The Vacuum Energy Density for Spherical and Cylindrical Universes

TL;DR

This paper calculates the vacuum energy density for scalar fields in spherical and cylindrical universes, demonstrating a regularization method's effectiveness and revealing key differences in energy densities across geometries and dimensions.

Contribution

It introduces a simple binomial expansion regularization method for vacuum energy calculations and compares it with more complex techniques, providing new insights into geometric and dimensional effects.

Findings

No poles at four dimensions for the studied geometries

Coincidence of results for 3d and 4d cylinders after pole subtraction

Vacuum energy density for cylinders is significantly smaller than for spheres of the same dimension

Abstract

The vacuum energy density (Casimir energy) corresponding to a massless scalar quantum field living in different universes (mainly no-boundary ones), in several dimensions, is calculated. Hawking's zeta function regularization procedure supplemented with a very simple binomial expansion is shown to be a rigorous and well suited method for performing the analysis. It is compared with other, much more involved techniques. The principal-part prescription is used to deal with the poles that eventually appear. Results of the analysis are the absence of poles at four dimensions (for a 4d Riemann sphere and for a 4d cylinder of 3d Riemann spherical section), the total coincidence of the results corresponding to a 3d and a 4d cylinder (the first after pole subtraction), and the fact that the vacuum energy density for cylinders is (in absolute value) over an order of magnitude smaller than for spheres of the same dimension.