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paralleltransport

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- TL;DR Summary
- Why are general relativity texts written so much more formally than other physics texts.

Hi all,

What I notice is that there's a significant difference in

Some example I've seen:

1. All GR texts I've seen make a big deal about Hausdorff vs. non-Hausdorff or mention it.

2. They all talk about topological space, open/closed set to start defining manifolds.

3. One defines

4. The descriptions of boundaries etc... often use abstract set notation.

5. Symmetry arguments try to use killing vector fields, orthonormal to foliations of manifold etc...

I've never used the fact about topological sets, and chart compatibility etc... on a single GR problem. Mostly calculus modified to the appropriate setting.

One could say QM's math involves functional analysis (operator theory on hilbert space, subtleties about infinite dimensional operators etc...) but very few QM text make a big deal out of it. Similarly, classical mechanics could involve the study of vector bundles, symplectic forms, and E&M/gauge theories could involve principal-G bundles but Jackson/Goldstein don't try to make a big deal out of it. I don't even know what math QFT would involve but it's highly non-trivial (that one I don't think mathematicians even figured out completely yet).

What's the main motivation behind this dichotomy? Why isn't GR taught the same way other physics subjects are taught (minimal math, and only introduce it if it's necessary)?

What I notice is that there's a significant difference in

*style*between the GR texts and the other textbooks. In particular, GR texts very much try hard to read like a math textbooks, emphasizing theorems and abstract definitions, which I'm not sure are practically useful (though beautiful they are). I have in mind for example something like Carroll's book or Wald's book, compared to equivalent undergraduate/graduate texts in other physics subjects (Jackson for E&M, Goldstein for mechanics, Sakurai/Griffiths for QM).Some example I've seen:

1. All GR texts I've seen make a big deal about Hausdorff vs. non-Hausdorff or mention it.

2. They all talk about topological space, open/closed set to start defining manifolds.

3. One defines

*differentiable*manifolds using abstract charts and maps and C∞C∞C∞C∞ invertible functions to stitch them together.4. The descriptions of boundaries etc... often use abstract set notation.

5. Symmetry arguments try to use killing vector fields, orthonormal to foliations of manifold etc...

I've never used the fact about topological sets, and chart compatibility etc... on a single GR problem. Mostly calculus modified to the appropriate setting.

One could say QM's math involves functional analysis (operator theory on hilbert space, subtleties about infinite dimensional operators etc...) but very few QM text make a big deal out of it. Similarly, classical mechanics could involve the study of vector bundles, symplectic forms, and E&M/gauge theories could involve principal-G bundles but Jackson/Goldstein don't try to make a big deal out of it. I don't even know what math QFT would involve but it's highly non-trivial (that one I don't think mathematicians even figured out completely yet).

What's the main motivation behind this dichotomy? Why isn't GR taught the same way other physics subjects are taught (minimal math, and only introduce it if it's necessary)?

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