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Builder-G: Mathematician's review

Reviewer role: PhD mathematician (derivations, notation, units, hidden assumptions). Document: G. Builder, "Ether and Relativity", Aust. J. Phys. 11, 279-297 (1958) [read: page images pp.2-17 myself, pp.1, 18-19 not re-read; Step 1 read them].

Source gap (repeated from Step 1): the PDF has no text layer. Equations were read from page images and I re-derived the ones that matter (6b, 7a, 7b, the light-clock footnote) by hand. Dingle's 1957 Nature letter and Builder's 1957a/b papers are not in the folder; statements about them rest on Builder's paraphrase or the web notes below.

Symbols and background (for the reader)

S, S': inertial frames (frames in uniform straight-line motion, no rotation). S0: Builder's postulated "absolute" inertial frame, at rest in the ether (the universe as a whole).
c: speed of light. u, v: speeds of two clocks A and B measured in S. u0, v0: the same speeds measured in S0. beta = v/c.
Proper time of a clock: the time it records along its own path, tau = integral of sqrt(1 - v^2/c^2) dt (Builder's eq 1-3).
Lorentz covariance: a law is Lorentz covariant if it keeps the same form in every inertial frame. This is what the "principle of relativity" demands. It does not demand that the numerical values of forces or energies be the same in every frame.

The mathematics in the paper, checked

Eq (1)-(3), clock retardation [read p.2]. dt_a/dt = sqrt(1-u^2/c^2) etc. is correct; the retardation t_a - t_b is a difference of two proper times between two shared events, so it is invariant. Correct.
Eq (4)-(5) [read p.4]. This is eq (1)-(3) with S replaced by S0. It is a relabelling, valid because S0 is "one of the inertial frames". Correct but vacuous as physics, since nothing in (4)-(5) can be evaluated (he concedes u0, v0 are unmeasurable).
Eq (6a), (6b) [read: page image p.6]. I checked (6b) by hand: the field of a uniformly moving magnetic pole, at the same S-instant, is B0[1 + beta^2/2 - (3/2)(v.r)^2/(r^2 c^2)] to second order, from the standard Lienard-Wiechert-type closed form B = g(1-beta^2) r / (r^2 (1-beta^2 sin^2 theta)^(3/2)) [memory: high]. The induced E on a charge at rest then gives exactly Builder's bracket. Correct to O(beta^2). (6a) is exact.
Eq (7a), (7b) [read: page image p.7]. (7a): same expansion for a moving point charge's Coulomb field at present position, (1-beta^2)/(1-beta^2 sin^2 theta)^(3/2) = 1 + beta^2/2 - (3/2)(v.r^)^2/c^2 + O(beta^4). Matches. (7b): the source is at rest so there is no magnetic force; pure Coulomb. Correct. I did not independently expand the cross term in the general eq (7) and (8); they have the right structure (a Coulomb factor times a magnetic-type term (1/c^2) v1 x (v2 x r^)) and are plausible to O(beta^2), but unverified. Tag: eq (7), (8) [read, structure only].
Footnote p.14, light clock along the motion. Round trip time in the ether frame is L/(c-v) + L/(c+v) = 2Lc/(c^2-v^2). With L = L0 sqrt(1-beta^2) this is 2L0/(c sqrt(1-beta^2)), so the clock rate drops by sqrt(1-beta^2). Correct for this clock orientation.

So there are no arithmetic or sign errors that I found. The paper's problems are logical, not computational.

Hidden assumptions that carry the argument

A1 (p.3): "physical explanation must be independent of the frame, so the clock must interact with something universal". The step from "the formula takes the same form in every frame" to "some single frame is physically causal" is a non sequitur. Invariance of form is exactly what one expects of a law with no preferred frame.
A2 (p.3): "each clock is isolated, so it interacts with the universe". Retardation depends on the path in spacetime, a geometric (Minkowski metric) fact. The proper time integral is the length of a curve, like the length of a road, and a road's length needs no "interaction with the universe". Builder treats a geometric quantity as a dynamical one.
A3 (pp.9-10): acceleration is absolute, hence there is a unique absolute inertial frame, hence absolute velocity. Mathematically, the Newtonian (Galilean) and Minkowski spacetimes both single out the class of inertial frames (a class of straight-line paths), but neither picks one member of that class. The group of boosts is a symmetry, so absolute acceleration is defined without absolute velocity. "Absolute acceleration implies absolute velocity" is false as a mathematical statement (Newton's own corollary V of the Principia says uniform velocity is undetectable [memory: high]). This is the key flaw.
A4 (pp.5-8): that force values in S being different for physically different configurations shows dependence on an absolute velocity. See claim 3.
A5 (p.16 p.17): Section VI assumes a clock slows by sqrt(1-v0^2/c^2) due to motion through the ether, a postulate built to reproduce SR, then calls the match to SR a "causal explanation".

Claims

Claim 1: SR clock retardation depends on individual speeds, so it demands a causal role for velocity relative to an absolute frame

a. Mathematics: the formulas are fine. The inference is not: (3) is a function of u, v measured in S, but those are coordinates in an arbitrary frame; the invariant is the proper-time difference, a function of the worldlines only (A2). Whether the "individual speeds" matter is frame choice, not physics. The same would be true of the kinetic energy m v^2/2 in Newtonian mechanics, which depends on individual speeds in a frame but gives no absolute velocity. Note also that Builder's step "retardation cannot be a function of relative velocity" is true only because two clocks that meet twice must have been accelerated, so a relative-velocity formula is not defined between meetings.
b. Experiments: the formula (3) itself is confirmed: Ives-Stilwell 1938 (Doppler shift of moving hydrogen ions), the CERN muon storage ring (Bailey et al. 1977), Hafele-Keating 1971 (flying atomic clocks: -59 +/- 10 ns eastward, +273 +/- 7 ns westward vs predicted -40 +/- 23 and +275 +/- 21) [web: https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment ; https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation]. These confirm the prediction, not the "causal role" interpretation.
c. Tested: the formula yes, many times, replicated. The causal-absolute-velocity interpretation makes no different prediction, so cannot be tested against SR. Builder concedes u0, v0 cannot be measured (p.4).
d. If true: nothing in the maths changes; mainstream would have to add an undetectable preferred frame (an interpretation change, a new ontology).
Verdict: split. Formula: Established. "Demands absolute velocity as cause": Not testable as stated (and the argument for the demand is a non sequitur, see A1, A2).

a. Internally consistent: yes. This is the Lorentz-Poincare theory; its Lorentz-transformed predictions equal SR's by construction (p.4, p.13: Einstein's derivation is "formally identical" with a stationary frame taken to be the ether frame). The math is just a relabelling of one frame.
b. Experiments: any experiment that SR passes, this passes, by construction. The unmeasurability of absolute velocity is itself the content of null results (Michelson-Morley; Kennedy-Thorndike 1932 requires length contraction and time dilation both [web: https://en.wikipedia.org/wiki/Kennedy%E2%80%93Thorndike_experiment]).
c. Tested: no test can distinguish; so "not tested" in the sense that no controlled test can exist. Lorentz-violation searches (modern, Robertson-Mansouri-Sexl frameworks) bound deviations; none found [memory: medium, not individually checked].
d. If true: nothing observable changes. A preferred frame would have to be added, and then explained why every dynamical law conspires to hide it. Occam's razor, not mathematics, is the mainstream objection.
Verdict: Plausible, untested in the weak form ("compatible", which is a true mathematical equivalence) and Not testable as stated for "sufficient and needed" claims, since no measurable difference exists. Builder's "tenability beyond question" (p.14) is true only in this weak sense.

Claim 3: relativistic electrodynamics depends on individual (absolute) velocities; Newton's third law fails

a. The formulas (6b), (7a), (7b) are correct to O(v^2/c^2) (see above). The conclusion is a logical error. The two cases (a), (b) are different configurations in S: in (b) the "same instantaneous positions" in S are not simultaneous in the rest frame of either particle, so "the same relative velocity" does not mean the same configuration. The force in S is not required to be invariant; only the form of the Maxwell-Lorentz equations plus the transformation law of force are. If you Lorentz-boost case (b) to the frame of the charge you recover the (a)-type description with a different separation r (length contraction) and a different simultaneity, and the numbers reconcile with (6b). So no absolute velocity is detected: any inertial observer computes the same physical result for the same physical situation. Moreover the asymmetry would be seen by every inertial observer, as Builder himself says (p.8), so it cannot pick out one frame.
Newton's third law: F1 and F2 at simultaneous S-instants are not equal and opposite because momentum is shared with the field, and the "force pair at equal time" is not a Lorentz-invariant notion. Total momentum (particles + field) is conserved. This is standard (e.g. Griffiths Ch. 8 on field momentum) [memory: high]. Builder attributes it to retardation of potentials, which is right to the extent that it is the finite speed of propagation.
b. Experiments: nothing in the equations conflicts with experiment, since they are standard electrodynamics. The claim that this shows ether dependence is not an experimental result.
c. Tested: the equations are standard and have been tested for a century, never as a test of absolute velocity.
d. If true: SR would be wrong in claiming no detectable preferred frame. It is not true: this reasoning would equally show "absolute velocity" in Newtonian mechanics (kinetic energy and momentum depend on individual velocities in a frame; Galilean relativity still holds).
Verdict: formulas Established (as O(v^2/c^2) approximations); inference to absolute velocity Internally inconsistent (it claims the relativity principle is respected while treating frame-dependent values as frame-independent facts; fails by its own definition of the principle on p.12, which asks only for same form of laws).

Claim 4: absolute acceleration implies absolute inertial system and absolute velocity; reject Mach; absolute velocity maybe kinematically measurable (starlight Doppler asymmetry)

a. Inconsistent in the middle step (A3). Inertial frames form a class related by boosts; absolute acceleration (deviation from that class) requires the class, not a member. "Absolute velocity = velocity relative to the universe" is then a matter of definition (p.11: relative to the universe or fixed stars), and this is just a relative velocity to a chosen reference matter distribution. By that definition he has no conflict with SR (the CMB dipole gives such a velocity, roughly 370 km/s for the Sun [memory: high; not re-checked here]), but then the "absolute" system is a frame chosen by the matter distribution, not an ether with causal power over clocks. Builder switches between the two meanings. On Doppler asymmetry: starlight Doppler measures motion relative to the stars, a relative velocity. On Mach: his rejection rests on "instantaneous action", but Mach's principle in Einstein's/Lense-Thirring/Barbour-Bertotti forms does not require instantaneous action; the argument rejects a strawman (p.15).
b. Experiments: no experiment is contradicted; none supports the "rejection of Mach". Gravitomagnetic frame dragging (Gravity Probe B, LAGEOS) is mainstream evidence that rotating matter influences inertial frames [memory: medium; not checked here and not needed for the verdict].
c. Tested: Doppler measurements exist, but they give a frame relative to matter, which is mainstream.
d. If true: SR with a preferred frame; the mainstream would have to accept non-Machian absolute space plus an ether.
Verdict: split. "Absolute acceleration is detectable, defines inertial frames": Established. "Hence a unique absolute velocity": Internally inconsistent (non sequitur A3, equivocation on "absolute"). "Mach's hypothesis must be rejected": Historically superseded but instructive (argument tied to action at a distance; the debate continued in GR). "Absolute velocity as velocity relative to the universe's matter": Plausible, untested in the sense that cosmological frames exist.

Claim 5: clock slowing by sqrt(1-v0^2/c^2) follows from FitzGerald-Lorentz contraction, one hypothesis not two for Michelson-Morley; ether presentation is pedagogically superior

a. The longitudinal light-clock derivation is correct (verified above). But the claim "one hypothesis" fails to cover the general case. (i) A light clock aligned across the motion is slowed with no contraction (round trip 2L0/(c sqrt(1-beta^2))), so the slowing of a transverse clock is the same factor but a separate calculation; the two orientations agree only if the contraction takes exactly the value he cites. So the slowing is consistent with contraction, rather than "following from it" for every clock. (ii) For clocks not based on light and electromagnetic bonds (nuclear, gravitational) Lorentz covariance of all forces must be assumed (Poincare's extension), which is an additional hypothesis. (iii) The Kennedy-Thorndike experiment (1932) is the experiment that needs both contraction and dilation as separate ingredients [web: Wikipedia link above]. So Broad's "two hypotheses" criticism is correct for the family of experiments as a whole, Builder's only for Michelson-Morley taken alone.
On "pedagogically superior": opinion. Not testable. The ether version needs the extra unobservable frame.
b. Experiments: Michelson-Morley, Kennedy-Thorndike, Ives-Stilwell and the muon, Hafele-Keating results above agree with the formula.
c. Tested: the formula yes, widely replicated. The "causal mechanism" no.
d. If true: nothing observable changes.
Verdict: split. Formula and Lorentz-FitzGerald-Larmor derivation for a longitudinal electromagnetic clock: Established (as a kinematic result). "One hypothesis suffices in general / Broad is wrong": Refuted by calculation only in the sense that Kennedy-Thorndike needs both factors (not Michelson-Morley alone); I would grade the general statement Internally inconsistent only if read literally, so better: "overstated". "Ether presentation superior": Not testable as stated. As a whole the paper is Historically superseded but instructive: a careful 1958 statement of the Lorentz-Poincare view and the reasons it was not adopted.

Summary of the mathematical assessment

Algebra is sound; the formulas are standard SR/electrodynamics and the approximations check out to O(beta^2).
The weak statement (an ether frame is consistent with the Lorentz transformations) is true and well known. The strong statement (SR demands it) fails at three places: confusing geometric invariants with causes (A2), identifying form-invariance with a hidden cause (A1), and passing from absolute acceleration to absolute velocity (A3).
The paper contains no data and no numerical example.

Corrections to Step 1

facts.json says "Whether (6b), (7), (8) are correct was not checked". I checked (6b), (7a) and (7b) and they are correct to O(v^2/c^2). (7) and (8) in general form were not re-derived here.
facts.json "mainstream_conflict" calls the asymmetries "an observable asymmetry" in Builder's claim. Builder claims they are observable phenomena in a frame, not a way to measure absolute velocity; the real error is not the formulas but the inference.
facts.json gives Dingle's letter as Nature 179:866; the web gives pp.865-866 [web: https://www.nature.com/articles/1791071a0 ; search result]. Minor.
facts.json has 7 key_claims, claims.md 5; consistent after dedup (Mach rejection merged into claim 4; "pedagogy" into claim 5). Fine.
claims.md claim 5 says the slowing is "a consequence of Lorentz-FitzGerald contraction". Builder says this only for the longitudinal light clock footnote and for electromagnetically bound bodies, so this is narrower than stated.
claims.md "all 19 pages read": I re-read pp.2-17 only.