Black holes’ stability: A review R. A. Konoplya DAMTP, University of Cambridge, UK Tokyo, Nov. 11 - Nov. 16, 2012 the 60th birthday of T. Futamase, H. Kodama, M. Sasaki
Content: Recent reviews on stability of BHs: in D > 4 A. Ishibashi, H. Kodama, Prog. Theor. Phys. Suppl. 189 (2011) 165-209 D ≥ 4 R. A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83 (2011) 793-836 Introduction
Content: Recent reviews on stability of BHs: in D > 4 A. Ishibashi, H. Kodama, Prog. Theor. Phys. Suppl. 189 (2011) 165-209 D ≥ 4 R. A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83 (2011) 793-836 Introduction Criteria of stability: analytical vs numerical
Content: Recent reviews on stability of BHs: in D > 4 A. Ishibashi, H. Kodama, Prog. Theor. Phys. Suppl. 189 (2011) 165-209 D ≥ 4 R. A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83 (2011) 793-836 Introduction Criteria of stability: analytical vs numerical (In)stability of 3+1 dimensional BHs
Content: Recent reviews on stability of BHs: in D > 4 A. Ishibashi, H. Kodama, Prog. Theor. Phys. Suppl. 189 (2011) 165-209 D ≥ 4 R. A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83 (2011) 793-836 Introduction Criteria of stability: analytical vs numerical (In)stability of 3+1 dimensional BHs Instability of D > 4 BHs: Gregory-Laflamme instability and not only
Content: Recent reviews on stability of BHs: in D > 4 A. Ishibashi, H. Kodama, Prog. Theor. Phys. Suppl. 189 (2011) 165-209 D ≥ 4 R. A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83 (2011) 793-836 Introduction Criteria of stability: analytical vs numerical (In)stability of 3+1 dimensional BHs Instability of D > 4 BHs: Gregory-Laflamme instability and not only Potential turbulent instabilities
Content: Recent reviews on stability of BHs: in D > 4 A. Ishibashi, H. Kodama, Prog. Theor. Phys. Suppl. 189 (2011) 165-209 D ≥ 4 R. A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83 (2011) 793-836 Introduction Criteria of stability: analytical vs numerical (In)stability of 3+1 dimensional BHs Instability of D > 4 BHs: Gregory-Laflamme instability and not only Potential turbulent instabilities Conclusions We shall discuss mainly (but not only) linear dynamical (in)stabilities
Motivations: Two main motivations to study gravitational stability of black holes:
Motivations: Two main motivations to study gravitational stability of black holes: • Criterium of existence (in D = 4 for alternative theories of gravity and in D > 4 owing to absence of uniqueness)
Motivations: Two main motivations to study gravitational stability of black holes: • Criterium of existence (in D = 4 for alternative theories of gravity and in D > 4 owing to absence of uniqueness) • gauge-gravity duality (instability corresponds to the phase transition in the dual theory)
Motivations: Two main motivations to study gravitational stability of black holes: • Criterium of existence (in D = 4 for alternative theories of gravity and in D > 4 owing to absence of uniqueness) • gauge-gravity duality (instability corresponds to the phase transition in the dual theory) • scenarios with extra dimensions (though experimental data on LHC gives no optimism: no large total transverse energy so far at 8 TEV: CMS collaboration claims that semiclassical BHs with mass below 6.1 TeV are excluded)
From linearized perturbations to a master wave equation • Step 1: Perturbations can be written in the linear approximation in the form g µν = g 0 µν + δ g µν , (1) � 1 � 2Λ δ R µν = κ δ T µν − D − 2 Tg µν + D − 2 δ g µν . (2) Linear approximation means that in Eq. (2) the terms of order ∼ δ g 2 µν and higher are neglected. The unperturbed space-time given by the metric g 0 µν is called the background.
From linearized perturbations to a master wave equation • Step 1: Perturbations can be written in the linear approximation in the form g µν = g 0 µν + δ g µν , (1) � 1 � 2Λ δ R µν = κ δ T µν − D − 2 Tg µν + D − 2 δ g µν . (2) Linear approximation means that in Eq. (2) the terms of order ∼ δ g 2 µν and higher are neglected. The unperturbed space-time given by the metric g 0 µν is called the background. • Step 2: decomposition of the perturbed space-time into scalar, vector and tensor parts
From linearized perturbations to a master wave equation • Step 1: Perturbations can be written in the linear approximation in the form g µν = g 0 µν + δ g µν , (1) � 1 � 2Λ δ R µν = κ δ T µν − D − 2 Tg µν + D − 2 δ g µν . (2) Linear approximation means that in Eq. (2) the terms of order ∼ δ g 2 µν and higher are neglected. The unperturbed space-time given by the metric g 0 µν is called the background. • Step 2: decomposition of the perturbed space-time into scalar, vector and tensor parts • Step 3: using the gauge invariant formalism (or fixing the gauge)
From linearized perturbations to a master wave equation • Step 1: Perturbations can be written in the linear approximation in the form g µν = g 0 µν + δ g µν , (1) � 1 � 2Λ δ R µν = κ δ T µν − D − 2 Tg µν + D − 2 δ g µν . (2) Linear approximation means that in Eq. (2) the terms of order ∼ δ g 2 µν and higher are neglected. The unperturbed space-time given by the metric g 0 µν is called the background. • Step 2: decomposition of the perturbed space-time into scalar, vector and tensor parts • Step 3: using the gauge invariant formalism (or fixing the gauge) • Step 4: Reducing the perturbation equations (after separation of angular variables) to a second order partial differential equation, termed master wave equation . For example, for static and some stationary BHs the master wave equation has the form: − d 2 R + V ( r , ω ) R = ω 2 R , (3) dr 2 ∗
Criteria of stability: analytical vs numerical • If the effective potential V eff in the wave equation (3) is positive definite, the differential operator A = − ∂ 2 + V eff (4) ∂ r 2 ∗ is a positive self-adjoint operator in the Hilbert space of square integrable functions L 2 ( r ∗ , dr ∗ ) . Then, there are no negative (growing) mode solutions that are normalizable, i. e., for a well-behaved initial data (smooth data of compact support), all solutions are bounded all of the time.
Criteria of stability: analytical vs numerical • If the effective potential V eff in the wave equation (3) is positive definite, the differential operator A = − ∂ 2 + V eff (4) ∂ r 2 ∗ is a positive self-adjoint operator in the Hilbert space of square integrable functions L 2 ( r ∗ , dr ∗ ) . Then, there are no negative (growing) mode solutions that are normalizable, i. e., for a well-behaved initial data (smooth data of compact support), all solutions are bounded all of the time. • Yet, in majority of cases A is not positive (negativeness of the effective potential in some regions, dependence of the potential on the complex frequencies ω )
Criteria of stability: analytical vs numerical • If the effective potential V eff in the wave equation (3) is positive definite, the differential operator A = − ∂ 2 + V eff (4) ∂ r 2 ∗ is a positive self-adjoint operator in the Hilbert space of square integrable functions L 2 ( r ∗ , dr ∗ ) . Then, there are no negative (growing) mode solutions that are normalizable, i. e., for a well-behaved initial data (smooth data of compact support), all solutions are bounded all of the time. • Yet, in majority of cases A is not positive (negativeness of the effective potential in some regions, dependence of the potential on the complex frequencies ω ) • Sometimes the situation can be remedied by the so-called S-deformation of the wave equation to the one with positive definite effective potential, in such a way that the lower bound of the energy spectrum does not change.
Criteria of stability: analytical vs numerical • If the effective potential V eff in the wave equation (3) is positive definite, the differential operator A = − ∂ 2 + V eff (4) ∂ r 2 ∗ is a positive self-adjoint operator in the Hilbert space of square integrable functions L 2 ( r ∗ , dr ∗ ) . Then, there are no negative (growing) mode solutions that are normalizable, i. e., for a well-behaved initial data (smooth data of compact support), all solutions are bounded all of the time. • Yet, in majority of cases A is not positive (negativeness of the effective potential in some regions, dependence of the potential on the complex frequencies ω ) • Sometimes the situation can be remedied by the so-called S-deformation of the wave equation to the one with positive definite effective potential, in such a way that the lower bound of the energy spectrum does not change. • Usually, it is difficult to find an ansatz for the S-deformation, so that numerical treatment of the wave equation is necessary.
• The numerical criteria of stability could be the evidence that all the proper oscillation frequencies of the black hole, termed the quasinormal modes are damped.
• The numerical criteria of stability could be the evidence that all the proper oscillation frequencies of the black hole, termed the quasinormal modes are damped. • Quasinormal modes are eigenvalues of the master wave equation with appropriate boundary conditions: purely ingoing waves at the horizon and purely outgoing waves at infinity or de Sitter horizon. For AdS BHs boundary condition at infinity is dictated by AdS/CFT and is usually the Dirichlet one Ψ = 0 , where Ψ is some gauge inv. combination.
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