On the level of detail required for GR math

[Shirish]

I'm reading 'A Most Incomprehensible Thing: Notes towards a very gentle introduction to the mathematics of relativity' by Collier and came across the following paragraph in the chapter on tensors:

Later on, we'll meet the rules of tensor algebra, including operations such as scaling [...] and contraction [...]. Differential geometry is the theoretical foundation to these rules. However, just as you don't need to be an automotive engineer in order to drive a car, you don't have to know all the underlying mathematics if you want to manipulate tensors - just a working knowledge of the rules of tensor manipulation, which you can more or less learn by rote. So, much of this section is 'under the bonnet' detail - useful but not essential.

I'm a bit skeptical about the "useful but not essential" part, but I'd still like to get the opinion of the members here. Let's say a student of GR (or other branches like QFT or quantum gravity candidate theories, etc.) simply memorizes the rules of tensor manipulation without paying much attention to the underlying differential geometry - would they struggle at any point? Would this handicap them in any way (or maybe I should ask how seriously would this handicap them)?

Personally I definitely intend to learn differential geometry, but I'm still curious on the above questions.

 

[etotheipi]

I would also be interested in some answers to this question. It's difficult to tell when studying whether you are actually gaining an understanding of the maths or whether you just become more familiar with the symbolic manipulations.

 

[robphy]

It depends... what specifically do you want to do?

GR is a broad field, with different specialized problems requiring different in-depth knowledge and skills. Tensor calculations would be one skill that’s useful for some problems and differential geometry is another skill (with some overlap) that’s also useful for some problems... but maybe they are not needed in depth for some other problems.

 

[Shirish]

It depends... what specifically do you want to do?

GR is a broad field, with different specialized problems requiring different in-depth knowledge and skills. Tensor calculations would be one skill that’s useful for some problems and differential geometry is another skill (with some overlap) that’s also useful for some problems... but maybe they are not needed in depth for some other problems.

Thanks! Since I'm just starting out, I don't specifically have an idea what I want to do. Still, I'd like to know what subfields (broadly speaking) in GR specifically need differential geometry and which ones don't. I'm not asking about "subfields" in the sense of very specific research problems - just broad topics (e.g. dark energy, Hawking radiation, specific candidates for quantum gravity), etc.

 

[PeroK]

Thanks! Since I'm just starting out, I don't specifically have an idea what I want to do. Still, I'd like to know what subfields (broadly speaking) in GR specifically need differential geometry and which ones don't. I'm not asking about "subfields" in the sense of very specific research problems - just broad topics (e.g. dark energy, Hawking radiation, specific candidates for quantum gravity), etc.

Are you self studying?

 

[Shirish]

Are you self studying?

Yep!

 

[PeroK]

Yep!

What stage have you reached? What topics would you say you have mastered?

 

[romsofia]

People definitely memorize tensor manipulations and get by in a 1st course in GR! Anything involving einstein's equations will need tensors, and anything involving calculating curvature of spacetimes will need differential geometry, but you could memorize all the steps without *true* understanding. But that's not the fun of GR, so I wouldn't recommend that. However, I sympathize. I think initially the majority of people get overwhelmed with the math in GR. So, take it slow. Try to find meaning of the math in your own words.

Also, I've seen your other threads, and since you said you're self-studying, let me recommend a good *formal* treatment from one of my idols in the field: https://www.amazon.com/dp/0199666466/?tag=pfamazon01-20

It has solution to problems, and enough material so that you don't get too bogged down in the math (GR is VERY computation heavy). If you're having success with your current book, keep going with it until you feel satisfied.

 

[PeroK]

Yep!

How much time do you have to study? Are you working, retired?

Your profile says you have a Master's degree. Is that Physics/Maths?

 

[Shirish]

People definitely memorize tensor manipulations and get by in a 1st course in GR! Anything involving einstein's equations will need tensors, and anything involving calculating curvature of spacetimes will need differential geometry, but you could memorize all the steps without *true* understanding. But that's not the fun of GR, so I wouldn't recommend that. However, I sympathize. I think initially the majority of people get overwhelmed with the math in GR. So, take it slow. Try to find meaning of the math in your own words.

Also, I've seen your other threads, and since you said you're self-studying, let me recommend a good *formal* treatment from one of my idols in the field: https://www.amazon.com/dp/0199666466/?tag=pfamazon01-20

It has solution to problems, and enough material so that you don't get too bogged down in the math (GR is VERY computation heavy). If you're having success with your current book, keep going with it until you feel satisfied.

You're right. The way I'm doing it now is getting an introductory understanding to differential geometry concepts, like tangent spaces, vectors are operators and dual vectors are differentials, and how they naturally lead to the component and basis transformation equations.

I don't like the idea of simply memorizing the transformation equations. But with that said, I'm also not, at least at this point, getting into the more advanced stuff right off the bat - as in at this point I don't understand words like 'open cover' or 'Hausdorff space', etc. If I find any result that's generally supposed to be taken for granted (like the transformation relations), I'll try to learn slightly more advanced concepts which allow me to understand those results' derivations or underlying meaning.

I'm trying to avoid the purely Physics route with frivolous math understanding, and also to avoid the hardcore math route which will probably end up with me getting way too lost in the math.

And thank you for the recommendation! The amazon reviews seem to say that the book requires advanced math pre-requisites beyond differential geometry though. What's your experience with it?

 

[Shirish]

How much time do you have to study? Are you working, retired?

Your profile says you have a Master's degree. Is that Physics/Maths?

Oh I'm working! I guess time to study per day varies. In some weeks it can be around 2-3 hrs a day. In other busy weeks it's about 4-5 hrs a week, so I try to catch up when I get the time. I have masters in Physics but I was an unmotivated, average student. I'm extremely motivated now, even though it's 10 years too late

 

[robphy]

It might be best to look through threads like
https://www.physicsforums.com/threads/book-recommendations-in-general-relativity.916947/
and similar ones at the bottom of that page.

Find a set of books that fit with your preparation, learning style, and goals.

Presumably, those books (in their prefaces, tables of contents, etc) will indicate the levels of mathematical detail needed.
What is “frivolous math” vs what is “hardcore math” is user-dependent.

 

[Shirish]

It might be best to look through threads like
https://www.physicsforums.com/threads/book-recommendations-in-general-relativity.916947/
and similar ones at the bottom of that page.

Find a set of books that fit with your preparation, learning style, and goals.

Presumably, those books (in their prefaces, tables of contents, etc) will indicate the levels of mathematical detail needed.
What is “frivolous math” vs what is “hardcore math” is user-dependent.

Thanks for referring me to that thread! And you're right - being mostly a beginner, the math I find difficult will definitely be frivolous for the more experienced users here.

 

[haushofer]

Well, one of the more formal stuff which I find very important to grasp (which, at first, I didn't fully), is the notion of a tensor as geometric objects which do not depend on your coordinate choice. This confused me, because wasn't the whole notion of relativity to tell you how stuff changes when changing perspective/coordinates? That blocked my understanding for a while.

 

[romsofia]

I'm trying to avoid the purely Physics route with frivolous math understanding, and also to avoid the hardcore math route which will probably end up with me getting way too lost in the math.

And thank you for the recommendation! The amazon reviews seem to say that the book requires advanced math pre-requisites beyond differential geometry though. What's your experience with it?

I came into it past my PhD Quals, so maybe my experience won't be as genuine, but everything in this book is presented well enough that you should be able to follow her logic *if* you do the problems. That's the key of the book, there are exercises inside each chapter (not just at the end). Her chapter on special relativity is probably the best I've seen because of all the little footnotes on the side of the pages. She also takes the time to talk about what's physical, how to model matter, etc in a way that I haven't seen done in other books.

There were techniques when I was younger that I didn't bother studying the true properties of (Lie stuff mainly) because I thought I'd never need them. Turns out, you can use them for differential equations, and now I regret not putting in more effort when I first encountered them! So, when you read these books, keep that in mind. These authors have been doing research and mentoring students for decades (well some), and have been through many moments like the one above.

I wish I knew a sweet spot of theorems you *need* to know in order to understand GR because it isn't differential geometry. A lot of GR (since it is a branch of physics) is solving differential equations, which is its own story.

At the end of the day, most people skip around when you study any branch of physics; if you're getting bored of the math, skip ahead, read some of the physics, and if you can follow (and solve some problems) then keep going IMO. Good luck!

 

[kent davidge]

It's difficult to tell when studying whether you are actually gaining an understanding of the maths or whether you just become more familiar with the symbolic manipulations

That's why they say students should work on problems. That's a way of seeing if the concepts sank in.

 

[sunrah]

Well, one of the more formal stuff which I find very important to grasp (which, at first, I didn't fully), is the notion of a tensor as geometric objects which do not depend on your coordinate choice. This confused me, because wasn't the whole notion of relativity to tell you how stuff changes when changing perspective/coordinates? That blocked my understanding for a while.

Yes, this is the important point imo. Relativity is a geometric theory that is independent of coordinates. That means when we boost from one set of coordinates to another we are actually (in the words of Carroll) rotating time and space into each other (this is exemplified very well in the first chapter of his lecture notes), this leaves the spacetime interval invariant - and is why it is said to be coordinate independent.

BTW I am yet to see a good first book on the subject other than the introductory book by Lasenby, Efstathiou, and Hobson. Sean Carroll's notes are great to read but not if you haven't used tensors before.

 

[Shirish]

BTW I am yet to see a good first book on the subject other than the introductory book by Lasenby, Efstathiou, and Hobson. Sean Carroll's notes are great to read but not if you haven't used tensors before.

Thanks so much for the reference! I'll go through it when I'm done with the mathematics part in Luscombe's book (which by itself is a pretty amazing text).

 

[George Jones]

BTW I am yet to see a good first book on the subject other than the introductory book by Lasenby, Efstathiou, and Hobson. Sean Carroll's notes are great to read but not if you haven't used tensors before.

Have you looked at the books:

1) "Gravity: An Introduction to Einstein's General Relativity" by James Hartle;

2) "A General Relativity Workbook" by Thomas Moore?

Hartle believes that tensors and differential geometry can wait until after substantial familiarity with general relativity has been built up. Consequently, Hartle does not introduce tensors until page 427, but when introduced, they are presented in “modern” style as multilinear maps for which the transformation properties are a derived concept. Moore uses tensors from the get-go, but they are only ever defined by transformation properties.

Hartle and Moore have both written articles on their contrasting philosophies for teaching general relativity:

https://arxiv.org/abs/gr-qc/0506075

https://www.aapt.org/doorway/TGRU/articles/Moore GRArticle.pdf