Canonical formulation of gravitating spinning objects at 3.5 post-Newtonian order

TL;DR

Extends the ADM canonical formalism to self-gravitating spinning bodies up to 3.5PN order, keeping linear-in-spin terms and enforcing global Poincaré invariance as the consistency condition. It derives a general interaction Hamiltonian between matter and the transverse-traceless metric $h^{\text{TT}}_{ij}$ and shows the resulting wave equation agrees with the Einstein equations, providing a robust cross-check. The authors compute the 1PN energy flux at the spin-orbit level and verify agreement with established results (e.g., Kidder), establishing confidence in the formalism and its potential for NNLO spin dynamics. Together, these results furnish a solid foundation for accurate gravitational-wave modeling of spinning binaries and facilitate comparisons between ADM, action-based, and GR formulations for higher-order spin effects.

Abstract

The 3.5 post-Newtonian (PN) order is tackled by extending the canonical formalism of Arnowitt, Deser, and Misner to spinning objects. This extension is constructed order by order in the PN setting by utilizing the global Poincare invariance as the important consistency condition. The formalism is valid to linear order in the single spin variables. Agreement with a recent action approach is found. A general formula for the interaction Hamiltonian between matter and transverse-traceless part of the metric at 3.5PN is derived. The wave equation resulting from this Hamiltonian is considered in the case of the constructed formalism for spinning objects. Agreement with the Einstein equations is found in this case. The energy flux at the spin-orbit level is computed.