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) Institute of Physics, Academia Sinica1

Outline 1.Introduction 2.Entropy principle in spherical case --radiation 3.Entropy principle in spherical case –perfect fluid 4.Entropy principle in static spacetime 5.Related works 6.Conclusions Institute of Physics, Academia Sinica2

1. Introduction Institute of Physics, Academia Sinica3 Mathematical analogy beween thermodynamics and black holes:

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Institute of Physics, Academia Sinica5 What is the relationship between ordinary thermodynamics and gravity? We shall study thermodynamics of self- gravitating fluid in curved spacetime.

fluid S: total entropy of fluid M: total mass of fluid N: total particle number 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: Institute of Physics, Academia Sinica7

Assume the metric of the spherically symmetric radiation takes the form The constraint Einstein equation yields Institute of Physics, Academia Sinica8

Since, the extrema of is equivalent to the Euler-Lagrange equation: Institute of Physics, Academia Sinica9

Using to replace,, we arrive at the TOV equation Institute of Physics, Academia Sinica10

3. Entropy principle in spherical case---general perfect fluid (Sijie Gao, arXiv: , Phys. Rev. D 84, ) 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: Institute of Physics, Academia Sinica11

Thus, we have the Gibbs-Duhem relation Institute of Physics, Academia Sinica12

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Note that Thus,

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4.Proof of the entropy principle for perfect fluid in static spacetimes arXiv: arXiv: 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 Institute of Physics, Academia Sinica17

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Institute of Physics, Academia Sinica19 Proof of Theorem 1

Institute of Physics, Academia Sinica20 The total entropy Its variation: Total number of particle: The constraint

Institute of Physics, Academia Sinica21 Then

Institute of Physics, Academia Sinica22 (Constraint Einstein equation)

Institute of Physics, Academia Sinica23 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, (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) ] 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, (2013)] proved the maximum entropy principle in the framework of Lovelock gravity Institute of Physics, Academia Sinica27

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 Institute of Physics, Academia Sinica28

Thank you! Institute of Physics, Academia Sinica29