About the Time Dilation Calculator
This time dilation calculator applies Einstein's special relativity to show how much a moving clock runs slow as seen by someone at rest. Enter a speed – as a percentage of the speed of light or in km/s, m/s, km/h or mph – and the time that passes on the traveller's own clock, and it returns the Lorentz factor γ, the time that passes for the stationary observer, the difference between the two, how much a moving object is length-contracted, and how far the traveller gets in the observer's frame.
Use it for physics homework, to sanity-check science-fiction travel times, to see why cosmic-ray muons reach the ground, or to estimate the tiny but measurable effect on satellites and airliners. The table compares γ at a range of speeds so you can see how sharply the effect grows near light speed.
The calculator covers special relativity only: constant speed, flat space-time, no acceleration phases and no gravitational time dilation from general relativity. Speeds must be below the speed of light, 299,792,458 m/s.
With the default inputs, the time for the stationary observer (same unit) is 1.666667. Change any value above to recalculate instantly.
How to use the time dilation calculator
- 1Enter the speed and choose its unit (% of c is easiest for space travel).
- 2Enter how much time passes on the traveller's clock and pick the unit.
- 3Optionally enter the rest length of the moving object.
- 4Read γ, the observer's time, the extra time and the contracted length.
- 5Use the table to compare other speeds.
Formula and method
In special relativity every observer measures the same speed of light, which forces clocks moving relative to you to tick slowly. If a traveller's own clock records a proper time Δt₀, an observer at rest measures a longer time Δt = γΔt₀, where the Lorentz factor γ = 1/√(1 − β²) and β = v/c. Lengths along the direction of motion shrink by the same factor: L = L₀/γ.
At low speeds γ − 1 is tiny, so the calculator evaluates it as β²/(√(1 − β²)(1 + √(1 − β²))) to keep full precision; that is how it can show the few microseconds per day relevant to satellites. The distance covered is speed × observer time. Acceleration and general-relativistic (gravitational) effects are ignored.
- γ
- Lorentz factor
- v
- Relative speed
- c
- Speed of light, 299,792,458 m/s
- Δt₀
- Proper time on the moving clock
- Δt
- Time measured by the stationary observer
- L₀, L
- Rest length and contracted length
Worked examples
One year aboard a ship at 80% of light speed
γ = 1/√(1 − 0.64) = 1/0.6 = 1.667. One shipboard year is 1.667 years on Earth, a 100 m ship looks 60 m long, and it covers 0.8 × 1.667 = 1.33 light-years.
Ten years at 99% of light speed
γ = 1/√(1 − 0.9801) = 7.089, so 10 years for the crew is 70.9 years at home – the twin paradox. The ship crosses 70.2 light-years in Earth's frame.
GPS satellite orbital speed, 3.874 km/s
β² = (3,874/299,792,458)² = 1.67 × 10⁻¹⁰, so γ − 1 = 8.35 × 10⁻¹¹ and the satellite clock loses 8.35 × 10⁻¹¹ × 86,400 s ≈ 7.2 µs per day from speed alone. Gravity makes it gain more than that, which GPS corrects for.
Cosmic-ray muon at 99.8% of c
A muon lives 2.197 µs on average in its own frame. With γ = 15.82 it lasts 34.8 µs as seen from the ground and travels about 10.4 km – enough to reach the surface from the upper atmosphere.
Frequently asked questions
What is the time dilation formula?+
Δt = γΔt₀ with γ = 1/√(1 − v²/c²). Δt₀ is the time on the moving clock and Δt the time measured by an observer at rest. At 87% of light speed γ is about 2, so moving clocks run at half speed.
What is the Lorentz factor?+
The Lorentz factor γ is 1/√(1 − v²/c²). It equals 1 at rest, about 1.15 at half the speed of light, 2.29 at 90%, 7.09 at 99% and grows without limit as speed approaches c.
Is time dilation real or just an illusion?+
It is real and measured. Atomic clocks flown on airliners (the 1971 Hafele–Keating experiment), particle accelerators where fast unstable particles live longer, and muons from cosmic rays reaching the ground all confirm it.
How does time dilation affect GPS?+
Orbital speed makes GPS clocks run about 7 µs per day slow, while weaker gravity at altitude makes them run about 45 µs per day fast (general relativity). The net 38 µs/day gain is corrected, or positions would drift by kilometres daily.
What is length contraction?+
Objects moving relative to you are measured shorter along their direction of motion by the factor γ: L = L₀/γ. A 100 m spaceship at 80% of light speed is measured as 60 m long by an observer at rest.
Can anything travel at the speed of light?+
Massless particles such as photons always travel at c. Anything with mass would need infinite energy to reach c, because γ – and therefore its energy γmc² – grows without limit as v approaches c.