The Everyday
Phenomena of Black
Hole Chemistry
Robert Mann
D. Kubiznak, N. Altimirano, S. Gunasekaran,
B. Dolan, D. Kastor, J. Traschen, Z. Sherkatgnad
D.Kubiznak, R.B. Mann JHEP 1207 (2012) 033
S. Gunasekaran, D.Kubiznak, R.B. Mann JHEP 1211 (2012) 110
B. Dolan, D. Kastor, D.Kubiznak, R.B. Mann, J. Traschen Phys. Rev. D87 (2013) 104017
N. Altimirano, D.Kubiznak, R.B. Mann Phys. Rev. D88 (2013) 101502
N. Altimirano, D. Kubiznak, Z. Sherkatgnad, R.B. Mann CQG 31 (2014) 042001
N. Altimirano, D. Kubiznak, Z. Sherkatgnad, R.B. Mann Galaxies 2 (2014) 89
Black Hole Thermodynamics
Thermodynamics
Gravity
Energy E « M Mass
k
Temperature T «
Surface gravity
2p
A
Entropy S «
Horizon Area
4
k
dE = TdS + VdP + work terms « dM =
dA + WdJ + FdQ
8p
First Law
First Law
Smarr Formula
L. Smarr PRL 30, 71 (1973)
[Err. 30, 521 (1973)].
2
dr
ds 2 = -Vdt 2 +
+ r 2 dW22
V
Schwarzschild Black hole V = 1-
r+
E=M =
2
1
T=
4p r+
2M
r
S = pr
2
+
M = 2TS
Smarr
2M r 2
Schwarzschild-AdS Black hole V = 1+ 2
r
l
l 2 + r+2
l 2 + 3r+2
E=M =
r+ T =
2
2
2l
4p r+l
S = pr
2
+
M ¹ 2TS
Smarr
Caldarelli/Cognola/Kl
emm, CQG 17, 399
(2000)
Scaling Arguments
Suppose
¶f
¶f
rf ( x, y ) = p
x+q
y
¶x
¶y
f (a x,a y ) = a f ( x, y )
p
q
r
S-AdS Black Hole
M = M ( A, L )
A
S=
4G
M µL
AµL
-2
LµL
¶M
¶M
( D - 3) M = ( D - 2 ) A - 2 L
¶A
¶L
D-3
k
¶M
T=
= 4G
2p
¶A
D - 2)
2
(
M=
TS VP
( D - 3)
( D - 3)
D-2
L ( D - 2 ) ( D - 1)
P==
8p
16p l 2
¶M
V = -8p
¶L
Pressure from the Vacuum?
Dolan CQG 28 (2011)
125020; 235017
Schwarzschild-AdS Black hole
l 2 + r+2
l 2 + 3r+2
E=M =
r+ T =
2
2
2l
4p r+l
M = 2 (TS - VP )
S = pr
2
+
(D = 4)
dE = TdS +VdP
First Law
Smarr
Provided
1
3 1
P=L=
2
8p
8p l
Thermodynamic Pressure
¶M 4p 3
V = -8p
=
r+
¶L
3
Thermodynamic Volume
The Chemistry of AdS Black Holes
Include gauge charges:
First Law
d M = Thd Sh + å (W - W )d J + Fhd Q +Vhd P
i
h
i
¥
i
i
Smarr Relation
D-3
D-3
2
i
i
i
M = Th Sh + å (Wh - W¥ )J +
FhQ PVh
D-2
D-2
D-2
i
Thermodynamic Potential: Gibbs Free Energy
G = M - TS = G(T, P, Ji ,Q)
• Equilibrium: Global minimum of Gibbs Free Energy
• Local Stability: Positivity of the Specific Heat
æ ¶S ö
CP = T ç ÷
>0
è ¶T ø P,Ji ,Q
Mass as Enthalpy
Thermodynamics
Gravity
Enthalpy H « M Mass
k
Temperature T «
Surface gravity
2p
A
Entropy S «
Horizon Area
4
k
dH = TdS + VdP + « dM =
dA + VdP +
8p
First Law
First Law
H = E + PV +
« M = E - rV
Mass
= Total Energy
- Vacuum
Contribution
(infinite)
Everyday AdS Black Hole
Thermodynamics
• Hawking Page Transition
• Van der Waals Fluid and Charged AdS Black
Holes
• Reentrant Phase Transitions
• Black Hole Triple Points  Solid/Liquid/Gas
Hawking-Page Transition
D-dim’l Schwarzschild-AdS Black hole
S.W. Hawking & D.N. Page
Comm Math. Phys. 87 (1983) 577
2
dr
M
r
ds 2 = -Vdt 2 +
+ r 2 dW2k,D-2 V = k - D-3 + 2
r
V
2
• AF black holes evaporate by
Hawking radiation
• AdS is like a confining box
 static black holes in thermal
equilibrium
T < Tmin Þ gas of particles
1st order transition
between gas of particles
and large black holes at Tc
ì 1 spherical
ï
k = í 0 planar
ï -1 hyperbolic
î
Tmin = 1
CP < 0
CP > 0
D - 3) l 2 + ( D -1) r+2
(
T=
4p r+l
Phase transition in dual
CFT (quark-gluon plasma)
Witten (1998)
2
Fluid interpretation:
solid/liquid PT
(infinite coexistence line)
D.
Kubiznak/RBM
arXiv
[1404.2126]
Equation of state
Pv = T -
æ 3V ö
v = 2l r = 2 ç ÷
è 4p ø
A
N= 2
lP
2
P +
1
3
V
=6
N
k
2
p
v
depends on the
horizon topology
Planar black holes
 ideal gas
Van der Waals Fluids
Pv - (kT + bP)v + a(v - b) = 0
3
2
Critical Point
¶P
¶2 P
= 0,
=0
¶v
¶v
8a
a
kTc =
, vc = 3b, Pc =
27b
27b 2
P
v
T
p= , n = , t =
Pc
vc
Tc
3ö
æ
8t = (3n -1) ç p + 2 ÷
è
n ø
law of
correspondin
g states
PV Diagram
for a VdW
Fluid
Pc vc 3
=
kTc 8
Maxwell’s
Equal Area Law
v
dP
=
0
ò
Gibbs Free Energy
G = G(T , P)
æ
é (v - b)T 3/2 ùö a
= -kT ç 1+ ln ê
- + Pv
ú
÷
F
è
ë
ûø v
–0.1
–0.2
G
G
–0.3
characteristic of
the gas
–0.4
0.04
P
P
0.1
G
G
0.02
0.4
0.3
0.2
T
T
0.1
0.1
0.2
0.3
0.4
0
0
0.5
T
T
–0.1
–0.2
First Law
dG = -SdT + v dP
P>Pc
–0.3
P=Pc
–0.4
P<Pc
–0.5
–0.6
Critical Exponents
Line of Coexistence
P
P
0.04
0.03
dP
dT
=
coexistence
0.02
0.01
Sg - Sl
Critical Point
Order
Parameter
vg - vl
LIQUID
Clausius-Clapeyron
equation
GAS
Isothermal
Compressibility
Critical
Isotherm
T
0
0.1
0.2
0.3
h = vg - vl µ| t - 1|b
1 ¶v
kT = µ| t - 1 |-g
v ¶P T
P - Pc µ v - vc
0.4
1
For a VdW Fluid a = 0 b =
2
P
v
T
, n= , t=
Pc
vc
Tc
¶S
Cv = T
µ| t - 1|-a
¶T v
Specific
Heat
0.05
p=
g =1 d =3
d
Charged AdS Black Holes
as Van der Waals Fluids
dr
2
2
ds = -Vdt +
+ r dW2
V
2
2
2
F = dA
Kubiznak/Mann
JHEP 1207
(2012) 033
2M Q 2 r 2
V = 1+ 2 + 2
r
r
l
Q
A = - dt
r
1 æ 3r+2 Q 2 ö
Temperature T = =
1+ 2 - 2 ÷
ç
b 4p r+ è
l
r+ ø
1
Entropy
Potential
A
S = = p r+2
4
Q
F=
r+
Pressure
L
3 1
P==
8p 8p l 2
Volume
4 3
V = p r+
3
M = 2 (TS - PV ) + FQ
Smarr Relation
dM = TdS +VdP + FdQ
First Law
æ 3V ö
r+ = ç
è 4p ÷ø
Equation of State
1 æ 3r+2 Q 2 ö
T=
1+ 2 - 2 ÷
ç
4p r+ è
l
r+ ø
L
3 1
P==
8p 8p l 2
Physical Equation of State
c
c
Press = 2 P Temp = T
lP
k
Van der
Waal’s
Equation
1/3
T
1
Q2
P=
+
2
2r+ 8p r+ 8p r+4
Thermodynamic
Volume
kTemp
Press =
+¼
2
2lP r+
T
1
2Q
P= + 4
2
v 2p v p v
2
v = 2lP2 r+
Gibbs Free Energy of AdS RN BH
1
I =16p
ò
M
6ö 1
æ
2
-g ç R - F + 2 ÷ è
l ø 8p
1
ò¶ M d x hK - 4p
3
1æ
8p 3 3Q 2 ö
G = G(T , P) = ç r+ Pr+ +
4è
3
r+ ÷ø
1.8
¶M
Fixed Charge
G
1.6
Just like a
VdW Fluid!
1.4
G
ò
d 3 x hna F ab Ab + I c
1.2
G
1
0.8
0.6
0.4
0.004
P
P
0.002
0.06
0.05
0.04
T
0.03
T
0.02
0.01
0
T
NB: Disagree with
Chamblin et.al. PRDD60 (1999) 064018; 104026
T
1
2Q
P= + 4
2
v 2p v p v
2
Critical Behaviour
¶P
¶2 P
=0
=0
¶v
¶v
p=
P
v
T
, n= , t=
Pc
vc
Tc
6
Tc =
18p Q
Pc vc 3
=
kTc 8
vc = 2 6Q
1
Pc =
96p Q 2
Just like a
VdW Fluid!
P
2ö 1
æ
8t = 3n ç p + 2 ÷ - 3
è
n ø n
law of
correspondin
g states
v
P
Just like a VdW Fluid!
P
0.004
Critical Point
0.003
0.002
0.001
SMALL BH
LARGE BH
0
0.01
0.02
1
a =0 b=
2
0.03
TT
0.04
0.05
0.06
g =1 d =3
govern volume, compressibility, specific heat, and
pressure near the critical point
Reentrant Phase Transitions
RPT: If a monotonic variation of
any thermodynamic quantity that
results in two (or more) phase
transitions such that the final
state is macroscopically similar to
the initial state.
First observed in nicotine/water
• multicomponent fluid
systems
• gels
• ferroelectrics
• liquid crystals
• binary gases
T. Narayanan and A. Kumar
Physics Reports 249 (1994) 135
C. Hudson
Z. Phys.
Chem. 47
(1904) 113.
Single-Rotation Black Holes
2
2
2
2
r
=
r
+
a
cos
q
é
ù
r
+
a
d
j
(
)
é
ù
D
asin q dj
S sin q
2
ds = - 2 ê dt +
ê adt ú
a2
a2
ú
2
2
r ë
X
r êë
X
û
úû S = 1- l 2 cos q X = 1- l 2
2
2
+
r
2
D
dr 2 +
r
2
S
2
2
2
æ r2 ö
D = (r + a ) ç 1+ 2 ÷ - 2mr 5-D
è l ø
2
dq 2 + r 2 cos 2 q dW D-2
w D-2 m æ (D - 4)X ö
M=
1+
÷ø
2 ç
4p X è
2
w D-2 ma
J=
4p X 2
æ 1
1
r+2
D - 3ö 1
T=
[r+ ( 2 + 1) ç 2 2 +
- ]
2 ÷
2p
l
2r+ ø r+
è a + r+
2
S=
2
a r+2 + l 2
WH = 2 2
,
2
l r+ + a
w D-2 (a 2 + r+2 )r+d-4
4
X
A
=
4
D-3
2VP
dM = TdS +VdP + WH dJ
M = TS + W H J D-2
D
2
Smarr Relation
First Law
Dimensional Dependence of G
D=5
D>5
G
G
Zeroth-Order
Phase Transition
T
T
Reentrant Phase Transitions in D>5
G
Zeroth-Order Phase
Transition
CP > 0
æ dS ö
CP = T ç ÷ < 0
è dT ø P
T
Reentrant Phase Transitions in D>5
Coexistence
Lines
Low T
P
M edium T
High T
mixed
Þ water/nicotine Þ mixed
Intermediate BH Þ Small BH Þ Large BH
T
Slow and Ultraspinning Limits
0.2
P
æ a 4ö
T
d-3
p (d -1)16 d J 2
Slow P = +
+ Oç 4 ÷
2
2
2(d-1)
v p (d - 2)v 4w d-2 [(d-2)v]
èl ø
Ultraspinning
P
0.15
P6 =
pJ2
5V 2
-
5w 4
16 p 3VT
+
9 5w 4 J 2
3200 p 3V 5T 5
+ O(X9 )
T=Tc
0.1
SLOW
EXACT
T<Tc
0.05
v
ULTRASPIN
0 0.5
1
1.5
v
2
2.5
3
J/T Phase Diagram
J
N. Altimirano, D. Kubiznak,
Z. Sherkatgnad, R.B. Mann
Galaxies 2 (2014) 89
RPTs do not require
variable cosmological
constant
T
Occurs in any d ³ 6
“two components”
BH vs.Blackbrane?
Triple Points in Multiply Rotating Black
Holes
Triple Point!
J2
= 0.05
J1
Pc1 = 0.259
Pc1 = 0.0956
N. Altimirano, D. Kubiznak, Z. Sherkatgnad, R.B. Mann
CQG 31 (2104) 042001 arXiV:1308.2672
Reentrant Phase transition
Ending in a Triple Point
J2
= 0.05
J1
The Black Hole Triple Point
N. Altimirano, D. Kubiznak, Z. Sherkatgnad, R.B.
Mann CQG 31 (2014) 042001
Variable L and AdS/CFT?
æ ö
4p N = ç ÷
è lP ø
4
4
æ ö
2
=
g
YM N
çè l ÷ø
s
Varying L Û varying N
CFT Þ gYM does not run
Do we get RG flow? • Continuous variation of N is probably OK.
• Classical gravity corresponds to
(similar to TD limit…can vary number of moles continuously)
Quantized N…quantum gravity effects?
• “Grand-canonical ensemble of stringy vacua” with conjugate
quantity playing role of “chemical potential”?
Summary
• Cosmological Pressure can be understood as a
Thermodynamic Quantity
– First Law is modified to include a pressure-volume term
• Smarr Law needs to be modified to respect scaling
• Mass becomes Enthalpy
• “Everyday Chemical Thermodynamics” is manifest
– Van der Waals Transitions
• Small/large  liquid/gas phase
• Critical Phenomena analogous
– Reentrant Phase Transitions
– Triple Points
These phenomena
do not require a
variable
cosmological
constant!
Open Questions
•
•
•
•
Meaning of Conjugate Volume?
Compressibility and Stability?
Gauge/Gravity Duality interpretation?
More exotic black holes?
Mo/Li/Liu; Wei/Liu; Ballik/Lake
– Lovelock Black Holes?
– Lifshitz Black Holes?
Dutta/Jain/Soni; Zhou/Zhang/Wang
Breton/Vergliaffa; Cai/Cao/Zhang;
Allahverdizadeh/Lemos/Sheykhi;
Lu/Pang/Pope/Vasquez-Portiz
• 1+1 Pressure-Volume?
• Meaning of de Sitter Thermodynamics?
• Other “everyday analogues”?
D. Kubiznak, N. Altimirano, W. Brenna, M. Park, F. Simovic,
Z. Sherkatgnad, J. Mureika, S. Solodukhin