Tolman Law in a Nutshell

The Tolman law describes how the temperature in a fixed gravitational field depends on the position (see https://arxiv.org/abs/1803.04106 for a pedagogic review). Here I present a simple general derivation of the Tolman law. It’s generally in the sense that it is applicable to any (thermalized) physical degrees of freedom for which one can define Lagrangian and Hamiltonian. It’s simple in the sense that, in the derivation, one does not need to care about any details of those physical degrees of freedom, while general relativity is applied at a rather elementary level. To achieve simplicity, at some points I use heuristic arguments based on physical intuition, thus avoiding mathematical rigor. After the derivation, I briefly discuss a few elementary examples.

Derivation of the Tolman law

In a static spacetime, the metric $g_{\mu\nu}$ satisfies $g_{0i}=0$, while $g_{00}(x)=g_{00}({\bf x})$ and $g_{ij}(x)=g_{ij}({\bf x})$ do not depend on the global coordinate time $t=x^0$. Consider a local static observer at the position ${\bf x}$. For any local physical quantity $A$, I introduce the notation $A_{\rm obs}$ denoting the value of $A$ as measured by such a local observer. In particular
$$dt_{\rm obs}=\sqrt{g_{00}({\bf x})} dt .$$
For any physical system (e.g. a system of particles or matter fields) the action can be written in a schematic form
$$S=\int dt_{\rm obs}\, L_{\rm obs}$$
where $L_{\rm obs}$ is the observable Lagrangian. (For instance, if the Lagrangian is equal to kinetic energy, then observable Lagrangian is the observable kinetic energy as measured by the local static observer.) Combining the two equations above, we have
$$S=\int dt\, L$$
where
$$L=\sqrt{g_{00}({\bf x})} L_{\rm obs} .$$
Hence the canonical Hamiltonian $H$ derived from $L$ has an analogous form
$$H(q,p)=\sqrt{g_{00}({\bf x})} H_{\rm obs}(q,p) $$
where $q,p$ are the variables of the physical system in the phase space. For an arbitrary Hamiltonian system in thermal equilibrium, Boltzmann has shown that probability distribution in the phase space is proportional to
$${\rm exp}\left(-\frac{H(q,p)}{kT}\right)$$
where $k$ is the Boltzmann constant and $T$ is the temperature. The temperature $T$ is a constant in the thermal equilibrium. By writing it in terms of the observable Hamiltonian $H_{\rm obs}$ as
$${\rm exp}\left(-\frac{H_{\rm obs}(q,p)}{kT/\sqrt{g_{00}({\bf x})}}\right)$$
we see that $T/\sqrt{g_{00}({\bf x})}$ is naturally interpreted as the observable temperature
$$T_{\rm obs}({\bf x})=\frac{T}{\sqrt{g_{00}({\bf x})}} .$$
This last equation is known as the Tolman law.

Tolman law expressed in terms of Newton potential

It is useful to rewrite the Tolman law in terms of the Newton potential $V({\bf x})$. In general, we have
$$g_{00}({\bf x})=1+\frac{2V({\bf x})}{c^2}$$
where $c$ is the speed of light. In the Newtonian limit we have $2V({\bf x})\ll c^2$, so the Taylor expansion gives
$$T_{\rm obs}({\bf x}) \approx T \left( 1-\frac{V({\bf x})}{c^2} \right) .$$
In particular, near the surface of the Earth we have $V({\bf x}) \approx V_0 +gz$, where $z$ is the vertical coordinate and $g$ is the acceleration of the free fall, so
$$T_{\rm obs}({\bf x}) \approx T \left( 1-\frac{V_0 +gz}{c^2} \right) .$$

Tolman law around the black hole

For the Schwarzschild black hole we have

$$g_{00}(r)=1-\frac{2GM}{c^2r}=1-\frac{r_S}{r}$$

where $r_s$ is the Schwarzschild radius. Hence the Tolman law gives

$$T_{\rm obs}(r)=\frac{T}{\sqrt{1-\frac{r_S}{r}}} .$$

In particular

$$T_{\rm obs}(\infty)=T, \;\;\; T_{\rm obs}(r_s)=\infty .$$

If the temperature arises from Hawking radiation by the black hole, then $T$ is the Hawking temperature

$$T=\frac{\hbar c^3}{8\pi Gk M}.$$