Nonlinear fate of perturbed extremal Kerr black holes
In plain words
At maximum spin, small disturbances do not die away at the horizon: some of their derivatives grow forever, an effect found in 2011 by the mathematician Stefanos Aretakis. Whether this growth destroys the extremal black hole in the full theory is unknown.
Precise statement
Linear fields on extremal Kerr ($a=M$) show the Aretakis instability (non-decay and growth of transversal derivatives along the horizon) and azimuthal instabilities (Gajic 2023). Determine the nonlinear evolution of vacuum data near extremal Kerr: whether solutions settle to a subextremal or extremal Kerr exterior, whether horizon instabilities produce singular behavior, and the codimension of the set of data that remain extremal. An answer is a theorem describing the asymptotic state.
What would settle it
A nonlinear theorem for vacuum perturbations of extremal Kerr describing the late-time state and horizon regularity.
Status in the literature
Unverified note
Nonlinear stability with the Aretakis horizon instability is proved for extremal Reissner-Nordstrom in spherical symmetry (Angelopoulos, Kehle, Unger, arXiv:2410.16234, 2024); vacuum extremal Kerr is open as of 2026.