General proof of the entropy principle for self-gravitating fluid in static

spacetimes高思杰 (Gao Sijie)

北京师范大学(Beijing Normal University)

23/4/19 2014 Institute of Physics, Academia Sinica 1

Outline

1. Introduction2. Entropy principle in spherical case --radiation 3. Entropy principle in spherical case –perfect

fluid4. Entropy principle in static spacetime5. Related works6. Conclusions.

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1. Introduction

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Mathematical analogy beween thermodynamics and black holes:

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What is the relationship between ordinary thermodynamics and gravity?

We shall study thermodynamics of self-gravitating fluid in curved spacetime.

fluid

,S M N，S: total entropy of fluidM: total mass of fluid N: total particle number

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There are two ways to determine the distribution of the fluid:

1. General relativity: Einstein’s equation gives Tolman-Oppenheimer-Volkoff (TOV ) equation:

2. Thermodynamics: at thermal equilibrium.

Are they consistent?

Consider a self-gravitating perfect fluid with spherical symmetry in thermal equilibrium:

2. Entropy principle in spherical case---radiation Sorkin, Wald, Zhang, Gen.Rel.Grav. 13, 1127 (1981)

In 1981, Sorkin, Wald, and Zhang (SWZ) derived the TOV equation of a self-gravitating radiation from the maximum entropy principle.

Proof: The stress-energy tensor is given by

The radiation satisfies:

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Assume the metric of the spherically symmetric radiation takes the form

The constraint Einstein equation yields

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Since , the extrema of is equivalent to the Euler-Lagrange equation:

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Using to replace , , we arrive at the TOV

equation

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3. Entropy principle in spherical case---general perfect fluid

(Sijie Gao, arXiv:1109.2804, Phys. Rev. D 84, 104023 )

• To generalize SWZ’s treatment to a general fluid, we first need to find an expression for the entropy density .

• The first law of the ordinary thermodynamics: Rewrite in terms of densities:

Expand: The first law in a unit volume:

s

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Thus, we have the Gibbs-Duhem relation

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Note that

Thus, 23/4/19 14

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4.Proof of the entropy principle for perfect fluid in static spacetimes

arXiv: 1311.6899

• In this work, we present two theorems relating the total entropy of fluid to Einstein’s equation in any static spacetimes.

• A static spacetime admits a timelike Killing vector field which is hypersurface orthogonal.

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a

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Proof of Theorem 1

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The total entropy

Its variation:

Total number of particle:

The constraint

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Then

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(Constraint Einstein equation)

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Integration by parts:

Integration by parts again and dropping the boundary terms:

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5. Related works• Proof for stationary case----in process• Stability analysis (1) Z.Roupas [Class. Quantum Grav. 30, 115018 (2013)] calculated the

second variation of entropy, showing that the stability of thermal equilibrium is equivalent to stability of Einstein’s equations.

(2) Wald et. al. [Class. Quantum Grav. 31 (2014) 035023 ] proved the equivalence of dynamic equibrium and thermodynamic equibrium for stationary asymtotically flat spacetimes with axisymmetry.

• Beyond general relativity: Li-Ming Cao, Jianfei Xu, Zhe Zeng [Phys. Rev. D 87, 064005 (2013)] proved

the maximum entropy principle in the framework of Lovelock gravity.

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6. Conclusions

• We have rigorously proven the equivalence of the extrema of entropy and Einstein's equation under a few natural and necessary conditions. The significant improvement from previous works is that no spherical symmetry or any other symmetry is needed on the spacelike hypersurface. Our work suggests a clear connection between Einstein's equation and thermodynamics of perfect fluid in static spacetimes.

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Thank you!

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