Compact binary systems in Einstein-Aether gravity: Direct integration of the relaxed field equations to 2.5 post-Newtonian order

The Einstein-Aether theory is an alternative theory of gravity in which the spacetime metric is supplemented by a long-range timelike vector field (the "aether" field). Here, for the first time, we apply the full formalism of post-Minkowskian theory and of the Direct Integration of the Relaxed Einstein Equations (DIRE), to this theory of gravity, with the goal of deriving equations of motion and gravitational waveforms for orbiting compact bodies to high orders in a post-Newtonian expansion. Because the aether field is constrained to have unit norm, a naive application of post-Minkowskian theory leads to contributions to the effective energy momentum tensor that are {\em linear} in the perturbative fields. We show that a suitable redefinition of fields using an array of "superpotentials" can eliminate such linear terms to any desired post-Newtonian order, resulting in flat spacetime wave equations for all fields, with sources consisting of matter terms and terms quadratic and higher in the fields. As an initial application of this new method, and as a foundation for obtaining the equations of motion for compact binaries, we obtain explicit solutions of the relaxed equations sufficient to obtain the metric in the near zone through 2.5 post-Newtonian order, or $O[(v/c)^5]$ beyond the Newtonian approximation.