# 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.

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Basically I think the answer is to the question is no. For instance, there are fundamental difficulties in defining temperature. See physicsforums.com/showthread.php?t=644884 –  Ben Crowell May 2 '13 at 22:58
There do exist books with both "relativity" and "thermodynamics" in the title, for example amzn.com/0486653838 (Relativity, Thermodynamics and Cosmology, Richard C Tolman, Dover) –  DarenW May 3 '13 at 4:20
I belive that the trick to get a relativistic version of entropy, temperature and etc is to define things in the proper reference frame, since you can extend it latter to other references keeping lorentz covariance. At least this is one way to arrive at relativistic hydrodynamics. About the QFT and GR: I don't have any idea –  Hydro Guy May 3 '13 at 12:11
I know you don't want to hear it but I think Wick Rotation is really the right thing. The connection between the classical world and the quantum one is the path integral, through the action. This connects the partition function to the generating function and you're done, baring the Wick rotation. Now in curved space I don't have an answer, because even the definition of the Hilbert space (a la Hawking) is tricky. But then, QFT doesn't play well with GR, so we wouldn't expect the connection to extend that far anyway. –  levitopher May 3 '13 at 16:03