Energy-momentum tensor for a scalar Casimir apparatus in a weak gravitational field: Neumann conditions

TL;DR

This paper calculates the energy-momentum tensor for a scalar Casimir system under weak gravity with Neumann boundary conditions, revealing gravity-dependent corrections to energy, pressure, and a tiny upward force.

Contribution

It provides the first calculation of the energy-momentum tensor for a scalar Casimir setup with Neumann conditions in a weak gravitational field, including novel gravity-dependent effects.

Findings

Gravity introduces corrections to Casimir energy and pressure.

A tiny upward force acts on the apparatus due to gravity.

Results are consistent with conservation laws and conformal invariance.

Abstract

We consider a Casimir apparatus consisting of two perfectly conducting parallel plates, subject to the weak gravitational field of the Earth. The aim of this paper is the calculation of the energy-momentum tensor of this system for a free, real massless scalar field satisfying Neumann boundary conditions on the plates. The small gravity acceleration (here considered as not varying between the two plates) allows us to perform all calculations to first order in this parameter. Some interesting results are found: a correction, depending on the gravity acceleration, to the well-known Casimir energy and pressure on the plates. Moreover, this scheme predicts a tiny force in the upwards direction acting on the apparatus. These results are supported by two consistency checks: the covariant conservation of the energy-momentum tensor and the vanishing of its regularized trace, when the scalar field is conformally coupled to gravity.