# Time Dilation Calculator

Created by Gabriela Diaz
Last updated: Aug 06, 2022

The time dilation calculator gives you a better idea of the time behavior according to special relativity.

Einstein's relativity theory has shown that time is relative. Time perceived by one observer in its frame of reference differs from that of another observer in another inertial frame. Please keep reading to find out what time dilation is and what happens to it when we approach to the speed of light ⌛

## What is time dilation? — Time is relative

One of Einstein's special relativity theory's implications is that time is not absolute but rather relative.

The example of two inertial observers A and B, each carrying a clock, is frequently used to illustrate this relativeness aspect of time. If observer A remains stationary while observer B travels relative to A, B will see A's clocks moving slower than theirs. Similarly, A will perceive B's clock to be going slower — Observers perceive that a clock moves slower when it moves relative to them.

This effect of the time slowing down is known as time dilation. And the faster the relative velocity between observers, the greater the time dilation is, becoming more evident when speed approaches values in the order of magnitude of the speed of light. In special relativity, the principle that the speed of light is constant for every observer is essential, as it explains why time has to dilate in order for the speed of light to remain the same regardless of the observer's own motion.

Another common thought exercise used to explain time dilation is the twin paradox. In this imaginary situation, one twin travels into space in a high-speed rocket while his sibling stays on Earth. The astronaut-twin moves at a speed closer to the speed of light and, after some years, returns to Earth. Once back on Earth, the astronaut-twin finds his Earth-twin has aged lots more than he has.

You may wonder what the paradox is in this situation. The paradox arises when we consider the astronaut-twin to be the one that's stationary and the Earth-twin the one that's relatively moving. In this case, the Earth-twin would be younger than the astronaut. So, why is it stated that it's Earth-twin who has aged more? The reason for this is that these situations are not symmetrical. The astronaut-twin is undergoing a non-inertial movement (accelerating and decelerating during his journey), while his brother is moving in a non-accelerated relative motion.

To learn more about relativity and the relationship between mass and energy, visit the e = mc² calculator!

💡 Did you know there's another form of time dilation known as gravitational time dilation?

## The time dilation formula

Time dilation is determined as the difference of time perceived by the moving observer and the stationary observer $\Delta t'$. The time dilation formula based on special relativity is:

\small \begin{aligned} \Delta t' &= \gamma \ \Delta t \\ \Delta t' &= \cfrac{\Delta t}{\sqrt{1- v^2/c^2}} \end{aligned}

where:

• $\Delta t'$ — Time that has passed as measured by the traveling observer (relative time);
• $\gamma$ — Lorentz factor, ${\sqrt{1- v^2/c^2}}$;
• $\Delta t$ — Time that has passed as measured by a stationary observer;
• $v$ — Speed of the traveling observer; and
• $c$ — Speed of light (299,792,458 m/s).

You can learn more about the Lorentz factor of a moving object with the Lorentz factor calculator.

This is the expression used by the time dilation calculator ⌛ Notice that for this time difference to be evident, the speed $v$ must be at least in the order of magnitude of the speed of light to obtain any substantial difference in times observed by moving and stationary observers.

Nevertheless, the time dilation effect has been proved at speeds many orders of magnitude lower than the speed of light. This is the case of an experiment conducted in 1971 using three sets of atomic clocks. One of them remained on Earth while the others flew aboard two different airplanes. The results showed a difference in times elapsed by all sets of clocks.

💡 Give the time dilation calculator a try! What happens to time dilation values when you enter a speed of $0.001 \ c$ and then $0.9 \ c$?

Gabriela Diaz
Time interval (Δt)
sec
Observer velocity (v)
km/s
Relative time (Δt')
sec
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