I am a physicist and GPS indeed takes relativistic effects into account. GPS would not just be slightly wrong if it didn't: it would be entirely worthless. See for example [1].

[1] http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps....

Fun fact: When designing the original system there was disbelief amongst some involved that the relativistic effects were either real or relevant so they built in a switch to be able to remotely disable those parts of the calculations. They never did disable them.

Actually, that gose down as the funnest fact that I've learned for at least a month, so thank you.

I don't remember for sure but I think I learned that from one of these lectures...

"Particle Physics for Non-Physicists: A Tour of the Microcosmos"

http://www.thegreatcourses.com/tgc/courses/course_detail.asp...

GPS onboard clock is corrected for relativity due to it's orbit height and speed so it's clock time is approximately correct

But measuring the simultaneity of an event between two different points with a moving clock is a lot trick.

That might have been an issue if they were measuring anything with the satellite clocks.

But they weren't. By my understanding, they were merely using the satellite signals to synchronize the ground clocks in the ground frame, and after that was done, the satellite's were never heard from again.

Synchronizing two clocks in different positions is very difficult since relativity always comes in as you are moving a third clock between them. It's essentially impossible if you don't know the gravity field encountered by the moving clock.