Reddit mentions of Spacetime and Geometry: An Introduction to General Relativity

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.

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.

Griffiths is the go-to for advanced undergraduate level texts, so you might consider his Introduction to Quantum Mechanics and Introduction to Particle Physics. I used Townsend's A Modern Approach to Quantum Mechanics to teach myself and I thought that was a pretty good book.

I'm not sure if you mean special or general relativity. For special, /u/Ragall's suggestion of Taylor is good but is aimed an more of an intermediate undergraduate; still worth checking out I think. I've heard Taylor (different Taylor) and Wheeler's Spacetime Physics is good but I don't know much more about it. For general relativity, I think Hartle's Gravity: An Introduction to Einstein's General Relativity and Carroll's Spacetime and Geometry: An Introduction to General Relativity are what you want to look for. Hartle is slightly lower level but both are close. Carroll is probably better if you want one book and want a bit more of the math.

Online resources are improving, and you might find luck in opencourseware type websites. I'm not too knowledgeable in these, and I think books, while expensive, are a great investment if you are planning to spend a long time in the field.

One note: teaching yourself is great, but a grad program will be concerned if it doesn't show up on a transcript. This being said, the big four in US institutions are Classical Mechanics, E&M, Thermodynamics/Stat Mech, and QM. You should have all four but you can sometimes get away with three. Expectations of other courses vary by school, which is why programs don't always expect things like GR, fluid mechanics, etc.

I hope that helps!

It sounds like you want to understand both GR and the standard model of particle physics. For an intro to GR, try Sean Carroll's book or lecture notes. For the standard model, try Srednicki's book (there is a preprint PDF available from the author).

You should have a solid understanding of both GR and the standard model if you want to try to explain the weakness of gravity through beyond-SM or beyond-GR theories. There are no compact extra dimensions in the SM or GR, so what you're talking about is already beyond-SM / beyond-GR. The type of model which tries to combine gravity and standard model forces via compact extra dimensions is called a Kaluza-Klein model and it's been around for a long time (1921!). The more modern ideas about explaining the weakness of gravity through extra dimensions are e.g. DGP models or RS models or cascading gravity ... they seem kind of contrived to me, but there's no accounting for taste.

Yes, he proved it, which is the entire reason why Einstein is so famous in the first place. Here's an excellent introductory textbook on the subject if you're not afraid of the math.

General relativity is the most iron-clad, battle-tested theory in all of science, along with quantum mechanics. None of its predictions have been proven false yet. Just last fall, one of the Einstein's predictions, gravitational waves, was proven to be true by an experiment called LIGO (Laser Interferometer Gravitational-Wave Observatory).

Spacetime and Geometry by Sean Carroll

Oh, PDF. Hmm...

For Statistical physics I would second the recommendation of Pathria. Huang is also good.

For electromagnetism the standard is Jackson. I think it is pedagogically terrible, but I was able to slowly make my way through it. I don't know of a better alternative, and once you get the hang of it the book is a great reference. The problems in this book border from insane to impossible.

So that's the basics. It's up to you where to go from there. If you do decide to learn QFT or GR, my recommendations are Itzykson and Carroll respectively.

Good luck to you!

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.

Hartle: http://www.amazon.com/Gravity-Introduction-Einsteins-General-Relativity/dp/0805386629/ref=sr_1_7?ie=UTF8&qid=1420630637&sr=8-7&keywords=general+relativity

Pretty introductory, not a ton of math but enough to satisfy most undergrads. Includes a section on introductory Tensor Calculus.

Carroll: http://www.amazon.com/Spacetime-Geometry-Introduction-General-Relativity/dp/0805387323/ref=sr_1_3?ie=UTF8&qid=1420630637&sr=8-3&keywords=general+relativity

Probably the best intermediate book, does GR at an intermediate level. Includes several chapters on the math needed.

Wald: http://www.amazon.com/General-Relativity-Robert-M-Wald/dp/0226870332/ref=sr_1_2?ie=UTF8&qid=1420630637&sr=8-2&keywords=general+relativity

Covers GR at a fairly advanced level. More rigorous books exist, but are not appropriate for a first course.

I just want to add on here that Sean Carroll is a highly, highly respected physicist too. His intro to general relativity is widely used as a graduate textbook.

https://www.amazon.com/Spacetime-Geometry-Introduction-General-Relativity/dp/0805387323

So yeah, this guy is a big deal. He knows his shit. Not saying you're implying the opposite, just a nice tidbit 🙂

Ahh... I like you!

Great question!

The answer to this is actually extremely complicated. In fact, energy conservation is actually not true in general relativity. If you are interested in reading about this, check out Sean Carroll's blog entry on the topic. He's a well known cosmologist at Caltech who wrote a textbook on general relativity.

My undergraduate GR course used Spacetime and Geometry by Sean Carroll, which has a discussion of gravitational lensing in section 8.6. The problem is that the discussion there is built on the rest of the book, which is on the mathematically rigorous side of things. Also, it is kind of expensive, but you might be able to find it in a library.

Ok, so I would recommend Carrol's Spacetime and Geometry http://www.amazon.com/Spacetime-Geometry-Introduction-General-Relativity/dp/0805387323

If you are feeling more up to snuff with tensor calculus and mathematical analysis and can wade your way through R_n analysis, (in terms of problem solving and approaches), then go for Wald's Genearl Relativity http://www.amazon.com/General-Relativity-Robert-M-Wald/dp/0226870332

edit: warning: both of those books are graduate level. Any GR is only taught at grad level, but I took GR with Wald (yep the guy himself) my 3rd year with similar background to yours. You will be fine, but its going to be a lot of head beating against the wall. Some of that stuff is really complex and will possibly require more than one source to understand. JUST the book may not be enough. I would even recommend you talk to your local GR prof and see if you can send him questions as you work through this; I cant imagine any good professor refuse to help you in this way, as long as you dont send a question every 5 minute and they are actually substantial.

also, anything else you would be stepping lower than carrol and i would advise against it if you wanted to get a good grasp of mathematical approaches and rigorous proofs (especially Wald in this case)