The GHP formalism, with applications to Petrov type III spacetimes

Sammanfattning: We give a review of the construction and application of spinor fields in general relativity and an account of the spinor-based Geroch-Held-Penrose (GHP) formalism. Specifically, we discuss using the GHP formalism to integrate Einstein's equations as suggested by Held  and developed by Edgar and Ludwig and discuss the similaritites with the Cartan-Karlhede classification of spacetimes. We use this integration method to find a one-parameter subclass and a degenerate case, for which the Cartan-Karlhede algorithm terminates at second order, of the Petrov type III, vacuum Robinson-Trautman metrics. We use the GHP formalism to find the Killing vectors, using theorems by Edgar and Ludwig. The one-parameter family admits exactly two Killing fields, whereas the degenerate case admits three and is Bianchi type VI. Finally we use the Cartan-Karlhede algorithm to show that our class, including the degenerate case, is equivalent to a subclass found by Collinson and French. Our degenerate case corresponds to an example metric given by Robinson and Trautman and is known to be the unique algebraically special vacuum spacetime with diverging rays and a three-dimensional isometry group.