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 $\mu_0$. We show here how an eclectic 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 is 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 non-spherical 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 $\ell=2$ and $\ell=4$ radiation, for different values of $\mu_0$, from the three different approaches to the problem.