IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 43, NO. 4, AUGUST 1994

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Relativity in the Future of Engineering
Neil Ashby

Absti-act-Improvements in clock technology make it possible to develop extremely accurate timing, ranging, navigation, and communications systems. Three relativistic effects, time dilation, the Sagnac effect, and gravitational frequency shifts, must be accounted for in order for modern systems to work properly. These effects are related in a nonmathematical way to fundamental relativity principles: constancy of the speed of light, and the principle of equivalence. Examples of current and future engineering applications are discussed, such as in the Global Positioning System, in time synchronizationsystems,communications, and geodesy.

1. INTRODUCTION

R ELATIVISTIC effects become important in applications requiring very accurate timing, time transfer, or syn-

chronization. Many engineering systems are beginning to rely

on modern atomic clocks which have fractional frequency

stabilities of the order of

or

An excellent

example is the Global Positioning System (GPS), in which

about a dozen relativistic effects must be accounted for in order

for the system to work properly. Atomic clock technology not only provides the basis for the definition of the second as the unit of time, but, in addition, this technology is

expected to improve rapidly in the future. Vessot et al. [l] have summarized potential future performance improvements

in several promising devices including cryogenic H-masers, Cs fountains, and trapped Hg ions: these predictions are

summarized in Fig. 1. Such analyses show there is hope that

fractional frequency stabilities in the range

to

can

be achieved. In this paper however, a conservative fractional

frequency stability figure of

is adopted as a guideline

for judging what relativistic effects might be important in the future.

11. CONSTANCY OF THE SPEED Of LIGHT
Relativity enters metrology in a most fundamental way through the so-called “Second Postulate” of the special theory of relativity, the principle of the constancy of the speed of light, c. This now widely accepted principle states that the speed of light in free space has the same value in all inertial systems, independent of the motion of 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 = 299 792 458 meterslsecond.

(1)

Manuscript received July 26,1993; revised December 6, 1993.Invited paper presented at IEEE Intemational Frequency Control Symposium, June 2-4, 1993, Salt Lake City, UT USA.
The author is with the Department of Physics, University of Colorado,
Boulder, CO 80309 USA.
IEEE Log Number 9402046.

.12

I I ----r I I - I

-18 I

I

I

I

I

I

I

I

o

1

2

3

4

5

6

7

log(Avar8glng tlma [seconds])

Fig. 1. Predicted Allan variance for future frequency standards [I]. (This assumes no systematic effects in Cs and Hg devices.)

In conjunction with the adopted unit of time, this value for

c defines the SI unit of length, the meter. In thinking about

the speed of light, a convenient rule of thumb is that c is

approximately equal to 30 centimeters (cm) per nanosecond

(1 nanosecond = 1 ns =

second). No signal can

transport energy or information at a speed greater than c. To

overcome this limitation, much of the development of present-

day computers has involved increasing component density so

that the distance signals must travel between components is

reduced.

In an inertial frame of reference, the principle of the

constancy of c provides a means for synchronizing 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 the presence of a clock at B.)

Now suppose a signal originates at clock A at time t A . The

time required for the signal to propagate in one direction from

A to B is L l c . The clock at B will then be synchronized with

that at A if the signal arrives at the time t g given by

This procedure is called “Einstein Synchronization,” and clocks distributed at rest in any inertial frame will be presumed to be synchronized by this or an equivalent procedure.
The above discussion of electromagnetic signals ignores quite a few practical difficulties. Signals must have sufficient spectral bandwidth that it is possible to reconstruct welldefined pulses in time. Noise in real clocks and frequency drifts due to environmental factors, etc., are not a concern here. Also being ignored are effects on propagation speed which might

0018-9456/94$04.00 0 1994 IEEE

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I -L

c

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

arise because the signals propagate through a medium rather Fig. 3. Fault location using constancy of c. than through a vacuum.

A . Navigation
Keeping these caveats in mind, the constancy of c leads to the following idealized conception of a navigational system. Referring to Fig. 2, suppose four transmitters, each with its own standard clock, are placed at known locations rj. Assume the clocks are synchronized by the Einstein procedure. There is a receiver at unknown position r carrying a standard clock which has not been synchronized. Let these transmitters rapidly transmit synchronizedpulses which are tagged with the transmitter’s position and time, so that a receiver can determine the time t j and the location rj 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:

It - rjl = c(t - t j ) ; j = 1 , 2 , 3 , 4 ,

(3)

for the unknowns r and t. These equations just express the
principle of the constancy of c in an inertial frame. Clearly a timing error of one nanosecond would lead to an error of about 30 cm in position determination.

-

vt ct

ct vt

Fig. 4. Thought experiment illustrating relativity of simultaneity.

respective ends of the line. A ground survey would give the total length L:

L = L1+ Lz,

(4)

whereas from the constancy of c, the times tl and t z are related to the time t by propagation delay equations:

+ + tl = t Ll/C, tz = t L2/c.

(5)

Solution of only two propagation delay equations, in conjunction with (4), gives the time and position of the fault. To locate the fault to within 30 cm requires synchronization to better than a nanosecond.

B. Event Detection
There is a kind of reciprocity in this situation which can be used for event detection: suppose that instead of transmitters at the locations rj 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 at the
measured times t j .Then by solving four propagation equations
of the form of (3), the position of the event and the time at which it occurs can be determined. If some information about the position of the event is available, it may be possible to locate the event by solving fewer than four propagation delay equations.
C . Fault Location
An example of event detection using only two synchronized clocks, is the problem of determining 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 the line are synchronized from some independent primary reference clock. A fault occurring at distances L1 and LZfrom the respective detectors at the ends of the lines sends out a
signal at time t which is received at times tl and t z at the

111. BREAKDOWONF SIMULTANEITY
The discussion above assumes that the clocks are at rest in some inertial reference frame. Usually, however, clocks are in motion; for example in Fig. 2 the transmitters could be orbiting the earth. Relative motion introduces subtle new effects; perhaps the most profound of these is the breakdown of the Newtonian concept of simultaneity. Events which appear to occur simultaneously in one inertial frame may not appear simultaneous to observers in some other inertial frame, which is moving with respect to the first. This is a direct consequence of the principle of the constancy of c.
In discussing measurements made by observers in two different, relatively moving inertial frames, one always imagines that each observer is equipped with hisher own measuring rods and standard clocks, that the clocks used by observers in one frame are at rest, and that they are synchronized by the Einstein procedure. In each of the inertial frames, any particular electromagnetic signal propagates with speed c.
Consider then as in Fig. 4 two events consisting of two lightning strokes which hit the two ends of a train of length L = 2x simultaneously as seen by observers on the ground. The train is assumed to be moving to the right at speed v relative to the ground. For ease of discussion, the ground is

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called the “rest” frame, and the train is called the “moving” frame. Observers on the ground (in the rest frame) can determine the midpoint between the two lightning strokes, a distance x from either end of the initial position of the train. They will then find that light signals from the two events will propagate along the tracks and collide at the midpoint. This has nothing to do with the motion of the train.
Now look at the sequence of events involving a moving observer, sitting at the midpoint of the moving train. As the train moves forward, this observer moves toward the approaching light emitted from the event at the front of the train, and recedes from the light signal emitted from the event at the back of the train. Therefore the moving observer will encounter light from the front event first, and will have to conclude that the event at the front of the train occurred first. 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.
To analyze this approximately is not difficult. Suppose the zero of time for observers in both the rest and the moving kames is set to occur at the instant the midpoint of the train encounters the signal from the lightning stroke at the front of the train. Primes denote quantities measured by the moving observer. Then to the moving observer, the time t’ of the stroke at the front of the train is
t’= _ _X e
To observers in the rest frame, however, the midpoint of the
+ train and the wavefront are approaching each other at the
relative speed c w, so to first order in U,
(7)
Therefore

Fig. 5. For a sequence of Einstein synchronization processes around the closed circuit on the rotating earth’s surface, the Sagnac effect is proportional to the shaded area, which is the area enclosed by the circuit, projected onto the earth’s equatorial plane.

center of the earth. The moving frame is a reference frame extending over a small portion of the rotating earth’s surface, having velocity v = WT relative to the rest frame, where r is the distance of the clocks from the rotation axis.
Now imagine two clocks fixed a small east-west distance 2 apart on the surface of the earth. Viewed from the nonrotating frame they will be moving with approximately equal speeds v = wr. If a clock synchronization process involving electromagnetic signals were carried out by two earth-fixed observers using Einstein synchronization in the moving frame, then the two clocks would not be synchronous when viewed from the nonrotating frame. The magnitude of the discrepancy is vx/c2= wrx/c2 = (2w/c2)(rx/2)I.f this synchronization process is performed successively all the way around the equator, then effectively the distance is x = 2xr, and the time discrepancy is thus

2w

At = - x T?,

(9)

C2

where 7rr2is the area enclosed by the path followed during the synchronization process. For example, synchronization around the earth’s equator involves a discrepancy

The term -vx/e2 is a relativistic correction for breakdown of simultaneity. The effect is proportional to the relative velocity and proportional to the distance x.
Putting in some numbers, suppose v = 1000 kmhour (typical for jet aircraft) and x = 3500 km. Then the correction is 108 ns. The negative sign in (8) means that of two events simultaneous in the rest frame, to the moving observer the event farther out in front, at the more positive x,occurs earlier.
IV. SAGNACEFFECT
The above discussion of the breakdown of simultaneity can be used to understand some peculiar physics in a slowly rotating coordinate frame. The prime engineering application is to time transfer and synchronization on the surface of the rotating earth. For numerical examples, therefore, the angular
velocity will be that of the earth’s rotation, w = 7.29 x
rad/s, and the radius will be the earth’s equatorial radius,
R = 6.378 x lo6 meters.
In this case the rest frame is a local nonrotating frame, with axes pointing toward the “fixed” stars, but with origin at the

upon arriving back at the starting point. This effect is known as the Sagnac effect, If the synchroniza-
tion path were westward around the earth rather than eastward, then the discrepancy would be of opposite sign. This means that Einstein synchronization 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 necessarily synchronized with C. In order to avoid difficulties with such nontransitivity it is best to adopt time in the nonrotating frame as the measure of time in the rotating frame. Thus one discards Einstein synchronization in the rotating frame.
To put it another way, if Einstein synchronization is used in the earth-fixed rotating frame, then it is necessary to apply additional “Sagnac corrections” to the readings of clocks on the rotating earth, in order to obtain an internally consistent “coordinate time” on the earth’s surface.
This is illustrated in Fig. 5 , which is a sketch of a flattened rotating earth. For a sequence of synchronization processes forming a closed circuit on the rotating earth, upon projecting

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 43. NO. 4, AUGUST 1994

I ro + V I - rs I = cr

Fig. 7. Propagation of a signal from a satellite at position rs to a receiver which moves a distance ut during the propagation time. The Sagnac effect will be automatically included if m i v e r motion due to earth rotation during signal propagation is accounted for.

Fig. 6. Distribution of synchronization for a communications network.
the path onto the equatorial plane of the earth one can determine the projected area A E .The Consultative Committee for the Definition of the Second and the International Radio Consultative Committee have agreed that, in order to obtain consistently synchronized clocks on the earth’s surface at the subnanosecond level, the correction term to be applied is of the form
At = - 2w x A E , C2
where AE 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 the electromagnetic signal pulse. The area AE 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 frame of the earth, then At must be subtracted from the measured time difference (east clock minus west clock) in order to synchronize the clocks so they will measure coordinate time on the rotating earth. They will effectively measure time in the local nonrotating frame attached to the earth’s center.
Lack of transitivity in synchronization has implications for devices which rely on accurate synchronization. Suppose a communications network distributes synchronization through a series of nodes, along two different paths, to the ends of a communication link as in Fig. 6. If the area enclosed by the path, projected onto the earth’s equatorial plane, is not zero, then problems with inconsistent synchronization can arise. For example, suppose one synchronization link goes from San Francisco directly to New York, while a second link goes from San Francisco to Miami and then to New York. The discrepancy in synchronization between these two paths due to the Sagnac effect is about 11 ns. While this is not significant if the signal is 60 Hz as in a power grid, in an optical communicationsnetwork operating at 1015 Hz the discrepancy
amounts to lo7 cycles of oscillation. Depending on the design
of the system this may become significant in the future.

Furthermore, if the trouble is taken to incorporate hardware delays to compensate for the Sagnac effect while sending in one direction, then if the communications were sent back the other way over the same link with the same delay, the effect will become twice as big. The effect is asymmetric. The same effect will occur in optical fiber communications networks where the speed of signal propagation may be significantly less than e. In the rotating reference frame the Sagnac effect is a property of space and time, not dependent on signal propagation speed. The S’agnac effect is the basis for laser gyroscopes.
An equivalent way of looking at this phenomenon is diagrammed in Fig. 7, which shows a signal transmitted from a satellite to a ground-based receiver. From the point of view of the nonrotating frame, the signal goes in a straight line with speed c, from the initial transmitter position rs to the final receiver position. If in this frame one accounts for the motion of the receiver during the propagation of the signal, then the Sagnac effect will be automatically accounted for. Thus if the initial position of the the receiver is ro, the velocity of the receiver is U , and the signal propagation time is t , constancy of e requires

lro +ut - r s ( = (et).

(12)

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

V. TIME DILATION
The previous section discusked two effects which are of first order in the velocity: the breakdown of simultaneity, and the Sagnac effect. This section discusses another famous effect, time dilation, which is of second order in the velocity. Imagine two inertial frames, a “rest” frame or laboratory frame, and a moving frame. A clock in the moving frame beats more slowly than clocks in the rest frame to which it is successively compared. The following thought experiment should readily convince the reader that this is a consequence of the principle of the constancy of e. 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

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Thought exwriment v i m d in ‘moving‘ frame.

The lower part of Fig. 8 actually gives the right answer, for by the principle of the constancy of c, the vertical component
of the light velocity in the rest frame is just d p .Thus
for observers in the rest frame, the time t required for the light to reach the upper end of the rod is just

-

t=L / d n ’ .

(14)

So the relationship between t’ and t obtained by eliminating L and L’ from (13) and (14), and using the equality L = L’, is

t’ = d-t.

(15)

Thought experiment viewed i n ’rest‘ frame.
t y’

Usually the ratio v/c is small, so the square root can be expanded, giving approximately
t‘ x (1 - 1v2
The fractional slowing is given by the correction w2/2c2 in the above equation. This correction is also commonly called the second-order Doppler shift, or transverse Doppler shift.
Some examples of the size of this effect are as follows. For a clock at rest on the earth’s equator, and viewed from the nonrotating frame,

Fig. 8. Thought experiment showing that “moving”clocks beat more slowly than clocks that remain “at rest.”

- _ - O2 x -1.2 x 10-12;
2 c=
this would accumulate to about 104 ns in one day. For a clock in a satellite orbiting the earth at 100 km altitude,

direction of relative motion is parallel to the z, 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 end to the other. The situation is diagrammed in Fig. 8. To simplify the discussion one assumes that the light starts out at the instant the origins of the two reference frames pass by each other.
The upper part of Fig. 8 shows the situation from the point
of view of observers in the moving frame. The time t’ required
for light to travel along the rod is simply

t‘ = L’/c.

(13)

--1 -0 2 x -3.4 x 10-l0.
2 c2 For a clock in a GPS satellite,

--12-12 x -8.34 x
2 c2

Keeping in mind that in the future the fractional frequency

stability of orbiting clocks may approach a part in

these

are very large effects. Even for clocks of frequency stability

1 x

as in the GPS Block II satellites, the second-order

Doppler shift for an earth-fixed clock is significant.

The clock faces on the upper part of Fig. 8 indicate time at the beginning and end of the experiment.
The lower part of Fig. 8 shows the experiment from the point of view of observers in the rest frame. Breakdown of simultaneity would create difficulties for measurements of lengths oriented parallel to the relative velocity. But since this rod is oriented perpendicular to the relative velocity, by symmetry it is not possible for the rod to appear changed in length. So this rod has length L = L’ as it moves through the rest frame. The rod moves to the right with speed v and the light travels along the rod, so there has to be a horizontal component of velocity of the light pulse equal to U. The vertical component of the velocity of the light certainly has to be less than c; therefore the time required for the light to reach the upper end of the rod certainly has to be. greater than L/c. This argument shows qualitatively that the clocks in the moving frame will beat more slowly than the sequence of clocks with which they are compared in the rest frame.

VI. GRAVITATIONAFLREQUENCY S H a S
The Sagnac effect and the second-order Doppler shift are effects which can be understood on the basis of the Special Theory of Relativity. A third effect, the gravitational frequency shift, occurs when signals are sent from one location to another having a different gravitational potential. The effect can be understood in an elementary way using the fundamental assumption of the General Theory of Relativity-Einstein’s Principle of Equivalence [2].
A. The Principle of Equivalence
Einstein’s Equivalence Principle states that over a small region of space 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 turning a comer in a vehicle has the same physical effects as a real gravitational field. An immediate consequence

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space, enter the side of the accelerated laboratory [near the top, in Fig. 9(b)]. The observer in this laboratory is 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 nonaccelerated laboratory on earth, so 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.

B. Time Delay

_ _ . - - - - - - - - - - - _ _ _ _- __-_--_
f ig TLd
Fig. 9. (a) Laboratory near the earth’s surface where the acceleration due to gravity is g. All objects fall with equal accelerations. (b) By the Equivalence Principle, experiments performed in an accelerated lab in free space have the same outcomes.
of the Equivalence Principle is that gravitational fields can be reduced to zero by transforming to a freely falling reference frame. The fictitious gravitational field 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 as experiments performed in a laboratory in free space which is accelerated in the opposite direction with acceleration a = -9. In Fig. 9(a) 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 and gravitational mass, a subject we shall not go into here [3].) In Fig. 9(b), a similar experiment is performed in a laboratory in free space which is being pulled upward with acceleration g. In this case a nonaccelerated observer sees that the apple and the lead ball have no forces exerted on them. They remain at rest with respect to each other and the laboratory is accelerated past them, whereas the observer in the accelerated frame sees the objects “fall” downward with identical accelerations g.
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

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 slowed down slightly with respect to the

portions of the wavefront farther away from the source. The

wavefront then tilts over and the beam is thereby deflected.

This means that of two beams of light passing near a massive

source, the one which passes closer will take longer to pass

by. Thus not only is light deflected, it is slowed down by a

gravitational field.

Time delays of signals in the neighborhood of the earth

can be a few tenths of a nanosecond. Such time delays are

determined by a complicated logarithmic function of signal

path parameters, times the quantity ~ G M E / c ~wh, ere G is

the Newtonian gravitational constant and ME the earth’s mass.

For earth GME/c’ = 0.443 cm, so the scale of such effects

near earth is

-~-G M E!z-1.77cm = 0.06 ns.

c e2

C

(20)

This is not enough to worry about at the present time but

could be significant in the future: a timing error of 0.1 ns in a

navigational system would give rise to a 3 cm error in position.

C . Gravitational Frequency Shifts
It follows from the Equivalence Principle that an electromagnetic signal passing upwards in a gravitational field will be redshifted. In Fig. 10 is a sketch of an experiment performed in an equivalent laboratory, a rocket having acceleration g upwards in free space. Imagine the situation from the point of view of a nonaccelerated frame. Suppose a signal leaves the accelerated transmitter at the initial instant, when the transmitter velocity is still zero. The signal travels upwards a distance L , and is received by the accelerated receiver. The time required for the signal to propagate from transmitter to receiver is

t = L/c.

(21)

During this time, the receiver has picked up a velocity

v = gt = gL/c.

(22)

To the receiver, the signal appears to come from a receding

source and is Doppler shifted. In a first approximation the

fractional frequency shift is A f l f = - v / c ; therefore the

fractional frequency shift in the “effective” gravitational field

g is

_Af -- -‘---sL

f

c c2 .

(23)

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Fig. 10. A signal traveling upwards in a gravitationalfield is shifted towards lower frequencies.

The quantity gL can be interpreted as the change in gravitational potential, Aq5, of the signal.
At the surface of the earth,

g/c2 = 1.09 x

per km,

(24)

which is very important for today’s time standards. For example, a signal of definite frequency originating at mean sea level would be redshifted by 1.79 parts in 1013 upon arriving at the altitude of the NIST frequency standards laboratory in Boulder, CO. Consequently the contribution of the NIST time standard to Universal Coordinated Time (UTC) requires that a paper correction of -15.5 ns/day be applied to the NIST clock before it can be compared to time standards at mean sea level.
For a clock in a satellite orbiting the earth at 100 km altitude compared to one on the geoid,

3= 1.08 x lo-’’ C2
Not only will these effects be large in the future when clock stabilities approach a part in 1015 or better, it will be necessary to compute them quite accurately. This will mean, for example, that there will be a need for improved accuracy of the ephemerides of clock-carrying satellites.

VII. THE GLOBALPOSITIONING SYSTEM
The best existing example of an engineering system in which relativity plays an essential role is the GPS. This consists of a constellation of perhaps 24 earth-orbiting satellites carrying atomic clocks which synchronously transmit navigation signals, much as described in the discussion of Fig. 2. The satellite orbits are at approximately 20200 km altitude. Therefore clocks in the satellites will be significantly blueshifted in rate, compared to clocks on the ground. The

second-order Doppler shift of such clocks was given in (20).

Also, if the orbits are not perfectly circular (and they almost

never are), the clocks’ yo-yo motions towards and away

from the earth will generate additional periodic gravitational

frequency shifts, and periodic second-order Doppler shifts.

Further, observers on the ground who wish to make use of

the navigational signals will experience the Sagnac effect due

to the earth’s rotation.

A complete discussion of all the significant relativistic ef-

fects, with analytical expressions for the necessary corrections,

can be found elsewhere [4], [5]. Here rough magnitudes of

some of the corrections are given.

First, consider ground-based clocks in receivers which are

at rest on the earth’s surface. 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 the earth’s mass; the fractional frequency shift is
about -7 x lo-’’. The earth’s oblateness is associated with

a quadrupole contribution to the gravitational potential which

cannot be neglected; the fractional frequency shift is about

-4 x

If earth-based clocks are not on the geoid they

suffer a gravitational frequency shift [see Eqs. (23) and (24)].

Finally there is a second-order Doppler shift due to the earth’s

rotation; the fractional frequency shift from this effect can be

as large as -1.2 x

[see (17)].

For GPS receivers in motion relative to the earth’s surface,

there is an additional second-order Doppler shift due to their

speed with respect to the ground; this can be of the order
of lo-’’ depending on the ground speed. Also, the Sagnac

effect, or motion of the receiver during propagation of the

navigation signal, may give rise to effects of several hundred

nanoseconds magnitude.

The transmitters themselves suffer a frequency shift due to

the earth’s gravitational potential, and a second-order Doppler

shift due to orbital motion; these effects are several parts in

10”. The additional frequency shifts due to orbital eccen-

tricities can be tens of nanoseconds; for a GPS satellite of

eccentricity e = 0.01, the maximum size of the effect is about

23 ns.

Signals propagating from transmitter to receivers are subject

to the Sagnac effect, involving relativistic corrections of up

to several hundred nanoseconds. Relativistic time delay of

signals or relativistic deflection of signals is a few tenths of a

nanosecond and is currently neglected in the GPS.

VIU. THE! CONCEPT 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 earth is inconsistent if the Einstein procedure is used. How is it possible to synchronize a network of distributed, rapidly moving clocks so that a navigational system will work as conceived in Fig. 2? What has been found to work extremely well in the GPS is to use the time in the hypothetical underlying local inertial frame, with origin attached to the earth but not spinning, as the measure of time. This time is not time

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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 in the system, is an example of coordinate time.
Thus, imagine an underlying nonrotating frame, or local inertial frame, unattached to the spinning earth, with its origin at the center of the earth. This frame is sometimes called the “Earth-Centered Inertial” frame, or ECI frame. In this frame, introduce a fictitious set of standard clocks available anywhere, all synchronized via the Einstein procedure, and running at agreed upon rates such that synchronization is maintained. Gravitational effects are incorporated by choosing one clock as a Master Clock and requiring that all other clocks be synchronized to the Master Clock by simple transmission of signals without any frequency 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 the earth. To each one of these standard clocks apply a set of systematic corrections, so that at each instant the standard clock as corrected agrees with the time on a fictitious standard clock, at rest in the ECI frame, with which it instantaneously coincides. The set of corrected standard clocks will therefore be keeping coordinate time. In other words, coordinate time is equivalent
to time measured by standard clocks in the ECI frame [4].
Time measured this way on coordinate clocks has two highly desirable properties. First, synchronization is reflexive: if A is synchronized with B, then B is synchronized with A. Second, synchronization is transitive: if A is synchronized with B, and B is synchronized with C , then A is synchronized with C. Intemal inconsistencies are thereby eliminated.
GPS time is an example of coordinate time. 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 0.4465 parts per billion, or about 39000 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 MHz reference frequency of satellite clocks is adjusted downward to 10.229 999 995 43 MHz. The adjustment is accomplished on the ground before the satellites are launched.
Also, if the orbit of the satellite clock is not perfectly circular, there will be additional gravitational and motional rate shifts which have to be accounted for. The additional correction required to achieve synchronization when the orbit eccentricity e is not zero is given by the expression [5]

of improved earthquake prediction capability; the potential impact on construction codes, building restrictions, etc., is considerable. In recent years the GPS has been successfully used to measure very long baselines between fiducial points on different crustal plates by a method described as “carrier phase double difference.” Two receivers are placed at the ends of a baseline of interest, and signals from two satellites are then “double differenced” in a manner to be described below. Differencing removes the need for some systematic corrections but as will be seen, there are residual relativistic effects which must be accounted for.
Referring to Fig. 7 and the propagation time t given in (12), let the satellite position at the instant of transmission t s be denoted by r S and the receiver or observer position at the same instant be denoted by 20. Let the coordinate time of arrival of the signal at the observer be denoted by to. Then solving (12) for the propagation time gives [4], [5]

where w is the earth’s angular velocity of rotation, K represents a possible time offset or error of the receiver’s clock, and the receiver velocity U = w x ro has been inserted. The last term is the Sagnac correction. The rate adjustment applied to satellite clocks means that the quantity ts will have the correct scale when received on the geoid. There is a further correction, from the noncircular motion of the satellite, given by (26). Thus when all relativistic effects are incorporated,

+ 4.428 x 10-lOe& sinE.

(28)

Let subscripts 1 and 2 denote the two different satellites and the two different observers. Suppose there are receivers at two different positions which receive a time signal originating from a single satellite. Upon taking the first difference of the arrival times, it is immediately seen that the eccentricity term cancels out, leaving the expression:

At = +4.428 x 10-lOe& sinE s,

(26)

where a is the semimajor axis in meters and E is the eccentric anomaly. Usually the software in the user’s receiver makes this correction.
Ix. APPLICATIONOF SATELLITE NAVIGATIOINN GEODESY The motivation to obtain accurate measurements of movements of the earth’s crustal plates is intense. Knowledge of these very slow motions is crucial to the development

The Sagnac correction is still needed. The time of transmission of the signal, ts, cancels out which lessens the impact of selective availability.
Now the same set of measurements is taken, at essentially the same time, using signals from a second satellite. Writing another equation similar to (29) for the second set of measurements 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

ASHBY RELATIVITY IN THE FUTURE OF ENGINEERING

513

7.5

d5
c

Y
J 2.5

e

VI

W

O

rl
P -2.5

E N 4orspace curvature:

-5

0

500

1000 1500 2000 2500

Baseline Distance in h.

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

corrections due to the rotation of the earth:

(to2- tolls2 - (to2 - t0l)Sl

-- Its2 - r02l - ITS2 - roll - lrs1 - r 0 2 l

p+ + lrSl-cerolI

- . $‘S2
2cw2

1 C
- rs1) x (r02 - ro1) .

(30)

The Sagnac correction is still 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. 11 are plotted some baseline measurement data taken

repeatedly on baselines in the Southwest Pacific, of lengths up

to 2500 km [6]. 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.

x. IMPACT ON FUNDAMENTAL METROLOGY
The previous sections were devoted almost exclusively to the impact of relativity on the measurements of time, with distance derived by multiplying by c. At the level of a centimeter or less, there are additional effects on the measurement of position which arise because space in the neighborhood of a massive body is distorted. Consider as in Fig. 12 an attempt to establish a system of spatial coordinates in the neighborhood of the earth, against which to measure the positions of the earth’s crustal plates. Suppose that we wish to measure angles in the usual Euclidean way, so that a circle of coordinate radius T centered on the earth would have a circumference 27rr, measured with standard rods or with the help of the constancy of c. Two such circles, of coordinate radii T I and r2, are indicated in Fig. 12. The standard distance from the inner circle straight out along a radius to the outer circle is not 7-2 - T I ; instead one finds the standard distance d is [7]

The correction due to space curvature is of the order of 1 cm.

Stmdard distance from A to B Fig. 12. Effect of spatial curvature on standard distance measurements.

More generally, the fact that c has a defined numerical value 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 at rest at infinity in this system by the factor [8]

1 - L = 1 - 1.55 x

(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 can be used to construct a quantity having the physical dimensions of a length, namely G M E / c ~H.owever, c has a defined value; this means that in TDB coordinates, G M E is numerically smaller than in SI units:

XI. SOMEREMARKABCALNECELLATIONS
The earth is actually an oblate ellipsoid; clocks near one pole will be closer to the center of the earth than clocks on the equator, and will therefore be subject to a gravitational redshift; on the other hand in the ECI frame such clocks are moving more slowly than clocks near the equator and are subject to less second-order Doppler shift. This is diagrammed in Fig. 13. Over the ages the earth’s surface has assumed the approximate shape of a hydrostatic equipotential in the rotating frame: the 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 motional Doppler shifts, 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.
Also, what about the possibility that the sun, moon, or other planets might contribute to gravitational frequency shifts? The earth’s orbit is not perfectly circular so one might expect a yo-yo effect on the rates of earth-orbiting clocks somewhat analagous to the correction given in (25) for GPS clocks.

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more gnvitstioml redshift
I

reference frame. These effects are not noise; they are wellunderstood, and can be corrected for to a high level of accuracy. As clock stability and accuracy continues to improve it will become increasingly important for system designers and practitioners to become familiar with these effects so they will be accounted for properly. Hopefully this paper will help in a small way to educate those for whom the mathematical apparatus of general relativity is excessively cumbersome.

\ more timekiation

REFERENCES

Fig. 13. On the oblate rotating earth’s geoid, changes in gravitation frequency shift are precisely compensated by second-order Doppler shifts.
For example, when a satellite is in the earth’s shadow its clock should be gravitationally blueshifted as compared to a satellite-borne clock between the sun and earth. For such a configuration, the fractional frequency shift between clocks in the two satellites, due to the sun, is about three parts in a trillion, which in an hour would cause a 12 ns timing error to build up. Fortunately we do not have to worry about this! This effect is cancelled to high precision by other relativistic effects arising because the entire system of earth plus satellites is in free fall around the sun. By the principle of equivalence, we should not be surprised that for a system in free fall, the gravitational effects of the sun are transformed 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 time in the ECI frame, with clocks synchronized by the Einstein procedure (modified by gravitational effects), the gravitational effects due to other solar system bodies will cancel to high accuracy. The residual gravitational effects are due to tidal potentials only, and are less than one part in
XII. CONCLUSIONS
In this paper, numerous examples of relativistic effects which are important for current and future navigation, timing, and communications systems have been discussed. Relativistic effects are always systematic, but depend on knowledge of the positions and velocities of the various clocks in the given

R. F. C. Vessot, E. M. Mattison, M. W. Levine, and R. L. Walsworth,
“Status of local oscillators for operating ultra-high resolution frequency discriminators as frequency standards,” presented at the 24th Annu. PTTI Applications and Planning Meeting, McLean, VA, Dec.r 1-3, 1992. A. Einstein, The Meaning of Relativity, 3‘d ed. Princeton, NJ: Princeton Univ. Press, 1950. C. W. Misner, K. S. ”home, and J. A. Wheeler, Gravitation. San Francisco, CA: Freeman, 1993. N. Ashby and D. W. Allan, “Practical implications of relativity for a global coordinate time scale,” Radio Sci., vol. 14, p. 649, 1979.
N. Ashby, “A tutorial on relativistic effects in the global positioning
system,” NIST Contract 40 RANB9B8112, Final Rep., Feb. 1990. Copies of this report are available from the author on request.
J. Braun, UNAVCO, private communication, May 1993. G. C. McVittie, General Relativity and Cosmology. Urbana, I L University of Illinois Press, 1965, p. 88; see also C. Mgller, The Theory of Relativity, 2nd ed. Oxford, England: Clarendon, 1972, p. 440.
R. W. Hellings, “Relativistic effects in astonomical timing measurements,”Astron. J., vol. 91, p. 650, 1986.
Neil Ashby was bom in Dalhart, TX, on March
5 , 1934. He received the B.A. degree (Summa Cum Laude) in physics from the University of Colorado,Boulder, in 1955, and the M.S. degree and Ph.D. degree in theoretical physics from Harvard University, Cambridge, MA, in 1956 and 1961, respectively.
He spent a year in Europe as a postdoctoral student,then joined the faculty of the Department of Physics at the University of Colorado in 1962. He has been a Professor of Physics at the University of Colorado since 1970. He has served as a consultant to the Time and Frequency Division of NIST since 1975, working on relativistic effects on clocks and global time synchronization. He was Chairman of Boulder Faculty Assembly during 1980-1982 and served as Chairman of the Department of Physics, University of Colorado at Boulder, during 1984-1988. Dr. Ashby has been a member of the Intemational Committee on General Relativity and Gravitation since 1989. He serves on several intemational working groups on relativistic effects in geodesy and in metrology.