The GPS system (Global Positioning System) is composed of at least 24 satellites orbiting around the Earth in six planes making an angle of 55 degrees with the equatorial plane. The position of the satellites is known at all time, and the clocks of the satellites are perfectly synchronized. The satellites send signals repeated periodically. The receiver is equipped with a clock and at least four channels. It receives the signals of at least four satellites and measures (on its clock!) the transit times of at least four signals. However, since its clock is not necessarily synchronized with the clock of the satellites, these times are fictitious transit times. This yields a system of four equations in four unknowns: the three coordinates of the receiver, and the time shift between the clock of the satellites and that of the register. This system of equations has two solutions, one of which is the actual location of the receiver.

In practice, it is more complicated. Indeed, the speed of the satellites is sufficiently large that all of the calculations must be adapted to account for the effects of special relativity. Since the clocks on the satellites are traveling faster than those on Earth, they run slower. Furthermore, the satellites are in relatively close proximity to the Earth, which has significant mass. General relativity predicts a small increase in the speed of the clocks on board the satellites. Modeling the Earth as a large nonrotating spherical mass without any electrical charge, the effect is relatively easy to compute using the Schwarzschild metric, which describes the effects of general relativity under these simplified conditions. Fortunately, this simplification is sufficient to capture the actual effect to high precision. The two effects must both be considered because even though they are in opposite directions, they only partially cancel each other out.

One source of errors in the computations is the estimation of the speed of the signal. Differential GPS allows obtaining a much higher precision, by estimating with a higher precision the speed of the signals.

GPS is used in geography to measure the height of mountains and measure their growth. They have been used to establish the official height of Mount Everest and confirm that it is indeed the highest mountain of the Earth. There are very interesting mathematics in the signal of the GPS. These will be explained in another vignette.