perturbation theory of black holes - ulisboablackholes/nrhep/... · 2011-09-04 · static black...
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Perturbation theory of black holes
Akihiro Ishibashi
( Kinki University, Osaka )
NR-HEP workshop 3 Sep. 2011 Madeira
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Introduction
Perturbation analysis (analytical/numerical)
• GW emission from a particle (plunging into or orbiting) around a BH
• Stability problemStable final state of gravitational collapse
Unstable New branch of solutions
• Information about the geometry: Quasi-Normal Modes • Insights into Uniqueness/non-uniqueness• Attempt to find new, approximate solutions
(by deforming an existing solution)
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Two major issues when formulating perturbation theory
• Fixing gauge ambiguity
Imposing suitable gauge conditions
or
constructing manifestly gauge-invariant variables
• Reduction of perturbation equations to a simple, tractable form (Master equations)
classifying perturbation variables into (to be) mutually decoupled groups
separating variables
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Master equations
4D Static asymptotically flat vacuum case Regge-Wheeler 57
Zerilli 70
charge case Moncrief
-- Stability Regge-Wheeler 57, Veshveshwara 70 …
asymptotically AdS/dS case Cardoso-Lemos
--- set of decoupled self-adjoint ODEs
Stationary Rotating vacuum (Kerr) case Teukolsky 72
--- Stability Press-Teukolsky 73 --- Whihting 89 …
Talk by Ferrari
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4D classification (Uniqueness)
• BH Thermodynamics:
Stationary black holes obey equilibrium thermodynamic law (i.e. 1st law)
described by merely a small numbers of parameters
(Israel-Carter-Robinson-Mazur-Bunting-Chrusciel)
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Physical implications
Uniqueness + Stability + Cosmic Censorship
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• D>4 General Relativity
No uniqueness like 4D GR
Many unstable black (rotating) objects
Classification Problem in Higher Dimensions
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Stability results so far
• 4D Static: vacuum, charged, Λ stableRotating: vacuum (Kerr) stable
Λ(<0) (Kerr-AdS) unstable ( )
• D>4 Static: vacuum stable charged (D=5…11) w. Λ can be unstable
Rotating: many unstable cases
Gregory-Laflamme instability Superradiant instability
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• Rotating BH case Not separable in general (Talk by Godazgar)still a long way from having a complete perturbation theory
Progress in some special cases cohomogeneity-one (odd-dim. )
Kunduri-Lucietti –Reall 07, Murata-Soda 08 Kundt spacetimes (e.g. Near-horizon geom)
Durkee-Reall 11
Numerical approach e.g. Talks by Shibata, Dias,
• Static BH case simpler and tractable: -- can reduce to a set of decoupled ODEs
Perturbations in Higher Dimensional General Relativity
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In this talk
-- focus on Static black holes in General Relativity
-- discuss Gauge-invariant formulation of linear perturbations (metric pert. Approach)
Master equations
Stability analysis
Generalizations
Open issues
See. e.g. latest review AI & Kodama 1103.6148
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Background geometry
: m – dim spacetime metric
: n – dim Einstein metric
e.g. n-sphere
This metric describe a fairly generic class of metrics
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FLRW universe
Static black hole
Myers-Perry black hole ( w/ single rotation)
Black-brane
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Cosmological perturbation theory
: scale factor
: homogeneous isotropic time-slice
: FLRW background metric
Perturbations are decomposed into 3 types according to its tensorial behaviour on time-slice
Tensor-type: transverse-traceless ( possible only when ) Gravitational Waves
Vectro-type: div-free vector couple to matter e.g. velocity perturbations
Scalar-type: scalar couple to matter e.g. density perturbations
Gauge-invariant formulation Bardeen 80 Kodama-Sasaki 84
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Brane-world cosmology
• AdS - (Black Hole)-Bulk spacetime
• Brane-world
Bulk perturbations induce brane-world cosmological perturbations--- need to develop a formula for AdS-Black Hole perturbations --- convenient to decompose bulk perturbations into
Tensor-, Vector-, Scalar-type wrt
Kodama – AI – Seto ‘00
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Background geometry
charge
ADM-mass
Cosmological constant
Static solutions of Einstein-Maxwell + cosmological constantin
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Basic strategy to derive master equations
(1) Mode-decompose as
Tensor-type new component in D > 4 case
Vector-type axial - mode in D = 4 case
Scalar-type polar - mode in D = 4 case
(2) Expand by tensor harmonics defined on
(3) Write the Einstein equations in terms of the expansion coefficientsin 2-dim. spacetime spanned by
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Decomposition theorem
(i) Any (dual) vector field on compact Riemaniann manifold can be uniquely decomposed as
(ii) Any 2nd-rank symmetric tensor on compact Einstein manifold
can be uniquely decomposed as
Metric perturbations : decomposed as
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Tensor-type perturbations
• : Transverse-Traceless harmonic tensor on
• is a gauge-invariant variable
• Einstein’s equations reduce to Master equation
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Vector-type perturbations
• : Div.-free vector harmonics on :
• Gauge-invariant variable:
• Einstein’s equations reduce to
*
There exists such that
Einstein’s equation reduces to Master equation
-- corresponds to the Regge-Wheeler equation in 4D
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Scalar-type perturbations
• Construct gauge-invariant variables: on
• After Fourier transf. wrt ‘ ’ Einstein’s equations reduce to
--- such a system can be reduced to a single wave equation
• For a certain linear combination of
-- corresponds to the Zerilli equation in 4D
• Expand by scalar harmonics on :
• Set of 1st –order ODEs for • A linear algebraic relation among them
Einstein’s equations reduce to
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Thus, for each type of perturbations, we obtain a single master wave equation :
: effective potential
• Degrees of freedom:
Tensor harmonics : -- indpdt. components ( )
Vector harmonics : *** ( )
Scalar harmonics : 1 ***
Total components
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• 4D case:
Intertwining between vector (axial) and scalar (polar) perturbations
• D>4 case:
No such intertwining among Tensor-, Vector-, and Scalar-type perturbations
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Stability analysis
• Master equation takes the form:
If “ ” is a positive self-adjoint operator, the master equation does not admit “unstable” solutions
--- The black hole is stable
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Boundary conditions
• Asymptotically flat case:
at infinity
at Horizon
at
infinity
Horizon (-- can be removed)
Show positivity of under the boundary condition
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If stable wrt pert. w/ Dirichlet condition at Horizon, then stable wrt pert. on extended Schwarzschild (Kruskal) spacetime
Kay-Wald 87
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Stability wrt Tensor-type
Stable
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Stability wrt Scalar-type
• Not obvious to see whether is positive or not …
The potential is NOT positive definite in D > 4
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Stability proof
• Define w. some function
where
Boundary terms vanish under our Dirichlet conditions
Task: Find that makes positive definite
Then, is uniquely extended to be a positive self-adjoint operator
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“OK” “Stable”
When the horizon manifold is maximally symmetric
WRT Tensor- and Vector-perturbations Stable over entire parameter range
WRT Scalar-perturbations ??? when
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Numerical study
• Schwarzschild-de Sitter in stable
• Charged (RN) de Sitter can be unstable in
when and large enough
Konoplya-Zhidenko 07 09
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Potential for Scalar-type pert. w. non-vanishing ,
Not fully studied yet
For extremal and near-extremal case, the potential becomes negative in the immediate vicinity of the horizon
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Extremal BHs
• No bifurcate surface: Kay-Wald’s theorem concerning boundary conditions at horizon does not apply
What boundary conditions are appropriate at near-horizon throat?
• The effective potential has a negative ditch near the horizon
• Numerical (QNMs) study indicates instability
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Some generalizations and open problems
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Static black holes in Lovelock theory
• Master equations in generic Lovelock theory Takahashi – Soda 10
in Gauss-Bonnet theory Dotti – Gleiser 05
• Asymptotically flat, small mass BHs are unstable wrtTensor-type perturbations (in even-dim.) Scalar-type perturbations (in odd-dim.)
• Instability is stronger in higher multipoles rather than low-multipoles
Higher curvature terms involved
Equations of motion contain only up to 2nd-order derivatives
If , then for sufficiently large
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c. f. Cohomogeneity-1 Myers-Perry BHs
( not in the class of metrics of )
Talk by Dias
Kunduri-Lucietti –Reall 07, Murata-Soda 08
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Rotating case: Cohomogeneity-2 Myers-Perry BHs
Numerical approach to stability analysis Talks by Shibata, Dias
--- include the ultra-spinning case
Talk by Dias
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Cohomogeneity-2 MP case: Analytic formulation?
Tensor-type perturbations: A single master scalar variable on
satisfy the same equation for a massless Klein-Gordon field
How about vector-type and scalar-type perturbations?
KK-reduction along Equations for massive vector/ tensor fields on : 4-dim. Kerr –type spacetime
For vector-type: 3 master scalar variables on
For scalar-type: 6 master scalar variables on ( )
Not known how to (analytically) deal with even in the standard 4-dim. Kerr background
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Metric perturbation approach in
Tensor-type: 1 master scalar variable
Vector-type: gauge-invariant scalar variables
Scalar-type: gauge-invariant scalar variables
--- intricately coupled on
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: Gauge-invariant variables in
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Perturbation analysis and Classification problem
• In 4D classification: Uniqueness + Stability of the Kerr metric important (astro-)physical implications
Uniqueness Theorem was established by assembling
Topology results, Symmetry (Rigidity) results, etc. and all comp. of the Einstein equations all together
• How about D>4 classification?
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Role of symmetry in Stability problem
• Stability of an extremal black hole and its near-horizon geometry
Examine perturbations of the near-horizon geometry that respect the symmetry of the full BH solution
• Symmetric perturbation at Topology changing point
c.f. critical behaviorand critical exponent
Durkee - Reall 11
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Summary: linear perturbation
• Static BHs: Complete formulation for perturbations
• Rotating BHs: -- Still a long way from having a complete formulation
-- Considerable progress recently made for some special cases
• Interesting: Interplay between Symmetry /Topology properties
and Perturbation analysis