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

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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

高思杰 (Gao Sijie)

北京师范大学

(Beijing Normal University)

2014 Institute of Physics, Academia Sinica

- 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

Mathematical analogy beween thermodynamics and black holes:

2014 Institute of Physics, Academia Sinica

2014 Institute of Physics, Academia Sinica

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

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?

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

Assume the metric of the spherically symmetric radiation takes the form

The constraint Einstein equation

yields

2014 Institute of Physics, Academia Sinica

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

2014 Institute of Physics, Academia Sinica

Using to replace , , we arrive at the TOV

equation

2014 Institute of Physics, Academia Sinica

- 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

Thus, we have the Gibbs-Duhem relation

2014 Institute of Physics, Academia Sinica

2014 Institute of Physics, Academia Sinica

Note that

Thus,

2014 Institute of Physics, Academia Sinica

2014 Institute of Physics, Academia Sinica

- 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

2014 Institute of Physics, Academia Sinica

Proof of Theorem 1

2014 Institute of Physics, Academia Sinica

The total entropy

Its variation:

Total number of particle:

The constraint

2014 Institute of Physics, Academia Sinica

Then

2014 Institute of Physics, Academia Sinica

(Constraint Einstein equation)

2014 Institute of Physics, Academia Sinica

Integration by parts:

Integration by parts again and dropping the boundary terms:

2014 Institute of Physics, Academia Sinica

2014 Institute of Physics, Academia Sinica

2014 Institute of Physics, Academia Sinica

2014 Institute of Physics, Academia Sinica

- 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

- 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

2014 Institute of Physics, Academia Sinica