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=== [[General relativity]] ===
Special relativity allows the comparison of clocks only in a flat [[spacetime]], which neglects gravitational effects on the passage of time. According to general relativity, the presence of gravitating bodies (like Earth) curves spacetime, which makes comparing clocks not as straightforward as in special relativity. However, one can often account for most of the discrepancy by the introduction of [[gravitational time dilation]], the slowing down of time near gravitating bodies. In case of the GPS, the receivers are closer to the center of Earth than the satellites, causing the clocks at the altitude of the satellite to be faster by a factor of 5×10<sup>−10</sup>, or about +45.8 μs/day. This gravitational frequency shift is measurable. During early development some{{who|date=January 2024}} believed that GPS would not be affected by general relativistic effects, but the [[Hafele–Keating experiment]] showed that it would be.
=== Combined kinetic and gravitational time dilations ===
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: <math> -8.349\times 10^{-11}\times 60\times 60\times 24\times 10^9\approx -7214 \text{ ns} </math>
That is, the satellites' clocks are slower than Earth's clocks by 7214 nanoseconds a day due to their velocity.
: Note that this speed of {{val|3874|u=m/s}} is measured relative to Earth's center rather than its surface where the GPS receivers (and users) are. This is because Earth's equipotential makes net time dilation equal across its geodesic surface.<ref>{{Cite web |last=S. P. Drake |date=January 2006 |title=The equivalence principle as a stepping stone from special to general relativity |url=http://www.phys.unsw.edu.au/einsteinlight/jw/2006AJP.pdf |website=Am. J. Phys., Vol. 74, No. 1 |pages=22–25}}</ref> That is, the combination of Special and General effects make the net time dilation at the equator equal to that of the poles, which in turn are at rest relative to the center. Hence we use the center as a reference point to represent the entire surface.
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: <math> 5.307\times 10^{-10}\times 60\times 60\times 24\times 10^9\approx 45850 \text{ ns} </math>
That is, the satellites' clocks gain 45850 nanoseconds a day due to gravitational time dilation.
==== Combined time dilation effects ====
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: (1 – {{val|4.472|e=-10}}) × 10.23 = 10.22999999543
That is, we need to slow the clocks down from 10.23 MHz to 10.22999999543 MHz in order to negate both time dilation effects.
=== Sagnac distortion ===
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