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Neil Ashby

Dept. of Physics, University of Colorado

National Institute of Standards and Technology Affiliate

Email: ashby@boulder.nist.gov

1. Navigation—why you need a clock

2. Brief history of relativity in the GPS

3. What the GPS is

4. Relativistic effects:

Relativity of synchronization;

Time dilation;

Gravitational frequency shifts;

Sagnac effect;

5. Observations: testing relativity

TOPEX;

Frequency jumps;

Unmodeled effects;

6. Applications

Relativity in Global Satellite

Navigation Systems

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Latitude

3

w

4

Moon, Jupiter & Satellites

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GPS RELATIVITY MEETINGS

• 1979 SAMSO Relativity Seminar (Boulder)

• 1985 JASON Study

• 1986 Air Force Studies Board

• 1988-98 Various Working Group Meetings

• 1995 ARL-Chapel Hill

• 1997 ICD-200 Relativity Review (Boulder)

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GPS RELATIVITY MEETINGS

• 1979 SAMSO Relativity Seminar

• 1985 JASON Study

• 1986 Air Force Studies Board

• 1988-98 Various Working Group Meetings

• 1995 ARL-Chapel Hill

• 1997 ICD-200 Boulder

Erroneous Reports

• 1977-83 Moses, Cohen, Rosenblum

• 1992 Deines

• 1996 Fliegel & DiEsposti (Aerospace Corp)

• 2000-2006 Hatch

• 2008 Beisner

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GPS Constellation

• 24 Satellites (now>30)

• 6 orbital planes, 55o inclination

• Period: half a sidereal day

• Several atomic clocks/satellite

• Several spare satellites

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Control Segment

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GPS IIR Satellite

IIF

III

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Block III satellite

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Block III GPS satellite

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Other GNSS Satellites

Beidou GALILEO

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Constellation Status-GPS

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GPS Constellation Status

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GALILEO Constellation Status 11/21/2016

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Planned Beidou Constellation

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Beidou Constellation Status

• As of November 2016: 20 operational satellites of 35 planned

– 6 satellites in geostationary orbits;

– 8 in 55-degree inclined geosynchronous orbits;

– 6 in medium earth orbits at altitude 21,500 km

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• Constancy of the speed of light– The speed of light, c, is a constant independent of the

motion of the source (or of the observer);

• Principle of Equivalence (“weak form”)– Over a small region of space and time, the fictitious

gravitational field induced by acceleration cannot be

distinguished from a real gravitational field due to a

mass.

Fundamental Principles

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Constancy of c

,1,2,3,4

j jc t t

j

r - r

Synchronization

Is the key!

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Reciprocity

j jc t t r -r

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Exploded Block IIR Satellite View

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Clock Improvement Since 1000 A.D.

*

1 ns/day=

10-14

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Why are atomic clocks needed?

To reduce the effect of clock error to < 2 meters,

the clock error must be less than 2/c = 6.7 x 10-9 sec.

Half a day = 43200 seconds, so the fractional clock

error must be less than:

(2 m)/(43200 s x c) = 1.5 x 10-13.

Only atomic clocks can achieve such stability.

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25

Constancy of the speed of light

implies time dilation

(These clocks are at

rest in the moving frame.)

(This clock at rest in “rest”

frame, coincides with upper

clock in moving frame.)

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Einstein’s Light Clock

2 2c v

2 2

2 2

2 2

// ;

1 /

' / 1 /

L ct L c v

v c

t L c v c t

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How Big is Time Dilation?

22 2

2

211

2

11 / 1 ;

2

4000m/s;

18.9 10

2

vv c

c

v

v

c

(about 8 microseconds per day)

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Accounting For Relativistic Effects

Example: Time Dilation:

2 2

1/ 22 2

2

2

2

2

path

1 / ;

1 /

11 .

2

11

2

d v c dt

dt v c d

vd

c

vt d

c

Elapsed Coordinate time:

Observed Proper Time

Note: 2 2 22 2 2 2 2(1 / )cd cdt v c cdt dx dy dz

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Fundamental line element

2 2 2 2 2

2 2 22 2 2

2 2

2 2 2 2

2 2 2 2 2

2 2

0 ( )

1( ) ( ) 1

( )

2 21 ( ) 1

0.path

ds cdt dx dy dz

dx dy dzds cd cdt

c dt

cdt dx dy dz

ds cdt dx dy dzc c

ds

For light:

Time dilation:

With gravity:

Motion of

Planets:

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Coordinate Time

• In special relativity:

To each real clock, corrections are applied such

that at each instant, the clock would read the same

as a hypothetical clock at rest at the same point

in the underlying inertial frame.

• When gravitational fields are present:

Additional corrections compensate for gravitational

frequency shifts relative to a reference on earth’s geoid.

• GPS time is an example of coordinate time, in which

the reference is on the earth’s rotating geoid.

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Sagnac Effect

2 2 2 2 2 2

2 22 2 2 2 2 2

2

( ) ( ( ))

1 ( ) 2

ds c dt dr r d t dz

rc dt r d dt dr r d dz

c

w

ww

2 2 2 2 2 2 2( ) ( ) ( )ECIcd ds cdt dr r d dz

For light: solving for dt to first order in

the ddt term gives rise to the

Sagnac effect.

,w

This is the Langevin metric.

In a rotating coordinate system such as one fixed to the earth, let the

axis representing the zero for the angle rotate with constant

angular speed:

2

2

d r ddt

c c

w

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Sagnac Effect on Synchronization

in a Rotating System

w

Sagnac Correction =2

2zA

c

w

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Over a small region of space and time,

a fictitious gravity field induced by

acceleration cannot be distinguished

From a gravity field produced by mass.

Equivalence principle and

gravitational frequency shifts

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Gravitational Frequency Shift

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Gravitational Frequency Shift

2 2

/ ;

;

.

t L c

gLv gt

c

f v gL

f c c c

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Relativity of Simultaneity

To an observer on the ground, let two lightning

strokes at the front and back of the train

be simultaneous.

The “moving” observer at the train’s midpoint finds

the event at front occurs first.

2 2'

vxt t t

c c

V r

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Principle of Equivalence

V

V+V

Acceleration

Local Inertial

Frame

r

2

2 2

2

2

2

2

Induced potential difference/c ;

Gravitational potential difference/c ;

Net potential difference/c = 0;

c Tc

c

c

A r V r

r

A r r

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Fundamental Scalar Invariant

2 2 2 2 2

2 2

2 2

2 1

2 2

2 2( ) 1 ( ) 1 ( )

3 11

2

c d c dt dx dy dzc c

J aGM z

r r r

For a clock near earth,

2

2 2

path

11

2

vt d

c c

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Earth-fixed Clock

0

2

path

2 2

0 2 1

2 2 2 2

1 1

10

10

1 ;

2 2

6.95348 .00376 .01203 10

6.96927 10

t dc

GMJ aGM

c c a c a c

w

This is the fractional frequency shift of an atomic clock

fixed on earth, relative to an atomic clock at infinity.

Note about centripetal term

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Clocks on earth’s geoid beat at equal rates

Clocks at rest on geoid

beat at equal rates, defining

International Atomic Time.

They are synchronized in

the underlying inertial frame.

Centripetal potential,

monopole potential,

quadrupole, and higher

potential terms conspire

to give an equipotential

in the rotating frame.

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Earth-based Time Scale

SI Second:

0

2

2 2 2 2 20

2 2

1

2( ) 2( ) 1 ( ) 1 ( )

t tc

cd cdt dx dy dzc c

(Basis for International Atomic Time, Universal Coordinated Time.)

(This number is now a defined quantity.)100

26.969290134 10 .GL

c

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Atomic Clock in a Satellite

122 2

0 0

2 2 2 2

0SV 2 2 2

path

10

path

2

2( ) ( )1 ; 1

2

3 2 1 11

2

[1 4.4647 10 ]

2sin (sec)

meter

v vd dt dt d

c c c c

GM GMt d

ac c c a r

d

GM ae E

c

21; a = semimajor axis

2 2

GMv

a

38.6 s/day

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Factory Frequency Offsets

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Frequency shifts cancel at this radius

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Three Important Effects (GPS)

#1: Scale correction to satellite clock:

10.23000000000 MHz 10.22999999543 MHz

#2: Receiver must implement the eccentricity correction:

104.4428 10 sin (sec)meter

ae E

#3: User must account for time required for signal propagation

(Sagnac effect) if relevant.

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SV#13 eccentricity effect

(TOPEX receiver)

0.013e

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GALILEO Satellites in unintended orbits

Normal radius of a GALILEO satellite: 29599.8 km

Eccentricity: 0

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Quasi-Zenith Satellite System (Japan)

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Frequency “Breaks”

Due to Orbit Adjustments

0

2 2 2

3 2 1 1.

2

f GM GM

f c a c c a r a

If eccentricity is small,

2 2

-13

-13

3;

2

FOR SV#43:

Measured: -1.85 10

Predicted: -1.77 10

Now implemented.

f GM a

f c a

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Unmodeled Relativistic Effect: Oblateness

Effect of Earth’s oblateness on satellite orbit:

2

2 2

2

3 2

3;

2J

GMJ z r

r r

Change in monopole potential:2 2

2 1

3

sincos2( ) ...

4

GM GMJ a If

r a w

Change in kinetic energy:

2 2 2

2 1

3

sincos2( ) ...

2 2

v GMJ a If

a w

Change in frequency:2 2

2 1

3 2

sincos2( ) ...

f GMJ a If

f a c w

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A Coincidence?

There are many terms in the perturbations arising

from Earth’s oblateness with coefficient

231 (sin )

2I

For GPS, this is nearly zero. ( 55 )I

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Shapiro delay SV to earth surface

2 1 1 2 2 1

3

1 2 2 1

logG e

gravity

r r L r r r rGMt

c c r r r r

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Spectrum of lunar tidal potential

The coefficients are functions of the eccentricities and inclinations of the SV and the

moon with respect to the equator. The phases are functions of the altitudes of perigee

and the angles of the lines of nodes.

There are significant contributions from many frequencies

in the neighborhood of 6 hours. (These correspond to )

The short-period terms are sufficiently close together that they can beat against

each other, reinforcing and cancelling. They can combine and have amplitudes

that are estimated to be greater than about

Detailed calculation of the lunar tidal potential gives perturbations in terms of

cos( )i i sat i moon i

i

A n t m tw w

6,... 8; 7,... 8i in m

152 10 .

2.in

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Unmodeled Relativistic Effects:

Lunar and Solar Tides

Lunar and solar tidal perturbations are estimated to affect the

fractional frequency shifts of GPS SV clocks in a predictable way

by about153.7 10

The principal periods with which this occurs are near 6 hours

but there are many nearly equal frequencies.

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Surface Plate Velocities

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Control of Monster Machines

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Autonomous Operation

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Precision Agriculture

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Surveying

Finding boundary markers

lost for a century.

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Animal

Tracking

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GNSS-other satellite navigation

systems

GLONASS-Russia

GALILEO-ESA

BEIDOU--China

IRNSS--INDIA

QZSS--JAPAN

AUGMENTATION SYSTEMS:

WAAS

EGNOS

QZSS

All use the same fundamental relativity concepts.

The GALILEO specs state “all relativity corrections are

the responsibility of the user.” ????

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Clock Coefficient a0 of

GPS Satellite clocks, 1992-2014

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References:

http://relativity.livingreviews.org/Articles/lrr-2003-1/index.html

“100 Years of Relativity,” World Scientific,

A. Ashtekar, ed., (2005), Chapter 10

“Handbook of Spacetime,” Springer,

Ashtekar and Petkov, eds, (2014), Chapter 24

“General Relativity: The Most Beautiful of Theories,”

Rovelli, ed., de Gruyter, (2015) pp 165-188

END

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GPS DEVELOPMENT KIT

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