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In relativity, we define proper time for a particle therefore can discuss about casuality-order of events preserved for it. For statistical mechanics in classical mechanics, macroscopic systems evolving by time follow the same time axis-hence the increase of entropy by time(a.k.a. the second law of thermodynamics) can be accepted 'naturally'. However, for a macroscopic system in equilibrium, can we define proper time? For example, for gases, if the comoving frame of the particles differ, the particles themselves can evolve through their own proper time-however what about the gas, the macroscopic system itself? Is there a well-defined time describing change for equilibrium statistical mechanics?

p.s.) The anomaly in definition of temperature in special relativity also led me to this frustration.

Source: http://kirkmcd.princeton.edu/examples/temperature_rel.pdf

Thank you for reading my question. If you have a clear answer, I would appreciate it if you let me know.

Jiwoo Hong
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  • A gas is not a Lorentz invariant system to begin with. Why would it have a proper time? – FlatterMann Apr 19 '23 at 11:16
  • It is true that it's hard to treat gases covariantly. However, to utilize statistical mechanics also in the relativistic realm, we should bring a measure for the change of macrosystem-acting as time. For point particle(or system constituted with elements holding the same comoving frame), we can use proper time for it-but what about a macrosystem with different inertial frames? Must relativistic statistical mechanics only deal with elements having the same velocity? – Jiwoo Hong Apr 19 '23 at 13:55
  • The macrosystem doesn't even have temperature. It will look red-shifted (colder) for an observer moving away from the gas and blue-shifted (hotter) for an observer moving towards it. Such an observer will not be in equilibrium with the gas, either. The Cherenkov effect even displays characteristics of superluminal motion when relativistic charges enter a medium with dielectric constant >0. Matter is simply not the same as vacuum. – FlatterMann Apr 19 '23 at 18:24

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Every macroscopically large box of gas that is not accelerating has a well-defined rest frame of reference and thus the proper time of that box of gas will be equivalent to the coördinate time in that rest frame of reference in Special Theory of Relativity.

There is no generally accepted solution if the box of gas is moving. Plenty of authors have argued their case, but you have those that argue it is going to transform by multiplying the Lorentz factor $\gamma$, others argue it should be a division, yet others say it should be Lorentz invariant, and some others entertain the square of the Lorentz factor in either multiplication or division. My personal belief is that temperature is simply only well-defined for systems at rest.

In General Theory of Relativity the concept is worse and I am not familiar with attempted solutions.

naturallyInconsistent
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  • Thank you for your response. I also agree this problem might be a hot potato. Then, could we only deal with only particular macrosystems in rest using equilibrium statistical mechanics? – Jiwoo Hong Apr 19 '23 at 14:03
  • not particular, but rather any macrosystem at rest. Again, I emphasise that this is my guess, and it is definitely possible, if but an extremely conservative retreat. If it is solved another way in the future, it would be really interesting. By that I mean the temperature for moving gas---the proper time for box of gas at constant overall velocity is just Lorentz transform of the coördinate time as is standard to basic SR. – naturallyInconsistent Apr 19 '23 at 14:07
  • Here is what it's probably the most reasonable point of view on the Lorentz transformation of temperature (aka "Ott-imbroglio"): The zeroth law of thermodynamics in special relativity (2020). Both Van Kampen (1968) and Israel (1981) argued that one must consider that thermodynamic systems in relative motion can exchange energy and momentum and therefore the result will depend on the exact circumstance of the experiment. The linked paper extends this "modern" point of view. – Quillo Oct 07 '23 at 14:27