Fermi Coordinates of an

Observer Moving in a Circle in

Minkowski Space: Apparent

Behavior of Clocks

ARL-TR-2211 May 2000

Thomas B. Bahder 

Approved for public release; distribution unlimited.
The findings in this report are not to be construed as an

official Department of the Army position unless so

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thereof.

Destroy this report when it is no longer needed. Do not

return it to the originator.
ARL-TR-2211 May 2000

Army Research Laboratory

Adelphi, MD 20783-1197

Fermi Coordinates of an

Observer Moving in a Circle in

Minkowski Space: Apparent

Behavior of Clocks

Thomas B. Bahder 

Sensors and Electron Devices Directorate

Approved for public release; distribution unlimited.
Abstract

Space-time coordinate transformations valid for arbitrarily long coordinate

time are derived from global Minkowski coordinates to the Fermi coordinates of an observer moving in a circle in three-dimensional space. The

metric for the Fermi coordinates is calculated directly from the tensor transformation rule. The Fermi coordinates are used in an examination (from the

observer’s reference frame) of the detailed behavior of ideal clocks.

A complicated relation exists between Fermi coordinate time and proper

time on stationary clocks (in the Fermi frame) and between proper time

on satellite clocks that orbit the observer. For portable clocks that orbit the

Fermi coordinate origin, an orbital Sagnac-like effect exists. The coordinate

speed of light is isotropic but is periodic in time and varies with Fermi coordinate position. In a numerical illustration of the magnitudes of these kinematic effects, this report uses parameters relevant to clocks carried aboard

satellites orbiting the Earth.

ii
Contents

1 Background 1

2 Introduction 3

3 Fermi-Walker Transport Differential Equations 5

4 Determination of the Tetrad 8

5 Fermi Coordinates 10

6 Metric in Fermi Coordinates 13

7 Apparent Behavior of Clocks in Fermi Coordinates 15

7.1 Stationary Clock . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.2 Clock in Circular Motion . . . . . . . . . . . . . . . . . . . . . 16

7.3 Sagnac-Like Effect for Portable Clocks . . . . . . . . . . . . . 18

8 Coordinate Speed of Light 20

9 Summary 23

Acknowledgments 23

References 24

Appendix. Conventions 27

Distribution 29

Report Documentation Page 31

iii
Figures

1 Observer’s circular 3-d path is shown against background of

global inertial Minkowski coordinates xi ............ 5

2 Observer’s world line C is shown with initial tetrad basis

vectors λi

(α) at s = 0 at point P0 ................. 7

3 Global inertial coordinate axes xi and Fermi coordinates X(a)

are shown in relation to coordinate position X . . . . . . . . 21

iv
Tables

1 Numerical values of constants . . . . . . . . . . . . . . . . . . 17

v
1. Background

The U.S. Army is developing precision-guided munitions intended to be

used against the threat of armored vehicles. These munitions have a coupled GPS (Global Positioning System) and inertial guidance system. The

GPS portion of the system is intended to give precise updates to the inertial system during mid-course flight. Such precision-guided munitions are

required to have a terminal accuracy of better than 1 m (corresponding to

the typical scale dimension of an armored vehicle). This means that the accuracy of GPS position updates must be significantly better than 1 m in

order for the total system to perform at 1 m.

The Department of Defense (DoD) and the civilian population have numerous other interests in precise navigation, such as aircraft instrument

approaches. Today, the stand-alone GPS is used routinely for nonprecision

instrument approaches. It would be a great advantage if the stand-alone

GPS could also be used for precision approaches.∗ Depending on the future course of development of the GPS, such approaches could be feasible

for unmanned as well as manned aircraft, if the GPS were sufficiently accurate and had a sufficient level of receiver autonomous integrity monitoring

(RAIM).†

In addition to these applications, DoD has numerous other needs for precise navigation, such as mapping safe corridors and positioning spacebased interferometers for surveillance of the ground. Another area where

the GPS needs ultra-high accuracy is for precise time transfer, which is necessary for applications in communications encryption.

The current stand-alone GPS does not meet the accuracies needed in many

of the above-mentioned areas. At present, it is well known that small anomalies exist in position and time computed from GPS data [1–3]. The origin of

these anomalies is not understood. In particular, GPS time transfer data

from the U.S. Naval Observatory (USNO) indicate that GPS time (for all

satellites) is periodic with respect to the Master Clock, which is the most

∗The FAA (Federal Aviation Administration) system known as WAAS (Wide Area Augmentation System), which is to provide precision approaches to most airports in the U.S.,

is a differential GPS system. I am referring to the possibility of a precision approach system

based upon the stand-alone GPS, with no differential GPS corrections.

†For example, one could obtain an improvement in integrity over the present standalone GPS by increasing the number of GPS satellites in view.

1
accurate source of official time for DoD. Furthermore, the time obtained

from all GPS satellites appears to speed up and slow down in-phase [3]. In

1997, the periodicity of this effect was approximately equal to the sidereal

day and had a peak-to-peak amplitude of ≈20 ns, which translates to 20 ft

(≈6 m) of error in light travel time [3]. The existence of these anomalies in

GPS data motivates the theoretical investigation reported here.

The purpose of this work is to explore the possibility that the periodicity

in time between the GPS and the USNO Master Clock is a relativistic kinematic effect that is due to the way in which the GPS is implemented. If such

an explanation were to prove correct, then this effect could be taken into account, and the accuracy of the GPS would be improved. The work reported

here is only a first step in the analysis, because gravitational effects have

been neglected. As elaborated below, the gravitational field is an essential

ingredient in the analysis, but the formalism is sufficiently complicated that

it is useful to first explore the effects without the contribution of gravity.

The GPS is implemented in an Earth-centered quasi-inertial (ECQI) reference frame. This reference frame is not a relativistic space-time coordinate

system (as is required by a fully relativistic theory); instead, it is a spatial reference system plus a time scale [4]. From the point of view of relativity theory, the GPS satellite clocks keep coordinate time in a modified

Schwarzschild metric [5], not the ECQI system of coordinates.

In this report, the ECQI frame coordinates are replaced by a Fermi-Walker

transported system of coordinates. The apparent behavior of real clocks

(i.e., the GPS satellites) is examined from the point of view of this coordinate system.

2
2. Introduction

It is well known that the choice of a particular space-time coordinate system can lead to apparent effects that vanish when an alternative coordinate

system is used [6–9]. In this work, I report a detailed calculation of the apparent behavior of ideal clocks as observed from a system of coordinates

whose origin moves in a circle in three-dimensional (3-d) space and whose

axes maintain their orientation with respect to the distant stars. Such a system of coordinates is similar to that used to implement an Earth-centered

inertial frame of reference, as in astrodynamics applications [10,11]. However, the case treated here differs from that of the Earth-centered inertial

frame, in that gravitational effects are neglected. In this work, the origin of

coordinates is kept in circular motion by a force (such as that provided

by rocket engines), so the system of coordinates is accelerated but nonrotating. I take a Fermi coordinate system as the closest relativistic analogue to a nonrotating system of local Cartesian coordinates used in experiments [6,10,11]. In particular, a Fermi coordinate system is a nonrotating

system of coordinates where Coriolis forces are absent and light travels essentially along a straight line. I take the Fermi coordinate origin to move

in a circle in 3-d space, and therefore, the observer’s tetrad is Fermi-Walker

transported along a helical time-like world line.

In the tetrad formalism, laboratory measurements are interpreted as projections of tensors on the tetrad basis vectors. These projections are invariant

quantities under transformations of the space-time coordinates. However,

these projections depend on the world line of the observer and the choice

of local Cartesian axes used by the observer [6]. The need for the tetrad

formalism to relate experiment to theory, as well as the problem of measurable quantities in general relativity, is extensively discussed by Pirani [12],

Synge [6], Soffel [10], Brumberg [11], and more recently Guinot [9].

The tetrad formalism was initially investigated for the case of inertial observers that move on geodesics [12–17]. However, many observers are terrestrially based or are based on noninertial platforms. The general theory

for the case of noninertial observers has been investigated by Synge [6],

who considered nonrotating observers moving along a time-like world line,

and by others [18–23], who considered accelerated, rotating observers. For

arbitrary observer motions, the effects seen by the observer are indeed

very complicated, and the general theory [6,12–23] gives limited physical

insight.

3
In this report, I work out in detail a simple model problem. I consider the

particular case of an observer moving in a circle in 3-d space. I use the tetrad

formalism to obtain the transformation from an inertial coordinate system

in Minkowski space-time to the accelerated, nonrotating system of Fermi

coordinates of an observer moving in a circle in 3-d space. I compute the

apparent behavior of ideal clocks as observed with respect to these Fermi

(laboratory) coordinates. The results are given here as expansions in low

velocity of the observer, compared with the speed of light. However, the

coordinate transformations derived here are not expanded in a power series in time, and hence they are valid for arbitrarily long times. Knowing

the correct long-time behavior of the coordinate transformations allows the

study of the apparent long-time behavior of clocks as observed with respect to the Fermi coordinates of the observer. In a numerical example of

the magnitude of the effects, I use parameters relevant to clocks carried

aboard GPS satellites orbiting the Earth [24].

4
3. Fermi-Walker Transport Differential Equations

Consider a time-like world line C of an observer moving in a circle in 3-d

space, given in Minkowski space-time coordinates by

x0 = x0

0 + u ,

x1 = x1

0 + a cos (ωu/c) ,

x2 = x2

0 + a sin (ωu/c) , (1)

x3 = x3

0 ,

where xi

0 are constants, u is a parameter along the world line, and c is the

speed of light in an inertial frame in vacuum. The observer travels in a circle

in the x-y plane (see fig. 1). Along his world line, the observer travels with

a 4-velocity ui = dxi

/ds and acceleration

wi = δui

δs = duids + Γi

jkuj

uk , (2)

Figure 1. Observer’s

circular 3-d path is shown

against background of

global inertial Minkowski

coordinates xi

. At proper

time s = 0, observer is at

point A. At proper time

s > 0, observer is at point

B. Origin of Fermi

coordinates is at current

position of observer.

Orientation of Fermi

coordinate axes X(a) is

shown schematically at a

given proper time s.

X

X

ω

A

B

a

x1

x2

(2)

(1)

5
where Γi

jk is the affine connection. The normalization of the 4-velocity,

ui

ui = −1, provides the relation between the arc length, s = cτ , where τ is

the proper time, and the parameter u = γs, where γ = (1−ν2)−1/2. (See the

appendix for the conventions used here.) The parameter ν is the dimensionless velocity, given by ν = aω/c. Since the affine connection components all

vanish in Minkowski space, the explicit expressions for the 4-velocity and

acceleration are

ui

(s) = γ



1, −aω

c

sin γω

c

s



,

aω

c

cos γω

c

s



, 0



, (3)

wi = −aγ2

ω

c

2



0, cos γω

c

s



,sin γω

c

s



, 0



. (4)

As the observer moves on the time-like world line C, he carries with him

an ideal clock and three gyroscopes. At some initial coordinate time x0,

the observer is at point P0 at proper time τ = s/c = 0. On his world line,

the observer carries with him three orthonormal basis vectors λi

(α), where

α = 1, 2, 3 labels the vectors and i = 0, 1, 2, 3 labels the components of

these vectors in Minkowski coordinates. These vectors form the basis for

his measurements [12] (see fig. 2). The orientation of each basis vector is

held fixed with respect to each of the gyroscopes’ axes of rotation [25]. The

fourth basis vector is taken to be the observer 4-velocity, λi

(0) = ui

. The four

unit vectors λi

(a) form the observer’s tetrad, which is an orthonormal set of

vectors at P0,

gij λi

(a) λj

(b) = η(ab) , (5)

where the matrix η(ab) = gab is the Minkowski metric (see the appendix).

At a later time s = cτ , the observer is at a point P. The observer’s orthonormal set of basis vectors is related to his tetrad basis at P0 by Fermi-Walker

transport. Fermi-Walker transport preserves the lengths and relative angles

of the transported vectors. For an arbitrary vector with contravariant components fi

, its components at P are related to its components at P0 by the

Fermi-Walker transport differential equations [6]:

δfi

δs = Wijfj , (6)

where

Wij = ui

wj − wiuj . (7)

6
Figure 2. Observer’s

world line C is shown with

initial tetrad basis vectors

λi

(α) at s = 0 at point P0.

Fermi transported tetrad

basis vectors at finite

proper time s are shown at

point P.

C

(1) λi

(1) λi

(2) λi

(2) λi

(3) λi

(3) λi

(0) λi

ui = λi

P ́

s

s = 0 P0

σ

P

μi

(0)

When we use equation (6) to transport a vector fi that is orthogonal to the

4-velocity, ui

fi = 0, the second term in equation (7) does not contribute.

We refer to transport of such space-like basis vectors as Fermi transport,

and Wij → W ̃ ij = ui

wj . For an arbitrary vector fi perpendicular to the

4-velocity, the explicit form of the Fermi transport differential equations for

the world line in equation (1) is

df 0

dξ = −γ3ν cos (γξ) f 1 − γ3ν sin (γξ) f 2 , (8)

df 1

dξ = γ3ν2 cos (γξ) sin (γξ) f 1 + γ3ν2 sin2 (γξ) f 2 , (9)

df 2

dξ = −γ3ν2 cos2 (γξ) f 1 − γ3ν2 cos (γξ) sin (γξ) f 2 , (10)

df 3

dξ = 0 , (11)

where the components fi are functions of ξ and the dimensionless proper

time is given by ξ = ωs/c. The differential equations (8) to (11) are identical

to the equations that describe the Thomas precession of an electron’s spin

vector as it moves in a circular orbit around the nucleus [26]. The solution

of equations (8) to (11) is given by

f 0 = −γνA cos 

γ2ξ + α



+ β , (12)

f 1 = A cos (γξ) cos 

γ2ξ + α



+ Aγ sin (γξ) sin 

γ2ξ + α

 , (13)

f 2 = A sin (γξ) cos 

γ2ξ + α



− Aγ cos (γξ) sin γ2ξ + α

 , (14)

f 3 = δ , (15)

where A, α, β, and δ are integration constants.

7
4. Determination of the Tetrad

At the point P0, the tetrad basis vectors satisfy equation (5). These equations constitute 12 relations (since eq (5) is symmetric in α and β) for 16

components λi

(a). However, the zeroth member of the tetrad is the 4-velocity,

which is assumed known,

λi

(0) = ui = γ



1, v1, v2, v3 , (16)

and is given by equation (3). Equation (5) can be rewritten as two equations:

gijλi

(α)λj

(0) = 0, α = 1, 2, 3 , (17)

gijλi

(α)λj

(β) = δαβ, α, β = 1, 2, 3 , (18)

where δαβ = 1 if α = β and 0 otherwise. Substituting the explicit form of

the Minkowski metric into equation (17) gives a condition on the zeroth

components of a tetrad vector if the spatial components are known (Greek

indices take values α, κ =1, 2, 3):

λ0

(α) = δβκvβλκ(α) . (19)

The use of equation (19) to eliminate the time components λ0

(α) from equation (18) leads to an orthogonality relation for only the spatial components

of the tetrad vectors:

δμνλμ

(α)

λν

(β) − δμνδκγvμvκλν(α)λγ

(β) = δαβ . (20)

The general idea is to solve equation (20) for the spatial components of the

tetrad at s = 0, and then to substitute the spatial components into equation

(19) to obtain the time components of each tetrad vector. Thus, I obtain the

tetrad vectors at the initial time corresponding to s = 0. For each tetrad

vector, I use these components as initial conditions in equations (12) to (15)

to determine the tetrad components at point P at finite s.

The exact solution of equation (20) can be obtained if we note that for vα =

0, the spatial components of the tetrad vectors are orthonormal, and the

solution is given by

λμ

(α) = δμ

α . (21)

8
I assume that the tetrad unit vectors are oriented approximately parallel

to the observer’s local Cartesian x, y, and z axes. For small vα  1, the

solution of equation (20) can be obtained as a triple power series in vα,

by iteration, starting with equation (21) as the first approximation. Having

solved for the tetrad components as a power series, it is easy to guess the

exact solution to be

λ0

(α) = γδαβvβ , (22)

λμ

(α) = δμ

α + γ − 1

ν2 δκαvκvμ , (23)

λi

(0) = ui = γ



1, v1, v2, v3 . (24)

where the components of dimensionless velocity vα are given in equation (3). Equations (22) to (24) give the components of the observer’s tetrad

basis vectors at point P0. At point P, the arc length is s > 0, and equations

(22) to (24) give the initial components for each tetrad vector. These initial

components are used in the general solution of a Fermi-Walker transported

vector, given in equations (12) to (15). The tetrad components at point P at

finite s are given by

λi

(0) = {γ, −γν sin(γξ), γν cos(γξ), 0} , (25)

λi

(1) = 

−γν sin(γ2ξ), cos(γξ) cos(γ2ξ) + γ sin(γξ) sin(γ2ξ), cos(γ2ξ) sin(γξ)

− γ cos(γξ) sin(γ2ξ), 0

	 , (26)

λi

(2) = 

γν cos(γ2ξ), −γ cos(γ2ξ) sin(γξ) + cos(γξ) sin(γ2ξ), γ cos(γξ) cos(γ2ξ)

+ sin(γξ) sin(γ2ξ), 0

	 , (27)

λi

(3) = {0, 0, 0, 1} . (28)

Equations (25) to (28) are valid for all proper times s, because they are exact

solutions of equations (8) to (11) and they satisfy the orthogonality relations

in equation (5) along the observer’s world line C.

9
5. Fermi Coordinates

Fermi coordinates are defined by the geometric construction shown in figure 2. Every event P in space-time has coordinates xi in the global inertial

coordinate system. According to the observer moving in circular motion

along a time-like world line, the same event has the Fermi coordinates X(a)

,

a = 0, 1, 2, 3. The first Fermi coordinate, X(0) = s, is just the proper time

(in units of length) associated with the event P

. The proper time for P is

defined as the value of arc length s such that a space-like geodesic from

point P passes through event P

, where the tangent vector of this geodesic,

μi

, is orthogonal to the observer 4-velocity at P:

μi ui|P = 0 . (29)

For the simple case of Minkowski space, the geodesic connecting the points

P and P is simply a straight line,

μi = N(xi − xi

(γs)) , (30)

where xi

(u) and xi are the coordinates of P and P, respectively, in the

global inertial frame, and N is a normalization constant that makes μi a

unit vector. The orthogonality condition in equation (29) is

gij μi

(s) λj

(0)(s)=0 (31)

and gives an implicit equation for s for a given P

. This orthogonality condition gives the first Fermi coordinate of the point P

,

X(0) = s , (32)

and equation (31) gives s as an implicit equation:

γs = x0  − x0

0 + ν




x1  − x1

0



sin γωs

c



− x2  − x2

0



cos γωs

c

 . (33)

In the limit of small speeds of the observer, ν → 0, equation (33) gives

s = x0  − x0

0. For ν  1, equation (33) can be solved for s by iteration,

resulting in a power series in ν:

10
X(0) = s = ∆x0 +



∆x1 sin 
∆x0ω

c



− ∆x2 cos 
∆x0ω

c

 ν − 1

2

∆x0ν2

− 1

2a



−a + 2∆x1 cos 
∆x0ω

c



+ 2∆x2 sin 
∆x0ω

c



×



∆x2 cos 
∆x0ω

c



− ∆x1 sin 
∆x0ω

c

 ν3 + O(ν)

4 , (34)

where ∆xi = xi − xi

0. For future reference, I label the coordinates of P,

P

, and P0 in the global inertial frame by P = (x0 , x1 , x2 , x3 ), P =

(x0, x1, x2, x3), and P0 = (x0

0, x10, x20, x30).

The contravariant spatial Fermi coordinates, X(α)

, α = 1, 2, 3, are defined

as [6]

X(α) = σμi

λ(α)

i = gij σ(s) μi

(s) η(αβ) λj

(β)

(s) , (35)

where I used the Minkowski definition of space-time distance σ, along the

space-like geodesic between P and P

:

σ2 = gij (xi − xi

)(xj  − xj) , (36)

where gij is the Minkowski metric and η(ij) = η(ij) = ηij is an invariant

matrix, which is numerically equal to the Minkowski metric. Since the parameter s in equation (35) is a function of P coordinates xi

, s must be

eliminated by equation (34), so that X(α) are explicit functions of the global

inertial coordinates xi

. I have calculated the resulting coordinate transformation correct to fourth order in ν. However, since the expressions are complicated, I write them here correct only to third order in ν:

X(1) = ∆x1 − a cos 
∆x0ω

c



+

1

4



∆x1 − ∆x1 cos 
2∆x0ω

c



− ∆x2 sin 
2∆x0ω

c

 ν2

+

1

2a

∆x0



−∆x2 + a sin 
∆x0ω

c

 ν3 + O(ν)

4 , (37)

X(2) = ∆x2 − a sin 
∆x0ω

c



+

1

4



∆x2 + ∆x2 cos 
2∆x0ω

c



− ∆x1 sin 
2∆x0ω

c

 ν2

+

1

2a

∆x0



∆x1 − a cos 
∆x0ω

c

 ν3 + O(ν)

4 , (38)

X(3) = ∆x3 . (39)

11
The inverse transformation can be obtained from equations (37) to (39) by

iteration and some tedious algebra:

x0  − x0

0 = X(0) +



X(2) cos 

ωX(0)

c



− X(1) sin 

ωX(0)

c

 ν +

1

2

X(0)ν2 + O(ν)

3 , (40)

x1  − x1

0 = X(1) + a cos 

ωX(0)

c



+

1

4



X(1) − X(1) cos 

2 ωX(0)

c



− X(2) sin 

2 ωX(0)

c

 ν2 + O(ν)

3 , (41)

x2  − x2

0 = X(2) + a sin 

ωX(0)

c



+

1

4



X(2) + X(2) cos 

2 ωX(0)

c



− X(1) sin 

2 ωX(0)

c

 ν2 + O(ν)

3 , (42)

x3  − x3

0 = X(3) . (43)

12
6. Metric in Fermi Coordinates

The space-time interval in the Fermi coordinate system of the observer is

ds2 = −G(ij) dX(i) dX(j) , (44)

where G(ij) are the metric tensor components when the X(i) are used as

coordinates. A direct calculation of the metric tensor components from the

tensor transformation rule,

G(ij) = gkl

∂xk

∂X(i)

∂xl

∂X(j) , (45)

using the transformation equations (40) to (43) and the Minkowski metric

for gij , gives G(αβ) = δαβ, G(0α) = 0, and

G(00) = −(1 + ζ)

2 , (46)

where ζ is given by

ζ =



1 − ∆x1

a

cos 
ω∆x0

c



− ∆x2

a

sin 
ω∆x0

c

 ν2 + O(ν4) . (47)

Using the transformation equations (40) to (43) to express ζ as a function of

the Fermi coordinates, I obtain

ζ = −1

a




X(1) cos ω

c

X(0)

+ X(2) sin ω

c

X(0) ν2 + O(ν4) . (48)

The result in equation (46) agrees with the general theory of Synge [6], specialized to flat space. Synge shows that

ζ = X(β) w(β) = η(αβ) X(α) w(β) = η(αβ) X(α) wi λ(β)

i . (49)

To third order in ν, equation (48) agrees with the general theory given by

Synge. The quantities X(β) = η(αβ) X(α) are the covariant Fermi coordinates, and w(β) = wi λ(β)

i are the components of the observer’s 4-acceleration

in the Fermi coordinate system. The 4-acceleration wi is related to the ordinary 3-d acceleration aβ, β = 1, 2, 3, by

wi = γ2

c2

d2xi

dt2 = γ2c2 ai . (50)

13
Using this relation, we can write ζ in terms of the global inertial coordinates

of P and P as

ζ = γ2

c2 δαβ 

xα  − xα(s)



aβ . (51)

For the circular motion treated here, from equation (48), the three spatial

components of the observer’s acceleration in Fermi coordinates are given

by

w(β) = −ν2

a



cos ω

c

X(0)

,sin ω

c

X(0)

, 0



(52)

where I have dropped fourth-order terms in ν. To third order in ν, the observer’s acceleration is just the classical centripetal acceleration for an observer moving in a circle.

The quantity c2ζ is a time-dependent potential that determines the rate of

proper time with respect to coordinate time in Fermi coordinates. Proper

time is kept by an ideal clock. Coordinate time depends on the definition of

the reference frame and coordinates used within that frame. The quantity

ζ, given in equation (51), depends on acceleration aβ and the difference in

coordinates, xα  − xα(s), from the spatial origin of the Fermi coordinates.

Away from the spatial origin of Fermi coordinates, the acceleration produces a periodic time-dependent effective gravitational potential field c2ζ,

which leads to a change in the relation of coordinate time to proper time,

through the metric components G(ij).

14
7. Apparent Behavior of Clocks in Fermi Coordinates

The significance of proper time is that it is the quantity that ideal clocks

keep, that it may be related to time on real clocks, and that it is closely

related to measurable quantities. The significance of coordinate time is that

it enters into the theory that defines the space-time grid. In order to analyze

experiments with clocks, we must relate what is measured (i.e., proper time

intervals between space-time events) to coordinate positions and times of

these events.

Consider an ideal clock whose world line is given in Fermi coordinates by

X(i) = X(i)

(X(0)). At coordinate times X(0)

A and X(0)B , the clock is at spatial

positions X(α)

A and X(α)B , respectively. The elapsed proper time between

these two events, ∆τ = ∆s/c, is given by the integral along the world line

of the clock:

∆s =

 B

A



−G(ij)

dX(i)

dX(0)

dX(j)

dX(0)1/2

dX(0) . (53)

7.1 Stationary Clock

For the case of a stationary clock, located at constant Fermi coordinate position X(α)

, we obtain the relation between proper time τ and coordinate

time X(0) by taking dX(α) = 0 in equation (44). The rate of proper time with

respect to coordinate time is then given by

ds

dX(0) =1+ ζ



X(0), X(α)



. (54)

Equation (54) is a special case of the well-known result that the rate of

proper time depends on the location of the clock [31]. For the simple case of

circular motion of the observer with ζ given by equation (48), substitution

in equation (54) and integration from X(0)

A to X(0)B gives

∆s = X(0)

B − X(0)A − aω

c



X(1) 


sin ω

c

X(0)

B



− sin ω

c

X(0)

A



− X(2) 


cos ω

c

X(0)

B



− cos ω

c

X(0)

A

 , (55)

where I have dropped terms of O(ν4). Equation (55) shows that there is

a complicated relation between elapsed proper time on a clock located at

15
Fermi coordinate position (X(1), X(2)) and Fermi coordinate time X(0). For

example, a clock on the X(1) axis (with X(2) = 0) has a periodic time with

respect to the observer’s Fermi coordinate time, given by ∆s = X(0)

B −

(aω/c) X(1) sin(ωX(0)

B /c). The amplitude and sign of this periodicity depend on X(1), the clock’s spatial Fermi coordinate. The initial conditions

in equation (55) determine complicated phase relations between proper

time and coordinate time. This complicated behavior can, of course, be

understood if we consider the clock’s complicated motion in the global

Minkowski coordinate system.

7.2 Clock in Circular Motion

Now consider an ideal clock that is a distance b from the origin of Fermi

coordinates and that executes circular motion in Fermi coordinates in the

X(1)-X(2) plane. I take the world line for this motion to be given by

X(1) = b cos 
Ω

c

X(0)

, (56)

X(2) = b sin 
Ω

c

X(0)

, (57)

X(3) = 0 . (58)

The clock moves in the X(1)-X(2) plane, which coincides with the plane

defined by the circular motion of the origin of Fermi coordinates (see fig. 2).

Substitution of the satellite clock’s world line into equation (53) leads to the

following relation between elapsed proper time observed on the clock and

Fermi coordinate time:

∆s =



1 − 1

2


bΩ

c

2

 

X(0)

B − X(0)A



− ab

c

ω2

Ω − ω



sin 
Ω − ω

c

X(0)

B



− sin 
Ω − ω

c

X(0)

A

 . (59)

The rate of proper time with respect to coordinate time is

ds

dX(0) = −1

2


bΩ

c

2

−

ω

c

2

ab cos 
Ω − ω

c

X(0)

. (60)

The proper time has a constant rate offset from coordinate time, represented by the first term in equation (59). In addition, the proper time has a

periodic component with respect to coordinate time, given by the second

term in equation (59).

16
To demonstrate the magnitude of these effects, I use parameters that correspond to the Earth orbiting the sun and to GPS satellites orbiting the

Earth [24]. I compute magnitudes of the following terms with the values

shown in table 1:

1

2


bΩ

c

2

= 8.348 × 10−11 , (61)

1

2


bΩ

c

2

×

2π

Ω = 3.597 × 10−6 s . (62)

Equation (61) gives the constant rate offset of the moving clock with respect

to Fermi coordinate time, due to time dilation. In the GPS, this term leads

to a net slowing down of GPS clocks by approximately 7 μs per day [5],

which results from multiplying the value given in equation (62) for a single

revolution by a factor of 2 (since the GPS satellites make approximately two

revolutions per Earth day).

The second term in equation (59) is a periodicity of the proper time with

respect to Fermi coordinate time, with amplitude

ab

c2

ω2

Ω − ω

= 12.05 × 10−9 s , (63)

which corresponds to an amplitude in the rate of

ab ω

c

2

= 1.755 × 10−12 . (64)

The proper time given by equation (59) is periodic with respect to Fermi coordinate time with the difference frequency Ω − ω, which is the difference

frequency between satellite and observer rotations. This is a kinematic effect due to the type of coordinate system used (Fermi coordinates) and the

fact that the observer (the Fermi coordinate frame) is moving along an arc

of a circle, and therefore experiences an acceleration. Note that the periodic

term vanishes when the observer angular velocity ω = 0. If the observer

were moving in a straight line and hence had zero acceleration, wi = 0,

Table 1. Numerical values

of constants. Constant Definition Value

a Earth orbital semi-major axis 1.496 × 1011 m

ω Earth orbital angular velocity 1.99238 × 10−7 s

−1

b GPS satellite semi-major axis 2.656177 × 107 m

Ω GPS satellite angular velocity 1.45842 × 10−4 s

−1

c Vacuum speed of light 2.997924 × 108 m/s

17
the periodic effect would also be absent, as can be seen from the general

equation (49). Although these kinematic effects are present in the GPS, one

must be careful in applying these results to GPS because gravitational effects have been neglected in this calculation and including them would

significantly modify the results [17,21].

7.3 Sagnac-Like Effect for Portable Clocks

Sagnac demonstrated that there is a phase shift between two counterpropagating light beams on a rotating platform and that this phase shift depends

on the angular frequency of rotation of the platform [27,28,30]. In the Fermi

coordinates of the observer moving in a circle, there is a Sagnac-like effect

for two portable clocks that are orbiting in opposite directions.

Consider a clock that moves eastward along a circle of radius b in the equatorial plane of the Fermi coordinates with world line given by equations

(56) to (58) (see fig. 2). The speed of the clock in Fermi coordinates is bΩ/c.

Assuming that the clock begins its journey at point A at time X(0)

A = 0 and

ends it at X(0)

B = 2πc/Ω (which corresponds to one revolution), the proper

time that elapses on the clock is given by equation (59) as

∆s+ =



1 − 1

2


bΩ

c

2



2π

Ω c +

ab

c

ω2

Ω − ω

sin 2π

ω

Ω



. (65)

The first term in equation (65) is the time dilation due to the speed bΩ/c of

the clock. The second term depends on the path traversed.

Next, consider an identical clock that starts at point A at time X(0)

A = 0 but

moves westward. The world line of this clock is then given by equations

(56) to (58) but with Ω = −|Ω|. The proper time on this clock is given by

∆s− =



1 − 1

2


bΩ

c

2



2π

|Ω|

c − ab

c

ω2

|Ω| + ω

sin 
2π

ω

|Ω|



. (66)

When the east-moving and west-moving clocks have each made one revolution and they have returned to their starting positions at point A, the

difference in their times is given by ∆τ+ − ∆τ− = (∆s+ − ∆s−)/c. For the

case b  a and ω  |Ω| (as is the case with Earth satellites), the difference

of proper time on the two clocks is given by

τ+ − τ− = 2

b

a

aω

c

2 |Ω|

Ω2 − ω2 sin 


2π

ω

|Ω|



= 4π

b

a

aω

c

2 ω

Ω2 + O


ω2

Ω2



, (67)

18
where in the last line I have dropped terms of O(ω2/Ω2). This Sagnac-type

effect is due to the noninertial nature of the Fermi coordinate system. The

origin of this effect can be understood from the point of view of the global

inertial coordinate system. If the Fermi coordinate origin moved along a

straight line in 3-d space (rather than a circle), then there would be no acceleration and ζ = 0. In this case, the east-moving and west-moving clocks

would have zero time difference after one revolution. However, since the

origin of the Fermi coordinates moves along a helix, there is an acceleration that picks out a direction in the Fermi coordinate space-time, and

the symmetry between the two clock world lines is broken. Consequently,

the elapsed proper time is different on the two clocks when they are compared after one revolution. Note that the time difference on the two clocks

increases with increasing angular velocity of rotation ω, analogous to the

effect observed by Sagnac [27]. This increase arises because the breaking

of symmetry is due to the magnitude of the acceleration w(β) (see eq (49)).

For values of the parameters appropriate to GPS satellites, given in table

1, I find the magnitude of the time difference on the two clocks after one

revolution to be

τ+ − τ− = 2.066 × 10−10 s . (68)

This is a small effect, which vanishes when the origin of Fermi coordinates

moves along a straight line or when ω = 0.

19
8. Coordinate Speed of Light

The 3-d space of Minkowski space-time is homogeneous and isotropic, and

the speed of light is independent of position and direction of propagation.

However, for the case of Fermi coordinates of an observer moving in a circle, the symmetry of the 3-d space is broken by an acceleration, and the

coordinate speed of light depends on position in the 3-d space.

For a general space-time metric gij , the coordinate speed of light vc in the

direction of a unit vector eα is given by Moller [29]:  ̈

vc =

√−g00

1 + γαeα c , (69)

where

γα = g0α √−g00

(70)

and the unit vector satisfies the relation γαβ eα eβ = 1, where the 3-d spatial

metric is given by [31]

γαβ = gαβ − g0α g0β

g00

. (71)

Equation (69) shows that for a general space-time metric, the speed of

light depends on direction eα and on position and time, through the metric components. A nonzero off-diagonal metric component g0α leads to

an anisotropy (directional dependence) of the coordinate speed of light.

The metric for Fermi coordinates given in equation (46) has vanishing offdiagonal terms, γα = 0, and consequently the coordinate speed of light is

isotropic. However, the metric component G(00) has a nontrivial position

and time dependence, and equation (69) gives the coordinate speed of light

as vc = c (1 + ζ), where ζ is a function of Fermi coordinate position and

time, given by equation (48). Therefore, the coordinate dependence of the

speed of light is

∆c

c = −ν2a




X(1) cos ω

c

X(0)

+ X(2) sin ω

c

X(0) + O(ν4) . (72)

This variable speed of light is a kinematic effect due to the noninertial

nature of the Fermi coordinate system. Note that at any given position

20
X = {X(1), X(2), X(3)}, the speed of light depends on time in a periodic

way, with the orbital period of the reference frame, ω. This expression can

be rewritten as

∆c

c = a

ω

c

2

X · nˆ + O(ν4) , (73)

where the unit vector nˆ = 

− cos ω

c X(0)

, − sin ω

c X(0)

, 0

 is in the observer’s orbital plane and points in the direction of acceleration, toward the

center of the circle (see fig. 3). As the Fermi reference frame moves around

the circle, the vector nˆ always points toward the circle’s center in Fermi coordinates. At any given time X(0), equation (73) shows that to second order

in ν, in the plane given by

X · nˆ = 0 , (74)

the speed of light does not differ from its vacuum inertial frame value c. In

the 3-d half-space containing positions X closer to the center of the circle,

the speed of light is increased. In contrast, in the 3-d half-space containing

positions X that are farther from the center of the circle, the speed of light

is decreased. As time X(0) increases, the dividing plane, given by equation

(74), rotates at angular frequency ω around the circle, with the origin of

Fermi coordinates remaining the point at which the plane is tangent to the

circle.

The magnitude of the inhomogeneous variation in the speed of light in

equation (72) is small. For the kinematics of Earth satellites such as GPS,

Figure 3. Global inertial

coordinate axes xi and

Fermi coordinates X(a) are

shown in relation to

coordinate position X, unit

vector nˆ, and plane

X · nˆ = 0, at a given

coordinate time X(0). X(1)

X(2)

n^

X

x1

x2

plane

21
with the numbers given in table 1 and the coordinates X(1), X(2) ∼ b, the

effect has a magnitude

ν2

a

b = 1.755 × 10−12 (75)

and has a complicated position and time dependence, given by equation

(72). Recently, GPS data have been used [32] in obtaining an upper bound

on the anisotropy of the speed of light: ∆c/c < 10−9. Given the present

accuracy of GPS [24], the small inhomogeneity of the speed of light given

by equation (72) is probably not measurable.

22
9. Summary

In this report I present a detailed calculation of the coordinate transformation from global Minkowski coordinates to the Fermi coordinates of an observer moving in a plane around a circle in 3-d space. The transformation is

valid for arbitrarily long coordinate time (not a power series in time). The

observer’s axes are Fermi-Walker transported along his helical world line. I

considered two cases: stationary clocks (with respect to Fermi coordinates)

and satellite clocks in circular orbit with respect to Fermi coordinates. The

proper time interval between two events on the stationary clock, given by

equation (55), is a complicated periodic function of the clock’s Fermi coordinate time and coordinate position in the plane perpendicular to the rotation axis. I considered next a clock that orbits the Fermi coordinate origin

(similar to a satellite) in the equatorial plane of rotation and computed the

relation of proper time to coordinate time. I find that the proper time on the

clock has time dilation (constant rate offset) plus a periodic dependence on

Fermi coordinate time at the difference frequency, Ω − ω, between satellite

clock orbital frequency Ω and Fermi coordinate frame orbital frequency ω.

Next, I looked at the time difference of two counter-orbiting clocks in the

equatorial plane of rotation, one west-moving and the other east-moving.

After one revolution, the difference in proper time on the clocks shows a

small Sagnac-like effect, increasing with the angular frequency of revolution of the Fermi frame, ω, given by equation (67) .

Finally, I computed the coordinate speed of light in the Fermi coordinates.

The speed of light is isotropic but depends on Fermi coordinate position

and time. Equation (73) gives the change in the speed of light from its vacuum value in an inertial frame. In order to illustrate the magnitude of these

effects, I have used numbers relevant to GPS satellite orbits and Earth orbital frequency.

Acknowledgements

The author thanks Tom Van Flandern for useful discussions.

23
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24
13. Fermi, E. (1922). Atti R. Accad. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 31,

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21. Fukushima, T. (1988). Celest. Mech. 44, 61.

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Math. Phys. 35, 6224.

23. Nelson, R. A. (1990). Gen. Rel. Grav. 22, 431.

24. Kaplan, E. D. (1996). In Understanding GPS: Principles and Applications

(Mobile Communications Series, Artech House, Boston).

25. Misner, C. W., K. S. Thorne, and J. A. Wheeler (1973). In Gravitation

(W. H. Freeman and Company, New York).

26. Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the Theory of Relativity (John Wiley and Sons, New York)

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Phys. Radium 5th Ser. 4, 177.

28. For a review, see Post, E. J. (1967). Rev. Mod. Phys. 39.

29. Moller, C. (1972). The Theory of Relativity (Second Edition, Clarendon

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25
31. Landau, L. D., and E. M. Lifshitz (1975). In Classical Theory of Fields

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26
Appendix. Conventions

I use Roman indices, i, a = 0, 1, 2, 3, on space-time coordinates xi and Greek

indices, α = 1, 2, 3, for spatial coordinates. The proper time interval between two events in Minkowski space-time is dτ = ds/c, where

ds2 = −gij dxi dxj ; (A-1)

here, g00 = −1, gαβ = δαβ, and g0α = 0. In Fermi coordinates, X(a)

, a =

0, 1, 2, 3, the interval is

ds2 = −G(ab) dX(a) dX(b) . (A-2)

27
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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Fermi Coordinates of an Observer Moving in a Circle

in Minkowski Space: Apparent Behavior of Clocks

May 2000 Interim, 3/98–10/98

Space-time coordinate transformations valid for arbitrarily long coordinate time are derived from

global Minkowski coordinates to the Fermi coordinates of an observer moving in a circle in

three-dimensional space. The metric for the Fermi coordinates is calculated directly from the

tensor transformation rule. The Fermi coordinates are used in an examination (from the

observer‘s reference frame) of the detailed behavior of ideal clocks.

A complicated relation exists between Fermi coordinate time and proper time on stationary

clocks (in the Fermi frame) and between proper time on satellite clocks that orbit the

observer. For portable clocks that orbit the Fermi coordinate origin, an orbital Sagnac-like

effect exists. The coordinate speed of light is isotropic but is periodic in time and varies

with Fermi coordinate position. In a numerical illustration of the magnitudes of these

kinematic effects, this report uses parameters relevant to clocks carried aboard

satellites orbiting the Earth.

GPS, navigation, time transfer, timing, relativity clocks, Fermi coordinates,

reference frame, Sagnac effect

Unclassified

ARL-TR-2211

0NENCC

611102.H47

AH47

61102A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release;

distribution unlimited.

38

Unclassified Unclassified UL

2800 Powder Mill Road

Adelphi, MD 20783-1197

U.S. Army Research Laboratory

Attn: AMSRL-SE-EP bahder@arl.mil

U.S. Army Research Laboratory

31

Thomas B. Bahder

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197
DEPARTMENT OF THE ARMY

U.S. Army Research Laboratory2800 Powder Mill RoadAdelphi, MD 20783-1197

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