Study of multi-black-hole and ring-singularity apparent horizons

We study critical black-hole separations for the formation of a common apparent horizon in systems of N black holes in a time-symmetric configuration. We study in detail the aligned equal mass cases for N=2, 3, 4, 5, 6, and relate them to the unequal mass binary black-hole case. We then study the apparent horizon of the time-symmetric initial geometry of a ring singularity of different radii. The apparent horizon is used as indicative of the location of the event horizon in an effort to predict a critical ring radius that would generate an event horizon of toroidal topology. We found that a good estimate for this ring critical radius is 20/(3π)M. We briefly discuss the connection of this two cases through a discrete black-hole 'necklace' configuration.