Many online resources make grandiose claims about how GPS would be useless without relativistic corrections.
Specifically, they outline that satellite clocks runs slower by 7μs/day due to special relativistic effects as the satellites are moving at ~ 14,000km/h. And also time runs faster by 45μs/day due to general relativity effects coming from the higher orbit.
This yields a 38μs/day net drift, which we're told would cause errors in the order of 11km/day if left uncorrected.
But if all satellites are orbiting at the same altitudes and speed, why do we care at all about time dilation between the satellites and the ground? My GPS receiver on the ground only cares about the differences between the timestamps it receives from the various satellites, right?
Specifically, receiving $[ (t_0, p_0) @ u_0, ..., (t_n, p_n) @ u_n ]$ gives me no additional information vs. receiving $[ (t_0-K, p_0) @ u_0, ..., (t_n-K, p_n) @ u_n ]$ for some unknown $K$ (where $t_i$ is the transmission time of the signal from the $i$-th satellite and $p_i$ is its position at that point and $u_i$ is the receiver stopwatch time (not an absolute clock)).
(There is one obvious way in which satellites care about time dilation, and it is that they would overestimate the earth's rotation and therefore all readings would drift west, but only by 37μs/day $\times$ 1700km/h ≈ 2cm/day! And this could be easily corrected by periodically having satellites verify their positions vs ground objects)
I'm not well versed in the theory of relativity, so surely I must be missing something. Please enlighten me!
(Other related questions:
Why does GPS depend on relativity? — unfortunately all the answers seem to regurgitate the same 10km/day, without explaining why time dilation vs ground clocks matter at all
That 10km/day error predicted if GPS satellite clocks not corrected for relativity attempts to specifically zoom in on that point but the answers seem tentative and provide no references)