In a currently running following Youtube video, the British Astrophysicist and Professor Brian Cox claims that there would be 10 km difference/error in position per day for GPS location, if Einstein’s relativity is not used to make a correction in the GPS: https://www.youtube.com/watch?v=Mpw68rvF4pc
The engineers. H.F. Fliegel and R. S. DiEsposti of the GPS Joint Program Office of the Aerospace Corporation involved in the development of the GPS system back in 1997 in a published paper came to the following conclusion: “Except for the leading γ [gamma] factor [in their final equation], it is the same formula derived in classical physics for the signal travel time from the GPS satellite to the ground station. As we have shown, introducing the γ factor makes a change of only 2 or 3 millimeters to the classical result. In short there are no ‘missing relativity terms.’ They cancel out.” General Relativity Theory is not needed”. : https://articles.adsabs.harvard.edu/pdf/1997ptti.conf..189F
At about the same time (1997), Peter Wolf and Gérard Petit in a published paper on “Satellite test of special relativity using the global positioning system”, wrote: “A test of special relativity has been carried out using data of clock comparisons between hydrogen maser clocks on the ground and cesium and rubidium clocks on board 25 global positioning system (GPS) satellites. The clocks were compared via carrier phase measurements of the GPS signal using geodetic receivers at a number of stations of the International GPS Service for Geodynamics (IGS) spread worldwide. In special relativity, synchronization of distant clocks by slow clock transport and by Einstein synchrony (using the transmission of light signals) is equivalent in any inertial frame. A violation of this equivalence can be modeled using the parameter δc/c, where c is the round-trip speed of light (c=299792458m/s in vacuum) and δc is the deviation from c of the observed velocity of a light signal traveling one way along a particular spatial direction with the measuring clocks synchronized using slow clock transport. In special relativity δc/c=0. Experiments can set a limit on the value of δc/c along a particular spatial direction (henceforth referred to as “direction of δc”). Within this model our experiment is sensitive to a possible violation of special relativity in any direction of δc, and on a nonlaboratory scale (baselines >~20000km). The results presented here set an upper limit on the value of δc/c