Time Dilation: Moving Clocks Run Slow

A clock built out of light

Picture the simplest clock imaginable: two mirrors facing each other, with a single flash of light bouncing straight up and down between them. Each round trip is one tick. Nothing about this clock is exotic — it just counts light bounces. But because the speed of light is the same for every observer (one of the two postulates of relativity), this humble clock is about to reveal something strange.

Watch it fly past: the light takes a longer path

If you ride along with the clock, the light goes straight up and down — a short path. But if the clock flies past you at speed v, you see the light travel a diagonal zig-zag, because the top mirror has moved sideways before the light arrives. A diagonal is longer than a straight up-down. Since light cannot speed up to cover the extra distance, it simply takes more of your time — so each tick of the passing clock lasts longer. That is time dilation.

  Clock at rest (you ride with it):    Clock flying past at speed v:

      [top mirror]                          [top]    [top]    [top]
          ^                                    \        |       /
          |   light goes                        \       |      /     diagonal
          |   straight up                        \      |     /      = longer
          |   and down                            \     |    /
          v                                         \   |   /
   [bottom mirror]                          [bot]----[bot]----[bot]--->  v

   short path = short tick               longer path = longer tick

The number that does the stretching: gamma

Working out exactly how much longer the diagonal is gives one tidy stretch factor, the Lorentz factor, usually written gamma. You just plug a speed into it:

    gamma = 1 / sqrt(1 - v^2/c^2)

    moving clock's tick = gamma * (proper-time tick)

1. At v = 0.10c (10% of light speed): gamma = 1.005 — clocks slow by half a percent. Barely noticeable.
2. At v = 0.50c: gamma = 1.155 — a moving second now takes about 1.15 of your seconds.
3. At v = 0.99c: gamma = 7.09 — the moving clock crawls, ticking seven times slower than yours.

import math

def gamma(v_over_c):
    return 1 / math.sqrt(1 - v_over_c**2)

for frac in (0.10, 0.50, 0.99, 0.999):
    print(f"v = {frac}c  ->  gamma = {gamma(frac):.3f}")

# v = 0.1c   -> gamma = 1.005
# v = 0.5c   -> gamma = 1.155
# v = 0.99c  -> gamma = 7.089
# v = 0.999c -> gamma = 22.366

It is real: muons and atomic clocks

This is not a thought-experiment trick. Muons — heavy cousins of the electron — are made when cosmic rays hit the upper atmosphere about 15 km up. A muon at rest decays in about 2 microseconds, far too fast to reach the ground even at near light speed. Yet detectors at sea level catch them in droves. The reason: they fly at roughly 0.99c, so their internal clocks run about 7 times slow (gamma ≈ 7), stretching their brief lives just long enough to finish the trip.

Engineers measure it directly too. Fly an atomic clock around the world on an airliner and it returns a few hundred nanoseconds behind a twin left at the airport — exactly as gamma predicts. The GPS in your phone must correct for this kind of effect every second, or its positions would drift by kilometers within a day.

Symmetric, not a contradiction

The symmetry holds as long as nobody turns around. When one twin actually accelerates to come back, that breaks the tie — but that is the famous twin paradox, a story for the next lessons.