Temperature Equilibrium in a Static Gravitational Field

Abstract

In the case of a gravitating mass of perfect fluid which has come to thermodynamic equilibrium, it has previously been shown that the proper temperature ${T}_{0}$ as measured by a local observer would depend in a definite manner on the gravitational potential at the point where the measurement is made. In the present article the conditions of thermal equilibrium are investigated in the case of a general static gravitational field which could correspond to a system containing solid as well as fluid parts. Writing the line element for the general static field in the form $d{s}^{2}={g}_{\mathrm{ij}}d{x}_{i}d{x}_{j}+{g}_{44}d{t}^{2} i,j=1,2,3,$where the ${g}_{\mathrm{ij}}$ and ${g}_{44}$ are independent of the time $t$ it is shown that the dependence of proper temperature on position at thermal equilibrium is such as to make the quantity ${T}_{0}\sqrt{{g}_{44}}$ a constant throughout the system.