  In this work, we investigate the consequences of the spontaneous breaking of
Lorentz symmetry, triggered by the bumblebee vector field, on the usual
Einstein-Hilbert theory. Specifically, we consider the Einstein-Hilbert action
modified by the bumblebee dynamic field, and evaluate the graviton propagator
using an extended basis of Barnes-Rivers tensor projectors, involving the
Lorentz-violating vector. Once the propagator is carried out, we proceed with
discussing the consistency of the model, writing the dispersion relations, and
analyzing causality and unitarity. We verify that this model possesses two
dispersion relations: one provides causal and unitary propagating modes, while
the second yields a causal but nonunitary mode which spoils the physical
consistency of the model.
