Time Dilation Calculator

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How to Use the Time Dilation Calculator

1. 1. Choose the mode. Lorentz factor, time dilation and length contraction works from a speed. Round trip gives the twin-paradox result for a distance and speed. Find the speed works backwards from γ or from a pair of times or lengths.
2. 2. Enter the speed. Use a fraction of c (0.8 means 80% of the speed of light), a percentage of c, or km/s, m/s, km/h or mph. It must be below c. The speed of light is exactly 299,792,458 m/s.
3. 3. Add a time or a length if you want them. Proper time is the time shown by a clock moving with the object; proper length is the length measured at rest. Leave either blank to skip it.
4. 4. Read the results and the working. A graph of γ against speed shows where your speed sits on the curve, and the working lists every equation with your numbers.

The Equations of Special Relativity

γ = 1 / √(1 − v²/c²) = 1 / √(1 − β²)

Δt = γ Δt₀

L = L₀ / γ

Einstein's 1905 theory rests on two postulates: the laws of physics are the same in every inertial frame, and the speed of light is the same for every observer. Together they mean that a clock moving relative to you ticks more slowly than your own, by the factor γ, and that an object moving past you is shorter along its direction of travel by the same factor. The clock's own reading is its proper time Δt₀, and a ruler's length at rest is its proper length L₀.

At everyday speeds γ is indistinguishable from 1, which is why we never notice. At 10% of c it is 1.005; at 50%, 1.155; at 90%, 2.294; at 99%, 7.089. It grows without limit as v approaches c, which is why no object with mass can reach the speed of light. The calculator computes 1 − β² as (1 − β)(1 + β) and γ − 1 as γ²β²/(γ + 1), so it stays accurate for speeds extremely close to c and extremely far below it. The derivations are in OpenStax University Physics, section 5.3 on time dilation and section 5.4 on length contraction.

Worked Examples

1. 0.8c. γ = 1/√(1 − 0.64) = 1.667. One year on the moving clock is 1.667 years for an observer watching it pass, and a 4 light-year length contracts to 2.400 ly.
2. A muon at 98% of c. At rest a muon lasts 2.197 µs on average. At 0.98c, γ = 5.025, so in the lab it lives 11.04 µs and covers 3.244 km, not the 0.6455 km that 2.197 µs alone would allow. That is how cosmic-ray muons made high in the atmosphere reach the ground.
3. A GPS satellite. At 3.874 km/s, γ − 1 = 8.349 × 10⁻¹¹, so its clock runs 7.214 µs behind per day. Gravity at altitude makes it run about 45 µs per day faster (a general-relativistic effect), so the net is about +38 µs per day, which GPS corrects.
4. A jet at 900 km/h. After ten hours a clock aboard lags by 12.52 ns: far too small to notice, but measurable with atomic clocks.
5. Finding the speed. If a trip takes 10 years by one clock and 6 by another, γ = 10/6 = 1.667, so v = 0.8c. A rod 4 ly long at rest and 2.4 ly in motion gives the same answer.

The Twin Paradox Trip

One twin flies to a star d light-years away and back at a constant speed while the other stays on Earth. By Earth's clocks the trip takes 2d/v; the traveller's clock runs slow by γ, so the traveller ages 2d/(γv). At 0.8c to a star 4 ly away, Earth waits 10.00 years and the traveller ages 6.000. In the traveller's frame the distance is contracted to d/γ = 2.400 ly, and 2 × 2.400/0.8 = 6 years confirms it.

The trip is not symmetrical: the traveller turns around, switching between inertial frames, and Earth does not. That is why only the traveller returns younger. Proxima Centauri (4.25 ly) at 0.9c takes 9.444 years on Earth and 4.117 on the ship; Sirius (8.6 ly) at 0.99c takes 17.37 years against 2.451; Vega (25 ly) at 0.999c takes 50.05 years against 2.238.

Assumptions and Limits

• Special relativity only. Constant velocity in flat spacetime. Gravity adds its own time dilation, from general relativity, which matters for GPS.
• One direction of contraction. Length shrinks only along the direction of motion.
• Symmetry of observers. Each of two inertial observers sees the other's clock run slow; the paradox disappears when one of them changes frame, as in the twin trip.
• Units. Light-years use the Julian year of 365.25 days.

Frequently Asked Questions

What is the time dilation formula?

Δt = γ Δt₀, where Δt₀ is the proper time on the moving clock and γ = 1/√(1 − v²/c²) is the Lorentz factor. At 0.8c, γ = 1.667, so one second on the moving clock corresponds to 1.667 seconds on yours.

What is the Lorentz factor?

It is γ = 1/√(1 − v²/c²), the factor by which time stretches and lengths shrink for something moving at speed v. It equals 1 at rest and rises without limit as v approaches the speed of light.

Has time dilation been confirmed by experiment?

Yes, many times. Fast-moving muons live longer than muons at rest, atomic clocks flown on aircraft in the 1970s lagged behind clocks on the ground by the predicted amounts, and particle accelerators routinely rely on the effect. GPS would drift by kilometres per day without relativistic corrections.

How does the twin paradox resolve?

The twins are not equivalent: the travelling twin accelerates and turns around, which means switching between two different inertial frames, while the twin on Earth stays in one. A calculation in either frame that includes this change agrees that the traveller is younger.

Is length contraction real?

Yes: lengths measured along the direction of motion are shorter by γ in the frame where the object moves. It is a consequence of the same postulates as time dilation, and the contraction and the muon's longer lifetime are two views of the same fact.

Why can't anything go faster than light?

As v approaches c, γ grows without limit, and so does the energy needed to accelerate a mass further. Reaching c would take infinite energy, so objects with mass can approach it but not reach it.