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208 Citations# Spacetime and Geometry: An Introduction to General Relativity

# Abstract

The ever growing relevance of general relativity to astrophysics and cosmology continues to motivate the publication of new textbooks which put the theory in a fresh perspective informed by recent developments. In the last few years we have witnessed the appearance of two new books which reflect this trend, and which stand proud among the classic relativity texts. While the 1970s were the decade of Weinberg [1] and Misner et al [2], and the 80s the decade of Schutz [3] and Wald [4], this is clearly the decade of Hartle [5] and Carroll. Hartle has introduced a novel pedagogical approach to teaching general relativity, which he convincingly argues [6] should be done in the standard undergraduate physics curriculum. His 'physics-first approach' emphasizes physical phenomena and minimizes mathematical formalism. Hartle achieves a lot by introducing only the spacetime metric and the geodesic equation, which are the main tools needed to explore curved spacetime and extract physical consequences. To be sure, to explain how the metric is obtained in the first place does require a background of differential geometry and the formulation of the Einstein field equations. But in Hartle's book this material is wisely presented at a later stage, after an ample sampling of the physics of curved spacetime has motivated the need for the advanced mathematics. Carroll follows instead the traditional route, what Hartle calls the 'math-first approach', in which one introduces first the required mathematical formalism and only then derives the physical consequences. He is, of course, in good company, as this is the method followed in all existing textbooks (with Hartle's being the sole exception). Carroll's approach may not be original, but it is tried and true, and the result of Carroll's efforts is an excellent introduction to general relativity. The book covers the standard topics that would be found in virtually all textbooks (differential geometry, the field equations, linearized theory, black holes, and cosmology), but in addition it contains topics (such as quantum field theory in curved spacetime) which can rarely be found in introductory texts. All these topics are presented with authority and in great pedagogical style. I enjoy the book's informal, even conversational, tone, which helps Carroll establish a good rapport with the reader. All in all, this is a very usable text that offers a modern, viable alternative to existing books. My favourite part of the book is the first three chapters on differential geometry. The presentation of the mathematical formalism is crystal clear and very enjoyable, and it comes with a large number of helpful (and attractive) diagrams. Carroll's presentation of differential geometry is sophisticated but completely accessible, and it is quite broad. It includes all the topics that might be considered elementary (such as vectors and tensors, parallel transport, geodesics and curvature), but also a number of topics that might be considered advanced (such as differential forms, nonmetric connections, torsion, Lie differentiation and Killing vectors). Another particularly successful chapter is the fourth, which presents the Einstein field equations. These are first motivated in the usual way (as the simplest tensorial generalization of Poisson's equation), but are then derived from a variational principle. (This is done in the absence of the action's boundary term, whose inclusion would complicate matters and require machinery that Carroll does not introduce.) What I like most about this chapter is that alternative theories of gravitation (such as scalar-tensor theories and higher-dimensional versions of general relativity) get a fairly detailed treatment. Alternatives to general relativity are hardly ever discussed in textbooks, and this is a welcome initiative. The book's next two chapters are devoted to black holes. Carroll's treatment of the Schwarzschild spacetime is very detailed and complete, but his discussion of the Reissner-Nordström and Kerr spacetimes is far more sketchy. I would have liked to see an equally detailed presentation of these spacetimes. Carroll also provides a good descriptive account of the general properties of black-hole spacetimes. The book's seventh chapter contains a very enjoyable discussion of the linearized approximation to general relativity. The traditional presentation of this topic makes immediate use of the Lorenz gauge condition, which tends to create the (wrong) impression that all components of the gravitational field are radiative. With his careful treatment of gauge transformations, and his exploration of different gauge conditions, Carroll achieves the best textbook presentation of linearized theory to date. The theory is applied to calculate the deflection of light in a weak static field, and to the propagation of gravitational waves in flat spacetime. Less successfully, however, it is also applied to the generation of gravitational waves. Carroll presents the usual derivation of the quadrupole formula but fails to mention that the linearized theory is not an adequate foundation in the context of self-gravitating systems. It is a pity that the application of the quadrupole formula to binary stars does not come with such an important warning. Carroll next moves on to cosmology, a field of research that evolves so rapidly that any new textbook runs the risk of becoming rapidly outdated. This coverage of cosmology is well informed by the recent spectacular developments (including the supernovae data which reveal an accelerated expansion and the mapping of the anisotropies of the cosmic microwave background radiation (CMBR) which reveals a spatially flat universe). Carroll's presentation also includes a pedagogical account of the inflation paradigm, which has become an integral part of the standard cosmological model. This chapter, however, more than any other, left me wanting for more. I am disappointed that it contains no discussion of cosmological perturbations; this is a surprising omission, since the presentation of the linearized theory in chapter 7 is so clearly inspired by the cosmological problem. I am equally disappointed not to find a detailed discussion of the CMBR anisotropies; this omission also is surprising, since the peak structure of their multipole moments makes such a compelling case for inflationary ideas. Given that Carroll is a working cosmologist, it is indeed a surprise to me that this chapter on cosmology happens to be so brief. The ninth and final chapter of Carroll's book is devoted to a topic that has never been covered in an introductory text: quantum field theory in curved spacetime. To include this was a truly inspired thought, and Carroll is to be congratulated for this initiative. Quantum-field processes play an essential role in the physics of structure formation in the early universe, and they give rise to the famous Hawking effect which causes a black hole to behave as a thermal body. A complete education in general relativity cannot exclude this important subject, and we now have a textbook which presents it in a clear, accessible way. In summary, I am positively impressed by this book, in spite of the fact that I find it to be flawed in certain places. I firmly believe that the book stands proud among the best relativity texts. Would I use it in a general relativity course? The answer is: surely, given the right group of students. In the past I have had the pleasure of teaching both an introductory course for undergraduates and an advanced course for graduate students. In my opinion, none of these student groups are a good match for Carroll's book. For the undergraduate course I would choose Hartle over Carroll, as I much favour the physics-first approach. For the graduate course I rely on an existing working knowledge of general relativity and I cover advanced topics that are not found in Carroll's text. The right target group, I imagine, would be graduate students enrolled in an introductory course on general relativity. These students would require more sophistication than can be found in Hartle's book, and they would likely be a great match for Carroll's text. References [1] Weinberg S 1972 Gravitation and cosmology: Principles and applications of the general theory of relativity (New York: Wiley) [2] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (San Francisco: Freeman) [3] Schutz B F 1985 A First Course in General Relativity (Cambridge: Cambridge University Press) [4] Wald R M 1984 General Relativity (Chicago, IL: Chicago University Press) [5] Hartle J B 2003 Gravity: An Introduction to Einstein's General Relativity (San Francisco: Addison Wesley) [6] Hartle J B 2005 General relativity in the undergraduate physics curriculum Preprint gr-qc/0506075

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