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Title: The Large Scale Structure of Space-Time by S. W. Hawking, G. F. R. Ellis, P. V. Landshoff, D. R. Nelson, D. W. Sciama, S. Weinberg ISBN: 0-521-09906-4 Publisher: Cambridge University Press Pub. Date: 27 February, 1975 Format: Paperback Volumes: 1 List Price(USD): $60.00 |
Average Customer Rating: 4.6 (5 reviews)
Rating: 5
Summary: The Large Scale structure of good science books (& spacetime
Comment: I think that this book has great depth, and is one of the best Stephen Hawking books I have read. My favourite remains 'A Brief History of Time', but still this book is extremely excellent. My compliments to the chef.
Rating: 5
Summary: A classic in mathematical general relativity
Comment: This book is now a classic and is written by two giants in mathematics and physics. It wil be used for many years to come and is certainly one of the most widely quoted in the subject.
The authors begin the book by a discussion of the role of gravity in physics and its role as determining the causal structure of the universe. They introduce the idea of a closed trapped surface, setting the stage for the goal of the book, namely the study of the conditions under which a space-time singularity must occur. Black holes and the beginning of the universe are cited as examples of these singularities. The authors also outline briefly the content of each chapter. A neat argument is given for the significance of focal points via the use of Raychaudhari's equation.
The second chapter is an overview of the background in differential geometry needed in the rest of the book. Although complete from an axiomatic point of view, the approach is much too formal for readers who do not have a knowledge of differential geometry. Such a reader should gain the necessary background elsewhere.
General relativity as a theory of gravitation is discussed in chapter 3. Spacetime is assumed to be a connected 4-dimensional smooth manifold on which is defined a Lorentz metric. The topology is assumed to be Hausdorff. Some of the more interesting or well-written parts of this chapter include the example of a spacetime that is not inextendible, the determination of the conformal factor for the spacetime metric, and the discussion of alternative field equations.
The authors discuss the physicial significance of curvature in chapter 4, namely its effect on families of timelike and null curves. The most important part of this chapter is the discussion on certain inequalities tht the energy-momentum tensor should satisfy from a physical viewpoint. These inequalities, called the weak energy condition and the dominant energy condition, allow the authors to prove the existence of singularities in a later chapter. The reader can see clearly the role of the Jacobi equation, and its solution, the Jacobi field, in measuring the separation of nearby geodesics. The existence of conjugate points is proven, and shown to imply the existence of self-intersections in families of geodesics. As a warm-up to showing the non-existence of geodesics of maximal length, the authors employ variational calculus to study how to vary non-spacelike curves connecting points in convex normal neighborhoods in spacetime, and between points and hypersurfaces. In particular, it is shown that a timelike geodesic curve from a hypersurface to a point is maximal iff there is no conjugate point to the hypersurface along the curve. In addition, the authors prove that two points joined by a non-spacelike curve which is not a null geodesic can be joined by a timelike curve.
The authors consider the exact solutions of the Einstein field equations in chapter 5. Most of the "usual" spacetimes are considered, including Minkowski, De Sitter, Anti-de-Sitter, Robertson-Walker, Schwarzschild, Reissner-Nordstrom, Kerr, Taub-Nut, and Godel. The emphasis in on the global properties of the spacetimes and the existence of singularities in them. The famous Penrose diagrams are used to "compactify" spacetimes in order to study their behavior at infinity and their conformal properties. The authors first introduce the concept of a future (past) Cauchy development here, so important in later developments in the book. The reader can see the tools developed in chapter 4 in play here; for example, the existence of a singularity in a spatially homogeneous cosmology is shown to follow directly from the Raychaudhuri equation. The existence of the singularity is proved to be independent of any acceleration or rotation of matter in such cosmologies.
In chapter 5, the authors consider the causal structure of spacetime, namely the study of its conformal geometry. The consideration of the set of all metrics conformal to the physical metric allows one to discuss "geodesic completeness" of spacetime, this concept forming the basis of a later definition of a singularity in spacetime. The more interesting topics discussed in this chapter include the causality conditions (there are no closed non-spacelike curves), and the Alexandrov topology and its connection with the strong causality condition (every neighborhood of a point contains a neighborhood of the point no non-separable curve of which intersects it more than once). When strong causality does hold, the Alexandrov topology is equivalent to the usual manifold topology, and thus the topology of spacetime can be determined by the observation of causal relationships. The discussion on the role of global hyperbolicity in showing the existence of a maximal geodesic is also very well-written.
The next chapter is pretty much independent of the rest, and was put in no doubt for the mathematician who desires to understand the Einstein equations as a set of nonlinear second-order hyperbolic partial differential equations with initial data on a 3-dimensional manifold, the famous Cauchy problem in general relativity.
Chapter 8 is the most important in the book, for its uses the constructions of earlier chapters to define the notion of a singularity in spacetime. The authors argue that singularities are points where physical laws break down and thus to characterize them one attempts to find out whether any such points have been removed, making spacetime "incomplete" in some sense. Such a notion of incompleteness is very meaningful in topological spaces with a positive definite metric, since in that case one can define completeness in terms of the convergence of Cauchy sequences. In spacetimes with a Lorentz metric, the authors discuss the notion of geodesic completeness for null and timelike geodesics. A very detailed treatment of the now famous singularity theorems is given, these theorems involving an inequality of the Ricci tensor. The last two chapters of the book are more physical in nature wherein the singularity problem is shown to have physical relevance via the occurence of black holes at the endpoint of evolution of massive stars.
Rating: 5
Summary: A wonderful, foundational work of mathematical physics.
Comment: The early seventies saw a revolution in cosmology; for the first time, modern mathematical methods were applied to the discipline, with intriguing results. This book was (along with Penrose's articles) the seminal work in global general relativity. Often overlooked is that the first half of Hawking & Ellis is devoted to traditional GR via the tensor calculus, and the q-form conception. However, trying to learn GR with this volume is not recommended (instead, cf. D'Inverno). The meat-and-potatoes of the book is the discussion of gravitational collapse, and the singularity theorems. They provide us with intuitively good reasons for believing in some very strange phenomenon. If you're interested in the frontiers of modern science, and have the appropriate mathematical background, this book cannot be recommended too highly. The little yellow book stands supreme in the hierarchy of works of modern physics.
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Title: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago Lectures in Physics) by Robert M. Wald ISBN: 0226870278 Publisher: University of Chicago Press Pub. Date: September, 1994 List Price(USD): $20.00 |
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Title: General Relativity by Robert M. Wald ISBN: 0226870332 Publisher: University of Chicago Press Pub. Date: July, 1984 List Price(USD): $37.00 |
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Title: Geometrical Methods of Mathematical Physics by Bernard Schutz ISBN: 0521298873 Publisher: Cambridge University Press Pub. Date: 28 January, 1980 List Price(USD): $36.95 |
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Title: Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll ISBN: 0805387323 Publisher: Pearson Addison Wesley Pub. Date: 20 June, 2003 List Price(USD): $80.00 |
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Title: Gravitation and Cosmology : Principles and Applications of the General Theory of Relativity by Steven Weinberg ISBN: 0471925675 Publisher: Wiley Text Books Pub. Date: July, 1972 List Price(USD): $113.95 |
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