Is there a relativity-compatible thermodynamics? I am just wondering that laws in thermodynamics are not Lorentz invariant, it only involves the $T^{00}$ component. Tolman gave a formalism in his book. For example, the first law is replaced by the conservation of energy-momentum tensor. But what will be the physical meaning of entropy, heat and temperature in the setting of relativity? What should an invariant Stephan-Boltzmann's law for radiation take shape? And what should be the distribution function?
I am not seeking "mathematical" answers. Wick rotation, if just a trick, can not satisfy my question. I hope that there should be some deep reason of the relation between statistical mechanics and field theory. In curved spacetime, effects like particle production seems very strange to me, since they originate from the ambiguity of vacuum state which reflects the defects of the formalism. The understanding of relativistic thermodynamics should help us understand the high energy astrophysical phenomena like GRB and cosmic rays.
 A: I don't know a definitive answer to your (really good) question, but here is a quote from an old textbook I have by Christian Moller ("The Theory of Relativity"):

Shortly after the advent of the relativity theory, Planck, Hassenoerl,
  Einstein and others advanced separately a formulation of the
  thermodynamical laws in accordance with the special principle of
  relativity.  This treatment was adopted unchanged including the first
  edition of this monograph.  However it was shown by Ott and indepently
  by Arzelies, that the old formulation was not quite satisfactory, in
  particular because generalized forces were used instead of the true
  mechanical forces in the description of thermodynamical processes.
The papers of Ott and Arzelies gave rise to many controversial
  discussions in the literature and at the present there is no
  generally accepted description of relativistic thermodynamics.

So at least at the time that was written it was unresolved.  I'd be interested if there are any more recent updates.
