PHYSICAL REVIEW A

VOLUME 56, NUMBER 6
ARTICLES

DECEMBER 1997

Satellite test of special relativity using the global positioning system
Peter Wolf1,2 and Ge´rard Petit1 1Bureau International des Poids et Mesures, Pavillon de Breteuil, F-92312 Se`vres Cedex, France 2School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, Great Britain
͑Received 20 November 1996; revised manuscript received 19 May 1997͒
A test of special relativity has been carried out using data of clock comparisons between hydrogen maser clocks on the ground and cesium and rubidium clocks on board 25 global positioning system ͑GPS͒ satellites. The clocks were compared via carrier phase measurements of the GPS signal using geodetic receivers at a number of stations of the International GPS Service for Geodynamics ͑IGS͒ spread worldwide. In special relativity, synchronization of distant clocks by slow clock transport and by Einstein synchrony ͑using the transmission of light signals͒ is equivalent in any inertial frame. A violation of this equivalence can be modeled using the parameter ␦c/c, where c is the round-trip speed of light ͑cϭ299 792 458 m/s in vacuum͒ and ␦c is the deviation from c of the observed velocity of a light signal traveling one way along a particular spatial direction with the measuring clocks synchronized using slow clock transport. In special relativity ␦c/cϭ0. Experiments can set a limit on the value of ␦c/c along a particular spatial direction ͑henceforth referred to as ‘‘direction of ␦c’’͒. Within this model our experiment is sensitive to a possible violation of special relativity in any direction of ␦c, and on a nonlaboratory scale ͑baselines у20 000 km͒. The results presented here set an upper limit on the value of ␦c/cϽ5ϫ10Ϫ9 when considering all spatial directions of ␦c and ␦c/cϽ2 ϫ10Ϫ9 for the component in the equatorial plane. ͓S1050-2947͑97͒06712-7͔
PACS number͑s͒: 03.30.ϩp, 06.20.Jr, 06.30.Ft, 95.40.ϩs

I. INTRODUCTION
The equivalence of distant clock synchronization in iner-
tial frames by slow clock transport and by Einstein synchrony ͓1͔ is fundamental to the theories of special and general relativity. It can be tested directly by comparing the one-way propagation times ͑using slow clock transport synchronization͒ of light signals along known paths, but in different spatial directions ͑in the past, often referred to as a test of the isotropy of the one-way speed of light͒. The only such test was carried out by Krisher et al. ͓2͔, who compared the phases of two hydrogen masers separated by a distance of 21
km and linked via an ultrastable fiber-optics link of the
NASA deep-space network. The sensitivity of this experiment, expressed as a limit on the parameter ␦c/c, was ␦c/cϽ3.5ϫ10Ϫ7. Riis et al. ͓3͔ tested the variation with spatial direction of the first-order Doppler shift of light emitted by an atomic beam ͑and indirectly thereby the synchrony equivalence͒ using fast-beam laser spectroscopy, obtaining what is currently the smallest limit, ␦c/cϽ3ϫ10Ϫ9 ͓4͔. Both of these experiments relied on the rotation of the Earth to change the direction of signal transmission and were therefore only sensitive to a component of the direction of ␦c that lies in the equatorial plane. The Gravity Probe A rocket experiment ͓5͔ can be used to measure the first-order Doppler shift of the link between the ground and on-board masers, giving a limit of ␦c/cϽ3.2ϫ10Ϫ9 in one particular spatial direction. The only experiment sensitive in any spatial direction of ␦c was carried out by Turner and Hill ͓6͔, who

measured the first-order Doppler shift in a Mo¨ssbauer rotor, obtaining a limit of ␦c/cϽ3ϫ10Ϫ8. We present here the results of a test of special relativity sensitive in any spatial direction of ␦c. Using the clocks on board the global positioning system ͑GPS͒ satellites ͑providing baselines у20 000 km͒, we obtain a limit of ␦c/cϽ5ϫ10Ϫ9 when considering all spatial directions and ␦c/cϽ2ϫ10Ϫ9 for the component that lies in the equatorial plane. These results, together with those obtained by previous experiments are summarized in Table I. In Sec. II the principle of the experiment is explained, Sec. III provides details about the experimental procedure and data treatment, and Sec. IV shows the results. We consider possible systematic effects in Sec. V with a final discussion and conclusion in Sec. VI.
II. PRINCIPLE OF THE EXPERIMENT
Satellites of the GPS constellation are distributed in six orbital planes, at an inclination of 55° in near circular orbits with a period corresponding to 0.5 sidereal days ͑718 min͒ ͓7͔. Each satellite is equipped with an on-board atomic clock and a dual-frequency signal transmission system.
The emission time of a signal as measured by the onboard clock ␶e and its reception time as measured by the ground-clock ␶r are recorded. The difference Tϭ␶rϪ␶e represents the transmission time of the signal plus some initial, unknown, phase difference between the clocks when these are synchronized using slow clock transport. Defining D as the distance along a straight line from the satellite ͑at the moment of emission͒ to the ground station ͑at the moment of

1050-2947/97/56͑6͒/4405͑5͒/$10.00

56 4405

© 1997 The American Physical Society

4406

PETER WOLF AND GE´ RARD PETIT

56

TABLE I. Tests of special relativity showing the limits they set on ␦c/c and their respective sensitivities to the possible spatial directions of ␦c.

Direct measurements T. P. Krisher et al. ͑1990͒ GPS test ͑this experiment͒ GPS test ͑this experiment͒
Indirect measurements E. Riis et al. ͑1987͒ R. F. C. Vessot et al. ͑1979͒ K. C. Turner and H. A. Hill ͑1964͒

Limits on ␦c/c
␦ c / c Ͻ 3.5ϫ 10Ϫ 7 ␦ c / c Ͻ 5 ϫ 10Ϫ 9 ␦ c / c Ͻ 2 ϫ 10Ϫ 9
␦ c / c Ͻ 3 ϫ 10Ϫ 9 ␦ c / c Ͻ 3 ϫ 10Ϫ 9 ␦ c / c Ͻ 3 ϫ 10Ϫ 8

Component in equatorial plane All spatial directions
Component in equatorial plane
Component in equatorial plane Component in one particular direction
All spatial directions

reception͒ in a geocentric, inertial ͑nonrotating͒ coordinate system, one can write

D

TϪ c ϭ⌬0,

͑1͒

where ⌬0 is a constant characterizing the initial phase difference of the two clocks. Special relativity requires that, for a series of measurements in different directions ͑e.g., during a complete passage of the satellite͒, TϪD/c should remain constant, after correction for the relative rate of the two
clocks due to the gravitational redshift, second-order Doppler shift and the intrinsic ͑proper͒ frequency difference of the clocks.
In the theoretical framework generally used for the inter-
pretation of similar tests of special relativity, the speed of
light as measured by distant clocks that are synchronized using slow clock transport is cϩ␦c in one direction and c Ϫ␦c in the opposite direction along a particular spatial axis ͑in an inertial frame͒. The experiments then determine whether the special relativistic postulate ␦cϭ0 is confirmed within the uncertainty of the experiment and set an upper limit on the parameter ␦c/c. A more sophisticated theoretical approach to all tests of special relativity was developed by Mansouri and Sexl ͓8͔. A detailed interpretation of our experiment in this framework is presented elsewhere ͓9͔; only the results are given here ͑see Sec. VI͒.
The effect of a nonzero value of ␦c on the transmission time of an individual link would be (␦c/c)(D/c)cos␣, where ␣ is the angle between the direction of ␦c and of the transmitted signal, resulting in a measurable variation of T as a
function of direction of signal transmission. However, a vio-
lation of special relativity might also affect the determination
of the satellite ephemerides, and therefore the value of D, leaving the difference TϪD/c unchanged. A meaningful test of special relativity using the above principle therefore re-
quires a method of satellite orbit determination that is insensitive to a nonzero value of ␦c.
This is the case for the GPS ephemerides obtained by the Center for Orbit Determination in Europe ͑CODE͒ of the IGS ͓10͔. The method used adjusts a post-Keplerian, nonrelativistic orbit model to doubly differenced GPS timing data ͑see Fig. 1͒. The effects of a nonzero value of ␦c on the individual links cancel ͑to first order͒ when the double differences are formed ͑see Fig. 1͒. The IGS-CODE method is used to simultaneously adjust a number of parameters, in-

cluding the satellite ephemerides and the ground station coordinates, thereby providing values of D that are unaffected by ␦c.
Additionally one has to ensure that corrections applied to the raw timing data used for orbit determination and the measurement of T do not presuppose the validity of special relativity. In fact, two corrections are routinely applied to GPS timing data, which are of relativistic origin and therefore do imply ␦cϭ0 ͓7͔: the correction for the gravitational redshift and the second-order Doppler shift of the rate of the satellite clock with respect to coordinate time, and the correction for the so-called Sagnac effect, which is due to the rotation of the Earth during signal transmission. Both of these are small corrections of order cϪ2; hence the effect of an error in these corrections due to a nonzero value of ␦c would be negligible with respect to the first-order effect on T ͑for a more detailed theoretical analysis see, e.g., ͓14͔͒.
Observation of GPS satellites in varying spatial directions thus provides a meaningful test of special relativity via relation ͑1͒, with D obtained using IGS-CODE ephemerides and station coordinates.
III. EXPERIMENTAL PROCEDURE The IGS is a global network of ground stations that continuously observe the GPS satellites for civil, geodetic pur-
FIG. 1. Double difference (Y ) for pairs of stations ͑A and B͒ and satellites ͑1 and 2͒, Y ϭ(T1BϪT1A)Ϫ(T2BϪT2A). The effect of a nonzero value of ␦c on Y , given by ͓(D1Bcosa1B ϪD1Acosa1A)Ϫ(D2Bcosa2BϪD2Acosa2A)͔␦c/c2, where ␣ is the angle between the directions of ␦c and of the transmitted signal, vanishes.

56

SATELLITE TEST OF SPECIAL RELATIVITY USING . . .

4407

poses ͓10͔. From the raw data the IGS processing centers calculate ͑among other parameters͒ precise satellite ephemerides and ground station coordinates. These, together
with the raw observations, are freely available through the internet via anonymous ftp ͓10͔. We use data from eight ground stations for our experiment: Brussels ͑Belgium͒, Algonquin ͑Canada͒, Yellowknife ͑Canada͒, Fairbanks ͑Alaska, USA͒, Kokee Park ͑Hawaii, USA͒, Fortaleza ͑Brazil͒, Santiago ͑Chile͒, and Hobart ͑Australia͒. The motivation for this choice of ground stations is to ensure global
coverage while providing maximum ground clock stability ͓for averaging times Ϸ6 h ͑one passage͔͒ by using only stations that are equipped with hydrogen-maser clocks. The
geodetic GPS receivers used provide raw phase measure-
ments of the two GPS carrier frequencies at a sample interval of 30 s. The data sets cover six days ͑1994, September 18 to 23͒ and contain observations of all 25 GPS satellites available at the time. During this period ͑coinciding with the military intervention in Haiti͒ all GPS signals were free of the intentional degradation ͓selective availability ͑SA͔͒ which is imposed by the US military. In general, this affects all but
two satellites, making them unusable for the experiment de-
scribed here. Of the 25 satellites used, 19 are equipped with
cesium clocks and six with rubidium clocks. From the raw data the differences TϪD/c are formed,
taking into account corrections for the variable part of the
gravitational redshift and second-order Doppler shift, the Sagnac effect, the ionospheric delay ͑using an ionosphere-free combination of the two frequencies͒, and the tropospheric delay ͓using the NATO Standardization Agreement ͑STANAG͒ tropospheric model ͓7͔͔. The amount of data is reduced by averaging over nine 30-s points, in order to save computer space and to make the data more manageable.
For a test of special relativity, one is interested in the variation of the difference TϪD/c during individual passages of the satellite over the ground station, i.e., variations over time scales of roughly six hours. Therefore we first filter the data, excluding all long-term variations (у5 d), effectively subtracting the relative rate of the two clocks. Then an arbitrary offset per passage is adjusted as only the variation of TϪD/c during the passage is of interest.
As is well known, measurements of the GPS carrier phase are subject to an unknown phase ambiguity error of an integer number of cycles. This does not present a problem for our purposes as long as the induced error remains constant during each passage, which is the case if the receiver stays locked onto the satellite over the complete passage. Therefore all passages that were incomplete ͑data gaps indicating a possible loss of the satellite͒ were excluded.
The 0.5 sidereal day period of the GPS satellites implies that a station ‘‘sees’’ each passage of a particular GPS satellite at the same time of day ͑in sidereal days͒ and in the same directions ͑in a geocentric nonrotating frame͒. So, to see the data more clearly, all passages can be projected onto the same day, by shifting them individually by an integer number of sidereal days. Figure 2 shows the residuals of a typical data set after filtering and adjustment of an offset per passage, and with all passages shifted onto the same day. The standard deviation of these residuals for the complete data set ͑all stations and satellites͒ is 2.2 ns.

FIG. 2. Residuals of (TϪD/c) after filtering and adjustment of an offset per passage. The graph shows six passages of GPS satellite vehicle 22 over Brussels shifted onto the same day.
IV. RESULTS Figure 3 shows the spatial directions of signal transmission for the individual links in a nonrotating geocentric frame. There are no links at colatitudes below Ϸ20° and above Ϸ163°, which is due to the 55° inclination of the satellite orbits and implies that the experiment was least sensitive in the N-S direction.
The effect of a nonzero value of ␦c on the transmission time T for a particular link is given by (D/c) (␦c/c)cos␣, where ␣ is the angle between the direction of signal transmission and the direction of ␦c. This model was fitted to the data using the least-squares method, adjusting the value of ␦c/c and an offset per satellite-station pair. The adjustment was performed for a range of directions of ␦c, spanning colatitudes and longitudes from 0 to ␲ in a grid of 0.1 radϫ0.1 rad. It is sufficient to cover half of all possible spatial directions, as opposing directions correspond to the same magnitude of ␦c/c with the opposite sign. Directions are given in the nonrotating geocentric frame that is coinci-
FIG. 3. Directions of signal transmission, in a geocentric nonrotating frame, for all satellite station links.

4408

PETER WOLF AND GE´ RARD PETIT

56

nithal tropospheric delay obtained from the STANAG model and the five-day averages of the IGS estimates do not exceed 200 ps for any station with the standard deviation of the averages Ͻ110 ps. For low elevations these differences can increase to maximum 1.1 ns ͑at elevations of 10°͒. We conclude that, for our purposes, the STANAG model is sufficient, as the dominant limitation of the sensitivity of our experiment is more likely to arise from satellite clock instabilities. All other, unmodeled, systematic effects, such as the residual ionospheric delay ͑in f 3͒ or the variation of the gravitational redshift due to the quadrupole moment of the Earth, should not induce uncertainties exceeding 100 ps.

FIG. 4. Adjusted values of ␦c/c as a function of direction of ␦c in a geocentric nonrotating frame.
dent with the Earth fixed frame of the International Earth Rotation Service ͑the ITRF͒ on March 7, 1995, 0 h 00 min universal time ͑UTC͒.
Figure 4 shows the adjusted values of ␦c/c as a function of their direction. The extremum value of ␦c/c is 4.9 ϫ10Ϫ9 at a colatitude of 2.9 rad and a longitude of 0.5 rad. In the equatorial plane ͑colatitude of 1.6 rad͒ the extremum value of ␦c/c is 1.6ϫ10Ϫ9 at a longitude of 0.6 rad. The formal statistical standard uncertainties of these values are below 3ϫ10Ϫ10, but the overall uncertainty of the experiment is largely dominated by systematic effects ͑Sec. V͒ that are responsible for the final uncertainty ͑Sec. VI͒.
V. SYSTEMATIC EFFECTS
The three main systematic effects that could affect the data are the satellite clock instability, ephemerides errors, and uncertainties in the modeled tropospheric delays. Prelaunch measurements of the relative frequency stability of the GPS cesium clock showed an instability of ␴y(␶)Ϸ1.5 ϫ10Ϫ13 ͑standard Allan deviation͒ for integration times of Ϸ6 h ͓11͔. On-orbit measurements showed satellite clock instabilities of order 10Ϫ13 for integration times of one day ͓12͔, which corresponds to ␴y(6 h)Ϸ2ϫ10Ϫ13 when extrapolated assuming a ␶Ϫ1/2 dependence. This translates into an accumulated time error over one passage of the satellite of
␦␶Ϸ␶␴y(␶)/ͱ6Ϸ2 ns. Beutler et al. ͓13͔ estimate the un-
certainty of the IGS-CODE satellite ephemerides to be 15 to 20 cm, corresponding to a timing error of ␦␶Ϸ0.7 ns.
The tropospheric delays were estimated using the standard STANAG model ͓7͔. More accurate values for these delays can be obtained through estimations in a regional or global network ͑as done, for example, by the IGS͒. We have studied such estimates ͑four values per day and station͒ for a period of five days in September 1995 and for six of the eight stations used ͑no IGS estimates were available for the remaining two stations͒. The differences between the ze-

VI. DISCUSSION AND CONCLUSION
Timing errors due to the systematic effects discussed in the previous section could give rise to values of ␦c/c of order 10Ϫ8 ͓␦c/cϷ␦␶/(D/c)͔, assuming that no cancellation takes place between the systematic effects of different satellite-station pairs. This is an order of magnitude larger than the maximum value observed. The experiment therefore cannot suggest a violation of special relativity.
It is unlikely that, in a global treatment, the systematic errors are correlated with the signature of a nonzero value of ␦c, as they are expected, in general, to correlate differently with the effect of ␦c/c for each satellite-station pair and, to some extent, for each passage. So the results from a number of randomly chosen subsets of the data can provide an estimate of the reliability of the results presented in Sec. IV. The least-squares adjustment was repeated for five different subsets of the complete data, removing observations from three to six satellites ͑i.e., 12% to 24% of the total data͒. The mean of the five adjusted extremum values of ␦c/c is 4.6ϫ10Ϫ9 when considering all spatial directions and 2.2ϫ10Ϫ9 for the component in the equatorial plane, with a standard deviation about the mean of 1.3ϫ10Ϫ9 in both cases. These values should be compared to ␦c/c of 4.9ϫ10Ϫ9 and 1.6ϫ10Ϫ9 obtained from the complete data set ͑cf Sec. IV͒. So, assuming no correlation ͑and resulting cancellation͒ between the systematic effects in a global treatment and the effect of a possible nonzero value of ␦c ͑for the reasons mentioned above͒, we can set a limit of ␦c/cϽ5ϫ10Ϫ9 when considering all spatial directions and ␦c/cϽ2ϫ10Ϫ9 for the component of the direction of ␦c that lies in the equatorial plane.
Interpretation of the experiment, using the test theory developed by Mansouri and Sexl ͓8͔, shows that the value of ␦c/c obtained here is related to the parameter ␣ of the test theory ͑in special relativity ␣ϭϪ1/2͒ by the well-known relationship ͑see ͓9͔ for details͒

␦c

v

c ϭ͑1ϩ2␣͒ c ,

͑2͒

where v is the velocity of the Earth with respect to the ‘‘universal frame’’ defined in the theory ͑the frame in which slow clock transport and Einstein synchronism are equivalent͒, and the parameter ␣ is dependent on the choice of Einstein synchronism in this universal frame ͓15͔. Additionally one obtains an explicit expression for the double difference Y ͑cf Fig. 1͒ that is independent of ␣ to first order in v/c ͓9͔,
which confirms the validity of the experimental principle in

56

SATELLITE TEST OF SPECIAL RELATIVITY USING . . .

4409

this test theory. Taking v as the velocity of the Earth with
respect to the ‘‘mean rest frame of the universe’’ (v Ϸ300 km/s) in the direction of the dipole anisotropy of the cosmic microwave background ͑declinationϷϪ1°, right ascensionϷ11 h͒ ͓16,17͔, we obtain a limit of ͉␣ϩ1/2͉Ͻ1 ϫ10Ϫ6 ͑␦c/cϽ2ϫ10Ϫ9 in this direction͒. This, to our knowledge, is the smallest limit for the parameter ␣ published to date ͓4͔.
In principle, any space mission flying a highly stable atomic clock that is equipped with a time transfer system could be used for a similar test of special relativity. An example is the experiment on timing, ranging and atmospheric sounding ͑ExTRAS͒ mission ͓18͔ that is currently ‘‘on hold.’’ This mission is of interest, as the space clock will be a hydrogen maser expected to be significantly more stable in the short term than the GPS clocks, and because the optical

two-way time transfer system planned should diminish the effects of the time transfer ͑in particular, tropospheric͒ and ephemerides uncertainties ͓18͔.
Finally we would like to emphasize that an experiment like the one described in this paper can be carried out at minimal cost by virtually anyone, as all the IGS data are freely available on the internet via anonymous ftp. This is of particular interest in view of a recent U.S. presidential decision ͓19͔ to switch off selective availability completely on all GPS satellites within the next ten years.
ACKNOWLEDGMENTS
We thank the IGS stations and centers for making their data and their products freely available, Claudine Thomas for helpful discussions, and D. A. Blackburn for revision of the manuscript.

͓1͔ A. Einstein, Ann. Phys. ͑Leipzig͒ 17, 891 ͑1905͒ ͓for a translation see, e.g., H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity ͑Dover, New York, 1923͒.
͓2͔ T. P. Krisher et al., Phys. Rev. D 42, R731 ͑1990͒. ͓3͔ E. Riis et al., Phys. Rev. Lett. 60, 81 ͑1988͒. ͓4͔ We do not take into account the experiment briefly mentioned
by G. R. Isaak in two paragraphs of his general review article ͓Phys. Bull. 21, 255, ͑1970͔͒ often cited by other authors ͑e.g., ͓2,8,14͔͒. No detailed account of the experiment ͑description, results, statistical analysis, systematic effects, etc.͒ is given; instead, the reader is directed to another publication ͓G. R. Isaak et al. Nature ͑to be published͔͒, which, to our knowledge, has not been published in Nature ͑cf. author index of Nature for 1970 and the following years͒ or elsewhere. ͓5͔ R. F. C. Vessot and M. W. Levine, Gen. Relativ. Gravit. 10, 181 ͑1979͒. ͓6͔ K. C. Turner and H. A. Hill, Phys. Rev. 134, B252 ͑1964͒. ͓7͔ Arinc Research Corporation Report No. 3659-01-01-4296, 1990 ͑unpublished͒. ͓8͔ R. Mansouri and R. U. Sexl, Gen. Relativ. Gravit. 8, 497 ͑1977͒; 8, 515 ͑1977͒; 8, 809 ͑1977͒.

͓9͔ P. Wolf, Ph.D. thesis, Queen Mary and Westfield College, London, 1997 ͑unpublished͒; Bureau International des Poids et Mesures, Monograph No. 97/1, 1997 ͑unpublished͒, available from the author on request.
͓10͔ Jet Propulsion Laboratory Report No. 95-18 9/95, 1995 ͑unpublished͒.
͓11͔ J. A. Wisnia, NASA Conference Publication No. 3218, 1992 ͑unpublished͒.
͓12͔ T. B. McCaskill et al., NASA Conference Publication No. 3159, 1991 ͑unpublished͒.
͓13͔ G. Beutler et al., Manuscr. Geodet. 19, 367 ͑1994͒. ͓14͔ P. Tourrenc, T. Melliti, and J. Bosredon, Gen. Relativ. Gravit.
28, 1071 ͑1996͒. ͓15͔ I. Vetharaniam and G. E. Stedman, Phys. Lett. A 183, 349
͑1993͒. ͓16͔ D. J. Fixsen, E. S. Cheng, and D. T. Wilkinson, Phys. Rev.
Lett. 50, 620 ͑1983͒. ͓17͔ P. M. Lubin, G. L. Epstein, and G. F. Smoot, Phys. Rev. Lett.
50, 616 ͑1983͒. ͓18͔ P. Wolf, Phys. Rev. A 51, 5016 ͑1995͒. ͓19͔ GPS World 7 ͑5͒, 50 ͑1996͒.