How Time Dilation Works: Clocks That Run Slow When Moving Fast

Time dilation is the claim that a clock moving relative to you ticks more slowly than your own clock. This sounds like science fiction, but it is measured every day. Muons from cosmic rays reach the ground in numbers that only make sense with time dilation. Satellite clocks must be corrected for it. Atomic clocks flown on airliners have shown it directly.

This post explains where time dilation comes from, how to calculate it, and what it means for everyday life.

The starting point: the speed of light is the same for everyone

Albert Einstein's special theory of relativity, published in 1905, begins with two principles. The laws of physics are the same in every frame of reference that moves at constant velocity. The speed of light in vacuum is the same for every observer, no matter how fast the observer or the light source is moving.

The second principle is the strange one. If you measure a light beam while you run toward it or away from it, you still get the same speed, about 299,792 kilometres per second. Your rulers and clocks must then adjust so that the speed comes out the same for everyone. Time dilation is one of the adjustments.

A light clock makes it visible

Physicists often explain time dilation with an imaginary light clock. Take two mirrors facing each other with a pulse of light bouncing between them. Each bounce counts as one tick. Now watch the same clock from a moving spaceship that flies past sideways.

To you, the light bounces straight up and down between the mirrors. To the moving observer, the light has to travel a diagonal path because the mirrors move sideways during each bounce. The light covers a longer distance along the diagonal, but it still moves at the same speed. A longer path at a constant speed means more time between bounces. The moving observer therefore sees fewer ticks per second on your clock than you see on your own.

The effect is symmetrical. You also see the moving observer's clock running slow. The two observers do not disagree about what happened. They simply disagree about the time between events, because the time between events depends on the motion of the measurer.

The Lorentz factor

The exact amount of time dilation depends on speed, and it is given by a number called the Lorentz factor, written as gamma (γ):

γ = 1 / sqrt(1 − v² / c²)

Here v is the relative speed and c is the speed of light. The time measured by a moving clock, called the proper time for a moving observer, is related to the time on your clock by:

Δt_moving = γ × Δt_rest

If γ is 2, then one hour on your clock corresponds to two hours on the moving clock. At everyday speeds, γ is almost exactly 1, which is why nobody notices time dilation while walking, driving or flying.

Table of time dilation factors

The table lists the Lorentz factor for several fractions of the speed of light. Each value means the moving clock ticks that many times more slowly than a stationary clock, or equivalently, a stationary observer sees the moving clock lag by that factor.

Speed as fraction of c	Lorentz factor (γ)	Time for 1 hour on the stationary clock
0.10 c	1.005	about 1 h 0.3 min
0.50 c	1.155	about 1 h 9 min
0.866 c	2.000	2 hours
0.90 c	2.294	about 2 h 18 min
0.99 c	7.089	about 7 h 5 min
0.999 c	22.37	about 22 h 22 min

The numbers grow slowly at first and then very quickly. Below half the speed of light, the effect is small. Above 90 percent of light speed, it becomes dramatic. As speed approaches c, γ grows without limit, which is why no object with mass can reach the speed of light.

Cosmic-ray muons

The clearest everyday test of time dilation involves muons. These unstable particles form when cosmic rays strike the upper atmosphere, roughly 15 kilometres above sea level. A muon at rest decays after an average lifetime of about 2.2 microseconds. At that lifetime, and travelling near the speed of light, a muon should travel only about 660 metres before it decays.

Yet muons are routinely detected at the ground, many kilometres below their creation point. Their speed is about 99.5 percent of light speed, which gives γ of about 10. Their clocks run ten times slower in our frame, so their average lifetime stretches to about 22 microseconds. In that time, they cover several kilometres. Without time dilation, the observed flux at the surface would be far too small. Particle physicists have confirmed the effect many times in both natural and laboratory settings.

Atomic clocks on airplanes

In 1971, Joseph Hafele and Richard Keating flew caesium atomic clocks around the world on commercial airliners, once eastward and once westward. They then compared the clocks with reference clocks left at the United States Naval Observatory. The flying clocks showed small differences that matched predictions combining special relativity, which accounts for speed, and general relativity, which accounts for altitude and Earth's rotation. The exact numbers depend on flight path and direction, so the test was a strong confirmation of both theories.

Airliners fly at a few hundred metres per second, which is tiny compared with light speed. The effects in that experiment were measured in nanoseconds, not hours. The lesson is that time dilation is real even at low speeds. It is just too small to notice in daily life.

GPS satellites need the correction

The Global Positioning System is a practical application of relativity. GPS satellites orbit at about 3.9 kilometres per second, and they sit higher above Earth, where gravity is weaker. Special relativity says their clocks should run slow because they move quickly. General relativity says they should run fast because they sit in weaker gravity. The net effect is that satellite clocks gain roughly 38 microseconds per day compared with clocks on the ground.

That may sound tiny, but light travels about 11 kilometres in 38 microseconds. If engineers ignored this drift, GPS positions would wander by kilometres per day. Every GPS receiver depends on the correction being built into the satellite timing system.

What time dilation does not mean

Time dilation does not mean that time itself flows differently in some vague sense, and it does not allow travellers to change the past. Each observer measures the time on their own clock as normal. The disagreement only appears when two observers compare clocks in different states of motion. Time dilation also does not mean that a moving person feels slower or that their heartbeat changes in a noticeable way. Their body and clock slow together, so they do not notice anything unusual inside their own frame.

The twin paradox is a common source of confusion. A twin who makes a fast round trip returns younger than the twin who stayed home. The asymmetry comes from the acceleration needed to turn around. Only the travelling twin changes inertial frames, so only that twin's clock experiences the full time dilation over the journey.

Key takeaways

Time dilation follows from the constant speed of light. A moving clock ticks more slowly than a clock at rest, as measured by an observer who sees it move.

The size of the effect is given by the Lorentz factor γ = 1 / sqrt(1 − v²/c²). Below about half the speed of light, the change is small. Near light speed, it grows very quickly.

Cosmic-ray muons, atomic clocks on airliners and GPS satellites all confirm the effect. GPS is the most practical example, because its clocks must be corrected every day to give accurate positions.

Time dilation is symmetrical in special relativity, and it does not allow changes to the past. The only asymmetry in the twin paradox comes from acceleration during the turnaround.