Title:

Symmetries in general relativity

The purpose of this thesis is to study those nonflat spacetimes in General Relativity admitting high dimensional Lie groups of motions, homotheties, conformals and affines, and to prove a theorem on the relationship between the first three of these. The basic theories and notations of differential geometry are set up first, and a useful theorem on firstorder partial differential equations is proved. The concepts of General Relativity are introduced, spacetimes are defined and a brief account of the wellknown Petrov and Segre classifications is given. The interplay between these classifications and the isotropy structure of the various Lie groups is discussed as is the socalled 'Schmidt method'. Generalised p.p. waves are studied, with a special study of the subclass of generalised plane waves undertaken, many different characterisations of these latter are found and their admitted symmetries are completely described. Motions, homotheties and affines are considered. A survey of symmetries in Minkowski space, and a summary of known results on spacetimes with high dimensional groups of motions is given. The problem of rdimensional groups of homotheties is studied. The r 6 cases are completely resolved, and examples in the r = 5 cases are given. All examples of nonflat spacetimes admitting the maximal group of affines are displayed, correcting an error in the literature. The thesis ends with a proof of the BilyalovDefriseCarter theorem, which states that for any non conformally flat spacetime there is a conformally related metric for which the original group of conformals is a group of homotheties (motions if not conformal to generalised plane waves). The proof given does not use Bilyalov's analyticity assumption, and is more geometric than DefriseCarter. The maximum size of the conformal group for a given Petrov type is found. An appendix gives a brief account of some REDUCE routines used to check some algebraic manipulations.
