On the Local Existence for the Characteristic Initial Value Problem in General Relativity

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere S with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of S in the future of S. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy-type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.