . TheauthorsacknowledgeDr.\nDouglasFregolentefora carefulreadingonthemanuscript.\nReferences\n[1] E.Fermi, On the Origin of theCosmic Radiation, Phys. Rev . 75 (1949) 1169.\n14101102103104105106107\n1−δ4×10-38×10-32×10-23×10-26×10-21×10-1V\nδ=0.99\nδ=0.996\nδ=0.999\nδ=0.9996\nδ=0.9999\n10-310-210-1100101102103104105\nn/(1−δ)z1×10-56×10-53×10-42×10-38×10-3V/(1−δ)σ(a)\n(b)\nFigure9: (a)Different curvesofthe Vforfivedifferent control parameters. (b)Theircollapse onto asinglea nd universal\nplot.\n[2] S.E.Sklarz,D.J.Tannor,N.Khaneja, Decoherence Contr olbyTrackingaHamiltonian Reference Molecule, Phys.\nRev. A69 (2004) 053408.\n[3] R.Gommers,S.Bergamini,F.Renzoni,Dissipation-Indu ced SymmetryBreakinginaDrivenOpticalLattice, Phys.\nRev. Lett. 95 (2005) 073003.\n[4] M. Steiner, M. Freitag, V. Perebeinos, J. C. Tsang, J. P. S mall, M. Kinoshita, D. Yuan, J. Liu, P. Avouris, Phonon\npopulations and electrical power dissipation in carbon nan otube transistors, Nature Nanotechnology 4 (2009) 320.\n[5] K. Nakamura and T.Harayama, Quantum Chaos and Quantum Do ts, Oxford University Press,Oxford, 2004.\n[6] D.G. 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Phys.57 (1985) 617.\n16"    },    {        "title": "2402.02940v1.Fractional_damping_induces_resonant_behavior_in_the_Duffing_oscillator.pdf",        "content": "Fractional damping induces resonant behavior in the Duffing\noscillator\nMattia Coccolo, Jes´ us M. Seoane, and Miguel A.F. Sanju´ an\nNonlinear Dynamics, Chaos and Complex Systems Group,\nDepartamento de F´ ısica, Universidad Rey Juan Carlos,\nTulip´ an s/n, 28933 M´ ostoles (Madrid), Spain\n(Dated: February 6, 2024)\nAbstract\nThe interaction between the fractional order parameter and the damping parameter can play a\nrelevant role for introducing different dynamical behaviors in a physical system. Here, we study\nthe Duffing oscillator with a fractional damping term. Our findings show that for certain values of\nthe fractional order parameter, the damping parameter, and the forcing amplitude high oscillations\namplitude can be induced. This phenomenon is due to the appearance of a resonance in the Duffing\noscillator only when the damping term is fractional.\n1arXiv:2402.02940v1  [nlin.CD]  5 Feb 2024I. INTRODUCTION\nNonlinear oscillators are dynamical systems that exhibit oscillatory behaviors but do not\nfollow a linear relationship between the restoring force and the displacement. These sys-\ntems often arise in various scientific and engineering fields, and their study has profound\nimplications in understanding complex phenomena and designing efficient devices. When\nconsidering damping in nonlinear oscillators, a commonly used approach is to introduce a\nfractional derivative in the damping term. The use of fractional derivatives in damping terms\nprovides a powerful tool to capture the memory effects and non-local behavior observed in\ncertain physical systems. Unlike the traditional integer-order derivatives that describe in-\nstantaneous rates of change, fractional derivatives involve a non-integer order and account\nfor past history in the damping process. This allows us a more accurate representation of\ndamping in systems with memory and hysteresis as occurs in some properties of complex\nmaterials. The presence of a fractional derivative damping term in nonlinear oscillators can\nlead to novel behaviors such as sub-diffusive or super-diffusive damping, where the decay\nof oscillations is slower or faster than in traditional damping scenarios. Moreover, the in-\nclusion of the fractional derivatives can alter the stability regions, bifurcation boundaries,\nand resonance responses of the oscillatory system, introducing rich and intricate dynamics.\nThe study of nonlinear oscillators with fractional derivative in the damping term finds ap-\nplications in disciplines such as mechanical engineering, physics, robotics, control systems,\nand even social sciences [1–10]. Understanding the behavior and control of these oscillators\nis crucial for optimizing energy dissipation, enhancing stability, and designing systems with\nspecific response characteristics.\nOn the other hand, nonlinear resonances [11] is a fascinating phenomenon that occurs\nwhen a system, subjected to an external force or perturbation, exhibits a response that\nis not proportional to the input. Unlike linear resonance, where the response is directly\nproportional to the forcing frequency, nonlinear resonances arise due to the intricate interplay\nbetween multiple frequencies and energy transfer mechanisms within the system, as in our\ncase the fractional order parameter, the damping parameter and the forcing amplitude. In\nnonlinear resonances, the system’s response can display a wide range of complex behaviors,\nincluding the generation of harmonics, sub-harmonics, chaotic oscillations, and bifurcations.\nAs stated before, the fractional derivatives are the key to explore and modeling materials\n2with memory effects and fractal geometry or diffusion processes, among others. In fact,\nfractional derivatives can play a crucial role on them. Therefore, it is relevant to analyze\nwhen the fractional order parameter can trigger complex behaviors and nonlinear resonance\nphenomena.\nHere, we analyze the Duffing oscillator with a fractional derivative in its damping term\nas in Ref. [12]. Specifically, we focus our study in the numerical analysis of the fractional\norder parameter effects on the oscillations amplitude of the system. We have found that the\nfractional order parameter can induce resonance phenomena. Firstly, we analyze, for smaller\nforcing amplitude values, beyond a certain threshold, the appearance of resonance peaks that\nare triggered by the high damping parameter values and specific values of the fractional\norder parameter. These high amplitude oscillations are related with a complex dynamics of\nthe oscillator in the phase space. Then, we find a region of resonances for higher forcing\namplitude values. Consequently, we characterize and analyze the appearance of resonance\nphenomena induced by the effect of the three main ingredients: high forcing amplitude and\ndamping parameter values, and some specific values of the fractional order parameter. It\nis relevant to emphasize that these complex behaviors and resonance phenomena are an\neffect of the fractional order parameter, and they only appear when a fractional derivative\ndamping term exists.\nThe organization of this paper is as follows. In Sec. II, we introduce the fractional\nDuffing oscillator and its dynamics. In Sec. III, we numerically present the effects on the\noscillations amplitude of the fractional order parameter in which a resonant-like behavior\nis uncovered. The impact of the forcing amplitudes due to the fractional order parameter\nis analyzed in Sec. IV, where we numerically characterize the birth of this resonance-like\nbehavior. Conclusions and a discussion of the main results of this paper are presented in\nSec. V.\nII. MODEL DESCRIPTION\nWe study a Duffing oscillator with a fractional derivative of fractional order parameter α\ninstead of the first derivative, i.e., ˙ x(t)→Dαx(t) =dαx(t)/dtαas a damping term. Then,\nthe Duffing oscillator with a fractional damping term is given by\nd2x\ndt2+µdαx\ndtα−x+x3=Fcosωt, (1)\n3where µis the damping parameter, Fandωare the forcing amplitude and forcing frequency,\nrespectively, and αis the fractional order parameter. In order to integrate the system, we\nuse the Gr¨ unwald-Letnikov fractional derivative, as in previous works [12, 13]. Consequently,\nthe system is composed by a set of three fractional differential equations that yield\n\n\nDαx=y,\nD1−αy=z,\nDz=µy+Fcos (ωt) +x−x3,(2)\nwhere zis an auxiliary component coming from the transformation into a fractional order\nsystem. The integration step used is π/2000, and the maximum time value is t= 300 that\nis large enough for the system to reach the steady state. We have carried out numerical\nsimulations with several initial conditions and, although the results are numerically different,\nthe general trends of the curves are qualitatively similar. So, we carry on the study with\nthe initial condition ( x0, y0, z0) = (0 .2,0.3,0.0), as in Ref. [12, 14]. We expect that our\nconclusions are of general validity and not specific for the considered boundary conditions.\nIn Fig. 1, we show some examples of trajectories of the system and the impact of the α\nvalue on its dynamics, when the µparameter varies. In fact, according to Fig. 1(a), where\nµ= 0.6, and Fig. 1(b), where µ= 0.9, we have to change the αparameter from α= 0.3\ntoα= 0.1, respectively, if we want to observe similar oscillations amplitudes. In Fig. 1(c),\nwe compare trajectories for µ= 0.9 and α= 0.3 (in blue) and for µ= 0.9 and α= 0.1 (in\nred). Having traced more trajectories than those shown, we have seen that the phenomenon\nis complex and worthy of study.\nIII. FRACTIONAL ORDER PARAMETER EFFECTS ON THE OSCILLATIONS\nAMPLITUDE\nAs we previously mentioned, some preliminary computations about the resonance induced\nby the fractional derivative in the Duffing damping term have been done in [12]. The authors\nhave disclosed that for a high value of the damping parameter µ= 0.8 and for distinctive\nvalues of the fractional order parameter α, some interesting peaks pop up when the forcing\namplitude is large enough, as shown in Fig. 2. Also, for higher αvalues we have seen that\nthe oscillations amplitude do not show those kind of peaks and that the curve stays smooth\n4FIG. 1: In this figure, we have plotted the asymptotic behaviors of the system trajectory in phase\nspace for (a) µ= 0.6 and α= 0.3, (b) µ= 0.9 and α= 0.1 and (c) µ= 0.9 and α= 0.1 (in red)\nandα= 0.3 (in blue). We fix the forcing amplitude F= 1.5 for all cases. Note the key role of the\nαandµparameters in the amplitude of the oscillations.\nunder an oscillations amplitude the of value 0 .5. We can see that two particular peaks stand\nout in the amplitude and the Q−factor plots, Fig. 2(a) and Fig. 2(c), respectively. However,\nthis fact was just mentioned, though not analyzed in detail. As stated before, the Q−factor\nprovides an idea of how much the signal is amplified by a certain parameter, in our case α.\nWe calculate it by computing the sine and cosine components:\nBs=2\nnTZnT\n0x(t) sinωtdt and Bc=2\nnTZnT\n0x(t) cosωtdt, (3)\nwhere T= 2π/ω andnis an integer. Then, we can find the dependence on αwith\nQ=p\nB2\ns+B2\nc\nα. (4)\nThose peaks are related with fractional order parameter values that suddenly give birth\nto aperiodic orbits, as we can see in Fig. 2(b). They are resonance-like peaks for the α\nparameter, as confirmed by the Q−factor, but it is worth of study the conditions for them\nto appear and the parameter values for which the gain in amplitude is higher. Our objective\n5now is to analyze in depth this resonant behavior. For this purpose, we plot the gradient of\nthe amplitude in the parameter set α−µfor two values of the forcing amplitude F= 0.01\nandF= 0.1 in Fig. 3. In the first case, Fig. 3(a), we can see that the gradient in the\nparameter set is smooth, the oscillations amplitude is not that high and the higher oscil-\nlations amplitude zone is very easy to spot. The only interesting thing is the intermediate\noscillations amplitude smudge for µ≳0.6. Even more interesting are the other two panels\nwhere Figs. 3(c) is a zoom of Fig. 3(b). When we reach the forcing amplitude value F= 0.1,\nthe parameter set becomes more complicated and the high and low oscillations amplitude\nintermingle.\nFrom values µ≳0.4, the structure that can be spotted in Fig. 2(a) and that ends at\nα≈0.15, starts to take form with lots of the resonance peaks, in which the oscillations span\nboth potential wells. In fact, in Fig. 4 we show the oscillations amplitude plots for different\nµvalues as a section of the 3 −Dfigure. There, we can see that for µ= 0.35 there is no\nsign of the structure, while it is visible for µ= 0.55. Clearly, for higher µvalues it becomes\neven more visible. In the 3 −Dfigure, it is possible to discern the built-up of such structure\nand how the higher oscillations amplitude peaks diminish when µgrows. This structure is\nthe result of the higher oscillations amplitude being attenuated by the dissipation term, so\nthat the peaks are the consequence of the fractional order parameter feat interacting with\nthe forcing amplitude. Hence, it seems that they are fractional-induced amplitude peaks\nthat behave like a resonance when only the fractional order parameter varies. In fact, if we\ncompute the Q−factor for the parameter µthe peaks do not show up in the plot.\nIn Fig. 4(e), we can see that the trajectories become bounded to one well when the dis-\nsipation is too strong, µ= 1, for the fractional order parameter to maintain the interwell\noscillations. Figure 3(c) has been plotted to have a better understanding of the higher am-\nplitude oscillations zone. In particular of the cross like patch centered at µ= 0.6. Here, we\ncan find a region of small oscillations amplitude embedded inside a high oscillations ampli-\ntude region. It is precisely there, where the already mentioned structure, well recognisable\nin Fig. 2(a), starts to take form.\n6FIG. 2: We plot here (a) the oscillations amplitude, (b) the asymptotic behaviors diagram and (c)\ntheQ−factor for F= 0.1, µ= 0.8. The two resonance peaks can be recognise in the panels.\nIV. THE FORCING AMPLITUDE EFFECT ON THE OSCILLATIONS AMPLI-\nTUDE\nTo carry on our analysis, we have decided to study the effect of the forcing amplitude F\non the oscillations amplitude since it will provide more easily clear light on the resonance-like\nphenomenon. The main result of this manuscript is shown in Fig. 5, where we have depicted\nthe gradient parameter set F−αof the oscillations amplitude for different values of µ. As\nexpected, the higher the damping term the lower the oscillations amplitude. Interestingly,\na high amplitude oscillations region starts to appear when µ >0.5, in particular it becomes\nmore evident at µ≈0.6. This region is centered around F= 1.5 and spans all or part\nof the parameter set, depending on the damping parameter. In fact, we can see there\nthat all along the panels of Fig. 5 that region shrinks along the αaxis together with the\noscillations amplitude. That means that for a high enough damping parameter value, the\nhigh oscillations amplitude is confined around a certain forcing amplitude value, that is,\nF= 1.5 and it is induced by a certain range of fractional order parameter values. These\nresults corroborate what was observed in Fig. 1.\n7FIG. 3: The figure shows the oscillations amplitude for F= 0.01 (a) and F= 0.1 (b). Panel (c) is\na zoom of panel (b). Here it is possible to appreciate how the difference in the amplitude forcing\ncomplicates the topology of the parameter set.\nBesides, the region shrinks spanning smaller values of the fractional order parameter\nwhen the dissipation grows. This phenomenon can be explained as in the previous section.\nThe interaction among high dissipation, certain values of the fractional order parameter and\nhigh forcing amplitude triggers a resonant behavior. The interesting fact is that the gain in\namplitude is confined to certain αvalues, so we can define the phenomenon as a resonance\ninduced by the fractional order parameter. The difference with F= 0.1 is that now the\nforcing amplitude is large enough to sustain the high oscillations amplitude inside a large\nregion of the parameter set and not just for some particular values of α.\nFor a better understanding and the characterization of the phenomenon, we have deep-\nened its analysis with the other figures below. We want to stress out that all the figures\nfrom now on are related with Fig. 5 and they provide more specific details of the features\n8FIG. 4: In panel (a), we show the 3 Dversion of Fig. 3(b). The other panels display 4 slices of the\npanel (a) along the αaxis and for (b) µ= 0.35, (c) µ= 0.55, (d) µ= 0.76, and (e) µ= 1. Here,\nthe formation of the central structure is clear. In fact, in panel (c) it is possible to see how the\nhigh oscillations amplitude of panel (b) decreases and start to form the central structure of panels\n(d) and (e).\nof the resonance phenomenon. Therefore, we start by showing in Fig. 6(a) the oscillations\namplitude versus the forcing amplitude Fforµ= 0.6. Every curve is associated with a\nfractional order parameter value as indicated there. It is possible to see that the amplitude\ncurves corresponding with the two values α= 0.1 and α= 0.3, for this damping parame-\nter, cross the high oscillations amplitude region of Fig. 5(b) and present the corresponding\npeaks, near F= 1.5. On the other hand, the curves related with the other two αvalues are\nsmooth after a first jump, at α≈0.5. In Fig. 6(b), we show the frequencies of the oscilla-\ntions, calculated with the Fast Fourier transform. It is possible to see that from F= 0.8\nthe calculated frequencies tend naturally to the forcing frequency ω= 1, but not near the\nforcing amplitude for which the resonance peaks are produced for the αvalues that cross\nthe high amplitude region of Fig. 5(b).\nIn order to have a deeper insight on the dynamics of the system, we plot Fig. 7 in which\nwe show the maxima-minima diagram for µ= 0.6 and for 4 values of the αparameter, along\n9FIG. 5: We show the oscillations amplitude in the parameter set F−α, for different µvalues. Here,\nit appears along the panels the formation of a well located region of high oscillations amplitude\nnearF= 1.5\nwith two orbits in the case of F= 1.5 and α= 0.1 (Fig. 7(e)) and α= 0.3 (Figs. 7(f)). These\nfractional order parameter values are the ones that cross the high oscillations amplitude in\nFig. 5(b). The other two values, see Figs. 7(c) and (d), do not cross that region as indicated\nby the maxima-minima diagram smoothness. One consideration can be made, we can see\nthat there is an important difference between the two plotted orbits, as seen in Figs. 7(e)\nand (f). The first one is not periodic, while the second one is periodic of period 3. It seems\nthat when the fractional order parameter grows, the large amplitude aperiodic trajectories\ndisappear becoming periodic as if the fractional order parameter contributed to introduce\norder into chaos.\nNow, we check the robustness of the resonance induced by the fractional order parameter\nwhen the forcing frequency ωchanges. For that purpose, we plot the Q−factor in Fig. 8,\nwhich shows the oscillations amplitude and the maxima-minima diagram in function of the\nforcing frequency and for F= 1.5 and µ= 0.6. In Figs. 8(a) and (b) it is possible to notice\nsome peaks in the Q−factor curve that assure us that they are related with the fractional\norder parameter induced resonance. This resonance phenomenon needs an interaction among\n10FIG. 6: The figure shows four slices of Fig. 5(b) overlapped in panel (a) and the oscillations\nfrequencies calculated with the Fast Fourier transform in panel (b). In the first panel, the curves\nrelated to fractional order values, α= 0.1 and α= 0.3, that cross the high amplitude oscillations\nregion are recognisable. The other two, after they grow at F∼0.5, become smooth. The frequen-\ncies in panel (b) for F≳0.9 tend to the forcing frequency ω= 1, except for the ones related with\nthe resonance peak.\nhigh forcing amplitude, high damping parameter and small values of the fractional order\nparameter. In Fig. 8(a), we have highlighted three peaks at α= 0.54, α= 1 and α= 2.01\nthat will be discussed later when we analyze higher µvalues. The peaks are corroborated\nby the maxima-minima diagram, as seen in Fig. 8(c). In this last figure, it is interesting to\nnotice that the higher peaks are related with aperiodic trajectories.\nNow, we analyze the situation in which the damping parameter has a large value and,\ntherefore, we are in the overdamped regime. In Fig. 9, we can find a similar plot as Fig. 6\nfor the damping parameter value µ= 0.9. Here, in Fig. 9(a), we can see that only for\nα= 0.1 there is a resonance peak, as expected from Fig. 5(e). The computed oscillations\nfrequencies, Fig. 9(b), behave as in the previous case. This result is confirmed by Fig. 10,\nwhere we can see that the resonance amplitude can be found only for α= 0.1, Fig. 10(a),\nwhile for the others fractional values the maxima-minima diagram is smooth for large forcing\n11FIG. 7: We show the behavior of the system for fixed values of αandµ. Comparing panel (a) and\n(b), we can observe that the system behaviors at F= 1.5 are different. For α= 0.1, in panel (e),\nthe oscillations are clearly not periodic, while for α= 0.3, in panel (f), the oscillations are periodic\nof period 3. In the other two panels, the higher oscillations amplitude enlargement is lacking.\namplitudes, Figs. 10(b)-(d). Interestingly, the high oscillations amplitude is related with\nperiodic trajectories, as we can see in Fig. 10(e), differently from what was happening in\nFig. 7(e). We also compare the trajectories for F= 1.5 at α= 0.1 and at α= 0.3 in\nFig. 10(f) and we can appreciate the gain in amplitude, due to the resonance. Then, we\nplot Fig. 11, by varying the forcing frequency ω. Here, we can spot the three peaks already\nmentioned before, as depicted in Fig. 8(a). Again, the interaction between high damping\nparameter, forcing and fractional order parameter induces the resonance. In this figure, we\ncan see that the Q−factor curves, Fig. 11(a) and Fig. 11(d), become smoother and that the\nthree peaks at ω= 0.5, ω= 1 and ω= 2 are more recognizable. Of course, those peaks are\nrelated with higher oscillations amplitude, as shown in Fig. 11(b) and Fig. 11(e). Then, in\nFig. 11(c) and Fig. 11(f), we can appreciate that the peaks match regions of non periodic\ndynamics of the system. Contrarily to the µ= 0.6 situation, in the µ= 0.9 and µ= 1\ncases, the resonance peaks are centered to specific values of the forcing frequency ω, related\namong them as ω0= 1,ω1/2=ω0/2 and ω2= 2ω0. So, we obtained the resonance frequency\n12FIG. 8: We show the Q−factor (a), the oscillations amplitude (b) and the maxima-minima diagram\nin function of the forcing frequency ωfor the following parameters: µ= 0.6,α= 0.1 and F= 1.5.\nIn the first panel, three peaks are emphasized with three vertical lines in order to compare them\nwith similar peaks in other forthcoming figures. The peaks in the panels are the result of the\nequilibrium reached by the high oscillations suppression due to the damping term, the effect of the\nforcing amplitude and the action of the fractional order parameter.\nand its harmonic and subharmonic. Also, in Fig. 11(c) and Fig. 11(f), we can see that the\ndissipation keeps doing its job by bringing more order into chaos.\nFinally, we present the oscillations amplitude gradient for F= 1 in the parameter set\nµ−α, Fig. 12(a) and for F= 2 in Fig. 12(b). As expected, in the both cases, the high\noscillations amplitudes are common in all the parameter set, although the higher ones are\nconcentrated at small µandαvalues, like in Fig. 3(b). Then, in the first case, F= 1.5,\nthe lower amplitude oscillations can be found in the exact opposite corner of the figure and\nscattered all around its upper center. Otherwise, in the second case, F= 2, as the damping\nparameter reaches the value µ= 0.6, the oscillations amplitude diminishes abruptly and for\nthe parameter values for which the resonance is triggered the oscillations amplitude becomes\nsmaller than in the F= 1.5 case. This outcome is corroborated by Fig. 12(c), where the sign\nof the difference between the oscillations amplitude at F= 2 and F= 1.5 is shown. The\n13FIG. 9: Panel (a) shows four slices of Fig. 5(e) overlapped. Panel (b) shows the oscillations\nfrequencies calculated with the fast Fourier transform. In panel (a), it is possible to see that the\ncurve for α= 0.1 is the only one showing the peak due to the high oscillations amplitude region of\nFig. 7. Also here, the oscillations frequencies tend to the forcing frequency ω= 1, except for the\nones related with the oscillations amplitude peak.\nyellow points indicate a positive difference AF=2> A F=1.5, the blue ones a negative difference\nAF=1.5> A F=2. The points in this plot are clearly intermingled, but the blue points generate\na more defined region for higher µvalues. This figure shows well that, although the forcing\namplitude is bigger in the F= 2 case, the interaction between the damping parameter and\nthe fractional order parameter can trigger trajectories with a higher oscillations amplitude,\ngiving birth to a resonance for F= 1.5. Then, in the other panels, we show three slices\nof the two gradient plots, shown in Fig. 12(a) and Fig. 12(b). Here, we can see that, for\nsmall damping parameter values ( µ= 0.18) the oscillations amplitude for F= 2 is generally\ncomparable with the oscillations amplitude for F= 1.5. When the damping parameter\ngrows, µ= 0.5 the oscillations amplitude for F= 1.5 are a little bit higher than the\nF= 2 case, but not for all the αvalues. Then, when the damping parameter grows further,\nµ= 0.8, the oscillations amplitude for F= 1.5 is definitively higher. Figures 12(d-f) show\nthe tendency of the oscillations amplitude for both forcing amplitudes. All of this corroborate\n14FIG. 10: The panels show the behavior of the system for fixed αandµ. Analyzing panel (a), we\ncan see that the system behavior at F= 1.5 is periodic of period 3, as shown in panel (e), unlike\nin Fig. 7(e). In the other three panels, the higher oscillations amplitude enlargement is lacking. In\nfact, in panel (f) we compare the orbits for α= 0.1 and α= 0.3 and we can appreciate the red\ntrajectory broadening with respect to the blue orbit, due to the resonance effect.\nthe results shown in Fig. 12(c). Again, at a high damping parameter value the F= 1.5 case\noscillates at higher amplitudes due to the resonance induced by the interaction among the\nhigh damping parameter, the fractional order parameter and the forcing amplitude.\n15FIG. 11: The panels show for µ= 0.9,α= 0.1 and F= 1.5 (a) the Q−factor, (b) the oscillations\namplitude and (c) the maxima-minima diagram in function of the forcing frequency ω. Panels\n(d), (e) and (f) show respectively the same plots for µ= 1. In panels (a) and (d) the already\ncommented three peaks are emphasized with three vertical lines in order to compare them with\nsimilar peaks in Fig. 8. Those peaks can be seen more clear, and others, that can be seen in Fig. 8,\nare disappearing as an effect of the dissipation growth. Also they are centered to certain values of\nthe forcing frequency ω= 0.5, ω= 1 and ω= 2.\n16FIG. 12: The figure shows (a) the gradient of the oscillations amplitude in the parameter set α−µ\nforF= 1, (b) the same gradient plot for F= 2, and (c) the sign of the difference AF=2−AF=1.5. In\npanel (c) the yellow points show the positive sign of the difference, while the blue points represent\nthe negative sign. The panels (d-f) show the oscillations amplitude at different values of the\ndamping parameter µand for the forcing amplitudes: F= 2 in blue and F= 1.5 in red. We can\nsee that as the damping parameter grows, the oscillations amplitude for F= 1.5 becomes higher\nthan in the case of F= 2.\n17V. CONCLUSIONS AND DISCUSSION\nWe have studied the fractional Duffing oscillator with a damping fractional term, using\nthe Gr¨ unwald-Letnikov integrator. For the parameter values used, we have find that the\nfractional order parameter can induce a resonance. When the forcing is relatively small, at\nF= 0.1, the peaks related with the resonance appear for particular values of the fractional\norder parameter, values that change in function of the damping parameter. For larger values\nof the forcing amplitude, say F= 1.5, the higher oscillations amplitude due to the resonance\nform a region in the parameter set ( F−α). In both cases the phenomenon is clear, the high\noscillations amplitude, that the oscillator exhibits at low damping parameter values, are\nattenuated at higher dissipation but the interaction between a particular forcing amplitude\nand certain low fractional order parameter values maintain them, generating a resonance.\nWe think that this phenomenon can be relevant to better understand the relation between\nthe damping parameter and the fractional order parameter. Besides, it shows the richness in\nthe dynamical behavior due to the presence of a fractional term in the nonlinear oscillator.\nThis last point can easily be observed in the overdamped case, where the dynamics of the\nsystem becomes still complex. Finally, we expect that this study can be useful to predict\nthe birth of complex or resonant behaviors in viscoelastic and porous materials or diffusion\nprocesses, that can be only modeled by using fractional derivatives.\nVI. ACKNOWLEDGMENTS\nThis work has been supported by the Spanish State Research Agency (AEI) and the\nEuropean Regional Development Fund (ERDF, EU) under Project No. PID2019-105554GB-\nI00 (MCIN/AEI/10.13039/501100011033).\n[1] Borowiec M, Litak G, Syta A. Vibration of the Duffing oscillator: effect of fractional damping.\nShock Vib. 2007; 14: 29-36.\n[2] Dafermos C. Dissipation in materials with memory. In: Lodge AS, Renardy M, Nohel JA,\neditors. Viscoelasticity and Rheology. Proceedings of a Symposium Conducted by the Math-\nematics Research Center, Wisconsin–Madison: Academic Press; 1985, p. 221-234.\n18[3] Lu Z, Wang Z, Zhou Y, Lu X. Nonlinear dissipative devices in structural vibration control: A\nreview. J. Sound Vib. 2018; 423: 18-49.\n[4] Chellaboina V, Haddad WM. Exponentially dissipative nonlinear dynamical systems: a non-\nlinear extension of strict positive realness. In: Jeltsema D, Scherpen J, editors. Proceedings\nof the 2000 American Control Conference, Chicago: IEEE Xplore; 2000, p. 3123-3127.\n[5] De S, Kunal K, Aluru NR. Nonlinear intrinsic dissipation in single layer MoS 2 resonators.\nRSC Adv. 2017; 7: 6403-6410.\n[6] Elliott SJ, Tehrani MG, Langley RS. Nonlinear damping and quasi-linear modelling. Philos.\nTrans. R. Soc. A 2015; 373: 20140402.\n[7] Horr AM, Schmidt LC. A fractional-spectral method for vibration of damped space structures.\nEng. Struct. 1996; 18: 947-956.\n[8] Ding Y, Liu X, Chen P, Luo X, Luo Y. Fractional-Order Impedance Control for Robot Ma-\nnipulator. Fractal fract. 2022; 6: 684.\n[9] Li X, Liu Z, Li J, Tisdell C. Existence and controllability for nonlinear fractional control\nsystems with damping in Hilbert spaces. Acta Math. Sci. 2019; 39; 229-242.\n[10] Strogatz SH. Nonlinear dynamics and chaos with student solutions manual: With applications\nto physics, biology, chemistry, and engineering. Boca Raton: CRC press; 2018.\n[11] Rajasekar S, Sanju´ an MAF. Nonlinear resonances. Switzerland: Springer International Pub-\nlishing; 2016.\n[12] Coccolo M, Seoane JM, Lenci S, Sanju´ an MAF. Fractional damping effects on the transient\ndynamics of the Duffing oscillator. Commun. Nonlinear Sci. Numer. Simul. 2023; 117: 106959.\n[13] Ortiz A, Yang J, Coccolo M, Seoane JM, Sanju´ an MAF. Fractional damping enhances chaos\nin the nonlinear Helmholtz oscillator. Nonlinear Dyn. 2020; 102: 2323-2337.\n[14] Syta A, Litak G, Lenci S, Scheffler M. Chaotic vibrations of the duffing system with fractional\ndamping. Chaos 2014; 24: 013107.\n19"    },    {        "title": "0710.1052v1.Channel_Adapted_Quantum_Error_Correction_for_the_Amplitude_Damping_Channel.pdf",        "content": "1\nChannel-Adapted Quantum Error Correction for the\nAmplitude Damping Channel\nAndrew S. Fletcher, Peter W. Shor, and Moe Z. Win Fellow, IEEE\nAbstract — We consider error correction procedures designed\nspecifically for the amplitude damping channel. We analyze am-\nplitude damping errors in the stabilizer formalism. This analysis\nallows a generalization of the [4;1]‘approximate’ amplitude\ndamping code of [1]. We present this generalization as a class of\n[2(M+ 1);M]codes forM\u00151and present quantum circuits\nfor encoding and recovery operations. We also present a [7;3]\namplitude damping code based on the classical Hamming code.\nAll of these are stabilizer codes whose encoding and recovery\noperations can be completely described with Clifford group\noperations. Finally, we describe optimization options in which\nrecovery operations may be further adapted according to the\ndamping probability \r.\nI. I NTRODUCTION\nIn the most common treatments, quantum error correction\n(QEC) is developed for a very generic error model. An\narbitrary error on a single qubit is correctable if both Pauli X\nandZare correctable on that qubit; the continuum of quantum\nerrors are thus reduced into a simple, discrete set. Using this\napproach, we may design error correction procedures that\napply to a wide variety of quantum noise processes – the\nchannel need only be well approximated by independent qubit\nerrors.\nThe general application of standard QEC comes with a\nprice in efficiency. Quantum error correcting codes require\na large number of redundant qubits; for short block lengths,\ngeneric codes are limited to low rates. While robust to arbitrary\nqubit errors, both error correction performance and efficiency\ncan be improved by adapting the encoding and recovery\noperations to the physical noise process. Such adaptation is\nreasonable since, for any particular device, the noise will have\na structure governed by the physical coupling of the system\nand the environment. Intuitively, we should be able to engineer\nimproved error correction by careful adaptation of both the\nencoding and recovery operations.\nThe concept of channel-adapted error correction is not new:\nearly work labeled ‘approximate’ quantum error correction\nwas presented in [1]. Much recent progress has been due\nto optimization efforts [2]–[7]. In each case, rather than\ncorrecting for arbitrary single qubit errors, the error recovery\nscheme was adapted to a model for the noise, with the goal\nThis paper is based on a thesis submitted in partial fulfillment of the\nrequirements for the degree of Doctor of Philosophy in the Department of\nElectrical Engineering and Computer Science at the Massachusetts Institute\nof Technology in June, 2007.\nA.S.F. would like to thank the Department of the Air Force, who sponsored\nthis work under AF Contract #FA8721-05-C-0002. All authors thank the Na-\ntional Science Foundation for support through grant CCF-0431787. Opinions,\ninterpretations, recommendations and conclusions are those of the authors and\nare not necessarily endorsed by the United States Government.to maximize the fidelity of the operation. In [2], a semidefi-\nnite program (SDP) was used to maximize the entanglement\nfidelity, given a fixed encoding and channel model. In [5]\nand [6], encodings and decodings were iteratively improved\nusing the performance criteria of ensemble average fidelity\nand entanglement fidelity, respectively. A sub-optimal method\nfor minimum fidelity, using an SDP, was proposed in [7].\nAn analytical approach to channel-adapted recovery based on\nthe pretty-good measurement and the average entanglement\nfidelity was derived in [8]. The main point of each scheme\nwas to improve error corrective procedures by adapting to the\nphysical noise process.\nThe optimization efforts cited above detail mathematical\nand algorithmic tools with general application. That is to say,\ngiven any model for the noise process and an appropriately\nshort code we can apply optimal [2], [5] and structured near-\noptimal [4] algorithms to provide channel-adapted encoding\nand recovery operations.\nIt is important to note that the aforementioned tools are\nnot, in themselves, complete solutions to the problem of\nchannel-adapted QEC. When designing an error correction\nprocedure, there is more to consider than whether an encoding\nor a recovery is physically legitimate. This motivated our\nexploration of near-optimal recovery operations [4], where\nwe imposed a projective syndrome measurement constraint\non recovery operations. Even given such a constraint, to\nimplement channel-adapted QEC efficiently we need to design\nencoding and decoding procedures with sufficiently simple\nstructure to allow efficient implementation. Furthermore, while\nthe optimization routines focus on the entanglement fidelity\nand ensemble average fidelity due to their linearity, we should\nstill like to understand the minimum fidelity, or worst case\nperformance.\nTo explore these issues in greater depth, we must consider\nchannel-adapted QEC for a specific channel model. We exam-\nine the amplitude damping channel, denoted Ea, given by the\noperation elements\nE0=\u0014\n1 0\n0p1\u0000\r\u0015\nandE1=\u00140p\r\n0 0\u0015\n:(1)\nAmplitude damping is a logical choice for several reasons.\nFirst of all, it has a useful physical interpretation: the param-\neter\rindicates the probability of decaying from state j1ito\nj0i(i.e.the probability of losing a photon). Second, amplitude\ndamping cannot be written with scaled Pauli matrices as the\noperator elements; thus Theorem 1 from [3] does not apply\nand the optimal recovery operation does not have a near-trivial\nform. Finally, due to its structure, the amplitude damping\nchannel can still be described with the stabilizer formalism,arXiv:0710.1052v1  [quant-ph]  4 Oct 20072\nR1j0Li(\u000bh0000j+\fh1111j) +j1Li(1p\n2h0011j+1p\n2h1100j)\nR2j0Li(\fh0000j\u0000\u000bh1111j) +j1Li(1p\n2h0011j\u00001p\n2h1100j)\nR3 j0Lih0111j+j1Lih0100j\nR4 j0Lih1011j+j1Lih1000j\nR5 j0Lih1101j+j1Lih0001j\nR6 j0Lih1110j+j1Lih0010j\nR7 j0Lih1001j\nR8 j0Lih1010j\nR9 j0Lih0101j\nR10 j0Lih0110j\nTABLE I\nOPTIMAL QER OPERATOR ELEMENTS FOR THE [4,1] CODE . OPERATORS\nR1ANDR2CORRESPOND TO THE “NO DAMPINGS ”TERME\n5\n0WHERE\u000b\nAND\fDEPEND ON\r.R3\u0000R6CORRECT FIRST ORDER DAMPINGS .\nR7\u0000R10PARTIALLY CORRECT SOME SECOND ORDER DAMPINGS ,\nTHOUGH AS ONLYj0LiIS RETURNED IN THESE CASES SUPERPOSITION IS\nNOT PRESERVED .\ngreatly aiding analysis.\nII. Q UALITATIVE ANALYSIS OF CHANNEL -ADAPTED QER\nFOR APPROXIMATE [4,1] CODE\nWe begin with a qualitative understanding of the [4;1]\n‘approximate’ code of [1] and its optimal channel-adapted\nrecovery operation. The logical codewords are given by\nj0Li=1p\n2(j0000i+j1111i) (2)\nj1Li=1p\n2(j0011i+j1100i): (3)\nConsider the optimal channel-adapted recovery for the [4,1]\n‘approximate’ code of [1] computed with the semidefinite\nprogram as presented in [2]. This is an example of a channel-\nadapted code, designed specifically for the amplitude damping\nchannel rather than arbitrary qubit errors. Its initial publication\ndemonstrated the utility of channel-adaptation (though without\nusing such a term) for duplicating the performance of standard\nquantum codes with both a shorter block length and while\nachieving a higher rate.\nIn [1], the authors proposed a recovery (decoding) circuit\nand demonstrated its strong performance in minimum fidelity.\nIt is interesting to note that the recovery operation (described\nin quantum circuit form in Fig. 2 of [1]) is not a projective\nsyndrome measurement followed by a unitary rotation as\nis standard for generic codes; instead it is a \r-dependent\noperation which includes a generalized (POVM) measurement.\nIn contrast, the optimal recovery (in terms of entanglement\nfidelity) obtained via convex optimization does conform to\nsuch a structure. The optimal recovery operation is given in\nTable I. We will analyze each of the operator elements in turn.\nFor clarity of presentation, we begin with first and second\norder damping errors and then we turn our attention to the\nrecovery from the ‘no damping’ term.\nA. Recovery for first and second order damping errors\nNeitherE0norE1in (1) is a scaled unitary matrix, but\nwe may understand the channel by considering E1the ‘error’event. Let us denote a first order damping error as E(k)\n1, which\nconsists of the qubit operator E1on thekthqubit and the\nidentity elsewhere. Consider now the effect of E(1)\n1on the\ncodewords of the [4;1]code:\nE1\nI\n3j0Li=p\rj0111i; (4)\nE1\nI\n3j1Li=p\rj0100i: (5)\nWe see that the code subspace is perturbed onto an orthogonal\nsubspace spanned by fj0111i;j0100ig.R3projects onto this\nsyndrome subspace and recovers appropriately into the logical\ncodewords. Recovery operators R4,R5, andR6similarly\ncorrect damping errors on the second, third, and fourth qubits.\nNotice that the first order damping errors move the information\ninto mutually orthogonal subspaces. It is therefore not hard\nto see that the set of errors fI\n4;E(k)\n1g4\nk=1satisfy the error\ncorrecting conditions for the [4;1]code. (That the [4;1]code\nsatisfies the error correcting conditions for damping errors was\npointed out in [9].)\nConsider now the subspace spanned by\nfj1010i;j0101i;j0110i;j1001ig. By examining the logical\ncodewords in (2) and (3), we see that this subspace can only\nbe reached by multiple damping errors. Unfortunately, in\nsuch a case we lose the logical superpositions as only j0Liis\nperturbed into this subspace. Consider, for example the two\ndamping error E(1)\n1E(3)\n1. We see that\nE(1)\n1E(3)\n1j0Li=\rj0101i; (6)\nE(1)\n1E(3)\n1j1Li= 0: (7)\nWhile we cannot fully recover from such an error, we rec-\nognize that these higher order errors occur with probability\n\r2. Furthermore, we see that operator elements R7\u0000R10\ndo recover thej0Liportion of the input information. This\ncontributes a small amount to the overall entanglement fidelity,\nthough would obviously not help the minimum fidelity case.\nIndeed,R7\u0000R10do not contribute to maintaining the fidelity\nof an inputj1Listate.\nWe should also note that only a subset of all second order\ndampings are partially correctable as above. We reach the\nsyndrome subspaces from R7\u0000R10only when a qubit from the\nfirst pair and a qubit from the second pair is damped, allowing\nthej0Listate to be recovered. If both the first and second\nqubits (or both the third and fourth qubits) are damped, the\nresulting states are no longer orthogonal to the code subspace.\nIn fact, these are the only errors that will cause a logical bit\nflip, recoveringj0Liasj1Liand vice versa.\nB. Recovery from the distortion of the ‘no damping’ case\nWe turn now to the recovery operators R1andR2. Together\nthese project onto the syndrome subspace with basis vectors\nfj0000i;j1111i;j1100i;j0011igwhich includes the entire\ncode subspace. We just saw that I\n4together with single qubit\ndampings are correctable, but E\n4\nadoes not have an operator\nelement proportional to I\n4. Instead, the ‘no dampings’ term\nis given by E\n4\n0which depends on the damping parameter\n\r. Indeed, consider the effect of the no damping term on the3\nlogical code words:\nE\n4\n0j0Li=1p\n2(j0000i+ (1\u0000\r)2j1111i) (8)\nE\n4\n0j1Li=1\u0000\rp\n2(j1100i+j0011i): (9)\nA standard recovery operation projects onto the code sub-\nspace. Consider the effect of such a recovery on an arbitrary\ninput stateaj0Li+bj1Li. The resulting (un-normalized) state\nis\na(1\u0000\r+\r2\n2)j0Li+b(1\u0000\r)j1Li: (10)\nThe extra term\r2\n2distorts the state from the original input.\nWhile this distortion is small as \r!0, both the original\nrecovery operation proposed in [1] and the optimal recovery\nseek to reduce this distortion by use of a \r-dependent opera-\ntion. We analyze the optimal recovery operation for this term\nand compare its efficacy with the simpler projection.\nWe see that R1projects onto a perturbed version\nof the codespace with basis vectors f(\u000bj0000i+\n\fj1111i);(1p\n2j0011i+1p\n2j1100i)gwhere\u000band\fare\nchosen to maximize the entanglement fidelity. We can use\nany of the numerical techniques of [2], [4] to compute\ngood values for \u000band\f, but we would like an intuitive\nunderstanding as well. \u000band\f(wherej\fj=p\n1\u0000j\u000bj2)\nadjust the syndrome measurement P1so that it is no longer\nj0Lih0Lj+j1Lih1Lj, the projector onto the code subspace.\nIf we choose them so that h0LjP1j0Li=h1LjP1j1Lithen\nwe will perfectly recover the original state when syndrome\nP1is detected for the no damping case. If syndrome P2is\ndetected, the no damping state will be distorted, but for small\n\r, the second syndrome is a relatively rare occurrence. It\ncould even be used as a classical indicator for a greater level\nof distortion.\nWe can see in Fig. 1 that the benefit of the optimal\nrecovery operation is small, especially as \r!0, though\nnot negligible. Furthermore, the standard projection onto the\ncode space is a simple operation while the optimal recovery\nis both\r-dependent and relatively complex to implement.\nFor this reason, it is likely preferable to implement the more\nstraightforward code projection, which still reaps most of the\nbenefits of channel-adaptation.\nIII. A MPLITUDE DAMPING ERRORS IN THE STABILIZER\nFORMALISM\nWe turn our attention to the stabilizer formalism [9], [10],\nto demonstrate its utility for interpreting the amplitude damp-\ning channel. In particular, understanding amplitude damping\nin terms of stabilizers allows a generalization of the [4;1]\ncode for higher rates. The stabilizer formalism provides an\nextremely useful and compact description for quantum error\ncorrecting codes. Code descriptions, syndrome measurements,\nand recovery operations can be understood by considering the\nn\u0000kgenerators of an [n;k]stabilizer code. In standard prac-\ntice, Pauli group errors are considered and if fXi;Yi;Zign\ni=1\nerrors can be corrected, we know we can correct an arbitrary\nerror on one of the qubits since the Pauli operators are a basis\nfor single qubit operators.\nFig. 1. Optimal vs. code projection recovery operations for the [4,1] code.\nWe compare the entanglement fidelity for the optimal recovery operation\nand the recovery that includes a projection onto the code subspace. For\ncomparison, we also include the original recovery operation proposed in [1]\nand the baseline performance of a single qubit. While the optimal recovery\noutperforms the code projector recovery, the performance gain is likely small\ncompared to the cost of implementing the optimal.\nLet’s consider the [4;1]code in terms of its stabilizer group\nG=hXXXX;ZZII;\nIIZZi. We can choose the logical Pauli operators \u0016X=\nXXII and\u0016Z=ZIZI to specify the codewords in (2) and\n(3). We saw in Sec. II that E(i)\n1damping errors together with\nI\n4are correctable errors. Since each of these errors is a linear\ncombination of Pauli group members:\nE(i)\n1=p\r\n2(Xi+iYi); (11)\nwe might presume that fI;Xi;Yig4\ni=1are a set of correctable\noperations and the desired recovery follows the standard sta-\nbilizer syndrome measurement structure. This is not the case.\nConsider that the operator X1X2(or equivalently XXII ) is in\nthe normalizer N(G)of the code stabilizer, and thus fX1;X2g\nare not a correctable set of errors.\nHow, then, can the [4;1]code correct errors of the form Xi+\niYi? Instead of projecting onto the stabilizer subspaces and\ncorrectingXiandYiseparately, we take advantage of the fact\nthat the errors happen in superposition and project accordingly.\nAs we saw, Xi+iYiandXj+iYjproject into orthogonal\nsubspaces when i6=jand we can recover accordingly. In\nfact, the correct syndrome structures can also be described in\nterms of stabilizers; understanding these syndromes enables\ndesign and analysis of other amplitude damping codes.\nLetG=hg1;:::;gn\u0000kibe the generators for an [n;k]\nstabilizer code. We wish to define the generators for the\nsubspace resulting from a damping error Xi+iYion the\nithqubit. First, we should note that we can always write the\ngenerators of Gso that at most one generator commutes with\nXiand anti-commutes with Yi(corresponding to a generator\nwith anXon theithqubit), at most one generator that anti-\ncommutes with both XiandYi(corresponding to a generator4\n1stsubspace\n-ZZII\nIIZZ\nZIII2ndsubspace\n-ZZII\nIIZZ\nIZII\n3rdsubspace\nZZII\n-IIZZ\nIIZI4thsubspace\nZZII\n-IIZZ\nIIIZ\nTABLE II\nSTABILIZERS FOR EACH OF THE DAMPED SUBSPACES OF THE [4;1]CODE .\nwith anZon theithqubit), and all other generators commute\nwith both operators. Let j i2C(G)be an arbitrary state in\nthe subspace stabilized by G. Ifg2Gsuch that [g;Xi] =\n[g;Yi] = 0 , then\n(Xi+iYi)j i= (Xi+iYi)gj i=g(Xi+iYi)j i:(12)\nFrom this we see that the ithdamped subspace is stabilized\nby the commuting generators of G. Now consider an element\nofGthat anti-commutes with XiandYi. Then\n(Xi+iYi)j i= (Xi+iYi)gj i=\u0000g(Xi+iYi)j i;(13)\nso\u0000gis a stabilizer of the ithdamped subspace. Finally,\nconsider agwhich commutes with Xibut anti-commutes with\nYi:\n(Xi+iYi)j i= (Xi+iYi)gj i=g(Xi\u0000iYi)j i:(14)\nWe see that neither gnor\u0000gis a stabilizer for the subspace.\nIt is, however, not hard to see that Ziis a generator:\nZi(Xi+iYi)j i= (iYi\u0000i2Xi)j i= (Xi+iYi)j i:(15)\nIn this manner, given any code stabilizer G, we can construct\nthe stabilizer for each of the damped subspaces.\nConsider now the stabilizer description of each of the\ndamped subspaces for the [4;1]code. These are given in Table\nII. Recall that two stabilizer subspaces are orthogonal if and\nonly if there is an element gthat stabilizes one subspace while\n\u0000gstabilizes the other. It is easy to see that each of these sub-\nspaces is orthogonal to the code subspace, as either \u0000ZZII or\n\u0000IIZZ is included. It is equally easy to see that the first and\nsecond subspaces are orthogonal to the third and fourth. To see\nthat the first and second subspaces are orthogonal, note that\n\u0000IZII stabilizes the first subspace, while IZII stabilizes the\nsecond. Equivalently, \u0000IIZI stabilizes the fourth subspace,\nthus making it orthogonal to the third.\nWe can now understand the optimal recovery operation in\nterms of the code stabilizers. Consider measuring ZZII and\nIIZZ . If the result is (+1;+1) then we conclude that no\ndamping has occurred and perform the non-stabilizer opera-\ntions ofR1andR2to minimize distortion. If we measure\n(\u00001;+1) we know that either the first or the second qubit\nwas damped. We can distinguish by measuring ZIII , with\n+1indicating a damping on the first qubit and \u00001a damping\non the second. If our first syndrome is (+1;\u00001), we can\ndistinguish between dampings on the third and fourth by\nmeasuringIIZI . If our first syndrome yields (\u00001;\u00001)weconclude that multiple dampings occurred. We could simply\nreturn an error, or we can do the partial corrections of R7\u0000R10\nby further measuring both ZIII andIIZI . It is worth\npointing out a feature of the stabilizer analysis highlighted by\nthis multiple dampings case. Each of the damping subspaces\nfrom Table II has three stabilizers and thus encodes a 2\ndimensional subspace. Consider applying E(1)\n1to the third\ndamped subspace, equivalent to damping errors on qubits 1\nand 3. Note that there is no generator with an Xin the first\nqubit; the resulting subspace is stabilized by\nh\u0000ZZII;\u0000IIZZ;IIZI;ZIII i: (16)\nAs this has four independent generators, the resulting subspace\nhas dimension 1. We saw this in the previous section, where\nfor multiple dampings the recovery operation does not preserve\nlogical superpositions but collapses to the j0Listate.\nStabilizer descriptions for amplitude damping-adapted\ncodes are quite advantageous. Just as in the case of standard\nquantum codes, the compact description facilitates analysis\nand aids design. While the recovery operations for the am-\nplitude damping codes are not quite as neatly described as\nthe standard stabilizer recovery, the stabilizer formalism facil-\nitates the description. Furthermore, by considering stabilizer\ndescriptions of the [4;1]code and its recovery operation, we\nmay design other channel-adapted amplitude damping codes.\nWe will rely on stabilizers throughout the remainder of the\npaper.\nIV. G ENERALIZATION OF THE [4,1] CODE FOR HIGHER\nRATES\nThe stabilizer analysis for the [4;1]code provides a ready\nmeans to generalize for higher rate code. Consider the two\ncodes given in Table III (A). Each of these is an obvious\nextension of the [4;1]code, but with a higher rate. Indeed the\ngeneral structure can be extended as far as desired generating\nan[2(M+ 1);M]code for all positive integers M. We can\nthus generate a code with rate arbitrarily close to 1=2.\nWhile the codes presented in Table III (A) have an obvious\npattern related to the [4;1]code, we will find it more conve-\nnient to consider the stabilizer in standard form as given in\nTable III (B). The standard form, including the choice of \u0016Xi\nand \u0016Zi, provides a systematic means to write the encoding\ncircuit. The change is achieved through a reordering of the\nqubits which, due to the symmetry of the channel, has no\neffect on the error correction properties.\nLet’s consider the form of the M+ 2 stabilizer group\ngenerators. Just as with the [4;1]code, the first generator has\nanXon every qubit. The physical qubits are grouped into\nM+ 1pairs; for each pair (i;j)there is a generator ZiZj.\nThe structure of the stabilizers makes it easy to see that\nfI\n2(M+1);E(k)\n1g2(M+1)\nk=1satisfy the error correcting condi-\ntions for the [2(M+1);M]code. To see this, we will show that\nthe damped subspaces are mutually orthogonal, and orthogonal\nto the code subspace. Consider a damping on the ithqubit,\nwhereiandjare a pair. The resulting state is stabilized by Zi,\n\u0000ZiZj;and the remaining Z-pair generators. We will call this\ntheithdamped subspace. This subspace is clearly orthogonal5\nj0i H\u000f\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n...\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nk1\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nk2\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n...\nkM \u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFig. 2. Circuit to encode the arbitrary state of Mqubits given in qubits\nk1\u0001\u0001\u0001kMinto2(M+ 1) physical qubits. This is the [2(M+ 1);M]code\nin standard form.\nto the code subspace due to the presence of the \u0000ZiZj\nstabilizer. For the same reason, the ithdamped subspace is\nclearly orthogonal to the kthdamped subspace for k6=j.\nFinally, the ithandjthdamped subspaces are orthogonal as\nwe see that Zistabilizes the ithand\u0000Zistabilizes the jth.\nBy writing the [2(M+1);M]codes in the standard form, it\nis easy to generate an encoding circuit. The circuit to encode\nthe arbitrary state j iin theMqubitsk1\u0001\u0001\u0001kMis given in\nFig. 2. The encoding circuit requires 3M+1CNOT operations\nand one Hadamard gate.\nLet’s write out the logical codewords of the [6;2]code given\nthe choice of \u0016Ziin Table III:\nj00Li=1p\n2(j000000i+j111111i) (17)\nj01Li=1p\n2(j001001i+j110110i) (18)\nj10Li=1p\n2(j000110i+j111001i) (19)\nj11Li=1p\n2(j110000i+j001111i): (20)\nEach codeword is the equal superposition of two basis states.\nWe can see by inspection that the damped subspaces are\nmutually orthogonal: E(k)\n1will eliminate one of the two basis\nstates from each codeword and the resulting basis states do\nnot overlap.\nA. Syndrome measurement\nWe begin the recovery by first measuring the Z-pair sta-\nbilizers. A\u00001result on the (i;j)-pair stabilizer indicates\na damping of either the ithorjthqubit. This holds true\neven if multiple Z-pair stabilizers measure \u00001. Such a result\nindicates multiple damped qubits. Once we have identified the\nqubit pair, we perform an additional stabilizer measurement to\ndetermine which of the qubits was damped. As an example, if\nthe(i;j)-pair was damped, we measure Zi, with a +1result\nindicating a damping on the ithqubit and a\u00001indicating a\ndamping on the jthqubit. We perform this measurement for\nall pairs which measure \u00001.\nIf multiple stabilizers yield a \u00001measurement then we have\nmultiple damped qubits. As before, this reduces by half the(A)[6;2]code\nXXXXXX\nZZ I I I I\nI I ZZ I I\nI I I I ZZ[8;3]code\nXXXXXXXX\nZZ I I I I I I\nI I ZZ I I I I\nI I I I ZZ I I\nI I I I I I ZZ\n(B)[6;2]standard form\nXXXXXX\nZZ I I I I\nI I Z I I Z\nI I I ZZ I\n\u0016X1=IIIXXI\n\u0016X2=IIXI IX\n\u0016Z1=ZII I Z I\n\u0016Z2=ZII I I Z[8;3]standard form\nXXXXXXXX\nZZ I I I I I I\nI I Z I I I I Z\nI I I Z I I Z I\nI I I I ZZ I I\n\u0016X1=III IXXI I\n\u0016X2=IIIXI IXI\n\u0016X3=IIXI I I IX\n\u0016Z1=ZII I I Z I I\n\u0016Z2=ZII I I I Z I\n\u0016Z3=ZII I I I I Z\nTABLE III\nSTABILIZERS FOR [6;2]AND[8;3]AMPLITUDE DAMPING CODES . IN(A),\nTHESE ARE WRITTEN IN A WAY TO ILLUSTRATE THE CONNECTION TO THE\n[4;1]CODE . IN(B), WE PRESENT THE CODE IN THE STANDARD FORM ,\nWHICH WE ACHIEVE MERELY BY SWAPPING THE CODE QUBITS AND\nCHOOSING THE LOGICAL OPERATORS SYSTEMATICALLY . THE STANDARD\nFORM PROVIDES A CONVENIENT DESCRIPTION FOR GENERATING\nQUANTUM CIRCUITS FOR ENCODING .\ndimension of the subspace and we cannot preserve all logical\nsuperpositions. For an example, examine the stabilizers for the\n[6;2]code when both the first and fifth qubits are damped:\nh\u0000ZZIIII;IIZIIZ; \u0000IIIZZI;ZIIIII;IIIIZI i:(21)\nThis subspace has 5 stabilizers and thus has rank 2. Further-\nmore, combining the last two stabilizers, we can see that\nZIIIZI =\u0016Z1stabilizes the subspace, indicating that the\nremaining logical information is spanned by fj01Li;j00Lig.\nIn general, for a [2(M+ 1);M]code, up to M+ 1dampings\ncan be partially corrected as long as the dampings occur on\ndistinct qubit pairs. If mis the number of damped qubits, then\nthe resulting subspace has dimension 2M+1\u0000m.\nIf allZ-pair measurements for the [2(M+ 1);M]code\nreturn +1, we determine that we are in the ‘no dampings’\nsyndrome and may perform some further operation to reduce\ndistortion as much as possible. As in the example of the [4;1]\ncode in Sec. II-B, we can choose to optimize this recovery\nwith a\r-dependent recovery or we can apply a stabilizer\nprojective measurement. In the former case, we may calculate\nan optimized recovery with a SDP or any of the near-optimal\nmethods of [3], [4]. If we choose a stabilizer measurement,\nwe simply measure the all- XgeneratorX\u0001\u0001\u0001Xwhere a +1\nresult is a projection onto the code subspace. A \u00001result can\nbe corrected by applying a Zioperation (in fact a Zon any\none of the qubits will suffice). This can be seen by noting\nthat the\u0000X\u0001\u0001\u0001Xstabilizer changes the logical codewords\nby replacing the +with a\u0000.6\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\n...\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\nk1\u000f\nk2\u000f\nk3\u000f\nk4 \u000f\n...\nkM+1 \u000f\nkM+2 \u000f\n...\nk2M+1 \u000f\nk2M+2 \u000f\n(A)\nj0iH\u000fHFE\r\r\r\nk1\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nk1\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n...\nk2M+2\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001bj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\n...\nki\u000f\n(B) (C)\nFig. 3. Syndrome measurement circuits for the [2(M+1);M]code. Circuit\n(A) measures each of the Z-pair stabilizers. If all of the measurements in (A)\nare+1, we are in the ‘no damping’ syndrome and we perform the syndrome\nmeasurement in (B). If the (i;j)-pair stabilizer measures \u00001, we perform the\nsyndrome measurement in (C).\nB. Stabilizer syndrome recovery operations\nIn the previous section, we described syndrome measure-\nments to determine which qubits were damped. We also\nexplained the extent to which multiple qubit dampings are\ncorrectable. We now present a straightforward set of Clifford\ngroup operations to recover from each syndrome.\nConsider a syndrome measurement in which we determine\nthatmqubitsi1;:::;imwere damped, where m\u0014M+ 1.\nWe recover from this syndrome via the following three steps:\n1) Apply a Hadamard gate Hi1on thei1qubit.\n2) With qubit i1as the control, apply a CNOT gate to every\nother qubit.\n3) Flip every damped qubit: Xi1\u0001\u0001\u0001Xim.\nThe procedure is illustrated as a quantum circuit for a two-\ndamping syndrome and the [6;2]code in Fig. 4.\nTo see that this is the correct syndrome recovery for the\n[2(M+ 1);M]code, we need to examine the effect of theH\u000fX\n\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001bX\n\u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFig. 4. Syndrome recovery circuit for the [6,2] code with the first and fifth\nqubits damped.\nthree gate operations on the damped subspace stabilizers. In\nthe syndrome where i1;:::;imare damped, we have three\ncategories of generators for the resulting stabilizer group: \u0000Z-\npair stabilizers for the damped pairs, +Z-pair stabilizers for\nthe non-damped pairs, and Zi1;:::;Zimfor each damped\nqubit. We need to see the effect of the recovery gate operations\non each of these generators. Fortunately, we can demonstrate\nall of the relevant cases with the example of the [6;2]code\nwith the first and fifth qubits damped:\n-ZZ I I I I\nI I Z I I Z\n-I I I ZZ I\nZ I I I I I\nI I I I Z I!H1-XZ I I I I\nI I Z I I Z\n-I I I ZZ I\nX I I I I I\nI I I I Z I!CNOT 1’s-YYXXXX\nI I Z I I Z\n-I I I ZZ I\nX XXXXX\nZ I I I Z I\n!X1X5YYXXXX\nI I Z I I Z\nI I I ZZ I\nXXXXXX\nZ I I I Z I=ZZ I I I I\nI I Z I I Z\nI I I ZZ I\nXXXXXX\nZ I I I Z I:(22)\nThe final two sets of stabilizers are equivalent since ZZIIII\nis the product of XXXXXX andYYXXXX . The first four\ngenerators of the resulting group are the code stabilizer. The\nlast generator is \u0016Z1which, as we saw before, indicates that\nthe recovered information is spanned by fj00Li;j01Ligwhile\nthe other two dimensions of information have been lost.\nWhile we have shown that the syndrome recovery operation\nreturns the information to the code subspace, it remains to\ndemonstrate that the information is correctly decoded. We\ncan demonstrate this by considering the syndrome recovery\noperation on each of the \u0016Ziof the code. By showing that each\nof these is correctly preserved, we conclude that the syndrome\nrecovery operation is correct.\nWe have chosen the \u0016Ziso that each has exactly two qubit\nlocations with a Zwhile the rest are I. There are, therefore,\nfive cases of interest. In case 1, neither of the damped qubits\ncorresponds to a location with a Z. In case 2, the first damped\nqubit (i1) corresponds to a location with a Z. In case 3, one\nof theZlocations corresponds to a damped qubit, but it is\nnoti1. In case 4, both of the Zlocations correspond to a\ndamped qubit, but neither is i1. Finally, case 5 is when both\nZlocations correspond to damped qubits and one is i1.\nWithout loss of generality, we can see the effect of each case\nby considering an example using ZIIIZI and appropriately\nselected damped qubits. Consider case 1 where we can let7\ni1= 2:\nZIIIZI!Hi1ZIIIZI!CNOT i1’sZIIIZI\n!Xi1\u0001\u0001\u0001XimZIIIZI: (23)\nIn case 2,i1= 1:\n\u0000ZIIIZI!Hi1\u0000XIIIZI!CNOT i1’s\u0000YXXXYX\n!Xi1\u0001\u0001\u0001XimYXXXYX: (24)\nNotice that this last is equivalent to ZIIIZI asXXXXXX\nis in the stabilizer.\nIn case 3,i1= 2 whilei2= 5:\n\u0000ZIIIZI!Hi1\u0000ZIIIZI!CNOT i1’s\u0000ZIIIZI\n!Xi1\u0001\u0001\u0001XimZIIIZI: (25)\nIn case 4, let i1= 2;,i2= 1, andi3= 5:1\nZIIIZI!Hi1ZIIIZI!CNOT i1’sZIIIZI\n!Xi1\u0001\u0001\u0001XimZIIIZI: (26)\nIn case 5,i1= 1 andi2= 5:\nZIIIZI!Hi1XIIIZI!CNOT i1’sYXXXYX\n!Xi1\u0001\u0001\u0001XimYXXXYX: (27)\nWe see that in all cases, the recovery procedure correctly\npreserves the geometry of the encoded information, even in\nthe case of multiple qubit dampings. It is worth emphasizing,\nhowever, that when multiple qubits are damped at least half\nof the information dimensions are lost.\nC. Performance comparison\nIt is useful to compare the performance of each of the\n[2(M+ 1);M]codes in terms of the damping parameter \r.\nConsider a comparison between the [4;1]code and the [6;2]\ncode. To make a valid comparison, we need to establish a\ncommon baseline. We do this by considering the encoding\nof two qubits with the [4;1]code. For the completely mixed\nstate\u001a=I=2, this is the equivalent of squaring the single\nqubit entanglement fidelity:\n\u0016Fe(\u001a\n\u001a;R\u000eE\nR\u000eE ) =\u0016Fe(\u001a;R\u000eE )2: (28)\nThis comparison is given in Fig. 5 (A). To compare multiple\ncodes, it is more straightforward to normalize each to a single\nqubit baseline. This can be done by computing \u0016F(1=k)\ne for an\n[n;k]code. The normalized performance for the [4;1],[6;2],\n[8;3]and[10;4]codes is given in Fig. 5 (B).\nIt is very interesting to note how comparably these codes\nmaintain the fidelity even as the code rate increases. This\nis particularly striking when noting that each code can still\nperfectly correct only a single damping error. Thus, the [4;1]\n4\ncan correct 4 dampings (as long as they occur on separate\nblocks) while the [10;4]code can only perfectly correct 1. Yet\nwe see that the normalized performance is quite comparable.\n1While this contradicts our statement that the lowest numbered qubit would\nbei1, the assignment of i1= 2 when the first qubit is also damped has no\nimpact on the argument.\n(A)\n(B)\nFig. 5. Performance omparison of generalized amplitude damping codes.\nIn (A) we compare the [6;2]code with the [4;1]repeated twice. In (B),\nwe compare the [4;1],[6;2],[8;3]and[10;4]codes. The entanglement\nfidelity has been normalized as 1=kwherekis the number of encoded qubits.\nNotice that despite the increasing rates, the normalized entanglement fidelity\nmaintains high performance.\nWe take a closer look at the performance of the [8;3]\ncode in Fig. 6. We see that, while most of the entanglement\nfidelity is supplied by correcting no damping and E(i)\n1terms,\na not insignificant performance benefit arises by partially\ncorrecting second order damping errors. In the case of the\n[4;1]recovery, we concluded that such contributions improved\nthe entanglement fidelity, but not the minimum fidelity as\nj1Liwas never preserved by such a recovery. This is not the\ncase for the higher rates. Two damping errors eliminate half\nof the logical space, but different combinations of damping\nerrors will divide the logical space differently. For example, an\ndamping error on the fifth and sixth qubits means the resulting\nspace is stabilized by \u0016Z1\u0016Z2thus eliminating logical states\nj01xLiandj10xLi(wherexindicates either 0or1). On the\nother hand, a damping on the fifth and seventh qubits results\nin a space stabilized by \u0016Z1\u0016Z3eliminating logical states j0x1Li\nandj1x0Li. Thus, correcting second order damping errors still\ncontributes to minimum fidelity performance.\nGiven their identical rates, it is reasonable to compare the\n[8;3]amplitude damping code presented here with the generic\n[8;3]stabilizer code due to Gottesman [9]. The stabilizers for\nthis code are presented in Table IV. This code can correct8\nFig. 6. Fidelity contributions for each order error of the [8;3]amplitude\ndamping code. We see that the no damping, first, and second order recovery\nsyndromes contribute to the entanglement fidelity of the recovery operation.\nGottesman [8;3]code\nXXXXXXXX\nZZZZZZZZ\nIXIXY ZY Z\nIXZY IXZY\nIYXZXZ IY\nTABLE IV\nSTABILIZERS FOR THE [8,3] CODE DUE TO GOTTESMAN [9].\nan arbitrary single qubit error, and thus can correct all first\norder amplitude damping errors, as well as the less probable\nZerrors. These are corrected with 25 stabilizer syndrome\nmeasurements (Pauli operators on each of the 8 qubits as well\nas the identity). This leaves an additional 7 degrees of freedom\nto correct for higher order errors. While typically these are\nnot specified, since we know the channel of interest is the\namplitude damping channel, we can do a small amount of\nchannel-adaptation by selecting appropriate recovery opera-\ntions for these syndromes. Since XandYerrors are the most\ncommon, we choose operators with 2 X’s or 2Y’s (or one of\neach).\nThe comparison between the rate 3=8codes is given Fig. 7.\nHere we see that the channel-adapted [8;3]code outperforms\nthe generic Gottesman code, but the effect is minor. The\nattention to higher order syndromes is seen to improve the\nperformance of the [8;3]code modestly. It should be pointed\nout that both recovery operations can be accomplished with\nClifford group operations, and neither is dependent on \r.\nV. L INEAR CODES FOR THE AMPLITUDE DAMPING\nCHANNEL\nThe channel-adapted codes of the previous section have\nsimilar corrective properties to the [4;1]code:fI;E(i)\n1gare\ncorrectable errors while fXi;Yigare not. It is actually quite\nsimple to design channel-adapted codes that correct both Xi\nandYierrors and thus can correct fI;E(i)\n1gas well. Consider\nFig. 7. Comparison of the amplitude damping [8;3]code and the generic\nrate[8;3]code due to Gottesman. We include both the Gottesman recovery\nwhere no attention is paid to second order recoveries, as well as a recovery\nwhere second order syndromes are chosen to adapt to the amplitude damping\nchannel.\n[7;3]linear code\nI I I ZZZZ\nI ZZ I I ZZ\nZ I Z I Z I Z\nXXXXXXX\nTABLE V\nAMPLITUDE DAMPING CHANNEL -ADAPTED [7;3]LINEAR CODE . LOOKING\nAT THE FIRST THREE GENERATORS ,THIS IS CLEARLY BASED ON THE\nCLASSICAL HAMMING CODE . THE FOURTH GENERATOR DIFFERENTIATES\nBETWEENXANDYSYNDROMES .\nthe[7;3]code presented in Table V. The first three stabilizers\ncan be readily identified as the classical [7;4]Hamming code\nparity check matrix (replacing 0 with Iand 1 withZ). They\nare also three of the six stabilizers for the Steane code. Measur-\ning these three stabilizers, an Xiwill result in a unique three\nbit measurement syndrome (M1;M 2;M 3). (In fact, a nice\nproperty of the Hamming code is that the syndrome, replacing\n+1with 0 and\u00001with 1, is just the binary representation of i,\nthe qubit that sustained the error.) Unfortunately, a Yierror will\nyield the same syndrome as Xi. We add the XXXXXXX\ngenerator to distinguish the two, resulting in 14 orthogonal\nerror syndromes for the fXi;Yig7\ni=1.\nAs in previous examples, we have a choice of recovery\noperations for the ‘no dampings’ syndrome. We can minimize\nthe ‘no damping’ distortion as was done in previous cases\nby computing the optimal or structured near-optimal recov-\nery within this subspace. This will result in a \r-dependent\nrecovery operation. Alternatively, we can simply measure\nXXXXXXX with a +1projecting onto the code subpsace\nand a\u00001requiring a correction of Zi. We compare these\nrecovery operations in Fig. 9.\nWe see in Fig. 10 that the [7;3]code slightly outperforms\nthe[8;3]code of Sec. IV. The [7;3]code perfectly corrects9\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\nj0i \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFE\r\r\r\nj0i H\u000fHFE\r\r\r\n\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\n\u000f \u001f\u001e\u001d\u001c \u0018\u0019\u001a\u001b\nFig. 8. Syndrome measurement circuit for the [7;3]amplitude damping\ncode.\nFig. 9. Optimal vs. code projection recovery operations for the [7,3] code. We\ncompare the entanglement fidelity for the optimal recovery operation and the\nrecovery that includes a projection onto the code subspace. For comparison,\nwe also include the baseline performance of three unencoded qubits. While\nthe optimal recovery outperforms the code projector recovery, the performance\nbenefit is likely small compared to the cost of implementing the optimal.\nfirst order dampings and does not correct any second order\ndampings while the [8;3]code partially corrects for higher\norder dampings. The performance advantage of the [7;3]code\narises from the decreased length: the probability of a higher\norder damping error decreases as only seven physical qubits\nare needed.\nGiven its structure, it is logical to compare the [7;3]\namplitude damping code to the [7;1]Steane code, as both are\nderived from the classical Hamming code. It was shown in\n[3], [4] that the Steane code is not particularly well adaptable\nto amplitude damping errors; despite its extra redundancy,\nthe channel-adapted [5;1]code significantly outperforms the\nchannel-adapted Steane code. This is particularly unfortunate\nas the Steane code can be implemented with such efficiency,\nFig. 10. Comparison of the [7;3]and[8;4]qubit amplitude damping codes.\nWe see that the [7;3]performance is slightly better, despite the higher rate.\nFig. 11. Comparison of the [7;1]Steane code and the [7;3]amplitude\ndamping code, normalized by1=k. We see that the [7;3]performance is very\nsimilar to the EigQER optimized recovery for the Steane code.\nwith particular value for fault tolerant quantum computing.\nThe[7;3]code provides a useful compromise position.\nWe see in Fig. 11 the performance comparison for [7;3]\ncode and the [7;1]code (with and without channel-adapted\nrecovery). It comes as no surprise that the [7;3]code out-\nperforms the [7;1]with standard stabilizer recovery: each\nperfectly corrects the first order damping errors, but the [7;3]\ncode has done so while preserving three times as much infor-\nmation. It is interesting to see how close the [7;3]performance\nis to the channel-adapted [7;1]. It was shown in [3], [4]\nthat the channel-adapted [7;1]at least partially corrects some\nsecond-order damping errors; the [7;3]code does not. This is\nmitigated by the higher rate of the [7;3]code as again, three\ntimes as much information is preserved.\nThe[7;3]code is not the only high rate linear code for10\namplitude damping errors. Consider any classical linear code\nthat can correct 1 error for which codewords have even parity.\nWe can convert this code to a quantum amplitude damping\ncode in the same way as the [7;3]code. IfHis the parity\ncheck matrix for an [n;k]classical linear code, then each row\ncan be made a quantum code stabilizer replacing 1’s with Z\nand 0’s with I. To distinguish XiandYierrors we include\nX\nnas a generator. Since the classical code has even parity,\nwe know that this generator commutes with the others. This\nconstruction yields a [n;k\u00001]quantum amplitude damping\ncode that corrects for single amplitude damping errors.\nThe[7;3]code we have presented here follows the structure\nproposed in [9] for amplitude damping codes; namely the code\nis a combination of an X-error correcting code and a Z-error\ndetecting code. It is not immediately clear how to generalize to\nterror correcting linear codes. Instead of a single generator to\ndistinguishXiandYierrors, we require an extra tgenerators\nas we must distinguish XiandYifor each corrected damping.\nVI. A MPLITUDE DAMPING ERRORS AND THE SHOR CODE\nWe now turn our attention to the [9;1]Shor code and\nits performance with a recovery operation channel-adapted to\namplitude damping errors. The structured near-optimal results\nin [3], [4] showed that the Shor code provides remarkably good\nprotection from the amplitude damping channel. Furthermore,\nthe structured recovery operation is essentially optimal. Thus,\nin this case, the optimal channel-adapted recovery operation\ncan be described as a projective syndrome measurement fol-\nlowed by a unitary operation. Given this intuitive structure, we\ncan analyze the amplitude damping channel-adapted recovery\noperation.\nWe first note that first order errors fE(k)\n1gare perfectly\ncorrectable. This comes as no surprise, since the Shor code\ncan correct an arbitrary single qubit operation. What may\nbe surprising is that second order errors fE(j)\n1E(k)\n1gare also\nperfectly correctable. This was pointed out in [9] and can be\nseen through the same kind of stabilizer analysis of damped\nsubspaces as we employed for the [2(M+ 1);M]codes.\nWe see in Table VI a few representative syndrome subspaces\nfor damping errors on the Shor code. From these subspaces, we\nsurmise that the first step in making a syndrome measurement\nis to measure the first 6 code stabilizers (each of which has\na pair ofZ’s). Depending on those outcomes, we can make a\nfurther stabilizer measurement.\nAs an example, consider when the first stabilizer returns a\n\u00001and the rest return +1. In that case, we can conclude that\neither the first qubit was damped, or both the second and third\nqubits were damped. These can be distinguished by measuring\nZ1with a +1indicating qubit one and a \u00001indicating both\nqubits two and three.\nIt is interesting to note that in this case, we will have only\nmeasured 7 stabilizers, and thus need one further measurement\nto achieve a 2 dimensional subspace. A natural choice would\nbe to measure IIIXXXXXX , but this is an opportunity\nfor a\r-dependent measurement instead. As before, such an\noperation can improve performance at the cost of circuit\ncomplexity. In most of this chapter, we have leaned towardQubit 1 damped\n-ZZI I I I I I I\nIZZI I I I I I\nIIIZZ I I I I\nIII I ZZ I I I\nIII I I I ZZ I\nIII I I I I ZZ\nIIIXXXXXX\nZII I I I I I IQubits 2 & 3 damped\n-ZZI I I I I I I\nIZZI I I I I I\nIIIZZ I I I I\nIII I ZZ I I I\nIII I I I ZZ I\nIII I I I I ZZ\nIIIXXXXXX\n-ZII I I I I I I\nQubits 1 & 7 damped\n-ZZIIIIIII\nIZZIIIIII\nIIIZZIIII\nIIIIZZIII\n-IIIIIIZZI\nIIIIIIIZZ\nZIIIIIIII\nIIIIIIZII\nTABLE VI\nSTABILIZERS FOR SEVERAL DAMPED SUBSPACE SYNDROMES FOR THE\nSHOR CODE .\nFig. 12. Channel-adapted stabilizer recovery vs. \r-dependent recovery for\nthe Shor code and the amplitude damping channel\nthe simpler operation, concluding that \r-dependent operations\nprovide some performance benefit but not enough to justify\nthe added complexity. We will see that in the case of the Shor\ncode, the performance gain may be sufficiently large to warrant\na\r-dependent operation.\nBefore this consideration, let’s turn to another syndrome for\nmultiple qubit dampings. We already examined an example\nwhere the second and third qubits are both damped. The Shor\ncode is divided into three blocks of three qubits each; this case\nextends exactly to all circumstances when both damped qubits\nfall on the same block. The third example in Table VI is an\nexample of two qubits damped from different blocks; in this\ncase the first and fifth stabilizers are both measured to be \u00001.\nWhile the most likely cause of this syndrome is a two-qubit\ndamping, we can further measure Z1andZ7to correct for a\nthree or four-qubit damping occurrence.\nThe preceding discussion of stabilizer subspaces provides11\ntwo alternative recovery operations. We may begin with a\nprojective syndrome measurement of the first 6 code gener-\nators. At that point, we may either make a set of stabilizer\nmeasurements to project onto the damped subspaces, or we\nmay make a \r-dependent syndrome recovery to minimize this\ndistortion. It turns out that the best \r-dependent syndrome\nrecovery has equivalent performance to the structured near-\noptimal recovery operation and is therefore essentially optimal.\nThe stabilizer recovery, while simple to implement with Clif-\nford group operations, has significantly weaker performance.\nWe compare the two recovery operations for various values of\n\rin Fig. 12.\nHow should we understand the extensive performance gains\nfor the\r-dependent recovery? Both are consequences of the\nE0distortion imparted onto the quantum state and the degrees\nof freedom in the code. The \r-dependent operation arises when\nwe have a remaining degree of freedom after determining the\nsyndrome. For the [2(M+ 1);M]codes and the [7;3]code,\nwe only have such freedom in the ‘no damping’ syndrome;\nin all of the damping syndromes, the syndrome measurement\nrequires a full set of stabilizer measurements. We saw that for\nthe Shor code first order dampings require only 7 stabilizer\nmeasurements to determine the syndrome, leaving one extra\ndegree of freedom. We also have an extra degree of freedom\nwhen two qubits from the same block are damped. These\nconstitute all of the first and some of the second order\nsyndromes, each of which can be optimized to minimize E0\ndistortion.\nVII. C ONCLUSION\nWe have developed several quantum error correcting codes\nchannel-adapted for the amplitude damping channel. All of\nthe encodings can be compactly described in the stabilizer\nformalism. While optimized \r-dependent recovery operations\nare possible, a much simpler recovery operation using only sta-\nbilizer measurements and Clifford group operations achieves\nnearly equivalent performance. The channel-adapted codes\nhave much higher rates (with short block lengths) than generic\nquantum codes.\nThe creation of straightforward codes for the amplitude\ndamping channel is a major step toward the still-open question\nof channel-adapted fault tolerant quantum computing. Intu-\nitively, channel-adaptation should be able to improve fault\ntolerant thresholds and reduce the necessary overhead for fault\ntolerance. Before such intuition is confirmed for the amplitude\ndamping channel, several obstacles must be overcome. As an\nexample, we must construct a universal set of operations on\none or more of the channel-adapted codes presented here.\nSuch challenges are acknowledged, but deferred for future\nconsideration.\nREFERENCES\n[1] D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y . Yamamoto, “Approx-\nimate quantum error correction can lead to better codes,” Phys. Rev. A ,\nvol. 56, pp. 2567–2573, 1997.\n[2] A. S. Fletcher, P. W. Shor, and M. Z. Win, “Optimum quantum error\nrecovery using semidefinite programming,” Phys. Rev. A , vol. 75, no. 1,\np. 012338, 2007.[3] A. S. Fletcher, “Channel-adapted quantum error correction,” Ph.D.\ndissertation, Massachusetts Institute of Technology, June 2007.\n[4] A. S. Fletcher, P. W. Shor, and M. Z. Win, “Structured near-optimal\nchannel-adapted quantum error recovery,” 2007, in preparation .\n[5] R. L. Kosut and D. A. Lidar, “Quantum error corretion via convex\noptimization,” 2006.\n[6] M. Reimpell and R. F. Werner, “Iterative optimization of error correcting\ncodes,” Phys. Rev. Lett. , vol. 94, p. 080501, 2005.\n[7] N. Yamamoto, S. Hara, and K. Tsumura, “Supobtimal quantum-error-\ncorrecting procedure based on semidefinite programming,” Phys. Rev.\nA, vol. 71, p. 022322, 2005.\n[8] H. Barnum and E. Knill, “Reversing quantum dynamics with near-\noptimal quantum and classical fidelity,” J. Math. Phys. , vol. 43, no. 5,\npp. 2097–2106, May 2002.\n[9] D. Gottesman, “Stabilizer codes and quantum error correction,” Ph.D.\ndissertation, California Institute of Technology, Pasadena, CA, 1997.\n[10] ——, “Class of quantum error-correcting codes saturating the quantum\nHamming bound,” Phys. Rev. A , vol. 54, p. 1862, 1996."    },    {        "title": "2212.13717v1.Atomic_decomposition_for_Morrey_Lorentz_spaces.pdf",        "content": "arXiv:2212.13717v1  [math.FA]  28 Dec 2022ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES\nNAOYA HATANO\nAbstract. In this paper, we consider the atomic decomposition for Morr ey-Lorentz spaces\nand applications. Morrey-Lorentz spaces, which have struc tures of Morrey spaces, Lorentz\nspaces and their weak-type spaces, are introduced by M. A. Ra gusa in 2012. Our study gave\nsome extension of the atomic decomposition to Morrey-Loren tz spaces. As an application,\nthe Olsen inequality can be obtained more sharpness.\nKeywords Morrey-Lorentzspaces,weakMorreyspaces,Hardy-Morrey- Lorentzspaces,atomic\ndecomposition, Olsen’s inequality.\nMathematics Subject Classifications (2010) Primary 42B35; Secondary 42B25\n1.Introduction\nThe purpose of this paper is to give the atomic decomposition for Mor rey-Lorentz spaces.\nMorrey-Lorentz spaces, which are an extension of Morrey and Lo rentz spaces, were introduced\nby Ragusa [32] in 2012. To introduce the Morrey-Lorentz spaces, we recall the definition of the\nLorentz spaces. In 1950, Lorentz introduced the Lorentz spac es in [26]. For α>0 andt >0,\nwe define the distribution function λfand the rearrangement function f∗by\nλf(α)≡ |{x∈Rn:|f(x)|>α}|, f∗(t)≡inf{α>0 :λf(α)≤t},\nrespectively. Then for 0 <p,q≤ ∞, theLorentz quasi-norm /ba∇dbl·/ba∇dblLp,qis defined by\n/ba∇dblf/ba∇dblLp,q≡\n\n/parenleftbiggˆ∞\n0(t1\npf∗(t))qdt\nt/parenrightbigg1\nq\n,0<p,q< ∞,\nsup\nt>0t1\npf∗(t), 0<p≤ ∞,q=∞,\nand the Lorentz space Lp,q(Rn) is the set of all measurable functions fwith the finite quasi-\nnorm/ba∇dbl· /ba∇dblLp,q. We remark that the Lorentz spaces Lp,p(Rn) andLp,∞(Rn) coincide with the\nLebesgue space Lp(Rn) and the weak Lebesgue space W Lp(Rn), respectively.\nIn addition, we will recall the definition of Morrey-Lorentz spaces. Let 0<q≤p<∞and\n0<r≤ ∞. We define the Morrey-Lorentz space Mp\nq,r(Rn) by\nMp\nq,r(Rn)≡/braceleftBigg\nf∈L0(Rn) :/ba∇dblf/ba∇dblMp\nq,r≡sup\nQ∈Q(Rn)|Q|1\np−1\nq/ba∇dblfχQ/ba∇dblLq,r<∞/bracerightBigg\n,\nwhere the class L0(Rn) is the set of all measurable functions defined on RnandQ(Rn) denotes\nthe family of all cubes with parallel to coordinate axis in Rn. We note that the Morrey-Lorentz\nspacesMp\np,r(Rn) andMp\nq,q(Rn) consist with the Lorentz space Lp,q(Rn) and the Morrey space\nMp\nq(Rn), respectively. Especially, Mp\nq,∞(Rn) correspondsto the weakMorreyspaceW Mp\nq(Rn)\nwhich the weak Morrey quasi-norm /ba∇dbl·/ba∇dblWMp\nqis defined by\n/ba∇dblf/ba∇dblWMp\nq≡sup\nλ>0λ/ba∇dblχ{x∈Rn:|f(x)|>λ}/ba∇dblMp\nq.\n12 NAOYA HATANO\nThe set of all dyadic cubes is denoted by D(Rn);\nDm(Rn)≡/braceleftBigg\nQmk≡n/productdisplay\ni=1/bracketleftbiggki\n2m,ki+1\n2m/parenrightbigg\n:k= (k1,...,k n)∈Zn/bracerightBigg\n,D(Rn)≡/uniondisplay\nj∈ZDj(Rn)\nform∈Z. Then by the simple consideration, we have that\n/ba∇dblf/ba∇dblMp\nq,r∼sup\nQ∈D(Rn)|Q|1\np−1\nq/ba∇dblfχQ/ba∇dblLq,r,\nfor any measurable function finMp\nq,r(Rn). ForE(Rn) =Lq,r(Rn) or WLq(Rn) with the\nquasi-norm /ba∇dbl·/ba∇dblE,\nEloc(Rn)≡ {f∈L0(Rn) :/ba∇dblfχK/ba∇dblE<∞, for all compact sets KinRn}.\nWe aim here to prove the following result which is extension of the atom ic decomposition to\nMorrey-Lorentz spaces.\nTheorem 1.1. Suppose that the parameters p,q,r,s,t,v satisfy\n0<q≤p<∞,0<r≤ ∞,0<t≤s<∞,0<v≤1,\nq<t, p<s, v< min(q,r).\nAssume that {Qj}∞\nj=1⊂ Q(Rn),{aj}∞\nj=1⊂WMs\nt(Rn), and{λj}∞\nj=1⊂[0,∞)fulfill\n/ba∇dblaj/ba∇dblWMs\nt≤ |Qj|1\ns,supp(aj)⊂Qj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r<∞.\nThen,f=/summationtext∞\nj=1λjajconverges a.e. and satisfies\n(1.1) /ba∇dblf/ba∇dblMp\nq,r/lessorsimilarp,q,r,s,t,v/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r.\nIn particular, f=/summationtext∞\nj=1λjajconverges in Lq,r\nloc(Rn)ifr <∞and inLp,r(Rn)ifp=qand\nr<∞.\nIn this theorem, we can take the atoms {aj}∞\nj=1from the large space, the weak Morrey space\nWMs\nt(Rn). The difference between Morreyspaces and weak Morreyspaces in [13]. In addition,\nwe can choose the parameter vfreely.\nIt is possible to rewrite Theorem 1.1 to Lorentz spaces and weak Mor rey spaces, respectively,\nas follows:\nCorollary 1.2. Suppose that the parameters p,r,s,t,v satisfy\n0<p<t≤s<∞,0<r≤ ∞,0<v≤1, v<min(p,r).\nAssume that {Qj}∞\nj=1⊂ Q(Rn),{aj}∞\nj=1⊂WMs\nt(Rn), and{λj}∞\nj=1⊂[0,∞)fulfill\n/ba∇dblaj/ba∇dblWMs\nt≤ |Qj|1\ns,supp(aj)⊂Qj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLp,r<∞.ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 3\nIfv<min(p,r), thenf=/summationtext∞\nj=1λjajconverges a.e. and satisfies\n/ba∇dblf/ba∇dblLp,r/lessorsimilarp,r,s,t,v/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLp,r.\nIn particular, f=/summationtext∞\nj=1λjajconverges in Lp,r(Rn)ifr<∞.\nCorollary 1.3. Suppose that the parameters p,q,s,t,v satisfy\n0<q≤p<∞,0<t≤s<∞,0<v≤1, p<s, v<q<t.\nAssume that {Qj}∞\nj=1⊂ Q(Rn),{aj}∞\nj=1⊂WMs\nt(Rn)and{λj}∞\nj=1⊂[0,∞)fulfill\n/ba∇dblaj/ba∇dblWMs\nt≤ |Qj|1\ns,supp(aj)⊂Qj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nWMp\nq<∞.\nThenf=/summationtext∞\nj=1λjajconverges a.e. and satisfies\n/ba∇dblf/ba∇dblWMp\nq/lessorsimilarp,q,s,t,v/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nWMp\nq.\nThe next assertionconcernsthe decomposition offunctions in Mp\nq,r(Rn). Hereafter, wewrite\nN0≡N∪{0}. Denote by PK(Rn) the set of all polynomial functions with degree less than or\nequal toK. The set PK(Rn)⊥denotes the set of all f∈L0(Rn) for which /an}b∇acketle{t·/an}b∇acket∇i}htKf∈L1(Rn) andˆ\nRnxαf(x)dx= 0 for any α∈Nn\n0with|α| ≤K, where/an}b∇acketle{t·/an}b∇acket∇i}ht= (1+|·|2)1\n2.\nTheorem 1.4. Let1< q≤p <∞,0< r≤ ∞,K∈N0, andf∈ Mp\nq,r(Rn). Then, there\nexists a triplet {λj}∞\nj=1⊂[0,∞),{Qj}∞\nj=1⊂ Q(Rn), and{aj}∞\nj=1⊂L∞(Rn)∩ P⊥\nK(Rn)such\nthatf=/summationtext∞\nj=1λjajinS′(Rn)and that, for all v>0,\n(1.2) |aj| ≤χQj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilarv/ba∇dblf/ba∇dblMp\nq,r.\nWe can also Theorem 1.4 to Lorentz spaces and weak Morrey spaces , respectively, as follows:\nCorollary 1.5. Let1< p <∞,0< r≤ ∞,K∈N0, letf∈Lp,r(Rn). Then there exists\na triplet {λj}∞\nj=1⊂[0,∞),{Qj}∞\nj=1⊂ Q(Rn)and{aj}∞\nj=1⊂L∞(Rn)∩ P⊥\nK(Rn)such that\nf=/summationtext∞\nj=1λjajinS′(Rn)and that, for all v>0,\n|aj| ≤χQj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLp,r/lessorsimilarv/ba∇dblf/ba∇dblLp,r.\nCorollary 1.6. Let1< q≤p <∞,K∈N0, letf∈WMp\nq(Rn). Then there exists a\ntriplet{λj}∞\nj=1⊂[0,∞),{Qj}∞\nj=1⊂ Q(Rn)and{aj}∞\nj=1⊂L∞(Rn)∩ P⊥\nK(Rn)such that\nf=/summationtext∞\nj=1λjajinS′(Rn)and that, for all v>0,\n|aj| ≤χQj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nWMp\nq/lessorsimilarv/ba∇dblf/ba∇dblWMp\nq.4 NAOYA HATANO\nTheorems 1.1 and 1.4 are special cases of Theorems 1.7 and 1.9 to follo w, respectively, which\nconcerns the decomposition of Hardy-Morrey-Lorentz spaces. Moreover, Theorem 1.8 stands\nfor the critical case t= 1 in Theorem 1.7. Recall that, for 0 <q≤p<∞and 0<r≤ ∞, the\nHardy-Morrey-Lorentz space HMp\nq,r(Rn) is defined to be the set of all f∈ S′(Rn) for which the\nquasi-norm /ba∇dblf/ba∇dblHMp\nq,r≡ /ba∇dblsupt>0|et∆f|/ba∇dblMp\nq,ris finite, where et∆fstandsforthe heat expansion\nofffort>0;\net∆f(x) =/angbracketleftBigg\n1/radicalbig\n(4πt)nexp/parenleftbigg\n−|x−·|2\n4t/parenrightbigg\n,f/angbracketrightBigg\n, x∈Rn.\nIn addition, HMp\nq,∞(Rn) coincides with the Hardy-weak Morrey space HWMp\nq(Rn), intro-\nduced by Ho in [20].\nTheorem 1.7. Suppose that the parameters p,q,r,s,t,v satisfy\n0<q≤p<∞,0<r≤ ∞,1<t≤s<∞,0<v≤1,\nq<t, p<s, v< min(q,r),\nWritedv≡[n(1/v−1)]. Assume that {Qj}∞\nj=1⊂ Q(Rn),{aj}∞\nj=1⊂WMs\nt(Rn)∩Pdv(Rn)⊥\nand{λj}∞\nj=1⊂[0,∞)fulfill\n/ba∇dblaj/ba∇dblWMs\nt≤ |Qj|1\ns,supp(aj)⊂Qj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r<∞.\nThenf=/summationtext∞\nj=1λjajconverges in S′(Rn)and satisfies\n(1.3) /ba∇dblf/ba∇dblHMp\nq,r/lessorsimilarp,q,r,s,t/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r.\nTheorem 1.8. Suppose that the parameters p,q,r,s,v satisfy\n0<q≤p<∞,0<r≤ ∞,1≤s<∞,0<v≤1, q<1, p<s, v< min(q,r).\nWrite anddv≡[n(1/v−1)]. Assume that {Qj}∞\nj=1⊂ Q(Rn),{aj}∞\nj=1⊂ Ms\n1(Rn)∩Pdv(Rn)⊥\nand{λj}∞\nj=1⊂[0,∞)fulfill\n/ba∇dblaj/ba∇dblMs\n1≤ |Qj|1\ns,supp(aj)⊂Qj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r<∞.\nThenf=/summationtext∞\nj=1λjajconverges in S′(Rn)and satisfies\n(1.4) /ba∇dblf/ba∇dblHMp\nq,r/lessorsimilarp,q,r,s/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r.\nTheorem 1.9. Suppose that the real parameters p,q,randKsatisfy\n0<q≤p<∞,0<r≤ ∞, K∈N0∩/parenleftbiggn\nq0−n−1,∞/parenrightbigg\n,\nwhereq0≡min(1,q). Letf∈HMp\nq,r(Rn). Then there exists a triplet {λj}∞\nj=1⊂[0,∞),\n{Qj}∞\nj=1⊂ Q(Rn)and{aj}∞\nj=1⊂L∞(Rn)∩P⊥\nK(Rn)such thatf=/summationtext∞\nj=1λjajinS′(Rn)andATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 5\nthat, for all v>0,\n|aj| ≤χQj,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilarv/ba∇dblf/ba∇dblHMp\nq,r.\nConcerning Mp\nq,r(Rn) andHMp\nq,r(Rn), we have the following assertion:\nProposition 1.10. Let1≤q≤p<∞and0<r≤ ∞.\n(1)Iff∈ Mp\nq,r(Rn)andq>1, thenf∈HMp\nq,r(Rn).\n(2)Assume that q >1orq= 1≥r. Iff∈HMp\nq,r(Rn), thenfcan be represented by a\nlocally integrable function and the representative belong s toMp\nq,r(Rn).\nRemark 1.11. (1) It is noteworthy that the case q= 1≥rin this proposition covers a\nresult of [18] as a special case of r= 1.\n(2) It is not clear that HMp\n1,r(Rn)֒→ Mp\n1,r(Rn) forr >1. Because the embedding\nHMp\n1,r(Rn)֒→L1\nloc(Rn) fails. Let n= 1. Here, we explain that L1,r(R) does not\nembed into L1\nloc(R). In fact, if we let\nf=∞/summationdisplay\nj=22j\njχ[2−j−1,2−j),\nthen the necessary and sufficiently condition of f∈L1,r(R) isr >1 (see [5, p. 56]).\nMeanwhile, for any k∈N, we see that\n/ba∇dblfχ[0,2−k)/ba∇dblL1=∞.\nHardy-Morrey-Lorentzspacesadmitacharacterizationbyusing thegrandmaximaloperator.\nTo formulate the result, we recall the following two fundamental no tions.\n(1) Topologize S(Rn) by norms {pN}N∈Ngiven by\npN(ϕ)≡/summationdisplay\n|α|≤Nsup\nx∈Rn(1+|x|)N|∂αϕ(x)|\nfor eachN∈N. Define FN≡ {ϕ∈ S(Rn) :pN(ϕ)≤1}.\n(2) Letf∈ S′(Rn). The grand maximal operator Mfis given by\nMf(x)≡sup{|t−nψ(t−1·)∗f(x)|:t>0,ψ∈ FN}, x∈Rn,\nwhere we choose and fix a large integer N.\nThe Hardy-Morrey-Lorentz quasi-norm /ba∇dbl·/ba∇dblHMp\nq,ris rewritten as follows.\nProposition 1.12. Let0<q≤p<∞and0<r≤ ∞. Then\n/ba∇dblMf/ba∇dblMp\nq,r∼ /ba∇dblf/ba∇dblHMp\nq,r\nfor allf∈ S′(Rn).\nIn 2009, H. Jia and H. Wang [24] provided the atomic decomposition fo r Morrey spaces. T.\nIida, Y. Sawano and H. Tanaka [23] also gave the decomposition and it s applications. Later,\nmany authors showed for various functional spaces; Lorentz sp aces [31, 1, 25] Lebesgue spaces\nwith variable exponent [27, 33, 8] Orlicz-Morrey spaces of the third kind [12], local Morrey\nspaces [4, 16], generalized Morrey spaces [3], weak Morrey spaces [2 0], mixed Morrey spaces\n[29], etc.6 NAOYA HATANO\nWe organize the remaining part of the paper as follows: In Section 2, we introduce the\nnecessary fundamental statements for Morrey-Lorentz spac es. In Section 3, we prove Proposi-\ntions 1.10 and 1.10. In Section 4, we prove Theorems 1.1, 1.7 and 1.8. I n Section 5, we prove\nTheorem 1.9. In Section 6, we give an application for our main results.\n2.Preliminaries\n2.1.Fundamental properties for function spaces. Accordingto the paper [22], the H¨ older\ninequality for Lorentz quasi-norms.\nLemma 2.1 ([22, Theorem 4.5]) .Assume that 0<p,p1,p2<∞and0<q,q1,q2≤ ∞satisfy\n1\np=1\np1+1\np2,1\nq=1\nq1+1\nq2.\nThen,\n/ba∇dblf·g/ba∇dblLp,q/lessorsimilar/ba∇dblf/ba∇dblLp1,q1/ba∇dblg/ba∇dblLp2,q2\nfor allf∈Lp1,q1(Rn)andg∈Lp2,q2(Rn).\nThe embedding relations for Morrey-Lorentz spaces are given by R agusa in [32].\nProposition 2.2 ([32, Theorem 3.1]) .The following assertions hold :\n(1)If0<q≤p<∞and0<r1≤r2≤ ∞, then\nMp\nq,r1(Rn)֒→ Mp\nq,r2(Rn).\n(2)If0<q2<q1≤p<∞and0<r1,r2≤ ∞, then\nMp\nq1,r1(Rn)֒→ Mp\nq2,r2(Rn).\nAlthough we cannot calculate /ba∇dblχE/ba∇dblMp\nq,rfor all measurable subsets E, we can do so for any\ncubeE.\nProposition 2.3. Let0<q≤p<∞and0<r≤ ∞, and letQ∈ Q(Rn). Then\n/ba∇dblχQ/ba∇dblMp\nq,r=/parenleftBigq\nr/parenrightBig1\nr|Q|1\np,\nwhere we assume that (q/r)1/r= 1forr=∞.\nProof.As is mentioned in [14, Example 1.4.8], for each measurable set E,\n/ba∇dblχE/ba∇dblLq,r=/parenleftBigq\nr/parenrightBig1\nr|E|1\nq.\nIt follows that\n/ba∇dblχQ/ba∇dblMp\nq,r=/parenleftBigq\nr/parenrightBig1\nrsup\nR∈Q(Rn)|R|1\np−1\nq|Q∩R|1\nq.\nThen, combining the estimates\nsup\nR∈Q(Rn)|R|1\np−1\nq|Q∩R|1\nq≤sup\nR∈Q(Rn)|Q∩R|1\np−1\nq|Q∩R|1\nq=|Q|1\np\nand\nsup\nR∈Q(Rn)|R|1\np−1\nq|Q∩R|1\nq≥sup\nR∈Q(Rn),R⊂Q|R|1\np−1\nq|R|1\nq=|Q|1\np,\nwe obtain the desired result. /square\nMorrey-Lorentz quasi-norms satisfy the Fatou property.ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 7\nLemma 2.4 (Fatou property for Morrey-Lorentz space) .Let0<q≤p<∞and0<r≤ ∞,\nand let{fj}∞\nj=1⊂L0(Rn)be a nonnegative collection such that f= limj→∞fjexists a.e. Then,\nwe have\n/ba∇dblf/ba∇dblMp\nq,r≤liminf\nj→∞/ba∇dblfj/ba∇dblMp\nq,r.\nProof.For eachQ∈ Q(Rn),\nfjχQ→fχQa.e.,\nand therefore, by the Fatou property for the Lorentz quasi-no rm/ba∇dbl · /ba∇dblLq,r(see [14, Exercise\n1.4.11]),\n/ba∇dblfχQ/ba∇dblLq,r≤liminf\nj→∞/ba∇dblfjχQ/ba∇dblLq,r.\nConsequently,\n/ba∇dblf/ba∇dblMp\nq,r≤sup\nQ∈Q(Rn)|Q|1\np−1\nqliminf\nj→∞/ba∇dblfjχQ/ba∇dblLq,r≤liminf\nj→∞sup\nQ∈Q(Rn)|Q|1\np−1\nq/ba∇dblfjχQ/ba∇dblLq,r\n= liminf\nj→∞/ba∇dblfj/ba∇dblMp\nq,r.\n/square\n2.2.Maximal operators. Let 0< η <∞and 0< θ≤ ∞. For a measurable function f\ndefined on Rn, define a function M(η,θ)fby\nM(η,θ)f(x)≡sup\nQ∈Q(Rn)χQ(x)/ba∇dblfχQ/ba∇dblLη,θ\n/ba∇dblχQ/ba∇dblLη,θ, x∈Rn.\nWhenθ=η, we abbreviate M(η,η)=M(η). The operator M(η,θ)is bounded on the Lorentz\nspaceLp,q(Rn).\nProposition 2.5. [19, Proposition 2] Let0<p,q≤ ∞,0<η<∞and0<θ≤ ∞. Ifη<p,\nthen the operator M(η,θ)is a bounded operator from Lp,q(Rn)to itself.\nThe critical case η=pin this proposition, which the operator M(p,q)is bounded from\nLp,q(Rn) to WLp(Rn) if and only if 0 <q≤p<∞, is given in [7, Theorem 6].\nThe maximal operator M(1,1)is denoted by M, which is called the Hardy-Littlewood max-\nimal operator. It is well known that the boundedness of the opera torMholds on Morrey\nspaces.\nProposition 2.6 ([6]).Let1≤q≤p<∞. Then, the following assertions hold :\n(1)For allf∈ Mp\n1(Rn),\n/ba∇dblMf/ba∇dblWMp\n1/lessorsimilar/ba∇dblf/ba∇dblMp\n1.\n(2)If1<q≤p<∞, for allf∈ Mp\nq(Rn),\n/ba∇dblMf/ba∇dblMp\nq/lessorsimilar/ba∇dblf/ba∇dblMp\nq.\nAs an extension of Proposition 2.6 (2), we now recall the boundedne ss of the operator Mon\nMorrey-Lorentz spaces.\nProposition 2.7 ([19, Proposition 4]) .Let1<q≤p<∞and0<r≤ ∞. Then we have\n/ba∇dblMf/ba∇dblMp\nq,r/lessorsimilar/ba∇dblf/ba∇dblMp\nq,r\nfor any measurable function fdefined on Rn. More generally, if 0<η<q≤p<∞, then\n/ba∇dblM(η)f/ba∇dblMp\nq,r/lessorsimilar/ba∇dblf/ba∇dblMp\nq,r\nfor any measurable function fdefined on Rn.8 NAOYA HATANO\nWe recall the Fefferman-Stein inequality for Morrey-Lorentz quas i-norm. The case r=∞\nin Proposition 2.8, below, is not contained in [19, Theorem 10 (1)], but, using Proposition 2.6\nin [20], we can check this case, similarly.\nProposition 2.8. (1)Let1<q≤p<∞,1<r≤ ∞and1<u<∞. Then\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(Mfj)u\n1\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1|fj|u\n1\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r\nfor all sequences of measurable functions {fj}∞\nj=1.\n(2)Let1<q≤p<∞and0<r≤ ∞. Then\n/vextenddouble/vextenddouble/vextenddouble/vextenddoublesup\nj∈NMfj/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddoublesup\nj∈N|fj|/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r.\n2.3.Lorentz-block spaces. It is known the predual space of Morrey-Lorentz space in [11,\nLemma 3.1] and [21, Theorem 3.5]. Let 1 < q≤p <∞and 1< r≤ ∞, and the conjugate\nnumberp′ofpis defined by1\np+1\np′= 1. Then the predual space Hp′\nq′,r′(Rn) of the Morrey-\nLorentz space Mp\nq,r(Rn) is given by\nHp′\nq′,r′(Rn) =\n\ng=∞/summationdisplay\nj=1µjbj:{µj}∞\nj=1∈ℓ1(N), eachbjis a (p′,q′,r′)-block\n\n.\nHere by “a ( p′,q′,r′)-block” we mean an Lq′,r′(Rn)-function supported on a cube Q∈ Q(Rn)\nwithLq′,r′(Rn)-norm less than or equal to |Q|1\nq′−1\np′. The norm of Hp′\nq′,r′(Rn) is defined by\n/ba∇dblg/ba∇dblHp′\nq′,r′= inf∞/summationdisplay\nj=1|µj|,\nwhere inf is over all admissible expressions above. We remark that th e norm equivalent\n/ba∇dblf/ba∇dblMp\nq,r∼sup/braceleftbiggˆ\nRn|f(x)g(x)|dx:/ba∇dblg/ba∇dblHp′\nq′,r′= 1/bracerightbigg\nis obtained. In particular, it is known that the predual space of the weak Morrey space\nWMp\nq(Rn) =Mp\nq,∞(Rn) isHp′\nq′,1(Rn) in [20, Theorem 3.6] and [36, Theorem 2.5].\n3.Proof of Propositions 1.10 and 1.12\n3.1.Proof of Proposition 1.10. Letwbe a locally integrable function, and recall that wis\nanA1-weight when Mw/lessorsimilarw. We define the weighted L1-spaceL1(Rn,w) by the space of all\nf∈L0(Rn) with finite norm\n/ba∇dblf/ba∇dblL1(w):=ˆ\nRn|f(x)|w(x)dx.\nTo prove Proposition 1.10, we use the following lemmas and the atomic d ecomposition for\nthe weighted Hardy space H1(Rn,w) given by Nakai and Sawano [28].\nLemma 3.1. (1)For allf∈ S(Rn),\n(3.1) et∆f(x)→f(x)ast↓0.ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 9\n(2)Letw∈A1, and letf∈L1(Rn,w). Then for each x∈Rnandt>0,\nˆ\nRn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/radicalbig\n(4πt)nexp/parenleftbigg\n−|x−y|2\n4t/parenrightbigg\nf(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledy<∞\nholds. Moreover, we have\n(3.2) |f| ≤sup\nt>0|et∆f|,a.e.\n(3)For allf∈ S′(Rn),\net∆f→finS′(Rn).\nProof.(1) Because\n|exp(−|z|2)f(x−2√\ntz)| ≤exp(−|z|2)/ba∇dblf/ba∇dblL∞∈L1(Rn),\nby the Lebesgue convergence theorem, we have\net∆f(x) =1√\nπnˆ\nRnexp(−|z|2)f(x−2√\ntz)dz→1√\nπnˆ\nRnexp(−|z|2)f(x)dz=f(x)\nast↓0. This proves (3.1).\n(2) To prove (3.2), it suffices to show that the set\nEk≡/braceleftBigg\nx∈Rn: limsup\nt↓0|et∆f(x)−f(x)|>1\nk/bracerightBigg\nis a null set for all k∈Nandf∈L1(Rn,w). Fixε>0, and take g∈ S(Rn) such that\n/ba∇dblf−g/ba∇dblL1(w)<ε.\nBecause the function\n(0,∞)∋λ/ma√sto−→ϕ(λ)≡exp(−λ)∈(0,∞)\nis positive and decreasing on (0 ,∞), we deduce from [9, Proposition 2.7] that\nsup\nt>0|et∆f|= sup\nt>0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/radicalbig\n(4πt)nϕ/parenleftbigg\n−|·|2\n4t/parenrightbigg\n∗f/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar1√πn/ba∇dblϕ(|·|2)/ba∇dblL1Mf.\nThus, by (3.1), we estimate\nw(Ek)≤w/parenleftBigg/braceleftBigg\nlimsup\nt↓0|et∆[f−g]|>1\n2k/bracerightBigg/parenrightBigg\n+w/parenleftbigg/braceleftbigg\n|f−g|>1\n2k/bracerightbigg/parenrightbigg\n/lessorsimilarw/parenleftbigg/braceleftbigg\nM[f−g]>1\n2k/bracerightbigg/parenrightbigg\n+w/parenleftbigg/braceleftbigg\n|f−g|>1\n2k/bracerightbigg/parenrightbigg\n.\nApplying the weak-type boundedness of MonL1(Rn,w) (see, e.g., [14, Theorem 7.1.9]) and\nChebyshev’s inequality, we conclude that\nw(Ek)/lessorsimilar[w]A14k/ba∇dblf−g/ba∇dblL1(w)<4kε.\nWe finish the proof of Lemma 3.1 because ε>0 andw(x)dxand dxare mutually absolutely\ncontinuous.\n(3) We omit the proof of this statement. See [35, Theorem 1.35] fo r the discrete case. A\nminormodificationsufficesforthecontinuouscase. Thesameargum entappliestotheGaussian,\nalthough the Gaussian is not compactly supported. /square\nIn addition, we use the following lemma:10 NAOYA HATANO\nLemma 3.2. Let1≤p<∞. Then the following assertions hold :\n(1)Ifδ∈(0,1)satisfiesδ>1−1/p,\nMp\n1(Rn)֒→L1(Rn,(Mχ[−1,1]n)δ).\n(2)There exists a sufficiently large number N∈Nsuch that\n(3.3) |/an}b∇acketle{tf,ϕ/an}b∇acket∇i}ht|/lessorsimilar/ba∇dblf/ba∇dblMp\n1·pN(ϕ)\nfor allf∈ Mp\n1(Rn)andS(Rn). In particular, the embedding\nMp\n1(Rn)֒→ S′(Rn)\nholds.\nProof.The proof of (1) is similar to [37, Proposition 285].\nFor convenience, we give the proof of (2). We estimate\n|/an}b∇acketle{tf,ϕ/an}b∇acket∇i}ht| ≤ˆ\nRn|f(x)ϕ(x)|dx≤ˆ\nRn|f(x)|\n(1+|x|)npn(ϕ)dx\n≤\nˆ\n[−1,1]n|f(x)|dx+∞/summationdisplay\nj=11\n(√n2j−1)nˆ\n[−2j,2j]n|f(x)|dx\npn(ϕ)\n/lessorsimilar\n/ba∇dblf/ba∇dblMp\n1+2(2−1\np)n\n√nn∞/summationdisplay\nj=12−jn\np/ba∇dblf/ba∇dblMp\n1\npn(ϕ)\n=/parenleftBigg\n1+2(2−1\np)n\n√nn2−n\np\n1−2−n\np/parenrightBigg\n/ba∇dblf/ba∇dblMp\n1·pn(ϕ).\nThen, (3.3) is proved. /square\nLetwbe a locally integrable function. We define the weighted L1-norm/ba∇dbl·/ba∇dblL1(w)by\n/ba∇dblf/ba∇dblL1(w)≡ˆ\nRn|f(x)|w(x)dx, f∈L0(Rn),\nand set\nH1(Rn,w)≡/braceleftBigg\nf∈ S′(Rn) :/ba∇dblf/ba∇dblH1(w)≡/vextenddouble/vextenddouble/vextenddouble/vextenddoublesup\nt>0|et∆f|/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1(w)<∞/bracerightBigg\n.\nWe use the following atomic decomposition for H1(Rn,w) with theA1-weightw.\nTheorem 3.3 ([28]).Letwbe anA1-weight, and let f∈H1(Rn,w). Then there exists a\ntriplet{λj}∞\nj=1⊂[0,∞),{Qj}∞\nj=1⊂ Q(Rn)and{aj}∞\nj=1⊂L∞(Rn)such thatf=/summationtext∞\nj=1λjaj\ninS′(Rn)and that\n|aj| ≤χQj,ˆ\nRnaj(x)dx= 0,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1λjχQj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1(w)/lessorsimilar/ba∇dblf/ba∇dblH1(w).\nIn particular, H1(Rn,w)embedded into L1(Rn,w).\nNow, we start the proof of Proposition 1.10.ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 11\nProof of Proposition 1.10 .(1) By embedding for Morrey-Lorentz spaces (see Proposition 2.2 )\nand Lemma 3.2 (2), we have\nf∈ Mp\nq,r(Rn)֒→ Mp\n1(Rn)֒→ S′(Rn).\nAs is described in [9, Proposition 2.7], we have a pointwise estimate\n|et∆f| ≤Mf.\nSinceMis shown to be bounded on the Morrey-Lorentz spaces Mp\nq,r(Rn) (Proposition 2.7),\nwe havef∈HMp\nq,r(Rn).\n(2) Letf∈HMp\nq,r(Rn). Using Proposition 2.2 and Lemma 3.2 (1), we have\nMp\nq,r(Rn)֒→ Mp\n1(Rn)֒→L1(Rn,(Mχ[−1,1]n)δ).\nSince (Mχ[−1,1]n)δis anA1-weight (see [9, Theorem 7.7]), by Theorem 3.3,\nHMp\nq,r(Rn)֒→H1(Rn,(Mχ[−1,1]n)δ)֒→L1(Rn,(Mχ[−1,1]n)δ).\nConsequently, applying Lemma 3.1 (1), we obtain\nHMp\nq,r(Rn)֒→ Mp\nq,r(Rn).\n/square\n3.2.Proof of Proposition 1.12. TheproofissimilartoHardyspaceswithvariableexponents\n[8, 27]. We content ourselves with stating two fundamental estimat es (3.4) and (3.5) below.\nSuppose that we are given an integer K≫1. We write\nM∗\nheatf(x)≡sup\nj∈Z/parenleftBigg\nsup\ny∈Rn|e2j∆f(y)|\n(1+4j|x−y|2)K/parenrightBigg\n, x∈Rn.\nThe next lemma stands for the pointwise estimate for M∗\nheatin terms of the usual Hardy-\nLittlewood maximal operator M.\nLemma 3.4 ([27, Lemma 3.2], [34, §4]).For0<θ<1, there exists Kθso that for all K≥Kθ,\nwe have\n(3.4) M∗\nheatf(x)/lessorsimilarM/bracketleftbigg\nsup\nk∈Z|e2k∆f|θ/bracketrightbigg\n(x)1\nθ, x∈Rn\nfor anyf∈ S′(Rn).\nIn the course of the proof of [27, Theorem 3.3], it can be shown tha t\n(3.5) Mf(x)∼sup\nτ∈FN,j∈Z|τj∗f(x)|/lessorsimilarM∗\nheatf(x)\nonce we fix an integer K≫1 andN≫1.\nApplying Proposition 2.7 with the fundamental pointwise estimates (3 .4) and (3.5), Propo-\nsition 1.12 can be proved with ease. We omit the details.12 NAOYA HATANO\n4.Proof of Theorems 1.1, 1.7 and 1.8\n4.1.Proof of Theorem 1.1. We employ the argument from the proof of Theorem 1.1 in [23].\nBy the decomposition of Qj, we may assume that each Qjis a dyadic cube. We may assume\nthat there exists N∈Nsuch thatλj= 0 whenever j≥N. In addition, let us assume that the\naj’s are non-negative. By the embedding ℓv(N)֒→ℓ1(N) and the duality argument, we note\nthat\n/ba∇dblf/ba∇dblv\nMp\nq,r≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1|λjaj|v/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nM˜p\n˜q,˜r∼sup\n\nˆ\nRn∞/summationdisplay\nj=1|λjaj(x)|v|g(x)|dx:/ba∇dblg/ba∇dblH˜p′\n˜q′,˜r′= 1\n\n,\nwhere we set ˜ p:=p/v, ˜q:=q/vand ˜r:=r/v. Then, we may assume that the aj’s are non-\nnegative and gis a non-negative(˜ p′,˜q′,˜r′)-block with associated dyadic cube Q. Then, we show\nthat\n(4.1)ˆ\nRn∞/summationdisplay\nj=1(λjaj(x))vg(x)dx/lessorsimilarp,q,r,s,t,v/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublev\nMp\nq,r.\nAssume first that each QjcontainsQas a proper subset. If we group the j’s such that all\nQjare identical, we can assume that each Qjis aj-th parent of Qfor eachj∈N. Then, by\nthe H¨ older inequality for Lorentz spaces (see Lemma 2.1),\nˆ\nRn∞/summationdisplay\nj=1(λjaj(x))vg(x)dx=∞/summationdisplay\nj=1λv\njˆ\nQaj(x)vg(x)dx\n/lessorsimilar∞/summationdisplay\nj=1λv\nj/ba∇dblajχQ/ba∇dblv\nLt,∞/ba∇dblg/ba∇dblL˜q′,˜r′|Q|v\nq−v\nt\n≤∞/summationdisplay\nj=1λv\nj/ba∇dblaj/ba∇dblv\nWMs\nt|Q|−v\ns+v\nt|Q|1\n˜q′−1\n˜p′|Q|v\nq−v\nt\n=∞/summationdisplay\nj=1λv\nj|Qj|v\ns|Q|v\np−v\ns.\nNote that by Proposition 2.3, for each J∈N,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBigg∞/summationdisplay\nk=1(λkχQk)v/parenrightBigg1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r≥λJ/ba∇dblχQJ/ba∇dblMp\nq,r∼λJ|QJ|1\np.\nConsequently, it follows from the condition p<sthat\nˆ\nRn∞/summationdisplay\nj=1(λjaj(x))vg(x)dx/lessorsimilar∞/summationdisplay\nj=1|Qj|v\ns−v\np|Q|v\np−v\ns·/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBigg∞/summationdisplay\nk=1(λkχQk)v/parenrightBigg1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublev\nMp\nq,r\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBigg∞/summationdisplay\nk=1(λkχQk)v/parenrightBigg1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublev\nMp\nq,r.ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 13\nConversely, assume that Qcontains each Qj. Then, again by the H¨ older inequality (see Lemma\n2.1),\nˆ\nRn∞/summationdisplay\nj=1(λjaj(x))vg(x)dx=∞/summationdisplay\nj=1λv\njˆ\nQjaj(x)vg(x)dx/lessorsimilar∞/summationdisplay\nj=1λv\nj/ba∇dblaj/ba∇dblv\nLt,∞/ba∇dblgχQj/ba∇dblL˜t′,1\n≤∞/summationdisplay\nj=1λj/ba∇dblaj/ba∇dblv\nWMs\nt|Qj|−v\ns+v\nt/ba∇dblgχQj/ba∇dblL˜t′,1\n≤∞/summationdisplay\nj=1λv\nj|Qj|v\nt/ba∇dblgχQj/ba∇dblL˜t′,1,\nwhere˜t:=t/v. Thus, in terms of the maximal operator M(˜t′,1), we obtain\nˆ\nRn∞/summationdisplay\nj=1(λjaj(x))vg(x)dx≤∞/summationdisplay\nj=1λv\nj|Qj|·inf\ny∈QjM(˜t′,1)g(y)\n≤ˆ\nRn\n∞/summationdisplay\nj=1(λjχQj(y))v\nχQ(y)M(˜t′,1)g(y)dy.\nHence,weobtain(4.1)bytheH¨ olderinequality(seeLemma2.1)andt heL˜q′,˜r′(Rn)-boundedness\nof the maximal operator M(˜t′,1)(see Proposition 2.5).\nIt remains to check the convergence of the sum. Here, when r<∞, by the estimate of (1.1),\nthe Lebesgue convergence theorem yields\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1λjaj−J/summationdisplay\nj=1λjaj\nχR/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLq,r≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=J+1|λjaj|\nχR/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLq,r→0\nasJ→ ∞for eachR∈ Q(Rn). Namely, f=/summationtext∞\nj=1λjajconverges in Lq,r\nloc(Rn). The case of\np=qandr<∞can be also dealt with by a similar approach.\n4.2.Lemmas for the proofs of Theorems 1.8 and 1.9. To prove Theorems 1.8 and 1.9,\nwe gave some estimate for atoms belonging to the critical cases Ms\n1(Rn).\nLemma 4.1. Let1≤s<∞,K∈N0, andQ∈ Q(Rn). Assume that a∈ Ms\n1(Rn)∩PK(Rn)⊥\nsatisfies\n(4.2) supp( a)⊂Q,/ba∇dbla/ba∇dblMs\n1≤ |Q|1\ns.\nThen, for all ϕ∈ S(Rn)andN >0,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nRna(x)ϕ(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarℓ(Q)n+K+1sup\ny∈Q1\n1+|y|N.\nHere, the implicit constant in /lessorsimilardepends on ϕ.\nProof.By the mean-value theorem, there exists θ∈(0,1) depending on x,K,Q, andϕsuch\nthat\nˆ\nRna(x)ϕ(x)dx=ˆ\nRna(x)\nϕ(x)−/summationdisplay\n|α|≤K1\nα!∂αϕ(c(Q))(x−c(Q))α\ndx\n=ˆ\nRna(x)/summationdisplay\n|β|=K+11\nβ!∂βϕ((1−θ)x+θc(Q))(x−c(Q))βdx.14 NAOYA HATANO\nThen, from (4.2)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nRna(x)ϕ(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarℓ(Q)K+1sup\ny∈Q1\n1+|y|Nˆ\nQ|a(x)|dx\n≤ℓ(Q)K+1sup\ny∈Q1\n1+|y|K/ba∇dbla/ba∇dblMs\n1·|Q|−1\ns+1\n/lessorsimilarℓ(Q)n+K+1sup\ny∈Q1\n1+|y|N,\nas desired. /square\nLemma 4.2. Let0< q≤p <∞,1≤s <∞, andK∈N0. Assume that {λQ}Q∈D(Rn)⊂\n[0,∞)and{aQ}Q∈D(Rn)⊂ Ms\n1(Rn)∩PK(Rn)⊥satisfy\nsupp(aQ)⊂3Q,/ba∇dbla/ba∇dblMs\n1≤ |Q|1\ns,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λQχQ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq<∞.\nIfq≤1andn+K+1>n/q, then\n(4.3)∞/summationdisplay\nm=1/summationdisplay\nQ∈Dm(Rn)λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λQχQ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq.\nProof.Fixm≥1. To prove (4.3), we use the fact that for each ˜ m∈Zn,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λQχQ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq/greaterorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈Dm(Rn),|c(Q)−˜m|≤nλQχQ\nχ˜m+[−n,n]n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLq\n= 2−mn\nq\n/summationdisplay\nQ∈Dm(Rn),|c(Q)−˜m|≤nλq\nQ\n1\nq\n≥2−mn\nq/summationdisplay\nQ∈Dm(Rn),|c(Q)−˜m|≤nλQ.\nIn particular, for all R∈ Dm(Rn),\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λQχQ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq/greaterorsimilar2−mn\nqλR.\nWe remark that for each m≥1 and ˜m∈Zn,\n♯{Q∈ Dm(Rn) :3Q∋0}= 4n.\nIt follows from Lemma 4.1 that\n/summationdisplay\nQ∈Dm(Rn),3Q∋0λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilar2−m(n+K+1)/summationdisplay\nQ∈Dm(Rn),3Q∋0λQ\n/lessorsimilar2−m(n+K+1−n\nq)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λQχQ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq.(4.4)\nIn addition, setting\nDm,˜m(Rn) :={Q∈ Dm(Rn) :|c(Q)−˜m| ≤n}ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 15\nfor eachm≥1 and ˜m∈Zn, we have\nDm(Rn) =/uniondisplay\n˜m∈ZnDm,˜m(Rn).\nThen, there exists a mapping ιm:Dm(Rn)→Znsuch thatQ∈ Dm,ιm(Q)(Rn); therefore\n/summationdisplay\nQ∈Dm(Rn),3Q/ne}ationslash∋0λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|\n/lessorsimilar2−m(n+K+1)/summationdisplay\n˜m∈Zn/summationdisplay\nQ∈Dm(Rn),ιm(Q)=˜mλQ·sup\ny∈3Q1\n1+|y|n+1\n/lessorsimilar2−m(n+K+1)/summationdisplay\n˜m∈Zn1\n1+|˜m|n+1/summationdisplay\nQ∈Dm,˜m(Rn)λQ\n/lessorsimilar2−m(n+K+1−n\nq)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λQχQ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq.(4.5)\nBecause\nn+K+1>n\nq,\nthen we obtain the desired result. /square\n4.3.Proofs of Theorems 1.7 and 1.8: convergence of f=/summationtext∞\nj=1λjaj.First, we prove\nthe convergence of\nf=∞/summationdisplay\nj=1λjajinS′(Rn).\nWe start with an important reduction. For each J∈N, we take any cube Q(J)∈ D(Rn)\nwith mimimal volume such that QJ⊂3Q(J), and we set\nEQ:={j∈N:Q=Q(j)}\nand\nλQ:=/summationdisplay\nj∈EQλj, aQ:=\n\n0, λ Q= 0,\n1\nλQ/summationdisplay\nj∈EQλjaj, λQ/ne}ationslash= 0.\nNote that {EQ}Q∈D(Rn)is pairwise disjoint. Then, {aQ}Q∈D(Rn)and{λQ}Q∈D(Rn)satisfy\n/ba∇dblaQ/ba∇dblMs\n1≤1\nλQ/summationdisplay\nj∈EQλj/ba∇dblaj/ba∇dblMs\n1≤1\nλQ/summationdisplay\nj∈EQλj|Qj|1\ns≤ |3Q|1\ns.\nTakingθ∈(1/v,∞), by the fact that χQ(J)/lessorsimilarnMχQJforJ∈N, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)(λQχQ)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)\n/summationdisplay\nj∈EQλj(MχQj)θ\nv\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)/summationdisplay\nj∈EQ/parenleftBig\nM/bracketleftBig\nλ1\nθ\njχQj/bracketrightBig/parenrightBigθv\n1\nθv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleθ\nMθp\nθq,θr.16 NAOYA HATANO\nThen, by Proposition 2.8,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)(λQχQ)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)/summationdisplay\nj∈EQ(λjχQj)v\n1\nθv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleθ\nMθp\nθq,θr\n=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nθv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleθ\nMθp\nθq,θr<∞.\nHence, we may assume\n{aQ}Q∈D⊂/braceleftBigg\nWMs\nt(Rn)∩Pdv(Rn)⊥,1<t≤s<∞,\nMs\n1(Rn)∩Pdv(Rn)⊥,1 =t≤s<∞,{λQ}Q∈D⊂[0,∞),\nwith supp(aQ)⊂3QforQ∈ D(Rn) instead of\n{aj}∞\nj=1⊂/braceleftBigg\nWMs\nt(Rn)∩Pdv(Rn)⊥,1<t≤s<∞,\nMs\n1(Rn)∩Pdv(Rn)⊥,1 =t≤s<∞,{λj}∞\nj=1⊂[0,∞).\nThen, it suffices to show that\n(4.6)/summationdisplay\nQ∈D(Rn)λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|=0/summationdisplay\nm=−∞/summationdisplay\nQ∈Dm(Rn)λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|+∞/summationdisplay\nm=1/summationdisplay\nQ∈Dm(Rn)λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|<∞.\nFirst, we estimate the first part of (4.6). Fix m≤0. For each Q∈ Dm(Rn),3Q/ne}ationslash∋0 implies\nthat|y| ≥ℓ(Q) for ally∈3Q, and then\n|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilarˆ\n3Q|aQ(x)|dxsup\ny∈3Q1\n1+|y|2n+1−n\ns\n/lessorsimilar/ba∇dblaQ/ba∇dblMs\n1·|Q|−1\ns+1sup\ny∈3Q1\n1+|y|2n+1−n\ns/lessorsimilar|Q|1\nssup\ny∈3Q1\n1+|y|n+1.\nIt follows that\n/summationdisplay\nQ∈Dm(Rn),3Q/ne}ationslash∋0λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilar/summationdisplay\nQ∈Dm(Rn),3Q/ne}ationslash∋0λQ·|Q|1\nssup\ny∈3Q1\n1+|y|n+1\n/lessorsimilar2−mn\ns/summationdisplay\n˜m∈Zn1\n1+|˜m|n+1/summationdisplay\nQ∈Dm(Rn),3Q/ne}ationslash∋0\n|c(Q)−˜m|≤nλQ.\nMeanwhile, if 3Q∋0, by Lemma 3.2 (2),\n|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilarϕ/ba∇dblaQ/ba∇dblMs\n1/lessorsimilar|Q|1\ns.\nThus,\n/summationdisplay\nQ∈Dm(Rn),3Q∋0λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilar/summationdisplay\nQ∈Dm(Rn),3Q/ne}ationslash∋0λQ·|Q|1\ns≤4n2−mn\ns+mn\npsup\nQ∈D(Rn)λQ|Q|1\np.\nNote that for each R∈ D(Rn),\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)(λQχQ)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r≥ /ba∇dblλRχR/ba∇dblMp\nq,r=/parenleftBigq\nr/parenrightBig1\nrλR|R|1\npATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 17\nby Proposition 2.3. We conclude that\n0/summationdisplay\nm=−∞/summationdisplay\nQ∈Dm(Rn)λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilar0/summationdisplay\nm=−∞2−mn\ns+mn\np/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)(λQχQ)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)(λQχQ)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r.(4.7)\nNote that\nn+dv+1−n\nq0>n+/parenleftBign\nv−n−1/parenrightBig\n+1−n\nq0=n\nv−n\nq0≥0.\nThus, there exists ε∈(0,q0) such that\nn+dv+1−n\nq0−ε>0,\nwhereq0:= min(1,q). Hence, by Lemma 4.2, the second part of (4.6) can be estimated a s\nfollows:\n∞/summationdisplay\nm=1/summationdisplay\nQ∈Dm(Rn)λQ|/an}b∇acketle{taQ,ϕ/an}b∇acket∇i}ht|/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λQχQ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq0−ε/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/summationdisplay\nQ∈D(Rn)(λQχQ)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r, (4.8)\nwhereinthelastinequality,weusedtheembedding Mp\nq,r(Rn)֒→ Mp\nq0−ε,q0−ε(Rn) =Mp\nq0−ε(Rn)\n(see Proposition 2.2). Combining these estimates (4.7) and (4.8), we finish the proof of (4.6).\n4.4.Proofs of Theorems 1.7 and 1.8: (1.3)and(1.4).To prove the estimates of (1.3) and\n(1.4) in Theorems 1.7 and 1.8, respectively, we use the following lemma, whose proof is similar\nto that of Lemma 4.1.\nLemma 4.3 ([27, (5.2)]) .If{aj}∞\nj=1satisfies the same assumptions as in Theorems 1.7and\n1.8, then\nMaj(x)/lessorsimilarχ3Qj(x)Maj(x)+MχQj(x)n+dv+1\nn, x∈Rn.\nLet us show Theorem 1.7. Using Proposition 1.12 and Lemma 4.3, we hav e\n/ba∇dblf/ba∇dblHMp\nq,r∼ /ba∇dblMf/ba∇dblMp\nq,r≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1λjMaj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1λj/parenleftBig\nχ3QjMaj+(MχQj)n+dv+1\nn/parenrightBig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1λjχ3QjMaj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1λj(MχQj)n+dv+1\nn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r=:I1+I2.\nFirst, we consider I1. We note that for each j∈N, owing to the W Ms\nt(Rn) =Ms\nt,∞(Rn)-\nboundedness of M(see Proposition 2.7), by applying Proposition 2.8 and Theorem 1.1 and18 NAOYA HATANO\nusing the fact that χ3Qj≤3nMχQjfor eachj∈N, we have\nI1/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχ3Qj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1λv\nj(MχQj)2\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r.\nNext, we consider I2. Set\nP:=n+dv+1\nnp, Q:=n+dv+1\nnq,andR:=n+dv+1\nnr.\nThen, by Proposition 2.8 and the embedding ℓv(N)֒→ℓ1(N), we obtain\nI2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1λj(MχQj)n+dv+1\nn\nn\nn+dv+1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen+dv+1\nn\nMP\nQ,R/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1λjχQj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r.\nThus, we obtain the desired result.\nSimilarly, because Msatisfies the weak-type boundedness on Ms\n1(Rn) (see Proposition 2.6)\nin the estimate of I1, we can prove Theorem 1.8.\n5.Proof of Theorem 1.9\nTo prove Theorem 1.9, we use a new approach provided in [29, Subsec tion 4.3] and the\nfollowing lemma, as given in [35, Exercise 3.34].\nLemma 5.1. Let0< q≤p <∞,0< r≤ ∞,K∈N, and0< v <∞, and letf∈\nHMp\nq,r(Rn)∩L1\nloc(Rn). Then, we can find {aj}∞\nj=1⊂L∞(Rn)∩ P⊥\nK(Rn)and a sequence\n{Qj}∞\nj=1of cubes such that\n(1) supp(aj)⊂Qj,\n(2)f=∞/summationdisplay\nj=1ajinS′(Rn), and\n(3)\n∞/summationdisplay\nj=1(/ba∇dblaj/ba∇dblL∞χQj)v\n1\nv\n/lessorsimilarMf.\nProof of Theorem 1.9.It suffices to prove the case of v= 1; the case of v >0 can be proved\nsimilarly. Let f∈HMp\nq,r(Rn). Fixt>0. Because D(Rn) is a countable set, applying Lemma\n5.1 toet∆f∈HMp\nq,r(Rn)∩L1\nloc(Rn) for\n{3Q}Q∈D(Rn),{λt\nQ}Q∈D(Rn),{λQat\nQ}Q∈D(Rn)ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 19\ninstead of\n{Qj}∞\nj=1,{/ba∇dblaj/ba∇dblL∞}∞\nj=1,{aj}∞\nj=1,\nrespectively, we can consider the decomposition et∆f=/summationtext\nQ∈D(Rn)λt\nQat\nQin the topology of\nS′(Rn), whereat\nQ∈ P⊥\nK(Rn),λt\nQ≥0, and\n|at\nQ| ≤χ3Q,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λt\nQχ3Q/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilar/ba∇dblM[et∆f]/ba∇dblMp\nq,r/lessorsimilar/ba∇dblMf/ba∇dblMp\nq,r.\nBy the weak-* compactness of the unit ball of L∞(Rn), there exists a sequence {tl}∞\nl=1that\nconverges to 0 such that both λQ= liml→∞λtl\nQandaQ= liml→∞atl\nQexist for all Q∈ Din the\nsense that\nlim\nl→∞ˆ\nRnatl\nQ(x)ϕ(x)dx=ˆ\nRnaQ(x)ϕ(x)dx\nfor allϕ∈L1(Rn). We claim that f=/summationtext\nQ∈D(Rn)λQaQin the topology of S′(Rn). Let\nϕ∈ S(Rn) be a test function. Then, by Lemma 3.1 (3), we have\n/an}b∇acketle{tf,ϕ/an}b∇acket∇i}ht= lim\nl→∞/an}b∇acketle{tetl∆f,ϕ/an}b∇acket∇i}ht= lim\nl→∞/summationdisplay\nQ∈D(Rn)λtl\nQˆ\nRnatl\nQ(x)ϕ(x)dx\nfrom the definition of convergence in S′(Rn). Once we fix m, we have\nλtl\nQ/lessorsimilar2mn\np/ba∇dblMf/ba∇dblMp\nq,rand/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nRnatl\nQ(x)ϕ(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ˆ\n3Q|ϕ(x)|dx.\nAdditionally, by the equation\n/summationdisplay\nQ∈Dm(Rn)2mn\np/ba∇dblMf/ba∇dblMp\nq,rˆ\n3Q|ϕ(x)|dx= 3n2mn\np/ba∇dblMf/ba∇dblMp\nq,r/ba∇dblϕ/ba∇dblL1<∞,\nwe can use Fubini’s theorem to obtain\n/summationdisplay\nm∈Zˆ\nRn\n/summationdisplay\nQ∈Dm(Rn)λtl\nQatl\nQ(x)\nϕ(x)dx=/summationdisplay\nm∈Z/summationdisplay\nQ∈Dm(Rn)λtl\nQˆ\nRnatl\nQ(x)ϕ(x)dx.\nHereinafter, we use also abbreviation\nam,l:=/summationdisplay\nQ∈Dm(Rn)λtl\nQˆ\nRnatl\nQ(x)ϕ(x)dx,\nand we fix 0 <ε≪1.\nWhenm∈Z∩(−∞,0], we see that\n|am,l|/lessorsimilar/summationdisplay\nQ∈Dm(Rn)2mn\np/ba∇dblMf/ba∇dblMp\nq,rˆ\n3Q|ϕ(x)|dx/lessorsimilarϕ2mn\np/ba∇dblMf/ba∇dblMp\nq,r\nby the previous argument, and therefore\n(5.1)0/summationdisplay\nm=−∞|am,l|/lessorsimilar/ba∇dblMf/ba∇dblMp\nq,r.\nIn addition, taking 0 <ε≪1 byK+1>n(1/(q0−ε)−1)>0, namely,\nn+K+1>n\nq0−ε,20 NAOYA HATANO\nby Lemma 4.2, we obtain\n(5.2)∞/summationdisplay\nm=1|am,l|/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nQ∈D(Rn)λt\nQχ3Q/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp\nq,r/lessorsimilar/ba∇dblMf/ba∇dblMp\nq,r.\nThus, by (5.1) and (5.2), we obtain\n∞/summationdisplay\nm=−∞|am,l|/lessorsimilar/ba∇dblMf/ba∇dblMp\nq,r.\nAs a consequence, applying the Lebesgue convergence theorem, we obtain\nlim\nl→∞∞/summationdisplay\nm=−∞am,l=∞/summationdisplay\nm=−∞lim\nl→∞am,l.\nHence, using Fubini’s theorem again, we have\n/an}b∇acketle{tf,ϕ/an}b∇acket∇i}ht=∞/summationdisplay\nm=−∞\nlim\nl→∞/summationdisplay\nQ∈Dm(Rn)λtl\nQˆ\nRnatl\nQ(x)ϕ(x)dx\n\n=∞/summationdisplay\nm=−∞\nlim\nl→∞ˆ\nRn\n/summationdisplay\nQ∈Dm(Rn)λtl\nQatl\nQ(x)\nϕ(x)dx\n\n=∞/summationdisplay\nm=−∞/summationdisplay\nQ∈Dm(Rn)lim\nl→∞/parenleftbiggˆ\nRnλtl\nQatl\nQ(x)ϕ(x)dx/parenrightbigg\n=∞/summationdisplay\nm=−∞/summationdisplay\nQ∈Dm(Rn)ˆ\nRnλQaQ(x)ϕ(x)dx=/angbracketleftBigg/summationdisplay\nQ∈D(Rn)λQaQ,ϕ/angbracketrightBigg\n.\nConsequently, we obtain the desired result. /square\n6.Application to the Olsen inequality\nAs an application of Theorems 1.1 and 1.4, we can reprove the following Olsen inequality\nabout the fractional integral operator Iα, 0<α<n, defined by\nIαf(x)≡ˆ\nRnf(y)\n|x−y|n−αdy\nfor a measurable function fdefined on Rn.\nTheorem 6.1. Let0< α < n ,1< p1≤p0<∞,0< p2≤ ∞,1< q1≤q0<∞,\n1<r1≤r0<∞and1<r2≤ ∞. Assume that\nr1<q1,1\nq0≤α\nn<1\np0,1\nr0=1\nq0+1\np0−α\nn.\nIf we suppose either (1)or(2)as follows :\n(1) 0<r2,p2<∞andr0\np0=r1\np1=r2\np2,\n(2)r2=p2=∞andr0\np0=r1\np1,\nthen we have\n/ba∇dblg·Iαf/ba∇dblMr0r1,r2/lessorsimilar/ba∇dblg/ba∇dblWMq0q1/ba∇dblf/ba∇dblMp0p1,p2,\nfor any measurable functions fandginMp0p1,p2(Rn)andWMq0q1(Rn), respectively.ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 21\nIf we rewrite by f/ma√sto→ |∇u|in this theorem, the following norm estimate with a weight, which\nis extension of the Fefferman-Phong inequality [10, p. 143], can be o btained.\nTheorem 6.2. Letn≥3and1<q≤n\n2<∞. Then there exists a constant K >0such that\nˆ\nRn|u(x)|2V(x)dx≤K/ba∇dblV/ba∇dbl\nWMn\n2qˆ\nRn|∇u(x)|2dx\nfor all measurable functions uwith∇u∈(L2(Rn))nand non-negative measurable functions\nV∈WMn\n2q(Rn).\nThis theorem follows the necessary condition of the positivity of the Schr¨ odinger operator\nL≡ −∆−Vfor the potential V≥0; when 0< K/ba∇dblV/ba∇dbl\nWMn\n2q≤1, using the integration by\nparts, we see that\n(Lu,u)L2=ˆ\nRn|∇u(x)|2dx−ˆ\nRn|u(x)|2V(x)dx≥/parenleftbigg\n1−K/ba∇dblV/ba∇dbl\nWMn\n2q/parenrightbiggˆ\nRn|∇u(x)|2dx≥0.\nOntheotherhand, consideringthe embedding Mq0q1(Rn)֒→WMq0q1(Rn)andthe case p2=p1\nandq2=q1in Theorem 6.1, we can obtain the original Olsen inequality given in [30]. In\nparticular, in the case (2) in Theorem 6.1, we can rewrite as follows:\nTheorem 6.3. Let0< α < n ,1< p1≤p0<∞,1< q1≤q0<∞and1< r1≤r0<∞.\nAssume that\nr1<q1,1\nq0≤α\nn<1\np0,1\nr0=1\nq0+1\np0−α\nn,r0\np0=r1\np1.\nThen we have\n/ba∇dblg·Iαf/ba∇dblWMr0r1/lessorsimilar/ba∇dblg/ba∇dblWMq0q1/ba∇dblf/ba∇dblWMp0p1,\nfor any measurable functions fandginWMp0p1(Rn)andWMq0q1(Rn), respectively.\nTo prove Theorem 6.1, the author employed the generalization for A dams theorem [2] and\ntwo lemmas mentioned in [23] as follows.\nProposition 6.4. [19, Proposition 3] Let0< α < n ,1< q≤p <∞,1< t≤s <∞and\n0<r,u≤ ∞. Assume that1\ns=1\np−α\nn. If we suppose either (1)or(2)as follows:\n(1) 0<r,u< ∞ands\np=t\nq=u\nr,\n(2)r=u=∞ands\np=t\nq,\nthen we have\n/ba∇dblIαf/ba∇dblMs\nt,u/lessorsimilar/ba∇dblf/ba∇dblMp\nq,r,\nfor any measurable function finMp\nq,r(Rn).\nIt is well known that Hardy and Littlewood [17] and Sobolev [38] prove d the boundedness of\na fractional integral operatoron Lebesgue spaces, which is called the Hardy-Littlewood-Sobolev\ninequality: if 0 <α<n and 1<p<s< ∞satisfies 1/s= 1/p−α/n, then\n(6.1) /ba∇dblIαf/ba∇dblLs/lessorsimilar/ba∇dblf/ba∇dblLp\nfor allf∈Lp(Rn) (see, e.g., [15, Theorem 1.2.3]).\nLemma 6.5 ([23, Lemma 4.1]) .There exists a constant depending only on nandαsuch that,\nfor every cube Q, we haveIαχQ(x)≥Cℓ(Q)αχQ(x)for allx∈Q.22 NAOYA HATANO\nLemma 6.6 ([23, Lemma 4.2]) .LetK= 0,1,2,.... Suppose that Ais anL∞(Rn)∩PK(Rn)⊥-\nfunction supported on a cube Q. Then,\n|IαA(x)| ≤Cα,K/ba∇dblA/ba∇dblL∞ℓ(Q)α∞/summationdisplay\nk=11\n2k(n+K+1−α)χ2kQ(x), x∈Rn.\nWe remark that Lemma 6.6 is proved by a method akin to Lemma 4.1.\nNow we start the proof of Theorem 6.1. We may assume that f≥0 because the integral\nkernel ofIαis positive. Now, we prove Theorem 6.1. Note that by Fatou’s lemma an d the\nFatou property for Morrey-Lorentz spaces (see Lemma 2.4),\n/ba∇dblg·Iαf/ba∇dblMr0r1,r2≤/vextenddouble/vextenddouble/vextenddoubleg·liminf\nm→∞Iαfm/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2≤liminf\nm→∞/ba∇dblg·Iαfm/ba∇dblMr0r1,r2\nfor allf∈ Mp0p1,p2(Rn) andg∈ Mq0q1,q2(Rn), where\nfm:=fχB(m)χ[0,m)(|f|)∈L∞\nc(Rn), m∈N.\nThen, we may assume that f∈L∞\nc(Rn) is a positive measurable function in view of the posi-\ntivity of the integral kernel. We decompose faccording to Theorem 1.4 with sufficiently large\nK≫1;f=/summationtext∞\nj=1λjajconverges in Lw(Rn) for allw∈(1,∞), where {Qj}∞\nj=1⊂ D(Rn),\n{aj}∞\nj=1⊂L∞(Rn)∩PK(Rn)⊥, and{λj}∞\nj=1⊂[0,∞) fulfill (1.2).\nHere, we claim that\n(6.2) Iαf(x) =∞/summationdisplay\nj=1λjIαaj(x),a.e.x∈Rn.\nIn fact, fixing w∈(1,n/α) and then choosing w∗∈(1,∞) satisfying 1 /w∗= 1/w−α/n, we\nhave/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleIαf−N/summationdisplay\nj=1λjIαaj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLw∗=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleIα\nf−N/summationdisplay\nj=1λjaj\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLw∗/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublef−N/summationdisplay\nj=1λjaj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLw→0\nasN→ ∞from the Hardy-Littlewood-Sobolev inequality (see (6.1)). We finish the proof of\n(6.2).\nThen, by Lemma 6.6, we obtain\n|g(x)Iαf(x)|/lessorsimilar∞/summationdisplay\nj,k=1λj\n2k(n+K+1−α)ℓ(Qj)α|g(x)|χ2kQj(x).\nTherefore, we conclude that\n/ba∇dblg·Iαf/ba∇dblMr0r1,r2/lessorsimilar/ba∇dblg/ba∇dblWMq0q1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj,k=1λjℓ(2kQj)α−n\nq0\n2k(n+L+1)·ℓ(2kQj)n\nq0\n/ba∇dblg/ba∇dblWMq0q1|g|χ2kQj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2.\nFor each (j,k)∈N×N, write\nκjk:=λjℓ(2kQj)α−n\nq0\n2k(n+L+1), bjk:=ℓ(2kQj)n\nq0\n/ba∇dblg/ba∇dblWMq0q1|g|χ2kQj.\nThen,\n∞/summationdisplay\nj,k=1λjℓ(2kQj)α−n\nq0\n2k(n+K+1)·ℓ(2kQj)n\nq0\n/ba∇dblg/ba∇dblWMq0q1|g|χ2kQj=∞/summationdisplay\nj,k=1κjkbjk,ATOMIC DECOMPOSITION FOR MORREY-LORENTZ SPACES 23\neachbjkis supported on a cube 2kQjand\n/ba∇dblbjk/ba∇dblWMq0q1≤ |2kQj|1\nq0.\nObserve that χ2kQj≤2knMχQj. Hence, if we choose v,θ∈Rsuch that\nK >α−n\nq0−1+θn−n, θ>1\nv≥1\nmin(r1,r2),0<v≤1,\nthen we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj,k=1(κjkχ2kQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj,k=1/parenleftBigg\nλjℓ(2kQj)α−n\nq0\n2k(n+K+1)χ2kQj/parenrightBiggv\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1/parenleftBig\nλjℓ(Qj)α−n\nq0(MχQj)θ/parenrightBigv\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2\n=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n\n∞/summationdisplay\nj=1/parenleftbigg\nM/bracketleftbigg/parenleftBig\nλjℓ(Qj)α−n\nq0χQj/parenrightBig1\nθ/bracketrightbigg/parenrightbiggθv\n\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2.\nBy virtue of Proposition 2.8, the Fefferman-Stein inequality for Morr ey-Lorentz spaces, along-\nsidefj= (λjℓ(Qj)α−n/q0χQj)1/θ, we can remove the maximal operator and obtain\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj,k=1(κjkχ2kQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1/parenleftBig\nλjℓ(Qj)α−n\nq0χQj/parenrightBigv\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2.\nWe distinguish two cases here:\n(1) Ifα=n/q0, thenp0=r0,p1=r1, andp2=r2. Thus, we can use (1.2).\n(2) Ifα>n/q 0, then by Proposition 6.4 and Lemma 6.5, we obtain\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1/parenleftBig\nλjℓ(Qj)α−n\nq0χQj/parenrightBigv\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nI/parenleftBig\nα−n\nq0/parenrightBig\nv\n∞/summationdisplay\nj=1/parenleftbig\nλjχQj/parenrightbigv\n\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj=1(λjχQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMp0p1,p2.\nThus, we can still use (1.2).\nConsequently, we obtain\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj,k=1(κjkχ2kQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2/lessorsimilar/ba∇dblf/ba∇dblMp0p1,p2<∞. (6.3)24 NAOYA HATANO\nObserve also that q0>r0andq1>r1. Thus, by Theorem 1.1 and (6.3), it follows that\n/ba∇dblg·Iαf/ba∇dblMr0r1,r2/lessorsimilar/ba∇dblg/ba∇dblWMq0q1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞/summationdisplay\nj,k=1(κjkχ2kQj)v\n1\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nMr0r1,r2/lessorsimilar/ba∇dblg/ba∇dblWMq0q1/ba∇dblf/ba∇dblMp0p1,p2.\nAcknowledgement\nThe authors would like to thank Professor Yoshihiro Sawano for his c areful reading of the\nmanuscript. The author was financially supported by a Research Fe llowship from the Japan\nSociety for the Promotion of Science for Young Scientists (21J121 29).\nReferences\n[1] W. Abu-Shammala and A. Torchinsky, The Hardy-Lorentz sp acesHp,q(Rn), Studia Math. 182 (2007),\nno. 3, 283–294.\n[2] D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1 975), no. 4, 765–778.\n[3] A. Akbulut, V. S. Guliyev, T. 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Partial Differ-\nential Equations 20(11–12), 2005–2055 (1995).\n[31] D. V. Parilov, Two theorems on the Hardy-Lorentz classe sH1,q, (Russian) Zap. Nauchn. Sem. S.-\nPeterburg. Otdel. Mat. Inst. Steklov. (POMI) 327 (2005), Is sled. po Line˘ ın. Oper. i Teor. Funkts. 33,\n150–167, 238; translation in J. Math. Sci. (N.Y.) 139 (2006) , no. 2, 6447–6456.\n[32] M. A. Ragusa, Embeddings for Morrey-Lorentz spaces, J. Optim. Theory Appl. 154 (2012), no. 2, 491–499.\n[33] Y. Sawano, Atomic decompositions of Hardy spaces with v ariable exponents and its application to bounded\nlinear operators, Integral Equations Operator Theory 77 (2 013), no. 1, 123–148.\n[34] Y. Sawano, A note on Besov-Morrey spaces and Triebel-Li zorkin-Morrey spaces, Acta Math. Sin. (Engl.\nSer.) 25 (2009), no. 8, 1223–1242.\n[35] Y. Sawano, Theory of Besov spaces, Developments in Math ematics, 56. Springer, Singapore, 2018.\n[36] Y. Sawano and S. R. El-Shabrawy, Weak Morrey spaces with applications, Math. Nachr. 291 (2018), no.\n1, 178–186.\n[37] Y. Sawano, G. Di Fazio and D. Hakim, Introduction and App lications to Integral Operators and PDE’s,\nVolume I, (Chapman & Hall/CRC Monographs and Research Notes in Mathematics).\n[38] S. L. Sobolev (1963), On a theorem of functional analysi s, Transl. Amer. Math. Soc., American Mathe-\nmatical Society Translations: Series 2, 34 (2): 39–68, doi: 10.1090/trans2/034/02, ISBN 9780821817346;\ntranslation of Mat. Sb., 4 (1938) pp. 471–497.\n(Naoya Hatano) Department of Mathematics, Chuo University, 1-13-27, Kasug a, Bunkyo-ku, Tokyo\n112-8551, Japan\nEmail address , Naoya Hatano: n.hatano.chuo@gmail.com"    },    {        "title": "1310.6386v1.Landau_damping_in_a_collisionless_dipolar_Bose_gas.pdf",        "content": "arXiv:1310.6386v1  [cond-mat.quant-gas]  23 Oct 2013Landau damping in a collisionless dipolar Bose gas\nStefan S. Natu∗\nCondensed Matter Theory Center, National Institute of Stan dards and Technology and Joint Quantum Institute,\nDepartment of Physics, University of Maryland, College Par k, Maryland 20742-4111 USA\nRyan M. Wilson\nJoint Quantum Institute, National Institute of Standards a nd Technology\nand University of Maryland, College Park, Maryland 20742-4 111 USA\nWe present a theory for the Landau damping of low energy quasi -particles in a collisionless,\nquasi-2D dipolar Bose gas and produce expressions for the da mping rate in uniform and non-\nuniform systems. Using simple energy-momentum conservati on arguments, we show that in the\nhomogeneous system, the nature of the low energy dispersion in a dipolar Bose gas severely inhibits\nLandau damping of long wave-length excitations. For a gas wi th contact and dipolar interactions,\nthe damping rate for phonons tends to decrease with increasing dipolar interactions; for strong\ndipole-dipole interactions, phonons are virtually undamp ed over a broad range of temperature. The\ndamping rate for maxon-roton excitations is found to be sign ificantly larger than the damping rate\nfor phonons.\nI. INTRODUCTION\nThe trapping and cooling of highly magnetic atoms\nsuch as52Cr,162Dy,164Dy and168Er [1–4] has al-\nlowed experimentalists to create Bose-Einstein conden-\nsates with strong dipole-dipole interactions [5]. The long\nrange character of these interactions leads to phenom-\nena such as geometry dependent mechanical stability\n[6], anisotropic superfluid velocity [7], novel topological\ndefects and vortex lattice configurations upon rotation\n[8, 9]. A key ingredient responsible for some of the new\nphysics of dipolar Bose gases is the low energy excita-\ntion spectrum, which has a phonon-roton character in\nreduced dimensions [10–12], resembling that of super-\nfluid He–4 [13–15]. Here we present a theory to describe\nthe damping of these excitations at finite temperature,\nwhere the damping is provided by interactions between\nlow lying excitations, mediated by the underlying Bose\ncondensate.\nThe dynamics of Bose condensed gases with short-\nrange interactions is well studied [16, 18–29] at zero and\nfinite temperature, in homogeneous as well as trapped\ngases. In the homogeneous case, there is a continuum\nof excitations characterized by the familiar Bogoliubov\ndispersion, which contains linearly dispersing phonons at\nenergies smaller than the chemical potential µ, and free-\nparticle-like excitations at energies larger than µ[30]. At\nlowtemperaturesandlowdensities,suchasthoserealized\nin cold atomic gases, the mean free path of the excita-\ntions is comparable to the system size and the system\nenters the “collisionless” regime, where the characteris-\ntic excitation frequency is much larger than the typical\nrelaxation rate due to collisions ωτ≫1. In this limit,\ninteractions between quasi-particles are primarily medi-\n∗Electronic address: snatu@umd.eduated by the Bose condensate. A given excitation can\ndecay into two excitations with lower energy, a process\nknown as Beliaev damping [16], which dominates at low\ntemperatures kBT≪µ[31]. At higher temperatures,\nan excitation can also decay by near resonant coupling\nto transitions between other elementary excitations, a\nmechanism known as Landau damping. In the context of\nthe dilute Bose gas, Landau damping was first discussed\nin the low temperature limit by Hohenberg and Martin\n[17], and at higher temperatures by Sz´ epfalusy and Kon-\ndor [18]. A systematic theory of Landau damping in a\ndilute Bose gas covering both temperature regimes was\nlater provided by Pitaevskii and Stringari [25]. In the\nopposite limit, where the typical collision time is small\nωτ≪1, the damping of collective excitations is governed\nby two-fluid hydrodynamics [32].\nIn trapped systems, the spectrum of low lying modes\nis discrete, and the frequencies obtained experimentally\nmatch theoretical predictions based on superfluid hy-\ndrodynamics [33–36]. At higher energies and temper-\natures, the dynamics are collision dominated, and the\ntemperature dependence of the excitation frequencies\nand the associated damping rates has been studied in\nRefs. [19, 27, 28, 37]. At lower energies, the discretiza-\ntionofthe collectivemodesforbidsBeliaevdampingalto-\ngether, and the temperature dependence of the damping\nrates observed experimentally [36, 37] has been found to\nbe consistent with Landau damping [21, 22, 24, 26].\nIn recent years, the focus has shifted to understand-\ning the properties of Bose condensed gases with long\nrangeinteractions. At lowtemperatures, the dynamicsof\nsuch a condensate is well described by a non-local Gross\nPitaevskii equation, which was successfully employed in\ndescribing the anisotropic collapse of a dipolar52Cr con-\ndensate [6]. The spectrum of excitations in the dipolar\ngas has been established in homogeneous and harmoni-\ncally confined systems [10–12, 38–40].\nIn this work, we study the Landau damping of ex-\ncitations in a quasi-2D dipolar Bose gas. We follow2\nthe time-dependent mean-field formalism developed by\nGiorgini [20, 21], which describes the coupled dynamics\nof the condensed and non-condensed clouds interacting\nwith short range interactions. Generalizing this formal-\nismtoincludelongrangeinteractions,weobtaintheLan-\ndau damping rate in a dipolar gas. We remark that the\nzero temperature versionof this theory yields the Beliaev\ndamping rate, which is studied in detail in Refs. [41, 42].\nWe find that the Landau damping of low energy, long\nwave-length excitations is exponentially suppressed at\nlow temperatures. This is in departure from the T2de-\npendence of the damping rate expected in 2D systems\nwith short-range interactions [23]. Surprisingly we find\nthat the phonon damping rate decreases with increas-\ning dipolar interactions, and for sufficiently large dipolar\ninteractions, phonons are virtually undamped at finite\ntemperature. By contrast, the damping rate for roton-\nmaxonexcitationsissignificantlyhigherthan the phonon\ndamping rate.\nThispaperisorganizedasfollows: InSecIIwedescribe\nthe time-dependent mean-field scheme used to compute\nthe damping rates, and obtain formulas for the Landau\ndampingrateinhomogeneousandtrappedsystemsinthe\npresenceoflongrangeinteractions. In SecIII,wespecial-\nize to the case of a homogeneous, quasi-2D dipolar gas,\nand compute the Landau damping rate for phonons at\nlow and high temperatures. To highlight the role of the\nlow energy dispersion in determining the damping rates,\nwecomparethephonondampingrateforapurelydipolar\ngas with that of a gas with purely contact interactions.\nWe then consider the case where both dipolar and con-\ntact interactions are present. In Sec IV we numerically\ncalculate the Landau damping rate at intermediate mo-\nmenta, where the low energy dispersion relation develops\na maxon-roton feature. We discuss the experimental im-\nplications of our work in Section V, and summarize our\nresults in Section VI.\nII. FORMALISM\nThe Hamiltonian for a Bose gas interacting with a po-\ntentialVtot(r−r′) is given by:\nH=/integraldisplay\ndrΨ†(r,t)/bracketleftBig\n−/planckover2pi12∇2\n2m−µ(r)/bracketrightBig\nΨ(r,t)+ (1)\n1\n2/integraldisplay\ndrdr′Vtot(r−r′)Ψ†(r,t)Ψ†(r′,t)Ψ(r′,t)Ψ(r,t)\nwhere Ψ( r,t) is the bosonic annihilation operator at po-\nsitionrand timet,mis the mass, and µ(r) is the\nspatially varying chemical potential. In the presence\nof a trapping potential U(r),µ(r) =µ0−U(r), where\nµ0is the chemical potential in the center of the trap.\nThe interaction potential is assumed to be a sum of\ntwo termsVtot(r−r′) =gδ(r−r′) +Vlr(r−r′) where\ng=4π/planckover2pi12a\nmparametrizes the contact (short-range) part\nof the potential with a 3D s-wave scattering length a,and all the long range terms are included in Vlr(r−r′).\nThroughout we will assume that the potential satisfies\nVtot(r) =Vtot(−r). For dipolar bosons with dipole mo-\nmentsd, that are oriented along the zaxis [5]:\nVlr(r−r′) =d21−3cos2(θ)\n|r−r′|3(2)\nwhereθis the angle between the vector r−r′and thez\naxis.\nThe equation of motion for Ψ is obtained using the\nHeisenberg prescription:\ni/planckover2pi1∂tΨ(r,t) =/bracketleftBigg/parenleftBig\n−/planckover2pi12∇2\n2m−µ(r)/parenrightBig\n+/integraldisplay\ndr′Vtot(r−r′)×(3)\n|Ψ(r′,t)|2/bracketrightBigg\nΨ(r,t).\nAt temperatures below the Bose-condensation tempera-\nture, the bosonic annihilation operator can be decom-\nposed as a sum of two parts: Ψ( r,t) =φ(r,t) +ψ(r,t)\nwhereφ(r,t) =/an}b∇acketle{tΨ(r,t)/an}b∇acket∇i}htis a complex order parameter\nrepresenting the condensate, and ψ(r,t) represents the\nnon-condensed atoms, which by definition have the prop-\nerty/an}b∇acketle{tψ(r,t)/an}b∇acket∇i}ht= 0. The averages here in general denote\nnon-equilibrium averages, as we allow for fluctuations of\nthe condensed and non-condensed atoms. Equilibrium\naverages will be denoted by /an}b∇acketle{t../an}b∇acket∇i}ht0.\nDecomposingthe term proportionalto Ψ†Ψ†Ψ in Eq. 3\ninto condensate and non-condensate contributions, the\nequation of motion for the condensate field reads:\ni/planckover2pi1∂tφ(r,t) =−/parenleftBig/planckover2pi12∇2\n2m+µ(r)/parenrightBig\nφ(r,t)+/integraldisplay\ndr′Vtot(r−r′)(4)\n×/bracketleftBig\n|φ(r′,t)|2φ(r,t)+φ(r′,t)n(r′,r,t)+\nφ∗(r′,t)m(r′,r,t)+n(r′,t)φ(r,t)/bracketrightBig\nwhere we have introduced normal and anomalous densi-\nties\nn(r′,r,t) =/an}b∇acketle{tψ†(r′,t)ψ(r,t)/an}b∇acket∇i}ht (5)\nm(r′,r,t) =/an}b∇acketle{tψ(r′,t)ψ(r,t)/an}b∇acket∇i}ht\nand we use the short-hand notation n(r,t) =\n/an}b∇acketle{tψ†(r,t)ψ(r,t)/an}b∇acket∇i}htandm(r,t) =/an}b∇acketle{tψ(r,t)ψ(r,t)/an}b∇acket∇i}ht. Setting\nnandmto zero in Eq. 4, one obtains a non-local Gross-\nPitaevskii equation (GPE), which describes the dynam-\nics of a Bose condensate at zero temperature interacting\nwith long range interactions [5, 6].\nIn general we are interested in small, time-dependent\nfluctuations of the condensate about its equilibrium\nvalue, which is obtained by solving the GPE in equilib-\nrium. To describe the fluctuations, we expand φ(r,t) =\nφ0(r)+δφ(r,t), whereφ0(r)istheequilibriumcondensate\ndensity, which we assume without any loss of generality3\nto be real [43], and δφ(r,t) corresponds to small, time-\ndependent fluctuations about the stationary state. At fi-\nnite temperature, the fluctuations of the condensate are\ncoupled to fluctuations of the non-condensate densities,\nwhichcanbe writtenin the form: n(r,r′,t) =neq(r,r′)+\nδn(r,r′,t) andm(r,r′,t) =δm(r,r′,t). Here we assume\nthat the anomalous density of the thermal cloud is zero\nin equilibrium ( meq(r,t) =/an}b∇acketle{tψ(r,t)ψ(r,t)/an}b∇acket∇i}ht0= 0), an ap-\nproximation due to Popov [31]. As discussed by Giorgini\n[20, 21], this approximation yields an inadequate result\nfor therealpart of the phonon frequency shift, but leaves\nthe imaginary part (which contributes to damping) un-\nchanged. A detailed description of the Popov approxi-\nmation and its implications can be found in Ref. [32].\nLinearizing Eq. 4 in the Popov approximation, the\nequation of motion for the fluctuations of the condensate\nreads:\ni∂tδφ(r,t) =ˆL0δφ(r,t)+/integraldisplay\ndr′Vtot(r−r′)×(6)\n/bracketleftBig\nφ0(r)φ0(r′)(δφ∗(r′,t)+δφ(r′,t))+\nn0(r′,r)δφ(r′,t)+φ0(r′)(δn(r′,r,t)+\nδm(r′,r,t))+φ0(r)δn(r′,t)/bracketrightBig\nwhere we define the operator\nˆL0≡ −/planckover2pi12∇2\n2m−µ(r)+/integraldisplay\ndr′Vtot(r−r′)ntot(r′) (7)\nandntot(r) =|φ0(r)|2+neq(r) is the total density. In\ngeneral we define ntot(r′,r) =φ0(r′)φ0(r)+neq(r′,r).\nThe corresponding Hamiltonian for the non-condensed\natoms can be decomposed into two parts:\nHnc=H0+Hδ (8)\nwhere\nH0=/integraldisplay\ndrdr′/braceleftBig\nψ†(r,t)ˆL0ψ(r,t)+Vtot(r−r′)ntot(r′,r) (9)\n×/bracketleftBig\nψ†(r′,t)ψ(r,t)+ψ†(r,t)ψ(r′,t)/bracketrightBig\n+\nφ0(r)φ0(r′)×/bracketleftBig\nψ†(r′,t)ψ†(r,t)+ψ(r,t)ψ(r′,t)/bracketrightBig/bracerightBig\nin Popov approximation. The Hamiltonian H0couples\nthe non-condensed atoms to the equilibrium condensate\nand non-condensate densities, and does not contain any\ncondensate fluctuations.\nThe latter are described by the Hamiltonian Hδ, con-\ntaining terms linear in δφ:\nHδ=1\n2/integraldisplay\ndrdr′Vtot(r−r′)/bracketleftBig\nφ0(r)/braceleftBig\n(δφ(r,t)+ (10)\nδφ∗(r,t))×ψ†(r′,t)ψ(r′,t)+δφ(r′,t)×\nψ†(r′,t)ψ(r,t)+δφ∗(r′,t)ψ†(r,t)ψ(r′,t)+\nδφ(r′,t)ψ†(r,t)ψ†(r′,t)+δφ∗(r′,t)×\nψ(r′,t)ψ(r,t)/bracerightBig\n+r↔r′/bracketrightBigwherer→r′above refers to interchanging randr′for\neveryterm in the preceding curly brackets. In the weakly\ninteracting gas, the condensate density greatly exceeds\nthe density of the non-condensed atoms ( |φ0|2≫neq).\nAs a result, interactions between the non-condensed\natomsaremainlymediated bythe condensate, and there-\nfore we have ignored terms involving only non-condensed\natoms (all terms in Hδcontaining four non-condensed\nfield operators).\nThe Hamiltonian H0given by Eq. 9 can be diagonal-\nized via the usual Bogoliubov transformation:\nψ(r,t) =/summationdisplay\niui(r)ai(t)+v∗\ni(r)a†\ni(t) (11)\nψ†(r,t) =/summationdisplay\niu∗\ni(r)a†\ni(t)+vi(r)ai(t)\nwhereai(t) denotes the bosonic quasi-particle annihila-\ntion operator at time t, and satisfies the usual bosonic\ncommutation relations [30]. The complex functions ui\nandviobey/integraltext\ndrdr′(u∗\ni(r)uj(r′)−v∗\ni(r)vj(r′)) =δijδ(r−\nr′) whereδijis the Kronecker delta symbol.\nThe resulting Hamiltonian describes a non-interacting\ngas of Bogoliubov quasi-particles:\nH0=/summationdisplay\niEia†\niai (12)\nwhereEiarethe quasi-particleenergiesobtained by solv-\ning the Bogoliubov equations:\nEiui(r) =ˆL[ui(r)]+ˆM[vi(r)] (13)\n−Eivi(r) =ˆL[vi(r)]+ˆM[ui(r)]\nwhere we define\nˆL[f(r)] =ˆL0f(r)+/integraldisplay\ndr′Vtot(r−r′)ntot(r′,r)f(r′)(14)\nˆM[f(r)] =/integraldisplay\ndr′Vtot(r−r′)φ0(r)φ0(r′)f(r′).\nA key difference between the Bogoliubov equations\nabove and those for a gas interacting with purely con-\ntact interactions arises in the non-local term in the defi-\nnition of ˆL[f(r)] andˆM[f(r)]. This term represents the\n“exchange” (Fock) contribution to the interaction. For a\ngas with short-range interactions, this term is identical\nto the “direct” (Hartree) contribution, and together they\ngive rise to a factor of 2 in the interaction term. In the\npresence of long range interactions, these contributions\nmust be treated separately, and the problem of solving\nthe Bogoliubov equations becomes computationally in-\ntensive [44–46].\nThe equation for the fluctuations of the condensate\nEq. 6 can be expressed in terms of the Bogoliubov oper-\natorsaianda†\niby expanding the normal and anomalous\ndensities as follows:\nn(r′,r,t) =/summationdisplay\niju∗\ni(r′)uj(r)fij+vi(r′)v∗\nj(r)fji+ (15)\nu∗\ni(r′)v∗\nj(r)g∗\nij+vi(r′)uj(r)gij+vi(r′)v∗\nj(r)4\nand\nm(r′,r,t) =/summationdisplay\nijui(r′)v∗\nj(r)fji+v∗\ni(r′)uj(r)fij+ (16)\nui(r′)uj(r)gij+v∗\ni(r′)v∗\nj(r)g∗\nij+ui(r′)v∗\nj(r),\nwhere it is convenient to define the functions:\nfij(t)≡ /an}b∇acketle{ta†\ni(t)aj(t)/an}b∇acket∇i}ht (17)\ngij(t)≡ /an}b∇acketle{tai(t)aj(t)/an}b∇acket∇i}ht.\nThe equation for the fluctuations of the condensate(Eq. 6) is thus coupled to the dynamics of the non-\ncondensed atoms via fijandgij. Physically, fijcorre-\nspondstothecreationandannihilationofaquasi-particle\nwith energy EiandEj, andcontributestoLandaudamp-\ning, whilegij(g∗\nij) corresponds to the annihilation (cre-\nation) of two quasi-particles with energies EiandEj,\nand contributes to Beliaev damping [20]. Henceforth we\nfocus on the dynamics of fijalone.\nThe equation for the fluctuations of the condensate\n(Eq. 6) can be written in terms of fij(t) as:\ni/planckover2pi1∂tδφ(r,t) =ˆL[δφ(r,t)]+ˆM[δφ∗(r)]+/summationdisplay\nijfij(t)/integraldisplay\ndr′Vtot(r−r′)/parenleftBig\nφ0(r′)/bracketleftBig\nu∗\ni(r′)uj(r)+vj(r′)v∗\ni(r)+ (18)\nv∗\ni(r)uj(r′)+v∗\ni(r′)uj(r)/bracketrightBig\n+φ0(r)/bracketleftBig\nu∗\ni(r′)uj(r′)+vi(r′)v∗\nj(r′)/bracketrightBig/parenrightBig\n+...\nThe...at the end of Eq. 18 refers to the terms propor-\ntional togijthat give rise to Beliaev damping, which we\ndo not study here.Using Heisenberg’s equation of motion, i/planckover2pi1∂t/an}b∇acketle{tˆO/an}b∇acket∇i}ht=\n/an}b∇acketle{t[O,Hnc]/an}b∇acket∇i}ht, the equation of motion for fijreads:\ni/planckover2pi1∂tfij(t) = (ǫj−ǫi)fij(t)+(nth\ni−nth\nj)\n2/integraldisplay\ndrdr′Vtot(r−r′)/bracketleftBig\n2φ0(r)(δφ(r,t)+δφ∗(r,t))×{u∗\nj(r′)ui(r′)+(19)\nv∗\ni(r′)vj(r′)}+2(φ0(r)δφ(r′,t)+φ0(r′)δφ∗(r,t)){u∗\nj(r′)ui(r)+vj(r′)v∗\ni(r)}+(φ0(r)δφ(r′,t)+φ0(r′)δφ(r,t))×\n{u∗\nj(r)vi(r′)+vi(r)u∗\nj(r′)}+(φ0(r)δφ∗(r′,t)+φ0(r′)δφ∗(r,t)){ui(r)v∗\nj(r′)+v∗\nj(r)ui(r′)}/bracketrightBig\nwhere we define the equilibrium density of the quasi-\nparticles at temperature T,nth\ni=/an}b∇acketle{ta†\niai/an}b∇acket∇i}ht0=1\neβEi−1\nwhereβ= 1/kBT.\nThus the fluctuations of the condensate are coupled\nto the fluctuations of the non-condensed atoms and vice\nversa. The full self-consistent solution of this system ofequations is highly non-trivial. In order to solve the cou-\npled system of equations we assume that the condensate\nfluctuates with frequency ωasδφ(r,t) =δφ1(r)e−iωtand\nδφ∗(r,t) =δφ2(r)e−iωtand take the Fourier transform of\nEq. 19 to get:\nfij(ω) =(nth\ni−nth\nj)\n2(/planckover2pi1ω−(ǫj−ǫi))/integraldisplay\ndrdr′Vtot(r−r′)/bracketleftBig\n2φ0(r)(δφ1(r)+δφ2(r))×{u∗\nj(r′)ui(r′)+ (20)\nv∗\ni(r′)vj(r′)}+2(φ0(r)δφ1(r′)+φ0(r′)δφ2(r)){u∗\nj(r′)ui(r)+vj(r′)v∗\ni(r)}+(φ0(r)δφ1(r′)+φ0(r′)δφ1(r))×\n{u∗\nj(r)vi(r′)+vi(r)u∗\nj(r′)}+(φ0(r)δφ2(r′)+φ0(r′)δφ2(r)){ui(r)v∗\nj(r′)+v∗\nj(r)ui(r′)}/bracketrightBig\n.\nIn the absence of any coupling between the condensed and non-co ndensed atoms, the fluctuations of the con-5\ndensate(Eq. 18) obeythe Bogoliubovequationswith real\nfrequencyω0:\n/planckover2pi1ω0/parenleftbigg\nδφ0\n1(r)\nδφ0\n2(r)/parenrightbigg\n=/parenleftbiggˆLˆM\n−ˆM∗−ˆL∗/parenrightbigg/parenleftbigg\nδφ0\n1(r)\nδφ0\n2(r)/parenrightbigg\n(21)\nand satisfy the normalization criterion/integraltextdr(|δφ0\n1(r)|2−\n|δφ0\n2(r)|2) = 1.\nIn the limit of small non-condensate density, we may\ntreat the condensate-non-condensed atom coupling as a\nperturbation, and write δφ1,2=δφ0\n1,2+δφ′\n1,2, where\nthe corrections to the oscillation amplitude are chosen\nto be orthogonal to those in the absence of condensate-\nnoncondensate coupling:\n/integraldisplay\ndr/parenleftBig\nδφ0\n1(r)δφ∗′\n1(r)−δφ0\n2(r)δφ∗′\n2(r)/parenrightBig\n= 0.(22)\nSimilarly, we write the perturbed eigenfrequency as\nω=ω0+δω−iΓLwhereδωis the correction to the real\npart of the normal mode eigenfrequency and Γ Ldenotes\nthe Landau damping coefficient.\nTo lowest order, the equation governing the fluctua-\ntions of the condensate in the presence of coupling to the\nnon-condensed atoms reads:\n/planckover2pi1ωδφ1(r) =ˆL[δφ0\n1(r)]+ˆM[δφ0\n2(r)]+/summationdisplay\nijfij(ω0)Aij(r)\n(23)\nwhere we have replaced all the δφandδφ∗terms in the\nRHS, includingin theexpressionfor fij(Eq.20) with δφ0\n1\nandδφ0\n2, andωwithω0. We alsointroducethe shorthand\nAij(r) =/integraldisplay\ndr′Vtot(r−r′)/parenleftBig\nφ0(r′)/bracketleftBig\nu∗\ni(r′)uj(r)+ (24)\nvj(r′)v∗\ni(r)+ui(r′)v∗\nj(r)+ui(r)v∗\nj(r′))/bracketrightBig\n+\nφ0(r)/bracketleftBig\nu∗\ni(r′)uj(r′)+vi(r′)v∗\nj(r′)/bracketrightBig/parenrightBig\n.\nTo obtain the damping rate, we multiply Eq. 23 and\nthe corresponding equation for δφ2byδφ∗\n1andδφ∗\n2re-\nspectively, take the difference of the equations and inte-\ngrate over space to find [20]:\n/planckover2pi1ω=/planckover2pi1ω0+/summationdisplay\nijfij(ω0)/integraldisplay\ndr/parenleftBig\nAij(r)δφ∗\n1(r)+A∗\nji(r)δφ∗\n2(r)/parenrightBig\n(25)\nThe damping rate, which is the focus ofthis workis given\nby the imaginary part of ω:\nΓL(ω0) =π\n/planckover2pi1/summationdisplay\nij(nth\ni−nth\nj)δ(/planckover2pi1ω0−(Ej−Ei))|Aij|2(26)\nwhere we have used Eqs. 20, 24 and 25 to express fijin\nterms ofAij, and we define the matrix element:\nAij=/integraldisplay\ndr/parenleftBig\nAij(r)δφ∗\n1(r)+A∗\nji(r)δφ∗\n2(r)/parenrightBig\n(27)The formula for the Landau damping rate given by\nEq. 26 is the main result of this section. We remind the\nreader that in deriving this result, we have not assumed\nanyparticularformofthetwo-bodyinteractionpotential,\nonly that it is symmetric with respect to the interchange\nof the two particles. Furthermore, our equations are gen-\neral and may be applied to study the Landau damping\nof collective modes in inhomogeneous systems, such as\ntrappedatomic gases.\nA. Landau damping in a homogeneous gas\nIn general, solving for the finite temperature damping\nrates of a trapped gas is numerically intensive [24, 27–\n29]. In the presence of long range interactions, this is\nfurther exacerbated by the exchange interaction, which\nisnon-localin nature. Asa result, calculatingthe Bogoli-\nubov co-efficients in a trapped dipolar gas is extremely\nchallenging, and practical only in certain simple cases\n[38, 44, 46].\nIn the rest of this work, we focus instead on a ho-\nmogeneous dipolar gas. In a homogeneous system, the\ncondensate density φ0(r) =√n0is constant throughout\nspace. Furthermore, in order to obtain semi-analytic re-\nsults, we will neglect the exchange contribution to the\nBogoliubov equations of motion arising from the non-\ncondensed atoms, and retain only the condensate con-\ntribution. In other words, we approximate ntot(r′,r)≈\nφ0(r)φ0(r′) =φ2\n0. We expect this approximation to be\nquantitatively accurate at temperatures kBT≤µ, where\nthe condensate is not significantly depleted. In this limit,\nthe homogeneous Bogoliubov equations of motion can be\nreadily solved in Fourier space, and we will use the re-\nsulting solutions in subsequent computations. For suffi-\nciently large systems, our calculation can be readily gen-\neralized to include the effects of a trap via a local density\napproximation [21].\nIn the homogeneous geometry, the fluctuations of\nthe condensate take the plane-wave form: δφ(r) =\n1√\nVe−ik·rukandδφ∗(r) =1√\nVe−ik·rvk;up=\n1√\nVupe−ip·r, andvp=1√\nVvpe−ip·r, whereukandvk\nare the Bogoliubov coherence factors which read:\nuk=/radicalBigg\n1\n2/parenleftBigǫk+Vtot(k)\nEk+1/parenrightBig\n(28)\nvk=−sgn(Vtot(k))/radicalBigg\n1\n2/parenleftBigǫk+Vtot(k)\nEk−1/parenrightBig\nwhere we denote Vtot(k) =/integraltext\ndre−ik·(r−r′)Vtot(r−r′) as\nthe Fourier transform of the two-body interaction po-\ntential. Here ǫk=/planckover2pi12k2/2mand the dispersion Ek=/radicalbig\nǫk(ǫk+2Vtot(k)n0).\nInserting the expressions for the fluctuations of the\ncondensate and the non-condensed atoms into Eqs. 24,\nand 26, and integrating over space, the formula for the6\nLandau damping rate for a quasi-particle with momen-\ntumkand energy /planckover2pi1ωkin a homogeneous system reads:\nΓL(k) =−πn0\n/planckover2pi1/summationdisplay\npq(nth\nq−nth\np)δ(/planckover2pi1ωk−Eq+Ep)×(29)\nAk\npqAk\nqpwhere\nAk\npq=δk,q−p/braceleftBig\nuk/bracketleftBig\nVtot(p)(upuq+vpuq)+Vtot(q)(vpuq+vpvq)/bracketrightBig\n+vk/bracketleftBig\nVtot(q)(upuq+vqup)+Vtot(p)(vpvq+(30)\nupvq)/bracketrightBig\n+(uk+vk)Vtot(k)(upuq+vpvq)/bracerightBig\n.\nWe denote by p,q, the momenta of the outgoing quasi-\nparticles, which are determined by the constraints of\nenergy-momentum conservation, enforced by the delta\nfunctions in Eqns. 29, 30. Here nth\nk=1\neβEk−1is the Bose\noccupation factor for a mode with energy Ek.\nB. Quasi- 2D dipolar gas\nA quasi-2D dipolar gas is created experimentally by\nemployingtightconfinementalongtheaxial( z)direction,\nU(r) =1\n2m(ω2\nxx2+ω2\nyy2+ω2\nzz2), whereωz≫ωx,ωy.\nHere we assume that the confining direction is the same\nas the direction in which the dipoles are polarized, al-\nthough our results can be readily generalized to in-\nclude an arbitrary polarization angle. In the limit of\ntight confinement, µ≪/planckover2pi1ωz, with no loss of generality,\nthe density can be expressed as n(r) =n2D(ρ)χ(z) =\n1√\nπl2zn2D(ρ)e−z2/l2\nz, whereρandzare the radial and ax-\nial coordinates respectively, and lzis a length-scale on\nthe order of the harmonic oscillator wave-length in the\naxial direction lz∼/radicalbig\n/planckover2pi1/mωz[39]. Integrating out the z-\ndirection, one obtains an effective quasi-two dimensional\ndescription, which depends on ρalone. For the homoge-\nneous case we consider here ( ωx=ωy= 0), the Fourier\ntransform of the resulting quasi-2D interaction potential\nreads [7, 39]:\nVq2D(k) =gq2D+gq2D\ndF/parenleftBigklz√\n2/parenrightBig\n(31)\nwhere we define a quasi two-dimensional interaction\nparameter gq2D=g√\n2πlz,k=/radicalBig\nk2x+k2yis the ra-\ndial momentum, gq2D\nd=8π\n3√\n2πlzd2, andF(x) = 1−\n3\n2√πxErfc(x)ex2, where Erfc( x) is the complimentary\nerror function. When the dipoles are aligned parallel to\none another, the dipole-dipole interaction only depends\non the magnitude of the radial momentum.\nThe Landau damping rate can be computed in\nquasi-2D simply by replacing Vtot(k) withVq2D(k)andn0withn2D\n0in Eqs. 29 and 30. The in-\ntegrals are now performed over two-dimensional\nk-space. The Bogoliubov eigenfrequencies and\ncoefficients are: Ek=/radicalbig\nǫk(ǫk+2Vq2D(k)n2D\n0)\nanduk=/radicalbigg\n1\n2/parenleftBig\nǫk+Vq2D(k)n2D\n0\nEk+1/parenrightBig\nandvk=\n−sgn(Vq2D(k))/radicalbigg\n1\n2/parenleftBig\nǫk+Vq2D(k)n2D\n0\nEk−1/parenrightBig\nrespectively.\nAs discussed by Fischer [39] and Santos et al.[10], the\nquasi-2D dipolar Bose gas is mechanically stable even in\nthe absence of short-range repulsive interactions ( gq2D=\n0). Below, we first consider Landau damping of phonons,\nandderiveanenergeticcriterionforLandaudampingina\npurely dipolar gas. We contrast the damping in a purely\ndipolar gas with analogous results in a gas with contact\ninteractions[18, 20, 22, 23, 25, 26]. We then calculate the\nLandau damping rate for a gas where both contact and\ndipolar interactions are present, as is the case for current\nexperiments on dipolar52Cr,162Dy,164Dy and168Er [1–\n4]. Finally, we discuss in detail the nature of Landau\ndamping for strong dipolar interactions, where the low\nenergy dispersion acquires a roton-maxon character.\nIII. LANDAU DAMPING OF PHONONS\nA. Criterion for Landau damping\nWe first consider the Landau damping of a phonon\nwith wave-vector kinto two quasi-particles with wave-\nvectorspandq. As shown by Eq. 29, Landau damping\nis only allowed provided the conditions for energy and\nmomentum conservation can be simultaneously satisfied.\nAt low energies, the dispersion of the incoming quasi-\nparticle takes the form Ek≈/planckover2pi1ck, wherecis the phonon\nvelocity in the gas, which will be defined subsequently.\nMomentum conservation (enforced by the delta func-\ntion in Eq. 30) implies that q=k+p, which can be\nexpanded as q=|q| ≈p+kcos(θ) in the long wave-\nlength limit /planckover2pi1k≪mc. Hereθis the angle between the\nvectorspwithk. Likewise, Eq−Ep≈/planckover2pi1vg(p)kcos(θ)7\nwherevg(p) = 1//planckover2pi1∂Ep/∂pis the group velocity of the\nexcitations [25]. Energetically, Landau damping can oc-\ncur only if\nc\nvg(p)≤1, (32)\ni.ethe velocity of the incoming quasi-particle has to be\nless than the group velocity of the decaying excitations.\nWe note that this criterion is not limited to damping\nof phonons in Bose condensates; rather it is completely\ngeneral, and also applies to Landau damping in Fermi\nliquids or plasmas [47].\nAt large values of /planckover2pi1p≫mc, the dispersion for both\ndipolar and contact interactions becomes free-particle-\nlike,Ep→/planckover2pi12p2/2mand thusvg(p)→/planckover2pi1p/m. Asp→\n∞, Eq. 32 is always satisfied, and Landau damping can\noccur.\nHowever, at small quasi-particle wave-vectors, the dis-\npersion of the dipolar gas is dramatically different from a\ngas with contact interactions. For a gas with purely con-\ntact interactions, the dispersion takes the form: Ep//planckover2pi1≈\ncp+/planckover2pi12p3\n8m2c, wherec=/radicalbig\ngq2Dn2D\n0/mis the phonon ve-\nlocity. Thus the criterion for energy conservation as\np/mc→0 reads:c/vg∼1− O(p2)<1 and Landau\ndamping is allowed. By contrast, the dipolar dispersion\nEpis given by:\nEp//planckover2pi1≈cdp−3\n4/radicalbiggπ\n2lzcdp2+/planckover2pi12p3\n8m2cd(33)\nwhere we have introduced a speed of sound associated\nwith the dipolar interaction cd=/radicalBig\ngq2D\ndn2D\n0/m. Asp→\n0,cd/vg∼1 +O(p)>1, and Landau damping is thus\nforbidden by energy-momentum conservation.Landau damping becomes possible when the cubic\nterm in the expansion of Eq. 33 dominates over the\nquadratic term. Equating the quadratic and cubic terms\ninEq.33, wefindthatLandaudampingoccursonlywhen\npexceeds a critical value:\np>pcrit=2√\n2πlzm2c2\nd\n/planckover2pi12=2\nlzEint\nEho(34)\nwhereEint=gq2D\ndn2D\n0is the mean-field interaction en-\nergy andEho=/planckover2pi12/ml2\nz=/planckover2pi1ωzis the axial harmonic\noscillator energy. In other words, a phonon with wave-\nvectorkcan only Landau damp into excitations with\nwave-vectors pandp+kforp>pcrit.\nB. Temperature dependence of damping rate in a\npurely dipolar gas\nTo obtain the temperature dependence of the phonon\ndamping rate, we expand Eq. 30 for small values of k.\nIntroducing dimensionless quantities for the interaction\nstrength, momentum and temperature, ˜ cd=/radicalbig\nEint/Eho,\n˜k=klz/√\n2, andτ=kBT/Ehorespectively, the Landau\ndamping ratefora phonon with momentum kand energy\nEk=/planckover2pi1cdkin a quasi-2D dipolar gas reads:\n/planckover2pi1ΓL\nEk=˜cd\n4πn2D\n0l2zI(τ) (35)\nwhere the integral I(τ) is given by:\nI(τ) =1\nτ/integraldisplay∞\n√\n2˜c2\nddx xcot(θc)1\n(eu/2τ−e−u/2τ)2×/parenleftBigx2+˜c2\ndF(x)\nu(1+F(x))−˜c2\nd\nuF2(x)sgn(F(x))+ (36)\n˜c4\ndF(x)\nv(x)sgn(F(x))/bracketleftBig−F(x)v(x)+u∂xF(x)\nu/radicalbig\nu2+˜c4\ndF(x)2/bracketrightBig/parenrightBig2\nwhereF(x) is defined in Eq.31, u=/radicalbig\nx2(x2+2˜c2\ndF(x)),\nv(x) =∂u/∂xandθc=√\n2˜cd/v(x). Foragaswithpurely\ncontact interactions, we simply set F(x) = 1, and ˜cd→˜c\nwhere ˜cis similarly defined, and the integral over xruns\nfrom 0 to ∞.\nIn Fig. 1 we plot the integral I(τ) obtained by numer-\nically integrating Eq. 36 at fixed ˜ cd. The corresponding\ndamping rate for a gas with purely contact interactions\n(˜c= 1) is also shown for comparison. The phonon damp-\ningrateinapurelydipolargasremainsconsistentlylower\nthan that for a gas with contact interactions at all tem-peratures. At high temperature ( τ >1), the damping\nrates appear to scale linearly with temperature in both\ncases, while at low temperature ( τ≪1), the damping\nrate in a dipolar gas drops significantly faster than the\ndamping rate for a gas with contact interactions. In par-\nticular, the dipolar damping rate is found to scale as\ne−1/τforτ≪1. This is in stark contrast with the cor-\nresponding damping rate in a gas with purely contact\ninteractions, which scales as T4in 3D [18, 20, 25, 26] and\nT2in 2D [23].8\n00.40.81.2012\nΤ/ScriptCapI/LParen1Τ/RParen1 00.10.200.060.12\nΤ/ScriptCapI/LParen1Τ/RParen1\nFIG. 1: The integral I(τ) (Eq. 36) plotted as a function of\nthe reduced temperature τ=kBT/Ehoin quasi-2D. The solid\nline is for apurelydipolar gas while the dashedline is for ag as\nwith contact interactions. We choose the dimensionless spe ed\nof sound for the dipolar and contact interactions ˜ cd= ˜c= 1\nto enable comparison between the two cases. At high temper-\natures the damping rates scales linearly with Tin both cases,\nwith the dipolar damping rate being consistently smaller th an\nthat for a gas with contact interactions. Inset is a zoom-in\nat low temperatures showing the dramatic suppression of the\ndipolar damping rate compared to that in a gas with contact\ninteractions.\nAs discussed previously, for a dipolar gas, modes with\nenergy less than Ecrit=/planckover2pi12p2\ncrit/2mdo not contribute to\nLandau damping. At low temperatures kBT≪Ecrit,\nthe occupation of modes with energy E > E critscales\nasnth\nk>kcrit∼e−βEk, and subsequently the damping is\nsuppressed. As shown in the inset of Fig. 1, the low\ntemperature damping rate in a purely dipolar gas grows\nmuch more slowly than the T2scaling for a gas with con-\ntact interactions [23]. Consequently, phonons in a purely\ndipolar gas are virtually undamped at low temperatures.\nAt high temperatures, kBT≫Ecrit, the damping rate\nscales linearly with temperature as in the purely contact\ncase, howeverthe magnitude of the damping rate at fixed\n˜cdis smaller than that of a gas with the purely contact\ninteractions, as phonons cannot damp into excitations\nwith momentum less than pcrit.\nC. Landau damping with contact and dipolar\ninteractions\nIn Fig. 2 we show the temperature dependence of the\nLandau damping rate for phonons in a gas with con-\ntact and dipolar interactions. Once again the damping\nrate can be expressed as/planckover2pi1ΓL\nEk=˜c\n4πn2D\n0l2zIf(τ), where\n˜c=/radicalbig\ngq2Dn2D\n0/Ehoparameterizes the contact part of\nthe interaction, Ek=/planckover2pi1ck, and the integrand If(τ) con-\ntains the temperature dependence. The full expression\nforIf(τ) is rather cumbersome, and is not shown here.\nWe also introduce a dimensionless ratio ˜ g=gq2D\nd/gq2D,\nwhich parametrizes the relative strength of the dipole-\ndipole and the contact interaction.\nInFig.2, ˜cisheldfixedandwevary ˜ g. Thecorrespond-00.40.81.200.511.5\nΤ/ScriptCapIf/LParen1Τ/RParen1\n00.40.81.20123\nk/LParen1unitsof 2/Slash1lz/RParen1E/LParen1k/RParen1/Slash1Ekin\nFIG. 2: (Top Plot) Temperature dependence of the phonon\ndamping rate plotted versus reduced temperature τ=\nkBT/Ehoin a quasi-2D gas with dipolar and contact inter-\nactions present. The contact interaction is held fixed (˜ c= 1)\nin all the curves and the dimensionless ratio ˜ g=gq2D\nd/gq2D\nis varied. From top to bottom: (thin dashed) ˜ g= 0.1, (thick,\nsolid) ˜g= 1, (thick, dashed) ˜ g= 2.5, (thick dotted) ˜ g= 5.\n(Bottom Plot) The energy momentum dispersion correspond-\ning to the same values of ˜ gand ˜cas in the top figure.\ning energy-momentum dispersion, normalized to Eho, is\nshown on the bottom. Somewhat counterintuitively, we\nfind that at fixed temperature, increasing the dipolar in-\nteraction decreases the phonon damping rate. As the\ndipolar interaction strength is increased, pcritincreases,\nand the number of available modes for Landau damping\ndecreases. When the interactions become large enough\nsuch thatEcrit=/planckover2pi12p2\ncrit/2m≫kBT, the damping rate\nis strongly suppressed as the available modes for Landau\ndampingarelargelyunoccupied. Forstrongdipole-dipole\ninteractions,phononsarevirtuallyundampedatlowtem-\nperatures. At higher temperatures τ∼1, the damping\nrate again scales linearly with T.\nBefore discussing the damping of the roton-maxon ex-\ncitations, we briefly comment on the validity of the re-\nsults presented in this section. Strictly speaking, the\nquasi-2D ansatz we employ here is only valid provided\nthatη=m(gq2D+gq2D\nd)/4π/planckover2pi12≪1. As the interaction\nstrength is increased, the Gaussian ansatz for the axial\nwave-function needs to be modified, for example by re-\nplacing the width of the Gaussian lzwith a variational\nparameter that depends on the interaction strength [48].\nWhile this will quantitatively change our results, it will\nnot affect the qualitative physics. We also expect our\nresults to hold as long as the temperature T≤µ. At\nhigher temperatures and strong interactions, the conden-\nsate becomes significantly depleted, our starting point of\ntreating the coupling between the condensed and non-\ncondensate atoms as a weak perturbation to Bogoliubov9\n00.40.81.200.40.8\nΤ/CapGammaL/LParen1Τ/RParen1/Slash1/CapGamma0\n00.40.81.21.6200.511.5\nk/LParen1unitsof 2/Slash1lz/RParen1E/LParen1k/RParen1/Slash1Ekin\nFIG. 3: (Top Plot) Temperature dependence of the Landau\ndamping rate at three different momenta corresponding to\nthe phonon (dotted), maxon (dashed) and roton (solid) for\na quasi-2D gas with contact and dipolar interactions. The\ndamping rate is normalized to /planckover2pi1Γ0=/radicalbig\n2/πEho˜c2(a/lz). The\ncontact interaction pararmetrized by ˜ c=/radicalbig\ngq2Dn2D\n0/Eho= 1\nand the dimensionless ratio ˜ g=gq2D\nd/gq2D= 7.2. (Bottom\nPlot) The dispersion for ˜ c= 1 and ˜ g= 7.2 plotted versus k.\nThe solid, dashed and dotted circles correspond to the value s\nof the momenta used in the Top plot.\ntheory is no longer valid. Moreover, exchange interac-\ntions between non-condensed atoms alter the Bogoliubov\nequations of motion. A systematic understanding of how\nthese effects modify the physics of the dipolar gas is cur-\nrently being developed [49].\nIV. LANDAU DAMPING IN THE\nMAXON-ROTON REGIME\nThus far we have focused on the damping of low en-\nergy phonons in a quasi-2D dipolar gas. One of the\nnovel features of a dipolar gas is that for sufficiently\nstrong dipole-dipole interactions, the dispersion develops\na roton-maxon character [10, 39, 51] at intermediate mo-\nmenta. We now discuss the Landau damping of these\nexcitations.\nIn Fig. 3 we plot the Landau damping rate at interme-\ndiate momenta for strong dipole-dipole interactions. The\ndamping rate is normalized to /planckover2pi1Γ0=/radicalbig\n2/πEho˜c2(a/lz).\nThe curves are obtained by numerically integrating\nEqn. 29 using Eq. 30. As before, we fix the contact part\nofthe interaction(˜ c= 1) and varythe ratio ˜ g. The corre-\nsponding phonon damping rate for the same value of ˜ gis\nshown for comparison. At the value of ˜ gchosen here, the\ncriticalmomentum atwhich Landau dampingofphonons\nisallowedis pcrit∼2/planckover2pi1√\n2/lz(Eq.34). Phonondampingis\ntherefore highly suppressed at strong dipole-dipole inter-00.10.20.30.40.5012345\nΤ/CapGammaL/LParen1Τ/RParen1/Slash1/CapGamma0\nFIG. 4: Temperature dependenceof theLandau dampingrate\nat ˜g= 7.2 andk= 1.45 (corresponding to the roton minimum\non the plot in Fig. 3 (normalized to /planckover2pi1Γ0=/radicalbig\n2/πEho˜c2(a/lz))\nfor a quasi-2D gas with contact and dipolar interactions. Th e\ndashed curve shows the damping rate for the same value of\nmomentum, but with no dipolar interactions ˜ g= 0. For our\nparameters, the roton instability (where the roton minimum\nis at zero energy) occurs at ˜ g= 7.5.\nactions as virtually none of the modes that are available\nfor Landau damping are thermally occupied.\nTo understand the damping rate at intermediate mo-\nmenta, first we show that rotons cannot scatter off of\nphonons. To see this, consider an incoming roton with\nmomentum k, scattering off of a phonon with momen-\ntump. The criterion for Landau damping now reads\nc/vr\ng(k)≤1 wherevr\ng(k) is the group velocity of the ro-\ntons. Near the roton minimum, the excitations are free-\nparticle like with a small gap, and the group velocity\nscales as |k−kr|wherekrcorresponds to the momentum\nat the roton minimum. As k→kr,vr\ng→0, and Landau\ndamping is thus forbidden.\nSolving the energy conservation criterion numerically,\nwe find that Landau damping of rotons turns on at in-\ntermediate momenta, pr\ncrit≥0.5√\n2/planckover2pi1/lz, which is consis-\ntent with the location of the maxon peak in the excita-\ntion spectrum. This critical momentum is considerably\nsmaller than pcritfor phonon damping, and as a result,\nat any given temperature, a larger number of modes con-\ntribute to the Landau damping of rotons. Hence, the\nrotondampingrateissignificantlyhigherthan the damp-\ning rate for phonons. The damping rate for the maxon\nmode lies between the damping rates for the phonon and\nroton modes.\nFinally, in Fig. 4, we compare the damping rate at\nfixed momentum for a gas with strong and weak dipole-\ndipole interactions. Recall that in the long wave-length\nlimit, we found that the damping rate decreases with\nincreasing dipole-dipole interactions over the tempera-\nture range 0 < τ/lessorsimilar1 (see Fig. 2). By contrast, here\nwe find that the damping rate at intermediate momenta\nincreases with increasing dipolar interactions, except at\nverylow temperatures. At very lowtemperatures, modes\nwith energies ∼Epr\ncritare unoccupied, and the rotons are\nundamped. At intermediate temperatures, the damping\nrate grows linearly with Tbut the slope is much larger\nfor the rotons (solid curve) than for the excitations in a\nnon-dipolar gas at the same momentum (dashed curve).10\nThe softening of the roton mode Ekr→0 is associated\nwith a divergence of the Bogoliubov amplitudes ukand\nvk. As the damping rate is directly proportional to the\nBogoliubov amplitudes, via the matrix elements Akqin\nEq. 30, the damping rate also increases near the roton\nminimum for strong dipole-dipole interactions. We note\nhowever that our approach breaks down well before the\nroton instability is reached, as the diverging Bogoliubov\namplitudes imply significant condensate depletion, and\nour assumption of ntot∼φ2\n0becomes invalid.\nBoudgemaa and Shlyapnikov [52] argue that for low\ntemperatures T≪∆r, where ∆ ris the roton gap, the\nBogoliubov approach is valid as long as η≪1. At\nhigher temperatures or stronger interactions where the\nroton gap is small T≫∆r, thermal effects begin to dom-\ninate. For the interaction parameters of Fig. 3, the roton\ngap ∆ r∼0.65Ekin(see bottom panel in Fig. 2). Some-\nwhat surprisingly, the damping rate for the roton mode\nis large even at temperatures smaller than ∆ rsuggesting\nthat roton excitations may be short-lived at finite tem-\nperatures.\nV. EXPERIMENTAL SIGNIFICANCE\nWe now briefly discuss the possibility of observing\nthe physics described above in experiments. Experi-\nmentalists can create a single 2D pancake trap by con-\nfining atoms in an optical dipole trap with frequencies\nωr∼2π×10 Hz, and ωz∼2π×103Hz [54]. For\nbosonic164Dy, which was recently Bose-condensed by\nLu and co-workers [2], this yields a transverse confine-\nment length lz=/radicalbig\n/planckover2pi1/mωz= 0.1µm. Experiments typi-\ncally operate at temperatures T∼10nK, which yields\naτ=kBT/Eho∼0.1, and corresponding 2D densi-\nties ofn2D\n0∼5×1014cm−2[54]. This implies that\n˜c=/radicalbig\ngq2Dn2D\n0/Eho∼1 can be achieved for scattering\nlengthsa/lz∼0.02≪1 [52]. Choosing a typical scatter-\ning length of a∼100aB,/planckover2pi1Γ0∼1−10Hz.\nDamping of excitations in the non-dipolar gas was ob-\nservedbyKatz et al.[53]whousedBraggspectroscopyto\nstudy the momentum dependence of the Beliaev damp-\ning rate of phonons. By tuning the energy difference and\nthe angle between the Bragg beams, experimentalists ex-\ncited quasi-particles at different energies and momenta.\nAfter a short time-of-flight, a halo ofscattered atomswas\nobserved at momenta intermediate between the conden-\nsate and the momentum of the atoms excited initially.\nBy measuring the ratio of the number of scattered atoms\nto the initial cloud of atoms excited by the probe beams,\nas a function of energy and momentum, Katz and co-\nworkers were able to extract the damping rate [53]. A\nsimilar technique may be used to detect Landau damp-\ning in the dipolar gas.\nThe energy resolution achieved in the experiment was\n∼kHz, and the momentum resolution was ∼2π/ζwhere\nζ∼0.25µm is the coherence length of the Bose con-densate. The roton mode is located at klz∼1.5, and\nhence lies above the typical momentum resolution in ex-\nperiments. In addition to spectroscopic probes, Landau\ndampingalsoinfluencesthedecayofcollectiveexcitations\nin trapped gases [35, 37].\nVI. SUMMARY AND CONCLUSIONS\nIn summary, we have developed a theory of damping\nof low energy excitations in a Bose gas interacting with\nlong range interactions, generalizing previous works on\nBose gases interacting with short range forces [20, 23].\nWe work in the collisionless limit, where the damping\nis provided by scattering of a quasi-particle with other\nexcitations, mediated by the Bose condensate. Following\nthe time-dependent mean-field approach of Giorgini [20],\nwe derive a coupled system of equations describing the\ndynamics of the condensed and non-condensed atoms,\nwithin the Popov approximation. We then solve these\nequations perturbatively in the interaction strength, to\nobtain the expression for the Landau damping rate in a\nBose gas with long range interactions.\nFocussing on the homogeneous gas and neglecting ex-\nchange interactions between non-condensed atoms, we\nfind that the nature of the low energy quasi-2D dipolar\ndispersion forbids a phonon from decaying into modes\nwith momenta lower than a certain threshold. This leads\ntoadramaticsuppressionofLandaudampingatlowtem-\nperatures. For stronger dipole-dipole interactions, we\nfind that phonons are virtually undamped over a large\ntemperature regime. By contrast, at intermediate mo-\nmenta and strong interactions, where the dispersion de-\nvelopsaroton-maxonfeature, wefindmuchhigherdamp-\ning rates.\nFinally, we remark that experiments on collective\nmodes in trapped gases typically also measure the tem-\nperature dependence of the frequency shift of the collec-\ntive modes, in addition to the damping rates [35, 37].\nThese frequency shifts are related to many body effects\nwhich go beyond the Popov approximation [15, 19, 21,\n27, 28]. A systematic study of these effects and how they\naffect the frequencies of the collective excitations in the\ndipolar gas has not yet been performed, and is a promis-\ning direction for future study [50].\nVII. ACKNOWLEDGEMENTS\nIt is a pleasure to thank Benjamin Lev, Kristian Bau-\nmann and Mingwu Lu for several discussions regarding\nthe experimental implications of the physics described\nabove. 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Phys 13075019 (2011)."    },    {        "title": "1607.07693v1.Nonminimal_Lorentz_violation.pdf",        "content": "arXiv:1607.07693v1  [hep-ph]  25 Jul 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nNonminimal Lorentz violation\nMatthew Mewes\nPhysics Department, California Polytechnic State Univers ity\nSan Luis Obispo, CA 93407, USA\nThis contribution to the CPT’16 meeting provides a brief ove rview of recent\nstudies of nonminimal Lorentz violation in the Standard-Mo del Extension.\nThe Standard-Model Extension (SME) provides a general field-th eoretic\ndescription of violations of Lorentz and CPT invariance in particle phy sics\nand in gravity and has facilitated hundreds of experimental tests o f these\nfundamentalsymmetries.1ALorentz-violatingterminthelagrangianofthe\nSME is constructed from a tensor coefficient contracted with a con ventional\ntensor operator, giving contributions to the action that schemat ically take\nthe form\nδS=/integraldisplay\nd4x(coefficient tensor) ·(tensor operator) . (1)\nThe coefficients for Lorentz violation give the vacuum a non-scalar s truc-\nture and act as Lorentz-violating background fields. The various v iolations\nare often classified according to the mass dimension dof the associated\noperator.\nEarly development of the SME focused largely on minimal violations —\nthose involving Lorentz-violating operators with renormalizable mas s di-\nmensions d= 3 and 4 — leading to the so-called minimal SME (mSME).\nPioneering works constructed the minimal modifications to the Stan dard\nModel of particle physics2followed by General Relativity.3While the\nmSME has served as the theoretical basis for a vast majority of Lo rentz\ntests performed to date,1more recent efforts aim at exploring nonminimal\nviolations associated with operators of mass dimensions d≥5. These in-\nclude the construction of the full nonminimal extensions for free p hotons,4\nneutrinos,5free Dirac fermions,6and linearized gravity.7–9\nThe study of Lorentz-violating operators of nonrenormalizable dim en-\nsions can be motivated through simple dimensional analysis. For oper atorsProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\nof dimension d, the coefficients for Lorentz violation have mass dimension\n4−d. Assuming a connection to Planck-scale physics, we might naively ex-\npect coefficients that are of order ∼M4−d\nP, whereMPis the Planck mass.\nThis type of suppression gives the SME the structure of a series ap proxima-\ntion, whichfitswellwiththenotionthatitrepresentsthelow-energ ylimitof\nsome underlying high-energytheory. The idea is that conventional Lorentz-\ninvariant physics (the Standard Model and General Relativity) cor responds\nto the dominant leading-order terms in the series. The Lorentz-vio lating\nterms in the SME give higher-order corrections, which decrease in s ize with\nhigherdimension d. Thismakessensefornonminimalviolationswith d≥5,\nbut is problematic for minimal violations, where d= 4 violations would be\nunsuppressed and d= 3 violations would be very large. Minimal violations\nwould be easily detected in this picture and precluded by their absenc e.\nThe lowest-order Lorentz violations would then be nonminimal ones w ith\nmass dimension d≥5.\nWhile the mSME has been tested extensively, the nonminimal parts of\nthe SME remain relatively unexplored, with most of the parameter sp ace\nunconstrained and completely open to future experimentation. At present,\nconstraintsonnonminimalLorentzviolationinphotonsincludetests involv-\ningvacuumbirefringence,4,10vacuumdispersion,11andresonantcavities.12\nIn neutrinos, there are constraints from oscillation experiments5and kine-\nmatical tests involving high-energy astrophysical neutrinos.5,13For heavier\nfermions, current constraints on nonminimal violations come from s tud-\nies of muonic atoms14and hydrogen-like atoms.15Bounds on nonminimal\nLorentz violation in gravityhavebeen obtained in tests ofshort-ra ngegrav-\nity,7,16from limits on gravitational ˇCerenkovradiationin cosmic rays,8and\nfrom limits on birefringence in gravitational waves.9\nThe different sectors of the nonminimal SME share some common fea -\ntures, which we illustrate here by considering the recently constru cted non-\nminimal extension for linearized General Relativity.9The lagrangian for\nthis extension, in term of the metric perturbation hµν, can be written\nL=1\n4ǫµρακǫνσβληκλhµν∂α∂βhρσ\n+1\n4hµν/hatwidesµνρσhρσ+1\n4hµν/hatwideqµνρσhρσ+1\n4hµν/hatwidekµνρσhρσ,(2)\nwhere the first term is the usual Lorentz-invariantpart. The ter m involving\nthe tensor operator /hatwidesµνρσcontains CPT-even Lorentz violations for even\nd≥4,/hatwideqµνρσgives CPT-odd violations with odd d≥5, and/hatwidekµνρσgives\nCPT-even violations with even d≥6. Each of the tensor operators repre-\nsents an infinite series of derivatives. For example, /hatwidesµρνσis given by theProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\nexpansion /hatwidesµρνσ=/summationtext\nds(d)µρǫ1νσǫ2ǫ3...ǫd−2∂ǫ1∂ǫ2∂ǫ3...∂ǫd−2. The expansion\ncoefficients s(d)µρǫ1νσǫ2ǫ3...ǫd−2are constant and control the Lorentz viola-\ntion. The differing symmetries of the coefficients in each class result in\ndifferent physical consequences.\nThe minimal version of the above theory is obtained by restricting at -\ntention to d= 4/hatwidesµνρσterms, which has the effect of demoting the operator\n/hatwidesµνρσto the constant s(4)µνρσ. In other sectors of the SME, going the\nother way, from the minimal to the nonminimal extension, is effective ly the\nreverse of this procedure. In almost every case, the introductio n of nonmin-\nimal violations can be viewed as the promotion of minimal d= 3,4 coeffi-\ncients from constants to operators.4–6The exception is linearized General\nRelativity, where the operators /hatwideqµνρσand/hatwidekµνρσlack minimal counterparts.\nThe power of the SME lies in its ability to give theoretically con-\nsistent predictions for almost any system. For example, with the th e-\nory in Eq. (2) we can examine the effects of general Lorentz violatio n\non gravitational waves. A key result is the modified dispersion relatio n,\nω=/parenleftbig\n1−ς0±/radicalbig\n(ς1)2+(ς2)2+(ς3)2/parenrightbig\n|p|, which predicts several unconven-\ntional features, including dispersion and birefringence. The ςaparameters\nare complicated momentum-dependent combinations of the coefficie nts for\nLorentz violation. However, as in many other practical applications of\nthe SME, the problem can be made tractable by performing a spheric al-\nharmonic decomposition. This aids in the cataloging of the many Loren tz-\nviolating effects that can arise. It also significantly simplifies the rota tions\nbetween noninertial laboratory frames and the inertial Sun-cent ered frame\nconventionally used for reporting results.17\nIn the current example, the spherical expansion of the ςaparameters is\nς0=/summationdisplay\nωd−4Yjm(−ˆp)k(d)\n(I)jm, ς3=/summationdisplay\ndjmωd−4Yjm(−ˆp)k(d)\n(V)jm,\nς1∓iς2=/summationdisplay\nωd−4±4Yjm(−ˆp)/parenleftbig\nk(d)\n(E)jm±ik(d)\n(B)jm/parenrightbig\n.(3)\nThe spherical coefficients for Lorentz violation k(d)\n(I)jm,k(d)\n(E)jm,k(d)\n(B)jm, and\nk(d)\n(V)jmcompletely characterize the leading-order effects of Lorentz viola -\ntion in gravitational waves. Currently, the k(d)\n(I)jmcoefficients have been\nbounded by the apparent absence of gravitational ˇCerenkov radiation in\ncosmic rays.8Thek(d)\n(E)jm,k(d)\n(B)jm, andk(d)\n(V)jmcoefficients are constrained\nby limits on birefringence in gravitational waves.9\nNote that the above expansion involves spin-weighted spherical ha r-\nmonics sYjm. These are similar to the familiar harmonics Yjm=0Yjm,Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n4\nbut carry spin weight s, which is equivalent to helicity, up to a sign. The\nappearance of the s=±4 harmonics in the above example stems from the\nspin-2 nature of gravitational waves. The violations associated wit h the\ns=±4 harmonics couple the ±2-helicity waves, leading to changes of ±4\nin helicity. Similarly, the s=±2 harmonics arise for spin-1 photons in the\nSME,4ands=±1 harmonics appear in the spherical expansions for spin-1\n2\nfermions.5,6\nAcknowledgments\nThis workis supported in part by the United States National Science Foun-\ndation under grant PHY-1520570.\nReferences\n1.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2016 edition, arXiv:0801.0287v9.\n2. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n3. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004); Q.G. Bailey and V.A.\nKosteleck´ y, Phys. Rev. D 74, 045001 (2006); V.A. Kosteleck´ y and J. Tasson,\nPhys. Rev. D 83, 016013 (2011).\n4. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. Lett. 99, 011601 (2007); Ap.\nJ.689, L1 (2008); Phys. Rev. D 80, 015020 (2009); Phys. Rev. Lett. 110,\n201601 (2013).\n5. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 85, 096005 (2012).\n6. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 88, 096006 (2013).\n7. Q.G. Bailey, V.A. Kosteleck´ y, and R. Xu, Phys. Rev. D 91, 022006 (2015).\n8. V.A. Kosteleck´ y and J.D. Tasson, Phys. Lett. B 749, 551 (2015).\n9. V.A. Kosteleck´ y and M. Mewes, Phys. Lett. B 757, 510 (2016).\n10. F.W. Stecker, Astropart. Phys. 35, 95 (2011).\n11. V. Vasileiou et al., Phys. Rev. D 87, 122001 (2013); F. Kislat and H.\nKrawczynski, Phys. Rev. D 92, 045016 (2015).\n12. S.R. Parker et al., Phys. Rev. Lett. 106, 180401 (2011); M. Mewes, Phys.\nRev. D85, 116012 (2012); Y. Michimura et al., Phys. Rev. D 88, 111101\n(2013); S.R. Parker et al., Phys. Lett. A 379, 2681 (2015).\n13. J.S. D´ ıaz, V.A. Kosteleck´ y, and M. Mewes, Phys. Rev. D 89, 043005 (2014);\nF.W. Stecker et al., Phys. Rev. D 91, 045009 (2015).\n14. A.H. Gomes, V.A. Kosteleck´ y, and A.J. Vargas, Phys. Rev . D90, 076009\n(2014).\n15. V.A. Kosteleck´ y and A.J. Vargas, Phys. Rev. D 92, 056002 (2015).\n16. C.-G. Shao et al., Phys. Rev. D 91, 102007 (2015); J.C. Long and V.A. Kost-\neleck´ y, Phys. Rev. D 91, 092003 (2015); C.-G. Shao et al., arXiv:1607.06095.\n17. R.Bluhm et al., Phys.Rev.Lett. 88, 090801 (2002); Phys. Rev.D 68, 125008\n(2003); V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 66, 056005 (2002)."    },    {        "title": "1010.1781v1.Vector_Superfields_and_Lorentz_Violation.pdf",        "content": "arXiv:1010.1781v1  [hep-ph]  8 Oct 2010NCF/005\nVector Superfields and Lorentz Violation\nDon Colladay∗and Patrick McDonald†\nNew College of Florida\n5800 Bay Shore, Rd,\nSarasota, FL 34243\n(Dated: December 1, 2018)\nWe extend Lorentz-violating Supersymmetry models to inclu de vector superfields. The CPT-\npreserving model generalizes easily, while the obvious att empt at generalizing the CPT-violating\nmodel meets serious obstructions. Generalizations of the C PT-preserving but Lorentz-Violating\nmodel to higher dimensions are also straightforward. Compa ctification is used to reduce the\nsix-dimensional theory to an N= 2 Lorentz-violating theory in four dimensions, while the t en-\ndimensional theory is used to produce an N= 4 Lorentz-violating theory. This may be useful in\nfuture constructions involving Ads/CFT correspondence wi th Lorentz violation.\nPACS numbers: 11.30.Cp, 11.30.Pb,\nI. INTRODUCTION\nThe conventional approach to applying supersymmet-\nric theories to nature is to first extend the Poincar´ e sym-\nmetry to a superalgebra, then break the supersymmetry\nto reproduce something resembling the standard model\nat low energies. Alternatively, one may consider preserv-\ning the supersymmetry and breaking the Lorentz sym-\nmetry of the theory. Attempts have been made in this\ndirection using two main approaches. The first involves\nperturbing the superalgebra sector involving QandPµ\noperators, while the second involves keeping this sector\nof the superalgebra unperturbed. In the second case, it\nis argued that it is not possible to include any operators\ninto the minimal supersymmetric extension of the Stan-\ndard Model with dimension four or less[1]. There is also\na third approach in which the SUSY is explicitly bro-\nken along with the Lorentz symmetry by the background\nfields present in the supermultilplets[2].\nThe first case in which the superalgebra is modified\nis explored in the current manuscript. Previous work\nalong these lines began with the chiral superfield model\nby Berger and Kosteleck´ y [3] in which the derivative\noperator was twisted in various ways to produce CPT-\npreserving and CPT-violating models of supersymmetry\nwith Lorentz Violation. In the current manuscript, these\nmodels are extended to the vector superfield in higher di-\nmensions, and nonabelian theories. The CPT-preserving\nmodelgeneralizeswithminimaldifficulty, whiletheCPT-\nviolating version of the theory meets serious impedi-\nments. Compactification of the higher-dimensional the-\nories is used to produce four-dimensional SUSY theories\nwithN= 2 and N= 4 extended supersymmetries. The\nN= 4 theory is particularly interesting as it provides\nthe first perturbed version of the conformal field theory\nthat may have a simple Ads/CFT dual.\n∗Also at Division of Natural Sciences, New College of Florida .\n†mcdonald@ncf.eduII. CPT PRESERVING MODEL\nThe CPT-preserving model that violates Lorentz sym-\nmetry is particularly simple to implement as there exists\na modified superspace implementation in terms of Qop-\nerators. ThisisnotthecasefortheCPT-violatingmodel.\nIntroduce the twisted derivative operator [3]\n˜∂µ=∂µ+kν\nµ∂ν, (1)\nthe general vector superfield is expanded in terms of the\ntwisted derivative as\nV(x,θ) =C(x)+iθγ5ω(x)−i\n2θγ5θM(x)−1\n2θθN(x)\n+1\n2θγ5γµθ˜Vµ−iθγ5θθ/bracketleftbigg\nλ(x)+i\n2˜/ne}ationslash∂ω(x)/bracketrightbigg\n+1\n4(θθ)2/parenleftbigg\nD(x)−1\n2˜∂µ˜∂µC(x)/parenrightbigg\n, (2)\nwhere˜Vµ=Vµ+kµνVνis inserted into the expan-\nsion. This turns out to be necessary to preserve conven-\ntional gauge invariance in the resulting supersymmetric\nlagrangian. It is also consistent with imposing a shift\nof the entire covariant derivative operator rather than\nsimply the partial derivative part. The supersymme-\ntry transformations are implemented using the operator\nδQV(x,θ) =−iǫQV(x,θ), where\nQ=i∂θ−γµθ˜∂µ. (3)\nNote that this operator satisfies the perturbed anticom-\nmuation relation\n{Q,Q}= 2γµ(Pµ+kµνPν). (4)\nMatching terms in the conventional way yields the SUSY\ntransformationpropertiesofthefields. Theonesthatwill\nbe useful in the present work are\nδλ=i\n2σµν˜Fµνǫ−iγ5Dǫ, (5)\nδ˜Vµ=−iǫ/parenleftBig\nγµλ+i˜∂µω/parenrightBig\n, (6)\nδD=ǫ˜/ne}ationslash∂γ5λ, (7)2\nwhere the quantity ˜Fis a twisted field strength (note\nthat˜Fdoes not have the conventional meaning of dual\nin this manuscript) given by\n˜Fµν=˜∂µ˜Vν−˜∂ν˜Vµ. (8)\nNotethatthe D-termstilltransformsasatotalderivative\nandcanthereforebeusedtoconstructSUSYlagrangians.\nA general super gauge transformation is implemented\nby adding a chiral superfield expressed in terms of the\ntwisted derivative. The result is the vector superfield in\nthe Wess-Zumino gauge\nV(x,θ) =1\n2θγ5γµθ˜Vµ−iθγ5θθλ(x)+1\n4(θθ)2D(x) (9)\nNotethat atwistedgaugetransformationon ˜Vµresultsin\nsymmetry under the conventional gauge transformation\nforVµ\nVµ→Vµ+∂µΛ. (10)\nThis is the reason that the original superfield was ex-\npanded in terms of ˜Vµrather than Vµ. The supercovari-\nant derivative can be used to calculate the spinor super-\nfieldWα= 1/4Dγ5DDαV, withD=i∂θ+γµθ˜∂µ. The\nleft component of Wis\nWL=λL(x+)−1\n2γµγνθL˜Fµν(x+)−iθRθL˜/ne}ationslash∂λR(x+)\n−iθLD(x+), (11)\nwhere the left-chiral coordinate x+is given by\nxµ\n+=xµ+i\n2θγ5γµθ+i\n2kµνθγ5γνθ.(12)\nThelagrangianinvariantunderthemodifiedSUSYtrans-\nformation is constructed using the θ2-term in the second-\norder function\nL=1\n2Re/parenleftbig\nWLγ5WL|θ2/parenrightbig\n=−1\n4˜F2+i\n2λ˜/ne}ationslash∂λ+1\n2D2.(13)\nNote that this construction works because the left-\nsuperfield satisfies the twisted chirality condition\nDRWL= 0 (14)\nwhereDR= 1/2(1+γ5)Dis the right-handed projection\nof the twisted supercovariant derivative operator. Any\nfunction of WLtherefore satisfies the twisted chirality\ncondition and hence leads to a SUSY covariant θ2-term,\nor ‘F-Term’ as it is commonly denoted. Explicit verifica-\ntion of this SUSY requires the use of the twisted Jacobi\nIdentity\nǫµναβ˜∂ν˜Fαβ= 0, (15)\nsatisfied by the twisted field strength due to simple sym-\nmetry principles. This Lagrangian can be put into theform of the Standard Model Extension (SME) Parame-\nters by expanding the Lagrangian in the form\nL=−1\n4(F2+kµναβ\nFFµνFαβ)+i\n2(λγµ∂µλ+λcµνγµ∂νλ)\n+1\n2D2, (16)\nwhere\nkµναβ\nF= 2(2kαµ+(k2)αµ)gβν+4(kµα+(k2)αµ)kνβ\n+(k2)αµ(k2)βν, (17)\nand\ncµν=kµν. (18)\nIn this expression, k2represents the square of the matrix\nfork\n(k2)µν=kµαkν\nα. (19)\nNote that in contrastto the scalarchiralsuperfield model\nwhere only second-order terms in kwere present, third-\nand fourth-order terms in kappear in the vector super-\nfield Lagrangian.\nIII. ATTEMPT AT CONSTRUCTION OF A CPT\nVIOLATING MODEL\nCPT violation can be incorporated into the chiral su-\nperfield using a phase redefinition of the fields involved\n[3]. When this same procedure is adopted for the vector\nsuperfield case, several problems emerge. Suppose that\nthe fermion field is redefined by\n˜λ=e−iγ5k·xλ (20)\nto induce CPT-violating terms as is done in the chiral\nsuperfield case. The phase cannot be absorbed into the\nreal vector field Vµas can be seen from the following\ndecomposition of the vector field supersymmetry trans-\nformation\nδVµ=−iǫγµ(e−ik·xλR+eik·xλL).(21)\nThis expression implies that it is impossible to absorb\nboth of the phase changes into a single redefinition of the\nvector field as was possible in the chiral superfield case.\nThis indicates that the CPT-violating generalization of\nthe vector superfield probably does not exist.\nIV.N= 2FROM SIX-DIMENSIONAL THEORY\nThe extension to higher dimensions and the follow-\ning compactification process uses the notation and tech-\nniques introduced in [4]. The definitions of LandRare\nswitched relative to the first half of this paper, this no-\ntation is kept in order to provide an easier comparison\nwith the unperturbed results of [4].3\nThe supersymmetricLagrangian(13) maybe extended\nto six dimensions by allowing the γmatrices and vector\nfield indices to run from 0 to 6 and setting the auxiliary\nfieldDto zero. It is not possible to impose the Majorana\ncondition in six dimensions, but it is possible to impose\nthe Weyl condition. The Lagrangian\nL=−1\n4˜F2+iλ˜/ne}ationslash∂λ. (22)\nis supersymmetric under the modified supersymmetry\ntransformations\nδλ=i\n2σµν˜Fµνǫ, (23)\nδ˜Vµ=−i/parenleftbig\nǫΓµλ−λΓµǫ/parenrightbig\n. (24)\nThe gamma matrices are chosen in a basis where Γµ=\nγµ⊗I2forµ= 0,1,2,3 and\nΓ4,5=γ5⊗iσ1,2, (25)\nΓ7=γ5⊗σ3. (26)\nIn this basis, the Weyl condition fixes λto havethe struc-\nture\nλ=/parenleftbigg\nLχ\nRχ/parenrightbigg\n, (27)\nwhereχis a four-dimensional Dirac spinor. The com-\nponents of the vector potential remain the same for\nµ= 0,1,2,3, while A4=S,A5=P, are relabeled\nas scalar and pseudoscalar fields. It is convenient to in-\nvestigate special cases for kµν. The first case is when the\nlorentz violation is restricted to lie in the four physical\ndimensions ( kµ4=kµ5= 0). The conventional dimen-\nsional reduction procedure yields the Lagrangian\nL=−1\n4˜F2+1\n2˜∂µS˜∂µS+1\n2˜∂µP˜∂µP+iχ/ne}ationslash˜∂χ.(28)\nThis lagrangian has an N= 2 supersymmetry under the\ntransformation laws\nδ˜Vµ=i(αγµχ−χγµα), (29)\nδP=χγ5α−αγ5χ, (30)\nδS=i(χα−αχ), (31)\nδχ=/parenleftbiggi\n2σµν˜Fµνα+i/ne}ationslash˜∂Pγ5− /ne}ationslash˜∂S/parenrightbigg\nα.(32)\nAnother interesting special case is when the Lorentz vi-\nolation is restricted to the compactified dimensions. In\nthis case, the gauge and fermion terms are unperturbed\nwhile the SandPfields are mixed by the kµνparameters\naccording to\n˜S= (1−k44)S−k45P, (33)\n˜P= (1−k55)P−k45S. (34)\nThe lagrangian in this case is the usual one with the\nreplacements S→˜SandP→˜P. In the free theory this\nis just a redefinition of fields, but if the theory is coupled\nto other sectors that involve the SU(2) symmetry, there\ncould be observable effects of the violation.V.N= 4FROM TEN-DIMENSIONAL THEORY\nThe supersymmetricLagrangian(13) maybe extended\nto ten dimensions by allowing the γmatrices and vector\nfield to have indices running from 0 to 9 and setting the\nauxiliary field Dto zero\nL=−1\n4˜F2+i\n2λ˜/ne}ationslash∂λ. (35)\nThe gamma matrices are 32 ×32 matrices in this di-\nmension and are denoted using Γµ. The first four\n(µ= 0,1,2,3) are chosen to reduce to the usual four-\ndimensional gamma matrices following compactification\nΓµ=γµ⊗I8. The rest of the gamma matrices are de-\nfined as linear combinations of a more convenient basis\nselected for its SU(4) properties\nΓ4= Γ14+Γ23, iΓ7= Γ14−Γ23,(36)\nΓ5= Γ24−Γ13, iΓ8= Γ24+Γ13,(37)\nΓ6= Γ34+Γ12, iΓ9= Γ34−Γ12.(38)\nwhere\nΓij=γ5⊗/parenleftbigg\n0ρij\nρij0/parenrightbigg\n, (39)\nare written in terms of the 4 ×4 matrices\n(ρij)kl=δikδjl−δjkδil, (40)\nand\n(ρij)kl=1\n2ǫijmn(ρmn)kl=ǫijkl. (41)\nThe product of all of the gamma matrices defines\nΓ11=γ5⊗/parenleftbigg\nI40\n0−I4/parenrightbigg\n, (42)\nand the charge conjugation matrix is\nC10=C⊗/parenleftbigg\n0I4\nI40/parenrightbigg\n, (43)\nwhereCis the usual four-dimensionalchargeconjugation\nmatrix. Thespinorisrequiredtobe aneigenstateofboth\nΓ11andC10therefore imposing the Majorana and Weyl\nconditions as in the standard case.\nThe previous construction of the perturbed theory still\nworks, provided that the twisted Jacobi Identity is mod-\nified appropriately to\nAναβ˜∂ν˜Fαβ= 0, (44)\nwhereAναβis a totally antisymmetric 3-tensor in 10 di-\nmensions. It is then possible to compactify the extra six\ndimensionsontoatorus[4]andobtainaperturbed N= 4\nsupersymmetric model that violates Lorentz Invariance\nthrough the twisted derivative. Any Lorentz violation4\npresent in the extra dimensions will show up as a vio-\nlation of the SU(4) symmetry. Such a model should be\ninteresting from the perspective of Ads/CFT theory as\nit provides the first example of an appropriate Lorentz-\nviolating theory that can be used in this context. In fact,\nstandard dimensional reduction procedures may be used\nto produce different SUSY theories in four dimensions\nthat may or may not violate Lorentz invariance in the\nfour physical dimensions.\nFor example, The N= 4 SUSY model with Lorentz\nviolation takes the form\nL=−1\n4˜F2+iχi˜/ne}ationslash∂Lχi+1\n4˜∂µφij˜∂µφij,(45)\nwhere the kµνbackground tensor was chosen to vanish\nalongthecompactifieddimensions,butbeofgeneralform\nin the remaining four dimensions. The complex scalar\nfieldsφijtransform as a 6ofSU(4) and are defined ac-\ncording to\nφi4=1√\n2(Ai+3+iAi+6), i= 1,2,3,(46)\nand\nφjk=1\n2ǫjklmφlm= (φjk)∗, (47)\nand the fermions χiform a4representation of SU(4).\nThe modified SUSY transformation laws of the fields are\nδ˜Aµ=−i(αiγµLχi−χiγµLαi) (48)\nδφij=−√\n2i(αjR˜χi−αiR˜χj+ǫijkl˜αkLχl) (49)\nδLχi=i\n2σµν˜FµνLαi−√\n2γµ˜∂µφijR˜αj(50)\nδR˜χi=i\n2σµν˜FµνR˜αi+√\n2γµ˜∂µφijLαj(51)\nPerturbations appear in all three sectors and they are\nconnected by the requirement of supersymmetry.\nAnother interesting special case occurs when the\nLorentz-violating coefficients are taken to vanish along\nthe four physical dimensions, but are allowed to be ar-\nbitrary in the compactified dimensions. In this case, the\nfour-dimensionaltheory is Lorentzinvariant, but violates\nSU(4) symmetry. The perturbations all appear in the\nscalar sector as the resulting Lagrangian takes the form\nL=−1\n4F2+iχi/ne}ationslash∂Lχi+1\n4∂µ˜φij∂µ˜φij,(52)\nwhere˜φij=φij+ Λijklφkland the matrix Λijklcon-\ntains the induced Lorentz violation from the higher di-\nmensions. The supersymmetry transformations for these\nfields are\nδAµ=−i(αiγµLχi−χiγµLαi) (53)\nδ˜φij=−√\n2i(αjR˜χi−αiR˜χj+ǫijkl˜αkLχl) (54)\nδLχi=i\n2σµνFµνLαi−√\n2γµ∂µ˜φijR˜αj(55)\nδR˜χi=i\n2σµνFµνR˜αi+√\n2γµ∂µ˜φijLαj.(56)In this case, identification of ˜φwith the physical\nscalar fields removes any effect of the higher-dimensional\nLorentz violation and restores SU(4) symmetry. How-\never, this is only possible if the fields φijdo not couple\nwith other sectors as their redefinition would affect their\nsymmetry properties in relation to the other sectors. If\nthis is the case, the above model demonstrates a possible\nmethod by which the SU(4) symmetry of an N=4 su-\npersymmetry model may be broken. Some fields can be\ndriven to irrelevant status by making their kinetic cou-\nplingverysmall. Notethatthe matrixΛijklin thismodel\ndoes not in fact have to be small since it does not induce\nLorentz violation in the four physical dimensions.\nVI. NON ABELIAN GENERALIZATION\nThe above constructions also work if the full covariant\nderivative is twisted by kµνaccording to\n˜Dµ=Dµ+kµνDν. (57)\nFor example, the non-Abelian ten-dimensional theory\nthen takes the form\nL= Tr/bracketleftbigg\n−1\n2˜F2+iλ˜/ne}ationslashDλ/bracketrightbigg\n, (58)\nwhereig˜Fµν= [˜Dµ,˜Dν], and the supersymmetry trans-\nformations are the same as before written in terms of\n˜F. The key fact that allows the supersymmetry trans-\nformation to succeed is the Bianchi Identity satisfied by\n˜Fµν\n[˜Dα,[˜Dβ,˜Dγ]]+[˜Dβ,[˜Dγ,˜Dα]]+[˜Dγ,[˜Dα,˜Dβ]] = 0.\n(59)\nThe first case considered is when the Lorentz-violating\ncouplings kµνare restricted to the four physical dimen-\nsions. The resulting Lagrangian is\nL= Tr/bracketleftbigg\n−1\n2˜F2+2iχi˜/ne}ationslashDLχi\n+√\n2g/parenleftbig\n˜χ[φij,Lχj]−χ[φij,R˜χj]/parenrightbig\n+1\n2˜Dµφij˜Dµφij+g2\n8[φij,φkl]2/bracketrightbigg\n.(60)\nThe case when kµνis restricted to the compactified di-\nmensions is simplest, the theory is the same as the con-\nventional one with the replacement\n˜φij=φij+Λijklφkl, (61)\nwith lagrangian\nL= Tr/bracketleftbigg\n−1\n2F2+2iχi/ne}ationslashDLχi\n+√\n2g/parenleftBig\n˜χ[˜φij,Lχj]−χ[˜φij,R˜χj]/parenrightBig\n+1\n2Dµ˜φijDµ˜φij+g2\n8[˜φij,˜φkl]2/bracketrightbigg\n.(62)5\nVII. NOTE ON COORDINATE\nTRANSFORMATIONS\nAn interesting question arises as to the nature of the\nkµνmodel regarding physical observability due to possi-\nble coordinate transformations that can alter, or remove\nLorentz-Violating terms[5]. To answer this question, one\nstarts with the conventional Lagrangian with no symme-\ntry violating terms in it and performs the appropriate\nchange in coordinates for\n∂′µ=∂µ+kµ\nν∂ν, (63)\ncorresponding to the derivative replacement used in the\nSUSY-LV formulas. The initial lagrangian is written in\nthe form\nL0=−1\n4gµαgνβFµνFαβ+i\n2gµνλγµ∂νλ+1\n2D2(64)\nwhere the metric gµν=ηµνis exhibited explicitly. This\nform ofthe SUSYlagrangianis written in geometricform\nso that it is covariant under the linear coordinate trans-\nformation consistent with Eq. (63) , xµ=x′µ−kµ\nνx′ν.\nA simple calculation to lowest-order in kµνyields a mod-\nified metric\ng′µν=ηµν+2kµν, (65)\nand the corresponding lagrangian in the new coordiates\nreads the same as Eq. (64) with primes on all of the\nquantities involved, including the gamma matrices which\nsatisfy the modified Clifford algebra relation\n{γ′µ,γ′ν}= 2g′µν= 2ηµν+4kµν.(66)\nExpanding the g′factors and expressing the primed\ngamma in terms of the conventional gamma yields the\nLagrangian\nL=−1\n4(ηµαηνβF′\nµνF′\nαβ+kµναβ\nFF′\nµνF′\nαβ)\n+i\n2(λ′ηµνγµ∂′\nνλ′+λ′kµνγµ∂′νλ′)+1\n2D′2.(67)\nThis lagrangian is formally equivalent to the supersym-\nmetric Lorentz-breaking theory Eq. ( 16), however, thevectors and spinors are resolved in a non-orthogonal co-\nordinate system This means that all manipulations using\nthe transformed lagrangian must take into account this\ndifferent property of the vector indices. For example,\nquantities like\nA′µA′\nµ=g′µνA′\nµA′\nν=ηµνA′\nµA′\nν+2kµνA′\nµA′\nν.(68)\nmust be used to compute lengths of vectors in the trans-\nformed system. Since the assumed metric in Eq. ( 16)\nis simply ηµν, no such complication in dealing with vec-\ntor components arises, so the theory is physically distinct\nfrom a conventional theory expressed in skewed coordi-\nnates, provided that the metric is accessible to the ex-\nperimentalist through using rods and clocks constructed\nof other, conventional Lorentz invariant fields.\nAnother way to put this is that it is possible to change\ncoordinates to remove kµνin the Lagrangian, but it will\nre-appear in the vector coordinate manipulations as a\npesky non-orthogonality of the vectors (and spinors) in-\nvolved in the calculations of physical observables. It will\nalsoappearin anyLorentz-invariantsectorsasanew per-\nturbation in those fields. This argument also works pro-\nvided there are at least two sectors with different valued\nkµνtensors present in the theory.\nVIII. SUMMARY\nThe main result of this paper is that the scalar CPT-\npreserving models with Lorentz Violation easily general-\nize to the vector superfield and can be extended to the\nusual higher-dimensional cases. In addition, compactifi-\ncation can be used to construct extended supersymmetry\nmodels in four-dimensions. The theory may be abelian,\nor non-abelian, the same basic technique works for both\ncases. On the other hand, the scalarCPT-violatingmod-\nels do not easily generalize to the vector superfield. The\nreason for this is connected with the inability to separate\nthe vector potential into left- and right-handed compo-\nnents.\nThe new N= 4 theories with Lorentz violation are\nparticularly interesting from a theoretical viewpoint as\nthe conventional theory plays a crucial role in Ads/CFT\ntheories. One of the main purposes of this paper is to\nconstruct the perturbed N= 4 theories so that they\nmay be studied in this context.\n[1] S. NibbelinkandM. Pospelov, Phys.Rev.Lett. 94, 081601\n(2005).\n[2] H. Belich, G. Dias, J. Helayel-Neto, F. Leal, and\nW. Spalenza(2010), arXiv:1009.1326.\n[3] M. Berger and A. Kosteleck´ y, Phys. Rev. D 65, 091701\n(2002).[4] L. Brink, J. Schwartz, and J. Scherk, Nucl. Phys. B 121,\n77 (1977).\n[5] D. Colladay and P. McDonald, J. Math. Phys. 43, 3554\n(2002)."    },    {        "title": "1210.6976v1.Gravitational_bremsstrahlung_in_ultra_planckian_collisions.pdf",        "content": "arXiv:1210.6976v1  [hep-th]  25 Oct 2012CCTP-2012-21\nCERN-PH-TH/2012-275\nGravitational bremsstrahlung in ultra-planckian collisi ons\nDmitry Gal’tsova, Pavel Spirinaand Theodore N. Tomarasb,c\naDepartment of Theoretical Physics, Moscow State Universit y, 119899, Moscow, Russia;\nbCERN, Theory Division;\ncDepartment of Physics and Crete Center for Theoretical Phys ics,\nUniversity of Crete, 71003, Heraklion, Greece.\n(Dated: January 12, 2021)\nA classical computation of gravitational bremsstrahlung i n ultra-planckian collisions of massive\npoint particles is presented in an arbitrary number dof toroidal or non-compact extra dimensions.\nOur method generalizes the post-linear formalism of Genera l Relativity to the multidimensional\ncase. The total emitted energy, as well as its angular and fre quency distribution are discussed\nin detail. In terms of the gravitational radius rSof the collision energy, the impact parameter b\nand the Lorentz factor in the CM frame, the leading order radi ation efficiency in the Lab frame\nis shown to be ǫ∼(rS/b)3(d+1)γcmford= 0,1 andǫ∼(rS/b)3(d+1)γ2d−3\ncmford/greaterorequalslant2, up to a\nknownd-dependent coefficient and a ln γcmfactor for d= 2, while the characteristic frequency of\nthe radiation is ω∼γ/b. The contribution of the low frequency part of the radiation (soft gravitons)\nto the total radiated energy is shown to be negligible for all values of d. The domain of validity of\nthe classical result is discussed. Finally, it is shown that within the region of validity of our approach\nthe efficiency can obtain unnatural values greater than one, w hich is interpreted to mean that the\nperipheral ultra-planckian collisions should be strongly radiation damped.\nPACS numbers: 11.27.+d, 98.80.Cq, 98.80.-k, 95.30.Sf\nContents\n1. Introduction 2\n2. General setting 4\nA. The model 4\nB. The iteration scheme 5\n1. The asymptotic behavior of the fields 7\nC. The energy-momentum of the gravitational radiation - Formulae 8\n1. The uncompactified case M1,D−1 8\n2. The compactified case M1,3×Td10\nD. Graviton polarizations 12\n3. The gravitational radiation source 13\nA. The local part1TMN+1T′\nMNof the source 14\nB. The non-local part SMNof the source 15\nC. Destructive interference of the beamed high-frequency comp onents 17\nD. The total radiation amplitude 17\n1. The beamed, high-frequency radiation amplitude 18\n2. Wide-angle, medium-frequency radiation amplitude 19\n3. The beamed, medium-frequency radiation amplitude 19\nE. Summary 20\n4. The emitted energy 20\nA. The total radiated energy 20\nB. The low frequency part of the spectrum 22\nC. Quantum constraints 24\n1. Theω∼γ2/bandω∼1/bregimes 24\n2. Theω∼γ/bregime 24\nD. Radiation emission with characteristic frequency ω∼γ/b 25\n1.d/greaterorequalslant3 25\n2.d= 0 26\n3.d= 1,2 26\nE. Extreme radiation efficiency for d/greaterorequalslant3 andω∼γ/b 262\n5. Conclusions 27\nAcknowledgements 28\nA. Definitions and variations 28\n1. Perturbation theory variation over gravitational constant in fl at background 28\n2. Useful kinematical formulae 29\nB. Momentum integrals 29\n1. Basic scalar integral 29\n2. Vectorial and tensorial integrals 31\n3. Derived integrals for stresses 32\nC. Integration over frequencies and angles 34\nD. Asymptotic behavior of retarded fields 34\nReferences 36\n1. INTRODUCTION\nThe scenario of TeV-scale gravity with large extra dimensions [1, 2] is currently being tested experimentally\nat the LHC [3]. In this class ofhigher-dimensional gravitytheories th e Planck mass can be of the order of a TeV,\nprovided that the number of extra dimensions is greater than two. If so, the LHC will probe ultra-planckian\nphysics where gravitynot only is the dominant force [4], but in addition it is believed to be adequately described\nby the classical Einstein equations [5]. This, presumably, allows one to make reliable theoretical predictions of\ngravitational effects without entering into the complications relate d to quantum gravity. One such prediction,\nnamely the possibility of black hole formation at the LHC was extensive ly discussed in the literature during the\npast ten years [3, 6–9].\nThe main focus ofthis paper is the study ofgravitationalbremsstr ahlungin ultra-planckianparticle collisions.\nIn ultra-planckian collisions with impact parameters smaller than the S chwarzschild radius of the center of mass\nenergy of the colliding particles, the creation of a black hole is a domina nt process and is accompanied by\nradiation losses at the level of 10-40% depending on the number of e xtra dimensions (see [10] generalizing\nearlier calculations by D’Eath and Payne [11, 12]). On the other hand, for impact parameters larger than\nthe gravitational radius, the black hole formation process is expon entially suppressed and the gravitational\nbremsstrahlung becomes the dominant inelastic classical effect, wh ich furthermore can be very strong due to\nthe huge number of light Kaluza-Klein states in the graviton spectru m. It is worth noting, that gravitational\nradiation may not be maximal in head-on collisions. For instance, radia tion from point particles falling into the\nblack hole is minimal for radial infall and grows initially with the impact parameter, a fact w hich is well known\nboth in four-dimensional general relativity and in models with extra d imensions. The way this is understood is\nthe following: ultra-relativistic particles hitting with some critical impa ct parameter can be captured into quasi-\ncircular orbits before they plunge into the hole. During these revolu tions they radiate with almost constant rate\nand lose energy more efficiently than during a radial infall.\nA large number of papers from various groups is devoted to the stu dy of gravitational bremsstrahlung in a\nvariety of physical set-ups and using different approaches. A rat her incomplete list includes the following: A\nclassical calculation relevant to zero and small impact parameter us ing the colliding wave picture was suggested\nlong ago by D’Eath [11, 12]. More recently this approach was extende d to the extradimensional theories [9, 10].\nThe approach of Smarr et al. in [13] was based on the low frequency approximation for the amplitude to\nestimate the full energy loss. Matzner and Nutku proposed the us e of the method of virtual gravitons [14] in\nanalogy with the Weizs¨ acker-Williams approach in quantum electrody namics.\nAnother approach consists in considering radiation in the linearized t heory from particles falling towards\na black hole [15, 16]. Gravitational bremsstrahlung was also studied in the Born approximation of quantum\ngravity (in four dimensions) [17], while the relationship between the Bo rn approximation and the post-linear\nformalism was discussed in [18]. More recently, a novel approach, ma king use of perturbative quantum gravity\nto derive effective equations of motion and to describe gravitationa l radiation using those, was proposed under\nthe name of “effective theory” approach in [19]. Estimates of radiat ion losses in the eikonal approximation\nof (multidimensional) quantum gravity were given in [5], while Amati, Ciafa loni and Veneziano in a series of\npapers [20] have considered the problem of radiation in ultra-high en ergy collisions of massless particles in the\ncontext of quantum string theory. Estimates of the gravitationa l bremsstrahlung in high energy collisions in the\nADD model were given in [21]. Finally, a lot of numerical work has also bee n devoted to this subject [22–27].3\nIn four-dimensional theories, in particular, gravitational bremss trahlung in small-angle ultrarelativistic scat-\ntering of point particles was calculated long ago, most notably by Kov acs and Thorne [28], in the framework of\nthe so-called “fast motion approximation” scheme, first proposed by Bertotti, Havas and Goldberg [29]. A simi-\nlar independent calculation, using momentum space perturbation th eory up to second order in the gravitational\nconstant, was performed in Ref. [18] leading to essentially the sam e results. The latter approach amounts to\nperforming a perturbation expansion of the metric around the Mink owski background up to second order in the\ngravitational constant and is applicable for arbitrary mass-ratios of the colliding particles. In the case of one\nmassMmuch larger than the other m, the same result had been obtained earlier by Peters, also using a line ar\napproximation but around the Schwarzschild background of the he avy mass [30].\nThe purpose of this paper is to study gravitational radiation losses in the case of ultra-relativistic collisions\nwith large impact parameters, for which the scattering angle is small and calculations can be performed reliably\nwithin the post-linear approximation scheme of General Relativity. T his paper continues a series of investiga-\ntions [31–33] devoted to bremsstrahlung in flat space-time arising u nder non-gravitational scattering of charged\nparticles [31] and scalar radiation in gravity-mediated particle collision s [32]. Here we focus on the problem of\ngravitational radiation in collisions of massive point particles interacting gravitationally. The computation is\npurely classical and iterative and, as such, it can only be reliable within a certain domain of validity in the space\nof parameters. Somewhat unexpectedly, it turns out that, for p arameter values within the region of validity of\nthe classical approximation the radiation efficiency , i.e. the fraction of the initial energy which is radiated away,\nis substantially enhanced compared to the four-dimensional case, contrary to earlier qualitative estimates [34].\nMoreover, for a number of extra dimensions greater than two, th is quantity grows with a dimension-dependent\npower of the Lorentz-factor γof the collision.\nSpecifically, for the leading order radiation efficiency in the laborator y frame we obtain (up to a dimension-\ndependent numerical coefficient, which however can be quite large, and an inessential ln γfactor ford= 2 extra\ndimensions)1\nǫ≡E\nE0∼/parenleftBigrS\nb/parenrightBig3(d+1)\nγcm, d= 0,1 (1.1)\nǫ∼/parenleftBigrS\nb/parenrightBig3(d+1)\nγ2d−3\ncm, d/greaterorequalslant2, (1.2)\nwherebis the impact parameter of the collision, rSis the Schwarzschild radius for the center-of-mass collision\nenergy\nrS=1√π/bracketleftBigg\n8Γ/parenleftbigd+3\n2/parenrightbig\nd+2/bracketrightBigg1\nd+1/parenleftbiggGD√s\nc4/parenrightbigg1\nd+1\n, (1.3)\nandE0=m(γ+1)≃mγis the initial energy in the Lab frame. For collisions of two equal mass p articles the\nLorentz factors in the Lab and in the center-of-mass frames are related byγ= 2γ2\ncm−1.\nIn the special case of d= 0 our results for the total emitted energy as well as for its angula r and frequency\ndistribution are in perfect agreement with the results of Kovacs an d Thorne [28].\nThe present paper is organized in five sections of which this Introdu ction is the first. In Section 2 the\nmodel, the basic formulae and the iterative procedure, that will be e mployed, are described. Also, a complete\northonormalsetofgravitonpolarizationtensorsinarbitrarydime nsions, appropriateforthe problemathandare\nexplicitly constructed. In Section 3 the expressions of the local an d non-local contributions to the gravitational\nradiation source are obtained. The phenomenon of destructive int erference of the leading local and non-local\nterms is shown. The total radiation amplitude is also obtained. Sectio n 4 contains the computation of the\nleading ultra-relativistic order emitted energy in the classical theor y in arbitrary dimensions. It also contains a\ndetailed discussion of the frequency and angular distributions of th e emitted radiation. The zero frequency limit\nof the radiationis analyzed in detail in all dimensions. In the same Sect ion the domain of validity of the classical\ncomputation is discussed. The final result for the radiation efficienc y is given and the unnatural possibility for\nit, to take values greater than one and even diverge in the massless limit ford/greaterorequalslant2 extra dimensions, is critically\nanalyzed. A summary together with a few final comments are given in the final Conclusion section.\n1Notice that these expressions differ from the generic formul aǫ∼(rS/b)3(d+1)γ2d+1\ncmgiven erroneously for all dimensions in [33].\nThe difference is due to an error in [33], related to the extent of the phenomenon of destructive interference discussed in the text.\nNevertheless, the qualitative conclusions of [33] are stil l correct and will be discussed further in Section 5 of the pre sent paper.4\n2. GENERAL SETTING\nWewillstudygravitationalbremsstrahlungbothinuncompactifiedM inkowskispace-time M1,D−1=M1,3×Rd\nand in the ADD model M1,3×Tdwith thedextradimensions compactifiedon a torus. In both casesthe collision\nwill be confined on a brane M1,3, and theD−dimensional cartesian coordinates are split as xM= (xµ;yi),\nxµ∈M1,3,yi∈RdorTd, respectively. The brane subspace M1,3is selected by the initial conditions on the\nparticle velocities and is fixed throughout the collision process, since the gravitational interaction cannot expel\nthe particles from the brane.\nA. The model\nConsider two point masses mandm′moving along the world-lines xM=zM(τ) andxM=z′M(τ′) (M=\n0,1,...,D−1) and interacting with the D−dimensional gravitational field. The corresponding action is\nS=−/summationdisplay1\n2/integraldisplay/parenleftbigg\negMN˙zM˙zN+m2\ne/parenrightbigg\ndτ−1\nκ2\nD/integraldisplay\nRD√−gdDx, (2.1)\nwhereκ2\nD≡16πGD,e(τ) is the ein-bein of the trajectory and the summation is over the two particles. The\nmetric signature is chosen to be (+ ,−,...,−) and our convention for the Riemann tensor is RBNRS≡ΓB\nNS,R−\nΓB\nNR,S+ΓA\nNSΓB\nAR−ΓA\nNRΓB\nAS, with ΓA\nNR= (1/2)gAB(gBR,N+gNB,R−gNR,B). Finally, the Ricci tensor and\ncurvature scalar are RMN≡δB\nARAMBNandR≡gMNRMN, respectively.\nThe action is invariant under the general coordinate reparametriz ation of spacetime, as well as under the\nindependent reparametrizations of the particle trajectories. Fo r the trajectory of particle mthe transformation\nisτ→˜τ= ˜τ(τ), under which all fields are scalars except for the einbein e(τ), which transforms according to\ne(τ)→˜e(˜τ) =e(τ)d˜τ/dτ.\nVaryingSwith respect to e(τ) andzM(τ) one obtains the equations\ne2gMN˙zM˙zN=m2(2.2)\nand\nd\ndτ/parenleftbig\ne˙zNgMN/parenrightbig\n=e\n2gNP,M˙zN˙zP, (2.3)\nrespectively. Two analogous equations are obtained by varying with respect toe′andz′M, while variation of\ngMNleads to the Einstein equations\nGMN=1\n2κ2\nDTMN, TMN=/summationdisplay\ne/integraldisplay˙zM˙zNδD(x−z(τ))√−gdτ. (2.4)\nStrictly speaking the notion of point-like particles is not compatible wit h full non-linear gravity. Nevertheless,\nit still makes sense in the context of the weak-field perturbation ex pansion around flat spacetime, which will be\nadopted here. In this approach one writes the metric as\ngMN=ηMN+κDhMN, (2.5)\nand expands all quantities in powersof hMN, usingηMNto raise and lowerthe indices. The flat background will\nbe eitherD-dimensional Minkowski space M1,D−1, or the product M1,3×Td(d=D−4) of four-dimensional\nMinkowski and a d−torus.\nIt is convenient to define the quantity\nψMN≡hMN−1\n2hηMN, h≡ηMNhMN, (2.6)\nand fix the general coordinate reparametrization symmetry by ch oosing to work in the flat-space harmonic\ngauge\n∂NψMN= 0. (2.7)\nThe reparametrization freedom of the particle trajectories will be dealt with later. The expansion of several\nrelevant quantities in powers of hMNis given in Appendix A. In particular, the Einstein tensor in the harmon ic\ngauge becomes\nGMN=−κD\n2✷ψMN−κ2\nD\n2SMN+NMN, (2.8)5\nwhere✷=∂M∂Mis the flat d’Alembert operator, SMNis theO(h2) part ofGMNgiven by\nSMN(h) =hP,Q\nM(hNQ,P−hNP,Q)+hPQ(hMP,NQ+hNP,MQ−hPQ,MN−hMN,PQ)−\n−1\n2hPQ\n,MhPQ,N−1\n2hMN✷h+1\n2ηMN/parenleftbigg\n2hPQ✷hPQ−hPQ,LhPL,Q+3\n2hPQ,LhPQ,L/parenrightbigg\n,(2.9)\nwhileNMNstands for all cubic and higher in hMNterms in the Einstein tensor.\nA few remarks are in order. Usually in the ADD scenario one assumes t hat matter is localized on the brane,\ni.e. restrictedto thesubspace M1,3. This assumptionisnon-contradictorywithin the linearizedgravity, sincethe\nmatter energy momentum in this case is purely flat-space tensor, w hose conservation does not involve gravity.\nIn the full non-linear gravity the D-dimensional Bianchi identity\nTN\nM;N=1√−g[TN\nM√−g],N−1\n2gNP,MTNP= 0 (2.10)\nimpliesD-dimensional geodesic equations for the particles (2.3) (and similarly for brane fields), which are\ngenerally inconsistent with the assumption of matter localization on t he brane. For this reason we have to\nconsider the world-lines in (2.1) from the beginning as D-dimensional curves. In general, without any extra\nforces ensuringconfinement, the particle motion can be localized on the brane only by suitable initial conditions.\nHowever, it will be shown that the gravitational interaction treate d perturbatively will not expel the colliding\nparticlesfromthebraneatleasttoleadingorderinouriterativesch eme. Thiswillbesufficientfortheconsistency\nof our computation of bremsstrahlung.\nAnother technical difference from the standard ADD approach, a lso related to the fact that our treatment\nappeals to the D-dimensional picture rather than to the four-dimensional one, will be the treatment of the\ngraviton as a D-dimensional massless particle, instead of as a set of four-dimensio nal massive fields of spins\n2,1,0 by suitable rearrangement of the components of the D-dimensional graviton [2]. In particular, it will be\nmore convenient here to choose directly the D(D−3)/2 polarization tensors of the graviton propagating in the\nbulk, rather than to split them into polarizations of massive particles in four dimensions.\nB. The iteration scheme\nOurapproachis identical tothe oneused in simpler models in [31] and [32 ]. The iterativesolutionofthe above\nfield equations and gauge fixing conditions amounts to formally expan ding and computing all fields step-by-step\nin the form\nΦ =0Φ+1Φ+2Φ+... , (2.11)\nwhere Φ denotes any of the fields zM(τ),z′M(τ′),e(τ),e′(τ′) andhMN(x) and with the left superscript labeling\nthe step of the iteration.\nThe zeroth order solution is trivial. It describes the two particles moving with constant velocitie s (uM=\n(uµ,0,...,0) andu′M= (u′µ,0,...,0), respectively in the M1,3subspace, whose indices are labeled by lower case\ngreek letters µ,ν= 0,1,2,3. Correspondingly, their trajectories0zM= (0zµ(τ),0...,0), and similarly for0z′M,\nare the straight lines\n0zµ(τ) =zµ(0)+uµτ,0z′µ(τ′) =z′µ(0)+u′µτ′, (2.12)\nin the absence of any gravitational field\n0hMN= 0. (2.13)\nThe Lagrange multipliers are chosen equal to the corresponding pa rticle masses\n0e=m,0e′=m′, (2.14)\nso that the trajectories are parametrized by the corresponding proper times and the four-velocities satisfy the\nnormalization conditions ηMNuMuN≡u2=u′2= 1.\nWe choose to work in the Lorentz frame in which the target particle m′is initially at rest and with the\nprojectilemmoving along the z-axis. Also, we introduce the space-like vector bM= (bµ,0,...,0) lying on the\nbrane, with:\nbµ=zµ(0)−z′µ(0). (2.15)6\nWith no loss of generality one can further assume ( bu) = (bu′) = 0 and choose the x-axis along b. Thus,\nuµ=γ(1,0,0,v), u′µ= (1,0,0,0), bµ= (0,b,0,0), (2.16)\nwhereγ= 1/√\n1−v2andbis the impact parameter.\nThe first order correction is obtained next. The zeroth order straight particle trajectories are sources of the\nfirst order gravitational field1hMN(1h′\nMN) of each particle, which in turn, causes the first order deviation of\nthe trajectory1z′M(1zM) of the other one. In the process, the first correction1e(1e′) of the einbein fields is\nalso obtained. Explicitly, from the zeroth order trajectories one o btains the zeroth order energy-momentum\ntensor\n0TMN=/summationdisplay\nm/integraldisplay\nuMuNδD(x−0z(τ))dτ, (2.17)\nwhich in this order has only {M,N}={µ,ν}components, and from the first order Einstein equations, given\nin the harmonic gauge by\n✷1ψMN=−κD0TMN, (2.18)\nthe first order correction1ψMNto the metric is obtained. Although only the1ψµνcomponents of1ψMNare\nnon-vanishing, the corresponding1hMNwill also have non-vanishing bulk components from the trace part in\nEq. (2.18).\nUsing1hMNand the zeroth order solution in equations (2.2) and (2.3) one obtain s for1eand1zMthe\nequations2\n1e=−m\n2/parenleftbig\nκD1hMNuMuN+2ηMNuM1˙zN/parenrightbig\n(2.19)\nand\nd\ndτ/parenleftbig1euM+m1˙zM/parenrightbig\n=−κDmHPQMuPuQ, H PQM=1hPM,Q−1\n21hPQ,M, (2.20)\nwhich upon elimination of1egive for1zM\nΠMNd\ndτ1˙zN=−κDΠMNHPQNuPuQ, (2.21)\nwhere\nΠMN=ηMN−uMuN(2.22)\nis the projector onto the subspace orthogonal to uM. Equations (2.18), (2.19) and (2.21), together with two\nsimilar equations for1e′and1z′M, form a complete set of equations in this order. The gauge fixing con dition\n(2.7) for1ψMNis a consequence of the conservation of0TMN.\nIt should be kept in mind that in the equation of motion (2.21) of partic lemthe gravitational field on the\nright hand side is the one due to m′. Singularities due to the action on mof its own gravitational field can be\nremoved in lowest order by classical renormalization of the affine par ameter of its trajectory [40]; this will be\nenough for our purposes. In what follows we will omit self-action ter ms and consider only mutual gravitational\ninteraction.\nIn the next order of iteration one obtains the leading contribution to gravitational radiation. It is due to the\naccelerated particles, as well as to the cubic gravitational self-int eraction. Indeed, the Einstein field equation\nreduces for the second order correction2ψMNof the gravitational field to\n✷2ψMN=−κDτMN, (2.23)\nwith the source term having three contributions:\nτMN=1TMN+1T′\nMN+SMN/parenleftbig1h/parenrightbig\n. (2.24)\n2Our gauge condition is gMN˙zM˙zN= 1. To this order it reduces to1e= 0.7\npkM\np\nT'MN'pkM\np\nTMN'p\npkM\nSMN'\nFigure 1: Schematic representation of the sources of gravit ational radiation described by (2.23). The interactions ar e\nmeant to all orders in the quantum perturbation expansion se nse. Solid (dotted) lines are colliding particles (gravito ns).\nThe first two terms of τMN(2.24) represent the particles’ contribution to radiation. As obta ined to this order\nfrom (2.4) they are3\n1TMN(x) =m/integraldisplay/bracketleftbigg\n21˙z(MuN)+κD/parenleftbigg\n2uP1hP(MuN)−1h\n2uMuN/parenrightbigg\n−uMuN1zP∂P/bracketrightbigg\nδD(x−0z(τ))dτ.(2.25)\nDitto form′withureplacedby u′. Torepeat, in (2.25)1his due tom′. Finally, the part SMNofτMNrepresents\nthe contribution of the gravitationalfield itself to gravitationalra diation. Given that SMNis quadratic in hMN,\nconsistency of the iteration to this order requires to substitute hMN→1hMNinsideS. Furthermore, just as the\nself-interaction terms were ignored in the particles’ equations of m otion, one has to keep in SMN(1h) only the\nproductsofthefirstordercorrections1hdue to different particles . Thevariouscontributionstothegravitational\nradiation to this order is shown schematically in Figure 1.\nIt is straightforward to verify on the basis of the equations of mot ion satisfied by the first order fields that\n∂NτMN= 0, (2.26)\nwhich guarantees the validity of the gauge fixing condition (2.7) to th is order.\nSoSMNconsidered as a quadratic form in1hMNconstitutes the non-local (in terms of the flat space picture)\nsource of gravitational radiation. This non-locality is due to the non -linearity of the theory and, as will be\nshown, leads to important differences in the radiation spectrum fro m linear theories like electromagnetism.\nMore detailed discussions of this point in four dimensions can be found in [18], as well as in its generalization\nto arbitrary dimension but in a simpler non-linear model in [32].\n1. The asymptotic behavior of the fields\nThesametensor SMNconsideredasaquadraticformin2hMNplaysthe roleofaneffectiveenergy-momentum\ntensor of gravitational waves [37]. This interpretation is based on t he conservation equation (2.26) in combi-\nnation with the asymptotic properties of the solutions of the flat-s pace d’Alembert equation and is valid in all\nspace-time dimensions.\nRecall, that in four dimensions1hµν, which according to (2.18) is due of a static or a uniformly moving mass ,\nfalls-off as 1 /r, and the corresponding “field strength” (derivatives1hµν,λ) behave as 1 /r2. On the other hand,\nthe fall-off of the radiative component of the retarded potential2hµνis 1/rand the same for its derivatives\n2hµν,λ. Correspondingly, in D-dimensional Minkowski space-time the behaviors of1hMNand its derivatives\n1hMN,Lare 1/rD−3and 1/rD−2, respectively, where ris the radial coordinate in the D−1-dimensional space.\nThe correspondingfall-off for the radiation field can be found from t he recurrencerelations for Green’s functions\nin neighboring dimensions. The derivation looks different in even and od d dimensions, but it leads to the same\nresult, namely (see Appendix D)\n2hMN∼2hMN,P∼1\nr(D−2)/2. (2.27)\nIn the case of the ADD model the behavior of the field of a uniformly m oving mass depends on the ratio of r\nto the compactification radius R. Forr≪Rit has theD-dimensional behavior 1 /rD−3, while for r≫Rit\n3Symmetrization over two indices is defined according to A(MN)≡(AMN+ANM)/2.8\ndecreases as 1 /r(withrbeing the radial coordinate in three-space). We assume that the o bservational wave-\nzone corresponds to this second condition. Then, the fall-off of th e radiation field in the wave-zone will be the\nsame as in four dimensions.\nSince the effective stress-tensor SMN(2h) is a homogeneous second order function of2hand its derivatives,\nits fall-off in D-dimensional Minkowski will be 1 /r(D−2)and in the ADD case 1 /r2. This is precisely what is\nneeded to get non-zeroflux of radiationin the wavezone through t heD−2-dimensionalsphere at spatial infinity\nofM1,D−1, or through the corresponding two-dimensional sphere in the ADD scenario. Note that the use of\npseudotensors to define the momentum density of gravitational r adiation leads to the same result. Also, note\nthat as a consequence of the above behavior the higher order non -linear terms contained in NMNof Eq.(2.8)\ndo not contribute to the radiation energy-momentum.\nC. The energy-momentum of the gravitational radiation - For mulae\nFollowing Weinberg [37], we introduce the gravitational energy-mom entumtMN≡GMN−G(1)\nMN. Combined\nwith the matter part TMN, the total energy-momentum of the system is conserved, i.e. it sa tisfies\n∂N(TMN+tMN) = 0, (2.28)\nwhich is a consequence of the identity ∂NG(1)MN= 0, and which to second order in the above iteration scheme\ncoincides with (2.26).\n1. The uncompactified case M1,D−1\nConsider the flat-space world-tube Wbounded by two space-like hypersurfaces Σ ±∞defined att→±∞,\nand the infinitely far away time-like Cylindrical surface C. Thus, the boundary ∂WofWis the union of Σ −∞,\nΣ+∞andC.\nUsing the energy-momentum conservation equation (2.28) one can write successivelyfor the emitted radiation\nmomentum ∆ PM\n∆PM=−/integraldisplay\nΣ+∞TMNdσN+/integraldisplay\nΣ−∞TMNdσN=−/integraldisplay\n∂WTMNdσN=−/integraldisplay\nW∂NTMNdDx\n=/integraldisplay\nW∂NtMNdDx=/integraldisplay\n∂WtMNdσN=/integraldisplay\nCtMNdσN. (2.29)\nTo justify the above steps, notice that the first equality is (with dσ0>0 in both Σ −∞and Σ +∞surfaces) the\nstatement that ∆ PMis minus the change of momentum of the matter system; for the sec ond one uses the fact\nthat the matter system is localized inside the volume and the contribu tion ofCon the surface integral vanishes.\nGauss’ theorem was used next, together with (2.28) to end up with the surface integral of tMN. Finally, for the\nlast equality one used the fact that the surface integrals of tMNon Σ−∞and Σ +∞are zero.\nIn particular, to the order of our computation in the previous subs ection, using Gauss’ theorem, equations\n(2.8), (2.9) and the gauge fixing condition, one may write instead of ( 2.29)\n∆PM=/integraldisplay\nCSMN(2h)dσN=/integraldisplay\nW∂NSMN(2h)dDx=1\n2/integraldisplay\nW2hPQ,M✷2ψPQdDx+/integraldisplay\nW2hPQ✷2ψMP,QdDx.\n(2.30)\nFurthermore, it is straightforward to show that the last integral on the right vanishes. Indeed,\n/integraldisplay\nW2hPQ✷2ψMP,QdDx=/integraldisplay\nC2hPQ✷2ψMPdσQ−/integraldisplay\nW2hPQ\n,Q✷2ψMPdDx\n=−1\n2/integraldisplay\nW2h,P✷2ψMPdDx\n=−1\n2/integraldisplay\nC2h✷2ψMPdσP+1\n2/integraldisplay\nW2h✷2ψMP,PdDx= 0, (2.31)\nwhere the first integrals of the first and third lines vanish as a conse quence of the asymptotic behavior of the\nfields given in (D.4), (D.5), and (2.27).9\nThus, the emitted momentum is\n∆PM=1\n2/integraldisplay\nW2hPQ,M✷2ψPQdDx. (2.32)\nUsing the Fourier transformed quantities\n2hMN(x) =1\n(2π)D/integraldisplay\n2hMN(k)e−ikQxQdDk, τMN(x) =1\n(2π)D/integraldisplay\nτMN(k)e−ikQxQdDk.(2.33)\none rewrites (2.32) in the form\n∆PM=−iκ2\nD\n2(2π)D/integraldisplay\nkMGret(k)τSN(k)τ∗\nLR(k)˜ΛSNLRdDk, (2.34)\nwhereτ∗\nMN(k) =τMN(−k) by the reality of τMN(x), and the tensor ˜ΛSNLRis given by\n˜ΛSNLR=1\n2/bracketleftbig\nηSLηNR+ηSRηNL/bracketrightbig\n−1\nD−2ηSNηLR. (2.35)\nThe retarded Green’s function is Gret=−P(1/k2)+iπǫ(k0)δ(k2). Its real part leads to an integrand, which\nis odd under parity kM→−kMand does not contribute to the integral. Thus, one writes equivalen tly\n∆PM=κ2\nD\n2(2π)D−1/integraldisplay\nθ(k0)kMτSN(k)τ∗\nLR(k)˜ΛSNLRδ(k2)dDk. (2.36)\nTaking into account the transversalityof τMN(kMτMN(k) = 0) and the on-shell condition k2= 0 of the emitted\nwave, one can replace the Minkowski metric in ˜ΛSNLRby\n∆MN≡gΠM\nLk′ΠLN=ηMN+kMkN−2(kg)k(MgN)\n(kg)2, (2.37)\nwith any time-like unit vector g, where gΠ = 1−g⊗gandk′Π = 1 +k′⊗k′/(kg)2are projectors onto\nsubspaces transverse to gandk′≡gΠk=k−(kg)g, respectively. Since ( k′g) = 0, the projectors gΠ and\nk′Π commute. Their product ∆MNis then a symmetric projector onto the subspace Mk,g, perpendicular to k\nandg.By construction, the projector ∆ is idempotent (∆2= ∆), thus on Mk,git acts as the unit operator. In\nwhat follows, we will conveniently choose gM=u′\nMand calculate the flux in the Lorentz frame (the Lab) with\nu′\nM= (1,0,...,0).\nThus,˜Λ can equivalently be replaced in (2.36) by\nΛSNLR=1\n2/bracketleftbig\n∆SL∆NR+∆SR∆NL/bracketrightbig\n−1\nD−2∆SN∆LR, (2.38)\nand then upon integration over |k|, one ends up with\n∆PM=κ2\nD\n4(2π)D−1∞/integraldisplay\n0ωD−3dω/integraldisplay\nSD−2dΩkMτSN(k)τ∗\nLR(k)ΛSNLR, (2.39)\nwhereω≡k0=|k|. It is easy to show that ∆ ≡ηMN∆MN=D−2. Furthermore, Λ satisfies the following\nrelations:\n(a) It is traceless on both pairs of indices\nηSNΛSNLR= 0, ηLRΛSNLR= 0, (2.40)\nand (b) it is a projection operator, since it is idempotent (Λ2= Λ)\nηPSηQTΛMNSTΛPQLR= ΛMNLR, (2.41)\nand projectsany second rank symmetric tensor, element of a spa ce we denote by K, onto a subspace KΛ. Acting\non aτMN(orτLR)4, it returns a new symmetric tensor τ′MN= ΛMNPQτPQwith two indices. Λ acts as a\n4The arguments below apply independently to the real and the i maginary parts of the complex-valued τMNthat enters in (2.39).10\nlinear operator on the space Kof symmetric tensors of rank two5. The properties of τ′determine the space\nKΛ.\nIn particular, the dimensionality of KΛis determined as follows: τ′\nMNis a symmetric D×Dmatrix and\nthus it has D(D+1)/2 elements. In addition it satisfies the independent conditions τ′\nMNu′N= 0,τ′\nMNkN= 0\nandηMNτ′\nMN= 0, which impose6D+(D−1)+1 = 2Dconditions, leaving a total of D(D−3)/2 indepen-\ndent elements in τ′. Thus, the dimensionality of KΛisD(D−3)/2, the same as the number of independent\npolarizations of the graviton in D−dimensions.\nGiven that Λ acts on KΛas the unit operator one can write it (decomposition of unity) in term s of an\northonormal basis tensors {εMN\nP}, withP= 1,2,...,D(D−3)/2 ofKΛ\nΛMNLR=/summationdisplay\nPεMN\nPεLR\nP. (2.42)\nUsing this expression for Λ, equation (2.39) takes the form\n∆PM=κ2\nD\n4(2π)D−1/summationdisplay\nP∞/integraldisplay\n0ωD−3dω/integraldisplay\nSD−2dΩkM/vextendsingle/vextendsingleτLR(k)εLR\nP/vextendsingle/vextendsingle2, (2.43)\ni.e. a sum of independent contributions one for each polarization.\n2. The compactified case M1,3×Td\nFollowing the same steps as in M1,D−1, one is led to a similar expression for the emitted momentum in the\nADD scenario. Consider the flat-space world-tube Wbounded by two space-like hypersurfaces Σ ±∞defined at\nt→±∞, and the infinitely distant time-like surface Sr→∞, defined for−∞< t <∞, and whose intersection\nwith the space-like hypersurfaces t= const will be denoted by Bt. According to our convention all quantities\nconstructed from the metric perturbation hMNare flat-space tensors, so the covariant derivatives acting on\nthem are simply partial derivatives. Starting with the energy-mome ntum conservation (2.28), and following the\nsame steps as above, one concludes, first of all, that the change o f the momentum of the system carried away\nby gravitational waves during the collision can be computed by integr ating the flux of momentum through Bt\nand also over time\n∆PM=∞/integraldisplay\n−∞dt/integraldisplay\nBtSMN(h)dD−2BN. (2.44)\nNote that the hypersurface Btis the product S2×Tdand the corresponding measure is dD−2BN=r2dΩ2ddy,\nwithrthe radial coordinate in R3. Thus, using Gauss’ theorem one may write:\n/integraldisplay\nW∂NSMNd4xddy=/integraldisplay\nΣ∞SM0d3xddy−/integraldisplay\nΣ−∞SM0d3xddy+ lim\nr→∞∞/integraldisplay\n−∞dt/integraldisplay\nTdddy/contintegraldisplay\nSMrr2dΩ2.(2.45)\nGiven that for t=±∞the metric is flat7, the contribution of the integrals over Σ ±∞is zero. Taking, in\naddition, into account the asymptotic behavior of2hand its derivatives (see Appendix D), one concludes that\n(2.44) takes the form:\n∆PM=/integraldisplay\nW∂NSMN/parenleftbig2h/parenrightbig\ndDx. (2.46)\nFinally, repeating the same steps and arguments as between equat ions (2.30) and (2.32) and taking into account\nthe asymptotic behavior relevant to the case at hand, one ends up with the same expression as in the M1,D−1\n5This becomes even more evident if one denotes the pair of indi ces (MN) asaand the pair ( LR) asb. Then the action of Λ on τ\ntakes the form τ′a= Λabτb.\n6The condition τ′\nMNkN= 0 givesD−1 constraints. This is due to the fact that k2= 0, which allows an arbitrariness in τ′of\nthe formαkMkN.\n7In this calculation we ignore the self-action of fields upon t he particles, but even if we did not, the change of (infinite) s elf-field\nmomenta would be zero.11\nFigure 2: The angles in lab frame used in the text.\ncase for the emitted momentum\n∆PM=1\n2/integraldisplay\nW2hPQ,M✷2ψPQdDx. (2.47)\nIn terms of the Fourier transformed quantities defined by\nhMN(x,y) =1\n(2π)4V/summationdisplay\nn/integraldisplay\nhMN\n(n)(kµ)e−ikµxµ+iniyi/Rd4k,\nτMN(x,y) =1\n(2π)4V/summationdisplay\nn/integraldisplay\nτMN\n(n)(kµ)e−ikµxµ+iniyi/Rd4k. (2.48)\n(Vis the volume of the d−dimensional torus) the emitted momentum in the transverse to the brane directions\ntakes the form\n∆Pi=/integraldisplay\nR3×Tdτi0d3xddy∼/integraldisplay\ndk0dk′0/summationdisplay\nn∈Zdei(k0−k′0)tk0\n[(k0)2−k2−k2\nT][(k′0)2−k2−k2\nT]ni, (2.49)\nwherekis a wave 3-vector on the brane and k2\nT≡n2/R2. ∆Pivanishes since the integrand is odd under\nparityn→−n. The emitted momentum tangential to the brane can be obtained alo ng the same lines as above\nwith the only difference of using the discrete Fourier transformatio n on the torus. Finally, the emitted energy\n(E≡∆P0) is\nE=κ2\nD\n32π3V/summationdisplay\nn∈Zd∞/integraldisplay\n0k2d|k|/integraldisplay\nS2dΩτSN(k)τ∗\nLR(k)ΛSNLR/vextendsingle/vextendsingle/vextendsingle\nk0=√\nk2+k2\nT, (2.50)\nwhereτMN(k) is the four-dimensional Fourier-transform of the total source evaluated at k0=/radicalbig\nk2+k2\nT.\nThewavevectoroftheemitted gravitationalwaveisagenuine D-dimensionalvectorwith quantizedtransverse\ncomponents. We denote\nkM= (kµ, ki\nT), ki\nT=ni\nR, ni∈Z. (2.51)\nThe notation and conventions for the angles used in this paper is sho wn in Figure 2. For the brane four-vector\nkµwe use the following parametrization ensuring zero square of kM:\nkµ=/parenleftbigg/radicalBig\n̟2+k2\nT, ̟sinθcosϕ, ̟sinθsinϕ, ̟cosθ/parenrightbigg\n. (2.52)\nFinally, using (2.42) the formula for the emitted energy becomes\nE=κ2\nD\n32π3V/summationdisplay\nP/summationdisplay\nn∈Zd∞/integraldisplay\n0̟2d̟/integraldisplay\nS2dΩ/vextendsingle/vextendsingleτSN(k)εSN\nP/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle\nk0=√\n̟2+k2\nT. (2.53)12\nD. Graviton polarizations\nThe computation of the emitted energy can be simplified considerably if one chooses appropriately the or-\nthonormal basis of the D(D−3)/2 graviton polarization tensors {εP\nMN}that appear in (2.43) and (2.53). The\nconstruction of such a convenient basis is presented here.\n•The first step towards their construction is to define D−2 space-like unit vectors orthogonal to kandu′\nand among themselves. The first D−4 vectorseM\nα,α= 3,...,D−2 we choose to be orthogonal to the collision\nthree-dimensional subspace spanned by the vectors uM, u′M, bMwhich lie on the brane and together with k\nfully characterize the kinematics of the problem under study. Thus , they are taken to satisfy the orthogonality\nproperties\neM\nαeβM=−δαβ,(eαk) = (eαu′) = (eαu) = (eαb) = 0. (2.54)\nAs mentioned above, in the lab frame u′M= (1,0,...,0). Also, with no loss of generality the vectors eαcan be\nchosen to have vanishing components along the brane.\nConsider next that the brane Cartesian coordinate frame {t,x,y,z}together with the above D−4 space-like\nunit vectors eM\nαform aD-dimensional right-handed basis{t,x,y,z,e 3,...,e D−2}and choose the D-dimensional\nLevi-Civita symbol so that ǫ0xyz3...(D−2)= 1. The two remaining space-like vectors e1ande2are defined as\nfollows:eM\n2is chosen orthogonal to both uM, u′M, namely\neM\n2=N−1ǫMM1M2...MD−1uM1u′\nM2kM3e3M4...e(D−2)MD−1, (2.55)\nwhereNis the normalization factor which can be obtained by squaring the abo ve expression. Specifically,\nmaking use of the relation k2= 0 one obtains\nN2=−[(u′k)u−(uk)u′]2. (2.56)\nAccording to this definition\n(e2u) = (e2u′) = (e2k) = (e2eα) = 0, α= 3,4,...,D−2. (2.57)\nFinally, the remaining vector e1is chosen orthogonalto u′and to all previously constructed polarizationvectors.\nIt is given by\neM\n1=N−1/bracketleftbigg\n(ku)u′M−(ku′)uM+/parenleftbigg\n(uu′)−(ku)\n(ku′)/parenrightbigg\nkM/bracketrightbigg\n, (2.58)\nwith the same normalization factor. Note that eM\n2is confined on the brane, while eM\n1has both brane and bulk\ncomponents due to kM.\n•The next step is to use the above basis of orthonormal polarization vectors to bu ild theNP=D(D−3)/2\ntraceless mutually orthogonal polarization tensors εMN\nP, i.e. satisfying:\nεMN\nPηMN= 0 andεMN\nPεP′MN=δPP′. (2.59)\nFirst, we build from e1ande2two tensors, the ones familiar from four-dimensional gravity:\nεMN\n(I)=eM\n1eN\n1−eM\n2eN\n2√\n2, εMN\n(II)=eM\n1eN\n2+eM\n2eN\n1√\n2, (2.60)\nor, equivalently, the corresponding chiral ones\nεMN\n±=εMN\n(I)±iεMN\n(II)√\n2=eM\n±eN\n±, eM\n±=eM\n1±ieM\n2√\n2. (2.61)\nThe remaining set of tensors is taken to have the same general for m as the type-I and type-II above, namely to\nhave (up to coefficients) the form of a sum of same index binomials eM\nieN\nifor the first kind, and different index\nbinomialse(M\nieN)\njfor tensors of the 2nd type. Thus, the third tensor involving e1, e2, contains the diagonal\nsum of the D−4 vectorseM\nα\nεMN\n(III)=/radicalbigg\n2(D−4)\nD−2/bracketleftBigg\n1\nD−4D−2/summationdisplay\nα=3eM\nαeN\nα−eM\n1eN\n1+eM\n2eN\n2\n2/bracketrightBigg\n. (2.62)13\nForD>4 one can build the larger set of ( D−2)(D−3)/2 tensors similar to ε(II), starting with the full set of\nunit vectors e1, e2, eα, namely\nεMN\n(ij)=eM\nieN\nj+eM\njeN\ni√\n2, i<j, j = 1,2,...,D−2. (2.63)\nFinally, the remaining D−5 polarization tensors εαare obtained from the linearly independent tensors\nfMN\nα=1/radicalbig\n(D−5)(D−4)\n(D−5)eM\nαeN\nα−/summationdisplay\nβ/negationslash=αeM\nβeN\nβ\n, α= 3,...,D−3. (2.64)\nbuilt from eα, after ortho-normalization to satisfy the conditions (2.59)8. ForD/lessorequalslant5 there are no such\npolarizations.\n•Finally, one can verify that ΛMNLRcan be written in terms of these D(D−3)/2 polarization tensors\nεP,P= I,II,III,(ij),α, in the form\nΛMNLR=NP/summationdisplay\nPεMN\nPεLR\nP. (2.65)\nAs advertised, the advantage of using the above special choice of polarization tensors in (2.43) and (2.53) is now\nevident. The indices of the energy-momentum tensor τMNare carried by the four independent vectors u,u′,b,k,\nof which the first three belong to the brane, while kMis multidimensional. These four entirely characterize the\nbremsstrahlung process. A consequence of the fact that the ve ctorseαare by construction orthogonal to those,\nis that only the three polarizations, namely with P=±and III contribute to (2.43) and (2.53). Moreover, the\neαterms ineIIIcan also be dropped and the latter simplifies to the “effective” polaris ation:\nεMN\n(III)=−/radicalBigg\nD−4\n2(D−2)/bracketleftbig\neM\n1eN\n1+eM\n2eN\n2/bracketrightbig\n=−/radicalbigg\n2(D−4)\nD−2e(M\n+eN)\n−. (2.66)\nIncidentally, notice that this effective εMN\n(III)vanishes, as it should, in the special case of four dimensions.\nHaving chosen the vectors eM\n1andeM\n2to be orthogonal to kMandu′M, only the scalar products of eM\n1and\neM\n2withuMandbMwill enter in the computation of the radiated energy momentum. Acc ording to (2.57),\n(e2u) vanishes. The remaining non-vanishing products ( e1b), (e1u) and (e2b) are given in the Appendix A2.\n3. THE GRAVITATIONAL RADIATION SOURCE\nTo linear order the metric perturbation is just the sum hMN+h′\nMNof the separate contributions of the\ntwo particles. They are for each particle the solution of (2.18) with t he corresponding zeroth order energy\nmomentum tensor (2.17) for its source. In Fourier space they are given by9\nhMN=/summationdisplay\nl∈Zdhl\nMN(q), hl\nMN(q) =2πκDm\nq2−q2\nTeiqbδ(qu)/parenleftbigg\nuMuN−1\nd+2ηMN/parenrightbigg\n; (3.1)\nh′\nMN=/summationdisplay\nl∈Zdh′l\nMN(q), h′l\nMN(q) =2πκDm′\nq2−q2\nTδ(qu′)/parenleftbigg\nu′\nMu′\nN−1\nd+2ηMN/parenrightbigg\n, (3.2)\nwhereqi\nT=li/R, the transverse components of the momentum, and q2\nT=/summationtext(qi\nT)2.\nThe deflection of the trajectory of particle mdue to its gravitational interaction with m′is obtained by\ninserting the above h′\nMNin (2.21) and solving for1zM. One obtains\n1zM(s) =−im′κ2\nD\n(2π)3V/summationdisplay\nl∈Zd/integraldisplay\nd4qδ(qu′)\n(q2−q2\nT)(qu)e−iqb/parenleftBig\ne−i(qu)s−1/parenrightBig/bracketleftbigg\nγu′M−1\nd+2uM−γ2\n∗\n2(qu)qM/bracketrightbigg\n,(3.3)\n8One could a priori take α= 3,4,...,D−2 for the label of the polarizations fα. However, they are not independent, since as one\ncan easily verifyD−2/summationdisplay\nα=3fMN\nα= 0.\n9It is worth noting that in four dimensions and in the center of mass frame the limit of hMNform→0 andγ→ ∞withE=mγ\nfixed, is the Aichelburg-Sexl metric [38].14\nwhereγ2\n∗≡γ2−1/(d+2). Similarly for1z′M. As expected the trajectory deviation in the transverse directio ns\nδyi∼/summationdisplay\nl∈Zd/integraldisplay\nd4qqi\nT\nq2−q2\nTδ(qu′)\n(qu)2e−iqb/parenleftBig\ne−i(qu)s−1/parenrightBig\n(3.4)\nvanishes, since the summand is odd under parity ( qi\nT→−qi\nT), and as expected the particles do not leave the\nscattering plane.\nA. The local part1TMN+1T′\nMNof the source\nThe contributions of the particle trajectories to the gravitationa l radiation source is given in (2.25). They\ncan equivalently be written in the form\n1TMN(k) =eikz(0)m/integraldisplay\ndτei(ku)τ/bracketleftbigg\ni(k1z)uMuN+2u(M1˙zN)−κDh′\n2uMuN+2κDuSh′\nS(MuN)/bracketrightbigg\n,(3.5)\nand\n1T′\nMN(k) =eikz′(0)m′/integraldisplay\ndτei(ku′)τ/bracketleftbigg\ni(k1z′)u′\nMu′\nN+2u′\n(M1˙z′\nN)−κDh\n2u′\nMu′\nN+2κDu′\nShS\n(Mu′\nN)/bracketrightbigg\n,(3.6)\nrespectively. With the specific choice of polarization tensors made a bove, the projection of1T′\nMN(k) on any of\nthem vanishes. Then, substituting the h′and1zgiven in (3.2) and (3.3), one obtains:10\n1TMN(k) =mm′κ2\n(2π)2Veikb/summationdisplay\nl∈Zd/bracketleftbigg/parenleftbigg\nγku′\nkuIl−γ2\n∗\n2(ku)2(k·Il)/parenrightbigg\nuMuN+γ2\n∗\n(ku)u(MIl\nN)/bracketrightbigg\n, (3.7)\nin terms of the integrals IlandIl\nM, defined by\nIl=/integraldisplayδ(pu′)δ(ku−pu)e−i(pb)\np2−p2\nTd4p, Il\nM=/integraldisplayδ(pu′)δ(ku−pu)e−i(pb)\np2−p2\nTpMd4p, (3.8)\nwithpi\nT=li/R. These were computed in [42] and expressed in terms of Macdonald f unctions\nIl=−2π\nγvK0(zl), Il\nM=−2π\nγvb2/parenleftbigg\nbzK0(zl)γu′\nM−uM\nγv+iˆK1(zl)bM/parenrightbigg\n, (3.9)\nwith argument\nzl≡(z2+p2\nTb2)1/2, z≡(ku)b\nγv, (3.10)\nwhile the hatted Macdonald functions are defined by ˆKν(x)≡xνKν(x).\nSubstituting back into (3.7) one obtains\n1TMN(k) =−mm′κ2\n4πVeikb\nγv3/summationdisplay\nl∈Zd/bracketleftBigg\n−iΓˆK1(zl)\nz2σMN−2ΓγK0(zl)u(Mu′\nN)+/parenleftbigg/parenleftbig\n2v2−Γ/parenrightbig\nγz′\nz−Γ/parenrightbigg\nK0(zl)uMuN/bracketrightBigg\n,\n(3.11)\nwhere\nσMN≡(kb)uMuN−2(ku)u(MbN)and Γ≡γ2\n∗\nγ2= 1−1\n(d+2)γ2. (3.12)\n10Note, incidentally, that as expected the partof1zMwhich corresponds to uniformmotion, i.e. the term −1 in the firstparenthesis,\ndoes not contribute to (3.7).15\nFinally, omitting the terms longitudinal in u′, since they give zero when contracted with any polarization tensor ,\nthe above expression simplifies to11\n1TMN(k) =−mm′κ2\n4πVeikb\nγv3/summationdisplay\nl∈Zd/bracketleftBigg\n−iΓˆK1(zl)\nz2σMN+/parenleftbigg/parenleftbig\n2v2−Γ/parenrightbig\nγz′\nz−Γ/parenrightbigg\nK0(zl)uMuN/bracketrightBigg\n.(3.13)\nAs in previous analogous cases [32, 42], the exponential fall-off of th e Macdonald functions for large values of\nthe argument zlleads to an effective natural cut-off Ninton the number of interaction modes lin the sum. One\ncan estimate the radius lintof the sphere in the {li}space, beyond which the modes can be neglected, by setting\n(lint/R)2b2∼1, (3.14)\nfrom which the number of contributing interaction modes is obtained\nNint∼/parenleftbiggR\nb/parenrightbiggd\n∼V\nbd. (3.15)\nForb≪R, which is the case of interest here, one has Nint≫1, and the mode-summation can be converted to\nintegration using [42] ( Z >0)\n1\nV/summationdisplay\nlˆKλ/parenleftBig/radicalbig\nZ2+pT2b2/parenrightBig\n≃1\n(2π)d/2bdˆKλ+d/2(Z), (3.16)\nto end up with\n1TMN(k) =−λei(kb)\nγv3/bracketleftBigg\n−iΓˆKd/2+1(z)\nz2σMN+/parenleftbigg/parenleftbig\n2v2−Γ/parenrightbig\nγz′\nz−Γ/parenrightbigg\nˆKd/2(z)uMuN/bracketrightBigg\n, (3.17)\nwhere\nλ≡κ2\nDm′m\n2(2π)d/2+1bd. (3.18)\nAngular and frequency characteristics. The local radiation source (3.17) above is expressed solely in terms o f\nMacdonald functions with argument z. As has been demonstrated in analogous instances before [32, 42], the\nexponential fall-off of these functions implies an effective cut-off of O(1) in the angle and frequency dependent\nvariablez, which eventually constrains the angular and frequency distributio n of the radiation. Below, it will\nbe shown that a similar situation arises as well with the non-local amplit udeSMN.\nB. The non-local part SMNof the source\nThe stress tensor SMNis defined in (A.5) and calculated in coordinate representation in (2.9) . Again, since\nits trace does not contribute to the emitted momentum, one may re strict himself to its traceless part. Write,\nas explained above, hMN+h′\nMNfor the leading metric perturbation and substitute in SMN, while omitting\nself-action terms, to obtain for its n-th mode (in analogy to (2.48))\nS(n)\nMN(xµ) =1\nV/summationdisplay\nl/bracketleftbigg\nh(n−l)A,B\nM/parenleftBig\nh′(l)\nNB,A−h′(l)\nNA,B/parenrightBig\n−1\n2h(n−l)\nAB,Nh′(l)AB\n,M−1\n2h(n−l)\nMN✷h′(l)+\n+/parenleftBig\nh(n−l)\nMA,NB+h(n−l)\nNA,MB−h(n−l)\nAB,MN−h(n−l)\nMN,AB/parenrightBig\nh′(l)AB/bracketrightbigg\n+{h←→h′},(3.19)\nand withS(n)\nMN(xµ) the four-dimensional Fourier-transform of S(n)\nMN(kµ). Upon substitution of the solutions\n(3.1) we have in momentum representation12\nS(n)\nMN(k) =mm′κ2\nDei(kb)\n4π2/bracketleftBig\nuMuN(ku′)2Jn(k)+2γ(ku′)u(MJn\nN)(k)+γ2\n∗Jn\nMN(k)/bracketrightBig\n. (3.20)\n11In what follows we will omit from the radiation source all ter ms, which give zero when contracted with all polarization te nsors,\nsince they do not contribute to the emitted energy-momentum .\n12All terms, which upon contraction with the polarization ten sors give zero, have been omitted from S(n)\nMN(k).16\nThe non-vanishing components of the scalar, vector and tensor in tegrals that appear in (3.20), namely\nJn(k)≡1\nV/summationdisplay\nl/integraldisplay\nd4pδ(pu′)δ(ku−pu)e−i(pb)\n(p2−p2\nT)[(k−p)2−(ki\nT−pi\nT)2], pM= (pµ,pi\nT), p2≡pµpν(3.21)\nJn\nM(k)≡1\nV/summationdisplay\nl/integraldisplay\nd4pδ(pu′)δ(ku−pu)e−i(pb)\n(p2−p2\nT)[(k−p)2−(ki\nT−pi\nT)2]pM, (3.22)\nand13\nJn\nMN(k)≡1\nV/summationdisplay\nl/integraldisplay\nd4pδ(pu′)δ(ku−pu)e−i(pb)\n(p2−p2\nT)[(k−p)2−(ki\nT−pi\nT)2]pMpN, (3.23)\nare computed in Appendix B. Using (B.8), (B.15) and (B.17) one obtain s14\nSMN(k) =λei(kb)\nv1/integraldisplay\n0/bracketleftbiggbMbN\nb2γΓˆKd/2+1(ζn)+/parenleftbigg\n2i(ku′)u(MbN)+Γ\nγv2/bracketleftBig\n2iγ2v2b(M˜NN)−uMuN/bracketrightBig/parenrightbigg\nˆKd/2(ζn)+\n+b2/parenleftbigg\nΓγ˜NM˜NN+1\nγ(ku′)2uMuN+2(ku′)u(M˜NN)/parenrightbigg\nˆKd/2−1(ζn)/bracketrightbigg\ne−i(kb)xdx, (3.24)\nwithζngiven in (B.7). Upon substitution of ˜NM≡−[(1−x)(ku)+γx(ku′)]uM/(γ2v2), one ends-up with the\nfully expanded expression\nSMN=λei(kb)\nv1/integraldisplay\n0/braceleftbiggbMbN\nb2γΓˆKd/2+1(ζn)+/bracketleftbigg\n2iu(MbN)/parenleftbigg\n(ku′)/parenleftbigg\n1−Γx\nv2/parenrightbigg\n−Γ\nγv2(1−x)(ku)/parenrightbigg\n−Γ\nγv2uMuN/bracketrightbigg\nˆKd/2(ζn)\n+/bracketleftbigg\n(ku′)2/parenleftbigg\n1−2x\nv2+Γx2\nv4/parenrightbigg\n−2(ku)(ku′)(1−x)\nγv2/parenleftbigg\n1−Γx\nv2/parenrightbigg\n+Γ(ku)2(1−x)2\nγ2v4/bracketrightbiggb2\nγuMuNˆKd/2−1(ζn)/bracerightbigg\ne−i(kb)xdx.\n(3.25)\nLike the local source (3.17), the non-local SMNin its effective form (3.25) is also confined on the brane.\nFurthermore, SMNis a linear combination of terms proportional to integrals of the form\nJ(σ,τ)≡Λd1/integraldisplay\n0xσe−i(kb)xˆKd/2+τ(ζn)dx, (3.26)\n(withσ= 0,1,2), defined in (B.19) and studied in Appendix B15. In [32] it was shown that these integrals can\nbe written as a sum of two terms. One, is expressed in terms of hatt ed Macdonald functions of only the variable\nzand the other hatted Macdonalds of another variable z′. So, one writes in an obvious notation\nJ(σ,τ)=J[z]\n(σ,τ)+J[z′]\n(σ,τ), z≡(ku)b\nγv, z′≡(ku′)b\nγv. (3.27)\nThe total radiation amplitude is thus naturally written as a sum of two terms. Both expressed in terms of\nMacdonald functions, but with argument zthe first and z′the second. As explained in [32] the characteristics\noftheMacdonaldfunctionstranslateeventuallyintoangularandfr equencydistributionpropertiesoftheemitted\nradiation.\n13Note that the bulk components Jn\niandJn\niMvanish by parity only if ki\nT= 0.\n14We passed again from summation into integration. Also, to si mplify notation a bit, the index ( n), labeling the number of emission\nKK-mode, is suppressed.\n15Note that the coefficient Λ d, which appeared naturally in the definition (B.19) of Append ix B, is kept here also, in order to\navoid the introduction of more symbols. Also, note that J(σ,τ)are clearly functions of kandn, which are suppressed in order to\nsimplify notation a bit.17\nC. Destructive interference of the beamed high-frequency c omponents\nAs explained in [31] and [32], if the radiation source consisted of the lo cal partTMNalone, which is expressed\nin terms of Macdonalds with argument z, the emitted radiation would be mainly z-type, i.e. beamed ϑ∼γ−1,\nwith high characteristic frequency ω∼γ2/b. Now, it will be shown, exactly like in scalar radiation discussed in\n[32], that in the sum TMN+SMNthe two leading powers of γin the ultra-relativistic expansions of TMNand\nSMNexactly cancel and the total amplitude is suppressed, compared t o each one separately, by two powers of\nγ. This cancelation is a consequence of the equivalence principle: geod esics of the ultrarelativistic particles and\nof the emitted massless quanta stay close together and enhance t he formation length of radiation [39].\nWe will not repeat here the derivation presented in [32]. Instead, w e will just highlight the main steps of the\nderivation in the present context.\nStep 1: Forz-type radiation, one has z∼O(1) andz′∼O(γ). Thus, the contribution of J[z′], which contains\nonly Macdonalds of z′are exponentially suppressed and can be neglected. Thus, the dom inant contribution to\nz-type radiation is due to J[z].\nStep 2: The integrals appearing in SMNwithσ= 1,2 are suppressed compared to the one with σ= 0.\nConsider for instance J(1,τ). Using (B.22, B.32) and (B.35) from Appendix B one can write for its lea ding terms\nJ[z]\n(1,τ)=Λd\nξ2/parenleftbigg\n−sin2ϑˆKd/2+τ+1(z)−(d+2τ+3)β2\na4ξ2ˆKd/2+τ+2(z)+β2\na4ξ2ˆKd/2+τ+3(z)/parenrightbigg\n+... (3.28)\nwitha,βandξdefined in Appendices A and B. This is of order of a−2≪1 compared to J[z]\n(0,τ)(given in\nAppendix B), since in the region of interest a=O(γ).\nStep 3: Since the destructive interference effect has to do with the two le ading terms in the local and non-local\namplitudes, we focus on exactly these two. Hence, we neglect ( ku)/γcompared to ( ku′), set Γ = 1 and v= 1\nand omitx−andx2−terms in the integrand of (3.25). Thus, the leading part of SMNreads\nSMN≃λei(kb)1/integraldisplay\n0/bracketleftbiggbMbN\nb2γˆKd/2+1(ζn)+/parenleftbigg\n2iu(MbN)(ku′)−1\nγuMuN/parenrightbigg\nˆKd/2(ζn)+z′2γˆKd/2−1(ζn)uMuN/bracketrightbigg\ne−i(kb)xdx.\n(3.29)\nStep 4: The quantities to be compared are the projections of SMNandTMNon the polarization tensors. It\nis easy to check, using the formulae of Appendix A for the contract ions ofuMandbMwith the polarization\nvectors, that at small angles ϑ/lessorsimilarγ−1no extra powers of γare introduced through such contractions. Thus,\ncomparing directly the coefficients of uMuN,u(MbN)andbMbN, one ends-up with the following expression for\nthe leading terms of SMN:\nSMN≃λei(kb)1/integraldisplay\n0/bracketleftBig\n2iu(MbN)(ku′)ˆKd/2(ζn)+z′2γˆKd/2−1(ζn)uMuN/bracketrightBig\ne−i(kb)xdx, (3.30)\nStep 5: Using (B.10) appropriately simplified for the range of parameters o f interest here, namely\n1/integraldisplay\n0dxe−i(kb)xˆKτ−1(ζn)≃1\nγzz′ˆKτ(z)−i(kb)\n(γzz′)2ˆKτ+1(z)+... ,\ninto (3.30), one obtains\nSMN≃λei(kb)/bracketleftbiggz′\nzˆKd/2(z)uMuN−i1\nγz2ˆKd/2+1(z)σMN/bracketrightbigg\n, (3.31)\nplus contributions to the real and the imaginary parts suppressed by two powers of γcompared to the terms\nkept in (3.31).\nStep 6: Making the same approximationsfor the local part1TMNin (3.17), one ends-up with1TMN=−SMN\ngiven in (3.31) above and proves the so called destructive interference of the leading terms of local and stress\namplitudes.\nD. The total radiation amplitude\nTo summarize, the total radiation source τMNwas naturally split into the sum of a function of zand a\nfunction of z′. The two leading terms in the expansion of the function of zin powers of γwere shown to cancel.18\nThe next step is to compute the leading terms of τMN, which survive destructive interference and use them\nto compute the emitted radiation energy to leading order. The discu ssion follows the steps described in [32]\nfor scalar radiation, only repeated three times because of the thr ee relevant polarizations. That is, for each\npolarization one first distinguishes three characteristic regimes of angular and frequency distribution of the\nemitted energy, namely (a) the beamed ( ϑ∼1/γ) high-frequency ( ω∼γ2/b) regime, (b) the beamed medium-\nfrequency ( ω∼γ/b) regime, and (c) the unbeamed ϑ∼1 medium-frequency regime, and then computes the\nemitted energy in each one of them.\n1. The beamed, high-frequency radiation amplitude\nIn this regime z′∼O(γ) and the contribution of Macdonald functions of z′is exponentially suppressed.\nPolarization I: Projecting (3.17) and (3.25) on εMN\nIand taking into account that only the eM\n1eN\n1/√\n2 part\nofεMN\nIcontributes, one obtains for1TI≡1TMNεMN\nIandSI≡SMNεMN\nI\n1TI(k) =−λei(kb)\n√\n2γv3/bracketleftBigg\n−iΓˆKd/2+1(z)\nz2(kb)[ψ2γ2−1]+/parenleftbigg/parenleftbig\n2v2−Γ/parenrightbig\nγz′\nz−Γ/parenrightbigg\nˆKd/2(z)γ2v2sin2ϑ/bracketrightBigg\n,(3.32)\nand\nSI=λγei(kb)\n√\n2v1/integraldisplay\n0/braceleftbigg\n(cos2φcos2ϑ−sin2φ)ΓˆKd/2+1(ζn)+/bracketleftbigg\n2ivcosϑ(kb)/parenleftbigg\n1−Γx\nv2−ψΓ(1−x)\nv2/parenrightbigg\n−Γsin2ϑ/bracketrightbigg\n×\n׈Kd/2(ζn)+/bracketleftbigg1\nψ2/parenleftbigg\n1−2x\nv2+Γx2\nv4/parenrightbigg\n−2(1−x)\nψv2/parenleftbigg\n1−Γx\nv2/parenrightbigg\n+Γ(1−x)2\nv4/bracketrightbigg\nz2v4sin2ϑˆKd/2−1(ζn)/bracerightbigg\ne−i(kb)xdx,(3.33)\nrespectively. The next step is to expand in powers of γ. For that one has to use the expansions of cos ϑ,v, Γ\nandψin Appendix A, the fact that integrals with extra xαin the integrand are suppressed by γ−2α, and the\nexpressions for the integrals J[z]\n(0,τ)andJ[z]\n(1,τ)given in Appendix B and (3.28). After a somewhat tedious but\nstraightforward calculation one ends-up with the leading part of τI≡τMNεMN\nI:\nτI≃λei(kb)\n√\n2sin2ϑ\nγψ/bracketleftBigg\nd+1\nd+2ˆKd/2(z)+(d+1)ˆKd/2+1(z)\nz2+/parenleftbiggγ2ψ2cos2φ\nsin2ϑ−sin2φ/parenrightbiggˆKd/2+2(z)\nz2/bracketrightBigg\n.(3.34)\nPolarization II: Similarly, projection onto the second polarization tensor leads (with an analogousnotation)\nto\n1TII=2iλei(kb)\n√\n2vˆKd/2+1(z)\nzγsinϑsinφ, (3.35)\nand\nSII=γλei(kb)\n√\n2v1/integraldisplay\n0/bracketleftbigg\nΓcosϑsin2φˆKd/2+1(ζn)−2izsinϑsinφ/parenleftbiggv2\nψ/parenleftbigg\n1−Γx\nv2/parenrightbigg\n−Γ(1−x)/parenrightbigg\nˆKd/2(ζn)/bracketrightbigg\ne−i(kb)xdx.\nFollowing the same steps as above, one obtains for the leading ultra- relativistic term in their sum\nτII≃λei(kb)\n√\n2γ/parenleftbig\nψ+sin2ϑ/parenrightbig\nsin2φˆKd/2+2(z)\nz2. (3.36)\nPolarization III: Repeating the same steps for the third polarization tensor, while ta king into account that\nonly thebMbN−binomial does not vanish upon projection on eM\n2eN\n2, one obtains:\n1TIII=−/radicalbigg\nd\nd+21TI, (3.37)\nand\nSIII=−/radicalbigg\nd\nd+2\nSI+λγei(kb)\n√\n21/integraldisplay\n02sin2φˆKd/2+1(ζn)e−i(kb)xdx\n. (3.38)19\nCombining these two and expanding, one has to leading order\nτIII=−/radicalbigg\nd\nd+2λei(kb)\n√\n2sin2ϑ\nγψ/bracketleftBigg\nd+1\nd+2ˆKd/2(z)+(d+1)ˆKd/2+1(z)\nz2+/parenleftbiggγ2ψ2\nsin2ϑ−sin2φ/parenrightbiggˆKd/2+2(z)\nz2/bracketrightBigg\n.(3.39)\nUp to the common phase factor and the factor λin front, all three amplitudes are real and of O(1/γ).\n2. Wide-angle, medium-frequency radiation amplitude\nIn this regime it is more convenient to consider directly the behavior o f the radiation source, before one\nprojects onto the three polarization tensors. Also, in this regime z∼γ, and consequently the Macdonald\nfunctions of zgive exponentially suppressed contributions and will be neglected. T hus, the leading contribution\nis due to the z′−dependent part of the non-local source SMN, which we compute next.\nIn Subsection 3C it was shown that for ω∼γ2/b, ϑ∼γ−1one hasJ(1,τ)∼J(0,τ)/γ2.This implied that only\nthe neighborhood of x= 0 contributes to these integrals.\nSimilarly,itwillbeshownthatintheregionofinteresthere,namely( ω∼γ/b, ϑ∼1),theleadingcontribution\nto these integrals comes from the neighborhood of x= 1. Indeed, consider the integral with (1 −x), i.e.\nJ(0,τ)−J(1,τ). Using (B.32) and subsequently (B.22) this difference takes the for m\nJ(0,τ)−J(1,τ)≃Λde−i(kb)\nξ2/bracketleftBigg/parenleftbigg\n1−2\nψ/parenrightbiggˆKd/2+τ+1(z′)\nγ2+cos2φ\na2ˆKd/2+τ+3(z′)+.../bracketrightBigg\n. (3.40)\nThis is of orderO(γ−2) with respect to both J(0,τ)andJ(1,τ). This implies that the integrals J(0,τ)andJ(1,τ)\nare almost equal, and the main contribution in them comes from the do mainx= 1−0, and consequently, that\nintegral with (1−x)2is even more suppressed.\nThus, it is natural to rearrange the terms of the integrand in (3.25 ) in powers of 1−xas well as in powers\nofγ, taking into account that each power of 1 −xcontributes a factor of O(1/γ2) to the integral. Taking in\naddition into account that in the region of interest here γ(ku′) is of the same order as ( ku), one obtains\nτMN≃λei(kb)1/integraldisplay\n0/braceleftBigg\nbMbN\nb2γˆKd/2+1(ζn)+/bracketleftbigg\n2iu(MbN)/parenleftbigg/parenleftbigg\n(ku′)−(ku)\nγ/parenrightbigg\n(1−x)−d+1\nd+2(ku′)\nγ2/parenrightbigg\n−uMuN\nγ/bracketrightbigg\nˆKd/2(ζn)\n−1\nd+2z′2\nγuMuNˆKd/2−1(ζn)/bracerightBigg\ne−i(kb)xdx. (3.41)\nUsing (e1u) =γvsinϑ, (e2u) = 0 and (e1b) =O(b) = (e2b), one can simplify (3.41) and write instead\nτMN≃λei(kb)\nγ1/integraldisplay\n0/bracketleftbiggbMbN\nb2γ2ˆKd/2+1(ζn)−uMuNˆKd/2(ζn)−1\nd+2z′2uMuNˆKd/2−1(ζn)/bracketrightbigg\ne−i(kb)xdx.(3.42)\nIt is straightforward to check that its projection onto any of the three polarization tensors gives to leading order\nthe same result as the projection of (3.41).\nFrom (B.22) one obtains to leading ultra-relativistic order\n1/integraldisplay\n0e−i(kb)xˆKτ(ζn)dx≃e−i(kb)\nz′2γ2ψˆKτ+1(z′), (3.43)\nusing which in (3.42), one finally ends-up with\nτMN≃λ\nz′2γ3ψ/bracketleftbiggbMbN\nb2γ2ˆKd/2+2(z′)−uMuNˆKd/2+1(z′)−1\nd+2z′2uMuNˆKd/2(z′)/bracketrightbigg\n.(3.44)\n3. The beamed, medium-frequency radiation amplitude\nStarting again from (3.17) and (3.25), one can observe that for ϑ∼γ−1the projection on polarization tensors\ndoes not introduce γ-factors, hence one can estimate the order in γdirectly from the coefficients of the tensor\nbinomialsuMuN,bMbNandu(MbN). In the regime of interest here one has z∼1/γ,z′∼1, using which\none can immediately conclude that both local and stress sources, a s well as their projections on the relevant\npolarizations are of the same order, namely TMN∼SMN∼τ∼O(γ) in any dimension.20\nE. Summary\nThe results of this section are summarized in Table I below, which show s the leading behavior of the corre-\nsponding amplitudes after projection on the polarization tensors.\n❅\n❅❅ϑωω∼1/b ω∼γ/b ω∼γ2/b\nγ−1no destructive interference\nτ∼T≫Sno destructive interference\nS[z]∼T∼S[z′]∼γdestructive interference: T≈−S[z]\nS[z′]∼exp(−γ),\nτ=O(T/γ2)∼1/γ\n1no destructive interference\nτ∼T∼Sdestructive interference:\nS[z]≈T∼exp(−γ)\nτ=S=S[z′]∼γ−1destructive interference\nT∼S∼τ∼exp(−γ)\nTable I. The leading behavior of the amplitudes (projected o n polarizations) in the various characteristic angular-\nfrequency regimes. S[z]andS[z′]stand for the contributions of integrals J[z]andJ[z′], respectively. The dependence of\nλonγis not included, i.e. all estimates are given in units λ= 1.\nNotice that the entries in Table I do not depend on the dimensionality d. Also, note that this leading\ndependence of the amplitudes on γis valid in the whole range [0 ,2π] of the azimuthal angle ϕ.\n4. THE EMITTED ENERGY\nThe emitted energy, obtained from (2.43) for M= 0 is\nE=κ2\nD\n4(2π)D−1/summationdisplay\nP∞/integraldisplay\n0ωD−2dω/integraldisplay\nSD−2dΩ/vextendsingle/vextendsingleτLR(k)εLR\nP/vextendsingle/vextendsingle2. (4.1)\nIt differs from the scalar radiation case studied in [32] by the summa tion over polarizations and the substitution\nof the scalar charge f→κDm. Thus, it is expected to lead up to numerical coefficients of order O(1) to the\nsame behavior in γas the energy emitted in the scalar case. On the basis of the qualitat ive arguments given\nin the beginning of Section 4 of [32] and the subsequent numerical co mputations one obtains16for the emitted\nenergy in the various angular-frequency domains the behavior sum marized in Table II.\n❍❍❍❍❍ϑωDωD≪γ/b ωD∼γ/b ωD∼γ2/b ωD≫γ2/b\nγ−1negligible\n(phase space)E∼γ3,fromTandSE∼γd+2,fromT+S[z]negligible radiation\n1negligible\n(phase space)E∼γd+1,fromS[z′]negligible radiation negligible radiation\nTable II. The relative contribution in the total emitted ene rgy from different characteristic regions of angle and\nfrequency. All estimates are given in units λ=b= 1.\nA. The total radiated energy\nAccording to Table II the frequency and angulardistribution ofthe dominant component ofradiationdepends\non the dimensionality d. Thus, like in the scalar case [32] the study of the emitted energy h as to proceed\nseparately for d/greaterorequalslant2 and ford= 0,1. Furthermore, the following analysis is a triple repetition of the sca lar\ncase, because of the three polarizations.\n16Up to a logarithmic overall factor for d= 1, discussed below.21\nd/greaterorequalslant2.\nFord/greaterorequalslant2 the dominant radiation is beamed with characteristic frequency ar oundω∼γ2/b. According to\nTable I, the emission amplitude is O(γ−1), dominated by its real part.\nForR≫bthe KK-summation is replaced by integration and the relevant formu la for the emitted energy is\n(2.43). One next substitutes the high-frequency amplitudes (3.34 , 3.36 and 3.39) corresponding to the three\npolarizations and neglects their behavior at smaller frequencies. Sq uaring each amplitude, one recognizes the\nappearance of products of Kd/2(z),ˆKd/2+1/z2andˆKd/2+2/z2with themselves, and also the presence of the\nfactorωd+2from phase space. Thus, one starts with\ndE\ndΩ=κ2\nD\n4(2π)D−1/summationdisplay\nP∞/integraldisplay\n0ωD−2dω/vextendsingle/vextendsingleτLR(k)εLR\nP/vextendsingle/vextendsingle2, (4.2)\nand upon integration over the angles except ϑone obtains\ndE\ndϑ=(κ3\nDmm′)2γd+1\n8(2π)2d+5b3d+3sind+3ϑ\nψd+32/summationdisplay\na,b=0C(d)\nabD(d)\nab(ϑ). (4.3)\nwhere\nC(d)\nab≡/integraldisplay\nˆKd/2+a(z)ˆKd/2+b(z)zd+2(δ0a+δ0b−1)dz, (4.4)\nD(d)\n00(ϑ) = 2(d+1)3\n(d+2)3sin2ϑ\nγ2ψ2D(d)\n01(ϑ) = (d+2)D(d)\n00(ϑ)\nD(d)\n02(ϑ) =d+1\n(d+2)2/bracketleftbigg\nd−(d+1)sin2ϑ\nγ2ψ2/bracketrightbigg\nD(d)\n11(ϑ) = (d+2)2D(d)\n00(ϑ)\nD(d)\n12(ϑ) = (d+2)D(d)\n02(ϑ) D(d)\n22(ϑ) =d+1\nd+2/parenleftbigg\n2γ2ψ2\nsin2ϑ+3\n4sin2ϑ\nγ2ψ2/parenrightbigg\n+2\nd+2+γ2sin2ϑ.(4.5)\n(all ofO(1) forϑ∼1/γ), while use was made of (C.5):\n/integraldisplay\nSd+1dΩd+1= Ωd+1,/integraldisplay\nSd+1cos2φdΩd+1=1\n2Ωd+1,/integraldisplay\nSd+1cos4φdΩd+1=3\n8Ωd+1. (4.6)\nUpon integration over ϑusing (C.2) one ends up with\nE=πd/2+1(κ3\nDmm′)2γd+2\n8(2π)2d+5Γ(d/2+1)b3d+32/summationdisplay\na,b=0C(d)\nabD(d)\nab, (4.7)\nwhere the constants D(d)\nabare given by\nD(d)\n00= 2d+5(d+1)3\n(d+2)3Γ/parenleftbigd+6\n2/parenrightbig\nΓ/parenleftbigd+4\n2/parenrightbig\nΓ(d+5)D(d)\n01= (d+2)D(d)\n00\nD(d)\n02=−2d+3(d+1)(2d+3)\n(d+2)2Γ/parenleftbigd+4\n2/parenrightbig\nΓ/parenleftbigd+2\n2/parenrightbig\nΓ(d+4)D(d)\n11= (d+2)2D(d)\n00 (4.8)\nD(d)\n12= (d+2)D(d)\n02 D(d)\n22= 2d−2(5d+6)(3d2+15d+20)Γ/parenleftbigd+2\n2/parenrightbig\nΓ/parenleftbigd\n2/parenrightbig\nΓ(d+4).\nFinally, after the summation in (4.7) one obtains for the total emitte d energy in the lab frame\nE≃Cd(κ3\nDmm′)2\nb3d+3γd+2(4.9)\nwithC2= 6.23×10−4,C3= 2.61×10−4,C4= 1.98×10−4,C5= 2.16×10−4,C6= 3.10×10−4.\nd= 1.\nThe emitted energy flux as well as its frequency and angular distribu tions ind= 0,1 were computed numer-\nically.22\n(a) (b)\nFigure 3: Frequency (a) and angular (b) distribution of the t otal emitted energy for d= 0 and γ= 103.\nAccording to Tables I and II in this case most of the radiation is beame dϑ/lessorsimilar1/γ, but, as it was described\nin [32] and immediately follows from the destructive interference, in fi ve dimensions the frequency distribution\nfalls as 1/ωin the region from O(γ/b) toO(γ2/b), hence the total emitted energy grows as γ3lnγ, while the\ndominant contribution comes from this entire region. From the know n expressions for the amplitudes in integral\nform, we compute the integral numerically for several values of γand conclude:\nE≃C1(κ3\n5mm′)2\nb6γ3lnγ , C 1= 1.66×10−4. (4.10)\nd= 0.\nAgain, according to Tables I and II in this case most of the radiation is beamed within ϑ/lessorsimilar1/γand with\nfrequencies around ω∼γ/b, with the contribution from higher frequencies decaying as 1 /ω2. In this domain\nthe local and non-local parts of the amplitude are equally important . Thus, one has to add (3.17) and (3.25)\nto compute the total radiation amplitude, project onto the two po larizationsǫ±, square each one of those, add\nthem up and multiply by the appropriate phase space factor. After the angular integration one obtains the\nfrequency (or z′) distribution of the emitted energy shown in Figure 3a, while the angu lar distribution of the\ntotal emitted energy (which carries the anisotropy) is given in Figur e 3b.\nIntegrating the frequency distribution numerically over ωfrom 0 to +∞one obtains for the total emitted\nenergy\nE≃C0(κ3\n4mm′)2\nb3γ3, C 0= 5.7×10−4. (4.11)\nB. The low frequency part of the spectrum\nA few comments, related to the low frequency part of the classical radiation spectrum, are in order here.\n(a) Ford= 0the distribution dE/dωof the emitted energy has non-vanishing finite zero-frequency limit .\nIndeed, comparing local (3.17) and stress (3.25) sources one con cludes that for ω→+0 the dominant part of\nthe amplitude is given by the imaginary part of the local piece17(omit the phase factor ei(kb)), whose behavior\nis\nτMN(k)≃−iλΓ\nγv3ˆK1(z)\nz2σMN∼1\nω. (4.12)\n17In the diagrammatic quantum mechanical treatment of soft gr avitational radiation there is a corresponding statement, namely\nthat the soft radiation emitted from internal lines is negli gible. Only gravitons attached on the external lines contri bute in the\nzero-frequency limit.23\nUpon contraction with the polarizations ε±, substitution of the finite zero-limit ˆK1(0) = 1 of the hatted\nMacdonald function, and setting v= 1,Γ = 1, one obtains\nτ±≃iλ\n2γsinϑ\nωb/bracketleftbigg\ncosφ/parenleftbigg1\nγ2ψ2−1/parenrightbigg\n±2isinφ\nψ/bracketrightbigg\n. (4.13)\nSubstitute this into (4.2) and integrate over the azimuthal angle φ, to obtain\ndE\ndω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=κ2\n4λ2γ2\n16(2π)2b2/integraldisplay\ndϑsin3ϑ/parenleftbigg\n1+4\nψ2+1\nγ4ψ4/parenrightbigg\n(4.14)\nplus subleading powers of γ. Note that in the low frequency limit allangles (0 /lessorequalslantϑ/lessorequalslantπ) contribute to dE/dω.\nFinally, using V3\n0≃4/3 (C.3),V3\n4≃4γ4/3 (C.2),V3\n2≃4(ln2γ−1) (C.4), and substituting the expression\nforλ, one ends-up with\ndE\ndω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=(κ3\n4mm′)2\n4(2π)4b2γ2/parenleftbigg\nln2γ−5\n6/parenrightbigg\n, (4.15)\nwhich for κ4=m=m′=b= 1 andγ= 103agrees with the value obtained numerically in Figure 3a18.\n(b) One can multiply the above result for dE/dω|ω=0by the range ωmax∼γ/bto obtain a rough estimate of\nthe total emitted energy. Its γdependence differs from (4.11) by a logarithmic factor.\nHowever, this simple method to obtain an order of magnitude estimat e cannot be used in d >0, because\naccording to Table II the dominant contribution to the emitted ener gy comes from the high frequency regime,\nwhich does not have overlap with the ω→0 domain.\n(c) Ford >0, one cannot replace at small frequencies the KK-mode summation w ith integration. One has\ninsteadtokeepseparatelythethree-dimensionalcoordinates ̟,θ,ϕandtheextra-dimensionaldiscretemomenta\nkT. In this case\nz=(ku)b\nγv=b\nv/parenleftbigg/radicalBig\n̟2+k2\nT−̟vcosθ/parenrightbigg\nandz′=(ku′)b\nγv=b\nγv/radicalBig\n̟2+k2\nT,\nwhile (kb) =−̟sinθcosϕ.\nThe real part inside the square bracket of (3.13) is regular or blows -up logarithmically at ̟= 0. Thus,\nit vanishes when multiplied by the volume measure. Consequently, the dominant contribution to the emitted\nenergy is due to the imaginary part, i.e. (ignoring the irrelevant phas e factor)\nτMN(k)≃1TMN(k)≃imm′κ2\nD\n4πV1\nγv3/summationdisplay\nl∈ZdˆK1(zl)\nz2σMN. (4.16)\nTaking into account that ˆK1(zl) is regular at all frequencies, the low-frequency behavior is deter mined by the\nfactorσMN/z2∼ω/z2. Fork2\nT>0 this limit is zero and does not contribute. For kT= 0 we have ̟=ω\nand thereby σMN/z2∼1/ωas in four dimensions. Thus, as expected, only the zeroth emission mode n= 0\ncontributes in the summation over nof (2.53) in the ω→0 limit.\nNow, forkT= 0z=ωbψ(θ)/vwithψ(θ)≡1−vcosθand\nτMN(k)≃mm′κ2\nD\n4πγV/summationdisplay\nl∈ZdˆK1(|pT|b)\nω2b2ψ2(θ)σMN, pi\nT=li\nR. (4.17)\nForb≪R, which is the case of interest here, the summation over the interac tion-KK modes can be restricted\nup to modes with |pT|b∼1. The result, which can also be obtained by the usual substitution o f the summation\nwith integration, is\nτMN(k)≃λ\nγ2d/2Γ(d/2+1)\nω2b2ψ2(θ)σMN, (4.18)\nand subsequently, using (2.53), one concludes that\ndE\ndω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=2d−2Γ2(d/2+1)(κ3\nDmm′)2\n(2π)d+4Vb2d+2γ2/bracketleftbigg\nln2γ+1\n6/parenleftbiggd\nd+2−5/parenrightbigg/bracketrightbigg\n. (4.19)\n18Note that this result differs from the one obtained in [18], wh ere it was claimed that dE/dωblows-up logarithmically as ω→0.24\nThe volume Vof the internal space survives in the denominator, because, as ex plained above, only the n= 0\nmode of radiation contributes. Of course, for d= 0 (4.19) coincides with (4.15).\n(d)InMD>4the frequency spectrum vanishes for ω→0. There are many ways one can convince himself\nabout this. First, it is obtained by taking the limit V→∞of (4.19). Alternatively, observe that in higher\ndimensional Minkowski the relative contribution on the various radia tion modes satisfies/summationtext\nn=0/slashbig/summationtext\nn2>0→0.\nTaking in addition into account that non-zero modes ndo not contribute in the limit ω→0, one concludes\nthatdE/dω|ω=0vanishes in higher dimensions. Finally, the same conclusion is reached b y taking the behavior\nτ∼1/ω, square and multiply by the measure ωd+2. Again, for d >0 the limit of the integrand for ω→0 is\nzero.\nC. Quantum constraints\nThe above results were obtained in the context of classical theory . However, quantum mechanics restricts\nthe region of validity of a classical computation. Thus, to ensure th e reliability of the above conclusions, apart\nfrom the condition b≪R, which allows one to replace the summations over KK modes with integr ations, and\nwhich will be assumed in the sequel, one has to justify (a) the use of t rajectories for the colliding particles,\n(b) the expansion around flat space for the gravitational field, an d (c) the use of classical field description of\nthe radiation. From the analysis of the elastic ultrarelativistic scatt ering problem [42] one concludes that the\nclassical computation around flat space-time of the cross-sectio n is identical to the eikonal approximation of the\nquantum answer, as long as the conditions\n√s= 2mγcm≫M∗, b>b γ≡rSγ1/(d+1)\ncm, b<b c≡rS(rS/λB)2/d(4.20)\nare satisfied19. For the radiation problem at hand, the weak particle-recoil condit ion due to the emission of\ngravitons with characteristic frequency ωis satisfied if the momenta of the emitted gravitons are much smaller\nthan the momentum transfer of the elastic collision. However, expe rience from analogous computations of the\ntotal energy of synchrotron radiation shows that this condition c an be relaxed and replaced, instead, by the\nweakerω≪E0, whereE0≃mγis the total energy of the colliding particles. When the emitted energ yEis of\norderE0, this condition also guarantees a large number of emitted quanta, a nd justifies further the description\nof radiation with a classical field20.\n1. The ω∼γ2/bandω∼1/bregimes\n•Radiation with characteristic frequency ω∼γ2/bdoes not satisfy the above constraints, and consequently\nmay not be reliably treated classically in the presence of extra dimens ions. Indeed, for ω∼γ2/bthe condition\nω <mγ leads tob>γ/m>b c, which contradicts the requirement b<bc. Thus, in the sequel we shall ignore\nthe radiation with characteristic frequency γ2/b, even though according to Table II it is classically dominant\nford/greaterorequalslant2.\n•On the contrary, as explained in [32] and shown in Table II the radiatio n emitted in the soft frequency\nregimeω/lessorsimilar1/b, satisfies all the above conditions, but its energy is negligible by phas e space in all dimensions,\nand will not be discussed further.\n2. The ω∼γ/bregime\nAccording to Table II the dominant radiation in this regime is beamed fo rd= 0,1, almost isotropic for d/greaterorequalslant3,\nwhile ford= 2 the two components are of comparable importance.\nForω∼γ/bthe quantum condition ω<mγ reduces tomb>1.\nThus, in terms of the rescaled dimensionless parameters\nE≡√s\nM∗, µ≡m\nM∗, β≡bM∗ (4.21)\n19For colliding particles with equal masses γ2\ncm= (1 +γ)/2∼γ/2\n20In what follows and in order to simplify notation we replace t he strong inequalities by simple ones.25\na reasonable set of requirements for the validity of the above class ical results in this intermediate frequency\nregime is21:\nE≫1, µ≪1, β >/parenleftbiggE2\nµ/parenrightbigg1/(d+1)\n, β <E2/d, βµ> 1, (4.22)\nwhich are equivalent to\nE≫1, µ≪1,1\nµ</parenleftbiggE2\nµ/parenrightbigg1/(d+1)\n<β <E2/d, (4.23)\nand can easily be satisfied in all dimensions22.\nThus, we shall next compute the energy emitted in the ω∼γ/bcharacteristic frequency regime23.\nD. Radiation emission with characteristic frequency ω∼γ/b\nAs explained above, the estimates in Table II were obtained under th e assumption that the frequency inte-\ngrations could be restricted to the appropriate regime correspon ding to each entry, i.e. from a fraction of the\ncorresponding characteristic frequency to a few times that freq uency. The quantum conditions discussed above\nrestrictωtoω/lessorequalslantωmax=mγ, which translates to z′\nmax=mb>1.\n1.d/greaterorequalslant3\nThus, ford/greaterorequalslant3, according to Tables I and II the leading contribution to the emitte d energy is obtained\nfrom (3.44) integrated over phase space with z′up toz′\nmax. But, the fact that the integrand is exponentially\nsuppressedbeyond z′∼1 , allowsone with negligibleerror(in the overallcoefficient) to extend thez′integration\nto +∞. As far as the lower limit is concerned it is a priori 1 /b, since (3.44) is valid for ω >O(b−1). However,\nsince the integrand in (4.4) is regular at z′= 0 one can shift the lower limit of integration to 0, the relative\nerror being of order 1 /γ2. This reproduces the power of γestimated in Table II. The purpose of the rest of this\nsubsection is essentially to obtain the overall coefficient.\nProjecting (3.44) onto ε±andεIIIone obtains\nτ±≃λ\n2z′2γψ/bracketleftbigg\n(cos2φcos2ϑ−sin2φ±isin2φcosϑ)ˆKd/2+2(z′)−sin2ϑ/parenleftbigg\nˆKd/2+1(z′)+1\nd+2z′2ˆKd/2(z′)/parenrightbigg/bracketrightbigg\nτIII≃−/radicalBigg\nd\n2(d+2)λ\nz′2γψ/bracketleftbigg\n(1−cos2φsin2ϑ)ˆKd/2+2(z′)−sin2ϑ/parenleftbigg\nˆKd/2+1(z′)+1\nd+2z′2ˆKd/2(z′)/parenrightbigg/bracketrightbigg\n.(4.24)\nOne then according to (2.43) has to square these, multiply with the p hase space factor and integrate over angles\nand frequencies.\nThe result of the integration over frequencies takes the form\ndE\ndΩd+2=(κ3\nDmm′)2γd+1\n16(2π)2d+5b3d+3ψ22/summationdisplay\na,b=0C(d)\nab¯D(d)\nab(ϑ,φ), (4.25)\nwhereC(d)\nabare given by (4.4) and\n¯D(d)\n00(ϑ,φ) =d+1\n(d+2)3sin4ϑ ¯D(d)\n01(ϑ,φ) = (d+2)¯D(d)\n00(ϑ,φ)\n¯D(d)\n02(ϑ,φ) =−sin2ϑ\nd+2/bracketleftbigg\ncos2φ+2\nd+2(cos2φsin2ϑ−1)/bracketrightbigg\n¯D(d)\n11(ϑ,φ) = (d+2)2¯D(d)\n00(ϑ,φ)\n¯D(d)\n12(ϑ,φ) =−(d+2)¯D(d)\n02(ϑ,φ) ¯D(d)\n22(ϑ,φ) =d+1\nd+2(1−cos2φsin2ϑ)2.(4.26)\n21The condition µ≪1 is necessary to justify the use of point-particle, instead of black-hole description of the colliding particles.\n22Ford= 0 there is no upper bound on the allowed values of β.\n23Incidentally, this is the frequency regime in which one woul d a priori expect to have dominant radiation: Due to the Loren tz\nsqueezing of the field lines along the direction of relative m otion of the colliding particles, the characteristic durat ion of the\ncollision in the lab frame becomes τ∼b/γ. Thus, the characteristic frequency of the emitted radiati on should be ω∼1/τ∼γ/b.26\nThe values of C(d)\nabare to be computed with help of (C.6).\nThe integration of (4.26, 4.25) over the angles except ϑ(i.e. over the sphere Sd+1) is easily performed using\nformulae (4.6), while the final integration over ϑis done with the help of (C.3)24. Making use also of (4.4), one\nfinally obtains for the emitted energy\nE=¯Cd(κ3\nDmm′)2\nb3d+3γd+1, (4.27)\nwith¯C3= 1.08×10−4,¯C4= 4.77×10−5,¯C5= 4.03×10−5and¯C6= 4.98×10−5.\n2.d= 0\nAs explained above, the z′integration extends a priori up to z′\nmax∼mb>1. It was checked numerically that\n(a) the estimate of the cubic power of γgiven in Table II is correct, and (b) that the coefficient is insensitive t o\nthe upper limit of integration, varying by a factor of two as one chan ges the upper limit of the z′integration\nbetween 1 and∞.\nThus, (4.11) as a reliable estimate.\n3.d= 1,2\nIn this case the estimation of the contribution to the emitted energ y from the ω∼γ/bpart of the spectrum\nwas done numerically. Specifically, one started with (3.17) and (3.25) and integrated numerically over ϑfrom 0\ntoπand overz′from 0 to something of order 1. For a reliable estimate it is enough to in tegrate up to z′= 1.\nThe result for d= 2 is\nE≃0.04(κ3\n6mm′)2\nb9γ3lnγ (4.28)\nand ford= 1\nE≃0.9×10−4(κ3\n5mm′)2\nb6γ3. (4.29)\nE. Extreme radiation efficiency for d/greaterorequalslant3andω∼γ/b\nConsider specifically the case m=m′, divide the emitted energy by the total energy E0=m+mγof\nthe colliding particles and use the definition of rSgiven in the Introduction, to write the radiation efficiency\nǫ≡E/E0in the form25\nǫ∼/parenleftBigrS\nb/parenrightBig3(d+1)\nγcm∼E4\nµβ3(d+1), d= 0,1 (4.30)\nand\nǫ∼/parenleftBigrS\nb/parenrightBig3(d+1)\nγ2d−3\ncm∼E2d\nβ3(d+1)µ2d−3, d/greaterorequalslant2. (4.31)\nIt should be pointed out that these expressions differ from the gen eric formula ǫ∼(rS/b)3(d+1)γ2d+1\ncmgiven\nerroneously in [33]. The difference is due to an error in [33] related to t he extent of destructive interference.\nNevertheless, the qualitative conclusions presented in [33] are still valid. Incidentally, the powers of γin the\nemitted energy differ from the scalar radiation studied in [32] only by t he substitution f→κDm.\n24At small angles expressions (4.24) are not valid. But as befo re, if we substitute integralπ/integraldisplay\nO(1/γ)dθbyπ/integraldisplay\n0dθ, the relative error is\nO(1/γ) since the integrand doesn’t blow up at ϑ= 0+O(1/γ). Thus we may apply (C.3).\n25Since it is not our intention to engage in numerology, we igno re in these formulae inessential d-dependent coefficients, e ven though\nthey are roughly of O(10d), or the factor ln γ, which is present for d= 2.27\nIn the parameterrangedefinedin (4.23) the radiationefficiencyis alw aysmuchsmallerthanone for d= 0,1,2.\nIndeed, consider the case d= 0. The maximum ǫis obtained for the minimum allowed value β∼(E2/µ)1/(d+1)\nof the impact parameter and is equal to\nǫmax(d= 0)∼/parenleftBigµ\nE/parenrightBig2\n≪1. (4.32)\nThus, ind= 0 a small fraction of the available energy is emitted in gravitational r adiation and vanishes in the\nmassless limit. However, one can convince her/himself that for d/greaterorequalslant3 the efficiency can take values arbitrarily\nclose to one or even become much greater than one for a wide range of parameters obeying the classicality\nconditions (4.23) stated above. For instance, in d= 4 the minimum value of ǫis obtained for β∼E2/dand is\nǫmin(d= 4)∼√\nE\nµ5≫1. (4.33)\nDitto for dimensions d >4. This is a priori a puzzling result. It leads for d/greaterorequalslant3 to unphysical, i.e. greater\nthan one, values of the radiation efficiency, and actually blows-up in t he massless limit ( µ→0). In relation\nto this we would like to offer the following comments: (a) The massless lim it is a very singular limit in our\napproach. The characteristic frequencies and angles all become s ingular in that limit. In particular, since in\nthat limitbc/bγ∼(E2/dµ)1/(d+1)→0, it is impossible to satisfy (4.23), even if one ignores the “quantum”\nconstraintβµ >1. Thus, the massless limit lies outside the domain of validity of our appr oximation. (b)\nWeinberg has shown [41] that in d= 0 there is no infrared divergence in the soft graviton emission rate from\nmassless colliding particles. This is in agreement with our results not on ly for the soft component in d= 0,\nbut also for soft as well as high frequency emission in d= 0,1,2. (c) The explicit computation presented here\nis rather involved, but, as explained, the powers of γin the emitted energy are determined on general grounds,\nbased on properties of Macdonald functions, number of KK interac tion and emission modes and dimensionality\nof phase space. (d) One may suspect that some classicality conditio n is missed, which might render the above\nformulae non-applicable. But the constraints used in (4.23) are quit e straightforward and based on well-known\nphysics. So, (e) it may well be the case, that the above results are reliable and the interpretation of their\npuzzling features is that in ultra-planckian particle collisions there is a lot of energy emitted in gravitational\nradiation. This will lead to strong radiation damping, which should be ta ken into account in the study of such\nultra-planckian scattering processes.\n5. CONCLUSIONS\nA detailed study was presented of classical gravitational radiation emitted in ultra-relativistic collisions of\nmassive point-particles interacting gravitationally. The space-time was assumed to have an arbitrary number\nof toroidal or non-compact extra dimensions and the post-linear a pproximation scheme of General Relativity\nwas employed for the computation. The angular and frequency dist ributions of radiation, as well as the total\nemitted energy were studied in detail to leading ultra-relativistic ord er.\nThree characteristic frequency regimes (1 /b,γ/bandγ2/b) of the emitted radiation were identified and the\ncharacteristics of the dominant contribution was determined in var ious dimensions.\nIn particular, in any number of dimensions the soft component of ra diation is mainly due to the scattered\nparticles, with negligible contribution coming from the cubic graviton in teraction term26. However, in contrast\nto the four-dimensional case, in any number of extra dimensions d>0 the frequency spectrum of the emitted\nradiation vanishes as ω→0 and the total emitted energy in soft gravitons is negligible.\nAlso, contrary to what happens with the soft radiation emission, th e cubic graviton interaction and the\nscattered particles themselves are equally important as sources o f radiation with high frequency. In fact it was\nshown that in any dimension they lead to partial cancellation ( destructive interference ) of the total beamed\nradiation amplitude in the high frequency domain, as a result of which t he emitted energy in the γ2/bfrequency\nregime is reduced by two powers of the Lorentz factor γin the Lab frame.\nThe relevance of the classical analysis to the full quantum radiation problem was also discussed. The classi-\ncality conditions , necessary for the classical treatment to be a good approximatio n to the full quantum problem\nwere derived and the radiation efficiency ǫ, i.e. the fraction of the initial energy which is emitted in gravita-\ntional radiation, was computed for parameter values inside the reg ion of validity of our classical computation.\nAlthough for d= 0,1,2,ǫis smaller than one in the whole allowed range of parameters, it can be a rbitrarily\nclose to one or even exceed unity in d/greaterorequalslant3. One possible interpretation of this “unphysical” result is that inde ed\n26This is a well known fact, verified easily also in the context o f Feynman diagram infrared graviton summation.28\nin such ultra-planckian collisions there is a lot of gravitational radiatio n, which will lead to strong damping,\nwhich should be included in a reliable treatment of the scattering proc ess.\nHowever, the implementation of this interpretation, the proper tr eatment of the massless limit, the extension\nof the region of validity to impact parametersoutside the range (4.2 3) and the comparison with quantum results\nbased, for example, on string theory, are currently under invest igation and, hopefully, will be the subject of a\nfuture publication.\nAcknowledgements\nWe are grateful to Dr. Georgios Kofinas for useful discussions an d to Mr. Yiannis Constantinou for help\nwith the numerical work. This work was supported in part by EU gran ts PERG07-GA-2010-268246, the EU\nprogram “Thales” ESF/NSRF 2007-2013 and grant 11-02-01371- a of RFBR. It has also been co-financed by\nthe European Union (European Social Fund, ESF) and Greek nation al funds through the Operational Program\n“Education and Lifelong Learning” of the National Strategic Refer ence Framework (NSRF) under “Funding of\nproposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes”. DG and\nPS are grateful to the Department of Physics of the University of Crete for its hospitality in various stages of\nthis work. PS is also grateful to the DAAD service for co-funding an d to the LMU, Munich, for its hospitality\nduring the final stage of the work.\nAppendix A: Definitions and variations\n1. Perturbation theory variation over gravitational const ant in flat background\nWe use the notation of Weinberg [37].\ng(1)\nMN=hMN Γ(1)R\nMN= (hR\nM,N+hR\nN,M−h,R\nMN)/2\nΓ(1)N\nMN=h,M/2 Γ(1)M\nNRηNR= 0. (A.1)\nThe definition of Riemann and Ricci tensors:\nRMNLR= ΓM\nNR,L−ΓM\nNL,R+ΓM\nSLΓS\nNR−ΓM\nSRΓS\nNL, R MN≡RLMLN. (A.2)\nThus\nR(1)\nMN=1\n2(hR\nN;MR+hR\nM,NR−✷hMN−h,MN) =−1\n2✷hMN\nR(1)=−✷h+h,MN\nMN−RMNhMN=−1\n2✷h\nG(1)\nMN=1\n2(−✷ψMN−ηMNξ,L\nL+ξM,N+ξN,M) =−1\n2✷ψMNξM=∂NψMN= 0.(A.3)\nThe second-order variations:\nΓ(2)M\nLR=−1\n2hMN(hLN,R+hRN,L−hLR,N)\nΓ(2)L\nML=−1\n2hLRhLR,M Γ(2)M\nNRηNR= 0\nR(2)\nMN=−1\n2/parenleftbigg\nhA,B\nM(hNB,A−hNA,B)−1\n2hAB\n,MhAB,N+hAB(hMA,NB+hNA,MB−hAB,MN−hMN,AB)/parenrightbigg\nR(2)= (gLRRLR)(2)=−hLRR(1)\nLR+ηLRR(2)\nLR=1\n2/parenleftbigg\n2hAB✷hAB−hAB,LhAL,B+3\n2hAB,LhAB,L/parenrightbigg\nG(2)\nMN=R(2)\nMN−1\n2ηMNR(2)−1\n2hMNR(1). (A.4)\nAs a result:\nSMN=−G(2)\nMN=1\n2/bracketleftbigg\nhAB(hMA,NB+hNA,MB−hAB,MN−hMN,AB)−1\n2hAB\n,MhAB,N−1\n2hMN✷h+\n+hA,B\nM(hNB,A−hNA,B)+1\n2ηMN/parenleftbigg\n2hAB✷hAB−hAB,LhAL,B+3\n2hAB,LhAB,L/parenrightbigg/bracketrightbigg\n.(A.5)29\n2. Useful kinematical formulae\nThe angles in the formulae below are defined in Fig.2.\nuµ≡γ(1,0,0,v), u′µ≡(1,0,0,0), ψ≡1−vcosθcosα= 1−vcosϑ, k Ti=ni\nR\n̟≡|k|,cosα=̟\nω, z′=(ku′)b\nγv=ωb\nγv, z=(ku)b\nγv=ωb\nvψ=z′γψ, z2\nl≡z2+b2pT2, pTi=li\nR\nξ2≡2γzz′−z2−z′2=̟2b2sin2θ+b2k2\nT= (ωbsinϑ)2= (γvz′sinϑ)2\n−(kb) =ξcosφ=γz′vsinϑcosφ=γz′vcosαsinθcosϕ=̟bsinθcosϕ\nβ≡γzz′−z2=̟2b2cosϑ(1−vcosϑ)\nvcos2α=γ2z′2ψ(1−ψ). (A.6)\nNext we compute non-vanishing products of polarization vectors: Using (2.16), (2.51), (2.52) and these\ndefinitions one finds the following scalar products\n(ku) =γ/parenleftbigg/radicalBig\n̟2+k2\nT−̟vcosθ/parenrightbigg\n,(ku′) =/radicalBig\n̟2+k2\nT,(uu′) =γ. (A.7)\nTo calculate the scalar product ( be2) one observes that ǫxz0τ3...D−2bxuzu′\n0kτe3...eD−2=−bγvk⊥,wherek⊥is\nthe length of the projection of kMonto the subspace orthogonal to t,x,z, sincekis transverse to all eα’s and\nthe formǫτ3...D−2kτe3...eD−2, remaining after factoring out the t,x,zsubspace, represents the volume of the\nrectangle. Thus\nk⊥=/radicalBig\n̟2sin2θsin2ϕ+k2\nT=ωsinφ, (A.8)\nand one finds\n(be2) =−b/parenleftbigg̟2sin2θsin2ϕ+k2\nT\n̟2sin2θ+k2\nT/parenrightbigg1/2\n=−b/parenleftbiggcos2αsin2θsin2ϕ+sin2α\ncos2αsin2θ+sin2α/parenrightbigg1/2\n=−bsinφ. (A.9)\nEvaluation of the scalar product ( e1b) is straightforward and leads to\n(be1) =−b\n2̟2sin2θcosϕ/radicalBig\n(̟2sin2θ+k2\nT)(̟2+k2\nT)=−b\nvcosφ/parenleftbigg\n1−(ku)\nγ(ku′)/parenrightbigg\n=(kb)\n(ku′)ctgϑ. (A.10)\nFinally, the remaining non-vanishing scalar product is\n(e1u) =γv/parenleftbigg̟2sin2θ+k2\nT\n̟2+k2\nT/parenrightbigg1/2\n=γv/radicalbig\ncos2αsin2θ+sin2α=γvsinϑ. (A.11)\nAppendix B: Momentum integrals\n1. Basic scalar integral\nThe local integrals for massless and massive modes are calculated in [3 1]. The main integral for stress (also\nfor massive modes) is calculated in [32].\nJ(k)≡1\nV/summationdisplay\nlJnl(k), Jnl(k) =/integraldisplay\nd4pδ(pu′)δ(ku−pu)e−i(pb)\n(p2−p2\nT)[(k−p)2−(ki\nT−pi\nT)2]. (B.1)\nω≡k0=/radicalbig\nk2+kT2andkis a 3−dimensional vector lying on the 3 −brane, while kTi=ni/RandpTi=li/R\nwith integers{ni},{li}, ared−dimensional discrete vectors, corresponding to the emission and in teraction\nmodes, respectively.\nUsing Feynman parametrization Jnltakes the form:\nJnl=1/integraldisplay\n0dxe−i(kb)x/integraldisplay\nd4pδ[(pu′)+(ku′)x]δ[(pu)−(1−x)(ku)]e−i(pb)\n[p2−(kTx−pT)2]2. (B.2)30\nIntegrating over p0and splitting pinto the longitudinal p||(with respect to u) and transversal p⊥parts,\nintegrate over p||. The corresponding covariant splitting of bµon temporal, longitudinal and transversal parts\nreads\nbµ= (bu′)u′µ−γ(bu′)−(bu)\nγ2v2(γu′µ−uµ)+bµ\n⊥\nand denote scalar impact parameter bas\nb≡/radicalBig\n−b2\n⊥=/parenleftbigg\n−b2−[(bu)u′−(bu′)u]2\nγ2v2/parenrightbigg1/2\n. (B.3)\nThen, introducing in p⊥the spherical coordinates d2p⊥=|p⊥|dΩ1d|p⊥|and integrating first over the angles\nand then over|p⊥|, one obtains\nJnl=πb2\nγv1/integraldisplay\n0dxe−i(Nb)ˆK−1(ζnl), (B.4)\nwith\nζ2\nnl(x) =z′2x2+2γzz′x(1−x)+z2(1−x)2+b2(kTx−pT)2\nNµ=1\nγ2v2/bracketleftbig\nx(ku′)u′\nµ−(1−x)(ku)uµ/bracketrightbig\n−1\nγv2/bracketleftbig\nx(ku′)uµ−(1−x)(ku)u′\nµ/bracketrightbig\n+xkµ. (B.5)\nSum up over l′s with rule (3.16) to get\nJn(k) = Λd1/integraldisplay\n0dxe−i(Nb)ˆKd/2−1(ζn); Λ d≡πb2−d\n(2π)d/2γv. (B.6)\nwith\nζ2\nn(x) =z′2x2+2γzz′x(1−x)+z2(1−x)2;ζn(0) =z, ζn(1) =z′. (B.7)\nFinally, fix ( bu′) = (bu) = 0 to get\nJn(k) = Λd1/integraldisplay\n0dxe−i(kb)xˆKd/2−1(ζn). (B.8)\nThus\nJn(k) =J[z]+J[z′]. (B.9)\nIn the frequency-angular region ̺∼ωbϑ≫1J[z]andJ[z′]are given by the series over small 1 /̺2:\nJ[z]=Λd\na2ξ2/parenleftbigg\nβˆKd/2(z)−i(kb)ˆKd/2+1(z)−(d+1)β\na2ˆKd/2+1(z)+βsin2φ\na2ˆKd/2+2(z)/parenrightbigg\n+Rz (B.10)\nJ[z′]= Λde−i(kb)/parenleftbiggξ2−β\nξ2a2ˆKd/2(z′)−icosφ\na2ξˆKd/2+1(z′)/parenrightbigg\n+Rz′, (B.11)\nwhere\na≡/radicalBigg\nβ2\nξ2+z2=z\nsinϑ.\nThusJn(k) can be split in two parts with drastically different spectral-angular b ehavior each.31\n2. Vectorial and tensorial integrals\nVectorial integral is defined by\nJn\nM(k)≡1\nV/summationdisplay\nlJnl\nM(k), Jnl\nµ(k) =/integraldisplay\nd4pδ(pu′)δ(ku−pu)e−i(pb)\n(p2−p2\nT)[(k−p)2−(ki\nT−pi\nT)2]pM.(B.12)\nThus\nJn\nM(k) =i∂J\n∂bM.\nSubstituting Jn(k) in the form (B.6) and differentiating, one gets taking (B.3) into acco unt\nJn\nM= Λd1/integraldisplay\n0dxe−i(Nb)/bracketleftBigg\nNMˆKd/2−1(ζn)+iˆbM\nb2ˆKd/2(ζn)/bracketrightBigg\n, (B.13)\nwith\nˆbM≡bM−(bu)uM+(bu′)u′M−γ[(bu)u′M+(bu′)uM]\nγ2v2, N M= (Nµ;xkTi), (B.14)\nwhere it is convenient to use the following properties of hatted Macd onalds:\nˆK′\n±n(x) =−xˆK±n−1(x).\nIn special frame\nJn\nM= Λd1/integraldisplay\n0dxe−i(kb)x/bracketleftbigg\nNMˆKd/2−1(ζn)+ibM\nb2ˆKd/2(ζn)/bracketrightbigg\n. (B.15)\nSome useful products\nN·u= (ku) N·u′=Jn·u′= 0 N·k=1\nb2[x/parenleftbig\nz2+z′2−2zz′γ/parenrightbig\n+/parenleftbig\nγzz′−z2/parenrightbig\n]\nJn·u= (ku)JnN·b=x(kb) N2=1\nb2/bracketleftbig\nx2/parenleftbig\nz2+z′2−2zz′γ/parenrightbig\n−z2/bracketrightbig\n.\nTensorial integral is defined\nJMN(k)≡1\nV/summationdisplay\nlJnl\nMN(k), Jnl\nMN(k) =/integraldisplay\nd4pδ(pu′)δ(ku−pu)e−i(pb)\n(p2−p2\nT)[(k−p)2−(ki\nT−pi\nT)2]pMpN.(B.16)\nThus\nJn\nMN(k) =i∂Jn\nM\n∂bN.\nSubstituting Jn\nMin the form (B.13) and differentiating, we have\nJn\nMN=Λd\nb21/integraldisplay\n0dxe−i(kb)x/bracketleftbigg\nb2NMNNˆKd/2−1(ζn)+2iN(MbN)ˆKd/2(ζn)−bMbN\nb2ˆKd/2+1(ζn)−\n−/parenleftbigg\nηMN+1\nv2γ2/bracketleftBig\nuMuN+u′\nMu′\nN−2γu(Mu′\nN)/bracketrightBig/parenrightbigg\nˆKd/2(ζn)/bracketrightbigg\n. (B.17)\nSome products:\nJn\nMNuN= (ku)Jn\nMJn\nMNu′N= 0JMNuMuN= (ku)2Jn. (B.18)\nHere we see that integrals Jn\nM, Jn\nMNrepresent the superposition of some scalar integrals of the type\nJ(σ,τ)≡Λd1/integraldisplay\n0xσe−i(kb)xˆKd/2+τ(ζn)dx, (B.19)32\n(withσ= 0,1,2): for instance\nJn\nM=1\nvb/bracketleftbigg/parenleftbigg1\nγ[z′u′\nM+zuM]−[z′uM+zu′\nM]+vbkM/parenrightbigg\nJ(1,−1)+z/parenleftbigg\nu′\nM−1\nγuM/parenrightbigg\nJ(0,−1)+ivbM\nbJ(0,0)/bracketrightbigg\n.\n(B.20)\nTwo integrals J(0,−1)andJ(0,0)(i.e. withσ= 0) are of the type of the basic scalar integral and thereby known ,\nwhile the derived one J(1,−1)is new. The computation of integrals with σ= 1,2 represents the goal of next\nsubsection.\n3. Derived integrals for stresses\nNow consider (B.19) with σ= 1,τ=−1 :\nJ(1,−1)= Λd1/integraldisplay\n0xe−i(kb)xˆKd/2−1(ζn)dx, (B.21)\nIf we’d know J(k) exactly (B.21) can be calculated by differentiation:\nJ(1,−1)=−i\nξ∂J(0,−1)\n∂cosφ,\nOur strategy consists in the reduction of J(σ,τ)into the superposition of J(0,τ′)(with some τ′s), for which\nwe already have calculated expressions (B.10, B.11). Note that for higherτwe keep Λ dand shift index of\nMacdonalds in (B.10, B.11): thereby (B.10, B.11) may be generalized in to\nJ[z]\n(0,τ−1)=Λd\na2ξ2/parenleftbigg\nβˆKd/2+τ(z)−i(kb)ˆKd/2+τ+1(z)−(d+2τ+1)β\na2ˆKd/2+τ+1(z)+βsin2φ\na2ˆKd/2+τ+2(z)/parenrightbigg\nJ[z′]\n(0,τ−1)≃Λde−i(kb)\na2ξ2/bracketleftBig\n(ξ2−β)ˆKd/2+τ(z′)−iξcosφˆKd/2+τ+1(z′)/bracketrightBig\n. (B.22)\nRepresenting x= [x−β/ξ2]+β/ξ2, we have\nJ(1,−1)=β\nξ2J(0,−1)+Λd\nξ1/integraldisplay\n0re−i(kb)xˆKd/2−1(ζn)dx=β\nξ2J(0,−1)+Λd\nξ¯J, (B.23)\nwith notation\n¯J=1/integraldisplay\n0dxr(x)e−i(kb)xˆKd/2−1(ζn). (B.24)\nRepresenting [43, 44]\nˆKν−µ−1/parenleftBig/radicalbig\na2−r2/parenrightBig\n=r−µaν∞/integraldisplay\n0yµ+1Iµ(ry)\n(y2+1)ν/2Kν(a/radicalbig\ny2+1)dy, (B.25)\nwe have\n¯J=∞/integraldisplay\n0dyaνyµ+1\n(y2+1)ν/2Kν(a/radicalbig\ny2+1)1/integraldisplay\n0dxe−i(kb)xr1−µIµ(ry). (B.26)\nFixµ= 1/2, thenν= (d+1)/2 = (D−3)/2>−1, and\nr1/2I1/2(ry)y3/2=y√\n2√πsinh(ry). (B.27)33\nTaking the internal integral over x\n1/integraldisplay\n0dxe−i(kb)xsinh(ξ[x−β/ξ2]y) =/summationdisplay\nj=0,1(−1)j−1e−i(kb)jyξcosh(ξdjy)+i(kb)sinh(ξdjy)\nξ2y2+(kb)2,\nwith\ndj=j−β/ξ2=δ1j−β/ξ2j= 0,1,\nwe arrive at\n¯J=√\n2√πa(d+1)/2/summationdisplay\nj=0,1(−1)j−1e−i(kb)j∞/integraldisplay\n0dyK(d+1)/2(a/radicalbig\ny2+1)\n(y2+1)ν/2y2ξcoshξdjy+iy(kb)sinhξdjy\nξ2y2+(kb)2.(B.28)\nSubstituting ( kb) =−ξcosφ(A.6) to get\n¯J=√\n2√πaν\nξ/summationdisplay\nj=0,1(−1)j−1e−i(kb)j∞/integraldisplay\n0dyK(d+1)/2(a/radicalbig\ny2+1)\n(y2+1)ν/2y2coshξdjy−iycosφsinhξdjy\ny2+cos2φ.\nIn the integrand numerator add and subtract cos2φtoy2:\n¯J=√\n2√πa(d+1)/2\nξ/summationdisplay\nj=0,1(−1)j−1e−i(kb)j∞/integraldisplay\n0dyK(d+1)/2(u/radicalbig\ny2+1)\n(y2+1)(d+1)/4/bracketleftbigg\ncoshξdjy−cos2φcoshξdjy+iycosφsinhξdjy\ny2+cos2φ/bracketrightbigg\nand then, integrating cosh ξdjyin a bracket with help of (B.25) and comparing the remainder with\nJ(0,0)=21/2a(d+1)\nπ1/2Λd\nξ/summationdisplay\nj=0,1(−1)j+1e−ij(kb)∞/integraldisplay\n0dyˆK−(d+1)/2/parenleftBig\na/radicalbig\ny2+1/parenrightBigysinh(ξδjy)−icosφcosh(ξδjy)\ny2+cos2φ\n[32, eqn.(3.28)], one concludes\n¯J=1\nξ/bracketleftBig\ne−i(kb)ˆKd/2(z′)−ˆKd/2(z)/bracketrightBig\n−icosφ\nΛdJ(0,0). (B.29)\nSubstituting into (B.23) one gets the recurrence relation:\nJ(1,−1)=β\nξ2J(0,−1)+Λd\nξ2/bracketleftBig\ne−i(kb)ˆKd/2(z′)−ˆKd/2(z)/bracketrightBig\n−icosφ\nξJ(0,0). (B.30)\nThe property is hold for all indices ( τ/greaterorequalslant−1), thus we can generalize:\nJ(1,τ)=β\nξ2J(0,τ)+Λd\nξ2/bracketleftBig\neiξcosφˆKd/2+τ+1(z′)−ˆKd/2+τ+1(z)/bracketrightBig\n−icosφ\nξJ(0,τ+1)=−i\nξ∂J(0,τ)\n∂(cosφ).(B.31)\nSplitting it onto z−andz′−parts gives\nJ[z]\n(1,τ)=β\nξ2J[z]\n(0,τ)−Λd\nξ2ˆKd/2+τ+1(z)−icosφ\nξJ[z]\n(0,τ+1)\nJ[z′]\n(1,τ)=β\nξ2J[z′]\n(0,τ)+Λd\nξ2eiξcosφˆKd/2+τ+1(z′)−icosφ\nξJ[z′]\n(0,τ+1). (B.32)\nDifferentiating (B.31) over cos φ, one gets\nJ(2,τ)=β\nξ2J(1,τ)+Λd\nξ2eiξcosφˆKd/2+τ+1(z′)−1\nξ2J(0,τ+1)−icosφ\nξJ(1,τ+1). (B.33)\nSubstituting (B.31)\nJ(2,τ)=β2\nξ4J(0,τ)+Λdβ\nξ2eixξcosφˆKd/2+τ+1(ζn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n0−2iβcosφ\nξ3J(0,τ+1)+Λd\nξ2eiξcosφˆKd/2+τ+1(z′)−\n−1\nξ2J(0,τ+1)−iΛdcosφ\nξ3eixξcosφˆKd/2+τ+2(ζn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n0−cos2φ\nξ2J(0,τ+2). (B.34)34\nSplitting,\nJ[z]\n(2,τ)=β2\nξ4J[z]\n(0,τ)−Λdβ\nξ2ˆKd/2+τ+1(z)−2iβcosφ\nξ3J[z]\n(0,τ+1)−1\nξ2J[z]\n(0,τ+1)+iΛdcosφ\nξ3ˆKd/2+τ+2(z)−cos2φ\nξ2J[z]\n(0,τ+2)\nJ[z′]\n(2,τ)=β2\nξ4J[z′]\n(0,τ)+Λd1+β\nξ2eiξcosφˆKd/2+τ+1(z′)−2iβcosφ\nξ3J[z′]\n(0,τ+1)−1\nξ2J[z′]\n(0,τ+1)−\n−iΛdcosφ\nξ3eiξcosφˆKd/2+τ+2(z′)−cos2φ\nξ2J[z′]\n(0,τ+2). (B.35)\nSubstituting (B.32) into (B.20) gives the full expansion of vectorial integral in terms of Macdonald functions.\nAppendix C: Integration over frequencies and angles\n(a) Here are computed the angular integrals of the general form\nVn\nm=π/integraldisplay\n0sinnϑ\n(1−vcosϑ)mdϑ (C.1)\nwith integers m,n, which were encountered in the text. For 2 m>n+1 one finds in the leading order [31]\nVn\nm=2m−1Γ/parenleftbign+1\n2/parenrightbig\nΓ/parenleftbig\nm−n+1\n2/parenrightbig\nΓ(m)γ2m−n−1. (C.2)\nFor 2m<n+1 one obtains\nVn\nm=2n−mΓ/parenleftbign+1\n2/parenrightbig\nΓ/parenleftbign+1\n2−m/parenrightbig\nΓ(n−m+1). (C.3)\nIn the case 2 m=n+1 the behavior of the integral is logarithmic. The only one of those n eeded here is the:\nV3\n2=2\nv3/bracketleftbigg\nln1+v\n1−v−2v/bracketrightbigg\n= 4(ln2γ−1)+O(γ−2). (C.4)\n(b) Integration over the remaining angles is performed next. Since , with our choice of the coordinate system\nonly one extra angle ϕenters the amplitudes, only the following formula is needed:\n/integraldisplay\nSD−3|sinϕ|NdΩ =2π(D−3)/2\nΓ/parenleftbigD−2+N\n2/parenrightbigΓ/parenleftbiggN+1\n2/parenrightbigg\n. (C.5)\n(c) Finally, computation of the integrals over the frequency or ove r the impact parameter involving two Mac-\ndonald functions of the same argument is performed using the form ula [44]:\n∞/integraldisplay\n0Kµ(cz)Kν(cz)zα−1dz=2α−3Γ/parenleftbigα+µ+ν\n2/parenrightbig\nΓ/parenleftbigα+µ−ν\n2/parenrightbig\nΓ/parenleftbigα−µ+ν\n2/parenrightbig\nΓ/parenleftbigα−µ−ν\n2/parenrightbig\ncαΓ(α). (C.6)\nAppendix D: Asymptotic behavior of retarded fields\nIn this Appendix we derive the asymptotic behaviors of the retarde d solutions of the gravitational wave\nequation in arbitrary dimensions and of their space-time derivatives .\nThederivationisbasedonthewellknownformula[45] forthe retar dedGreen’sfunction ofthe D−dimensional\nd’Alembert operator\nGD(x) =\n\n1\n2πD/2−1θ(x0)δ(D/2−2)(x2), D =even ;\n1\n2π(D−1)/2θ(x0)/parenleftbiggd\ndx2/parenrightbiggD−3\n2/parenleftbiggθ(x2)\n(x2)1/2/parenrightbigg\n, D=odd ,(D.1)\nwithδ(−1)(x2)≡θ(x2) andx2= (x0)2−(x1)2−...−(xD−1)2.35\nFirst, suppose we have one point particle, moving along the traject ory Γ, parametrized by zM(τ) and\nproperly normilized (˙ z2= 1). The corresponding source of gravitational the field on flat ba ckground is\nTMN(x) =m/integraltext\n˙zM(τ)˙zN(τ)δ(x−z(τ))dτ.The solution of the d’Alembert equation ✷ψMN\nret=−κDTMNis\nthe convolution ψMN(x) =Gret∗TMN. Let us denote by ˆ τthe retardation point on the smooth particle-\nworldline Γ corresponding to a given observation point xM= (t,x). GivenxMand the time-like trajectory\nzM(τ), ˆτis the unique solution (with x0>z0(ˆτ)) of the equation ( x−z(τ))2= 0, and specifies the intersection\nof Γ with the past light-cone of the observation point xM. Using hats to denote all corresponding kinematical\nquantities (e.g. ˆ˙z≡˙z(ˆτ)), the Li´ enard-Wiechert solution becomes27\nψMN\nret(x) =−κDm\n4πD/2−1/parenleftbigg1\n2ρ∂\n∂ˆτ/parenrightbiggD−4\n2ˆ˙zMˆ˙zN\nρ, (D.2)\nwhere\nρ(x)≡−1\n2∂\n∂ˆτ(x−ˆz)2=ˆ˙z·(x−ˆz)>0.\nNote that in the co-moving at the retardation moment ˆ τLorentz frame, the quantity ρ(x) coincides with the\nspatial distance|x−ˆz|between the observation and the retardation points.\nNow, introduce the null vector cM= (xM−ˆzM)/ρ. It represents the properly normalized vector from the\nretardation point towards the observation point. Introduce in ad dition the vector n≡c−ˆ˙z, which satisfies\nn2=−1 andn·ˆ˙z= 0, as a consequence of ˆ˙z·c= 1 and ˆ˙z2= 1. Thus, cMis the sum of the orthogonal unit\nvectors: ˆ˙zM, which specifies a time direction, and nMwhich is space-like and determines a spatial direction,\ni.e. a point on the sphere SD−2, which is the “basis” of a null-cone with axis along ˆ˙zMand apex at ˆ zM. Thus,\nfor an arbitrary point xMon this light cone, ρis the distance of the apex to the center of the sphere which\ncontainsxM, and is also equal to the radius of that sphere. Thus, going to the w ave-zone of radiation emitted\nby the accelerated particle corresponds to considering the limit ρ→+∞keeping fixed the quantities n,ˆ˙z,ˆ¨z,\netc, which are related only to direction and refer to the apex ˆ zof the light-cone.\nNow with the help of the relations [46]\n∂Mˆτ=cM ˙ρ= (ˆ¨zc)ρ−1\n∂Mρ=ˆ˙zM+ ˙ρcM ∂McN=1\nρ/parenleftBig\nδN\nM−ˆ˙zMcN−cMˆ˙zN−˙ρcMcN/parenrightBig\n∂Mˆ˙zN=cMˆ¨zN,and similar ones for higher derivatives,(D.3)\nand consecutive-differentiation rule (D.2), one can check explicitly t hat in even dimensions the asymptotic\nbehavior of ψMNand its derivatives is\n•ψMN\nD=ψMN\nRad\nρD/2−1+...+ψMN\nNewt\nρD−3, withψMN\nRad,...,ψMN\nNewt– tensors depending on c,ˆ˙z,ˆ¨z,...\n•ψMN,P\nD=ΨMNP\nRad\nρD/2−1+...+ΨMNP\nNewt\nρD−2with ΨMNP\nRad, ...,ΨMNP\nNewt– tensors depending on c,ˆ˙z,ˆ¨z,.... The same\nasymptotic behavior holds for higher order space-time derivatives , but not for ✷ψMN, for which one\nobtains instead\n•✷ψMN\nD=˜˜ψMN\nRad\nρD/2+...+˜˜ψMN\nNewt\nρD−1.\nIn odd dimensions, on the other hand, the solutions of d’Alembert eq uations are not known in closed form\nand one has to rely on approximate asymptotic expansions. In part icular, one can use the expansion (proven\nfor scalar and vector fields in [47] and conjectured for the tenso r fieldψMN)\nψMN\nD=ψMN\nRad\nρD/2−1+O(ρ−D/2), (D.4)\n27The square root derivative, which appears in odd-dimension s, is defined as the convolution\n(d/dx)1/2f(x) =−1\n2√π/integraldisplayx\n0dt\nt3/2f(x−t).36\nwhereψMN\nRadagain depends on cMand on the retarded quantities ˆ˙z,ˆ¨zetc, but is expressed as an integral along\nthe particle trajectory from τ′=−∞toτ′= ˆτ. Taking into account that cMenters into ψMN\nRadeither in the\nproduct (ˆz−z(τ′))·cor with a free index, one can apply (D.3) to the integral and, using c2= 0 to obtain the\nsame behavior as in even dimensions, namely28:\n✷ψMN\nRad\nρD/2−1=O(ρ−D/2) and ✷ψMN\nD=O(ρ−D/2). (D.5)\nFor full gravity these properties do not hold exactly due to non-line arities, but to the order discussed here we\nhave effective linear theory with linear operator ✷and sources1T,1T′andS(1h,1h′), respectively. The first\ntwo are local currents and satisfy the assumptions made above. S(1h), on the other hand, is not a current\ngenerated by the point-like source, but, expressed explicitly with t he help of (A.5), and taking into account\nthe field equation (2.23), one can easily check that this part of ✷2ψMNfalls-off much faster at large distances,\nnamely as the product of two Coulomb fields,\n✷2ψMN(S) =−κDSMN∼1h1h′=O(ρ−(2D−6)). 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Quant ization of space-time symmetry\ngroups (Lorentz, Poincare, the conformal group) has remained a problem for incorporating ana\napplying the structure of quantum groups into physical systems. A deformation of the Lorentz\ngroup has been studied in [5] and a six-generator deformed Lorent z algebra has been found in\n[?] in terms of the chiral SL(2)×SL(2) generators.\nIn this paper we consider a realization of deformed Lorentz algebra commutation relations\nobeyed by the generators - rotations and boosts - and construc t its representations which are\nthe exact quantum analogue of the classical Lorentz group repre sentations.\nThe quantum group is generally defined as a q-deformation of the universal enveloping\nalgebra of the underlying classical group. Thus when introducing a q uantum group one should\nfirst construct a deformed associative with a non-cocommutative Hopf algebra structure; this\nis usually done in terms of generators obeying deformed Lie commuta tion relations.\nWe recall that the Lorentz group contains the SU(2) subgroup of rotations Mi,i= 1,2,3,\nand the boost generators Ni,i= 1,2,3 form an irreducible SU(2) vector representation. One\ncan form two chiral SL(2) subgroups IL\ni=Mi+iNiandIR\ni=Mi−iNiwhich act only on\nspinors with undotted and dotted indices. The rotation SU(2) subgroup is the diagonal in\nSLL(2)×SLR(2). The matrix elements of the two-dimensional fundamental rep resentation\ngenerators (and of the conjugated one) satisfy /angbracketleftNi/angbracketright=∓i/angbracketleftMi/angbracketright, the latter expressing that IL,R\ni\nvanish when acting on functions of only dotted and undotted indices respectively.\nIn defining a deformed quantum Lorentz group we wish to generalize the properties of the\nclassical Lorentz group to the q-case. We shall determine the deformed commutation rela-\ntions as relations imposed on the matrix elements of the fundamenta l representation generators\nacting on two-dimensional spinors with undotted indices. We assume a full analogy with the\nclassical case, namely that the quantum Lorentz groupcontains t heSUq(2) subgroup, the boost\noperatorstransformunder SUq(2)ascomponents ofirreducible tensor operators, andthefund a-\nmental representation has the properties of the two-dimensiona l classical spinor representation.\nLetM±,M3satisfy the deformed commutation relations of SUq(2) [7]. In the deformed\ncase the SUq(2) generators do not transform under the irreducible tensor re presentation. The\nq-analogue of an irreducible SUq(2) tensor operator has been defined [8, 9] as the set of 2 l+1\n1components Tl\nm,m=−l,...,lobeying\n[M3,Tl\nm] =mTl\nm, (1)\n[M±,Tl\nm]q−m/2= [l∓m]1/2[l±m+1]1/2Tl\nm±1qM3/2\nwhere [A,B]α\nq=AB−qαBA. There is an alternative definition to (1), obtained by q→q−1.\nAccordingly one can construct two q-vector operators SandT:\nS±=±q∓M±q−M3/2(2)\nS0= [2]−1/2(q−1/2M−M+−q1/2M+M−)\nand\nT±=±q±M±qM3/2(3)\nT0= [2]−1/2(q1/2M−M+−q−1/2M+M−)\nWe identify the operators −iq−1/4M+q−M3/2,−iq−1/4M−qM3/2and−i[M3]q−M3/2with the gen-\neratorsN+,N−andN3respectively, of a two-dimensional q-deformed Lorentz boost transfor-\nmation, expressed in a q-tensor form. We identify further the operators M±in aq-tensor form,\ni.e.M±≡q−1/4M±q∓M3/2, and the operator M3with the rotation generators of a q-deformed\nLorentz transformation. The commutation relations of a deforme d Lorentz algebra are im-\nposed as the relations obeyed by the generators of the two-dimen sional Lorentz transformation.\nNamely, the rotations M±,M3satisfy the Lie brackets of the deformed Uq(su(2)); the commu-\ntation relations between rotations and boosts are determined as t he action of the Uq(su(2))\ngenerators on the irreducible suq(2) tensor operators (2) and (3); the commutators between\nthe boosts are such, that in the limit 1 →1 the classical two-dimensional boost generators are\nrecovered. The deformed Lorentz algebra has the form\nM+M−−M−M+= [2M3], (4)\nM3M±−M±M3=±M±,\nN+N−−N−N+=−[2M3]\nN3N+q1/2−q−1/2N+N3=−M+,\n˜N3N−q1/2−q−1/2N−˜N3=M−,\nM3N±−N±M3=±N±,\nM+N−q−1/2−q1/2N−M+= [2]˜N3+(q1/2−q−1/2)C′q\n2,\nM−N+q−1/2−q1/2N+M−=−[2]N3+(q1/2−q−1/2)C′q\n2,\nM+˜N3q1/2−q−1/2˜N3M+=−N+,\nM−N3q1/2−q−1/2N3M−=N−\nand all other (usual) commutators vanish. The element\n2C′q\n2=M+N−q−1/2−N−M+q1/2+M−N+q−1/2−N+M−q1/2\nq1/2−q−1/2(5)\n2is central in the algebra and is the quantum analogue of the second o rder Lorentz Casimir C′\n2=\n−MiNi,i= 1,2,3. In the limit q→1 the deformed Lorentz algebra (4) contracts to the Lie\nalgebra of the classical Lorentz group. One can show that the ope ratorsM±,M3,N±(q−1),N3\nsatisfy the q-deformed Lorentz algebra (4) with qreplaced by q−1andN↔˜N3. Then a\nq-adjoint involution on the deformed Lorentz group can be defined\n(M±)∗=M∓, M∗\n3=M3, (6)\n(N∗\n±=N∓(q−1), N∗\n3=N3\nTheq-deformed Lie-brackets (4) and the corresponding ones with qreplaced by q−1and\nN3↔˜N3define a six-generator quantum Lorentz group.\nWe note an interesting novelty, the appearance of the central ele ment in the defining com-\nmutation relations of the deformed Lorentz algebra (16). It is a pr operty of the deformed\nrelations obeyed by the generators of the quantized universal en veloping algebra which is an\nassociative algebra with a unit and with a Poincare-Birkoff-Witt basis.\nTo construct the irreducible representations of the deformed Lo rentz algebra we follow the\nclassical procedure. Namely, the representations are realized in t he space of the Uq(su(2))\nirreducible representations with canonical basis |j,m/angbracketrightqwithjinteger or half integer and m=\n−j,...,j. The action of the rotation and boost generators on the basis vec tors is given by:\nM±|j,m/angbracketrightq= [j∓m]1/2[j±m+1]1/2q−1/4q∓m/2|j,m±1/angbracketrightq (7)\nM3|j,m/angbracketrightq=m|j,m/angbracketrightq (8)\nN±|j,m/angbracketrightq=±cj[j∓m]1/2[j∓m−1]1/2q−1/4q−(j±m)/2|j−1,m±/angbracketrightq (9)\n−aj[j∓m]1/2[j±m+1]1/2q−1/4q∓m/2|j,m±1/angbracketrightq\n±cj+1[j±m+1]1/2[j±m+2]1/2q1/4q(j∓m)/2|j+1,m±1/angbracketrightq\nN3|j,m/angbracketrightq=cj[j−m]1/2[j+m]1/2q−m/2|j−1,m/angbracketrightq (10)\n−aj[m]q−m/2|j,m/angbracketrightq\n−cj+1[j+m+1]1/2[j−m+1]1/2q−m/2|j+1,m/angbracketrightq\nThe action of the operator C′q\n2on the basis vectors is given by\nC′q\n2|j,m/angbracketrightq=i[l0][l1]|j,m/angbracketrightq (11)\nThe coefficients aj,cjcan be determined by using the commutators between the generat orsN+\nandN−andN±, andN3which results in the pair of difference equations\n(aj+1[j+2]−aj[j])cj+1= 0, (12)\nc2\nj[2j−1]−a2\nj−c2\nj+1[2j+3] = 1 (13)\nWe first note that since j≥0 there is a minimal (integer or half integer) value jmin=l0and\nhencej=l0+1,l0+2,.... Assuming that the coefficient cl0= 0 we have two possibilities, either\ncl0= 0, cl0+1/negationslash= 0,...cl0+n/negationslash= 0, cl0+n+1= 0, (14)\n3and the representation is finite-dimensional, or\ncl0= 0, cj/negationslash= 0for any j > l 0, (15)\nand the representation is infinite-dimensional. Eqs.(12, 13) for the coefficient can be easily\nsolved and the result, being dependent on two constants l0,l1is\naj=i[l0][l1]\n[j][j+1], cj=i\n[j]/radicalBigg\n([j]2−[l0]2)([j]2−[l1]2)\n[2j−1][2j+1](16)\nfor anyj > l0. Sincej=l0+n, wherenis a natural number, the representation will be\nfinite-dimensional if for some n\n[l1]2= [l0+n+1]2(17)\nDue to the property of the quantity [ A] the above equation is satisfied for ±l1=±(l0+n+1).\nThe parameter l1is in general a complex number, but l0+n+1 is a real positive number, so\nthat the representation series will terminate if, for some real l1,\n|l1|=l0+n+1. (18)\nhencethespincontentoftheirreduciblefinite-dimensionalrepres entationLorentz q-representation\nis determined by\nj=l0,l0+1,...,|l1|−1, m=−j,−j+1,...,j. (19)\nWe summarize the result: The irreducible representation of the def ormed Lorentz algebra is\ndetermined by the pair ([ l0],[l1], wherel0is a non-negative integer or half-integer real number\nandl1is a complex number. The irreducible representation corresponding to a given pair\n([l0],[l1]) intheUq(su(2))canonical basis |j,m/angbracketrightqis given by formulae (7-11)with the coefficients\n(16).\nIf, for some natural number n,\n[l1]2= [l0+n+1]2(20)\nthen the representation is finite-dimensional with the possible value s ofjandmgiven by (19).\nIf\n[l1]2/negationslash= [l0+n+1]2, (21)\nthen the representation is infinite-dimensional. In the limit q→1, (7-11) and (16) reproduce\nexactly the irreducible Lorentz group [10] representations.\nWe now consider the conditions under which the representations of the deformed Lorentz\nalgebra are unitary. Since the generator N3is self-q-adjoint, according to (6), then\n/angbracketleftj,m|N3|j,m/angbracketright=/angbracketleftj,m|N∗\n3|j,m/angbracketright, (22)\n/angbracketleftj−1,m|N3|j−1,m/angbracketright=/angbracketleftj−1,m|N∗\n3|j−1,m/angbracketright\nHenceaj= ¯ajandcj=−¯cj. From the first of the formulae (16), it follows that the condition\naj= ¯ajis satisfied if either [ l1] is arbitrary and [ l0] = 0, or [ l0] = 0 is arbitrary and i[l1] is real.\nThe second possibility with qreal means that l1should be pure imaginary\nl1=iρ, (23)\n4withρreal. The condition cj=−¯cjfor the second of the formulae (16) means that the\nexpression under the square root must be positive, and this is obvio usly the case if, only\n[j]2−[l1]2>0 (24)\nWe have to consider two possibilities:\n(a)l0/negationslash= 0 and [ l1] pure imaginary, which coincides with (23).\n(b) [j]2≥[l1]2with [l1] real.\nThe latter expression has to be satisfied for all jand this is only possible if [ l1]2≤[1]. Hence\nthe possible values of [ l1] are\n0<|l1| ≤1 (25)\nThe relations between N±and their q-adjoint yield the same values for l0andl1.\nWe thus conclude: The irreducible representations of the deforme d Lorentz algebra deter-\nmined by the pair [ l0],[l1] is unitary if either l1is pure imaginary and l0is an arbitrary non-\nnegative integer or half-integer, or l0= 0 and l1is a real number in the interval 0 <|l1| ≤1.\nIn the limit q→1 the corresponding representations (7-11) reproduce exactly the infinite-\ndimensional Lorentz group representations [10] of the principal a nd complementary series re-\nspectively.\nTo analyze the Hopf structure of the quantum lorentz group we ne ed to generalize to the\nq-case the classical picture of forming two SL(2) groups from Lorentz rotations and boosts.\nFor this purpose we consider the operators\nIL\n±=M±+iN±, (26)\nIR\n±M±−iN±,\nIL\n3= [M3]q−M3/2+iN3,\nIR\n3= [M3]q−M3/2−iN3,\n˜IL\n3= [M3]qM3/2+i˜N3\n˜IR\n3= [M3]qM3/2−i˜N3\nThese operators satisfy the algebra\nIL\n+IL\n−−IL\n−IL\n+= 2(IL\n3+˜IL\n3), (27)\nIL\n3IL\n+q1/2−q−1/2IL\n+IL\n3= 2IL\n+,\n˜IL\n3IL\n−q1/2−q−1/2IL\n−˜IL\n3=−2IL\n−,\n˜IL\n3IL\n+q−1/2−q1/2IL\n+˜IL\n3= 2IL\n+,\nIL\n3IL\n−q−1/2−q1/2IL\n−IL\n3=−2IL\n−,\nIR\n+IR\n−−IR\n−IR\n+= 2(IR\n3+˜IR\n3),\nIR\n3IR\n+q1/2−q−1/2IR\n+IR\n3= 2IR\n+,\n˜IR\n3IR\n−q1/2−q−1/2IR\n−˜IR\n3=−2IR\n−,\n˜IR\n3IR\n+q−1/2−q1/2IR\n+˜IR\n3= 2IR\n+,\nIR\n3IR\n−q−1/2−q1/2IR\n−IR\n3=−2IR\n−,\n5The generators IL\n±,IL\n3,˜IL\n3simply commute with IR\n±,IR\n3,˜IR\n3. It seems that the algebra (27) has\nmore than six generators. Due to the relations\n1+α˜IL\n3+α˜IL\n3= (1−αIL\n3−αIR\n3)−1, (28)\n˜IL\n3−˜IR\n3= (1+α˜IL\n3+α˜IL\n3)(IL\n3−IR\n3)\nwithα= (q1/2−q−1/2)/2, the quantum algebra (27) is, in fact, generated by two raising, t wo\nlowering and two diagonal operators.\nThere exists a q-adjoint involution in the algebra (27), defined as\n(IL\n±(q))∗=IR\n∓(q−1), (29)\nIL∗\n3=IR\n3,˜IL∗\n3=˜IR\n3\nDenoting the two-dimensional q-deformed Lorentz representations\n|1/2,m;l0= 1/2,l1= 3/2/angbracketrightq=τ1/2, (30)\n|1/2,m;l0= 1/2,l1=−3/2/angbracketrightq= ˜τ1/2,\nwe observe that IL\n±,IL\n3) (respectively IR\n±,IR\n3)) vanish when acting on ˜ τ1/2(respectively τ1/2).\nThe algebra (27)is the q-analogueof the chiral decomposition of the classical Lorentz gro up,\nwhich is exactly reproduced in the limit q→1.\nWe further introduce the shifted diagonal generators\nTL,R\n3= 2−(q1/2−q−1/2)IL,R\n3, (31)\n˜TL,R\n3= 2+(q1/2−q−1/2)˜IL,R\n3\nwhich does not change the structure of the algebra (27). The co- product for the deformed\nalgebra (27) is given by which amounts to a non-cocommutative Hopf algebra structure.\n∆(IL\n+) =IL\n+⊗1+TL\n3⊗IL\n+, (32)\n∆(IL\n−) =IL\n−⊗˜TL\n3+1⊗IL\n−,\n∆(IR\n+) =IR\n+⊗˜TR\n3+1⊗IR\n+,\n∆(IR\n−) =IR\n−⊗1+TL\n3⊗IR\n−,\n∆(TL,R\n3) =TL,R\n3⊗TL,R\n3,\n∆(˜TL,R\n3) =˜TL,R\n3⊗˜TL,R\n3.\nTo summarize, we have defined aquantized Lorentzalgebra andfou ndthatevery irreducible\nclassical Lorentz group representation labelled by l0andl1can beq-deformed to an irreducible\nq-representation of the deformed Lorentz algebra labelled by [ l0] and [l1]. The possible values\nofl0andl1are exactly the same as in the classical case.\nTheauthorisgratefulforthesupportandhospitalityoftheTheo ryDivisionatCERNwhere\nmost of this work was completed. Partial support of the National F oundation for Scientific\nResearch under contract Φ-11 is acknowledged.\n6References\n[1] V.B.Drinfeld, Quantun groups, in:Proc.ICM (MCRI, Berkeley, CA, 1 986)\n[2] M.Jimbo, Lett.Math.Phys.10 (1985)63\n[3] M.Jimbo, Lett.Math.Phys.11 (1985) 247\n[4] M.Jimbo, Comm.Math.Phys. 102, (1986) 537\n[5] P.Podles and S.L.Woronowicz, Comm.Math.Phys. 130 (1990) 381\n[6] O.Ogievetsky, W.B.Schmidke, J.Wess and B.Zumino, Lett.Math.Phys.2 3 (1991) 233\n[7] L.C.Biedenharn, Lett.Math.Phys.20 (1990) 271\n[8] V.Rittenberg and V.Scheunert, J.Math.Phys.33 (1992) 436\n[9] Yu.F.Smirnov, V.N.Tolstoy and Yu.I.Kharitonov, Sov.J.Nucl.Phys.53 (1 991) 593\n[10] M.A.Naimark, Lineaar representations of the Lorentz group (P ergamon, Oxford, 1964)\n7"    },    {        "title": "2104.13513v1.High_Order_Explicit_Lorentz_Invariant_Volume_preserving_Algorithms_for_Relativistic_Dynamics_of_Charged_Particles.pdf",        "content": "High Order Explicit Lorentz Invariant\nVolume-preserving Algorithms for Relativistic\nDynamics of Charged Particles\nYulei Wanga,∗, Jian Liub, Yang Hec\naSchool of Astronomy and Space Science, Nanjing University, Nanjing 210023, Peoples’\nRepublic of China\nbDepartment of Engineering and Applied Physics, University of Science and Technology of\nChina, Hefei, 230026, Peoples’ Republic of China\ncSchool of Mathematics and Physics, University of Science and Technology Beijing, Beijing,\n100083, Peoples’ Republic of China\nAbstract\nLorentz invariant structure-preserving algorithms possess reference-independent\nsecular stability, which is vital for simulating relativistic multi-scale dynamical\nprocesses. The splitting method has been widely used to construct structure-\npreserving algorithms, but without exquisite considerations, it can easily break\nthe Lorentz invariance of continuous systems. In this paper, we introduce a\nLorentz invariant splitting technique to construct high order explicit Lorentz\ninvariant volume-preserving algorithms (LIVPAs) for simulating charged parti-\ncle dynamics. Using this method, long-term stable explicit LIVPAs with dif-\nferent orders are developed and their performances of Lorentz invariance and\nlong-term stability are analyzed in detail.\nKeywords: Lorentz Invariant Volume-preserving Algorithms, Relativistic\nCharged Particles, High Order Explicit Schemes, Reference-independent\nSecular Stability\n∗Corresponding author\nEmail address: wyulei@nju.edu.cn (Yulei Wang)\nPreprint submitted to Journal Of Computational Physics April 29, 2021arXiv:2104.13513v1  [physics.plasm-ph]  28 Apr 20211. Introduction\nStructure-preservingalgorithmspossessinglong-termstabilitiesplaykeyroles\nin simulations of various fields [1–7]. In plasma studies, these advanced schemes\nhave been applied in subjects including the secular dynamical simulations of\ncharged particles [8–10], the long-term analysis of plasma kinetic processes [10–\n15], the analysis of magnetohydrodynamical phenomena [16–18], and the simu-\nlations of nonlinear processes [19, 20]. The basic idea of constructing structure-\npreserving algorithms is to keep the fundamental geometry structures of ordi-\nnary differential equations (ODEs) or partial differential equations (PDEs) dur-\ning discretization [3, 21, 22]. In other words, the one-step maps, approximating\nthe continuous exact solutions, should inherit mathematical natures of origi-\nnal continuous systems, such as the volume-preserving property of source-free\nsystems and the symplectic-preserving property of Hamiltonian systems [21].\nThese properties work as constrains for numerical systems, which can limit the\nglobal accumulations of numerical errors during long-term iterations [21, 22].\nExplicit schemes, compared with implicit schemes, are more convenient for\nimplementations and more efficient especially in cases of complex vector fields.\nBecause the symplectic Runge-Kutta methods are often implicit [21], the Hamil-\ntonian splitting technique has been applied widely to build explicit structure-\npreserving algorithms for charged particle dynamics. Written in canonical co-\nordinates, the Hamiltonian equation of charged particles can be expressed as\nJ˙Z=∇ZH, where Zis the canonical coordinate, Jis the symplectic struc-\nture, andHis the Hamiltonian. Through the sum-split procedure of H, one\ncan obtain canonical symplectic subsystems which can be discretized via stan-\ndard symplectic methods, like the generating function method [23, 24]. Then\nthe composites of subsystem algorithms are the canonical symplectic algorithm\nfor the original system [25]. On the other hand, it has been find that the\nLorentz force equation of charged particles possesses non-canonical symplectic\nstructure, which can be written as K˙z=∇zH, where zis phase space coordi-\nnate andKdenotes the K-symplectic structure. Similarly, the sum-split proce-\n2dure of Hamiltonian produces the explicit K-symplectic algorithms [22, 26, 27].\nCompared with the symplectic methods, volume-preserving algorithms (VPAs)\nimpose looser constrains on discrete systems. However, VPAs still possess sig-\nnificant long-term stability and have been used widely in Particle-in-Cell sim-\nulations [28–32]. The volume-preserving systems can be expressed as ˙z=V,\nwhere the source-free vector field Vsatisfies∇z·V= 0. Through splitting\nVinto several source-free sub-vectors, explicit VPAs can also be conveniently\nconstructed [31–36].\nFor relativistic dynamical systems, the Lorentz invariance is another fun-\ndamental property, which should also be kept after discretization. When the\neffects of the special relativity are considered, simulating a physical process in\ndifferent Lorentz inertial frames can minimize the range of time and space scales\nand thus reduce the cost of calculation [37, 38]. In such cases, the Lorentz in-\nvariance of algorithms becomes very important. In 2008, Vay has pointed out\nthe importance of Lorentz invariance for relativistic particles and constructed\na new scheme that preserves the E×Bvelocity in different frames [38]. In\n2017, Higuera and Cary improved Vay’s method and built a VPA that also\npreserves E×Bvelocity [31]. The reference-independency of the numerical\nE×Bvelocity can be treated as one key outcome of Lorentz invariance of\nalgorithms. Strictly speaking, the Lorentz invariant algorithms (LIAs) should\nproduce reference-independent discrete numerical results when solving the same\nprocess in arbitrary Lorentz inertial frames. Equivalently, the difference equa-\ntions of LIAs keep unchanged after all observables are transformed to that in\nanother Lorentz inertial frame. Therefore, the definition of LIAs can be given\nas follows [39]. For a Lorentz invariant system F, an algorithm,A, is Lorentz\ninvariant if it satisfies that\nDA◦TLF=TL◦DAF (1)\nwhere,DArepresentsnumericaldiscretizationoperationusing A,TLdenotesthe\ntransformation operation on variables based on the Lorentz matrix L, and \"◦\" is\nthe composite operator. As an element of Lorentz group, LsatisfiesLgLT=g,\n3wheregis the Lorentz metric tensor [40].\nThe Lorentz invariance and the structure-preservation are two independent\naspects for constructions of advanced algorithms. Symplectic-preserving or\nvolume-preserving algorithms sometimes break the Lorentz invariance, while\ntraditional algorithms like the Newton method and the Runge-Kutta method\ncan produce Lorentz invariant schemes if they are applied to Lorentz invariant\ndynamical equations written in the 4-dimensional spacetime. Furthermore, if\nwe combine the benefits of the structure-preservation and the Lorentz invari-\nance, the resulting structure-preserving LIAs will possess reference-independent\nsecular stability, which is very important for simulating relativistic multi-scale\nprocesses. To construct Lorentz invariant algorithms, one straightforward way\nis directly discretizing the Lorentz invariant dynamical equations in which all\nvariables are written as geometric objects in 4-dimensional spacetime [40]. The\nintegrity of these geometric objects, such as the spacetime coordinates and 4-\ndimensional tensors, should be kept during discretization [39]. For example,\nfor the 4-dimensional Lorentz invariant Hamiltonian equation of charged parti-\ncles [40], the symplectic-Euler method gives an explicit 1-order symplectic LIA\n[39]. Similarly, symplectic Runge-Kutta methods, such as the implicit mid-point\nsymplectic scheme, also generate symplectic LIAs when applied to this system.\nAlthough symplectic LIAs with different orders can be constructed using the\nsymplectic Runge-Kutta methods, they are all implicit and their usage are not\nconvenient. On the other hand, although Hamiltonian splitting technique can\nproduce explicit high-order symplectic algorithms, it has to divide the Hamilto-\nnian into several pieces to construct symplectic schemes for subsystems [23–26].\nThe fine-grained splitting breaks the Lorentz invariance. Therefore, in this\npaper, we relax the constrain of symplectic-preservation and focus on the con-\nstructions of high order explicit Lorentz invariant volume-preserving algorithms\n(LIVPAs). Higher order schemes mean faster convergence rate, but, unfortu-\nnately, moreCPUtimeconsumedforone-stepiteration. Generallyspeaking, the\nbalance between the complexity and the convergence rate should be considered\nduring practical simulations. Though it’s not necessary to build algorithms with\n4arbitrary high orders, schemes with order higher than 1 are still needed. From\nthe view point of applications, the convergence rates of 1-order algorithms are\nsometimes too slow to be applied. Therefore, although a little complex than 1-\norder algorithms, 2-order and 4-order schemes, like the Crank-Nicolson method\n(2-order), the Boris method (2-order), and the 4-order Runge-Kutta method,\nhave been widely applied in plasma studies. Especially, for simulations of pro-\ncesses with small scales, such as turbulence process and magnetic reconnection\nprocesses, high temporal and spatial resolutions are necessary. Though cost-\ning more CPU time for one-step iteration, high order schemes, with larger step\nlength, can be cheaper than low order schemes.\nWithout exquisite considerations, the splitting technique can easily break\nthe Lorentz invariance of the original system, which will be discussed in detail\nin Sec.2. In this paper, however, we find a Lorentz invariant splitting procedure\nfor constructing LIVPAs for charged particle dynamics. The splitting method\nis chosen for our kernel technique for two main reasons. First, through different\ncomposite of subsystem algorithms, high-order explicit schemes can be easily\nconstructed [21]. Second, compared with the original systems, finding Lorentz\ninvariant algorithms for subsystems is much easier, and it is also readily to\nunderstand that the composed scheme is Lorentz invariant if algorithms of all\nsubsystemsareLorentzinvariant. Althoughwesplittheoriginalsystemintosev-\neral pieces, the Lorentz invariance of algorithms still remains. Explicit LIVPAs\nwith different orders are constructed and tested in detail. Compared with the\nexplicit symplectic algorithms constructed via Hamiltonian splitting method,\nthese LIVPAs can give reference-independent numerical results [24]. Compared\nwith the Vay [38] and the Higuera-Cary [31] schemes that preserve the E×B\nvelocity, LIVPAs show Lorentz invariance that is independent with the config-\nurations of step length. Moreover, because of the preservation of phase space\nvolume, LIVPAs show better secular stability than the Runge-Kutta method.\nThe rest part of this paper is organized as follows. In section 2, basic prop-\nerties of Lorentz invariant algorithms and their relations with splitting method\nare discussed by use of a simple system. In section 3, the invariant form of the\n5Lorentz force equation is analyzed. We introduce the Lorentz invariant splitting\nmethod and construct the LIVPAs in Sec.4. In section 5, LIVPAs are applied\nin a typical field configuration and their numerical performances are studied.\nFinally, we conclude this paper in Sec.6.\n2. LIAs and The Splitting Technique\nInthissection, wegiveageneralpictureofLIAs. Tosimplifytheexpressions,\nhere we use a simple 2-dimensional Lorentz invariant system F2,\ndx\ndτ=u, (2)\nwhere Lorentz invariant vectors x= (t, x)Tandu= (γ, px)Tdenotes respec-\ntively the space-time coordinate and momentum. The proper time, τ, is invari-\nable under Lorentz transformations. xanduare functions of τ.\nSuppose there are two Lorentz inertial reference frames. One frame Ois\nresting relative to lab, and another frame O/primemoves with constant speed β\nrelative toO. In this case, the Lorentz matrix is\nL2=∂x/prime\n∂x= Γ\n1−β\n−β1\n, (3)\nwhere Γ = 1//radicalbig\n1−β2, and the superscript \"/prime\" denotes physical quantities ob-\nserved inO/prime. The Lorentz invariance of Equation 2 means its form keeps un-\nchanged after xanduare transformed into the moving frame, namely, TL2F2\ngives\ndx/prime\ndτ=u/prime. (4)\nOne way of discretization F2is using the 1-order forward Euler difference\nA1, which does not break the integrity of xandu. In frameO, the difference\nequation,DA1F2, is\nxk+1=xk+ ∆τuk, (5)\nwherekdenotes \"kth\" step and ∆τis the step length. This algorithm can be\nproven to be Lorentz invariant as follows. First, replace xanduin Eq.5 with\n6L−1\n2x/primeandL−1\n2u/prime, respectively, which gives L−1\n2x/primek+1=L−1\n2x/primek+ ∆τL−1\n2u/primek.\nThen, we can left-multiply L2on both side and obtain TL2◦DA1F2,\nx/primek+1=x/primek+ ∆τu/primek. (6)\nOn the other hand, because the original system is Lorentz invariant, it is read-\nily to see that the difference equation DA1◦TL2F2can be directly obtained by\ndiscretizing Eq.4 with algorithm A1, which has the same form as Eq.6. Conse-\nquently, for algorithm A1and system F2, we haveTL2◦DA1F2=DA1◦TL2F2.\nTherefore, the 1-order forward Euler difference is Lorentz invariant. Suppose\nthat we set the initial condition as x0inO, the corresponding initial condi-\ntion inO/primeisx/prime0=L2x0. Because Eq.6 is directly obtained from the Lorentz\ntransformation of Eq.5, numerical solutions x/primekof Eq.6 can also be calculated\nfromxkgiven by Eq.5 via Lorentz transformation. Therefore, for simulations\nof the same process, the difference equation of algorithm A1gives reference-\nindependent numerical results in arbitrary Lorentz inertial frames.\nTo illustrate the effects of the splitting procedure on Lorentz invariance, we\ndivide uinto two parts, u=u1+u2= (γ,0)T+(0,px)T. Then we discretize the\ncorresponding subsystems by different schemes, which is of common occurrence\nwhen constructing algorithms using splitting technique. We calculate u1atkth\nstep while u2at(k+ 1)th step. After composing, the final algorithm DA2F2is\ntk+1=tk+ ∆τγk, (7)\nxk+1=xk+ ∆τpk+1\nx. (8)\nNow we deriveTL◦DA2F2. According to x=L−1\n2x/primeandu=L−1\n2u/prime, we\ncan directly substitute t= Γ (t/prime+βx/prime),x= Γ (βt/prime+x/prime),γ= Γ (γ/prime+βp/prime\nx), and\npx= Γ (βγ/prime+p/prime\nx)into Eqs.7 and 8. The resulting algorithm, TL2◦DA2F2, is\nt/primek+1=t/primek+ ∆τΓ2/parenleftbig\nγ/primek−β2γ/primen+1+βp/primek\nx−βp/primek+1\nx/parenrightbig\n, (9)\nx/primek+1=x/primek+ ∆τΓ2/parenleftbig\nβγ/primek+1−βγ/primen+p/primek+1\nx−β2p/primek\nx/parenrightbig\n,(10)\n7which is of different form compared with DA2◦TL2F2, namely,\nt/primek+1=t/primek+ ∆τγ/primek, (11)\nx/primek+1=x/primek+ ∆τp/primek+1\nx. (12)\nBecauseTL2◦DA2F2/negationslash=DA2◦TL2F2,A2is not Lorentz invariant. Even though\nwe set the same initial condition, numerical results calculated by Eqs.11-12 in\nframeO/primecannot be directly converted to results of Eqs.7-8 via Lorentz trans-\nformation. In other words, for the same process, the difference equation of\nalgorithmA2gives reference-dependent solutions in different Lorentz inertial\nframes. One result of reference-dependency is the inconsistent numerical E×B\nvelocity in different frames [31, 38].\nAlthough the system discussed above is very simple and we won’t use al-\ngorithms likeA2in practical simulations, we can clearly see the effects of the\nsplitting method on the Lorentz invariance of original systems. The break of\nLorentz invariant objects like xanducan, though not always, easily cause the\nviolation of Lorentz invariance of numerical schemes.\n3. The Lorentz Force Equation of Charged Particles\nBefore constructing the LIVPAs, we first analyze the Lorentz invariant form\nof Lorentz force equation [40], namely,\ndxα\ndτ=Uα,\ndpα\ndτ= ˜qFαβUβ, (13)\nwherexα= (t,x,y,z )Tis the coordinate in 4-dimensional spacetime, Uα=\n(γ,γvx,γvy,γvz)Tis the 4-velocity, γ=/radicalbig\n1 +p2is the Lorentz factor, pα=\n(γ,px,py,pz)Tis the 4-momentum, and\nFαβ=\n0−Ex−Ey−Ez\nEx 0−BzBy\nEyBz 0−Bx\nEz−ByBx 0\n(14)\n8is the electromagnetic tensor which is the function of xα.Ex,Ey,Ez,Bx,By,\nandBzdenote the space components of the electric field and magnetic field,\nrespectively. ˜qdenotes the charge sign, namely, ˜q= 1for positive charged par-\nticles and ˜q=−1for negative charged particles. In this paper, without special\ninstructions, all physical quantities are normalized according to the units given\nin Tab.1 and the space components of vectors and tensors are written in the\nCartesian coordinate system. The superscripts and subscripts written in Greek\nalphabetsdenote\"contravariant\"and\"covariant\"componentsrespectively. They\ncan convert to each other through the Lorentz metric tensor gαβsatisfying\ngαβ=gαβ=\n1 0 0 0\n0−1 0 0\n0 0−1 0\n0 0 0−1\n. (15)\nFor example, xα=gαβxβ= (t,−x,−y,−z)T. Einstein’s convention for the\nsummation over repeat indices is applied in this paper.\nPhysical quantities Symbols Units\nTime t,τ m0/qB 0\nSpace xαm0c/qB 0\nMomentum pαm0c\nVelocity Uα,β c\nElectric strength EB0c\nMagnetic strength B B0\nVector field A m0c/q\nScalar field φ m0c2/q\nEnergy H,H,γ m0c2\nTable 1: Units of all the physical quantities used in this paper. m0is the rest mass of a\nparticle,qis the norm of charge carried by a particle, cis the speed of light, and B0is the\nreference strength of magnetic field.\nNowweorganizeEq.13intoaformofordinarydifferentialequationsinterms\n9ofxαandpα. Consideringthatthenormalized4-velocityand4-momentumhave\nthesamevalue, namely, pα=Uα= (γ,px,py,pz)T. Eq.13canthusberewritten\nas\ndxα\ndτ=pα,\ndpα\ndτ= ˜qFα\nβpβ, (16)\nwhere\nFα\nβ=Fαξgξβ=\n0ExEyEz\nEx 0Bz−By\nEy−Bz 0Bx\nEzBy−Bx 0\n. (17)\nDefiningZ= (xα,pα)T, we can transform Eq.16 to a compact form\n˙Z=V(Z), (18)\nwhere the dot operator denotes the full derivative d/dτ, and the vector field on\nright sideV(Z) =/parenleftBig\npα,˜qFα\nβpβ/parenrightBigT\ncan also be represented by the Lie derivative\nXV=pα∂\n∂xα+ ˜qFα\nβpβ∂\n∂pα. (19)\nTherefore, Eq.18 becomes ˙Z=XVZ, whose analytical solution can be gener-\nated by the one-parameter Lie group exp (τXV)asφτ:= exp (τXV)Z(0)[41].\nConsidering that\n∇Z·V=∂pα\n∂xα+∂\n∂pα/parenleftbig\n˜qFα\nβpβ/parenrightbig\n= 0, (20)\nVis a source-free vector field, and, the solution φτof Eq.16 is a volume-\npreserving map.\n4. Constructions of LIVPAs\nIn this section, we construct explicit high-order LIVPAs of Eq.16 by using\nsplitting technique. To keep the Lorentz invariance of subsystems, we want to\n10keep the integrity of invariant objects, xα,pα, andFα\nβ, and the first choice of\nsplittingV(Z)seems to be\nV(Z) =Vp+VF=\npα\n0\n+\n0\n˜qFα\nβpβ\n. (21)\nHowever, for arbitrary Fα\nβ, it’s hard to find the exact solution or volume-\npreserving approximations for the subsystem of VF. Therefore, we have to\nseek other proper ways of splitting Fα\nβ, which should satisfy that\n1. the subsystems can be solved by explicit volume-preserving numerical\nschemes;\n2. the algorithms of subsystems should be Lorentz invariant.\nIn the following three subsections, we first exhibit the Lorentz invariant split-\ntingprocedurewhichgeneratesvolume-preservingLorentzinvariantsubsystems.\nSecond, we construct the LIVPAs for subsystems. Third, we construct the final\nLIVPAs with different orders.\n4.1. The Lorentz invariant volume-preserving splitting procedure\nInthissection, threeLorentzinertialframes, Or,O, andO/prime, willbeinvolved.\nTo give clear expressions, here we describe the definitions of the three frames\nand several symbols that will be frequently used. We call Orthe splitting\nreference frame (SRF), which works as a medium linking the numerical results\ninarbitraryLorentzinertialreferenceframesthroughLorentztransformation. O\nandO/primeare two arbitrary Lorentz inertial frames. Symbols of main observables\nwe use in the three frames are listed in Tab.2. And the definitions of three\nLorentz transformation matrices, L,M, andN, are listed in Tab.3. In O, the\nelectromagnetic tensor is Fα\nβ, and, if observed in frame Or(transformed back\nto frameOr), it becomes [40]\nFα\nβ=L−1Fα\nβL=\n0Er\nxEr\nyEr\nz\nEr\nx 0Br\nz−Br\ny\nEr\ny−Br\nz 0Br\nx\nEr\nzBr\ny−Br\nx 0\n. (22)\n11InOr, the kinetic tensor Kα\nβand the rotation tensor Rα\nβare respectively the\nelectric and magnetic parts of Fα\nβ, namely,\nKα\nβ=\n0Er\nxEr\nyEr\nz\nEr\nx0 0 0\nEr\ny0 0 0\nEr\nz0 0 0\n, (23)\nRα\nβ=\n0 0 0 0\n0 0 Br\nz−Br\ny\n0−Br\nz 0Br\nx\n0Br\ny−Br\nx 0\n. (24)\nKα\nβcorresponds to the change of kinetic energy, while the Rα\nβpart corresponds\nto the rotation of momentum.\nObservables inOrinOinO/prime\nCoordinate xr,αxαx/primeα\nMomentum pr,αpαp/primeα\nElectric Field ErE E/prime\nmagnetic Field BrB B/prime\nElectromagnetic tensor Fα\nβFα\nβF/primeα\nβ\nKinetic tensor Kα\nβKα\nβK/primeα\nβ\nRotation tensor Rα\nβRα\nβR/primeα\nβ\nSquare of Kinetic tensor Sα\nβSα\nβS/primeα\nβ\nSquare of Rotation tensor Pα\nβPα\nβP/primeα\nβ\nTable 2: List of observable symbols in Or,O, andO/prime. The detailed definitions of kinetic\nand rotation tensors are given by Eqs.23 and 24. To simplify the expressions, we defines\nSα\nβ=/parenleftbig\nKα\nβ/parenrightbig2andPα\nβ=/parenleftbig\nRα\nβ/parenrightbig2.\nNow we introduce the splitting procedure of the original system. In frame\n12Lorentz matrix Definition Description\nL ∂xα/xr,βOrelative toOr\nM ∂x/primeα/xβO/primerelative toO\nN ML =∂x/primeα/xr,βO/primerelative toOr\nTable 3: Definitions of the Lorentz transformation matrices L,M, andN. To simplify the\nexpressions, in this paper, we do not explicitly write the upper and lower indexes of L,M,\nandN.\nO, we split the vector field Vinto three parts, namely,\nV(Z) =\npα\n˜qFα\nβpβ\n=Vp+VK+VR=\npα\n0\n+\n0\n˜qKα\nβpβ\n+\n0\n˜qRα\nβpβ\n,\n(25)\nwhereKα\nβ=LKα\nβL−1,Rα\nβ=LRα\nβL−1. There are three main reasons for\nsplittingVin this way. First, Vp,VK, andVRare source-free vector fields,\nwhich generate volume-preserving subsystems. Second, as will be shown later,\nbecauseKα\nβandRα\nβare related to the kinetic and rotation tensors observed in\nthe SRFOr, the Lorentz invariance of the subsystems and the corresponding\nalgorithms can be easily kept. Third, such splitting technique can significantly\nsimplify the expressions of final difference equations.\nThe three parts of Vcan be proven to be source free vector fields as follows.\nForVp,∇Z·Vp=∂pα/∂xα+ 0 = 0; forVK, considering that Kα\nβis a similar\nmatrix ofKα\nβ,∇Z·VK= 0 + ˜q∂/parenleftBig\nKα\nβpβ/parenrightBig\n/∂pα= ˜qKα\nα= ˜qKα\nα= 0; forVR,\nRα\nβandRα\nβare also similar matrices, thus ∇Z·VR= 0 + ˜q∂/parenleftBig\nRα\nβpβ/parenrightBig\n/∂pα=\n˜qRα\nα= ˜qRα\nα= 0. Therefore, after splitting, we get three volume-preserving\n13subsystems,\nsτ\np:=\n\ndxα\ndτ=pα,\ndpα\ndτ= 0,(26)\nsτ\nK:=\n\ndxα\ndτ= 0,\ndpα\ndτ= ˜qKα\nβpβ,(27)\nsτ\nR:=\n\ndxα\ndτ= 0,\ndpα\ndτ= ˜qRα\nβpβ.(28)\nNow we prove that sτ\np,sτ\nK, andsτ\nRare Lorentz invariant systems. Because\nproofs of the three subsystems are similar, here we only give the proof of sτ\nK.\nWe need to derive the expression of sτ\nKin another Lorentz frame O/prime. For the\nequation of xα, left-multiply Mon both side gives dx/primeα/dτ= 0. For equa-\ntion ofpα, left-multiply Mon left side gives dp/primeα/dτ, and on right side gives\n˜qMKα\nβpβ= ˜qMLKα\nβL−1M−1Mpβ= ˜qNKα\nβN−1p/primeβ= ˜qK/primeα\nβp/primeβ. The resulting\nexpression of sτ\nKin frameO/primeis\ns/primeτ\nK:=\n\ndx/primeα\ndτ= 0,\ndp/primeα\ndτ= ˜qK/primeα\nβp/primeβ,(29)\nwhich has the same form as Eq.27. Therefore, sτ\nKis Lorentz invariant.\nUsing the properties of Kα\nβandRα\nβ, the analytical solutions of sτ\np,sτ\nK, and\n14sτ\nRcan be directed obtained as\nφτ\np:=\n\nxα(τ) =τpα+xα(0),\npα(τ) =pα(0),(30)\nφτ\nK:=\n\nxα(τ) =xα(0),\npα(τ) = exp/parenleftBig\nτ˜qKα\nβ/parenrightBig\npβ(0)\n=/bracketleftBig\nIα\nβ+sinh(τ˜qEr)\nErKα\nβ+cosh(τ˜qEr)−1\n(Er)2Sα\nβ/bracketrightBig\npβ(0),(31)\nφτ\nR:=\n\nxα(τ) =xα(0),\npα(τ) = exp/parenleftBig\nτ˜qRα\nβ/parenrightBig\npβ(0)\n=/bracketleftBig\nIα\nβ+sin(τ˜qBr)\nBrRα\nβ+1−cos(τ˜qBr)\n(Br)2Pα\nβ/bracketrightBig\npβ(0),(32)\nwhere exp (·)is the exponential map, Iα\nβdenotes the 4-dimensional identity\nmatrix,and Er=/radicalBig\n(Erx)2+/parenleftbig\nEry/parenrightbig2+ (Erz)2andBr=/radicalBig\n(Brx)2+/parenleftbig\nBry/parenrightbig2+ (Brz)2\nare respectively the strength of electric and magnetic fields observed in Or. One\nshould notice that the value of ErandBrare evaluated at xr,α=L−1xα. For\ndetailed derivations of Eqs.31-32, please refer to the appendix section.\n4.2. LIVPAs for subsystems\nBased on the exact solutions φτ\np,φτ\nK, andφτ\nR, the volume-preserving algo-\nrithms for corresponding subsystems can be obtained as,\nΦ∆τ\np:=\n\nxα,k+1=xα,k+ ∆τpα,k,\npα,k+1=pα,k,(33)\nΦ∆τ\nK:=\n\nxα,k+1=xα,k,\npα,k+1=/bracketleftbigg\nIα\nβ+sinh(∆τ˜qEr,k)\nEr,kKα,k\nβ+cosh(∆τ˜qEr,k)−1\n(Er,k)2Sα,k\nβ/bracketrightbigg\npβ,k,\n(34)\nΦ∆τ\nR:=\n\nxα,k+1=xα,k,\npα,k+1=/bracketleftbigg\nIα\nβ+sin(∆τ˜qBr,k)\nBr,kRα,k\nβ+1−cos(∆τ˜qBr,k)\n(Br,k)2Pα,k\nβ/bracketrightbigg\npβ,k,\n(35)\n15whereKα,k\nβ,Sα,k\nβ,Rα,k\nβ, andPα,k\nβdenote the value of Kα\nβ,Sα\nβ,Rα\nβ, andPα\nβ\nevaluated at xα,k.Er,kandBr,kare the value of ErandBrcalculated at\nxr,α,k=L−1xα,k.\nMeanwhile, the implicit midpoint scheme for subsystem sτ\nRcan be proven\nto be volume-preserving [35]. The difference equation is\nΦ∆τ\nRc :=\n\nxα,k+1=xα,k,\npα,k+1=/bracketleftBig\nIα\nβ−∆τ\n2˜qRα\nβ/bracketrightBig−1/bracketleftBig\nIα\nβ+∆τ\n2˜qRα\nβ/bracketrightBig\npβ,k\n=/bracketleftBig\nIα\nβ+2a\n1+a2(Br)2Rα\nβ+2a2\n1+a2(Br)2Rα\nβ/bracketrightBig\npβ,k,(36)\nwherea= ∆τ˜q/2and the detailed derivations of the compact form are shown\nin appendix. The implicit midpoint scheme is equivalent to the Cayley trans-\nformation. For a matrix A, its Cayley transformation is defined as cay (A) =\n(I−A/2)−1(I+A/2), which is the 2-order approximation of exp (A). IfAis\nantisymmetric, then the determinant of cay (A)equals 1[21]. One should notice\nthat∆τ˜qRα\nβisanantisymmetricmatrix,whichthusmeans det/bracketleftBig\ncay/parenleftBig\n∆τ˜qRα\nβ/parenrightBig/bracketrightBig\n≡\n1. Because ∆τ˜qRα\nβ= ∆τ˜qLRα\nβL−1is the similar matrix of ∆τ˜qRα\nβ, it\nis readily to see that det/bracketleftBig\ncay/parenleftBig\n∆τ˜qRα\nβ/parenrightBig/bracketrightBig\n≡1. Therefore, considering that\n∂pα,k+1/∂pβ,k= cay/bracketleftBig\n∆τ˜qRα\nβ/bracketrightBig\n, we have/vextendsingle/vextendsingle∂pα,k+1/∂pβ,k/vextendsingle/vextendsingle≡1. Consequently,\nthe algorithm is volume-preserving [35].\nIt can be proven that Φ∆τ\np,Φ∆τ\nK,Φ∆τ\nR, and Φ∆τ\nRcare all Lorentz invariant.\nSimilar to the discussion in Sec.2, we need to check the expressions of these\ndifference equations after transformed into another Lorentz frame O/prime. Here,\nas an example, we only give the proof of Φ∆τ\nK. For the difference equation\nxα,k+1=xα,k, left-multiply Mon both sides gives x/primeα,k+1=x/primeα,k. For the\ndifference equation of pαinΦ∆τ\nK, it is of compact form which origins from\npα,k+1= exp/parenleftBig\n∆τ˜qKα\nβ/parenrightBig\npβ,k. Left-multiply Mon both sides of this exponential\n16difference equation gives,\np/primeα,k+1=Mexp/parenleftbig\n∆τ˜qKα\nβ/parenrightbig\nM−1p/primeβ,k(37)\n= exp/parenleftbig\n∆τ˜qMKα\nβM−1/parenrightbig\np/primeβ,k= exp/parenleftbig\n∆τ˜qK/primeα\nβ/parenrightbig\np/primeβ,k\n=/bracketleftBigg\nIα\nβ+sinh/parenleftbig\n∆τ˜qEr,k/parenrightbig\nEr,kK/primeα\nβ+cosh/parenleftbig\n∆τ˜qEr,k/parenrightbig\n−1\n(Er,k)2S/primeα\nβ/bracketrightBigg\np/primeβ,k,\nwhereEr,kandBr,karethevalueof ErandBratxr,α,k=N−1x/primeα,k=L−1xα,k.\nThe simplification process of the exponential map is the same as Eq.53 in the\nappendix section. Therefore, after Lorentz transformation, in O/prime,Φ∆τ\nK, becomes\nΦ/prime∆τ\nK:=\n\nx/primeα,k+1=x/primeα,k,\np/primeα,k+1= exp/parenleftBig\n∆τ˜qK/primeα\nβ/parenrightBig\np/primeβ,k\n=/bracketleftbigg\nIα\nβ+sinh(∆τ˜qEr,k)\nEr,kK/primeα,k\nβ+cosh(∆τ˜qEr,k)−1\n(Er,k)2S/primeα,k\nβ/bracketrightbigg\np/primeβ,k,\n(38)\nwhich has the same form as Eq.34. It can also be noticed that, in an arbitrary\nLorentz frame, the value of Er,kis defined and evaluated in the SRF Or.Er,k\nworks as a link that connecting the difference equations in different Lorentz\nframes, which does not break the Lorentz invariance. This can also reflect the\nreason for choosing a SRF. Similarly, Φ∆τ\np,Φ∆τ\nR, and Φ∆τ\nRccan also be proven\nto be Lorentz invariant.\n4.3. High order LIVPAs\nConsidering that for Lorentz invariant volume-preserving sub-maps, com-\nposite algorithms of them are also LIVPAs. Therefore, we can compose them\nto obtain explicit algorithms with different orders [34]. The 1-order LIVPA can\nbe obtained as\nΦ∆τ\n1= Φ∆τ\nR◦Φ∆τ\nK◦Φ∆τ\np. (39)\nBased on the symmetric composite, the 2-order symmetric scheme can be built\nas follows,\nΦ∆τ\n2= Φ∆τ\n2p◦Φ∆τ\n2\nK◦Φ∆τ\nR◦Φ∆τ\n2\nK◦Φ∆τ\n2p. (40)\n17For higher order schemes, the 2 (l+ 1)-order schemes can be obtain by [34]\nΦ∆τ\n2(l+1)= Φul∆τ\n2◦Φwl∆τ\n2◦Φul∆τ\n2, (41)\nwhereul=/parenleftbig\n2−21/(2l+1)/parenrightbig−1andwl= 1−2u<0. For example, we can get the\n4-order LIVPA as\nΦ∆τ\n4= Φu1∆τ\n2◦Φw1∆τ\n2◦Φu1∆τ\n2, (42)\nwhereu1=/parenleftbig\n2−3√\n2/parenrightbig−1andw1=−3√\n2//parenleftbig\n2−3√\n2/parenrightbig\n. Notice that we can also\nreplace Φ∆τ\nRin Eqs.39, 40, and 42 by Φ∆τ\nRc, and new types of LIVPAs can be\nobtained as\nΦ∆τ\n1c= Φ∆τ\nRc◦Φ∆τ\nK◦Φ∆τ\np, (43)\nΦ∆τ\n2c= Φ∆τ\n2p◦Φ∆τ\n2\nK◦Φ∆τ\nRc◦Φ∆τ\n2\nK◦Φ∆τ\n2p, (44)\nΦ∆τ\n4c= Φu1∆τ\n2c◦Φw1∆τ\n2c◦Φu1∆τ\n2c. (45)\nBecause Φ∆τ\nRis constructed from the exact solution while Φ∆τ\nRcis a 2-order\napproximation solution, algorithms using Φ∆τ\nRare theoretically more accurate\nthan that using Φ∆τ\nRc.\nFinally, we summarize the key procedures for using the LIVPAs.\n1.Preparation Step : Choose a SRFOrand calculateFα\nβusingFα\nβand\nL, namely,Fα\nβ=L−1Fα\nβL. Get the expressions for Er,Br,Kα\nβ, and\nRα\nβ. Note that the SRF should not change after determined.\n2.Iteration Step 1 : Usexα,kto obtainKα,k\nβandRα,k\nβ. Substitute L−1xα,k\ninto expressions of ErandBrto getEr,kandBr,k.\n3.Iteration Step 2 : Substitute Kα,k\nβ,Rα,k\nβ,Er,k, andBr,kinto the differ-\nence equations of LIVPAs to get xα,k+1andpα,k+1.\n185. Numerical Experiments\nIn this section, we study the numerical performances of LIVPAs in a typical\nstatic axisymmetric electromagnetic field, namely,\nA=B0R2\n3R0ˆeθ, (46)\nϕ=E0R2\n0\nR, (47)\nB=B0R\nR0ˆez, (48)\nE=E0R2\n0\nR2ˆeR, (49)\nwhereR=/radicalbig\nx2+y2,θ, andzare cylindrical coordinates, ˆeR,ˆeθ, and ˆez\nare the unit vectors of cylindrical coordinates, B0andE0respectively denote\nthe strength of magnetic and electric fields, and R0is the space parameter of\nthe field. In this section, we keep several fundamental parameters unchanged.\nExpressed in SI, the field parameters are set as B0= 1 T,E0= 10 V/m, and\nR0= m 0c/eB0≈1.69×10−3m, where eis the unit charge. The charge of the\nparticle is set as ˜q= 1.\n5.1. Lorentz invariance\nAs discussed in Sec.4, through performing Lorentz transformation on differ-\nence equations, we can theoretically test the Lorentz invariance of algorithms.\nMeanwhile, according to the definition of Lorentz invariant algorithms, we can\nalso directly use numerical results to examine the Lorentz invariance. We use\nthe symbol ΦAto express the difference equation of an algorithm Ain frameO.\nReplacing all variables in ΦAby variables observed in another Lorentz frame O/prime,\nwe can get the difference equation in O/prime, denoted by ΘA, which has the same\nform asA. The initial condition Z0inOcorresponds to the initial condition\nZ/prime0=MZ0inO/primefor the same process. The numerical solutions of ΦAand\nΘAcalculated inOandO/primeare thus denoted by ΦAZ0andΘAZ/prime0, respectively.\nTo compare the results in different frames, we can transform ΘAZ/prime0to frame\nO, namely, calculating TM−1◦ΘAZ/prime0. If neglecting machine errors, Lorentz\ninvariant algorithms should satisfy that ΦAZ0=TM−1◦ΘAZ/prime0.\n19DuringtheconstructionsofLIVPAs, therearethreeLorentzframesinvolved,\nOr,O,andO/prime. Sincethechoiceof Orisarbitrary,inthissection,weset Or=O,\nwhich means L=Iα\nβandN=M. The expressions of electromagnetic field in\nOis given by Eqs.46-49. Meanwhile, without loss of generality, we consider M\nis a subset of the Lorentz group, namely, the Lorentz boost matrix,\nL=∂x/primeα\n∂xβ=\nΓ−Γβ1−Γβ2−Γβ3\n−Γβ11 +(Γ−1)β2\n1\nβ2(Γ−1)β1β2\nβ2(Γ−1)β1β3\nβ2\n−Γβ2(Γ−1)β1β2\nβ2 1 +(Γ−1)β2\n2\nβ2(Γ−1)β2β3\nβ2\n−Γβ3(Γ−1)β1β3\nβ2(Γ−1)β2β3\nβ2 1 +(Γ−1)β2\n3\nβ2\n,(50)\nwhereβ=|β|, and Γ = 1//radicalbig\n1−β2is the Lorentz factor. The initial condition\nof the charged particle in rest reference frame Ois set asZ0=/parenleftbig\nxα,0,pα,0/parenrightbig\n,\nwherexα,0= (0,0,2,0)andpα,0=/parenleftbig√\n2,0,1,0/parenrightbig\n. We set the speed of frame O/prime\nrelative toOasβcor= (0.5,0,0). Initially, the local time of the OandO/primeare\nboth set as 0, and the origin points of OandO/primecoincide in 4-spacetime.\nIn Figure 1, the results of the 2-order LIVPA Φ2given by Eq.40 are de-\npicted. According to Fig.1a and Fig.1b, we can see that the orbit calculated\nin two frames are consistent. This can be further explained by Fig.1c. Dx, the\ndifference of x-coordinates in Fig.1a and Fig.1b, is on the order of 10−14m,\nwhich approaches to the machine precision. Meanwhile, according to Figs.1d-\n1i, if we increase the step length ∆τ, though the accuracy of orbits decreases,\nthe reference-independence of numerical results still holds. The magnitude of\nDxfor different ∆τkeeps unchanged.\nIn figure 1, we plot the first turn of the orbit and Dxis on the order of\nmachine precision. The differences between the results obtained in different\nLorentz inertial frames by LIVPAs result from the machine truncation errors.\nAs the increase of iteration times, the accumulation of machine errors cannot\nbe avoided. Here, we analyze the accumulation of machine errors by studying\nthe magnitude of |Dx|/R0in terms of iteration step number, see Fig.2. The\nstep length is set as ∆τ= 0.1. The same processes are calculated by using\n2-order LIVPA, 4-order LIVPA, and the 2-order implicit midpoint canonical\n20symplectic algorithm (IMCSA). The IMCSA is obtained by discretizing the 4-\ndimensional Lorentz invariant Hamiltonian equation of charged particles [40]\nusing the implicit mid-point symplectic scheme. For both cases, the value of\n|Dx|/R0is on the magnitude of machine error at the beginning and finally\nreaches 10−5which is still a negligible value after 106iterations.\nTo compare with LIVPAs, we calculate the same process by use of the 2-\norder explicit canonical symplectic algorithm (ECSA) given in Ref.[24]. This\nalgorithm is built based on generating function method during which the in-\nvariant Hamiltonian is divided into 7 parts. As we have discussed in Sec.2, the\nsplitting method can easily break the Lorentz invariance of continuous systems.\nIn Figure 3, the results in different frames of ΦECSA 2are plotted. When we\nset the step length as ∆τ= 0.01, the differences of x-coordinate calculated by\n2-order ECSA in OandO/primeare on the order of R0, see Fig.3c. Especially, as we\nincrease ∆τto0.1, ECSA becomes unstable in O/primeand gives incorrect results,\nsee Fig.3e. Figure 3 implies that the 2-order ECSA is not Lorentz invariant\nand one should be very careful to use ECSA in moving frames even though it\npossesses secular stability in the rest frame.\nThe Vay scheme [38] and the Higuera-Cary scheme [31] can preserve the\nE×Bvelocity in different Lorentz frames. Both methods show excellent secular\nstabilities for simulating relativistic charged particles [30]. Here, we use them\nto solve the same process in Fig.1. In OandO/prime, long-term stable orbits can\nbe obtained by both algorithms. Therefore, in figure 4, we only depict the first\nmajor turn. The time step is denoted by ∆tand∆t/primeinOandO/prime, respectively.\nIn the case of ∆t= ∆t/prime= 0.1, regardless of the small deviations, the results in\ndifferent frames are consistent, see Figs.4a and c. As the step length increases\nto0.5, however, we can see significant differences between the red and blue lines.\nThe results show that, Vay’s and Higuera-Cary’s method have better Lorentz\ninvariance for small step length. As the step length increases, even though the\nglobal stability still holds, the Lorentz invariance can be broken, which might\nresultfromthegrowthofhigherordertermsmentionedinRef.[31]. ForLIVPAs,\nhowever, the reference-independent property is not affected by the step length,\n21see Fig.1.\n5.2. Long-term stability\nThrough conserving the volume of phase space, the algorithms perform bet-\nter in secular simulations compared with traditional algorithms like the Newton\nmethod, the Runge-Kutta method [33–35, 41, 42]. Here we compare the secular\nstabilities of 2-order LIVPA Φ2with the Lorentz invariant 4-order Runge-Kutta\n(RK4) method. The Lorentz invariant RK4 method is obtained by discretizing\nEq.13 using the 4-order Runge-Kutta method. Because the discretization keeps\nthe integrity of all the 4-dimensional geometric objects, the resulting method is\nLorentz invariant, which can also be proven numerically in figure 5.\nFor a charged particle moving in static field, there are two important invari-\nants, namely, the mass-shell\nH=gαβpαpβ=γ2−p2≡1, (51)\nand the energy of the particle\nH=γ+ϕ=/radicalbig\n1 +p2+ϕ, (52)\nIn figure 6, the long-term error evolutions of the mass shell H(Z)and the\nparticle energy H(Z)simulated by Φ2and RK4 are depicted. In the case of\nthe mass shell in Fig.6a, the error of mass shell grows significantly to 1 for\nRK4, while the error given by 2-order LIVPA is limited near 0. In the case\nof the particle energy in Fig.6b, the energy calculated by RK4 decreases 30%\nafter 5×106steps, but the energy obtained by Φ2conserves well. Therefore,\nthe long-term stability of 2-order LIVPA is better than RK4, even though its\norder is smaller. We also depict the orbits at different moments obtained by\n2-order LIVPA and RK4 in figure 7. The orbits calculated by 2-order LIVPA\nare depicted in Fig.7a, which remains stable after 5×106iterations. For RK4,\nhowever, because of the accumulations of numerical errors of RK4, the radius\nof rotation on minor period keeps shrinking and incorrect orbits are obtained\nafter long-term simulation, see Fig.7b.\n225.3. Convergence rate of LIVPAs\nIn Sec.4, we have constructed several LIVPAs with different orders via dif-\nferent composing procedures. In order to verify their orders, we perform conver-\ngence analysis on the 1-order LIVPA Φ1, 2-order LIVPA Φ2, and 4-order LIVPA\nΦ4. The results are depicted in figure 8. The sequences of energy error ∆Hare\ncalculated from τ= 0toτ= 10, and the infinity norm of ∆H/H 0is used to\nplot the convergence rate. The red line is the convergence rate of Φ1. It has the\nsame slope with the function 10−10∆τ, which implies Φ1is a 1-order algorithm.\nSimilarly, Φ2, plotted by the green line, can be proven to be a 2-order algorithm\nvia compared with the reference function 10−10∆τ2.Φ4is a 4-order scheme ac-\ncording to the blue line. Because the value of ∆H/H 0reaches machine errors,\nthe convergence rate of Φ4slows down as the step length becomes very small,\nsee the last point of the blue line at ∆τ= 10−2.\n6. Conclusion\nInthispaper,westudytheconstructionsofexplicitLorentzinvariantvolume-\npreserving algorithms for relativistic charged particle dynamics. Through intro-\nducing a splitting reference frame (SRF), we prove that the corresponding split-\nting operation can avoid breaking the Lorentz invariance of the original system.\nBy use of this procedure, we build explicit LIVPAs with different orders. The\nLorentz invariant properties of LIVPAs are tested in a typical electromagnetic\nfield configuration, which shows that LIVPAs possess reference-independent sec-\nular stabilities. It is proven that the Hamiltonian splitting technique for con-\nstructing the explicit symplectic algorithms [24] breaks the Lorentz invariance.\nCompared with the Vay scheme [38] and the Higuera-Cary scheme [31], the\nbenefits of LIVPAs are also obviously reflected. The reference-independency\nof LIVPAs is not affected by the configurations of step length, while both Vay\nand Higuera-Cary methods show the decreases of accuracy in different frames\nwhen the step length grows. Meanwhile, long-term behaviors of LIVPAs show\nbetter than the Runge-Kutta method in resolving the motion constants such\n23as the mass-shell and the particle energy. Therefore, LIVPAs have better per-\nformances when simulating nonlinear multi-scale processes. We also provide\nthe numerical convergence analysis to LIVPAs, which proves the ability of the\nLorentz invariant splitting method in constructing high order explicit schemes.\nThe work in this paper extends the method introduced in Ref.[39] in which\nthe integrity of geometric objects is necessary to construct Lorentz invariant\nalgorithms. We generalize the idea to a more flexible level, that even though\nthe geometric objects in 4-dimensional spacetime are broken, the Lorentz in-\nvariance of algorithms still holds. The combination of the splitting technique\nand the Lorentz invariant method provides a convenient way to build advanced\nalgorithms with high orders. Additionally, generally speaking, as the orders of\nalgorithms increase, it will be more difficult to implement adaptive time step op-\ntimization. Especially, constructing structure-preserving algorithms with adap-\ntive time step is still a difficult task. Some works have been done by use of the\ntime transformation method [21, 43] which should transform the original system\ninto a new form. The results given by symplectic algorithms of the new system\nin which the step length are still fixed are equivalent to results by adaptive step\nlength in the original system. In future work, more works will be done to study\nand apply the LIVPAs to key physical problems in different areas, and the usage\nof LIVPAs in Particle-in-Cell codes will also be studied.\nAcknowledgements\nThisresearchissupportedbyNaturalScienceFoundationofChina(Nos.11805203,\n11775222, 11505185), and National Magnetic Confinement Fusion Energy R&D\nProgram of China (2017YFE0301700).\nAppendix\nIn this appendix, we provide the detailed derivations of Eqs.31, 32, and 36.\nAccording to the definitions of Kα\nβandRα\nβ, it is readily to find that/parenleftBig\nKα\nβ/parenrightBig3\n=\n(Er)2Kα\nβand/parenleftBig\nRα\nβ/parenrightBig3\n=−(Br)2Rα\nβ. Therefore, considering that Erand\n24Brare scalar fields, we have/parenleftBig\nKα\nβ/parenrightBig3\n=L/parenleftBig\nKα\nβ/parenrightBig3\nL−1=L(Er)2Kα\nβL−1=\n(Er)2LKα\nβL−1= (Er)2Kα\nβ. One should notice that, Eris a function in frame\nOrwhileKα\nβis the matrix evaluated in frame O. When we calculate the value\nof/parenleftBig\nKα\nβ/parenrightBig3\natxαin frameO, the input of value of Ershould beL−1xα. From\nthe viewpoint of manifold, xαandL−1xαare different coordinates of the same\npoint on manifold. Similarly, we can also obtain/parenleftBig\nRα\nβ/parenrightBig3\n=−(Br)2Rα\nβ.\nWe first prove that exp/parenleftBig\nτ˜qKα\nβ/parenrightBig\n=Iα\nβ+sinh(τ˜qEr)\nErKα\nβ+cosh(τ˜qEr)−1\n(Er)2Sα\nβin\nEq.31. Using/parenleftBig\nKα\nβ/parenrightBig2\n=Sα\nβand/parenleftBig\nKα\nβ/parenrightBig3\n= (Er)2Kα\nβ, we have\nexp/parenleftbig\nτ˜qKα\nβ/parenrightbig\n=Iα\nβ+τ˜qKα\nβ+/parenleftBig\nτ˜qKα\nβ/parenrightBig2\n2!+/parenleftBig\nτ˜qKα\nβ/parenrightBig3\n3!+···\n=Iα\nβ+τ˜qKα\nβ+(τ˜q)2\n2!Kα\nβ+(τ˜q)3(Er)2\n3!Kα\nβ+(τ˜q)3(Er)2\n4!Sα\nβ+···\n=Iα\nβ+1\nEr/parenleftBigg\nτ˜qEr+(τ˜qEr)3\n3!+(τ˜qEr)5\n5!+···/parenrightBigg\nKα\nβ\n+1\n(Er)2/parenleftBigg\n−1 + 1 +(τ˜qEr)2\n2!+(τ˜qEr)4\n4!+···/parenrightBigg\nSα\nβ\n=Iα\nβ+1\nEreτ˜qEr−e−τ˜qEr\n2Kα\nβ+1\n(Er)2/parenleftbigg\n−1 +eτ˜qE+e−τ˜qE\n2/parenrightbigg\nSα\nβ\n=Iα\nβ+sinh (τ˜qEr)\nErKα\nβ+cosh (τ˜qEr)−1\n(Er)2Sα\nβ. (53)\nThederivationofEqs.32issimilar. Theequation exp/parenleftBig\nτ˜qRα\nβ/parenrightBig\n=Iα\nβ+sin(τ˜qBr)\nBrRα\nβ+\n1−cos(τ˜qBr)\n(Br)2Pα\nβcan be proven using/parenleftBig\nRα\nβ/parenrightBig2\n=Pα\nβand/parenleftBig\nRα\nβ/parenrightBig3\n=−(Br)2Rα\nβ\n25as follows,\nexp/parenleftbig\nτ˜qRα\nβ/parenrightbig\n=Iα\nβ+τ˜qRα\nβ+/parenleftBig\nτ˜qRα\nβ/parenrightBig2\n2!+/parenleftBig\nτ˜qRα\nβ/parenrightBig3\n3!+···\n=Iα\nβ+τ˜qRα\nβ+(τ˜q)2\n2!Pα\nβ−(τ˜q)3(Br)2\n3!Rα\nβ−(τ˜q)3(Br)2\n4!Pα\nβ+···\n=Iα\nβ+1\nBr/parenleftBigg\nτ˜qBr−(τ˜qBr)3\n3!+(τ˜qBr)5\n5!−···/parenrightBigg\nRα\nβ\n−1\n(Br)2/parenleftBigg\n−1 + 1−(τ˜qBr)2\n2!+(τ˜qBr)4\n4!−···/parenrightBigg\nPα\nβ\n=Iα\nβ+sin (τ˜qBr)\nBrRα\nβ+1−cos (τ˜qBr)\n(Br)2Pα\nβ. (54)\nThe explicit form of Cayley transformation in Eq.36 can be proven as follows,\ncay/parenleftbig\n∆τ˜qRα\nβ/parenrightbig\n=/parenleftbig\nIα\nβ−aRα\nβ/parenrightbig−1/parenleftbig\nIα\nβ+aRα\nβ/parenrightbig\n=/bracketleftBig\nIα\nβ+aRα\nβ+/parenleftbig\naRα\nβ/parenrightbig2+/parenleftbig\naRα\nβ/parenrightbig3+···/bracketrightBig/parenleftbig\nIα\nβ+aRα\nβ/parenrightbig\n=/bracketleftBig\nIα\nβ+/parenleftBig\n1−(aBr)2+ (aBr)4−···/parenrightBig\naRα\nβ\n+/parenleftBig\n1−(aBr)2+ (aBr)4−···/parenrightBig\na2Pα\nβ/bracketrightBig/parenleftbig\nIα\nβ+aRα\nβ/parenrightbig\n=/bracketleftBigg\nIα\nβ+a\n1 + (aBr)2Rα\nβ+a2\n1 + (aBr)2Pα\nβ/bracketrightBigg\n/parenleftbig\nIα\nβ+aRα\nβ/parenrightbig\n=/parenleftBigg\nIα\nβ+2a\n1 + (aBr)2Rα\nβ+2a2\n1 + (aBr)2Pα\nβ/parenrightBigg\n. (55)\nReferences\nReferences\n[1] K. Feng, Difference schemes for hamiltonian formalism and symplectic ge-\nometry, Journal of Computational Mathematics 4 (3) (1986) 279–289.\n[2] E. Forest, R. D. 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Controlled Fusion 57 (5) (2015) 054007.\n[43] Y. Shi, Y. Sun, Y. Wang, J. Liu, Study of adaptive symplectic methods for\nsimulating charged particle dynamics, Journal of Computational Dynamics\n6 (2) (2019) 429.\n30-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n02468\nt(s)×10-10-20246Dx(m)×10-14\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n02468\nt(s)×10-10-101Dx(m)×10-14\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n02468\nt(s)×10-10-2-1012Dx(m)×10-15(c) (a)ΦLIV P A 2Z0 TM−1◦ΘLIV P A 2Z/prime0\nΦLIV P A 2Z0 TM−1◦ΘLIV P A 2Z/prime0\nΦLIV P A 2Z0 TM−1◦ΘLIV P A 2Z/prime0Δτ=0.01 Δτ=0.01Δτ=0.01\nΔτ=0.1 Δτ=0.1Δτ=0.1\nΔτ=0.5 Δτ=0.5Δτ=0.5(d)(b)\n(e) (f)\n(g) (h) (i)Figure 1: Orbits given by 2-order LIVPA Φ2in different Lorentz frames. The results in (a),\n(b) and (c) are simulated with the step-length ∆τ= 0.01, results in d, e and f are calculated\nby∆τ= 0.1, and results in g, h and i are calculated by ∆τ= 0.5. The differences between\nthe results calculated in rest and moving frames are on the order of machine precision.\n31101102103104105106\nSteps10-2010-1510-1010-5100|Dx|/R0(m)2-order IMCSA\n2-order LIVPA\n4-order LIVPAFigure 2: Evolution of |Dx|/R0as the increase of iteration steps. The step-length is set as\n∆τ= 0.1. The accumulations of the 2-order LIVPA (blue line), the 4-order LIVPA (red line),\nand the implicit midpoint canonical symplectic algorithm (IMCSA, green line) have the same\ntrend. After 106iterations, the order of Dxis still ignorable compared with R0. The IMCSA,\nis also a LIA, is obtained by discretizing the 4-dimensional Lorentz invariant Hamiltonian\nequation of charged particles [40] using the implicit mid-point symplectic scheme.\n32-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n02468\nt(s)×10-10-2-10123Dx(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n02468\nt(s)×10-10-10-8-6-4-202Dx(m)×10-3\nTM−1◦ΘECSA 2Z/prime0TM−1◦ΘECSA 2Z/prime0ΦECSA 2Z0\nΦECSA 2Z0(c) (b) (a)\n(d) (e) (f)Δτ=0.01 Δτ=0.01Δτ=0.01\nΔτ=0.1 Δτ=0.1Δτ=0.1Figure 3: Orbits given by 2-order ECSA given in Ref.[24] in different Lorentz frames. The\nresults in (a), (b) and (c) are simulated with the step-length ∆τ= 0.01, while results in d, e\nand f are calculated by ∆τ= 0.1. The position difference in two frames is comparable to R0\nwhen ∆τ= 0.01, and ECSA becomes unstable in the frame O/primefor∆τ= 0.1.\n33-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505\ny(m)×10-3\nΦVa yZ0\nTM−1◦ΘVa yZ/prime0\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505\ny(m)×10-3\nΦHCZ0\nTM−1◦ΘHCZ/prime0Δt=Δt/prime=0.1 Δt=Δt/prime=0.5\nΔt=Δt/prime=0.1 Δt=Δt/prime=0.5Vay\nHiguera-Cary Higuera-CaryVay\n(a) (b)\n(d) (c)Figure 4: The orbits calculated by Vay’s method [38] and the Higuera-Cary method [31].\nResults of Vay’s method are depicted in (a) and (b), while results of the Higuera-Cary method\nare plotted in (c) and (d). Blue lines give the orbit obtained in frame Oand red lines show the\nresults in frameO/prime. Both methods show better reference-independency for small step length.\nAs the step length increases, the red lines show obvious deviation from blue lines, which shows\nthat Vay’s and Higuera-Cary’s method are not Lorentz invariant. However, both algorithms\npossess long-term stability in different step length.\n34-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n02468\nt(s)×10-10-10-505Dx(m)×10-15\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n02468\nt(s)×10-10-1-0.500.511.5Dx(m)×10-15\nTM−1◦ΘRK4Z/prime0TM−1◦ΘRK4Z/prime0ΦRK4Z0\nΦRK4Z0(c) (b) (a)\n(d) (e) (f)Δτ=0.01 Δτ=0.01Δτ=0.01\nΔτ=0.1 Δτ=0.1Δτ=0.1Figure 5: Orbits given by 4-order Lorentz invariant Runge-Kutta method in different Lorentz\nframes. The results in (a), (b) and (c) are simulated with the step-length ∆τ= 0.01, while\nresults in d, e and f are calculated by ∆τ= 0.1. The results show that the 4-order Runge-\nKutta method is Lorentz invariant if it is directly used to discrete the invariant form of the\nLorentz force equation, namely, Eq.13.\n012345\nSteps×10600.20.40.60.81ΔH/H0\n2-order LIVPA\nRK4\n012345\nSteps×106-0.3-0.25-0.2-0.15-0.1-0.0500.05ΔH/H 0\n2-order LIVPA\nRK4(a) (b)\nFigure6: Long-termerrorevolutionsofthemassshell(a)andtheparticleenergy(b)calculated\nby 2-order LIVPA and 4-order Runge-Kutta. The results of 2-order LIVPA are depicted by\nblue lines, while the results of Runge-Kutta method are plotted by red lines. The total number\nofstepsis 5×106. Thedefinitionsoferrorsare ∆H=H(n∆τ)−H 0and∆H=H(n∆τ)−H0.\nThe step length is ∆τ= 0.1.\n35-5 0 5\nx(m)×10-3-5-4-3-2-1012345y(m)×10-3\nstep=0 step=2.5e6 step=5e6\n-5 0 5\nx(m)×10-3-5-4-3-2-1012345y(m)×10-3\nstep=0 step=2.5e6 step=5e62-order LIVPA RK4 (a) (b)Figure 7: Orbits calculated by 2-order LIVPA (a) and 4-order Runge-Kutta (b) at different\nmoments. The total number of steps is 5×106and the step length is set as ∆τ= 0.1.\n10-210-1100\nΔτ10-2010-1810-1610-1410-1210-1010-8max (ΔH/H 0)\n1-order LIVPA\n2-order LIVPA\n4-order LIVPA\n10−10Δτ\n10−11Δτ2\n10−12Δτ4\nFigure 8: Convergence rates of 1-order LIVPA Φ1, 2-order LIVPA Φ2, and 4-order LIVPA Φ4.\nThe sequences of energy error ∆Hare calculated from τ= 0toτ= 10, and the infinity norm\nof∆H/H 0is used to plot the convergence rate. The three black lines with different slope are\nreference functions with 1-order, 2-order, and 4-order convergence rates respectively.\n36"    },    {        "title": "1309.3019v1.Magnetohydrodynamics_dynamical_relaxation_of_coronal_magnetic_fields__IV__3D_tilted_nulls.pdf",        "content": "arXiv:1309.3019v1  [astro-ph.SR]  12 Sep 2013Astronomy& Astrophysics manuscriptno.20346 c/circleco√yrtESO 2021\nJuly26,2021\nMHDdynamicalrelaxationofcoronal magnetic fields\nIV.3D tiltednulls\nJorge Fuentes-Fern´ andez and Clare E.Parnell\nSchool of Mathematics andStatistics,Universityof StAndr ews, NorthHaugh, StAndrews, Fife,KY16 9SS,Scotland\nABSTRACT\nContext.Therearevarioustypesofreconnectionthatmaytakeplacea t3Dmagneticnullpoints.Eachdi fferentreconnectionscenario\nmust be associated witha particular type of current layer.\nAims.A range of current layers may form because the topology of 3D n ulls permits currents to form by either twisting the field\nabout the spine of the null or by folding the fan and spine into each other. Additionally, the initial geometry of the field c an lead to\nvariationsinthecurrentsthatareaccumulated.Here,west udycurrentaccumulationsinso-called3D‘tilted’nullsfo rmedbyafolding\nof the spine and fan. A non-zero component of current paralle l to the fan is required such that the null’s fan plane and spin e are not\nperpendicular. Ouraimsaretoprovide validmagnetohydros tatic equilibriaandtodescribe thecurrentaccumulations invarious cases\ninvolving finiteplasma pressure.\nMethods. To create our equilibrium current structures we use a full, n on-resistive, Magnetohydrodynamics (MHD) code so that no\nreconnectionisallowed.Aseriesofexperiments areperfor medinwhichaperturbed3Dtiltednullrelaxestowards anequ ilibriumvia\nreal,viscous damping forces. Changes tothe initialplasma pressure and tomagnetic parameters areinvestigated syste matically.\nResults.Aninitiallytiltedfanisassociatedwithanon-zeroLorent zforcethatdrivesthefanandspinetocollapsetowardseach other,\nin a similar manner to the collapse of a 2D X-point. In the final equilibrium state for an initially radial null with only the current\nperpendicular to the spine, the current concentrates along the tilt axis of the fan and in a layer about the null point with a sharp peak\nat the null itself. The continued growth of this peak indicat es that the system is in an asymptotic regime involving an infi nite time\nsingularityatthenull.Whenthe initialtiltdisturbance ( currentperpendicular tothespine) iscombined withaspira l-type disturbance\n(current parallel tothe spine), the final current density co ncentrates inthree regions: one on the fan along its tilt axi s and two around\nthe spine, above and below the fan. The increased area of curr ent accumulation leads toa weakening of the singularity for med at the\nnull. The3D spine-fan collapse withgeneric current studie d here provides the ideal setupfor non-steady reconnection studies.\nKey words. Magnetohydrodynamics (MHD) –Sun:corona – Sun:magnetic to pology –Magnetic reconnection\n1. Introduction\nMagnetic null points (locations where all three components of\nthe magnetic field are zero) have been associated with sites o f\nmagnetic reconnectionever since the notion of magnetic rec on-\nnectionwasfirstconceived(Parker1957;Sweet1958).Itisn ow\nknown that, in three dimensions (3D), they are not the only\npossible location for reconnection (e.g., Schindleret al. 1988;\nHesse &Schindler 1988; Hornig&Priest 2003; Priest et al.\n2003), which may also take place in magnetic separa-\ntors (Longcope& Cowley 1996; Longcope 2001; Hayneset al.\n2007; Parnellet al. 2008, 2010a,b) and quasi-separatrix la y-\ners (Priest &D´ emoulin 1995; Demoulinet al. 1996, 1997;\nAulanieret al. 2006; Restanteet al. 2009; Wilmot-Smithet a l.\n2009).However,3Dnullpointsstill remainkeysitesforcur rent\naccumulationandenergyrelease.UsingSOHOMDI(Michelson\nDopplerImager)magnetogramdata,Longcope&Parnell(2009 )\nestimate thattherearearound20,000nullsabove1.5Mmin th e\nsolar atmosphere during solar minimum. Moreover, due to the\nrapid fall offin complexityof the magneticfield abovethe pho-\ntosphere, more nulls are likely to occur below 1.5 Mm than are\nabove.\nOver the years 3D magnetic null points have been studied\nin detail on a number of di fferent levels. The first set of pa-\npers considers the basic structure of nulls (e.g. Cowley 197 3;\nFukao& Tsuda 1973) with a comprehensiveaccount of the lin-\near structure of all possible 3D nulls provided in Parnellet al.(1996). The general geometry of 3D magnetic null points in-\nvolvestwoimportantfeatures.Thefieldlinesthatextendin to/out\nofthenullitself lie eitheralongapairoflines,knownas spines,\nor on a surface, known locally as the fan. The field lines in the\nfan are all directed either away from the null, forming what i s\nknown as a positive (or B-type) null, or into the null, creati ng a\nnegative (A-type) null. In the case of a positive null the spi nes\naredirectedintothenullandforanegativenulltheyaredir ected\naway.\nA second set considers the nature of possible reconnection\nscenarios at 3D nulls driven by plasma flows of varying types,\nwhich may depend on the behaviour of the flow pattern rela-\ntive to the spine and fan. These perturbations may be caused\nby a local driver (Rickard& Titov 1996; Bulanovet al. 2002;\nPontinetal. 2004, 2005, 2007a,b; Pontin&Galsgaard 2007;\nPriest &Pontin 2009; Massonetal. 2009; Wyper& Jain 2010;\nAl-Hachami&Pontin2010;Pontinet al.2011)orbyanindirec t\nperturbation (Pariatet al. 2009, 2010; Edmondsonetal. 200 9;\nMassonet al. 2009, 2012) far away from the null. In particula r,\nflow patterns that twist the field about the spine, known as tor -\nsional reconnection regimes, dissipate currents that lie p arallel\nto thespine andresult in theslippageofthe fieldlinesabout the\nspine. Flow patternsacrossthe fan or spine,knownasfan-sp ine\nreconnectionregimes,areassociatedwithcurrentsperpen dicular\nto thespineandcause a shearingofthe spineand /orfanleading\ntoreconnectionat thenullitself.\n1Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\nIn generalterms,reconnectionstudiesconsiderpotential 3D\nnulls that are drivenby some external force initiating reco nnec-\ntion. However, an arguably more realistic scenario is that fl ows\nin the system generate currents resulting in the collapse of the\nnull in a particular way, since non-potential nulls are gene ri-\ncally unstable (Parnellet al. 1997), and leading to the accu mu-\nlation of current at, or in the vicinity of, the null. Papers t hat\nlook at the accumulation of currents and their associated eq ui-\nlibria form a third set of papers (e.g. Pontin& Craig 2005;\nFuentes-Fern´ andez& Parnell 2012). This paper belongs to t his\nset.\nCurrent accumulations due to spiral nulls in which\nthe field twists about the spine were studied in\nFuentes-Fern´ andez& Parnell (2012), the previous paper to\nthis one in the series on ‘Dynamical relaxation of coronal\nmagnetic fields’. We looked at the formation of a static non-\nforce-free equilibrium due to a torsional-type disturbanc e at\na 3D null point. In that paper, current accumulations were\nfound due to the viscous non-resistive MHD relaxation of a\nnon-potential null with a spine-aligned current density. I n the\npresent paper, we use the same approach, but here we study\ncurrent accumulations due to a shearing-type disturbance t o a\n3D null in which the current and spine fold towards on another .\nHereweconsideranon-potentialnullwithafan-alignedcur rent\ndensity. The current accumulations formed must be di fferent to\nthose found in Fuentes-Fern´ andez& Parnell (2012) since th e\nreconnectionin thetwo casesis completelydi fferent.\nPontin&Craig (2005) has considered the case of a such a\nperturbation, associated with a component of the current th at\nis strictly perpendicular to the spine. They find a collapse o f\nthe fan and spine towards each other and the formation of a\ncurrent singularity at the location of the null in a non-forc e-\nfree equilibrium. In their paper, they consider the nature o f this\nsingularity and find that it scales with the numerical resolu -\ntion in an equivalent manner to the 2D singularities studied by\nCraig& Litvinenko (2005). Also, they find that increasing th e\npressure doesnot stop the formationof the singularity,but does\nsignificantly reduce its strength. Both of these studies use d a\nfrictional Lagrangian code with a fictitious damping term, −κv,\nadded to the momentum equation to enable the field to relax to\nan equilibrium. This model imposes a natural constraint, si nce\ntheirdampingtermisnotphysical.Indeed,thepolytropicm odel\nwithp≈ργis used for pressure, which imposes a condition of\nadiabaticitytothe relaxationprocess.\nFuentes-Fern´ andezet al. (2011) considered very simi-\nlar, but subtly different, experiments to Craig& Litvinenko\n(2005) involving the collapse of non-potential 2D X-points\nembedded within a plasma. One key di fference is that\nFuentes-Fern´ andezet al. (2011) were able to followthe dyn am-\nical evolution of the system and the energetics, by using an\nMHD relaxation driven by viscous forces, which naturally ha s\nan associated heating term in the energy equation. Thus, the\nsystemdoesnotevolveadiabatically.WhileCraig& Litvine nko\n(2005) focuson the scaling laws for the formationof the sing u-\nlarity, Fuentes-Fern´ andezet al. (2011) focus on the actua l non-\nforce-freeequilibriumobtained due to the presence of a pla sma\npressure and the nature of the singularity as it grows in time .\nFuentes-Fern´ andezet al. (2011) show that the field left aft er the\nviscous relaxation has finished is in a “quasi-equilibrium” state\nwhere the central current layer, which stretches out a short way\nalong the separatrices, slowly changes its shape becoming t hin-\nner and shorter. Indeed, the system is found to have entered a n\nasymptotic regime and is heading towards an infinite time sin -\ngularity.The approach of Fuentes-Fern´ andezet al. (2011) is\nquite different to the frictional, adiabatic relaxation of\nCraig& Litvinenko (2005). Similarly, the study we carry\nout here uses a di fferent relaxation to the frictional, adia-\nbatic approach of Pontin&Craig (2005) because it involves\nthe viscous, non-resistive, non-adiabatic relaxation of 3 D\nnulls. Furthermore, the model we consider here di ffers from\nFuentes-Fern´ andez& Parnell (2012) because here the colla pse\nofthenullandcurrentaccumulationisassociatedwithacur rent\nthat is purely perpendicular to the spine as opposed to paral lel\nto the spine. Additionally, here we also consider the MHD\nrelaxation associated with a more general current that has\ncomponentsboth perpendicularand parallel to the spine, un like\neitherofthepreviouspapers.\nMore specifically, the aim of this paper is to investigate the\nnatureof currentaccumulationsat 3D radial magneticnulls fol-\nlowing a collapse due to the presence of an initial homoge-\nneous current density whose component perpendicular to the\nspine is non-zero. This may be thought of as the result of\na shearing-fan disturbance. The characteristics of any infi nite\ntime singularities will be determined, if they arise. We eva lu-\nate the effects of the plasma pressure in the evolution, while\nboththe initial disturbanceandthe backgroundplasma pres sure\nare changed systematically. This work is a continuation of t he\nwork carried out in Fuentes-Fern´ andezetal. (2010, 2011) a nd\nFuentes-Fern´ andez& Parnell (2012) on non-resistive MHD r e-\nlaxationofmagneticfieldsembeddedinnon-zerobetaplasma s.\nThe paper is structured as follows. In Sec. 2, we present the\nequations that define the initial configuration and the detai ls of\nthe numerical experiments. The results for 3D tilted nulls t hat\nare initially radial and have a fan-aligned current are pres ented\nin Sec. 3, while, in Sec. 4, the more general results for a radi al\nnull with components of current that are both parallel and pe r-\npendicularto the spine can be found.Finally, we conclude wi th\na generaloverviewoftheproblemin Sec.5.\n2. Initialmagneticfieldconfigurationsand\nnumericalscheme\nThe initial magnetic field is chosen to be that of a pos-\nitive linear 3D null point located at the origin of a con-\nstatnt current and associated with it. Like the initial field\nin Fuentes-Fern´ andez& Parnell (2012), the spine of the\nnull is chosen to be along the z-axis, however unlike\nFuentes-Fern´ andez& Parnell (2012) the initial magnetic fi eld\nhere has a component of current perpendicular to the spine\n(along the x-axis and parallel to the fan), rather than parallel to\nthe spine. In the first set of experiments, the current is pure ly\nperpendicularto the spine, andin the secondset it hasa gene ral\ncurrent with components both parallel and perpendicular to the\nspine.\nThecurrentdensityischosentobeconstantandcanbewrit-\ntenintermsofthecomponentparalleltothefan, jf,andparallel\ntothe spine, jsp,as\nj=1\nµ(jf,0,jsp). (1)\nFollowingParnelletal. (1996) themagneticfield, B, is\nB=/parenleftBigg\nx−jsp\n2y,jsp\n2x+by,jfy−(b+1)z/parenrightBigg\n, (2)\nwhereb>0inordertoensurethatthespineliesalongthe z-axis\nandthatthenullispositive.\n2Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\nFig.1.Magneticconfigurationfor the initial non-equilibriumsta te with homogeneousfan-alignedcurrent,forthe case with jf=1\nandp0=1, showing (a) the 3D configurationwith field lines above and b elow the fan in purple and orange, respectively. The fan\nplane is outlined in dashed black, and the spine is represent ed in green, with projections onto the xz-plane and yz-plane (dashed\ngreenlines).In (b),thespine (blue)and thesection ofthe f an plane(pink)are plottedin the x=0 plane,and thegreyarrowsshow\nthedirectionoftheinitial Lorentzforces.\nFig.2.Magnetic configurationfor the final equilibriumstate for th e radial tilted null shown in Fig. 1, showing (a) the 3D configu -\nrationwithfield linesaboveandbelowthe faninpurpleandor ange,respectively.Thefan planeisoutlinedin dashedblac kandthe\nspine is representedin green,with projectionsonto the xz-plane and yz-planein dashed greenlines. In (b), the spine (blue)and the\nsectionofthefanplane(pink)areplottedinthe x=0 plane.\nAs in Fuentes-Fern´ andez& Parnell (2012), for the numeri-\ncalexperimentsweusedLare3D,astaggeredLagrangian-rem ap\ncode that solves the full MHD equations with user-controlle d\nviscosity (see Arberetal. 2001). The resistivity is set to z ero\nfor our experiments, and the numerical domain is a 3D uni-\nform grid of 5123points. The boundary conditions are closed,\nthe magnetic field lines are line-tied, and the MHD equations\nare normalised following a standard procedure. For further de-\ntailsontheboundaryconditionsandnormalisation,seeSec .3of\nFuentes-Fern´ andez& Parnell(2012).\nThus,thenormalisedidealMHD equationsareasfollows:\n∂ρ\n∂t+∇·(ρv)=0, (3)\nρ∂v\n∂t+ρ(v·∇)v=−∇p+(∇×B)×B+Fν, (4)\n∂p\n∂t+v·∇p=−γp∇·v+Hν, (5)∂B\n∂t=∇×(v×B), (6)\nwhereFνandHνaretheviscousforceandviscousheatingterms,\nand the internal energy, ǫ, is given by the ideal gas law, p=\nρǫ(γ−1),withγ=5/3.\n3. Fan-parallelcurrentdensity\n3.1. Initialstate\nWe first lookat therelaxationofinitial configurationsofa m ag-\nnetic null point with constant current density everywhere i n the\ndirection parallel to the fan surface, but perpendicular to the\nspine (i.e., along the x-axis), of the form ( jf,0,0). For simplic-\nity, we assume b=1 in Eq. (2), so the field lines have no pre-\nferred direction on the fan plane, but expand regularly thro ugh\ntheplane.\nOurinitial magneticfieldis thengivenby\n(Bx,By,Bz)=(x,y,jfy−2z). (7)\n3Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\nFig.3.Contour plots of the di fferent forces acting in the final equilibrium state in the x=0 plane, for the same experiment as in\nFig. 1. Showing, from left to right, the magnitude of the pres sure force (−|∇p|), of the Lorentz force and of the total force. Values\narenormalisedto themaximumforceofthe initialstate. It c anbe observedthat thepressureandLorentzforcesbalancee achother\nout,creatingaclearlynon-force-freeequilibrium.\nFig.4.Contourplot (left) andcorrespondingsurface(right)ofth e magnitudeofthe electric currentdensityin the final equil ibrium\nstate, forthesame experimentasinFig.1,in acrosssection inthex=0 plane.\nInitially, the spine line lies along the z-axis, and the\nfan plane is not perpendicular to the spine (unlike in\nFuentes-Fern´ andez& Parnell (2012)). Instead, it is tilte d about\nthex-axis (the direction of the current density), and lies in the\nplane\nz=jf\n3y, (8)\n(Parnellet al. 1996). We ran various experiments with di fferent\ninitialplasmapressuresandcurrentdensities.Figure1sh owsthe\nmagneticconfigurationoftheinitialstate,for jf=1andp0=1.\nInitially,forsuch a non-potentialnull,the Lorentzforce ,j×\nB, has theyandzcomponents that are non-zero. The forces in\ntheinitialstateactareshowninFig.1b,pushingthespinet othe\nright above the fan and to the left below it, and pushing the fa n\nup for positive values of yand down for negative values of y.\nTherefore,the spine and fan will collapse into each other in the\ndirectionoftheoriginaltilt ofthefan.3.2. Finalequilibriumstate\nWeinitiallyconcentrateonthecaseshowninFig.1,with jf=1\nandp0=1. The numerical simulations finish at about 300 fast\nmagnetosonictraveltimes(definedasthetimeforafastmagn e-\ntosonic wave to travel from the null to one of the boundaries) .\nThe magnetic field configurationin the final state shown in Fig .\n2. In comparison to the initial state (Fig. 1), the relaxatio n col-\nlapses the spine towards the fan in the region near the null, b ut\nthe fan plane is only slightly perturbed. The total energy of the\nevolution is checked to be conserved throughout the relaxat ion\ntowithinanerrorof0.02%.Thisconservationofenergydemo n-\nstrates that numerical di ffusion does not play a significant role\nin the relaxation. Less than 2% of the initial magnetic energ y\nis converted to internal energy via viscous damping during t he\nrelaxation.\nThefinalstateisanon-force-freeequilibriumwherethenon -\nzero Lorentz forces are globally balanced by non-zero plasm a\npressure forces. In Fig. 3, we show the di fferent forces in the\nverticalx=0plane.Inthisexperiment,thesystemhasreacheda\n4Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\nFig.5.Plots of the magnitude of the current density for three\ndifferentcuts, alongthespineline (black),alongthe tilt-axi s(x-\naxis) of the fan (dashed blue) and along the y-axis on the fan\nsurface(dashedorange).\nFig.6.Contour plot of the equilibrium plasma pressure for the\nsame experiment as in Fig. 1, for a vertical cross section in t he\nx=0plane.\nnon-force-freeequilibriumwiththestrongestpressurefo rceand\nLorentz force concentrated around the spine line and about t he\nfan plane, and the field is close to force-free everywhere els e.\nWeak residual total forces remain near the fan away from the\nnull. These forces are the consequence of the line tiding of t he\nfield (of the fan plane in particular) at the boundaries, and t hey\nwould take much longer to completely disappear. Nonetheles s,\ntheydonotinfluencetheresultsofthepresentstudy.\nThe final distribution of the magnitude of the current den-\nsity in the x=0 plane is shown in Fig. 4. Not surprisingly,\nthe current accumulations appear at the same locations as th e\nstrongest pressure forces and Lorentz forces shown in Fig. 3 ,\nnamely, on the fan plane and along the spine. The current den-\nsity is approximately zero away from the fan and the spine. It\nshows a weak accumulation along the spine line, but is about\none order of magnitude stronger on the fan plane. At the loca-\ntion of the null we find a pronounced peak of the current den-\nsity. The whole picture is reminiscent of the 2D X-point col-\nlapse (Fuentes-Fern´ andezet al. 2011), in which the curren t ac-cumulatesalongthefourseparatricesandthesystemevolve sto-\nwards an infinite time singularity at the location of the null (but\ncontrasts with Fuentes-Fern´ andez&Parnell (2012) where t he\ncurrent accumulates in the vicinity of the spines). Parnell et al.\n(1997) proved that the magnetic field locally about a non-\npotential3Dnullpoint,i.e. thelinearfieldabouta non-pot ential\nnull, produces a Lorentz force that cannot be balanced by a\nplasma pressure force, and so there are no static equilibriu m\nmodels of linear non-potential or force-free nulls. Theref ore,\neven if the system is in global equilibrium in the final state, the\ncollapse of the fan and spine towards each other does not al-\nlowapotentialconfigurationatthenull,andlocally,thepr essure\nforcescannot balance the Lorentzforces there, which gives rise\nto the formationofa singularity.Unfortunately,the same a naly-\nsis of forces about the null defined in Fuentes-Fern´ andezet al.\n(2011) to evaluate the formation of the singularity cannot b e\nmade here, due to the relatively low grid resolution that pla ces\nthe small local region in which these forces act into just a fe w\ngridpoints.Theformationofacontinuouslygrowingsingul arity\nat thelocationofthenullisevaluatedinSec. 3.3.\nThe final current layer around the location of the null can\nbe better appreciated in Fig. 5. Here, we show three di fferent\ncuts of the currentdensity,along the line of the spine, alon g the\ntilt axis of the fan surface ( x-axis), and along the y-axis of the\nfan surface. We see that the current density on the fan plane i s\nenhanced over ten times that of the initial background curre nt\ndensity, and it is higher in a thin layer along the x-axis, which\nis the initial tilt axis of the fan. The amplitude of the curre nt\nlayer is greatest at the location of the null, corresponding to the\npronouncedpeak observedin Fig. 4. The distribution of curr ent\ndensity observed in Fig. 5 agree qualitatively with the resu lts\nof Pontin&Craig (2005), who considerthe collapse of actual ly\nthe same initial field, using a di fferent numerical method. The\ncomparisonshows that the way the current density accumulat es\nabout the null is not an artefact of the numerical method, but a\nphysicalresult.\nIn Fig. 6, we show the final distribution of plasma pressure\nin thex=0 plane. The plasma pressure is enhanced inside the\nregionswhere the spine and fan have collapsed towardsone an -\nother,but decreased in the regionsoutside. The highest res idual\ngradientsin pressureoccurat the locationsof the fan andsp ine,\nand the gradient of plasma pressure from positive to negativ e y,\nfar from the null point (or far from the spine line in general) , is\nnotassharpasitisatthex =0plane(theoneinthefigure).Once\nagain,thisresultbringsoutthefactthatthecollapseofa3 Dnull\npointsresemblethe2D collapsein di fferentways.\n3.3. Singularcurrent\nFinally, for this setup, we want to evaluate the formation of the\nsingularityatthelocationofthenull,bylookingatthetim eevo-\nlution of the maximum current, at the location of the null, fo r\nexperimentswith di fferentinitial plasma pressuresand di fferent\ninitial currents (inclinations of the fan plane). In partic ular, we\nwant to see if there is a current singularity being formed, an d if\nthere is, whether its growth rate implies an infinite or finite rate\nofformation.\nKlapper (1997) rigorously demonstrated that, for 2D ideal\nincompressibleplasmas,a singularityofthecurrentdensi tywill\ntake an infinite amount of time to develop, unless driven by a\npressure singularity occurringoutside the neighbourhood of the\nnull point. Fuentes-Fern´ andezet al. (2011) showed numeri cally\nthatthisisalsothecaseforthe2Dcompressiblecollapseof mag-\n5Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\nFig.7.Time evolution of the peak current for five experiments\nwith different initial plasma pressures and an initial current of\njf=1. When a good asymptotic regime is achieved, a fit of\njmax=C+Dtisoverplotted(reddashedline).\nFig.8.ThesameasFig.7,butforfourexperimentswithdi fferent\ninitialcurrentdensitiesandaninitial plasmapressureof p0=1.\nnetic X-points, and here, we aim to extend these results for 3 D\nnullswith fan-alignedcurrent.\nThe ideal MHD relaxation of a 3D null point with an ini-\ntial homogeneouscurrent density parallel to the fan plane c aus-\ning (or caused by) a tilt of the fan, results in the collapse of the\nfan and the spine towards each other. The current density acc u-\nmulates weakly along the spine and on the fan plane forming\na ridge of current along the tilt axis of the fan (axis of the in i-\ntial current)which spikes at the null itself. The continuou scon-\ncentration of current at the location of the null is a natural con-\nsequence from the results of (Parnellet al. 1997), accordin g to\nwhichtheLorentzforcescannotbebalancedbyplasmapressu re\nforces about the 3D null. These forces feed current at the loc a-\ntions of the null point slowly, forming an infinite time singu lar-\nity.Figures7and 8show thetime evolutionofthispeakcurre nt\nforvaryinginitialplasmapressuresandcurrentdensities ,respec-\ntively. In Fig. 7, the current is set to jf=1 and the pressure\nvaries from p0=0.5 top0=1.5. After the rapid viscous relax-\nation,thesystementersaregimeinwhichthemaximumcurren t\nfollows a linear function of the form jmax=C+Dt, where the\ngrowthrate of the singularity, D, decreasesas the initial plasma\npressure, p0,increases,aswasfoundinthecompressible2Dnull\nFig.9.3D magnetic configurationof the initial non-equilibrium\nstate with genericcurrentdensity,forthecase with jsp=jf=1\nandp0=1, showingthesamefeaturesasinFig.1.\ncollapseexperimentsofFuentes-Fern´ andezet al.(2011). Forthe\nexperimentswiththelowestinitialplasmapressures,afito fthis\nformcannotbemade,asthenumericaldi ffusionlimitisreached\nbeforeawell-definedlinearregimehasbeenachieved.\nSimilar linear behaviour is found in Fig. 8, where the pres-\nsure is set to the constant value p0=1, and the current density\nis varied from jf=0.25 tojf=1. When the initial current of\nthesystemisdecreasedthennaturallythecurrentinthesin gular-\nityalsodecreasesasdoestherateatwhichthesingularityf orms\n(Fig. 8). A higher initial current density produces a bigger col-\nlapse of the fan andspine towardseach other,hencean increa se\ninthemagnitudeandgrowthrateofthesingularity.Indeed, when\nthe initial current is particularly weak then the collapse i s prac-\nticallynegligible,soit istherate ofgrowthofthe singula rity.\n4. Genericcurrentdensity\nIn the previous section, we considered the MHD relaxation of\nan initial radial null with a component current in the fan pla ne,\nperpendiculartothespine.Insuchaninitialfieldthefanpl aneis\ntiltedwiththecurrentalignedwiththeaxisoftilt.Inthis section,\nwe considerthe numericalequilibriumresulting fromthe MH D\nrelaxationofaninitialradialnullthathasagenericcurre nt,with\nnon-zero components along both the spine and the fan. To our\nknowledgethecollapseofthesetypesofnullshavenotbeenc on-\nsideredbefore.Here,themagneticfieldoftheinitialnullc onsists\nof a tilted fan,as in the previouscase, but thistime with twi sted\nfieldlinesaroundthespine.Fuentes-Fern´ andez& Parnell( 2012)\nconsidered the dynamical relaxation of a null with only a com -\nponentofcurrentparalleltothespine,whichgivesrisetot wisted\nfieldaboutthespine,butmaintainsthefanandspineatright an-\ngles.\n4.1. Initialstate\nA 3D null configuration of a radial null with a generic current\ndensityoftheform j=1\nµ(jf,0,jsp) isconsidered.FromEq.(2),\nthemagneticfield oftheinitialstate isgivenby\n(Bx,By,Bz)=(x−jsp\n2y,jsp\n2x+y,jfy−2z). (9)\n6Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\nFig.10.3Dmagneticconfigurationforthefinalequilibriumwith\ngeneric current density, for the same experiment as in Fig. 9 ,\nshowingthesame featuresasinFig. 1.\nFig.11.Contour plot of the magnitude of the electric current\ndensityinthefinalequilibriumstate,forthesameexperime ntas\ninFig. 9, ina crosssectioninthe x=0plane.\nHere again we have assumed b=1 for simplicity. For the case\nstudiedherewe havechosenthe backgroundplasmapressuret o\nbep0=1 and,forthecurrentdensity, jsp=jf=1.Thus,in the\ninitialstate,thecurrentdensityvectorliesinthe xz-plane,at45◦\ntoboththe xandthezaxis,anditsmagnitudeis |j|=√\n2every-\nwhere.Initially,thespinelineliesalongthe z-axis,thefanplane\ntilts about the x-axis following the function z=jf\n3y, and the\nmagnetic field lines show a homogeneoustwist about the spine\n(seeFig.9).TheLorentzforcesthatactinthesysteminitia llyare\na combination of two contributions. The first is the one shown\nin Fig. 1b, which will drive the collapse of the fan and spine\ntowards one another. The second is a magnetic tension force,\nasdescribedin Fuentes-Fern´ andez&Parnell(2012),which will\nremovethetwist fromthefansurface.\n4.2. Finalequilibriumstate\nThe resultsof the ideal relaxationfromthis initial configu ration\nshowninFig. 9 combinethefeaturesfromtherelaxationofsp i-\nFig.12.Time evolution of the peak current for five experiments\nwith different initial current densities involving jfandjsp, and\nan initial plasma pressure of p0=1. When a good asymptotic\nregime is achieved, a fit of jmax=C+Dtis overplotted (red\ndashed).\nralnulls(Fuentes-Fern´ andez&Parnell2012) andthose oft ilted\nnulls, from Sec 3. The magnetic field lines in the final equilib -\nrium state are shown in Fig. 10. The relaxation drives the col -\nlapse of the fan and the spine towards each other at the same\ntimeasitconcentratestheinitialtwistofthefieldlinesab outthe\nspine line, above and below the fan surface. This can be more\nclearly seen in Fig. 11, where we plot the magnitudeof the cur -\nrent density for the final equilibriumstate, in the x=0 plane. It\nconcentratesinthreedi fferentregions:(i)onthefansurface,due\ntothespine-fancollapse;(ii)aroundthespineabovethefa n;and\n(iii) aroundthe spine below the fan. Clearly, accumulation (i) is\nthesameaswhatisseeninFig.4andisthereforeassociatedw ith\nthecomponentofcurrentperpendiculartothespine,wherea sthe\nlast two accumulationsresult from a concentrationof the in itial\ntwistofthefieldlinesaboutthespineandresemblethehourg lass\nconfiguration obtained in Fuentes-Fern´ andez&Parnell (20 12).\nThey are thusassociated with the componentof currentparal lel\ntothe spine.\nWe now evaluate the consequencesof the two deformations\n(twist and tilt) on the formationand growthrate of the singu lar-\nityatthenull.InFig.12weshowthetimeevolutionofthepea k\ncurrent density (the current density at the null) for five di fferent\nexperiments with a fixed background plasma pressure ( p0=1)\nand varying both jfandjspindependently. When jspis varied\nandjfis fixed, the growth rate of the singularity does not vary\nsignificantly,asthetorsionaldisturbanceleadstoacurre ntaccu-\nmulation about the spines, not at the null. However, a decrea se\ninjffor fixed jspslows down the formation of the singularity,\nasdiscussedbelow.\nWhenthe initial currentdensityhasthe form j=(jf,0,jsp),\nwherejsp=jf=1, both contributions of current density ini-\ntially parallel and perpendicular to the spine are the same a s in\neach of these independentcases studied before: the case jf=1\nandjsp=0 studiedpreviouslyin thispaper,andthecase jf=0\nandjsp=1studiedinFuentes-Fern´ andez&Parnell(2012).The\nmagnitude and growth rate of the singularity for jf=1 and\njsp=1 (D=0.09)is to be comparedwith the resultsfrom Sec.\n3.3 for the case jf=1 andjsp=0 (D=0.1).The additionof a\nspine-aligned component of the current density slows down t he\nformationofthesingularity.\n7Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\nFig.13.Plots of the yz-plane at x=0 showing the spine and\nthecutalongthefanplane,for(a)theexperimentswithfanp ar-\nallel current density only and (b) the experiments with gene ric\ncurrent density. In both plots, the non-zero components of t he\ninitialcurrentdensityarevariedsystematically.\nThe decrease in the growth rate of the singularity may be\nexplained as follows. The fan-aligned component of the cur-\nrent density, jf, curves the spine line making it longer within\nthe boundaries of the domain. This way, the relaxation of the\ntwistcomponentofthemagneticfield(identifiedwiththespi ne-\nalignedcomponentofthecurrentdensity)requiresmorecur rent\nto be allocated along and around the spine than in the absence\nofjf(when the spine line is straight), thereby decreasing the\namount of current that is available for the formation of the s in-\ngularity.\nThe curvature of the spine line may be appreciated in Fig.\n13. Here we show plots of the yz-plane at x=0, indicating the\nspinesandthecutsalongthefanplanefordi fferentexperiments.\nFigure 13a corresponds to the experiments of Sec. 3 with fan-\naligned current density. Clearly, as jfincreases, the spines be-\ncomemorecurvedandgetlonger.Figure13bcorrespondstoth e\nexperimentsdiscussedinthepresentsectionwithgenericc urrent\ndensity.We notethat changesin jsp,forfixed jf, donot involve\na change in the shape of the spine line or in the fan plane, i.e.\nthey all lie under the solid lines of the jsp=jf=1 case. On\nthe contrary,increasingvaluesof jf, for fixed jsp, lead to larger\ndeformationsofthe spineandthefan,henceofthecurrentla yer\naboutthenull.Thisisduetothegreaterinitialinclinatio nofthe\nfan plane that, upon the ideal MHD relaxation of the system, i s\ntransferred into a greater folding of the spine towards the f an\nplane.\nOver all, when a current perpendicular and parallel to the\nspine are combined together, the resulting non-force-free equi-\nlibrium is such that it combines the results from a strictly t ilted\nnull with that from a strictly spiral null. Te spiraling of th e field\nlines along the spine does not concentrate any current at the lo-\ncation of the null (Fuentes-Fern´ andez&Parnell 2012), so i t is\nnot a direct contribution to the formation of the singularit y. In\ncontrast,the additionof a currentdensity parallelto the s pine is\nfoundto slowdownthe formationofthesingularityitself.\n5. Summaryand conclusions\nThe 3D relaxation of tilted magnetic null points with an init ial\nnon-zero component of the current density perpendicularto the\nspinehasbeeninvestigatedundernon-resistivecondition s,inall\ncasesresultinginanon-force-freeequilibrium,whereall theve-locities have been damped out by viscous forces. In the final\nequilibriumstates, theforcesduetotheplasmapressurean dthe\nmagneticfield,whichareofaboutthesamestrengthastheini tial\nnon-equilibrium Lorentz forces, balance each other out, re sult-\ning in a genuine non-force-free equilibrium, save at the loc a-\ntionofthenullwhereaninfinitetime singularityoccurs,si milar\nto the 2D X-point collapse studied by Fuentes-Fern´ andezet al.\n(2011).\nDuring the relaxation, for the case with an initial radial nu ll\nand constant current density purely parallel to the fan (i.e ., a\ntilted fan on whichfield lines expandradially),the field evo lves\nby warping the fan and the spine so that, locally about the nul l,\ntheycollapse towardseach other,concentratingthe curren tden-\nsity on the fan surface, but leaving it weak along the spine. T he\namplitude of the resulting current density on the fan is larg est\nalongthe x-axis,theinitialtiltaxisofthefanalongwhichtheini-\ntial and final currentdensities are directed, and it shows a s harp\npeakat thelocationofthenull.\nAfter the viscousrelaxationhas finished, the system hasen-\ntered an asymptotic regime in which the current at the locati on\nofthenullincreaseslinearlywithtime.Thisbehaviourisi denti-\nfiedwiththeslowformationofaninfinitetimesingularitywh ose\ngrowth rate depends directly on the values of the initial pla sma\npressure and current density (tilt of the fan). A decrease in the\nplasma pressure at a fixed current density or an increase in th e\nelectriccurrentdensityatafixedplasmapressureproducea nin-\ncrease in the growth rate of the singularity being formed at t he\nlocationofthenull.\nOverall,thestructureoftheequilibriumisverysimilarto the\n2D X-pointcollapsestudied in Fuentes-Fern´ andezet al. (2 011);\nin particular,(i) the currentdensity accumulationsalong the fan\nand spine are equivalent to the accumulations found along th e\nseparatrices in the 2D X-point case, (ii) the collapse and de for-\nmation of these features towards each other locally about th e\nnull is observed in both cases, as is (iii) the formation of an in-\nfinite time singularity at the null. The results also agree wi th\npast studies of compressible 3D nulls with current purely pe r-\npendiculartothespinethatcollapsetoformcurrentsingul arities\n(Pontin& Craig2005).However,wehavebeenabletostudythe\nexact distribution of the current in the final equilibrium st ate in\nmoredetail.Furthermore,ournumericalcodeallowsustode ter-\nmine the full dynamical evolution of the system, and therefo re\nwe are able to get a numerical growth rate for the singularity ,\nwhichdirectlydependsontheinitialplasmapressure.\nIn Fuentes-Fern´ andez&Parnell (2012) where the initial\nmagnetic field was twisted about the spine owing to current\nbeing parallel to the spine (along the z-axis), (i) the spine and\nfanremainedperpendicularanddidnotcollapse,(ii) thecu rrent\n(which remained parallel to the spine) accumulated in a pair of\nconeshapefeaturesaboutthespine,and(iii)nocurrentsin gular-\nity wasfoundatthe nullin theradialnullcase. Thusthecurr ent\naccumulationsare totallydi fferentinthese twoscenarios,asex-\npected.\nTheeffectsofamoregenericcurrentwithcomponentsalong\nboth the fan and spine have not been considered before the\npresent paper. In the cases where the initial current densit y has\nbothcomponentsparallelandperpendiculartothespine,th eini-\ntialfieldshowsatiltedfanonwhichfieldlinesexpandinspir als\naround the null. The resulting non-force-free equilibrium com-\nbines the two contributions, from the tilted fan and the twis ted\nfield lines, creating a current density that concentrates on the\nfan plane, building a singularity at the location of the null , and\nalong the spine line, concentrating the twist of the field lin es\nthere. In principle, the results are a simple combination of the\n8Jorge Fuentes-Fern´ andez and ClareE.Parnell:MHD dynamic al relaxation ofcoronal magnetic fields\ntwisted and tilted cases, and so, the current density should be\nevenlyditributedforbothcontributions,eventhoughthem agni-\ntudeand growthrate of thesingularityhasbeendecreased.T his\nis because, since the spine is longer now than in the jf=0 and\njsp=1case(becauseitisbended),therelaxationofthetwistre-\nquiresabitmorecurrent,therebyslightlydecreasingthea mount\nof current available for the singularity at the null. The add ition\nofaspine-alignedcomponentofthecurrentdensityslowsdo wn\ntheformationofthesingularity.\nIn every case considered here, we set b=1 in the mag-\nnetic field. This means that the two eigenvalues that are asso ci-\nated with the fan plane have the same magnitude, and thus, the\nfan plane has no major and minor axes, so the field lines have\nno preferreddirection in the fan plane (Parnellet al. 1996) . The\npresenceofmajororminoraxesinafanplanemaywellhavean\neffectontheequilibriaobtainedfollowingthecollapseofsuc ha\nnullwitha non-zerocomponentofcurrent.\nMagneticreconnectionisanimportantprocessofenergyre-\nlease in scenarios like the solar corona and the Earth’s mag-\nnetotail. In the second case, it provides a mechanism for par -\nticle acceleration and for the global aurora. In the first cas e,\nit is the mechanism for particle acceleration, solar flares, and\ncoronal mass ejections, and is highly likely to provide an im -\nportant source of energy for the high temperatures observed in\nthe corona. In the present paper, we have described a valid in i-\ntial equilibrium field for the study of spontaneous magnetic re-\nconnection,initiatedbymicroinstabilitiesata3Dmagnet icnull.\nSuchastudyisanaturalcontinuationoftheworkcarriedout by\nFuentes-Fern´ andezet al. 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We prove that if the dampi ng coefficient function is strictly\npositive near the origin then this equation possesses a glob al attractor.\n1.INTRODUCTION\nIn this paper, we consider the following Cauchy problem:\nutt+σ(u)ut−uxx+λu+f(u) =g(x),(t,x)∈(0,∞)×R, (1.1)\nu(0,x) =u0(x), u t(0,x) =u1(x), x ∈R, (1.2)\nwhereλis a positive constant, g∈L1(R)+L2(R) and nonlinear functions f(·) andσ(·) satisfy the\nfollowing conditions:\nf∈C1(R),f(u)u≥0,/notturnstileu∈R, (1.3)\nσ∈C(R),σ(0)>0,σ(u)≥0,/notturnstileu∈R. (1.4)\nApplying standard Galerkin’s method and using techniques of [6, Pro position 2.2], it is easy to\nprove the following existence and uniqueness theorem:\nTheorem 1. Assume that the conditions (1.3)-(1.4) hold. Then for any T >0and(u0,u1)∈\nH:=H1(R)×L2(R)the problem (1.1)-(1.2) has a unique weak solution u∈C([0,T];H1(R))∩\nC1([0,T];L2(R))∩C2([0,T];H−1(R))on[0,T]×Rsuch that\n/bardbl(u(t),ut(t))/bardblH≤c(/bardbl(u0,u1)/bardblH),/notturnstilet≥0,\nwherec:R+→R+is a nondecreasing function. Moreover if v∈C([0,T];H1(R))∩C1([0,T];L2(R))∩\nC2([0,T];H−1(R))is also weak solution to (1.1)-(1.2) with initial data (v0,v1)∈ H, then\n/bardblu(t)−v(t)/bardblL2(R)+/bardblut(t)−vt(t)/bardblH−1(R)≤\n≤/tildewidec(T,/tildewideR)/parenleftBig\n/bardblu0−v0/bardblL2(R)+/bardblu1−v1/bardblH−1(R)/parenrightBig\n,/notturnstilet∈[0,T],\nwhere/tildewidec:R+×R+→R+is a nondecreasing function with respect to each variable an d/tildewideR=\nmax{/bardbl(u0,u1)/bardblH,/bardbl(v0,v1)/bardblH}.\nThus, by the formula ( u(t),ut(t)) =S(t)(u0,u1), the problem (1.1)-(1.2) generates a weak con-\ntinuous semigroup {S(t)}t≥0inH, whereu(t,x) is a weak solution of (1.1)-(1.2), determined by\nTheorem 1.1, with initial data ( u0,u1).\nThe attractors for equation (1.1) in the finite interval were studie d in [2], assuming positivity of\nσ(·). For two dimensional case, the attractors for the wave equatio n with displacement dependent\ndamping were investigated in [7] under conditions\nσ∈C1(R), 0< σ0≤σ(u)≤c(1+|u|q),/notturnstileu∈R, 0≤q <∞,\nand\n|σ′(u)| ≤c[σ(u)]1−ε,/notturnstileu∈R, 0< ε <1, (1.5)\non the damping coefficient. Recently, in [3], condition (1.5) has been im proved as\n|σ′(u)| ≤cσ(u),/notturnstileu∈R.\n2000Mathematics Subject Classification. 35B41, 35L05.\nKey words and phrases. Attractors, wave equations.\n12 A.KH.KHANMAMEDOV\nFor the three dimensional bounded domain case, the existence of a global attractor for the wave\nequation with displacement dependent damping was proved in [6] when σ(·) is a strictly positive\nand globally bounded function. In this case, when σ(·) is not globally bounded, but is equal to a\npositive constant in a large enough interval, the existence of a globa l attractor has been established\nin [4].\nIn the articles mentioned above, the existence of global attracto rs was proved under positivity or\nstrict positivity condition on the damping coefficient function σ(·). In this paper, we study a global\nattractor for (1.1)-(1.2) under weaker conditions on σ(·) and prove the following theorem:\nTheorem 2. Under conditions (1.3)-(1.4) a semigroup {S(t)}t≥0generated by (1.1)-(1.2) possesses\na global attractor in H.\n2.Proof of Theorem 1.2\nTo prove this theorem we need the following lemma:\nLemma 1. Let conditions (1.3)-(1.4) hold and let Bbe a bounded subset of H. Then for any ε >0\nthere exist T0=T0(ε,B)>0andr0=r0(ε,B)>0such that\n/bardblS(t)ϕ/bardblH1(R\\(−r0,r0))×L2(R\\(−r0,r0))< ε,/notturnstilet≥T0,/notturnstileϕ∈B. (2.1)\nProof.Let (u0,u1)∈BandS(t)(u0,u1) = (u(t),ut(t)). Multiplying (1.1) by utand integrating\nover (0,t)×Rwe obtain\n/bardblut(t)/bardbl2\nL2(R)+/bardblu(t)/bardbl2\nH1(R)+t/integraldisplay\n0/integraldisplay\nRσ(u(τ,x))u2\nt(τ,x)dxdτ≤c1,/notturnstilet≥0.(2.2)\nLetη∈C1(R), 0≤η(x)≤1,η(x) =/braceleftbigg0,|x| ≤1\n1,|x| ≥2,ηr(x) =η(x\nr) and Σ( u) =u/integraltext\n0σ(s)ds.\nMultiplying (1.1) by η2\nrΣ(u), integrating over (0 ,t)×Rand taking into account (2.2) we have\nt/integraldisplay\n0/integraldisplay\nRη2\nr(x)σ(u(τ,x))u2\nx(τ,x)dxdτ+λt/integraldisplay\n0/integraldisplay\nRη2\nr(x)Σ(u(τ,x))u(τ,x)dxdτ≤\n≤c2(1+√\nt+t\nr+t/bardblg/bardblL1(R\\(−r,r))+L2(R\\(−r,r))),/notturnstilet≥0,/notturnstiler >0. (2.3)\nBy (1.4), there exists l >0, such that\nσ(0)\n2≤σ(s)≤2σ(0),/notturnstiles∈[−l,l]. (2.4)\nUsing embedding H1\n2+ε(R)⊂L∞(R) and taking into account (2.2) and (2.4) we find\nt/integraldisplay\n0/integraldisplay\nRη2\nr(x)u2(τ,x)dxdτ≤2\nσ(0)t/integraldisplay\n0/integraldisplay\n{x:|u(τ,x)|≤l}η2\nr(x)Σ(u(τ,x))u(τ,x)dxdτ+\n+c3t/integraldisplay\n0/integraldisplay\n{x:|u(τ,x)|>l}η2\nr(x)|u(τ,x)|dxdτ≤2\nσ(0)t/integraldisplay\n0/integraldisplay\n{x:|u(τ,x)|≤l}η2\nr(x)Σ(u(τ,x))u(τ,x)dxdτ+\n+2c3\nσ(0)lt/integraldisplay\n0/integraldisplay\n{x:|u(τ,x)|>l}η2\nr(x)Σ(u(τ,x))u(τ,x)dxdτ\nand consequently\nt/integraldisplay\n0/bardblηru(τ)/bardbl5\nL∞(R)dτ≤c4t/integraldisplay\n0/bardblηru(τ)/bardbl2\nL2(R)dτ≤c5t/integraldisplay\n0/integraldisplay\nRη2\nr(x)Σ(u(τ,x))u(τ,x)dxdτ, (2.5)GLOBAL ATTRACTORS 3\nforr≥1. So by (2.2), (2.3) and (2.5), we get\nt/integraldisplay\n0/bracketleftbigg/vextenddouble/vextenddouble/vextenddoubleη2rσ1\n2(u(τ))ut(τ)/vextenddouble/vextenddouble/vextenddouble2\nL2(R)+/vextenddouble/vextenddouble/vextenddoubleη2rσ1\n2(u(τ))ux(τ)/vextenddouble/vextenddouble/vextenddouble2\nL2(R)+λ/vextenddouble/vextenddouble/vextenddoubleη2rσ1\n2(u(τ))u(τ)/vextenddouble/vextenddouble/vextenddouble2\nL2(R)+\n+/bardblηru(τ)/bardbl5\nL∞(R)/bracketrightBig\ndτ≤c6(1+√\nt+t\nr+t/bardblg/bardblL1(R\\(−r,r))+L2(R\\(−r,r))),/notturnstilet≥0,/notturnstiler≥1.(2.6)\nNow denote Φ r(u(t)) :=1\n2/bardblηrut(t)/bardbl2\nL2(R)+1\n2/bardblηrux(t)/bardbl2\nL2(R)+µ/an}bracketle{tηrut(t), ηru(t)/an}bracketri}ht+λ\n2/bardblηru(t)/bardbl2\nL2(R)+\n/an}bracketle{tηrF(u(t)), ηr/an}bracketri}ht+/an}bracketle{tηrg, ηru(t)/an}bracketri}ht, where µ= min/braceleftbigg/radicalBig\nλ\n2,σ(0)\n5,λ\n2σ(0)/bracerightbigg\n,/an}bracketle{tu, v/an}bracketri}ht=/integraltext\nRu(x)v(x)dxand\nF(u) =u/integraltext\n0f(s)ds.By (2.4) and (2.6), it follows that for any δ >0 there exist Tδ=Tδ(B)>0,\nr1,δ=r1,δ(B)>1 and for any r≥r1,δthere exists t∗\nδ,r∈[0,Tδ] such that\nΦr(u(t∗\nδ,r))< δ,/notturnstiler≥r1,δ. (2.7)\nAgain by (2.2), we have\n/bardblηru(t)/bardblL2(R)≤/vextenddouble/vextenddoubleηru(t∗\nδ,r)/vextenddouble/vextenddouble\nL2(R)+t/integraldisplay\nt∗\nδ,r/bardblηrut(s)/bardblL2(R)ds≤/vextenddouble/vextenddoubleηru(t∗\nδ,r)/vextenddouble/vextenddouble\nL2(R)+c7(t−t∗\nδ,r)\nand consequently\n/bardblηru(t)/bardbl3\nL∞(R)≤c8/bardblηru(t)/bardblL2(R)≤c9(Φ1\n2r(u(t∗\nδ,r))+/bardblg/bardbl1\n2\nL1(R\\(−r,r))+L2(R\\(−r,r))+t−t∗\nδ,r)<\n< c9(δ1\n2+/bardblg/bardbl1\n2\nL1(R\\(−r,r))+L2(R\\(−r,r))+t−t∗\nδ,r),/notturnstilet≥t∗\nδ,r,/notturnstiler≥r1,δ.\nDenoting T∗\nδ,r=t∗\nδ,r+l3\n3c9and choosing δ∈(0,l6\n9c2\n9), by the last inequality, we can say that there\nexistsr2,δ≥2r1,δsuch that\n/bardblu(t)/bardblL∞(R\\(−r2,δ,r2,δ))< l,/notturnstilet∈[t∗\nδ,r,T∗\nδ,r]. (2.8)\nNow multiplying (1.1) by η2\nr(ut+µu), integrating over Rand taking into account (2.4) and (2.8) we\nobtain\nd\ndtΦr(u(t))+c10Φr(u(t))≤c11(1\nr+/bardblg/bardblL1(R\\(−r,r))+L2(R\\(−r,r))),/notturnstilet∈[t∗\nδ,r,T∗\nδ,r],\nand consequently\nΦr(u(t))≤Φr(u(t∗\nδ,r))e−c10(t−t∗\nδ,r)+c11(1\nr+/bardblg/bardblL1(R\\(−r,r))+L2(R\\(−r,r)))1−e−c10(t−t∗\nδ,r)\nc10,(2.9)\nforr≥r2,δ. By (2.7) and (2.9), there exists r3,δ≥r2,δsuch that\nΦr(u(t))< δ,/notturnstiler≥r3,δ,/notturnstilet∈[t∗\nδ,r,T∗\nδ,r].\nHence denoting by nδthe smallest integer number which is not less than3c9Tδ\nl3and applying above\nprocedure at most nδtime, we find\nΦr(u(Tδ))< δ,/notturnstiler≥r4,δ,\nfor some r4,δ≥2nδr1,δ. From the last inequality it follows that for any ε >0 there exist /hatwideTε=\n/hatwideTε(B)>0 and/hatwiderε=/hatwiderε(B)>0 such that/vextenddouble/vextenddouble/vextenddoubleS(/hatwideTε)ϕ/vextenddouble/vextenddouble/vextenddouble\nH1(R\\(−/hatwiderε,/hatwiderε))×L2(R\\(−/hatwiderε,/hatwiderε))< ε,/notturnstileϕ∈B.\nSince, by (2.2), B0=∪\nt≥0S(t)Bis a bounded subset of H, for any ε >0 there exist T0=T0(ε,B)>0\nandr0=r0(ε,B)>0 such that\n/bardblS(T0)ϕ/bardblH1(R\\(−r0,r0))×L2(R\\(−r0,r0))< ε,/notturnstileϕ∈B0.\nTaking into account positively invariance of B0, from the last inequality we obtain (2.1). /square4 A.KH.KHANMAMEDOV\nBy (2.1) and (2.4), for any bounded subset BofHthere exist /hatwideT0=/hatwideT0(B)>0 and/hatwider0=/hatwider0(B)>0\nsuch that\nσ(u(t,x))≥σ(0)\n2,/notturnstilet≥/hatwideT0,/notturnstile|x| ≥/hatwider0. (2.10)\nHence using techniques of [5] one can prove the asymptotic compac tness of the semigroup {S(t)}t≥0,\nwhich is included in the following lemma:\nLemma 2. Assume that conditions (1.3)-(1.4) hold and Bis bounded subset of H. Then every\nsequence of the form {S(tn)ϕn}∞\nn=1,{ϕn}∞\nn=1⊂B,tn→ ∞, has a convergent subsequence in H.\nBy (2.10) and the unique continuation result of [8], it is easy to see tha t problem (1.1)-(1.2) has a\nstrict Lyapunov function (see [1] for definition). Thus according t o [1, Corollary 2.29] the semigroup\n{S(t)}t≥0possesses a global attractor.\nRemark 1. We note that, for the problem considered in [2], from compact embedding H1\n0(0,π)⊂\nC[0,π], it immediately follows that σ(u(t,x))≥σ(0)\n2,/notturnstilet≥0,/notturnstilex∈[0,ε]∪[π−ε,π]for some\nε∈(0,π). So a global attractor still exists if one replaces the posit ivity condition on σ(·)by the\nσ(0)>0.\nReferences\n[1] I. Chueshov, I. Lasiecka, Long-time behavior of second o rder evolution equations with nonlinear damping, Memoirs\nof AMS, 195 (2008).\n[2] S. Gatti, V. Pata, A one-dimensional wave equation with n onlinear damping, Glasg. Math. J. , 48 (2006), 419–430.\n[3] A. Kh. Khanmamedov, Global attractors for 2-D wave equat ions with displacement dependent damping, Math.\nMethods Appl. Sci. , 33 (2010) 177-187.\n[4] A. Kh. Khanmamedov, A strong global attractor for 3-D wav e equations with displacement dependent damping,\nAppl. Math. Letters , 23 (2010) 928-934.\n[5] A. Kh. Khanmamedov, Global attractors for the plate equa tion with a localized damping and a critical exponent\nin an unbounded domain, J. Diff. Eqs., 225 (2006) 528-548.\n[6] V. Pata, S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping,\nMath. Methods Appl. Sci. , 29 (2006), 1291–1306.\n[7] V. Pata, S. Zelik, Attractors and their regularity for 2- D wave equations with nonlinear damping, Adv. Math. Sci.\nAppl., 17 (2007), 225–237.\n[8] A. Ruiz, Unique continuation for weak solutions of the wa ve equation plus a potential, J. Math. Pures Appl. , 71\n(5) (1992) 455–467.\nDepartment of Mathematics, Faculty of Science, Hacettepe U niversity, Beytepe 06800 ,\nAnkara, Turkey\nE-mail address :azer@hacettepe.edu.tr"    },    {        "title": "2101.02794v2.Mechanisms_behind_large_Gilbert_damping_anisotropies.pdf",        "content": "Mechanisms behind large Gilbert damping anisotropies\nI. P. Miranda1, A. B. Klautau2,∗A. Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nA method with which to calculate the Gilbert damping parameter from a real-space electronic\nstructure method is reported here. The anisotropy of the Gilbert damping with respect to the\nmagnetic moment direction and local chemical environment is calculated for bulk and surfaces\nof Fe 50Co50alloys from first principles electronic structure in a real space formulation. The size\nof the damping anisotropy for Fe 50Co50alloys is demonstrated to be significant. Depending on\ndetails of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several\nmicroscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where\nin particular interface/surface effects stand out, as do local distortions of the crystal structure.\nAlthough theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev.\nLett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge\ndamping anisotropies.\nIntroduction: Magnetic damping has a critical impor-\ntanceindeterminingthelifetime,diffusion,transportand\nstability of domain walls, magnetic vortices, skyrmions,\nand any nano-scale complex magnetic configurations [1].\nGiven its high scientific interest, a possibility to obtain\nthis quantity by means of first-principles theory [2] opens\nnew perspectives of finding and optimizing materials for\nspintronic and magnonic devices [3–8]. Among the more\npromising ferromagnets to be used in spintronics devices,\ncobalt-iron alloys demonstrate high potentials due to the\ncombination of ultralow damping with metallic conduc-\ntivity [4, 9].\nRecently, Li et al.[10] reported an observed, gi-\nant anisotropy of the Gilbert damping ( α) in epitaxial\nFe50Co50thin films (with thickness 10 −20nm) reach-\ning maximum-minimum damping ratio values as high as\n400%. TheauthorsofRef. [10]claimedthattheobserved\neffect is likely due to changes in the spin-orbit coupling\n(SOC) influence for different crystalline directions caused\nby short-range orderings that lead to local structural dis-\ntortions. This behaviour differs distinctly from, for ex-\nample, pure bcc Fe [11]. In order to quantitatively pre-\ndict the Gilbert damping, Kambersky’s breathing Fermi\nsurface (BFS) [12] and torque-correlation (TC) [13] mod-\nels are frequently used. These methods have been ex-\nplored for elements and alloys, in bulk form or at sur-\nfaces, mostly via reciprocal-space ab-initio approaches,\nin a collinear or (more recently) in a noncollinear con-\nfiguration [14]. However, considering heterogeneous ma-\nterials, such as alloys with short-range order, and the\npossibility to investigate element specific, non-local con-\ntributions to the damping parameter, there are, to the\nbest of our knowledge, no reports in the literature that\nrely on a real space method.\nIn this Letter, we report on an implementation of ab\ninitiodamping calculations in a real-space linear muffin-tin orbital method, within the atomic sphere approxi-\nmation (RS-LMTO-ASA) [15, 16], with the local spin\ndensity approximation (LSDA) [17] for the exchange-\ncorrelation energy. The implementation is based on the\nBFSandTCmodels, andthemethod(SupplementalMa-\nterial - SM, for details) is applied to investigate the re-\nported, huge damping anisotropy of Fe50Co50(100)/MgO\nfilms [10]. A main result here is the identification of\na microscopic origin of the enhanced Gilbert damping\nanisotropy of Fe50Co50(100) films, and the intrinsic rela-\ntionships to the local geometry of the alloy. Most signifi-\ncantly, wedemonstratethatasurfaceproducesextremely\nlarge damping anisotropies that can be orders of magni-\ntude larger than that of the bulk. We call the attention\nto the fact that this is the first time, as far as we know,\nthat damping values are theoretically obtained in such a\nlocal way.\nResults: We calculated: i)ordered Fe50Co50in theB2\nstructure (hereafter refereed to as B2-FeCo) ii)random\nFe50Co50alloysinbccorbctstructures, wherethevirtual\ncrystal approximation (VCA) was applied; iii)Fe50Co50\nalloys simulated as embedded clusters in a VCA matrix\n(host). In all cases VCA was simulated with an elec-\ntronic concentration corresponding to Fe50Co50. The ii)\nandiii)alloys were considered as in bulk as well as in\nthe (001) surface, with bcc and bct structures (here-\nafter correspondingly refereed as VCA Fe50Co50bcc,\nVCA Fe50Co50bct, VCA Fe50Co50(001) bcc and VCA\nFe50Co50(001) bct). The effect of local tetragonal distor-\ntions was considered with a localc\na= 1.09ratio (SM for\ndetails). All data for cluster based results, were obtained\nfrom an average of several different configurations. The\ntotal damping for a given site iin real-space ( αt, Eqs. S6\nand S7 from SM) can be decomposed in non-local, αij\n(i/negationslash=j), and local (onsite), αonsite(orαii,i=j) contri-\nbutions, each of them described by the tensor elements2\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(1)\nwheremiis the total magnetic moment localized in the\nreference atomic site i,µ,ν={x,y,z},ˆTis the torque\noperator, and η(/epsilon1) =∂f(/epsilon1)\n∂/epsilon1the derivative of the Fermi\ndistribution. The scalar αijparameter is defined in the\ncollinear regime as αij=1\n2(αxx\nij+αyy\nij).\nTo validate our methodology, the here obtained total\ndamping for several systems (such as bcc Fe, fcc Ni, hcp\nand fcc Co and B2-FeCo) were compared with estab-\nlished values available in the literature (Table S1, SM),\nwhere an overall good agreement can be seen.\nFig. 1 shows the non-local contributions to the damp-\ning for bcc Fe and B2-FeCo. Although the onsite contri-\nbutions are around one order of magnitude larger than\nthe non-local, there are many αijto be added and total\nnet values can become comparable. Bcc Fe and B2-FeCo\nhave very different non-local damping contributions. El-\nement resolved αij, reveal that the summed Fe-Fe in-\nteractions dominate over Co-Co, for distances until 2a\ninB2-FeCo. We observe that αijis quite extended in\nspace for both bcc Fe and B2-FeCo. The different con-\ntributions to the non-local damping, from atoms at equal\ndistance arises from the reduced number of operations in\nthe crystal point group due to the inclusion of SOC in\ncombination with time-reversal symmetry breaking. The\nB2-FeCo arises from replacing every second Fe atom in\nthe bcc structure by a Co atom. It is interesting that this\nreplacement (i.e. the presence of Co in the environment)\nsignificantly changes the non-local contributions for Fe-\nFe pairs , what can more clearly be seen from the Insetin\nFig. 1, where the non-local damping summed over atoms\nat the same relative distance for Fe-Fe pairs in bcc Fe\nandB2-FeCo are shown; the non-local damping of Fe-Fe\npairs are distinctly different for short ranges, while long\nranged (further than ∼2.25 Å) contributions are smaller\nand more isotropic.\nThe damping anisotropy, i.e. the damping change,\nwhen the magnetization is changed from the easy axis\nto a new direction is1\n∆αt=/parenleftBigg\nα[110]\nt\nα[010]\nt−1/parenrightBigg\n×100%, (2)\nwhereα[110]\ntandα[010]\ntare the total damping obtained\nfor magnetization directions along [110]and [010], re-\nspectively. Analogousdefinitionalsoappliesfor ∆αonsite.\n1We note that this definition is different to the maximum-\nminimum damping ratio, defined asα[110]\nt\nα[010]\nt×100%, from Ref.\n[10].We investigated this anisotropy in surfaces and in bulk\nsystems with (and without) tetragonal structural distor-\ntions. Our calculations for VCA Fe50Co50bcc show a\ndamping increase of ∼13%, when changing the magne-\ntization direction from [010]to[110](Table S2 in the\nSM). The smallest damping is found for the easy magne-\ntization axis, [010], which holds the largest orbital mo-\nment (morb) [18]. For VCA Fe50Co50bcc we obtained\na small variation of ∼2%for the onsite contribution\n(α[010]\nonsite = 8.94×10−4andα[110]\nonsite = 8.76×10−4),\nwhat implies that the anisotropy comes mostly from\nthe non-local contributions, particularly from the next-\nnearest neighbours. For comparison, ∆αt∼3%(with\n∆αonsite∼0.4%) in the case of bcc Fe, what corrobo-\nrates the reported [11] small bcc Fe anisotropy at room\ntemperature, andwiththebulkdampinganisotropyrates\n[19].\nWe also inspected the chemical inhomogeneity influ-\nence on the anisotropy, considering the B2-FeCo alloy,\nwhere the weighted average damping (Eq. S7 of SM)\nwas used instead. The B2-FeCo bcc (∼7%) and VCA\nFe50Co50bcc (∼13%) anisotropies are of similar magni-\ntudes. Both B2structure and VCA calculations lead to\ndamping anisotropies which are significantly lower than\nwhatwasobservedintheexperiments, anditseemslikely\nthatthepresenceofdisorderincompositionand/orstruc-\ntural properties of the Fe/Co alloy would be important\nto produce large anisotropy effects on the damping.\n\nα\nij\n\t\n×\n\t\n10\n-4\n−3\n−2\n−1\n0\n1\n2\n3\n4\n\nNormalized\n\t\ndistance\n1.0\n1.5\n2.0\n2.5\n3.0\n\n\t\nB2\n\t\nFe-Co\n\t\nB2\n\t\nFe-Fe\n\t\nB2\n\t\nCo-Co\n\t\nbcc\n\t\nFe-Fe\n\t\n−10\n0\n10\n20\n\n\t\n1.0\n1.5\n2.0\n2.5\n3.0\nFigure 1. (Color online) Non-local damping contributions,\nαij, in (Fe-centered) bulk B2-FeCo and bcc Fe, as a function\nof the normalized distance in lattice constant units a.Inset:\nNon-local contributions from only Fe-Fe pairs summed, for\neach distance, in bcc Fe bulk (empty blue dots) and in the\nB2-FeCo (full red dots). The onsite damping for Fe (Co) in\nB2-FeCo isαFe\nonsite = 1.1×10−3(αCo\nonsite = 0.8×10−3) and for\nbcc Fe it is αFe\nonsite = 1.6×10−3. The magnetization direction\nisz([001]). Lines are guides for the eyes.\nWeanalyzedtheroleoflocaldistortionsbyconsidering3\na hypothetical case of a large, 15%(c\na= 1.15), distortion\non thez-axis of ordered B2-FeCo. We found the largest\ndamping anisotropy ( ∼24%) when comparing the results\nwith magnetization in the [001](α[001]\nt= 10.21×10−3)\nand in the [010](α[010]\nt= 7.76×10−3) directions. This\nconfirms that, indeed, bct-like distortions act in favour of\nthe∆αtenhancement (and therefore, of the maximum-\nminimum damping ratio), but the theoretical data are\nnot large enough to explain the giant value reported ex-\nperimentally [10].\nNevertheless, in the case of an alloy, the local lattice\ndistortions suggested in Ref. [10] are most to likely occur\nin an heterogeneous way [20], with different distortions\nfor different local environments. To inspect this type\nof influence on the theoretical results, we investigated\n(Table S3, SM) clusters containing different atomic con-\nfigurations embedded in a VCA Fe50Co50matrix (with\nFe bulk lattice parameter); distortions were also consid-\nered such that, locally in the clusters,c\na= 1.15(Ta-\nble S4, in the SM). Moreover, in both cases, two types\nof clusters have to be considered: Co-centered and Fe-\ncentered. The αtwas then computed as the sum of the\nlocal and non-local contributions for clusters with a spe-\ncific central (Fe or Co) atom, and the average of Fe-\nand Co-centered clusters was taken. Fe-centered clus-\ntershaveshownlargeranisotropies, onaverage ∼33%for\nthe undistorted (∼74%for the distorted) compared with\n∼8%fortheundistortedCo-centeredclusters( ∼36%for\nthe distorted). Although these results demonstrate the\nimportance of both, local distortions as well as non-local\ncontributions to the damping anisotropy, they are not\nstill able to reproduce the huge observed [10] maximum-\nminimum damping ratio.\nWe further proceed our search for ingredients that\ncould lead to a huge ∆αtby inspecting interface effects,\nwhich are present in thin films, grain boundaries, stack-\ning faults and materials in general. Such interfaces may\ninfluence observed properties, and in order to examine\nif they are relevant also for the reported alloys of Ref.\n[10], we considered these effects explicitly in the calcu-\nlations. As a model interface, we considered a surface,\nwhat is, possibly, the most extreme case. Hence, we per-\nformed a set of αtcalculations for the Fe50Co50(001),\nfirst on the VCA level. Analogous to the respective bulk\nsystems, we found that the onsite contributions to the\ndamping anisotropy are distinct, but they are not the\nmain cause ( ∆αonsite∼18%). However, the lack of in-\nversion symmetry in this case gives a surprisingly large\nenhancement of ∆αt, thus having its major contribution\ncoming from the non-local damping terms, in particular\nfrom the next-nearest neighbours. Interestingly, negative\nnon-local contributions appear when αtis calculated in\nthe[010]direction. These diminish the total damping\n(the onsite contribution being always positive) and gives\nrise to a larger anisotropy, as can be seen by comparisonof the results shown in Table I and Table S5 (in the Sup-\nplemental Material). In this case, the total anisotropy\nwas found to be more than ∼100%(corresponding to a\nmaximum-minimum damping ratio larger than 200%).\nA compilation of the most relevant theoretical results\nobtained here is shown in Fig. 2, together with the ex-\nperimental data and the local density of states (LDOS)\natEFfor each magnetization direction of a typical atom\nin the outermost layer (data shown in yellow). As shown\nin Fig. 2, the angular variation of αthas a fourfold ( C4v)\nsymmetry, with the smallest Gilbert damping occurring\nat 90◦from the reference axis ( [100],θH= 0◦), for both\nsurface and bulk calculations. This pattern, also found\nexperimentally in [10], matches the in-plane bcc crys-\ntallographic symmetry and coincides with other mani-\nfestations of SOC, such as the anisotropic magnetoresis-\ntance [10, 21]. Following the simplified Kambersky’s for-\nmula [13, 22], in which (see SM) α∝n(EF)and, there-\nfore, ∆α∝∆n(EF), we can ascribe part of the large\nanisotropy of the FeCo alloys to the enhanced LDOS dif-\nferences at the Fermi level, evidenced by the close corre-\nlation between ∆n(EF)and∆αtdemonstrated in Fig. 2.\nThus, as a manifestation of interfacial SOC (the so-called\nproximity effect [23]), the existence of ∆αtcan be under-\nstoodintermsofRashba-likeSOC,whichhasbeenshown\nto play an important role on damping anisotropy [24, 25].\nAnalogous to the bulk case, the higher morboccurs where\nthe system presents the smallest αt, and the orbital\nmoment anisotropy matches the ∆αtfourfold symme-\ntry with a 90◦rotation phase (see Fig. S3, SM). Note\nthat a lower damping anisotropy than Co50Fe50(001) is\nfound for a pure Fe(001) bcc surface, where it is ∼49%\n(Table S2, SM), in accordance with Refs. [7, 26], with\na dominant contribution from the onsite damping val-\nues (conductivity-like character on the reciprocal-space\n[19, 27]).\nThe VCA surface calculations on real-space allows to\ninvestigate the layer-by-layer contributions (intra-layer\ndamping calculation), as shown in Table I. We find that\nthemajorcontributiontothedampingsurfaceanisotropy\ncomes from the outermost layer, mainly from the differ-\nence in the minority 3dstates around EF. The deeper\nlayers exhibit an almost oscillatory ∆αtbehavior, simi-\nlar to the oscillation mentioned in Ref. [28] and to the\nFriedel oscillations obtained for magnetic moments. The\ndamping contributions from deeper layers are much less\ninfluenced by the inversion symmetry breaking (at the\nsurface), as expected, and eventually approaches the typ-\nicalbulklimit. Therefore, changesintheelectronicstruc-\ntureconsiderednotonlytheLDOSoftheoutermostlayer\nbut a summation of the LDOS of all layers (including the\ndeeper ones), which produces an almost vanishing differ-\nence between θH= 0◦andθH= 45◦(also approaching\nthe bulk limit). The damping anisotropy arising as a sur-\nface effect agrees with what was observed in the case of\nFe [7] and CoFeB [29] on GaAs(001), where the damping4\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.10.20.3\nΔn(EF)\n(st./Ry−at.)\n0.0050.0100.015\nαt\nθH\nFigure 2. (Color online) Total damping and LDOS difference\natEF,∆n(EF), as a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCAFe 50Co50(001)bcc. Triagles: (greenfull)averageover32\nclusters (16 Fe-centered and 16 Co-centered), with bcc struc-\nture at the surface layers (SM) embedded in a VCA medium;\n(gray open) similar calculations, but with a local lattice dis-\ntortion. Circles: (yellow open) ∆n(EF)betweenθH= 0◦and\nthe current angle for a typical atom in the outermost layer\nof VCA Fe 50Co50(001) bcc; (blue full) experimental data [10]\nfor a 10-nm Fe 50Co50/Pt thin film; (purple full) average bulk\nVCA Fe 50Co50bcc; and (brown full) the B2-FeCo bulk. Lines\nare guides for the eyes.\nanisotropy diminishes as the film thickness increases.\nTable I. Total intra-layer damping ( αt×10−3) and anisotropy,\n∆αt(Eq. 2), of a typical (VCA) atom in each Fe 50Co50(001)\nbcc surface layer for magnetization along [010]and[110]di-\nrections. In each line, the sum of all αijin the same layer is\nconsidered. Outermost (layer 1) and deeper layers (2-5).\nLayerαt[010]αt[110] ∆αt\n1 7.00 14.17 +102.4%\n2 1.28 1.16 −9.4%\n3 2.83 3.30 +16.6%\n4 2.18 1.99 −8.7%\n5 2.54 2.53 −0.4%\nWe also studied the impact of bct-like distortions in\nthesurface, initiallybyconsideringtheVCAmodel. Sim-\nilartothebulkcase,tetragonaldistortionsmaybeimpor-\ntant for the damping anisotropy at the surface, e.g. when\nlocal structural defects are present. Therefore, localized\nbct-like distortions of the VCA medium in the surface,\nparticularly involving the most external layer were inves-\ntigated. The structural model was similar to what was\nused for the Fe50Co50bulk, consideringc\na= 1.09(see\nSM). Our calculations show that tetragonal relaxations\naround a typical site in the surface induce a ∆αt∼75%,\nfromα[010]\nt= 8.94×10−3toα[110]\nt= 15.68×10−3. Themain effect of these distortions is an enhancement of the\nabsolute damping values in each direction with respect to\nthe pristine (bcc) system. This is due to an increase on\nαonsite, fromα[010]\nonsite = 7.4×10−3toα[010]\nonsite = 9.5×10−3,\nand fromα[110]\nonsite = 8.7×10−3toα[110]\nonsite = 11.7×10−3;\nthe resulting non-local contributions remains similar to\ntheundistortedcase. Theinfluenceofbct-likedistortions\non the large damping value in the Fe50Co50surface is in\nline with results of Mandal et al.[30], and is related to\nthe transition of minority spin electrons around EF.\nWe then considered explicit 10-atom Fe50Co50clusters\nembedded in a VCA FeCo surface matrix. The results\nfrom these calculations were obtained as an average over\n16 Fe-centered and 16 Co-centered clusters. We con-\nsidered clusters with undistorted bcc crystal structure\n(Fig. 2, yellow open circles) as well as clusters with lo-\ncal tetragonal distortions (Fig. 2, black open circles). As\nshown in Fig. 2 the explicit local tetragonal distortion\ninfluences the damping values ( α[010]\nt= 10.03×10−3and\nα[110]\nt= 14.86×10−3)andtheanisotropy, butnotenough\ntoreproducethehugevaluesreportedintheexperiments.\nA summary of the results obtained for each undis-\ntorted FeCo cluster at the surface is shown in Fig. 3:\nCo-centered clusters in Fig. 3(a) and Fe-centered clusters\nin Fig. 3(b). A large variation of αtvalues is seen from\nclustertocluster, dependingonthespatialdistributionof\natomic species. It is clear that, αtis larger when there is\na larger number of Fe atoms in the surface layer that sur-\nroundsthecentral, referenceclustersite. Thiscorrelation\ncan be seen by the numbers in parenthesis on top of the\nblue symbols (total damping for each of the 16 clusters\nthat were considered) in Fig. 3. We also notice from the\nfigure that the damping in Fe-centered clusters are lower\nthan in Co-centered, and that the [010]magnetization di-\nrection exhibit always lower values. In the Insetof Fig. 3\nthe onsite contributions to the damping, αonsite, and the\nLDOS atEFin the central site of each cluster are shown:\na correlation, where both trends are the same, can be ob-\nserved. The results in Fig. 3 shows that the neighbour-\nhood influences not only the local electronic structure at\nthe reference site (changing n(EF)andαonsite), but also\nmodifies the non-local damping αij, leading to the cal-\nculatedαt. In other words, the local spatial distribution\naffects how the total damping is manifested, something\nwhich is expressed differently among different clusters.\nThis may open up for materials engineering of local and\nnon-local contributions to the damping.\nConclusions: We demonstrate here that real-space\nelectronic structure, based on density functional theory,\nyield a large Gilbert damping anisotropy in Fe50Co50al-\nloys. Theory leads to a large damping anisotropy, when\nthe magnetization changes from the [010]to the [110]di-\nrection, which can be as high as ∼100%(or200%in the\nminimum-maximum damping ratio) when surface calcu-\nlations are considered. This is in particular found for5\n\u0001\u0001\u0002\u0003\u0004\u0005\u0006\u0007\n\u0005\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\n\u0001\u0001\n\u000b\f\u0001\r\u000e\u0006\u000f\u0010\u0011\u0002\u0010\u0012\u0010\u0013\u0001\u0002\u0001\u0014\u0005\u0004\u0005\u0015\u0001\u0002\u0001\u0014\u0004\u0004\u0005\u0015\u0001\u0001\u0016\u0016\u0003\u0004\u0005\u0006\u0007\u0005\u0004\u0005\b\u0005\u0007\u0005\n\u0001\u0002\u0003\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\b\t\u0017\u0018\u0004\u0005\u0004\b\u0004\t\u0004\u0017\u0001\u0006\u0007\u0007\b\t\n\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\n\u000b\n\f\u0006\u0007\u0007\b\t\u000b\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\u000b\u000b\n\f\n\u0001\u0001\u0002\u0003\u0001\u0004\u0005\u0006\u0007\n\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\n\u000b\f\r\u0002\u000e\u000f\u0001\u0010\f\u0011\u0012\u000e\u000f\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001\u0016\u0012\u0017\u0001\u0018\u000e\u0006\u0019\u000e\u0010\u0002\u000e\u000f\u000e\u001a\u0001\u0001\u001b\u001b\u0003\u0001\u0004\u0005\u0006\u0007\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\u0002\u0003\u0004\u0005\u0004\u0014\u0004\u0015\t\u0005\t\t\n\u0001\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001(0)(1)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(3)(3)(4)(4)\n(1)(1)(2)(2)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(4)(5)\nFigure 3. Damping for the [010](open circles) and [110](full\ncircles) magnetization directions for distinct types of 10-atom\nFe50Co50bcc clusters, embedded in VCA Fe 50Co50(001) bcc\nand without any distortion around the reference atom (for\nwhichαtandαonsiteare shown). (a) Co-centered and (b)\nFe-centered clusters. The quantity of Fe atoms in the surface\nlayers (near vacuum) are indicated by the numbers in paren-\nthesis and the results have been ordered such that larger val-\nues are to the left in the plots. Insets:αonsitefor the [010]\n(red open circles) and [110](blue filled circles) magnetization\ndirections, and corresponding local density of states, n(EF),\nat the Fermi level (green filled and unfilled triangles) at the\ncentral atom (placed in the outermost layer) for both types\nof clusters. Lines are guides for eyes.\ncontributions from surface atoms in the outermost layer.\nHence the results presented here represents one more ex-\nample, in addition to the well known enhanced surface\norbital moment [31], of the so-called interfacial spin-orbit\ncoupling. This damping anisotropy, which holds a bcc-\nlike fourfold ( C4v) symmetry, has a close relation to the\nLDOS difference of the most external layer at EF(ma-\njorly contributed by the minority dstates), as well as\nto the orbital moment anisotropy with a 90◦phase. As\na distinct example of an interface, we consider explicitly\nthe Fe50Co50cluster description of the alloy. In this case,\nbesides an onsite contribution, we find that the damp-\ning anisotropy is mostly influenced by non-local next-\nnearest-neighbours interactions.\nSeveral Gilbert damping anisotropy origins are also\ndemonstrated here, primarily related to the presence of\ninterfaces, alloy composition and local structural distor-\ntions (as summarized in Table S6, in the SM [32]). Pri-\nmarily we find that: ( i) the presence of Co introduces anenhanced spin-orbit interaction and can locally modify\nthe non-local damping terms; ( ii) the randomness of Co\nin the material, can modestly increase ∆αtas a total ef-\nfect by creating Co-concentrated clusters with enhanced\ndamping; ( iii) at the surface, the spatial distribution of\nFe/Co, increases the damping when more Fe atoms are\npresent in the outermost layer; and ( iv) the existence\nof local, tetragonal distortions, which act in favour (via\nSOC) of the absolute damping enhancement, by modify-\ning theαonsiteof the reference atom, and could locally\nchange the spin relaxation time. Furthermore, in rela-\ntionship to the work in Ref. [10], we show here that bulk\nlike tetragonal distortions, that in Ref. [10] were sug-\ngested to be the key reason behind the observed huge\nanisotropy of the damping, can in fact not explain the\nexperimental data. Such distortions were explicitly con-\nsidered here, using state-of-the-art theory, and we clearly\ndemonstrate that this alone can not account for the ob-\nservations.\nAlthough having a similar trend as the experimen-\ntal results of Ref. [10], we do not reproduce the most\nextreme maximum-minimum ratio reported in the ex-\nperiment,∼400%(or∆αt∼300%). The measured\ndamping does however include effects beyond the intrin-\nsic damping that is calculated from our electronic struc-\nturemethodology. Other mechanismsare knownto influ-\nence the damping parameter, such as contributions from\neddy currents, spin-pumping, and magnon scattering, to\nname a few. Thus it is possible that a significant part\nof the measured anisotropy is caused by other, extrin-\nsic, mechanisms. Despite reasons for differences between\nobservation and experiment on films of Fe50Co50alloys,\nthe advancements presented here provide new insights on\nthe intrinsic damping anisotropy mechanisms, something\nwhich is relevant for the design of new magnetic devices.\nAcknowledgements: H.M.P. and A.B.K. acknowledge\nfinancial support from CAPES, CNPq and FAPESP,\nBrazil. The calculations were performed at the computa-\ntional facilities of the HPC-USP/CENAPAD-UNICAMP\n(Brazil), at the National Laboratory for Scientific Com-\nputing (LNCC/MCTI, Brazil), and at the Swedish Na-\ntional Infrastructure for Computing (SNIC). I.M. ac-\nknowledge financial support from CAPES, Finance Code\n001, process n◦88882.332894/2018-01, and in the Insti-\ntutional Program of Overseas Sandwich Doctorate, pro-\ncess n◦88881.187258/2018-01. 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Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nI. Theory\nThe torque-correlation model, first introduced by\nKamberský [1], and later elaborated by Gilmore et al.\n[2], can be considered as both a generalization and an\nextended version of the breathing Fermi surface model,\nwhich relates the damping of the electronic spin orienta-\ntion, with the variation in the Fermi surface when the\nlocal magnetic moment is changed. In this scenario,\nand considering the collinear limit of the magnetic or-\ndering, due to the spin-orbit coupling (SOC), the tilting\nin magnetization ˆmby a small change δˆmgenerates a\nnon-equilibrium population state which relaxes within a\ntimeτtowards the equilibrium. We use an angle θ, to\nrepresent the rotation of the magnetization direction δˆm.\nIftheBlochstatesofthesystemsarecharacterizedbythe\ngenericbandindex natwavevector k(withenergies /epsilon1k,n),\nit is possible to define a tensorfor the damping, that has\nmatrix elements (adopting the isotropic relaxation time\napproximation)\nανµ=gπ\nm/summationdisplay\nn,mdk\n(2π)3η(/epsilon1k,n)/parenleftbigg∂/epsilon1k,n\n∂θ/parenrightbigg\nν/parenleftbigg∂/epsilon1k,m\n∂θ/parenrightbigg\nµτ\n~\n(S1)\nwhich accounts for both intraband ( n=m, conductivity-\nlike) and interband ( n/negationslash=m, resistivity-like) contribu-\ntions [2]. Here µ,νare Cartesian coordinate indices,\nthat will be described in more detail in the discussion\nbelow, while η(/epsilon1k,n) =∂f(/epsilon1)\n∂/epsilon1/vextendsingle/vextendsingle/vextendsingle\n/epsilon1k,nis the derivative of\nthe Fermi distribution, f, with respect to the energy\n/epsilon1, andn,mare band indices. Therefore, the torque-\ncorrelation model correlates the spin damping to vari-\nations of the energy of single-particle states with respect\nto the variation of the spin direction θ, i.e.∂/epsilon1k,n\n∂θ. Us-\ning the Hellmann-Feynmann theorem, which states that\n∂/epsilon1k,n\n∂θ=/angbracketleftψk,n|∂H\n∂θ|ψk,n/angbracketright, and the fact that only the spin-\norbit Hamiltonian Hsochanges with the magnetization\ndirection, the spin-orbit energy variation is given by\n∂/epsilon1k,n(θ)\n∂θ=/angbracketleftψk,n|∂\n∂θ/parenleftbig\neiσ·ˆnθHsoe−iσ·ˆnθ/parenrightbig\n|ψk,n/angbracketright(S2)\nin which σrepresents the Pauli matrices vector, and\nˆnis the direction around which the local moment hasbeen rotated. The expression in Eq. S2 can be eas-\nily transformed into∂/epsilon1k,n(θ)\n∂θ=i/angbracketleftψk,n|[σ·ˆn,Hso]|ψk,n/angbracketright\nand we call ˆT= [σ·ˆn,Hso]thetorqueoperator. In\nview of this, it is straightforward that, in the collinear\ncase in which all spins are aligned to the zdirection,\nσ·ˆn=σµ(µ=x,y,z), originating the simplest {x,y,z}-\ndependent torque operator ˆTµ. Putting together the in-\nformation on Eqs. S1 and S2, and using the fact that\nthe imaginary part of the Greens’ functions can be ex-\npressed, in Lehmann representation, as Im ˆG(/epsilon1±iΛ) =\n−1\nπ/summationtext\nnΛ\n(/epsilon1−/epsilon1n)2+Λ2|n/angbracketright/angbracketleftn|, then it is possible to write in\nreciprocal-space [3]:\nανµ=g\nmπ/integraldisplay /integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTνImˆGˆTµImˆG/parenrightBig\nd/epsilon1dk\n(2π)3.(S3)\nIn a real-space formalism, the Fourier transformation\nof the Green’s function is used to find a very similar ex-\npression emerges for the damping element ανµ\nijrelative to\ntwo atomic sites iandj(at positions riandrj, respec-\ntively) in the material:\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(S4)\nwhere we defined mi= (morb+mspin)as the total mag-\nnetic moment localized in the reference atomic site iin\nthe pair{i,j}. The electron temperature that enters into\nη(ε)is zero and, consequently, the energy integral is per-\nformed only at the Fermi energy. In this formalism, then,\nthe intraband and interband terms are replaced by onsite\n(i=j) and non-local ( i/negationslash=j) terms. After calculation of\nall components of Eq. S4 in a collinear magnetic back-\nground, we get a tensor of the form\nαij=\nαxxαxyαxz\nαyxαyyαyz\nαzxαzyαzz\n, (S5)\nwhich can be used in the generalized atomistic Landau-\nLifshitz-Gilbert (LLG) equation for the spin-dynamics\nof magnetic moment on site i[4]:∂mi\n∂t=mi×/parenleftBig\n−γBeff\ni+/summationtext\njαij\nmj·∂mj\n∂t/parenrightBig\n. Supposing that all spins are\nparallel to the local zdirection, we can define the scalar2\nαvalue as the average between components αxxandαyy,\nthat is:α=1\n2(αxx+αyy).\nOnce one has calculated the onsite ( αonsite) and the\nnon-local (αij) damping parameters with respect to the\nsite of interest i, the total value, αt, can be defined as\nthe sum of all these α’s:\nαt=/summationdisplay\n{i,j}αij. (S6)\nIn order to obtain the total damping in an heteroge-\nneous atomic system (more than one element type), such\nas Fe50Co50(with explicit Fe/Co atoms), we consider the\nweighted average between the different total local damp-\ning values ( αi\nt), namely:\nαt=1\nMeff/summationdisplay\nimiαi\nt, (S7)\nwheremiis the local magnetic moment at site i, and\nMeff=/summationtext\nimiis the summed total effective magnetiza-\ntion. This equation is based on the fact that, in FMR ex-\nperiments, the magnetic moments are excited in a zone-\ncentered, collective mode (Kittel mode). In the results\npresented here, Eq. S7 was used to calculate αtofB2-\nFeCo, both in bcc and bct structures.\nII. Details of calculations\nThe real-space linear muffin-tin orbital on the atomic-\nsphere approximation (RS-LMTO-ASA) [5] is a well-\nestablished method in the framework of the DFT to de-\nscribetheelectronicstructureofmetallicbulks[6,7], sur-\nfaces [8, 9] and particularly embedded [10] or absorbed\n[11–14] finite cluster systems. The RS-LMTO-ASA is\nbased on the LMTO-ASA formalism [15], and uses the\nrecursion method [16] to solve the eigenvalue problem\ndirectly in real-space. This feature makes the method\nsuitable for the calculation of local properties, since it\ndoes not depend on translational symmetry.\nThe calculations performed here are fully self-\nconsistent, and the spin densities were treated within\nthe local spin-density approximation (LSDA) [17]. In all\ncases, we considered the spin-orbit coupling as a l·sterm\nincluded in each variational step [18–20]. The spin-orbit\nis strictly necessary for the damping calculations due to\nits strong dependence on the torque operators, ˆT. In\nthe recursion method, the continued fractions have been\ntruncated with the Beer-Pettifor terminator [21] after 22\nrecursion levels ( LL= 22). The imaginary part that\ncomes from the terminator was considered as a natural\nchoice for the broadening Λto build the Green’s func-\ntions ˆG(/epsilon1+iΛ), which led to reliable αparameters in\ncomparison with previous results (see Table S1).To account for the Co randomness in the experimental\nFe50Co50films [22], some systems were modeled in terms\nof the virtual crystal approximation (VCA) medium of\nFe50Co50, considering the bcc (or the bct) matrix to have\nthe same number of valence electrons as Fe50Co50(8.5\ne−). However, we also investigated the role of the Co\npresence, as well as the influence of its randomness (or\nordering), by simulating the B2(CsCl) FeCo structure\n(a=aFe). The VCA Fe50Co50andB2-FeCo bulks were\nsimulated by a large matrix containing 8393 atoms in\nreal-space, the first generated by using the Fe bcc lat-\ntice parameter ( aFe= 2.87Å) and the latter using the\noptimized lattice parameter ( a= 2.84Å). Thisachoice\nin VCA Fe50Co50was based on the fact that it is eas-\nier to compare damping results for Fe50Co50alloy and\npure Fe bcc bulk if the lattice parameters are the same,\nand the use of the aFehas shown to produce trustwor-\nthyαtvalues. On the other hand, bct bulk structures\nwithc\na= 1.15(B2-FeCo bct and VCA Fe50Co50bct) are\nbased on even larger matrices containing 49412 atoms.\nThe respective surfaces were simulated by semi-matrices\nof the same kind (4488 and 19700 atoms, respectively),\nconsidering one layer of empty spheres above the outer-\nmost Fe50Co50(or pure Fe) layer, in order to provide a\nbasis for the wave functions in the vacuum and to treat\nthe charge transfers correctly.\nWe notice that the investigations presented here are\nbased on a (001)-oriented Fe50Co50film, in which only a\nsmall lattice relaxation normal to the surface is expected\nto occur (∼0.1%[23]).\nDamping parameters of Fe-centered and Co-centered\nclusters, embedded in an Fe50Co50VCA medium, have\nbeen calculated (explicitly) site by site. In all cases,\nthese defects are treated self-consistently, and the po-\ntential parameters of the remaining sites were fixed at\nbulk/pristine VCA surface values, according to its envi-\nronment. When inside the bulk, we placed the central\n(reference) atom of the cell in a typical site far away\nfrom the faces of the real-space matrix, avoiding any un-\nwanted surface effects. We considered as impurities the\nnearest 14 atoms (first and second nearest neighbours,\nup to 1a) from the central atom, treating also this sites\nself-consistently, in a total of 15 atoms. We calculated\n10 cases with Fe and Co atoms randomly positioned: 5\nwith Fe as the central atom (Fe-centered) and 5 with Co\nas the central atom (Co-centered). An example (namely\ncluster #1 of Tables S3 and S4), of one of these clusters\nembedded in bulk, is represented in Fig. S1(a). As the\nself-consistent clusters have always a total of 15 atoms,\nthe Fe (Co) concentration is about 47% (53%) or vice-\nversa. On the other hand, when inside the surface, we\nplaced the central (reference) atom of the cluster in a\ntypical site of the most external layer (near vacuum),\nsince this has shown to be the layer where the damping\nanisotropy is larger. Therefore, we considered as impu-\nrities the reference atom itself and the nearest 9 atoms3\n(up to 1a), in a total of 10 atoms (and giving a perfect\n50% (50%) concentration). An example of one of these\nclusters embedded in a surface is shown in Fig. S1(b).\n(a)\n(b)\nFigure S1. (Color online) Schematic representation of an ex-\nampleof: (a)Fe-centered15-atomclusterembeddedinaVCA\nFe50Co50bcc bulk medium; (b) Co-centered 10-atom cluster\nembedded in a VCA Fe 50Co50(001) bcc surface medium. Yel-\nlow and blue spheres represent Fe and Co atoms, respectively,\nwhile gray atoms represent the VCA Fe 50Co50sites (8.5 va-\nlence e−). The Fe(Co) concentration in the clusters are: (a)\n53% (47%) and (b) 50% (50%) . The total number of atoms\nincluding the surrounding VCA sites are: (a) 339 and (b) 293.\nThey were all accounted in the sum to obtain αtat the central\n(reference) Fe (a) and Co (b) site.\nTo simulate a bct-like bulk distortion, the 8 first neigh-\nbours of the central atom were stretched in the cdi-\nrection, resulting in ac\na= 1.15ratio. On the other\nhand, when embedded on the Fe50Co50(001) bcc surface,\nthe central (reference) atom is placed in the outermost\nlayer (near vacuum), and we simulate a bct distortion\nby stretching the 4 nearest-neighbours (on the second\nlayer) to reproduce ac\na= 1.09ratio (the maximum per-\ncentage that the atoms, in these conditions, could be\nmoved to form a bct-like defect). In this case, a total\nof 10 atoms (the nearest 9 atoms from the central one\n– up to 1a– and the reference atom) were treated self-\nconsistently, analogous to as shown in Fig. S1(b). As in\nthe case of the pristine bcc Fe50Co50clusters embedded\nin the VCA surface, we considered a total of 32 10-atom\nclusters with different Fe/Co spatial distributions, being\n16 Fe-centered, and 16 Co-centered.III. Comparison with previous results\nTheab-initio calculation of the Gilbert damping, in\nthe collinear limit, is not a new feature in the literature.\nMainly, the reported theoretical damping results are for\nbulk systems [2, 4, 24–28], but, some of them even stud-\nied free surfaces [29]. Therefore, in order to demonstrate\nthe reliability of the on-site and total damping calcula-\ntions implemented here in real-space, a comparison of\nthe presently obtained with previous (experimental and\ntheoretical) results, are shown in Table S1. As can be\nseen, our results show a good agreement with previously\nobtainedαvalues, including some important trends al-\nreadypredictedbefore. Forexample, thereducedGilbert\ndamping of Co hcp with respect to the Co fcc due to\nthe reduction of the density of states at the Fermi level\n[24, 28], (∼10.92states/Ry-atom in the hcp case and\n∼16.14states/Ry-atom in the fcc case).\nIV. Details of the calculated damping values\nThe damping values obtained for the systems studied\nhere are shown in Tables (S2-S5). These data can be use-\nful for the full understanding of the results presented in\nthe main text. For easy reference, in Table S2 the αtof a\ntypical atom in each system (bulk or surface) for different\nspin quantization axes are shown. These data are plot-\nted in Fig. 2 of the main text. The obtained values show\nthat, indeed, for bulk systems the damping anisotropies\nare not so pronounced as in the case of Fe50Co50(001)\nbcc surface.\nAs observed in Table S2, the increase in αtwhen\nchanging from the bcc Fe50Co50(c\na= 1) to the bct\nFe50Co50bulk structure (c\na= 1.15) is qualitatively con-\nsistent to what was obtained by Mandal et al.[33] (from\nαt= 6.6×10−3in the bcc to αt= 17.8×10−3in the\nbct, withc\na= 1.33[33]).\nTables S3 and S4 refer to the damping anisotropies\n(∆αt) for all Fe-centered and Co-centered clusters stud-\nied here, with different approaches: ( i) bcc clusters em-\nbedded in the VCA medium (Table S3) and ( ii) bct-like\nclusters embedded in the VCA medium (Table S4).\nIn comparison with bct-like clusters, we found larger\nabsoluteαtvalues but lower damping anisotropies. In\nall cases, Fe-centered clusters present higher ∆αtper-\ncentages.\nIn Table S5 the onsite damping anisotropies ( ∆αonsite)\nfor each layer of the Fe50Co50(001) bcc surface (\"1\" repre-\nsents the layer closest to vacuum) are shown. In compar-\nison with the total damping anisotropies (Table I of the\nmain text), much lower percentages are found, demon-\nstrating that the damping anisotropy effect comes ma-\njorly from the non-local damping contributions.\nThe most important results concerning the largest\ndampinganisotropiesaresummarizedinTableS6, below.4\nTable S1. Total damping values ( ×10−3) calculated for some bulk and surface systems, and the comparison with previous\nliterature results. The onsite contributions are indicated between parentheses, while the total damping, αt, are indicated\nwithout any symbols. All values were obtained considering the [001]magnetization axis. The VCA was adopted for alloys,\nexcept for the Fe 50Co50bcc in theB2structure (see Eq. S7). Also shown the broadening Λvalue considered in the calculations.\nBulks a(Å) This work Theoretical Experimental Λ(eV)\nFe bcc 2.87 4.2(1.6) 1 .3[2]a/(3.6)[4] 1.9[30]/2.2[31]\nFe70Co30bcc 2.87 2.5(0.7) − 3−5[32]d\nFe50Co50bcc 2.87 3.7(1.0)[VCA]/ 2.3(1.0)[B2]1.0[25]c[VCA]/ 6.6[33] [B2] 2.3[27]\nNi fcc 3.52 27.8(57.7) 23 .7[34]/( 21.6[4])b26.0[31]/24.0[35]\nNi80Fe20(Py) fcc 3.52 9.8(12.1) 3 .9[25]c8.0[30]/5.0[35]\nCo fcc 3.61 [3] 3.2(5.3) 5 .7[28]/(3.9[4])b11.0[30]∼5×10−2\nCo hcp 2.48/4.04 [28] 2.1(6.2) 3 .0[28] 3.7[31]\nCo85Mn15bcc [36] 2.87 [28] 6.2(4.2) 6 .6[28] −\nCo90Fe10fcc 3.56 [37] 3.6(4.2) − 3.0[35]/4.8[37]\nSurfaces a(Å) This work Theoretical Experimental\nFe(001) bcc [110] 2.87 5.8(5.4)e− 7.2[38]h/6.5[39]i\nFe(001) bcc [100] 2.87 3.9(4.4)f∼4[29]g4.2[40]j\nNi(001) fcc 3.52 80.0(129.6)∼10[29]g/12.7[41]m22.1[42]l\nPdFe/Ir(111) [43] fcc 3.84 3.9(2.7)n− −\nPdCo/Ir(111) [44] fcc 3.84 3.2(14.7)o− −\naWith Λ∼2×10−2eV.\nbWith Λ = 5×10−3eV.\ncWith Λ∼1.4×10−4eV.\ndFor a 28%Co concentration, but the results do not significantly change for a 30%Co concentration. Range including results before and\nafter annealing.\neOf a typical atom in the more external surface layer (in contact with vacuum), in the [110] magnetization direction.\nfOf a typical atom in the more external surface layer (in contact with vacuum), in the [100] magnetization direction.\ngFor a (001) bcc surface with thickness of N= 8ML (the same number of slabs as in our calculations), and Λ = 10−2eV.\nhAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001) thin film (sample S2 in Ref. [38]) in the [110] magnetization direction.\niAnisotropic damping obtained for a 1.14 nm Fe/InAs(001) thin film in the in-plane [110] hard magnetization axis.\njFor a 25-nm-thick Fe films grown on MgO(001).\nkFor epitaxial Fe(001) films grown on GaAs(001) and covered by Au, Pd, and Cr capping layers.\nlIntrinsic Gilbert damping for a free 4×[Co(0.2 nm)/Ni(0.6 nm)](111) multilayer. Not the same system as Ni(001), but the nearest system\nfound in literature.\nmFor a Co | Ni multilayer with Ni thickness of 4 ML (fcc stacking).\nnOf a typical atom in the Fe layer.\noOf a typical atom in the Co layer.\nThe alloys with short-range orders (SRO) are described\nas FeCo clusters (with explicit Fe and Co atoms) embed-\nded in the Fe50Co50VCA medium – with and without\nthe bct-like distortion. In this case, the damping is cal-\nculated as a weighted average (Eq. S7). As discussed in\nthe main text, it can be seen from Table S6 that distor-\ntions and disorder can increase the anisotropy but the\nmajor effect comes from the surface. We notice that the\nnumber of clusters considered is limited in the statistical\naverage.\nIV. Kambersky’s simplified formula\nInordertoconnecttheanisotropyoftheGilbertdamp-\ning to features in the electronic structure, we consider in\nthe following Kambersky’s simplified formula for Gilbert\ndamping [47, 48]α=1\nγMs/parenleftBigγ\n2/parenrightBig2\nn(EF)ξ2(g−2)2\nτ.(S8)\nHere,γis the gyromagnetic ratio, n(EF)represents the\nLDOS at the Fermi level, ξis the SOC strength, τis the\nelectron scattering time, Msis the spin magnetic mo-\nment, andgis the spectroscopic g-factor [35, 49]. Note\nthat Eq. S8 demonstrates the direct relation between\nαandn(EF), often discussed in the literature, e.g., in\nRef. [27]. Our first principles calculations have shown\nno significant change in ξ, upon variation of the mag-\nnetization axis, for the FeCo systems ( ξCo= 71.02meV\nandξFe= 53.47meV). Hence, we can soundly relate the\ndamping anisotropy ∆αtto∆n(EF).\nFigure S2 shows how the LDOS difference (per atom)\n∆n(E)between the [010]and [110]magnetization di-\nrections is developed in pure Fe(001) bcc and in VCA\nFe50Co50(001) bcc surfaces, respectively. In both cases,5\nTableS2. Totaldamping( αt×10−3)ofatypicalatomin\neach system for the spin quantization axes [010](θH=\n90◦) and [110](θH= 45◦); also shown for the [001]and\n[111]. Bulk and surface bct systems are simulated with\nc\na= 1.15.\nBulks\nBulk αt[010]αt[110] ∆αt\nFe bcc 4.18 4.31 +3.1%a\nB2-FeCo bcc 2.28 2.44 +7.2%\nB2-FeCo bct 7.76 8.85 +12.4%\nVCA Fe 50Co50bcc 3.70 4.18 +13.0%\nVCA Fe 50Co50bct 4.69 5.10 +8.7%\nαt[010]αt[001] ∆αt\nB2-FeCo bct 7.76 10.21 +24.1%\nVCA Fe 50Co50bct 4.69 5.75 +22.6%\nαt[010]αt[111] ∆αt\nFe bcc 4.18 4.56 +9.1%b\nSurfaces\nSurface αt[010]αt[110] ∆αt\nFe(001) bcc 3.85 5.75 +49.4%\nFe/GaAs(001) bcc [38] 4.7(7) 7.2(7) +53(27)%c\nFe/MgO(001) bcc [45] 3.20(25) 6.15(20) +92(14)%d\nVCA Fe 50Co50(001) bcc 7.00 14.17 +102.4%\nVCA Fe 50Co50(001) bct 15.20 14.80−2.6%\nαt[010]αt[001] ∆αt\nVCA Fe 50Co50(001) bct 15.20 15.56 +2.4%\nVCA Fe 50Co50(001) bcc 7.00 9.85 +40.7%\naMankovsky et al.[24] find a damping anisotropy of ∼12%\nfor bulk Fe bcc at low temperatures ( ∼50K) between\n[010] and [011] magnetization directions. For this result,\nthe definition α=1\n2(αxx+αyy)was used.\nbThis result agrees with Gilmore et al.[46], which find\nthat the total damping of pure Fe bcc presents its higher\nvalue in the [111] crystallographic orientation and the\nlower value in the [001] direction, except for high scatter-\ning rates. Also agrees with Mankovsky et al.[24] results.\ncAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001)\nthin film (sample S2 in Ref. [38]) in the [010] and [110]\nmagnetization directions.\ndFor a Fe(15 nm)/MgO(001) film at T= 4.5K in the high-\nest applied magnetic field, in which only intrinsic contri-\nbutions to the anisotropic damping are left.\nthe chosen layer,denoted as first, is the most external\none (near vacuum). the VCA Fe50Co50(001) bcc we also\ncalculated ∆n(E)for all layers summed (total DOS dif-\nference).\nAs can be seen, although in all cases the quantity\n∆n(E)exhibits some oscillations, differently from what\nwe observe forthe pureFe(001) surface case, at the Fermi\nenergy, there is a non-negligible difference in the minor-\nity spin channel ( 3dstates) for the VCA Fe50Co50(001).\nConsidering the results presented in Table I (main text)\nthe larger contribution to the damping anisotropy comes\nfrom the most external layer. The results by Li et al.\n[22] indicate a small difference (for two magnetization\ndirections) of the total density of states at the FermiTable S3. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin-\nquantization axis [010]and [110], considering the 15-atom\nFeCo cluster together with the VCA medium in the summa-\ntion for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 10.11 9.65 4.8%\n2 8.09 6.96 16.2%\n3 7.81 7.02 11.3%\n4 7.11 7.02 1.3%\n5 7.48 6.88 8.7%\nAverage 8.12 7.51 8.1%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.68 2.03 32.0%\n2 2.49 2.05 21.5%\n3 2.56 1.86 37.6%\n4 2.45 1.79 36.9%\n5 2.76 2.01 37.3%\nAverage 2.59 1.95 32.8%\nTable S4. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin\nquantization axis [010]and [110], with bct-like distortions/parenleftbigc\na= 1.15/parenrightbig\n, considering the 15-atom FeCo cluster together\nwith the VCA medium in the summation for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 5.85 4.37 33.9%\n2 5.95 4.21 41.3%\n3 5.88 4.35 35.2%\n4 5.90 4.41 33.8%\n5 5.86 4.34 35.0%\nAverage 5.89 4.34 35.7%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.36 1.39 69.8%\n2 2.27 1.32 72.0%\n3 2.22 1.26 76.2%\n4 2.25 1.26 78.6%\n5 2.42 1.38 75.4%\nAverage 2.30 1.32 74.2%\nTable S5. Onsite damping ( αonsite×10−3) of a typical atom\nin each layer of the VCA Fe 50Co50(001) bcc for the spin quan-\ntization axis [010]and[110].\nLayerαonsite[010]αonsite[110] ∆αonsite\n1 7.36 8.70 +18.2%\n2 0.63 0.69 +9.5%\n3 1.41 1.44 +2.1%\n4 0.87 0.86−1.1%\n5 0.99 0.97−2.0%6\nTableS6. SummaryofthemainFe 50Co50dampinganisotropy\nresults for: pure ordered ( B2) alloy; pure random (VCA) bulk\nalloy; bcc bulk together with short-range order (SRO) clus-\nters (see Table S3); bulk together with bct-like distorted clus-\nters inside (see Table S4); surface calculations, in the pristine\nmode and with explicit bct-like clusters embedded (surface +\ndistortion). The maximum-minimum ratio according to Ref.\n[22] isα[110]\nt\nα[010]\nt×100%.\nStructure ∆αtMax-min ratio\nOrdered alloy bcc 7.2% 107.2%\nOrdered alloy bct 24.1% 124.1%\nRandom alloy bcc 13% 113%\nRandom alloy bct 22.6% 122.6%\nRandom alloy + SRO 14.9% 114.9%\nRandom alloy + SRO + Distortion 47.2% 147.2%\nSurface (external layer) 102.4% 202.4%\nSurface (ext. layer) + Distortion 75.4% 175.4%\n10-nm Co 50Fe50/Pt [22] (exp.) 281.3% 381.3%\n−2−1 0 1 2\n−0.02 −0.01  0  0.01  0.02Δn(E)[010]−[110]  (states/Ry−atom)\nEnergy (E−EF) (Ry)Fe(001) bcc (first)\nVCA Fe50Co50(001) bcc (first)\nVCA Fe50Co50(001) bcc (all)\nFigure S2. LDOS difference (per atom), ∆n(E), between the\n[010]and[110]magnetization directions, for both spin chan-\nnels (full lines for majority spin and dashed lines for minority\nspin states), in the outermost layer in pure Fe(001) bcc (in\nblack); outermost layer in VCA Fe 50Co50(001) bcc (in blue);\nand all layers summed in VCA Fe 50Co50(001) bcc (in red).\nlevel,N(EF), what the authors claim that could not ex-\nplain the giant maximum-minimum damping ratio ob-\nserved. So, in order to clarify this effect in the VCA\nFe50Co50(001) bcc, ∆n(E)was also calculated for the all\nlayers summed, what is shown in Fig. S2 (in red). This\ndifference is in fact smaller if we consider the DOS of\nthe whole system, with all layers summed. However, if\nwe consider only the most external layer, then the LDOS\nvariation is enhanced. This is consistent with our theo-\nretical conclusions. As we mention in the main text, this\ndo not rule out a role also played by local (tetragonal-\nlike) distortions and other bulk-like factors in the damp-\ning anisotropy.For the outermost layer of Fe(001) bcc, the calculated\nLDOSatEFis∼20.42states/Ry-atominthe[110]direc-\ntion and∼20.48states/Ry-atom in the [010] direction,\nwhich represents a difference of ∼0.3%and agrees with\nthe calculations performed by Chen et al.[38].\nV. Correlation with anisotropic orbital moment\nBesides the close relation exhibited between ∆αt\nand∆n(EF), we also demonstrate the existence of an\nanisotropic orbital moment in the outermost layer, in\nwhich the fourfold symmetry ( C4v) matches the damp-\ning anisotropy with a 90◦phase. Fig. S3 shows this\ncorrelation between ∆αtand∆morbfor two situations:\n(i) for a typical atom in the outermost layer of VCA\nFe50Co50(001) bcc (blue open dots); and ( ii) for a typi-\ncal atom in the VCA Fe50Co50bcc bulk, considering the\nsame ∆morbscale. For case ( i) we find orbital moments\ndifferencesmorethanoneorderofmagnitudehigherthan\ncase (ii).\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.51.52.5\nΔmorb\n(µB/atom × 10−3)\n0.0050.0100.015\nαt\nθH\nFigure S3. (Color online) Total damping and orbital moment\ndifference, ∆morbas a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCA Fe 50Co50(001) bcc. Circles: (blue open) morbdifference\nbetweenθH= 90◦and the current angle for a typical atom in\nthe outermost layer of VCA Fe 50Co50(001) bcc; and (yellow\nfull) samemorbdifference but for a typical atom in the VCA\nFe50Co50bcc bulk (in the same scale). Lines are guides for\nthe eyes.\nVI. Contribution from next-nearest-neighbours\nFinally, we show in Fig. S4 the summation of all non-\nlocal damping contributions, αij, for a given normalized\ndistance in the outermost layer of VCA Fe50Co50(001)7\nbcc. As we can see, the next-nearest-neighbours from a\nreference site (normalized distanced\na= 1) have very dis-\ntinctαijcontributions to αtfor the two different mag-\nnetization directions ( [010]and[110]), playing an impor-\ntant role on the final damping anisotropy. We must note,\nhowever, that these neighbours in a (001)-oriented bcc\nsurface are localized in the same layer as the reference\nsite, most affected by the interfacial SOC. Same trend is\nobserved ford\na= 2, however less intense. This is con-\nsistent with our conclusions, about the relevance of the\noutermost layer on ∆αt.\n−3−2−1 0 1 2\n 0.5  1  1.5  2  2.5  3  3.5  4∑αij × 10−3\nNormalized distance[110]\n[010]\nFigure S4. 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Rev. 76, 743 (1949)."    },    {        "title": "1309.4247v1.BPS_Maxwell_Chern_Simons_like_vortices_in_a_Lorentz_violating_framework.pdf",        "content": "arXiv:1309.4247v1  [hep-th]  17 Sep 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n1\nBPS MAXWELL-CHERN-SIMONS-LIKE VORTICES IN A\nLORENTZ-VIOLATING FRAMEWORK\nR. CASANA,∗M.M. FERREIRA JR., E. DA HORA, and A.B.F. NEVES\nDepartamento de F´ ısica, Universidade Federal do Maranh˜ a o\nS˜ ao Lu´ ıs, Maranh˜ ao 65080-805, Brazil\n∗E-mail: rodolfo.casana@gmail.com\nWe have analyzed Maxwell-Chern-Simons-Higgs BPS vortices in a Lorentz-\nviolating CPT-odd context. The Lorentz violation induces p rofiles with a coni-\ncal behavior at the origin. For some combination of the coeffic ients for Lorentz\nviolation there always exists a sufficiently large winding nu mber for which the\nmagnetic field flips its sign.\nLorentz-violating (LV) theories have caught much attention since the pro-\nposaloftheStandard-ModelExtension(SME),1whosepropertieshavebeen\nintensively scrutinized in many areas of contemporary physics. The study\nof topological defects in LV scenarios was conducted initially for kink s,2\nmonopoles.3Also, the formation of topological defects (monopoles) was an-\nalyzed in a broader framework of field theories endowed with tensor fields\nthat spontaneously break Lorentz symmetry.4Recently, the existence of\nBPS vortex of type Abrikosov-Nielsen-Olesen has been shown in the pres-\nence of CPT-even LV terms of the SME.5\nThe aim of this contribution is to study Chern-Simons-like BPS vortice s\nin an LV framework attained via the dimensional reduction of the Max well-\nCarroll-Field-Jackiw-Higgs model.6Such dimensional reduction provides a\nMaxwell-Chern-Simons-Higgs (MCSH) model modified by LV terms,\nL=−1\n4FµνFµν+1\n4sǫνρσAνFρσ+|∂µφ−ieAµφ|2+1\n2∂µψ∂µψ\n−e2ψ2|φ|2−U(|φ|,ψ)−1\n2ǫµρσ(kAF)µ(Aρ∂σψ−ψ∂ρAσ),(1)\nwhere the LV parameter splays the role of a Chern-Simons coupling, and\n(kAF)µis the (1+2)-dimensional Carroll-Field-Jackiw (CFJ) background.\nThe potential U(|φ|,ψ) = (ev2−e|φ|2−sψ)2/2 provides the BPS vortices.Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n2\nThe corresponding Gauss law is\n∂j∂jA0−sB−ǫij(kAF)i∂jψ= 2e2A0|φ|2. (2)\nTo explicitly attain the vortex solutions, we use the ansatz φ=\nvg(r)einθ, Aθ=−(a(r)−n)/er, A 0=ω(r), ψ=∓ω(r). The functions\na(r),g(r) andω(r) satisfy the following boundary conditions:\ng(0) = 0,a(0) =n,ω′(0) =±ω0(kAF)θ/2, (3)\ng(∞) = 1,a(∞) = 0,ω(∞) = 0, (4)\nwherenis the winding number of the vortex configuration, ( kAF)θis the\nangular component of ( kAF)iwhereas the constant ω0=ω(0) depends on\nthe boundary conditions and it is numerically determined. In this ansa tz\nthe magnetic field is given by B=−a′/er. Then, the stationary energy\ndensity associated with the model (1) is written as\nE=1\n2/bracketleftbig\nB∓/parenleftbig\nev2/parenleftbig\n1−g2/parenrightbig\n±sω/parenrightbig/bracketrightbig2+v2/parenleftBig\ng′∓ag\nr/parenrightBig2\n±ev2B+∂aJa,(5)\nwhere the condition ( kAF)0= 0 was adopted to insure positiveness of the\nenergy density. The energy is minimized by requiring\ng′=±ag\nr, B=±ev2/parenleftbig\n1−g2/parenrightbig\n+sω, (6)\nwhich are the BPS equations, where the upper (lower) sign corresp onds to\nn>0 (n<0). These and the Gauss law now written as\nω′′+ω′\nr−sB∓(kAF)θ/parenleftBig\nω′+ω\n2r/parenrightBig\n−2e2v2g2ω= 0 (7)\ndescribe topological vortices in this LV MCSH model. For ( kAF)θ= 0 one\nrecoversthe MCSH system. Under BPS equations and boundary co nditions\n(3)-(4), the Eq. (5) provides the BPS energy EBPS=±2πv2n, which is\nproportional to the quantized magnetic flux, Φ B= 2πn/e.\nFigure 1 depicts the profiles of the magnetic field obtained by numeric al\nintegration of Eqs. (6)-(7) for s= 1, and some values of ( kAF)θandn. For\nlarge values of radial coordinate, the profiles behave in a similar way t o\nthe MCSH ones but with mass scales ( β) satisfying β(kAF)θ>0<β(kAF)θ=0<\nβ(kAF)θ<0. The role played by the LV parameter becomes more pronounced\nat the origin, where it generates a conical structure absent in the MCSH\nprofiles. The numerical analysis shows that for n>0 (analogous results are\nobtained for n <0), a fixeds, and (kAF)θ>0,there are always two well\ndefined regions with positive magnetic flux, occurring no magnetic fie ld\nreversion. On the other hand, for fixed s, and (kAF)θ<0,there alwaysProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n3\nFig. 1. Magnetic field B(r). Heres= 1,e=v= 1. Dark gray lines, ( kAF)θ=−1; black\nlines, (kAF)θ= 0, MCSH; slate gray lines, ( kAF)θ= +1; dotted lines, n= 1; dashed\nlines,n= 6; and solid lines, n= 15.\nexists a sufficiently large winding number n0such that for all n > n 0the\nmagnetic field reverses its sign. Consequently, there are always tw o well\ndefined regions with opposite magnetic flux. The flipping of the magne tic\nflux represents another remarkable feature induced by Lorentz violation.\nAcknowledgments\nThe authors thank FAPEMA, CNPq and CAPES for financial support .\nReferences\n1. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n2. M.N. Barreto, D. Bazeia, and R. Menezes, Phys. Rev. D 73, 065015 (2006).\n3. N.M. Barraz Jr., J.M. Fonseca, W.A. Moura-Melo, and J.A. H elay¨ el-Neto,\nPhys. Rev. D 76, 027701 (2007).\n4. M.D. Seifert, Phys. Rev. Lett. 105, 201601 (2010); Phys. Rev. D 82, 125015\n(2010).\n5. R. Casana, M.M. Ferreira Jr., E. da Hora, and C. Miller, Phy s. Rev. D 86,\n065011 (2012); Phys. Lett. B 718, 620 (2012).\n6. H. Belich, M.M. Ferreira Jr, and J.A. Helay¨ el-Neto, Eur. Phys. J. C 38, 511\n(2005)."    },    {        "title": "1508.07830v1.The_synchrotron_self_Compton_spectrum_of_relativistic_blast_waves_at_large_Y.pdf",        "content": "Mon. Not. R. Astron. Soc. 000, 000–000 (2008) Printed 8 November 2021 (MN L ATEX style file v2.2)\nThe synchrotron-self-Compton spectrum of relativistic blast waves\nat large Y\nMartin Lemoine1?\n1Institut d’Astrophysique de Paris, CNRS, UPMC, 98 bis boulevard Arago, F-75014 Paris, France\nABSTRACT\nRecent analyses of multiwavelength light curves of gamma-ray bursts afterglows point to\nvalues of the magnetic turbulence well below the canonical \u00181% of equipartition, in agree-\nment with theoretical expectations of a micro-turbulence generated in the shock precursor,\nwhich then decays downstream of the shock front through collisionless damping. As a direct\nconsequence, the Compton parameter Ycan take large values in the blast. In the presence\nof decaying micro-turbulence and/or as a result of the Klein-Nishina suppression of inverse\nCompton cooling, the Yparameter carries a non-trivial dependence on the electron Lorentz\nfactor, which modifies the spectral shape of the synchrotron and inverse Compton compo-\nnents. This paper provides detailed calculations of this synchrotron-self-Compton spectrum in\nthis largeYregime, accounting for the possibility of decaying micro-turbulence. It calculates\nthe expected temporal and spectral indices \u000band\fcustomarily defined by F\u0017/t\u0000\u000b\nobs\u0017\u0000\fin\nvarious spectral domains. This paper also makes predictions for the very high energy photon\nflux; in particular, it shows that the large Yregime would imply a detection rate of gamma-ray\nbursts at>10GeV several times larger than currently anticipated.\nKey words: acceleration of particles – shock waves – gamma-ray bursts: general\n1 INTRODUCTION\nThe afterglow radiation of gamma-ray bursts, which spans the\nrange from the radio domain to the X-ray and possible higher fre-\nquencies, with a characteristic decrease in time of the peak flux,\nis nicely interpreted in terms of the synchrotron radiation of ultra-\nrelativistic electron that have been accelerated at the forward ultra-\nrelativistic collisionless shock wave of the outflow, see e.g. Piran\n(2004) for a detailed review.\nThis model thus opens a beautiful connection between as-\ntronomical data and the microphysics of the rather extreme phe-\nnomena that relativistic collisionless shocks represent. Zooming\nin on microphysical scales reveals the shock front as a micro-\nturbulent magnetic barrier which isotropizes the incoming back-\nground plasma, see e.g. Moiseev & Sagdeev (1963) for pioneering\nstudies and Kato & Takabe (2008), Spitkovsky (2008a) for detailed\nnumerical simulations. This micro-turbulent barrier itself results\nfrom the build-up of electromagnetic micro-instabilities ahead of\nthe shock front, in the shock precursor where supra-thermal parti-\ncles mix with the incoming background plasma, e.g. Medvedev &\nLoeb (1999), Wiersma & Achterberg (2004), Lyubarksy & Eichler\n(2006), Achterberg & Wiersma (2007), Achterberg et al. (2007),\nBret et al. (2010), Lemoine & Pelletier (2010, 2011), Rabinak et al.\n(2011), Shaisultanov et al. (2012), Lemoine et al. (2014a, b).\nThe resulting micro-turbulence, with typical length scale\n?e-mail: lemoine@iap.fr\u0015\u000eB\u0018c=!pi(!pithe ion plasma frequency) and typical strength1\n\u000fB\u00180:01, thus emerges as a key ingredient for the micro-\nphysics of collisionless shocks. Actually, it also represents a key\nrequisite for the relativistic Fermi process, since this latter can take\nplace (in ideal conditions) only when an intense micro-turbulence,\nwith power on scales smaller than the gyroradius of the acceler-\nated particles, is able to unlock the particles off the background\nmagnetic field lines, by scattering them faster than a gyration\ntime in this background field (Lemoine et al. 2006, Niemiec et\nal. 2006, Pelletier et al. 2009). In terms of magnetization \u001b=\nB2=\u0000\n8\u00194\u00002\nbnmpc2\u0001\n, withBthe background field expressed in\nthe downstream frame, this condition amounts to \u001b\u001c\u000f2\nB\n(Lemoine & Pelletier 2010, 2011, Lemoine et al. 2014a), a con-\ndition which is indeed satisfied for the forward shock of gamma-\nray bursts, since \u001b\u001810\u00009in the interstellar medium. This point\nof view has been confirmed by particle-in-cell simulations (e.g.\nSpitkovsky 2008b, Martins et al. 2009, Nishikawa et al. 2009,\nSironi & Spitkovsky 2009, 2011, Sironi et al. 2013, Haugbølle\n2011).\nFinally, these micro-instabilities may also build the magne-\ntized turbulence in which the electrons eventually produce the af-\nterglow radiation in a synchrotron-like process (Medvedev & Loeb\n1999) – jitter effects are expected to be weak in the conditions\n1\u000fB\u0011\u000eB2=\u0000\n8\u00194\u00002\nbnmpc2\u0001\nrepresents the magnetic energy fraction\nof equipartition, if \u0000b\u001d1represents the relative Lorentz factor between\nupstream and downstream, nthe upstream proper density.\nc\r2008 RASarXiv:1508.07830v1  [astro-ph.HE]  31 Aug 20152 M. Lemoine\ntypical of those shock waves – provided it survives collisonless\ndamping downstream of the shock front (Gruzinov & Waxman\n1999). Recent analyses of the damping of this Weibel-type micro-\nturbulence yield a damping rate /k3, wherekdenotes a turbulent\nwavenumber, indicating that small scales are dissipated first, but\nthat large scales may survive longer (Chang et al. 2008, Lemoine\n2015). In turn, this implies that the turbulence strength, or \u000fB,\nshould decay as a power-law in (proper) time (or distance) down-\nstream of the shock front, with an index which itself depends on\nthe (unknown) micro-turbulent power spectrum at the shock front.\nInterestingly, this time dependence of \u000fBturns out to be encoded in\nthe multiwavelength synchrotron spectrum, since electrons of dif-\nferent Lorentz factors cool at different times, hence in regions of\ndifferent magnetic field strengths (Rossi & Rees 2003, Derishev\n2007, Lemoine 2013).\nAlthough one cannot exclude that other external instabilities\nwould pollute the blast with magnetized turbulence, it is tempt-\ning to consider that the generation of micro-turbulence could be\nresponsible at the same time for the formation of the shock, for\nthe acceleration of particles and for the radiation of these parti-\ncles. It is furthermore tempting to follow this thread to use the\nmultiwavelength spectrum of gamma-ray bursts as a tomograph of\nthe magnetized turbulence. As a matter of fact, the recent detec-\ntions of extended emission of gamma-ray bursts in the >100MeV\nband by the Fermi-LAT instrument do point to a net decay of the\nmicro-turbulence behind the shock front (Lemoine et al. 2013),\nwith\u000fB/(t!pi=100)\u000btand\u00000:5.\u000bt.\u00000:4. This argu-\nment can be recapped as follows: if the accelerated particles scatter\nin a micro-turbulence, the maximal synchrotron photon energy is\nlimited to a few GeV at an observer time of 100s (Kirk & Re-\nville 2010, Plotnikov et al. 2013, Wang et al. 2013, Sironi et al.\n2013); this maximal energy scales as the square root of \u000fB+, i.e.\nthe magnitude of the turbulence in the vicinity of the shock front,\nand the above value assumes \u000fB+= 0:01; therefore, the inter-\npretation of this extended >100MeV emission as a synchrotron\nprocess points to the existence of a strong micro-turbulence close\nto the shock front; on the other hand, multiwavelength fits of the af-\nterglows for these Fermi-LAT gamma-ray bursts indicate that low-\nenergy photons are produced in regions of rather low \u000fB\u0000, of the\norder of 10\u00006\u000010\u00005; this discrepancy between \u000fB\u0000and\u000fB+is\nnaturally interpreted as the decay of the micro-turbulence through\ncollisionless damping. As discussed in Lemoine et al. (2013), the\ndecay rate\u000bt\u0018 \u0000 0:5 to\u00000:4further matches the results of a\ndetailed numerical experiment reported in Keshet et al. (2009).\nThe decay of micro-turbulence has also been proposed as\na possible solution to the abnormal spectral indices observed in\nthe prompt emission phase (Derishev 2007), but admittedly, the\nphysics of mildly relativistic shock waves may well differ from that\nof ultra-relativistic shock waves; in particular, the extended precur-\nsor size in mildly relativistic shocks opens the way to other insta-\nbilities operating on larger length scales. Although the present con-\nsiderations can also be generalized to the case of internal shocks,\nall of the discussion that follows will focus on the external ultra-\nrelativistic shock front.\nAn interesting consequence of a decaying micro-turbulence,\nor more generally, of a low average value of \u000fBas measured in the\nFermi-LAT and other bursts (e.g. Kumar & Barniol-Duran 2009,\n2010, He et al. 2011, Santana et al. 2014, Barniol-Duran 2014), is\na large Compton parameter Y, at least if one omits the influence\nof Klein-Nishina effects, which actually depend on electron en-\nergy hence on observed frequency, see below. Recall indeed that\nin the standard model, assuming that electrons cool through in-verse Compton interactions on their synchrotron spectrum in the\nThomson limit (i.e. neglecting Klein-Nishina – KN – effects),\nY\u0018(\u0011\u000fe=\u000fB)1=2atY\u001d1, with\u0011'min\u0002\n1;(\rc=\rm)2\u0000p\u0003\nthe cooling efficiency of electrons, \rcdenoting the Lorentz factor\nof electrons which cool on a dynamical timescale, \rmthe minimum\nLorentz factor of the injected electron power-law and \u0000pthe index\nof this power-law, e.g. Panaitescu & Kumar (2000), Sari & Esin\n(2001), Piran (2004). This Yparameter also reflects the power of\nthe inverse Compton component relatively to the synchrotron com-\nponent, therefore the ratio of emission at very high energies to that\nin the X-ray range. The possibility of a large Yparameter should\nthus increase the chances of observating gamma-ray bursts at very\nhigh energies with upcoming ˇCerenkov telescopes. The observa-\ntion of this inverse Compton component would then open a new\nspectral domain with which one could study the microphysics of\nthe turbulence in the blast.\nHowever, the computation of the synchrotron-self Compton\n(SSC) spectrum of the blast in the large Yregime is not triv-\nial because the shape of the synchrotron spectrum influences the\ncooling history of the electrons, which itself determines the syn-\nchrotron spectral shape. In particular, the KN suppression of the\ninverse Compton cross-section may itself modify this cooling his-\ntory, hence modify the very shape of the synchrotron spectrum, see\nNakar et al. (2009), Wang et al. (2010); see also Barniol Duran &\nKumar (2011) for the particular case of GRB090902B and Bo ˇsjnak\net al. (2009), Daigne et al. (2011) for related discussions in the con-\ntext of gamma-ray burst prompt emission. In the case of decaying\nmicro-turbulence, this issue is more acute, because the synchrotron\npower\u0017F\u0017may be rising with frequency, due to the fact that lower\nfrequency photons are emitted by electrons of longer cooling time,\nin regions of lower magnetic field strength (Lemoine 2013). There-\nfore, the Lorentz factor-dependent KN limit determines the radia-\ntion intensity on which an electron can cool. Finally, so far the in-\nfluence of decaying micro-turbulence on the synchrotron spectrum\nhas been studied for a fixed Yparameter, independent of electron\nLorentz factor, hence the general shape of the SSC spectrum is not\nknown in this physically relevant case.\nThese considerations motivate the present study, which calcu-\nlates the SSC spectrum for a relativistic blast wave in the large Y\nregime. Although the prime objective is to understand how the de-\ncaying micro-turbulence affects the SSC spectrum, the present dis-\ncussion also addresses the case of a uniform, low value of \u000fB. The\nresults are applied to the case of the GRB afterglows, by calculating\nthe spectrum at various observer time and by plotting, in particular,\nthe spectral slopes in various frequency windows. The spectrum is\nalso evaluated in the multi-GeV range, with a proper account of\nsynchrotron and SSC contributions, to make clear predictions for\nfuture ˇCerenkov observatories such as HAWK and CTA.\nThis paper is organized as follows: Sec. 2 presents an analyti-\ncal description of the spectrum in the slow-cooling limit, account-\ning for KN effects, and introduces a fast algorithm to compute this\nspectrum in the fast cooling regime; Sec. 3 then computes the light\ncurves and plots the temporal and spectral indices \u000band\f, com-\nmonly defined by F\u0017/t\u0000\u000b\nobs\u0017\u0000\f, and gives predictions for the\nvery high energy photon flux; finally, Sec. 4 summarizes these re-\nsults and provides some conclusions.\nc\r2008 RAS, MNRAS 000, 000–000SSC spectra at large Y 3\n2 SSC SPECTRUM\n2.1 Set-up\nThe set-up is as follows: electrons are swept-up by a relativistic\nshock front propagating in a density profile n/r\u0000k, then acceler-\nated on a short time-scale to a power-law dNe;0=d\r/\r\u0000p\u0002(\r\u0000\n\rm)above a minimal Lorentz factor \rm. The minimal Lorentz fac-\ntor is defined as usual by \rm\u0011\u000fe\u0000bmp=me(p\u00002)=(p\u00001),\nwith\u000fe\u00180:1. Cooling takes place on much longer timescales.\nWhen describing the cooling history, the total inverse Compton\ncross-section is modelled as a top-hat, with \u001b=\u001bTfor\u0017 <~\u0017(\r)\nand zero otherwise, ~\u0017(\r)denoting the frequency of photons with\nwhich electrons of Lorentz factor \rinteract at the KN limit (Nakar\net al. 2009, Wang et al. 2010), i.e.\n~\u0017\u0011\u0000bmec2\nh\r(1 +z): (1)\nAll frequencies are written in the observer rest frame; as mentioned\nabove, \u0000bdenotes the Lorentz factor of the blast in the source rest\nframe. The Compton parameter can then be approximated as\nY(\r)'Urad[\u0017 <~\u0017(\r)]\nUB(\r); (2)\nwhere\nUrad[<~\u0017(\r)] =Z~\u0017(\r)\n0d\u0017U\u0017(\u0017) (3)\nrepresents the comoving radiation energy density at frequencies\n\u0017 < ~\u0017(\r), on which the electron of Lorentz factor \rcan cool. Of\ncourse, if ~\u0017 >\u0017 peak at which the differential energy density U\u0017(\u0017)\nreaches its maximum, then Urad[\u0017 <~\u0017(\r)]\u0018\u0017peakU\u0017peak\u0018\nUradand it does not depend on \r(Uraddenotes here the total co-\nmoving energy density). In principle, Uradincludes all form of ra-\ndiation, synchrotron and inverse Compton alike; however, multi-\nple Compton scattering can be neglected for standard gamma-ray\nbursts parameters, hence Uradis hereafter determined with the syn-\nchrotron spectrum only.\nThe quantity UB(\r)represents the energy density contained\nin the magnetic field and it depends on \rif the turbulence de-\ncays in (proper) time behind the shock, because electrons of dif-\nferent Lorentz factors then cool in regions of different magnetic\nfield strengths. In this work, the decay law of the turbulence takes\nthe power-law form \u000fB(t)/t\u000btfar from the shock front and\n\u000fB\u0018\u000fB+= 0:01close to the shock front. How far is ex-\npressed in terms of the proper time tsince the plasma element was\ninjected through the shock, i.e. t\u0011x=\f din terms of the down-\nstream (comoving) distance to the shock front xand\fd, the shock\nfront velocity relative to the downstream rest frame. According to\nPIC simulations, the characteristic (temporal) scale \u0001separating\nfar from close to the shock front is \u0001\u0018102\u00003!pi, see Chang et\nal. (2008), Keshet et al. (2009) for simulations, as well as Lemoine\n(2013), Lemoine (2015) for discussions of this issue; for reference,\n!\u00001\npi\u00187:5\u000210\u00004n\u00001=2\n0 s for a relativistic blast wave propa-\ngating in a medium of proton density n0cm\u00003. The uncertainty\non the value of \u0001does not really influence the results presented\nbelow, because it can be embedded in that associated to the de-\ncay power-law exponent \u000bt. Recent work mentioned above suggest\n\u000bt\u0018 \u0000 0:5!\u0000 0:4for a decay law \u000fB\u0018\u000fB+(t!pi=100)\u000bt,\nhence the following adopts \u0001 = 100!\u00001\npiand\u000bt=\u00000:4. This\ndecay implies that the minimum magnetic field in the blast, close to\nthe contact discontinuity, is characterized by the equipartition frac-\ntion\u000fB\u0000=\u000fB+(tdyn=\u0001)\u000btin terms of the dynamical timescaletdyn=r=(\u0000bc). Typical values are tdyn\u0018105\u0000106s at an ob-\nserver time of 104s (e.g. \u0000b\u001820\u000030andr\u00181017\u00001018cm),\nwith a mild dependence on the model parameters, leading to val-\nues of the order of \u000fB\u0000\u001810\u00005for\u000bt\u0018 \u0000 0:5. Note that\ntdyn=\u0001/t(5\u00002k)=(8\u00002k)\nobs evolves slowly as a function of observer\ntime, hence so does \u000fB\u0000.\nIn this work, it is assumed that a particle emits its synchrotron\nradiation at the location at which it cools, if it cools on a dynam-\nical timescale; in the opposite limit, it is assumed that it emits its\nsynchrotron photons in the magnetic field close to the contact dis-\ncontinuity, of strength \u000eB\u0000(associated to the parameter \u000fB\u0000). The\njustification of this approximation is as follows: the energy emitted\nin synchrotron photons by a particle up to time t, along its trajec-\ntory downstream of the shock, can be written as\nEsyn=1\n6\u0019\u001bTcZt\n0d\u001c\u000eB2(\u001c)\r2\ne(\u001c)\f2\ne(\u001c) (4)\nas a function of the time dependent Lorentz factor \re(\u001c)of the\nparticle along its cooling trajectory, with \fe(\u001c)\u00181the particle\nvelocity in units of c. One can then show that Esynis dominated\nby the contribution at t\u0018tcool(\r), where\rdenotes the initial\nvalue\re(0), as follows. At early times \u001c\u001ctcool(\r),\re(\u001c)\u0018\n\rand the integrand scales as UB(\u001c)/\u001c\u000bt, hence the integral\nis dominated by the large time behavior if \u000bt>\u00001, which is\nindeed an explicit assumption of the present work. At late times\n\u001c\u001dtcool(\r), one can obtain an upper bound on how fast \re(\u001c)\ndecreases by considering synchrotron losses only, i.e. by neglecting\ninverse Compton losses. The integration of\nd\re\nd\u001c=\u00004\n3\u001bTcUB(\u001c)\nmec2\r2\ne(\u001c)\fe(\u001c)2(5)\nleads to\re(\u001c)\u0018\r[\u001c=tcool(\r)]\u00001\u0000\u000btfor\u001c\u001dtcool(\r).\nTherefore, the integrand in Eq. (4) behaves as \u001c\u00002\u0000\u000btfor\u001c >\ntcool(\r), henceEsynis indeed dominated by the contribution at\nt\u0018tcool(\r)provided\u000bt>\u00001.\nThis thus supports the above approximation that a particle\nwith initial Lorentz factor \r > \r cemits its synchrotron radia-\ntion on a magnetic field of strength \u000eB[tcool(\r)], withtcool(\r)the\ncooling time of the particle. Of course, if tcool(\r)> t dyn, the\nparticle does not actually cool, and Esynis then dominated by the\nupper bound t\u0018tdyn, i.e. particles radiate synchrotron radiation\nin the relaxed magnetic field at the back of the blast, of strength\n\u000eB\u0000.\nSincetcool(\rc) =tdynby definition, and since \u000eB(tdyn) =\n\u000eB\u0000, also by definition,\nUB(\r)'UB\u0000max\u0014\n1;\u0012tcool(\r)\ntdyn\u0013\u000bt\u0015\n(6)\nwithUB\u0000\u0011\u000eB2\n\u0000=(8\u0019) =UB(\rc).\nIn line with the above discussion, electrons of Lorentz factor\n\rradiate their energy through synchrotron at a typical frequency\n\u0017syn(\r)/\u000eB[tcool(\r)]\r2'\u0017c\u0014tcool(\r)\ntdyn\u0015\u000bt=2\u0012\r\n\rc\u00132\n:\n(7)\nHere,\u0017c\u0011\u0017syn(\rc)denotes the synchrotron peak frequency for\nparticles of Lorentz factor \rc.\nFinally, the cooling timescale can be written\ntcool(\r)'tdyn1 +Yc\n1 +Y(\r)UB\u0000\nUB(\r)\rc\n\r;\n'tdyn\u00141 +Y(\r)\n1 +Yc\u0015\u00001=(1+\u000bt)\u0012\r\n\rc\u0013\u00001=(1+\u000bt)\n:(8)\nc\r2008 RAS, MNRAS 000, 000–0004 M. Lemoine\nThe second equality is obtained by replacing UB(\r)=UB\u0000with its\nvalue given in Eq. (6), assuming \r > \r c.\nEquations (2), (6), (7) and (8) then allow to derive the follow-\ning scalings for \r > \r c:\nUB(\r)\nUB\u0000'\u00141 +Y(\r)\n1 +Yc\u0015\u0000\u000bt=(1+\u000bt)\u0012\r\n\rc\u0013\u0000\u000bt=(1+\u000bt)\n;(9)\n\u0017syn(\r)\n\u0017c'\u00141 +Y(\r)\n1 +Yc\u0015\u0000\u000bt=[2(1+\u000bt)]\u0012\r\n\rc\u00132\u0000\u000bt=[2(1+\u000bt)]\n;\n(10)\nY(\r) [1 +Y(\r)]\u0000\u000bt=(1+\u000bt)\nYc(1 +Yc)\u0000\u000bt=(1+\u000bt)'Urad(\u0017 <~\u0017)\nUrad(\u0017 <~\u0017c)\u0012\r\n\rc\u0013\u000bt=[(1+\u000bt)]\n:\n(11)\nYcstands forY(\rc).\nOnce the ratio Urad(\u0017 <~\u0017)=Urad(\u0017 <~\u0017c)– which captures\nKN effects – has been specified, it is possible to derive the scalings\nofY(\r)as a function of \r, hence of\u0017syn(\r). One should empha-\nsize that in the above expressions, \rrepresents the initial Lorentz\nfactor of the particle, after acceleration has shaped the power-law,\nbut before cooling has effectively taken place.\n2.2 Synchrotron spectrum\nStandard calculations of the synchrotron spectrum of a blast wave\ngenerally derive the stationary electron distribution in the blast, by\nsolving a transport equation in momentum space, averaged over the\ndepth of the blast, accounting for injection of the power-law at the\nshock and for cooling downstream (e.g. Sari et al. 1998, Sari & Esin\n2001); this yields a standard broken power-law shape dNe=d\re\nwith indices\u00002for\rc< \r < \r mor\u0000pfor\rm< \r < \r c\nand\u0000p\u00001formax(\rc;\rm)< \r . However, one can also de-\nrive the synchrotron spectrum by a direct mapping of the energy\ninitially stored into the electron population to that radiated in syn-\nchrotron; this approach is simpler in the present case and it works\nas follows.\nElectrons injected with Lorentz factor \r > max(\rm;\rc)\nemit a fraction [1 +Y(\r)]\u00001of their energy in synchrotron ra-\ndiation; the energy density contained in such electrons (within\nan interval d ln\r) is itself a fraction (p\u00002) (\r=\rm)2\u0000pd ln\r\nof the energy density Uecontained in electrons immediately be-\nhind the shock front, before cooling has started to take place;\nUe=\u000fe4\u00002\nbnmpc2. Consequently, the synchrotron flux received\nat frequency \u0017=\u0017syn(\r)can be written:\n\u0017F\u0017;syn'1\n4\u0019D2\nL4\n3\u00002\nb4\u0019r2c(p\u00002)Ue\n1 +Y(\r)\u0012\r\n\rm\u00132\u0000pd ln\r\nd ln\u0017\n(12)\nThe above assumes p > 2, but it can be generalized to any in-\ndexp > 1; indeed, the above picture assumes that particles with\n\r > max(\rm;\rc)radiate their energy at \u0017syn(\r)then do not con-\ntribute anymore to the synchrotron spectrum, which is a reasonable\napproximation if p > 1; in contrast, if p < 1, the radiation of the\nhigh energy particles during their complete cooling history domi-\nnates that of the lower energy ones.\nSimilarly, one can write down the flux for \rm> \r > \r cor\n\rc> \r > \r m, whichever occurs, as follows:\n\u0017F\u0017;syn'\u0017mF\u0017m;syn8\n>><\n>>:\r\n\rm1 +Ym\n1 +Y(\rm> \r > \r c)\n\u0012\u0017\n\u0017m\u0013(3\u0000p)=2\n(\rc> \r > \r m)\n(13)withYm\u0011Y(\rm)and\u0017m=\u0017syn(\rm). The spectrum at\n\u0017 < \u0017 cis indeed unchanged with respect to the standard case,\nalthough the value of UB\u0000must be used to compute the character-\nistic frequencies. Regarding the fast cooling limit \rc< \r < \r m,\nthe factor can be understood by noting that all injected particles\nwith\r > \r mshift from\rto\rcduring their cooling history, and\nthat at each point along this cooling trajectory they radiate (in syn-\nchrotron) a fraction (\r=\rm) (1 +Ym)=(1 +Y)of the energy ra-\ndiated (in synchrotron) by a particle of Lorentz factor \rm. Note\nalso that in the limits \u000bt!0,Y(\r)\u001c1, one recovers the usual\nscaling\u0017F\u0017;syn/\u00171=2since\u0017/\r2.\nProvided\rcandYcare given, one can derive \u0017syn(\r)and\nY(\r), hence\r(\u0017)andY(\u0017)by using Eqs. (9), (10) and (11),\nthen\u0017F\u0017;synusing Eqs. (12) and (13). Consider as an example\nthe caseY\u001d1, omitting KN effects, i.e. ~\u0017!+1for all\n\u0017: for\r > \r c, one derives Y(\r)/\r\u000btfrom Eq. (11), hence\n\u0017p/\r2\u0000\u000bt=2from Eq. (10), hence \u0017F\u0017;syn/\r2\u0000p=(1+Y)/\n\u0017(2\u0000p\u0000\u000bt)=(2\u0000\u000bt=2)from Eq. (12). This latter scaling matches the\nexplicit calculation of Lemoine (2013), which computes the inte-\ngrated cooling history of the electron power-law in the decaying\nmagnetic field. This example also confirms that \u0017F\u0017;synrises with\n\u0017if\u000bt.2\u0000p, hence KN effects should not be omitted in an\naccurate calculation of \u0017F\u0017;syn.\nThe above provides the tools needed to calculate the syn-\nchrotron spectrum; explicit calculations in both the slow and the\nfast cooling regimes are provided in the following sections.\n2.3 Inverse Compton spectrum\nIn order to calculate the inverse Compton emissivity, the syn-\nchrotron spectral density F\u0017;synmust be folded over the Compton\ncross-section and the cooled particle distribution, integrated over\nthe blast. The following derivation of the cooled distribution func-\ntion relies on the observations that dtcool(\r)=d\r < 0and that the\ncooling history \rcool(t)at timest > t 0does not depend on the\nhistory at times t < t 0, i.e. electrons of various Lorentz factor\nfollow a universal cooling trajectory \rcool(t).\nOne writes d_Ne;0the (comoving) rate at which electrons are\nswept-up and accelerated into a power-law in a Lorentz factor in-\nterval d\r. One also defines dNe, which represents the total number\nof electrons in the blast in a Lorentz factor interval d\re; note the\ndifference of notation: \rerefers to a value of the Lorentz factor of\nthe blast averaged distribution, while \rcorresponds to the Lorentz\nfactor of the injection distribution, i.e. before cooling has occurred.\nThe injection of particles with Lorentz factor \rpopulates the\ndownstream with particles of Lorentz factor \re=\rover a frac-\ntiontcool=tdynof the depth of the blast. In contrast, particles in-\njected with\rc> \r > \r mretain their Lorentz factor (no cooling)\nand populate the whole blast. Therefore, one can write the average\ndistribution for \re> \r mas:\ndNe\nd\re'd_Ne;0\nd\r(\r=\re) min [tdyn;tcool(\r)] (\r >\r m):\n(14)\nNote also that tdynsets the scale for adiabatic losses, while tcool\nsets the scale for radiative losses.\nIn the fast cooling regime, all electrons injected with Lorentz\nfactor\r > \r mcool down to \rc, following a same cooling history\nc\r2008 RAS, MNRAS 000, 000–000SSC spectra at large Y 5\n\rcool(\u001c), therefore\ndNe\nd\re'Ztdyn\n0d\u001cZ+1\n\rmd\rd_Ne;0\nd\r\u000e[\re\u0000\rcool(\u001c)]\n' _Ne;0tcool(\re)\n\re(15)\nwhere _Ne;0=R+1\n\rmd\rd_Ne;0=d\rrepresents the total injection\nrate of electrons. The last expression in Eq. (15) follows from the\ndefinitiontcool(\re) =\rejd\re=dtj\u00001after interchanging the inte-\ngrals.\nUsing the scaling of tcool(\r), one then derives\ndNe\nd\re'_Ne;0tdyn\n\rm\u0012\re\n\rm\u0013\u0000p\n(\rm<\re<\rc)\ndNe\nd\re'_Ne;0tdyn\n\rm\u0012\re\n\rm\u0013\u00001\u0012\re\n\rc\u0013\u00001=(1+\u000bt)\n\u0002\u00141 +Y(\re)\n1 +Yc\u0015\u00001=(1+\u000bt)\n(\rc<\re<\rm)\ndNe\nd\re'_Ne;0tdyn\n\rm\u0012\re\n\rm\u0013\u0000p\u0012\re\n\rc\u0013\u00001=(1+\u000bt)\n\u0002\u00141 +Y(\re)\n1 +Yc\u0015\u00001=(1+\u000bt)\n[max(\rc;\rm)<\re]\n(16)\nIn the limit \u000bt! 0, one recovers dNe=d\re/\n[1 +Y(\re)]\u00001\r\u00002\nefor\rc< \re< \r mordNe=d\re/\n[1 +Y(\re)]\u00001\r\u0000p\u00001for\rm< \r c< \re; both match the ex-\npressions derived in Nakar et al. (2009) and in Wang et al. (2010)\nfor a homogeneous (non-decaying) turbulence.\nIt is also instructive to show that the above average electron\ndistribution, when associated with the appropriate (i.e. Lorentz fac-\ntor dependent) magnetic field, leads to the same scalings for the\nsynchrotron spectrum as Eqs. (12) and (13). Consider for instance\nthe regime\r > max (\rc;\rm): the average synchrotron power\nof the blast scales as \u0017F\u0017;syn/UB(\re)\r2\nedNe=d ln\re; using\nthe scaling for UB(\re)given in Eq. (9), one recovers \u0017F\u0017;syn/\n[1 +Y(\re)]\u00001\r2\u0000p\ne indicated in Eq. (12). One can proceed simi-\nlarly for the other two regimes, noting in particular that for \rm<\n\re< \r c(slow cooling), UB=UB\u0000and it no longer depends on\n\re.\nFinally, using the above electron distribution, one can compute\nthe inverse Compton component using standard formulae:\nF\u0017;IC(\u0017IC)'\u001cICZ\nd\re1\nNedNe\nd\reZ1\n0dq(1\u0000u)gKN(q)\n\u0002F\u0017;syn\u0014\u0017IC\n4\r2q(1\u0000u)\u0015\n\u0002 (1\u0000u);\n(17)\nwithu\u0011(1 +z)h\u0017IC=\u0000\n\u0000b\remec2\u0001\n; the function gKN(q) =\u0002\n2qlnq+ (1 + 2q)(1\u0000q) +G2(1\u0000q)=[2(1 +G)]\u0003\n, withG\u0011\nu=(1\u0000u)characterizes the energy dependence of the Klein-\nNishina cross-section, see Blumenthal & Gould (1970) for details.\nThe prefactor \u001cICdefines the optical depth to Compton scat-\ntering; an explicit calculation leads to\n\u001cIC= 3\u001bTNe\n4\u0019r2= 12\u001bT\u0000bnctdyn (18)\nNote that this expression uses Ne= 4\u0019r2ctdyn4\u0000bn, i.e. it only\nincludes the electrons that have been swept up in the last dynamicaltimescale, because electrons injected at earlier times have been adi-\nabatically cooled to Lorentz factors < \r cand therefore do not par-\nticipate in shaping the inverse Compton spectrum. With the above\nvalue of\u001cIC, one can verify that the energy density of the radi-\nation associated to the inverse Compton component is correctly\nnormalized to Yctimes that contained in the synchrotron com-\nponent, as can be verified by calculatingR\nd\u0017ICF\u0017IC;ICin terms\nofR\nd\u0017synF\u0017syn;syn(neglecting the influence of Klein-Nishina ef-\nfects).\n2.4 Slow cooling\n2.4.1 General procedure\nAs discussed in detail in Nakar et al. (2009) and Wang et al. (2010),\nthe cooling Lorentz factors and Compton parameters \rcandYccan\nbe obtained in the slow cooling regime \rm< \r cfrom the system\nYc(1 +Yc)'\u000fe\n\u000fB\u0000\u0012\rc\n\rm\u00132\u0000pUsyn(<~\u0017c)\nUsyn(<\u0017c)\n\rc'\rc;syn\n1 +Yc(19)\nwith\rc;syn\u0011(3=4)mec=(\u001bTUB\u0000tdyn)the cooling Lorentz fac-\ntor in the relaxed turbulence, in the absence of inverse Compton\nlosses.\nWriteCc\u0011Usyn(<~\u0017c)=Usyn(\u0017c)the term entering the first\nequation. If ~\u0017c< \u0017 c,Cc<1because the peak of the synchrotron\nflux lies at\u0017cor above (see below); in this limit, Klein-Nishina sup-\npression inhibits the cooling of \rcelectrons. Assuming that such\nelectrons cool by interacting with the \u0017min< \u0017 < \u0017 cpart of\nthe synchrotron spectrum, which is generically the case, one can\nwriteCc= (~\u0017c=\u0017c)(3\u0000p)=2/\r\u00003(3\u0000p)=2\nc and the system can be\nsolved easily.\nIn the opposite limit, ~\u0017c> \u0017 c; one may have Cc\u00181if the\npeak of the synchrotron flux is located at \u0017c, orCc&1if the peak\nlies at higher frequencies. In this latter case, Cc'(~\u0017c=\u0017c)1\u0000\f,\nwhere\fis the synchrotron spectral index defined by F\u0017;syn/\n\u0017\u0000\fin the spectral range above \u0017c, calculated thereafter. The ana-\nlytical calculation proposed here uses Eq. (27) below to derive this\n\f, but the result depends little on this choice, because in all cases\nconsidered, 1\u0000\fis small, meaning that the peak flux is not very\ndifferent from the flux \u0017F\u0017at\u0017c.\nThe next step is to determine the critical Lorentz factors b\rc\u0011\nb\r(\u0017c)andb\rm\u0011b\r(\u0017m)with the general definition (Nakar et al.\n2009):\nb\r(\u0017) =\u0000bmec2\n(1 +z)h\u0017; (20)\nwhich corresponds to the Lorentz factor for which electrons inter-\nact with photons of frequency \u0017at the onset of the Klein-Nishina\nregime. Note that \u0017mand\u0017care to be calculated in the relaxed\nturbulence of strength \u000eB\u0000in this slow cooling regime. Using\nEqs. (9), (10) and (11), one may then calculate the Compton pa-\nrametersY(b\rc)andY(b\rm). Note also that the slow cooling limit\n\rm<\rcimpliesb\rc<b\rm.\nAnother critical Lorentz factor is \r0, for whichY(\r0) = 1 . If\nY(b\rm)>1, thenb\rm<\r0, and\r0can be obtained by solving\nY(\r0) [1 +Y(\r0)]\u0000\u000bt=(1+\u000bt)\nY(b\rm) [1 +Y(b\rm)]\u0000\u000bt=(1+\u000bt)=\u0014~\u0017(\r0)\n\u0017m\u00154=3\u0012\r0\n\rm\u0013\u000bt=(1+\u000bt)\n(21)\nwhich derives from Eq. (11). This latter equation assumes that\nc\r2008 RAS, MNRAS 000, 000–0006 M. Lemoine\n~\u0017(\r0)lies in the spectral range between \u0017a(synchrotron self-\nabsorption frequency) and \u0017m, in general a very good approxi-\nmation; it can be generalized to other cases without difficulty. If\nY(b\rm)<1, then\r0<b\rm; one can repeat the above exercise\nto derive\r0, replacing the 4=3exponents with (3\u0000p)=2, which\ncharacterizes the spectral dependence of \u0017F\u0017;synbetween\u0017mand\n\u0017c.\nFor improved accuracy, one may also determine the next order\ncritical Lorentz factor bb\rc\u0011\u0000bmec2=[(1 +z)h\u0017syn(b\rc)]. Fol-\nlowing a procedure similar to the above, one can derive \u0017(bb\rc)and\nY(bb\rc).\nThe following makes use of the short-hand notation: b\u0017=\n\u0017syn(b\r)for various critical Lorentz factors; similarly, \u0017syn(\r0)is\nwritten\u00170for commodity.\nOne can derive the power-law 1\u0000\f0index of\u0017F\u0017;synas fol-\nlows. Below \u0017c, the scaling remains unchanged compared to the\nstandard case because the cooling history does not influence the\nsynchrotron spectrum, and the results will not be repeated here.\nAbove\u0017c, the spectral slope is determined by the scalings of Y(\r),\n\u0017(\r)and\u0017F\u0017;syn/(1 +Y)\u00001\r2\u0000p[Eq. (12)]. In particular, if\n~\u0017(\r)lies in a spectral domain in which \u0017F\u0017;syn/\u00171\u0000\f, below\nthe peak of the synchrotron energy flux, then electrons of Lorentz\nfactor\rcool by inverse Compton interactions with that portion of\nthe synchrotron spectrum if Y(\r)\u001d1, in which case Eqs. (9),\n(10) and (11) lead to\nY/\r\u000bt\u0000(1\u0000\f)(1+\u000bt)\n\u0017/\r2\u0000\f\u000bt=2\ntcool/\r\u0000\f(22)\nAssuming\u0017F\u0017;syn/\u00171\u0000\f0in the range of interest around \u0017(\r),\nand using the above scalings, one obtains directly\n1\u0000\f0=2\u0000p\u0000\u000bt+ (1\u0000\f)(1 +\u000bt)\n2\u0000\u000bt=2 + (1\u0000\f)\u000bt=2: (23)\nThe limit\f!1recovers the case in which ~\u0017lies above the peak\nof the synchrotron flux, discussed previously.\n2.4.2 Power-law segments\nAs mentioned above, the synchrotron spectrum at \u0017 < \u0017 cre-\nmains unaffected and it is not discussed here. At the upper end\nof the spectral range, i.e. \u00170< \u0017 ,Y(\r)<1because of the\nKlein-Nishina suppression of electron cooling, which implies that\n\u0017F\u0017;syn/\u00171\u0000\f0with\n1\u0000\f0=2\u0000p\n2\u0000\u000bt=[2(1 +\u000bt)][\u00170< \u0017] (24)\nas can be derived from Eq. (12) with 1 +Y'1. This scaling also\nmatches that derived by a full computation of the electron cooling\nhistory with negligible inverse Compton losses in Lemoine (2013).\nFor\u000bt= 0, one recovers of course the standard fast cooling index\n\f0=p=2.\nIn the rangeb\u0017m< \u0017 < \u0017 0, if it exists, the synchrotron\nspectrum is shaped by electrons with Lorentz factor \rsuch that\nb\rm< \r < \r 0, which thus cool by interacting with photons in the\nrange\u0017 < \u0017 m; one can therefore use Eq. (23) with 1\u0000\f= 4=3,\nso that\n1\u0000\f0'5\u00003p=2 +\u000bt=2\n3 +\u000bt=4(b\u0017m< \u0017 < \u0017 0) (25)\nThe limit\u000bt!0gives\f0=\u00002=3 +p=2, which fits the results\nof Nakar et al. (2009) in this frequency range.In the range max (\u0017c;b\u0017c)< \u0017 < min (b\u0017m;\u00170), the Lorentz\nfactor of electrons shaping that part of the spectrum satisfies\nmax (\rc;b\rc)< \r < min (b\rm;\r0), hence one can use Eq. (23)\nwith1\u0000\f= (3\u0000p)=2, leading to\n1\u0000\f0'7\u00003p+ (1\u0000p)\u000bt\n4 + (1\u0000p)\u000bt=2\n[max (\u0017c;b\u0017c)< \u0017 < min (b\u0017m;\u00170)] (26)\nNote thatb\u0017c< \u0017 cis equivalent to ~\u0017c< \u0017 c. Here as well, one\nrecovers the index \f0=\u00003=4 + 3p=4derived in Nakar et al.\n(2009) in the limit \u000bt!0.\nIf~\u0017c< \u0017 c, meaningb\rc< \r c, the above completes the\ndescription of the spectrum. If, however, \u0017c<~\u0017c, one needs to\ndescribe the intermediate range \u0017c< \u0017 <b\u0017c. This range can\nactually be decomposed into two sub-ranges, as follows.\nForbb\u0017c< \u0017 <b\u0017c, the corresponding Lorentz factor verifies\nbb\rc< \r <b\rc, hence\u0017c<~\u0017 <b\u0017c. Consequently, the particle\ncools on the spectral range of \u0017F\u0017;synthat it contributes to through\nsynchrotron emission, so that one can use Eq. (23) with \f=\f0,\ngiving\n1\u0000\f0'\u00002 + 3\u000bt+\u0002\n4 + (4\u00008p)\u000bt+\u000b2\nt\u00031=2\n2\u000bt\u0010\nbb\u0017c< \u0017 < \u0017 c\u0011\n(27)\nand1\u0000\f0'0:12for\u000bt=\u00000:5,p= 2:3, i.e. a slowly rising\n\u0017F\u0017;syn.\nFinally, in the remaining range \u0017c< \u0017 <bb\u0017c, the particle\ncools on photons of frequency in the range b\u0017c<~\u0017 < ~\u0017c, for\nwhich 1\u0000\fis given by Eq. (26) above. This leads to\n1\u0000\f0'15\u00007p+\u000bt(1\u0000p) (10\u0000p+\u000bt)=2\n8 +\u000bt(1\u0000p)(10 +\u000bt)=4(28)\nOne finds 1\u0000\f0'0:09for\u000bt=\u00000:5andp= 2:3.\n2.4.3 Inverse Compton component\nIn principle, one can derive analytical approximations to the inverse\nCompton component using the above broken power-law approxi-\nmations, folded over the particle distribution, as in Eq. (17) above.\nHowever, this appears rather intricate, given the number of poten-\ntial power-law segments, in regards of the quality of the approxima-\ntion that one can obtain; indeed, as discussed in detail in Sari & Esin\n(2001), folding over the distribution function generally introduces\nlogarithmic departures, which smooth out the power-law segments\nand breaks. One can nevertheless describe the general features of\nthis inverse Compton component in the slow cooling regime as fol-\nlows.\nBelow\u0017IC;m= 2\r2\nm\u0017m, the slope is that of \u0017F\u0017;synbelow\n\u0017m, i.e. 4/3. For \u0017IC;m< \u0017 < \u0017 IC;cwith\u0017IC;c= 2\r2\nc\u0017c, the\nslope of\u0017F\u0017;ICis(3\u0000p)=2, reflecting that of \u0017F\u0017;synin the corre-\nsponding range. Above \u0017IC;c, the inverse Compton component re-\nflects, up to the afore-mentioned logarithmic corrections, the slope\nof\u0017F\u0017;synabove\u0017c, which is generally close to flat or gently ris-\ning, see above.\nConsequently, if the synchrotron spectrum has an extended\nrange above \u0017cwhere it is close to flat (i.e. \f\u00181), then one\nmight have a close to flat inverse Compton component, at least up\nto the cut-off frequency defined as\n\u0017IC;KN=\r2\nc~\u0017c (29)\nc\r2008 RAS, MNRAS 000, 000–000SSC spectra at large Y 7\ncorresponding the boosting of ~\u0017cphotons at the onset of the Klein-\nNishina regime by \rcelectrons. One can define another cut-off fre-\nquency, as follows:\n\u0017IC;bc=b\r2\nc\u0017c (30)\nwhich corresponds to the boosting of \u0017cphotons by electrons of\nLorentz factor b\rc, at the onset of the Klein-Nishina regime. The\nratio\u0017IC;bc=\u0017IC;KNcan be written as b\rc=\rc, hence the ordering of\none with respect to the other depends on whether \r <b\rcor not.\nIn any case, the actual cut-off occurs at \u0017IC;KN, while the pres-\nence of\u0017IC;bcmay lead to a feature (e.g. softening) in the inverse\nCompton spectrum. This can be understood by noting that in the\npresent case, the synchrotron (energy) flux generically peaks above\n\u0017c, while the particle distribution function falls steeply beyond \rc,\nhence the peak of the inverse Compton component is determined\nby the boosting of ~\u0017cphotons by electrons of Lorentz factor \rc.\n2.4.4 Comparison to numerical calculations\nThe above analytical broken power-law model of the synchrotron\nspectrum is compared to a full numerical calculation (with the algo-\nrithm described in Sec. 2.5 thereafter) in Fig. 1, in two different rep-\nresentative cases: upper panel, observer time tobs= 104s, blast\nenergyE= 1053ergs, external density n= 0:01cm\u00003; lower\npanel,tobs= 3\u0002104s,E= 1054ergs,n= 1035r\u00002cm\u00003\n(wind profile with shock radius rexpressed in cm); for both,\n\u000fe= 0:1,p= 2:3,\u000fB+= 0:01and\u000bt=\u00000:4, assuming\n\u000fB=\u000fB+\u0002\nt=(100!\u00001\npi)\u0003\u000bt. For a decelerating adiabatic Bland-\nford & McKee (1976) solution, the value of the blast Lorentz fac-\ntor at these observer times are \u0000b'33for the upper panel and\n\u0000b'29in the lower panel. For the above decay law of the mag-\nnetic field, one finds for the first scenario \u000fB\u0000= 3:2\u000210\u00005\n(tdyn= 1:3\u0002106s,!\u00001\npi= 7:6\u000210\u00003s), and in the sec-\nond scenario \u000fB\u0000= 2:1\u000210\u00005(tdyn= 1:7\u0002106s,!\u00001\npi=\n3:5\u000210\u00003s).\nThe critical frequencies are indicated with arrows. The thick\nsolid line corresponds to the numerical calculation (synchrotron in\nblue, inverse Compton component in orange) while the dashed line\nshows the analytical estimates, which clearly provides a faithful\nmatch in both cases.\nIn the first scenario (upper panel), b\rc'60while\rc'107:\nKlein-Nishina effects are therefore particularly strong; there are ac-\ntually so strong that \u00170< \u0017 c, which means that the Klein-Nishina\nsuppression of electron cooling reduces the Compton parameter\nto below unity at \rc. Consequently, the dependence of Yon\r\ndoes not affect the spectrum above \u0017c. Noting that in this region\n\u0017F\u0017/\r2\u0000p[Eq. (12)] and \u0017/\r2\u0000\u000bt=2(1+\u000bt)[Eq. (10)], one\nfinds\u0017F\u0017/\u00171\u0000\fwith1\u0000\f= (2\u0000p)=[2\u0000\u000bt=2(1 +\u000bt)],\nwhich matches Eq. (24). As mentioned earlier, the spectrum re-\nmains unaffected with respect to the standard synchrotron spectrum\nbelow\u0017cbecause the electrons shaping that part of the spectrum do\nnot cool on a dynamical timescale.\nIn this first scenario, the method proposed in Eqs. (19) over-\nestimates\rcby a factor 2:5, hence\u0017cby a factor 6and\u0017IC;KN=\n~\u0017c\r2\ncby a factor 2:5as well. Taking into account this overestimate,\nthe numerical calculation indicates that the suppression of the in-\nverse Compton flux becomes noticeable at a factor \u00185below the\ntheoretical value of \u0017IC;KN, as calculated with the correct \rc. As\nanticipated, the characteristic frequency \u0017IC;bcleads to a soft soft-\nening in the inverse Compton component, but not to a cut-off.\nIn the second scenario (lower panel), \rc<b\rchence Klein-Nishina suppression of the inverse Compton cooling becomes ef-\nfective atb\u0017conly. The (analytical) synchrotron spectrum is thus\nclose to flat in the region \u0017c.\u0017.b\u0017c, as indicated by Eqs. (28),\n(27) above, then rising with 1\u0000\f'0:14corresponding to\nEq. (26) above for the range b\u0017c< \u0017 < \u0017 0, because\u00170< \u0017 m.\nAbove\u00170, Eq. (24) applies and gives the same high energy spectral\nslope as for the previous scenario.\nIn this second case, the analytical calculations underestimate\n\rcby a factor 1:8. There is nevertheless a broad satisfactory agree-\nment between the analytical synchrotron spectrum and the numeri-\ncal calculation.\n2.4.5 Comparison to non-decaying scenarios\nFigure 2 provides a numerical comparison of the spectra shown\nin the lower panel of Fig. 1 with two calculations for the same\nparameters but a homogeneous (non-decaying) turbulence: one in\nwhich\u000fB=\u000fB+= 0:01, another one in which \u000fB=\u000fB\u0000=\n2:1\u000210\u00005, which corresponds to the value of \u000fB(tdyn), i.e. close\nto the contact discontinuity, in the above decaying micro-turbulence\nmodel.\nAs expected the SSC spectrum with decaying microturbu-\nlence merges with that corresponding to uniform \u000fB\u0000at frequen-\ncies below\u0017c, since electrons of Lorentz factor \r < \r cthen cool\nin magnetized turbulences of equal strength in both models. The\nsynchrotron spectrum for decaying micro-turbulence also merges\nwith the synchrotron spectrum for uniform \u000fB+at the highest fre-\nquencies, since the cooling time for those emitting electrons be-\ncomes shorter than \u0001, hence the particles effectively cool in a\nmagnetic field characterized by \u000fB+. However, the integrated pow-\ners for these two models differ, because the cooling efficiencies\n\u0018(\rc=\rm)2\u0000pdiffer.\nIn this regard, the slow cooling synchrotron spectrum for de-\ncaying micro-turbulence is a hybrid of the spectra for uniform high\nand low\u000fB, transiting from \u000fB\u0000at low frequencies to \u000fB+at high\nfrequencies. This justifies the use of a two zone model, one with\nlow\u000fB\u0000and one for high \u000fB+, to compute an approximated spec-\ntrum in wavebands at respectively low and high frequencies.\nIn Fig. 2, the synchrotron spectra have been arbitrarily con-\ntinued at very high frequencies, albeit with a thin line, but they\nshould of course cut-off at some maximal energy where the acceler-\nation timescale becomes of the same order as the cooling timescale.\nSince this depends on acceleration physics, see the discussion in\nLemoine (2013), the lines have been turned from thick to thin at\nan ad-hoc location corresponding to a synchrotron photon energy\nof1GeV . This estimate is discussed in Plotnikov et al. (2013),\nLemoine (2013), Wang et al. (2013), Sironi et al. (2013); it depends\non the afterglow parameters and, in particular, on observer time.\nFor reference, one notes the critical frequencies:\n\u0017IC;c= 2\r2\nc\u0017c\n'7:3\u00021023HzE54\u000f\u00007=2\nB\u0000;\u00005A\u00009=2\n?;11:7t2\nobs;4:5z\u00003\n+Y\u00004\nc;2\n\u0017IC;KN =1\n1 +z\u0000b\rcmec2\n'4\u00021025HzE1=2\n54\u000f\u00001\nB\u0000;\u00005A\u00003=2\n?;11:7t1=2\nobs;4:5z\u00003=2\n+Y\u00001\nc;2\n(31)\nwith the notations z+= (1 +z)=2,Yc;2= (1 +Yc)=100and\nA?;11:7=A?=(5\u00021011g=cm2). For reference, in the scenario of\nFigs. 2 and 3, A?'0:3andYc'50for\u000fB\u0000= 2:1\u000210\u00005. Note\nthe strong dependence of \u0017IC;cand\u0017KNon the external density.\nc\r2008 RAS, MNRAS 000, 000–0008 M. Lemoine\n10-1510-1410-1310-1210-1110-1010-910-8nFn@ergcm2sHzDnm nc nc n`m n`c n0\nnIC,m nIC,c nIC,KN nIC,c`\n1010101110121013101410151016101710181019102010211022102310241025102610271028102910301031103210-1510-1410-1310-1210-1110-1010-910-8\nn@HzDnFn@ergcm2sHzDnm ncnc n`m n`c n0\nnIC,m nIC,cnIC,KN\nnIC,c`\nFigure 1. Comparison of the analytical calculation (dashed red line) of the synchrotron spectrum in the slow cooling regime to a numerical calculation (solid\nblue line), for two representative cases: upper panel, observer time tobs= 104s, blast energy E= 1053ergs, external density n= 0:01cm\u00003; lower panel,\ntobs= 3\u0002104s,E= 1054ergs,n= 1035r\u00002cm\u00003; in both cases, \u000fe= 0:1,p= 2:3and\u000fB=\u000fB+h\nt=(100!\u00001\npi)i\u00000:4\n. The analytical estimates\nof the characteristic frequencies are indicated with arrows. The solid orange line represents the numerical calculation of the inverse Compton component.\neB=10-2,at=0\nat=-0.4\neB=2.110-5,at=0\n10101011101210131014101510161017101810191020102110221023102410251026102710281029103010-1410-1310-1210-1110-1010-910-8\nn@HzDnFn@ergscm2sD\nFigure 2. Synchrotron and inverse Compton spectra at tobs= 3\u0002104s, representative of the slow cooling regime, for a blast with energy E= 1054ergs\nimpinging on a progenitor wind with density n= 1035r\u00002cm\u00003(rin cm), assuming \u000fe= 0:1andp= 2:3, for three microphysical models, as\nindicated: homogeneous (non-decaying) \u000fB=\u000fB+= 0:01(dashed red line); decaying \u000fB=\u000fB+h\nt=(100!\u00001\npi)i\u00000:4\n(solid blue); homogeneous\n\u000fB=\u000fB\u0000= 2:1\u000210\u00005(dash-dotted orange), the value of \u000fB\u0000being representative of \u000fBclose to the contact discontinuity in the decaying \u000fBmodel.\nThe synchrotron model predictions have been thinned beyond an ad-hoc maximal synchrotron photon energy of 1 GeV (see text). Characteristic frequencies\nare: for\u000bt=\u00000:4and for\u000fB=\u000fB\u0000,\u0017m'5:6\u00021011Hz,\u0017c'1:2\u00021016Hz and\u0017IC;KN'2\u00021026Hz; for\u000fB=\u000fB+,\u0017m'1:2\u00021013Hz,\n\u0017c'4:2\u00021013Hz and\u0017IC;KN'2\u00021024Hz.\nc\r2008 RAS, MNRAS 000, 000–000SSC spectra at large Y 9\nAs discussed above, the peak of the inverse Compton component\nis expected to occur at \u0017IC;KN, although the numerical calculation\nsuggests that the turn-over becomes manifest a factor \u00185below\nthe above theoretical value.\nThe VERITAS collaboration has recently been able to observe\nthe exceptional GRB130427A and to put stringent upper limits on\nthe emission at &100GeV (Aliu et al. 2014). The absence of de-\ntection of this energy range suggests that, for this burst at least, the\ninverse Compton component has cut-off below '100GeV , while\nthe Fermi detection of multi-GeV photons up to a day or so sug-\ngests that this cut-off lied above 1\u000010GeV . Such a cut-off energy\nfits well with the above estimates for \u0017IC;KNfor a low average \u000fB.\n2.5 Fast cooling\nThe fast cooling regime involves a susbtantial variety of syn-\nchrotron spectra, with multiple breaks and indices, as discussed in\ndetail in Nakar et al. (2009) and Wang et al. (2010) for the case\nof uniform\u000fB. One key difference with the slow-cooling regime is\nthat particles with Lorentz factors \r < max (\rc;\rm)may have a\nnon-trivial cooling history while in the slow-cooling regime, such\nparticles do not cool. As a consequence, it is difficult to even derive\nthe cooling Lorentz factor and the Compton parameter Ycwhen\nKlein-Nishina effects become significant.\nIn this fast-cooling regime, it is actually more efficient to com-\npute the spectrum numerically, using the following simple and ef-\nficient algorithm. One starts with a template synchrotron spectrum,\nfor instance that corresponding to a homogeneous magnetized tur-\nbulence. One can then derive a first approximation to \rcandYc,\neither using standard formulae (e.g Panaitescu & Kumar 2000) –\nwhich ignore KN effects – or through an explicit determination of\n\rcas the Lorentz factor for which cooling takes place on a dy-\nnamical timescale, using a radiation energy density inferred from\nthe template spectrum. With \rcandYc, one can solve Eqs. (9),\n(10) and (11) to compute the frequencies and Compton parame-\nter as a function of the initial Lorentz factor of an electron; one\ncan then use Eqs. (12) and (13) to compute an improved version\nof the synchrotron spectrum, properly taking into account the cool-\ning history of the electrons in the decaying turbulence as well as\nall relevant Klein-Nishina effects. This latter spectrum remains an\napproximation, because it relies on a guessed value for \rcandYc.\nNevertheless, iterating the above process, using each time as a tem-\nplate the previously computed synchrotron spectrum, one obtains\nafter\u001810iterations a self-consistent synchrotron spectrum, with\n\rcandYcdetermined to high accuracy. Finally, one can derive the\ninverse Compton spectrum using Eq. (17).\nThe above algorithm provides a self-consistent estimate of the\nsynchrotron and inverse Compton spectra with a normalization ac-\ncuracy of order unity. This accuracy can be checked by calculating\na posteriori the integrated synchrotron and inverse Compton pow-\ners and comparing to the total electron power injected through the\nshock: in the fast cooling regime, these should match. The error\nis of order 10\u000040% for the SSC spectrum of a decaying micro-\nturbulence as shown in Fig. 3, depending on observer time; it is less\nthan10\u000020% for\u000bt= 0 and\u000fB=\u000fB+, but it becomes a factor\n.2for\u000bt= 0 and\u000fB=\u000fB\u0000. Most of the error results from the\nbroken power-law normalization of the flux in Eq. (12) and from\nthe treatment of the Klein-Nishina cross-section as a step function\nin the calculation of the synchrotron cooling history. The resulting\nuncertainty remains nevertheless satisfactory given the uncertainty\nassociated for instance to the definition of tdyn(hence\rc) in the\nabsence of a realistic description of the blast energy profile.A detailed example of the SSC spectrum of the blast, for the\nsame parameters as in Fig. 2, is presented in Fig. 3 at an observer\ntimetobs= 30 s; assuming a Blandford & McKee (1976) decel-\nerating solution in a wind profile, the Lorentz factor of the blast at\nthat time is \u0000b'160. The value of \u000fB\u0000in this case is 4:2\u000210\u00005.\nAs expected, the spectrum corresponding to a decaying turbulence\nmerges with the spectrum for uniform \u000fB=\u000fB+above a fre-\nquency\u0017\u00181023Hz, since the electrons that emit in that range\ncool fast, in a region where \u000fB'\u000fB+. In this fast cooling regime,\nthe total integrated energy densities of the three SSC spectra cor-\nrespond to the injected electron energy density. The spectrum for\ndecaying micro-turbulence does not merge with that for uniform\n\u000fB=\u000fB\u0000at low frequencies, since the cooling Lorentz factors\ndiffer for both. In this fast cooling regime, one cannot therefore de-\nscribe accurately the synchrotron spectrum at low frequencies with\na spectrum computed for uniform low \u000fB: an explicit calculation\nbecomes necessary.\nFinally, note that the inverse Compton component in the fast\ncooling regime is expected to peak at \r2\nm\u0017m, if the synchrotron flux\npeaks at\u0017mand if one omits KN effects. The Klein-Nishina sup-\npression implies a turn-over of the IC component at most at \r2\nm~\u0017m,\nfor reasons analog to those discussed in the slow cooling regime,\nsee also Nakar et al. (2009). If \u0017m<~\u0017m, the slow rise of the\nsynchrotron flux above \u0017min the case of low \u000fB(due to the KN\nsuppression of electron cooling) or decaying microturbulence im-\nplies a comparable behavior of the IC component between \r2\nm\u0017m\nand\r2\nm~\u0017m. This feature is not clearly seen in Fig. 3 due to the (rel-\native) proximity of these two frequencies, \r2\nm~\u0017m'6\u00021025Hz\nand\r2\nm\u0017m'1024Hz.\n3 SPECTRA AND LIGHT CURVES\n3.1 Spectral and temporal behaviors\nThe spectra shown in Figs. 2 and 3 illustrate how a complete and\nsimultaneous spectral coverage would allow to tomograph the evo-\nlution of the micro-turbulence behind the relativistic shock. The\neffects are most noticeable in the X-ray and MeV regions, as one\nwould expect: in this region of the spectrum, the emitting electrons\nfeel the decaying turbulence, while at the highest frequencies, they\ncool in regions with \u000f\u0018\u000fB+, and at frequencies \u0017 < \u0017 cthey\ncool in a magnetic field characterized by \u000fB\u0000, the value of which\nevolves slowly in time.\nAfterglow models often rely on the spectral and temporal\nslopes in various domains and their so-called closure relations to\nmake comparison to observations. Figure 4 therefore presents the\nspectral slopes \fdefined byF\u0017/\u0017\u0000\f(whereF\u0017sums the syn-\nchrotron and inverse Compton fluxes), for the optical, X-ray, high\nenergy ( 0:1\u000010 GeV) and very high energy ( >10 GeV ) ranges\n(note that no attenuation in extra-galactic background radiation has\nbeen assumed for the latter range). This figure compares the spec-\ntral slopes for the three previous representative models: ( \u000bt= 0,\n\u000fB= 0:01),\u000bt=\u00000:4and (\u000bt= 0,\u000fB= 10\u00005) with other-\nwise same parameters as in Figs. 2, 3, except k, which takes values\n0(constant density profile) or 2(stellar wind).\nIn the optical, \fhas been calculated at a reference frequency\nof4:7\u00021014Hz (R band); in the X-ray, \fis calculated as the\naverage slope over the interval of energies 0:3\u000010keV; at high\nenergy, it is calculated as the average of the energy interval 0:1\u0000\n10GeV and at very high energy, over >10GeV .\nAccording to Fig. 4, the most robust signature of a decay-\ning micro-turbulence appears to be a slightly harder slope in the\nc\r2008 RAS, MNRAS 000, 000–00010 M. Lemoine\neB=10-2,at=0\nat=-0.4\neB=4.210-5,at=0\n10101011101210131014101510161017101810191020102110221023102410251026102710281029103010-1110-1010-910-810-710-610-5\nn@HzDnFn@ergscm2sD\nFigure 3. Same as Fig. 2 at observer time tobs= 30 s, representative of the fast cooling regime. Characteristic frequencies are: for \u000bt=\u00000:4,\u0017c'1012Hz,\n\u0017m'3\u00021016Hz and\r2\nm~\u0017m'6\u00021025Hz; for\u000fB=\u000fB\u0000,\u0017c'2\u00021013Hz,\u0017m'3\u00021016Hz and\r2\nm~\u0017m'6\u00021025Hz; for\u000fB=\u000fB+,\n\u0017c'4\u00021011Hz,\u0017m'4\u00021017Hz and\r2\nm~\u0017m'6\u00021025Hz.\nX-ray,\f'0:9[panels (b) and (e)] vs \f'1:15for uniform\n\u000fBin the first hours [panels (a), (c), (d) and (f)]. The latter value\ncorresponds to the fast cooling regime \f=p=2, therefore it de-\npends onp; however, one does not expect it to go below 1, because\np>2is a generic prediction of relativistic shock acceleration, e.g.\nBednarz & Ostrowski (1998), Kirk et al. (2000), Achterberg et al.\n(2001), Lemoine & Pelletier (2003), Sironi et al. (2013). Current\ndata do not allow to distinguish between these limits; in particular,\nthe Swift data lead to \f'1\u00060:1(Evans et al. 2009) in afterglows\nwith standard power-law decay. Interestingly, even in the case of a\nhomogeneous turbulence, the X-ray slope hardens at late times be-\ncause of the emergence in the X-ray range of the inverse Compton\ncomponent; for instance, in panels (a) and (c), one can see \ftransit\nto values of order 0.7, corresponding to the low energy extension\nof this inverse Compton component with slope \f= (p\u00001)=2.\nIn the optical range, the slope is comparable to that in the X-\nray range for the decaying micro-turbulence scenario at observer\ntimes\u0018103s, but significantly harder at earlier times when the\noptical falls in the range \u0017c\u0000\u0017m: Fig. 4 indicates values \f\u00180:3,\nharder than expected (1=2)in the standard fast cooling regime in\nthis range of frequencies. At an observer time tobs= 30 s, for\nk= 2corresponding to panel (e) as well as to Fig. 3, ~\u0017m'1:4\u0002\n1018Hz, which lies below the peak of the synchrotron component\n(see Fig. 3). This implies that Klein-Nishina effects are significant,\nand that\rmelectrons cool by interacting with the segment in the\nrange 1017\u00001021Hz, whose index \f\u00180:8. In this case, one can\ncompute the expected index \f0of the segment below \u0017m, using a\nvariant of Eq. (23): one notes that \u0017F\u0017;syn/\r=(1+Y)[Eq. (13)],\nwhich gives\n1\u0000\f0=1\u0000\u000bt+ (1\u0000\f)(1 +\u000bt)\n2\u0000\u000bt=2 + (1\u0000\f)\u000bt=2'0:3 (32)\nthe last equality applying for \u000bt=\u00000:4and\f= 0:8. This\nexplains the values of \u000btfound in the optical range. Such values\ndo depart from the standard synchrotron spectra, although Klein-\nNishina effects may also cause values \f\u00180:3below\u0017minin the\ncase of homogeneous turbulence scenarios, see Nakar et al. (2009),\ntheir figures 2 and 3 for example. Therefore, it is not clear at presentwhether one can consider such values of \fas a clear signature of a\ndecaying micro-turbulence.\nAll in all, an accurate measurement of \fin the X-ray range, or\nbetter, in the MeV range if the MeV afterglow could be detected,\nwould provide the best probe of \u000bt, see also Fig. 2 and 3 for illus-\ntrations of these effects.\nAnother quantity of interest for a general description of the\nafterglow is the temporal slope, defined by F\u0017/t\u0000\u000b\nobs. Of course,\nthese temporal slopes directly depends on the time evolution of the\nvarious parameters, through the assumed evolutionary law for \u0000b\nand the evolution of randn, in contrary to the spectral slopes \fat\nany given time. Here \u0000bis assumed to decrease as in the Blandford\n& McKee (1976) adiabatic solution, i.e. \u0000b/t(k\u00003)=[2(4\u0000k)].\nThe values of \u000bfor the same models and intervals as Fig. 4 are\nreported in Fig. 5. The largest differences between constant (high)\nand decaying \u000fBresult from the different transit times between the\nslow and fast cooling regimes, but these times depend in turn on\nother parameters that are a priori unknown. The light curves other-\nwise present similar features, without a clear trend distinguishing\none from the other.\nFigures 4 and 5 thus indicate that \u000band\fare by themselves\nweakly sensitive probes of the dynamics of the magnetized turbu-\nlence in the blast and that it is not possible at present to distinguish\na decaying micro-turbulence from a uniform low- or high \u000fBon the\nbasis of these data. It appears much more effective to try to probe\n\u000fBthrough a multiwavelength fit of the afterglow light curves, us-\ning not only the temporal and spectral slopes, but also the ratio of\nfluxes between different spectral windows, as done in Lemoine et\nal. (2013), Liu et al. (2013) for Fermi-LAT bursts.\n3.2 Emission at very high energy\nFinally, an interesting consequence of a decaying micro-turbulence\nis the generic prediction of substantial emission at the highest en-\nergies, due to the large value of the Compton parameter. Naive\nestimates,Y\u0018p\n\u000fe=\u000fB\u0000, indicate values of several hundreds\nforY, although they neglect Klein-Nishina effects which depend\non the electron energy, therefore on observed frequency; further-\nc\r2008 RAS, MNRAS 000, 000–000SSC spectra at large Y 11\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè\nààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòììììììììììììììììììììììììììììììHaLeB=0.01,k=0\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV-0.250.000.250.500.751.001.251.50b\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè ààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòììììììììììììììììììììììììììììììHbLat=-0.4,k=0\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV-0.250.000.250.500.751.001.251.50b\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèàààààààààààààààààààààààààààààà\nòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòììììììììììììììììììììììììììììììHcLeB=10-5,k=0\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV-0.250.000.250.500.751.001.251.50b\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè\nàààààààààààààààààààààààààààààà\nòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòììììììììììììììììììììììììììììììHdLeB=0.01,k=2\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV-0.250.000.250.500.751.001.251.50b\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèàààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòììììììììììììììììììììììììììììììHeLat=-0.4,k=2\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV-0.250.000.250.500.751.001.251.50b\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèàààààààààààààààààààààààààààààà\nòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòììììììììììììììììììììììììììììììHfLeB=10-5,k=2\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV\n102103104-0.50.00.51.01.5\ntobs@sDb\nFigure 4. Flux density index \f, withF\u0017/\u0017\u0000\f, as a function of time,\nin various wavebands, for different external density profiles: k= 2 (wind,\nn= 1035r\u00002cm\u00003) andk= 0 (constant density, n= 1cm\u00003), assum-\ning uniform\u000fB= 0:01,\u000fB= (100!pit)\u000btor uniform\u000fB= 10\u00005.\nBlast energy, jet Lorentz factor and \u000feare as in Fig. 2. Wavebands are as in-\ndicated: for reference, optical (circles) \u0017= 4:7\u00021014Hz, X-ray (squares)\nintegrated from 0.3 to 10 keV (i.e. 0:72\u000024\u00021017Hz), 0.1-10 GeV (up-\nward triangles, in frequency 0:24\u000024\u00021023Hz) and>10GeV (dia-\nmonds, in frequency >24\u00021023Hz).\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè\nààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò\nììììììììììììììììììììììììììììììHaLeB=0.01,k=0\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV0.00.51.01.52.0a\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèàààààààààààààààààààààààààààààà\nòòòòòòòòòòòòòò\nòòòòòòòòòòòòòòòò\nìììììììììììììì\nììììììììììììììììHbLat=-0.4,k=0\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV0.00.51.01.5a\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèàààààààààààààààààààààààààààààà\nòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò\nììììììììììììììììììììììììììììììHcLeB=10-5,k=0\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV0.00.51.01.5a\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè\nààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòò\nòòòò\nìììììììììììììììììììììììììì\nììììHdLeB=0.01,k=2\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV0.00.51.01.5a\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè\nàààààààààààààààààààààààààààààà òòòòòòòòòòòòòòò\nòòòòòòòòòòòòòòò\nìììììììììììììì\nì\nìììììììììììììììHeLat=-0.4,k=2\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV0.00.51.01.5a\nèè\nè\nèèèèèèèèèèèèèèèèèèèèèèèèèèèàà\nà\nàà\nààààà\nààààààààààààààààààààòò\nò\nòò\nòòòò\nò\nòòòòòòòòòòòòòòòòòòòòìì\nì\nìì\nìììì\nì\nììììììììììììììììììììHfLeB=10-5,k=2\nèoptical\nà X-ray\nò 0.1-10GeV\nì >10GeV\n102103104-1.0-0.50.00.51.01.52.0\ntobs@sDaFigure 5. Observer time decay index of the flux density, i.e. F\u0017/t\u0000\u000b\nobs,\nfor various models, as indicated in Fig. 4. Blast energy and \u000feare as in\nFig. 2.\nc\r2008 RAS, MNRAS 000, 000–00012 M. Lemoine\nò\nòòòòòòòòòòòòòòòòòòòòòòòòòòòòòó\nóó\nóó\nóó\nó\nó\nóóóóóóóóóóóóóóóóóóóóóææææææææææææææææææææææææææææææççççççççççççççççççççççççççççççààààààààààààààààààààààààààààààáááááááááááááááááááááááááááááá\n102103104100101\ntobs@sDFH>30GeVLFthH>30GeVL\nFigure 6. Ratio of the total energy flux above 30GeV to a theoretical flux\nobtained by matching the calculated total flux at 0:1GeV and extrapolating\nthis flux to higher energies with a power-law F\u0017;th/\u0017\u00001:1, as in Inoue\net al. (2013). The symbols correspond to the six models studied in Figs. 4\nand 5, as follows: red circles \u000fB=\u000fB+and\u000bt= 0 ; blue squares\n\u000bt=\u00000:4; orange triangles \u000fB=\u000fB\u0000and\u000bt= 0 . Filled symbols\ncorrespond to k= 2 and open symbols to k= 0; other parameters are as\nin Fig. 2.\nmore, the non-trivial spectral shape above \u0017cin the case of decay-\ning micro-turbulence modifies the ratio of the inverse Compton to\nthe synchrotron component. As discussed in Wang et al. (2013),\nemission above 10GeV is most likely of inverse Compton origin,\nbecause the maximal synchrotron photon energy is more likely of\nthe order of 1GeV or so at 100\u00001000 sec observer time. This\nemission is a prime target for future gamma-ray telescopes such as\nHAWK (Mostafa 2013) or CTA (Inoue et al. 2013). For the partic-\nular case of CTA, Inoue et al. (2013) have investigated the detec-\ntion rates of gamma-ray bursts above 30GeV by assuming that the\nspectrum continues beyond 1GeV with a spectral index \f= 1:1\n(corresponding to the standard fast cooling regime \f=p=2with\np= 2:2) and scaling the flux at 1GeV to that measured by the\nFermi-LAT instruments. Their simulations lead to about one de-\ntection per year. This rate is rather low, therefore any improvement\nwould be quite valuable, given the potential impact of a high energy\ndetection.\nOne can use the calculations of Sec. 2 to study how a decay-\ning micro-turbulence or low \u000fB\u0000affects these predictions. In or-\nder to do so, one calculates the ratio of the total (synchrotron +\ninverse Compton) energy flux above 30GeV ,F(>30 GeV) =R\n30 GeVF\u0017d\u0017to a theoretical reference flux Fth(>30 GeV) . As\nin Inoue et al. (2013), Fth(>30 GeV) is obtained by extrapolating\nthe total flux measured at a reference energy, here 0:1GeV , with an\nindex\f'1:1, i.e.\nFth(>30 GeV) = F\u0017(0:1 GeV)\n\u0002Z+1\n30 GeV=hd\u0017\u0012h\u0017\n0:1 GeV\u0013\u00001:1\n(33)\nwithF\u0017=F\u0017;syn+F\u0017;ICthe total flux. The reference energy\nchosen here is smaller than that in Inoue et al. (2013), because the\ninverse Compton flux is already prominent at 1GeV in the scenar-\nios studied, as shown in Fig. 2 and 3 for example. However, given\nthe value of the index \f, the theoretical flux \u0017F\u0017is roughly flat\nabove 1GeV , hence this should not affect the statistics of detection.\nThe results are shown in Fig. 6, which carries out this evalua-\ntion for the six models shown in Figs. 4 and 5. This figure indicatesthat a decaying micro-turbulence with \u000bt=\u00000:4(or a low aver-\nage\u000fB) increases by a factor of a few, up to an order of magnitude,\ndepending on \u000bt,kandtobs, the prospects of observing the after-\nglows at energies >30GeV , relatively to statistics computed for a\nmodel with\u000fB= 0:01, as in Inoue et al. (2013). This certainly\nbrings the number of potential detections by instruments such as\nCTA in a more confortable range.\nFurthermore, it is important to note that in most models stud-\nied here, the inverse Compton component contributes to a signifi-\ncant fraction of the total flux at 0:1GeV , therefore the above ratio\nactually is an underestimate of the ratio of the inverse Compton flux\nat high energies to the synchrotron flux at GeV energies.\n4 CONCLUSIONS\nThis paper has discussed the spectral shapes of the synchrotron-\nself-Compton spectrum of a relativistic blast wave, including all\nrelevant Klein-Nishina effects, and their evolution in time. A par-\nticular emphasis has been put on the impact of a decaying micro-\nturbulence behind the shock front, which is motivated by theoretical\nanalysis (Chang et al. 2008, Lemoine 2015) and observational in-\nference (Lemoine et al. 2013, Liu et al. 2013). However, the results\nare fully applicable to the case of a uniformly magnetized blast,\npossibly with a low value of the average \u000fB.\nA decaying micro-turbulence and/or a low average value of\n\u000fBboth lead to a large YcCompton parameter, with interesting\nphysical and observational consequences. From a more theoreti-\ncal point of view, the Klein-Nishina suppression of inverse Comp-\nton cooling has a strong effect on the synchrotron spectral shape,\nas noted elsewhere for the case of a uniformly magnetized blast\n(Nakar et al. 2009, Wang et al. 2010). In the case of a decaying\nmicro-turbulence, the modification is not trivial to compute, and\nthe present paper has described a simple algorithm which allows\nto compute the full SSC spectrum with satisfactory accuracy, at a\nmodest numerical cost. Among the interesting phenomenological\nconsequences, one may point out the slight deviations in the spec-\ntral and temporal slopes induced by the decaying micro-turbulence,\nor by KN effects in uniformly magnetized blasts at low \u000fB. A multi-\nwavelength coverage of the afterglow would, in principle, allow one\nto tomograph the dynamics of this magnetized turbulence, through\nits influence on the light curves in various wavebands. However,\nas expressed in terms of the spectral \fand temporal \u000bslopes, de-\nfined customarily through F\u0017/t\u0000\u000b\nobs\u0017\u0000\f, the deviations are rela-\ntively weak and not currently distinguishable through observations.\nA multiwavelength fit of the afterglow, which also relies on the flux\nratios between various wavebands, seems to provide a more sensi-\ntive probe of the dynamics of the micro-turbulence.\nFinally, a large Ycparameter also implies a large inverse\nCompton flux at multi-GeV energies, relatively to the lower energy\nsynchrotron flux, with direct consequences for the detectability of\ngamma-ray bursts afterglows by future gamma-ray telescopes. 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In this paper, we are interested in analyzing the asymptotic profiles of solutions to\nthe Cauchy problem for linear structurally damped σ-evolution equations in L2-sense. Depending\non the parameters σandδwe would like to not only indicate approximation formula of s olutions\nbut also recognize the optimality of their decay rates as wel l in the distinct cases of parabolic like\ndamping and σ-evolution like damping. Moreover, such results are also di scussed when we mix\nthese two kinds of damping terms in a σ-evolution equation to investigate how each of them affects\nthe asymptotic profile of solutions.\n1.Introduction\nLet us consider the following Cauchy problem for structural ly damped σ-evolution equations:\n#\nutt` p´∆qσu`ap´∆qδ1ut`bp´∆qδ2ut“0,\nup0,xq “u0pxq, utp0,xq “u1pxq,(1)\nwhereσě1, 0 ăδ1ăσ{2ăδ2ăσanda,b“0,1 with pa,bq ‰ p0,0q.\nAt first, we recall some recent results related to the study of (1) withσ“1 and pa,bq “ p1,0q\norpa,bq “ p0,1q, the so-called structurally damped wave equations, in the f ollowing form:\n#\nutt´∆u`µp´∆qδut“0,\nup0,xq “u0pxq, utp0,xq “u1pxq,(2)\nwithδP p0,1qandµą0. The authors in [ 6] succeeded in obtaining some of sharp pL1XL2q ´L2\nestimates to ( 2), i.e. the mixing of additional L1regularity for the data on the basis of L2´L2\nestimates. A direct application of these estimates is to pro ve the global (in time) existence of\nsmall data energy solutions in low space dimensions to the co rresponding semi-linear structurally\ndamped wave models with power nonlinearties |u|p. The point in the cited paper is that from the\nview of decay estimates they proposed to separate ( 2) into “parabolic like models” with δP p0,1\n2q\nand “hyperbolic like models” with δP p1\n2,1q. This comes from the properties of solutions to ( 2)\nwhich change completely from the former case to the latter ca se. After that, in the quite recent\npaper [11] the asymptotic profile of solutions to ( 2) and some of their optimal decay estimates have\nbeen explored so well. In particular, the authors provided t he different approximation formulas of\nsolutions by a constant multiple of a special function for la rgetě1 corresponding to the cases\nδP p0,1\n2q,δ“1\n2andδP p1\n2,1q.\nConcerning the more general cases of σě1 to (1), the following so-called structurally damped\nσ-evolution equations have been well-studied in several rec ent papers (see, for example, [ 1,2,3,4,\n7,8]):#\nutt` p´∆qσu`µp´∆qδut“0,\nup0,xq “u0pxq, utp0,xq “u1pxq,(3)\n2010Mathematics Subject Classification. Primary: 35B40, 35L30; Secondary: 35G10, 35M11.\nKey words and phrases. σ-evolution equations; structural damping; parabolic like models;σ-evolution like models;\nasymptotic profile.\n12 TUAN ANH DAO\nwhereσě1,δP p0,σqandµą0. Namely, the results for decay rates of solutions to ( 3) in the\nL2´L2theory by assuming additional L1regularity for the data were derived in [ 4]. Quite recently,\ntaking into considerations some of decay estimates for solu tions to ( 3) basing on the Lq´Lqtheory\nfor anyqP p1,8qthe authors in [ 3,7,8] have investigated pLmXLqq ´LqandLq´Lqestimates\nwithqP p1,8qandmP r1,qq. More in detail, to establish this, they applied two main str ategies\nincluding the theory of modified Bessel functions combined w ith Fa` a di Bruno’s formula and the\nMikhlin-H¨ ormander multiplier theorem. By using the obtai ned decay estimates, the novelty of the\ncited papersare to prove the global (in time) existence of sm all data Sobolev solutions from suitable\nfunction spaces basing on Lqspaces and to determine critical exponents as well to some se mi-linear\nmodels with power nonlinearties |u|por|ut|p. However, one may realizes that the asymptotic\nprofiles of solutions and the optimality of their decay rates have not been indicated in the above\nmentioned references clearly. For this reason, one of the ma in goals of this paper is to report such\nresults for solutions to ( 3).\nAccording the classification of ( 3) proposed in [ 2,3,7,8], here we want to distinguish ( 1) into\nthree main models depending on the parameters aandb. In particular, the first model of our\nconsiderations is the σ-evolution equations with parabolic like structural dampi ng corresponding\ntothecase pa,bq “ p1,0q. Inthismodel, wearegoingtoshowthattheasymptoticprofil eofsolutions\nto (1) is the same as that to the following anomalous diffusion equat ions (see later, Theorem 1.1):\nvt` p´∆qσ´δ1v“0, v p0,xq “v0pxq, (4)\nfor a suitable choice of data v0. The second one is the model with σ-evolution like structural\ndamping corresponding to the case pa,bq “ p0,1q, the so-called “hyperbolic like models” in the\ncaseσ“1. We recognize that some kind of wave structure appears and o scillations come into\nplay from the asymptotic profile of solutions in this model (s ee later, Theorem 1.2). This means\nthe above mentioned diffusion phenomenon does not happen. Our interest is the last model with\nmixing two distinct kinds of structural damping including p arabolic like damping and σ-evolution\nlike damping corresponding to the case pa,bq “ p1,1q, the so-called double damping terms (see, for\ninstance, [ 5,9,10]). This connection brings some interesting properties for solutions to ( 1) in the\ncaseδ1`δ2ąσwhich inherit from the two former models (see later, Theorem 1.3). More precisely,\nby the presence of parabolic like damping, on the one hand, th e asymptotic profile of solutions\nto (1) is also the same as that to ( 4). On the other hand, the solutions to ( 1) possess the same\nregularity as that to the second model by the presence of σ-evolution like damping. Analyzing\nthese properties is the second main goal of the this paper.\n1.1.Notations.\n‚We write fÀgwhen there exists a constant Cą0 such that fďCg, andf«gwhengÀfÀg.\n‚As usual, Haand 9Ha, withaě0, denote Bessel and Riesz potential spaces based on L2spaces.\nHere/angbracketleftbig\nD/angbracketrightbigaand |D|astand for the pseudo-differential operators with symbols/angbracketleftbig\nξ/angbracketrightbigaand |ξ|a,\nrespectively.\n‚We denote pwpt,ξq:“FxÑξ`\nwpt,xq˘\nas the Fourier transform with respect to the space variable\nof a function w“wpt,xq.\n‚We put rss`:“maxts,0uas the positive part of sPR. Moreover, we fix the constant m0:“2m\n2´m,\nthat is,1\nm0“1\nm´1\n2withmP r1,2q.\n‚Letχ“χp|ξ|qbe aC8\n0pRnqcut-off nonnegative function equal to 1 for small |ξ|and vanishing\nfor large |ξ|. We decompose a function w“wpt,xqinto two parts localized separately to low and\nhigh frequencies as follows:\nwpt,xq “wlowpt,xq `whighpt,xq,\nwhere\nwlowpt,xq “F´1`\nχp|ξ|qpwpt,ξq˘\nandwhighpt,xq “F´1´`\n1´χp|ξ|q˘\npwpt,ξq¯\n.ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 3\n‚For later convenience, we denote the following quantity:\nP1:“ż\nRnu1pxqdx.\n1.2.Main results. The following results describe the large time behavior of so lutions to ( 1).\nTheorem 1.1 (a“1andb“0).Letj“0,1andsě0. We assume the condition ną4δ1and\nthe initial data\npu0,u1q PA1\n0ˆA1\n1:“ pL1XHs`jσq ˆ pL1XHrs`pj´1qσs`q.\nThen, the Sobolev solutions to ( 1) satisfy the following estimates for large tě1:\n›››Bj\nt|D|s´\nupt,¨q ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q¯›››\nL2“o´\nt´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1¯\n.(5)\nMoreover, if P1‰0, then the following estimates hold for large tě1:\nC1t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1ď››Bj\nt|D|supt,¨q››\nL2ďC2t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1, (6)\nwhereC1andC2are some suitable positive constants.\nTheorem 1.2 (a“0andb“1).Letj“0,1andsě0. We assume the condition ną2σand\nthe initial data\npu0,u1q PA2\n0ˆA2\n1:“ pL1XHs`2jδ2q ˆ pL1XHrs`2pj´1qδ2s`q.\nThen, the Sobolev solutions to ( 1) satisfy the following estimates for large tě1:\n›››|D|s´\nupt,¨q ´P1F´1´\ne´1\n2t|ξ|2δ2sinpt|ξ|σq\n|ξ|σ¯\npt,¨q¯›››\nL2“o´\nt´n\n4δ2´s\n2δ2`σ\n2δ2¯\n, (7)\n›››Bt|D|supt,¨q ´P1|D|sF´1´\ne´1\n2t|ξ|2δ2cospt|ξ|σq¯\npt,¨q›››\nL2“o´\nt´n\n4δ2´s\n2δ2¯\n. (8)\nMoreover, if P1‰0, then the following estimates hold for large tě1:\nC1t´n\n4δ2´s`pj´1qσ\n2δ2ď››Bj\nt|D|supt,¨q››\nL2ďC2t´n\n4δ2´s`pj´1qσ\n2δ2, (9)\nwhereC1andC2are some suitable positive constants.\nTheorem 1.3 (a“1andb“1).Letj“0,1andsě0. Let us assume δ1`δ2ąσ. We suppose\nthe condition ną4δ1and the initial data\npu0,u1q PA3\n0ˆA3\n1:“ pL1XHs`2jδ2q ˆ pL1XHrs`2pj´1qδ2s`q.\nThen, the Sobolev solutions to ( 1) satisfy the following estimates for large tě1:\n›››Bj\nt|D|s´\nupt,¨q ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q¯›››\nL2“o´\nt´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1¯\n.(10)\nMoreover, if P1‰0, then the following estimates hold for large tě1:\nC1t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1ď››Bj\nt|D|supt,¨q››\nL2ďC2t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1,(11)\nwhereC1andC2are some suitable positive constants.\nThe organization of this paper is as follows: In Section 2, we present preliminary knowledge\nas the representation of solutions, pointwise estimates in Fourier space and some of decay estimates\nfor solutions to ( 1) in Sections 2.1,2.2and2.3, respectively. Then, we prove main results for the\nasymptotic profile of solutions to ( 1) and indicate their optimal decay estimates as well in Secti on\n3. In particular, the proofs in Sections 3.1,3.2and3.3correspond to the cases pa,bq “ p1,0q,\npa,bq “ p0,1qand pa,bq “ p1,1q.4 TUAN ANH DAO\n2.Preliminaries\n2.1.Representation of solutions. At first, using partial Fourier transformation to ( 1) we obtain\nthe following Cauchy problem:\nputt``\na|ξ|2δ1`b|ξ|2δ2˘\nput` |ξ|2σpu“0,pup0,ξq “xu0pξq,putp0,ξq “xu1pξq.(12)\nThe characteristic roots are\nλ1,2“λ1,2pξq “1\n2´\n´`\na|ξ|2δ1`b|ξ|2δ2˘\n˘b`\na|ξ|2δ1`b|ξ|2δ2˘2´4|ξ|2σ¯\n.\nThe solutions to ( 12) are presented by the following formula (here we assume λ1‰λ2):\npupt,ξq “λ1eλ2t´λ2eλ1t\nλ1´λ2xu0pξq `eλ1t´eλ2t\nλ1´λ2xu1pξq “:xK0pt,ξqxu0pξq `xK1pt,ξqxu1pξq.\nNow depending on the parameters aandbwe shall decompose the above representation formula of\nsolutions to ( 12) into several parts as follows:\n‚a“1 andb“0,1:\npupt,ξq “`xK1\n0pt,ξq `xK2\n0pt,ξq˘\nxu0pξq ``xK1\n1pt,ξq `xK2\n1pt,ξq˘\nxu1pξq,\nwhere\nxK1\n0pt,ξq “´λ2eλ1t\nλ1´λ2,xK2\n0pt,ξq “λ1eλ2t\nλ1´λ2,\nxK1\n1pt,ξq “eλ1t\nλ1´λ2,xK2\n1pt,ξq “´eλ2t\nλ1´λ2.\n‚a“0 andb“1:\npupt,ξq “`zKcos\n0pt,ξq `zKsin\n0pt,ξq˘\nxu0pξq `xK1pt,ξqxu1pξq,\nwhere\nzKcos\n0pt,ξq “e´1\n2t|ξ|2δ2cos`\nt|ξ|σfp|ξ|q˘\n,zKsin\n0pt,ξq “e´1\n2t|ξ|2δ2|ξ|2δ2sin`\nt|ξ|σfp|ξ|q˘\n2|ξ|σfp|ξ|q,\nxK1pt,ξq “e´1\n2t|ξ|2δ2sin`\nt|ξ|σfp|ξ|q˘\n|ξ|σfp|ξ|q,\nwith\nfp|ξ|q “$\n&\n%b\n1´1\n4|ξ|4δ2´2σfor small |ξ|,\nib\n1\n4|ξ|4δ2´2σ´1 for large |ξ|.\n2.2.Pointwise estimates in Fourier space. Taking account of the cases of small and large\nfrequencies separately we have the asymptotic behavior of t he characteristic roots as follows:\n1. a“1 andb“0 :λ1„ ´|ξ|2pσ´δ1q, λ 2„ ´|ξ|2δ1, λ 1´λ2„ |ξ|2δ1for small |ξ|,\nandλ1,2„ ´|ξ|2δ1˘i|ξ|σ, λ 1´λ2„i|ξ|σfor large |ξ|,\n2. a“0 andb“1 :λ1,2„ ´|ξ|2δ2˘i|ξ|σ, λ 1´λ2„i|ξ|σfor small |ξ|,\nandλ1„ ´|ξ|2pσ´δ2q, λ 2„ ´|ξ|2δ2, λ 1´λ2„ |ξ|2δ2for large |ξ|,\n3. a“1 andb“1 :λ1„ ´|ξ|2pσ´δ1q, λ 2„ ´|ξ|2δ1, λ 1´λ2„ |ξ|2δ1for small |ξ|,\nandλ1„ ´|ξ|2pσ´δ2q, λ 2„ ´|ξ|2δ2, λ 1´λ2„ |ξ|2δ2for large |ξ|.ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 5\n2.2.1.The case a“1andb“0.\nLemma 2.1. Letsě0andj“0,1. Then, the following estimates hold:\n|ξ|sχp|ξ|qˇˇBj\ntxK1\n0pt,ξqˇˇÀe´c0t|ξ|2pσ´δ1q|ξ|s`2jpσ´δ1q,\n|ξ|sχp|ξ|qˇˇBj\ntxK2\n0pt,ξqˇˇÀe´c0t|ξ|2δ1|ξ|s`2jδ1`2pσ´2δ1q,\n|ξ|sχp|ξ|qˇˇBj\ntxK1\n1pt,ξqˇˇÀe´c0t|ξ|2pσ´δ1q|ξ|s`2jpσ´δ1q´2δ1,\n|ξ|sχp|ξ|qˇˇBj\ntxK2\n1pt,ξqˇˇÀe´c0t|ξ|2δ1|ξ|s`2pj´1qδ1,\nand\n|ξ|s`\n1´χp|ξ|q˘ˇˇBj\ntxK0pt,ξqˇˇÀe´c0t|ξ|2δ1|ξ|s`jσ,\n|ξ|s`\n1´χp|ξ|q˘ˇˇBj\ntxK1pt,ξqˇˇÀe´c0t|ξ|2δ1|ξ|s`pj´1qσ,\nwherec0is a suitable positive constant.\n2.2.2.The case a“0andb“1.\nLemma 2.2. Letsě0andj“0,1. Then, the following estimates hold:\n|ξ|sχp|ξ|qˇˇBj\ntzKcos\n0pt,ξqˇˇÀe´c0t|ξ|2δ2|ξ|s`jσ,\n|ξ|sχp|ξ|qˇˇBj\ntzKsin\n0pt,ξqˇˇÀe´c0t|ξ|2δ2|ξ|s`pj´1qσ`2δ2,\n|ξ|sχp|ξ|qˇˇBj\ntxK1pt,ξqˇˇÀe´c0t|ξ|2δ2|ξ|s`pj´1qσ,\nand\n|ξ|s`\n1´χp|ξ|q˘ˇˇBj\ntxK0pt,ξqˇˇÀe´c0t|ξ|2pσ´δ2q|ξ|s`2jpσ´δ2q`e´c0t|ξ|2δ2|ξ|s`2jδ`2pσ´2δ2q,\n|ξ|s`\n1´χp|ξ|q˘ˇˇBj\ntxK1pt,ξqˇˇÀe´c0t|ξ|2pσ´δ2q|ξ|s`2jpσ´δ2q´2δ2`e´c0t|ξ|2δ2|ξ|s`2pj´1qδ2,\nwherec0is a suitable positive constant.\n2.2.3.The case a“1andb“1.\nLemma 2.3. Letsě0andj“0,1. Then, the following estimates hold:\n|ξ|sχp|ξ|qˇˇBj\ntxK1\n0pt,ξqˇˇÀe´c0t|ξ|2pσ´δ1q|ξ|s`2jpσ´δ1q,\n|ξ|sχp|ξ|qˇˇBj\ntxK2\n0pt,ξqˇˇÀe´c0t|ξ|2δ1|ξ|s`2jδ1`2pσ´2δ1q,\n|ξ|sχp|ξ|qˇˇBj\ntxK1\n1pt,ξqˇˇÀe´c0t|ξ|2pσ´δ1q|ξ|s`2jpσ´δ1q´2δ1,\n|ξ|sχp|ξ|qˇˇBj\ntxK2\n1pt,ξqˇˇÀe´c0t|ξ|2δ1|ξ|s`2pj´1qδ1,\nand\n|ξ|s`\n1´χp|ξ|q˘ˇˇBj\ntxK0pt,ξqˇˇÀe´c0t|ξ|2pσ´δ2q|ξ|s`2jpσ´δ2q`e´c0t|ξ|2δ2|ξ|s`2jδ2`2pσ´2δ2q,\n|ξ|s`\n1´χp|ξ|q˘ˇˇBj\ntxK1pt,ξqˇˇÀe´c0t|ξ|2pσ´δ2q|ξ|s`2jpσ´δ2q´2δ2`e´c0t|ξ|2δ2|ξ|s`2pj´1qδ2,\nwherec0is a suitable positive constant.\n2.3.Decay estimates.6 TUAN ANH DAO\n2.3.1.The case a“1andb“0.\nLemma 2.4. Letsě0andj“0,1. Then, the following estimates hold for mP r1,2q:\n››Bj\nt|D|s`\nK1\n0lowpt,xq ˚u0pxq˘\npt,¨q››\nL2À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j}u0}Lm,(13)\n››Bj\nt|D|s`\nK2\n0lowpt,xq ˚u0pxq˘\npt,¨q››\nL2À p1`tq´n\n2δ1p1\nm´1\n2q´s\n2δ1´j´σ´2δ1\nδ1}u0}Lm,(14)\nfor any space dimensions ně1, and\n››Bj\nt|D|s`\nK1\n1lowpt,xq ˚u1pxq˘\npt,¨q››\nL2À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j`δ1\nσ´δ1}u1}Lm,(15)\n››Bj\nt|D|s`\nK2\n1lowpt,xq ˚u1pxq˘\npt,¨q››\nL2À p1`tq´n\n2δ1p1\nm´1\n2q´s\n2δ1´j`1}u1}Lm, (16)\nfor any space dimensions ną2m0δ1. Moreover, the following estimates hold for mP r1,2s:\n››Bj\nt|D|s`\nK0high pt,xq ˚u0pxq˘\npt,¨q››\nL2Àe´ctt´n\n2δ1p1\nm´1\n2q´s`jσ´a\n2δ1}u0}Ham, (17)\n››Bj\nt|D|s`\nK1high pt,xq ˚u1pxq˘\npt,¨q››\nL2Àe´ctt´n\n2δ1p1\nm´1\n2q´s`jσ´a\n2δ1}u1}Hra´σs`\nm, (18)\nfor allaě0and for any space dimensions ně1, wherecis a suitable positive constant.\nProof.First, we shall prove ( 13). By the first estimate in Lemma 2.1, we apply Parseval-Plancherel\nformula and H¨ older’s inequality to obtain the following es timate:\n››Bj\nt|D|s`\nK1\n0lowpt,xq ˚u0pxq˘\npt,¨q››\nL2“››|ξ|sχp|ξ|qBj\ntxK1\n0pt,ξqxu0pξq››\nL2\nÀ››e´ct|ξ|2pσ´δ1q|ξ|s`2jpσ´δ1q››\nLm0}xu0}Lm1.\nThanks to the Hausdorff-Young inequality, we can control }xu0}Lm1by}u0}Lm. Hence, we have only\nto control the Lm0norm of the above multiplier. Using Lemma 3.1gives\n››e´ct|ξ|2pσ´δ1q|ξ|s`2jpσ´δ1q››\nLm0À p1`tq´n\n2m0pσ´δ1q´s\n2pσ´δ1q´j“ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j.\nThis completes the proof of ( 13). In the same way we may arrive at the estimates from ( 14) to (16).\nThen, in order to indicate ( 17) and (18), we repeat the proof of ( 13) by using a suitable regularity\nof the data u0andu1. Indeed, by Lemma 2.1we get\n››Bj\nt|D|s`\nK0high pt,xq ˚u0pxq˘\npt,¨q››\nL2“››|ξ|s`\n1´χp|ξ|q˘\nBj\ntxK0pt,ξqxu0pξq››\nL2\nÀ››e´c0t|ξ|2δ1|ξ|s`jσxu0pξq››\nL2Àe´c0\n2t››e´c0\n2t|ξ|2δ1|ξ|s`jσxu0pξq››\nL2\nÀe´ct››e´c0\n2t|ξ|2δ1|ξ|s`jσ´a››\nLm0››|ξ|axu0pξq››\nLm1,wherec:“c0\n2\nÀe´ctt´n\n2δ1p1\nm´1\n2q´s`jσ´a\n2δ1}u0}Ham.\nThis completes the proof of ( 17). By an analogous argument we may also conclude ( 18). Therefore,\nLemma2.4is proved. /square\nHence, we obtain decay properties of Sobolev solutions to ( 1).\nProposition 2.1. Letsě0andj“0,1. LetmP r1,2q. We assume the condition ną2m0δ1\nand the initial data\npu0,u1q P pLmXHs`jσq ˆ pLmXHrs`pj´1qσs`q.\nThen, the Sobolev solutions to ( 1) satisfy the following pLmXL2q ´L2estimates:\n››Bj\nt|D|supt,¨q››\nL2À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j}u0}LmXHs`jσ\n` p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j`δ1\nσ´δ1}u1}LmXHrs`pj´1qσs`.ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 7\n2.3.2.The case a“0andb“1.\nLemma 2.5. Letsě0andj“0,1. Then, the following estimates hold for mP r1,2q:\n››Bj\nt|D|s`\nKcos\n0lowpt,xq ˚u0pxq˘\npt,¨q››\nL2À p1`tq´n\n2δ2p1\nm´1\n2q´s`jσ\n2δ2}u0}Lm, (19)\n››Bj\nt|D|s`\nKsin\n0lowpt,xq ˚u0pxq˘\npt,¨q››\nL2À p1`tq´n\n2δ2p1\nm´1\n2q´s`pj´1qσ\n2δ2´1}u0}Lm, (20)\nfor any space dimensions ně1, and\n››Bj\nt|D|s`\nK1lowpt,xq ˚u1pxq˘\npt,¨q››\nL2À p1`tq´n\n2δ2p1\nm´1\n2q´s`pj´1qσ\n2δ2}u1}Lm, (21)\nfor any space dimensions nąm0σ. Moreover, the following estimates hold for mP r1,2s:\n››Bj\nt|D|s`\nK0high pt,xq ˚u0pxq˘\npt,¨q››\nL2\nÀe´ct´\nt´n\n2pσ´δ2qp1\nm´1\n2q´s´a\n2pσ´δ2q´j`t´n\n2δ2p1\nm´1\n2q´s´a\n2δ2´j`2δ2´σ\nδ2¯\n}u0}Ham, (22)\n››Bj\nt|D|s`\nK1high pt,xq ˚u1pxq˘\npt,¨q››\nL2\nÀe´ct´\nt´n\n2pσ´δ2qp1\nm´1\n2q´s´a\n2pσ´δ2q´j`t´n\n2δ2p1\nm´1\n2q´s´a\n2δ2´j¯\n}u0}Hra´2δ2s`\nm, (23)\nfor allaě0and for any space dimensions ně1, wherecis a suitable positive constant.\nProof.Following the proof of Lemma 2.4we may conclude all the desired estimates in Lemma 2.5\nby the aid of the statements in Lemma 2.2. /square\nHence, we obtain decay properties of Sobolev solutions to ( 1).\nProposition 2.2. Letsě0andj“0,1. LetmP r1,2q. We assume the condition nąm0σand\nthe initial data\npu0,u1q P pLmXHs`2jδ2q ˆ pLmXHrs`2pj´1qδ2s`q.\nThen, the Sobolev solutions to ( 1) satisfy the following pLmXL2q ´L2estimates:\n››Bj\nt|D|supt,¨q››\nL2À p1`tq´n\n2δ2p1\nm´1\n2q´s`jσ\n2δ2}u0}LmXHs`2jδ2\n` p1`tq´n\n2δ2p1\nm´1\n2q´s`pj´1qσ\n2δ2}u1}LmXHrs`2pj´1qδ2s`.\n2.3.3.The case a“1andb“1.\nLemma 2.6. Letsě0andj“0,1. Then, the following estimates hold for mP r1,2q:\n››Bj\nt|D|s`\nK1\n0lowpt,xq ˚u0pxq˘\npt,¨q››\nL2À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j}u0}Lm,(24)\n››Bj\nt|D|s`\nK2\n0lowpt,xq ˚u0pxq˘\npt,¨q››\nL2À p1`tq´n\n2δ1p1\nm´1\n2q´s\n2δ1´j´σ´2δ1\nδ1}u0}Lm,(25)\nfor any space dimensions ně1, and\n››Bj\nt|D|s`\nK1\n1lowpt,xq ˚u1pxq˘\npt,¨q››\nL2À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j`δ1\nσ´δ1}u1}Lm,(26)\n››Bj\nt|D|s`\nK2\n1lowpt,xq ˚u1pxq˘\npt,¨q››\nL2À p1`tq´n\n2δ1p1\nm´1\n2q´s\n2δ1´j`1}u1}Lm, (27)\nfor any space dimensions ną2m0δ1. Moreover, the following estimates hold for mP r1,2s:\n››Bj\nt|D|s`\nK0high pt,xq ˚u0pxq˘\npt,¨q››\nL2\nÀe´ct´\nt´n\n2pσ´δ2qp1\nm´1\n2q´s´a\n2pσ´δ2q´j`t´n\n2δ2p1\nm´1\n2q´s´a\n2δ2´j`2δ2´σ\nδ2¯\n}u0}Ham, (28)\n››Bj\nt|D|s`\nK1high pt,xq ˚u1pxq˘\npt,¨q››\nL2\nÀe´ct´\nt´n\n2pσ´δ2qp1\nm´1\n2q´s´a\n2pσ´δ2q´j`t´n\n2δ2p1\nm´1\n2q´s´a\n2δ2´j¯\n}u0}Hra´2δ2s`\nm, (29)8 TUAN ANH DAO\nfor allaě0and for any space dimensions ně1, wherecis a suitable positive constant.\nProof.Following the proof of Lemma 2.4we may conclude all the desired estimates in Lemma 2.6\nby the aid of the statements in Lemma 2.3. /square\nHence, we obtain decay properties of Sobolev solutions to ( 1).\nProposition 2.3. Letsě0andj“0,1. LetmP r1,2q. We assume the condition ną2m0δ1\nand the initial data\npu0,u1q P pLmXHs`2jδ2q ˆ pLmXHrs`2pj´1qδ2s`q.\nThen, the Sobolev solutions to ( 1) satisfy the following pLmXL2q ´L2estimates:\n››Bj\nt|D|supt,¨q››\nL2À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j}u0}LmXHs`2jδ2\n` p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j`δ1\nσ´δ1}u1}LmXHrs`2pj´1qδ2s`.\n3.Proofs\n3.1.The case a“1andb“0.In order to prove Theorem 1.1, the following auxilliary results\ncome into play.\nProposition 3.1. Letsě0andj“0,1. LetmP r1,2q. Then, the following estimates hold:\n›››Bj\nt|D|s´´\nK1\n0lowpt,xq ´F´1´\ne´t|ξ|2pσ´δ1qχpξq¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´2δ1\nσ´δ1}u0}Lm, (30)\nfor any space dimensions ně1, and\n›››Bj\nt|D|s´´\nK1\n1lowpt,xq ´F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1χpξq¯¯\n˚u1pxq¯\npt,¨q›››\nL2(31)\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´3δ1\nσ´δ1}u1}Lm, (32)\nfor any space dimensions ną2m0δ1.\nProof.In order to indicate Proposition 3.1, at first it is reasonable to prove the following estimates:\n|ξ|sχp|ξ|qˇˇˇBj\nt´xK1\n0pt,ξq ´e´t|ξ|2pσ´δ1q¯ˇˇˇ\nÀe´ct|ξ|2pσ´δ1q`\nt|ξ|s`2p2σ´3δ1q`2jpσ´δ1q` |ξ|s`2pσ´2δ1q`2jpσ´δ1q˘\n, (33)\nand\n|ξ|sχp|ξ|qˇˇˇBj\nt´xK1\n1pt,ξq ´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯ˇˇˇ\nÀe´ct|ξ|2pσ´δ1q`\nt|ξ|s`4pσ´2δ1q`2jpσ´δ1q` |ξ|s`2pσ´3δ1q`2jpσ´δ1q˘\n, (34)\nwherecis a suitable positive constant. Then, following the proof o f Lemma 2.4we may conclude\nall the desired statements. In the first step, let us consider (33) in the case j“0 to presentxK1\n0pt,ξq\nas follows:\nxK1\n0pt,ξq “´λ2eλ1t\nλ1´λ2“eλ1t´λ1eλ1t\nλ1´λ2.\nBy the mean value theorem we obtain\neλ1t´e´t|ξ|2pσ´δ1q“t`\nλ1` |ξ|2pσ´δ1q˘\nepωλ1´p1´ωq|ξ|2pσ´δ1qqt,ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 9\nwhereωP r0,1s. Consequently, we deriveˇˇˇeλ1t´e´t|ξ|2pσ´δ1qˇˇˇďtˇˇˇ´λ1´ |ξ|2pσ´δ1qˇˇˇe´mint´λ1,|ξ|2pσ´δ1qut.\nUsing Newton’s binomial theorem we re-write ´λ1for small frequencies in the form\n´λ1“1\n2|ξ|2δ1´\n1´b\n1´4|ξ|2pσ´2δ1q¯\n“1\n2|ξ|2δ1´\n1´´\n1´2|ξ|2pσ´2δ1q´2|ξ|4pσ´2δ1q´o`\n|ξ|4pσ´2δ1q˘¯¯\n“ |ξ|2pσ´δ1q` |ξ|2p2σ´3δ1q`o`\n|ξ|2p2σ´3δ1q˘\n.\nTherefore, we getˇˇˇeλ1t´e´t|ξ|2pσ´δ1qˇˇˇÀt|ξ|2p2σ´3δ1qe´t|ξ|2pσ´δ1q. (35)\nThanks to the asymptotic behavior of characteristic roots f or small frequencies, we may arrive at\nthe following estimates:\n|ξ|sχp|ξ|qˇˇˇxK1\n0pt,ξq ´e´t|ξ|2pσ´δ1qˇˇˇÀe´t|ξ|2pσ´δ1q`\nt|ξ|2p2σ´3δ1q` |ξ|2pσ´2δ1q˘\n. (36)\nHence, the estimate ( 33) is true for j“0. Now in oder to estimate ( 34) in the case j“0, we can\nre-write\nxK1\n1pt,ξq “eλ1t\nλ1´λ2“eλ1t\n|ξ|2δ1´1\n|ξ|2δ1´\n1´|ξ|2δ1\nλ1´λ2¯\neλ1t.\nHence, we have\nxK1\n1pt,ξq ´e´t|ξ|2pσ´δ1q\n|ξ|2δ1“eλ1t´e´t|ξ|2pσ´δ1q\n|ξ|2δ1´1\n|ξ|2δ1´\n1´|ξ|2δ1\nλ1´λ2¯\neλ1t.\nMoreover, we notice that it holds\n1´|ξ|2δ1\nλ1´λ2“1´|ξ|2δ1\na\n|ξ|4δ1´4|ξ|2σ“´4|ξ|2σ\na\n|ξ|4δ1´4|ξ|2σ`a\n|ξ|4δ1´4|ξ|2σ` |ξ|2δ1˘„ ´|ξ|2pσ´2δ1q.\nUsing again the estimate ( 35) and the asymptotic behavior of characteristic roots for sm all fre-\nquencies we may conclude\n|ξ|sχp|ξ|qˇˇˇxK1\n1pt,ξq ´e´t|ξ|2pσ´δ1q\n|ξ|2δ1ˇˇˇÀe´ct|ξ|2pσ´δ1q`\nt|ξ|s`4pσ´2δ1q` |ξ|s`2pσ´3δ1q˘\n.(37)\nTherefore, the estimate ( 34) is true for j“0. By analogous arguments we also obtain the following\nestimates for j“1:\n|ξ|sχp|ξ|qˇˇˇBt´xK1\n0pt,ξq ´e´t|ξ|2pσ´δ1q¯ˇˇˇÀe´t|ξ|2pσ´δ1q`\nt|ξ|2p3σ´4δ1q` |ξ|2p2σ´3δ1q˘\n, (38)\n|ξ|sχp|ξ|qˇˇˇBt´xK1\n1pt,ξq ´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯ˇˇˇÀe´ct|ξ|2pσ´δ1q`\nt|ξ|s`2p4σ´5δ1q` |ξ|s`4pσ´2δ1q˘\n.(39)\nThus, it is obvious that all the estimates from ( 36) to (39) imply immediately ( 33) and (34). This\ncompletes our proof. /square\nProposition 3.2. Letsě0andj“0,1. LetmP r1,2q. Then, the following estimates hold:\n›››Bj\nt|D|s´´\nK1\n0pt,xq ´F´1´\ne´t|ξ|2pσ´δ1q¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´2δ1\nσ´δ1}u0}Lm`e´ctt´n\n2δ1p1\nm1´1\n2q´s`jσ´a1\n2δ1}u0}Ha1m1\n`e´ctt´n\n2pσ´δ1qp1\nm2´1\n2q´s´a2\n2pσ´δ1q´j}u0}Ha2m2(40)10 TUAN ANH DAO\nfor any space dimensions ně1, and\n›››Bj\nt|D|s´´\nK1\n1pt,xq ´F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯¯\n˚u1pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´3δ1\nσ´δ1}u1}Lm`e´ctt´n\n2δ1p1\nm1´1\n2q´s`jσ´a1\n2δ1}u1}Hra1´σs`\nm1\n`e´ctt´n\n2pσ´δ1qp1\nm2´1\n2q´s´a2\n2pσ´δ1q´j}u1}Hra2´2δ1s`\nm2(41)\nfor any space dimensions ną2m0δ1. Herea1, a2ě0,m1, m2P r1,2sandcis a suitable positive\nconstant.\nProof.At first, let us re-write the expression in the L2norm of ( 40) as follows:\nBj\nt|D|s´´\nK1\n0pt,xq ´F´1´\ne´t|ξ|2pσ´δ1q¯¯\n˚u0pxq¯\n“ Bj\nt|D|s´´\nK1\n0lowpt,xq ´F´1´\ne´t|ξ|2pσ´δ1qχpξq¯¯\n˚u0pxq¯\n` Bj\nt|D|s`\nK1\n0high pt,xq ˚u0pxq˘\n` Bj\nt|D|s´\nF´1´\ne´t|ξ|2pσ´δ1q`\n1´χpξq˘¯\n˚u0pxq¯\n.\nWe notice that it holds\n|ξ|s`\n1´χpξq˘ˇˇBj\nte´t|ξ|2pσ´δ1qˇˇÀ |ξ|s`2jpσ´δ1qe´t|ξ|2pσ´δ1q.\nFollowing the proof of Lemma 2.4we derive\n›››Bj\nt|D|s´\nF´1´\ne´t|ξ|2pσ´δ1q`\n1´χpξq˘¯\n˚u0pxq¯\npt,¨q›››\nL2Àe´ctt´n\n2pσ´δ1qp1\nm2´1\n2q´j´s´a2\n2pσ´δ1q}u0}Ha2m2.(42)\nTherefore, combining ( 17), (30) and (42) we may arrive at ( 40). In an analogous way to get ( 42)\nwe also obtain\n›››Bj\nt|D|s´\nF´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1`\n1´χpξq˘¯\n˚u1pxq¯\npt,¨q›››\nL2Àe´ctt´n\n2pσ´δ1qp1\nm2´1\n2q´s´a2\n2pσ´δ1q´j}u1}Hra2´2δ1s`\nm2.(43)\nHence, combining ( 18), (32) and (43) we may conclude ( 41). Summurizing, the proof of Proposition\n3.2is completed. /square\nProof of Theorem 1.1.In order to prove the asymptotic profile of solutions to ( 1), we may estimate\n›››Bj\nt|D|s´\nupt,¨q ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q¯\npt,¨q›››\nL2\nÀ››Bj\nt|D|s`\nK1\n0pt,xq ˚u0pxq˘\npt,¨q››\nL2`››Bj\nt|D|s`\nK2\n0pt,xq ˚u0pxq˘\npt,¨q››\nL2\n`››|D|s`\nK2\n1pt,xq ˚u1pxq˘\npt,¨q››\nL2`›››Bj\nt|D|s´´\nK1\n1pt,xq ´F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯¯\n˚u1pxq¯\npt,¨q›››\nL2\n`›››Bj\nt|D|s´\nF´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\n˚u1pxq ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯¯\npt,¨q›››\nL2\n“:I1`I2`I3`I4`I5.\nCombining ( 13) and (17), (14) and (17), (16) and (18) we get\nI1À p1`tq´n\n4pσ´δ1q´s\n2pσ´δ1q´j}u0}A1\n0,\nI2À p1`tq´n\n4δ1´s\n2δ1´j´σ´2δ1\nδ1}u0}A1\n0,\nI3À p1`tq´n\n4δ1´s\n2δ1´j`1}u1}A1\n1,ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 11\nrespectively. By ( 41) we obtain\nI4À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´3δ1\nσ´δ1}u1}A1\n1.\nTo control I5, we shall apply Lemma 3.2. Indeed, at first it is clear that using Parseval-Plancherel\nformula and the change of variables |ξ| “t´1\n2pσ´δ1q|η|gives\n›››Bj\nt|D|sF´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q›››\nL2“›››|ξ|s`2jpσ´δ1q´2δ1e´t|ξ|2pσ´δ1q›››\nL2\n“t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1´ż8\n0|η|2s´4δ1`4jpσ´δ1qe´2|η|2pσ´δ1qdη¯1\n2\n“Ct´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1 (44)\nwith the constant C:“`ş8\n0|η|2s´4δ1`4jpσ´δ1qe´2|η|2pσ´δ1qdη˘1\n2ą0, where we used the condition\nną4δ1. Then, we employ Lemma 3.2to derive\nI5“o´\nt´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1¯\nastÑ 8.\nTherefore, from all the above estimates for Ikwithk“1,¨ ¨ ¨,5 and paying attention that it holds\n´n\n4δ1´s\n2δ1`1ă ´n\n4pσ´δ1q´s\n2pσ´δ1q`δ1\nσ´δ1with the condtion ną4δ1we may conclude immediately\n(5). Then, from ( 5) and (44) we may arrive at the desired upper bound in the following way :\n››Bj\nt|D|supt,¨q››\nL2ď›››Bj\nt|D|s´\nupt,¨q ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q¯›››\nL2\n` |P1|›››Bj\nt|D|sF´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q›››\nL2\nďC|P1|t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1`o´\nt´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1¯\nďC2t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1astÑ 8,\nwhereC2is a suitable positive constant. Moreover, to indicate the l ower bound, we can proceed as\nfollows:\n››Bj\nt|D|supt,¨q››\nL2ě |P1|›››Bj\nt|D|sF´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q›››\nL2\n´›››Bj\nt|D|s´\nupt,¨q ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q¯›››\nL2\něC|P1|t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1´o´\nt´n\n4pσ´δ1q´j´s\n2pσ´δ1q`δ1\nσ´δ1¯\něC1t´n\n4pσ´δ1q´s\n2pσ´δ1q´j`δ1\nσ´δ1astÑ 8,\nwhereC1is a suitable positive constant. Summurizing, Theorem 1.1is proved. /square\n3.2.The case a“0andb“1.In order to prove Theorem 1.2, the following auxilliary results\ncome into play.\nProposition 3.3. Letsě0andmP r1,2q. Then, the following estimates hold:\n›››|D|s´´\nKcos\n0lowpt,xq ´F´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘\nχpξq¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2`σ\n2δ2}u0}Lm, (45)12 TUAN ANH DAO\n›››|D|s´´\nBtKcos\n0lowpt,xq `F´1´\n|ξ|σe´t|ξ|2δsin`\nt|ξ|σ˘\nχpξq¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2}u0}Lm, (46)›››|D|s´´\nBtK1lowpt,xq ´F´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘\nχpξq¯¯\n˚u1pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2`σ\n2δ2}u1}Lm, (47)\nfor any space dimensions ně1, and\n›››|D|s´´\nK1lowpt,xq ´F´1´\ne´t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σχpξq¯¯\n˚u1pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2`σ\nδ2}u1}Lm, (48)\nfor any space dimensions nąm0σ.\nProof.To prove Proposition 3.3, we only need to show the following estimates:\n|ξ|sχp|ξ|qˇˇˇzKcos\n0pt,ξq ´e´1\n2t|ξ|2δ2cospt|ξ|σqˇˇˇÀte´ct|ξ|2δ2|ξ|s`4δ2´σ, (49)\n|ξ|sχp|ξ|qˇˇˇxK1pt,ξq ´e´1\n2t|ξ|2δ2sinpt|ξ|σq\n|ξ|σˇˇˇÀe´ct|ξ|2δ2`\nt|ξ|s`4δ2´2σ` |ξ|s`4δ2´3σ˘\n,(50)\nand\n|ξ|sχp|ξ|qˇˇˇBtzKcos\n0pt,ξq `e´1\n2t|ξ|2δ2|ξ|σsinpt|ξ|σqˇˇˇÀe´ct|ξ|2δ2`\nt|ξ|s`4δ2` |ξ|s`2δ2˘\n, (51)\n|ξ|sχp|ξ|qˇˇˇBtxK1pt,ξq ´e´1\n2t|ξ|2δ2cospt|ξ|σqˇˇˇÀe´ct|ξ|2δ2`\nt|ξ|s`4δ2´σ` |ξ|s`2δ2´σ˘\n,(52)\nwherecis a suitable positive constant. Then, following the proof o f Lemma 2.4we may conclude\nall the desired statements. Indeed, at first let us consider ( 49) to re-wirte\nzKcos\n0pt,ξq ´e´1\n2t|ξ|2δ2cospt|ξ|σq “e´1\n2t|ξ|2δ2`\ncos`\nt|ξ|σfp|ξ|q˘\n´cospt|ξ|σq˘\n. (53)\nUsing the mean value theorem we have\ncos`\nt|ξ|σfp|ξ|q˘\n´cospt|ξ|σq “ ´t|ξ|σ`\nfp|ξ|q ´1˘\nsin`\nt|ξ|σ`\nω1fp|ξ|q `1´ω1˘˘\n,(54)\nwhereω1P r0,1s. Moreover, it holds for small frequencies\nfp|ξ|q ´1“ ´|ξ|4δ2´2σ\n4`\n1`b\n1´1\n4|ξ|4δ2´2σ˘. (55)\nFrom (53) to (55), we imply immediately ( 49). In order to indicate ( 50), we get\nxK1pt,ξq ´e´1\n2t|ξ|2δ2sinpt|ξ|σq\n|ξ|σ“e´1\n2t|ξ|2δ2´sin`\nt|ξ|σfp|ξ|q˘\n|ξ|σfp|ξ|q´sinpt|ξ|σq\n|ξ|σ¯\n“e´1\n2t|ξ|2δ2\n|ξ|σ´sin`\nt|ξ|σfp|ξ|q˘\n´sinpt|ξ|σq\nfp|ξ|q´sinpt|ξ|σqfp|ξ|q ´1\nfp|ξ|q¯\n.(56)\nApplying again the mean value theorem leads to\nsin`\nt|ξ|σfp|ξ|q˘\n´sinpt|ξ|σq “t|ξ|σ`\nfp|ξ|q ´1˘\ncos`\nt|ξ|σ`\nω2fp|ξ|q `1´ω2˘˘\n,(57)ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 13\nwhereω2P r0,1s. Hence, combining from ( 55) to (57) gives (50). Next, to show ( 51), we notice\nthat it holds\nBtzKcos\n0pt,ξq `e´1\n2t|ξ|2δ2|ξ|σsinpt|ξ|σq\n“ ´e´1\n2t|ξ|2δ2´1\n2|ξ|2δ2cos`\nt|ξ|σfp|ξ|q˘\n` |ξ|σfp|ξ|qsin`\nt|ξ|σfp|ξ|q˘\n´ |ξ|σsinpt|ξ|σq¯\n“ ´e´1\n2t|ξ|2δ2´1\n2|ξ|2δ2cos`\nt|ξ|σfp|ξ|q˘\n` |ξ|σfp|ξ|q`\nsin`\nt|ξ|σfp|ξ|q˘\n´sinpt|ξ|σq˘\n` |ξ|σ`\nfp|ξ|q ´1˘\nsinpt|ξ|σq¯\n.(58)\nFrom (55), (57) and (58), we may arrive at ( 51). Finally, by using the relation\nBtxK1pt,ξq ´e´1\n2t|ξ|2δ2cospt|ξ|σq\n“e´1\n2t|ξ|2δ2´\n´ |ξ|2δ2´σsin`\nt|ξ|σfp|ξ|q˘\nfp|ξ|q``\ncos`\nt|ξ|σfp|ξ|q˘\n´cospt|ξ|σq˘¯\nand combining ( 54), (55) we may conclude ( 52). This completes the proof of Proposition 3.3./square\nProposition 3.4. Letsě0and LetmP r1,2q. Then, the following estimates hold:\n›››|D|s´´\nKcos\n0pt,xq ´F´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2`σ\n2δ2}u0}Lm`e´ctt´n\n2pσ´δ2qp1\nm1´1\n2q´s´a1\n2pσ´δ2q}u0}Ha1m1\n`e´ctt´n\n2δ2p1\nm2´1\n2q´s´a2\n2δ2`2δ2´σ\nδ2}u0}Ha2m2`e´ctt´n\n2δ2p1\nm3´1\n2q´s´a3\n2δ2}u0}Ha3m3, (59)\n›››|D|s´´\nBtKcos\n0pt,xq `F´1´\n|ξ|σe´t|ξ|2δ2sin`\nt|ξ|σ˘¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2}u0}Lm`e´ctt´n\n2pσ´δ2qp1\nm1´1\n2q´1´s´a1\n2pσ´δ2q}u0}Ha1m1\n`e´ctt´n\n2δ2p1\nm2´1\n2q´1´s´a2\n2δ2`2δ2´σ\nδ2}u0}Ha2m2`e´ctt´n\n2δ2p1\nm3´1\n2q´s´a3\n2δ2´σ\n2δ2}u0}Ha3m3,(60)\n›››|D|s´´\nBtK1pt,xq ´F´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘¯¯\n˚u1pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2`σ\n2δ2}u1}Lm`e´ctt´n\n2pσ´δ2qp1\nm1´1\n2q´1´s´a1\n2pσ´δ2q}u1}Hra1´2δ2s`\nm1\n`e´ctt´n\n2δ2p1\nm2´1\n2q´1´s´a2\n2δ2}u1}Hra2´2δ2s`\nm2`e´ctt´n\n2δ2p1\nm3´1\n2q´s´a3\n2δ2}u1}Ha3m3, (61)\nfor any space dimensions ně1, and\n›››|D|s´´\nK1pt,xq ´F´1´\ne´t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯¯\n˚u1pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2`σ\nδ2}u1}Lm`e´ctt´n\n2pσ´δ2qp1\nm1´1\n2q´s´a1\n2pσ´δ2q}u1}Hra1´2δ2s`\nm1\n`e´ctt´n\n2δ2p1\nm2´1\n2q´s´a2\n2δ2}u1}Hra2´2δ2s`\nm2`e´ctt´n\n2δ2p1\nm3´1\n2q´s´a3\n2δ2}u1}Hra3´σs`\nm3,(62)\nfor any space dimensions nąm0σ. Herea1, a2, a3ě0,m1, m2, m3P r1,2sandcis a suitable\npositive constant.14 TUAN ANH DAO\nProof.At first, let us re-write the expression in the L2norm of ( 59) as follows:\n|D|s´´\nKcos\n0pt,xq ´F´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘¯¯\n˚u0pxq¯\n“ |D|s´´\nKcos\n0lowpt,xq ´F´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘\nχpξq¯¯\n˚u0pxq¯\n` |D|s`\nK0high pt,xq ˚u0pxq˘\n` |D|s´\nF´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘`\n1´χpξq˘¯\n˚u0pxq¯\nIt holds that\n|ξ|s`\n1´χpξq˘ˇˇe´t|ξ|2δ2cos`\nt|ξ|σ˘ˇˇÀ |ξ|se´t|ξ|2δ2.\nFollowing the proof of Lemma 2.4we obtain\n›››|D|s´\nF´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘`\n1´χpξq˘¯\n˚u0pxq¯\npt,¨q›››\nL2Àe´ctt´n\n2δ2p1\nm3´1\n2q´s´a3\n2δ2}u0}Ha3m3.(63)\nTherefore, combining ( 22), (45) and (63) we may arrive at ( 59). In analogous ways to get ( 42) we\nderive the following estimates:\n›››|D|s´\nF´1´\ne´t|ξ|2δ2sin`\nt|ξ|σ˘`\n1´χpξq˘¯\n˚u0pxq¯\npt,¨q›››\nL2Àe´ctt´n\n2δ2p1\nm3´1\n2q´s´a3\n2δ2}u01}Ha3m3,(64)\n›››|D|s´\nF´1´\ne´t|ξ|2δ2cos`\nt|ξ|σ˘`\n1´χpξq˘¯\n˚u1pxq¯\npt,¨q›››\nL2Àe´ctt´n\n2δ2p1\nm3´1\n2q´s´a3\n2δ2}u0}Ha3m3,(65)\nand\n›››|D|s´\nF´1´\ne´t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ`\n1´χpξq˘¯\n˚u1pxq¯\npt,¨q›››\nL2Àe´ctt´n\n2δp1\nm3´1\n2q´s´a3\n2δ2}u1}Hra3´σs`\nm3.(66)\nThen, combining ( 22), (46) and (64) we may conclude ( 60). Combining ( 23), (47) and (65) we may\nconclude ( 61). Combining ( 23), (48) and (66) we may conclude ( 62). Summurizing, the proof of\nProposition 3.4is completed. /square\nProof of Theorem 1.2.In order to prove the asymptotic profile of solutions to ( 1), we may estimate\n›››|D|s´\nupt,¨q ´P1F´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯\npt,¨q¯\npt,¨q›››\nL2\nÀ››|D|s`\nKcos\n0pt,xq ˚u0pxq˘\npt,¨q››\nL2`››|D|s`\nKsin\n0pt,xq ˚u0pxq˘\npt,¨q››\nL2\n`›››|D|s´´\nK1pt,xq ´F´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯¯\n˚u1pxq¯\npt,¨q›››\nL2\n`›››|D|s´\nF´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯\n˚u1pxq ´P1F´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯¯\npt,¨q›››\nL2\n“:J1`J2`J3`J4.\nCombining ( 19) and (22), (20) and (22) we obtain\nJ1À p1`tq´n\n4δ2´s\n2δ2}u0}A2\n0,\nJ2À p1`tq´n\n4δ2´s\n2δ2´1`σ\n2δ2}u0}A2\n0,\nrespectively. By ( 62) we derive\nJ3À p1`tq´n\n2δ2p1\nm´1\n2q´1´s\n2δ2`σ\nδ2}u1}A2\n1.\nTo control J4, we shall apply Lemma 3.2. First, thanks toˇˇsin`\nt|ξ|σ˘ˇˇď1, we employ Parseval-\nPlancherel formula and Lemma 3.1to get\n›››|D|sF´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯\npt,¨q›››\nL2“›››|ξ|s´σe´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘›››\nL2ďCt´n\n4δ2´s\n2δ2`σ\n2δ2,(67)ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 15\nwhere we used the condition ną2σandCis a suitable positive constant. Then, by Lemma 3.2\nwe imply immediately\nJ4“o´\nt´n\n4δ2´s\n2δ2`σ\n2δ2¯\nastÑ 8.\nTherefore, from all the above estimates for Jkwithk“1,2,3,4 we may arrive at ( 7). Next, from\n(7) and (67) we may estimate the desired upper bound in the following way :\n››|D|supt,¨q››\nL2ď›››|D|s´\nupt,xq ´P1F´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯¯\npt,¨q›››\nL2\n` |P1|›››|D|sF´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯\npt,¨q›››\nL2\nďC|P1|t´n\n4δ2´s\n2δ2`σ\n2δ2`o´\nt´n\n4δ2´s\n2δ2`σ\n2δ2¯\nďC2t´n\n4δ2´s\n2δ2`σ\n2δ2astÑ 8,\nwhereC2is a suitable positive constant. Moreover, to indicate the l ower bound, using again\nParseval-Plancherel formula and the change of variables |ξ| “t´1\n2δ2|η|we have\n›››|D|sF´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯\npt,¨q›››\nL2“´ż\nRn|ξ|2ps´σqe´t|ξ|2δ2sin2`\nt|ξ|σ˘\ndξ¯1\n2\n“´ż\nRn|ξ|2ps´σqe´t|ξ|2δ2dξ´1\n2ż\nRn|ξ|2ps´σqe´t|ξ|2δ2cos`\n2t|ξ|σ˘\ndξ¯1\n2\n“t´n\n4δ2´s\n2δ2`σ\n2δ2´ż8\n0|η|2ps´σq`n´1e´|η|2δ2d|η| ´1\n2ż8\n0|η|2ps´σq`n´1e´|η|2δ2cos`\n2t1´σ\n2δ2|η|σ˘\nd|η|¯1\n2.\nApplying Lemma 3.3leads toż8\n0|η|2ps´σq`n´1e´|η|2δ2cos`\n2t1´σ\n2δ2|η|σ˘\nd|η| Ñ0 astÑ 8.\nIt follows immediately\n›››|D|sF´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯\npt,¨q›››\nL2ěC˚t´n\n4δ2´s\n2δ2`σ\n2δ2astÑ 8,\nwhereC˚isasuitablepositiveconstant. Herewenoticethattheinte gralş8\n0|η|2ps´σq`n´1e´|η|2δ2d|η|\nis a positive constant because of the condition ną2σ. Thus, we conclude\n››|D|supt,¨q››\nL2ě |P1|›››|D|sF´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯\npt,¨q›››\nL2\n´›››|D|s´\nupt,xq ´P1F´1´\ne´1\n2t|ξ|2δ2sin`\nt|ξ|σ˘\n|ξ|σ¯¯\npt,¨q›››\nL2\něC˚|P1|t´n\n4δ2´s\n2δ2`σ\n2δ2´o´\nt´n\n4δ2´s\n2δ2`σ\n2δ2¯\něC1t´n\n4δ2´s\n2δ2`σ\n2δ2astÑ 8,\nwhereC1is a suitable positive constant. Summurizing, Theorem 1.2is proved. /square\n3.3.The case a“1andb“1.In order to prove Theorem 1.3, the following auxilliary results\ncome into play.\nProposition 3.5. Letsě0andj“0,1. Let us assume δ1`δ2ąσ. Then, the following estimates\nhold formP r1,2q:›››Bj\nt|D|s´´\nK1\n0lowpt,xq ´F´1´\ne´t|ξ|2pσ´δ1qχpξq¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´2δ1\nσ´δ1}u0}Lm, (68)16 TUAN ANH DAO\nfor any space dimensions ně1, and\n›››Bj\nt|D|s´´\nK1\n1lowpt,xq ´F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1χpξq¯¯\n˚u1pxq¯\npt,¨q›››\nL2(69)\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´3δ1\nσ´δ1}u1}Lm, (70)\nfor any space dimensions ną2m0δ1.\nProof.The proof of this proposition is similar to the proof of Propo sition3.1. For this reason, we\nonly present the steps which are different. Then, we shall repe at some of the arguments as we did\nin the proof of Proposition 3.1to conclude the desired estimates.\nIndeed, following the proof of Proposition 3.1it is sufficient to prove the following estimate:\n|ξ|sχp|ξ|qˇˇˇxK1\n0pt,ξq ´e´t|ξ|2pσ´δ1qˇˇˇ\nÀe´ct|ξ|2pσ´δ1q`\nt|ξ|s`2p2σ´3δ1q`2jpσ´δ1q` |ξ|s`2pσ´2δ1q`2jpσ´δ1q˘\n, (71)\nwherecis a suitable positive constant. Recalling the characteris tic rootλ1we re-write as follows:\n´λ1“1\n2´`\n|ξ|2δ1` |ξ|2δ2˘\n´b`\n|ξ|2δ1` |ξ|2δ2˘2´4|ξ|2σ¯\n“p´\n1´a\n1´q2¯\n,\nwhere we introduce p:“1\n2`\n|ξ|2δ1` |ξ|2δ2˘\nandq:“2|ξ|σ\n|ξ|2δ1`|ξ|2δ2. It is clear that qă1 for small\nfrequencies. For this reason, applying Newton’s binomial t heorem gives\n´λ1“p´\n1´´\n1´1\n2q2´1\n8q4´o`\nq4˘¯¯\n“p´1\n2q2`1\n8q4`o`\nq4˘¯\n“|ξ|2σ\n|ξ|2δ1` |ξ|2δ2`|ξ|4σ\np|ξ|2δ1` |ξ|2δ2q3`o´|ξ|4σ\np|ξ|2δ1` |ξ|2δ2q3¯\n.\nHence, a standard calculation leads to\n´λ1´ |ξ|2pσ´δ1q“ ´|ξ|2σ`2pδ2´δ1q\n|ξ|2δ1` |ξ|2δ2`|ξ|4σ\np|ξ|2δ1` |ξ|2δ2q3`o´|ξ|4σ\np|ξ|2δ1` |ξ|2δ2q3¯\n“|ξ|2σ\n|ξ|2δ1` |ξ|2δ2´|ξ|2σ\np|ξ|2δ1` |ξ|2δ2q2´ |ξ|2pδ2´δ1q¯\n`o´|ξ|4σ\np|ξ|2δ1` |ξ|2δ2q3¯\n“|ξ|2σ\n|ξ|2δ1` |ξ|2δ2|ξ|2σ´ |ξ|2pδ1`δ2q`\n1` |ξ|2pδ2´δ1q˘2\np|ξ|2δ1` |ξ|2δ2q2`o´|ξ|4σ\np|ξ|2δ1` |ξ|2δ2q3¯\ną0\nfor small frequencies, where the condition δ1`δ2ąσplays an important role. This implies\nimmediately the following relations:\nmin/visualspace\n´λ1,|ξ|2pσ´δ1q(\n“ |ξ|2pσ´δ1qandˇˇˇ´λ1´ |ξ|2pσ´δ1qˇˇˇď|ξ|4σ\np|ξ|2δ1` |ξ|2δ2q3ď |ξ|2p2σ´3δ1q.\nBy analogous arguments as in the proof of Proposition 3.1we may arrive at ( 71). Therefore, the\nproof of Proposition 3.5is completed. /square\nFollowing the proof of Proposition 3.2we may conclude the following result by using Proposition\n3.5and Lemma 2.6.ASYMPTOTIC PROFILE OF SOLUTIONS TO STRUCTURALLY DAMPED σ-EVOLUTION EQUATIONS 17\nProposition 3.6. Letsě0andj“0,1. Let us assume δ1`δ2ąσ. Then, the following estimates\nhold formP r1,2q:\n›››Bj\nt|D|s´´\nK1\n0pt,xq ´F´1´\ne´t|ξ|2pσ´δ1q¯¯\n˚u0pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´2δ1\nσ´δ1}u0}Lm`e´ctt´n\n2pσ´δ2qp1\nm1´1\n2q´s´a1\n2pσ´δ2q´j}u0}Ha1m1\n`e´ctt´n\n2δ2p1\nm2´1\n2q´s´a2\n2δ2´j`2δ2´σ\nδ2}u0}Ha2m2`e´ctt´n\n2pσ´δ1qp1\nm3´1\n2q´s´a3\n2pσ´δ1q´j}u0}Ha3m3(72)\nfor any space dimensions ně1, and\n›››Bj\nt|D|s´´\nK1\n1pt,xq ´F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯¯\n˚u1pxq¯\npt,¨q›››\nL2\nÀ p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´3δ1\nσ´δ1}u1}Lm`e´ctt´n\n2pσ´δ2qp1\nm1´1\n2q´s´a1\n2pσ´δ2q´j}u1}Hra1´2δ2s`\nm1\n`e´ctt´n\n2δ2p1\nm2´1\n2q´s´a2\n2δ2´j}u1}Hra2´2δ2s`\nm2`e´ctt´n\n2pσ´δ1qp1\nm3´1\n2q´s´a3\n2pσ´δ1q´j}u1}Hra3´2δ1s`\nm3(73)\nfor any space dimensions ną2m0δ1. Herea1, a2, a3ě0,m1, m2, m3P r1,2sandcis a suitable\npositive constant.\nProof of Theorem 1.3.In order to prove the asymptotic profile of solutions to ( 1), we will follow\nthe proof of Theorem 1.1. At first, as in the proof of Theorem 1.1we recall the following estimates:\n›››Bj\nt|D|s´\nupt,¨q ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\npt,¨q¯\npt,¨q›››\nL2\nÀ››Bj\nt|D|s`\nK1\n0pt,xq ˚u0pxq˘\npt,¨q››\nL2`››Bj\nt|D|s`\nK2\n0pt,xq ˚u0pxq˘\npt,¨q››\nL2\n`››|D|s`\nK2\n1pt,xq ˚u1pxq˘\npt,¨q››\nL2`›››Bj\nt|D|s´´\nK1\n1pt,xq ´F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯¯\n˚u1pxq¯\npt,¨q›››\nL2\n`›››Bj\nt|D|s´\nF´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯\n˚u1pxq ´P1F´1´e´t|ξ|2pσ´δ1q\n|ξ|2δ1¯¯\npt,¨q›››\nL2\n“:I1`I2`I3`I4`I5.\nCombining ( 24) and (28), (25) and (28), (27) and (29) we derive\nI1À p1`tq´n\n4pσ´δ1q´s\n2pσ´δ1q´j}u0}A3\n0,\nI2À p1`tq´n\n4δ1´s\n2δ1´j´σ´2δ1\nδ1}u0}A3\n0,\nI3À p1`tq´n\n4δ1´s\n2δ1´j`1}u1}A3\n1,\nrespectively. By ( 73) we arrive at\nI4À p1`tq´n\n2pσ´δ1qp1\nm´1\n2q´s\n2pσ´δ1q´j´σ´3δ1\nσ´δ1}u1}A3\n1.\nThen, we shall repeat some of the arguments as we did in the pro of of Theorem 1.1to conclude\nthe desired estimates. Summurizing, Theorem 1.3is proved. /square\nAcknowledgments\nThe PhD study of the second author is supported by Vietnamese Government’s Scholarship (Grant\nnumber: 2015/911).\nAppendix A\nA.1. Useful lemmas18 TUAN ANH DAO\nLemma 3.1. Letně1,cą0,αą0andβPRsatisfyn`βą0. The following estimates hold\nfortą0:ż\n|ξ|ď1|ξ|βe´c|ξ|αtdξÀ p1`tq´n`β\nαandż\n|ξ|ě1|ξ|βe´c|ξ|αtdξÀt´n`β\nα.\nThe proof of this lemma can be found in [ 5].\nLemma 3.2. Letaě0. Let us assume v“vpxq PL1andφ“φpt,xqbe a smooth function\nsatisfying››|D|aφpt,¨q››\nL2Àt´αand››|D|a`1φpt,¨q››\nL2Àt´α´β,\nfor some positive constants α, β ą0. Then, the following estimate holds:\n›››|D|a´\nφpt,xq ˚vpxq ´´ż\nRnvpyqdy¯\nφpt,xq¯\npt,¨q›››\nL2“o`\nt´α˘\nastÑ 8,\nfor all space dimensions ně1.\nOne can be found the proof of this lemma in [ 11].\nLemma 3.3 (A variant version of Riemann-Lebesgue theorem) .Iff“fprq PL1andsuppfP\np0,8q, then it holds:\nż8\n0fprqe´zrdrÑ0as|z| Ñ 8within the half-plane Rezě0,\nthat is,ż8\n0fprqcosprτqdrÑ0andż8\n0fprqsinprτqdrÑ0asτÑ 8.\nReferences\n[1] P.T. Duong, Some results on the global solvability for st ructurally damped models with special nonlinearity,\nUkrainian Math. J. ,70(2019), 1395–1418.\n[2] M. D’Abbicco, M.R. Ebert, A classifiation of structural d issipations for evolution operators, Math. Methods Appl.\nSci.,39(2016), 2558–2582.\n[3] M. D’Abbicco, M.R. Ebert, A new phenomenon in the critica l exponent for structurally damped semi-linear\nevolution equations, Nonlinear Anal. ,149(2017), 1–40.\n[4] P.T. Duong, M. Kainane Mezadek, and M. Reissig, Global existence for semi-linear structurally damped σ-\nevolution models , J. Math. Anal. Appl., 431(2015), 569–596.\n[5] T.A. Dao, H. Michihisa, Study of semi-linear σ-evolution equations with frictional and visco-elastic da mping,\npreprint, arXiv:1906.04471, 2019.\n[6] M. D’Abbicco, M. Reissig, Semilinear structural damped waves,Math. Methods Appl. Sci. ,37(2014), 1570–1592.\n[7] T.A. Dao, M. Reissig, An application of L1estimates for oscillating integrals to parabolic like semi -linear struc-\nturally damped σ-evolution models , J. Math. Anal. Appl., 476(2019), 426–463.\n[8] T.A. Dao, M. Reissig, L1estimates for oscillating integrals and their application s to semi-linear models with\nσ-evolution like structural damping , Discrete Contin. Dyn. Syst. A, 39(2019), 5431–5463.\n[9] R. Ikehata, H. Michihisa, Moment conditions and lower bo unds in expanding solutions of wave equations with\ndouble damping terms, Asymptot. Anal. ,114(2019), 19–36.\n[10] R.Ikehata, A.Sawada, Asymptoticprofile ofsolutions f or waveequationswith frictional andviscoelastic damping\nterms,Asymptot. Anal. ,98(2016), 59–77.\n[11] R. Ikehata, H. Takeda, Asymptotic profiles of solutions for structural damped wave equations , J. Dyn. Differ.\nEqu.,31(2019), 537–571.\nTuan Anh Dao\nSchool of Applied Mathematics and Informatics, Hanoi Unive rsity of Science and Technology, No.1\nDai Co Viet road, Hanoi, Vietnam\nFaculty for Mathematics and Computer Science, TU Bergakade mie Freiberg, Pr ¨uferstr. 9, 09596,\nFreiberg, Germany\nE-mail address :anh.daotuan@hust.edu.vn"    },    {        "title": "2312.03070v1.Rescuing_the_Unruh_Effect_in_Lorentz_Violating_Gravity.pdf",        "content": "Rescuing the Unruh Effect in Lorentz Violating Gravity\nF. Del Porro,1, 2, 3, ∗M. Herrero-Valea,4,†S. Liberati,1, 2, 3, ‡and M. Schneider1, 2, 3, §\n1SISSA, Via Bonomea 265, 34136 Trieste, Italy\n2INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy\n3IFPU - Institute for Fundamental Physics of the Universe\nVia Beirut 2, 34014 Trieste, Italy\n4Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology,\nCampus UAB, 08193 Bellaterra (Barcelona) Spain\n(Dated: December 7, 2023)\nWhile the robustness of Hawking radiation in the presence of UV Lorentz breaking is well-\nestablished, the Unruh effect has posed a challenge, with a large literature concluding that even\nthe low-energy restoration of Lorentz invariance may not be sufficient to sustain this phenomenon.\nNotably, these previous studies have primarily focused on Lorentz-breaking matter on a conven-\ntional Rindler wedge. In this work, we demonstrate that considering the complete structure of\nLorentz-breaking gravity, specifically the presence of a hypersurface orthogonal æther field, leads to\nthe selection of a new Rindler wedge configuration characterized by a uniformly accelerated æther\nflow. This uniform acceleration provides a reference scale for comparison with the Lorentz-breaking\none, thus ensuring the persistence of the Unruh effect in this context. We establish this by calcu-\nlating the expected temperature using a Bogolubov approach, and by analyzing the response of a\nuniformly accelerated detector. We suggest that this resilience of the Unruh effect opens interesting\npossibilities towards future developments for using it as a tool to constrain Lorentz breaking theories\nof gravity.\n∗fdelporr@sissa.it\n†mherrero@ifae.es\n‡liberati@sissa.it\n§mschneid@sissa.itarXiv:2312.03070v1  [gr-qc]  5 Dec 20232\nI. INTRODUCTION\nThermodynamical aspects of gravity have become a cornerstone in our understanding of the most intimate\nnature of the fabric of space-time [1]. The idea that these are ingrained in the fine features of gravitation is\nby now so rooted that the possible loss of these features within alternative theories of gravity is sometimes\nused as a selection requirement for disfavouring them, if not ruling them out completely.\nIn this sense, in the past years there has been an intense debate on the possibility that the Unruh effect\nmight be lost in the context of Lorentz breaking theories [2–4], in open contrast with the by now acquired\ncommon wisdom of the robustness of Hawking radiation within the same framework [5–8]. Of course, such\ndistinctly different behavior can appear suspicious at first sight, given the deep link between these two effects.\nAll in all, a Schwarzschild static observer hovering above a very large black hole would perceive nothing else\nthan a Rindler wedge, and experience a thermal bath at a temperature that is in agreement with the Unruh\none.\nHowever, the different fate of the two phenomena when dealing just with ultra-violet (UV) Lorentz breaking\nmatter can be readily understood in terms of separation of scales: while the Hawking effect is characterized\nby an objective scale provided by the surface gravity of the black hole, which in turns is determined by the\nconserved charges of the black hole solution (e.g. mass and angular momentum for a Kerr black hole), no\nsuch scale is present in the Unruh effect, as the Rindler wedge temperature can always be rescaled to be\n1/2π[9]. The proper acceleration of a given Rindler hyperbola marks instead the equivalent of a Tolman\nfactor for the Hawking temperature. The absence of an intrinsic scale (akin to the black hole surface gravity\nκ) to be contrasted to the UV Lorentz breaking scale, say Λ, is what prevents the scale separation ( κ≪Λ)\nso crucial in preserving the Hawking effect for example in analog models of gravity [10, 11].\nSo it was no surprising that a stream of papers on the subject concluded that the question concerning\nthe robustness of Unruh radiation in the presence of UV Lorentz breaking matter had to be answered in\nthe negative [2–4]. Note that technically, the main culprit of such an apparent wipe out of the effect can be\ntraced down to the breakdown of the KMS condition of the Wightman function [3, 12]. However, the latter\nis a sufficient, but not necessary, condition for the detection of thermal features [13], leaving the possibility\nthat the analyses carried out so far might have been incomplete.\nIndeed, there is an important point missing in the literature. Within Riemannian geometry, describing a\nmodified dispersion relation requires an æther – i.e. a vector breaking local Lorentz invariance (LLI) – so\nthat matter fields can be coupled to it. In other words, one must provide a mechanism for the emergence of\nLorentz-breaking interactions. However, the dynamics of the æther was never taken into account in previous\ninvestigations, an approach that is not consistent with the assumption of background independence, which\nrequires a dynamical framework for the metric andthe æther, and could have an important effect when\ndiscussing the dynamics of matter within these settings.\nIn the present work we shall advocate this alternative point of view, i.e. that a full discussion of the Unruh\neffect within a quantum gravitational framework, possibly entailing the UV breakdown of local space-time\nsymmetries, has necessarily to include a full understanding of the gravitational sector as well1. In particular,\nwe shall consider the Unruh effect while taking into account the dynamics of the æther within a well defined\nmodel of quantum gravity based on UV Lorentz breaking, i.e. Hoˇ rava gravity.\n1See e.g. [14] for a similar point of view albeit in a different framework from the one discussed here3\nHoˇ rava–Lifshitz gravity [15] is constructed by appending the space-time manifold with a preferred foliation\nin spatial hypersurfaces. A preferred foliation a priori breaks LLI and diffeomorphism invariance, but in turn\nallows for operators in the action containing higher orders in spatial derivatives, which modify the graviton\npropagator and lead to power-counting renormalizability, while keeping only two time derivatives, avoiding\nthe presence of Ostrogradsky ghosts. Furthermore, the theory can be still described via a diffeomorphism-\ninvariant action by introducing a hypersurface orthogonal, unit-normalized, timelike vector field U: the\naforementioned æther [16, 17]. For a recent review of the many succeses and challenges of the theory see\n[18].\nThe most general power counting renormalizable Lagrangian compatible with Hoˇ rava’s proposal contains\nup to six derivatives and can be ordered by their number: L=L2+L4+L6. The higher derivative terms\nL4andL6are weighted by a UV scale Λ UV< M P, where MPis the Planck mass. Hence, at low energies\nE≪ΛUV, one can truncate the theory by retaining only L2. Noticeably, such low-energy limit of Hoˇ rava\ngravity – known as khronometric gravity [17] – coincides with a particular case of Einstein-æther (EA)\ngravity [19–21] (where the æther is restricted to be hypersurface orthogonal) whose Lagrangian is the most\ngeneral one for a metric and a unit timelike vector field, containing only up to two derivatives.\nMatter coupled to Hoˇ rava gravity is endowed with the same derivative structure in the UV, as a conse-\nquence of the presence of the preferred foliation. Assuming only CPT invariant terms [22], this leads to\nsuperluminal dispersion relations at high energies, above a scale Λ, which is neither necessarily related to\nΛUVnor to MP. This justifies truncating the gravitational action while retaining all terms in the matter ac-\ntion coupled to it. The presence of superluminal propagation immediately highlights how different the notion\nof causality may be in such a framework. Killing Horizons (KH) lose their meaning as causal boundaries, so\none can expect, for instance, that black hole phenomenology will change significantly.\nHowever, since all physical trajectories must respect the causal structure set by the foliation, it is possible\nto introduce a new notion of causal boundary. This is realized in particular for static black holes, where it\nwas found that there exists a particular slice of constant preferred time which is also a constant-radius leaf.\nIn this case, escaping the interior of the sphere defined by such a leaf becomes impossible for signals with\nany speed, as this would require to propagate backwards in preferred time. Such a causal boundary is called\naUniversal Horizon (UH) [23]. Recently, it has been proven that UHs admit thermodynamical properties,\nleading to Hawking radiation in a similar – but nonetheless different – fashion as in the standard general\nrelativistic case [5, 7, 8, 24]. Furthermore, it was also noted that low energy modes leaving the UH are always\nreprocessed in such a way that, for large black holes, the observed radiation at infinity will closely mimic –\nmodulo sub-leading corrections – the standard Hawking effect [6].\nGiven these recent insights about how and why Hawking radiation survives in Hoˇ rava–Lifshitz gravity,\nleads naturally to the question of the fate of the Unruh effect. In what follows, we consider a uniformly\naccelerated observer in flat space-time, accompanied by the corresponding æther obtained as a solution of the\nlow energy action of Hoˇ rava–Lifshitz gravity. We shall see that the dynamics of the æther implies dramatic\nconsequences for the Unruh effect, and foremost its survival – in close analogy with the Hawking effect.\nIndeed, we will see that, contrary to naive expectations, the metric-æther solution consistent with the\npresence of an accelerated detector coupled to the field leads to a non-trivial space-time, endowed with a\nUH. This imitates the near Killing horizon limit of a black hole geometry in Hoˇ rava gravity – as the Rindler\nwedge does for the Schwarzschild black hole in general relativity. We shall see then that the æther does carry\nan intrinsic scale – the UH surface gravity – which makes the Unruh effect robust against Lorentz breaking\neffects, again in close analogy with the Hawking effect.4\nTo connect our results to the Unruh effect in relativistic scenarios, we will also study the response of\nan Unruh-DeWitt detector in two different limits: the decoupling limit for the æther, when diffeomorphism\ninvariance is recovered; and the case when the field itself becomes relativistic, and its evolution is determined\nby the standard Klein-Gordon equation.\nThe paper is organized as follows. In section II, we discuss the causal structure implied by Hoˇ rava–Lifshitz\ngravity, review the relativistic Rindler wedge, and derive a non-relativistic analog of the latter, by solving\nthe equation of motion of the khronon clock, and later checking the satisfaction of the full set of equations.\nSection III, constitutes the main body of our work, where we shortly review the standard Unruh effect and\nderive its Lorentz-breaking version via Bogolubov coefficients relating the Minkowski vacuum with the one\nmeasured by an accelerated observer. We go further by studying the response of an Unruh-DeWitt detector\nin this situation, and the relativistic limit of the setting. Finally, we draw our conclusion in section IV.\nThroughout the analysis, we use natural units, mostly plus metric signature, and the tensor (index-free)\nnotation, such that g(X, Y) =gabXaYbdenotes the scalar product between two vectors XandY– but\ndivert from this whenever it seems instructive.\nII. GEOMETRICAL SETUP\nGeneral relativity introduces the mathematical model of space-time as a collection of events described by\nthe pair ( M, g) where Mis a connected, para-compact, smooth Hausdorff manifold, and ga Lorentz metric\nonM. Albeit being a very basic and generic construction, this might be challenged in some approaches\nto quantum gravity, where various cherished properties of general relativity are deemed emergent beyond\nthe Planck scale. This emergence opens up an immense playground to study the numerous proposals for a\nquantum theory of gravity by their phenomenological implications; the Unruh effect represents one of them.\nA. Causal Structure\nIn this article, we abandon the notion of Lorentz symmetry, which results as a prediction of several quantum\ngravity proposals [25], and study the existence of an effect analogous to the Unruh effect. Amongst all\npossibilities to break local Lorentz symmetry [26], we choose the introduction of a preferred frame. Canonical\nquantum gravity features a vast variety of proposals with preferred frames; one promising example is e.g.\nHoˇ rava-Lifshitz gravity [15], being unitary and power-counting renormalizable. For low energies, Hoˇ rava-\nLifshitz gravity is described by khronometric gravity [17], which is a subclass of Einstein-Æther gravity\n[19]. The latter introduces a preferred time direction that can be utilized to define a physical foliation of a\nmanifold into spatial leafs Σ. However, a more accurate point of view is the converse. That is, the manifold\nis assembled from spcelike submanifolds that are ordered along a preferred time direction. Effectively this\namounts to the introduction of an hypersurface orthogonal æther unit one-form Uthat defines the folitation\nsuch that we can vicariously work with Einstein-Æther gravity [27].\nBeing hypersurface orthogonal, the æther can be described in terms of a universal clock Θ, dubbed khronon,\nsuch that\nU=dΘ\n∥dΘ∥, (1)5\nwhere ∥dΘ∥=p\ng−1(dΘ,dΘ). From a geometrical point of view, Udefines a global time-orientation\nand fulfills a well defined equation of motion. Hence, we shall refine our definition of space-time as being\nthe triplet ( M, g, U ) with M=R×Σ an oriented foliated manifold, and Σ ∋⃗ xthe spatial submanifold\nwith induced metric γ=g+U⊗U. Although Einstein-Æther gravity admits a fully covariant formulation,\nsolutions will preserve this property, due to the foliation itself being a physical object. Indeed, this fact\nrestricts the allowed transformations Diff→FDiff , the so called foliation preserving diffeomorphisms\n(Θ, ⃗ x)7→(Θ′(Θ), ⃗ x′(Θ, ⃗ x)). (2)\nRequiring Θ′(Θ) to be monotonous preserves the orientation of the time direction, while x′(Θ, ⃗ x) acts as a\nfull, time-dependent diffeomorphism on the spatial submanifold.\nNote that losing the merits of Diff, we are facing a modified causal structure due to the possibility of\nsuperluminal propagation, i.e. physically allowed causal curves which would be considered to be space-like\nin terms of general relativity. Lorentz-violating theories feature causal cones that are widened up to the\nextent that they become a subset of Σ. In this extreme case, the causal structure simplifies brutally in the\nsense that the qualification “timelike” describes any future directed curves from one hypersurface to the\nnext, disregarding the inclination of their tangent vector. Only motions with tangent vector confined on TpΣ\nwill be called “spacelike”, while “not causal” captures any past directed motion [28]. In short, the opening\nof the lightcone leads to a Newtonian causal structure.\nThis fundamentally challenges the status of Killing horizons that are defined through ∥χ∥= 0, with χa\ntimelike Killing vector. In the specific case of the Unruh effect, Killing horizons mark the closures of the\nrelativistic Rindler wedge, and hence a proper understanding of the effect in the presence of Lorentz violations\nnecessarily passes by the understanding of the underlying causal structure. In the case under study, and\ndespite the possibility of superluminal propagation speed, there exist additionally universal horizons , defined\nthrough\ng(U, χ) = 0 ,as well as g(A, χ)̸= 0, (3)\nacting as a universal, outermost trapping surface [28, 29]. Note that the acceleration of the æther is given by\nA=LUUwithLUbeing the Lie derivative with respect to the aether. The first condition defines a surface\nof simultaneity which will not be attached to i0. As such, classical signals traveling with infinite speed will\nbe able to straddle but never escape from this surface, beyond which lurks a different folium of space-time.\nThe second condition ensures that the surface gravity κ=−g(A, χ)/2 of the horizon is non-zero, i.e. the\nhorizon remains non-degenerate, which guarantees that the æther can be integrated through this surface.\nThis is all realized in a simple but crucial way in Minkowski space-time, where the metric reads\ngM=−dt⊗dt+ dx⊗dx+ dE2 (4)\nand the uniform æther UM= dt. Here, we have chosen a single spatial direction xand denoted the remaining\nline element of the two-dimensional flat Euclidean plane by d E2for later convenience. This constitutes the\nfoliated Minkowski space-time ( M, gM, UM).\nIn the ( t, x) plane, stacking the leafs along Umotivates also the introductions of a spatial vector Sthat\nspans the x-direction on Σ and fulfills g(U, S) = 0 and g(S, S) = 1, thus S∈TpΣ. Together with U, the\nspatial vector Sdefines a preferred frame accustomed to the physical foliation. Note, the spatial vector\nadmits an ambiguity since its orientation can be chosen to point either inwards or outwards, with respect to\nspatial infinity i0; we fix the orientation to outwards pointing at spatial infinity.6\nB. The Rindler patch in Lorentz breaking gravity\nAfter the above brief summary of the causal structure induced by the metric and the æther, we focus now\non constructing the geometrical arena of the Unruh effect, i.e. the Rindler wedge. It will be evident to the\ncareful reader that such a concept is far from obvious, given the dramatic modifications of the space-time’s\ncausal structure we just discussed. First of all, one is forced to refute the idea of a wedge limited by the\nKilling horizons, due to the occurrence of infinite speed signals. Nevertheless, we still want to deal with a\nset up which is as close as possible to the standard Unruh effect, and which can be smoothly reduced to it\nin some suitable relativistic limit, where the æther is decoupled from the uniformly accelerated detector and\nthe test field. In order to do so, we wish the space-time and its foliation to admit a boost Killing vector, or\nin other words, they need to be invariant under application of the boost vector. That is, we shall ask the\nmetric to be the Minkowski metric with a boost invariant æther. Then, we shall also place our ideal detector\non the usual orbit of the boost Killing vector and investigate the vacuum it experiences.\n1. Rindler Wedge: recap of the relativistic construction\nLet us start by briefly reviewing the relativistic Rindler wedge to extract its internal geometrical logic\nwhich can then be utilized to identify an according patch of space-time in Lorentz-violating gravity. The\nrelativistic Rindler space-time ( R, gR) is a subset of Minkowski space-time, experienced by a uniformly\naccelerated observer. In Rindler coordinates, its metric takes the form\ngR=e2aξ(−dη⊗dη+ dξ⊗dξ) + dE2 (5)\nwhere ξdescribes the boost direction, and ais a bookkeeping parameter with the dimension of an acceleration.\nFor all our purposes, the problem at hand effectively reduces to a two-dimensional problem, and therefore\nwe can relinquish the Euclidean plane E2in our subsequent analysis.\nThe metric gRis static and admits a globally timelike Killing vector χ=∂ηwhich in Minkowski space-time\ncan be easily recognized to be the boost Killing vector χ=a(x∂t+t∂x). Because of the relativistic causal\nstructure, the observer is only able to chart a certain region, i.e. gR, of the full Minkowski space-time gM.\nThe two metrics are related by the coordinate transformation (where E2transforms via the identity map)\nη(t, x) =1\naartanh\u0012t\nx\u0013\n, ξ(t, x) =1\n2aln\u0000\na2(x2−t2)\u0001\ny=y;z=z. (6)\nThe coordinates ( η, ξ) are adapted to an observer on an orbit of the boost Killing vector in flat space-time,\nand they are only defined on a restricted region due to their logarithmic nature. Indeed, it is easy to see\nthat in Minkowski space-time, the trajectory of an observer at ξ= const is x2−t2=a−2\np, i.e. a hyperbolic\ntrajectory with proper acceleration ap=a e−aξ.\nSince the Rindler space-time is just a section of Minkowski, it is Riemann-flat and its asymptotic regions\ncorrespond to the future null infinity I+, and past null infinity I−, because the space-time is asymptotically\nsimple and empty. Hence, the Penrose diagram for the Rindler dissection of Minkowski space-time – cf. figure\n1 – comprises of four regions, the left and right wedges, RandL, and a future and past wedge, Fand\nP. Those regions are separated by a Killing horizon, that is, a bifurcating, non-degenerate, null 3-surface\ndefined by the Killing vector χbecoming null. In the coordinate patch (5), this condition holds at\ngR(χ, χ) = 0 ⇔ g00χ0χ0=−e2aξ→0⇔ ξ→ −∞ , (7)7\nℐ−ℐ+i0i+\ni−ℛℋ+ℋ−ℛℒ𝒫ℱℳ\nFIG. 1: Penrose diagram for the right Rindler wedge R(colored area in the Penrose diagram of the full\nMinkowski manifold M) with the hyperbolic trajectory of the relativistic accelerated observer ranging from\ni−toi+and rapidity aη. The future and past horizons H+andH−determine the closures of the part of\nthe manifold that borders to FandP. Each point in this figure represents a Euclidean flat plane.\nsuch that the horizon creates an asymptotic boundary and, therefore, determines the closure of the Rindler\nwedge. The feature that a coordinate patch asymptotes to the Killing horizon is familiar from Schwarzschild\nspace-time in tortoise coordinates.\nAfter analyzing the construction of the relativistic Rindler patch, we extract the following properties:\nthe Rindler patch is a g-complete, globally hyperbolic Riemann-flat space-time that admits a boost Killing\nvector. These conditions will serve as a blueprint to construct the Rindler patch in a Lorentz breaking\nsetting.\n2. The non-relativistic Rindler patch\nIn contrast to the general relativistic case, in Lorentz-breaking theories we face a Newtonian causal struc-\nture, in which the Killing horizon, as a null surface, becomes permeable in both ways for signals that travel\nwith propagation speed cs>1. Hence, the non-relativistic version of the Rindler wedge should be larger\nthan the corresponding relativistic one, and include the latter. However, the foliated Rindler patch Rwill\nbe a subset of the foliated Minkowski manifold Mand will fail to cover it completely, as we will see.\nLet us start by recalling the space-time triplet ( R, gR, UR) in Einstein-Æther theory from II A, and de-\nmanding the following properties to be satisfied by a non-relativistic version of the Rindler wedge\n•Boost invariance: LχgR≡0, and LχUR=LχSR≡0\n•Riemann flatness: Riem = 0\nMoreover, we demand the pair ( gR, UR) to solve the equations of motion of Einstein-Æther gravity [19].\nTo derive the ingredients of our space-time, we use the formulation of Einstein-æther theory in terms of8\nirreducible representations of the SO(3) Lorentz subgroup that leaves the æther invariant [21]\nSeæ=−1\n16πG⋆Z\nMd4xp\n−det(g)\u0000\nR+cθθ2+cσσ2+cAA2\u0001\n, (8)\nwhere Rdenotes the Ricci scalar curvature, ciare dimensionless couplings, and we introduced the expansion\nof the aether θ=∇U, its shear σ=∇ ⊗U−θγ, and its acceleration A=LUU. Note that the “bare”\ngravitational constant G⋆here is not a priori equal to the observed Newton constant Gn, since the two are\nrelated via a combination of the couplings (see e.g. [21]). In principle, this action would also admit a term\ncωω2, however, we choose to work within the Hoˇ rava gravity framework, that is taking the limit cω→ ∞ ,\nwhich implies a vanishing twist and a coincidence of the Einstein-æther action with L2[21]. This action\ndefines the so called khronometric gravity theory.\nWe can now deduce the equations of motion via coupling the action (8) to matter in a minimal way, hence\n[30, 31]\nGab= 8πG⋆(Tm\nab+TΘ\nab), (9)\nwhere Gabdenotes the Einstein tensor, Tmabis the stress energy tensor of the matter fields, and TΘabis the\nkhronon’s stress-energy tensor\nTΘ\nab=∇c(gcdU(bFa)d−gcdFd(aUb)−F(ab)Uc)+cAAaAb+(Ud∇cFcd−cAAcAc)UaUb+1\n2gabLeæ+2E(aUb),\n(10)\nwhere we have defined Leæto be the Lagrange density of (8) and\nFab=cθθgab+cσ∇bUa+cAAbUa,\nEa=γab(∇cFcb−cAAc∇bUc).(11)\nThe variation of the action (8) with respect to Θ leads to the scalar field equation2\n∇a\u0012Ea\n∥dΘ∥\u0013\n= 0. (12)\nNow, we can use the properties attributed to the relativistic Rindler space-time ( R, gR) in order to construct\nthe Lorentz-violating space-time ( R, gR, UR). Our strategy focusses on solving (12), which is a simple scalar\nequation, and later we study whether the resulting space-time satisfies the full set of equations of motion\n(9) via evaluation of the stress-energy tensor of the æther. Here, we only sketch the results and refer to\nAppendix A for explicit calculations.\nSince all relevant physics takes place in a (1 + 1)-dimensional submanifold, we adopt the following ansatz\nto determine our space-time’s building blocks\ngW=W2(τ, ρ)(−dτ⊗dτ+ dρ⊗dρ) + dE2, U W=W(τ, ρ)dτ, (13)\nwith W(τ, ρ) a conformal factor. This ansatz reflects the dimensionality of the physical setup: since the\nobserver’s trajectory is embedded in a (1 + 1)-dimensional submanifold spanned by UandS, we use an\nadapted coordinate system {τ, ρ}such that Uassumes the form in (13) and S=W(τ, ρ)dρ. This is\n2Note that due to Bianchi identities, (9) and (12) are not independent, see e.g. [31].9\ncomplemented with the statement that all two-dimensional metrics are conformally flat. Hence, the (1 +\n1)-dimensional submanifold containing the trajectory of the observer can be decomposed into the {U, S}\northonormal basis as\ng=−U⊗U+S⊗S+ dE2. (14)\nImposing boost invariance and Riemann flatness, the khronon equation of motion (12) leads to a single\nsolution (and its time correspondent reversal)\nW±(τ, ρ) =1\n¯a(ρ±τ), (15)\nwhich we can insert into (13) to arrive at\ngW=−dτ⊗dτ+ dρ⊗dρ\n¯a2(ρ±τ)2+ dE2, U W=dτ\n¯a(ρ±τ). (16)\nAs required, this solution is Lie dragged with respect to the Killing vector χ=τ∂τ+ρ∂ρ, that is, the boost\nKilling vector3, see below. In (15), ¯ aarises as an integration constant, but it is straightforward to see that\nit encodes a geometrical meaning, corresponding to the norm of the æther acceleration ∥A∥= ¯a, as well as\nto its expansion θ=∇U= ¯a.\nThe metric (16) is Riemann flat, and therefore it is always possible to introduce a coordinate change\n(τ, ρ)→(t, x) to the Minkowski metric (4) taking the form\nτ(t, x) =x−t\n2+1\n2¯a2(t+x),and ρ(t, x) =t−x\n2+1\n2¯a2(t+x). (17)\nIn the chart parametrized by ( t, x), the Killing vector χ=τ∂τ+ρ∂ρbecomes, as anticipated, the usual boost\ngenerator χ= ¯a(x∂t+t∂x). From that, the æther can be easily deduced given the shape of W(τ(t, x), ρ(t, x)).\nTo draw a closer comparison between this non-relativistic Rindler manifold Rand the relativistic R, it is\nconvenient to perform the coordinate transformation (6) to the chart ( η, ξ), in which the metric is given by\n(5). The resulting geometry is\ngR=e2¯aξ(−dη⊗dη+ dξ⊗dξ) + dE2, U R=−e2¯aξ+ 1\n2dη+e2¯aξ−1\n2dξ , (18)\nwhile the boost generator becomes χ=∂η, and the Killing horizon is placed at ξ→ −∞ . As we shall\nsee, this space-time incorporates the usual Rindler wedge fully. However, its different causal structure\nallows for trajectories crossing the Killing horizon in both ways and, as such, the foliation extends into\nthe neighboring regions of R. Here, we stress again the role of the parameter ¯ a. While it is usually just\na bookkeeping parameter, in this case, it represents a physical scale which arises from the gravitational\nbackground, associated with the expansion and acceleration of the aether.\nFinally, let us point to an alternative derivation of this geometry, by focusing on the near-horizon limit of\nan Einstein-Æther-Schwarzschild black-hole [23] with metric and æther\ngS=−\u0012\n1−2M\nr\u0013\ndt⊗dt+dr⊗dr\n1−2M\nr+r2dS2, U S=\u0012\n1−M\nr\u0013\ndt+M\nr−2Mdr , (19)\n3Note also, that such solution admits an additional pair of Killing vectors K±=∂τ±∂ρwhich generate the past and future\nKilling horizons.10\nwhere dS 2is the line-element of the two sphere. Retaining the leading order in an ( r−2M)-expansion, and\nrelabelling afterwards r−2M= 2Me2¯aξandt=η, then (19) becomes\ngS=e2¯aξ(−dη⊗dη+ dξ⊗dξ) + 4M2dS2, U S=−e2¯aξ+ 1\n2dη+e2¯aξ−1\n2dξ (20)\nwhich reduces to (18) in the large mass limit, that implies S 2→E2.\n3. The Rindler patch\nComing back to (18), we introduce at this point a change of variables through 2¯ aϵ(ξ) =e2¯aξ, in order to\ncover the region beyond the Killing horizon Fas well, finding\ngR= 2¯aϵ(−dη⊗dη+ dξ⊗dξ) + dE2, U R=−2¯aϵ+ 1\n2dη+2¯aϵ−1\n2dξ, (21)\nThe extension from RintoFthus requires a sign change in ϵ\n2¯aϵ(ξ) =\n\n+e2¯aξinR,\n−e2¯aξinF.(22)\nTo determine the extent of this manifold into the region F, it is convenient to regard ϵas a spatial coordinate.\nWe can now verify the existence of a universal horizon at ξofor the accelerated æther solution with respect\ntoϵ(ξ)\ngR(χ, U) =−1 + 2¯ aϵ(ξo)\n2= 0, (23)\nwhich admits the solution ϵ(ξo) =−1/(2¯a). Associated to the universal horizon, we also compute its surface\ngravity\nκuh=−1\n2gR(A, χ)\f\f\f\f\nuh=¯a\n2, (24)\nwhich is again characterized by the æther’s acceleration ¯ a, being the only scale in the problem.\nSince ϵ <0 atξo, the universal horizon lies in F, which means in turn that the foliation of Rextends\ninto the relativistic wedge F– since the previous condition can never be met in R. To substantiate this\nresult, we solve (1) for the khronon fields coordinating the foliation in both patches, RandL, in Minkowski\ncoordinates – foliation leafs corresponding to constant khronon surfaces. We find\nΘR(t, x) =−1\n¯aln\u0012¯a2(x2−t2) + 1\n¯a(x+t)\u0013\n,ΘL(t, x) =−1\n¯aln\u0012\n−¯a2(x2−t2) + 1\n¯a(x+t)\u0013\n. (25)\nThe khronon leafs accumulate exactly at the hyperbola t2−x2= 1/¯a2, which corresponds to the location of\nthe universal horizon. The first observation to emphasize here is the existence of two solutions that constitute\nthe foliation of the right and the left Rindler patches ( RandL, respectively) until the universal horizon, as\nwell as the future and past patches ( FandP). Second, due to the sign difference in the argument of the\nlogarithm, both foliations are oriented in opposite directions. If, for instance, Θ Ris future oriented, then Θ L\nis past oriented, and vice versa. More colloquially speaking, the clock Θ Rticks in opposite direction with\nrespect to Θ L. Since the lapse of the foliation NR=−gR(χ, U) flips sign across the universal horizon, Fis11\n-505-505𝕃𝔽\n-10-50510-10-50510ℝℙ\nFIG. 2: Foliated area of Minkowski space-time through an accelerated æther. The right, green panel shows\nthe lines of constant Θ R, which generate a future directed æther URand charts RandP. The left, red\npanel depicts the folium generated by constant Θ Llines, and covers the regions LandF. The\ncorresponding æther ULis past-directed with respect to UR.\nfoliated according to Θ LandPwith respect to Θ R. As can be seen in Figure 2, foliation leafs accumulate\nfrom both sides at the universal horizon, describing a natural closure that limits the Rindler patches.\nFigure 2 also shows that the accelerated æther cuts the space-time into four pieces. We observe that the\nfoliation of Rextents into the region F, yielding a very particular form of the right patch that resembles the\nshape of the exterior foliation of an Einstein-Æther-Schwarzschild black-hole [6]. Similar to the black hole\ninterior, the change of sign in gR(χ, U) ensures that the manifold is C2(R) across the horizon. Noticeably,\nthis property is necessary if one wants to define a consistent quantum field theory on this space-time [24].\nIt should be emphasized that the space-time, and thus the foliation, are invariant under the action of the\nboost vector χ. For foliated manifolds, this implies that the æther itself is invariant. As a consequence,\ngR(χ, U) is independent of the time coordinate ηand the æther is Lie dragged with respect to ∂η. This\ndetermines our foliation uniquely and constitutes the Rindler patch.\nAn analog patch Lcan be found by considering the outside red part in Figure 2 with adjacent region P\ngiven by the lower green part. Altogether, these four parts cover all of Minkowski space-time like in the\nrelativistic scenario. The corresponding Penrose diagram can be seen in Fig. 3.\n4. The Stress-Energy Tensor\nAfter obtaining a solution to equation (12) and studying its properties, we focus now on ensuring that this\nspace-time is also a solution of the full set of gravitational equations of motion (9). A priory, the coincidence\nof our æther flow solution with the one obtained in the near-horizon limit of the Schwarzschild black-hole\nsolution appeared to guarantee this automatically because of the linearity in ϵof the latter. However, this is\nactually not true. The non-linear dependence on Uleads to a non-zero result when retaining only the first\norder in the near horizon expansion of U. Indeed, one can easily check that although Riem= 0, TΘ̸= 0 and12\nℐ−ℐ+i0i+\ni−ℝℋ∞ℐ−ℝ𝕃ℙ𝔽𝕄\nFIG. 3: Penrose diagram for the right Rindler patch R(colored area in M) with hyperbolic trajectory of\nthe relativistic accelerated observer ranging from i−toi+with rapidity ¯ aη. The future universal horizon\nH∞determines the closures of the part of the manifold that borders to F. The Killing horizon is displayed\nby the dotted null line while the constant khronon leafs are the dashed purple lines. It is visible that\nR⊂Rfor identical hyperbolae. The left Rindler patch Lcan be obtained by reflecting Rat the point\nwhereI+meets the Killing horizon.\nhence (9) seem to be not satisfied.\nIn the case of the relativistic Rindler wedge R, we face a very similar issue regarding the space-time\ngeometry sourced by the stress-energy tensor. If one assumes matter fields in R, their stress-energy tensor is\nfound to admit a nonzero vacuum expectation value [32]. This apparent tension is resolved when considering\nthe stress-energy tensor in the left Rindler wedge Las well. Then, one can show that the value of the\nstress-energy tensor in Ris exactly compensated by the one in L. In doing so, one introduces non-localities\ninto the system, that tell the observer in Rabout the existence of Lin the full manifold.\nIn the non-relativistic scenario, our space-time comprises of an additional field, the æther, even in the\npurely gravitational case. If we consider the action Seæin (8), the stress-energy tensor for the gravitational\nsector can be derived as usual by varying Seæwith respect to the inverse metric, hence, we find\nδSeæ=−1\n16πG⋆Z\nMd4xp\n−det(g) (G−TΘ)δg−1, (26)\nwhere we have neglected a possible matter contribution, thus working on vacuum.\nThe equation of motion (12) is, as usual, given by setting δSeæ= 0. For M=M, the Minkowski manifold,\nwe find that this is trivially zero because we have a uniform æther accompanying a flat metric. However,\nthis is not the case anymore for the uniformly accelerated æther which sources the non-relativistic Rindler\nwedge, since TΘ̸= 0. Naively, this seems to indicate that Riem= 0 is not a valid solution anymore, and that\nthe construction through this work is thus non-sense.\nNevertheless, this is not sure, because having two Rindler patches requires splitting the integration of the\nfull manifold into two separate integrals, over the RandLregions respectively, each with non-zero individual13\nstress-energy tensor. The variation principle yields a global statement, and therefore, we need to combine\nboth integrals associated to the variation w.r.t. the inverse metric g−1in each wedge\n0 =δSeæ=Z\nRd4x NRp\ndet(γR) (GR−TR\nΘ)δg−1\nR+Z\nLd4x NLp\ndet(γL) (GL−TL\nΘ)δg−1\nL, (27)\nwhere we introduced the lapse NR=−gR(χ, U) inRandNL=−gL(χ, U) inL. Since both regions RandL\nare geometrically identical, we require δg−1\nL≡δg−1\nR. However, as mentioned before, the æther’s orientation\nchanges while preserving the same flow, i.e. NR=−NL. We can then conflate in one the two integrals. If\none now considers that both patches are Riemann flat, and that GR=GL≡0, the equations of motion\nreduce to\nTL\nΘ(−x)−TR\nΘ(x) = 0 . (28)\nWe recognize again that the split into two disjoint regions triggers non-localities which highlights the fact\nthat the system in Rrequires the counterpart L. This allows to reinstate the total energy balance and\nensures that the non-relativistic Rindler wedge is a solution of the full set of equations of motion.\nIII. THE UNRUH EFFECT\nAfter having constructed the Rindler patch R, we are ready to investigate the Unruh effect. The relativistic\ncalculation involves a Bogolubov transformation that compares the vacuum state defined by an inertial\nobserver with the one defined by an accelerated observer. In other words, the vacuum state on the full\nMinkowski manifold Mis contrasted with the vacuum restricted to R. For the sake of clarity, we will briefly\nreview the relativistic calculation, following mainly [33], before we analyze the non-relativistic Unruh effect\nalong similar lines.\nA. The Unruh Effect in General Relativity\nThe Unruh effect consists in the detection of a thermal bath by a uniformly accelerated (Rindler) observer\nin Minkowski vacuum. Its derivation usually follows from the confrontation of the vacuum associated to\nan inertial observer in Minkowski with that associated to the second quantization of the field in a basis of\nmodes appropriated for the Rindler observer. More specifically, the vacuum |0⟩Mis defined by the inertial\nobserver that lives in Minkowski space-time ( M, gM) through ˆ ak|0⟩M= 0, with ˆ akbeing the annihilation\noperator for a mode with momentum k. Together with the creation operator ˆ a†\nk, this defines the number\noperator ˆNM\nk= ˆa†\nkˆakthat can be used to define the Fock space for the modes uk(t) solution of the field\nequation □ϕ(t, x) = 0\nϕ(t, x) =Z\nRdk\n2πuk(t)eikx, (29)\nwhere ∂2\ntuk(t)+k2uk(t) = 0. Of course, in Minkowski space-time the mode functions are just given by plane\nwaves uk(t) =ei|k|t/p\n2|k|, where the normalization is performed via the Wronskian.\nIn the same manner, we can quantize the system in the accelerated observer frame. To quantize ϕin the\nRindler wedge, we take advantage of the fact that the relevant physics takes place in a (1 + 1)-dimensional14\nsubmanifold and restrict our treatment correspondingly. It has been shown that including E2will not spoil\nthis analysis [34].\nWorking in a (1 + 1)-dimensional submanifold offers various advantages, such as the conformal flatness of\nthe metric, as well as the conformal covariance of the minimally coupled massless scalar field. Altogether, this\nimplies that the spectrum of the scalar field will also satisfy a Klein-Gordon equation in the Rindler wedge,\nalbeit in a different coordinate system. The latter, starting from the Klein-Gordon equation in Minkowski\ncoordinates, is defined by nothing else than the inverse of the coordinate transformation we introduced via\nEq. (6)\nt(η, ξ) =eaξ\nasinh( aη), x (η, ξ) =eaξ\nacosh( aη). (30)\nNote that this change of coordinates is well defined for real values of ηandξin the portion x >|t|,\nthat is, in R. We thus define the vacuum state |0R⟩through the annihilation operator ˆbR,kwith conjugate\ncreation operator ˆb†\nR,k. Using the mode solutions of the Klein-Gordon equations in the Rindler wedge,\nvR,k(η, ξ) =ei|k|η+ikξ/p\n2|k|, the field can be decomposed in this case as\nˆϕ(η, ξ)|R=X\nkˆbR,kvR,k(η, ξ) +ˆb†\nR,kv∗\nR,k(η, ξ). (31)\nHowever, this right wedge basis is incomplete, as it cannot cover the whole Minkowski space-time, due to\nthe support of the coordinates ( η, ξ). As such, it requires to be completed by a similar set of modes on the\ncomplementary left Rindler wedge Lin order to be compared with the Minkowski modes via Bogolubov\ntransformations. To achieve this, one can employ that boost Killing vector χdefines an additional timelike\nsymmetry in the left Rindler wedge, where x <−|t|. Indeed, in L, we can quantize the field analogously,\nbut now using modes vL,k(η, ξ) that live exclusively in the left wedge\nˆϕ(η, ξ)|L=X\nkˆbL,kvL,k(η, ξ) +ˆb†\nL,kv∗\nL,k(η, ξ). (32)\nNow, a crucial point in the standard derivation of the Unruh effect lies in recognizing that the above\nmodes are not analytical on the two Killing horizons [35]. Following Unruh’s original article [36] one can\nnonetheless use a specific complex combination of left and right modes which then becomes analytical across\nthe Killing horizon. Those modes define positive and negative frequency modes with support in R∪F(an\nanalog combination holds for L∪P)\nψ+\nk(η, ξ) =eπω\n2avR,k(η, ξ) +e−πω\n2av∗\nL,−k(η, ξ), ψ−\nk(η, ξ) =e−πω\n2av∗\nR,−k(η, ξ) +eπω\n2avL,k(η, ξ), (33)\nwhere we have used the relativistic dispersion relation ω=|k|. Moreover, it can be shown that ψ+\nk(η, ξ) is a\nsuperposition of only positive frequency Minkowski modes (see e.g. [37, 38]). In analogy, ψ−\nk(η, ξ) is shown\nto be a superposition of negative frequency Minkowski modes.\nOne can then express the creation and annihilation operators within the basis ψ+\nk(η, ξ) through a Bogol-\nubov transformation and act on |0⟩M. To this aim, we first invert (33) by writing ˆbR,kin terms of ˆ akand\nˆa†\nk, before we compute the occupation number measured by the Rindler observer to be\n⟨0|ˆNR\nk|0⟩M=⟨0|ˆb†\nR,kˆbR,k|0⟩M=1\ne2πω\na−1, (34)\nwhich results in a Bose–Einstein distribution. From (34), we can read off the temperature of the right Rindler\nwedge, TR≡a/2π.15\nAs a final remark, let us notice that TRis exclusively governed by the horizon H+, but not by the\nparticular observer. In fact, the bookkeeping parameter acan always be set to one in (30) without loss of\ngenerality. However, the proper temperature measured by an accelerated observer, travelling along a specific\nhyperbola of constant ξ, can be derived from the wedge temperature by the appropriate Tolman factor\nTp=TR√−g00=aeaξ\n2π=ap\n2π. (35)\nWe hence see that the observed temperature of a given Rindler observer depends on its proper acceleration\nap, and coincides with TRon the special hyperbola ξ= 0.\nB. The Unruh Effect in Lorentz-Violating Gravity\nIn this subsection we aim to elicit a comparable mode structure within the previously designed Rindler\npatch defined by (18) and (19). We do this by coupling Einstein-Aether gravity to a scalar field enjoying the\nsame UV scaling, the Lifshitz scalar field\nSϕ=−1\n2Z\nMd4xp\n−det(g)\ng(∇ϕ,∇ϕ) +nX\nj=2a2j\nΛ2j−2ϕ(−∆)jϕ\n, (36)\nwhere ∆ = γ(∇,∇) is the Laplace-Beltrami operator on the spatial submanifold Σ. The Lorentz-breaking\nscale is denoted by Λ, and a2j∈Rare dimensionless coupling constants with a2n≡1. Variation with respect\nto the scalar field yields the Lifshitz-Klein-Gordon equation\n□ϕ−nX\nj=2a2j\nΛ2j−2(−∆)jϕ= 0. (37)\nThe critical exponent nis typically chosen to match that of the gravitational ultraviolet action of Hoˇ rava\ngravity ( n= 3 in four space-time dimensions). In this way, the full action combining gravity and matter will\nbe invariant under an anisotropic scaling in the ultraviolet, which ensures power-counting renormalizability.\nAlthough we have ignored higher derivative terms in the gravitational action, we include them in the action\nfor the scalar field, which is justified as long as Λ ≪MP. We will also keep narbitrary in what follows.\n1. Mode Solutions\nThe equation of motion (37), unlike the Klein-Gordon equation, cannot be decoupled along the natural\ntime and space directions UandS, since none of them are Killing vectors. In order to proceed further we\nthus adopt a WKB ansatz for the field\nϕ(x) =ϕo(x)ei\nℏS(38)\nwhere ϕo(x) is a real, slowly varying function (with respect to the phase) that for convenience has been chosen\nto be constant ϕo(x) =ϕo. The principal function Sis expanded in orders of the smallness parameter ℏ. To\nleading order, we have S0which describes a point particle action\nS0=Z\nMdS0=Z\nM(−ω U+k S) (39)16\nwhere we have projected the one-form d S0onto the preferred frame, and defined the preferred frequency ω\nand wavenumber kthrough [6]\nLUϕ=−iωϕ , LSϕ=ikϕ. (40)\nPlugging this ansatz in the equation of motion (37) and using the eikonal approximation, yields the\ndispersion relation\nω2=k2+nX\nj=2a2jk2j\nΛ2j−2(41)\nwhere we extracted the Λ-independent term that constitutes the relativistic limit k≪Λ. Note that due to\nthe eikonal approximation, derivatives of the wavenumber are neglected.\nEven though our definition of preferred frequency and wavenumber seems coherent with the notion of the\ntemporal and spatial integral lines, both quantities fail to be constants of motion. Nevertheless, one can\nextract the space-time dependence of ωandkfrom (41) using the conserved energy Ω given by the Killing\nfield χ(remember that both, the metric and the æther, are Lie dragged by χ) [39]:\nΩ =ω g(U, χ)−k g(S, χ). (42)\nA full solution requires to find the roots of a polynomial of degree 2 n, which is a challenging task. However,\nwe can infer two main limits in the solution space by performing an asymptotic expansion of the dispersion\nrelation, corresponding to ω≃Ω and ω≃Λ, which we call soft modes andhard modes respectively. While\nthe first describe fluctuations of low energy Ω ≪Λ that behave close to a relativistic field, the latter captures\nthose which are dominated by the higher order terms in the dispersion relation. An accelerated observer\nperceives consequently an admixture of high-energetic Lorentz-violating fluctuations to the usual Rindler\nbath of relativistic fluctuations. For the ease of notation, we again introduce the lapse N=−g(χ, U), and\na similar quantity related to the spatial vector V=g(χ, S) in the following.\nLet us briefly elaborate on the soft modes by solving the dispersion relation near the universal horizon in\nthis regime using (42), that is\nΩ2\nN2=\u0012\n1−V2\nN2\u0013\nk2+2V kΩ\nN2+O\u0012k\nΛ\u0013\n, (43)\nwhich leads to\nk≃ ±Ω\nV, (44)\nat leading order in the limit N→0, representing the universal horizon. Note that Vassumes a finite value\nat the position of the universal horizon, i.e. V→ −1, such that the momentum as well as the energy stay\nfinite.\nFrom (42) we infer that the wavenumber behaves similarly to the preferred energy. If for instance ω→ ∞ ,\nwhich happens for the hard modes at the universal horizon, k→ ∞ by the same degree of divergence, such\nthat Ω remains finite and conserved [5, 39]. Thus, we find for the hard modes\nΩ2\nN2+V2\nN2k2=2V kΩ\nN2+(V k)2n\nΛ2n−2+O\u0000\nk2n−2\u0001\n(45)17\nwhere we kept the k-dependent terms that involve Nin order to perform a consistent limit towards the\nuniversal horizon, since those carry a meaningful contribution. Under consideration of (42), we can find the\nsolution for the preferred energy as well as the wavenumber to be\nω≃ ±nV2\nn−V2Ω\nN+ΛVn\nn−1\nNn\nn−1, k≃ ±Λ\n(V N)1\nn−1+VΩ\nn−V2. (46)\nIn the limit N→0, both the wavenumber and the preferred energy diverge which suggest that the modes\nget clenched at the universal horizon, being unable to cross it in a classical sense.\n2. Bogolubov Coefficients\nIn order to calculate the temperature of the Rindler patch, we shall follow a procedure similar to the\nrelativistic case previously reviewed. I.e. we shall relate the modes in the Minkowski space-time ( M, gM, UM)\nwith those that only have support in an analytical extension of the right Rindler patch modes in ( R, gR, UR),\nvia a Bogolubov transformation. In this sense, we shall investigate if there is a thermal effect induced by\nthe space-time closure, i.e. the universal horizon H∞, from the perspective of a Rindler observer.\nOur starting point requires to define the symplectic product on the classical phase space. The presence\nof a physical foliation suggests to work in the canonical phase space Γ canwith elements ( ϕ, π) where the\nvariable conjugated to the field ϕisπ=p\n−det(g)LUϕ. The symplectic product is then the closed form ι\n[40]\nι(ϕ1, ϕ2) =iZ\nΣ(ϕ∗\n1π2−ϕ2π∗\n1) (47)\nwhere we defined the current J(ϕ1, ϕ2) =i(ϕ∗\n1∇ϕ2−ϕ2∇ϕ∗\n1) on the phase space, and the integration runs\nover a surface Σ orthogonal to U. Note that orthogonality in the previous expression eliminates all terms\nfrom higher spatial derivatives. As such, the inner product coincides with the Klein-Gordon inner product in\nrelativistic theories, which simplifies the derivation of the Bogolubov coefficients tremendously (cf. appendix\nB for details). However, for any time-orientation other than along U, the higher-derivative terms may\nreappear.\nRecall that vΩare the modes defined by the Rindler observer, while u¯Ωdescribe the Minkowski modes.\nAs such, we define the Bogolubov coefficients for vΩassociated to the Rindler vacuum in terms of u¯Ω. The\nBogolubov transformation is the bounded linear transformation ( ˆb†,ˆb)7→(ˆa†,ˆa) that is given through the\ncoefficients\nαΩ¯Ω=ι(u¯Ω, vΩ) and βΩ¯Ω=−ι(u¯Ω, vΩ)∗, (48)\nwhich fulfill the relationsR\ndΩ′(|αΩΩ′|2− |β¯ΩΩ′|2) =δ(Ω−¯Ω) andR\ndΩ′(αΩΩ′β¯ΩΩ′−βΩΩ′α¯ΩΩ′) = 0.\nIt is worth stressing that there appears to be here an ambiguity in the question of which inner product\nshould be used. Usually, one uses the inner product associated to the space-time perceived by the observer\nthat performs the final measurement. In other words, the inner product shall correspond to the Hilbert\nspace we map into. However, since the inner product is invariant under conformal transformations, we are\nable to chose here the Minkowski inner product also for the Rindler observer in the (1 + 1)-dimensional case\nwithout loss of generality.18\nAs emphasized, soft modes are analytical when approaching the universal horizon, while hard modes\nasymptote; geometrically the khronon leafs accumulate at the universal horizon. We can see that this setup\nis similar to the relativistic case: the support of the modes is divided into two complementary regions –\nleft and right patches, both featuring a universal horizon. The only missing step that extracts the thermal\nproperties of the horizon is to perform an analytic continuation of the hard modes across the universal\nhorizon. For this, it is sufficient to perform our analysis within its neighborhood.\nTo understand the non-analytic behavior at the horizon, we once again draw some intuition from the\nSchwarzschild space-time. Let us then consider (19), treating the function ϵas a coordinate we find\ndη\ndϵ=2\n1 + 2¯ aϵ, (49)\nat leading order around the horizon. Integrating this relation, we get\nη= 2Zdϵ\n1 + 2¯ aϵ=1\n¯aln(1 + 2¯ aϵ) +η0. (50)\nDue to this behavior, on the submanifold H∞(and its adjacent leafs), the WKB modes in the Rindler\npatchR(the same holds for L) acquire the form\nvR,Ω(η, ϵ) =vo\nRexp\u0012\n−iΩη+iΩ\n¯aln(1 + 2¯ aϵ)\u0013\n, (51)\nwhere vo\nRis constant. From here, it becomes obvious that (50) is the equation for the constant-phase contours\nof the mode. Note that for the horizon H−∞inL, the argument of the exponential picks up an additional\nminus sign.\nBecause of the logarithmic exponent, it is understood that the modes (51) are only defined within R.\nNevertheless they can be extended into Fby analytically continuing the logarithmic term [37] along the lines\nof [35, 36] yielding\nΦ+\nΩ(η, ϵ) =vL,Ω(η, ϵ) +e−πΩ\n¯a(vR,Ω(η, ϵ))∗, Φ−\nΩ(η, ϵ) =vR,Ω(η, ϵ; Ω) + eπΩ\n¯a(vL,Ω(η, ϵ))∗(52)\nwhere Φ+\nΩ(η, ϵ) and Φ−\nΩ(η, ϵ) contain only the modes ei¯Ωϵande−i¯Ωϵrespectively, which are defined on the\nfull Minkowski space-time.\nThe analogy with (33) is so striking that after changing basis from vR/L,Ω(η, ϵ) to Φ±\nΩ(η, ϵ) with a proper\nnormalization, we obtain the Bogolubov coefficient’s squared norm |βΩ¯Ω|2by performing the inner product\nbetween the analytically continued basis Φ±\nΩ(η, ϵ) and vR,Ω(η, ϵ). Through an energy integration, we find the\nfollowing number of created particles [6]\n⟨ˆNΩ⟩M=Z\nd¯Ω|βR\n¯ΩΩ|2=1\ne2πΩ\n¯a−1(53)\nwhere we have defined the particle number operator ˆNΩ=ˆb†\nΩˆbΩ, and the expectation value has been evaluated\nwithin the Minkowski vacuum state. We find that the number of measured particles follows a Bose-Einstein\ndistribution from which we can read off the associated patch-temperature\nTR=¯a\n2π=κuh\nπ. (54)\nNote, we identified ¯ a= 2κuhwhere κuhis the surface gravity calculated from the expansion, which is related\nto the peeling surface gravity [33] at the universal horizon.19\nAgain, the above temperature is purely set on the basis of geometrical considerations related to the\nuniversal horizon induced by the æther flow of the Rindler patch. In this sense, we derived the equivalent\nof the Rindler wedge temperature TR≡a/2π. However, this is not the temperature that an observer will\ndetect while moving on a specific orbit of the boost Killing vector, the equivalent of (35) for our case.\nOne might contemplate applying the usual Tolman factor to get the proper acceleration, but this would\nnot do: indeed the Tolman factor is purely metric dependent and would not capture the relevance of the\nobserver motion with respect to the preferred frame set by the æther. So, let us now push our investigation\nfurther by considering the above mentioned observers and the response of an Unruh-DeWitt detector.\n3. Unruh-DeWitt Detector\nTo understand what an actual Rindler observer would measure, we consider a simple model describing a\npoint-like Unruh-DeWitt detector [41]. We will again work within the (1 + 1)-dimensional submanifold that\ncontains the world-line of the detector. This treatment is in line with [35], which also shows the generalization\nto the full (1 + 3)-dimensional setup.\nThe detector is composed out of a Hermitian operator ˆ µwhich acts on a two-dimensional Hilbert space\nHµ≃C2spanned by the orthonormal basis {|E0⟩,|E1⟩}where |E0⟩denotes the ground-state and |E1⟩the\nexcited state. These states are designed to be the eigenstates of the free Hamilton operator ˆHµ, that is, they\nfulfill ˆHµ|E0/1⟩=E0/1|E0/1⟩with E0< E 1. To detect field excitations, we couple the detector to a scalar\nfield ˆϕvia the interaction term [35, 41, 42]\nˆW=bZ∞\n−∞dτ χ(τ)ˆµ(τ)ˆϕ[y(τ)] (55)\nwith coupling strength b∈R, and switching function χ(τ), that has support only on the time interval of the\nmeasurement [4]. The evolution parameter τis always adapted to the Cauchy problem; and it is typically\nchosen to be the proper time of the detector on its world-line. Here, we choose τto be the preferred time of\nthe foliation, dictated by the khronon Θ. This aligns the Hamiltonian flow with the direction of the preferred\nclock and yields a consistent Schr¨ odinger evolution.\nSince our detector is constantly accelerated, its domain of dependence will only cover the right Rindler\npatchR, as previously argued, whose khronon is given by (25). The Hamiltonian flow must then be tangent\ntoUand the Schr¨ odinger operator iLU−ˆHµacquires an additional dependence on the lapse. Colloquially\nspeaking, the detector evolution is determined by the preferred time rescaled by the lapse as\nˆµ(s) =e−iNˆHµτ(s)ˆµ0eiNˆHµτ(s). (56)\nNote that we have introduced the proper time of the detector sviaτ(s) =se¯aξso that we can relate our\nresult to the relativistic Unruh setup that is naturally parametrized by s, the detector clock.\nThe Hamilton operator [41] that acts on states in the Hilbert space Hϕis given by\nˆHϕ(τ) =Z\nΣτd3yp\n−det(γ)n\nˆΠ(τ, ⃗ y)LUˆϕ(τ, ⃗ y)−L[ˆϕ,∇ˆϕ](τ, ⃗ y)o\n, (57)\nwhere the Lagrange density L[ˆϕ,∇ˆϕ](τ, ⃗ y) is given by (36). As usual, ˆΠ(τ, y) denotes the canonical momen-\ntum conjugated to the field ˆϕ(τ, y). Then the total Hilbert space is H=Hµ⊗ H ϕand the total Hamilton\noperator ˆH=ˆHµ⊗ˆidϕ+ˆidµ⊗ˆHϕ+ˆW. To avoid infrared divergences [35], we introduce a regularization scale20\nmthat can be interpreted as a fiducial mass for the field, which then obeys ( □−Pn\nj=2a2j\nΛ2j−2(−∆)j−m2)ϕ= 0.\nAlso, by doing this the comparison with the relativitic case as discussed in [35] becomes immediate. Note\nthat the quantum field ˆϕmust be evaluated on the detector’s trajectory y(s) such that ˆϕ(τ, ⃗ y)→ˆϕ[y(s)].\nAs an operator, ˆϕis represented through a positive frequency basis, such that\nˆϕ(y) =Z\nRdk\n2πn\nˆakuk(y) + ˆa†\nku∗\nk(y)o\n(58)\nwhere uk(y) is a solution to the equations of motion and ˆ ak|0⟩= 0 the destruction operator annihilating the\nvacuum state of the quantum field |0⟩.\nIn general, the outcome of a measurement is given by acting with ˆWonto a given state. This can\neither be the ground state |0⟩=|0⟩ ⊗ |E0⟩, or the corresponding excited state |1⟩=|k⟩ ⊗ |E1⟩which\ndescribes a particle with momentum kthat has excited the detector – a clicking event. With this, ˆW=R\nRds{ˆid⊗ˆµ(s) +ˆϕ[y(s)]⊗ˆid}, since the total Hilbert space is given by a tensor product, and we can define\nthe excitation rate over a probing-time interval ∆ sas\nR=1\n∆sZ\nRdk|Ak|2, (59)\nwhere Ak=i⟨1|ˆW|0⟩and ∆ s=R∞\n−∞χ(s)ds. Note that we tacitly assumed kto be conserved. This is not\ntrue in the Lorentz violating case, but let us assume it so for the moment. We will come back to this point\nlater when we particularize the setup to our Gedankenexperiment (for a general discussion, cf. [35]).\nHere, we are interested in the question of what a detector in Minkowski vacuum measures on a uniformly\naccelerated trajectory. Let us start with the solution space to (37) in a foliated Minkowski vacuum – hence\nwith a homogeneous æther. We find for the mode\nuk[y(s)] =eik[y(s)]\np\n2ωΛ(k), (60)\nwhere k[y(s)] =ωΛ(k)t(s)−kx(s) and ωΛ(k) =q\nm2+k2+k4\nΛ2denotes the dispersion relation in Minkowski\nspace-time.\nIn the (foliated) Minkowski space-time, the hyperbola described by the accelerated detector can be\nparametrized as usual\nt(s) =1\napsinh( aps), x (s) =1\napcosh( aps), (61)\nwhere apis the proper acceleration. Using the mode (60), we find that the amplitude in (59) factorizes as\nfollows\n⟨1|ˆW|0⟩=Z\nIds⟨E1|⊗⟨k|ˆµ(s)ˆϕ[y(s)]|0⟩⊗|E0⟩=Z\nIds⟨E1|ˆµ(s)|E0⟩Z\nRdkp\n8π2ωΛ(k)ei\nap(kcosh( aps)−ωΛ(k) sinh( aps)),\n(62)\nwhere I= [s0, s1] is the time-interval of the measurement determined by the indicator function χ(s). The\nfirst of these factors depends on the specifics of the detector, while the second, usually called the response\nfunction, describes how the field is perceived on the hyperbola.\nThe evaluation of ˆ µ(s) requires requires to consider the Hamiltonian flow. From the coordinate transfor-\nmations (6), we derive the function τ(s) on a fixed ξtrajectory from (25) to be τ(s) =e−¯aξs. Note how,21\nvia this relation, the æther acceleration just came into play. The consequences for the rate are immediate.\nConsidering (56) and the fact that |E0/1⟩form a basis of Hµ, we find that\n⟨E1|ˆµ(s)|E0⟩=q eiN∆Eτ(s)(63)\nwhere we defined q=⟨E1|ˆµ0|E0⟩, and ∆ E=E1−E0.\nAfter squaring the amplitude, we are thus facing the following integral for the rate\nR=|q|2\n4πap∆sZ∞\n−∞dk\nωΛ(k)Z\nIdsZ\nIds′\u001a\neiNe−¯aξ∆E(s−s′)eiωΛ(k)\naph\nsinh\u0000\naps\u0001\n−sinh\u0000\naps′\u0001i\neik\naph\ncosh\u0000\naps′\u0001\n−cosh\u0000\naps\u0001i\u001b\n(64)\nIn the following, we will choose the detector to be measuring over the whole timespan, i.e. χ(s) = 1 ,∀s∈R,\nsuch that I= (−∞,∞). To deal with this integral, let us define σ=s−s′and 2 ζ=s+s′. Using the\nproperties of hyperbolic functions we can rewrite (64) as\nR=|q|2\n2πap∆sZ∞\n−∞dk\nωΛ(k)Z∞\n−∞dσZ∞\n−∞dζn\neiNe−¯aξ∆Eσei2\napsinh(apσ\n2)[ωΛ(k) cosh( apζ)−ksinh(apζ)]o\n. (65)\nTheσ-integral can be transformed into a soluble form after another change of variables λ= exp( apσ/2) so\nthat we find\nR=|q|2\n2πap∆sZ∞\n−∞dk\nωΛ(k)Z∞\n−∞dζZ∞\n0dλ\nλ\u001a\nλi2Ne−¯aξ\nap∆Eei\nap(λ−1\nλ)[ωΛ(k) cosh( apζ)−ksinh(apζ)]\u001b\n=e−πNe−¯aξ\nap∆E×R,\n(66)\nwhere Ris the so called response function of the detector, explicitly shown below in (73).\nTo extract the temperature from this calculation, we need to compute also the de-excitation rate. While\n(66) is given by the k-integral of |Ak|2, where Akis the matrix element Ak=i⟨1|ˆW|0⟩given in (59), the\nde-excitation rate ¯Ris given by the probability of the inverse process to occur\n¯R=1\n∆sZ\nRdk|¯Ak|2. (67)\nwhere ¯Ak=i⟨0|ˆW|1⟩.\nIn practical terms, this requires to compute the de-excitation probability of the detector\n⟨E0|ˆµ(s)|E1⟩=q∗e−iN∆Eτ(s), (68)\nwhere q∗=⟨E0|ˆµ0|E1⟩is the complex conjugate of q. Additionally we need to compute the |k⟩ → | 0⟩matrix\nelement of ˆϕ[y(s)] which is given by\n⟨0|ˆϕ[y(s)]|k⟩=Z\nRdkp\n8π2ωΛ(k)e−i\nap(kcosh( aps)−ωΛ(k) sinh( aps))=\u0012\n⟨k|ˆϕ[y(s)]|0⟩\u0013∗\n. (69)\nThe computation now goes along the same lines as that previously shown for R. We arrive to\n¯R=1\n∆sZ\nRdk|¯Ak|2=eπNe−¯aξ\nap∆E×R, (70)\nwhere Ris the same response function, containing the remaining ζ-integral, appearing in (66).\nUsing (66) and (70) we can see that the ratio\nR\n¯R=e−2πNe−¯aξ\nap∆E, (71)22\nis a Boltzmann factor which then allows us to read off the temperature that is measured by the detector\nTdet=ape¯aξ\n2πN=¯a\n2πN. (72)\nThis turns out to have the same value as in the relativistic version of the Unruh effect, albeit being rescaled\nby the factor Ne−aξ.\nAlternatively, one can interpret (72) as the wedge temperature rescaled by Ninstead of√−g00. With\nhindsight this is not a surprising result. Indeed, the conversion factor linking the Rindler patch temperature\nto the observed one on a given hyperbola is nothing else that the rescaling factor between the preferred and\nthe Killing time. In the relativistic case, one can get (35) from the wedge temperature just by looking at\nthe proportionality factor between the proper time sof the observer and the Killing time η: on a ξ= const .\nhyperbola we have d s=√−g00dη, telling us the different rates at which the two times pass. Similarly, if\none computes the same quantity for the preferred time τon the same hyperbola, one gets U= dτ=Ndη.\nFrom that, we can directly read the new proportionality factor – corresponding to the lapse N– which we\nhave found in (72).\nBeyond computing the temperature, let us also focus on the shape of the response function Rin (66). The\nλ-integral can be performed analytically, yielding the modified Bessel function of the second kind Kν(x) (or\nMacdonald function)\nR=|q|2\n2πap∆sZ∞\n−∞dk\nωΛ(k)Z\ndζ Kiν\u00122\nap{ωΛ(k) cosh( apζ)−ksinh( apζ)}\u0013\n(73)\nwhere we defined ν:=2Ne−aξ\nap∆E. Note that this does not exhibit any thermal behavior, and therefore\ncontributes only to gray-body modifications to the rate, as argued in [35] for the relativistic response function.\n4. Relativistic Limit\nWhat is left to discuss is whether the relativistic result of the detection process is in any form recovered\nwhen the Lorentz violating dynamics are lifted. In practice, this can be achieved by suppressing the momen-\ntum of the excitations of the field with respect to the Lorentz-violating scale of the matter sector k≪Λ.\nThis can be argued perturbatively. Due to the exponential in (60), we can easily expand ωΛ(k) for k≪Λ\nand find non-relativistic corrections to (59). In the limit Λ → ∞ , the dispersion relation becomes relativistic\nωΛ(k)→√\nk2+m2, and we expect to recover the usual form of the relativistic response function as derived\nin [35].\nTo substantiate this picture, we examine the argument of the integral R=R\nRAkdk. Here Akdescribes\nthe momentum distribution that determines the rate when integrated over the full k-space. We plotted this\nquantity evaluated on the central hyperbola for several values of the Lorentz-violating scale Λ in Fig.4. As\nit can be seen, the maximum coincides for all distributions, and in particular with that of the relativistic\nlimit Λ → ∞ (in blue). However, while the latter decays monotonically towards larger values of k, the rest\nof the distributions show an oscillatory tail, connected to the zeros of the Bessel function, that starts earlier\nfor lower values of Λ. This behavior seems to point to the emergence of Lorentz symmetry at low energies.\nAs long as the condition k≪Λ can be trusted, the effect of the Lorentz-violating operators, and thus\nthe oscillatory behavior, can be safely neglected. Once the approximation breaks down, we start observing\nmodifications in Akthat differ from the relativistic case.23\n𝔄k\n∞−∞k0\nFIG. 4: The distribution Akin the response function integral Rfor different values of Λ: the Lorentz\nbreaking amplitudes are shown for different values of Λ ∈ {10,102,103,104}(in units of ¯ a) in orange, green,\nred, and purple, while the relativistic result obtained from the integral in [35] is indistinguishable from the\npurple curve (but shows no oscillations whatsoever). We have set the particular hyperbola for which ap= ¯a\nfor this plot, and identified the mass in the result of [35] to equal m. We have furthermore chosen m= ¯a.\nTo develop a deeper understanding of what happens when Lorentz-symmetry is reinstated, let us discuss\nthe behavior of Akin the non-relativistic case further. First of all, we place the observer onto the hyperbola\nwhere ap= 1 for simplicity and rewrite\nAk∝Z∞\n−∞dζKiν(2{ωΛ(k) cosh( ζ)−ksinh( ζ)})\nωΛ(k)=Z∞\n−∞dζKiν\u0000\n[ωΛ(k)−k]eζ+ [ωΛ(k) +k]e−ζ\u0001\nωΛ(k).(74)\nNow, with a change of variable ζ→ζ+ ln( ωΛ(k)−k), which is valid for any k∈R, we get\nAk∝Z∞\n−∞dζKiν\u0000\neζ+ [ω2\nΛ(k)−k2]e−ζ\u0001\nωΛ(k). (75)\nEq. (75) is illuminating in several aspects. First of all, it is clear that a relativistic dispersion relation will\ndecouple the ζ-integration and the k-integration in (66) as already observed in [35]. Since in the relativistic\ncase we have\nAk∝Z∞\n−∞dζKiν\u0000\neζ+m2e−ζ\u0001\n√\nk2+m2, (76)\nwhere mis the mass of the field, it becomes obvious that the two integrals factorize. Then, the shape of Ak\nwill be controlled by the√\nk2+m2in the denominator. Let us point out that this fact is intimately linked\nwith the boost invariance of the dispersion relation. This can be deduced by noticing that the argument of\nthe Bessel function in (74) is just the result of a boost of ( ωΛ(k), k) with rapidity ζ. In [35] it has been shown\nthat a change of variable k→k′(k, ζ) in the k-integration of (73) while applying the inverse boost, leaves\nthe measure unchanged (so d k′/ω′= dk/ωfor the relativistic case), thus factorizing the ζ- and k- integrals.24\nKiν(eζ+ϖ(k)e−ζ)\nζ\nFIG. 5: Shape of Kiν(eζ+ϖ(k)e−ζ) for ϖ(k) = 2 and ν= 4πas a function of ζ\nWithout this symmetry, however, we cannot disentangle the two integrals, since the coefficient ( ω2\nΛ(k)−k2)\nremains k-dependent. This explains why, for k= 0, the relativistic and non relativistic values of Akboth\ngive\nA0∝Z∞\n−∞dζKiν\u0000\neζ+m2e−ζ\u0001\nm, (77)\nwhile they depart for other values of k. In other words, in the infrared region, our detector enjoys the\nsame response function regardless of Lorentz-symmetry while high energy measurements differ significantly.\nIn fact, in the deep ultraviolet region, where k→ ∞ , we notice that the non-relativistic Akis strongly\nsuppressed with respect to the relativistic one. This is a consequence of the ultraviolet behavior of ωΛ(k).\nWhile in the latter case ω∝ |k|at large k, the former case leads to ω∝ |k|n/Λn−1, so that Akis suppressed\nby a power law.\nLet us define for convenience the quantity ϖ(k) =ω2\nΛ(k)−k2. In the intermediate region, we notice the\npresence of a finite number of oscillations in the non-relativistic Ak. Mathematically this can be explained\nby looking at the shape of Kiν(eζ+ϖ(k)e−ζ) before the ζ-integration, at fixed k, as shown in Fig.5. There,\nwe observe a finite number of oscillations, while the tails decay very rapidly4. Note that for large values\nofϖ(k), the Bessel function stops oscillating [43]. The same happens for the value of the mass, which\nacts here as an IR regulator effectively cutting-off the distribution at low energies. In particular, since\nϖ(k) =ω2\nΛ(k)−k2≥m2, we can show, by computing its minimum, that eζ+ϖ(k)e−ζ≥2p\nϖ(k)≥2m,\nand the number of zeros of Kiνis governed by mand ∆ E. For large values of the mass m, no oscillation is\npresent, and no bumps appear in Ak.\nIt should be mentioned that the detector still couples to the æther even in the decoupling limit of the field\nΛ→ ∞ . Due to this, the previous limit will not impact the value of the temperature in (72). Reinstalling\n4For large values of the argument, we have Kiν(x)∼e−x/√x25\nLorentz symmetry in the gravitational sector is much less trivial, since the actual mechanism for it is\nunknown. However, in the low energy limit, the influence of quantum gravity modifications should cease, such\nthat our detector must somehow decouple from the æther, and the relativistic causal structure reappears.\nAlthough we have not explored a smooth way to implement this, we can argue that, in this limit, the\ntrajectory of the detector must be measured with its proper time τ(s)≡s, instead of the khronon-clock5.\nThis automatically fixes N=√−g00in (72), and the correct relativistic result is recovered.\nIV. DISCUSSION AND CONCLUSIONS\nThe thermodynamical aspects of space-time in the framework of Lorentz breaking gravity are a fascinating\nsubject, not only as a challenge to our intuition, but also because they let us explore the real foundations of\nthese tantalizing features of gravitation. In this work, we have dealt with the Unruh effect, a phenomenon\nthat, in the past literature, has been often deemed non-robust against a UV breakdown of special relativity.\nWe have seen here that this is not the case as long as the preferred frame set by the æther dynamics is\nsuitably taken into account. This leads us to identify what is the correct Rindler patch (metric and æther\nflow) to be considered in the context of Einstein-æther gravity. Such patch is unique and enforces the æther\nto have a constant acceleration, that then provides a physical scale allowing for the robustness of the Unruh\neffect. Moreover, this patch coincides with the near Killing horizon limit of a large Schwarzschild black hole\nin Einstein-æther gravity, nicely preserving the link between the two geometries (Rindler and Schwarzschild)\npresent in the relativistic case.\nIndeed, we saw that the so found Rindler patch is endowed with a temperature TR=¯a\n2π=κuh\nπset by the\ncausal structure of the solution (metric and æther flow). We can see here that this is basically the usual\nformula for the Rindler wedge temperature with the crucial difference that the usual bookkeeping parameter\nais replaced by the physical, constant acceleration (and expansion) of the æther, ¯ a.\nFurthermore, we have also computed the response of an Unruh-DeWitt detector carried on along an orbit of\nthe boost Killing vector (the trajectory of the usual constantly accelerated observers in the Unruh effect) and\ncoupled both to the æther and to the same non-relativistic field that we had considered in the computation\nof the Rindler patch temperature via Bogolubov coefficients. We found that the temperature perceived by\nthe detector is related to the wedge temperature by a factor keeping into account the relative acceleration\nwith respect to the preferred frame such that Tdetect =¯a\n2πN.\nNoticeably, this result shows that the temperature is insensitive to the dispersive properties of the non-\nrelativistic field (namely to the Lorentz violating scale Λ). However, the detector response function is sensitive\nto these properties, and in a very interesting way, i.e. by showing a series of oscillations in the distribution Ak\ncharacterizing the response function integral R. Such features probably deserve further investigations as they\nmight appear also in experimental contexts, such as analog gravity experiments where modified dispersion\nrelations are unavoidable and would lead to some corrections to the expected low frequency, relativistic,\nbehavior of the Unruh effect6(see e.g. [44]).\n5This is because within these settings, the construction of Cauchy data is intimately tied to the foliation. Whenever we use\nthe preferred time direction to constitute the Hamiltonian flow in non-relativistic setups, we collect a lapse function from the\næther. However, when we restore Lorentz symmetry, the phase space becomes relativistic, and we find (35), where the factor√−g00originates from normalizing the tangent vector to the Hamiltonian flow.\n6The careful reader might wonder how the Unruh effect might be robust in analog gravity given that no dynamical æther is\npresent in that context for providing the invariant scale ¯ a. The answer resides in the lacking of diffeomorphism invariance of\nthat setting that does not allow to rescale at will the proper acceleration of a Rindler observers given that this will be the\nactual acceleration of the latter in the laboratory, fundamental, frame.26\nSo in conclusion, the outcome of our investigations can be summarized in a few lessons:\n•The Unruh effect does survive within the Lorentz breaking gravity context.\n•This, together with the recently found results concerning the Hawking effect [6], supports the growing\nevidence that the whole horizon thermodynamics setting should be exportable to the above theoretical\nframework7.\n•The Unruh effect is closely similar to the relativistic one, and reduces to it in the appropriate limit,\nnonetheless present, especially in the response function, a series of signatures that should be possible\nto seek for in future experiments.\nFinally, let stress that we have considered here an eternal configuration of the the geometry. However, in a\nrealistic setting one would like to describe how our background solution can be attained from a Minkowski\nspace-time with a uniform æther. This might be related to the backreaction of the detector on the æther.\nWe hope that these issues will be further explored in next coming works.\nACKNOWLEDGMENTS\nWe wish to dedicate this paper to the memory of R. Parentani, in particular S.L. for stimulating discussions\nand insights about the possible robustness of the Unruh effect. We are also grateful to D. Mattingly for\ndiscussions. M. H-V. wants to thank the APP department at SISSA for their hospitality during the early\nstages of this work. M.S. wants to thank Steven Carlip for his thoughts on this topic and, in particular,\nfor bringing our attention to Unruh-DeWitt detectors. The work of F. D. P., S. L., and M. S. has been\nsupported by the Italian Ministry of Education and Scientific Research (MIUR) under the grant PRIN\nMIUR 2017-MB8AEZ. The work of M. H-V. has been supported by the Spanish State Research Agency\nMCIN/AEI/10.13039/501100011033 and the EU NextGenerationEU/PRTR funds, under grant IJC2020-\n045126-I; and by the Departament de Recerca i Universitats de la Generalitat de Catalunya, Grant No 2021\nSGR 00649. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.\nAppendix A: Conformal factor\nHere we give a detailed derivation of the solution (15) to the equation of motion (12). We will assume\ninvariance under boosts and local flatness – i.e. Riem = 0. For the ease of notation, we work with the\n(1 + 1)-dimensional submanifold of the metric in (13), thus neglecting the sub-manifold E2, which will not\ncontribute in any case.\nLet us start with the first condition, that is that the æther as well as the metric are Lie dragged with\nrespect to the boost vector. To this aim we have to find out the explicit form of the Killing vector which\n7In particular, now that we have an appropriate generalization of the Rindler wedge to Lorentz breaking theories, it would be\ninteresting to see if the space-time thermodynamics program [45] can be made to comprise this class of theories.27\nobeys Lχg= 0. Using the conformal metric, we find the following set of equations\n∂τχτ−χτ∂τln(W(τ, ρ))−χρ∂ρln(W(τ, ρ)) = 0 , (A1)\n∂ρχρ−χτ∂τln(W(τ, ρ))−χρ∂ρln(W(τ, ρ)) = 0 , (A2)\nχρ∂ρln(W(τ, ρ)) +W(τ, ρ)∂τln(χτW(τ, ρ)) = 0 (A3)\nwhich leads immediately to the relation ∂τχτ=∂ρχρ. Formally, the khronon equation (12) is not independent\nof (9), since it can be obtained from the latter by using the Bianchi identities implied by FDiff invariance (cf.\n[31] for details). However, for our purposes we use (12) as a consistency check for our solution. Moreover, the\nspace-time should not depend on the time associated to the timelike Killing vector field, such that Lχg(U, χ) =\n0 and Lχg(S, χ) = 0 respectively. Together with the Killing equation we find that the components of the\nKilling vector read\nχτ=f1(τ)W2(τ, ρ) and χρ=f2(ρ)W2(τ, ρ) (A4)\nwith until now, arbitrary functions f1(τ) and f2(ρ). Using again the space-time independence of the Killing\ntime, and plugging in the above components, we find that ∂τf1(τ) =∂ρf2(ρ)∀τ, ρwhich implies that the\nderivatives must equal to a constant c0so that the arbitrary functions f1andf2take the form\nf1(τ) =c0τ+c1,and f2(ρ) =c0ρ+c2. (A5)\nNow, using the equations above, we can find a functional relation between W(τ, ρ) and f1(τ) and f2(ρ)\nW(τ, ρ) =h\u0010\nf1(τ)\nf2(ρ)\u0011\nf1(τ)(A6)\nwhere his a function to determine. We do this by imposing local flatness through Riem = 0. We cast the\ntwo-dimensional Ricci scalar curvature in the ( τ, ρ) coordinate system and find\nR=−2(−∂2\nτ+∂2\nρ)W(τ, ρ) + 2(∂ρW(τ, ρ))2−(∂τW(τ, ρ))2\nW(τ, ρ)= 0, (A7)\nwhich leads to the following solution\nW(τ, ρ) =F1(τ+ρ)F2(τ−ρ), (A8)\nwith, again, arbitrary functions F1andF2.\nTo simplify f1(τ) and f2(ρ), we impose the coordinate shift τ→τ−c1\nc0as well as ρ→ρ−c2\nc0such that\nf1(τ) =c0τandf2(ρ) =c0ρ. Relating the two forms (A6) and (A7) for the conformal factor and their τ-\nandρ-derivatives we arrive at\n1 +τ∂τln(F1(τ+ρ))−τ∂τln(F2(τ−ρ)) +ρ∂ρln(F1(τ+ρ)) +ρ∂ρln(F2(τ−ρ)) = 0 . (A9)\nSince we find a differential equation for the functions F1(τ+ρ) and F2(τ−ρ), we find a unique family of\nsolutions for both that lead to the following conformal factor\nW(τ, ρ) =\u0012ρ+τ\nρ−τ\u0013αC\nρ+τ(A10)28\nwhith the integration constants C, α∈R.\nTo determine αwe need to fulfill the equation (12). However, since our solution is hypersurface orthogonal,\nthe equations of motion of khronometric gravity coincide with those of in Einstein-Æther gravity. This\nimplies that we can proceed to solve the simpler equation E= 0 with Egiven in (11) instead of (12), and\nhypersurface orthogonality will ensure the equivalence of solutions. Inserting our ansatz for gW, we find that\nforEto vanish, every individual ters in Ehas to be zero independently, since the couplings are arbitrary.\nFrom this, it follows that LSθ= 0, which simplifies to ∂ρθ= 0. Since θ=∇U=∂τW−1(τ, ρ) we have\n∂τ∂ρW−1(τ, ρ) = 0, which is solved by\nW(τ, ρ) =1\nA(τ) +B(ρ), (A11)\nfor two generic functions A(τ) and B(ρ). This restricts the coefficient α∈ {0,1}, and after setting C=1\n¯a\nfor convenience, we find the two solutions\nW±(τ, ρ) =1\n¯a(ρ±τ). (A12)\nas given in (15). Let us notice that the conformal patch we used here is only the easiest way for deriving\nthe solution. However, it is easy to see that the two-dimensional problem of finding a flat boost-invariant\nsolution of (12) is anyway overdetermined. One could have proceeded in the following way: the flatness of the\nsolution ensures that the metric can be written as the Minkowski metric, in the right system of coordinates.\nThen, the normalization condition for a two dimensional vector Ulinks the two components through the\nrelation g(U, U) =−1. Finally, boost-invariance makes the problem one-dimensional, transforming (12) into\nan ODE for one of the two components of the aether vector, which leads to the same the solution found\nthrough the conformal method.\nAppendix B: Inner product\nIn (47) we have defined the symplectic product on the space of solutions. This inner product is the same\nthat follows from the relativistic Klein-Gordon equation (see e.g. [35, 46]). Hence, we find accordingly the\nconserved current\nJKG(ϕ2, ϕ1) =−i(ϕ1∇ϕ∗\n2−ϕ∗\n2∇ϕ1) (B1)\nfor solutions ϕi. The standard symplectic product is defined with the symmetries of general relativity, i.e.\nit remains invariant under choosing a time. In theories with physical, preferred foliations, the hypersurfaces\nare determined and so is the orthogonal derivative which in our case is along U. The associated conserved\ncurrent reads then\nJLV\na=−i\u0014\nϕ1\u0012\n∇a−γab∇bnX\nj=22jα2j\nΛ2j−2∆j−1\u0013\nϕ∗\n2−ϕ∗\n2\u0012\n∇a−γab∇bnX\nj=22jα2j\nΛ2j−2∆j−1\u0013\nϕ1\u0015\n. (B2)\nHence, the symplectic product for the Lifshitz field can be constructed using the preferred frame\nιτ(ϕ1, ϕ2) =Z\nΣτ(U(JLV)) =−iZ\nΣτ(U(ϕ1∇ϕ∗\n2−ϕ∗\n2∇ϕ1)), (B3)29\nwhere we have taken into account that U⊥S. The integration is performed on a spatial submanifold Σ τon\nwhich τis constant.\nWhat is left to show is that the flux of JLVis zero in a volume that is enclosed by two surfaces, one at\nτ=τiand the other at τ=τo. Using Gauß’s theorem we get:\nιτo(ϕ1, ϕ2)−ιτi(ϕ1, ϕ2) =Z\nvoliod2x(√−gdJLV), (B4)\nFurthermore, the equation of motion for ϕ1andϕ2implies some internal cancellations between individual\nterm such that we remain with\nιτo(ϕ1, ϕ2)−ιτi(ϕ1, ϕ2) =\n\u0012Z\nΣi−Z\nΣo\u0013 \u0014\nγ(U,∇)ϕ1nX\nj=22jα2j\nΛ2j−2∆j−1ϕ∗\n2−γ(U,∇)ϕ∗\n2nX\nj=22jα2j\nΛ2j−2∆j−1ϕ1\u0015\n= 0.(B5)\nHence, we conclude that ιτo(ϕ1, ϕ2) =ιτi(ϕ1, ϕ2), or in other words, the inner product is still independent\nfrom the particular choice of the leaf Σ. For any other timelike vector t̸=±U, the last line of eq.(B5)\nyields a nonzero result because any timelike vector which is not tangent to Ufails to be orthogonal to S,\nwhatsoever.\nIt is interesting to notice that (B3) is invariant under the conformal rescaling\nU=Uadxa→UW=W(x)Uadxa, S =Sadxa→SW=W(x)Sadxa(B6)\nfor any arbitrary function W(x). 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Parentani, Physical Review D 91(2015), 10.1103/physrevd.91.124049."    },    {        "title": "1003.4681v1.Dynamical_shift_condition_for_unequal_mass_black_hole_binaries.pdf",        "content": "arXiv:1003.4681v1  [gr-qc]  24 Mar 2010Dynamical shift condition for unequal mass black hole binar ies\nDoreen M¨ uller, Jason Grigsby, Bernd Br¨ ugmann\nTheoretical Physics Institute, University of Jena, 07743 J ena, Germany\n(Dated: August 19, 2021)\nCertain numerical frameworks used for the evolution of bina ry black holes make use of a gamma\ndriver, whichincludesadampingfactor. Suchsimulations t ypicallyuseaconstantvaluefor damping.\nHowever, it has been found that very specific values of the dam ping factor are needed for the\ncalculation of unequal mass binaries. We examine carefully the role this damping plays, and provide\ntwo explicit, non-constant forms for the damping to be used w ith mass-ratios further from one. Our\nanalysis of the resultant waveforms compares well against t he constant damping case.\nPACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx\nI. INTRODUCTION\nThe ability to simulate the final inspiral, merger, and\nring-down of black hole binaries with numerical relativ-\nity [1–3] plays a key role in understanding a source of\ngravitational waves that may one day be observed with\ngravitational wave detectors. While initial simulations\nfocused on binaries of equal-mass, zero spin, and quasi-\ncircular inspirals, there currently is a large effort to ex-\nplore the parameter space of binaries, e.g. [4–7]. A key\npart of studying the parameter space is to simulate bina-\nries with intermediate mass-ratios.\nTodate, themassratiofurthestfromequalmassesthat\nhas been numerically simulated is 10:1 [8, 9]. These sim-\nulations use the Baumgarte-Shapiro-Shibata-Nakamura\n(BSSN) formulation [10–12] with 1+log slicing, and the\n˜Γ driver condition for the shift [13, 14]. In [8], it was\nnoted that the stability of the simulation is sensitive to\nthe damping factor, η, used in the ˜Γ driver condition,\n∂2\n0βi=3\n4∂0˜Γi−η∂0βi. (1)\nHere,βiistheshiftvectordescribinghowthecoordinates\nmove inside the spatial slices, ∂0≡∂t−βi∂i, and˜Γiis\nthe contraction of the Christoffel symbol, ˜Γi\njk, with the\nconformal metric, ˜ γjk.\nThe standard choice for ηis to set it to a constant\nvalue, which works well even for the most demanding\nsimulations as long as the mass ratio is sufficiently close\nto unity. In binary simulations, a typical choice is a con-\nstant value of about 2 /M, withMthe total mass of the\nsystem. This choice, however, leads to instabilities for\nthe mass ratio 10:1 simulation [8], although stability was\nobtained for η= 1.375/M. The value of ηis chosen to\ndamp an outgoing change in the shift while still yield-\ning stable evolutions. As we will show, if ηis too small,\nthere are unwanted oscillations, and values that are too\nlarge lead to instabilities. By itself, this observation is\nnot new, see e.g. [15–18]. The key issue for unequal\nmasses is that, as evident from (1), the damping factor η\nhas units of inverse mass, 1 /M. Therefore, the interval\nof suitable values for ηdepends on the mass of the blackholes. For unequal masses, a constant ηcannot equally\nwell accommodateboth black holes. Aconstantdamping\nparameter implies that the effective damping near each\nblack hole is asymmetric since the damping parameter\nhas dimensions 1 /M. For large mass ratios, this asym-\nmetry in the grid can be large enough to lead to a failure\nof the simulations because the damping may become too\nlarge or too small for one of the black holes. To cure this\nproblem, we need a position-dependent damping param-\neter that adapts to the local mass. In particular, we want\nit to vary such that, in the vicinity of the ithpuncture\nwith massMi, its value approaches 1 /Mi.\nA position-dependent ηwas already considered when\nthe˜Γdriverconditionwasintroduced[8,19–22], but such\nconstructions were not pursued further because for mod-\nerate mass ratios a constant ηworks well. Recently, we\nrevived the idea of a non-constant ηfor moving punc-\nture evolutions in order to remove the limitations of a\nconstantηfor large mass ratios. In [23], we constructed\na position-dependent ηusing the the conformal factor,\nψ, which carries information both about the location of\nthe black holes, and about the local puncture mass. The\nform ofηwas chosen to have proper fall-off rates both\nat the punctures and at large distance from the binary.\nIn [9], this approach was used successfully for mass ratio\n10:1. (We note in passing that damping is useful in other\ngauges as well, e.g. in [24] the modified harmonic gauge\ncondition includes position-dependent damping by use of\nthe lapse function.)\nIn the present work, we examine one potential short-\ncoming of the choice of [23], which leads us to suggest\nan alternative type of position-dependent η. Using [23],\nwe find large fluctuations in the values of that η, and\nthis might lead to instabilities in the simulation of larger\nmass-ratio binary black holes. To address this, we have\ntested two new explicit formulas for the damping factor\ndesignedto havepredictablebehaviorthroughoutthe do-\nmain of computation. We find the new formulas to pro-\nduce only small changes in the waveforms that diminish\nwithresolution,andthereisagreatdealoffreedominthe\nimplementation. Independently of our discussion here, in\n[16] the stability issues for large ηare explained, and a\nnon-constant ηis suggested (although not yet explored2\nin actual simulations), that, in its explicit coordinate de-\npendence, is similar to one of our suggestions.\nThe paperisorganizedasfollows. We firstdescribethe\nreasons for the damping factor and some of the reasons\nforlimitingitsvalueinSecII. InSec.III, wediscusssome\nprevious forms of ηthat have been used. We also present\ntwo new definitions and why we investigated them. In\nSec. IV, we find that these new definitions agree well\nwith the use of constant ηin the extracted gravitational\nwaves for mass ratios up to 4:1. Finally, in Sec. V, we\ndiscuss further implications of this work.\nII. MOTIVATION\nIn order to define a position-dependent form for η, it\nis important to determine what this damping parameter\naccomplishes in numerical simulations. For this reason,\nwe examine the effects of running different simulations\nwhile varying ηbetween runs. First we use evolutions of\nsingle non-spinning black-holes to identify the key phys-\nical changes. Then we examine equal-mass binaries to\ndetermine specific values desired in ηat both large and\nsmall radial coordinates.\nA. Numerics\nFor all the work in this paper, we have used the BAM\ncomputer codedescribed in [17, 25, 26]. It uses the BSSN\nformalism with 1+log slicing and ˜Γ driver condition in\nthe moving puncture framework [2, 27]. Puncture ini-\ntial data [28] with Bowen-York extrinsic curvature [29]\nhave been used throughout this work, solving the Hamil-\ntonian constraint with the spectral solver described in\n[30]. For binaries, parameters were chosen using [31] to\nobtain quasi-circularorbits, while the parametersfor sin-\ngle black holes were chosen directly. We extract waves\nvia the Newman-Penrose scalar Ψ 4. The wave extraction\nprocedure is described in detail in [17]. We perform a\nmode decomposition using spin-weighted spherical har-\nmonics with spin weight −2,Y−2\nlm, as basis functions and\ncalculate the scalar product\nΨlm\n4=/parenleftbig\nY−2\nlm,Ψ4/parenrightbig\n=/integraldisplay2π\n0/integraldisplayπ\n0sinθdθdϕY−2\nlmΨ4.(2)\nWe furthersplit Ψlm\n4intomode amplitude Almandphase\nφlmin order to cleanly separate effects in these compo-\nnents,rex·Ψlm\n4=Almeiφlm. In this paper, we focus on\none of the most dominant modes, the l=m= 2 mode,\nand report results for this mode unless stated otherwise.\nThe extraction radius used here is rex= 90M.B. Single, non-spinning puncture with constant\ndamping\nThedamping factor, η, in Eq.(1), isincluded to reduce\ndynamics in the gauge during the evolution. To examine\nthe problem brought up in the introduction, we compare\nresults of a single, non-spinning puncture with mass M.\nWe use a Courant factor of 0 .5 and 9 refinement levels\ncentered around the puncture. The resolution on the\nfinest grid is 0 .025M, and the outer boundary is situated\nat 256M. Varying the damping constant between 0 .0/M\nand4.5/M,twomainobservationscanbemade. First, as\ndesigned, a non-zero ηattenuates emerging gauge waves\nefficiently. Second, an instability develops for values of η\nthat are too large.\nFigs. 1, and 2 illustrate the first observation. Both\nfigures show the x-component of the shift along the x-\ncoordinate using η∈ {0.0/M,1.5/M,3.5/M}. Apart\n0 10 20 30 40\nx/M00.050.10.150.2βxη = 0.0/M\nη = 1.5/M\nη = 3.5/M\nFIG. 1: The x-component of the shift, βx, for a single non-\nspinning puncture of mass Mat timet= 15.2M. The three\nlines were taken for different values of the damping factor η.\nThe solid line (black) is for η= 0.0/M. The dashed line (red)\nis forη= 1.5/Mand the dotted-dashed line (green) is for\nη= 3.5/M. This shows the beginning of a pulse in βxfor\nsmaller values of η.\nfrom the usual shift profile, Fig. 1 shows the beginnings\nof a pulse in the η= 0.0/Mcase (solid line) at x≈10M\nafter 15.2Mof evolution. Examining Fig. 2, where we\nzoom in at a later time, t= 30.4M, one can see that\nthe pulse has started to travel further out (solid line).\nLooking carefully, one can also see a much smaller pulse\nin theη= 1.5/Mline (dashed). Lastly, by examination,\none can find almost no traveling pulse in the η= 3.5/M\ncurve (dotted-dashed line). The observed pulse in the\nshift travels to regions far away from the black hole and\neffects the gauge of distant observers. This might have3\n0 10 20 30 40 50\nx/M-0.0100.010.020.030.04βxη = 0.0/M\nη = 1.5/M\nη = 3.5/M\nFIG. 2: The x-component of the shift, βx, for a single non-\nspinning puncture of mass Mat timet= 30.4M. The three\nlines were taken for different values of the damping factor η\nwith the same line type and color scheme as in Fig. 1. Here\nit is clear a pulse radiates outward in the shift with smaller\nvalues of η.\nundesirable implications for the value of such numerical\ndata when trying to understand astrophysical sources.\nFor values of ηlarger than 3 .5/M, an instability\narises in the shift at larger radius. Fig. 3 shows the x-\ncomponent of the shift vector using damping constants\nη= 3.5/M(solid line), η= 4.0/M(dashed line) and\nη= 4.5/M(dotted-dashed line). The plots show an in-\nstability in simulations with η >3.5/Mdeveloping in\nβi, which eventually leads to a failure of the simulations.\nContrary to this, the simulation using η= 3.5/Mdoes\nnot show this shift related instability. In test runs we\nfound that by decreasing the Courant factor used, we\ncould increase the value of the damping factor and still\nget stable evolutions. This agrees with [16] where it was\nshown that the gamma driver possesses the stiff prop-\nerty, which limits the size of the time-step in numerical\nintegration based on the value of the damping.\nFigures 1, 2, and 3 make clear how the choice of the\ndamping factor affects the behavior of the simulations.\nThe value we choose for ηshould be non-zero and not\nlargerthan 3 .5/Mto allow for effective damping and sta-\nble simulations. The exact cutoff value between stable\nand unstable simulations is not relevant here since the\nposition dependent form we develop in Sec. III gives us\nthe flexibility we need to obtain stable simulations.0 50 100 15000.511.520 50 100 15000.511.52\nPSfrag replacementsx/M\nx/Mη= 3.5/M\nη= 3.5/Mη= 4.0/M\nη= 4.0/Mη= 4.5/Mβxβxt= 16M\nt= 30.4M\nFIG. 3: The x-componentof theshift vectorin the x-direction\nfor a single non-spinning puncture of mass Mat times t=\n16.0Mandt= 30.4M. The three different lines mark three\nvalues of the damping constant η. The solid line (black) is\nforη= 3.5/M, the dashed line (red) for η= 4.0/Mand the\ndotted-dashed line (green) for η= 4.5/M. Att= 16M, the\nsimulation using η= 4.5/Mdevelops an instability in the\nshift vector and fails soon afterward, the same happens for\nη= 4.0/Matt= 30.4M. In the simulation using η= 3.5,\nno such instability develops (not shown).\nC. Equal mass binary with constant damping\nTo examine the effect of ηon the extraction of gravi-\ntational waves, we compare the results from simulations\nof an equal mass binary with total mass Min quasi-\ncircular orbits with initial separation D= 10M, using\nη∈ {0.0/M,0.5/M,2.0/M}. Again, the Courant factor\nis chosen to be 0 .5 and we use, in the terminology of [17],\nthe grid configuration χ[6×56 : 5×112 : 6] with a finest\nresolution of 0 .013M. Here, the extraction radius rexis\nchosen to be 90 M.\nFor vanishing η, we find a lot of noise in the the real\npart of the 22-mode of rexΨ4, shown in the solid curve of\nFig. 4. A small, but non-vanishing ηsuffices to suppress\nthis noise, as seen in the dashed curve of this figure. The\ndotted-dashedcurvein this plot is the result forusing the\nvalueη= 2.0/M. We see a difference in time between\npeak amplitudes of the three curves due to the change\nof coordinates that the alternation of ηintroduces. We\ndid, however, find that by decreasing the Courant factor\nthosedifferencesbetween peakamplitudes summarilyde-\ncreased.\nTo understand the noise in the waves for η= 0.0/M,\nwe look at the shift vector at different times. The first\npanel of Fig. 5 shows the x-component of the shift over4\n0 200 400 600 800-0.008-0.006-0.004-0.00200.0020.004\nPSfrag replacementsη= 0.0/M\nη= 0.5/M\nη= 2.0/MRe{Ψ224}rex·M\nt/M0 200 400 600 800 1000 1200-0.06-0.04-0.0200.020.040.06\nPSfrag replacements\nηs= 0.0M\nηs= 0.5M\nηs= 2.0M\nRe{Ψ224}rex·M\nt/M\nFIG. 4: Real part of the 22-mode of Ψ 4over time for equal\nmass simulations using different values for η. The inset shows\nthe full waveform until ringdown. The solid curves (black)\nare forη= 0, the dashed curve (red) mark η= 0.5/Mand\nthe dotted-dashed curves (green) are for η= 2.0/M. Without\ndamping in the shift, the extracted waves are noisy at times\nwhen the amplitude is still small (black, solid curve).\nx, againforη∈ {0.0/M,0.5/M,2.0/M},shortlyafter the\nbeginning of the simulation. The fourth panel shows the\nsame at a time shortly before the merger, and the two\npanels in the middle represent intermediate times. We\nsee clear gauge pulses in the earliest time panel for all\nthree curves. We also observe the amplitude of this pulse\ndecreasing with increasing η. As time goes on, the gauge\npulse travelsoutwardsas in the case for a single puncture\nin section IIB. For vanishing η(solid line), the shift\nbecomes more and more distorted, and the distortions\ndo not leave the grid. For non-zero η, the amplitude of\nthe gauge pulse decreases when traveling outwards, and\nthe shape of βxis not distorted. There is, compared to\nη= 2.0/M(dotted-dashed line), only a small bump left\nin theη= 0.5/Mcase (dashed line), that changes its\nshape slightly during the simulation, but does not travel\nto large distances from the punctures. The coordinates\nare disturbed in the case where no damping is used, and\nthus the noise in rexRe{Ψ22\n4}is not surprising.\nIn this series, using a Courant factor of 0.5, we only\nobtained stable evolutions for η <3.5/Mwhich agrees\nwith thelimits foundin sectionIIB. Ifwechosethe value\nofηtoo large, the same kind of instability in the shift\nvectorwefoundtheredevelopsintheequalmasscaseand\nthe simulations fail. The failure occurs relatively early,\nbefore 50Mof evolution time, whereas the stable runs\nlasted about 1200 M,including mergerand ringdown(we\nstopped the runs after ringdown).\nIII. POSITION-DEPENDENT FORMS OF η\nInsectionIIB, wesawthatasufficientlevelofdamping\nis needed to limit gauge dynamics, and too much damp--200 -100 0 100 200-0.004-0.00200.0020.004\n-200 -100 0 100 200-0.004-0.00200.0020.004\n-200 -100 0 100 200-0.004-0.00200.0020.004\n-200 -100 0 100 200-0.004-0.00200.0020.004PSfrag replacementsη= 0.0/M\nη= 0.5/M\nη= 2.0/Mt= 64M\nt= 170 M\nt= 456 M\nt= 751 Mβxβxβxβx\nx/Mx/Mx/Mx/M\nFIG. 5: x-component of the shift vector, βx, for three differ-\nent choices of ηat four different times during the simulation.\nThe physical system is the same as in Fig. 4. The merger\ntakes place at approximately t= 1000M. In the η= 0/M\ncase (black, solid curve), the shift vector is not damped and\ntherefore, a pulse travels outwards and distorts the shift o ver\nthe whole grid. The amplitude of this pulse is considerably\ndamped when using a non-vanishing ηand therefore the dis-\ntortions are reduced. For η= 0.5/M(red, dashed curve),\nthere are still small bumps traveling out which are reduced\nby using η= 2/M(green, dotted-dashed curve).\ning can lead to numerical instabilities. In section IIC,\nwe saw the positive effect that sufficient damping has on\nthe resultantwaveformforequalmassbinaries. While we\nstill need damping in the gamma driver in the unequal\nmass case, a constant value may not fulfill the require-\nments of limiting gauge dynamics and permitting stable\nevolutions. Rather, we need a definition for the damping\nthat adjusts the value to the local mass-scale.\nWe will examine definitions, that naturally track the\nposition, and mass of the individual black holes. The\nchoice ofηshould provide a reasonable value both near\nthe individual black holes, and at large distance from\nthe binary. We will start by examining some previous5\nwork, that has used non-constant forms of the damping\nparameter, and why it may be necessary to use other\nformulas. We will then present the two new formulas for\nη, which we designed for this work.\nA. Previous dynamic damping parameters\nA position dependent damping was introduced some\nyears ago by the authors of [20], and was later used in\n[21]. That formula reads\nη=ηpunc−ηpunc−η∞\n1+(ψ−1)2(3)\nwithηpunc,η∞being constants, and assuming ψ=\n1 +M1/(2r1) +M2/(2r2) (riis the distance to the ith\npuncture). Thisformulawasusedtodampgaugedynam-\nics while using excision for equal-mass head-on collisions.\nIt has since been found that using the moving puncture\nframework allows for constant damping in the approxi-\nmately equal mass case. We are looking for a formula\nwhich is suitable for the quasicircular inspiral of inter-\nmediate mass-ratio binaries.\nPreviously [23], we used the formula\nη(/vector r) =ˆR0/radicalbig\n˜γij∂iψ−2∂jψ−2\n(1−ψ−2)2, (4)\nfor determining a position dependent damping coeffi-\ncient instead of using a constant η. WithˆR0taken to be\na unitless constant, it can be seen that Eq. (4) has units\nof inverse mass. The dependence on the BSSN variable,\nψ, naturally tracks the position, and mass of the black\nholes. The application of Eq. (4) gave good values for\nthe damping both at the punctures, and at the outer\nboundary, and was even found to somewhat decrease the\ngrid-size of the larger black hole. The latter point could\nhave positive effects on how the individual black holes\nare resolved on the numerical grid. It even had the addi-\ntional effect of keeping the horizon shapes roughly circu-\nlar, even close to merger - something that doesn’t hold in\nthe constant ηcase. Most importantly, the simulations\nremained stable, without significantly changing the grav-\nitational waves. The formula was later used successfully\nfor the 10:1 mass-ratio in [9].\nDespite all this, Eq. (4) provides reason for concern.\nFig.6showstheformof ηusingEq.(4)foranon-spinning\nbinary of equal mass in quasicircular orbits starting at a\nseparation of D= 10Mat four different times in the\nsimulation. As can be seen, noise travels out from the\norigin as time progresses. This leaves steady features on\nthe form of ηwhich could spike to higher and lower val-\nues than the range determined in Sec. IIB. Additionally,\nthese sharp features may lead to unpredictable coordi-\nnate drifts, and could, in some cases, affect the long-term\nstability of the simulation.\nTo illuminate the origin of the disturbances in η(/vector r),\nwe looked at the development of η(/vector r) in simulations of0 200 400 600 80000.511.522.53\n0 200 400 600 80000.511.522.53\n0 200 400 600 80000.511.522.53\n0 200 400 600 80000.511.522.53\nPSfrag replacements\nx/Mη(x)·M η (x)·M η (x)·M η (x)·Mt= 0M\nt= 79M\nt= 165 M\nt= 300 M\nFIG. 6: Damping factor, η, along the x-axis using Eq. (4).\nThe simulated configuration is an equal mass binary with ini-\ntial separation D= 10Mand orbits lying in the ( x,y)-plane.\nShown are four different times during the simulation.\na single, non-spinning puncture, and a single, spinning\npuncture (Sz/M2= 0.25). The result for the spinning\ncase is plotted in Fig. 7 at two different times over the\nx-axis. Again, we see a pulse traveling outwards. Only\nthis time, it does not leave much noise on the grid. The\nfact that this pulse travels at a speed which is roughly\n1.39 (in our geometric units where c=G= 1) in both\nthe spinning and non-spinning scenario indicates that it\nis related to the gauge modes traveling at speed√\n2 in\nthe asymptotic regionwhere α≃1 (see [32] and [19] for a\ndiscussion of gauge speeds). In contrast to gauge pulses\nin the lapse, α, or shift vector, βi, the pulse in η(x) is\namplified as it walks out. We found the same result in\nthe single puncture simulation without spin. We believe\nthe reason for this behavior is that as the distance to the\npuncture increases,the conformalfactor, ψ, getscloserto\nunity. Therefore, the denominator in Eq. (4) approaches\nzero, and the gauge disturbances in the derivatives of ψ\nare magnified. We further observed reflections at the re-6\nfinement boundaries as this pulse passes through them.\nThismayexplainthe fluctuationsin η(x) shownin Fig.6.\nWhile one could continue to fine-tune a formula depen-\ndent on the conformalfactorto deal with these problems,\nwe looked in a different direction to determine the form\nof the damping parameter.\n-200 -100 0 100 20000.511.522.53\nPSfrag replacements\nx/Mη(x)·Mt= 50 .56M\nt= 101 .25M\nFIG. 7: Form of η(/vector r) for a single spinning puncture sitting\natx= 0 using Eq. (4) after simulation time t= 50.56M\n(solid black line) and t= 101.25M(dashed red line) over\nx−direction.\nB. Formulas for ηwith explicit dependence on the\nposition and mass of the punctures\nSince we always know the location of a puncture, and\nwe know what its associated mass, we chose a form\nof damping that uses this local information throughout\nthe domain. To address the demands and concerns dis-\ncussed in Section II and IIIA, we designed two position-\ndependent forms of η. The two forms we tested are\nη(/vector r) =A+C1\n1+w1(ˆr2\n1)n+C2\n1+w2(ˆr2\n2)n,(5)\nand\nη(/vector r) =A+C1e−w1(ˆr2\n1)n+C2e−w2(ˆr2\n2)n.(6)\nInEqs.(5) and(6), w1andw2arerequiredtobe positive,\nunitless parameters which can be chosen to change the\nwidth of the functions. The power nis taken to be a\npositive integer which determines the fall-off rate. The\nconstantsA,C1, andC2are then chosen to provide the\ndesired values of ηat the punctures, and at at infinity.\nLastly, ˆr1and ˆr2are defined as ˆ ri=|/vector ri−/vector r|\n|/vector r1−/vector r2|, whereiis\neither one or two, and /vector riis the position of the i’th black\nhole.\nThe definition of ˆ riis chosen to naturally scale the fall-\nofftothe separationofthe blackholes. w1,w2, andncan\nbe chosen to change the overallfall-off. Our work focuseson the choice w1=w2=wandn= 1. Following [23],\nwe construct the damping factor to have units of inverse\nmass. We choose A= 2/Mtot, whereMtot≡M1+M2\nis defined as the sum of the irreducible masses. We then\ntakeCi= 1/Mi−A. It is then evident that both Eqs. (5)\nand (6) will give a constant value of η= 2/Mtotin the\nequal mass case.\nWe designed the two formulas for ηin order to test\nthe value of using fundamentally different functions. In\nour simulations, we found little noticeable difference in\nthe application of one compared to the other. In the\nabsence of such a difference, it becomes more beneficial\nto use Eq. (5), as Gaussians are computationally more\nexpensive. It should be pointed out that Eq. (5) is very\nsimilar to Eq. (13) suggested in [16], and we believe the\nfollowing results are very similar to what would be found\nusing that form for the damping. Going into the present\nwork, we haveno ansatz which might suggestthese forms\nof damping yield wave forms which are any better than\nthe use of any previous form of η. However, as will be\nseen in the results sections, the waveforms we get from\nunequal mass binaries show noticeable improvement over\nthe constant ηcase.\nIV. RESULTS\nFor data analysis purposes, we are mainly interested in\nthe properties of the emitted gravitational waves of the\nblack hole binary systems under study. Hence, it is im-\nportant to check how the changes in the gauge alter the\nextracted waves. In the context of gravitational wave ex-\ntraction, Ψ 4is only first orderinvariant under coordinate\ntransformations. In addition, we haveto chosean extrac-\ntion radius rexfor the computation of modes, which is\nalso coordinate dependent. Although the last point can\nbe partly addressed by extrapolation of rex→ ∞, it is\na priori not clear how much a change of coordinates af-\nfects the gravitational waves. Furthermore, a change of\ncoordinates implies an effective change of the numerical\nresolution, and for practical purposes we have to ask how\nmuch waveforms differ at a given finite resolution.\nA. Waveform comparison using formula (5)\nThe results in the following section refers to the use of\nEq. (5). We compare numerical simulations using three\ndifferent grid configurations, which correspond to three\ndifferent resolutions. In the terminology of [17], the grid\nset-upsareφ[5×64 : 7×128 : 5],φ[5×72 : 7×144: 5], and\nφ[5×80 : 7×160: 5], which correspondsto resolutionson\nthe finest grids of 3 M/320 (N= 64),M/120 (N= 72)\nand 3M/400 (N= 80), respectively. When referring\nto results from different resolutions, we will from here\non use the number of grid points on the finest grid, N,\nto distinguish between them. In this subsection, we use\nw1=w2= 12 andn= 1 in Eq. (5). As test system we7\nuse an unequal mass black hole binary with mass ratio\nm2/m1= 4 andan initial separationof D= 5Mwithout\nspins in quasi-circular orbits.\nFor orientation, Fig. 8 shows the amplitude of the 22-\nmode,A22, computed with the standard gauge η= 2/M\n(displayed as solid lines) and with the new η(/vector r) using\nEq. (5) (displayed as non-solid lines). The three differ-\nent colors correspond to the three resolutions. The inset\nshows a larger time range of the simulation, while the\nmain plot concentrateson the time frame around merger.\nThe plotgivesacourseviewofthe closenessofthe results\nwe obtain with standard and new gauges.\nIn Fig. 9, we plot the relative differences between\nthe amplitudes at low and medium (solid lines), and\nmedium and high resolution (non-solid lines) obtained\nwithη= 2/M(light gray lines) as well as η(/vector r) (Eq. (5))\n(black lines). Here, we find the maximum error be-\ntween the low and medium resolution of the series using\nη= 2/Mamounts to about 12% (solid gray curve). Be-\ntween medium and high resolution (dashed gray curve),\nwe find a smaller relative error, but it still goes up to\n7% at the end of the simulation. Employing Eq. (5),\nthe maximum amplitude error between low and medium\nresolution (solid black line) is only about 4%, and there-\nfore even smaller than the error between medium and\nhigh resolution for the constant damping case. Between\nmedium and high resolution, the relative amplitude dif-\nferences for Eq. (5) are in general smaller than the ones\nbetween low and medium resolution, although the maxi-\nmum error is comparable to it (dot-dashed black line).\nWe repeat the previous analysis for the phase of the\n22-mode,φ22. Again, we comparethe errorsbetween res-\nolutions in a fixed gauge. Figure 10 shows that the error\nbetween lowest and medium resolution using η= 2/M\n(solid gray line) grows up to about 0.31 radians. For the\ndifferences between medium and high resolution (dashed\nline) we find a maximal error of 0.2 radians for η= 2/M.\nForη(/vector r) following Eq. (5), the phase error between low\nand medium resolution is only about 0.19 radians (solid\nblack line) and decreases to 0.1 radians between medium\nand high resolution (dot-dashed line). Again, employ-\ning the position dependent form of η, Eq. (5), the error\nbetween lowest and medium resolution is lower than the\none we obtain for constant ηbetween medium and high\nresolution. The results for amplitude and phase error\nsuggest that we can achieve the same accuracy with less\ncomputationalresourcesusingaposition-dependent η(/vector r).\nB. Waveform comparison using formula (6)\nWe repeated the analysis of Sec. IVA with the wave-\nforms we obtain using Eq. (6) (with w1=w2= 12 and\nn= 1). We use the same initial conditions (mass ratio\n4 : 1,D= 5M, no spins), and compare the amplitudes\nand phases of the 22-mode of Ψ 4with the results of the\nη= 2/M-runs. The grid configurations remain the same.\nThe results are very similar to the ones we obtained in160 165 170 175 180 185 1900.0150.0200.0250.0300.035\nt/Slash1MA22/DotMatΗM\n100 120 140 160 180 2000.0000.0050.0100.0150.0200.0250.0300.035\nt/Slash1MA22/DotMatΗMN/Equal64\nN/Equal72\nN/Equal80\nFIG. 8: Amplitude of the 22-mode of Ψ 4of a binary with\nmass ratio 4:1 and initial separation D= 5M. The different\ncolors correspond to three different resolutions according to\nthe grid setup described in the text. The solid lines are resu lts\nforη= 2/M, the dashed, dotted and dot-dashed ones are for\nη(/vector r) (Eq. (5)). The inset shows the simulation from shortly\nafter the junk radiation passed, in the main plot we zoom into\nthe region of highest amplitude (near the merger).\n100 120 140 160 180 200/Minus0.050.000.050.100.15\nt/Slash1M/CapDelta/CapAlpΗa22/Slash1/CapAlpΗa22/LParen164/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen180/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen164/Minus72/RParen1,Η/Equal2/Slash1M\n/LParen180/Minus72/RParen1,Η/Equal2/Slash1M\nFIG. 9: Relative differences of the amplitude of the 22-mode\nof Ψ4between resolutions N= 64 and N= 72 (gray solid\ncurve) as well as N= 72 and N= 80 (gray dashed curve)\nwhen using η= 2/M. The same for η(/vector r) (Eq. (5)) between\nN= 64 and N= 72 (black solid curve) and N= 72 and\nN= 80 (black dot-dashed curve). The physical situation is\nthe same as in Fig. 8. The maximum differences are above\n10%, comparing low and medium resolution of the constant\nηsimulations (gray solid line).\nFigs. 9 and 10, and we therefore do not show them here.\nAlthough Eqs. (5) and (6) result in different shapes for\nη(/vector r), Ψ22\n4is very similar. Therefore, the comparison to\nη= 2/Mnaturally gives very similar results, too. The\nphase differences between results from Eqs. (5) and (6)\nat a given resolution are shown in Fig. 11. These are,\nwith a maximum phase error of 0.004 radians, very small\ncomparedto the phase errorsbetween resolutions, which,\nat minimum, are about 0.1 radian (see Fig. 10). Fig. 12\ncomparesthe phase errorbetween low and medium (solid8\n100 120 140 160 180 2000.00.10.20.3\nt/Slash1M/CapDeltaΦ22/LParen164/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen180/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen164/Minus72/RParen1,Η/Equal2/Slash1M\n/LParen180/Minus72/RParen1,Η/Equal2/Slash1M\nFIG. 10: Phase differences between lowest and medium reso-\nlution for the series using η= 2/M(solid gray line) and η(/vector r)\n(Eq. (5)) (solid black line) as well as between medium and\nhigh resolution for η= 2/M(dashed gray line) and for η(/vector r)\n(Eq. (5)) (dot-dashed black line). The physical situation i s\nthe one of Fig. 8.\n100 120 140 160 180 200/Minus0.004/Minus0.0020.0000.0020.004\nt/Slash1M/CapDeltaΦ22N/Equal64,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1\nN/Equal72,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1\nN/Equal80,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1\nFIG. 11: Phase differences between waveforms obtained\nwith Eq. (5) and Eq. (6) in three different resolutions (solid ,\ndashed, dotted-dashed lines) for mass ratio 4:1, D= 5M.\nlines), and medium and high resolution (dotted-dashed\nand dashed line) of Eq. (6) (gray) to the ones of Eq. (5)\n(black). For comparison, the error between medium and\nhigh resolution is also plotted for Eq. (4) in this figure\n(dotted line). The plot indicates that the errors between\nresolutions are in good agreement for the different posi-\ntion dependent formulas of η.\nC. Behavior of the shift vector\nIn [23], we found an unusual behavior of the shift vec-\ntor. This is illustrated in Fig. 13, where we plot the\nx-component of the shift, βx, in thex-direction after\n160Mof evolution (this means approximately 80 Maf-\nter merger) for all four versions of the damping constant\nwe used for comparison in this paper before, and for the\nsame binary configuration as the one used in Secs. IVA100 120 140 160 180 2000.000.050.100.150.20\nt/Slash1M/CapDeltaΦ22/LParen180/Minus72/RParen1,ΗEq./LParen14/RParen1\n/LParen164/Minus72/RParen1,ΗEq./LParen15/RParen1\n/LParen180/Minus72/RParen1,ΗEq./LParen15/RParen1\n/LParen164/Minus72/RParen1,ΗEq./LParen16/RParen1\n/LParen180/Minus72/RParen1,ΗEq./LParen16/RParen1\nFIG. 12: Phase difference between waveforms at low and\nmedium resolution (solid lines) and medium and high reso-\nlution (dotted-dashed and dashed line) using either Eq. (5)\n(black lines) or Eq. (6) (gray lines) for mass ratio 4:1, D=\n5M. For comparison, we also show the phase difference ob-\ntainedwith Eq. (4) between mediumandhigh resolution (dot-\nted line).\n0 100 200 300 400 5000.0000.0010.0020.0030.004\nx/Slash1MΒxΗ/Equal2/Slash1M\nEq./LParen14/RParen1\nEq./LParen15/RParen1\nEq./LParen16/RParen1\nFIG. 13: x-component of the shift vector in x-direction after\n160Mof evolution of the system with mass ratio 4 : 1 and\nD= 10M. The black, dot-dashed line refers to the use of a\nconstant damping η, while the black, solid line uses Eq. (4).\nThe gray, dashed line is for the use of Eq. (6) and the gray,\ndotted one for Eq. (5). Except for the constant η(black, dot-\ndashed line), the results in this plot are indistinguishabl e.\nand IVB. Like in [23], we find that using Eq. (4) results\nin a shift which falls off to zero too slowly towards the\nouter boundary, and which develops a “bump” (black,\nsolid line), while the constant damping case (black, dot-\ndashed line) falls off to zero quickly. Employing Eqs. (5)\nor (6) avoids this undesirable feature. After merger, the\nshift falls off to zero when going awayfrom the punctures\nas it does in the constant damping case (gray dashed and\ndotted lines). Using Eq. (5) or (6) prevents unwanted co-\nordinate drifts at the end of the simulations.9\n0 10 20 30 40 500102030405060\nPSfrag replacements\nt/MAAH,1/AAH,2\nn= 1, w= 0.1\nn= 1, w= 0.5 n= 1, w= 6.0\nn= 1, w= 12.0\nn= 1, w= 16.0n= 1, w= 200 .0\nn= 3, w1= 0.01,\nw2= 0.001\nη= 2/M\nηfrom Eq. (4)\nFIG. 14: Shown is the time dependence of the ratio between\nthe coordinate areas of the apparent horizons of both black\nholes in a simulation with mass ratio 4 : 1 with initial separa -\ntionD= 5M. The black, blue and red lines use η(/vector r), Eq. (5)\nwithvaryingvaluesofthewidthparameter w. Theorange line\n(dash-dot-dot) uses the constant damping η= 2/Mand the\ngreen (dash-dot) one refers to the result of [23] with Eq. (4) .\nUsing Eq. (5), the coordinate areas can be varied with re-\nspect to each other depending on the choice of w. A ratio of\n1 means the black holes have the same size on the numerical\ngrid.\nV. DISCUSSION\nIn this work, we examined the role that the damping\nfactor,η, plays in the evolution of the shift when using\nthe gamma driver. In particular, we examined the range\nof values allowed in various evolutions, and what effects\nshowed up because of the value chosen. We then de-\nsigned a form of ηfor the evolution of binary black holes\nwhich provides appropriate values both near the individ-\nual punctures and far away from them with a smooth\ntransition in between.\nIn Sec. IV, we directly examined the waveformsfor the\ncaseusingEq.(5), where w1=w2= 12andn= 1. While\nthe form of ηis predictable, and can be easily adjusted\nfor stability, we also saw that the waveforms produced\nusing this definition showed less deviation with increas-\ning resolution than using a constant η. When examining\nthe waveforms produced using Eq. (6), we found simi-\nlar results. In the absence of a noticeable difference in\nthe quality of the waveforms, Eq. (5) is computationally\ncheaper, and, as such, is our preferred definition for the\ndamping.\nWe have already pointed out a certain freedom to\npick parameters in Eqs. (5) and (6). We did perform\nsome experimentation along this line where we varied\nw=w1=w2to see if we could get a useful effect of the\ncoordinate size of the apparent horizons on the numer-\nical grid. In [17, 33], it was noticed that the damping\ncoefficient affects the coordinate location of the apparent\nhorizon, and therefore the resolution of the black hole onthe numerical grid. Fig. 14 plots the ratio of the grid-\narea of larger apparent horizon to the smaller apparent\nhorizon as a function of time for w-values of 0 .1, 0.5 and\nfor 200, all with n= 1. Also plotted is the relative co-\nordinate size for the same binaries using a constant η\nin dashed, double-dotted line, and for using Eq. (4) in\na blue dashed-dotted line. All the evolutions show an\nimmediate dip, and then increase in the grid-area ratio\nduring the course of the evolution. While a very low ra-\ntio was found using Eq. (4), the orange dotted line was\nlater found for the choices of n= 3 withw1= 0.01 and\nw2= 0.0001 with Eq. (5). Due to this freedom in the\nimplementation of our explicit formula for the damping,\nit may be possible to further reduce the relative grid size\nof the black holes. This effect could be important in eas-\ning the computational difficulty of running a numerical\nsimulation for unequal mass binaries.\nHaving a form of ηthat leads to stable evolutions for\nany mass-ratio is an important step towards the numer-\nical evolution of binary black holes in the intermediate\nmass-ratio. We believe the form given in Eq. (5) pro-\nvides such a damping factor at a low computational cost,\nalthough the test results presented are limited to mass\nratio 4 : 1. We plan to examine larger mass ratios in\nfuture work. The new method should allow binary sim-\nulations for mass ratio 10 : 1, or even 100 : 1. It remains\nto be seen whether other issues than the gauge are now\nthe limiting factor for simulations at large mass ratios.\nAcknowledgments\nItisapleasuretothankZhoujianCaoandErikSchnet-\nter for discussions. This work was supported in part\nby DFG grant SFB/Transregio 7 “Gravitational Wave\nAstronomy” and the DLR (Deutsches Zentrum f¨ ur Luft\nund Raumfahrt). D. M. was additionally supported by\nthe DFG Research Training Group 1523 “Quantum and\nGravitational Fields”. Computations were performed on\nthe HLRB2 at LRZ Munich.10\n[1] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr-\nqc/0507014.\n[2] M. Campanelli, C. O. Lousto, P. Marronetti, and\nY. Zlochower, Phys. Rev. Lett. 96, 111101 (2006), gr-\nqc/0511048.\n[3] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and\nJ. van Meter, Phys. Rev. Lett. 96, 111102 (2006), gr-\nqc/0511103.\n[4] F. 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Gundlach and J. M. Martin-Garcia, Phys. Rev. D 74,\n024016 (2006), gr-qc/0604035.\n[33] F. Herrmann, D. Shoemaker, and P. Laguna (2006), gr-\nqc/0601026."    },    {        "title": "1009.3046v2.A_discontinuous_Galerkin_method_for_the_Vlasov_Poisson_system.pdf",        "content": "A discontinuous Galerkin method for the\nVlasov-Poisson system\nR. E. Heath,aI. M. Gamba,bP. J. Morrison,cand C. Michlerd\naApplied Research Laboratories, University of Texas at Austin, TX 78758, USA.\nPresent Address: GE Global Research, 1 Research Circle, Niskayuna, NY 12309;\nEmail: heathr@ge.com\nbDepartment of Mathematics and ICES, University of Texas at Austin, TX 78712,\nUSA; Email: gamba@math.utexas.edu\ncDepartment of Physics and Institute for Fusion Studies, University of Texas at\nAustin, TX 78712, USA; Email: morrison@physics.utexas.edu\ndInstitute for Computational Engineering and Sciences (ICES) The University of\nTexas at Austin Austin, TX 78712 U.S.A., Present Address: Computing\nLaboratory, The University of Oxford, Wolfson Building, Parks Road Oxford OX1\n3QD, U.K.; Email: christian.michler@comlab.ox.ac.uk\nAbstract\nA discontinuous Galerkin method for approximating the Vlasov-Poisson system of\nequations describing the time evolution of a collisionless plasma is proposed. The\nmethod is mass conservative and, in the case that piecewise constant functions are\nused as a basis, the method preserves the positivity of the electron distribution\nfunction and weakly enforces continuity of the electric \feld through mesh interfaces\nand boundary conditions. The performance of the method is investigated by com-\nputing several examples and error estimates associated system's approximation are\nstated. In particular, computed results are benchmarked against established theoret-\nical results for linear advection and the phenomenon of linear Landau damping for\nboth the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are\nconsidered: nonlinear Landau damping and a version of the two-stream instability\nare computed. For the latter, \fne scale details of the resulting long-time BGK-like\nstate are presented. Conservation laws are examined and various comparisons to\ntheory are made. The results obtained demonstrate that the discontinuous Galerkin\nmethod is a viable option for integrating the Vlasov-Poisson system.\nKey words: Discontinuous Galerkin method, Vlasov-Poisson system, Landau\ndamping, Lorentz distribution, two-stream instability, BGK states.\nPreprint submitted to Elsevier 27 October 2018arXiv:1009.3046v2  [physics.plasm-ph]  1 Oct 20111 Introduction\nThe single species Vlasov-Poisson system is a nonlinear kinetic system that\nmodels time evolution of a collisionless plasma consisting of electrons and a\nuniform background of \fxed ions under the e\u000bects of a self-consistent elec-\ntrostatic \feld and possibly an externally supplied \feld. The Vlasov equation\nmodels the transport of the electrons that are coupled to the electrostatic\npotential through Poisson's equation.\nThe unknown electron distribution function, a phase space density, is denoted\nbyf=f(x;v;t );where the independent variables x;v;t are position, velocity,\nand time, respectively. For a given t, the quantity f(x;v;t )dxdv denotes the\nnumber of electrons contained in the in\fnitesimal phase space volume element\ndxdv centered about ( x;v) at timet. Upon a proper renormalization, fcan\nbe interpreted as a probability distribution function for the electrons over the\nphase space.\nThe Vlasov-Poisson system has solutions that can exhibit a variety of dynam-\nical phenomena [69,7,44]. One of the most well-known e\u000bects is \flamentation\nor `phase mixing' as it is sometimes called, which occurs when di\u000berent char-\nacteristics surfaces associated to the nonlinear transport (Vlasov) equation\nwrap in the phase space. This e\u000bect results in sti\u000b gradients, since fgenerally\ntakes on disparate values along di\u000berent characteristics.\nRelated to \flamentation is the interesting phenomenon of Landau damping\n[42]: electrostatic disturbances can be interpreted as the interaction between\nplasma waves and electrons with a resulting net energy transfer from the wave\nto electrons leading to an exponential collisionless damping of the electric \feld\nmodes in time. Because of such phenomena, the Vlasov-Poisson equation can\nbe challenging to simulate numerically.\nExisting numerical techniques for solving the Vlasov-Poisson system can be\ndivided into two groups: (i) those that approximate the system in the phase\nspace directly and (ii) those that transform the system into a di\u000berent coordi-\nnate space. The numerical approaches that treat the phase space directly do\nnot, however, usually involve discretizing the Vlasov equation. Rather most\nof these methods take advantage of the characteristic structure of the Vlasov\nequation, which implies that the plasma particles evolve along trajectories that\nsatisfy a given set of ordinary di\u000berential equations. The most widely used par-\nticle scheme is the Particle-in-Cell (PIC) method [9,38,27], which represents\nensembles micro-particles as a \fnite number of macro-particles. Each macro-\nparticle is then assumed to evolve along a characteristic trajectory, where the\nelectric \feld de\fning the trajectory is computed via any standard scheme.\nThe PIC method seems to give reasonable results in cases where the tail of\n2the distribution is negligible and a large number of particles is not necessary.\nOtherwise, the method su\u000bers from numerical noise that is proportional to\n1=pn;wherenis the number of particles.\nOther methods based on the discretization of the phase space have also been\nproposed. In [11], an operator splitting method was introduced and shown\nto be both e\u000ecient and accurate for solving a wide range of problems. A\ncontinuous \fnite element method was developed in [72,73] and was shown to\nachieve results similar to those obtained in [11]. A positive and mass conserva-\ntive scheme was employed in [33] to solve both linear and nonlinear damping\nproblems in one- and two-dimensional physical space. This method is de\fned\nat a given time step by \frst building a piecewise constant approximation over\na mesh of the phase space using the approximation obtained from the previ-\nous time step and two correction terms whose values are found by solving two\n\fxed point problems. The piecewise approximation is then used in conjunction\nwith a slope limiter to reconstruct a local polynomial approximation of ffor\neach cell in the mesh.\nTransform methods based on Fourier or Hermite series have also been used,\ne.g. in [36,3,29] and more recently in [31]. In [40,41], the phase space was\ntransformed using a Fourier transform and a splitting method was employed to\nadvance the approximation in time. This method also included a \flamentation\n\fltering step for the purpose of smoothening the \flaments. The numerical\nresults obtained using this method seem reasonable only for problems in which\n\flamentation is not a dominant e\u000bect, where perhaps the nonlinearity either\nslows down or prevents the onset of \flamentation. However, this method may\nbe inadequate for problems where the physics of interest depends upon the\n\flamentary nature of the distribution, such as is the case for Landau damping.\nThe objective of this paper is to propose a coupled Upwind Penalty Galerkin\n(UPG) method for the approximation of the Vlasov-Poisson system and to\nevaluate its numerical e\u000ecacy. Our UPG method gives a uni\fed approach for\napproximating both the hyperbolic and elliptic parts of the Vlasov-Poisson\nsystem. Speci\fcally, the Vlasov equation is discretized using the standard Up-\nwind Galerkin (UG) scheme for conservation laws [24,23,22] and the Poisson\nequation is discretized using one of the three Discontinuous Galerkin (DG)\ninterior penalty available schemes [70,59,63]. Stability and convergence esti-\nmates for the UPG method were presented in [37] for the six-dimensional\nphase space In the same reference, the method was shown to be both locally\nand globally mass conservative.\nMore speci\fcally, the semi-discrete UPG approximation fhto the electron\ndistribution function is de\fned to be the solution of a \frst-order, nonlinear,\nordinary di\u000berential equation (ODE) system. Moreover, it has been shown that\nthe method preserves positivity of fhwhen piecewise constants basis functions\n3are used to approximate the solution to the Vlasov-Poisson system.\nIn this manuscript we show the numerical e\u000ecacy of the DG method by\nperforming accuracy, convergence, and conservation tests on computed UPG\napproximate solutions to a variety of linear and nonlinear problems where\nsu\u000eciently data or information of the true nonlinear solutions has been es-\ntablished. The computed results for these problems are benchmarked against\nknown theoretical results and are compared to results obtained using estab-\nlished methods.\nFor computing plasma problems the UPG method o\u000bers several advantages.\nIn particular, the local nature of the method facilitates adaptive mesh re\fne-\nments with easy adaptation to parallelization techniques. By taking advantage\nof these bene\fts, regions of the phase space where the electron distribution\nexperiences strong \flamentation or boundary layer e\u000bects can be resolved by\nlocal mesh re\fnements. The discontinuous nature of the method also helps\nto resolve the sti\u000b gradients associated with \flamentation, since requiring the\napproximation to be continuous in these cases can be too restrictive and typ-\nically lead to excessive numerical di\u000busion and oscillatory behavior. Due to\nthe fact that the method imposes boundary conditions weakly, a variety of\nboundary conditions can easily be accommodated.\nRecently, an alternative DG formulation for the Boltzmann-Poisson system\nwas introduced in [12,13,14,15,16] for simulations of hot electron transport for\none and two space dimensional nanometer scale devices, where kinetic correc-\ntions are known to be very signi\fcant. In [20] a DG scheme was constructed for\nthe Vlasov-Boltzmann equation by means of a maximum principle satisfying a\nlimiter for conservation laws [74,75], and it was shown to be high order accu-\nrate and positivity-preserving, not only for piecewise constant basis functions\nbut also for higher order polynomial approximations. Consequently a new ap-\nproximation of the Vlasov system based on a UPG scheme for higher order\npolynomial basis functions is currently being developed and tested [18,17].\nThus, the DG method is well-suited to approximate a range of plasmas span-\nning from the collisionless to the highly collisional regimes.\nFor completeness of this introduction we include some historical remarks re-\ngarding DG methods. The initial development of these numerical approximat-\ning schemes for hyperbolic and elliptic equations occurred independently, but\nnearly simultaneously. In 1973, the \frst DG scheme for linear hyperbolic equa-\ntions was introduced by Reed and Hill for approximating a neutron transport\nequation [57]. This work was followed by Lesaint and Raviart [43] in 1974,\nwhere a priori error estimates were proved for the DG method applied to\ntwo-dimensional, linear hyperbolic problems. The DG schemes for hyperbolic\nproblems were further studied in the series of papers [24,23,22], which culmi-\nnated in the introduction of the local discontinuous Galerkin (LDG) method\n4[25]. The generality of the LDG method was further extended to the multi-\ndimensional setting under more relaxed assumptions on the data in [21]. One\nof the \frst DG schemes for approximating the solutions to second-order ellip-\ntic equations was introduced in 1971 by Nitsche [55] where Dirichlet bound-\nary conditions were enforced weakly rather than strongly through the use\nof a penalty term. Shortly thereafter, applications of the penalty method to\nLaplace's equation were proposed by Babu\u0014 ska et al. in [4,5,6].\nThe use of penalty terms across interior faces as a means of enforcing continu-\nity among adjacent elements was introduced in [70] and [71] using a symmetric\ninterior penalty Galerkin (SIPG) \fnite element method. A non-symmetric in-\nterior penalty method (NIPG) similar to the SIPG method was proposed and\nanalyzed in [59]. The incomplete interior penalty method (IIPG) was intro-\nduced in [63,28,64] and is very similar to the SIPG and NIPG methods.\nThe outline of this paper is as follows. In Section 2, we describe the Vlasov-\nPoisson system and give a brief discussion of Landau damping. In Section\n3, the UPG method for the approximation of the Vlasov-Poisson system is\nderived and the error estimates associated to the approximation of the system\nare stated. In Section 4, several numerical simulation results are presented and\nanalyzed, including a study of the free streaming operator (simple advection),\nLandau damping for Maxwellian and Lorenztian equilibria, strong nonlinear\nLandau damping and a careful study of a symmetric two-stream instability,\nall for the case of repulsive potential forces. In Section 5, we comment on the\ne\u000eciency of the UPG method, draw some conclusions, and remark on future\nwork. Finally, an Appendix dedicated to the analysis of dispersion relations\nfor the Lorentzian and two-stream equilibria is included.\n2 The Vlasov-Poisson System\nThe Vlasov-Poisson system considered in this work has been scaled by the\nusual characteristic time and length scales, i.e., time is scaled by the inverse\nplasma frequency !\u00001\np, length by the Debye length \u0015D, and velocity accordingly\nby a thermal velocity vth=!p\u0015D.\nUsing this nondimensionalization, the Vlasov-Poisson problem is as follows:\nFor the divergence free vector in ( x;v) phase space\n\u000b(x;v;t ) = (v;\u0000E(v;t)) for (x;v;t )2\n\u0002[0;T]; (1)\n\fnd the electron distribution function f(x;v;t ) and the electric \feld E(x;t)\n5with corresponding electrostatic potential \b( x;t) such that, for \fxed T >0,\n0 =@tf+ div(\u000bf) =@tf+v\u0001rxf\u0000E\u0001rvf; in \n\u0002(0;T];(2a)\nE=\u0000rx\b ;\u0000\u0001x\b = 1\u0000Z\nRnfdv; in \nx\u0002(0;T];(2b)\nsubject to an initial condition on fand boundary conditions on fand \b:The\ndomain of de\fnition of the initial boundary value problem \n := \n x\u0002Rn,\nwhere the physical domain \n x\u001aRncan either be bounded or all of Rnwith\nn= 1;2;3. The boundary condition given for fdepends on \n x:If \nx=Rn;\nthen the condition f!0 must be enforced both as jxj!1 and asjvj!1:\nIf \nxis bounded, then a condition must be imposed on falong the in\row\nboundary \u0000I;de\fned by\n\u0000I=f(x;v)2@\nx\u0002Rnjv\u0001\u0017x<0g; (3)\nwith\u0017xbeing the unit outward normal vector to @\nx:OftenfIis given in\nsome parts of the boundary and may be periodic in other parts of the boundary\nregion \u0000I. In this manuscript we assume periodic boundary conditions in space\nand the decaying boundary condition in velocity.\nThe Poisson equation must also be endowed with spatial boundary conditions,\neither Dirichlet, Neumann, Robin, or periodic, on di\u000berent disjoint regions of\nthe boundary @\nx. We denote the Dirichlet portion of the boundary by @\nx;D.\nIf the measure of the Dirichlet boundary is zero, i.e. j@\nx;Dj= 0, and one has\nhomogeneous or periodic boundary conditions such thatR\n@\nxr\b\u0001\u0017x= 0,\nthen in order to maintain a well-posed problem that keeps the existence and\nuniqueness of the corresponding Poisson boundary value problem, one needs\nto add to the solution space the compatibility (neutrality) conditionR\n\nx(1\u0000R\nRnfdv)dx= 0, or equivalentlyR\n\nxR\nRnfdxdv =j\nxjon each connected\ncomponent of the spatial domain \n.\nMacroscopic \ruid quantities of interest are easily computed from f. The elec-\ntron density \u001a=\u001a(x;t), current density j=j(x;t), kinetic energy density\nEk(x;t) and electrostatic energy density Ee(x;t) are de\fned by\n\u001a(x;t) =Z\nRnf(x;v;t )dv; (4)\nj(x;t) =Z\nRnvf(x;v;t )dv; (5)\nEk(x;t) =1\n2Z\nRnjvj2f(x;v;t )dv; (6)\nEe(x;t) =1\n2jE(x;t)j2: (7)\nThese quantities satisfy a number of respective conservation laws (see e.g.\n6[58]). In particular, it is well-known that the Vlasov-Poisson system conserves\ntotal particle number, momentum, energy, and the Casimir invariants, which\nare given, respectively, by\nN=Z\n\nx\u0002Rnf(x;v;t )dxdv =Z\n\nx\u001a(x;t)dx; (8)\nP=Z\n\nx\u0002Rnvf(x;v;t )dxdv =Z\n\nxj(x;t)dx; (9)\nH=1\n2Z\n\nx\u0002Rnjvj2f(x;v;t )dxdv +1\n2Z\n\nxjE(x;t)j2dx (10)\n=Z\n\nx\u0010\nEk(x;t) +Ee(x;t)\u0011\ndx;\nC=Z\n\nx\u0002RnC(f)dxdv; (11)\nThe notationC(f) in (11) refers to an arbitrary function of fand includes the\n`enstrophy' whenC(f) =f2, entropy whenC(f) =\u0000flnf, or particle number,\nas in (8), whenC(f) =f. We will check the invariance of these quantities in\nour nonlinear computations, particularly in Section 4.2.2.\nInteresting properties of the Vlasov-Poisson system result by considering a\nlinear perturbation \u000ef(x;v;t ) to an equilibrium distribution feq(v) over the\n2D-phase space [0 ;L]\u0002(\u00001;1); L > 0:Speci\fcally, suppose that f=\nfeq+\u000ef, wherefeqis a given equilibrium probability distribution, \u000efand \b\nareL-periodic in x, and the initial average value of \u000efover \n is zero. Equations\n(2a)-(2b) imply that \u000efsatis\fes\n@t(\u000ef) +v(\u000ef)x\u0000E(\u000ef)v=Ef0\neq [0;L]\u0002(\u00001;1)\u0002(0;T];(12a)\nE=\u0000\bx [0;L]\u0002(0;T];(12b)\n\bxx=Z\nRn\u000efdv [0;L]\u0002(0;T]:(12c)\nSupposingjE(\u000ef)vj<<1 and dropping this term from (12a) leads to\n@t(\u000ef) +v(\u000ef)x=Ef0\neq [0;L]\u0002(\u00001;1)\u0002(0;T]; (13a)\nE=\u0000\bx [0;L]\u0002(0;T]; (13b)\n\bxx=Z\nRn\u000efdv [0;L]\u0002(0;T]: (13c)\nThe linear system of (13a)-(13c) was analyzed by using the Laplace transform\nin the famous paper of Landau [42], by expansion in terms of continuum eigen-\nfunctions in [69], and by a tailored integral transform introduced in [51,52].\nLandau showed that an electric \feld mode Ek(!;t) decays exponentially in\nthe long-time limit, which we investigate in Section 4.1.2 for two well-known\nequilibria: the Maxwellian\nfM= (2\u0019T)\u0000n=2e\u0000v2=2T; (14)\n7and the Lorentzian,\nfL(v) =1\n\u0019\r\nv2+\r2: (15)\n3 Method Formulation\nIn this section, we derive the UPG method for the Vlasov-Poisson system. The\nderivation proceeds by \frst discretizing the Vlasov equation using the stan-\ndard UG discretization for transport equations [21]. Here it is assumed that\nthe electric \feld is given and hence the divergence-free \row \feld \u000b(x;v;t ) =\n(v;\u0000E(x;t)), de\fned in (1) for the Vlasov equation (2a), is known. Afterwards,\nthe DG discretization for the Poisson equation is considered.\nIt is assumed that any mesh for the phase space, where by mesh we mean a par-\ntitioning of the phase space into convex sets called elements, is the Cartesian\ncross product of a mesh for the physical domain and a mesh for the velocity\ndomain. Under this assumption, the physical and velocity domains can be in-\ndependently re\fned. Given a mesh for the phase space, the UG method then\nde\fnes an approximate solution to the true Vlasov solution in such a way that\nat any given time the approximate solution restricted to each element of the\nmesh is a polynomial function. However, the approximate solution is not re-\nquired to be continuous across the intersections of any two adjacent elements,\nso it is a piecewise de\fned polynomial function with respect to the mesh at\nany given time.\nIn order to compute the approximation to the electrostatic potential from\nPoisson's equation (2b), for a given distribution function, one may use one\nof three interior penalty methods that weakly enforce both approximate con-\ntinuity across the interior mesh faces and Dirichlet boundary regions. These\nthree alternative methods, symmetric interior penalty Galerkin (SIPG) [70,71],\nnon-symmetric interior penalty Galerkin (NIPG) [59], and incomplete interior\npenalty Galerkin (IIPG) [63,64] are discussed in detail and a priori error\nestimates for each of them are given in the respective references. The only dif-\nference among the three methods is in the value of one speci\fc parameter that\narises in the weak formulation that is common to each of them. Thus, for a\ngiven space charge function, each penalty Galerkin method de\fnes a piecewise\npolynomial approximation of the true solution to the Poisson equation using\na mesh in the spatial domain.\nThe spatial domain mesh used in the discretization of Poisson's equation is\nrequired to be the same as that used in the UG discretization of the Vlasov\nequation. This requirement is a practical one, both in terms of analysis and\nimplementation. However, the polynomial degree of the potential approxima-\n8tion on a given element of the spatial mesh is not required to equal the degree,\nwith respect to x, of the polynomial approximation of the distribution f.\nThus, the UPG method of approximation to the Vlasov-Poisson system is de-\n\fned by coupling together the UG method of approximation to the Vlasov\nequation with interior penalty methods of approximation to the Poisson equa-\ntion. This nonlinear semi-discrete approximation results in a \frst-order non-\nlinear ODE system, the solution of which determines the approximation fhof\nf. The resulting ODE system is readily solved using an explicit conservative\ntime-integrator such as the Runge-Kutta method. Moreover, in the process of\ncomputing fh, both the approximation Ehof the electric \feld Eand the ap-\nproximation \b hof the potential \b are computed by one of the penalty methods\nfor the elliptic equation.\n3.1 Preliminaries\nWe assume the computational domain in velocity space is a bounded set \n v\nand that the approximate solution to f(x;v;t ) is assumed to vanish in @\nv.\nIt is then, implicitly for our simulation, assumed that the velocity support\nof the approximation to the true solution fof the Vlasov-Poisson system is\ncontained in \n vfor all times. This is a reasonable assumption for problems with\nspatial periodic boundary conditions, as it is expected that most of the density\nassociated with the approximation to fwill be contained in a su\u000eciently large\n\fxed set \n v. The error due to this assumption only depends on the density of\nthe true solution fcomputed on the complement of \n vinRn.\nFurther, the conservative nature of the transport equation (2a) with the space\ndependent divergence-free \row \feld \u000b, motivates us to choose the computa-\ntional scheme as follows:\nLetfThxghx>0be a sequence of successively re\fned meshes of the bounded\ndomain \n x\u001aRnand letfThvghv>0be a sequence of successively re\fned meshes\nof the bounded domain \n v\u001aRn, wheren= 1;2;3:Given the meshes Thx=\nfKjxgNhx\njx=1andThv=fKjvgNhv\njv=1, the elements KjxandKjvcomprising each\nof the respective meshes are sets of the following types: intervals, if n= 1;\ntriangles or quadrilaterals, if n= 2; and tetrahedra, prisms or hexahedra, if\nn= 3. The corresponding spatial re\fnement level hxand velocity re\fnement\nlevelhvare de\fned by hx= maxjxfdiam(Kjx)gandhv= maxjvfdiam(Kjv)g,\nrespectively.\nA sequence of successively re\fned meshes fThgh>0of the, now, computational\ndomain \n = \n x\u0002\nvis generated by de\fning each mesh Th=fKjgNh\nj=1to\nbeTh=Thx\u0002Thv;where the re\fnement level is h= (hx;hv). Thus, for any\ngiven element Kj2Ththere exists a unique pair of elements Kjx2Thxand\n9Kjx2Thxsuch thatKj=Kjv\u0002Kjv, which is equivalent to the existence\nof an invertible mapping from j2f1;:::;Nhgto (jx;jv)2f1;:::;Nhxg\u0002\nf1;:::;Nhvg, whereNh=NhxNhv.\nThe derivation of the following UPG method requires the use of the broken\nSobolev space HsfThg,s>1=2, which is de\fned as follows:\nHsfThg=fw2L2(\n)jwjKj2HsfKjg;8Kj2Thg; (16)\ni.e.,HsfThgis the space of those functions that have elementwise weak deriva-\ntives up to, and including, the order s. Then for nonnegative integers rxand\nrv, the discontinuous approximation space Drx;rv(Th)\u001aHsfThgis given by\nDrx;rv(Th) =fw2HsfThgjwjKj2QrxfKjxg\u0002QrvfKjvg;8Kj2Thg;\n(17)\nwhere Qr(K) denotes the space of polynomials on a set Kwith degree less\nthan or equal to rin each variable. Thus, Pr(K)\u001aQr(K), where Pr(K)\ndenotes the space of polynomials satisfying that the sum of the degrees of all\nthe variables is less than or equal to r.\nThe choice of Qr(K) for basis functions is suitable for Cartesian meshes in\nbothx-space and v-space, respectively, where trace and inverse inequalities\nthat are derived by mapping to the reference element in the approximating\nframework are possible, as \frst introduced in [37].\nHowever, one may use triangles in two-dimensions, and prisms or hexahedra in\nthree-dimensions, for both x-space and v-space, for which the natural choice\nof polynomial space would be PrxfKjxg\u0002PrvfKjvg. We point out that these\nselections of approximating spaces is consistent with the divergence free, linear,\nand conservation form of the Vlasov equation, and the fact that the choice\nof the mesh associated with the computational domain \n = \n x\u0002\nvis a set\nof product mesh elements Kj=Kjv\u0002Kjv, forj2f1;:::;Nhg, (jx;jv)2\nf1;:::;Nhxg\u0002f 1;:::;NhvgandNh=NhxNhv. Such is clearly preserved by\nthe Vlasov \row and the corresponding approximation results for Pr(K) will\nbe valid. This choice may be preferable in higher dimensions since the number\nof degrees of freedom of the basis functions in Pr(K) isrn+1, while for Qr(K)\nis (r+ 1)n, forn-dimensional calculations.\nThe discontinuous nature of the space HsfThgneeds the introduction of\nmesh faces. If Kjis a boundary element, then Fk=@Kj\\@\n is called\na boundary mesh face. If K1andK2are two intersecting elements whose\ncommon intersection lies in the interior of \n, then Fk=@K1\\@K2is said\nto be an interior mesh face. The set of all mesh faces is denoted by Fh=\nfF1;:::;FPh;FPh+1;:::;FMhg, whereFkis an interior face if 1 \u0014k\u0014Phand\na boundary face if Ph+ 1\u0014k\u0014Mh. Each face Fk2Fhis associated with a\nunit normal vector \u0017k. Fork>Ph,\u0017kis chosen to be the outward unit normal\n10to@\n. For 1\u0014k\u0014Ph, we \fx\u0017kto be one of the two unit normal vectors to\nFk:For every interior face Fk;the elements K1andK2will always be used to\ndenote the two unique elements such that Fk=K1\\K2:Moreover, it is always\nassumed that K1satis\fes\u0017K1=\u0017konFk, where\u0017K1denotes the outward unit\nnormal to@K1:Then\u0017K2=\u0000\u0017konFk.\nThe fact that each mesh This the product of a spatial mesh Thxand a velocity\nmeshThvgives a speci\fc structure to the boundaries of the elements and to the\nset of mesh facesFh. It follows that for Kj=Kjx\u0002Kjvwe have@Kj= (@Kjx\u0002\nKjv)[(Kjx\u0002@Kjv). We denote the set of mesh faces for ThxandThvbyFhx=\nfFx\n1;:::;Fx\nPhx;Fx\nPhx+1;:::;Fx\nMhxgandFhv=fFv\n1;:::;Fv\nPhv;Fv\nPhv+1;:::;Fv\nMhvg,\nrespectively, where Fx\nkxis an interior face if 1 \u0014kx\u0014Phxand a boundary face\nifPhx+ 1\u0014kx\u0014Mhx, andFv\nkvis an interior face if 1 \u0014kv\u0014Phvand a\nboundary face if Phv+ 1\u0014kv\u0014Mhv. Then, given any arbitrary Fk2Fh,\neither there exist an Fx\nkx2Fhxand aKjv2Thvsuch thatFk=Fx\nkx[Kjvor\nthere exist a Kjx2Thxand anFv\nkv2Fhvsuch thatFk=Kjx[Fv\nkv.\nWhen considering functions in HsfThg, we use the usual average and jump\noperators, respectively, de\fned for w2HsfThgalong an interior face Fkby\nw=1\n2\u0010\n(wjK1)jFk+ (wjK2)jFk\u0011\n; [w] = (wjK1)jFk\u0000(wjK2)jFk:(18)\nThe above de\fnitions are also valid for vector-valued functions w2[HsfThg]2n,\nin which case it follows that [ w]\u0001\u0017k= (wjK1)jFk\u0001\u0017K1+ (wjK2)jFk\u0001\u0017K2.\n3.2 Upwind Galerkin Approximation of the Vlasov Equation\nHere we describe the UG scheme for the Vlasov equation in full generality for\na 2n-dimensional ( n= 1;2 or 3) phase space with in\row boundary conditions\nand piecewise polynomials of arbitrary degree approximating f. The simpler\nderivation of the method for a two-dimensional phase space with periodic\nboundary conditions in xand a piecewise constant approximation to fis\ngiven explicitly in Section 3.5, since this method was used for the numerics of\nSection 4. Note, for this simpler version the derivation does not require use of\nthe set of mesh faces Fh.\nFor a given time T > 0 and data trio ( \u000b;f0;fI), the Vlasov equation along\nwith the corresponding initial and boundary conditions are\n@tf+\u000b\u0001rf= 0 \n\u0002(0;T]; (19a)\nf(t= 0) =f0 \n; (19b)\nf=fI \u0000I\u0002(0;T]; (19c)\n11where\u000b= (v;\u0000E)2R2n,E2Rnis assumed given, r= (rx;rv), and\n\n = \nx\u0002\nv. It is assumed that \n x= \u0005n\ni=1[0;Xi] and \nv= [\u0000Vc;Vc]n;where\nX1;:::;Xn;Vc>0 are \fxed. Following (3), we also de\fne (keeping the same\nnotation without loss of generality) the computational in\row boundary \u0000I,\nassociated with the computational domain \n, by\n\u0000I=f(x;v)2@\nj\u000b\u0001\u0017 <0g; (20)\nwhere\u0017is the outward unit normal to @\n. Then, \u0000O=@\nn\u0000I.\nFor domains K\u001aR2n, let (\u0001;\u0001)Kdenote the L2(K)-inner product. To distin-\nguish integration over domains K\u001aR2n\u00001, we use the notation h\u0001;\u0001iK. A weak\nformulation for (19a)-(19c) is derived by multiplying equation (19a) by an ar-\nbitrary test function w2H1(Th) and integrating by parts over an arbitrary\nKj2Th. This yields\n(@tf;w)Kj\u0000(f;\u000b\u0001rw)Kj+hfw;\u000b\u0001\u0017Kji@Kj= 0;8t2(0;T]; (21)\nwith\u0017Kjbeing the outward unit normal to @Kjandw\u0000denoting the interior\ntrace ofKj, i.e.,w\u0000(x;v) = lims#0\u0000w\u0010\n(x;v) +s\u0017Kj\u0011\n.\nUpon summing equation (21) over all Kjand weakly enforcing the in\row\nboundary condition, we get\n(@tf;w)\n\u0000NhX\nj=1(f;\u000b\u0001rw)Kj+PhX\nk=1hf[w];\u000b\u0001\u0017kiFk+X\nFk2\u0000Ohfw;\u000b\u0001\u0017kiFk\n=\u0000X\nFk2\u0000IhfIw;\u000b\u0001\u0017kiFk:(22)\nApproximating fby a function fh, which may be discontinuous across the\ninterior faces, requires the introduction of a numerical \rux fu. A standard\ntechnique is to replace fby its \\upwind value\" fuon the interior faces [21],\nwherefualong each interior face Fkis de\fned by the given advecting \row\n\feld\u000b(x;v;t ) according to\nfu(v;v;t ;\u000b) = lim\ns#0f((x;v;t ) +s\u000b(x;v;t ))) =\n=8\n><\n>:fjK1(x;v;t );if\u000b(x;v;t )\u0001\u0017k\u00150;\nfjK2(x;v;t );if\u000b(x;v;t )\u0001\u0017k<0:\nConsistency follows from condition (23), since fu(\u000b) =fonFkwhenever\nfis continuous across each interior face Fk. We also note that fudepends\nnonlinearly on \u000bas, in general, if \u000b1and\u000b2are two \row \felds having di\u000berent\nvalues onFkand ifg2HsfThgis discontinuous across Fk, thengu(\u000b1+\u000b2)6=\ngu(\u000b1) +gu(\u000b2). Replacing fon the interior faces in (22) by its upwind value\n12fuleads to\n(@tf;w)\n\u0000NhX\nj=1(f;\u000b\u0001rw)Kj+PhX\nk=1hfu[w];\u000b\u0001\u0017kiFk+X\nFk2\u0000Ohfw;\u000b\u0001\u0017kiFk\n=\u0000X\nFk2\u0000IhfIw;\u000b\u0001\u0017kiFk;\nwhich is the UG scheme for the Vlasov equation. Finally, de\fning the bilinear\noperatorAby\nA(f;w;\u000b) =\u0000NhX\nj=1(f;\u000b\u0001rw)Kj+PhX\nk=1hfu[w];\u000b\u0001\u0017kiFk+X\nFk2\u0000Ohfw;\u000b\u0001\u0017kiFk\n(23)\nand the corresponding linear operator L, depending on the in\row data fI, by\nL(w;\u000b;fI) =\u0000X\nFk2\u0000IhfIw;\u000b\u0001\u0017kiFk; (24)\nyields a variational formulation for the semi-discrete problem of \fnding the\nfh2C1([0;T];Drx;rv(Th)) approximation to f, satisfying,\n(@tfh;wh)\n+A(fh;wh;\u000b) =L(wh;\u000b;fI)8t2(0;T]; (25)\n(fh(x;v;0);wh)Kj= (f0;wh)Kj8Kj2Th; (26)\nfor allw2Drx;rv(Th), withf0andfIapproximations of the initial data\nf(x;v;0) and the in\row boundary data on \u0000 I, respectively.\nWe note that (25) produces a \frst-order ODE system to be described below.\nIndeed, for each Ki=Kix\u0002Kiv2Th;let i\n1;:::; i\nnbbe a basis for Qrx(Kix)\u0002\nQrv(Kiv), wherenb= (rx+ 1)n\u0002(rv+ 1)nand then extend the domain\nof these functions to \n by de\fning each to be identically zero in \n nKi.\nIf\f(t) = (\f1\n1(t);:::;\f1\nnb(t);:::;\fNh\n1(t);:::;\fNhnb(t)) denotes the unique vector\nsuch that the UG approximation fhsatis\fes\nfh(x;v;t ) =NhX\nj=1nbX\nm=1\fj\nm(t) j\nm(x;v); (27)\nthenfhjKj=nbP\nm=1\fj\nm j\nm. Inserting (27) into (25) yields\nNhX\nj=1nbX\nm=1_\fj\nm(t) ( j\nm;wh)\n+NhX\nj=1nbX\nm=1\fj\nm(t)A( j\nm;wh;\u000b)\n=L(wh;\u000b;fI);8wh2Drx;rv(Th):(28)\n13Finally, sincef i\npgnb;Nh\np=1;i=1is a basis for Drx;rv(Th), then (28) is equivalent to\nnbX\nm=1_\fi\nm(t) ( i\nm; i\np)Ki+X\nj2N(i)nbX\nm=1\fj\nm(t)A( j\nm; i\np;\u000b)\n=L( i\np;\u000b;fI);8i2f1;:::;Nhg;8p2f1;:::;nbg;(29)\nwhereN(i) contains the indices of all neighboring elements of Ki.\nEquation (29) is seen to generate an equivalent matrix system, where nbrows\nof the matrix are generated at a time by sequentially taking ito equal 1;:::;Nh\nand for each isequentially taking pto equal 1;:::;nbin Eq. (29). This proce-\ndure results in the matrix ODE system\nA1_\f(t) +A2(\u000b)\f(t) =L(\u000b;fI); (30)\nwhereA1is a constant matrix and A2(\u000b) is the corresponding sparse matrix,\nboth of which are of dimension nbNh\u0002nbNh, andL(\u000b;fI) is a vector of length\nnbNh.\nSince the support of the functions  i\n1;:::; i\nnbisKi,8i2f1;:::;Nhg, it follows\nthatA1is a block-diagonal matrix, where each block is an nb\u0002nbmatrix. This\nmeans that the inverse of A1is easily computed. Thus, the UG approximation\nfhis equivalently de\fned to be the unique solution to\n_\f(t) =\u0000A\u00001\n1A2(\u000b)\f(t) +A\u00001\n1L(\u000b;fI); (31)\nwhere the initial condition \f(0) is uniquely determined by (26). To solve\nthis system in time, a conservative explicit time stepping method such as\nthe Runge-Kutta method can be used.\n3.3 Interior Penalty Approximations of the Poisson Equation\nIn order to make this manuscript self contained, we also discuss the interior\npenalty formulations for the Poisson equation that weakly enforces approxi-\nmate continuity across interior mesh faces and Dirichlet boundary regions. As\nalready noted, the mesh Thxused to discretize the Poisson equation must be\nthe same as that for the spatial domain used for the Vlasov equation, but the\npolynomial degree rxfor the Poisson equation need not be equal to the degree\ninxforfh.\nHence, for a given i) source G2L2(\nx), ii) boundary data \bD2L2(@\nx;D)\nin the portion of the boundary referred as the Dirichlet boundary @\nx;D, and\niii)r\b\u0001\u0017= 0 (homogeneous Neumann) or periodic boundary conditions on\n@\nxn@\nx;D, the boundary complement of the Dirichlet region, then the more\n14general form of Poisson equation with a positive de\fnite permittivity term a\ndiscretized using the symmetric interior penalty (SIPG) method [70,71], in-\ncomplete interior penalty (IIPG) method [63,64,28], or non-symmetric interior\npenalty (NIPG) [59] method is\n\u0000rx\u0001(arx\b) =G \nx; (32a)\nBx\b = \bD@\nx;D: (32b)\nwherea(x) is any positive-de\fnite continuous function in ( C1(\nx))n\u0002n.\nForK\u001aRn,n=1,2, or 3, recall ( \u0001;\u0001)Kis theL2(K)-inner product with inte-\ngration over K\u001aRn\u00001, while the notation h\u0001;\u0001iKis for boundary integrals. In\naddition, we use the identity\nNhxX\njx=1harx\b\u0001\u0017Kjx;\u0012\u0000i@Kjx=PhxX\nkx=1harx\b [\u0012] + [rx\b]a\u0012;\u0017kxiFkx\n+X\nFkx2@\nxhrx\b\u0001\u0017kx;\u0012iFkx; (33)\nwhere\u0017Kjxis the outward unit normal to @Kjx.\nThus, the corresponding schemes SIPG, IIPG, and NIPG are all derived by\nmultiplying (32a) by an arbitrary test function \u00122HsfThxg,s >1=2, inte-\ngrating by parts on each Kjx2Thx;and summing each of the resulting local\nequations. Whence, one obtains the non-symmetric variational formulation\ngiven by\nNhxX\njx=1(arx\b;rx\u0012)Kjx\u0000PhxX\nkx=1harx\b [\u0012] + [arx\b]\u0012;\u0017kxiFkx\n\u0000X\nFkx2@\nxharx\b\u0001\u0017kx;\u0012iFkx= (G;\u0012)\nx;(34)\nwhere the identity (33) was used to represent the inner boundary integrals.\nIn particular, since the true solution for a bounded right-hand-side has at least\nthe regularity \b2H1(\nx)\\H2fThxg, it follows that [\b] \u00110 and [rx\b]\u00110\nalong every interior face Fkx[32]. Thus, In order to get a good approximation\nfor a regular solution satisfying these two jump conditions, one adds `interior'\npenalty terms that vanishes on the true solution \b. These terms are of the\nform\ncsPhxX\nkx=1harx\u0012[\b];\u0017kxiFkx; (35)\nwhere the values selected for csfor the SIPG, IIPG and NIPG methods are\n-1, 0 and 1, respectively. Similarly, such penalization is also required for the\nDirichlet boundary region @\nx;Das follows.\n15Indeed, we add to the bilinear form the interior penalty terms\nPhxX\nkx=1\u001b\n(hjx)n=2h[\b];[\u0012]iFkx; (36)\nwhich are now completed by weakly enforcing both approximate continuity\nacross the interior mesh faces and Dirichlet boundary regions @\nx:D, by in-\ncluding the penalization\nX\nFkx2@\nx;D\u001b\n(hjx)n=2h\b\u0000\bD;\u0012iFkx; (37)\nwhere\u001b > 0 is an arbitrary penalty parameter that is usually set equal to\nunity. Note that homogeneous or periodic boundary conditions vanish on\nboundary terms of the corresponding bilinear structure. Consequently, they\ndo not require the boundary penalization term.\nUsually the penalization terms are denoted by the following non-symmetric\nbilinear form\nJ\u001b(\b;\u0012) =PhxX\nkx=1\u001b\n(hjx)n=2h[\b];[\u0012]iFkx+X\nFkx2@\nx;D\u001b\n(hjx)n=2h\b;\u0012iFkx: (38)\nAdding both penalty terms and (35) to the left-hand-side of (34) results in\nNhxX\njx=1(arx\b;rx\u0012)Kjx\u0000PhxX\nkx=1harx\b [\u0012] +csarx\u0012[\b];\u0017kxiFkx\n\u0000X\nFkx2@\nxharx\b\u0001\u0017kx;\u0012iFkx+PhxX\nkx=1\u001b\n(hjx)n=2h[\b];[\u0012]iFkx\n+X\nFkx2@\nx;D\u001b\n(hjx)n=2h\b;\u0012iFkx= (G;\u0012)\nx+X\nFkx2@\nx;D\u001b\n(hjx)n=2h\bD;\u0012iFkx:(39)\nEquation (39) completely de\fnes each of the three interior penalty schemes.\nSettingAcs(\b;\u0012) equal to the \frst four terms on the left-hand-side of (39),\nwhere dependence on the parameter csfrom (35) is noted as a subscript, we let\nthe bilinear operator Bcs(\b;\u0012) :=Acs(\b;\u0012) +J\u001b(\b;\u0012) and the linear operator\nH(\u0012;G;\bD) be equal to the two terms on the right-hand-side of (39). Then\nthe function \b h2Drx(Thx) is the corresponding interior penalty Galerkin\napproximation to the Poisson solution \b, if\nBcs(\bh;\u0012h) =H(\u0012h;G;\bD);8\u0012h2DrxfThxg: (40)\nNote,Bcsis positive de\fnite (or coercive) even though each of Acs(\b;\u0012) and\n16J\u001b(\b;\u0012) separately are only positive semi-de\fnite [28,37,59,63,64]. In partic-\nular,Bcsgenerates an equivalent norm for the Hilbert space H1(Th),\nk\u0012k2\nNIPG =Acs(\u0012;\u0012) +J\u001b(\u0012;\u0012); \u00122H1(Th); (41)\nif the measure of the Dirichlet boundary j@\nx;Dj>0:In the case of periodic\nor homogeneous Neumann boundary conditions we need the compatibility\nconditionR\n\nxR\nRnfdxdv =j\nxj, for each connected component \n xof the\nspatial domain. In fact, the approximation to the Vlasov equation is done\nwith a conservative UPG scheme and high order Runge-Kutta schemes, and\nin particular, for the case of periodic boundary conditions the compatibility\ncondition at the numerical level is satis\fed as well.\nFinally it is possible to see that the bilinearity and positive de\fniteness of\nBcsfor either a portion of Dirichlet or full Neumann or periodic boundary\nconditions, implies that (40) is equivalent to a uniquely solvable matrix system\ndescribed next. Let nb= (rx+ 1)nand\u0016=f\u00161\n1;:::;\u00161\nnb;:::;\u0016Nhx\n1;:::;\u0016Nhxnbg\nbe the unique vector such that\n\bh=NhxX\njx=1nbX\nm=1\u0016jx\nm\u0012jx\nm(x): (42)\nUpon substituting this representation into (40), we conclude that (40) is equiv-\nalent to\nB\u0016=H; (43)\nwhereBis an (nb+1)Nhx\u0002(nb+1)Nhxinvertible sparse matrix and H(G;\bD)\nis a vector of length ( nb+ 1)Nhx. However, B is not block diagonal, as was the\ncase forA1in the Vlasov ODE system. Thus, if the spatial domain is two- or\nthree-dimensional, then using an iterative solver is in general the most e\u000ecient\nmeans of computing the solution \u0016in (43). However, for the one-dimensional\nspatial domain, \u0016can be computed by using an LU-matrix decomposition\nalgorithm to factor B. For convenience, we write \u0016=B\u00001H, even though in\npractice\u0016might be computed using an iterative method.\n3.4 Discontinuous Galerkin Approximation of the Vlasov-Poisson System\nThe UPG method for the Vlasov-Poisson system results from combining the\nUG approximation of the Vlasov equation together with the interior penalty\napproximation of the Poisson equation. Thus, the approximation fh(t) to the\nsolutionf(x;v;t ) of the Vlasov-Poisson system (2a)-(2b), at time t, results\nfrom an iteration as follows.\nLet ~fh(t) be given, where ~fh(0) is the approximation of the initial distribu-\ntion function. Then, an approximation \u000bh(t) = (v;\u0000Eh(t)) to\u000b(x;v;t ) =\n17(v;\u0000E(x;t)) is determined by computing the corresponding approximation\nto\u0000Eh(t), using one of the interior penalty approximation schemes to com-\npute an approximation \b h(t) to the potential \b D(x;t) via the formula \u0016=\nB\u00001H(G;\bD), whereGis de\fned to be 1 \u0000R~fh(t)dv.\nNext, the approximate local \feld is computed by taking the local spatial po-\ntential gradients ( Eh)jKjx=\u0000rx(\bh)jKjxon eachKjx, which implies that\nEh(t) =Eh(~fh(t)) is discontinuous across the interior faces of Thx. Conse-\nquently, it follows that the approximate \row \feld \u000bh(t) =\u000bh(~fh(t)), or equiv-\nalently\u000bh(t) =\u000bh(\f(t)), where\f(t) is a well de\fned given function.\nSummarizing, for any given ~fh(t), \frst compute the approximate \u000bh(t), and\nthen the approximate solution fh(t) to the Vlasov-Poisson system by solv-\ning the ODE system (31) with \u000b(t) replaced by \u000bh(\f(t)). This leads to the\nfollowing de\fnition:\nDe\fnition 1 The semi-discrete function fh(\f(t))2C1([0;T];Drx;rv(Th))is\nthe UPG approximation to Vlasov-Poisson solution f, iffh(\f(t)) :=fhas\nde\fned in (27) where\f(t)satis\fes the nonlinear system of ODEs\n(i) (fh(\f(0));wh)Kj= (f0;wh)Kj;8Kj2Th;8w2Drx;rv(Th);\n(ii)\u000bh(\f(t)) = (v;rx\bh(\u0016(t))) with \u0016(t) =B\u00001H(\f(t));\n(iii)_\f(t) =\u0000A\u00001\n1A2(\u000bh(\f(t)) )\f(t) +A\u00001\n1L(\u000bh(\f(t));fI); (44)\nfor allt2(0;T].\nSystem (44) can be solved for \f(t) using any explicit time-integrator, following\na classical Gummel map type iteration, where at any given approximation\n\fn\u00001, the\f(tn\u00001) step (ii) is \frst computed since it does not involve time\nvariation. Thus, one obtains \u000bh(\f(tt\u00001)), which allows for the calculation of\n\fn, the approximation to \f(tn) by step (iii). In our simulations we use a\nconservative high order Runge-Kutta time integrator.\nThis very same iteration scheme was previously proposed in [14] for the calcu-\nlation of Boltzmann-Poisson solvers for semiconductors, and related work by\nthe same authors cited below. There the calculation of the Poisson equation\nis done by a LDG scheme.\nWe also point out that an error estimate for this nonlinear scheme can be\nfound in [37], which we state here in a concise form. Let ( f;\b) be a solution\npair of the Vlasov Poisson system (2b), with boundary and initial conditions\nas described above, potential \b( t;\u0001)2H\u0016s(\nx) for \nx\u001aRn, and distribution\nfunctionf2C1([0;T];H2s(Th)) for \n\u001aR2n, for both \u0016s;s > n . Also, letFk\ndenote the interior faces associated with element Kkin \n, withFx\nkxdenoting\nthe corresponding one associated with the elements in the x-space \n x, as it\n18was de\fned at the end of Section 3.1.\nFurther, recall Qr(K) is the space of polynomials on a set Kof degree less than\nor equal to r(cf. (16) and (17)), and rxandrvthe degrees in x-space and v-\nspace, respectively. Let the parameter \u0015be the \frst eigenvalue to the Poisson\nequation in \n x,~\bhbe the distributional solution to the perturbed Poisson\nequation for a source term \u001ah, the charge density ( v-average) associated with\nfh, and let\u0016xand\u0016vbe de\fned by\n\u0016x= minfrx+ 1;\u0016sgand\u0016v= minfrv+ 1;sg: (45)\nThen, we obtained the following error estimate for the 2 n-dimensional semi-\ndiscrete formulation of the Vlasov Poisson system in terms the di\u000berence of\nsuitable norms of potentials \b \u0000\bh, \feldsE\u0000Eh=\u0000r\b+r\bh, and particle\ndistribution functions f\u0000fh:\nk\b\u0000\bhk2\nNIPG\u0014\u0015\u00001k\u001a\u0000\u001ahk2\nL2(\nx)+ch2\u0016x\u00002\nr2\u0016s\u00002\nxk~\bhk2\nL2(\nx);\nkr\b\u0000r\bhk2\nL2(\nx)+PhxX\nkx=1rv\u001b\njhjxjn=2k\b\u0000\bhk2\nL2(Fkx)+X\nFkx2\nx;Drx\u001b\njhjxjn=2k\b\u0000\bhk2\nL2(Fkx)\n\u0014\u0015\u00001k\u001a\u0000\u001ahk2\nL2(\nx)+ch2\u0016x\u00002\nr2\u0016s\u00002\nxk~\bhk2\nL2(\nx); (46)\nkf(T)\u0000fh(T)k2\nL2(\n)+ZT\n0PhX\nk=1kj\u000bh\u0001\u0017kj1=2[f\u0000fh]k2\nL2(Fk)\n+ZT\n0kj\u000bh\u0001\u0017kj1=2[f\u0000fh]k2\n0;\u00000+ZT\n0kj\u000bh\u0001\u0017kj1=2[f\u0000fh]k2\n0;\u0000I\n\u0014Ch2\u0016v\u00001+ofh;\u0016x;\u0016vg(h2\u0016v\u00001);\nwhere\u001bis the penalization parameter of (36) and (37) and the k\u0001k2\nNIPG\nwas de\fned in the previous subsection at (41). In addition, for a su\u000eciently\nsmooth potential \b( t;\u0001)2H\u0016s(\nx) for \nx\u001aRnand distribution function\nf(t;\u0001)2H2s(\n) for \n\u001aR2n, where the order of smoothness is given by the\nparameters sand \u0016s, this estimate is optimal.\nWe close this subsection by noting that very recently our iteration scheme\nwas reproduced in [1,2] with a di\u000berent Poisson solver. These authors perform\nerror estimates for quadratic basis functions that preserve energy, but do not\npreserve the positivity of f. Numerical simulations have yet to be performed\nfor their scheme and the amount of degradation cause by the lack of positivity\nremains to be ascertained.\n193.5 Two-Dimensional Phase Space with Piecewise Constant Approximation\nWe end this section with a description of the simpli\fed scheme for a two-\ndimensional phase space using piecewise constant approximations to f. As\nnoted above, this is a positivity preserving (monotone) scheme that was used\nfor the plasma simulations presented in Section 4. Such a piecewise constant\nbasis function scheme can be easily extended to higher dimensions. We point\nout that, in work currently under preparation [18,19], we extend the positivity\ncondition to higher order basis functions by new limiter techniques inspired by\n[20,74,75], which are maximum principle preserving and can be applied to both\nthe Vlasov-Poisson and Vlasov-Maxwell systems. It remains a challenge to \fnd\na proper scheme that would preserved positivity and higher order moments,\nlike momentum and energy. In the future we hope to compare our approach\nwith extensive existing work on Vlasov-Ma xwell system [45,68,67].\nHere, the simpli\fed spatial and velocity domains are \n x= [0;L] and \nv=\n[\u0000Vc;Vc], with mesh points 0 = x0< x 1< ::: < x Nhx\u00001< xNhx=Land\n\u0000Vc=v0< v 1< ::: < v Nhv\u00001< vNhv=Vc, whereNhx;Nhv2N. Then\nforjx= 1;:::;Nhxandjv= 1;:::;Nhv, takeThx=fKjxgNhx\njx=1andThv=\nfKjvgNhv\njv=1by de\fning each spatial element Kjx= [xjx\u00001;xjx] of sizehjx=\nxjx\u0000xjx\u00001, and each velocity element Kjv= [vjv\u00001;vjv] of sizehjx=vjv\u0000\nvjv\u00001, respectively. A mesh Th=fKjgNh\nj=1of the phase space domain \n is now\ngenerated according to Kj=Kjx\u0002Kjv, where the index jis de\fned by the\nelement ordering j= (jv\u00001)Nhx+jx, forjx= 1;:::;Nhxandjv= 1;:::;Nhv,\nso thatThcontains a total of Nh=NhxNhvelements. The corresponding\npiecewise basis function is then given by setting \u0012ix(x) = 1, forx2Kix, and\n\u0012ix(x) = 0, otherwise, for ix= 1;:::;Nhx. Similar basis is also constructed for\n\u001fiv(v),iv= 1;:::;Nhv. Then, taking  i(x;v) =\u0012ix(x)\u001fiv(v), fori= 1;:::;Nh,\ngenerates the approximating space D0;0(Th) = spanf 1;:::; Nhg.\nThe corresponding upwind function fude\fned in (23) on @Kin\u0000 is now\nwritten in the simpler form\nfu(x;v;t ;\u000b) =8\n><\n>:f\u0000(x;v;t );if\u000b(x;v;t )\u0001\u0017Ki\u00150;\nf+(x;v;t );if\u000b(x;v;t )\u0001\u0017Ki<0;(47)\nforf\u0006(x;v;t ) = lims!0\u0006f((x;v) +s\u0017Ki;t) and the outward unit normal to\nKjdenoted by \u0017Ki(x;v) is simply de\fned by (0 ;\u00001) forv=viv\u00001, (0;1) for\nv=viv, (1;0) forx=xixand (\u00001;0) forx=xix\u00001.\n20Therefore, the corresponding lowest order UG scheme is\n(@tf;w)\n+PhX\nk=1hfu[w];\u000b\u0001\u0017kiFk+X\nFk2\u0000Ohfw;\u000b\u0001\u0017kiFk\n=\u0000X\nFk2\u0000IhfIw;\u000b\u0001\u0017kiFk:\nIn particular, for the piecewise constant UG approximation, fh(x;v;t ) =\nNhP\nj=1\fj(t) j(x;v) tof(x;v;t ), clearly one obtains ( fh)jKj=\fj(t), which implies\n@t\u0012Z\nKjfhdvdx\u0013\n=hjxhjv_\fj(t); (48)\nand the corresponding semi-discrete UG approximation fh=PNh\nj=1\fj(t) j\nis the unique function in C1([0;T];D0;0(Th)) satisfying the initial conditionR\nKifh(x;v;0) =R\nKif0,8i2f1;:::;Nhgand8t2(0;T], and\nhjxhjv_\fi(t) +Z\n@Ki=@\n(fh)u(\f(t))\u000b\u0001\u0017KidS+Z\n@Ki\\\u0000Ofh(\f(t))\u000b\u0001\u0017KidS\n+Z\n@Ki\\\u0000I(fh(\f(t)) )I\u000b\u0001\u0017KidS= 0; fori= 1;:::;Nh:\n(49)\nThis last identity is a linear ODE system for any given electric \feld E, where\nthe integration along the interior faces @Kin\u0000 in (49) ( i.e., @Ki\\@\n =;),\nis simply\nZ\n@Ki=@\nf\u000b\u0001\u0017KidS=Zxix\nxix\u00001E(x;t) (f(x;viv\u00001;t)\u0000f(x;viv;t))dx\n+Zviv\nviv\u00001v(f(xix;v;t)\u0000f(xix\u00001;v;t))dv (50)\nand the integrations along @Ki\\\u0000Oand@Ki\\\u0000Isatisfy\nZ\n@Ki\\\u0000Ofh\u000b\u0001\u0017KidS=Z\n@Ki\\\u0000I(fh)I\u000b\u0001\u0017KidS= 0: (51)\n4 Numerical Results\nIn this section numerical results are presented for six examples chosen to test\nthe accuracy and convergence of the proposed DG method. The examples\nchosen are typical for testing Vlasov-Poisson algorithms (see e.g. [11]), but we\nhave also included some atypical, more extensive comparison to theory. Four\n21of the examples test the linear dynamics and its associated \fne structure (\fl-\namentation) in phase space, while two examine the nonlinear evolution. The\nlinear results are presented in Section 4.1: in Section 4.1.1 the ability of the\nDG method to solve the advection equation, the Vlasov equation with the\nelectric \feld set to zero, is considered for both Maxwellian and Lorentzian\nequilibria, while in Section 4.1.2 the method is applied to the Landau problem\nand numerical results are extensively compared with the theoretical results of\nlinear Landau damping, also for both Maxwellian and Lorentzian equilibria.\nThe nonlinear results are presented in Section 4.2: in Section 4.2.1 nonlin-\near Landau damping is considered while in 4.2.2 the nonlinear two-stream\ninstability problem is computed.\nFor all examples, piecewise constants are used to approximate the distribution\nf, piecewise quadratic polynomials are used to approximate the potential \b,\nand time is discretized using a conservative fourth-order Runge-Kutta method.\nFor all but the \frst two linear advection examples, the NIPG penalty method\nis used to approximate the Poisson system, and the linear system that results\nfrom using the NIPG method is solved using an LU-decomposition algorithm.\nThroughout this section, it is assumed that the distribution function fhas\nthe form\nf(x;v;t ) =feq(v) +\u000ef(x;v;t ); (52)\nand the initial and boundary conditions used in all of the examples are of the\nform\n\u000ef(x;v;0) =Acos(kx)feq(v); (53a)\n\u000ef(0;v;t) =\u000ef(L;v;t ); (53b)\n\b(0;t) = \b(L;t); (53c)\nfor (x;v;t )2[0;L]\u0002[\u0000Vc;Vc]\u0002(0;T);whereVc>0; L > 0;andT > 0 are\ngiven. The constant Vcis the cuto\u000b velocity and is chosen large enough so that\nthe values of fare negligibly small when jvj=Vc:It follows that each example\nis completely determined by specifying the governing equations along with the\nparameters feq(v); A; k; L; V c; T. For both linear and nonlinear dynamics\nthe initial condition is denoted by f0(x;v) =f(x;v;0) =feq+\u000ef(x;v;0).\n4.1 Linear Results\nBoth the linear advection example of Section 4.1.1 with the initial condition\nf0=feq+\u000ef(x;v;0), where\u000ef(x;v;0) is given by (53a), and the Landau\ndamping example of Section 4.1.2, governed by (13a)-(13c), require the speci-\n\fcation offeq. For both examples, the two choices for feqintroduced in Section\n222 are considered: the Maxwellian equilibrium fMof (14) and the Lorentz equi-\nlibriumfLof (15).\nBecause it is most common to consider the Maxwell equilibrium, we explicate\nhere several reasons for considering the Lorentz equilibrium, which to our\nknowledge has not been numerically tested previously in the literature.\n(1) Naturally occurring plasmas are sometimes not Maxwellian but posses\nkappa distributions [39,47] that have power law tails in v. The Lorentz\nequilibrium is in a sense an extreme case of these in that it has v\u00002\ndecay at in\fnity, with the existence of the particle density but not kinetic\nenergy. In any event, distributions with power law tails are of physical\ninterest and thus worth studying in their own right. (See also e.g. [65,66].)\n(2) Because of the slow decay in v, the e\u000bect of truncating the velocity domain\nis ampli\fed and a greater velocity domain is needed. This makes the\nLorentz equilibrium a more stringent test for a numerical algorithm.\n(3) The linear dynamics of Vlasov theory is dominated by phase mixing,\nthe mechanism that underlies Landau damping (cf. Section 4.1.2). For\nthe advection problem, the Lorentz equilibrium gives decay of the form\nexp (\u0000kt), as opposed to the Maxwell equilibrium that gives decay of\nthe form exp (\u0000k2t2) (cf. Section 4.1.1), and this suggests it might be\na better test for getting Landau damping right. In fact, the reason for\nthis exponential decay is that linear advection with the Lorentz equilib-\nrium shares the same analytic structure as that of the Landau damping\nproblem (cf. Section 4.1.2), while linear advection with the Maxwell equi-\nlibrium does not. The essence of Landau damping can be traced to the\nRiemann-Lebesgue lemma [62], which states that charge density integrals\nof the form\nlim\nt!1Z\ndvg(v)eivt= 0\nprovidedg2L1, i.e.Rdvjg(v)j<1. The rate of this temporal decay\ndepends on the nature of the function g(v).\nThe underlying reason for exponential decay in both the advection\nproblem with the Lorentz equilibrium and the Landau damping problem\nwith either equilibrium, is that the function g(v) for these problems is\nanalytic in a strip in the complex v-space, and the damping rate is deter-\nmined by the pole closest to the real vaxis. Thus, the basic mechanism\nof Landau damping is tested in the simpler advection problem when fis\ngiven by (52) with \u000efgiven by (53a) and feqbeing the Lorentz equilib-\nrium.\n(4) With the Lorentz equilibrium one can use residue calculus to explicitly\nobtain expressions for the damping rates (see Appendix A). Although for\nthe advection problem this is also true for the Maxwellian, this is not the\ncase for the Landau damping problem of Section 4.1.2.\n234.1.1 Advection Results\nThe advection equation,\nft+vfx= 0; (54)\nis a natural test for assessing Vlasov algorithms because the Poisson equation\nis removed from the calculation and the focus is placed on the resolution of\nphase space. With a Maxwellian equilibrium this example has been treated in\nmany works, for example in [11,33,54,56]. In [56] four standard Vlasov solvers\nare compared.\nAfter computing f, the long time behavior of the solution can be checked\nby comparing the computational results with the known theoretical damping\nbehavior due to phase mixing. To this end the net charge density \u001atotis given\nby\n\u001atot(x;t) = 1\u0000Z1\n\u00001dvf(x;v;t )\n= 1\u0000Z1\n\u00001dvf 0(x\u0000vt;v)\n=\u0000AZ1\n\u00001dvcos [k(x\u0000vt)]feq(v); (55)\nwhere the second equality follows because the solution to (54) is given by\nf(x;v;t ) =f0(x\u0000vt;v), and the third upon substitution of (53a). According\nto the Riemann-Lebesgue lemma, lim t!1\u001atot= 0 under mild requirements on\nfeq. Below we give explicit forms for the decay for the Maxwell and Lorentz\nequilibria.\nAlthough it is common to consider the linear advection problem for test-\ning numerical algorithms, it is not so well-known that there is an intimate\nrelationship between the solution of the advection problem and the actual\nLandau damping problem. In fact, there is a one-to-one correspondence be-\ntween solutions of the two. In [51,52] an invertible linear integral transform, a\ngeneralization of the Hilbert transform, called the G-transform, was explicitly\nconstructed that maps (13a) into the advection equation (54). Thus, given an\ninitial condition for the advection equation, there exists an initial condition\nfor (13a) that transforms into the same solution. For the Lorentz equilibrium\none can use residue calculus to obtain explicit expressions. In particular, the\nG-transform of the Lorentz equilibrium fLof (15) is\nG[fL] =1\n\u00191\n1 +v2\"\n1 +1\nk2(1\u00003v2)\n(1 +v2)2#\n; (56)\nwhere the procedure is done mode by mode and kis the mode number. This\n24means that a solution to the linear advection problem with the initial condition\nf0=Acos(kx)fL (57)\nis equivalent to the solution of the linear Vlasov-Poisson system with the initial\ncondition\nf0=Acos(kx)G[fL]: (58)\nThe di\u000berence in long time decay between the advection problem and that\nof Landau damping can be traced to poles that occur in the integral trans-\nform. One could use the integral transform to further test the veracity of a\nnumerical algorithm by comparing the solutions of the advection and Lan-\ndau problems, but we will not do so here. However, given the understanding\nprovided by the integral transform, it is quite natural to examine the advec-\ntion problem with initial conditions that are meromorphic in velocity like the\nLorentz equilibrium.\nLinear advection with a Maxwell equilibrium\nFor our \frst example (54) is solved for feq=fMgiven by (14). We choose\nA= 0:1;k= 0:5;L= 4\u0019;VC= 5 andT= 40. For this particular case, it\nis easily shown by elementary methods that the net charge density of (55) is\ngiven by\n\u001atot(x;t) =\u0000Acos (kx)e\u0000k2t2=2: (59)\nThis implies that max xj\u001atot(x;t)j= 0:1e\u0000t2=8, sincek= 0:5.\nTo test the accuracy and convergence of the UG method, max xj\u001atot(x;t)jis\ncomputed numerically and the results are plotted in Fig. 1. The numerical\nresults were generated using the \fve uniform meshes ( Nhx;Nhv) = (500;400),\n(1000;800), (2000 ;1600), (4000 ;400), (8000 ;400), where NhxandNhvdenote\nthe number of partitions of the x-axis and the v-axis, respectively.\nThe \frst three meshes are such that hx\u0019hv, whereas the fourth and \ffth\nmeshes are such that hx\u0019hv=8 andhx\u0019hv=16, respectively. The motivation\nfor using the last two meshes comes from the fact that the problem being ap-\nproximated involves only advection in the x-direction. Hence, it is reasonable\nto assume that mesh re\fnements in xwill improve the numerical accuracy as\nmuch as performing re\fnements in both xandv, provided the re\fnement level\ninvis su\u000eciently small so that the error is almost entirely due to the re\fne-\nment level in x. Figure 1 clearly shows that the UG method is both accurate\nand numerically convergent under mesh re\fnements.\nOur results compare with those of [11,33,54,56] for early times where solutions\nare accurate, but unlike the others we do not obtain the later time recurrence\nthat arises from periodicity of f0(x\u0000vt;v) in its \frst argument and the velocity\n25mesh size used in evaluating the integral of (55). This is because of the \fneness\nof our mesh and because of the speci\fc dissipative properties of the DG method\ngive monotonic error. With a mesh of Nhv= 400 the recurrence time is TR=\n2\u0019k=\u0001v=\u0019Nhv=(2Vc)\u0019126.\nLinear advection with a Lorentz equilibrium\nIn this second example, we consider the advection problem with the Lorentz\nequilibrium (15). Numerical results are given for four di\u000berent values of the\nwavenumber k. We will revisit these cases in Section 4.1.2, where we consider\nthe actual Landau damping problem with the electric \feld not taken to be\nzero. Speci\fcally, here Eq. (54) is solved for feq=fL=\u0019\u00001=(v2+1),A= 0:01,\nand the large value Vc= 30 for each of the wavenumbers k= 1=8, 1/6, 1/4,\nand 1/2. The corresponding values for LandTfork=1/8, 1/6, 1/4, and 1/2\nareL= 16\u0019, 12\u0019, 8\u0019and 4\u0019andT= 75;75, 50, and 50, respectively. The\nuniform mesh ( Nx;Nv) = (1000;2000) was employed in each of the four cases.\nFor this case, it is easily shown using residue calculus that the charge density\nof (55) is given by\n\u001atot(x;t) =\u0000Acos (kx)e\u0000kt: (60)\nThis implies that max xj\u001atot(x;t)j= 0:01e\u0000kt.\nThe computed results for each of the four wavenumbers kare shown in Fig. 2\nalong with the exact result of (60). From the \fgure it is seen that for early\ntimes the computations match the theoretical result (60). At late times the\ncomputations diverge and, as anticipated, it is more di\u000ecult to resolve cases\nwith larger k, i.e. with \fner spatial structure.\nFig. 1. (Linear advection with Maxwell equilibrium) Plot of log10(max xj\u001atot(x;t)j) =\n\u0000(log10e)t2=8 + 1 vs.t: analytic solution (solid) , (Nhx;Nhv) = (500,400) (dot) ,\n(1000,800) (dash-dot) , (2000,1600) (dash-dot-dot) , (4000, 400) (short dash) , (8000,\n400) (long dash) .\n26Fig. 2. (Linear advection with Lorentz equilibrium) Plot of log10(max xj\u001atot(x;t)j) =\n\u0000(log10e)kt + 2 vs. t:k=1/8 (top left) ,k=1/6 (top right) ,k=1/4 (bottom left) and\nk=1/2 (bottom right) ; analytic solution (dash) , numerical solutions (solid) .\n4.1.2 Linear Landau Damping Results\nThe next two examples test the ability of the method to reproduce results\nconsistent with the theoretical results of linear Landau damping. We empha-\nsize that it is not enough to merely show exponential decay, but to believe an\nalgorithm one must compare the decay rates and the parametric dependence\nof the theoretical rates. As above, both Maxwellian and Lorentzian equilibria\nare considered. For the Maxwellian equilibrium, which were previously treated\nin [11,33,36,54,73,76], results are computed using four successively re\fned uni-\nform meshes in order to demonstrate that the numerical decay rates converge\nto the theoretical decay rate under mesh re\fnement. For the Lorentz equilib-\nrium, we will see that the UG method is robust in the sense that it produces\nthe correct decay rates for di\u000berent wavenumbers k.\nAs noted in Section 2, Landau showed the electric \feld decays exponentially\nin the time-asymptotic limit (for more rigor see [46] for the linear case and\nthe recent nonlinear results of [53], of which Ref. 37 is an early version of the\npresent work). More speci\fcally, if we write the frequency as !(k) =!R(k) +\ni\r(k), where!R(k) and\r(k) are real-valued, then in the time-asymptotic limit\nEk(!;t) decays at a rate \r(k) and oscillates at a frequency !R(k).\n27Besides the Maxwellian equilibrium of (14), the Lorentzian equilibrium fLof\n(15) gives rise to exponential damping of the electric \feld. In this case, the\ndecay rate\r(k) for then-th electric \feld mode with k= 2\u0019n=L is given by\n\r(k) =\u0000k; (61)\nwhere\r <0 implies damping, and the corresponding frequency of the electric\n\feld oscillations satis\fes\n!R(k) = 1: (62)\nThe derivation of (61)-(62) is described in Appendix A. It is important to note\nthat formulae (61) and (62) are explicit, which directly results from using the\nLorentz equilibrium, whereas, as noted above, the formula for \r(k) and!R(k)\nwhen a Maxwell equilibrium is used are implicitly de\fned [42].\nLinear Landau damping with a Maxwell equilibrium\nWe solved the linear system (13a)-(13c) with feq=fM= (2\u0019)\u00001=2e\u0000v2=2,\nA= 0:01,k= 0:5,L= 4\u0019,Vc= 4:5 andT= 80. For this problem, the\ntheoretical decay rate and frequency of oscillations to the third decimal digit\nare respectively equal to \r=\u00000:153 and!R= 1:415 [11]. Approximate\nsolutions are computed using the four uniform meshes ( Nhx;Nhv) = (250;400),\n(500;800), (1000 ;1600) and (2000 ;1600).\nPhase-space contour plots and cross-sectional plots in vof the approximate\nsolutionfhfor (Nhx;Nhv) = (500;800) andt= 0;t= 25,t= 50, andt= 75 are\ndisplayed in Fig. 3. These sequential plots show the increase in \flamentation\noffhas time elapses.\nThe convergence of numerical decay rates of the dominant electric \feld Fourier\nmode is demonstrated in Fig. 4. The decay rate resulting for each mesh was\ncomputed by calculating the slope of the straight line plotted in the \fgure. In\neach case, the line was de\fned by the point occurring at the peak of the third\noscillation and the point occurring at the peak of the ninth oscillation. A time\nstep of \u0001t= 0:001 was used in order to ensure that the points de\fning each\nof the lines were actual computed data points rather than points that were\ndetermined using some interpolation of the data. Under mesh re\fnement, the\nnumerical decay rate is seen to converge, up to three decimal-digit accuracy, to\nthe theoretical decay rate of -0.153. In all four cases, the numerical frequency\nis observed to correspond to the theoretical frequency up to three decimal-\ndigit accuracy. We also note that, upon re\fning the mesh, the decay of the\ndominant mode is sustained for longer times before leveling o\u000b. Our results\ncompare favorably with those of previous works [11,33,36,54,73,76], which were\nobtained by various other methods. Because of the \fneness of our mesh we\nwere able to proceed to longer times than all but [76] which achieved machine\nzero for small perturbations.\n28As with the advection results above, in contrast to all other works, we do not\nsee recurrence. It is important to note that recurrence is in fact an indication\nof numerical error. Recurrence is a general phenomenon in \fnite-dimensional\ndynamical systems with time advancement maps that are measure preserving,\none-one, onto, bicontinuous, and have a bounded phase space. Poincar\u0013 e proved\nrecurrence for \fnite-dimensional Hamiltonian systems, although it can hold\nfor other systems as well. The Vlasov-Poisson system is an in\fnite-dimensional\nHamiltonian system [49], and there is no general recurrence theorem for such\nsystems.\nNumerical truncation procedures generally are not Hamiltonian, and indeed\nwe have shown this to be the case for the DG method used here. However,\nit has been shown that the DG method, although not Hamiltonian, does give\na \fnte-dimensional system that preserves phase space volume [18] and has\na bounded energy. Thus our semi-discrete system has all the ingredients for\nmaking a general estimate for the recurrence time for the distribution function\nin terms of the number of degrees of freedom, like that done by Boltzmann for\ngas dynamics. This results is a very long time for meshes with any signi\fcant\nresolution. Note, this approach di\u000bers from a result that is often quoted in nu-\nmerical works that follows for the method of [11]; viz. for a given Fourier mode\nand a mesh of size \u0001 vthe advection equation was shown to have a recurrence\ntime for the electric \feld of TR= 2\u0019k=\u0001v. Many Vlasov algorithms see recur-\nrence at a value that is close to this advection value TR. It is important to note\nthat this does not mean that the distribution function is recurring in phase\nspace on this time scale. In [18] we present detailed recurrence calculations for\nour DG algorithm with piecewise constant and higher order polynomials.\nIn any event, recurrence is not physical and the same can be said for the non-\nrecurrent \rattening of the electric \feld decay rate that is seen in our results\nwith the DG algorithm at late times. The lack of recurrence in our results\nis due to both the very \fne mesh size as well as the dissipative nature of\nDG algorithms, which is monotonic in nature. However, from Fig. 4 (and also\nFig. 5 below) we would argue that DG can do a good job.\nLinear Landau damping with a Lorentz equilibrium\nThe ability of the UPG method to achieve accurate damping results across\ndi\u000berent wavenumbers is now investigated. As noted above, the Lorentz equi-\nlibrium is used in this example because explicit formula for the electric \feld\ndamping rates and the frequencies of the damped oscillations are easily ob-\ntained (see Appendix A). Moreover, this example also tests the ability of the\nmethod to produce accurate results for an equilibrium that has a much heavier\ntail invthan does the Maxwell equilibrium. The heavy (algebraic) tail of the\nLorentz equilibrium leads to a faster rate of \flamentation because there are\nmore resonant electrons than for the Maxwell equilibrium.\n29The linear system (13a)-(13c) is solved with feq=fL=\u0019\u00001=(v2+1),A= 0:01,\nandVc= 30:for each of the wavenumbers k= 1=8, 1/6, 1/4, and 1/2. Note,\nVcneeds to be large because of the algebraic tail of fL. The corresponding\nvalues forLandTfork=1/8, 1/6, 1/4, and 1/2, are L= 16\u0019, 12\u0019, 8\u0019and\n4\u0019andT= 75;75, 50, and 50, respectively. Uniform meshes were employed\nin all four cases, where in each case ( Nx;Nv) = (1000;2000).\nIn Fig. 5, log plots of \u001atotfor the four cases are given. The observed damping\nfork= 1=2 lasts for a shorter time duration than does the damping for the\nother three wavenumbers, even though the mesh for k= 1=2 has the \fnest\nresolution in x. This result is due to the fact that the \flamentation in the\nvelocity direction develops more rapidly than it does in the other three cases.\nFrom (61), obtained in Appendix A, it follows that for a given wavenumber k\nthe fundamental mode of the electric \feld damps exponentially in time at a\nrate equal to \r(k) =\u0000kand the frequency for all of the damped oscillations is\n!(k) = 1. Therefore, the theoretical damping rates corresponding to k= 1=8,\n1/6, 1/4, and 1/2 are \r(k) =\u00001=8, -1/6, -1/4, and -1/2. In all of the four cases\nshown in Fig. 5, the numerical damping rates and frequencies of oscillation\nare equal to the theoretical values up to the \frst two decimal digits.\n30Fig. 3. (Linear Landau damping with Maxwell equilibrium) Contour plots ( left)\nand cross-sectional plots ( right),x= 2\u0019, for\u000efatt= 0,t= 25,t= 50,t= 75\n(descending order ). 31Fig. 4. (Linear Landau damping with Maxwell equilibrium) Time decay plots\nof fundamental mode (with arbitrary ordinate origin) under mesh re\fnement:\n(Nhx;Nhv) =(250, 200) (top left) , (500, 400) (top right) , (1000, 800) (bottom left)\nand (2000, 1600) (bottom right) . The numerical decay rate converges to the theoret-\nical value of -0.153 to within three decimal-digit accuracy; similarly, the numerical\noscillation frequency agrees with the theoretical frequency of !R= 1:415 to within\nthree decimal digits.\n32Fig. 5. (Linear Landau damping with Lorentz equilibrium) Time decay plots of\nfundamental modes (with arbitrary ordinate origin): k=1/8 (top left) ,k=1/6 (top\nright) ,k=1/4 (bottom left) andk=1/2 (bottom right) .\n334.2 Nonlinear Results\nTwo nonlinear calculations are performed, an example of strong nonlinear\nLandau damping and a version of the two-stream instability that we examine\nin greater detail. In the former case, in addition to early time damping, we\nexpect to see motions on the bounce time scale due to particle trapping, while\nin the latter we expect to see trapping and an asymptotic approach to a BGK\nstate.\n4.2.1 Nonlinear Landau Damping\nFollowing [11,26,31,33,41,54,73,76] we consider an initial condition near a\nMaxwellian and of the form given by (52) and (53a){(53c) with feq=fM=\n(2\u0019)\u00001=2e\u0000v2=2,A= 0:5,k= 0:5,L= 4\u0019,Vc= 5:0 and a run time of T= 120.\nUnlike the case considered in Section 4.1.2 we evolve under the full non-\nlinear Vlasov equation. Solutions are computed using a uniform mesh with\n(Nhx;Nhv) = (1000;750).\nBecauseAhas been increased to 0 :5, higher modes are excited at very early\ntimes and the damping rate signi\fcantly exceeds the linear rate of \r=\u00000:153.\nThis is because energy leaves the \frst mode as the higher modes get excited;\ni.e., in addition to the phase mixing process there is an energy transfer process\ncaused by the nonlinear interaction of the modes. Figure 6 shows the ampli-\ntudes of the \frst four modes as a function of time, with a form similar to\nprevious calculations. These plots are obtained from our mesh data by using\nthe following `log Fourier Mode' function:\nlogFMk(t) = log10\u0012q\njEs;k(t)j2+jEc;k(t)j2=L;\u0013\n(63)\nwith\nEs;k(t) :=ZL\n0dxE(x;t) sin 2\u0019kx\nL!\n(64)\nand\nEc;k(t) :=ZL\n0dxE(x;t) cos 2\u0019kx\nL!\n; (65)\nwherekis the mode number sought. From Fig. 6 it is seen that mode-one\nreaches its minimum value at around t\u001915 and then all modes grow until\nthey reach their maxima at t\u001940, consistent with previous calculations.\nUsing the maximum amplitude of mode-one, the bounce time is calculated to\nbeTB\u001920 and this is also in agreement with previous results.\nExamination of Fig. 7 reveals that we obtain a damping rate for the \frst mode\nof about\r=\u00000:287, a value consistent with the \u00000:281 obtained by [11] and\n34\u00000:243 by [73], given the di\u000berent ways authors have used to make this kind\nof \ft. Also, given that we have a \fner mesh, some deviation could be due to\nour more precise coupling to the higher modes. Examination of Fig. 8 shows\nthere is signi\fcant entropy [(11) with C=\u0000flnf)] dissipation at short times,\nas also seen by [33], and this introduces some error. Also note, we have run to\nT= 120, which is signi\fcantly longer than previous calculations and a small\ndecay in all four of the amplitudes is seen. This could be due to transfer to\nhigher modes, cascading, or due to dissipation in the algorithm. Over the full\nlength of the run the total energy Hof (10) is c onserved to within a few\npercent, and in the later part of the run entropy is well conserved. This is\nimproved in the next section, where we treat the two-stream instability, by\ndecreasing the mesh size.\nIn Fig. 9 we plot the spatial average of the distribution function. Like other\nauthors, we obtain early plateau formation in the vicinity of the phase velocity\nof the wave, seen in panel (b), which broadens as the higher order modes\nare excited. At around t= 40, approximately the time of the \frst bounce\nmaximum, signi\fcant smoothing takes place and the system settles into a\nnearly constant average state with a persistent electron hole. The smoothing\natt\u001940 can also be seen in the results of [11,33]. Finally, we note the\npresence a small periodic dimpling behavior at the maximum that persists for\nlate times.\n35(a)k= 0:5.\n (b)k= 1:0.\n(c)k= 1:5.\n (d)k= 2:0.\nFig. 6. (Nonlinear Landau damping with Maxwell equilibrium) Amplitudes of the\n\frst four modes versus time. Mesh size ( Nx;Nv) = (1000;750).\n36Fig. 7. (Nonlinear Landau damping with Maxwell equilibrium) Initial damping of\ndominant mode followed by growth due to particle trapping.\n(a) Energy.\n (b) Entropy.\nFig. 8. (Nonlinear Landau damping with Maxwell equilibrium) The total energy H\nof (10), kinetic plus electrostatic, and entropy, (11) with C=\u0000flnf, versus time.\n374.2.2 Nonlinear Two-Stream Instability\nIn this subsection, we numerically study the long-time nonlinear evolution of\nthe two-stream instability, a standard example that has been used for demon-\nstrating the e\u000ecacy of a variety of numerical algorithms [11,29,36,41,54,56,73].\nLike these previous calculations, we consider the equilibrium state\nfTS(v) =1p\n2\u0019v2e\u0000v2=2(66)\nand apply our algorithm to the nonlinear system (12a)-(12c) with data of\nthe form given by (52) and (53a){(53c). We choose parameters values to be\nA= 0:05,k= 0:5,L= 4\u0019,Vc= 5, andT= 100, a uniform mesh de\fned by\n(Nx;Nv) = (1800;1400), and a time step \u0001 t= 0:00125.\nQualitative Behavior\nFigures 10 and 11 show 3D and 2D contour plots of the distribution function\nfin phase space at di\u000berent instances of time. The initial data consists of two\nsymmetric, counter-streaming, warm beams with the small sinusoidal pertur-\nbation superimposed, as described above. We obtain the qualitative behavior\nexpected: the linear two-stream instability grows exponentially until nonlin-\nearity becomes important and trapping occurs, with an eventual long time\nasymptotic Bernstein-Greene-Kruskal (BGK) state [8] with an apparent elec-\ntron hole-like structure (cf. e.g. [60]). As the nonlinear system evolves, linear\nmodes grow and saturate as shown in Fig. 12, the phase space hole forms as\na portion of the electron distribution becomes trapped and exhibits \flamen-\ntation. Over time, the sharp contour variation associated with \flamentation\nis smoothed out, a consequence of nonlinearity and the numerical dissipation\nof the UPG method. (We note, here for aesthetic reasons some smoothing\nwas also done by the plotting routine.) Our numerical results are indicative\nof a very stable computational scheme. This is partly due to the fact that our\nDG method is monotone and mass conservative. The particle number error\naccumulation in time is due to the nonlinear coupling.\nEarly Growth\nAs noted above, Fig. 12 shows the time evolution of the \frst four modes, which\nhave wavenumber values k= 0:5,k= 1,k= 1:5, andk= 2, fromt= 0 to\nt=T= 100. (Recall, our domain is of size 4 \u0019.) At early times, 0 \u0014t<20, the\nbehavior illustrated in Fig. 12 is consistent with the results of previous authors\nwhere the initialized mode-one with k= 0:5 dominates the other modes that\nare not initially excited. All modes reach a maximum at t\u001918 with growth\nrates on the order of a couple of tenths of !pwhich is typical of linear theory\nfor this problem. During early times, noise in the system and the growth of\nmode-one excites the other modes as was the case for the nonlinear Landau\n38damping of Section 4.2.1. A comparison was made in the calculations of [29]\nusing results from [36], but we give some further details here.\nThe plasma dispersion function from Vlasov linear theory is given by\n\u000f= 1\u00001\nk2Z1\n\u00001f0\n0(v)dv\nv\u0000!=k; (67)\nwhere!will be o\u000b the real axis as is known for the two-stream instability. We\nevaluate this \u000fwith the two-stream equilibrium of (66) in order to obtain the\nlinear growth rates. In Appendix A we show that \u000fcan be rewritten in the\nform\n\u000f(z;k) = 1\u00001\nk2h\n1\u00002z2+ 2Z(z)(z\u0000z3)i\n; (68)\nwherez=!=(kp\n2) andZis the usual plasma dispersion function as de\fned\nin [34]. Figure 13 is a plot of the growth rate obtained from \u000f= 0 adapted to\nour domain with L= 4\u0019(cf. [36] Fig. 3) con\frming that our early growth is\nabout right.\nConservation Properties\nFigure 14 shows plots of the invariants of Section 2 as functions of time, while\nFig. 15 depicts the relative error of the total energy, ( H\u0000H0)=H0, and simi-\nlarly the enstrophy. The top left panel of Fig. 14 shows that the total particle\nnumber is conserved quite well, with an error no larger than 0 :01% over the\nfull 100 units of time of our simulations. Actually, the DG discretization and\nRunge-Kutta (RK) method used for time advancement perfectly conserves this\nquantity for the transport equation. However the nonlinear iteration generates\na monotone in time error for the total particle number of order of 10\u00004in 100\ntime units. The top right panel of Fig. 14 shows the evolution of the total\nmomentum, P. This quantity is exactly conserved because the DG method\ncannot break symmetry, and the same is true for the time advancement al-\ngorithm. In fact the method perfectly conserves all odd velocity moments of\nf. The middle left panel of Fig. 14 depicts the evolution of the enstrophy\nCasimir invariant,Rdxdvf2, a q uantity that appears to be seldom-monitored\nin Vlasov codes ([33] being an exception). From the inset of Fig. 15, it is seen\nto be conserved to within about 10%, which we suspect is comparatively good,\nand can be considerably improved by re\fnement and an increase in polynomial\norder. Conservation of this quantity arises because the solution of the Vlasov\nequation is a rearrangement, a property that we discuss below in more detail.\nIt is violated because of small scale error and the di\u000busive nature of numerical\nalgorithms. Note, like total particle number N, the error in the enstrophy is\nmonotonic in time. The middle right and bottom left panels of Fig. 14 show\nthe evolution and error of the kinetic and electrostatic energies (6) and (7),\nthe \frst and second terms of (10), respectively. Individually these quantitie\ns are not conserved by the Vlasov equation, as is clearly evident from the\n39\fgures, but their sum, H, shown in the bottom left panel, is conserved with a\nrelative error less than 4% over the entire course of the run. The oscillations\nin the kinetic and electrostatic energies, indicative of the trapping process,\ncancel upon addition and give an error that increases monotonically in time.\nIn Fig. 15 the relative error of the enstrophy and total energy are depicted. In\nother algorithms the error in His oscillatory in nature, typically with small\ntemporal mean. Even though this mean error could be small, it is important\nto realize that the oscillations amount, in a sense, to successively reinitializing\nthe code, and the cumulative error of the solution for such a process is hard\nto assess.\nIn addition to the conserved quantities of (8), (9), (10), and (11), the Vlasov\nequation possesses topological conservation . As mentioned above, it is well-\nknown that the solution of the Vlasov equation is a rearrangement [35]. This\nmeans that the solution at any time tcan be obtained formally from its initial\nvalue as follows:\nf(x;v;t ) =f0(x0(x;v;t );v0(x;v;t )) =f0\u000ez0; (69)\nwherez0= (x0(x;v;t );v0(x;v;t )) is obtained upon inverting the solution of\nthe characteristic equations z=z(z0;t). Because the characteristic equations\nhave Hamiltonian form they preserve volume (here area), and\n@(x0;v0)\n@(x;v)= 1: (70)\nA consequence of this is the family of Casimir invariants of (11), whose invari-\nance follows directly upon e\u000becting the coordinate change z0$zand making\nuse of (69) and (70):\nZ\ndx0Z\ndv0C(f0(x0;v0) =Z\ndxZ\ndvC(f(x;v;t ); (71)\nwhere we have suppressed limits of integration.\nUnder continuity assumptions, the level sets of f0are topologically equivalent\nto the composition f0\u000ez0=f. This is what is meant by topological conser-\nvation. It follows that the number and nature of extrema, the values of fat\nthese extrema, and the kinds of separatrices connecting saddle points must\ncorrespond to those of f0. In addition, although not a topological property,\nthe area between any two contours of fmust also be conserved, a consequence\nof the area preservation property of the characteristics. Since DG is a weak\nformulation with lack of continuity, the extent to which level sets are actually\ntopologically conserved remains to be seen. Casimir invariants such as enstro-\nphy and entropy may be conserved well and be consistent with rearrangement\ninequalities such as Jensen's inequality without the continuity assumption.\n40Because the initial condition has the form f0(x;v) =fTS(v)[1+0:05 cos(x=2)],\nthe initial phase space has nearly imperceptible initial `tears' in the \frst\npanels of Figs. 10 and 11, that should be consistent with asymptotic state\natt=T= 100. Extrema of f0(x;v) occur at points ( x;v): (x;0), for all\nx2f0;4\u0019g, (0;\u0006p\n2), and (2\u0019;\u0006p\n2). Thus there is a trough along the line\nv= 0. The points (0 ;\u0006p\n2) = (4\u0019;\u0006p\n2) atop the beams are maxima strad-\ndled by thin separatrices entering and emerging from the saddle points lo-\ncated at (2\u0019;\u0006p\n2). The following extremal values of f0remain \fxed in time\nunder Vlasov dynamics, although their locations can change: f0(0;0) = 0,\nf0(2\u0019;0) = 0,f0(0;\u0006p\n2) = 0:308, andf0(2\u0019;\u0006p\n2) = 0:279. A glance at\nFig. 10 shows some degradation of the value of fatop the ridge-like struc-\ntures, while the vanishing value of fat the points (0 ;0) and (2\u0019;0) is rigor-\nously maintained. A measure of error in an algorithm is the extent to which\nit \flls in the hole, i.e. returns values of f0(2\u0019;0)6= 0.\nIn our simulations the value of fremains zero at the minimum (2 \u0019;0) and it\nremains zero to high accuracy at (0 ;0). The values of fat the extrema and\nthe associated separatrices atop the beams are never very discernible, their\nexistence due to only a few percent in the variation of f, and are washed\nout by numerical dissipation as the contours of the distribution function wrap\naround and `trap particles' over the course of the simulation. However, the\ntrough atv= 0 is structurally unstable and separatrices emerge form here\nand connect the saddle point at v= 0 andx= 0 = 4\u0019that straddles the\nminimum at (2 \u0019;0), the center of the electron hole. The eventual boundaries\nthat delineate the trapped and untrapped particle populations, as the potential\nsaturates into what appears to be a \fnal BGK-like state, is what we discuss\nnext.\nBGK Saturation\nExamination of Figs. 10, 11, and 12 shows the initial linear growth phase,\nfollowed by a particle trapping phase, and eventually a strong indication of a\nsaturated state with clean contours of f, resulting at least in part from the\nsmall scale averaging inherent in any algorithm. It is generally believed that\nthis evolution saturates, in some weak sense, to a BGK mode, although no\nproof exists. To test this belief we check to see if the contours of fare aligned\nwith contours of the particle energy, E=v2=2\u0000\b(x;t), which is well-known\nto be the case for an equilibrium state of the Vlasov equation (e.g. [8]).\nTo this end we plot f100(E(x;v)) :=f(x;v;100), where inE(x;v) =v2=2\u0000\n\b(x;100), the particle energy at t= 100. (Note, we suppress the time variable.)\nFigure 16 clearly indicates that a saturated BGK state has been achieved.\nHere, in the left panel, we have plotted fversusE(x;v), att= 100, for all\n9 million pairs ( x;v) of the uniformly distributed mesh over our phase space.\nObserve that to within the thickness of the line, f100appears as a graph over\n41E(x;v) and that this is true even for large values of E, wheref100>0 is\nmaintained. Green and red dots correspond to positive and negative values\nof the velocity, respectively, and there does not seem to be any systematic\ndirectional bias. Because the computation was done for piecewise constant\nelement functions, it is known that DG ensures f(x;v;t ) remains positive for\nall times a nd this is then the case for f100. The right panel of Fig. 16 is a plot\nof the electrostatic potential as a function of position for several instances in\ntime as indicated in the \fgure. The curve labeled by t= 100 is taken to be\nthe saturated electrostatic potential that was used in the calculation of f100.\nIn Figs. 18 and 19 high resolution details of f100are depicted. In Fig. 18(a)\na cusp is seen at the trapping boundary that occurs at E= 0. A magni\fed\nview of this is shown in Fig. 18(b). In the course of the evolution the presence\nof the electrostatic force produces a trapped particle population with E<\n0. It is conceivable that trapping, like scooping ice cream, is a non-analytic\nprocess and that this cusp represents something real about the mathematical\nnature of the dynamics. However, it also may be attributed to the choice of\nthe numerical scheme, yet discontinuities have previously been proposed at\ntrapping boundaries: in [60] a discontinuity is used in making electron hole\nmodels and in [30] two kinds of discontinuities are proposed for adiabatic and\nsudden trapping. None of these discontinuities match the cusp like feature\nthat we observe, but the traveling wav e state treated in [30] is not the same\nas that reached by our simulation and it is possible that an analysis similar to\nthat of [30] could produce a cusp. Further studies of this e\u000bect are currently\nbeing investigated. Figures 18(c) and 18(d) examine f100near the minimum\nvalue ofE. Note the clear steepening of f100as it approaches its minimum\nvalue. It is possible that there is a universal nature to both this steepening\nand to the cusp at the trapping boundary, but we leave an investigation of\nthis to future work. Figures 18(e) and 18(f) examine f100near its maximum\nvalue. The spread here as well as the other panels of Fig. 18 give a sense of\nhow close the system has relaxed to an equilibrium state.\nFigure 19 examines the details of f100for higher values of E. In Fig. 19(b) a\nsplitting is seen to occur at around E= 1:3. In Figs. 19(c) and 19(d) we see\nthat this small splitting persists to large values of E. It is interesting to note\nthat because the red and green dots are mixed within each band, the splitting\nis not an e\u000bect of velocity direction. The splitting is within the resolution of\nthe code and is believed to be a real e\u000bect, one that seems to indicate that\ncomplete saturation has not occurred.\nThe above tests provide strong indication that the code has relaxed to near a\nBGK state. As further evidence we test to see if the charge density associated\nwithf100is consistent with that for a \fnal BGK state. To this end we \frst\nobserve that f100can be \ft reasonably well by a model distribution function\n42of the following form:\nf\ft=a(E+ \bM)(E+E\u0003)e\u0000\fE: (72)\nHere \bMis the maximum value of \b, and because E=v2=2\u0000\b we see\u0000\bM\nis the smallest possible value of E. This and positivity of fimplyE\u0003>\bM.\nObviously (72) does not account for the \fne features discussed above, but it\nwill be su\u000ecient for our purpose.\nThe quantities a, \bM,E\u0003, and\fare \ftting parameters that can be matched to\nthe code output. We obtain a rough idea of the degree of self-consistency, i.e.\nthe degree to which Eq. (72) produces the correct electrostatic potential when\nsubstituted into Poisson's equation, by using the following values extracted\nfrom the code output: \b M= 1:06 andE\u0003= 1:59. For convenience we choose\n\f= 1. More signi\fcant \fgures in a, viz.a= 0:1148, are required to ensure\nthat the total net charge is zero. In Fig. 17 we see these choices provide a\npretty good \ft to the data. However, insu\u000ecient charge near the maximum\nis spread to higher values of E. In a forthcoming publication we will examine\nthis sort of modeling in much greater detail.\nWe note, there is a relationship between EM, the value at which fobtains its\nmaximum, and the other parameters. That is, f0(EM) = 0 implies\nEM=2\u0000\f(\bM+E\u0003) +q\n4 +\f2(\bM\u0000E\u0003)2\n2\f(73)\nor\nE\u0003=\bM\u0000\fEM\bM+ 2EM\u0000\fE2\nM\n\fEM+\f\bM\u00001; (74)\nwhere the latter is useful when EMis extracted from data. In fact, the value\nofE\u0003= 1:59 used above follows from EM= 0:71 (cf. Fig. 18).\nGivenf\ft, the charge density is given by\n\u001a\ft(\b) = 1\u0000Z1\n\u00001dvf \ft= 1\u0000Z1\n\u0000\bdEf\ft(E)q\n2(E+ \b); (75)\nwhere in the second equality the formula E=v2=2\u0000\b is used to change the\nintegration variable from vtoE. With the choice (72) this quantity can be\nexplicitly integrated to give\n\u001a\ft= 1\u0000p\n2\u0019a\n\f5=2\"3\n4\u0000\f2(\bM\u0000\b)(\b\u0000E\u0003) +\f\n2(\bM\u00002\b +E\u0003)#\ne\f\b:(76)\nDe\fning a pseudopotential Vby@V=@\b =\u001a\ft, and integrating Poisson's equa-\n43tion yields\nE2\n2+V(\b) =V0; (77)\nwhereV0is a constant and\nV(\b) = \b (78)\n\u0000p\n2\u0019a\n\f7=2\u001415\n4\u0000\f2(\bM\u0000\b)(\b\u0000E\u0003) +3\n2\f(\bM\u00002\b +E\u0003)\u0015\ne\f\b\nis the expression for the pseudopotential. Here E2=2 acts as a pseudo kinetic\nenergy. Thus we can interpret the BGK state by comparison to a particle\ndropped in the potential Vat zero kinetic energy. Such a \fctitious particle\nreturns to a state of zero kinetic energy as xtraverses the spatial domain,\nwhich must be the case if there is zero net charge.\nIn Fig. 20,E2is plotted against \b for our simulation data. In light of (77) and\n(78), we expect E2to be a graph over \b at time t= 100, and from the \fgure\nthis indeed appears to be the case. Also, since at t= 0, \b00=Acos(x=2), it is\neasy to see that E2(x;t= 0)=2 =A\b(x;t= 0)\u0000\b(x;t= 0)2=8. This follows\nfrom the choice of ground for \b and the absence of net charge which gives\nvia Gauss' theorem E(x= 4\u0019) =E(x= 0). With our boundary conditions\nE(x= 0) = 0. Thus we expect the parabolic curve in Fig. 20 labeled by\nt= 0 with maximum \b occurring \b M= 8A= 8\u00020:05 = 0:40. However, it is\nremarkable that at intermediate times E2is also a graph over \b. This can be\nshown to be true if \b maintains symmetry about x= 2\u0019. This suggests that\n\b can serve as a good spatial coordinate, an idea we will discuss further in a\nforthcoming publication.\nThe data of Fig. 20 can be compared to the model of Eq. (77) with (78).\nChoosingV0so thatE2(\b = 0) = 0 and the parameter values of Fig. 17,\nyields the plot of Fig. 21. Here we see a reasonable \ft to the data at t= 100.\nNote,E2(\bM)\u00190 and the maximum value of E2is a close \ft. We note, a\ncertain level of accuracy is needed in the parameters because E2is a sensitive\nfunction of \b. We have not optimized the \ft, but the result of Fig. 20 is roughly\nwhat one might expect for the expansion of f100with only three terms of a\nLaguerre series, and thus gives a fair indication of self-consistency which was\nour goal. As noted above, we will revisit this again in great detail in a future\npublication.\n44(a)t= 0.\n (b)t= 12:5.\n(c)t= 31:5.\n (d)t= 43:75.\n(e)t= 75.\n (f)t= 112:5.\nFig. 9. (Nonlinear Landau damping with Maxwell equilibrium) Spatial average of\nthe distribution function at the times indicated.\n45Fig. 10. (Nonlinear two-stream instability) 3D plots of the solution fatt= 0,\nt= 20,t= 40,t= 60,t= 80 andt= 100.\n46Fig. 11. (Nonlinear two-stream instability) Contour plots of the solution fatt= 0,\nt= 20,t= 40,t= 60,t= 80 andt= 100.\n47Fig. 12. (Nonlinear two-stream instability) Time series plots of the \frst four modes:\nk= 1=2(top left) ,k= 1 (top right) ,k= 3=2(bottom left) , andk= 2 (bottom\nright) .\nFig. 13. Plot of growth rate \rvs.kfor the equilibrium distribution function of (66).\n48Fig. 14. (Nonlinear two-stream instability) Temporal evolution of the total particle\nnumberN(top left) , momentum P(top right) , enstrophy (middle left) , and kinetic\nenergy (middle right) , electrostatic energy (bottom left) , and the total energy H\n(bottom right) . See Section 2 for de\fnitions of these quantities.\n49(a)\n (b)\nFig. 15. (Nonlinear two-stream instability) Relative error of the enstrophy (left) and\nthe total energy H(right) .\n.\nFig. 16. (Nonlinear two-stream instability) Plot of the distribution function at time\nt= 100,f100, as a function of the particle energy E=v2=2\u0000\b(x;t= 100). Green\ndots correspond to positive velocities and red dots to negative velocities, v(left) .\nRight panel depicts the potential \b( x;t) at the times indicated. \b( x;t= 100) was\nused inEfor the plot of f100(right) .\n50Fig. 17. (Nonlinear two-stream instability) Model distribution function f\ftof\nEq. (72) with \b M= 1:06,E\u0003= 1:59,\f= 1, anda= 0:1148 (solid blue) com-\npared to code results at t= 100 (red dots).\n51(a)\n (b)\n(c)\n (d)\n(e)\n (f)\nFig. 18. (Nonlinear two-stream instability) Fine scale details of the distribution\nfunctionf100(E), depicting a cusp formation near the trapping boundary at E= 0\n(top left and right) . Near the minimum value of E,f100is observed to greatly steepen\n(middle left and right) . The maximum of f100is achieved at about E= 0:71(bottom\nleft and right) . In all plots red dots correspond to negative velocities and green dots\nto positive velocities, v. No asymmetry in the sign of vis evident.\n52(a)\n (b)\n(c)\n (d)\nFig. 19. (Nonlinear two-stream instability) Plot of f100(E) nearE= 1:3 indicating\nmild splitting (top left) , with stronger splitting seen at E \u0019 1:5(top right) . The\nsplitting continues to larger values of E(bottom left and right) . Red dots corre-\nspond to negative velocities and green dots to positive velocities v, with no evident\nasymmetry in the sign of v.\n53Fig. 20. (Nonlinear two-stream instability) Plot of E2versus \b for the code output\nat the times indicated. Note, E2is a graph over \b even for the unsaturated states,\ni.e. timest\u0014100.\nFig. 21. (Nonlinear two-stream instability) Plot of E2vs. \b as given by (77) with\n(78) for the model f\ftof (72) with the parameters of Fig. 12, and code output (red),\nboth at time t= 100.\n545 Conclusions\nIn this paper we have demonstrated the viability of the DG method with up-\nwinding for solving the Vlasov-Poisson system. The method was described in\ndetail for general polynomial bases of elements. Several examples were com-\nputed to demonstrate the utility of the method using a piecewise uniform\nbasis. A convergence study was performed for the simple advection problem,\nindicating the degree to which \flamentation can be resolved, and high res-\nolution computations of the linearized Vlasov-Poisson system were also per-\nformed. For linearization about a Maxwellian equilibrium, computation results\nwere seen to compare very well with theoretical calculations of Landau damp-\ning. Computations for Lorentzian equilibria were also presented and the wave\nnumber dependence of the Landau damping rate was veri\fed, apparently for\nthe \frst time in a simulation. The problems of nonlinear Landau damping\nand the symmetric two-stream instability were then considered, and results\nwere compared to previous calculations. In both cases, constants of motion\nwere monitored and the error was seen to be monotonic, due to nonlinear\ncoupling. Also, reasons for the lack of numerical error in the form of recur-\nrence in our results were discussed. High resolution features of the distribution\nfunction were displayed for the long time BGK state that was reached in the\ntwo-stream calculations. Comparisons between theory and code results were\nmade, particularly for the end BGK state, in an attempt to provide a high\nstandard for truth in numerical code work.\nThere are many future directions suggested by this work, some of which are\nongoing. In a couple of upcoming publications [18,17] the authors will report\non computations using higher order polynomial bases with an improved tem-\nporal integrator and a limiter that maintains positivity of the distribution\nfunction, as well as a study of numerical recurrence for these schemes. Indeed,\n[18] contains a thorough study of dependence of recurrence times and recur-\nrence amplitudes on the mesh size in x;vand the time step \u0001 t, for Vlasov\nlinear advection, as well as dependence on the order, type, and choice of ba-\nsis functions for the DG scheme. It is quite remarkable, that for the lowest\norder polynomial space of piecewice constant functions, one can prove that re-\ncurrence occurs, but the recurrence amplitude will decay, thereby suggesting\nvalue in choosing lower order functions. This fact signi\fcantly improves the\nperformance criteria originally developed in [11].\nAs noted in the Introduction, the DG method can be run easily on nonuniform\nmeshes, and in work not reported here we have seen this to be the case. This is\na \frst step toward producing an adaptive code. Similarly, parallel implemen-\ntation awaits. There are many physical applications within reach, such as the\ntreatment of a plasma diode which requires physical boundary conditions and\nthe inclusion of various collision operators of relevance to plasma dynamics.\n55Since the Vlasov-Poisson system serves as a test bed for more sophisticated\nkinetic theories, it is our opinion that the DG method proposed here is an\nattractive alternative for many plasma physics computations.\nAcknowledgment\nR. E. Heath was supported by a Research and Development Grant from Ap-\nplied Research Laboratories, The University of Texas at Austin, Austin, TX;\nI. M. Gamba was supported by NSF grants DMS-0807712 and FRG-0757450,\nwhile P. J. Morrison was supported by U.S. Dept. of Energy Contract #\nDE-FG05-80ET-53088. Support from the Institute from Computational En-\ngineering and Sciences at the University of Texas at Austin is also gratefully\nacknowledged. We all thank Clint Dawson for his technical help and Wenqi\nZhao for her technical support with data post-processing. I. M. Gamba and\nP. J. Morrison thank Cedric Villani for interesting discussions related to his\nrecent work on Landau damping.\nA Appendix: Dispersion Relations\nHere we analyze the dispersion relations for the Lorentzian and two-stream\nequilibria given by (15) and (66), respectively. With the assumption that\nf(x;v;t ) =feq(v) +\u000ef(x;v;t ) and\u000ef(x;v;t )\u0018exp(ikx\u0000i!t) and the scaling\nof variables described in Section 2, the plasma `dispersion relation' is given by\n\u000f(k;!) = 1\u00001\nk21Z\n\u00001f0\neq(v)\n(v\u0000!=k)dv; (A.1)\nwherekis assumed to be real and positive and for bounded systems k=\n2\u0019n=L withnan integer. Landau damping arises upon analytically continuing\nthis expression into the lower half !-plane and thus deforming the contour\nof integration [42]. For integration along the real axis, stable and unstable\neigenmodes (and embedded modes if they exist) satisfy\n\u000f(k;!) = 0; (A.2)\nand it is this quantity we wish to investigate for both discrete modes and\nLandau `modes', the latter obtained by analytic continuation into the lower\nhalf!-plane. In the latter case, the solution !(k) of (A.2) characterizes the\nasymptotic-time behavior of mode Ek(!;t) of the electric \feld. Speci\fcally, if\n!=!R+i\r, where!Rand\rare real-valued, then \r <0 is the time-asymptotic\nrate of decay of the mode Ek(!;t) and!Ris the frequency of oscillation.\n56For the Lorentz equilibrium distribution function of (15),\nf0\neq(v) =\u00002\n\u0019v\n(v2+ 1): (A.3)\nUpon de\fning u=!=kand substituting (A.3) into (A.1), we obtain\n\u000f(k;ku ) = 1 +2\n\u0019k2Z1\n\u00001v\n(v2+ 1)2(v\u0000u)dv; (A.4)\nwhich expresses \u000fas a function uandk. Evaluation of this integral is a straight-\nforward application of Cauchy's theorem. Assuming uis in the upper half\nplane, poles exist at v=iandv=u, with corresponding residues of the\nintegrand being Ri=i=[4(u\u0000i)2] andRu=u=(u2+ 1)2. Summing over the\nresidues gives the dispersion function\n\u000f(k;ku ) = 1\u00001\nk2(u+i)2: (A.5)\nUpon setting \u000f= 0, since!=ku, we obtain\n!=\u00061\u0000ik=!R+i\r; (A.6)\nwhich demonstrates, contrary to our assumption, that there are no discrete\nmodes in the upper half u-plane. This is consistent with the well-known result\nthat equilibrium distribution functions that are monotonic in v2possess no\ndiscrete growing or damped modes. However, continuing (A.5) into the lower\nhalf plane gives Landau damping with \r(k) =\u0000kandj!R(k)j= 1 for the\ntime asymptotic behavior.\nFor the two-stream equilibrium of (66), (A.1) gives rise to instability, i.e., for\nthis case there is in fact a root with uin the upper half plane. For compu-\ntational reasons it is convenient to write \u000fin terms of the plasma Z-function\nwhich is related to both the Hilbert transform and the error function (see e.g.\n[34]). Upon inserting (66) and performing some manipulation, (A.1) can be\nwritten as\n\u000f= 1\u00002\nk2[J1(z) +J2(z)]; (A.7)\nwhere\nJ1(z) =1p\u0019Z1\n\u00001e\u0000w2wdw\nw\u0000z; J 2(z) =1p\u0019Z1\n\u00001e\u0000w2w3dw\nw\u0000z; (A.8)\nandz=u=p\n2. With the standard de\fnition of the plasma Z-function,\nZ(z) =1p\u0019Z1\n\u00001e\u0000w2dw\nw\u0000z= 2ie\u0000z2Ziz\n\u00001e\u0000t2dt (A.9)\n57where in the \frst expression =(z)>0 and the value of Zfor=(z)<0 is\nobtained by analytic continuation, while the second expression is valid for all\ncomplexz. The second expression is desirable for computations. Also, some-\ntimes it is convenient to use the derivative formula\nZ0=\u00002(1 +zZ); (A.10)\nwhich is valid for all z. 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The observe d time series data with the Atmospheric\nImaging Assembly (AIA)telescopeonNASA’s Solar Dynamics Observatory (SDO)satelliteon2015March2,\nconsisting of 400 consecutive images with 12 seconds cadence in the 171˚A pass band is analyzed for evidence\nof transversaloscillations along the coronal loops by Lomb-Scarg leperiodgram. In this analysis signatures of\ntransversal coronal loop oscillations that are damped rapidly were found with dominant oscillation periods\nin the range of P = 12 .25−15.80 minutes. Also, damping times and damping qualities of transversal coronal\nloop oscillations at dominant oscillation periods are estimated in the ran ge ofτd= 11.76−21.46 minutes\nandτd/P = 0.86−1.49, respectively. The observational results of this analysis show t hat damping qualities\ndecrease slowly with increasing the amplitude of oscillation, but period s of oscillations are not sensitive\nfunction of amplitude of oscillations. The order of magnitude of the d amping qualities and damping times\nare in good agreement with previous findings and the theoretical pr ediction for damping of kink mode\noscillationsby dissipation mechanism. Furthermore, oscillationof loop segmentsattenuate with time roughly\nast−αthat magnitude values of αfor 30 different segments change from 0.51 to 0.75.\nKeywords: Sun, corona; Sun, damping of transversal coronal loop oscillation s; Sun, flare\n1. Introduction\nCoronal seismology was first introduced in the 20th century by Uch ida (1968, 1970). Then, it has been\nsuccessfully used to probe of physical parameters in the solar cor ona. There are now many studies on\nobservations and the theory of solar coronal seismology (see, e.g.Erd´ elyiet al.2003; Wang 2004; As-\nchwanden 2004, 2006; Roberts 2004; Nakariakov and Verwichte 2 005; Andries et al.2009; Taroyan and\nErd´ elyi 2009; Ruderman and Erd´ elyi 2009). Observations of so lar corona with space telescopes such as\nHinode, Yohkoh, Transition Region AndCoronal Explorer (TRACE), Solar Terrestrial Relations Observatory\n(STEREO), Solar and Heliospheric Observatory (SoHO) and SDO have helped us to study in detail trans-\nverseoscillationsof coronalloops. Today, there are many observ ationsoftransversemotions ofcoronalloops.\nFor example, Nakariakov et al.(1999), Aschwanden et al.(1999) and Verwichte et al.(2004) observed fast\nkink mode in coronal loops by TRACE. Roberts et al.(1984), Asai et al.(2001), Melnikov et al.(2005) and\nAschwanden et al.(2004) observed fast sausage mode oscillations in radio wavelength s. Morton et al.(2011)\nobservedsymmetricsausagemodeinacoronalloopwith Rapid Oscillations in the Solar Atmosphere (ROSA)\ninstrument. Kohutova and Verwichte (2016) studied the kinematic s and oscillations of the coronal rain by,\nSOT/Hinode, Interface Region Imaging Spectrograph (IRIS) and AIA/SDO. They detected two distinct\ntransverseoscillationsregimes: large-scaleoscillations and small-sc ale oscillations. The large-scaleand small-\nscale oscillations were exited by a transient mechanism and a continuo us drive mechanism, respectively.\nDepartment of Physics, University of Qom, Qom University Bl vd Alghadir, P.O. Box\n3716146611, Qom, I. R. Iran.\nemail:a.abedini@qom.ac.ir\nemail:aabedini448@yahoo.com\np. 1A. Abedini\nGoddard et al.(2016) estimated the physical parameters of 120 individual kink os cillations of coronal loops\nandfoundthatperiodofoscillationscaleswiththecoronallooplengt h,andtheinitialamplitudeofoscillation\nis determined by the initial displacement of coronal loop. Also, they f ound that the magnitude of kink speed\nof coronal loops are in the range C k= (800−3300) kms−1. Nakariakov et al.(1999), Aschwanden and\nSchrijver 2002, and Aschwanden et al.(2002) observed an important characteristic of transversal co ronal\nloop oscillations that the amplitude of oscillations were damped strong ly. A large number of theoretical\nstudies investigating the damping time of the transverse oscillations in a coronal loop have focused on the\neffects of phase mixing, resonant absorption, gravitational stra tification, magnetic field divergence (see, e.g.\nAschwanden et al.2003; Safari et al.2007; Ebrahimi and Karami 2016). In general, phase mixing and\nresonant absorption are shown to be the main physical mechanisms for damping of the standing transverse\noscillationsofcoronalloops.Themajorityofpreviousstudiesontr ansversalcoronallooposcillationsassumed\nthat coronal loop is in a stationary state. In addition, dynamics of a coronal loop mainly were determined\nby the normal modes (see, e.g.Edwin and Roberts 1982; 1983; Spruit 1982; Cally 1986, 2003). How ever, the\nnormal modes of a coronal loop are not the full picture of the loop d ynamics. Only a few limited studies have\nbeen done about the time-dependent problem of the excitation and damping of coronal loop oscillations. For\nexample, Murawski and Roberts (1993a) studied the time evolution of driven waves in coronal loops which\nwere approximated by smoothed slabs, and they investigated some properties of sausage and kink modes\nof these loops. Also, they studied the temporal signatures of impu lsively generated fast waves in the solar\ncoronalloops by means of numericalsimulations (1993b,1993c).R ae and Roberts (1982), Uralov (2003) and\nTerradas et al.(2005) investigated the role of a traveling pulse generated by a loca lized event on coronal loop\noscillations. These authors claimed that the wake of the propagatin g pulse could be responsible for decay\nof the observed oscillations. Many continuous observations of cor onal loops with satellite telescopes reveal\nthat the transversal coronal loop oscillations are accompanied by a nearby flare in an active region (see,\ne.g.Schrijver and Brown 2002; Hudson and Warmuth 2004; Aschwande net al.1999, 2002; Aschwanden\nand Schrijver 2011; Nistic` o et al.2013; Verwichte et al.2013; Anfinogentov et al.2015; Hindman and\nJain 2014; Pascoe et al.2016). As a result, some authors claimed that excitation and dampin g of transversal\noscillation of coronal loops may be caused by the wake of a travelingd isturbance from nearby flares (see, e.g.\nTerradas et al.2005). Recently, Zimovets and Nakariakov (2015) studied the exc itation of kink oscillations\nof coronal loops by flares and lower coronal eruptions/ejections (LCEs) of some unstable configuration of\ncoronal magnetised plasma, such as a filament, a system of corona l loops, a magnetic flux rope and a\ncoronal mass ejection from a nearby flare site by analyzing 169 kink oscillation loops in 58 events. In this\nanalysis, it was established that nearby lower coronal eruptions is t he most probable mechanism for exciting\ndecaying kink oscillations of coronal loops compared to a blast shock wave from a flare. In this paper,\nI attempted to provide more insight into the excitation, damping and temporal evolution of transversal\noscillations in solar coronal loops, and also made an attempt to deduc e a quantitative relation between\ndamping time, damping quality, oscillation amplitude, dissipation mechan ism and the wake phenomenon\nthat is produced by a traveling disturbance in solar coronal loops. F or this purpose, the nature of transversal\ncoronal loop oscillations (as observed with AIA/SDO) is analyzed by h istogram equalization technique and\nLomb-Scargleperiodgramalgorithmthat showsa clearlybetter det ection accuracyandefficiency foranalysis\nofnoisytime seriesdata.The magnitudevaluesofphysicalparamet ersoftransversalcoronallooposcillations\nsuch as period of oscillation, decay time and damping quality are extra cted in 171 ˚A passband. Moreover,\nboth observationally and theoretically the quantitative dependenc e of damping of transversal coronal loop\noscillation on dissipation mechanisms and the wake phenomenon are st udied. The rest of this paper is\norganized as follows. In Section 2 the observations are presented , in Section 3 the method of data analysis\nare described, in Section 4 the physical parameters of transvers al coronal loop oscillation are estimated, and\nthe theoretical considerations are given in Section 5. Finally, discus sion and conclusions are presented in\nSection 6.\n2. Observations\nThe dataused for this study areEUVimagesofthe solarcoronalloo ps,takenby AIA/SDO 171 ˚A. The initial\nraw images had to be cleaned and calibrated before using. The images used here are at level 1.5. For images\nwith level 1.5, co-alignment, flat-fielding,vignetting, filter and bad-p ixel/cosmic-raycorrections have already\np. 2Observations of excitation and damping of transversal osci llation in coronal loops\nSolar-X(arcsec)-1200 -1000 -800Solar-Y(arcsec)\n-400-300-200\n2015-03-02,09:42:23 UT,AIA/SDO 171 channel\nSolar-X(arcsec)-1200 -1000 -800Solar-Y(arcsec)\n-400-300-200\nFlare center\nOscillating Loop\nFigure 1. A snapshot of the flare and near-by loop, like a semi circle, ob served by AIA/SDO in 171 ˚A on 2015 March 2,\n09:42:23 UT. The locations of the flare and loop are shown with yellow arrows in the left panel. To obtain the transversal\ncoronal loop oscillations, the part of area between two circ ular arcs containing the loop is subdivided into 30 sectoria ls that are\nperpendicular to the visible loop strands. These sectorial areas are then joined by successive cross sections, three pi xels wide\n(see one of these sectorials (zoomed) in the bottom of the rig ht panel).\nbeen applied. Also, these images have been centered, rotated so t hat Sun’s North Pole is up in the image and\nrescaleddowntoaplatescaleof0.6arcsec/pixel.Additionally,thediff erentialrotationeffectisalsocorrected.\nThe data under analysis here is taken on 2015 March 2, from 09:00:00 UT until 10:30:00 UT and it consists\nof a series of 1100 ×600 pixels, sun-centered, subfield images on the 171 ˚A (Fe IX) passband with a time\ncadence of 12 seconds. The continuous observations of the selec ted active region loop and its environment\nmonitored with Helioviewer, JHelioviewer, Large Angle and Spectrometric Coronagraph (LASCO) onboard\nthe SoHO, CME catalogue and CACTus CME list show that there is a flar e and a LCE in the same parental\nactive region. The solar flare that has been located at position (x flare,yflare) = (−960′′,−192′′) near the loop\nis started around 2015 March 2, 09:09:00 UT and ended around 2015 March 2, 09:31:12 UT with a peak\nflux of 1182.3 erg /(cm2s) that is occurred at 09:17:12 UT. Also, the LCE that has been locat ed at position\n(xLCE,yLCE) = (−1006.8′′,−195.6′′) near the loop started around 2015 March 2, 08:21:07 UT and ended\naround 2015 March 2, 09:01:07 UT. The left panel of Figure 1 shows a snapshot of the flare and a nearby\nloop in 171 ˚A on 2015 March 2, 09:24:23 UT that we are analyzing in this paper. The locations of the flare\nand loop are shown with yellow arrows. To obtain the transversal co ronal loop oscillations, the part of area\nbetween two circular arcs containing the semi circle-like loop is subdiv ided many distinct sectorials that are\nperpendicular to the visible loop strands (Figure 1, right panel). The se sectorial areas are then joined by\nsuccessive cross sections, three pixels wide. Also, one of the sect orial segments is zoomed in the lower part of\nthe right panel. Figure 1 shows thirty distinct sectorial areas that are analyzed in detail in the next sections.\n3. Data Analysis\nIn order to extract the oscillatory nature of transversal coron al loop oscillation, the histogram of images is\nequalized into histogram of first image by Image Processing Toolbox o f Matlab. Histogram equalization is\na technique for adjusting image intensities to enhance contrast. T ypical examples of time −distance plots\np. 3A. Abedini\nFigure 2. Typical examples of time −distance plot of original intensity (a) for sectorial 10 are shown in the top panel before\nhistogram equalization. Enhanced time −distance plot of intensity (b) for sectorial 10 are shown in t hebottom panel after\nhistogram equalization. Location of loop enclosed by a red rectangle in the panels.\nof original (a) and histogram equalization (b) intensities time series o f images for sectorial 10 are shown in\nthe top and bottom panels of Figure 2, respectively. Location of th e loop enclosed by a red rectangle in the\npanels. The part of area between two circular arcs containing the s emi circle-like loop is subdivided into 30\nsectorial areas that are perpendicular to the visible loop strands. Moreover, each of these sectorial areas are\njoined by successive cross sections, about three pixels wide. The m ean intensity of cross sections as function\nof time is calculated by dividing the total intensity of cross sections in to the number of pixels at each cross\nsections. A background intensity must be subtracted from the or iginal intensity of cross sections to enhance\nthe contrast of time −distance plot and loop axis displacement. Different enhancement algo rithms are used\nto remove the background for an improved visualization of transve rsal displacements of the loop axis in a\ntime−distance plot (see, e.g.Aschwanden and Schrijver 2011; Yuan and Nakariakov 2012; Thre lfallet al.\n2013; Abedini 2016). Here, following Aschwanden and Schrijver (2 011), a running-minimum difference in\nthe form\ni(nc,sk,ti) =I(nc,sk,ti)−Ib, (1)\nIb= min[I(nc,sk,ti−j),...,I(nc,sk,ti+j)],\nis used to remove the background intensity of the time −distance plots. This running-minimum algorithm\nsubtracts a running-minimum obtained within a time interval with a leng th of 2jfrom intensity time series\nof macropixels that are located in the time −distance plots of the sectorial areas. Where ncis the number of\nsectorial areas, skis the location of the kth macropixel at a specific sectorial area, tiis time of ith frame and\nti−j,...,ti+jrepresent a time interval of 2 jframes that symmetrically are placed around ith frame in every\ntime slice ( nc= 1,k= 7,i= 2 corresponds to the seventh macropixel along the sectorial 1 a tt= 12s). The\nwidth of selected arcs on the loop are varied between three to six ma cropixels. An appropriate background\nis subtracted from the original intensities. Sufficiently enhanced tim e−distance plots of sectorial areas are\nfound by setting the j= 9,...,12. For example, time −distance plots of normalized background-subtracted\nintensities after histogram equalization ( in(sk,ti) =i(sk,ti)/max[i(s1,ti),...,i(send,ti)]) for thirty sectorial\nareas by running-minimum algorithm are shown in Figures 3 and 4. In or der to improve visualization of\ntransversal loop oscillation, low oscillation periods (P ≤36s) are filtered from intensities after running-\nminimum subtraction in the thirty time −distance plots. The time −distance plots of intensities along the\nsectorial areas show that transversal coronal loop oscillations a re occurred in the range of 13 to 43 minutes.\np. 4Observations of excitation and damping of transversal osci llation in coronal loops\n                               \n                               \nnc=1,j=11-AIA/SDO 171 channel \n20 40 60 80Distance(Mm)5101520\nnc=2,j=12-AIA/SDO 171 channel\n20 40 60 805101520\nnc=3,j=12-AIA/SDO 171 channel\n20 40 60 805101520\nnc=4,j=12-AIA/SDO 171 channel\n20 40 60 80Distance(Mm)5101520\nnc=5,j=11-AIA/SDO 171 channel\n20 40 60 805101520\nnc=6,j=12-AIA/SDO 171 channel\n20 40 60 805101520\nnc=7,j=9-AIA/SDO 171 channel\n20 40 60 80Distance(Mm)5101520\nnc=8,j=9-AIA/SDO 171 channel\n20 40 60 805101520\nnc=9,j=13-AIA/SDO 171 channel\n20 40 60 805101520\nnc=10,j=11-AIA/SDO 171 channel\n20 40 60 80Distance(Mm)1020\nnc=11,j=9-AIA/SDO 171 channel\n20 40 60 805101520\nnc=13,j=9-AIA/SDO 171 channel\nTime(min)20 40 60 80Distance(Mm)5101520\nnc=15,j=9-AIA/SDO 171 channel\nTime(min)20 40 60 805101520\nnc=12,j=9-AIA/SDO 171 channel\n20 40 60 805101520\nnc=14,j=9-AIA/SDO 171 channel\nTime(min)20 40 60 805101520\nFigure 3. Running-minimum algorithm to visualize the transversal co ronal loop oscillations in time −distance plots of inten-\nsities (in(sk,ti) =i(sk,ti)/max[i(s1,ti),...,i(send,ti)]) in 171 ˚Achannel of AIA for the sectorial areas 1-15 (time range is 201 5\nMarch 2, 9:00:00-10:30:00 UT). In order to improve visualiz ation of transverse displacements of loop low oscillation p eriods\n(P≤36 s) are filtered from intensities after running-minimum su btraction in the time −distance plots. Clearly, transversal\ncoronal loop oscillations is occurred in the panels in the ra nge of 15 to 45 minutes. The number of sectorial area ( nc) and\nmagnitude value of jare shown on top of each panel.\nClearly, the enhanced time −distance plots of sectorial areas are found by setting the j= 9,...,12. From\nFigures 3 and 4 it can be seen that the histogram equalization techniq ue has enhanced the contrast of\ntransverse displacements of loop compared to the previous studie s (see,e.g.Aschwanden and Schrijver 2011;\nGoddard et al.2016). Moreover, in these figures damping and the start times of t ransverse displacements\nare clearly visible.\n4. Estimation of the physical parameters of transversal coronal loop oscillatio n\nA coronal loop can be oscillated in various directions. A basic type of c oronal loop oscillation is called\ntransverse oscillation. The transverse oscillation can be caused by sausage and kink type modes or it can be\ncausedbythe wakeofatravelingdisturbancefrom nearbyflares. Here,the physicalparametersoftransversal\ncoronal loop oscillations that are powerful diagnostic tools for cor onal seismology are estimated.\n4.1. The measurement of transversal coronal loop oscillati on periods\nIn this subsection, oscillation periods of transversal motion of loop segments are estimated. Transversal\ndisplacements of the loop have been interpreted in terms of the tra nsverse standing kink mode or the wake\nof a traveling disturbance of nearby events, since they trigger tr ansversal displacements in the loop axis.\nIn the Figure 5, intensity time series, Lomb-Scargle power spectra l and probability of power spectral of\nmacropixel (13,10) are displayed in the panels a), b) and c), respec tively. Intensities of macropixels may be\nvaried by the transversal and/or the longitudinal oscillations. So, in order to infer the transversal coronal\np. 5A. Abedini\nnc=16,j=9,  AIA/SDO 171  channel\n20 40 60 80Distance(Mm)5101520\nnc=17,j=9,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=18,j=11,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=19,j=9,  AIA/SDO 171  channel\n20 40 60 80Distance(Mm)5101520\nnc=20,j=9,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=21,j=9,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=22,j=9,  AIA/SDO 171  channel\n20 40 60 80Distance(Mm)5101520\nnc=23,j=9,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=24,j=11,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=25,j=11,  AIA/SDO 171  channel\n20 40 60 80Distance(Mm)5101520\nnc=26,j=11,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=27,j=11,  AIA/SDO 171  channel\n20 40 60 805101520\nnc=28,j=11,  AIA/SDO 171  channel\nTime(min)20 40 60 80Distance(Mm)5101520\nnc=29,j=11,  AIA/SDO 171  channel\nTime(min)20 40 60 805101520\nnc=30,j=11,  AIA/SDO 171  channel\nTime(min)20 40 60 805101520\nFigure 4. The same as in Figure 3 but for sectorial areas 16-30.\nTime(min)0 10 20 30 40 50 60 70 80i(arbitrary units) -101a)Intensity\nperiodicity(min)0 2 4 6 8 10 12 14 16 18 20 22 24Power\n00.51b) Lomb-Scargle Power Spectrum\nPeriodicity(min)0 2 4 6 8 10 12 14 16 18 20 22 24Probablity00.51d) Probabilites\nFigure 5. a) Temporal variation of histogram-equalized and backgrou nd-subtracted intensity time series of macropixel (13,10)\nare shown as function of time. b) Lomb-Scargle power spectra l density of time series intensity is shown as function of osc illation\nperiods. c) The true-alarm probability of peaks in the Lomb- Scargle power spectra are shown as function of oscillation p eriods.\nloop oscillations as function of time in each time −distance plots, a Gaussian function of the form\nG(s,t) =A0(t)exp[−(s−µ(t))2\n2σs(t)2]+B0(t), (2)\np. 6Observations of excitation and damping of transversal osci llation in coronal loops\nPeriodicity(min)0 2 4 6 8 10 12 14 16 18 20 22 24Power\n00.51c) Lomb-Scargle Power Spectrum\nPeriodicity(min)0 2 4 6 8 10 12 14 16 18 20 22 24Probability00.51d) ProbabilitiesDistance(Mm)8 9 10 11 12 13 14 15 16 17 18 19 20 21is/ismax\n00.511.5a) nc=10\nTime(min)0 10 20 30 40 50 60 70 80D(t)(Mm)\n-6-4-20246b) Transversal displacements of the loopA0=1.04,µ=14.17(Mm),\nσs=0.80(Mm),B0=0.02t=20(min)\nFigure 6. a) Histogram-equalized, background-subtracted and norma lized intensity profile after running-minimum subtraction\n(is(s,t)/ismax(s,t)) along the sectorial 10 at t=20 min are plotted versusthe projected distance. This profile is fitted by an\nGaussian function, G(s,t) =A0(t)exp(−(s−µ(t))2\n2σs(t)2) +B0(t) (red line), and fit parameters are shown on the top part of the\npanel. b) Transversal displacements of the loop axis D(t) = (µ(t)−< µ(t)>) for sectorial 10 are shown as function of oscillation\nperiods. c)Lomb-Scargle power spectral density of D(t) isshown as function of oscillation periods. d) The true-al arm probability\nof peaks in the Lomb-Scargle power spectra are shown as funct ion of oscillation periods.\nis fitted to the histogram-equalized, background-subtracted an d normalized intensities along the paths by\nMatlab Statistics Toolbox Curve Fitting and Distribution Fitting. This fi tting yields the peak of intensity\nprofileA0(t), location of the loop axis µ(t), standard deviation σs(t), and mean background intensity B0(t)\nalong a specific path in each time, respectively. The oscillation periods of loop axis are calculated by\nLomb-Scargle power spectral densities of D(t) = (µ(t)−< µ(t)>) time series. For example, histogram-\nequalized, background-subtracted and normalized intensity profi le relative to the peak of intensity profile\n(is(s,t)/ismax(s,t)) for sectorial 10 at t= 20 minutes are plotted versus the projected distance in the pane l\na) of Figure 6. The normalized intensity profiles (blue plus signs) is fitt ed by an Gaussian function (red\nline), and fit parameters are displayed in the top part of the panel. T ransversal displacements of the loop\naxis (D(t)) for sectorial 10 are shown as function of oscillation periods in the panel b) of Figure 6. There\nare a number of techniques that allow to estimate the statistical sig nificance of the detected periodicity,\nfor example the well-known Lomb-Scargle method. This periodgram a lgorithm gives a better detection\naccuracy and efficiency for analysis of noisy time series data. Also, t he probability of peaks in the power\nspectraldensityofnoisytimeseriesdatahappening bychancecan be calculated.Furthermore,this algorithm\navoidspossibleincorrectresultsthat mayoccurfromreplacement ofmissingtime seriesdatabyinterpolation\nmethods. For example, the Lomb-Scargle power spectral densitie s of transversal displacements for sectorial\n10 are shown as function of oscillation periods in the panel c) of Figur e 6. Lomb-Scargle power spectral\ndensities of transversal displacements show that there are indee d several statistically dominant peaks of\npower spectra in the range of P = 2 −20 minutes. The true-alarm probability of significant peaks of power\np. 7A. Abedini\nspectrain the paneld) showthat onlyP ≈3,5and14minutes aresignificantoscillationperiod.Comparisons\nof probability of peaks before (panel c) of Figure 5) and after run ning-minimum subtraction (panel d) of\nFigure 6) reveal that significant peaks are quite similar. Furthermo re, probability of significant peaks in\n30 different sectorial areas reveal three or four significant oscilla tion periods. One dominant peak around\nP = 14 minutes is present in all Lomb-Scargle power spectral densitie s of sectorial areas. But, other peaks\nare completely different in the power spectral densities.\n4.2. The measurement of the damping of transversal loop osci llation\nHere, damping times of transversal loop oscillation are measured. A lso, the dependence of damping times\non amplitude and also on its periodicity are investigated. Lomb-Scarg le power spectral densities of D(t)\nshow that spectral components of the transversal displacemen t along the thirty different sectorial areas are\nsignificant at some specified oscillation periods in the range of P = 2 −20 minutes. But, oscillation periods\naround the P = 14 minutes are almost dominant in all power spectra. A ccordingly, the damping time of loop\noscillationalongthe30differentsegmentsofcoronallooparemeasu redonlyforcommondominantoscillation\nperiods. So, in the Lomb-Scargle power spectral density dominant components of loop axis displacement at\neach sectorial areas are selected by multiplying each of the compon ents into the Gaussian filter of\na′\nj=Nf/summationdisplay\ni=1aiexp[−(νi−νj)2\n2σ2ν], (3)\nwheretheindices iandjaretherankoftheharmonics, Nfisthetotalnumberofthespectralcomponents, σν\nis standard deviation, νiandνjare the frequency of the ith andjth harmonics, aianda′\njare theith andjth\nFourier coefficient before and after transformation, respective ly. Two different kinds of theoretical damping\nmechanisms have been proposed for transversal coronal loop os cillation: (1) The damping of transversal\ncoronal loop oscillations may be produced by the dissipation mechanis m. Generally, resonant absorption\nis found to be the dominant damping mechanism compared to the othe r dissipation mechanism (see, e.g.\nAschwanden et al.2003). (2) The damping may be produced by the wake of the travelin g disturbance (see,\ne.g.Rae and Roberts 1982; Uralov 2003; Terradas, Oliver, and Balleste r 2005). A localized disturbance\nfrom a solar flare propagates toward the loop. As a consequence, coronal loop axis oscillates transversally\nat the external cutoff frequency, and it is damped approximately w ith time as t−1/2by the wake of the\ntraveling disturbance. To investigate the damping nature of trans versal coronal loop oscillation, the vari-\nation of filtered displacement ( D(t)) with time are fitted by two different forms of damped sine function ,\nf1(t) = a 1exp(−(t−t0)/τd)sin(2π(t−t0)/P1−φ10) and f 2(t) = a 2(t−t0)−αsin(2π(t−t0)/P2−φ02). For\nexample, time −distance plot for sectorial 10 is displayed in the top panel of Figure 7 . Also, the extracted\namplitudes (blue plus signs) at a dominant oscillation period (P = 14 min) a re fitted by two different forms\nof damped sine functions and fit parameters are written in the top a nd bottom part of panel of Figure\n7, respectively. The extracted physical quantities such as period of oscillation (P), damping time ( τd),\ndamping quality ( τd/P), phase angle ( φ) and magnitude values of α, for filtered background-subtracted\nintensity at a sequence of 400 images with a time cadence of 12 secon ds are listed in Table 1 for 30 different\nsegments of the coronal loop. The results of this analysis reveal t hat the magnitude of damping time and\ndamping quality for dominant oscillation periods of segments (P = 12 .25−15.80 ) along the coronal loop\nare within the range of τd≃11.76−21.46 min. Oscillation of loop segments attenuates with time roughly\nast−αthat the magnitude of αchange from 0.55 to 0.75. The damping quality of loop segments are in\nthe range of τd/P≃0.86−1.49 which clearly indicates that the damping regime is strong for oscillat ions.\nAlso, these results show that damping times increase with increasing oscillation period. Damping properties\ndecrease slowly with increasing amplitude, but they are not sensitive to the oscillation periods. This range\nof oscillation periods and damping times of loop segments are in good ag reement with previous findings (see,\ne.g.Nakariakov et al.1999; Aschwanden et al.2002; Verwichte et al.2004; Wang and Solanki 2004)\n4.3. The measurement of speed of the flare-generated disturb ance and the lower coronal\neruption/ejection\nAssociation of transversal coronal loop oscillations with a hypothe tical driver of loop oscillations such as a\nflare, an LCE or a CMEs must be investigated in order to identify a rela tionship between the excitation\np. 8Observations of excitation and damping of transversal osci llation in coronal loops\nTime(min)15 20 25 30 35 40 45 50 55 60 65 70 75 80d(Mm)\n-10-8-6-4-20246810\nnc=10,j=11-AIA/SDO 171 channel\nTime(min)10 20 30 40 50 60 70 80Distance(Mm)510152025\na1 = 4.92 (3.15, 5.93), τd1= 16.54 (14.82,19.26)\nP1= 14.36 (13.80,14.68), φ01 = 3.28(2.88,3.32)\na 2= 4.77 (3.26, 6.83),  α= 0.52(0.37, 0.66)\nP2 = 14.28  (13.91,14.44),   φ02= 3.30(3.08,3.27)\nFigure 7. Template time −distance plot for sectorial 10 is shown in the top panel . The variation of filtered transver-\nsal displacement ( D(t)) with time for sectorial 10 at a dominant oscillation perio d P = 14 min is shown as\nfunction of time in the bottom panel. Also, amplitudes are fit ted by two different forms of damped sine function,\nf1(t) = a 1exp(−(t−t0)/τd)sin(2π(t−t0)/P1−φ01)(redline),f 2(t) = a 2(t−t0)−αsin(2π(t−t0)/P2−φ02)(green line ),and\nfit parameters are shown on the top and bottom parts of panel, r espectively.\nand damping of transversal coronal loop oscillations with these phe nomena. For this purpose, the possible\nsolar flare, LCE, or CME associated with the oscillating loop were chec ked using Helioviewer, JHelioviewer,\nXRS/GOES, LASCO/SoHO, CME catalogue and CACTus CME list. By usin g these tools, a flare was found\nnear the selected loop at position (x flare,yflare) = (−960′′,−192′′).It started around 2015 March 2, 09:09:00\nUT and ended around 2015 March 2, 09:31:12 UT with a peak flux of 118 2.3 erg/(cm2s) at 09:17:12 UT.\nAlso, an LCE was detected near the limb at position (x LCE,yLCE) = (−1006.8′′,−195.6′′).This LCE started\naround 2015 March 2, 08:21:07 UT and ended around 2015 March 2, 0 9:01:07 UT. Moreover, in the vicinity\nof the loop, flare and LCE, CME was not detected one hour before a nd after of these events start times.\nOnly one CME was found near the oscillating loop with LASCO/SoHO. It s tarted around 2015 March 2,\n12:09:05UT havingaradialvelocity269km s−1, an angularwidth 20degreeand aposition angle118degree,\nrespectively. So, the speeds required for a hypothetical driver o f transversal loop oscillations to reach the\noscillation loop segments sites from the starting point were estimate d. Following Zimovets and Nakariakov\n(2015), these travel speeds were found in the range of v flare= 195−520km s−1and vLCE= 14−35km s−1\nby the ratio of the distances between the oscillating loop segments a nd sites of these events to the time\ndifferences between the starting times of LCE and flare with the osc illation loop segments. This result shows\nthat oscillation in a coronal loop cannot be excited by a low-speed sho ck wave of flare, since the values of the\nAlfv´ en speed are mostly more than 800 kms−1(see,e.g.Nakariakov and Ofman 2001; Stepanov et al.2012;\nDe Moortel and Nakariakov 2012; Zimovets and Nakariakov 2015).\n5. Theoretical considerations\nRecently,space-basedtelescopeobservationshaveshownthat transversalcoronallooposcillationsarestrongly\ndamped, with damping times of only 1-2 oscillation periods. Theoretica lly, two different kinds of physical\np. 9A. Abedini\nTable 1. Intensity profiles along 30 different sectorial areas of the c oro-\nnal loop for 400 images are fitted with two different forms of da mped\nsine function, f 1(t) = a 1exp(−(t−t0)/τd)sin(2π(t−t0)/P1−φ10) and\nf2(t) = a 2(t−t0)−αsin(2π(t−t0)/P2−φ02). Magnitudes of fit parameters, such\nas periodicity, damping time ( τd), damping quality ( τd/P), magnitude values of αand\nphase angle ( φ) of 30 sectorial areas that they are perpendicular to the vis ible loop\nstrands observed in the AIA/SDO 171 ˚A passband.\nnca1τd1P1τd1/P1φ01a2αP2φ02\n(Mm) (min) (min) (rad) (Mm) (min) (rad)\n13.0618.40 13.86 1.33 3.05 3.230.5513.26 3.04\n23.9816.99 14.59 1.17 3.19 4.400.6914.45 3.14\n33.8620.06 13.02 1.54 3.04 4.250.5913.31 3.10\n43.4221.46 14.68 1.460 3.32 4.450.6714.05 3.24\n53.3518.58 15.15 1.22 3.26 4.400.6515.13 3.17\n64.4616.67 14.32 1.16 3.22 5.440.7514.16 3.18\n74.6419.65 13.16 1.49 3.07 5.010.6213.25 3.14\n85.1018.09 13.85 1.31 3.09 5.170.5914.27 3.13\n95.1114.94 14.43 1.03 3.21 5.140.6415.03 3.14\n104.9216.54 14.36 1.15 3.29 4.770.5214.28 3.30\n116.2413.52 14.36 0.91 3.13 7.580.7514.67 3.14\n126.3212.16 14.14 0.86 3.05 7.440.7215.02 3.12\n136.7615.58 14.58 1.07 3.09 6.700.6014.27 3.14\n146.3614.63 15.20 0.96 3.01 6.570.6215.43 3.12\n155.6518.65 15.73 1.18 3.19 6.900.6415.75 3.16\n166.9815.70 15.20 1.03 3.29 6.450.6314.49 3.14\n176.6016.26 14.33 1.13 3.23 6.170.6512.99 3.14\n18- - - - - ----\n19- - - - - ----\n205.6416.97 12.29 1.38 3.18 6.180.5812.81 3.15\n215.7312.34 12.38 0.99 3.12 5.320.6612.25 3.14\n225.7315.48 13.97 1.11 3.29 6.250.7013.93 3.24\n236.1711.76 12.25 0.87 3.16 6.240.6712.12 3.14\n246.8015.00 12.56 1.17 3.22 5.900.6212.39 3.13\n256.1317.63 15.80 1.16 3.33 5.510.6415.13 3.14\n266.1616.30 13.92 1.17 3.06 5.740.6514.99 3.14\n276.6613.40 13.65 0.98 3.14 5.550.6114.13 3.14\n284.2316.39 13.91 1.18 3.26 3.200.5113.16 3.14\n294.4420.12 14.51 1.39 3.23 4.260.6114.20 3.14\n304.6420.29 13.91 1.46 3.28 3.800.6314.32 3.14\nmechanisms have been offered for damping of transversal corona l loop oscillations: damping by dissipation\nmechanism, and damping by the wake of a traveling disturbance.\n5.1. Dissipation mechanism\nSeveral dissipation mechanisms have been proposed for damping of transversal coronal loop oscillations.\nTheoretical studies investigating the decay of transverse oscillat ions of coronal loops have concentrated\non the effects of phase mixing, resonant absorption, lateral and f oot point wave leakage, gravitational\nstratification and magnetic field divergence. Generally, resonant a bsorption is found to be the relevant\np. 10Observations of excitation and damping of transversal osci llation in coronal loops\ndispassion mechanism for decay of kink mode oscillations (see, e.g.Aschwanden et al.2003 and references\ntherein). The damping quality of a kink mode oscillating loop with a thin bo undary layer due to resonant\nabsorption is given by (see, e.g.Goossens et al.2002; Ruderman and Roberts 2002; Van Doorsselaere et\nal.2004)\nτd/P = C(rloop\nlskin)(1+ρe\nρi\n1−ρe\nρi), (4)\nwhere C is a constant that depends on the radial density profile, r loopis the loop radius, l skinis the skin\ndepth of the coronal loop, P is the mode period 2 L/(2n−1)CK,nis harmonic number of the nth harmonic,\nρiandρeare the internal and external plasma density of corona loop, resp ectively.\n5.2. Wake of the traveling disturbance\nUralov (2003) and Terradas et al.(2005) studied the effect of a localized disturbance from a solar flar e on\ntransverse motion of a loop. In the zero- βplasma limit, they found that dynamics of coronal loop and its\nenvironment is governed by the Klein-Gordon equation with cutoff fr equency ωc= 2πcA/2Lby assuming\nthat loop and its environment with a uniform magnetic field is extended between z=±L/2. These authors\npointed out that transversal displacement of coronal loop decay s approximately with time as t−1/2by the\nwake of a traveling disturbance. The wakefield model does not take into account the field-aligned structuring\nof the corona, in other words, it does not have loops. But, the str ucturing could affect the wave behaviour\nquite significantly, as it was shown by Yuan et al.(2015). Decay of transversal loop oscillations due to\nthe wake of a disturbance is not related to any dissipation mechanism such as resonant absorption, phase\nmixing,etc. Also, these transversal displacement of coronal loops are not n ecessarily related to the kink\nmode oscillations, although standing kink oscillations can occur after the decay process by means of the\nwave packet energy.\n6. Discussion and Conclusions\nIn this paper, it is attempted to provide more insight into the excitat ion, damping and temporal evo-\nlution of transversal displacements of the loop axis near flares, an d also an attempt is made to deduce\na quantitative relation between oscillation period, damping time, damp ing quality, oscillation amplitude,\ndissipation mechanism, lower coronal eruption of some unstable cor onal magnetic field configuration and\nthe wake phenomena that is produced by a traveling disturbance in s olar coronal loops. For this purpose,\nthe nature of transversal oscillation of a coronal loop ( as observ ed with AIA/SDO) is analyzed by the\nhistogram-equalizationmethod and Lomb-Scargle periodgram algor ithm method that reveal a clearly better\ndetection accuracy and efficiency for analysis of noisy time series da ta. The values of physical parameters of\ntransversal corona loop segments are extracted in the 171 ˚A passband. Observational results obtained can\nbe summarized in the following main points:\n•The observational results of this study show that histogram equa lization technique has enhanced the\ncontrast of transversal coronal loop oscillations in the time −distance plots (s ee Figures 3 and 4)\ncompared to the previous studies (see, e.g.Aschwanden and Schrijver 2011; Goddard et al.2016).\n•The values of the physical parameters of transversal displaceme nts of the loop segments that have been\nestimatedin this analysisareingoodagreementwith previousfindings (see,e.g.Nakariakov et al.1999;\nAschwanden and Schrijver 2002; Aschwanden et al.2002; Verwichte et al.2004; Wang and Solanki\n2004; Goddard and Nakariakov 2016).\n•The Lomb-Scargle algorithm shows that spectral components of t he transversal displacement along the\n30 different sectorial areas measured are dominant at some specifi ed oscillation period in the range of\nP =2−20 minutes. But, probability of the observed peaks in the power spe ctral densities reveal that\noscillation periods around the P = 14 minutes are the most dominant co mponent in all power spectral\ndensities of segments.\np. 11A. Abedini\n•Themagnitudeofdampingtimesanddampingqualitiesofthedominanto scillationperiodofloopsegments\n(P = 12.25−15.80 minutes) that were determined for the exponential damping are measured in the\nrange of τd= 11.76−21.46 minutes and τd/P = 0.86−1.49,respectively.\n•The magnitude of damping qualities clearly indicate that damping regime is strong for oscillations. Also,\nthe observational results of this analysis reveal that damping qua lities decrease slowly with increasing\namplitude of the oscillation, but they are not sensitive to the oscillatio n periods.\n•Oscillation of loop segments attenuate with time roughly as that t−αvalues of αfor 30 different segments\nchange from 0.51 to 0.76. These values are in good agreement with th eoretical predictions. But, the\nestimated probability of peaks in Lomb-Scargle power spectral den sities of oscillation segments reveal\nthat there is indeed only one dominant peak in each power spectral d ensity. So, considering that the\ndamping of transversal coronal loop oscillation is induced by the wak e of a traveling disturbance is not\nphysically compatible with the dissipation mechanism.\n•Finally,consideringthatdecayingkink oscillationsareexcitedbyalowe rcoronaleruptionofsomeunstable\nconfiguration of coronal magnetised plasma is acceptable as it is com patible with a blast shock wave\ndue to a nearby flare.\nAcknowledgments The author thanks the anonymous referee for the very helpful comments and suggestions. This work\nwas carried out with the support of the University of Qom.\nDisclosure of Potential Conflicts of Interest The authors declare that they have no conflicts of interest.\nReferences\nAbedini, A.: 2016, Astrophys. Spa. 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J.: 2004, SOHO 13 Waves, Oscillations and Small-Sca le Transients Events in the Solar Atmosphere, ed. H. Lacoste\n(ESA SP- 547; Noordwijk: ESA, ESTEC). ADS.\nZimovets, I. V., Nakariakov, V. M.: 2015, Astron. Astrophys. 577, A4.DOI.ADS.\np. 13"    },    {        "title": "2309.08281v1.On_the_formation_of_singularities_for_the_slightly_supercritical_NLS_equation_with_nonlinear_damping.pdf",        "content": "arXiv:2309.08281v1  [math.AP]  15 Sep 2023ON THE FORMATION OF SINGULARITIES FOR THE SLIGHTLY\nSUPERCRITICAL NLS EQUATION WITH NONLINEAR DAMPING\nPAOLO ANTONELLI AND BORIS SHAKAROV\nAbstract. We consider the focusing, mass-supercritical NLS equation augmented with a nonlin-\near damping term. We provide sufficient conditions on the nonl inearity exponents and damping\ncoefficients for finite-time blow-up. In particular, singula rities are formed for focusing and dissipa-\ntive nonlinearities of the same power, provided that the dam ping coefficient is sufficiently small.\nOur result thus rigorously proves the non-regularizing effe ct of nonlinear damping in the mass-\nsupercritical case, which was suggested by previous numeri cal and formal results.\nWe show that, under our assumption, the damping term may be co ntrolled in such a way that\nthe self-similar blow-up structure for the focusing NLS is a pproximately retained even within the\ndissipative evolution. The nonlinear damping contributes as a forcing term in the equation for\nthe perturbation around the self-similar profile, that may p roduce a growth over finite time inter-\nvals. We estimate the error terms through a modulation analy sis and a careful control of the time\nevolution of total momentum and energy functionals.\n1.Introduction\nIn this work, we consider the NLS equation with nonlinear damping\n(1)/braceleftBigg\ni∂tψ+∆ψ+|ψ|2σ1ψ+iη|ψ|2σ2ψ= 0,\nψ(0) =ψ0∈H1(Rd),\nwhereψ:R+×Rd→Candη>0 is the damping coefficient. More precisely, our goal is to investi-\ngate the formation of singularities in finite time. Equation (1) arises a s an effective model in various\ncontexts, see for instance [13, 7, 6].\nThe nonlinear damping appearing in (1) is usually introduced as a regula rizing term of the singu-\nlar dynamics provided by the focusing NLS equation. It is well known t hat the undamped NLS\nequation (1) with η= 0 andσ1≥2\ndmay experience the formation of singularity in finite time, see\n[21, 24]. From the modeling point of view, this means that the NLS effec tive description fails to be\naccurate close to the blow-up time and further effects, that were neglected in the derivation of the\nNLS equation, become relevant near the singularity. Several phen omena may be taken into account\nwithin the NLS description, see for instance [12] for a quite general overview. In this work, we focus\non nonlinear damping terms as in (1).\nA relevant question in this perspective is to determine whether the n onlinear damping truly acts as\na regularization in the vanishing dissipation regime. More precisely, giv en the singular dynamics for\nη= 0, we are interested in determining whether the regularized equat ion (1) has a global solution\nfor anyη>0, no matter how small.\nThe eventual (weak) limit as η→0 of such solutions (if it exists) may be seen as a possible criterion\nto continue the solution beyond the singularity, in the same spirit of v anishing viscosity limits for\nconservation laws [8]. To our knowledge, the only related rigorous re sults available in the literature\nare due to Merle [30, 29], where Hamiltonian-type regularization is ado pted, see also [28] for a re-\nlated result for the generalized KdV equation. On the other hand, s everal numerical simulations\n12 P. ANTONELLI AND B. SHAKAROV\nwere performed to investigate this issue, see for instance [39, 14, 15, 17] and the references therein.\nThe regularizing property of nonlinear damping is already established in some cases. For2\nd≤σ1<\nσ2≤2\n(d−2)+[4, 3] and for2\nd=σ1=σ2[11], the Cauchy problem (1) is globally well-posed in H1\nfor anyη >0. On the other hand, there are also other cases where the dampin g does not act as a\nregularization and the dynamics remains singular for sufficiently small values ofη >0. This is the\ncase for instance of a linear damping σ2= 0 [40, 37]. Moreover, in [11] it is also shown that, for\n0≤σ2< σ1=2\nd, it is possible to determine an open set of initial data that develop a sin gularity\nin finite time. This is achieved by adapting the analysis developed in [33, 3 2, 31, 34, 38] by Merle\nand Rapha¨ el for the mass-critical NLS, where they study the st ability of blow-up in a self-similar\nregime.\nIn the physically relevant case2\nd< σ1=σ2≤2\n(d−2)+, this question remains unanswered in the\ngeneral case. In [3] the authors prove global well-posedness of ( 1) inH1only for sufficiently large η,\nnamely by requiring η≥min(σ1,√σ1). It is not clearwhether this result is sharp, howevernumerical\nsimulations [39, 14, 15] suggest that finite time blow-up still occurs f or small values of η.\nThis work provides a rigorous answer to this question in the slightly su percritical case, confirming\nthe numerical findings of [15]. In particular, we prove that for2\nd<σ1=σ2<2\nd+δ∗, whereδ∗>0\nis sufficiently small, it is possible to provide an open set of initial data tha t develop a singularity in\nfinite time in a self-similar regime. Our result shows that the self-similar blow-up regime studied in\n[36] remains unaltered even under the action of the dissipative effec ts encoded in (1). In fact, we are\ngoing to prove a more general theorem on finite time blow-up for a lar ger class of nonlinear damping\nterms, see Theorem 1.1 below.\nWe first introduce the following notations. Let sc=sc(σ1) be the Sobolev critical exponent associ-\nated withσ1,\n(2) sc=d\n2−1\nσ1,\nnamelyscdetermines the critical regularity ˙Hscfor the well-posedness of equation (1) with η= 0.\nWe also define\n(3) σ∗=2sc\nd−2sc=scσ1, σ∗=2\nd−2−σ1.\nMoreover, we also set\n(4) σ2,max=/braceleftbiggσ1ford≤3,\nσ∗ford≥4.\nOur main result is stated as follows.\nTheorem 1.1. There exists σcrit>2\nd, such that for any2\nd< σ1< σcrit,σ∗< σ2≤σ2,maxthe\nfollowing holds true. There exists η∗=η∗(σ1)>0such that for any 0< η≤η∗, there exists an\nopen set O ⊂H1(Rd)such that if ψ0∈ Othen the corresponding solution ψ∈C([0,Tmax),H1(Rd))\nto(1)develops a singularity in finite time, that is Tmax<∞and\nlim\nt→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞.\nA more precise statement of our blow-up result is provided later in Th eorem 1.2. In particular,\nthe explicit blow-up rate is given there. As previously said, our proof follows the strategy developed\nin [36] for the Hamiltonian dynamics, given by (1) with η= 0, which in turn exploited previous\nresults by the same authors related to the formation of singularitie s in the mass-critical case [31, 32].\nMore precisely, we construct a set of initial data whose evolution is a lmost self-similar. By a fine\ncontrol of the modulation parameters entering the description of the self-similar regime, it is thenBLOW-UP OF THE DAMPED NLS EQUATION 3\npossible to show the occurrence of finite-time blow-up.\nLet us emphasize that the introduction of a dissipative term in the dy namics introduces further\nmathematicaldifficulties. First ofall, the self-similarprofileweconside rin ouranalysisisdetermined\nby the undamped equations, see also (6) below, for instance. At pr esent, it is not even clear whether\nit would be possible to determine a profile that takes into account also the dissipative term. A\ngeneralized notion of dissipative solitons is present in the literature [ 1, 2, 23], where the profiles are\ndetermined not only by the balance between dispersion and focusing effects but also between gain\nand loss terms. In (1) the sole presence of a nonlinear damping cann ot be balanced by other effects.\nThe fact that the self-similar profile is determined by the Hamiltonian p art of the equation, yields\na non-trivial forcing term in the equation for the perturbation, giv en by the nonlinear damping\nitself. Through a careful modulation analysis, we determine conditio ns under which this forcing can\nbe controlled, so that the perturbation is shown to be sufficiently sm all with respect to the self-\nsimilar profile. In particular, in the case σ1=σ2, the control is determined by imposing a smallness\ncondition on η>0, as stated in the main theorem above.\nMoreover,asecondmaindifficultyisthatinourcasethefunctionalsr elatedtothephysicalquantities\nsuch as total mass, momentum and energy are - straightforward ly - not conserved in time anymore.\nWe thus need a suitable control on the time evolution of these quant ities that will in turn provide\nthe necessary bounds on the perturbation and the modulation par ameters.\nFord≥4, the restriction σ2≤σ∗<σ1prevents us to consider the case σ2=σ1. This is a technical\ncondition, needed to ensure the validity of the Sobolev embedding H1(Rd)֒→L2(σ1+σ2+1)(Rd) in\n(17), (18) below, see also Section 5. On the other hand, the condit ionσ∗<σ2is motivated by the\nfact that we cannot control Sobolev norms ˙Hsnorms with s < scin the self-similar regime, see\n(103). We remark that for the undamped dynamics (1) with η= 0, it was proved in [35] that all\nradially symmetric blowing-up solutions leave the critical space ˙Hscat blow-up time.\nLet us further remark that the smallness condition η≤η∗is only necessary when σ2=σ1. In\nthe caseσ1> σ2, it is also possible to show that for any η, there exists a set of initial conditions\ndepending on η,σ1andσ2whose corresponding solutions blow-up in finite time. We will further\ndiscuss this point in Section 6 and throughout this work.\nMoreover, as it will be clear in our analysis, the smallness of η∗is determined in terms of the\nsmallness of sc, which is related to σ1through identity (2). Indeed, we will see that η∗∼s3\nc. For\nthis reason, with some abuse of notation, in what follows we often wr iteη∗(sc) instead of η∗(σ1),\nwherescandσ1are related by identity (2).\nFinally, ourresultprovidesadditionalevidencethatthemasssuper criticalself-similarcollapseisvery\ndifferent from the one occurring in the mass critical case. In the lat ter case, indeed, any damping\ncoefficientη>0 regularizes the dynamics and prevents the formation of finite time blow-up [11, 4].\nIn Section 6 we will provide an argument explaining why the solution esc apes the self-similar regime\nin the case σ1=σ2=2\ndfor anyη>0. We will now present a road map of how Theorem 1.2 will be\nproved.\n1.1.Singularity formation. We start with an initial condition which can be decomposed as a\nsoliton profile and a small perturbation\n(5) ψ0(x) =λ−1\nσ1\n0/parenleftbigg\nQb0/parenleftbiggx−x0\nλ0/parenrightbigg\n+ξ0/parenleftbiggx−x0\nλ0/parenrightbigg/parenrightbigg\neiγ0,\nwhere 0< b0,λ0≪1 are small parameters, and Qb0is roughly a localized radial solution to the\nstationary equation\n(6) ∆ Qb0−Qb0+ib0/parenleftbigg1\nσ1Qb0+x·∇Qb0/parenrightbigg\n+|Qb0|2σ1Qb0= 0.4 P. ANTONELLI AND B. SHAKAROV\nIt is known (see, for instance, [39, 36]) that self-similar blowing-up s olutions of the supercritical NLS\nfocus as a zero energy solution to (6) plus a non-focusing radiation . For any initial value Qb0(0) and\nanyb0∈R, nontrivial solutions to (6) exist [41, 9], but any zero energy solutio n does not belong\ntoL2(Rd) [25] and thus we employ a suitable localization in space. Moreover, th e parameter b0is\nchosen to be close to a value b∗=b∗(sc)>0. By continuity, we may find a time interval where the\nsolution can be decomposed as\n(7) ψ(t,x) =1\nλ1\nσ1(t)/parenleftbigg\nQb(t)/parenleftbiggx−x(t)\nλ(t)/parenrightbigg\n+ξ/parenleftbigg\nt,x−x(t)\nλ(t)/parenrightbigg/parenrightbigg\neiγ(t)\nwhere the parameter bis still close enough to b∗andλandξare still small enough. By perturbation\ntechniques, we choose the parameters b,λ,x,γ so thatξsatisfies four suitable orthogonality condi-\ntions.\nNext, we will use a local virial law and a suitable Lyapunov functional t o prove that the parame-\nterb(t) is trapped around the value b∗for all times. This yields the following law for the scaling\nparameter\nλ(t)∼/radicalBig\n−2b∗t+λ2\n0.\nIn particular, there exists a time Tmax(sc,λ0)>0 such that λ(t)→0 ast→Tmax. Consequently,\nthe kinetic energy of the solution diverges\nlim\nt→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞.\nWe will use the coercivity property stated in Proposition2.12 below to provethe dynamical trapping\nof the parameter b. In order to control the six negative directions on the right-hand side of (37), we\nwill use four orthogonality conditions implied by the selection of the pa rametersb,λ,x,γ and two\nalmost orthogonal conditions which come from suitable bounds on th e energy and the momentum\nof the solution.\nIn ourcase, therearetwomoredifficulties thatdo notarisewhen η= 0. First, since Qbapproximates\na solution ofthe undamped NLS, one issue will be to show that the con tribution ofthe dampingterm\nin (1) can be considered of a smaller order with respect to the rest o f the dynamics. in particular,\nthe damping term generates a forcing in the equation of the remaind er, which will be controlled\nusing the smallness of ηandλ. Second, the presence of the damping implies that the energy\nE[ψ(t)] =/integraldisplay1\n2|∇ψ(t)|2−1\n2σ+2|ψ(t)|2σ+2dx\nand the momentum\nP[ψ(t)] =/parenleftbig\n∇ψ(t),iψ(t)/parenrightbig\nare not conserved. But to show the dynamical trapping of the par ameterbaroundb∗, we needE\nandPto remain small enough to control two negative directions in the coe rcivity (37) below. In\ncomparison with the undamped case, this is not a trivial consequenc e implied by the smallness of the\nenergy and momentum of the initial condition. Thus, we will study the ir time evolution and show\nthat they remain sufficiently small until the blow-up time under the as sumptiond≤3,σ2≤σ1and\nη≤η∗(sc) ord≥4 andσ2<σ∗. Finally, we recall that the localized profile Qbis close inH1(Rd)\nto the ground state of the undamped NLS which is the unique real-va lued solution Q∈H1(Rd)\n[27, 18] to the elliptic equation\n(8) ∆ Q−Q+|Q|2σ1Q= 0.\nIn light of the discussion above, Theorem 1.1 will be the consequence of the following result.BLOW-UP OF THE DAMPED NLS EQUATION 5\nTheorem 1.2. There exists s∗\nc>0such that for any 0<sc<s∗\ncandσ∗<σ2≤σ1whend≤3or\nσ∗<σ2<σ2,maxwhend≥4, there exists η∗(sc)>0such that if η≤η∗, then there exists an open\nsetO ⊂H1(Rd)such that if ψ0∈ O, then for any t∈[0,Tmax)the corresponding maximal solution\nto(1)can be written as\nψ(t,x) =1\nλ1\nσ1(t)/parenleftbigg\nQ/parenleftbiggx−x(t)\nλ(t)/parenrightbigg\n+ζ/parenleftbigg\nt,x−x(t)\nλ(t)/parenrightbigg/parenrightbigg\neiγ(t),\nwhereλ,γ∈C1([0,Tmax);R),x∈C1([0,Tmax),Rd),\nlim\nt→Tmaxλ(t) = 0,lim\nt→Tmaxx(t) =x∞∈Rd,\nthere exists 0<δ(sc) =δ≪1such that\n/ba∇dbl∇ζ/ba∇dblL∞([0,Tmax),L2)≤δ,\nand the blow-up rate is given by\n/ba∇dbl∇ψ(t)/ba∇dbl2\nL2∼(Tmax−t)−(1−sc).\nMore precisely, we will show that if we define\n(9) b∗=−π(1+δ)\nln(sc)\nthen the law of the scaling parameter will satisfy the following bounds\n(10) λ2(0)−2(1+δ)b∗t≤λ2(t)≤λ2(0)−2(1−δ)b∗t.\nRemark 1.3.Recently, in [5], the authors provided a complete description of a ze ro energy solution\nto (6) forσ1∈/parenleftbig2\nd,2\nd+ε/parenrightbig\nandεsmall enough.\n2.Preliminaries\nWe start this section with a list of notations that we are going to use t hroughout this work.\nWe use the symbol A∼B, to indicate the fact that there exist two constants C1,C2>0 such that\nC1B≤A≤C2B.\nFor anyf∈H1(Rd), we define the operators\n(11) Λ f=1\nσ1f+x·∇f, Df=d\n2f+x·∇f.\nWe notice that\n(12) Λf=Df−scf.\nMoreover, by integrating by parts we have\n(13) ( f,Λg) =−2sc(f,g)−(g,Λf)\nand\n(14) ∆Λ f= 2∆f+Λ∆f.\nWe use the following notation\n1\n(d−2)+=/braceleftBigg\n∞,ford≤2,\n1\n(d−2),ford≥3.\nNext, we recall some preliminary results.6 P. ANTONELLI AND B. SHAKAROV\nDefinition 2.1. We say that a pair ( q,r) is admissible if 2 ≤q,r≤ ∞, (q,r,d)/\\e}atio\\slash= (2,∞,2) and\n2\nq=d/parenleftbigg1\n2−1\nr/parenrightbigg\nWe will exploit the following Strichartz estimates [20, 26].\nTheorem 2.2. For everyφ∈L2(Rd)and every (q,r)admissible, there exists a constant C >0\nsuch that for any t>0,\n/ba∇dbleit∆φ/ba∇dblLq((0,t),Lr)≤C/ba∇dblφ/ba∇dblL2.\nMoreover, if f∈Lγ′((s,t),Lρ′(Rd))where(γ,ρ)is an admissible pair, and\nN(f) =/integraldisplayt\nsei(t−τ)∆f(τ)dτ,\nthen there exists a constant C >0such that for any (q,r)admissible, we have\n/ba∇dblN(f)/ba∇dblLq((s,t),Lr)≤C/ba∇dblf/ba∇dblLγ′((s,t),Lρ′).\nByusingthe Strichartzestimates, itis possibletoprovethe localex istenceofsolutionstoequation\n(1) (see [3, Proposition 2 .3]).\nTheorem 2.3. Letη∈R,σ1,σ2<2/(d−2)ifd≥3. Then for any ψ0∈H1(Rd)there exists\nTmax>0and a unique solution ψto(1)such that for any (q,r)admissible, we have\n(15) ψ,∇ψ∈C/parenleftbig\n[0,Tmax),L2(Rd)/parenrightbig\n∩Lq\nloc/parenleftbig\n(0,Tmax),Lr(Rd)/parenrightbig\n.\nMoreover, either Tmax=∞orTmax<∞and\nlim\nt→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞.\nThe usual physical quantities (the total mass, the total energy and the momentum) are governed\nby the following time-dependent functions [10].\nTheorem 2.4. Letψ0∈H1(Rd)andψ∈C/parenleftbig\n[0,Tmax),H1(Rd)/parenrightbig\nthe corresponding solution to (1).\nThen the total mass, the total energy and the momentum satisf y\n(16) M[ψ(t)] =/integraldisplay\n|ψ(t)|2dx=M[ψ0]−2η/integraldisplayt\n0/ba∇dblψ(s)/ba∇dbl2σ2+2\nL2σ2+2ds,\n(17)E[ψ(t)] =/integraldisplay1\n2|ψ(t)|2−1\n2σ1+2|ψ(t)|2σ1+2dx\n=E[ψ0]+η/integraldisplayt\n0/integraldisplay\n|ψ|2σ1+2σ2+2−|ψ|2σ2|∇ψ|2\n−2σ2|ψ|2σ2−2Re/parenleftbig¯ψ∇ψ/parenrightbig2dxds,\nand\n(18) P[ψ(t)] =/parenleftbig\n∇ψ(t),iψ(t)/parenrightbig\n=P[ψ0]−2η/integraldisplayt\n0/integraldisplay\n|ψ|2σ2Im(¯ψ∇ψ)dxds,\nrespectively.\nRemark 2.5.The dissipative terms appearing in (16), (17) and (18) are always we ll defined because\nof Strichartz estimates in Theorem 2.3, see (15).BLOW-UP OF THE DAMPED NLS EQUATION 7\nWe also notice that if σ2<σ1<1\nd−2, then it follows\n4σ2+2≤2(σ1+σ2+1)≤4σ1+2<2d\n(d−2)+.\nThus, by the Sobolev embedding theorem, the positive term on the r ight-hand side of (17) can be\nbounded by/integraldisplayt\ns/integraldisplay\n|ψ|2(σ1+σ2+1)dxdτ≤(t−s)/ba∇dblψ/ba∇dbl2(σ1+σ2+1)\nL∞((s,t),H1).\nAnalogously, we can also estimate the term on the right-hand side of (18) as\n/integraldisplayt\ns/integraldisplay\n|ψ|2σ2Im(¯ψ∇ψ)dxdτ/lessorsimilar(t−s)/ba∇dblψ/ba∇dbl2σ2+1\nL4σ2+2/ba∇dbl∇ψ/ba∇dblL2/lessorsimilar(t−s)/ba∇dblψ/ba∇dbl2σ2+2\nH1.\nThese computations will be exploited in Section 5. Indeed, they will be crucial to control uniformly\nin time the right-hand side of (17) and (18).\n2.1.Construction of the Approximated Soliton Core. In this subsection, we are going to\nconstructasuitablelocalizedsolutiontothe stationaryequation(6 ), thatwillconstitutethe blowing-\nup soliton core in the self-similar regime. This construction already ap peared in [36, Section 2],\nhowever for the sake of completeness we are going to present the main related results and properties\nhere.\nLet us notice that the damping term does not enter into the constr uction of the approximated blow-\nup core.\nLetρ∈(0,1),b>0, we define the following radii\nRb=2\nb/radicalbig\n1−ρ, R−\nb=/radicalbig\n1−ρRb.\nLetφb∈C∞(Rd) be a radially symmetric cut-off function such that\n(19) φb(x) =/braceleftBigg\n0,for|x| ≥Rb,\n1,for|x| ≤R−\nb,0≤φb(x)≤1.\nWe also denote the open ball in Rdof radiusRband centered at the origin by\nB(0,Rb) ={x∈Rd:|x|<Rb}.\nWe recall that scdenotes the Sobolev critical exponent defined in (2). We start with the following\nlemma.\nLemma 2.6. There exists σ(1)\nc>2\ndandρ(1)>0, such that for any2\nd< σ1< σ(1)\nc,ρ∈(0,ρ(1)),\nthere exists b(1)(ρ)>0, such that for any 0< b≤b(1), there exists a unique radial solution\nPb∈H1\n0(B(0,Rb))to the elliptic equation\n/parenleftbigg\n−∆+1−b2|x|2\n4/parenrightbigg\nPb−P2σ1+1\nb= 0,\nwithPb>0inB(0,Rb). Moreover Pb∈C3(B(0,Rb))and\n/ba∇dblPb−Q/ba∇dblC3→0\nasb→0, whereQthe unique positive solution to (8). Finallyb(1)(ρ)→0asρ→0.8 P. ANTONELLI AND B. SHAKAROV\nThis lemma was stated in [36, Proposition 2 .1] and its proof is a straightforward adaptation of\nthat given for [31, Proposition 1] in the mass-critical case. To prov e the existence, one shows that\nPbis a suitably scaled minimizer of the functional\nF(u) =/integraldisplay\nB(0,Rb)|∇u|2+/parenleftbigg\n1−b2|x|2\n4/parenrightbigg\n|u|2dx\nin the set\nU=/braceleftbig\nu∈H1\n0,rad(B(0,Rb)) :/ba∇dblu/ba∇dblL2σ1+2= 1/bracerightbig\n,\nwhereH1\n0,rad(B(0,Rb))isthesubsetof H1\n0(B(0,Rb))containingradiallysymmetricfunctions. Notice\nthat the functional Fis bounded from below in B(0,Rb) and−∆+1−b2|x|2\n4is an uniformly elliptic\noperator. In particular, both the maximum principle and standard r egularity results for elliptic\nequations are satisfied, see [19] for instance.\nNext, we define the function\nˆQb=φbe−ib|x|2\n4Pb,\nin the whole Rdspace, where the cut-off φbis defined in (19). Notice that the profile ˆQb∈H1(Rd)\nsatisfies the equation\n(20) ∆ ˆQb−ˆQb+ib/parenleftbiggd\n2ˆQb+x·∇ˆQb/parenrightbigg\n+|ˆQb|2σ1ˆQb=−˜Ψb,\nwhere the remainder term ˜Ψbcomes from the localization in space and is defined by\n(21) ˜Ψb=/parenleftbig\n2∇φb·Pb+Pb(∆φb)+(φ2σ1+1\nb−φb)P2σ1+1\nb/parenrightbig\ne−ib|x|2\n4.\nWe recall some properties of the profile ˆQbwhich were shown in [36, Proposition 2 .1].\nProposition 2.7. For any polynomial pand anyk= 0,1, there exists a constant C >0such that\n(22)/vextenddouble/vextenddouble/vextenddouble/vextenddoublepdk\ndxk˜Ψb/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞≤e−C\n|b|.\nMoreover, the function ˆQbsatisfies\nP[ˆQb] =/parenleftBig\n∇ˆQb,iˆQb/parenrightBig\n= 0,/parenleftBig\nΛˆQb,iˆQb/parenrightBig\n=/parenleftBig\nx·∇ˆQb,iˆQb/parenrightBig\n=−b\n2/ba∇dblxˆQb/ba∇dbl2\nL2,\nwhereΛis defined in (11)and\nd\ndb2/parenleftbigg/integraldisplay\n|ˆQb|2dx/parenrightbigg\n|b=0=C(σ1),\nwithC(σ1)→C >0asσ1→2\nd.\nNotice that heuristically ˆQbprovides an approximating solution to (6) in the ball B(0,Rb). In\nthe complementary region, Rd\\B(0,Rb) nonlinear effects become negligible, hence it is sufficient\nto study the outgoing radiation defined in the next lemma, that first appeared in [32, Lemma 15].\nThe results in the lemma below will be exploited, in particular, to estimat e the mass flux leaving\nthe collapsing core in Lemma 4.6 below.\nLemma 2.8. There exists ρ2>0such that for any 0<ρ<ρ 2there exists b2(ρ)>0such that for\nany0<b<b 2, there exists a unique radial solution ζb∈˙H1(Rd)to\n(23) ∆ ζb−ζb+ib/parenleftbiggd\n2ζb+x·∇ζb/parenrightbigg\n=˜Ψb,BLOW-UP OF THE DAMPED NLS EQUATION 9\nwhere˜Ψbis defined in (21). Moreover there exists C >0such that\n(24) Γ b= lim\n|x|→+∞|x|d|ζb(x)|2/lessorsimilare−C\nb,\nand there exists c>0such that for |x| ≥R2\nb\n(25) e−(1+cρ)π\nb≤4\n5Γb≤ |x|d|ζb(x)|2≤e−(1−cρ)π\nb.\nFurthermore, the following estimates hold\n(26) /ba∇dbl∇ζb/ba∇dbl2\nL2≤Γ1−cρ\nb,\n/vextenddouble/vextenddouble/vextenddouble|y|d\n2(|ζb|+|y||∇ζb|)/vextenddouble/vextenddouble/vextenddouble\nL∞(|y|≥Rb)≤Γ1\n2−cρ\nb.\nFinally, we have that/vextenddouble/vextenddouble/vextenddouble|ζb|e−|y|/vextenddouble/vextenddouble/vextenddouble\nC2(|y|≤Rb)≤Γ3\n5\nb,/ba∇dbl∂bζb/ba∇dblC1≤Γ1\n2−cρ\nb.\nRemark 2.9.We observe that the profile ζb∈˙H1\nrad(Rd) is not inL2(Rd) because of the logarithmic\ndivergence implied by (24), (25).\nWe now want to suitably modify the profile ˆQbto obtain an approximating solution to\n(27) i ∂tQb+∆Qb−Qb+ib/parenleftbigg1\nσ1Qb+x·∇Qb/parenrightbigg\n+|Qb|2σ1Qb= 0,\nwhereb=b(t) is a function depending on time. Notice that if we suppose that ˙b(t) = 0 anddσ1= 2,\nthenˆQbis already an approximating solution to (27), see (20) and (22). In o ur case, we suppose\nthat there exists β(b)>0 such that ˙b(t) =βsc. In other words, we search for localized solutions to\nthe following equation\n(28) i scβ∂bQb+∆Qb−Qb+ib/parenleftbigg1\nσ1Qb+x·∇Qb/parenrightbigg\n+|Qb|2σ1Qb= 0.\nWe find a solution to this equation as a suitable perturbation of the pr ofileˆQb. We make the\nfollowing ansatz on Qb,\nQb=ˆQb+scTb,\nby requiring that it solves equation (28) inside the ball |x| ≤R−\nb, up to an error of order s1+C\ncfor\nsomeC >0. Such a solution was already studied in [36, Proposition 2 .6].\nProposition 2.10. There exists ˜σc>2\ndandρ3>0such that for any2\nd< σ1<˜σc,ρ∈(0,ρ3)\nthere exists b3(ρ)>0such that for any 0<b<b 3there exists a radial function Tb∈C3(Rd)and a\nconstantβ >0such thatQb=e−ib|y|2\n4(Pbφb+scTb)satisfies\n(29) iscβ∂bQb+∆Qb−Qb+ibΛQb+|Qb|2σ1Qb=−Ψb\nwhere\n(30) Ψb=˜Ψb+Φb,\nand we have\n(31)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledk\ndxkΦb/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞(|x|≤R−\nb)/lessorsimilars1+C\nc,\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubledk\ndxkΦb/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞(|x|≥R−\nb)/lessorsimilarsc,10 P. ANTONELLI AND B. SHAKAROV\nfork= 0,1andC >0. Moreover, Qbsatisfies\n(32)|E[Qb]|/lessorsimilarΓ1−cρ\nb+sc,\nP[Qb] = (∇Qb,iQb) = 0,\n(ΛQb,iQb) = (x·∇Qb,iQb) =−b\n2/ba∇dblxQ/ba∇dbl2\nL2(1+δ1(sc,b)),\n/ba∇dblQb/ba∇dbl2\nL2=/ba∇dblQ/ba∇dbl2\nL2+O(sc)+O(b2),\nwhereδ1(sc,b)→0as asb,sc→0andcρ≪1is defined in (26)andQis a solution to (8). The\nprofileQbsatisfies also the uniform estimate\n(33)/vextendsingle/vextendsingle/vextendsingle/vextendsingleP(x)dk\ndxkQb(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−(1+c)|x|\nfor any polynomial P(x). Finally, we have that\n(34)/vextenddouble/vextenddouble/vextenddoublee(1−ρ)π\n4|x|Tb/vextenddouble/vextenddouble/vextenddouble\nC3+/vextenddouble/vextenddouble/vextenddoublee(1−ρ)π\n4|x|∂bTb/vextenddouble/vextenddouble/vextenddouble\nC2+|∂bβ(b)|/lessorsimilar1.\nWe will now show a useful Pohozaev-type estimate for the profile Qb.\nProposition 2.11. LetQbbe a solution to (29). Then we have\n(35) 2 E[Qb]−(iscβ∂bQb+Ψb,ΛQb) =sc/parenleftbig\n2E[Qb]+/ba∇dblQb/ba∇dbl2\nL2/parenrightbig\n.\nProof.We take the scalar product of equation (29) with Λ Qb\n0 = (∆Qb−Qb+ibΛQb+|Qb|2σ1Qb,ΛQb)+(iscβ∂bQb+Ψb,ΛQb).\nNow by integration by parts, we observe that\n(∆Qb+|Qb|2σ1Qb,1\nσ1Qb+x·∇Qb) = 2(sc−1)E[Qb],\nand\n−(Qb,ΛQb) =sc/ba∇dblQb/ba∇dbl2\nL2.\nas a consequence, we obtain that\n0 = 2(sc−1)E[Qb]+sc/ba∇dblQb/ba∇dbl2\nL2+(iscβ∂bQb+Ψb,ΛQb)\nwhich is equivalent to (35). /square\nFinally, by applying the operator Λ to (29), we see that Λ Qbsatisfies the following equation\n(36)iβscΛQb+Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb\n+2σ1Re(¯Qbx·∇Qb)|Qb|2σ1−2Qb=−ΛΨb,\nthat will be used in the Appendix.\n2.2.Coercivity Property. We conclude the section by discussing the coercivity properties of t he\nlinearized operator around the ground state in the mass-critical c ase. Since our analysis deals with\nthe slightly mass-supercritical case, we derive a similar property by perturbative arguments. Let us\nfirst define the following quadratic form associated with the linearize d operator around the ground\nstate in the mass-critical case,\n(37)H(f,f) =/ba∇dbl∇f/ba∇dbl2\nL2+2\nd/parenleftbigg\n1+4\nd/parenrightbigg/integraldisplay\nQ4\nd−1\nc(y·∇Qc)Re(f)2dy\n+2\nd/integraldisplay\nQ4\nd−1\nc(y·∇Qc)Im(f)2dy.BLOW-UP OF THE DAMPED NLS EQUATION 11\nwhereQcis the ground state profile for the mass-critical NLS, namely is the u nique positive solution\nto\n(38) ∆ Qc−Qc+|Qc|4\ndQc= 0.\nWe recall that\nDQc=2\ndQc+y·∇Qc.\nWe statethefollowingcoercivitypropertyforthe quadraticform H. Letusremarkthat, eventhough\nwe present it as a proposition, the following result was proved rigoro usly only in the one-dimensional\ncase [33], while for the multi-dimensional case d≤10 it was proved only numerically [16, 42].\nProposition 2.12. For anyf∈H1(Rd)there exists c>0such that\n(39)H(f,f)≥c/parenleftbigg\n/ba∇dbl∇f/ba∇dbl2\nL2+/integraldisplay\n|f|2e−|y|dy/parenrightbigg\n−c/parenleftbigg\n(f,Qc)2+(f,|y|2Qc)2+(f,yQc)2+(f,iDQc)2\n+(f,iD(DQc))2+(f,i∇Qc)2/parenrightbigg\n.\n3.Setting up the Bootstrap\nIn this section, we are going to define the set of initial conditions tha t we consider for our analysis.\nThese data are suitable perturbations of the scaled self-similar solit on profile defined in Proposition\n2.10 and are going to form singularities in finite time. In what follows, we also discuss how to\nestimate the solution in such a way that it retains its self-similar struc ture along the evolution.\nFinally, we are going to show that there exists a finite time when the sc aling parameter λ(t), related\nto the size of the solution, goes to zero, thus exhibiting the format ion of a singularity. The idea is\nto find a set that is almost invariant under the dynamics of equation ( 1) and to use a bootstrap\nargument to show that the solution remains inside this set until the s caling parameter becomes zero\nand consequently the solution blows up.\nWe use the quantity ρ3and the profile Qbdefined in Proposition 2.10, and the constant Γ bdefined\nin (24). Moreover, we denote by\n˜sc=d\n2−1\n˜σc,\nwhere ˜σcis determined in Proposition 2.10. In the next definition and subseque nt propositions, it\nwill be useful to write the assumptions on σ1by exploiting identity (2). This means that every time\na condition on scis found, such as for instance 0 < sc< Scfor some constant Sc, it should be\ninterpreted as a condition on the exponent σ1<σ1(Sc) whereσ1(Sc) is given by\nσ1(Sc) =2\nd−2Sc.\nDefinition 3.1. Let 0<sc<˜scand 0< ρ<ρ 3. We define the set O ⊂H1(Rd) as the family of\nall functions φ∈H1such that there exist λ0,b0>0,x0∈Rd,γ0∈Randξ0∈H1(Rd) such that\n(40) φ(x) =λ−1\nσ1\n0/parenleftbigg\nQb0/parenleftbiggx−x0\nλ0/parenrightbigg\n+ξ0/parenleftbiggx−x0\nλ0/parenrightbigg/parenrightbigg\neiγ0,\nwith the following conditions: there exists 0 <ν(ρ)≪1 such that\n(41) Γ1+ν10\nb0≤sc≤Γ1−ν10\nb0,12 P. ANTONELLI AND B. SHAKAROV\nthe scaling parameter satisfies\n(42) 0<λ0<Γ100\nb0,\nthe initial momentum and energy are bounded as\n(43) λ2−2sc|E[φ]|+λ1−2sc|P[φ]|<Γ50\nb0,\nthere exists s=s(σ1,σ2) in the interval\n(44) sc<s<min/parenleftbiggdσ1\n2σ1+2,dσ2\n2σ2+2,1\n2/parenrightbigg\nsuch that the following inequality is verified\n(45)/integraldisplay\n||∇|sξ0|2+|∇ξ0|2+|ξ0|2e−|y|dy<Γ1−ν\nb0,\nwhere\ny=x−x0\nλ.\nFinally, the remainder ξ0satisfies also the orthogonality conditions\n(46) ( ξ0,|y|2Qb0) = (ξ0,yQb0) = (ξ0,iΛ(ΛQb0)) = (ξ0,iΛQb0) = 0.\nIn this definition, the constant ν=ν(ρ) is chosen so that\n(47) Γ1−cρ\nb0≤Γ1−ν50\nb0,\nwherec>0 is the constant in (25). Note that from the definition of Γ bin (25) and from condition\n(41), it follows that\nsc∼Γb∼e−(1+cρ)π\nb\nthat is\nb0∼b∗=−π(1+δ)\nln(sc)\nnamelyb0is chosen to be close to the value b∗(sc)>0 defined in (9). The constant s=s(σ1,σ2) is\nchosen so that the L2σ1+2andL2σ2+2norms ofξmay be controlled by interpolating between ˙Hs\nand˙H1, see (78) and (103) below. Finally, we recall that the set Ois non-empty, see [36, Remark\n2.10]. We remark that this is the same set defined in [36], whose evolution s develop a singularity in\nfinite time. In this work, we are going to prove that the same initial da ta produce singular solutions\nalso under the dissipative dynamics (1). Thus, our result implies that the nonlinear damping is not\nable to regularize the dynamics and to prevent the formation of sing ularities, in the cases under\nour consideration. This is not in contradiction with the global well-pos edness result proven in [3],\nwhere the condition η≥min(σ1,√σ1) was needed. In fact, here we prove our main result under a\nsmallness assumption on η, namely we require\n(48) η≤s3\nc.\nMoreover, in Section 6 we are going to provide an alternative definitio n of the set O, in the case\nσ2< σ1. In particular, hypothesis (48) will no longer be needed, even thou gh the set will depend\nonσ1,σ2andη.\nNow, let us consider an initial datum ψ0∈ Oand letψ∈C([0,Tmax),H1(Rd)) be the corresponding\nsolution to (1), where Tmax≤ ∞is its maximal time of existence. By continuity of the solution,\nthere exists a time interval [0 ,T1), withT1≤Tmax, where the inequalities in Definition 3.1 are\nstill valid but with slightly larger bounds. Moreover, by standard per turbation techniques, we can\nalso preserve the orthogonal conditions in (46) for any t∈[0,T1) by modulational analysis, see forBLOW-UP OF THE DAMPED NLS EQUATION 13\ninstance [31, Lemma 2] where this was proved for the Hamiltonian dyn amics. The proof of a similar\nproperty for solutions to the dissipative dynamics (1) follows straig htforwardly.\nProposition 3.2. There exists 0< T1≤Tmax,b,λ,γ∈C1([0,T1),R),x∈C1([0,T1),Rd)and\nξ∈C([0,T1),H1(Rd))such that for all t∈[0,T1), the solution ψto(1)may be decomposed as\n(49) ψ(t,x) =λ(t)−1\nσ1/parenleftbig\nQb(t)(y)+ξ(t,y)/parenrightbig\neiγ(t),\nwith\n(50) y=x−x(t)\nλ(t),\nwhereξsatisfies the following orthogonal conditions:\n(51) ( ξ(t),|y|2Qb(t)) = (ξ(t),yQb(t)) = (ξ(t),iΛ(ΛQb(t))) = (ξ(t),iΛQb(t)) = 0,\nand we have\nΓ1+ν2\nb(t)≤sc≤Γ1−ν2\nb(t), (52)\n0≤λ(t)≤Γ10\nb(t), (53)\nλ2−2sc|E[ψ(t)]|+λ1−2sc(t)|P[ψ(t)]| ≤Γ2\nb(t), (54)\n/integraldisplay\n||∇|sξ(t)|2dy≤Γ1−50ν\nb(t), (55)\n/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−20ν\nb(t). (56)\nWe notice that (52) and the smallness of νandscimply that Γ b(t)<1 for anyt∈[0,T1). Our\nmain goal is to prove that the solution remains in this self-similar regime untilTmax, that is the\ndecomposition of ψ, along with the properties of the modulation parameters as stated in Proposition\n3.2 are valid until the maximal time of existence. This is achieved by usin g a bootstrap argument.\nWe show that the bounds in Proposition 3.2 can be improved and thus t he self-similar regime may\nbe extended in time. This amounts to finding a dynamical trapping of t he parameter bto improve\n(52) and (56), to find a suitable differential equation for λto improve (53), and to prove that the\nenergy and the momentum of the solution remain small enough to impr ove (54). The trapping of the\ncontrol parameter bwill be achieved in Section 4 by finding a lower bound for ˙busing a virial-type\nargument, and an upper bound with a monotonicity formula. The equ ation satisfied by the scaling\nparameterλwill be a direct consequence of the dynamical trapping of b(t) aroundb∗, see Section 5.\nFinally, the controls on EandPwill be obtained in Section 5 as a consequence of the choice of the\ndamping parameter ηin (48). In fact, we will show the following bootstrap result.\nProposition 3.3. There exists s∗\nc>0, such that for any sc< s∗\nc, there exist smax(sc)> sc, and\nν∗(sc)>0such that for any sc<s<s max(sc)and0<ν <ν∗and for any t∈[0,T1),the following\ninequalities are true:\nΓ1+ν4\nb(t)≤sc≤Γ1−ν4\nb(t), (57)\n0≤λ(t)≤Γ20\nb(t), (58)\nλ2−2sc|E[ψ(t)]|+λ1−2sc|P[ψ(t)]| ≤Γ3−10ν\nb(t), (59)\n/integraldisplay\n||∇|sξ(t)|2≤Γ1−45ν\nb(t), (60)\n/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−10ν\nb(t). (61)14 P. ANTONELLI AND B. SHAKAROV\nAs a consequence of this proposition, we see that the solution satis fies improved bounds with\nrespect to those stated in Proposition 3.2. A standard continuity a rgument then implies that the\nsame bounds are satisfied in the whole interval [0 ,Tmax).\nLet us remark that the same dynamical trapping argument to show self-similar blow-up was already\nexploited in [36]. In particular, this implies that a similar argument to [36] works also in our case\nfor the damped NLS, under the assumptions of Theorem 1.2. On the other hand, dealing with the\ndissipative dynamics (1) entails new mathematical difficulties with resp ect to [36]. First of all, we\nnotice that the self-similar profile Qb, given by equation (29), does not determine an approximate\nsolution to (1), not even close to the collapsing core. In particular, this implies that the equation\nfor the perturbation ξbears a forcing term of order one, depending on Qb. In the case under our\nconsideration here, we can see that the forcing term produces an error that becomes non-negligible\non a range of times of order/parenleftBig\nηλ2−2σ2\nσ1/parenrightBig−1\n.\nWe overcome this difficulty by choosing the parameters (and in partic ular, the damping coefficient\nin the case σ1=σ2) in such a way that Tmax≪/parenleftBig\nηλ2−2σ2\nσ1/parenrightBig−1\n.\nThe second mathematical difficulty induced by the dissipative dynamic s is that the global quantities,\nsuch as total momentum and total energy, are not conserved alo ng the flow of (1), see (18) and (17),\nrespectively. Consequently, while the bootstrap condition (54) is s traightforwardly satisfied in the\nHamiltonian case, a fundamental step in our analysis would be to cont rol the time evolution of these\nquantities.\nIn Section 6 we will also show that in the case σ2< σ1, it is possible to deal with these issues by\nchoosing the scaling parameter λsmall enough and depending on σ1,σ2andηso that the initial\ncondition is very close to the blow-up point and the damping term is not strong enough to force the\nsolution out of the self-similar regime before the blow-up time.\nAs the last step, we will show that inside the self-similar regime, the sc aling parameter λ(t) goes to\nzero.\nIn particular, the following Corollary can be inferred from Propositio n 3.2.\nCorollary 3.4. For anysc<s∗\ncwheres∗\ncis defined in Proposition 3.3, the bounds in Proposition\n3.2 are valid for any t∈[0,Tmax). Moreover, there exists a constant 0< C=C(ν,sc)≪1such\nthat for any t∈[0,Tmax),\n(62) λ2\n0−2(1+C)t≤λ2(t)≤λ2\n0−2(1−C)t.\nAs a consequence of the corollary above, there exists Tmax(ν,sc,λ0)<∞such thatλ→0 as\nt→Tmax(ν,sc,λ0) andλbehaves like\nλ2(t)∼Tmax−t.\nMoreover, the estimate (62) also provides a blow-up rate for the L2-norm of the gradient of the\nsolution, as we have\n/vextenddouble/vextenddouble∇ψ(t)/vextenddouble/vextenddouble2\nL2=λ2(sc−1)(t)/vextenddouble/vextenddouble∇(Qb(t)+ξ(t))/vextenddouble/vextenddouble2\nL2∼(Tmax−t)−(1−sc).\n4.Estimates on the Modulation Parameters\nIn this section, we start our analysis of solutions emanating from init ial conditions in the set O,\ndefined in Definition 3.1. Let ψ0∈ O, we denote by ψ∈C([0,Tmax),H1(Rd)) its corresponding\nmaximal solution, with Tmax≤ ∞. Then we decompose the solution as the sum of a soliton core\nand remainder as in (49). By Proposition 3.2, the bounds (52) - (56) are satisfied for all t∈[0,T1],\ntogether with the four orthogonal conditions in (51).\nThe purpose of this section is to obtain preliminary estimates on the p arameters that are requiredBLOW-UP OF THE DAMPED NLS EQUATION 15\nto prove the dynamical trapping of bin the next section. A crucial step will be to use the coercivity\nproperty stated in Proposition 2.12. Even though this proposition is related to the mass critical\ncase, we will use the smallness of scand property (12) to obtain a similar coercivity property in the\nslightly super-critical case. We will then show that the negative par t of the right-hand side of (39)\nis controlled by the positive part. When computing the quadratic for m (39) on the perturbation ξ,\nwe see that four of the negative terms appearing in are small enoug h because of the orthogonality\nconditions (51), the smallness of scand property (12). We now show how to control the two\nremaining negative terms.\nBy plugging the decomposition (49) into equation (1), we obtain the f ollowing equation for the\nperturbation ξ,\ni∂τξ+Lξ+i(˙b−βsc)∂bQb−i/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛ(Qb+ξ)−i˙x\nλ·∇(Qb+ξ)\n−(˙γ−1)(Qb+ξ)+iηλ2−2σ2\nσ1|Qb+ξ|2σ2(Qb+ξ)+R(ξ)−Ψb= 0,(63)\nwhere the scaled time τis defined according to\n(64) τ(t) =/integraldisplayt\n01\nλ2(v)dv,\nand we recall that\nΛf=1\nσ1f+y·∇f.\nIn (63) the dot denotes the derivative with respect to τand the space derivatives are intended to be\nperformed with respect to the scaled space variable ywhere\n(65) y=x−x(t)\nλ(t).\nMoreover, we used the notation Lfor the linearized operator around Qb,\nLξ= ∆ξ−ξ+ibΛξ+2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb+|Qb|2σ1ξ, (66)\nandRcontains all nonlinear terms in ξ,\n(67)R(ξ) =|Qb+ξ|2σ1(Qb+ξ)−|Qb|2σ1Qb−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ,\nwhereas Ψ bis defined in (30).\nWe notice that the damping term yields a forcing in the equation for ξ, given by\n(68) i ηλ2−2σ2\nσ1|Qb|2σ2Qb\ndepending on Qb,ηandλ. One of the main goals in our subsequent analysis is to exploit the\nsmallness of the product ηλ2−2σ2\nσ1in order to obtain a suitable control on the nonlinear damping\nterm so that the self-similar regime is maintained along the evolution. T his task is achieved by the\nselection of ηfor instance as in (48) or by the smallness of λforσ2<σ1, see Section 6. Let us notice\nthat the assumption σ∗<σ2≤σ2,max, see (3) and (4), implies that\n0≤2−2σ2\nσ1<2(1−sc)\nford≤3, while\n0<4(d−2)σ1−1\n(d−2)σ1≤2−2σ2\nσ1<2(1−sc),\nford≥4.\nIn what follows, we will use the following estimates on scalar products .16 P. ANTONELLI AND B. SHAKAROV\nLemma 4.1. For anyt∈[0,T1), any polynomial P(y)and any integers k∈ {0,1,2,3}andn∈\n{0,1}, we have\n(69)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nξ,Pdk\ndykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay\n|ξ|2e−|y|dy/parenrightbigg1\n2\n,\n(70)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nξ,Pdk\ndyk∂bQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n,\n(71)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggdk\ndykQb,Pdn\ndynΨb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nξ,Pdn\ndynΨb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1−ν\nb,\n(72)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n∂bQb,Pdk\ndykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n|y|2Qb,Pdk\ndykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar1.\nThese inequalities have been proven in [31, Lemma 4]. They follow from t he Cauchy-Schwarz\ninequalityand theuniform estimate(33). In the next twoLemmas4.2 and4.3weprovethe smallness\nof the scalar products ( ξ(t),Qb(t)) and (ξ,i∇Qb(t)), so to conclude our control of the negative terms\nappearing in (39) up to additional terms controlled by the smallness o fsc. This will be achieved by\nexploiting the balance laws satisfied by the total momentum and ener gy (18) and (17), respectively,\ntogether with the bounds (54).\nLemma 4.2. There exists ν1>0such that for any ν <ν1andt∈[0,T1), we have\n(73) |(ξ(·,t),Qb(t))|/lessorsimilar/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy+Γ1−2ν\nb(t).\nLet us remark that, by combining (73) with the bound (56), we would obtain the following\nestimate\n(74) |(ξ(·,t),Qb(t))|/lessorsimilarΓ1−20ν\nb(t).\nAlthough this rougher bound is sufficient to show the coercivity of th e linearized operator, the\nestimate (73) will be crucial to prove Proposition 3.3.\nProof.We plug the decomposition of the solution (49) into the energy equat ion to obtain\n(75)2λ2−2scE[ψ] = 2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2\nL2−2(∆Qb,ξ)\n−1\nσ1+1/integraldisplay\n|Qb+ξ|2σ1+2−|Qb|2σ1+2dy.\nBy using equation (29) satisfied by Qbin the expression above, we get\n2λ2−2scE[ψ] = 2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2\nL2\n+2(iscβ∂bQb−Qb+ibΛQb+Ψb,ξ)−/integraldisplay\nR(2)(ξ)dy,\nwhereR(2)(ξ) is defined by\n(76) R(2)(ξ) =1\nσ1+1/parenleftbig\n|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)/parenrightbig\n.\nEquivalently, we write the identity above as\n2(Qb,ξ) =−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2\nL2\n+2(iscβ∂bQb+ibΛQb+Ψb,ξ)−/integraldisplay\nR(2)(ξ)dy.(77)BLOW-UP OF THE DAMPED NLS EQUATION 17\nNow we bound the terms on the right-hand side of (77). For the firs t term, we use the control on\nthe energy (54)/vextendsingle/vextendsingle−2λ2−2scE[ψ]/vextendsingle/vextendsingle≤2Γ2\nb.\nFor the second term, we use the properties of Qblisted in (32), inequality (47) and the control (52)\nonscto obtain that/vextendsingle/vextendsingleE[Qb]/vextendsingle/vextendsingle≤Γ1−cρ\nb+sc≤Γ1−ν50\nb+Γ1−ν2\nb.\nFor the fourth term, we use again the control (52) on sc, estimates (70), (71) and (56) to get\n|sc(iβ∂bQb,ξ)+(Ψ b,ξ)|/lessorsimilar/parenleftBigg\nsc/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n+Γ1−ν\nb/parenrightBigg\n/lessorsimilar/parenleftBig\nΓ1−ν2\nbΓ1\n2−10ν\nb+Γ1−ν\nb/parenrightBig\n.\nMoreover, we also observe that (i bΛQb,ξ) = 0 from (51). Finally, we use estimates (69) and (56)\nto control the remainder term R(2)(ξ) defined in (76) as follows,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR(2)(ξ)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy+/ba∇dblξ/ba∇dbl2σ1+2\nL2σ1+2.\nIn order to bound the L2σ1+2-norm ofξ, we observe that since\nsc<s<d\n2−d\n2σ1+2=dσ1\n2σ1+2=s(σ1)\nthen by Sobolev embedding and subsequent interpolation between H1(Rd) andHs(Rd), we have\n(78) /ba∇dblξ/ba∇dbl2(σ1+1)\nL2(σ1+1)/lessorsimilar/ba∇dbl|∇|s(σ1)ξ/ba∇dbl2(σ1+1)\nL2≤ /ba∇dblξ/ba∇dbl2θ(σ1+1)\n˙H1/ba∇dblξ/ba∇dbl2(1−θ)(σ1+1)\n˙Hs,\nfor someθ=θ(s)∈(0,1), wheresis determined in Definition 3.1, see (44). Now by using the\ncontrols (56) and (55) we obtain\n/ba∇dbl∇ξ/ba∇dbl2θ(σ1+1)\n˙H1/ba∇dbl∇ξ/ba∇dbl2(1−θ)(σ1+1)\n˙Hs ≤Γ(1−20ν)θ(σ1+1)\nbΓ(1−50ν)(1−θ)(σ1+1)\nb\n≤Γ(1−50ν)(σ1+1)+30 νθ(1+σ1)\nb.\nBy collecting everything together, we obtain that\n|(ξ,Qb)|/lessorsimilarΓ2\nb+Γ1−ν50\nb+Γ1−ν2\nb+/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy\n+Γ1−ν2\nbΓ1\n2−10ν\nb+Γ1−ν\nb+Γ(1−50ν)(σ1+1)+30 νθ(1+σ1)\nb.\nBy choosing νsmall enough, we have thus obtained estimate (73). /square\nWe also exploit the equation of the momentum (18) to derive a bound o n (ξ(·,t),i∇Qb(t)), as in\nthe following lemma.\nLemma 4.3. There exists ν(1)>0such that for any 0<ν <ν(1)and anyt∈[0,T1), we have\n(79) |(ξ(·,t),i∇Qb(t))| ≤Γ1−50ν\nb(t).\nProof.We plug decomposition (49) into the momentum (18) to obtain\n2(ξ,i∇Qb) =−λ1−2scP[ψ]+P[Qb]+P[ξ].\nThe first term on the right-hand side is bounded by (54),\n/vextendsingle/vextendsingle−λ1−2scP[ψ]/vextendsingle/vextendsingle≤Γ2\nb.18 P. ANTONELLI AND B. SHAKAROV\nNext, we observe that by the properties of Qb, we haveP[Qb] = 0, see (32). For the last term, since\ns<1\n2, we use (44) to estimate\n|P[ξ]|/lessorsimilar/ba∇dblξ/ba∇dbl2\n˙H1\n2/lessorsimilar/ba∇dblξ/ba∇dbl2θ\n˙H1/ba∇dblξ/ba∇dbl2(1−θ)\n˙Hs\nwhere\nθ=2−4s\n2−s>0.\nNow we use estimates (56) and (55) to obtain\n/ba∇dblξ/ba∇dbl2θ\n˙H1/ba∇dblξ/ba∇dbl2(1−θ)\n˙Hs≤Γ(1−20ν)θ\nbΓ(1−50ν)(1−θ)\nb= Γ(1−50ν)+30νθ\nb.\nBy combining all previous estimates, we get\n|(ξ,i∇Qb)|/lessorsimilarΓ2\nb+Γ(1−50ν)+30νθ\nb.\nInequality (79) thus follows by choosing νsufficiently small. /square\nOur next step is to obtain suitable estimates on the modulational par ameters defined in the\ndecomposition (49). This is accomplished by exploiting the equation (6 3) satisfied by the remainder\nξ, and the orthogonality conditions listed in (51).\nLemma 4.4. For anyt∈[0,T1), we have\n(80)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙λ(τ)\nλ(τ)+b(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|˙b(τ)|/lessorsimilarΓ1−20ν\nb(τ),\nand/vextendsingle/vextendsingle/vextendsingle/vextendsingle(˙γ(τ)−1)−1\n/ba∇dblΛQb(τ)/ba∇dbl2\nL2(ξ(τ),LΛ(ΛQb(τ))/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙x(τ)\nλ(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilarδ2/parenleftbigg/integraldisplay\n|∇ξ(τ)|2+|ξ(τ)|2e−|y|dy/parenrightbigg1\n2\n+Γ1−20ν\nb(τ),(81)\nwhereδ2=δ2(sc)>0. Moreover, δ2(sc)→0, assc→0.\nProof.The lemma is proved by taking the scalar product of equation (63) wit h suitable terms that\nallow us to exploit the orthogonality conditions (51). The same appro ach was already used in [36],\nsee Lemma 3.1 and Proposition 3.3 therein, see also [38, Appendix A], to study the undamped\ndynamics. For this reason we write equation (63) as\n(82) U(ξ)+iηλ2−2σ2\nσ1|Qb+ξ|2σ2(Qb+ξ) = 0,\nso that, in what follows, we exploit the analysis already developed in [36 ].\nLet us first consider the bound on |˙b|. We take the scalar product of (82) with Λ Qb. Following the\ncomputations in Appendix A, we use estimates (69), (74), (79), (5 5) and (56) to obtain\n|˙b|/lessorsimilar/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy+Γ1−11ν\nb+ηλ2−2σ2\nσ1/vextendsingle/vextendsingle/parenleftbig\ni|Qb+ξ|2σ2(Qb+ξ),ΛQb/parenrightbig/vextendsingle/vextendsingle.\nSimilarly, we obtain the estimate for other parameters\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙λ\nλ+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy+Γ1−11ν\nb+ηλ2−2σ2\nσ1/vextendsingle/vextendsingle/parenleftbig\n|Qb+ξ|2σ2(Qb+ξ),|y|2Qb/parenrightbig/vextendsingle/vextendsingle.BLOW-UP OF THE DAMPED NLS EQUATION 19\nand\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(˙γ−1)−1\n/ba∇dblΛQb/ba∇dbl2\nL2(ξ,LΛ(ΛQb)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙x\nλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarδ2/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n+/integraldisplay\n|∇ξ|2dy+Γ1−11ν\nb\n+ηλ2−2σ2\nσ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbig\n|Qb+ξ|2σ2(Qb+ξ),yQb/parenrightbig\n+/parenleftbig\ni|Qb+ξ|2σ2(Qb+ξ),Λ(ΛQb)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nLet us now control the contributions coming from the damping term . We write\n|Qb+ξ|2σ2(Qb+ξ) =|Qb|2σ2Qb+R(1)(ξ).\nWe use (72) to obtain that\n(83)/vextendsingle/vextendsingle/parenleftbig\ni|Qb|2σ2Qb,ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightbig/vextendsingle/vextendsingle/lessorsimilar1.\nOn the other hand, by using (69) and (56), we also have that\n/vextendsingle/vextendsingle/vextendsingle/parenleftBig\nR(1)(ξ),ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay\n|∇ξ|2dy+/integraldisplay\n|ξ|2e−|y|dy/parenrightbigg1\n2\n.\nThus it follows that\n(84)ηλ2−2σ2\nσ1/vextendsingle/vextendsingle/parenleftbig\ni|Qb+ξ|2σ2(Qb+ξ),ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightbig/vextendsingle/vextendsingle\n/lessorsimilarηλ2−2σ2\nσ1/parenleftbigg\n1+/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n/lessorsimilarΓ3−3ν2\nbΓ20−20σ2\nσ1\nb≤Γ2\nb,\nwhere we used the hypothesis on η(48), (52) and (53). /square\nOne can see that the estimates (81) and (80) are the same as thos e in [36, Lemma 3 .1] for the\nundamped case η= 0. This is because our choice of ηandσ2≤σ1imply that the damping term is\nof lower order with respect to other terms in equation (63). in part icular, the scalar products with\nthe forcing term defined in (68) are well-controlled by the smallness o fηλ2−2σ2\nσ1.\n4.1.Local Virial Law. We will now derive suitable inequalities on ˙bto prove that bis trapped\naround the value b∗defined in (9). We first obtain a lower bound, see (85) below. This est imate\nis connected with the local virial law for the remainder ξ. For details, see [33, Section 3] and [31,\nSection 4].\nLemma 4.5. There exists s(2)\nc>0such that for any sc<s(2)\ncand for any t∈[0,T1), there exists\nC >0such that\n(85) ˙b(t)≥C/parenleftbigg\nsc+/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy−Γ1−ν6\nb(t)/parenrightbigg\n.20 P. ANTONELLI AND B. SHAKAROV\nProof.By taking the scalar product of (63) with Λ Qbwe obtain\n0 = (i∂τξ,ΛQb)+˙b(i∂bQb,ΛQb)−(iβsc∂bQb+Ψb,ΛQb)+(Lξ+R(ξ),ΛQb)\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛQb+i˙x\nλ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛξ+(˙γ−1)ξ+i˙x\nλ·∇ξ,ΛQb/parenrightBigg\n+ηλ2−2σ2\nσ1(i|Qb+ξ|2σ2(Qb+ξ),ΛQb).\nNotice that only the last term in this equation depends on the damping . The computation involving\nthe other terms will be shown in Appendix A following the exposition of [ 36, Proposition 3 .3]. These\ncomputations yield the following inequality\n˙b≥C/parenleftbigg\nsc+/integraldisplay\n|∇ξ|2dy+/integraldisplay\n|ξ|2e−|y|dy−Γ1−ν6\nb/parenrightbigg\n−ηλ2−2σ2\nσ1/vextendsingle/vextendsingle(i|Qb+ξ|2σ2(Qb+ξ),ΛQb)/vextendsingle/vextendsingle,\nfor someC >0. On the other hand, the contribution of the damping term is negligib le as we control\nit with the calculations in the previous lemma (see (84))\nηλ2−2σ2\nσ1/vextendsingle/vextendsingle(i|Qb+ξ|2σ2(Qb+ξ),ΛQb)/vextendsingle/vextendsingle/lessorsimilarΓ2\nb.\n/square\n4.2.Refined virial estimate. In this subsection, we derive an upper bound for ˙b. We will study\nthe mass flux escaping the self-similar soliton core region. The outgo ing radiation ζbdefined in\nLemma 2.8 will play a central role in this task. Let φ∈C∞\nc(Rd) be a radial cut-off defined by\nφ(r)∈[0,1], φ(r) =/braceleftBigg\n1 forr∈[0,1),\n0 forr≥2,\nWe also define\nφA(r) =φ/parenleftBigr\nA/parenrightBig\nwhereA=A(t) is determined by\n(86) A(t) = Γ−a\nb(t)\nanda=a(ν)>0 will be chosen later. We denote by ˜ζb=φAζbthe localization of the outgoing\nradiation. We notice that the equation for the outgoing radiation (2 3) implies that ˜ζbsatisfies the\nequation\n(87) ∆ ˜ζb−˜ζb+ib/parenleftbiggd\n2+y·∇/parenrightbigg\n˜ζb=φA˜Ψb+F=˜Ψb+F\nwhere\n(88) F= (∆φA)ζb+2∇φA·∇ζb+iby·∇φAζb,\nand˜Ψbwas defined in (21). In particular, by recalling that suppΨ b⊂B(0,Rb)⊂B(0,2\nb) and by\nthe definition of A(t), we have that φA˜Ψb=˜Ψb. Moreover, from the properties of ζbestablished in\nLemma 2.8, we infer that ˜ζb∈H1\nradand, by suitably choosing a=a(ν)>0, we also have\n(89) /ba∇dbl˜ζb/ba∇dbl2\nH1≤Γ1−cρ\nb.BLOW-UP OF THE DAMPED NLS EQUATION 21\nWe define the following refined soliton core and remainder,\n˜Qb=Qb+˜ζband˜ξ=ξ−˜ζb,\nso that we decompose the solution as\n(90) ψ(t,x) =λ(t)−1\nσ1/parenleftbigg\n˜Qb(t)/parenleftbiggx−x(t)\nλ(t)/parenrightbigg\n+˜ξ/parenleftbigg\nt,x−x(t)\nλ(t)/parenrightbigg/parenrightbigg\neiγ(t).\nWe now claim that also the new soliton core ˜Qbis an approximating solution to (28). By using (87)\nand (29), we obtain\niβsc∂b˜Qb+∆˜Qb−˜Qb+ibΛ˜Qb+|˜Qb|2σ1˜Qb= iβsc∂bQb+∆Qb−Qb\n+ibΛQb+|Qb|2σ1Qb\n+∆˜ζb−˜ζb+i/parenleftbiggd\n2˜ζb+y·∇˜ζb/parenrightbigg\n−iscb˜ζb\n−iscb˜ζb+iscβ∂b˜ζb\n+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb\n=−Ψb+˜Ψb+F−iscb˜ζb+iscβ∂b˜ζb\n+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb.\nWe observe that by using (30), we get −Ψb+φA˜Ψb= Φb. Thus the new profile ˜Qbsatisfies\n(91) i βsc∂b˜Qb+∆˜Qb−˜Qb+ibΛ˜Qb+|˜Qb|2σ1˜Qb=−˜Φb+F\nwhere\n(92) ˜Φb=−Φb−iscb˜ζb+iβ∂b˜ζb+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb,\nandFis defined in (88). By using the equation (91) for ˜Qband decomposition (90), we derive the\nequation for ˜ξ,\ni∂τ˜ξ+˜L˜ξ+i(˙b−βsc)∂b˜Qb−i/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛ(˜Qb+˜ξ)−i˙x\nλ·∇(˜Qb+˜ξ)\n−(˙γ−1)(˜Qb+˜ξ)+iηλ2−2σ2\nσ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ)+˜R(˜ξ)−˜Φb+F= 0,(93)\nwhere\n˜L˜ξ= ∆˜ξ−˜ξ+ibΛ˜ξ+2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb+|˜Qb|2σ1˜ξ,\nand\n˜R(˜ξ) =|˜Qb+˜ξ|2σ1(˜Qb+˜ξ)−|˜Qb|2σ1˜Qb−2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb−|˜Qb|2σ1˜ξ.\nWe recall again that the scaled time and space denoted by τandyare defined in (64) and (65),\nrespectively. With some abuse of notations, in what follows we often explicit the τ−dependence of\nfunctions. For instance, we denote ˜ξ(τ), to actually mean ˜ξ(t−1(τ)).\nByexploitingagainthe equation(93) forthe remainder ˜ξ, the controls(81) and(80)and estimates\nin Proposition 3.2, we obtain the following bounds.\nLemma 4.6. There exists C >0such that any t∈[0,T1), there exists C >0such that\nC2/parenleftbigg/integraldisplay\n|∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy+Γb(τ)/parenrightbigg\n≤C/parenleftbiggd\ndτf(τ)+sc/parenrightbigg\n+/integraldisplay\n{A(τ)≤|y|≤2A(τ)}|ξ(τ)|2dy,(94)22 P. ANTONELLI AND B. SHAKAROV\nwhere\n(95) f(τ) =−1\n2(i˜Qb(τ),y·∇˜Qb(τ))−(i˜ξ(τ),Λ˜Qb(τ)).\nProof.As for Lemmas 4.4 and 4.5, we conveniently write equation (93) as\nU(1)(˜ξ)+iηλ2−2σ2\nσ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ) = 0,\nso that forU(1)((˜ξ) we again exploit the analysis already presented in [36, Lemma 3 .5], [34, Lemma\n6] and reported in Appendix B . By taking the scalar product of equa tion (93) with Λ ˜Qb, we obtain\n(96) ( P(˜ξ),Λ˜Qb)+(iηλ2−2σ2\nσ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb) = 0.\nExploiting the computations in Appendix B yields the following inequality\nd\ndτf≥C/parenleftbigg/integraldisplay\n|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg\n−1\nC/parenleftBigg\nsc+/integraldisplay2A\nA|ξ|2dy/parenrightBigg\n−ηλ2−2σ2\nσ1/vextendsingle/vextendsingle/vextendsingle(i|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb)/vextendsingle/vextendsingle/vextendsingle.\nWe will now show that in this inequality, the contribution of the damping term can be considered\nnegligible.\nFrom (89) and (72) it follows that\n(97)/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ni|˜Qb|2σ2˜Qb,Λ˜Qb/parenrightBig/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ni|Qb+˜ζb|2σ2(Qb+˜ζb),Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/parenleftbig\ni|Qb|2σ2Qb,ΛQb/parenrightbig/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ni|Qb|2σ2Qb,Λ˜ζb/parenrightBig/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ni|Qb+˜ζb|2σ2(Qb+˜ζb)−i|Qb|2σ2Qb,Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle.\nWe already know from (83) that/vextendsingle/vextendsingle/parenleftbig\ni|Qb|2σ2Qb,ΛQb/parenrightbig/vextendsingle/vextendsingle/lessorsimilar1.\nFor the other terms in (97), we use (89) and (47) to obtain\n/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ni|Qb|2σ2Qb,˜ζb/parenrightBig/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ni|Qb+˜ζb|2σ2(Qb+˜ζb)−i|Qb|2σ2Qb,Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1−ν\nb.\nThus we obtain that\nηλ2−2σ2\nσ1/vextendsingle/vextendsingle/vextendsingle(i|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarηλ2−2σ2\nσ1≤Γ2\nb\nwhere we used again (48), (52) and (53). Inequality (94) follows st raightforwardly from (166) and\n(52). /square\nNotice that the control (94) is the same as that in [36, Lemma 3 .5]. This is again because the\ncontribution of the damping term is negligible in the self-similar regime du e to the choice of ηand\nthe fact that σ2≤σ1.\nWe proceed by finding a suitable bound for the mass flux term/integraldisplay\n{A(τ)≤|y|≤2A(τ)}|ξ|2dy\nthat appears on the right-hand side of (94). To do this, we introdu ce a further radial, smooth cut-off\nχ∈[0,1] withχ(r) = 0 forr≤1\n2, andχ(r) = 1 forr≥3, withχ′≥0 andχ′(r)∈/bracketleftbig1\n4,1\n2/bracketrightbig\nfor\n1≤r≤2. Let\n(98) χA(r) =χ/parenleftBigr\nA/parenrightBigBLOW-UP OF THE DAMPED NLS EQUATION 23\nwhereA=A(t) is defined in (86). In the following lemma, we will choose s>0 depending also on\nσ2in order to be able to control the L2σ2+2-norm of the remainder ξ, see (103) below.\nLemma 4.7. For anyt∈[0,T1), we have\n(99)b(τ)/integraldisplay\n{A(τ)≤|y|≤2A(τ)}|ξ(τ)|2dy/lessorsimilarλ−2sc(τ)d\ndτ/parenleftbigg\nλ2sc(τ)/integraldisplay\nχA|ξ(τ)|2dy/parenrightbigg\n+Γ3\n2−10ν\nb(τ)+Γ2a\nb(τ)/ba∇dbl∇ξ(τ)/ba∇dbl2\nL2.\nProof.We take the scalar product of equation (1) with i χA/parenleftBig\nx−x(t)\nλ(t)/parenrightBig\nψ(t,x) and obtain\n1\n2d\ndt(ψ,χAψ)−(ψ,(∂tχA)ψ)−(∇xψ,i(∇xχA)ψ)+η(|ψ|2σ2ψ,χAψ) = 0.\nBy using decomposition (49) and the scaled space and time variables, we rewrite the equation above\nas\n(100)0 =1\n2λ2scd\ndτ/parenleftbig\nλ2sc(ξ,χAξ)/parenrightbig\n−(ξ,(∂τχA)ξ)−(∇ξ,i(∇χA)ξ)\n+ηλ2−2σ2\nσ1(|ξ|2σ2ξ,χAξ)+R(1)(Qb,ξ),\nwhere the gradient and the scalar products are taken with respec t to the variable\ny(τ) =x−x(τ)\nλ(τ)\nand\nR(1)(Qb,ξ) =1\n2λ2scd\ndτ/parenleftbig\nλ2sc((Qb,χAQb)+2(Qb,χAξ))/parenrightbig\n−(Qb,(∂τχA)Qb)\n−2(Qb,(∂τχA)ξ)−(∇Qb,i(∇χA)Qb)−2(∇Qb,i(∇χA)ξ)\n+ηλ2−2σ2\nσ1/parenleftbig\n(|Qb+ξ|2σ2(Qb+ξ),χA(Qb+ξ))−(|ξ|2σ2ξ,χAξ)/parenrightbig\n.\nWe now estimate the reminder term R(1)(Qb,ξ). Let us recall that for |y| ≥2\nbwe haveQb=scTb,\nwhereTbis exponentially decreasing, see Proposition 2.10. We claim that, for t he terms depending\nonly onQb, we have\n/vextendsingle/vextendsingle/vextendsingleR(2)(Qb)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2λ2scd\ndτ/parenleftbig\nλ2sc(Qb,χAQb)/parenrightbig\n−(Qb,(∂τχA)Qb)−(∇Qb,i(∇χA)Qb)\n+ηλ2−2σ2\nσ1/parenleftbig\n|Qb|2σ2Qb,χAQb)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilars2\nc+ηλ2−2σ2\nσ1s2σ2+2\nc.\nHere we used the bound\nd\ndτ/parenleftbig\nλ2sc(Qb,χAQb)/parenrightbig\n=s2\ncd\ndτ/parenleftbigg\nλ2sc/integraldisplay\nχA|Tb|2dy/parenrightbigg\n/lessorsimilars3\ncλ2sc/parenleftBigg˙λ\nλ−b+b/parenrightBigg\n/ba∇dblQb/ba∇dbl2\nL2+s2\ncλ2sc˙b∂b/ba∇dblQb/ba∇dbl2\nL2\n/lessorsimilars3\ncλ2sc/parenleftbigg\nΓ1\n2−10ν\nb−c\nlnsc/parenrightbigg\n+s2\ncλ2scΓ1\n2−10ν\nb/lessorsimilars2\nc.24 P. ANTONELLI AND B. SHAKAROV\nthat follows from the properties of Qb(32). All other terms may be estimated by\n/vextendsingle/vextendsingle/vextendsingleR(1)(Qb,ξ)−R(2)(Qb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarsc/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n.\nTo prove the two claims above, we use the following property\n|(Tb,ξ)| ≤/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n,\nthat can be proven similarly to (69) and (70). Thus by using (52), (5 3) and (48), we obtain that\n/vextendsingle/vextendsingle/vextendsingleR(1)(Qb,ξ)/vextendsingle/vextendsingle/vextendsingle/lessorsimilars2\nc+ηλ2−2σ2\nσ1s2σ2+2\nc+sc/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n/lessorsimilarΓ3\n2−10ν\nb.\nIn particular, from (100) we get the following inequality\n1\n2λ2scd\ndτ/parenleftbig\nλ2sc(ξ,χAξ)/parenrightbig\n≥(ξ,(∂τχA)ξ)+(∇ξ,i(∇χA)ξ)\n−ηλ2−2σ2\nσ1(|ξ|2σ2ξ,χAξ)−Γ3\n2−10ν\nb.\nFrom straightforward computations, we see that\n∂τχA(τ)/parenleftbiggx−x(τ)\nλ(τ)/parenrightbigg\n=−1\nA/parenleftBigg\n˙x\nλ+/parenleftBigg˙λ\nλ+˙A\nA/parenrightBigg\ny/parenrightBigg\n·∇χ/parenleftbiggx−x(τ)\nλ(τ)A(τ)/parenrightbigg\n.\nand consequently, we can rewrite the inequality above as\n(101)1\n2λ2scd\ndτ/parenleftbig\nλ2sc(ξ,χAξ)/parenrightbig\n≥b(ξ,y·∇χAξ)\n−/parenleftBigg\nξ,1\nA/parenleftBigg\n˙x\nλ+/parenleftBigg˙λ\nλ+b/parenrightBigg\ny+˙A\nAy/parenrightBigg\n·(∇χ)ξ/parenrightBigg\n+(∇ξ,i(∇χA)ξ)−ηλ2−2σ2\nσ1/integraldisplay\nχA|ξ|2σ2+2dy−Γ3\n2−10ν\nb.\nThe definition of χA(98) implies the following chain of estimates\n(102)1\n8/integraldisplay\n{A≤|y|≤2A}|ξ|2dy≤1\n2/integraldisplay\n{A≤|y|≤2A}χ′(|y|\nA)|ξ|2dy≤1\n2/integraldisplay\n{A≤|y|≤2A}|y|\nAχ′(|y|\nA)|ξ|2dy\n=1\n2/integraldisplay\n{A≤|y|≤2A}y\nA·∇χ(|y|\nA)|ξ|2dy.\nWe will now exploit it to bound the terms in inequality (101). For the firs t term, we can easily infer\nthe following bound\nb/integraldisplay\ny·∇χA|ξ|2dy≥b\n8/integraldisplay\n{A≤|y|≤2A}|ξ|2dy.\nFor the second term, the definition of A(t) (86) and estimate (25) for Γ ballow us to infer\n˙A\nA=−ac˙b\nb2,\nwhich yields\n1\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg˙λ\nλ+b+˙A\nA/parenrightBigg/integraldisplay\ny·∇χ|ξ|2dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓa\nbΓ1−20ν\nb/integraldisplay\nA≤|y|≤2A|ξ|2dy,BLOW-UP OF THE DAMPED NLS EQUATION 25\nwhere we have used (80) and (102). By using (81) we may analogous ly estimate\n1\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay˙x\nλ·∇χ|ξ|2dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓa\nb/parenleftbigg\n(/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy)1/2+Γ1−20ν\nb/parenrightbigg/integraldisplay\nA≤|y|≤2A|ξ|2dy\n/lessorsimilarΓa\nbΓ1\n2−10ν\nb/integraldisplay\nA≤|y|≤2A|ξ|2dy,\nwhere we used (56) in the last inequality.\nFor the third term on the right-hand side of (101) we use Young’s ine quality and get\n1\nA|(∇ξ,i(∇χ)ξ)| ≤1\nA/ba∇dbl∇ξ/ba∇dblL2/parenleftBigg/integraldisplay\nA≤|y|≤2A|ξ|2dy/parenrightBigg1/2\n≤40\nbA2/ba∇dbl∇ξ/ba∇dbl2\nL2\n+b\n40/integraldisplay\nA≤|y|≤2A|ξ|2dy.\nFinally, we consider the contribution coming from the nonlinear dampin g. By recalling that σ2>\nσ∗= 2sc/(d−2sc), we have\nsc<d\n2−d\n2σ2+2=dσ2\n2σ2+2.\nThis implies that we can choose ssuch that\nsc<s<dσ2\n2σ2+2=s(σ2).\nConsequently, we can interpolate the space ˙Hs(σ2)(Rd) between ˙H1(Rd) and˙Hs(Rd) and obtain\nthat\n(103) /ba∇dblξ/ba∇dbl2σ2+2\nL2σ2+2/lessorsimilar/ba∇dbl|∇|s(σ2)ξ/ba∇dbl2σ2+2\nL2/lessorsimilar/ba∇dblξ/ba∇dblθ(2σ2+2)\n˙H1/ba∇dblξ/ba∇dbl(1−θ)(2σ2+2)\n˙Hs\nfor someθ(s)∈(0,1). Now from (56) and (55) it follows that\n/ba∇dblξ/ba∇dblθ(2σ2+2)\n˙H1/ba∇dblξ/ba∇dbl(1−θ)(2σ2+2)\n˙Hs ≤Γ(1−50ν)(σ2+1)+30 νθ(1+σ2)\nb.\nThus, by collecting everything, we have that\n(104)1\n2λ2scd\ndτ/parenleftbig\nλ2sc(ξ,χAξ)/parenrightbig\n≥/parenleftbiggb\n8−cΓa\nbΓ1\n2−10ν\nb−b\n40/parenrightbigg/integraldisplay\n{A≤|y|≤2A}|ξ|2dy\n−Γ2a\nb/ba∇dbl∇ξ/ba∇dbl2\nL2−Γ3\n2−10ν\nb\n−ηλ2−2σ2\nσ1Γ(1−50ν)(σ2+1)+30 νθ(1+σ2)\nb.\nInequality (99) is a simple consequence of the choice of η(48). /square\nLet us emphasize that the assumption σ2> σ∗, see (3), is required to obtain the bound (103).\nAs already remarked, this assumption is related to the fact that it is not possible to control Sobolev\nnorms ofξrougher than the critical norm ˙Hsc. Consequently, our argument cannot be applied for\ninstance to the linearly damped NLS equation (equation (1) with σ2= 0) since we would need to\nestimate the term\nηλ2/integraldisplay\nχA|ξ|2dy.\nBy using estimates (94) and (99), it is possible to define a Lyapunov f unctional that provides an\nupper bound for ˙b. In the next lemma, it will be fundamental to exploit the decay of the total mass.\nWe notice that this fact suggests that a different regularization of the focusing NLS dynamics could26 P. ANTONELLI AND B. SHAKAROV\nnot yield the same result.\nIn what follows we define\n(105)J(τ) =/integraldisplay\n(1−χA(τ))|ξ(τ)|2dy+/ba∇dblQb(τ)/ba∇dbl2\nL2−/ba∇dblQ/ba∇dbl2\nL2+2(ξ(τ),Qb(τ))\n−b(τ)f(τ) +/integraldisplayb(τ)\n0f(v)dv,\nwherefis defined in (95) and Qis the unique positive solution to (8).\nLemma 4.8. There exist a1=a1(ν)>0such that for any a < a1and anyt∈[0,T1), there exist\nc>0such that\n(106)d\ndτJ(τ)/lessorsimilarb(τ)sc+Γ2a\nb(τ)/ba∇dbl∇ξ(τ)/ba∇dbl2\nL2\n−b(τ)/parenleftbigg\nΓb(τ)+/integraldisplay\n|∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy/parenrightbigg\n.\nProof.By multiplying inequality (94) by b, we obtain\n(107) C2b/parenleftbigg/integraldisplay\n|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg\n≤Cb/parenleftbiggd\ndτf+sc/parenrightbigg\n+b/integraldisplay\n{A≤|y|≤2A}|ξ|2dy.\nThe last term on the right-hand side of (107) may be estimated by (9 9), so that\n(108) b/integraldisplay\n{A(τ)≤|y|≤2A(τ)}|ξ|2dy/lessorsimilarλ−2scd\ndτ/parenleftbigg\nλ2sc/integraldisplay\nχA|ξ|2dy/parenrightbigg\n+Γ3\n2−10ν\nb+Γ2a\nb/ba∇dbl∇ξ/ba∇dbl2\nL2.\nIn order to find a satisfactory bound on the first term on the right -hand side of the inequality above,\nwe exploit the fact that the total mass is non-increasing. By writing\nd\ndτ/ba∇dblψ/ba∇dbl2\nL2=d\ndτ/parenleftbigg\nλ2sc/integraldisplay\n(χA+(1−χA))|ξ|2+|Qb|2dy+2λ2sc(ξ,Qb)/parenrightbigg\n≤0,\nwe have that\n(109)λ−2scd\ndτ/parenleftbigg\nλ2sc/integraldisplay\nχA|ξ|2dy/parenrightbigg\n≤ −d\ndτ/parenleftbigg/integraldisplay\n(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg\n−2sc˙λ\nλ/parenleftbigg/integraldisplay\n(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg\n.\nFrom the definition of χAgiven in (98) and by Hardy’s inequality, we have\n/integraldisplay\n(1−χA)|ξ|2≤/integraldisplay\n|y|≤3A|y|2\n|y|2|ξ|2dy/lessorsimilarA2/ba∇dbl∇ξ/ba∇dbl2\nL2.\nSimilarly, we can obtain a comparable bound in dimensions one and two, s ee [34, Appendix C]. In\nparticular, for any dimension we conclude that\n(110)/integraldisplay\n(1−χA)|ξ|2/lessorsimilarA2/integraldisplay\n|∇ξ|2+|ξ|e−|y|dy.BLOW-UP OF THE DAMPED NLS EQUATION 27\nThus, by using estimates (80), (74) and (56) we can bound the sec ond term on the right-hand side\nof (109) by/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc˙λ\nλ/parenleftbigg/integraldisplay\n(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc/parenleftBigg˙λ\nλ+b/parenrightBigg/parenleftbigg/integraldisplay\n(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsinglescb/parenleftbigg/integraldisplay\n(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilarsc/parenleftbig\nΓ1−20ν\nb+b/parenrightbig/parenleftbigg\n(1+A2)/integraldisplay\n|∇ξ|2+|ξ|e−|y|dy+/ba∇dblQb/ba∇dbl2\nL2/parenrightbigg\n.\nNow we want that\n(111) A2/integraldisplay\n|∇ξ|2+|ξ|e−|y|dy≤A2Γ1−20ν\nb= Γ−2a+1−20ν\nb/lessorsimilar1,\nwhere we used the definition of Ain (86) and (56). That is amust satisfy the following inequality\na≤1−20ν\n2.\nThis implies that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc˙λ\nλ/parenleftbigg/integraldisplay\n(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarscb.\nBy combining (108) with (109), we obtain\nb/integraldisplay\n{A≤|y|≤2A}|ξ|2dy/lessorsimilar−d\ndτ/parenleftbigg/integraldisplay\n(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg\n+scb+Γ3\n2−10ν\nb+Γ2a\nb/ba∇dbl∇ξ/ba∇dbl2\nL2.\nBy plugging the above inequality into (107), we infer\n(112)b/parenleftbigg/integraldisplay\n|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg\n/lessorsimilarbd\ndτf\n−d\ndτ/parenleftbigg/integraldisplay\n(1−χA)|ξ|2dy+/ba∇dblQb/ba∇dbl2\nL2+2(ξ,Qb)/parenrightbigg\n+scb+Γ3\n2−10ν\nb+Γ2a\nb/ba∇dbl∇ξ/ba∇dbl2\nL2.\nFinally, let us consider f=f(τ) as defined in (95). By the monotonicity property of b, see (85) for\ninstance, we denote - by some abuse of notation - f=f(b(τ)). In this way, we may write\nbd\ndτf=d\ndτ(bf)−d\ndτ/integraldisplayb\n0f(v)dv.\nLet us now recall the definition of J(τ) given in (105), we have\nJ(τ) =/integraldisplay\n(1−χA(τ))|ξ(τ)|2dy+/ba∇dblQb(τ)/ba∇dbl2\nL2−/ba∇dblQ/ba∇dbl2\nL2+2(ξ(τ),Qb(τ))\n−b(τ)f(τ)+/integraldisplayb(τ)\n0f(v)dv.28 P. ANTONELLI AND B. SHAKAROV\nBy using the previous identities and (52), we see that (112) implies th e following estimate\n(113) b/parenleftbigg/integraldisplay\n|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg\n/lessorsimilar−d\ndτJ+bsc+Γ2a\nb/ba∇dbl∇ξ/ba∇dbl2\nL2,\nwhich readily gives (106). /square\nLet us now discuss how the functional Jis related to the control parameter b. First, we define\nK(τ) as\n(114) K(τ) =/ba∇dblQb(τ)/ba∇dbl2\nL2−/ba∇dblQ/ba∇dbl2\nL2−b(τ)f(τ) +/integraldisplayb(τ)\n0f(v)dv\nwhereQis the ground state profile to (1), see (8). In this way, we obtain th at\nJ(τ)−K(τ) =/integraldisplay\n(1−χA(τ))|ξ(τ)|2dy+2(ξ(τ),Qb(τ)).\nThus, by using (110) and (73) we get\n(115) J(τ)−K(τ)/lessorsimilar(1+A2)/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy+Γ1−ν\nb.\nWe stress that to provide a satisfactory upper bound on the differ enceJ−K, it is not sufficient to\nconsider the weaker bound (74) in the inequality above. Indeed, in t he proof of the bootstrap in the\nnext section, we will first show that the control (56) is not satisfa ctory to obtain the better bound\n(61).\nTo obtain a lower bound we observe that, by using (77), we have\nJ−K=/integraldisplay\n(1−χA)|ξ|2dy−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2\nL2\n+2(iscβ∂bQb+ibΛQb+Ψb,ξ)−/integraldisplay\nR(2)(ξ)dy\nwhereR(2)(ξ) is defined in (76). We rewrite the equation above as\n(116)J−K=−2λ2−2scE[ψ]+2E[Qb]+2(iscβ∂bQb+Ψb,ξ)\n+(L(1)ξ,ξ)−/integraldisplay\nχA|ξ|2dy−1\nσ1+1/integraldisplay\nR(3)(ξ)dy\nwhere\n(117) L(1)ξ=−∆ξ+ξ−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ\nand\nR(3)(ξ) =|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)\n−(2σ1+2)|Qb|2σ1−2/parenleftbig\n|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig\n.\nWe recall the coercivity property\n(L(1)ξ,ξ)−/integraldisplay\nχA|ξ|2dy≥C/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy−Γ3\n2−10ν\nb\nwhich was proved in [34, Appendix D]. Then in (116) by further using (54), (32), (70) and (71) to\nbound the rests as /vextendsingle/vextendsingle/vextendsingle/vextendsingle2(iscβ∂bQb+Ψb,ξ)−1\nσ1+1/integraldisplay\nR(3)(ξ)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓbBLOW-UP OF THE DAMPED NLS EQUATION 29\nwe obtain\n(118) J−K/greaterorsimilar/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy−Γb−sc.\nBy combining (118) and (115), we obtain that\n(119)/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy−Γb−sc/lessorsimilarJ−K/lessorsimilar(1+A2)/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy+Γ1−ν\nb.\nNow we want to prove that K(τ) is a small perturbation of b2(τ). By slightly abusing notations\nagain, we write K(τ) =K(b(τ)). The following lemma can be found in [36, Section 4].\nLemma 4.9. There exists b(1)>0such that for any 0<b<b 1<b(1), we have\n(120) K(b)−K(b1)/lessorsimilarsc\nand\n(121) b2−sc/lessorsimilarK(b)/lessorsimilarb2+sc.\nThe proofof this lemma follows from the properties of Qbstated in (32), the definition of fin (95)\nthe decomposition (90), and the smallness of the outgoing radiation (89). Observe that by collecting\ntogether (119) and (121), we have that J(τ) is close to b2(τ) up to smaller order corrections and\nup to the term A2/integraltext\n|ξ|2e−|y|+|∇ξ|2dy. Consequently, if aandbare small enough, we obtain the\nfollowing inequality\n(122)/vextendsingle/vextendsingleJ(τ)−db2(τ)/vextendsingle/vextendsingle/lessorsimilarΓ1−20ν\nb(τ)+A2/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy≤νb2(τ).\n5.Proof of the Bootstrap\nIn this section, we are going to prove Proposition 3.3. We proceed in t he following order.\n(1) Using the monotonicity properties (85) and (106), we refine th e control over the remainder\nξas in (61).\n(2) Inequalities (85) and (106) also imply the dynamical trapping of b(57). In particular, bis\nalmost constant and close to the value b∗(sc)>0 defined in (9).\n(3) From (80), we obtain the equation for the scaling parameter\n˙λ\nλ∼ −b+Γb.\nThe previous points yield (58) and a precise law for λ.\n(4) By finding suitable bounds on the time derivatives of the energy a nd the momentum, we\nwill prove (59).\n(5) Finally, we will deduce the ˙Hs-norm control of ξin (60).\nWe shall stress that the first three points of our scheme are cons equences of the local virial law\n(85) and the monotonicity formula (106) and their proofs are very similar to those in [36]. On the\nother hand, in the undamped case, the fourth point comes natura lly from the conservation of the\nenergy and the momentum. In our case, we will show that the growt h of the time-dependent energy\nand momentum (see equations (18) and (17)) can be controlled unt il the blow-up time by our choice\nofη. Finally, with respect to the proof in [36], the fifth point requires an a dditional change to treat\nthe newdissipative term. Forthe convenienceofthe reader, we will restatethe bootstrapproposition\nbelow.30 P. ANTONELLI AND B. SHAKAROV\nProposition 5.1. There exists s∗\nc>0,s∗>s∗\nc,ν∗>0anda∗(ν∗)>a∗(ν∗)>0, such that for any\nsc<s∗\nc,sc<s<s∗,ν <ν∗anda∗<a<a∗and for any t∈[0,T1),the following inequalities are\ntrue:\nΓ1+ν4\nb(t)≤sc≤Γ1−ν4\nb(t), (123)\n0≤λ(t)≤Γ20\nb(t), (124)\nλ2−2sc|E[ψ(t)]|+λ1−2sc|P[ψ(t)]| ≤Γ3−10ν\nb(t), (125)\n/integraldisplay\n||∇|sξ(t)|2≤Γ1−45ν\nb(t), (126)\n/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−10ν\nb(t). (127)\nProof.We chooses∗\ncto be the minimum of all the conditions for scfound in previous sections. Then\nwe fixsc<s∗\ncand we choose ssuch that\nsc<s<min/parenleftbiggdσ1\n2σ1+2,dσ2\n2σ2+2,1\n2/parenrightbigg\nsuch that (103), (78) and (79) are true. Now we choose an initial c onditionψ0∈ O, where the set O\nis defined in 3.1. Consequently, there exists a time interval [0 ,T1] where the estimates in Proposition\n3.2 are true. In particular, by choosing scsmall enough we obtain that b(t) is small for any t∈[0,T1]\nand also that Γ b(t)<1.\nWe start by proving inequality (127). We proceed by contradiction. First, suppose that there exists\nτ0∈[0,τ−1(T1)) such that\n/integraldisplay\n|∇ξ(τ0)|2+|ξ(τ0)|2e−|y|dy>Γ1−10ν\nb(τ0).\nNotice that the initial condition satisfies (45), and in particular, sinc e Γb<1, for anyt∈[0,T1), see\n(52), we have/integraldisplay\n|∇ξ(0)|2+|ξ(0)|2e−|y|dy<Γ1−7ν\nb(0),\nThen by continuity of ξ,band Γb, there exists a time interval [ τ1,τ2]⊂[0,τ0) such that\n(128)/integraldisplay\n|∇ξ(τ1)|2+|ξ(τ1)|2e−|y|dy= Γ1−7ν\nb(τ1),\n/integraldisplay\n|∇ξ(τ2)|2+|ξ(τ2)|2e−|y|dy= Γ1−10ν\nb(τ2),\nand for any τ∈[τ1,τ2]\n(129)/integraldisplay\n|∇ξ(τ)|2+|ξ(τ)|2e−|y|dy≥Γ1−7ν\nb(τ).\nThe virial estimate (85), controls (52), (53) and (54) imply that fo r anyτ∈[τ1,τ2], we have\n˙b/greaterorsimilarsc+/integraldisplay\n|∇ξ(τ)|2+|ξ(τ)|2e−|y|dy−Γ1−ν6\nb/greaterorsimilarΓ1+ν2\nb+Γ1−7ν\nb−Γ1−ν6\nb>0,\nforνsmall enough and hence\n(130) b(τ2)≥b(τ1).\nWe notice that the definition of Γ b(25) implies also that\n(131) Γ b(τ2)≥Γb(τ1).BLOW-UP OF THE DAMPED NLS EQUATION 31\nOn the other hand, from the smallness of ˜ζb(89), we obtain the inequality\n(132)/integraldisplay\n|∇˜ξ|2+|˜ξ|2e−|y|dy≥1\n2/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy−Γ1−ν\nb\nwhich combined with the second monotonicity estimate (106) and (52 ) yields\nd\ndτJ/lessorsimilarbsc+Γ2a\nb/ba∇dbl∇ξ/ba∇dbl2\nL2−bΓb−b/integraldisplay\n|∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy\n/lessorsimilarb/parenleftbigg\nsc−Γb−1\n2/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy+Γ1−ν\nb+Γ2a\nb\nb/ba∇dbl∇ξ/ba∇dbl2\nL2/parenrightbigg\n/lessorsimilarb/parenleftBig\nΓ1−ν2\nb−Γb−Γ1−7ν\nb+Γ1−ν\nb/parenrightBig\n≤0,\nforνsmall enough where we have chosen a>0 so that\nΓ2a\nb\nb≤1\n4.\nThus we obtain that\n(133) J(τ2)≤J(τ1).\nNext, using (119) and (133) we get\nK(τ2)+/integraldisplay\n|ξ(τ2)|2e−|y|+|∇ξ(τ2)|2dy−Γb(τ2)−sc/lessorsimilarJ(τ2)≤J(τ1)\n/lessorsimilarK(τ1)+(1+A2)/integraldisplay\n|ξ(τ1)|2e−|y|+|∇ξ(τ1)|2dy+Γ1−ν\nb(τ1).\nEquivalently, by using (128) we get\n/integraldisplay\n|ξ(τ2)|2e−|y|+|∇ξ(τ2)|2dy= Γ1−10ν\nb(τ2)\n/lessorsimilarK(τ1)−K(τ2)+sc+Γb(τ2)+(1+A2)Γ1−7ν\nb(τ1)+Γ1−ν\nb(τ1).\nOn the right-hand side of the equation above, we use (120) to get\n|K(τ1)−K(τ2)|/lessorsimilarsc.\nThus using the inequality sc≤Γ1−ν2\nb(τ1)(52) and the definition of A(86), we obtain\nΓ1−10ν\nb(τ2)/lessorsimilarΓ1−ν2\nb(τ1)+Γb(τ2)+Γ1−7ν−2a\nb(τ1)+Γ1−ν\nb(τ1).\nNow we suppose that a≤ν\n2. This implies that there exists C >0 such that\nΓ1−10ν\nb(τ2)≤CΓ1−8ν\nb(τ1).\nBy exploiting (131), and for bsmall enough, this implies that\nΓ1−10ν\nb(τ2)≤CΓ1−8ν\nb(τ1)≤CΓ1−8ν\nb(τ2)=CΓν\nb(τ2)Γ1−9ν\nb(τ2)≤Γ1−9ν\nb(τ2),\nwhich is a contradiction because Γ b(τ2)<1.\nAs our next step, we will prove (123). Assume by contradiction tha t there exists τ0∈[0,τ−1(T1)]\nsuch thatsc>Γ1−ν4\nb(τ0).By continuity, this implies that there exists τ1<τ0such that\nsc= Γ1−ν5\nb(τ1)\nand for any τ∈[τ1,τ0],\nsc>Γ1−ν5\nb(τ).32 P. ANTONELLI AND B. SHAKAROV\nIn particular, we have that\nd\ndτΓb(τ1)<0\nwhich is equivalent to ˙b(τ1)<0 from the definition of Γ bin (25). On the other hand, the local virial\ninequality (85) implies that\n˙b(τ1)/greaterorsimilarsc−Γ1−ν6\nb(s6)/greaterorsimilarΓ1−ν5\nb(s6)−Γ1−ν6\nb(s6)>0,\nwhich is a contradiction.\nSimilarly, suppose by contradiction that there exists τ0∈[0,τ−1(T1)) such that sc<Γ1+ν4\nb(τ0).Then,\nby using (122), we get that\nsc≤Γ1+ν5\n/radicalBig\nJ(τ0)\nd\nNow letτ1∈[0,τ0] be the largest time such that\nsc= Γ1+ν6\n/radicalBig\nJ(τ1)\nd\nThus, by definition of τ1and Γb(25) we obtain that\nd\ndτJ(τ1)≥0,\nwhile from the monotonicity formula (106) and (132) we obtain\nd\ndτJ(τ1)/lessorsimilarb(τ1)/parenleftBigg\nsc−Γb(τ1)−1\n2/integraldisplay\n|∇ξ(τ1)|2+|ξ(τ1)|2e−|y|dy+Γ1−ν\nb(τ1)+Γ2a\nb(τ1)\nb(τ1)/ba∇dbl∇ξ(τ1)/ba∇dbl2\nL2/parenrightBigg\n/lessorsimilarb(τ1)/parenleftBigg\nΓ1+ν6\n/radicalBig\nJ(τ1)\nd−Γ1+ν2\n/radicalBig\nJ(τ1)\nd/parenrightBigg\n≤0,\nwhich is a contradiction.\nThe next step will be to proveinequality (124). From the control (1 23), it followsthat the parameter\nbis dynamically trapped around b0, that is, for any t∈[0,T1), we have the following upper and\nlower bounds\n1−ν4\n1+ν10b0≤b(t)≤1+ν4\n1−ν10b0.\nThus, we notice that the control on the parameter (80), and ineq uality (61) imply that for any\nt∈[0,T1)\n0<1−2ν4\n1+ν10b0≤ −˙λ\nλ=−1\n2d\ndtλ2≤1+2ν4\n1−ν10b0,\nwhere we used that\nd\ndτλ=˙λ=λ2d\ndtλ\nfrom (64). Equivalently, we obtain the law of the parameter λ\n(134) λ2\n0−2/parenleftbigg1+2ν4\n1−ν10/parenrightbigg\nb0t≤λ2(t)≤λ2\n0−2/parenleftbigg1−2ν4\n1+ν10/parenrightbigg\nb0t.\nin particular, λ(t) is a non-increasing function of time. This estimate and the dynamica l trapping\nof the parameter b(123) imply inequality (124).BLOW-UP OF THE DAMPED NLS EQUATION 33\nThe next step is to obtain the bound on the total energy Eand momentum P(125). We start by\nderiving a suitable bound for the energy functional. We use (17) to o btain that\nE[ψ(t)] =E[ψ0]+η/integraldisplayt\n0/integraldisplay\n|ψ|2(σ1+σ2+1)−|ψ|2σ2|∇ψ|2−2σ2|ψ|2σ2−2Re/parenleftbig¯ψ∇ψ/parenrightbig2dxdv\n≤E[ψ0]+η/integraldisplayt\n0λ2sc−2σ2\nσ1−2/integraldisplay\n|Qb+ξ|2(σ1+σ2+1)dydv.\nWe notice that if d≤3 andσ2≤σ1<1\n(d−2)+or ifd≥4, andσ2< σ∗, thenH1(Rd)֒→\nL2(σ1+σ2+1)(Rd). Thus, for scsmall enough we can use the Jensen inequality, (61), (55) and inter -\npolation\n/ba∇dblξ/ba∇dbl2(σ1+σ2+1)\nL2(σ1+σ2+1)/lessorsimilar/ba∇dblξ/ba∇dbl2θ(σ1+σ2+1)\n˙H1 /ba∇dblξ/ba∇dbl2(1−θ)(σ1+σ2+1)\n˙Hs /lessorsimilar1\nfor someθ(σ1,σ2,s)∈(0,1), to obtain\n/ba∇dblQb+ξ/ba∇dbl2(σ1+σ2+1)\nL2(σ1+σ2+1)≤C/parenleftBig\n/ba∇dblQb/ba∇dbl2(σ1+σ2+1)\nL2(σ1+σ2+1)+/ba∇dblξ/ba∇dbl2(σ1+σ2+1)\nL2(σ1+σ2+1)/parenrightBig\n≤C\nwhereC=C(σ1,σ2,b∗)>0. If follows that\nλ2−2scE[ψ(t)]≤λ2−2scE[ψ0]+Cηλ2−2sc/integraldisplayt\n0λ2sc−2σ2\nσ1−2dv. (135)\nNow we observe that, from (134), there exists a constant 0 < c=c(ν,b0)≪1 such that, for any\nt∈[0,T1),\nλ2\n0−2(1+c)t≤λ2(t)≤λ2\n0−2(1−c)t.\nIntegrating λin time, it is straightforward to see that for any α∈R, we have\n(136)/integraldisplayt\n0λ(τ)αdv/lessorsimilarλ(t)α+2.\nThus, from (135), it follows that\n(137) λ2−2scE[ψ(t)]≤λ2−2scE[ψ0]+Cηλ2−2σ2\nσ1.\nFrom (58) and (43) we have\nλ2−2sc|E[ψ0]| ≤Γ50\nb.\nMoreover, we bound the last term using the smallness of η(48), (57) and (57)\nCηλ2−2σ2\nσ1≤Cs3\nc≤Γ3−9ν\nb\nforνsmall enough. This implies the bootstrapped control on the energy\nλ2−2sc|E[ψ(t)]| ≤Γ3−10ν\nb.\nWe use the same procedure to bound the momentum functional P. From (18), we have\nP[ψ(t)] =P[ψ0]+2η/integraldisplayt\n0/integraldisplay\n|ψ|2σ2Im(¯ψ∇ψ)dxdτ\n=P[ψ0]+2η/integraldisplayt\n0λ−2σ2\nσ1+2sc−1/ba∇dblQb+ξ/ba∇dbl2σ2+1\nL4σ2+2/ba∇dbl∇(Qb+ξ)/ba∇dblL2dv.\nAgain, ifd≤3,scis small enough and σ2≤σ1or ifd≥4, andσ2<σ∗, then we use use the Jensen\ninequality, (61), (55), interpolation and (136) to prove that ther e existsC=C(σ2,b∗)>0 such that\n(138) λ1−2scP[ψ(t)]≤λ1−2scP[ψ0]+Cηλ2−2σ2\nσ1.34 P. ANTONELLI AND B. SHAKAROV\nConsequently, (43), (48) and (57) imply that\nλ1−2scP[ψ]≤Γ3−10ν\nb,\nforνsmall enough. This concludes the proof of (59).\nThe last step is to obtain the ˙Hs-norm control (126). We define\nˆQ(t,x) =λ−1\nσ1Qb/parenleftbiggx−x(t)\nλ/parenrightbigg\neiγ,\nˆξ(t,x) =λ−1\nσ1ξ/parenleftbigg\nt,x−x(t)\nλ/parenrightbigg\neiγ,\nwe decompose the solution as\nψ=ˆQ+ˆξ.\nFrom (1), we see that the function ˆξsatisfies the equation\n(139) i ˆξt+∆ˆξ=−E(ˆQ)−N1(ˆξ)−iηN2(ˆξ),\nwhereEdoes not depend on ˆξ\nE(ˆQ) = i∂tˆQ+∆ˆQ+|ˆQ|2σ1ˆQ+iη|ˆQ|2σ2˜Q\nand\nN1(ˆξ) =|ˆQ+ˆξ|2σ1(ˆQ+ˆξ)−|ˆQ|2σ1ˆQ,\nN2(ˆξ) =|ˆQ+ˆξ|2σ2(ˆQ+ˆξ)−|ˆQ|2σ2ˆQ.\nUsing the equation satisfied by Qb(29), we obtain that\nE(ˆQ) =1\nλ2+1\nσ1eiγ/parenleftBigg\n−Ψb+i(˙b−βsc)∂bQb−i/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛQb−i˙x\nλ·∇Qb\n−(˙γ−1)Qb+ iηλ2−2σ2\nσ1|Qb|2σ2Qb/parenrightBig\n.\nWe notice that inequality (126) is equivalent to\n(140)/integraldisplay\n||∇|sˆξ|2dx≤λ2(sc−s)Γ1−45ν\nb\nsince /integraldisplay\n||∇|sˆξ|2dx=λ2(sc−s)/integraldisplay\n||∇|sξ|2dy,\nand thus we will now prove (140). We define for j= 1,2 where\n(141) rj=d(2σj+2)\nd+2sσj, γj=4(σj+1)\nσj(d−2s),2\nγj=d\n2−d\nrj.\nWe notice that ( γj,rj) are Strichartz admissible pairs (see Definition 2.1). Now we write equ ation\n(139) in the integral form\nˆξ(t) =ei∆tˆξ(0)+/integraldisplayt\n0ei(t−τ)∆/parenleftBig\nE(Qb)+N1(ˆξ)+iηN2(ˆξ)/parenrightBig\ndτ.\nand we use the Strichartz estimates in Theorem 2.2 to obtain\n(142)/ba∇dbl|∇|sˆξ/ba∇dblL∞([0,T1],L2)/lessorsimilar/ba∇dbl|∇|sˆξ0/ba∇dblL2+/ba∇dbl|∇|sE/ba∇dblL1([0,T1],L2)\n+/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1+η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,t],Lr2).BLOW-UP OF THE DAMPED NLS EQUATION 35\nFrom the initial bound (45), we have\nλ2(s−sc)/integraldisplay\n||∇|sˆξ(0)|2dx=/integraldisplay\n||∇|sξ0|2dy≤Γ1−ν\nb0,\nand thus the bootstrapped controls on b, see (57) and λ, see (58) imply\n(143) /ba∇dbl|∇|sˆξ0/ba∇dbl2\nL2≤Γ1−ν\nb0λ2(sc−s).\nWe claim that the remaining terms in (142) are bounded as\n(144) /ba∇dbl|∇|sE/ba∇dbl2\nL1([0,T1],L2)≤Γ1−15ν\nbλ2(sc−s),\nand\n(145) /ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1([0,T1],Lr1)+η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,T1],Lr2)≤Γ1\n2(1−41ν)\nbλsc−s.\nNotice that (144) and the bound on N1(ˆξ) in (145) has been already proven in [36, Section 4], up\nto the term coming from the damping in E. In particular, from the estimates on the parameters\n(81), (80), the bound on Ψ b(31), and the bootstrap bounds for b,λandξ, see (57), (58) and (61)\nrespectively, and from the smallness condition on ηin (48) there holds for any t∈[0,T1)\n/ba∇dbl|∇|sE(ˆQ)/ba∇dblL2/lessorsimilar1\nλ2+s−sc/parenleftbigg/integraldisplay\n|∇ξ|2dy+/integraldisplay\n|ξ|2e−|y|dy+Γ1−11ν\nb+ηλ2−2σ2/σ1/parenrightbigg1\n2\n/lessorsimilarΓ1\n2(1−12ν)\nb\nλ2+s−sc.(146)\nWe use inequality (136) to integrate (146) in time which yields\n/integraldisplayt\n0/ba∇dbl|∇|sE(ˆQ)/ba∇dblL2dτ/lessorsimilar/integraldisplayt\n0Γ1\n2(1−12ν)\nb0\nλ2+(s−sc)dτ/lessorsimilarΓ1\n2(1−12ν)\nb0\nλs−sc.\nLastly, we will show inequality (145). The main ingredient is the inequalit y\n(147) /ba∇dbl|∇|sNj(ˆξ)/ba∇dblLr′\nj/lessorsimilarλ(2σj+1)(sc−s+2/γj)/ba∇dbl|∇|s+2/γjξ/ba∇dblL2\nfor anyj= 1,2 which has been proven in [36, Appendix] and will be shown in Appendix C. Notice\nthat since\ns+2/γj=s+(d−2sc)σj\n2σj+2,\nfor anyj= 1,2, then if we choose sto be close enough to sc, we haves<s+2/γj<1. In particular,\nwe can interpolate and use (61) and (55) to obtain\n/ba∇dbl|∇|s+2/γjξ/ba∇dbl2\nL2/lessorsimilar/ba∇dbl|∇|sξ/ba∇dbl2θj\nL2/ba∇dbl∇ξ/ba∇dbl2(1−θj)\nL2≤Γ1−(10+40θj)ν\nb\nwhere\nθj=1−s−2/γj\n1−s.\nForj= 1, we choose sc≪1 small enough and sclose enough to scto have\n/ba∇dbl|∇|s+2/γjξ/ba∇dblL2/lessorsimilarΓ1\n2(1−40ν)\nb0.\nIt follows that\n/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1/lessorsimilarΓ1\n2(1−40ν)\nb0/parenleftbigg/integraldisplay\nλγ′\n1(2σ1+1)(sc−s+2/γj)/parenrightbigg1\nγ′\n1.\nSince\nγ′\n1(2σ1+1)(s+2/γ1) = 2+γ′\n1(s−sc),36 P. ANTONELLI AND B. SHAKAROV\nwe can use (136) to integrate in time and obtain the first inequality in ( 145)\n/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1/lessorsimilarΓ1\n2(1−41ν)\nb0\nλ(s−sc).\nMoreover, we make the same computations for j= 2 and use the control on η(48) and (57) to\nconclude that\n(148) η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,T1],Lr2)≤ηΓ1−(10+40θ2)ν\nbλsc−s≤Γ2−50ν\nbλsc−s.\n/square\nThus Proposition 3.3 has been proven for any t∈[0,T1). Consequently, we can extend by\ncontinuity the time interval of the self-similar regime, that is the time interval where the bounds in\nProposition 3.2 are true. Recursively, we extend the self-similar reg ime to the whole time interval\n[0,Tmax) whereTmax(ψ0) is the maximal time of existence of the solution stemming from ψ0. Now\nwe show that ψexperiences a collapse in finite time.\nCorollary 5.2. There exists a time Tmax<∞such that\nlim\nt→Tmaxλ(t) = 0.\nMoreover, there exists x∞∈Rdsuch that\nlim\nt→Tmaxx(t)→x∞,\nwherex(t)is defined in (50).\nProof.Letψ0∈ OwhereOis defined in 3.1 and let Tmax≤ ∞be the maximal time of existence\nof the corresponding solution to (1). Then Proposition 3.3 implies tha t for anyt∈[0,Tmax), the\nbounds on the scaling parameter in (134) are true, namely, there e xists 0< c=c(ν,b0)≪1 such\nthat\n(149) λ2\n0−2(1+c)t≤λ2(t)≤λ2\n0−2(1−c)t.\nConsequently, there exists a time 0 <T=T(λ0,ν,b0)<∞such that lim t→Tλ(t) = 0. Furthermore,\nby using decomposition (49) we also obtain that\nlim\nt→T/ba∇dbl∇ψ(t)/ba∇dbl2\nL2= lim\nt→Tλ2(sc−1)(t)/ba∇dbl∇(Qb(t)+ξ(t))/ba∇dbl2\nL2=∞.\nIndeed, in the limit above, the exponent 2( sc−1) is negative because sc<1, and for any t∈[0,T],\nQb(t)∈H1(Rd),/vextenddouble/vextenddouble∇Qb(t)/vextenddouble/vextenddouble\nL2≥C >0,\nwhile (123) and (127) imply that\n/vextenddouble/vextenddouble∇ξ(t)/vextenddouble/vextenddouble2\nL2≤Γ1−10ν\nb/lessorsimilars1−10ν\n1+ν4\nc.\nin particular, the maximal time of existence is given by T <∞. Finally, we prove the convergence\nto a blow-up point. By exploiting (81) and (149), we get\n/vextendsingle/vextendsinglex(T)−x(t)/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\ntdx\ndsds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplayT\nt1\nλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλd\ndτx/vextendsingle/vextendsingle/vextendsingle/vextendsingleds\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1\n2\n+Γ1−20ν\nb(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([0,T])/integraldisplayT\nt/parenleftbig\nλ2\n0−2(1+c)s/parenrightbig−1\n2ds.BLOW-UP OF THE DAMPED NLS EQUATION 37\nWe use (57) and (61) to obtain\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1\n2\n+Γ1−20ν\nb(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([0,T])/lessorsimilar1.\nMoreover, we compute the integral as follows\n/integraldisplayT\nt/parenleftbig\nλ2\n0−2(1+c)s/parenrightbig−1\n2ds=−2\n2+c(λ2\n0−2(1+c)s)1\n2|T\nt.\nThis implies that\nlim\nt→T/vextendsingle/vextendsinglex(T)−x(t)/vextendsingle/vextendsingle/lessorsimilarlim\nt→T(λ2\n0−2(1+c)s)1\n2|T\nt= 0.\n/square\n6.The case σ2<σ1\nIn this section, we discuss the case where the exponent of the dam ping term is strictly smaller\nthan that of the power-type nonlinearity.\nFirst, we observe that the condition which we have chosen for the d amping parameter η≤s3\ncis not\nstrictly necessary. In fact, this condition was used to prove that the damping term is of smaller order\nwith respect to other terms while the solution is in the self-similar regim e. In particular, it was used\nfor instance in (104), (84), (146), (148), (137) and (138), whe re the damping term contributions can\nbe always bounded by\nηλ2−2σ2\nσ1(t)/parenleftBigg\nC(Qb(t))+c/parenleftbigg/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1\n2/parenrightBigg\n,\nwhereC(Qb)>0 is a constant depending only on Qb. We know that in the self-similar regime, from\n(56), we have that\n/parenleftbigg/integraldisplay\n|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1\n2\n/lessorsimilarΓ1\n2−10ν\nb(t)/lessorsimilarC(Qb(t))\nforb(t) small enough. Thus, in order to prove that the damping term is sma ller than the leading\norder terms in the equation referred to above, which are of the siz e Γb, we shall have for example\nthat\nηλ2−2σ2\nσ1(t)≤Γ2\nb(t)\nfor anyt∈[0,T1). This is equivalent to asking that the scaling parameter satisfies th e inequality\nλ(t)<η−1Γσ1\nσ1−σ2\nb(t).\nNotice that the right-hand side of this inequality degenerates for σ2→σ1, and consequently, for\nσ2=σ1, the supposition on the smallness of ηbecomes necessary to carry on with the bootstrap.\nOn the other hand, for σ2<σ1, we can use this condition to proceed with the bootstrap instead of\nthat in (53). We observe that in this case, the set of initial condition s would depend on η,σ1and\nσ2. For instance, it would be necessary to choose the initial condition s atisfying, say\nλ0<η−1Γ10σ1\nσ1−σ2\nb(t)\ninstead of the weaker bound (42). In particular, the bigger ηis and the closer σ2is toσ1, the more\nfocused the initial condition needs to be to experience a collapse in fin ite time. This implies the\nfollowing.38 P. ANTONELLI AND B. SHAKAROV\nTheorem 6.1. There exists s∗\nc>0such that for any 0< sc< s∗\nc, anyσ∗< σ2< σ1whend≤3\norσ∗< σ2< σ∗whend≥4and anyη >0there exists a set Oη,σ1,σ2⊂H1(Rd)such that if\nψ0∈ Oη,σ1,σ2then the corresponding solution ψ∈C([0,Tmax),H1(Rd))to(1)develops a singularity\nin finite time.\nWe emphasize againthat ourargumentworksonlywhen σ1>σ2. Whenσ1=σ2, it is crucialthat\nσ1>2\ndandηto be small enough. On the other hand, it is already known that when σ1=σ2=2\nd,\nsolutions are global in time. This means that a solution escapes the se lf-similar regime no matter\nhow smallηandλ0are. Heuristically, this is explained by the fact that, in the self-similar regime,\nwhenσ1=2\nd, the control parameter converges to zero in time, that is b(t)→0 fort→Tmax(see\n[34] for example). Consequently, for σ2=σ1=2\nd, there would exist a time T=T(η,b0,λ0)>0\nsuch that for some t>T, the critical inequality η/lessorsimilarΓb(t)is not true. Thus the damping term would\nbecome of the leading order inside the self-similar dynamics and interr upt it.\nAs our final remark, we notice that if σ2> σ1, then heuristically the self-similar regime would be\ninterrupted. Indeed the convergence of λto zero implies that contributions of damping terms will\neventually be of the leading order in the dynamics because\nλ2−2σ2\nσ1(t)→ ∞.\n7.Appendix\n7.1.Appendix A. In this appendix we will report the proof of [36, Proposition 3 .3] that contains\ncomputations used in Lemma 4.5. We state the result below for the re ader’s convenience.\nLemma 7.1. Suppose that η= 0. Then there exists s(1)\nc>0such that for any sc< s(1)\ncand for\nanyt∈[0,T1), there exists C >0such that\n(150) ˙b≥C/parenleftbigg\nsc+/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy−Γ1−ν6\nb/parenrightbigg\n.\nWe now present some preliminary computations. We start with the fo llowing.\nLemma 7.2. LetQbbe a solution to (29). Then\n(151) (i ∂bQb,ΛQb) =−1\n4/ba∇dblxQ/ba∇dbl2\nL2(1+δ1(sc,b))+sc(iQb,∂bQb),\nwhereδ1(sc,b)→0assc,b→0andQis the unique positive solution to (8).\nProof.We recall the third property in (32)\n(152) (iQb,x·∇Qb) = (iQb,ΛQb) =−b\n2/ba∇dblxQ/ba∇dbl2\nL2(1+δ1(sc,b))\nwhereδ1(sc,b)→0 assc,b→0. By taking the derivative of (152) with respect to bwe obtain\nd\ndb(iQb,ΛQb) = (i∂bQb,ΛQb)+(iQb,Λ∂bQb) =−1\n2/ba∇dblxQ/ba∇dbl2\nL2(1+δ1(sc,b)).\nLet us also recall identity (14),\n(iQb,Λ∂bQb) =−2sc(iQb,∂bQb)+(i∂bQb,ΛQb).\nBy plugging it into the previous formula, we obtain\n(i∂bQb,ΛQb) =−1\n4/ba∇dblxQ/ba∇dbl2\nL2(1+δ1(sc,b))+sc(iQb,∂bQb).\n/squareBLOW-UP OF THE DAMPED NLS EQUATION 39\nNext, we recall the definition of the linearized operator around Qb, see (66)\n(153) Lξ= ∆ξ−ξ+ibΛξ+2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb+|Qb|2σ1ξ,\nand the nonlinear remainder term in the equation (63) for the pertu rbation, namely\nR(ξ) =|Qb+ξ|2σ1(Qb+ξ)−|Qb|2σ1Qb−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ.\nLemma 7.3. We have that\n(154)(Lξ+R(ξ),ΛQb) =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb)\n−2λ2−2scE[ψ]+2E[Qb]+H(ξ,ξ)+E(ξ,ξ)\n+1\nσ1+1/integraldisplay\nR(3)(ξ)dy+(R3(ξ),ΛQb).\nwhereHis the quadratic form defined in (37),E(ξ,ξ)satisfies the following bound\n|E(ξ,ξ)| ≤δ2(sc)/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy,\nwithδ2(sc)→0assc→0and the remainder terms R(3)(ξ),R3(ξ), are given by\n(155)R(3)(ξ) =|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)\n−(2σ1+2)|Qb|2σ1−2/parenleftbig\n|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig\n,\nand\n(156) R3(ξ) =R(ξ)−2σ1|Qb|2σ1−2/parenleftbigg\nRe(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg\n.\nProof.We observe that by using properties (13), (14) we have\n(∆ξ,ΛQb) = 2(ξ,∆Qb)+(ξ,Λ∆Qb)\nand\n(ibΛξ,ΛQb) =−2scb(iξ,ΛQb)+(ξ,ibΛ(ΛQb)).\nConsequently, by using the equation (36) satisfied by Λ Qband the definition (153), we obtain that\n(Lξ,ΛQb) = 2(ξ,∆Qb)−2scb(iξ,ΛQb)\n+(ξ,Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb+2σ1Re(¯QbΛQb)|Qb|2σ1−2Qb)\n= 2(ξ,∆Qb)−2scb(iξ,ΛQb)+2σ1(ξ,Re(¯Qb,1\nσ1Qb)|Qb|2σ1−2Qb)\n+(ξ,Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb+2σ1Re(¯Qby·∇Qb)|Qb|2σ1−2Qb)\n= 2(ξ,∆Qb+|Qb|2σ1Qb)−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb).\nNow for the first term on the right-hand side of the equation above we notice that (75) implies\n2(∆Qb+|Qb|2σ1Qb,ξ) =−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2\nL2−/integraldisplay\nR(2)(ξ)dy,\nwhere\nR(2)(ξ) =1\nσ1+1/parenleftbig\n|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)/parenrightbig\n.\nThus we arrive to the preliminary equation\n(Lξ+R(ξ),ΛQb) =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb)\n−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2\nL2−/integraldisplay\nR(2)(ξ)dy+(R(ξ),ΛQb).40 P. ANTONELLI AND B. SHAKAROV\nIt remains to extract the quadratic terms in −/integraltext\nR(2)(ξ)dy+(R(ξ),ΛQb). We write\n/integraldisplay\nR(2)(ξ)dy= 2/integraldisplay\n|Qb|2σ1−2/parenleftbig\n|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig\n+1\nσ1+1/integraldisplay\nR(3)(ξ)dy\nwhereR(3)(ξ) is the rest defined in (155). Then we also write that\n(R(ξ),ΛQb) = (R3(ξ),ΛQb)\n+2σ1/parenleftbigg\n|Qb|2σ1−2/parenleftbigg\nRe(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg\n,\n1\nσ1Qb+y·∇Qb/parenrightbigg\nwhereR3(ξ) is defined in (156). In the equation above, we notice that\n2σ1/parenleftbigg\n|Qb|2σ1−2/parenleftbigg\nRe(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg\n,1\nσ1Qb/parenrightbigg\n= 2/integraldisplay\n|Qb|2σ1−2/parenleftbig\n|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig\ndy\nthat is we can write that\n−/integraldisplay\nR(2)(ξ)dy+(R(ξ),ΛQb) =−1\nσ1+1/integraldisplay\nR(3)(ξ)dy+(R3(ξ),ΛQb)\n+2σ1/parenleftbigg\n|Qb|2σ1−2/parenleftbigg\nRe(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb\n+Qb|ξ|2/parenrightbigg\n,y·∇Qb/parenrightbigg\n.\nConsequently, by replacing Qbwith the ground state Qgenerating an error which we denote by\nE(ξ,ξ), one obtains that\n/ba∇dbl∇ξ/ba∇dbl2\nL2−/integraldisplay\nR(2)(ξ)dy+(R(ξ),ΛQb) =H(ξ,ξ)+E(ξ,ξ)\n−1\nσ1+1/integraldisplay\nR(3)(ξ)dy+(R3(ξ),ΛQb)\nwhereH(ξ,ξ) is the quadratic form defined in (37). For scsmall enough, we also obtain that E(ξ,ξ)\ncan be bounded by\n(157) |E(ξ,ξ)| ≤δ2(sc)/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy,\nwhereδ2(sc)→0 assc→0. /square\nFor details of the proof above, we refer to [36, Proposition 3 .3]. We are now ready to prove the\nlocal virial property 4.5 when η= 0.BLOW-UP OF THE DAMPED NLS EQUATION 41\nProof of Lemma 7.1. By taking the scalar product of equation (63) with Λ Qb, we obtain\n(158)0 = (i∂τξ,ΛQb)+˙b(i∂bQb,ΛQb)+(2E[Qb]−(iβsc∂bQb+Ψb,ΛQb))\n+(Lξ+R(ξ),ΛQb)−2E[Qb]\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛ(Qb+ξ)+i˙x\nλ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛξ+(˙γ−1)ξ+i˙x\nλ·∇ξ,ΛQb/parenrightBigg\n.\nWe will study the contributions of the terms in (158) separately. Fo r the first term, we observe that\n(159) (i ∂τξ,ΛQb) =d\ndτ(iξ,ΛQb)−(iξ,∂τΛQb) =d\ndτ(iξ,ΛQb)−˙b(iξ,Λ∂bQb).\nFor the second term, we use (151) to get\n˙b(i∂bQb,ΛQb) =−˙b1\n4/ba∇dblxQ/ba∇dbl2\nL2(1+δ1(sc,b))+sc(iQb,∂bQb).\nMoreover, from (70) and (56) , we bound the second term on the r ight-hand side of (159) as\n|(iξ,Λ∂bQb)|/lessorsimilar/parenleftbigg/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy/parenrightbigg1\n2\n≤Γ1\n2−10ν\nb,\nwhich implies that there exists c>0 such that\n˙b((i∂bQb,ΛQb)−(iξ,Λ∂bQb)) =˙bsc(∂bQb,iQb)−˙b/parenleftbigg/ba∇dblyQ/ba∇dbl2\nL2\n4(1+δ1(sc,b))−cΓ1\n2−10ν\nb/parenrightbigg\n.\nFor the third term in (158), we use the Pohozaev-type estimate (3 5) to obtain\n2E[Qb]−(iβsc∂bQb+Ψb,ΛQb) =sc(2E[Qb]+/ba∇dblQb/ba∇dbl2\nL2).\nFor the fourth term, we use (154) to obtain that\n(Lξ+R(ξ),ΛQb)−2E[Qb] =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb)\n−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)\n−1\nσ1+1/integraldisplay\nR(3)(ξ)dy+(R3(ξ),ΛQb)\nwhereR(3)is defined in (155). For the fifth term in (158), by straightforward computations we get\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛQb+i˙x\nλ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg\n=sc(˙γ−1)/ba∇dblQb/ba∇dbl2\nL2.\nBy combining the previous computations, we have\n(160)−˙b((i∂bQb,ΛQb)−(iξ,Λ∂bQb)) =d\ndτ(iξ,ΛQb)\n+sc/parenleftbig\n−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2\nL2+(˙γ−1)/ba∇dblQb/ba∇dbl2\nL2/parenrightbig\n−(ξ,ΛΨb)−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)+1\nσ1+1/integraldisplay\nR(3)(ξ)dy\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛξ+(˙γ−1)ξ+i˙x\nλ·∇ξ,ΛQb/parenrightBigg\n,42 P. ANTONELLI AND B. SHAKAROV\nor equivalently\n(161)˙b/parenleftbigg/ba∇dblyQ/ba∇dbl2\nL2\n4(1+δ1(sc,b))−cΓ1\n2−10ν\nb/parenrightbigg\n≥d\ndτ(iξ,ΛQb)\n+sc/parenleftBig\n˙b(∂bQb,iQb)−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2\nL2+(˙γ−1)/ba∇dblQb/ba∇dbl2\nL2/parenrightBig\n−(ξ,ΛΨb)−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)+1\nσ1+1/integraldisplay\nR(3)(ξ)dy\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛξ+(˙γ−1)ξ+i˙x\nλ·∇ξ,ΛQb/parenrightBigg\n.\nWe will now study term by term the right-hand side of (161). The firs t term vanishes for any\nt∈[0,T1) because of one orthogonality condition in (51). Furthermore, by choosingscsmall\nenough, we notice that\n/parenleftbigg/ba∇dblyQ/ba∇dbl2\nL2\n4(1+δ1(sc,b))−cΓ1\n2−10ν\nb/parenrightbigg\n≥/ba∇dblyQ/ba∇dbl2\nL2\n8.\nFor the second term, we observe that by using estimates (52), (5 6) and (80), we obtain that\nsc/parenleftBig\n˙b(∂bQb,iQb)−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2\nL2+(˙γ−1)/ba∇dblQb/ba∇dbl2\nL2/parenrightBig\n≥sc/ba∇dblQb/ba∇dbl2\nL2\n2,\nforbsmall enough. Furthermore, we use (70) to bound the term\n|(ξ,ΛΨb)|/lessorsimilarΓ1−ν\nb,\nand the control on the energy (54) to bound\n/vextendsingle/vextendsingle2λ2−2scE[ψ]/vextendsingle/vextendsingle/lessorsimilarΓ2\nb.\nAll the rests are bounded using (157) and (69)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleE(ξ,ξ)−1\nσ1+1/integraldisplay\nR(3)(ξ)dy+(R3(ξ),ΛQb)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ4(sc)/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy\nfor someδ4(sc)>0 withδ4(sc)→0 assc→0. For the last term on the right-hand side of (160),\nwe notice that using (13) and two of the orthogonality conditions in ( 51), we have\n(iΛξ,ΛQb) =−(iξ,Λ(ΛQb))−2sc(iξ,ΛQb) = 0.\nMoreover, by using (81), Cauchy-Schwartz inequality and (56), w e obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\ni˙x\nλ·∇ξ,ΛQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dbl∇ξ/ba∇dblL2/parenleftBigg\nδ2(sc)/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n+/integraldisplay\n|∇ξ|2dy+Γ1−11ν\nb/parenrightBigg\n/lessorsimilarδ2(sc)Γ1−20ν\nb+Γ3\n2−31ν\nb.\nIn the same way, we observe that\n−((˙γ−1)ξ,ΛQb) =−/parenleftbigg/parenleftbigg\n(˙γ−1)−1\n/ba∇dblΛQb/ba∇dbl2\nL2(ξ,LΛ(ΛQb)/parenrightbigg\nξ,ΛQb/parenrightbigg\n−/parenleftbigg/parenleftbigg1\n/ba∇dblΛQb/ba∇dbl2\nL2(ξ,LΛ(ΛQb)/parenrightbigg\nξ,ΛQb/parenrightbiggBLOW-UP OF THE DAMPED NLS EQUATION 43\nand we use again (81) to obtain that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/parenleftbigg/parenleftbigg\n(˙γ−1)−1\n/ba∇dblΛQb/ba∇dbl2\nL2(ξ,LΛ(ΛQb)/parenrightbigg\nξ,ΛQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓb.\nNow we use the coercivity property in Proposition 2.12 to obtain that there exists c>0 such that\nH(ξ,ξ)−((˙γ−1)ξ,ΛQb)≥c/integraldisplay\n|ξ|2e−|y|+|∇ξ|2dy\n−c/parenleftbig\n(ξ,Q)2+(ξ,|y|2Q)2+(ξ,yQ)2\n+(ξ,iDQ)2+(ξ,iD(DQ))2+(ξ,i∇Q)2/parenrightbig\n,\nwhereDξis defined in (11). By using property (12), (56) and the closeness o fQbwithQ, we have\nthat there exists a constant δ5(sc)>0 such that\n(ξ,Q)2+(ξ,|y|2Q)2+(ξ,yQ)2+(ξ,iDQ)2+(ξ,iD(DQ))2+(ξ,i∇Q)2\n≥(ξ,Qb)2+(ξ,|y|2Qb)2+(ξ,yQb)2+(ξ,iΛQb)2+(ξ,iΛ(ΛQb))2+(ξ,i∇Qb)2\n−δ5(sc)(sc+Γb)\nwhereδ5(sc)→0 assc→0. Finally, we use the orthogonality conditions (51), inequalities (74) and\n(79) to obtain\n/vextendsingle/vextendsingle(ξ,Qb)2+(ξ,|y|2Qb)2+(ξ,yQb)2+(ξ,iΛQb)2+(ξ,iΛ(ΛQb))2+(ξ,i∇Qb)2/vextendsingle/vextendsingle/lessorsimilarΓ3\n2−20ν\nb.\nThus (160) implies (150). /square\n7.2.Appendix B. In this appendix we report the computations needed in Lemma 4.6, ba sed on\n[36, Section 3 .3] and [34, Lemma 6] and stated in Lemma 7.4 below. We recall that the remainder\nterm˜ξin (90) satisfies\ni∂τ˜ξ+˜L˜ξ+i(˙b−βsc)∂b˜Qb−i/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛ(˜Qb+˜ξ)−i˙x\nλ·∇(˜Qb+˜ξ)\n−(˙γ−1)(˜Qb+˜ξ)+iηλ2−2σ2\nσ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ)+˜R(˜ξ)−˜Φb+F= 0,(162)\nwhere\n˜L˜ξ= ∆˜ξ−˜ξ+ibΛ˜ξ+2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb+|˜Qb|2σ1˜ξ,\n˜R(˜ξ) =|˜Qb+˜ξ|2σ1(˜Qb+˜ξ)−|˜Qb|2σ1˜Qb−2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb−|˜Qb|2σ1˜ξ.\n(163) F= (∆φA)ζb+2∇φA·∇ζb+iby·∇φAζb,\nand\n(164) ˜Φb=−Φb−iscb˜ζb+iβ∂b˜ζb+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb.\nHere Φ bis defined in (30), ζbis the outgoing radiation of Lemma 2.8, ˜ζb=φAζbis the localized\noutgoing radiation where φAis a smooth cut-off defined in the beginning of Subsection 4.2. By\nconstruction, ˜ζbsatisfies the following inequality\n(165)/vextenddouble/vextenddouble/vextenddouble(1+|y|)10(|˜ζb|+|∇˜ζb|2/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/vextenddouble(1+|y|)10(|∂b˜ζb|+|∇∂b˜ζb|2/vextenddouble/vextenddouble/vextenddouble2\nL2≤Γ1−cρ\nb,\nwherecρ≪1 is defined in Lemma 2.8. By exploiting equation (162) and reproducing the steps in\nLemma 7.1, we obtain the following.44 P. ANTONELLI AND B. SHAKAROV\nLemma 7.4. Letη= 0. Then for any t∈[0,T1), there exists C >0such that\nd\ndτf(τ)≥C/parenleftbigg/integraldisplay\n|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg\n−1\nC/parenleftBigg\nsc+/integraldisplay2A\nA|ξ|2dy/parenrightBigg\n, (166)\nwhere\n(167) f=−1\n2Im/parenleftbigg/integraldisplay\ny·∇˜Qb˜Qbdy/parenrightbigg\n−/parenleftBig\n˜ζb,iΛ˜Qb/parenrightBig\n+(ξ,iΛ˜ζ).\nProof.By taking the scalar product of equation (162) with Λ ˜Qb, we obtain that\n(168)0 = (i∂τ˜ξ,Λ˜Qb)+˙b(i∂b˜Qb,Λ˜Qb)+/parenleftBig\n2E[˜Qb]−/parenleftBig\niβsc∂b˜Qb+˜Φb−F,Λ˜Qb/parenrightBig/parenrightBig\n+(˜L˜ξ+˜R(˜ξ),Λ˜Qb)−2E[˜Qb]\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛ(˜Qb+˜ξ)+i˙x\nλ·∇˜Qb+(˙γ−1)˜Qb,Λ˜Qb/parenrightBigg\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛ˜ξ+(˙γ−1)˜ξ+i˙x\nλ·∇˜ξ,Λ˜Qb/parenrightBigg\n.\nNow we repeat the steps in Lemma 7.1, and arrive to the preliminary es timate\n(169)−˙b/parenleftBig\n(i∂b˜Qb,Λ˜Qb)−(i˜ξ,Λ∂b˜Qb)/parenrightBig\n−d\ndτ(i˜ξ,Λ˜Qb)\n=sc/parenleftBig\n−2b(i˜ξ,Λ˜Qb)−β(˜ξ,iΛ∂b˜Qb)+2E[˜Qb]+/ba∇dbl˜Qb/ba∇dbl2\nL2+(˙γ−1)/ba∇dbl˜Qb/ba∇dbl2\nL2/parenrightBig\n−(˜ξ,Λ(˜Φb−F))−2λ2−2scE[ψ]+H(˜ξ,˜ξ)+E(˜ξ,˜ξ)+1\nσ1+1/integraldisplay\nR(3)(˜ξ)dy\n−/parenleftBigg\ni/parenleftBigg˙λ\nλ+b/parenrightBigg\nΛ˜ξ+(˙γ−1)˜ξ+i˙x\nλ·∇˜ξ,Λ˜Qb/parenrightBigg\n.\nThe most important difference between (169) and (160) is the leadin g order term ( ˜ξ,Λ(˜Φb−F))\ninstead of ( ˜ξ,ΛΨb). First, by integration by parts and by using (13), we have that\n(i∂b˜Qb,Λ˜Qb) =∂b(i˜Qb,Λ˜Qb)−(i˜Qb,Λ∂b˜Qb)\n=∂b(i˜Qb,y·∇˜Qb)+(iΛ˜Qb,∂b˜Qb)+2sc(i˜Qb,∂b˜Qb),\nwhich equivalently implies that\n−˙b(i∂b˜Qb,Λ˜Qb) =−1\n2d\ndτ(i˜Qb,y·∇˜Qb)−˙bsc(i˜Qb,∂b˜Qb).\nThus we can write the left-hand side of (169) as\n−˙b/parenleftBig\n(i∂b˜Qb,Λ˜Qb)−(i˜ξ,Λ∂b˜Qb)/parenrightBig\n−d\ndτ(i˜ξ,Λ˜Qb) =d\ndτf(τ)+˙b(i˜ξ,Λ∂b˜Qb)−˙bsc(i˜Qb,∂b˜Qb)\nwhere\nf(τ) =−1\n2(i˜Qb,y·∇˜Qb)−(i˜ξ,Λ˜Qb).BLOW-UP OF THE DAMPED NLS EQUATION 45\nWe notice that by using the estimate on |˙b|(80), the smallness of ˜ζb(165), (69) and the controls\n(56), (52) we obtain\n−˙b(i˜ξ,Λ∂b˜Qb) =−˙b(i(ξ−˜ζb),Λ∂b(Qb+˜ζb))\n/greaterorsimilarΓ1−20ν\nb/parenleftBigg/parenleftbigg/integraldisplay\n|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1\n2\n+Γ1−cρ\nb/parenrightBigg\n/greaterorsimilarΓ3\n2−31ν\nb.\nNext, all the terms on the right-hand side of (169) are bounded in t he same way as in Lemma 7.1\nexcept for the scalar product ( ˜ξ,Λ(˜Φb−F)). In this way, we obtain that there exists a constant\nC >0 such that\n(170)d\ndτf≥C/integraldisplay\n|˜ξ|2e−|y|+|∇˜ξ|2dy−C(sc+Γ3\n2−50ν\nb)−(˜ξ,Λ(˜Φb−F)).\nand Φ bis defined in (29) and Fin (88). We notice that from the definition of ˜Φbin (92) and the\nbound on Φ bin (31) and the control on the outgoing radiation (165) we get\n/vextenddouble/vextenddouble/vextenddouble(1+|y|10)(|˜Φb|+|∇˜Φb|)/vextenddouble/vextenddouble/vextenddouble2\nL2≤Γ1+ν4\nb+sc,\nwhich implies that/vextendsingle/vextendsingle/vextendsingle(˜ξ,Λ˜Φb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1+ν4\nb.\nFinally, it remains to study the last term ( ˜ξ,ΛF) in (170). We observe that\n(˜ξ,ΛF) = (ξ−˜ζb,ΛF) = (ξ,ΛF)−(˜ζb,ΛF).\nFor the second term on the right-hand side of the equation above, we notice that\n−(˜ζb,ΛF) =−(˜ζb,DF)+sc(˜ζb,F)\nand from (165)/vextendsingle/vextendsingle/vextendsinglesc(˜ζb,F)/vextendsingle/vextendsingle/vextendsingle≤scΓ1−cρ≤Γ3\n2,\nwhile using (25)\n−(˜ζb,DF)≥C1Γb\nfor someC1>0. Finally, we use the Young inequality and (165) to obtain\n|(ξ,DF)|/lessorsimilar/parenleftBigg/integraldisplay2A\nA|ξ|2dy/parenrightBigg1\n2/parenleftBigg/integraldisplay2A\nA|F|2/parenrightBigg1\n2\n≤1\nC2/integraldisplay2A\nA|ξ|2dy+C1\n2Γb.\nfor someC2>0. Collecting everything we see that from (170) we obtain\nd\ndτf≥C/integraldisplay\n|˜ξ|2e−|y|+|∇˜ξ|2dy−C(sc+Γ1+ν2\nb)−(˜ξ,Λ(˜Φb−F))\n≥C/integraldisplay\n|˜ξ|2e−|y|+|∇˜ξ|2dy+C1Γb−C1\n2Γb−C3/parenleftBigg\nsc+/integraldisplay2A\nA|ξ|2dy/parenrightBigg\nfor someC3>0. This is equivalent to (166) /square46 P. ANTONELLI AND B. SHAKAROV\n7.3.Appendix C. In this appendix, we will report the proof inequality (147) which has a lready\nbeen done in the appendix of [36].\nProof.We give a proof only for the term N1(ξ), since that for N2(ξ) is equivalent. We define the\nfunctionF:C→Cas\nF(z) =|z|2σ1z.\nFrom the definition of r1(141) it follows that\n/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLr′/lessorsimilar1\nλ(2σ1+1)(˜s−sc)/ba∇dbl|∇|s(F(Qb+ξ)−F(Qb))/ba∇dblLr′\nwhere\n(171) ˜ s=s+d\n2−d\nr.\nWe claim the following estimate:\n/ba∇dbl|∇|s(F(Qb+ξ)−F(Qb))/ba∇dblLr′/lessorsimilar/ba∇dbl|∇|˜sξ/ba∇dblL2.\nIndeed observe that\n(172) F(Qb+ξ)−F(Qb) =/parenleftbigg/integraldisplay1\n0∂zF(Qb+τξ)dτ/parenrightbigg\nξ+/parenleftbigg/integraldisplay1\n0∂¯zF(Qb+τξ)dτ/parenrightbigg\n¯ξ.\nBoth terms on the right-hand side of (147) are treated in the same way. We define q∈Rsuch that\n(173)1\nq=1\nr′−1\nr.\nFor any function h, by the definition of r(141), ˜s(171) and by Sobolev embedding we have\n(174) /ba∇dblh/ba∇dblL2σ1q/lessorsimilar/ba∇dbl|∇|sh/ba∇dblLr/lessorsimilar/ba∇dbl|∇|˜sh/ba∇dblL2.\nBy the fractional Leibniz rule (see for example [22]), it follows that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/parenleftbigg\nξ/integraldisplay1\n0∂zF(Qb+τξ)dτ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLr′/lessorsimilar/ba∇dbl|∇|sξ/ba∇dblL2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay1\n0∂zF(Qb+τξ)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLq(175)\n+/ba∇dblξ/ba∇dblL2σ1q/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/integraldisplay1\n0∂zF(Qb+τξ)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLu,\nwhere\n(176)1\nu=1\nr′−1\n2σ1q.\nThen, from (174), we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/parenleftbigg\nξ/integraldisplay1\n0∂zF(Qb+τξ)dτ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLr′/lessorsimilar/ba∇dbl|∇|˜sξ/ba∇dblL2/parenleftbigg/integraldisplay1\n0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ (177)\n+/integraldisplay1\n0/ba∇dbl|∇|s∂zF(Qb+τξ)/ba∇dblLudτ/parenrightbigg\n,\nNow it remains to prove that\n(178)/parenleftbigg/integraldisplay1\n0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ+/integraldisplay1\n0/ba∇dbl|∇|s∂zF(Qb+τξ)/ba∇dblLudτ/parenrightbigg\n/lessorsimilar1.\nBy homogeneity\n∀τ∈[0,1],|∂zF(Qb+τξ)|/lessorsimilar|Qb|2σ1+|ξ|2σ1,BLOW-UP OF THE DAMPED NLS EQUATION 47\nand so\n/integraldisplay1\n0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ/lessorsimilar/integraldisplay1\n0(|Qb|2σ1+|ξ|2σ1\nL2σ1q)dτ/lessorsimilar/integraldisplay1\n0(|1+|ξ|2σ1\nL2)dτ/lessorsimilar1.\nMoreover, recall [22] the equivalent definition of the homogeneous Besov norm:\n∀0<˜s<1,/ba∇dblu/ba∇dbl2\n˙B˜s\nq,2∼/integraldisplay∞\n0/parenleftbig\nR−˜ssup\n|y|≤R|u(.−y)−u(.)|Lq/parenrightbig21\nRdR.\nRecall also that /ba∇dbl|∇|˜sψ/ba∇dblLq/lessorsimilar/ba∇dblψ/ba∇dbl˙B˜s\nq,2.\nObserve that, for 1 ≤d≤3, 2σ1>2 and it follows by homogeneity,\n|∂zF(u)−∂zF(v)|/lessorsimilar|u−v|(|u|2σ1−1+|v|2σ1−1).\nWe define\nhτ=Qb+τξ,0≤τ≤1.\nWe estimate from H¨ older and (174)\n|∂zF(hτ)(.−y)−∂zF(hτ)(.)/ba∇dblLu/lessorsimilar/vextenddouble/vextenddouble((hτ)(.−y)−(hτ)(.)|)(|hτ(.−y)|2σ1−1+|hτ(.)|2σ1+1)/vextenddouble/vextenddouble\nLu\n/lessorsimilar/ba∇dblhτ(.−y)−hτ(.)/ba∇dblLr/ba∇dblhτ/ba∇dbl2σ1−1\nL2σ1q\n/lessorsimilar/ba∇dblhτ(.−y)−hτ(.)/ba∇dblLr/ba∇dbl|∇|˜shτ/ba∇dbl2σ1−1\nL2,\nand so we have that\n/ba∇dbl|∇|s∂zF(hτ)/ba∇dblLu/lessorsimilar/integraldisplay∞\n0/parenleftbig\nR−˜ssup\n|y|≤R|∂zF(hτ)(.−y)−∂zF(hτ)(.)|Lu/parenrightbig21\nRdR\n/lessorsimilar/ba∇dbl|∇|˜shτ/ba∇dbl2σ1−1\nL2/ba∇dbl|∇|shτ/ba∇dblLr/lessorsimilar/ba∇dbl|∇|˜shτ/ba∇dbl2σ1\nL2,\nand this concludes the proof of (147). /square\nReferences\n[1] V.V. 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Soc. , 19(1):37–90, 2006.\n[35] F. Merle and P. Rapha¨ el. Blow up of the critical norm for some radial L2super critical nonlinear Schr¨ odinger\nequations. Amer. J. Math. , 130(4):945–978, 2008.\n[36] F. Merle, P. Rapha¨ el, and J. Szeftel. Stable self-simi lar blow-up dynamics for slightly L2super-critical NLS\nequations. Geom. Funct. Anal. , 20(4):1028–1071, 2010.\n[37] M. Ohta and G. Todorova. Remarks on global existence and blowup for damped nonlinear Schr¨ odinger equations.\nDiscrete Contin. Dyn. Syst. , 23(4):1313–1325, 2009.\n[38] P. Rapha¨ el. Stability of the log-log bound for blow up s olutions to the critical non linear Schr¨ odinger equation.\nMath. Ann. , 331(3):577–609, 2005.\n[39] C. Sulem and P.-L. Sulem. The nonlinear Schr¨ odinger equation , volume 139 of Applied Mathematical Sciences .\nSpringer-Verlag, New York, 1999. Self-focusing and wave co llapse.BLOW-UP OF THE DAMPED NLS EQUATION 49\n[40] M. Tsutsumi. Nonexistence of global solutions to the Ca uchy problem for the damped nonlinear Schr¨ odinger\nequations. SIAM J. Math. Anal. , 15(2):357–366, 1984.\n[41] X. P. Wang. On singular solutions of the nonlinear Schroedinger and Zak harov equations . ProQuest LLC, Ann\nArbor, MI, 1990. Thesis (Ph.D.)–New York University.\n[42] K. Yang, S. Roudenko, and Y. Zhao. Blow-up dynamics and s pectral property in the L2-critical nonlinear\nSchr¨ odinger equation in high dimensions. Nonlinearity , 31(9):4354–4392, 2018.\nGran Sasso Science Institute, viale Francesco Crispi, 7, 67 100 L’Aquila, Italy\nEmail address :paolo.antonelli@gssi.it\nGran Sasso Science Institute, viale Francesco Crispi, 7, 67 100 L’Aquila, Italy\nEmail address :boris.shakarov@gssi.it"    },    {        "title": "2307.02352v2.Optimal_damping_of_vibrating_systems__dependence_on_initial_conditions.pdf",        "content": "Optimal damping of vibrating systems: dependence on initial conditions\nK. Lelasa,∗, I. Nakićb\naFaculty of Textile Technology, University of Zagreb, Croatia\nbDepartment of Mathematics, Faculty of Science, University of Zagreb, Croatia\nAbstract\nCommon criteria used for measuring performance of vibrating systems have one thing in common: they\ndo not depend on initial conditions of the system. In some cases it is assumed that the system has zero\ninitial conditions, or some kind of averaging is used to get rid of initial conditions. The aim of this paper\nis to initiate rigorous study of the dependence of vibrating systems on initial conditions in the setting of\noptimal damping problems. We show that, based on the type of initial conditions, especially on the ratio of\npotential and kinetic energy of the initial conditions, the vibrating system will have quite different behavior\nand correspondingly the optimal damping coefficients will be quite different. More precisely, for single degree\nof freedom systems and the initial conditions with mostly potential energy, the optimal damping coefficient\nwillbeintheunder-dampedregime, whileinthecaseofthepredominantkineticenergytheoptimaldamping\ncoefficient will be in the over-damped regime. In fact, in the case of pure kinetic initial energy, the optimal\ndamping coefficient is +∞! Qualitatively, we found the same behavior in multi degree of freedom systems\nwith mass proportional damping. We also introduce a new method for determining the optimal damping of\nvibrating systems, which takes into account the peculiarities of initial conditions and the fact that, although\nin theory these systems asymptotically approach equilibrium and never reach it exactly, in nature and in\nexperiments they effectively reach equilibrium in some finite time.\nKeywords: viscous damping, optimal damping, multi-degree of freedom, initial conditions\n1. Introduction\nIf we have an multi-degree of freedom (MDOF) linear vibrating system, i.e. a system of coupled damped\noscillators, how to determine damping coefficients that ensure optimal evanescence of free vibrations? In\nthe literature one finds several different criteria, typically based on frequency domain analysis of the system,\nalthough there are also approaches based on time domain analysis [1]. The tools used for designing the\ncriteria include modal analysis [2], transfer functions [3], H2andH∞norms coming from systems theory\n[4, 5] and spectral techniques [6]. A general overview of the optimization tools for structures analysis can\nbe found in e.g. [7]. Another optimization criterion used is to take as optimal the damping coefficients that\nminimize the (zero to infinity) time integral of the energy of the system, averaged over all possible initial\nconditions corresponding to the same initial energy [8]. This criterion was investigated widely, mostly by\nmathematicians in the last two decades, more details can be found, e.g., in references [8, 9, 10, 11, 12].\nHowever, what is common to all these criteria is that they implicitly or explicitly ignore the dependence\nof the dynamics of the system on the initial conditions. Sometimes this is suitable, e.g. for systems with\ncontinuous excitation, but in some cases it make sense to study the free vibrations of the system with non-\nzero initial conditions. A prominent example where this is the case is the vibration control of buildings\nsubjected to earthquake excitation [13, 14]. Indeed, depending on the initial conditions, MDOF systems can\n∗Corresponding author\nEmail addresses: klelas@ttf.unizg.hr (K. Lelas), nakic@math.hr (I. Nakić)\nPreprint submitted to Journal of Sound and Vibration January 31, 2024arXiv:2307.02352v2  [physics.class-ph]  30 Jan 2024exhibit oscillatory or non-oscillatory response [15], so it is clear that initial conditions can play an important\nrole in the overall dynamics of the system.\nImplicitly, dependence of the behavior of system on initial conditions has been investigated in the context\nof time-optimal vibrations reduction [16] and transient response [17] in terms of computationally efficient\nmethods for the calculation of the system response. Our aim with this paper is to start more systematic\ninvestigation of the role of initial conditions in the study of linear vibrating systems. Specifically, the\ndependence of the energy integral on the initial conditions has not been investigated, as far as we are aware,\nand therefore it is not clear how much information about the behavior of vibrating systems is lost by taking\nthe average over all initial conditions or by assuming zero initial conditions and it is not clear how well the\noptimal damping obtained in this way works for a specific choice of initial conditions, e.g. for an experiment\nwith initial conditions such that the initial energy consists only of potential energy, etc. We have chosen to\nstudy the particular criterion of minimizing time integral of the energy as in this case it is straightforward to\nmodify it to take into account the initial conditions: instead of averaging over all possible initial conditions,\nwe study the dependence of the time integral of the energy of the system on initial conditions. Specifically,\nfor criteria based on frequency domain approach, which are designed for forced vibrations, it is not clear\nhow to take into account the non-zero initial conditions in a systematic way.\nWe will explore this dependence by considering free vibrations of single degree of freedom (SDOF),\ntwo-degree of freedom (2-DOF) and MDOF vibrating systems with mass proportional damping (MPD). In\nparticular, for a SDOF, averaging over all initial conditions gives the critical damping as optimal [8, 10], and\nwe show, by considering the minimization of the energy integral without averaging over initial conditions,\nthat damping coefficients approximately 30%less than critical to infinite are obtained as optimal, depending\non the initial conditions. We systematize all our results with respect to the relationship between initial\npotential and initial kinetic energy, e.g., for initial conditions with initial potential energy grater than initial\nkinetic energy the optimal damping coefficient is in the under-damped regime, while for initial conditions\nwith initial kinetic energy grater than initial potential energy we find the optimal damping deep in the\nover-damped regime. We also consider the minimization of the energy integral averaged over a subset of\ninitial conditions and obtain a significant dependence of the optimal damping coefficient on the selected\nsubset. Qualitatively, we find the same behavior in 2-DOF and MDOF systems as well.\nFurthermore, we show that the minimization of the energy integral for certain types of initial conditions\ndoes not give a satisfactory optimal damping coefficient. Specifically, for SDOF systems, the obtained\noptimal damping coefficient does not distinguish between two initial states with the same magnitude of\ninitial displacement and initial velocity, but which differ in the relative sign of initial displacement and\ninitial velocity. These initial conditions differ significantly in the rate of energy dissipation as a function of\nthe damping coefficient, i.e. it is not realistic for one damping coefficient to be optimal for both of these\ninitial conditions. The same is true for each individual mode of MDOF systems with respect to the signs of\ninitial displacements and velocities, expressed via modal coordinates. Another disadvantage of this criterion\nisthat, forinitialconditionswithpurelykineticinitialenergy, itgivesaninfiniteoptimaldampingcoefficient,\nwhich is not practical for experiments. Also, the energy integral is calculated over the entire time, due to\nthe fact that these systems asymptotically approach equilibrium and never reach it exactly, but in nature\nand experiments they effectively reach equilibrium in some finite time.\nWe introduce a new method for determining the optimal damping of MDOF systems, which practically\nsolves the aforementioned problems and gives optimal damping coefficients that take into account the pecu-\nliarities of each initial condition and the fact that these systems effectively reach equilibrium in some finite\ntime. We take that the system has effectively reached equilibrium when its energy drops to some small\nfraction of the initial energy, e.g., to the energy resolution of the measuring device with which we observe\nthe system. Our method is based on the determination of the damping coefficient for which the energy of\nthe system drops to that desired energy level the fastest.\nIn this paper we focus on mass proportional damping so that we could analytically perform a modal\nanalysis and present ideas in the simplest possible way, but, as we briefly comment at the end of the paper,\neverything we have done can be done in a similar fashion analytically for the case of Rayleigh damping [18]\nas well as for tuned mass damper [19, 20]. Also, it is possible to carry out this kind of analysis numerically\n2for systems with damping that does not allow analytical treatment. This will be the subject of our further\nresearch.\nThe rest of the paper is organized as follows: Section 2 is devoted to SDOF systems, in particular\nminimization of the energy integral and optimal damping is studied for the chosen set of initial conditions.\nIn Section 3 we analyze 2-DOF systems with MPD. MDOF systems with MPD are the subject of Section\n4. In Section 5 we propose a new optimization criterion and analyze its properties. Section 6 summarizes\nimportant findings of the paper.\n2. SDOF systems\nFree vibrations of a SDOF linear vibrating system can be described by the equation\n¨x(t) + 2γ˙x(t) +ω2\n0x(t) = 0 , x(0) = x0,˙x(0) = v0, (1)\nwhere x(t)denotes the displacement from the equilibrium position (set to x= 0) as a function of time,\nthe dots denote time derivatives, γ > 0is the damping coefficient, ω0stands for the undamped oscillator\nangular frequency (sometimes called the natural frequency of the oscillator) and (x0, v0)encode the initial\nconditions [21, 22]. The physical units of the displacement x(t)depend on the system being considered. For\nexample, for a mass on a spring (or a pendulum) in viscous fluid, when it is usually called elongation , it is\nmeasured in [m], while for an RLC circuit it could either be voltage, or current, or charge. In contrast, the\nunits of γandω0are[s−1]for all systems described with the SDOF model. The form of the solution to\nthe equation (1) depends on the relationship between γandω0, producing three possible regimes [21, 22]:\nunder-damped ( γ < ω 0), critically damped ( γ=ω0) and over-damped ( γ > ω 0) regime.\nHere we would like to point out that, although it is natural to classify the solution of SDOF into three\nregimes depending on the value of γ, we can actually take one form of the solution as a unique solution valid\nfor all values of γ >0,γ̸=ω0,\nx(t) =e−γt\u0012\nx0cos(ωt) +v0+γx0\nωsin(ωt)\u0013\n, (2)\nwhere ω=p\nω2\n0−γ2is the (complex valued) damped angular frequency. In order to describe the critically\ndamped regime, one can take the limit γ→ω0of the solution (2) to obtain the general solution of the\ncritically damped regime\nxc(t) =e−ω0t(x0+ (v0+ω0x0)t). (3)\nTherefore, in order to calculate the energy and the time integral of the energy, we do not need to perform\nseparate calculations for all three regimes, but a single calculation using the displacement given by (2) and\nthe velocity given by\n˙x(t) =e−γt\u0012\nv0cos(ωt)−γv0+ω2\n0x0\nωsin(ωt)\u0013\n. (4)\nFor simplicity, in this section we will refer to the quantity\nE(t) = ˙x(t)2+ω2\n0x(t)2(5)\nas the energyof the system, and to the quantities EK(t) = ˙x(t)2andEP(t) =ω2\n0x(t)2as the kinetic energy\nandpotential energy of the system respectively. The connection of the quantity (5) to the usual expressions\nfor the energy is straightforward, e.g., for a mass mon a spring in viscous fluid\nE(t) =m\n2E(t), (6)\nand similarly for other systems described with the SDOF model. Using (2) and (4) in (5), we obtain\nE(t) =e−2γt\u0012\nE0cos2(ωt) +γ\u0000\nω2\n0x2\n0−v2\n0\u0001sin(2ωt)\nω+\u0000\nE0(ω2\n0+γ2) + 4ω2\n0γx0v0\u0001sin2(ωt)\nω2\u0013\n(7)\n3for the energy of the system, where E0=v2\n0+ω2\n0x2\n0is the initial energy given to the system at t= 0.\nAccordingly, E0K=v2\n0istheinitialkineticenergyand E0P=ω2\n0x2\n0istheinitialpotentialenergy. Expression\n(7) is valid for both under-damped and over-damped regimes, and to obtain the energy of the critically\ndamped regime we take the γ→ω0limit of the energy (7), and obtain\nEc(t) =e−2ω0t\u0000\nE0+ 2ω0\u0000\nω2\n0x2\n0−v2\n0\u0001\nt+ 2ω2\n0(E0+ 2ω0x0v0)t2\u0001\n. (8)\n2.1. Minimization of the energy integral and optimal damping in dependence of initial conditions\nWe consider the SDOF system with initially energy E0. All possible initial conditions that give this\nenergy can be expressed in polar coordinates with constant radius r=√E0and angle θ= arctan\u0010\nv0\nω0x0\u0011\n,\ni.e. we have\nω0x0=rcosθ\nv0=rsinθ .(9)\nIn Fig. 1 we sketch the circle given by (9), i.e. given by all possible initial conditions with the same energy\nE0. For clarity of the exposition, here we comment on a few characteristic points of the circle presented in\nFig. 1:\n•Initial conditions ω0x0=±√E0andv0= 0, i.e. with purely potential initial energy (and zero initial\nkinetic energy), correspond to two points on the circle with θ={0, π}.\n•Initial conditions ω0x0=±p\nE0/2andv0=±p\nE0/2, i.e. with initial potential energy equal to initial\nkinetic energy, correspond to four points on the circle with θ={π/4,3π/4,5π/4,7π/4}.\n•Initial conditions ω0x0= 0andv0=±√E0, i.e. with purely kinetic initial energy (and zero initial\npotential energy), correspond to two points on the circle with θ={π/2,3π/2}.\nθ r=√E0(ω0x0, v0)\nω0x0v0\nFigure 1: Sketch of all possible initial conditions with the same initial energy E0in the (ω0x0, v0)coordinate system. Square\nof the coordinates corresponds to initial potential energy E0P=ω2\n0x2\n0and initial kinetic energy E0K=v2\n0respectively. This\nrepresentation gives us a useful visualization, e.g.: all initial conditions with E0P> E 0Kare represented by two arcs, i.e.\npoints with θ∈(−π/4, π/4)∪(3π/4,5π/4)(blue dotted arcs); initial conditions with E0K=E0andE0P= 0are represented\nby two points on a circle with θ={π/2,3π/2}(two red filled circles); etc.\nUsing (9) in (7) and (8), we obtain the energy of the under-damped and over-damped regime\nE(t, θ) =E0e−2γt\u0012\ncos2(ωt) +γcos 2θsin(2ωt)\nω+\u0000\nω2\n0+γ2+ 2ω0γsin 2θ\u0001sin2(ωt)\nω2\u0013\n,(10)\nand the energy of the critically damped regime\nEc(t, θ) =E0e−2ω0t\u0000\n1 + 2 ω0(cos 2 θ)t+ 2ω2\n0(1 + sin 2 θ)t2\u0001\n, (11)\n40.1 0.71 1 2 3012345\nγ/ω 0I(γ, θ)ω0/E0θ= 0\nθ=π/4\nθ=π/2\nFigure 2: Integral (13) for three initial conditions θ={0, π/4, π/2}.\nas functions of θ, instead of x0andv0. Now we integrate energy (10) over all time, i.e.\nI(γ, θ) =Z∞\n0E(t)dt , (12)\nand obtain\nI(γ, θ) =E0\n2ω0\u0012ω2\n0+γ2\nγω0+γ\nω0cos 2θ+ sin 2 θ\u0013\n. (13)\nIntegral (13) is valid for all three regimes, i.e. for any γ >0.\nWe note here that the energy (see (7) and (8)) is invariant to a simultaneous change of the signs of the\ninitial conditions, i.e. to the change (x0, v0)→(−x0,−v0)(but not to x0→ −x0orv0→ −v0separately).\nThis change of signs corresponds to the change in angle θ→θ+π, therefore, functions (10), (11) and (13)\nare all periodic in θwith period π.\nIn Fig. 2 we show the integral (13) for γ∈[0.1ω0,3ω0]for three different initial conditions, i.e. for\nθ={0, π/4, π/2}. We can see that I(γ, θ= 0)(red solid curve), with purely potential initial energy and\nzero initial kinetic energy, attains minimum for γ= 0.707ω0(rounded to three decimal places), i.e. well in\nthe under-damped regime. For the initial condition with equal potential and kinetic energy, I(γ, θ=π/4)\n(black dotted curve) attains minimum for γ=ω0, i.e. at the critical damping condition. Interestingly,\nfor the initial condition with purely kinetic energy and zero potential energy, I(γ, θ=π/2)(blue dashed\ncurve) has no minimum in the displayed range of damping coefficients, therefore here we explicitly show this\nfunction\nI(γ, θ=π/2) =E0\n2γ, (14)\nand it is clear that (14) has no minimum. This is easy to understand from a physical point of view, i.e. if\nall the initial energy is kinetic, the higher the damping coefficient, the faster the energy dissipation will be.\nIf we consider the optimal damping as the one for which the integral (13) is minimal, we can easily\ndetermine the optimal damping coefficient γopt(θ)from the condition\n∂I(γ, θ)\n∂γ\f\f\f\f\nγopt= 0, (15)\nand we obtain\nγopt(θ) =r\n1\n2 cos2θω0. (16)\nIn Fig. 3 we show the optimal damping coefficient (16) for θ∈[0,2π](function (16) has a period π, but\nhere we choose this interval for completeness), and here we comment on the shown results with respect to\nthe relationship between initial potential energy ( E0P=ω2\n0x2\n0) and initial kinetic energy ( E0K=v2\n0) for any\ngiven initial condition, i.e. for any θ:\n50 0.25 0.50.75 1 1.25 1.51.75 201234\nγopt=ω0\nθ/πγopt(θ)/ω0\nFigure 3: Optimal damping coefficient (16) (solid red curve) as a function of all possible initial conditions, i.e. for θ∈[0,2π].\nBelow the dashed horizontal line, optimal damping coefficients are in the under-damped regime, above the line in the over-\ndamped regime, and in the critically damped regime at the crossing points of the line and the solid red curve.\n•Initial conditions with E0P> E 0Kcorrespond to the set θ∈(−π/4, π/4)∪(3π/4,5π/4). For these\ninitial conditions, optimal damping coefficients (16) are in the under-damped regime, i.e. γopt∈\u0002√\n2ω0/2, ω0\u0001\n, with the minimum value γopt=√\n2ω0/2 = 0 .707ω0(rounded to three decimal places)\nobtained for θ={0, π}, i.e. for two initial conditions with E0=E0PandE0K= 0.\n•Initial conditions with E0P=E0Kcorrespond to four points θ={π/4,3π/4,5π/4,7π/4}with optimal\ndamping coefficient (16) equal to critical damping, i.e. γopt=ω0.\n•Initial conditions with E0K> E 0Pcorrespond to the set θ∈(π/4,3π/4)∪(5π/4,7π/4). For these\ninitial conditions, optimal damping coefficients (16) are in the over-damped regime, i.e. γopt∈(ω0,∞),\nwhere γoptdiverges for θ={π/2,3π/2}, i.e. for two initial conditions with E0K=E0andE0P= 0.\nBefore closing this subsection, we would like to point out two more ways in which we can write relation\n(16) that will prove useful when dealing with MDOF systems. The ratio of the initial potential energy to\nthe initial total energy is\nβ=E0P\nE0= cos2θ , (17)\nwhere we used first of the relations (9) and E0P=ω2\n0x2\n0. Using (17), optimal damping coefficient (16) can\nbe written as a function of the fraction of potential energy in the initial total energy, i.e.\nγopt(β) =r1\n2βω0. (18)\nThus, from (18) one can simply see that γoptis in the under-damped regime for β∈(1/2,1], in the critically\ndamped regime for β= 1/2and in the over-damped regime for β∈[0,1/2). Using β=ω2\n0x2\n0/E0in (18) we\ncan express the optimal damping coefficient in yet another way, as a function of the initial displacement x0,\ni.e.\nγopt(x0) =s\nE0\n2x2\n0=s\nv2\n0+ω2\n0x2\n0\n2x2\n0, (19)\nwhere x0∈[−√E0/ω0,√E0/ω0]and for v0the condition v2\n0=E0−ω2\n0x2\n0holds. One of the benefits of\nrelation (19) is that it can be seen most directly that the optimal damping coefficient does not distinguish\ninitial conditions (±x0,±v0)and(±x0,∓v0), which is a shortcoming of this optimization criterion, because\nthe energy as a function of time is not the same for those two types of initial conditions (see (7) and (8))\nand the energy decay may differ significantly depending on which of those initial conditions is in question.\nWe will deal with these and other issues of energy integral minimization as an optimal damping criterion in\nthe subsection 4.2.\n60.1 0.781 1.66 2 3012345\nγ/ω 0I(γ, ϕ1, ϕ2)ω0/E0ϕ1=−π/4,ϕ2=π/4\nϕ1=π/4,ϕ2= 3π/4\nϕ1=−π/4,ϕ2= 3π/4\nFigure 4: Averaged integral (21) for three sets of initial conditions.\n2.2. Minimization of the energy integral averaged over a set of initial conditions and optimal damping in\ndependence of the chosen set\nNow we calculate the average of the integral (12) over a set of initial conditions with θ∈[ϕ1, ϕ2], i.e.\nI(γ, ϕ1, ϕ2) =1\nϕ2−ϕ1Zϕ2\nϕ1I(γ, θ)dθ , (20)\nand we obtain\nI(γ, ϕ1, ϕ2) =E0\n2ω0\u0012ω2\n0+γ2\nγω0+γ\n2ω0(ϕ2−ϕ1)(sin 2 ϕ2−sin 2ϕ1) +1\n2(ϕ2−ϕ1)(cos 2 ϕ1−cos 2ϕ2)\u0013\n.(21)\nIn Fig. 4 we show averaged integral (21) for three different sets of initial conditions. For the set of initial\nconditions with ϕ1=−π/4andϕ2=π/4, i.e. with E0P≥E0K(where the equality holds only at the end\npoints of the set), minimum of the averaged integral (solid red curve) is at γ= 0.781ω0(rounded to three\ndecimal places). For the set of initial conditions with ϕ1=π/4andϕ2= 3π/4, i.e. with E0K≥E0P(where\nthe equality holds only at the end points of the set), minimum of the averaged integral (dashed blue curve)\nis atγ= 1.658ω0(rounded to three decimal places). For the set of mixed initial conditions with ϕ1=−π/4\nandϕ2= 3π/4, i.e. with E0P> E 0KandE0K> E 0Ppoints equally present in the set, minimum of the\naveraged integral (dotted black curve) is at the critical damping condition γ=ω0.\nIf we consider the optimal damping as the one for which the averaged integral (21) is minimal, we can\neasily determine the optimal damping coefficient γopt(ϕ1, ϕ2)form the condition\n∂I(γ, ϕ1, ϕ2)\n∂γ\f\f\f\f\nγopt= 0, (22)\nand we obtain\nγopt(ϕ1, ϕ2) =s\n2(ϕ2−ϕ1)\n2(ϕ2−ϕ1) + sin 2 ϕ2−sin 2ϕ1ω0. (23)\nWe note here that averaged integral (21) and optimal damping coefficient (23) are not periodic functions in\nvariables ϕ1andϕ2, if we keep one variable fixed and change the other. But they are periodic, with period\nπ, if we change both variables simultaneously.\nIn Fig. 5 we show the optimal damping coefficient (23) as a function of ϕ2with fixed ϕ1= 0, and the\nresults shown can be summarized as follows:\n•Forϕ1= 0andϕ2∈[0, π/2)∪(π,3π/2), the optimal damping coefficient (23) is in the under-damped\nregime. In this case, integral (20) is averaged over sets that have more points corresponding to initial\nconditions with E0P> E 0K, in comparison to the points corresponding to initial conditions with\nE0K> E 0P.\n70 0.25 0.50.75 1 1.25 1.51.75 20.70.80.911.1\nγopt=ω0\nϕ2/πγopt(0, ϕ2)/ω0\nFigure 5: Optimal damping coefficient (23) (solid red curve) as a function of ϕ2∈[0,2π]for fixed ϕ1= 0. Below the dashed\nhorizontal line, optimal damping coefficients are in the under-damped regime, above the line in the over-damped regime, and\nin the critically damped regime at the crossing points of the line and the solid red curve.\n•Forϕ1= 0andϕ2={π/2, π,3π/2,2π}, the optimal damping coefficient (23) is equal to critical damp-\ning. In this case, integral (20) is averaged over sets that have equal amount of points corresponding\nto initial conditions with E0P> E 0Kand initial conditions with E0K> E 0P.\n•Forϕ1= 0andϕ2∈(π/2, π)∪(3π/2,2π), the optimal damping coefficient (23) is in the over-damped\nregime. In this case, integral (20) is averaged over sets that have more points corresponding to initial\nconditions with E0K> E 0P, in comparison to the points corresponding to initial conditions with\nE0P> E 0K.\n3. 2-DOF systems with MPD\nFigure 6: Schematic figure of a 2-DOF system.\nHere we consider 2-DOF system shown schematically in Fig. 6. The corresponding equations of motion\nare\nm1¨x1(t) =−c1˙x1(t)−k1x1(t)−k2(x1(t)−x2(t)),\nm2¨x2(t) =−c2˙x2(t)−k3x2(t) +k2(x1(t)−x2(t)).(24)\nWe will consider MPD [23], i.e. masses {m1, m2}, spring constants {k1, k2, k3}, and dampers {c1, c2}can\nin general be mutually different but the condition c1/m1=c2/m2holds. In this case we can use modal\nanalysis [2, 21] and the system of equations (24) can be written via modal coordinates [21] as\n¨q1(t) + 2γ˙q1(t) +ω2\n01q1(t) = 0\n¨q2(t) + 2γ˙q2(t) +ω2\n02q2(t) = 0 ,(25)\nwhere qi(t)andω0i, with i={1,2}, denote the modal coordinates and undamped modal frequencies of\nthe two modes, while γ=ci/2miis the damping coefficient. In the analysis that we will carry out in this\n8subsection, we will not need the explicit connection of modal coordinates qi(t)and mass coordinates, i.e.\ndisplacements xi(t), and we will deal with this in the next subsection when considering a specific example\nwith given masses, springs and dampers. Similarly as in Section 2 (see (2)), the general solution for the i-th\nmode can be written as\nqi(t) =e−γt\u0012\nq0icos(ωit) +˙q0i+γq0i\nωisin(ωit)\u0013\n, (26)\nwhere ωi=p\nω2\n0i−γ2is the damped modal frequency, and qi(0)≡q0iand ˙qi(0)≡˙q0iare the initial condi-\ntions of the i-th mode. Thus, the reasoning and the results presented in Section 2, with some adjustments,\ncan by applied for the analysis of the 2-DOF system we are considering here.\nThe energy of the system is\nE(t) =2X\ni=1mi˙xi(t)2\n2+k1x1(t)2\n2+k3x2(t)2\n2+k2(x1(t)−x2(t))2\n2, (27)\nand we take that the modal coordinates are normalised so that (27) can be written as\nE(t) =2X\ni=1Ei(t) =2X\ni=1\u0000\n˙qi(t)2+ω2\n0iqi(t)2\u0001\n(28)\nwhere Ei(t)in (28) denotes the energy of the i-th mode. Total energy at t= 0, i.e. the initial energy, is\ngiven by\nE0=2X\ni=1E0i=2X\ni=1(E0Ki+E0Pi) =2X\ni=1\u0000\n˙q2\n0i+ω2\n0iq2\n0i\u0001\n, (29)\nwhere E0idenotes the initial energy of the i-th mode, E0Ki= ˙q2\n0iandE0Pi=ω2\n0iq2\n0idenote initial kinetic\nand initial potential energy of the i-th mode.\nAll possible initial conditions with the same initial energy (29) can be expressed similarly as in the SDOF\ncase (see (9) and Fig. 1) but with two pairs of polar coordinates, one pair for each mode. For the i-th mode\nwe have radius ri=√E0iand angle θi= arctan\u0010\n˙q0i\nω0iq0i\u0011\n, i.e. we can write\nω0iq0i=ricosθi\n˙q0i=risinθi.(30)\nThus, each initial condition with energy E0=E01+E02can be represented by points on two circles with\nradii r1=√E01andr2=√E02, for which condition r2\n1+r2\n2=E0holds, and with angles θ1andθ2that tell\nus how initial potential and initial kinetic energy are distributed within the modes. Using relation (10) for\nSDOF systems, we can write the energy of the i-th mode in polar coordinates (30) as\nEi(t) =E0ie−2γt\u0012\ncos2(ωit) +γcos 2θisin(2ωit)\nωi+\u0000\nω2\n0i+γ2+ 2ω0iγsin 2θi\u0001sin2(ωit)\nω2\ni\u0013\n(31)\nfor the under-damped ( γ < ω 0i) and over-damped ( γ > ω 0i) regime, and the energy of the i-th mode in the\ncritically damped regime is obtained analogously using the relation (11).\nConsequently, the integral of the energy (28) over the entire time, for some arbitrary initial condition, is\nsimply calculated using relation (13) for each individual mode, we obtain\nI(γ,{E0i},{θi}) =2X\ni=1Z∞\n0Ei(t)dt=2X\ni=1E0i\n2ω0i\u0012ω2\n0i+γ2\nγω0i+γ\nω0icos 2θi+ sin 2 θi\u0013\n. (32)\nFurthermore, initial energy of the i-th mode can be written as E0i=a2\niE0, where coefficient a2\ni∈[0,1]\ndenotes the fraction of the initial energy of the i-th mode in the total initial energy. Coefficients of the two\n9modes satisfy a2\n1+a2\n2= 1and therefore can be parameterized as\na1= cos ψ\na2= sin ψ,(33)\nwhere ψ∈[0, π/2]. Taking (33) into account, we can write (32) as\nI(γ, ψ, θ 1, θ2) =E02X\ni=1a2\ni\n2ω0i\u0012ω2\n0i+γ2\nγω0i+γ\nω0icos 2θi+ sin 2 θi\u0013\n. (34)\nIf we consider the optimal damping coefficient as the one for which the integral (34) is minimal, we can\neasily determine the optimal damping coefficient form the condition\n∂I(γ, ψ, θ 1, θ2)\n∂γ\f\f\f\f\nγopt= 0, (35)\nand we obtain\nγopt(ψ, θ1, θ2) =s\nω2\n01ω2\n02\n2ω2\n02cos2ψcos2θ1+ 2ω2\n01sin2ψcos2θ2. (36)\nIt is easy to see that, for any fixed ψ, the function (36) has smallest magnitude for cos2θ1= cos2θ2= 1,\nwhich corresponds to the initial conditions with initial energy comprised only of potential energy distributed\nwithin the two modes, i.e E0=E0P1+E0P2. In that case we can write the denominator of (36) as\nf(ψ) =q\n2ω2\n02cos2ψ+ 2ω2\n01sin2ψ=q\n2(ω2\n02−ω2\n01) cos2ψ+ 2ω2\n01, (37)\nwhere we used sin2ψ= 1−cos2ψ. Since ω01< ω 02, the function (37) has maximum for ψ= 0. Thus,\nthe minimum value of the optimal damping coefficient (36) is√\n2ω01/2, and it is obtained for ψ= 0and\nθ1={0, π}, which corresponds to the initial conditions with initial energy comprised only of potential energy\nin the first mode, i.e. E0=E0P1. On the other hand, for any fixed ψ, the function (36) has singularities\nforcos2θ1= cos2θ2= 0, which corresponds to the initial conditions with initial energy comprised only of\nkinetic energy. Thus, the range of the optimal damping coefficient (36) is\nγopt∈h√\n2ω01/2,+∞\u0011\n. (38)\nNow we calculate the average of the integral (34) over a set of all initial conditions, we obtain\nI(γ) =1\n2π3Zπ/2\n0dψZ2π\n0dθ1Z2π\n0dθ2I(γ, ψ, θ 1, θ2) =E0\n42X\ni=1\u0012ω2\n0i+γ2\nγω2\n0i\u0013\n, (39)\nand from the condition\n∂I(γ)\n∂γ\f\f\f\f\nγopt= 0, (40)\nwe find that the optimal damping coefficient with respect to the averaged integral (39) is given by\nγopt=s\n2ω2\n01ω2\n02\nω2\n01+ω2\n02. (41)\nInordertomoreeasilyanalyzethebehaviorofthedampingcoefficient(36)withregardtothedistribution\nof the initial potential energy within the modes and its relationship with the damping coefficient (41),\n10similarly as in subsection 2.1 (see (17) and (18)), we define the ratio of the initial potential energy of the\ni-th mode and the total initial energy, i.e.\nβi=E0Pi\nE0. (42)\nSince the initial potential energy satisfies E0P=E0P1+E0P2≤E0, we have βi∈[0,1]and the condition\n0≤(β1+β2)≤1holds. Taking E0Pi=E0icos2θi(see (30)) and E0i=a2\niE0with (33) into account, we\nhave\nβ1= cos2ψcos2θ1\nβ2= sin2ψcos2θ2.(43)\nUsing (43), relation (36) can be written as\nγopt(β1, β2) =s\nω2\n01ω2\n02\n2ω2\n02β1+ 2ω2\n01β2. (44)\nFor clarity, we will repeat briefly, the minimum value of (44) is√\n2ω01/2, obtained for β1= 1andβ2= 0\n(or in terms of the angles in (36), for ψ= 0andθ1={0, π}), while γopt→+∞forβ1=β2= 0(or in\nterms of the angles in (36), for any ψwith θ1={π/2,3π/2}andθ2={π/2,3π/2}). The benefit of relation\n(44) is that we expressed (36) through two variables instead of three, i.e. this way we lost information about\nthe signs of the initial conditions and about distribution of initial kinetic energy within the modes, but the\noptimal damping coefficient (36) does not depend on those signs anyway, due to the squares of trigonometric\nfunctions in variables θ1andθ2, and, for a fixed distribution of initial potential energy within the modes,\nthe optimal damping coefficient (36) is constant for different distributions of initial kinetic energy within\nthe modes. By looking at relations (44) and (41), it is immediately clear that γopt(β1, β2) =γoptfor\nω2\n02β1+ω2\n01β2=ω2\n01+ω2\n02\n4, (45)\nwhile γopt(β1, β2)<γoptif the left hand side of relation (45) is greater than the right hand side, and\nγopt(β1, β2)>γoptif the left hand side of relation (45) is smaller than the right hand side.\nAgain, similarly as in subsection 2.1 (see (19)), using βi=ω2\n0iq2\n0i/E0we can express the optimal damping\ncoefficient (44) as a function of the initial modal coordinates as well, i.e.\nγopt(q01, q02) =s\nE0\n2q2\n01+ 2q2\n02, (46)\nwhere q0i∈[−√E0/ω0i,√E0/ω0i]and the condition 0≤(ω2\n01q2\n01+ω2\n02q2\n02)≤E0holds. We can express\ncondition (45) in terms of initial modal coordinates, i.e. γopt({q0i}) =γoptfor\nq2\n01+q2\n02\nE0=ω2\n01+ω2\n02\n4ω2\n01ω2\n02, (47)\nwhile γopt({q0i})<γoptif the left hand side of relation (47) is greater than the right hand side, and\nγopt({q0i})>γoptif the left hand side of relation (47) is smaller than the right hand side.\nWe note here that we did not use explicit values of the undamped modal frequencies ω01andω02in\nthe analysis so far, and relations presented so far are valid for any 2-DOF system with MPD. In the next\nsubsection, we provide a more detailed quantitative analysis using an example with specific values of modal\nfrequencies.\n113.1. Quantitative example\nHere we consider the 2-DOF system as the one shown schematically in Fig. 6, but with m1=m2=m,\nk1=k2=k3=kandc1=c2=c. The corresponding equations of motion are\nm¨x1(t) =−c˙x1(t)−kx1(t)−k(x1(t)−x2(t)),\nm¨x2(t) =−c˙x2(t)−kx2(t) +k(x1(t)−x2(t)).(48)\nFor completeness, we will investigate here the behavior of the optimal damping coefficient given by the\nminimization of the energy integral for different initial conditions, and its relationship with the optimal\ndampingcoefficientgivenbytheminimizationoftheaveragedenergyintegral, inallthreecoordinatesystems\nthat we introduced in the previous subsection and additionally in the coordinate system defined by the initial\ndisplacements of the masses. System of equations (48) can be easily recast to the form (25) with the modal\ncoordinates\nq1(t) =rm\n4(x1(t) +x2(t))\nq2(t) =rm\n4(x1(t)−x2(t)),(49)\nand with the natural (undamped) frequencies of the modes ω01=ω0andω02=√\n3ω0, where ω0=p\nk/m.\nNormalisation factorsp\nm/4in (49) ensure that our expression (28) for the energy of the system corresponds\nto energy expressed over the displacements and velocities of the masses, i.e.\nE(t) =2X\ni=1\u0000\n˙qi(t)2+ω2\n0iqi(t)2\u0001\n=2X\ni=1\u0012m˙xi(t)2\n2+kxi(t)2\n2\u0013\n+k(x1(t)−x2(t))2\n2. (50)\nUsing the specific values of undamped modal frequencies of this system, relations (36), (41) and (44)\nbecome\nγopt(ψ, θ1, θ2) =s\n3\n6 cos2ψcos2θ1+ 2 sin2ψcos2θ2ω0, (51)\nγopt=√\n6\n2ω0, (52)\nγopt(β1, β2) =r3\n6β1+ 2β2ω0. (53)\nSince ω01=ω0, the range of (53) is γopt∈[√\n2ω0/2,+∞)(see (38)).\nAs examples of the behavior of the damping coefficient (51) as a function of the angles {ψ, θ1, θ2}and\nits relationship with the damping coefficient (52), in Fig. 7 we show γopt(ψ, θ1, θ2)/γoptforψ={π/3, π/6}\nandθi∈[0, π]. In Fig. 8 we show ratio of the damping coefficient (53) and the damping coefficient (52), i.e.\nγopt(β1, β2)/γopt.\nIf the initial energy is comprised only of potential energy, in terms of initial modal coordinates we have\nE0=ω2\n0q2\n01+ 3ω2\n0q2\n02, thus, the initial modal coordinates satisfy\nq01s\nω2\n0\nE0∈[−1,1],\nq02s\nω2\n0\nE0∈h\n−√\n3/3,√\n3/3i\n,\n0≤ω2\n0\nE0\u0000\nq2\n01+ 3q2\n02\u0001\n≤1.(54)\n12Figure 7: Ratio γopt(ψ, θ1, θ2)/γoptof the optimal damping coefficients (51) and (52) for ψ={π/3, π/6}andθi∈[0, π]. (a)\nForψ=π/3, the total initial energy E0is distributed within the modes as E01=E0/4andE02= 3E0/4. (b) For ψ=π/6,\nthe total initial energy is distributed within the modes as E01= 3E0/4andE02=E0/4. Singularities for θ1=θ2=π/2\nare indicated by infinity symbols, and the points around singularities for which γopt(ψ, θ1, θ2)/γopt>3are removed on both\nfigures (central white areas). Black lines, on both figures, indicate the points for which γopt(ψ, θ1, θ2)/γopt= 1. On both\nfigures, ratio attains minimum for the corner points, i.e. for (θ1, θ2) ={(0,0),(0, π),(π,0),(π, π)}.\nFigure 8: Ratio γopt(β1, β2)/γoptof the optimal damping coefficients (53) and (52) for β1∈[0,1],β2∈[0,1]and the constraint\n0≤(β1+β2)≤1. Singularity for β1=β2= 0is indicated by the filled red circle, and the points near singularity, for which\nγopt(β1, β2)/γopt>3, are removed, thus, a small white triangle is formed with the right angle at the origin. Black line indicates\nthe points for which γopt(β1, β2)/γopt= 1. The minimum value of the ratio is at the point (β1, β2) = (1 ,0).\nFurthermore, we can write the optimal damping coefficient (46) as\nγopt(q01, q02) =s\nE0\n2ω2\n0(q2\n01+q2\n02)ω0, (55)\nand the condition (47) as\nω2\n0\nE0\u0000\nq2\n01+q2\n02\u0001\n=1\n3. (56)\n13In Fig. 9(a) we show the ratio of (55) and (52), i.e. γopt(q01, q02)/γopt. The domain of this function\nconsists of points inside and on the ellipse, i.e. it is given by (54). Similarly as before, singularity at\n(q01, q02) = (0 ,0)is indicated by the infinity symbol, and the points for which γopt(q01, q02)/γopt>3are\nremoved. For points inside the circle we have γopt(q01, q02)/γopt>1, and for points outside the circle\nwe have γopt(q01, q02)/γopt<1. Minimum values of this ratio are√\n3/3≈0.58, obtained for the points\n(q01, q02) ={(−√E0/ω0,0),(√E0/ω0,0)}.\nFigure 9: (a) Ratio γopt(q01, q02)/γoptof the optimal damping coefficients (55) and (52). (b) Ratio γopt(x01, x02)/γoptof the\noptimal damping coefficients (57) and (52). Singularities, at points (0,0)on the both figures, are denoted by infinity symbols,\nand the points near singularities, for which γopt/γopt>3, are removed. Black circles on both figures indicate the points for\nwhich γopt/γopt= 1.\nUsing (49) we can write the optimal damping coefficient (55) in terms of initial displacements xi(0)≡x0i\nas\nγopt(x01, x02) =s\nE0\nm(x2\n01+x2\n02)=s\nE0\nmω2\n0(x2\n01+x2\n02)ω0. (57)\nIf the initial energy is comprised only of potential energy, in terms of initial displacements we have E0=\nmω2\n0(x2\n01+x2\n02−x01x02), thus, the initial displacements of the masses satisfy\nx0is\nmω2\n0\nE0∈[−1,1],\n0≤mω2\n0\nE0\u0000\nx2\n01+x2\n02−x01x02\u0001\n≤1,(58)\nand the condition (56) is now\nmω2\n0\nE0\u0000\nx2\n01+x2\n02\u0001\n=2\n3. (59)\nIn Fig. 9(b) we show the ratio of (57) and (52), i.e. γopt(x01, x02)/γopt. The domain of this function\nconsists of points given by (58). Similarly as before, singularity at (x01, x02) = (0 ,0)is indicated by the\ninfinity symbol, and the points for which γopt(x01, x02)/γopt>3are removed. For points inside the circle\nγopt(x01, x02)/γopt>1, and for points outside the circle γopt(x01, x02)/γopt<1. Minimum values of this\nratio are√\n3/3≈0.58, obtained for the points (x01, x02) =\u0010\n±q\nE0\nmω2\n0,±q\nE0\nmω2\n0\u0011\n.\n14Figure 10: Schematic figure of a MDOF system with Ndegrees of freedom.\n4. MDOF systems with MPD\nHere we consider the MDOF system with Ndegrees of freedom shown schematically in Fig. 10. As in\nthe Section 3, we will consider MPD, i.e. masses {m1, m2, ..., m N}, spring constants {k1, k2, ..., k N+1}, and\ndampers {c1, c2, ..., c N}can in general be mutually different but the condition ci/mi= 2γholds for any\ni={1, ..., N}, where γis the damping coefficient. Therefore, the reasoning we presented in Section 3 can\nbe applied here, with the main difference that now the system has Nmodes instead of two. Again, we can\nwrite each initial condition over polar coordinates, as in the 2-DOF case (see (30)), only now we have N\npairs of polar coordinates instead of two.\nThe energy of each mode is given by (31), and consequently, the integral of the total energy over the\nentire time, for some arbitrary initial condition, is simply calculated similarly as in (32), i.e.\nI(γ,{E0i},{θi}) =NX\ni=1Z∞\n0Ei(t)dt=NX\ni=1E0i\n2ω0i\u0012ω2\n0i+γ2\nγω0i+γ\nω0icos 2θi+ sin 2 θi\u0013\n, (60)\nwhere, again, Ei(t)is the energy of the i-th mode, E0iis the initial energy of the i-th mode. Thus, each\ninitial condition with energy E0=PN\ni=1E0iis represented by points on Ncircles with radii ri=√E0i, for\nwhich conditionPN\ni=1r2\ni=E0holds, and with angles θithat tell us how initial potential and initial kinetic\nenergy are distributed within the modes.\nSimilarly as before, initial energy of the i-th mode can be written as E0i=a2\niE0, where coefficient\na2\ni∈[0,1]denotes the fraction of the initial energy of the i-th mode in the total initial energy E0, and the\ncondition\nNX\ni=1a2\ni= 1 (61)\nholds. Relation (61) defines a sphere embedded in N-dimensional space and we can express the coefficients\naiover N-dimensional spherical coordinates ( N−1independent coordinates, i.e. angles, since the radius\nis equal to one), but for the sake of simplicity we will not do that here and we will stick to writing the\nexpressions as a functions of the coefficients ai. Thus, we can write (60) as\nI(γ,{ai},{θi}) =NX\ni=1Z∞\n0Ei(t)dt=E0NX\ni=1a2\ni\n2ω0i\u0012ω2\n0i+γ2\nγω0i+γ\nω0icos 2θi+ sin 2 θi\u0013\n.(62)\nWe differentiate relation (62) by γand equate it to zero and get\nγopt({ai},{θi}) = NX\ni=12a2\nicos2θi\nω2\n0i!−1/2\n(63)\nas the optimal damping coefficient for which integral (62) is minimal. For any fixed set of coefficients\n{ai}, the smallest magnitude of the function (63) is obtained for cos2θi= 1∀i, which corresponds to the\n15initial conditions with initial energy comprised only of potential energy distributed within the modes, i.e\nE0=PN\ni=1E0Pi. In that case the denominator of (63) is\nf({ai}) = NX\ni=12a2\ni\nω2\n0i!1/2\n(64)\nand using a2\n1= 1−PN\ni=2a2\ni(see (61)) we can write (64) as\nf({ai}) = \n2\nω2\n01+NX\ni=22a2\ni\u00121\nω2\n0i−1\nω2\n01\u0013!1/2\n. (65)\nSince ω01< ω 0ifor any i≥2, each term in the sum of relation (65) is negative, and we can conclude that\nthe function (65) has maximum for the set {ai}={1,0, ...,0}. Thus, the minimum value of the optimal\ndamping coefficient (63) is√\n2ω01/2, and it is obtained for a1= 1andθ1={0, π}, which corresponds to the\ninitial conditions with initial energy comprised only of potential energy in the first mode, i.e. E0=E0P1.\nOn the other hand, for any fixed set {ai}, the function (63) has singularities for cos2θi= 0∀i. Thus, the\nrange of the optimal damping coefficient (63) is\nγopt∈h√\n2ω01/2,+∞\u0011\n. (66)\nIn Appendix A we have calculated the average of the integral (62) over a set of all initial conditions and\nobtained\nI(γ) =E0\n2NNX\ni=1\u0012ω2\n0i+γ2\nγω2\n0i\u0013\n. (67)\nWe differentiate relation (67) by γand equate it to zero and obtain\nγopt=N1/2 NX\ni=11\nω2\n0i!−1/2\n(68)\nas the optimal damping coefficient with respect to the averaged integral (67).\nSince the ratio of the initial potential energy of the i-th mode and the total initial energy is\nβi=E0Pi\nE0=a2\nicos2θi, (69)\nwhere βi∈[0,1]and the condition 0≤PN\ni=1βi≤1holds, we can write (63) as a function of the distribution\nof the initial potential energy over the modes, i.e.\nγopt({βi}) = NX\ni=12βi\nω2\n0i!−1/2\n. (70)\nThe minimum value of (70) is√\n2ω01/2, obtained for β1= 1andβi= 0fori≥2, while γopt→+∞for\nβi= 0∀i. Using βi=ω2\n0iq2\n0i/E0, we can write (63) as a function of initial modal coordinates as well, i.e.\nγopt({q0i}) =s\nE0\n2PN\ni=1q2\n0i, (71)\nwhere q0i∈[−√E0/ω0i,√E0/ω0i]and the condition 0≤PN\ni=1ω2\n0iq2\n0i≤E0holds.\n164.1. Quantitative example\nHere we consider the MDOF system as the one shown schematically in Fig. 10 but with mi=m,ci=c\nfori={1, ..., N}, and with ki=kfori={1, ..., N + 1}. Such a system without damping, i.e with ci= 0∀i,\nis a standard part of the undergraduate physics/mechanics courses [22]. Therefore, for the MDOF system\nwith Ndegrees of freedom we are considering here, the undamped modal frequencies are [21, 22]\nω0i= 2ω0sin\u0012iπ\n2(N+ 1)\u0013\n,with i={1, ..., N}, (72)\nand where ω0=p\nk/m. In Fig. 11(a) we show undamped modal frequencies ω01,ω0Nand damping\ncoefficient γopt, i.e. (68), calculated with (72), as functions of N. We clearly see that the coefficient γoptis\nin the over-damped regime from the perspective of the first mode, and in the under-damped regime from\nthe perspective of highest mode, for any N > 1, and in the case N= 1all three values match. In Fig. 11(b)\nwe show ratios γopt/ω01andω0N/γoptand we see that both ratios increase with increasing N.\nFigure 11: (a) Undamped modal frequencies ω01(blue circles), ω0N(red x’s) and the damping coefficient γopt(black squares)\nas functions of the number of the masses N. (b) Ratios γopt/ω01(blue line) and ω0N/γopt(red line), shown as solid lines due\nto the high density of the shown points.\nWe show in Appendix B that the following limits hold\nlim\nN→+∞γopt(N) = 0 , (73)\nlim\nN→+∞γopt(N)\nω01(N)= +∞, (74)\nlim\nN→+∞ω0N(N)\nγopt(N)= +∞. (75)\nWe note here that these limit values do not correspond to the transition from a discrete to a continuous\nsystem, but simply tell us the behavior of these quantities with respect to the increase in the number of\nmasses, i.e. with respect to the increase in the size of the discrete system.\nFrom everything that has been said so far, it is clear that the damping coefficient γopt, obtained by\nminimizing the energy integral averaged over all initial conditions that correspond to the same initial energy,\ncannot be considered generally as optimal and that, by itself, it says nothing about optimal damping of the\nsystem whose dynamics started with some specific initial condition. Damping coefficient (63), which is given\nby the minimization of the energy integral for a specific initial condition, is of course a better choice for\noptimal damping of an MDOF system, than the damping coefficient γopt, if we want to consider how the\nsystem dissipates energy the fastest for a particular initial condition, but, as we argue in the subsection 4.2,\nthis damping coefficient also has some obvious deficiencies.\n174.2. Issues with the minimum of the energy integral as a criterion for optimal damping\nWe can ask, for example, whether in an experiment, with known initial conditions, in which an MDOF\nsystem is excited to oscillate, a damping coefficient (63) would be the best choice if we want that the system\nsettles down in equilibrium as soon as possible? Here, in three points, we explain why we think the answer\nto that question is negative:\n•From relation (62), we see that, due to the term sin 2θi, the energy integral is sensitive to changes\nθi→ −θiandθi→π−θi, which correspond to changes of initial conditions (q0i,˙q0i)→(q0i,−˙q0i)and\n(q0i,˙q0i)→(−q0i,˙q0i). When we differentiate (62) to determine γfor which the energy integral has a\nminimum, the term sin 2θicancels and as a result the coefficient (63) is not sensitive to this change\nin initial conditions. Such changes in the initial conditions lead to significantly different situations.\nFor example, if q0i>0and ˙q0i>0, the i-th mode in the critical and over-damped regime (i.e. for\nγ≥ω0i) will never reach the equilibrium position, while for q0i>0and ˙q0i<0, and i-th mode\ninitial kinetic energy grater than initial potential energy, it can go through the equilibrium position\nonce, depending on the magnitude of the damping coefficient, and there will be the smallest damping\ncoefficient in the over-damped regime for which no crossing occurs and for which the solution converges\nto equilibrium faster than for any other damping coefficient [24]. Therefore, the damping coefficient\nconsidered optimal would have to be sensitive to this change in initial conditions.\n•Damping coefficient (63) has singularities for cosθi= 0∀i, i.e. for initial conditions for which all\ninitial energy is kinetic. For such initial conditions, the higher the damping coefficient, the higher and\nfaster the dissipation. In other words, the higher the damping coefficient, the faster the energy integral\ndecreases. Therefore, coefficient (63) diverges for that type of initial conditions. This would actually\nmean that, for this initial conditions, it is optimal to take the damping coefficient as high as possible,\nbut in principle this corresponds to a situation in which all modes are highly over-damped, i.e. all\nmasses reach their maximum displacements in a very short time and afterwards they begin to return\nto the equilibrium position almost infinitely slowly. Figuratively speaking, it is as if we immersed the\nsystem in concrete. This issue has recently been addressed in the context of free vibrations of SDOF\n[24] and was already noticed in [25]. Therefore, simply taking the highest possible damping coefficient,\nas suggested by relation (63) for this type of initial conditions, is not a good option.\n•The damping coefficient (63) is determined on the basis of the energy integral over the entire time and\ntherefore it does not take into account that in nature and experiments these systems effectively return\nto the equilibrium state for some finite time.\nBecause of the above points, in the next section we provide a new approach to determine the optimal\ndamping of MDOF systems.\n5. Optimal damping of an MDOF system: a new perspective\nFrom a theoretical perspective, systems with viscous damping asymptotically approach the equilibrium\nstate and never reach it exactly. In nature and in experiments, these systems reach the equilibrium state\nwhich is not an exact zero energy state, but rather a state in which the energy of the system has decreased\nto the level of the energy imparted to the system by the surrounding noise, or to the energy resolution of\nthe measuring apparatus. Following this line of thought, we will define a system to be in equilibrium for\ntimes t > τsuch that\nE(τ)\nE0= 10−δ, (76)\nwhere E(τ)is the energy of the system at t=τ,E0is the initial energy, and δ > 0is a dimensionless\nparameter that defines what fraction of the initial energy is left in the system. This line of thought has\nrecently been used to determine the optimal damping of SDOF systems [24], and here we extend it to\nMDOF systems. Therefore, in what follows, we will consider as optimal the damping coefficient for which\nthe systems energy drops to some energy level of interest, e.g. to the energy resolution of the experiment,\nthe fastest and we will denote it with ˜γ.\n185.1. Optimal damping of the i-th mode of a MDOF system with MPD\nHere we will consider the behavior of the energy of the i-th mode of the MDOF system with MPD and\ndetermine the optimal damping coefficient ˜γiof the i-th mode with respect to criterion (76). For any MDOF\nsystem with N≥1degrees of freedom with MPD, each mode behaves as a SDOF system studied in Section\n2, with the damping coefficient γand the undamped (natural) frequency ω0i. Thus (see relation (31)), the\nratio of the energy of the i-th mode, Ei(γ, t), and initial energy of the i-th mode, E0i, is given by\nEi(γ, t)\nE0i=e−2γt\u0012\ncos2(ωit) +γcos 2θisin(2ωit)\nωi+\u0000\nω2\n0i+γ2+ 2ω0iγsin 2θi\u0001sin2(ωit)\nω2\ni\u0013\n(77)\nfor the under-damped ( γ < ω oi) and over-damped ( γ > ω 0i) regime. We will repeat here briefly for clarity,\nωi=p\nω2\n0i−γ2is the damped angular frequency and θiis the polar angle which determines the initial\nconditions q0iand ˙q0iof the i-th mode and the distribution of the initial energy within the mode, i.e.\ninitial potential and initial kinetic energy of the i-th mode are E0Pi=E0icos2θiandE0Ki=E0isin2θi\nrespectively. Energy to initial energy ratio for the i-th mode in the critically damped regime ( γ=ω0i) is\nsimply obtained by taking γ→ω0ilimit of the relation (77), and we obtain\nEi(γ=ω0i, t)\nE0i=e−2ω0it\u0000\n1 + 2 ω0i(cos 2 θi)t+ 2ω2\n0i(1 + sin 2 θi)t2\u0001\n. (78)\nIn relations (77) and (78), we explicitly show that the energy depends on the damping coefficient and time,\nbecause in what follows we will plot these quantities as functions of these two variables for fixed initial\nconditions, i.e. fixed θi. We will investigate the behavior for several types of initial conditions, which of\ncourse will not cover all possible types of initial conditions, but will give us a sufficiently clear picture of the\ndeterminationandbehavioroftheoptimaldampingwithrespecttotheinitialconditionsandtheequilibrium\nstate defined with condition (76).\n5.1.1. Initial energy of the i-th mode comprised only of potential energy\nIn Fig. 12 we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition\nθi= 0, which corresponds to the initial energy of the i-th mode comprised only of potential energy. Four\nblack contour lines denote points with Ei(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, as indicated by\nthe numbers placed to the left of each contour line. Each contour line has a unique point closest to the γ\naxis, i.e. corresponding to the damping coefficient ˜γifor which that energy level is reached the fastest. As an\nexample, we draw arrow in Fig. 12 that points to the coordinates (γ, t) = (0 .840ω0i,5.15ω−1\n0i), i.e. to the tip\nof the contour line with points corresponding to Ei(γ, t) = 10−4E0i. Thus, for the initial condition θi= 0,\n˜γi= 0.840ω0iis the optimal damping coefficient for the i-th mode to reach this energy level the fastest, and\nit does so at the instant τi= 5.15ω−1\n0i. In Table 1 we show results for other energy levels corresponding\nto contour lines shown in Fig. 12. Here, and in the rest of the paper, we have rounded the results for the\ndamping coefficient to three decimal places, and for the time to two decimal places.\nEi(γ, t)/E0i˜γi[ω0i]τi[ω−1\n0i]\n10−30.769 4.18\n10−40.840 5.15\n10−50.885 6.16\n10−60.915 7.20\nTable 1: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the\ninitial condition θi= 0.\nConsider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates\nonly in the first mode and that all initial energy was potential, i.e. E01=E0andθ1= 0. Furthermore,\n19Figure 12: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi= 0. For this initial\ncondition, initial energy of the i-th mode is comprised only of potential energy. Four black contour lines denote points with\nEi(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, as indicated by the numbers placed to the left of each contour line. As\nan example of determining the optimal damping for which the system reaches the desired energy level the fastest, i.e. with\nrespect to the condition (76), we draw the arrow that points to the coordinates (γ, t) = (0 .840ω0i,5.15ω−1\n0i)for which the i-th\nmode reaches the level Ei(γ, t)/E0i= 10−4the fastest. Thus, ˜γi= 0.840ω0iis the optimal damping coefficient to reach this\nenergy level the fastest. Optimal values for other energy levels, denoted with contour lines, are given in Table 1.\nsuppose that the system has effectively returned to equilibrium when its energy drops below 10−6E0, due\nto the resolution of the measuring apparatus. It is clear form the Table 1 that ˜γ1= 0.915ω01would be\noptimal in such a scenario. In the same scenario, optimal damping coefficient given by the minimization of\nthe energy integral, i.e. (63), would be γopt=√\n2ω01/2 = 0 .707ω01, thus, a very bad choice in the sense\nthat this damping coefficient would not be optimal even in an experiment with a significantly poorer energy\nresolution (see Table 1). This simple example illustrates that, from a practical point of view, one has to take\ninto account both the initial conditions and the resolution of the measuring apparatus in order to determine\nthe optimal damping coefficient.\n5.1.2. Initial energy of the i-th mode comprised only of kinetic energy\nIn Fig. 13(a) and (b) we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial\ncondition θi=π/2, which corresponds to the initial energy of the i-th mode comprised only of kinetic\nenergy. In Fig. 13(b) we show results for larger data span than in Fig. 13(a), and only contour line for points\ncorresponding to Ei(γ, t) = 10−3E0i. The left arrow in Fig. 13(b) indicates the same coordinates as the\narrow in Fig. 13(a), and the right arrow in Fig. 13(b) points to the coordinates (γ, t) = (13 .316ω0i,4.66ω−1\n0i)\nwith Ei(γ, t) = 10−3E0i. Thus, for γ >13.316ω0ithe system comes sooner to the energy level 10−3E0ithan\nforγ= 0.722ω0i, but these highly over-damped damping coefficients would correspond to restricting the\nsystem to infinitesimal displacements from equilibrium, after which the system returns to the equilibrium\nstate practically infinitely slowly [24]. Thus, for this initial condition we take the damping coefficient in the\nunder-damped regime, i.e. ˜γi= 0.722ω0i, as optimal for reaching the level Ei(γ, t) = 10−3E0ithe fastest.\nFor all energy levels the behaviour is qualitatively the same, and the results are given in Table 2.\nConsider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates\nonly in the first mode and that all initial energy was kinetic, i.e. E01=E0andθ1=π/2. Furthermore,\nsuppose that the system has effectively returned to equilibrium when its energy drops below 10−6E0, due\nto the resolution of the measuring apparatus. It is clear form the Table 2 that ˜γ1= 0.892ω01would be\noptimal in such a scenario. In the same scenario, optimal damping coefficient given by the minimization of\nthe energy integral, i.e. (63), would be γopt= +∞.\n20Figure 13: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=π/2. For this initial\ncondition, initial energy of the i-th mode is comprised only of kinetic energy. (a) Four black contour lines denote points with\nEi(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, andthearrowpointstothecoordinates (γ, t) = (0 .722ω0i,4.66ω−1\n0i), with\nEi(γ, t)/E0i= 10−3, for which this level of energy is reached in shortest time for the shown data span. (b) Contour line for\npoints with Ei(γ, t) = 10−3E0iis shown for larger data span, left arrow points to the coordinates (γ, t) = (0 .722ω0i,4.66ω−1\n0i),\nand the right arrow to the coordinates (γ, t) = (13 .316ω0i,4.66ω−1\n0i), both with Ei(γ, t)/E0i= 10−3. Thus, for γ >13.316ω0i\nenergy level 10−3E0iis reached faster than for γ= 0.7223ω0i. See text for details.\nEi(γ, t)/E0i˜γi[ω0i]τi[ω−1\n0i]\n10−30.722 4.66\n10−40.794 5.50\n10−50.852 6.42\n10−60.892 7.40\nTable 2: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the\ninitial condition θi=π/2.\nHere we note that if in such an experiment we can set the damping coefficient to be in the over-damped\nregime in the first part of the motion, i.e. when the system is moving from the equilibrium position to\nthe maximum displacement, and in the under-damped regime in the second part of the motion, i.e. when\nthe system moves from the position of maximum displacement back towards the equilibrium position, then\nthe fastest way to achieve equilibrium would be to take the largest experimentally available over-damped\ncoefficient in the first part of the motion, and the under-damped coefficient optimised like in 5.1.1 in the\nsecond part of the motion, with the fact that we have to carry out the optimization with respect to the\nenergy left in the system at the moment when the system reached the maximum displacement and with\nrespect to the energy resolution of the experiment.\n5.1.3. Initial energy of the i-th mode comprised of potential and kinetic energy\nIn Fig. 14(a) we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition\nθi=π/3, whichcorrespondstothe initialenergyofthe i-thmodecomprisedofkineticenergy E0Ki= 3E0i/4\nand potential energy E0Pi=E0i/4, with both initial normal coordinate and velocity positive, i.e. with\nq0i>0and ˙q0i>0. The results for optimal damping are obtained by the same method as in 5.1.1 and are\ngiven in Table 3 for data shown in Fig. 14(a), and in Table 4 for data shown in Fig. 14(b). We see that\nthe energy dissipation strongly depends on the relative sign between q0iand ˙q0i. It was recently shown,\n21for free vibrations of SDOF, that for an initial condition with initial kinetic energy greater than initial\npotential energy and opposite signs between x0andv0, an optimal damping coefficient can be found in the\nover-damped regime [24], thus, the same is true when we consider any mode of a MDOF system with MPD.\nFigure 14: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), (a) for initial condition θi=π/3, and (b) for initial\ncondition θi=−π/3. For both initial conditions, initial energy of the i-th mode is comprised of kinetic energy E0Ki= 3E0i/4\nand potential energy E0Pi=E0i/4. For θi=π/3initial normal coordinate and velocity are of the same signs, i.e. q0i>0and\n˙q0i>0. For θi=−π/3initial normal coordinate and velocity are of the opposite signs, i.e. q0i>0and ˙q0i<0.\nEi(γ, t)/E0i˜γi[ω0i]τi[ω−1\n0i]\n10−30.751 4.66\n10−40.825 5.58\n10−50.875 6.55\n10−60.908 7.58\nTable 3: Optimal damping coefficient ˜γifor which the energy\nof the i-th mode drops to the level 10−δE0ithe fastest, with\nthe initial condition θi=π/3.Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1\n0i]\n10−31.075 1.87\n10−41.112 2.42\n10−51.135 3.02\n10−61.145 3.64\nTable 4: Optimal damping coefficient ˜γifor which the energy\nof the i-th mode drops to the level 10−δE0ithe fastest, with\nthe initial condition θi=−π/3.\nConsider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates\nonly in the first mode and that 75%of initial energy was kinetic and 25%of initial energy was potential,\nand with q01>0and ˙q01>0, i.e. E01=E0andθ1=π/3. Furthermore, suppose that the system\nhas effectively returned to equilibrium when its energy drops below 10−6E0, due to the resolution of the\nmeasuring apparatus. It is clear form the Table 3 that ˜γ1= 0.908ω01would be optimal in such a scenario. In\nthe same scenario, but with q01>0and ˙q01<0, i.e. for θ1=−π/3, we see from Table 4 that ˜γ1= 1.145ω01\nwould be optimal. Optimal damping coefficient given by the minimization of the energy integral, i.e. (63),\nis insensitive to the change of the sign of ˙q01, and it would be γopt=√\n2ω01= 1.414ω01in both cases.\nWe note here, that for the initial conditions of the i-th mode with initial kinetic energy much grater than\ninitial potential energy, i.e. E0Ki>> E 0Pi, and with opposite signs of initial displacement and velocity,\ni.e.sgn(q0i)̸= sgn( ˙ q0i), the optimal damping coefficient is going to be deep in the over-damped regime and\ndissipation of initial energy will happen in a very short time. If, for any reason, this is not desirable in\nsome particular application, one can always find damping coefficient in the under-damped regime, with that\nsame initial condition, which can serve as an alternative. As an example of such a situation, in Fig. 15 we\nshow the base 10 logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=−9π/20, which\ncorresponds to the initial energy of the i-th mode comprised of kinetic energy E0Ki≈0.98E0iand potential\nenergy E0Pi≈0.02E0i, with q0i>0and ˙q0i<0. In Fig. 15 we see that the i-th mode will reach the energy\n22level 10−6E0ithe fastest for γ= 3.222ω0i, and in case, e.g., that such damping coefficient is difficult to\nachieve experimentally, another choice for the optimal damping coefficient can be γ= 0.883ω0i.\nFigure 15: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=−9π/20. Black contour\nline denotes the points with Ei(γ, t) = 10−6E0i. Left arrow points to the coordinates (γ, t) = (0 .883ω0i,7.30ω−1\n0i)for which\nlevel 10−6E0iis reached the fastest in the under-damped regime, and the right arrow points to the coordinates (γ, t) =\n(3.222ω0i,0.87ω−1\n0i)for which the same level is reached the fastest in the over-damped regime.\n5.2. Optimal damping of a MDOF system with MPD\nIf all modes of a MDOF system with Ndegrees of freedom are excited, the ratio of the energy of the\nsystem, E(γ, t), and initial energy of the system, E0, is given by\nE(γ, t)\nE0=NX\ni=1E0i\nE0e−2γt\u0012\ncos2(ωit) +γcos 2θisin(2ωit)\nωi+\u0000\nω2\n0i+γ2+ 2ω0iγsin 2θi\u0001sin2(ωit)\nω2\ni\u0013\n,(79)\nwhere the set of all initial energies of the modes, i.e. {E0i}, and the set of all polar angles, i.e. {θi},\ndetermines the initial condition of the whole system. Since for MPD the damping of the system as a whole\nis determined by only one damping coefficient γ, we can calculate the base 10logarithm of the ratio (79),\nbut using a unique units for γ,tandω0ifor all modes, and from these data determine the optimal damping\ncoefficient ˜γ, for which the system will come to equilibrium in the sense of the condition (76) the fastest,\nin the same way as in subsubsections 5.1.1-5.1.3 where we showed how to determine the optimal damping\nof individual modes. One practical choice for the units might be ω01forγand for ω0i∀i, and ω−1\n01fort.\nThis way, we have the easiest insight into the relationship between the first mode and the optimal damping\ncoefficient that we want to determine, in the sense that we can easily see whether the first mode is under-\ndamped, over-damped or critically damped in relation to it, which is important since the first mode is often\nthe dominant mode. If we apply this to the 2-DOF system studied in 3.1, we obtain\nE(γ, t)\nE0=2X\ni=1E0i\nE0e−2γt\u0012\ncos2(ωit) +γcos 2θisin(2ωit)\nωi+\u0000\nω2\n0i+γ2+ 2ω0iγsin 2θi\u0001sin2(ωit)\nω2\ni\u0013\n,(80)\nwhere ω01=ω0,ω02=√\n3ω0,ω1=p\nω2\n0−γ2,ω2=p\n3ω2\n0−γ2and we take that the damping coefficient\nis in ω0units, while the time is in ω−1\n0units. We are now in a position to determine the optimal damping\nof this 2-DOF system for different initial conditions. Again, we will not investigate all possible types of the\n23initial conditions, but two qualitatively different ones, one with initial energy comprised only of potential\nenergy, and the other with initial energy comprised only of kinetic energy. These two examples will give us\na picture of the procedure for determining the optimal damping coefficient ˜γfor this 2-DOF system. The\nsame procedure for determining the optimal damping can be in principle carried out for any MDOF system\nwith MPD, with any initial condition.\n5.2.1. Optimal damping of the 2-DOF system with initial energy comprised only of potential energy\nHerewechooseinitialconditionwith E01=E02=E0/2andθ1=θ2= 0, i.e. withinitialpotentialenergy\ndistributed equally between the two modes and zero initial kinetic energy. In Fig. 16 we show the base 10\nlogarithm of the ratio (80), i.e. log (E(γ, t)/E0), for the chosen initial condition. In Table 5 we show results\nfor other energy levels corresponding to contour lines shown in Fig. 16. For this initial condition, optimal\ndamping coefficient given by the minimization of the energy integral, i.e. (63), is γopt=p\n3/4ω0= 0.866ω0.\nFigure16: Thebase 10logarithmoftheratio(80), i.e. log (E(γ, t)/E0), forinitialcondition E01=E02=E0/2andθ1=θ2= 0.\nFor this initial condition, initial energy of the 2-DOF system is comprised only of potential energy distributed equally between\nthe modes. Four black contour lines denote points with E(γ, t)/E0={10−3,10−4,10−5,10−6}respectively, as indicated by\nthe numbers placed to the left of each contour line. As an example of determining the optimal damping for which the system\nreaches the desired energy level the fastest, i.e. with respect to the condition (76), we draw the arrow that points to the\ncoordinates (γ, t) = (0 .859ω0,5.37ω−1\n0)for which the energy of the system reaches the level E(γ, t)/E0= 10−4the fastest.\nThus, ˜γ= 0.859ω0is the optimal damping coefficient to reach this energy level the fastest.\nE(γ, t)/E0˜γ[ω0]τ[ω−1\n0]\n10−30.817 4.36\n10−40.859 5.37\n10−50.893 6.27\n10−60.924 7.55\nTable 5: Optimal damping coefficient ˜γfor which the energy of the system drops to the level 10−δE0the fastest, with the\ninitial condition E01=E02=E0/2andθ1=θ2= 0.\n245.2.2. Optimal damping of the 2-DOF system with initial energy comprised only of kinetic energy\nHere we choose initial condition with E01=E02=E0/2andθ1=θ2=π/2, i.e. with initial kinetic\nenergy distributed equally between the two modes and zero initial potential energy. In Fig. 17(a) and (b)\nwe show the base 10logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for the chosen initial condition. In\nTable 6 we show results for other energy levels corresponding to contour lines shown in Fig. 17(a). For this\ninitial condition, optimal damping coefficient given by the minimization of the energy integral, i.e. (63), is\nγopt= +∞.\nFigure 17: The base 10logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for initial condition E01=E02=E0/2andθ1=\nθ2=π/2. For this initial condition, initial energy of the 2-DOF system is comprised only of kinetic energy distributed equally\nbetween the modes. (a) Four black contour lines denote points with E(γ, t)/E0={10−3,10−4,10−5,10−6}respectively, and\nthearrowpointstothecoordinates (γ, t) = (0 .783ω0,4.60ω−1\n0), with E(γ, t)/E0= 10−3, forwhichthislevelofenergyisreached\nin shortest time for the shown data span. (b) Contour line for points with E(γ, t) = 10−3E0is shown for larger data span, left\narrow points to the coordinates (γ, t) = (0 .783ω0,4.60ω−1\n0), and the right arrow to the coordinates (γ, t) = (15 .927ω0,4.60ω−1\n0),\nboth with Ei(γ, t)/E0i= 10−3. Thus, for γ >15.927ω0energy level 10−3E0is reached faster than for γ= 0.783ω0.\nE(γ, t)/E0˜γ[ω0]τ[ω−1\n0]\n10−30.783 4.60\n10−40.838 5.72\n10−50.861 6.47\n10−60.909 7.78\nTable 6: Optimal damping coefficient ˜γfor which the energy of the system drops to the level 10−δE0the fastest, with the\ninitial condition E01=E02=E0/2andθ1=θ2=π/2.\n6. Conclusion and outlook\nThe main message of this paper is that the dissipation of the initial energy in vibrating systems signifi-\ncantly depends on the initial conditions with which the dynamics of the system started, and ideally it would\nbe optimal to always adjust the damping to the initial conditions. We took one of the known criteria for\noptimal damping, the criterion of minimizing the (zero to infinity) time integral of the energy of the system,\naveraged over all possible initial conditions corresponding to the same initial energy, and modified it to take\ninto account the initial conditions, i.e. instead of averaging over of all possible initial conditions, we studied\n25the dependence of the time integral of the energy of the system on initial conditions and determined the\noptimal damping as a function of the initial conditions. We found that the thus obtained optimal damping\ncoefficients take on an infinite range of values depending on the distribution of initial potential energy and\ninitial kinetic energy within the modes. We also pointed out the shortcomings of the thus obtained optimal\ndamping coefficients and introduced a new method for determining optimal damping. Our method is based\non the determination of the damping coefficients for which the energy of the system drops the fastest be-\nlow some energy threshold (e.g. below the energy resolution of the experiment). We have shown that our\nmethod gives, both quantitatively and qualitatively, different results from the energy integral minimization\nmethod. In particular, the energy integral minimization method gives infinite optimal damping for initial\nconditions with purely kinetic energy, i.e. this method overlooks the region of underdamped coefficients\nfor which strong energy dissipation occurs with this type of initial conditions, while this region is clearly\nseen and taken into account if one looks the energy behaviour directly, as we did. Furthermore, the energy\nintegral minimization method gives the optimal damping which does not depend on the signs of the initial\nconditions, and we have shown that energy dissipation can strongly depend on them, which is taken into\naccount in our method.\nAlthough the paper is dedicated to the case of mass-proportional damping, the new method we propose\nfor determining the optimal damping can be applied to the types of damping we did not study in this paper.\nFor example, in the case of a system with Rayleigh damping, the energy can be determined analytically\nusing modal analysis, and based on that analytical expression, it can be numerically investigated for which\nvalues of the mass and stiffness proportionality constants the energy of the system drops the fastest below\nsome energy threshold which effectively corresponds to the equilibrium state. In the case of a system with\ndamping that does not allow analytical treatment, energy, as a function of time and magnitudes of individual\ndampers, can be determined numerically, e.g. by studying the vibrating system as a first order ordinary\ndifferential equation with matrix coefficients and using modern numerical methods for finding a solution of\nsuch an equation. This approach allows one to numerically solve systems with many degrees of freedom.\nThus, we can numerically analyze the time evolution of the energy and find a set of damping parameters for\nwhich the energy drops to a desired energy threshold the fastest. Of course, this approach can be applied\nonly for systems with a moderate number of degrees of freedom and a small number of dampers, due to\nthe rapid growth of the parameter space that needs to be searched. Despite these limitations, we believe\nthat our approach to optimal damping can be useful because, as we have shown, it can provide insights\nthat other approaches overlook. Therefore, in future work we will investigate in detail the application of\nour approach to systems with damping that does not allow modal analysis. Furthermore, real systems can\nrespond to many different initial conditions in operating conditions. We envision that our approach can be\nused to provide an overall optimal damping with respect to all initial conditions or with respect to some\nexpected range of initial conditions. For this purpose, one could consider the energy averaged over the\ninitial conditions and find the damping for which this averaged energy drops to a desired energy threshold\nthe fastest. This will be the topic of our next work.\n7. Acknowledgments\nWe are grateful to Bojan Lončar for making schematic figures of 2-DOF and MDOF systems, i.e. Fig.\n6 and 10, according to our sketches. This work was supported by the QuantiXLie Center of Excellence, a\nprojectco-financedbytheCroatianGovernmentandEuropeanUnionthroughtheEuropeanRegionalDevel-\nopment Fund, the Competitiveness and Cohesion Operational Programme (Grant No. KK.01.1.1.01.0004).\nThe authors have no conflicts to disclose.\nAppendix A. Average of the integral (62)over a set of all initial conditions\nFor reader’s convenience, we will repeat the integral (62) here\nI(γ,{ai},{θi}) =E0NX\ni=1a2\ni\n2ω0i\u0012ω2\n0i+γ2\nγω0i+γ\nω0icos 2θi+ sin 2 θi\u0013\n. (A.1)\n26In order to calculate the average of (A.1) over a set of all initial conditions, one has to integrate (A.1)\nover all coefficients ai, which satisfyPN\ni=1a2\ni= 1andai∈[−1,1], and over all angles θi∈[0,2π]. Due toR2π\n0cos 2θidθi=R2π\n0sin 2θidθi= 0, terms with sine and cosine functions don’t contribute to the average of\n(A.1). Integration over all possible coefficients aiamounts to calculating the average of a2\niover a sphere\nof radius one embedded in Ndimensional space. If we were to calculate the average of the equation of a\nspherePN\ni=1a2\ni= 1over a sphere defined by that equation, we would get\nNX\ni=1a2\ni= 1, (A.2)\nwhere a2\nidenotes the average of a2\niover a sphere. Due to the symmetry of the sphere and the fact that we\nare integrating over the whole sphere, contribution of each a2\niin the sum (A.2) has to be the same, so we\ncan easily conclude that\na2\ni=1\nN, (A.3)\nfor any i. Thus, the average of (A.1) over all possible initial conditions is\nI(γ) =E0\n2NNX\ni=1\u0012ω2\n0i+γ2\nγω2\n0i\u0013\n. (A.4)\nAppendix B. Limit values (73),(74)and(75)\nFor reader’s convenience, we repeat here (68) and (72)\nγopt=N1/2 NX\ni=11\nω2\n0i!−1/2\n(B.1)\nω0i= 2ω0sin\u0012iπ\n2(N+ 1)\u0013\n,with i={1, ..., N}. (B.2)\nUsing (B.2), we can write (B.1) as\nγopt= 2ω0N1/2 NX\ni=11\nsin2ζi!−1/2\n, (B.3)\nwhere ζi=iπ\n2(N+1). Using the fact that sinx < xfor0< x < π/ 2, we obtain\nγopt<2ω0N1/2 \n4(N+ 1)2\nπ2NX\ni=11\ni2!−1/2\n= 2ω0N1/2 π\n2(N+ 1) NX\ni=11\ni2!−1/2\n.\nNow taking N→ ∞and using the well-known formulaP∞\ni=11\ni2=π2\n6, we obtain (73).\nNow we focus on the limit (74). We will use the following well-known inequality sinx > x/ 2for0< x <\nπ/2(this can be easily seen by, e.g. using the fact that sinis a concave function on [0, π/2]). From (B.3) it\nfollows\nγopt\nω01=N1/2(sinζ1)−1 NX\ni=11\nsin2ζi!−1/2\n> N1/2ζ−1\n1·1\n2 NX\ni=11\nζ2\ni!−1/2\n=1\n2N1/2 NX\ni=11\ni2!−1/2\n,\nhence we obtain (74).\n27The limit (75) is also easy to prove. Since\nlim\nN→+∞ω0N= lim\nN→+∞2ω0sin\u0012Nπ\n2(N+ 1)\u0013\n= 2ω0 (B.4)\nand we already showed (73), it is easy to conclude that\nlim\nN→+∞ω0N\nγopt= +∞, (B.5)\ni.e. the limit (75) holds.\nReferences\n[1] R. A. Rojas, E. Wehrle, R. 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Truhar, Mixed control of vibrational systems, ZAMM-Journal of Applied Mathematics and\nMechanics/Zeitschrift für Angewandte Mathematik und Mechanik 99 (9) (2019) e201800328.\n28"    },    {        "title": "1208.0318v1.Artificial_Neural_Network_Based_Prediction_of_Optimal_Pseudo_Damping_and_Meta_Damping_in_Oscillatory_Fractional_Order_Dynamical_Systems.pdf",        "content": " \n   \n 1 \nArtificial Neural Network Based Prediction of \nOptimal Pseudo-Damping and Meta-Damping in \nOscillatory Fractional Order Dynamical Systems \nSaptarshi Das 1, Indranil Pan 1,2  \n1. Department of Power Engineering, Jadavpur \nUniversity, Salt-Lake Campus, LB-8, Sector 3, Kolka ta-\n700098, India. Email: saptarshi@pe.jusl.ac.in  \n2. MERG, Energy, Environment, Modelling and Mineral s \n(E 2M2) Research Section, Department of Earth Science \nand Engineering, Imperial College London, Exhibitio n \nRoad, London SW7 2AZ, UK. Email:  \ni.pan11@imperial.ac.uk , indranil.jj@student.iitd.ac.in  Khrist Sur 1,3 , Shantanu Das 4 \n3. Center for Soft Computing Research, Indian Stati stical \nInstitute, 203 Barrackpore Trunk Road, Kolkata-7001 08, \nIndia. Email: khrist@ieee.org , khrist@isical.ac.in  \n4. Reactor Control Division, Bhabha Atomic Research  \nCentre, Mumbai-400085, India. \nEmail: shantanu@magnum.barc.gov.in \n \n \nAbstract —This paper investigates typical behaviors like \ndamped oscillations in fractional order (FO) dynami cal systems. \nSuch response occurs due to the presence of, what i s conceived as, \npseudo-damping and meta-damping in some special cla ss of FO \nsystems. Here, approximation of such damped oscilla tion in FO \nsystems with the conventional notion of integer ord er damping \nand time constant has been carried out using Geneti c Algorithm \n(GA). Next, a multilayer feed-forward Artificial Ne ural Network \n(ANN) has been trained using the GA based results t o predict the \noptimal pseudo and meta-damping from knowledge of t he \nmaximum order or number of terms in the FO dynamica l system. \nKeywords— Artificial Neural Network (ANN); fraction al order \nlinear systems; meta-damping; pseudo-damping; Genet ic Algorithm   \nI.   INTRODUCTION  \nFractional order dynamical systems which are govern ed by \nfractional order differential equations have got re newed interest \nin the science and engineering community in recent past for its \nhigher capability and flexibility in modeling of na tural \nprocesses [1], [2]. It is well known from basics of  control \ntheory that second order stable oscillatory dynamic al systems \nor higher order stable oscillatory systems, which c an also be \napproximated as second order transfer function mode ls, decay \nwith an exponential envelope upon step or impulse t ype \nexcitation [3]. In other words, a physical system, governed by \nsecond order differential equation of the form (1),  with \nexcitation ()u t and response ()y t shows oscillatory time \nresponse for ()0,1 ξ∈. \n() ()( ) ( )\n( ) ( )( ) ( )2\n2 2 \n2\n2\n2\n22\n2d y t dy t y t u t dt dt \nd y t dy t y t u t dt dt ξω ω ω \nτ ξτ + + = \n⇒ + + =             (1)  \nHere, parameters {}, , ξ ω τ represent the system’s damping \nratio, natural frequency and time constant respecti vely with 1τ ω =. For step and impulse type excitation, the dynamic al \nsystem governed by (1) exhibit damped time response s with an \nexponential envelope, represented by (2) and (3) re spectively. \n( )2\n2 1 \n211 sin 1 tan \n1tey t t ξω ξω ξ ξ ξ−\n−    −    = − − +     −          (2) \n( ) ()2\n2sin 1 \n1tey t t ξω ωω ξ \nξ−\n= − \n−            (3) \nIn contrast, a dynamical system, governed by a two term \nfractional order differential equation (4) can also  show \noscillatory damped time response for ()1,2 α∈, although there \nis no explicit damping term, containing ξin (4). This typical \nbehavior of fractional order systems lead to the co ncept of \n“Pseudo-damping” which can not be visualized with t he \nconventional theory of integer order calculus for d escribing the \ndynamics of physical systems. \n()()()taD y t by t u t α+ =              (4) \nLaplace transform of (4) with zero initial conditio n gives the \nsystem’s transfer function as (5) which again produ ces its \nimpulse response as the Green’s function (6) upon i nverse \nLaplace transformation [1] for the two-term FO syst em (4). \n( )()\n( ) ( )21 1 1 Y s G s U s as b a s b a α α   = = =     + +             (5) \n( )1\n2 , 1 bg t t E t a a α α \nα α −  = −                 (6) \nIn (6), ,Eα β represents the two-parameter Mittag-Leffler \nfunction which is a higher transcendental, encompas sing a \nlarge family of conventional transcendental functio ns like \ntrigonometric, inverse circular, exponential, logar ithmic, \nhyperbolic etc. [1]-[2]. The series representation of two  \n   \n 2 \nparameter Mittag-Leffler function is given by (7) w hich is a \ngeneralized template and reduces to an exponential function for \n1, 1 α β = = [1]-[2]. Also, from (6) it can be observed that the  \nenvelope is guided by a power law instead of an exp onential \none in (3). In this system, a Mittag-Leffler type o scillation \ntakes place in contrast to the sinusoidal oscillati on in (3). \n( )( ),\n0: , 0, 0 k\nkzE z kα β α β α β ∞\n== > > Γ + ∑           (7) \nThe present paper firstly attempts to approximate t he \noscillations, produced due to step excitation of (4 ) with an \nequivalent template given by (1) using GA i.e. find ing optimal \npseudo/meta (FO)-damping or time constants, associa ted with \nthe oscillatory time response of the FO system. Nex t the \noptimal FO damping and time constants are predicted  using a \nmultilayer feed-forward ANN. This approach reduces the \ncomputational load, associated with running GA ever y time for \nfinding out the equivalent optimal FO damping for a ny \narbitrary FO system within this range and such an a pplication is \njustified from the point that multilayer feed-forwa rd ANN is \ngenerally very good function approximator [4]-[5]. In [6], the \nconcept of optimal fractional order damping was fir st proposed \nwith a specific need for faster stabilization of os cillatory \nsystems using the concept of FO damping with respec t to some \nintegral performance indices like Integral of Squar ed Error \n(ISE), Integral Time weighted Squared Error (ITSE) etc. This \npaper gives a new concept of finding the optimal in teger order \nequivalence of FO damping using ISE/ITSE as perform ance \nindices and also proposes their ANN based predictio n. \nRest of the paper is organized as follows. Section II briefly \nintroduces the basics of pseudo-damping and meta-da mping in \nsome special class of oscillatory FO dynamical syst em. Section \nIII describes time domain simulation of pseudo/meta -damping \nand their GA based optimal time domain approximatio n. \nSection IV presents the ANN based training and pred iction \nperformance for these optimal FO-damping. The paper  ends \nwith the conclusion as section V, followed by the r eferences. \nII.  CONCEPT OF PSEUDO AND META -DAMPING IN \nFRACTIONAL ORDER LINEAR DYNAMICAL SYSTEMS  \nIn [1]-[2], it has been reported that time and freq uency \ndomain representations of few special functions, re lated to \nfractional calculus like R-function and G-function are given by: \n()\n( )( )1 1 \n1\n01n v v n \nns a t \ns a n v α\nαα+ − − ∞\n−\n=  =  − Γ + −   ∑ L                    (8) \n( )( )( ) ( ){ } ( )( )\n( ) ( )( )1\n1\n01 1 \n1jr j v v\nr\njr r j r a t s\nj r j v s a α\nα α+ − − ∞\n−\n=  − − − − − −   =  Γ + Γ + − −  ∑⋯L  (9) \nIf Laplace transform of excitation and response of a \nfractional order transfer function (FOTF) ()G s be ()U s and \n()Y s respectively, then simple treatments as in (10) giv es its \nstep and impulse response. Therefore, to find out t he step and \nimpulse response for few classes of fractional orde r systems, \n1v= − and 0v=need to be considered respectively in (8) and (9). Time domain simulation for FO systems using (8 )-(9) \nneeds evaluation of few convergent infinite series at each \ndiscrete time step ( t). For numerical implementation the \ninfinite series have been evaluated in MATLAB at ea ch t, with \nthe order of accuracy being 0.001. \n()()()\n( ) ( ) ( )\n( )\n( )1\n1for step response \nfor impulse response Y s G s U s \ny t g t u t \nG s s \nG s −\n−=\n⇒= ∗ \n    =    L\nL        (10) \nUnder this condition, expressions (8) and (9) repre sents the step \nresponse of stable FOTFs with the replacement of ( a) by unity \nand ( b) by ( b−) in structures like (4). i.e. \n( )1 2 1 1 ,rP P s b s b αα= = + +          (11) \nIt has been seen from (4) that the step response gi ves \nsustained oscillation for 0α=. But the response becomes \ndamped for 1 2 α< < . Hence, a fractional order system of the \nform (4) having no explicit damping term in it, als o exhibits \ndamped oscillation in time response for 1 2 α< < . Such an \noscillation has been approximated by using a second  order \nsystem of the form (1) while minimizing few integra l error \nindices. The FO system of the structure (11) with 1 2 α< < , \ncan be modified with normalized frequency to unity ( 1b=) as: \n1 2 2 1 1 \n1 2 1 Ps s s ατ τξ =+ + + ɶ≃                  (12) \nwhere, {},τ ξ can be termed as the optimal pseudo-time \nconstant and pseudo-damping respectively with respe ct to some \nintegral error index. \nThe second class of FO systems in (11) exhibits dif ferent \ntype of oscillations if different combinations of o rders are used \nin the expansion of the polynomials, although the h ighest order \nof the models are same and only the number of fract ional order \nterms varies in the model. It is well known that or der of a FO \nLTI system is determined by the maximum order prese nt in the \ndenominator polynomial. If it be assumed that 2rα=, the \nsystem governed by (9) becomes a different class of  fractional \nsecond order system (13) which can again be represe nted by \nequivalent second order approximation with {},τ ξ being the \noptimal meta-time constant and meta-damping respect ively. \nSimilar treatment of normalizing the frequency to u nity yields: \n( )2 2 2 2 1 1 \n2 1 1Ps s sα ατ τξ =+ + +ɶ≃                 (13) \nThe following examples put more light on the behavi or of \nsuch systems with meta-damping. Simple modification  of (13) \ngives first and second order transfer functions lik e (14). It is \ninteresting to note that though the leading order r emains one \nand two in these models, the number of fractional o rder \nelements increase upon binomial expansion for the t erms with \nhigher powers. These additional number of FO terms puts extra  \n   \n 3 \ndamping to the FO system which is defined as the me ta-\ndamping in FO dynamical system of the form (13). Th us a FO \nsystem represented by (14) is distinctly characteri zed by the \nnumber of FO elements present in it and not by the leading FO \norder unlike (12). This typical behavior is the mot ivation \nbehind defining two different class of FO damping i .e. pseudo-\ndamping for system (12) and meta-damping for system  (13). \n( ) ( )( )( )( )\n( ) ( )( )( )( )1 5 4 2 10 0.2 0.25 0.5 0.1 \n2 10 8 5 4 0.2 0.25 0.4 0.5 1 1 1 1 1 , , , , \n1 1 1 1 1 \n1 1 1 1 1 , , , , \n1 1 1 1 1 s s s s s \ns s s s s αα\nαα=\n+ + + + + \n=\n+ + + + + ⋯\n⋯(14) \nIt is therefore clear that the pseudo-damping is as sociated \nwith the reduction in the highest order of a FO sys tem whereas \nmeta-damping is associated with the increase in fra ctional order \nelements within a FO model though the highest order  of the \nplant remains the same. \nIII.  TIME DOMAIN SIMULATION OF FRACTIONAL ORDER \nSYSTEMS WITH PSEUDO -DAMPING AND META -DAMPING  \nMATLAB based codes have been developed using the \ninfinite series representations of such FOTF i.e. ( 8)-(9) under \nimpulse/step excitation. Time domain simulation usi ng (8)-(9) \noften gives poor result above 30 seconds. This is d ue to the fact \nthat gamma function in the denominator of (8)-(9) a pproaches \ntowards a very large value which can not be compute d using \nmost of the scientific programming languages, due t o buffer \noverflow. Thus it is recommended to reliably use ex pression \n(8)-(9) for time domain simulation of the special c lass of FO \nsystems only up to 30 seconds. Simulation of first order system \nwith meta-damping and 0.9 α<also becomes computationally \ninfeasible due to blowing up of the associated gamm a \nfunctions. Similarly, second order systems with met a-damping \nand 0.9 α<gives reliable time response up to 25 seconds, \nbelow which the results are unreliable as also repo rted in \nHartley and Lorenzo [6] in the context of optimal F O damping. \nA.  Step Response Characteristics \n \nFigure 1.  Step response of FO system with pseudo-damping. \nFO systems given in (12)-(13) have now been subject ed to \nstep input excitations and evaluated at each discre te time step using (8)-(9) and shown in Fig. 1-3. Fig. 1 shows t hat the \noscillations become more damped with decrease in th e order \n(α) of FO system (12). Similar behaviors can be found  for FO \nsystems with meta-damping with leading order being 2 (Fig. 2) \nand 1 (Fig. 3) respectively. It is interesting to n ote that even \nfirst order systems in the presence of other FO ele ments may \nexhibit oscillations as shown in Fig. 3. In Fig. 1 the decaying \nenvelope may be guided by a power law as reported i n (6), but \nthe nature of oscillations for meta-damping in Fig.  2-3 are more \ncomplex to be represented as closed form solutions unlike (6). \n \nFigure 2.  Step response characteristics of fractional second order system \nwith.meta-damping. \n \nFigure 3.  Step response characteristics of fractional first o rder system with \nmeta-damping. \nB.  Impulse Response Characteristics \nThe impulse responses have been shown next for the above \ndiscussed three classes of FO systems. As expected in Fig. 6, \nthe oscillations start from a value of unity for fi rst order \nsystems (with additional FO elements causing meta-d amping). \nFig. 5 shows the oscillations starting from zero co nfirming the \npreservation of the second order behavior of the FO  system. In \nFig. 4 showing FO systems with pseudo-damping mixed  \nbehaviors can be observed regarding the initial val ue of the \nimpulse response which indicates that the classical  notion of \njudging the order of the system by looking only at the impulse \nresponse characteristics is not valid for pseudo-da mping. \n  \n   \n 4 \n \nFigure 4.  Impulse response of FO system with pseudo-damping. \n \nFigure 5.  Impulse response characteristics of fractional seco nd order system \nwith.meta-damping. \n \nFigure 6.  Impulse response characteristics of fractional firs t order system \nwith meta-damping. \nC.  Genetic Algorithm Based Approach for Finding Optima l \nPseudo-Damping and Meta-Damping \nGenetic algorithm (GA) is a stochastic optimization  process \nwhich can be used to minimize a chosen objective fu nction. A \nsolution vector is initially randomly chosen from t he search \nspace and undergoes reproduction, crossover and mut ation, in \neach iteration to give rise to a better population of solution vectors in the next iteration. Reproduction implies  that \nsolution vectors with higher fitness values can pro duce more \ncopies of themselves in the next generation. Crosso ver refers \nto information exchange based on probabilistic deci sions \nbetween solution vectors. In mutation a small rando mly \nselected part of a solution vector is occasionally altered, with a \nvery small probability. This way the solution is re fined \niteratively until the objective function is minimiz ed below a \ncertain tolerance level or the maximum number of it erations \nare exceeded. In the present study the number of po pulation \nmembers in GA is chosen to be 20. The crossover and  \nmutation fraction are chosen to be 0.8 and 0.2 resp ectively for \nminimization of the following objective functions. \n( ) ( )2 2 \n0 0 ,ISE ITSE J e t dt J t e t dt ∞ ∞ \n= = ⋅ ∫ ∫         (15) \nEvaluation of the objective functions have been don e in each \ngeneration of GA within the finite time horizon of 25 seconds \nas discussed earlier. Minimization of error index ( 15) between \nthe respective FO systems and a second order approx imation \nas in (13) and (14) gives the optimum values of pse udo/meta-\ndamping and time constant. The ISE and ITSE based \noptimization results have been reported in Table I- III for the \nthree test cases, as in Fig. 1-3. \nTABLE I.  GA  BASED RESULTS FOR OPTIMAL PSEUDO -DAMPING FOR \nFRACTIONAL ORDER SYSTEMS  \nISE Based ITSE Based Fractional \nOrder ( α) Jmin  τ ξ J min  τ ξ \n1.1 0.0054 0.3485 1.3152 0.0235 0.47 0.9848 \n1.2 0.0168 0.5246 0.8094 0.0647 0.6467 0.6887 \n1.3 0.0287 0.6587 0.596 0.0979 0.7659 0.5537 \n1.4 0.0379 0.7634 0.4672 0.1153 0.8457 0.4635 \n1.5 0.0429 0.8432 0.374 0.1176 0.8998 0.3878 \n1.6 0.0429 0.9029 0.2965 0.1085 0.9374 0.3146 \n1.7 0.0378 0.9463 0.2247 0.0927 0.9647 0.239 \n1.8 0.0277 0.976 0.1527 0.074 0.9837 0.1597 \n1.9 0.0127 0.993 0.0771 0.0448 0.9947 0.0786 \nTABLE II.  GA  BASED RESULTS FOR OPTIMAL META -DAMPING FOR \nFRACTIONAL ORDER SYSTEMS WITH LEADING ORDER = 1 \nISE Based ITSE Based Fractional \nOrder ( α) Jmin  τ ξ J min  τ ξ \n1.1 0.0057 0.3463 1.1946 0.02 0.4006 1.0538 \n1.2 0.0147 0.342 1.067 0.0512 0.4933 0.7747 \n1.3 0.0273 0.4182 0.7884 0.0769 0.5761 0.6247 \n1.4 0.0415 0.4793 0.6322 0.0956 0.6399 0.5336 \n1.5 0.0571 0.5317 0.5308 0.1105 0.6906 0.4667 \n1.6 0.0748 0.5996 0.4393 0.1271 0.7352 0.4091 \n1.7 0.097 0.6597 0.3708 0.1568 0.7784 0.3529 \n1.8 0.13 0.7233 0.3086 0.2382 0.8253 0.2913 \n1.9 0.1996 0.7959 0.2422 0.5806 0.8828 0.2131  \n   \n 5 \nTABLE III.  GA  BASED RESULTS FOR OPTIMAL META -DAMPING FOR \nFRACTIONAL ORDER SYSTEMS WITH LEADING ORDER = 2 \nISE Based ITSE Based Fractional \nOrder ( α) Jmin  τ ξ J min  τ ξ \n1.1 0.005 1.0426 0.7299 0.0462 1.0837 0.712 \n1.2 0.0138 1.0499 0.574 0.1053 1.0939 0.5732 \n1.3 0.0215 1.0452 0.4668 0.1359 1.0809 0.4809 \n1.4 0.0266 1.0362 0.3845 0.1407 1.0609 0.4066 \n1.5 0.0291 1.0263 0.3153 0.1315 1.0409 0.3395 \n1.6 0.029 1.0171 0.2529 0.117 1.0241 0.2741 \n1.7 0.0266 1.0094 0.1931 0.1038 1.012 0.2083 \n1.8 0.0217 1.0042 0.1325 0.0935 1.0047 0.1405 \n1.9 0.0119 1.0012 0.0679 0.0657 1.0012 0.0701 \nIV.  ANN  BASED PREDICTION OF OPTIMAL PSEUDO -\nDAMPING AND META -DAMPING  \nA.  Multi-Layer Feedforward Neural Network Architecture  \nThe standard neural network architecture consists o f an \ninput layer, one or more hidden layers with multipl e \nperceptrons and an output layer. The number of perc eptrons in \nthe hidden layer and the number of hidden layers ar e generally \nproblem specific and depend on the choice of the us er. In the \npresent study, the number of hidden layers is varie d from 1 to 2 \nand for each case the number of neurons in each lay er is varied \nfrom 5 to 25 in incremental steps of 5. The ANN is fully \nconnected, i.e., the output from each input and hid den neuron is \ndistributed to all the neurons of the subsequent la yer. Also a \nfeed-forward architecture is used, i.e. the data fl ows and is \nprocessed sequentially through the input, hidden an d output \nlayers and are not fed-back to the previous layers,  unlike the \nrecurrent ANN structure.  Hyperbolic tangent sigmoid ( tansig ) \nand logarithmic sigmoid ( logsig ) type activation functions and \ntheir combinations are used to create different ANN  \narchitectures for comparing the relative effectiven ess of these \nstructures at capturing the nonlinear relationship between the \ninput ( α) and output ( ,τ ξ ) data. \nB.  Training Performance of ANN and Time Domain \nPerformance of the Predicted Outputs \nMultilayer feed-forward ANN has now been employed t o \npredict the optimal pseudo/meta damping/time consta nts from \nthe knowledge of the fractional order of the dynami cal system. \nTables IV-VI gives the GA based optimal pseudo/meta  \ndamping ( ξ) and time constant ( τ) of the FO systems in terms \nof equivalent second order systems considering the ITSE \ncriterion, since ITSE puts more penalties on the er ror at later \nstages unlike ISE producing better accuracy. The AN Ns are \ntrained with Levenberg-Marquardt back-propagation a lgorithm \nwhich is a gradient based method and often gets stu ck in local \nminima. To check the consistency of the ANNs for ca pturing \nthe input-output relationship, the Mean Squared Err or (MSE) \nof 25 independent runs has been chosen as the perfo rmance \nmeasure. This is justified from the fact that often  large size \nmultilayer ANNs accurately establishes arbitrary no nlinear \nrelation between any input-output data but they mig ht not show consistency in the mapping [4]-[5]. Hence there is always a \ntrade-off between size of the ANN and the predictio n accuracy. \nTABLE IV.  TRAINING PERFORMANCE FOR VARIOUS ANN  \nCONFIGURATIONS FOR THE PREDICTION OF PSEUDO -DAMPING FOR 25  \nINDEPENDENT RUNS  \nNumber of \nlayers Number of neurons \nin each hidden layer Activation \nfunction Average \nMSE \ntansig 0.009 5 \nlogsig 0.0061 \ntansig 0.07 10 \nlogsig 0.0326 \ntansig 0.1009 15 \nlogsig 0.026 \ntansig 0.1609 20 \nlogsig 0.0418 \ntansig 0.1216 1 \n25 \nlogsig 0.0559 \ntansig/tansig 0.0185 \ntansig/logsig 0.0148 \nlogsig/tansig 0.0238 5 \nlogsig/logsig 0.0153 \ntansig/tansig 0.0453 \ntansig/logsig 0.0169 \nlogsig/tansig 0.0218 10 \nlogsig/logsig 0.0133 \ntansig/tansig 0.0714 \ntansig/logsig 0.0245 \nlogsig/tansig 0.0779 15 \nlogsig/logsig 0.0346 \ntansig/tansig 0.1075 \ntansig/logsig 0.0269 \nlogsig/tansig 0.1492 20 \nlogsig/logsig 0.0375 \ntansig/tansig 0.141 \ntansig/logsig 0.0341 \nlogsig/tansig 0.2229 2 \n25 \nlogsig/logsig 0.0357 \n \nFigure 7.  ANN prediction of pseudo-damping for FO system (12) .  \n   \n 6 \nTABLE V.  TRAINING PERFORMANCE FOR VARIOUS ANN  \nCONFIGURATIONS FOR THE PREDICTION OF META -DAMPING IN FO  SYSTEMS \nWITH LEADING ORDER = 1 FOR 25  INDEPENDENT RUNS  \nNumber of \nlayers Number of neurons \nin each hidden layer Activation \nfunction Average \nMSE \ntansig 0.0109 5 \nlogsig 0.0038 \ntansig 0.0487 10 \nlogsig 0.0232 \ntansig 0.0942 15 \nlogsig 0.0257 \ntansig 0.1113 20 \nlogsig 0.026 \ntansig 0.1002 1 \n25 \nlogsig 0.0694 \ntansig/tansig 0.0126 \ntansig/logsig 0.0117 \nlogsig/tansig 0.0086 5 \nlogsig/logsig 0.0074 \ntansig/tansig 0.0385 \ntansig/logsig 0.021 \nlogsig/tansig 0.0392 10 \nlogsig/logsig 0.0121 \ntansig/tansig 0.0763 \ntansig/logsig 0.0257 \nlogsig/tansig 0.0414 15 \nlogsig/logsig 0.0314 \ntansig/tansig 0.0985 \ntansig/logsig 0.0201 \nlogsig/tansig 0.0934 20 \nlogsig/logsig 0.0596 \ntansig/tansig 0.0991 \ntansig/logsig 0.0303 \nlogsig/tansig 0.0973 2 \n25 \nlogsig/logsig 0.0281 \n \n \nFigure 8.  ANN based prediction of meta-damping for fractional  second order  \nsystem. TABLE VI.  TRAINING PERFORMANCE FOR VARIOUS ANN  \nCONFIGURATIONS FOR THE PREDICTION OF META -DAMPING IN FO  SYSTEMS \nWITH LEADING ORDER = 2 FOR 25  INDEPENDENT RUNS  \nNumber of \nlayers Number of neurons \nin each hidden layer Activation \nfunction Average \nMSE \ntansig 0.0032 5 \nlogsig 0.0014 \ntansig 0.0274 10 \nlogsig 0.0091 \ntansig 0.0686 15 \nlogsig 0.0093 \ntansig 0.0431 20 \nlogsig 0.019 \ntansig 0.0654 1 \n25 \nlogsig 0.0436 \ntansig/tansig 0.0058 \ntansig/logsig 0.0059 \nlogsig/tansig 0.0038 5 \nlogsig/logsig 0.0043 \ntansig/tansig 0.014 \ntansig/logsig 0.0099 \nlogsig/tansig 0.0126 10 \nlogsig/logsig 0.0057 \ntansig/tansig 0.0451 \ntansig/logsig 0.006 \nlogsig/tansig 0.0353 15 \nlogsig/logsig 0.0154 \ntansig/tansig 0.0451 \ntansig/logsig 0.0304 \nlogsig/tansig 0.0339 20 \nlogsig/logsig 0.0139 \ntansig/tansig 0.0372 \ntansig/logsig 0.0137 \nlogsig/tansig 0.0351 2 \n25 \nlogsig/logsig 0.0206 \n \n \nFigure 9.  ANN based prediction of meta-damping for fractional  first order \nsystem.  \n   \n 7 \nTABLE VII.  ANN  BASED PREDICTED VALUES OF PSEUDO /M ETA -\nDAMPINGS AND TIME CONSTANTS FOR THREE CLASS OF FO  SYSTEMS  \nType of the \nFO System Minimum \nMSE Fractional \nOrder ( α) τ ξ \n1.1 0.469965 0.984699 \n1.2 0.69043 0.673145 \n1.3 0.760275 0.595913 \n1.4 0.819033 0.531421 \n1.5 0.898188 0.387535 \n1.6 0.937233 0.314444 \n1.7 0.966328 0.242967 \n1.8 1.009048 0.130724 with pseudo-\ndamping 6.1938×10 -4 \n1.9 0.986992 0.061456 \n1.1 0.400597 1.053798 \n1.2 0.495297 0.751861 \n1.3 0.576102 0.624702 \n1.4 0.639898 0.5336 \n1.5 0.688546 0.46988 \n1.6 0.7352 0.4091 \n1.7 0.779042 0.350727 \n1.8 0.8253 0.2913 First order \nwith meta-\ndamping 9.6422×10 -5 \n1.9 0.916861 0.207589 \n1.1 1.091173 0.707394 \n1.2 1.088813 0.573404 \n1.3 1.076325 0.4878 \n1.4 1.062211 0.402436 \n1.5 1.040513 0.339679 \n1.6 1.019453 0.278533 \n1.7 1.012378 0.208497 \n1.8 1.006116 0.13578 Second order \nwith meta-\ndamping 1.4428×10 -5 \n1.9 1.002982 0.070353 \n \nIn IV-VI the ANN with 5 neurons in the single hidde n layer \nwith logsig  activation function has been found to produce \nconsistently good prediction of the optimal FO damp ing in \nterms of minimum average MSE for 25 runs. Subsequen t \nincrease in number of layers or different combinati ons of \nactivation functions may result in lower MSE in som e \nparticular cases. But considering the consistency o f ANNs, \nthese complicated structures have been found to pro duce \nalways a higher value of average MSE. Table VII rep orts the \npredicted values and MSE of three different classes  of FO \nsystems for different fractional orders while consi dering single \nhidden layer containing 5 neurons and logsig  activation \nfunction. The simulations and results justify the a rgument that \ncomplicated ANN topology may establish arbitrary ma pping \nbut the consistency of such mapping reduces with la rge size of \nthe network [4]-[5]. Time response curves for the s ystems with \nthe predicted pseudo-meta damping and time constant s \ncorresponding to that presented Table VII and the G A based \noptimum results in Tables I-III has been shown in F ig. 7-9 for \nstep input excitation. In Fig 7-9 it is clear that the ANN based \npredicted results (dashed lines) are very close to the GA based \noptimum values (continuous lines). The prediction i s very accurate for first and second order meta-damping th an the \npseudo-damping and also for low value of fractional  order ( α). \nIn contemporary literatures like [7]-[9], the conce pt of \npseudo-damping was first proposed with a different second \norder like structure of the FO systems with commens urable \norders. In the present paper, pseudo-damping refers  to the \ndamping introduced in the system with decrease in i ts leading \norder which is different from that reported in [6]- [9]. The \ncontribution of the present paper is firstly to giv e systematic \ndefinition of pseudo and meta-damping in FO systems  and \nsecondly their ANN based prediction from the GA bas ed ITSE \noptimum results. \nV.  CONCLUSION  \nIn this paper, the concepts of pseudo-damping and m eta-\ndamping are introduced for some special class of fr actional \norder dynamical systems. Genetic algorithm is used to obtain \nthe equivalent second order damping characteristics  of these \nFO systems. An ANN based approach is used next to m odel \nthis arbitrary (nonlinear) relationship and elimina te the \nrequirement of running the computationally intensiv e GA every \ntime. Extensive parametric study have been done to find out the \nmultilayer feed-forward ANN architecture that is ca pable to \noptimally capture the nonlinearity, by minimizing t he MSE and \nsimultaneously avoiding the pitfall of over-fitting . Consistency \nof ANN structures for mapping the fractional order to optimal \nFO damping and time constants are judged by conside ring \naverage MSE of 25 independent runs. The time domain  \ncomparison of the GA based optimum results with the  ANN \nbased predicted results is found to be very close. \nREFERENCES  \n[1]  Igor Podlubny, “Fractional order differential equat ions”, Academic \nPress, 1999. \n[2]  Shantanu Das, “Functional fractional calculus”, Spr inger-Verlag, 2011. \n[3]  Katsuhiko Ogata, “Modern control engineering”, Pren tice-Hall, 1997. \n[4]  Moshe Leshno, Vladimir Ya. Lin, Allan Pinkus, and Sh imon Schocken, \n“Multilayer feedforward networks with a nonpolynomi al activation \nfunction can approximate any function”, Neural Networks , vol. 6, no. 6, \npp. 861-867, 1993. \n[5]  Franco Scarselli and Ah Chung Tsoi, “Universal Appro ximation Using \nFeedforward Neural Networks: A Survey of Some Existi ng Methods, \nand Some New Results”, Neural Networks , vol. 11, no. 1, pp. 15-37, Jan. \n1998. \n[6]  Tom T. Hartley and Carl F. Lorenzo, “A frequency-domai n approach to \noptimal fractional-order damping”, Nonlinear Dynamics , vol. 38, no. 1-\n4, pp. 69-84, 2004. \n[7]  Rachid Malti, Xavier Moreau, Firas Khemane, and Ala in Oustaloup, \n“Stability and resonance conditions of elementary f ractional transfer \nfunctions”, Automatica , vol. 47, no. 11, pp. 2462-2467, Nov. 2011. \n[8]  Rachid Malti, Xavier Moreau, and Firas Khemane, “Re sonance and \nstability conditions for fractional transfer functi ons of the second kind”, \nIn: Dumitru Baleanu, Ziya Burhanettin Guvenc, and J.A.  Tenreiro \nMachado, New Trends in Nanotechnology and Fractional  Calculus \nApplications , Springer-Verlag, pp. 429-444, 2010.  \n[9]  Jocelyn Sabatier, Olivier Cois, and Alain Oustaloup , “Modal placement \ncontrol method for fractional systems: application to a testing bench”, \nProceedings of DETC ’03, ASME 2003 Design Engineering Te chnical \nConferences and Computers and Informatics in Enginee ring \nConference , Sept. 2003, Chicago, Illinois, USA. \n "    },    {        "title": "1007.5505v1.Matrix_Models_and_Lorentz_Invariance.pdf",        "content": "arXiv:1007.5505v1  [hep-th]  30 Jul 2010Matrix Models and Lorentz Invariance\nJens Hoppe\nDepartment of Mathematics, Royal Institute of Technology\n100 44 Stockholm, Sweden\nAbstract\nThe question of Lorentz invariance in the membrane matrix model is a ddressed.\n1 Introduction\nIt is generally believed that the matrix model Hamiltonian that was der ived in [1] (con-\ncerningthebosonictheory)and[2],[3](concerning thesupersymme tric theory; seealso[4])\nis not Lorentz invariant (not even classically). When trying to either prove, or disprove,\nthis belief, one finds quite a number of subtleties, and interesting co nnections, involving\nnot only the (re)construction of ζ(conventionally called x−) but also the fluid dynamics\npoint of view [5], the no-go theorem for particle relativistic interactio ns [6], as well as\nsum-rules involving eigenvalues of (continuous and discrete) Laplac ians, and dualities in\ntheappearance of theLaplacianoperatorsontheparameter-sp ace, respectively embedded\nmembrane.\nOne should mention that Goldstone [7] proved Lorentz invariance of the continuous\n(bosonic) theory in the mid-eighties (introducing some Green’s func tion that allows to\nexpressζin terms of the transverse degrees of freedom), and that de Wit e t al [8] - while\nextending Goldstone’s analysis of the continuous theory to the sup ersymmetric one- came\nto the conclusion that the regularized theory is not Lorentz invaria nt.\nTwo important new structures intimately related to Lorentz-invar iance were found in [10].\n2 Canonical Light-Cone Description of M(em)branes\nBosonic membranes can be described by ([1],[9])\nH[/vector x,/vector p;η,ζ0] =1\n2η/integraldisplay/vector p2+g\nρd2ϕ (1)\n/integraldisplay\nfa/vector p·∂a/vector xd2ϕ= 0 whenever ∂a(ρfa) = 0 (2)\nwhere\ng= det(∂a/vector x·∂b/vector x) (3)\n1andρis a non-dynamical density of unit weight (/integraltext\nρ(ϕ)d2ϕ= 1). (1) and (2) can be\nthought of as arising from\n/tildewideH[/vector x,/vector p;ζ,π] :=/integraldisplay/vector p2+g\n−2πdMϕ (4)\nCa:=π∂aζ+/vector p·∂a/vector x= 0 (a= 1,...,M) (5)\nvia Hamiltonian reduction (using that ˙ π=−δ/tildewideH\nδζ= 0, cp.[1],[11]),\nπ(ϕ) =ρ(ϕ)/integraldisplay\nπdMϕ=−ρ(ϕ)η, (6)\n(and choosing M=2), resp. by comparison with the Lagrangian equa tions of motion\n1√\nG∂α(√\nGGαβ∂βxµ) = 0, µ= 0,1,...,D−1 (7)\nin the gauge\nϕ0=x0+xD−1\n2:=τ, G 0a=∂aζ−˙/vector x∂a/vector x= 0, (8)\nin which (7), then implying\n∂\n∂τ/radicalbiggg\n2˙ζ−˙/vector x2= 0, (9)\nreduces to having to solve\n¨/vector x=1\n/tildewideρ∂a(ggab\n/tildewideρ∂bxµ), (10)\n(the equation for ζ:=x0−xD−1automatically follows), with /tildewideρ(ϕ)(∼=−π(ϕ)) =ηρ(ϕ)\nbeing the time-independent density appearing in (9). ζ0(mentioned in (1) as part of the\ncanonical variables) is equal to the integral over ζ(a time-dependent constant that is not\ndetermined by (8)), and has the time-evolution (cp. (9))\n˙ζ0=−δH\nδη=H\nη. (11)\nFinally, the mass-squared of the (internal degrees of freedom pa rt of the) M(em)brane,\nM2=/integraldisplay/vector p2+g\nρ−(/integraldisplay\n/vector p)2=D−2/summationdisplay\ni=1∞/summationdisplay\nα=1piαpiα+V (12)\npiα:=/integraldisplay\npiYαdMϕ, x jβ:=/integraldisplay\nxjYβρdMϕ\n/integraldisplay\nYαYβρdMϕ=δαβ,∞/summationdisplay\nα=1Yα(ϕ)Yα(/tildewiderϕ) =δ(ϕ,/tildewideϕ)\nρ(ϕ)−1\nthen takes a very simple form (cp.[1]), with the interaction Vdetermined entirely in terms\nof the structure constants of the Lie-algebra of diffeomorphisms ,ϕa→/tildewideϕa, with unit\nJacobian (which is the residual diffeomorphism symmetry group left a fter gauge fixing).\n2ζ(seemingly having disappeared) is needed both geometrically (to rec onstruct the world\nvolume swept out by the membrane in space-time) and for thePoinca re’-invariance of (12)\n(aswellasreturnsasthecentralobjectinthefluiddynamicdescr iption, afterahodograph\ntransformation [5] ) Obviously, M2commutes with Hand/vectorP=/integraltext\n/vector pdMϕ=/vector p0( which\ngenerate light-cone time- and space translations), as well as with /vectorP+=η=−/integraltext\nπdMϕ,\nζ0η, andη/integraltext\nxiρdMϕ, while\nMi−=/integraldisplay\n(xi/vector p2+g\n2ηρ−ζpi)dMϕ (13)\nnecessitates the reconstruction of ζ(in terms of the other degrees of freedom) from (cp.\n(8), (9))\nη∂aζ=/vector p\nρ∂a/vector x (14)\nη2˙ζ=1\n2/vector p2+g\nρ2, (15)\nconsistent provided that the equations of motion (cp. (10)/(1))\nη˙/vector x=/vector p\nρ, η˙/vector p\nρ=1\nρ∂a(ggab\nρ∂b/vector x) :=∆/vector x, (16)\n( and the constraint (2)) hold.\n3 Reconstructing ζ\nThe reconstruction of ζfrom\nη∂aζ=/vector p\nρ∂a/vector x,2η2˙ζ=/vector p2+g\nρ2(∗)\nis [7]\nζ:=η(ζ−ζ0) =/integraldisplay\nG(ϕ,/tildewideϕ)/tildewide∇a(/vector p\nρ/tildewide∂a/vector x)ρdM/tildewideϕ (17)\nwhere\nG(ϕ,/tildewideϕ) :=∞/summationdisplay\nα=1−1\nµαYα(ϕ)Yα(/tildewideϕ) (18)\n(conveniently choosing the Yαas eigenfunctions of ∆, ∆ Yα=−µαYα) satisfies\n∆G(ϕ,/tildewideϕ) =δ(ϕ,/tildewideϕ)\nρ(ϕ)−1, (19)\nand/vector xand/vector pmust satisfy\n/integraldisplay\nfa/vector p·∂a/vector xdMϕ= 0 whenever ∇afa= 0. (20)\n3To rewrite (17) in a form that suggests a canonical discrete analog ue ofζone just needs\nto note that\n∇au∇av=1\n2(∆(uv)−u∆v−v∆u); (21)\nso (17) can be written as (note the 3 typos in eq.(7) of [10], and one m issingρin eq.(5)\nof [12])\n2ζ=/vector p·/vector x\nρ−/integraldisplay/vector p·/vector x\nρ+/integraldisplay\nG(ϕ,/tildewideϕ)(/vector p\nρ∆/vector x−/vector x∆/vector p\nρ)ρdM/tildewideϕ (22)\nfrom which it follows that (cp.(16))\n2η˙ζ=/vector p2\nρ2−/integraldisplay/vector p2\nρ2+/integraldisplay\nG(ϕ,/tildewideϕ)(∆/vector x∆/vector x−∆/vector x∆/vector x)ρdM/tildewideϕ+/vector x∆/vector x−/integraldisplay\n/vector x∆/vector x.(23)\n(implying that the terms not involving /vector pgiveg/ρ2−/integraltext\ng/ρ2).\nInserting (18) and using that ∆ /vector x=−µβ/vector xβYβetc. one finds that\n2ζ=/summationdisplay\nα,β,γµα+µβ−µγ\nµα/vector pγ·/vector xβdαβγYα(ϕ)+2/vectorP(/vector x−/vectorX), (24)\nimplying e.g. that (for general M, {...}being totally antisymmetric, and linear in the\nderivatives of the M entries)\nη/summationdisplay\nα,βdαβγµα+µβ−µγ\nµα˙/vector pγ·/vector xβ=1\nM!dαβγ{xi1,...,xiM}β{xi1,...,xiM}γ,(25)\nrather nontrivial identities involving the eigenvalues of the Laplacian ∆ and the totally\n(anti-)symmetric structure constants\ndαβγ:=/integraldisplay\nYαYβYγρdMϕ, g αα1...αM:=/integraldisplay\nYα{Yα1,...,YαM}ρdMϕ; (26)\nnote that (16) can be written in terms of the multilinear brackets (c p.[1]) as\nη˙/vector p\nρ=1\n(M−1)!{xi1,...,xiM−1,{xi1,...,xiM−1,/vector x}}. (27)\n4 Matrix Analogues\nOne possibility to define (for M=2) a matrix ζN, that could then be used for\n2ηMi−= Tr(Xi(/vectorP2+W)−2ηζNPi) (28)\nwould be to use (24), i.e. define\n2ζN:=N2−1/summationdisplay\n1µa+µb−µc\nµa/vector pc·/vector xbd(N)\nabcT(N)\na+2/vector p·(/vectorX−/vector x). (29)\n4(denoting the transverse zero-modes now by x and p, and the mat rices by X and P).\nWhile it is convenient to assume\n1\nNTr(T(N)\naT(N)\nb) =δab,∆NT(N)\na=−µ(N)\naT(N)\na (30)\nd(N)\nabc:=1\n2NTr({Ta,Tb}Tc),∆N=−1\n/planckover2pi12[Tn,[Tn,·]],\ncorresponding to the matrix regularization\nYα(ϕ)→T(N)\na. (31)\nit is in practice often simpler to have complex eigenfunctions Yαand non-hermitean T(N)\na,\n(for the sphere, e.g., see [1], [13]; a= (lm) andµ(N)\nl<N,m=µlm, independent of N,/planckover2pi1=\n2√\nN2−1,T(N)\nl≥N,m≡0 ).\nThe problem with (29) (which looks like a nice definition) is that it is degen erate for\nN= 2 (where also the invariance requirement that the dynamical Poiss on-bracket of\nLabc/vector xb/vector pcandǫa′b′c′/vector xb′/vector pc′should equal ǫaa′eLedf/vector xd/vector pfon the constrained phase-space does\nnot have any non-trivial solution) , while for higher N difficult to test.\nAlready in the (much simpler) continuous case, it is quite non-trivial ( even for the\nstring, and torodial membranes) to verify (24) directly, i.e. withou t going back to (17).\nDifferentiating (17), on the other hand, with respect to (light-con e) time, one trivially\ngets/vector p2/(2ηρ2)−/integraltext\n(/vector p2/(2ηρ2)) and due to\n2∇a(1\nρ∂b(ggbc\nρ∂c/vector x)∂a/vector x) =∇a(2\nρ∂b(δb\nag/ρ)−1\nρ2∂ag) = ∆(g/ρ2) (32)\nη/integraldisplay\nG(ϕ,/tildewideϕ)/tildewide∇a(˙/vector p/ρ/tildewide∂a/vector x)dM/tildewideϕ=g\n2ρ2−/integraldisplayg\n2ρ2=−/integraldisplay\n∇aG∂a/vector x∆/vector x (33)\n(indirectly proving (25)). When trying to use eq.(27) for the proof of (33), one can make\nuse of an identity, that for M=2 reads\n{xj,{xj,/vector x}}{/vector x,Yβ}=1\n4{{xi,xj}2,Yβ} ∀β (34)\nand whose discrete analogue,\n[Xj,[Xj,/vectorX]][/vectorX,Tb] =1\n4[[Xi,Xj]2,Tb] (35)\ndoes not hold in general - but would need to hold at least for someTb(those entering the\nLaplacian on the discrete base-manifold). While this makes discrete P oincare’ invariance\nrather unlikely, it does not prove that there is not perhaps anothe r way to define a matrix\nζNfor which (restricting for the moment to just finding quantities con stant in time)\nTr(ηXi˙ζN−ζNPi). (36)\nUsing the discrete equations of motion,\nη˙Xi=Pi, η˙Pi=−[Xj,[Xj,Xi]] :=∆NXi (37)\n5one finds that the only requirement for (36) to be time-independen t is that ζN, just as\n/vectorX(hence: as it should), satisfy the discrete analogue of what follows from (14) and (15)\n(cp. (10)),\nη2¨ζ=∆Nζ. (38)\nA solution of (38), in terms of /vectorXand/vectorPsatisfying (37) and the usual ”Gauss-law” matrix\nmodel constraint\n[/vectorX,/vectorP] = 0, (39)\ncan be constructed when trying to discretize (14) by noting that in the continuous case\none certainly has\nη{ζ,xi}=/vector p\nρ{/vector x,xi} (40)\nη{ζ,pi}=/vector p\nρ{/vector x,pi} (41)\n– whose canonical discrete analogues would be\n2η[ζN,Xi] =/vectorP[/vectorX,Xi]+[/vectorX,Xi]/vectorP (42)\n2η[ζN,Pi] =/vectorP[/vectorX,Pi]+[/vectorX,Pi]/vectorP. (43)\nUsing (42), (43) and (39), it easily follows that ζNdoes satisfy (38), provided\n2η˙ζN=/vectorP2−1\n2[Xj,Xk]2, (44)\nwhose commutator with Xiis consistent with what one gets from differentiating (42),\nusing (43). The problem with this solution however is that in contrast to the continuous\ncase, (42) and (43) (even one of the two by itself) are not necess arily consistent, i.e., for\nconsistency require further identities to be satisfied by the Xi(τ) andPj(τ).\nE.g., commuting (42) with Pi, and subtracting (43) commuted with Xi(then summing\nover i) one obtains (using (39))\n2[Pi,Xj][Xi,Pj] = [Pi,Pj][Xi,Xj]+[Xi,Xj][Pi,Pj]. (45)\nThe difference between discrete and continuous cases becomes mo st drastic by noting that\nη[2ζN,Ta] =/vectorP[/vectorX,Ta]+[/vectorX,Ta]/vectorP (46)\n- in a way, the most natural generalization of\nη∂aζ=/vector p\nρ∂a/vector x,resp.η{ζ,Yα}=/vector p{/vector x,yα}, (47)\nby commutating again with Ta(and noting that [ Ta,[Ta,·]] is the adjoint Casimir, χN)\nwould actually imply that\n2ηζN=/vectorP·/vectorX+/vectorX·/vectorP+1\nχN([/vectorP,Ta][/vectorX,Ta]+[/vectorX,Ta][/vectorP,Ta]) (48)\n6would be symmetric under /vectorX↔/vectorP, in sharp contrast to (22). Note that in the continuous\ncase a similar argument is prevented by the value of the correspond ing ”Casimir” (for\nthe infinite -dimensional Lie algebra) being infinite. What one coulddo, however, is to\ndemand (46) onlyfor someTa, namely those entering the Laplacianof thediscretized base\nmanifold; for the 2-sphere e.g. (cp. [1],[13]) these would be the 3 gene ratorsTl=1,m=−1,0,+1,\nthus obtaining\n2η∆NζN=/vectorP∆N/vectorX+∆N/vectorX/vectorP−1\n/planckover2pi12([Tm,/vectorP][Tm,/vectorX]+[Tm,/vectorX][Tm,/vectorP]).(49)\nThis approach also leads to (29).\nFinally, one could try to directly solve\n2η˙ζN=/vectorP2+W\nη, W=−1\n2[Xi,Xj]2(50)\nbut one should be aware of possible renormalizations.\nIn any case it would be interesting to see a critical dimension when qua ntizing.\nAcknowledgement\nI would like to thank the students of my KTH course “Dynamics of Str ings and Mem-\nbranes” (which this note, written in March 2010, is based on) for th eir interest.\nNote Added\nThe original equations determining ζ(cp.(14),(15)) are invariant under diffeomorphisms\nhaving unit Jacobian. Therefore (22)/(24) should, on the constr ained phase space, also\nhave this property, despite the dependence of ∆ (hence its eigenv alues) on the metric\nchosen to obtain ζexplicitely. Indeed, one can check by an explicit (somewhat non-\ntrivial) calculation (for simplicity I chose M=2) that the dynamical Pois son-bracket of\nζαwithgβγǫ/vector xγ/vector pǫ, i.e. the β-component of φ:={/vector p\nρ,/vector x}=ǫab\nρ∂a(/vector p\nρ)∂b/vector x, isgαβγζγon the\nconstrained phase space (implying in particular that Mi−is well defined, i.e. commutes\nwith the constraint φ). I thank Joakim Arnlind, Martin Bordemann, Ki-Myeong Lee\nand Piljin Yi for discussions related to this point.\nReferences\n[1] Jens Hoppe Quantum Theory of a Massless Relativistic Surface and ... Ph.D. Thesis,\nMIT 1982, http://dspace.mit.edu/handle/1721.1/15717 ; see also Membranes and\nMatrix Models ,arXiv:hep-th/0206192 , IHES/P/02/47, and references therein.\n[2]M.Claudson,M.B.Halpern; Nucl.Phys.B250(1985)689. R.Flume, Ann.Ph ys.164(1985)\n189. M.Baake, P.Reinicke, V.Rittenberg; J.Math.Phys.26 (1985) 1070\n[3] B.deWit, J.Hoppe, H.Nicolai, Nucl.Phys.B305 (1989) 545.\n[4] T.Banks, W.Fischler, S.Shenker, L.Susskind, Phys.Rev.D55 (1997) 6189\n7[5] M.Bordemann, J.Hoppe; Phys.Lett.B325 (1994) 359\n[6] D.G. Currie, T.F. Jordan, E.C.G. Sudarshan; Rev. Mod. Phys. 35 ( 1965) 350.\nH. Leutwyler; Nuovo Cim. 37 (1965) 556.\n[7] J. Goldstone, unpublished notes ( passed on to the authors of [8 ] in 1987/88 )\n[8] B.deWit, U.Marquard, H.Nicolai, Comm.Math.Phys.128 (1990) 39\n[9] J.Goldstone; unpublished\n[10] J.Hoppe, arXiv:1003.5189\n[11] J.deWoul, J.Hoppe, D.Lundholm; arXive:1006.4714\n[12] J.Hoppe; “Quantum Reconstruction Algebras”\nhttp://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-13591\n[13] J.Hoppe, S.T.Yau; Comm.Math.Phys.195 (1998) 67.\n8"    },    {        "title": "0906.1592v2.A_Tail_of_a_Quark_in_N_4_SYM.pdf",        "content": "arXiv:0906.1592v2  [hep-th]  3 Sep 2009A Tail of a Quark in N= 4SYM\nMariano Chernicoff∗, J. Antonio Garc´ ıa†and Alberto G¨ uijosa‡\nDepartamento de F´ ısica de Altas Energ´ ıas, Instituto de Ci encias Nucleares\nUniversidad Nacional Aut´ onoma de M´ exico\nApdo. Postal 70-543, M´ exico D.F. 04510\nAbstract\nWe study the dynamics of a ‘composite’ or ‘dressed’ quark in s trongly-\ncoupled large- NcN= 4 super-Yang-Mills, making use of the AdS/CFT cor-\nrespondence. We show that the standard string dynamics nice ly captures the\nphysics of the quark and its surrounding non-Abelian field co nfiguration, mak-\ning it possible to derive a relativistic equation of motion t hat incorporates the\neffects ofradiationdamping. Fromthisequationonecandeduc eanon-standard\ndispersion relation for the composite quark, as well as a Lor entz covariant for-\nmula for its rate of radiation. We explore the consequences o f the equation in\na few simple examples.\n1 Introduction and Summary\nWhen a charge radiates, energy conservation dictates that it mus t be subjected to a\nreactive force originating from its self-field, that tends to damp its motion. In the\ncontext of classical electrodynamics, the study of this damping or radiation reaction\nforce began over a century ago [1, 2, 3, 4], and continues to this d ay [5, 6]. Reviews\nand additional references on the subject may be found in [7, 8, 9, 1 0].\nIn a non-relativistic approximation, the dynamics of an electron tha t is modeled\nas a vanishingly small spherically symmetric charge distribution is cont rolled by the\nAbraham-Lorentz equation [1, 2]\nm/parenleftBig\n¨/vector x−te...\n/vector x/parenrightBig\n=/vectorF , (1)\nwhere˙≡d/dtandte≡2e2/3mc3is a timescale set by the classical electron radius.\nIn this equation, the damping force (the second term in the left-ha nd side) is seen to\n∗e-mail: mariano@nucleares.unam.mx\n†e-mail: garcia@nucleares.unam.mx\n‡e-mail: alberto@nucleares.unam.mx\n1be proportional to the jerk /vector≡˙/vector a≡¨/vector v. The search for a Lorentz-covariant version of\n(1) led to the (Abraham-)Lorentz-Dirac equation [4],\nm/parenleftbigg\n˚ ˚xµ−te/bracketleftbigg\n˚ ˚ ˚xµ−1\nc2˚ ˚xν˚ ˚xν˚xµ/bracketrightbigg/parenrightbigg\n=Fµ, (2)\nwith ˚≡d/dτ,τthe proper time (defined such that ˚ xµ˚xµ=−c2) andFµ≡\nγ(/vectorF·/vector v/c,/vectorF) the four-force. The second term within the square brackets (p roportional\ntothesquareoftheproperacceleration)isthenegativeofthera teatwhichenergyand\nmomentum is carried away from the charge by radiation, according t o the covariant\nLienard(-Larmor-Heaviside-Abraham) formula. So, strictly spea king, it is only this\nterm that can properly be called radiation reaction. The first term w ithin the square\nbrackets, usually called the Schott term, and whose spatial part y ields the damping\nforceof(1)inthenon-relativisticlimit, isknowntoarisefromtheeffe ctofthecharge’s\n‘near’ or ‘bound’ (as opposed to radiation) field [11, 7].\nThe appearance of third-order terms in (1) and (2) leads to unphy sical behavior,\nincluding pre-accelerating and self-accelerating (or ‘runaway’) solu tions. These defi-\nciencies are known to originate from the assumption that the charg e is pointlike.1For\na charge distribution of small but finite size l, the above equations can be shown to be\ntruncations of expressions that involve an infinite number of deriva tives (ld/dt)n(and\ngenerally include terms that are non-linear in these derivatives), bu t are physically\nsound as long as l > cteand [7, 8, 9].\nOf course, one should keep in mind that the unphysical behavior implie d by (1)\nand (2) would be visible only for time and distance scales smaller than th e Compton\nwavelength λC≡/planckover2pi1/mof the charge, and thus lies outside of the actual range of\nvalidity of classical electrodynamics. How the preceding story gene ralizes to the case\nof fully quantum electrodynamics (QED) has been studied from differ ent angles in\n[12, 13, 14, 15] and references therein. In particular, in [12] it was shown that, for\na pointlike non-relativistic electron, QED leads to an equation of motio n with an\ninfinite number of higher derivatives, implying that the electron acqu ires an effective\nsizel=λCdue to its surrounding cloud of virtual particles.\nGoing further to non-Abelian gauge theories is a serious challenge.2paper to show\nthat the AdS/CFT correspondence [18, 19, 20] allows us to examine this question\nrather easily in quantum strongly-coupled non-Abelian gauge theor ies. The essence\nof the matter is that in the context of this duality the quark corres ponds to the tip\nof a string, whose body codifies the profile of the non-Abelian (near and radiation)\nfields sourced by the quark. In other words, the quark has a tail, a nd it is this tail\nthat is responsible for the damping force. Indeed, this mechanism h as already been\nseen at work in the recent computations of the drag force exerte d on the quark by a\n1This assumption leads to the further complication of an infinite electr omagnetic self-energy,\nwhich has already been been absorbed within the renormalized mass mshown in (1) and (2). An\nalternative approach which circumvents this divergence has been p roposed recently in [5].\n2See respectively [16] and [17] for work on radiation within classical an d (weakly-coupled) quan-\ntum Yang-Mills theory.\n2thermal plasma, which is described in dual language in terms of a strin g living on a\nblack hole geometry [21, 22]. Our analysis makes it clear that, irrespe ctive of whether\na spacetime black hole is present or not, the body of the string plays the role of an\nenergy sink, as befits its identification as the embodiment of the gluo nic degrees of\nfreedom.3\nWe expect this basic story to apply generally to all examples of the ga uge/string\nduality, including cases with finite temperature or chemical potentia ls, but for sim-\nplicity we will concentrate on the case of quark motion in the vacuum o fN= 4\nsuper-Yang-Mills (SYM), where, building upon previous work [30, 25], we can achieve\nfull analytic control. As has been remarked several times in the pas t, it is interesting\nthat even in this non-confining theory the gluonic field configuration can be encoded\nin a ‘QCD’ string, albeit one that lives in a curved higher dimensional spa cetime.\nOur results are a direct consequence of this amazing fact.\nThe paper is organized as follows. In Section 2, we explain how the sta ndard\nstring dynamics is mapped by the AdS/CFT correspondence onto th e dynamics of\nan extended radiating particle. We begin by setting up our problem in S ection 2.1,\nreviewing the relevant context and emphasizing some key features —most notably, the\nfact that the quark with finite mass that the correspondence put s at our disposal is\nautomatically ‘dressed’ or ‘composite’, as discussed around Eq. (9) . We then attack\nthe problem in Section 2.2, where we derive an equation of motion for t his quark,\nEq. (28), which constitutes ourmainresult.4Asexplained atlengthintheparagraphs\nthat follow it, this equation is a nonlinear generalization of (2), which in corporates\nthe effects of radiation damping, but has no pre-accelerating or se lf-ac celerating\nsolutions. From this equation one can read off a non-standard dispe rsion relation\nfor the quark, Eq. (34), as well as a Lorentz covariant formula fo r its radiation rate,\nEq. (35). An interesting novel feature in these expressions is the ir dependence on the\nexternal force exerted on the quark, which is a reflection of its ex tended, and hence\ndeformable, nature. We close Section 2 by commenting on our failure to rewrite (28)\nin terms of an action principle.\nIn Section 3, we explore some of the physics implied by (28), specializin g to a\nfew simple examples. For the case of one-dimensional motion, we sho w in Section 3.1\nthat even though, as expected on physical grounds, zero exter nal force implies zero\nacceleration, the converse is not true: the quark will not accelera te when subjected\nto an external force that takes the specific form (41) (which is ide ntically zero only\nwhen the parameter t0→ ±∞). More generally, for each given quark trajectory there\nis a one-parameter family of possible external forces. This again is a manifestation\nof the fact that, because of the extended character of the qua rk, the energy supplied\nto it can not only increase its velocity, but also modify its associated g luonic field\nprofile. We end the paper by studying the nonrelativistic limit of (28) in Section 3.2,\n3On the other hand, energy loss via the string does turn out to be clo sely associated with the ap-\npearanceof a worldsheet horizon, as noticed initially in [23, 24] at finite temperature and empha sized\nin [25] for the zero temperature case. This association has been fu rther studied in [26, 27, 28, 29].\n4A brief report of this derivation was given in the recent letter [31].\n3where the linearized form of the expressions allows us to make direct contact with\nthe energy analysis of ([30, 25]) and to easily write down an action prin ciple.\nAll in all, then, we have in (28) a physically sensible and interesting desc ription\nof the dynamics of a composite quark in N= 4 SYM. This result serves, on the one\nhand, to illustrate the power of the AdS/CFT correspondence, an d on the other, to\nshed some light on the largely uncharted terrain of radiation in stron gly-coupled non-\nAbelian gauge theories. It would be interesting to extend this analys is in a number\nof directions. In particular, it seems worthwhile to explore the mann er in which the\nsplit between intrinsic and radiated energy (and momentum) of the q uark achieved\nin [30, 25] and the present paper, via examination of the string world sheet, manifests\nitself in the gluonic field profile, by directly computing the expectation value of the\nenergy-momentum tensor or similar local operators [32]. It is also na tural to try to\ncarry over some of the present methods to the finite temperatur e context [33], where\none sh ould be able to make contact with previous AdS/CFT analyses o f energy\nloss in a thermal plasma (a rather large body of work detonated by t he seminal\nworks [21, 22, 34, 35]), including the interesting recent studies of B rownian motion\n[36, 37, 38].\n2 From Strings to Quarks\n2.1 Basic setup\nItisbynowwell-known thatstrongly-coupled N= 4SU(Nc)SYMwithcoupling gYM\nis dual to Type IIB string theory on a background that asymptotic ally approaches\nthe AdS 5×S5geometry5\nds2=GMNdxMdxN=R2\nz2/parenleftbig\n−dt2+d/vector x2+dz2/parenrightbig\n+R2dΩ5, (3)\nR4\nl4\ns=g2\nYMNc≡λ\n(with a constant dilaton and Ncunits of Ramond-Ramond five-form flux through the\nfive-sphere), where lsdenotes the string length [18]. The radial direction zis mapped\nholographically into a variable length scale in the gauge theory [39]. The directions\nxµ≡(t,/vector x) are parallel to the AdS boundary z= 0 and are directly identified with the\ngauge theory directions. The state of IIB string theory describe d by the unperturbed\nmetric (3) corresponds to the vacuum of the N= 4 SYM theory, and the closed string\nsector describing (small or large) fluctuations on top of it fully capt ures the gluonic\n(+adjointscalar and fermionic) physics.\nFrom the gauge theory perspective, the introduction of an open s tring sector\nassociated with a stack of NfD7-branes in the geometry (3) is equivalent to the\naddition of Nfhypermultiplets in the fundamental representation of the SU(Nc)\n5From this point on we work in natural units c= 1 =/planckover2pi1.\n4gauge group, breaking the supersymmetry down to N= 2 [40]. These are the degrees\nof freedom that we refer to as ‘quarks,’ even though they include b oth spin 1 /2 and\nspin 0 fields. For Nf≪Nc, the backreaction of the D7-branes on the geometry can\nbe sensibly neglected; in the field theory this corresponds to workin g in a ‘quenched’\napproximation which disregards quark loops (as well as the positive b eta function\nthey would generate). The D7-branes cover the four gauge theo ry directions, and\nextend along the radial AdS direction up from the boundary at z= 0 to a position\nz=zmwhere they ‘end’ (meaning that the S3⊂S5that they are wrapped on shrinks\ndown to zero size), which is inversely proportional to the quark mas s,\nzm=√\nλ\n2πm. (4)\nAn isolated quark is dual to an open string that extends radially from the D7-\nbranes to the AdS horizon at z→ ∞. The string dynamics follows as usual from the\nNambu-Goto action\nSNG=−1\n2πα′/integraldisplay\nd2σ/radicalbig\n−detgab≡/integraldisplay\nd2σLNG, (5)\nwheregab≡∂aXM∂bXNGMN(X) (a,b= 0,1) denotes the induced metric on the\nworldsheet. In our work the entire string will be taken (consistent ly with the corre-\nsponding equations of motion) to lie at the ‘North Pole’ of the S5(the point where\ntheS3⊂S5that the D7-branes are wrapped on collapses to zero size), so the angular\ncomponents of the metric (which are associated with the orientatio n of the gauge\ntheory fields in the internal SU(4) symmetry group) will not play any role, and the\nlower endpoint of the string will necessarily lie at z=zm.\nWe can exert an external force /vectorFon the string endpoint by turning on an electric\nfieldF0i=Fion the D7-branes. This amounts to adding to the Nambu-Goto actio n\nthe usual minimal coupling\nSF=/integraldisplay\ndτ Aµ(X(τ,zm))∂τXµ(τ,zm),\nor, in terms of the quark worldline,\nSF=/integraldisplay\ndτ Aµ(x(τ))˚xµ(τ). (6)\nNotice that the string is being described (as is customary) in first-q uantized lan-\nguage, and, as long as it is sufficiently heavy, we are allowed to treat it semiclassically.\nIn gauge theory language, then, we are coupling a first-quantized quark to the gluonic\n(+ other SYM) field(s), and then carrying out the full path integra l over the strongly-\ncoupled field(s) (the result of which is codified by the AdS spacetime) , but treating\nthe path integral over the quark trajectory xµ(τ) in a saddle-point approximation.\n5Variationofthestringaction SNG+SFimplies thestandardNambu-Gotoequation\nof motion for all interior points of the string, plus the standard bou ndary condition\n[41]\nΠz\nµ(τ)|z=zm=Fµ(τ)∀τ , (7)\nwhere\nΠz\nµ≡∂LNG\n∂(∂zXµ)=√\nλ\n2π/parenleftBigg\n(∂τX)2∂zXµ−(∂τX·∂zX)∂τXµ\nz2/radicalbig\n(∂τX·∂zX)2−(∂τX)2(1+(∂zX)2)/parenrightBigg\n(8)\nis the worldsheet (Noether) current associated with spacetime mo mentum, and we\nhaver recognized Fµ=−Fνµ∂τxν= (−γ/vectorF·/vector v,γ/vectorF) as the Lorentz four-force.\nFor the interpretation of our results it will be crucial to keep in mind t hat the\nquark described by this string is not ‘bare’ but ‘composite’ or ‘dress ed’. This can be\nseenmostclearlybyworking outtheexpectationvalueofthegluonic fieldsurrounding\na static quark located at the origin6[43],\n1\n4g2\nYM∝angbracketleftTrF2(x)∝angbracketright=√\nλ\n16π2|/vector x|4\n1−1+5\n2/parenleftBig\n2πm|/vector x|√\nλ/parenrightBig2\n/parenleftbigg\n1+/parenleftBig\n2πm|/vector x|√\nλ/parenrightBig2/parenrightbigg5/2\n. (9)\nForm→ ∞(zm→0), this is just the Coulombic field expected (by conformal\ninvariance) for a pointlike charge. For finite mthe profile is still Coulombic far away\nfrom the origin but in fact becomes non-singular at the location of th e quark,\n1\n4g2\nYM∝angbracketleftTrF2(x)∝angbracketright=√\nλ\n128π2/bracketleftBigg\n15/parenleftbigg2πm√\nλ/parenrightbigg4\n−35\n|/vector x|4/parenleftbigg2πm|/vector x|√\nλ/parenrightbigg6\n+.../bracketrightBigg\nfor|/vector x|<√\nλ\n2πm.\nAs seen in these equations, the characteristic thickness of this no n-Abelian charge\ndistribution is precisely the length scale zmdefined in (4). This is then the size of the\ngluonic cloud that surrounds the quark, or in other words, the ana log of the Compton\nwavelength for our non-Abelian source.\nIt is interesting to note that the string can be alternatively viewed a s a Born-\nInfeld string, i.e., a soliton of the gauge and scalar fields on the D7-br ane [44]. Since\nthe small fluctuations of these fields (corresponding to microscop ic open strings) are\nknown to be dual to mesons, the composite quark itself can be thou ght of as a soliton\nconstructed by aligning a large number of mesons [45]. The cloud sur rounding our\nquark is then best thought of as ‘mesonic’ rather than ‘gluonic’. Mes ons are indeed\nknown to be the lightest states in the spectrum of the strongly-co upled gauge theory,\nwith masses of order mmes≡1/zm= 2πm/√\nλ≪m[46], and form factors with size\nset byzm[47].\n6More precisely, the operator in the left-hand side of (9) is the dual of the dilaton field, and\nincludes not only the standard Yang-Mills term but also scalar and fer mion contributions that can\nbe found in [42], and which we suppress for notational simplicity.\n6So, to summarize, zmcan properly be called the quarkCompton wavelength in-\nsofar as it gives the size of the cloud of virtual particles surroundin g the quark, but\none should bear in mind that it is given not by 1 /mbut by 1/mmes, and in this sense\nit could also be referred to as the mesonCompton wavelength.\n2.2 Equation of motion for the quark\nThe first analysis of an accelerating quark via the AdS/CFT corresp ondence was\ncarried out in [48], which used tools developed in [49] to study the dilato nic waves\ngiven off by small fluctuations on a radial string in AdS 5, and infer from them the\nprofile of the gluonic field ∝angbracketleftTrF2(x)∝angbracketrightin the presence of a quark undergoing small\noscillations. The results of [48] painted an interesting picture of the propagation of\nnonlinear waves in N= 4 SYM, but did not allow a definite identification of waves\nwith the 1 /|/vector x|falloff associated with radiation. (More recently, this falloff has been\nsuccessfully detectedinthesamesetupthroughacalculationofth eenergy-momentum\ntensor∝angbracketleftTµν∝angbracketright[50].)\nIn[48]it wasnotedthatfor zm→0andinthelinearizedapproximation, thestring\naction(5)correctlyimpliestheexpectedactionforanordinarynon -relativisticparticle\nof massm→ ∞. In the present section we will obtain the relativistic generalization\nof this result retaining the full non-linear structure of the Nambu- Goto string, and\nthen further extend the analysis to the case of finite m.\nWe will take as our starting point the results obtained in a remarkable paper\nby Mikhailov [30], which we now briefly review (a more detailed explanation can be\nfound in [25]). This author considered an infinitely massive quark, and was able to\nsolve the equation of motion for the dual string on AdS 5, for anarbitrary timelike\ntrajectory of the string endpoint. In terms of the coordinates u sed in (3), his solution\nis\nXµ(τ,z) =zdxµ(τ)\ndτ+xµ(τ), (10)\nwithxµ(τ) the worldline of the string endpoint at the AdS boundary— or, equiv -\nalently, the worldline of the dual, infinitely massive, quark— parametr ized by its\nproper time τ.\nCombining (3) and (10), the induced metric on the worldsheet is foun d to be\ngττ=R2\nz2(z2˚ ˚x2−1), g zz= 0, g zτ=−R2\nz2, (11)\nimplyinginparticularthattheconstant- τlinesarenull, afactthatplaysanimportant\nrole in Mikhailov’s construction. The solution (10) is ‘retarded’, in the s ense that the\nbehavior at time t=X0(τ,z) of the string segment located at radial position z\nis completely determined by the behavior of the string endpoint at an earliertime\ntret(t,z) obtained by projecting back toward the boundary along the null lin e at fixed\nτ. An analogous ‘advanced’ solution built upon the same endpoint/qua rk trajectory\ncan be obtained by reversing the sign of the first term in the right-h and side of (10).\n7In gauge theory language, this choice of sign corresponds to the c hoice between a\npurely outgoing or purely ingoing boundary condition for the waves in the gluonic\nfield at spatial infinity. Both on the string and the gauge theory side s, more general\nconfigurations should of course exist, but obtaining them explicitly is difficult due to\nthe highly non-linear character of the system. Henceforth we will f ocus solely on the\nretarded solutions, which are the ones that capture the physics o f present interest,\nwith influences propagating outward from the quark to infinity.\nFrom the µ= 0 component of (10), parametrizing the quark worldline by x0(τ)\ninstead of τ, and using dτ=√\n1−/vector v2dx0, where/vector v≡d/vector x/dx0, the relation that defines\nthe retarded time follows as\nt=z1√\n1−/vector v2+tret, (12)\nwhere the endpoint velocity /vector vis meant to be evaluated at tret. In these same terms,\nthe spatial components of (10) can be formulated as\n/vectorX(t,z) =z/vector v√\n1−/vector v2+/vector x(tret) = (t−tret)/vector v+/vector x(tret). (13)\nUsing (12) and (13), Mikhailov was able to rewrite the total string en ergy in the form\nE(t) =√\nλ\n2π/integraldisplayt\n−∞dtret/vector a2−[/vector v×/vector a]2\n(1−/vector v2)3+Eq(/vector v(t)), (14)\nwhere of course /vector a≡d/vector v/dx0. The first term codifies the accumulated energy lostby\nthe quark over all times prior to t, and is surprisingly seen to have precisely the same\nform as the standard Lienard formula from classical electrodynam ics.7The second\nterm in the above equation arises from a total derivative on the str ing worldsheet,\nand gives the expected Lorentz-covariant expression for the en ergy intrinsic to the\nquark [25],\nEq(/vector v) =√\nλ\n2π/parenleftbigg1√\n1−/vector v21\nz/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglezm=0\n∞=γm . (15)\nFor the spatial momentum, [30, 25] similarly find\n/vectorP(t) =√\nλ\n2π/integraldisplayt\n−∞dtret/vector a2−[/vector v×/vector a]2\n(1−/vector v2)3/vector v+/vector pq(/vector v(t)), (16)\nwith\n/vector pq=√\nλ\n2π/parenleftbigg/vector v√\n1−/vector v21\nz/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglezm=0\n∞=γm/vector v . (17)\nWe see then that, in spite of the non-linear nature of the system, M ikhailov’s pro-\ncedure leads to a clean separation between the tip and the tail of th e string, i.e.,\n7Possible experimental implications of this result have been explored in [51].\n8between the quark (including its near field) and its gluonic radiation fie ld. We will\nnow exploit this separation to study in more detail the dynamics of th e quark.\nOur initial observation is that, when we regard the Nambu-Goto act ion as a\nfunctional of the quark trajectory xµby plugging (11) back into (5)+(6), we can\nexplicitly carry out the integral over zto obtain\nSNG+SF=−R2\n2πα′/integraldisplay\ndτ/integraldisplay∞\nzm→0dz\nz2+/integraldisplay\ndτ Aµ(x(τ))˚xµ(τ) (18)\n=−m/integraldisplay\ndτ+/integraldisplay\ndτ Aµ(x(τ))˚xµ(τ),\nwhich is evidently the standard action for a pointlike externally force d relativistic\nparticle (with mass m→ ∞). Notice that the associated equation of motion does\nnotinclude a damping force, which is just as one would expect for an infinit ely\nmassive charge, because the coefficient te∝1/mof the damping terms in (1) and (2)\napproaches zero as m→ ∞.\nLet us now consider the more interesting case of a quark with finite m ass,zm>0,\nwhere, as we emphasized in the previous subsection, our non-Abelia n source is no\nlonger pointlike but has size zm. In this case the string endpoint is at z=zm, and\nwe must again require it to follow the given quark trajectory, xµ(τ). As before, this\nconditionby itself doesnotpick out aunique string embedding. Just lik e wediscussed\nfor the infinitely massive case below (11), we additionally require the s olution to be\n‘retarded’ or ‘purely outgoing’, in order to focus onthe gluonic field c ausally set up by\nthe quark. As in [25], we can inherit this structure by truncating a su itably selected\nretardedMikhailovsolution. Theembeddings ofinterest touscanth usberegardedas\nthez≥zmportions of the solutions (10), which are parametrized by data at t he AdS\nboundary z= 0.8Henceforth we will use tildes to label these (now merely auxiliary)\ndata, and distinguish them from the actual physical quantities (ve locity, proper time,\netc.) associated with the endpoint/quark at z=zm, which will be denoted without\ntildes.\nIn this notation, (10) reads\nXµ(˜τ,z) =zd˜xµ(˜τ)\nd˜τ+ ˜xµ(˜τ). (19)\nRepeated differentiation of this equation with respect to ˜ τand evaluation at z=zm\n(where we can read off the quark trajectory xµ(˜τ)≡Xµ(˜τ,zm)) leads to the recursive\n8That this direct truncation indeed retains the retarded structur e of the solutions is manifestly\nconfirmed in our final rewriting (29), where information is seen to pr opagate upward along the body\nof the string, i.e., from the UV to the IR of the gauge theory.\n9relations\ndxµ\nd˜τ=zmd2˜xµ\nd˜τ2+d˜xµ\nd˜τ,\nd2xµ\nd˜τ2=zmd3˜xµ\nd˜τ3+d2˜xµ\nd˜τ2,\n...\ndnxµ\nd˜τn=zmdn+1˜xµ\nd˜τn+1+dn˜xµ\nd˜τn. (20)\nAdding these equations respectively multiplied by ( −zm)n−1, we can deduce that\nd˜xµ\nd˜τ=dxµ\nd˜τ−zmd2xµ\nd˜τ2+z2\nmd3xµ\nd˜τ3−... , (21)\nand, upon further differentiation,\nd2˜xµ\nd˜τ2=d2xµ\nd˜τ2−zmd3xµ\nd˜τ3+z2\nmd4xµ\nd˜τ4−... . (22)\nThis last expression already takes us halfway towards the equation we are after, but\nwe still need to find a relation between d˜τand the endpoint/quark proper time dτ,\nand similarly rewrite d2˜xµ/d˜τ2in terms of quantities at the actual string boundary\nz=zminstead of the auxiliary data at z= 0.\nThe first task is easy: from (19) it follows that\ndXµ=dzd˜xµ\nd˜τ+d˜τ/parenleftbigg\nzd2˜xµ\nd˜τ2+d˜xµ\nd˜τ/parenrightbigg\n,\nwhich evaluated at fixed z=zmimplies\ndxµ=d˜τ/parenleftbigg\nzmd2˜xµ\nd˜τ2+d˜xµ\nd˜τ/parenrightbigg\n,\nand therefore\ndτ2≡ −dxµdxµ=d˜τ2/bracketleftBigg\n1−z2\nm/parenleftbiggd2˜x\nd˜τ2/parenrightbigg2/bracketrightBigg\n. (23)\nTo arrive at this last equation, we have made use of the fact that ˜ τis by definition\nthe proper time for the auxiliary worldline at z= 0, so (d˜x/d˜τ)2=−1 and (d˜x/d˜τ)·\n(d2˜x/d˜τ2) = 0.\nWhat remains then is to express d2˜xµ/d˜τ2as a function of quark data. For this\nwe note first that, upon substituting the solution (19), the momen tum current (8)\n(with appropriate tildes) evaluated at z=zmsimplifies to\n2π√\nλ˜Πz\nµ=1\nzmd2˜xµ\nd˜τ2+/parenleftbiggd2˜x\nd˜τ2/parenrightbigg2d˜xµ\nd˜τ. (24)\n10To avoid possible confusion, we should note that the tilde in the left-h and side does\nnot indicate evaluation at z= 0 (as all other tildes do), but the fact that this current\nis defined as charge (momentum) flow per unit ˜ τ. The corresponding flow per unit\nτis clearly just9Πz\nµ= (∂˜τ/∂τ)˜Πz\nµ, and it is this object which according to (7) must\nequal the external force Fµ. Using this, (23) and the first equation of (20) in (24),\none can deduce (after some straightforward algebra) that\nd2˜xµ\nd˜τ2=1/radicalbig\n1−z4\nm/F2/parenleftbigg\nzm/Fµ−z3\nm/F2dxµ\ndτ/parenrightbigg\n, (25)\nwhere we have used the abbreviation /Fµ≡(2π/√\nλ)Fµ. SinceFµdxµ/dτ= 0 (which\nis merely the statement that no work is done on the quark in its instan taneous rest\nframe), this implies that ( d2˜x/d˜τ2)2=z2\nm/F2, which allows (23) to be simplified into\nd˜τ=dτ/radicalbig\n1−z4\nm/F2. (26)\nUsing (25) and (26), we can finally rewrite (22) in the form\nzm/Fµ\n/radicalbig\n1−z4m/F2=/parenleftbiggz3\nm/F2\n1−z4m/F2/parenrightbiggdxµ\ndτ+/radicalbig\n1−z4m/F2d\ndτ/bracketleftbigg/radicalbig\n1−z4m/F2dxµ\ndτ/bracketrightbigg\n(27)\n−zm/radicalbig\n1−z4m/F2d\ndτ/bracketleftbigg/radicalbig\n1−z4m/F2d\ndτ/bracketleftbigg/radicalbig\n1−z4m/F2dxµ\ndτ/bracketrightbigg/bracketrightbigg\n+... .\nThis equation of motion for the quark contains an infinite number of d erivatives of\nxµ, precisely as one would expect for an extended color charge distrib ution, based on\nthe classical or quantum electrodynamic analogs [7, 8, 12] mentione d in the Intro-\nduction. Notice that to arrive at this result we have made no assump tion about the\nprofile of the charge distribution. The dual string dynamics automa tically incorpo-\nrates the physics of this profile, which is codified by the slope /vector s≡∂z/vectorX(zm,t) [25]. It\nwould be interesting to explore this connection in more detail throug h a calculation\nof∝angbracketleftTrF2(x)∝angbracketrightand similar observables for an accelerating quark in vacuum [48, 50, 32].\nA more manageable form of the equation of motion can be obtained by going back\nto the second equation in (20),\nd2xµ\nd˜τ2=d2˜xµ\nd˜τ2+zmd3˜xµ\nd˜τ3.\nThrough (25), (26) and (4), this can be reexpressed as\nd\ndτ\nmdxµ\ndτ−√\nλ\n2πmFµ\n/radicalBig\n1−λ\n4π2m4F2\n=Fµ−√\nλ\n2πm2F2dxµ\ndτ\n1−λ\n4π2m4F2, (28)\n9The transformation rules for Πa\nµunder more general reparametrizations can be found in, e.g.,\n[52].\n11which is the equation we advertised in the Introduction. Notice that it involves only\nthe four-velocity and four-acceleration of the quark, so, in going from (27) to (28), we\nhave traded aninfinite number ofhigher derivatives for a somewhat more complicated\nFµdependence. This is to some extent analogous to the possibility of tr ading (1), (2)\nor its non-pointlike generalizations for an integro-differential (non local in the force)\nequation with derivatives of xµonly up to second order [7, 8]. What is different is\nthat our end result, equation (28), does not involve any nonlocality . (It is also highly\nnonlinear, because it includes effects that are neglected for simplicit y in nearly all\nprevious analyses of radiation damping.)\nBefore proceeding with the analysis of (28), we would like to note for future use\nthat the information we have gathered in the process of its derivat ion, and more\nspecifically, equations (20) and (25), allow Mikhailov’s solution (10) to be rewritten\npurely in terms of z=zmdata as\nXµ(τ,z) =/parenleftBigg\nz−zm/radicalbig\n1−z4m/F2/parenrightBigg/parenleftbiggdxµ\ndτ−z2\nm/Fµ/parenrightbigg\n+xµ(τ). (29)\nA first check on (28) is to note that it correctly reduces to md2xµ/dτ2=Fµin\nthe pointlike limit m→ ∞(where the Compton wavelength zm→0). We will now\nperform a more substantial check by showing that it also makes firm contact with the\nresults of [25] at finite m.\nIn [25], two of us generalized the analysis of Mikhailov [30] to the case o f a quark\nwith finite mass, concentrating for simplicity on motion along one dimen sion. We\nshowed that under such circumstances the total string (= field + q uark) energy E\nand momentum Pare no longer given by (14) and (16), but become\nE(t) =√\nλ\n2π/integraldisplayt\n−∞dtF2\nm2/parenleftBigg\n1−√\nλ\n2πm2Fv\n1−λ\n4π2m4F2/parenrightBigg\n+1−√\nλ\n2πm2Fv/radicalBig\n1−λ\n4π2m4F2γm , (30)\nP(t) =√\nλ\n2π/integraldisplayt\n−∞dtF2\nm2/parenleftBigg\nv−√\nλ\n2πm2F\n1−λ\n4π2m4F2/parenrightBigg\n+v−√\nλ\n2πm2F/radicalBig\n1−λ\n4π2m4F2γm ,\nwithFthe external force. These expressions show that for m <∞the rate (seen\ninside the integrals) at which energy/momentum is radiated by the qu ark differs from\nthe Lienard result, and the formulas for the intrinsic energy Eqand momentum pqof\nthe quark (given by the terms that follow the integrals) are similarly n on-standard.\nNow, the total momentum Pof the string (= field + quark) changes only due to the\nforce that we exert on the endpoint (= quark), so dP/dt=F, or, using (30),\n√\nλ\n2πF2\nm2/parenleftBigg\nv−√\nλ\n2πm2F\n1−λ\n4π2m4F2/parenrightBigg\n+d\ndt\nmγv−√\nλ\n2πmγF/radicalBig\n1−λ\n4π2m4F2\n=F . (31)\nLet us now compare this against our equation of motion (28). For line ar motion\nalong direction x1, we have dxµ/dτ=γ(1,v) andFµ=γ(Fv,F), so the µ= 1\n12component of (28) reads\nd\ndt\nmγv−√\nλ\n2πmγF/radicalBig\n1−λ\n4π2m4F2\n=F−√\nλ\n2πm2F2v\n1−λ\n4π2m4F2. (32)\nThis is in precise agreement with (31). Similarly, it is easy to see that th eµ= 0\ncomponent of (28) yields the expected result dE/dt=FvwithEas in (30).\nWe have thus verified that our quark equation of motion reproduce s the en-\nergy/momentum split between quark and radiation field previously de duced in [25]\nfor the case of one-dimensional motion. It becomes clear then tha t (28) encodes the\ncovariant generalization of this split to the case of arbitrary motion — a result that\nwould have been extraordinarily difficult to obtain using the non-cova riant approach\nof [25]. To make this generalization explicit, we rewrite our equation in t he form\ndPµ\ndτ≡dpµ\nq\ndτ+dPµ\nrad\ndτ=Fµ, (33)\nrecognizing\npµ\nq=mdxµ\ndτ−√\nλ\n2πmFµ\n/radicalBig\n1−λ\n4π2m4F2(34)\nas the intrisic four-momentum of the quark, and\ndPµ\nrad\ndτ=√\nλF2\n2πm2/parenleftBigg\ndxµ\ndτ−√\nλ\n2πm2Fµ\n1−λ\n4π2m4F2/parenrightBigg\n(35)\nas the rate at which four-momentum is carried away from the quark by chromo-\nelectromagnetic radiation.\nUsing again the fact that F ·∂τx= 0, we can immediately deduce from (34) the\nmass-shell condition p2\nq=−m2, which shows in particular that pµ\nqis indeed a four-\nvector, and so the split Pµ=pµ\nq+Pµ\nraddefined in (33)-(35) is correctly Lorentz covari-\nant. As we indicated above, Pµ\nradrepresents the portion of the total four-momentum\nstored at any given time in the purely radiative part of the gluonic field set up by\nthe quark. The remainder, pµ\nq, includes the contribution of the nearfield sourced by\nour particle, or in quantum mechanical language, of the gluonic cloud surrounding\nthe quark, which gives rise to the deformed dispersion relation seen in (34). In other\nwords,pµ\nqis the four-momentum of the ‘dressed’ or ‘composite’ quark. All of this is\ncompletely analogous to the classical electromagnetic case that we briefly reviewed in\nthe Introduction, and in particular, to the covariant splitting of th e Maxwell tensor\nachieved in [11]. It is truly remarkable that the AdS/CFT correspond ence grants us\nsuch direct access to this piece of strongly-coupled non-Abelian ph ysics.\nNow that we have performed some checks on (28) and understood its proper\nphysical interpretation, we should consider its implications. As notic ed already in\n[25] for the case of linear motion, a salient feature of the equation o f motion (28),\n13as well as the dispersion relation (34) and radiation rate (35), is the presence of a\ndivergence when F2=F2\ncrit, where\nF2\ncrit=4π2m4\nλ(36)\nis the critical value at which the force becomes strong enough to nu cleate quark-\nantiquark pairs (or, in dual language, to create open strings) [24 ].\nLet us now examine the behavior of a quark that is sufficiently heavy, or is forced\nsufficiently softly, that the condition/radicalbig\nλ|F2|/2πm2≪1(i.e.,|F2| ≪ |F2\ncrit|) holds. It\nisthennaturaltoexpand theequation ofmotionina power series int hissmall param-\neter. To zeroth order in this expansion, we have the pointlike result md2xµ/dτ2=F,\nas we had already mentioned above. If we instead keep terms up to fi rst order, we\nfind\nmd\ndτ/parenleftBigg\ndxµ\ndτ−√\nλ\n2πm2Fµ/parenrightBigg\n=Fµ−√\nλ\n2πm2F2dxµ\ndτ. (37)\nIn theO(√\nλ) terms it is consistent, to this order, to replace Fµwith its zeroth order\nvalue, thereby obtaining\nm/parenleftBigg\nd2xµ\ndτ2−√\nλ\n2πmd3xµ\ndτ3/parenrightBigg\n=Fµ−√\nλ\n2πd2xν\ndτ2d2xν\ndτ2dxµ\ndτ. (38)\nInterestingly, (38) coincides exactlywith the Lorentz-Dirac equation (2)! As ex-\npected from the discussion in the preceding paragraphs, on the lef t-hand side we find\nthe Schott term (associated with the near field of thequark) arisin g fromthe modified\ndispersion relation (34). On the right-hand side we see the radiation reaction force\ngiven by the covariant Lienard formula, as expected from the resu lt (14) [30], which\nis the pointlike limit of the radiation rate (35). Moreover, by comparin g (2) and (38)\nwe learn that it is zm=√\nλ/2πmthat plays the role of characteristic time/size te\nfor the composite quark. As we discussed around (9), zmis the Compton wavelength\n(i.e., size of the gluonic cloud) of the quark, which is indeed the natura l quantum\nscale of the problem.\nIf we continue withtheexpansion of(28), theresult at second ord er canbewritten\nas\nm˚ ˚xµ−√\nλ\n2π( ˚ ˚ ˚xµ−˚ ˚xν˚ ˚xν˚xµ)+λ\n4π2m/parenleftbigg\n˚ ˚ ˚ ˚xµ−3˚ ˚xν˚ ˚ ˚xν˚xµ−3\n2˚ ˚xν˚ ˚xν˚ ˚xµ/parenrightbigg\n=Fµ.\nWe can of course continue this expansion procedure to arbitrarily h igh order in/radicalbig\nλ|F2|/2πm2, and at order nin this parameter, we would obtain an equation with\nderivatives up to order n+ 2. Now, it is interesting to note that, in the case of\nnon-relativistic QED, there are actually two different scales that ap pear in the cor-\nresponding equation of motion [12]: while the terms with derivatives of fourth and\nhigher order are all characterized by the quantum (Compton) radius λC=/planckover2pi1/m, the\nthird-derivative (i.e., Abraham-Lorentz) term involves only the classical electron ra-\ndiuscte=e2/mc2≪λC. In our strongly-coupled non-Abelian setting, the analogs\n14of these two scales happen to coincide. On the one hand, zmis analogous to λCin\nthat it gives the size of the cloud of virtual particles surrounding th e quark, which as\nwe explained at the end of Section 2, is set by the meson (and not the quark) mass,\nzm= 1/mmes. Ontheother hand, zmisalso analogousto cteinthatit gives theradius\nof a charge distribution whose chromoelectrostatic energy equals the quark mass m,\nif we take into account the strong-coupling form of the potential V(L)∝√\nλ/L[53].\nAs promised in the Introduction, our full equation (28) is thus reco gnized as an\nextension of the Lorentz-Dirac equation that automatically incorp orates the size zm\nof our non-classical, non-pointlike and non-Abelian source. The pas sage from (38) to\n(28), which can be viewed intuitively as the addition of an infinite numbe r of higher\nderivative terms ( zmd/dτ)n, has a profound impact on the space of solutions. Here\nwe will limit ourselves to two general observations, leaving the searc h for specific\nexamples of solutions to the next section. The first is to notice that , unlike its\nclassical electrodynamic counterparts (1) and (2), our composit e quark equation of\nmotion has no pre-accelerating or self-accelerating solutions. Tha t is to say, the\nbehavior of the quark at any given time τdoes not depend on Fµ(τ′) atτ′> τ, and,\nin the (continuous) absence of an external force, (28) uniquely p redicts that the four-\nacceleration of the quark must vanish. Our second observation, h owever, is that the\nconverse to this last statement is not true: constant four-veloc ity does not uniquely\nimply a vanishing force. We will expand on this in the examples of the nex t section.\nNotice that, in the systematic approach from our result (28) to th e Lorentz-Dirac\nequation, the pathologies that afflict the latter appear only in the ve ry last step, when\nthe˚ ˚ ˚xµterm is introduced upon approximating (37) by (38). Indeed, it has often been\nadvocated to eliminate these pathologies by a ‘reduction of order’ p rocedure (see,\ne.g., [54, 5]), which treats the damping force as a perturbative corr ection and thereby\njustifies the replacement of (38) by (37). (A closely related line of r easoning can be\nfound in [55].) It is therefore satisfying to see that the dressed qua rk equation of\nmotion predicted by AdS/CFT, Eq. (28), automatically comes out in ‘r educed order’\nform.\nIt is curious to note that (28), which incorporates the effect of ra diation damping\non the quark, has been obtained from (10), which does notinclude such damping for\nthe string itself. The supergravity fields set up by the string are of order 1/N2\nc, and\ntherefore subleading at large Nc. Even more curious [48] is the fact that it is precisely\nthese suppressed fields that encode the gluonic field profile genera ted by the quark, as\nhas been explored in great detail (mostly at finite temperature) in r ecent years (see,\ne.g., [56] and references therein). It would be interesting to explor e how the split into\nnear and radiation fields is achieved from this perspective, but we lea ve this problem\nto future work [32].\nIt is natural to inquire whether the equation of motion (28) can be e ncoded in\na variational principle. Since the complete system includes the compo site quark in\ninteraction with its radiation field, and the four-momentum of the lat ter is given by\nan integral over the quark worldline, at least naively we would expect that, if it is\nat all possible to write down an action, it should depend bilocally on the w orldline.\n15This would rule out the obvious candidate action, namely S=SNG+SFwith the\nclassical solution (19) plugged in. Indeed, the latter procedure sim ply leads again\nto (18) (with τin the Nambu-Goto term relabeled to ˜ τ), which (23) allows to be\nconverted into\nS=−m/integraldisplaydτ/radicalBig\n1−λ\n4π2m4F2+/integraldisplay\ndτ Aµ(x(τ))˚xµ(τ) (39)\nUnlike what happened in the pointlike (and no radiation damping) limit m→ ∞, in\nthis case variation of Swith respect to xµholdingFµfixed does notyield the correct\nequation of motion (28). Of course, Sis by definition the correct on-shell action for\nthe system, in the sense that it yields the right number when evaluat ed on a given\nworldline (using the Fµ(τ) obtained by solving (28)), but it would somehow need to\nbe rewritten in bilocal form to explicitly show all relevant xµdependence and thus\nconstitute the desired variational principle.\n3 Examples\n3.1 One-dimensional motion\nWe have already seen in Section 2.2 that, in the case of one-dimension al motion, our\ngeneral equation of motion (28) reduces to (32). The latter can b e further simplified\nto\na=zm/ F(1−v2)3/2\n/radicalBig\n1−z4m/ F2+z2\nm˙/ F(1−v2)\n1−z4m/ F2. (40)\nwhere we have preferred to express the prefactors in terms of t he quark Compton\nwavelength zminstead of its mass m, and as in (25) we have used the abbreviation\n/ F≡(2π/√\nλ)F. Here we see directly that, in contrast with the usual case, the\nacceleration at any given time depends not only on the value of the ap plied force but\nalso on its rate of change.\nIt follows from (40) that a quark that is free for any extended per iod of time will\nnot accelerate. In other words, as we had already indicated in the g eneral discussion\nbelow (38), there are no self-accelerating solutions, which is just a s one would have\nexpected given the extended nature of the charge. On the other hand, the very fact\nthat the charge has a ‘deformable’ internal structure implies that there is more than\none way to get it to follow any given trajectory. Indeed, for any ch oice ofv(t), (40)\nfixes the applied force / F(t) not algebraically, but through a differential equation that\ninevitably gives rise to a one-parameter family of solutions (differing b y their initial\nconditions).\nThe simplest example of this non-uniqueness of the external force is the case of\nconstant velocity, where (40) can be easily seen to imply that\n/ F(t) =±1\nz2msech/parenleftbigg√\n1−v2\nzm(t−t0)/parenrightbigg\n, (41)\n16witht0an integration constant. Only for t0→ ±∞does one recover the simple\nresultF(t) = 0. For all finite values of the integration constant, the force st arts out\nat zero at asymptotically early times, rises steadily until it attains th e critical value\nF=√\nλ/2πz2\nmgiven by (36) at t=t0, and then approaches zero again as t→ ∞.\nIt is certainly peculiar that one can continually apply a force to the qu ark and still\nhave it move at constant velocity. Nonetheless, it is easy to verify t hrough numerical\nintegration that indeed the application of the force (41) to the end point of a string\nwhose initial profile is chosen in accord with (29) produces no acceler ation. The\nenergy provided to the system by F(t) does not translate into an increase of the\nendpoint velocity, but into a continuous modification of the string ta il, or, in gauge\ntheory language, a change of the gluonic field profile. In fact, with t he formulas\nderived in the previous section, we can make a much more precise sta tement: (34)\nand (35) reduce to [25]\nEq=/parenleftBigg\n1−z2\nm/ Fv/radicalbig\n1−z4\nm/ F2/parenrightBigg\nγm ,dErad\ndt=z2\nm/ F2/parenleftbigg1−z2\nm/ Fv\n1−z4m/ F2/parenrightbigg\n,\nwhere we see that application of the force (41) results in\nEq=/braceleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoth/parenleftbiggt−t0\nγzm/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∓v/vextendsingle/vextendsingle/vextendsingle/vextendsinglecsch/parenleftbiggt−t0\nγzm/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg\nγm ,\ndErad\ndt= csch2/parenleftbiggt−t0\nγzm/parenrightbigg/bracketleftbigg\n1−vsech2/parenleftbiggt−t0\nγzm/parenrightbigg/bracketrightbigg\n,\nwhich show precisely what fraction of the energy goes into (or come s out from) re-\narranging the near gluonic field, and what fraction goes into radiatio n. In the limit\nt0→ ±∞, no force is applied and we of course recover Eq=γm,dErad/dt= 0 at all\ntimes. Similar conclusions can be drawn about the linear quark moment um.\nNotice that the width of the time interval over which the force (41) differs appre-\nciablyfromzeroisjusttheComptonwavelength ofthequark, zm(withanappropriate\nLorentz dilation factor), which serves to emphasize that this pecu liar phenomenon is\nmade possible only due to the non-pointlike nature of our non-Abelian source.\nThis non-uniqueness of the force is equally present for arbitrary t rajectories. An-\nother example that is easy to study is the case of constant force. If we assume that\nF(t) =constant, (40) reduces to d(γv)/dt=zm/ F//radicalbig\n1−z4m/ F2, which has the same\nform as the standard equation of motion for a particle with constan t proper acceler-\nation, except that the force here is non-linearly rescaled. The solu tion is thus\nx(t) =x0±/radicalBigg\n1\nz2m/ F2−z2\nm+(t−t0)2,\nwherex0andt0are integration constants. But if we run the argument in re-\nverse, and ask what type of force would lead to the hyperbolic motio nx(t) =\nx0±/radicalbig\nC2+(t−t0)2, we see again that the differential equation (40) admits solutions\nother than the obvious / F(t) = 1/zm/radicalbig\nC2+z2\nm.\n17Before closing this subsection, we would like to make one additional ob servation.\nAfter some straightforward algebra, it is easy to see that (35) ca n be rewritten in the\nform\ndPµ\nrad\ndτ=√\nλ\n2πm2\nF2\n/radicalBig\n1−λ\n4π2m4F2\npµ\nq. (42)\nPlugging this into (33), one obtains a first order differential equatio n forpµ\nqwith the\nsame structure as the Langevin equation, but with a friction coeffic ient that depends\non the external force,\ndpµ\nq\ndτ=−µ(F)pµ\nq+Fµ, (43)\nwhereµ(F)≡(√\nλ/2πm2)F2//radicalbig\n1−λ/4π2m4F2. In the case of motion in one spatial\ndimension, this equation can be solved analytically,\npq= exp\n−√\nλ\n2πm2/integraldisplaytF2(x)/radicalBig\n1−λ\n4π2m4F2(x)dx\n\n×\nA+/integraldisplayt\nexp/parenleftBigg√\nλ\n2πm2/integraldisplayyF2(x)/radicalBig\n1−λ\n4π2m4F2(x)dx/parenrightBigg\nF(y)dy\n,(44)\nwithAan integration constant. We should stress that this solution is valid o nly\nin the case of one dimensional motion, where F2=/vectorF2. In two or three spatial\ndimensions we have not been able to construct an explicit solution, be cause in that\ncaseF2involves thevelocity ofthequark, andremainsundetermined until w e findthe\nquark trajectory. Nevertheless, it is interesting to note that, in the case of constant\nF2(which is notthe same as constant /vectorF2), equation (43) can be interpreted as a\nLangevin equation of motion for a particle with momentum pq(up to a constant\nterm). The force in terms of the physical time tis just the force needed to move the\nquark in a ‘dissipative medium’ characterized by the constant frictio n coefficient µ.\nItwouldbeinteresting toexplorethedynamicsofthedressed quar kintwoorthree\nspatial dimensions. However, as we have just remarked, in that ca se (28) is highly\nnonlinear not only in the external force (which it was already in one dim ension), but\nalso in the velocity of the quark. For this reason, it is unfortunately very difficult to\nfind analytic solutions to it. In the heavy quark (or small force) app roximation where\n(28) linearizes and reduces to the Lorentz-Dirac equation (38), w e could of course\ncarry over to our setting the various solutions that have been wor ked out in the\npast. For instance, in the first reference of [7], Rohrlich was able to find an analytic\nsolution for a central force problem using the Frenet equations. S tarting from this\nand perturbing the solution it should be possible to obtain (at least nu merically)\nthe first correction beyond Lorentz-Dirac predicted by our fram ework, and deduce\nfor instance the r ate of synchrotron radiation. Such a calculation might shed some\nadditional light on the physics behind these extended objects, but is not central for\nthe purposes of our analysis here, so we prefer to leave it for futu re work.\n183.2 Nonrelativistic limit\nLet us now choose a specific Lorentz frame ( /vector x,t), and restrict attention to motions\nsuch that the quark velocity /vector v≡d/vector x/dt, acceleration /vector a≡d2/vector x/dt2, jerk/vector≡d3/vector x/dt3,\nand all higher derivatives, as well as the force / Fand its rate of change d / F/dt, are\nsmall in units of the Compton wavelength zm. Under such conditions dτ≃dt, and\nthe spatial components of (27) adopt the linearized form\nd2/vector x\ndt2−zmd3/vector x\ndt3+z2\nmd4/vector x\ndt4−...=zm/vector/ F . (45)\nAddingtothisexpression itstimederivativemultipliedby zm, weobtainthesimplified\nform\nd2/vector x\ndt2=zm/vector/ F+z2\nmd/vector/ F\ndt, (46)\nwhich is the linearized version of (28). We see here very directly that , as explained\nin Section 2.2, the unusual dependence on the rate of change of th e force encodes an\ninfinite number of higher derivatives of the quark trajectory /vector x(t), derivatives which\nin turn reflect the extended nature of the quark.\nIt is also useful to note that in this linearized limit, the string embeddin g (29) can\nbe rewritten in the form\n/vectorX(t,z) =/vector x(tret)+(z−zm)∞/summationdisplay\nl=1(−zm)l−1dl/vector x\ndtl(tret), tret≡t−z+zm,(47)\nwhichinvolvesonlythequarkworldline /vector x(t)andnottheforce /vectorF(t). Ifweplugthisinto\nthe(quadraticversionofthe)totalstringenergy E(t)≡ −/integraltext\ndzΠt\nt=/integraltext\ndz(˙/vectorX2+/vectorX′2)/2,\nand imitate Mikhailov’s procedure [30] (see Section 2.2), the express ion naturally\nsplits into\nE(t) =/integraldisplayt\n−∞dtretdErad\ndtret+Eq(t),\nwith\nEq(t) =1\n2m/parenleftBigg∞/summationdisplay\nl=1(−zm)l−1dl/vector x\ndtl(t)/parenrightBigg2\nand\ndErad\ndtret=/parenleftBigg∞/summationdisplay\nl=2(−zm)l−2dl/vector x\ndtl(tret)/parenrightBigg2\n+zm\n(t−tret+1)2d/vector x\ndt(tret)·/parenleftBigg∞/summationdisplay\nl=2(−zm)l−2dl/vector x\ndtl(tret)/parenrightBigg\n.\nUsing (45), it is easy to check that these expressions indeed coincid e with the non-\nrelativistic limit of the time component of (34) and (35). The spatial c omponents\ncan be verified in a similar manner. Notice that, in the relativistic case a nalyzed\nin [25] and Section 2.2 of the present work, the explicit presence of t he force Fµin\n19the solution (29) prevents us from deriving the split (33) as we did he re, directly\nimitating Mikhailov’s procedure. It is therefore comforting to see th at, at least in\nthe non-relativistic case, the direct result at z=zmindeed agrees with what we had\npreviously inferred indirectly from Mikhailov’s unambiguous split for th e auxiliary\ndata atz= 0.\nJust like in the fully relativistic setting, if we directly substitute (47) in to the\n(quadratic version of the) string action (5) + (6), we do notarrive at a variational\nprinciple that correctly encodes the equation of motion (45) or (46 ). Here, however,\nit is easy to write down an action that does the right job. In fact, we have not one\nbut (at least) two options: we can evidently get (45) from\nSnr=−m\n2/integraldisplay\ndt/bracketleftBigg/parenleftbiggd/vector x\ndt/parenrightbigg2\n+/parenleftbiggd2/vector x\ndt2/parenrightbigg2\n+/parenleftbiggd3/vector x\ndt3/parenrightbigg2\n+.../bracketrightBigg\n+/integraldisplay\ndt/vectorF·/vector x ,\nand (46) evidently follows from\nS′\nnr=−m\n2/integraldisplay\ndt/parenleftbiggd/vector x\ndt−zm/integraldisplayt\ndt′/vector/ F(t′)−z2\nm/vector/ F(t)/parenrightbigg2\n.\nIt is curious to note that the action Snrin terms of higher order derivatives of\nthe position vector with respect to τis reminiscent of the dynamical description of\na particle in noncommutative symplectic mechanics given in [57], where it was found\nthat the action can be written in terms of higher order derivatives o f the position\nvector despite the fact that the equations of motion are of secon d order.\nLet us now explore some solutions. Either from S′\nnror directly from the equation\nof motion (46), one immediately has a first integral of the motion,\nd/vector x\ndt=zm/integraldisplayt\n−∞dt′/vector/ F(t′)+z2\nm/vector/ F(t)+/vector v−∞.\nThe case with zero acceleration corresponds to //vectorF+zm˙//vectorF= 0, whose solution is\n/vector/ F(t) =/vector/ F0exp[−(t−t0)/zm]. This is of course simply the non-relativistic limit of\n(41). Since we are working now in the linearized approximation, this so lution of the\nhomogeneous equation of motion can be added to any solution of the inhomogeneous\nequation (46) to yield another solution, so it is very clear that the no n-uniqueness\nof the force is present for any quark trajectory whatsoever. 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