Energy-momentum tensor of a Casimir apparatus in a weak gravitational field: scalar case

TL;DR

This paper investigates the energy-momentum tensor of a massless scalar field in a Casimir setup within a weak gravitational field, showing conservation and absence of trace anomaly under conformal coupling, simplifying previous electromagnetic case analysis.

Contribution

It provides a simplified analysis of the scalar Casimir effect in gravity, demonstrating covariant conservation and no trace anomaly with conformal coupling, unlike the electromagnetic case.

Findings

Energy-momentum tensor is covariantly conserved.

Trace anomaly vanishes with conformal coupling.

Casimir energy and pressure are finite under conformal coupling.

Abstract

Recent work in the literature had evaluated the energy-momentum tensor of a Casimir apparatus in a weak gravitational field, for an electromagnetic field subject to perfect conductor boundary conditions on parallel plates. The Casimir apparatus was then predicted to experience a tiny push in the upwards direction, and the regularized energy-momentum tensor was found to have a trace anomaly. The latter, unexpected property made it compelling to assess what happens in a simpler case. For this purpose, the present paper studies a free, real massless scalar field subject to homogeneous Dirichlet conditions on the parallel plates. Working to first order in the constant gravity acceleration, the resulting regularized and renormalized energy-momentum tensor is found to be covariantly conserved, while the trace anomaly vanishes if the massless scalar field is conformally coupled to gravity. Conformal coupling also ensures a finite Casimir energy and finite values of the pressure upon parallel plates.