I am currently taking an undergraduate modern physics course that offers a brief overview of relativity. Let us consider a scenario where we have one clock on earth in a rest frame, and one clock traveling in space at a constant speed v. we have some sort of mechanism for viewing the traveling clock in real time. Relativity tells us that the two clocks will display different times. Now, if we bring both clocks to rest and compare them side by side, experimental evidence tells us they will still display different times. I cant fathom how once they are both in the same frame, two standard mechanical watches would display different times. It makes since for an atomic clock to display different times, but I don't quite understand how a regular mechanical watch would be affected by the velocity differences. In other words, I am asking what is the underlying process that causes the hands on a mechanical watch to move at different speeds.