Tensors, metrics, differential geometry, and all that

[disknoir]

I'm looking to learn general relativity, but I'm having a hard time. Frankly, I can't find any textbooks that I can understand.

There seems to be a gap between the maths I did at uni, and the maths of general relativity.

I've done vector calculus, differential equations, linear algebra and complex analysis etc, but I just can't seem to find any resources to bridge the gap.

While at uni, I mostly used the k.a. stroud engineering mathematics books.

Do I need to re-visit some of this from a more rigorous angle? I'm getting quite frustrated, as I no-longer have a tutor or class mates to discuss things with.

I'm starting a course in general relativity in February, and really want to get a head start.

I'd like to know what maths I need to learn to fill in the gaps in my knowledge, and in what order I should learn them.

Advice/book recommendations would be great.
Thanks

 

[yicong2011]

My favorite GR introductory books are Wolfgang Rindler's "Relativity: Special, General and Comological" and Ray de'Inverno's "Introducing Einstein's Relativity". Ray de'Inverno's book provides you lots of exercises on tensor calculus. Most of his exercises are straightforward, easy-to-do, but can help you to build up skills. Bernard Schutz's "A First Course in General Relativity" is also nice: many undergraduate GR books shun to discuss "differential forms" , but Bernard Schutz does not omit this.

 

[disknoir]

Thanks.

Luckily, I have access to an academic library; I'll have a look for them.

 

[Daverz]

For GR, Hartle takes a physics first approach that will get you into applications faster.

https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20

If you want a book that explains the mathematics thoroughly, you could try Dodson & Poston.

https://www.amazon.com/dp/354052018X/?tag=pfamazon01-20

 

[rezeew]

for your interest in learning about general relativity! It's a fascinating and challenging field of study, and it's understandable that you're feeling frustrated with the lack of accessible resources for learning about the mathematical foundations of the theory.

First, let me assure you that you are on the right track with your background in vector calculus, differential equations, linear algebra, and complex analysis. These are all essential tools for understanding general relativity.

However, as you have noticed, there is a significant leap from the standard mathematics used in undergraduate courses to the mathematics used in general relativity. This is because general relativity is built upon the framework of differential geometry, which is a more advanced and abstract branch of mathematics.

To fill in the gaps in your knowledge, I would recommend starting with a textbook on differential geometry, such as "Introduction to Smooth Manifolds" by John M. Lee or "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo. These books will introduce you to the concepts of tensors, metrics, and differential geometry, which are crucial for understanding general relativity.

It may also be helpful to revisit some of the mathematics you learned in your undergraduate courses from a more rigorous perspective. This will give you a deeper understanding of the concepts and prepare you for the more abstract mathematics of general relativity.

In terms of the order in which to learn these topics, it may be best to start with differential geometry and then move on to more specific topics in general relativity, such as the Einstein field equations and the Schwarzschild solution.

Additionally, there are many online resources available, such as lecture notes and video lectures, that can supplement your textbook learning and provide a platform for discussing and clarifying concepts with others.

I would also recommend reaching out to your future course instructor or a mentor in the field for personalized recommendations and guidance.

Don't get discouraged if it takes time and effort to fully grasp the mathematical foundations of general relativity. It is a complex subject, but with determination and perseverance, you can achieve a solid understanding and excel in your course. Best of luck in your studies!