Thermodynamic equilibrium in general relativity

The thermodynamic equilibrium condition for a static self-gravitating fluid in the Einstein theory is defined by the Tolman-Ehrenfest temperature law, $T\sqrt{g_{00}(x^i)}=\text{constant}$, 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, $\mu \sqrt{g_{00}(x^i)}=\text{constant}$. In this paper 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.

Figure 1

The Tolman-Ehrenfest effect. The plot shows the proper temperature for different central temperature, $T_0=\frac{m{c}^2}{k{b}_0}$. Left panel: the red line represents the solution in the case $b_0=1$, while the black line, green, and yellow lines the case $b_0=2$, $b_0=3.23$, and $b_0=350$ respectively. The quantity $b_0=\frac{m{c}^2}{k{T}_0}$, is fundamentally the inverse of the temperature, while the normalizing factor on the $x$ axis, $r_{*}=1/\sqrt{4\pi G{\rho }_{0}mk{T}_0}$, where ${\rho }_0$ is the central density. Right panel: the case $b_0=350$ using a different scale.

Figure 2

The Tolman-Ehrenfest effect. The red line represents the solution in the case $b_0=1$, while the black line and green line the case $b_0=2$, and $b_0=3.23$ respectively. The quantity $b_0=\frac{m{c}^2}{k{T}_0}$, is fundamentally the inverse of the temperature, while the normalizing factor on $x$ axis, $r_{*}=1/\sqrt{4\pi G{\rho }_{0}mk{T}_0}$, where ${\rho }_0$ is the central density.