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From what I understand, GPS localization is based on the difference in the reception time from the time of emission of a signal from different GPS satellites whose positions are known. For this we need very precisely synchronized clocks in the satellites.

Due to the relative motion of the satellites, we need special relativity to make sure to keep the clocks in sync with each other (and with a reference clock on earth, but absolute time on earth doesn't play a role). Due to the change in gravitational potential, we would need general relativity to keep the clocks in sync with a reference clock on earth.

In the answers to other questions linked to in the comments, a computation is made in which the difference in time between a precision clock on earth and one in a GPS satellite in increased by about 38 $\mu s$ per day. The conclusion that is then drawn is that light travels 11 km in that time, so that we would get that error for each day that passes.

However, for GPS to work, we don't need to measure time difference with a clock on earth (we don't have a precision clock in our GPS receiver, and I don't think we connect to one either), only time differences between satellite clocks, so it would seem to me that the clocks in the satellites only have to be in sync with each other. Since the satellites are at the same height, the gravitational time contraction should be very similar for all of them.

Am I misunderstanding or overlooking something?

EDIT

My question is not about how general relativity is used in GPS, but rather if that is really needed? Supposing we wouldn't be aware of general relativity, we might scratch our heads over why the clocks keep getting out of sync w.r.t. our local clocks in spite of the special relativistic corrections, but wouldn't the clocks on the satellites still be in sync such that the computed positions would be essentially correct? This means that the answer linked to in a duplicate vote is the opposite of an answer to this question. The computation there seems correct, but doesn't seem relevant for the position computation (as absolute earth time doesn't play a role if I correctly understand how GPS works).

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    $\begingroup$ Possibly dupe of physics.stackexchange.com/q/1061 $\endgroup$ – HDE 226868 Dec 5 '15 at 22:25
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    $\begingroup$ @HDE226868 it is asked to explain how general relativity is used for GPS to function properly, which is explained in the answers. My question is if we really need it. I'll edit to make that clearer. $\endgroup$ – doetoe Dec 5 '15 at 22:28
  • $\begingroup$ More on GR+GPS. $\endgroup$ – Qmechanic Dec 5 '15 at 22:31
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    $\begingroup$ @Danu If you read the only correct answer on that page you'll note that the necessity of GR for GPS is a complete and utter myth. $\endgroup$ – user10851 Dec 5 '15 at 22:57
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    $\begingroup$ Nice document discussing the GR-Needed-in-GPS-fallacy: alternativephysics.org/book/GPSmythology.htm $\endgroup$ – DilithiumMatrix Dec 6 '15 at 1:06