Casimir Effect for a Massive Scalar Field in Lorentz-Violating Aether Compactification

This work investigates the influence of Lorentz symmetry breaking, introduced by an aether-like field $\alpha_\phi$, on the Casimir effect within a five-dimensional flat spacetime. By considering a quasiperiodic condition regulated by the parameter $\beta$ and an extra dimension compactified at scale $b$, we derive closed-form expressions for the Casimir energy and the resulting force between two parallel plates under Neumann boundary conditions. Our results demonstrate that $\beta$ acts as a crucial control parameter, enabling a continuous transition between attractive and repulsive regimes, with a characteristic symmetry around $\beta = 0.5$. We show that the Lorentz-violating parameter $\alpha_\phi$ functions as an enhancement factor, significantly amplifying the vacuum interaction, while the geometric ratio $a/b$ proves decisive for system stabilization. Specifically, we find that the high-compactification regime leads to a plateau in the Casimir force, effectively stabilizing the interaction. Furthermore, we analyze the mass spectrum of the field, recovering standard geometric forms in the massless limit and demonstrating that while light fields ($M \ll 1$) exhibit subtle quadratic corrections, heavy fields ($M \gg 1$) lead to an exponential suppression of the Casimir effect. The interplay between Lorentz violation and extra-dimensional compactification provides a rich mechanism with potential applications in the modulation of vacuum-induced interactions at micro and nano scales.