Extended Geroch-Held-Penrose formalism

Abstract

An extension is presented of the well-known Geroch-Held-Penrose (GHP) formalism, itself an extension of the still better known Newman-Penrose (NP) formalism. The extended formalism given here uses only quantities that transform properly under all diagonal transformations of the spin frame, that is, not only under boost-rotations, but also under conformal rescalings. Full use is made of the formalism's symmetry under all discrete operations, that is, under conjugation, the prime operation, and the (modified) Sachs transformations. Just as the GHP formalism is considerably simpler than the NP formalism in the case where a spacelike surface is singled out in a natural way, so the present formalism leads to further simplification when a conformal spacelike surface can be singled out. This is the case, for example, in considerations of future null infinity. In general situations all three formalisms are on an equal footing.