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 T₀ 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 ds^2=g₈₉dx₈dx₉+g₄₄dt^2 i, j=1, 2, 3, where the g₈₉ and g₄₄ 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 T₀g₄₄ a constant throughout the system.