TITLE: Principal bundle approach to general relativity
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: 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.