Reissner-Nordstr\"om Black Holes at second post-Minkowskian order from Scattering Amplitudes

Abstract

We employ one-loop scattering amplitudes in Einstein-Maxwell theory to compute the classical Hamiltonian of a binary system of two charged, non-spinning compact objects. The Hamiltonian is valid to all orders in velocity and up to second post-Minkowskian order (2PM), i.e. $\mathcal{O}(G^2)$. The classical interaction potential is extracted via matching to a non-relativistic classical effective field theory. We also provide the scattering angle at 2PM order. We perform several cross checks on our results and find full agreement with existing results in the literature. Finally, we also briefly discuss a comparison for the scattering angle, the binding energy and the periastron shift of a bound system up to the second post-Newtonian order.