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Near-horizon instability of rapidly rotating black holes

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Aretakis' discovery of a horizon instability of extremal black holes came as something of a surprise given earlier proofs that individual frequency modes are bounded. Is this kind of instability invisible to frequency-domain analysis? The answer is no: We show that the horizon instability can be recovered in a mode analysis as a branch point at the horizon frequency. We use the approach to generalize to nonaxisymmetric gravitational perturbations and reveal that certain Weyl scalars are unbounded in time on the horizon. We will also discuss new results showing how the instability manifests for *nearly* extremal black holes: long-lived quasinormal modes collectively give rise to a transient period of growth near the horizon. This period lasts arbitrarily long in the extremal limit, reproducing the Aretakis instability precisely on the horizon. We interpret these results in terms of near-horizon geometry and discuss potential astrophysical implications.