Non-vacuum gravitational effective action

Abstract

Curvature expansion for the heat kernel trace and the one-loop effective action is built for the wave operator of the theory in the quasi-thermal setup of a nonvacuum quantum state. This setup implies a non-static and non-stationary Euclidean gravitational background with periodic boundary conditions of the period $\beta=1/T$, where $T$ plays the role of effective global temperature to be locally rescaled by the metric gravitational potential. The results are obtained in the approximation quadratic in metric perturbations on top of flat Euclidean space and covariantized in terms of spacetime curvature. Covariantization includes a special vector field $\xi^\mu(x)$ which generalizes the Killing vector of static geometries with time translation isometry to the case of a generic arbitrarily inhomogeneous metric subject to timelike periodicity condition. This vector field is obtained as a covariant metric functional to quadratic order in metric perturbations and gives rise to the local function $T/\sqrt{\xi^2(x)}$, $\xi^2(x)=g_{\mu\nu}(x)\xi^\mu(x)\xi^\nu(x)$, reducing to Tolman temperature $T/\sqrt{g_{00}(x)}$ on stationary manifolds with Killing symmetry. High ``temperature'' asymptotic behavior of the nonlocal formfactors -- operator coefficients of the curvature tensor structures in the heat kernel and effective action -- are obtained and possible cosmological applications of these results are discussed.