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

Understand

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.