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Abstract

This work is devoted to the study of Dirac fields and of their evolution on globally hyperbolic asymptotically flat space-times by means of $3+1$ decomposition techniques. The principles of the $3+1$ decomposition are explained and used to define classes of space-times for which the regularity and fall-off at infinity of the metric are precisely specified. Dirac's equation is expressed and its $3+1$ decomposition is described both in terms of Dirac spinors and in the framework of the two-spinor formalism. The global Cauchy problem is then solved on three types of space-times. First, we work on the classes of space-times defined earlier. For weak regularity and fall-off of the metric, we solve in Sobolev and weighted Sobolev spaces the Cauchy problem for general symmetric hyperbolic systems with weakly regular coefficients. We apply these results to Dirac's equation. Then, we consider Schwarzschild and Kerr black holes. The geometry of these space-times is described in details. Depending on the choice of observer field, the general results above are either applied directly or adapted.