The book 'Gravitation' and supporting materials

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I'm a math major, but I really want to learn about general relativity (the 'real' GR, not the watered-down version that gets into popular science books) on my own. I'm not in any rush to learn it--i.e., I'm willing to work through a big tome like Gravitation over the course of, say, the next year, just in my spare time, as 'hobby reading'.

I think my physics background is slightly lacking for Gravitation, though. For instance, the section on differential forms uses de Broglie waves, about which I am not knowledgeable enough to follow the discussion.

So I'm looking for supplementary material on the following topics:

1) special relativity from a mathematically modern perspective (using Minkowski space);
2) a nice review of electromagnetism; and
3) something that will bring me up to speed on such topics as de Broglie waves.

More specifically, I'm looking for concise and to-the-point books on these topics. I really don't want to get a thick volume for each of them. I'm thinking that I wouldn't need to read more than 100 pages for each of the three categories--I do have some physics background, but I'm needing a refresher and I need to see some things (like SR) from a more modern perspective.

Any recommendations?

[As a side-note, Penrose's The Road to Reality sounded ideal to me, and I got it a few years ago. But I have to admit I found the explanations really hard to follow, even when he was discussing something I was already intimately familiar with--I've covered almost all the mathematical material in a classroom setting, for instance, but still find his writing on differential forms beyond confusing. I appreciate his effort, but am not convinced he succeeded.]
 

bcrowell

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I assume that you're referring to Misner, Thorne, and Wheeler?

Penrose's book is a strange, flawed masterpiece. Sometimes I find it very useful as a reference, but I doubt that there is a human being in the world who could read it from cover to cover and understand it, without already having encountered most of the ideas already.

When you refer to de Broglie waves, that's a quantum-mechanical term. You don't need to know anything about quantum mechanics to understand GR, which is a classical theory. If MTW mentions quantum issues, just skip that part.

The best book on SR that I know of is Taylor and Wheeler, Spacetime Physics.

The best E&M intro in my opinion is Purcell, Electricity and Magnetism. You should be able to find a copy if you have borrowing privileges from a good university library. Any E&M book will do lots of topics that are irrelevant for your purposes (circuits, electromagnetic properties of materials), so just skip those.

Actually, I think you should refine you choice of GR book a little. The choices aren't as black and white as pop-science versus a graduate textbook. There are lots of GR books that are aimed at people with a lower level of physics and math background, e.g., Hartle, Gravity: An Introduction to Einstein's General Relativity. My own book, which is free online http://www.lightandmatter.com/genrel/ , is also aimed at that level. There's also Rindler.
 
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I was able to get through Penrose's first hundred pages or so just fine..and I have been away from mathematics for a loooong time..... and then hit the same wall I think is referred to above...small sections later in the book are comprehensible, but overall it's either at a very advanced or rather opaque level....

If you search the forums here for previous threads on special relativity...you'll pick up some useful insights and some recommended references as well....
 

bcrowell

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I was able to get through Penrose's first hundred pages or so just fine..and I have been away from mathematics for a loooong time..... and then hit the same wall I think is referred to above...small sections later in the book are comprehensible, but overall it's either at a very advanced or rather opaque level....
There are certain places where it's great. For example, it has the best description of quaternions that I know of, and a great explanation of sectional curvature as opposed to the curvature measured by the Ricci tensor. Even though it doesn't really work as a from-scratch intro to physics, it at least *attempts* to explain it all from scratch, avoiding unnecessary technicalities and giving clear physical motivation and interpretation for everything. It's the first place I go when I want an explanation of a specific topic, and if it does explain that topic, usually it does it better than any other book -- but you will have to go somewhere else to get any mathematical calculations, for example.
 

Ben Niehoff

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I find MTW a bit longwinded at times, and its exposition of the mathematical material is nonstandard, tiresome and confusing. But one benefit is that it covers a lot of physical phenomena. For that reason it is good to have on your shelf.

For learning more of the differential geometry, I recommend Carrol's GR text, and Nakahara's text on differential geometry and topology. Nakahara uses more formal language, and probably defines differential forms in a way you are familiar with (and certainly makes no mention of de Broglie waves, which are completely irrelevant).
 

atyy

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For a math major, I'd recommend Ludvigsen, which I got to via Penrose. It's very short, very geometrical, and very physical (ie. points out which experimental results correspond to which mathematical axioms). There are a few holes in the chain of reasoning, which one can look up in MTW or Rindler later.
 

bcrowell

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MTW is also 40 years out of date. It was a great book for young physicists of two generations ago, but at this point I don't think it's a sensible starting point for learning GR. If the OP really wants a graduate text rather than an undergraduate one (why???), then Carroll seems pretty good to me, and there is a free online version.
 

pervect

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Wald, "General Relataivity" is probably dated by now, but it'd be a good choice for a mathemetician I think. Concise, and rigorous - at the expense of perhaps sometimes being not very friendly to one's intuition. I haven't read Caroll's, but the price is right. I found having more than one text helpful, sometimes one text wouldn't explain something well but the other text would.
 
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I would also not recommend MTW's monster as an introduction.

I can't think of any GR books that review E&M in a way useful to the uninitiated. I would suggest Feynman's coverage of E&M in volume 2 of The Feynman Lectures (in fact, the whole set is good for background reading) or Schwartz, Principles of Electrodynamics. Something more recent which may fit your 100 page criterion (well, 144 pages), but which I have not read, is A Student's Guide to Maxwell's Equations.

https://www.amazon.com/dp/071670336X/?tag=pfamazon01-20 is the best way to get a strong grounding in SR.

Of GR books, Schutz does a good job introducing SR. Half of Rindler is an excellent and thorough intro to SR.

Taylor & Wheeler wrote a sort of sequel to Spacetime Physics titled Exploring Black Holes, which avoids differential geometry. A similar book is Ellis & Williams, Flat and Curved Space-Times. Not the full monty, but may be satisfying enough.

Some comments on textbooks:

Hartle, Gravity. Aimed at undergraduates. Tries to do as much Physics as possible before introducing the full apparatus of Riemannian geometry. Really great on recent experiments and observations.

Rindler, Relativity: Special, General, and Cosmological. Excellent on Special Relativity. Often eccentric, but also offers much insight and has many clever derivations. May frustrate the mathematically inclined, though. Rindler also wrote a book on Special Relativity only that has a little more material than is here.

Schutz, A First Course in General Relativity. More traditional (Riemann tensor before physics), but gentler than the older texts.

Ohanian, Gravitation and Spacetime, 2nd ed.. Very physical.

Carroll, Spacetime and Geometry. Probably the best modern graduate text.

Wald, General Relativity. More precise mathematically. Sometimes to a fault, I think.
 
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For E&M, I've discovered Vol. 2 of the Course of Theoretical Physics, The Classical Theory of Fields. This is, more or less, exactly what I need. It is the best treatment of this material I've ever seen. The only downside I can find is that its view of tensors is antiquated (in fact, I would contend that the view of a tensor as being 'a set of quantities that transform in such-and-such way' is the worst possible view of tensors one could ever have). Nevertheless, the rest of the book is absolutely outstanding, and MTW thankfully uses a more modern conception of tensors (even though the explanations aren't rigorous).

I've been so inspired by Vol. 2 of this series that I'm now planning on reading the first three volumes, including the first on mechanics, despite my familiarity with the calculus of variations.

What's generally thought of this series of books (the Course of Theoretical Physics)? These books seem to me to avoid all the problems that devastate much of physics education and which actually drove me away from the study of physics to mathematics. They feel a bit like Feynman's lectures except without Feynman holding himself back from doing some actual math. They're weak on problem-solving, but you don't learn problem-solving from a book--you learn it by doing it; there is no other way. On the other hand, unless you're Emmy Noether, you're not going to discover that the conservation laws can be derived from symmetries on your own without a book.
 
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For mathematics people, the language may be more familiar to you if you use Barrett O'Neill's book "Semi-Riemannian Geometry: with Applications to Relativity".
 

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