Quantization of Entropy spectra of black holes ( 黑洞谱熵量子化 )

Transcription

1 Quantization of Entropy spectra of black holes ( 黑洞谱熵量子化 ) Soonkeon Nam ( 南淳建 ) Kyung Hee University ( 庆熙大 ) (In collaboration with Yongjoon Kwon) , Hefei (Based on Class.Quant.Grav.27: (2010), (2010) & Class.Quant.Grav.28: (2011) )

2 Quantum Gravity ( 量子引力理论 ) We do not have a definite theory of quantum gravity yet. So we would like to go back to the period of early quantum physics, before the birth of Quantum Mechanics and Schoedinger equation. - We had Bohr s quantization, Sommerfeld s semiclassical quantization, and many more... We would like to be in that mind set, for the quantization of Black hole spectra. 2

3 0. Pioneering work in black hole quantization J.D.Bekenstein 1974, See also [arxiv: gr-qc/ ] Q) What is an adiabatic invariant? A) Horizon area of black hole How is it quantized? where is an undetermined dimensionless constant

4 Quantization of black hole spectrum Demystifying BH s entropy proportional to area The quantization of horizon area in equal steps brings to mind an horizon formed by patches of equal area This patchwork horizon can be regarded as having many degrees of freedom, one for each patch. Since the patches are all equivalent, each will have the same number of quantum states, k. Provided k is regarded as an effective number of equally probable states

5 Quantization Adiabatic Invariant : A physical quantity that is almost constant when changes are made very slowly (applied to Plasma physics..)-chandrasekhar Ehrenfest Theorem : Any classical adiabatic invariant corresponds to a quantum entity with discrete spectrum Bekenstein (1974) : Black hole horizon area can be regarded as an adiabatic invariant 5

6 6

7 So the horizon area is an adiabatic invariant. 7

8 Ⅰ. Quasinormal modes S.Hod Black hole has the characteristic modes called quasinormal modes. [arxiv: gr-qc/ ] Based on Bohr s correspondence principle, when, the real part of the asymptotic quasinormal modes can be regarded as quantum transition frequency in the semiclassical limit. Energy of the quantum emitted by BH changes BH mass For Schwarzschild BH,

9 9

10 10

11 11

12 12

13 13

14 14

15 15

16 Perturbation around a black hole : Schroedinger like wave equation with complex frequency Radial boundary conditions Asymptotically flat BH : ingoing at horizon, outgoing at infinity Asymptotically AdS BH : ingoing at horizon, vanishes at infinity Example : Schwarzschild BH 16

17 Methods for QNM (Konoplya and Zhdenko, RMP 83 (2011) 793) The Mashhoon method: Approximation with the Poschl- Teller potential Chandrasekhar-Detweiler and shooting methods WKB method, Integration of the wavelike equations Fit and interpolation approaches Frobenius method Method of continued fractions, Nollert improvement Horowitz-Hubeny method Limit of high damping : Monodromy method

18 G.Kunstatter [arxiv: gr-qc/ ] Given a system with energy E and vibrational frequency invariant is given by, an adiabatic Bohr-Sommerfeld quantization : (In the semiclassical limit) For Schwarzschild BH M.R.Setare [arxiv: hep-th/ ] Non-rotating BTZ BH : nonequally spaced area spectrum and dependent of cosmological constant Quasinormal modes [D.Birmingham (hep-th/ ) / V. Cardoso and J.P.S. Lemos (gr-qc/ )]

19 M.R.Setare & E.C.Vagenas [arxiv: hep-th/ ] Modified adiabatic invariant for rotating black holes Quasinormal modes of Kerr black holes : Horizon area spectrum of the Kerr BH is not equally spaced, inconsistent with Bekenstein s proposal [S.Hod (gr-qc/ )] What is wrong? Quasinormal modes are wrong. M.Maggiore In the semiclassical limit (large n), i.e. for highly damped modes, identify the transition frequency with For Schwarzschild BH, [arxiv: ], where

20 20

21 Old Quantum Physics Adiabatic invariant is a quantity of physical system which remains constant when changes are made very slowly Any classical adiabatic invariant corresponds to a quantum entity with discrete spectrum - Ehrenfest principle For simple harmonic oscillator, Hamiltonian H is given by The area of the ellipse Adiabatic invariant

22 Old Quantum Physics Sommerfeld s semiclassical quantization For simple harmonic oscillator, from For Hydrogen atom

23 A.J.M.Medved / E.C.Vagenas [arxiv: / ] For Kerr BH, using the modified adiabatic invariant [U.Keshet and A.Neitzke (arxiv: )], where Nonequally spaced Y.K & S.Nam [arxiv: ] The Modified Kunstatter s formula is not correctly derived. Have to apply the Bohr-Sommerfeld quantization to not an adiabatic invariant but an action variable, since not every adiabatic invariant is an action variable although an action variable is adiabatic invariant Quantum black hole with the transition frequency in the semiclassical limit can be regarded as a classical system of periodic motion with oscillation frequency (by Bohr s correspondence principle)

24 Identify an action variable Consider an action-variable in a system of one degree of freedom with oscillation frequency of periodic motion Action variable of the classical system in periodic motion From the Hamilton equations where is the oscillation frequency of the periodic motion An action variable for the classical periodic system with oscillation frequency via Bohr-Sommerfeld quantization This form is nothing but the Kunstatter s formula and now it can be applied even for rotating black holes

25 BTZ black holes Non-rotating BTZ BH Metric Rotating BTZ BH Metric 25

26 (1) Non-rotating BTZ BH Quasinormal modes : (Cardoso, et.al & Birmingham, et. al. (2001)) (2) Rotating BTZ BH Quasinormal modes :

27 Black holes in Topologically Massive Gravity E.O.M :, where BH Solutions 1, BTZ with CS-term : vanishing Cotton tensor 2. Warped AdS3 BH : nonvanishing Cotton tensor 27

28 ADT mass [Abbott, Deser, Tekin] ADT mass as a conserved charge written in terms of ADM mass M and angular momentum J in the absence of the Chern-Simons term (Moussa, et.al (2003)) 28

29 (1) BTZ black hole [YK & S.Nam (arxiv: )] Area spectra : Non-equally spaced and dependent of coupling constant Entropy spectrum : Equally spaced and independent of coupling constant This is same as rotating BTZ black hole in pure Einstein gravity Therefore it implies that entropy is more fundamental than horizon area

30 (2) Warped AdS black hole Metric : Two families of quasinormal modes from vanishing Dirichlet boundary condition at infinity gives two transition frequencies [ Bin Chen, X. Zu ] Area spectra and entropy spectrum are non-equally spaced. Q : Is this correct? For warped AdS black hole, the effective potential in the wave equation is finite at spatial infinity, contrary to BTZ case with divergent effective potential at infinity for which vanishing Dirichlet boundary condition should be imposed.

31 ADT mass for Warped AdS BH ADT mass as a conserved charge written in terms of ADM mass M and angular momentum J in the absence of the Chern-Simons term (Moussa, et.al (2003)) ADT mass for Warped AdS BH (Annonis, et.al (2008)) 31

32 Ⅱ. Other characteristic modes of black holes Non-quasinormal modes or holographic quasinormal modes [D.Birmingham and S.Carlip (hep-th/ )] [K.S.Gupta and S.Sen(hep-th/ )] Holographic quasinormal modes are obtained from the monodromy condition without imposing the boundary conditions by using the monodromies at horizons. For BTZ black holes, holographic quasinormal modes are exactly same as quasinormal modes. [D.Birmingham and S.Carlip (hep-th/ )] Now, we will consider holographic quasinormal modes rather than conventional quasinormal modes. Holographic quasinormal modes for warped AdS black hole [K.S.Gupta and et.al. (arxiv: )] From these two holographic quasinormal modes

33 Area quantization : Non-equally spaced and dependent of coupling constant The area spectrum is not equally spaced Entropy Entropy spectrum : Equally spaced and independent of coupling constant This is same as for rotating BTZ BH in TMG as well as Einstein gravity. Universality that the entropy spectrum of a black hole is equally spaced.

34 Ⅲ. Resonance modes as characteristic modes of black holes Consider the wave equation in scattering problem Boundary condition : QNMs correspond to the poles of transmission and reflection amplitudes Other resonance modes : Total Transmission Modes (TTM) Total Reflection Modes (TRM) Kerr black hole [U.Keshet and A.Neitzke (arxiv: )] Three kinds of resonance modes : Relation among them :

35 For highly damped modes Where is monotonically (but very slightly) increased in angular momentum And can be written in terms of elliptic integrals Consider three kinds of resonance modes as characteristic modes of black holes since all they only depend on black hole parameters Propose that TTM and TRM also carry some information about quantum structure, so that they should play the same role as QNM in quantization of black hole [Y.K and S.Nam (arxiv: )] Area and entropy spectra : The first relation among resonance modes implies that [U.Keshet and A.Neitzke (arxiv: )]

36 For slowly rotating case Quantization conditions : Check the relation Area and entropy spectra Same entropy spectra from TRM for Schwarzschild and charged black holes The universal behavior of the entropy spectrum which is equally spaced is still observed.

37 Area spectra of black holes : Equally or non-equally spaced 37

38 38

39 39

40 40

41 41

42 42

43 43

44 44

45 45

46 46

47 47

48 48

49 49

50 50

51 Entropy spectra of black holes : Equally spaced 51

52 IV. Conclusion and Comments??! Universal behavior that the entropy spectra of various black holes are equally spaced, i.e.,, not horizon area. It does not matter if the gravity theory does not satisfy the Bekenstein -Hawking area law, e.g., black holes in TMG. Even for the black holes in New Massive Gravity, e.g., BTZ, warped AdS black holes, new type black holes, and etc. Considered characteristic modes of black holes which are holographic quasinormal modes (non-quasinormal modes) and three resonance modes (TTM and TRM as well as QNM). Which one does really have the information of quantum structure of black holes? If all they are the characteristic modes of black holes, the quantization results from them should be consistent. What kind of relation can exist among them? It is believed that the quantization of black holes in these attempts may play a significant role to shed some light on the nature of quantum gravity.

53 53

54 Appendix

55 Quantization of black hole Bekenstein (1974) : horizon area can be considered as an adiabatic invariant For RN BH, consider a classical point particle of charge e with conserved total energy adjusted to which shoots in radially. The particle s motion has a turning point at the horizon. Because of this, the assimilation of particle by the black hole takes place especially slowly; considered as an adiabatic process. Small change For non-extremal BH, because and Adiabatic invariant

56 Monodromies In complex plane, as we integrate a function along some contours, a change in the function is called monodromy. In our case, we need to solve the wave equation with proper boundary condition and the behavior of the solution can be considered, so that QNM can be obtained. Through analytic continuation, we have complex r-plane. Usually the solution has exp[iwx], therefore we want to contour integral along the Stokes line(im(wx)=0) which means Re(x)=0 in highly damped modes. From the wave equation, for the solution at near horizon the tortoise coordinate is given by The logarithm of complex number is multivalued.

57 Monodromies Chosen m=1, along the contour around the horizon counterclockwise the monodromy of the tortoise coordinate is given by Since the solution of wave equation is given by exp[iwx], the monodromy of the tortoise coordinate gives the monodromy of the solution as follows: By matching the monodromy along the other contour which can be deformed without crossing the singularity, we find the QNM (i.e. from monodromy condition)

58 Non-quasinormal modes Quasinormal mode boundary are associated with the behavior of the wave function at horizon and infinity. This has been mysterious, since it is not clear why the black hole quantization should care about infinity. Without imposing the boundary condition at infinity, by using monodromy condition same QNMs are obtained. For BTZ, ingoing solution at horizon is given by Therefore the monodromy of the solution along the contour around the horizon We have monodromies of the two linearly independent solutions at inner horizon Demand monodromy condition from each contour can be deformed each other

59 Holographic quasinormal modes Monodromy condition is from the geometric interpretation. BTZ black hole has hyperbolic structure, so that BTZ is locally isometric to H3 Sullivan s theorem : one to one correspondence between conformal structure and hyperbolic structure Boundary of BTZ black hole is topologically torus T2. Torus has SL(2,Z) modular group which is generated by T and S operators. Monodromy should be invariant under the action of the modular group Therefore Using, we obtain the condition similar as monodromy condition which give same QNMs For the warped AdS BH, same condition is derived. This is obtained without imposing any boundary condition at infinity.

60 Action variable in our proposal Action variable of the classical system in periodic motion Hamiltonian of the classical system : Therefore, since Angle variable which is the generalized coordinate conjugate to Hamilton equation of motion Since change of for one cycle of the periodic motion is Therefore is the oscillation frequency of the periodic motion From Hamilton equation, we obtain

61 Application of Maggiore s proposal For the Kerr black hole, If the previous Hod result of quasinormal modes and the modified adiabatic invariant were used, With transition frequency

62 Charges If Lagrangian in a theory has a symmetry, the theory has a conserved charge corresponding to the symmetry by Nother theorem For example, consider a theory, guage invariant (symmetry under gauge transformation) Conserved (divergenceless) current is given by where is the electromagnetic field strength. Charge passing through a spacelike hypersurface is given by

63 63