INTEGRATION IN THE GHP FORMALISM .1. A COORDINATE APPROACH WITH APPLICATIONS TO TWISTING TYPE N SPACES

The compacted spin coefficient (GHP) formalism is clearly more concise and efficient than the older Newman-Penrose formalism. Yet few people use it when integration of the field equations is involved, Held bein g the notable exception. However, to most workers in the field, Held's approach seems far removed from the usual Newman-Unti (NU) type integ ration procedure. This paper and a subsequent one are concerned with i ntegration within the GHP formalism. In this first paper we develop a GHP coordinate-style integration procedure modelled closely on the NU procedure whereas in the second paper we present a GHP operator-style integration procedure along the lines suggested by Held. For simplicit y of illustration we restrict the discussion to algebraically special vacuum spacetimes. We show clearly the similarities and differences be tween the two approaches, and compare their respective efficiencies. T o deal with a concrete example, we illustrate the two methods by once more considering the problem of twisting type N vacuum solutions to Ei nstein's field equations. The GHP approach enables us to have a compre hensive overview of this much discussed problem and gain new insight i nto the relationship between various results derived in a number of di fferent formalisms.