Nearly extremal apparent horizons in simulations of merging black holes

Abstract

The spin angular momentum $S$ of an isolated Kerr black hole is bounded by\nthe surface area $A$ of its apparent horizon: $8\\pi S \\le A$, with equality for\nextremal black holes. In this paper, we explore the extremality of individual\nand common apparent horizons for merging, rapidly spinning binary black holes.\nWe consider simulations of merging black holes with equal masses $M$ and\ninitial spin angular momenta aligned with the orbital angular momentum,\nincluding new simulations with spin magnitudes up to $S/M^2 = 0.994$. We\nmeasure the area and (using approximate Killing vectors) the spin on the\nindividual and common apparent horizons, finding that the inequality $8\\pi S <\nA$ is satisfied in all cases but is very close to equality on the common\napparent horizon at the instant it first appears. We also introduce a\ngauge-invariant lower bound on the extremality by computing the smallest value\nthat Booth and Fairhurst's extremality parameter can take for any scaling.\nUsing this lower bound, we conclude that the common horizons are at least\nmoderately close to extremal just after they appear. Finally, following\nLovelace et al. (2008), we construct quasiequilibrium binary-black-hole initial\ndata with "overspun" marginally trapped surfaces with $8\\pi S > A$ and for\nwhich our lower bound on their Booth-Fairhurst extremality exceeds unity. These\nsuperextremal surfaces are always surrounded by marginally outer trapped\nsurfaces (i.e., by apparent horizons) with $8\\pi S<A$. The extremality lower\nbound on the enclosing apparent horizon is always less than unity but can\nexceed the value for an extremal Kerr black hole. (Abstract abbreviated.)\n