Late-time asymptotics for the wave equation on extremal Reissner-Nordstr\"om backgrounds

TL;DR

This paper establishes precise late-time decay rates for wave solutions on extremal Reissner-Nordström black holes, revealing how initial data influence asymptotic behavior through novel physical space techniques.

Contribution

It introduces a new method using physical space techniques, including weighted energy hierarchies and singular time inversion, to derive detailed asymptotics and horizon charges.

Findings

Derived explicit late-time asymptotics for wave solutions.

Introduced a new horizon charge relevant to extremal black holes.

Extended previous numerical and heuristic analyses.

Abstract

We derive the precise late-time asymptotics for solutions to the wave equation on extremal Reissner-Nordstr\"om black holes and explicitly express the leading-order coefficients in terms of the initial data. Our method is based on purely physical space techniques. We derive novel weighted energy hierarchies and develop a singular time inversion theory, which allow us to uncover the subtle contribution of both the near-horizon and near-infinity regions to the precise asymptotics. We introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes. Our work confirms, and in some cases extends, the numerical and heuristic analysis of Lucietti-Murata-Reall-Tanahashi, Ori-Sela and Blaksley-Burko.