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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

Chern
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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.

Zo the Relativist
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