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== Relativity ==
[[Special Relativity]] (SR) and [[General Relativity]] (GR) are two separate and distinct theories under the title of the [[Theory of Relativity]]. SR and GR make different (opposite) predictions when it comes to the clocks on-board GPS satellites. Note the opposite signs (plus and minus) due to the different effects.▼
[[File:Orbit times.svg|thumb|right|Satellite clocks are slowed by their orbital speed but sped up by their distance out of the Earth's gravitational well.]]
A number of sources of error exist due to [[Theory of relativity|relativistic]] effects<ref>Webb (2004), p. 32.</ref> that would render the system useless if uncorrected. Three relativistic effects are time dilation, gravitational frequency shift, and eccentricity effects. Examples include the relativistic time ''slowing'' due to the speed of the satellite of about 1 part in 10<sup>10</sup>, the gravitational time dilation that makes a satellite run about 5 parts in 10<sup>10</sup> ''faster'' than an Earth-based clock, and the [[Sagnac effect]] due to rotation relative to receivers on Earth. These topics are examined below, one at a time.
=== Special
SR
▲[[Special Relativity]] (SR) and [[General Relativity]] (GR) are two separate and distinct theories under the title of the [[Theory of Relativity]]. SR and GR make different (opposite) predictions when it comes to the clocks on-board GPS satellites. Note the opposite signs (plus and minus) due to the different effects.
=== General Relativity (GR) ===
GR
=== Combined SR and GR ===
▲SR ([[Special Relativity]]) predicts that the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground clocks by a factor of <math>{v^{2}}/{2c^{2}}\approx 10 ^{-10}</math>, or result in a delay of about -7.2 μs/day, where the orbital velocity is v = 4 km/s, and c = the speed of light. The SR effect is to their constant movement and height relative to the Earth-centered, non-rotating approximately inertial [[special relativity#Reference frames, coordinates and the Lorentz transformation|reference frame]]. In short, the clocks on the satellites are slowed down by the velocity of the satellite. This [[time dilation]] effect has been measured and verified using the GPS.
▲GR ([[General Relativity]]) has the opposite effect. The effect of gravitational frequency shift on the GPS due to [[general relativity]] is that a clock closer to a massive object will be slower than a clock farther away. Applied to the GPS, the receivers are much closer to Earth than the satellites, causing the GPS clocks in the satellites to be faster by a factor of 5×10<sup>−10</sup>, or about +45.9 μs/day. This gravitational frequency shift is measurable. In fact, during early development, it was believed that GPS would not be affected by GR effects but [[Hafele–Keating_experiment]] showed it did.
▲Combining SR and GR, the discrepancy is about +38 microseconds per day. This is a difference of 4.465 parts in 10<sup>10</sup>.<ref>Rizos, Chris. [[University of New South Wales]]. [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm GPS Satellite Signals] {{Webarchive|url=https://web.archive.org/web/20100612004027/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm |date=2010-06-12}}. 1999.</ref> Without correction, errors of roughly 11.4 km/day would accumulate in the position.<ref>{{Cite book |last=Faraoni |first=Valerio |url=https://books.google.com/books?id=NuS9BAAAQBAJ |title=Special Relativity |publisher=Springer Science & Business Media |year=2013 |isbn=978-3-319-01107-3 |edition=illustrated |page=54}} [https://books.google.com/books?id=NuS9BAAAQBAJ&pg=PA54 Extract of page 54]</ref> This initial pseudorange error is corrected in the process of solving the [[GPS#Navigation equations|navigation equations]]. In addition, the elliptical, rather than perfectly circular, satellite orbits cause the time dilation and gravitational frequency shift effects to vary with time. This eccentricity effect causes the clock rate difference between a GPS satellite and a receiver to increase or decrease depending on the altitude of the satellite.
<CENTER>
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| SR (Special Relativity) || -7.2 μs/day || Clocks slowed in Satellites due to Velocity
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| GR (General Relativity) || +45.
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| Total (Combined) || +38.6 μs/day || GR is larger effect than SR
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To compensate for the discrepancy, the frequency standard on board each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543 MHz instead of 10.23 MHz.<ref name="Nelson">[http://www.aticourses.com/global_positioning_system.htm The Global Positioning System by Robert A. Nelson Via Satellite] {{Webarchive|url=https://web.archive.org/web/20100718150217/http://www.aticourses.com/global_positioning_system.htm |date=2010-07-18 }}, November 1999</ref> Since the atomic clocks on board the GPS satellites are precisely tuned, it makes the system a practical engineering application of the scientific theory of relativity in a real-world environment.<ref>Pogge, Richard W.; [http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html "Real-World Relativity: The GPS Navigation System"]. Retrieved 25 January 2008.</ref> Placing atomic clocks on artificial satellites to test Einstein's general theory was proposed by [[Friedwardt Winterberg]] in 1955.<ref>{{Cite web |date=1956-08-10 |title=Astronautica Acta II, 25 (1956). |url=http://bourabai.kz/winter/satelliten.htm |access-date=2009-10-23}}</ref> The conclusion is that the GPS satellites must compensate for GR, the physics of [[black holes]] and extreme gravity.
===
To calculate the amount of daily time dilation experienced by GPS satellites relative to Earth we need to separately determine the amounts due to [[special relativity]] (velocity) and [[general relativity]] (gravity) and add them together.
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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
: 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> \Delta \left(\frac{1}{\gamma }\right) \approx 5.307\times 10^{-10} </math>
This represents the fraction by which the satellites' clocks move faster than Earth's. It is then multiplied by the number of nanoseconds in a day:
: <math> 5.307\times 10^{-10}\times 60\times 60\times 24\times 10^9\approx 45850 \text{ ns} </math>
That is ==== Combined SR and GR ====
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: (1 – {{val|4.472|e=-10}}) × 10.23 = 10.22999999543
That is
=== Sagnac distortion ===
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