The characteristic Cauchy problem for Dirac fields on curved backgrounds

Details The characteristic Cauchy problem for Dirac fields on curved backgrounds The characteristic Cauchy problem for Dirac fields on curved backgrounds

2009-03-03

arxiv.org/abs/0903.0515v2

Mathematics

Analysis of PDEs

39 pages. An error in lemma 3.1 in the first version has been corrected. Moreover, the treatment of the constraints (restricti

Scientific paper

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.

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