What Math Books Should I Read to Understand General Relativity?

[whyevengothere]

This is kinda good: Koks, Don Explorations in mathematical physics. The concepts behind an elegant language

 

[haushofer]

Nakahara. The golden road between math.rigour and physical applications.

 

[atyy]

I liked Crampin and Pirani https://www.amazon.com/dp/0521231906/?tag=pfamazon01-20 and Fecko https://www.amazon.com/dp/0521187966/?tag=pfamazon01-20 (very weird, but very fun and good explanations). My background is a biologist reading for fun, so I think these are pretty approachable like Nakahara, which I like also.

A great free set of notes on GR is Blau's http://www.blau.itp.unibe.ch/GRLecturenotes.html.

And another free set on more advanced topics is Winitzki's https://sites.google.com/site/winitzki/index/topics-in-general-relativity.

 

[Sunnyocean]

Thank you very much atyy :)

 

[bolbteppa]

Even ignoring manifolds and point set topology you guys can't really ignore field theory & action principles, i.e. classical field theory (electromagnetism) & the calculus of variations, the way you can ignore manifolds.

If you just jump into GR with a modest background you could ruin the subject for yourself - if you see an action or the EM Lagrangian you can only hope the 5 pages of preparation (if at all) they spent on these was enough, also it'd be pretty awful to start dealing with field tensors without action principles & the theory as a whole would seem pretty divorced from the rest of physics what with all the Riemann's & Christoffel's taking up your time... In other words, the fun naive feeling you have in your head as to what GR is could be totally shattered, you could be left with that awful feeling of confusion, possibly be tricked into thinking that not knowing manifolds and dual spaces is the reason why you're having problems, or just generally have feeling that you can't understand something because you skipped something else that you should have done.

There is a nice way to actually study the structure of GR by studying the prerequisites properly. Look at the table of contents of Landau vol. 2: chapter's 1 - 9 are SR & EM (special relativity and electromagnetism) done absolutely from scratch using the (multi-variable) calculus of variations, 10 - 14 are general relativity. If you look closely you'll notice:
1, 2, 3 ~ 10
4 ~ 11,
5 ~ 12,
6, 7 ~ 13 (& bits of 9)
That is, chapters 1, 2 & 3 are an easy version of chapter 10, i.e. the structure of GR in chapter 10 closely parallel's what you already did in 1, 2 & 3 taking special relativity and electromagnetism as your example, what you do in chapter 4 is analogous to what you do in chapter 11, etc...

By studying this book you are already studying general relativity on chapter 1, just an easy version of one part of it in such a way that when you get to chapter 10 you'll really appreciate how radical a change GR is in context, you are doing it honestly in a way you won't have to un-learn, developing the necessary prerequisites, & you see the absolute unity of physics at your feet as a by product. No manifolds, no stupid vector calculus EM (which you don't need to know, better not to tbh), no memorizing Maxwell's equations as if they're god-given or rely on their heuristic derivations, and no need to resort to books like these.

The only potential roadblocks standing in your way of being able to pick this book up are the (multi-variable) calculus of variations and classical mechanics (based on single variable calculus of variations), & Landau's hard so you'll probably have to look in other sources at times, but it's always worth more to struggle towards understanding what he's saying than running away, though you might need tools in other books to see it in a nice way, e.g. Noether in Gelfand, Tensor's in Borisenko, remembering some crazy EM derivations via differential forms in MTW, fun things from Padmanabhan. Do whatever you have to do but "http://www.researchgate.net/post/Any_recommended_reading_for_physics_undergraduate_student"

Following the above approach is to think of GR as a natural extension of the calculus you know and love, but it is a local approach (local means you always assume a coordinate system and basis, though arbitrary). Taking the manifolds approach to GR is basically taking the point set topology (think Kelley) and Lebesgue measure approach to calculus, or at least the metric space & Riemann-Stieltjes approach to real analysis (still paying lip service to the Kelley & Lebesgue approach) so that you can take a global approach to GR (global means you don't impose any coordinate system or vector space basis, you do things before laying these out). The benefits are generality, nice global geometric intuition, and more... The cost is that you'll instantly be told to work with a special kind of second-countable Hausdorff topological space out of the blue & to impose conditions that are obvious in, or naturally fall out of, the local approach - if we're going to get so pedantic why not go back to axiomatic set theory and do it properly, taking your time to first do single & multi-variable calculus, using these as practice with your tools so that you'll have some intuition by the time you get to GR? It's chancy to take unfamiliar tools and apply them to something as notoriously difficult as GR when you can do GR using tools you have or are relatively easily accessible. If you didn't do it for calculus, why arbitrarily do it for GR & ignore doing GR in a way you can do based on what you've already done?

To learn GR directly through the language of manifolds without properly developing those tools is to learn a subject unintuitively paying lip service to special cases you have little to no appreciation for while using tools you have little/no intuition for leaving yourself open to committing some basic basic topological error should you stray too far from the path your book sets out. A hint at what you're doing is that you're shifting the emphasis from thinking of your metric as g_{\mu \nu} to thinking of it as g_{\mu \nu} = g(\hat{e}_{\mu},\hat{e}_{\nu}) and giving a set-theoretical & topological foundation to the notion of vectors on a curved surface, what a surface itself is, what it means to define g over the whole surface and then playing a lot of necessary games to do what is very natural using the coordinate approach. There's a lot of value to this approach, but if you didn't have the urge to go back to point set topology in your study of calculus (or axiomatic set theory like I did ) don't think it's absolutely necessary in order to understand GR when it's not.

 

[Sunnyocean]

Thank you bolbteppa, while I cannot by far claim that I understood everything you said, this is my feeling too: that I need a very solid understanding of all the mathematics involved if I am to really understand General Relativity.

 

[dextercioby]

I don't know, when I think about it, I only truly like the textbook by Ray d'Inverno, so that the level of mathematical sophistication is minimal, as said above by WBN, too much mathematics overshadows the physical content of a theory (though at some points may be useful).

 

[aleazk]

Re, Wald discussion:

To be honest, I consider Wald's 'General Relativity' one of the most beautiful textbooks I ever read. I remember when I was first studying the subject (by myself, before taking any GR course), I didn't like any of the textbooks I found. And then I randomly found Wald and it was like some kind of 'religious experience', lol. All, all of those doubts, questions, etc., that I just felt, but couldn't even formulate in words (remember, I was just a beginner in the topic), I found them beautifully formulated and answered in this book. I simply couldn't believe it. It was the first time I found a confirmation of my own mentality in a physics textbook, and I think Wald is the reason I stayed in physics rather than pursuing a pure math degree. So, as you may guess, my relation to that book is, for good and bad, personal. You can imagine my joy when I discovered that the relativity group in my university (I'm not from the US) had strong ties with the Chicago group of Wald and Geroch. Some years ago he came to my university and I thanked him (yes, I know I'm sounding a little silly with all this).

So, with all that rambling, you now have an idea of my mentality. I simply can't even think in the idea of making any calculation without knowing first, in a solid way, both the mathematical and conceptual foundations. Ok, I'm exaggerating a little, but you get the idea. And Wald's textbook gave me precisely that. Even his chapter on spinors has valuable insights.

But, of course, you can't base all of your study of a topic in one single book, GR and Wald are certainly not the exception. As I advanced in my understanding of the topic, I also studied from other textbooks, particularly for some of the most practical aspects. For example, in the second course I took on GR, we used mainly 'practical' books, like Poisson, Padmanabhan, Rindler, and many others.

But I also needed to supplement Wald with geometry books, like the wonderful textbooks by J.M.Lee. Even in the advanced topics (like causality theory and the singularity theorems), I found Wald a little sketchy sometimes. I often turned to Hawking&Ellis, Penrose's Techniques of differential topology, and others.

But all that said, the very solid conceptual basis that Wald gave me was of vital help for undertaking all of this.

One has to learn from a textbook the things at which that textbook is good. This may sound trivial and obvious; but, of course, if one tries to do the opposite, the chosen textbook is not going to be helpful. And I would say that this happens more often than not in some courses.

 

[Sunnyocean]

Re, Wald discussion:

So, with all that rambling, you now have an idea of my mentality. I simply can't even think in the idea of making any calculation without knowing first, in a solid way, both the mathematical and conceptual foundations. Ok, I'm exaggerating a little, but you get the idea. And Wald's textbook gave me precisely that. Even his chapter on spinors has valuable insights.

Couldn't agree more. That's me as well: before I add 1+1, I need to know all the algebraic axioms which constitute the foundation of addition. The same goes for all kinds of mathematics: geometry, calculus etc. It's the *only* way in which science should be done. Especially physics. And I stress the word "science". If you are doing accounting or are working in industry, you might not need all those axioms, of course.

Though accidents may occur if engineers don't have a sound knowledge of the physics or chemistry they apply. God forbid! So engineers too may need to know the very fine nuances of the physical concepts they use.

Also, thank you for the very useful and passionate answer.

 

[vanhees71]

Then I'd recommend Weinberg's "Gravitation and Cosmology" or Landau/Lifshits (Ricci calculus) or Misner, Thorne, Wheeler (modern Cartan form calculus). Another mathematically more advanced book is Straumann's General Relativity.

 

[robphy]

aleazk,

You might appreciate Bob Geroch's notes on relativity (and other topics)
http://home.uchicago.edu/~geroch/Links_to_Notes.html
which he scanned and posted above. (In the preface of his text, Wald acknowledges influence from Geroch.)

Recently, the Minkowski Institute Press (run by Vesselin Petkov) started getting some of those notes typeset in LaTeX
https://www.amazon.com/dp/0987987178/?tag=pfamazon01-20

You might also appreciate some of the notes of David Malament
http://www.lps.uci.edu/lps_bios/dmalamen
who was also at Chicago.

Geroch and Malament were part of the Conceptual Foundations of Science program at Chicago.

 

[aleazk]

Couldn't agree more. That's me as well: before I add 1+1, I need to know all the algebraic axioms which constitute the foundation of addition. The same goes for all kinds of mathematics: geometry, calculus etc. It's the *only* way in which science should be done. Especially physics. And I stress the word "science". If you are doing accounting or are working in industry, you might not need all those axioms, of course.

Though accidents may occur if engineers don't have a sound knowledge of the physics or chemistry they apply. God forbid! So engineers too may need to know the very fine nuances of the physical concepts they use.

Also, thank you for the very useful and passionate answer.

haha, yes, that's the spirit! and, as you say, God forbid! A friend of mine, a 'convert to physics', started his days in Geology. He said that he decided to go to physics because he was tired that in the chemistry classes, the teacher always started the lecture with the following statement: "this is the atom, this is Schrödinger's equation of QM for this atom... now, only God knows what happens with all this, but these are the results", and then he quoted some basic QM results and formulas. My friend, who has a very formal way of thinking, was furious. Of course, he soon realized that he was in the wrong place if he wanted to know more about "what only God knows". Anyway, I'm guessing it can be pretty hard to give a QM lecture for Geology students. For questions of time, background, etc., you simply can't be 100% precise. Certainly, it's not something I would like to do, I would suffer!

aleazk,

You might appreciate Bob Geroch's notes on relativity (and other topics)
http://home.uchicago.edu/~geroch/Links_to_Notes.html
which he scanned and posted above. (In the preface of his text, Wald acknowledges influence from Geroch.)

Recently, the Minkowski Institute Press (run by Vesselin Petkov) started getting some of those notes typeset in LaTeX
https://www.amazon.com/dp/0987987178/?tag=pfamazon01-20

You might also appreciate some of the notes of David Malament
http://www.lps.uci.edu/lps_bios/dmalamen
who was also at Chicago.

Geroch and Malament were part of the Conceptual Foundations of Science program at Chicago.

Thank you very much for these notes, robphy! I knew some of them because Geroch send them to my GR teacher by email, or he made a copy when he was in Chicago, can't remember (he did his phd with Geroch in the 80s*), but some of them are new to me. I definitely will check them. Yes, Geroch is certainly behind many of the novel approaches found in Wald (the abstract index notation, with Penrose, the approach to the covariant derivative, etc.)

*A nice Wald anecdote from this: my teacher told me that he often went to Geroch's office in order to ask him some questions about the work, but often Geroch was travelling, etc., and so the office was occupied only by a very young Wald. He said that, before addressing the actual question, Wald always preferred to very carefully establish both the mathematical and conceptual background of the question, thing which often was a little time consuming, but he did it very kindly nevertheless. Many times, my teacher says, the question was actually related to some very subtle misunderstanding of something in this background and that thanks to Wald and his approach he was able to solve it.