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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. Outline. Introduction Entropy principle in spherical case --radiation Entropy principle in spherical case –perfect fluid - PowerPoint PPT Presentation
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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
23/4/19 2013中国科学技术大学交叉中心 1
Outline
1. Introduction2. Entropy principle in spherical case --radiation 3. Entropy principle in spherical case –perfect
fluid4. Entropy principle in static spacetime5. Related works6. Conclusions.
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1. Introduction
General Relativity Black hole mechanics(Bekenstein, Bardeen,1973)
Hawking radiation (1974) Black hole thermodynamics
thermodynamics General Relativity
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Ted Jacobson (1995) assumed the first law holds for local Rindler horizons. Then the Einstein equation can be derived.
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In 1965, W.J.Cocke (Ann. Inst. Henri Poincare, 2, 283) proposed a maximum entropy principle for self-gravitating fluid.
fluid
MS,
Tolman-Oppenheimer-Volkoff (TOV ) equation:
S: total entropy of fluidM: total mass of fluid
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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
R
The radiation satisfies:
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Assume the metric of the radiation takes the form
The constraint Einstein equation yields
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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:
s
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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.
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a
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Integration by parts:
Integration by parts again and dropping the boundary terms:
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.
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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.
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