Mathematically-oriented Treatment of General Relativity

Question

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 1

The Physics work in this field is rigorous enough. Hawking and Ellis is a standard reference, and it is perfectly fine in terms of rigor.

Digression on notation

If you have a tensor contraction of some sort of moderate complexity, for example:

$$ K_{rq} = F_{ij}^{kj} G_{prs}^i H^{sp}_{kq}$$

and you try to express it in an index-free notation, usually that means that you make some parenthesized expression which makes

$$ K = G(F,H)$$

Or maybe

$$ K = F(G,H) $$

Or something else. It is very easy to prove (rigorously) that there is no parentheses notation which reproduces tensor index contractions, because parentheses are parsed by a stack-language (context free grammar in Chomsky's classification) while indices cannot be parsed this way, because they include general graphs. The parentheses generate parse trees, and you always have exponentially many maximal trees inside any graph, so there is exponential redundancy in the notation.

This means that any attempt at an index free notation which uses parentheses, like mathematicians do, is bound to fail miserably: it will have exponentially many different expressions for the same tensor expression. In the mathematics literature, you often see tensor spaces defined in terms of maps, with many "natural isomorphisms" between different classes of maps. This reflects the awful match between functional notation and index notation.

Diagrammatic Formalisms fix Exponential Growth

Because the parenthesized notation fails for tensors, and index contraction matches objects in pairs, there are many useful diagrammatic formalisms for tensorial objects. Diagrams represent contractions in a way that does not require a name for each index, because the diagram lines match up sockets to plugs with a line, without using a name.

For the Lorentz group and general relativity, Penrose introduced a diagrammatic index notation which is very useful. For the high spin representations of SU(2), and their Clebsch-Gordon and Wigner 6-j symbols, Penrose type diagrams are absolutely essential. Much of the recent literature on quantum groups and Jones polynomial, for example, is entirely dependent on Penrose notation for SU(2) indices, and sometimes SU(3).

Feynman diagrams are the most famous diagrammatic formalism, and these are also useful because the contraction structure of indices/propagators in a quantum field theory expression leads to exponential growth and non-obvious symmetries. Feynman diagrams took over from Schwinger style algebraic expressions because the algebraic expressions have the same exponential redundancy compared to the diagrams.

Within the field of theoretical biology, the same problem of exponential notation blow-up occurs. Protein interaction diagrams are exponentially redundant in Petri-net notation, or in terms of algebraic expressions. The diagrammatic notations introduced there solve the problem completely, and give a good match between the diagrammatic expression and the protein function in a model.

Within the field of semantics within philosophy (if there is anything left of it), the ideas of Frege also lead to an exponential growth of the same type. Frege considered a sentence as a composition of subject and predicate, and considered the predicate a function from the subject to meaning. The function is defined by attaching the predicate to the subject. So that "John is running" is thought of as the function "Is running"("John").

Then an adverb is a function from predicates to predicates, so "John is running quickly" means ("quickly"("Is running"))("John"), where the quickly acts on "is running" to make a new predicate, and this is applied to "John".

But now, what about adverb modifiers, like "very", as in "John is running very quickly"? You can represent these are functions from adverbs to adverbs, or as functions from predicates to predicates, depending on how you parenthesize:

(("very"("quickly"))("Is running"))("John")

vs.

(("very")(("quickly")("Is running"))("John")

Which of these two parenthetization is correct define two schools of semantic philosophy. There is endless debate on the proper Fregian representation of different parts of speech. The resolution, as always, is to identify the proper diagrammatic form, which removes the exponential ambiguity of parenthesized functional representation. The fact that philosophers have not done this in 100 years of this type of debate on Fregian semantics shows that the field is not healthy.

Answer 2

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.

Answer 3

I recommend Yvonne Chocquet-Bruhat, Géométrie différentielle et systèmes extérieurs because it is so short and has exercises and is in the notation you want. I recommend it most highly (even if you can't get it with her inscribed autograph.)

I also recommend very strongly her much longer (but try to get the first edition, which is still plenty long) Analysis, Manifolds and Physics by Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, and Margaret Dillard-Bleick which has many exercises and much more Physics...but is too long. Goodness, it even includes Brownian Motion and path integrals....

That said, Dirac and Schroedinger have good and very short Physics books on the subject, I recommend those too even though they are not quite what you asked for.

Bob Geroch's book is valuable too.

Answer 4

The other answers already mention many good textbooks about mathematical aspects of general relativity. However, two books still seem to be missing, which in my opinion are also quite good on this subject. Let me add them here:

• M. Kriele: Spacetime. Foundations of General Relativity and Differential Geometry. volume 59 of Lecture Notes in Physics Monographs. Springer, Berlin, Heidelberg, 1999.
• R. Sachs and H.-H. Wu: General Relativity for Mathematicians. volume 48 of Graduate Texts in Mathematics. Springer, New York, 1977.

Answer 5

Maybe this one: A Mathematical Introduction to General Relativity

The prerequisites are minimal: A second/third year undergraduate student in Mathematics with no prior knowledge of differential geometry or Physics beyond a standard first year college course on General Physics should be able to follow this book. There are complete solutions to all the exercises included in the book.