The characteristic Cauchy problem for Dirac fields on curved backgrounds

TL;DR

This paper establishes existence and uniqueness results for the characteristic Cauchy problem of Dirac fields on curved spacetimes, using energy estimates and analyzing light-ray dynamics near the cone vertex.

Contribution

It provides a rigorous framework for solving the Dirac equation on light-cones in curved backgrounds, including explicit data spaces and constraint analysis.

Findings

Proves existence and uniqueness of solutions in a geodesically convex neighborhood.

Defines explicit data spaces on the light-cone for solutions in H^1.

Analyzes light-ray dynamics near the cone vertex using Geroch-Held-Penrose formalism.

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 neighbourhood of the vertex. This is done for data in $L^2$ and we give an explicit definition of the space of data on the light-cone producing a solution in $H^1$. The method is based on energy estimates following L. H\"ormander (J.F.A. 1990). 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.