Spin-squared Hamiltonian of next-to-leading order gravitational interaction

TL;DR

The paper addresses the unresolved static part of the next-to-leading order (NLO) spin-squared Hamiltonian $H^{NLO}_{S1^2}$ for binary black holes within the ADM/ADMTT formalism. It employs a three-dimensional covariant ansatz for the source terms in the Hamilton constraint and fixes the coefficients by matching to the Kerr metric, yielding $c_1=-1/2$, $c_2=0$, $c_3=0$, and $c_4=0$, and thereby completing the NLO spin interactions up to quadratic order. The resulting Hamiltonian, expressed in ADMTT gauge, combines with previously known nonstatic pieces to give the full PN NLO spin dynamics up to $S^2$ for BBH, consistent with the test-mass limit. This advances gravitational-wave template accuracy and clarifies the relationship with prior results such as PR08, noting a spin-frame dependent discrepancy that can be resolved by a spin-variable redefinition.

Abstract

The static, i.e., linear momentum independent, part of the next-to-leading order (NLO) gravitational spin(1)-spin(1) interaction Hamiltonian within the post-Newtonian (PN) approximation is calculated from a 3-dim. covariant ansatz for the Hamilton constraint. All coefficients in this ansatz can be uniquely fixed for black holes. The resulting Hamiltonian fits into the canonical formalism of Arnowitt, Deser, and Misner (ADM) and is given in their transverse-traceless (ADMTT) gauge. This completes the recent result for the momentum dependent part of the NLO spin(1)-spin(1) ADM Hamiltonian for binary black holes (BBH). Thus, all PN NLO effects up to quadratic order in spin for BBH are now given in Hamiltonian form in the ADMTT gauge. The equations of motion resulting from this Hamiltonian are an important step toward more accurate calculations of templates for gravitational waves.