Perturbations around Black Hole solutions

1. Perturbations around Black Hole solutions Elcio Abdalla

2. Classical (non-relativistic) Black Hole • The escape velocity is equal to the velocity of light • Therefore,

3. The Schwarzschild Black Hole • Birckhoff Theorem: a static spherically Symmetric solution must be of the form • Schwartzschild solution in D=d+1 dimensions (d>2):

4. Properties of the BH solution • Considering the solution for a large (not too heavy) cluster of matter (i.e. radius of distribution > 2M, G=1=c). In this case one finds the Newtonian Potential • For heavy matter (namely highly concentrated) with radius R< 2M there is a so called event horizon where the g00 vanishes. To an outsider observer, the subject falling into the Black Hole takes infinite time to arrive at R=2M

5. Properties of the BH solution • Only the region r> 2M is relevant to external observers. • Law of Black Hole dynamics: BH area always grows • Quantum gravity: BH entropy equals 1/4 of BH area • No hair theorem: BH can only display its mass (attraction), charge (Gauss law) and angular momentum (precession of gyroscope) to external observers

6. Reissner-Nordstrom solution • For a Black Hole with mass M and charge q, in 4 dimensions, we have the solution

7. Cosmological Constant • Einstein Equations with a nonzero cosmological constant are • Λ>0 corresponds to de Sitter space • Λ<0 corresponds to Anti de Sitter space

8. Lovelock Gravity

9. Lovelock Gravity

10. Black Holes with nontrivial topology

11. Black Holes with nontrivial topology

12. Black Holes with nontrivial topology

13. 2+1 dimensional BTZ Black Holes • General Solution where J is the angular momentum

14. 2+1 dimensional BTZ Black Holes • AdS space • where -l2 corresponds to the inverse of the cosmological constant Λ

15. Quasi Normal Modes • First discovered by Gamow in the context of alpha decay • Bell ringing near a Black Hole • Can one listen to the form of the Black Hole? • Can we listen to the form of a star?

16. Quasi normal modes expansion QNMs were first pointed out in calculations of the scattering of gravitational waves by Schwarzschild black holes. Due to emission of gravitational waves the oscillation mode frequencies become complex, the real part representingthe oscillation the imaginary part representing the damping.

17. 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

18. Excitation of the black hole oscillation • Collapse is the most frequent source for the excitation of BH oscillation. Many stars end their lives with a supernova explosion. This will leave behind a compact object which will oscillate violently in the first few seconds. Huge amounts of gravitational radiation will be emitted. • Merging two BHs • Small bodies falling into the BH. • Phase-transition could lead to a sudden contraction

19. Detection of QNM Ringing • GW will carry away information about the BH • The collapse releases an enormous amount of energy. • Most energy carried away by neutrinos. This is supported by the neutrino observations at the time of SN1987A. • Only 1% of the energy released in neutrinos is radiated in GW • Energy emitted as GW is of order

20. Sensitivity of Detectors • Amplitude of the gravitational wave for stellar BH for galactic BH Where E is the available energy, f the frequency and the r is the distance of the detector from the source. Anderson and Kokkotas, PRL77,4134(1996)

21. Sensitivity of Detectors • An important factor for the detection of gravitational wave consists in the pulsation mode frequencies. The spherical and bar detectors 0.6-3kHz The interferometers are sensitive within 10-2000kHz For the BH the frequency will depend on the mass and rotation: 10 solar mass BH 1kHz 100 solar mass BH 100Hz Galactic BH 1mHz

22. Quasi-normal modes in AdS space-time AdS/CFT correspondence: The BH corresponds to an approximately thermal state in the field theory, and the decay of the test field corresponds to the decay of the perturbation of the state. 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)

23. Quasi normal modes in RN AdS • We consider the metric

24. Quasi Normal Modes • We can consider several types of perturbations • A scalar field in a BH background obeys a curved Klein-Gordon equation • An EM field obeys a Maxwell eq in a curved background • A metric perturbation obeys Zerilli’s eq.

25. Quasi normal modes in RN AdS • We use the expansion

26. Quasi normal modes in RN AdS

27. Quasi normal modes in RN AdS • Decay constant as a function of the Black Hole radius

28. Quasi normal modes in RN AdS • Dependence on the angular momentum (l)

29. Quasi normal modes in RN AdS • Solving the numerical equation

30. Quasi normal modes in RN AdS • Solving the numerical equation

31. Quasi normal modes in RN AdS • Result of numerical integration

32. Quasi normal modes in RN AdS • Approaching criticality

33. Quasi normal modes in AdS topological Black Holes

34. Quasi normal modes in AdS topological Black Holes

35. Quasi normal modes in AdS topological Black Holes

36. Quasi normal modes in AdS topological Black Holes

37. Quasi normal modes in AdS topological Black Holes

38. Quasi normal modes in AdS topological Black Holes

39. Quasi normal modes in AdS topological Black Holes

40. Quasi normal modes in AdS topological Black Holes

41. Quasi normal modes in AdS topological Black Holes

42. Quasi normal modes in AdS topological Black Holes

43. Quasi normal modes in AdS topological Black Holes

44. Quasi normal modes in 2+1 dimensions • For the AdS case

45. Quasi normal modes in 2+1 dimensional AdS BH 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.

46. Perturbations in the dS spacetimes We live in a flat world with possibly a positive cosmological constant Supernova observation, COBE satellite Holographic duality: dS/CFTconjecture A.Strominger, hep-th/0106113 Motivation: Quantitative test of the dS/CFT conjecture E.Abdalla,B.Wang et al, PLB 538,435(2002)

47. Perturbations in dS spacetimes • Small dependence on the charge of the BH • Characteristic of space-time (cosmological constant)

48. 2+1-dimensional dS spacetime The metric of 2+1-dimensional dS spacetime is: The horizon is obtained from

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

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