Transcript Comments on Higher-Dimensional C

Perturbations and Stability of
Higher-Dimensional Black Holes
Hideo Kodama
Cosmophysics Group
Institute of Particle and Nuclear Studies
KEK
Lecture at 4th Aegean Summer School, 17-22 September 2007
Contents

Introduction



Perturbations of Static Black Holes




Overview of the BH stability issue
Linear perturbations
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 Issue
Four-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 ( l h <1, rh ¿ l) [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 Perturbations
Perturbation 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 method
This 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 nonlocally 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 gauge
In 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 dn2=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 twodimensional 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.
Perturbations
Gauge 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, ai

Spatial tensor: hij, ij

Decomposition of vectors
A vector field vi on K can be decomposed as

Decomposition of tensors
Any 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 Equations
Only 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 transformations
For 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 modes
From 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 Perturbations
Scalar 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 transformations
For the scalar-type gauge transformation
the perturbation variables transform as
Linear Perturbations

Gauge invariants
From the gauge transformation law
we find the following gauge-invariant combinations.
Linear Perturbations

Einstein equations

 Gab :
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 <
rh4 on the basis of the argument by Hawking, Reall 1999.
Slowly Rotating AdS-Kerr

Everywhere Time-like Killing Vector
For slowly rotating black hole, there exists a Killing vector that is
everywhere timelike in DOC: for example, when ai2 l2< rh4 (i=1,2)
for D=5, or when a12 l2< rh4, 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 Conjecture
On the basis of this observation, Hawking and Reall conjectured
that AdS-Kerr black holes with slow rotations such that ai2 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 <
r h4 .

Effective Potential

In the effective potential
both U0 and U1 are positive for a2l 2 < rh4 .



For a2l 2 > rh4 , 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 Schrodingertype 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
0
y=
x
x=x 1
n
tio
ra
le
ce zon
Ac ori
H
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.
x=x 2 BH
=0 case
in
y=
fin
ity x1
around the north pole x=x1 (=), where
ity
fin
in
If 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
Ac
H
y= o ce
x riz ler
1
on at
io
n
Conical String Singularity
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-befound 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 Perturbation
When 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
Solution
If 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 Constraint
For 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.