Updated: Sep 21, 2022
It is a common myth that GPS uses---nor needs---special relativity. Van Flandern tried to bury the myth by what would happen if a series of satellites were to use "Einstein synchronization" when a closed circuit was completed and the signal return to the satellite which originally emitted it. Pure chaos would result. Although he tries to push Lorentz's theory in favor of Einstein's, for which time dilation and space contraction are "real" occurrences, neither is need. Nor is the term -vx/c^2 in the time transformation, usually referred to as "time slippage", as in the article by Neil Ashby, "Relativity and the Global Positioning System."
The GPS system is much, much more, simpler! To make a calculation the GPS receiver (a common quartz clock) needs to know two things: The location of at least 3 satellites, and the distance between you and the satellites. GPS analyzes high-frequency, low-power radio signals from satellites. Like other forms of electromagnetic radiation, radio waves travel at the speed of light. So the distance to a satellite is the time the signal takes to arrive times the speed of light. Of course, the signal does not travel in vacuum due to the earth's ionosphere, etc., but these factors can be easily corrected.
Why at least 3 satellites are necessary? Consider a two-dimensional case for simplicity. Suppose a traveler, who has no idea where he or she is, is told that he/she is x miles from A. This is really not much help for he/she could be anywhere on a radius of x miles from A. Now, he/she is told that he/she is y miles from B. Since x and y are general not equal, the circles will intersect at two points. So instead of a radius of possibilities, we are down to two possibilities. A third person will then tell he/she that he/she is z miles from C. The third sphere can intersect the other two circles in a maximum of two points. Then adjustments can be made so that all three circles can intersect in one, and only one, point, Sync in the diagram below.
Based on this, the receiver resets its clock to by in synchronization with the satellites' more sophisticated atomic clocks. This is does in continuation. Moreover, it makes no difference if all the distances are in error, since the errors will be common to all measurements, and will be only proportionally incorrect.
Moving off the plane, circles are replaced by spheres, and the process is known as trilateration. Earth can act as the fourth sphere, so that only one of the two points will be on the surface of the planet. Thus, the one in space is easily eliminated.
All this can be easily found on the internet, like: https://electronics.howstuffworks.com/gadgets/travel/gps3.htm
Then, one asks why GPS is camouflaged as being a product of special relativity? The speed implicated in this is the speed of the satellite moving relative to the earth's frame. But where on earth does this speed come in the above explanation of the functioning of GPS? It doesn't, neither does Lorentz's transform, which was misinterpreted by Einstein as representing something physical, which it doesn't!
Lorentz belatedly admits that his transform, which he began using in 1904, was originally conceived by Voigt, some quarter of a century before. The Wikipedia article on the subject claims that whereas the Michelson-Morley and Kennedy-Thorndike interferometer experiments could not distinguish between the two transformations due to the constancy of in all inertial frames, the vindication of the Lorentz transform had to wait until 1938 until Ives and Stilwell made their experiment, which amounted to a second-order Doppler effect. They achieved this by averaging forward and backward motions which left a second-order term. Ives, to his dying day (he was a devoted anti-relativist like Louis Essen, the inventor of the cesium clock), refused to acknowledge that he gave crucial support to special relativity, attributing his result to a peculiar effect that clocks show when they are in motion: they tend to run slower. This is again a two-way measurement and can make no statement about the one-way speed of light. Einstein's savoir was in the fact that all measurements till today give the two-way speed of light exactly equal to c. Should that change in the future would lead to the crumbling of special relativity as we know it today.
Even Ives himself was chained to the Lorentz transformation wherein he considered "slow" clock transport with regard to a preferential frame for the speed of light. For in Lorentz's theory any attempt at the synchronization of clocks with respect to an ether wind will create biases that will cancel out making it literally impossible to make a measurement of a speed of light with respect to any spatial frame, even though no assumption is made with regard to the speed of light in different directions. In other words, Lorentz's theory is a two-timer theory as witnessed by the Lorentz space Lorentz contraction, and time dilation factors. This is why effects like "time-slippage" are irrelevant to the operation of GPS, although many authors will disagree.
Hence, there is no "competition" between Lorentz and Einstein transforms. Lorentz acknowledges reference to a particular frame, which has subsequently been identified as the cosmic background radiation, whereas Einstein considers all reference frames equivalent that move with the same speed. Since the speed of light is a universal constant, the time it takes for a signal to arrive at point B when it was transmitted from A and immediately reflected back to A, when it arrives at B, is simply the arithmetic mean of the time when the signal is sent out and the time when it arrives back at A.
But, where is this convention made use of in GPS? It isn't! Einstein's convention is position-independent, whereas GPS is! Moreover, GPS is velocity-independent. Never once have we mentioned, nor taken into account, the relative speed of the satellite with respect to the earth's frame. In other words, Einstein's convention that two observers are required to have the same speed if they are to agree on which events are simultaneous, wheres GPS merely requires the observers to be at the same location, formed from the unique point of the intersection of three spheres.
The advantage of position-dependent synchronization, over velocity-dependent synch, is that it leaves the one-way speed of light completely independent. As we have seen in our last blog, it singles out the harmonic mean over that of the arithmetic mean. Since the latter is in general larger than the former, unless all speeds are the same making v=0, the arithmetic mean can be greater than the speed of light (like the geometric mean which is also greater than the harmonic mean. But if v=0, what is the goal of the Michelson-Morley experiment which sought to determine the speed of the earth through the luminiferous ether? And, if the speed is zero, why then was it necessary for FitzGerald and Lorentz (separately) to introduce the hypothesis that somehow the arm of the interferometer is shortened in the direction of motion? The velocity v does not show up in the direction of motion in which case the speed would be c+v, yet it shows up in some knived contraction factor. It is neither logical, nor necessary.