Fokker action of non-spinning compact binaries at the fourth post-Newtonian approximation

TL;DR

This work derives the Fokker action for non-spinning compact binaries at 4PN in harmonic coordinates, combining dimensional regularization for UV divergences with a finite-part scheme for IR tails. It provides a comprehensive PN construction, including the non-local tail contribution, and performs a detailed comparison with ADM-based Hamiltonians, finding agreement with self-force constraints once an ambiguity parameter is fixed, but noting unresolved discrepancies with prior ADM/Damour–Jaranowski–Schaefer results. The authors demonstrate Lorentz invariance, remove accelerations to obtain an ADM-like Lagrangian, and compute the circular-orbit energy from both Lagrangian and Hamiltonian perspectives, highlighting the importance of properly treating non-local tail effects. The work clarifies regularization and matching procedures, paving the way for precise 4PN templates while signaling that further work is needed to fully reconcile different formalisms for the complete 4PN dynamics.

Abstract

The Fokker action governing the motion of compact binary systems without spins is derived in harmonic coordinates at the fourth post-Newtonian approximation (4PN) of general relativity. Dimensional regularization is used for treating the local ultraviolet (UV) divergences associated with point particles, followed by a renormalization of the poles into a redefinition of the trajectories of the point masses. Effects at the 4PN order associated with wave tails propagating at infinity are included consistently at the level of the action. A finite part procedure based on analytic continuation deals with the infrared (IR) divergencies at spatial infinity, which are shown to be fully consistent with the presence of near-zone tails. Our end result at 4PN order is Lorentz invariant and has the correct self-force limit for the energy of circular orbits. However, we find that it differs from the recently published result derived within the ADM Hamiltonian formulation of general relativity [T. Damour, P. Jaranowski, and G. Schäfer, Phys. Rev. D 89, 064058 (2014)]. More work is needed to understand this discrepancy.