Approximate methods for black hole collisions

The head-on collision of two nonrotating black holes, initially falling toward each other with speed of approach equal to escape velocity, is studied. The initial-value problem for the two-black-hole configuration is solved by giving the extrinsic curvature on a Euclidean space slice, with the equal masses of the two holes and their separation as free parameters. An approximating space-time is then constructed by stacking up a whole sequence of initial-value slices with decreasing separation between the black holes. The space-time singularities at the centers of the two black holes collide at some instant of time, and after that time the space-time so constructed is just the Schwarzschild solution with mass equal to the total mass of both black holes. The approximating space-time describes the coalescence of the event horizons, but contains no gravitational waves.