1. Bin Wang Fudan University Shanghai, China Perturbations around Black Holes

2. Outline  Perturbations in Asymptotically flat spacetimes  Perturbations in AdS spacetimes  Perturbation behaviors in SAdS, RNAdS etc. BH backgrounds  Testing ground of AdS/CFT, dS/CFT correspondence  QNMs and black hole phase transition  Detect extra dimension from the QNMs  Conclusions and Outlook

3. 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.

4.  As this gas forms a flattened disk, it swirls toward the companion. Friction caused by collisions between the particles in the gas heats them to extreme temperatures and they produce X-rays that flicker or vary in intensity within a second. Many bright X-ray binary sources have been discovered in our galaxy and nearby galaxies. In about ten of these systems, the rapid orbital velocity of the visible star indicates that the unseen companion is a black hole. (The figure at left is an X-ray image of the black hole candidate XTE J1118+480.) The X-rays in these objects are produced by particles very close to the event horizon. In less than a second after they give off their X-rays, they disappear beyond the event horizon.galaxy

5. 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.

6. Quasinormal Modes Why it is called QNM?  They are not truly stationary, damped quite rapidly  They seem to appear only over a limited time interval, NMs extending from arbitrary early to late time. What’s the difference between QNM of BHs and QNM of stars?  Stars: fluid making up star carry oscillations, Perturbations exist in metric and matter quantities over all space of star  BH: No matter could sustain such oscillation. Oscillations essentially involve the spacetime metric outside the horizon.

7. 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/9909058

8. 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

9. 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

10. The perturbation equations  There are two classes of tensor spherical harmonics (polar and axial). The differences are their parity under space inversion. Function acquires a factor refering to polar perturbation, and axial with a factor  The radial component of perturbation outside the BH satisfy

11. The perturbation equations  For axial perturbation :  For polar perturbation:

12. The perturbation equations  The perturbation is described by Incoming wave transmitted reflected wave wave

13. 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

14. Main results of QNM in asymptotically flat spacetimes

15. Tail phenomenon of a time- dependent case  Hod PRD66,024001(2002) V(x,t) is a time-dependent effective curvatue potential which determines the scattering of the wave by background geometry

16. 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)

17. QNM in time-dependent background  M with t, ω i The decay of the oscillation becomes slower

18. QNM in time-dependent background  M ( ) with t, the oscillation period becomes longer (shorter)

19. Detectable by ground and space- based instruments Needs accurate waveforms produced by GR community Schutz, CQG(96)

20. 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)

21. QNM in Schwarzschild AdS BHs Horowitz et al PRD(99)  D-dimensional SAdS BH metric: R is the AdS radius, is related to the BH mass is the area of a unit d-2 sphere. The Hawking temperature is

22. 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

23. QNM in SAdS BHs  In the absence of the BH  In the absence of the BH, r* has only a finite range and solutions exist for only a discrete set of real w.  Once BH is added  Once BH is added, w may have any values.  Definition of QNM in AdS BHs  Definition of QNM in AdS BHs: QNMs are defined to be modes with only ingoing waves near the horizon. Exists for only a discrete set of complex w We want modes with behavior near the horizon

24. QNM in SAdS BHs  It is convenient to set and work with the ingoing Eddington coordinates. Radial wave equation reads We wish to find the complex values of w such that Eq. has a solution with only ingoing modes near the horizon and vanishing at infinity.

25. QNM in SAdS BHs - Results  For large BH (r+>>R) , r+. Additional symmetry: depend on the BH T (T~r+/R^2)  For intermediate & small BH do not scale with the BH T r+ 0, ∝

26. 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

27. QNM in RN AdS BHs  Besides r+, R, it has another parameter Q. It possesses richer physics to be explored.  In the extreme case,

28. QNM in RN AdS BH  Consider the massless scalar field obeying Using, the radial function satisfies where

29. QNM in RN AdS BH  Solving the numerical equation Price et al PRD(1993) Wang, Lin, Molina, PRD(2004)

30. QNM in RN AdS BH - Results  With additional parameter Q, neither nor linearly depend 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

31. QNM in RN AdS BH - Results  Q>Qc: 0  Q>Qc: changes from increasing to decreasing Exponential decay Q Qmax Power-law decay

32. QNM in RN AdS BH - Results  Higher modes: Asymptotically flat spacetime const., while with large With some (not clear yet) correspondence between classical and quantum states, assuming this constant just the right one to make LQG give the correct result for the BH entropy. Whether such kind of coincidence holds for other spacetimes? In AdS space ? For the same value of the charge, both real and imaginary part of QN frequencies increases with the overtone number n. Hod. PRL(98)

33. QNM in RN AdS BH - Results  Higher modes: For the large black hole regime the frequencies become evenly spaced for high overtone number n.  For lowly charged RNAdS black hole, choosing bigger values of the charge, the real part in the spacing expression becomes smaller, while the imaginary part becomes bigger. Call for further Understanding from CFT?

34. QNM in BH with nontrivial topology Wang, Abdalla, Mann, PRD(2003)

35. Quasi normal modes in AdS topological Black Holes QNM depends on curvature coupling & spacetime topology

36. Support of (A)dS/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

37. QNM in 2+1 dimensional BTZ BH  General Solution where J is the angular momentum

38. QNM in 2+1 AdS BH  For the AdS case Exact agreement: QNM frequencies & location of the poles of the retarded correlation function of the corresponding perturbations in the dual CFT A Quantitative test of the AdS/CFT correspondence. [Birmingham et al PRL(2002)]

39. 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)

40. 2+1-dimensional dS spacetime The metric of 2+1-dimensional dS spacetime is: The horizon is obtained from

41. Perturbations in the dS spacetimes Scalar perturbations is described by the wave equation Adopting the separation The radial wave equation reads

42. Perturbations in the dS spacetimes Using the Ansatz The radial wave equation can be reduced to the hypergeometric equation

43. Perturbations in the dS spacetimes  For the dS case

44. Perturbations in the dS spacetimes Investigate the quasinormal modes from the CFT side: For a thermodynamical system the relaxation process of a small perturbation is determined by the poles, in the momentum representation, of the retarded correlation function of the perturbation

45. Perturbations in the dS spacetimes Define an invariant P(X,X ’ )associated to two points X and X ’ in dS space The Hadamard two-point function is defined as Which obeys

46. Perturbations in the dS spacetimes We obtain where The two point correlator can be got analogously to hep-th/0106113; NPB625,295(2002)

47. Perturbations in the dS spacetimes Using the separation : The two-point function for QNM is

48. 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 Abdalla et al PRD(02)

49. 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)

50. QNM – way to detect extra dimensions  QNM behavior: 4D: The late time signal-simple power-law tail Black String: High frequency signal persists

51. QNM – way to detect extra dimensions Brane-world BH – Read Extra Dimension:  Hawking Radiation? -LHC  QNM? –GW Observation? (Chen&Wang PLB07) (Shen&Wang PRD06) Black String Stability (Thermodynamical =?Dynamical)

52. QNM-black hole phase transition  Topological black hole with scalar hair

53. QNM-black hole phase transition Can QNMs reflect this phase transition? Martinez etal, PRD(04)

54. QNM-black hole phase transition  Perturbation equation MTZ TBH Above critical valueBelow critical value Koutsoumbas et al(06), Shen&Wang(07)

55. QNM-black hole phase transition  ADS BLACK HOLES WITH RICCI FLAT HORIZONS ON THE ADS SOLITON BACKGROUND AdS BH with Ricci flat horizon AdS soliton Flat AdS BH perturbation equation DECAY Modes AdS Soliton perturbation equation NORMAL Modes Hawking-Page transition Surya et al PRL(01) Shen & Wang(07) Question: Ricci flat BH and Hawking-Page phase Transition in GB Gravity&dilaton Gravity Cai, Kim, Wang(2007)

56. 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 test BHs’ phase transition  More??

57. Thanks!