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Observer-‐Dependent Entropy in Black Hole Thermodynamics?
Robert Mann
Area Law
What happens if you pour a cup of tea into a black hole?
• Tea is hot – has entropy• Black hole absorbs everything and has no structure• Where does the entropy go?Bekenstein: Tea has mass à will increase mass of black hole
à area must increase
A(area) S (entropy)
Bekenstein PRD 7 (1973) 2333
Energy E↔ M Mass
Black Holes as Chemical Systems?
Thermodynamics Gravity
Entropy S↔ A4!
Horizon Area Temperature T ↔ !κ
2π Surface gravity
dE = TdS +VdP + work terms ↔ dM = κ8π
dA +ΩdJ +ΦdQ
First Law First Law
Dolan CQG 28 (2011) 125020; 235017
Kubiznak/Mann/Teo CQG 34 (2017) 063001
Scaling Arguments
f α px,α qy( ) =α r f x, y( )Suppose
rf x, y( ) = p ∂ f∂x
x + q ∂ f∂y
y
M = M A,Λ( )AdS Black Holes M ∝ LD−3 A ∝ LD−2 Λ ∝ L−2
D − 3( )M = D − 2( ) ∂M∂A
A − 2 ∂M∂Λ
Λ
M = D − 2( )D − 3( )TS −
2D − 3( )VP
S = A4G
T = κ2π
= 4G ∂M∂A
P = − Λ8π
=D − 2( ) D −1( )16π l2
V = −8π ∂M∂Λ
Caldarelli/Cognola/Klemm, CQG 17 (2000) 399
Creighton/Mann, PRD53 (1995) 4569Padmanabhan, CQG 19 (2002) 5387
Kastor/Ray/Traschen CQG 26 (2009) 195011 Dolan CQG 28 (2011) 125020; 235017
Enthalpy H ↔ M Mass
Pressure from the Cosmic VacuumThermodynamics Gravity
Entropy S↔ A
4! Horizon Area
Temperature T ↔ κ
2π Surface gravity
dH = TdS +VdP + !↔ dM = κ8π
dA +VdP + !
First Law First Law
H = E + PV + ↔ M = E − ρVMass
= Total Energy- Vacuum
Contribution (infinite)
Pressure P↔− Λ8πG
Cosmological Constant
C. Teitelboim PLB 158 (1984) 293 J. Creighton and R.B. Mann, PRD 52 (1995) 4569 Caldarelli/Cognola/Klemm CQG 17 (2000) 399; T. Padmanabhan, CQG 19 (2002) 5387
Kastor/Ray/TraschenCQG 26 (2009) 195001
Cosmological constant: Einstein’s biggest blunder? In order to obtain static Universe Einstein introduced:
Dark Energy: Λ > 0 (cosmic tension)String Theory: Λ < 0 (cosmic pressure)
Λ
R = (D −1)VωD−2
⎛⎝⎜
⎞⎠⎟
1 D−1ωD−2A
⎛⎝⎜
⎞⎠⎟1 D−2
≤1V
A
Q: What is the smallest area that encloses a given Euclidean volume V?
A: A spherical surface
Cvetic/Gibbons/Kubiznak/Pope PRD84 (2011) 024037 RBH ≥1Conjecture: All black holes obey theReverse Isoperimetric Inequality
ωD−2 =2π
D−12
Γ D −12
⎛⎝⎜
⎞⎠⎟
à For a given thermodynamic volume, the entropy of a black hole is maximized by the Schwarzschild-‐AdS sol’n
Meaning of Thermodynamic Volume?
RBH ≡(D −1)VBH
ωD−2
⎛⎝⎜
⎞⎠⎟
1 D−1ωD−2ABH
⎛⎝⎜
⎞⎠⎟
1 D−2
Black Holes:
4D Kerr-AdS BH
RBH ≡3V4π
⎛⎝⎜
⎞⎠⎟1 3 4π
A⎛⎝⎜
⎞⎠⎟1 2
= r+lr+2 + a2
l2 − a21+
a2 r+2 + l2( )
2r+2 (l2 − a2 )
⎛
⎝⎜
⎞
⎠⎟3
l2 − a2
r+2 + a2
≥1 →1a = 0
The Chemistry of AdS Black Holes
D − 3D − 2
M = ThSh + (Ωhi −Ω∞
i )i∑ J i + D − 3D − 2ΦhQ −
2D − 2
PVh
δM = ThδSh + (Ωhi −Ω∞
i )i∑ δ J i +ΦhδQ +VhδP
First Law
Smarr Relation
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
CP = T∂S∂T
⎛⎝⎜
⎞⎠⎟ P,Ji ,Q
> 0
Results from Black Hole Chemistry• Hawking Page Transition
– solid/liquid phase transition with infinite coexistence line• Black Holes as Van der Waals Fluids
– Complete correspondence between intrinsic and extrinsic variables
• Reentrant Phase Transitions– Change from one phase to another and back again as one parameter (eg. temperature) monotonically changes
• Black Hole Triple Points ßà Solid/Liquid/Gas
N. Altimirano, D. Kubiznak, Z. Sherkatgnad, R.B. Mann
Galaxies 2 (2014) 89
N. Altimirano, D. Kubiznak, Z. Sherkatgnad, R.B. Mann
CQG 31 (2104) 042001
Kubiznak/Mann CJP 93 999 (2016)
Kubiznak/MannJHEP 1207 (2012) 033
• Multiple re-‐entrant phase transitions– Higher curvature gravity
• Reverse VdW phase transitions– Exhibit similar phenomena but in lower dimensions
• Isolated Critical Points– Isotherms cross at a particular value of the volume V– Some black holes are like polymers: they do not have standard critical exponents
• Heat Engines– Can compare black hole efficiencies through various engine cycles
Frassino/Kubiznak/Mann/SimovicJHEP 1409 (2014) 080
Brenna/Hennigar/MannJHEP 1507 (2015) 077
B. Dolan, A. Kostouki, D.Kubiznak, R.B. Mann, CQG
31 (2014) 242001
Hennigar JHEP 1710 (2017) 082
Johnson CQG 31 (2014) 205002 Henningar/McCarthy/Ballon/Mann
CQG 34 (2017) 175005
• Van der Waals black holes– Black holes + exotic matter yield exact VdW equation
• Lifshitz Black Holes– Standard Smarr Formula holds in all cases– Can use this to define mass (otherwise intractable)
• Holographic Smarr Relation– Higher curvature è non-‐planar loops
• Superfluid Black Holes– Black holes with scalar “hair” can exhibit a superfluid phase transition analogous to 4He
A.Rajagopal, D. Kubinzank, R.B. Mann, PLB B737 (2014) 277
T. Delsate, , R.B. Mann, JHEP 1502(2015) 070
W.G. Brenna, M. Park, R.B. Mann, PRD 92 (2015) 044015
E. Tjoa, R. Hennigar, R.B. Mann PRL 118 (2017) 021301 E. Tjoa, R. Hennigar, R.B. Mann JHEP 1702 (2017) 040H. Dykaar, R. Hennigar, R.B. Mann JHEP 1705 (2017 ) 045
Karch/Robinson JHEP 1512 (2015) 073 Sinamuli/Mann PRD 96 (2017) 068008
Van der Waals fluid
Parameter a measures the attraction between particles (a>0) and b corresponds to “volume of fluid particles”.
Critical point:
Chamblin/Emparan/Johnson/Myers Phys.Rev. D60 (1999) 064018Cvetic/Gubser-‐JHEP 9904 (1999) 024
Kubiznak/Mann JHEP 1207 (2012) 033Gunasekaran/Kubiznak/Mann JHEP 1112 (2012) 110
Charged AdS black holes as Van der Waals fluids
Charged AdS Black Hole
Critical point:
Kubiznak/Mann JHEP 1207 (2012) 033Gunasekaran/Kubiznak/Mann JHEP 1112 (2012) 110
Charged AdS black holes as Van der Waals fluids
Phase diagrams
Coexistence line
• MFT critical exponents
govern specific heat, volume, compressibility and pressure at the vicinity of critical point.
• Clausius-Clapeyron and Ehrenfest equations are satisfied
Kubiznak/Mann JHEP 1207 (2012) 033Gunasekaran/Kubiznak/Mann JHEP 1112 (2012) 110
Phase diagrams
Coexistence line
• MFT critical exponents
govern specific heat, volume, compressibility and pressure at the vicinity of critical point.
• Clausius-Clapeyron and Ehrenfest equations are satisfied
Small/large black hole phase transition
A system undergoes an RPT if a monotonic variation of any thermodynamic quantity results in two (or more) phase transitions such that the final state is macroscopically similar to the initial state. C. Hudson
Z. Phys. Chem. 47 (1904) 113.First observed in nicotine/water
T. Narayanan and A. Kumar Physics Reports 249 (1994) 135
• multicomponent fluid systems
• gels• ferroelectrics• liquid crystals• binary gases
And later in many other systems:
And recently in Black Holes!
Reentrant Phase Transitions
Coexistence Lines
N. AltimiranoD. KubiznakR.B. Mann
PRD 88 (2013) 101502
Example: Re-‐entrant Phase Transition in Rotating AdS Black Holes
P
T
Pz
Tz
Pt
Tt
INTERMEDIATE BH
SMALL BH
LARGE BH
0.054
0.056
0.058
0.06
0.23 0.235 0.24
Low T Medium T High T mixed ⇒ water/nicotine ⇒ mixedIntermediate BH ⇒ Small BH ⇒ Large BH
P
T
large/small/large black hole phase transition
Altimirano/Kubiznak/Sherkatgnad/Mann Galaxies 2 (2014) 89
Takes place in many examples
Surprises in Black Hole Entropy
• Super-‐entropic Black holes– Entropy exceeding the reverse isoperimetric inequality
• Entropy of de Sitter Black Holes–With and without a cavity
• Accelerating Black Holes– Thermodynamics with Radiation?
R = (D −1)VωD−2
⎛⎝⎜
⎞⎠⎟
1 D−1ωD−2A
⎛⎝⎜
⎞⎠⎟1 D−2
Cvetic/Gibbons/Kubiznak/Pope PRD84 (2011) 024037 R ≥1
Conjecture: All black holes obey theReverse Isoperimetric Inequality
ωD−2 =2π
D−12
Γ D −12
⎛⎝⎜
⎞⎠⎟
Implication: For a given thermodynamic volume, the entropy of a black hole is maximized by the Schwarzschild-‐AdS sol’n
Utility? Works for most (charged, rotating) black holes
Counterexample! Super-‐entropic Black Holes
Black Hole Volume and EntropyRecall: Isoperimetric ratio
Super-‐Entropic Black Holes• New ultraspinning limit to the class of Kerr black hole metrics• Non-‐compact horizons with finite area • Asymptotically AdS, but with boundary rotating at the speed of light• Obtained in the context of Black Hole Chemistry• First counterexamples to the Reverse Isoperimetric Inequality
à Super-‐entropic!• Many other examples exist
ds2 = − ΔΣ
dt − l sin2θdψ⎡⎣ ⎤⎦2+ ΣΔdr2 + Σ
sin2θdθ 2 + sin
4θΣ
ldt − (r2 + l2 )dψ⎡⎣ ⎤⎦2
A − qrΣ
dt − l sin2θdψ( )Σ = r2 + l2 cos2θ
Δ = (l + r2
l)2 − 2mr + q2
D. Klemm PRD 89(2014) 048007
Hennigar/Kubiznak/MannPhys Rev Lett 115
(2015) 031101
Hennigar/Kubiznak, /Muskoe/Mann JHEP 1506 (2015) 096
Entropy Volume
S = µω d−24
(l2 + r+2 )r+
d−4 = A4
V = r+Ad −1
R =d −1( )Vµω d−2
⎛⎝⎜
⎞⎠⎟
1d−1 µω d−2
A⎛⎝⎜
⎞⎠⎟
1d−2
= r+2
l2 + r+2
⎛⎝⎜
⎞⎠⎟
1(d−1)(d−2)
Kerr-‐CFT Correspondence
Kerr-‐AdS Black Hole
Super-‐Entropic Black Hole
Kerr-‐CFT Limit
Super-‐Entropic Kerr-‐CFT
1) ψ = φ /Ξ 2) a→ l3) Compactify ψ ~ψ + µ
1) ψ = φ /Ξ 2) a→ l3) Compactify ψ ~ψ + µ
t = r0tε
r = r+ + εr0r
φ = φ +Ξω hr0t / ε
t = r0tε
r = r+ + εr0r
φ =ψ
C.M. Sinamuli, R.B. Mann JHEP 1608 (2016) 148 arXiv:1512.07597
ds2 = − ΔaΣa
dt − asin2θ
Ξdφ⎡
⎣⎢
⎤
⎦⎥
2
+ ΣaΔa
dr2 + ΣaSdθ 2 + S sin
2θΣa
adt − r2 + a2
Ξdφ⎡
⎣⎢
⎤
⎦⎥
2
A = − qrΣa
dt − asin2θ
Ξdφ⎛
⎝⎜⎞⎠⎟
Σa = r2 + a2 cos2θ , Ξ = 1− a
2
l2,
S = 1− a2
l2cos2θ Δa = (r
2 + a2 )(1+ r2
l2)− 2mr + q2
Kerr-Newman AdS Black Hole
Γ(θ ) = l2
2x2 + cos2θ
1+ 3x2 α (θ ) = 2
sin2θ(1+ 3x2 ) γ (θ ) = l2 sin4θ (1+ x
2 )2
x2 + cos2θ
r02 = l
2
21+ x2
1+ 3x2 k = x
(1+ x2 )(1+ 3x2 ) f (θ ) = q (1+ x
2 )x
x2 − cos2θx2 + cos2θ
x = r+l
ds2 = Γ(θ )[−r2dt 2 + dr2
r2+α (θ )dθ 2 + γ (θ )
Γ(θ )(dψ + krdt)2 ]
Super-‐Entropic Kerr-‐CFT Limit
A = f (θ )(dψ + krdt)
c = 3kµ2π
Γ(θ )γ (θ )α (θ )∫ dθ =3kµπ
l2 (1+ x2 )Central Charge
CardyFormula SCFT =
π 2
3cLTL S=
µπ2(l2 + r+
2 )
TR = 0
TL =1
2πk
CFT TemperaturesWorks in any
dimension and for Gauged-‐SUGRA
• What is the significance of the Reverse Isoperimetric Inequality Under what conditions does it hold?
• What are the underlying degrees of freedom of super-‐entropic black holes? Are there “more” degrees of freedom than we expect?
• Not all super-‐entropic black holes violate the Reverse Isoperimetric Inequality
à What is the meaning of the transition?
x = r+l
y = bl
R
NOT Super-‐entropicReverse Isoperimetric Inequality obeyed
Super-‐entropicExample: Doubly-‐spinning super-‐entropic Black Hole
horizon sizeorthogonalspin
Entropic Questions
Rq=0
12 = 127
⎛⎝⎜
⎞⎠⎟
(3x2 + y2 − 2x2y2 )3
x2 (1− y2 )2 (x2 +1)(x2 + y2 )
Thermodynamics of de Sitter Black Holes
P = − Λ
8π= − 3
8π1ℓ2
Negative Pressure (tension) in de Sitter Spacetime?
δM = ThδSh + (Ωhi −Ω∞
i )i∑ δ J i +VhδP
D − 3D − 2
M = ThSh + (Ωhi −Ω∞
i )i∑ J i − 2D − 2 PVh
SmarrRelation
First Law
Black Holes
δM = −TcδSc + (Ωci −Ω∞
i )i∑ δ J i +VcδP
D − 3D − 2
M = −TcSh + (Ωci −Ω∞
i )i∑ J i − 2D − 2 PVc
SmarrRelation
First Law
dS Horizon
Thermodynamics of Kerr de Sitter Black Holes
ds2 = −W (1− r2
ℓ2)dt 2 + 2m
UWdt − aiµi
2dϕiΞii=1
N∑
⎛⎝⎜
⎞⎠⎟
2
+ r2 + ai
2
Ξii=1N∑ µi
2dϕi2 + dµi
2( )
+ Udr2
X − 2m+ εr2dν 2 + 1
W (ℓ2 − r2 )r2+ai
2
Ξii=1N∑ µidµi+εr
2νdν⎛⎝⎜
⎞⎠⎟
2
W = µi
2
Ξii=1N∑ + ν 2
X = rε−2 (1− r
2
ℓ2) (r2 + ai
2 )i=1
N∏
2Λ = (D −1)(D − 2)
ℓ2
U = Zℓ
2
ℓ2 − r21− ai
2µi2
r2 + ai2
i=1
N∑
⎛⎝⎜
⎞⎠⎟
Ξi = 1+
ai2
ℓ2
= 1 D=even
0 D=odd ⎧⎨⎩
µi2
i=1
N∑ + ν 2 = 1
• Multiply-‐rotating Kerr de Sitter Black hole in D dimensions• 2 horizons at different temperatures
Cosmological Horizon Black Hole Horizon
Even Dim’l Kerr-‐dS Black HolesM = mωD−2
4π Ξ jj∏
1Ξii
∑ , Ji =maiωD−24πΞi Ξ j
j∏
Sc =ωD−24
rc2 + ai
2
Ξii∏ =
Ac4
Sh =ωD−24
rh2 + ai
2
Ξii∏ =
Ah4
Tc = −
rc2πℓ2
(ℓ2 − rc2 )
rc2 + ai
2i∑ +
ℓ2 + rc2
4πrc ℓ2
Th =
rh2πℓ2
(ℓ2 − rh2 )
rh2 + ai
2i∑ −
ℓ2 + rh2
4πrhℓ2
Ωc
i = (ℓ2 − rc
2 )aiℓ2 rc
2 + ai2( )
Ωhi = (ℓ
2 − rh2 )ai
ℓ2 rh2 + ai
2( )Vc =
rcAcD −1
+ 8π(D −1)(D − 2)
aii∑ Ji . Vh =
rhAhD −1
+ 8π(D −1)(D − 2)
aii∑ Ji .
2m = 1
ℓ2rc(ℓ2 − rc
2 ) (rc2 + ai
2 )i∏ =
1ℓ2rh
(ℓ2 − rh2 ) (rh
2 + ai2 )
i∏
Cosmological Horizon Black Hole Horizon
Odd Dim’l Kerr-‐dS Black HolesM = mωD−2
4π Ξ jj∏
1Ξii
∑ −12
⎛⎝⎜
⎞⎠⎟
Ji =maiωD−2
4πΞi Ξ jj∏
Sc =ωD−24rc
rc2 + ai
2
Ξii∏ =
Ac4
Sh =ωD−24rh
rh2 + ai
2
Ξii∏ =
Ah4
Tc = −
rc2πℓ2
(ℓ2 − rc2 )
rc2 + ai
2i∑ +
12πrc
Th =rh2πℓ2
(ℓ2 − rh2 )
rh2 + ai
2i∑ −
12πrh
Ωc
i = (ℓ2 − rc
2 )aiℓ2 rc
2 + ai2( )
Ωhi = (ℓ
2 − rh2 )ai
ℓ2 rh2 + ai
2( )Vc =
rcAcD −1
+ 8π(D −1)(D − 2)
aii∑ Ji . Vh =
rhAhD −1
+ 8π(D −1)(D − 2)
aii∑ Ji .
2m = 1
ℓ2rc(ℓ2 − rc
2 ) (rc2 + ai
2 )i∏ =
1ℓ2rh
(ℓ2 − rh2 ) (rh
2 + ai2 )
i∏
(Reverse) Isoperimetric Inequality?
R = (D −1)VωD−2
⎛⎝⎜
⎞⎠⎟
1 D−1ωD−2A
⎛⎝⎜
⎞⎠⎟1 D−2
R ≤1VA
Kerr-(A)dS Black Hole
Vh =
rhAhD −1
+ 8π(D −1)(D − 2)
aii∑ Ji =
rhAhD −1
1+ ℓ2 ± rh
2
(D − 2)ℓ2rh2
ai2
Ξii∑
⎡
⎣⎢
⎤
⎦⎥
Ah =
ωD−2rh1−ε
rc2 + ai
2
Ξii∏
RD−1 = rh 1+z
D − 2⎡⎣⎢
⎤⎦⎥1rh1−
(rh2 + ai
2 )Ξii
∏⎡
⎣⎢
⎤
⎦⎥
12−D
= 1+ zD − 2
⎡⎣⎢
⎤⎦⎥
(rh2 + ai
2 )rh2Ξii
∏⎡
⎣⎢
⎤
⎦⎥
12−D
≥ 1+ zD − 2
⎡⎣⎢
⎤⎦⎥
2D −1
1Ξii
∑ +ai2
rh2Ξii
∑⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥
D−14−2D
= 1+ zD − 2
⎡⎣⎢
⎤⎦⎥1+ 2z
D −1⎡⎣⎢
⎤⎦⎥
D−14−2D
≡ F(z)
F(0) = 1 dF(z)dz
> 0 F(z) ≥1 R ≥1
Cvetic/Gibbons/Kubiznak/Pope PRD84 (2011) 024037
Cosmic Volume
Can we understand cosmic volume without black hole volume?Yes! With cosmic solitons!
Clarkson/MannPRL 96 (2006) 051104
• Geometry depends on relative size of the soliton and the• No black hole horizon! • Can now have a cosmological horizon surrounding soliton• Obtained a number of results depending on mass/energy of the soliton and its size relative to the cosmic horizon
Mbarek/MannPLB 765 (2017) 352
Soliton: a bubble in spacetime!
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