Conducting a Crucial Experiment of the Constancy of the Speed of Light Using GPS: Comments on Ashby's ?Relativity and the Global Positioning System?: Difference between revisions
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==Abstract== | ==Abstract== | ||
''Proceedings of the ION 58th Annual Meeting & CIGTF 21st Guidance Test Symposium'', 24-26 June 2002, pp 495-505. Contrary to the assertion of Special Relativity, the speed of light is not always constant relative to a moving observer. The Global Positioning System (GPS) shows that the speed of light in the Earth Centered Inertial (ECI) non-rotating frame remains at c relative to the | ''Proceedings of the ION 58th Annual Meeting & CIGTF 21st Guidance Test Symposium'', 24-26 June 2002, pp 495-505. Contrary to the assertion of Special Relativity, the speed of light is not always constant relative to a moving observer. The Global Positioning System (GPS) shows that the speed of light in the Earth Centered Inertial (ECI) non-rotating frame remains at c relative to the frame—but not relative to an observer or receiver moving in that frame. When a GPS receiver changes its translation speed relative to the ECI frame, the speed of light measured relative to the receiver changes. A crucial experiment of the constancy of the speed of light relative to a moving receiver could be conducted in the following way: Let two GPS satellites and two airplanes be positioned in a straight line. Let the two airplanes travel at the same speed directly toward one of the two satellites and directly away from the other satellite. The travel time differences of GPS signals arriving at the two airplanes is measured and recorded with the airplanes flying first toward one of the satellites and then flying the opposite direction toward the other satellite. The travel time differences obtained as the airplanes fly in opposite directions are compared. If the travel time difference is the same when the velocity of the airplanes is changed, then the speed of light is indeed constant relative to the moving airplanes, otherwise it is not. The calculation using the GPS range equation and the results of a Real-Time Kinematic (RTK) differential GPS test have shown that the constancy of the speed of light relative to moving airplanes is not correct. The change of the time difference could reach about 10 ns for subsonic airplanes and 30 ns for supersonic airplanes. The result of this crucial experiment is not only important scientifically, but also indicates the possibility of a new way to directly measure vehicle speed relative to the ECI frame. | ||
==Overview== | |||
This is a working paper by two navigation engineers — [[Ruyong Wang]] of St. Cloud State University and [[Ronald R Hatch]] of NavCom Technology, then President of the Institute of Navigation — delivered at the ION's 58th Annual Meeting in June 2002. It is framed as a direct reply to Neil Ashby's ''Physics Today'' article of May 2002, "Relativity and the Global Positioning System," and it has two halves: a critique of Ashby's account of the [[Sagnac Effect]] in [[GPS]], and the design of an airborne experiment intended to settle the question by measurement. | |||
The authors' position is carefully bounded. They do not dispute that light propagates isotropically at ''c'' in the Earth-Centred Inertial frame; they insist on it. What they dispute is the further step of calling that the constancy of the speed of light. "The assertions that the speed of light is constant in just one inertial frame, the ECI frame, really is not what the constancy of the speed of light means." Their conclusion is that the ECI frame is a ''preferred'' frame near the Earth, that the one-way Sagnac effect is caused by ''any'' receiver motion rather than by rotation specifically, and that a moving receiver therefore does not see an isotropic light speed. | |||
==The argument== | |||
===Sound in air as the template=== | |||
The GPS propagation-delay equation is |'''r'''<sub>''r''</sub>(''t''<sub>''r''</sub>) − '''r'''<sub>''s''</sub>(''t''<sub>''s''</sub>)| = ''c''(''t''<sub>''r''</sub> − ''t''<sub>''s''</sub>), where the source position is taken at transmission and the receiver position at reception. Ashby calls this "an apparently simple application of the second postulate"; Wolf and Petit had concluded that if the equation is correct, special relativity is correct. | |||
Wang and Hatch answer with an exact analogue: in a frame at rest in the air, sound obeys |'''r'''<sub>''r''</sub>(''t''<sub>''r''</sub>) − '''r'''<sub>''s''</sub>(''t''<sub>''s''</sub>)| = ''a''(''t''<sub>''r''</sub> − ''t''<sub>''s''</sub>) with ''a'' the speed of sound. "Do we have a principle of the constancy of the speed of sound because of the constant speed of sound ''a'' appearing in the propagation delay equation? No, we do not." They quote Hecht's textbook example of two ships and a sound-emitting buoy: for the ship steaming toward the buoy, the blast crosses from bow to stern in less time, because the stern advances to meet it, and the wave speed measured aboard is faster. | |||
===The bow-to-stern calculation=== | |||
The same geometry is then run with GPS receivers at bow and stern, a differential GPS station on the same meridian, and separation ''L''. At rest, the two receivers get the signal at ''t''<sub>1</sub> = ''t''<sub>0</sub> + ''l''/''c'' and ''t''<sub>2</sub> = ''t''<sub>0</sub> + (''l''+''L'')/''c'', so bow-to-stern takes ''L''/''c''. Under way at speed ''v'', the same equation gives ''t''′<sub>1</sub> = ''t''′<sub>0</sub> + ''l''′/(''c''+''v'') and ''t''′<sub>2</sub> = ''t''′<sub>0</sub> + (''l''′+''L'')/(''c''+''v''), so the traverse takes ''L''/(''c''+''v''). The difference is | |||
Δ''t'' = ''L''/''c'' − ''L''/(''c''+''v'') = ''vL''/''c''<sup>2</sup> | |||
to first order. Since the bow-to-stern distance aboard is a fixed ''L'', "measured by this observer, the speed of light is not constant." | |||
Two standard objections are pre-empted. Relativity of simultaneity is said not to matter because the clocks need not be synchronized at all: giving them arbitrary biases δ''t''<sub>1</sub> and δ''t''<sub>2</sub> adds the same constant to both cases, so it cancels from Δ''t''; and since both clocks share the same motion, their rate changes cancel too. Lorentz contraction is said not to matter because Δ''L'' = (''L''/2)(''v''/''c'')<sup>2</sup> is second order while the effect sought is first order. The Michelson–Morley null result is likewise held to be no obstacle: it is a two-way, second-order experiment, its light path is defined by a structure moving with the apparatus, and "the Michelson–Morley experiment has never been conducted in a lab moving relative to the earth." | |||
===Against Ashby on the Sagnac effect=== | |||
Three of Ashby's statements are contested. Against "the fundamental principle on which GPS navigation works is... the constancy of ''c''," the authors reply that the range equation depends on constancy relative to the ECI frame, not relative to the receiver. Against "observers in the non-rotating ECI frame would not see a Sagnac effect; instead, they would see that receivers are moving while a signal is propagating," they reply that receiver motion during transit ''is'' the Sagnac effect, so this concedes the point — "a bit of a sophistry." Against "if one works entirely in the nonrotating ECI frame there is no Sagnac effect," they offer operational evidence: NavCom uses JPL software which computes entirely in ECI, and after investigating discrepancies against their own Earth-centred Earth-fixed solution they found the measured and theoretical ranges agreed precisely in both frames, "indicating that the Sagnac correction had been applied in each frame." The JPL deep-space equations of Moyer, worked in the solar-system barycentric frame and accounting for receiver motion during transit, are cited to the same end. | |||
They add a scaling argument against attributing the effect to curvature. At Los Angeles the Earth turns about 27 metres during the nominal 70 ms satellite-to-receiver transit, and the departure of that 27 m arc from its chord is a few tens of microns. "It certainly seems incredible that a 35 micron deviation from a straight line could induce a 27 meter change in the measured range." A carrier-phase test is reported in support: with a stationary reference site, the remote antenna was raised and lowered 32 cm over eight seconds; the Sagnac correction was still required, and the rms residuals stayed at a few millimetres rather than rising to metres. | |||
===The crucial experiment=== | |||
Two atomic clocks with transmitters, reflectors and receivers are mounted at points A and B, separation ''L'', on a vehicle flying due south at speed ''v'' (due south to remove the Earth's rotation). A signal goes A→B→A, and the nominal one-way difference Δ''t''<sub>1</sub> = [''t''′<sub>1</sub>(A) − ''t''<sub>1</sub>(B)] − [''t''<sub>1</sub>(B) − ''t''<sub>1</sub>(A)] is formed from raw clock readings. The vehicle then turns and flies north at the same speed, giving Δ''t''<sub>2</sub>. The prediction is that the two differ by | |||
Δ''t''<sub>2</sub> − Δ''t''<sub>1</sub> = 4''vL''/''c''<sup>2</sup>, | |||
a first-order effect immune to Lorentz contraction. If instead the two speeds differ, the expected difference is 2''L''(''V''<sub>1</sub>+''V''<sub>2</sub>)/''c''<sup>2</sup>. | |||
The simplification that makes the experiment practical is to drop the airborne transmitters. If two receivers A and B lie on the straight line between two satellites S<sub>1</sub> and S<sub>2</sub> on opposite horizons, then the A-to-B propagation time is just the difference of the two satellite-to-receiver times, so existing satellite transmitters do the work. Two aircraft carrying GPS receivers fly south at separation ''L'', record arrival times from both satellites, turn, fly north and record again. Baseline is set by the horizon: ''L'' = 3,572[√''h''<sub>1</sub> + √''h''<sub>2</sub>] metres, or 7,144√''h'' for equal heights. The predicted signals are tabulated: | |||
{| class="wikitable" | |||
! Height !! 5 km !! 10 km !! 20 km | |||
|- | |||
| ''L'' || 500 km || 700 km || 1,000 km | |||
|- | |||
| 4''vL''/''c''<sup>2</sup>, ''v'' = 300 m/s || 6.7 ns || 9.3 ns || 13.3 ns | |||
|- | |||
| 4''vL''/''c''<sup>2</sup>, ''v'' = 700 m/s || 15.6 ns || 21.8 ns || 31.1 ns | |||
|} | |||
The error budget is the paper's strongest section. A transverse offset Δ''h'' of receiver A from the line changes the satellite range by only about Δ''h''<sup>2</sup>/2''S''<sub>1</sub>''A'', with ''S''<sub>1</sub>''A'' ≈ 26,000 km — 0.2 mm for 100 m, 2 m for 10 km — so alignment is not critical below a few kilometres. Satellite motion over a 30-second run is small. Because the measurement is a multiple difference — between two receivers, two propagation directions and two motion states — satellite position bias, satellite clock bias, ionospheric and tropospheric delays all cancel, and the low elevation angle that would normally disqualify a satellite becomes harmless. The Appendix works the four cases through the full pseudorange equation ρ = ''c''(''T''<sub>''r''</sub>−''T''<sub>''s''</sub>) + ''c''δ<sub>''r''</sub> − ''c''δ<sub>''s''</sub> + Δ''D'' + ''c''Δ''I'' + ''c''Δ''T'', showing every bias term dropping out and leaving ''c''Δ''t'' = ''Lc''/(''c''−''v'') − ''Lc''/(''c''+''v'') = 2''Lv''/''c'' for one satellite, doubled for two. The acknowledgements record that the University of Calgary simulated the scenario but that the transmit-time precision was insufficient to resolve the effect. | |||
==Assessment== | |||
This is the most professionally executed paper of its kind on this wiki, and its quality shows in the error budget rather than the polemic. Hatch and Wang are engineers who work with these signals daily, the proposed measurement is concrete and affordable, the differencing structure genuinely does cancel the dominant GPS error sources, and the authors state in advance what result would ''verify'' the constancy of the speed of light. That last point deserves emphasis: the experiment is offered as decidable either way. | |||
Every number in the paper that can be checked, checks. Δ''t'' = ''L''/''c'' − ''L''/(''c''+''v'') = ''vL''/''c''(''c''+''v'') ≈ ''vL''/''c''<sup>2</sup>. The horizon formula is right: √(2''R''<sub>⊕</sub>''h'') = 3,570√''h'' metres. The baselines follow — 7,144√5000 = 505 km, 7,144√10000 = 714 km, 7,144√20000 = 1,010 km. Every entry of the timing table reproduces: 4(300)(5×10<sup>5</sup>)/''c''<sup>2</sup> = 6.68 ns, 4(700)(10<sup>6</sup>)/''c''<sup>2</sup> = 31.2 ns, and the four in between. The transverse-offset table reproduces exactly from Δ''h''<sup>2</sup>/2''S''<sub>1</sub>''A''; and ''S''<sub>1</sub>''A'' is right, since a satellite on the horizon is √(26,560<sup>2</sup> − 6,371<sup>2</sup>) = 25,780 km away. The Los Angeles figure is right too: the surface speed at 34° N is 465 cos 34° = 386 m/s, and 386 × 0.070 s = 27 m. The Appendix algebra is correct. This is careful work. | |||
Two small numerical points. The sagitta of a 27 m arc on the latitude circle at Los Angeles (radius 5,280 km) is ''s''<sup>2</sup>/8''R'' = 17 µm, and the departure from the tangent at one end is 69 µm; the quoted 35 µm sits between the two, and nothing in the argument turns on which is meant. More substantially, the satellite-motion estimate is internally inconsistent: 0.25° of orbital arc at the GPS radius of 26,560 km is about 116 km, not the 27 km stated. Carried through the authors' own formula, the resulting line-of-sight deviation is roughly 3 km rather than 0.74 km — still inside the 5 km tolerance they set, so the conclusion survives, but with far less margin than the paper suggests. | |||
The real difficulty is not arithmetic but interpretation, and it is the classic one. The 4''vL''/''c''<sup>2</sup> prediction is what special relativity predicts as well. The authors work in ECI coordinates throughout, and their clocks A and B are explicitly ''not'' synchronized in the aircraft frame; they are free-running clocks whose offsets from ECI coordinate time are treated as constants. Under exactly that convention, special relativity gives ''L''/(''c''−''v'') one way and ''L''/(''c''+''v'') the other, and 4''vL''/''c''<sup>2</sup> for the two-state difference — because ECI is an inertial frame in which light travels at ''c'', and the receivers move during transit. The paper's own equations are the relativistic ones. A positive result would therefore confirm the standard treatment rather than refute it. | |||
The dismissal of relativity of simultaneity is where this becomes visible. Constant biases do cancel, as the authors show. But the quantity special relativity says is convention-dependent is not a fixed offset; it is the ''v''-dependent term ''vx''/''c''<sup>2</sup> relating ECI-synchronized clocks to clocks Einstein-synchronized aboard the aircraft — and that term is precisely ''vL''/''c''<sup>2</sup>, the very quantity being measured. Saying "the synchronization of the clocks is not needed here" does not remove the issue; it fixes the synchronization to the ECI convention and then reports the anisotropy that convention entails. What the experiment would measure is a genuine, real, first-order effect — the one-way Sagnac term that GPS already corrects for on every signal path — but it does not discriminate between "there is a preferred frame" and "we chose to synchronize to one frame." | |||
On the substantive dispute with Ashby, the authors are largely right on the physics and wrong about who it favours. The Sagnac effect in GPS ''is'' receiver motion during transit, it is not specifically rotational, and the JPL barycentric equations do treat light speed as constant with respect to the chosen frame. Ashby would not dispute any of this; it is standard practice, and it is why the effect is corrected in ECI as well as in Earth-fixed coordinates. The curvature argument is aimed at a position Ashby is not obliged to hold. | |||
The reported carrier-phase test is the weakest evidence offered. Raising an antenna 32 cm over eight seconds adds a radial velocity of 4 cm/s to a site already moving at 386 m/s with the Earth's rotation; the receiver's path in ECI remains, to four decimal places, the same rotational arc it always was. That the Sagnac correction was still needed shows only that the site rotates, which was never in doubt. The test does not isolate straight-line motion, and cannot bear the weight placed on it. | |||
Finally, the paper's conclusion that the ECI frame is "preferred near the earth" is doing less work than it appears. ECI is preferred for GPS in the practical sense that the system defines a common coordinate time in it — but any inertial frame would serve, at the cost of transforming every clock rate and every synchronization offset. The paper does not explain how the experiment would distinguish ECI from, say, the barycentric frame it elsewhere cites JPL as using, and the two are in relative motion at 30 km/s. | |||
==See also== | |||
* [[Ruyong Wang]] | |||
* [[Ronald R Hatch]] | |||
* [[GPS]] | |||
* [[Sagnac Effect]] | |||
* [[Georges Sagnac]] | |||
* [[Speed of Light]] | |||
* [[Special Relativity]] | |||
* [[Simultaneity]] | |||
* [[Michelson–Morley experiment]] | |||
* [[Length Contraction]] | |||
* [[Herbert E Ives]] | |||
* [[Time Dilation]] | |||
* [[Galilean Electrodynamics]] | |||
* [[Aether]] | |||
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Latest revision as of 13:26, 21 July 2026
| Scientific Paper | |
|---|---|
| Title | Conducting a Crucial Experiment of the Constancy of the Speed of Light Using GPS: Comments on Ashby's ?Relativity and the Global Positioning System? |
| Read in full | Link to paper |
| Author(s) | Ruyong Wang, Ronald R Hatch |
| Keywords | Speed of Light, GPS, Ashby, Relativity |
| Published | 2002 |
| No. of pages | 11 |
| Pages | 495-505 |
Read the full paper here
Abstract
Proceedings of the ION 58th Annual Meeting & CIGTF 21st Guidance Test Symposium, 24-26 June 2002, pp 495-505. Contrary to the assertion of Special Relativity, the speed of light is not always constant relative to a moving observer. The Global Positioning System (GPS) shows that the speed of light in the Earth Centered Inertial (ECI) non-rotating frame remains at c relative to the frame—but not relative to an observer or receiver moving in that frame. When a GPS receiver changes its translation speed relative to the ECI frame, the speed of light measured relative to the receiver changes. A crucial experiment of the constancy of the speed of light relative to a moving receiver could be conducted in the following way: Let two GPS satellites and two airplanes be positioned in a straight line. Let the two airplanes travel at the same speed directly toward one of the two satellites and directly away from the other satellite. The travel time differences of GPS signals arriving at the two airplanes is measured and recorded with the airplanes flying first toward one of the satellites and then flying the opposite direction toward the other satellite. The travel time differences obtained as the airplanes fly in opposite directions are compared. If the travel time difference is the same when the velocity of the airplanes is changed, then the speed of light is indeed constant relative to the moving airplanes, otherwise it is not. The calculation using the GPS range equation and the results of a Real-Time Kinematic (RTK) differential GPS test have shown that the constancy of the speed of light relative to moving airplanes is not correct. The change of the time difference could reach about 10 ns for subsonic airplanes and 30 ns for supersonic airplanes. The result of this crucial experiment is not only important scientifically, but also indicates the possibility of a new way to directly measure vehicle speed relative to the ECI frame.
Overview
This is a working paper by two navigation engineers — Ruyong Wang of St. Cloud State University and Ronald R Hatch of NavCom Technology, then President of the Institute of Navigation — delivered at the ION's 58th Annual Meeting in June 2002. It is framed as a direct reply to Neil Ashby's Physics Today article of May 2002, "Relativity and the Global Positioning System," and it has two halves: a critique of Ashby's account of the Sagnac Effect in GPS, and the design of an airborne experiment intended to settle the question by measurement.
The authors' position is carefully bounded. They do not dispute that light propagates isotropically at c in the Earth-Centred Inertial frame; they insist on it. What they dispute is the further step of calling that the constancy of the speed of light. "The assertions that the speed of light is constant in just one inertial frame, the ECI frame, really is not what the constancy of the speed of light means." Their conclusion is that the ECI frame is a preferred frame near the Earth, that the one-way Sagnac effect is caused by any receiver motion rather than by rotation specifically, and that a moving receiver therefore does not see an isotropic light speed.
The argument
Sound in air as the template
The GPS propagation-delay equation is |rr(tr) − rs(ts)| = c(tr − ts), where the source position is taken at transmission and the receiver position at reception. Ashby calls this "an apparently simple application of the second postulate"; Wolf and Petit had concluded that if the equation is correct, special relativity is correct.
Wang and Hatch answer with an exact analogue: in a frame at rest in the air, sound obeys |rr(tr) − rs(ts)| = a(tr − ts) with a the speed of sound. "Do we have a principle of the constancy of the speed of sound because of the constant speed of sound a appearing in the propagation delay equation? No, we do not." They quote Hecht's textbook example of two ships and a sound-emitting buoy: for the ship steaming toward the buoy, the blast crosses from bow to stern in less time, because the stern advances to meet it, and the wave speed measured aboard is faster.
The bow-to-stern calculation
The same geometry is then run with GPS receivers at bow and stern, a differential GPS station on the same meridian, and separation L. At rest, the two receivers get the signal at t1 = t0 + l/c and t2 = t0 + (l+L)/c, so bow-to-stern takes L/c. Under way at speed v, the same equation gives t′1 = t′0 + l′/(c+v) and t′2 = t′0 + (l′+L)/(c+v), so the traverse takes L/(c+v). The difference is
Δt = L/c − L/(c+v) = vL/c2
to first order. Since the bow-to-stern distance aboard is a fixed L, "measured by this observer, the speed of light is not constant."
Two standard objections are pre-empted. Relativity of simultaneity is said not to matter because the clocks need not be synchronized at all: giving them arbitrary biases δt1 and δt2 adds the same constant to both cases, so it cancels from Δt; and since both clocks share the same motion, their rate changes cancel too. Lorentz contraction is said not to matter because ΔL = (L/2)(v/c)2 is second order while the effect sought is first order. The Michelson–Morley null result is likewise held to be no obstacle: it is a two-way, second-order experiment, its light path is defined by a structure moving with the apparatus, and "the Michelson–Morley experiment has never been conducted in a lab moving relative to the earth."
Against Ashby on the Sagnac effect
Three of Ashby's statements are contested. Against "the fundamental principle on which GPS navigation works is... the constancy of c," the authors reply that the range equation depends on constancy relative to the ECI frame, not relative to the receiver. Against "observers in the non-rotating ECI frame would not see a Sagnac effect; instead, they would see that receivers are moving while a signal is propagating," they reply that receiver motion during transit is the Sagnac effect, so this concedes the point — "a bit of a sophistry." Against "if one works entirely in the nonrotating ECI frame there is no Sagnac effect," they offer operational evidence: NavCom uses JPL software which computes entirely in ECI, and after investigating discrepancies against their own Earth-centred Earth-fixed solution they found the measured and theoretical ranges agreed precisely in both frames, "indicating that the Sagnac correction had been applied in each frame." The JPL deep-space equations of Moyer, worked in the solar-system barycentric frame and accounting for receiver motion during transit, are cited to the same end.
They add a scaling argument against attributing the effect to curvature. At Los Angeles the Earth turns about 27 metres during the nominal 70 ms satellite-to-receiver transit, and the departure of that 27 m arc from its chord is a few tens of microns. "It certainly seems incredible that a 35 micron deviation from a straight line could induce a 27 meter change in the measured range." A carrier-phase test is reported in support: with a stationary reference site, the remote antenna was raised and lowered 32 cm over eight seconds; the Sagnac correction was still required, and the rms residuals stayed at a few millimetres rather than rising to metres.
The crucial experiment
Two atomic clocks with transmitters, reflectors and receivers are mounted at points A and B, separation L, on a vehicle flying due south at speed v (due south to remove the Earth's rotation). A signal goes A→B→A, and the nominal one-way difference Δt1 = [t′1(A) − t1(B)] − [t1(B) − t1(A)] is formed from raw clock readings. The vehicle then turns and flies north at the same speed, giving Δt2. The prediction is that the two differ by
Δt2 − Δt1 = 4vL/c2,
a first-order effect immune to Lorentz contraction. If instead the two speeds differ, the expected difference is 2L(V1+V2)/c2.
The simplification that makes the experiment practical is to drop the airborne transmitters. If two receivers A and B lie on the straight line between two satellites S1 and S2 on opposite horizons, then the A-to-B propagation time is just the difference of the two satellite-to-receiver times, so existing satellite transmitters do the work. Two aircraft carrying GPS receivers fly south at separation L, record arrival times from both satellites, turn, fly north and record again. Baseline is set by the horizon: L = 3,572[√h1 + √h2] metres, or 7,144√h for equal heights. The predicted signals are tabulated:
| Height | 5 km | 10 km | 20 km |
|---|---|---|---|
| L | 500 km | 700 km | 1,000 km |
| 4vL/c2, v = 300 m/s | 6.7 ns | 9.3 ns | 13.3 ns |
| 4vL/c2, v = 700 m/s | 15.6 ns | 21.8 ns | 31.1 ns |
The error budget is the paper's strongest section. A transverse offset Δh of receiver A from the line changes the satellite range by only about Δh2/2S1A, with S1A ≈ 26,000 km — 0.2 mm for 100 m, 2 m for 10 km — so alignment is not critical below a few kilometres. Satellite motion over a 30-second run is small. Because the measurement is a multiple difference — between two receivers, two propagation directions and two motion states — satellite position bias, satellite clock bias, ionospheric and tropospheric delays all cancel, and the low elevation angle that would normally disqualify a satellite becomes harmless. The Appendix works the four cases through the full pseudorange equation ρ = c(Tr−Ts) + cδr − cδs + ΔD + cΔI + cΔT, showing every bias term dropping out and leaving cΔt = Lc/(c−v) − Lc/(c+v) = 2Lv/c for one satellite, doubled for two. The acknowledgements record that the University of Calgary simulated the scenario but that the transmit-time precision was insufficient to resolve the effect.
Assessment
This is the most professionally executed paper of its kind on this wiki, and its quality shows in the error budget rather than the polemic. Hatch and Wang are engineers who work with these signals daily, the proposed measurement is concrete and affordable, the differencing structure genuinely does cancel the dominant GPS error sources, and the authors state in advance what result would verify the constancy of the speed of light. That last point deserves emphasis: the experiment is offered as decidable either way.
Every number in the paper that can be checked, checks. Δt = L/c − L/(c+v) = vL/c(c+v) ≈ vL/c2. The horizon formula is right: √(2R⊕h) = 3,570√h metres. The baselines follow — 7,144√5000 = 505 km, 7,144√10000 = 714 km, 7,144√20000 = 1,010 km. Every entry of the timing table reproduces: 4(300)(5×105)/c2 = 6.68 ns, 4(700)(106)/c2 = 31.2 ns, and the four in between. The transverse-offset table reproduces exactly from Δh2/2S1A; and S1A is right, since a satellite on the horizon is √(26,5602 − 6,3712) = 25,780 km away. The Los Angeles figure is right too: the surface speed at 34° N is 465 cos 34° = 386 m/s, and 386 × 0.070 s = 27 m. The Appendix algebra is correct. This is careful work.
Two small numerical points. The sagitta of a 27 m arc on the latitude circle at Los Angeles (radius 5,280 km) is s2/8R = 17 µm, and the departure from the tangent at one end is 69 µm; the quoted 35 µm sits between the two, and nothing in the argument turns on which is meant. More substantially, the satellite-motion estimate is internally inconsistent: 0.25° of orbital arc at the GPS radius of 26,560 km is about 116 km, not the 27 km stated. Carried through the authors' own formula, the resulting line-of-sight deviation is roughly 3 km rather than 0.74 km — still inside the 5 km tolerance they set, so the conclusion survives, but with far less margin than the paper suggests.
The real difficulty is not arithmetic but interpretation, and it is the classic one. The 4vL/c2 prediction is what special relativity predicts as well. The authors work in ECI coordinates throughout, and their clocks A and B are explicitly not synchronized in the aircraft frame; they are free-running clocks whose offsets from ECI coordinate time are treated as constants. Under exactly that convention, special relativity gives L/(c−v) one way and L/(c+v) the other, and 4vL/c2 for the two-state difference — because ECI is an inertial frame in which light travels at c, and the receivers move during transit. The paper's own equations are the relativistic ones. A positive result would therefore confirm the standard treatment rather than refute it.
The dismissal of relativity of simultaneity is where this becomes visible. Constant biases do cancel, as the authors show. But the quantity special relativity says is convention-dependent is not a fixed offset; it is the v-dependent term vx/c2 relating ECI-synchronized clocks to clocks Einstein-synchronized aboard the aircraft — and that term is precisely vL/c2, the very quantity being measured. Saying "the synchronization of the clocks is not needed here" does not remove the issue; it fixes the synchronization to the ECI convention and then reports the anisotropy that convention entails. What the experiment would measure is a genuine, real, first-order effect — the one-way Sagnac term that GPS already corrects for on every signal path — but it does not discriminate between "there is a preferred frame" and "we chose to synchronize to one frame."
On the substantive dispute with Ashby, the authors are largely right on the physics and wrong about who it favours. The Sagnac effect in GPS is receiver motion during transit, it is not specifically rotational, and the JPL barycentric equations do treat light speed as constant with respect to the chosen frame. Ashby would not dispute any of this; it is standard practice, and it is why the effect is corrected in ECI as well as in Earth-fixed coordinates. The curvature argument is aimed at a position Ashby is not obliged to hold.
The reported carrier-phase test is the weakest evidence offered. Raising an antenna 32 cm over eight seconds adds a radial velocity of 4 cm/s to a site already moving at 386 m/s with the Earth's rotation; the receiver's path in ECI remains, to four decimal places, the same rotational arc it always was. That the Sagnac correction was still needed shows only that the site rotates, which was never in doubt. The test does not isolate straight-line motion, and cannot bear the weight placed on it.
Finally, the paper's conclusion that the ECI frame is "preferred near the earth" is doing less work than it appears. ECI is preferred for GPS in the practical sense that the system defines a common coordinate time in it — but any inertial frame would serve, at the cost of transforming every clock rate and every synchronization offset. The paper does not explain how the experiment would distinguish ECI from, say, the barycentric frame it elsewhere cites JPL as using, and the two are in relative motion at 30 km/s.