Vacuum fluctuation force on a rigid Casimir cavity in a gravitational field

doi:10.1016/S0375-9601(02)00445-0

Physics Letters A

Volume 297, Issues 5–6, 20 May 2002, Pages 328-333

https://doi.org/10.1016/S0375-9601(02)00445-0 Get rights and content

Abstract

We discuss the possibility of verifying the equivalence principle for the zero-point energy of quantum electrodynamics, by evaluating the force, produced by vacuum fluctuations, acting on a rigid Casimir cavity in a weak gravitational field. The resulting force has opposite direction with respect to the gravitational acceleration; the order of magnitude for a multi-layer cavity configuration is derived and experimental feasibility is discussed, taking into account current technological resources.

Introduction

Although much progress has been made in the evaluation and experimental verification of effects produced by vacuum energy in Minkowski space–time [1], [2], [3], [4], [5], [6], it remains unclear why the observed universe exhibits an energy density much smaller than the one resulting from the application of quantum field theory and the equivalence principle [7], [8]. Various hypotheses have also been put forward on the interaction of virtual quanta with the gravitational field. For instance, some arguments seem to suggest that virtual photons do not gravitate [8], while other authors have suggested that Casimir energy contributes to gravitation [9]. So far it seems fair enough to say that no experimental verification that vacuum fluctuations can be treated according to the equivalence principle has been obtained as yet, even though there are expectations, as we agree, that this should be the case.

Motivated by all these considerations, our Letter computes the effect of a gravitational field on a rigid Casimir cavity, evaluating the net force acting on it. The order of magnitude of the resulting force, although not allowing an immediate experimental verification, turns out to be compatible with the current extremely sensitive force detectors, actually the interferometric detectors of gravitational waves. The Casimir cavity is rigid in that its shape and size remain unchanged under certain particular external conditions such as, for example, the absence of accelerations or impulses. We evaluate the force acting on this non-isolated system at rest in the gravitational field of the earth by studying the regularized energy-momentum tensor. The associated force turns out to have opposite direction with respect to gravitational acceleration. Orders of magnitude are discussed bearing in mind the current technological resources as well as experimental problems, and physical relevance of the analysis is stressed.

Section snippets

Evaluation of the force

In order to evaluate the force due to the gravitational field let us suppose that the cavity has geometrical configuration of two parallel plates of proper area A=L² separated by the proper length a. The system of plates is taken to be orthogonal to the direction of gravitational acceleration $g →$ .

In classical general relativity, the force density can be evaluated according to [10] $f_{ν} =− 1 − det g ∂ ∂x^{μ} − det g T_{ν}^{μ} + 12 ∂g_{ρσ} ∂x^{ν} T^{ρσ},$ where T^μν is the energy-momentum tensor of matter, representing the energy

Problems of experimental verification

In considering the possibility of experimental verification of the extremely small forces linked to this effect we point out that such measurements cannot be performed statically; this would make it necessary to compare the weight of the assembled cavity with the sum of the weights of its individual parts, a measure impossible to perform. On the contrary, the measurements we are interested in should be performed dynamically, by modulating the force in a known way; the effect will be detected if

Conclusions

The relation of the Casimir energy with the geometry of bounding surfaces has been properly understood only very recently, thanks to the outstanding work in Ref. [24], and another open problem of modern physics, i.e., the cosmological constant problem, results from calculations which rely on the application of the equivalence principle to vacuum energy [7], and this adds interest to our calculations, that we have performed by focusing on a Casimir apparatus.

Our original contributions are given

Acknowledgements

The INFN financial support is gratefully acknowledged. The work of G. Esposito has been partially supported by the Progetto di Ricerca di Interesse Nazionale SINTESI 2000. We are indebted to Bernard Kay and Serge Reynaud for correspondence, and to Giuseppe Marmo for conversations. More important, we acknowledge the lovely support of Cristina, Michela and Patrizia.

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Vacuum fluctuation force on a rigid Casimir cavity in a gravitational field

Abstract

Introduction

Section snippets

Evaluation of the force

Problems of experimental verification

Conclusions

Acknowledgements

Phys. Rep.

Phys. Rep.

Phys. Rev. D

Aspects of Quantum Field Theory in Curved Space-Time

Phys. Rev. Lett.

Phys. Rev. Lett.

Science

Rev. Mod. Phys.

Quantum Mechanics and Path Integrals

Astrophys. J.

The Theory of Relativity

Phys. Rev.

Gravitation