Time Dilation and Length Contraction

1Muons that should not be here

Cosmic-ray protons from space smash into air molecules about 15 km above your head and create showers of particles, among them muons, heavy cousins of the electron. A muon at rest lives on average only 2.2 μs before it decays. Even at nearly the speed of light, in 2.2 μs it should cover only about 660 m, so almost none should survive the 15 km journey to the ground. Yet detectors at sea level count roughly one muon per square centimetre every minute, passing through you right now. The explanation is that the muons' clocks, seen from the ground, run slow.

2Time dilation: the light clock

Imagine a clock that ticks by bouncing a pulse of light between two mirrors a distance $LL$ apart. In the clock's own frame, one tick (up and back) takes

Now let the clock move sideways at speed $vv$. Seen from the ground, the mirrors move while the light is in flight, so the light follows a longer, zigzag path.

By the second postulate, the light still travels at $cc$ in the ground frame. If one tick takes $\mathrm{\Delta }t\backslash Delta\; t$ in the ground frame, then in half a tick the light travels $\frac{c\mathrm{\Delta }t}{2}\backslash frac\{c\backslash Delta\; t\}\{2\}$ along the slanted path while the clock moves $\frac{v\mathrm{\Delta }t}{2}\backslash frac\{v\backslash Delta\; t\}\{2\}$. By Pythagoras,

where the Lorentz factor is

A moving clock runs slow by the factor $\gamma \backslash gamma$: this is time dilation. Every clock, mechanical, atomic or biological, must behave the same way, or the principle of relativity would let you detect your motion by comparing them. The time $\mathrm{\Delta }{t}_0\backslash Delta\; t\_0$ measured by a clock present at both events (here, at rest relative to them) is the proper time, and it is always the shortest.

3Length contraction

How does the journey look to the muon? In its own frame it lives only 2.2 μs; it is the Earth and its atmosphere that rush towards it at $0.995c0.995c$. For both observers to agree that the muon reaches the ground, the muon must find the atmosphere much thinner than 15 km. Moving objects are shortened along the direction of motion.

Let a rod have length $L_{0}L\_0$ in its own rest frame: its proper length. An observer moving past it at speed $vv$ measures the time $\mathrm{\Delta }{t}_0\backslash Delta\; t\_0$ for the rod to pass on her own clock (proper time for her, since both ends pass the same point, her), so she measures the length as $L=v\mathrm{\Delta }{t}_{0}L\; =\; v\backslash Delta\; t\_0$. In the rod's frame the same passage takes $\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0\backslash Delta\; t\; =\; \backslash gamma\backslash Delta\; t\_0$, so $L_0=v\mathrm{\Delta }t=\gamma v\mathrm{\Delta }{t}_0=\gamma LL\_0\; =\; v\backslash Delta\; t\; =\; \backslash gamma\; v\backslash Delta\; t\_0\; =\; \backslash gamma\; L$. Hence

This is length contraction. It happens only along the direction of motion; lengths at right angles to the motion are unchanged.

4Evidence from everyday technology

Time dilation has been measured many times: by flying atomic clocks round the world on airliners (Hafele and Keating, 1971), by timing muons circulating in a storage ring at CERN, whose lifetimes were stretched by the predicted factor, and every day by the Global Positioning System. GPS receivers find their position by timing radio signals from satellites, so the satellites' atomic clocks must agree with ground clocks to within a few nanoseconds.