Cosmology Course: General Relativity     

The first lecture sketched all the mathematical notions necessary for the understanding of Einstein's theory of general relativity. Unfortunately, there was not enough time to cover examples or even to present simple computations. John Baez has a very nice tutorial that covers a large fraction of what I talked about. The following problems from the Problem Book (see reference below) are recommended to help with further understanding.

1. Metrics: Problems 6.7, 6.10.
2. Covariant Differentiation and Geodesics: Problems 7.1, 7.2, 7.5, 7.6, 7.7, 7.10, 7.20.
3. Curvature: Problems 9.2, 9.7, 9.8, 9.11, 9.12, 9.13, 9.15, 9.16.
4. Symmetries: Problems 10.2, 10.3, 10.11.
5. Einstein Equations: Problem 13.4
6. Cosmology: Problems 19.1, 19.3, 19.7, 19.10.

The solutions to these problems are fairly short and should provide enough background for later parts of the course. More problems will be suggested as the course proceeds.

The limitations of time imposed by the format of only one lecture a week are such that it is impossible to present the required general relativity in anything but the form of caricature. The references below provide all the information to be covered in the lectures. For those eager to get to cosmological issues directly, the book by Peebles is recommended. Dirac, Landau and Lifshitz, and Weinberg are recommended for those who want to take a deeper, though still non-geometric, path. The geometric approach to relativity, while elegant and powerful, is not required for the vast majority of cosmological applications. Knowledge of ordinary tensor calculus is adequate. Still, those desiring to step beyond these boundaries should sample Schutz, MTW, and Wald.

1. General Theory of Relativity, Paul A.M. Dirac (Princeton, 1996). The most compact introduction to general relativity; very clear and uses only ordinary calculus. Has a nice discussion of the action principle. This slim paperback, though old-fashioned, is probably the single best book for a quick self-study. For applications, one has to look elsewhere.
2. The Classical Theory of Fields, Lev D. Landau and Evgenii M. Lifshitz (Pergamon, 1985). This remarkable text contains an introduction to the general theory in Landau and Lifshitz's inimitable style. For those allergic to modern differential geometry this represents the next step after Dirac. Not very complete, however.
3. Problem Book in Relativity and Gravitation, Alan P. Lightman, William H. Press, Richard H. Price, and Saul A. Teukolsky (Princeton, 1975). A superb collection of problems with solutions. The fastest way to get used to relativity in action.
4. Gravitation, Charles W. Misner, Kip S. Thorne, and John A. Wheeler (Freeman, 1973). A classic reference, the `phone book' contains much not to be found anywhere else. The geometric spirit of general relativity shines throughout despite some idiosyncracies of presentation. It is now showing its age, however, and badly needs an update.
5. Principles of Physical Cosmology, Phillip J.E. Peebles (Princeton, 1993). All the introductory general relativity needed for cosmology is presented here with an excellent sense of connection to observations. The presentation is straightforward and non-geometric.
6. A First Course in General Relativity, Bernard F. Schutz (Cambridge, 1985). Probably the best modern introductory text available, assumes only a physics undergraduate level knowledge of calculus. A good combination is to read this book with the one below as a reference.
7. Geometrical Methods of Mathematical Physics, Bernard F. Schutz (Cambridge, 1980). A concise and accessible introduction to differential geometric concepts, methods, and tools. An almost unique reference.
8. General Relativity, Robert M. Wald (Chicago, 1984). The standard `modern' presentation of the subject, Wald's book is complete and rigorous. Not recommended for the nonspecialist as it tends towards the dry and formal: dip into it only as needed.
9. Gravitation and Cosmology, Steven Weinberg (Wiley, 1972). Presented with the clarity that is the hallmark of Weinberg's writings, this book contains excellent discussions of several cosmological topics. The approach to general relativity is that of an empiricist, not a geometer. Like MTW, unfortunately a bit too long in the tooth.

The URLs given below provide a wealth of information. One has to be careful using the Web as a source of information: it's probably a good idea not to stray too far!

A short history of the early development of general relativity.

A very useful collection of sites: Relativity on the World Wide Web.

Ned Wright's excellent Cosmology FAQ.

Sean Carroll's course notes on general relativity.

Salman Habib / LANL / habib@lanl.gov / revised January 02