GPS and Relativity

You were trying to drive to Melbourne, Australia, yet you ended up in Melbourne, Florida. Apart from somehow being able to drive over water, you also probably mindlessly followed the instructions of a GPS device.

Practically ubiquitous these days, the Global Positioning System (GPS) is responsible for the ever decreasing ability of people to use maps and their own orientation.

The GPS is a network of satellites orbiting at an altitude of 20000 km and moving at a speed of 14000 km/h (many people think that they are geostationary satellites, but they are not).

The way they work is that each satellite continuously transmits two pieces of information:

  • The position of the satellite
  • The time at which the position was reported

A GPS receiver will collect this information from a satellite, and using the speed of light will calculate how long the message has taken to reach it and hence the distance from the satellite. Having 4 of these measurements allows the receiver to calculate its position in a couple of instants and to an accuracy of 5-10 m.

GPS satellites have very very accurate atomic clocks on board to enable them to be incredibly precise when sending time information. However, things are not as simple as just having very accurate clocks.

GPS and relativity

No Einstein, no party GPS

We might think that relativity only applies in the domain of objects moving close to the speed of light, or near black holes, but in fact, if GPS satellites were not adjusted for relativity the position measurements would quickly become inaccurate.

To an observer on earth, the satellites are moving, and they’re moving rather quickly. According to special relativity, the clocks on the satellites should tick more slowly.

Secondly, because the satellites are high up, far away from the earth, the curvature of the spacetime due to the mass of the Earth is less compared to that at the surface. General relativity says that clocks closer to a massive object will appear to tick more slowly than those located further away. Hence, to an observer on the Earth’s surface, the clocks on the satellites will appear to be ticking faster.

The effects of special and general relativity slightly cancel out, but the satellite clocks still have to be adjusted back by about 38 microseconds a day. Which doesn’t seem like much, but, as GPS requires nanosecond accuracy, if unaccounted for, the accuracy of the system would decrease by about 10 kilometres a day.

Francesco

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