A spinor approach to general relativity

Abstract

A calculus for general relativity is developed in which the basic role of tensors is taken over by spinors. The Riemann-Christoffel tensor is written in a spinor form according to a scheme of Witten. It is shown that the curvature of empty space can be uniquely characterized by a totally symmetric four-index spinor which satisfies a first order equation formally identical with one for a zero rest-mass particle of spin two. However, the derivatives used here are covariant, so that on iteration, instead of the usual wave equation, a nonlinear “source” term appears. The case when a source-free electromagnetic field is present is also considered. (No quantization is attempted here.) The “gravitational density” tensor of Robinson and Bel is obtained in a natural way as a striking analogy with the spinor expression for the Maxwell stress tensor in the electromagnetic case. It is shown that the curvature tensor determines four gravitational principal null directions associated with flow of “gravitational density”, which supplement the two electromagnetic null directions of Synge. The invariants and Petrov type of the curvature tensor are analyzed in terms of these, and a natural classification of curvature tensors is given. An essentially coordinate-free method is outlined, by which any analytic solution of Einstein's field equations may, in principle, be found. As an elementary example the gravitational and gravitational-electromagnetic plane wave solutions are obtained.