Center-of-Mass Equations of Motion and Conserved Integrals of Compact Binary Systems at the Fourth Post-Newtonian Order

TL;DR

This work delivers a self-consistent 4PN center-of-mass formulation for non-spinning compact binaries, deriving CM equations of motion and a CM Lagrangian, along with conserved energy and angular momentum that include the non-local tail contributions. It provides explicit expressions for instantaneous CM dynamics, and a thorough treatment of gravitational-wave tails at 4PN, including their DC and AC components via Fourier analysis, as well as dissipative radiation-reaction terms up to 4PN. The circular-orbit reductions yield closed-form relations for the orbital frequency, energy, angular momentum, and periastron advance, incorporating tail effects and matching gravitational self-force results in the small mass-ratio limit. The results enable precise comparisons with numerical relativity and self-force calculations and improve gravitational-wave template accuracy by incorporating non-local 4PN tail dynamics into both conservative and dissipative sectors.

Abstract

The dynamics of compact binary systems at the fourth post-Newtonian (4PN) approximation of general relativity has been recently completed in a self-consistent way. In this paper, we compute the ten Poincaré constants of the motion and present the equations of motion in the frame of the center of mass (CM), together with the corresponding CM Lagrangian, conserved energy and conserved angular momentum. Next, we investigate the reduction of the CM dynamics to the case of quasi-circular orbits. The non local (in time) tail effect at the 4PN order is consistently included, as well as the relevant radiation-reaction dissipative contributions to the energy and angular momentum.