How applicable is graduate-level Differential Geometry to GR?

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I'm an undergraduate student who is trying to decide whether to focus on mathematics or physics. I'd like to know how much Differential Geometry is applicable to GR? If I were to take rigorous courses in Differentiable Manifolds and Differential Geometry, will these courses allow a deeper understanding of GR compared to a physics student who has not taking such rigorous mathematics?

Furthermore, could you also apply this question to rigorous math in general? How much will courses such as Lie Algebras and Lie Groups, and Algebraic Topology and Geometry apply to pursuits of Theoretical Physics?

I guess the most basic question I'm trying to ask is: is it necessary to have a deep understanding of these topics to make important/original contributions in physics, or can these insights be achieved with just a basic understanding of the mathematics that I mentioned?
 

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bcrowell
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IMO taking a previous course in differential geometry is not likely to help you at all in GR. They are likely to spend huge amounts of time on curves and surfaces embedded in a three-dimensional Euclidean space. Most likely they will have little or nothing to say about semi-Riemannian spaces. All of this means that virtually nothing you learn will be applicable to GR.

Of course, becoming a better mathematician is always a good thing, but you you could become a better mathematician in lots of different ways.
 
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Not being a mathematician has means I'm reluctant to learn any new maths that isn't directly applicable, but ...

I have found that studying the Cartan structure equations, differential forms and the exterior algebra has been very rewarding and given me a much better insight into GR. To make physical sense of GR it is necessary to use frame fields which tell us about the experience of specific observers in spacetimes and Cartan geometry is the way to do this.

Understanding covariant derivatives and parallel transport I think requires some differential geometry, in which terms the meaning of affine connections becomes clearer ( for instance).

But, I don't think a graduate level course is required. A freely available text which concentrates on the Riemannian manifold is "Riemannian Manifolds : An introduction to Curvature" by John M. Lee.

Group theory and Lie algebras are important in physics, where symmetries are a great help in understanding kinematics and dynamics, but I doubt if deep understanding of these topics is necessary ( or much fun :smile:).

Most physicists are pretty good at cherry-picking from maths only what they need to solve the problem at hand.
 
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I guess the most basic question I'm trying to ask is: is it necessary to have a deep understanding of these topics to make important/original contributions in physics, or can these insights be achieved with just a basic understanding of the mathematics that I mentioned?

Depends on the topic. One reason that I recommend that people interested in GR have a look at Shapiro and Teukolsky's "Black Holes, White Dwarfs, and Neutron Stars" or Kip Thorne's the Membrane Paradigm is that it gives students a flavor of how GR is used in astrophysics.

What happens is that you have someone that has deep knowledge of GR write some equations and figure out a model that people with less knowledge of GR but more knowledge about something else can use in their models. QFT works the same way. In the things I'm interested in I have to understand enough GR to understand what the specialists are doing and maybe tweak it, but I don't have to understand it enough so that I can reproduce what they are doing from scratch.

Also, what you find in supernova theory is that you take a paper on GR, put a GR model into your supernova model, figure out that GR is completely irrelevant in supernova explosions, and then turn off GR so that your computer models run faster :-) :-) :-)
 
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I have found that studying the Cartan structure equations, differential forms and the exterior algebra has been very rewarding and given me a much better insight into GR. To make physical sense of GR it is necessary to use frame fields which tell us about the experience of specific observers in spacetimes and Cartan geometry is the way to do this.

Same here. I found Wald's text on GR to be easier for me to understand than Weinberg's because he uses more elegant math. Also Misner tries to explain one-forms in his "telephone book" but I found a more abstract presentation to be easier to understand. Other people find the opposite.

Also, I've found that courses and textbooks that are "pure mathematics" (i.e. Theorem 1. Let X be something followed by a two page proof) to be difficult to read. The books I find useful are "proof light/picture heavy" books like Schutz's Geometrical methods of mathematical physics.

Most physicists are pretty good at cherry-picking from maths only what they need to solve the problem at hand.

Also physicists use math in a way that makes mathematicians go mad.
 
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Group theory and Lie algebras are important in physics, where symmetries are a great help in understanding kinematics and dynamics, but I doubt if deep understanding of these topics is necessary ( or much fun :smile:).

Not the I know enough physics to make proper use of Group Theory in a physics context, but group theory and abstract algebra as a whole is a ton of fun. Probably my favorite topic in math (geometry being my second favorite).

Abstract algebra is beautiful!
 
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