Vacuum solutions in the Einstein-Aether Theory

The Einstein-Aether (EA) theory belongs to a class of modified gravity theories characterized by the introduction of a time-like unit vector field called aether. In this scenario, a preferred frame arises as a natural consequence of a broken Lorentz invariance. In the present work, we have obtained and analyzed some exact solutions allowed by this theory for two particular cases of a perfect fluid, both with Friedmann-Lemahre-Robertson-Walker symmetry: (i) a fluid with constant energy density (p = -rho(o)) and (ii) a fluid with zero energy density (rho(o) = 0) corresponding to the vacuum solution with and without cosmological constant (Lambda), respectively. Our solutions show that the EA and general relativity (GR) theories only differ in coupling constants. This difference is clearly shown because of the existence of singularities that are not in GR theory. This characteristic appears in the solutions with p = -rho(o) as well as with rho(o) = 0, where this last one depends only on the aether field. Furthermore, we consider the term of the EA theory in the Raychaudhuri equation and discuss the meaning of the strong energy condition in this scenario and found that this depends on the aether field. The solutions admit an expanding or contracting system. Bounce, singular, constant, and accelerated expansion solutions were also obtained, exhibiting the richness of the EA theory from the dynamic point of view of a collapsing system or of a cosmological model. The analysis of energy conditions, considering an effective fluid, shows that the term of the aether contributes significantly for the accelerated expansion of the system for the case in which the energy density is constant. On the other hand, for the vacuum case (rho(o) = 0), the energy conditions are all satisfied for the aether fluid.