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Einstein-Hilbert Graviton modes modified by the Lorentz-violating bumblebee Field

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.