Thermodynamics of Kerr-AdS Black Holes
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Transcript of Thermodynamics of Kerr-AdS Black Holes
Thermodynamics of Kerr-AdS Black HolesThermodynamics 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:
(1) 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
(2) 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:
2 2 2 2 22 2 2 ( )
( )a Sin aSin r a
ds dt dtd
2 2 2 2 22 2 2 2( )r a a Sin
Sin d dr d
where2 2 2r a Cos 2 2 2r a Mr
There are two Killing vectors:
( )a a
t
( )a a
These two Killing vectors obey equations:
; [ : ]a b a b
; ;b b
a b a b
;a b a bb bR ;a b a b
b bR
; [ ; ]a b a b
with conventions: cab acbR R 1
;[ ] 2a
d bc adbcv R v
(J.M. Bardeen, B. Carter and S. Hawking, CMP 31,161 (1973))
S
S B S
B
S
;a b a bb bR
Consider an integration for
over a hypersurface S and transfer the volume on the left to an integral over a 2-surface bounding S.S
Note the Komar Integrals:
;18
a bab
S
J d
measured at infinity
;a b a bab b a
S S
d R d
;14
a bab
S
M d
;14(2 )b b a a b
a a b ab
S B
M T T d d
Then we have
where 12 8ab ab abR Rg T
Similarly we have
;18
a b a bb a ab
S B
J T d d
For a stationary black hole, is not normal to the black holehorizon, instead the Killing vector does, Where
is the angular velocity.
( )a a
t
a a aH
|tH H
g
g
;14(2 ) 2b b a a b
a a b H H ab
S B
M T T d J d
where;1
8a b
H ab
B
J d
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 .
[ ]ab a bd n dA
B
an
1aan
B
;1 14 4
a bab
B B
d dA
where is constant over the horizon;
a ba bn
4(2 ) 2b b aa a b H H
S
M T T d J A
42 H HM J A
For Kerr Black Holes: Smarr Formula
2 4 2 1/ 2
4 2 1/ 2
2 4 2 1/ 2
2 4 2 1/ 2
2 ( ( ) )
( )
2 ( ( ) )
8 ( ( ) )
HH
H
H
H
H
J
M M M J
M J
M M M J
A M M J
where
Integral mass formula
8H HM J A
The Differential Formula: first law
H HdM dJ TdS / 2T
/ 4S A
Bekenstein, Hawking
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 0r
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 AdSmetric in non-standard coordinates.
Making coordinate transformation:
The background is
Then one has
1/T
' ' 'dM TdS dJ
However, Gibbons et al. showed recently that
Gibbons et al.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
2( / / )E Q t al
' ( / )E Q t
That is,
2'E E al J
where2al
angular velocity of boundary
Five dimensional Kerr-AdS black holes (given in hep-th/9811056):
where
Gibbons et al.:
Hawking et al.:
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:
whereindependent rotation parameter number, defining mod 2, so that
Moreover
The horizon is determined by equation: V-2m=0.
The surface gravityand 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 resideon the boundary of bulk spacetime.
Suppose the boundary locates at with spatial volume V, Rescale the coordinates so that the CFTs resides on
r R r
2 2 2sds dt ds
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
For the Kerr-AdS Black Holes:
2' (2 ' ' )
2 c CFT c
RS E E E
n
The prescription of Hawking et al
The prescription of Gibbons et al
Conclusions:
Thanks