# observable quantities are gauge invariant?

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.

Best regards

@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

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Could you please be more clear what "gauge symmetry" you are talking about? Yang-Mills? Diffeomorphisms? Local Lorentz? Something else? –  Luboš Motl Dec 3 '12 at 12:42
@LubošMotl Thanks for you comment. Basically I am working on GR so the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. –  Chern Dec 3 '12 at 14:15
Spatial Velocity is not invariant even in SR, not speaking about GR in which frame of references has special treatment. –  TMS Dec 3 '12 at 14:42
Even if we're talking about four-velocity, it's a local observable, and as such is not going to be Diff invariant. Only stuff like integrals of contractions of the curvature tensor are Diff invariants. See Lubos' answer to physics.stackexchange.com/questions/4359 –  twistor59 Dec 3 '12 at 17:09
@twistor59 Thanks for your comments, Now the question is that In what sense,"observable quantity is gauge invariant" is correct? –  Chern Dec 3 '12 at 21:34