Moving clocks run slow
Einstein’s special relativity (1905) states that the speed of light is the same for every observer. A surprising consequence is that a clock moving relative to you ticks more slowly than your own. The effect is tiny at everyday speeds but grows dramatically close to the speed of light.
The formulas
- Lorentz factor: γ = 1 ÷ √(1 − v²/c²)
- Time dilation: Δt = γ × Δt₀, where Δt₀ is the proper time measured by the moving traveler
- Length contraction: L = L₀ ÷ γ, along the direction of motion
c = 299,792.458 km/s.
Worked example at 80% of c
- v/c = 0.8, so v²/c² = 0.64.
- γ = 1 ÷ √0.36 = 1 ÷ 0.6 ≈ 1.6667.
- One year on the traveler’s clock lasts 1.6667 years for an observer at rest: a difference of 0.6667 years, about 8 months.
- A 100 m spaceship is measured as 60 m long by the observer.
- The speed is about 239,834 km/s.
Lorentz factor by speed
| Speed (% of c) | γ |
|---|---|
| 0.001% (3 km/s) | 1.00000000005 |
| 10% | 1.005 |
| 50% | 1.155 |
| 80% | 1.667 |
| 90% | 2.294 |
| 99% | 7.089 |
| 99.9% | 22.37 |
| 99.99% | 70.71 |
Below about 10% of the speed of light, relativity changes results by less than 0.5%.
Evidence
- Muons created by cosmic rays 10 to 15 km up live only about 2.2 microseconds at rest, too short to reach the ground. Many arrive because their clocks run slow.
- Hafele and Keating flew atomic clocks around the world in 1971 and measured the predicted differences.
- GPS satellites move at about 3.9 km/s, which slows their clocks by about 7 microseconds per day. General relativity (weaker gravity) speeds them up by about 45, so engineers correct a net 38 microseconds per day. Without it, positions would drift by about 10 km a day.
The twin paradox
A twin who travels to a star at high speed and returns is younger than the twin who stayed home. The situation is not symmetric, because the traveler accelerates and changes direction, which is why the paradox is resolved.
Limits of the calculator
The formulas assume constant speed in flat space-time. Gravity also affects time, as described by general relativity, and acceleration phases of a real journey need a more detailed calculation.