General proof of the entropy principle for self gravitating fluid in static spacetimes
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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). Outline. Introduction Entropy principle in spherical case --radiation Entropy principle in spherical case –perfect fluid

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

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

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)

2014 Institute of Physics, Academia Sinica


Outline

Outline

  • Introduction

  • Entropy principle in spherical case --radiation

  • Entropy principle in spherical case –perfect fluid

  • Entropy principle in static spacetime

  • Related works

  • Conclusions.

2014 Institute of Physics, Academia Sinica


1 introduction

1. Introduction

Mathematical analogy beween thermodynamics and black holes:

2014 Institute of Physics, Academia Sinica


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

2014 Institute of Physics, Academia Sinica


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

What is the relationship between ordinary thermodynamics and gravity?

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

2014 Institute of Physics, Academia Sinica


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

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

S: total entropy of fluid

M: total mass of fluid

N: total particle number

fluid

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?


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

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:

2014 Institute of Physics, Academia Sinica


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

Assume the metric of the spherically symmetric radiation takes the form

The constraint Einstein equation

yields

2014 Institute of Physics, Academia Sinica


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

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

2014 Institute of Physics, Academia Sinica


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

Using to replace , , we arrive at the TOV

equation

2014 Institute of Physics, Academia Sinica


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

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:

2014 Institute of Physics, Academia Sinica


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

Thus, we have the Gibbs-Duhem relation

2014 Institute of Physics, Academia Sinica


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

2014 Institute of Physics, Academia Sinica


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

Note that

Thus,


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

2014 Institute of Physics, Academia Sinica


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

2014 Institute of Physics, Academia Sinica


4 proof of the entropy principle for perfect fluid in static spacetimes arxiv 1311 6899

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

2014 Institute of Physics, Academia Sinica


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

2014 Institute of Physics, Academia Sinica


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

Proof of Theorem 1

2014 Institute of Physics, Academia Sinica


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

The total entropy

Its variation:

Total number of particle:

The constraint

2014 Institute of Physics, Academia Sinica


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

Then

2014 Institute of Physics, Academia Sinica


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

(Constraint Einstein equation)

2014 Institute of Physics, Academia Sinica


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

Integration by parts:

Integration by parts again and dropping the boundary terms:

2014 Institute of Physics, Academia Sinica


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

2014 Institute of Physics, Academia Sinica


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

2014 Institute of Physics, Academia Sinica


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

2014 Institute of Physics, Academia Sinica


5 related works

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.

2014 Institute of Physics, Academia Sinica


6 conclusions

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.

2014 Institute of Physics, Academia Sinica


Thank you

Thank you!

2014 Institute of Physics, Academia Sinica


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