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