This work aims to solve Einstein Equation in a scenario with Lorentz symmetry violation for
gravitational waves polarization study. It will be commented about Standard Model Extension,
spontaneous and explicit violation of a symmetry as well as their consequences. The bumblebee
model will be used for the study of Lorentz violation, where the terms that breaks off the symmetry are included in the lagrangean of the gravitational theory. The Euler-Lagrange equations are
used for determination of modified graviton propagator, where we expend the lagrangean up to
second order of the pertubation field. We conclude that, in this scenario, the dispersion relation
of graviton is different of the usual one, where we have a term that selects a preferred direction in
the spacetime. Besides that, the graviton still with two degrees of freedom, despite the existence
of bumblebee field. Then, the modified wave equation for perturbation field is solved and we
compare the polarization states of the gravitational wave solution modified with the usual case.
We show that for a bumblebee field being timelike or in the same direction of wave momentum
we have no changes in the polarization tensor. But for a bumblebee field being in another diretion
we have a modified polarization tensor for the graviton