Time dilation explained: why moving clocks run slow

Time dilation is the finding from Einstein's relativity that time does not pass at the same rate for everyone. A clock moving relative to you ticks slower, by a factor called the Lorentz factor, γ = 1 ÷ √(1 − v²/c²), where v is the clock's speed and c the speed of light. Clocks deeper in a gravitational field tick slower too. At everyday speeds the effects are tiny, but they are measurable, and the GPS satellite system would drift by kilometres a day if it ignored them. The time dilation calculator computes both effects.

Why moving clocks run slow

Special relativity, which Einstein published in 1905, rests on two ideas: the laws of physics are the same for everyone moving at a steady velocity, and light in a vacuum travels at the same speed, 299,792,458 m/s, for all of them. A thought experiment shows the consequence. Imagine a clock made of a pulse of light bouncing between two mirrors. To someone travelling with the clock, each tick is a straight trip up and down. To someone watching it fly past, the pulse follows a longer diagonal path, but still at the speed of light, so each tick must take longer. Working through the geometry with Pythagoras gives exactly the Lorentz factor. The effect is symmetrical: each observer sees the other's clock running slow, which stops being a contradiction once you account for who changes direction.

The Lorentz factor in numbers

• 10% of light speed: γ ≈ 1.005, a slowdown of half a per cent.
• 50%: γ ≈ 1.155.
• 80%: γ ≈ 1.667.
• 90%: γ ≈ 2.294.
• 99%: γ ≈ 7.09.
• 99.9%: γ ≈ 22.4.

γ grows without limit as v approaches c, one way of seeing why nothing with mass can reach light speed. At everyday speeds it is astonishingly close to 1. An airliner at 250 m/s (900 km/h) loses only about 1.25 nanoseconds per hour relative to the ground from its speed alone. An astronaut on the International Space Station, moving at about 7.66 km/s, loses roughly 28 microseconds a day from speed, slightly offset by the weaker gravity at that height. Scientific notation explained, with e-notation and engineering notation helps with numbers this small.

The twin paradox

Suppose one twin stays on Earth while the other travels to a distant star and back at 80% of light speed, and 10 years pass at home. With γ = 1.667, the traveller ages only 10 ÷ 1.667 = 6 years and comes home four years younger than their sibling. The apparent paradox is that each twin sees the other moving, so why is the traveller the younger one? Because the situations are not symmetrical: the traveller accelerates, turns round and comes back, changing reference frame, while the twin at home does not. Distances along the direction of motion shrink by the same factor γ, so to the traveller the journey itself is shorter, which is how both points of view agree. For a sense of the real distances to stars, the light-year converter gives travel times at the speeds of actual spacecraft.

The evidence

Time dilation is not just theory. Muons, unstable particles created when cosmic rays hit the upper atmosphere, live about 2.2 microseconds on average at rest. Even at 99.5% of light speed that would let a typical muon travel only about 660 m before decaying, yet muons reach the ground in large numbers, because at that speed their clocks run about ten times slower from our point of view. In a muon storage ring at CERN, muons circulating with γ ≈ 29.3 lived about 64.4 microseconds, matching the prediction precisely. In 1971 Joseph Hafele and Richard Keating flew caesium atomic clocks around the world on commercial airliners, once eastwards and once westwards, and found time differences in line with relativity once both speed and altitude were taken into account.

Gravity and GPS

General relativity, from 1915, adds a second effect: clocks lower in a gravitational field run slower than clocks higher up. GPS satellites orbit about 20,200 km above Earth at about 3.87 km/s, and both effects act on their atomic clocks:

• their speed makes them lose about 7 microseconds a day relative to clocks on the ground;
• the weaker gravity at their altitude makes them gain about 46 microseconds a day;
• the net result is a gain of about 38 microseconds a day.

That sounds negligible, but GPS works by timing radio signals that travel at the speed of light, and light covers about 11 km in 38 microseconds. Left uncorrected, the errors would build up at roughly 11 km a day. To compensate, the satellite clocks are set before launch to run slightly slow, at 10.22999999543 MHz instead of 10.23 MHz, so that in orbit they tick at the right rate as seen from the ground. The calculator's GPS panel reproduces this sum, the orbital period calculator shows why that altitude gives an orbit of just under 12 hours, and Escape velocity explained: how fast you need to go to leave a planet covers more of the gravity side.