Covariant formulation of the post-1-Newtonian approximation to General Relativity

TL;DR

This paper develops a coordinate-independent, geometric formulation of the post-1-Newtonian approximation to general relativity, generalizing Newton-Cartan theory without relying on a fundamental spacetime metric.

Contribution

It introduces a novel, coordinate-free geometric framework for post-1-Newtonian gravity, extending Newton-Cartan theory to include relativistic corrections.

Findings

Recover standard post-Newtonian equations in asymptotically flat spacetimes

Provides a geometric, metric-free formulation of post-Newtonian gravity

Facilitates coordinate-independent analysis of relativistic gravitational effects

Abstract

We derive a coordinate-independent formulation of the post-1-Newtonian approximation to general relativity. This formulation is a generalization of the Newton-Cartan geometric formulation of Newtonian gravity. It involves several fields and a connection, but no spacetime metric at the fundamental level. We show that the usual coordinate-dependent equations of post-Newtonian gravity are recovered when one specializes to asymptotically flat spacetimes and to appropriate classes of coordinates.