soft question - What are the prerequisites to studying general relativity?

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?

Answer

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.