1. 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) 2015-10-272014 Institute of Physics, Academia Sinica1

2. 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. 2015-10-272014 Institute of Physics, Academia Sinica2

3. 1. Introduction 2015-10-272014 Institute of Physics, Academia Sinica3 Mathematical analogy beween thermodynamics and black holes:

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5. 2015-10-272014 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.

6. fluid S: total entropy of fluid M: total mass of fluid N: total particle number 2015-10-276 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:

7. 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: 2015-10-272014 Institute of Physics, Academia Sinica7

8. Assume the metric of the spherically symmetric radiation takes the form The constraint Einstein equation yields 2015-10-272014 Institute of Physics, Academia Sinica8

9. Since, the extrema of is equivalent to the Euler-Lagrange equation: 2015-10-272014 Institute of Physics, Academia Sinica9

10. Using to replace,, we arrive at the TOV equation 2015-10-272014 Institute of Physics, Academia Sinica10

11. 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: 2015-10-272014 Institute of Physics, Academia Sinica11

12. Thus, we have the Gibbs-Duhem relation 2015-10-272014 Institute of Physics, Academia Sinica12

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14. Note that Thus, 2015-10-2714

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17. 4.Proof of the entropy principle for perfect fluid in static spacetimes arXiv: 1311.6899 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. 2015-10-272014 Institute of Physics, Academia Sinica17

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19. 2015-10-272014 Institute of Physics, Academia Sinica19 Proof of Theorem 1

20. 2015-10-272014 Institute of Physics, Academia Sinica20 The total entropy Its variation: Total number of particle: The constraint

21. 2015-10-272014 Institute of Physics, Academia Sinica21 Then

22. 2015-10-272014 Institute of Physics, Academia Sinica22 (Constraint Einstein equation)

23. 2015-10-272014 Institute of Physics, Academia Sinica23 Integration by parts: Integration by parts again and dropping the boundary terms:

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27. 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. 2015-10-272014 Institute of Physics, Academia Sinica27

28. 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. 2015-10-272014 Institute of Physics, Academia Sinica28

29. Thank you! 2015-10-272014 Institute of Physics, Academia Sinica29