Black holes, star clusters, and naked singularities: numerical solution of Einstein’s equations

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Abstract
We describe a new method for the numerical solution of Einstein’s equations for the dynamical evolution of a collisionless gas of particles in general relativity. The gravitational field can be arbitrarily strong and particle velocities can approach the speed of light. The computational method uses the tools of numerical relativity and N -body particle simulation to follow the full nonlinear behaviour of these systems. Specifically, we solve the Vlasov equation in general relativity by particle simulation. The gravitational field is integrated by using the 3 + 1 formalism of Arnowitt, Deser and Misner. Physical applications include the stability of relativistic star clusters the binding energy criterion for stability, and the collapse of star clusters to black holes. Astrophysical issues addressed include the possible origin of quasars and active galactic nuclei via the collapse of dense star clusters to supermassive black holes. The method described here also provides a new tool for studying the cosmic censorship hypothesis and the possibility of naked singularities. The formation of a naked singularity during the collapse of a finite object would pose a serious difficulty for the theory of general relativity. The hoop conjecture suggests that this possibility will never happen provided the object is sufficiently compact (≤ M ) in all of its spatial dimensions. But what about the collapse of a long, non-rotating, prolate object to a thin spindle? Such collapse leads to a strong singularity in newtonian gravitation. Using our numerical code to evolve collisionless gas spheroids in full general relativity, we find that in all cases the spheroids collapse to singularities. When the spheroids are sufficiently compact the singularities are hidden inside black holes. However, when the spheroids are sufficiently large there are no apparent horizons. These results lend support to the hoop conjecture and appear to demonstrate that naked singularities can form in asymptotically flat space-times.