We consider a self-interacting scalar field theory in a slowly varying gravitational background field. Using $\zeta$-function regularization and heat-kernel techniques, we derive the one-loop effective Lagrangian up to second order in the variation of the background field and up to quadratic terms in the curvature tensors. Specializing to different spacetimes of physical interest, the influence of the curvature on the phase transition is considered.

• Received 21 April 1993

DOI:https://doi.org/10.1103/PhysRevD.48.2813