Head-on collision of two black holes: Comparison of different approaches

A benchmark problem for numerical relativity has been the head-on collision of two black holes starting from the ''Misner initial data,'' a closed form momentarily stationary solution to the constraint equations with an adjustable closeness parameter μ0. We show here how an electric mixture of approximation methods can provide both an efficient means of determining the time development of the initial data and a good understanding of the physics of the problem. When the Misner data are chosen to correspond to holes initially very close together, a common horizon surrounds both holes and the geometry exterior to the horizon can be treated as a nonspherical perturbation of a single Schwarzschild hole. When the holes are initially well separated the problem can be treated with a different approximation scheme, ''the particle-membrane method.'' For all initial separations, numerical relativity is in principle applicable, but is costly and of uncertain accuracy. We present here a comparison of the different approaches. We compare waveforms, for l=2 and l=4 radiation, for different values of μ0, from the three different approaches to the problem.