# 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!

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."

• Thank you! The idea of temperature as "speed of time" is intriguing. I wonder if that implies a boost of one system with respect to another, which will change the "speed of time" with accordance to time dilation, also produce such an effect. In an ideal gas this is somewhat logical, as V would be contracted by a factor of γ so PV=NkT suggests the temperature will change. In this case as well as in other systems this could be seen as an effect of the change in observed energy + the equipartition principle. Is such an effect really observed- does relative velocity affect observed temperature? – A. Ok Jun 19 '17 at 15:52

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.

• The Ehrenfest-Tolman effect is to the Unruh effect what Bekenstein's finite temperature and entropy result for black holes is to Hawking;s radiation. – Lawrence B. Crowell Jun 26 '17 at 18:56
• @LawrenceB.Crowell How so? Can it be derived holographically? – A. Ok Jun 26 '17 at 19:46
• Probably the best way to quantize this is to take the Poisson bracket in the expression $\frac{d\phi(O)}{ds}~=~\frac{1}{kT}\{O,~H\}$ and replace it with a quantum commutator for operators. – Lawrence B. Crowell Jun 27 '17 at 20:06