Gauged motion in general relativity and in Kaluza-Klein theories

In a recent paper [1] a new generalization of the Killing motion, the {\it gauged motion}, has been introduced for stationary spacetimes where it was shown that the physical symmetries of such spacetimes are well described through this new symmetry. In this article after a more detailed study in the stationary case we present the definition of gauged motion for general spacetimes. The definition is based on the gauged Lie derivative induced by a threading family of observers and the relevant reparametrization invariance. We also extend the gauged motion to the case of Kaluza-Klein theories.Comment: 42 pages, revised version, typos correction along with some minor changes, Revtex forma