Third post-Newtonian dynamics of compact binaries: Equations of motion in the center-of-mass frame

TL;DR

This work completes the conservative $3$PN dynamics of compact binaries in the center-of-mass frame by deriving explicit center-of-mass expressions in harmonic coordinates and connecting them to the ADM Hamiltonian via a unique contact transformation. It provides a $3$PN center-of-mass Lagrangian, the associated Noetherian conserved energy and angular momentum, and confirms their equivalence with the ADM formulation while clarifying gauge and regularization ambiguities. The paper also derives a gauge-invariant stability criterion for circular orbits, linking the COM and ADM approaches and enabling precise predictions for the innermost stable circular orbit across mass ratios. Together with the treatment of radiation-reaction corrections, these results enhance the accuracy of compact-binary waveform modeling at high post-Newtonian order.

Abstract

The equations of motion of compact binary systems and their associated Lagrangian formulation have been derived in previous works at the third post-Newtonian (3PN) approximation of general relativity in harmonic coordinates. In the present work we investigate the binary's relative dynamics in the center-of-mass frame (center of mass located at the origin of the coordinates). We obtain the 3PN-accurate expressions of the center-of-mass positions and equations of the relative binary motion. We show that the equations derive from a Lagrangian (neglecting the radiation reaction), from which we deduce the conserved center-of-mass energy and angular momentum at the 3PN order. The harmonic-coordinates center-of-mass Lagrangian is equivalent, {\it via} a contact transformation of the particles' variables, to the center-of-mass Hamiltonian in ADM coordinates that is known from the post-Newtonian ADM-Hamiltonian formalism. As an application we investigate the dynamical stability of circular binary orbits at the 3PN order.