Charged Instabilities of Extremal Black Holes

One of the biggest open questions in General Relativity concerns the stability of black holes to external perturbations. Extremal black holes – which are maximally charged or spinning – represent a particularly interesting piece of this puzzle because they are known to develop a horizon instability in the presence of scalar matter. In this talk, I will discuss how this picture extends to charged perturbations, which can trigger new instabilities via both gravitational and electromagnetic interactions. In particular, I will demonstrate that on a fixed background metric, charged matter can accumulate on the event horizon of an extremal black hole without decaying, thus posing a new challenge to black hole stability. In demonstrating this result, I will discuss our numerical formalism for solving the equations of motion in double-null coordinates, which allows us to capture the dynamics of the charged fields at the boundaries of the spacetime. I will conclude by presenting updates on the non-linear problem, where metric backreaction is included and may be able to both sustain and mitigate the instabilities.