Perturbations around Black Holes Bin Wang Fudan University Shanghai, 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.

Wave dynamics in the asymptotically flat space-time Schematic Picture of the wave evolution:  Shape of the wave front (Initial Pulse)  Quasi-normal ringing Unique fingerprint to the BH existence Detection is expected through GW observation  Relaxation K.D.Kokkotas and B.G.Schmidt, gr-qc/ B.Wang, gr-qc/

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  ω i 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 (ω i,n ~ 1/M), shorter for higher- order modes (ω i,n+1 > ω i,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 perturbation The 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: 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 geometry will 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 correspondence The 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/ 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 dimensions Gravity 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 ChenSongbai Chen, Bin Wang, Ru-Keng Su Physics Letters B 647, 282 (2007)Bin WangRu-Keng Su Masato NozawaMasato Nozawa, Tsutomu Kobayashi, Phys. Rev. D 78, (2008)Tsutomu Kobayashi Usama A. al-BinniUsama A. al-Binni, George Siopsis, arXiv: George SiopsisarXiv: 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 transition S=A/4 QNM-way to see phase transition

KoutsoumbasKoutsoumbas, Musiri, Papantonopoulos, Siopsis, JHEP(06)Musiri PapantonopoulosSiopsis Shen, Wang, Lin, Cai, Su, JHEP(07)ShenWangLinCaiSu

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 thermodynamics Stability 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!