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

TL;DR

This paper calculates advanced spin-orbit effects at 3.5PN order for compact binary systems, enhancing the precision of gravitational wave models involving spinning black holes and neutron stars.

Contribution

It introduces a novel computation of next-to-next-to-leading order spin contributions using a pole-dipole stress-energy tensor and confirms consistency with previous Hamiltonian results.

Findings

Derived 3.5PN order spin-orbit equations of motion

Validated conservation of energy and Lorentz invariance

Established equivalence with ADM Hamiltonian formalism

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.