Reddit reviews Geometrical Methods of Mathematical Physics

Anti-disclaimer: I do have personal experience with all the below books.

I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.

It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.

If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.

I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .

Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.

Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.

I learned from Baez & Muniain; Gauge Fields, Knots, and Gravity.

Toward the end of the course, I met Brian Greene at a public talk, and he recommended Schutz; Geometrical Methods of Mathematical Physics.

I found Geometrical methods of mathematical physics by Bernard Schutz to be helpful, though it doesn't have many problems and it doesn't go into much depth on covariant differentiation. But it is good about discussing the modern view of tensors.

I would recommend watching the first half of the International Winter School on Gravity & Light (check the YouTube channel as well ) if you are interested in learning tensor calculus for use in differential geometry for GR.

I learned tensor calculus in bits from several different courses and texts, so I'm not sure what the best ones that are actually dedicated to the subject might be. In any case, I think you'll have a lot of fun learning the subject.

I quite like Schutz’s book:

Geometrical Methods of Mathematical Physics

https://www.amazon.co.uk/dp/0521298873/

Have you had a look at Carroll's general relativity notes? Chapters 2 and 3 are predominantly about developing the mathematics behind GR, and are very good introductions to this. I have a copy of Carroll's book and I can promise you that those chapters are almost unchanged in the book as compared to the lecture notes. This is my main suggestion really, as the notes are freely available, written by an absolute expert and a joy to read. I can't recommend them (and the book really) enough.

Most undergraduate books on general relativity start with a "physics first" type approach, where the underlying material about manifolds and curvature is developed as it is needed. The only problem with this is that it makes seeing the underlying picture for how the material works more difficult. I wouldn't neccessarily say avoid these sort of books (my favourite two of this kind would be Cheng's book and Hartle's.) but be aware that they are probably not what you are looking for if you want a consistent description of the mathematics.

I would also say avoid the harder end of the scale (Wald) till you've at least done your course. Wald is a tough book, and certainly not aimed at people seeing the material for the first time.

Another useful idea would be looking for lecture notes from other universities. As an example, there are some useful notes here from cambridge university. Generally I find doing searches like "general relativity site:.ac.uk filetype:pdf" in google is a good way to get started searching for decent lecture notes from other universities.

If you're willing to dive in a bit more to the mathematics, the riemannian geometry book by DoCarmo is supposed to be excellent, although I've only seen his differential geometry book (which was very good). As a word of warning, this book might assume knowledge of differential geometry from his earlier book. The book you linked by Bishop also looks fine, and there is also the book by Schutz which is supposed to be great and this book by Sternberg which looks pretty good, although quite tough.

Finally, if you would like I have a dropbox folder of collected together material for GR which I could share with you. It's not much, but I've got some decent stuff collected together which could be very helpful. As a qualifier, I had to teach myself GR for my undergrad project, so I know how it feels being on your own with it. Good luck!

Absolutely. The stress tensor is a (2, 0) tensor (called contravariant in the physicists definition), which means that it takes two vector inputs to produce a real number.

If you input a vector, say e1 (this may be x-hat, the unit vector in the x direction in Cartesian coordinates), it will return a vector which represents the force per unit area in that direction. It actually returns a 1-form (covariant vector), but in the case of the stress tensor, which is a Cartesian tensor, covariant vectors are the same as contravariant vectors, their duals.

This operation is called tensor contraction, where the tensor only acts on one input and returns another tensor of rank (n-1, m-1), or in the case of the stress tensor, it returns a (1, 0) tensor which is just a covariant vector, or in the case of cartesian tensors, it is just a vector (contravariant).

I encourage anybody who is interested in this stuff to read Schutz's Geometrical Methods of Mathematical Physics, as this book describes tensors fully in the newer language (my definition number 2), and does so with applications to physics. Most tensors in physics are taught in the old indices/transformation law language, and can be quite confusing for first timers.