On the Thermodynamical Equilibrium of Fluids in Gravitational Fields

It is shown that the thermodynamical equilibrium condition for a fluid at rest in a static gravitational field is, apart from the Tolman condition $T{({g}_{44})}^{\frac{1}{2}}=\mathrm{const}.$, governed by the condition $\ensuremath{\alpha}{({g}_{44})}^{\frac{1}{2}}=\mathrm{const}.$, where $\ensuremath{\alpha}$ is the chemical potential of the fluid (including mass energy) regarded as a function of the space coordinates. This relation constitutes the relativistic generalization of the well-known Gibbs condition for the equilibrium in a gravitational field.