Nearly extremal apparent horizons in simulations of merging black holes

The spin $S$ of a Kerr black hole is bounded by the surface area $A$ of its apparent horizon: $8\pi S \le A$. We present recent results (arXiv:1411.7297) for the extremality of apparent horizons for merging, rapidly rotating black holes with equal masses and equal spins aligned with the orbital angular momentum. Measuring the area and (using approximate Killing vectors) the spin on the individual and common apparent horizons, we find that the inequality $8\pi S < A$ is satisfied but is very close to equality on the common apparent horizon at the instant it first appears---even for initial spins as large as $S/M^2=0.994$. We compute the smallest value $e_0$ that Booth and Fairhurst's extremality parameter can take for any scaling of the horizon's null normal vectors, concluding that the common horizons are at least moderately close to extremal just after they appear. We construct binary-black-hole initial data with marginally trapped surfaces with $8\pi S > A$ and $e_0>1$, but these surfaces are always surrounded by apparent horizons with $8\pi S