Moving clocks run slow.
Einstein's special relativity says a clock moving relative to you ticks slower than yours — by nanoseconds a day on a jet, by microseconds a day in orbit, and dramatically once speed nears light itself. GPS satellites have to correct for exactly this, every single day.
The triangle proportions are exact. Near c, both clocks shrink together so the rapidly lengthening path stays in view.
For one half-tick, (ct)² = L² + (vt)². Solving gives t = L/[c√(1−v²/c²)] = γL/c, so the moving tick takes γ times as long.
Speed v
v → c ⇒ γ → ∞
≈0denser resolution as v → c0.999 c
1.00 year on the moving clock =
negligiblesmallmeasurablelarge
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Where γ actually comes from
The light travels at the same fixed speed c in both clocks. The moving clock’s photon takes the longer diagonal path, so one tick takes longer.
The Lorentz factor γ = 1/√(1−v²/c²) comes from special relativity, the theory Albert Einstein published in 1905. A clock moving at speed v ticks slow by a factor of 1/γ as measured by a stationary observer — so over τ seconds of the moving clock's own (proper) time, the stationary clock counts γ·τ, an extra (γ−1)·τ. This page only models that velocity term. GPS satellites' real onboard clock correction (about +38 µs/day, running fast overall) also includes general-relativity gravitational time dilation — clocks run faster higher up in a weaker gravity well — which is a separate, larger effect this page does not model; don't read this page's velocity-only −7 µs/day figure as the real GPS correction on its own. The "1 year" reference duration is a Julian year (365.25 days = 31,557,600 s). Orbital speeds (ISS, GPS) are treated as constant for this estimate, using the standard instantaneous-velocity time-dilation formula even though orbital motion is technically accelerating, not inertial.
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