A path for learning General Relativity formally

Question

I was a physics major a couple years ago and took a few undergrad and grad general relativity classes, got decently good at it, but changed majors and have forgotten most of the stuff. I wanted to delve into the subject again, but this time look more formally into the math and physics, if that makes sense. So I wanted to go from the ground up on learning the mathematical and physical foundations for the subject, before picking up one or two books on GR itself.

This is the path as I see it right now, but I wanted to check in with people much more advanced as myself to see if this makes sense

1. Analysis on $\mathbb{R}$ and $\mathbb{R}^n$ (probably little and then big Rudin)
2. Point-set topology (I haven't studied the subject so no idea on books, would love suggestions)
3. Differential Geometry (same as before, all I knew about DG was what Wald taught me lol)
4. Classical Mechanics (my intuition is to pick up Landau or Goldstein since I remember Taylor, the one I used for the course I took, not being quite as formal. I've also heard stuff about Landau and Goldstein not being super formal, so any other suggestions would be good)
5. Electromagnetism (it's everyone's introduction to fields, right? I don't think I've found a book that is as formal as I'd want for this project, so any suggestions are welcome. If Griffiths is the best that's fine, I'll just go with it)
6. Finally, General Relativity! Wald is my go-to, but I expect to have to read a bunch to get a good understanding.

I'd love to hear suggestions on how to complement/change this path. Since I'm now a PhD student in a completely different area I don't have a specific timeframe I need to do this in, so I have time. Book suggestions, subject suggestions, everything is welcome!

Answer 1

I don't think that looks like a good plan! The math books basically constitute 70% of an undergraduate degree and part of a graduate degree in mathematics, so unless that's what you're going for I don't recommend it. If it is and you're interested in mathematical general relativity, then math.stackexchange.com would probably better serve you.

If you're only partly interested in the math, read Manifolds Tensors and Forms by Paul Renteln. You can even skip or skim chapters 1,2,4,5,6, and 9 if you want to get straight to the differential geometry (chapters 3 and 8 are huge chapters). Then pick whatever general relativity book interests you the most, probably from this list. The one by Bernard Schutz has a really gentle difficulty ramp (you could start reading it today without even reading Renteln), and the one by Carroll goes into more detail. From there you could proceed to more advanced and math-ey books like the one by Hawking and Ellis, or have a good foundation to start reading existing physics publications.

Answer 2

I don't know if this is the best way to go about this but here are some comments:

If your objective is to learn general relativity, going as back as studying analysis and point-set topology from dedicated books may be too much. Don't get me wrong, these subject are important to understanding the theory. However, it might be better to start with a formal account of differential geometry which treats these topics. I would suggest this playlist https://www.youtube.com/playlist?list=PLmsIjFudc1l2wDQ_ekx6iLtqcWJQQvOsw of lectures by Prof. Schuller. He does use some results from multivariate analysis, which can be found in https://www.youtube.com/playlist?list=PL5I-Eyk8l9FHdJUd9UujGcvumjCFPHbrd. If you are interested in a more profound and general account of differential geometry for different areas of physics, delving into Schuller's lecture series https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic next would be a good idea.

I don't think this is however enough for doing research in GR. It is certainly enough to do research in things happening in curved spacetimes but I wouldn't know how to proceed in order to do research in GR proper. Hope these are useful!