Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry

We perform a numerical study of the critical regime at the threshold of black hole formation in the spherically symmetric, general relativistic collapse of collisionless matter. The coupled Einstein-Vlasov equations are solved using a particle-mesh method in which the evolution of the phase-space distribution function is approximated by a set of particles (or, more precisely, infinitesimally thin shells) moving along geodesies of the spacetime. Individual particles may have nonzero angular momenta, but spherical symmetry dictates that the total angular momentum of the matter distribution vanish. In accord with previous work by Rein, Rendall, and Schaeffer, our results indicate that the critical behavior in this model is type I; that is, the smallest black hole in each parametrized family has a finite mass. We present evidence that the critical solutions are characterized by unstable, static spacetimes, with nontrivial distributions of radial momenta for the particles. As expected for type I solutions, we also find power-law scaling relations for the lifetimes of near-critical configurations as a function of the parameter-space distance from criticality.