Time Dilation Calculator

What the time dilation calculator does

One of the strangest predictions of relativity is that time itself runs slower for things moving fast. This calculator works out that effect, taking a time interval and a velocity, and finding how much that interval stretches as the speed climbs toward the speed of light.

Below is what time dilation is, the equation behind it, why we never notice it in daily life, and a worked example.

How to use it

1. Enter the time interval, the duration measured by the moving clock, with its unit.
2. Enter the velocity, as a speed or a fraction of the speed of light, which cannot exceed light speed.
3. Press Calculate for the dilated time, or Reset to clear it.

What time dilation is

Time dilation is the slowing of time for an object in motion, as seen by an observer it is moving relative to. It is a direct consequence of Einstein's special relativity, published in 1905, which rests on the startling fact that the speed of light is the same for everyone, no matter how they move. To keep that fact consistent, space and time themselves must bend, and one result is that a moving clock ticks more slowly than a stationary one. The faster it moves, the slower it runs.

This is not an illusion or a mechanical quirk of clocks; it is a property of time itself. A fast-moving particle genuinely experiences less time passing than a stationary one, and a traveller moving near the speed of light would age more slowly than those left behind. The effect is utterly negligible at the speeds of everyday life, which is why it defies our intuition, but it grows without limit as the speed approaches that of light. This calculator quantifies exactly how much time stretches at any given speed.

The equation it uses

The dilated time is the original interval divided by a factor that depends on the speed:

Δt' = Δt ÷ √(1 − v² ÷ c²)

Here Δt is the time interval measured by the moving clock, Δt' is the dilated interval seen by the observer, v is the velocity, and c is the speed of light. The denominator is the famous square root that appears throughout relativity; its reciprocal is called the Lorentz factor. When v is small compared with c, the denominator is almost exactly one and there is virtually no dilation. As v approaches c, the denominator shrinks toward zero, and the dilated time grows enormously. The calculator evaluates this to give the stretched interval.

Why we never notice it

If time dilation is real, why does nobody ever notice their watch running slow on a fast train or a plane? The answer lies in how small everyday speeds are compared with the speed of light. Light travels at about 300,000 kilometres per second, while even a jet airliner manages only a fraction of a kilometre per second. At such speeds, the dilation factor differs from one by less than a trillionth, far too tiny to perceive or for an ordinary clock to register.

The effect only becomes appreciable at a substantial fraction of the speed of light, speeds reached by subatomic particles and imagined for futuristic starships, but never by trains, planes, or rockets as we know them. This is why relativity seems so counterintuitive: our intuitions were built in a slow world where time dilation is utterly invisible. The calculator makes the hidden effect visible by letting you dial the speed up toward light speed and watch the stretching of time emerge, gently at first and then dramatically.

Where time dilation is real and measured

Far from being mere theory, time dilation is measured every day and even has to be corrected for in technology we rely on. The satellites of the global positioning system carry precise clocks moving fast in orbit, and relativity makes those clocks tick differently from clocks on the ground. If the effect were not accounted for, GPS positions would drift by kilometres within a day, so engineers build the correction in. Your phone's navigation works partly because relativity is taken seriously.

Subatomic particles show the effect even more vividly. Muons, created high in the atmosphere by cosmic rays, decay so quickly that they should never reach the ground, yet they do, because their tremendous speed dilates their internal clock and stretches their short lifetime enough for the journey. In particle accelerators, fast-moving particles live measurably longer than slow ones, exactly as the formula predicts. The famous twin paradox, in which a space-traveller returns younger than their stay-at-home sibling, is the same effect taken to an extreme. The calculator captures the principle behind all of these.

Units and precision

The calculator takes the time interval in units from seconds to years, and the velocity as a speed or directly as a fraction of the speed of light, refusing any value above light speed since nothing can travel faster. It returns the dilated time in your chosen unit. The calculation is an exact application of the relativistic formula, and results carry several significant figures, which is useful since the effect is minute at low speeds.

A worked example

Suppose a clock moves at about 87 percent of the speed of light, and we follow one second on it.

The dilation factor at that speed works out to almost exactly two, so the dilated interval is Δt' = Δt ÷ √(1 − v² ÷ c²) ≈ 2 seconds. In other words, while one second passes for the fast-moving clock, two seconds pass for the stationary observer: time runs at half speed. At a more modest 60 percent of light speed, the stretch is gentler, about 25 percent, and at 99 percent it becomes dramatic, stretching time more than sevenfold.

Questions people ask

How do you calculate time dilation?

Divide the time interval by √(1 − v²/c²), where v is the speed and c is the speed of light. The faster the motion, the more time stretches.

Is time dilation real or just theoretical?

Real and measured. GPS satellites must correct for it, cosmic-ray muons reach the ground because of it, and fast particles in accelerators live longer, all matching the formula.

Why don't we notice time dilation in daily life?

Because everyday speeds are a tiny fraction of light speed, making the effect smaller than a trillionth. It only becomes noticeable at a large fraction of the speed of light.

What is the twin paradox?

A thought experiment where a twin travels near light speed and returns younger than the twin who stayed home. It is a real consequence of time dilation taken to an extreme.

References

A quick note on where the physics comes from. Time dilation and special relativity are standard physics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The constancy of the speed of light follows the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 3, Section 5.4, Time Dilation. https://openstax.org/books/university-physics-volume-3/pages/5-4-time-dilation
2. HyperPhysics, Time Dilation. http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/tdil.html
3. National Institute of Standards and Technology (NIST), Fundamental Physical Constants, Speed of light in vacuum. https://physics.nist.gov/cgi-bin/cuu/Value?c