Time Dilation CalculatorΔt = γ Δt₀ · γ = 1 ÷ √(1 − v²/c²)
In special relativity, a moving clock runs slow: Δt = γΔt₀, where the Lorentz factor γ = 1 ÷ √(1 − v²/c²). At 80% of light speed, γ = 1.667, so 1 year on board equals 1.667 years on Earth. Enter a speed and a time interval to get the dilated time and length contraction, or plan a twin-paradox trip to a star and compare the ages of the traveller and the twin who stays home.
Formula verified against OpenStax — University Physics, Volume 3 · NIST — CODATA 2022 Fundamental Physical Constants
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Speed & time
Enter speed as a fraction of c, or in km/s. It must be below the speed of light.
Step-by-step working
What is time dilation?
Time dilation is the effect, predicted by Einstein’s special relativity, in which a clock moving relative to an observer ticks more slowly than one at rest with that observer. The faster the motion, the stronger the effect.
It is not a mechanical flaw in clocks: time itself passes at different rates in different frames. It has been confirmed with atomic clocks, particle accelerators and the GPS satellite network.
Quick answers
How do you calculate the Lorentz factor?
γ = 1 ÷ √(1 − v²/c²). At 90% of light speed γ = 2.294; at 99% it is 7.09.
How much slower does time run on the ISS?
From its 7.66 km/s orbital speed alone, γ − 1 ≈ 3.3 × 10⁻¹⁰, about 10 ms per year. Gravity pushes slightly the other way, so the net effect is a bit smaller.
What is the twin paradox?
A twin who travels at high speed and returns is younger than the twin who stayed home. The traveller switches inertial frames at turnaround, which breaks the symmetry.
Does time dilation affect GPS?
Yes. Speed slows satellite clocks by about 7 µs/day and weaker gravity speeds them up by about 45 µs/day. Uncorrected, GPS positions would drift by about 10 km a day. See the orbital velocity calculator for satellite speeds.
Time dilation formula
Δt₀ is the proper time, measured by a clock moving with the object.
| Symbol | Meaning | SI unit |
|---|---|---|
| γ | Lorentz factor | — |
| v | Relative speed | m/s |
| c | Speed of light | m/s |
| Δt₀ | Proper time (moving clock) | s |
| Δt | Dilated time (stationary observer) | s |
| L₀, L | Rest length and contracted length | m |
How to calculate time dilation
- Express the speed as β = v ÷ c.
- Compute γ = 1 ÷ √(1 − β²).
- Multiply the proper time by γ for the stationary observer’s time.
- Divide lengths by γ for length contraction along the motion.
Worked example: travelling at 0.8c
A spaceship travels at 80% of light speed. How much time passes on Earth for each year on board?
Δt = 1.667 × 1 yr = 1.667 years
A one-way trip to Proxima Centauri (4.24 ly) at 0.8c takes 5.3 years in Earth’s frame but only 3.18 years for the crew. Muons from cosmic rays reach the ground for the same reason.
Lorentz factor at different speeds
| Speed | γ | 1 year on board = Earth time |
|---|---|---|
| 0.1c | 1.005 | 1.005 years |
| 0.5c | 1.155 | 1.155 years |
| 0.8c | 1.667 | 1.667 years |
| 0.9c | 2.294 | 2.294 years |
| 0.99c | 7.089 | 7.089 years |
| 0.999c | 22.37 | 22.37 years |
Common mistakes to avoid
- Swapping proper and dilated time. The moving clock shows the shorter (proper) time.
- Thinking it’s an illusion. Time dilation is measured with atomic clocks on aircraft and in GPS satellites.
- Ignoring gravity. GPS satellites run fast by about 45 µs/day from weaker gravity, against 7 µs/day slow from speed.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Time Dilation Calculator: frequently asked questions
The slowing of a moving clock as seen by a stationary observer, predicted by special relativity: Δt = γΔt₀.
Δt = Δt₀ ÷ √(1 − v²/c²), where Δt₀ is the proper time on the moving clock.
Yes. The Hafele–Keating atomic clock flights (1971), the extended lifetimes of fast muons and particle accelerators all confirm it.
It becomes large, over 15%, above about half the speed of light. At everyday speeds it only shows up with atomic clocks.
Not anything with mass. γ goes to infinity, so it would take infinite energy.
References & sources
- OpenStax — University Physics, Volume 3 — Ch. 5.3–5.4, Time Dilation and Length Contraction
- NIST — CODATA 2022 Fundamental Physical Constants — c = 299,792,458 m/s (exact)
- Hafele & Keating (1972), Science 177 — atomic clocks around the world
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.