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Abstract

In this brief note, we revisit the study of the leading order late time decay
tails of massless scalar perturbations outside an extreme Reissner-Nordstr\"om
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 [arXiv:1510.06169] using Fourier methods. Next, we consider
related initial data on $t=\mbox{const}$ hypersurfaces, and present decay
results at timelike infinity, near future null infinity, and near the future
horizon. Along the way, using the $r_* \to -r_*$ inversion symmetry of the
extreme Reissner-Nordstr\"om spacetime, we relate the higher multipole Aretakis
and Newman-Penrose constants for a massless scalar in this background.