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

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.