\zeta function renormalization of one-loop stress tensors in curved spacetime: a check on the method in the conical manifold and other cases

TL;DR

This paper validates a local ζ-function method for renormalizing one-loop stress tensors in curved spacetime, confirming its accuracy across various geometries including conical singularities and non-closed manifolds.

Contribution

It demonstrates the effectiveness of the ζ-function approach in diverse curved spacetime scenarios, extending its applicability beyond initial assumptions.

Findings

Complete agreement with other methods in all tested cases

Method works for manifolds with conical singularities

Validates the ζ-function approach for open and singular spacetimes

Abstract

A previously introduced method to renormalize the one-loop stress tensor and the one-loop vacuum fluctuations in a curved background by a direct local $\zeta$-function approach is checked in some thermal and nonthermal cases. First the method is checked in the case of a conformally coupled massless field in the static Einstein universe where all hypotheses initially requested by the method hold true. Secondly, dropping the hypothesis of a closed manifold, the method is checked in the open static Einstein universe. Finally, the method is checked for a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e. Rindler spacetimes/ large mass black hole manifold/cosmic string manifold). In all cases, a complete agreement with other approaches is found. Concerning the last case in particular, the method is proved to give rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. In the last case, comments on the measure employed in the path integral, the use of the optical manifold and the different approaches to renormalize the Hamiltonian are made.