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I don't know if I am allowed to posit a hypothetical situation where it was attempted to find a model for these effects without being able to use the clock that Einstein used - a light clock?A: Was it necessary to use a time measuring apparatus (in the model not the experiments) with moving parts that matched the fastest possible speed?B:(what I am especially interested to know) Was it possible to create the Special Relativity model -or one extremely similar -if the only time measuring apparatus one had at ones disposal was an atomic clock which ,it is said has no moving parts ?Am I trying to find such a model by unreasonably tying one hand behind my back ? Can it be done anyway?
Although the light clock is used a great deal in illustrating SR it isn't part of the derivation. Einstein's paper which introduced SR dealt with how we view the symmetry of a wire moving through a magnetic field vs a stationary wire in a varying magnetic field and how this related to the properties of electromagnetic fields as described by Maxwell's equations. The solution to the problem was that it doesn't matter whether you consider the wire stationary or the field stationary (it's all relative) and that the speed of light does not depend on the speed of the observer (unlike the speed of sound and other waves travelling in a medium).All the rest, light clocks, observers on moving trains vs platform observers, were secondary and used to explain the consequences of SR. In reality you can use any clock, mechanical, atomic, light, or even the ageing rate of the human body, all will show the same time dilation effects.
Try "Relativity" by Albert Einstein. It used to be available as a fairly small paperback and says it all very clearly. George Gamow does the same with Mr Tompkins but some people are put off by his presentation as a novel rather than a straight scientific text. What makes these explanations digestible is the use of one or two spatial dimensions rather than a full 3D generalisation, so the illustrations are very comprehensible.