The structure of tetrad formalisms in general relativity: The general case

Abstract

The sets of equations that form the basis for the tetrad formalism approach in general relativity contain considerable redundancy. Papapetrou has determined this redundancy explicitly in the form of three sets of identities and employed these in investigations of the Newman-Penrose tetrad formalism. In this paper Papapetrou's work is reviewed and some of his results that do not seem to be well known are emphasized, along with some general implications. The main new result that is established concerns the Geroch-Held-Penrose formulation of the tetrad formalism. When the sets of equations that are usually used in this formulation are considered in the light of Papapetrou's identities, it is found that certain formal simplifications can be made and that the Geroch-Held-Penrose formulation can be presented more concisely. It is emphasized that the results in this paper apply in the most general case only. Any special cases (e.g., simplified tetrad and/or Riemann tensor) need to be considered separately.