Why moving clocks run slow
The short answer
Light moves at the same speed for every observer. In a clock where light bounces between two mirrors, a moving clock's light must travel a longer, slanted path to keep up, so each tick takes longer. At 0.6 times the speed of light the path is 5 units for every 4 of the gap, so the moving clock ticks 4 times while one at rest ticks 5. The effect is mutual, and it is real: GPS satellite clocks fall about 7 microseconds a day behind from their speed alone.
Chapters and transcript
- 0:00
A clock made of light: up and back is one tick.
Light bounces between two mirrors. Each round trip, up and back, is one tick.
- 0:07
Two of them, side by side, tick together.
Two identical light clocks at rest tick in step.
- 0:14
Send one past at 0.6 times light speed.
One clock moves off at 0.6 times the speed of light. In the film light moves at a fixed speed on screen, and every photon position is computed from it.
- 0:19
To keep up, its light has to zigzag.
Seen from the ground, the moving clock's light must travel a slanted path to hit the mirrors as they move.
- 0:25
Same light speed for everyone: the longer path takes longer.
Light moves at the same speed for every observer, so the longer slanted path takes more time. At 0.6 c the path is 5 units for every 4 of the gap while the clock moves 3.
- 0:31
So it ticks 4 times while ours ticks 5.
Each tick of the moving clock takes 1.25 times as long, so it ticks 4 times while the clock at rest ticks 5.
- 0:40
The faster it goes, the slower it ticks.
The slowing grows with speed by the factor 1 / √(1 − v²/c²).
- 0:50
At 0.99 c: 14 ticks for every 100 of ours.
At 0.3 c a clock ticks 95 times for every 100 ticks of a clock at rest, at 0.6 c 80 times, at 0.8 c 60 times and at 0.99 c 14 times.
- 1:03
Ride along: its light just goes up and down.
Riding with the moving clock, its light simply goes up and down, and it ticks normally.
- 1:12
From there, our clock is the slow one.
From the moving clock, it is the other clock that moves and whose light zigzags, so that clock is the slow one. The effect is the same both ways.
- 1:22
GPS satellites fly at 3.9 km/s.
GPS satellites orbit about 26,561 km from Earth's centre at about 3.87 km/s, figures the film computes from Ashby (2003).
- 1:28
Their clocks fall 7 µs behind each day.
Their speed alone makes their clocks fall about 7.2 microseconds behind each day. Weaker gravity at that height has the opposite and larger effect: Ashby gives a net gain of about 38.6 microseconds a day.
- 1:35
That is 2 km of light. GPS corrects for it.
In 7.2 microseconds, light travels about 2.2 km, so GPS satellite clocks are adjusted for these effects.
- 1:44
Time depends on how you move.
Time passes at different rates for clocks moving differently.
Sources
Every figure in the film comes from these. The animation is illustrative.