Null Aether Theory: $pp$-Wave and AdS Wave Solutions

General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the aether. In this work, we study plane wave metrics in such a theory. For this purpose, we assume that the aether field is a null vector field satisfying certain conditions--we refer to the theory constructed in this way as Null Aether Theory (NAT). Assuming the Kerr-Schild form for such metrics we show that the theory admits exact plane wave solutions in any dimension $D\geq3$. The field equations are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. Specifically, when the background metric is flat, i.e. for the $pp$-wave spacetimes, these equations decouple and we obtain one Laplace equation and one massive Klein-Gordon equation in $(D-2)$-dimensional Euclidean flat space. Depending on the value of the parameter $c_3$ and the form of the solution of the Laplace equation, we show that plane waves--subclass of $pp$-waves--are also solutions to the theory. In the case of AdS background, we are able to solve explicitly the coupled differential equations in three dimensions and thereby construct exact AdS-plane wave solutions in the theory. We also show that this latter solution can be generalised to higher dimensions under the assumption that the AdS wave is homogeneous along the transverse $D-3$ coordinates.