GPS satellites don’t correct for relativistic effects. They just set the Earth as the preferred frame, which is a great choice for a terrestrial positioning system.
I’ll see if I can find the paper it’s quite interesting.
Edit: I can't find the specific paper I'm thinking of. It's one that discusses how when other orbiting bodies want to use GPS they have to make a bunch of complex corrections that terrestrial users don't. But a search for Earth Centerered Inertial Frame will find you plenty of papers discussing the general concept.
GPS satellites don't compensate for relativistic effects (except for the huge offset built into their atomic clocks), but receivers certainly do. The satellite orbits are slightly elliptical which leads to varying gravitational time dilation. If this is not taken into account, positioning is off by around 10 m [1]. This is an effect from general relativity, so it might be left out in simple textbook explanations.
Too late to edit, but here is the paper that I meant: https://apps.dtic.mil/sti/pdfs/ADA516975.pdf. At a mere 12 pages it's an easy read. Needless to say, the military is almost certainly the highest quality source for info on the actual engineering practicalities of GPS.
First sentence of the introduction:
The Operational Control System (OCS) of the Global Positioning System (GPS) does not include the rigorous transformations between coordinate systems that Einstein's general theory of relativity would seem to require
I believe there is a correction for the onboard satellite atomic clocks, which run faster than ground/ECI clocks due to a combination of special and general relativistic effects. The correction is a simple one - they just calibrate the clock with a small polynomial. IIRC nothing but the linear and constant terms are even required or used.
"Relativity in the Global Positioning System" by Neil Ashby of the University of Colorado is excellent. Chapter 6 discusses relativistic corrections for satellites using GPS.
Quote from the paper:
“Relativistic principles and effects which must be considered include the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and relativity of synchronization.“
So many factors to unpack yet the solution is right there.
Interestingly while all those points factor in it would be possible to have a working GPS system without "understanding" the error or having any grasp of relativity.
To expand on that, consider if the satellites went up (with ideas of basic triangulation from beacons orbiting) and then the drifting error for a supposedly fixed ground position was noticed what could be done?
The 'unknown cause' error function over time twixt fixed position and uncorrected GPS calculation can be fed into a Kalman filter to derive weights that eliminate the error.
Typically what happens in many instrumentation applications is models are created to derive functions to emit answers, errors are noticed, people think hard to add extra terms to account for errors and eventually either all errors are accounted for or some residual 'wobble' remains which can be smoothed away by an adaptive error model.
To this day in high precision GPS applications post processing runs are used to improve accuracy that account for relativity factors, atmospheric twinkle factors, (other factors I'd have to look up), and still there's a bit or error left over that can be sweep away (for a time) with a Kalman filter.
I’ll see if I can find the paper it’s quite interesting.
Edit: I can't find the specific paper I'm thinking of. It's one that discusses how when other orbiting bodies want to use GPS they have to make a bunch of complex corrections that terrestrial users don't. But a search for Earth Centerered Inertial Frame will find you plenty of papers discussing the general concept.