A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics (英語) ペーパーバック – 2007/11/5
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This 2004 textbook fills a gap in the literature on general relativity by providing the advanced student with practical tools for the computation of many physically interesting quantities. The context is provided by the mathematical theory of black holes, one of the most elegant, successful, and relevant applications of general relativity. Among the topics discussed are congruencies of timelike and null geodesics, the embedding of spacelike, timelike and null hypersurfaces in spacetime, and the Lagrangian and Hamiltonian formulations of general relativity. Although the book is self-contained, it is not meant to serve as an introduction to general relativity. Instead, it is meant to help the reader acquire advanced skills and become a competent researcher in relativity and gravitational physics. The primary readership consists of graduate students in gravitational physics. It will also be a useful reference for more seasoned researchers working in this field.
'… an elegant, thoughtful, useful and altogether commendable publication.' Contemporary Physics
'The author puts emphasis on training the readers and equipping them with the relevant skills of a working relativist. The text reaches a high pedagogical standard … In this way the author succeeds in closing a gap in the existing text book literature especially for a readership mainly oriented towards physics.' Monatshefte für Mathematik
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This book is not for the beginner in relativity. The reason is that it takes you through a lot about GR which can (should, I say) be avoided in a first pass of GR. It is difficult to get the larger picture if you learning GR for the first time from this book. I therefore suggest, you should have had at least one course on GR. For those already adept in GR (say, by reading Carroll), this book is a gold mine.
Rather than focusing on the physics behind GR (which it occasionally does though), it concerns itself with solving Einstein's equations and the various ways it can be set up. Thus, the book is exactly as it describes itself to be. It is a mathematical toolkit for people wishing to compute things in GR. What I like about it is that it works through a lot of dirty equations that other books avoid and relegate to references. This, for me is the most important factor.
The first chapter quickly goes through some basic mathematics required. If you are reading this book, most of chapter 1 should already be familiar to you. Ch 2 discusses various geometrical aspects of curved spacetimes, such as the Raychaudhuri equations, the energy conditions, etc. Chapter 3 is a thorough discussion of hypersurfaces and its geometries. Ch 4 uses a lot of Ch. 3. Here the ADM formalism of GR is discussed. The final chapter (which I am yet to read) discussed black holes.
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