Perturbations around Black Holes

Bin Wang Fudan UniversityShanghai, China

Outline

Perturbations in Asymptotically flat spacetimes Perturbations in AdS spacetimes

Testing ground of AdS/CFT, dS/CFT correspondence

Perturbations- way to see extra dimension way to relate dynamics and thermodynamics

Conclusions and Outlook

Searching for black holes

Study X-ray binary systems. These systems consist of a visible star in close orbit around an invisible companion star which may be a neutron star or black hole. The companion star pulls gas away from the visible

star.

Do black holes have a characteristic “sound”?

Yes.Yes.

During a certain time interval the evolution of initial perturbation is dominated by damped single-frequency oscillation.

Relate to black hole parameters, not on initial perturbation.

IR i

Wave dynamics in the asymptotically flat space-time

Schematic Picture of the wave evolution: Shape of the wave front (Initial Pulse) Quasi-normal ringingUnique fingerprint to the BH existenceDetection is expected through GW observation RelaxationK.D.Kokkotas and B.G.Schmidt, gr-qc/9909058B.Wang, gr-qc/0511133

The perturbation equations Introducing small perturbation

In vacuum, the perturbed field equations simply reduce to

These equations are in linear in h

For the spherically symmetric background, the perturbation is forced to be considered with complete angular dependence

The perturbation equations

Different parts of h transform differently under rotations

“S” transform like scalars, represented by scalar spherical harmonics

Vectors and tensors can be constructed from scalar functions

The perturbation equations

The perturbation is described by

Incoming wave

transmitted reflected wave wave

The perturbation equations

For axial perturbation:

For polar perturbation:

Main results of QNM in asymptotically flat spacetimes

ωi always positive damped modes The QNMs in BH are isospectral (same ω for different perturbations eg axial or polar)

This is due to the uniqueness in which BH react to a perturbation

(Not true for relativistic stars)

Damping time ~ M (ωi,n ~ 1/M), shorter for higher-order modes (ωi,n+1 > ωi,n)

Detection of GW emitted from a perturbed BH direct measure of the BH mass

Main results of QNM in asymptotically flat spacetimes

QNM in time-dependent background

Vaidya metric

In this coordinate, the scalar perturbation equation is

Where x=r+2m ln(r/2m-1) […]=ln(r/2m -1)-1/(1-2m/r)

Xue, Wang, Abdalla MPLA(02)Shao, Wang, Abdalla, PRD(05)

QNM of BH absorbing DE

With the accretion of DE onto the BH

Babichev et al, PRL (2004)

In the universe filled with DE modeled as scalar field, the action has the form

Varying the action with respect to

where the ′+′ sign describes the the phantom field while the ′−′ sign describes the quintessence field X.He, B.Wang et al,

PLB(09)

QNM of BH absorbing DE

The QNM results discussed here are

sufficient to illustrate the possibility

to distinguish whether DE lies

above or below the w=-1

Late time evolution of the perturbationThe black hole does not disappear peacefully, it will explode after getting enough phantom energy. The result is consistent with the Big Rip scenario.

X.He, B.Wang et al, PLB(09)

Quasi-normal modes in AdS space-time

AdS/CFT correspondence:A large static BH in AdS spacetime corresponds to an

(approximately) thermal state in CFT.

Perturbing the BH corresponds to perturbing this thermal state, and the decay of the perturbation describes the return to thermal equilibrium.

The quasinormal frequencies of AdS BH have direct interpretation in terms of the dual CFT

J.S.F.Chan and R.B.Mann, PRD55,7546(1997);PRD59,064025(1999)G.T.Horowitz and V.E.Hubeny, PRD62,024027(2000);CQG17,1107(2000)B.Wang et al, PLB481,79(2000);PRD63,084001(2001);PRD63,124004(2001);

PRD65,084006(2002)

QNM in SAdS BHs The minimally coupled scalar wave equation

If we consider modes

where Y denotes the spherical harmonics on

The wave equations reads

QNMs are defined to be modes with only ingoing waves near the horizon.

QNM in SAdS BHs - Results For large BH (r+>>R) , r+. Hubeny, Horowitz PRD(99)

Additional symmetry: depend on the BH T (T~r+/R^2)

For intermediate & small BH do not scale with the BH T

r+ 0,

∝

QNM in SAdS BHs - Results SBH has only one dimensionful parameter-T must be multiples of this T Small SAdS BH do not behave like SBHs Decay at very late time SBH: power law tail SAdS BH: exponential decay Reason:Reason: The boundary conditions at infinity are changed. Physically, the late time behavior of the field is affected

by waves bouncing off the potential at large r

QNM in RN AdS BHs

Besides r+, R, it has another parameter Q. It possesses richer physics to be explored.

In the extreme case,

QNM in RN AdS BH - Results With additional parameter Q, neither nor

linearly depends on r+ as found in SAdS BH. For not big Q: Q , ,

If we perturb a RNAdS BH with high Q, the

surrounding geometrywill not ring as much and as

long as that of BH with small Q

QNM in RN AdS BH - Results Q>Qc: 0

Q>Qc: changes from increasing to decreasing

Exponential decay

Q Qmax

Power-law decay

QNM in BH with nontrivial topology

Wang, Abdalla, Mann, PRD(2002)

Quasi normal modes in AdS topological Black Holes

QNM depends on curvature coupling & spacetime topology

Support of AdS/CFT from QNM

AdS/CFT correspondenceThe decay of small perturbations of a BH at

equilibrium is described by the QNMs.

For a small perturbation, the relaxation process is completely determined by the poles, in the momentum representation, of the retarded correlation function of the perturbation.

？QNMs in AdS BH Linear response theory in FTFT [Birmingham et al PRL(2002)]

Perturbations in the dS spacetimes

We live in a flat world with possibly a positive cosmological constant

Supernova observation, COBE satellite

Holographic duality: dS/CFT conjecture A.Strominger, hep-th/0106113

Motivation: Quantitative test of the dS/CFT conjecture E.Abdalla, B.Wang et al, PLB (2002)

Perturbations in the dS spacetimes

The poles of such a correlator corresponds exactly to the QNM obtained from the wave equation in the bulk.

These results provide a quantitative test of the dS/CFT correspondence

This work has been extended to four-dimensional

dS spacetimes E.Abdalla, B.Wang et al, PLB (2002) E. Abdalla et al PRD(02)

QNM – way to detect extra dimensions

String theory makes the radial prediction:Spacetime has extra dimensionsGravity propagates in higher dimensions.

Maarten et al (04)

QNM – way to detect extra dimensions

QNM behavior:

4D: The late time signal-simple power-law tail

Black String: High frequency signal persists

Other investigations on QNM to detect extra dimensional effects:

Songbai Chen, Bin Wang, Ru-Keng Su Physics Letters B 647, 282 (2007) Masato Nozawa, Tsutomu Kobayashi, Phys. Rev. D 78, 064006 (2008) Usama A. al-Binni, George Siopsis, arXiv:0708.3363 S.B.Chen & B. Wang, PRD(08)

Stability of black string, black ring etc.

Gregory etal, PRL(93), Hirayama & Kang, PRD(01)B. Wang et al, PRD(08)

Black hole phase transition:

EBH can be got from NEBH through phase transitionS=A/4

QNM-way to see phase transition

QNM-way to see phase transition

Koutsoumbas, Musiri, Papantonopoulos, Siopsis , JHEP(06)

Shen, Wang, Lin, Cai, Su, JHEP(07)

QNM-way to relate dynamics and thermodynamics

Davis’s point of heat capacity: e.g.RN

The heat capacity diverges when Q -> Qc: thermal instability

Whether this thermal instability has some dynamical signature?

Reflected by QNM?

QNM-way to relate dynamics and

thermodynamics

Q~Qc, Re(w),Im(w) start to have oscillations, and the complex w plan start to exhibit spiral-like shape.

For RN BH, Jing, Pan, PLB(08)For charged KK BH with squashed horizon, X. He and B.Wang et al, PLB(08)

QNM-way to relate dynamics and

thermodynamicsStability of the BTZ black string against fermonic perturbation

and gravitational perturbation : The BTZ black string can be dynamically stable provided that

the , which is determined by the compactification of the extra dimension, is over a threshold value.

The BTZ black string can be unstable and pinch-off to form a black hole if is smaller than this threshold value .

The BTZ black string is not a privileged stable phase.

Agrees with thermodynamical argument (Emparan, Horowitz, Myers, JHEP (2000)

:L.Liu, B.Wang, PRD(08)

Conclusions and Outlook Importance of the study in order to foresee gravitational

waves accurate QNM waveforms are needed

QNM in different stationary BHs QNM in time-dependent spacetimes QNM around colliding BHs

Testing ground of Relation between AdS space and Conformal Field Theory Relation between dS space and Conformal Field Theory

Possible way to detect extra-dimensions Possible way to relate dynamics to thermodynamics……

Thanks!