The explanations for time dilation that I have seen all use the thought experiment of a photon bouncing between two parallel mirrors, which are themselves moving in a direction perpendicular to the photon’s motion. From the Pythagorean theorem, along with the formula velocity * time = distance, and assuming a constant speed of light, one can easily derive the Lorentz transformation and the phenomenon of time dilation.
What I don’t understand is this: it seems to me that in order for an observer, who is presumed to be “at rest”, to perceive such an effect, that photon would have to be in two places at the same time: bouncing back and forth forever between the mirrors, and also in the eye of the observer (or external measuring apparatus). This seems quite impossible. Either that, or there would have to be another photon – a stream of photons – emanating out from the bouncing photon itself back to the eye of the observer or his measuring device. This seems just as impossible. It would destroy the simple geometry of the intended thought experiment. In other words, either one thing would have to be in two places at the same time, or there would have to be two things, in two different places, that are “really” one thing.
It seems to me that the explanatory power of such a thought experiment depends on a “slippage”, or conflation, of two points of view: one is the point of view of an outsider – the all-knowing theorist who creates this diagram (or those who read it and understand it) and computes the Lorentz gamma factor. The other is an observer who is presumed to be located inside the thought experiment and is subject to its posited parameters. The theorist is outside, not subject to any constraints, and knows everything that is going to happen. The observer who is said to witness the time dilation is inside, subject to the effects that are a result of the geometry and the postulate of the constant speed of light.
In other words, it seems to me that the thought experiment illegitimately claims that an actual situated observer would be able to perceive something which an external, omniscient theorist can only conceive.