Gravity: An Introduction to Einstein’s General Relativity James B. Hartle Addison-Wesley, San Francisco, 2003. $59.80 (582 pp.). ISBN 0-8053-8662-9
One of the illustrations in Alice’s Adventures in Wonderland shows Alice holding a large flamingo. Alice seems to be considering the questions, “What is this?” and “Now that I’ve got it, what do I do with it?” Students beginning the study of general relativity likely ask themselves the same questions about the spacetime metric, the covariant derivative operator, and the curvature tensor. Sean Carroll’s Spacetime and Geometry: An Introduction to General Relativity and James Hartle’s Gravity: An Introduction to Einstein’s General Relativity are two new textbooks that present different pedagogical approaches to answering such questions, that is, to teaching general relativity.
Carroll’s text adds a new option to the list of solid general relativity texts appropriate for graduate students planning to do research in gravity, high-energy theory, or both. Carroll presents a middle ground between a formal, mathematical introduction to general relativity and a utilitarian particle-physics presentation, the latter sidestepping the differential geometry involved in gravity. The student is led from manifolds through the curvature tensor to Einstein’s field equations in a manner that is both correct and accessible. The exposition is aided by many examples that illustrate phenomena of interest, such as the expanding universe. A number of sections that treat more advanced topics appear in the first part of the text. For example, students read about causality in chapter 2. Rather than include several thorough chapters on causality, which would provide the background needed to read advanced papers or do research on the subject, Carroll discusses some of the basic concepts in a few accessible pages. His approach allows lecturers to mention the topic and, later in the course, give the definition of a black hole.
The second part of Spacetime and Geometry is devoted to a wide range of advanced topics, such as rotating black holes and perturbation theory. The chapter on cosmology details our current understanding of what the universe looks like, which has changed considerably since most of the standard general relativity texts were written in the 1970s and 1980s. Carroll’s presentation is appropriate to current questions in cosmology; for example, the book contains discussions on gravitational lensing and inflation. For professors who manage to save some time at the end of the semester, the book has a chapter on quantum field theory in flat and curved spacetime, which leads up to a discussion on black-hole evaporation.
Of course, every teacher has a different opinion about the amount of foundational mathematics to include in a general relativity class that has to fit into one semester. Carroll has successfully steered a middle course, and I believe his text will be very useful to students.
Jim Hartle’s Gravity is a gem that offers a novel approach to general relativity pedagogy. It is written for senior-level undergraduate physics students, but I expect it will be useful for a broader audience. The writing throughout is clear, methodical, and elegant, spiced with the author’s characteristic dry sense of humor. Hartle’s strategy is not to start, in the usual way, with the definition of a manifold and the development of differential geometry. Instead, the bulk of the text uses only calculus and basic differential equations; a streamlined treatment of differential geometry is given at the end of the text. The primary analytical tools developed are how to extract information from metrics and how to study geodesic motion in a given spacetime. Geodesics and Christoffel symbols are introduced using Lagrangian techniques. Given a spacetime metric, the primary analytical tools are then sufficient for extracting an enormous amount of interesting physics.
Key concepts, such as the notion of invariant geometrical quantities that underlie the differential geometric formalism of general relativity, are presented early on. The text begins with the best nuts-and-bolts discussion of gravity as spacetime geometry that I have seen. That discussion enables students to take on the presented calculations with a “full GR attitude.” Hartle’s reversal of the usual ordering of a general relativity syllabus takes some getting used to, but it offers the potential for real success. Instead of struggling through the first half of the course to get to the Einstein equation, many students—undergraduate and graduate—are likely to develop a better grasp of the physics of general relativity, and how to extract it from basic calculations with the spacetime metric, through Hartle’s method.
Hartle’s textbook makes an interesting read, even for more senior relativists. It is packed with calculations of observable general relativistic effects and accompanied by excellent diagrams and descriptions of actual experiments. The author takes calculations one step further than most general relativity texts and gives the reader the physics punch line. One such example concerns unbound orbits in the Schwarzschild geometry. First, readers are led through the routine analysis of particle motion in the relativistic potential. Then sample orbits are displayed as two-dimensional plots. In one picture, a massive particle comes in from infinity, completely loops the central mass (crossing its own path), and heads off to infinity. Such behavior is very non-Newtonian! Another example that comes to mind (perhaps because gyroscopes seemed so confusing in first-year physics) is the clear calculation of the precession of a gyroscope in the field of a slowly rotating body. That example includes a short discussion about NASA’s Gravity Probe B. The chapters on cosmology are excellent and include a survey of current observations, especially useful for those of us who need to update our cosmology notes. The diagrams and plots in these chapters are densely packed and useful.
Carroll and Hartle are to be commended for their care in avoiding gender and ethnicity bias within the text and drawings of their books. I am particularly fond of Hartle’s drawing of the pole vaulter fitting the long pole into the short barn. Both books are fine contributions that extend in useful and different ways the range of pedagogical choices available to instructors.
