Casimir densities for two concentric spherical shells in the global monopole spacetime

TL;DR

This paper analyzes quantum vacuum effects for a scalar field between two concentric spherical shells in a global monopole spacetime, deriving explicit formulas for vacuum densities, stresses, and forces, with applications to various geometric limits.

Contribution

It provides new analytical expressions for vacuum expectation values and Casimir forces in a global monopole background with spherical boundaries, including interference effects and asymptotic behaviors.

Findings

Interaction forces are finite and attractive for Dirichlet boundary conditions.

Vacuum densities exhibit specific asymptotic behaviors in various geometric limits.

Results generalize known Minkowski space Casimir effects to curved global monopole spacetime.

Abstract

The quantum vacuum effects are investigated for a massive scalar field with general curvature coupling and obeying the Robin boundary conditions given on two concentric spherical shells with radii $a $ and $b$ in the $D+1$-dimensional global monopole background. The expressions are derived for the Wightman function, the vacuum expectation values of the field square, the vacuum energy density, radial and azimuthal stress components in the region between the shells. A regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of combinations of the cylinder functions. This formula allows us to extract from the vacuum expectation values the parts due to a single sphere on background of the global monopole gravitational field, and to present the "interference" parts in terms of exponentially convergent integrals, useful, in particular, for numerical evaluations. The vacuum forces acting on the boundaries are presented as a sum of the self--action and interaction terms. The first one contains well known surface divergences and needs a further regularization. The interaction forces between the spheres are finite for all values $a<b$ and are attractive for a Dirichlet scalar. The asymptotic behavior of the vacuum densities is investigated (i) in the limits $a\to 0$ and $b\to \infty $, (ii) in the limit $a,b\to \infty $ for fixed value $b-a$, and (iii) for small values of the parameter associated with the solid angle deficit in global monopole geometry. We show that in the case (ii) the results for two parallel Robin plates on the Minkowski bulk are rederived to the leading order.