Local vs. global temperature under a positive curvature condition

TL;DR

This paper compares global and local temperature concepts for a scalar field in curved space-time, establishing conditions under which local temperature is well-defined and related monotonically to the global temperature.

Contribution

It proves the monotonic relationship between local and global temperature in certain space-times and identifies conditions for the local temperature's existence using the positive mass theorem.

Findings

Local temperature is a continuous, increasing function of global temperature.

Local temperature is well-defined in ultra-static space-times with non-negative scalar curvature.

Counter-examples show the necessity of assumptions for local temperature definition.

Abstract

For a massless free scalar field in a globally hyperbolic space-time we compare the global temperature T, defined for the KMS states $\omega^T$, with the local temperature $T_{\omega}(x)$ introduced by Buchholz and Schlemmer. We prove the following claims: (1) Whenever $T_{\omega^T}(x)$ is defined, it is a continuous, monotonically increasing function of T at every point x. (2) $T_{\omega}(x)$ is defined when the space-time is ultra-static with compact Cauchy surface and non-trivial scalar curvature $R\ge 0$, $\omega$ is stationary and a few other assumptions are satisfied. Our proof of (2) relies on the positive mass theorem. We discuss the necessity of its assumptions, providing counter-examples in an ultra-static space-time with non-compact Cauchy surface and R<0 somewhere. We interpret the result in terms of a violation of the weak energy condition in the background space-time.