On late time tails in an extreme Reissner–Nordström black hole: frequency domain analysis

Abstract

In this paper, we revisit the study of the leading order late time decay tails of massless scalar perturbations outside an extreme Reissner–Nordström black hole. Previous authors have analysed this problem in the time domain; we analyse the problem in the frequency domain. We first consider initial perturbations with generic regular behaviour across the horizon on characteristic surfaces. For this set-up, we reproduce some of the previous results of Sela (2016 Phys. Rev. D 93 024054) using Fourier methods. Next we consider related initial data on hypersurfaces, and present decay results at timelike infinity, near future null infinity, and near the future horizon. Along the way, using the inversion symmetry of the extreme Reissner–Nordström spacetime, we relate the higher multipole Aretakis and Newman–Penrose constants for a massless scalar in this background.