Thermodynamic Equilibrium in General Relativity

TL;DR

This paper generalizes the conditions for thermodynamic equilibrium in static self-gravitating fluids within general relativity, showing a unified relation between temperature and chemical potential that extends beyond the classical Tolman-Ehrenfest law.

Contribution

It proves a more general relation linking temperature and chemical potential in equilibrium, valid for any equation of state, and clarifies the conditions under which the original law applies.

Findings

Unified relation between temperature and chemical potential in equilibrium.

Original Tolman-Ehrenfest law applies only when chemical potential vanishes.

Derived temperature expressions for isothermal gas and neutron star models.

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.