How to Learn both Differential Geometry and Relativity?

In summary, it is possible to learn differential geometry simultaneously while learning the relativity and gravitation. 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.
  • #1
bacte2013
398
47
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.
 
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  • #2
bacte2013 said:
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.
 
  • #3
Simon Bridge said:
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?
 
  • #4
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?
 

1. What is the best way to approach learning Differential Geometry and Relativity?

The best way to approach learning Differential Geometry and Relativity is to first have a solid understanding of calculus and linear algebra. This will provide a strong foundation for understanding the geometric concepts and mathematical equations involved in both fields. It is also helpful to have some background knowledge in physics and mechanics.

2. How much time should I dedicate to studying Differential Geometry and Relativity?

The amount of time needed to fully understand Differential Geometry and Relativity will vary for each individual. However, it is recommended to dedicate several hours a week consistently to studying and practicing problems. It is also important to review and reinforce previous concepts before moving on to more advanced topics.

3. Is there a specific order in which I should learn Differential Geometry and Relativity?

It is generally recommended to first learn Differential Geometry before diving into Relativity. This is because Differential Geometry provides the mathematical framework and tools needed to understand the concepts of Relativity. However, some basic concepts of Relativity can be learned concurrently with Differential Geometry.

4. Are there any online resources available for learning Differential Geometry and Relativity?

Yes, there are many online resources available for learning Differential Geometry and Relativity. Some popular options include online courses, video lectures, and interactive tutorials. It is important to do some research and find resources that best fit your learning style and level of understanding.

5. How can I apply my knowledge of Differential Geometry and Relativity in real-world situations?

Differential Geometry and Relativity have many applications in various fields such as physics, engineering, and astronomy. Understanding these concepts can also provide a deeper understanding of the laws of the universe and help in solving complex problems. Additionally, knowledge of Differential Geometry is essential for understanding Einstein's theory of General Relativity, which has numerous practical applications in modern technology.

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