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Relativity and the Global Relativity and the Global Positioning SystemPositioning System
Theresa RowlandTheresa RowlandSenior SeminarSenior Seminar
Spring 2006Spring 2006
OutlineOutline
What is GPS?What is GPS?HistoryHistoryHow it worksHow it worksTrilaterationTrilaterationSources of ErrorSources of ErrorDifferential GPSDifferential GPSRelativistic effectsRelativistic effects
What is GPS?What is GPS?
www.spaceandtech.com/.../navstar-gps_consum.shtml
Navigation Signal Navigation Signal Timing and Timing and Ranging Global Ranging Global Positioning SystemPositioning System
Space SegmentSpace Segment
www.foehn-aventure.com/indlandsis/GPS
24 Satellites at 24 Satellites at specific locations specific locations around the eartharound the earth
Control SegmentControl Segment
www.freeflightsystems.com/gps_control.htm
User SegmentUser Segment
ssro.ee.uec.ac.jp/lab_tomi/.../gps-images.htm
GPS HistoryGPS History1949: First atomic clock1949: First atomic clock1956: 1956: AtomichronAtomichron developed by developed by
ZachariasZacharias and National Co.and National Co.1973: 1973: NavstarNavstar GPS development GPS development
approved by approved by DoDDoD1974: First test satellite launched1974: First test satellite launched1977: More modern test satellites 1977: More modern test satellites
launched with rubidium clockslaunched with rubidium clocks19891989--93: 6 satellites launched per year93: 6 satellites launched per year2000: Higher GPS accuracy available for 2000: Higher GPS accuracy available for
nonnon--military usemilitary use
How it works:How it works:Coarse Acquisition Coarse Acquisition Code: repeats Code: repeats every 1023 bitsevery 1023 bits
PP--code (precise code (precise code): repeats code): repeats every 7 daysevery 7 days
Why so Why so complicated?complicated?
www.sl.wikipedia.org
TrilaterationTrilateration
Accuracy to 10Accuracy to 10--20 20 meters for most meters for most handhand--held devices, held devices, but can be within but can be within millimeters for millimeters for more complex more complex systemssystems
www.ascproceedings.org
TrilaterationTrilateration
Start with three spheres:
2 2 2 2r x y z= + +
2 2 2 22 ( )r x d y z= − + +
2 2 2 23 ( ) ( )r x i y j z= − + − +
www.wikipedia.org
2 2 21 2
2r r dx
d− −
=
2 2 2 22 2 2 1 2
1 2
( )4
r r dy z rd
− ++ = −
2 2 2 21r x y z= + + 2 2 2 2 2 2 2
1 3 1 22
( ) ( )2 2 8
r r x i r r djyj d j
− + − − += + −2 2 2 2
2 ( )r x d y z= − + +
2 2 2 23 ( ) ( )r x i y j z= − + − +
2 2 21z r x y= − −
Differential GPSDifferential GPS
Sources of ErrorSources of Error
AtmosphereAtmosphereReflection of signal from nearby Reflection of signal from nearby objectsobjectsOrbital CorrectionsOrbital CorrectionsCheap receiversCheap receiversUntil 2000, deliberate errors Until 2000, deliberate errors programmed into the systemprogrammed into the systemRelativity!Relativity!
Clock Differences Due to HeightClock Differences Due to Height
SchwarzchildSchwarzchild Metric:Metric:2
2 2 2 22(1 ) 21
M drd dt r dMRr
τ φ= − − − −−
2 2 2 22 2( ) (1 ) ( ) (1 )d M d Mr vdt r dt rτ φ
= − − = − −
2
2
2
2(1 )( ) 21
ss s
ee
e
M vdt r
Mdt vr
− −=
− −
Size of Relativistic EffectsSize of Relativistic EffectsIgnore motions of clocks for now…
12
12
2(1 )
2(1 )
s s
e
e
Md t rd t M
r
−≈
−
(1 ) 1nd nd+ ≈ + If n and nd are small
1s
e s e s e
dt M M M Mdt r r r r
≈ − + −
M = 0.444 cm
rs = 2.66 x 107 cm
re = 6.39 x 108 cm
224.83 10s e
M M xr r
−≈
1 1e
M Mr r
s
e s
dtdt
≈ − + ≈ − .00000266
Over a full 24-hour day, this can result in a location error of about 798 meters
Clock VelocitiesClock VelocitiesWith respect to Earth’s center, the velocity of a clock on it’s surface is:
286,400
er vs
π= =2x10-6 c=463.239 m/s
Satellite Velocity:
2
s
var
=
2
2s s
GMm mvr r
=V=.000013c
12 2 2'[1 ( ') / ]b
a
t
AB tdt v t cτ = −∫
2
21 vd d tc
τ = −
=.9999999936t
= 19.2 seconds per day
This could result in a location error of about 168 meters over a full day
SagnacSagnac EffectEffect
Light Travelling around a rotating loop
SagnacSagnac EffectEffect2 2( ) ( )j jc t t r r− = −
2 2 2 2 2 2( )ds cdt dr r d dzφ= + + +
''' ''
t tr r
tz zφ φ ω
=== +=
2 22 2 2 2
2
'(1 )( ') 2 ' ' ' ( ')rds cdt r d dt dc
ω ω φ σ− =− − + +
22 22 ' '( ')( ') ( ') 0r d cdtcdt d
cω φ φ− − =
2 4( ')2
b b accdta
− ± −=
2' '' 'r dcdt dc
ω φ σ= +
'2
' 2' zpath path path
ddt dAc cσ ω
= +∫ ∫ ∫=207.4 ns
Distance error of about 62.2 meters per day
Summary of Relativistic Summary of Relativistic CorrectionsCorrections
Clock Differences due to HeightClock Differences due to Height•• 798 meters per day error798 meters per day error
Clock Velocities due to Time DilationClock Velocities due to Time Dilation•• Up to 168 meters per day errorUp to 168 meters per day error
SagnacSagnac EffectEffect•• 62.2 meters per day error62.2 meters per day error
SummarySummary
What is GPSWhat is GPSHistoryHistoryHow it worksHow it worksTrilaterationTrilaterationDifferential GPSDifferential GPSRelativistic EffectsRelativistic Effects
Thank You For Coming!Thank You For Coming!
References:References:Ashby, N. “Relativity and the Global Positioning System”, livingReviews Vol. 6, 2003
Wikipedia, “Trilateration”, http://en.wikipedia.org/trilateration
Taylor, John F, and Wheeler, John Archibald, “Exploring Black Holes, and introduction to General Relativity” Addison Wesley Longman, 2000
“The Sagnac Effect” www.mathpages.com/rr/s2-07.htm
www.navstar.gps.gov/