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As we know, the first law of thermodynamic is $ dE=TdS-PdV+\mu dN $. is it true for a generic metric? what is the covariant form of this equation? for example, I think chemical potential and number density will be corrected as follow: \begin{equation} \epsilon=Ts-P+A_{\mu}J^{\mu} \end{equation} where $A_{0}$ is chemical potential.

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