What are some good books, videos, websites for getting started with general relativity? I would prefer mathematically rigorous sources.
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I can only recommend textbooks because that's what I've used, but here are some suggestions:
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.
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.
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.
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.
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.
A modern discussion of gravitational collapse for physicists. (That is, it's not a hardcore mathematical physics monograph, but also not handwave city.)
While technically an introduction, because the reader need not know anything about relativity to read this, it's quite mathematically sophisticated.
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.
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).
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.
A standard reference for spinor analysis in GR, the Cauchy problem in GR, and Bondi mass.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
A standard high-level introduction to Riemannian geometry. The inclusion of topics like holonomy and analytic aspects of the theory is appreciated.
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.
Texts that will elucidate the topological aspects of GR and geometry.
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.
A standard introduction to differential topology. Some results useful for GR include the Poincare-Hopf theorem and the Jordan-Brouwer theorem.
The classical introduction to Morse theory, which is used explicitly in Beem, Ehrlich & Easley and Cheeger & Ebin and implicitly and Hawking & Ellis and others.
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.
Texts on general differential geometry.
This is the standard reference for connections on principal and vector bundles.
The first three chapters of this text cover manifolds, lie groups, forms, bundles and connections in great detail, with very few proofs omitted. The rest of the book is on functorial differential geometry, and is seriously advanced. That material is not needed for GR.
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.
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.
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.
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.
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.
Look here for an intuitive yet rigorous (the author is Russian) explanation of Lagrangian and Hamiltonian mechanics and differential geometry.
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.
The standard graduate level introduction to partial differential equations.
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.
Adding two more in the list...
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:
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.
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.
I recommend you those books from the excellent Chicago Physics Bibliography:
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.
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"
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):
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:
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.
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.
It all depends on your background. The recent translation to English of Grøn/Næss Norwegian GR book is a very easy and plesant read:
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.
To get a first idea of what GR is all about, with loads of solved exercises, try General Relativity Without Calculus.
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:
To have fun while reading these books, you can enjoy "The Einstein Theory of Relativity: A trip to the fourth dimension", by Lillian Lieber.