Thermodynamic Equilibrium in General Relativity

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.

7 pages, 2 figures, PRD accepted