Time Dilation Calculator
Calculate relativistic time dilation using Einstein's special relativity
Enter the velocity of the moving frame, the proper time experienced by the moving observer, and the speed of light to calculate the dilated time observed from the stationary frame, the Lorentz factor γ, and the time difference.
Time Dilation Calculator
Calculate relativistic time dilation using Einstein's special relativity
About the Time Dilation Calculator
Time dilation is one of the most counterintuitive and experimentally well-confirmed predictions of Einstein's special theory of relativity, published in 1905. It tells us that time is not absolute: the rate at which a clock ticks depends on how fast it is moving relative to an observer. A clock moving at velocity v relative to a stationary observer runs slow by the Lorentz factor γ = 1 / √(1 − v²/c²), where c is the speed of light in vacuum.
The key formula is t′ = γ × t₀, where t₀ is the proper time — the time recorded by the moving clock itself — and t′ is the coordinate time recorded by the stationary observer. Because γ ≥ 1 always, the stationary observer always measures a longer time interval than the moving clock. The time difference Δt = t′ − t₀ is zero at v = 0 and grows without bound as v approaches c.
At everyday velocities — even the 7.9 km/s of the International Space Station — the Lorentz factor differs from 1 only in the tenth decimal place, making the effect imperceptible in daily life. But in the world of precision metrology and satellite navigation, these tiny discrepancies matter enormously. GPS satellites orbit at about 3.87 km/s; special relativity causes their onboard clocks to lose approximately 7 microseconds per day relative to Earth-based clocks. Without correction, GPS position errors would accumulate at a rate of about 2 km per day.
At higher velocities, the effect becomes dramatic. At 86.6% of the speed of light, γ = 2 and the moving clock runs at half the rate of the stationary one. At 99% c, γ ≈ 7.1; at 99.9% c, γ ≈ 22.4. This dilation is directly observed in particle physics: muons created in the upper atmosphere by cosmic rays have a rest-frame half-life of only 2.2 microseconds, which would allow them to travel at most about 660 metres before decaying. Yet muons are routinely detected at sea level after travelling 15 km because their half-life, as observed from Earth, is dilated by a factor of γ ≈ 22 to about 48 microseconds.
This calculator allows you to explore time dilation across the full range of velocities from zero to near-c, making it a useful educational and engineering tool for physics students, aerospace engineers, and anyone curious about the nature of time and relativity.
Time Dilation Examples
These examples illustrate time dilation at velocities ranging from satellite orbits to relativistic particles.
| Scenario | Dilated Time | Notes |
|---|---|---|
| GPS satellite: v = 3 874 m/s, t₀ = 86 400 s (1 day) | t′ ≈ 86 400.000 002 s (Δt ≈ 2 μs/day from special relativity only) | GPS satellites orbit at ~3.87 km/s. Special-relativistic time dilation alone causes satellite clocks to lose about 7 μs/day. General-relativistic effects (altitude) add +45 μs/day, giving a net gain of ~38 μs/day that is pre-corrected in GPS firmware. |
| Spaceship at 10% c: v = 29 979 246 m/s, t₀ = 3 600 s | t′ ≈ 3 618 s, γ ≈ 1.005 | At 10% the speed of light, the Lorentz factor is only 1.005, so time dilation is small but measurable — about 18 extra seconds in an hour. |
| Spaceship at 90% c: v = 269 813 212 m/s, t₀ = 1 s | t′ ≈ 2.294 s, γ ≈ 2.294 | At 90% the speed of light the effect becomes dramatic — a proper second inside the ship appears as 2.29 seconds to a stationary observer. |
| Muon at 99.5% c: v = 298 344 295 m/s, t₀ = 2.2 μs | t′ ≈ 22 μs, γ ≈ 10 | Cosmic-ray muons are created in the upper atmosphere and survive to sea level because their 2.2 μs half-life is dilated to ~22 μs in the Earth frame, letting them travel ~6.6 km. |
How to use the time dilation calculator
- Enter the velocity of the moving object or frame in metres per second in the Velocity field. For a fraction of light speed, multiply the fraction by 299 792 458.
- Enter the proper time t₀ — the time interval measured by a clock that travels with the moving object — in seconds.
- The speed of light c defaults to 299 792 458 m/s (exact SI value). You can change it to explore hypothetical scenarios or use different units.
- Click Calculate to see the Lorentz factor γ, the velocity as a fraction of c (β = v/c), the dilated time t′ = γ × t₀, and the time difference t′ − t₀.
- Use the example buttons to load real-world scenarios including a GPS satellite, a spaceship at 10% light speed, and a relativistic particle.
Time Dilation FAQ
What is time dilation?
Time dilation is a consequence of Einstein's special theory of relativity. It states that a clock moving relative to a stationary observer ticks more slowly than an identical clock at rest. The faster the moving clock travels, the more slowly it ticks. This is not a mechanical effect — it is a fundamental property of spacetime. From the moving clock's own perspective, time passes normally; the dilation is only apparent when comparing the two clocks after reuniting them.
What is the Lorentz factor and how does it work?
The Lorentz factor γ = 1 / √(1 − v²/c²) quantifies the magnitude of relativistic effects. At low velocities γ ≈ 1 and relativistic effects are negligible. As v approaches c, γ increases rapidly and diverges to infinity at v = c — which is why massive objects cannot reach the speed of light. The dilated time is t′ = γ × t₀, where t₀ is the proper time (time in the moving frame) and t′ is the coordinate time (time in the stationary frame).
Is time dilation experimentally confirmed?
Yes — time dilation has been confirmed by numerous experiments. The Hafele–Keating experiment in 1971 flew atomic clocks on aircraft and measured time differences matching relativistic predictions. Muons created in the upper atmosphere by cosmic rays reach sea level despite their short half-life only because their lifetime is dilated in Earth's frame — a confirmation confirmed to high precision at particle accelerators. GPS satellites require both special- and general-relativistic corrections to maintain centimetre-level accuracy.
What is proper time versus coordinate time?
Proper time (t₀) is the time measured by a clock that travels with the moving object — it is the 'natural' time experienced by the moving observer. Coordinate time (t′) is the time measured by a stationary observer watching the moving clock. Special relativity tells us t′ = γ × t₀, so the stationary observer always measures a longer time interval than the moving clock shows. This asymmetry is at the heart of the famous twin paradox.
What is the twin paradox?
The twin paradox describes a scenario where one twin stays on Earth while the other travels at relativistic speed and returns. The travelling twin ages less because they experienced less proper time. The apparent paradox — 'but from the traveller's perspective, the Earth was moving, so shouldn't Earth twin be younger?' — is resolved by the fact that the traveller must decelerate and turn around, breaking the symmetry. Acceleration introduces a difference between the two frames, and the traveller is always the younger one when they reunite.
Does this calculator include gravitational time dilation?
No — this calculator computes only special-relativistic (velocity-based) time dilation using the Lorentz factor. Gravitational time dilation, described by general relativity, occurs near massive objects: clocks closer to a gravity source tick more slowly. For GPS satellites, both effects apply: the satellites move fast (special relativity slows their clocks by ~7 μs/day) and they are farther from Earth (general relativity speeds their clocks by ~45 μs/day), giving a net gain of ~38 μs/day that must be corrected.