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Unformatted text preview: RELATIVITY AND THE GLOBAL
POSITIONING SYSTEM It’s been almost a century
since Einstein intro—
duced the special theory of
relativity. All observa
tional tests to date conﬁrm
both the special and the
general theory. These tests We need general relativity to understand
extreme astrophysical realms. But the
theory also turns out to be essential for
the many mundane activities that
nowadays rely on the precision of signal reaches the ground,
its intensity is only about
3 X 10—14 W/mz. To process
such faint signals, GPS
receivers must implement
very special techniques.
Figure 2 shows the new have ranged from sensitive the GPS Block IIR satellites on an
laboratory experiments in ' assembly line. volving optics, atoms, nu Data transmitted by
clei, and subnuclear parti— ‘ . the satellites are continu—
cles to the observation of Nell ASth ously monitored by receiv orbiting clocks, planets,
and objects far beyond the Solar System. The general theory of relativity will soon be tested
with high precision by Stanford University’s Satellite Test
of the Equivalence Principle (STEP), 1 and observations by
the worldwide array of gravitationalwave detectors
presently under construction are expected to test the the
my in the extreme realm of strong gravitational ﬁelds and
high velocities (see the articles by Clifford Will and by
Barry Barish and Rainer Weiss in PHYSICS TODAY, Octo
ber 1999, pages 38 and 44, respectively). Numerous relativistic issues and effects play roles in
the global positioning system, on which millions of driv
ers, hikers, sailors, and pilots depend to ﬁnd out where
they are. The GPS system is, in effect, a realization of Ein
stein’s View of space and time. Indeed, the system cannot
function properly without taking account of fundamental
relativistic principles. That is the subject of this article. The global positioning system
The orbiting component of the GPS consists of 24 satellites  (plus spares): four satellites in each of six different planes inclined 55" from Earth’s equatorial plane. The satellites
are positioned within their planes so that, from almost any
place on Earth, at least four are above the horizon at any
time. Orbiting about 20 000 km above Earth’s surface, all
the satellites have periods of 11 hours and 58 minutes.
Because that’s half a sidereal day, a ﬁxed observer on the
ground will see a given satellite at almost exactly the same
place on the celestial sphere twice each day. Each satellite
carries one or more very stable atomic clocks, so that the
satellites can transmit synchronous timing signals. The
signals carry coded information about the transmission
time and position of the satellite. Figure 1 shows one of the new generation of GPS
orbiters (called Block IIR satellites) that have recently
begun replacing the older generation. Its antenna array
efﬁciently beams rightcircularly polarized radio signals
toward Earth’s surface. By the time the spreading radio NEIL ASHBY is a professor ofpbysz'cs at the University of Colorado in
Bowl . Since 1974, be has been a consultant to NIST, Boulder; on rela tivistic gym on clocks. __
, e 200:. American Insstuzﬁf Physics, soonszzsszosozoz ing stations around the
globe and forwarded to a master control station, where
satellite orbits and clock performance are computed. The
resulting orbital and clock data are then uploaded to the
satellites for retransmission to users. The fundamental principle on which GPS navigation
works is an apparently simple application of the second
postulate of special relativity—namely, the constancy of c,
the speed of light. Referring to ﬁgure 3, suppose that a
receiver, on or near Earth’s surface, simultaneously
receives signal pulses from four satellites, transmitted at
times t, fromsatellites at positions r,. Then the position r
of the receiver and the time t on its clock when the four _ signals arrive can be determined by solving four simulta neous equations .
Ir — ril = C(t — ti); i= 1, 2, 3, 4.  (1) These propagationdelay equations, strictly valid in an
inertial frame, are the basis for position and time deter
mination by the GPS receivers. Accurate navigation with the GPS is made possible by
the phenomenal performance of modern atomic clocks.2 If
navigation errors of more than a meter are to be avoided,
an atomic clock must deviate by less than about 4 nanosec _onds from perfect synchronization with the other satellite clocks. That amounts to a fractional time stability of bet
ter than a part in 1013. Only atomic clocks can do that.
Even so, the system requires frequent uploads of clock cor
rections to the satellites. The reference for GPS time is a composite clock based
on the US Naval Observatory’s ensemble of about 50
cesiumbeam frequency standards and a dozen hydrogen
masers. Clock times on GPS satellites usually agree with
the observatory’s ensemble to within about 20 ns. Relativistic effects are much larger than a part in 1013.
For example, satellite speeds v are about 4 kin/s. Time dila
tion then causes the moving clocks’ frequencies to be slow by Aﬂf = 112/2c2 210'”. Gravitational effects are even ' larger. In fact, relativistic effects are about 10 000 times
too large to ignore. Suppose one wanted to improve GPS spatial precision ' so that receiver positions could be determined with an
uncertainty of only a centimeter. A radio wave travels 1 cm
in 0.03 ns. So one would have to account for all temporal MAY 2002 PHYSICS TODAY 41 FIGURE 1. GLOBAL POSITIONING SYSTEM _ .,
satellite of the new Block IIR generation '
that has begun replacing the older genera _
tion. Several dozen GPS satellites are in '
orbit at any one time. The antenna array ,
broadcasts circularly polarized L—band 1
microwave signals to Earth. Each satellite
carries one or more atomic clocks
The extended planes are solar
photovoltaic panels relativistic effects down
to a few hundredths of
a nanosecond. But the
secondorder Doppler
shift of an orbiting
atomic clock, if it were
not taken into account,
would cause an error
this large to build up in
less than half a second.
An effect of compara—
ble size is contributed
by the gravitational blueshift, which results when a pho
tonaor a clock—moves to lower altitude. If these rela
tivistic effects were not corrected for, satellite clock errors
building up in just one day would cause navigational
errors of more than 11 km, quickly rendering the system
useless. Selfconsistent synchronization Clocks moving along different trajectories in space and on
Earth undergo different gravitational and motional fre
quency shifts. The “proper times” recorded by all these
clocks in their own rest frames quickly diverge. Therefore
one needs some reasonable means of synchronization, in
order that equations 1 have their intended meaning—
expressing signal propagation at speed c in straight lines
in an inertial frame. The times t, at which the transmis
sions originate must be established by a selfconsistent
Synchronization scheme. In Earth’s neighborhood, the ﬁeld equations of general
relativity involve only a single overall time variable. While
there is freedom in the theory to make arbitrary coordi
nate transformations, the simplest approach is to use an approximate solution of the ﬁeld equations in which ' Earth’s mass gives rise to small corrections to the simple
Minkowski metric of special relativity, and to choose coor
dinate axes originating at the planet’s center of mass and
pointing toward ﬁxed stars. In this Earthcentered iner
tial (ECI) reference frame, one can safely ignore relativis
tic eﬁ'ects due to Thomas precession or Lense—Thirring
drag. The gravitational effects on clock frequency, in this
frame, are due to Earth’s mass and its multipole moments. In the E01 frame, the fundamental invariant space
time interval ds2 of general relativity can be written in the
approximate form d52 = —(1 +.2i‘l)(cdt')2 + ( C2 1—32 CZ )(clx2 + (13/2 + (122), (2) where (I) < 0 is the Newtonian gravitational potential. For
the GPS, we can ignore terms of order smaller than c”?
The variable t’ in the equation is called the coordinate 42 MAY 2062 PHYSICS TODAY time. In general relativity, one can construct a consistent
spacetime coordinate system for a “patch” that encom
passes Earth and its GPS satellites without having to
resort to more than the one such time variable. One can
think of this coordinate time as the proper time on an
atomic clock at rest far away from Earth‘s gravity. imam oasuxoor However, the rate of International Atomic Time (TM) ' is based on atomic clocks resting essentially at sea level,
where they are subject to secondorder Doppler shifts due
to Earth’s rotation and gravitational redshifts relative to
clocks 20000 km higher up. The two different time vari
ables can be reconciled by sealing the rate of coordinate
time so that it matches the rate of TM. The time variable
it actually used in the GPS is related to the coordinate time
t' of equation 2 by t’ = t(1 — U/cz), where the constant
parameter U includes motional effects due to Earth’s rota
tion and gravitational effects from its mass distribution. It is very useful that Earth’s geoid—the planet’s ide
alized sealevel surface—is a surface of constant effective
gravitational potential U in an Earthﬁxed rotating refer
ence frame, so that all atomic clocks at rest on the geoid
tick at the same rate. That’s a nontrivial consequence of a
combination of effects arising from time dilation and the
multipole expansion of Earth’s nonspherical mass distri
bution.“ To an approximation good enough for the GPS,
the constant U can be calculated in terms of Earth’s mass,
its quadrupole moment, and its rotational angular veloc
ity 0E. Then the metric can be written as d32 = — (1 + M) (cdt)2 + c2
(3) ( — 3%de + dy2 + ea),
C2 and the proper rate of all atomic clocks at rest on the geoid
will be given by dt = ds/c. For an atomic clock moving along some arbitrary path,
one can envision measuring the clock’s proper time incre ment ds/c, solving equation 3 for dt, and then integrating
dt along the path to get the elapsed coordinate time t. http://www.physicstodeiyorg Thus, for each atomic clock, the GPS generates a “paper
clock” that reads t. All coordinate clocks generated in this
way would be self~consistently synchronized if one brought
them together—assuming that general relativity is cor—
rect. That, in essence, is the procedure used in the GPS?4 In equation 3, the leading contribution to the gravita
tional potential <13 is the simple Newtonian term —GME Ir.
The picture is Earthcentered, and it neglects the presence
of other Solarsystem bodies such as the Moon and Sun.
That they can be neglected by an observer sufﬁciently close
to Earth is a manifestation of general relativity’s equiva
lence principle.5 In the ECI frame, the only detectable effects of distant
masses are their residual tidal potentials. Tidal effects on
orbiting GPS clocks due to the Moon and Sun amount to less
than a part in 1015. Currently they are ignored. But tidal
potentials do have a signiﬁcant effect on satellite orbits. GPS receivers The GPS system transfers transmission coordinate times
t, to a receiver in a very sophisticated manner. The prin—
cipal signal currently used by nonmilitary receivers is
the socalled L1 signal at 1575.42 MHz. This frequency is
an integral multiple of 10.23 MHz, a fundamental fre
quency synthesized from an atomic clock aboard each
satellite. The satellite’s transmitter impresses upon this
Sinusoidal carrier wave a unique digital code sequence
(the coarseacquisition or C/A code), repeated once each
millisecond. The bits are encoded by reversing the phase of the car
rier wave for a 1, and leaving the phase unchanged for a
0. This choice of encoding mode is important, because the
phase of an electromagnetic wave is a relativistic scalar.
The phase reversals correspond to physical points in
spacetime at which—for all observers—the electric and
magnetic ﬁelds vanish. For the L1 signal, each bit lasts 1540 carrier cycles.
This rather large number of cycles is not wasted; much of
it is used for a very fast encrypted military code, 90° out of phase with the CIA code. Civilian navigation informa _ tion is encoded on top of the CIA code at 50 bits per sec http://www.phy;icstoday.org IGURE 2. THE NEW BLOCK IIR global posi ioning system satellites on the assembly line ; at Lockheed Martin. A third generation is on
the drawing boards. end. The navigation data
include information from
which the satellite’s posi
tion and clock time can be
accurately determined, and
an almanac from which
approximate positions of
other GPS satellites can be computed. The timing signal
corresponds to a phase reversal at a particular place in the
navigation code sequence. Every civilian GPS receiver carries circuitry that lets
the receiver generate code sequences corresponding to the
CIA code sequences from all the satellites. Many such
sequences can be generated in parallel, depending on the
sophistication of the receiver. Because the satellite is moving with respect to the
receiver, there is a ﬁrstorder Doppler shift of the received
carrier signal, of order v/c : 105. A receiver may incorpo
rate hundreds, or even thousands, of correlators that search
in parallel through different frequency shifts and time off
sets by comparing its own code sequence with those it
receives. When an apprOpriately high correlation is found,
the receiver locks onto the signal. The uniqueness of the
transmitted code sequences lets the receiver identify which
satellite a signal is coming from. Firstorder Doppler shifts,
sometimes measured to within a few hertz, are used by some
receivers to aid in extrapolating navigation solutions for
ward in time. With the receiver locked onto a signal and the Doppler
shift matched, timing information is obtained by compar
ing the receiver’s clock time tr'with the time ticks encoded
in the signal, thus measuring the “pseudoranges” C(t, — t,),
which are simply related to the right side of equations 1. The relativistic fractional frequency shifts that con
cern us most—for example, the secondorder Doppler
shifts due to the motion of the orbiting clocks relative to
the receivers—are a few parts in 101". These clocks are also
very high up in Earth’s gravity field and therefore suffer
a gravitational frequency shift, given by —=—: (4) where Ad) is the gravitational potential difference between the satellite and the geoid. This gravitational shift causes ' clocks in GPS satellites to run faster than otherwise iden
tical clocks on the ground by about 5 X 10‘”. Furthermore,
because none of the orbits is perfectly circular, a satellite MAY 2002 PHYSICS TODAY 43 ' FIGURE 3. THE FUNDAMENTAL PRINCIPLE of the global posi
tioning system is the constancy of the speed of light c in an
inertial frame. If a receiver on the ground simultaneously
receives signals from {our GPS satellites above its horizon, the
distance D to each is given by 03:, where AI is the time interval
between transmission and reception. But because the satellites
and the receiver are moving through the local inertial frame
and are at different gravitational potentials, their clocks cannot
be synchronized without taking account of relativistic effects. speeds up, or slows down, to conserve angular momentum
as its distance from Earth varies along its orbit. That Kep
lerian variation periodically changes the second—order
Doppler shift, while changing the gravitational frequency
shift in the same sense. Diurnal rotation and the Sagnac effect Computations of satellite orbits, signal paths, and rela
tivistic effects appear to be most convenient in an ECI
frame. But navigation must generally be done relative to
Earth’s surface. So GPS navigation messages must allow
users to compute satellite positions in an Earthﬁxed,
rotating coordinate system, the socalled WGS84 refer
ence frame.6 The navigation messages provide ﬁctitious orbital ele~
ments from which a user can calculate the satellite’s posi—
tion in the rotating WGS84 frame at the instant of its sig
nal transmission. But this creates some subtle conceptual
problems that must be carefully sorted out before the most
accurate position determinations can be made. For exam
ple, the principle of the constancy of c cannot be applied in
a rotating reference frame, where the paths of light rays
are not straight; they spiral. One of the most confusing relativistic effects—the
Sagnac effect—appears in rotating reference frames.7 (See
PHYSICS TODAY, October 1981, page 20.) The Sagnac effect
is the basis of the ringlaser gyroscopes now commonly
used in aircraft navigation. In the GPS, the Sagnac effect
can produce discrepancies amounting to hundreds of
nanoseconds. Observers in the nonrotating ECI inertial frame
would not see a Sagnac effect. Instead, they would see that
receivers are moving while a signal is propagating.
Receivers at rest on Earth are moving quite rapidly (465
m/s at the equator) through the ECI frame. Correcting for
the Sagnac effect in the Earthﬁxed frame is equivalent to
correcting for such receiver motion in the ECI ﬁ'ame. Sup—
pose one sends a radio wave in a circle around the equa
tor, from west to east, in an attempt to synchronize clocks
along the path, invoking the constancy of c. Observers in
the ECI frame see this wave propagating eastward a dis
tance x in time x/c. Clocks in the signal’s path move away
from the wavefront with speed (1,, R, where 0,, is Earth’s
angular velocity and R is its radius. The distance such a
clock moves in a time sale is OE Rx/c, and it takes an addi
tional time OE Ric/c2 for the beam to catch up. For one com
plete circuit of the equator, this additional time is about ' 200 ns. Sending the signal in the opposite direction
reverses the effect’s sign. The Sagnac effect also occurs if an atomic clock is
moved slowly from one reference station on the ground to
another. For slow clock transport, the effect can be viewed
in the ECI frame as arisingfrom a difference between the
time dilation of the portable clock and that of a reference
clock whose motion is solely due to Earth’s rotation.
Observers at rest on the ground, seeing these same asym  44 _MAY 2002s PHYSICS TODAY metric effects, attribute them instead to gravitomagnetic
effects—that is to say, the warping of spacetime due to
spacetime terms in the generalrelativistic metric tensor.
Such terms arise when one transforms the invariant ds2
from a nonrotating reference frame to a rotating frame.7 Thus, attempts to establish a network of synchronized
clocks on Earth’s surface are subject to asymmetric, path
dependent effects arising from the planet’s rotation. When
atomic clocks became accurate enough for these effects to
be signiﬁcant, various proposals were made to deal with
the Sagnac effect. One such proposal involved placing a
discontinuity in TAI at the International Date Line. But
such a scheme would not avoid pathdependent effects. Synchronization should be an equivalence relation. To
make it so, one could use the coordinate time. To achieve
consistently synchronized clocks on Earth’s surface at the
subnanosecond level, the Consultative Committee for the
Deﬁnition of the Second and the International Radio Con
sultative Committee have agreed that the correction term
to be applied for the Sagnac effect should be ZOEAE/cz,
where AE is the projected area on Earth’s equatorial plane
swept out by a vector from Earth’s center to the position
of the portable clock or signal pulsed7 (AE is taken to be
positive if the head of the vector moves eastward.) The Sagnac effect is particularly important when GPS
signals are used to compare times of primaryreference
cesium clocks at national standards laboratories far from
each other. Because their locations are very precisely
known, each laboratory needs only one of the equations 1
to obtain GPS time from a satellite. The measurements are
made in “comm...
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