SL 2,C symmetry of the gravitational field

Abstract

In a previous paper the symmetry of the gravitational-field dynamical variables was discussed. A scheme was given according to which the spin coefficients and the Riemann tensor were represented in the form of linear combinations of the infinitesimal generators of the groupSL2,C, similar to the way Yang and Mills write their dynamical variables in terms of the Pauli spin matrices, where the spin coefficients take the role of the Yang-Mills-like potentials and the Riemann tensor takes the role of the fields. In this paper we use this representation of the gravitational-field dynamical variables in order to write the field equations of general relativity, and obtain a simple and attractive representation for the field equations of Newman and Penrose. Comparison between the present method and that of Utiyama, based on the Yang-Mills theory, is discussed and the possibility of using it as a basis for quantization is briefly mentioned.