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PostPosted: Tue Jun 12, 2007 4:48 am 
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.
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. [url "http://www.metaresearch.org/cosmology/gps-relativity.asp#footnotes"][1][/url] 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.
4. Is the speed of light constant?
Other studies using GPS data have placed far more stringent limits than we will here. But our goal here is not to set the most stringent limit on possible variations in the speed of light, but rather to determine what the maximum possible variation might be that can remain consistent with the data. The GPS operates by sending atomic clock signals from orbital altitudes to the ground. This takes a mere 0.08 seconds from our human perspective, but a very long (although equivalent) 80,000,000 ns from the perspective of an atomic clock. Because of this precision, the system has shown that the speed of radio signals (identical to the "speed of light") is the same from all satellites to all ground stations at all times of day and in all directions to within ±12 meters per second (m/s). The same numerical value for the speed of light works equally well at any season of the year.
Technical note: Measuring the one-way speed of light requires two clocks, one on each end of the path. If the separation of the clocks is known, then the separation divided by the time interval between transmission and reception is the one-way speed of the signal. But measuring the time interval requires synchronizing the clocks first. If the Einstein prescription for synchronizing clocks is used, then the measured speed must be the speed of light by definition of the Einstein prescription (which assumes the speed of light is the same in all inertial frames). If some other non-equivalent synchronization method is used, then the measured speed of the signal will not be the speed of light. Clearly, the measured signal speed and the synchronization prescription are intimately connected.
Our result here merely points out that the measured speed does not change as a function of time of day or direction of the satellite in its orbit when the clock synchronization correction is kept unchanged over one day. As for seasonal variations, all satellite clocks are "steered" to keep close to the U.S. Naval Observatory Master Clock so as to prevent excessive build up of errors from random rate changes over long time periods. So we cannot make direct comparisons between different seasons, but merely note that the same value of the speed of light works equally well in any season.
5. What is a "GPS clock"?
Cesium atomic clocks operate by counting hyperfine transitions of cesium atoms that occur roughly 10 billion times per second at a very stable frequency provided by nature. The precise number of such transitions was originally calibrated by astronomers, and is now adopted by international agreement as the definition of one atomic second.
GPS atomic clocks in orbit would run at rates quite different from ground clocks if allowed to do so, and this would complicate usage of the system. So the counter of hyperfine cesium transitions (or the corresponding phenomenon in the case of rubidium atomic clocks) is reset on the ground before launch so that, once in orbit, the clocks will tick off whole seconds at the same average rate as ground clocks. GPS clocks are therefore seen to run slow compared to ground clocks before launch, but run at the same rate as ground clocks after launch when at the correct orbital altitude.
We will refer to a clock whose natural ticking frequency has been pre-corrected in this way as a "GPS clock". This will help in the discussion of SR effects such as the twins paradox. A GPS clock is pre-corrected for relativistic rate changes so that it continues to tick at the same rate as Earth clocks even when traveling at high relative speeds. So a GPS clock carried by the traveling twin can be used to determine local time in the Earth's frame at any point along the journey -- a great advantage for resolving paradoxes.
6. Is acceleration an essential part of resolving the "twins paradox"?
If the traveling twin carries both a natural clock and a GPS clock on board his spacecraft, he can observe the effects predicted by SR without need of any acceleration in the usual twins paradox. That is as it should be because cyclotron experiments have shown that, even at accelerations of 1019 g (g = acceleration of gravity at the Earth's surface), clock rates are unaffected. Only speed affects clock rates, but not acceleration per se.
Suppose that the traveling twin is born as his spaceship passes by Earth and both of his on-board clocks are synchronized with clocks on Earth. The natural on-board clock ticks more slowly than the GPS on-board clock because the rates differ by the factor gamma that SR predicts for the slowing of all clocks with relative speed v. [gamma = 1/sqrt(1-v2/c2)] But everywhere the traveling twin goes, as long as his speed relative to the Earth frame does not change, his GPS clock will give identical readings to any Earth-synchronized Earth-frame clock he passes along the way. And his natural clock will read less time elapsed since passing Earth by the factor gamma. His biological processes (including aging), which presumably operate at rates comparable to the ticking of the natural clock, are also slowed by the factor gamma.
Since this rate difference is true at every instant of the journey beginning with the first, there are no surprises if the traveling twin executes a turn-around without change of speed and returns to Earth. He will find on journey's completion what he has observed at every step of the journey: His natural clock and his biological age are slower and younger by the factor gamma than that of his Earth-frame counterparts everywhere along his journey, including at its completion. The same would have been true if he had not turned around, but merely continued ahead. He would be younger than his peers on any planet encountered who claim to have been born at the same time that the traveler was born (i.e., when he passed Earth) according to their Earth-frame perspective.
Clearly, acceleration or the lack thereof has no bearing on the observed results. If acceleration occurs, it is merely to allow a more convenient comparison of clocks by returning to the starting point. But since the traveler can never return to the same point in space-time merely by returning to the same point in space, the results of a round-trip comparison are no different in kind from those made anywhere along the journey. The traveler always judges that his own aging is slower than that in any other frame with a relative motion.
Then why isn’t the traveler entitled to claim that he remained at rest and the Earth moved? The traveler is unconditionally moving with respect to the Earth frame and therefore his clocks unconditionally tick slower and he ages less as judged by anyone in the Earth frame. However, if the traveler makes the same judgment, the result will depend on whether he values his natural clock or his GPS clock as the better timekeeper. If he takes readings on the GPS clock to represent Earth time, his inferences will always agree with those of Earth-frame observers. If he instead uses the results of the exchange of light signals to make inferences of what time it is at distant locations, he will conclude that the Earth-bound twin is aging less than himself because of their relative motion. But on the occasion of any acceleration his spaceship undergoes, the traveler will infer a discontinuity in the age of his Earth-bound counterpart, which can be either forward or backward in time depending on which direction the traveler accelerates. At the end of any round trip after any number of such accelerations, the traveler and Earth-bound twins will always agree about who should have aged more.
7. Does the behavior of GPS clocks confirm Einstein SR?
To answer this, we must make a distinction between Einstein SR and Lorentzian Relativity (LR). Both Lorentz in 1904 and Einstein in 1905 chose to adopt the principle of relativity discussed by Poincare in 1899, which apparently originated some years earlier in the 19th century. Lorentz also popularized the famous transformations that bear his name, later used by Einstein. However, Lorentz’s relativity theory assumed an aether, a preferred frame, and a universal time. Einstein did away with the need for these. But it is important to realize that none of the 11 independent experiments said to confirm the validity of SR experimentally distinguish it from LR -- at least not in Einstein's favor.

thanx to meta research ................

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