A Class of Dust-Like Self-Similar Solutions of the Massless Einstein–Vlasov System

In this paper the existence of a class of self-similar solutions of the Einstein–Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein–Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein–Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point P0 and converges to a stationary solution P1 as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes. The spacetimes constructed do not constitute a counterexample to cosmic censorship since they are not asymptotically flat. They should be seen as the first step on a possible path towards constructing solutions of importance for understanding the issue of the formation of naked singularities in general relativity.