Poincare-Henri: Primer

Source note (carried from Steps 1 and 3). The project folder holds one document: George Bruce Halsted's English translation of Henri Poincare's lecture "The New Mechanics", printed in The Monist vol. 23 no. 3 (July 1913), pp. 385-395. It has a full text layer and no equations; it was read in full, with page images of pp. 2 and 6 [read: p.1-11]. There is no source gap for the essay itself. Everything about its context (the date, Poincare's 1906 technical memoir, his longer Lille lecture of the same year, de Sitter's 1911 calculation, Bucherer's 1908 experiment, Lorentz's 1900 gravity paper) comes from the Investigator's web notes and the reviewers' web checks, not from files in the project folder; research/files/ is empty. Page numbers "p.N" are PDF pages; journal page = 384 + N.

Framing. This is not an outsider's rejected theory. It is a popular lecture given at Gottingen on 28 April 1909 (one of six Wolfskehl lectures) by the leading mathematician-physicist of his day, published in English a year after his death in 1912 [web: https://en.wikisource.org/wiki/The_New_Mechanics_(G%C3%B6ttingen); Lebon bibliography via research/2026-10-06-google.md]. Most of what it says became standard physics. The parts that did not (an ether that alone is "inert", all mass electromagnetic, a Lorentz-type gravity law, planets radiating their way into the Sun) were replaced by calculation and later experiment. No record of blocked funding, hidden data or dismissal without testing was found by any reviewer [Arbiter]. Studying it is a way to watch special relativity being explained by one of its co-creators, in the ether language that was about to be dropped, and to see which of his steps were physics and which were interpretation.

Tags used below: [read] from the essay; [web] from an Internet source (URL given, usually via the Investigator's notes); [calc] arithmetic done for this primer in example.py; [memory] recalled, with confidence. [Arbiter] means the final verdict in arbiter.md.

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1. The idea in one paragraph

Poincare tells a general audience that Newton's mechanics, which seemed eternal, is "greatly shaken" but not overthrown. He names Lorentz and Max Abraham as the physicists whose work has shaken it ("destroyers" of the old mechanics) [read: p.1]; they were not co-authors of one theory but rivals, whose competing electron models were set against each other by experiment (Abraham's lost to Lorentz's in Bucherer's 1908 data). The "new mechanics" he describes is Lorentz's, and it rests on three hypotheses. First, no body can be pushed past the speed of light: each equal push adds less speed than the one before, as if the body's mass grew with its speed, and experiments on the fast particles shot out by radium show this. Second, the principle of relativity holds without exception: no experiment can tell whether you are "really" moving, and observers carried along through the ether (the medium then thought to carry light) can only set their clocks to an "apparent" local time, never a true common time. Third, a moving body is squashed slightly along its direction of motion, the Earth by about one part in 200 million, and Michelson's light experiments are said to confirm it. He then pictures matter as made of electrons whose inertia really belongs to the ether around them ("there is no matter, but only gaps in the ether"), says accelerated planets should radiate energy and very slowly fall into the Sun, and reports that Lorentz's new velocity-dependent law of gravity pushes Mercury's orbit in the right direction but explains only a small part of its known anomaly. He ends by insisting that Newtonian mechanics will always remain the mechanics of everyday speeds and must still be taught first [read: p.1-11].

2. What you need to know first

Galilean relativity and velocity addition. In Newtonian mechanics, if a train moves at speed V and a ball is thrown forward on it at speed u, the ground observer sees u + V. The laws of mechanics look the same in any frame moving at constant velocity (an "inertial frame"); you cannot tell by mechanical experiments inside a smoothly moving ship whether it moves. Poincare's boat passenger [read: p.3] is this principle. Read: Kleppner and Kolenkow, An Introduction to Mechanics, the chapters on relative motion and on special relativity [memory, high on the book; chapter numbers differ between editions].

Waves in a medium, and the ether. Sound travels at a fixed speed relative to the air. Nineteenth-century physicists assumed light, an electromagnetic wave predicted by Maxwell's equations to travel at c = 1/sqrt(mu0 eps0) (about 300,000 km/s; mu0 and eps0 are the magnetic and electric constants), must likewise travel at c relative to some medium, the "luminiferous ether". If so, an observer moving through the ether at speed v should measure light going c - v one way and c + v the other. Detecting that difference would reveal "absolute" motion. Read: Griffiths, Introduction to Electrodynamics, the chapter on electromagnetic waves, and the opening of the chapter "Electrodynamics and Relativity" (chapter 12 in the 3rd and 4th editions) [memory, high].

Interference and fringes. When one light beam is split in two, the halves sent along different paths, and recombined, they add or cancel depending on the difference in path length. A path difference of one wavelength lambda (lambda, the distance between wave crests; about 500 nm for green light) shifts the pattern of bright and dark bands ("fringes") by one fringe. So a change in light travel time Delta t shows up as a fringe shift N = c Delta t / lambda. This is the principle of the Michelson interferometer. Read: Hecht, Optics, the chapter on interference (Michelson interferometer section) [memory, high].

Momentum and impulse. An impulse is a force F acting for a time Delta t; it changes the momentum p by F Delta t. In Newton's mechanics p = m v with constant mass m, so equal impulses give equal speed increments, without limit [read: p.1]. Read: any introductory mechanics text.

Radiation from accelerated charges. A charge q moving at constant velocity carries its field along; an accelerating charge radiates electromagnetic waves. The radiated power (Larmor formula, in SI units, the International System of metres, kilograms, seconds and amperes) is P = q^2 a^2 / (6 pi eps0 c^3), where a is the acceleration. Antennas, X-ray tubes and glowing hot bodies all work by this. Poincare's examples (wireless telegraphy, flames, incandescent solids, the Crookes tube) [read: p.8-9] are all this effect. Read: Griffiths, the chapter on radiation ("Point charges" section) [memory, high].

Orbits, perihelion and arcseconds. A planet under Newton's inverse-square gravity alone follows a closed ellipse with the Sun at one focus. The perihelion is the point nearest the Sun. Pulls from other planets make the ellipse slowly rotate ("perihelion advance" or "precession"). Angles that small are measured in arcseconds: 1 arcsecond (written 1") = 1/3600 of a degree = 4.85 x 10^-6 radian. Read: Kleppner and Kolenkow, the chapter on central-force motion [memory, high].

What this primer introduces from scratch. The Lorentz factor gamma, length contraction, time dilation and the relativity of simultaneity are explained in Section 3 with nothing more than algebra. No tensors and no general relativity (GR, Einstein's 1915 theory of gravity as curved spacetime) are needed; where GR's results are quoted (Mercury, gravitational waves), they are quoted as formulas with every symbol defined, not derived.

3. The textbook view

Special relativity (Einstein 1905; Lorentz, Poincare and Minkowski contributed much of the mathematics) rests on two postulates: the laws of physics are the same in every inertial frame, and light in vacuum travels at the same speed c in every inertial frame [memory, high]. Everything below follows from these.

The Lorentz factor. For a speed v, define beta = v/c and

gamma = 1 / sqrt(1 - beta^2).

gamma is 1 at rest and grows without bound as v approaches c. For everyday speeds gamma - 1 is about beta^2/2: for a car at 100 km/h this is 4.3 x 10^-15, for the Earth in orbit (29.78 km/s) 4.9 x 10^-9, for Mercury (47.9 km/s) 1.3 x 10^-8 [calc: example.py Part B; Arbiter]. That is the whole content of Poincare's last claim: Newton is the small-beta limit.

Momentum and the speed limit (Poincare's first hypothesis). Momentum is p = gamma m v, where m is the "rest mass" (the mass measured at rest; modern usage calls it simply "the mass"). Solving for v gives v/c = (p/mc) / sqrt(1 + (p/mc)^2), which approaches 1 but never reaches it however large p becomes. Equal impulses add equal amounts of p, so they add less and less v. Numerically: if the first impulse brings a body to 0.5c, the second brings it to 0.756c, the third to 0.866c, the fourth to 0.918c; the gains are 0.5, 0.256, 0.110, 0.052 c. Newton would give 0.5, 1.0, 1.5, 2.0 c [calc: example.py B1]. This is exactly the shrinking-segment diagram on Poincare's p.2 [read: page image p.2].

Because the response to a force depends on direction, the 1909 language had two "masses": a force along the motion meets a "longitudinal mass" gamma^3 m, a force across it a "transverse mass" gamma m [memory, high]. Poincare mentions this in one clause ("the angle this velocity makes with the direction of the accelerative force") [read: p.8]. For his radium rays at 100,000 km/s (beta = 0.334), gamma = 1.061 and gamma^3 = 1.194, a 12% difference [calc: B2]. Today we keep m fixed and say that momentum and energy, not mass, grow with speed; "mass increases" is the older idiom, not a different physics.

Time and simultaneity (the second hypothesis). Clocks can only be synchronised by sending signals. If two clocks a distance d apart are synchronised by light in their own frame, an observer for whom they move along the line joining them at speed v finds them out of step by about v d / c^2. Lorentz called the shifted time coordinate "local time"; Poincare showed in 1900 that it is the time read by clocks synchronised with light signals by observers who assume light takes equal times both ways [memory, medium-high]. A moving clock also runs slow by the factor 1/gamma ("time dilation").

Velocities do not simply add. Two speeds u1 and u2 combine as

u = (u1 + u2) / (1 + u1 u2 / c^2),

which never exceeds c. Poincare's two observers each moving at 200,000 km/s in opposite directions do not see 400,000 km/s, but 276,923 km/s (with his c = 300,000 km/s) [calc: B3].

Length contraction (the third hypothesis). An object of length L0 in its own rest frame, moving at speed v along its length, has length

L = L0 sqrt(1 - beta^2) = L0 / gamma

in the frame where it moves; for small beta the fractional shortening is about beta^2/2. It applies only along the motion. Lorentz and FitzGerald proposed it independently (1889-1892) to explain the Michelson-Morley null result; in special relativity it follows from the two postulates [memory, high; Poincare's 1906 memoir credits "Lorentz and FitzGerald" by name, web: https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)].

The three effects (contraction, time dilation, offset of simultaneity) together make the Lorentz transformation, which in units where c = 1 Poincare wrote in his 1906 memoir as x' = k l (x + e t), t' = k l (t + e x), with e the speed and k = 1/sqrt(1 - e^2) (gamma) and l a scale factor he showed must be 1 [web: same URL]. The quantity x^2 + y^2 + z^2 - t^2 is the same for all inertial observers.

Electromagnetic mass. The field around a charge stores energy, so pushing a charge means also changing its field; this adds inertia. For a sphere of charge e and radius r the field mass is of order e^2/(4 pi eps0 r c^2); setting it equal to the electron's mass gives the "classical electron radius", 2.8 x 10^-15 m [memory, high]. Around 1900 Abraham, Lorentz and others hoped that all of the electron's mass was of this kind. Two problems followed. A purely electromagnetic electron gives a mass from its field momentum that is 4/3 of the mass from its field energy, and it would fly apart under its own repulsion; Poincare himself showed in 1906 that a non-electromagnetic pressure (now called "Poincare stress") is needed to hold it together [web: https://en.wikipedia.org/wiki/Electromagnetic_mass; Investigator note on the 1906 memoir]. Today the pure electromagnetic-mass hypothesis is rejected; field energy does contribute inertia (E = mc^2 for any energy), and most of the mass of a proton or neutron is the energy of the strong-force field inside it, not the rest mass of the smaller particles (quarks) it is built from [memory, high].

Gravity and Mercury. Newton's gravity acts instantly. Any law consistent with relativity must propagate at finite speed and depend on the bodies' velocities. Mercury's orbit shows an advance of its perihelion that the pulls of the other planets do not explain: Le Verrier in 1859 put this excess at about 38" per century, Newcomb in 1882 at about 43" per century [web: https://en.wikipedia.org/wiki/Tests_of_general_relativity]. General relativity (Einstein, 1915) gives an advance per orbit of

Delta phi = 6 pi G M / (c^2 a (1 - e^2)),

where G is Newton's gravitational constant, M the Sun's mass, a the orbit's semi-major axis (mean radius) and e its eccentricity. For Mercury (a = 5.79 x 10^10 m, e = 0.2056, 415 orbits per century) this is 42.98" per century [calc: B5]. Velocity-dependent Lorentz-covariant gravity laws of 1905-1908 (Poincare's, Minkowski's) give about 7" per century: de Sitter computed 7.15" in 1911, Wacker 7.2", and Lorentz quoted 6.69" in 1910 [web: https://en.wikisource.org/wiki/On_the_bearing_of_the_Principle_of_Relativity_on_Gravitational_Astronomy]. Lorentz's own earlier 1900 law gives only about 1.4" per century, and Lorentz said it "cannot account for the observed inequality" [web: https://en.wikisource.org/wiki/Considerations_on_Gravitation; calc: B5].

Gravitational radiation. In general relativity orbiting masses do radiate, but gravitational waves, not light. For the Earth and Sun the radiated power is about 196 W and the time to spiral in is about 10^23 years [calc: B6, from the standard circular-orbit formulas of Peters (1964), memory, high]. The effect has been observed in the shrinking orbit of the Hulse-Taylor binary pulsar and in the detections of merging black holes by LIGO (the Laser Interferometer Gravitational-Wave Observatory, a pair of 4 km Michelson-type interferometers in the United States) [memory, high].

4. Where Poincare departs

Poincare's departures are mostly not departures from the physics we now use. They are differences of interpretation (the ether) and of three specific hypotheses that later failed. Side by side:

The exact point of departure. The physics of the first three hypotheses is the physics of special relativity. The departure is one assumption Poincare keeps that Einstein dropped: that there is a medium, the ether, with a preferred rest frame and a "true" time, and that the relativity principle holds because nature compensates exactly (contraction of rulers, local time of clocks) for motion through it. This is the Lorentz ether theory. It is fair to Poincare to state clearly what the reviewers agreed on: the Lorentz ether theory makes the same predictions as special relativity for every measurement; it was not refuted by experiment, it lost on economy, because the ether frame by its own principle can never be found [Arbiter, claim 2; physicist; contrarian]. Poincare's own wording reflects this: he says observers cannot know true time, not that it does not exist [read: p.5].

His Lille version of the lecture gives his own worked example of this compensation. Comparing the annual aberration (the small yearly shift in a star's apparent direction caused by the Earth's motion) of two stars on opposite sides of the sky should, on the ether picture, reveal the Sun's absolute speed. Poincare says he is "absolument sur d'avance" (absolutely sure in advance) that no difference will be found, because the astronomers' graduated circles are themselves deformed so as to give "une parfaite compensation" [web: https://fr.wikisource.org/wiki/La_M%C3%A9canique_nouvelle_(Lille), via research/2026-10-06-fr-wikisource.md item 5]. This supports the point the mathematician and contrarian make, which the arbiter keeps: the lecture treats the principle as certain in advance, while his 1906 memoir calls the same postulate open to experimental test [Arbiter, claim 2 and QE revision]. No verdict changes: the principle is Established.

From that ether picture come the three hypotheses that did fail, and each rests on real physics:

1. "Only the ether is inert." The real physics: a moving charge's field stores energy and momentum, and this gives inertia (you can feel it as self-induction; Poincare's Lille version of the lecture makes exactly this comparison, "c'est ce que les electriciens appellent la self-induction" [web: https://fr.wikisource.org/wiki/La_M%C3%A9canique_nouvelle_(Lille)]). The step that failed: from "some inertia is electromagnetic" to "all of it is". The experiments he cites measure how inertia changes with speed, the shape of m(v), not what share of it is "constant", so they cannot show the constant part is zero [mathematician; Arbiter]. And his own 1906 memoir needed a non-electromagnetic stress, which sits badly with "only the ether is inert" [web: Investigator note on the memoir].

2. Gravity as an electrodynamic force. The real physics: a relativistic theory must make gravity propagate at finite speed and depend on velocities, and Poincare had shown in 1905-06 how to do this consistently with the Lorentz transformation [web: memoir, section 9]. In his Lille lecture he also answered Laplace's old argument that gravity must travel at least ten million times faster than light, pointing out that Laplace had assumed a particular aberration-like force: "Le resultat de Laplace ne prouve rien contre la theorie de Lorentz" [web: Lille]. That reply is correct in outline [Arbiter]. The step that failed: the specific laws give only a sixth of Mercury's anomaly, which Poincare reported honestly ("hardly decisive") [read: p.10].

3. Planets radiating into the Sun. The real physics: accelerated charges radiate (Larmor). In Lorentz's 1900 picture gravity is a leftover of slightly unequal electric attraction and repulsion between charges ("Franklin's hypothesis", in Poincare's Lille account) [web: Lille; Lorentz 1900], so in that picture accelerated planets should radiate something. The step that cannot be checked: no calculation is given for the time, and the picture behind it was abandoned [Arbiter, claim 5b]. Read literally, as radiation from a planet's ordinary electric charge, the claim fails by a wide margin: even a generous charge on the Earth gives an infall time near 5 x 10^34 years, not 10^17 (Section 5 table; example.py B7) [Arbiter, QE revision].

An earlier rescue of the same anomaly: Leverrier's planet. Just before reporting the new gravity's result, Poincare tells how the 19th century tried to explain Mercury: "Leverrier attributed this anomaly to a planet not yet discovered and an amateur astronomer thought he observed its passage across the sun. Since then no one else has seen it and it is unhappily certain that this planet perceived was only a bird" [read: p.10]. (The planet was later called Vulcan, and the amateur was Lescarbault, in 1859 [memory, high; not in the text].) It is worth setting beside the contraction hypothesis. Both were proposed to save an existing theory from one awkward observation: an unseen planet to save Newton's gravity, a contraction of matter to save the ether from the Michelson-Morley null. The planet made a prediction (it should be seen) and failed. The contraction survived, but only as part of a larger structure, the Lorentz transformation, that made new predictions of its own (time dilation, tested by Kennedy-Thorndike and Ives-Stilwell; Section 5). The lesson is not that saving hypotheses are always wrong, but that one earns its keep only when it predicts something beyond the observation it was made to explain [Arbiter, QE revision item 4].

Where the essay overstates its evidence (all three reviewers and the arbiter agree; the record is kept here because the reader should weigh it): - "Thus we have ascertained" (p.3): in April 1909 the beta-ray data were disputed. Kaufmann, whose experiments the essay cites, concluded from his own 1905-06 data that the Lorentz-Einstein formula was disproved [web: https://en.wikipedia.org/wiki/Kaufmann%E2%80%93Bucherer%E2%80%93Neumann_experiments]. Bucherer's 1908 measurements, at beta = 0.32 to 0.69, found e/m0 (charge over rest mass) constant at 1.695-1.706 x 10^7 (later corrected to 1.730) under Lorentz's formula while Abraham's formula drifted from 1.676 to 1.578, and Bucherer declared "the decision turns out in favor of the Lorentz-Einstein theory" [web: https://en.wikisource.org/wiki/Translation:Measurements_of_Becquerel_rays]. In the printed discussion Bestelmeyer said he was "not convinced" the sources of error played no role, estimating that about 10% of the rays passed at off-nominal speed and that the Lorentz-Abraham difference was "only a few tenths of a millimeter" [web: same]. Bucherer, the author of the confirming experiment, also listed what he saw as errors in Kaufmann's work; that is a competitor's characterization of a rival's data. Zahn and Spees argued in 1938 that the early experiments showed only a qualitative mass increase and could not decide between theories [web: Kaufmann-Bucherer-Neumann page]. Poincare's later Lille text names both Bucherer and Kaufmann [web: Lille]. - "Kaufmann's experiments show that the constant real mass ... may be considered as null" (p.8) uses Kaufmann's data for a conclusion Kaufmann himself did not draw about Lorentz's theory. - "Perfectly verified" (p.7): see Section 5. - Mercury "runs over about 100 kilometers ... per second" (p.3): the mean speed is 47.9 km/s [calc: B5]; harmless to his point. - "38"" and "32"" (p.10) carry no time unit. They are per century: 38" is Le Verrier's figure, and Poincare's Lille text says the new mechanics gives "un mouvement de 6''" leaving "une marge de 32''" [web: Lille]. With Newcomb's 43", the unexplained part would be about 36-37" [Arbiter]. - Einstein, Minkowski and FitzGerald are not named anywhere in the essay [read: deep read]; the 1906 memoir does credit Lorentz and FitzGerald [web]. Why is not shown by the record (see Section 7, "What the record does not say").

5. Worked example: the Earth's "deformation" and what Michelson's experiment can and cannot show

Poincare's one checkable number on relativity is: "The earth, borne along in its revolution through its orbit, is deformed about 1/200 000 000" [read: p.6; page image p.6 confirms the figure]. He then says Michelson's experiments show the "third hypothesis is perfectly verified" [read: p.7]. We check the number, see why that contraction is exactly what the Michelson experiment needs, and then see why the experiment cannot on its own "verify" it. All steps are reproduced in example.py, Part A.

Step 1. The number. Take the Earth's mean orbital speed v = 29.78 km/s [memory, high] and c = 299,792 km/s.

beta = v/c = 29.78 / 299,792 = 9.934 x 10^-5 beta^2 = 9.868 x 10^-9 fractional shortening = 1 - sqrt(1 - beta^2) ≈ beta^2/2 = 4.934 x 10^-9 = 1 / (2.027 x 10^8).

Poincare's 1/200,000,000 is right to the two digits he gave [calc: A1]. Applied to the Earth's diameter, 12,742 km x 4.934 x 10^-9 = 0.063 m: about 6.3 cm shortening along the direction of motion [calc].

A hidden assumption (pointed out by the mathematician, kept by the arbiter): this uses the orbital speed as the Earth's speed through the ether, as if the Sun were at rest in it. The Sun moves at about 370 km/s relative to the cosmic microwave background [memory, high]; with that speed the figure would be 7.6 x 10^-7, about 150 times larger [calc: A1]. By the principle Poincare himself asserts, neither number can be measured by an observer moving with the Earth: "the meter we could use would shorten like the length to be measured" [read: p.6].

Step 2. Light times in the two arms. In a Michelson interferometer a beam is split; one half goes along arm 1 (length L1, parallel to the motion through the ether), the other along arm 2 (length L2, perpendicular); both reflect back and recombine. Work in the ether frame, with the apparatus moving at speed v.

• Arm 1, outward the light gains on the receding mirror at relative speed c - v; on the way back it meets the beam splitter at c + v: t_par = L1/(c - v) + L1/(c + v) = 2 L1 c / (c^2 - v^2) = (2 L1 / c) gamma^2.
• Arm 2: the light must aim slightly forward so as to hit the moving mirror; its speed across the arm is sqrt(c^2 - v^2), so t_perp = 2 L2 / sqrt(c^2 - v^2) = (2 L2 / c) gamma.

Both results are checked symbolically in example.py [calc: A2].

Step 3. Equal arms, no contraction. Set L1 = L2 = L. Expanding for small beta (gamma^2 ≈ 1 + beta^2, gamma ≈ 1 + beta^2/2):

Delta t = t_par - t_perp ≈ (2L/c)(beta^2 - beta^2/2) = L beta^2 / c = L v^2 / c^3.

Rotating the instrument by 90 degrees swaps the arms, so the path difference c Delta t changes by 2 L beta^2. The expected fringe shift is

N = 2 L beta^2 / lambda.

With the 1887 Michelson-Morley values, an effective arm length L = 11 m (the light was folded back and forth by mirrors) and lambda = 500 nm [memory, high]:

N = 2 x 11 x 9.868 x 10^-9 / (5.00 x 10^-7) = 0.43 fringe [calc: A3].

Michelson and Morley reported a shift of at most about one twentieth of this, about 0.02 fringe [memory, medium: their 1887 paper; not web-checked]. The "null result".

Step 4. Contract the parallel arm. Now let arm 1 shorten to L / gamma (Poincare's deformation). Then

t_par = (2 L / (gamma c)) gamma^2 = (2L/c) gamma = t_perp, exactly.

The difference is zero to all orders in beta, not merely approximately [calc: A2, "exact cancellation"]. The contraction factor sqrt(1 - beta^2) is precisely what is needed, and its value at the Earth's orbital speed is the 1/200,000,000 of Step 1. So far Poincare is right: the deformation hypothesis and the null result fit each other perfectly.

Step 5. Why "perfectly verified" is too strong. The contraction was invented (by FitzGerald and Lorentz) to explain this null result, so the null result cannot also be an independent test of it. Other explanations of 1887 gave the same null: an ether dragged along by the Earth; emission theories in which light's speed adds to its source's; or no ether at all [physicist; mathematician]. And the contraction alone is not enough once the two arms differ in length. With L1 ≠ L2 and arm 1 contracted:

Delta t = 2 (L1 - L2) gamma / c,

which still depends on v. As the Earth's speed through a hypothetical ether changes over the year, the fringes should drift. Counted in fringes, the phase difference is nu x Delta t, where nu is the frequency of the light on the apparatus. If, in addition, moving clocks and the atoms that emit the light run slow by 1/gamma (time dilation, so nu = nu0 / gamma with nu0 the rest frequency), then

nu Delta t = (nu0 / gamma) x 2 (L1 - L2) gamma / c = 2 nu0 (L1 - L2) / c,

independent of v [calc: A4]. As an illustration only: with arms differing by 0.16 m (roughly the Kennedy-Thorndike value [memory, medium]), green light at 546 nm, and a speed through the ether swinging between 340 and 400 km/s over half a year, contraction alone predicts a seasonal drift of about 0.15 fringe; contraction plus time dilation predicts none [calc: A4]. Kennedy and Thorndike's 1932 experiment, with unequal arms, found no such drift, which length contraction alone cannot explain and contraction together with time dilation does [web: https://en.wikipedia.org/wiki/Kennedy%E2%80%93Thorndike_experiment]. Ives and Stilwell (1938) then measured time dilation directly [memory, high].

What the example teaches. Poincare's number is correct, and his deformation is exactly what Michelson's arrangement requires. What the experiment supports is the whole Lorentz transformation (contraction together with time dilation and local time), not a free-standing squashing of matter. That is why the arbiter splits the verdict: the arithmetic is correct and length contraction is established as part of the Lorentz transformation, while "Michelson's experiments verify the deformation" is, as stated, an overstatement [Arbiter, claim 3]. In fairness, Poincare's own lecture introduces local time and contraction together as the "bases of the new mechanics" [read: p.7]; it is the claim that one experiment "perfectly verified" one hypothesis that goes too far.

Other numbers in the essay, checked the same way (example.py Part B):

6. Evidence table

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

7. What a learner should take away

What is solid. - The speed limit, the growth of momentum as gamma m v, and the direction-dependent response to force (gamma^3 versus gamma) are standard physics, and Poincare's verbal argument and diagram describe them correctly. - The relativity principle and the impossibility of detecting absolute motion: established to extraordinary precision. - His numbers on relativity are right: 1/200,000,000 for the Earth, and the clock and velocity-addition examples reconstruct correctly. - His closing point is a lesson in itself: a new theory contains the old one as a limit, and you cannot understand the new one without the old. - His honesty about Mercury: he reported that the new gravity got only the direction right and left 32", and called the result "hardly decisive". That was confirmed by three independent calculations within two years. - Two side remarks that later became standard physics (none of the reviewers judged them; the arbiter's QE revision records them as plausible pictures in 1909 and established in their modern forms [memory, high; not web-checked]): - X-rays from cathode-ray impact. In a Crookes tube the cathode rays (electrons) "violently strike the anticathode", giving "an electromagnetic agitation which many physicists identify with the Rontgen rays" [read: p.9]. Today: X-rays are produced when fast electrons are slowed sharply in a target ("bremsstrahlung", German for braking radiation), the Larmor effect of Section 2 at large deceleration. - Electrons in a hot body. Lorentz's picture of an incandescent solid: free electrons fly about inside it, "a swarm of gnats enclosed in a jar and striking with their wings against the walls of their prison", and each impact emits a wave [read: p.8-9]. Today: the free-electron (Drude-Lorentz) model of metals, the first step toward the modern theory of conduction. - A historical note: the essay describes the neutral atom as "a positive central mass around which circulate a great number of negatively charged electrons", a "central nucleus" [read: p.7], two years before Rutherford's scattering experiments established the nuclear atom in 1911 [memory, high]. In 1909 it was one model among several ("Some represent ..."); it was not yet tested.

What is wrong, and why the mistake is instructive. - Treating one null result as verification of one hypothesis. The Michelson-Morley null was the reason the contraction was proposed; it cannot then confirm it independently. Section 5 shows that contraction alone fails as soon as the arms are unequal. Lesson: ask what else would give the same result, and what experiment would separate them. - Reading a model into data. The beta-ray experiments measured how inertia changes with speed. They could not say how much of it was "constant real mass". Lesson: a fit to a formula tells you about the formula's shape, not about the hidden parts of the model behind it. - Overstating the state of evidence. In 1909 the electron-mass data were disputed by the very experimenters he relied on. Lesson: popular lectures often round up the evidence; check the original papers and their printed discussions. - All mass as field energy. An attractive unifying idea that failed for the electron (the 4/3 problem, the need for Poincare's own stress) but left a true residue: field energy does carry inertia, and most of the mass of ordinary matter is strong-field energy. - Small factual slips: Mercury's speed doubled; a perihelion figure without its unit; the older 38" instead of Newcomb's 43".

What is genuinely open, or not settled by the record. - Ether versus no ether is not an open experimental question in the usual sense: the Lorentz ether theory and special relativity predict the same measurements, so only a detected preferred-frame effect (a "Lorentz violation") could separate them; none has been found to about 1 part in 10^17 [web: Tests of special relativity]. Whether you call the ether "refuted" or "redundant" matters to how you read Poincare; the reviewers agree it is redundant, not refuted. - Does mass as attraction grow with speed, as mass as inertia does? Poincare himself recorded this as open. At Lille he distinguished "la masse, coefficient d'inertie" (mass as resistance to acceleration) from "la masse, coefficient d'attraction" (mass as the source and receiver of gravity), and said whether the second grows with velocity "est la une question que nous n'avons aucun moyen de decider" (is a question we have no means of deciding) [web: https://fr.wikisource.org/wiki/La_M%C3%A9canique_nouvelle_(Lille), via research/2026-10-06-fr-wikisource.md item 6]. This is the question that later became the equivalence principle (inertial mass equals gravitational mass), on which Einstein built general relativity [memory, medium]. The arbiter's verdict for the 1909 question: Never properly tested at the time; later tested by Eotvos-type torsion-balance experiments [memory, high; not web-checked] [Arbiter, QE revision item 1]. It is a caution in Poincare's favour: he did not claim more than he knew here. - Related work: Weber's law. Lorentz's 1900 paper says that Wilhelm Weber's velocity-dependent force law (a 19th-century law of electric force that depends on the charges' relative speed) can account "to some extent" for Mercury's perihelion motion, where his own law cannot [web: https://en.wikisource.org/wiki/Considerations_on_Gravitation, via research/2026-10-06-en-wikisource.md section 4]. Weber-type gravity laws belong to the same family as the laws in this essay and were set aside by mainstream physics. The arbiter keeps it as a lead; it has not yet been researched in this review [Arbiter, QE revision item 2; sites.md]. - How good were the 1900s beta-ray data? A modern historical reanalysis exists ("Hazy Spots on Photographic Plates", Historical Studies in the Natural Sciences, 2023) but was not opened in this review [sites.md]. - What Poincare actually computed for planetary radiation: not in the Monist text; perhaps in the Lille lecture or the 1908 Science et methode chapter "La mecanique nouvelle et l'astronomie" [Arbiter]. - Which gravity law the essay's "Lorentz's law" corresponds to: Lorentz's 1900 law (about 1.4") or a Lorentz-invariant law of Poincare's or Minkowski's (6-7"). Poincare proposed two laws; S. Walter's study is the lead [sites.md; Arbiter].

What the record does not say (absence of evidence, as analysed by the arbiter; nothing here points to suppression, but the gaps are real): - The "6"" and the Bucherer passage appear in Poincare's Lille lecture (3 August 1909) but not in the Monist translation of the Gottingen lecture. Competing explanations: the Gottingen French never had them (Poincare added them later; but Bucherer's paper predates April 1909); Halsted or the Monist editor condensed the text; the Monist's French source differed; Poincare left them out for a general audience (yet he kept the 38" and 32"). The deciding record is the French text in Sechs Vortrage (1910), pp. 49-58, not yet consulted [Arbiter]. - Einstein, Minkowski and FitzGerald are not named. Possible readings: Poincare habitually credited the new mechanics to Lorentz's school [memory, medium; not checked]; priority was not the lecture's subject; or a deliberate omission. The same pattern appears in the Lille text. No record of motive exists; the omission is stated and left to the reader [Arbiter]. Credit was in any case spread wide: Voigt had written down a form of the transformation in 1887, and the rival electron models were Abraham's rigid, Lorentz's deformable and Langevin's constant-volume electron (Minkowski called the rigid electron "a monster") [web via the Investigator, sites.md Leads item 3; Arbiter, QE revision item 6]. This widens the possible readings without deciding between them. - No letters to Halsted, Kaufmann or Bucherer appear in the Poincare Papers correspondence calendar (letters from Hilbert arranging the Gottingen week, and Poincare's May 1905 letters to Lorentz, "Mes resultats confirment pleinement les votres", are there) [web: https://henripoincarepapers.univ-nantes.fr/chp/]. Halsted translated after Poincare's death, so no letter need exist; the calendar may be incomplete. Silence here says nothing about the claims. - A Halsted offprint titled "The New Mechanics" at Truman State University (Greenwood Papers) is catalogued "approximately 1902", which conflicts with 1913. It may be a cataloguer's estimate, a different Halsted translation with the same title, or a mistitle; the offprint's printed volume and page line would decide [web: https://manuscripts.truman.edu/repositories/2/archival_objects/16255]. - The 1905-1911 relativistic-gravity programme goes quiet after 1911. This one has a documented explanation: general relativity explained the full anomaly in 1915, and Minkowski (1909) and Poincare (1912) had died [Arbiter]. - Later use of the text. A modern geocentrist book cites the essay's remark that "reactionary French journals have made me prove that the sun turns around the earth" [read: p.4] in support of geocentrism [web: archive.org snippet via Investigator]. That is the book's characterization. Poincare rejected that reading himself at Lille: "je n'ai jamais eu une telle pensee ; c'est bien pour la verite que Galilee combattait" [web: Lille].

8. Check yourself

1. Using beta^2/2, what fractional contraction would Poincare's hypothesis give for a body moving at 300 km/s? How does it compare with his figure for the Earth?
2. In a Michelson interferometer with equal arms, the parallel arm is contracted by sqrt(1 - beta^2). Show that the two light times become equal. Then explain in one or two sentences why this does not, by itself, prove that matter contracts.
3. A body at rest receives equal impulses, the first bringing it to 0.6c. What speed does it have after the second? (Hint: p = gamma m v.)
4. Poincare says the new mechanics leaves "a margin of 32"" for Mercury out of 38". If the true anomaly is 43" per century and the Lorentz-invariant law gives 7.15", what fraction is explained, and how large is the margin?
5. Kaufmann's experiments are cited for "the constant real mass of the electron is negligible". What did the experiments actually measure, and why can that not settle the question?

9. Further reading

Poincare's own works - "The New Mechanics" (Gottingen, 28 April 1909), tr. G. B. Halsted, The Monist 23 (1913) 385-395. The project document; also on Wikisource [web: https://en.wikisource.org/wiki/The_New_Mechanics_(G%C3%B6ttingen)]. French original in Sechs Vortrage uber ausgewahlte Gegenstande aus der reinen Mathematik und mathematischen Physik (1910), pp. 49-58. - "La Mecanique nouvelle" (Lille, at the congress of the AFAS, the Association francaise pour l'avancement des sciences, 3 August 1909): the longer twin of the lecture, with the 6", Bucherer, inertial versus gravitational mass and Laplace [web: https://fr.wikisource.org/wiki/La_M%C3%A9canique_nouvelle_(Lille)]. - "Sur la dynamique de l'electron", Rendiconti del Circolo matematico di Palermo 21 (1906) 129-176: the mathematics behind the essay (Lorentz group, invariant interval, Poincare stress, relativistic gravity) [web: https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)]. - "The Measure of Time" (1898), on clock synchronisation by signals; Science and Hypothesis (1902), The Value of Science (1905) and Science and Method (1908, book III "The New Mechanics") [web: Wikisource author page, via Investigator; memory, high on the books].

Contemporary sources named in this primer - H. A. Lorentz, "Considerations on Gravitation" (1900) [web: https://en.wikisource.org/wiki/Considerations_on_Gravitation]. - A. H. Bucherer, "Measurements of Becquerel rays" (1908), with the printed discussion [web: https://en.wikisource.org/wiki/Translation:Measurements_of_Becquerel_rays]. - W. de Sitter, "On the bearing of the Principle of Relativity on Gravitational Astronomy", Monthly Notices of the Royal Astronomical Society (MNRAS) 71 (1911) 388-415 [web: https://en.wikisource.org/wiki/On_the_bearing_of_the_Principle_of_Relativity_on_Gravitational_Astronomy].

Mainstream textbooks and histories [memory: high on titles and authors; chapters named only where certain] - E. F. Taylor and J. A. Wheeler, Spacetime Physics: special relativity with algebra only, ideal for this reader. - D. J. Griffiths, Introduction to Electrodynamics, chapter 12 "Electrodynamics and Relativity" (3rd and 4th editions). - R. P. Feynman, The Feynman Lectures on Physics, vol. I chapters 15-16 (special relativity; relativistic energy and momentum) and vol. II chapter 28 "Electromagnetic Mass" (the 4/3 problem and Poincare stress, explained for undergraduates). - A. I. Miller, Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1981): detailed history including Kaufmann, Bucherer and Poincare. - A. Pais, Subtle is the Lord (1982): the scientific biography of Einstein, with chapters on Lorentz and Poincare. - C. M. Will, Was Einstein Right? (popular) and Theory and Experiment in Gravitational Physics (technical): the Mercury test and its history. - S. Walter, "The Four-Dimensional Movement in Gravitation, 1905-1910" (Springer): Poincare's two gravity laws [web: lead in sites.md, not opened].

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Answers

1. beta = 300 / 299,792 = 1.0 x 10^-3, so beta^2/2 = 5.0 x 10^-7, one part in 2 million. That is about 100 times Poincare's Earth figure ((300/29.78)^2 = 101), because the effect grows as v^2.

2. With L1 contracted to L/gamma: t_par = (2 (L/gamma) / c) gamma^2 = 2 L gamma / c, and t_perp = 2 L gamma / c; they are equal. It does not prove contraction because the contraction was introduced to produce exactly this result, other hypotheses (ether drag, emission theory, no ether) also give a null, and with unequal arms contraction alone still predicts a speed-dependent signal that is absent unless time dilation is added (Kennedy-Thorndike).

3. At 0.6c, gamma = 1.25, so p1/(mc) = 1.25 x 0.6 = 0.75. After the second impulse p/(mc) = 1.5, so v/c = 1.5 / sqrt(1 + 2.25) = 1.5 / 1.803 = 0.832. Newton would give 1.2c.

4. 7.15 / 43 = 0.166, about one sixth; the margin is about 36" per century. (With Poincare's numbers, 6 / 38 = 0.158 and the margin 32".) General relativity's 42.98" closes it.

5. They measured the deflection of beta rays in electric and magnetic fields, which gives the charge-to-mass ratio as a function of speed, i.e. the shape of m(v). In Lorentz's formula m(v) = gamma m0, the rest mass m0 is carried through whole; splitting it into a "constant real" part and an "ether" part is a model choice that the shape of the curve cannot decide. Kaufmann himself concluded in 1906 that his data disproved the Lorentz-Einstein formula, so the essay uses his data for a conclusion he did not draw.