On Late Time Tails in an Extreme Reissner-Nordström Black Hole: Frequency Domain Analysis

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-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 [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 $r_{\ast }\to -r_{\ast }$ 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.

22 pages; 2 figures