Seeking advice for mathematics books before GR

[PhysicsMajorLeo]

Hi Everyone, I am a physics undergraduate students who intended to study General Relativity. I have planned to purchase or borrow one of the following books：

1.A first course in General Relativity by Schutz
2.Space time and Geometry by Carroll (I have heard that it is an advance textbook rather than an introductory textbook, is that true?)

I have been searching for maths skills that are required for studying General Relativity through the internet, and one of the suggestion is to study tensor through the first three chapter of the book by Schutz. Is this suggestion true? Or are there any others textbooks explain tensor in more details? As for the remaining mathematical skills, I have no idea where to start with. I would like to seek for advice on the appropriate mathematics book that would have strengthen by mathematical background to understand General Relativity in those introductory book, and the materials in the course. Thank you!

As for the Mathematics background, I have currently complete mathematics course including Calculus, Multivariable Caculus, Vector Calculus, Linear Algebra, ODE, PDE. For physics background, I have completed Classical Mechanics which include Lagrangian and Hamiltonian Mechanics.

 

[jedishrfu]

There are the Schaum's Outlines on Vector Analysis and Tensor Analysis which should cover everything you need and then some.

Since you've covered so many courses already, tensors should be mostly notational to you except for the notions of curvature and the Frenet-Serret formulae.

https://en.wikipedia.org/wiki/Frenet–Serret_formulas

https://en.wikipedia.org/wiki/Curvature_of_Riemannian_manifolds

Probably some book on differential geometry would be good too. I used an old one authored by McConnell published as a Dover book but I'm sure someone here knows a much better one.

Once you get past this stuff, you might also look as differential forms as well.

 

[PhysicsMajorLeo]

Thanks for the advice, I have a better picture on what I should read before getting into General Relativity. However, I am a bit confused that whether I should go through the whole Introductory Differential Geometry, or I could just focus on applications of Riemannian Geometry？Thank you！

 

[vanhees71]

I'd recommend to get a good introductory book on GR. Usually these books cover the necessary differential geometry quite well. I like Landau, Lifshitz vol. 2 and Weinberg, Gravitation and Cosmology (1971) best. For the more modern approach via Cartan calculus Misner, Thorne, and Wheeler is fine.

 

[PhysicsMajorLeo]

Thank you for the advice！In fact，I have checked out that the cost for Gravitation and Cosmology by Weinberg from amazon is quite reasonable，which makes me a bit interested in purchasing this book。However，I would like to ask，when comparing the GR book by Schutz and that by Carroll，which one would be more suitable for introductory level and which one would have focus on developing basic mathematical skills？Thanks！

 

[vanhees71]

There's a nice script by Carroll online for free. Maybe the book is similar:

https://arxiv.org/abs/gr-qc/9712019

 

[PhysicsMajorLeo]

Thank you for the link！I think I would get a look on the link and also the book you suggested！:D