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GPS and relativity

Space and time · level 8, lesson 3 of 6

A question to start

Your phone finds where you are from the signals of satellites 20,000 km up. Why would it be kilometres out within a day if nobody had allowed for Einstein?

The idea

A GPS receiver works out its position from the times signals take to arrive from several satellites. Light goes 30 cm in a nanosecond, so the satellite clocks must agree to within nanoseconds.

Each satellite orbits about 20,000 km up, at about 14,000 km/h, once every 12 hours or so. Two effects change its clock rate, as seen from the ground:

  • Speed (special relativity): the moving clock runs slow, by about 7 µs a day.
  • Height (general relativity): the clock sits higher in Earth's gravity, so it runs fast, by about 45 µs a day.

Together the satellite clocks run fast by about 38 µs a day. The clocks are adjusted for this. Without it, positions would be wrong after only about 2 minutes, and errors would grow by about 10 km every day.

speed: 7 µs a day slowheight: 45 µs a day fasttotal: 38 µs a day fastorbit about 20,000 km up

Worked examples

1. The speed effect. \(v \approx 3{,}870\) m/s. \(\frac{v^2}{2c^2} = \frac{1.50 \times 10^7}{1.80 \times 10^{17}} \approx 8.3 \times 10^{-11}\). Times 86,400: about 7.2 µs a day slow.

2. The height effect. Earth's \(GM = 3.986 \times 10^{14}\) m³/s². From the ground (\(R = 6.371 \times 10^6\) m) to the orbit (\(r = 2.657 \times 10^7\) m from the centre): \(\frac{GM}{c^2}\left(\frac{1}{R} - \frac{1}{r}\right) \approx 4.43 \times 10^{-3} \times 1.19 \times 10^{-7} \approx 5.3 \times 10^{-10}\). Times 86,400: about 45.7 µs a day fast.

3. The position error. 38 µs times the speed of light: \(38 \times 10^{-6} \times 3 \times 10^8 \approx 11{,}000\) m, about 11 km in a day.

Try it

  1. Which effect makes GPS clocks run slow? Which makes them run fast?
  2. Which is bigger?
  3. Why do nanoseconds matter for GPS?
  4. A navigation satellite clock is out by 1 µs. How big a position error could that give?
Answers
  1. Speed makes them slow; being higher in Earth's gravity makes them fast.
  2. The height effect, about 45 µs a day against 7.
  3. Light goes 30 cm a nanosecond, so a few nanoseconds of error is a metre of position error.
  4. About \(10^{-6} \times 3 \times 10^8 = 300\) m.

Common mistake

Using the plain \(g h\) rule for 20,000 km. Gravity weakens a lot over that height. Use \(\frac{GM}{c^2}\left(\frac{1}{R} - \frac{1}{r}\right)\) instead.

Where it leads