r

9*" \

NBS TECHNICAL NOTE 659

U. S. DEPARTMENT OF COMMERCE /National Bureau of Standards

AN EARTH-BASED

COORDINATE CLOCK NETWORK
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

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3 Part of the Center for Radiation Research.

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* Part of the Center for Building Technology.
HAL BUREAU OF STANDARD;

i 1 9 \m

;;:

;.. AN EARTH-BASED

' COORDINATE CLOCK NETWORK

15

Neil Ashby

Professor of Physics and Astrophysics

University of Colorado

Boulder, Colorado 80302

* ' i

j^(ikMA£a§ JAj^e )A0. (oS^

Work performed for the Time and Frequency Division

under contract #CST-8335

/

i^

U.S. DEPARTMENT OF COMMERCE,

NATIONAL BUREAU OF STANDARDS. Richard W Roberts Director Issued April 1975
National Bureau of Standards Technical Note 659

Nat. Bur. Stand. (U.S.), Tech Note 659, 34 pages (April 1975)

CODEN: NBTNAE

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. 013.46:659) $ .85
Contents

Page

1. Introduction 1

2

.

Summary of Results 3

A; A coordinate clock system measuring global

coordinate time 3

B) A coordinate clock system with a discontinuity 4

C) Investigation of other effects 4

3. Synchronization in a Static Gravitational Field 5

A) Synchronization (Initialization) using light signals 6

B) Slow transport of standard clocks 8

4. Effect of Gravitational Fields of the Sun and Moon 11

5. Clock Synchronization on the Rotating Earth 15

A) Synchronization of coordinate clocks by

slow transport of standard clocks 19

B) Inf initesimally slow transport on the geoid 21

C) Global coordinate clock synchronization using

light signals 21

D) "Hafele-Keating" discontinuity 23

E) Use of satellites to construct a coordinate

time network 24

6. Miscellaneous Effects 25

A) Mass of the atmosphere 25

B) Effect of flattening of the earth on discontinuity 26

C) Retardation 26

References and Footnotes 28

Appendix—Latitude and Polar Angle in Ellipsoidal Geometry

An Earth-Based Coordinate Clock Network

Neil Ashby

This paper investigates some of the possible operational procedures for

synchronizing clocks at fixed sites spread around on the earth's surface,

to within a 1 nanosecond level of accuracy. Since a common synchronization procedure is by transport of standard clocks in commercial jet airline flights, and most of the effects we shall discuss are fractional

corrections to the elapsed time, as a criterion of whether an effect is

significant at the 1 nanosecond level we take for comparison purposes

an elapsed time T =10 hours. This is a typical time for an intercontinental airplane flight. Analysis of a number of effects which might

affect clock synchronization is carried out within the framework of

general relativity. These effects include the gravitational fields of

the earth, sun, and moon, and orbital motion, rotation, and flattening

of the earth. It is shown that the only significant effects are due

to the gravitational field and rotation of the earth, and motion of

the transported clocks. Operational procedures for construction of a synchronized coordinate clock network using light signals and trans- ported standard clocks are discussed and compared.

Key words: Clocks; coordinate time; general relativity; time

Introduction

The purpose of this paper is to investigate some of the possible operational procedures

for synchronizing clocks at various fixed sites spread around on the earth's surface. There

are a number of effects which could conceivably play a role in the synchronization of

clocks—among these are the gravitational fields of the earth, moon, and sun, and the orbital motions, rotation, and flattening of the earth. Since most of the effects we shall

discuss turn out to be fractional corrections to elapsed time (that is, the corrections

to the time are proportional to the time) , as a criterion of whether an effect is significant at the level of 1 nanosecond (1 ns) , we shall take for comparison purposes an elapsed time T = 10 hours, which corresponds to a typical time for an intercontinental

airplane flight. This "comparison flight" will, where appropriate, be assumed to occur

at an altitude of h = 12,000 meters and at a speed of 450 meter /sec. The reason for

selection of these parameters is that a fairly common synchronization procedure will be

by transport of standard clocks in commercial jet airline flights pL]

The analysis will be carried out within the framework of the general theory of relativity; we shall not discuss any modification of procedure which might be necessary in some other gravitational theory such as that due to Brans and Dicke [2 J

.
The basic assumptions and results of the general theory of relativity, which

we shall repeatedly use, are as follows (we use the notation and sign conventions of

Weber [3]):

1) There exists a metric tensor g (U,v = 1, 2, 3) such that the space-time

interval ds defined by

-ds

2 = g dxMdxV

(1)

is invariant with respect to arbitrary coordinate transformations.

2) Propagation of ligh

by the vanishing of ds:

2) Propagation of light rays along the path element dx is described

= g dxydx . (2)

3) The proper time elapsed on a standard clock transported along the space-time

path element dx is

i ds = i Jc c Jj —a VA v (3)

5 dx dx . 3 yv

4) Of particular significance to this investigation is the interpretation of

the coordinate x , which is a global coordinate time having the units of

length. It is assumed that at each point of space it is possible to equip an observer with a clock (a coordinate clock) which can be used to measure the

coordinate time of any event occurring at that point. The importance of

coordinate time may be appreciated by considering, for example, the phenomenon

of the redshift of a clock in a static gravitational field. Standard clocks

(e.g., atomic clocks) will have different rates depending on the gravitational

potential at the position of the clock. In contrast, a coordinate clock will

always have the same rate independent of position in the gravitational field.

5) The metric coefficients g may be calculated from the field equations

of general relativity. Furthermore, because in the vicinity of the earth's

surface all gravitational fields are weak, it will be sufficient to work in

the linearized approximation to general relativity in which

gyv = V + V C4)

with n = (-1, 1, 1, 1) the metric tensor of flat space and

h « 1

so that squares and products of h are negligible.
2. Summary of Results

We have investigated two principal schemes for obtaining a worldwide network of synchronized coordinate clocks. These are: A) a coordinate clock network measuring the global

coordinate time x , and B) a coordinate clock network wd Keating" discontinuity) along some line between the poles.coordinate time x , and B) a coordinate clock network with a discontinuity (the "HafeleA) A coordinate clook system measuring global coordinate time x

In this scheme, it is assumed that the metric tensor is known theoretically in terms

of the earth's mass and rotation. One may then, with some selected point on the geoid (the

geoid is the equipotential surface at mean sea level) as reference, transport standard

clocks from point to point on the earth's surface. The elapsed global coordinate time Ax during the process may be calculated from the equation

oath L

c 2c c J Ax =path

where ds is the elapsed proper time as measured on the transported clock; g is the

acceleration of gravity; v is the ground velocity of the clock having an eastward

component v„; h is the altitude above the geoid; oo is the rotational angular velocity

E of the earth; a.. is the earth's equatorial radius, and <j) is the geographical latitude.

The standard clock readings must thus be corrected for redshift, ground speed, and earth

rotation before using them to measure coordinate time.

Note that for a clock at rest on the earth, Equation (5) reduces to

(3)

A

. Ax = As 1 -^2 c

(6)

so that a standard clock at rest on the earth's surface must have its rate corrected for

redshift before being used to measure elapsed coordinate time x at that point.

Light signals may be used to perform an equivalent coordinate clock synchronization

by using the following formula for the coordinate time elapsed during propagation which

involves a correction due to rotation of the earth:

Ax = I da f<

path

wa, c„ cos

1 + - 1 E

2 c

(7)

In the above expression, da is the element of proper distance along the path of the light

ray, assumed to lie within about 12,000 meters of the geoid, and c is the eastward

component of the light velocity. The coordinate clock network obtained by this scheme has

the following features and disadvantages:

1) Use of transported clocks and light signals as corrected by Equations (5) and (7)

will give a mutually consistent network of coordinate clocks.

2) The synchronization procedure gives path-independent results with respect to either transport of clocks or use of light signals.
3) The procedure has the property of transitivity: if clocks A and B are synchronized, then A and C will be synchronized.

4) The coordinate clocks directly measure the global coordinate time x .

5) There is no discontinuity in synchronization.

6) A disadvantage is that corrections must be applied to standard clock readings due

to redshift (whether the clock is at rest or is transported), ground velocity, and

earth rotation.

Equation (5) may also be applied to standard clocks while in motion—for example, while

being transported in aircraft.

B) A coordinate clock system with a discontinuity

This scheme has been mentioned by implication in a paper by Schlegel [4] . Here

one transports clocks or uses light signals without applying corrections for the

earth's rotation. Thus Equation (5) for standard clock transport becomes

<- /*.[] I

is V-f^i]

- (8)

path

And for light propagation, Equation (7) becomes

"*- /

da. (9)

path

This scheme has the following features and advantages:

1) There must be, due to the earth's rotation, a discontinuity in synchronization

along some line between the poles given by [51

2 2 2 2

27Toja 1 cos cj>/c = 207.4 cos <f> ns. (10)

The existence of this discontinuity somewhat cancels the advantage of not having

to correct in Equations (8) and (9) for the earth's rotation.

2) The use of clocks or light signals will give a mutually consistent network

of synchronized coordinate clocks as long as the discontinuity is not crossed

and transport is along a parallel of latitude.

3) The synchronization procedure is in general path-dependent.

4) The procedure is in general non-transitive.

5) The coordinate clocks do not measure the global coordinate time x of

relativity theory.

6) Calculated corrections must be applied, for ground velocity and redshift,

to the readings of transported clocks.

C) Investigation of other effects

Quite a number of effects which influence these synchronization procedures

were investigated and found to be insignificant at the 1 ns level. These effects

will be discussed in detail later, but it is of interest to mention some of them

here. They include variations of the gravitational field strength with latitude and

4
altitude, gravitational anomalies, flattening of the earth due to rotation, mass of

the atmosphere, gravitational fields due to the sun and moon and orbital motion of the

earth. These latter effects are individually of order 10 ns during a typical flight but

cancel very precisely.

3. Synchronization in a Static Gravitational Fie,ld

To provide a foundation for later discussion, in this section we shall describe

the operational procedure by which one could set up a series of coordinate clocks,

measuring the global coordinate time x , at rest in space. This procedure is

worth some discussion because it is more usual in theoretical work to assume the

coordinate clocks exist and then to imagine using them to observe and to compare what

happens to standard clocks and other devices.

The real situation is different. We actually have available not coordinate clocks

but standard clocks whose rate depends on both position and velocity. Furthermore,

we know to high precision the positions and motion of the earth, sun, and moon which

are the only significant sources of gravitational fields in the neighborhood of the

earth's surface. We may, therefore, regard g as known theoretically and the elapsed

time on a moving standard clock as experimentally observable. Equation (3) then allows

us to compute the elapsed coordinate time at the position of a transported standard

clock and hence to set up a coordinate clock at that position.

We now consider this process in detail. We stress the assumption that there

exists a global coordinate time x . We shall also assume the metric tensor is static,

-^-0, (11) Bx°

and time orthogonal,

80k

=

°' (k = 1 ' 2 ' 3) - (12 >

2 We work entirely within the framework of general relativity, in which the invariant ds

is now given by

-ds

2 = g 00

(dx°)

2 + gk£

dxk

dx

£

. (13)

where repeated latin indices are summed from 1 to 3. The path of a light ray is

characterized by

ds

2 = (14)

and the distance between two events as measured with standard measuring rods is given by

da = (gk£dx

kdxV/2

. (15)
From an operational viewpoint, it is possible at any point of space to place a standard clock at rest. One way of achieving this (at least in principle) has been

described as follows T61 : Observers using radar arrange to move along the coordinate

world lines x = const., (k = 1, 2, 3). They do this by adjusting their velocities until each finds that the radar echoes from his neighbors require the same round-trip time at each repetition. Equivalently, each returning echo must show

zero Doppler shift. Let As be the invariant proper time between beats of a standard clock placed at

rest. This will be the same number no matter where the clock is placed. Then since

the clock is at rest, the coordinate time Ax between beats is given by

Ax "*// '00

' (16)

This result can be used to determine the rate of a coordinate clock, which measures

the coordinate time x , placed at the position of the standard clock. In this

expression we may imagine that gnn

is a function of position determined from general

relativity in terms of the distribution of mass. Thus since standard clocks and

numerical values of g„_ are available, Equation (16) provides an operational procedure for setting the rates of coordinate clocks.

The next problem is to initialize the coordinate clocks at different points in

space so that the coordinate clock readings provide us with the desired global coordinate

times. We shall consider two alternate (but equivalent) procedures for accomplishing

this: A) use of light signals, and B) slow transport of standard clocks.

A) Synchronization (Initialization) using light signals

In Figure 1, the lines A and B represent the world lines of two coordinate

k k k

clocks at rest at positions x , x + dx which we wish to synchronize.

Figure 1

6
A light signal leaves A at coordinate time -x. , coincides with B at x_ , and

A. a returns to A at x. . The coordinate times x. and x. are measured and the A A A

coordinate clock at A halfway between the events x. and x. reads A

.

A f. 1 f

XA

=1/ 2 (V \

] (1?)

The event at x. should then have the same coordinate time as that at x„ , a a so the coordinate clock at B (or at A) is zeroed so that

To see the basis for this procedure starting from the expression for the

metric, we have for the coordinate time for light to go from A to B ,

and to come back from B to A ,

& 00

where do is the standard distance from A to B. Thus,

5 0o(

dxgo)

2+ d°

2 =

° (19)

(dx° Y + da

2 = (20)

y come J

/^-goo dx°

go = dx°come

= da

/*/ /|-gnn00

and

t + Aal{ X

B X

A. T uu /t 5 00

X -xJ + doU-g \ ~ X

B T QU /\"8 00

because the metric is static. Thus,

XA

= 1/2

(

XA

f + XA±)

= ^

(

X

B + d°/V^o" + X

B

- dCT /f^ = X

B ' (22>

and using Equation (18) gives the desired synchronization procedure.

Equivalently, the above procedure can be used to synchronize a series of

coordinate clocks along a path taken by a light ray by writing

'/{ dx

u = da /J-g00

(23)
and integrating along the path

Ax -jda/yf^t 00 (24)

path

This procedure gives a resulting synchronization which is independent of the path

along which the light ray travels; otherwise it would contradict the hypothesis that there

exists a global coordinate time. It is also independent of time, since the positions of the

coordinate clocks and the numerical values of g , are time-independent.

B) Slow transport of standard clocks

We next establish the fact that if a standard clock is moved sufficiently slowly

over a small distance past a stationary standard clock and then brought slowly back into

coincidence with the stationary clock, the two clocks will undergo the same elapsed proper

time. To prove this, we observe that the stationary clock will undergo an elapsed proper

time

As v- '00 Ax (25)

The moving clock will undergo the elapsed proper time

As f{-§00 dx

path

1 +

, 2 ,„ N 2 do g 0Q

(dx )

i/:

~-{> 00 Ax - 1/2/•

dx

V-8

(26)

00

If the velocity is sufficiently small the second term is negligible. If v is replaced

by its maximum value, v , the condition that the second term in Equation (26) be negli- max ^ ° gible in comparison with the contribution from h

nn , the deviation of g„ n

2 v max << 1 + 3 00

from -1 , is

(27)

Then in the first term of Equation (26) for inf initesimally short paths, g.,- may be evaluated

at any position along the path. This argument establishes the statement in the first para

graph of this section. Next consider the process of synchronization by means of slow transport of standard

clocks; in Figure 2, the solid lines with arrows represent light signals used to synchronize coordinate clocks at A and B repeatedly. We choose

0" 0' 0' XA

XA

XA

XA

(28)

and then set the coordinate clocks at point

that

, an infinitesimal distance away from A , so
X

B XA

' 0' 0' X

B XA

' 0" 0" X

B

= XA

(29)

Figure 2

Now starting at coordinate time x. , cause a standard clock to be transported slowly

and uniformly from A to B and back again (dotted line in Figure 2) , arriving at B

0' 0" at x_ (measured by the coordinate clock at B) , and at A at x. . The coordinate time

D A

required to go from A to B is

0' 0' dx X

B XA

X

A

i/i, °" °^ XA

= 1/2(XA

" XA

} (30)

0' Note that x_ - x, is a combination of measurements made by coordinate clocks at difB A

' ferent locations. In terms of proper time elapsed on a standard clock at A ,

dx° = 1/2

S

Vg oo

(A) (31)

If the elapsed proper time on the moving standard clock is As , then we have seen already that As = s" - s. . Thus,

dx = 1/2 As/ t/-g00(A) If (32)

By symmetry, 1/2 As is the proper time elapsed on the slowly moving standard clock in

going from A to B , which we shall denote by ds . Thus,

Adx = Ads K- '00 (33)

where to first order in infinitesimal quantities, g„ n can be evaluated at any point

from A to B . To extend this procedure to clocks which are arbitrarily distant from each other,

imagine carrying a standard clock very slowly from one point to another along a path

for which g nn

is known. Then
AAx = path

ds

(34)

The following argument establishes the equivalence of the two methods of coordinate

clock synchronization discussed above. In Figure 3, let coordinate clocks at A and

B , a finite distance apart, be synchronized by means of light signals and let the

dotted line represent the path of a standard clock carried from A to B . Synchronization signals are

V 8

Figure 3

sent out from B at intervals Ax and arrive back after reflection from the slowly

e moving clock at intervals Ax . At some point along the path, bounded by reflections

a

of signals which were emitted from B at coordinate times x and x and which

e12e arrive back at B at coordinate times x and x , the coordinate time interval

a a

2 between the reflections at P, and P„ will be given by

dxu =1/2 C + X - X - x

e

2 a2 el al

= 1/2

r o o X - X

L

6

2 el K + Axa]

, + x - x

a

2 al

= 1/2 |Ax° + Ax (35)

where the right side of this expression contains only quantities measured by the

coordinate clock at B . However, we have already shown that

Adx = Ads F00

(36)

10
where ds is the increment of proper time on the transported clock. Thus, integrating along the path,

Ax° = f —

= 1/2 T (Ax° + Ax°)

J, / *~> v e a pathV-g()0

= 1/2 |~x° (last) -x° (first) + x° (last) - x° (first)] , (37)

where the arguments "last" and "first" refer to the last and first synchronization

signals sent out from B . Then since x (last) = x (last) = x , the arrival

a e arr time of the transported clock at B , we have

Ax° = x° - 1/2 (x° (first) + x° (first)) arr a e

= x -xT) =x - X

,

arr B arr A

/

ds (38)

path yj-g00

Thus by either method, a consistent set of mutually synchronized coordinate clocks

is obtained. This synchronization scheme has the following properties:

1) Reflexivity. If A is synchronized with B , B is synchronized with

A. This follows from time- independence of the metric and the fact that the

integral of do/*/ -gnn

is independent of direction along a path.

2) Transitivity. If A is synchronized with B , and B is synchronized

with C , then A is synchronized with C.

3) Time-independence, because the metric is static.

4) Path-independence, otherwise a global coordinate time would not exist, contrary

to hypothesis.

4. Effect of Gravitational Fields of the Sun and Moon

Since it has been suggested in the literature fl, 4 1 that the gravitational potentials of the sun and moon play a role in processes of clock synchronization, the purpose

of this section is to estimate the magnitude of such effects. In the neighborhood of the

earth, the gravitational potentials of the sun and moon are small, so it is clear that we can work in a linear approximation in which squares and products of h can be

neglected. Consider the term h in a heliocentric coordinate system, and consider

yv first of all the effect of the (static) gravitational potential of the sun on the earth- moon system. We can think of the center of mass of the earth-moon system as freely falling

towards the sun. Consequently, at this point we may erect a coordinate system whose origin

11
moves along with the center of mass of the earth-moon system and whose orthonormal axes

are obtained „by parallel transport of some initially chosen orthonormal tetrad of basis

vectors. (We ignore for the moment the gravitational fields of the earth and moon.) Such

a coordinate system is termed a normal fermi system and has been discussed by Manasse

and Misner[7]. In normal fermi coordinates x

M

, the metric takes the form (see[7] for derivation)

_ i

. o ^ m (39) 800 K

0£.0m X X

g . = 2/3 R „. xV" (40)

g.. =6.. + 1/3 R. . xV (41)

where R. . is the Riemann-Christof f el curvature tensor, evaluated at the origin of the

normal fermi system, along the freely falling orbit. The distances x are proper

distances measured from the origin.

The components of the curvature tensor can, however, be evaluated in the heliocentric

system of coordinates. Then in transforming them to normal fermi coordinates, the trans- formation coefficients will be of the form

Sx^ = *U j. „_„ „* w#,„ JGM L2 = i 6M + terms of order V GM / c R

V 1 V " si s dx

where R is the radius of the orbit and M is the sun's mass. Since we are retaining

s s only first-order terms, the transformation coefficients are Kronecker delta-functions, and

the curvature tensor is an invariant.

To see this in more detail, the typical transformation formula for a coordinate will

be of the form

x = x' + R cos (tot'

)

s

where to is the angular velocity of the earth in its orbit. Then

R w R

dx s s v _ v . o' c c R c 3x s

But since the velocity is determined by

GM 2

s _ v_

R

2 "r s s

we obtain

v -^ / GM

s

c Vc2R

s

12
Furthermore, in a linear approximation the curvature tensor may be expressed as

Ra3y6

= h

a6,3Y~

h

86,aY~

h

cry,B6 + h$y,a& (43)

which thus involves second derivatives of the metric tensor. The largest terms in h in normal fermi coordinates will thus be of the form

-h

00,£m

X^

m

(44)

where hl n is calculated in heliocentric coordinates in which pi

2GM

h

00 " -I

3- (45)

c r

where r is the distance from the sun to the observation point, and M is the sun's

1

S mass. Since x ~ a.. , the earth's radius, we have

GM a

2

. h

oo = —

O

(46)

c R

s Then using

M = 1.970 x 10

33 gm,

s °

R = 1.495 x 10

10 cm

s

a = 6.3781 x 10 cm

we get

h

0Q

= 4 x 10

17

(47)

which is negligibly small when computing elapsed times. This term is not negligible when

computing forces on particles—it is responsible for tidal forces.

In a similar manner, all other contributions from the sun's mass to h in normal

yv fermi coordinates may be shown to be negligibly small. This conclusion is valid only in

the neighborhood of the earth.

These conclusions may be explained qualitatively as follows. A standard clock at

a radius slightly greater than that of the freely falling orbit (which for simplicity is

assumed circular) will be blue-shifted in the sun's gravitational field since it is further

2 from the origin. The fractional amount by which it is blue-shifted is g Ar/c where

2 s g = GM /R is the gravitational field strength due to the sun at the earth's orbit. Thus

s s s

2 2 the fractional blue-shift is GM Ar/c R

s s However, since the clock is at a greater radius than one at the origin, its velocity

in a heliocentric system will be greater and it will be redshifted more due to time dilation. Let v„ be the orbital velocity of the earth and v = vn (1 + Ar/R ) be the

s

13
velocity of the clock at radius R + Ar . This clock would be essentially at rest in the

local inertial frame. The fractional redshift compared to a clock at the origin will be

/ v 2

/„[—?v >. vo* \/l -— -A ,

v c V c c R

s But since the orbit is determined by the condition

R

2 ~ R

s s

the redshift cancels the blue-shift to terms linear in the distance from the origin.

Hof fman[3] has previously discussed the effect of this cancellation on clock rates. Thus to high accuracy, in a system of local inertial coordinates erected at the center

of mass of the earth-moon system, the effect of the sun's gravitational field can be neglected.

The above qualitative argument shows this is because the center of mass is in free fall toward

the sun. We can now imagine that starting from a heliocentric system of coordinates, a coordinate transformation is made to normal fermi coordinates whose origin is at the center of mass of the earth-moon system. In this system, over distances of the order of the earth's

radius, the effect of the sun may be neglected while calculating elapsed times. The

actual distance from the center of the earth to the origin of these normal fermi coordinates is

M

R = R „ ? „ = 4.64 x 10

8 cm

e m M + M

m e

where R is the earth-moon distance, and M and M are the masses of the moon and

earth, respectively. m

This distance is somewhat

me

less than the earth's radius. We can now ask what the effect of the moon's gravitational field on the metric tensor is. In this coordinate system, the earth is in free fall toward the moon. Hence by another coordinate transformation to a second set of normal fermi coordinates whose origin is at the center of the mass of the earth, it is evident that the effect of the moon can be similarly

transformed away.

In any case, even if the second coordinate transformation is not made, the effect

of the moon can be seen to be negligible by means of the following argument. Due to the

free fall of the earth and moon toward their mutual center of mass, the earth executes a slow orbital motion about the center of mass. The velocity of a point on the earth's surface

due to this motion is less than

3

oj n (a n + R ) = 2.9 x 10 cm/sec

1 e

where

oj„ = 2.6 x 10 sec

14
io

-14

is the angular velocity of the earth-moon axis. The fractional redshift due to time dilation of a standard clock due to this motion may be expected to be of order

2

T

c

and during a comparison flight of duration T , would contribute less than .36 ns to the

elapsed time, which is negligible. In actuality, there will be an additional blue-shift

contribution due to the moon's gravitational field which will nearly cancel exactly the

time-dilation effect. Therefore, we conclude that the gravitational fields of the sun

and the moon may be neglected, for our purposes, in the vicinity of the earth's surface.

5 . Clock Synchronization on the Rotating Earth

We have shown that because the earth is freely falling in the gravitational fields

of the sun and the moon, there exists a locally minkowskian inertial frame whose origin is at the center of the earth and in which these fields can be ignored. The only remaining

gravitational field is that due to the (rotating) earth itself.

In local inertial (normal fermi) coordinates with origin at the center of mass of the

earth, the metric tensor now takes the form

g 0Q

= - 1 - 2W(r")/c

2 (4g)

g i0

= (49)

g.. = 5 (1 - 2W(?)/c

2

) (50)

where W(r) is the potential due solely to the earth's gravitational field. We now transform to rotating coordinates (x' ,y ' , z

' , t

' ) by means of

x' = x cos wt + y sin cot (51)

y' = y cos cot - x sin cot (52)

z' = z (53)

t' = t or x = x . (54)

where the angular velocity of rotation of the earth is

co = 7.2921 x 10~ 5 sec"

1 (55)

15
The metric then becomes in linear approximation

g oo

= _1 ~h [

w(^

- 1/2 w2(x ' 2 + y,2)

] c

J

-y'w/c '01 6 10

'02 =

8 20

= X

' W/C

5 ij = 6 ij

(1 " 2W(r)/c }

Thus keeping only linear terms

-ds 1 + \ (W(r) - 1/2 w

2 (x'

2 + y'

2

)) (dx°)

2 + 2(u> x ?') - dr 1 dx°'/c

+ [l - 2W(r")/c

2

] [(dx')

2 + (dy')

2 + (dz 1 ) 2 ].

(56)

(37)

(58)

(59)

(60)

The neglected terms are, for propagation of light, at most

GM toa 2 e 1 , . N 2 —^ (Ax ) 2 a, c

c 1

and over a path length 10,000 km are completely negligible. For objects moving with

speed v , the neglected terms would be of order v/c smaller and hence negligible.

Henceforth we drop primes on the coordinates and take the metric to be:

-ds k- (W(r) - 1/2 oj 2 (x

2 + y

2 )) (A(dxV)

0, + 2to x r • drdx /c + (l - -| W(?) ) ((dx

2 + dy

2 + dy2 \ (61)

The potential due to the earth's flattened distribution of mass may, with accuracy sufficient

for our purposes, be taken as [9, 10] :

.2 W(r) = M G

e

1 —J C

2P

2 (cos 6) (62)

where 9 is the polar angle measured from the rotation axis and

90 } ? M G = 3.986003 x 10 cm /sec

-3

C„ = - 1.08270 x 10

a = 6.378139 x 10 cm

?

2 (cos 9) = 1/2 [3 cos 9 - l]

16

(63)

(64)

(65)

(66)
To express this potential in more convenient form, in terms of geographical latitude ({> , and altitude h above mean sea level, we use the following expansions

a f sin (j> 4- h (67)

where f is the flattening of the earth, and

2 2 ~ 2 sin ({) (1 - 4f cos (J)) = sin cf>. (68)

The derivations of Equations (67) and (68) are given in the Appendix. Then if we put

V(r) = W(r) - 1/2 oj 2 (x

2 + y2

) (69)

we have

v(2) = - M G

e

C

2 a

l 2 1 - -— (1-3 cos 9) 2r

- 1/2 w

2 r2 (1 - cos

2 9). (70)

The geoid is the equipotential at altitude h = , or r = a.. (1-f sin cj))

.

The value of this potential is, to sufficient accuracy,

M G

1 - 1/2 C (1 - 3 cos 9) 1/2 (/a2 (1 - cos

2 6)

2 3

and this may be solved for r to give

1 - 1/2 C

2

(1 - 3 sinZ

<t>)j - 1/2 a/a^ (1 - sinZ

<$>)/V

Q Sit

M G

e

M G

e

1 -

2 3 ,_ \

u> a

± 3C

2

. m^-—> Sln

(71)

(72)

Identifying the above expression for the radius of the geoid as a function of angle

with (see Equation (67))

r = a (1-f sin (J>) (73)

we obtain the following expression for the flattening f :

2 3

. Wal 3C 2- 1 2M G 2 298.08

e

(74)

17
To obtain an expression for the acceleration of gravity, we calculate the expansion of

V(r) to the first order in h:

. 9V

8 = +

37 r = a (1 - f sin <}>)

M G , M GC. 9 —e 3e2 2 2 . — —= w a

x + sin 4, a

l a

l

2M Gf „ M GC„

e ,9 e 2

2 2 2 a

l a

l

+ u a,

978.027 + 5.192 sin

2

<j> cm/sec

2

. (75)

We thus obtain for g 00'

3 00 (

1+ "t ( V + %Wh A (76)

The term 2V /c is of order

2M G -9

e 1.4 x 10

2 c a.

For a standard clock at rest at mean sea level (where h = , the geoid in this model)

,

ds = / 1 M. G \

2 c a, dx (77)

And during a flight of duration T , the term in V would contribute a correction

c c

M

eGTc = 50,000 ns.

2

c a

x

(78)

which is certainly a large effect but which is the same for all clocks, since all

available standard clocks are in fact placed near the geoid. Therefore, if we redefine

the rates of all coordinate clocks so as to absorb this term, we may write

o. - - [

i

+

*T] • (79)

The metric is then to a sufficient accuracy

- ds = I + Milhj (dx

o

} 2 + A _ 2WiS,\ (dx

2 + dy2 + dy

2

} ** * o + 2o) x r • dr dx /c. (80)

18
It should be noted that the expression g((J)) appearing in Equations (75) and (80) includes

rotation.

With the aid of Equation (80) , we are now in a position to analyze various processes

of clock synchronization. Consider first a series of standard clocks at rest at altitudes

h = (on the geoid) . The proper time elapsed will be given by:

ds = dx° . (81)

Thus we have shown that by distributing a system of standard clocks about on the geoid, at rest, all such clocks will beat at the same rate, equal to the rate at which the global

coordinate clocks will beat. This result was discussed by Cocke Cll~J . We still have the

problem of initializing these clocks.

A) Synchronization of coordinate clocks by slow transport of standard clocks Let a standard clock be set in motion and carried from one point to another along

a path near the geoid, so that Equation (80) holds. It is easy to see from the metric,

Equation (80), that the proper time elapsed on the standard clock is path-dependent. In *-*-*

particular, the term 2u!xi> drdx /c is responsible for the well-known 207 .4 ns east-west asymmetry [lj in equatorial around-the-globe flights. Nevertheless, since all the coefficients in the above expression for the metric are known when the altitude, course, and

speed of plane carrying the clock are known, the metric can be inverted to calculate

the elapsed coordinate time in terms of the elapsed proper time on the transported clock:

Ax° = /"ds jl - ^

+1/2 4 +

" X

f

' *

path

(82)

Hence by selecting one location as a reference, all coordinate clocks (whether on the

geoid or not) may be synchronized by slow transport of standard clocks. In Equation (82), v is the ground speed (measured on the flattened earth) and r is the vector from the center of mass of the earth to the position of the moving clock measured in the rotating

coordinate system. The coordinate clock synchronization scheme based on the above equation,

because of the existence in general relativity theory of a global coordinate time, is both

path-independent and transitive.

With the aid of equation (82), we can estimate the magnitude of a number of effects.

a) Effect of latitude variations in g((j>) on contributions from redshift

2 2 This term, during a comparison flight, would be of order 5.2 sin <{)h T /c . For a flight of duration 10 hours and altitude 12,000 meters, the contribution will be of order

0.25 ns and is, therefore, negligible.

b) Effect of altitude variations in g((j>) on contributions from redshift

2 Although a term proportional to h is not explicitly included in our expansion of

V(r), Helmert's Equation[l2] for the acceleration of gravity is, to the accuracy of terms considered here,

g(<(>) = 978 + 5.2 sin2

<j> - 3.086 x

10~ 6 h cm/sec

2 (83)

19
(with h in cm) . This gives a potential term in g__ of order

1.5 x 10~6 h

2 /c2

And for a flight of duration T and altitude h = 1.2 x 10 cm , the correction due to

c c altitude variations in g is approximately 0.09 ns , which is negligible.

c) Effect of gravitational anomalies on redshift

According to Caputo (~9~l , undulations of the geoid from the calculated ellipsoidal

shape are 50 meters or less. A height variation of Ah = 50 meters would give rise to

2 ~ corrections to the elapsed time of order gAhT /c = 0.2 ns , which is negligibly small.

d) Errors in altitude measurement

The calculation in paragraph c above shows that if an accuracy level of 1 ns is

desired, h must be measured during transport to an accuracy of better than 250 meters.

This is not a very severe restriction.

Next let us examine the term which depends explicitly on the rotation of the earth.

Using Equation (67) for r , the altitude contribution would be of order

u)hT vy- = 0.08 ns. (84) 2c

where we have assumed a maximum ground speed of v = 1,000 mi/hr = 4.5 x 10 cm/ sec. This contribution is, therefore, negligible.

e) Effect of flattening of earth

The flattening contribution in Equation (67) would affect the contribution from the

rotational term in the amount

ua f T v — 2

C = 0.28 ns.

c

which is negligible.

With these results, the rotational term can be simplified as follows:

-»-»-»

a) x r • v = coa,IEv„ cos d> (85)

where v is the eastward component of ground velocity and § is the latitude. (v is

£i E negative if the clock is transported westward.) Thus the final expression for elapsed

coordinate time is

2 wa-, v-r, cos Ax" = f

is

V

ath L

/ . l _£H + ^: +

a

i ve

(86)

2 . 2 2 c 2c c path

In developing Equation (85), flattening of the earth was neglected. Referring to the

Appendix, the relation between polar angle and latitude is

2 cos = sin <p (1 - 2f cos (}>) (87)

20
so the error introduced into Equation (85) is of order f . This error would contribute

about the same amount as that calculated in section e above, .28 ns , and is negligible.

B) Inf initesimally slow transport on the geoid

In the limit as v -* , and if h = , we have

Ax° = /= ds (1 + a n v^ cos (jj/c 2

)

:; 1 E wa r

= As + —„— / ds v,, cos <J) . (89)

2 J E

We consider several examples.

1) Transport along a meridian: then v = and Ax = As . We could

E use this result to synchronize coordinate clocks over the entire surface of

the earth by selecting one location as reference and transporting standard

clocks slowly along a meridian to one of the poles and then back along

another meridian to an arbitrary position on the earth's surface. The

proper time elapsed on the transported clock will read coordinate time.

2) Transport along a parallel of latitude: then v = v and

/ds v cos (f) •= cL cos <j) where L is the distance traversed eastwards. The

coordinate time elapsed will be

wa

i Ax = As 4- L cos <J). (90)

Note that the correction term is independent of velocity, provided only

that the velocity is small. This result implies that the proper time

elapsed on a standard clock carried eastward around the globe will be

2 2 2 2

2Tiu>a cos <p/c = 207.4 cos (J) ns less than that on a standard clock

which remains at rest, while a standard clock carried slowly westward in

2 the geoid would lead a standard clock, which remained at rest, by 207.4 cos (f> ns.

C) Global coordinate clock synchronization using light signals

0.0,2

2 The propagation of a light ray is described by ds =0. This gives

n o / n a>t 2a)a i cos <J> c c ( dx ) \

= (dx°)

2 (i +

2jL(|)h _ _^ _E

J

\ c c

2 c /l--| W(r)j ((dx)

Z + (dy)

Z + (dz)Z

/. (91)

where c is the eastward component of velocity of the light ray. For motions in the

E geoid, and in linear approximation, W(r) can be replaced by its approximate value

M G MG M G M , ,

' . 2 N ~ e . . 7T .-k e _ e (1 + f sxn 4> - h) = . (92) W(r) = — = a

r a 1

The fractional errors in coordinate time of propagation of a light signal introduced are

as follows:

21
2 2 2 A) The altitude terms in the coefficient of dx + dy + dz are of order

2M

eGh

- -12

= 3 x 10

2 2 c a

±

which for a light signal traversing a path of length 10,000 km on the earth's surface,

-13

would amount to only about 10 sec , which is negligible.

B) The term arising from neglect of flattening contributes for a light path of

this length

2M Gf nn9 e 10 ~ _ c —x x = .05 ns.

2 c

c a..

which is also negligible. The term arising from neglect of the angularly-dependent

->- part of W(r) can from Equations (62) and (74) be seen to be of similar order of

magnitude. The remaining term is constant, and is

2M G ~ -9

e = 1.4 x 10

2 c a.

Solving Equation (91) for dx , in linear approximation, we have

dx ^dx + dy + dy

M G

e

1 + 2

c a,

... wa c cos

g(<i>)h 1 E

2 2 c c (93)

_9

The first term in brackets in Equation (93) is of order 10 , but it is the same

for all propagation paths within about 12,000 meters of the earth's surface and we may

incorporate it into measurement of proper distance. The proper distance da between two points on the geoid will be obtained by integrating

da = /l + ^-Vdx2 + dy2 4dZ

2)

1/2

.

2 -12

Furthermore, the term gh/c in Equation (93) is of order 10 which is negligibly

small. Then approximately,

dx = da 1 +

uan c„ cos <b IE

(94)

(95)

The only significant effect is due to rotation of the earth. In the rotational term in

Equation (95), altitude and flattening effects may be safely neglected over 10,000-meter

path lengths at the 1 ns level of accuracy.

Then in using one-way light signals to synchronize the coordinate clocks placed on

the geoid, one may select a reference clock and send out light signals within 12,000 meters

of the earth's surface; the coordinate time elapsed is

22
n r

wa

i r

Ax =

j da + —| / cE cos 4 do. (96)

path

during the propagation. It should be noted that for 1 ns accuracy, the proper distance

along the path in the first term, Ida , must be known to an accuracy of 30 cm or better.

We now consider some examples.

1) Light signals sent along a meridian: For this case, c = and the elapsed

coordinate time is just

A x° =

path.

/

2) Signals sent along a parallel of latitude: For this case,

f

wa

i f Ax = J da + cos <j> J da • path path

Thus if we adopt the global coordinate time x as a basis for a synchronized

coordinate clock nerwork on the surface of the earth, then light propagation times

must be corrected for the earth's rotation according to Equation (95). The resulting

synchronization will agree with that based on transport of standard clocks as expressed

by Equation (86) and will be path-independent and transitive.

D) The "Hafele-Keating" discontinuity

It has been suggested [4] that one way to achieve a synchronized coordinate

time net on the surface of the rotating earth is to select a meridian or other line

2 between the poles along which one allows for a 207.1 cos <j> ns discontinuity in

synchronization of the coordinate clock network. The network could be synchronized

by slow transport of clocks or by light signals without correction for earth rotation

as long as the discontinuity is not crossed and transport is along parallels of

latitude. A clock traversing the discontinuity from west to east would have to be

2 2 2 advanced by 2Tra .to cos <J)/c , while one traversing the discontinuity from east to west would have to be retarded by the same amount. Suppose we adopt on the earth's surface a coordinate time net x_ with a discontinuity related to the global coordinate time x

u by

ma^ cos 4> L. *D

= X

"

c

where L is the distance eastward from some reference meridian

(97)

- Tra cos <j> < L < ira

1 cos <J> (98)

23
of the clock in question and where, for convenience, we have chosen the discontinuity

at the meridian on the opposite side of the globe from the reference meridian on which L = . Then from Equation (89), use of slowly transported standard clocks without

correction for earth rotation will give the required time x of Equation (97):

1/

wa

As - Ax° - f dsv cos

only if the transport is carried out along a parallel of latitude . This is because

of the appearance of the factor cos <{) in the integrand. Similarly from Equation

(96) , use of light signals without correction for rotation will give an equivalent

synchronization procedure:

wa. da = Ax r— / da c„ cos

only if the light path is along a parallel of latitude , since again a factor cos $ appears

in the integrand.

Hence we have proved that use of light signals and standard clocks, uncorrected for rotation, can be used to construct equivalent synchronized coordinate clock nets with a discontinuity for only a limited class of paths. This scheme has several disadvantages:

1) The effects of redshift and time dilation on transported clocks must still

be included.

2) Only a limited class of paths can be used; otherwise the synchronization procedure is path-dependent and intransitive.

3) The time readings obtained do not represent the global coordinate time, although

the transformation to global coordinate time (Equation (97)) is fairly simple.

4) The discontinuity may be a nuisance for clocks which must traverse it.

E) Use of satellites to construct a coordinate time network

The use of satellites to carry reference clocks and to send synchronization pulses

to earth-based clocks could be analyzed by use of Equation (93) . The term involving

2 g(<j))h/c for satellite heights up to 1,200 km is still negligibly small. Altitude cor- rections to the rotational term are, however, of order —coh ~ -7

= 3 x 10

c

9 -2 which in a propagation time of t ~ 10 /c = 3 x 10 sec would be of order 10 ns and would

be appreciable. Thus the rotational term must be written as in the following expression for

the elapsed global coordinate time along the path of the pulse.

, , u). x r • c

path

24
where r is the radius vector from the earth's center to the path of the pulse of

velocity c . The above equation, Equation (99) , could be used in principle to synchronize any

receiving station using reference pulses from a satellite which were sent out at regular

coordinate time intervals. To obtain a clock on a satellite which measures coordinate

time x requires more analysis.

Returning to the expressions (48, 49, 50) for the metric tensor before introducing

the transformation to rotating coordinates, we write approximately W(r) = -M G/r , so

-ds

2 =

/ 2M G\ . / 2GM \

(

i -^H + r-^)

2 2 2 (dx + dy + dy ) (100)

and for a satellite moving in a circular orbit at low speed,

ds

2 = (2GM 2 \

c r c /

(101)

Thus in terms of elapsed proper time on a standard clock carried in the satellite,

2GM 2

,, o.2 2 (dx ) = ds 1 + _J1 + X_

2 2 (102)

and since the orbit is determined by

GM

we have

dx = ds 1 +

3GM

e 2c r (103)

And from this equation or its generalization for an elliptical orbit, equal intervals

of coordinate time on the satellite can be determined by calculation L-13J assuming that

the positions r (x ) of the satellite can be determined and the integral of Equation (99)

performed for clock synchronization. Since in 1 ns light travels 30 cm , positional

accuracies of the satellite and of the receiving stations must be better than ± 30 cm , for 1 ns synchronization accuracy.

In addition, variations in propagation time due to dispersion of the electromagnetic waves in the ionosphere must be considered. These variations may be of

the order of a few nanoseconds fl4j .

6. Miscellaneous Effects

A) Mass of the atmosphere

The mass of the atmosphere is about 10 that of the earth. As the mass

varies, the gravitational potential will vary, but neglect of the variation due

25
to the atmosphere's mass could at most introduce an error of 10 times the

redshift due to the earth's mass and is, therefore, negligible.

B) Effect of flattening of the earth on discontinuity

The "Hafele-Keating" discontinuity in its exact form is

2

ojr sin 8L _ 2Trco(r sin 8)

2 "

2

From the Appendix, at zero altitude, we have

(r sin 9)

2 2

a.^ cos <j

2 x 2 2 cos (J) + (1-f) sin

(104)

Thus, keeping only corrections of first order in f , the discontinuity is

2TTcoa^ cos

2

<j> (1 + 2f sin

2

(J>)/c 2 = 207.4 cos

2

<J> (1 + 2f sin2

cj)) ns. (105)

2 2 Since the maximum value of sin <j> cos (J) is 1/4, the correction from flattening is

less than

2f(207.4)/4 = 0.35 ns.

and is, therefore, negligible.

C) Retardation

One may ask whether the finite propagation time of gravitational signals from

the moon affects the gravitational field in which the earth is freely falling and

in which normal fermi coordinates are finally erected at the center of the

earth's orbit. Consider the terms h in a coordinate system whose origin is uv ' a at the center of mass of the earth-moon system. Weber [3l has given explicit

expressions for h in the linear approximation. These expressions for a

" -*- * -> source mass m at r' , moving with velocity v , and with observation point r , are as follows. Defining

t> = h - 1/2 5 h

uv uv uv a

(106)

we have

4Gm/c 00 v^^-^ffft)

(107)

26
- 4Gmv!/c

3 |r - r' | ^oi

=

x z z—^—

(108)

1

\ c\t-P\)

i ,, i / _ 4

1

4GMv!v:/c |r - r* '«

^ — (109)

' c |r - r' I /

If we apply the above expressions to the case of the moon, the leading term in <J> nn

2i-> -> will be proportional to the ordinary newtonian potential, 4Gm/c |r - r |. The

leading correction term is due to retardation and will be of order

4Gm v

c |r - r |

The velocity of the moon is such that v/c ~ 3 x 10 . Estimating the magnitude of

this retardation correction in the vicinity of the earth where

|r1 - r' = R = 3.84 x 10 cm , we obtain

' m

4Gm v ~ 10-18

c |r - r |

—1 8 This retardation term contributes a term of order 10 to h _. . In a typical flight

-14

lasting 10 hours , such a term would contribute no more than about 4 x 10 sec to the elapsed coordinate time and may, therefore, be neglected. Thus, we conclude

that retardation corrections to the metric from motion of the moon may be neglected.

27
References and Footnotes

[I] J. C. Hafele and R. E. Keating, Science 177, pp. 166, 168 (1972).

[ 2] C. H. Brans and R. H. Dicke, Phys. Rev. 124, p. y25 (.1961) .

[3j J. Weber, General Relativity and Gravitational Waves , Interscience Publishers, Inc. New York (1961)

.

[4] R. Schlegel, Amer. J. Phys. 4_2, p. 183 (1974).

[5J Schlegel 's expression for this discontinuity contains an error.

[ 6j C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation , W. H. Freeman and Co., San Francisco, p. 596 (ly73)

.

["7] F. K. Manasse and C. W. Misner, J. Math. Phys. 4_, p. /35 (1963).

[8] B. Hoffman, Phys. Rev. L2JL, 337 (1961).

[9] M. Caputo, The Gravity Field of the Earth , Academic Press, New York, p. 32 (1967;.

[lO] G. D. Garland, The Earth's Shape and Gravity , Pergamon Press, New York (1965);

R. H. Rapp, Geophys. Res. Letters 1, 35 (1974).

[II] W. J. Cocke, Phys. Rev. Letters 16, 662 (1966); 779 (1966); 1233 (1966).

[l2 J Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., 32nd Edition,

p. 2508 (1954).

[l3 J Note that this equation does not take account of the redefinition of coordinateclock rates mentioned in connection with Equation (77)

.

[14 J J. Barnes, private communication.

28
APPENDIX

Latitude and Polar Angle in Ellipsoidal Geometry

In this paper, the rotating earth is considered an ellipsoid of revolution with

equatorial radius a.. and polar radius a„ , where

a = a.Al f) (Al)

with f = 1/300 the flattening of the earth. In this appendix, we wish to derive relations between the polar angle measured from the rotation axis and

the geographic latitude <J> , measured along a plumb line which is normal to the

ellipsoid. The problem is, given the position of an observer in terms of altitude h and latitude <j) , to express r and 6 in terms of h and <j> . Let X,Y be the cartesian coordaintes of the point P . Then

X = r sin Y = r cos (A2)

We next express the coordinates (x,y) of the point P' on the surface which is

the projection of P along a plumb line, in terms of tj) . Write

r , sin d>' = y = S sin

P (A3)

r , cos d>' = x = C cos

P

Equations (A3) and (A4) define new quantities S and C . We have

(A4)

tan <}>' = y/x (A5)

and from analytic geometry,

2 2 tan $ = ya /xa , (A6)

29
and hence

2 C _ tan <}> 1 1

( A7 \ S tan <j>' 2 ,- -.2

a (1-f)

S = (l-f)

2C . (A8)

Since the equation of the ellipsoid is

we obtain

2 2 ^2+^=1 (A9)

a

l a

3

C

2 cog2(P . (l-f)

4 C

2 sin2

(j> (A1Q)

2 2 a

l a

3

C = , l ' (All)

2 2 T

cos <j> + (1-f) sin

Now we may write the coordinates of P as

a cos q> X = h cos $ + ; a o (A12)

2 2

2~ cos <j> + (1-f) sin "

(1-f) 2 a sin $ Y = h sin $ + | a (A13)

J cos <j> + (1-f) sin (j>

Then

2 2 2 r = X + Y^ (A14)

and after some algebra, we find that retaining only first-order corrections to a. in flattening and altitude,

2 r = a - a f sin (J) + h (A15)

30
2 Also, neglecting altitude corrections which can be shown to be of order (h/a ) and hence negligible, we find for the polar angle

2

2 y 2 2 cos = —-r = sin $ (l-4f cos <)))

.

(A16)

31
NBS-114A (REV. 7-731

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLK A I ION OK REPORT NO.

NBS TN-659

2. (xiv't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE 5. Publication Date April 1975

An Earth-Eased Coordinate Clock Network

6. Performing Organization Code

7. AUTHOR(S) 8. Performing Organ. Report No. Neil Ashby

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2770108

11. Contract/Grant No.

CST-8335

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as item 9

13. Type of Report & Period Covered

Fmai

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This paper investigates some of the possible operational procedures for

synchronizing clocks at fixed sites spread around on the earth's surface, to within a 1 nanosecond level of accuracy. Since a common synchronization procedure is by transport of standard clocks in commercial jet

airline flights, and most of the effects we shall discuss are fractional

corrections to the elapsed time, as a criterion of whether an effect is

significant at the 1 nanosecond level we take for comparison purposes an elapsed time T = 10 hours. This is a typical time for an intercontinental

airplane flight. Analysis of a number of effects which might affect

clock synchronization is carried out within the framework of general

relativity. These effects include the gravitational fields of the

earth, sun, and moon, and orbital motion, rotation, and flattening of

the earth. It is shown that the only significant effects are due to the

gravitational field and rotation of the earth, and motion of the transported clocks. Operational procedures for construction of a synchronized

coordinate clock network using light signals and transported standard

clocks are discussed and compared.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Clocks; coordinate time; general relativity; time

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