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