Spin-multipole effects in binary black holes and the test-body limit

TL;DR

This paper explores how the spin-multipole structure influences binary black hole dynamics within general relativity, deriving a comprehensive Hamiltonian that captures all orders in spin and reveals a deep connection to Kerr spacetime geodesics.

Contribution

It provides a complete derivation of a binary black hole Hamiltonian valid for all spin orders, extending previous results and establishing a link to test-body limits and Kerr geodesics.

Findings

Confirmed patterns in spin effects up to fourth order

Derived a Hamiltonian valid for all spin orders

Established equivalence with test black hole in Kerr spacetime

Abstract

We discuss the effects of the black holes' spin-multipole structure in the orbital dynamics of binary black holes according to general relativity, focusing on the leading-post-Newtonian-order couplings at each order in an expansion in the black holes' spins. We first review previous widely confirmed results up through fourth order in spin, observe suggestive patterns therein, and discuss how the results can be extrapolated to all orders in spin with minimal information from the test-body limit. We then justify this extrapolation by providing a complete derivation within the post-Newtonian framework of a canonical Hamiltonian for a binary black hole, for generic orbits and spin orientations, which encompasses the leading post-Newtonian orders at all orders in spin. At the considered orders, the results reveal a precise equivalence between arbitrary-mass-ratio two-spinning-black-hole dynamics and the motion of a test black hole in a Kerr spacetime, as well as an intriguing relationship to geodesic motion in a Kerr spacetime.