Time Dilation Calculator

What time dilation means

Special relativity says that the rate at which a clock ticks depends on how it moves relative to whoever is watching it. A clock moving past you runs slower than an identical clock sitting beside you. This is time dilation, and it is not an illusion or a fault in the clock: both the moving clock and the biological processes of anyone riding with it really do advance less between two events.

The effect follows from two postulates: the laws of physics are the same in every inertial frame, and the speed of light in vacuum has the same value for every observer. From those it follows that the time between two events measured by someone who is present at both of them, the proper time, is the shortest time any observer will record.

The time dilation formula

Δt = γ × Δt₀Δt₀ is the proper time on the moving clock; Δt is the time measured by the observer who sees it move.

γ = 1 ÷ √(1 − v² ÷ c²)γ is the Lorentz factor, v the relative velocity and c = 299,792,458 m/s.

Because v is always smaller than c, the square root is smaller than one and γ is at least 1. The ratio v/c is often written β, which makes the factor γ = 1 ÷ √(1 − β²). The calculator reports the dilated time Δt and the percentage difference (γ − 1) × 100.

How to use the calculator

1. Enter the relative velocity and choose its unit: fractions of light speed (c), miles per hour, kilometres per hour, miles per second, kilometres per second, metres per second, yards per second, feet per second or knots.
2. Enter the time interval as measured by the traveller, the person or clock moving with the object, and choose seconds, minutes, hours, days, weeks or years.
3. Press Calculate. Relative time is how long that interval lasts for the stationary observer, in the same unit you chose. Time dilation (%) is how much longer it is.

A velocity equal to or above the speed of light is rejected, since the formula has no real value there. At ordinary speeds the percentage rounds to 0% and the relative time equals the input, because the effect is far below what six decimal places can show.

Worked examples

A fast spacecraft at 0.8c. Here β = 0.8, so γ = 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 = 5/3. A trip that takes 3 days on the ship's clock lasts 3 × 5/3 = 5 days for an observer left behind, which is 66.67% longer.

100,000 km/s. This is about one third of light speed: β = 100,000 ÷ 299,792.458 = 0.33356, giving γ = 1.060752. One year aboard corresponds to 1.060752 years outside, a dilation of 6.08%.

Close to light speed. At 0.99c, γ = 7.0888. Ten years of travel time for the crew correspond to roughly 70.9 years back home, which is the usual textbook picture of a round trip that ends with the traveller much younger than the twin who stayed.

The time dilation curve

γ grows very slowly at first and then climbs steeply. Up to half of light speed the correction is at most about 15%; beyond 0.9c each extra nine in 0.99, 0.999, 0.9999 multiplies γ by roughly three.

At everyday speeds the difference is tiny. An airliner at 250 m/s gains or loses about 3.5 × 10⁻¹³ of each second, which adds up to roughly 12.5 nanoseconds over a ten-hour flight. The Earth's orbital speed of 29.78 km/s accounts for about 0.16 seconds over a whole year.

Limits of the calculation

• Special relativity only. Gravity also changes clock rates. Satellite navigation systems have to correct for both the velocity effect calculated here and a gravitational effect, and the two do not cancel.
• Constant relative velocity. The formula applies between inertial frames. Acceleration, turning around or circling are not modelled, which is exactly why the twin paradox needs a more careful treatment.
• Symmetry. Each observer sees the other's clock run slow. The asymmetry in a round trip comes from the fact that only one twin changes frames.
• The interval you enter must be the proper time. If you instead know the time measured by the stationary observer, divide by γ rather than multiplying.

Time dilation has been confirmed many times: cosmic-ray muons, whose rest-frame lifetime is about 2.2 microseconds, survive long enough to reach the ground because their clocks run slow, and atomic clocks flown on aircraft in the 1970s showed shifts matching the predictions of relativity.