The relation between the Unruh effect and the Ehrenfest-Tolman effect 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):
T=a/2π
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:
T||ξ||=const
||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?
Thank you!
 A: The Ehrenfest-Tolman effect is a sort of “temperature = speed of time” physics. The physics is based around the Killing vector $K^a$ with $|K|~=~\sqrt{g_{ab}K^aK^b}$.  Temperature is then $T|K|~=~const$. This physics then works for spacetimes that permit Killing vector fields.
To think about this we consider the Schwarzschild black hole with $K^t\partial_t$ $=~\sqrt{1~-~r_s/r}$, with $r_s~=~2GM/c^2$. Now consider the gradient of the temperature $\nabla T~=$ $\frac{1}{2}|K|^{-1}$ and we can see that
$$
\frac{\nabla T}{T}~=~\frac{1}{2}\frac{1}{1~-~r_s/r}\frac{r_s}{r^2}~=~g/c^2,
$$ 
where $g$ is the gravity. This is the same result as the result on page 121 of Wald's book  This gives the Newtonian result for gravity with $r~>>~r$.
The result $\frac{\nabla T}{T}~=~g/c^2$ is the distance of the horizon $d~=~g/c^2$. We can think of this thermodynamic result as an expression of time dilation. The Shannon-Khinchin formula $S~=~-k\sum_n\rho_nlog(\rho_n)$ defines the statistical thermal state $\Omega$. This is easily seen if $\rho_n~=~1/n$ then
$$
S~=~-k\sum_{n=1}^N\frac{1}{n}log\frac{1}{n}~=~k~log(N),
$$
where $N$ is the statistical ensemble state $\Omega$. For observables $O~\in~\cal O$ we define a flow $\phi:{\cal O}~\rightarrow~{\cal O}$ according to
$$
\frac{d\phi(O)}{ds}~=~\{S,~O\}~=~\{O,~log(\Omega)\},
$$
such that $\Omega~=~e^{-H/kT}$. The evolution equation may now be written according to the Hamiltonian $H$ with
$$
\frac{d\phi(O)}{ds}~=~\{O,~log(\Omega)\}~=~\frac{1}{kT}\{O,~H\},
$$
which tells us that $\frac{d}{ds}~=~\frac{1}{kT}\frac{d}{dt}$. This connects the proper time, which we see is also a thermal time, $s$ with a Hamiltonian time $t$.
So the Unruh-Hawking effect and the Tolman-Ehrenfest results are closely related to each other. They both involve the connection between general relativity and temperature. The Tolman-Ehrenfest result ties this in with the idea of "speed of time."
A: I think that there might not be a way to derive Unruh effect (classically!) from the Eherenfest-Tolman effect.
The main approach I have tried is to use the Eherenfest-Tolman relation $T||ξ||=const$ and apply it to a uniformly accelerated (WLOG, accleration $a$ in the x direction) system. such a system can be described by Rindler (accelerated Minkowski) coordinates, $\ ds^2=-a^2x^2dt^2+dx^2+dy^2+dz^2.$
The Killing vector fields are $\partial_t, \partial_y, \partial_z, $ and $y\partial_z-z\partial_y$, as well as others (generating rotations and boosts). 
This is where I get stuck- I am not sure how to calculate $||ξ||=\sqrt{g_{ab}ξ^aξ^b}$. I would very much appreciate some help with that part. 
However, my incompetence is not the reason I believe there isn't such a derivation- there is a much more crucial problem. The Unruh effect gives (in non-natural units) a factor of $\hbar,$ namely $T=\frac{\hbar a}{2\pi c k_B},$ and I just can't fathom how this factor will sprout in a non-quantum analysis of this problem.
Although I still believe these two phenomena are highly related, I believe there isn't a derivation that will give Unruh effect directly from the Eherenfest-Tolman effect.
EDIT: I realized $\hbar$ can just be a part of the const in the derivation, so I am back to wondering if one effect is derivable from the other. I actually think my approach mentioned above, might be the right approach.
