# Where do I start with Non-Euclidean Geometry?

I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many other resources start out with something like "well space is just a non-euclidean manifold with a Ricci tensor defined as follows:", which would be cool if I understood non-Euclidean manifolds or Ricci tensors.

Unfortunately, when I try to crack open Wikipedia articles on non-Euclidean articles on the Ricci tensor I have trouble making sense of all the foreign terms.

Is non-euclidean geometry a good place to start if I want to understand general relativity? What's a good introductory resource for non-Euclidean geometry for someone who's only ever dealt with Euclidean?

(Note: I understand the basic principles of general relativity, i.e. how acceleration and gravity are different perspectives about the same thing and how clocks move slower when higher in a gravity field, but I want to understand the math and how it was derived.)

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I personally recommend books.google.com/books/about/… – DJBunk May 18 '12 at 18:01
– Qmechanic May 18 '12 at 19:13

It really depends on your background in mathematics, what references are appropriate. In my oppinion you need at least a solid understanding of linear algebra and vector calculus, as most books on general relativity will assume.

A very nice although quite long book on general relativity is "Gravitation" by Misner, Thorne and Wheeler. It is very pedagocical and has intuitive descriptions of all geometric concepts you need to understand. In this regard it is better than many books on differential geometry. The only drawback is that a proper development of those foundations takes a lot of time and they take quite a few detours, so it is probably not suitable for someone, who is impatient to learn about general relativity.

On the other hand the less thourough books usually reduce the mathematical part to cooking recipes and give you little to no idea, why they actually work. If you have had no formal education in mathematics on university level, I suspect it will be hard to understand general relativity on more than a superficial level. As a general rule it is usually not a good idea to rush to the subjects like general relativity immediatly. It makes understanding them unneccessarily hard, compared to an approach where one tries to understand the fundamentals first.

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A high school student can learn GR from Schutz, it's first principles. Misner Thorne and Wheeler, despite the pedigree, is too chatty to be useful. – Ron Maimon May 18 '12 at 19:13
It depends on why you want to learn general relativity. Their book contains a lot of insights, that you won't find in any other book. It would probably help to read it concurrently to a lecture or lecture notes. To get an idea what the bare essentials are. – orbifold May 18 '12 at 19:16
I disagree that a high school student can understand GR in any deep way. Sure you can understand computations for the schwarzschild metric and similiar concrete examples, after being taught how christoffel symbols are to be computed and how one determines geodesics. But being able to perform computations is a long way from actually understanding what they signify. – orbifold May 18 '12 at 19:19
I already "know" GR (as in, I can apply the formulas and understand the results), but I want to understand how they work and why. I have a pretty strong math background, so I can move the numbers around and get the right answers, but I don't yet have an intuition as to where the equations come from in the first place, or what they really "mean". Sounds like "Gravitation" might be a good book for me. – So8res May 18 '12 at 19:24
@orbifold: I tried to slog through MTW, and it's too long, how can you have 1200 pages? Are you writing the bible? GR is a small subject, and is well served by a small book. There isn't 1200 pages of unique content in there. I learned GR over summer break between junior and senior year from Schutz and Dirac. It's not difficult for a still-active research subject in physics, because unlike quantum field theory, renormalization, or string theory, GR is well explained in textbooks. Schutz provides very good physical intuition, as good as any presentation, or better. – Ron Maimon May 19 '12 at 1:35

I wouldn't start with learning the maths. Mathematicians take a very different approach from physicists and I doubt it would help much.

You just need the right textbook. I strongly recommend "A first course in general relativity" by Bernard F. Schutz. This seems to me to strike the right balance between understanding the physics and understanding the maths. Note however that even a "first course" is pretty hard work - to get through the book will require much sweat, and I speak from experience :-)

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I read this book, it's really good +1, the author knows the material, and it gets the point across with no previous background in physics required, beyond stuff in the Feynman Lectures. The only drawback is that it doesn't show you how to calculate curvature, which is the main calculational problem in GR. You can see this answer for how to do that: physics.stackexchange.com/questions/14136/… . – Ron Maimon May 18 '12 at 19:15

I recomended to study non-euclidan geometry from the book "Non-Euclidian Geometry" by Stefan Kulczycky (Transleted from Polish) - Pergamon Press Haim Reouven-Israel

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As per our Resource-Recommendation policy, could you please sketch what topics the book is about and why you find it particularly suited? Thank you! – Martin Jan 23 '15 at 13:01