Relativistic Effects on Satellite Clock as Seen from Earth

The general theory of relativity implies time contraction due to the Earth's gravitational field. A clock on a satellite appears to run faster when observed from the surface of the Earth.

The special theory of relativity predicts time dilation. A clock on a fast-flying satellite appears to run slower when observed from a stationary position on Earth.

It is possible to observe the combined effect of general and special relativity on an atomic clock on board a GPS satellite.

General and special relativity effects are stronger near the Earth's surface than they are on an orbiting satellite. Also the speed of the satellite must be greatest at its minimum height, so that centrifugal force can cancel the stronger gravitational attraction. As the height increases, both effects decrease nonlinearly according to the Lorenz factor from special relativity and following the Schwarzschild solution of the Einstein field equations from general relativity. But the two effects are opposite, so there is a height (about 3200 km above the Earth) where they cancel; the clock is seen to be working correctly at this point when observed from the Earth's surface. A GPS (Global Positioning System) satellite system is an example and practical proof of both of Einstein's theories. GPS receivers are made to receive 10.23 MHz code. But GPS satellites must broadcast this code on the frequency 10.22999999543 MHz to cancel relativistic effects. The effect (a time difference about 38 μs/day) is apparently insignificant, but it must be taken into account or there would be a 11.5 km/day position measurement error. The errors are marked at the approximate heights of International Space Station (ISS), GPS satellites (GPS), and geostationary satellites (GEO) in the plot.