Time Dilation Formula
Einstein's time dilation formula calculates how time slows at high speeds.
Learn the Lorentz factor formula with worked examples.
The Formula
Time dilation is a consequence of Einstein's special theory of relativity, published in 1905. It states that time passes more slowly for an object moving at high speed relative to a stationary observer.
The factor γ = 1/√(1 - v²/c²) is called the Lorentz factor. At everyday speeds, γ is essentially 1, so time dilation is undetectable. But as v approaches the speed of light c, γ grows rapidly, and time differences become significant.
This effect has been confirmed experimentally. In 1971, Hafele and Keating flew atomic clocks on commercial jets and measured time differences matching the predictions of relativity. GPS satellites also account for time dilation to maintain positional accuracy.
Variables
| Symbol | Meaning |
|---|---|
| Δt' | Dilated time — time measured by the stationary observer (in seconds) |
| Δt | Proper time — time measured by the moving observer (in seconds) |
| v | Relative velocity of the moving object (in m/s) |
| c | Speed of light ≈ 3 × 10⁸ m/s |
| γ | Lorentz factor = 1/√(1 - v²/c²) |
Example 1
A spacecraft travels at 0.8c (80% of the speed of light). If 10 years pass on the spacecraft, how much time passes on Earth?
Identify the values: Δt = 10 years, v = 0.8c
Calculate v²/c² = (0.8c)²/c² = 0.64
Δt' = 10 / √(1 - 0.64) = 10 / √0.36 = 10 / 0.6
Δt' ≈ 16.67 years on Earth
Example 2
A muon particle created in the upper atmosphere has a rest-frame lifetime of 2.2 μs. It travels at 0.99c. What is its observed lifetime?
Identify the values: Δt = 2.2 μs, v = 0.99c
Calculate v²/c² = 0.9801
Δt' = 2.2 / √(1 - 0.9801) = 2.2 / √0.0199 = 2.2 / 0.1411
Δt' ≈ 15.6 μs
When to Use It
Time dilation becomes important whenever velocities are a significant fraction of the speed of light.
- Particle physics — predicting how long unstable particles survive
- Space travel scenarios — calculating time differences for astronauts
- GPS satellite corrections — compensating for relativistic time shifts
- Thought experiments in special relativity