Physicist review: Kelly-AG ("The Sagnac Effect and Uniform Motion")

Reviewer: PhD physicist (Step 2). Source tags: [read] = the document or saved file; [web] = URL given; [memory] = recalled, with confidence; [calc] = my own computation (Python venv, this session).

Source scope: one 13-page undated paper by A. G. Kelly (Kelly_2.pdf, text layer read in full by me), plus the Investigator's saved files. The Wang et al. 2003 paper was read in full from research/files/. Equations garbled in extraction (pp.2-3, 6, 11-12) were checked against the algebra rather than page images.

0. Physics you need first (plain language)

The Sagnac effect. Send two light beams round the same closed loop in opposite directions, with the source and detector fixed to the loop. If the loop rotates, the beam travelling with the rotation takes slightly longer than the one against it. For a loop enclosing area A that rotates at angular speed w (radians per second), the time difference is

dt = 4 A w / c^2 (c = speed of light, 3.0e8 m/s)

The general form is dt = (2/c^2) times the loop integral of (v . dl). Here v is the velocity of the loop's material at each point and dl is a small step along the light path. The notation "v . dl" is the dot product, meaning the component of v along the path. For a rotating rigid loop, v = w x r, and Stokes' theorem (the vector-calculus theorem turning a loop integral into a surface integral) turns the integral into 2 A w. That gives the formula above.

Where the effect comes from. In a non-rotating (inertial) frame, light travels at c in all directions. A source on the rotating loop moves while the light is in flight. One beam has to "catch up" with a receding detector, and the other meets an approaching one. Rotation is detectable locally, with no outside reference, which is why gyroscopes and ring lasers work.

Worked example (1 fringe). One fringe corresponds to dt = lambda/c. For a circle, v = r w and A = pi r^2, so the shift is 4 pi r v/(c lambda). Setting it equal to 1 with lambda = 550 nm gives r v = c lambda/(4 pi) = 13.1 m^2/s [calc]. At r = 1 m the rim speed is 13.1 m/s. At r = 10 m only 1.3 m/s is needed. Kelly writes "13 m/s per metre of radius" (p.4), which gets the scaling backwards (the paper's own eq. 13 gives 13.13 m^2/s).

Mainstream view in one paragraph. Special relativity (SR) treats the Sagnac effect fully and predicts dt = 4Aw/c^2 to first order in v/c. Max von Laue gave the SR derivation in 1911, two years before Sagnac's experiment (Wikipedia, https://en.wikipedia.org/wiki/Sagnac_effect [web]). Einstein rebutted Harzer's 1914 claim that Harress's rotating-prism result contradicted SR (same page [web]). Langevin (1921) and Post (1967) gave rotating-frame treatments [web: same page; Post cited by Kelly]. GPS engineers routinely apply a "Sagnac correction" of up to about 133 ns for Earth rotation [memory, high; Ashby, Living Reviews in Relativity 2003]. The effect measures rotation relative to local inertial frames, and uniform translation of a closed loop gives no shift.

Corrections to Step 1

The deep read and claims.md are accurate on what the paper says. The corrections below are mostly additions.

1. Algebra. I confirm eqs. (1)-(12) are correct. The circular result in eq. (6), dt = 4Aw/(c^2 - v^2), is exact in the inertial frame. By hand: 2 pi r [1/(c-v) - 1/(c+v)] = 4 pi r v/(c^2 - v^2) [calc].
2. Sagnac's area is wrong twice in the paper. Kelly gives 0.86 m^2 (p.1) and 666 cm^2 (p.11), for a disc he calls 50 cm in diameter (area 0.196 m^2, so 0.86 m^2 is impossible). Step 1 flagged only "0.86 vs 0.0866". The value 866 cm^2 reproduces Sagnac's observed ~0.07 fringe (two directions of spin) at 2 rev/s and 436 nm: 0.033 fringe each way, 0.067 when reversed [calc; 866 cm^2, 436 nm from memory, medium]. The error does not touch the argument but shows poor numerical care.
3. Ring-laser "accuracy better than 1 part in 10^20" (claim 4, p.7) is a misreading. Stedman, Li and Bilger, Applied Optics 34, 5375 (1995) report a ring-laser frequency resolution of 1 part in 3e21 and a rotation resolution of 300 prad [web: https://opg.optica.org/ao/abstract.cfm?uri=ao-34-24-5375, via search summary]. This is resolution relative to the optical frequency (the beat is about 70 Hz against 4.7e14 Hz optical [calc]). It is not accuracy of the Sagnac formula, which is verified at far lower relative precision. Step 1 reported the figure as the paper's claim without comment. The "12 orders of magnitude over M&G" is likewise a comparison of frequency resolutions, not of formula tests.
4. Other numbers [calc]. - Earth-spin shift for a 1 m disc: about 1e-6 (r = 0.5 m) to 4e-6 (r = 1 m) fringe at 45 degrees latitude, against Kelly's "about 1e-5". - Distance of Michelson-Gale (M&G) from the Earth's axis: R cos(lat) = 4750 km, not 4250 km. - Orbital angle during M&G: 7e-11 degrees, not 2.3e-10. - Orbital angle during one equatorial GPS signal circuit: 1.5e-6 degrees, not 2.9e-6. - Bilger's 8750 km "diameter" is also slightly off (9240 km at 43.5 degrees S). - The orbital-angle arguments of pp.9-10 are conceptually irrelevant (see claim 1c). - The M&G prediction 0.236 is reproduced: 4 A W sin(lat)/(c lambda) = 0.236 using A = 207,840 m^2 and lambda = 570 nm, matching Marett.
5. Step 1's note that claims 1 and 5 rest on a digest only is resolved. I read p.10, and Wang et al. 2003 in full. Wang does not show what Kelly says (claim 1).
6. Mischaracterised experiment. Kelly (p.10) says Wang sent light "out ... and back again" along a straight line. Wang used a closed fibre loop (a belt over two wheels, with semicircular ends) with source and detector co-moving, and states that "uniform motion [of the whole FOG] did not cause any phase shift because the FOG is only sensitive to the rotational movement" [read: Wang 2003 p.2].
7. "Novel" claims. Kelly calls the M&G projected-area explanation "novel" and the polygon proof "the first general derivation ... which does not assume an ether" (pp.5, 10). Both are standard. Michelson-Gale's own analysis uses the component of the Earth's rotation normal to the loop, A W sin(lat). The line-integral form for arbitrary shape is Post's (1967), which Kelly cites. Von Laue (1911) did the derivation within SR [web: Wikipedia above].

Claim-by-claim

Claim 1: The Sagnac effect applies to uniform straight-line motion

a. Mathematically consistent? Split. - The segment formula dt = 2 v dl/c^2 is a correct piece of the line integral (2/c^2) times the integral of (v . dl). - Kelly's extension fails. For a closed path on a body translating at constant velocity V, the integral of (V . dl) around the closed loop is V . (integral of dl) = 0. His own derivation (p.11, D = integral of u dl) therefore yields zero for rigid translation. - His "large-disc limit" argument (p.10) does not give a translation limit. As the radius grows at fixed rim speed, the full-circle result 4 pi r v/c^2 grows without bound. It does not tend to a translation result. The circulation of the velocity field around the loop is what counts, and it stays non-zero however flat a short arc looks.

b. Consistent with experiment? - The segment result is confirmed. Wang et al. measure 0.03229 +/- 0.00365 rad per (m^2/s) against the theoretical 0.03200 (4 pi/(c lambda)), a ~11% test [read: Wang 2003 p.3]. Their zero-area and figure-8 geometries show the phase scales with fibre length times speed, not area. - The whole-loop translation result is null in experiment: Wang's FOG riding the conveyor shows no shift [read]. A fibre-optic gyroscope on a moving train or aircraft reports only rotation. Earth's orbital speed (30 km/s) produces no Sagnac signal in any ring laser or gyroscope [memory, high]. - The "inertial frame" argument (pp.9-10) misses the point. An inertial frame is identified by zero local rotation (a gyroscope, a Foucault pendulum), not by how many degrees the frame's centre has swept around the Sun. Rotation rate is locally measurable (Earth spin 7.3e-5 rad/s); the orbital rate (2e-7 rad/s) is ~365 times smaller, and no uniform-translation signal exists at all. GPS is not "biased": it works in the Earth-centred inertial frame and adds the Sagnac term when converting to the rotating Earth frame [memory, high]. - Post's "disturbing conceptual discontinuity" (quoted p.9) is resolved by the line-integral form: the Sagnac phase measures circulation of the frame velocity, which is zero for translation.

c. Tested? Wang, Zheng, Yao, Langley (Phys. Lett. A 312, 7, 2003) and Wang, Zheng, Yao (PRL 93, 143901, 2004) tested the segment form with ~150 citations each (Investigator [web]). Hollow-core fibre (vacuum) test: proposed, no result found. The ~11% precision is modest. Replicated? No independent replication found in my web check or the Investigator's; follow-ups (Huang 2021, sensor designs) are design and theory papers [web: Investigator]. The Investigator correctly noted that source and detector co-move and the loop is closed. This is a "moving fibre with co-moving source/detector" test, not a uniform-translation test.

d. If true (strong form), what changes? Every inertial-frame experiment would show a first-order shift against absolute velocity, contradicting the Michelson-Morley family of null results, ring-laser Earth-rotation measurements, and the Galilean/Lorentz principle of relativity. Nothing in mainstream physics would need to change for the weak form.

Verdict (split). - Weak form (a moving-medium segment contributes 2 v dl/c^2; Sagnac is a line-integral effect): Established (SR-consistent, Wang et al. to ~11%; not independently replicated at higher precision). - Strong form (a closed light path on a rigidly, uniformly translating body shows a Sagnac shift, as the abstract and conclusion say): Refuted by calculation (the closed integral of V . dl is zero) and contradicted by experiment, including Wang's own null for translating the whole FOG.

Claim 2: Light travels at c relative to the laboratory, independent of source speed, and this one assumption gives the Sagnac result for any path shape

a. Mathematically consistent? The two circular derivations (pp.2-3) are the same equations; the first assumes "c +/- v relative to the disc", the second "c in the lab", which are the same statement. The polygon proof (pp.11-12) is correct to first order: t' - t'' = 2 u l/c^2 per element, and the u^2 terms cancel. It matches the exact circle result 4Aw/(c^2 - v^2), whose correction is relative v^2/c^2. The summation of D = integral u dl = 2Aw is right (the text drops the pi in "4nA", which should be 4 pi n A, likely an extraction garble; I did not render p.12). The "proof" on p.3 ("if the assumption gives the precise result, this proves the light behaves this way") is affirming the consequent: any frame-dependent theory with light speed c in an inertial frame (SR, Lorentz ether, Sagnac's own ether) gives the same first-order answer, as Kelly himself says on pp.10-11.

b. Consistent with experiment? Source independence of the speed of light is well supported. Sagnac-type results with co-rotating source and detector rule out simple emission (ballistic, Ritz) theories, where light speed is c relative to the source and the effect would vanish [memory, high]. Other tests I recall: Alvager et al. 1964 (pion decay at 0.99975c) [memory, medium; not web-checked]. Where Kelly is wrong is "the laboratory". For M&G and Bilger, the laboratory is the rotating Earth. Light does not travel at c relative to a rotating laboratory (that is the Sagnac effect itself). His successful M&G and ring-laser calculations use the Earth's angular velocity, i.e. a non-rotating frame. The Investigator's finding fits: Monograph 2 says light follows the Earth's orbit and gravity but not its spin, which is an Earth-centred non-rotating frame, not "the laboratory" [web, Investigator]. The paper uses two different meanings of "laboratory".

c. Tested? The general-shape result was derived by Post (1967) and confirmed in Sagnac, Harress, Pogany, Dufour-Prunier, ring-laser and fibre-gyro experiments [read: Kelly refs; memory, high]. The vacuum/air case is verified by ring lasers.

d. If true: Nothing changes. The statement is the SR second postulate restricted to an inertial frame, and it is what Post's derivation uses.

Verdict: Established as a derivation and as a statement about inertial frames (not original with Kelly). His wording ("relative to the laboratory") is internally inconsistent for Earth-fixed experiments. The claim that the correct answer "proves" the assumption is a logical fallacy rather than a physics error.

Claim 3: SR has no role in the Sagnac effect

a. The order-of-magnitude argument is correct: first-order Sagnac versus v^2/c^2 SR corrections. Kelly's inertial-frame calculation also shows the exact lab-frame result 4Aw/(c^2 - v^2); the clock on the disc then adds the SR time-dilation factor, so the proper-time shift is 4Aw g/c^2 with g = 1/sqrt(1 - v^2/c^2) [memory, high; Post 1967 / Landau-Lifshitz]. For v = 13 m/s that factor differs from 1 by ~1e-15. Kelly's "SR [does] not apply" is true only in the sense that the first-order result does not need the Lorentz factor. His speed claim (visible at 13 m/s, so relativity "cannot possibly have any function") is a non-sequitur: SR is not a high-speed-only theory. It also predicts the low-speed effect in the inertial-frame derivation.

b. SR also supplies the framework that treats rotation and synchronisation correctly: GPS uses SR plus the Sagnac term and agrees with experiment to nanosecond level [memory, high].

c. Tested? Kelly's "SR gives different orders of magnitude" is accurate as arithmetic. The extension that Sagnac "contradicts SR" is argued for in Kelly 2001 (Infinite Energy 39, 24; not read). No test distinguishes his position from SR, since both give the same prediction.

d. If true: If SR had no role, then SR would be silent about rotating-frame optics; the actual SR/GR literature (von Laue 1911, Langevin 1921, Post 1967) shows it is not.

Verdict: Split. "The Sagnac time difference is first order in v/c and does not need relativistic corrections" is Established. "Sagnac contradicts or lies outside SR" is Refuted by calculation: SR (von Laue 1911) predicted the effect, and the inertial-frame calculation Kelly uses is SR's own. Where mainstream and Kelly agree, the disagreement is mostly about framing.

Claim 4: Michelson-Gale is an ordinary Sagnac test (projected area, sin latitude); ring lasers confirm the formula to better than 1 part in 10^20

a. Mathematically consistent? Yes for the M&G part. The Earth-rotation component normal to the horizontal loop is W sin(lat), equivalent to projecting the area onto the equatorial plane. With A = 207,840 m^2, lat = 41 degrees 46', lambda = 570 nm the prediction is 0.236 [calc], matching Kelly and Marett. The "1 in 10^20" figure is a misreading (Correction 3). The arithmetic on p.6 (0.23 x 5700e-10 / 3e-18) is garbled in the text layer; the quoted result 4.37e-16 s is right if c = 3e8 [calc].

b. Consistent with experiment? M&G: mean 0.230 against 0.236 predicted [read: p.5]. Doug Marett recomputes 0.2364 versus 0.230 +/- 0.005 [web: http://www.conspiracyoflight.com/Michelson-Gale/Michelson-Gale.html, Investigator]. Ring lasers detect the Earth's rotation at the expected rate and sidereal period (Schreiber et al., per Marett [web, Investigator]). Marett is an amateur experimenter without a commercial stake, whose analysis agrees with Kelly's.

c. Tested? Yes: Michelson and Gale (1925), ring lasers (Bilger et al. 1993/1995, Stedman et al.), fibre gyros. Replicated many times in the ring-laser and gyroscope literature. Spread in the M&G data (+0.55 to -0.05 per Kelly) is large; modern ring lasers are far more precise [read; memory, medium on modern precision].

d. If true: Nothing changes. This is the SR/GR-consistent picture of Earth rotation. Marett's reading that it supports an Earth-centred preferred frame is an interpretation, not a test (the Investigator reports it as his view).

Verdict: Established for the M&G and ring-laser agreement with 4Aw sin(lat)/c^2. The precision claim "1 part in 10^20" is a misstatement (frequency resolution, not formula accuracy). The "novel" label is incorrect (Correction 7).

Claim 5: Light does not travel at c relative to a moving disc along straight segments, so Sagnac experiments contradict constancy of light speed in the moving frame

a. The inertial-frame closing speed of light and a rotating observer's point is c +/- v: that is arithmetic and it is the same in SR. The mismatch is in "speed relative to the disc". SR says the speed measured with local rulers and clocks in a local inertial frame is c. It says nothing forces coordinate speed in a rotating frame (with a particular synchronisation) to be c. Landau-Lifshitz treat this: the rotating frame's coordinate light speed varies and clocks cannot be Einstein-synchronised all the way round a closed loop (the Sagnac time gap) [memory, high]. The Sagnac experiment measures a phase difference at one detector with one clock; it does not measure one-way speed on any segment.

b. Local isotropy of the speed of light has been tested directly on rotating apparatus, e.g. Brillet and Hall (1979), and Michelson-Morley-type experiments on rotating tables, with limits of a part in 10^15 and better [memory, medium; not web-checked]. If light failed to travel at c relative to local rulers and clocks on the disc, these would see it.

c. Tested? The Sagnac observations are real and agree with the formula. The conclusion Kelly draws from them (contradiction of SR) has not been supported by any test; SR users have predicted these results since 1911.

d. If true: SR's local-invariance postulate would fail, contradicting rotating Michelson-Morley tests and the entire GPS and ring-laser engineering base.

Verdict: Split. The observation (a rotating observer's coordinate speed is c +/- v) is Established and is part of SR/GR practice. The inference "contradicts SR" is Refuted by calculation (SR predicts it) and rests on conflating coordinate speed with locally measured speed.

Claim 6: Einstein never mentioned Sagnac or Michelson-Gale despite their bearing on the speed of light

a. Not a mathematical or physical claim. The quotations are Silberstein (1921) and Turner (1979), the latter from a book generally regarded as hostile to relativity [memory, medium].

b/c. Record check. Einstein did write on a Sagnac-type experiment: his 1914 rebuttal of Harzer's reanalysis of Harress's rotating-prism result [web: https://en.wikipedia.org/wiki/Sagnac_effect]. That is a rotating-ring test of the same family. Kelly's wording is a limited "this author has located no reference", and Kelly did not find the 1914 note (not stated either way). Whether Einstein named Sagnac or Michelson-Gale specifically I have not verified (the Investigator left it open). Von Laue, in Einstein's circle, treated Sagnac in SR in 1911, and Langevin in 1921 [web, same page]. Silence by one author is evidence neither of a gap nor of suppression; the Sagnac effect was treated within SR by others without waiting for Einstein.

d. Nothing in physics depends on it.

Verdict: Not testable as stated as a physics claim (an authorial silence cannot refute or support a theory). As history it is partly contradicted by Einstein's 1914 note on Harress, and the surrounding record shows the Sagnac effect was integrated into SR early.

Investigator findings weighed (kept in the record)

• Kelly's other work (IEI Monographs 1-6, 1995-98; Hafele-Keating critique 2000; book "Challenging Modern Physics", 2005; patent DE1528915A1): same programme. The Hafele-Keating raw data (East -196/-54/+166/-97 ns; West +413/-44/+101/+26 ns, against published averages -59/+273 ns) is real USNO data Kelly obtained [web: Investigator, 2000 paper]. Individual clocks scatter at the hundred-ns level, comparable to the effect. This is a legitimate observation on that experiment's statistics, though later flights (Alley, Hafele-Keating's own later analysis, GPS) are better tests of time dilation [memory, medium]. It is outside this paper's claims; I did not review it in depth.
• Vigier (Phys. Lett. A 234, 75, 1997) built a non-zero-photon-mass interpretation of Sagnac citing Kelly's analysis [web: Investigator]. Evidence that a professional physicist took the analysis seriously; it is not endorsement of Kelly's anti-SR conclusions.
• Cowan's memoriam (GED 16(6), 2005): "essentially correct" refers to Kelly's analyses of Sagnac; Cowan is a sympathetic colleague by his own account. I find the analyses correct to first order (claims 2, 4); his conclusions against SR do not follow (claims 3, 5).
• Irish Times 1996 correspondence: reported opposition and Kelly's open offer of a debate "never taken up", from one sympathetic source (Cowan). The letters themselves are not located. I cannot weigh the quality of the opposing arguments.
• Marett supports claim 4 and shares the Earth-centred preferred-frame interpretation. His fibre-gyro experiments are his own; I did not review them.
• Wang et al. follow-ups (Huang 2021; "one-way interferometer"; GSJ analysis): same moving-fibre line-integral physics, not translation of a closed loop.
• Absence of evidence: no post-2005 Kelly publications are expected after his death on 30 July 2005. The undated paper was generated as PDF in 2011; its place in his bibliography remains unknown.

Summary table