Casimir effect in hemisphere capped tubes

TL;DR

This paper analyzes the Casimir effect in a (2+1)-dimensional spacetime with a cylindrical tube capped by a hemisphere, calculating vacuum energies and stresses for a scalar field with various curvature couplings.

Contribution

It provides a detailed calculation of vacuum expectation values and Casimir forces in a mixed cylindrical-hemispherical geometry, including effects of curvature coupling.

Findings

Vacuum energy density is negative on the cylindrical part for conformal coupling.

Energy density varies near the boundary, being negative near the top of the hemisphere.

Net Casimir force on the boundary is zero, indicating equilibrium.

Abstract

In this paper we investigate the vacuum densities for a massive scalar field with general curvature coupling in background of a (2+1)-dimensional spacetime corresponding to a cylindrical tube with a hemispherical cap. A complete set of mode functions is constructed and the positive-frequency Wightman function is evaluated for both the cylindrical and hemispherical subspaces. On the base of this, the vacuum expectation values of the field squared and energy-momentum tensor are investigated. The mean field squared and the normal stress are finite on the boundary separating two subspaces, whereas the energy density and the parallel stress diverge as the inverse power of the distance from the boundary. For a conformally coupled field, the vacuum energy density is negative on the cylindrical part of the space. On the hemisphere, it is negative near the top and positive close to the boundary. In the case of minimal coupling the energy density on the cup is negative. On the tube it is positive near the boundary and negative at large distances. Though the geometries of the subspaces are different, the Casimir pressures on the separate sides of the boundary are equal and the net Casimir force vanishes. The results obtained may be applied to capped carbon nanotubes described by an effective field theory in the long-wavelength approximation.