A note on instabilities of extremal black holes under scalar perturbations from afar

TL;DR

This paper demonstrates that extremal black holes exhibit instabilities under scalar perturbations even when initial data is supported far from the horizon, extending previous results to more general initial conditions.

Contribution

It removes the restriction on initial data support, showing instabilities develop from afar, and applies to both extremal Kerr and Reissner-Nordstrom black holes.

Findings

Instabilities occur even with initial data supported far from the horizon.

Higher order derivatives become unstable compared to previous cases.

Results align with numerical analyses by Lucietti et al.

Abstract

In previous work of the author it was shown that instabilities of solutions to the wave equation develop asymptotically along the event horizon of extremal Kerr provided a certain expression H of the initial data is non-trivial on the horizon. In this note we remove this restriction by showing that instabilities develop even from initial data supported arbitrarily far away from the horizon (for which, in particular, H=0). The latter instabilities concern one order higher derivatives compared to the case where H is non-zero. The result also applies to extremal Reissner-Nordstrom. This note was motivated by numerical analysis of Lucietti, Murata, Reall and Tanahashi.