Why GPS needs relativity: 38 microseconds a day

A GPS satellite carries an atomic clock, and that clock does not keep the same time as an identical clock on the ground. Left alone, it would gain about 38 millionths of a second every day. Two pieces of relativity cause the drift, and they push in opposite directions.

What is being claimed

A GPS receiver works out where it is from time. Each satellite broadcasts the time on its clock, and the receiver turns the delay into a distance by multiplying by the speed of light. Light covers about 30 centimetres in a nanosecond, so a clock that is off by a microsecond puts a range off by about 300 metres. That makes the satellite clocks the heart of the system. Neil Ashby notes in his 2003 review that a good cesium clock left alone for a day should be correct to within about 5 parts in 10¹⁴, or 4 nanoseconds. “Relativistic effects are huge compared to this.”

Why it is worth knowing

Some popular accounts say relativity makes the satellite clocks run slow, or add the two effects together. That is wrong. The net effect makes them run fast compared with a clock on the ground. Only one of the two effects runs slow, and it is the smaller one. GPS is an everyday machine that has to get both effects right, with the right signs, to work at all.

Two effects, opposite signs

Gravity makes the satellite clock run fast. General relativity says a clock deeper in a gravitational field ticks more slowly. The ground sits deep in Earth's gravity, and the satellite orbits at a radius of 26,562 km, where gravity is weaker. Seen from the ground, the satellite clock gains. In his 2006 teaching article Ashby puts the gain at a fraction +5.288 × 10⁻¹⁰ of the clock's rate. Over a day of 86,400 seconds that is +45.7 microseconds (our arithmetic).

Speed makes the satellite clock run slow. Special relativity says a moving clock ticks more slowly than clocks at rest. The satellite moves at close to 4,000 metres per second. The reference clock on the equator moves too, at about 465 metres per second as Earth turns, so what counts is the difference. Ashby gives −8.228 × 10⁻¹¹, which is −7.1 microseconds a day (our arithmetic).

Add them and gravity wins by a factor of about 6.4. The net fraction, from Ashby, is +4.4647 × 10⁻¹⁰: the satellite clock gains about 38.6 microseconds a day (our arithmetic). (The two printed terms add to 4.4652; the small difference is rounding in them.) Ashby calls the total “about 10,000 times too large to ignore”.

What “kilometres a day” means

Ashby turns each rate into a distance: the gravity term “would translate into a navigational error of 13.7 km” in a day, and the speed term would contribute “of order 2.13 km/day”. The net comes to about 11.6 km a day (our arithmetic). These numbers are the clock error multiplied by the speed of light. They are not a promise that your position would be 11 km wrong after a day. A receiver also solves for its own clock, and an error shared by every satellite is partly absorbed there. The distance is a clear way to show how big the timing error is, not a forecast of where you would end up.

How it is fixed

Most of the fix happens in the satellite, not in your phone. In older satellites the clock was tuned on the ground before launch so that it ran slow by the net amount. In Ashby's 2003 review the 10.23 MHz reference frequency becomes 10.22999999543 MHz. Once in orbit, gravity and speed bring it up to 10.23 MHz as seen from the ground. Ashby's 2006 article says newer rubidium clocks are measured after they reach orbit, and the corrections go out in the navigation message. What does fall to the receiver is a smaller wobble: no orbit is perfectly circular, and the leftover term “can give rise to an error of as much as 75 ns if not accounted for”.

The prediction has been tested. The first cesium clock in orbit flew on the NTS-2 satellite, launched on 23 June 1977. Its measured rate was +442.5 parts in 10¹² against a predicted +446.5, which Ashby calls about “a 1% verification”. The balance also depends on the orbit. For a low orbiter such as the Space Shuttle the speed effect wins, and Ashby finds that the two cancel at an orbit radius of about 9,545 km.

In short

A GPS satellite clock gains about 45.7 microseconds a day because gravity is weaker up there, and loses about 7.1 because it is moving fast. Its net drift is a gain of about 38.6 microseconds a day. The clock is set slow before launch, or corrected by message, so that it keeps the ground's time.