Relativity is required to explain the time dilation, but it's not needed to notice that the GPS clocks run 40µs too fast. It wouldn't take too long for them to measure enough data to create an empirical skew-versus-altitude table, which should then be enough for them to correct the time skew, even without understanding what was going on.

I've always wondered - if special and general relativity weren't known, could the GPS engineers have just measured the error and empirically accounted for it? I don't know relativity or GPS or engineering well enough to answer.

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