Probing Lorentz symmetry violation via the Casimir effect in rectangular cavities

We investigate the Casimir effect as a probe of Lorentz symmetry violation for a real scalar field confined to a rectangular waveguide with Dirichlet boundary conditions. The field dynamics is governed by a Lorentz-violating extension of the Klein-Gordon theory involving a fixed background four-vector $u_\mu$. Focusing on four representative configurations in which the background is aligned with the temporal direction or with one of the spatial axes of the cavity, we derive the modified mode spectra and the corresponding vacuum energies. We show that these configurations induce anisotropic modifications of the dispersion relations that depend explicitly on the orientation of the background vector relative to the cavity geometry, while still preserving mode separability. The resulting Casimir energy acquires characteristic direction-dependent corrections that encode the breaking of Lorentz symmetry, without altering the universal functional structure of the spectral kernel. Our analysis provides a controlled and transparent framework for isolating Lorentz-violating effects in confined geometries and highlights Casimir systems as sensitive probes of anisotropic physics and fundamental spacetime symmetries.