3rd post-Newtonian higher order Hamilton dynamics for two-body point-mass systems

TL;DR

This paper derives the third post-Newtonian order Hamiltonian for two-body point-mass systems, revealing potential limitations of the binary point-mass model at this order due to regularization ambiguities.

Contribution

It provides a higher order Hamiltonian formulation for two-body systems up to 3PN, highlighting issues with the binary point-mass approximation at this level.

Findings

Unique Hamiltonian terms obtained except for one coefficient.

Regularization procedures used to handle distributional derivatives.

Results suggest the binary point-mass model may be invalid at 3PN.

Abstract

The paper presents the conservative dynamics of two-body point-mass systems up to the third post-Newtonian order ($1/c^6$). The two-body dynamics is given in terms of a higher order ADM Hamilton function which results from a third post-Newtonian Routh functional for the total field-plus-matter system. The applied regularization procedures, together with making use of distributional differentiation of homogeneous functions, give unique results for the terms in the Hamilton function apart from the coefficient of the term $(\nu p_{i}{\pa_{i}})^2r^{-1}$. The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order.