GPS and Relativity: An Engineering Overview
| Scientific Paper | |
|---|---|
| Title | GPS and Relativity: An Engineering Overview |
| Read in full | Link to paper |
| Author(s) | Henry P Fliegel |
| Keywords | relativity, GPS |
| Published | 1996 |
| No. of pages | 11 |
| Pages | 189-199 |
Read the full paper here
Abstract
We give and explain in detail the formulas for the relativistic corrections to be implemented in high-speed aircraft, or when using other satellites in connection with GPS, or when using GPS from another satellite. We explain how to use these formulas in various scenarios, give numerical examples, and itemize the pitfalls to be avoided by (for example) receiver manufacture.
Overview
This is a working paper by Henry F. Fliegel and Raymond S. DiEsposti of the GPS Joint Program Office at The Aerospace Corporation, delivered to the Precise Time and Time Interval meeting. It is not a dissident paper: it is the engineering authority most often cited on both sides of the long-running argument about whether GPS confirms, refutes or ignores relativity, and it is archived here because so much of that argument turns on what it actually says.
Its starting point is a concession that has been quoted in both directions. The Operational Control System (OCS) of GPS "does not include the rigorous transformations between coordinate systems that Einstein's general theory of relativity would seem to require" — no rigorous transformations between the space vehicles, the Monitor Stations, the rotating Earth and the geocentric inertial frame. The authors give the reason directly: those effects, where they differ from classical predictions, "are too small to matter — less than one centimeter, for users on or near the earth." The paper exists because a new class of users — satellites taking time and position from GPS — cannot live with the approximations, and because "because those approximations have not been publicly analyzed and presented, there is much confusion in the GPS literature."
The argument
The three effects
Fliegel and DiEsposti sort the corrections by cause: relative motion, difference of gravitational potential, and acceleration.
The velocity effect is special relativistic. Where classical physics gives the Doppler-shifted frequency in terms of vcosA/c, the relativistic expression multiplies by γ = secP where tanP = v/c. The authors stress the consequence for engineers: even at cosA = 0, where naive intuition gives no Doppler shift, the received frequency is still low by the γ factor — the transverse Doppler effect. They note that γ behaves like a secant, being very nearly one for small v/c and increasing with the square of the angle. The angle A must be corrected for aberration, since the cosine has to be evaluated in the observer's frame; the transformation from the ECI-frame angle A′ to the observer-frame angle A is given exactly, and the authors point out that the γ factor cancels in the expression for cosA although it would appear in those for sinA and tanA.
The gravitational effect is derived heuristically: a photon falling into a potential well gains frequency, its energy hf playing the role of mc2, so that the fractional frequency shift equals the potential difference divided by c2.
The acceleration effect is shown to require no separate term. By the equivalence principle the product gr in an accelerating chamber corresponds to a potential difference, but the resulting −gr/c2 is simply the change in receiver velocity during the signal's flight — which is already included if the Doppler shift is evaluated using the relative velocity at the time of reception. Only a user labelling events at transmission time need add it explicitly, and "among GPS users, hardly anyone does so."
What the hardware actually does
Because receivers work in the time domain rather than the frequency domain, the corrections are implemented differently. Each SV clock is offset from its nominal rate by about −4.45 × 10−10, or −38 microseconds per day. Of this, about −45 µs/day is due to the gravitational potential difference between the satellite at its mean radius and the Earth's surface, and +7 µs/day to the mean orbital speed of about 3.87 km/s. Each receiver then adds the orbital-eccentricity term of ICD-GPS-200, which the authors note is appropriate for users on or near the surface but not for users in space, who should instead apply the frequency-domain equations and integrate.
The "sticky wickets"
The second half addresses specific disputes.
Is the OCS Newtonian? It computes the velocity effect only for the mean orbital speed, not for the speed of each satellite relative to each Monitor Station. The authors show this is tolerable for two reasons. Treated as range, the γ-factor foreshortening at 3.87 km/s is γ − 1 = 8.33 × 10−11, which over a 30,000 km range is 2.5 millimetres — "close enough for government work." Treated as accumulated Doppler it would be worse, because the Earth's rotation (up to 0.465 km/s at the equator) adds vectorially to the satellite speed, making γ − 1 vary by ±1.88 × 10−11; over an hour that variable part would alias into range as about 20.3 metres. The saving grace is the Kalman filter: Monitor Station time is estimated jointly with the SV clocks and is effectively updated continuously, so the relevant interval is the signal propagation time of about 0.1 second, which again gives about 2.5 millimetres. The authors add a warning that this "happy result depends entirely on the Kalman filter" being "springy"; retuned to a long time constant, the neglected γ factor would produce errors of many metres. Their verdict is even-handed: "In principle, the critics of GPS in the relativity debate have not been completely wrong. The neglected γ factor could hurt us. The OCS software should be reformulated. Nevertheless, in practice, neglect of relativity does not now contribute measurably to the GPS error budget."
Does GPS work in an ECI frame? Only partly. GPS uses a mixed system: spatial coordinates are ECI, but the time rate is that of the rotating Earth. The paper invokes the standard theorem that ideal frequency standards anywhere on the rotating geoid run at the same rate, because the slowing of equatorial clocks by rotation is exactly cancelled by their higher gravitational potential. GPS time is steered to UTC(USNO) and hence to TAI, whose unit is the SI second on the geoid; a satellite user who naively mapped everything into a true ECI frame with Geocentric Coordinate Time would incur an error of roughly 60 µs/day. The ECI does enter implicitly, however, because comparing clocks on different continents requires the Sagnac correction, which needs the rotation rate of the ECEF frame relative to the ECI.
Are there missing relativity terms? Steven Deines argued in 1992 that ground-based receivers carry uncompensated Earth-rotation terms. Working through Nelson's transformation between a rotating, accelerated frame and a non-accelerating one, and substituting the acceleration and velocity of a station fixed to the surface, Fliegel and DiEsposti show — as Neil Ashby had — that the extra terms cancel identically by a vector identity, leaving "just what one would expect by a Lorentz transformation from the center of rotation to the instantaneous rest frame of the accelerated origin." Their conclusion: "there are no 'missing relativity terms.' They cancel out."
Assessment
The paper's arithmetic is correct in every place it can be checked. A GPS orbit radius of 26,560 km gives an orbital speed of √(GM/r) = 3,874 m/s, matching the 3.87 km/s used throughout. At that speed γ − 1 = ½(v/c)2 = 8.35 × 10−11, agreeing with the quoted 8.33 × 10−11; times 30,000 km this is 2.50 mm, and times 0.1 second times c it is again 2.50 mm. The clock offset checks: 4.45 × 10−10 × 86,400 s = 38.4 µs/day, and −45 + 7 = −38. The equatorial speed of 0.465 km/s is right (2π × 6,378 km / 86,164 s). The variation of γ − 1 works out to vΔv/c2 = 2.0 × 10−11, close to the quoted 1.88 × 10−11 — the small difference is what one expects for Monitor Stations away from the equator — and 1.88 × 10−11 × 3,600 s × c = 20.3 m exactly as stated. The 60 µs/day TCG–TT figure matches the IAU constant LG = 6.969 × 10−10 times 86,400 s = 60.2 µs/day. There is no unit slip, no stray factor and no fitted parameter anywhere in it.
Its lasting value is that it separates two claims that are constantly conflated in the wider argument. The large mean offset — 38 µs/day, worth about 11 km of position error per day if left uncorrected — is a relativistic correction, it is built into every GPS satellite clock, and it is essential. The residual terms beyond that offset and the eccentricity correction are what the OCS neglects, and those are at the millimetre level for ground users. Both the claim that "GPS proves relativity every day" and the claim that "GPS ignores relativity entirely" can be supported by quoting half of this paper; only reading the whole of it gives the correct picture, which is that the first-order relativistic effects are large, applied and verified, and the higher-order ones are below the noise.
Two cautions are worth recording for readers of this wiki. First, the paper's frank admission that "the OCS software should be reformulated" is an engineering remark about a specific Kalman filter tuning, not a concession that relativity is in doubt; the same paragraph says the neglected effects are unmeasurable as the system is configured. Second, the geoid result invoked here — that ideal clocks anywhere on the rotating geoid tick at the same rate — is sometimes presented in the dissident literature as a surprising coincidence between rotation and potential. It is not a coincidence but a theorem: the geoid is defined as a surface of constant effective (gravitational plus centrifugal) potential, and the clock rate depends on exactly that combination, so equality of rates on it follows by definition rather than by accident. Anyone building an argument on the apparent cancellation should note that the cancellation is what "geoid" means.
The paper's genuine limitations are those of its genre. It is an overview, so the derivations are referred out to Landau and Lifshitz, to the Explanatory Supplement, and to Ashby and Spilker; the heuristic derivation of the gravitational shift by treating hf as mc2 is a physical argument, not a proof, and the authors do not pretend otherwise. And the paper does not settle the question raised in discussion by Gernot Winkler and Carroll Alley, namely whether residual frequency-offset error propagates into uploaded predictions between Monitor Station observations; Fliegel's own answer is that the effect repeats day after day and so probably does not badly damage the predictions, but he concedes "it may" and that the clock error "has just [not] disappeared."
Note on this archive copy: the PDF is an OCR'd scan and the text layer is damaged. Several equations are unreadable and constants appear mangled (γ printed as "7", mc2 as "m 2", "formulas" as "fonnuh"). The numerical values quoted above have been checked independently against the underlying physics; the equations have been described rather than transcribed. Readers wanting the exact expressions should consult the original PTTI proceedings or Ashby and Spilker.