Right now, above your head, a fleet of GPS satellites is racing round the Earth at about 14{,}000\ \text{km/h}. The atomic clocks they carry are the most precise timekeepers ever flown — and they do not keep the same time as the identical clocks sitting in the control room below. A clock in motion ticks a little slower than one at rest. This is not a defect, not a wiring fault, not a trick of the signal: it is how time itself behaves. The effect is called time dilation, and it is the first and most startling consequence of special relativity.

The whole page rests on one sentence, worth reading twice: a clock moving past you runs slow compared with your own. Not because moving clocks are badly made — an ideal, perfect clock does exactly the same — but because the two of you are measuring time along different paths through spacetime. Below we will build the effect from a single thought experiment, pin down the precise factor by which the moving clock slows, and confront the puzzle that trips up almost everyone: if motion is relative, whose clock is really the slow one?

A real clue: muons that shouldn't reach the ground

Nature runs this experiment for us millions of times a second. When cosmic rays slam into the top of the atmosphere, roughly 15\ \text{km} up, they create showers of muons — heavy cousins of the electron. A muon is unstable: left alone it survives on average only 2.2\ \mu\text{s} (millionths of a second) before decaying.

Even travelling at almost the speed of light, c \approx 3\times 10^8\ \text{m/s}, in 2.2\ \mu\text{s} a muon should cover only

Just 660 metres — it should die high in the sky, and almost none should ever reach a detector on the ground. Yet detectors at sea level are drenched in muons. How do they cross 15\ \text{km} when their little internal clock allows for barely half a kilometre? Because that internal clock — like every moving clock — runs slow as seen from the ground. From our frame the muon's 2.2\ \mu\text{s} of life is stretched by a large factor, giving it easily enough time to make the trip. The muon is a clock falling out of the sky, and it is ticking in slow motion.

The light clock: where the slowdown comes from

Einstein's second postulate is the lever that moves everything: light travels at the same speed c for every observer, no matter how fast the source or the observer is moving. To see what that forces, build the simplest possible clock. Put two mirrors a distance L apart and bounce a single pulse of light between them; each round trip is one "tick".

At rest (in the clock's own frame) the light just goes straight up and straight back, a distance 2L. Since it moves at c, one tick takes a time we call the proper time \tau:

Now watch the same clock glide past you at speed v. While the light is climbing from the bottom mirror to the top, the whole clock has moved sideways, so from your seat the pulse follows a diagonal, and coming back down it follows another diagonal — a stretched-out zig-zag. The light has further to travel. But it can't travel any faster to make up the difference — its speed is stuck at c for you too. A longer path at the same speed can only mean one thing: more time per tick. Reveal the figure step by step to watch the vertical path stretch into the longer diagonal.

From the picture to the formula

Let t be the time you measure for one tick of the moving clock, and \tau the proper time it measures for itself. Look at just the first half-tick — the light going from the bottom mirror up to the top. In that half-tick:

• the pulse rises a height L = c\tau/2 (the vertical leg);
• the clock slides sideways by v\,t/2 (the horizontal leg);
• so the light itself travels the diagonal c\,t/2 (the hypotenuse).

Those three lengths form a right triangle, so Pythagoras ties them together:

Multiply through by 4 and gather the t terms:

Divide by c^2, take the square root, and solve for t:

The quantity multiplying the proper time appears so often it gets its own name and symbol, the Lorentz factor \gamma (gamma):

Getting a feel for γ

The whole size of the effect lives in that one factor \gamma. The graph plots it against \beta = v/c, from a standstill up to 99\% of the speed of light.

Two things jump out. First, near \beta = 0 the curve is almost flat at \gamma = 1: at everyday speeds \gamma differs from 1 by less than a part in a trillion, which is exactly why you have never noticed your watch slowing when you run for a bus. Second, as \beta \to 1 the curve turns almost vertical and shoots to infinity — at the speed of light a clock would freeze entirely, and it would take an infinite factor (and infinite energy) to get a massive object there. That is why nothing with mass can reach c.

Some handy values to carry around:

• \beta = 0.6 gives \gamma = 1.25 (clocks run 25% slow);
• \beta = 0.866 gives \gamma = 2 (time runs at half speed);
• \beta = 0.99 gives \gamma \approx 7.09;
• \beta = 0.999 gives \gamma \approx 22.4.

Worked examples

Example 1 — a fast spaceship. A ship flies past Earth at \beta = 0.8 (that is v = 0.8c). The pilot's clock ticks off \Delta\tau = 1\ \text{year} of proper time. How much time passes on Earth? First the Lorentz factor:

The pilot ages one year while Earth ages one year and eight months. The moving clock (the pilot's) recorded the smaller, proper time.

Example 2 — the muon, quantified. Take a muon moving at \beta = 0.999, so \gamma \approx 22.4. In the ground frame its average lifetime is stretched from 2.2\ \mu\text{s} to

In that stretched time, at nearly c, it travels roughly c\,\Delta t \approx (3\times 10^8)(49\times 10^{-6}) \approx 15\ \text{km} — precisely the depth of atmosphere it needs to cross. The muons make it to the ground because, from our point of view, their clocks are running more than twenty times slow.

Example 3 — working backwards. An unstable particle is seen to live for \Delta t = 5.0\ \mu\text{s} in the lab, but its true (proper) lifetime is \Delta\tau = 2.0\ \mu\text{s}. How fast was it going? Its Lorentz factor is \gamma = \Delta t/\Delta\tau = 2.5. Invert \gamma = 1/\sqrt{1-\beta^2}:

It was moving at about 0.92c.