1001.0019.txt

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.
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