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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 – 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

If (as your question somewhat implies) your gauge group is the Lorentz group, absolute velocity is gauge covariant--it changes with your Lorentz transform in the way proscribed by the Lorentz transforms. It is definitely not invariant since, as intuition tells you, it changes from Lorentz frame to Lorentz frame.

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