Some examples of space-time curvature collineations (Q1886171)

Many of the difficulties encountered in studying curvature collineations in general relativity are due, firstly, to the fact that the associated vector fields may constitute an infinite-dimensional vector space and, secondly, to the fact that their existence is intimately related to the rank of the curvature tensor when the latter is regarded in the well-known way as a \(6\times 6\) matrix function. Problems of this nature are seldom encountered in the study of other symmetries and this, together with the inherent complications of the curvature tensor, mean that new techniques must be sought in such an investigation. The aim of this note is to briefly review the known results which isolate a class of space-times which contain the only possible examples where proper curvature collineations can exist and then to show, by means of examples, rather variable properties of the metric within this class. These examples help to answer the questions whether the vector space of such vector fields is finite- or infinite-dimensional, and if finite-dimensional, it may or may not equal the Killing or homothetic algebra of a space-time. A simple theorem although restrictive, is given which is helpful in constructing such examples.