Theory and applications of the classification of second order symmetric tensors in Einstein's general theory of relativity
The purpose of this thesis is to present some new ideas related to the classification of second order symmetric tensors in general relativity theory. In the introductory chapter, the tensors which are to be classified are defined and their role in Einstein's theory discussed. An introduction to tetrads, null rotations and the theory of bivectors as well as a summary of some of the main contributions to classification theory and its applications are also provided in the first chapter. The second chapter contains brief descriptions of the known methods of classifying the Weyl and Ricci tensors. In addition, new methods of classifying the E tensor and of classifying bivectors by Segre type are discussed. The contribution due to the E tensor when a spherically symmetric cloud of test particles is scattered by a gravitational field is analysed, thus extending some known results. As an example of the classification of second order tensors, the Segre type of the energy-momentum tensor for a moving charged particle is calculated. Chapter Three is concerned with spaces admitting symmetries and in particular the restrictions imposed on the Ricci tensor by locally isotropic space-times. A short proof is presented of a known result concerning a symmetry in an Einstein-Maxwell space-time. The classifications of the Ricci tensor due to Ludwig and Scanlon and Penrose are investigated in the final chapter. The correspondence between these schemes and the Segre classification is examined using an entirely vector approach. The projective model constructed by Penrose is discussed in detail and some ideas pertaining to this model are extended.