Time dilation and length contraction

The light clock

Einstein used thought experiments to work out what his two postulates imply. Imagine a light clock: a pulse of light bouncing between two mirrors a distance $D$ apart. Each round trip is one "tick".

On board a spaceship, the light goes straight up and down. One tick (up only, for simplicity) takes $t_0 = D/c$.

An observer on Earth watches the ship pass at speed $v$. During the tick the ship moves sideways a distance $vt$, so the light follows a diagonal path. By postulate 2 the light still travels at $c$, so the diagonal has length $ct$. By Pythagoras,

$$(ct)^2 = (ct_0)^2 + (vt)^2$$

Solving for $t$:

$$t = \frac{t_0}{\sqrt{1 - \dfrac{v^2}{c^2}}}$$

The Earth observer measures each tick to take longer than the ship observer does. The moving clock runs slow. This is time dilation, and it applies to every process on the ship: atomic clocks, heartbeats, ageing. It is not a fault in the clock; time itself passes differently in the two frames.

Proper time and the Lorentz factor

$t_0$ is the proper time: the time between two events measured by an observer for whom both events happen at the same place, so a single clock can time them. Here, both ends of a tick happen at the same place on the ship. Any other observer measures a longer time $t$, the dilated time.

The factor that appears everywhere in relativity is the Lorentz factor,

$$\gamma = \frac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}$$

so $t = \gamma t_0$. At everyday speeds $\gamma$ is 1.000 000 000 000 to many decimal places, which is why Newton's physics works so well. It is 1.15 at half the speed of light, 2.3 at 0.9$c$, 7.1 at 0.99$c$, and becomes infinite as $v$ approaches $c$.

Length contraction

If the astronauts' journey takes less time, but both observers agree on the relative speed, the astronauts must measure a shorter distance. A moving object, or a moving distance, is contracted along the direction of motion:

$$l = l_0\sqrt{1 - \frac{v^2}{c^2}}$$

$l_0$ is the proper length: the length measured by an observer at rest relative to the object (or to the two end points). Every observer moving relative to it measures a shorter length $l$. Only lengths parallel to the motion contract; perpendicular dimensions are unchanged.

The evidence

Cosmic-ray muons. Bruno Rossi and David Hall (1941) compared muon counts on a mountain in Colorado and lower down, and found far more surviving than expected. David Frisch and James Smith (1963) did it precisely: on Mount Washington (1910 m) they counted 563 muons per hour in a particular speed range. Without time dilation only about 27 per hour should have reached sea level. They measured about 410, matching time dilation with $\gamma \approx 8.8$.

Particle accelerators. In 1977 at CERN, muons circling a storage ring at $\gamma = 29.3$ lived for 64.4 µs instead of 2.2 µs, agreeing with time dilation to within 0.1%. Accelerators such as the Large Hadron Collider are designed around relativistic effects every day.

Atomic clocks in aircraft. In 1971 Joseph Hafele and Richard Keating flew caesium atomic clocks around the world on commercial flights, eastward and then westward, and compared them with clocks at the US Naval Observatory. Relative to the ground clocks, the eastward clocks lost 59 ± 10 ns and the westward clocks gained 273 ± 7 ns, agreeing with predictions within the uncertainty. (These figures include a second effect from general relativity: clocks higher in Earth's gravity run faster. Special relativity alone predicts the moving clocks run slow.) In 2010 optical atomic clocks detected time dilation at speeds of just 10 m/s.

GPS. GPS satellites orbit at about 3.9 km/s. Special relativity makes their clocks run slow by about 7 µs per day, while weaker gravity at their altitude makes them run fast by about 45 µs per day. The net 38 µs per day is corrected for in the satellites. Without it, position errors would grow by about 10 km each day.