Next-to-next-to-leading order spin-orbit effects in the equations of motion of compact binary systems

TL;DR

This work delivers the next-to-next-to-leading order spin-orbit corrections at $3.5$PN for the equations of motion of compact binaries, focusing on maximally spinning objects. Using a pole-dipole stress-energy description and a PN expansion in harmonic coordinates, the authors derive the spin evolution and the conservative spin-orbit acceleration, employing Hadamard regularization and selectively dimensional regularization. Key validations include a conserved energy up to $3.5$PN, exact Lorentz invariance in the harmonic framework, and correct test-mass limits that reproduce Kerr geodesics and Papapetrou motion, plus an explicit equivalence with ADM results via a contact transformation. The results strengthen the theoretical underpinning of gravitational-wave templates and pave the way for 3.5PN spin-orbit contributions to the energy flux and waveform phase, with implications for high-precision GW data analysis and parameter estimation.

Abstract

We compute next-to-next-to-leading order spin contributions to the post-Newtonian equations of motion for binaries of compact objects, such as black holes or neutron stars. For maximally spinning black holes, those contributions are of third-and-a-half post-Newtonian (3.5PN) order, improving our knowledge of the equations of motion, already known for non-spinning objects up to this order. Building on previous work, we represent the rotation of the two bodies using a pole-dipole matter stress-energy tensor, and iterate Einstein's field equations for a set of potentials parametrizing the metric in harmonic coordinates. Checks of the result include the existence of a conserved energy, the approximate global Lorentz invariance of the equations of motion in harmonic coordinates, and the recovery of the motion of a spinning object on a Kerr background in the test-mass limit. We verified the existence of a contact transformation, together with a redefinition of the spin variables that makes our result equivalent to a previously published reduced Hamiltonian, obtained from the Arnowitt-Deser-Misner (ADM) formalism.