Time Dilation Calculator
A moving clock ticks slower than one at rest, an effect of Einstein’s special relativity. Enter a proper time and a velocity (or pick a real-world preset) and this calculator returns the dilated time t = t0 / √(1 − v²/c²), the Lorentz factor gamma, the length contraction the traveller sees, and how far they go. A reverse mode solves the velocity needed for a target slowdown.
Formula
Worked example
A clock moving at v = 0.8c for a proper time t₀ = 1 s: β = 0.8, so γ = 1 / √(1 − 0.8²) = 1 / √0.36 = 1.6667. The dilated time is t = 1 × 1.6667 = 1.6667 s, a 1 m rod contracts to 1 ÷ 1.6667 = 0.6 m, and the moving clock runs 40% slower.
How time dilation is calculated
Special relativity says that a clock moving relative to you ticks more slowly than a clock at rest beside you. The relationship is t = t0 / √(1 − v²/c²), where t0 is the proper time (the interval read off the moving clock), v is the relative velocity, and c is the speed of light, 299,792,458 metres per second. The square-root term defines the Lorentz factor gamma, γ = 1 / √(1 − v²/c²), so the equation is often written simply as t = γ·t0. Because gamma is always greater than or equal to 1, the dilated time t is never shorter than the proper time. This calculator lets you enter the velocity as a fraction of c, in m/s, km/s, mi/s or km/h, or pick a real-world preset such as the ISS or a GPS satellite. You can also choose the time unit, from seconds up to years, so the dilated interval comes back in the same unit you entered.
Length contraction, distance and the reverse mode
The same Lorentz factor that stretches time also shrinks length along the direction of motion: a rod that is L0 long at rest measures L = L0 / gamma to an observer it flies past, so this calculator reports how short a one metre rod becomes. It also reports how far the traveller covers in the rest frame over the dilated interval, which for fast trips is naturally given in kilometres or light years. The reverse mode flips the problem around: instead of giving a speed, you give a target slowdown (the Lorentz factor you want), and the calculator inverts γ = 1 / √(1 − β²) to return β = √(1 − 1/γ²), the exact fraction of light speed required. For example, to make a clock run exactly twice as slow you must travel at about 0.866c, roughly 259,627 km/s.
Why the effect only shows up near light speed
For ordinary speeds the ratio v/c is so tiny that gamma is indistinguishable from 1 and time dilation is unmeasurable in daily life. A passenger jet at 250 m/s has a Lorentz factor that differs from 1 by less than one part in a trillion. The effect grows slowly until the velocity climbs past roughly 10% of the speed of light, then accelerates sharply: at 0.8c gamma is 1.67, at 0.99c it is about 7.09, and at 0.999c it reaches roughly 22.4. As v approaches c the denominator √(1 − v²/c²) approaches zero and gamma diverges toward infinity, which is one way of seeing why no massive object can ever be pushed all the way to the speed of light.
Where time dilation matters in the real world
Time dilation is not a thought experiment, it is engineered around every day. The GPS network depends on it: the satellites move fast enough that special-relativistic dilation slows their onboard clocks by about 7 microseconds per day, a shift that must be corrected or positions would drift by kilometres. Particle physicists rely on it too, fast-moving muons created in the upper atmosphere survive long enough to reach the ground only because their internal clocks run slow in our frame, which is why the muon preset sits at 0.998c. The famous twin paradox is the same effect: a twin who travels out and back at high speed returns younger than the one who stayed home, because the traveller’s proper time is shorter. This calculator covers the special-relativity (velocity) part of the effect only, gravitational time dilation from differences in gravitational potential is a separate calculation.
Lorentz factor at common speeds
| Speed (fraction of c) | Lorentz factor γ | 1 s of proper time becomes | 1 m rod becomes | Effect |
|---|---|---|---|---|
| 0.001c (~300 km/s) | 1.0000005 | ~1.0000005 s | ~1.0000000 m | Negligible |
| 0.1c | 1.0050 | 1.0050 s | 0.9950 m | Small |
| 0.5c | 1.1547 | 1.1547 s | 0.8660 m | Moderate |
| 0.8c | 1.6667 | 1.6667 s | 0.6000 m | Large |
| 0.99c | 7.0888 | 7.0888 s | 0.1411 m | Extreme |
| 0.999c | 22.366 | 22.366 s | 0.0447 m | Extreme |
How the Lorentz factor gamma grows as velocity approaches the speed of light c.
Frequently asked questions
What is proper time and which clock does it belong to?
Proper time (t₀) is the time interval measured by a clock that is at rest in the moving frame, the clock travelling along with the object. It is the shortest possible interval between two events. The dilated time t is what a stationary observer measures for the same pair of events. Pick a unit from seconds up to years, and the dilated time comes back in that same unit.
How do I enter the velocity, or use a preset?
Keep the preset on Custom and use the unit selector beside the velocity field to type a value in m/s, km/s, mi/s, km/h, or as a fraction of c (for example 0.8 means 80% of light speed). Or choose a real-world preset such as the ISS, a GPS satellite, Voyager 1, the Parker Solar Probe or a cosmic-ray muon to autofill its speed. The calculator converts everything to a fraction of the speed of light, c = 299,792,458 m/s, before applying the formula.
How does the reverse mode (speed from a target slowdown) work?
Switch the first selector to "Speed needed" and enter the slowdown you want as a Lorentz factor: 2 means the moving clock runs twice as slow, 10 means ten times. The calculator inverts the Lorentz formula, β = √(1 − 1/γ²), to return the exact fraction of light speed and the speed in km/s. A factor of 2 needs about 0.866c; a factor of 10 needs about 0.995c.
What is length contraction and how is it related?
The same Lorentz factor that stretches time also contracts length along the direction of motion: a rod of rest length L0 measures L = L0 / γ to an observer it passes. At 0.8c a one metre rod looks 0.6 m long, and at 0.99c it shrinks to about 0.14 m. Time dilation and length contraction are two faces of the same geometry of spacetime, so this calculator shows both.
Can the velocity equal or exceed the speed of light?
No. As v approaches c the Lorentz factor γ = 1 / √(1 − v²/c²) grows without bound, and at v = c it is undefined because you would be dividing by zero. No object with mass can reach the speed of light, so the calculator returns an empty result if you enter a speed at or above c. Keep the fraction of c strictly below 1 to get a finite answer.