Mathematical Skills for Understanding General and Special Relativity

[Aaronvan]

I'm curious as to the highest level of mathematical skill required for adequate understanding of Einstein's theories. For example, how deep into PDE's do the field equations require?

 

[bcrowell]

GR can be understood at many different levels. If all you knew was algebra, you could understand Relativity Simply Explained, by Gardner. If all you know is a semester's worth of calculus, you can understand Exploring Black Holes, by Taylor and Wheeler. With a little more background, you can understand upper-division undergrad books like Hartle.

Sophistication in physics is much more important, and much harder to acquire, then sophistication in mathematics if you want to understand GR.

 

[xienohp]

I think you can divide the learning process into three levels.
1. SR: You need the transformation of coordinates and basic physical minds.
2. GR, Introductory level: Some deviation from differential geometry textbook has already naturally make Riemann tensor and revalent tools falling out. Most textbooks starts from this, esp for the classic book Gravity.
3. GR, make use of differential form rather than covarient formulation: make writings simpler, but far far abstract than 2nd one. It is used together with the tensors to make physics simpler and merge things in diff. fields together.

 

[tom.stoer]

I think we should ask what we mean by "understanding GR".

If GR is used as a framework to generate and study stellar and cosmological solutions I would say tensor calculus & PDEs are required. If we go one step further and try to understand GR as a structure describing the foundations of spacetime, its symmetry principles tec., then differential geoemtry and topology are required.

 

[Aaronvan]

Thanks, I was thinking about GR at the grad level, I guess. The reason this question came up is because I read (some time ago) an interview with a physicist who stated something to the effect that “Einstein’s field equations are very [for him] difficult to grasp.” That got me thinking what it would take to have a clear elementary grasp of both theories.

The Taylor & Wheeler text appears very accessible. Thanks for recommending it. (Years ago I had a small book called “The Search for Gravity Waves” and I recall at the time having difficulty with the mathematics).