The GPS system accounts for both special and general relativity effects to ensure accurate positioning and time synchronization. While the satellites in the GPS system are indeed moving relative to each other, the system takes into consideration the relativistic effects to maintain accurate time synchronization.
Special relativity predicts that time dilation occurs due to relative motion between observers. If we consider two satellites moving relative to each other at high speeds, special relativity would suggest that their clocks would exhibit a time dilation effect. However, in the GPS system, these effects are compensated for.
The GPS satellites have highly accurate atomic clocks on board, which are synchronized to Earth-based clocks. Additionally, the GPS system incorporates general relativity corrections to account for the time dilation caused by the Earth's gravitational field. The combination of both effects—special relativity due to relative motion and general relativity due to gravity—allows the GPS system to maintain precise time synchronization.
To compensate for the relative motion time dilation, the GPS satellites' clocks are adjusted to run slightly faster than clocks on Earth. This adjustment accounts for the predicted special relativity time dilation effect between the moving satellites and the Earth's surface.
Additionally, the gravitational time dilation caused by the Earth's gravitational field is also taken into consideration. Clocks closer to a massive object, such as the Earth, will run slightly slower compared to clocks at a higher altitude. Therefore, the GPS system incorporates general relativity corrections to account for the difference in gravitational time dilation between the satellites and the Earth's surface.
By combining these adjustments, the GPS system ensures that the clocks on the satellites remain synchronized with Earth-based clocks, allowing for accurate positioning calculations. This synchronization accounts for both relative motion time dilation and gravitational time dilation effects, enabling precise navigation and timing information for GPS users.