THE CHARACTERISTIC CAUCHY PROBLEM FOR DIRAC FIELDS ON CURVED BACKGROUNDS

Abstract

On arbitrary spacetimes, we study the characteristic Cauchy problem for Dirac fields on a light-cone. We prove the existence and uniqueness of solutions in the future of the light-cone inside a geodesically convex neighborhood of the vertex. This is done for data in L2 and we give an explicit definition of the space of data on the light-cone producing a solution in H1. The method is based on energy estimates following Hörmander [A remark on the characteristic Cauchy problem, J. Funct. Anal.93(2) (1990) 270–277]. The data for the characteristic Cauchy problem are only a half of the field, the other half is recovered from the characteristic data by integration of the constraints, consisting of the restriction of the Dirac equation to the cone. A precise analysis of the dynamics of light rays near the vertex of the cone is done in order to understand the integrability of the constraints; for this, the Geroch–Held–Penrose formalism is used.