Higher-order-in-spin interaction Hamiltonians for binary black holes from Poincaré invariance

TL;DR

This work uses space-time Poincaré invariance within the ADM formalism in generalized isotropic coordinates to derive higher-order spin Hamiltonians for binary black holes. It provides explicit expressions for the S^3p and S^2p^2 interactions at linear order in G and demonstrates the absence of quartic S^4 terms at 2PN by Kerr-expansion analysis, with multiple coefficient constraints arising from the Poincaré algebra and cross-checks via the test-particle limit. A static-source calculation fixes remaining canonical ambiguities, yielding consistent results with prior approaches and solidifying the high-spin dynamics essential for accurate gravitational-wave templates. The paper also outlines the remaining G^2 spin terms to be addressed in future work, completing the 2PN spin-interaction landscape for binary black holes.

Abstract

The fulfillment of the space-asymptotic Poincaré algebra is used to derive new higher-order-in-spin interaction Hamiltonians for binary black holes in the Arnowitt-Deser-Misner canonical formalism almost completing the set of the formally $1/c^4$ spin-interaction Hamiltonians involving nonlinear spin terms. To linear order in $G$, the expressions for the $S^3p$- and the $S^2p^2$-Hamiltonians are completed. It is also shown that there are no quartic nonlinear $S^4$-Hamiltonians to linear order in $G$.