Complete Hamiltonian for spinning binary systems at first post-Minkowskian order

TL;DR

The authors derive an exact, closed-form 1PM Hamiltonian for spinning binary systems using on-shell amplitude techniques. A key advance is the precise Thomas–Wigner rotation factor, which, together with a dressed 1PM amplitude built from spin-generating functions $W(\tau)$, yields a master potential that factorizes spin and momentum at each spin order. The work reproduces known 1PN results via canonical transformations and matches the Kerr EOB mapping, validating the approach and exposing hidden simplifications such as the isotropic gauge and the absence of $S\cdot p$ terms. The results pave the way for higher-order extensions (2PM) and provide a robust bridge to PN and EOB formalisms, with clear implications for accurate modeling of spinning compact binaries in GR.

Abstract

Building upon recent progress in applying on-shell amplitude techniques to classical observables in general relativity, we propose a closed-form formula for the conservative Hamiltonian of a spinning binary system at the 1st post-Minkowskian (1PM) order. It is applicable for general spinning bodies with arbitrary spin multipole moments. The formula is linear in gravitational constant by definition, but exact to all orders in momentum and spin expansions. At each spin order, our formula implies that the spin-dependence and momentum dependence factorize almost completely. We expand our formula in momentum and compare the terms with 1PM parts of the post-Newtonian computations in the literature. Up to canonical transformations, our results agree perfectly with all previous ones. We also compare our formula for black hole to that derived from a spinning test-body near a Kerr black hole via the effective one-body mapping, and find perfect agreement.

Figures (1)

• Figure 1: The graviton exchange diagram between source $a$ and $b$ that yields the leading $1/q^2$ singularity, which is responsible for the classical potential.