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What are some good books, videos, websites for getting started with general relativity? I would prefer mathematically rigorous sources.

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

15 Answers 15

up vote 57 down vote accepted

I can only recommend textbooks because that's what I've used, but here are some suggestions:

  • Gravity: An Introduction To General Relativity by James Hartle is reasonably good as an introduction, although in order to make the content accessible, he does skip over a lot of mathematical detail. For your purposes, you might consider reading the first few chapters just to get the "big picture" if you find other books to be a bit too much at first.
  • A First Course in General Relativity by Bernard Schutz is one that I've heard similar things about, but I haven't read it myself.
  • Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll is one that I've used a bit, and which goes into a slightly higher level of mathematical detail than Hartle. It introduces the basics of differential geometry and uses them to discuss the formulation of tensors, connections, and the metric (and then of course it goes on into the theory itself and applications). It's based on these notes which are available for free.
  • General Relativity by Robert M. Wald is a classic, though I'm a little embarrassed to admit that I haven't read much of it. From what I know, though, there's certainly no shortage of mathematical detail, and it derives/explains certain principles in different ways from other books, so it can either be a good reference on its own (if you're up for the detail) or a good companion to whatever else you're reading. However it was published back in 1984 and thus doesn't cover a lot of recent developments, e.g. the accelerating expansion of the universe, cosmic censorship, various results in semiclassical gravity and numerical relativity, and so on.
  • Gravitation by Charles Misner, Kip Thorne, and John Wheeler, is pretty much the authoritative reference on general relativity (to the extent that one exists). It discusses many aspects and applications of the theory in far more mathematical and logical detail than any other book I've seen. (Consequently, it's very thick.) I would recommend having a copy of this around as a reference to go to about specific topics, when you have questions about the explanations in other books, but it's not the kind of thing you'd sit down and read large chunks of at once. It's also worth noting that this dates back to 1973, so it's out of date in the same ways as Wald's book (and more).
  • Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven Weinberg is another one that I've read a bit of. Honestly I find it a bit hard to follow - just like some of Weinberg's other books, actually - since he gets into such detailed explanations, and it's easy to get bogged down in trying to understand the details and forget about the main point of the argument. Still, this might be another one to go to if you're wondering about the details omitted by other books. This is not as comprehensive as the Misner/Thorne/Wheeler book, though.
  • A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics by Eric Poisson is a bit beyond the purely introductory level, but it does provide practical guidance on doing certain calculations which is missing from a lot of other books.
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Also, Sean Carroll's notes on GR: preposterousuniverse.com/grnotes (which later turned into his book) – nibot Nov 8 '10 at 18:15
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I would vote for Schutz. It is mathematically rigorous enough. – Gordon Feb 2 '11 at 16:45
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I am used to Schutz. It is excellent, but it does no mention to electrodynamics, so it is a good idea to start first with an excerpt from, say, Jackson about SR, and then use Schutz. – Eduardo Guerras Valera Nov 28 '12 at 16:17
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Wald and MTW are extremely out of date at this point. Carroll makes more sense as a modern first graduate-level text in GR, and the fact that it's available in a free version is a nice bonus. – Ben Crowell Sep 18 '13 at 15:43
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@DavidZ: For example, they predate the discovery of cosmological acceleration and the entire modern era of high-precision cosmology. They're 30-40 years out of date on recent theoretical progress in, e.g., numerical relativity, semiclassical gravity, and cosmic censorship. – Ben Crowell Sep 18 '13 at 22:46

I think D'Inverno's "Introducing Einstein's Relativity" is a good text for a rigorous primer in GR.

The following link could be useful for you:

http://www.desy.de/user/projects/Physics/Administrivia/rel_booklist.html

To have fun while reading these books, you can enjoy "The Einstein Theory of Relativity: A trip to the fourth dimension", by Lillian Lieber.

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I recommend you those books from the excellent Chicago Physics Bibliography:

  • Schutz, B., A First Course in Generaly Relativity

    Schutz's book is a really nice introduction to GR, suitable for undergraduates who've had a bit of linear algebra and are willing to spend some time thinking about the math he develops. It's a good book for audodidacts, because the development of the theory is pedagogical and the problems are designed to get you used to the basic techniques. (Come to think of it, Schutz's book is not a bad place to learn about tensor calculus, which is one of the handiest tools in the physics toolkit.) Concludes with a little section on cosmology.

  • Dirac, P.A.M., General Relativity

    You might have heard that Paul Dirac was a man of few words. Read this book to find out how terse he could be. It develops the essentials of Lorentzian geometry and of general relativity, up through black holes, gravitational radiation, and the Lagrangian formulation, in a blinding 69 pages! I think this book grew out of some undergrad lectures Dirac delivered on GR; they are more designed to show what the hell theory is all about than to teach you how to do calculations. I actually didn't like them all that much; they were a little too dry for my taste. It's amusing though, to put Dirac's book next to the book of Misner, Thorne, and Wheeler.

  • D'Inverno, R., Introducing Einstein's Relativity

    I think that D'Inverno is the best of the undergraduate texts on GR (an admittedly small group). It's a tad less elementary than Schutz, and it has a lot more detail and excursions into interesting topics. I seem to remember that it's development of necessary mathematics struck me as somehow lacking, but unfortunately I don't remember what exactly annoyed me. But for physics, I don't think you can beat it. Just be careful: you might find that there's a bit too much here.

  • Misner, C., Thorne, K., & Wheeler, J.A., Gravitation

    Gravitation has a lot of nicknames: MTW, the Phonebook, the Bible, the Big Black Book, etc,... It's over a thousand pages in length, and probably weighs about 10 pounds. It makes a very effective doorstop, but it would be a shame to use it as one. MTW was written in the late 60's/early 70's by three of the best gravitational physicists around--Kip Thorne, Charles Misner, and John Wheeler--and it's a truly great book. I'm not sure I'd recommend it for first time buyers, but after you know a little about the theory, it's about the most detailed, lucid, poetic, humorous, and comprehensive exposition of gravity that you could ask for. Poetic? Humorous? Yep. MTW is laden with stories and quotations. Detailed? Lucid? Oh yes. The theory of general relativity is all laid out in loving detail. You will not find a better explanation of the physics of gravitation anywhere. Comprehensive? Well, sorta. MTW is a little out of date. MTW is good for the basics, but there's actually been quite a bit of work done in GR since it's publication in 1973. See Wald for details.

  • Wald, R., General Relativity

    My favorite book on relativity. Wald's book is elegant, sophisticated, and highly geometric. That's geometric in the sense of modern differential geometry, not in the sense of lots of pictures, however. (If you want pictures, read MTW.) After a concise introduction to the theory of metric connections & curvature on Lorentzian manifolds, Wald develops the theory very quickly. Fortunately, his exposition is very clear and supplemented by good problems. After he's introduced Einstein's equation, he spends some time on the Schwarzchild and Friedman metrics, and then moves on into a collection of interesting advanced topics such as causal structure and quantum field theory in strong gravitational fields.

  • Stewart, J., Advanced General Relativity

    Stewart's book is often for sale at Powell's, which is why I've included it in this list. It's coverage of differential geometry is very modern, and useful if you want some of the flavor of modern geometry. But it's topics are all covered in Wald's book and more clearly to boot.

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I've been trying to teach myself GTR for about the last twelve months. I stopped my formal maths/physics education when I was 18, many years ago.

IMveryveryHO you could do worse than starting with the twelve video lectures by Leonard Susskind of Stanford University. They're on YouTube but there's a general link here http://www.cosmolearning.com/courses/modern-physics-general-relativity/ They really are excellent.

I find all of the textbooks hard going! But I liked Lambourne (Relativity, Gravitation and Cosmology) - about the most accessible of the bunch, I found. I bought Lambourne after spending a lot of time trying to understand Schutz, which is quite rigorous enough for me and a good reference book for my level. He takes you through the maths quite carefully, but it's not easy and big chunks go straight over my head. I liked it enough to buy a copy though.

I also like Foster and Nightingale which is nice and concise and which I got cheap second hand.

I bought D'Inverno second hand but I wish I hadn't bothered. Much too difficult, though I do occasionally look at it.

I tried Relativity Demystified but it didn't.

Carroll has put a complete course of notes online as well. See http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html

You might also want to take a look at A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity by Collier. According to the blurb:

This book is aimed at the enthusiastic general reader who wants to move beyond the maths-lite popularisations in order to tackle the essential mathematics of Einstein's fascinating theories of special and general relativity ... the first chapter provides a crash course in foundation mathematics. The reader is then taken gently by the hand and guided through a wide range of fundamental topics, including Newtonian mechanics; the Lorentz transformations; tensor calculus; the Schwarzschild solution; simple black holes (and what different observers would see if someone was unfortunate enough to fall into one). Also covered are the mysteries of dark energy and the cosmological constant; plus relativistic cosmology, including the Friedmann equations and Friedmann-Robertson-Walker cosmological models.

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+1 for Collier's Most Incomprehensible Thing – Chris Degnen Apr 4 at 11:06

This list is extensive, but not exhaustive. I am aware that there are more standard GR books out there such as Hartle and Schutz, but I don’t think these are worth mentioning. Books with stars are, in my opinion, “must have” books. (I) denotes introductory, (IA) denotes advanced introductory, i.e. the text is self-contained but it would be very helpful to have experience with the subject and (A) denotes advanced.

Special Relativity

  • E. Gourgoulhon (2013), Special Relativity in General Frames. (A) $\star$

This is a rigorous and encyclopedic treatment of special relativity. It contains pretty much everything you will ever need in special relativity, like the Lorentz factor for a rotating, accelerating observer. It is not an introduction, the author does not bother to motivate the Minkowski metric structure at all.

Introductory General Relativity

These books are "introductory" because they assume no knowledge of relativity, special or general. Additionally, they do not require the reader to have any knowledge of topology or geometry.

  • S. Carroll (2004), Spacetime and Geometry. (I) $\star$

A standard first book in GR. There isn't much to say here, it's an excellent, accessible text that gently introduces differential and Riemannian geometry.

  • A. Zee (2013), Einstein Gravity in a Nutshell. (I) $\star$

This is one of the best physics books ever written. This can be comfortably read by anyone who knows $F=ma$, vector calculus and some linear algebra. Zee even completely develops the Lagrangian formalism from scratch. The math is not rigorous, Zee focuses on intuition. If you can't handle a book talking about Riemannian geometry without the tangent bundle, or even charts, this isn't for you. It's rather large, but manages to go from $F=ma$ to Kaluza-Klein and Randall-Sundrum by the end. Zee frequently comments on the history or philosophy of physics, and his comments are always welcome. The only weakness is that the coverage of gravitational waves is simply bad. Other than that, simply fantastic. (Less advanced than Carroll.)

Advanced General Relativity

These books either require previous knowledge of relativity or geometry/topology.

  • Y. Choquet-Bruhat (2009), General Relativity and the Einstein Equations. (A)

A standard reference for the Cauchy problem in GR, written by the mathematician who first proved it is well-posed.

-S.W. Hawking and G.F.R. Ellis (1973), The Large Scale Structure of Space-Time. (A) $\star$

The classic book on spacetime topology and structure. The chapter on geometry is really meant as a reference, not everything is given a proper proof. They present GR axiomatically, this is not the place to learn the basics of the theory. This text greatly expands upon chapters 8 through 12 in Wald, and Wald constantly references this in those chapters. Hence, read after Wald. For mathematicians interested in general relativity, this is a major resource.

  • P. Joshi (2012), Gravitational Collapse and Spacetime Singularities. (A)

A modern discussion of gravitational collapse for physicists. (That is, it's not a hardcore mathematical physics monograph, but also not handwave city.)

  • M. Kriele (1999), Spacetime. (IA)

While technically an introduction, because the reader need not know anything about relativity to read this, it's quite mathematically sophisticated.

  • R. Penrose (1972), Techniques of Differential Topology in Relativity. (A)

This is a proof graveyard. Some of the proofs here are not found anywhere else. If you're willing to skip 70 pages of pure math and take the results on faith, skip this. It overlaps with Hawking & Ellis a lot.

  • E. Poisson (2007), A Relativist’s Toolkit. (A) $\star$

This is really a toolkit, you're assumed to know basic GR coming in, but will leave with an idea of how to do some of the more complicated computations in GR. Includes a very good introduction to the Hamiltonian formalism in GR (ADM).

  • R.K. Sachs and H. Wu (1977), General Relativity for Mathematicians. (A)

This is an extremely rigorous text on GR for mathematicians. If you don't know what "let $M$ be a paracompact Hausdorff manifold" means, this isn't for you. They do not explain geometry (Riemannian or otherwise) or topology for you. Put aside the strange notation and (sometimes stupid) comments on physics vs. mathematics and you have a solid text on the mathematical foundations of GR. It would be most helpful to learn GR from a physicist before reading this.

  • J. Stewart (1991), Advanced General Relativity. (A)

A standard reference for spinor analysis in GR, the Cauchy problem in GR, and Bondi mass.

  • N. Straumann (2013), General Relativity. (IA) $\star$

A mathematically sophisticated text, thought not as much as Sachs & Wu. The coverage of differential geometry is rather encyclopedic, it's hard to learn it for the first time from here. If you're a mathematician looking for a first GR book, this could be it. Besides the overall "mathematical" presentation, notable features are a discussion of the Lovelock theorem, gravitational lensing, compact objects, post-Newtonian methods, Israel's theorem, derivation of the Kerr metric, black hole thermodynamics and a proof of the positive mass theorem.

  • R.M. Wald (1984), General Relativity. (IA) $\star$

The standard graduate level introduction to general relativity. Personally, I'm not a fan of the first four chapters, the reader is much better off reading Wald with a basic understanding of GR and geometry. However, the rest of the text is excellent. If you can only read one text in the "advanced" list, it should be Wald. Some topology would be good, the appendix on it is not very extensive.

General Relativity Reference Texts

These are some canonical reference texts.

  • S. Chandrasekhar (1983), The Mathematical Theory of Black Holes. (A)

Pages and pages of calculations. More pages of calculations. This book has derivations of all black hole solutions, geodesic trajectories, perturbations, and more. Not something you would sit down and read for fun.

  • C.W. Misner, K.S. Thorne, and J.A. Wheeler (1973), Gravitation. (I)

The most cited text in the field. It is absolutely massive and covers so much. Be warned, it's somewhat out-of-date and the notation is generally terrible. The best use for MTW is to look up a result every now and then, there are better books to learn from.

  • H. Stephani, et al. (2009), Exact Solutions of Einstein’s Field Equations. (A)

If an exact solution of the Einstein equations was found before 2009, it is in this book and is likely accompanied by a derivation, a sketch of the derivation and some references.

  • S. Weinberg (1972), Gravitation and Cosmology. (I)

Weinberg takes an interesting philosophical approach to GR in this book, and it's not good for an introduction. It was the standard reference for cosmology in the 70s and 80s, and it's not unheard of to reference Weinberg in 2016.

Riemannian and Pseudo-Riemannian Geometry

Texts focused entirely on the geometry of Riemannian and Pseudo-Riemannian manifolds. These all require knowledge of differential geometry beforehand, save for O'Neil.

  • J.K. Beem, P.E. Ehrlich, and K.L. Easley (1996), Global Lorentzian Geometry. (A)

A very advanced text on the mathematics of Lorentzian geometry. The reader is assumed to be familiar with Riemannian geometry. Hawking & Ellis, Penrose and O'Neil are crucial, this book builds on the material in those texts (and the authors tend to not repeat proofs that can be found in those three). The sprit of the book is to see how many results from Riemannian geometry have Lorentzian analogues. The actual applications to physics are speculative.

  • J. Cheeger and D.G. Ebin (1975), Comparison Theorems in Riemannian Geometry. (A)

An advanced text on Riemannian geometry, the authors explore the connection between Riemannian geometry and (algebraic) topology. Many of the concepts and proofs here are used again in Beem and Ehrlich.

  • M.P. do Carmo (1992), Riemannian Geometry. (I) $\star$

A terrific introduction to Riemannian geometry. The presentation is leisurely, it's a joy to read. Notable topics covered are global theorems like the sphere theorem.

  • J.M. Lee (1997), Introduction to Riemannian Manifolds. (I)

A standard introduction to Riemannian geometry. When I don't understand a proof in do Carmo or Jost, I look here. It covers somewhat less material than do Carmo, though they are similar in spirit.

  • J. Jost (2011), Riemannian Geometry and Geometric Analysis. (IA)

An advanced "introduction" to Riemannian geometry that covers PDE methods (for instance, the existence of geodesics on compact manifolds is proved using the heat equation), Hodge theory, vector bundles and connections, Kähler manifolds, spin bundles, Morse theory, Floer homology, and more.

  • P. Petersen (2016), Riemannian Geometry. (IA)

A standard high-level introduction to Riemannian geometry. The inclusion of topics like holonomy and analytic aspects of the theory is appreciated.

  • B. O’Neil (1983), Semi-Riemannian Geometry with Applications to Relativity. (I) $\star$

A somewhat standard introduction to Riemannian and pseudo-Riemannian geometry. Covers a surprising amount of material and is quite accessible. The sections on warped products and causality are very good. Since large parts of the book do not fix the signature of the metric, one can reliably lift many results from O'Neil into GR.

Topology

Texts that will elucidate the topological aspects of GR and geometry.

  • G.E. Bredon (1993), Topology and Geometry. (IA) $\star$

A good introduction to general topology and differential topology if you have a strong analysis background. Most, if not all, theorems of general topology used in GR are contained here. Most of the book is actually algebraic topology, which is not so useful in GR.

  • V. Guillemin and A. Pollack (1974), Differential Topology. (I)

A standard introduction to differential topology. Some results useful for GR include the Poincare-Hopf theorem and the Jordan-Brouwer theorem.

  • J. Milnor (1963), Morse Theory.

The classical introduction to Morse theory, which is used explicitly in Beem, Ehrlich & Easley and implicitly and Hawking & Ellis and others.

  • N.E. Steenrod (1951), The Topology of Fiber Bundles.

Most advanced GR books contain the following: "The manifold $M$ admits a Lorentzian metric if and only if (a) $M$ is noncompact, (b) $M$ is compact and $\chi (M)=0$. See Steenrod (1951) for details." This book contains the most fundamental topological theorem of GR, that, to my knowledge, is not proved anywhere else.

Differential Geometry

Texts on general differential geometry.

  • S. Kobayashi and K. Nomizu (1963), Foundations of Differential Geometry (Vol. 1, 2). (A)

This is the standard reference for connections on principal and vector bundles.

  • J.M. Lee (2009), Manifolds and Differential Geometry. (IA)

A somewhat advanced introduction to differential geometry. Connections in vector bundles are explored in depth. Some advanced topics, like the Cartan-Maurer form and sheaves, are touched upon. Chapter 13, on pseudo-Riemannian geometry, is quite extensive.

  • J.M. Lee (2013), Introduction to Smooth Manifolds. (I) $\star$

A very well-written introduction to general differential geometry that doubles as an encyclopedia for the subject. Most things you need from basic geometry are contained here. Note that connections are not discussed at all.

  • R.W. Sharpe (1997), Differential Geometry. (A)

An advanced text on the geometry of connections and Cartan geometries. It provides an alternative viewpoint of Riemannian geometry as the unique (modulo an overall constant scale) torsion-free Cartan geometry modeled on Euclidean space.

  • G. Walschap (2004), Metric Structures in Differential Geometry. (IA)

A very rapid (and difficult) introduction to differential geometry that stresses fiber bundles. Includes an introduction to Riemannian geometry and a lengthy discussion of Chern-Weil theory.

Misc.

  • S. Abbot (2015), Understanding Analysis. (I)

A gentle introduction to real analysis in a single variable. This is a good text to "get your feet wet" before jumping into advanced texts like Jost's Postmodern Analysis or Bredon's Topology and Geometry.

  • V.I. Arnold (1989), Mathematical Methods of Classical Mechanics. (IA) $\star$

Look here for an intuitive yet rigorous (the author is Russian) explanation of Lagrangian and Hamiltonian mechanics and differential geometry.

  • K. Cahill (2013), Physical Mathematics. (I)

This book starts from the basics of linear algebra, and manages to cover a lot of basic math used in physics from a physicist's point of view. A handy reference.

  • L.C. Evans (2010), Partial Differential Equations.

The standard graduate level introduction to partial differential equations.

  • J. Jost (2005), Postmodern Analysis. (A)

An advanced analysis text which goes from single-variable calculus to Lebesgue integration, $L^p$ spaces and Sobolev spaces. Contains proofs of theorems such as Picard-Lindelöf, implicit/inverse function and Sobolev embedding, which are ubiquitous in geometry and geometric analysis.

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Small comment: G&P is not really a standard introduction to topology, IMO. Note that it does not have any of the basic definitions etc. that e.g. Munkres (Topology) has. It's more of an exposition of the authors' view of diff.top. with unusual focus on the notion of transversality (and the authors say so in the introduction/preface). However, one can of course argue that diff.top. does not really have any alternative standard textbooks that deal with the smooth setting only. – Danu Apr 16 at 13:51
    
@Danu I said it's a standard introduction to diff.top, not topology in general. "The" standard intro would probably be Hirsch. – 0celo7 Apr 16 at 14:54

To me there a two sides to understanding GR. For the conceptual side you cannot do any better than getting it straight from the horses mouth (i.e. Einstein):

http://www.bartleby.com/173/

The other side of the coin is the mathematical apparatus. I got a lot of mileage out of this introduction to tensor calculus for GR:

http://web.mit.edu/edbert/GR/gr1.pdf

Really focuses on the bare-bones of the math while not omitting the coordinate free treatment. Only prerequisites are calculus and linear algebra.

Then as an additional reference I find L. D. Landau's text book on theoretical physics Vol 2 very useful.

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A second recommendation for the A zee book. I'd say GRAVITATION is the goal,but I'd get there by:

"Exploring Blackholes" by Wheeler, nice intro, stops at Schwartzchild.

then the soft introduction provided by piccioni, which exists in many places (amazon, nook, oyster) but not in print, oddly. "General Relativity" 1-3. The other books in the series might be worth your time also.

"Einstein Gravity in a nutshell" A. Zee. Zee's stuff is always accessible and insightful, this a wonderful way to get GR into your head, along with some glorious connections to fundamental physics. If you were going to go with a single book, I'd do this one.

From here, maybe, possibly, you can start and finish the glory that is GRAVITATION. I'm terrible at math (for a physicist) so I may have taken a few more books to get my tensors in a row before I could hit the big book.

While we're here, "A general relativity workbook" is an excellent resource.

See also: http://www.desy.de/user/projects/Physics/Administrivia/rel_booklist.html http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html#intro_gr

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It all depends on your background. The recent translation to English of Grøn/Næss Norwegian GR book is a very easy and plesant read:

Einstein's Theory: A Rigorous Introduction for the Mathematically Untrained

Still, it is rigorous (it even says so in the title!). They don't go very far, but do touch upon some solutions (e.g. Schwarzschild) and cosmology.

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While I myself like Ludvigsen "General Relativity: A geometric approach", it probably doesn't fall into the category of being "mathematicallly rigorous". If you really want Mathematical rigour, then Wald "General Relativity"

enter image description here

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This has already been mentioned. – MBN Jul 28 '13 at 18:01
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@MBN: Ooops, didn't notice . – centralcharge Jul 29 '13 at 1:46

One key title appears missing from the answers provided so far: Einstein Gravity in a Nutshell by Tony Zee. This new book (published 2013) provides a mathematically rigorous treatment, yet is colloquial in tone and very accessible. I own Wald, Schutz, and Hartle, but Zee's book has quickly developed into my favorite text on General Relativity.

Those who have read Zee's Quantum Field Theory in a Nutshell know what to expect. The two 'Nutshell titles' combined give an amazingly accessible and complete introductory overview of modern physics.

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I learned my GR from Landau and Lifshitz Classical Theory of Fields, 2nd edition. Even at 402 (4th Edition) pages it is kind of breathless.

The interesting thing about it is the first half is special relativity and electrodynamics which dovetails into the 2nd half which is GR. One has to persivere because it's terse but not too terse. Like Weinberg it has a more 'physics feel' to than a 'math' one. It is just the basics but done with rigor. Alas , as far as I know,there has been no update since 1974, not sure why. An amusing take on GR is Zel'dovich, Ya. B. and Novikov, I. D. Relativistic Astrophysics, Vol. 1: Stars and Relativity.

With a lot of quirky side streets still not treated in other books , alas also not updated since 1971... tho Frolov and Novikov's 1998 Black Hole Physics: Basic Concepts and New Developments is kind of a sequel with more GR off shoots.

Russian books that seem to be just about Black Holes usually have a good introduction to GR, and are kind of quirky to my amusement with their diversions!

If you want real 'brain burn Chandrasekhar's The Mathematical Theory of Black Holes is totally comprehensive, if exhausting ,another book like MTW for one's shelf as a reference.

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This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that do not yet have commentary.

  • Lillian Lieber: Einstein's Theory of Relativity.

  • You can't beat a bit of Hobson.

  • The lecture notes by Geroch. Including notes on general relativity.
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I would suggest it really is worth reading Misner, Thorne, and Wheeler (MTW). Its the only textbook I have managed to find which really explains things so I can understand each line and also covers the main advanced aspects of the theory. I would also definitely suggest you should have read a good book on special relativity before tackling MTW.

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Adding two more in the list...

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Hi raj. Could you add more explaining why you recommend these books? See "How should I answer a resource recommendation question?" in our policy linked above. – Chris White Apr 16 at 18:11
    
These are 'mathematically rigorous' with lots of exercises and projects with hints to many of them. In my opinion these may be a good start for GR and its applications. – raj Apr 17 at 19:54

To get a first idea of what GR is all about, with loads of solved exercises, try General Relativity Without Calculus.

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13  
The OP asked for "Mathematically rigorous" references; I expect this falls a bit short. – AdamRedwine Sep 7 '11 at 20:56
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"Without Calculus"? Seriously? . – centralcharge Jul 28 '13 at 7:55
    
Although the OP asked for a mathematically rigorous answer, having also a math-less answer will be useful for popularize science. Thus, I voted it up to try to save from a possible deletion. – peterh Apr 10 at 19:15

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