So I am working on Relativistic Thermodynamics and I have skimmed through a few papers by Hamity, Callen, etc. People are not in agreement as far as I know, but I don't understand why people don't derive the transformation laws from first principles. So for example for the entropy I would go about it this way (correct me if I am wrong): The entropy is defined as the logarithm of number of states which have a given Energy E for a particular Hamiltonian (omitting the logarithm): $$\int\,dp\,dx\,\,\delta(H(x,p)-E)$$ Isn't it obvious from this that the Entropy defined in this way cannot be Lorentz Invariant? Because $$dp\rightarrow \gamma dp$$ $$dx\rightarrow \gamma dx$$ and $$H(x,p)-E\rightarrow \gamma(H(x,p)-E)$$ So the Dirac Delta is not affected. Am I thinking wrong here?
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