Higher-order-in-spin interaction Hamiltonians for binary black holes from source terms of Kerr geometry in approximate ADM coordinates

TL;DR

This work constructs an approximate Kerr geometry in ADM-compatible coordinates by transforming the Kerr metric from quasi-isotropic and harmonic forms into ADMTT coordinates, keeping terms up to $1/r^4$ and $a^2$. It then derives distributional source terms for the constraint equations and uses them to obtain new higher-order spin interaction Hamiltonians for binary black holes, valid to leading order in the momenta. The authors present explicit spin-spin and spin-orbit couplings, including nonlinear spin terms (e.g., $H_{S_1^2S_2^2}$, $H_{SO}^{(a)}$, $H_{SO}^{(b)}$), and demonstrate angular-momentum conservation for the total system. These results extend the ADM-based analytic framework for spinning binaries and provide refined ingredients for gravitational-wave modeling and comparison with numerical relativity.

Abstract

The Kerr metric outside the ergosphere is transformed into ADM coordinates up to the orders $1/r^4$ and $a^2$, respectively in radial coordinate $r$ and reduced angular momentum variable $a$, starting from the Kerr solution in quasi-isotropic as well as harmonic coordinates. The distributional source terms for the approximate solution are calculated. To leading order in linear momenta, higher-order-in-spin interaction Hamiltonians for black-hole binaries are derived.