What types of math do I need to know to understand the mathematics of the General Theory of Relativity?

Question

I've been a self-studying physics student and have understood the concepts involving the Special and General Theories of Relativity for a while now. I want to go to the next level now by taking on the mathematics. Thank you for your help!

I've been a self-studying physics student and have understood the concepts involving the Special and General Theories of Relativity for a while now. I want to go to the next level now by taking on the mathematics.

I understand there are a lot of different mathmatical disciplines involved, but I'm sure there's a starting place. Since I am a self-study I need to know what math books to get from Barnes & Nobles. (I got a gift card for Christmas.) Thank you.

I'm a beginner in math. I'm no dummy though so don't roll your eyes! Get real basic with me like "Algebra > Calculus > I don't know after that. :("

Thank you all for your help. Seems like I'm finally going to have to take up calculus. I kept putting it off; now it's time I guess. I'll start out with Calculus For Dummies. I hear that's what Newton used to get started so it must be good. Thanks Again.

Answer 1

You will need to know Riemannian geometry and other topics in differential geometry. Here's some suggested reading:

First, learn multivariable calculus and get very comfortable with it. Then read do Carmo's "Differential Geometry of Curves and Surfaces". From there, you could look at Do Carmo's graduate level book on Riemannian geometry. I believe his Riemannian geometry book is the most easily accessible compared to other books on this topic.

For a super fast course in Riemannian geometry, you might consult the relevant chapter on that topic in John Milnor's book on Morse theory.

Finally, to learn just what you need for general relativity, check out "A First Course In General Relativity" by Bernard Schutz. The mathematics in this book is at a much more elementary (and physics oriented) level than the mathematics in the other books. However, I found that the low level presentation of the mathematics in that book made it more difficult to learn.

Unfortunately, that's the way it is when it comes to differential geometry and tensor calculus. It takes about a year to become fully comfortable with the ideas of the subject. Without the pressure of having to take tests and turn in homework assignments, it may take even longer. Just remember that there's no one good book on the subject and hang in there.

Some other reading material:

John M. Lee, Introduction to Smooth Manifolds (1st year grad students love it, professors tend to dislike it)

Theodore Frankel, The Geometry of Physics

Answer 2

well you got to build on up quite a bit, and unfortunatly I don't think your at that point yet of self studying your way into it. If you were an advanced undergrad i'd say it could be possible to self study your way through at least the basic mathematics of it, but really I think trying to get there like this is going to be pretty difficult.
But i'll try to guide ya as best I can.
I'll assume your at least ready to start out calculus.

Your going to have to start out in calculus and really build from there you must learn differentation, integration and just about anything you can pull out of those books. by the time your ready to move on to other things you should be able to do just about any calculus problem in any of those books. Concentrate heavily on vector calculus in 3 dimensions.
From there your going to probably want to study linear algebra and that by yourself is going to be ugly. Start with matrix mathematics and learn those operations and then move into learning about solving linear systems. Know the terms and theorms of linear algebra.
At the same time as that you can start your voyage into solving some applied problems in Differential equations. This is where your calculus is going to come into play a bit more and getting Diff EQ down is going to help quite a bit.
After you feel pretty comfortable with those 2 your going to have to find some sort of mathematical text on logic. Your going to at least have to give yourself a crash course on mathematical logic and the creation of proofs.... Definatly the hardest part of mathematics.
Well if you get this far on your own i'd be pretty impressed and pretty curious to see how good you are, but you got a ways to go. You still need to study Analysis, you probably would want to get some modern algebra in there somewhere... which by yourself i would say is damn near impossible. Then you would need to study fourier analysis, some pretty advanced calculus, and some applied mathematics. I would say 5 years would be about the right timetable to expect some good results.

Answer 3

Tensor analysis and differential geometry - hopefully as a physics student you've already studied at least three semesters of calculus as well as differential equations.

Try: algebra > trig > precalculus > calculus (differentiation, integration) > differential equations > tensor analysis > differential geometry. It's a good 4 years of college and grad level math. Good luck!

Answer 4

Experimental fact: the linked fee of light measured does not rely on your state of action, nor on the action of the source. This probably harmless fact means that in case you're in a vehicle vacationing at 10% of the linked fee of light (0.10 c), and somebody on the line at the back of you factors a flash gentle when you, then in case you degree the linked fee of the gentle beam that passes you, you will no longer degree 0.ninety c as you could assume from Newtonian concept ( c - 0.10c = 0.ninety c) yet you will degree c!!!! this could strike you as ABSURD, even nonetheless this is actual! Now how can it incredibly is defined that inspite of your action relative to the guy with the flash gentle, you and he agree on the linked fee of light? for sure on your length of speed (which of path means measuring a distance and a time) your ruler and your clock ought to be "replaced" a manner or the different to make certain which you to receive an analogous type c. for this reason for observers in relative action lengths and time ought to act in diverse techniques.

Answer 5

Unfortunately, lots. Differential geometry and tensor calculus are among the subjects that you will need to understand.

Answer 6

Well Basicly it means:

Energy=Mass*Speed of Light*Speed of Light