Lorentz violating scalar Casimir effect for a $D$-dimensional sphere

TL;DR

This paper studies how Lorentz symmetry breaking affects the scalar Casimir effect inside a D-dimensional sphere, revealing how the Casimir stress depends on the Lorentz-violating parameter and the dimension, including a transition from repulsive to attractive forces.

Contribution

It provides analytical expressions for the Casimir stress under Lorentz violation in a spherical geometry, including a critical parameter value where the force changes nature.

Findings

Casimir stress factorizes in the timelike case as Lorentz-invariant result times (1+λ)^(-1/2)

Analytical expression for Casimir stress in the spacelike case without simple factorization

Existence of a critical λ_c where the Casimir force switches from repulsive to attractive in D>2, especially detailed for D=3

Abstract

We investigate the Casimir effect, due to the confinement of a scalar field in a $D$-dimensional sphere, with Lorentz symmetry breaking. The Lorentz-violating part of the theory is described by the term $\lambda (u \cdot \partial \phi) ^{2}$, where the parameter $\lambda$ and the background vector $u^{\mu}$ codify the breakdown of Lorentz symmetry. We compute, as a function of $D$, the Casimir stress by using Green's function techniques for two specific choices of the vector $u ^{\mu}$. In the timelike case, $u ^{\mu} = (1,0,...,0)$, the Casimir stress can be factorized as the product of the Lorentz invariant result times the factor $(1 + \lambda) ^{-1/2}$. For the radial spacelike case, $u ^{\mu} = (0,1,0,...,0)$, we obtain an analytical expression for the Casimir stress which nevertheless does not admit a factorization in terms of the Lorentz invariant result. For the radial spacelike case we find that there exists a critical value $\lambda _{c} = \lambda _{c} (D)$ at which the Casimir stress transits from a repulsive behavior to an attractive one for any $D> 2$. The physically relevant case $D = 3$ is analyzed in detail where the critical value $\lambda _{c}|_{\small D=3} = 0.0025$ was found. As in the Lorentz symmetric case, the force maintains the divergent behavior at positive even integer values of $D$.