On-shell electric-magnetic duality and the dual graviton

Using on-shell amplitude methods, we explore four-dimensional electric-magnetic duality and its double copy. We show explicitly that the on-shell scattering amplitudes know about "dual" photons (and dual gravitons), that the off-shell photon propagator double-copies to the graviton propagator, and that the magnetic part of the propagator is essential for the double copy to hold. We also show that there is an equivalent gravitomagnetic part of the graviton propagator, which is essential in giving rise to solutions with either angular momentum or NUT charge. Furthermore, we comment on the so-called Weinberg paradox, which states that scattering amplitudes involving the mixing of electric and magnetic monopoles cannot be Lorentz invariant, and would seem to preclude the existence of the 't Hooft-Polyakov (topological) monopole. We trace this paradox to the magnetic part of the propagator, showing that it can be eliminated if one restricts the scenario to proper orthochronous Lorentz transformations. Finally, we compute the fully relativistic cross section for arbitrary spin dyons using the recently formulated on-shell duality transformation, and we show that this is always fully Lorentz invariant.