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I want to delve into Cartan's formulation of General Relativity but I don't really know where to look. Does anyone have any book suggestions?

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Friedrich Hehl has written several pedagogical reviews on gauge aspects of gravity. I recommend the following:

On the Gauge Aspects of Gravity

F. Gronwald, F.W. Hehl

International School of Cosmology and Gravitation 14 (1996): 148-198.

https://arxiv.org/abs/gr-qc/9602013

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IMHO, the most student-friendly exposition of Einstein-Cartan gravity is A. Zee's fantastic book: Einstein Gravity in a Nutshell.

  • Chapter IX7 gives an introduction to differential forms and vielbeins/tetrads.

  • Chapter IX8 guides you through step-by-step calculations of the
    expanding universe, spherically symmetric static spacetime, etc. using tetrads and spin-connections. You usually won't find such detailed example calculations with Einstein-Cartan gravity in other GR books.

Like his other nutshell books, A. Zee's style is casual and to the point. It's always fun to read.

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