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GPS - RELATIVITY AND THE GLOBAL POSITIONING SYSTEM It’s...

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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 gravitational-wave 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 right-circularly 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, soonszzsszos-ozoz 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,- from-satellites 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 propagation-delay 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 cesium-beam 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 second-order 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. Self-consistent 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 self-consistent 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 Earth-centered 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 second-order 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 sea-level surface—is a surface of constant effective gravitational potential U in an Earth-fixed 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 Earth-centered, and it neglects the presence of other Solar-system 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 so-called 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 coarse-acquisition 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 first-order 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. First-order 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 second-order 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 Earth-fixed, rotating coordinate system, the so-called WGS-84 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 WGS-84 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 ring-laser 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 Earth-fixed 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 general-relativistic 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 path-dependent 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 primary-reference 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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