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

TL;DR

The paper advances the conservative two-body problem at 3PN order using the ADM canonical formalism and a Routh reduction to derive an autonomous Hamiltonian for point masses. It employs Hadamard regularization and Riesz-kernel techniques to handle singular field sources and TT-decompositions, yielding explicit 3PN contributions up to $1/r^4$ with a single unresolved ambiguity coefficient $\omega$ multiplying a specific 3PN term. The analysis reveals a finite but non-unique 3PN term $-(\nu p_i \partial_i)^2 r^{-1}$, interpreted as a quadrupole-tidal interaction scaled by Schwarzschild radii, which suggests the binary point-mass model may be ill-defined at 3PN and points toward the necessity of extended-body sources. The work includes detailed regularization procedures, checks against the test-particle and static limits, and a discussion of the physical implications of the ambiguity, highlighting the limits of the point-mass idealization at this perturbative order. Overall, the paper provides a thorough 3PN framework with rigorously implemented regularization, while identifying a fundamental unresolved issue in the binary point-mass description that motivates further research with extended bodies.

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 $(νp_{i}{\pa_{i}})^2r^{-1}$. The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order.