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.

This post imported from StackExchange MathOverflow at 2016-03-03 10:43 (UTC), posted by SE-user Igor Khavkine