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I am curious if there is any literature (texbooks, mainly, but articles will do too, though I don't have easy access to any paid journal) that deals with general relativity by using Ehresmann connections on the (orthonormal) frame bundle in a rigorous manner, rather than Koszul connections on the tangent bundle, and develops calculus directly on the frame bundle, rather than on spacetime itself.

Basically, I am interested in this, for the sake of being interested in it, however I have hopes that I might be able to use this formalism to attack some problems in my research.

To clarify a bit more:

  • I am not looking for local tetrad formalism, however I am hoping that if what I am asking for exists, it will resemble local tetrad formalism a lot, but with globally defined quantities, as opposed to just local ones.
  • Although I'll take any materials on the subject gladly, I really would prefer if the resource treated lagrangian formalism. I am mostly aware of how Ehresmann connections on the frame bundle work, but I have absolutely no clue how to do langrangian formalism with it, and this is an absolute necessity for my work.
  • I have Kobayashi & Nomizu for necessary extra mathematical details, but I'd prefer this resource to be generally self-contained.
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  • $\begingroup$ One place to look maybe texts on quantum gravity and quantum GR; one approach there is to take the Yang-Mills analogy seriously and start quantization that way, which would be close to what you are thinking about. (But in terms of the analysis, isn't it the case that a lot of meaningful computations really need fixing a local trivialization, hence reduce to a tetrad formalism?) $\endgroup$
    – Willie Wong
    Feb 24, 2016 at 15:35
  • $\begingroup$ That said, I don't pretend to fully understand your question. So maybe someone else will give better ideas. $\endgroup$
    – Willie Wong
    Feb 24, 2016 at 15:38
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    $\begingroup$ This is most easily done using moving frames and Cartan's formulation of the geometric invariants (e,g., connection and curvature) in terms of differential forms. One usually fixes a local orthonormal frame of tangent vectors and works with the dual 1-forms, this can be reformulated using lifted Maurer-Cartan forms on the principal SO(3,1)-bundle. A paper on the overall approach (no mention of GR) is: Griffiths, P. On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41 (1974), 775–814. $\endgroup$
    – Deane Yang
    Feb 24, 2016 at 19:01
  • $\begingroup$ One nice aspect of the Cartan approach is that the differential forms are both global and canonical. $\endgroup$
    – Deane Yang
    Feb 24, 2016 at 19:09
  • $\begingroup$ @WillieWong I have been looking into loop quantum gr literature for some time, since they seem to be the ones doing what you said, but it all seems terribly local for me :/ . Maybe I was looking in the wrong place. I'll edit the question soon, which might clarify things. $\endgroup$
    – Bence Racskó
    Feb 25, 2016 at 14:10

2 Answers 2

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A first remark is that in many spacetimes of interest, it is possible to choose a global tetrad (or frame field). So the need to lift everything from the spacetime to the frame bundle to have globally defined objects disappears. This is the case, for example, on any globally hyperbolic spacetime where the Cauchy surface is a prallelizable manifold. All compact orientable 3-manifolds and even many non-compact ones are parallelizable.

The above observation might explain why most references don't bother going beyond the local tetrad formalism. However, I do know of at least two references that bother going through the exercise of lifting all the relevant objects to the frame bundle:

  • Frédéric Hélein, Dimitri Vey, Curved space-times by crystallization of liquid fiber bundles [arXiv:1508.07765]
  • Kartik Prabhu, The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom [arXiv:1511.00388]

Both papers are rather extensive and only some of the early sections might be relevant for what you are interested in.

While both these references are quite recent, I'm sure that the method of working directly on the frame bundle has been known for a long time. I don't know though who might have been the first to go through a similar exercise in the literature.

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  • $\begingroup$ Thank you, I'll plow thru the papers when I get the chance. And yeah I'd also imagine it has been known for some time, yet I have been looking for this for about 2 months and I have only found literature detailing the local tetrad formalism. I'll also edit my question soon to clarify what I am looking for, since I'd imagine it is actually quite elementary. $\endgroup$
    – Bence Racskó
    Feb 25, 2016 at 14:08
  • $\begingroup$ I'll add to Igor and say not so elementary, but more that most of the people who work in GR aren't interested, firstly because of the "failure" of things like Kaluza-Klein and secondly because of Global Hyperbolicity (Igor is spot on). $\endgroup$
    – Ben Whale
    Mar 29, 2016 at 23:48
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You might enjoy Bleeker's book "Gauge Theory and Variational Principles" (http://www.amazon.com/Gauge-Theory-Variational-Principles-Physics/dp/0486445461). His focus is definitely more on particle theory than relativity, though there is a good section on it towards the end of the book. He spends a great deal of time dealing with Lagrangians in the abstract with the occasional example (Electromag and QED I think).

It's what I read after Kobiyashi and Nomizu to pick up the physical side of principal fiber bundles. And it is one of my favorites. It is very dense, but (almost) everything is spelled out is gratuitous detail (including a lot of mappings that are left implied by others).

You might also like to take a look at Kaluza-Klein theory (https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory), if you haven't already.

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