On Late Time Tails in an Extreme Reissner-Nordstr\"om Black Hole: Frequency Domain Analysis

TL;DR

This paper analyzes late time decay tails of scalar perturbations around an extreme Reissner-Nordström black hole using frequency domain methods, reproducing and extending previous time domain results.

Contribution

It introduces frequency domain analysis for late time tails in extreme Reissner-Nordström black holes, relating constants via spacetime symmetries and extending decay results to various regions.

Findings

Reproduced previous decay tail results using Fourier methods.

Derived decay behaviors at different spacetime regions.

Linked Aretakis and Newman-Penrose constants through spacetime symmetry.

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.