Hong Life CalculatorsUnderstanding the time dilation calculator — concept and principle
Special relativity says a moving clock runs slow compared with one at rest. This calculator takes a speed and the time that passes on the moving clock and gives the Lorentz factor, the time an observer at rest measures, the extra time and how much lengths along the motion contract. It works from near light speed down to everyday speeds without losing precision.
How it is calculated
γ = 1 ÷ √(1 − v²/c²)
Δt = γ × Δτ
Extra time = (γ − 1) × Δτ
Length contraction: L = L₀ ÷ γ
- v is the speed, entered as % of c, km/s or km/h; c = 299,792,458 m/s exactly.
- Δτ is the proper time on the moving clock, in the unit you choose; the extra time is also converted to seconds (1 year = 365.25 days).
- γ − 1 is computed as β² ÷ (√(1 − β²)(1 + √(1 − β²))), which avoids rounding loss at low speeds.
The principle
Einstein's postulate is that light has the same speed for every observer. Picture a clock made of light bouncing between two mirrors. To someone watching it fly past, the light follows a longer diagonal path, yet moves at the same speed c, so each tick takes longer. Pythagoras on that triangle gives exactly the factor γ. Nothing is wrong with the moving clock: all processes on board, including ageing, run at the slower rate as seen from the rest frame. The effect is symmetric, and the twin "paradox" is resolved because the travelling twin changes direction.
Worked example
Default inputs: speed 90 % of c, 1 year on the moving clock.
- β = 0.9, so 1 − β² = 1 − 0.81 = 0.19.
- γ = 1 ÷ √0.19 = 1 ÷ 0.43589 = 2.29416.
- Time for the observer at rest = 2.29416 × 1 = 2.29416 years.
- Extra time = 1.294 years = 1.294 × 31,557,600 s = 4.084 × 10⁷ s.
- Lengths along the motion shrink to 1 ÷ 2.29416 = 43.59 %; speed as a fraction of c = 0.9.
How to read the result
Below about 10 % of light speed γ stays within 0.5 % of 1, so effects are tiny but measurable with atomic clocks. At 90 % γ is about 2.3, at 99 % about 7.1, and it grows without limit as v approaches c. GPS satellites must correct for this effect and for gravitational time dilation from general relativity, which works in the opposite direction.
Common mistakes
- Swapping proper time and observer time; the moving clock always shows less time.
- Entering a speed at or above light speed, which has no physical meaning here.
- Expecting this to include gravity; it is special relativity only.
Frequently asked questions
Does time really slow down, or is it an illusion?
It is real and measured: fast-moving particles live longer, and clocks flown on aircraft show the predicted differences.
How much slower does time run on an airliner?
At about 900 km/h the moving clock loses only around 30 nanoseconds per day from speed alone.
Why do GPS satellites need relativity?
Without correcting for both speed and gravity effects, position errors would build up quickly.
Sources: OpenStax, University Physics Volume 3 (2016), §5.3 "Time Dilation" and §5.4 "Length Contraction" · BIPM, The International System of Units (SI Brochure), 9th edition (2019)