  A review of the original thermodynamic formulation of the Tolman-Ehrenfest
effect prescribing the temperature profile of uncharged fluid at thermal
equilibrium forming stationary configurations in curved space-time is proposed.
A statistical description based on relativistic kinetic theory is implemented.
In this context the Tolman-Ehrenfest relation arises in the Schwarzschild
space-time for collisionless uncharged particles at Maxwellian kinetic
equilibrium. However, the result changes considerably when non-ideal fluids,
i.e., non-Maxwellian distributions, are treated, whose statistical temperature
becomes non-isotropic and gives rise to a tensor pressure. This is associated
with phase-space anisotropies in the distribution function, occurring both for
diagonal and non-diagonal metric tensors, exemplified by the Schwarzschild and
Kerr metrics respectively. As a consequence, it is shown that for these systems
it is not possible to define a Tolman-Ehrenfest relation in terms of an
isotropic scalar temperature. Qualitative properties of the novel solution are
discussed.
