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
2017/7/14
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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.
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1. Introduction
General Relativity
Black hole mechanics
(Bekenstein, Bardeen,1973)
Hawking radiation
(1974)
thermodynamics
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Black hole 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.
S: total entropy of fluid
M: total mass of fluid
S ,M
fluid
Tolman-Oppenheimer-Volkoff
(TOV ) equation:
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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 R . The
stress-energy tensor is given by
The radiation satisfies:
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Assume the metric of the radiation takes the form
The constraint Einstein equation
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yields
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Since
Euler-Lagrange equation:
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, the extrema of
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is equivalent to the
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Using
to replace
,
, we arrive at the TOV
equation
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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 s .
• The first law of the ordinary thermodynamics:
Rewrite in terms of densities:
Expand:
The first law in a unit volume:
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Thus, we have the Gibbs-Duhem relation
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Note that
Thus,
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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.
a

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Proof of Theorem 1
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The total entropy
Its variation:
Total number of particle:
The constraint
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Then
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(Constraint Einstein equation)
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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, 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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Thank you!
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