Covariant extrinsic curvature expansion of the nonlocal effective action for a massless scalar field on a manifold with boundary

TL;DR

This paper derives a covariant, nonlocal effective action expansion for a massless scalar field on manifolds with boundaries, extending previous results to more general geometries using heat-kernel methods.

Contribution

It introduces a geometric framework for the nonlocal effective action that generalizes earlier Monge-patch results to arbitrary surfaces with boundary.

Findings

Derived a covariant expansion of the nonlocal effective action to quadratic order in extrinsic curvature.

Extended previous results to more general surfaces without a global Monge-patch.

Computed particle-creation rates for oscillating deformed geometries in 2+1 and 3+1 dimensions.

Abstract

We study the nonlocal effective action of a massless scalar field defined on a flat manifold with a curved boundary. Using a heat-kernel approach, we derive a covariant expansion of the nonlocal contribution to quadratic order in the extrinsic curvature tensor. Our construction provides a geometric framework that both reproduces earlier results obtained for Monge-patch embeddings and extends them to more general surfaces that need not admit a global Monge-patch description. The expansion is valid in the regime where gradients of the extrinsic curvature dominate over nonlinear curvature effects. As an application, we compute the particle-creation rate for an oscillating deformed ring in $2+1$ dimensions and an oscillating deformed sphere in $3+1$ dimensions.