General relativistic dynamics of compact binaries at the third post-Newtonian order

TL;DR

This paper derives the third post-Newtonian order relativistic equations of motion for binary systems of point masses, incorporating advanced regularization techniques, and discusses their invariance, energy conservation, and implications for gravitational wave observations.

Contribution

It provides the first detailed derivation of 3PN equations of motion using a Lorentzian Hadamard regularization, addressing non-linear potentials and regularization ambiguities.

Findings

Equations are invariant under Lorentz transformations.

A conserved energy expression is identified at 3PN order.

Results are relevant for gravitational wave data analysis.

Abstract

The general relativistic corrections in the equations of motion and associated energy of a binary system of point-like masses are derived at the third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian expansion of the metric in harmonic coordinates at the 3PN approximation. The metric is parametrized by appropriate non-linear potentials, which are evaluated in the case of two point-particles using a Lorentzian version of an Hadamard regularization which has been defined in previous works. Distributional forms and distributional derivatives constructed from this regularization are employed systematically. The equations of motion of the particles are geodesic-like with respect to the regularized metric. Crucial contributions to the acceleration are associated with the non-distributivity of the Hadamard regularization and the violation of the Leibniz rule by the distributional derivative. The final equations of motion at the 3PN order are invariant under global Lorentz transformations, and admit a conserved energy (neglecting the radiation reaction force at the 2.5PN order). However, they are not fully determined, as they depend on one arbitrary constant, which reflects probably a physical incompleteness of the point-mass regularization. The results of this paper should be useful when comparing theory to the observations of gravitational waves from binary systems in future detectors VIRGO and LIGO.