Today we have a solid theoretical understanding of the dynamics of black holes, as predicted by classical general relativity, for the typical binary merger expected as an astrophysical gravitational wave source. However, in more "extreme" situations, namely, black holes that collide with velocities close to the speed of light and non-linear perturbations of (near-)extremal black holes, less is known, in some respects even qualitatively. In these lectures I will discuss some of these open questions, describe some recent results, and speculate about possible answers.

 

Extremal black holes are those with the maximum amount of charge and/or angular momentum allowed by general relativity, and are characterized by having zero surface gravity (zero temperature in the thermodynamic analogue

description). Results from linear perturbation theory show that extremal holes can behave very differently from their subextremal counterparts, including the fact that exactly extremal black holes are unstable (the celebrated Aretakis instability), and that in the limit of extremality a subset of the black hole's quasi-normal modes approach zero damping. This has inspired some to argue that turbulent-like dynamics may occur on the horizons of perturbed near-extremal black holes, and that the Aretakis instability survives at the non-linear level with sufficiently fine-tuned perturbations that could furthermore exhibit some form of critical phenomena.

 

In this lecture I will describe recent work studying the non-linear dynamics of (near-) extremal charged black holes, albeit restricted to spherical symmetry. Though in this setting we cannot address the turbulence question, we

can address aspects of putative fine-tuned critical behavior.