Gravitational spin Hamiltonians from the S matrix
Varun Vaidya
TL;DR
The paper develops a framework that uses generalized unitarity and BCFW recursion within an effective field theory to compute spin-dependent gravitational interactions in inspiraling binaries at successive post-Newtonian orders. By treating gravity as exchange of a massless spin-2 field on flat space and matching amplitudes to a worldline EFT, it derives spin-orbit terms to 2.5PN and higher-spin contributions (S^2 at 3PN, S^3 at 3.5PN, S^4 at 4PN) for arbitrary objects, with BHs corresponding to minimal coupling. It further connects these PN results to the Kerr metric by a PN expansion and demonstrates the universal structure of spin interactions across spins through a small set of Wilson coefficients, including guidance on stellar multipoles. The approach offers substantial computational advantages over Feynman-diagram methods and provides a coherent bridge between scattering amplitudes and classical gravity relevant for gravitational-wave template modeling.
Abstract
We utilize generalized unitarity and recursion relations combined with effective field theory(EFT) techniques to compute spin dependent interaction terms for inspiralling binary systems in the post newtonian(PN) approximation. Using these methods offers great computational advantage over traditional techniques involving feynman diagrams, especially at higher orders in the PN expansion. As a specific example, we reproduce the spin-orbit interaction up to 2.5 PN order as also the leading order $S^2$(3PN) hamiltonian for an arbitrary massive object. We also obtain the unknown $S^3$(3.5PN) spin hamiltonian for an arbitrary massive object in terms of its low frequency linear response to gravitational perturbations, which was till now known only for a black hole. Furthermore, we derive the missing $S^4$ Hamiltonian at leading order(4PN) for an arbitrary massive object and establish that a minimal coupling of a massive elementary particle to gravity leads to a black hole structure. Finally, the Kerr metric is obtained as a series in $G_N$ by comparing the action of a test particle in the vicinity of a spinning black hole to the derived potential.