Accumulation of Charge on an Extremal Black Hole's Event Horizon

Extremal black holes are known to develop a horizon instability in the presence of external scalar fields. In this talk, I will discuss the extension of this Aretakis instability to charged perturbations, demonstrating that a linear instability manifests in the electric field. In doing so, I will discuss a new gauge-invariant formalism for computing the dynamics of the charged fields, and I will introduce a compactified coordinate system that allows us to numerically evolve the equations of motion to null infinity, thus capturing the appropriate power-law tails at the boundaries of the spacetime. I will conclude by discussing how we expect our results to generalize to the non-linear setting, where metric backreaction is included and may be able to mitigate the instabilities.