Gravity from a reduction of the structure group of a frame bundle $FX$ to a Lorentz group $SO(1,3)$

Question

According to https://en.wikipedia.org/wiki/World_manifold, gravity can be understood as follows:

In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.e., a structure group of a frame bundle $FX$ must be reduced to a Lorentz group $SO(1,3)$. The corresponding global section of the quotient bundle $FX/SO(1,3)$ is a pseudo-Riemannian metric $g$ of signature $(+,−−−)$ on $X$. It is treated as a gravitational field in General Relativity and as a classical Higgs field in gauge gravitation theory.

A Lorentzian structure need not exist. Therefore, a world manifold is assumed to satisfy a certain topological condition. It is either a noncompact topological space or a compact space with a zero Euler characteristic. Usually, one also requires that a world manifold admits a spinor structure in order to describe Dirac fermion fields in gravitation theory. There is the additional topological obstruction to the existence of this structure. In particular, a noncompact world manifold must be parallelizable.

To me this approach is easy to understand because it makes gravity feel almost inevitable. However, I have not been able to find any introductory material on this approach.

Specifically, I am interested in understanding this specific sentence:

It is treated as a gravitational field in General Relativity

What are the next steps to get, say the Hilbert-Einstein Lagrangian or the Einstein field equations from the above?

Answer 1

The author of that article (writing as User:Gsard) was Gennady Sardanashvily, a Russian gravitational physicist who died in 2016. Possibly you can find helpful material in his papers.

Answer 2

It should be noted that once you unpack it, there really isn't anything different in the text than the usual formulation of GR, albeit a bit more careful about some formalities such as the possibility of global existence of a non-degenerate metric $g_{\mu\nu}$. That is, the wikipedia article seems to be written by someone who simply talks about GR in a somewhat different language, but it is really doing nothing special (and I do not quite understand why the "World manifold" exists as a separate wikipedia article written in this way, to be honest).

As noted in another answer, the writer of the article seems to be Gennady Sardanashvily. One of the main references on the wiki is his 2011 paper on this topic "Classical gauge gravitation theory". I believe you will find more about gravity written in this language, even though many of the things in the paper seem to be just other rewordings of known results. I would also recommend to keep in touch with the variety of ways the mainstream talks about gravity and its various formulations. A very useful summary is given in the 2020 monograph by Krasnov Formulations of General Relativity.