Comment Separation of Math and Physics (Score 1) 358
I have to disagree with a lot of the posts out here on this subject so far. Yes, general relativity is about the physics, but as a physics grad student, I've had some of my greatest frustrations just trying to "learn the math through the physics". I think that it's very helpful (at least it has been to me) to learn the math on it's own from mathematics text and then, once the math is understood, pick up a physics text so you can focus solely on the physics ideas. Again, this is all my own humble opinion, but it is what has worked for me, and many of my friends in the field have related similar stories.
To answer your question more specifically, here is what I would recommend:
On the math side:
1) Review single variable calculus
2) Multivariable calculus
3) Linear Algebra (check out Axler's "Linear Algebra Done Right" - this book is amazing)
4) Differential Equations
5) Differential Geomotery
General relativity is all differential geometery, so understanding this is what you're shooting for. However, just knowing the math isn't enough; you'll need to get up to speed on physics as well. So, assuming you've had an introduction to physics somewhere (high school or undergrad):
1) Intermediate mechanics (The book by Taylor is brilliant)
2) Electricity & Magnetism (Griffiths is the way to go here, no question)
3) Special Relativity
4) General Relativity
I know you know technically need a course in E&M to understand general relativity, but a lot of Einstein's work on special relativity was motivated/inspired by ideas from E&M (and a lot of his work on general relativity was inspired by his own work on special relativity...).
Finally, on last word of warning: relativity is something you'll have to approach multiple times before you fully understand it. First, try to understand special relativity on a very simple like (Feynman has a very simple exposition on this). This doesn't take any math beyond algebra. Look at the equations for the Lorentz transformations and do some problems on time dilation and such. Next, try to understand special relativity from a more advanced point of view using Minkowski space and all the fancy linear algebra and calculus that comes with it. Then try to understand general relativity as a generalization of this, where space-time is curved by the matter in it. It's a very incremental process. If you get an undergrad degree in physics, you would probably see relativity, in some level of increasing complexity/subtlety, at least three times on your way from Newtonian mechanics to general relativity.
Though let me stress again: I think it's worth it to learn the math for the sake of the math which will free you to really focus on the physics when you go to tackle the actual ideas within. I've seen so many people get disheartened as they struggle to understand both at the same time and end up strangling themselves on the twisted mess that you get when you try to do both at once.
To answer your question more specifically, here is what I would recommend:
On the math side:
1) Review single variable calculus
2) Multivariable calculus
3) Linear Algebra (check out Axler's "Linear Algebra Done Right" - this book is amazing)
4) Differential Equations
5) Differential Geomotery
General relativity is all differential geometery, so understanding this is what you're shooting for. However, just knowing the math isn't enough; you'll need to get up to speed on physics as well. So, assuming you've had an introduction to physics somewhere (high school or undergrad):
1) Intermediate mechanics (The book by Taylor is brilliant)
2) Electricity & Magnetism (Griffiths is the way to go here, no question)
3) Special Relativity
4) General Relativity
I know you know technically need a course in E&M to understand general relativity, but a lot of Einstein's work on special relativity was motivated/inspired by ideas from E&M (and a lot of his work on general relativity was inspired by his own work on special relativity...).
Finally, on last word of warning: relativity is something you'll have to approach multiple times before you fully understand it. First, try to understand special relativity on a very simple like (Feynman has a very simple exposition on this). This doesn't take any math beyond algebra. Look at the equations for the Lorentz transformations and do some problems on time dilation and such. Next, try to understand special relativity from a more advanced point of view using Minkowski space and all the fancy linear algebra and calculus that comes with it. Then try to understand general relativity as a generalization of this, where space-time is curved by the matter in it. It's a very incremental process. If you get an undergrad degree in physics, you would probably see relativity, in some level of increasing complexity/subtlety, at least three times on your way from Newtonian mechanics to general relativity.
Though let me stress again: I think it's worth it to learn the math for the sake of the math which will free you to really focus on the physics when you go to tackle the actual ideas within. I've seen so many people get disheartened as they struggle to understand both at the same time and end up strangling themselves on the twisted mess that you get when you try to do both at once.