How Does GPS Correct for Relativity? The 38 Microseconds a Day

Your phone finds you by timing radio signals from satellites about 20,000 km overhead. That only works if the clocks on those satellites and the clocks on the ground agree to within a few tens of nanoseconds. Relativity says they cannot agree on their own. Our sticker character Orbitick is the satellite clock that has been told, before launch, to tick slightly slow so that it arrives in orbit telling the right time.

Why does GPS need relativity at all?

A GPS receiver works out its distance to each satellite from how long the signal took to arrive, multiplied by the speed of light, c = 299,792,458 m/s (an exact defined value; NIST CODATA^[1]). Light covers about 30 cm in a nanosecond, so timing errors become position errors very quickly. Richard Pogge of Ohio State University notes that the satellite clock ticks must be known to 20–30 nanoseconds for the system to deliver its accuracy (Pogge, Ohio State^[2]).

Neil Ashby, whose review in Living Reviews in Relativity is the standard reference, lists the relativistic effects GPS must handle: the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts and the relativity of synchronisation (Ashby 2003^[3]).

Where does +38 microseconds per day come from?

Two effects pull in opposite directions, and it matters to keep them separate.

Special relativity: moving clocks run slow (about −7 µs/day)

A clock moving relative to you ticks slower by the factor √(1 − v²/c²). GPS satellites orbit at roughly 14,000 km/h, so special relativity predicts that their clocks fall behind ground clocks by about 7 microseconds per day (Pogge^[2]). This is the same time dilation that lets fast cosmic-ray muons reach the ground; Muonaut is that story in pixels.

General relativity: clocks higher up run fast (about +45 µs/day)

Clocks deeper in a gravitational well tick more slowly. Satellites sit much higher in Earth’s gravitational potential than we do, so their clocks run faster. General relativity predicts a gain of about 45 microseconds per day (Pogge^[2]).

The net: about +38 µs/day

Add them: +45 − 7 ≈ +38 microseconds per day. Pogge spells out what ignoring this would mean: a navigational fix would be wrong after about 2 minutes, with position errors accumulating at roughly 10 kilometres a day (Pogge^[2]). Note the split: the larger effect is gravitational, not the satellite’s speed, which surprises many people.

How do engineers actually correct for it?

The main fix is astonishingly simple: change the clock rate before launch. Ashby explains that the satellite clocks are adjusted lower in frequency on the ground so that, seen from the geoid, they beat at the nominal 10.23 MHz once in orbit (Ashby^[3]). The net rate offset general relativity predicts, as quoted by Ashby for the NTS-2 test below, is +446.5 parts in 10¹²; multiplied by the 86,400 seconds in a day, that is about 38.6 µs per day, consistent with the rounded +38.

A few smaller corrections are applied in software:

• Eccentricity correction. GPS orbits are slightly elliptical, so the speed and height change around each orbit. Ashby notes that this periodic term is handled in the receiver or folded into the clock correction sent with the navigation message (Ashby^[3]).
• Sagnac effect. Because Earth rotates while the signal is in flight, synchronisation in an Earth-fixed frame needs a correction; Ashby notes it can amount to hundreds of nanoseconds depending on geometry (Ashby^[3]).

How do we know the correction is right?

This is not just theory baked into an engineering spec; it was measured.

• NTS-2 (1977). Ashby recounts that the Navigation Technology Satellite 2 carried a caesium clock that was run for about 20 days before its frequency synthesiser was switched on. Its measured rate offset was +442.5 parts in 10¹², against a general-relativistic prediction of +446.5 parts in 10¹², agreement within the accuracy of the clock (Ashby^[3]).
• Clocks on airliners (1971). Joseph Hafele and Richard Keating flew four caesium clocks around the world on commercial flights. Relative to the U.S. Naval Observatory time scale, the clocks lost 59 ± 10 ns on the eastward trip and gained 273 ± 7 ns on the westward trip, in good agreement with predictions (Hafele & Keating, observed^[4]; predictions in Hafele & Keating, predicted^[5]).
• Light climbing gravity (1960). Robert Pound and Glen Rebka measured the gravitational frequency shift of gamma rays in the laboratory, in a paper titled “Apparent Weight of Photons” (Pound & Rebka 1960^[6]).
• Tabletop relativity (2010). NIST researchers using two aluminium-ion optical clocks saw time dilation at relative speeds below 10 m/s, and gravitational time dilation from a height change of less than 1 metre (Chou et al. 2010^[7]).

Both effects go back to Einstein: special relativity in his 1905 paper on the electrodynamics of moving bodies (Einstein 1905^[8]), and gravitational time dilation in general relativity. For why c is the speed limit in the first place, read our pillar post Why Is the Speed of Light the Limit?.

Common misconceptions

“GPS proves relativity, so without it GPS would be off by 38 µs.”

Close, but the drift is cumulative: 38 µs per day, growing every day, unless corrected. That is why the fix is a permanent rate change, not a one-off reset.

“It is mostly because the satellites move fast.”

No. The speed effect (−7 µs/day) is the smaller one. The gravitational effect (+45 µs/day) dominates.

Wear the correction

Orbitick and Muonaut also come together on the On Cosmic Time sticker sheet. For tees, browse relativity and cosmos. If you prefer your time travel to stay theoretical, see Physics Myths Busted.

FAQ

How much faster do GPS clocks run?

About 38 microseconds per day relative to ground clocks: roughly +45 µs from weaker gravity minus 7 µs from orbital speed.

How is it corrected?

Mainly by setting the satellite clocks slightly slow before launch, plus smaller corrections (orbit eccentricity, Sagnac effect) in the navigation software.

Was it ever tested in orbit?

Yes. In 1977 the NTS-2 satellite measured a rate offset of +442.5 parts in 10¹² against a predicted +446.5 parts in 10¹².

References

1. NIST CODATA, speed of light in vacuum. https://physics.nist.gov/cgi-bin/cuu/Value?c
2. Richard W. Pogge (Ohio State University), “Real-World Relativity: The GPS Navigation System”. https://www.astronomy.ohio-state.edu/pogge.1/Ast162/Unit5/gps.html
3. N. Ashby (2003), “Relativity in the Global Positioning System,” Living Rev. Relativ. 6, 1 (open full text). https://pmc.ncbi.nlm.nih.gov/articles/PMC5253894/
4. Hafele & Keating (1972), “Around-the-World Atomic Clocks: Observed Relativistic Time Gains,” Science 177, 168. https://doi.org/10.1126/science.177.4044.168
5. Hafele & Keating (1972), “Around-the-World Atomic Clocks: Predicted Relativistic Time Gains,” Science 177, 166. https://doi.org/10.1126/science.177.4044.166
6. Pound & Rebka (1960), “Apparent Weight of Photons,” Phys. Rev. Lett. 4, 337. https://doi.org/10.1103/PhysRevLett.4.337
7. Chou, Hume, Rosenband & Wineland (2010), “Optical Clocks and Relativity,” Science 329, 1630. https://doi.org/10.1126/science.1192720
8. A. Einstein (1905), “Zur Elektrodynamik bewegter Körper,” Ann. Phys. 17, 891. https://doi.org/10.1002/andp.19053221004

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.