GPS clock corrections
Alan Grayson
Our previous calculation of the timekeeping error of a satellite clock would have been fine had we not overlooked an important fact: We pretended as though the clocks were at rest on Earth at mean sea level. So, let’s see what relativity has to say about clocks in 20,000-kilometer-high circular orbits around Earth. The satellite orbits are not perfectly circular, or identical, but for now let’s pretend that they are. We call that modeling. The clocks would move at a rate of about 4 kilometers per second and exist in an environment where Earth’s gravity is only about one-fourth that at sea level.
According to the theory of special relativity, a moving clock ticks more slowly when compared with one that’s stationary at sea level. A clock aboard a GPS satellite will lose about 7 microseconds per day. That is three orders of magnitude larger than our budget for satellite clock error discussed earlier, therefore we can’t simply ignore it.
According to the theory of general relativity, on the other hand, a clock in a weaker gravitational field will tick faster than one that’s stationary at sea level. Apparently, gravity weighs down time, too. A clock aboard a GPS satellite in a medium Earth orbit will gain about 45 microseconds per day over a clock that’s at sea level on the earth.
The net effect: A GPS satellite clock will gain about 38 microseconds per day over a clock at rest at mean sea level. This effect is secular, meaning the time offset will grow from day to day