instabilities and quasinormal modes of black holes in the ... · as was shown in [g. dotti, r. j....

Instabilities and quasinormal modes of black holes in the

Einstein-Gauss-Bonnet theory

Roman Konoplya

Theoretical Astrophysics, Eberhard-Karls University of Tubingen, Tubingen 72076, Germany

Oxford U., January 17, 2017based on

R. K., A. Zhidenko arXiv:1701.01652;M. Cuyubamba, R. K., A. Zhidenko Phys.Rev. D93 (2016) 104053;

R. K., A. Zhidenko Phys.Rev. D77 (2008) 104004

Roman Konoplya Instabilities and quasinormal modes of black holes in the Einstein-Gauss-Bonnet theory

Introduction and Motivations

Observation of quasinormal modes, apparently, of black holes in theLIGO/VIRGO experiments: though quasinormal modes are detected withreasonable accuracy, the black-hole mass and angular momentum areknown with an error of quite a few tens of percents. Within such a largeuncertainty an alternative theory of gravity can comfortably "survive" (R.K., A. Zhidenko, Phys.Lett. B756 (2016) 350-353).

Although in D = 4 the GB term is invariant, it can show up when coupledto dilaton, though, here we shall consider only D > 4 theories, for whichthere are several motivations and peculiarities:

a) Gauss-Bonnet black holes in AdS space (and higher curvaturecorrection in general) may help to explain what happens in the holographydescirption in the regime of nite t'Hooft coupling (S. Grozdanov, et.al. JHEP 1607, 151 (2016); S. Waeber, et. al. JHEP 1511, 087 (2015);T. Andrade, et. al arXiv:1610.08987)


Introduction and Motivations

b) Higher curvature corrected theories of gravity are also importantalternatives to the Einstein gravity, as they appear in the low-energy limitof the heterotic string theory.

c) The pure AdS space-time is known to be non-linearly unstable [P.Bizon, A. Rostworowski, Phys. Rev. Lett. 107, 031102 (2011); P. Bizon,M. Maliborski, A. Rostworowski, Phys. Rev. Lett. 115, no. 8, 081103(2015)], while it has been claimed that the GB term is treating thisinstability [N. Deppe et. al PRL114, 071102 (2015)].

d) Even relatively small -correction to the black hole geometry leads toeect of considerable suppression of the Hawking evaporation (by quite afew orders) [R. K., A. Zhidenko Phys.Rev. D82 (2010) 084003].

Therefore, here we discuss stability and quasinormal modes of black hole in theEinstein-Gauss-Bonnet theory, allowing for a non zero lambda-term. We shallchoose the branch of solutions which has the Schwazrschild limits when the GBcoupling goes to zero, and thus, discuss the three cases: = 0, > 0 and < 0.


= 0

The Lagrangian of the Einstein-Gauss-Bonnet action is

I =1

16GD

ZdDx

pgR + 0Z

dDxpg(RabcdR

abcd 4RcdRcd + R2): (1)

Here 0 is a positive coupling constant.The metric has the form,

ds2 = f (r)dt2 dr 2

f (r) r 2d2

D2; (2)

f (r) = 1+r 2

(D 3)(D 4)(1 q(r)) ; q(r) =

s1 +

4(D 3)(D 4)

(D 2)rD1;

where = 16GD0. In order to measure all the quantities in terms of the

black hole horizon r0 radius we parameterize the black hole mass as

=(D 2)rD3

0

4

2 +

(D 3)(D 4)

r 20

: (3)


= 0

As was shown in [G. Dotti, R. J. Gleiser, Phys. Rev. D 72, 044018 (2005),Phys. Rev. D 72, 124002 (2005)], the gravitational perturbations of aGauss-Bonnet black hole can be decoupled from their angular part and reducedto the wave-like equation of the form

@2

@t2 @2

@r 2?+ V (r)

(t; r) = 0; dr? =

dr

f (r): (4)

with the eective potentials which have VERY cumbersome form. Forasymptotically at or de Sitter background quasinormal modes satisfy thefollowing boundary conditions: purely incoming waves at the BH horizon andpurely outgoing waves at innity/cosmological horizon, i.e. no re ections fromboth sides. We study the ringing of GB black hole using a numericalcharacteristic integration method [C. Gundlach, R. H. Price and J. Pullin,Phys. Rev. D 49, 883 (1994)], that uses the light-cone variables u = t r? andv = t + r?. In the characteristic initial value problem, initial data are speciedon the two null surfaces u = u0 and v = v0. The discretization scheme weused, is

(N) = (W )+(E)(S)2V (W )(W ) + V (E)(E)

8+O(4) ; (5)

where we have used the following denitions for the points:N = (u +; v +), W = (u +; v), E = (u; v +) and S = (u; v).


= 0

`

8 1:34610 1:27412 1:22716 1:17020 1:13632 1:08740 1:07050 1:05864 1:047

Threshold as a function of the inverse multipole number `. Tensor type ofgravitational perturbations D = 6. The points ` = 16; 20; 32; 40; 50; 64 were tby the line = 2:627`1 + 1:005. The theoretical result is t 1:006.


= 0

`

8 0:26810 0:25812 0:25016 0:24120 0:23532 0:22540 0:22250 0:219

Threshold as a function of the inverse multipole number `. Scalar type ofgravitational perturbations D = 5.The points ` = 16; 20; 32; 40; 50 were t by the line = 0:517`1 +0:209. Thetheoretical result is t 0:207.


= 0

Although the eective potential has small negative gap for perturbation ofscalar type (D = 6, l = 2, = 0:3), this gap does not yet lead to theinstability. It causes exponentially damping \tails" to appear just after theinitial outburst. It means that the purely imaginary, non-oscillating modedominates at some values of the parameters in the stability region.


= 0

Conclusions of the st part.

All small asymptotically at Einstein-Gauss-Bonnet black holes areunstable in D = 5 and 6 space-times.

The instability "developes" at high multipole numbers ` for D = 5 in thescalar channel and for D = 6 in the tensor channel

The instability leads to the absence of convergence by `, i.e. to theabsence of a well-posed initial values problem.

The same instability for higher than D = 6 black holes was also found forthe Lovelock black holes in T. Takahashi, J. Soda, Prog. Theor. Phys. 124, 911 (2010); T. Takahashi, J. Soda, Prog. Theor. Phys. 124, 711 (2010).


> 0

The Einstein-Gauss-Bonnet Lagrangian takes the following form

L = 2 + R + (RR 4RR

+ R2); (6)

where = 1=2`2s is a positive coupling constant. An exact static vacuumsolution of the Einstein-Gauss-Bonnet equations can be written in the form

ds2 = f (r)dt2 + 1

f (r)dr 2 + r 2 d2

n (7)

where d2n is the (n = D 2)-dimensional line element and

f (r) = 1 r 2 (r); (8)

W [ ] n(n 1)(n 2)

4 2 +

n

2

n + 1=

rn+1; (9)

where is a positive constant proportional to the black-hole mass.

(r) =1

(n 1)(n 2)

s1 +

4(n 1)(n 2)

n

rn+1+

n + 1

1

!(10)


> 0

In the de Sitter spacetimes the span of the spatial coordinate is limited by thecosmological horizon rC > rH , which we use in order to parametrize thecosmological constant as

=n(n + 1)

2

rn1C

rn1H

rn+1C

rn+1H

+(n 1)(n 2)

2

rn3C

rn3H

rn+1C

rn+1H

!; (11)

In the limit rC ! rH we obtain the extremal value of the cosmological constant,which is given as follows

extr =n(n 1)

2 r 4H

r 2H +

(n 2)(n 3)

2

: (12)

Limit rC !1 corresponds to the asymptotically at spacetime ( = 0).Hereafter, we measure all the quantities in units of the event horizon, weintroduce dimensionless parameters, 0 rH

rC< 1 and

r2H

0, while frequencies

are measured in the units of inverse horizon radius r1H

.


> 0

Left: Stability and instability regions for scalar-type gravitational perturbationsin 5 dimensions. Upper right corner corresponds to the -instability, while thelower right corner corresponds to the eikonal instability. The overlap of regionsof both types of instability produce the instability region for D = 5 case. Right:Stability and instability regions for gravitational perturbations in D = 6 asoverlap of the -instability in the scalar channel and eikonal instability in thetensor channel.


> 0

D -instability eikonal instability

5 scalar-type (` = 2) scalar-type

6 scalar-type (` = 2) tensor-type

7 scalar-type (` = 2)

8 scalar-type (` = 2)

Table: Summary of instabilities of Einstein-Gauss-Bonnet-de Sitter black holes: eachtype of instability implies its parametric region. For the eikonal instability this regionexpands as ` increases, so that the instability region in this case corresponds to thelimit `!1.

We suspect that the "tradiational" instability which occurs at the lowest `owing to the non-zero -term and does not take place for asymptotically atEinstein-Gauss-Bonnet black hole is similar in nature with the one found in [R.K. , A. Zhidenko Phys.Rev.Lett. 103 (2009) 161101] for Reissner-Nordstrom-deSitter black holes, though the role of the -term in this instability is not clearyet.


> 0

Several years ago Shahar Hod formulated an interesting proposal [S. Hod Phys.Rev. D 75, 064013 (2007)] stating that the damping rate of the fundamentalquasinormal frequency of any black hole in nature is constrained by the value ofits Hawking temperature in a specic way. Namely, he argued that

jIm!j TH ; (13)

where TH is the Hawking temperature. For static spherically symmetric blackholes

TH =H2

=f 0(rH)

4: (14)

Using numerical and analytical results for quasinormal modes of four- andhigher-dimensional Schwarzschild, Schwazrschild-dS and Schwarzschild-AdSblack holes, Hod illustrated that his inequality is fullled for asymptotically atblack holes as well as for nonasymptotically at ones. The arguments werebased on semiclassical consideration and thermodynamic ideas.


> 0

We analyzed the case of Einstein-Gauss-Bonnet-dS black holes and found theperturbations of the scalar channel does not obey the proposal, but of thevector and tensor ones, do obey. As we do not believe that in a real (generallyspeaking non-linear) physical process it is possible to excite only the scalarmodes, we conclude that no violation of the Hod's proposal was found in ourcase.

Right: Variation of the imaginary part of the dominant quasinormal mode withrespect to rH=rC for vector- and tensor-type perturbations in 5 dimensions for = 0:2r 2H . Left: Variation of the real (left panel) and imaginary (right panel)parts of the dominant quasinormal mode as functions of rH=rC for scalar-typeperturbations in 5 dimensions.


> 0

Conclusions of the second part.

In addition to the eikonal instability owing to the non-zero GB coupling,there is a "usual" instability owing to a non-zero -term.

The Hod' conjecture is not violated for this case.

The eikonal instability is accompanied by the absence of the well-posedinitial value problem, apparently, similar to that found in H. Reall at. all.,Class.Quant.Grav. 31 (2014) 205005. This is not the case of the other -instability which occurs at the lowest `.


< 0

ds2 = f (r)dt2 + f 1(r)dr 2 + r 2 d2n; where d

2n is a (n = D 2)-dimensional

sphere, and f (r) = 1 r 2 (r); where (r) satises

W [ ] n (1 + e )=2 =(n + 1) = rn1: (15)

Here e (n 1)(n 2)=2 and is a constant, proportional to mass; therelevant solution is

(r) =

qn2 + 8en (rn1 + =(n + 1)) n

=2en: (16)

We shall express as a function of the event horizon rH :

2 = n rn1H (1 + er2

H 2r 2H=n(n + 1)): (17)

Here we shall measure in units of the AdS radius R (dened by relation (r !1) = 1=R2). Then,

= n(n + 1)(1 e=R2)=2R2; (18)

implying e < R2.


< 0

For asymptotically AdS case the quasinormal boundary conditions are in asense dierent from those of the asymptotically at/de Sitter case:

purely incoming wave at the event horizon (19)

0 at inifnity(so

D=5 is exceptional case)(20)For AdS case we are using the shooting method, i.e. compare the asymptoticexpression of the solution of the wave equation for any given ! with the resultof numerical integration from the horizon.For large ` from [T. Takahashi, J. Soda Prog. Theor. Phys. 124, 711 (2010)],we have

Vt(r) =`2f (r)T 00(r)

(n 2)rT 0(r)+O(`); (21)

Vv (r) =`2f (r)T 0(r)

(n 1)rT (r)+O(`); (22)

Vs(r) =`2f (r)(2T 0(r)2 T (r)T 00(r))

nrT 0(r)T (r)+O(`); (23)

whereT (r) = rn1dW =d = nrn1(1=2 + e (r)): (24)


< 0

The eikonal instability occurs when the dominant term above has a negativegap, so that the potential is negatively dominant for suciently large `.Analytical proof of suciency of a negative gap in the dominant term for AdSblack-hole instability is similar to that for the asymptotically at case [T.Takahashi, J. Soda Prog. Theor. Phys. 124, 711 (2010)].

-0.3 -0.2 -0.1 0.0 0.10.0

0.5

1.0

1.5

2.0

ΑR2

r HR

-0.6 -0.4 -0.2 0.0 0.20.0

0.5

1.0

1.5

2.0

ΑR2r HR

Left: Parametric regions of the eikonal instability for tensor-type perturbations(red, middle) and scalar-type perturbations (cyan) of D = 5 (n = 3) GB-AdSblack holes. Right: Parametric regions of the eikonal instability for tensor-typeperturbations of D = 6 (n = 4) GB-AdS black holes.


< 0

Vector channel. No eikonal instability.

Scalar channel. For n = 3 the eikonal instability occurs at

e > R2r 2H(p2 1)=2(

p2r 2H + R2) (e 6= R2=2) (25)

and, for small black holes (r 4H < R4=2), at

e < (p2 + 1)R2r 2H=2(R2

p2r 2H):

The case 3 + 2e = 0 is special, in which black holes of arbitrary size donot have the eikonal instability. It corresponds to e = R2=2 ( = 3=R2)for which (16) takes the formf (r) = (r=R)2 + 1p4=3R2 = (r 2 r 2H)=R

2:

Tensor channel. For n = 3 the eikonal instability occurs at

R2r 2H2

p3 +

p2p

3R2 p2r 2H

< e < R2r 2H2

p3p2p

3R2 +p2r 2

H

for small black holes (r 4H < 3R4=2) and at

e < R2r 2H(p3

p2)=2(

p3R2 +

p2r 2H) (26)

for large black holes (r 4H 3R4=2).


< 0

0 1 2 3 4rHR0

1

2

3

4

5ImHΩLR

The imaginary part of ! at ` = 2; 3; 4; 10 (from left to right) for = 0:4R2,D = 5, scalar channel. The value of rH=R at which the instability occurs growslinearly with ` which means that all GB black holes are unstable at = 0:4R2

-0.4 -0.2 0.2 0.4ReHΩLR

-2.80

-2.75

-2.70

-2.65

-2.60

ImHΩL

The hydrodynamic \mode" goes o the imaginary axis for very large negative, i.e. in the regime of instability: =R2 = 2:56;2:57;2:58;2:59;2:60(the real part increases with ); rH = 10R, ` = 2, D = 5.


< 0

In [S. Grozdanov, et. al. JHEP 1607, 151 (2016)] an interesting phenomenonwas observed: the vector (hydrodynamic) modes go o the imaginary axis forhigher curvature corrected black holes with planar horizon: initially, purelyimaginary modes acquired nonzero real part and, thereby, \duplicate", whatmust indicate the breakdown of the hydrodynamic regime. Here we haveobserved this phenomenon occurs only for suciently large negative ,corresponding to the unstable sector. In the stable sector, i.e. for small , thedominant vector modes are roughly linear with respect to and, for example,for ` = 2;D = 5; rH = 10R obey the formula ! i(0:126=R 0:376=R3):We nd that planar congurations (n = 3) should be unstable at

e < (p3p2)R2=2p2 0:112372R2

and, from (25), at

e > (p2 1)R2=2

p2 0:146447R2

e 6= R2=2:

We suppose, therefore, that such an instability could be seen for the planarGB-AdS black holes at high values of the momentum.


< 0

Conclusions of the third part.

The eikonal instability and its exact parametric regions for asymptoticallyAdS black holes in the Einstein-Gauss-Bonnet theory were found.

The instability is "driven" by the purely imaginary modes which do nothave analogues when the Gauss-Bonnet coupling is zero.

In [N. Deppe et. al PRL114, 071102 (2015)] it was shown that, if theenergy content of the AdS spacetime is not suciently large, a (small)black hole cannot be formed at a nonzero GB coupling. This fact waspreviously used to suppose the stabilization of the pure AdS space-time bythe GB-term. However, it could be very well explained by the found herelinear instability of any small asymptotically AdS black holes inGauss-Bonnet theory: the perturbation cannot \condense" into a smallblack hole, simply because the latter is unstable. Thus, no assumption ofstabilization of the pure AdS space by the GB term becomes necessary insuch a picture.


The End

Conclusions for all the three parts

All suciently small black holes with = 0; < 0; > 0 are gravitationallyunstable and it is an eikonal type of instability developing at highmultipoles `.

For > 0 there is also an instability owing to the non-zero -term at thelowest multipole ` = 2.

The black hole approaches the threshold of the eikonal instability"through" the "new" branch of purely imaginary modes which appearsonly at a non-zero Gauss-Bonnet coupling.

Instability of small Gauss-Bonnet-AdS black holes may explain the gap inthe formation of small black holes when considering critical collapse in thepure AdS space-time in GB theory.


instabilities and quasinormal modes of black holes in the ... · as was shown in [g. dotti, r. j....

Documents