|
== Relativity ==
[[Theory of Relativity]] introduces several effects that need to be taken into account when dealing with precise time measurements. First, according to [[special relativity]] time passes differently for objects in relative motion. That is known as "kinetic" [[time dilation]]: in an inertial reference frame, the faster an object moves, the slower its time appears to pass
[[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) below due to the different effects.
(as measured by the frame's clocks). [[General relativity]] takes into account also the effects that gravity has on the passage of time. In the context of GPS the most prominent correction introduced by general relativity is [[gravitational time dilation]]: the clocks located deeper in the gravitational potential well (i.e. closer to the attracting body) appear to tick slower.
[[File:Orbit times.svg|thumb|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 Relativity]] (SR) ===
SR predicts that clocksas slowthe velocity of an object increases (in a given frame), it's time slows down (as velocitymeasured increases.in that Thatframe). isFor instance, 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> where the orbital velocity is v = 4 km/s and c = the speed of light. The result is an error of about -7.2 μs/day in the satellite. The SR effect is due to theirthe constant movement andof heightGPS clocks 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.
=== [[General Relativity]] (GR) ===
GRSR hasallows theto oppositecompare effect.clocks only GRin predictsa thatflat clocks[[spacetime]], speedwhich upneglects asgravitational theyeffects geton furtherthe awaypassage fromof atime. massiveAccording objectto GR, (the Earthpresence inof thisgravitating casebodies (like Earth). curves Thespacetime, effectwhich ofmakes gravitationalcomparing frequencyclocks shiftnot onas thestraightforward GPSas duein toSR. [[generalHowever, relativity]]one iscan thatoften aaccount clockfor closermost toof athe massivediscrepancy objectby willthe beintroduction slowerof than[[gravitational atime clockdilation]], fartherthe slowing down of time near gravitating awaybodies. AppliedIn tocase of the GPS, the receivers are much closer to Earth than the satellites, causing the GPSlocks clocksat inthe altitude of the satellitessatellite 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 it wassome believed that GPS would not be affected by GR effects, but the [[Hafele–Keating experiment]] showed it would be.
=== Combined kinetic and gravitational time dilations ===
The first sentence could be improved by recognizing that GR includes time dilation for relative motion AND gravitational time dilation, the following replacement for the first sentence is suggested;
WhenCombined, combiningthese SRsources andof GR,time dilation cause the discrepancyclocks ison aboutthe satellites count extra +38 microseconds per day , compared to the clocks on the ground. 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. ▼In addition to time dilation of SR for relative motion GR also includes gravitational Time dilation
=== Combined SR and GR ===
▲When 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.
{| class="wikitable" style="margin:1em auto;"
|+ SR and GR combined
|-
! TheoryTime dilation !! Value !! Notes
|-
| SR (Special Relativity)Kinetic || -7.2 μs/day || Clocks slowed in Satellitessatellites due to Velocity
|-
| GR (General Relativity)Gravitational || +45.8 μs/day || Clocks sped up in Satellitessatellites due to lowerhigher Gravityaltitude
|-
| Total (Combined) || +38.6 μs/day || GR is larger effect than SR
|}
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.
=== Calculations ===
To calculate the amount of daily time dilation experienced by GPS satellites relative to Earth we need to separately determine the amounts due to [[specialsatellite's relativity]] (velocity) and [[general relativity]] (gravity)altitude, and add them together.
==== SpecialKinetic Relativitytime (SR)dilation ====
The amount due to velocity will be determined using the [[Lorentz transformation]]. ThisThe willtime bemeasured by an object moving with velocity <math>v</math> changes by (the inverse of) the [[Lorentz factor]]:
: <math> \frac{1}{\gamma } = \sqrt{1-\frac{v^2}{c^2}} </math>
For small values of ''v/c'', by using [[binomial expansion]] this approximates to:
: <math> \frac{1}{\gamma } \approx 1-\frac{v^2}{2 c^2} </math>
The GPS satellites move at {{val|3874|u=m/s}} relative to Earth's center.<ref name="Nelson" /> We thus determine:
: <math> \frac{1}{\gamma } \approx 1-\frac{3874^2}{2 \left(2.998\times 10^8\right)^2} \approx 1-8.349\times 10^{-11} </math>
This difference below 1 of {{val|8.349|e=-11}} represents the fraction by which the satellites' clocks movetick slower than Earth'sthe stationary clocks. It is then multiplied by the number of nanoseconds in a day:
: <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 lose 7214 nanoseconds a day due to SRtheir effectsvelocity.
: 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.
==== GeneralGravitational Relativitytime (GR)dilation ====
The amount of dilation due to gravity will be determined using the [[gravitational time dilation]] equation:
: <math> \frac{1t_r}{t_\gamma infty} =\sqrt{1-\frac{2G M}{r c^2}} </math>
where <math>t_r</math> is the time passed at a distance <math>r</math> from the center of the Earth and <math>t_\infty</math> is the time passed for a far away observer.
For small values of ''M<math>GM/r'', by using [[binomial expansion]](rc^2)</math> this approximates to:
: <math> \frac{1t_r}{t_\gamma infty} \approx 1-\frac{G M}{r c^2} </math>
Determine the difference <math>\Delta t</math> between the satellite's time <math>t_{r_{\text{GPS}}}</math> and Earth time <math>t_{r_{\text{Earth}}}</math>:
We are again only interested in the fraction below 1, and in the difference between Earth and the satellites. To determine this difference we take:
: <math> \Delta t \leftapprox (1-\frac{1G M}{r_{\gammatext{GPS}} c^2}\right) \approx -(1-\frac{G M_M}{r_{\text{earthEarth}} c^2})= \frac{G M}{R_r_{\text{earthEarth}} c^2}-\frac{G M_{\text{earth}}M}{R_r_{\text{gpsGPS}} c^2} </math>
Earth has a radius of 6,357 km (at the poles) making ''R<submath>earthr_{\text{Earth}}</submath>'' = 6,357,000 m and the satellites have an altitude of 20,184 km<ref name="Nelson" /> making their orbit radius ''R<submath>gpsr_{\text{GPS}}</submath>'' = 26,541,000 m. Substituting these in the above equation, with Earth mass ''M<sub>earth</sub>'' = {{val|5.974|e=24}}, ''G'' = {{val|6.674|e=-11}}, and ''c'' = {{val|2.998|e=8}} (all in [[International System of Units|SI]] units), gives:
: <math> \Delta \left(\frac{1}{\gamma }\right)t \approx 5.307\times 10^{-10} </math>
This represents the fraction by which the clocks at satellites' clocksaltitude movetick faster than on the surface of the 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 the satellites' clocks gain 45850 nanoseconds a day due to GRgravitational effectstime dilation.
==== Combined SRtime anddilation GReffects ====
These effects are added together to give (rounded to 10 ns):
: (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 thetime SR and GRdilation effects of relativity.
=== Sagnac distortion ===
GPS observation processing must also compensate for the [[Sagnac effect]]. The GPS time scale is defined in an [[inertial]] system but observations are processed in an [[ECEF|Earth-centered, Earth-fixed]] (co-rotating) system, a system in which [[Relativity of simultaneity|simultaneity]] is not uniquely defined. A coordinate transformation is thus applied to convert from the inertial system to the ECEF system. The resulting signal run time correction has opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east–west error on the order of hundreds of nanoseconds, or tens of meters in position.<ref>Ashby, Neil [http://www.ipgp.fr/~tarantola/Files/Professional/GPS/Neil_Ashby_Relativity_GPS.pdf Relativity and GPS]. [[Physics Today]], May 2002.</ref>
== Natural sources of interference ==
| 