Thermal equilibrium in general relativity The Newtonian condition for thermal equilibrium for a static system is $T = \mathrm{const}$. 
In this homework I'm asked to show that it's curved space generalization is $T(-g_{00})^{\frac{1}{2}} = \mathrm{const}$, where $g_{\mu \nu}$ is the metric (assumed stationary). But I have no idea how to start.
 A: The clearest and most general derivation of this result is given in MTW's Gravitatiom p. 567-568 using the laws of thermodynamics in GR. 
A simplified argument is in the "Problem book in relativity and gravitation" by Lightman et al. as (solved) exercise 14.2
The physical argument goes like this: imangine a black body higher up in a static gravitational field connected with a optical fiber to a lower black body. It emits a certain number of (thermal) fotons in a certain time with a certain frequency. They descend to the lower black body and arrive with a higher frequency (gravitational blueshift) with a different arrival time. The lower black body also emits a certain number of fotons with this higher frequency and if you calculate these quantities with GR and Planck's law you can see that a higher temperature is necessary in the lower black body (to equate the number of incoming/outgoing fotons) where the ratio of the temperatures is the same as that of the frequencies (look at the exponent in the denominator of Plank's law). 
anyway, in my opinion this homework seems to me rather difficult if you have not been  given the appropriate background during the course (thermodynamics in GR).
A: Well T is an average energy and in general relativity, there is no guarantee of energy conservation if the metric isn't stationary, so I guess that may be the problem? You could have a system at equilibrium but the evolution of the metric could change the energy.
A: I don't know exactly where to start either but one could think of the black body radiation coming from a source far away that was emitted by a source at temperature $T_1$ where $g_{00}$ had some value say $g_{00}^1$.
Now,if we make use of the fact that $g_{00}$ influences the photon frequency via gravitational Doppler effect, then imagining balance of energy fluxes with another body at temperature $T_2$ and $g_{00}^2$ probably leads to the relation you have to find.
Sorry it's a bit hand waving for now...
