  Using the Newman-Penrose formalism we study the characteristic initial value
problem in vacuum General Relativity. We work in a gauge suggested by Stewart,
and following the strategy taken in the work of Luk, demonstrate local
existence of solutions in a neighbourhood of the set on which data are given.
These data are given on intersecting null hypersurfaces. Existence near their
intersection is achieved by combining the observation that the field equations
are symmetric hyperbolic in this gauge with the results of Rendall. To obtain
existence all the way along the null-hypersurfaces themselves, a bootstrap
argument involving the Newman-Penrose variables is performed.
