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I tried to check this in google scholar but didn't find a paper explicitly focused on this topic. Do anyone know of some references on this issue? I do not mean the thermodynamics in curved spacetime which is well covered by Tolman's book ``Relativity, Thermodynamics and Cosmology''.

Qmechanic
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Wein Eld
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    can you please elaborate why you think the curvature of space-time will at all effect the statistics? I mean locally every curved space is euclidean. So local equilibrium will not be affected at all. – Ari Mar 08 '16 at 14:41
  • I am mostly interested in the merit of general covariance and statistical physics. Since we know we need the energy spectrum or Hamilton to write down the partition function(functional if for fields) which must based on a 3+1 decomposition. – Wein Eld Mar 08 '16 at 14:51
  • Related: http://physics.stackexchange.com/q/110763/2451 and links therein. – Qmechanic Mar 08 '16 at 14:52
  • @Ari, Furthermore, for the very early universe, we can not take the view that every small piece of space is almost flat. – Wein Eld Mar 08 '16 at 14:52
  • @Qmechanic, Thanks very much, Qmechanic. The post you recommended is also useful to me. But it is still different from my question anyway. – Wein Eld Mar 08 '16 at 14:59
  • @WeinEld: That's exactly the view we are taking because there is no data indicating otherwise. We aren't "suffering" from having to explain a "rough" early universe, but from the opposite: it seems to have been rather smooth. In any case... statistics is statistics. It has nothing to do with metric smoothness. – CuriousOne Mar 08 '16 at 15:01
  • @WeinEld: Consider looking into finite temperature field theory in curved spacetime. – Qmechanic Mar 08 '16 at 15:19
  • @CuriousOne, I did not mean we will have a bumpy spacetime for the early universe. I just mean that the curved spacetime may play its role for the very early universe. And the spacetime is always smooth as long as we do not care about quantum gravity. Statistics, as a mathematical concept, is of course irrelevant of metric, but as for the statistical PHYSICS, it is important to understand how it can reconcile with general covariance. – Wein Eld Mar 08 '16 at 15:40

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