Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics Vol 47)

Abstract

This book is concerned with mathematical topics related to general relativity. Chapters 1-6 are expositions of a number of parts of mathematics which are important for relativists. The main subjects covered arelinear algebra, general topology, manifolds, Lie groups and, in particular detail, the Lorentz group. Chapters 7-13 analyse aspects ofthe geometrical structure of spacetimes. They focus on symmetries ofvarious types and on properties of curvature. Subjects covered includethe Petrov classification, holonomy groups, the relation between metric and curvature, affine vector fields, conformal symmetries, projectivesymmetries and curvature collineations. This part of the book is atreatise on the (mainly local) geometry of four-dimensional Lorentz manifolds, with attention to energy-momentum tensors of interest ingeneral relativity.In the first six chapters the author has concentrated on givingdefinitions and statements of theorems, the proofs being left tothe references that are quoted. He has clearly put much effort intoproducing a very smooth exposition which is easy to follow and has succeeded in giving us a readable and informative account ofthe mathematics covered. At the same time mathematical rigour is strictly observed. He also takes the time to carefullydiscuss many of the subtleties which arise. This part of thebook has the character of a textbook suitable for students ofgeneral relativity but experienced researchers will also findit a useful reference and are likely to come across interesting facts they have not met elsewhere.The remaining chapters are more like a research monograph and are influenced by the author's own research interests. More proofsare included. Much of the material in this part will be ofinterest to a narrower audience of relativists than that of thefirst part. It should, however, be of interest to those working on exact solutions of the Einstein equations and related topics.It is also the case that most relativists are concerned with detailed properties of symmetries and curvature in some phases of their work and may find information which can help them in this book.Two examples of useful things to be found in this book for whichit is difficult to find a comparable account elsewhere are thediscussions of covering spaces and the subgroups of the Lorentz group. Section 3.10 treats covering spaces in a general topological context while section 4.14 shows how these ideascan be adapted to the smooth category. Section 6.4 lists allthe connected subgroups of the Lorentz group and collectsbackground information about these subgroups.I have no doubt that this book will be valuable for studentstaking courses in general relativity and for people preparingtheir PhD in the subject.