Context for this question: There is a famous thought experiment used to explain time dilation that uses two mirrors and a photon to set up a clock. The two mirrors are placed parallel to one another, and a photon is sent travelling perpendicular to the planes of the mirror, bouncing back and forth. Each time the photon hits a mirror, the clock ticks. When it is viewed by an observer travelling at relative velocity perependicular to the direction of the photon, the mirror-clock ticks more slowly due to the apparent zigzagging motion of the light. I started wondering why this is true in general, and not merely a feature of this particular type of clock. I found this earlier question which asked just that. This question is a follow-up to the answer provided by robphy on that post.
Robphy states that all clocks must experience the same phenomenon by invoking the principle of relativity:
An inertial observer carries both a light clock and a mechanical wristwatch, which agree when all are at rest. If they don't agree when the inertial observer is moving [with nonzero constant velocity] carrying these clocks, then that observer can distinguish being at rest from traveling with nonzero constant velocity.
I don't understand this answer. Why wouldn't the two clocks agree agree? If the inertial observer is moving with nonzero constant velocity carrying the clocks, wouldn't the situation be identical to the observer being in the rest frame for both clocks? So why would the clocks have different measurements at all?
Best Answer
You seem to think that Robphy uses the principle of relativity as a kind of self-evident universal truth (like 1+1=2), which you could come up by yourself if you think hard enough. He then seems to deduce something even more obvious from it (equivalence of light clocks and everyday clocks). But you believe in neither truth and ask us how Robphy came to think that way.
The problem is that the principle of relativity is not something trivial like some axioms of math, but it is a law of nature that has been repeatedly confirmed by observation. We could well have lived in a different world, where it were possible to distinguish between absolute rest and motion. One possibility for such a distinction would be if light and everyday clocks went different when moving at various velocities.
But even if all clocks always go synchronously in all systems, that is not a "proof" of the principle of relativity. The principle of relativity tells us that there have never been found any experiments whatsoever (either with clocks or anything else), that allow us to distinguish between absolute rest and movement. Possibly, we have not searched hard enough and in 500 years from now we could find such an experiment, but at the moment that is the state of affairs (and to be sure, broad consensus is that it is pretty unlikely that special relativity will ever be broken). In that sense, finding such clocks or conditions would invalidate the principle of relativity, because the principle of relativity specifically says that there are no such clocks.
So we can't answer your question "why", we can only confirm that none of us knows any way to distinguish between absolute rest and motion, of which differently going clocks depending on velocity would be one example. In the same sense, I don't know of anybody who can cancel gravitation. That doesn't mean that anti-gravitation doesn't exist, nor does it automatically imply that we just have to search hard enough to find anti-gravitation.