Black Hole Uniqueness Theorems

It is nearly 30 years since Werner Israel published his uniqueness theorem for the Schwarzschild black hole. This result, which surprised the cognoscenti at first, was quickly recognized as being of major importance. It soon initiated a flurry of research on uniqueness and no-hair theorems and encouraged subsequent developments such as the formulation of black hole thermodynamics. Research related to the initial uniqueness theorems continues today, with earlier results being made more complete and rigorous and new theorems for a variety of matter couplings being proved. The application of new techniques has enabled new proofs of old theorems to be developed and new results to be obtained, and now there is a substantial body of work in this area. This book is a timely addition to the literature which surveys these interesting and important results. The book aims to provide a self-contained introduction to the theory of stationary black holes and the uniqueness theorems. It deals primarily with the mathematical and formal aspects of black hole theory and contains little discussion of the physics or astrophysics of the systems. Only the theory of classical black holes in four dimensions is considered and the author presents the material in a fairly formal way via definitions, propositions and theorems. He makes extensive use of differential forms and the index-free approach to differential geometry. A good working knowledge of the latter is assumed. The first chapters are devoted to a discussion of static and stationary spacetimes, the Killing vector fields associated with their symmetries, and the vacuum and electrovacuum Einstein field equations. The Kerr and Kerr - Newman metrics are derived, the properties of stationary black holes are discussed and the four laws of black hole physics are presented. While the first eight chapters contain material about stationary spacetimes and background information about stationary black holes, the last four chapters are devoted to proofs of uniqueness theorems for rotating and non-rotating stationary black holes. The author emphasizes the `modern' approach to the theorems for vacuum and electrovacuum black holes. The proofs of the theorems for static black holes are based on conformal transformations and the positive-energy theorem and the proofs for stationary black hole theorems emphasize the non-linear sigma model formulation. Alternative approaches to the theorems are noted and well referenced. The last two chapters deal with scalar field couplings and include topics which the author has worked on in recent years. Here harmonic mappings, the Skyrme model and conformal scalar fields are amongst the subjects discussed. These final chapters, in particular, lead into areas of current research activity. This work in the Cambridge Lecture Notes in Physics series should be useful to everyone interested in stationary black hole systems. It contains copious references and provides easy access to a large amount of information about these systems as well as details of the uniqueness theorems themselves. On occasion I found the style somewhat too formal and the notation less than transparent. I also felt that some brief discussion of matters not dealt with in the text (such as the ongoing discussions of the significance of `hair' and `no-hair', higher-dimensional black holes, string theory and other related topics of current interest) would have helped to place the results in a broader context; however, these are minor caveats. In general this book is a welcome addition to the literature on black holes.