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Thermodynamics of Kerr-AdS Black Holes

Thermodynamics of Kerr-AdS Black Holes. Rong-Gen Cai ( 蔡荣根） Institute of Theoretical Physics Chinese Academy Of Sciences ICTS-USTC, 2005,6.3. Main References:. S.W. Hawking, C.J. Hunter and M.M. Taylor-Robinson ROTATION AND THE ADS / CFT CORRESPONDENCE :

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Thermodynamics of Kerr-AdS Black Holes

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  1. Thermodynamics of Kerr-AdS Black Holes Rong-Gen Cai (蔡荣根） Institute of Theoretical Physics Chinese Academy Of Sciences ICTS-USTC, 2005,6.3

  2. Main References: • S.W. Hawking, C.J. Hunter and M.M. Taylor-Robinson • ROTATION AND THE ADS / CFT CORRESPONDENCE: • Phys.Rev.D59:064005,1999; hep-th/9811056 • G.W. Gibbons, M. Perry and C.N. Pope • THE FIRST LAW OF THERMODYNAMICS FOR KERR-ANTI-DE • SITTER BLACK HOLES: hep-th/0408217 • (3) R.G. Cai, L.M. Cao and D.W. Pang • THERMODYNAMICS OF DUAL CFTS FOR KERR-ADS BLACK HOLES: • hep-th/0505133

  3. Outline: First Law of Kerr Black Hole Thermodynamics First Law of Kerr-AdS Black Hole Thermodynamics Thermodynamics of Dual CFTs for Kerr-AdS Black Holes

  4. 1. First Law of Kerr Black Hole Thermodynamics Kerr Solution: where There are two Killing vectors:

  5. These two Killing vectors obey equations: with conventions: (J.M. Bardeen, B. Carter and S. Hawking, CMP 31,161 (1973))

  6. Consider an integration for S over a hypersurface Sand transfer the volume on the left to an integral over a 2-surface bounding S. measured at infinity Note the Komar Integrals:

  7. Then we have where Similarly we have

  8. For a stationary black hole, is not normal to the black hole horizon, instead the Killing vector does, Where is the angular velocity. where Angular momentum of the black hole

  9. Further, one can express where is the other null vector orthogonal to , normalized so that and dA is the surface area element of . where is constant over the horizon

  10. For Kerr Black Holes: Smarr Formula Integral mass formula where Bekenstein, Hawking The Differential Formula: first law

  11. 2. First Law of Kerr-AdS Black Hole Thermodynamics Four dimensional Kerr-AdS black hole solution (B. Carter,1968): where The horizon is determined by

  12. Defining the mass and the angular momentum of the Black hole as: Hawking et al. hep-th/9811056 where and are the generators of time translation and rotation, respectively, and one integrates the difference between the generators in the spacetime and background over a celestial sphere at infinity.

  13. The background: M=0 Kerr-AdS solution,which is actually an AdS metric in non-standard coordinates. Making coordinate transformation: The background is

  14. Then one has

  15. However, Gibbons et al. showed recently that Gibbons et al. hep-th/0408217 The results in hep-th/0408217:

  16. Hawking et al. Gibbons et al. (hep-th/9811056) (hep-th/0408217)

  17. In fact, the relationship between the mass given by Hawking et al. and that by Gibbons et al. is That is, where angular velocity of boundary

  18. Five dimensional Kerr-AdS black holes (given in hep-th/9811056): where

  19. Gibbons et al.: Hawking et al.:

  20. D>4 Kerr-AdS black hole solutions with the number of maximal rotation parameters (Gibbons et al. hep-th/0402008), with a single rotation parameter (Hawking et al. hep-th/9811056) In the Boyer-Linquist coordinates: where independent rotation parameter number, defining mod 2, so that

  21. Moreover The horizon is determined by equation: V-2m=0. The surface gravity and horizon area

  22. They satisfy the first law of black hole thermodynamics:

  23. In the prescription of Hawking et al.

  24. 3. Thermodynamics of Dual CFTs for Kerr-AdS Black Holes According to the AdS/CFT correspondence, the dual CFTs reside on the boundary of bulk spacetime. Suppose the boundary locates at with spatial volume V, Rescale the coordinates so that the CFTs resides on Recall

  25. The relationship between quantities on the boundary and those in bulk Other quantities, like angular velocity and entropy, remain unchanged For the CFT, the pressure is

  26. When D=odd, When D=even We find ( hep-th/0505133) (In the prescription of Hawking et al.) (in the prescription of Gibbons et al.)

  27. As a summary The prescription of Hawking et al. The prescription of Gibbons et al.

  28. Further Evidence: Cardy-Verlinde Formula Consider a CFT residing in (n-1)-dimensional spacetime described by Its entropy can be expressed by (E. Verlinde, 2000) where

  29. For the Kerr-AdS Black Holes: The prescription of Hawking et al The prescription of Gibbons et al Conclusions:

  30. Thanks

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