Late-time decay of perturbations outside extremal charged black hole

We analyze the late-time decay of scalar perturbations in extremal Reissner-Nordstrom spacetime. We consider individual spherical-harmonic modes l of a test massless scalar field, restricting our attention to the initial data of compact support, with generic regular behavior across the horizon. We obtain a decay rate proportional to t(-(2l+3)) (just like in Schwarzschild) for incident waves scattered by the black hole. However, for waves originating at the horizon's neighborhood, we obtain a slightly slower decay, proportional to t(-(2l+2)). We discuss relations to previous works.