How to Calculate Relativistic Time Dilation

Time dilation has a reputation for being exotic, and the arithmetic is genuinely a single equation. What’s counter-intuitive isn’t the maths — it’s how completely nothing happens until you’re moving extraordinarily fast, and then how quickly everything does.

The Lorentz factor

γ = 1 ÷ √(1 − v²/c²)

Dilated time is proper time multiplied by γ. That’s the whole calculation.

The behaviour of that expression is the interesting part:

At 0.1c — 30,000 km/s, about 100 times faster than any spacecraft has travelled — γ is 1.005. Half a percent.

At 0.5c it’s 1.155. At 0.87c it’s 2.03, so one year aboard is two years outside. At 0.99c it’s 7.09. At 0.999c it’s 22.4.

The curve is almost flat, then it isn’t. That’s why relativity was undetectable for the whole of physics before the twentieth century — nothing anyone could observe moved fast enough for γ to differ measurably from 1.

The other effect

Gravitational time dilation is separate. Clocks run slower deeper in a gravitational well — closer to a mass — and the effect depends on the gravitational potential rather than on velocity.

GPS is the standard example because both effects apply simultaneously and in opposite directions. Satellite clocks run slow by about 7 microseconds a day from their orbital velocity, and fast by about 45 microseconds a day from being in weaker gravity. The net is roughly 38 microseconds a day of gain.

That sounds negligible. It isn’t: an error of 38 microseconds corresponds to about 11 kilometres of positioning error, accumulating daily. GPS satellites carry a deliberate frequency offset to compensate, which makes satellite navigation one of the few pieces of everyday infrastructure that would fail without general relativity.

Why people get stuck here

• Expecting a linear effect. It’s flat then explosive, which defeats intuition.
• Asking whose clock is “really” right. Both. Each measures proper time along its own path, and there’s no privileged frame.
• The twin paradox. Resolved by noting that the travelling twin accelerates and the stay-at-home one doesn’t, breaking the symmetry.
• Mixing the two effects. Velocity and gravitational dilation are different calculations, and in GPS they pull in opposite directions.

Common mistakes to avoid

• Using speed in km/h without converting to a fraction of c.
• Applying the special-relativistic formula to a situation where gravity dominates.
• Assuming the effect is symmetric in the twin scenario — the acceleration matters.
• Treating dilation as an illusion or a measurement artefact. It’s been measured directly with atomic clocks on aircraft and in laboratories.

How to do it with Time Dilation Calculator

The Time Dilation Calculator reports the Lorentz factor alongside the result, which is the number that makes it intuitive.

1. Enter a velocity as a fraction of the speed of light.
2. Enter the elapsed proper time you want converted.
3. Read γ as well as the dilated time — γ is what shows why the effect vanishes at ordinary speeds.
4. For the gravitational case, use mass and radius instead.

The Hafele-Keating experiment, which flew atomic clocks around the world in 1971, is the classic direct measurement. Other physics tools are in the tools directory.

Frequently asked questions

Why is GPS the standard example?

Because satellite clocks experience both effects at once: slowed by velocity, sped up by weaker gravity, with the gravitational effect winning. Without correcting for both, positions would drift by kilometres a day.

Does time dilation apply at everyday speeds?

Yes, but immeasurably for most purposes — an aircraft passenger gains tens of nanoseconds on a long flight. It’s real and it has been measured with atomic clocks.

Whose clock is correct?

Both. Each observer measures proper time along their own worldline, and there’s no privileged clock. The difference only becomes comparable when the paths meet again.

Final thought

Look at γ, not the dilated time. It’s the number that shows why relativity is invisible below about a tenth of light speed and inescapable above it.

Try the free Time Dilation Calculator