Post-Minkowski expansion of general relativity

A post-Minkowski approximation of general relativity is described as a power series expansion in $G$, Newton’s gravitational constant. Material sources are hidden behind boundaries, and only the vacuum Einstein equations are considered. An iterative procedure is outlined which, in one complete step, takes any approximate solution of the Einstein equations and produces a new approximation which has the error decreased by a factor of $G$. Each step in the procedure consists of three parts: first the equations of motion are used to update the trajectories of the boundaries; then the field equations are solved using a retarded Green function for Minkowski space; finally, a gauge transformation is performed which makes the geometry well behaved at future null infinity. Differences between this approach to the Einstein equations and similar ones are that we use a general (nonharmonic) gauge and formulate the procedure in a constructive manner which emphasizes its suitability for implementation on a computer.