Special Relativity in 14 Easy (Hyper)steps

4. Quantitative description of time dilation

Let's calculate how much longer it takes a moving light clock to tick, compared to a stationary light clock.

The idea is simple: light always travels about a foot per nanosecond. The light beam inside a moving light clock travels farther per "tick" than the the light beam inside the stationary clock. We can determine the time between ticks once we know the distances involved.

The bottom-mirror-to-top-mirror trip only takes  t = D/c  in the stationary clock.

Let's say the corresponding trip takes  t'  in the moving clock.  During time  t',  the clock moves to the right a distance  vt'.  The light beam travels along the hypotenuse of right triangle whose other sides have lengths  vt'  and D.

Since light always travels at c, we must have .
We can solve for t' by squaring both sides and doing some algebra. The result is .

Take a look at what we've found: the moving clock takes a factor of   longer to tick than does the stationary clock, since its light beam travels farther per tick.

When  the moving clock can take a very long time to tick compared to the stationary clock.  Moving clocks (and everything else that's moving with respect to an observer) seem to be running too slowly!

Fast-moving people think slowly, respond slowly, get jokes slowly,... relative to the observers who watch them.