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In regard to atomic clocks, would all clocks be a little off since you carried it home from the store?
“In your case the two men would still observe the events differently (i.e. different intervals of time between the two lightning strikes) due to the fact that the train is moving.”
Yes, but the man on the platform will not see the strikes at the same time. This should prove that it is not only movement which can change your perception.
Also, if the man on the train had super long hands and one was in front of the train and and the other in back, he would feel both strikes at the same time. this should show that time has not slowed. And even though he sees them at different times, that’s because of the fact that the back light hasn’t hit him, but it’s still there.