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arXiv · 1911.09060

Thermodynamic Equilibrium in General Relativity

J. A. S. Lima·A. Del Popolo·A. R. Plastino
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Abstract

The thermodynamic equilibrium condition for a static self-gravitating fluid in the Einstein theory is defined by the Tolman-Ehrenfest temperature law, $T{\sqrt {g_{00}(x^{i})}} = constant$, according to which the proper temperature depends explicitly on the position within the medium through the metric coefficient $g_{00}(x^{i})$. By assuming the validity of Tolman-Ehrenfest "pocket temperature", Klein also proved a similar relation for the chemical potential, namely, $\mu {\sqrt {g_{00}(x^{i})}} = constant$. In this letter we prove that a more general relation uniting both quantities holds regardless of the equation of state satisfied by the medium, and that the original Tolman-Ehrenfest law form is valid only if the chemical potential vanishes identically. In the general case of equilibrium, the temperature and the chemical potential are intertwined in such a way that only a definite (position dependent) relation uniting both quantities is obeyed. As an illustration of these results, the temperature expressions for an isothermal gas (finite spherical distribution) and a neutron star are also determined.

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J. A. S. Lima, A. Del Popolo, A. R. Plastino. 2019-11-20. Thermodynamic Equilibrium in General Relativity. https://doi.org/10.1103/physrevd.100.104042

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