A. A. Araújo Filho
Abstract We study the propagation and emission of gravitational waves in the metric–affine formulation of the bumblebee model, where spontaneous Lorentz symmetry breaking arises from a vector field acquiring a nonvanishing vacuum expectation value. Working in the geometric–optics limit of the linearized theory, we derive the modified dispersion relation governing the graviton modes and show that it depends on the orientation of the wave vector relative to the background vector. The polarization sector is examined for timelike and spacelike configurations of the Lorentz-violating vacuum. In both cases only two independent tensor modes propagate, although their propagation properties and tensor structure depend on the orientation of the background field. We then construct the retarded Green function associated with the modified wave operator and determine the radiation-zone produced by localized sources. In the timelike configuration the Lorentz-violating effects appear through a modified propagation speed and an overall amplitude renormalization, leading to a shifted retarded time while preserving the quadrupole structure of the waveform. In contrast, the spacelike sector introduces anisotropic corrections to the quadrupole amplitude together with an additional contribution proportional to the third time derivative of the quadrupole moment. As an astrophysical application, the gravitational radiation emitted by a circular binary black-hole system is evaluated, allowing observational constraints on the Lorentz-violating combination $$\xi b^{2}$$ ξ b 2 to be estimated using multimessenger bounds from GW170817/GRB 170817A and waveform consistency requirements from gravitational wave observations.