Temperature Equilibrium in a Static Gravitational Field

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 $d{s}^2=g_{\mathrm{ij}}d{x}_{i}d{x}_j+g_{44}d{t}^2 i,j=1,2,3,$