Casimir effect of a rough membrane in an aether-like Lorentz-violating scenario

In this paper, we explore the Casimir effect of a rough membrane within the framework of theories that break Lorentz symmetry. We consider two constant aether vectors: one time-like and other space-like, simultaneously. We employ an appropriate change of coordinates such that the membrane assumes a completely flat border and the remaining terms associated with the roughness are considered as part of the potential. Quantum fluctuations are induced by a scalar quantum field subject to Dirichlet boundary conditions. The spectrum is obtained through perturbation theory and regularized using the [Formula: see text]-function method. We provide an explicit example of a membrane with periodic boundaries. The presence of aether vectors has a significant impact on the dominant term of the Casimir effect, while roughness only affects the secondary terms. Additionally, we examine the finite-temperature case.