einstein and gps - uni-osnabrueck.de · einstein and gps peterhertel universityofosnabrück...

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




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.





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.





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.





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.





The principle

You see the lightning immediatelybut thunder requires 3 seconds for one kilometer





The principle

You see the lightning immediately

but thunder requires 3 seconds for one kilometer





The principle

You see the lightning immediatelybut thunder requires 3 seconds for one kilometer





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





One satellite

t1r1 = c(t̄1 − t1)

Signal is emitted at time t1and is received at time t̄1.

It propagates with speed of light c





One satellite

t1r1 = c(t̄1 − t1)

Signal is emitted at time t1and is received at time t̄1.It propagates with speed of light c





Two satellites

t1r1 = c(t̄1 − t1)

t2 r2 = c(t̄2 − t2)

The receiver must fulfill both conditions





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





Three satellites


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





Three satellites


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





Three satellites


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





Three satellites


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





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.





Four satellites


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.





Four satellites


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.





Four satellites


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.





Four satellites


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.





Four satellites


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.





Four satellites


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.





Four satellites


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.





Four satellites


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.





24 satellites

circular orbits (we shall see why)20200 km above earth2032 kg1.14 kW





24 satellites

circular orbits (we shall see why)

20200 km above earth2032 kg1.14 kW





24 satellites

circular orbits (we shall see why)20200 km above earth

2032 kg1.14 kW





24 satellites

circular orbits (we shall see why)20200 km above earth2032 kg

1.14 kW





24 satellites

circular orbits (we shall see why)20200 km above earth2032 kg1.14 kW





Replacement

Block II replacement 12





Orbits

6 orbits inclined 55 degrees with respect to ecliptic plane4 equally spaced satellites on each orbit





Orbits

6 orbits inclined 55 degrees with respect to ecliptic plane

4 equally spaced satellites on each orbit





Orbits

6 orbits inclined 55 degrees with respect to ecliptic plane4 equally spaced satellites on each orbit





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





NIST-F1 cesium fountain atomic clock

accuracy is 0.5× 10−15





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





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





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





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





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





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





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





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





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




Technical Problems

Speed of signal is not exactly c (ionosphere and troposphere)

IntensitiesMoon and Sun perturb satellite orbitsShadowing. . . and many more




Technical Problems

Speed of signal is not exactly c (ionosphere and troposphere)Intensities

Moon and Sun perturb satellite orbitsShadowing. . . and many more




Technical Problems

Speed of signal is not exactly c (ionosphere and troposphere)IntensitiesMoon and Sun perturb satellite orbits

Shadowing. . . and many more




Technical Problems

Speed of signal is not exactly c (ionosphere and troposphere)IntensitiesMoon and Sun perturb satellite orbitsShadowing

. . . and many more




Technical Problems

Speed of signal is not exactly c (ionosphere and troposphere)IntensitiesMoon and Sun perturb satellite orbitsShadowing. . . and many more




Special relativityGeneral relativityCorrections matter!

Albert Einstein, 1879-1955





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





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





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






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







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







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






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







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







{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







{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







{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





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!







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!







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!







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!







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!







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!





Any Questions?

Make it as simple as possible, but not simpler!




Black HolesGravitational lensCosmic Background Radiation


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 . . .






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 . . .






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 . . .






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 . . .






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 . . .


Black Hole at Center of Milky Way - 4 Mio sun masses




Gravitational lens

A very massive galaxy cluster (Abell 1689) at a distance of 2×109

light years produces a strong curvature of space.


Cosmic 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.






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.






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.






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.





Thanks for your attention!

[email protected]


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