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We compute next-to-next-to-leading order spin contributions to thepost-Newtonian equations of motion for binaries of compact objects, such asblack holes or neutron stars. For maximally spinning black holes, thosecontributions are of third-and-a-half post-Newtonian (3.5PN) order, improvingour knowledge of the equations of motion, already known for non-spinningobjects up to this order. Building on previous work, we represent the rotationof the two bodies using a pole-dipole matter stress-energy tensor, and iterateEinstein's field equations for a set of potentials parametrizing the metric inharmonic coordinates. Checks of the result include the existence of a conservedenergy, the approximate global Lorentz invariance of the equations of motion inharmonic coordinates, and the recovery of the motion of a spinning object on aKerr background in the test-mass limit. We verified the existence of a contacttransformation, together with a redefinition of the spin variables that makesour result equivalent to a previously published reduced Hamiltonian, obtainedfrom the Arnowitt-Deser-Misner (ADM) formalism.

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