Lorentz-HA: Primer

Source note (carried from Steps 1 and 3). The project folder holds one document: H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat, "a course of lectures delivered in Columbia University, New York, in March and April 1906", second edition, Leipzig: B. G. Teubner, 1916. The text is that of the first edition (1909), with footnotes and Notes marked "[1915]" [read: title page; deep read]. It is a 361-page scan with an OCR text layer. There is no source gap: the central Chapter V (PDF pp.196-239) and Notes 72*, 75* and 86 were read from page images in the deep read. After QE (2026-10-08) four more deep reads, from page images, cover Chapter I §§26-36 (Kaufmann, the electron's mass, charge and size), Chapter II §§58-76 (the long-wave radiation law and Jeans's difficulty), Chapter V §§146-165 (Stokes's dragged ether, Fresnel's drag coefficient) and Notes 16*, 38-42*, 65-69*. The rest of Chapters I-IV (Zeeman effect, Lorentz-Lorenz law, dispersion, Faraday rotation) still rests on digests; only claim 7 (background) depends on it [Arbiter, QE revision]. The lectures themselves: the Columbia series began on 23 March 1906, and Chapter V ("Optical Phenomena in Moving Systems") was given on 26-27 April 1906 [web: Columbia Spectator archive, https://archive-publications.library.columbia.edu/, via the Investigator]; Dutch newspapers noticed the trip (Algemeen Handelsblad 31-01-1906; Zutphensche Courant 23-04-1906) [web: Delpher, https://www.delpher.nl/nl/kranten, snippets, via the Investigator]. Page numbers "p.N" are PDF pages; the printed book page is PDF page minus 9.

Framing. Lorentz (Nobel Prize 1902) is not an outsider. His transformation is the one at the heart of special relativity. What makes this book worth studying is that it shows the other road to relativity: an ether at rest, real shrinking of moving bodies caused by forces, real slowing of moving clocks, and an electron built as a tiny charged shell. Lorentz worked this out in full, in his own hand, and then said plainly where it fell short of Einstein's version. The book also records an honest scientific argument in real time: in 1906 the best electron-mass experiment (Kaufmann) went against what is now textbook physics, and Lorentz said so in print. No funding or publication blockage was found [Arbiter]. Most of the record is scholarly and archival.

Tags: [read] from the book; [read: page image p.N] from a rendered page; [web] from an Internet source (URL given, usually via the Investigator's notes or a reviewer's check); [calc] computed for this primer in example.py; [memory] recalled, with confidence. [Arbiter] marks the final verdict in arbiter.md. No new verdicts are given here.

Run the worked example with: .venv/bin/python review/Lorentz-HA/example.py

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1. The idea in one paragraph

Light is a wave, and in 1900 nearly everyone thought a wave needs a medium, the "ether". If the earth moves through a still ether, light should travel at slightly different speeds in different directions relative to the earth, and careful experiments should show it. They did not: Michelson and Morley's interferometer, and the searches by Rayleigh and Brace for a direction effect in transparent bodies, all came out empty. Lorentz's answer was that the laws of electricity and magnetism themselves hide the motion. He found a change of variables (now called the Lorentz transformation) that leaves those laws unchanged for a moving system. He then argued that if the forces that hold matter together are electrical in nature, a moving body must shrink slightly along its motion, its clocks must run slightly slow, and its electrons must get heavier with speed in a precise way. All of these effects together hide the motion exactly. He ended up with the same formulas as Einstein (1905), but kept the ether as a real, if undetectable, background, and regarded the moving observer's lengths and times as distorted. He judged in 1909 that Einstein "simply postulates what we have deduced", and in 1915 that his own "chief cause of failure" was treating the moving observer's time as a mere mathematical aid.

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2. What you need to know first

2.1 Maxwell's equations, and Lorentz's units. Maxwell's equations describe how electric and magnetic fields are made by charges and currents and how they change in time. Lorentz writes them in Heaviside-Lorentz units, which put the 4 pi into Coulomb's law instead of into Maxwell's equations: div d = rho, curl h = (1/c)(dd/dt + rho v), curl d = -(1/c) dh/dt, div h = 0. Here d is the electric field (in empty space Lorentz's "dielectric displacement" equals the electric field), h the magnetic field, rho the charge density, v the velocity of the charge, and c the speed of light. "rot" in the book means curl. To convert a Heaviside-Lorentz charge squared to SI, use e^2(HL) = e^2(SI)/epsilon0, where epsilon0 = 8.854e-12 F/m is the electric constant [memory, high]. Read: Griffiths, Introduction to Electrodynamics, the chapters on electrostatics and Maxwell's equations, and its appendix on units.

2.2 Galilean and Lorentz transformations. A transformation tells you how to turn the coordinates (x, t) one observer uses into those (x', t') of another moving at speed w along x. Newton's (Galilean) rule is x' = x - wt, t' = t. Maxwell's equations do not keep their form under it. The Lorentz transformation does: x' = gamma(x - wt), y' = y, z' = z, t' = gamma(t - wx/c^2), with gamma = 1/sqrt(1 - w^2/c^2). Lorentz writes gamma as k, and beta = w/c. For the earth's orbital speed (about 30 km/s), beta is about 1e-4 and beta^2 about 1e-8, so k differs from 1 by only 5e-9. Read: Griffiths, the chapter on electrodynamics and relativity; Taylor and Wheeler, Spacetime Physics.

2.3 Interferometers and "second-order" effects. An interferometer splits a light beam, sends the halves along two paths and recombines them. A shift of the bright and dark fringes measures a difference in travel time of a fraction of one light period. An effect proportional to beta is "first order"; one proportional to beta^2 is "second order". First-order ether effects cancel in many experiments for simple reasons; the Michelson-Morley experiment was designed to be sensitive to beta^2. Read: Hecht, Optics, the chapter on interference.

2.4 Momentum in a field, and "electromagnetic mass". A moving charge carries a magnetic field with it, and the electric and magnetic fields together carry momentum (density d x h/c in Lorentz's units). Pushing a charge faster means building up more field momentum, which feels like extra inertia: "electromagnetic mass". If all of an electron's mass were of this kind, the electron's mass would depend on its speed in a way set by its shape. Read: The Feynman Lectures on Physics, vol. II, ch. 28 ("Electromagnetic mass") [memory, high]; Griffiths, the section on momentum in fields.

2.5 Longitudinal and transverse mass. With Newton's law in the form force = rate of change of momentum, F = dG/dt, and a momentum G(v) that is not simply proportional to v, a push along the motion and a push across it meet different resistance. The transverse mass is m'' = G/v (sideways pushes only turn the momentum), and the longitudinal mass is m' = dG/dv (pushes along the motion change its size). For relativity's momentum G = m0 v/sqrt(1 - v^2/c^2) these are m'' = k m0 and m' = k^3 m0. Today we use only the momentum formula and the rest mass m0; the two "masses" are a historical way of writing the same physics.

2.6 How a transparent material slows light (the electron-oscillator model). In Lorentz's picture each molecule contains electrons tied by spring-like forces. Light of frequency nu shakes them, they re-radiate, and the sum slows the wave. In the simplest form (one kind of electron, no damping, local-field correction left out) the refractive index n obeys n^2 - 1 = N e^2/(f - m nu^2), with N the number of electrons per volume, e their charge, f the spring constant and m the electron mass. The mass enters only through the product m nu^2. That small fact is the key to the worked example. "Dispersion" means n depends on nu; for water n rises from 1.3312 (red) to 1.3371 (blue) [memory, medium-high]. Read: Griffiths, the section on dispersion; Hecht, the chapter on the propagation of light.

2.7 Double refraction (birefringence). In some crystals light vibrating in one direction travels at a different speed from light vibrating at right angles. The difference of the two indices, Delta n, measures the effect. A liquid such as water is normally singly refracting (Delta n = 0). Rayleigh (1902) and Brace (1904) asked whether the earth's motion through the ether would make water slightly doubly refracting. Read: Hecht, the chapter on polarization.

2.8 Taylor series to second order, and round-off. Most of the book's numbers come from expanding a formula in powers of beta and keeping the beta^2 term: sqrt(1 + x) is about 1 + x/2, 1/sqrt(1 - beta^2) is about 1 + beta^2/2. At beta = 1e-4 the effects are parts in 10^9 or smaller, which is close to the limit of ordinary double-precision arithmetic on a computer (about 16 digits). Section 5 shows a case where a direct computer evaluation fails for that reason.

2.9 "Same predictions, different story": empirical equivalence. Two theories are empirically equivalent when every experiment gives the same number in both. Lorentz's ether theory with a full transformation and Einstein's special relativity are a standard example. Choosing between such theories is done on grounds such as simplicity, not by experiment. Read: J. S. Bell, "How to teach special relativity" (1976), in Speakable and Unspeakable in Quantum Mechanics [memory, high], a modern physicist's defence of teaching relativity Lorentz's way.

2.10 Potential flow (needed for section 5d). A fluid flow is irrotational when curl u = 0, where u is the fluid velocity. Then u = grad phi for some scalar phi, the velocity potential. If the fluid is also incompressible (div u = 0), phi obeys Laplace's equation, div grad phi = 0, the same equation as the electrostatic potential in empty space. A sphere moving through such a fluid fixes the flow completely once you require that, on the sphere's surface, the fluid's velocity perpendicular to the surface equals the sphere's. The velocity along the surface is then whatever the solution gives, and in general the fluid slides over the sphere. Read: any fluid-mechanics text on potential flow past a sphere (e.g. Batchelor, An Introduction to Fluid Dynamics) [memory, high].

2.11 Kirchhoff's law of radiation (needed for check-yourself question 6). At a given wavelength and temperature, a body's emissivity E (light emitted) divided by its absorption coefficient A (fraction of incident light absorbed) is the same for every body: it equals the emissivity of a perfectly black body (A = 1) at that temperature. So a good mirror (small A) is a poor emitter. The black-body emissivity per wavelength is given by Planck's law; at long wavelengths it reduces to the Rayleigh-Jeans law, which is proportional to the absolute temperature T. Jeans proposed that the T-proportional law holds at every wavelength, with real bodies simply slow to reach it. Read: any modern-physics text on black-body radiation (e.g. Eisberg and Resnick, Quantum Physics) [memory, high].

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3. The textbook view

3.1 The transformation and the invariance of Maxwell's equations. The Lorentz transformation of 2.2, together with the field rule E'_y = gamma(E_y - beta B_z), E'_z = gamma(E_z + beta B_y), B'_y = gamma(B_y + beta E_z), B'_z = gamma(B_z - beta E_y) (in Lorentz's units, E = d, B = h), leaves Maxwell's equations exactly the same in the primed variables, with charges and currents included, provided charge density and current are transformed as a four-vector [memory, high]. It preserves x^2 + y^2 + z^2 - c^2 t^2, so a light flash spreads as a sphere at speed c for both observers. Replacing w by -w gives the exact inverse (reciprocity). Precision tests: isotropy of light speed to about 1e-17 in rotating-cavity experiments [web: https://arxiv.org/pdf/1002.1284], and time dilation to a few parts in 10^9 in storage-ring tests [web: https://arxiv.org/pdf/1409.7951].

3.2 Why Michelson-Morley expected a fringe shift. Suppose light goes at c in an ether and the apparatus moves at w along one arm of length L. The round trip along the motion takes L/(c - w) + L/(c + w) = (2L/c) k^2; across the motion it takes (2L/c) k. The difference is about (L/c) beta^2. In fringes, 2 L beta^2/lambda for a 90-degree turn of the whole apparatus. For the 1887 apparatus (total light path about 22 m, so L about 11 m), beta^2 = 1e-8 and lambda = 589 nm: 22 x 1e-8 / 589e-9 = 0.37 fringe. Lorentz gives "0.4" [read: p.201-204; calc]. The 1887 result was certainly below one twentieth of that (at most 0.02 of the 0.40 fringe expected) and probably below one fortieth, and Morley and Miller (1905) put any effect below one hundredth of the computed value [web: https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment; the 1887 paper's own wording, "certainly less than the twentieth part ... and probably less than the fortieth part", is memory, high, not re-fetched; read: p.204]. (Earlier versions of this primer said "about one fortieth (0.02 against 0.40)"; 0.02/0.40 is one twentieth, so that wording was wrong [Arbiter, QE revision (a)].) A null result needs the arm along the motion to be shorter by the factor 1/k (the FitzGerald-Lorentz contraction).

3.3 Contraction is not enough: time dilation. If the arms have different lengths, contraction alone does not cancel the effect; moving clocks (and light frequencies) must also run slow by k. Kennedy and Thorndike (1932) did this test and found no effect, which requires both [web: research/wiki-Kennedy-Thorndike_experiment.txt]. Ives and Stilwell (1938) measured the slowing of moving atomic clocks directly [web: https://en.wikipedia.org/wiki/Ives%E2%80%93Stilwell_experiment]. A different kind of test: Trouton and Rankine (1908) looked for an effect of the earth's motion on an electrical measurement made inside the moving laboratory (the reviewers' secondary accounts describe a wire's resistance and a charged capacitor), of the kind a real contraction might cause, and found none [web: https://en.wikipedia.org/wiki/Trouton%E2%80%93Rankine_experiment, as reported by the physicist and contrarian; primary not read]. Such experiments probe whether contraction shows up in electrical properties measured inside the moving system, which is where a dynamical contraction might have differed from relativity.

3.4 Relativistic momentum and the "masses". Special relativity gives momentum p = gamma m0 v for any particle, whatever it is made of, from the requirement that momentum conservation keep its form under the Lorentz transformation [memory, high]. Hence m'' = k m0 and m' = k^3 m0. These are confirmed to high precision; the first electron-deflection experiment the literature treats as decisive (to about 1%) is Rogers, McReynolds and Rogers, Phys. Rev. 57, 379 (1940) [web: Wikipedia KBN page, saved as research/pages/wiki-kaufmann-bucherer-neumann.txt], and particle accelerators rely on it daily [memory, high].

3.5 What an electron is today. Modern physics treats the electron as a point particle whose mass is a measured number, not as a ball of charge whose mass comes from its field. A classical shell of charge has known problems: the field's energy and momentum do not transform together unless something else (a non-electrical stress) holds the shell together. This is the "4/3 problem" [memory, medium]. Section 5c shows the numbers.

3.6 Double refraction from motion. In special relativity a moving piece of water is just water at rest seen from another frame; the physics in its own rest frame cannot depend on how it moves relative to anything. So no motion-induced double refraction is expected, and none has been found (Rayleigh 1902, Brace 1904) [read: p.205, p.228; web: research/wiki-Experiments_of_Rayleigh_and_Brace.txt].

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4. Where Lorentz departs

The real physics Lorentz starts from is solid. Maxwell's equations are invariant under his transformation (Voigt had applied an equivalent transformation to the free-ether equations in 1887, which Lorentz concedes in a footnote: the paper "to my regret has escaped my notice all these years" [read: p.207 fn.1]). Who the book credits, as a matter of record [read: pp.8, 207, 222-224, index p.351; QE search of the text layer]: the transformation to Voigt (p.207 footnote); the reading of the transformation as a statement about space and time to Einstein (sections 189-194); Poincaré only for the electron's stress and its stability (sections 181-182, his 1906 Palermo paper), not for the transformation, its group property or the principle of relativity; Larmor only once, in the 1909 preface, among authors whose work "is not mentioned". The book gives no reasons, so this is a fact about the book, not a ruling on priority [Arbiter, QE revision (b)]. The force between moving charges does change with motion as his relation (300) says; this is ordinary electrostatics of a moving charge. And the forces that hold solids together are in fact almost entirely electromagnetic, as quantum chemistry later showed [memory, high]. So his central step, "if molecular forces transform like electric forces, bodies must contract", is not a trick: it is a physical argument from a reasonable premise, and modern writers (Bell 1976; H. R. Brown, Physical Relativity, 2005 [memory, high]) teach relativity this way. The departures are in what he added, what he kept, and how strongly he concluded.

Where the inferences hold. Every piece of algebra the reviewers checked holds: the invariance of the free-ether equations for any constant l, the force relation, the masses (305) and (313), Poincare's stress energy, the numbers in section 185, the round-trip times and the light-signal clock synchronisation, and the reciprocity of (328)-(330) [Arbiter; calc in example.py Parts A, B, F, G]. Lorentz's moving observer really would measure light speed c and find his clocks synchronised to "local time" (example.py Part G). Lorentz also reports evidence against himself and states his weak points openly: Kaufmann's data were "decidedly unfavourable to the idea of a contraction, such as I attempted to work out", and "perhaps, after all, we are wholly on the wrong track when we apply to the parts of an electron our ordinary notion of force" [read: p.222, p.224].

Where they fall short. 1. "Deduced" versus "postulated". Lorentz's deduction needs the hypotheses that all molecular and elastic forces obey (300), that masses vary as (305), and that l = 1. In Note 75 (1915) he shows that relativity "requires a general law of force" that every force must obey, and that "attractive or repulsive forces depending only on the distances are found not to follow the general law" [read: p.337-340]. The same note draws a testable consequence: "the principle implies that all forces are propagated with the velocity of light", gravitation included [read: deep read, pp.337-340]. No reviewer weighed it and it carries no verdict here, since the speed of gravity lies outside the book's evidence [Arbiter, QE revision (d)]. So his hypotheses are, in effect, Lorentz covariance of all forces, which is Einstein's postulate stated for each force separately. The deduction is honest and instructive, but it is not independent of the postulate [Arbiter, claim 6]. 2. "Proves". The Michelson-Morley null result is an upper bound (in 1887 certainly below 1/20 of the expected shift and probably below 1/40; in 1905 below 1/100), and it fixes only the ratio of the two arm lengths. Lorentz's "proves" is conditional on an immovable ether and nothing else compensating, as he says ("so long as we persist in regarding the ether as immovable") [read: p.204-205; Arbiter, claim 2]. 3. The electron. The shell model needs a non-electrical stress of unknown origin, and Lorentz concedes it is unstable to changes of shape and orientation [read: p.223-224]. Its mass is "wholly electromagnetic" only for momentum: with the stress, a quarter of the rest energy is stress energy (section 5c) [Arbiter, item 6]. 4. The ether.* By Lorentz's own results, no experiment can tell which observer moves: "there would be no reason for preferring the times and lengths measured by the one to those determined by the other" [read: p.238]. A preferred frame that no experiment can reveal is not a testable claim [Arbiter, claim 6].

Lorentz's later statements (Investigator's finds, kept). In Das Relativitätsprinzip (Haarlem lectures, 1913; Teubner 1914) Lorentz calls the abolition of the ether "to a certain degree a dispute about words" and says he finds "a certain satisfaction" in an ether with some substantiality, sharp separation of space and time, and absolute simultaneity [web: https://archive.org/details/bub_gb_89PPAAAAMAAJ, via the Investigator's cleaned translation; not re-read by any reviewer]. At the 1927 Mount Wilson conference he is quoted: "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, ApJ 68 (1928) 341; the primary scan was not read, so the wording is unchecked]. Einstein, in his 1920 Leiden address, said the Lorentz ether had kept only one mechanical property, immobility [web: https://en.wikisource.org/wiki/Ether_and_the_Theory_of_Relativity]; Einstein is an interested party in the priority and interpretation question.

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5. Worked example: a rigid electron would make moving water doubly refracting by about 2e-11

This is Lorentz's own calculation (sections 184-185, PDF pp.226-228), re-derived step by step. It is the argument he used in 1906 against Abraham's rigid electron at the very moment Kaufmann's experiment seemed to favour it. Everything is reproduced in example.py Parts A-D, which ran cleanly.

The setup. Two rival electrons: Max Abraham's rigid sphere (it never changes shape) and Lorentz's deformable electron (it flattens by k along its motion). Lorentz's theorem of corresponding states says a moving body behaves exactly like a body at rest if the ratio of longitudinal to transverse mass is m'/m'' = k^2. If the ratio is anything else, the moving body corresponds to a rest body whose electrons have different masses for vibrations along and across the motion, and such a body would be doubly refracting [read: page image p.227].

Step 1. The size of the motion. The earth's orbital speed is about 29.8 km/s [memory, high], so beta = 9.93e-5. Lorentz rounds to beta = 1e-4 ("the velocity of the earth is one ten-thousandth part of the speed of light") [read: page image p.228]. So beta^2 = 1e-8.

Step 2. The masses of the two electrons, from their momentum. For Abraham's sphere the field momentum is G = m0 v (3/(4 beta^2)) [ ((1 + beta^2)/(2 beta)) ln((1 + beta)/(1 - beta)) - 1 ] [memory, high; its series is checked against the book in the next line]. Using m'' = G/v and m' = dG/dv (section 2.5) and expanding in beta (with ln((1+beta)/(1-beta)) = 2(beta + beta^3/3 + beta^5/5 + ...)): m''/m0 = h'' = 1 + (2/5) beta^2, m'/m0 = h' = 1 + (6/5) beta^2. These are exactly the factors Lorentz prints for "a spherical electron" [read: page image p.227; calc: example.py Part A]. For Lorentz's electron G = m0 k v, so m'' = k m0 = (1 + (1/2) beta^2 + ...) m0 and m' = k^3 m0. The ratios: Abraham m'/m'' = 1 + (4/5) beta^2; Lorentz m'/m'' = k^2 = 1 + beta^2. The rigid sphere misses the needed ratio by (1/5) beta^2 [read: page image p.219; calc].

Step 3. A mass change acts like a frequency change. In the oscillator formula of 2.6, n^2 - 1 = N e^2/(f - m nu^2), multiplying the mass by alpha gives f - alpha m nu^2 = f - m (sqrt(alpha) nu)^2. So a mass alpha m at frequency nu behaves exactly like mass m at frequency sqrt(alpha) nu. Lorentz: "all depends on the product m n^2, and ... a change of m, say in the ratio of 1 to alpha, will have the same effect as a change of n in the ratio of 1 to alpha^(1/2)" (Lorentz writes the light frequency as n) [read: page image p.226; calc: example.py Part B].

Step 4. The effective frequencies for the two polarizations. The moving body corresponds to a rest body whose electrons have mass (h'/k^3) m0 for vibrations along the motion (OX) and (h''/k) m0 across it (OZ). Light travelling along OY therefore behaves as if its frequency were along OX: sqrt(h'/k^3) n = (1 + (3/5) beta^2)(1 - (3/4) beta^2) n = (1 - (3/20) beta^2) n, along OZ: sqrt(h''/k) n = (1 + (1/5) beta^2)(1 - (1/4) beta^2) n = (1 - (1/20) beta^2) n. The difference is (1/10) beta^2 n = 1e-9 n [read: page image p.227-228; calc]. Check for the deformable electron: with h' = k^3 and h'' = k both factors are exactly 1, so no double refraction at any speed (example.py Part B).

Step 5. From a frequency difference to an index difference. A relative frequency change d(nu)/nu changes the index by dn = (nu dn/dnu)(d(nu)/nu) = (-lambda dn/dlambda)(d(nu)/nu). For water near 20 C, n = 1.33713 at 486.1 nm, 1.33299 at 589.3 nm and 1.33115 at 656.3 nm [memory, medium-high]. Fitting n = A + B/lambda^2 + C/lambda^4 through these three points and differentiating at 589.3 nm gives nu dn/dnu = 0.0186. (The plain chord between 486 and 656 nm, multiplied by 589 nm, gives 0.0207; the chord's slope belongs to a shorter wavelength, so it is a little high.) Then Delta n = 0.0186 x 1e-9 = 1.9e-11 (2.1e-11 with the chord). Lorentz: "In the case of water, and for yellow light, this change of frequency would produce a change in the index of refraction of about 2 x 10^-11" [read: page image p.228; calc: example.py Part C]. The physicist and the arbiter got 2.0e-11 and the mathematician 1.8e-11 with slightly different slope estimates [Arbiter]; all agree with "about 2e-11".

Step 6. Against Brace's sensitivity. Lorentz: "Brace's means of observation were so sensitive that 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]. 1.9e-11/1e-12 = 19, close to Lorentz's "twentieth part". (Earlier review files read 10^-13 from the garbled text layer; the arbiter's page image settled it as 10^-12 [Arbiter, correction 1].) Brace's own 1904 paper states a sensitivity of about 7.8e-13 and says his apparatus was "about 1600 times greater than the smallest effect" expected from his own, different calculation; the two calculations should not be mixed. Brace gives the light path as "this 30 metres of water" (Wikipedia's 28.5 m is a secondary figure) [web: https://en.wikisource.org/wiki/On_Double_Refraction_in_Matter_moving_through_the_Aether; the arbiter first had only a model summary, and QE then fetched the text and confirmed both numbers in Brace's paper; Arbiter, QE revision (e)].

Step 7. A computer trap. If you type Abraham's formula into a normal floating-point program at beta = 9.93e-5, the bracket is a number of about 1e-8 made by subtracting two numbers near 1, and then it is multiplied by 3/(4 beta^2), about 8e7. The round-off error ends up at a few times 1e-5, about ten thousand times the effect being sought; example.py prints m''/m0 - 1 = -2.5e-5 instead of +3.9e-9. Using the series, or 40-digit arithmetic, gives the split 9.8675e-10, matching (1/10) beta^2 to five figures (Part D). Lorentz's hand series were the right tool.

What the example shows, and what it does not. - It shows that, within Lorentz's dispersion model (electron mass entering only through m nu^2, all electrons affected alike), the rigid electron predicts a double refraction about 20 times larger than Brace could detect, and Brace saw none. The constant-volume electron of Bucherer and Langevin gives an effect "even a little stronger" [read: page image p.229]. Verdict on that claim: Tested, not supported (margin about 20, model-dependent) [Arbiter, claim 4]. - It does not prove the deformable electron. Any theory in which a moving body corresponds exactly to a body at rest (any Lorentz-covariant theory) gives zero; the null result is consistent with all of them [Arbiter]. - Lorentz hedged himself: "we have based our 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]. A factor of 20 is a clear margin, but not so large that a different model of the molecule could not change it.

5b. Why Kaufmann's beta rays found the question hard

Kaufmann measured how electric and magnetic fields bend fast electrons (beta rays from radium) and inferred their transverse mass. How different are the two predictions? From the exact formulas (example.py Part E):

The physicist's figures (1.03 at 0.5, 1.08 at 0.7, 1.26 at 0.9) are reproduced [Arbiter; calc]. Radium beta rays reached about 0.9c [web: Wikipedia KBN page, saved in research/pages]. So a few-percent systematic error in field strengths or plate positions could decide the contest either way, except at the highest speeds. Lorentz's own Chapter I gives a feel for the precision of the period: he lists measured values of e/m (in units of 10^7, with his factor c sqrt(4 pi)) of Simon 1.878, Kaufmann 1.823 (worked out with Abraham's formulae), Bestelmeyer 1.72, Bucherer 1.763 and Wolz 1.767, a spread of about 9% around the 1.77 he adopts [read: page image p.53; Arbiter, QE revision (f)]. Compare that 9% with the 3% (beta = 0.5) to 8% (beta = 0.7) differences in the table. That is why the record is so turbulent: Kaufmann (1906) fitted Abraham; Planck's reanalysis found the data not fully decisive; Bucherer (1908) favoured Lorentz-Einstein, and Bestelmeyer disputed his method; Zahn and Spees (1938) and Faragó and Jánossy (1957) later argued that the early experiments showed only a qualitative mass increase; Rogers et al. (1940) first fitted the Lorentz-Einstein curve to about 1% [web: Wikipedia KBN page, as checked by the arbiter; primary papers not read]. Between them, Glitscher (1917) tested the same mass law in a different way, through the fine structure of the hydrogen spectrum, and found it supported [web: https://en.wikipedia.org/wiki/Kaufmann%E2%80%93Bucherer%E2%80%93Neumann_experiments, as reported by the mathematician and contrarian; restored by Arbiter, QE revision (d)].

Two positions in one book. In Chapter I (pp.51-52) Lorentz reads the same 1906 Kaufmann data as fitting Abraham's formula, adopts "Kaufmann's conclusion, or hypothesis" that "the negative electrons have no material mass at all", and calls it "one of the most important results of modern physics"; he adds that a material mass of about 1/100 of the electromagnetic one cannot be excluded and drops it "with a view to simplicity". In section 179 the same data are "decidedly unfavourable" to his deformable electron. The 1915 edition reverses section 179 in a footnote ("This can no longer be said now") and Note 86, but leaves the Chapter I passage without a 1915 note [read: pp.51-52, 221-222, 348; Arbiter, QE revision (f)]. So the 1906 data supported "no material mass" under both electron models; they separated the models only through which formula fitted. The arbiter's verdict on "no material mass": Historically superseded but instructive. That conclusion also assumed that all electrons carry one common unit charge, and in 1915 Lorentz recorded that this was not settled (section 7, open question 6). His numbers in section 35 (electron mass 7e-28 g, radius 1.5e-13 cm, read from the page image p.55) are not exactly consistent with each other: from m = 7e-28 g, his own formula (72), written in ordinary Gaussian units as R = 2e^2/(3mc^2) with e = 4.5e-10 esu (his 1.5e-20 c sqrt(4 pi) converted), gives 2.1e-13 cm. That is a factor of 1.4, within his "not far from the truth"; no verdict depends on it [Arbiter, QE revision (f); calc].

5c. The shell electron in numbers

Lorentz's shell (radius R, charge e on its surface) has field momentum G = (e^2/(6 pi c^2 R)) k w (311), so its rest mass is m0 = e^2/(6 pi c^2 R) [read: page image p.220]. The condition m' = k^3 l m0 then forces dl/dw = 0, so l = 1 [read: page image p.221; calc: example.py Part F]. The field's energy at rest is e^2/(8 pi R), which is only three quarters of m0 c^2 = e^2/(6 pi R). Poincaré's inward stress S = e^2/(32 pi^2 R^4) times the shell volume (4/3) pi R^3 sqrt(1 - beta^2) supplies the rest: e^2/(24 pi R) sqrt(1 - beta^2), exactly the extra energy (316) that section 180 requires [read: page image p.222-223; calc]. So (rest energy)/(field energy) = 4/3, and a quarter of the rest energy is stress energy [Arbiter, item 6]. For a real electron, converting to SI, R = e^2/(6 pi epsilon0 m c^2) = 1.88e-15 m, two thirds of the "classical electron radius" 2.82e-15 m [calc, with standard constants: memory, high]. Scattering experiments find no size for the electron down to far smaller distances (below about 1e-18 m) [memory, medium; not web-checked; no verdict rests on it].

5d. The road not taken: Stokes's dragged ether, in numbers

Readers of alternative science will meet the idea that the earth drags the ether along with it, so that no "ether wind" is felt at the surface. Lorentz weighs exactly this idea in the book (sections 146-149, Notes 66-67), and his numbers can be checked (example.py Part H). This was added after QE; no Step 2 reviewer covered it, and the verdicts below are the arbiter's [Arbiter, claim 8 and QE revision (h)].

Stokes (1845). The ether near the earth moves with the earth, with no slip at the surface. Aberration of starlight then follows at once; Lorentz says it would explain everything observed "without any mathematical formula", "if only we could reconcile with each other its two fundamental assumptions" [read: p.185].

The difficulty (Note 66). Stokes also needs the ether flow to be irrotational (section 2.10). For an incompressible ether, a sphere of radius R moving at speed w along x has the velocity potential phi = -(1/2) R^3 w x / r^3, with r the distance from the sphere's centre [read: page image p.325]. At the sphere's equator (x = 0, r = R), the fluid's velocity is d phi/dx = -(1/2) w: backwards, while the surface moves forward at w. The slip is 1.5 w [read: page image p.325; calc: example.py Part H]. For the earth (about 30 km/s) that is about 45 km/s of slip [calc, deep reader]: no dragging at all. So the two assumptions cannot both hold. The arbiter's verdict on this form: Internally inconsistent (shown by Lorentz's own calculation).

Planck's repair, worked out by Lorentz (1898-99). Let the ether be a gas obeying Boyle's law (density proportional to pressure), pulled toward the earth by a force falling as 1/r^2. It then piles up near the earth: density k = k0 e^(mu omega / r), where k0 is the density far away and mu and omega are constants of the ether's "gas law" and its attraction. With X = mu omega / R, Lorentz finds a flow that obeys the continuity equation and no-flow-through-the-surface; the slip at the equator is (X^3/4) e^(-X) / (1 - (X^2/2 + X + 1) e^(-X)) of the stream speed [read: page image p.326; formula as recomputed by the deep reader]. Numbers: - X = 10: density near the earth e^10 = 22,026 times that in space; slip 0.011 of the stream speed (recomputed 0.01138). - X = 11: density e^11 = 59,874 times; slip 0.0056 (recomputed 0.00556), Lorentz's "no more than one half percent" [read: pp.182, 326; calc: example.py Part H]. So the slip can be made as small as one likes, at the cost of an ether tens of thousands of times denser near the earth, and the further assumption that this dense ether does not change the speed of light at all.

Lorentz's judgement, in his words. "We must be careful not rashly to reject a new idea, and in making his suggestion Planck has certainly done a good thing. Yet I dare say that this assumption of an enormously condensed ether, combined, as it must be, with the hypothesis that the velocity of light is not in the least altered by it, is not very satisfactory." He adds that Planck himself "is inclined to prefer the unchangeable and immovable ether of Fresnel" [read: pp.182-183]. That is a judgement of plausibility, not a refutation, and the constants mu and omega are given no physical justification in the note. The pages do not test the dragged ether against Fizeau's partial drag or any later experiment. The arbiter's verdict on the condensed-ether form: Plausible, untested as a model, marked provisional, because no review file has yet weighed it against later tests of ether drag near large masses (Lodge, Michelson-Gale, Sagnac-type experiments) [Arbiter, claim 8; memory, medium on those later tests, not web-checked].

A related residual effect (section 177). Lorentz's own stationary-ether theory leaves one second-order effect for celestial sources: if the sun is at rest in the ether, the sodium D-lines in sunlight and in a terrestrial flame should differ in relative frequency by a factor of order w^2/c^2 (about 1e-8), which he calls "of a purely theoretical interest" [read: deep read p.218; Arbiter, QE revision (d)]. No review file weighs it, and it carries no verdict here.

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6. Evidence table

Verdicts are the arbiter's final verdicts, on the Step 2 scale.

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7. What a learner should take away

What is solid. - The Lorentz transformation, the contraction together with time dilation, the measured light speed c, light-signal synchronisation giving "local time", reciprocity, and the mass law p = k m0 v. All of this is in the book, derived with care, and all of it is established physics. - Lorentz's picture of why rods shrink and clocks slow (because the forces that build them are electromagnetic and change with motion) is not wrong. It gives the same predictions as special relativity once every force is assumed to obey the same transformation, and it is taught today as a valid way in (Bell; Brown). - His conduct is a model: he printed Kaufmann's result against his own theory, he conceded Voigt's priority for the transformation, he named his hypotheses, and in 1915 he named his own "chief cause of failure". A smaller example: in section 50 he notes that his more rigorous electron theory of metals makes the ratio of thermal to electrical conductivity agree worse with experiment than Drude's cruder one, and says the earlier agreement may be "a fortuitous coincidence"; a 1915 footnote names Kamerlingh Onnes's low-temperature results as unexplained [read: digest pp.75-76; digest only]. In Note 42* (1915) he also records the reversal of his own 1909 hope that equipartition might simply fail: it "in all probability" holds for ordinary dynamics, so "a profound modification of fundamental principles" is needed [read: p.298]. - He also gave a calculation a reader can redo by hand against Jeans's radiation law (Note 42, check-yourself question 6), and he weighed the dragged-ether alternative fairly before setting it aside (section 5d).

What is wrong, and why the mistake is instructive. - "Michelson's experiment proves the contraction." A null result in one apparatus is consistent with the contraction but cannot prove it; Kennedy-Thorndike later showed contraction alone is not enough. Lesson: a null result constrains a combination of effects, not each effect alone. - Treating t' as "only auxiliary". Lorentz had the right formula but the wrong status for it, and so did not get exactly invariant equations for matter. Lesson: deciding what a mathematical quantity means physically can matter as much as getting the formula. - The shell electron. It needed a non-electrical stress and was unstable. Lesson: a model can give exactly the right formula (the mass law) for reasons that do not survive; the formula outlived the model. - Confidence in the 1908 evidence. Lorentz reversed his view in 1909 on Bucherer's experiment alone, which later reviewers judged less decisive than he thought. The formula was right, but the evidence at the time was thinner than the text suggests. Lesson: being right and being justified are different things; check how much the data can discriminate (section 5b).

What is genuinely open (from the arbiter's list). 1. How good, quantitatively, was the 1908-1915 evidence for the mass law, and why exactly did Lorentz accept it in 1909? The records that would tell: Zahn-Spees (1938), Faragó-Jánossy (1957), Potters (2023, PhilSci-Archive 22988), and Lorentz's own calculations and letters in Noord-Hollands Archief, archive 364 (items 135, 486, 501; letters from Bucherer 1908-09 and Kaufmann 1904) [web: research/2026-10-08-noord-hollands-archief.md]. None has been read. 2. Is the factor-20 margin against Brace robust to other models of how the electron mass enters optics? Brace's 1904 paper (Phil. Mag. (6) 7, 317) read in full would show his own calculation. 3. Does any experiment separate a physical ether frame from special relativity? Current bounds show no preferred frame; the only concrete proposal found is Bischoff van Heemskerck's modified Ives-Stilwell test (untested). 4. What did Lorentz write in his undated treatise defending the ether (NHA archive 364, item 398), and when? And what do Voigt's 1888 letters to him say, given that Lorentz wrote Voigt's 1887 paper "escaped my notice all these years"? Both are unread records, not evidence either way. 6. Are electric charges always whole multiples of one elementary charge? In Note 16* (1915) Lorentz writes: "Millikan has found values for e which can be considered as multiples of a definite elementary charge. Ehrenhaft, however, has been led to the conclusion that in some cases the charges are no multiples of the elementary one and may even be smaller than it. The question cannot be said to be wholly elucidated" (references: Ehrenhaft, Wiener Sitzungsber. IIa 128 (1914) p.58; Millikan, Phys. Zeitschr. 11 (1910) p.1097) [read: pp.259-260]. So in 1915 the author recorded a contested experimental anomaly as open. It bears on claim 3: Chapter I turns e/m into an electron mass and radius by assuming one common unit charge, so a different e would scale them (m in proportion to e, R in proportion to e) [Arbiter, QE revision (c)]. How the Millikan-Ehrenhaft dispute later went, and how the sub-electron results were later explained, has not been researched in this review [memory, medium]; it is a target for the Investigator, and this entry should not be read as a statement of how it ended. 7. Does any experiment after 1906 bear on the condensed-ether form of the dragged ether (section 5d)? The arbiter names this as unreviewed; the records would be the ether-drag experiments near large masses (Lodge, Michelson-Gale, Sagnac-type tests), with sources [Arbiter, claim 8]. 5. Dayton Miller's small positive ether drift (about 10 km/s against about 30 km/s expected), which Lorentz discussed in correspondence (1906, 1921, 1926) and lectures (1925-26), was not replicated, and a reanalysis (Roberts 2006) found it statistically insignificant [web: https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment, as reported]. The record went quiet because the result was set aside, not because of any documented suppression [Arbiter]; Miller's original data and Shankland's 1955 analysis are the records that would let a reader judge.

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8. Check yourself

1. Lorentz's rival electrons give m'/m'' = 1 + (4/5) beta^2 (Abraham) and 1 + beta^2 (Lorentz). At the earth's speed (beta = 1e-4), by how much do they differ, and why does such a tiny difference still matter for an optical experiment?
2. Using the oscillator formula n^2 - 1 = N e^2/(f - m nu^2), show that raising the electron mass by 1% acts like raising the light frequency by about 0.5%.
3. In the 1887 Michelson-Morley apparatus (light path about 22 m), what fringe shift does a stationary ether predict for yellow light (589 nm), and what does a null result below one twentieth of that (at most 0.02 fringe) tell you, and not tell you?
4. Lorentz's moving observer has a rod that is "really" L/k long and a clock that runs k times slow. Show that he measures the round-trip light time along his rod as 2L/c.
5. Why is "the electron's mass is wholly electromagnetic" true for Lorentz's shell in one sense and false in another?
6. (Lorentz's Note 42.) Polished silver reflects about 90% of yellow light. If Jeans were right that black-body emission at a fixed wavelength is proportional to absolute temperature at all wavelengths, how bright would a silver plate at 15 °C be, in yellow light, compared with a black body glowing at 1200 °C? Use Kirchhoff's law (section 2.11). What do we actually see, and what does that say about Jeans's law?

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9. Further reading

The inventor's own key works. - H. A. Lorentz, The Theory of Electrons (Teubner 1909; 2nd ed. 1916, the project's document). The first edition is on the Internet Archive as cu31924005244615 [web]. - H. A. Lorentz, "Electromagnetic phenomena in a system moving with any velocity smaller than that of light", Proc. KNAW 6 (1904), 809-831, the original of Chapter V sections 169-178 [web: https://dwc.knaw.nl/DL/publications/PU00014148.pdf]. - H. A. Lorentz, Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem (Teubner 1914) [web: https://archive.org/details/bub_gb_89PPAAAAMAAJ]. - H. A. Lorentz, remarks at the Conference on the Michelson-Morley Experiment, Astrophysical Journal 68 (1928) 341ff. [web: https://ui.adsabs.harvard.edu/scan/manifest/1928ApJ....68..341M]. - H. A. Lorentz, Lectures on Theoretical Physics, vol. I ("Aether theories and aether models") and vol. III ("The principle of relativity for uniform translations") [web, per the Investigator; vol. III not yet located online]. - A. J. Kox (ed.), The Scientific Correspondence of H. A. Lorentz, vols. 1 (2008) and 2 (2018); A. J. Kox and H. F. Schatz, A Living Work of Art (biography) [web, per the Investigator].

Mainstream textbooks and histories. - D. J. Griffiths, Introduction to Electrodynamics: Maxwell's equations, momentum in fields, dispersion, and the chapter on electrodynamics and relativity. - E. F. Taylor and J. A. Wheeler, Spacetime Physics: the Lorentz transformation from scratch. - R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, vol. II, ch. 28 "Electromagnetic mass" (the 4/3 problem and Poincaré stresses) [memory, high]. - E. Hecht, Optics: interference, polarization, dispersion. - J. S. Bell, "How to teach special relativity" (1976), reprinted in Speakable and Unspeakable in Quantum Mechanics [memory, high]: the Lorentzian, dynamical route explained by a modern physicist. - H. R. Brown, Physical Relativity: Space-time Structure from a Dynamical Perspective (Oxford 2005) [memory, high]: a modern defence of the view that contraction and dilation are dynamical effects. - A. I. Miller, Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1981) [memory, high]: detailed history of Kaufmann, Abraham, Bucherer and Lorentz. - O. Darrigol, Electrodynamics from Ampère to Einstein (2000) [memory, high]. - M. Janssen and M. Mecklenburg, "From classical to relativistic mechanics: electromagnetic models of the electron" (2007) [memory, medium].

Experiments and later work (via the Investigator and reviewers). - D. B. Brace, "On double refraction in matter moving through the aether", Phil. Mag. (6) 7 (1904) 317 [read: cited p.228; web: Wikisource]; Rayleigh, "Does motion through the aether cause double refraction?", Phil. Mag. (6) 4 (1902) 678 [read: cited p.228]. - W. Kaufmann, "Über die Konstitution des Elektrons", Ann. Phys. 19 (1906) 487 [read: cited p.221]; A. H. Bucherer, Phys. Zeitschr. 9 (1908) 755 [read: cited p.348]. - C. T. Zahn and A. H. Spees, Phys. Rev. 53 (1938) 511; P. S. Faragó and L. Jánossy, Nuovo Cimento 5 (1957); M. M. Rogers et al., Phys. Rev. 57 (1940) 379; J. Potters, HSNS 53 (2023), PhilSci-Archive 22988 [web]. - Neo-Lorentzian literature (related work; Wikipedia calls such views "unfounded within the mainstream", its characterization): L. Jánossy, Theory of Relativity Based on Physical Reality (1971); S. J. Prokhovnik, The Logic of Special Relativity (1967); H. E. Ives's papers; the Robertson-Mansouri-Sexl test theory; S. V. M. Clube, "Does neo-lorentzian relativity exist?" (1980); I. Schmelzer (2012), a proposed extension of the Lorentz ether to gravity; M. Arminjon's scalar-ether theory of gravitation (papers 1998-2011, e.g. Found. Phys. 2004) [web: research/2026-10-08-google-search.md and research/2026-10-08-wikipedia.md, snippets; none read in full]; F. Bischoff van Heemskerck, F1000Research 12:407 (2023) [web: https://pmc.ncbi.nlm.nih.gov/articles/PMC11318248/]. - Trouton and Rankine (1908); Glitscher (1917) [web: Wikipedia pages on the Trouton-Rankine and Kaufmann-Bucherer-Neumann experiments; primaries not read]. - P. Zeeman, Proc. Amsterdam Acad. 17 (1914) 445 and 18 (1915) 398, the repetition of Fizeau's experiment [read: cited p.329]. H. A. Lorentz, "Stokes's theory of aberration in the supposition of a variable density of the aether", Proc. Amsterdam Acad. 1898-99, p.448 [read: cited p.182]. F. Ehrenhaft, Wiener Sitzungsber. IIa 128 (1914) 58; R. A. Millikan, Phys. Zeitschr. 11 (1910) 1097 [read: cited p.260]. - E. B. Wilson, review of The Theory of Electrons, 2nd ed., Bull. Amer. Math. Soc. 24 (1917) [web: https://www.ams.org/journals/bull/1917-24-02/S0002-9904-1917-03025-X; landing page only, not read].

Related primers in this review: Poincare-Henri (the Michelson-Morley and Kennedy-Thorndike calculation), Ives (the 1938 time-dilation plates).

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Corrections to earlier documents

QE revision (2026-10-08), applied in this file: 1. Michelson-Morley 1887 bound. This primer (sections 3.2, 4 point 2, evidence table row 2, check-yourself question 3 and its answer) said "about one fortieth (0.02 fringe observed against 0.40 expected)". 0.02/0.40 is one twentieth, so the two halves of that sentence disagreed. Corrected to "certainly below 1/20 (at most 0.02 of 0.40), probably below 1/40", the 1887 paper's own two-level wording [qe.md check 7; Arbiter, QE revision (a)]. 2. Evidence table row 7 quoted Zeeman's 0.826 +/- 0.007 from a digest that called the table "partly garbled". It is now read from the page image p.329 and confirmed, with the calculated values and caveats added [Arbiter, QE revision (f)]. 3. Brace's figures (section 5, step 6) were cited from a model summary of Brace's paper; they are now confirmed in the paper's text (qe.md check 6). 4. Source note: the digest-only status is narrowed to the parts of Chapters I-IV not covered by the four QE deep reads.

Earlier note, still valid: One note for readers comparing files: the predicted index difference in claim 4 is quoted as 2.0e-11 (physicist, arbiter) and 1.8e-11 (mathematician). This primer gets 1.9e-11 with a three-term Cauchy fit at 589 nm and 2.1e-11 with the F-C chord; the spread comes from how water's dispersion slope is estimated, and all agree with Lorentz's "about 2e-11". The margin against Brace is therefore about 18-21, "about twenty" as Lorentz says.

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Answers

1. The difference is (1/5) beta^2 = 2e-9. It matters because the theorem of corresponding states needs the ratio to be exactly k^2; any departure makes the moving body correspond to a rest body whose electrons have different masses along and across the motion, which is a doubly refracting body. Optical interference can detect index differences of 1e-12, so a 1e-9 frequency effect, scaled by water's dispersion (about 0.02), gives about 2e-11, which is within reach.
2. f - (1.01 m) nu^2 = f - m (sqrt(1.01) nu)^2, and sqrt(1.01) = 1.005. So the electrons respond as if the light frequency were 0.5% higher.
3. Shift = 2 L beta^2/lambda with L about 11 m: 22 x 1e-8/589e-9 = 0.37 fringe (Lorentz: 0.4). A null below one twentieth (0.02 fringe or less; probably below one fortieth, 0.01 fringe) means the ratio of the two arm lengths must equal 1/k to within that precision, that is, the contraction (or something with the same effect) is present. It does not tell you the cause (forces, or kinematics), and it does not test time dilation, which needs unequal arms (Kennedy-Thorndike).
4. True time: (L/k)(1/(c + w) + 1/(c - w)) = (L/k)(2c/(c^2 - w^2)) = (2L/c)(k^2/k) = 2kL/c. His clock runs k times slow, so it shows 2kL/c divided by k = 2L/c, and he concludes the speed of light is c [read: page image p.233-234; calc: example.py Part G].
5. True for momentum: all of the shell's momentum, G = m0 k v with m0 = e^2/(6 pi c^2 R), is field momentum. False for energy: the field's rest energy is e^2/(8 pi R), only three quarters of m0 c^2; the remaining quarter is the energy of Poincaré's non-electrical stress.
6. The silver's absorption is A = 1/10, so by Kirchhoff its emission is 1/10 of a black body's at 15 °C (288 K). On Jeans's law a black body at 288 K emits 288/1473 = 0.196, about 1/5, of one at 1200 °C (1473 K). So the silver would emit (1/10)(1/5) = 1/50 (0.0196) of a white-hot black body. Lorentz: it "ought certainly to be visible in the dark" [read: page image p.296]. It is not: cold silver is dark. So Jeans's law cannot hold at short wavelengths, and Lorentz concludes the only formula that accounts for the facts is Planck's [read: p.297]. With Planck's law at 580 nm the same ratio is about 8e-32, which is nothing visible [calc: example.py Part I; the Planck comparison is the deep reader's and this primer's, not Lorentz's].