Evolutions from extremality

We examine the evolution of extremal spherically symmetric black holes, developing both general theory as well as the specific cases of (charged) null dust and massless scalar field spacetimes. As matter accretes onto extremal marginally trapped tubes, they generically evolve to become non-extremal with the initial extremal horizon bifurcating into inner and outer non-extremal horizons. At the start of this process arbitrarily slow matter accretion can cause a geometrically invariant measure of horizon growth to jump from zero to infinity. We also consider dynamical horizons that are extremal throughout their evolution and see that such spacetimes contain two extremal black hole horizons: an inner isolated one and an outer dynamical one. We compare these extremal dynamical horizons with the dynamical extreme event horizon spacetimes of Murata, Reall and Tanahashi.