Statistical Mechanics, Gravity, and Euclidean Theory

TL;DR

This paper reviews methods for computing free energy in gravitational backgrounds, extending spectral theory techniques to non-linear problems, and explores their implications for Euclidean quantum gravity and thermodynamics near horizons.

Contribution

It introduces a novel method to handle non-linear spectral problems in gravitational settings and connects canonical and Euclidean definitions of free energy in complex backgrounds.

Findings

Extended heat kernel asymptotics to non-linear spectral problems

Derived high-temperature asymptotics for free energy and stress-energy tensor

Clarified relations between canonical and Euclidean approaches in spacetimes with horizons

Abstract

A review of computations of free energy for Gibbs states on stationary but not static gravitational and gauge backgrounds is given. On these backgrounds wave equations for free fields are reduced to eigen-value problems which depend non-linearly on the spectral parameter. We present a method to deal with such problems. In particular, we demonstrate how some results of the spectral theory of second order elliptic operators, such as heat kernel asymptotics, can be extended to a class of non-linear spectral problems. The method is used to trace down the relation between the canonical definition of the free energy based on summation over the modes and the covariant definition given in Euclidean quantum gravity. As an application, high-temperature asymptotics of the free energy and of the thermal part of the stress-energy tensor in the presence of rotation are derived. We also discuss statistical mechanics in the presence of Killing horizons where canonical and Euclidean theories are related in a non-trivial way.