I have a simply question, that is whether spatial velocity is gauge invariant. It is seems that under a infinitesimal coordinate transformation the velocity is just transform as other vectors, and it is not invariant. on the other hand the velocity surely can be measured.
@Chern it might be adding a sentence containing this issue to your question, so the essence of the question is "I've read that observable quantities are gauge invariant, but if GR is a gauge theory under Diff(M), then why aren't things like particle velocities counterexamples?" – twistor59 yesterday