mathematical physics - Mathematically-oriented Treatment of General Relativity

Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would

1. 
   

   Prove all theorems used.

   
   


2. 
   

   Use modern "mathematical notation" as opposed to "physics notation", especially with respect to linear algebra and differential geometry.

   
   


3. 
   

   Have examples that illustrate both computational and theoretical aspects.

   
   


4. 
   
   

   Have a range of exercises with varying degrees of difficulty, with answers.

   
   

An ideal text would read a lot more like a math book than a physics book and would demand few prerequisites in physics. Bottom line is that I would like a book that provides an axiomatic development of general relativity clearly and with mathematical precision works out the details of the theory.

Addendum (1): I did not intend to start a war over notation. As I said in one of the comments below, I think indicial notation together with the summation convention is very useful. The coordinate-free approach has its uses as well and I see no reason why the two can't peacefully coexist. What I meant by "mathematics notation" vs. "physics notation" is the following: Consider, as an example, one of the leading texts on smooth manifolds, John Lee's Introduction to Smooth Manifolds. I am very accustomed to this notation and it very similar to the notation used by Tu's Introduction to Manifolds, for instance, and other popular texts on differential geometry. On the other hand, take Frankel's Geometry of Physics. Now, this is a nice book but it is very difficult for me to follow it because 1) Lack of proofs and 2)the notation does not agree with other math texts that I'm accustomed to. Of course, there are commonalities but enough is different that I find it really annoying to try to translate between the two...

Addendum (2): For the benefit of future readers, In addition to suggestions below, I have found another text that also closely-aligns with the criteria I stated above. It is, Spacetime: Foundations of General Relativity and Differential Geometry by Marcus Kriele. The author begins by discussing affine geometry, analysis on manifolds, multilinear algebra and other underpinnings and leads into general relativity at roughly the midpoint of the text. The notation is also fairly consistent with the books on differential geometry I mentioned above.

Answer

I agree with Ron Maimon that Large scale structure of space-time by Hawking and Ellis is actually fairly rigorous mathematically already. If you insist on somehow supplementing that:

• For the purely differential/pseudo-Riemannian geometric aspects, I recommend Semi-Riemannian geometry by B. O'Neill.



• For the analytic aspects, especially the initial value problem in general relativity, you can also consult The Cauchy problem in general relativity by Hans Ringström.


• For a focus on singularities, I've heard some good things about Analysis of space-time singularities by C.J.S. Clarke, but I have not yet read that book in much detail myself.


• For issues involved in the no-hair theorem, Markus Heusler's Black hole uniqueness theorems is fairly comprehensive and self-contained.


• One other option is to look at Mme. Choquet-Bruhat's General relativity and Einstein's equations. The book is not really suitable as a textbook to learn from. But as a supplementary source book it is quite good.

If you are interested in learning about the mathematical tools used in modern classical GR and less on the actual theorems, the first dozen or so chapters of Exact solutions of Einstein's field equations (by Stephani et al) does a pretty good job.