  This paper focuses on how the production and polarization of gravitational
waves are affected by spontaneous Lorentz symmetry breaking, which is driven by
a self-interacting vector field. Specifically, we examine the impact of a
smooth quadratic potential and a non-minimal coupling, discussing the
constraints and causality features of the linearized Einstein equation. To
analyze the polarization states of a plane wave, we consider a fixed vacuum
expectation value (VEV) of the vector field. Remarkably, we verify that a
space-like background vector field modifies the polarization plane and
introduces a longitudinal degree of freedom. In order to investigate the
Lorentz violation effect on the quadrupole formula, we use the modified Green
function. Finally, we show that the space-like component of the background
field leads to a third-order time derivative of the quadrupole moment, and the
bounds for the Lorentz-breaking coefficients are estimated as well.
