Casimir densities for a spherical brane in Rindler-like spacetimes

TL;DR

This paper calculates the quantum vacuum effects for a scalar field near a spherical brane in a Rindler-like spacetime, relevant for black hole near-horizon geometries, using zeta regularization and analyzing various limits.

Contribution

It provides explicit expressions for vacuum expectation values of the field and energy-momentum tensor in a higher-dimensional Rindler-like spacetime with a spherical brane, including boundary effects.

Findings

Vacuum expectation values are separated into boundary-free and brane-induced parts.

Explicit formulas are derived for the Wightman function and vacuum densities.

Various limiting cases of the brane-induced effects are analyzed.

Abstract

Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor are evaluated for a scalar field obeying mixed boundary condition on a spherical brane in (D+1)-dimensional Rindler-like spacetime $Ri\times S^{D-1}$, where $Ri$ is a two-dimensional Rindler spacetime. This spacetime approximates the near horizon geometry of (D+1) -dimensional black hole in the large mass limit. The vacuum expectation values are presented as the sum of boundary-free and brane-induced parts. Further we extract from the Wightman function for the boundary-free geometry the corresponding function in the bulk $R^{2}\times S^{D-1}$. For the latter geometry the vacuum expectation values of the field square and the energy-momentum tensor do not depend on the spacetime point. For the renormalization of these quantities we use zeta regularization technique. Various limiting cases of the brane-induced vacuum expectation values are investigated.