GR benefits from a large number of possible books. I strongly recommend that you use multiple books throughout the course and therefore I do not insist on any one of them. The two which I think are best suited to the course, and that you might want to consider buying are:
- Hobson, M.P., Efstathiou, G., Lasenby, A.N., General Relativity: An Introduction for Physicists, CUP, 2006. A fairly complete book which contains more than this course can cover, but at a reasonable level for the most part. The approach is quite mathematical in its approach which may suit some students well. I strongly recommend you look at both this and the next book.
- Schutz, B.F., A first course in general relativity. This is a good, fairly short and readable book. This is the one which the course follows most closely in structure. However, do not think that this will be all you need; no single book can be. NB In 2009 a second edition of Schutz's book appeared; try to get this rather than the first edition.
Other books worth perusing are:
- Hartle, Gravity: An Introduction to Einstein's General Relativity. An accessible book, but the physics-before-maths approach is not quite to my taste, although it may suit some, so try to give it a look.
- Zee, A., Einstein Gravity in a Nutshell. Another recent book taking the physics-first approach. Readable, and with some interesting diversions into theoretical areas this course doesn't cover.
- Rindler, W., Relativity: special, general and cosmological, 2nd edition. (NB not 'Essential Relativity' which has much less on GR.) This book is mostly focussed on special relativity, but it is certainly worth a look.
- Foster, J., Nightingale, J.D., A short course in general relativity. Quite a good book for a quick introduction to GR, although it is a bit too short to satisfy fully. Introductory chapter is worth a read.
- Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. I have always found the notation used in this book a little confusing, and some of it is much too advanced, but I list it as a good example of the coordinate transform approach to tensors favoured especially in older books, and it is a classic of the field.
- Misner, Thorne and Wheeler, Gravitation. Another classic. Too advanced to recommend and the style of presentation (as well as its weight) takes some getting used to, but worth a skim late on in the course if only to improve the muscle tone of your arms.
- Wald, General Relativity is often recommended although it is rather advanced. I list it here for completeness.
GR can be confusing and it can help a great deal to see how different authors cover the same material, so at the risk of being a bore please look at more than one book. Something to be aware of from the start is that there is no consensus on the signs for various quantities and that notation can vary as well. This is usually a minor issue, but be aware of it when you read the books. I will use a metric consistent with that of Hobson et al. and Rindler, but different from Schutz.
The other difference you will find at the start of books is the degree of emphasis on the index notation versus "frame invariant" approaches, similar to the difference between vector components and symbols for vectors without components. This is a discussion that has taken place for many years in contexts other than GR. Think how much simpler EM appears in vectorial rather than component form (4 Maxwell's equations rather than 10), and yet if you want to solve them in particular contexts, there is no avoiding the components. In GR, the index notation has always tended to win out, although some have emphasized the frame invariant approach, most notably Misner, Thorne and Wheeler. In the end you cannot avoid the index notation, but the frame invariant approach can be illuminating. Schutz emphasizes the geometrical, frame-invariant approach more than Hobson. It is worth reading Schutz and at least one other to see this difference in emphasis.