The Post Minkowskii Expansion of General Relativity

TL;DR

This paper introduces a constructive post-Minkowskii approximation method for general relativity, using an iterative process in a non-harmonic gauge that improves solutions by reducing errors with each step, suitable for computational implementation.

Contribution

It presents a novel iterative procedure for approximating solutions to Einstein's equations in vacuum, emphasizing a non-harmonic gauge and computational feasibility.

Findings

Error decreases by a factor of G with each iteration

Method applicable to computer implementation

Uses a non-harmonic gauge for flexibility

Abstract

We describe a post-Minkowskii approximation of general relativity 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 described 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's function for Minkowskii 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 (non-harmonic) gauge and formulate the procedure in a constructive manner which emphasizes its suitability for implementation on a computer.