Thermal time and Tolman-Ehrenfest effect: 'temperature as the speed of time'

The notion of thermal time has been introduced as a possible basis for a fully general-relativistic thermodynamics. Here we study this notion in the restricted context of stationary spacetimes. We show that the Tolman-Ehrenfest effect (in a stationary gravitational field, temperature is not constant in space at thermal equilibrium) can be derived very simply by applying the equivalence principle to a key property of thermal time; at equilibrium, temperature is the rate of thermal time with respect to proper time—the 'speed of (thermal) time'. Unlike other published derivations of the Tolman-Ehrenfest relation, this one is free from any further dynamical assumption, thereby illustrating the physical import of the notion of thermal time.