Robert Scott's notes for self-study in university undergraduate and postgraduate level math and physics:

Also see:

Einstein's geometric theory of gravitation -- General Relativity

Recommendations: If you are a physics undergraduate or have a technical degree like engineering, so you are comfortable with basic differential calculus and linear algebra, start with Schutz (2009) or perhaps Hobson et al. (2006). If you are a graduate student in physics, you might start with Schutz (2009), Hobson et al. (2006) or Carroll (2004). A maths postgrade might start with Wald(1984). To deepen your knowledge and understanding of GR at any level, Rindler(2001) is quite valuable. Misner, Thorne and Wheeler (1973) of course provides a wealth of supplementary information for the advanced student of GR.

Schutz (2009) "A First Course in General Relativity", This is a wonderful introductory book on Einstein's general theory of relativity. I wrote a student's study guide based on Schutz's book, submitted to CUP October 2014. You'll find complete detailed answers to about 1/2 of the problems. I include new exercises (most of them are easier) with answers to the odd ones. (Answers to almost all even ones are provided to instructors only.) The book will also come with Maple worksheets. For those looking for my "Notes on Schutz" (sorry!) I have temporarily removed them.
Are you also writing an undergraduate physics textbook? I am happy to read your draft manuscript and check for errors, unclear points, etc., in exchange for offering feedback on my book. Last updated June, 2013.

General Relativity: An introduction for physicists, by Hobson, Efstathiou, Lasenby (2009) Another introductory book but perhaps at a slightly more advanced level than the book by Schutz (2009). I found this book easier to follow in some cases than Schutz's book, perhaps because they go into more detail for subjects like Riemann manifolds. But they spend less time developing the idea of a tensor and place more demands on the reader in some places. My course notes follow Hobson et al. rather closely.
Gravitation, by Misner, Thorne, and Wheeler (1973). This is the classic reference on gravity and Einstein's general theory of relativity. This book is not for beginners -- you should know Special Relativity presented from the formalism of 4-vectors. If you're in doubt, have a look at the 2nd paragraph of section 2.1 of Chapter 2. If you cannot claim to be familar with everything they mention, then I recommend either of the two books above. I found the mathematics was not explained clearly. I recommend Schutz (2009) or Hobson et al.(2006) for beginners and those not familar with tensors and the distinction between covariant and contravariant vectors (or one-forms and vectors). I'm currently working on a solution manual in the same style as the one I wrote for Schutz(2009). Updated Dec 25, 2014.

Notes on General Relativity, by Robert Wald (1984). This book is not for beginners. Schutz (2009) places Wald's book in the most advanced catagory that ``make heavy demands of the student". This book aims to be mathematically rigorous. For instance he introduces manifolds in Chapter 2 and proves that the tangent vector space has the same dimension as the manifold. I find the book much less readable than the references above. On the other hand when I read the GR literature, I most often turn to Wald's book to make sense of it. Here's a little test to see if this book is for you. Compare the definition of stationary space-times given by Wald and those of introductory books (like Schutz and Hobson et al.). While Wald states: "A spacetime is said to be stationary if there exists a one-parameter group of isometries, phi_t, whose orbits are timelike curves." Schutz and other introductory books state something like: For a stationary spacetime the time derivative of all components of the metric tensor are zero. If you find these two statements equally enlightening perhaps you can skip the introductory books and jump into the deep-end with Wald. A good background in differential geometry and analysis, and some knowledge of topological spaces would be very helpful. Updated October 25, 2012.

Notes on Relativity and Differential Geometry, by Richard Faber (1983). See description below under mathematics books.

SPACETIME AND GEOMETRY: An introduction to General Relativity, by Sean Carroll (2004) This is a graduate student level introduction to general relativity written in a clear and folksy style. The pace is more rapid then Schutz(2009) or Hobson et al.(2006); the section on tensor calculus is surprisingly brief. But the benefit is that he covers more advanced topics, e.g. a very nice section on classical field theory that forms the basis of so much of modern physics. And Carroll has a real gift for presentation, as is clear from his popular level documentary film (in the style of video taped lectures) on dark energy and dark matter for the Teaching Company. For more on Carroll and his publications, see Carroll's homepage .

Relativity: Special, General and Cosmological, by Wolfgang Rindler (2001) This is an introduction for math and physics students at a similar level to Schutz(2009) and Hobson et al. (2006). Unlike the latter two references who start with a single chapter on Special Relativity, Rindler devotes 6 chapters to special relativity, not starting GR until page 165. In the spirit of Einstein himself, there is a strong emphasis on connecting mathematical ideas with physical experiments. For example the Cartesian coordinates are introduced with the thought experiment (due to Einstein 1905) of synchronizing the clocks by passing light signals between them. And there are some gems of differential geometry not normally covered in introductory GR texts. This makes for great supplementary reading for the student of GR at any level.

Undergraduate Pure and Applied Mathematics

Notes on "Mathematical methods in the physical sciences" by Boas(1983)This is a wonderful book on basic and useful applied mathematics. You'll find my notes on Chapter 1 on infinite series, and Chapter 9 on Calculus of Variations, including answers to most of the problems.

Notes on Relativity and Differential Geometry, by Richard Faber (1983). This is an introductory reference on Einstein's general theory of relativity at a similar level to Schutz (2009) or Hobson et al. (2006) but written by a mathematician. The mathematics is explained very clearly. One needs only basic knowledge of vector calculus. Some of the exercises are very challenging, while most are quite doable. This is not the best book from which to learn GR. The most serious limitation is that Faber avoids defining tensors! I soon got tired of typing out all solutions. (Apparently a solution manual exists but it's out of print and difficult to get hold of.) Rather than type out solutions for every one, in my notes you will find my complete solutions to some of the more interesting exercises. Email me if you'd like to see a particular solution that I haven't presented. I've completed Part I (on differential geometry) but have no immediate plans to do Parts II or III. Last update May, 2012. You'll also find my extended Index of Symbols and Terms and a List of Definitions, Theorems, and Lemmas.

A first course in Real Analysis, by Protter and Morrey (1994) If you are a physicist that's frustrated that you cannot read mathematics books, this book may help. It's a first book on rigorous (pure) mathematics if you like. It assumes very little background and introduces the foundations of analysis. It's a prerequisit for, for example: functional analysis, dynamical systems, measure theory, and much more.
Mathematical Physics: A modern introduction to its foundations, by Sadri Hassani(1999) If you are a physicist, the more math you know the better. But most of the mathematics literature is written for mathematicians and to understand it you have to start at the very beginning (e.g. proving that a x 0 = 0 for all a in the reals, theorem 1.3 of Protter and Morrey (1994) ). This can be frustratingly slow and perhaps you'd like to jump ahead to the good stuff. But be careful because if you pickup almost any postgraduate level mathematics book you'll have a lot of trouble if you haven't put in the hours learning the basics. Hassani's book provides an extremely useful alternative. Here in one single (rather large) textbook you'll find an introduction to much of the mathematics useful to physicists: operator theory, Hilbert spaces, manifolds and differential geometry, Lie algebras, and on and on. And it's accessible to physics undergraduates! What really sold me on Hassani's book was comparing the material covered in this single book to the course context of an undergraduate mathematics degree programme in the UK; if you slug your way through all 1025 pages of this book you'll have a solid introduction to most of the course material of a mathematics degree. This should be contrasted to many "math for physics students" books (e.g. Boas(1983)), which focus on learning a bunch of techniques for solving problems, but do almost nothing to bridge the communication barrier between mathematicians and physcists. In short, highly recommended for the physicist!
Geometrical methods of mathematical physics, by Bernard Schutz (1980) This book has the same primary goal and benefit of Hassani(1999): it bridges the gap between physics and mathematicans. It is also very well written and assessible to the advanced physics undergraduate. Its scope is much more limited and consequently, at 250pp, it's much easier to fit in your backpack. However the mathematical scope is still broader than Schutz(2009) on GR, and makes for useful reading to the GR student concurrently with or after Schutz(2009). Highly recommended.