Newtonian and post-Newtonian approximations are asymptotic to general relativity

A precise definition of the Newtonian and post-Newtonian hierarchy of approximations to general relativity is given by studying a $C^{\infty }$ sequence of solutions to Einstein's equations that is defined by initial data having the Newtonian scaling property: $v^i\sim \epsilon$, $\rho \sim {\epsilon }^2$, $p\sim {\epsilon }^4$, where $\epsilon$ is the parameter along the sequence. We map one solution in the sequence to another by identifying them at constant spatial position $x^i$ and Newtonian dynamical time $\tau =\epsilon t$. This mapping defines a congruence parametrized by $\epsilon$, and the various post-Newtonian approximations emerge as derivatives of the relativistic solutions along this congruence. We thereby show for the first time that the approximations are genuine asymptotic approximations to general relativity. The proof is given in detail up to first post-Newtonian order, but is easily extended. The results will be applied in the following paper to radiation reaction in binary star systems, to give a proof of the validity of the "quadrupole formula" free from any divergences.