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arXiv:1001.0019v1 [gr-qc] 30 Dec 2009On the instability of Reissner-Nordstr¨ om black holes in de Sitter backgrounds Vitor Cardoso∗ CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal & Department of Physics and Astronomy, The University of Miss issippi, University, MS 38677-1848, USA Madalena Lemos†and Miguel Marques‡ CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (Dated: November 3, 2018) Recent numerical investigations have uncovered a surprisi ng result: Reissner-Nordstr¨ om-de Sitter black holes are unstable for spacetime dimensions larger th an 6. Here we prove the existence of such instability analytically, and we compute the timescal e in the near-extremal limit. We find very good agreement with the previous numerical results. Our res ults may me helpful in shedding some light on the nature of the instability. PACS numbers: 04.50.Gh,04.70.-s I. INTRODUCTION In physics, stability of a given configuration (solution of some set of equations), is a useful criterium for rele- vance of that solution. Unstable configurations are likely not tobe realizablein practice, and representaninterme- diate stage in the evolution of the system. Nevertheless, the instability itself is of great interest, since an under- standing of the mechanism behind it may help one to better grasp the physics involved. In particular, it is of interest to be able to predict which other systems display similar instabilities, or even have a deeper understanding of the physics behind the instability (why is the system unstable? is there some fundamental principle behind the instability?). In General Relativity, the Kerr family exhausts the blackhole solutionsto the electro-vacEinstein equations. Kerr black holes are stable, and can therefore describe astrophysicalobjects. However,there aremanyinstances of instabilities afflicting objects with an event horizon, such as the Gregory-Laflamme [1], the ultra-spinning [2] or superradiant instabilities [3] and other instabilities of higher-dimensional black holes in alternative theories [4, 5](for a review see Ref. [6]). Konoplya and Zhidenko (hereafter KZ) recently stud- ied small perturbations in the vicinity of a charged black hole in de Sitter background, a Reissner-Nordstr¨ om de Sitter black hole (RNdS) [7]. Their (numerical) results show that when the spacetime dimensionality D >6, the spacetime is unstable, provided the charge is larger than agiventhreshold, determined byKZforeach D. Because ∗Electronic address: vitor.cardoso@ist.utl.pt †Electronic address: madalena.dal@gmail.com ‡Electronic address: miguel.e.marques@gmail.comthe results are so surprising (the mechanism behind it is not yet understood), we set out to to investigate this in- stability and hopefully understand it better. Our results can be summarized as follows: (i) we can prove analyti- cally the existence of unstable modes for charge Qhigher thanacertainthreshold. (ii)inthenear-extremalregime, we are able to find an explicit solution for the unstable modes, determining the instability timescale analytically. We hope that our incursion in this topic helps to better understand the physics at work. II. EQUATIONS This work focuses on the higher dimensional RNdS ge- ometry, described by the line element ds2=−f dt2+f−1dr2+r2dΩ2 n, (1) wheredΩ2 nis the line element of the nsphere and f= 1−λr2−2M rn−1+Q2 r2n−2. (2) the background electric field is E0=q/rn, withqthe electric charge. The quantities MandQare related to the physical mass M and charge qof the black hole [8], andλto the cosmological constant. The spacetime di- mensionality is D=n+2. The above geometry possesses three horizons: the black-holeCauchyhorizonat r=ra, the black hole event horizon is at r=rband the cosmological horizon is at r=rc, whererc> rb> ra, the only real, positive zeroes off. For convenience, we set rb= 1, i.e., we measure all quantities in terms of the event horizon rb. We thus get 2M= 1+Q2−λ, (3)2 Furthermore, we can also write λ=r−4−n c(rn+2 c−r3 c)(rn+2 c−Q2r3 c) rn+2c−rc.(4) For a fixed rcand spacetime dimension D, the existence ofaregulareventhorizonimposesthatthecharge Qmust be smaller than a certain value Qext. With our units this maximum charge is Q2 ext=rn c/parenleftbig −2rc+(n+1)rn c−(n−1)rn+2 c/parenrightbig −rc/parenleftbig rc(n+1)−2nrnc+(n−1)r2n+1c/parenrightbig.(5) Gravitational perturbations of this spacetime couple to the electromagnetic field, and were completely character- ized by Kodama and Ishibashi [8]. They can be reduced to a set of two second order ordinary differential equa- tions of the form, d2 dr2∗Φ±+/parenleftbig ω2−VS±/parenrightbig Φ±= 0, (6)where the tortoise coordinate r∗and the potentials VS± are defined through r∗≡/integraldisplay f−1dr, V S±=fU± 64r2H2 ±.(7) We have H+= 1−n(n+1) 2δx, (8) H−=m+n(n+1) 2(1+mδ)x, (9) and the quantities U±are given by U+=/bracketleftbig −4n3(n+2)(n+1)2δ2x2−48n2(n+1)(n−2)δx −16(n−2)(n−4)]y−δ3n3(3n−2)(n+1)4(1+mδ)x4 +4δ2n2(n+1)2/braceleftbig (n+1)(3n−2)mδ+4n2+n−2/bracerightbig x3 +4δ(n+1)/braceleftbig (n−2)(n−4)(n+1)(m+n2K)δ−7n3+7n2−14n+8/bracerightbig x2 +/braceleftbig 16(n+1)/parenleftbig −4m+3n2(n−2)K/parenrightbig δ−16(3n−2)(n−2)/bracerightbig x +64m+16n(n+2)K, (10) U−=/bracketleftbig −4n3(n+2)(n+1)2(1+mδ)2x2+48n2(n+1)(n−2)m(1+mδ)x −16(n−2)(n−4)m2/bracketrightbig y−n3(3n−2)(n+1)4δ(1+mδ)3x4 −4n2(n+1)2(1+mδ)2/braceleftbig (n+1)(3n−2)mδ−n2/bracerightbig x3 +4(n+1)(1+ mδ)/braceleftbig m(n−2)(n−4)(n+1)(m+n2K)δ +4n(2n2−3n+4)m+n2(n−2)(n−4)(n+1)K/bracerightbig x2 −16m/braceleftbig (n+1)m/parenleftbig −4m+3n2(n−2)K/parenrightbig δ +3n(n−4)m+3n2(n+1)(n−2)K/bracerightbig x +64m3+16n(n+2)m2K. (11) The variables x,yand parameters µ,mare defined through x≡2M rn−1, y≡λr2, (12) µ2≡M2+4mQ2 (n+1)2, m≡k2−nK,(13) andthe quantity δis implicitly givenby µ= (1+2mδ)M. Note that the following relations holds Q2= (n+ 1)2M2δ(1+mδ). Note also that for the spacetime considered in this pa- perK= 1, whichmeansthatthe eigenvalues k2aregivenbyk2=l(l+n−1), where lis the angular quantum number, that gives the multipolarity of the field. The behavior of the potentials varies considerably over the range of parameters. In Fig. 1 we show V−forD= 8, rc= 1/0.95,l= 2andthreedifferentvaluesofthecharge, Q= 0.2,0.35,0.44. III. A CRITERIUM FOR INSTABILITY A sufficient (but not necessary) condition for the exis- tence of an unstable mode has been proven by Buell and3 /s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51 /s48/s44/s48/s52 /s48/s44/s48/s53/s45/s50/s48/s50/s52/s54 /s49/s48/s52 /s32/s86 /s45/s49/s48/s52 /s32/s86 /s45 /s32/s32/s86 /s45 /s114/s45/s49/s32/s81/s61/s48/s46/s50/s48 /s32/s81/s61/s48/s46/s51/s53 /s32/s81/s61/s48/s46/s52/s52/s49/s48/s51 /s32/s86 /s45 FIG. 1: Behavior of V−for different parameters, for D= 8. Here we fix the event horizon at rb= 1, and the cosmological horizon at rc= 1/0.95. We consider l= 2 modes and three different charges, Q= 0.2,0.35,0.44. Shadwick [9] and is the following, /integraldisplayrc rbV fdr <0. (14) The instability region is depicted in figure 2 for several /s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s32 /s32/s81/s47/s81 /s101/s120/s116 /s114 /s98/s47/s114 /s99 FIG. 2: The parametric region of instability in Q/Qext−rb/rc coordinates, according to criterim (14), for l= 2. Top to bottom, D= 7,8,9,10,11. spacetime-dimension D, which can be compared with the numerical results by KZ, their figure 4. It is apparent that condition (14) very accurately describes the numer- ical results for rb/rc∼1, a regime we explore below in Section IV. As one moves away from extremality cri- terium (14) is just too restrictive. An improved analysis and refined criterium would be necessary to describe the whole rangeofthe numericalresults. Nevertheless, figure2 is very clear: higher-dimensional ( D >6) RNdS black holes are unstable for a wide range of parameters. IV. AN EXACT SOLUTION IN THE NEAR EXTREMAL RNDS BLACK HOLE Let us now specialize to the near extremal RNdS black hole, which we define as the spacetime for which the cos- mological horizon rcis very close (in the rcoordinate) to the black hole horizon rb, i.e.rc−rb rb≪1. The wave equationin this spacetime can be solvedexactly, in terms of hypergeometric functions [10]. The key point is that the physical region of interest (where the boundary con- ditions are imposed), lies between rbandrc. Thus, f∼2κb(r−rb)(rc−r) rc−rb, (15) where we have introduced the surface gravity κbassoci- ated with the event horizon at r=rb, as defined by the relationκb=1 2df/drr=rb. For near-extremal black holes, it is approximately κb∼(rc−rb)(n−1) 2r2 b/parenleftbig 1−nQ2/parenrightbig .(16) In this limit, one can invert the relation r∗(r) of (7) to get r=rce2κbr∗+rb 1+e2κbr∗. (17) Substituting this on the expression (15) for fwe find f=(rc−rb)κb 2cosh(κbr∗)2. (18) As such, and taking into account the functional form of the potentials for wave propagation, we see that for the near extremal RNdS black hole the wave equation (6) is of the form d2Φ(ω,r) dr2∗+/bracketleftBigg ω2−V0 cosh(κbr∗)2/bracketrightBigg Φ(ω,r) = 0,(19) with V0=(rc−rb)κb 2VS±(rb) f(20) The potential in (19) is the well known P¨ oshl-Teller po- tential [11]. The solutions to (19) were studied and they are of the hypergeometric type, (for details see Refs. [12, 13]). Itshouldbesolvedunderappropriateboundary conditions: Φ∼e−iωr∗, r∗→ −∞ (21) Φ∼eiωr∗, r∗→ ∞. (22)4 These boundary conditions impose a non-trivial condi- tion onω[12, 13], and those that satisfy both simultane- ously are called quasinormal frequencies. For the P¨ oshl- Teller potential one can show [12, 13] that they are given by ω=κb/bracketleftBigg −/parenleftbigg j+1 2/parenrightbigg i+/radicalBigg V0 κ2 b−1 4/bracketrightBigg , j= 0,1,.... (23) We conclude therefore that an instability is present TABLE I: The threshold of instability for near-extremal RNdS black holes (i.e., black holes for which the cosmologic al and event horizon almost coincide) for l= 2 modes. We show the prediction from the exact, analytic expression obtaine d in the near extremal limit (24), which we label Q/QN extand the one from criterium (14) which we label as Q/QV ext. Both these results are compared to the numerical results by KZ. D 7 8 9 10 11 D→ ∞ Q/QN ext0.913 0.774 0.683 0.617 0.567p 2/D Q/QV ext0.913 0.775 0.684 0.618 0.568p 2/D Q/QNum ext0.94 0.78 0.68 0.61 0.55 — whenever V0is negative. The threshold of stability in the near-extremal regime is therefore given by VS±(rb) f= 0, (24) The expression for VS±(rb)/fis lengthy, and we won’t presentit here. Thevaluesofthe charge Q/Qextthat sat- isfy the condition above are given in Table I (for l= 2), and compared to the prediction from the analysis in Sec- tion III, criterium (14). The agreement is excellent. Fur-thermore, we compare these predictions against the nu- merical results by KZ, extrapolated to the extremal limit (ρ= 1 in KZ notation). The agreement is remarkable. V. CONCLUSIONS We have shown analytically that charged black holes in de Sitter backgrounds are unstable for a wide range of charge and mass of the black hole, confirming previous numerical studies [7]. The stability properties of the ex- tremalD= 6 black hole remain unknown. Our methods and results and inconclusive at this precise point, further dedicated investigations would be necessary. Ouranalyticalresultinthenear-extremalregimecould be used to investigate further the nature of this instabil- ity, something we have not attempted to do here. A possible refinement concerns the large- Dlimit of the in- stability, where it couldbe possible to find an analytical expression throughout all range of parameters. We have inmind resultsandtechniquessimilartothoseofKoland Sorkin [14]. It would also be interesting to investigate the stability properties, using this or other techniques, of near-extremal Kerr-dS black holes, which have recently been conjectured to have an holographic description [15]. Acknowledgements We warmly thank Roman Konoplya and Alexander Zhidenko for useful correspondence and for sharing their numerical results with us. This work was partially funded by Funda¸ c˜ ao para a Ciˆ encia e Tecnologia (FCT)- Portugal through projects PTDC/FIS/64175/2006, PTDC/ FIS/098025/2008,PTDC/FIS/098032/2008and CERN/FP/109290/2009. [1] R. Gregory and R. Laflamme, Phys. Rev. Lett. 70, 2837 (1993); H. Kudoh, Phys. Rev. D 73, 104034 (2006); V. Cardoso and O. J. C. Dias, Phys. Rev. Lett. 96, 181601 (2006). [2] R. Emparan and R. C. Myers, JHEP 0309, 025 (2003); O. J. C. Dias, P. Figueras, R. Monteiro, J. E. Santos and R. Emparan, arXiv:0907.2248 [hep-th]. [3] V. Cardoso, O. J. C. Dias, J. P. S. Lemos and S. Yoshida, Phys. Rev. D 70, 044039 (2004) [Erratum-ibid. 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