Next-to-leading order spin-orbit and spin(a)-spin(b) Hamiltonians for n gravitating spinning compact objects

TL;DR

This work provides a complete derivation of the conservative next-to-leading order spin-orbit and spin(a)-spin(b) Hamiltonians for an arbitrary number $n$ of gravitating spinning compact objects using the ADM formalism in the ADMTT gauge. It reveals genuine three-body spin interactions at this order and furnishes explicit two- and three-body contributions, checked against the Poincaré algebra and via an independent spin-precession-based derivation. The results extend known binary spin Hamiltonians to the $n$-body case and enable accurate investigations of hierarchical triples, Kozai resonances, and potential chaotic dynamics in relativistic three-body systems. Mathematica source files for the three-body case accompany the publication to facilitate practical applications and cross-checks.

Abstract

We derive the post-Newtonian next-to-leading order conservative spin-orbit and spin(a)-spin(b) gravitational interaction Hamiltonians for arbitrary many compact objects. The spin-orbit Hamiltonian completes the knowledge of Hamiltonians up to and including 2.5PN for the general relativistic three-body problem. The new Hamiltonians include highly nontrivial three-body interactions, in contrast to the leading order consisting of two-body interactions only. This may be important for the study of effects like Kozai resonances in mergers of black holes with binary black holes.