The Moon-High Clock

A GPS satellite carries an atomic clock — and to a clock, height and speed are not free. Drag one up from the launchpad and watch two of Einstein's effects fight for it. The answer to how GPS works is the truce they reach.

Companion film — 2:30 Drag a satellite up its orbit and watch the net clock-rate curve climb from below the line to above it. Below the crossing — the ISS at 408 km — speed wins and the clock runs slow (−24.6 µs/day; Scott Kelly came home ≈−8.4 ms younger than his identical twin, that being this engine's ISS rate × 340 days). Above it — GPS at 20,189 km — height wins and the clock runs fast (+38.5 µs/day; ≈11.5 km/day of drift if uncorrected). They cancel at exactly 1.5 Earth-radii — a number that falls out of the algebra, not a fit. Every figure on screen is recomputed live from c, GM⊕ and R⊕ by the same rates.mjs the verifier (9/9) imports; the soundtrack is two clock voices that beat apart and fall into unison at the crossing — a craft sonification, not a measured tone. Source: research/gps-relativity/film.

Your phone finds itself by listening to the clocks on satellites overhead and timing how long each signal took to arrive. Light covers about 30 cm per nanosecond, so a clock that is wrong by a millionth of a second throws your position off by hundreds of metres. The satellites' clocks are wrong, predictably, for two reasons — and GPS only works because it corrects for both.

Altitude · orbital speed
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Special relativity (motion)
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General relativity (height)
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Net clock rate
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If this clock ran a whole day with no relativity correction, GPS would place you —.

Two things change as the satellite climbs. It slows down — a high orbit is a lazy one — and special relativity says a moving clock ticks slow, so this effect shrinks as you rise. Meanwhile it climbs out of Earth's gravity, and general relativity says a clock higher in a gravitational field ticks fast, so this effect grows. One falls, one rises. Somewhere they must cross.

They cross at a radius of exactly 1.5 Earth-radii — about 3,186 km up. Below it (the ISS lives here) motion wins and the orbiting clock runs slow; astronauts age a sliver slower than we do. Above it gravity wins. GPS, at 20,180 km, is far above the line: its clocks gain about 38 microseconds every day. Light travels about 11 km in 38 microseconds — so an uncorrected GPS would walk you off the map within hours. The engineers knew this before the first satellite flew: the onboard clocks are deliberately set to tick at the wrong rate on the ground, so they run right once in orbit.

The check

For a circular orbit of radius r around Earth, with GM = 3.986004418×10¹⁴ m³/s², c = 299,792,458 m/s, and Earth's mean radius R⊕ = 6,371 km, the fractional clock rates are dSR = −GM/(2rc²) (motion, always slow) and dGR = (GM/c²)(1/R⊕ − 1/r) (height, always fast). Setting their sum to zero gives r = 1.5 R⊕ exactly — no fitting, no constants tuned. Every figure below is recomputed by these two lines, live, and matches research/gps-relativity/compute.mjs to the displayed digits:

orbitSR (µs/day)GR (µs/day)net (µs/day)

Standard reference values: GR +45 µs/day, SR −7 µs/day, net +38 µs/day for GPS (Ashby 2003; Pogge / Ohio State). The ~11 km/day figure is the light-travel distance of the accumulated clock offset — the scale of the ranging error; the full navigation solution differences several satellites, so the realised position error differs in detail, but the size is right and the system genuinely fails without the correction.

So how does GPS work? It is a constellation of flying clocks that have been told the truth about height and speed. Relativity is not a rounding error here — it is the difference between a map and a guess.