Relativity in the Global Positioning System
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Abstract
The Global Positioning System (GPS) uses accurate, stable atomic clocks in satellites and on the ground to provide worldwide position and time determination. These clocks have gravitational and motional frequency shifts which are so large that, without carefully accounting for numerous relativistic effects, the system would not work. This paper discusses the conceptual basis, founded on special and general relativity, for navigation using GPS. Relativistic principles and effects which must be considered include the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and relativity of synchronization. Experimental tests of relativity obtained with a GPS receiver aboard the TOPEX/POSEIDON satellite will be discussed. Recently frequency jumps arising from satellite orbit adjustments have been identified as relativistic effects. These will be explained and some interesting applications of GPS will be discussed.
1 Introduction
The Global Positioning System (GPS) can be described in terms of three principal “segments”: a Space Segment, a Control Segment, and a User Segment. The Space Segment consists essentially of 24 satellites carrying atomic clocks. (Spare satellites and spare clocks in satellites exist.) There are four satellites in each of six orbital planes inclined at 55° with respect to earth’s equatorial plane, distributed so that from any point on the earth, four or more satellites are almost always above the local horizon. Tied to the clocks are timing signals that are transmitted from each satellite. These can be thought of as sequences of events in spacetime, characterized by positions and times of transmission. Associated with these events are messages specifying the transmission events’ spacetime coordinates; below I will discuss the system of reference in which these coordinates are given. Additional information contained in the messages includes an almanac for the entire satellite constellation, information about satellite vehicle health, and information from which Universal Coordinated Time as maintained by the U.S. Naval Observatory — UTC(USNO) — can be determined.
The Control Segment is comprised of a number of groundbased monitoring stations, which continually gather information from the satellites. These data are sent to a Master Control Station in Colorado Springs, CO, which analyzes the constellation and projects the satellite ephemerides and clock behaviour forward for the next few hours. This information is then uploaded into the satellites for retransmission to users.
The User Segment consists of all users who, by receiving signals transmitted from the satellites, are able to determine their position, velocity, and the time on their local clocks.
The GPS is a navigation and timing system that is operated by the United States Department of Defense (DoD), and therefore has a number of aspects to it that are classified. Several organizations monitor GPS signals independently and provide services from which satellite ephemerides and clock behavior can be obtained. Accuracies in the neighborhood of 5–10 cm are not unusual. Carrier phase measurements of the transmitted signals are commonly done to better than a millimeter.
GPS signals are received on earth at two carrier frequencies, L1 (154 × 10.23 MHz) and L2 (120 × 10.23 MHz). The L1 carrier is modulated by two types of pseudorandom noise codes, one at 1.023 MHz — called the Coarse/Acquisition or C/Acode — and an encrypted one at 10.23 MHz called the Pcode. Pcode receivers have access to both L1 and L2 frequencies and can correct for ionospheric delays, whereas civilian users only have access to the C/Acode. There are thus two levels of positioning service available in real time, the Precise Positioning Service utilizing Pcode, and the Standard Positioning Service using only C/Acode. The DoD has the capability of dithering the transmitted signal frequencies and other signal characteristics, so that C/Acode users would be limited in positioning accuracy to about ±100 meters. This is termed Selective Availability, or SA. SA was turned off by order of President Clinton in May 2000.
The plot for Cesium, however, characterizes the best orbiting clocks in the GPS system. What this means is that after initializing a Cesium clock, and leaving it alone for a day, it should be correct to within about 5 parts in 10^{14}, or 4 nanoseconds. Relativistic effects are huge compared to this.
The purpose of this article is to explain how relativistic effects are accounted for in the GPS. Although clock velocities are small and gravitational fields are weak near the earth, they give rise to significant relativistic effects. These effects include first and secondorder Doppler frequency shifts of clocks due to their relative motion, gravitational frequency shifts, and the Sagnac effect due to earth’s rotation. If such effects are not accounted for properly, unacceptably large errors in GPS navigation and time transfer will result. In the GPS one can find many examples of the application of fundamental relativity principles. These are worth careful study. Also, experimental tests of relativity can be performed with GPS, although generally speaking these are not at a level of precision any better than previously existing tests.
The timing pulses in question can be thought of as places in the transmitted wave trains where there is a particular phase reversal of the circularly polarized electromagnetic signals. At such places the electromagnetic field tensor passes through zero and therefore provides relatively moving observers with sequences of events that they can agree on, at least in principle.
2 Reference Frames and the Sagnac Effect
Almost all users of GPS are at fixed locations on the rotating earth, or else are moving very slowly over earth’s surface. This led to an early design decision to broadcast the satellite ephemerides in a model earthcentered, earthfixed, reference frame (ECEF frame), in which the model earth rotates about a fixed axis with a defined rotation rate, ω_{E} = 7.2921151467 × 10^{−5} rad s^{−1}. This reference frame is designated by the symbol WGS84 (G873) [19, 3]. For discussions of relativity, the particular choice of ECEF frame is immaterial. Also, the fact the the earth truly rotates about a slightly different axis with a variable rotation rate has little consequence for relativity and I shall not go into this here. I shall simply regard the ECEF frame of GPS as closely related to, or determined by, the International Terrestrial Reference Frame established by the International Bureau of Weights and Measures (BIPM) in Paris.
It should be emphasized that the transmitted navigation messages provide the user only with a function from which the satellite position can be calculated in the ECEF as a function of the transmission time. Usually, the satellite transmission times t_{j} are unequal, so the coordinate system in which the satellite positions are specified changes orientation from one measurement to the next. Therefore, to implement Eqs. (1), the receiver must generally perform a different rotation for each measurement made, into some common inertial frame, so that Eqs. (1) apply. After solving the propagation delay equations, a final rotation must usually be performed into the ECEF to determine the receiver’s position. This can become exceedingly complicated and confusing. A technical note [10] discusses these issues in considerable detail.
Although the ECEF frame is of primary interest for navigation, many physical processes (such as electromagnetic wave propagation) are simpler to describe in an inertial reference frame. Certainly, inertial reference frames are needed to express Eqs. (1), whereas it would lead to serious error to assert Eqs. (1) in the ECEF frame. A “Conventional Inertial Frame” is frequently discussed, whose origin coincides with earth’s center of mass, which is in free fall with the earth in the gravitational fields of other solar system bodies, and whose zaxis coincides with the angular momentum axis of earth at the epoch J2000.0. Such a local inertial frame may be related by a transformation of coordinates to the socalled International Celestial Reference Frame (ICRF), an inertial frame defined by the coordinates of about 500 stellar radio sources. The center of this reference frame is the barycenter of the solar system.
In the ECEF frame used in the GPS, the unit of time is the SI second as realized by the clock ensemble of the U.S. Naval Observatory, and the unit of length is the SI meter. This is important in the GPS because it means that local observations using GPS are insensitive to effects on the scales of length and time measurements due to other solar system bodies, that are timedependent.
The time transformation t = t′ in Eqs. (3) is deceivingly simple. It means that in the rotating frame the time variable t′ is really determined in the underlying inertial frame. It is an example of coordinate time. A similar concept is used in the GPS.
This should be compared with Eq. (7). Pathdependent discrepancies in the rotating frame are thus inescapable whether one uses light or portable clocks to disseminate time, while synchronization in the underlying inertial frame using either process is selfconsistent.
Eqs. (7) and (11) can be reinterpreted as a means of realizing coordinate time t′ = t in the rotating frame, if after performing a synchronization process appropriate corrections of the form +2ω_{E} ƒ_{path} dA′_{z}/c^{2} are applied. It is remarkable how many different ways this can be viewed. For example, from the inertial frame it appears that the reference clock from which the synchronization process starts is moving, requiring light to traverse a different path than it appears to traverse in the rotating frame. The Sagnac effect can be regarded as arising from the relativity of simultaneity in a Lorentz transformation to a sequence of local inertial frames comoving with points on the rotating earth. It can also be regarded as the difference between proper times of a slowly moving portable clock and a Master reference clock fixed on earth’s surface.
This was recognized in the early 1980s by the Consultative Committee for the Definition of the Second and the International Radio Consultative Committee who formally adopted procedures incorporating such corrections for the comparison of time standards located far apart on earth’s surface. For the GPS it means that synchronization of the entire system of groundbased and orbiting atomic clocks is performed in the local inertial frame, or ECI coordinate system [6].
GPS can be used to compare times on two earthfixed clocks when a single satellite is in view from both locations. This is the “commonview” method of comparison of Primary standards, whose locations on earth’s surface are usually known very accurately in advance from groundbased surveys. Signals from a single GPS satellite in common view of receivers at the two locations provide enough information to determine the time difference between the two local clocks. The Sagnac effect is very important in making such comparisons, as it can amount to hundreds of nanoseconds, depending on the geometry. In 1984 GPS satellites 3, 4, 6, and 8 were used in simultaneous common view between three pairs of earth timing centers, to accomplish closure in performing an aroundtheworld Sagnac experiment. The centers were the National Bureau of Standards (NBS) in Boulder, CO, PhysikalischTechnische Bundesanstalt (PTB) in Braunschweig, West Germany, and Tokyo Astronomical Observatory (TAO). The size of the Sagnac correction varied from 240 to 350 ns. Enough data were collected to perform 90 independent circumnavigations. The actual mean value of the residual obtained after adding the three pairs of time differences was 5 ns, which was less than 2 percent of the magnitude of the calculated total Sagnac effect [4].
3 GPS Coordinate Time and TAI
In the GPS, the time variable t′ = t becomes a coordinate time in the rotating frame of the earth, which is realized by applying appropriate corrections while performing synchronization processes. Synchronization is thus performed in the underlying inertial frame in which selfconsistency can be achieved.
The Earth’s geoid. In Eqs. (12) and (15), the rate of coordinate time is determined by atomic clocks at rest at infinity. The rate of GPS coordinate time, however, is closely related to International Atomic Time (TAI), which is a time scale computed by the BIPM in Paris on the basis of inputs from hundreds of primary time standards, hydrogen masers, and other clocks from all over the world. In producing this time scale, corrections are applied to reduce the elapsed proper times on the contributing clocks to earth’s geoid, a surface of constant effective gravitational equipotential at mean sea level in the ECEF.
Universal Coordinated Time (UTC) is another time scale, which differs from TAI by a whole number of leap seconds. These leap seconds are inserted every so often into UTC so that UTC continues to correspond to time determined by earth’s rotation. Time standards organizations that contribute to TAI and UTC generally maintain their own time scales. For example, the time scale of the U.S. Naval Observatory, based on an ensemble of Hydrogen masers and Cs clocks, is denoted UTC(USNO). GPS time is steered so that, apart from the leap second differences, it stays within 100 ns UTC(USNO). Usually, this steering is so successful that the difference between GPS time and UTC(USNO) is less than about 40 ns. GPS equipment cannot tolerate leap seconds, as such sudden jumps in time would cause receivers to lose their lock on transmitted signals, and other undesirable transients would occur.
To account for the fact that reference clocks for the GPS are not at infinity, I shall consider the rates of atomic clocks at rest on the earth’s geoid. These clocks move because of the earth’s spin; also, they are at varying distances from the earth’s center of mass since the earth is slightly oblate. In order to proceed one needs a model expression for the shape of this surface, and a value for the effective gravitational potential on this surface in the rotating frame.
Better models can be found in the literature of geophysics [18, 9, 15]. The next term in the multipole expansion of the earth’s gravity field is about a thousand times smaller than the contribution from J_{2}; although the actual shape of the geoid can differ from Eq. (20) by as much as 100 meters, the effects of such terms on timing in the GPS are small. Incorporating up to 20 higher zonal harmonics in the calculation affects the value of Φ_{0} only in the sixth significant figure.
Observers at rest on the geoid define the unit of time in terms of the proper rate of atomic clocks. In Eq. (19), Φ_{0} is a constant. On the left side of Eq. (19), dτ is the increment of proper time elapsed on a standard clock at rest, in terms of the elapsed coordinate time dt. Thus, the very useful result has emerged, that ideal clocks at rest on the geoid of the rotating earth all beat at the same rate. This is reasonable since the earth’s surface is a gravitational equipotential surface in the rotating frame. (It is true for the actual geoid whereas I have constructed a model.) Considering clocks at two different latitudes, the one further north will be closer to the earth’s center because of the flattening — it will therefore be more redshifted. However, it is also closer to the axis of rotation, and going more slowly, so it suffers less secondorder Doppler shift. The earth’s oblateness gives rise to an important quadrupole correction. This combination of effects cancels exactly on the reference surface.
4 The Realization of Coordinate Time
It is obvious that Eq. (24) contains within it the wellknown effects of time dilation (the apparent slowing of moving clocks) and frequency shifts due to gravitation. Due to these effects, which have an impact on the net elapsed proper time on an atomic clock, the proper time elapsing on the orbiting GPS clocks cannot be simply used to transfer time from one transmission event to another. Pathdependent effects must be accounted for.
On the other hand, according to General Relativity, the coordinate time variable t of Eq. (24) is valid in a coordinate patch large enough to cover the earth and the GPS satellite constellation. Eq. (24) is an approximate solution of the field equations near the earth, which include the gravitational fields due to earth’s mass distribution. In this local coordinate patch, the coordinate time is singlevalued. (It is not unique, of course, because there is still gauge freedom, but Eq. (24) represents a fairly simple and reasonable choice of gauge.) Therefore, it is natural to propose that the coordinate time variable t of Eqs. (24) and (22) be used as a basis for synchronization in the neighborhood of the earth.
The relativistic effect on the clock, given in Eq. (27), is thus corrected by Eq. (28).
Suppose for a moment there were no gravitational fields. Then one could picture an underlying nonrotating reference frame, a local inertial frame, unattached to the spin of the earth, but with its origin at the center of the earth. In this nonrotating frame, a fictitious set of standard clocks is introduced, available anywhere, all of them being synchronized by the Einstein synchronization procedure, and running at agreed upon rates such that synchronization is maintained. These clocks read the coordinate time t. Next, one introduces the rotating earth with a set of standard clocks distributed around upon it, possibly roving around. One applies to each of the standard clocks a set of corrections based on the known positions and motions of the clocks, given by Eq. (28). This generates a “coordinate clock time” in the earthfixed, rotating system. This time is such that at each instant the coordinate clock agrees with a fictitious atomic clock at rest in the local inertial frame, whose position coincides with the earthbased standard clock at that instant. Thus, coordinate time is equivalent to time that would be measured by standard clocks at rest in the local inertial frame [7].
When the gravitational field due to the earth is considered, the picture is only a little more complicated. There still exists a coordinate time that can be found by computing a correction for gravitational redshift, given by the first correction term in Eq. (28).
5 Relativistic Effects on Satellite Clocks
For atomic clocks in satellites, it is most convenient to consider the motions as they would be observed in the local ECI frame. Then the Sagnac effect becomes irrelevant. (The Sagnac effect on moving groundbased receivers must still be considered.) Gravitational frequency shifts and secondorder Doppler shifts must be taken into account together. In this section I shall discuss in detail these two relativistic effects, using the expression for the elapsed coordinate time, Eq. (28). The term Φ_{0} in Eq. (28) includes the scale correction needed in order to use clocks at rest on the earth’s surface as references. The quadrupole contributes to Φ_{0} in the term GM_{E}J_{2}/2a_{1} in Eq. (28); there it contributes a fractional rate correction of −3.76 × 10^{−13}. This effect must be accounted for in the GPS. Also, V is the earth’s gravitational potential at the satellite’s position. Fortunately, the earth’s quadrupole potential falls off very rapidly with distance, and up until very recently its effect on satellite vehicle (SV) clock frequency has been neglected. This will be discussed in a later section; for the present I only note that the effect of earth’s quadrupole potential on SV clocks is only about one part in 10^{14}, and I neglect it for the moment.
There is an interesting story about this frequency offset. At the time of launch of the NTS2 satellite (23 June 1977), which contained the first Cesium atomic clock to be placed in orbit, it was recognized that orbiting clocks would require a relativistic correction, but there was uncertainty as to its magnitude as well as its sign. Indeed, there were some who doubted that relativistic effects were truths that would need to be incorporated [5]! A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by general relativity, then the synthesizer could be turned on, bringing the clock to the coordinate rate necessary for operation. After the Cesium clock was turned on in NTS2, it was operated for about 20 days to measure its clock rate before turning on the synthesizer [11]. The frequency measured during that interval was +442.5 parts in 10^{12} compared to clocks on the ground, while general relativity predicted +446.5 parts in 10^{12}. The difference was well within the accuracy capabilities of the orbiting clock. This then gave about a 1% verification of the combined secondorder Doppler and gravitational frequency shift effects for a clock at 4.2 earth radii.
Additional small frequency offsets can arise from clock drift, environmental changes, and other unavoidable effects such as the inability to launch the satellite into an orbit with precisely the desired semimajor axis. The navigation message provides satellite clock frequency corrections for users so that in effect, the clock frequencies remain as close as possible to the frequency of the U.S. Naval Observatory’s reference clock ensemble. Because of such effects, it would now be difficult to use GPS clocks to measure relativistic frequency shifts.
When GPS satellites were first deployed, the specified factory frequency offset was slightly in error because the important contribution from earth’s centripetal potential (see Eq. (18) had been inadvertently omitted at one stage of the evaluation. Although GPS managers were made aware of this error in the early 1980s, eight years passed before system specifications were changed to reflect the correct calculation [2]. As understanding of the numerous sources of error in the GPS slowly improved, it eventually made sense to incorporate the correct relativistic calculation. It has become common practice not to apply such offsets to Rubidium clocks as these are subject to unpredictable frequency jumps during launch. Instead, after such clocks are placed in orbit their frequencies are measured and the actual frequency corrections needed are incorporated in the clock correction polynomial that accompanies the navigation message.
It is not at all necessary, in a navigation satellite system, that the eccentricity correction be applied by the receiver. It appears that the clocks in the GLONASS satellite system do have this correction applied before broadcast. In fact historically, this was dictated in the GPS by the small amount of computing power available in the early GPS satellite vehicles. It would actually make more sense to incorporate this correction into the time broadcast by the satellites; then the broadcast time events would be much closer to coordinate time — that is, GPS system time. It may now be too late to reverse this decision because of the investment that many dozens of receiver manufacturers have in their products. However, it does mean that receivers are supposed to incorporate the relativity correction; therefore, if appropriate data can be obtained in raw form from a receiver one can measure this effect. Such measurements are discussed next.
6 TOPEX/POSEIDON Relativity Experiment
A report distributed by the Aerospace Corporation [14] has claimed that the correction expressed in Eqs. (38) and (39) would not be valid for a highly dynamic receiver — e.g., one in a highly eccentric orbit. This is a conceptual error, emanating from an apparently official source, which would have serious consequences. The GPS modernization program involves significant redesign and remanufacturing of the Block IIF satellites, as well as a new generation of satellites that are now being deployed — the Block IIR replenishment satellites. These satellites are capable of autonomous operation, that is, they can be operated independently of the groundbased control segment for up to 180 days. They are to accomplish this by having receivers on board that determine their own position and time by listening to the other satellites that are in view. If the conceptual basis for accounting for relativity in the GPS, as it has been explained above, were invalid, the costs of opening up these satellites and reprogramming them would be astronomical.
There has been therefore considerable controversy about this issue. As a consequence, it was proposed by William Feess of the Aerospace Corporation that a measurement of this effect be made using the receiver on board the TOPEX satellite. The TOPEX satellite carries an advanced, sixchannel GPS receiver. With six data channels available, five of the channels can be used to determine the bias on the local oscillator of the TOPEX receiver with some redundancy, and data from the sixth channel can be used to measure the eccentricity effect on the sixth SV clock. Here I present some preliminary results of these measurements, which are to my knowledge the only explicit measurements of the periodic part of the combined relativistic effects of time dilation and gravitational frequency shift on an orbiting receiver.
A brief description of the pseudorange measurement made by a receiver is needed here before explaining the TOPEX data. Many receivers work by generating a replica of the coded signal emanating from the transmitter. This replica, which is driven through a feedback shift register at a rate matching the Dopplershifted incoming signal, is correlated with the incoming signal. The transmitted coordinate time can be identified in terms of a particular phase reversal at a particular point within the code train of the signal. When the correlator in the receiver is locked onto the incoming signal, the time delay between the transmission event and the arrival time, as measured on the local clock, can be measured at any chosen instant.
Let the time as transmitted from the jth satellite be denoted by t′_{j}. After correcting for the eccentricity effect, the GPS time of transmission would be t′_{j} + (Δtr)_{j} . Because of SA (which was in effect for the data that were chosen), frequency offsets and frequency drifts, the satellite clock may have an additional error b_{j} so that the true GPS transmission time is t_{j} = t′_{j} + (Δt_{r})_{j}  b_{j}.
The purpose of the TOPEX satellite is to measure the height of the sea. This satellite has a sixchannel receiver on board with a very good quartz oscillator to provide the time reference. A radar altimeter measures the distance of the satellite from the surface of the sea, but such measurements play no role in the present experiment. The TOPEX satellite has orbit radius 7,714 km, an orbital period of about 6745 seconds, and an orbital inclination of 66.06° to earth’s equatorial plane. Except for perturbations due to earth’s quadrupole moment, the orbit is very nearly circular, with eccentricity being only 0.000057. The TOPEX satellite is almost ideal for analysis of this relativity effect. The trajectories of the TOPEX and GPS satellites were determined independently of the onboard clocks, by means of Doppler tracking from ≈ 100 stations maintained by the Jet Propulsion Laboratory (JPL).
The receiver is a dual frequency C/A and Pcode receiver from which both code data and carrier phase data were obtained. The dualfrequency measurements enabled us to correct the propagation delay times for electron content in the ionosphere. Close cooperation was given by JPL and by William Feess in providing the dualfrequency measurements, which are ordinarily denied to civilian users, and in removing the effect of SA at time points separated by 300 seconds during the course of the experiment.

ECI centerofmass position and velocity vectors for 25 satellites, in the J2000 Coordinate system with times in UTC. Data rate is every 15 minutes; accuracy quoted is 10 cm radial, 30 cm horizontal.

ECI position and velocity vectors for the TOPEX antenna phase center. Data rate is every minute in UTC; accuracy quoted is 3 cm radial and 10 cm horizontal.

GPS satellite clock data for 25 satellites based on ground system observations. Data rate is every 5 minutes, in GPS time; accuracy ranges between 5 and 10 cm.

TOPEX dual frequency GPS receiver measurements of pseudorange and carrier phase for 25 satellites, a maximum of six at any one time. The data rate is every 10 seconds, in GPS time.
During this part of 1995, GPS time was ahead of UTC by 10 seconds. GPS cannot tolerate leap seconds so whenever a leap second is inserted in UTC, UTC falls farther behind GPS time. This required highorder interpolation on the orbit files to obtain positions and velocities at times corresponding to times given, every 300 seconds, in the GPS clock data files. When this was done independently by William Feess and myself we agreed typically to within a millimeter in satellite positions.
The L1 and L2 carrier phase data was first corrected for ionospheric delay. Then the corrected carrier phase data was used to smooth the pseudorange data by weighted averaging. SA was compensated in the clock data by courtesy of William Feess. Basically, the effect of SA is contained in both the clock data and in the pseudorange data and can be eliminated by appropriate subtraction. Corrections for the offset of the GPS SV antenna phase centers from the SV centers of mass were also incorporated.
7 Doppler Effect
Since orbiting clocks have had their rate adjusted so that they beat coordinate time, and since responsibility for correcting for the periodic relativistic effect due to eccentricity has been delegated to receivers, one must take extreme care in discussing the Doppler effect for signals transmitted from satellites. Even though secondorder Doppler effects have been accounted for, for earthfixed users there will still be a firstorder (longitudinal) Doppler shift, which has to be dealt with by receivers. As is well known, in a static gravitational field coordinate frequency is conserved during propagation of an electromagnetic signal along a null geodesic. If one takes into account only the monopole and quadrupole contributions to earth’s gravitational field, then the field is static and one can exploit this fact to discuss the Doppler effect.
8 Crosslink Ranging
Consider next the process of transferring coordinate time from one satellite clock to another by direct exchange of signals. This will be important when “Autonav” is implemented. The standard atomic clock in the transmitter satellite suffers a rate adjustment, and an eccentricity correction to get the coordinate time. Then a signal is sent to the second satellite which requires calculating a coordinate time of propagation possibly incorporating a relativistic time delay. There is then a further transformation of rate and another “e sin E” correction to get the atomic time on the receiving satellite’s clock. So that the rate adjustment does not introduce confusion into this analysis, I shall assume the rate adjustments are already accounted for and use the subscript ‘S’ to denote coordinate time measurements using rateadjusted satellite clocks.
This result contains all the relativistic corrections that need to be considered for direct time transfer by transmission of a timetagged pulse from one satellite to another. The last term in Eq. (52) should not be confused with the correction of Eq. (40).
9 Frequency Shifts Induced by Orbit Changes
Improvements in GPS motivate attention to other small relativistic effects that have previously been too small to be explicitly considered. For SV clocks, these include frequency changes due to orbit adjustments, and effects due to earth’s oblateness. For example, between July 25 and October 10, 2000, SV43 occupied a transfer orbit while it was moved from slot 5 to slot 3 in orbit plane F. I will show here that the fractional frequency change associated with a change da in the semimajor axis a (in meters) can be estimated as 9.429×10^{−18}da. In the case of SV43, this yields a prediction of −1.77×10^{−13} for the fractional frequency change of the SV43 clock which occurred July 25, 2000. This relativistic effect was measured very carefully [12]. Another orbit adjustment on October 10, 2000 should have resulted in another fractional frequency change of +1.75×10^{−13}, which was not measured carefully. Also, earth’s oblateness causes a periodic fractional frequency shift with period of almost 6 hours and amplitude 0.695 × 10^{−14}. This means that quadrupole effects on SV clock frequencies are of the same general order of magnitude as the frequency breaks induced by orbit changes. Thus, some approximate expressions for the frequency effects on SV clock frequencies due to earth’s oblateness are needed. These effects will be discussed with the help of Lagrange’s planetary perturbation equations.

the effect of earth’s mass on gravitational frequency shifts of atomic reference clocks fixed on the earth’s surface relative to clocks at infinity;

the effect of earth’s oblate mass distribution on gravitational frequency shifts of atomic clocks fixed on earth’s surface;

secondorder Doppler shifts of clocks fixed on earth’s surface due to earth rotation;

gravitational frequency shifts of clocks in GPS satellites due to earth’s mass;

and secondorder Doppler shifts of clocks in GPS satellites due to their motion through an EarthCentered Inertial (ECI) Frame.
Epstein et al. [12] suggested that the above frequency shift was relativistic in origin, and used precise ephemerides obtained from the National Imagery and Mapping Agency to estimate the frequency shift arising from secondorder Doppler and gravitational potential differences. They calculated separately the secondorder Doppler and gravitational frequency shifts due to the orbit change. The NIMA precise ephemerides are expressed in the WGS84 coordinate frame, which is earthfixed. If used without removing the underlying earth rotation, the velocity would be erroneous. They therefore transformed the NIMA precise ephemerides to an earthcentered inertial frame by accounting for a (uniform) earth rotation rate.
Lagrange perturbation theory. Perturbations of GPS orbits due to earth’s quadrupole mass distribution are a significant fraction of the change in semimajor axis associated with the orbit change discussed above. This raises the question whether it is sufficiently accurate to use a Keplerian orbit to describe GPS satellite orbits, and estimate the semimajor axis change as though the orbit were Keplerian. In this section, we estimate the effect of earth’s quadrupole moment on the orbital elements of a nominally circular orbit and thence on the change in frequency induced by an orbit change. Previously, such an effect on the SV clocks has been neglected, and indeed it does turn out to be small. However, the effect may be worth considering as GPS clock performance continues to improve.
To see how large such quadrupole effects may be, we use exact calculations for the perturbations of the Keplerian orbital elements available in the literature [13]. For the semimajor axis, if the eccentricity is very small, the dominant contribution has a period twice the orbital period and has amplitude 3J_{2}a _{1} ^{2} sin^{2} i_{0}/(2a_{0}) ≈ 1658 m. WGS84 (837) values for the following additional constants are used in this section: a_{1} = 6.3781370 × 10^{6} m; ω_{E} = 7.291151467 × 10^{−5} s^{−1}; a_{0} = 2.656175×10^{7} m, where a_{1} and a_{0} are earth’s equatorial radius and SV orbit semimajor axis, and ω_{E} is earth’s rotational angular velocity.
The oscillation in the semimajor axis would significantly affect calculations of the semimajor axis at any particular time. This suggests that Eq. (33) needs to be reexamined in light of the periodic perturbations on the semimajor axis. Therefore, in this section we develop an approximate description of a satellite orbit of small eccentricity, taking into account earth’s quadrupole moment to first order. Terms of order J_{2} × e will be neglected. This problem is nontrivial because the perturbations themselves (see, for example, the equations for mean anomaly and altitude of perigee) have factors 1/e which blow up as the eccentricity approaches zero. This problem is a mathematical one, not a physical one. It simply means that the observable quantities — such as coordinates and velocities — need to be calculated in such a way that finite values are obtained. Orbital elements that blow up are unobservable.
These effects were considered by Ashby and Spilker [9], pp. 685–686, but in that work the effect of earth’s quadrupole moment on the term GM_{E}/r was not considered; the present calculations supercede that work.
Summary. We note that the values of semimajor axis reported by Epstein et al. [12] differ from the values obtained by averaging as outlined above, by 200–300 m. This difference arises because of the different methods of calculation. In the present calculation, an attempt was made to account for the effect of earth’s quadrupole moment on the Keplerian orbit. It was not necessary to compute the orbit eccentricity. Agreement with measurement of the fractional frequency shift was only a few percent better than that obtained by differencing the maximum and minimum radii. This approximate treatment of the orbit makes no attempt to consider perturbations that are nongravitational in nature, e.g., solar radiation pressure. The work was an investigation of the approximate effect of earth’s quadrupole moment on the GPS satellite orbits, for the purpose of (possibly) accurate calculations of the fractional frequency shifts that result from orbit changes.
10 Secondary Relativistic Effects
There are several additional significant relativistic effects that must be considered at the level of accuracy of a few cm (which corresponds to 100 picoseconds of delay). Many investigators are modelling systematic effects down to the millimeter level, so these effects, which currently are not sufficiently large to affect navigation, may have to be considered in the future.
Phase wrapup. Transmitted signals from GPS satellites are right circularly polarized and thus have negative helicity. For a receiver at a fixed location, the electric field vector rotates counterclockwise, when observed facing into the arriving signal. Let the angular frequency of the signal be ω in an inertial frame, and suppose the receiver spins rapidly with angular frequency Ω which is parallel to the propagation direction of the signal. The antenna and signal electric field vector rotate in opposite directions and thus the received frequency will be ω + Ω. In GPS literature this is described in terms of an accumulation of phase called “phase wrapup”. This effect has been known for a long time [17, 20, 21, 24], and has been experimentally measured with GPS receivers spinning at rotational rates as low as 8 cps. It is similar to an additional Doppler effect; it does not affect navigation if four signals are received simultaneously by the receiver as in Eqs. (1). This observed effect raises some interesting questions about transformations to rotating, spinning coordinate systems.
Effect of other solar system bodies. One set of effects that has been “rediscovered” many times are the redshifts due to other solar system bodies. The Principle of Equivalence implies that sufficiently near the earth, there can be no linear terms in the effective gravitational potential due to other solar system bodies, because the earth and its satellites are in free fall in the fields of all these other bodies. The net effect locally can only come from tidal potentials, the third terms in the Taylor expansions of such potentials about the origin of the local freely falling frame of reference. Such tidal potentials from the sun, at a distance r from earth, are of order GM_{⊙}r^{2}/R^{3} where R is the earthsun distance [8]. The gravitational frequency shift of GPS satellite clocks from such potentials is a few parts in 10^{16} and is currently neglected in the GPS.
11 Augmentation Systems
Navigation based on GPS can fail in many different ways. Transmitted power is low, leading to ease of jamming and loss of signal under forest canopies or in urban canyons. Clock failures in satellites can go undetected for hours if a monitor station is not in view, leading to unreliable signal transmissions. Among nations other than the United States, there is an element of distrust of military control of the GPS. Such disadvantages have led to a number of socalled “augmentations” of GPS designed to provide users with additional GPSlike signals, or correction signals, that increase the reliability of GPS navigation. In addition, there are several new independent Global Satellite Navigation Systems being developed and deployed. We shall describe these developments since the implementation of relativistic effects differs from one system to the next.
WAAS (WideArea Augmentation System) provides improved reliability and accuracy over the continential U.S.A. system of 24 receivers at precisely known locations continually monitors signals from GPS satellites and computes corrections that are uploaded to leased geosynchronous satellites for retransmission to users who have WAASenabled receivers. No new relativity effects are involved; the corrections account primarily for clock drifts and ionospheric and tropospheric delays. EGNOS (European Geostationary Navigation Overlay System) is a similar system for improving navigation over Europe. MTSAT is a Japanese augmentation system.
The Japanese QZSS (QuasiZenith Satellite System) is a satellitebased augmentation system consisting of three satellites in geosynchronous orbits (a = 42, 164 km, but with large eccentricity, e ≈ 0.1). The ground tracks of the satellites describe a figure 8 on earth’s surface. At apogee, where the satellites are moving most slowly, the satellites spend more time above Japan. For atomic clocks in such satellites, relativistic effects would cause a fractional frequency shift of about −5.39 × 10^{−10} (see Figure 2). Also, the eccentricity effect is much larger than in GPS for two reasons: both the semimajor axis a and the eccentricity are larger than in GPS. The eccentricity effect, given by Eq. (38), has an amplitude of about 290 ns. Although the satellites carry atomic clocks the system is termed an augmentation system since it is not globally available.
12 Global Navigation Systems
From a practical point of view, data from additional satellites can provide improved navigation performance. Also, political considerations have led to development and deployment of satellite navigation systems that are alternatives to GPS. When such systems are made interoperable with GPS, “GNSS” results (the Global Navigational Satellite System). Here we discuss briefly how relativistic effects are incorporated into these new systems.
GLONASS is a Russian system that is very similar to GPS. The satellites are at slightly lower altitudes, and orbit the earth 17 times while the GPS satellites orbit 16 times. Figure 2 shows that the factory frequency clock offset is slightly less than that for GPS. Although a full constellation of 24 satellites was originally envisioned, for many years no more than a dozen or so healthy satellites have been available.
GALILEO is a project of the European Space Agency, intended to put about 30 satellites carrying atomic clocks in orbit. In contrast to GPS which is free to users, the GALILEO system ultimately will be funded by user fees. Information released in 2006 by the GALILEO project [25] states that relativistic corrections will be the responsibility of the users (that is, the receivers). This means that GNSS devices capable of receiving both GPS and GALILEO signals will have to contain additional relativity software to process GALILEO signals. Since no “factory frequency offset” is applied to atomic clocks in the GALILEO satellites, relativity effects will cause satellite clock time to ramp away from TAI and will require large correction terms to be transmitted to users.
BEIDOU is a satellite navigation system being developed and deployed by the People’s Republic of China. In its early stages, there were three satellites capable of transponding timing signals between a master control station and receivers on the ground. Timed pulses are sent from the control station, to the satellites, and then to groundbased receivers, which sends them back through the satellites to the control station. With the timing information, and topographic information (the receivers have to be on earth’s surface), the receiver position can be computed and relayed back to the receiver. Since receivers must also transmit, they are bulky. The principal relativistic correction involved here is the Sagnac effect, which can amount to several hundred nanoseconds.
BEIDOU is intended to develop into a global satellite navigation system that is independent, yet interoperable with GALILEO. Very little information is currently available about the structure of this system.
13 Applications
The number of applications of GPS have been astonishing. It would take several paragraphs just to list them. Accurate positioning and timing, other than for military navigation, include synchronization of power line nodes for fault detection, communications, VLBI, navigation in deep space, tests of fundamental physics, measurements on pulsars, tests of gravity theories, vehicle tracking, search and rescue, surveying, mapping, and navigation of commercial aircraft, to name a few. These are too numerous to go into in much detail here, but some applications are worth mentioning. Civilian applications have overtaken military applications to the extent that SA was turned off in May of 2000.
The Nobelprizewinning work of Joseph Taylor and his collaborators [16, 23] on the measurement of the rate of increase of the binary pulsar period depended on GPS receivers at the Arecibo observatory, for transferring UTC from the U.S. Naval Observatory and NIST to the local clock. Time standards around the world are compared using GPS in commonview; with this technique SA would cancel out, as well as do many sources of systematic errors such as ionospheric and tropospheric delays. Precise position information can assist in careful husbandry of natural resources, and animal and vehicle fleet tracking can result in improved efficiency. Precision agriculture makes use of GPS receivers in realtime application of pesticides or fertilizers, minimizing waste. Sunken vessels or underwater ruins with historically significant artifacts can be located using the GPS and archeologists can return again and again with precision to the same location. Monster ore trucks or earthmoving machines can be fitted with receivers and controlled remotely with minimal risk of collision or interference with other equipment. Disposable GPS receivers dropped through tropical storms transmit higher resolution measurements of temperature, humidity, pressure, and wind speed than can be obtained by any other method; these have led to improved understanding of how tropical storms intensify. Slight movements of bridges or buildings, in response to various loads, can be monitored in real time. Relative movements of remote parts of earth’s crust can be accurately measured in a short time, contributing to better understanding of tectonic processes within the earth and, possibly, to future predictions of earthquakes. With the press of a button, a lost hiker can send a distress signal that includes the hikers’ location.
These and many other creative applications of precise positioning and timing are leading to a rapid economic expansion of GPS products and services. Manufacturers produce hundreds of different GPS products for commercial, private, and military use and the number and variety of products is increasing. The number of receivers manufactured each year is in excess of two million, and different applications are continually being invented. Marketing studies predict sales of GPS equipment and services exceeding $30 billion per year; revenue for the European Galileo system is projected to be 10 billion Euros per year.
14 Conclusions
The GPS is a remarkable laboratory for applications of the concepts of special and general relativity. GPS is also valuable as an outstanding source of pedagogical examples. It is deserving of more scrutiny from relativity experts.
Alternative global navigation systems such as GLONASS, GALILEO, and BEIDOU are all based on concepts of clock synchronization based on a locally inertial reference system freely falling along with the earth. This concept, fundamentally dependent on a relativistic view of space and time, appears to have been amply confirmed by the success of GPS.
Plans are being made to put lasercooled clock(s) having stabilities of \(5 \times {10^{  14}}/\sqrt \tau \) and accuracies of 1 × 10^{16}, on the International Space Station. This will open up additional possibilities for testing relativity as well as for making improvements in GPS and in other potential navigational satellite systems.
Footnotes
 1.
^{1} WGS84 (G873) values of these constants are used in this article.
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