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/