Abstract
We evaluate the Wightman function, the mean field squared and the vacuum expectation value of the energy–momentum tensor for a scalar field with the Robin boundary condition on a spherical shell in the background of a constant negative curvature space. For the coefficient in the boundary condition there is a critical value above which the scalar vacuum becomes unstable. In both the interior and the exterior regions, the vacuum expectation values are decomposed into the boundary-free and sphere-induced contributions. For the latter, rapidly convergent integral representations are provided. In the region inside the sphere, the eigenvalues are expressed in terms of the zeros of the combination of the associated Legendre function and its derivative and the decomposition is achieved by making use of the Abel–Plana type summation formula for the series over these zeros. The sphere-induced contribution to the vacuum expectation value of the field squared is negative for the Dirichlet boundary condition and positive for the Neumann one. At distances from the sphere larger than the curvature scale of the background space the suppression of the vacuum fluctuations in the gravitational field corresponding to the negative curvature space is stronger compared with the case of the Minkowskian bulk. In particular, the decay of the vacuum expectation values with the distance is exponential for both massive and massless fields. The corresponding results are generalized for spaces with spherical bubbles and for cosmological models with negative curvature spaces.
Highlights
In the absence of a reliable theory for quantum gravity, the influence of the gravitational field on quantum matter is investigated within the framework of semiclassical theory.In the latter, the gravitational field is considered as the classical background and the back-reaction of quantum effects is described by quasiclassical Einstein equations with the expectation value of the energy–momentum tensor for quantum fields in the right-hand side
We evaluate the Wightman function, the mean field squared and the vacuum expectation value of the energy–momentum tensor for a scalar field with the Robin boundary condition on a spherical shell in the background of a constant negative curvature space
The gravitational field is considered as the classical background and the back-reaction of quantum effects is described by quasiclassical Einstein equations with the expectation value of the energy–momentum tensor for quantum fields in the right-hand side
Summary
In the absence of a reliable theory for quantum gravity, the influence of the gravitational field on quantum matter is investigated within the framework of semiclassical theory. An interesting topic in the investigations of the Casimir effect is the dependence of the physical characteristics of the vacuum, like the energy density and stresses, on the geometry of the background spacetime The evaluation of these characteristics requires the knowledge of a complete set of modes for a quantum field and exact results can be provided for highly symmetric bulk and boundary geometries only. As local characteristics of the vacuum state we shall investigate the vacuum expectation values of the field squared and the energy–momentum tensor The latter serves as a source in the right-hand side of semiclassical Einstein equations and plays an important role in considerations of the back-reaction from quantum effects on the background geometry. In Appendix B, by making use of the generalized Abel–Plana formula, a summation formula is derived for the series of this type
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