Physicist review: Lorentz-HA (H. A. Lorentz, The Theory of Electrons, 2nd ed. 1916)

Reviewer: PhD physicist (Step 2). Date 2026-10-08. Tags: [read] from the book (PDF page numbers; printed page = PDF page minus 9), [web] with URL, [memory] with confidence.

How I worked, and a source note

• I read facts.json, claims.md, the whole Chapter V deep read, sites.md "Leads for review" and the research notes (Wikipedia/Wikisource copies, the Internet Archive note, misc sites). I did not open the Chapter V pages again except where noted. I re-read from the page images: PDF p.221 (eqs. 313, Kaufmann passage) and p.228 (Brace passage). I grepped the extracted text for the Kaufmann, Einstein, "substantiality" and Note 72* quotations: they are in the book as quoted [read: lines in the text layer for PDF pp.222, 239, 330].
• I re-did the algebra myself with SymPy/NumPy (numbers below are mine). Chapters I to IV and Notes 1-61 rest on digests only (Step 1 says so); claim 7 therefore rests on digests plus my own knowledge, and I mark that.
• Web checks: 2 searches (modern tests; electron-mass experiments). The Wikipedia pages in research/ are secondary; I use them only for what they cite.
• Units: Lorentz uses Heaviside-Lorentz units (no 4 pi in Maxwell's equations). beta = w/c. k = 1/sqrt(1 - beta^2) is today's gamma.

Corrections to Step 1

1. Brace's sensitivity is 10^-12, not 10^-13. The deep read (section 7), claims.md claim 4 and facts.json (Ch. V part 2, key claim 5; summary) all say "10^-13". The page image of PDF p.228 reads: "a difference between the principal refractive indices of 10^-12 could not have escaped him. This is about the twentieth part of the value which we have just computed" [read: page image p.228]. The OCR text layer has "10-'?" (garbled). The "twentieth part" confirms it: 2e-11 / 20 = 1e-12, whereas 1e-13 would be one two-hundredth. Consequence: Brace's margin over Lorentz's rigid-electron prediction is a factor 20, not 200. The verdict on claim 4 is unchanged, but the number must be fixed in claims.md, facts.json and the deep read, and in every downstream file. (Orchestrator: search for "10^-13", "1e-13", "10⁻¹³".)
2. Claim 3 overstates "must". claims.md says "The electron must be a deformable (contracting) charged shell". Lorentz does not say must. He says the idea "looked very tempting", that Kaufmann's data were "decidedly unfavourable" so that "it seems very likely that we shall have to relinquish this idea altogether", and later that it is "quite legitimate to maintain the hypothesis" and that "perhaps ... we are wholly on the wrong track when we apply to the parts of an electron our ordinary notion of force" [read: PDF pp.221-224]. Better wording: "Lorentz works out a deformable electron as the way to get his masses; he concedes it is not shown stable."
3. "His equations for moving axes are not exactly form invariant" (facts.json key claim 8; deep read section 11) needs a precise reading. The transformation itself (l = 1) is exact: with it the free-ether Maxwell equations keep their form, and the transformation forms a group (the moving observer's inverse is w -> -w, eq. 330) [read: pp.236-237]. What fails in 1909 is the description of matter: the transformed equations (292) contain the factor (1 - w u'_x/c^2) and extra terms in the force (293), because his auxiliary velocity u' and density rho' are not the relativistic velocity and charge density. Note 72* supplies the relativistic ones [read: pp.330-334]. The sentence in the book is Lorentz's own and is correct as he states it; the summary should not let a reader think the transformation is only approximately Lorentz-invariant.
4. Accept the Investigator's correction on claim 3 (Note 87 of the 1909 edition). Lorentz's statement that Bucherer's 1908 experiment confirms (313) is already in the first edition; 1915 added Hupka, Schaefer-Neumann, Guye-Lavanchy [web: archive.org cu31924005244615, via research/2026-10-08-internet-archive.md; I did not re-fetch]. claims.md should say "1909 (Bucherer 1908), widened in 1915". This matters for the history: Lorentz changed his mind about the deformable electron on a single new experiment, within about two years of Kaufmann's 1906 negative.
5. Claim 2 wording. "Michelson's experiment proves the contraction" is a quote of Lorentz (p.205) and is rightly flagged in claims.md as his words. It must not be read as established physics: see claim 2 below.
6. Step 1's own corrections to the digests (m', m'' at p.221; 1e-9 n at p.228; "iota" = l; no 4 pi in 297) are right; I confirmed (313) on the page image. Section numbering note (two "176" in the text layer) is also right.

Claim 1. The exact transformation x' = kl(x - wt), t' = kl(t - wx/c^2)

a. Mathematics. Internally consistent. With l = 1 it is the Lorentz transformation; the free-ether Maxwell equations keep their form [read: pp.206-208]. Lorentz's extra scale factor l is shown by Note 75 to be forced to 1 once the mass law (305) and F = dG/dt are used [read: pp.337-340]. I checked by SymPy that the inverse of (328) is (330) and that the light-cone form x^2 - c^2t^2 is preserved up to l^2. Units fine. One hidden assumption: the transformation is applied to the free-ether equations; matter needs the extra hypotheses of claims 2 and 3. b. Experiment. Fully consistent with the modern record: Michelson-Morley-type resonator tests (isotropy of light speed to about 10^-17 [web: https://arxiv.org/pdf/1002.1284, Herrmann et al. rotating cavity]), Kennedy-Thorndike, Ives-Stilwell and storage-ring tests (time-dilation factor verified to +/-2.3e-9 at beta = 0.338 [web: https://arxiv.org/pdf/1409.7951]). c. Tested? Yes, extensively, by many groups and with ever better precision, as the Lorentz-Einstein transformation, not specifically as "Lorentz's ether version". Voigt (1887) wrote an equivalent transformation; Lorentz concedes this in a footnote [read: p.207 fn.1]. Poincare (1905-06) completed the group property [memory, high]; the priority question is outside the physics verdict. d. If true. Nothing in mainstream physics changes: it is mainstream. Verdict: Established* (the transformation and its invariance of free-ether electrodynamics).

Claim 2. Contraction by k follows if molecular forces transform like electrostatic forces; Michelson-Morley null result

a. Mathematics. Consistent given the hypotheses. The argument (300) -> equilibrium maps to equilibrium -> L2/L1 = k is valid if (i) molecular forces obey (300) and (ii) the equilibrium configuration is unique. Lorentz flags both [read: pp.210-211]. I checked the first-order numbers: expected Michelson shift, with path about 22 m and beta^2 = 1e-8, is 22 m x 1e-8 / 589 nm = 0.37 fringe, matching Lorentz's 0.4 [read: pp.201-204]. b. Experiment. The contraction (with time dilation) is consistent with all modern tests. But Lorentz's own sentence that Michelson's experiment "proves" the contraction is wrong as logic: a null result in one apparatus is consistent with contraction, and also with other compensations, so it cannot prove it. Kennedy-Thorndike (1932, unequal arms) shows contraction alone is not enough and time dilation is also needed [web: research/wiki-Kennedy-Thorndike_experiment.txt, Wikipedia]. Trouton-Rankine (1908) found no effect in a charged-condenser/resistance setup (secondary account). c. Tested? Michelson-Morley 1881, 1887; Morley-Miller 1905 (effect "less than one hundredth" of the computed value) [read: p.204]. Kennedy-Thorndike 1932; modern cavity tests at 10^-17 [web: arXiv 1002.1284]. Replicated many times. Miller reported a non-null drift [memory, high]; his data are a disputed record that this review does not weigh further (see the contrarian). d. If true. Nothing; but the reading matters: in special relativity length contraction is a kinematic consequence of the transformation, not a force effect on matter. Lorentz's dynamic derivation needs all binding forces (including nuclear, unknown in 1906) to transform as electromagnetic ones. Note 75 concedes forces depending only on distance do not, and that gravitation must propagate at c [read: p.340]. This is a real extra assumption. Verdict: split. (i) The contraction itself, together with time dilation: Established. (ii) Lorentz's dynamical derivation from "molecular forces transform like electrostatic forces": Plausible, untested as a derivation (it is not independently testable; it holds if all forces are Lorentz-covariant, which is what relativity assumes). (iii) "Michelson's experiment proves the contraction": Historically superseded but instructive*: a reasonable 1906 position, but a null result alone proves nothing, and the full set of null experiments was what later forced the principle of relativity.

Claim 3. Deformable electron, Poincare stress, masses m' = k^3 m0, m'' = k m0; Kaufmann vs Bucherer

a. Mathematics. Consistent, and I confirmed it. dG/dv for G = m0 v/sqrt(1 - v^2) gives (1 - v^2)^(-3/2) (SymPy residue 0), and U = S v with S = e^2/(32 pi^2 R^4), v = (4/3) pi R^3 sqrt(1 - beta^2) equals e^2/(24 pi R) sqrt(1 - beta^2) exactly (residue 0), which is the extra non-electromagnetic energy of (316). Abraham's ratio m'/m'' = 1 + (4/5) beta^2 versus k^2 = 1 + beta^2: confirmed. Internal consistency of the model is another matter: Lorentz concedes the shell is unstable to shape and orientation changes [read: pp.223-224]. The quasi-stationary momentum assumption (electron follows velocity changes fast) is also an assumption [read: pp.225-226]. b. Experiment. The mass formulas (313) are the relativistic ones and are confirmed to high precision by accelerators today [memory, high]. The electron model is not: "mass wholly electromagnetic" is refuted. Setting e^2/(4 pi epsilon0 R) ~ m c^2 gives a classical radius of about 2.8e-15 m, while scattering shows the electron is point-like to below about 1e-18 m [memory, medium; not web-checked; the verdict does not turn on it because the contradiction stands on the 4/3 problem and the Poincare stress, which Lorentz himself states]. Modern theory treats the electron as a point particle whose mass is a parameter. c. Tested? Kaufmann 1901-1906 (beta-rays; photographic plates). In 1906 Lorentz read these data as "decidedly unfavourable" to his deformable electron and in agreement with Abraham's rigid sphere [read: p.221-222; I checked the page image]. I computed the ratio of the relativistic to the Abraham transverse mass: 1.03 at beta = 0.5, 1.08 at 0.7, 1.26 at 0.9, 1.44 at 0.95, so discrimination needs good data at beta above about 0.8. Bucherer 1908 (also in Note 87 of the 1909 edition); Hupka 1909/10; Schaefer-Neumann 1913; Guye-Lavanchy 1915 [read: p.348]. How good these were is disputed: Zahn and Spees (1938) and Farago-Janossy (1954/57) argued that assumptions in the early experiments were wrong or imprecise and they showed only a qualitative mass increase [web: Wikipedia KBN page; the APS listing exists: Phys. Rev. 53, 511 (1938), https://journals.aps.org/pr/issues/53/7; I did not read the paper]. Rogers, McReynolds and Rogers (Phys. Rev. 57, 379 (1940), cited in a 1957 review as per the search) is the first deflection measurement the literature treats as decisive [web: https://en.wikipedia.org/wiki/Tests_of_relativistic_energy_and_momentum, partly checked]; a 2023 history of the plates is J. Potters, HSNS 53 (PhilSci-Archive 22988) [web; abstract only]. So Lorentz's 1909/1915 confidence rested on experiments of limited discriminating power. Later evidence (Glitscher 1917 fine structure; accelerators) supports the formula. Replicated in the sense of repeated: yes, by several groups, with disagreement over their precision. d. If true. The mass formulas: nothing changes. The deformable shell: would require non-electromagnetic stresses (Poincare) of unknown origin, and stability that Lorentz did not show. The 4/3 mismatch between electromagnetic energy and momentum is resolved in modern treatments only by accounting for these stresses (momentum and energy then transform as a four-vector) [memory, medium]. Verdict: split. Speed dependence m' = k^3 m0, m'' = k m0 (momentum p = k m0 v): Established (decisive tests came after 1906, in 1940 and in accelerator physics). Deformable-shell electron held by a Poincare stress, with mass wholly electromagnetic: Historically superseded but instructive. The 1906 reading of the Kaufmann data was reasonable but the data were less decisive than they looked; Lorentz's 1909 reversal came on Bucherer alone.

Claim 4. Rigid and constant-volume electrons predict double refraction ~2e-11 in water; Brace saw none

a. Mathematics. Consistent. Frequency shift (1/10) beta^2 n = 1e-9 n for beta = 1e-4 [read: p.228; I reproduced the series]. I converted it to an index change using water's dispersion (n = 1.3371 at 486 nm, 1.3312 at 656 nm, from memory, medium; used to estimate slope only): delta n = -lambda (dn/dlambda) x 1e-9 = 2.0e-11. This matches Lorentz's "about 2e-11". The constant-volume case (318), l = k^(-1/3), also checked by Step 1's SymPy. b. Experiment. The null results are consistent with a theory in which the moving body corresponds to a rest body (any Lorentz-covariant theory). Correction: Brace's reach was 10^-12, a factor 20 below the prediction, not 10^-13 (see Corrections 1). Lorentz himself hedges: the calculation "rested on certain assumptions that could be changed", and Brace "made the calculation in a different manner" [read: p.228]. The prediction is model-dependent: it needs Lorentz's mapping of electron mass to refractive index through the term m n^2 in (201) and a uniform effect of the shape of the electron on all electrons in a molecule. c. Tested? Rayleigh 1902 (76 cm tube) and Brace 1904 (4.13 m water; Brace first read his null as a refutation of contraction) [web: research/wiki-Experiments_of_Rayleigh_and_Brace.txt, Wikipedia; Lorentz and Larmor showed in 1904 it follows with the full transformation]. Two experiments, by different people, same null. I know of no later repetition on this specific form [memory, low; not checked]. d. If true. Nothing: it is a null result consistent with relativity. It disfavours Abraham-type electrons within a stated model; it does not prove the deformable electron. Verdict: Tested, not supported (for rigid-sphere and constant-volume electrons within Lorentz's model, with the caveat that the predicted size depends on his assumptions). The null result itself is not evidence for the deformable electron beyond consistency.

Claim 5. Moving observer's clocks slow by k, light speed c, light-signal synchronisation gives local time; reciprocity

a. Mathematics. Consistent, and I verified the pieces: round-trip time 2cL/(k(c^2 - w^2)) = 2kL/c (327); the clock reading 1/k t - (w/c^2) x' from the optical synchronisation equals k(t - wx/c^2); the inverse transformation is (330). The step that the clock rates are slowed by k depends on the forces in the clockwork obeying (300) [read: p.233]. Lorentz's phrase "A's clocks will go k times slower than those of A0" is true in his ether frame only; the reciprocal statement that A0's clocks run slow as seen by A (pp.235-238) is also correct once A's synchronisation convention is used. Both statements are consistent, not contradictory, because they compare different sets of events (the usual resolution). b. Experiment. Time dilation: Ives-Stilwell 1938 and modern storage-ring versions to a few parts in 10^9 [web: https://arxiv.org/pdf/1409.7951]; airborne clocks and muon lifetimes [memory, high]. Speed of light: cavity tests [web, above]. c. Tested? Yes, many times, independently. Replicated. d. If true. No change. Interpretation matters: Lorentz regards the moving observer as measuring distorted rods and clocks ("errors of his rod and his clocks", p.235), Einstein as making correct measurements in his own frame. Verdict: Established (clock slowing, c, light-signal synchronisation, reciprocal descriptions). Lorentz's added claim that the ether frame's times are the "true" ones: see claim 6.

Claim 6. Lorentz's comparison with Einstein; the substantial ether; Note 72*

a. Mathematics. Consistent. Lorentz's statement that Einstein "simply postulates what we have deduced, with some difficulty and not altogether satisfactorily" is fair in its own terms, with one reservation. Lorentz's deduction needs the additional hypotheses (300), (305) and the electron model; the deduction closes only because those hypotheses are the Lorentz-covariance of all forces in disguise (Note 75 says so: "relativity requires a general law of force"). So the deduction is not independent of the postulate. Note 72 is an accurate account of the main structural point: treating t' as auxiliary hid that t' plays the same role as t [read: p.330]. b. Experiment. No experiment separates "ether with undetectable motion, all Lorentz-covariant" from special relativity. Lorentz's own results make the ether frame unobservable (reciprocity, pp.238-239). The neo-Lorentzian interpretations (Bell's teaching approach; the Robertson-Mansouri-Sexl test theory with an ether frame and free contraction and dilation parameters; Ives; Janossy; Prokhovnik) are in this class and are experimentally equivalent to special relativity if they keep the full transformation (the Wikipedia page that the Investigator used says so, and also that the mainstream calls such views "unfounded", its characterisation; research/2026-10-08-wikipedia.md). Test-theory bounds on the parameters (the 10^-17 cavity tests; Ives-Stilwell) constrain departures from them [web, above]. c. Tested? Not distinguishable by test where the full transformation is kept. Lorentz later called the ether-versus-Einstein difference "to a certain degree a dispute about words" and said the preference depends on one's habits of thought; he found "a certain satisfaction" in an ether with "some substantiality", sharp separation of space and time and absolute simultaneity, while holding a reservation about the claim that nothing can exceed light speed [web: Lorentz 1913 Das Relativitätsprinzip, https://archive.org/details/bub_gb_89PPAAAAMAAJ, via the Investigator; I did not re-read it]. At the 1927 Mount Wilson conference he is quoted as saying "I considered my time transformation only as a heuristic working hypothesis. So the theory of relativity is really solely Einstein's work" [web via Wikipedia; the ApJ 68 (1928) 341 text itself was not read by anyone; the wording is unchecked]. Einstein in 1920 said the Lorentzian ether had kept only immobility, and relativity removed that too [web: Wikisource, research/ws-einstein-1920-ether.txt]. A modern variant (Bischoff van Heemskerck, F1000Research 12:407, 2023) drops reciprocity and proposes a modified Ives-Stilwell test on which it differs from special relativity; the author states it matches the standard tests and that its different predictions are untested [web: https://pmc.ncbi.nlm.nih.gov/articles/PMC11318248/; I read only the abstract quoted in the research note, so I have not examined the model]. That is a case of a testable departure, which is different from Lorentz's own position. d. If true. If the substantial ether with a preferred frame were physically real and detectable, special relativity's postulate that no inertial frame is privileged would be false; but Lorentz's own equations make the frame undetectable, so nothing observable changes. Verdict: split. (i) Lorentz's historical self-assessment (Einstein's postulate gives what he deduced with difficulty; t' was the physical time): Historically superseded but instructive. (ii) The claim that an ether frame exists whose rods and clocks are the "natural" standard, when by his own results no experiment can reveal it: Not testable as stated. (iii) Neo-Lorentzian versions that give up reciprocity: Never properly tested, if and where they make predictions that differ (the Investigator found one, which I have not evaluated; it is not dismissed without testing, it is simply new and unchecked).

Claim 7. Background: Zeeman effect, Lorentz-Lorenz refractivity, dispersion, Faraday rotation, long-wave black-body law

Rests on digests of Chapters I-IV (not on a deep read, as Step 1 states) plus my own knowledge [memory, high]. a. Standard classical derivations; digests are consistent with them. b. The normal Zeeman triplet, the Lorentz-Lorenz formula (n^2 - 1)/(n^2 + 2) proportional to density, and the damped-oscillator dispersion are experimentally valid within their regimes. The complex (anomalous) Zeeman patterns and the electron-oscillator picture need spin and quantum mechanics; Lorentz himself says his atomic models are illustrations. The long-wave black-body result is Rayleigh-Jeans; it fails at short wavelengths and Lorentz notes Jeans's difficulty (Note 42 adds Planck). c. Zeeman (1896-97) and later experiments confirmed the triplet and e/m. d. Nothing. Verdict: Established for the normal Zeeman effect, Lorentz-Lorenz law and classical dispersion (as limiting cases); Historically superseded but instructive* for the classical-oscillator electron model of atoms.

Investigator's findings I keep in the record (with my weighing)

• 1909 edition Note 87 and the "change of mind" date (claim 3). Kept; changes the narrative, not the verdict.
• Electron-mass experiments and their later critique (Zahn-Spees, Farago-Janossy, Potters 2023). Stated beside Lorentz's reading of them; I read none of the primary papers, so I cannot say who is right on the plates, only that the discriminating power is contested.
• Kennedy-Thorndike and Trouton-Rankine. Kept: they separate contraction from time dilation (claims 2 and 5).
• Lorentz's 1913 and 1927 statements, the NHA archive 364 holdings (letters from Kaufmann 1904, Morley 1904, Miller 1906, Bucherer 1908-09, Voigt 1888; items 398, 495, 501, 135, 486, 528), Lorentz 1904 (KNAW), Kox's correspondence volumes. These are records not read: they could show what Lorentz thought about the ether and his reaction to Kaufmann and Bucherer in real time. No verdict can rest on them.
• Neo-Lorentzian literature (Bell, RMS, Janossy, Prokhovnik, Ives, Clube 1980, Schmelzer 2012, Arminjon) and Bischoff van Heemskerck 2023. Related work; the mainstream calls parts of it unfounded (its characterisation). Kept as related work whose testable members have been or can be tested.
• Dayton Miller's ether-drift work and Lorentz's correspondence with him (1906, 1921, 1926). Miller reported a positive drift [memory, high]. The later explanation (thermal artefacts, Shankland et al. 1955) is [memory, medium] and unchecked; not needed for any verdict here.
• Absence-of-evidence note: no silent record in this project (Lorentz was a Nobel laureate, publicly engaged); the main unread items are archive letters and the 1927 conference text.

Summary table