perturbations and stability of higher-dimensional black holes
DESCRIPTION
Perturbations and Stability of Higher-Dimensional Black Holes. Hideo Kodama Cosmophysics Group Institute of Particle and Nuclear Studies KEK. Lecture at 4 th Aegean Summer School, 17-22 September 2007. Contents. Introduction Overview of the BH stability issue Linear perturbations - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/1.jpg)
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
![Page 2: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/2.jpg)
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
![Page 3: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/3.jpg)
Chapter 1
Introduction
![Page 4: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/4.jpg)
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]
![Page 5: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/5.jpg)
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]
![Page 6: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/6.jpg)
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
![Page 7: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/7.jpg)
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
![Page 8: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/8.jpg)
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,
![Page 9: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/9.jpg)
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.
![Page 10: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/10.jpg)
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
![Page 11: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/11.jpg)
Chapter 2
Perturbations of Static Black Holes
![Page 12: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/12.jpg)
Background Solution
Ansatz Spacetime
Metric
where dn2=ij dxidxj is an n-dimensional Einstein space Kn
satisfying the condition
Energy-momentum tensor
![Page 13: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/13.jpg)
Background Solution
Einstein equations Notations
Einstein tensors
Einstein equations
![Page 14: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/14.jpg)
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
![Page 15: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/15.jpg)
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.
![Page 16: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/16.jpg)
PerturbationsGauge transformations
For the infinitesimal gauge transformation
the metric perturbation hMN= gMN transforms as
and the energy-momentum perturbation MN= TMN transforms as
![Page 17: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/17.jpg)
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
![Page 18: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/18.jpg)
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.
![Page 19: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/19.jpg)
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.
![Page 20: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/20.jpg)
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.
![Page 21: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/21.jpg)
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.
![Page 22: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/22.jpg)
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
![Page 23: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/23.jpg)
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|,
![Page 24: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/24.jpg)
Vector Perturbations
Vector Harmonics Definitions
Harmonic tensors
Exceptional modes:The following harmonic vectors correspond to the Killing vectors and are exceptional:
![Page 25: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/25.jpg)
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.
![Page 26: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/26.jpg)
Vector Perturbation
Perturbation equations Harmonic expansion
Gauge transformationsFor the vector-type gauge transformation
the perturbation variables transform as
Gauge invariants
![Page 27: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/27.jpg)
Perturbation equations
Einstein equations Generic modes
Exceptional mode: k2=(n-1)K(¸0)
![Page 28: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/28.jpg)
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
![Page 29: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/29.jpg)
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.
![Page 30: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/30.jpg)
Codimension Two Case
Potentials
![Page 31: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/31.jpg)
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:
![Page 32: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/32.jpg)
Scalar PerturbationsScalar Harmonics Definition
Harmonic vectors
Harmonic tensors
Exceptional modes
![Page 33: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/33.jpg)
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,…
![Page 34: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/34.jpg)
Scalar Perturbation
Linear Perturbations Harmonic expansion
Gauge transformationsFor the scalar-type gauge transformation
the perturbation variables transform as
![Page 35: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/35.jpg)
Linear Perturbations
Gauge invariantsFrom the gauge transformation law
we find the following gauge-invariant combinations.
![Page 36: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/36.jpg)
Linear Perturbations
Einstein equations G
![Page 37: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/37.jpg)
Linear Perturbations
Gai :
Tracefree part of Gij :
Gii :
![Page 38: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/38.jpg)
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
![Page 39: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/39.jpg)
Codimension Two Case
Potentials
![Page 40: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/40.jpg)
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.
![Page 41: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/41.jpg)
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
![Page 42: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/42.jpg)
Summary
![Page 43: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/43.jpg)
Chapter 3
Application to Other Systems
![Page 44: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/44.jpg)
Flat Black Branes
ASS4Lecture.dvi
![Page 45: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/45.jpg)
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.
![Page 46: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/46.jpg)
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]
![Page 47: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/47.jpg)
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.
![Page 48: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/48.jpg)
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.
![Page 49: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/49.jpg)
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]
![Page 50: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/50.jpg)
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
![Page 51: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/51.jpg)
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.
![Page 52: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/52.jpg)
C-metric
Schwarzschild Limit G(x) can be factorised as
where
In terms of the variables
the C-metric can be written
where
![Page 53: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/53.jpg)
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
![Page 54: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/54.jpg)
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
![Page 55: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/55.jpg)
Global structure
0<x · x2 x1 · x <0
x=0
![Page 56: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/56.jpg)
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)
![Page 57: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/57.jpg)
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.
![Page 58: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/58.jpg)
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
![Page 59: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/59.jpg)
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.
![Page 60: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/60.jpg)
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
![Page 61: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/61.jpg)
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
![Page 62: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/62.jpg)
Solution
The original perturbation variables are expressed as
where
![Page 63: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/63.jpg)
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.
![Page 64: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/64.jpg)
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
![Page 65: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/65.jpg)
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.
![Page 66: Perturbations and Stability of Higher-Dimensional Black Holes](https://reader036.vdocuments.mx/reader036/viewer/2022062520/568157e7550346895dc56271/html5/thumbnails/66.jpg)
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.