Topology and Differential Geometry texts for General Relativity

[CJ2116]

Hi everyone, I was wondering if I could some advice from anyone who has some experience with higher level general relativity. Any help would be greatly appreciated!

Some background:

I'm currently working through Robert Wald's General Relativity and am struggling a lot with the "advanced topics" chapters. For example, starting with the chapter on causality he begins to introduce notions from topology that I'm not really familiar with. The only topology that I have seen is from the second chapter of Rudin's Principles of Mathematical Analysis and that was from a course that I took two years ago in analysis. The appendix on it doesn't really help me any.

In fact, I'm also finding it hard to follow every appendix and the chapter on spinors. In other words, everything after chapter 6! Even reading the same topics in the appendices in Sean Carrolls book leaves me scratching my head. It's pretty clear that I don't have the proper background in topology and differential geometry (which I took a semester of, but at a lower level than used in these books.)

I eventually want to move onto more advanced books (Hawking & Ellis, Penrose & Rindler, De Felice & Clarke etc). However, I don't want to approach them with a shaky (and even sketchy) understanding of the mathematics involved. My goal in working through Wald is to understand everything in it thoroughly and patch up things that I missed in Schutz' and Carroll's books.

With this in mind, I want to set aside GR for the time being and focus more on becoming competent with the mathematical machinery. My question is: What books on topology and differential geometry (and even group theory) focus on what has the most application to GR? I see a lot of people recommend Munkres, but he doesn't seem to cover very much on manifolds. I have Barret O'Neill's Elementary Differential Geometry, but again there doesn't seem to be a whole lot overlap between it and what is covered in GR.

I'm looking for something at the advanced undergraduate or graduate level with a lot of worked problems (I can't emphasize that enough!) The only books I may have found so far are the ones by John M. Lee (Introduction to Topological Manifolds and Introduction to Smooth Manifolds.) Does anyone have any experience with them?

Sorry that this post became so long - I thought that if I listed specifics of what I want to study in GR, it would be easier to recommend an appropriate text!

Thanks!

 

[homeomorphic]

Something like Munkres is okay to start with because you have to walk before you can run. It may not cover manifolds much, but it does cover connectedness and compactness, which are very important properties that a manifold might have. So, I would take a look at Chapters 0-4, maybe skipping most of chapter 4 to start out. Then, you can try Guillemin and Pollack for differential topology, maybe along with Milnor's differential topology book.

I don't think you want to necessarily want to restrict yourself to stuff that has the most application to GR at first, particularly if you don't want a sketchy understanding, because you have to know the basics in order to get to a lot of it.

You won't find a lot of "worked problems" in topology/geometry books, for good reason. You have to do that part for yourself.

Eventually, you should look at Penrose's book, Techniques of Differential Topology in Relativity. I haven't had time to read it, but it looked really cool when I flipped through it.

 

[CJ2116]

Thanks for the reply.

I have a copy of Munkres, but haven't really read much beyond the first chapter on set theory. I'll definitely have to give it a closer look. I do remember it being very readable.

I don't think you want to necessarily want to restrict yourself to stuff that has the most application to GR at first, particularly if you don't want a sketchy understanding, because you have to know the basics in order to get to a lot of it.

This is an important point that I haven't really considered. I'm still in the college mindset that I have to rush through a select few number of topics in 4 months!

I didn't actually know that Penrose wrote a book on differential topology, guess I shouldn't be surprised, though!

 

[micromass]

I highly recommend the books by Lee: "Introduction to topological manifolds" and "Introduction to smooth manifolds". They approach topology from a differential geometry point-of-view and they are wonderful to read.

If you choose to do Lee, then you should know that it is not necessary to go through his entire topological manifolds. Many chapters can safely be omitted. The essential are the first 4 chapters, the stuff on homotopy, the fundamental group, the circle and covering maps. The rest is very nice, but not immediately necessary.

 

[CJ2116]

I bought a copy of the second edition of Introduction to Topological Manifolds last week and I really like what I have read of it so far, especially the fact that he put exercises in the middle of the text.

So, in other words I would should cover chapters 1-4, 7-8 and 11?

 

[micromass]

I bought a copy of the second edition of Introduction to Topological Manifolds last week and I really like what I have read of it so far, especially the fact that he put exercises in the middle of the text.

So, in other words I would should cover chapters 1-4, 7-8 and 11?

Well, you should start with the appendix. Work through appendix A and B first because they are very important. Then you can start by reading the text. The bare minimum is chapter 1-4. If you did those, then you can already start read his smooth manifolds texts and you will understand most of it.
However, Lee occasionally uses covering spaces and homotopies in his text (in a non-essential way though), so if you want a deeper understanding, then you can do 7-8 and 11-12.
For a complete understanding, you can do the entire book, but that's not really necessary.

 

[CJ2116]

Excellent, thanks for the clarification! His smooth manifold book is the one that I really want to get through!

 

[dextercioby]

I've on the internet that there's an errata to this book, so pay great attention to his writing.

 

[CJ2116]

He has errata on his webpage. Definitely a lot less than some other books I have used!

I really like this book so far! I have only read the first two appendices and parts of the second chapter and I already understand this stuff more than when I saw it in real analysis!