Next-to-next-to-leading order post-Newtonian spin(1)-spin(2) Hamiltonian for self-gravitating binaries

TL;DR

This work derives the NNLO post-Newtonian spin(1)-spin(2) Hamiltonian for two self-gravitating spinning compact objects, a regime where spin-spin interactions can rival 4PN effects for fast rotators. The Hamiltonian, formulated within the canonical ADM framework, is explicitly given with complex spin-momentum couplings and is validated by enforcing the approximate global Poincaré algebra, uniquely fixing the center-of-mass vector via a high-dimensional ansatz. The calculation leverages advanced symbolic tools and shows cancellations of intermediate UV contributions, underscoring its consistency and usefulness for waveform modeling and EOB calibration in spinning binaries. While the 4PN point-mass Hamiltonian remains unknown, this NNLO spin-spin result informs late-inspiral dynamics and can improve analytic template banks for gravitational-wave observations of compact binaries.

Abstract

We present the next-to-next-to-leading order post-Newtonian (PN) spin(1)-spin(2) Hamiltonian for two self-gravitating spinning compact objects. If both objects are rapidly rotating, then the corresponding interaction is comparable in strength to a 4PN effect. The Hamiltonian is checked via the global Poincare algebra with the center-of-mass vector uniquely determined by an ansatz.