Integration in the GHP formalism I: 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 being the notable exception. However, to most workers in the field, Held's approach seems far removed from the usual Newman-Unti (nu) type integration procedure. This paper and a subsequent one are concerned with integration within theghp formalism. In this first paper we develop aghpcoordinate-style integration procedure modelled closely on thenu procedure whereas in the second paper we present aghpoperator-style integration procedure along the lines suggested by Held. For simplicity of illustration we restrict the discussion to algebraically special vacuum spacetimes. We show clearly the similarities and differences between the two approaches, and compare their respective efficiencies. To deal with a concrete example, we illustrate the two methods by once more considering the problem of twisting typeN vacuum solutions to Einstein's field equations. TheGhp approach enables us to have a comprehensive overview of this much discussed problem and gain new insight into the relationship between various results derived in a number of different formalisms.