zotero-db/storage/8E8F9T3X/.zotero-ft-cache

1993 IEEE INTERNATIONAL FREQUENCY CONTROL SYMPOSIUM
RELATIVITY IN THE FUTURE OFENGINEERING
Neil Ashby Department of Physics, Campus Box 390
University of Colorado Boulder, CO 80309-0390

Abstract
Improvements inclock technology make itpossible to develop extremely accurate timing, ranging,navigation, and communications systems. Three relativistic effects-time dilation,theSagnac effect, andgravitational frequency shifts-must be accounted for in order for modern systems t o work properly. These effects will be related in a non-mathematical way to fundamental relativity principles: constancy of the speed of light, and the principle of equivalence. Examples of current and future engineering applications will be discussed, such as in the Global Positioning System, in time synchronization systems, geodesy, and communications.

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log(Averaglng llma [secondrl)

Atomic Clocks

Relativistic effects become important in applica-

tions requiring very accurate timing, time transfer, or

synchronization. Many engineeringsystems are be-

ginning to rely onmodernatomic clockswhich have

fractional frequency stabilities of the order of lo-’’ or

An excellent example is the Global Positioning

System (GPS), inwhich about a dozen relativistic ef-

fects must be accounted for in order for the system t o

w o r k properly. Atomic clock technology not only pro-

vides the basis for the definition of the second as the

unit of time,this technology is expected to improve

rapidly in the future. Vessot e t al.’ have summarized

potential future performance improvements in several

promising devices including cryogenic H-masers,CS

fountains, and trapped Hg ions;thesepredictions are

summarized in Fig. 1. Theseanalyses show there is

some hope that fractional frequency stabilities in the

range

to lo-’’ can be achieved. For thispaper I

shall however adopt a conservative fractional frequency

stability figure of

as a guideline for determining

what relativistic effects might be important in the fu-

ture.

Fig. 1. Predicted Allan variance for future frequency standards. (This assumes no systematic effects in CS and Hg devices.)
Constancy of the SDeed of Light

Relativity enters metrology in a most fundamental way through the so-called ‘Second Postulate’ of the specialtheory of relativity,the principle of the constancy of the speed of light, c. This nowwidelyacceptedprinciple states that the speed of light in free space has the same value in all inertial systems, independent of the motionof the source. The speed of light is also independent of the motion of the observer. The numerical value of c has been defined by convention:

c = 299 792 498 meters/second.

(1)

In conjunction with the adoptedunit, of time, thisvalue for c defines the SI unit of length, the meter. In thinking about the speed of light, a convenient alternative rule of thumb is that c is approximat,ely equal t o one footpernanosecond (1 nanosecond = 1 ns = lo-’ second).
In an inertial frame of reference, the principle of the constancy of c provides a means of synchronizing

2 0-7803-0905-7/93 $3.00 0 1993 IEEE

remotely placed clocks. Consider two standard clocks, A and B , placed at rest a distance L (meters) apart. (The distance L could be found by measuring the time on clock A required for a light signal to propagate from A to B and back, and multiplying by c / 2 . This would not depend on thepresence of a clock a t B.) Now suppose a signaloriginates at clock A at time t A . The time required for the signal to propagate in one direction from A to B is L / c . The clock at B will then be synchronized with that atA if the signal arrives at the time t B given by

+ t B = t A L / c .

(2)

with the transmitter’s position and time, so that a receiver can determine the time t j and the location r, of the pulse from transmitter j . The receiver’s position r and clock time t can then be determined by solving four simultaneous propagation delay equations:
(r - rj)’ = c z ( t - t.31’., j = 1,213141 (3)
for the unknownsr and t . These equationsjust express the principle of the constancyof the speed of light inan inertial frame. Clearly a timing error of one nanosecond would lead t o a an error of about a foot in position determination.

This procedure is called the ‘Einstein Synchronization

Procedure’ andclocks distributed at rest in any inertial

frame will be presumed to be synchronized by this or

an equivalent procedure.

Clearly in discussing electromagnetic signals as I

havedoneabove, I am ignoring quite a few practical

difficulties. Signals must have sufficient spectral band-

width thatit is possible to reconstruct well-defined

pulses in time. Noise in real clocks, frequency drifts

due t o environmentalfactors, etc., are not a concern

here. Also I’m usually going to ignore effects on prop-

agationspeed which mightarisebecause the signals

pr0pagat.e throughamediumratherthanthrough

a

vacuum.

Transmission from rj at tj

Event Detection
There is a kind of reciprocity in thissituation which canbe used for event detection:supposethat instead of transmitters at thelocations r, there are receivers, tied to synchronized standard clocks. Suppose that an event occurs at the position r at time t causing a signal to be transmitted, which is received at the four receivers at the respective known positions rj a t themeasuredtimes t j . Then by solving four propagation equations of the form of Eqs. ( 3 ) , the position of the event and the time at which itoccurs can be determined. If someinformationaboutthe position of the event is available, it maybe possible to locate the event by solving fewer than four propagation delay equations.

Reception

Fault Location

Fig. 2. Idealized conception of a navigation and time transfer system.
Navigation

Keeping these caveats in mind, the constancy of c

leads to the following idealized conception of a naviga-

tional system. Referring to Fig. 2, suppose four trans-

mitters, each with its own standard clock, are placed

a t known locations rj. Assume the clocks are synchro-

nized by the Einstein procedure. There is a receiver a t

unknown position r carrying a standard clockwhich

llas not been synchronizedL. etthese

transmitters

ra.pidly transmit synchronized pulses which are tagged

As an example of event detection using only two

synchronized clocks, consider the problem of determin-

ing the location and time of a fault that occurs in a

power line stretching between two detectors a distance

L apart. In Fig. 3 , clocks at the ends of thelineare

synchronized fromsome independent primaryreference

clock. A fault occurring at distances L1, L2 from the

respective detectors at the endsof the lines sends out a

signal at time t which is received a t times t l , t 2 at the

respective ends of the line. A previoussurvey would

give

+ L = L1 Lz,

(4)

whereasfrom the constancy of c , the times t l , 22 are related to the time t by propagation delay equations:

+ + t l = t Ll/C, 12 = 1 & / c .

(5)

Solution of onlytwopropagation delay equations, in

conjunction with Eq. (4)’gives the time and position

of thefault.Tolocatethefault

to withinone foot

requires synchronization to better thana nanosecond.

3

Primary Reference
Source

train. As thetrain moves forward,this observer approaches the oncoming light emittedfrom the event at the front of the train, and recedes from the light signal emitted from theevent at the back of the train. Therefore the moving observer will encounter light from the fronteventfirst, and will have to conclude that the event at the front of the train occurredfirst. By the principle of the constancy of c , light must travel with speed c no matter what the value of the relative speed v is. So if light from event A arrives before that from event B, which is the same distance away, then event A must occur first.

Fig. 3. Fault location using consta.ncy of c Breakdown of Simultaneitv

The discussion above assumes that the clocks are at, rest in some inertial reference frame. Usually howcvcr clocks are in motion; for example i n Fig. 2 t,he tra.nsmitt,erscould be orbiting the earth. Relative motion introducessubtle neweffects; perhapsthemost profound of these is the breakdown of the concept of simultaneity.Events which appear to occursimult,a~~c:ousilny one inertial frame may not appear simultall(:ous to observers in some other inertial frame, which i3 moving with respect tothefirst.This is a direct cmrlscqrrence of the principle of the constancy of c.
I n discussing measurements made by observers in t,wo different, relatively movinginertialframes,one always imagines that each observer is equippedwith Ilis/her own measuring rods and standard clocks, that tile clocksusedby observers in one frame are at rest, n ~ ~tthlatthey are synchronized by the Einstein procctlure.In each of theinertialframes, any particular c>lcctromagneticsignal propagates with speed c.
Consider then as i n Fig. 4 . twoeventsconsisting of two lighting strokeswlliclr hit the two endsof a train o f length L = 22 simult.aneously as seen by observers 071 !/re ground. The train is assumed to bemoving to the right at speed v relative to the ground. For ease of discussion, I’llrefer to the ground as the ‘rest’ frame, and the train as the ‘moving’frame. Observers on the ground (in the rest frame) can determine the midpoint het.ween thetwo lightning strokes, a distance x from cit,ller end of the initial position of the train. They will t I l c n find that lightsignalsfromthe two events will propgate along the tracks andcollide at themidpoint.. ‘I‘llis has nothing to do with the motion of t.lre train.
Now look at the sequence of event.s involving a Ilroving observer, sitting at the midpointof t.lle moving

Fig. 4. Thought experiment illustrating relativity of simultaneity.
To analyzethisapproximately is notdificult. Suppose the zero of time for observers i n both the rest. alld the moving frames is set to occur at the instant t.lie midpoint of the train encounters the signal from the lightning stroke at the front of the train. I’ll use primes to denote quantities measured by themoving observer. Then to the moving observer, the time 1’ of the stroke at the front of the train is
2’ = X
C
To observers i n the rest frame, however. the midpoint, of thetrain is approachingthe signal at the relative
speed c + v, so to first order i n v ,
Therefore
‘I’he term - v x / c 2 is a relativistic correction for breakdown of simultaneity. The effect is proport,ional t80the relative velocity and proportional t,o the dist,ance x.
Putting i n somenumbers,suppose IJ = 1000 km per hour (typical for a jet aircraft) and x = 3500 km. Then the correction is l08 11s. The negative sign in Eq.

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( 8 ) means that of two events simultaneous in the rest frame, to themoving observer the event farther out in front, at the more positive x, occurs earlier.

Samac Effect

The above discussion of the breakdown of simul-

taneity can be used to understand thepeculiar physics

on the edge of a slowly rotating disc. The prime engi-

neering application is to time transfer andsynchroniza-

tion on the surface of the rotating earth. For purposes

of illustration, therefore, I’ll use the angular velocity

of rotation of theearth, W = 7.29 x

rad/sec,and

for the equatorial radiusof the earth, R = 6.378 X lo6

meters.

In this case the rest frame is a local non-rotating

frame, with axes pointing toward the fixed stars, but

withorigin at the center of theearth.The moving

frame is a reference frame extending over a small por-

tion of therotatingearth’ssurface,having velocity

v = W T relative to the rest frame, where r is the dis-

tance of the clocks from the rotation axis.

Now imaginetwo clocks fixed a small east-west

distance I apart on the equator of the earth. Viewed

from the nonrotating frame they will be moving with

approximatelyequalspeeds v = w r . If a clock syn-

chronization process involving electromagnetic signals

were carried out by twoearth-fixed observers using Ein-

stein synchronization in the moving frame, then the

two clocks would not be synchronouswhen viewed from

the nonrotating frame. The magnitude of the discrep-

ancy is v x / c 2 = w r x / c 2 = ( 2 w / c 2 ) ( r z / 2 ) .If this syn-

chronization process isperformed successively all the

way around the circle, then effectively the distance I

is z = ~ K T a,nd the time discrepancy is thus

best to adopt time in the non-rotating frame as the measure of time in the rotating frame. Thus one discards Einstein synchronization in the rotating frame.
To put it anotherway, if Einstein synchronization is used in the earth-fixed rotating frame, then it insecessary to apply a ‘Sagnaccorrection’ to the readings of clocks on the rotating earth, inorder the obtain an internally consistent ‘coordinate time’ on earth’s surface.
This is illustrated in Fig5. , where there is a sketch of a flattened rotating earth.For a sequence of synchronization processes forming a closed circuit on the rotating earth, upon projecting the path onto the equatorial plane of the earth onecan determinethe projected area A E . The Consultative Committeefor the Definition of the Second and the International Radio Consultative Committee have agreed that, in order to obtain consistently synchronized clocks on the earth’ssurface at the subnanosecond level, the correction term to be applied is of the form

At = 2w/c2 X A E ,

(11)

where A E is the projected area on the earth’s equatorial plane swept out by the vector whose tail is at the center of the earth and whose head is at the position of theelectromagneticsignal pulse. Thearea A E is taken as positive if the head of the vector moves in the eastward direction. If two clocks located on the earth’s surface are compared by using electromagnetic signals in the rotating frameof the earth, thenAt must be subtracted from the measured time difference (east clock minus west clock) inorder to synchronize the clocks so they will measure coordinate time on the rotating earth. They willeffectively measure time in the local non-rotating frame attached to the earth’s center

At = 2w/c2 x r( 9r 2) ,

where TT’ isthearea enclosed by the path followed

during the synchronizationprocess. For example, syn-

chronization around the earth’s equatorinvolves a dis-

crepancy

At

=

2w -xR2

x 208 ns.

(10

C2

upon arriving back at the starting point. This effect is known as the Sagnac e & A . If the
synchronization path were westward around the earth rather than eastward, then thediscrepancy would be of opposite sign. This means that Einsteinsynchronization in a rotating reference frame is not self-consistent: If A is synchronized with B and B is synchronized with C, then A is not necessary synchronized with C. In order to avoid difficulties with such non-transitivity it is

Sagnac Correction:

At

=

2~A,y ,2

L

Fig. 5. Projected area for a sequence of Einsteinsynchronization processed forming a closed circuit on the rotating earth’s surface.

5

Lack of transitivity in synchronization has impli-

cations for devices which rely on accurate synchroniza-

tion.Suppose a communications network distributes

synchronizationthrough a series of nodes,alongtwo

different paths, to the ends of a communication link

as in Fig. 6. If thearea enclosed by thepath, pro-

jected ontotheearth'sequatorialplane,isnot

zero,

thenproblemswithinconsistentsynchronization can

a.rise.For example, suppose one synchronization link

goes from San Francisco directly to New York, while a

second link goes from SanFrancisco to Miami and then

to New York. The discrepancy in synchronization be-

tween these two paths dueto the Sagnac effect is about

11 ns. While this is notsignificant if the signalis 60 Hz

as in a power grid, in an optical communications net-

work operating at 1015 Hz the discrepancy amounts to

IO7 cycles of oscillation.Depending 011 the design of

I he system this may become significant, i n the future.

nal propagation may besignificantly less than c. In the rotating reference frame the Sagnaceffect is a property of space and time, not dependent on signal propagation speed.

Synchronization Link Synchronization Link

Irg + vt - rA12 = (ctp

Synchronization Link
r
Synchronization Link

Synchronization Link
P
Data Link

Fig. 6. Distribution of synchronization for a communications network.
Furthermore, if the trouble is taken to incorporate hardwaredelays to compensatefortheSagnac effect while sending in one direction, then when the communications aresent back theother wayover thesame link the effectwill become twice as big. The effect is asymmetric. The same effectwill occur in optical fiber communications networks where the speedof sig-

Fig. 7. The Sagnac effectwill be automatsicallyincluded if receiver motiondue to earth rotation during signal propagatioisnaccounted for.

An equivalent way of looking at this phenomenon

is diagrammed in Fig. 7, which shows a signalt,rans-

mitted from a satelliteto a ground-ba.sed recciver.

From the point of view of the nonrotating frame, the

signalgoesin a straight line withspeed c, from the

initial transmitter position 1'A to the final receiver po-

sit.ion. If in this frame one accounts for the mo(.ion of

the receiver during the propagation of the signal, then

the Sagnac effectwill be automatically accounted for.

This if the initial position of the the receiver is 1-8t,he

velocity of the receiver is v , and the signalpropagat.ion

time is t , constancy of c requires

+ I I'D vi! - ' A

(Ci!)2.

(12)

Iterative solution of Eq. (12) for t is equivalent to calculating the Sagnac correction.

Time Dilation

Inthe previoussection I discussed two effcct,s which are of first order in the velocity-the breakdown of simultaneity, and the Sagnac effect. In this seci.ion I shall discuss anotherfamous effect-time dila(.ionwhich is of second orderin the velocity. Imaginetwo

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inertial frames, a ‘rest’ frameor laboratory frame, and a moving frame. A clock in the moving frame beats more slowly than clocks in the rest frame with which it is successively compared. The following thought experiment will readily convince anyone that the principle of the constancy of the speed of light requires the ‘moving’ clock to beat more slowly. A prime denotes quantities measured in the moving frame.
Suppose that observers in the two inertial frames each possess a set of rectangular Cartesian coordinate axes which they orient so that the x,x‘ and y, y‘ axes are parallel. The direction of relative motionis parallel to thex,x’ axes. The moving observer orients a rod of length L’ along the y‘ axis, and sends a light signal up along this rod from one endto the other. The situation is diagrammed in Fig. 8. To simplify the discussion one assumes that thelight starts out at the instant the origins of the two reference frames pass by each other.

Thought experrmenc viewed

in ‘ r e s t ’ frame.

Y

ly ‘

I

The timet’ required for light to travel along the rod is

simply

t‘ = L / c .

(13)

The clock faces on the lower part of Fig. 8 indicate time at thebeginning and end of the experiment.
The upper part of Fig. 8 shows the experiment from the point ofviewof observers in the rest frame. Breakdown of simultaneity would create difficulties for measurements of lengths orientedparallel to the relative velocity. But since this rod is orientedperpendicular to the relative velocity, by symmetry it is not possible for the rod to appear changed in length. So this rod haslength L = L’ as it moves throughthe rest frame. The rod is moving to the right with speed v and the light travelsalong the rod, so there has to be a horizontal component of velocity of the light equal to v. The vertical component of the velocity of the light certainly h a s to be less than c; therefore the time required for the light to reach to upperend of the rod certainly h a s tobe greater than L/c. Thisargument shows qualitatively that theclocks in the moving frame will beat more slowly than the sequence of clocks with which they are compared in the rest frame.
The top part of Fig. 8 actually gives the right answer, for by the principle of the constancy of the speed of light, the vertical component of the light velocity in
the rest frame is just d n . Thus for observers in
the rest frame, the timet required for the light toreach
the upper end of the rodis just .

I Thought experiment vieved

so the relationship between t’ and t obtained by eliminating L from Eqs. (13) and (14) and L = L’ is:

1’ = J-t.

Usually the ratio v / c is small, so the square root can be expanded, giving approximately

g) t’ x (1 -

1.

Fig. 8. Thoughet xperiment showing that ‘moving’ clocks beat more slowly than clocks that remain ‘at rest’.
The lower part of Fig. 8 shows the situation from the point ofviewof observers in the moving frame.

The fractionalslowing is given by the correction v2/2c2 in the above equation.This correction is also commonly called the second-order Doppler shift, or transverse Doppler shift.
Some examplesof t,he size of this effect are as follows.For a clock a t rest on the earth’s equator, and viewed from the nonrotating frame,
--1-v2 x -1.2 x 10-12; 2 c2

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this would accumulate to about 104 ns in one day. For a clock in a satellite orbiting the earth at 100 km altitude.

For a clock in a GPS satellite,

--1-v2 x -8.34 x lo-”
2 c2

Keeping in mind that in the future the fractional fre-

quency stability of orbiting clocks may approach a part

in 1015, these are very large effects. Even for clocks of

frequency stability 1 x

as in thGe PS Block I1

satellites, the second-order Doppler shift for an earth-

fixed clock is significant.

Gravitational Frequency Shifts

and gravitational mass, a subject we shall not go into here.3)InFig. 9b, a similar experiment is performed in a laboratory in free space which is being pulled upward withacceleration g. A non-accelerated observer sees that the apple and the leadball have no forces exerted on them so remain at rest with respect to each otherandthelaboratory is accelerated pastthe objects, whereas the observer in the accelerated frame sees the objects ‘fall’ downward with identical accelerations g.

TheSagnac effect andthe second-order Doppler shift are effects which can be understood on the basis of the SpecialTheory of Relativity. A third effectthegravitational frequency shift-occurs when signals are sentfromonelocation toanother having a different gravitationalpotential.The effectcan be understood in anelementary way using thefundamental assumption of the GeneralTheory of RelativityEinstein’s Principle of Equivalence.2
The PrinciDle of Equivalence

Einstein’s Equivalence Principle states that over a small region ofspace and time, a fictitious gravitational field induced by acceleration cannot be distinguished from a gravitational field produced by mass. Thus the fictitious centrifugal force one feels in turninga corner in a vehicle h a s the same physical effects as a real gravitational field. An immediate consequence of the Equivalence Principle is that gravitationalfields can be reduced to zero by transforming to a freely falling reference frame. The fictitious gravitationalfield due to the acceleration then exactly cancels the real gravitational field.
All experiments performed in a real gravitational field, such as in a laboratory on the surface of the earth where there is a gravitational field g, will have the same results at experiments performed in a laboratory i n free space which is accelerated in the oppositedirection with acceleration a = -g. In Fig. Sa are sketched some experiments performed in a laboratory fixed on the earth’s surface. For example two objects of different compositions are observed to fall downward with equal accelerations g. (This is related to the deep experimental fact of the strict proportionality of inertial

Fig. 9a. All objects fall with equal accelerations in a laboratory near the earth’s surface.
The equivalence of the two laboratories implies that a beam of light is deflected toward the source of the gravitational field. Let a beam of light-which travels in a straight line in free space-enter the side of the accelerated laboratory (near the top, in Fig. 9b). The observer in this laboratoryis accelerated past the light, so it must appear to fall down just as do the massive objects. The experiment must have the same outcome in the non-accelerated laboratory on earths, o to an observer in a real gravitational field light must fall down. A beam of light passing near any massive body will be deflected towards the body.
Time Delav
If one imagines the wavefronts in a beam of light as the beam is deflected toward the massive source of a gravitational field, then one can picture the portions of the wavefront nearest the mass being sloweddown

8

slightly with respect to the portions of the wavefront farther away from the source. T h e wavefront then tilts over andthebeamisthereby deflected. Thismeans that of twobeams of lightpassingnear a massive source, the one which passes closer will take longer to get by. Thus not onlyislight deflected, it is slowed down by a gravitational field.
I
A
L

1
9

I

1 Free Space

Fig. 9b. By the Equivalence Principle, experiments performed in an acclerated lab in free space have the sameoutcomes.

Time delays of signals in the neighborhood of the earth can be a few tenths of a nanosecond. Such time delays aredetermined by a complicatedlogarithmic function of signalpathparameters,timesthequantity ~ G M E / c ~w,here G is the Newtonian gravitationalconstantand M E theearth'smass. For earth G M E / c =~ 0.443 cm, so the scale of such effects near earth is

-4-G M E

1.77 cm
%-

=

.06

ns.

(20

c c2

C

Fig. 10. A signal travelling upwardsin a gravitational field is shifted towards lower frequencies.
Gravitational Freauencv Shifts

It follows from the Equivalence Principle that an

electromagneticsignalpassingupwards in a gravita-

tional fieldwill beredshifted.In Fig. 10 is a sketch

of anexperiment performed in an equivalentlabora-

tory, a rocket havingacceleration g upwards in free

space.Imaginethesituationfromthe

point ofview

of anon-accelerated frame.Suppose a signal leaves

the accelerated transmitter at the initial instant, when

thetransmitter velocity is still zero. The signal up-

wards a distance L , and is receivedby the accelerated

receiver. The timerequired for the signal to propagate

from transmitter to receiver is:

This is not enough to worry about at thepresent time but could be significant in the future-a timing errorof .l nsin a navigational system wouldgiverise t o a 3 cm error in position.

During this time, the receiver has picked up a velocity

9

,l,o the receiver, the signal appears to come from a receding sourceand is Doppler shifted.To a first approx-
imation the fractionalfrequency shift isAf/f = - v / c ;
t.herefore the fractionalfrequency shift in the ‘effective’ gravitational field g is

The quantity gL can be interpreted as the change in gravitational potential, Ad, of the signal.
At the surface of the earth.

g / c 2 = 1.09 x 1 0 - l ~per km,

(24)

wIIicI1 is very important for today’stimest,andards.

For example a signal of definite frequency originating

lnean sea levelwouldberedshift.edby1.79

parts

i n 10l3 upon arriving at the altitude of the NIST fre-

quency standardslaboratory i n Boulder, CO. Conse-

quently the contributionof the NET time standard to

Universal Coordinated Time (UTC)requires that a pa-

per correct.ion of -15.5 ns/day he applied to the NIST

clock Iwfore it can be compared to time standards at.

tlleiln sea level.

For a clock in a satellite orbiting the earth at 100

km altitude compared to one on the geoid.
* = 1.08 x lo-”.
C2

Not only will these effects be large in the future whenclock stabilities approach a part i n l o t 5 or bett.cr, it will be necessary to compute them quite accurat.cly. This will mean: for example,thatthere will I)c r? nccd for improved precision of the ephemerides of clock-carrying satellites.

The Global Positioning System

The best existing example of an engineering syst.eln i n which relativityplays an essential role is the
GI’s. This consists of a constellation of perhaps 24
earth-orbitingsatellites carrying atomic clockswhich synchronously transmit navigation signals, muchas described in the discussion of Fig. 2. The satellite orbits are at approximately 20,200 kmaltitude. Therefore clocks in the satellites will be significantly blueshifted i n late, compared to clocks on the ground. Thesecondorder Doppler shiftof such clocks was given inEq. (20). Also, if the orbits are not perfectly circular (and they almost never are), the clocks’yo-yo motionstowards andawayfrom the earth will generate periodic addilional gravitational frequency shifts, and second-order Dopplershifts. Further, observers on the ground who

wish to make use of the navigational signalswill experience the Sagnac effect due to earth’s rotation.

A complete discussion of all the significant rela-

tivistic effects, with analytical expressions for the nec-

essarycorrections,canbefound

e l ~ e w h e r e .H~ere I

shall just indicate roughly the magnitudes of some of the corrections.

First, consider ground-based clocks inreceivers

which are at restontheearth’ssurface.Standard

clocks on the geoid are used to define the unit of time; however, from the point of view of a local, nonrotating frame, there is a frequency shift due to earth’s mass; the fractional frequency shift is about -7 x 10-l’. The earth’s oblateness is associated with a quadrupole contribution to thegravitational potent.ial which cannot

be neglected;thefractional frequency shift is about

-4 x lO-I3. If earth-based clocks are not, on the geoid they suffer a gravitational frequency shift (see Eqs. (23-24)). Finally there is a second-order Doppler shift due to theearth’srotation;thefractional frequency

shift from this effect can be as large as -1.2 x lo-’’

(see Eq. 17).

For GPS receivers in motion relativeto theea.rth’s

surfa.ce, there is anadditional second-orderDoppler shiftdue to theirspeedwithrespect totheground; thiscanbe of the order of lo-’? depending on t,he

ground speed. Also, the Sagnac effect-or motion of the receiver duringpropa.gation of the navigationsignal

may give rise to effects of several hundred nanoseconds

magnitude.

Thetransmitters themselves suffer a frequency

shift due to the earth’s gravitational potential, and a

second-order Doppler shift due t.o orbit,al motion; these

effects are several parts i n

Thaedditional fre-

quency shifts due to orbitaecl centricities can be tensof na.noseconds;for a GPS satellite of eccentricity e = .01, the maximum size of the effect is about 23 tis.

Propagation of signalsfromtransmitter to receivers are subject to the Sagnac effect, involving relativistic corrections of up t,o severa.1 hundred nanoseconds. Relativistic time delay of signals or relativistic deflection of signals is a few tenths of nanoseconds and is currently neglected in the GPS.

The ConceDt of Coordinate Time

With so many significant relativistic effects occurring on earth-fixed and earth-orbiting clocks, the problem of synchronization of the clocks becomes an acute one. Rates are affected by motional and gravitational effects; synchronization on the spinning earthis inconsistent, if the Einstein procedure is used. How is it passible to synchronize a network of distributed, rapidly

10

moving clocks so that a navigational system will work as conceived in Fig. 2? What has been found t o work extremely well in the GPS is to use the time in a hypothetical underlying local inertial frame, with origin attached to the earth but not spinninga,s the measure of time. This time is not time on any standard clock orbiting the earth; instead one makes use of general relativity to correct the readings of such clocks so they would agree with hypothetical clocks at rest in the local inertial frame. The time obtained by so correcting all the clocks i n the system, is an examploef Coordinate
time.
Thus, imagine an underlyingnonrotatingframe, or local inertialframe,unattached to thespinning earth, with with its origin at the center of the earth. This frameis sometimes called the “Earth-Centered Inertial” frame, or ECI frame. In this frame, introducea fictitious set of standard clocks available anywhere, all synchronized via the Einsteinprocedure, and running a t agreed upon rates such that synchronization is maintained. Gravitional effects are incorporated bychoosing one clock as a Master Clock and requiring that all other clocks be syntonized to the Master clock by simple transmission of signals without anyfrequency shift corrections. The resulting time scale is called coordinate time.
Now introduce a set of standard clocks distributed around the surface of the rotating earth, or orbiting Lhe earth. To each one of these standard clocks a set of systematiccorrections is applied, so that at each instant the standardclock as corrected agrees with the time on a fictitious standard clock, at rest in the ECI frame, withwhich it instantaneouslycoincides. The set of corrected standard clockswill therefore be keeping coordinatetime.Inother words,coordina.t,e time is equivalent to t.ime measurcd by standard clocks in the ECI frame.
Time measured oncoordinate clocks has two highly desirableproperties.First, synchronization is reflexive:if A is synchronized with B , then B is synchronized with A . Second,synchronization is transitive: if A issynchronizedwith E , and B is synchronized with C, then A is synchronized with C. Internal inconsistencies are therely eliminated.
GPS time is an example of coordinatetime. To an observer on the earth’s geoid, a standard clock in a GPS satellite in a nominally circular orbit would appear to be blueshifted by ,4465 parts per billion, or about 39,000 ns per day; this is a net effect of gravitational frequency shifts and motional Doppler shifts of satellite clocks relative to reference clocks fixed on the ground. To compensate for this,the 10.23 MHzreference frequency of satellite clocks is adjusted down-

ward to 10.229 999995 43 MHz. Theadjustment is accomplished on the ground before the satellitesare launched.
Also, if the orbit of the satellite clock is not perfectlycircular,there will beadditionalgravitational and motional rate shifts which have to beaccounted for. The additional correction required to achieve synchronization when the orbit eccentricity is not zero is given by the expression4
At = +4.428 X 10-”e& sinE sec, (26)
where a is the semi-major axis in meters and E is the eccentric anomaly. Usually the software in the user’s receiver makes this correction.
Application of Satellite Navigation in Geodesy
The motivation to obtain accurate measurement>s of movements of the earth’s crustal plates is intense. Knowledge of these very slow motionsis crucial to the development of improved earthquake prediction capability;thepotentialimpacton construction codes, buildingrestrictions,etc., is considerable. In recent years the GPS has been successfullyused to measure very long baselines between fiducial points on differentcrustalplates by a method described as “carrier phasedouble difference.” Two receivers are placed at the ends of a baseline of interest, and signalsfrom two satellites are then “double differenced” in a manner to be described below. Differenceing removes the need for some systematic corrections but as will be seen, there are residual relativisticeffects which must beaccounted for.
Referring to Fig. G andthepropagationtime t given inEq. (12), let thesatellitepositionatthe instant of transmission t s be denoted by rs and the receiver or observer position at the same instant be denoted by ro. Let the coordinate time of arrival of the signal at the observer be denoted by t o . Then solving Eq. (12) for the propagation time gives
‘The lastterm is theSagnac correction and I< represents a possible time offset or error of the receiver’s clock. The rate adjustment applied to satellite clocks means that the quantity t s will have the correct scale when received on the geoid. Thereis a further correction, from the non-circular motion of the satellite, given by Eq.(26). Thus when all relativistic effects

l1

are incorporated,

l .5

+ 4.428 x 10-"e& sinE.

5

Let subscripts 1 and 2 denote the two different satellites and thetwo different observers. Suppose there are receivers at two different positions which receive a t,ime signal originating from a single satellite. Upon taking the first differece of the arrival times, it is immediat.ely seen that theeccentricity termcancels out, leaving the expression:

2.5
5
Q
-2.5
-5

(291
The Sagnac correction is still needed. Thetime of transmission of the signal,t s , cancels out which lessens the impact of selective availability.
Now the sameset of measurements is taken, at. essentially the same time, using a second satellite. Writing another equation similarto Eq. (29) for the second satellite and taking the difference, it can immediately be seen that even the clock offsets in the receivers cancel out, leaving only the usual propagation delay terms with relativistic corrections due to the rotation of the earth:
The Sagnac correction is d l 1 necessary. In this application the correction is largest when the baseline is at right angles to the line between the satellites; it can be several hundred nanoseconds.
In Fig. l i are plotted some baseline measurement datatakenrepeatedly on baselines in theSouthwest Pacific, of lengths u p to 2500 km.5 Only the length of the baseline is shown here. The vertical scatter in the plotted points gives a measure of the errors involved. For the 2500 km baseline the spread is only a few cm.

500.

1000.

1500.

2000.

2500.

'

Baseline Distance in km.

Fig. 11. Scatter in baseline lengths for several different baselines measuredduringthe Southwest Pacific 1992 GPS Campaign. The data were provided by UNAVCO.

ImDact on Fundamental Metrologv

The previoussections have been devoted almost

exclusively to the impact of relativity on the measure-

ments of time, with dist'ancederived by multiplying by

c . A t the levelof a centimeter or less, therearead-

ditional effects on the nleasurement of position which

arise because space i n t,he neighborhood of a massive

body is distorted. Consider as in Fig. 12 an attempt t,o

establish a system of spatial coordinates in tlle neigh

borhood of earth, aga.inst which to measure the posi-

tions of theearth'scrustalplates.Supposethat

we

wish to measureanglesin the usua.1 Euclidean way,

so that a circle of coordinate radius v centered on t,he

earth would have a circumference 2rr, measured with

standard rods or with the help of the constancy of c .

Two such circles, of coordinate radii q and 1'2 are indi-

cated in Fig. 12. The standard distance from theinner

circle straight out along a radius to the outer circle is not r2 - T I ; instead one finds the standard distance d

IS

The correction due to space curvature is of the order of 1 cm.
More generally, the fact that c has a definednu-

12

mericalvalue means that the physical unit of length depends on the clock used to define the unit of time. For example, in Barycentric Dynamical Time (TDB), the unit of time is the same as that of clocks on earth, in orbit around the sun, and the point of view taken is that of an observer in a reference frame at rest with respect to the solar system barycenter. The clocks on earth beat more slowly than clocks a t rest at infinity in this system by the factor6

I - L = 1 - 1.55 X l o v 8 .

(32)

‘Therefore, the meter is physically longer, so the length of a physical object is numerically smaller by this factor. The mass of the earth canbe used to construct :L quantity having the physical dimensions of a lengt.11, na.me1y G M E I c ’ . However c has a definecl value; this rneans that in T D B coordinates, GME is numerically smaller than in SI units:

and will therefore besubjectto a gravitational redshift; on the other hand in the ECI frame such clocks are moving more slowly than clocks near the equator and are subjecto less second-order Doppler shift. This is diagrammedinFig. 13. Over the ages theearth’s surface has assumed the approximate shaopfea hydrostatic equipotential in the rotating framethe average shape of the ocean’s surface defines the geoid. it is a remarkable fact that on the geoid, there is a very precise cancellation of gravitational frequency shifts and motionalDopplershifts, so that all clocks at rest on the geoid beat at the same rate! Therefore it is possible to construct a network of standard clocks on the earth’s geoid, all beating at the same rate. However, to synchronize these clocks consistently it is necessary to correct for the Sagnac effect, due to the earth’s rotation.
more gravitational redshift

more time dilation

Standard distance fromA to B:
- - r2 r1 (.888 cm) In(r2/rl)

Fig. 13. On the oblate rotating earth’s geoid, changes in gravitation frequency shift areprecisely compensated by second-orderDoppler shifts.

Fig. 12. Effecotf spatiacl urvature ddarisdtanmceeasurcnlents.

So far I have also ignored the possibility that, the

sun, moon, or other planets might contribute to gravi-

on stan-tational

frequencyshift,s. Also, theearth’sorbit is not

perfect,ly circular so omnieght expect a yoe-fyfoect on

the rates of earth-orbiting clocks s0mewha.t analagons

SomReemarkablCeancellations
So far in this discussion I have ignored thefact t,llat theearth is actually a n oblate ellipsoid; clocks near onepole will be closer to the center of the earth

to the correction given in Eq. (25) for GPS clocks. For example, when a satellite is in earth’s shadow its clock shouldbe gravitationally blneshifted ascompared to a satellite-borne clock between thesunandearth. For
such a configuration, the fract(iona1frequency shift be-

13

tweenclocks in the twosatellites, due to the sun, is about three parts in a trillion, which in an hour would cause a 12 ns timing errorto build up. Fortunately we do nothave to worry about this! Thiseffect is cancelled to high precision by other relativistic effects arising because the entire systemof earth plus satellites isin free fall around the sun. By theprinciple of equivalence, we should not be surprised that for a systemin free fall, the gravitationaleffects of the sun arteransformed away. Detailed analysis of this situation is rather delicate; when comparing clocks in the ECI frame-which is falling around the sun-with clocks in the solar system center-of-mass frame, there is disagreement about the meaning of simultaneity in the two frames. Using coordinate timein the ECI framew, ith clocks synchronized by the Einstein procedure (modified by gravitationaleffects), thegravitational effects due t,o other solar system bodies will cancel to high accuracy. The residual gravitational effects are dueto tidal potentials only, and are less than a part in

[4] N . Ashby, “A Tutorial onRelativistic Effects i n the Global Positioning System,” NIST Contract No. 40 RANB9B8112, Final Report, Feb. 1990. Copies of this report are available from the author on request.
[5] JohnBraun, UNAVCO, privatecommunication.
[6] R. W . Hellings, “Relativistic Effects in Ast,onomical TimingMeasurements,” Astron J . 91, 650 (1986).

Conclusion

In this paper, numerous exampleosf relativistic effects which are important for current and future navigation, timing, and communications systemhsave been discussed. Relativistic effects are always systematic, but depend on knowledge of the positions and velocities of the various clocks in the given reference frame. T l m e effects are not noise; they are well-understood, and can be corrected for to a high level of accuracy. As clock stability and accuracycontinues to improve it will become increasingly important for system designers and practitioners tobecome familiar with these effects so they will beaccountedforproperly.Ihope thispaperhelpsin a small way to educate those for whom the mathematical apparatus of general relativity is excessively cumbersome.

References

[l]R. F. C. Vessot, E. M. Mattison, M.W . Levine, and R. L . Walsworth, “Statusof local oscillators for operating ultra-high resolution frequency discriminators as frequency standards,” 241h Annllal PTTI ApplicationsandPlanning Meeting,McLean, VA, December 1-3, 1992.
[a] A . Einstein, The Meaning of Relativity, 3‘d ed.,
Princeton Univ. Press, Princeton, N.J, (1950).

[3] C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W . €1. Freeman & Co., San Francisco, CA (1993), p. 13 ff.

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