Gravitational collapse in the vicinity of the extremal black hole critical point

We study the threshold of gravitational collapse in spherically symmetric spacetimes governed by the Einstein-Maxwell-Vlasov equations. We numerically construct solutions describing a collapsing distribution of charged matter that either forms a charged black hole or eventually disperses. We first consider a region of parameter space where the solutions at the threshold of black hole formation are stationary, horizonless shells. These solutions terminate at a critical point, with their charge-to-mass ratio approaching unity from below, and the instability timescale diverging. Beyond the critical point, we find a new region of parameter space where the threshold solution is an extremal black hole. We measure the scaling of the dynamical time period of the near threshold solutions and discuss how they are connected in the two regimes. If a similar picture to the one found here holds for known families of stationary solutions of rotating matter that approach the exterior of an extremal Kerr spacetime, they could provide a route to forming an extremal spinning black hole.