Scalar Casimir effect for D-dimensional spherically symmetric Robin boundaries

TL;DR

This paper analyzes the vacuum energy and stress components of a massive scalar field with Robin boundary conditions on spherical surfaces in D-dimensional space, revealing detailed behaviors and energy decompositions.

Contribution

It derives explicit formulas for vacuum energy densities and stresses in D-dimensional spherical geometries with Robin boundaries, including energy decomposition into volume and surface parts.

Findings

Vacuum densities near the sphere and at large distances are characterized.

The Casimir energy is decomposed into volumic and surface contributions.

Mode sum energy differs from volume integral energy due to surface energy contributions.

Abstract

The vacuum expectation values for the energy-momentum tensor of a massive scalar field with general curvature coupling and obeying the Robin boundary condition on spherically symmetric boundaries in D-dimensional space are investigated. The expressions are derived for the regularized vacuum energy density, radial and azimuthal stress components (i) inside and outside a single spherical surface and (ii) in the intermediate region between two concentric spheres. Regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of cylinder functions. Asymptotic behavior of the vacuum densities near the sphere and at large distances is investigated. A decomposition of the Casimir energy into volumic and surface parts is provided for both cases (i) and (ii). We show that the mode sum energy, evaluated as a sum of the zero-point energies for each normal mode of frequency, and the volume integral of the energy density in general are different, and argue that this difference is due to the existence of an additional surface energy contribution.