Scalar Casimir effect between two concentric spheres

TL;DR

This paper introduces a new method to calculate the Casimir energy between two concentric spheres for a massless scalar field, demonstrating its effectiveness by deriving results consistent with known parallel plate configurations.

Contribution

A novel approach for calculating the Casimir energy with spherical boundary conditions, applicable to concentric spheres and half spheres.

Findings

Casimir energy matches parallel plate results in the small separation limit.

New method simplifies calculations for spherical geometries.

Demonstrated efficiency through calculations for concentric and half spheres.

Abstract

The Casimir effect giving rise to an attractive force between the closely spaced two concentric spheres that confine the massless scalar field is calculated by using a direct mode summation with contour integration in the complex plane of eigenfrequencies. We devoleped a new approach appropriate for the calculation of the Casimir energy for spherical boundary conditions. The Casimir energy for a massless scalar field between the closely spaced two concentric spheres coincides with the Casimir energy of the parallel plates for a massless scalar field in the limit when the dimensionless parameter {\eta}, ({\eta}=((a-b)/(\surd(ab))) where a (b) is inner (outer) radius of sphere), goes to zero. The efficiency of new approach is demonstrated by calculation of the Casimir energy for a massless scalar field between the closely spaced two concentric half spheres. PACS number(s): 03.70.+k, 12.20.DS, 11.10.Gh