Real-World Relativity: GPS

Einstein’s theory of relativity is not as abstract as we may think. An example we often use is GPS, a system that as we know uses a method called trilateration, a method that depends on the precision of the satellite internal clocks to calculate absolute positions.

The problem is that time is relative, and though we all recognize it, probably most of us imagine that we needed extreme conditions to make it happen (I was one of them). But all we need is a satellite at about 20.000 km from the ground, orbiting at about 14.000 km/h and a process requiring 20 to 30 nanoseconds precision, and there you have it, you’ll have to take the theories of relativity into account:

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion.

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

The combination of these two relativitic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day!

Cool article! But you know what’s cooler? That these theories were almost like pure creation: it’s not like Einstein developed a GPS system, failed to make it work, to finally theorize: what if time would be relative? Naturally he did start from an empty canvas and no purpose at all, but in face of the problem the scientific community had identified, his leap was so extraordinary that after 100 years we still find it hard to explain. Thank you, Albert.