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

TL;DR

This paper introduces a novel Hamiltonian framework for two spinning compact bodies at next-to-leading order spin-orbit coupling, derived from the 2PN metric of spinless bodies in ADM coordinates and expressed through a conserved constant-magnitude 3D spin vector. The spin-orbit Hamiltonian is written as $H_{so} = \sum_a \boldsymbol{\Omega}_a \cdot \mathbf{S}_a$, with $\boldsymbol{\Omega}_a$ obtained from the angular velocity of spin precession in the 2PN metric, enabling orbital equations of motion without solving full spin-dependent field equations. The authors prove Poincaré invariance by constructing ten phase-space generators that realize the Poincaré algebra and determine the NLO spin-orbit boost terms via undetermined coefficients. They demonstrate equivalence with harmonic-coordinate results (FBB06/BBF06) by providing explicit transformations between ADM and harmonic variables for spins and orbital coordinates, validating the approach. The work lays a rigorous, invariant Hamiltonian basis for modeling spin effects in gravitational-wave sources and offers a pathway to integrate these results into effective-one-body formalisms.

Abstract

A Hamiltonian formulation is given for the gravitational dynamics of two spinning compact bodies to next-to-leading order ($G/c^4$ and $G^2/c^4$) 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 ${\bf S}_1$, ${\bf S}_2$. We prove the Poincaré invariance of our Hamiltonian by explicitly constructing ten phase-space generators realizing the Poincaré 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.