Hoop conjecture for colliding black holes

Abstract

We study the collision of black holes in the Kastor-Traschen space-time, at present the only such analytic solution. We investigate the dynamics of the event horizon in the case of the collision of two equal black holes, using the ray-tracing method. We confirm that the event horizon has trouser topology and show that its set of past end points (where the horizon is nonsmooth) is a spacelike curve resembling a seam of trousers. We show that this seam has a finite length and argue that twice this length be taken to define the minimal circumference $C$ of the event horizon. Comparing with the asymptotic mass $M,$ we find the inequality $C<4\ensuremath{\pi}M$ supposed by the hoop conjecture, with both sides being of similar order, $C\ensuremath{\sim}4\ensuremath{\pi}M.$ This supports the hoop conjecture as a guide to general gravitational collapse, even in the extreme case of head-on black-hole collisions.