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.
Highlights
The observation of the present accelerated expansion of the universe catalyzed a large number of theoretical speculations, looking for natural explanations for the acceleration.The aim of many theoretical constructions is to modify general relativity in the infrared, in order to produce an effective cosmological constant at the late stages of the universal evolution
We present the calculation of the effective action for weak gravitational fields in Section II, where we obtain explicit expressions for the nonlocal kernels
We have found a covariant, nonlocal expression for the in − in effective action for stationary metrics, considering a quantum massless, minimally coupled scalar field in thermal equilibrium
Summary
The observation of the present accelerated expansion of the universe catalyzed a large number of theoretical speculations, looking for natural explanations for the acceleration. The usual in-out effective actions obtained after the integration of quantum matter fields produce nonlocal and noncausal equations of motion for the metric [9]. This problem is generally avoided in the nonlocal cosmological models by using a procedure suggested many years ago by Barvinsky and Vilkovisky [8], which consists in computing the effective action in Euclidean spacetime, obtaining the field equations, and replacing there the Euclidean propagators by retarded propagators.
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