Time dilation calculator

What is time dilation?

A moving clock runs slow as seen by an observer it moves past. The time Δt the observer measures is longer than the proper time Δτ on the moving clock: Δt = γΔτ, with γ = 1 ÷ √(1 − v²/c²) (OpenStax University Physics).

Example: muons

A muon lives 2.20 microseconds in its own frame. Moving at 0.950c, γ = 3.20, so an observer on Earth measures a lifetime of 7.05 μs — long enough for many muons to reach the ground.

Is it noticeable at everyday speeds?

Barely. At 5,830 m/s — faster than any car or jet — one second on the moving clock lasts 1.000000000189 seconds for the observer.

What is γ (gamma)?

The Lorentz factor. It is 1 at rest, 1.01 at about 0.14c, 2 at about 0.87c, and grows without limit as the speed approaches the speed of light.

Can anything reach the speed of light?

Not anything with mass — γ would become infinite. The calculator accepts speeds below c = 299,792,458 m/s.

What speed gives a particular Lorentz factor?

Solve γ = 1 ÷ √(1 − v²/c²) for the speed: v = c·√(1 − 1/γ²). So γ = 2 is 0.866c, γ = 10 is 0.9950c and γ = 30 is 0.99944c.

Why is the traveling twin the younger one?

At γ = 30.0, a 2.00-year trip by the traveler takes 60.0 years on Earth (OpenStax University Physics, Volume 3, section 5.3). Each twin sees the other’s clock run slow, but the situation is not symmetric: the traveler accelerates and changes inertial frames to turn around, while the Earth twin stays in one. The calculator gives the same ratio: 60.0 years for 2.00 years of proper time.

Has time dilation been measured?

Yes. Cosmic-ray muons, whose half-life at rest is 1.52 µs, reach the ground in large numbers because their lifetime is longer in the Earth’s frame. In 1971 Hafele and Keating flew atomic clocks on commercial airliners and measured time differences to an accuracy of a few nanoseconds, in agreement with relativity once both special and general relativity were included (OpenStax University Physics, Volume 3, section 5.3).

The formula and what it assumes

For two events that happen at the same place in the moving frame, the elapsed time on that one clock is the proper time Δτ, the shortest interval between the events. An observer who sees the clock move at a constant speed v measures the longer time Δt = γΔτ, where γ = 1 ÷ √(1 − v²/c²) (OpenStax University Physics, Volume 3, section 5.3, Eq. 5.1–5.2). The calculator uses the exact c = 299,792,458 m/s and accepts speeds below c. It describes motion at a constant velocity between inertial frames; acceleration and gravity need more than this formula.

Worked example: ten years aboard a ship at 0.8c

A speed of 0.8c is 239,834 km/s. Then γ = 1 ÷ √(1 − 0.8²) = 1.6667. A traveler who ages 10 years (the proper time) is seen by an observer at rest to take 10 × 1.6667 = 16.67 years.

Common mistakes

Entering the speed as a percentage or as a speed in the wrong unit is the first slip: in the “fraction of c” mode 0.8 means 80% of the speed of light, and 80 is rejected as faster than light. The second is mixing up the two times. The proper time is the one measured by the clock that moves with the events, and the dilated time is always the longer one; multiplying by γ turns proper time into observer time, and dividing by γ does the reverse. The third is assuming the effect matters at everyday speeds. At 5,830 m/s OpenStax finds only 1.89 × 10⁻¹⁰ s of difference in a second (Example 5.1), and a detailed comparison of real clocks on aircraft needs gravity as well.

Sources: OpenStax University Physics, Volume 3, section 5.3 (time dilation, Eq. 5.1–5.2; Examples 5.1–5.3; the twin paradox and the Hafele–Keating experiment); the speed of light is exact by definition (NIST, SI). Related: escape velocity calculator, speed calculator, wavelength calculator and orbital period calculator.