Builder-G primer: "Ether and Relativity" (1958)

Document studied: G. Builder, "Ether and Relativity", Australian Journal of Physics 11, 279-297 (1958; manuscript received 20 November 1957). The paper gives no first name. It is the only document in the folder.

Source gap (carried from earlier steps): the PDF has no text layer. Every page was read from rendered images, and the equations were read by eye [read: page images pp.1-19]. Two things are missing and were not read by anyone in this review: H. Dingle's 1957 Nature letter that Builder is answering ("The 'Clock Paradox' of Relativity", Nature 179, 865-866), and Builder's two earlier papers (Aust. J. Phys. 10:246 and 10:424). Everything said here about Dingle's argument comes from Builder's paraphrase or from a web summary, and the two do not agree (see Section 7).

Page numbers below are PDF pages. Journal page = PDF page + 278.

Companion script: example.py in this folder reproduces every number in Section 5. Run it with .venv/bin/python review/Builder-G/example.py.

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1. The idea in one paragraph

In 1957 the astronomer Herbert Dingle argued that the "twin" or clock effect of special relativity shows that motion is absolute, so relativity must be wrong (this is how Builder reports him). Builder, a physicist at the University of Sydney, agreed with Dingle's starting point and rejected his conclusion. He argued that relativity itself forces us to give each clock's speed relative to the universe as a whole (an "ether") a real causal role, and that this does no harm, because no mechanical or electrical experiment can measure that speed. He backs this up three ways. First, the formula for how much a clock falls behind uses each clock's own speed. Second, the forces between moving charges depend on each charge's own velocity, not only on their relative velocity. Third, acceleration can be detected, so there must be a "true" standard of rest. He ends by saying the older ether version of relativity, due to Lorentz and Poincare, is the better way to teach it. The paper has no experiment and no data, and it predicts exactly what standard relativity predicts. Builder says so himself [read: pp.1-4, 15-18].

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2. What you need to know first

Inertial frames and the Galilean transformation. An inertial frame is a coordinate system in which a free body moves in a straight line at constant speed. In Newtonian mechanics two inertial frames moving at relative velocity V are related by x' = x - Vt, t' = t. A boost (a change to a moving frame) changes every velocity by V and leaves every acceleration unchanged. That last fact is the whole answer to one of Builder's arguments (Claim 4). Read: any classical mechanics text, e.g. Taylor, Classical Mechanics (chapter on Newton's laws and frames).

Special relativity (SR): postulates and Lorentz transformation. SR rests on two postulates: the laws of physics take the same form in every inertial frame, and light travels at c in every inertial frame. From these come the Lorentz transformation, time dilation, length contraction and the relativity of simultaneity (events simultaneous in one frame are not simultaneous in another). Read: Taylor and Wheeler, Spacetime Physics; Griffiths, Introduction to Electrodynamics, chapter 12 ("Electrodynamics and Relativity") [memory: high].

Proper time. The time a clock itself records along its path. It is the spacetime analogue of the length of a curve, and every observer computes the same value for it. Read: Taylor and Wheeler, Spacetime Physics.

Fields of a moving charge. A charge moving at constant velocity has an electric field that is no longer spherically symmetric. It is squashed toward the plane perpendicular to the motion, and the charge also makes a magnetic field. Read: Griffiths, chapters 10 and 12; Purcell and Morin, Electricity and Magnetism, the chapter on fields of moving charges.

Momentum in the electromagnetic field. Fields carry momentum. For two moving charges, the forces on them need not be equal and opposite (Newton's third law fails for the particles alone), but total momentum, particles plus field, is conserved. Read: Griffiths, Section 8.2 [memory: high].

Michelson-Morley and Kennedy-Thorndike experiments. Michelson-Morley (1887) compared light round-trip times along two perpendicular arms and found no dependence on orientation. Kennedy-Thorndike (1932) used arms of unequal length and looked for changes as Earth's velocity changes over the year, and found none. Together they test length contraction and time dilation [web: https://en.wikipedia.org/wiki/Kennedy%E2%80%93Thorndike_experiment]. Read: any modern-physics text, e.g. Serway, Moses and Moyer, Modern Physics, the relativity chapter.

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3. The textbook view

3.1 Clock retardation as proper time

Take an inertial frame S. A clock moves with speed u(t) as measured in S, where t is S's time coordinate and c = 2.998 x 10^8 m/s is the speed of light. The clock's own elapsed time (its proper time, written tau) is

d tau / dt = sqrt(1 - u^2/c^2),        tau = integral from t1 to t2 of sqrt(1 - u(t)^2/c^2) dt.

This is Builder's equation (1)-(2) [read: page image p.2]. For two clocks A and B that meet at S-times t1 and t2, the difference tau_A - tau_B is his equation (3). The textbook point is that tau is the length of the clock's worldline (its path through spacetime), computed with the spacetime interval

c^2 d tau^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2.

Every inertial observer gets the same tau, the same way every surveyor gets the same length for a road whatever grid they use. The speed u appears only because we chose to describe the path in frame S. Pick another frame and the speeds change, but the integral does not.

Example: clock B leaves clock A, travels at 0.6c, and comes back, taking 10 years of A's time. Then gamma = 1/sqrt(1 - 0.36) = 1.25, and B records 10/1.25 = 8 years. A records 10. The 2-year difference is agreed by everyone (example.py Part 5). B is the one that turned around, which an accelerometer on B detects. Clocks flown around the world (Hafele-Keating, 1971) and muons in the CERN storage ring confirmed this kind of effect. All three reviewers cite these results [web, via reviewers; see physicist.md].

3.2 Force between two moving charges

Write k = 1/(4 pi epsilon_0) = 8.988 x 10^9 N m^2/C^2 in SI units. Builder uses Gaussian units, in which k = 1. Let charge q2 move at constant velocity v, and let R be the vector from q2's present position to the field point. Let beta = v/c and theta = angle between v and R. The exact electric field is

E = k q2 (1 - beta^2) R / [ R^3 (1 - beta^2 sin^2 theta)^(3/2) ].

(Griffiths derives this from the Lienard-Wiechert potentials [memory: high].) Two special directions are worth memorising:

• side-on (theta = 90 degrees): E = gamma k q2 / R^2, stronger than Coulomb by gamma;
• head-on (theta = 0): E = k q2 / (gamma^2 R^2), weaker by 1/gamma^2.

Here gamma = 1/sqrt(1 - beta^2). The moving charge also makes a magnetic field B = (v x E)/c^2 in SI units, and a charge q1 moving at v1 feels F = q1 (E + v1 x B).

3.3 How forces transform between frames

Suppose a particle is momentarily at rest in frame S' and feels force F' there. In a frame S where the particle moves at speed v, the force components are

F_parallel (S) = F'_parallel,        F_perp (S) = F'_perp / gamma,

where "parallel" and "perpendicular" are relative to v. (This follows from F = dp/dt and the Lorentz transformation of momentum and time; Griffiths, ch. 12 [memory: high].) A force is not the same in every frame. Asking for the force "in frame S" is like asking for a velocity "in frame S": the number depends on the frame, and the physics does not.

3.4 Lorentz ether theory (LET)

Before Einstein, Lorentz, Larmor and Poincare built a theory with a real ether at rest in one frame. Bodies moving through it at speed v0 shrink along the motion by sqrt(1 - v0^2/c^2) (the Lorentz-FitzGerald contraction), and clocks moving through it slow by the same factor. With exact factors, LET makes the same predictions as SR for every experiment. The ether frame cannot be found, because every measuring device is affected in a compensating way [web: https://en.wikipedia.org/wiki/Lorentz_ether_theory]. The mainstream dropped LET because the ether does no work: it adds an unobservable object, and the exact compensation has to be assumed rather than explained. LET was never refuted by experiment. John Bell later argued that teaching in a "Lorentzian" style has real value [web: same URL].

3.5 The cosmic rest frame

The cosmic microwave background (CMB) is the leftover radiation from the early universe. It looks slightly hotter in one direction and slightly cooler in the opposite one (the "dipole"). From this the Sun's velocity relative to the CMB is 369.82 +/- 0.11 km/s [web: https://en.wikipedia.org/wiki/Cosmic_microwave_background]. This is a velocity relative to the matter and radiation of the universe. It is a perfectly good reference frame. No law of physics has been found to depend on it (Section 6).

3.6 One-way light speed and the synchronisation convention

How does an ether theory, in which light moves at c only in the ether frame, end up with every frame measuring c? Builder answers it himself. He writes that "true one-way measurements of light velocity are impossible (so that we can measure only the average value of the light velocity over go-and-return paths)" [read: OCR p.16]. In his conclusion he calls this "the only other important factor that has to be taken into account" besides motion through the ether [read: OCR p.18].

Here is why that matters. To time light over a one-way path from A to B you need two clocks, one at A and one at B, and they must already be synchronised. The usual way to synchronise them is to send a light signal from A to B and back and assume it took equal times each way. That assumption is the Einstein synchronisation convention (Reichenbach called it a convention, because no experiment can check it independently) [memory: high; not web-checked]. Only the round-trip speed can be measured without such an assumption.

Worked numbers (c = 1, beta = 0.6, gamma = 1.25). A rod of rest length L0 = 1 light-second moves through the ether, so it is contracted to 0.8. In ether time, light takes 0.8/(1 - 0.6) = 2.0 s to go from A to B and 0.8/(1 + 0.6) = 0.5 s to come back: 2.5 s in all. The rod's clocks run slow by 0.8, so A's clock records 2.5 x 0.8 = 2.0 s for the round trip, and the round-trip speed is 2 L0 / 2.0 s = 1 = c. The moving observer, using Einstein's convention, sets B's clock so that the outward trip took 1.0 s. In ether terms it really took 2.0 x 0.8 = 1.6 s of A-clock time. So B's clock is set 0.6 s behind the ether's idea of "the same moment". That 0.6 s is exactly v L0 / c^2, the term in the Lorentz transformation t' = gamma (t - v x / c^2) that makes simultaneity relative.

So, in Builder's account: contraction plus slow clocks make every round trip come out at c, and the convention that one-way speeds are equal makes every frame call the one-way speed c too. The price is that each frame's "simultaneous" differs from every other frame's. That is the relativity of simultaneity, and in LET it is an artefact of synchronising clocks with light that does not really move at c relative to the rod. Builder's "only other important factor" therefore amounts to adopting a synchronisation convention. The ether frame keeps its own "true" simultaneity, but no experiment can tell it apart from any other frame's, because every measurement you could make can be described equally well with Einstein-synchronised clocks in any frame. The arbiter rates this point Established (it is standard physics) and Builder's claim that the ether presentation is easier to teach Not testable as stated (an opinion) [arbiter.md, Claim 6 and QE revision item 2].

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4. Where Builder departs

Builder's mathematics is standard throughout. All three reviewers checked his equations (6b), (7a) and (7b), and two checked (7) and (8) as well. They are correct to order v^2/c^2 [read: page images pp.6-7; checked in physicist.md, contrarian.md, arbiter.md]. He departs in what he infers from them.

Builder's light clock, checked (example.py Part 5). The philosopher C. D. Broad (Scientific Thought, 1923) had objected that the ether theory needs "two independent hypotheses", length contraction and clock-rate reduction, to explain the Michelson-Morley null result and the measured constancy of the speed of light. Builder's p.16 footnote is aimed at Broad: it says the contraction "entails the clock-rate reduction, and the latter is not an independent hypothesis" [read: OCR p.16; Broad's initials and title from Builder's reference list, p.19]. A mirror pair with rest separation L0 moves at beta = 0.6 through the ether, with c = 1. If the arm points along the motion and is contracted to 0.8 L0, the round trip takes 0.8/(1 - 0.6) + 0.8/(1 + 0.6) = 2.5 L0, against 2 L0 at rest. The rate ratio is 2/2.5 = 0.8 = 1/gamma, so Builder is right for this geometry. Without contraction the same arm would take 3.125 L0. But an arm across the motion is not contracted, and its round trip is 2/sqrt(1 - 0.36) = 2.5 L0 too. So the slowing does not come from contraction in general. Contraction is needed to make the two orientations agree (Michelson-Morley). Kennedy-Thorndike then shows that the atomic clocks used as references must slow by the same gamma. That is a second, independent requirement, unless you assume in advance that every clock and every binding force obeys the Lorentz factor, which is the relativity postulate itself [arbiter.md, Claim 5]. So on the evidence of Kennedy-Thorndike, Broad's "two hypotheses" is the fairer count for the family of experiments as a whole.

Builder's appeal to Einstein (1920) and Mach (1883). To show the ether is respectable, Builder quotes Einstein's 1920 Leiden lecture (published in English in Sidelights on Relativity): "More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny the existence of an ether ... According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time" [read: OCR pp.14-15]. The quote is accurate as Builder gives it, and it does show that Einstein after 1920 was not hostile to the word "ether". What it does not show is support for a Lorentz ether. Einstein's ether is the gravitational field of general relativity: space with physical properties, but with no state of motion, so nothing can move relative to it and it cannot supply an absolute velocity. Builder himself quotes Einstein drawing the contrast: "What is fundamentally new in the ether of the general theory of relativity as opposed to the ether of Lorentz consists in this, that the state of the former is at every place determined by connections with the matter and the state of the ether in neighbouring places ... whereas the state of the Lorentzian ether in the absence of electromagnetic fields is conditioned by nothing outside itself, and is everywhere the same" [read: OCR p.15]. Builder's own footnote adds that Einstein used "absolute" of space only to mean that its properties are not affected by anything [read: OCR p.13]. So the 1920 quote is evidence about Einstein's vocabulary, not evidence for the frame Builder needs [arbiter.md, QE revision items 3 and 4; the comment on what the quote implies is the arbiter's reading, memory: medium]. Builder also points out that Mach (1883) had discussed a medium filling all space and acting on bodies by contact, held it "admissible", recognised that it "would be sufficient", and preferred his own distant-matter hypothesis only as more "expedient provisionally" [read: OCR p.13]. That shows the ether was an option Mach left open; it does not close the field version of Mach's principle that Builder sets aside (table row "Mach" above).

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5. Worked example: "same relative velocity, different force"

This is Builder's most concrete argument (eq. 7a vs 7b, p.7), and it can be settled with a pencil. The setup below follows his.

Builder's two cases, both described in frame S (q1 and q2 are charges, r is the separation vector from q2 to q1 at one S-instant, and r-hat is the unit vector along it):

• Case (a): q1 at rest, q2 moving with velocity v. He gives (eq. 7a) F1 = (q1 q2 / r^2) [1 - (3/2)(v . r-hat)^2/c^2 + (1/2) v^2/c^2] r-hat.
• Case (b): q1 moving with velocity -v, q2 at rest. He gives (eq. 7b) F1 = (q1 q2 / r^2) r-hat, pure Coulomb.

In both, the velocity of q2 relative to q1 is v. The forces differ, so Builder concludes the force "cannot be expressed as a function of the relative velocity" and depends on individual (absolute) velocities [read: page image p.7].

Step 1: real numbers

Take two electrons (q = e = 1.602 x 10^-19 C) side by side, so the separation is perpendicular to v (theta = 90 degrees, v . r-hat = 0). Let r = 1 nm and beta = 0.6, so gamma = 1.25.

Plain Coulomb force:

F0 = k e^2 / r^2 = (8.988e9)(1.602e-19)^2 / (1e-9)^2 = 2.307 x 10^-10 N.

• Case (a), exact: side-on field enhanced by gamma (Sec. 3.2), so F1 = 1.25 x F0 = 2.884 x 10^-10 N.
• Case (a), Builder's (7a): [1 + (1/2)(0.36)] F0 = 1.18 F0 = 2.722 x 10^-10 N. This is the order-beta^2 approximation to gamma. At beta = 0.6 it is 5.6% low. At beta = 0.1 it gives 1.005000 against the exact 1.005038, so his formula is fine for small speeds as claimed. example.py Part 2 shows symbolically that his bracket is exactly the beta^2 expansion of the exact field.
• Case (b), Builder's (7b): F1 = F0 = 2.307 x 10^-10 N.

So far Builder is right. In frame S, the two cases give different forces on q1.

Step 2: look at case (b) from q1's own rest frame

Call q1's rest frame S'. In case (b), q1 moves at -v in S, so in S' charge q1 is at rest and q2 moves at +v. The separation is perpendicular to the motion, and perpendicular distances do not change under a boost, so it is still 1 nm. That is exactly case (a). In S', q1 feels the case-(a) force, gamma x F0 = 2.884 x 10^-10 N.

Step 3: transform that force back to S

In S, q1 moves at speed v. By Sec. 3.3 a perpendicular force is divided by gamma:

F1 (S) = 2.884e-10 / 1.25 = 2.307 x 10^-10 N = F0.

This is exactly Builder's (7b).

Step 4: what this shows

Cases (a) and (b) are one physical situation described from two frames. "q2 moves past a resting q1" and "q1 moves past a resting q2" are the same event pair, seen by two observers. The force on q1 is gamma F0 in the frame where q1 is at rest and F0 in the frame where it moves, for the same reason a particle's momentum or energy differs between frames. Nothing in the calculation refers to an absolute frame. The difference comes entirely from the standard transformation rules.

The general angle (example.py Part 1, done symbolically in SymPy for any angle and any beta < 1, and checked numerically at random points). When the separation has a component along v, a second effect appears. Builder specifies "the same instantaneous positions" in S, but "instantaneous" means simultaneous in S, and simultaneity is relative. In q1's rest frame the separation along v becomes x0/gamma, where x0 is that component in S. With that corrected separation, the exact moving-charge field and the force transformation give back (7b) exactly. The difference between the boosted result and (7b) simplifies to zero. Builder's sentence "the instantaneous positions are the same in the two cases" (p.6) quietly assumes that simultaneity is absolute, which is the very thing at issue. Section 3.6 shows why that assumption cannot be checked: with one-way light speed unmeasurable, "the same instant" in two places depends on how you synchronise the clocks, and Builder's own account (p.16) concedes as much.

Step 5: Newton's third law (eq. 8), with the same numbers

In case (a), q1 is at rest, so it makes only an electrostatic field, and the force on q2 is plain Coulomb, F0 = 2.307 x 10^-10 N. The force on q1 is gamma F0 = 2.884 x 10^-10 N. They are not equal and opposite, and the mismatch is 5.77 x 10^-11 N (example.py Part 4). Builder is right that the third law fails for the two particles. The textbook response is that the electromagnetic field carries momentum, and the total of particles plus field is conserved (Griffiths Sec. 8.2 [memory: high]). This is a property of fields with a finite propagation speed. It needs no ether.

Step 6: the magnet and the conductor (eqs. 6a, 6b), and Einstein's 1905 example

Einstein opened his 1905 paper with a magnet and a conducting loop [memory: high; Builder refers to it (p.7) but does not quote it]. Move the magnet and keep the loop still: the changing B induces an electric field (Faraday's law, curl E = -dB/dt), and that E drives the current. Move the loop and keep the magnet still: there is no E, but the charges in the wire move through B and feel the magnetic force q v x B (motional emf). Two different explanations, yet the current is the same whenever the relative motion is the same. Einstein took this as a hint that only relative motion matters (Griffiths discusses the same example in ch. 7 on Faraday's law and at the start of ch. 12 [memory: medium]). Builder rejects it [read: pp.6-7]. He takes a magnetic point pole m and a point charge q (Gaussian units, so the magnetic force carries a 1/c; in SI it is just q v x B). In case (a) the pole is at rest and the charge moves at v, so F = (1/c) q v x B0, where B0 is the pole's field at the charge (eq. 6a). In case (b) the charge is at rest and the pole moves at -v, and the force is the same expression multiplied by the bracket {1 - (3/2)(v . r)^2/(r^2 c^2) + (1/2) v^2/c^2} (eq. 6b), the bracket you met in (7a). Same relative velocity, different force, so the induction phenomena "must depend in part on the individual velocities", and Einstein's view, true only when "transmission delays, and the consequent retardation of the potentials, can be neglected", is a practical approximation for the laboratory, but "that it is a fundamental characteristic of the phenomena, must be rejected" [read: OCR pp.6-7]. His formula is right: a deep read derived (6b) from the field of a moving pole and checked it numerically, with the leftover error shrinking like beta^4 (relative error 8.3e-3 at v = 0.4c, 1.7e-6 at 0.05c), while (6a) alone is off by 11% at 0.4c [deep/Ether-And-Relativity-1957-51e9ba.md, sections 6-7]. But Steps 2-4 apply word for word. Boost from the pole's rest frame to the charge's rest frame and case (a) becomes case (b): one two-body system seen from two frames. The extra factor is the gamma by which a perpendicular force changes between frames (Sec. 3.3), plus the changed longitudinal separation, not a sign of absolute velocity. And Einstein's claim was about the current in the loop, which is the same in every frame; force on a charge is not. Builder says the loop case "can readily be verified" but shows no calculation [read: p.6]. Verdict (arbiter, QE revision item 1): equations (6a)/(6b) Established; the inference that they reveal an absolute frame, and so refute Einstein's relative-motion view, Refuted by calculation. His side remark that Einstein never used the example as a premise of the theory (p.7) is historically fair but does not affect the physics.

Verdict this example supports (arbiter, Claim 3): the equations are Established. The claim that the force difference reveals a privileged frame is Refuted by calculation.

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6. Evidence table

Verdicts are the arbiter's final verdicts (arbiter.md). "Tested?" refers to the specific part of the claim in that row.

No suppression is on record. The paper was published in a mainstream journal, and nothing in the sources says it was blocked or dismissed without a hearing [arbiter.md].

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7. What a learner should take away

What is solid. - Builder's mathematics. The proper-time integrals, the moving-charge forces (6b), (7), (7a), (7b) and (8), and the longitudinal light clock are all correct to the order he states. - His observation that "relativity just means relative velocity" is a bad slogan. Forces between moving charges really do depend on each charge's velocity in a given frame, and Newton's third law really does fail for the particles alone. Many popular accounts gloss over this, and he was right to press it. - His defence of LET as a consistent alternative. It is consistent, and it makes the same predictions as SR. Anyone who says "experiment refuted the ether" is wrong: experiment refuted a detectable ether, and LET's ether is undetectable by design. - His definition of "absolute" as "relative to the universe as a whole" (p.11). It anticipated something real: the CMB rest frame exists and our speed relative to it is measured.

What is wrong, and why the mistakes are instructive. - Treating frame-dependent numbers as clues to an absolute frame. Section 5 is the lesson. A force, like a velocity or an energy, takes different values in different frames. The test of whether physics picks out a frame is whether some frame-independent observable depends on the choice. Builder never produces one. He even says (p.8) that every observer in S sees the same asymmetry, which means it cannot single out any one frame. - Quietly assuming absolute simultaneity. "The same instantaneous positions in S" sounds harmless. But the instant you compare two frames, simultaneity becomes the crux. This slip shows up in many arguments for and against relativity. Builder's own pp.16 and 18 supply the fix: one-way light speed cannot be measured, so which events count as simultaneous is a synchronisation convention (Section 3.6). - Inferring absolute velocity from absolute acceleration. Newton's own mechanics has absolute acceleration and no absolute velocity. Builder gets the entailment only by defining acceleration as a change in absolute velocity, which is circular. - Counting hypotheses. "One hypothesis" works for one light-clock orientation. For all clocks, you must assume that every force in nature obeys the same Lorentz factor, and that assumption is the relativity principle itself. A useful habit: when someone says "X explains Y with no extra assumption", list what has to be true of all matter for it to work. - False dichotomy on Mach. He rejects instantaneous action at a distance, which is reasonable, and then jumps to "only the ether remains". He skips the field option that his own footnote mentions.

What is genuinely open. - Interpretation. LET and SR agree on every prediction, so choosing between them is a matter of philosophy and taste, not experiment. Bell's view that the Lorentzian route has teaching value is a respected minority position [web: https://en.wikipedia.org/wiki/Lorentz_ether_theory]. Whether it actually teaches better has never been tested. - Preferred-frame physics at tiny levels. Searches for Lorentz violation keep going, partly because some quantum-gravity ideas allow a preferred frame [memory: medium]. So far all results are null, with bounds in the 1e-8 to 1e-12 range for the parameters quoted in Section 6. One stronger figure the physicist cited (about 1e-18) could not be verified and is not used here. - Mach's principle (field version). Frame dragging is confirmed only to about 20%. - What Dingle actually argued in 1957. Builder paraphrases Dingle as saying "retardation depends on absolute velocity". A web summary of Dingle's later campaign says he claimed SR requires each of two clocks to run slower than the other, a contradiction [web: https://en.wikipedia.org/wiki/Herbert_Dingle]. These may be different stages of Dingle's argument. Reading Nature 179, 865-866 would settle it.

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8. Check yourself

1. Clock B makes a round trip at 0.8c lasting 15 years of clock A's time (A stays inertial). How much does B record? Would an observer in a frame moving at 0.5c relative to A get a different answer for the difference?
2. Using Section 3.2, compute the force on a resting electron from a second electron passing head-on (theta = 0) at beta = 0.6 and distance 1 nm. Compare with F0.
3. In Section 5 the force on q1 is 2.884 x 10^-10 N in one frame and 2.307 x 10^-10 N in another. Name a quantity in that problem that every frame agrees on.
4. Give a one-line counterexample to "absolute acceleration entails absolute velocity" using a Galilean boost.
5. In the light-clock calculation of Section 4, why does showing that contraction gives the right slowing for the longitudinal arm not prove that all clocks slow by gamma?

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9. Further reading

Mainstream - D. J. Griffiths, Introduction to Electrodynamics: ch. 12 (Electrodynamics and Relativity); Sec. 8.2 (momentum in fields, Newton's third law) [memory: high]; the field of a uniformly moving point charge (ch. 10). - E. M. Purcell and D. J. Morin, Electricity and Magnetism: the chapter on fields of moving charges (gives the side-on gamma enhancement used in Section 5). - E. F. Taylor and J. A. Wheeler, Spacetime Physics: proper time as the length of a worldline; the twin effect. - J. S. Bell, "How to teach special relativity", reprinted in Speakable and Unspeakable in Quantum Mechanics (Cambridge) [memory: high]. A modern argument for the Lorentzian route Builder favoured. - Wikipedia, "Lorentz ether theory": https://en.wikipedia.org/wiki/Lorentz_ether_theory - Wikipedia, "Kennedy-Thorndike experiment": https://en.wikipedia.org/wiki/Kennedy%E2%80%93Thorndike_experiment - Wikipedia, "Test theories of special relativity" (Robertson-Mansouri-Sexl framework and current bounds): https://en.wikipedia.org/wiki/Test_theories_of_special_relativity

The researcher and the debate - G. Builder, "Ether and Relativity", Aust. J. Phys. 11, 279-297 (1958). This is the document in the folder. - G. Builder, "The resolution of the clock paradox", Aust. J. Phys. 10, 246 (1957), and "The clock retardation problem", Aust. J. Phys. 10, 424 (1957) [read: titles from his reference list; papers not read]. - H. Dingle, "The 'Clock Paradox' of Relativity", Nature 179, 865-866 (1957) [web: https://www.nature.com/articles/179865a0; not read]. - Wikipedia, "Herbert Dingle" (his later campaign and how it was received): https://en.wikipedia.org/wiki/Herbert_Dingle

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Answers

1. gamma = 1/sqrt(1 - 0.64) = 1/0.6 = 1.667, so B records 15/1.667 = 9 years, a difference of 6 years. Every inertial frame gets the same 9 and 15, because proper time is the frame-independent length of each worldline. The intermediate speeds differ in the new frame, but the integrals do not.
2. Head-on: E = k e / (gamma^2 r^2), so F = F0/gamma^2 = 2.307e-10 / 1.5625 = 1.477 x 10^-10 N, weaker than Coulomb by 0.64. (Builder's (7a) gives [1 - 1.5(0.36) + 0.5(0.36)] = 0.64 exactly here, a coincidence at theta = 0: 1 - beta^2 is the exact factor.)
3. Several work: the charges, the proper (rest-frame) separation, the relative speed 0.6c, and any outcome you could record, such as where q1 ends up hitting a detector after a given proper time. The force itself is not one of them.
4. In frame S a ball's velocity goes from 0 to 2 m/s in 1 s, so a = 2 m/s^2. In frame S' moving at 5 m/s, it goes from -5 to -3 m/s, and a = 2 m/s^2. The acceleration is agreed, and the velocity is not, so there is no absolute velocity.
5. The transverse arm slows by the same gamma with no contraction at all, so the slowing does not come from contraction. And an atomic clock is not a light clock. Its rate depends on the forces binding the atom. For it to slow by exactly gamma, every such force must obey Lorentz covariance. Kennedy-Thorndike tests exactly this, and it is an extra assumption, not a consequence of contraction.