The relation between the Unruh effect and the Ehrenfest-Tolman effect

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."