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Mathematician review: Poincare, "The New Mechanics" (Step 2)

Reviewer: PhD mathematician. Basis: the full essay text [read: p.1-11 of _text/TheNewMechanics-7c45ca.txt], deep/TheNewMechanics-7c45ca.md, facts.json, claims.md, and the Investigator's sites.md, research notes (en/fr Wikisource, Google, Henri Poincare Papers, Truman). research/files/ is empty. All arithmetic below was done by me (Python) from the stated inputs. Tags: [read] document; [web] Investigator's URL, not re-fetched by me; [memory] my recall, confidence given. I did not run separate web checks; every [memory] fact below is high confidence unless marked, and no verdict turns on a medium one.

Key fact about the document: it has NO equations (one line diagram only) [read: p.1-11]. "Checking the mathematics" therefore means (i) checking its numbers, (ii) reconstructing the mathematics it asserts in words, (iii) checking it against Poincare's own formal papers found by the Investigator.

Corrections to Step 1

Date and identity (facts.json "about 1910", [memory, medium]): wrong. The text is Halsted's translation of the Gottingen Wolfskehl lecture of 28 April 1909, printed in The Monist 23(3), July 1913, pp. 385-395, after Poincare's death [web: https://en.wikisource.org/wiki/The_New_Mechanics_(G%C3%B6ttingen)]. Accept the Investigator's correction. The Lille lecture of 3 Aug 1909 is a longer sibling text, not this one [web: fr.wikisource La_Mécanique_nouvelle_(Lille)].
Step 1 never checked the essay's numbers. One is wrong: "Mercury ... also runs over about 100 kilometers ... per second" [read: p.3]. Mercury's mean orbital speed is 47.9 km/s (59 km/s at perihelion) [memory, high]; check: 2pia/T = 2pi5.791e10 m / (87.969 d) = 47.9 km/s. The error is a factor of about 2 and does not affect any argument (his point is that Mercury is still far below c). Step 1 repeated the figure without comment in the deep read (section 1: "about 100 kilometers ... per second").
Unit of the Mercury figures. Step 1 correctly noted that "38"" and "32"" carry no time unit in the text. The Investigator shows these are per century (Lille text says 38 - 6 = 32; de Sitter 7".15 per century, Wacker 7".2, Lorentz 6".69) [web]. Accept, but note the reading is inferred from other sources, not from this document. My own check agrees: the anomaly cannot be per year (that would be 3800"/century, about 70% of Mercury's whole observed precession of roughly 5600"/century [memory, high]) and cannot be per orbit.
claims.md item 5 bundles two independent claims (Mercury perihelion; planets spiral into the sun). I split them below (5a, 5b). Claim 4 also bundles "mass is velocity-dependent" (shared with claim 1) with "real mass is null"; I judge only the second under claim 4.
claims.md item 3 says "Michelson's experiments are said to verify this." Correct as a description. The logical status of "verified" is the issue (see claim 3).
facts.json "mainstream_conflict" says general relativity "later accounted for the Mercury excess". Correct (my calculation: 6piGM/(c^2 a (1-e^2)) per orbit gives 42.98"/century for Mercury [memory, high: GR formula; inputs standard]). Keep.

Claim 1. Mass grows with velocity; the speed of light cannot be exceeded; radium-ray experiments show it

a. Mathematics. The essay's argument is a verbal one: equal impulses give ever smaller velocity gains. Reconstructed: impulse = change in momentum p = gammamv, gamma = 1/sqrt(1-v^2/c^2). Then v = p c / sqrt(m^2 c^2 + p^2), which tends to c but never reaches it. Worked example: first impulse gives v = 0.5c (p1 = 0.577 mc). Equal second and third impulses give p = 1.155 mc and 1.732 mc, hence v = 0.756c and 0.866c. The gains are 0.5c, 0.256c, 0.110c: shrinking exactly as the p.2 diagram shows (segments n'1n'2, n'2n'3 getting shorter) [read: page image p.2]. So the picture is internally consistent. Two notes: (i) the limit c is introduced as an assumption ("we assume it to be impossible") [read: p.2], so the argument "mass increases because c is a limit" is a postulate-driven one, not a derivation; (ii) "mass" here is the resistance to acceleration, which is not one number: later [read: p.8] he says mass depends on the angle between velocity and force. With the standard results, longitudinal mass = gamma^3 m and transverse mass = gamma m. At v = 1e5 km/s (the radium rays, beta = 0.334) gamma = 1.061 and gamma^3 = 1.194, so the two differ by 12%. The essay mentions this in one clause and never quantifies it; the Lille text gives it more room [web, Investigator]. Numbers in the passage: 100,000 km/s is one-third of 300,000 km/s: correct.

b. Experiments. Consistent with beta-ray deflection experiments (Kaufmann 1901-06, Bucherer 1908) [web: https://en.wikisource.org/wiki/Translation:Measurements_of_Becquerel_rays]. Bucherer at beta = 0.32-0.69 finds e/m0 constant (1.695-1.706 x 10^7 in his units) under the Lorentz formula while the Abraham formula drifts (1.676 to 1.578) [web]. Later accelerator work (for example Bertozzi 1964, and all modern accelerators) [memory, high] makes this one of the best-tested results in physics. Dissent kept on record: Bestelmeyer in the 1908 discussion doubted that the sources of error were negligible (about 10% of rays off nominal speed; Lorentz-Abraham difference only a few tenths of a millimetre); Kaufmann himself in 1906 concluded relativity was disproved [web, same page]. The essay cites no experimenter at this point [read: p.3].

c. Tested? Yes, repeatedly: Kaufmann, Bucherer, Neumann (1914), Guye-Lavanchy (1916), later high-precision work [memory, high on the names; details medium and not needed]. Replicated: yes.

d. If true: Newtonian F = ma with constant m must be replaced. That change was made.

Verdict: Established (velocity limit and velocity-dependent inertia, in the Lorentz-Einstein form, with gamma and gamma^3). Wording caveat: modern usage keeps invariant mass and puts the velocity dependence into momentum; "mass increases" is the 1909 idiom.

Claim 2. The relativity principle has absolute value; observers carried through the ether know only "apparent" (local) time

a. Mathematics. Two worked examples in words; both reconstruct correctly. - Paris-Berlin synchronisation [read: p.4-5]. Distance about 880 km [memory, medium; I use it only as an illustration], Earth's orbital speed 29.78 km/s [memory, high]. If the apparent one-way light times are taken as equal when they are not, the first-order error is v d cos(theta)/c^2 = 29.78e3 * 880e3 / c^2 = 2.9e-7 s (0.29 microsecond) at worst. The essay's required accuracy is "the thousand-millionth of a second" (1e-9 s) [read: p.4]. So the effect he says is undetectable is about 290 times larger than his own stated precision: the example is numerically sound. Round-trip light times differ only at order v^2/c^2, which is why a first-order ether-drift signal cannot be seen by crossing signals [reconstruction; standard]. - Two observers at 200,000 km/s each way [read: p.5-6]. He says A would naively see 400,000 km/s, "a velocity he cannot attain", and the clock offsets rescue the principle. The correct composition is u = (v1+v2)/(1+v1 v2/c^2) = 4e5/(1+4/9) = 276,923 km/s, below c (taking c = 300,000 km/s as in the essay). The essay never gives this number and does not show that the corrected speed stays below c for all inputs, but the qualitative claim is right. One imprecision: "B's watch went too fast" folds time dilation and the synchronisation offset into one phrase. - Hidden assumption: that observers are moving "with reference to the ether" at all. The mathematics (Lorentz transformation) is the same whether or not the ether exists; the ether adds a preferred frame that, by the principle itself, no experiment can identify. So "true time" (in the ether frame) and "apparent time" differ by an unobservable quantity. The essay frames the difficulty as one of knowing the true time [read: p.5], not of its non-existence. - Tension with Poincare's own formal paper: the essay says the principle has "an absolute value" and "admits of no restriction" [read: p.4]; the 1905/6 Palermo memoir calls it a "Postulate of Relativity" admitted "without restriction" but one that "may be either confirmed or disproved later by more precise experiments" [web: https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)]. The popular essay is stronger than the technical memoir. Not a mathematical contradiction, a difference in claimed status.

b. Experiments. The principle is consistent with Michelson-Morley (1887), Kennedy-Thorndike (1932), Trouton-Noble (1903), Ives-Stilwell (1938), and modern cavity and clock tests [memory, high]. All give null results for absolute motion. The essay does not name these ("delicate experiments ... which I shall not describe") [read: p.4].

c. Tested? Yes, many times, by many groups, with ever tighter nulls. Note the Investigator's observation that Poincare's own technical paper treated it as open to test; it has since been tested and held.

d. If true: the ether loses any role (true: standard relativity discards it). Newtonian absolute time must go. The Lorentz-Poincare version (ether plus unobservable true time) and Einstein's version (no ether) make identical predictions for every measurable quantity, so the difference between them is a matter of interpretation.

Verdict, split. (i) Principle of relativity as a null result for detection of absolute motion: Established. (ii) The additional claim that there is an ether frame with a "true" time that observers cannot know: Not testable as stated (no measurable difference from the version without it), and historically superseded but instructive.

Claim 3. A moving body is deformed along its motion (Earth about 1/200,000,000); Michelson's experiments verify this

a. Mathematics. The number is correct. Contraction factor sqrt(1 - beta^2) is about 1 - beta^2/2. For the Earth's orbital speed, beta = 29.78 km/s / c = 9.93e-5, so beta^2/2 = 4.93e-9 = 1/2.03e8. The page image shows "1/200 000 000" [read: page image p.6]; Poincare rounded 1/2.03e8 to 1/2e8. In length: 12,742 km x 4.93e-9 = 6.3 cm shortening of the Earth's diameter along its direction of motion. Hidden assumption: that the Earth's speed relative to the ether is its orbital speed, i.e. that the Sun is at rest in the ether. The Sun moves at about 370 km/s relative to the cosmic microwave background [memory, high]; if that were the speed through the ether the number would be 7.6e-7, about 150 times larger. The essay does not say this, and the principle of relativity he asserts makes the number unmeasurable anyway. A second point: the Earth is accelerating in orbit (not inertial), so the formula is an approximation; the effect of this is negligible at this order.

Why the contraction is needed: in Michelson's interferometer the arm parallel to the motion has round-trip time (2L/c)/(1 - beta^2); the perpendicular arm (2L/c)/sqrt(1 - beta^2). Shortening the parallel arm by sqrt(1 - beta^2) makes the parallel time (2L/c)/sqrt(1 - beta^2), equal to the perpendicular one: the null result is restored exactly at second order. This is correct as stated. The expected shift without it for L = 11 m, lambda = 500 nm, v = 29.78 km/s is 2 L beta^2/lambda = 0.43 fringe [memory: the 11 m effective arm and 500 nm are standard Michelson-Morley values, high], versus the observed shift of under about 0.02 fringe [memory, high].

Logical status. "The third hypothesis is perfectly verified" [read: p.7] overstates what the experiment shows. A null result is explained by the contraction but also by no ether, by an ether dragged along, and (at that time) by emission theories; and the contraction alone is not enough. Kennedy-Thorndike (1932) tests the speed-dependence of interferometer arms of unequal length and needs a time-dilation factor as well [memory, high]. The essay presents contraction as the third of three ad hoc hypotheses (relativity postulate, apparent time, contraction) [read: p.6-7]; in the 1906 memoir it is a consequence of the Lorentz group and is named "Lorentz contraction" credited to Lorentz and FitzGerald [web, Investigator]. The essay names nobody for it [read: p.6].

b. Experiments. Michelson-Morley 1887 and later repetitions (Michelson-Morley 1887; Miller's contested positive claims 1920s; Kennedy-Thorndike 1932; modern optical-resonator nulls) [memory, high]. Consistent.

c. Tested? The null result has been repeated many times. The contraction as a physically distinct effect cannot be measured separately by a comoving ruler (the essay itself says so: "the meter we could use would shorten like the length to be measured" [read: p.6]); in standard relativity the length contraction is measured only as a relation between frames, and is tested indirectly through the Lorentz transformation as a whole.

d. If true: Euclidean rigid bodies do not exist; in the Lorentz-Poincare reading the molecular forces must themselves transform as the electromagnetic ones do (Poincare 1906 [web]); in Einstein's reading it is a kinematic effect of simultaneity. No change beyond what standard relativity already made.

Verdict, split. (i) Length contraction as part of the Lorentz transformation: Established (indirectly, through the whole of relativity). (ii) The arithmetic 1/200,000,000: correct. (iii) "Michelson verified the contraction": overstated; the experiment is a null result compatible with it, Tested in the weak sense only. (iv) The reading of contraction as a real deformation of matter moving through an ether: Historically superseded but instructive.

Claim 4. Matter's constant mass vanishes: electron inertia is the ether's inertia; Kaufmann shows real mass is null

a. Mathematics. The essay gives no formula. Reconstruction: the electromagnetic (field) mass of a charged sphere of radius r is of order m_em = e^2/(4 pi eps0 r c^2) (up to a numerical factor of 2/3 to 4/3 depending on the model). Setting m_em equal to the electron mass gives the classical electron radius, 2.8e-15 m [memory, high]. This is why Abraham's rigid electron and Lorentz's deformable electron were attractive: the whole mass could be field energy.

Problems that a mathematician flags: - The 4/3 problem. For a spherical shell the field momentum gives m = (4/3) U/c^2 while the field energy U gives U/c^2, so the field mass is inconsistent with E = mc^2 unless something else adds a stress. Poincare's own 1906 memoir finds that a non-electromagnetic "constant external pressure" is needed to hold the deformable electron together [web, Investigator, en.wikisource Translation:On_the_Dynamics_of_the_Electron_(July)]. That is incompatible with the essay's "Only the ether, and no longer matter, is inert" [read: p.8]: a non-electromagnetic force means not all of the electron's dynamics is the ether's. The essay omits this. (Cross-document tension, not an internal contradiction of the essay alone; the essay does not mention the stress.) - Wrong inference from the data. What Kaufmann- and Bucherer-type experiments measure is the shape of m(v), not the share of m that is "constant". In the Lorentz formula m(v) = gamma m0 the whole of m0 is carried through unchanged; the separation into "constant real mass" and "ether inertia" is a model, not a measurement. The essay's "Kaufmann's experiments show that the constant real mass ... may be considered as null" [read: p.8] is therefore a statement about Kaufmann's (Abraham-type) fit of 1902-03, not a result the later Lorentz-Einstein formula supports [memory, medium-high on Kaufmann's early interpretation; web: Bucherer says Kaufmann concluded relativity was disproved by his 1906 data]. Poincare cites Kaufmann for one conclusion and not for his rejection of Lorentz's theory. Both are on record; keep both. - Modern values: for an electron the classical field mass at any known size is not the whole story; the electron is point-like to current limits, and most of the mass of a proton or neutron is the energy of the strong-interaction field, not of its quarks' rest masses [memory, high]. So "all mass is field energy" is true in spirit for nucleons and not for the electron's mass.

b. Experiments. The velocity dependence agrees with experiment (claim 1). "Constant mass is null" is not demonstrated by them (above). "Gaps in the ether" has no observable consequence as stated.

c. Tested? The velocity dependence: yes. The EM-only mass hypothesis: tested indirectly and abandoned (Abraham's rigid electron was disfavoured by Bucherer 1908 and by the success of relativity [web: Bucherer; memory high]; the ether was dropped).

d. If true: no particle would have intrinsic mass; mass would be field energy. This is a different programme from standard relativity (m0 is a rest-frame energy, E = mc^2, any origin). In modern physics part of that idea survives (mass from field energy in nucleons; Higgs mechanism for others) but with no ether.

Verdict, split. (i) Velocity-dependent electron inertia: Established (see claim 1). (ii) "Constant mass is null; only the ether is inert; no matter, only gaps": Historically superseded but instructive; the experiment cited does not discriminate the claim, and Poincare's own memoir needs a non-electromagnetic stress.

Claim 5a. Lorentz's velocity-dependent attraction law improves Mercury's perihelion in the right direction; 38" observed, 32" left unexplained

a. Mathematics. No formula is given [read: p.9-10]. Independent calculations in the Investigator's notes [web: de Sitter, MNRAS 71, 1911, https://en.wikisource.org/wiki/On_the_bearing_of_the_Principle_of_Relativity_on_Gravitational_Astronomy] give +7".15 per century for Mercury under the Poincare/Minkowski law (Wacker 7".2; Lorentz 1910, 6".69 "a first approximation"), and smaller values for Venus (1".43) and Earth (0".639). This supports: - the essay's direction ("the sense of the error") and size (the Lille text says 6", leaving 32" [web]); - his statement that the difference grows with planetary speed and is largest for Mercury [read: p.10]: the effect scales like (v/c)^2 times the orbital frequency, so Mercury has the largest per-century value; de Sitter's table agrees; - the arithmetic 38 - 6 = 32 (not printed in the Monist text, printed in Lille). Caution on attribution: the essay says the law is Lorentz's, and that "Lorentz" gets the right sense. Lorentz's own 1900 law gives only a few arcseconds for all the elements, which the Investigator computes as about 1.4"/century for the perihelion (162e-10 per revolution x 415 revolutions x 2.06e5 arcsec/rad: I reproduced 1.39") and which Lorentz says "cannot account for the observed inequality" [web: https://en.wikisource.org/wiki/Considerations_on_Gravitation]. The 6-7" figure belongs to the later Lorentz-invariant (Poincare/Minkowski) laws. So "Lorentz's law" is not one fixed number; the essay's 6" is not from Lorentz 1900. Also the essay says "velocity of Mercury" where perihelion motion is meant [read: p.10].

Comparison with general relativity. My calculation from GM = 1.327e20 m^3/s^2, a = 5.791e10 m, e = 0.2056, T = 87.97 days gives 42.98"/century (6 pi GM/(c^2 a (1-e^2)) per orbit) [memory, high on the formula]. The relativistic velocity-dependent gravity of 1909-11 gives 7.15", which is 1/6.01 of that. The observed excess is therefore about 43" [Newcomb 1882, memory, high] rather than Le Verrier's 38" [memory, medium: the Le Verrier figure is 38"/century in 1859; Newcomb's later value is 43"; I did not re-check the web, no verdict depends on this].

The essay's phrase "results coincide ... to within the approximation of the measurements" for other planets [read: p.10] is plausible for Jupiter and beyond but I do not have the 1909 observational uncertainties for Venus and Earth (1".4 and 0".64 per century) and say unknown.

b. Experiments. Mercury's anomalous precession is a real observation. The Lorentz-type law accounts for about a sixth of it. The remainder was explained by general relativity (1915) and then confirmed by later radar and ephemeris work [memory, high].

c. Tested? The calculation was independently done by de Sitter, Wacker and Lorentz [web]; it was not "tested" as a gravity theory against alternatives except through Mercury and the Moon, where it failed to give the full effect. Poincare says so himself: "hardly decisive in favor of the new mechanics" [read: p.10].

d. If true: gravity would be a Lorentz-invariant field with finite propagation speed, which general relativity later became in another form. The essay is honest about the shortfall.

Verdict: Historically superseded but instructive. The calculation is sound (about 6-7"/century, right direction, largest for Mercury); the law fails to explain Mercury as a whole, as the essay itself says; general relativity supplies the rest. Part (i) "the direction is right": Established (calculation reproduced in independent sources). Part (ii) "this law is the explanation of Mercury": Tested, not supported.

Claim 5b. Accelerated celestial bodies radiate and the planets fall into the Sun in "some millions of milliards of centuries"

a. Mathematics. "Millions of milliards of centuries" = 1e6 x 1e9 x 100 yr = 1e17 yr [reading of the French idiom milliard = 1e9]. No derivation is shown in the essay [read: p.9]. The Lille text says the planets' accelerations are about a billion times smaller than at a Crookes tube's anticathode [web, Investigator]; I cannot check that. There is a conflict of mechanism: planets are electrically neutral to high accuracy [memory, high]. In the electromagnetic picture the effect only exists if gravity is derived from unequal electric forces (Lorentz 1900 / "Franklin's hypothesis" [web, Lille]), and then dipole radiation vanishes if charge-to-mass ratios are equal, leaving a weaker multipole term.

My comparison figure: in general relativity, quadrupole gravitational radiation gives an inspiral time T = (5/256) c^5 a^4 / (G^3 m1 m2 (m1+m2)). For Mercury and Sun: 4.3e22 yr = 4.3e20 centuries; for Earth and Sun: 1.1e23 yr = 1.1e21 centuries (my calculation, standard formula [memory, high]). Poincare's figure of 1e15 centuries is smaller by a factor of about a million; but both are astronomically long, so his conclusion ("this prospect can hardly frighten us") stands under either. Tidal and relativistic effects aside, orbital decay of planets by radiation has not been observed.

b. Experiments. The existence of radiation from accelerated mass is confirmed in the gravitational case (Hulse-Taylor pulsar decay, LIGO, [memory, high]). Radiation from accelerated charges (antennas, incandescence, Crookes tube) is established.

c. Tested? The planetary figure was not tested and cannot be.

d. If true: nothing new in mainstream physics beyond gravitational radiation, which general relativity predicts with a different mechanism and much smaller rate.

Verdict: Plausible, untested for the stated mechanism and number (no calculation shown; the number is not reproduced by the one mechanism I can compute); the qualitative claim "accelerated masses radiate" is Established only in the general-relativistic form, which the essay does not have.

Claim 6. Classical mechanics is not finally condemned; it remains the mechanics of small velocities; the new one cannot be understood without it

a. Mathematics. Correct and checkable. The relative correction is gamma - 1 = beta^2/2: car at 100 km/h: beta = 9.3e-8, gamma - 1 = 4.4e-15; Mercury at 47.9 km/s: 1.3e-8; Earth at 29.8 km/s: 4.9e-9; radium ray at 1e5 km/s: 0.061 (6% for gamma, 19% for gamma^3). The limit v << c recovers Newton's laws exactly (correspondence principle). His own statement [read: p.8] that for stationary or quasi-stationary motions "the new mechanics does not differ ... from the Newtonian" is the same point.

b. Experiments. Every engineering and astronomical use of Newtonian mechanics at low speeds [memory, high]. One exception worth recording: precision GPS needs relativistic corrections at about 1e-10 relative level [memory, high].

c. Tested? Constantly.

d. If true: nothing changes; this is the standard view. The "pedagogic danger" he predicts [read: p.10-11] is a prediction about teachers, not a physics claim; I do not judge it.

Verdict: Established.

Investigator's findings I kept (whatever the verdict)

Exact identity and date (Gottingen 1909, Monist July 1913) and the French original in Sechs Vortrage (1910) pp. 49-58 [web, JSTOR snippet]. The question whether the "6"" and Bucherer passage were in the Gottingen French or cut in translation is open (volume not yet checked).
Lille text gives 6" and 32" outright, names Bucherer and Kaufmann, separates inertial and gravitational mass ("aucun moyen de decider" whether the attraction coefficient depends on velocity) and answers Laplace's speed-of-gravity bound [web: fr.wikisource]. Poincare also rejected the geocentric reading of his relativity remark at Lille ("je n'ai jamais eu une telle pensee") [web], relevant to the later geocentrist use of the Monist text [web, archive.org snippet].
Palermo memoir (1906): names the Lorentz contraction, needs a non-electromagnetic stress, proposes gravity at light speed "mainly compensated" so as to agree with astronomy [web].
de Sitter 7".15, Wacker 7".2, Lorentz 6".69 / 1".4 (1900 law) [web].
Bucherer 1908 results and the printed dissent (Bestelmeyer; Kaufmann's own conclusion; Minkowski's "monster"; Voigt's 1887 transformation) [web].
Truman State: Halsted offprint titled "The New Mechanics", catalogued "approximately 1902", conflicts with 1913; unresolved [web: https://manuscripts.truman.edu/repositories/2/archival_objects/16255]. Does not affect any mathematics here.
Not reached by the Investigator (time limit) and bearing on my claims: Sechs Vortrage text; Wacker dissertation; Lorentz Phys. Z. 11 (1910); the 2023 study of Kaufmann/Bucherer data quality ("Hazy Spots on Photographic Plates"); S. Walter on Poincare's two gravity laws. Absence of these is a gap in my checks, not evidence either way.

Summary of verdicts

Mass-velocity law and light-speed limit: Established.
Relativity principle: Established (null detection of absolute motion); ether plus "true time": Not testable as stated / superseded.
Contraction 1/200,000,000: arithmetic correct; Lorentz contraction Established as part of relativity; "Michelson verifies" overstated (null result only); real deformation in an ether superseded.
Constant mass null / only ether inert: Historically superseded but instructive (the data cited do not discriminate it; memoir needs a non-EM stress). 5a. Mercury: calculation sound (about 6-7" per century); fails to explain the full anomaly (Tested, not supported as the explanation); superseded by GR. 5b. Planets spiral into the Sun: Plausible, untested as stated; magnitude unsupported; GR figure about 1e6 times longer.
Classical mechanics remains valid at low speed: Established.