Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

The zeroth law of black hole thermodynamicsrevisited

Christine Gruberin collaboration with A. Bravetti, C. Lopez-Monsalvo, F. Nettel

RTG-Workshop, JUB BremenFeb. 2017

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Zeroth law of thermodynamics

... defines a notion of equilibrium:

... and of temperature.(intensive quantity!)

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Zeroth law of thermodynamics

... defines a notion of equilibrium:

... and of temperature.(intensive quantity!)

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Black hole thermodynamics I

Analogy to ordinary TD:energy... U ↔ M ... mass

temperature... T ↔ κ ... surface gravityentropy... S ↔ A ... horizon area

..... ↔ .....

First law of BH TD:dM = TdS + ΩdJ + ΦdQ

Smarr relation: [1]

(D− 3)M = (D− 2)TS + (D− 2)ΩJ + (D− 3)ΦQ

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Black hole thermodynamics II

Conjugate variables: intensive↔ extensive!

T =∂M∂S

, Ω =∂M∂J

, Φ =∂M∂Q

Fundamental relation:

S(M, J,Q) = π(

2M2 − Q2 + 2√

+M4 −M2Q2 − J2)

=⇒ entropy representation:S,M, J,Q,

1T,−Ω

T,−Φ

T

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Black hole thermodynamics: AdS extension

AdS black holes: [1, 2]

pressure... P ↔ Λ ... cosmological constantvolume... V ↔ Vth ... thermal volume of the bh

First law: derived from geometric properties.

dM = TdS + ΩdJ + ΦdQ + VdP ≡ dH

=⇒ mass equals enthalpy: M = H = U + PV

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Phase transitions in black hole systems?

Hawking-Page phase transition: [3, 4]

between thermal AdS space and large black hole solution.Calculating the free energy!

Kerr-AdS: [1, 2]

Van der Waals-typephase transitionin P-V-diagram!Swallow tail inGibbs free energy.

0 100 200 300 400 5000.000

0.002

0.004

0.006

0.008

V

P(V,T)

T=0.035

T=0.037

T=0.04

T=0.043

T=0.045

T=0.05

0 100 200 300 400 500

2.452.502.552.602.652.702.75

V

P(V,T)[per103]

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Questions:

Is the P-V-diagram at constant T transferable toblack hole thermodynamics?

mIs T a valid intensive quantity to measure

thermodynamic equilibrium?

Suppose:S... thermodyn. potential, qi... extensive vars., pi... intensive vars.

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

(Quasi-)Homogeneity in thermodynamics

Quasi-homogeneity: [5, 6] of degree r and type β

S(λβ1q1, ..., λβnqn) = λrS(q1, ..., qn)

Homogeneity of degree 1: S(λU, λV, λN) = λS(U,V,N)

Intensive thermodynamic variables:

pi(qj) ≡ ∂S

(qj)

∂qi =⇒ pi

(λβjqj

)= λr−βipi(qj)

... quasi-homogeneous of degree r − βi.

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Euler’s theorem

For quasi-homogeneous functions holds:

rS(qj) =

n∑i=1

βi[qipi(qj)

]

Smarr relation for black holes ≡ Euler relation

S =12

1T

M − 12

Ω

TJ − Φ

TQ +

12

PT

V

⇓read off degree and type!

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Gibbs-Duhem relation

... obtained by combining Smarr relation with first law.Usually:

Ud(

1T

)− Vd

(PT

)+ Nd

(µT

)= 0

Generalized Gibbs-Duhem:n∑

i=1

[(βi − r)pi(qj)dqi︸ ︷︷ ︸

(1)

+βiqidpi(qj)︸ ︷︷ ︸(2)

]= 0

=⇒ (1) Variation of extensive quantities qi

=⇒ (2) Usual variation of intensive quantities pi

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Mathematical inconsistency

Is the generalized Gibbs-Duhem relation fulfilledin the quasi-homogeneous case?

Definition of equilibrium:

dpi = 0 =⇒ (2) = 0

Homogeneous TD: βi = r = 1 =⇒ (1) = 0Quasi-homogeneous TD:

βi, r 6= 1 =⇒ (1) 6= 0 E(1) related to latent heat in phase transitions!

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Generalized intensities

Redefine equilibrium!

pi(qj) ≡[(

qi)βi−r]1/βi

pi(qj)

... obtained by solving the generalized Gibbs-Duhem relation.

... one particular solution, more general intensities possible.

... quasi-homogeneous of degree 0.

... reduce to pi in the case βi = r = 1.

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Generalized Gibbs-Duhem relation

Rewrite the generalized Gibbs-Duhem relation:n∑

i=1

[(βi − r) pi(qj)dqi + βiqidpi(qj)

]=

≡n∑

i=1

βi(qi)r/βi dpi(qj)︸ ︷︷ ︸

=0

= 0

Fulfilled by new definition of equilibrium!

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

The simplest case: not quite significant?

Schwarzschild entropy: S(M) = 4πM2 ⇒ r = 2, β = 1

Usual temperature:1T

=∂S∂M

= 8πM

... homogeneous of degree 1! Not truly intensive!

New definition of equilibrium:

1T=(Mβ−r)1/β 1

T=

1TM

= 8π

... trivially intensive!15

Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

The popular case: Kerr-AdS

Fundamental potential: H(S,P, J = 1)"Intensive" quantities T, P not (quasi-)homogeneous ofdegree 0.Explicit calculation: generalized Gibbs-Duhem not fulfilledPhase transition: swallow tail in Gibbs free energyG(T,P, J) = U − TS + PV

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

The popular case: Kerr-AdS

Introduce new intensive quantities:

T = TU and P = PUV1/3

... quasi-homogeneous of degree 0!

No PT!0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

V

P˜(V,T˜)

T˜=0.2

T˜=0.3

T˜=0.035

T˜=0.4

T˜=0.5

T˜=0.6

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Another interesting case: Hawking-Page?

PT in AdS black holes: J = 0

But: S and V are not independent variables!(3V4π

)2

=

(Sπ

)3

=⇒ P(V,T) and G(T,P) cannot be calculated!

PT structure has been found in different way!Method not applicable?

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

Conclusions

=⇒ Introduction of true intensities is importantin quasi-homogeneous thermodynamics.

Revise notion of TD equilibrium in general?Thermodynamic response functions?Extend methodology?Fundamental meaning?

Thanks! PLB 774, 417-424 (2017)

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Introduction & Motivation Quasi-homogeneous thermodynamics Generalized intensities Examples

References I

B P Dolan.Where is the PdV term in the fist law of black hole thermodynamics?arXiv preprint arXiv:1209.1272, 2012.

D Kastor, S Ray, and J Traschen.Enthalpy and the mechanics of AdS black holes.Classical and Quantum Gravity, 26(19):195011, 2009.

S. W. Hawking and D. N. Page.Thermodynamics of black holes in anti-de sitter space.Commun. Math. Phys., 87:577–588, 1983.

T. Hartman.Gr lecture notes.2015.

F Belgiorno.Quasi-homogeneous thermodynamics and black holes.Journal of Mathematical Physics, 44(3):1089–1128, 2003.

F Belgiorno and SL Cacciatori.General symmetries: From homogeneous thermodynamics to black holes.The European Physical Journal Plus, 126(9):1–19, 2011.

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