Tolman’s Law describes how temperature in a fixed gravitational field depends on position (an educational overview can be found at https://arxiv.org/abs/1803.04106). Here I present a simple general derivation of Tolman’s Law. It is general in the sense that it is applicable to all (thermized) physical degrees of freedom for which Lagrange and Hamilton can be defined. It is simple in the sense that one does not have to worry about details of these physical degrees of freedom in deriving it, while general relativity is applied on a more elementary level. For the sake of simplicity, in some places I use heuristic arguments based on physical intuition to avoid mathematical rigor. After the derivation, I will briefly discuss some elementary examples.

1. Derivation of Tolman’s law

In a static spacetime, the metric ## g _ { mu nu} ## ## 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 each local physical quantity ## A ## I add the notation ## A _ { rm obs} ##, which denotes the value of ## A ## that is measured by such a local observer. Certain
$$ dt _ { rm obs} = sqrt {g_ {00} ({ bf x})} dt. $$
For each physical system (e.g. a system of particles or fields of matter) the action can be written in schematic form
$$ S = int dt _ { rm obs} , L _ { rm obs} $$
## L _ { rm obs} ## is the observable Lagrange. (For example, if the Lagrange equals the kinetic energy, then the observable Lagrange is the observable kinetic energy measured by the local static observer.) If we combine the above two equations 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
$$ H (q, p) = sqrt {g_ {00} ({ bf x})} H _ { rm obs} (q, p) $$
## q, p ## are the variables of the physical system in phase space. For any Hamilton system in thermal equilibrium, Boltzmann has shown that the probability distribution in phase space is proportional to
$$ { rm exp} left (- frac {H (q, p)} {kT} right) $$
## k ## is the Boltzmann constant and ## T ## is the temperature. The temperature is a constant in thermal equilibrium. By writing in the form of the observable Hamiltonian ## H _ { rm obs} ## as
$$ { rm exp} left (- frac {H _ { rm obs} (q, p)} {kT / sqrt {g_ {00} ({ bf x})}} right) $ $
we see that ## T / sqrt {g_ {00} ({ bf x})} ## is of course interpreted as an observable temperature
$$ T _ { rm obs} ({ bf x}) = frac {T} { sqrt {g_ {00} ({ bf x})}}. $$
This last equation is known as Tolman’s law.

2. Tolman’s law expressed as Newton’s potential

It is useful to rewrite Tolman’s law in terms of the Newton potential ## V ({ bf x}) ##. In general we have
$$ g_ {00} ({ bf x}) = 1+ frac {2V ({ bf x})} {c ^ 2} $$
## c ## is the speed of light. In the Newtonian limit we have ## 2V ({ bf x}) ll c ^ 2 ##, so the Taylor expansion results
$$ T _ { rm obs} ({ bf x}) approx. T left (1- frac {V ({ bf x})} {c ^ 2} right). $$
In particular, near the earth’s surface we have ## V ({ bf x}) roughly V_0 + gz ##, where ## z ## is the vertical coordinate and ## g ## is the acceleration of the free fall, so
$$ T _ { rm obs} ({ bf x}) approximately T left (1- frac {V_0 + gz} {c ^ 2} right). $$

3. Tolman’s law about the black hole

For the Schwarzschild black hole we have

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

## r_s ## is the Schwarzschild radius. Hence the Tolman law there

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

Certain

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

If the temperature is created by Hawking radiation through the black hole, ## T ## is the Hawking temperature

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

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