time dilation
Time dilation is special relativity's most famous surprise: a clock that moves past you ticks more slowly than one sitting still beside you. Not because the moving clock is broken or badly built — every clock does it, including the ticking of atoms, the swing of a pendulum, and the ageing of a living body. Time itself, measured in a moving frame, runs slow as seen from yours. The catch that makes it real (and not a trick) is that this slowing only becomes noticeable at speeds approaching that of light.
Precisely: if a clock moving at speed v measures a time interval delta-t_0 between two ticks that happen at the same place in its own frame (this is the proper time), then you, watching it fly past, measure a longer interval delta-t = gamma times delta-t_0, where gamma = 1 / sqrt(1 - v^2/c^2) is the Lorentz factor. Since gamma is always 1 or more, the moving clock's ticks are stretched out — dilated. At everyday speeds gamma is so close to 1 that the effect is utterly invisible; at 87% of c, gamma = 2 and moving clocks run at half speed. It follows directly from the constancy of c, via the simple picture of a 'light clock' whose beam has to travel a longer, slanted path when the clock moves.
This is not science fiction — it is engineering fact. GPS satellite clocks must be corrected for time dilation every day or navigation would drift by kilometres. Fast-moving unstable particles (muons made in the upper atmosphere) survive far longer than their at-rest lifetime, reaching the ground only because their internal clocks run slow relative to ours.
A muon created 15 km up lives only about 2.2 microseconds at rest — far too short to reach the ground even at nearly c. Yet swarms of them do reach us, because at v = 0.998c their gamma is about 16, so their internal clock runs 16 times slow and they cross the atmosphere before decaying.
Fast muons reach the ground only because their moving clocks run slow — time dilation, verified daily.
Time dilation is symmetric: each inertial observer sees the OTHER'S clock as the slow one, and both are right, because they are comparing different pairs of events. The asymmetric ageing of the twin paradox appears only when one twin turns around and changes frames.