How to Learn both Differential Geometry and Relativity?

[bacte2013]

Dear Physics Forum personnel,

Is it possible to learn differential geometry simultaneously while learning the relativity and gravitation? I have been reading Weinberg's book (currently in Chapter 02), but I believe that modern research in relativity is heavily based on the differential geometry and algebraic topology, which I did not master yet. I would like to learn them alongside with relativity, which will help me to prepare for upcoming theoretical research in relativity. I was thinking about studying differential geometry first, but it would be inefficient in time-wise as I am not convinced that I need thorough knowledge in it, at least from my experience with Weinberg.

I have strong background in set-theoretic topology (Engelking) and algebra (Lang, Aluffi, Isaacs), but my memory of single-variable analysis and multilinear algebra are quite foggy...Also, I only have practical knowledge in vector calculus.

 

[Simon Bridge]

Is it possible to learn differential geometry simultaneously while learning the relativity and gravitation?

Yes. It is harder in many ways - but there is an advantage in that you can use the GR to focus the differential geometry whereas a math paper will be more general.

 

[bacte2013]

Yes. It is harder in many ways - but there is an advantage in that you can use the GR to focus the differential geometry whereas a math paper will be more general.

Thank you for your advice. I have a lot of time to devote myself to physics as my current medical treatment forced me to take only few official courses on this semester. I have been searching books that are more mathematically-inclined and also covering manifolds, and I found some books like Sachs/Wu, Hawking/Ellis, etc. Some people said that reading mathematically-inclined books in relativity will hurt the understanding, which makes me worry too. Weinberg is not quite heavy in mathematics, particularly in differential geometry, and I found that physical explanation is causing me a lot of confusions. Maybe I do not have a sufficient background?

 

[Simon Bridge]

Some people said that reading mathematically-inclined books in relativity will hurt the understanding, which makes me worry too.

To be "physics", you need to be able to relate the maths to something you can measure in Nature. It is possible to study the mathematics of relativity asa concept though and lots of theoretical physicists seem to do just that.

Weinberg is not quite heavy in mathematics, particularly in differential geometry, and I found that physical explanation is causing me a lot of confusions. Maybe I do not have a sufficient background?

It's just practise ... perhaps where you get confused you can try describing the problem in another thread?