Relativistic stellar systems with spindle singularities

We develop a method for constructing axisymmetric star clusters in general relativity. The method yields a self-consistent numerical solution of the relativistic Vlasov equation coupled to Einstein's equations for the gravitational field. We apply the method to construct equilibrium sequences of prolate relativistic clusters. The limiting members of these sequences contain spindle singularities. Because the clusters are equilibria, we can be certain that no black holes are present. The limiting behavior of these sequences resembles the singular endpoint found in recent dynamical simulations of prolate cluster collapse. In the dynamical simulations, one could not be absolutely certain that there were no black holes, although there were no apparent horizons. The existence of equilibrium prolate clusters arbitrarily close to a singular state without black holes makes the spindles formed in dynamical collapse more compelling candidates for naked singularities.