High Temperature Massless Scalar Partition Function on General Stationary Backgrounds

TL;DR

This paper computes the high-temperature equilibrium partition function for a massless real scalar field nonminimally coupled to the scalar curvature on generic stationary backgrounds, up to second order in the derivative expansion and second order in curvatures. It employs covariant perturbation theory and heat-kernel methods to obtain both local and nonlocal contributions, including a nonlocal piece with form factors depending on the Matsubara spectrum and the Tolman temperature gradient. For conformal coupling, the Weyl anomaly is derived at fourth order in derivatives and second order in curvatures, with results consistent between the derivative and curvature analyses. The findings generalize ultrastatic results, reveal the role of vorticity and temperature gradients in the nonlocal sector, and offer tools for constructing hydrodynamic equilibrium partition functions with finite-temperature corrections.

Abstract

The high temperature equilibrium partition function of a massless real scalar field nonminimally coupled to the scalar curvature is computed at second order in the derivative expansion on a generic stationary background. Using covariant perturbation theory, the expression of the thermal partition function at second order in powers of curvatures is also obtained, including its nonlocal contributions. For conformal coupling, the Weyl anomaly at fourth order in derivatives and second order in curvatures is evaluated using both expansions and the results found to be consistent.