Comment on "Casimir effect in a weak gravitational field: Schwinger's approach"

TL;DR

This paper critiques a previous claim about gravitational corrections to the Casimir effect, clarifying the proper calculation method and showing that the effect remains unchanged in a weak gravitational field.

Contribution

It corrects the methodology used in prior work by emphasizing the proper eigenfunction approach, demonstrating that gravitational corrections cancel out in the Casimir effect.

Findings

Proper time Hamiltonian should be traced using covariant D'Alembertian eigenfunctions

Gravitational corrections to Casimir energy cancel out in weak fields

The vacuum energy density remains consistent with Minkowski space results

Abstract

We show that the statement in F. Sorge [Class. Quant. Grav. 36, no.23, 235006 (2019)] that the Casimir effect receives second order corrections due to gravity is not consistent. We remark especially on the tracing of the proper time Hamiltonian, where the correct procedure is to use the eigenfunctions and eigenvalues of the covariant DAlembertian. After some cancellations we find that the value of the functional W [0] is the same as obtained by Sorge. However, we argue that the proper vacuum energy density carries extra space-time volume terms that cancel over the gravitational correction, returning to the same expression as in Minkowski space-time.