Self studying abstract mathematics

[Cruz Martinez]

Hello everyone,

I have been trying to teach myself GR for sometime now, and while I started to learn through books like Schutz and Hartle, I reached a point where the motivation behind the mathematics they used didn't satisfy me nor did it appeal to me.

I think the worst I remember is when Schutz defines a manifold along the lines of: manifold is a fancy word for space. I could only wonder just exactly what kind of space that was.
While this approach might prove beneficial during an introduction, I never felt I was learning, well basically anything at all regarding the mathematics.

I have been trying to teach myself the differential geometry along with the physics of GR, I've used a number of sources but currently I'm working through the mathematical chapters of "Relativity on curved manifolds" by De Felice et. al., and the appendices in General Relativity by Wald.

This is my first time dealing with abstract math, I took a linear algebra course before but it never got so much into the abstract side.
Which brings me to my question: How do I make the transition from more applied math such as boundary value problems and ODE's to the more abstract thing? I think this is especially difficult since I am self studying.

What I gather so far from differential geometry is that the whole thing is about starting with primitive notions such as sets, and giving them different kinds of structures, such as a differentiable structure or a metric structure via a metric tensor. And it seems that abstract mathematics is pretty much like that, identifying and isolating the essential and building upon that.
I still struggle reading proofs and sometimes it takes me a while to get what the authors mean when they make mathematical statements, perhaps I should start with something lighter to get used to the general spirit of abstract math?

 

[WannabeNewton]

I wouldn't start with De Felice et al if you haven't already mastered GR because that book is meant for people who already have a solid understanding of GR. The introductory mathematical chapters in it are rather cursory and just meant as swift review, not to mention the book has no problem sets.; don't get me wrong, it's a brilliant GR book if you're ready for it.

Honestly I think self-studying pure math is quite difficult because there are a lot of subtleties, methods, tricks, and insights that you can really only get from a good teacher. Furthermore your question is quite amorphous. With that being said, what pure math books have you tried to learn from so far? You haven't listed your background in pure math so it's hard to gauge what kind of help or advice would be appropriate for you.

As an aside, I have found the following set of GR notes to have an extremely lucid, careful, and explicit introduction to the differential geometry one would need to be firmly grounded in GR: http://www.socsci.uci.edu/~dmalamen/courses/FndsofGR/GR.pdf

 

[micromass]

Your question is a bit vague. I agree that Schutz' definition of a manifold is simply horrible. It is confusing even to people who already know precisely what a manifold is!

Doing Wald or De Felice is too much at this point. I wouldn't do that at all. To be honest, you don't need all that much abstract mathematics to be able to study GR. You certainly don't need to go into the intricacies of differential geometry. I guess you just need some lucid definitions of the concepts.

At first I think you might want to focus a bit on more rigorous linear algebra, that is always fun to know. I highly recommend the excellent book "linear algebra done wrong", which is freely available: http://www.math.brown.edu/~treil/papers/LADW/LADW.html Especially Chapter 8 would be very useful for you. Malament is a great book too of course, so definitely try that.

 

[Cruz Martinez]

Your question is a bit vague.

Oh sorry about that, let me try to clarify.
By making a transition I mean a mental transition, not a real life transition such as changing majors and the like.
How to switch from learning applied things, where you can always relate what you're learning to a real world situation, to the abstract things where it just feels you somehow have to find it within your mind and make things fit together and make sense? Is this clearer?
I just feel like I'm having a hard time adapting to this type of thing without the advantages of having a teacher and classmates to talk to about the material.

To answer WannabeNewton's question, I tried to learn the differential geometry before from Carroll and also from Bishop and Goldberg's tensor analysis on manifolds, lecture notes on differentiable manifolds by Gerardo del Castillo, etc.
Apart from that my background is only in physics, up to electrodynamics, quantum mechanics, statistical mechanics and thermodynamics, no pure math classes taken besides linear algebra.

 

[micromass]

Oh sorry about that, let me try to clarify.
By making a transition I mean a mental transition, not a real life transition such as changing majors and the like.
How to switch from learning applied things, where you can always relate what you're learning to a real world situation, to the abstract things where it just feels you somehow have to find it within your mind and make things fit together and make sense? Is this clearer?

Yes, this is clear. But to be honest, if you're reading abstract things which you cannot relate to a real world situation, then you are typically doing something too advanced. In principle, the reason for the abstraction should be clear when you read it. That means that you should have enough prerequisite knowledge in order to say where the abstraction comes exactly and why we make the abstraction. For example, there are many many good reasons for making the abstraction to manifolds, but sadly enough these reasons typically require some prerequisite knowledge. This knowledge does not need to be mathematical, it can be physical as well.

WBN might disagree with me, but I feel that in order to really understand manifolds, you need to be comfortable with the easier situation of metric spaces and curves and surfaces in R^3. Without knowing about these things, manifolds will feel like an extremely abstract entity.

The question is, how do you get to the stage where manifolds are easy for you. I'm a pure mathematician, so I think the best way is to read some pure mathematics. The disadvantage to this, is that understanding pure mathematics is very slow. Additionally, pure mathematicians has the habit to go very deep into things that aren't really relevant to physicists (or which are so incredibly intuitively obvious that physicists like to take these as fact). So pure math might not be the easiest route towards understanding manifolds and the math of GR.

 

[homeomorphic]

You might try a book like Elementary Differential Geometry by O'Neill or something similar that covers curves and surfaces, first.