Gravitational spin Hamiltonians from the S matrix

We utilize generalized unitarity and recursion relations combined with effective field theory techniques to compute spin-dependent interaction terms for an inspiralling binary system 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 (up to 2.5PN order) and the leading-order S-2 (2PN) Hamiltonian for a binary system with one of the massive objects having nonzero spin using the S-matrix elements of elementary particles. For the same system, we also obtain the S-3 (3.5PN) spin Hamiltonian for an arbitrary massive object, which was until now known only for a black hole. Furthermore, we derive the missing S-4 Hamiltonian at leading order (4PN), again for an arbitrary massive object and establish that the minimal coupling of an elementary particle to gravity automatically captures the physics of a spinning black hole. 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.