Special Relativity: Time Dilation

Time dilation is one of the most counterintuitive predictions of Einstein's 1905 special theory of relativity: moving clocks run slower than stationary ones. Not because they're broken, not because of some mechanical effect — time itself passes more slowly for a moving object relative to a stationary observer.

The time dilation formula is:

$\Delta t = \gamma \Delta t_{0}$

where $\Delta t_{0}$ is the "proper time" (time measured by the moving clock, in its own rest frame), $\Delta t$ is the time measured by the stationary observer, and

$\gamma = \dfrac{1}{\sqrt{1 - v^{2}/c^{2}}}$

is the Lorentz factor. For $v = 0.9c$: $\gamma = 1/\sqrt{1 - 0.81} = 1/\sqrt{0.19} = 2.294$.

So 1 second on the ship = 2.294 seconds on Earth. The ship's clock ticks at 43.6% the rate of Earth's clock.

Worked examples at different speeds

• $v = 0.1c$: $\gamma = 1.005$. Time dilation: 0.5%. Unmeasurable without atomic clocks.
• $v = 0.5c$: $\gamma = 1.155$. 1 year ship time = 1.155 years Earth time.
• $v = 0.9c$: $\gamma = 2.294$. 1 year ship time = 2.3 years Earth time.
• $v = 0.99c$: $\gamma = 7.089$. 1 year ship time = 7.1 years Earth time.
• $v = 0.999c$: $\gamma = 22.37$. 1 year ship time = 22.4 years Earth time.
• $v = 0.9999c$: $\gamma = 70.71$. 1 year ship time = 70.7 years Earth time.

At speeds approaching $c$, time dilation becomes extreme. A muon created in the upper atmosphere traveling at $0.998c$ ($\gamma = 15.8$) lives 15.8× longer than its rest-frame lifetime of 2.2 μs, allowing it to reach the ground — direct evidence of time dilation.

Experimental confirmation

• Muon experiments (1941+): Cosmic ray muons reach the ground despite their short lifetime, exactly as predicted by time dilation.
• Hafele-Keating experiment (1971): Atomic clocks flown around the world on commercial jets showed time differences matching relativity (including both special and general relativistic effects) to within experimental error.
• GPS satellites: GPS clocks run fast by ~38 μs/day due to general relativistic effects and slow by ~7 μs/day due to special relativistic time dilation. Without correcting for both, GPS would drift by ~10 km/day.
• Particle accelerators: Unstable particles in accelerators live far longer than their rest-frame lifetimes, exactly as $\gamma$ predicts.

Common mistakes

• Thinking time dilation is just an "illusion." It's not — it's a real physical effect. The traveling twin ages less (the twin paradox). When they reunite, there's a genuine age difference.
• Applying time dilation to both observers symmetrically. In the standard setup, the "stationary" observer sees the moving clock run slow, AND the moving observer sees the stationary clock run slow. This seems contradictory but isn't — they're measuring different things (different notions of "simultaneous").

Try it in the visualization

Drag the speed slider from 0 to 0.999c and watch the Earth clock speed up relative to the ship clock. At $v = 0$, both tick together. At $v = 0.9c$, the ship clock ticks at less than half the rate. The γ factor curve shows the dramatic nonlinearity — almost no effect below 0.5c, then a steep rise as you approach c.