[PPT] - Black holes, the Van der Waals gas, compressibility and the speed PowerPoint Presentation

SLIDE 1

Black holes, the Van der Waals gas, compressibility and the speed of sound

Brian P . Dolan

Dept. of Mathematics, Heriot-Watt University, Edinburgh

and Maxwell Institute for Mathematical Sciences, Edinburgh

EMPG, 25th Sept. 2013

Brian Dolan Black hole compressibility 0/20

SLIDE 2

Outline

Review of black hole thermodynamics Entropy and Temperature 1st and 2nd laws Hawking radiation Smarr relation Pressure and Enthalpy Enthalpy and the 1st law Volume Equation of state Critical behaviour Compressibility Speed of sound Conclusions

Brian Dolan Black hole compressibility 1/20

SLIDE 3

Entropy and Temperature

Entropy: S ∝

A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 4

Entropy and Temperature

Entropy: S ∝

A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 5

Entropy and Temperature

Entropy: S ∝

A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 6

Entropy and Temperature

Entropy: S ∝

A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 7

Entropy and Temperature

Entropy: S ∝

A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 8

Entropy and Temperature

Entropy: S ∝

A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 9

Entropy and Temperature

Entropy: S ∝

A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 10

Entropy and Temperature

Entropy: S = 1

4 A ℓ2

Pl (ℓ2

Pl = G/c3, G = c = 1).

Bekenstein (1972)

Temperature, T = κ

2π: κ =surface gravity.

Hawking (1974)

Schwarzschild black-hole: κ =

1 4M

T =

8πM .

Solar mass black hole: T = 6 × 10−8 K , S ≈ 1078.

Internal energy U(S): T = ∂U

∂S .

Identify M = U(S) ⇒ dM = TdS. Schwarzschild: rh = 2M, A = 4πr 2

h = 16πM 2,

dA = 32πMdM, TdA

ℓ2

Pl = 4dM. Brian Dolan Black hole compressibility 2/20

SLIDE 11

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 12

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 13

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 14

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 15

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 16

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 17

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 18

First and Second laws

More generally M = U(S, J, Q),

(angular momentum, electric charge),

First Law of Black Hole Thermodynamics

dM = T dS + ΩdJ + ΦdQ.

Can extract useful work in a Penrose process.
Kerr black hole (Q = 0):

T = 1 8πM

1 − 4π2 J 2

S2

.
Extremal: Jmax =

S 2π ⇒ T = 0.

Reduce J ⇒ extract energy.
Maximum efficiency for fixed S: η = 1 −

1 √ 2 ≈ 29%.

Second Law of Black Hole Thermodynamics

∆S ≥ 0.

Brian Dolan Black hole compressibility 3/20

SLIDE 19

Hawking radiation

Heat capacity: C = ∂U

∂T = −T ∂2F ∂T 2 .

Free energy, F (T ), is Legendre transform of U(S)

F = U − TS = M − κA 8π .

Schwarzschild:

F = M 2 =

16πT .
Heat capacity:

C = −8πM 2

= −2S < 0.

Negative!

Radiates with power P ∼ AT 4

3 ∼

M2 .

Lifetime: τ ∼ M

P ∼ M3 , M ∼ 1012 kg ⇒ τ ∼ 1010 years.

Fixing J or Q = stabilises the black hole.

Brian Dolan Black hole compressibility 4/20

SLIDE 20

Hawking radiation

Heat capacity: C = ∂U

∂T = −T ∂2F ∂T 2 .

Free energy, F (T ), is Legendre transform of U(S)

F = U − TS = M − κA 8π .

Schwarzschild:

F = M 2 =

16πT .
Heat capacity:

C = −8πM 2

= −2S < 0.

Negative!

Radiates with power P ∼ AT 4

3 ∼

M2 .

Lifetime: τ ∼ M

P ∼ M3 , M ∼ 1012 kg ⇒ τ ∼ 1010 years.

Fixing J or Q = stabilises the black hole.

Brian Dolan Black hole compressibility 4/20

SLIDE 21

Hawking radiation

Heat capacity: C = ∂U

∂T = −T ∂2F ∂T 2 .

Free energy, F (T ), is Legendre transform of U(S)

F = U − TS = M − κA 8π .

Schwarzschild:

F = M 2 =

16πT .
Heat capacity:

C = −8πM 2

= −2S < 0.

Negative!

Radiates with power P ∼ AT 4

3 ∼

M2 .

Lifetime: τ ∼ M

P ∼ M3 , M ∼ 1012 kg ⇒ τ ∼ 1010 years.

Fixing J or Q = stabilises the black hole.

Brian Dolan Black hole compressibility 4/20

SLIDE 22

Hawking radiation

Heat capacity: C = ∂U

∂T = −T ∂2F ∂T 2 .

Free energy, F (T ), is Legendre transform of U(S)

F = U − TS = M − κA 8π .

Schwarzschild:

F = M 2 =

16πT .
Heat capacity:

C = −8πM 2

= −2S < 0.

Negative!

Radiates with power P ∼ AT 4

3 ∼

M2 .

Lifetime: τ ∼ M

P ∼ M3 , M ∼ 1012 kg ⇒ τ ∼ 1010 years.

Fixing J or Q = stabilises the black hole.

Brian Dolan Black hole compressibility 4/20

SLIDE 23

Hawking radiation

Heat capacity: C = ∂U

∂T = −T ∂2F ∂T 2 .

Free energy, F (T ), is Legendre transform of U(S)

F = U − TS = M − κA 8π .

Schwarzschild:

F = M 2 =

16πT .
Heat capacity:

C = −8πM 2

= −2S < 0.

Negative!

Radiates with power P ∼ AT 4

3 ∼

M2 .

Lifetime: τ ∼ M

P ∼ M3 , M ∼ 1012 kg ⇒ τ ∼ 1010 years.

Fixing J or Q = stabilises the black hole.

Brian Dolan Black hole compressibility 4/20

SLIDE 24

Hawking radiation

Heat capacity: C = ∂U

∂T = −T ∂2F ∂T 2 .

Free energy, F (T ), is Legendre transform of U(S)

F = U − TS = M − κA 8π .

Schwarzschild:

F = M 2 =

16πT .
Heat capacity:

C = −8πM 2

= −2S < 0.

Negative!

Radiates with power P ∼ AT 4

3 ∼

M2 .

Lifetime: τ ∼ M

P ∼ M3 , M ∼ 1012 kg ⇒ τ ∼ 1010 years.

Fixing J or Q = stabilises the black hole.

Brian Dolan Black hole compressibility 4/20

SLIDE 25

Hawking radiation

Heat capacity: C = ∂U

∂T = −T ∂2F ∂T 2 .

Free energy, F (T ), is Legendre transform of U(S)

F = U − TS = M − κA 8π .

Schwarzschild:

F = M 2 =

16πT .
Heat capacity:

C = −8πM 2

= −2S < 0.

Negative!

Radiates with power P ∼ AT 4

3 ∼

M2 .

Lifetime: τ ∼ M

P ∼ M3 , M ∼ 1012 kg ⇒ τ ∼ 1010 years.

Fixing J or Q = stabilises the black hole.

Brian Dolan Black hole compressibility 4/20

SLIDE 26

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 27

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 28

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 29

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 30

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 31

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 32

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 33

Smarr relation

Smarr (1973)

Ordinary thermodynamics: U(S, V , n) (n = number of

moles) is a function of extensive variables. U is also extensive ⇒ λdU(S, V , n) = U(λdS, λdV , λdn) ⇒ U = S ∂U ∂S + V ∂U ∂V + n ∂U ∂n Euler equation ⇒ U = ST − VP + nµ (µ = chemical potential) ⇒ G = U + VP − ST = nµ . Gibbs-Duhem relation

Black hole in D dimensions, angular momenta Ji:

(Q = 0) S → λD−2S, Ji → λD−2Ji, M → λD−3M ⇒ λD−3M(S, Ji) = M(λD−2S, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST + (D − 2)Ji Ωi.

Brian Dolan Black hole compressibility 5/20

SLIDE 34

Smarr relation

Doesn’t work when there’s a Cosmological Constant!
Solution:

(Kastor, Ray+Traschen [0904.2765])

include Λ as a thermodynamic variable, Λ → λ−2Λ,

(Henneaux+Teitelboim (1984)).

λD−3M(S, Λ, Ji) = M(λD−2S, λ−2Λ, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S − 2Λ∂M ∂Λ + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST − 2ΛΘ + (D − 2)Ji Ωi Θ := ∂M

∂Λ

S,J.

Brian Dolan Black hole compressibility 6/20

SLIDE 35

Smarr relation

Doesn’t work when there’s a Cosmological Constant!
Solution:

(Kastor, Ray+Traschen [0904.2765])

include Λ as a thermodynamic variable, Λ → λ−2Λ,

(Henneaux+Teitelboim (1984)).

λD−3M(S, Λ, Ji) = M(λD−2S, λ−2Λ, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S − 2Λ∂M ∂Λ + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST − 2ΛΘ + (D − 2)Ji Ωi Θ := ∂M

∂Λ

S,J.

Brian Dolan Black hole compressibility 6/20

SLIDE 36

Smarr relation

Doesn’t work when there’s a Cosmological Constant!
Solution:

(Kastor, Ray+Traschen [0904.2765])

include Λ as a thermodynamic variable, Λ → λ−2Λ,

(Henneaux+Teitelboim (1984)).

λD−3M(S, Λ, Ji) = M(λD−2S, λ−2Λ, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S − 2Λ∂M ∂Λ + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST − 2ΛΘ + (D − 2)Ji Ωi Θ := ∂M

∂Λ

S,J.

Brian Dolan Black hole compressibility 6/20

SLIDE 37

Smarr relation

Doesn’t work when there’s a Cosmological Constant!
Solution:

(Kastor, Ray+Traschen [0904.2765])

include Λ as a thermodynamic variable, Λ → λ−2Λ,

(Henneaux+Teitelboim (1984)).

λD−3M(S, Λ, Ji) = M(λD−2S, λ−2Λ, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S − 2Λ∂M ∂Λ + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST − 2ΛΘ + (D − 2)Ji Ωi Θ := ∂M

∂Λ

S,J.

Brian Dolan Black hole compressibility 6/20

SLIDE 38

Smarr relation

Doesn’t work when there’s a Cosmological Constant!
Solution:

(Kastor, Ray+Traschen [0904.2765])

include Λ as a thermodynamic variable, Λ → λ−2Λ,

(Henneaux+Teitelboim (1984)).

λD−3M(S, Λ, Ji) = M(λD−2S, λ−2Λ, λD−2Ji) (D − 3)M = (D − 2)S ∂M ∂S − 2Λ∂M ∂Λ + (D − 2)Ji ∂M ∂Ji ⇒ (D − 3)M = (D − 2)ST − 2ΛΘ + (D − 2)Ji Ωi Θ := ∂M

∂Λ

S,J.

Brian Dolan Black hole compressibility 6/20

SLIDE 39

Enthalpy

Pressure P = − Λ

8π ⇒ mass is the enthalpy:

Enthalpy

M = H(S, P, J)

(Kastor, Ray+Traschen [0904.2765])

Internal energy U(S, V , J) is the Legendre transform:

U = H − PV where V := ∂M

∂P is a thermodynamic volume.

First Law of thermodynamics

dU = T dS − P dV + ΩdJ

For Λ < 0, Penrose process can be up to 51.8% efficient

(29% for Λ = 0).

(BPD [1008.5023])

Brian Dolan Black hole compressibility 7/20

SLIDE 40

Enthalpy

Pressure P = − Λ

8π ⇒ mass is the enthalpy:

Enthalpy

M = H(S, P, J)

(Kastor, Ray+Traschen [0904.2765])

Internal energy U(S, V , J) is the Legendre transform:

U = H − PV where V := ∂M

∂P is a thermodynamic volume.

First Law of thermodynamics

dU = T dS − P dV + ΩdJ

For Λ < 0, Penrose process can be up to 51.8% efficient

(29% for Λ = 0).

(BPD [1008.5023])

Brian Dolan Black hole compressibility 7/20

SLIDE 41

Enthalpy

Pressure P = − Λ

8π ⇒ mass is the enthalpy:

Enthalpy

M = H(S, P, J)

(Kastor, Ray+Traschen [0904.2765])

Internal energy U(S, V , J) is the Legendre transform:

U = H − PV where V := ∂M

∂P is a thermodynamic volume.

First Law of thermodynamics

dU = T dS − P dV + ΩdJ

For Λ < 0, Penrose process can be up to 51.8% efficient

(29% for Λ = 0).

(BPD [1008.5023])

Brian Dolan Black hole compressibility 7/20

SLIDE 42

Enthalpy

Pressure P = − Λ

8π ⇒ mass is the enthalpy:

Enthalpy

M = H(S, P, J)

(Kastor, Ray+Traschen [0904.2765])

Internal energy U(S, V , J) is the Legendre transform:

U = H − PV where V := ∂M

∂P is a thermodynamic volume.

First Law of thermodynamics

dU = T dS − P dV + ΩdJ

For Λ < 0, Penrose process can be up to 51.8% efficient

(29% for Λ = 0).

(BPD [1008.5023])

Brian Dolan Black hole compressibility 7/20

SLIDE 43

Enthalpy

Pressure P = − Λ

8π ⇒ mass is the enthalpy:

Enthalpy

M = H(S, P, J)

(Kastor, Ray+Traschen [0904.2765])

Internal energy U(S, V , J) is the Legendre transform:

U = H − PV where V := ∂M

∂P is a thermodynamic volume.

First Law of thermodynamics

dU = T dS − P dV + ΩdJ

For Λ < 0, Penrose process can be up to 51.8% efficient

(29% for Λ = 0).

(BPD [1008.5023])

Brian Dolan Black hole compressibility 7/20

SLIDE 44

Thermodynamic Volume

In 4 − D the thermodynamic volume evaluates to

V = 1 3

rhAh + 4πJ2

M

,

where Ah is area of the event horizon and rh the radius

(Cveti˘ c et al. [1012.2888], BPD [1106.6260]).

Reverse Isometric inequality,
As J → 0, Ah → 4πr 2

h and

V → V0 := 4π 3 r 3

h = 4

3 S3 π 1/2 , As J → 0, V and S are not independent.

Brian Dolan Black hole compressibility 8/20

SLIDE 45

Thermodynamic Volume

In 4 − D the thermodynamic volume evaluates to

V = 1 3

rhAh + 4πJ2

M

,

where Ah is area of the event horizon and rh the radius

(Cveti˘ c et al. [1012.2888], BPD [1106.6260]).

Reverse Isometric inequality,
As J → 0, Ah → 4πr 2

h and

V → V0 := 4π 3 r 3

h = 4

3 S3 π 1/2 , As J → 0, V and S are not independent.

Brian Dolan Black hole compressibility 8/20

SLIDE 46

Thermodynamic Volume

In 4 − D the thermodynamic volume evaluates to

V = 1 3

rhAh + 4πJ2

M

,

where Ah is area of the event horizon and rh the radius

(Cveti˘ c et al. [1012.2888], BPD [1106.6260]).

Reverse Isometric inequality,
As J → 0, Ah → 4πr 2

h and

V → V0 := 4π 3 r 3

h = 4

3 S3 π 1/2 , As J → 0, V and S are not independent.

Brian Dolan Black hole compressibility 8/20

SLIDE 47

Thermodynamic Volume

In 4 − D the thermodynamic volume evaluates to

V = 1 3

rhAh + 4πJ2

M

,

where Ah is area of the event horizon and rh the radius

(Cveti˘ c et al. [1012.2888], BPD [1106.6260]).

Reverse Isometric inequality,
As J → 0, Ah → 4πr 2

h and

V → V0 := 4π 3 r 3

h = 4

3 S3 π 1/2 , As J → 0, V and S are not independent.

Brian Dolan Black hole compressibility 8/20

SLIDE 48

Equation of state (D = 4, Λ < 0)

Critical point (J = 0):

(PJ)crit ≈ 0.002857 ,

T

√ J

crit ≈ 0.04175 ,

V J3/2

crit

≈ 115.8

Define

t := T − Tc Tc , v := V − Vc Vc , p := P − Pc Pc . Expand the equation of state about the critical point: p = 2.42t − 0.81tv − 0.21v 3 + o(t 2, tv 2, v 4).

cf. Van der Waals gas: p = 4t − 6tv − 3

2v 3 + o(t 2, tv 2, v 4).

(Gunasekaran et al. [1208.6251], BPD [1209.1272]).

Brian Dolan Black hole compressibility 9/20

SLIDE 49

Equation of state (D = 4, Λ < 0)

Critical point (J = 0):

(PJ)crit ≈ 0.002857 ,

T

√ J

crit ≈ 0.04175 ,

V J3/2

crit

≈ 115.8

Define

t := T − Tc Tc , v := V − Vc Vc , p := P − Pc Pc . Expand the equation of state about the critical point: p = 2.42t − 0.81tv − 0.21v 3 + o(t 2, tv 2, v 4).

cf. Van der Waals gas: p = 4t − 6tv − 3

2v 3 + o(t 2, tv 2, v 4).

(Gunasekaran et al. [1208.6251], BPD [1209.1272]).

Brian Dolan Black hole compressibility 9/20

SLIDE 50

Equation of state (D = 4, Λ < 0)

Critical point (J = 0):

(PJ)crit ≈ 0.002857 ,

T

√ J

crit ≈ 0.04175 ,

V J3/2

crit

≈ 115.8

Define

t := T − Tc Tc , v := V − Vc Vc , p := P − Pc Pc . Expand the equation of state about the critical point: p = 2.42t − 0.81tv − 0.21v 3 + o(t 2, tv 2, v 4).

cf. Van der Waals gas: p = 4t − 6tv − 3

2v 3 + o(t 2, tv 2, v 4).

(Gunasekaran et al. [1208.6251], BPD [1209.1272]).

Brian Dolan Black hole compressibility 9/20

SLIDE 51

Equation of state (D = 4, Λ < 0)

Critical point (J = 0):

(PJ)crit ≈ 0.002857 ,

T

√ J

crit ≈ 0.04175 ,

V J3/2

crit

≈ 115.8

Define

t := T − Tc Tc , v := V − Vc Vc , p := P − Pc Pc . Expand the equation of state about the critical point: p = 2.42t − 0.81tv − 0.21v 3 + o(t 2, tv 2, v 4).

cf. Van der Waals gas: p = 4t − 6tv − 3

2v 3 + o(t 2, tv 2, v 4).

(Gunasekaran et al. [1208.6251], BPD [1209.1272]).

Brian Dolan Black hole compressibility 9/20

SLIDE 52

P − V diagram

0.003 8 0.002 0.001 6 4 2 0.005 10 0.004

P as a function of

3V

4π

1/3 , curves of constant T for J = 1.

Brian Dolan Black hole compressibility 10/20

SLIDE 53

Gibbs Free Energy

Gibbs Free Energy, G(T , P, J) = H(S, P, J) − TS: (J = 1)

T P G Brian Dolan Black hole compressibility 11/20

SLIDE 54

Critical exponents

CV = T / ∂T

∂S

V ,J ∝ t −α;
At fixed p < 0, v> − v< ∝ |t|β;
Isothermal compressibility, κT = − 1

V

∂V

∂P

T ,J ∝ t −γ;
At t = 0, |p| ∝ |v|δ;

Mean Field Exponents

α = 0, β = 1 2, γ = 1, δ = 3. Same as Van der Waals gas.

Brian Dolan Black hole compressibility 12/20

SLIDE 55

Critical exponents

CV = T / ∂T

∂S

V ,J ∝ t −α;
At fixed p < 0, v> − v< ∝ |t|β;
Isothermal compressibility, κT = − 1

V

∂V

∂P

T ,J ∝ t −γ;
At t = 0, |p| ∝ |v|δ;

Mean Field Exponents

α = 0, β = 1 2, γ = 1, δ = 3. Same as Van der Waals gas.

Brian Dolan Black hole compressibility 12/20

SLIDE 56

Critical exponents

CV = T / ∂T

∂S

V ,J ∝ t −α;
At fixed p < 0, v> − v< ∝ |t|β;
Isothermal compressibility, κT = − 1

V

∂V

∂P

T ,J ∝ t −γ;
At t = 0, |p| ∝ |v|δ;

Mean Field Exponents

α = 0, β = 1 2, γ = 1, δ = 3. Same as Van der Waals gas.

Brian Dolan Black hole compressibility 12/20

SLIDE 57

Critical exponents

CV = T / ∂T

∂S

V ,J ∝ t −α;
At fixed p < 0, v> − v< ∝ |t|β;
Isothermal compressibility, κT = − 1

V

∂V

∂P

T ,J ∝ t −γ;
At t = 0, |p| ∝ |v|δ;

Mean Field Exponents

α = 0, β = 1 2, γ = 1, δ = 3. Same as Van der Waals gas.

Brian Dolan Black hole compressibility 12/20

SLIDE 58

Critical exponents

CV = T / ∂T

∂S

V ,J ∝ t −α;
At fixed p < 0, v> − v< ∝ |t|β;
Isothermal compressibility, κT = − 1

V

∂V

∂P

T ,J ∝ t −γ;
At t = 0, |p| ∝ |v|δ;

Mean Field Exponents

α = 0, β = 1 2, γ = 1, δ = 3. Same as Van der Waals gas.

Brian Dolan Black hole compressibility 12/20

SLIDE 59

Kerr-Reissner-Nordström-AdS

Large and small black holes Unique black hole

J Q

Line of second order critical points

Reissner-Nordström anti-de Sitter (J = 0, Q = 0).

(Chamblin, Emparan, Johnston + Myers [hep-th/9902170]; Caldarelli, Gognola+Klemm [hep-th/9908022]; BPD [1209.1272].)

Brian Dolan Black hole compressibility 13/20

SLIDE 60

Compressibility

Asymptotically AdS Myers-Perry in D-dimensions:

(rotation parameters ai, for Λ = 0, ai = D−2

2 Ji M ).

V = 1 D − 1

rhAh +

8π (D − 2)

i=1

aiJi

(Cveti˘

c et al. [1012.2888], BPD [1106.6260]).

Brian Dolan Black hole compressibility 14/20

SLIDE 61

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 62

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 63

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 64

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 65

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 66

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 67

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 68

Compressibility

Adiabatic compressibility: κ = − 1

V ∂V ∂P

S,J.
Rotating black-hole in D-dimensions (Myers-Perry).

Dimensionless angular momenta, Ji := 2πJi

S ,

Constraint: T ≥ 0 ⇒

i 1 1+J 2

i ≥

1

2

even D 1

dd D.

Compressibility, Λ → 0

κ = 16πr 2

h

(D − 1)(D − 2)2

(D − 2)

i J 4 i − ( i J 2 i )2

D − 2 +

i J 2 i

,

(BPD [1308.5403])

0 ≤ κ < ∞.
e.g. 4-D with P = 0, κmax = 2.6 × 10−38

M M⊙

2 m s2 kg−1.

cf. neutron star, M ≈ M⊙, degenerate Fermi gas

⇒ κ ≈ 10−34m s2 kg−1.

Brian Dolan Black hole compressibility 15/20

SLIDE 69

Example of compressibility, D = 6, SO(5): J1, J2,

2
4
4

j_2 4 2 2

2

4 j_1

4
2

j_2 2 4

4
2

2 j_1 4 2 4 6 8 10 12 Brian Dolan Black hole compressibility 16/20

SLIDE 70

Speed of sound

Define ρ := M

V , then the thermodynamic speed of sound is

c2

s = ∂P

∂ρ

S,J

, ⇒ c−2

s

= 1 + κρ.

Speed of sound

c2

s =

1 (D − 2)

D − 2 +

i J 2 i

2

D − 2 + 2

i J 2 i + i J 4 i

.

1

D−2 ≤ c2 s ≤ 1.

Brian Dolan Black hole compressibility 17/20

SLIDE 71

Speed of sound

Define ρ := M

V , then the thermodynamic speed of sound is

c2

s = ∂P

∂ρ

S,J

, ⇒ c−2

s

= 1 + κρ.

Speed of sound

c2

s =

1 (D − 2)

D − 2 +

i J 2 i

2

D − 2 + 2

i J 2 i + i J 4 i

.

1

D−2 ≤ c2 s ≤ 1.

Brian Dolan Black hole compressibility 17/20

SLIDE 72

c2

s in 6-dimensions:

4
2

J_1 2

4

J_2

2

4 0.2 2 0.4 4 0.6 0.8 1 Brian Dolan Black hole compressibility 18/20

SLIDE 73

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 74

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 75

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 76

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 77

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 78

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 79

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 80

Asymptotically de Sitter

BPD, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen [1301.5926]

P = − Λ

8π < 0.

Two event horizons: black hole rbh; cosmological, rc.
Two different temperatures, Tbh = Tc, in general.
M(Sbh, P, J) = M(Sc, P, J).

Vbh = ∂M

∂P

Sbh,J , Vc = ∂M

∂P

Sc,J .
Vbh = rbhAbh

3

+ 4π

3 J2 M ,

Vc = rcAc

3

+ 4π

3 J2 M .

For fixed Vc the cosmological horizon entropy, Sc = Ac

4 , is

maximized by Schwarzschild- de Sitter space-time (J = 0).

Volume between horizons: V = Vc − Vbh = rcAc

3

− rbhAbh

3

.

For fixed V the total entropy, S = Sbh + Sc, is minimized if

there is no black hole.

Brian Dolan Black hole compressibility 19/20

SLIDE 81

Conclusions

Λ = 0

⇒ P dV term in black hole 1st law.

Black hole mass is identified with enthalpy, H(S, P, Ji):

dM = dH = T dS + V dP + ΩiJi , dU = T dS − P dV + ΩiJi .

Hawking temperature: T =
∂H

∂S

P.
“Thermodynamic” volume: V =
∂H

∂P

T .
PdV term affects Penrose processes — more efficient in

asymptotically AdS space-times.

D = 4: Van der Waals type equation of state.
Compressibility, 0 ≤ κ < ∞, with κ → ∞ for some Ji → ∞.
Instability of ultra-spinning black-holes.
Speed of sound:

1 D−2 ≤ c2 s ≤ 1.

Brian Dolan Black hole compressibility 20/20

SLIDE 82

Ripples on the horizon

Ultra-spinning black-holes are unstable

(Emparan + Myers hep-th/0308056).

Λ = 0, D ≥ 6:

(Emparan, Harmark, Niarchos, Obers + Rodriguez [hep-th:0708.2181]).

Brian Dolan Black hole compressibility 21/20

SLIDE 83

Ripples on the horizon

Ultra-spinning black-holes are unstable

(Emparan + Myers hep-th/0308056).

Λ = 0, D ≥ 6:

(Emparan, Harmark, Niarchos, Obers + Rodriguez [hep-th:0708.2181]). J S

Brian Dolan Black hole compressibility 21/20