Tolman-like temperature gradients in stationary spacetimes

TL;DR

This paper explores the concept of temperature gradients in stationary spacetimes, emphasizing the importance of the choice of 4-velocity for heat baths, especially in rotating black hole environments, and extends previous analyses to more general cases.

Contribution

It clarifies the appropriate 4-velocity fields for describing heat baths in stationary spacetimes beyond static cases, including rotating black holes.

Findings

Different 4-velocity fields are suitable for heat bath descriptions in stationary spacetimes.

The heat bath for Hawking radiation from rotating black holes is best described by a gradient flow normal to spacelike hypersurfaces.

Buchdahl's 1949 analysis is only part of the broader picture of temperature gradients in such spacetimes.

Abstract

It is (or should be) well known that specification of a heat bath requires both a temperature and a 4-velocity, the rest frame of the heat bath. In static spacetimes there is a very natural and unique candidate for the 4-velocity of the heat bath, the normalized timelike Killing vector. However in stationary non-static spacetimes the situation is considerably more subtle, and several different "natural" 4-velocity fields suitable for characterizing the rest frame of a heat bath can be defined - thus Buchdahl's 1949 analysis for the Tolman temperature gradient in a stationary spacetime is only part of the story. In particular, the heat bath most suitable for describing the Hawking radiation from a rotating black hole is best described in terms of a gradient flow normal to the spacelike hypersurfaces, not in terms of Killing vectors.