Abstract
The influence of a spherical boundary on the vacuum fluctuations of a massive scalar field is investigated in the background of a ( D + 1 ) -dimensional Milne universe, assuming that the field obeys Robin boundary conditions on the sphere. The normalized mode functions are derived for the regions inside and outside the sphere and different vacuum states are discussed. For the conformal vacuum, the Hadamard function is decomposed into boundary-free and sphere-induced contributions and an integral representation is obtained for the latter in both the interior and exterior regions. As important local characteristics of the vacuum state, the vacuum expectation values (VEVs) of the field squared and of the energy-momentum tensor are investigated. It is shown that the vacuum energy-momentum tensor has an off-diagonal component that corresponds to the energy flux along the radial direction. Depending on the coefficient in Robin boundary conditions, the sphere-induced contribution to the vacuum energy and the energy flux can be either positive or negative. At late stages of the expansion and for a massive field the decay of the sphere-induced VEVs, as functions of time, is damping oscillatory. The geometry under consideration is conformally related to that for a static spacetime with negative constant curvature space and the sphere-induced contributions in the corresponding VEVs are compared.
Highlights
In constructing quantum field theories in background geometries different from the Minkowski spacetime, among the most important points is the selection of a physically meaningful vacuum state
The Milne universe is described by the Friedmann–Robertson–Walker type line element with negative curvature spatial sections and with the scale factor being a linear function of the time coordinate
We have investigated the influence of a spherical boundary on the local properties of the vacuum state for a scalar field in the Milne universe
Summary
In constructing quantum field theories in background geometries different from the Minkowski spacetime, among the most important points is the selection of a physically meaningful vacuum state. For a massive scalar field with the Robin boundary condition the VEVs of the energy-momentum tensor inside and outside a spherical shell and in the region between two concentric spherical boundaries in (D + 1)-dimensional Minkowski spacetime have been investigated in [51]. In [67] the two-point function and the VEVs are investigated for a scalar field in a spherically symmetric static background geometry with two distinct metric tensors inside and outside a spherical boundary In this setup the exterior and interior geometries can correspond to different vacuum states of the same theory. In Appendix A we present a summation formula for series over the scalar eigenmodes inside the spherical shell that is used to derive an integral representation for the Hadamard function
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