Kelly-AG: Primer

Project: A. G. Kelly, "The Sagnac Effect and Uniform Motion", 13 pages, undated (Kelly_2.pdf). Author line: "A. G. Kelly, HDS Energy, Kells Co. Meath, Ireland" [read: p.1]. Alphonsus G. Kelly (died 30 July 2005) was an Irish electrical engineer with a long career at the Electricity Supply Board (ESB) and a PhD from 1966 [web: Irish Times obituary, https://www.irishtimes.com/news/pioneering-engineer-who-helped-transform-the-esb-1.482657, via the Investigator].

Source gap (repeated from Steps 1 and 3). The project folder holds this one paper. It has no date; it cites Wang et al. 2003, so it was written in 2003 or later, and our PDF copy was made on 16 Apr 2011, after Kelly's death [read: p.13; arbiter]. Kelly's other works (the Institution of Engineers of Ireland (IEI) Monographs 1, 3 and 4; Infinite Energy 39, 2001; Physics Essays 2000; the 2005 book Challenging Modern Physics) were not obtained. Monograph 2 (1996), his 2000 Hafele-Keating paper and the two Wang et al. papers were found by the Investigator and saved in research/files/. Pages 1-5 and 12 were read in depth with page images. After the quality check (qe.md), pages 6-12 were also deep-read, with page images of pp.6, 7, 8 and 12 (deep/Kelly-2-1a1021.md, QE sections); page 13 (references) was read from the text layer. For this primer I re-read pp.9-13 in the text layer and the Wang et al. 2003 paper, and re-derived every number in section 5 myself (example.py). This version follows the arbiter's "QE revision" section, which governs where it differs from the arbiter's first pass.

Source tags: [read] = from Kelly's paper or a saved file (page given); [web] = Internet source with URL; [memory] = general knowledge, confidence stated; [calc] = my computation in example.py. Verdicts are the arbiter's; this primer adds none.

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1. The idea in one paragraph

Send two light beams round a closed loop in opposite directions and bring them back together. If the loop is spinning, the beam going with the spin takes slightly longer than the one going against it, and the two beams no longer line up. This is the Sagnac effect (Georges Sagnac, 1913). It is real and used every day in aircraft navigation gyroscopes and in the timing of the Global Positioning System (GPS). Kelly argues three things from it. First, the simplest explanation is that light moves at a fixed speed relative to the stationary laboratory, whatever the motion of its source, and this one assumption gives the right answer for any loop shape. Second, Einstein's special relativity (SR) plays no part in explaining the effect, because the effect is far larger than any relativistic correction. Third, and this is the title claim, the effect is not special to rotation: it appears for objects moving in a straight line at constant speed too, as he says an experiment by Ruyong Wang and colleagues (2003) showed. If that were true, it would break the principle that no experiment done inside a steadily moving laboratory can detect the motion. The physics he starts from is sound, and his main derivation is correct. The question is which of his conclusions actually follow from it.

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2. What you need to know first

Interference and fringes. Light is a wave. When two beams from the same source recombine, they add up if their crests arrive together and cancel if a crest meets a trough, giving a pattern of bright and dark bands ("fringes"). If one beam is delayed by a time dt, the pattern slides by N = c dt / λ fringes, where c is the speed of light (2.998×10⁸ m/s) and λ (lambda) is the wavelength. Equivalently, the phase shift is Δφ = 2π c dt / λ radians. One fringe of shift means a delay of one wave period, about 1.8×10⁻¹⁵ s for green light (λ = 550 nm, nanometres). Interferometers are therefore clocks that can see delays of a thousandth of a femtosecond. Read: Hecht, Optics, the chapter on interference; Young and Freedman, University Physics (Kelly's own textbook reference [read: p.13]), the chapter on interference.

Reference frames, inertial and rotating. A reference frame is a choice of coordinates and clocks from which you describe motion. An inertial frame is one in which a free object moves in a straight line at constant speed; a frame fixed to a spinning turntable is not inertial, because free objects appear to curve (centrifugal and Coriolis effects). Whether a frame rotates can be checked from inside it, with a gyroscope or a Foucault pendulum, without looking outside. Steady straight-line motion cannot be detected from inside: that is Galileo's principle of relativity, and it is the principle Kelly's title claim challenges. Read: John R. Taylor, Classical Mechanics, ch. 9 ("Mechanics in Noninertial Frames") [memory, high].

Special relativity in two postulates. (i) The laws of physics take the same form in every inertial frame. (ii) Light moves at c in every inertial frame, whatever the speed of its source. Consequences: moving clocks run slow by the factor √(1 − v²/c²) (v the clock's speed); and clocks that are synchronised in one frame are not synchronised in another (relativity of simultaneity). All of these effects are "second order": they differ from Newtonian physics by amounts of size v²/c². At v = 13 m/s, v²/c² is about 2×10⁻¹⁵. Read: Taylor and Wheeler, Spacetime Physics (written for beginners); Griffiths, Introduction to Electrodynamics, ch. 12 [memory, high].

One-way versus round-trip speed of light. To measure how fast light goes from A to B you need a clock at A and a clock at B that agree. Agreeing means synchronising them, and the usual way (Einstein's) is to send light signals and assume the speeds out and back are equal. So the one-way speed depends partly on a convention; only the round-trip speed (out and back to the same clock) is measured without one. This matters for Kelly's claim 5. Read: Taylor and Wheeler, chapter on the synchronisation of clocks; the Wikipedia article "One-way speed of light" is a fair introduction.

Line integrals and Stokes' theorem. For a vector field u (here a velocity at each point) and a closed path, the line integral ∮ u · dl adds up the component of u along each small step dl of the path. Stokes' theorem turns it into a surface integral: ∮ u · dl = ∫ (∇ × u) · dA over any surface bounded by the loop. For a rigid rotation, u = ω × r (ω the angular velocity vector, r the position from the axis), and ∇ × u = 2ω, so ∮ u · dl = 2ωA, where A is the area enclosed (projected onto the plane perpendicular to ω). For a uniform velocity V (same everywhere), ∮ V · dl = V · ∮ dl = 0, because the steps of a closed path add up to zero displacement. These two facts decide the project. Read: any vector calculus text, e.g. Schey, Div, Grad, Curl, and All That.

Binomial expansion. For small x, (1 − x)⁻¹ = 1 + x + x² + ..., so 1/(c − u) = (1/c)(1 + u/c + u²/c² + ...). Kelly uses this to separate first-order from second-order terms.

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3. The textbook view

3.1 The Sagnac formula

Take a circular light path of radius r on a disc spinning at angular speed ω (radians per second). The light source, the beam splitter and the detector all ride on the disc. Work in the laboratory frame, which is inertial, and use the textbook rule that light moves at c in an inertial frame. A point on the rim moves at v = ωr.

• The beam going against the rotation meets the returning source early. It covers 2πr − v t′ in time t′, so c t′ = 2πr − v t′, giving t′ = 2πr / (c + v).
• The beam going with the rotation must chase the source: c t″ = 2πr + v t″, giving t″ = 2πr / (c − v).
• The difference: dt = t″ − t′ = 4πrv / (c² − v²) = 4Aω / (c² − v²), with A = πr².

Dropping v² against c² (relative error about v²/c²):

dt = 4Aω / c² (Sagnac's formula; Kelly's eq. 1 [read: p.1]).

The shape of the loop does not matter, nor where the axis is, only the area enclosed. In fringes, N = 4Aω / (cλ). The fully general first-order form, for a light guide whose every bit moves with velocity u, is

dt = (2 / c²) ∮ u · dl.

For rigid rotation this is (2/c²)(2Aω) = 4Aω/c². This general form was in E. J. Post's 1967 review (Reviews of Modern Physics 39, 475), which Kelly cites [read: p.13, ref. 2].

3.2 How SR handles it

The derivation above is the SR derivation, done in the inertial laboratory frame. Max von Laue gave it in 1911 [web: https://en.wikipedia.org/wiki/Sagnac_effect]. SR adds one thing: the clock on the rim moves at speed v, so it runs slow, and it records the time difference as 4Aω/(c² − v²) × √(1 − v²/c²). That differs from the lab-time value by a fraction of order v²/c², invisible in any Sagnac experiment so far. So SR does not "explain" the Sagnac effect by any special relativistic trick; it predicts it with the same first-order kinematics anyone would use in an inertial frame. Seen from a rider on the disc, light takes different times to go round each way, and the rider's coordinate speed of light along the rim is c + v one way and c − v the other. SR does not forbid that, because the rotating frame is not inertial and its clocks cannot be synchronised all the way round the loop [memory, high].

3.3 Michelson-Gale 1925: the Earth as the turntable

A loop fixed to the ground rotates with the Earth (Ω = 7.292×10⁻⁵ rad/s). Only the component of the rotation perpendicular to the loop counts. For a horizontal loop at latitude φ, that is Ω sin φ, so N = 4AΩ sin φ / (cλ). Michelson and Gale built a 2010 ft × 1113 ft rectangle of evacuated pipe at latitude 41° 46′ N and predicted 0.236 fringe; they measured a mean of 0.230 [read: p.4-5]. Modern ring lasers and fibre-optic gyroscopes (FOGs) measure Earth's rotation routinely in the same way [memory, high].

3.4 Uniform motion in the textbook view

If the whole apparatus (loop, source, detector) moves in a straight line at constant velocity V, then u = V everywhere and ∮ V · dl = 0: no shift. That is also what the principle of relativity requires. Post wrote exactly this: "the Sagnac interferometer and the ring laser cannot lead to a fringe shift or a beat frequency if the equipment is in uniform translational motion" (quoted by Kelly [read: p.9]).

But if only part of a closed light guide moves, as in a conveyor belt where the fibre slides along a straight run while the source and detector ride along with it, then u is not the same everywhere, ∮ u · dl ≠ 0, and the formula predicts a shift from each moving segment: dt = 2vΔl/c² for a segment of length Δl moving along itself at speed v.

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4. Where Kelly departs

The exact point of departure is the step from "a straight segment moving inside a closed loop contributes a shift" to "uniform straight-line motion produces the Sagnac effect". The line-integral formula Kelly derives himself on p.12 is a sum over the whole closed path. A segment adds 2uΔl/c², but in a rigid apparatus moving in a straight line, every segment with u > 0 is matched by return segments with u < 0, and the total is exactly zero. In Wang's conveyor the total is not zero, because the fibre on the two straight runs moves in opposite directions: the fibre circulates round the loop, like a belt, while the gyroscope holding the source and detector moves in a straight line with the conveyor (it does not go round the loop). That is not uniform motion of the apparatus.

The large-disc argument fails in a related way. If a loop of fixed size rides near the rim of a disc of radius R_d at a fixed speed v, then ω = v/R_d. The shift 4Aω/c² falls as 1/R_d and goes to zero as R_d grows. The "infinite disc" limit gives the translation answer, zero, not a Sagnac shift.

4.1 Kelly's two other experimental arguments for "c relative to the laboratory"

Michelson 1913, rotating mirrors (Astrophys. J. 37, 190-193; Kelly's ref. [9]). Kelly writes that light was bounced off mirrors turning at 1800 rpm (revolutions per minute) and that Michelson "proved from the resulting fringe shift" three things: (1) light does not bounce like a tennis ball off a moving racquet; (2) the reflected light does not take up the mirror's motion; (3) "the light actually ignores the whole rotating apparatus. It moves solely with respect to the laboratory". He calls it "a clear confirmation" of his paper, gives the accuracy as 1:50, and adds that Michelson made a factor-of-2 error cancelled by a second omission, an error "uncovered by a colleague who has been corresponding with this author" [read: page image p.7]. The factor-of-2 point is Kelly's report of private correspondence with an unnamed colleague; no document or calculation is given, and no second source was found.

What the experiment tested: the emission (ballistic) theory of light says light reflected from a mirror moving at speed v should leave at c + 2v. Michelson concluded that "within the limit of error of experiment (say 2 per cent), the velocity of a moving mirror is without influence on the velocity of light reflected from its surface" [web: https://en.wikisource.org/wiki/Effect_of_Reflection_from_a_Moving_Mirror_on_the_Velocity_of_Light, via the arbiter, search summary]. That 2 percent matches Kelly's 1:50. So conclusions (1) and (2) are what the experiment showed, and they support the source-independence half of Kelly's premise. But SR assumes the same source-independence, so the result does not separate Kelly's picture from SR. Conclusion (3) goes further than an experiment on reflection from a moving mirror can reach. Arbiter: an established experimental result (about 2 percent), consistent with claim 2, non-discriminating.

Ring-laser standing waves. A ring laser is a closed loop of mirrors with the laser gain medium inside it, so light circulates both ways at once and forms a standing wave (a fixed pattern of bright and dark points, "nodes"). Kelly writes that the standing waves "stay stationary in relation to the laboratory" (citing Martin, IEEE Spectrum 1986); that when the ring rotates the nodes pass an observation post at a rate proportional to the rotation rate, a rate that differs for a square or triangle from a circle; and that "This phenomenon is further confirmation of the proposal in this paper, that light, in small scale experiments, travels relative to the laboratory" (citing Anderson, Scientific American 254, 1986) [read: page image p.7]. The page image confirms his beat formula, Δf = 4AΩ/(λP), where Δf is the beat frequency, Ω the rotation rate and P the perimeter [read: page image p.7]. The standard textbook picture of a ring-laser gyroscope does describe the standing wave as fixed in a non-rotating frame while the ring turns under it [memory, medium; not web-checked]. So this is the same situation as Michelson 1913: consistent with Kelly, equally predicted by SR, and not a test between them. Kelly also says Sagnac saw his 1913 result only because vibration in the apparatus prevented "lock-in" (the two beams of a ring laser locking to one frequency at low rotation). No source is given; lock-in belongs to active ring lasers, and Sagnac's was a passive interferometer [memory, medium], so this stays in the record as an unsupported claim by Kelly, not a refuted one.

4.2 Kelly's charge of inconsistency: the GPS frame and the "orbital angle"

Kelly's own words [read: p.9]: "The non-rotating centre of the earth is acceptable by adherents of the theory of SR as a suitable place to set up a satisfactory Inertial Frame ... But the surface of the earth (the laboratory in this case) is not considered as an Inertial Frame for the Bilger et al. test ... Pretending that SR does not apply to rotation, while at the same time applying it daily to experiments like the Global Positioning System, that has a far greater amount of rotation than the Sagnac experiment, is indefensible ... a very biased selection of what is termed an 'Inertial Frame'." To back this he tabulates the "orbital angle" the Earth's centre sweeps round the Sun during each experiment: 1.6° during the 40-hour Hafele-Keating clock flights, 2.9×10⁻⁶ ° for GPS signals, and far smaller angles for Michelson-Gale, Bilger's ring laser and Sagnac [read: p.9-10]. His point: the frame mainstream physics calls inertial for GPS "rotates" millions of times more during the measurement than the lab frame it refuses to call inertial.

There is a fair core here, and it is the one place where Kelly accuses his critics of inconsistency rather than of an error in physics. The reply (arbiter, QE revision 5) has two parts. First, mainstream practice does not say "SR cannot touch rotation": GPS works in an Earth-centred non-rotating frame and applies the Sagnac term as the correction for the rotating Earth-fixed frame [memory, high]. Second, and decisive: whether a frame rotates is judged by its rotation rate against gyroscopes or light loops, not by the angle it sweeps during a measurement. The Earth's spin is 7.3×10⁻⁵ rad/s and its orbital turning about 2×10⁻⁷ rad/s [memory, high]; the very ring lasers in question measure the spin. An angle depends on how long the experiment lasts (40 hours against nanoseconds), so comparing angles compares durations, not rotation. Several of Kelly's printed angles also do not recompute, and his ratio "5×10⁶" conflicts with his own figures (they give 1.7×10⁷), though the ordering of the angles survives [calc by the deep reader]. Arbiter: the orbital-angle argument is Refuted by calculation (wrong quantity compared); the "5×10⁶" ratio is Internally inconsistent with his printed figures; the wider charge that frames are chosen unevenly is Not testable as stated (no author is named) and is answered by the rate criterion.

4.3 Kelly's charge of selective quotation of Einstein

Kelly quotes three Einstein passages [read: page image p.8]: the 1905 paper, which extends the moving-clock result to polygonal and closed curved paths ("the travelled clock ... will be 1/2 t v²/c² second slow"); 1916, where "special" means restriction to uniform translation; and 1919, "The rotating observer notes very well that of his two equivalent clocks, that placed on the circumference runs slower than that placed at the centre." He concludes that "authors can selectively quote Einstein's writings, whenever there is a seemingly insurmountable difficulty with SR theory", and that many references wrongly say SR cannot be applied to circular motion, "but Einstein applied that theory to those situations in the basic paper on SR".

The arbiter weighs this fairly to both sides (QE revision 3). Kelly is right that Einstein in 1905 applied SR clock-slowing to curved and closed paths, and a blanket statement that "SR cannot be applied to rotation" is wrong; SR handles rotation as seen from an inertial frame. The 1916 sentence defines which frames are equivalent; reading it as Einstein changing his mind is Kelly's interpretation. The 1919 quotation has no reference in Kelly's list and could not be traced; it is unverified, not shown false. No author is named in the charge, so it cannot be checked. One consequence for claim 3: if SR clock-slowing applies to a clock on the rim, SR does have a (second-order) role, and Kelly's p.12 statement that time on a spinning disc is identical with lab time sits in tension with his own use of Post's dilation factor on p.10 [deep read]. Claim 3's verdict is unchanged.

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5. Worked example: one fibre loop, moved three ways

Question. Take a fibre loop like the one in Wang, Zheng, Yao and Langley (Phys. Lett. A 312, 7, 2003). Move it in three different ways at the same speed and compute the Sagnac time difference each way. Which motions give a shift, and does Wang's measurement agree?

The numbers below come from example.py:

.venv/bin/python review/Kelly-AG/example.py

Wang's setup [read: research/files/wang-2003..., p.2-3]. A fibre ribbon runs round two wheels of 30 cm diameter, driven by a 1.5 m conveyor. The source (a 1310 nm light-emitting diode) and detector are in a fibre-optic gyroscope that moves in a straight line with the conveyor (it does not go round the loop). Loop perimeters were 2.5, 4.0, 8.0 and 16.0 m, with three turns of fibre; speeds were 3 to 9 cm/s. Wang et al. write that the gyroscope "travelled with the mechanical conveyor and its uniform motion did not cause any phase shift because the FOG is only sensitive to the rotational movement" [read: p.2]. That is part of the design, stated with a reason, not a separately reported null measurement.

Geometry I chose from those values. A "stadium" loop: two semicircles of radius R = 0.15 m joined by two straight runs. Perimeter P = 8.0 m, so each straight run is Ls = (8.0 − 2π × 0.15)/2 = 3.529 m. Enclosed area A = πR² + 2R·Ls = 0.0707 + 1.0588 = 1.129 m². Three turns, so total fibre length L = 24 m. Speed v = 6 cm/s. Wavelength λ = 1310 nm.

Formula used. dt = (2/c²) D per turn, with D = ∮ u · dl, and phase Δφ = 4πD/(cλ) per turn. I computed D numerically by stepping round the loop in 200,000 pieces and adding u · dl, so nothing is assumed about which motions should give zero.

Motion 1: spin the whole loop rigidly about its centre, at ω = v/R = 0.06/0.15 = 0.4 rad/s. u = ω × r at each point. Stokes says D = 2Aω = 2 × 1.129 × 0.4 = 0.9035 m²/s; the numerical sum gives the same, 0.90345 [calc]. dt = 3 × 2 × 0.9035 / (2.998×10⁸)² = 6.03×10⁻¹⁷ s; Δφ = 0.0867 rad. This is the ordinary Sagnac effect.

Motion 2: run the fibre round the loop like a belt (Wang's conveyor) at v = 6 cm/s. Every piece of fibre moves along the path at speed v, so u · dl = v dl and D = vP = 0.06 × 8.0 = 0.480 m²/s [calc]. Of this, the two straight runs supply 2v·Ls = 0.4235 m²/s, which is 88 percent. On the straight runs the fibre moves in a straight line at constant speed, and that is where most of the shift comes from. dt = 3 × 2 × 0.48 / c² = 3.20×10⁻¹⁷ s; Δφ = 0.0461 rad. Wang's prediction for a belt is Δφ = 4πvL/(cλ). The slope 4π/(cλ) at 1310 nm is 0.03200 rad per m²/s [calc], matching the paper's predicted 0.03200. Wang measured 0.03229 ± 0.00365 over all 24 arrangements, and 0.03200 ± 0.00528 for the zero-area arrangement, in which the two runs are pressed together so the loop encloses almost no area [read: Wang p.3]. The area formula 4Aω/c² would predict almost nothing for that arrangement; the line integral, and the data, say the shift depends on fibre length times speed. On this point Kelly's line-integral picture and Wang's data agree, and the enclosed area is not the fundamental quantity.

Motion 3: carry the whole loop in a straight line at V = 6 cm/s (any direction). u = V everywhere, so D = V · ∮ dl. The numerical sum gives D ≈ 5×10⁻¹⁷ m²/s, which is round-off; the exact value is zero [calc]. No shift. That is the uniform motion of Kelly's title. Wang et al. state that the FOG's own uniform motion produced no phase shift, which is what the line integral predicts.

Kelly's infinite disc. Mount the rigid stadium on a disc of radius R_d and turn the disc so the loop moves at 6 cm/s, i.e. ω = v/R_d. For a rigid rotation D = 2Aω wherever the loop sits on the disc [calc]:

As the motion becomes straighter, the shift goes to zero, the translation value [calc].

Is the zero only an approximation? Kelly works to first order in v/c. I also did the translation case exactly, in Kelly's own picture (light at c in the lab, apparatus moving at any speed V). A beam crossing a side of vector L that moves with velocity V takes the time t with |L + Vt| = ct. The beam going the other way crosses −L. The difference for one side is exactly 2(L · V)/(c² − V²) (SymPy), and the sum of L · V round a closed polygon is zero. For a triangle with 1 m sides moving at (0.3c, 0.1c), the computed difference is 0.0 s [calc, Part C]. So the strong form fails even inside Kelly's own absolute-frame assumption, at every speed. (A side note: Kelly's p.12 element formula keeps only the velocity component along the path. Applied to this case it would leave a spurious third-order remainder, 1×10⁻¹⁰ s here, because it ignores the sideways motion of the segment. That approximation does not matter for rotation at ordinary speeds.)

What the example shows. Both Kelly's derivation and the textbook come down to dt = (2/c²)∮u·dl. Under that one formula: - rotation of a rigid loop gives a shift (Sagnac); - a belt circulating round a loop gives a shift, most of it from the straight runs (Wang; Kelly's weak form is right); - straight-line motion of the whole apparatus gives exactly zero (Kelly's strong form is wrong, by his own formula).

Checks on the other numbers in the paper [calc, Part D]:

Where Kelly and SR really differ [calc, Part E]. Both give the same lab-time difference. Kelly says the rim clock shows lab time; SR says it shows lab time × √(1 − v²/c²). The fractional difference is 9.6×10⁻¹⁶ at 13 m/s, 5×10⁻¹³ at 300 m/s, 5×10⁻¹¹ at 3 km/s. A Sagnac measurement that fixed the size of the shift to better than those fractions could tell the two apart; first-order Sagnac data cannot.

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6. Evidence table

Verdicts are the arbiter's (arbiter.md, summary table and "QE revision" section). "Kelly" in the result column means the paper's own report.

Material kept in the record although outside the paper's Sagnac argument: - Kelly's Hafele-Keating critique (2000, saved in research/files). From the 1971 U.S. Naval Observatory (USNO) report he lists per-clock results: eastward −196, −54, +166, −97 ns; westward +413, −44, +101, +26 ns, against the published averages −59 and +273 ns [web: research/files/kelly-2000-hafele-keating-did-they-prove-anything.htm, via the Investigator]. The scatter between individual clocks is as large as the effect, a legitimate point about that experiment's statistics. These numbers have not been checked against the USNO report by any reviewer. Later clock experiments are the standard reply [memory, medium; not web-checked]. - Kelly's GPS-frame and "selective quotation of Einstein" charges (pp.8-9). Stated in his words, with the replies, in sections 4.2 and 4.3. - Unreferenced and unchecked citations. Kelly cites "the M & G (600m) or Saburi (0.9m) tests" as paths too short to tell from a straight line [read: p.10]; Saburi is not in his reference list and a web search did not identify the work (arbiter). The 1919 Einstein quotation has no reference either. - Doppler point (p.12, item 3). Kelly notes that Sagnac's camera rode on the disc and so saw no Doppler shift, and that a lab observer would see a negligible one; this matches the Post figures he quotes (Doppler about v/c of the effect) [read: p.12; arbiter]. - Essen's earlier Hafele-Keating critique. Essen (1977), Creation Research Society Quarterly 14, 46, cited by Kelly in his 2000 Hafele-Keating paper as reaching a similar conclusion earlier [web: research/2026-10-04-cartesio-episteme.md, via the Investigator]; listed in the contrarian's review. Not obtained. - Kelly's own experiments on unipolar induction (IEI Monographs 5 and 6, 1998, combined as "Faraday's Final Riddle; Does the Field Rotate with a Magnet?", saved in research/files/). His own apparatus and measurements; peripheral to Sagnac and not reviewed here. - Reception. Jean-Pierre Vigier, a professional physicist, cited Kelly's 1995 monograph in a photon-mass reading of the Sagnac effect (Phys. Lett. A 234, 75, 1997) [web, via the Investigator]. Ian Cowan (National Standards Authority of Ireland), a sympathetic colleague, wrote in Kelly's memoriam (Galilean Electrodynamics 16(6), 2005) that his analyses were "essentially correct", and that a months-long 1996 Irish Times letters controversy ended with Kelly's offer of a public debate "never taken up". That account is documented in one published source; the letters themselves have not been located. The Irish Times obituary independently says Kelly provoked "much academic debate, some of it acrimonious" [web: Irish Times link above]. - Why the record of the dispute is thin (arbiter's analysis). Explanations considered: letters pages that are unindexed and behind a paywall; standard textbook replies that left no separate trace; experts not engaging in a lay forum; suppression. No evidence of suppression was found; Kelly published in Infinite Energy, Galilean Electrodynamics, Physics Essays and the IEI monographs. The 1996 letters and the Irish company registers (for HDS Energy) would decide.

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7. What a learner should take away

Solid. - Kelly's circular and polygon derivations are correct. If light moves at a fixed speed in a non-rotating frame, a rotating loop gives dt = 4Aω/c², with a v²/c² correction. This is a clean way to see the Sagnac effect without any relativity, and it is worth working through yourself. - The effect is first order in v/c and appears at walking-pace speeds. No relativistic correction is needed to see it. Kelly is right about that. - Michelson-Gale is just a Sagnac interferometer turned by the Earth, with the sin(latitude) projection. Kelly explains this well, and the numbers agree. - The line-integral form dt = (2/c²)∮u·dl, which Kelly derives, is the deeper statement than the area formula. Wang's conveyor results, especially the zero-area loop, show it directly: the shift tracks fibre length times speed, not area. Kelly was right to see this experiment as important.

Wrong, and why the mistake is instructive. - Uniform motion of the whole apparatus gives no shift. The formula is a sum round a closed path, and a uniform velocity sums to zero round any closed path. The conveyor works because the fibre circulates: its two straight runs move in opposite directions while the source and detector go round with it. A useful habit: before concluding "a part does X, so the whole does X", check whether the parts cancel. - Limits must be taken carefully. "A huge disc looks like a straight line" is true of the path, but the Sagnac shift depends on ω, and ω = v/R_d goes to zero. - A correct prediction does not prove the assumption behind it, especially when a rival theory (here SR in the inertial lab) uses the same assumption and makes the same prediction. - "The laboratory" has to be inertial. On the spinning Earth, light does not move at c relative to the ground lab; that is exactly what Michelson-Gale measured. Kelly's own 1996 monograph uses the right frame (non-rotating, Earth-centred). - Consistent is not the same as discriminating. Michelson's 1913 mirrors and the ring-laser standing wave both agree with Kelly's premise, but SR predicts the same results, so they cannot decide between the two (section 4.1). Evidence for a theory has to be something its rival would not also predict. - Rotation is a rate, not an angle. Kelly's "orbital angle" table compares how far the Earth turns during each experiment, which mostly measures how long each experiment lasts. Whether a frame counts as non-rotating is decided by its rotation rate, which gyroscopes and the Sagnac loops themselves measure (section 4.2). - One-way speeds need synchronised clocks. A round-trip result cannot by itself show that the one-way speed on a segment contradicts SR.

Genuinely open. - Wang's conveyor result has been reported by one group only, at about 11 percent precision. A second group, and a vacuum (hollow-core fibre) version, which Wang proposed, would strengthen it. No report of either was found. - Kelly's wider programme (absolute time, light carried along with the Earth's orbit but not its spin) differs from SR at second order. Sagnac experiments do not test it; clock-rate and Doppler experiments do. His raw Hafele-Keating per-clock numbers deserve a check against the USNO report. - What Kelly's critics in 1996 actually argued, and whether his debate offer reached them, is unknown until the letters are found. - Whether Einstein ever named Sagnac or Michelson-Gale is unverified; a search of the Collected Papers of Albert Einstein would settle it.

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8. Check yourself

1. A square loop of side 1 m (area 1 m²) spins at 1 revolution per second, with 633 nm light. What is the Sagnac fringe shift? Does it change if the axis is moved to one corner?
2. Using dt = (2/c²)∮u·dl, show in one line why carrying a Sagnac interferometer in a straight line at 100 m/s gives no shift.
3. In a Wang-type conveyor, the two straight runs are 5 m long each, and the belt moves at 0.1 m/s; ignore the end wheels. What time difference do the straight runs produce for one turn of fibre? Why does it not cancel the way it does in question 2?
4. Kelly writes that one fringe needs "about 13 m/s per meter of radius". What rim speed is needed at r = 0.25 m (λ = 550 nm)?
5. Kelly says the correct Sagnac result proves that light travels at c relative to the laboratory. Give the logical objection, and say what kind of experiment could separate his picture from SR.

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9. Further reading

Mainstream - E. J. Post, "Sagnac effect", Reviews of Modern Physics 39, 475-493 (1967). The classic review, cited by Kelly; it contains the general line-integral form and the statement about uniform translation. - G. B. Malykin, "The Sagnac effect: correct and incorrect explanations", Physics-Uspekhi 43, 1229 (2000) [memory, high]. Surveys the many explanations, including flawed ones. - N. Ashby, "Relativity in the Global Positioning System", Living Reviews in Relativity 6, 1 (2003) [memory, high]. How the Sagnac correction is used in GPS, in the Earth-centred non-rotating frame. - R. Wang, Y. Zheng, A. Yao, D. Langley, Phys. Lett. A 312, 7-10 (2003), and R. Wang, Y. Zheng, A. Yao, PRL 93, 143901 (2004). Saved in research/files/. - E. F. Taylor and J. A. Wheeler, Spacetime Physics; D. J. Griffiths, Introduction to Electrodynamics, ch. 12; J. R. Taylor, Classical Mechanics, ch. 9. - A. A. Michelson, "Effect of Reflection from a Moving Mirror on the Velocity of Light", Astrophys. J. 37, 190-193 (1913), the experiment Kelly offers in support (section 4.1): https://en.wikisource.org/wiki/Effect_of_Reflection_from_a_Moving_Mirror_on_the_Velocity_of_Light. - Wikipedia, "Sagnac effect" (history: von Laue 1911, Harress, Einstein's 1914 notes, Langevin): https://en.wikipedia.org/wiki/Sagnac_effect.

Kelly's own works - "The Sagnac Effect and Uniform Motion" (this project's paper, undated, after 2003). - A New Theory on the Behaviour of Light, IEI Monograph 2 (1996), saved in research/files/. States his frame (Earth's orbit, not its spin) most clearly. - "Hafele & Keating Tests; Did They Prove Anything?" (2000), saved in research/files/. - Experiments on the Relative Motion of Magnets and Conductors and Faraday's Final Riddle: Does the Field Rotate with a Magnet?, IEI Monographs 5 and 6 (1998), his own unipolar-induction experiments; a combined copy is saved in research/files/. - "Sagnac effect contradicts special relativity", Infinite Energy 39, 24-28 (2001) [read: ref. 24]. Not obtained. - Challenging Modern Physics (2005); Physics Essays 13, 616 (2000); IEI Monographs 1, 3, 4. Not obtained.

Related, by others (outside the mainstream view) - D. Marett, conspiracyoflight.com: a Michelson-Gale recomputation and his own large-area fibre Sagnac experiments, read in a preferred-frame interpretation. - F. Selleri, absolute-simultaneity arguments; R. Klauber, rotating-frame analyses (both via the Investigator, sites.md). - Essen (1977), Creation Research Society Quarterly 14, 46: an earlier critique of the Hafele-Keating clock flights, which Kelly built on (via the Investigator, sites.md). Not obtained.

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Answers

1. N = 4AΩ/(cλ) with Ω = 2π rad/s: 4 × 1 × 6.283 / (2.998×10⁸ × 6.33×10⁻⁷) = 0.132 fringe (dt = 4AΩ/c² = 2.8×10⁻¹⁶ s). Moving the axis changes nothing: only the area enclosed and ω enter (Stokes' theorem; Sagnac showed this experimentally [read: p.1-2]).
2. u = V is the same everywhere, so ∮V·dl = V·∮dl = V·0 = 0, and dt = 0.
3. Each run gives 2vΔl/c² = 2 × 0.1 × 5 / (2.998×10⁸)² = 1.11×10⁻¹⁷ s; two runs give 2.2×10⁻¹⁷ s. It does not cancel because the two runs move in opposite directions while the light also traverses them in opposite directions, so u·dl has the same sign on both: the fibre circulates round the loop. In question 2 the apparatus moves one way, so the out and back legs of the light path cancel.
4. v = 13.12 m²/s / r = 13.12 / 0.25 = 52.5 m/s, not 3.3 m/s.
5. Affirming the consequent: "if A then B; B; therefore A" is invalid when another premise (SR in the inertial lab frame) also gives B, and here it gives the same first-order formula. The pictures differ at second order, in what a moving clock reads: experiments on clock rates and second-order Doppler shifts (Ives-Stilwell type, clock flights, rotating-table isotropy tests such as Brillet and Hall 1979), or a Sagnac measurement precise to better than about v²/c² of the shift.