Horizon Instability of Extremal Black Holes
Stefanos Aretakis
TL;DR
The paper demonstrates that axisymmetric extremal horizons exhibit linear scalar instabilities, with non-decay of $|Y\psi|$ and blow-up of higher derivatives $|Y^{k}\psi|$ along the horizon $\mathcal{H}^{+}$, governed by local horizon geometry through horizon-conservation laws. The author develops a general geometric framework under $\mathbb{R}\times\mathbb{T}^{1}$ symmetry, introduces adapted coordinates, and proves a horizon-conserved quantity $H[\psi](\tau)$ for solutions of $\Box_{g}\psi=0$, plus a spherical-harmonic hierarchy yielding conserved quantities $H_{l}[\psi]$ in symmetric settings. These results apply to Majumdar–Papapetrou multi-black-hole spacetimes and extremal Kerr (with or without a cosmological constant), yielding non-decay and pointwise and energy blow-up of higher derivatives; for extremal Kerr, axisymmetric decay results combine with the horizon laws to establish a robust scalar instability. The work identifies a purely local mechanism for horizon instability, contrasts with subextremal decay, and has influenced subsequent developments extending the conservation laws to other perturbations and dimensions, as well as exploring nonlinear instabilities. Overall, the findings have significant implications for the stability and dynamics of extremal black holes in general relativity.
Abstract
We show that axisymmetric extremal horizons are unstable under linear scalar perturbations. Specifically, we show that translation invariant derivatives of generic solutions to the wave equation do not decay along such horizons as advanced time tends to infinity, and in fact, higher order derivatives blow up. This result holds in particular for extremal Kerr-Newman and Majumdar-Papapetrou spacetimes and is in stark contrast with the subextremal case for which decay is known for all derivatives along the event horizon.