One of the most popular everyday ‘proofs’ of relativity comes from the Global Positioning System (GPS). GPS is used to pinpoint locations on the Earth’s surface and relies on radio signals sent from satellites in space. The signals carry coded information about the satellite’s location and the time the signal was sent. A GPS receiver on Earth collects this information from three or four satellites simultaneously and calculates the distance to each satellite. The receiver then calculates where these distances intersect to determine its location in three-dimensional space. The coordinates of longitude, latitude and altitude are given in reference to a three-dimensional mathematical model of the Earth’s ellipsoid shape (a slightly squashed sphere) called the ‘Conventional Inertial Frame’ or ‘World Geodetic System 1984’ (WGS 84)^{1} (figure 2).

The success of the positioning system relies on the ability of radio signals to transmit extremely precise information. To this end, GPS satellites carry caesium atomic clocks that are correct to less than 5 parts in 10^{14}, or about 4 billionths of a second per day.^{2} As the satellites are orbiting 20,184 km above the Earth, they are in a much weaker gravitational field compared to clocks on the Earth, and general relativity predicts that the satellite clocks will tick more quickly by 45 microseconds per day.^{3}

Since the satellite clocks are moving relative to receivers on Earth, special relativity predicts the satellite clocks will tick more slowly by some amount compared to ground-based clocks. Satellite orbital speeds are cited as 3,874 m/s and thus satellite atomic clocks are reported to experience a time dilation of about 7 microseconds per day.^{3}

When the slowing effect of special relativity on a GPS satellite clock rate is subtracted from the speeding-up effect of general relativity, the result is about 38 microseconds of increase per day (45-7). To correct for this time increase in satellite atomic clocks, GPS engineers adjust the clock rates before they are placed into orbit. The clocks are given a rate offset of 4.465 parts in 10^{10} from their nominal frequency of 10.23 MHz so that on average they appear to run at the same rate as a clock on the ground. The actual frequency of the satellite clocks before launch is thus 10.22999999543 MHz.^{3} In other words, the clocks are pre-tuned to count a different number of caesium oscillations per second compared to the standard on Earth, so that in space they measure the same duration of time for one second as on Earth.

Physicist Ron Hatch (1938 – 2019) was a co-inventor of GPS and one of the world’s foremost experts on GPS. Over his fifty-year career, he wrote many technical papers outlining innovative techniques for GPS navigation satellites and held over 30 patents. He also served as both the Chair and President of the Satellite Division of the Institute of Navigation (ION). In 1994, Hatch received the Johannes Kepler Award for his significant contribution to satellite navigation. In 2000 he was awarded the Thomas L. Thurlow Award and was elected a Fellow of the ION.

Hatch also published several papers to show that GPS has nothing to do with relativity, and in his 1992 book *Escape From Einstein*, he presented GPS data that provided evidence against special relativity.

Calculations of special relativistic time dilation are not performed in GPS operation. In special relativity, time dilation can only be calculated using the relative velocity strictly along the line of sight between two frames of reference; no other reference frames are relevant. However, physicists have calculated the time dilation for satellite atomic clocks using the satellite’s orbital velocity.^{3} The problem here is that orbital velocity is not a velocity along the line of sight between the satellite and a receiver on Earth. The orbital velocity is in a direction that is perpendicular to the radius between the satellite and a non-rotating point at the center of the Earth.

To illustrate this problem, consider the example of a satellite orbiting in sync with the Earth’s rotation. It remains at a fixed point above a certain location on Earth (weather and TV satellites utilize geostationary orbits of this kind). From the perspective of an observer on Earth, this satellite appears to remain fixed in the sky. To calculate time dilation due to special relativity in this case, the satellite’s velocity along the line of sight with a receiver on Earth is effectively zero. If we then calculate time dilation using this satellite’s orbital velocity of say 4,000 m/s (in a direction perpendicular to its orbit radius) we get an incorrect result.

The two reference frames needed to calculate the component of time dilation in special relativity are each continuously changing. GPS satellites orbit the Earth about twice a day and so they are continuously sweeping across from one horizon to the other. Furthermore, the reference frame of the receiver, your phone for example, even if it is ‘stationary’ relative to the surface of the Earth, is moving relative to the GPS satellite’s orbit due to the rotation of the Earth about its axis. The rotational speed of a particular point on the Earth’s surface will depend on its latitude, ranging from approximately 460 m/s for points along the equator to 0 m/s at the north or south poles. In other words, unless the receiver is at one of the Earth’s poles, it will be moving in a direction tangential to a satellite’s line of sight.

In addition to the time variations attributed to the effects of special relativity, there are other, well-documented time delays inherent in the GPS system. The signal transmitted from a satellite is itself subject to time delays on its way to a ground-based receiver. These delays include: slowing of the radio signal (Shapiro delay), Doppler effects, interference with the signal and the eccentricity of satellite orbits. Given these inherent delays, scientists admit that ‘it would be difficult’ to use GPS clocks to actually measure the relativistic effects.^{3}

In practice, these complexities are accounted for by approximating all motion with reference to a third, independent frame, that is, the WGS 84 Reference Frame.^{3} Thus, not only are calculations of special relativistic time dilation not performed in GPS, they are not *necessary*.

Furthermore, the overall correction factors do not need to be calculated by individual receivers on Earth for the system to work. This is because a GPS receiver’s position is determined solely by comparing the time signals it receives from several different satellites with each other, not with the clock in the receiver itself. In other words, as long as the satellite clocks are in sync relative to each other, the clock rate on Earth becomes redundant. To keep satellite signals in sync with each other and the ground, data transmitted by the satellites is continuously monitored by receiving stations around the globe and forwarded to a master control station at the US Naval Observatory.^{3} In this way, satellite clocks are periodically synchronized with a ground-based reference clock.

In addition to corrections for time differences, ongoing corrections for position are also needed. As noted, GPS receivers calculate accurate position coordinates with respect to the WGS 84 reference frame. The accuracy of this quasi-inertial frame is continually monitored and updated to account for factors such as the Earth’s crustal motion, plate tectonics, and other geophysical changes. The WGS 84 is aligned with Earth’s center of mass, which in turn is oriented to the ‘International Celestial Reference Frame’, a point centered at the center of mass (barycentre) of the solar system.^{3} In this way, the ongoing accuracy of GPS positioning is dependent on a reference point that is fixed with respect to the solar system.

It is evident that relativity is not needed to explain the satellite time (atomic clock frequency) offset. Alternative explanations of the same time increase onboard GPS satellites have been provided based on a variable speed of light and quantum mechanics.^{4} In a weaker gravitational field, atoms increase their frequency of oscillation and consequently atomic clocks run faster. This approach ties in with the observations of Einstein in 1911 and Dicke in 1957 that light speed is variable, and affords a more intuitive, mechanistic understanding of the gravitational redshift effect.

An experiment to test the effect of gravitational field strength on clock rate as well as the speed of light was proposed in the late 1990s. The speed of light was to be measured onboard the International Space Station using a new-and-improved atomic clock with a very stable super-cooled chamber.^{5} The aim was to compare the stable atomic clock’s microwave frequency with that of a regular atomic clock (to an accuracy of 1 x 10^{-17}) as a function of position and gravitational potential. It was also intended to detect any direction-dependent changes in the velocity of light.^{6} This particular mission, to have been funded by NASA, was unfortunately canceled to make way for something else.

In summary, whichever method is employed to describe the gravitational effects on satellite atomic clock rates, in practice the clock rates are: (1) adjusted on the ground before take-off as per their intended altitude and (2) adjusted periodically with reference to a ground-based clock. Satellite positions are also corrected periodically with reference to an Earth-centered reference frame, which is itself calibrated to the center of mass of the solar system.

### References

- Ashby, N. (2002) Relativity and the Global Positioning System. Physics Today, May 2002, vol 41
- Ashby, N. (2003) Relativity in the Global Positioning System. Living Reviews. Relativity, 6, 2003
- Nelson, R. (2013) The Global Positioning System – A National Resource
- Chang, D. (2018) A quantum mechanical interpretation of gravitational redshift of electromagnetic wave. Optik Vol. 174, Dec 2018, pp.636–641
- Buchman, S. Turneaure, J. P. Lipa, J. A. Dong, M. Cumbennack, K. M. and Wang, S. (1998) A Superconducting Microwave Oscillator Clock for Use on the Space Station. IEEE international frequency control symposium
- Buchman, S. Dong, M. Moeur, W. Wang, S. Lipa, J. A. and Tumeaure J. P. (2000) A Space-Based Oscillator Clock. Adv. Space Res. Vol. 25, No. 6, pp.1251–1254