Introduction to General Relativity, Black Holes, and Cosmology , Yvonne Choquet-Bruhat, Oxford U. Press, 2015. $99.95 (279 pp.). ISBN 978-0-19-966645-4 Buy at Amazon
Yvonne Choquet-Bruhat is a giant of mathematical general relativity. In 1952 she proved a groundbreaking theorem on the existence of solutions to the Einstein field equations. Since then she has obtained many results in general relativity, partial differential equations, classical field theory, and fluid dynamics. Now in her nineties, she shows no signs of slowing down, and even finds time to write textbooks.
Her latest, Introduction to General Relativity, Black Holes, and Cosmology, consists of two parts: “Fundamentals” and “Advanced Topics.” Unlike other areas of physics, general relativity needs the mathematics of differential geometry, so Choquet-Bruhat starts with that subject and explains manifolds, tensor fields, metrics, connections, and curvature. After introducing those tools, she moves on to the flat metric of special relativity, the curved metrics of general relativity, and the Einstein field equations, which relate the curvature of spacetime to the energy density of matter.
The main applications of general relativity are to the physics of black holes and to cosmology, and the author next turns to those topics. Nonrotating black holes—indeed, the exteriors of all nonrotating spherical objects—are described by the Schwarzschild metric. Using that metric, Choquet-Bruhat presents calculations of some of the effects used to test general relativity: the gravitational bending of light, the precession of the perihelion of Mercury, and the redshift of light in a gravitational field. She also discusses rotating black holes, which are described by the Kerr metric. In the last chapter in the “Fundamentals” section, she covers cosmology; she introduces the symmetries of cosmological metrics and shows how the Einstein field equations yield equations relating the expansion of the universe to the types of matter and energy that it contains.
The “Advanced Topics” section is primarily focused on the Cauchy problem for the Einstein field equations, both without matter and with various types of matter. For a given set of equations of motion, the Cauchy problem is the task of proving that, given a choice of initial data, a unique solution of the equations exists and that the solution depends continuously on the data. In the process of solving the Cauchy problem, one finds the true degrees of freedom of the theory by finding what initial data one is allowed to specify. For example, Maxwell’s equations without charges or currents include equations of motion for the electric and magnetic fields. However, the electric and magnetic fields cannot be chosen arbitrarily; they must have zero divergence.
In general relativity, the initial data are the metric of space and a geometric quantity called the extrinsic curvature—essentially, the derivative with respect to time of the space metric. Those fields also cannot be chosen arbitrarily; rather, they should satisfy a set of nonlinear constraint equations. Any sensible physical theory must have equations of motion for which the task of the Cauchy problem is successfully completed, so one can regard the mathematicians’ work in that area as checking the physical sensibleness of each proposed theory. That work is more important than ever these days, when new theories are frequently proposed and specified with no more than an expression for the theory’s Lagrangian.
The word “introduction” in the title of the book suggests that it would be an ideal first book in general relativity—and so it is, if the reader is a mathematician. For such a reader, the book provides an elementary introduction to the physics of general relativity and a beautiful exposition of how that physics relates to the differential geometry of spacetime and to the hyperbolic nonlinear partial differential equations that describe the evolution of the gravitational field. However, I would not recommend it as a first book in general relativity for a physicist.
Most physics students first learning general relativity are somewhat put off by the amount of new mathematics that needs to be learned, and the problem is not likely to be alleviated when the material is presented from the point of view of mathematics—even when, as in the present case, the author has a deep understanding of both physics and mathematics. Nonetheless, for a reader who already knows some general relativity, Choquet-Bruhat’s book is an ideal introduction to the mathematical approach. Nowadays, research in general relativity is done by members in both physics departments and mathematics departments. Without a bridge between those two groups, such as the one provided by this book, there is a danger that they will lack a common language and will end up talking past each other.
The French have a saying, “L’appétit vient en mangeant”; roughly translated, it means, “Eating a little makes you hungry for more.” Introduction to General Relativity, Black Holes, and Cosmology necessarily has an abbreviated treatment of some topics, often with the statement that the topic is “outside the scope of this book.” In particular, the reader might be left hungry for more details on the astrophysics of black holes, the history of the early universe and its relation to particle physics, the ongoing search for gravitational waves, and the efforts to find a quantum theory of gravity. On the mathematical side, the reader might want to know more about singularity theorems, global existence and stability theorems, and black hole uniqueness theorems. However, the author has wisely chosen to write a short book that gives a taste of the subject. Bon appétit.
David Garfinkle is a professor of physics at Oakland University in Rochester, Michigan, and a visiting research scientist at the University of Michigan in Ann Arbor. His research is in general relativity.
