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This question recently appeared on Slashdot:

Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade.

It seems like something that would be a good addition to this site: I think it's specific enough to be answerable but still generally useful. The textbook aspect is covered pretty well by Book recommendations, but beyond that: What college-level subjects in physics and math are prerequisites to studying general relativity in mathematical detail?

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closed as off topic by David Z Jun 9 '13 at 3:08

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It depends sort of on what you mean by "prerequisites". If you just mean to follow the logic and comprehend the computations, then I agree with nibot that surprisingly little is required. But having a thorough background like colin suggests definitely will lead to better appreciation of the subject. – Willie Wong Aug 29 '11 at 14:36
@Ron Maimon: You're not going to get very far without at least calculus. And a grounding in classical mechanics/E&M, on at least some level, is really damn close to essential. – Jerry Schirmer Sep 8 '11 at 20:25
@Jerry: Ok, read Serge Lang's calculus book, and read the mechanics and EM sections of the Feynman lectures. Then you can read Schutz self-contained undergraduate GR book. – Ron Maimon Sep 8 '11 at 20:48
up vote 20 down vote accepted

First general relativity is typically taught at a 4th year undergraduate level or sometimes even a graduate level, obviously this presumes a good undergraduate training in mathematics and physics. Personally, I'm more of the opinion that one should go and learn other physics before tackling general relativity. A solid background in classical mechanics with exposure to Hamiltonians, Lagrangians, and action principles at least. A course in electromagnetism (at the level of Griffiths) I think is also a good thing to have.

Mathematically, I think the pre-reqs are a bit higher and since the question asks about mathematical detail, I'll focus on that. I learnt relativity from a very differential geometry centric viewpoint (I was taught by a mathematician) and I found that my understanding of differential geometry was very helpful for understanding the physics. I've never been a fan of Hartle's book which I think is greatly lacking on the mathematical details but is good for physical intuition. However having worked in relativity for some time now I think it's better to teach from a more mathematical point of view so you can easily pick up the higher level concepts.

Additionally, I think you really need to understand what is going on mathematically to understand why we must construct things the way we do. I'm going to have to disagree with nibot here and say that you'll need more then just linear algebra and college calculus. Calculus you must have at least seen up to vector calculus and be familiar with it. Linear algebra is something you should have a very good understanding considering that we are dealing with vectors. A good course in more abstract algebra dealing with vector spaces, inner products/orthogonality, and that sort of thing is a must. To my knowledge this is normally taught in a second year linear algebra course and is typically kept out of first year courses. Obviously a course in differential equations is required and probably a course in partial differential equations is required as well.

I don't think a course in analysis is required, however since the question is more about the mathematical aspect, I'd say having a course in analysis up to topological spaces is a huge plus. That way if you're curious about the more mathematical nature of manifolds, you could pick up a book like Lee and be off to the races. If you want to study anything at a level higher, say Wald, then a course in analysis including topological spaces is a must. You could get away with it but I think it's better to have at the end of the day.

I'd also say a good course in classical differential geometry (2 and 3 dimensional things) is a good pre-req to build a geometrical idea of what is going on, albeit the methods used in those types of courses do not generalise.

Of course, there is also the whole bit about mathematical maturity. It's a funny thing that is impossible to quantify. I, despite having the right mathematical background, did not understand immediately the whole idea of introducing a tangent space on each point of a manifold and how $\{\partial_{i}\}$ form a basis for this vector space. It took me a bit longer to figure this out.

You can always skip all this and get away with just the physicists classical index gymnastics (tensors are things that transform this certain way) however I think if you want to be a serious student of relativity you'd learn the more mathematical point of view.

EDIT: On the suggestion of jdm, a course in classical field theory is good as well. There is a nice little Dover book appropriately titled Classical Field Theory that gets to general relativity right at the end. However I never took a course and I don't think many universities offer it anyway unfortunately. Also a good introduction if you want to go learn quantum field theory.

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I'd also recommend a course/book on classical field theory. There you can learn about co- and contravariant vectors, covariant derivatives, etc.. It usually also includes elasticity theory, where the stress-energy tensor is introduced first. You can learn a lot of the basic mathematics needed there without some of the peculiarities of GR. – jdm Aug 28 '11 at 21:52
Of course, I agree that this is the ideal preparation. – nibot Aug 28 '11 at 22:07
Oh, I completely forgot that jdm. I never took a course on classical field theory however it is a good pre-req. I'd also say that fluid mechanics is a good thing to have as well but not necessary. – colin Aug 28 '11 at 23:32
@jdm: honestly, I kind of wish that we started retiring graduate E&M for a classical field theory course. – Jerry Schirmer Sep 8 '11 at 20:26
@Jerry: I'm not familiar with the US course abbreviations... Is E&M electromagnetism / electodynamics, Maxwell eqn.s with boundary conditions and so on? We never had a proper graduate-level course in that (but that was more due to a lack of personnel), and classical field theory was voluntary. I agree that such a course would be very useful. – jdm Sep 9 '11 at 14:50

The subject is surprisingly self-contained, perhaps mostly because physicists typically learn the physical theory and the associated mathematics simultaneously. I'd say the main prerequisite is a little "mathematical maturity" and physical intuition. Any exposure to differential geometry and abstract mathematics can only help. A course in analytical classical mechanics would put you in the right mindset.

If you've taken college calculus, linear algebra, and have some understanding of special relativity, then I suggest you jump right in and ask questions when you encounter difficulties.

Colin's excellent answer is right on target in describing a more thorough background. Just don't be too shy to dig into Sean Carroll's notes immediately and see how much you understand and how much you need to review.

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Calculus, apparently, is not even a prerequisite for some presentations. There's a new book out from Springer called General Relavitiy without Calculus. Whether it actually works, I don't know. – Willie Wong Sep 2 '11 at 13:49

Can I recommend this set of videos from Stanford university as given by Leonard Susskind.

Very thorough and very haelpful for me inthe summer before I took the course.

Stanford Uni GR course

Also it wouldn't hurt to look at some none GR cosmology courses, these will cover some of the physical phenomena that GR is necessary for to describe completely

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Some reponses here are close to "do a mathematics degree, then a physics degree". I think this is not what you expected. Learning all that subjects is factible and natural while you are at the university, but trying to adquire all that knowledge alone on your own in your free time, with no professors, no classes, no press to do exams in certain dates... that is almost impossible.

Having said that, I think I understand exactly what you want, because I have had the same question once. I am a physicist, but after my degree, I realized that my understanding of GR from a mathematical point of view was only superficial. So I researched on the question and finally designed a "step by step" bibliography that I am still following in my free time. Here you have it:

1) Start with the old edition of the small Schaum book 'Vector Calculus' by Murray M. Spiegel. It starts with the very basic definition of vectors from high school, and ends with Christoffel Symbols and Geodesics. Every chapter has minimalist description of the essentials, followed by solved exercises. Read only the descriptive part of chapters 1 through 6 (the first 3 or 4 chapters will surely be completely known for you, but it is good to refresh), and then work on the complete chapters 7 (curvilinear coordinates) and 8 (tensor calculus), I mean: study chapters 7 and 8 with the solved exercises too.

Specially work ALL the solved and unsolved exercises of chapter 8.

This can't do miracles (i.e. it cannot be a substitute for a complete degree in mathematics) but it will give you very useful basic mathematical tools in a very short time. If you can do a partial derivate but don't know what means "derive the Christoffel symbols in spherical orthogonal coordinates", this is the book you have to start with.

2) After the Schaum, study the book "The Meaning of Relativity" (1922) by Einstein. It is a book based on a series of lectures he gave in 1921 in Princeton, with progressive explanations from tensor calculus to the Friedman cosmology, including special and general relativity. It is intended to be self-explanatory in the mathematics, but it will have much more meaning to you if you have worked out the Schaum book first. It will require also some short looks at wikipedia if your background in physiscs is not good (i.e. the Poisson equation or Maxwell equations and their meaning when you encounter them) but nothing difficult. The only problem with this book is sometimes the old-fashioned notation or, from time to time, some details you will have to guess (for example, he assumes c=1 for special relativity and it can be very confusing if you haven't noticed). But it is very stimulating to be learning from Einstein itself, and the modern books are in general either too basic, or too biased in one direction.

3) After having worked points 1 and 2, I have now jumped to learning chapters 1 to 6 from the book "General Relativity" by Wald, including the exercises (this is very important) that are solved somewhere in the internet (google for it). This is however quite a hard book, and I sometimes regret not to have used first another text. So I recommend you take here the book "Spacetime and Geometry: An Introduction to General Relativity" by Sean Carroll. It is not as hard as Wald but is rigorous and well explained, and the selection of topics is very interesting.

Another quite direct approach to learn relativity from the beginning may be the book "A First Course in General Relativity" from Schutz. This book is unique in its kind, because it developes a geometrical, rigorous approach, yet progressive and easy, to General Relativity and its mathematical machinery, assuming the target reader barely knows at the beginning how to do a partial derivate and little less more. It has many exercises whose detailed solutions are easy to find in the internet. The very last chapters about black holes and cosmology are only introductory, but if you reach them, you will be in a good position to start more ambitious projects (Carroll, Weinberg, etc). In fact, I am seriously thinking about giving up Wald for the moment, and come back to this book, that I partially used in my degree, and work it out from the beginning. I am sure Roger Penrose holds Wald in one hand and reads in a distracted manner while eating its Corn Flakes in the morning, as you and me do with a newspaper, but for me it is still too abstract...

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I read most of Schutz's first course straight out of high school and before university. I had done mathematics and physics A levels (standard for 18 year olds in the UK intending to go on to mathematics at university). It requires understanding what a partial derivative is and basic linear algebra, as well as a lack of fear. It helped that I'd previous read lots of pop science on special relativity beforehand, but it does start with a derivation of SR making the book pretty self-contained.

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As for special relativity not much mathematics is needed. Here it is one of the first courses in the undergraduate curriculum. Some linear algebra might be useful, but is not strictly necessary. It depends a bit on the text that you will use.

General relativity is considerably more difficult and requires a stronger background in mathematics, in particular on differential geometry. It depends on your own preferences if you like to study it in a physicists fashion (i.e., everything in local coordinates and with tensor indices) or mathematics (coordinate free). A good book on general relativity would introduce the necessary mathematics. I liked Wald's General Relativity, but with your background it is not very suitable I think. A good understanding of multi-dimensional calculus is certainly a prerequisite.

Edit: this was a response to a question that was merged with this one.

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Special Relativity is the first prerequisites, obviously. That should include the necessary linear algebra, group theory and classical field theory, because GR is itself a field theory.

After that the main thing you need to understand is calculus up to and including the chain rule for partial derivatives which is crucial for the composition of co-ordinate transformations on tensor fields.

That is all that is essential to learn the basics of general relativity. Perhaps you also need a good geometric intuition. If you have it, GR can be taught or learnt much earlier than it normally is.

There are some other things that would help, such as understanding the principle of least action which uses the calculus of variations. Electromagnetics up to the formulation of Maxwell's equations in SR is also worth getting to know first.

Then you pick up the smallest book on GR you can find (which would be Dirac's) and work through it. Later you can move to more modern treatments and larger texts with more details.

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Aside from the Feynman lectures, my favorite is Discovering Relativity for yourself by Sam Lilley. He spent many years teaching relativity in evening classes to students like housewives and tradesmen who had little background but strong determination. He proceeds through special relativity to general relativity in very modest steps, and is careful to make sure people do not get left behind. He starts with very little math, and carefully introduces the math as he goes along, until at the end he's being quite mathematical, but he's brought everyone with him so they understand it. At every step of the way, he makes it intuitive and interesting, especially by using a little cardboard device called a "slot". By sliding this slot across the space-time diagrams you can clearly understand what is happening.

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