Casimir energy for spherical shell in Schwarzchild black hole background

TL;DR

This paper calculates the Casimir energy of a massless scalar field around a spherical shell in Schwarzschild spacetime, using zeta function regularization and renormalization to obtain a finite result.

Contribution

It introduces a method to compute the Casimir energy in a curved spacetime with boundary conditions, combining zeta regularization and renormalization techniques.

Findings

Renormalized Casimir energy expression derived for the spherical shell

Divergent contributions identified and canceled via renormalization

Method applicable to similar problems in curved spacetime backgrounds

Abstract

In this paper, we consider the Casimir energy of massless scalar field which satisfy Dirichlet boundary condition on a spherical shell. Outside the shell, the spacetime is assumed to be described by the Schwarzschild metric, while inside the shell it is taken to be the flat Minkowski space. Using zeta function regularization and heat kernel coefficients we isolate the divergent contributions of the Casimir energy inside and outside the shell, then using the renormalization procedure of the bag model the divergent parts are cancelled, finally obtaining a renormalized expression for the total Casimir energy.