Tolman-Ehrenfest's criterion of thermal equilibrium extended to conformally static spacetimes

TL;DR

This paper extends the Tolman-Ehrenfest criterion of thermal equilibrium to conformally static spacetimes, explaining how temperature scales with the conformal factor and applying it to cosmological and black hole contexts.

Contribution

The paper generalizes the Tolman-Ehrenfest criterion to conformally static geometries, providing a unified framework for understanding temperature scaling in various cosmological and black hole spacetimes.

Findings

Temperature scales inversely with the conformal factor.

Reproduces cosmic microwave background evolution.

Describes Hawking temperature in conformally rescaled black holes.

Abstract

With insight from examples and physical arguments, the Tolman-Ehrenfest criterion of thermal equilibrium for test fluids in static spacetimes is extended to local thermal equilibrium in conformally static geometries. The temperature of the conformally rescaled fluid scales with the inverse of the conformal factor, reproducing the evolution of the cosmic microwave background in Friedmann universes, the Hawking temperature of the Sultana-Dyer cosmological black hole, and a heuristic argument by Dicke.