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What is the fate of an unstable star cluster with a highly relativistic core and an extensive Newtonian halo? The answer to this question may throw light on the origin of supermassive black holes in quasars and AGNs, and hence is astrophysically important. From a computational standpoint, however, ascertaining the answer poses a formidable challenge. It calls for the combined machinery of numerical relativity and N-body particle simulations. We solve the problem in this paper. The collapse of an extreme core-halo cluster leads to the formation of a black hole. The value of the black hole mass is the most significant physical result of our study. Determining this value is numerically difficult because of the large dynamic range that characterizes such a cluster. After the initial implosion, one must allow enough time to elapse so that the configuration can settle down to a stationary state. The new dynamical equilibrium state consists of a central black hole embedded in a stable, extended cluster. The spacetime must be evolved to a stationary state before the appearance of numerical or physical singularities associated with black hole formation. We show how, by making an appropriate choice of space and time coordinates and by working with dynamical variables that "freeze" at late times near the center, the complete evolution of such a cluster can be determined. We find that even for very centrally condensed configurations an appreciable fraction of the total cluster mass - much larger than the initial core mass - ultimately collapses to a black hole. Several coordinate systems have recently been proposed for doing numerical relativity. We have compared these in detail during our simulations of the collapse of very extreme configurations of collisionless matter. These comparisons, made in spherical symmetry, yield insights for future multidimensional work.
Shapiro et al. (Fri,) studied this question.