Horizon Hair from Inversion Symmetry

Abstract: The extreme Reissner-Nordstr\"om solution has a discrete conformal isometry that maps the future event horizon to future null infinity and vice versa, the Couch-Torrence inversion isometry. We study the dynamics of a probe Maxwell field on the extreme Reissner-Nordstr\"om solution in light of this symmetry. We present a gauge fixing that is compatible with the inversion symmetry. The gauge fixing allows us to relate the gauge parameter at the future horizon to future null infinity, which further allows us to study global charges for large gauge symmetries in the exterior of the extreme Reissner-Nordstr\"om black hole. Along the way, we construct Newman-Penrose and Aretakis like conserved quantities along future null infinity and the future event horizon, respectively, and relate them via the Couch-Torrence inversion symmetry.