Understanding Differential Forms: Torsion, Spin & Tetrad

[gnnmartin]

TL;DR Summary

I want a tutorial to help me understand the meaning and manipulation of (for examples) the torsion form, the spin connection, and the tetrad form, elements (I gather) of a differential form notation.

I recently came across a paper (referenced below) containing the statement that:"The differential form notation is much more concise and elegant than the tensor notation, but both contain the same information.", and the paper left me with a desire to understand the notation of differential forms, and the underlying maths. Is there a good book, or better still a good tutorial available on the web, that I should read?

The paper was by Evans & Eckardt, "The Bianchi identity of differential geometry", and can be found at
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.518.4525&rep=rep1&type=pdf

 

[PeterDonis]

The paper was by Evans & Eckardt, "The Bianchi identity of differential geometry",

I'm not sure about the claim in the abstract of the paper that "the second Bianchi equation used by Einstein and Hilbert is incomplete". I've never seen that claim in any other source. However, it doesn't seem like your question is about that claim specifically, so whether or not that claim is correct might not affect the discussion in this thread.

I learned about differential forms and differential geometry as they are used in General Relativity from the classic textbook by Misner, Thorne, and Wheeler (the one that is thick and heavy enough to make you wonder if it might undergo gravitational collapse and become a black hole ). Their treatment, I think, is considered to be somewhat idiosyncratic and might not be the best general introduction. Sean Carroll's online lecture notes on GR discuss differential geometry, but rather briefly, and I don't think he talks much about differential forms, but uses tensor notation.

 

[pervect]

I think the point the author is making (not too clearly), is that Einstein and others considered a torsion-free connection, while the "complete" identity as described in the paper apparently includes torsion.

I'm not familiar at all with the "tetrad form", nor the torsion form (though I've seen non-form discussions of torsion). I believe I've seen references to Cartan's forms and the spin connection form in MTW's "Gravitation".

While GR doesn't incorporate torsion, some other theories do, See for instance https://en.wikipedia.org/wiki/Einstein–Cartan_theory

 

[gnnmartin]

I'm not sure about the claim in the abstract of the paper that "the second Bianchi equation used by Einstein and Hilbert is incomplete". I've never seen that claim in any other source. However, it doesn't seem like your question is about that claim specifically, so whether or not that claim is correct might not affect the discussion in this thread.

I learned about differential forms and differential geometry as they are used in General Relativity from the classic textbook by Misner, Thorne, and Wheeler (the one that is thick and heavy enough to make you wonder if it might undergo gravitational collapse and become a black hole ). Their treatment, I think, is considered to be somewhat idiosyncratic and might not be the best general introduction. Sean Carroll's online lecture notes on GR discuss differential geometry, but rather briefly, and I don't think he talks much about differential forms, but uses tensor notation.

Thanks. I have that book: I didn't think to look there!

 

[vanhees71]

I think MTW is very good. It makes the abstract Cartan calculus available to physicists in always showing the connection to the physics concepts. The only problem I have with the book is this Track-1/Track-2 structure, which brings some disorder into the presentation.

For a short and more condensed treatment of Einstein-Cartan spaces and the necessity to introduce them in connection with spin, see

P. Ramond, Field Theory: A Modern Primer,
Addison-Wesley, Redwood City, Calif., 2 edn. (1989).

 

[gnnmartin]

Thanks.

 

[haushofer]

Maybe this insight also helps:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

 

[gnnmartin]

Maybe this insight also helps:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

Thanks. I think that this requires more basic understanding before I can follow this. I'l see how I get on ith Misner etc. and then may come back to this.

Thanks to all, I have enough to think about now.

 

[romsofia]

This is probably the best introduction I know of: http://sites.science.oregonstate.edu/physics/coursewikis/GDF/book/gdf/start

And if you want the physics side of it: http://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/start (especially this section http://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/nutshell.html )

I, personally, bought the complete book a few years ago: https://www.amazon.com/dp/1466510005/?tag=pfamazon01-20

Two other honorable mentions are: https://arxiv.org/pdf/0904.0423.pdf (if you already know GR and just want to see how you use forms) and https://www.amazon.com/dp/8847026903/?tag=pfamazon01-20 for the appendix (and some chapters, great book overall!).

 

[romsofia]

This is probably the best introduction I know of: http://sites.science.oregonstate.edu/physics/coursewikis/GDF/book/gdf/start

And if you want the physics side of it: http://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/start (especially this section http://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/nutshell.html )

I, personally, bought the complete book a few years ago: https://www.amazon.com/dp/1466510005/?tag=pfamazon01-20

Two other honorable mentions are: https://arxiv.org/pdf/0904.0423.pdf (if you already know GR and just want to see how you use forms) and https://www.amazon.com/dp/8847026903/?tag=pfamazon01-20 for the appendix (and some chapters, great book overall!).

Oh, and https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20 is a good book to look at, he has a GR book as well, but i haven't work through it as much as I would have liked, so can't recommend it, maybe someone else can input there.

 

[gnnmartin]

Thanks.