Instability of gravitational and electromagnetic perturbations of extremal Reissner-Nordstr\"om spacetime

TL;DR

This paper investigates the linear stability and instability of gravitational and electromagnetic perturbations in extremal Reissner-Nordström spacetime, revealing significant instability phenomena along the event horizon, especially for negative spin solutions.

Contribution

It extends the stability framework to extremal cases, proving decay and blow-up estimates for gauge-invariant quantities and uncovering new instability results for negative spin perturbations.

Findings

Negative spin solutions exhibit non-decay and blow-up along the horizon.

Instability results hold for a set of gauge-invariant quantities.

Extreme curvature component lphaar does not decay asymptotically.

Abstract

We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, $ |\mathcal{Q}|=M, $ Reissner-Nordstr\"om spacetime, as a solution to the Einstein-Maxwell equations. Our work uses and extends the framework \cite{giorgi2019boundedness,giorgie2020boundedness} of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon $ \mathcal{H}^+ $. In particular, we prove decay, non-decay, and polynomial blow-up estimates asymptotically along $ \mathcal{H}^+ $, the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component $ \underline{\alpha} $ not decaying asymptotically along the event horizon $ \mathcal{H}^+, $ a result previously unknown in the literature.