Perturbations and Stability of Higher-Dimensional Black Holes

Hideo KodamaCosmophysics GroupInstitute of Particle and Nuclear StudiesKEK

Lecture at 4th Aegean Summer School, 17-22 September 2007

Contents

Introduction Overview of the BH stability issue Linear perturbations

Perturbations of Static Black Holes Background solution Tensor/Vector/Scalar perturbations Summary

Applications to Other Systems Flat black branes Rotating black holes Accelerated black hole

Chapter 1

Introduction

Present Status of the BH Stability IssueFour-Dimensional Black Holes Stable

Static black holes Schwarzschild black hole [Vishveshwara 1970; Price 1972; Wald 1979,1980] Reissner-Nordstrom black hole [Chandrasekhar 1983] AdS/dS (charged) black holes [Ishibashi, Kodama 2003, 2004] Skyrme black hole (non-unique system) [Heusler, Droz, Straumann 1991,1992; Heusler,

Straumann, Zhou 1993] Kerr black hole [Whiting 1989]

Unstable YM black hole (non-unique system) [Straumann, Zhou 1990; Bizon 1991; Zhou,

Straumann 1991] Kerr-AdS black hole ( h <1, rh ¿ ) [Cardsoso, Dias, Yoshida 2006]

Unknown Kerr-Newman black hole Conjecture: large Kerr-AdS black holes are stable, but small ones are

SR unstable [Hawking, Reall 1999; Cardoso, Dias 2004]

Higher-Dimensional Black Objects

Stable Static black holes

AF vacuum static (Schwarzschild-Tangherlini) [Ishibashi, Kodama 2003] AF charged static (D=5,6-11) [Kodama, Ishibashi 2004; Konoplya, Zhidenko 2007] dS vacuum static (D=5,6,7-11), dS charged static (D=5,6-11) [IK 2003, KI

2004;Konoplya, Zhidenko 2007] BPS charged black branes (in type II SUGRA) [Gregory, Laflamme 1994:Hirayama,

Kang, Lee 2003] Unstable

Static black string (in AdS bulk), black branes (non-BPS) [Gregory, Laflamme 1993, 1995; Gregory 2000; Hirayama, Kang 2001: Hirayama, Kang, Lee 2003; Kang; Seahra, Clarkson, Maartens 2005; Kudoh 2006]

Rapidly rotating special GLPP (Kerr-AdS) bh [Kunhuri, Lucietti, Reall 2006] Unknown

Static black holes AF charged static (D>11), AdS (charged) static (D>4), dS (charged) static (D>11)

Rotating black holes/rings Conjecture:

Black rings are GL unstable. Rapidly rotating MP black holes are GL unstable [Emparan, Myers 2003] Doubly spinning black rings are SR unstable [Dias 2006] Kerr black brane Kerr4£ Rp is SR unstable [Cardoso, Yoshida 2005]

Linear PerturbationsPerturbation equations

When the spacetime metric (and matter fields/variables) is expressed as the sum of a background part and a small deviation as

in terms of the variables

the linearlised Einstein equations can be written as

where ML is the Lichnerowicz operator defined by

Linear Perturbations

Gauge problems Gauge freedom

In order to describe the spacetime structure and matter configuration as a perturbation from a fixed background (M,g,), we introduce a mapping

and define perturbation variables on the fixed background spacetime as follows:

F

Gauge Problems

For a different mapping F', these perturbation variables change their values, which has no physical meaning and can be regarded as a kind of gauge freedom.

The corresponding changes of the variables are identical to the transformation of the variables with respect to the transformation f=F‘ -1F. In the framework of linear perturbation theory,

To be explicit,

Gauge Problems

Two methods to remove the gauge freedom

Gauge fixing methodThis method is direct, but it is rather difficult to find relations between perturbation variables in different gauges in general.

Gauge-invariant method This method describe the theory only in terms of gauge-invariant quantities. Such quantities have non-local expressions in terms of the original perturbation variables in general.

These two approaches are mathematically equivalent, and a gauge-invariant variable can be regarded as some perturbation variable in some special gauge in general. Therefore, the non-locally of the gauge-invariant variables implies that the relation of two different gauges are non-local.

Gauge Problems Harmonic gauge

In this gauge, the perturbation equations read

and the gauge transformation is represented as

This gauge has residual gauge freedom

Synchronous gaugeIn the synchronous gauge in which

there exist the residual gauge freedom given by

For example, in the cosmological background

this produces a suprious decaying mode represented by

Chapter 2

Perturbations of Static Black Holes

Background Solution

Ansatz Spacetime

Metric

where dn2=ij dxidxj is an n-dimensional Einstein space Kn

satisfying the condition

Energy-momentum tensor

Background Solution

Einstein equations Notations

Einstein tensors

Einstein equations

Background Solutions

Examples Robertson-Walker universe: m=1 and K is a constant curvature

space.

Brane-world model: m=2 (and K is a constant curvature space). For example, the metric of AdSn+2 spacetime can be written

HD static Einstein black holes: m=2 and K is an Einstein space. K=Sn for the Schwarzschild-Tanghelini black hole. In general, the generalised Birkhoff theorem says that the electrovac solutions satisfying the ansataz with m=2 are exhausted by the Nariai-type solutions and the black hole type solution

Examples

Black branes: m=2+k and K=Einstain space. In this case, the spacetime factor Nm is the product of a two-dimensional black hole sector and a k-dimensional brane sector:

One can also generalise this background to introducing a warp factor in front of the black hole metric part.

HD rotating black hole (a special Myers-Perry solution): m=4 and K=Sn

where all the metric coefficients are functions only of r and .

Axisymmetric spacetime: m is general and n=1.

PerturbationsGauge transformations

For the infinitesimal gauge transformation

the metric perturbation hMN= gMN transforms as

and the energy-momentum perturbation MN= TMN transforms as

Perturbations

Tensorial Decomposition Algebraic tensorial type

Spatial scalar: hab, ab Spatial vector: hai, a

i Spatial tensor: hij, i

j

Decomposition of vectorsA vector field vi on K can be decomposed as

Decomposition of tensorsAny symmetric 2-tensor field on K can be decomposed as

Tensorial Decopositions

Irreducible types

In the linearised Einstein equations, through the covariant differentiation and tensor-algebraic operations, quantities of different algebraic tensorial types can appear in each equation.

However, in the case in which Kn is a constant curvature space, perturbation variables belonging to different irreducible tensorial types do not couple in the linearised Einstein equations, because there exists no quantity of the vector or the tensor type in the background except for the metric tensor.

The same result holds even in the case in which Kn is an Einstein space with non-constant curvature, because the only non-trivial background tensor other than the metric is the Weyl tensor that can only tranform a 2nd rank tensor to a 2nd rank tensor.

Tensor Perturbations

Tensor Harmonics Definition

where the Lichnerowitcz operator on K is defined by

When K is a constant curvature space, this operator is related to the Laplace-Beltrami operator by

Hence, Tiij satisfies

We use k2 in the meaning of L-2nK from now on when K is an Einstein space with non-constant sectional curvature.

Tensor Harmonics

Properties Identities:

For any symmetric 2-tensor on a constant curvature space satisfying

the following identities hold:

Spectrum: Let Mn be a n-dimensional constant curvature compact space with sectional curvature K. Then, the spectrum of k2 for the symmetric rank 2 harmonic tensor satisfies the condition

Sn: k2=l(l+n-1)-2, l=2,3,.. 2-dim case

In this case, a tensor hamonic represents an infinitesimal deformation of the moduli parameters.

In particular, there exists no tensor harmonics on S2.

Tensor Perturbations

Perturbation Equations Harmonic expansion

Gauge-invariant variables

Einstein EquationsOnly the (i,j)-component of the Einstein equations has the tensor-type component:

Here, ¤=DaDa is the D'Alembertian in the m-dimensional spacetime N.

Tensor Perturbations

Applications to the static Einstein black hole Master Equation

A static Einstein black hole corresponds to the case m=2 and

For this background, the perturbation equation without source

which can be written

where

Applications to a Static BH

Stability For the Schwarzschild black hole, we can show that Vt¸0.

Hence, it is stable. However, Vt is not positive definite in general, and the stability is not so obvious.

Energy integral

From the equation for HT, we find

Hence, in the case K is a constant curvature space, the stability of tensor perturbations results from k2¸ n|K|,

Vector Perturbations

Vector Harmonics Definitions

Harmonic tensors

Exceptional modes:The following harmonic vectors correspond to the Killing vectors and are exceptional:

Vector Harmonics

Properties Spectrum:

From the identities

We obtain the general bound the spectrum

Sn: k2=l(l+n-1)-1, l=1,2,… Exceptional modes:

The exceptional modes exist only for K¸0. For K=0, such modes exist only when K is isomorphic to TN£ Cn-N, where Cn-N is a Ricci flat space with no Killing vector.

Vector Perturbation

Perturbation equations Harmonic expansion

Gauge transformationsFor the vector-type gauge transformation

the perturbation variables transform as

Gauge invariants

Perturbation equations

Einstein equations Generic modes

Exceptional mode: k2=(n-1)K(¸0)

Vector Perturbation

Codimension Two Case Master equation

Generic modesFrom the energy-momentum conservation, one of the perturbation equation can be written

This leads to the master variable

in terms of which the remaining perturbation equation can be written

Exceptional modes

Codimension Two Case

Static black hole Master equation

where

This equation is identical to the Regge-Wheeler equation for n=2, K=1 and =0.

Codimension Two Case

Potentials

Codimension Two Case

Stability In the 4D case with n=2, K=1, =0, we have

In higher-dimensional cases, although the potential becomes negative near the horizon, we can prove the stability in terms of the energy integral because mv¸0:

Scalar PerturbationsScalar Harmonics Definition

Harmonic vectors

Harmonic tensors

Exceptional modes

Scalar Harmonics

Properties Spectrum:

For Qij defined by

We have the identity

From this we obtain the following bound on the spectrum

For Sn: k2= l(l+n-1), l=0,1,2,…

Scalar Perturbation

Linear Perturbations Harmonic expansion

Gauge transformationsFor the scalar-type gauge transformation

the perturbation variables transform as

Linear Perturbations

Gauge invariantsFrom the gauge transformation law

we find the following gauge-invariant combinations.

Linear Perturbations

Einstein equations G

Linear Perturbations

Gai :

Tracefree part of Gij :

Gii :

Scalar Perturbation

Codimension Two Case Master equation

For a static Einstein black hole, in terms of the master variable

the perturbation equations for a scalar perturbation can be reduced to

where

Codimension Two Case

Potentials

Codimension Two Case

Stability For n=2, K=1, =0, the master equation coincides with the

Zerilli equation and the potential is obviously positive definite:

where m=(l-1)(l+2). In higher dimensions, we have an conserved energy

integral,

We cannot conclude stability using this integral because Vs is not positive definite in general.

Codimension Two Case S-deformation

Let us deform the energy integral with the help of partial integrations as

where

Then, the effective potential changes to

For example, for

we obtain

where

Summary

Chapter 3

Application to Other Systems

Flat Black Branes

ASS4Lecture.dvi

Rotating Black Holes

Simple AdS-Kerr: a1=a, a2=…=aN=0

In this case, the metric is U(1)£ SO(n+1) symmetric with n=D-4.

For D¸ 7, the harmonic amplitude HT for tensor-type metric perturbations obeys the equation

This equation is exactly identical to the equation for the harmonic amplitude for a minimally-coupled massless scalar field in the same background!Therefore, we can apply the results on stability/instability of a massless scalar field to the tensor modes.

In particular, we can conclude that tensor perturbations are stable for a2 l2 < rh

4 on the basis of the argument by Hawking, Reall 1999.

Slowly Rotating AdS-Kerr

Everywhere Time-like Killing VectorFor slowly rotating black hole, there exists a Killing vector that is everywhere timelike in DOC: for example, when ai

2 l2< rh4 (i=1,2)

for D=5, or when a12 l2< rh

4, a2=…=aN=0. Energy Conservation Law

In this case, no instability occurs for a matter field satisfying the dominant energy condition [Hawking, Reall 1999]

where n T k is non-negative everywhere on .

Stability ConjectureOn the basis of this observation, Hawking and Reall conjectured that AdS-Kerr black holes with slow rotations such that ai

2 l2< rh4

will be stable against gravitational perturbations as well. At the same time, they also conjecture that rapidly rotating AdS-Kerr black holes will be unstable. This conjecture was proved for D=4 and rh¿ l [Cardoso, Dias, Yoshida 2006]

Energy Integral for Tensor Perturbations of Simple AdS-Kerr:In the coordinates in which the metric is written

for (t,r,x) defined by

the following energy integral is conserved:

where , F and U0 are always positive outside horizon, while U1 is positive definite only for a2 l 2 < rh

4.

Effective Potential In the effective potential

both U0 and U1 are positive for a2l 2 < rh

4 . For a2l 2 > rh

4 , however, U1 becomes negative in some range of r at x=-1, and the negative dip of the potential becomes arbitrarily deep as m increases.

Hence, it is highly probable that simple AdS-Kerr black holes in dimensions higher than 6 are unstable for tensor perturbations.

If we take ! 0 ( l ! 1) limit with fixed a and rh, the above stability condition is violated. This may suggest the instability of MP black holes unless the growth rate of instability vanishes at this limit.

Equally Rotating AdS-Kerr: a1==aN=a with D=2N+1.

In this case, the angular part of the metric has the structure of a twisted S1 bundle over CPN-1.

For a special class of tensor perturbations, the metric perturbation equation can be reduced to a Schrodinger-type ODE that has the same structure as that for a massless free scalar field.

It is claimed on the basis of analysis utilising the WKB approximation that such tensor perturbations satisfying the “superradiant condition” =m h are unstable if h l > 1, i.e., if there does not exist a global timelike Killing vector.[Kunhuri, Lucietti, Reall 2006]

Accelerated Black Hole

C-metirc Metric

C-metric is a Petrov type D static axisymmetric vacuum solution to the Einstein equations with cosmological constant.

The special case of the most general type D electrovac solution by Plebanski JF, Demianski M 1976

C-metric

Flat Limit

For = -1, M=0 and K=1, in terms of the variables

with

the C-metric can be written

This represents the Minkowski spacetime in the Rindler coordinates, and, each curve with constant x, y, has a constant acceleration.

The covered region has an acceleration horizon at y=-1, and the spatial infinity corresponds to x=y=-1.

C-metric

Schwarzschild Limit G(x) can be factorised as

where

In terms of the variables

the C-metric can be written

where

Conical String SingularityIf we choose the angle variable so that the metric is regular at the south pole x=x2 (=0), then the angular part of the metric is conformal to

around the north pole x=x1 (=), where

This implies that the metric has a conical singularity along the z-axis connecting the black hole horizon and the spatial infinity. This singularity corresponds to a string with positive tension 2=2.

0 case

Accele

ratio

n

Horizo

n

Acceleration

Horizon

y=x 1

y=x 1

y=x 0

x=x1x=x2

infini

ty

infinity

BH

Braneworld Black Hole AdS C-metric

Let us consider the special AdS C-metric

corresponding to

In the limit =0, in terms of the variables

The above AdS C-metric can be written

Global structure

0<x · x2 x1 · x <0

x=0

Black Hole in the 4D Braneworld

The extrinsic curvature of the timelike hypersurface x=0 is homogeneous and isotropic:

Hence, we can cut off the x>0 part of the solution and put the critical vacuum Z2 brane at the boundary x=0. This surgery provides a regular localised black hole in the 4D braneworld.

Emparan R, Horowitz GT, Myers RC (2000)

5D C-Metric as Braneworld BH ?

4D C-metric suggests that the yet-to-be-found localised BH solution in the 5D braneworld model may be given by an accelerated BH solution in the 5D AdS. However,

This solution should not represent a asymp. AdS regular black hole spacetime with a compact horizon because of the uniqueness theorem of the static AdS bh.(Cf. Chamblin, Hawking, Reall 1999; Kodama 2002)

The solution may not be singular in contrast to the 4D case, because the string in the 4d space has the codimension 3.

Hence, it is expected that the string source is

surrounded by a tubular horizon extending to infinity.

Perturbative Approach Static perturbations of a black hole

Background metric

Static scalar perturbation

Gauge-invariant variables

Master equation

where

Kodama, Ishibashi(2003) PTP110:701, (2004)PTP111:21; Kodama(2004) PTP112:249

Perturbative Approach

4D C-metric as a PerturbationWhen the acceleration MA' is small, the C-metric can be expressed as

This can be regarded as a scalar-type perturbation to the Schwarzschild solution. In the harmonic expansion, the gauge-invariant amplitudes are

From this, we find that this perturbation is produced from the source

This is consistent with the line density of a string, 2 = 8 , determined from the deficit angle.

Higher-Dimensional Analogue

Let us require that the source is localised on the half-infinite string:

Then, the l-dependence of the harmonic expansion coefficients of TMN

is completely determined as

Inserting these into the EM conservation law,

we obtain

SolutionIf we require that the solution for Y(l) is bounded at x=0 (r=1), then it is determined up to a constant A as

where

If we further require that Y(l) is bounded at x=1 (horizon), then A is determined as

Solution

The original perturbation variables are expressed as

where

Asymptotic Behaviour

At large r

where

At r ' rh

This indicates that the horizon is formed around ρn-2 ～ μ, and is consistent with the picture that the horizon at the central part of size r =rh is connected to a tubular horizon of radius » 1/(n-2) extending to infinity along the z-axis.

Brane ConstraintFor the exact Schwarzschild black hole, the hyperplane crossing the horizon at the equator is the only brane satisfying the junction condition

Hence, for the perturbative C-metric, the brane crosses the horizon near the equator: =/2 + (r).Then, the perturbation of the junction condition can be expressed as

which determines (r) as

and gives additional constraints on the metric perturbation at =/2

Kodama(2002) PTP108:253

Can We Get a Braneworld BH?

The condition for the existence of a Z2 vacuum brane configuration crossing the black hole horizon is given by

This gives a functional equation for the single function s(x) specifying the source distribution completely.

Cf. D. Karasik et al, PRD69(2004)064022; PRD70(2004)064007 Perturbative approach to the boundary value problem in the braneworld model. It concluded that the solution behaves badly at infinity if regular at horizon.

Summary

If the localised static black hole in the braneworld model can be obtained from a black hole accelerated by a string, its existence and uniqueness can be reduced to a functional equation for a string source function s(r) in the small mass limit of the bh.

The corresponding solution s(r) cannot be constant for the bulk dimension D>4. This implies that EOS of the string does not satisfy p=-in constrast to the case of D=4.

For D>4, the string is enclosed by a tubular horizon, because the spatial codimension is greater than 2.

It is likely that the brane condition for s(r) has a unique solution, but we have not succeeded in proving it yet.