From the 'common sense' point of view, the most remarkable properties of special relativity arise from the relativisation of time. The very concept of simultaneity - events taking place at the same moment - depends on the relative speed of the observer and is not, as Newton would have it, an absolute concept.
Even in the Newtonian world with a finite speed of light, one never sees the world right now because light does not enter your eyes instantaneously, but only as fast as the speed of light. When you glance at a watch, what you see is slightly 'out of date'; owing to the time it takes for light to get to the retina of your eye from the watch face and then for the signal to be transmitted by making a suitable correction one could still reconstruct an absolute time for events as they occur, which all observers would agree on. But Einstein's relativity prohibits this.
Equally startling is the phenomenon in special relativity called time dilation. (Moving clocks tick more slowly that those at rest.)
Time dilation has been experimentally verified many times. Certain elementary particles called muons, which are created in the Earth's atmosphere at altitudes of 10 kilometers by the impact of exceedingly fast particles known as cosmic rays, undergo radioactive decay so rapidly (in their own rest frame) that most would never reach us were it not for the fact that their decay time is dilated in our own frame of reference. If their clocks ticked as if they were at rest, the muons would only travel about 600 meters before decay: they do not live longer from their own perspective yet they survive up to nine times as long according to ours.
Many similar experiments performed by physicists with the aid of particle accelerators also demonstrate that the lifetimes of muons can be extended by accelerating them to very high velocities.
I guess the genius of Einstein was in this thought experiment!
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