Learning general relativity
Warning : this question may be borderline with physics.stackexchange, but I would like a mathematician's point of view.
Lately, I've been working quite a lot with (among others) hyperbolic spaces, action of Lie groups, etc. which required me to buff up my knowledge in the area. I still do not feel comfortable with the subject, but at least I am starting to have some intuition.
So, I thought it could be a nice idea to learn a little about general relativity. The first reason is that I've wanted to learn its basics for quite a time, but have never found the "right" occasion. Now, I can add a second reason : to get more familiar with the underlying mathematical objects.
So, what would would be a nice reference to learn the general theory of relativity? In my case, this would entail:
to be self-contained (i.e. for the physics side, no theoretical requirement beyond classical mechanics and electromagnetism) ;
to contain a high-level mathematical exposition.
Bonus points if the history of the theory and some important experiments are explained. Exercises are nice, too.
Carrol's "Spacetime and Geometry: an Introduction to General Relativity" is a great complement to the mentioned book by Wald in the comments. While some proofs are relegated toward the appendix, it's great to get an overall picture. Plus it's more modern and hence mentions some recent results in cosmology, in particular on the subject of gravitational waves. I would say, turn to Wald for the more rigorous exposition but stick to Carrol as it's much easier to read.
For something at a more leisurely pace, I highly recommend Hartle's book "Gravity" which provides a fantastic amount of down to earth explanations while still being fairly rigorous. I would especially recommend it for its section on explaining how the moon affects the tides.
One of the best books on both graduate differential geometry and general relativity is Barrett O'Neil's Semi-Riemannian Geometry With Applications to Relativity. Not only does it have a completely self-contained graduate course in differential geometry that requires only basic topology and algebra to read, it contains a fairly thorough and self contained course in general relativity using the language of forms and manifolds developed in the previous chapters. It's crystal clear and beautifully written,it's one of my favorite books on any subject. Sadly, it's out of print and very expensive. But it's worth the hassle if you're interested in a modern mathematical presentation of relativity.
I have written a book called A Mathematical Introduction to General Relativity.
It is aimed at advanced undergraduate mathematics students, and is written in the definition-theorem-proof format. The physics background needed is a basic university-level general physics course (so that Newton's laws of motion and the Newtonian gravitational law are familiar). A pdf file of the preface and the first chapter can be found on the publisher's website above, and a google preview can be found at here .