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

TL;DR

This paper explores the concept of thermal time in stationary spacetimes and derives the Tolman-Ehrenfest effect by linking temperature to the rate of thermal time relative to proper time, emphasizing its physical significance.

Contribution

It provides a simple derivation of the Tolman-Ehrenfest effect using thermal time without additional dynamical assumptions, highlighting the physical meaning of thermal time in relativistic thermodynamics.

Findings

Derived the Tolman-Ehrenfest relation from thermal time principles.

Showed temperature as the rate of thermal time relative to proper time.

Demonstrated the physical importance of thermal time in stationary spacetimes.

Abstract

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.