Symmetries in general relativity
The purpose of this thesis is to study those non-flat space-times 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 first-order partial differential equations is proved. The concepts of General Relativity are introduced, space-times are defined and a brief account of the well-known 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 so-called '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 space-times with high dimensional groups of motions is given. The problem of r-dimensional groups of homotheties is studied. The r 6 cases are completely resolved, and examples in the r = 5 cases are given. All examples of non-flat space-times admitting the maximal group of affines are displayed, correcting an error in the literature. The thesis ends with a proof of the Bilyalov-Defrise-Carter theorem, which states that for any non conformally flat space-time 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 Defrise-Carter. 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.