thermodynamics of kerr-ads black holes
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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 : - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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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
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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))
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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
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;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
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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
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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
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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
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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
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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.
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The background: M=0 Kerr-AdS solution,which is actually an AdSmetric in non-standard coordinates.
Making coordinate transformation:
The background is
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Then one has
1/T
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' ' 'dM TdS dJ
However, Gibbons et al. showed recently that
Gibbons et al.hep-th/0408217
The results in hep-th/0408217:
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Hawking et al. Gibbons et al.(hep-th/9811056) (hep-th/0408217)
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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
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Five dimensional Kerr-AdS black holes (given in hep-th/9811056):
where
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Gibbons et al.:
Hawking et al.:
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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
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Moreover
The horizon is determined by equation: V-2m=0.
The surface gravityand horizon area
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They satisfy the first law of black hole thermodynamics:
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In the prescription of Hawking et al.
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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
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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
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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.)
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As a summary
The prescription of Hawking et al. The prescription of Gibbons et al.
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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
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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:
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Thanks