A remark about wave equations on the extreme Reissner-Nordstrom black hole exterior

We consider a massless scalar field propagating on the exterior of the extreme Reissner-Nordstrom black hole. Using a discrete conformal symmetry of this spacetime, we draw a one-to-one relationship between the behavior of the field near the future horizon and near future null infinity. In particular, we show that the polynomial growth of the second and higher transversal derivatives along the horizon, recently found by Aretakis, reflects well-known facts about the retarded time asymptotics at null infinity. We also observe that the analogous relationship holds true for an axially symmetric massless scalar field propagating on the extreme Kerr-Newman background.