Relativistic mechanics of Casimir apparatuses in a weak gravitational field

TL;DR

This paper develops relativistic equations for the equilibrium of bodies in gravitational fields and applies them to analyze the weight of Casimir energy in weak gravitational fields, confirming it behaves as gravitational mass.

Contribution

It introduces general relativistic Cardinal Equations for extended bodies and applies them to Casimir apparatuses, clarifying their gravitational behavior in weak fields.

Findings

Casimir energy acts as gravitational mass $E_C/c^2$.

Supported rigid cavities exhibit weight equal to Casimir energy divided by $c^2$.

Separate supported plates show complex force interactions not deducible by simple arguments.

Abstract

This paper derives a set of general relativistic Cardinal Equations for the equilibrium of an extended body in a uniform gravitational field. These equations are essential for a proper understanding of the mechanics of suspended relativistic systems. As an example, the prototypical case of a suspended vessel filled with radiation is discussed. The mechanics of Casimir apparatuses at rest in the gravitational field of the Earth is then considered. Starting from an expression for the Casimir energy-momentum tensor in a weak gravitational field recently derived by the authors, it is here shown that, in the case of a rigid cavity supported by a stiff mount, the weight of the Casimir energy $E_C$ stored in the cavity corresponds to a gravitational mass $M=E_C/c^2$, in agreement with the covariant conservation law of the regularized energy-momentum tensor. The case of a cavity consisting of two disconnected plates supported by separate mounts, where the two measured forces cannot be obtained by straightforward arguments, is also discussed.