In General Relativity Theory, there is a great freedom in the choice of space-time coordinates. As long as two coordinate systems can be related by a diffeomorphism, it seems that they both serve to describe the same physical facts. Although gravity could never be formulated as a Yang-Mills field, could we identify the "gauge group" of gravity with the diffeomorphism group of space-time?
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1$\begingroup$ This question has been asked many times here, e.g: physics.stackexchange.com/q/71476 physics.stackexchange.com/q/46324 $\endgroup$– EletieCommented Feb 3, 2022 at 22:42
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1$\begingroup$ Does this answer your question? To which extent is general relativity a gauge theory? $\endgroup$– EletieCommented Feb 3, 2022 at 22:42
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Your idea is not too far off. The diffeomorphism symmetry is more of a global symmetry, while gauge symmetries are local. So if you think about local symmetry in GR you realize that the equivalence principle basically indicates that GR is locally equivalent to SR. So the appropriate local symmetry is Poincare symmetry.
You can take Poincare symmetry and make it a local gauge symmetry. When you do so, you get a gravitational interaction. However, it is slightly more general than GR. It is Einstein-Cartan gravity, which includes torsion in the presence of matter with spin. In vacuum it matches GR, so currently I am unaware of any experiment or observation which would distinguish the two.
Here is a brief introduction to this concept.
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$\begingroup$ True about diffeomorphisms; however local diffeomorphisms would be a fitting substitute in the above question. In fact linear frames are precisely the linearizations of the aforementioned group. Therefore we can consider GR to be the gauging of the linearization of local diffeomorphisms (i.e. local charts on spacetime). Which begs the question: "what happens when we gauge local diffeomorphisms beyond linear order?" $\endgroup$ Commented Oct 2 at 6:55