Extremal black holes

The extremal Kerr case |a|=M is subject to an instability discovered by Stefanos Aretakis

Theorem (Aretakis 2012). For generic solutions of the wave equation on an extremal Kerr background, the first derivatives fail to decay along the event horizon as advanced time tends to infinity, and moreover, the second derivatives blow up at a polynomial rate.

The above fact is in sharp contrast to the subextremal case, where in joint work with Igor Rodnianski, we have shown that arbitrary derivatives of solutions to the scalar wave equation decay polynomially.

Key to Aretakis's instability result is a remarkable series of conservation laws along the event horizon. These were derived first for extremal Reissner-Nordstrom, see here and here. In subsequent work, see this link, Aretakis showed these conservation laws (and the resulting instability) to hold for the wave equation on general axisymmetric extremal black holes, not necessarily vacuum. In particular, the instability holds for extremal Kerr.

The Aretakis instability has been further generalised by Harvey Reall and James Lucietti (see here) to several new settings: (1) to higher dimensions, (2) dropping the requirement of axisymmetry and perhaps most interestingly, (3) so as to also to the Teukolsky equation. 

Thus, we have the remarkable conclusion that all extremal black holes are unstable to gravitational perturbation along their event horizon.

It would be very interesting to understand the long time ramifications of this in the full non-linear theory.