Abstract
We calculate the finite vacuum energy density of the scalar and electromagnetic fields inside a Casimir apparatus made up of two conducting parallel plates in a general weak gravitational field. The metric of the weak gravitational field has a small deviation from flat spacetime inside the apparatus and we find it by expanding the metric in terms of small parameters of the weak background. We show that the found metric can be transformed via a gauge transformation to the Fermi metric. We solve the Klein-Gordon equation exactly and find mode frequencies in Fermi spacetime. Using the fact that the electromagnetic field can be represented by two scalar fields in the Fermi spacetime, we find general formulas for the energy density and mode frequencies of the electromagnetic field. Some well-known weak backgrounds are examined and consistency of the results with the literature is shown.
Highlights
The Casimir effect arises when there is a boundary in our problem and it predicts a force between two uncharged conducting metals in the presence of a quantum field
A purpose of this paper is to generalize the above analysis for a scalar field finding an exact solution of the Klein–Gordon equation in a general weak gravitational field
The electromagnetic field has two physical degrees of freedom and it is well known that in the Rindler spacetime the electromagnetic field can be represented in terms of two scalar fields satisfying the Klein–Gordon equation separately [31]
Summary
The Casimir effect arises when there is a boundary in our problem and it predicts a force between two uncharged conducting metals in the presence of a quantum field. We will expand the metric up to first order in terms of the parameters of the general weak gravitational field. We use the linearized weak field regime of general relativity and change the variables with the aid of the following gauge transformation: gμν = ημν + hμν , |hμν |
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