Learning General Relativity

July 6, 2009 6:06 AM

What textbook can I use to learn General Relativity, including the associated math?

What I have so far

I'm pretty firm on calculus and basic physics (and of course Euclidean geometry, etc). Less firm or polished, but technically familiar with vector calculus. I recognize words such as "laplacian" from electromagnetics, but really only have a vague understanding of the actual concepts.

What I'm interested in

I'd like to learn GR. And of course to learn GR, I need to learn some of the underlying mathematics such as differential geometry. I'm also interested in learning perhaps 10% more diff geo than is strictly required for GR.

The field

I have access to a largish technical library covering stuff from math to programming to astronomy to thermodynamics to engineering. "Classics" are probably in the catalog, but may be permanently checked out.

The requirements

Requirement 0 is really more of a guideline: I'd like something that teaches both GR and the needed math together. However, if there are N books that otherwise qualify and teach the subjects separately, I'm open to the idea.

1) Not overly formalized. I prefer a conversational, readable textbook.
2) Good problem sets. (I.e. not just one or two per chapter)
3) Answer key.
4) Not a "bible" or an "elegant reformulation". I need to be able to learn from it, not marvel at the comprehensiveness or elegance from a position of already knowing the subject.

An example of a nearly great suggestion is Gravity: An Introduction to Einstein's General Relativity by James Hartle. Why only "nearly" great? No answer key. Unfortunately, as perfect as the book otherwise appears, this renders it useless to me.

Some examples of actually great suggestions, but on different topics. If you know (of) these books, you will know the kind of thing I'm looking for:

Fundamentals of Physics by Halliday, Resnick and Walker
Div, Grad, Curl And All That by Schey.

posted by DU to science & nature (22 comments total)

Not kidding about that Hartle thing. I've actually tried using it 3 times but can't get past the first problem set. Each time I think "pfff...who needs answers" but then I do a problem I'm shaky on and there's no feedback on correctness. Not a firm foundation for the next problem set...
posted by DU at 6:09 AM on July 6

Use Hartle and rather than picking the problems you feel like doing, use the syllabus for a GR course that also uses Hartle as the main text. Many professors post the solutions to the problems online afterward so if you use the website from the previous semesters course you should be able to get your hands on the solutions as needed.
posted by Loto at 6:22 AM on July 6

An an example: http://www.physics.queensu.ca/~phys414/

Lectures and problem sets from an undergraduate GR course.
posted by Loto at 6:27 AM on July 6

I'm just going to append to Loto's example from Queen's physics: Prof. Widrow is quite a good prof, he taught me a course or two (not this one) and was not as scattershot as some other profs.
And along those lines, have you tried, well, emailing a random prof who teaches the course? I know that my math profs would get a couple of emails a year from curious outsiders asking them for reading lists, and they always seemed willing to help / amused [admittedly, they were in smaller fields than GR, which helped, but still].
posted by Lemurrhea at 6:37 AM on July 6

I found those worked problem sets. Unfortunately, I need to grind through more than 3 or 4 problems to really get something under my belt. For my "great" suggestions, I did all the problems I could, i.e. all the ones that had answers in the back. Halliday typically has 60-100 problems per chapter, so that's about 30-50 problems * 50 chapters = in the neighborhood of 2000 physics problems.

Also, I should have mentioned this before: I already did quite a bit of googling to see if I could find a pirated key to Hartle. I found claims of such a key but no actual key.
posted by DU at 6:50 AM on July 6

For my intro course in diff.geom, we used Elementary Differential Geometry by Barrett O'Neill. Just took a quick glance: in chapter 4 (calculus on a surface), there are 8 sections, each with approx. 15 problems, half of which are solved in the back, so about 50-odd questions/chapter. All on differential geometry.

I found the book good - not perfect¹ - but worth checking out as an introduction.

1. I did diff.geom after taking a course about Lie Groups & Algebras, which is backwards, so my opinion on the book will be a little diminished.

As a side note, I'm going to recommend to you Enumerative Geometry, although for the future, not yet. It's high-end geometry, the math behind Calabi-Yau manifolds and other string-theory concepts. This particular book is fantastically-written, although light on the problem sets.
posted by Lemurrhea at 7:12 AM on July 6 [1 favorite has favorites]

No suggestion, but as a word of advice if you didn't already know: differential geometry doesn't make sense the first two times you see it or so. Or at least, that seems to be the consensus amongst all the math grad students I know, as well as some professors.
posted by TypographicalError at 7:14 AM on July 6

The solution manual for Hartle does exist; I used it when I was a TA for an undergraduate GR course. If you want to get your hands on it, you might try seeing if any local universities use Hartle in their intro GR courses (course websites are good for this), and then contact the professor and see if he or she would be willing to help you out.

Beyond Hartle, the best book I can think of for intro GR is A First Course in General Relativity, by Bernard Schutz. I'm only familiar with the first edition (in fact, until I looked it up on Amazon just now I wasn't aware there was a second edition), but it has a fairly mainstream approach to the subject, 20-40 exercises per chapter, and an Appendix containing "Hints and Answers to Selected Exercises."
posted by Johnny Assay at 7:21 AM on July 6 [1 favorite has favorites]

"Conversational and readable" means you want Spacetime and Geometry by Sean Carroll. I enjoyed reading this book more than I enjoyed any other physics textbook. His derivations are generally rigorous, but there's enough connective tissue in between the derivations to put everything into context and explain why the particular derivation matters.

I know this doesn't answer your key point about an answer key, but if that requirement turns out to be infeasible I strongly recommend Carroll's book.

(Side note: Sometimes you can find answers to one author's GR problems in another author's worked examples. Convenient when it works out, but doesn't really suit your needs.)
posted by kiltedtaco at 7:23 AM on July 6 [1 favorite has favorites]

A Working Link to Carroll's Book might be nice.
posted by kiltedtaco at 7:24 AM on July 6 [1 favorite has favorites]

Yes, I've found Carroll's book next to Hartle's on the shelf. Another nearly great suggestion, sadly.

Even more sadly, the library doesn't appear to have O'Neill or Schutz. I even remember searching for Schutz last time I went looking for this. Grrrrr.
posted by DU at 7:28 AM on July 6

It appears, upon further research, that the second edition of Schutz does not contain the solutions as the first edition does; what I assume is the equivalent of the first edition's Appendix B is now available on the publisher's website, but only to instructors. You'll probably still be able to acquire a copy of the first edition via a used bookstore, but be aware that some of the chapters on the applications of GR are rather out of date, having been written in 1985. In particular, the first edition's chapter on cosmology should only be read in concert with another text written within the past five years. The current state of searches for gravitational waves is also greatly different from that described in the first edition.
posted by Johnny Assay at 7:38 AM on July 6

I own Foster & Nightingale but also have Carroll, and prefer and would recommend the latter.
posted by edd at 8:28 AM on July 6 [1 favorite has favorites]

Oh god, please don't use Hartle. It's awful.

We had a fourth-year course in GR from that book, and while I'll admit that it could have been more the presentation than the book itself, not a one of us in that class learned any GR from it. It really felt like a very ad-hoc presentation, explaining none of the hows-and-whys, going through a few examples without really helping build any of the intuition.

I suppose, to be fair, that we really only used the first third (or maybe half? I forget) of the textbook, and it's possible that had we covered the entire book we would have came out better. Maybe. I'd suggest looking elsewhere though.
posted by vernondalhart at 8:51 AM on July 6 [1 favorite has favorites]

OHanian was my main textbook for GR, but I believe there were only selected answers in my version. Also, it did not teach you the tensors required. So cross that one off of your list, or use it as a supplementary text.
posted by adipocere at 8:57 AM on July 6 [1 favorite has favorites]

When I had a similar desire some years ago, I picked up a copy of Gravitation, by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler.

It goes out of it's way to explain the math, the illustrations are profuse and wonderful, and the context of the material being explained in the greater body of physics is never neglected.

You can see quite a lot of it with the preview function in Google Books, as I've linked.

When I saw you had 12 answers, I made a little bet with myself at least three of them would be about this book. Shows how out of touch I am.
posted by jamjam at 9:05 AM on July 6 [1 favorite has favorites]

Yeah, you can't really effectively use Hartle without delving into Part III, which explains the notions behind curvature and the Einstein equation. I can see how a course that taught directly from the first 30-50% of the book would be frustrating.

What the hey, here's my capsule opinions on the other GR books I'm familiar with:

• Foster and Nightingale (mentioned above) is OK, but a little confusingly organized.
• Wald is an excellent text once you already understand the majority of it, but it is very formal and not a good text for a first pass at the material.
• Misner, Thorne, and Wheeler is a 1279-page behemoth, chock-full of good information, but I suspect you were specifically trying to rule it out with criterion #4 above.
• Weinberg is unnecessarily dismissive of the geometrical formulation of GR, much to (in my opinion) his text's detriment. It does treat some topics that can't easily be found elsewhere, but (again) nothing that you would need on a first pass.

posted by Johnny Assay at 9:13 AM on July 6 [1 favorite has favorites]

Yes, I had Thorne in mind when I wrote #4, but perhaps in error. Extra explanations, good illustrations and wider connections are all good things pedagogically. If it's also a "bible" I don't mind at all. The library has it (as it damn well should, but I'm sarcastically surprised due to other things being missing) so I guess I'll glance over it. Too bad there isn't a 3 volume version...
posted by DU at 9:24 AM on July 6

Confirmed: I was unfair to Misner/Thorne when writing #4. Also, I really like the "two tracks" format that lets the student dabble in the more advanced stuff without requiring it. However, it still fails via #3, much to my disappointment.
posted by DU at 9:43 AM on July 6

When I had a similar desire some years ago, I picked up a copy of Gravitation, by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler.

As a bonus, it weighs enough to demonstrate gravitational lensing.
posted by atrazine at 2:18 AM on July 7

I did this with Schutz.
posted by fantabulous timewaster at 4:28 AM on July 7

I was also taught with Misner, Thorne and Wheeler. I knew I'd also find a comment with the old joke about how the MTW book itself exerts a significant influence on the gravitational field. :)

My earlier Mefi Post on web resources recommended by 't Hooft leads to:

- His own lectures and exercises.

- Carroll's lecture notes

- Pope's lectures on Geometry and Group theory. (link is to PDF)
posted by vacapinta at 7:28 AM on July 7 [1 favorite has favorites]