Thermodynamics, Euclidean Gravity and Kaluza-Klein Reduction

TL;DR

This paper explores the relationship between Euclidean gravity's one-loop effective action and the free energy in stationary space-times, using Kaluza-Klein reduction to connect these concepts and analyze heat-kernel coefficients.

Contribution

It establishes explicit conditions linking the effective action and free energy in stationary space-times via Kaluza-Klein reduction, including new relations between heat-kernel coefficients.

Findings

Derived explicit relation between $W_E$ and $F$ in stationary space-times.

Discovered connection between heat-kernel coefficients in different dimensions.

Provided a framework for analyzing Wick rotation in Euclidean gravity.

Abstract

The aim of this paper is to find out a correspondence between one-loop effective action $W_E$ defined by means of path integral in Euclidean gravity and the free energy $F$ obtained by summation over the modes. The analysis is given for quantum fields on stationary space-times of a general form. For such problems a convenient procedure of a "Wick rotation" from Euclidean to Lorentzian theory becomes quite non-trivial implying transition from one real section of a complexified space-time manifold to another. We formulate conditions under which $F$ and $W_E$ can be connected and establish an explicit relation of these functionals. Our results are based on the Kaluza-Klein method which enables one to reduce the problem on a stationary space-time to equivalent problem on a static space-time in the presence of a gauge connection. As a by-product, we discover relation between the asymptotic heat-kernel coefficients of elliptic operators on a $D$ dimensional stationary space-times and the heat-kernel coefficients of a $D-1$ dimensional elliptic operators with an Abelian gauge connection.