# What the Global Positioning System Tells Us about Relativity

//What the Global Positioning System Tells Us about Relativity

## What the Global Positioning System Tells Us about Relativity

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.

1. 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.

1. 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.

1. 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.

1. 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.

1. 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.

1. 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 19thcentury. 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.

 Experiment Description Year Bradley Discovery of aberration of light 1728 Fresnel Light suffers drag from the local medium 1817 Airy Aberration independent of the local medium 1871 Michelson-Morley Speed of light independent of Earth’s orbital motion 1881 De Sitter Speed of light independent of speed of source 1913 Sagnac Speed of light depends on rotational speed 1913 Kennedy-Thorndike Measured time also affected by motion 1932 Ives-Stilwell Ions radiate at frequencies affected by their motion 1941 Frisch-Smith Radioactive decay of mesons is slowed by motion 1963 Hafele-Keating Atomic clock changes depend on Earth’s rotation 1972 GPS Clocks in all frames continuously synchronized 1997

Table 1. Independent experiments bearing on Special RelativitySeveral of the experiments bearing on various aspects of SR (see Table 1) gave results consistent with both SR and LR. But Sagnac in 1913, Michelson following the Michelson-Gale confirmation of the Sagnac effect for the rotating Earth in 1925 (not an independent experiment, so not listed in Table 1), and Ives in 1941, all claimed at the time they published that their results were experimental contradictions of Einstein SR because they implied a preferred frame. In hindsight, it can be argued that most of the experiments contain some aspect that makes their interpretation simpler in a preferred frame, consistent with LR. In modern discussions of LR, the preferred frame is not universal, but rather coincides with the local gravity field. Yet, none of these experiments is impossible for SR to explain.

For example, Fresnel showed that light is partially dragged by the local medium, which suggests a certain amount of frame-dependence. Airy found that aberration did not change for a water-filled telescope, and therefore did not arise in the telescope tube. That suggests it must arise elsewhere locally. Michelson-Morley expected the Earth’s velocity to affect the speed of light because it affected aberration. But it didn’t. If these experimenters had realized that the aether was not a single entity but changed with the local gravity field, they would not have been surprised. It might have helped their understanding to realize that Earth’s own Moon does not experience aberration as the distant stars do, but only the much smaller amount appropriate to its small speed through the Earth’s gravity field.

Another clue came for De Sitter in 1913, elaborated by Phipps [3], both of whom reminded us that double star components with high relative velocities nonetheless both have the same stellar aberration. This meant that the relative velocity between a light source and an observer was not relevant to stellar aberration. Rather, the relative velocity between local and distant gravity fields determined aberration. In the same year, Sagnac showed non-null results for a Michelson-Morley experiment done on a rotating platform. In the simplest interpretation, this demonstrated that speeds relative to the local gravity field do add to or subtract from the speed of light in the experiment, since the fringes do shift. The Michelson-Gale experiment in 1925 confirmed that the Sagnac result holds true when the rotating platform is the entire Earth’s surface.

When Ives and Stilwell showed in 1941 that the frequencies of radiating ions depended on their motion, Ives thought he had disposed once and for all of the notion that only relative velocity mattered. After all, the ions emitted at a particular frequency no matter what frame they were observed from. He was unmoved by arguments to show that SR could explain this too because it seemed clear that nature still needed a preferred frame, the motion relative to which would determine the ion frequencies. Otherwise, how would the ions know how often to radiate? Answers to Ives dilemma exist, but not with a comparable simplicity.

Richard Keating was surprised in 1972 that two atomic clocks traveling in opposite directions around the world, when compared with a third that stayed at home, showed slowing that depended on their absolute speed through space — the vector sum of the Earth’s rotation and airplane speeds — rather on the relative velocities of the clocks. But he quickly accepted that astronomers always use the Earth’s frame for local phenomena, and the solar system barycentric frame for other planetary system phenomena, to get results that agreed with the predictions of relativity. Being unaware of LR, he did not question the interpretation at any deeper level.

 Experiment Type Notes on Reciprocity Bradley Aberration Moon exempt Fresnel Fresnel drag Existence of aether Airy Existence of aether Water in ‘scope ignored Michelson-Morley No universal aether Aether “entrained”? De Sitter c independent of source Double star aberration Sagnac c depends on rotation Local gravity field non-rotating Kennedy-Thorndike Clocks slow Motion w.r.t. local gravity field Ives-Stilwell Ions slow “ Frisch-Smith Mesons live longer “ Hafele-Keating Clocks depend on rotation Preferred frame indicated GPS Universal synchronization Preferred frame = local gravity

Table 2. Independent experiments bearing on Special Relativity            Table 2 summarizes what the various experiments have to say about a preferred frame. These experiments confirm the original aether-formulated relativity principle to high precision. However, the issue of the need for a preferred frame in nature is, charitably, not yet settled. Certainly, experts do not yet agree on its resolution. But of those who have compared both LR and SR to the experiments, most seem convinced that LR more easily explains the behavior of nature.

1. How does the resolution of the “twins paradox” compare in LR and SR?

In LR, the answer is simple: The Earth frame at the outset, and the dominant local gravity field in general, constitutes a preferred frame. So the high-speed traveler always comes back younger, and there is no true reciprocity of perspective for his or other frames.

In SR, the answer is not so simple; yet an explanation exists. The reciprocity of frames required by SR when Einstein assumed that all inertial frames were equivalent introduces a second effect on “time” in nature that is not reflected in clock rates alone. We might call this effect “time slippage” so we can discuss it. Time slippage represents the difference in time for any remote event as judged by observers (even momentarily coincident ones) in different inertial frames.

For example, we would argue that, if it is 9/1998 here and now, it is also 9/1998 “now” at Alpha Centauri. But an observer here and now with a sufficiently high relative motion (say, 99% of c; gamma = 7) might judge that it is 9/1994 at Alpha Centauri “now” (meaning that he just left there one month of Earth time ago, and it was 8/1994 then). Or he might judge that it is 9/2002 at Alpha Centauri “now” (meaning that he will arrive there in one month of Earth-elapsed time, and will find the time to be 10/2002). These differences of opinion about what time it is at remote locations are illustrations of time slippage effects that appear only in Einstein SR to preserve the frame independence of its predictions.

So as a traveler passes Earth in 8/1994 at a speed of 0.99c , time slippage effects begin to build up. Seven months later by his natural clock, the traveler arrives at Alpha Centauri. His own GPS clock shows four years of elapsed time, and indeed Alpha Centauri residents who think they are calendar-synchronized with Earth agree that the twin arrives in 9/1998. But the traveler is convinced by Einstein SR that only one month of Earth time has elapsed since he passed Earth and noted the time as 8/1994. The traveler, upon arriving at Alpha Centauri, claims that the time is “now” 9/1994 on Earth. Alpha Centauri residents claim it is “now” 9/1998 on Earth. The difference is the time slippage predicted by SR.

If the traveler orbits Alpha Centauri at a speed of 0.99 c, then whenever he is headed in the direction of Earth his opinion changes to Earth time “now” is 9/2002. And whenever he is again headed away from Earth, Earth time is once again 9/1994. Earth time “now” changes continually, according to SR, because of these time slippage effects needed to retain frame reciprocity. Earth residents — even the ones who died in 1998 — are oblivious to their repeated passages into the future and past of the traveling twin, with concomitant deaths and resurrections.

So when the traveler finally does return, he will indeed find that time on Earth is 10/2002, just as his GPS clock shows. He accounts for this as two months of elapsed time on Earth’s slow-running clocks during his own 14-month (by his natural clock) journey, plus 8 years of “time slippage” when the traveler changed frames. There is no logical or mathematical inconsistency in this resolution, which is why SR remains a viable theory today.

We are, of course, free to question whether or not this mathematical theory retains a valid basis under the principles of causality. For those of us who answer “yes”, LR is unnecessary, and inelegant because it depends on a preferred frame. For those of us who answer “no”, LR is then the better descriptor of nature, requiring the sacrifice of symmetry (“covariance”) to retain causality.

1. What physical consequences arise from the differences between LR and SR?

In SR, speed causes changes in time and space themselves, not just in clocks and rulers. Rest mass remains unchanged, but resistance to acceleration increases toward infinity as speed approaches c. There is no absolute time or space in the universe. The time at remote locations depends on what frame one observes from. All frames are equivalent.

In LR, speed relative to the preferred frame (the local gravity field) causes clocks to slow and rulers to contract. Electromagnetic-based forces become increasingly less efficient with increasing speed relative to the preferred frame, and approach zero efficiency as speed approaches c. There are natural, physical reasons why these things should be so. [2] The frame of the local gravity field acts as a preferred frame. Universal time and remote simultaneity exist.

The single most important difference is that, in SR, nothing can propagate faster than c in forward time. In LR, electromagnetic-based forces and clocks would cease to operate at speeds of c or higher. But no problem in principle exists in attaining any speed whatever in forward time using forces such as gravity that retain their efficiency at high speeds.

[1] Alley, C.O. and Van Flandern, T. (1998). “Absolute GPS to Better Than One Meter”, preprint not yet submitted for publication.

[2] Van Flandern, T. (1993). Dark Matter, Missing Planets and New Comets, North Atlantic Books, Berkeley, CA.

[3] Phipps, T. (1989). “Relativity and Aberration”, Amer.J.Phys. 57, 549-551.

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)