What Math Books Should I Read to Understand General Relativity?

[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.