On the commutator equations in the extended GHP formalism

Abstract

Ludwig has recently developed an extension of the GHP formalism (EGHP formalism) which contains fewer variables and fewer equations than the original GHP formalism; on the other hand, the EGHP commutator equations are more complicated than their GHP counterparts, having explicit conformally-weighted parts as well as explicit spin- and boost-weighted parts which also occur in the GHP commutator equations. To extract all the information from the EGHP commutator equations, one would expect to have to apply them to at least seven real, appropriately weighted quantities. However it is shown---because of the redundancy inherent in tetrad formalisms---that, alongside the EGHP Bianchi and Ricci equations, it is always sufficient to apply all the EGHP commutator equations to only six real (three complex) quantities, four of which should be zero-weighted, functionally independent scalars while the other two should have non-zero spin and boost weight but any conformal weight. Furthermore, it is shown that, alongside the EGHP Bianchi and Ricci equations, it is almost always sufficient to apply all the EGHP commutator equations to only four real (two complex), zero-weighted, functionally independent scalars.