Nonlocal effective actions in semiclassical gravity: thermal effects in stationary geometries

TL;DR

This paper derives a covariant nonlocal effective action for quantum scalar fields in thermal equilibrium within stationary gravitational fields, highlighting the influence of initial states and thermal effects on semiclassical gravity.

Contribution

It provides a new covariant expression for the nonlocal effective action incorporating thermal effects and initial state considerations in stationary spacetimes.

Findings

Derived a covariant nonlocal effective action in thermal equilibrium.

Connected ultrastatic and static metrics via conformal transformations.

Analyzed thermal corrections to semiclassical Einstein equations.

Abstract

We compute the gravitational effective action by integrating out quantum matter fields in a weak gravitational field, using the Schwinger-Keldysh (in-in) formalism. We pay particular attention to the role of the initial quantum state in the structure of the nonlocal terms in the effective action, with an eye to nonlinear completions of the theory that may be relevant in astrophysics and cosmology. In this first paper we consider a quantum scalar field in thermal equilibrium, in a stationary gravitational field. We obtain a covariant expression for the nonlocal effective action, that can be expressed in terms of the curvature tensor, the four-velocity of the thermal bath and the local Tolman temperature. We discuss the connection between the results for ultrastatic and static metrics through conformal transformations, and the main features of the thermal corrections to the semiclassical Einstein equations.