Third post-Newtonian constrained canonical dynamics for binary point masses in harmonic coordinates

TL;DR

This paper develops a constrained canonical framework for the third post-Newtonian dynamics of binary point masses in harmonic coordinates, ensuring Poincaré invariance and extending previous approximations.

Contribution

It introduces a canonical formulation of 3pN binary dynamics using Dirac brackets and constructs an approximate Poincaré algebra within this framework.

Findings

Constructed a representation of the Poincaré algebra with Dirac brackets.

Ensured the generators act as standard Poincaré transformations on coordinates.

Extended the approach to second post-Coulombian approximation in electrodynamics.

Abstract

The conservative dynamics of two point masses given in harmonic coordinates up to the third post-Newtonian (3pN) order is treated within the framework of constrained canonical dynamics. A representation of the approximate Poincar\'e algebra is constructed with the aid of Dirac brackets. Uniqueness of the generators of the Poincar\'e group resp. the integrals of motion is achieved by imposing their action on the point mass coordinates to be identical with that of the usual infinitesimal Poincar\'e transformations. The second post-Coulombian approximation to the dynamics of two point charges as predicted by Feynman-Wheeler electrodynamics in Lorentz gauge is treated similarly.