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/
G.W. Gibbons, M. Perry and C.N. Pope
THE FIRST LAW OF THERMODYNAMICS FOR KERR-ANTI-DE
SITTER BLACK HOLES: hep-th/
(3) R.G. Cai, L.M. Cao and D.W. Pang
THERMODYNAMICS OF DUAL CFTS FOR KERR-ADS BLACK HOLES:
hep-th/

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
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:

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/
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
hep-th/
The results in hep-th/ :

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

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

21. 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

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

30. Thanks