On the Lichnerowicz operator in traversable wormhole spacetimes

TL;DR

This paper addresses the challenge of calculating Casimir energies in traversable wormhole spacetimes by transforming the Lichnerowicz operator into a transverse-traceless form, facilitating stability analysis.

Contribution

It introduces a method to gauge the modified Lichnerowicz operator into a transverse-traceless form for arbitrary backgrounds, enabling eigenvalue computation.

Findings

Casimir energies can be computed using the gauged transverse-traceless operator.

The method simplifies stability analysis of traversable wormholes.

Eigenvalue determination is feasible for complex curved backgrounds.

Abstract

The evaluation of Casimir energies in curved background spacetimes is an essential ingredient to study the stability of traversable wormholes. In practice one has to calculate the contribution of the transverse-traceless component of the metric perturbation on a curved spacetime background. This implies the study of an eigenvalue equation involving a modified form of the Lichnerowicz operator. For arbitrary background spacetimes, however, such an operator does not display transverse-traceless properties, a fact that impedes the determination of the eigenvalues. Against this background, we show that the problem can be circumvented. Casimir energies can be calculated by gauging the original form of the modified Lichnerowicz operator into a transverse-traceless one.