In one minute

Jessica Santiago and Matt Visser start from the point that a heat bath is specified by both a temperature and a four-velocity, which fixes its rest frame. In static spacetimes the natural and unique choice is the normalised timelike Killing vector. In stationary but non-static spacetimes several different natural four-velocity fields can each define the bath's rest frame, which makes the problem much more subtle. Buchdahl's 1949 analysis of the Tolman temperature gradient is therefore only part of the story. For Hawking radiation from a rotating black hole, the best-suited heat bath follows a gradient flow normal to spacelike hypersurfaces rather than Killing vectors.

Tolman-like temperature gradients in stationary spacetimes