Stress-Energy Tensor of a Scalar Field on a Product Spacetime with a Time-Dependent Compact Dimension

TL;DR

This paper calculates the vacuum stress-energy tensor of a scalar field in a spacetime combining an FLRW universe with a time-varying compact dimension, extending regularization techniques to analyze dynamic Casimir effects.

Contribution

It introduces a modified adiabatic regularization method to derive analytic expressions for stress-energy tensor in dynamic compactified spacetimes for dimensions 3 and 4.

Findings

Results align with known Casimir energies in static limits.

Higher-order corrections match FLRW results when scale factors coincide.

Analytic expressions are obtained for massless, conformally coupled scalar fields.

Abstract

We compute the vacuum expectation value of the stress-energy tensor of a scalar field on a product spacetime composed of an FLRW background times a compact dimension ($\mathcal{M}^{1, \,d-1} \times \mathcal{S}^1$), where the size of the latter is allowed to vary with time. We modify the standard adiabatic regularization prescription to obtain analytic expressions for both $d=3$ and $d=4$. In the massless and conformally coupled limit, the leading order time-dependent results are consistent with known time-independent Casimir contributions. Furthermore, in this limit the higher-order time-dependent corrections, when the FLRW and compact-dimension scale factors coincide, match known results for ($1+d$)-dimensional FLRW spacetime.