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

1. General proof of the entropy principle for self-gravitating fluid in static spacetimes 高思杰 北京师范大学(Beijing Normal University) Cooperated with 房熊俊 中国科学技术大学交叉中心

2. Outline • Introduction • Entropy principle in spherical case --radiation • Entropy principle in spherical case –perfect fluid • Entropy principle in static spacetime • Related works • Conclusions. 中国科学技术大学交叉中心

3. 1. Introduction Black hole mechanics (Bekenstein, Bardeen,1973) General Relativity Hawking radiation (1974) Black hole thermodynamics ? General Relativity thermodynamics 中国科学技术大学交叉中心

4. Ted Jacobson (1995) assumed the first law holds for local Rindler horizons. Then the Einstein equation can be derived. 中国科学技术大学交叉中心

5. In 1965, W.J.Cocke (Ann. Inst. Henri Poincare, 2, 283) proposed a maximum entropy principle for self-gravitating fluid. S: total entropy of fluid M: total mass of fluid fluid Tolman-Oppenheimer-Volkoff (TOV ) equation: 中国科学技术大学交叉中心

6. 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. Consider a box of radiation (photon gas) confined within radius . The stress-energy tensor is given by The radiation satisfies: 中国科学技术大学交叉中心

7. Assume the metric of the radiation takes the form The constraint Einstein equation yields 中国科学技术大学交叉中心

8. Since , the extrema of is equivalent to the Euler-Lagrange equation: 中国科学技术大学交叉中心

9. Using to replace , , we arrive at the TOV equation 中国科学技术大学交叉中心

10. 3. Entropy principle in spherical case---general perfect fluid (Sijie Gao, arXiv:1109.2804 ) • 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: 中国科学技术大学交叉中心

11. Thus, we have the Gibbs-Duhem relation 中国科学技术大学交叉中心

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13. Note that Thus,

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16. 4.Proof of the entropy principle for perfect fluid in static spacetimes • 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. 中国科学技术大学交叉中心

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18. Proof of Theorem 1 中国科学技术大学交叉中心

19. The total entropy Its variation: Total number of particle: The constraint 中国科学技术大学交叉中心

20. Then 中国科学技术大学交叉中心

21. (Constraint Einstein equation) 中国科学技术大学交叉中心

22. Integration by parts: Integration by parts again and dropping the boundary terms: 中国科学技术大学交叉中心

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26. 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. 中国科学技术大学交叉中心

27. 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. 中国科学技术大学交叉中心

28. Thank you! 中国科学技术大学交叉中心