Mathematics book before General Relativity

[Whitehole]

Hi, I'm new here and I'm trying to learn GR. I wanted to know the math books that I need to tackle GR properly, so far the books that I came across are:
Tensor Analysis on Manifolds by Bishop and Goldberg
Tensors, Differential Forms, and Variational Principles by Lovelock and Rund

I have a good background in standard undergraduate mathematics for physicist (Calculus, Linear Algebra, Differential Equations, etc). Can anyone comment about the books that I cited above? Also, what is the difference between tensor analysis and differential geometry? Some search in google gave me the idea that tensor analysis belongs to differential geometry, and other posts say that tensor analysis is just an extension of linear algebra. I'm confused. What do I really need in order to tackle GR "properly"? I have studied SR already so don't recommend me to study it first. Thanks!

 

[mpresic]

I have a copy of both books and I think both are good. I think Hartle's book or Carroll's book are self-contained, though

 

[Cruz Martinez]

Hi, I'm new here and I'm trying to learn GR. I wanted to know the math books that I need to tackle GR properly, so far the books that I came across are:
Tensor Analysis on Manifolds by Bishop and Goldberg
Tensors, Differential Forms, and Variational Principles by Lovelock and Rund

I have a good background in standard undergraduate mathematics for physicist (Calculus, Linear Algebra, Differential Equations, etc). Can anyone comment about the books that I cited above? Also, what is the difference between tensor analysis and differential geometry? Some search in google gave me the idea that tensor analysis belongs to differential geometry, and other posts say that tensor analysis is just an extension of linear algebra. I'm confused. What do I really need in order to tackle GR "properly"? I have studied SR already so don't recommend me to study it first. Thanks!

The book by bishop and Goldberg is very terse. By way of example, if you never did topology before from a more extensive sourse like e.g. Munkres or somethiing similar, it will be very difficult to absorb the knowledge from the chapter devoted to topology in Bishop et al. This is because this section is not meant as a pedagogical text, but as a reference for people who already know the material.

 

[Whitehole]

I have a copy of both books and I think both are good. I think Hartle's book or Carroll's book are self-contained, though

I know Hartle is self-contained but it does not use the math extensively and so is suitable for undergraduates but when I tried reading Carroll, there are too many terms that he does not explain really well assuming the reader does not have any background on the mathematics he is spitting out. This is one of the reason I want to study the math needed in order to study GR properly.

 

[Whitehole]

The book by bishop and Goldberg is very terse. By way of example, if you never did topology before from a more extensive sourse like e.g. Munkres or somethiing similar, it will be very difficult to absorb the knowledge from the chapter devoted to topology in Bishop et al. This is because this section is not meant as a pedagogical text, but as a reference for people who already know the material.

So what route do you recommend I take given my situation? I don't have any background in the tensor and differential geometry stuff. How about the other book that I stated?

 

[Cruz Martinez]

So what route do you recommend I take given my situation? I don't have any background in the tensor and differential geometry stuff. How about the other book that I stated?

This is where you have to figure out how well you want to know your math. I myself am taking the long route, it is a long route however, but if youre like me, very mathematically oriented this will be very rewarding in the end.
You can start with some fundamental topology as in the first four chapters on John Lee's "Introduction to Topological Manifolds". This covers topological spaces, subspaces, product spaces and quotient spaces, connectedness and compactness. This is a cool book because it develops topology with an eye on manifolds the whole time.
After this you can do smooth and pseudoriemannian manifolds. This pretty much covers the math of GR.

On the other hand, it is possible to have a good working knowledge of GR without mastering all the intricacies of the math behind it. This will take up less time if youre willing to sacrifice a little bit of rigour. I must admit however this is a path about which i cannot comment much. It is up to you to figure out what path you like better.

Edit: i have not read the second book you mentioned, so we will have to wait for someone else to give their opinion on it :)

 

[Whitehole]

This is where you have to figure out how well you want to know your math. I myself am taking the long route, it is a long route however, but if youre like me, very mathematically oriented this will be very rewarding in the end.
You can start with some fundamental topology as in the first four chapters on John Lee's "Introduction to Topological Manifolds". This covers topological spaces, subspaces, product spaces and quotient spaces, connectedness and compactness. This is a cool book because it develops topology with an eye on manifolds the whole time.
After this you can do smooth and pseudoriemannian manifolds. This pretty much covers the math of GR.

On the other hand, it is possible to have a good working knowledge of GR without mastering all the intricacies of the math behind it. This will take up less time if youre willing to sacrifice a little bit of rigour. I must admit however this is a path about which i cannot comment much. It is up to you to figure out what path you like better.

Edit: i have not read the second book you mentioned, so we will have to wait for someone else to give their opinion on it :)

I prefer the long route but I'm a beginner so I'm not sure how to do it. What is the prerequisite for reading Lee's book?

 

[Cruz Martinez]

I prefer the long route but I'm a beginner so I'm not sure how to do it. What is the prerequisite for reading Lee's book?

I'd say you need to be more or less fluent in naive set theory(just enough to understand the proofs) and perhaps a little metric space theory for intuition.

 

[Whitehole]

I'd say you need to be more or less fluent in naive set theory(just enough to understand the proofs) and perhaps a little metric space theory for intuition.

Thanks. Can you also point out the difference between tensor analysis and differential geometry? I mean I know that diff geom is the study of curves, etc. and tensor analysis is the language of GR but some authors tend to join those two in a single book. I mean what exactly are the books that I stated above?I'm confused.

 

[Cruz Martinez]

Thanks. Can you also point out the difference between tensor analysis and differential geometry? I mean I know that diff geom is the study of curves, etc. and tensor analysis is the language of GR but some authors tend to join those two in a single book. I mean what exactly are the books that I stated above?I'm confused.

It is not really a matter of finding a differemce between tensor analysis and differential geometry. Tensor analysis is done on a smooth manifold, so tensor analysis is part of differential geometry. I will take a look at the book by Rund et al. so i can compare them.

 

[Whitehole]

It is not really a matter of finding a differemce between tensor analysis and differential geometry. Tensor analysis is done on a smooth manifold, so tensor analysis is part of differential geometry. I will take a look at the book by Rund et al. so i can compare them.

Can you also give me a guide on what to study and read? Given I have already studied calculus, linear algebra, differential equations, undergraduate physics courses. Thanks for your suggestions.

 

[Cruz Martinez]

The book i did before studying Lee's topological manifolds was set theory and metric spaces by Irving kaplansky. You could try that too. Another alternative would be to read a set theory book such as halmos naive set theory, this set theory stuff will be useful in mathematical physics even if you don't wish to studythe math very deeply. If you think this will take too long you could try reading the appendices on Lee's top manifolds, again, this might be too terse because it is meant as reference.