Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity

TL;DR

The paper addresses the conservative dynamics of a two-body system at the $4\mathrm{PN}$ level in general relativity, reviewing and cross-validating approaches such as ADM Hamiltonian, Delaunay averaging, EOB, and SF theories. It demonstrates that the complete $4\mathrm{PN}$ dynamics obtained by Damour, Jaranowski, and Schäfer, including a nonlocal tail contribution, is confirmed piecewise by independent methods, and that SF data strongly support the ADM-Delaunay results, especially for the linear-in-$\nu$ (1SF) sector. It then scrutinizes Bernard et al.’s harmonic-Fokker action results, revealing significant inconsistencies with SF benchmarks and showing that a correct treatment of nonlocal tail effects requires careful energy definitions and additional infrared ambiguities beyond the single parameter proposed. The authors propose minimal corrections and argue for the necessity of extra IR ambiguity parameters to reconcile harmonic-coordinate results with SF data, highlighting the importance of a consistent nonlocal-to-local reduction. Overall, the work reinforces the reliability of the ADM-Delaunay-EOB framework for $4\mathrm{PN}$ conservative dynamics and clarifies subtleties in nonlocal tail treatments, with direct implications for high-precision gravitational-wave modeling.

Abstract

The fourth post-Newtonian (4PN) two-body dynamics has been recently tackled by several different approaches: effective field theory, Arnowitt-Deser-Misner Hamiltonian, action-angle-Delaunay averaging, effective-one-body, gravitational self-force, first law of dynamics, and Fokker action. We review the achievements of these approaches and discuss the complementarity of their results. Our main conclusions are: (i) the results of the first complete derivation of the 4PN dynamics [T.Damour, P.Jaranowski, and G.Schäfer, Phys. Rev. D 89, 064058 (2014)] have been, piecewise, fully confirmed by several subsequent works; (ii) the results of the Delaunay-averaging technique [T.Damour, P.Jaranowski, and G.Schäfer, Phys. Rev. D 91, 084024 (2015)] have been confirmed by several independent works; and (iii) several claims in a recent Fokker-action computation [L.Bernard et al., arXiv:1512.02876v2 [gr-qc]] are incorrect, but can be corrected by the addition of a couple of ambiguity parameters linked to subtleties in the regularization of infrared and ultraviolet divergences.