Apparent horizons of N black holes and black hole rings

We first study systems of N Schwarzschild black holes in a time-symmetric spacelike hypersurface with axial symmetry. Apparent horizons are found by numerically solving a non linear system of 3 coupled ODEs using Mathematica. The location of the apparent horizon is calculated in each system in order to find the critical separation between the black holes that creates an encompassing apparent horizon. A method for approximating the critical separations of N black holes by representing them as an effective system of two black holes is developed. Next, we study black hole rings of different mass. The apparent horizon is used as an approximation to the event horizon in an effort to predict a critical ring radius that generates an event horizon of toroidal topology. We found that a good estimate for this ring radius is 20/(3pi)M.