Scalar vacuum densities on Beltrami pseudosphere

TL;DR

This paper studies how spatial curvature and topology influence the vacuum state of a charged scalar field on a (2+1)-dimensional Beltrami pseudosphere, revealing divergent local characteristics and significant topological effects on stresses.

Contribution

It provides a detailed analysis of vacuum expectation values for a charged scalar field on a curved, topologically nontrivial surface, including asymptotic behaviors and the impact of topology on stresses.

Findings

Topological contributions to VEVs are finite and analyzed numerically.

VEVs decay as a power-law at large distances.

Nontrivial topology significantly affects vacuum stresses, especially for conformally coupled massless fields.

Abstract

We investigate the combined effects of spatial curvature and topology on the properties of the vacuum state for a charged scalar field localized on the (2+1)-dimensional Beltrami pseudosphere, assuming that the field obeys quasiperiodicity condition with constant phase. As important local characteristics of the vacuum state the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor are evaluated. The contributions in the VEVs coming from geometry with an uncompactified azimuthal coordinate are divergent, whereas the compact counterparts are finite and are analysed both numerically and asymptotically. For small values of proper radius of the compactified dimension, the leading terms of topological contributions are independent of the field mass and curvature coupling parameter, increasing by a power-law. In the opposite limit, the VEVs decay following a power-law in the general case. In the special case of a conformally coupled massless field the behavior is different. Unlike the VEV of field squared and vacuum energy density, the radial and azimuthal stresses are increasing by absolute value. As a consequence, the effects of nontrivial topology are strong for the stresses in this case at small values of radial coordinate.