Critical Collapse of Collisionless Matter in Spherical Symmetry

Abstract

We perform a numerical study of the critical regime for the general relativistic collapse of collisionless matter in spherical symmetry. The evolution of the matter is given by the Vlasov equation (or Boltzmann equation) and the geometry by Einstein's equations. This system of coupled differential equations is solved using a particle-mesh (PM) method. This method approximates the distribution function which describes the matter in phase space with a set of particles moving along the characteristics of the Vlasov equation. The individual particles are allowed to have angular momentum different from zero but the total angular momentum has to be zero to retain spherical symmetry. In accord wih previous work by Rein, Rendall and Schaeffer, our results give some indications that the critical behaviour in this model is of Type I (the smallest black hole in each family has a finite mass). For the families of initial data that we have studied it seems that in the critical regime the solution is a static spacetime with non-zero radial momentum for the individual particles. We have also found evidence for scaling laws for the time that the critical solutions spend in the critical regime.