How Time Dilation Is Calculated
Time dilation is a direct consequence of Einstein's special theory of relativity: a clock moving relative to an observer runs slower, from that observer's point of view, than an identical clock at rest with them. The proper time (Δt₀) is the interval measured by a clock traveling with the moving object itself — for example, the elapsed time on an astronaut's wristwatch. The dilated time (Δt) is the longer interval measured by a stationary observer watching that clock go by. The two are related by Δt = Δt₀ / √(1 − v²/c²) = γΔt₀, where v is the relative velocity between the two frames and c is the speed of light in a vacuum (299,792,458 m/s).
The Lorentz factor (γ)
The quantity γ = 1 / √(1 − v²/c²) is called the Lorentz factor. It depends only on the ratio β = v/c, not on the direction of motion, and it is always greater than or equal to 1. At everyday speeds — cars, planes, even orbital spacecraft — β is a tiny fraction of a percent, so γ is indistinguishable from 1 and the dilation is far too small to notice without atomic clocks. As v climbs toward c, γ grows nonlinearly: at β = 0.6 (60% of c), γ = 1.25; at β = 0.99, γ ≈ 7.09; at β = 0.9999, γ ≈ 70.7. Because γ blows up as β → 1, no object with mass can ever reach or exceed the speed of light — the formula becomes undefined (division by zero) exactly at v = c.
Proper time vs. dilated time in practice
This calculator takes the proper time Δt₀ (the time the moving object experiences) and the relative velocity v, then computes γ and the dilated time Δt that a stationary observer would measure. The effect is real and measured routinely: cosmic-ray muons created in the upper atmosphere travel at roughly 0.998c and, thanks to time dilation, survive long enough in their own decaying "clock" to reach the ground — without dilation, their short particle lifetime would let them travel only a fraction of that distance. Particle accelerators push electrons and protons to γ factors in the thousands, and the effect has been confirmed directly with atomic clocks flown on aircraft. Note that this tool covers kinematic (velocity-based) time dilation only; time dilation caused by gravity — relevant to GPS satellites and clocks at different altitudes — follows a different formula from general relativity.