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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!

Slereah
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Brunnun
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2 Answers2

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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.

David
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    Gotcha! I guess I can see that it may be too much ahahaha it's just that whenever I found myself looking over Differential Geometry textbooks after I coasted through an analysis course in undergrad it felt like I was missing a lot, so I thought it may be necessary to go back a bit. Maybe I was just trying to learn DG from the wrong angle, though. I certainly care more about the physics, but I wanted to know the math deeply so I can understand the physics effectively, you know? I'll look into Paul Renteln's book, thank you for the recommendation! – Brunnun Jul 30 '20 at 16:36
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    @Brunnun Hope it's useful! I was browsing the differential geometry section in the bookshelves at UCSD when I stumbled across it, and it's an absolute gem. There's some pretty amazing linear algebra stuff in chapters 1 and 2 as well (like problem 1.46 on symmetric polynomials, an identity by Newton, and the expansion of the determinant in terms of traces!). – David Jul 30 '20 at 18:35
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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!

Ivan Burbano
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  • Unless my life changes a lot I probably will not be doing research in GR very soon, though it's not something I'd be sad if it happened. In any case, thank you for the recommendation! I can see how going all the way back may be too much, I'm just constantly feeling like I'm lacking some understanding when looking into DG and thought the solution could be going back to basics. I'll take a look at this lecture series though - maybe an approach on DG from a different angle will help me understand it in a way I didn't when I tried the subject 2 years ago. Thank you!! – Brunnun Jul 30 '20 at 16:38
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    I have to say, I took analysis and point-set topology before I started learning differential geometry and I didn't really understood anything until I saw these lectures. – Ivan Burbano Jul 30 '20 at 16:39
  • Well that's a very fair point against my planned path lol do you still think these lectures are a good entry point, though? I took analysis more than three years ago (and never took topology) and haven't done much with the subject since, so my main worry is to jump into a DG course/textbook and be very overwhelmed – Brunnun Jul 30 '20 at 16:42
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    I think they are a very good entry point. They include a refresher on topology. In fact, the last set of lectures of Schuller goes as far back as to introduce logic and proofs. These are really intended to get from 0 to 100 in geometry for physics in a quick but rigorous fashion. – Ivan Burbano Jul 30 '20 at 16:49
  • Yeah I can see it! I'm always paranoid on "am I getting the most formal account I could???" but your description does look like the kind of thing I'd be interested in. So I guess if I wanted to go from the ground up in DG to pick up GR later I would go through the last set of lectures you linked, and maybe the textbook the other answerer suggested? That all sounds fair. Thank you for the help! – Brunnun Jul 30 '20 at 16:54