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Originally Posted by madman
but einstein then assumes (without proof) that the light will travel from one mirror to the other in the same amount of time regardless of whether the clock is still or in motion.
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That is not what Einstein assumed. He assumed that the speed of light is a constant in all inertial reference frames. And he
did not make that assumption "without proof", he made it because it is a hard requirement of
Maxwell's equations of electromagnetism. In short, forced to choose between Newton & Maxwell, Einstein chose Maxwell.
So the observer sitting at rest with respect to the clock sees the light pulse make a round trip that is twice the distance between the mirrors. But another observer sees the light pulse travel a longer distance, made up of the two legs of the triangle defined by the starting position of the light pulse, the point where it makes its first bounce, and the final position (i.e., bottom mirror -> top mirror -> bottom mirror). That distance is longer than twice the mirror separation, but both observers are in inertial frames, and therefore see the light pulse travel at the same speed. That being the case, the pulse must take longer to cover the 2nd trajectory, and so the 2nd observer will say that the moving clock ticks slower than does his own clock, at rest with resepct to him.