I seem to remember an experiment at the usno where they put an atomic clock on a plane and did a bunch of air travel, then compared it to a clock that remained at the observatory and there was a measurable shift. As you go faster, time slows to compensate for the constant speed of light. This is the explanation for that famous paradox of looking into a mirror while riding a train. Since light has a fixed speed, it can't go faster than the speed of light plus the speed of the train, so you shouldn't be able to see yourself in the mirror. So, something has to compensate, and that is time. Or something like that.