Primer: Richard C. Tolman, The Theory of the Relativity of Motion (1917)

Study document for the Tolman-R project. It is built on arbiter.md (the final verdicts), the three reviews (physicist.md, mathematician.md, contrarian.md), the deep reads in deep/, and the Investigator's findings (sites.md, research/). It gives no new verdicts.

Source tags: [read] = from the 1917 book (project scan, PDF page numbers "p.N"; printed page = PDF page minus 17) or from a project file named in the tag; [web] = Internet source with URL; [memory] = recalled background knowledge, with confidence; [calc] = recomputed in example.py.

Source note (carried over from claims.md and arbiter.md). The only project document is one 249-page OCR scan of the 1917 book. Its text layer garbles nearly every equation. These parts were read from page images in the deep reads (deep/), with the equations checked by SymPy or by hand: Chapters III-IV (PDF pp.45-65, 70), Chapter V (pp.66-78), Chapter VI sections 72-78 (pp.96-104), Chapter VIII (pp.119-137), Chapters X-XI (pp.154-180), Chapter XII sections 154-161 and 165-172 (pp.184-190, 193-204) and Chapter XIII (pp.204-238). No printed mathematical error was found in any of them; the only slips are roundings (10^21 for 8.99 x 10^20 erg). The reviewers also used a clean Project Gutenberg re-typesetting (research/files/Tolman-1917-Gutenberg-32857.pdf) for sections 21-23, 64, 77, 103-105, 109-111, 117-120, 146-150, 151-167 and 172. Still at digest level only: Chapter I, Chapter II outside sections 21-23, Chapter VI sections 59-71 (apart from 64), Chapter VII, Chapter IX outside sections 109-111, sections 162-164 and the appendices [read: claims.md source note]. What this primer says about those parts rests on digests and on [memory] of standard physics. The digests contain known number errors (corrected from the page images in deep/); where they differ, the deep reads govern. Use the scan for any equation you rely on: equation (229) has u_x on the scan's page image [read: page image p.186, arbiter.md and deep/], while the Gutenberg PDF has u_z [read: Gutenberg text layer, eq. 229], and the scan's own OCR text layer confuses the two as well [read: deep/, Chapter XII sections 154-161].

What kind of project this is. Unlike most projects in this collection, this is a mainstream textbook by a mainstream physicist. Richard Chace Tolman (1881-1948) took his PhD at MIT in 1910, taught at Berkeley when he wrote this book, moved to Caltech in 1922, and later served as vice chairman of the National Defense Research Committee and adviser on the Manhattan Project [web: https://oac.cdlib.org/findaid/ark:/13030/tf4w1003fs/]. The book was one of the first English-language textbooks of special relativity [memory, medium-high]. So the question here is not "is this fringe physics right?" but two separate questions the arbiter keeps apart: (A) is the physics right by today's knowledge, and (B) was the evidence in 1917 as strong as the book says it was?

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1. The idea in one paragraph

If no experiment can tell whether you are "really" moving, and if light always travels at the same speed whatever the motion of its source, then space and time cannot be the fixed stage that Newton assumed. Moving clocks must run slow, moving rods must be shorter along their motion, clocks that one observer has set together must look out of step to another, and velocities cannot simply add. Tolman takes these two starting points and works out what they force on the rest of physics: mass must grow with speed, energy must have mass, and a body under stress must carry hidden momentum because the stress is passing energy from one end to the other. That last idea leads to the book's most surprising results: a moving lever that needs a turning force to keep it from turning (yet does not turn), and a long argument about what "temperature" means for a body that moves past you. Most of the book is now standard physics. A few parts were, and still are, matters of definition or of how strong the 1917 evidence really was.

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2. What you need to know first

2.1 Newtonian mechanics and Galilean relativity. Newton's laws hold in "inertial frames": frames that are not accelerating. If frame S' moves at speed V along x relative to S, Galileo's transformation is x' = x - Vt, y' = y, z' = z, t' = t. Velocities add: a ball thrown forward at u' in S' moves at u = u' + V in S. Mechanics alone cannot tell which inertial frame is "at rest"; that is Galilean relativity. Tolman's Chapter I is this refresher. Read: John R. Taylor, Classical Mechanics (University Science Books, 2005), Chapter 1 [memory, high].

2.2 Lagrangian mechanics. Instead of forces, describe a system by one function, the Lagrangian (Tolman calls it the "kinetic potential" H), which at low speed is kinetic energy minus potential energy, L = T - V. Choose any set of numbers that fix the state of the system ("generalized coordinates" q, such as positions, a volume, or the strains in a solid). The equation of motion for each is d/dt(∂L/∂q̇) - ∂L/∂q = 0, or with an outside force Q acting on that coordinate, = Q. The partial derivative ∂L/∂q̇ is the "generalized momentum". The useful point for this book: once you pick L, every force, momentum and energy follows mechanically. So whatever is assumed in choosing L is carried into every result. Read: Taylor, Classical Mechanics, Chapter 7 [memory, high]; or Goldstein, Classical Mechanics, Chapters 1-2 [memory, high].

2.3 Stress and strain (no tensors needed). In a solid under load, cut an imaginary small cube. On each face there is a force per unit area. The face whose outward normal points along y can feel a push along x (a shear) as well as along y (a pressure or tension). Write t_xy for "force along x per unit area on the face whose normal is along y". With three force directions and three face orientations you get a 3 x 3 table of nine numbers, the stress. (A "tensor" here just means this table plus a rule for how its entries change when you rotate or move the coordinates.) In ordinary mechanics the table is symmetric, t_xy = t_yx, because otherwise a tiny cube would spin up without limit. Strain is the matching table of fractional stretches and shears. Read: any engineering mechanics-of-materials text; Feynman, Lectures on Physics Vol. II, chapters on elasticity [memory, medium on chapter numbers, so none given].

2.4 Momentum and energy flow in fields. In electromagnetism, energy flows (the Poynting vector S = E x B / μ0, in watts per square metre), and a field that carries energy flow S also carries momentum density S/c² (kg per square metre per second). This is why light exerts pressure. Tolman's Chapter X extends the same rule to every kind of energy flow, including energy passed along by stresses in a solid. Read: David J. Griffiths, Introduction to Electrodynamics, Chapter 8 (conservation laws, momentum in fields) [memory, high].

2.5 Thermodynamics: entropy and temperature. Entropy S counts (the logarithm of) how many microscopic states fit a macroscopic one: S = k ln W, with k Boltzmann's constant. Temperature is defined from it: 1/T = ∂S/∂E at fixed volume, or equivalently T = ∂E/∂S. Heat added reversibly is dQ = T dS. For black-body radiation in a box (a "hohlraum"), energy E = a v T⁴ and pressure p = (a/3) T⁴, with v the volume and a the radiation constant (Tolman's "Stefan's constant"). Read: Daniel V. Schroeder, An Introduction to Thermal Physics, Chapters 2-3 and 7 [memory, high].

2.6 Special relativity basics. The two postulates (no detectable absolute motion; light speed c the same for all inertial observers), the factor γ = 1/sqrt(1 - V²/c²) (Tolman writes κ), and β = V/c. If you have never met relativity, read the first chapters of Taylor and Wheeler, Spacetime Physics (2nd ed., 1992), before this book. Griffiths, Introduction to Electrodynamics, Chapter 12, covers relativity and its link to electromagnetism [memory, high].

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3. The textbook view (as it stands today)

Everything in this section is standard physics [memory, high] unless tagged otherwise. Symbols: c = speed of light (299 792 458 m/s); V = speed of frame S' relative to S along x; u = speed of a body; β = V/c; γ = 1/sqrt(1 - β²); m0 = rest mass.

3.1 Lorentz transformation. Replacing Galileo:

x' = γ (x - V t),   y' = y,   z' = z,   t' = γ (t - V x / c²)

It keeps x² + y² + z² - c²t² the same in both frames. Consequences: - Time dilation: a clock moving at V ticks slow by the factor 1/γ as judged by S. - Length contraction: a rod moving along its length is shorter by the factor 1/γ; transverse lengths are unchanged. - Relativity of simultaneity: clocks synchronised in S', a distance l' apart along the motion, look out of step by l'V/c² in S (rear clock ahead). - Velocity addition: u = (u' + V)/(1 + u'V/c²). Tolman's example: c/2 + c/2 gives 0.8c, not c [read: page image p.70].

3.2 Mass, momentum and energy. Momentum p = γ_u m0 u, energy E = γ_u m0 c², with γ_u = 1/sqrt(1 - u²/c²). The change of kinetic energy is c² times the change of γ_u m0. In 1917 the quantity γ_u m0 was called the mass m of the moving body ("relativistic mass"). Modern texts mostly keep the word "mass" for m0 alone; the physics is the same, only the naming differs.

3.3 Momentum density is energy-flux density over c². In modern language every form of energy and momentum is collected into a symmetric 4 x 4 table, the stress-energy tensor T^{μν}. You do not need the tensor machinery for this book; you need one consequence of its symmetry: the momentum density g (momentum per unit volume) equals the energy-flux density (energy per unit area per second) divided by c², for every kind of energy flow, including energy carried by stresses. For a body moving at u along x and carrying a shear stress t_xy, the shear forces do work t_xy·u per second per unit area on one face and pass it to the opposite face, so energy flows in the y direction, and therefore

g_y = t_xy u / c²

even though the body moves only along x. This is Tolman's equation (172) [read: page image p.159], and the modern symmetric tensor gives the same thing [read: mathematician.md, physicist.md; arbiter.md item 3].

3.4 Angular momentum of a moving stressed body. Angular momentum about a fixed point is M = ∫ r x g dV. If g is not parallel to the velocity u, then even in straight-line motion at constant speed,

dM/dt = ∫ u x g dV ≠ 0

and an outside torque of exactly this size is needed to keep the motion steady. This is Tolman's (192) [read: page image p.168]. Nothing spins: the angular momentum grows because a sideways momentum is being carried further and further from the fixed reference point.

3.5 Thermodynamics of moving bodies. What all sides agree on: entropy is the same in every frame (S = S0; it counts states, which do not depend on who looks), and pressure is the same in every frame (p = p0). The energy and momentum of a moving body at rest-frame energy E0, pressure p0 and rest volume v0, held by outside pressure, are (Planck 1907-08, Tolman (204)-(205)) [read: page images pp.178-179]:

E = γ (E0 + p0 v0) - p v,      G = γ (E0 + p0 v0) u / c²,      v = v0/γ

The extra p0v0 terms come from the work the outside pressure does at the moving walls. For a complete closed system (box plus the walls that hold it) they cancel and E = γE0, G = γE0 u/c² (von Laue 1911 [memory, high]; Tolman's own (193) [read: Gutenberg text, section 137]).

What is not agreed is a single law for the temperature of a moving body. Three answers are in the literature: Planck-Einstein T = T0/γ (the moving body looks colder; Tolman's choice), Ott-Arzelies T = γT0 (hotter; some authors add Eddington and Møller to the name), and Landsberg T = T0 (unchanged) [web: https://arxiv.org/abs/2204.12572; https://arxiv.org/abs/2606.00521]. Much modern work (relativistic hydrodynamics) defines temperature only in the body's own rest frame, and then carries it into other frames bundled with the body's velocity as a "four-vector" [memory, medium-high]. A four-vector is just a set of four numbers that change from frame to frame by the same Lorentz transformation rule as (ct, x, y, z) in section 3.1; nothing more about it is needed here. See section 4.

3.6 Electromagnetism. Maxwell's equations keep their form under the Lorentz transformation; electric and magnetic fields mix when you change frame; electric charge is the same in every frame; field energy E0 carries inertia E0/c². This is Tolman's Chapter XII and is standard [read: arbiter.md claim 5; memory, high].

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4. Where the inventor departs

Tolman's book departs from Newtonian physics everywhere; that is its subject. The table below instead lists where the book departs from what a modern textbook says or from what the 1917 evidence could support. Each row gives the arbiter's final reading.

Three of these deserve a closer look.

4.1 The temperature law: one result, three definitions. Tolman gets T = T0/γ in two lines. He treats temperature as the generalized force conjugate to entropy in his Lagrangian: T = -∂H/∂S. With H = -E0 sqrt(1 - u²/c²) and S = S0, the derivative brings out the factor sqrt(1 - u²/c²) = 1/γ, so T = T0/γ [read: page image p.176]. The factor comes entirely from the choice of H. Tolman says honestly that H "was not absolutely uniquely determined" [read: p.172].

Why does the choice matter? Add a little heat to a moving photon gas at fixed rest volume, raising its rest energy by dE0, at β = 0.6 (γ = 1.25). The lab energy rises by dE = 1.40 dE0, because the pressure, and so the pv terms, rise too. But if you say "part of that energy rise is the work needed to give the extra energy its extra momentum, u dG", and subtract it, the "heat" is dE - u dG = 0.80 dE0 = dE0/γ. That is exact for any substance: E - uG = E0/γ [calc: example.py Part 5; arbiter.md item 4]. Divide "heat" by the unchanged dS: - heat = dE - u dG gives T = T0/γ = 240 K for T0 = 300 K (Planck-Einstein, Tolman); - heat = γ dE0 (the energy rise if the pressure term is left out) gives T = γT0 = 375 K (Ott-Arzelies); - temperature defined only in the rest frame gives 300 K (Landsberg); - a fourth, operational choice, the temperature that a black body would need to have the moving gas's lab energy density, gives 345 K [calc: example.py Part 5].

None of these is an algebra mistake. They answer different questions. A real thermometer moving through black-body radiation does not even see a single temperature: it sees T(θ) = T0 sqrt(1 - β²)/(1 - β cos θ), depending on direction θ, which is what the cosmic microwave background dipole looks like [memory, high]. The arbiter's ruling is that Tolman's law is a theorem about his definition, not a physical prediction that an experiment could falsify (verdict: Not testable as stated; the arbiter's QE revision dropped an earlier "Historically superseded" label because the law was never replaced, only joined by rival conventions), and that it remained the majority view into the 1960s and is still "supported by many authors" according to Heras and Osorno 2022 [web: https://arxiv.org/abs/2204.12572]. A 2026 paper (Pouryazdan and Vakili, Phys. Lett. A 590, 131860) defines an effective temperature that rises with speed and argues against Tolman's form [web: https://arxiv.org/abs/2606.00521]. Both papers are theoretical; neither reports a measurement. A computer-simulation paper (Cubero et al., PRL 99, 170601, 2007) favours an observer-independent statistical temperature [web, abstract level, per contrarian.md].

4.2 The electron evidence of 1917. The book calls the electron-deflection experiments a "triumph" [read: p.56]. The wider record is more mixed. Kaufmann's own 1905-06 reading of his data favoured Abraham's rigid-electron model, not Lorentz-Einstein; Hupka (1910) himself remarked that his experiment was not a definitive result (Wikipedia's paraphrase; Hupka's paper was not opened); Zahn and Spees (1938) argued the Bucherer and Neumann experiments showed only a qualitative mass increase; Rogers et al. (1940) agreed with Lorentz-Einstein to 1% and could tell the theories apart [web: https://en.wikipedia.org/wiki/Kaufmann%E2%80%93Bucherer%E2%80%93Neumann_experiments]. Tolman himself had been more cautious in 1910, writing that the electron experiments "may not really indicate an increase in the mass" if forces at high speed differ from Maxwell's law [web: https://en.wikisource.org/wiki/The_Second_Postulate_of_Relativity]. The physics is now established beyond doubt; the 1917 wording overstated the 1917 evidence.

4.3 Emission theories and Ritz. An emission theory says light leaves its source at c relative to the source, so the source's speed adds to light's speed. That is perfectly compatible with the first postulate; it rejects only the second. Tolman took this idea seriously early on. In 1910 he ran his own test, comparing light from the approaching and receding limbs of the sun with a Lloyd's-mirror interferometer; he expected about 2 fringes of shift on the emission hypothesis and saw none [web: https://en.wikisource.org/wiki/The_Second_Postulate_of_Relativity]. In the same paper he left a loophole open himself: he noted that the possibility that the source-speed difference is destroyed near the earth "is not entirely excluded" [web: same URL, as recorded in research/2026-10-04-wikisource.md; the arbiter asks that the exact wording be rechecked against the paper before wider quotation]. That is the same kind of objection (light losing its source's speed on the way) that Fox later pressed with the extinction theorem (below). In 1912 he argued that canal-ray Doppler data ruled out two emission variants, but wrote that Ritz's theory "does not seem to have received the critical attention which it deserves" and called it "the only emission theory which has been developed with any completeness"; he proposed a Michelson-Morley test fed with sunlight and called a decision between Ritz and Einstein "a matter of the highest importance" [web: https://en.wikisource.org/wiki/Some_Emission_Theories_of_Light; quotations as recorded in research/2026-10-04-wikisource.md]. With M. La Rosa (Nuovo Cimento, 1912) he proposed the sunlight interferometer test; it was carried out in 1924 by R. Tomaschek (with starlight, Heidelberg) and D. C. Miller (with sunlight, Cleveland), whose results Martinez says "seemed to agree clearly" with Einstein and not Ritz [read: research/2026-10-04-martinez-ritz.md, from Martinez 2004]. Meanwhile R. D. Carmichael's textbook The Theory of Relativity (1913) wrote that there was then no undoubted experimental evidence for or against the light postulate, "despite Tolman's inclinations" [read: same note]. By 1917 the book cites only the astronomical argument of Comstock and de Sitter against Ritz, and states that optical experiments deciding between Einstein and Ritz "have never been performed" [read: Gutenberg text, section 22].

The historian Alberto Martinez (2004) argues that the early evidence was "misconstrued as telling against it more conclusively than actually was the case", and reports J. G. Fox's critique (1962-65): by the Ewald-Oseen extinction theorem, light passing through any medium (air, glass, a star's atmosphere, interstellar gas) is re-emitted and soon travels at c relative to that medium, forgetting its source's speed. On that basis Fox called interferometer tests "like those of Tomaschek or Miller or those using Tolman's apparatus" irrelevant to the source-speed question and argued de Sitter's binary-star evidence was not decisive either [read: research/2026-10-04-martinez-ritz.md, from research/files/Martinez-2004-Ritz-Einstein-Emission-Hypothesis.pdf]. Fox's own papers were not opened by anyone in this review; his argument is carried as Martinez's account. Martinez also records that "Ehrenfest and Tolman called for unambiguous empirical evidence to test Ritz's emission theory, but neither spent any effort in extending it"; Ritz died in 1909. Later tests designed to get round extinction did come: Alvager et al. (1964) found that photons from fast-decaying pions do not take on the source speed, and Brecher (1977) repeated the binary-star test with X-ray sources "under consideration of the extinction theorem, ruling out a source dependence as well" [web: https://en.wikipedia.org/wiki/Tests_of_special_relativity]. So the arbiter's split reading: in 1917 the Ritz variant was Never properly tested (not dismissed without testing; Tolman tried); today it is Tested, not supported. On the history, the arbiter finds no evidence of suppression (Tolman published his own emission papers in Physical Review) and reads Tolman's own 1910 "not entirely excluded" as a sign that he saw the loophole that made early null results uninformative [read: arbiter.md QE revision, retention items].

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5. Worked example: the right-angled lever

This is the book's most famous paradox. It first appeared in Lewis and Tolman (Philosophical Magazine, 1909) [memory, high; credited the same way by Shapiro 2012, research/files/arXiv-1206.4487-Shapiro-2012-right-angle-lever.pdf], and Tolman gives it as section 136 [read: page images pp.169-170; Gutenberg text, section 136]. Every number below is reproduced by example.py:

.venv/bin/python review/Tolman-R/example.py

Set-up (Tolman's Fig. 14). An L-shaped lever turns on a pivot B. Arm 1, length l1, runs from B straight "down" (along y) to point A. Arm 2, length l2, runs from B along x to point C. A force F1 pushes at A along +x; a force F2 pushes at C along -y. Put B at the origin: A = (0, -l1), C = (l2, 0).

Torque about B (z-component of r x F): - from F1 at A: τ1 = x·F_y - y·F_x = 0 - (-l1)(F1) = +F1 l1 - from F2 at C: τ2 = l2·(-F2) - 0 = -F2 l2

Step 1: the rest frame S0. Choose l1° = l2° = 0.5 m and F1° = F2° = 200 N. Then F1° l1° - F2° l2° = 100 - 100 = 0. The lever is in balance. (The pivot supplies whatever force makes the net force zero.)

Step 2: the frame S in which the lever moves along x at speed V. Four facts from earlier chapters [read: page image pp.169-170]: - l1 = l1°: arm 1 is perpendicular to the motion, so it does not contract. - l2 = l2° sqrt(1 - V²/c²): arm 2 lies along the motion, so it is Lorentz-contracted. - F1 = F1°: a force along the motion, on a body at rest in S0, is the same in both frames (Tolman's eq. 61). - F2 = F2° sqrt(1 - V²/c²): a force across the motion is smaller by 1/γ (Tolman's eq. 62).

Net torque in S:

F1 l1 - F2 l2 = F1° l1° - F2° l2° (1 - V²/c²) = F1° l1° V²/c² = F1 l1 V²/c²

The forces that balance the lever in its own frame do not balance in S. Tolman: "a moving lever will be in equilibrium only if the external forces have a definite turning moment" [read: p.170].

Step 3: put in numbers.

(a) The lever rides on the Earth, V = 30 km/s. β = 3.0 x 10⁴ / 2.998 x 10⁸ = 1.0 x 10⁻⁴, so β² = 1.0 x 10⁻⁸. Net torque = 100 N·m x 1.0 x 10⁻⁸ = 1.0 x 10⁻⁶ N·m [calc]. The changes to l2 and F2 are in the ninth significant figure.

(b) A fast case, V = 0.6c. sqrt(1 - 0.36) = 0.8. So l2 = 0.4 m and F2 = 160 N in S. F1 l1 = 200 x 0.5 = 100 N·m; F2 l2 = 160 x 0.4 = 64 N·m. Net torque = 36 N·m, which is 100 x 0.36 as the formula says [calc].

So why does the lever not start spinning in S? If it spun in S, it would spin in S0 too (rotation is not something a change of inertial frame can create or remove), and in S0 it is balanced.

Step 4: where the torque goes, route 1 (energy flow). In S, point A moves at V along x and the force F1 also points along x, so F1 does work on the lever at the rate

P = F1 V    (watts)

At C, F2 points along y while C moves along x, so F2 does no work. At the pivot, the reaction force has an x-component -F1, so energy leaves at B at the same rate F1 V. Steady state: a constant energy current I = F1 V flows up arm 1 from A to B, along +y.

By the rule of section 3.3, a flow of energy carries momentum. Energy entering per second, F1 V, has mass F1 V/c² (Tolman's words: "this new energy which enters at A each second has the mass F1V/c²" [read: p.170]). Summed over the whole arm, the momentum stored in the moving energy is

p_y = (energy current) x (length it flows along) / c² = F1 V l1 / c²

This momentum points along y, at right angles to the lever's motion.

Numbers: (a) at 30 km/s, I = 200 x 3 x 10⁴ = 6 x 10⁶ W and p_y = 6 x 10⁶ x 0.5 / (8.99 x 10¹⁶) = 3.3 x 10⁻¹¹ kg·m/s. (b) at 0.6c, I = 3.6 x 10¹⁰ W, and p_y = 2.0 x 10⁻⁷ kg·m/s [calc].

Step 5: where the torque goes, route 2 (angular momentum bookkeeping). Angular momentum about a point fixed in S: M_z = x p_y - y p_x. The momentum p_y is stored in arm 1, whose x-coordinate grows as x = Vt. So

dM_z/dt = V p_y = V (F1 V l1 / c²) = F1 l1 V²/c²

which is exactly the net torque of Step 2. Numbers: (a) 3 x 10⁴ x 3.3 x 10⁻¹¹ = 1.0 x 10⁻⁶ N·m; (b) 1.8 x 10⁸ x 2.0 x 10⁻⁷ = 36 N·m [calc]. The torque does not spin the lever up; it feeds the steadily growing angular momentum of a sideways momentum that is being carried away from the reference point. This is Tolman's (192), dM/dt = ∫ u x g dV, done for one arm [read: page image p.168]. The same answer follows from integrating u x g with g_y = t_xy u/c² over the stressed arm (mathematician.md's cross-check uses ∫ t_xy dV = F1 l1).

A sense of scale: after one year at 30 km/s, the angular momentum about a fixed point has grown by 1.0 x 10⁻⁶ N·m x 3.16 x 10⁷ s = 32 kg·m²/s [calc]. That sounds large, but it is just a tiny momentum (3 x 10⁻¹¹ kg·m/s) multiplied by a huge lever arm (the 9.5 x 10¹¹ m travelled in a year). Nothing about the lever itself changes.

Step 6: units check. Energy current F1 V: N x m/s = W. Divided by c²: W/(m²/s²) = kg/s (mass per second). Times l1: kg·m/s, a momentum. Times V: kg·m²/s² = N·m, a torque. Each step is consistent [calc].

Step 7: a stressed solid in general (Tolman's eq. 172). For a shear stress t_xy = 10⁹ Pa (near the strength of good steel) in a body moving at 1 km/s, g_y = t_xy u/c² = 10⁹ x 10³ / 8.99 x 10¹⁶ = 1.1 x 10⁻⁵ kg·m⁻²·s⁻¹: about one hundred-thousandth of a kg·m/s per cubic metre [calc]. This is why Tolman wrote that "at present we have no experiments of sufficient accuracy" to test his elasticity theory against older ones [read: p.172], and why no one has measured it directly since.

What the example teaches. 1. Momentum is not always parallel to velocity once energy can flow through a body. The textbook formula p = m v is the special case of a body with no internal energy currents. 2. "Torque" and "angular acceleration" are not interchangeable in relativity: a nonzero net torque can be balanced by the growth of angular momentum without any rotation. 3. The outcome (no rotation) is not in dispute. The best explanation is still argued about: Laue 1911 (energy current, Tolman's route), Tolman-Epstein 1911 (relativistic force and acceleration need not be parallel) [web: https://en.wikipedia.org/wiki/Trouton%E2%80%93Noble_experiment], Shapiro 2012 (angular momentum and relativity of simultaneity), and "hidden momentum" accounts [read: research/2026-10-04-lever-paradox.md]. Shapiro calls the history "extensive discussion and indeed controversy" [read: research/files/arXiv-1206.4487-Shapiro-2012-right-angle-lever.pdf, p.1]. 4. The electrical twin is the Trouton-Noble experiment (1901-1903): a charged capacitor carried along with the Earth should, on an ether picture, twist to line up with its motion. It did not; later repeats by Tomaschek (1925-26), Chase (1926-27) and Hayden (1994) agree [web: https://en.wikipedia.org/wiki/Trouton%E2%80%93Noble_experiment]. Tolman lists Trouton and Noble among the null ether-drift experiments [read: p.38].

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6. Evidence table

Verdicts are the arbiter's final verdicts (arbiter.md). "Replicated" means independent repetition on record in the sources used.

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7. What a learner should take away

Solid. - The whole kinematic core (Chapters III-V) is right, and Tolman's light-clock and colliding-sphere arguments are still good ways to learn it. Every equation the deep reads checked from page images in Chapters III-V is free of mathematical error, including the Doppler, aberration and Fresnel-drag formulas of Chapter V [read: deep/, sections 4.9 and Chapter V checks]. - The book made a real, falsifiable prediction and told experimenters how to test it. Section 46: "We may regard the atoms which are moving in these rays as little clocks ... the frequency of the emitted light ought to be less than that of light from stationary atoms of the same kind ... It would of course be necessary to view the canal rays at right angles to their direction of motion" [read: p.68]. At a canal-ray speed of 0.65% of c the predicted drop is 2.1 x 10⁻⁵ of the frequency [calc: example.py Part 6a]. Ives and Stilwell (1938) measured this same factor with canal rays, using light sent forward and backward along the beam, whose average cancels the ordinary (first-order) Doppler shift [web: https://en.wikipedia.org/wiki/Ives%E2%80%93Stilwell_experiment]. Why not Tolman's right-angle view? A deep reader's own estimate: a viewing-angle error of only 0.01 radian lets in an ordinary Doppler shift of 6.5 x 10⁻⁵, three times the effect sought [calc: deep/, Chapter V; example.py Part 6a]. Kept in the record: Ives read his own result as support for a Lorentz-Larmor ether, not for Einstein's interpretation [web: same URL], which shows that the measurement confirms the formula but not, by itself, one picture of what lies behind it. - Momentum density = energy flux / c² for every kind of energy, including stress. This is one of the deepest and least taught facts of relativity, and Tolman's Chapter X is a clear early statement of it. - Entropy and pressure are the same for all observers; the energy and momentum formulas for a moving thermodynamic system (Planck's) are right, once you count the work done by whatever holds the system.

Wrong, or overstated, and why that is instructive. - The 1917 text is more confident than its evidence. Tolman's own 1910 and 1912 papers were open about the emission hypothesis and cautious about the electron data; the book calls the electron work a "triumph" and gives Ritz less room. The record read in this review does not say why the tone changed (arbiter open question 6). The lesson: separate "this theory is right" (it is) from "this experiment proved it at the time" (often it did not yet). - "All energy has mass" is argued in the book by generalising from kinetic energy, and Tolman says so ("every reason to believe", "pragmatic"). A derivation that covers one case and an induction to all cases are different things, even when the induction turns out true. - The Lorentz transformation is found "by trial". A correct guess that is checked is a valid method, but the claim that the three conditions are sufficient needs an extra assumption (linearity) that the book does not state. - Numerical slips are small: "1 gram = 10^21 ergs" is 11% high (the exact value from his own c is 8.99 x 10^20), although section 77 gives 9 x 10^20 [calc: example.py Part 6c]. - The moving-body temperature law was presented as a result; it is really a consequence of a chosen definition (verdict Not testable as stated). That is a lesson in checking which parts of a derivation are physics and which are conventions.

Genuinely open. - Temperature of a moving body: is there any measurement that distinguishes the Planck-Einstein, Ott-Arzelies and Landsberg laws, or is the question only one of definition? Papers are still appearing (2022, 2026), all theoretical. - Was Wilson's moving-dielectric experiment ever repeated with a high-permeability material, as Tolman asked in 1917? The two theories differ only by (μ - 1)(V/c)H_y, so only a magnetic dielectric can tell them apart [calc: example.py Part 6d]. A deep reader recalls that Wilson and Wilson (1913) did this with a magnetic dielectric and found agreement with Tolman's (εμ - 1) form [memory, medium; the arbiter's web checks failed]. If true, the test existed when the book was written and supported it; if not, the term remains untested. The 1913 paper would settle it. - What exactly do Fox's 1962 and 1965 papers argue, and does extinction in stellar atmospheres and interstellar gas actually matter for de Sitter's binaries? Only a secondary account has been read here. (The physics question is settled by the 1964 and 1977 tests; the historical one is not.) - What is in Tolman's papers at Caltech (12 boxes, including course lecture notes and manuscripts) [web: https://collections.archives.caltech.edu/repositories/2/accessions/7775]? Possibly drafts behind the 1917 book. The 1952 NAS Biographical Memoir by Kirkwood, Wulf and (probably) Epstein was located but not read [web: https://www.nasonline.org/wp-content/uploads/2024/06/tolman-richard.pdf]. - Chapters V, VIII and XIII and sections 165-172 have now been checked from page images with no printed error found. Still at digest level: Chapter I, Chapter II outside sections 21-23, Chapter VI sections 59-71 (apart from 64), Chapter VII, Chapter IX outside sections 109-111, sections 162-164 and the appendices (see the source note).

Other things in the book worth knowing. None of these needs a verdict; they are kept because they are pertinent to this collection or instructive. - Homopolar dynamos (section 159, PDF p.189). Tolman's whole section, three sentences: "We also see that the 'electromotive' force which acts on a charge moving through a magnetic field finds its interpretation as an 'electric' force provided we make use of a system of coordinates which are themselves stationary with respect to the charge. Such considerations throw light on such questions, for example, as to the seat of the 'electromotive' forces in 'homopolar' electric dynamos where there is relative motion of a conductor and a magnetic field" [read: p.189]. In plain terms: a charge moving through a magnetic field feels the force q v x B; but an observer riding with the charge sees it at rest, so for that observer the same push must be an ordinary electric field (the field transformation of equation (227) produces it). So the "seat" of the EMF is local, in each moving piece of the conductor, judged in that piece's own rest frame. Limits flagged by the deep reader [read: deep/, Chapter XII sections 154-161]: Tolman's equations hold between non-accelerated frames, while a Faraday disc rotates (the standard fix, using a momentarily co-moving frame for each element, is not stated by Tolman); he names no disc and draws no conclusion; and the famous puzzle of a rotating, axially symmetric magnet is not addressed. Treat the section as a pointer, not a solution. The homopolar (Faraday-disc) question recurs across this Energy/Researchers collection, and this passage is the only one in the book that touches it; it is a candidate reason the book is in the collection, not an established one [read: arbiter.md QE revision]. - Interatomic energy (section 77, PDF pp.101-102). A footnote: "our theory points to the presence of enormous stores of interatomic energy which are still left in substances cooled to the absolute zero" [read: p.102]. Reason: cooling a body removes only a negligible part of its mass, so almost all of the mass-energy E = mc² must stay locked inside even at 0 K. In the same section Tolman suggests that radioactive decay may release enough energy to make "measurable differences in mass", which "might account for the lack of exact relations between the atomic weights of the successive products of radioactive decomposition"; he gives no numbers, no isotopes and no experimenters [read: p.102, via deep/ Chapter VI]. For scale: Cockcroft and Walton's 1932 nuclear-reaction check of E = mc² came fifteen years later [memory, high]. - A gas of massless particles (section 105, PDF pp.134-135, book pp.117-118). For particles whose rest mass is negligible, Tolman finds the average energy is 3kT per particle, E = 3nRT per mole (n = number of moles, R = gas constant), twice the Newtonian (3/2)nRT. Since energy has mass, such a mole at 300 K "would have a mass of 3 × 8.31 × 10^7 × 300 / 10^21 = 7.47 × 10^-11 grams", which "suggests a field of fascinating if profitless speculation" [read: page image, book p.118, via deep/ Chapter VIII; Gutenberg text]. Check: 3 x 8.31 x 10⁷ erg/(mol·K) x 300 K = 7.48 x 10¹⁰ erg; divided by Tolman's rounded c² = 10²¹ cm²/s² gives 7.48 x 10⁻¹¹ g (he printed 7.47); with the true c² = 8.99 x 10²⁰ it is 8.3 x 10⁻¹¹ g [calc: example.py Part 6b]. (The digest's "7.47 x 10⁻⁸ g" is a digest error.) This is an early treatment of what is now called an ultra-relativistic gas. - Faster than light and causality (section 52, PDF p.72). If a cause at A could send a signal to B faster than light, then for some moving observers the effect at B would come before its cause. Tolman is careful: this "might not be a logical impossibility; nevertheless its extraordinary nature might incline us to believe that no causal impulse can travel with a velocity greater than that of light". He also shows that things which carry no energy or cause, such as a row of blocks each set to fall by its own alarm clock, can "move" at any speed [read: p.72, via deep/ Chapter V]. A plausibility argument, not a proof, and stated as one. - What the book leaves out (Preface, PDF p.21). Accelerated frames and "Einstein's latest work on the relation between gravity and acceleration" (general relativity) are deliberately excluded [read: p.21, per qe.md]. - Chapter XIII method. The four-dimensional chapter uses the Wilson-Lewis "real-time" non-Euclidean geometry rather than Minkowski's trick of an imaginary time coordinate ict; its results agree with the earlier chapters [read: deep/, Chapter XIII].

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8. Check yourself

1. A lever like the one in section 5 has F1 = 50 N, l1 = l2° = 2 m, and moves at V = 0.8c along arm 2. What are l2 and F2 in the lab frame, what is the net torque, and what sideways momentum p_y is stored in arm 1?
2. Why does force F2 (at C, along -y) do no work on the moving lever, while F1 (at A, along +x) does?
3. Tolman says a body moving in a straight line at constant speed can have increasing angular momentum. Using M_z = x p_y - y p_x, explain in one or two sentences how that can be true without any rotation.
4. Tolman's derivation of T = T0 sqrt(1 - u²/c²) uses T = -∂H/∂S with H = -E0 sqrt(1 - u²/c²). Which single assumption would you change to get T = T0 instead? (No algebra needed.)
5. In 1917 Tolman wrote that optical experiments deciding between Einstein's and Ritz's theories "have never been performed". What did the arbiter conclude about that claim's status in 1917 and today, and what was Fox's objection to the early tests?

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9. Further reading

The inventor's own work - R. C. Tolman, The Theory of the Relativity of Motion (University of California Press, 1917). Project scan (equations garbled in the text layer); clean re-typesetting at Project Gutenberg #32857 [web: https://www.gutenberg.org/files/32857/32857-pdf.pdf], downloaded to research/files/. Check any equation you rely on against the scan's page images (eq. 229 has u_z in the Gutenberg PDF, u_x on the scan). - R. C. Tolman, "The Second Postulate of Relativity", Physical Review 31 (1910) 26-40 [web: https://en.wikisource.org/wiki/The_Second_Postulate_of_Relativity]. - R. C. Tolman, "Some Emission Theories of Light", Physical Review 35 (1912) 136-143 [web: https://en.wikisource.org/wiki/Some_Emission_Theories_of_Light]. - G. N. Lewis and R. C. Tolman, "The principle of relativity, and non-Newtonian mechanics", Philosophical Magazine (1909): the origin of Chapter III and the lever [memory, high]; other Tolman papers listed at [web: https://en.wikisource.org/wiki/Author:Richard_Chace_Tolman]. - R. C. Tolman, Relativity, Thermodynamics and Cosmology (Oxford, 1934): his later, fuller treatment, which kept T = T0/γ [memory, medium-high]. - R. C. Tolman and P. Ehrenfest (1930): the Tolman-Ehrenfest relation, T sqrt(g00) = constant, for temperature in a gravitational field (g00 is the time-time entry of the metric; in plain terms, a body in equilibrium is warmer deeper in a gravitational well). It is the general-relativity partner of the T = T0/γ convention, as Pouryazdan and Vakili (2026) discuss [read: research/2026-10-04-arxiv-temperature.md; mathematician.md claim 4d; memory, medium-high for the 1930 dates].

Mainstream and secondary sources - E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (W. H. Freeman, 1992): the gentlest serious introduction to special relativity. - D. J. Griffiths, Introduction to Electrodynamics: Chapter 8 (momentum in fields) and Chapter 12 (electrodynamics and relativity) [memory, high]. - W. Rindler, Relativity: Special, General, and Cosmological (Oxford, 2006): includes a discussion of the right-angled lever [memory, medium]. - J. A. Shapiro, "Torques without Rotation: the Right-Angle Lever", arXiv:1206.4487 (2012) [web: https://arxiv.org/abs/1206.4487]; in research/files/. - The American Journal of Physics thread on the lever: J. C. Nickerson and R. T. McAdory, "Right-angle lever paradox" (1975), and the comment by G. Cavallieri, O. Grøn, G. Spavieri and G. Spinelli, Am. J. Phys. 46(1), 108 (1978) [read: references of Shapiro 2012, via research/2026-10-04-lever-paradox.md; the 1975 date is the Investigator's memory, medium; not opened]. - H. E. Ives and G. R. Stilwell, "An Experimental Study of the Rate of a Moving Atomic Clock", J. Opt. Soc. Am. 28, 215 (1938): the canal-ray measurement of the time-dilation factor Tolman proposed testing in section 46 [memory, medium-high for the citation; overview at web: https://en.wikipedia.org/wiki/Ives%E2%80%93Stilwell_experiment]. - J. A. Heras and M. G. Osorno, "A note on the relativistic temperature", arXiv:2204.12572 (2022): its reference list is a bibliography of the whole temperature debate [web: https://arxiv.org/abs/2204.12572]; in research/files/. - S. Pouryazdan and B. Vakili, "Relativistic transformation of temperature revisited", Phys. Lett. A 590 (2026) 131860 [web: https://arxiv.org/abs/2606.00521]; in research/files/. - A. A. Martinez, "Ritz, Einstein, and the Emission Hypothesis", Physics in Perspective 6 (2004) 4-28 [web: https://www.martinezwritings.com/m/SiteMap_files/RitzEinstein.pdf]; in research/files/. To obtain next: J. G. Fox's papers (Am. J. Phys. 1962, 1965) [memory, medium on details]. - Wikipedia overviews used by the reviewers (secondary): "Tests of special relativity", "Kaufmann-Bucherer-Neumann experiments", "Trouton-Noble experiment".

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Answers to "Check yourself"

1. sqrt(1 - 0.64) = 0.6. l2 = 2 x 0.6 = 1.2 m; F2 = 50 x 0.6 = 30 N. F1 l1 = 100 N·m, F2 l2 = 36 N·m, net torque = 64 N·m = 100 x 0.8². V = 0.8c = 2.398 x 10⁸ m/s; energy current F1 V = 1.20 x 10¹⁰ W; p_y = F1 V l1/c² = 1.20 x 10¹⁰ x 2 / 8.99 x 10¹⁶ = 2.67 x 10⁻⁷ kg·m/s. Check: V p_y = 2.398 x 10⁸ x 2.67 x 10⁻⁷ = 64 N·m.
2. Power is force dotted with velocity. Every point of the lever moves along x at V. F2 points along y, so F2·V = 0. F1 points along x, parallel to the motion of A, so it delivers F1 V watts.
3. The body stores a constant momentum p_y at right angles to its motion. As the body moves along x, the x-coordinate of that stored momentum grows as Vt, so x p_y grows at the rate V p_y. The angular momentum about a fixed point grows because the lever arm grows, not because anything spins.
4. The choice of kinetic potential (equivalently, the definition of heat in the moving frame). Defining temperature only in the rest frame, so that T is not taken from the moving-frame H at all, gives T = T0 (Landsberg). Tolman's factor sqrt(1 - u²/c²) comes entirely from his H; he admits H "was not absolutely uniquely determined".
5. In 1917 the Ritz variant was Never properly tested (not dismissed without testing; Tolman ran a sun-limb test in 1910 and proposed a sunlight Michelson-Morley test in 1912). Today it is Tested, not supported (Alvager 1964; Brecher 1977, designed with extinction in mind). Fox's objection, as reported by Martinez: by the Ewald-Oseen extinction theorem, light that passes through matter is re-emitted at c relative to that matter, so tests whose light went through glass, air or stellar atmospheres could not reveal the source's speed.