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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). 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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  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) 2014 Institute of Physics, Academia Sinica

  2. 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

  3. 1. Introduction Mathematical analogy beween thermodynamics and black holes: 2014 Institute of Physics, Academia Sinica

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  5. 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

  6. 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?

  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: 2014 Institute of Physics, Academia Sinica

  8. Assume the metric of the spherically symmetric radiation takes the form The constraint Einstein equation yields 2014 Institute of Physics, Academia Sinica

  9. Since , the extrema of is equivalent to the Euler-Lagrange equation: 2014 Institute of Physics, Academia Sinica

  10. Using to replace , , we arrive at the TOV equation 2014 Institute of Physics, Academia Sinica

  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: 2014 Institute of Physics, Academia Sinica

  12. Thus, we have the Gibbs-Duhem relation 2014 Institute of Physics, Academia Sinica

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  14. Note that Thus,

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  17. 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

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  19. Proof of Theorem 1 2014 Institute of Physics, Academia Sinica

  20. The total entropy Its variation: Total number of particle: The constraint 2014 Institute of Physics, Academia Sinica

  21. Then 2014 Institute of Physics, Academia Sinica

  22. (Constraint Einstein equation) 2014 Institute of Physics, Academia Sinica

  23. Integration by parts: Integration by parts again and dropping the boundary terms: 2014 Institute of Physics, Academia Sinica

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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. 2014 Institute of Physics, Academia Sinica

  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. 2014 Institute of Physics, Academia Sinica

  29. Thank you! 2014 Institute of Physics, Academia Sinica

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