# Why speaking about diffeomorphisms for change of coordinates?

We often use diffeomorphisms to change coordinates on a smooth manifold $$(M,A)$$. But, from what I've seen, "changing coordinate" is simply a function $$\psi\circ\phi^{-1}$$, where $$\phi$$ and $$\psi$$ are charts.

It is clear that a diffeomorphism induces a change of coordinates but is the inverse also true? I don't see how to make a global transformation from all the $$\psi_j\circ\phi^{-1}_i$$'s wich don't necessarily agree on intersections.

This question is related to the fact that I don't really understand why GR is a gauge theory with gauge group Diff($$M$$). For me, invariance under coordinates transformations is more general than invariance under Diff($$M$$).

This question is the follow up of this question I posted in the math network but I didn't really get any answers.

• I do not know what you have in mind when you say coordinate change as opposed to diffeomorphism. The common understanding of these words in physics is as the same set of objects. Furthermore, GR is a gauge theory, but it is not a Yang-Mills theory. There is a way (frame fields) to turn GR into something more like a YM theory, but with a local Lorentz symmetry. Nov 27 '20 at 3:10

The key distinction between a coordinate transformation, in my opinion, is that for a diffeo the coordinates don't change, therefore the volume element $$d^4x$$ doesn't change, but $$\sqrt{-g} d^4x$$ would. Conversely, we know for a coordinate transformation it's the other way round.