I am interested in the relation and, perhaps, the equivalence between two effects to do with temperature and gravity.
The first one, the Unruh effect, states that the background black-body radiation, and thus the observed temperature associated with the vacuum, depends on the acceleration of the observer such that (in natural units):
a being the local acceleration, and T the observed vacuum temperature. Of course, using the equivalence principle, it is immediately deduced that this effect also exists in a garvitational field.
The second one, the Ehrenfest-Tolman effect, states that the temperature of a system in thermal equilibrium varies with the curvature of space time, such that:
||ξ|| being the norm of the timelike Killing vector field, and T the local temperature of the system.
These two effects both regard the behaviour of temperature in the vicinity of a gravitational field (or, equivalently, an accelerating system), and so I asked myself if they were related. Namely, is there a way to derive one from the other? Are they equivalent in a sense? If so, how are these two equivalent? Is there any other effect that resembles them or is equivalent to them? Is there a different model that produces them? If not, why are they different, and do they produce different predictions?
In short- what is the relation between the Unruh effect and the Eherenfest-Tolman effect?