Post-Newtonian Approximation in Maxwell-Like Form

TL;DR

This paper summarizes and extends the Maxwell-like formalism of the first post-Newtonian approximation in general relativity, including gravitational momentum and stress, aiding physical intuition in numerical relativity.

Contribution

It consolidates Damour, Soffel, and Xu's Maxwell-like formalism and extends it to include gravitational momentum density, flux, and conservation laws.

Findings

Extended Maxwell-like formalism to include gravitational momentum and stress.

Facilitated physical intuition for numerical relativity simulations.

Provided a summarized and accessible version of DSX formalism.

Abstract

The equations of the linearized first post-Newtonian approximation to general relativity are often written in "gravitoelectromagnetic" Maxwell-like form, since that facilitates physical intuition. Damour, Soffel and Xu (DSX) (as a side issue in their complex but elegant papers on relativistic celestial mechanics) have expressed the first post-Newtonian approximation, including all nonlinearities, in Maxwell-like form. This paper summarizes that DSX Maxwell-like formalism (which is not easily extracted from their celestial mechanics papers), and then extends it to include the post-Newtonian (Landau-Lifshitz-based) gravitational momentum density, momentum flux (i.e. gravitational stress tensor) and law of momentum conservation in Maxwell-like form. The authors and their colleagues have found these Maxwell-like momentum tools useful for developing physical intuition into numerical-relativity simulations of compact binaries with spin.