Here I asked a question. In one curious comment, I see a statement that gravity is a gauge theory. However, my definition (based on what I read till date) of a gauge theory is a field theory which is invariant under a (Lie) group of transformations where the group parameters are made local (functions of spacetime). And you have two kinds of fields. For example, you have a matter field $\psi$ and a gauge field $A_\mu$ for QED.

  • If the claim in the comment is true, what Lie group of transformations are we talking about here?

  • If the claim is true, is it possible to derive a Lagrangian where $g_{\mu\nu}$ pops up automatically (as happens in QED or Yang-Mills) by demanding some kind of "gauge invariance" principle to a Lagrangian containing matter fields?

Apologies to the experts if you feel I am blathering.