Superradiance of a charged scalar field coupled to the Einstein-Maxwell equations

We consider the Einstein-Maxwell-Klein-Gordon equations for a spherically symmetric scalar field scattering off a Reissner-Nordström black hole in asymptotically flat spacetime. The equations are solved numerically using a hyperboloidal evolution scheme. For suitable frequencies of the initial data, superradiance is observed, leading to a substantial decrease of mass and charge of the black hole. We also derive a Bondi mass loss formula using the Kodama vector field and investigate the late-time decay of the scalar field.