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

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.

💡 Research Summary

The paper presents a systematic derivation of the next‑to‑leading order (NLO) conservative spin‑orbit and spin(a)‑spin(b) Hamiltonians for an arbitrary number n of compact, spinning bodies within the post‑Newtonian (PN) framework. Using the Arnowitt‑Deser‑Misner (ADM) canonical formalism in the transverse‑tracefree (TT) gauge, the authors decompose the spatial metric and its conjugate momentum into scalar (φ), TT‑tensor (h_{ij}^{TT}), and longitudinal (˜π_i) potentials. The Hamiltonian constraint and momentum constraint are expanded in powers of 1/c, and the matter source terms are expressed in terms of the canonical masses m_a, momenta P_a^i, and spin tensors S_a^{ij}.

The key technical steps include: (i) solving the constraint equations order‑by‑order to obtain φ and ˜π_i up to the required PN orders (φ up to O(c^{-8}), ˜π_i up to O(c^{-5})); (ii) expanding the matter Hamiltonian density H_matter and momentum density H_matter^i to include linear spin contributions up to O(c^{-8}) and O(c^{-5}) respectively; (iii) performing integration‑by‑parts to eliminate mixed terms such as ˜π_{ij}^{(3)}π_{ij}^{(5)TT} and to rewrite higher‑order scalar potentials in terms of lower‑order quantities; (iv) converting the resulting expression into a Routhian, i.e., a Hamiltonian for particle degrees of freedom plus a Lagrangian for the TT field, and finally eliminating the TT field by inserting its approximate solution obtained from the inverse Laplacian.

The resulting NLO spin‑orbit Hamiltonian completes the 2.5PN description of the general relativistic three‑body problem. Unlike the leading‑order (LO) spin‑orbit term, which only contains pairwise (two‑body) interactions, the NLO term includes genuine three‑body contributions that couple three distinct bodies a, b, and c. Similarly, the NLO spin(a)‑spin(b) Hamiltonian, which is of 3PN order, also features three‑body interaction pieces. Both Hamiltonians are linear in each individual spin, consistent with the linear‑spin approximation used throughout.

Cross‑checks are performed in two independent ways: (1) by reproducing known two‑body results from earlier literature (e.g., Refs.