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

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 $\lambda (u\cdot \partial \varphi {)}^2$, 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 ${\lambda }_c{|}_{D=3}=0.0025$ was found. As in the Lorentz symmetric case, the force maintains the divergent behavior at positive even integer values of $D$.

V2: 14 pages, 5 figures; minor changes and clarifications. Accepted for publication in Physical Review D

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