Trapped Surfaces and the Development of Singularities

The study of singularities in general relativity was given a strong impetus by a topological approach due to Penrose and others, and powerful theorems concerning their existence have been developed. In particular a theorem by Penrose states that under certain conditions the existence of a trapped surface in a space-time guarantees that singularities will develop. Using the spin coefficient formalism we generalize from the Schwarzschild solution and prove the existence of a wide class of solutions possessing such trapped surfaces by displaying the solutions to terms linear in a certain null coordinate. Then, using an asymptotic procedure, the method is generalized to include a class of solutions possessing ``asymptotically trapped surfaces.''