ObscuripediaThe Free Obscure EncyclopediaThe corrections described in this article have already been applied, several times, by the device you are reading it on.
| Satellites | 24 or more, in six planes |
|---|---|
| Altitude | 20,184 km |
| Orbital speed | 3,874 m s−1 |
| Inclination | 55°; not polar |
| Clock gain, gravitational | +45,850 ns/day |
| Clock loss, velocity | −7,214 ns/day |
| Net | +38,640 ns/day |
| Corrected by | Detuning the clocks before launch |
| Error if not corrected | ≈11.6 km per day |
| Also requires | A correction for the Earth's rotation[1] |
| Cited as evidence for | A stationary Earth |
The Global Positioning System determines a receiver's position by timing signals from satellites whose positions are known, and it does so to within a few metres. To achieve that it must account for the curvature of spacetime, the velocity of the satellites, and the rotation of the Earth beneath the signal, and it must do all three before it can tell anyone which street they are on.[2]
It is, in consequence, the most thoroughly relativistic apparatus in ordinary civilian use, carried in every pocket. It is also produced, regularly, as evidence that relativity is a fiction and that the Earth does not move.[3]
Each satellite carries an atomic clock and broadcasts its own position and the time. A receiver picks up four or more such signals, and from the differences in their arrival times computes where it must be for all four to be consistent. Nothing is transmitted back; the receiver only listens.
The whole method rests on timing, and the tolerances are severe. Light travels about thirty centimetres in a nanosecond, so a clock error of a millionth of a second puts you three hundred metres from where you are. A system that hopes for metres needs its clocks right to a few billionths of a second and needs them to stay right, which is a great deal to ask of anything, and rather more to ask of something in a vacuum at four kilometres a second with nobody to wind it.
They do not stay correct on their own.
Four corrections are applied, in three different places, and each of the arguments below works by naming one of them and not the others.
| Correction | For | Size | Applied |
|---|---|---|---|
| Gravitational | Height in the potential | +45,850 ns/day | In the factory, by detuning the clock |
| Velocity | Orbital speed | −7,214 ns/day | |
| Eccentricity | Non-circular orbit | Periodic, small | By the receiver, as −2R·v/c² |
| Sagnac | The Earth turning in flight | Up to ~200 ns | By the receiver, every fix |
Two of them act on the satellite's own clock, in opposite directions and by different amounts.
The general-relativistic term is the larger, and it has the opposite sign to the special-relativistic one, so the two do not cancel and cannot be made to. What remains is a gain of about 38.6 microseconds a day.[4]
That is a small number until it is multiplied by the speed of light. Thirty-eight microseconds of clock error is eleven and a half kilometres of position error, accumulating daily. An uncorrected system would be useless within an hour and absurd within a week.
The clock correction is not applied by software after the fact. It is built into the hardware, on the ground, before the satellite goes up.
A GPS clock's nominal frequency is 10.23 MHz. The clocks are set instead to 10.22999999543 MHz, low by a fraction of
which is 38.6 microseconds a day: exactly the amount the clock will gain once it is in orbit.[5] The clock is deliberately built to keep the wrong time on Earth, so that it will keep the right time in space.
There is a reason to be confident about all this that does not depend on trusting anybody, and it is that the question was put to a vote of the apparatus in 1977.
NTS-2 carried the first caesium clock ever flown. Not everyone on the programme believed the relativistic effects were real, and rather than argue it out, the engineers built the clock with a frequency synthesiser that could be switched on from the ground after launch. If the clock in orbit ran at the predicted rate, the synthesiser would be turned on and bring it to the rate the system needed. If it did not, it would not.
The satellite went up in June 1977 and the clock was left alone for about twenty days, simply running, while its rate was measured against clocks on the ground. It ran fast by 442.5 parts in 1012. The prediction and the measurement differed by 3.97 parts in 1012, which is under one per cent of the effect.[16]
They turned the synthesiser on.
This is the exact opposite of the thing alleged. The correction was not smuggled in by theorists and defended afterwards; it was doubted, by the men building the hardware, who settled it the way engineers settle things, which is by flying it and reading the dial. Every satellite since has had the offset built in on the ground, the question having been asked once, in orbit, with a switch.
The remaining two are applied not in the factory but in the receiver, thousands of times a day, and the first of them is the most directly awkward for the use the system is put to.
The receiver's position is computed in a frame fixed to the Earth, and the Earth turns during the signal's flight. In the fifty to eighty milliseconds a signal takes to arrive, a receiver at the equator is carried some thirty metres eastward. The range equation must therefore include a Sagnac term; omitting it produces an east–west error of hundreds of nanoseconds, which is tens of metres.[1]
The correction depends on the rotation rate of the Earth. It is applied continuously, by every receiver, using a value for that rotation which does not vary and is not in dispute.
The fourth is smaller and is mentioned by almost nobody. Because the orbits are not exactly circular, each satellite's clock rate varies a little around every circuit, and the receiver applies a further term for it, written in the system's own interface specification as .[15] It is relativistic, it is periodic, it is in every receiver on the planet, and it is derived from the shape of an orbit the argument holds does not exist.
The most serious form of the objection is not that the corrections are absent but that they are Newtonian. GPS, it is argued, works in a single common time shared by every satellite and receiver, transforms between frames without difficulty, and therefore uses absolute time and space after all.
The premise is largely right and the conclusion does not follow. A coordinate time is a bookkeeping convention with a stated origin and rate, which is exactly what the world's civil time is and has been since 1967; it is not a claim that there is one true instant everywhere.
GPS does use one coordinate time, realised in a frame centred on the Earth and not rotating with it, and the transformation from that frame to the rotating one is written with – which is to say, in the Galilean form. Ashby says so plainly, and the objection is generally sourced to his saying it.[6]
What the objection omits is where the discarded content went. Setting does not delete the relativistic physics; it moves it. The term the Lorentz transformation would have carried reappears as the Sagnac correction, which is applied separately and which the objection has already conceded is applied. GPS takes the simpler bookkeeping and pays for it with an extra term, because a rotating frame is not inertial and dragging a Lorentz transformation into it is more trouble than it is worth. The transformation looks Galilean because the relativity is standing next to it in a separate line of the range equation.
The intuition offered alongside this is sonar: GPS is said to work exactly as sonar does in a fixed body of water, the ECI frame playing the part of the pool.
The analogy holds until the one place it needs to hold. A swimmer with a stopwatch can find the water's rest frame: sound goes faster downstream than up, and measuring the difference gives his motion through the medium. That measurement is the whole of sonar and it is available to anyone in the pool.
The corresponding measurement for light has been attempted repeatedly since 1887 and returns nothing. Local light speed is the same in every direction whatever the receiver is doing, which is why the ECI frame has to be chosen rather than found. Hatch's own formulation is exact on the point: the effect arises whenever the receiver moves with respect to the chosen isotropic light-speed frame. A frame one selects is not a medium one detected.[10]
A separate observation is made, and it is correct: no GPS satellite passes over the poles. The constellation is inclined at 55 degrees and the ground tracks stay between about 55° north and 55° south.
This is true, and it is a design decision made for money: a navigation constellation needs four satellites in view from everywhere at once, and inclined orbits manage it with fewer vehicles than polar ones would. The constellation is 55 degrees because 55 degrees was cheaper. Polar orbits are used freely where they suit the task – weather satellites, Iridium at 86°, and every Earth-observation satellite, which are put in polar orbits precisely because the planet turns underneath and hands them the whole surface in a day.[7]
The inference drawn from it is that something is being hidden at the poles. The satellites that do go over the poles are in the same public catalogues as the ones that do not.
The name most often produced in support of the argument is Ronald R. Hatch, and it is produced for a good reason: he was not a bystander.
Hatch, who died in 2019 at eighty, took a degree in physics and mathematics in 1962, worked at the Johns Hopkins Applied Physics Laboratory, at Boeing and at Magnavox, co-founded NavCom Technology, held more than thirty patents in satellite navigation, and served the Institute of Navigation as a Fellow and as its president. The Hatch filter – smoothing the noisy pseudorange with the precise but ambiguous carrier phase – is his, is standard, and is running in receivers today.[8]
He also spent three decades arguing that Einstein was wrong, and he published on it: Escape from Einstein (1992), which sets out an alternative he called a modified Lorentz ether gauge theory, and a run of papers with titles such as "Relativity and GPS" and "Instances of Relativistic Illogic (with a GPS assist)", the latter carrying its argument in its subtitle. He worked at times with Ruyong Wang, whose fibre-optic conveyor supplies the other half of the case.[11]
His principal opponent in this was Neil Ashby, whose account of relativity in GPS remains the standard reference and is the source of most of the figures on this page. The two men disagreed for years about a system they both understood in detail, which is a better class of disagreement than most.
Two things follow, and they are habitually run together.
His navigation work is genuinely his, genuinely good, and independent of the question. Carrier smoothing is a signal-processing technique concerning the statistics of two measurements of the same distance; it would work identically under any theory of space and time, and its correctness transfers to nothing.
And a Lorentzian ether theory, properly constructed, is empirically equivalent to special relativity. Same transformations, same predictions, same numbers to the last decimal. Lorentz's own was, which is why the choice between them was never settled by experiment and never could be: it is settled, so far as it is settled, on the ground that the preferred frame does no work and cannot be found. A man may prefer the ether formulation and be committed to no error whatever.[9]
What the preference cannot do is show relativity to be wrong, since the two accounts agree about every reading any instrument will give. Hatch's satellites and Einstein's satellites are detuned before launch by the same 4.46 parts in ten billion.
A stronger claim is now circulated on his behalf: that GPS did not work until Hatch supplied it with a frame equivalent to absolute space, and would not work today without him. This is not what he did. The Earth-centred inertial frame is ordinary orbital mechanics and long predates him; Hatch's contribution was the filter, which is a method of combining two noisy measurements of the same distance and has no reference frame in it at all. He was an important man in satellite navigation for reasons that have nothing to do with the ether, and the claim made in his name diminishes the work he actually did.[12]
The system described here was built for the American military and would have stayed there. That it is in every pocket is the consequence of a navigational failure.
On 1 September 1983 Korean Air Lines Flight 007, bound for Seoul, drifted progressively north of its intended track into Soviet airspace and was shot down by an interceptor near Sakhalin. All 269 people aboard were killed.
The drift is now attributed to the inertial navigation system never having captured the programmed route. If the units were armed while the aircraft was more than about seven and a half nautical miles from that track, they would not acquire it, and the autopilot would hold the heading it already had, indefinitely, without indicating that anything was wrong.[17] The aircraft was flying accurately. It was flying accurately along the wrong line.
On 16 September, a fortnight later, the American president directed that GPS be made available for civil aviation without charge once it was complete.[18] The most thoroughly relativistic instrument in ordinary use is public property because an aeroplane could not tell that it had never been on course.
The system is offered, then, as evidence against the theory it was built on, by an argument that arrives on a phone.
Its strongest form holds that the two relativistic effects cancel one another and are therefore not implemented at all, the real corrections being classical.
They do not cancel. They have opposite signs and different magnitudes, and the residue is the 38.6 microseconds a day that every satellite clock is built to lose. The point is not in dispute in the literature and it is not in dispute in the engineering paper the argument itself most often cites: Fliegel and DiEsposti's 1996 overview, written from inside the programme, states that each satellite clock is preset to run slow by about 4.45 × 10−10, which is the 38 microseconds a day, and gives the same two components – roughly 45 from the gravitational potential and 7 the other way from the orbital speed of 3.87 km s−1.[14]
The document produced to show that the correction is not applied specifies the size of the correction and the frequency it is applied at.
A related form holds that the corrections are Sagnac rather than relativistic, the Sagnac term being the larger. They correct different things and are both applied: one is arithmetic done in the receiver, the other a decision taken in a factory, and observing that the first is bigger does not remove the second any more than a large bill removes a small one.
A third holds that applying both a gravitational and a velocity correction refutes the equivalence principle, which holds that no local experiment distinguishes gravitation from acceleration: the principle is local, and says so. A satellite twenty thousand kilometres up and a receiver on the ground are not in one another's local neighbourhood, which is the entire reason the two clocks disagree and the entire reason the correction has two terms.[13]
GPS does work. It works for the four reasons in the table above, and a system produced as evidence for a stationary Earth cannot tell anyone where they are without first being told how fast the Earth is turning.
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