Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling

A Hamiltonian formulation is given for the gravitational dynamics of two spinning compact bodies to next-to-leading order ($G\mathrm{/}{c}^4$ and $G^2\mathrm{/}{c}^4$) in the spin-orbit interaction. We use a novel approach (valid to linear order in the spins), which starts from the second-post-Newtonian metric (in ADM coordinates) generated by two spinless bodies, and computes the next-to-leading order precession, in this metric, of suitably redefined ``constant-magnitude’’ 3-dimensional spin vectors ${\mathbf{S}}_1$, ${\mathbf{S}}_2$. We prove the Poincar'e invariance of our Hamiltonian by explicitly constructing ten phase-space generators realizing the Poincar'e algebra. A remarkable feature of our approach is that it allows one to derive the {\it orbital} equations of motion of spinning binaries to next-to-leading order in spin-orbit coupling without having to solve Einstein’s field equations with a spin-dependent stress tensor. We show that our Hamiltonian (orbital and spin) dynamics is equivalent to the dynamics recently obtained by Faye, Blanchet, and Buonanno, by solving Einstein’s equations in harmonic coordinates.