Initial data and black holes for matter models

To observe the dynamic formation of black holes in general relativity, one essentially needs to prove that closed trapped surfaces form during evolution from initial data that do not already contain trapped surfaces. We discuss the recent development of the construction of such admissible initial data for matter models. In addition, we extend known results for the Einstein equations coupled to perfect fluids in spherical symmetry and with linear equation of state to unbounded domains. Polytropic equations of state and regularity issues with the direct application of the singularity theorems in general relativity are discussed briefly.