Open a map on your phone and a blue dot appears, usually within a few metres of where you are standing. It seems effortless. Behind it is one of the most remarkable machines humans have ever built: a few dozen satellites, each carrying atomic clocks, hurtling round the Earth twice a day, and a trick of geometry that turns time into distance.
Start with the trick. Suppose someone tells you that you’re exactly five kilometres from the station. You could be anywhere on a circle five kilometres around it. Add a second fact, that you’re three kilometres from the temple, and that second circle crosses the first at just two points. Add a third distance, and only one point fits all three. This is called trilateration. In space, with three dimensions, the circles become spheres, but the idea is the same.
So how does your phone know how far away a satellite is? It times a signal. Each navigation satellite constantly broadcasts radio messages that say, in effect, “I am satellite number twelve, I am exactly here, and the time on my clock is exactly this.” Your phone receives the message, notes the time on its own clock, and works out how long the signal took to arrive. Radio waves travel at the speed of light, about three hundred thousand kilometres every second, so multiply the travel time by that speed and you have the distance. The satellites of the American GPS system orbit about twenty thousand kilometres up, so the signal takes around seventy thousandths of a second to reach you.
That speed is also the problem. Light covers about three hundred metres in a millionth of a second. If your timing is off by a single millionth of a second, your distance is off by three hundred metres. To find you within a few metres, the whole system has to measure time to within a few billionths of a second.
The satellites manage this with atomic clocks, which keep time by counting the incredibly regular vibrations of atoms such as caesium or rubidium, and stay accurate to within billionths of a second over days. Your phone, though, does not contain an atomic clock. Its clock is cheap and drifts. So how can it time anything to a billionth of a second?
Here’s the clever bit. The phone doesn’t try. It treats the error in its own clock as one more unknown to solve for. Its position has three unknowns, east-west, north-south and height, and its clock error makes four. Four unknowns need four equations, so the phone listens to at least four satellites at once. Only one combination of position and clock error makes all four distances agree, and the phone solves for it. As a bonus, every phone ends up knowing the time almost as precisely as the atomic clocks in orbit. In practice, phones listen to many more satellites than four, from several systems at once, including India’s own NavIC, Europe’s Galileo and Russia’s GLONASS, and use the extras to sharpen the answer.
And then there’s Einstein. According to his theory of special relativity, a clock that is moving fast runs slow compared with one at rest, and GPS satellites move at nearly four kilometres a second. That makes their clocks lose about seven millionths of a second every day compared with clocks on the ground. But according to his theory of general relativity, clocks deeper in a gravitational field run slower, and the satellites, twenty thousand kilometres up, are much higher in the Earth’s gravity than we are. That makes their clocks gain about forty-five millionths of a second a day. The two effects pull in opposite directions, and the gain wins: overall, each satellite’s clock runs about thirty-eight millionths of a second a day fast.
Thirty-eight millionths of a second sounds like nothing. But remember, a millionth of a second is three hundred metres. If engineers ignored relativity, the errors in the system would build up by around ten kilometres every single day. Within a week, your blue dot would be in the next city. So the satellite clocks are deliberately set to tick slightly slower before launch, to cancel the effect, and the system’s software accounts for the rest.
So the next time the blue dot finds you, it’s worth remembering what made it possible: clocks counting atoms, signals travelling at the fastest speed there is, a fourth satellite to fix your phone’s cheap clock, and two of the strangest ideas in physics, quietly checked billions of times a day by people looking for the nearest chai stall.