Some Remarks on Fourth-Order Tensor Fields on Space-Times

This paper is a contribution to Einstein’s general relativity theory and is mostly a review of known work. It concentrates attention on four fourth-order tensors which arise on the space-time manifold describing this theory and which are very useful. These are the (Riemann) curvature tensor, the Weyl conformal tensor, the “E” tensor and the Weyl projective tensor. The first of these, the curvature tensor, plays an important role in the formulation and interpretation of Einstein’s theory. Next, the Weyl conformal tensor is introduced and its conformal properties described and with it, the Petrov classification of gravitational fields which arises from this tensor. This, in turn, gives rise to the Bel criteria for distinguishing Petrov types at a point by an alignment of certain null directions at that point. The third of these tensors, the “E” tensor, is an important tensor in calculations due to its close connection to the Ricci tensor. The fourth tensor, the Weyl projective tensor, is then described together with its properties relating to the geodesic structure of space-time. As examples of the combined usefulness of these tensors, pp-waves and generalised pp-waves are discussed and related, and a review of the geodesic structure of vacuum metrics is given.