What is the time dilation?
Type how fast a clock moves and how long an event lasts on that clock. The time dilation calculator gives the Lorentz factor γ and how long the same event lasts for an observer at rest, from special relativity.
- Time for an observer at rest
- 1.666666667 yr
At 80% of the speed of light, 1 yr on the moving clock lasts 1.666666667 yr for an observer at rest.
- Lorentz factor (γ)
- 1.666666667
- Extra time
- 0.666667 yr
- Speed (% of c)
- 80
- Speed (m/s)
- 239,833,966.4
Time for an observer at rest: 1.666666667 yr. At 80% of the speed of light, 1 yr on the moving clock lasts 1.666666667 yr for an observer at rest.
How to calculate
Works out special-relativity time dilation: how long a time on a moving clock lasts for an observer at rest, Δt = γΔτ, with the Lorentz factor γ, from a speed in % of light speed or in m/s, km/h or mph.
Example with the default inputs (Speed as % of light speed, Speed (% of c) 80%, Time on the moving clock 1 yr): At 80% of the speed of light, 1 yr on the moving clock lasts 1.666666667 yr for an observer at rest.
Method: Δt = γ × Δτ, γ = 1 ÷ √(1 − v²/c²), c = 299,792,458 m/s.
- Special relativity: the clock moves at a constant speed relative to the observer; gravity (general relativity) is left out.
- The speed of light is exactly 299,792,458 m/s; 1 mph = 0.44704 m/s; 1 km/h = 1/3.6 m/s; a year is 365 days.
Worked examples
Each example is checked against the calculator on every build.
- Speed as % of light speed, Speed (% of c) 95%, Time on the moving clock 0.00000000000006976 yr gives Time for an observer at rest 0.000007045638767 s, Lorentz factor (γ) 3.202563.Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (Example 5.3: Δt = 7.05 μs)
- Speed as % of light speed, Speed (% of c) 80%, Time on the moving clock 1 yr gives Time for an observer at rest 52,560,000 s, Lorentz factor (γ) 1.666667, Extra time 21,024,000 s.Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation
- Speed as Speed, Speed 5,830 m/s, Time on the moving clock 0.00000003171 yr gives Time for an observer at rest 1 s, Extra time 1.89089 × 10⁻¹⁰ s.Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (Example 5.1: Δt = 1 s + 1.89 × 10⁻¹⁰ s)
- Speed as % of light speed, Speed (% of c) 60%, Time on the moving clock 0.0001142 yr gives Time for an observer at rest 4,500 s, Lorentz factor (γ) 1.25, Speed (m/s) 179,875,474.8.Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation
How it works
Type the speed of the moving clock, as a percent of the speed of light or as a speed, and the time on that clock (Δτ, the proper time).
- Speed as a fraction of light speed: β = v ÷ c, with c = 299,792,458 m/s exactly. A percent p gives β = p ÷ 100.
- Lorentz factor: γ = 1 ÷ √(1 − β²).
- Dilated time: Δt = γ × Δτ, the time an observer at rest measures for the same event.
- Extra time: Δt − Δτ = Δτ × β² ÷ (√(1 − β²) × (1 + √(1 − β²))). This is the same value written so that a small speed does not lose its digits.
Speed units: 1 km/h = 1/3.6 m/s; 1 mph = 0.44704 m/s. Time units: 1 min = 60 s, 1 h = 3,600 s, 1 d = 86,400 s, 1 yr = 365 d.
Exact arithmetic. The speed and the time are read as the exact decimals you typed. 1 − β² is worked out exactly as (1 − β)(1 + β); γ is exact when its square root is a fraction (β = 0.6 or 0.8), and a 64-bit float otherwise.
Output format. The dilated time shows 10 significant figures and the extra time 6, both in the unit of the time you typed. γ, the speed in % of c and in m/s show 10 significant figures. Values of 10¹⁵ or more, or below 10⁻⁶, show in scientific form.
When there is no answer.
- A speed at or above the speed of light (the boxes take 0 up to, not including, 100% or 299,792,458 m/s).
- A speed so small that the extra time is below the smallest number a computer can hold (about 5 × 10⁻³²⁴ s).
Assumptions
- Special relativity: a constant speed relative to the observer; gravity is left out.
- The time on the moving clock is more than 0 and at most 10¹⁵ s.
Worked examples by hand
A muon at 0.950c (OpenStax Example 5.3). γ = 1 ÷ √(1 − 0.9025) = 1 ÷ √0.0975 = 3.20256. Δt = 3.20256 × 2.20 μs = 7.04564 μs.
One year at 80% of c. γ = 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 = 5/3. Δt = 5/3 × 365 d = 608.333 d = 1.66667 yr; the extra time is 2/3 yr.
A craft at 5,830 m/s for 1 s (OpenStax Example 5.1). β = 5,830 ÷ 299,792,458 = 1.94468 × 10⁻⁵; β² = 3.78178 × 10⁻¹⁰; extra time ≈ β² ÷ 2 = 1.89089 × 10⁻¹⁰ s, so Δt = 1.000000000189 s.
One hour at 60% of c. γ = 1 ÷ √(1 − 0.36) = 1 ÷ 0.8 = 1.25, so 1 h becomes 1.25 h. The speed is 0.6 × 299,792,458 = 179,875,474.8 m/s.
Other questions people ask
What is the time dilation formula?
Δt = γ × Δτ, where Δτ is the time on the moving clock (the proper time), Δt is the time an observer at rest measures, and γ = 1 ÷ √(1 − v²/c²) is the Lorentz factor. At 80% of light speed, γ = 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 = 5/3, so 1 year on the moving clock is 1.667 years for the observer.
What is the Lorentz factor?
The Lorentz factor γ = 1 ÷ √(1 − v²/c²) says how much a moving clock runs slow. It is 1 at rest, 1.25 at 60% of light speed, 3.2 at 95%, and grows without limit as the speed nears c.
Why does a muon reach the ground?
A muon lives 2.20 μs on its own clock. At 0.950c, γ = 3.20, so in the Earth frame it lives 7.05 μs and travels about three times farther than it could without time dilation (OpenStax Example 5.3).
Is there time dilation at everyday speeds?
Yes, but it is tiny. A craft at 5,830 m/s gains only 1.89 × 10⁻¹⁰ s per second (OpenStax Example 5.1). The page works out that small difference without rounding it away.
Can anything reach the speed of light?
No object with mass can. At v = c the formula divides by zero, so the page takes speeds from 0 up to, but not including, 100% of c.
Does this include gravitational time dilation?
No. This page uses special relativity only: a clock moving at a constant speed. Clocks deeper in a gravity field also run slow (general relativity), which GPS satellites must correct for, but that is a different formula.