On the equations of motion of point-particle binaries at the third post-Newtonian order

TL;DR

This paper derives the third post-Newtonian equations of motion for point-particle binaries in general relativity, addressing regularization issues and identifying an undetermined coefficient indicating incompleteness at this order.

Contribution

It provides the 3PN equations of motion using Hadamard regularization and discusses the implications of an undetermined coefficient on the formalism's completeness.

Findings

Derived 3PN equations of motion for point particles

Identified an undetermined coefficient affecting the equations

Presented equations in the center-of-mass frame for circular orbits

Abstract

We investigate the dynamics of two point-like particles through the third post-Newtonian (3PN) approximation of general relativity. The infinite self-field of each point-mass is regularized by means of Hadamard's concept of ``partie finie''. Distributional forms associated with the regularization are used systematically in the computation. We determine the stress-energy tensor of point-like particles compatible with the previous regularization. The Einstein field equations in harmonic coordinates are iterated to the 3PN order. The 3PN equations of motion are Lorentz-invariant and admit a conserved energy (neglecting the 2.5PN radiation reaction). They depend on an undetermined coefficient, in agreement with an earlier result of Jaranowski and Schaefer. This suggests an incompleteness of the formalism (in this stage of development) at the 3PN order. In this paper we present the equations of motion in the center-of-mass frame and in the case of circular orbits.