The Newtonian Limit on Asymptotically Null Foliations

By suitably re-scaling the conformal Einstein's equations we are able to apply recent results in the theory of PDE, and prove that they possess slow solutions in a future neighborhood of an initial surface reaching ${\cal I}^+$. The structure of the equations obtained allows to split (up to any given order) the initial data into those generating slow solutions, i.e., those driven by the sources, and those generating fast solutions, i.e., those which represent gravitational radiation with no relation to the sources. Thus effectively resulting in a proposal to prescribe initial data for solutions with no extra radiation up to the order needed for each given application.