Double Phase Transitions of Kerr-AdS Black Hole
Yu Dai Tsai
NTHU PHYS
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Outline
This talk is based on my resent work with
Professor Wu, X. N. and Professor Yang, Yi.
(Arxiving, please wait), which will be presented
in the following manner.
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Outline
This talk is based on my resent work with
Professor Wu, X. N. and Professor Yang, Yi.
(Arxiving, please wait), which will be presented
in the following manner.
Introduction
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Outline
This talk is based on my resent work with
Professor Wu, X. N. and Professor Yang, Yi.
(Arxiving, please wait), which will be presented
in the following manner.
Introduction
Double Phase Structures of Kerr-AdS Black
Hole
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Outline
This talk is based on my resent work with
Professor Wu, X. N. and Professor Yang, Yi.
(Arxiving, please wait), which will be presented
in the following manner.
Introduction
Double Phase Structures of Kerr-AdS Black
Hole
Discussion
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Reference
“Double Phase Transition of Kerr-AdS Black Hole” follows
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Reference
“Double Phase Transition of Kerr-AdS Black Hole” follows
hep-th/9904197 (Chamblin et al.,1999): Holography,
thermodynamics and fluctuations of charged AdS black holes
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Reference
“Double Phase Transition of Kerr-AdS Black Hole” follows
hep-th/9904197 (Chamblin et al.,1999): Holography,
thermodynamics and fluctuations of charged AdS black holes
Phys.Rev.,D62,124023(Wu,2000): Multicritical phenomena of
Reissner-Nordstrom anti-de Sitter black holes
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Reference
“Double Phase Transition of Kerr-AdS Black Hole” follows
hep-th/9904197 (Chamblin et al.,1999): Holography,
thermodynamics and fluctuations of charged AdS black holes
Phys.Rev.,D62,124023(Wu,2000): Multicritical phenomena of
Reissner-Nordstrom anti-de Sitter black holes
JHEP,04,2010,118 (Sahay et al.,2010): Thermodynamic Geometry
and Phase Transitions in Kerr-Newman-AdS Black Holes
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Reference
“Double Phase Transition of Kerr-AdS Black Hole” follows
hep-th/9904197 (Chamblin et al.,1999): Holography,
thermodynamics and fluctuations of charged AdS black holes
Phys.Rev.,D62,124023(Wu,2000): Multicritical phenomena of
Reissner-Nordstrom anti-de Sitter black holes
JHEP,04,2010,118 (Sahay et al.,2010): Thermodynamic Geometry
and Phase Transitions in Kerr-Newman-AdS Black Holes
Your works here? contact: s9822703@nthu.edu.tw
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The Black Hole Thermodynamics
mechanical laws ↔ thermodynamics laws
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The Black Hole Thermodynamics
mechanical laws ↔ thermodynamics laws
the Hawking radiation confirmed its physical
significance
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The Black Hole Thermodynamics
mechanical laws ↔ thermodynamics laws
the Hawking radiation confirmed its physical
significance
sort of a “Deja vu” of the physics history
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The Black Hole Thermodynamics
mechanical laws ↔ thermodynamics laws
the Hawking radiation confirmed its physical
significance
sort of a “Deja vu” of the physics history
J
0.026
T<Tc1
0.025
0.024
T=Tc1
0.023
T>Tc1
0.022
0.021
0.1
W
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure: The critical isotherms near the critical temperature Tc1 .
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The Kerr-AdS metric
The Kerr-AdS metric
2
2
Σ 2
∆
a
sin
θ
r
dφ +
dr
ds 2 = −
dt −
Σ
Ξ
∆r
2
Σ 2 ∆θ sin2 θ
r 2 + a2
+ dθ +
adt −
dφ
(, 1)
∆θ
Σ
Ξ
where
2
∆r = r + a
∆θ = 1 −
2
r2
1+ 2
l
a2
cos2 θ,
2
l
Yu Dai Tsai
a2
− 2Mr , Ξ = 1 − 2 ,
l
Σ = r 2 + a2 cos2 θ. (2)
Double Phase Transitions of Kerr-AdS Black Hole
The Kerr-AdS Thermodynamic quantities
Gibbons et al. in arXiv:hep-th/0408217v3 (2006)
The “physical” mass (or energy) E and angular
momentum J is defined as
M
Ma
E = 2, J = 2 .
(3)
Ξ
Ξ
κ
The Hawking temperature and the entropy T = 2π
and S = A4 . The angular velocity of horizon Ω,
gtφ
aΞ[(r 2 + a2 )∆θ − ∆r ]
Ω=−
=
, ∆r → 0, (4)
gφφ
Γ
where Γ = (r 2 + a2 )2 ∆θ − ∆r a2 sin2 θ.
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
the first law of the black hole thermodynamics
The relation between each quantity gives us the first
law of the black hole thermodynamics, which is
dE = TdS + ΩδJ .
Yu Dai Tsai
(5)
Double Phase Transitions of Kerr-AdS Black Hole
The van der Waals system
a: attraction between each particle, b: the volume
of the particle
n2 a
(p + 2 )(V − nb) = nRT
V
Figure: The critical isotherms near the critical temperature Tc1 .
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The Kerr-AdS black hole
J
0.026
T<Tc1
0.025
0.024
T=Tc1
0.023
T>Tc1
0.022
0.021
0.1
W
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure: The critical isotherms near the critical temperature Tc1 .
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
critical isotherm:
Ω − Ωc = Aδ |J − Jc |δ sign(J − Jc ), (T = Tc ).
(6)
order parameter:
η1 = −Aβ (T − Tc )β , (T > Tc ).
(7)
heat capacity
CJ =
0
Aα0 {−(T − Tc )}−α , (T < Tc )
.
Aα0 {+(T − Tc )}−α , (T < Tc )
isothermal compressibility
0
Aγ 0 {−(T − Tc )}−γ , (T < Tc )
κT =
.
Aγ {+(T − Tc )}−γ , (T > Tc )
Yu Dai Tsai
(8)
(9)
Double Phase Transitions of Kerr-AdS Black Hole
The result is identical to the Weiss magnet
system(actually, classical mean field systems)
β=
1
2
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The result is identical to the Weiss magnet
system(actually, classical mean field systems)
β=
1
2
δ=3
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The result is identical to the Weiss magnet
system(actually, classical mean field systems)
β=
1
2
δ=3
α = α0 = 0
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The result is identical to the Weiss magnet
system(actually, classical mean field systems)
β=
1
2
δ=3
α = α0 = 0
γ = γ0 = 1
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
the second second order phase transition
J
0.4
0.3
0.2
0.1
T<Tc2
T=Tc2
T>Tc2
W
0.32
0.34
0.36
0.38
0.40
0.42
Figure: The critical isotherms
the Phase
critical
temperature
Tc2Hole
.
Yu Dai Tsai nearDouble
Transitions
of Kerr-AdS Black
The free energy F near the critical point:
Fs (, ω) = c 2 + cω ω 4/3 .
Thus we have the scaling symmetry of the free
energy near the critical point,
Fs (Λ1/2 , Λ3/4 ω) = ΛFs (, ω),
with all the scaling symmetry follows.
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Write the critical exponents in terms of p and q as
2p − 1
,
p
1−q
β=
,
p
2q − 1
γ=
,
p
q
.
δ=
1−q
α=
Yu Dai Tsai
(10)
Double Phase Transitions of Kerr-AdS Black Hole
From the equations above, the critical exponents
satisfy the following relations,
α + 2β + γ = 2 ,
α + β(δ + 1) = 2 ,
γ(δ + 1) = (2 − α)(δ − 1) ,
γ = β(δ − 1).
Yu Dai Tsai
(11)
Double Phase Transitions of Kerr-AdS Black Hole
Results and Discussion
The preliminary step to set the restriction of the
microscopic theory of gravity.
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Results and Discussion
The preliminary step to set the restriction of the
microscopic theory of gravity.
Applications on CMT
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Results and Discussion
The preliminary step to set the restriction of the
microscopic theory of gravity.
Applications on CMT
Extension to other black hole or other gravity
theory
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
Results and Discussion
The preliminary step to set the restriction of the
microscopic theory of gravity.
Applications on CMT
Extension to other black hole or other gravity
theory
Mean field gravity?
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The rotating holographic superconductor
Hortnoll (hep-th/0803.3295): Building a
Holographic Superconductor
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The rotating holographic superconductor
Hortnoll (hep-th/0803.3295): Building a
Holographic Superconductor
Julian Sonner (hep-th/0903.0627): a Rotating
Holographic Superconductor
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole
The rotating holographic superconductor
Hortnoll (hep-th/0803.3295): Building a
Holographic Superconductor
Julian Sonner (hep-th/0903.0627): a Rotating
Holographic Superconductor
jobs for string theorists
Yu Dai Tsai
Double Phase Transitions of Kerr-AdS Black Hole