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

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 (3.5 PN) 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 formalism.