Finite-temperature corrections to the energy-momentum tensor at one loop in static spacetimes

Finite-temperature corrections to the effective potential and the energy-momentum tensor of a scalar field are computed in a perturbed Minkoswki spacetime. We consider the explicit mode decomposition of the field in the perturbed geometry and obtain analytical expressions in the nonrelativistic and ultrarelativistic limits to first order in scalar metric perturbations. In the static case, our results are in agreement with previous calculations based on the Schwinger-DeWitt expansion which indicate that thermal effects in a curved spacetime can be encoded in the local Tolman temperature at leading order in perturbations and in the adiabatic expansion. We also study the shift of the effective potential minima produced by thermal corrections in the presence of static gravitational fields. Finally, we discuss the dependence on the initial conditions set for the mode solutions.