Reddit reviews A First Course in General Relativity

Don't bother, just pick up a GR textbook like Hartle or Schutz. Those books teach the math as they go.

Carroll

Carroll, course notes (free, I think it may be a preprint of the book)

Schutz

Wald

MTW (Some call it the GR bible)

They're all great books, Schutz I think is the most novice friendly but I believe they all cover tensor calculus and differential geometry in some detail.

The basic theme of differential geometry is to take calculus in R^(n) and do it in a more general n-dimensional spaces (called manifolds) that are "locally" like R^(n). For example, think of a sphere: when you look at it close enough (like when you're living on the Earth), it looks flat, and you can do calculus with lines on the ground and everything works out. On a larger scale, though, things get messed up when you look at scales large enough for the curvature of the Earth to make a difference. So you always have to look infinitesimally close (that's where the "differential" part comes in). Feel free to ask more about that.

As for notes, I mostly learned from this guy, whose notes on differential geometry are available online. I also really like this book. If you'd prefer a more easygoing, computational approach, take a look at this book, or some other gentle introduction to GR.

ETA: If you'd like to think about non-Euclidean geometry using only basic linear algebra, take a look at these notes.

I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.

If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.

Basics

I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.

Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.

From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.

This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.

Intermediate

At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.

This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.

Advanced

Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.

Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.

Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.

For both subjects you'll need a solid mathematical background. You'll need calculus and linear algebra. I recommend starting with it if you haven't learned yet. I really can't stress enough the importance of mathematics in both fields.

For basic quantum mechanics: Quantum Mechanics - David Griffiths (https://www.amazon.com/Introduction-Quantum-Mechanics-David-Griffiths/dp/1107179866) or Fundamentals of Modern Physics - Robert Eisberg, the later being just an introduction to Q.M.

For general relativity: Bernard Schutz's A First Course in General Relativity (https://www.amazon.com/First-Course-General-Relativity/dp/0521887054).

• First Course in GR by Bernard Schutz
  

I've currently not a lot of time so i'm not able to give a thoughtfull answer but there are plenty of books which could teach you special relativity (Carroll takes it pretty much as a prerequisite).
Maybe one of the following helps (but don't be surprised it take a lot of hard work to get some knowledge about it...):
https://www.youtube.com/watch?v=BAurgxtOdxY and following

Spacetime Physics - Edwin F. Taylor, John Archibald Wheeler should be quite nice (i've heard)

http://www.amazon.com/A-First-Course-General-Relativity/dp/0521887054 maybe this is a good starting point.

Take one book after another till one suits you. I think the only important point is that they have equations inside.

For an undergrad I would recommend

http://www.amazon.com/A-First-Course-General-Relativity/dp/0521887054

before moving on the MTW.

I would actually suggest NOT trying to learn about these subjects, at least not on their own. Put in the time to really learn tensors, then co- and contravariance will makes loads more sense!

I found the first 3 or 4 chapters of Schutz's "First Course on General Relativity" to be a great place for teaching these things to myself. You could also take a math methods course that covers tensors.

EDIT: This is the book I'm talking about:
http://www.amazon.com/A-First-Course-General-Relativity/dp/0521887054/ref=sr_1_1?ie=UTF8&qid=1345315215&sr=8-1&keywords=schutz+general+relativity

I learned the basics from Shutz (which I liked a lot). I don't remember it requiring Lagrangian/Hamiltonian stuff.

Griffith's Electrodynamics has a decent introduction to special relativity. Otherwise, Hartle's book is geared towards the advanced undergrad. Also, Schultz is good too.

This is the only one I have experience with. I like it just fine. Google it yourself and you'll find the full text and can skim it if you want to. The problems I've done were actually pretty interesting.

You're welcome!

To be honest, I went out of my way to take courses in Tensor Analysis and Differential Geometry before I started learning GR, and I can't say it was that useful. It didn't hurt, but if your interest is just in learning GR, then most introductory GR textbooks teach you what you need to know. I'd recommend Schutz as a good book with tons of exercises, or Carroll ,partly because his discussion of differential geometry is more modern than that of Schutz.

I have addressed every point you've made that wasn't complete nonsense. Buy this, and read it. If you need stronger fundamentals to understand it, buy books on those and read them too. You cannot spontaneously acquire pre-formed knowledge of physics, you must study it.

> If you define space as having structure, then what is holding that structure?

Space and space-time are already very well-defined, and this whole thread started with you simply denying that. Surely you don't doubt the existence of three spatial dimensions, commonly denoted X, Y, and Z. That is structure. The fact there are exactly three dimensions (plus time) in our universe in the first place is already pretty interesting in and of itself. Whether or not something exists to "hold" that structure or it exists spontaneously is also a fairly interesting question, but its answer is irrelevant. Space does exist, it has structure, and that structure can be described.

The structure of our space is the reason "left", "up", and "forward" are all mutually perpendicular, and it's impossible to point your finger in a fourth direction that isn't already composed of those three. It's the reason moving "towards" a black hole is the opposite of moving "away" from it, but this only holds true in flat space, and our space is only flat when it is "empty", or devoid of mass and energy. Whether something is an "object" or not is irrelevant, what matters is mass-energy. The presence of mass-energy changes the essence of directionality. Within the event horizon of a black hole, "away" ceases to exist entirely. Every possible direction points closer to the center, hence, nothing can escape, no matter how fast it goes, or whether it has mass or not.

Less intense gravitational fields behave similarly. Light always travels in a straight line, but the very meaning of "straight" changes in the presence of mass and energy. The structure of our space defines how distances are measured, and "straight" is simply the shortest path between two points. It doesn't always look "straight" in the traditional sense. So a photon will curve around massive objects as if it were pulled by gravity, despite having no mass of its own.

That's the question I was answering in the first place. Maybe space doesn't actually curve but light simply behaves as if it does for some other reason, as observed through the "sense faculties" which, incidentally, relies on light behaving in a predictable fashion. If whatever you are trying to posit as an alternative cannot account for that, then it is neither a meaningful nor useful distinction to make.

Energy isn't conserved. It can be -- and is -- created and destroyed. This is a direct result of general relativity, and it's been known for close to 100 years, but somehow word never really got out about it except to cosmologists (probably because it is very close to conserved on "small" scales like the Milky Way).

Source. Another source is page 348 of this textbook (search for "loses energy").