RankingsPrimerArbiterPhysicistMathematicianContrarianQESites logClaimsSourcesResearch notes

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

Reviewer: PhD mathematician (Step 2). Sources: facts.json, claims.md, deep/Kelly-2-1a1021.md, the extracted text of Kelly_2.pdf (I read pp.6-13 myself, plus p.1-5 through the deep read), sites.md "Leads for review", research notes, and Wang et al. 2003 (research/files). My own numerics used the project .venv (NumPy). Tags: [read] document, [web] URL, [memory] recalled (confidence stated).

Source gap

Only one primary document (Kelly_2.pdf, undated, after 2003; PDF produced 2011). Monograph 2 (1996) and Wang et al. were read only in part for the Investigator's leads. Kelly's own earlier derivations (Infinite Energy 2001, monographs 1, 3, 4, book) were not available. Conclusions about those rest on the Investigator's summaries.

Notation and the maths in one place (for the reader)

The Sagnac effect is the difference in arrival time of two light beams sent round a closed loop in opposite directions, when the loop (with source and detector attached) moves. In the laboratory (non-rotating) frame, to first order in v/c:

dt = (2/c^2) times the loop integral of (u . dl)

where dl is a small piece of the loop, u is the velocity of that piece, and the loop integral means "add up round the whole loop". For a rigid disc spinning at angular speed omega (radians per second), u = omega x r (cross product: speed omegar, tangent to the circle), and the loop integral equals 2omegaA, where A is the enclosed area (this is Stokes' theorem, the same curl-to-area theorem from vector calculus). Hence dt = 4Aomega/c^2. Worked number [read: Kelly p.4, my recomputation]: for a circle of radius 1 m and green light (550 nm), one fringe (a shift of one full interference band) needs v = clambda/(4pir) = 13.1 m/s at the rim.

Corrections to Step 1

  1. claims.md and deep read are mostly accurate; I confirm the algebra of eqs (1)-(12) and the p.12 binomial step (second-order terms cancel; first neglected term is third order).
  2. The deep read lists the p.12 labelling slip; I add that p.12 and p.3 use t', t'' for opposite directions (noted by deep read) and that the loop integral on p.12 is the right general result.
  3. Deep read item 4 (Sagnac's area "0.86 m^2", possible tenfold slip) needs refinement: Kelly gives TWO different areas for the same experiment. p.1: "0.86 m^2"; p.11: "enclosed area ... 666 cm^2" (= 0.0666 m^2) with a 50 cm disc [read: p.1, p.11]. These contradict each other. A web source gives the 1913 interferometer area as about 0.0860 m^2 (860 cm^2), 2 Hz, indigo mercury light 436 nm, fringe shift 0.07 ± 0.01 [web: https://www.idc-online.com/technical_references/pdfs/mechanical_engineering/The_Sagnac_effect.pdf and search result summary]. My recomputation: A = 0.0866 m^2 gives 0.067 fringe on reversal at 436 nm (matches); A = 0.86 m^2 would give 0.52; A = 0.0666 m^2 would give 0.051. So p.1 "0.86 m^2" is probably a factor-of-10 slip, and p.11 "666 cm^2" a digit slip (866 -> 666 probable) [web + my calculation]. Neither is load-bearing for the claims.
  4. Deep read says the "4250 km" figure is wrong; I confirm (distance from axis at 41.77 N is 4752 km; I computed 4751.7).
  5. The deep read's "Claims 1, 5 and ring-laser and Wang details rest on digest only". I have now read pp.6-13 and Wang et al. in full (see below). New defects found in those pages (section "New defects").
  6. Investigator's lead on Wang (claim 1): "in a fibre conveyor the loop is still closed, with source and detector co-moving". I confirm from Wang's text [read: Wang 2003 p.2: "in our FOC ... the light source and the detector are co-moving with the medium"], and add that the effect they measured is a loop sum, not a property of a single straight segment (below).

Claim 1. The Sagnac effect applies equally to uniform straight-line motion (conclusion, p.13)

a. Internal consistency: Partly. Kelly's own argument for it (pp.9-10) is a limit: a disc of radius approaching infinity "approximates a straight line", so the result "applies to all objects moving at constant speed". The maths does not support this limit. For a rigid loop, dt = (2/c^2) times the loop integral of u.dl. If the whole loop moves with one constant velocity V, then u = V everywhere and the loop integral of V.dl = V . (loop integral of dl) = 0, because a closed loop returns to its start. Zero shift. Kelly's quotation of Post (1967) says the same thing: a Sagnac interferometer in "uniform translational motion" cannot give a fringe shift [read: p.9]; Kelly takes the sentence as a "problem" to be resolved but gives no calculation that makes translation give a shift. In the large-disc limit, dt = 4pirv/c^2 grows with r only because the loop is still closed and the far side moves the other way relative to the light path; the "straight" rim section is cancelled by the opposite rim section for the part that is pure translation. Local straightness of a piece of the loop does not give the global result. What Wang et al. actually measured: a belt-like fibre loop. The fibre material circulates round a fixed track (two wheels), carrying source and detector with it. Here u is tangent to the path everywhere, so the loop integral is vL (L = fibre length), and dt = 2vL/c^2 [read: Wang 2003 pp.1-3]. Their 24 arrangements fit phase = 0.0320 rad per (m^2/s) vs the FOG prediction 4pi/(clambda) = 0.03200 (I recomputed 0.031997 for 1310 nm) [read: Wang p.3]. This is a good agreement (about 10% scatter in the fitted slope, ±0.0037 on 0.0323). The result is the loop integral formula above applied to a circulating medium, which is the same formula as for rotation. The zero-area and figure-8 configurations show the effect does not need an enclosed area (a real and interesting point against the "Aomega" form of the formula), but the loop is closed and the integral is over the whole circuit. The authors themselves call their per-segment statement an inference from subtracting loops ("a segment of uniformly moving fibre ... contributes", p.3) [read]. Also, Kelly p.10 describes Wang as "sending out light in a straight line portion of the light path and back again ... reversing a light beam sent out on a straight line on a moving platform". That is not what was done: beams went round a closed fibre loop in opposite directions, and the mechanical conveyor carrying the whole fibre-gyro in uniform motion "did not cause any phase shift" [read: Wang p.2]. So Wang's own paper contains the Post null result for pure translation. Kelly misdescribes the experiment. b. Experiments: consistent with Wang 2003/2004 for circulating media; inconsistent with uniform translation of a rigid loop (never observed; Post's statement; standard fibre gyros riding on moving vehicles show no translation signal) [memory, high for the null on translation; Wang p.2 for the conveyor null]. c. Tested: Wang et al. (St. Cloud State, 2003; NRL-loaned fibre gyro), 24 arrangements; follow-up PRL 2004 [web via Investigator: arXiv physics/0609222, physics/0609235]. Replication: Investigator reports about 150 citing works each, a sensor proposal (Huang 2021); I found no independent repetition and did not search further [memory, low]. d. If true as Kelly states it (translation alone gives a shift): invariance of physics under uniform motion (principle of relativity, for any theory, not only Einstein's) would fail, since a uniformly moving lab could detect its velocity by a closed light loop. Wang's actual claim (circulation of a medium gives dt = 2vL/c^2 independent of area) is already within standard theory: dt = (2/c^2) times the loop integral of u.dl. [memory, high] Verdict: split. Weak form (a moving medium or segment adds 2vdl/c^2 to a closed-loop sum; no enclosed area needed): Tested (Wang et al.), supported, consistent with the standard formula. Strong form as Kelly states it and argues it (a body in uniform straight-line motion, as a whole, shows the Sagnac effect; large-disc limit): Refuted by calculation (the loop integral of a constant vector over a closed loop is zero; Post's own sentence and Wang's own null on the conveyor), and rests on a mistaken reading of Wang.

Claim 2. Light travels at c relative to the laboratory, independent of source speed; this alone reproduces the Sagnac result for any shape; the correct result "proves" the assumption (pp.2-3, 11-13)

a. Maths: The derivations are algebraically correct (my checks agree with the deep read: eqs (1)-(12); binomial expansion on p.12; circulation 2omegaA). Two defects of logic. (i) The two "methods" on p.3 are the same equations under two descriptions, so the second does not check the first (deep read, which I confirm). (ii) "This proves that the light behaves in this way" (p.3): this is the converse fallacy. A formula derived from an assumption agreeing with data does not prove the assumption unless rival assumptions give a different formula. They do not: special relativity, with c in the inertial laboratory frame and source independence (Einstein's second postulate), gives the same lab-frame dt = 4Aomega/(c^2 - v^2). Kelly's assumption is part of SR [memory, high]. The rival that does differ is the emission theory (light speed c +/- v relative to the source); Sagnac-type results do rule that out for the path as a whole, a real point, but Kelly does not set it up as a discriminating test. Hidden assumptions: (1) time and distance on the disc equal lab time and distance (p.12 item 5 per deep read); this is accurate only to first order in v/c; (2) the light path in the lab is the fixed circle or polygon traversed at speed c (true for mirrors or guided light in vacuum; for fibre the Fresnel drag factor happens to cancel, as Wang notes), (3) the "t'" in p.11-12 is held at constant u over each element (adequate at first order). Kelly's own claim "this is the first general derivation of the Sagnac effect which does not assume an ether" (p.10) is incorrect as a statement about the literature: the standard textbook (Post 1967, which he cites) derives the same loop integral without an ether [memory, high; Kelly himself cites Post]. b. Experiments: Michelson-Gale, ring lasers, fibre gyros confirm the formula. Kelly's p.7 use of Michelson's 1913 rotating-mirror test as proof that "light ignores the whole rotating apparatus" is an interpretation I could not check; Kelly himself notes a factor-2 error in that paper [read: p.7]. [memory, low] on details. c. Tested: the Sagnac formula, extensively (Sagnac 1913, Harress/Knopf 1911/1920, Pogany 1926/28, Michelson-Gale 1925, ring lasers). The source-independence part is tested separately (astronomical and laboratory tests, e.g. de Sitter-type binary star arguments, Alvager 1964) [memory, medium]. d. If "c relative to the lab" were a different principle from SR's: for a lab on the spinning Earth the lab is not an inertial frame; Kelly's monograph 2 (per Investigator) says c in an Earth-centred non-spinning frame, which conflicts with the "laboratory" wording of the project paper [read: sites.md lead; claim 4]. Nothing in mainstream physics would need to change for the weak form (it is SR's postulate in an inertial frame). Verdict: split. The derived formula: Established (algebra correct; confirmed by experiment). The inference that the correct result "proves" lab-frame light speed, and that this differs from SR: Internally inconsistent (non sequitur; same assumption as SR's second postulate; the "first non-ether derivation" claim is wrong).

Claim 3. Special Relativity has no role; SR corrections are order v^2/c^2; the effect appears at rim speeds of about 13 m/s per metre of radius (pp.4, 10-11)

a. Maths: The magnitude comparison is correct: the first-order Sagnac term is v/c-linear; the gamma factor changes the result at v^2/c^2 (about 1.9e-15 at 13 m/s; I recomputed). The deep read's point (rim speed for one fringe is 13.13 m^2/s divided by r, not "per metre of radius") is confirmed: 26.2 m/s at 0.5 m, 13.1 at 1 m, 6.6 at 2 m. The "13 m/s per metre" wording has the scaling backwards (inverse to the radius). Kelly's text on p.10, that "some publications apply the gamma factor to the denominator and later also in the numerator, thus cancelling out", is an assertion about unnamed papers. Kelly's statement that SR "has no part in any explanation" is a claim about what theory explains the effect. The first-order formula is the same in SR and in Kelly's treatment; at second order SR gives the lab-frame difference 4Aomega/(c^2 - v^2) (Kelly's own eq (6)) and says a rim clock records it reduced by sqrt(1 - v^2/c^2) [memory, high]. So the two do not conflict, and "no role" is only true in the sense that nothing beyond first-order kinematics is needed to see the effect. The framing as a contradiction of SR rests on claim 5. Also on p.4, Kelly takes v -> c limits ("dt' becomes t0/2, speed relative to the observer becomes 2c") in the same paper that says relativity cannot matter at low speed; a lab-frame closing rate of 2c is fine in SR, a rider's measured speed of 2c is not (per deep read). b. Experiments: no experiment distinguishes "Kelly's first-order treatment" from "SR's first-order treatment"; the second-order term (relative 1e-15) is far below what any Sagnac test has resolved [memory, medium]. c. Hafele-Keating critique: Kelly's other programme (Kelly 2000, from the 1971 USNO report): per-clock raw readings East -196/-54/+166/-97 ns, West +413/-44/+101/+26 ns against published averages -59 and +273 ns [web: Investigator, research/files/kelly-2000-...htm]. These numbers are quoted from Kelly's page and I did not verify them against the USNO report. They bear on SR's clock tests (a different claim from the Sagnac formula) and should stay in the record. The spread among individual clocks (hundreds of ns) is as large as the effect, which is a legitimate point about clock noise; later flights (Alley 1970s, Gravity Probe, GPS) are independent time-dilation tests [memory, medium]. d. If true (SR plays no role in light propagation round loops): nothing changes in Sagnac-based GPS handling, because GPS uses the geocentric inertial frame, which is SR/GR consistent. Verdict: split. Magnitude comparison (SR corrections are second order): Established. "13 m/s per metre": a wording error; numerically right at r = 1 m. "SR has no role / Sagnac upsets SR" as a conclusion: Not testable as stated in the first-order regime, and where it is testable (second-order term) it gives no difference; the argument for a contradiction depends on claim 5.

Claim 4. Michelson-Gale 1925 is an ordinary Sagnac test using the projected area, giving sin(latitude); ring lasers confirm the formula to better than 1 part in 10^20 (pp.5-8)

a. Maths: The first half is correct and well explained. The shift is 4(A . Omega)/(clambda) with A the vector area; for a horizontal loop, A . Omega = AOmegasin(latitude). My computation: A = 207,836 m^2, Omega = 7.292e-5 rad/s, latitude 41 deg 46', lambda = 570 nm gives 0.2363 fringe; Kelly's 0.236 and 0.230 observed [read: p.5-6]; the time-difference arithmetic (0.23 x 570 nm / c = 4.37e-16 s) is right (my 4.373e-16). The "within 3%" is 2.5%. The drawing "area turns upside down after 12 hours" is a correct picture of why the centre of rotation does not matter. Errors: (i) "1 m disc fringe shift from Earth rotation about 1e-5 at mid-latitudes" (p.6): my value is about 2.5e-7 (A = 0.785 m^2, 45 deg, 550 nm); Kelly is a factor of about 40 too high. (ii) Distance 4250 km (p.5) is wrong (4752 km). (iii) Bilger et al. "accuracy better than 1 part in 10^20" (p.7): the Bilger/Stedman ring-laser work reports a frequency resolution of 1 part in 3e21 of the optical frequency, with rotation resolution 300 prad [web: https://opg.optica.org/ao/abstract.cfm?uri=ao-34-24-5375, Stedman, Li, Bilger, Applied Optics 34, 5375 (1995)]. That is the smallest beat-frequency change detectable relative to the light frequency (about 4.7e14 Hz), and not the accuracy with which the Sagnac formula was verified; the ratio of 10^20 to the formula is a misreading of a resolution figure. The G-0 Sagnac beat is a few hundred Hz [web search summary, same source]. (iv) "The area is less by a factor of 277,000" vs M&G: 207,836 / 0.75 = 277,115 [my check]: correct. (v) Ring-laser lock-in is used to say history "would have taken a different turn had the Sagnac test given zero" (p.7-8): lock-in is a property of ring lasers, whose two counter-rotating modes couple through backscatter, not of Sagnac's passive interferometer (separate beams from a lamp), so the story does not apply to Sagnac's 1913 apparatus [memory, high]. (vi) Kelly's "orbital angle" comparison (p.9): the Earth turned through 2.3e-10 degrees "of its orbit" during the M&G test, 1e10 times less than during Hafele-Keating, so (he argues) the lab surface should count as inertial if the geocentre does. This compares the wrong quantity. What decides whether a frame is non-rotating is its rotation rate relative to local free-fall gyroscopes/light loops, not the angle it turns through relative to the Sun. The Earth-centre frame falls freely round the Sun (the Sun's pull is removed locally), so it does not rotate with respect to local inertial frames at a measurable level, while the surface rotates at 7.29e-5 rad/s, which is what these very interferometers measure. The argument is therefore a category mistake and is circular as a defence of "lab is inertial" [memory, high]. b. Experiments: consistent (the experiment itself). Marett's independent recomputation of M&G gives 0.2364 against 0.230 ± 0.005 [web: http://www.conspiracyoflight.com/Michelson-Gale/Michelson-Gale.html, via Investigator]; he is an amateur with his own interpretation (an ECI preferred frame) and no commercial stake. c. Tested: M&G 1925 (Clearing, Illinois, a single campaign; individual fringe measurements from +0.55 to -0.05 [read: p.5]; the wide spread is Kelly's own observation); ring lasers (Bilger 1995, Schreiber and others) test it far more precisely; the sidereal-day result of Schreiber 2003 is cited by Marett. Replication: repeatedly, by many ring lasers. d. Nothing in mainstream physics changes; the standard account is the same. Verdict: the claim "M&G and ring lasers are ordinary Sagnac tests with the sin(latitude) projection": Established, with the arithmetic checks above. The sub-claims "10^-20 accuracy" and "1e-5 fringe" are Refuted by calculation (misread resolution; arithmetic). The "orbital angle" argument that the lab is as inertial as the geocentre: Internally inconsistent (compares the wrong quantity).

Claim 5. Light does not travel at c relative to the moving disc on straight segments of polygonal paths, so Sagnac, Harress and Pogany experiments contradict constancy of light speed in the moving frame (pp.8-9)

a. Maths: Eqs (3),(5),(10),(11) say the speed of the beam in the co-moving description is c +/- v, with the lab-frame constant c. The statement is a claim about coordinate speed in a rotating frame, where the one-way speed round the loop is anisotropic. SR never asserts light has speed c relative to a non-inertial frame over an extended path. SR's postulate is for inertial frames; for a co-moving observer measuring locally (an instantaneous inertial frame at the rim), light is measured at c. The Sagnac time difference equals the failure of Einstein clock synchronisation to close consistently round the loop (the same loop integral, divided by c^2) [memory, high]. Kelly's argument using Einstein 1905 (p.8) is: Einstein said a clock moved along a polygon gives the same slowing as a straight line, so a polygon is as good as a straight line, so SR must apply to the straight sections. That is a quotation of the clock-dilation result at second order (1/2 v^2/c^2), which is true, and does not address clock synchronisation along the path, which is what Sagnac depends on (first order). The step "straight segment on a rotating disc = inertial straight line" fails because the segment is accelerating (centripetal acceleration v^2/r) and the frame attached to the disc rotates, so no single inertial frame covers all of it; local intervals are fine, global comparison is not. Kelly's claim therefore conflicts with a postulate SR does not make. Kelly quotes Einstein 1916 ("The word special is meant to intimate...") and says authors "selectively quote" (p.8). Reported as his opinion. b. Experiments: Sagnac 1913, Harress 1911 (published 1920), Pogany 1926/28 all show the loop shift; they show the shift, not light speed relative to the disc; nobody has measured one-way light speed on a disc independent of clock synchronisation [memory, medium]. c. Tested: the observed shifts, by many; the SR-based loop-integral formula predicts them (Post 1967; standard). Neutral controlled test of Kelly's specific alternative (Galilean light speed relative to the lab with absolute time): same predictions to first order; second-order difference never resolved. d. If true: SR's postulate would have to be extended to say the speed of light is c (locally) only in inertial frames, which is already the case. Verdict: split. Observational part (a loop on a moving disc shows a direction-dependent time difference): Established. "Contradicts SR": Not testable as stated / rests on a misreading of SR's domain, and where predictions differ (second order) Never properly tested.

Claim 6. Einstein never mentioned the Sagnac or Michelson-Gale tests, supported by Silberstein (1921) and Turner (1979) (pp.4-5)

a. Maths: not applicable; a historical statement. Two internal inconsistencies: Dufour and Prunier "repeated the Sagnac test forty years after" but cites are 1937-42, 24-29 years after (deep read; confirmed [read: p.4, p.13]). Silberstein 1921 can only speak for Einstein up to 1921, but Michelson-Gale is 1925; Silberstein's quote is about "details of ... rotation", not the Sagnac test by name [read: p.4]. b-c. Historical record. Wikipedia: Einstein was aware of Harress's 1911 experiment, and rebutted Harzer's 1914 reanalysis of Harress (Astronomische Nachrichten 199) [web: https://en.wikipedia.org/wiki/Sagnac_effect, via Investigator]. Harress's experiment is now classed as a Sagnac-type rotating-ring experiment. This is evidence that Einstein did write about a rotating-interferometer experiment; it does not show he named Sagnac, and the Investigator notes that is still open. Kelly's source (Turner 1979, ed. Hazelett, a collection of Ives's relativity-critical papers) is an interested source; report it as its characterisation. Kelly's phrase is "This author has located no reference by Einstein", which is a report of his own search; it is stated honestly as such and cannot be refuted by silence elsewhere. Einstein also visited Miller's site in 1921 per Kelly [read: p.4] (not checked). d. Nothing in physics changes. Verdict: Not testable as stated in the sense of physics; as a historical statement, "Never properly tested": a narrow form (Einstein never named Sagnac or M&G) is unverified; the broad form (Einstein ignored rotating-interferometer experiments) is contradicted by the 1914 Harress note.

New defects found in pp.6-13 (not in the deep read)

  • p.10: Wang et al. misdescribed (out-and-back on a straight line) [read: p.10 vs Wang pp.1-3].
  • p.6: Earth-rotation fringe on a 1 m disc is about 2.5e-7, not 1e-5.
  • p.7: "1 part in 10^20" is a frequency resolution, not formula accuracy [web].
  • p.7-8: lock-in story misapplied to a passive interferometer.
  • p.9: "orbital angle" comparison of frames is a category error.
  • p.11 vs p.1: two different Sagnac areas (666 cm^2 vs 0.86 m^2).
  • p.10: "first general derivation without ether" overstated; the loop-integral derivation is standard, and Post (cited by Kelly) gives it.

Summary of verdicts

  1. Uniform motion gives the Sagnac effect: weak form (moving medium, closed loop) Tested, supported (Wang et al.); strong form (rigid body in uniform translation) Refuted by calculation (loop integral of constant velocity is zero).
  2. Light at c relative to the lab reproduces Sagnac: formula Established; "proves" and "differs from SR" Internally inconsistent.
  3. SR has no role: size comparison Established; the conclusion Not testable as stated (no resolved difference).
  4. Michelson-Gale as projected-area Sagnac: Established; "10^-20" and "1e-5 fringe" figures Refuted by calculation; "orbital angle" argument Internally inconsistent.
  5. Light not at c relative to the disc contradicts SR: observation Established; contradiction Not testable as stated / misreads SR's domain.
  6. Einstein never mentioned them: Never properly tested (narrow form open; broad form contradicted by the 1914 Harress note).

Pertinent Investigator findings kept in the record (whatever the verdict)

  • Kelly's biography: Alphonsus Gabriel Kelly (1926-2005), ESB Chief Project Engineer and Director; PhD 1966 on a siphon column; Irish Times obituary confirms "much academic debate, some acrimonious" [web: Investigator]. Vigier (Phys. Lett. A 234, 75, 1997) built on Kelly's 1995 monograph. Cowan's memoriam calls Kelly's analyses "essentially correct" (sympathetic colleague).
  • 1996 Irish Times correspondence and Kelly's unaccepted debate offer: one sympathetic source (Natural Philosophy wiki and Cowan); the letters themselves not found.
  • Monograph 2 (1996): light follows the Earth's orbit and gravity field but not its spin. Differs from "laboratory" in the project paper (conflict for a lab on the spinning Earth); worth keeping, as it is the better-stated version (a non-rotating Earth-centred frame, which matches the ECI practice of GPS).
  • Marett's experiments and Wang et al. follow-ups (Huang 2021, etc.): related work; keep. Marett's ECI preferred-frame reading is an interpretation consistent with SR's predictions for these experiments.
  • Hafele-Keating raw-clock numbers (Kelly 2000): kept; not verified against the USNO report (to do).
  • Absence of evidence: no documented replication or refutation of Kelly's specific claims by name; no experiment distinguishes his treatment from SR at first order. The undated paper's PDF is from 2011 (after his death), so the first-order publishing history is unknown.