How the Schwarzschild-de Sitter horizons remain in thermal equilibrium at vastly different temperatures

TL;DR

This paper generalizes the thermal equilibrium criterion to stationary heat conduction in Schwarzschild-de Sitter spacetime, demonstrating that the horizons remain in equilibrium with an explicit temperature profile.

Contribution

It extends the Tolman-Ehrenfest criterion to stationary conditions and applies it to show horizons in Schwarzschild-de Sitter spacetime stay in thermal equilibrium.

Findings

Horizons act as thermostats maintaining equilibrium

Explicit static temperature profile between horizons

Hawking radiation temperature interpolates smoothly

Abstract

The Tolman-Ehrenfest criterion of thermal equilibrium for a static fluid in a static spacetime is generalized to stationary heat conduction, in the approximation in which backreaction is negligible. Applying this generalized criterion to the Hawking radiation in the Schwarzschild-de Sitter geometry shows that the two horizons (which act as thermostats) remain in thermal equilibrium. The temperature of the radiation fluid interpolates between the temperatures at the horizons, with a static analytic profile that is given explicitly.