Lorentz-HA: Mathematician review (Step 2)

Reviewer: PhD mathematician. Project: H. A. Lorentz, The Theory of Electrons (2nd ed. 1916; Columbia lectures 1906; text of 1909 with [1915] notes). Date 2026-10-08.

Tags: [read] from the document (PDF page; printed page = PDF page minus 9); [read: page image p.N] from a rendered page; [calc] my own SymPy/NumPy check (script in the session scratchpad, not a deliverable); [web] with URL; [memory] with confidence. No source gap: the OCR text is readable and the central pages were read from images by the deep reader. I re-read the text layer of PDF pp.221-222 and the page images of pp.207 and 228 myself. The deep read agrees with them. Chapters I-IV and Notes 1-61 rest on digests only, which is fine for these claims.

0. Symbols and the mathematics in plain language (for the reader)

Lorentz uses Heaviside-Lorentz units (Maxwell's equations without 4 pi). Here c is the speed of light, w the speed of a system through the ether, beta = w/c, and k = 1/sqrt(1 - w^2/c^2) (today's gamma). l is an unknown scale factor with l = 1 at w = 0. d and h are the electric and magnetic field. Primed symbols (x', t', d', h') are "effective" quantities, meaning what the moving system's own measurements would give. "Local time" t' is the clock reading that follows from the transformation.

Lorentz's exact transformation is x' = k l (x - w t), y' = l y, z' = l z, t' = k l (t - w x / c^2) (eqs. 287-288) [read: page image p.206-207]. With l = 1 it is the modern Lorentz transformation. The whole chapter is a chain: (i) the transformation keeps the free-ether field equations the same; (ii) add hypotheses so that matter and electrons obey the same transformation; (iii) deduce a null result for all ether-drift experiments.

1. Mathematical audit (all steps I re-did)

Hidden assumptions, stated by Lorentz and a few he does not state. 1. Molecular and elastic forces obey (300), and equilibrium is unique [read: p.210-211]. Lorentz flags both. This is an assumption about all of matter, not a derivation. In Note 75 he says other forces "we cannot positively assert … obey the general law" and that distance-only forces "are found not to follow" it [read: p.337-340]. 2. Quasi-stationary electromagnetic momentum for an accelerating electron, with the shape following the velocity instantly. §183 admits a lag and offers no calculation [read: p.225-226]. 3. Lorentz's primed velocity and density (289), (290) were dropped to first order in the internal velocity u (§173 "skip over the difficulty"). So (292) still contains the factor (1 - w u'_x/c^2) and (293) has extra terms. The transformed equations are not exactly the rest-frame equations for moving charges [read: page image p.207; p.239; Note 72, p.330-334]. Lorentz says so himself. In the free ether (no sources) the transformation is exact. 4. The electron is a thin shell with Poincaré stress. Lorentz concedes it is unstable to shape change and that he has not shown stability [read: p.223-224]. 5. Not stated by Lorentz, my observation [calc]: with the Poincaré stress included, (m0 = e^2/6 pi c^2 R) is 4/3 of the field's rest energy over c^2 (e^2/8 pi R c^2). A quarter of the rest energy is stress energy. So the mass is not "wholly electromagnetic" in the energy sense, although the momentum is entirely field momentum. The stress behaves like a Lorentz-invariant term (energy S v, no momentum), which is why the sum is consistent. The "4/3 problem" is thereby resolved at the price of a non-electromagnetic ingredient. 6. The unobservable part: because the theory reproduces every ether-frame-independent prediction, the ether frame cannot be found by any experiment that the theory itself describes.

2. Per-claim review (questions a-d, verdict)

Claim 1. Exact transformation (l = 1 gives the modern Lorentz transformation; free-ether Maxwell equations keep their form)

a. Mathematics consistent? Yes for the free ether, for any l [calc]. For charged matter the form is only approximate in Lorentz's version, because of (289)-(290); Lorentz concedes this and says Einstein's variables fix it [read: p.239; Note 72]. Two things are not Lorentz's own: reciprocity needs l = 1 (the group property, as I showed above), and Lorentz fixes l = 1 only through the electron model (Note 75, §179). Voigt (1887) had already published an equivalent transformation, which Lorentz acknowledges in a footnote [read: p.207 fn.1]. b. Consistent with experiment? The transformation is the one that every precision test of special relativity supports: Michelson-Morley, Kennedy-Thorndike, Ives-Stilwell, particle accelerators [memory, high; Kennedy-Thorndike confirmed [web] https://en.wikipedia.org/wiki/Kennedy–Thorndike_experiment]. Fresnel drag follows [read: p.199-200]. c. Tested? Indirectly, by the experiments above, by many groups, and replicated. Not tested by Lorentz himself beyond Fizeau and Michelson-Morley drag. d. What would change if true? Nothing; mainstream physics adopted it.

Verdict: Established (the mathematical statement, with l = 1). Caveat: Lorentz's own 1904-1909 version is only first-order exact for sources; the exact form is Einstein/Poincaré. [Investigator note, kept: Lorentz's Proc. KNAW 6 (1904) original [web] https://dwc.knaw.nl/DL/publications/PU00014148.pdf.]

Claim 2. Contraction by k, so Michelson-Morley must be null; "with an immovable ether, Michelson's experiment proves the contraction"

a. Consistent? Two parts. (i) Given a stationary ether and constant c in it, the round-trip time along the motion is 2 L k^2 / c and across is 2 L k / c, so a null result needs L_parallel/L_perp = 1/k to all orders [calc]. This ratio is necessary. (ii) The claim that this contraction follows from (300) is valid arithmetic (F(S) = F(S0) × coefficients, equilibrium maps to equilibrium) but rests on the hypothesis that all binding forces behave like electrostatic ones and that equilibrium is unique [read: p.211]. Lorentz states both hypotheses openly. The word "proves" is too strong as mathematics. The experiment fixes only the ratio of the two arm lengths and the constancy of round-trip light time. It does not fix an absolute contraction or tell us what causes it. The quoted sentence is conditional: "so long as we persist in regarding the ether as immovable" [read: p.204-205]. b. Experiment? Michelson-Morley 1887, Morley and Miller 1905 (effect "less than one hundredth part of the computed value") [read: p.204]. Contraction alone cannot explain the Kennedy-Thorndike null result (1932), which also needs time dilation [web] https://en.wikipedia.org/wiki/Kennedy–Thorndike_experiment. Lorentz's theory contains time dilation too (§§177, 189-190: clocks go k times slow if clockwork obeys (300)), so it passes. It was not in Lorentz's text as a separate hypothesis in 1906. c. Tested? The null result was tested repeatedly by others (Michelson-Morley, Morley-Miller, Kennedy-Thorndike, later optical-resonator work [memory, high]). A direct test of the contraction as an independent length change is not possible, because every ruler contracts. Dayton Miller reported a non-null drift [memory, high] (Investigator lead, kept); later analyses attribute it to temperature effects [memory, medium; not checked on the web, because no verdict here depends on it]. d. What would change? If true as a physical effect of motion through a preferred frame, nothing observable changes. The mainstream reading is that the "ether" frame has no measurable role.

Verdict (split). Contraction as a necessary ratio for a null result: Established (calculation plus experiment). "Michelson's experiment proves the contraction" as a physical cause: Not testable as stated (compatible with the relativistic reading, no measurable difference). Forces-obey-(300) as a derivation: Plausible, untested for non-electromagnetic forces, and Historically superseded but instructive (Note 75* shows the principle must be imposed on every force, which rules out simple distance-only forces).

Claim 3. The electron as a deformable shell with Poincaré stress; masses m' = k^3 m0, m'' = k m0; Kaufmann "decidedly unfavourable", 1915 reversal

a. Consistent? Algebra: (311)-(316), the Poincaré stress and the mass formulas all check out [calc]. The unresolved item is dynamic stability. Lorentz concedes it ("This is certainly true", §182) and speculates that "we are wholly on the wrong track when we apply to the parts of an electron our ordinary notion of force" [read: p.223-224]. The model also needs a non-electromagnetic ingredient (item 5 above), so "wholly electromagnetic mass" is only true for momentum. Not internally inconsistent, but incomplete. b. Experiment? The mass-velocity formula itself, m'' = (1 - beta^2)^(-1/2) m0, is the relativistic one and is supported by later work: Neumann 1914, Guye and Lavanchy 1915, Glitscher 1917, Rogers et al. 1940 and modern accelerators [web] https://en.wikipedia.org/wiki/Kaufmann–Bucherer–Neumann_experiments. The shell model as a physical electron is not supported: it is not needed, and electrons are treated as point-like. c. Tested? Kaufmann (1901-1906, Ann. Phys. 19, 487): agreed with Abraham, "decidedly unfavourable" to the contraction [read: p.221-222]. Max Planck's reanalysis found Kaufmann's data not fully decisive [web, same URL]. Bucherer 1908 agreed with Lorentz-Einstein; Bestelmeyer criticised the method (non-compensated rays, error analysis). Neumann 1914 improved on it [web, same URL]. Zahn and Spees (1938) and Faragó and Jánossy (1957) argued the early experiments were qualitative only and could not decide between theories (Investigator lead, kept, and confirmed in the same web source). So "confirmed in 1915" is Lorentz's reading of experiments that later reviewers judged less decisive than he thought; the later precision came from other experiments. Lorentz's note changes from "decidedly unfavourable" (1906) to "confirmed" (1909 Note 87 on Bucherer 1908, widened in 1915 Note 86) [read: p.348; Investigator, web: archive.org cu31924005244615]. d. What would change? Nothing now: the mass law is textbook. The shell would require new physics for stability.

Verdict (split). Relativistic mass formulas: Established. The deformable-shell-plus-Poincaré-stress electron as a model: Historically superseded but instructive. Lorentz's 1906 reading of Kaufmann: Tested, not supported at the time, later reversed.

Claim 4. Rigid (Abraham) and constant-volume (Bucherer-Langevin) electrons predict about 2e-11 double refraction in water, which Brace (sensitivity 1e-13) did not see

a. Consistent? Yes. The frequency-shift argument (§184: mass change by factor alpha equals frequency change alpha^(1/2)) is sound, and the number reproduces [calc]. Lorentz's hedge: "calculations on certain assumptions that could be changed for others, and Brace himself has made the calculation in a different manner" [read: page image p.228]. For the constant-volume electron the text says double refraction is "even a little stronger", not exactly 2e-11 (see Corrections, item 2). Lorentz also says the molecular-motion treatment would need modification for the rigid electron [read: page image p.228]. b. Experiment? Rayleigh (1902) and Brace (1904) found nothing [read: p.205, 228]. I did not read the original papers. The sensitivity figure is Lorentz's statement of Brace's work, as [read], and I have not verified it independently. The Investigator found Rayleigh and Brace discussed in Wikipedia [web, per sites.md]. c. Tested? By Rayleigh and Brace. No independent replication of Brace's experiment is documented in the review files; the verdict rests on the null results as reported by Lorentz. d. What would change? If the rigid electron were true, Lorentz's contraction would be false and the mass-velocity law would differ; mainstream physics would need Abraham's dynamics.

Verdict: Tested, not supported (for rigid and constant-volume electrons, on Lorentz's account of Brace; I have not independently verified Brace's own paper).

Claim 5. Moving-observer picture: clocks slow by k, light speed c, synchronisation gives local time, reciprocity

a. Consistent? Yes. §190's round-trip time 2kL/c, with a clock slowed by k reading 2L/c, is correct. The synchronisation yields t' = t/k - (w/c^2) x' = k(t - w x/c^2) with x = L/k + w t [calc agrees with the deep read]. Reciprocity: inverse transformation equals the w -> -w transformation [calc]. As noted, reciprocity needs l = 1. A subtle point: Lorentz's derivation assumes the clock-work and rods obey (300); it is a conditional deduction. The conclusion that "no experiment shows who moves" is inside the model only if every physical process has the same transformation; Note 75 says this is an assumption for non-electromagnetic forces. b. Experiment? Time dilation: Ives-Stilwell and Kennedy-Thorndike [memory, high; KT [web] as above]. Isotropy of c: Michelson-Morley and successors. c. Tested? Yes, by many groups, replicated. d. What would change?* Nothing.

Verdict: Established. (As a statement about observable consequences. The interpretation that the ether is the "true" frame is claim 6.)

Claim 6. Lorentz on Einstein: "simply postulates what we have deduced, with some difficulty and not altogether satisfactorily"; keeps a substantial ether; 1915 "failure" because he treated t' as auxiliary

a. Consistent? The statements are Lorentz's own and are correctly quoted [read: p.238-239, 330]. As mathematics: "deduced" is conditional on the added hypotheses (300), (305) and l = 1 (and exactness only to the first order in u, item 3). Einstein's route is shorter because he postulates the principle of relativity and constancy of c and reaches the transformation without a model of matter, so what Lorentz calls a deficiency is also a gain. Lorentz concedes this: "Einstein's theory has another marked advantage" [read: p.239]. b. Experiment? No experiment distinguishes the ether interpretation from the Einstein interpretation; this is Lorentz's own point: "no reason for preferring the times and lengths measured by the one" [read: p.238]. In 1913 (Haarlem lectures) he calls the dispute "to a certain degree a dispute about words" [web, archive.org bub_gb_89PPAAAAMAAJ, per the Investigator; not re-read by me]. In 1927 he said "the theory of relativity is really solely Einstein's work" [web via Wikipedia, per the Investigator; primary not read]. c. Tested? Neo-Lorentzian theories (Prokhovnik, Jánossy, Ives, Bell's Lorentzian pedagogy, the Robertson-Mansouri-Sexl test theory, Bischoff van Heemskerck 2023) are reported by the Investigator as variants that match standard tests. Mainstream calls some of them unfounded. This is Wikipedia's characterisation, as reported by the Investigator, and the mathematics of the fully equivalent variants is not in dispute: equivalence is exactly what makes them untestable against SR. Where a variant departs from reciprocity (Bischoff van Heemskerck proposes a modified Ives-Stilwell test) it is an untested prediction and Never properly tested. d. What would change? If a preferred frame were measurable (a failure of Lorentz invariance), SR would have to change. Existing bounds on such violations [memory, medium] are tight; I did not check them.

Verdict (split). Historical account (Lorentz's comparison and later self-assessment): Historically superseded but instructive. Ether-fixed rods and clocks as the "true" description with the same observable content: Not testable as stated. Variants with modified predictions: Never properly tested (and the standard tests so far constrain them).

Claim 7. (Background) Electron theory reproduces Zeeman effect, Lorentz-Lorenz law, dispersion, Faraday rotation, long-wave black-body law

I read none of this beyond digests, and these topics are marginal to the central chain. a. The mathematics was not checked by me. b-c. These are textbook results of classical electron theory (Zeeman effect with Lorentz's 1897 theory, Rayleigh-Jeans limit) [memory, high]. d. The theory is superseded by quantum theory for these effects. Verdict: Established for the classical results as stated, with the note that the classical theory fails for the black-body spectrum at short wavelength and was replaced by quantum theory (Historically superseded but instructive for the electron model itself). Confidence limited: digest-level reading only.

3. Investigator's findings, as weighed here

Kept in the record (pertinent whatever the verdict): - First edition (1909) Note 87 already reports Bucherer 1908 as confirming (313); the [1915] Note 86 widens it [web, archive.org cu31924005244615]. See Corrections. - Lorentz 1913 Haarlem lecture (ether as "a dispute about words"; "a certain satisfaction" in absolute simultaneity) and Lorentz 1927 Mount Wilson remark [web, as reported, not re-read by me]. - NHA archive 364 (Lorentz papers): items on effective times (495), mass and Kaufmann (501), Bucherer (135), electron energy and Poincaré (486), ether (398); Voigt letters 1888. Records unread; contents unknown. A reader could settle what Lorentz knew of Voigt's 1887 paper (his footnote says it "escaped my notice all these years") by reading the Voigt letters. - Experiments after 1906: Bucherer, Wolz, Hupka, Neumann, Guye and Lavanchy, Glitscher, Rogers, and the dissent from Zahn and Spees (1938) and Faragó and Jánossy (1957). Potters (HSNS 2023) on the photographic-plate precision [web, per Investigator]. - Kennedy-Thorndike (1932), Trouton-Rankine (1908), Rayleigh and Brace, Einstein's Leiden 1920 remark that the Lorentz ether lost every mechanical property but immobility [web, per Investigator]. - Neo-Lorentzian literature and Dayton Miller's non-null ether-drift results (Lorentz corresponded with Miller 1906, 1921, 1926) [web, per Investigator; Miller [memory, high]]. - Columbia lecture dates (series began 23 March 1906; Chapter V given 26-27 April 1906) [web, Columbia Spectator, per Investigator]. These are historical context with no mathematical bearing.

4. Corrections to Step 1

1. claims.md claim 3 and facts.json (Note 86 entry): incomplete, as the Investigator found. The 1909 first edition (Note 87) already has the sentence on Bucherer 1908; 1915 Note 86 adds Hupka, Schaefer and Neumann, and Guye and Lavanchy. The note number differs between editions. I accept this correction [web, via the Investigator; the first edition was not re-read by me]. I would add: Lorentz's 1906 text said only "decidedly unfavourable"; later reviewers (Zahn and Spees, Faragó and Jánossy) judged the early confirming experiments less decisive than Lorentz states [web: https://en.wikipedia.org/wiki/Kaufmann–Bucherer–Neumann_experiments].
2. claims.md claim 4 says both rigid and constant-volume electrons predict "about 2e-11". Lorentz gives 2e-11 only for the rigid (Abraham) sphere; for the constant-volume electron the double refraction is "even a little stronger" [read: page image p.229]. Correct to "at least about 2e-11".
3. Brace sensitivity ratio: Lorentz writes that 1e-13 "is about the twentieth part of the value" 2e-11 [read: page image p.228]. Arithmetically 1e-13 is 1/200 of 2e-11, not 1/20. Either Lorentz's "twentieth" is a slip (in the 1909 text and kept in 1916), or the intended comparison is different. I cannot tell which; the OCR text and the page image agree on "twentieth". Either way the prediction exceeds the sensitivity by at least 20 times, so the conclusion stands. facts.json and the deep read quote the sentence without noting the factor.
4. claims.md claim 1 states the transformation "leaves the free-ether Maxwell equations in the same form". True and verified [calc]. But it should add that with sources (charges and currents) Lorentz's 1906 version is not exactly form-invariant (Note 72*, factor (1 - w u'_x/c^2) in (292)), and that reciprocity and the modern form need l = 1, which Lorentz fixes only through the electron model.
5. claims.md claim 3 "wholly electromagnetic": the deep read repeats Kaufmann's phrase. The rest energy with the Poincaré stress is 4/3 of the field's rest energy; the electromagnetic field alone gives only three quarters of it [calc].
6. Deep read table row (292): fine. No other transcription errors found in the passages I re-read (pp.207, 221-222, 228). Deep-read and my own SymPy checks agree throughout.

5. Summary of verdicts

1. Transformation: Established.
2. Contraction: Established as a necessary ratio; "proves" as a cause is Not testable as stated; (300) for non-electromagnetic forces is Plausible, untested.
3. Electron: relativistic masses Established; deformable shell with Poincaré stress Historically superseded but instructive.
4. Rigid-electron prediction versus Brace: Tested, not supported.
5. Moving-observer picture: Established.
6. Lorentz versus Einstein: Historically superseded but instructive; ether-preference Not testable as stated; variants with changed predictions Never properly tested.
7. Background: Established (classical results), digest-level only.