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: 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

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

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

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

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

Then we have where Similarly we have

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

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

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

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

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.

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

Then one has

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

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

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

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

Gibbons et al.: Hawking et al.:

In the Boyer-Linquist coordinates: 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

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

They satisfy the first law of black hole thermodynamics:

In the prescription of Hawking et al.

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

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

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.)

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

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

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

Thanks