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Einstein and GPS
Peter Hertel
University of OsnabrückGermany (EU)
September 14, 2009
A lecture delivered at Xi’an Institute of Optics and Precision Mechanics, China
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
What is GPS
Global positioning system
a system for determining positions at the globeoperated by US Department of Defense, partially available forcivil purposesused for navigation (ships, airplanes, cars, hikers. . . ) andautomatic toll collectionbut also for geodesics, earthquake warnings etc.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
What is GPS
Global positioning systema system for determining positions at the globe
operated by US Department of Defense, partially available forcivil purposesused for navigation (ships, airplanes, cars, hikers. . . ) andautomatic toll collectionbut also for geodesics, earthquake warnings etc.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
What is GPS
Global positioning systema system for determining positions at the globeoperated by US Department of Defense, partially available forcivil purposes
used for navigation (ships, airplanes, cars, hikers. . . ) andautomatic toll collectionbut also for geodesics, earthquake warnings etc.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
What is GPS
Global positioning systema system for determining positions at the globeoperated by US Department of Defense, partially available forcivil purposesused for navigation (ships, airplanes, cars, hikers. . . ) andautomatic toll collection
but also for geodesics, earthquake warnings etc.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
What is GPS
Global positioning systema system for determining positions at the globeoperated by US Department of Defense, partially available forcivil purposesused for navigation (ships, airplanes, cars, hikers. . . ) andautomatic toll collectionbut also for geodesics, earthquake warnings etc.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
The principle
You see the lightning immediatelybut thunder requires 3 seconds for one kilometer
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
The principle
You see the lightning immediately
but thunder requires 3 seconds for one kilometer
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
The principle
You see the lightning immediatelybut thunder requires 3 seconds for one kilometer
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
One satellite
t1r1 = c(t̄1 − t1)
Signal is emitted at time t1
and is received at time t̄1.It propagates with speed of light c
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
One satellite
t1r1 = c(t̄1 − t1)
Signal is emitted at time t1and is received at time t̄1.
It propagates with speed of light c
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
One satellite
t1r1 = c(t̄1 − t1)
Signal is emitted at time t1and is received at time t̄1.It propagates with speed of light c
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Two satellites
t1r1 = c(t̄1 − t1)
t2 r2 = c(t̄2 − t2)
The receiver must fulfill both conditions
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Two satellites
t1r1 = c(t̄1 − t1)
t2 r2 = c(t̄2 − t2)
The receiver must fulfill both conditions
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Three satellites
In three-dimensional space
every satellite defines a sphere of possible locationsintersection of two yields a circleintersection with third defines two locationsone of which is usually far away from earth’s surface
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Three satellites
In three-dimensional space
every satellite defines a sphere of possible locations
intersection of two yields a circleintersection with third defines two locationsone of which is usually far away from earth’s surface
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Three satellites
In three-dimensional space
every satellite defines a sphere of possible locationsintersection of two yields a circle
intersection with third defines two locationsone of which is usually far away from earth’s surface
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Three satellites
In three-dimensional space
every satellite defines a sphere of possible locationsintersection of two yields a circleintersection with third defines two locations
one of which is usually far away from earth’s surface
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Three satellites
In three-dimensional space
every satellite defines a sphere of possible locationsintersection of two yields a circleintersection with third defines two locationsone of which is usually far away from earth’s surface
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differhowever, the receiver clock may be adjustedsuch that the four locations coincideimportant: receiver does not require a precise clock!GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurements
which will differhowever, the receiver clock may be adjustedsuch that the four locations coincideimportant: receiver does not require a precise clock!GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differ
however, the receiver clock may be adjustedsuch that the four locations coincideimportant: receiver does not require a precise clock!GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differhowever, the receiver clock may be adjusted
such that the four locations coincideimportant: receiver does not require a precise clock!GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differhowever, the receiver clock may be adjustedsuch that the four locations coincide
important: receiver does not require a precise clock!GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differhowever, the receiver clock may be adjustedsuch that the four locations coincideimportant: receiver does not require a precise clock!
GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differhowever, the receiver clock may be adjustedsuch that the four locations coincideimportant: receiver does not require a precise clock!GPS device may adjust its internal clock. . .
. . . if it receives at least four satellites. . .
. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differhowever, the receiver clock may be adjustedsuch that the four locations coincideimportant: receiver does not require a precise clock!GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .
. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Four satellites
With four satellites the system of equations is over-determined
leaving out one of the four satellites results in four locationmeasurementswhich will differhowever, the receiver clock may be adjustedsuch that the four locations coincideimportant: receiver does not require a precise clock!GPS device may adjust its internal clock. . .. . . if it receives at least four satellites. . .. . . and more than four is even better.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
24 satellites
circular orbits (we shall see why)20200 km above earth2032 kg1.14 kW
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
24 satellites
circular orbits (we shall see why)
20200 km above earth2032 kg1.14 kW
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
24 satellites
circular orbits (we shall see why)20200 km above earth
2032 kg1.14 kW
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
24 satellites
circular orbits (we shall see why)20200 km above earth2032 kg
1.14 kW
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
24 satellites
circular orbits (we shall see why)20200 km above earth2032 kg1.14 kW
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Replacement
Block II replacement 12
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Orbits
6 orbits inclined 55 degrees with respect to ecliptic plane4 equally spaced satellites on each orbit
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Orbits
6 orbits inclined 55 degrees with respect to ecliptic plane
4 equally spaced satellites on each orbit
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Orbits
6 orbits inclined 55 degrees with respect to ecliptic plane4 equally spaced satellites on each orbit
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Period, velocity
ω =2πT
mω2r = G Mmr2
T = 2π
√r3
GM = 11.97 h
v = ωr = 3874 m/s
exactly two revolutions per (sidereal) day
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Period, velocity
ω =2πT
mω2r = G Mmr2
T = 2π
√r3
GM = 11.97 h
v = ωr = 3874 m/s
exactly two revolutions per (sidereal) day
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
NIST-F1 cesium fountain atomic clock
accuracy is 0.5× 10−15
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
NIST-F1 cesium fountain atomic clock
accuracy is 0.5× 10−15
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Even more accurate
Physics Nobel prize 2005 for Theodor Hänsch, Max PlanckInstitute for Quantum Optics, Munich, Germany
Optical frequency comb technologyclock accuracy of 10−18 is feasible
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Even more accurate
Physics Nobel prize 2005 for Theodor Hänsch, Max PlanckInstitute for Quantum Optics, Munich, GermanyOptical frequency comb technology
clock accuracy of 10−18 is feasible
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Even more accurate
Physics Nobel prize 2005 for Theodor Hänsch, Max PlanckInstitute for Quantum Optics, Munich, GermanyOptical frequency comb technologyclock accuracy of 10−18 is feasible
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Signals
L1 at 1575.42 Mhz, L2 at 1227.60 Mhz
Each satellite provides its own pseudo-random codeSignal comprises time and ephemeris which allow to locate thesatellite within 10 cmreceiver recognises signal and synchronizes, therebydetermining time delayL1 is no more spoiled artificially by noise (Clinton)L2 accessible for US military only
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Signals
L1 at 1575.42 Mhz, L2 at 1227.60 MhzEach satellite provides its own pseudo-random code
Signal comprises time and ephemeris which allow to locate thesatellite within 10 cmreceiver recognises signal and synchronizes, therebydetermining time delayL1 is no more spoiled artificially by noise (Clinton)L2 accessible for US military only
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Signals
L1 at 1575.42 Mhz, L2 at 1227.60 MhzEach satellite provides its own pseudo-random codeSignal comprises time and ephemeris which allow to locate thesatellite within 10 cm
receiver recognises signal and synchronizes, therebydetermining time delayL1 is no more spoiled artificially by noise (Clinton)L2 accessible for US military only
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Signals
L1 at 1575.42 Mhz, L2 at 1227.60 MhzEach satellite provides its own pseudo-random codeSignal comprises time and ephemeris which allow to locate thesatellite within 10 cmreceiver recognises signal and synchronizes, therebydetermining time delay
L1 is no more spoiled artificially by noise (Clinton)L2 accessible for US military only
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Signals
L1 at 1575.42 Mhz, L2 at 1227.60 MhzEach satellite provides its own pseudo-random codeSignal comprises time and ephemeris which allow to locate thesatellite within 10 cmreceiver recognises signal and synchronizes, therebydetermining time delayL1 is no more spoiled artificially by noise (Clinton)
L2 accessible for US military only
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
How does it work?The satellitesThe clocksSignals
Signals
L1 at 1575.42 Mhz, L2 at 1227.60 MhzEach satellite provides its own pseudo-random codeSignal comprises time and ephemeris which allow to locate thesatellite within 10 cmreceiver recognises signal and synchronizes, therebydetermining time delayL1 is no more spoiled artificially by noise (Clinton)L2 accessible for US military only
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Technical Problems
Speed of signal is not exactly c (ionosphere and troposphere)
IntensitiesMoon and Sun perturb satellite orbitsShadowing. . . and many more
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Technical Problems
Speed of signal is not exactly c (ionosphere and troposphere)Intensities
Moon and Sun perturb satellite orbitsShadowing. . . and many more
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Technical Problems
Speed of signal is not exactly c (ionosphere and troposphere)IntensitiesMoon and Sun perturb satellite orbits
Shadowing. . . and many more
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Technical Problems
Speed of signal is not exactly c (ionosphere and troposphere)IntensitiesMoon and Sun perturb satellite orbitsShadowing
. . . and many more
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Technical Problems
Speed of signal is not exactly c (ionosphere and troposphere)IntensitiesMoon and Sun perturb satellite orbitsShadowing. . . and many more
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
Albert Einstein, 1879-1955
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
Special Relativity
A moving clock (velocity v) runs slower than a resting clock(beat τ).
τ ′ = τ√1−(v/c)2
Before satellite launch the atomic clock must be speeded upby (1− δSR) = 1−
√1− (v/c)2
For v = 3874 m/s this amounts to δSR = 0.835 × 10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
Special Relativity
A moving clock (velocity v) runs slower than a resting clock(beat τ).τ ′ = τ√
1−(v/c)2
Before satellite launch the atomic clock must be speeded upby (1− δSR) = 1−
√1− (v/c)2
For v = 3874 m/s this amounts to δSR = 0.835 × 10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
Special Relativity
A moving clock (velocity v) runs slower than a resting clock(beat τ).τ ′ = τ√
1−(v/c)2
Before satellite launch the atomic clock must be speeded upby (1− δSR) = 1−
√1− (v/c)2
For v = 3874 m/s this amounts to δSR = 0.835 × 10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
Special Relativity
A moving clock (velocity v) runs slower than a resting clock(beat τ).τ ′ = τ√
1−(v/c)2
Before satellite launch the atomic clock must be speeded upby (1− δSR) = 1−
√1− (v/c)2
For v = 3874 m/s this amounts to δSR = 0.835 × 10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 1
Gravitational field changes beat of clocks too.
. . . and therefore the frequency ν of a photonsum of photon energy hν and potential energy U remainsconstantWith ν ′ at satellite distance r and ν at earth surface R weobtainhν ′ − G Mm
r = hν − G MmR
where mc2 ≈ hν ′ ≈ hν
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 1
Gravitational field changes beat of clocks too.. . . and therefore the frequency ν of a photon
sum of photon energy hν and potential energy U remainsconstantWith ν ′ at satellite distance r and ν at earth surface R weobtainhν ′ − G Mm
r = hν − G MmR
where mc2 ≈ hν ′ ≈ hν
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 1
Gravitational field changes beat of clocks too.. . . and therefore the frequency ν of a photonsum of photon energy hν and potential energy U remainsconstant
With ν ′ at satellite distance r and ν at earth surface R weobtainhν ′ − G Mm
r = hν − G MmR
where mc2 ≈ hν ′ ≈ hν
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 1
Gravitational field changes beat of clocks too.. . . and therefore the frequency ν of a photonsum of photon energy hν and potential energy U remainsconstantWith ν ′ at satellite distance r and ν at earth surface R weobtain
hν ′ − G Mmr = hν − G Mm
Rwhere mc2 ≈ hν ′ ≈ hν
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 1
Gravitational field changes beat of clocks too.. . . and therefore the frequency ν of a photonsum of photon energy hν and potential energy U remainsconstantWith ν ′ at satellite distance r and ν at earth surface R weobtainhν ′ − G Mm
r = hν − G MmR
where mc2 ≈ hν ′ ≈ hν
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 1
Gravitational field changes beat of clocks too.. . . and therefore the frequency ν of a photonsum of photon energy hν and potential energy U remainsconstantWith ν ′ at satellite distance r and ν at earth surface R weobtainhν ′ − G Mm
r = hν − G MmR
where mc2 ≈ hν ′ ≈ hν
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 2
This results in ν ′/ν = 1− GMc2
{1R −
1r
}
τ ′/τ = 1 + GMc2
{1R −
1r
}Before satellite launch the clock must be slowed down by afactor (1 + δGR) in order to be τ in orbitδGR = GM
c2
{1R −
1r
}= 5.283 ×10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 2
This results in ν ′/ν = 1− GMc2
{1R −
1r
}τ ′/τ = 1 + GM
c2
{1R −
1r
}
Before satellite launch the clock must be slowed down by afactor (1 + δGR) in order to be τ in orbitδGR = GM
c2
{1R −
1r
}= 5.283 ×10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 2
This results in ν ′/ν = 1− GMc2
{1R −
1r
}τ ′/τ = 1 + GM
c2
{1R −
1r
}Before satellite launch the clock must be slowed down by afactor (1 + δGR) in order to be τ in orbit
δGR = GMc2
{1R −
1r
}= 5.283 ×10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
General Relativity 2
This results in ν ′/ν = 1− GMc2
{1R −
1r
}τ ′/τ = 1 + GM
c2
{1R −
1r
}Before satellite launch the clock must be slowed down by afactor (1 + δGR) in order to be τ in orbitδGR = GM
c2
{1R −
1r
}= 5.283 ×10−10
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
GPS: Corrections matter!
Before launching a GPS satellite, its clock must be sloweddown by a factorδ = δGR − δSR = 4.446 ×10−10
such that it has the same beat as its counterpart on earth’ssurfaceWithout correction there would be an error which grows withtime. . .7.8 km in 12 hours (one revolution)No GPS without EinsteinNo good physics and technology without Theory!
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
GPS: Corrections matter!
Before launching a GPS satellite, its clock must be sloweddown by a factorδ = δGR − δSR = 4.446 ×10−10
such that it has the same beat as its counterpart on earth’ssurface
Without correction there would be an error which grows withtime. . .7.8 km in 12 hours (one revolution)No GPS without EinsteinNo good physics and technology without Theory!
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
GPS: Corrections matter!
Before launching a GPS satellite, its clock must be sloweddown by a factorδ = δGR − δSR = 4.446 ×10−10
such that it has the same beat as its counterpart on earth’ssurfaceWithout correction there would be an error which grows withtime. . .
7.8 km in 12 hours (one revolution)No GPS without EinsteinNo good physics and technology without Theory!
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
GPS: Corrections matter!
Before launching a GPS satellite, its clock must be sloweddown by a factorδ = δGR − δSR = 4.446 ×10−10
such that it has the same beat as its counterpart on earth’ssurfaceWithout correction there would be an error which grows withtime. . .7.8 km in 12 hours (one revolution)
No GPS without EinsteinNo good physics and technology without Theory!
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
GPS: Corrections matter!
Before launching a GPS satellite, its clock must be sloweddown by a factorδ = δGR − δSR = 4.446 ×10−10
such that it has the same beat as its counterpart on earth’ssurfaceWithout correction there would be an error which grows withtime. . .7.8 km in 12 hours (one revolution)No GPS without Einstein
No good physics and technology without Theory!
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
GPS: Corrections matter!
Before launching a GPS satellite, its clock must be sloweddown by a factorδ = δGR − δSR = 4.446 ×10−10
such that it has the same beat as its counterpart on earth’ssurfaceWithout correction there would be an error which grows withtime. . .7.8 km in 12 hours (one revolution)No GPS without EinsteinNo good physics and technology without Theory!
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Special relativityGeneral relativityCorrections matter!
Any Questions?
Make it as simple as possible, but not simpler!
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
General Relativity 3
Effect on clocks and measuring rods by earth’s gravitationalfields is tiny . . .
. . . but may be tremendous for large massesIf mass is too large, light is trapped (black hole)Huge masses act as gravitational lensesBig Bang, Expanding Universe, cosmic background radiationand more . . .
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
General Relativity 3
Effect on clocks and measuring rods by earth’s gravitationalfields is tiny . . .. . . but may be tremendous for large masses
If mass is too large, light is trapped (black hole)Huge masses act as gravitational lensesBig Bang, Expanding Universe, cosmic background radiationand more . . .
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
General Relativity 3
Effect on clocks and measuring rods by earth’s gravitationalfields is tiny . . .. . . but may be tremendous for large massesIf mass is too large, light is trapped (black hole)
Huge masses act as gravitational lensesBig Bang, Expanding Universe, cosmic background radiationand more . . .
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
General Relativity 3
Effect on clocks and measuring rods by earth’s gravitationalfields is tiny . . .. . . but may be tremendous for large massesIf mass is too large, light is trapped (black hole)Huge masses act as gravitational lenses
Big Bang, Expanding Universe, cosmic background radiationand more . . .
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
General Relativity 3
Effect on clocks and measuring rods by earth’s gravitationalfields is tiny . . .. . . but may be tremendous for large massesIf mass is too large, light is trapped (black hole)Huge masses act as gravitational lensesBig Bang, Expanding Universe, cosmic background radiationand more . . .
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
Gravitational lens
A very massive galaxy cluster (Abell 1689) at a distance of 2×109
light years produces a strong curvature of space.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
Message to take home
Experiments and applications should be guided by theory. . .
. . . but theory without experiments which may lead toapplications is good for nothingEvery physics student should study theoretical physics. . .. . . but teachers should make it fun, not stress.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
Message to take home
Experiments and applications should be guided by theory. . .. . . but theory without experiments which may lead toapplications is good for nothing
Every physics student should study theoretical physics. . .. . . but teachers should make it fun, not stress.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
Message to take home
Experiments and applications should be guided by theory. . .. . . but theory without experiments which may lead toapplications is good for nothingEvery physics student should study theoretical physics. . .
. . . but teachers should make it fun, not stress.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
Message to take home
Experiments and applications should be guided by theory. . .. . . but theory without experiments which may lead toapplications is good for nothingEvery physics student should study theoretical physics. . .. . . but teachers should make it fun, not stress.
Peter Hertel Einstein and GPS
Global Positioning SystemTechnical Problems
Basic ProblemsEinstein is right
Black HolesGravitational lensCosmic Background Radiation
Thanks for your attention!
Peter Hertel Einstein and GPS