Relativity and the Global
Positioning System
Theresa Rowland
Senior Seminar
Spring 2006
Outline
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What is GPS?
History
How it works
Trilateration
Sources of Error
Differential GPS
Relativistic effects
What is GPS?
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Navigation Signal
Timing and
Ranging Global
Positioning System
www.spaceandtech.com/.../
navstar-gps_consum.shtml
Space Segment
24 Satellites at
specific locations
around the earth
www.foehnaventure.com/indlandsis/G
PS
Control Segment
www.freeflightsystems.com/gps_control.htm
User Segment
ssro.ee.uec.ac.jp/lab_tomi/.../
gps-images.htm
GPS History
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1949: First atomic clock
1956: Atomichron developed by
Zacharias and National Co.
1973: Navstar GPS development
approved by DoD
1974: First test satellite launched
1977: More modern test satellites
launched with rubidium clocks
1989-93: 6 satellites launched per year
2000: Higher GPS accuracy available for
non-military use
How it works:
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www.sl.wikipedia.org
Coarse Acquisition
Code: repeats
every 1023 bits
P-code (precise
code): repeats
every 7 days
Why so
complicated?
Trilateration
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Accuracy to 10-20
meters for most
hand-held devices,
but can be within
millimeters for
more complex
systems
www.ascproceedings.org
Trilateration
Start with three spheres:
r =x +y +z
2
2
2
2
r = (x − d ) + y + z
2
2
2
2
2
r = ( x − i) + ( y − j ) + z
2
3
2
2
2
www.wikipedia.org
r −r −d
x=
2d
2
1
2
2
2
2
2
2 2
(
r
r
d
)
−
+
2
2
2
1
2
y + z = r1 −
4d 2
r12 = x 2 + y 2 + z 2
r22 = ( x − d ) 2 + y 2 + z 2
r32 = ( x − i ) 2 + ( y − j ) 2 + z 2
r12 − r32 + ( x − i ) 2 j (r12 − r22 + d 2 ) 2
+ −
y=
2j
2
8d 2 j
z= r −x −y
2
1
2
2
Differential GPS
Sources of Error
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Atmosphere
Reflection of signal from nearby
objects
Orbital Corrections
Cheap receivers
Until 2000, deliberate errors
programmed into the system
Relativity!
Clock Differences Due to Height
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Schwarzchild Metric:
2
2
M
dr
− r 2 dφ 2
dτ 2 = −(1 −
)dt 2 −
2M
R
1−
r
dτ 2
2M
2M
2 dφ 2
( ) = (1 −
) − r ( ) = (1 −
) − v2
dt
r
dt
r
2M
(1 −
− vs2 )
dts 2
rs
( ) =
2M
dte
1−
− ve2
re
Size of Relativistic Effects
Ignore motions of clocks for now…
d ts
d te
2 M
rs
≈
2 M
(1 −
re
(1 −
(1 + d ) ≈ 1 + nd
n
)
)
1
2
1
2
If n and nd are small
dt s
M M M M
≈ 1−
+
−
dt e
rs
re
rs re
M = 0.444 cm
rs = 2.66 x 107 cm
re = 6.39 x 108 cm
M M
−22
≈ 4.83 x10
rs re
dts
M M
≈ 1− +
≈ 1 − .00000266
dte
rs
re
Over a full 24-hour day, this can result in
a location error of about 798 meters
Clock Velocities
With respect to Earth’s center, the velocity
of a clock on it’s surface is:
2π re
= v =463.239 m/s
86, 400 s
=2x10-6 c
Satellite Velocity:
v 2
a =
rs
GMm mv
=
2
rs
rs
2
V=.000013c
tb
1
2 2
τ AB = ∫ dt '[1 − v (t ') / c ]
2
ta
dτ = dt
v
1 −
c
2
2
=.9999999936t
= 19.2 seconds per day
This could result in a location error of about
168 meters over a full day
Sagnac Effect
Light Travelling around a rotating loop
Sagnac Effect
c (t − t j ) = (r − rj )
2
2
ds = (cdt ) + dr + r dφ + dz
2
2
2
2
2
2
t = t'
r = r'
φ = φ '+ ω t '
z = z'
−ds =−(1−
2
ω r'
2
2
2
c
)(cdt ') + 2ωr ' dφ ' dt '+ (dσ ')
2
2
2
2
2
ω
r
'
dφ '(cdt ')
2
− (dφ ') 2 = 0
(cdt ') −
c
−b ± b − 4ac
(cdt ') =
2a
2
cdt ' =
ω r '2 dφ '
c
+ dσ '
dσ ' 2ω
'
dt
'
dA
=
+
∫path
∫path c c 2 ∫path z
=207.4 ns
Distance error of about 62.2 meters per day
Summary of Relativistic
Corrections
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Clock Differences due to Height
• 798 meters per day error
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Clock Velocities due to Time Dilation
• Up to 168 meters per day error
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Sagnac Effect
• 62.2 meters per day error
Summary
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What is GPS
History
How it works
Trilateration
Differential GPS
Relativistic Effects
Thank You For Coming!
References:
Ashby, N. “Relativity and the Global Positioning System”, living
Reviews 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/