What the Global Positioning System Tells Us about Relativity

Tom Van Flandern, Univ. of Maryland & Meta Research From the book ‘Open Questions in Relativistic Physics’ (pp. 81-90), edited by Franco Selleri, published by Apeiron, Montreal (1998) 1. What is the GPS? The Global Positioning System (GPS) consists of a network of 24 satellites in roughly 12-hour orbits, each carrying atomic clocks on board. The orbital radius of the satellites is about four Earth-radii (26,600 km). The orbits are nearly circular, with a typical eccentricity of less than 1%. Orbital inclination to the Earth’s equator is typically 55 degrees. The satellites have orbital speeds of about 3.9 km/s in a frame centered on the Earth and not rotating with respect to the distant stars. Nominally, the satellites occupy one of six equally spaced orbital planes. Four of them occupy each plane, spread at roughly 90-degree intervals around the Earth in that plane. The precise orbital periods of the satellites are close to 11 hours and 58 minutes so that the ground tracks of the satellites repeat day after day, because the Earth makes one rotation with respect to the stars about every 23 hours and 56 minutes. (Four extra minutes are required for a point on the Earth to return to a position directly under the Sun because the Sun advances about one degree per day with respect to the stars.) The on-board atomic clocks are good to about 1 nanosecond (ns) in epoch, and about 1 ns/day in rate. Since the speed of light is about one foot per nanosecond, the system is capable of amazing accuracy in locating anything on Earth or in the near-Earth environment. For example, if the satellite clocks are fully synchronized with ground atomic clocks, and we know the time when a signal is sent from a satellite, then the time delay for that signal to reach a ground receiver immediately reveals the distance (to a potential accuracy of about one foot) between satellite and ground receiver. By using four satellites to triangulate and determine clock corrections, the position of a receiver at an unknown location can be determined with comparable precision. 2. What relativistic effects on GPS atomic clocks might be seen? General Relativity (GR) predicts that clocks in a stronger gravitational field will tick at a slower rate. Special Relativity (SR) predicts that moving clocks will appear to tick slower than non-moving ones. Remarkably, these two effects cancel each other for clocks located at sea level anywhere on Earth. So if a hypothetical clock at Earth’s north or south pole is used as a reference, a clock at Earth’s equator would tick slower because of its relative speed due to Earth’s spin, but faster because of its greater distance from Earth’s center of mass due to the flattening of the Earth. Because Earth’s spin rate determines its shape, these two effects are not independent, and it is therefore not entirely coincidental that the effects exactly cancel. The cancellation is not general, however. Clocks at any altitude above sea level do tick faster than clocks at sea level; and clocks on rocket sleds do tick slower than stationary clocks. For GPS satellites, GR predicts that the atomic clocks at GPS orbital altitudes will tick faster by about 45,900 ns/day because they are in a weaker gravitational field than atomic clocks on Earth’s surface. Special Relativity (SR) predicts that atomic clocks moving at GPS orbital speeds will tick slower by about 7,200 ns/day than stationary ground clocks. Rather than have clocks with such large rate differences, the satellite clocks are reset in rate before launch to compensate for these predicted effects. In practice, simply changing the international definition of the number of atomic transitions that constitute a one-second interval accomplishes this goal. Therefore, we observe the clocks running at their offset rates before launch. Then we observe the clocks running after launch and compare their rates with the predictions of relativity, both GR and SR combined. If the predictions are right, we should see the clocks run again at nearly the same rates as ground clocks, despite using an offset definition for the length of one second. We note that this post-launch rate comparison is independent of frame or observer considerations. Since the ground tracks repeat day after day, the distance from satellite to ground remains essentially unchanged. Yet, any rate difference between satellite and ground clocks continues to build a larger and larger time reading difference as the days go by. Therefore, no confusion can arise due to the satellite clock being located some distance away from the ground clock when we compare their time readings. One only needs to wait long enough and the time difference due to a rate discrepancy will eventually exceed any imaginable error source or ambiguity in such comparisons. 3. Does the GPS confirm the clock rate changes predicted by GR and SR? The highest precision GPS receiver data is collected continuously in two frequencies at 1.5-second intervals from all GPS satellites at five Air Force monitor stations distributed around the Earth. An in-depth discussion of the data and its analysis is beyond the scope of this paper. [1] This data shows that the on-board atomic clock rates do indeed agree with ground clock rates to the predicted extent, which varies slightly from nominal because the orbit actually achieved is not always precisely as planned. The accuracy of this comparison is limited mainly because atomic clocks change frequencies by small, semi-random amounts (of order 1 ns/day) at unpredictable times for reasons that are not fully understood. As a consequence, the long-term accuracy of these clocks is poorer than their short-term accuracy. Therefore, we can assert with confidence that the predictions of relativity are confirmed to high accuracy over time periods of many days. In ground solutions with the data, new corrections for epoch offset and rate for each clock are determined anew typically once each day. These corrections differ by a few ns and a few ns/day, respectively, from similar corrections for other days in the same week. At much later times, unpredictable errors in the clocks build up with time squared, so comparisons with predictions become increasingly uncertain unless these empirical corrections are used. But within each day, the clock corrections remain stable to within about 1 ns in epoch and 1 ns/day in rate. The initial clock rate errors just after launch would give the best indication of the absolute accuracy of the predictions of relativity because they would be least affected by accumulated random errors in clock rates over time. Unfortunately, these have not yet been studied. But if the errors were significantly greater than the rate variance among the 24 GPS satellites, which is less than 200 ns/day under normal circumstances, it would have been noticed even without a study. So we can state that the clock rate effect predicted by GR is confirmed to within no worse than ±200 / 45,900 or about 0.7%, and that predicted by SR is confirmed to within ±200 / 7,200 or about 3%. This is a very conservative estimate. In an actual study, most of that maximum 200 ns/day variance would almost certainly be accounted for by differences between planned and achieved orbits, and the predictions of relativity would be confirmed with much better precision. 12-hour variations (the orbital period) in clock rates due to small changes in the orbital altitude and speed of the satellites, caused by the small eccentricity of their orbits, are also detected. These are observed to be of the expected size for each GPS satellite’s own orbit. For example, for an orbital eccentricity of 0.01, the amplitude of this 12-hour term is 23 ns. Contributions from both altitude and speed changes, while not separable, are clearly both present because the observed amplitude equals the sum of the two predicted amplitudes.

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