Why does time slow down near the speed of light?

1a light clock

First, a correction: there's no master clock that slows. Each of you sees the other's clock run slow. To see why, build a clock from a light pulse bouncing between two mirrors, a distance h apart.

2moving clock

Now watch that same clock fly past you. While the pulse is in flight the clock slides forward, so you see a longer diagonal path. But light's speed is the same for everyone.

3the triangle

Half a tick makes a right triangle. Up is the height h. Along is v times t over two. And the slanted light path is c times t over two. Pythagoras does the rest.

4the factor

Solve it: the moving tick t equals the rest tick t zero over the square root of one minus v squared over c squared. That factor, gamma, is never below one.

5muons

It's real. Cosmic-ray muons are born about fifteen kilometres up and live just two point two microseconds. At nearly light speed that's six hundred sixty metres, yet about a quarter reach the ground.

6GPS

And your phone uses it. A GPS satellite's clock runs about seven microseconds a day slow from its speed alone, so the system corrects for relativity every day.