You move an object around a path in spacetime, and it comes back with a different size. Since size is determined by elementary particle masses, the dilation operator would presumably act on the particle masses. Since nature likes quanta, the mass changes should occur in jumps - so you may be able to think of a muon as a fat electron, and a tauon as a fat muon. These particles decay into electrons and neutrinos, so one can think of the electron as the favored ground state of the family.

-drl