E. T. Whittaker: waves, potentials and an undulatory gravity: a primer

Project folder: Researchers/Whittaker-ET. It holds works by Sir Edmund Taylor Whittaker (1873-1956): the 1903 Mathematische Annalen paper "On the partial differential equations of mathematical physics" (scan, no text layer), the 1904 Proceedings of the London Mathematical Society paper on "two scalar potential functions", Volume 1 of A History of the Theories of Aether and Electricity (1951 edition) and its 1910 first edition, and Volume 2 of the History (The Modern Theories 1900-1926, 1953, added to the folder on 2026-10-07), plus the project README. The Investigator added Whittaker's own 1902 note in Monthly Notices of the Royal Astronomical Society (MNRAS), G. Johnstone Stoney's 1903 critique, obituaries and later work by others (research/ and sites.md).

Source note (repeated from claims.md and arbiter.md, updated after the second QE revision). The 1903 paper has no text layer, so every equation in it was read from page images. Every page has now been read that way, including section 5 parts 2-4 (journal pp. 347-353, the Helmholtz equation and generalized Bessel functions), which are pure mathematics and make no physical claim. In the 1951 History, Chapters V, VIII, IX and XIII and the scattered gravity passages were deep-read (page images where equations or exact wording mattered); the rest is digest only. The 1910 edition was compared with 1951 on the gravity and aether passages only. Page convention: the 1951 PDF holds two book pages per PDF page, so citations give the PDF page ("PDF p.110") and, where useful, the printed book page ("bk 207"). History Vol. 2 (1953) is now read: all 15 chunks digested; Chapter V "Gravitation" (PDF pp. 155-207, bk 144-196; in that chapter PDF = bk + 11) deep-read from the text layer with page images for the key sentences and equations; four other passages (Cunningham's aether, Planck and Nernst on zero-point energy, Einstein "with some amplifications", Hargreaves' potentials) checked by text search and context reading only; the rest of the volume (radioactivity, most of Chapter II on relativity, the quantum chapters) rests on digests. The further deep reads QE asked for (the Vol. 2 index, Chapter II, the preface, and Faraday in Vol. 1) were not run, so the Chapter II framing in section 4.9 rests on text search and context reading, and the Faraday item in section 4.8 on the text layer (read for this primer, not from page images). Earlier versions of this primer said Vol. 2 was "not obtained" and "unread"; that was true when written and is now out of date. Not obtained by anyone: the 1927 gravity paper beyond its first section, the 1928 lecture "The influence of gravitation on electromagnetic phenomena", the 1944 Endeavour article "Old and new ideas of gravitation", From Euclid to Eddington (1949), any reply by Whittaker to Stoney, E. T. Copson's 1928 paper on a point charge in a gravitational field, the 1902-03 newspaper coverage, and Whittaker's papers at the Fisher Library, Toronto (MS COLL 00219). Where a statement rests on [memory] or on web notes alone, it says so. The gravity passage (journal pp. 354-355) was read from page images by four separate readers, and their readings agree.

Source tags: [read] = from the project's documents (page given where useful); [web] = Internet source, URL or the research/ note that holds it; [memory] = recalled, with confidence; [calc] = computed for this primer or by a reviewer and reproduced in example.py.

The hypothesis put to the review that brought this project in (follow-up Q07): is gravity a longitudinal standing wave in the same medium that carries light? The question has two historical homes, and this primer keeps them apart (sections 4, 5 and 7). Whittaker's 1902-03 gravity note is the origin of "gravity as waves in the light medium". The 19th-century pulsation models of Bjerknes, Korn and Larmor, which Whittaker's own History records, are the origin of "gravity from in-phase longitudinal oscillation of a medium". The Q07 phrase is closer to the second (arbiter, QE revision). Whittaker's own History Vol. 2 (1953) adds a third, exact sense of "the same medium": for light, a static gravitational field behaves like a medium whose permittivity and permeability vary from place to place (section 4.9, section 5 Part H). That sense is standard general relativity; it involves no longitudinal waves and no speed above light's.

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1. The idea in one paragraph

In 1902-03 Whittaker, then a young Cambridge mathematician and Secretary of the Royal Astronomical Society, proved two elegant theorems. First, every solution of Laplace's equation (the equation obeyed by a static gravitational or electric potential in empty space) can be built by adding up very simple pieces, each depending on space through one combined variable. Second, every solution of the wave equation can be built by adding up plane waves travelling in every direction. From the second theorem he drew a physical suggestion. A steady gravitational field can be written as a sum of waves whose ups and downs cancel, leaving a field that does not change in time. Each of those waves would be longitudinal: its force points along the direction the wave travels, unlike light, whose forces lie across the direction of travel. He thought it "probable that an identical property of the medium" carries both light and these gravity waves, and that the gravity waves might travel "enormously" faster than light. He was careful to say this "does not explain the cause of gravity". A year later he showed that the whole electromagnetic field of moving electrons can be built from just two scalar functions instead of the usual four. The mathematics is correct and still used (it is an ancestor of twistor theory). The gravity suggestion was criticised within months by Stoney, was never developed further in print by Whittaker as far as we know, and in the forms that can be tested today it does not fit measurements of gravitational waves. Fifty years later, in Volume 2 of his own History (1953), Whittaker wrote that the old reasons for a gravity speed far above light's "do not to twentieth-century minds seem very plausible", that relativity requires gravity to travel at the speed of light, and that gravitational waves do so. He never mentions his own 1902-03 suggestion there: he neither repeats it nor withdraws it. In the same book he describes Einstein's curved space-time as FitzGerald's aether under a new name, and shows that for light a static gravitational field acts exactly like a transparent medium whose properties change from place to place. Later writers (Barrett, Bearden, Titleman) read these papers as describing "standing" longitudinal waves; that word is theirs, not Whittaker's. Whittaker's own History describes older models in which bodies pulsating in step in a fluid attract each other by an inverse-square law, and some 19th-century physicists used them for gravity. Those models are the nearest historical relatives of a "longitudinal standing wave" picture of gravity.

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2. What you need to know first

Laplace's equation and potentials. A potential V(x, y, z) is a scalar function whose negative gradient gives a force per unit mass (gravity) or per unit charge (electrostatics). Away from all masses or charges it obeys Laplace's equation, ∇²V = ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 0. The potential of a point mass, V = −GM/r, is the basic example (G is Newton's constant, M the mass, r the distance). Solutions of Laplace's equation are called harmonic functions. Read: Griffiths, Introduction to Electrodynamics, Ch. 2 and 3 (electrostatic potential, Laplace's equation).

The wave equation. ∇²V = (1/c²) ∂²V/∂t², where c is the wave speed. Whittaker writes it ∇²V = k² ∂²V/∂t², so his k is 1/(wave speed). Any function of the form F(n·r − ct), with n a unit vector and r = (x, y, z), is a solution: a shape F that slides along n at speed c without changing. That is a plane wave of arbitrary profile. Spherical waves have the form F(r − ct)/r. Read: French, Vibrations and Waves (MIT Introductory Physics Series); Griffiths Ch. 9.

Fourier analysis. Almost any reasonable function can be written as a sum (series) or integral of sines and cosines of different frequencies; the weights form its spectrum. A sudden step has a spectrum falling off as 1/(frequency). If you cut the spectrum off sharply at a maximum frequency, the rebuilt step overshoots by about 18 percent near the jump and rings: the Gibbs phenomenon. Two integrals carry the whole worked example: ∫₀^∞ sin(μa)/μ dμ = (π/2) sgn(a), where sgn(a) is +1 for a > 0, −1 for a < 0 and 0 at a = 0; and its cut-off version, ∫₀^M sin(μa)/μ dμ = Si(Ma), where Si is the sine integral function. Read: Boas, Mathematical Methods in the Physical Sciences, the chapter on Fourier series and transforms.

Longitudinal versus transverse waves. In a longitudinal wave the quantity that oscillates points along the direction of travel (sound in air: the air moves back and forth along the ray). In a transverse wave it points across the direction of travel (a wave on a string; light, where the electric and magnetic fields lie in the wave front). Read: French, Vibrations and Waves.

Standing versus travelling waves. Two equal waves running in opposite directions add to a standing wave: sin(kx − ωt) + sin(kx + ωt) = 2 sin(kx) cos(ωt). It has fixed nodes (places that never move) and every point oscillates in time. Conversely a travelling wave is the sum of two standing patterns a quarter-period apart: sin(kx − ωt) = sin(kx)cos(ωt) − cos(kx)sin(ωt). Here k (the wavenumber) is 2π divided by the wavelength and ω (the angular frequency) is 2π times the frequency; for a wave of speed c, ω = ck. Read: French, Vibrations and Waves.

Electromagnetic potentials and gauge. The electric field E and magnetic field B can be written with a scalar potential φ and a vector potential A: E = −∇φ − ∂A/∂t, B = ∇×A. Different (φ, A) pairs related by a gauge transformation give exactly the same fields, so the potentials are a bookkeeping device with some freedom in them; only E and B are measured. Read: Griffiths Ch. 10 (potentials and fields).

What general relativity (GR) says about gravitational waves, in plain words. Einstein's theory describes gravity as the curvature of space and time, not as a force carried by a medium. Its waves travel at the speed of light, and they are transverse and tensor: a passing wave stretches space in one direction across its path while squeezing it in the perpendicular direction, then the reverse. "Tensor" here just means the effect needs a small table of numbers (stretch along each pair of directions) to describe, not one number. A "scalar" or "breathing" mode would expand and shrink a ring of particles uniformly; a "longitudinal" mode would stretch along the direction of travel. GR has neither. Read: Schutz, Gravity from the Ground Up (no tensor background needed); Will, Was Einstein Right?

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3. The textbook view

Symbols used throughout. V: a potential; Whittaker also uses V for the wave speed in his gravity example (it appears only in the product Vt), and this primer follows him there. c: the speed of light. r: distance from the source. t: time. μ (mu): the wavenumber of one constituent wave (2π/wavelength). a = Vt − r: how far behind the wave front a point lies (positive inside the front). M: a maximum wavenumber (a cutoff). G, M_sun: Newton's constant and the Sun's mass; GM_sun = 1.327 × 10²⁰ m³/s² [memory, high]. g: gravitational acceleration.

3.1 Static gravity. Newton: a point mass M produces potential V = −GM/r and acceleration g = GM/r² toward it. V satisfies Laplace's equation everywhere except at the mass. At the Earth's distance from the Sun, 1 astronomical unit (AU) = 1.496 × 10¹¹ m, the Sun's pull is g = 5.93 × 10⁻³ m/s² [calc, example.py Part 5].

3.2 A static field obeys the wave equation for every speed. If V does not depend on t, then ∂²V/∂t² = 0, so ∇²V = 0 implies ∇²V = k²∂²V/∂t² for any k at all. Whittaker uses exactly this ("à fortiori ... where k is any constant") [read: 1903 p.354]. The consequence: a static field contains no information about the speed of any waves it might be built from. Only changing or moving sources can reveal a speed.

3.3 Every scalar-potential wave is "longitudinal". For a plane wave V = F(n·r − ct), the force −∇V = −F′ n points along n, the direction of travel. This holds for any F and any speed [calc, example.py Part 1]. So "longitudinal" is not a discovery about a medium; it is what you get whenever a force is the gradient of a single scalar. Light is transverse because its fields come from curls of vector potentials, not from the gradient of a scalar.

3.4 The modern view of gravity's speed and waves. In GR, changes in gravity propagate at c, and freely travelling gravitational waves are transverse tensor waves. Three measurements bear directly on Whittaker's suggestion: - Speed. The neutron-star merger GW170817 and its gamma-ray burst arrived about 1.7 s apart after travelling about 130 million years, giving −3 × 10⁻¹⁵ ≤ (v_gw − c)/c ≤ +7 × 10⁻¹⁶ [web: https://arxiv.org/abs/1710.05834]. - Polarization. The LIGO-Virgo-KAGRA (LVK) collaboration's tests on its third catalogue (GWTC-3) found no evidence for non-tensor modes [web: https://arxiv.org/abs/2112.06861]. Ground interferometers respond only weakly to a pure breathing mode, so a small scalar admixture is still allowed (Jin and Zheng, 2026 preprint, not peer reviewed) [web: https://arxiv.org/abs/2610.08588]. - Radiated energy. The Hulse-Taylor binary pulsar PSR B1913+16 loses orbital energy at 0.9983 ± 0.0016 of the GR prediction for tensor (quadrupole) radiation [web: https://arxiv.org/abs/1606.02744].

3.5 The old aberration argument, and its answer. Laplace modelled gravity as the push of a "gravific fluid" streaming toward the Sun, which gives an aberration-type drag on a moving planet. From the Moon's secular motion he "found that the velocity of the gravific fluid must be at least a hundred million times greater than that of light", that is, at least 10⁸ c. Whittaker's comment: "The assumptions made by Laplace are evidently in the highest degree questionable" [read: History 1951, PDF p.110, bk 207]. A figure of about 7 × 10⁶ c is sometimes quoted for Laplace; that is [memory, medium], unchecked, and not Whittaker's number. Modern versions of the aberration argument for Newtonian gravity give above 2 × 10¹⁰ c. Carlip (2000) showed that in GR the aberration is almost exactly cancelled by velocity-dependent terms, so c_g = c is allowed [web: https://arxiv.org/abs/gr-qc/9909087]. Whittaker's "may be enormously greater" fits the pre-relativity side of that argument.

Whittaker's own later view is on record. In Vol. 2 (1953) he writes, introducing Lorentz's 1900 paper: "the assumptions of Laplace's investigation, which led him to conclude that the velocity of propagation of gravitation must be vastly greater than that of light, do not to twentieth-century minds seem very plausible" [read: Vol. 2 bk 149, PDF p.160, page image]. A footnote files a second 19th-century finite-speed author, R. Lehmann-Filhés (1895), under the same remark. Two pages later: "From 1904 onwards ... Poincaré ... pointed out that if relativity theory were true, gravity must be propagated with the speed of light, and ... showed that this supposition was not contradicted by the results of observation, as Laplace had supposed it to be" [read: bk 151, PDF p.162, page image]. Poincaré's point, in modern terms, is the same as Carlip's: velocity-dependent terms in a law that respects relativity cancel the first-order aberration [memory, high]. Whittaker also cites Paul Gerber's velocity-dependent law (1898, 1917) as one possible account of Mercury's perihelion (bk 148), and states that no signal travels faster than c: the signal velocity "is the group velocity, which is never greater than c" (bk 159-160) [read]. Note what the Laplace sentence rejects: Laplace's assumptions (a gravity whose direction aberrates), not every possible superluminal speed.

3.6 Other tests that a "gravity in a medium" picture must pass. All materials fall alike to about 10⁻¹⁵ (MICROSCOPE satellite) [web: https://arxiv.org/abs/2209.15487]. Light bends near the Sun exactly as GR says (Cassini: γ = 1 + (2.1 ± 2.3) × 10⁻⁵; γ = 1 is GR) [web: https://doi.org/10.1038/NATURE01997]; a pure scalar gravity coupled to mass, such as Nordström's 1913 theory, gives no light bending [web: https://en.wikipedia.org/wiki/Scalar_theories_of_gravitation]. The inverse-square law holds down to 52 micrometres (Eöt-Wash torsion balances) [web: https://arxiv.org/pdf/2002.11761].

Why a single scalar potential was given up, as Whittaker tells it. Whittaker's 1902-03 gravity is exactly one scalar potential V written as a sum of waves, so the history of scalar gravity is the history his idea would have had to face. His Vol. 2 (1953) tells it [read: bk 151-157, PDF pp. 162-168, page images]: - Abraham (1911-12) let the speed of light c depend on the gravitational potential and took "the negative gradient of c" as the direction of the gravitational force. Einstein (1912) wrote a static theory of the same kind, with the local light speed c(x, y, z) playing the part of the potential. - Nordström (1912-14) is named with references only; Whittaker gives no equations and does not call the theory "scalar". The "scalar" label and the no-bending result above come from the [web] source, not from Whittaker. - Einstein and Grossmann (1913) replaced the one potential by ten functions g_pq, which describe the geometry of space-time: "a tremendous innovation, because it implied the abandonment of the time-honoured belief that a gravitational field can be specified by a single scalar potential-function." - Einstein's stated reason: compare black-body radiation in a movable mirror-walled box with the same radiation held in a fixed vertical shaft between two pistons joined by a rod. On the single-scalar theory "the work necessary to raise the radiation upwards against the force of gravity would in the second system be only one-third of the work required in the first system: a conclusion which was obviously wrong." Einstein added that his "strongest reason" was his conviction that the laws of physics should keep their form under a much wider group of coordinate changes (general covariance).

Where the "one-third" comes from [memory, medium; not in Whittaker's text]: the pressure of radiation is one-third of its energy density, p = u/3. A scalar theory couples gravity to a single number built from energy and pressure, and for radiation that number is not the full energy u. (For light the combination u − 3p is exactly zero, which is why Nordström's theory does not bend light at all.) Whatever holds the radiation, its weight ought to be its full energy divided by c², as Planck had argued in 1907 ("all energy must gravitate", Whittaker bk 151 [read]). So one scalar potential cannot be all of gravity if light and energy are to fall like everything else.

In the same chapter Whittaker also records models that do reproduce the classic tests with one scalar potential in ordinary flat space (section 4.9), and he warns that "It is unwise to accept a theory hastily on the ground of agreement between its predictions and the results of observation in a limited number of instances" (bk 182) [read].

3.7 Maxwell's energy obstacle (1864). Whittaker's own History records that Maxwell tried to put gravity into a medium the way he had put electricity and magnetism there, and gave up: because like masses attract, the energy stored in the gravitational field is negative (in modern form, −g²/(8πG) per unit volume [memory, high]), so a medium with positive energy would need "an enormously great" intrinsic energy wherever gravity is weak. "I cannot go any further in this direction in searching for the cause of gravitation" [read: History Vol. 1, Ch. VIII]. Whittaker notes that the positivity of energy was an assumption "everybody in Maxwell's day" made [read: same].

3.8 The two-potential result in textbook language (1904). Source-free electromagnetic fields have two independent polarizations, so two scalar functions are the minimum needed to describe them; the usual four potentials (φ, A_x, A_y, A_z) carry gauge redundancy. The modern names for Whittaker's construction are Hertz vectors or Debye potentials [memory, high]. The price of using only two is that they are tied to a chosen axis.

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4. Where Whittaker departs, side by side with the textbook

Whittaker's mathematics does not depart from textbook physics at all; it became textbook mathematics. The departure is one physical reading, set out in his 1902 note and repeated in 1903.

4.1 The two theorems (1903) [read: deep read, page images]. - Laplace: V = ∫₀^{2π} f(z + ix cos u + iy sin u, u) du, with f arbitrary. The reason it works: the combination s = z + ix cos u + iy sin u has gradient (i cos u, i sin u, 1), whose "length squared" is −cos²u − sin²u + 1 = 0, so ∇²f(s) = f″(s) × 0 = 0 [calc, example.py Part 1]. The arguments must be complex: no real direction has zero length. So the Laplace building blocks are not real waves. - Wave equation: V = ∫₀^{2π}∫₀^π f(x sin u cos v + y sin u sin v + z cos u + t/k, u, v) du dv. Here the building blocks are real plane waves travelling in every direction (u, v); the space part of the gradient has length 1 and the time part has length 1, and they cancel because space and time enter the equation with opposite signs [calc]. - Both are proved degree by degree for polynomial solutions; the step to the full solution (summing over degrees) is asserted, and the result is local, near a point where V is smooth enough to have a convergent Taylor series. The 1/r identity on journal p.334 holds only above the mass (z > c) and gives −1/r below it [calc, deep read]. Several misprints are listed in the deep read; none affects the theorems. Whittaker says "general finite solution", and the arbiter's verdict is Established in that local sense.

4.2 The gravity note, in his words. There are two texts: the 1902 MNRAS note (pp. 619-620) [web: https://academic.oup.com/mnras/article/62/9/617/1015465; research/2026-10-07-oup-mnras.md] and the 1903 paper (journal pp. 354-355) [read: 1903 page images pp.23-24]. Most of the argument is shared, but the two are not identical, so the wording is given for each where they differ.

Shared by both texts: - "distributions of simple waves exist such that, although each individual wave is periodic with respect to the time, yet the total disturbance at any point does not vary with the time" (quoted from 1902; 1903 makes the same point). - Example: a particle sends out spherical waves; the band of wavenumbers between μ and μ + dμ contributes (2 dμ / πμ) sin μ(Vt − r)/r, where V is the wave speed. "After the waves have reached the point r, so that (Vt − r) is positive", the sum is (2/πr)∫₀^∞ (sin y / y) dy = 1/r, the gravitational potential of the particle. - The field can be analysed "by a 'spectrum analysis' as it were, into an infinite number of constituent fields", each "of an undulatory character". - Longitudinal: "the varying vector is a gravitational force perpendicular to the wave-front", against light, "in which the varying vectors consist of electric and magnetic forces parallel to the wave-front". 1903 then states, in italics: "the waves will be longitudinal". - Medium: "it seems probable that an identical property of the medium ensures their transmission through space." The sentence before it differs: 1902 says the waves "are in other respects of the same type in the two cases"; 1903 says they "are in other respects exactly similar". - Speed: gravity would be "propagated with a finite velocity, which however need not be the same as that of light, and may be enormously greater" (1903 wording; 1902 is the same apart from punctuation).

The closing sentence, which is where the two texts differ most: - 1902: "this investigation does not explain the cause of gravity; all that is done is to show that the propagation across space of forces which vary as the inverse square of the distance does not require for its explanation any other property of the medium than one which has long been known and accepted." - 1903: "Of course, this investigation does not explain the cause of gravity; all that is done is to shew that in order to account for the propagation across space of forces which vary as the inverse square of the distance, we have only to suppose that the medium is capable of transmitting, with a definite though large velocity, simple periodic undulatory disturbances, similar to those whose propagation by the medium constitutes, according to the electromagnetic theory, the transmission of light."

So 1903 makes two things explicit that 1902 leaves implicit: the speed is "definite though large" (finite), and the property in question is the one that carries light "according to the electromagnetic theory", not an elastic-solid mode of the older aether. Neither text uses the word "standing" [read; web]. The 1903 section heading says gravitation is "explained as a mode of wave-disturbance", which claims more than the closing sentence does (arbiter R7). A contemporary reviewer, Wangerin, summarised the 1903 section in the Jahrbuch über die Fortschritte der Mathematik as gravity from "longitudinal waves ... with very great speed in a medium" [web: zbMATH JFM 34.0827.01, research/2026-10-07-zbmath.md], so readers of the time took it the same way.

4.3 Side by side.

4.4 What the reviewers added that Whittaker did not say [calc; physicist, mathematician and contrarian agree; arbiter confirms]. 1. The full sum is sgn(Vt − r)/r: +1/r inside the front (r < Vt) and −1/r outside it. The field is steady only inside a sphere that grows at speed V. 2. The source is a step. At r → 0 the strength multiplying 1/r is (2/π)∫ sin(μVt)/μ dμ = sgn(t): the source changes sign from −1 to +1 at t = 0 (a static −1/r plus a switched-on 2/r). The weights 2/(πμ) are simply the spectrum of a step. 3. The time-varying parts of the constituents do not vanish; they pile up into a thin shell at the moving front, where the force has a spike 2δ(Vt − r)/r (δ is the Dirac delta, an infinitely thin unit spike). 4. "Longitudinal" follows from using a scalar potential (section 3.3), and "any speed" follows from the field being static (section 3.2).

4.5 Stoney's 1903 objection, and the answer to it. G. Johnstone Stoney (the physicist who named the electron) praised the mathematics as "striking and most useful" and "a singularly elegant proof", then objected on physical grounds (MNRAS 63, 424-428) [web: research/2026-10-07-oup-mnras.md]. Each constituent force has a part falling as 1/r² and a part falling as 1/r, both "flickering in intensity" at the wave's frequency, and they add to a steady force only for "one definite set of coefficients". "The improbability of this assumption is so overwhelming that no physicist can feel himself justified in seriously arguing from the supposition that it represents the state of things which exists in nature, however much he may be impressed by the mathematical ingenuity of the investigation." His arithmetic is right (it is reproduced in section 5). The reviewers' answer is that the "special set" is just the spectrum of a step, which is not exotic; the remaining physical cost is a source whose strength flips from −1 to +1 at t = 0 (equivalently a static −1/r plus a switched-on 2/r), with a front expanding at speed V. A postscript records that someone at the RAS meeting raised Mercury's orbit as a sign that Laplace's equation needs changing for moving bodies. No reply by Whittaker has been found.

4.6 The two-potential paper (1904) does not contain the gravity idea. The 1904 paper builds the field of any moving electrons from two scalar functions F and G, built from retarded electron positions and tied to a chosen axis, and the deep reader checked numerically that it reproduces the standard (Liénard-Wiechert) field of an accelerating electron to 12 digits [read: deep read]. The word "longitudinal" does not occur, and gravity is not mentioned. Each plane-wave term gives an ordinary transverse electromagnetic wave [calc: mathematician]. In 1951 Whittaker himself described F and G as corresponding "to two plane-polarised components" of a light wave [read: History Vol. 1, Ch. XIII, PDF p.212]. The reading that 1904 shows light as "the interference of two longitudinal waves" comes from Bearden, Barrett (1993) and Titleman (arXiv:2205.08309) [web: research/2026-10-07-qeios.md]. It is their interpretation and stays in the record as theirs; the arbiter judges it Not testable as stated and at odds with the paper's text.

That reading has a 19th-century root, which Whittaker himself records [read: History 1951, PDF p.152, bk 291, page image]. In his 1861 lectures Riemann noticed that his potentials satisfy ∂φ/∂t + c div a = 0 (φ the scalar potential, a the vector potential, div the divergence), the same condition as L. Lorenz's potentials. "This appeared to Riemann to indicate that φ might represent the density of an aether, of which ca represents the velocity." Compare the continuity equation of a compressible fluid, ∂ρ′/∂t + ρ₀ div u = 0 (ρ′ a small change of density, ρ₀ the mean density, u the fluid velocity): with φ as density and ca as velocity, Riemann's equation says the same thing. On that reading a wave in φ is a compression (longitudinal) wave of the aether, which is what the modern density-wave reading of Whittaker's potentials asserts. Whittaker adds the difficulty: "the electric and magnetic forces correspond to second derivates of the displacement", so the field energy is hard to match to the model's energy. A modern limit [memory, high]: the condition is today called the Lorenz gauge condition, and since the potentials can be changed by a gauge transformation without changing E and B, a "density" read off φ depends on the gauge chosen. Riemann's φ is also a different object from Whittaker's 1904 F and G, which Whittaker calls plane-polarised components. A published critique of the modern reading also exists: Trovon de Carvalho and Rodrigues (2001), "The Non Sequitur Mathematics and Physics of the 'New Electrodynamics' Proposed by the AIAS Group", criticises the AIAS (Evans, Bearden and others) use of Whittaker's papers. It is known here only through Titleman's citation [web: research/2026-10-07-qeios.md; not read], and its title is those authors' characterization. It stands in the record beside Titleman's view, not in place of it.

4.7 Where the Q07 "longitudinal standing wave" hypothesis fits. There are three honest ways to state it, and they behave very differently: - (a) Spectrum form. Whittaker's steady field can be rewritten exactly as a sum of standing spherical waves of all frequencies, because each travelling wave is two standing patterns a quarter-period apart (section 2). Section 5 Part D shows that, inside the front, the whole 1/r comes from one family of standing waves. This form is mathematically correct, but it reproduces Newton exactly, so no static measurement can see it. - (b) Single-mode form. A real standing wave at one frequency, with nodes, makes g oscillate in time. Section 5 Part E shows how slowly it would have to oscillate to mimic the inverse-square law in the solar system, and why a pure standing wave cannot be all of gravity. - (c) Pulsation form (Bjerknes, Korn, Larmor; section 4.8). Every gravitating body pulsates (changes volume) in step with every other, and a medium carries the compressions between them. This is not Whittaker's idea. It is the older, separate home of "gravity from longitudinal oscillation", and the Q07 phrase is closer to it than to Whittaker's travelling waves. The field it needs is an oscillating near-field with a common phase, not a textbook standing wave with fixed nodes: a node between two bodies would reverse the force. Section 5 Part G puts numbers on its built-in condition (attraction only within a quarter wavelength).

4.8 What Whittaker's own History records: the 19th-century relatives of the idea. These are other people's hypotheses, reported by Whittaker with references but with no test and (except where shown) no numbers. They are included because they are the closest historical company for both Whittaker's note and the Q07 hypothesis; none of them is evidence for gravity being a wave. All [read: History 1951, pages given; deep reads in deep/; the passages are substantially the same in the 1910 edition except where noted]. - Riemann 1853: gravity from the aether's resistance to compression (PDF p.127, bk 240). In a fragment published after his death, Riemann proposed an aether whose elements resist compression and also resist changes of orientation. "The former property he conceived to be the cause of gravitational and electrostatic effects, and the latter to be the cause of optical and magnetic phenomena." Whittaker says it "was apparently not developed further". A compressional (longitudinal) stiffness of the light medium as the seat of gravity is the most direct ancestor of Whittaker's 1902 suggestion. - Maxwell 1864 (PDF p.136): tried to put gravity in the medium and gave up over the sign of the energy (section 3.7). - Kelvin 1896: condensational waves (PDF p.140, bk 267). Kelvin had "some inclination to speculate that alterations of electrostatic force due to rapidly changing electrification are propagated by condensational waves in the luminiferous aether". "Condensational" means compressional, that is, longitudinal. FitzGerald, writing privately to Heaviside on 11 June 1896, said he "tried hard to knock down Lord K." on this. That is one party's account of a private discussion. This is electrostatics, not gravity, but it is the same kind of longitudinal mode, proposed by Kelvin six years before Whittaker's note. - Riemann 1861: the scalar potential as aether density (PDF p.152). See section 4.6. - Bjerknes: pulsating spheres (PDF p.149, bk 284-285, page image). "Two spheres which are immersed in an incompressible fluid, and which pulsate (i.e. change in volume) regularly, exert on each other (by the mediation of the fluid) an attraction, determined by the inverse square law, if the pulsations are concordant; and ... a repulsion ... if the phases of the pulsations differ by half a period. If the phases differ by a quarter period, there is no action." (The quarter-period sentence is new in 1951.) Then the key condition: "It is necessary to suppose that the medium is incompressible, so that all pulsations are propagated instantaneously; otherwise attractions would change to repulsions and vice versa at distances greater than a quarter wave-length." Whittaker cites W. M. Hicks (Proc. Camb. Phil. Soc., 1879-80) for the mathematics; Hicks was not read in this review. Note the sign: like phases attract, which is gravity's sign and the opposite of electric charges. - Leahy 1884: the elastic-medium reversal (PDF p.149, bk 285). For spheres pulsating in an elastic medium, "the wave-length of the disturbance being supposed large in comparison with the distance between the spheres", Bjerknes' results are "reversed": like phases repel and unlike attract, still by the inverse square. This is a separate effect from the quarter-wave sign change. It happens at short range and comes from the kind of medium. So any pulsation theory has to say what the medium is (incompressible fluid, compressible fluid or elastic solid) before it can even give the sign of the force. - Korn 1898: a pulsation theory of gravitation (PDF p.149, footnote; also in 1910). "A theory of gravitation has been based by Korn on the assumption that gravitating particles resemble slightly compressible spheres immersed in an incompressible perfect fluid; the spheres execute pulsations, whose intensity corresponds to the mass of the gravitating particles, and thus forces of the Newtonian kind are produced between them" (Korn, Eine Theorie der Gravitation und der elektrischen Erscheinungen, Berlin 1898; not read). Because Korn's fluid is incompressible, the action is instantaneous: this is a pulsation theory, not a wave theory. - Larmor 1895: gravity between pulsating electrons (PDF p.150, bk 286; new in 1951). "Larmor suggested that gravitation between electrons might be explained if the electrons are vacuous and have free periods of vibration in the fluid aether, the explanation following the hydrodynamical pulsatory theory of Bjerknes" (Phil. Trans. 186, 1895, p.697; not read). Larmor was the leading Cambridge aether theorist of Whittaker's time [memory, high], so pulsation pictures of gravity were current in Whittaker's own milieu when he wrote his 1902 note. - How the older aether handled its longitudinal mode (Ch. V, PDF pp.71-81). In every elastic-solid aether, light (transverse) and a compressional (longitudinal) wave are two modes of one medium, set by two elastic constants. Poisson's speeds are √(n/ρ) for the transverse wave and √((k + 4n/3)/ρ) for the longitudinal one, with n the rigidity, k the resistance to compression and ρ the density. Because no longitudinal light was ever seen, theorists made the mode harmless in one of three ways. They made it infinitely fast and uncoupled (Fresnel's "indefinitely great" speed, Stokes, Green's nearly incompressible aether "like jelly"). They made its speed zero, so it carries no energy (Cauchy's and Kelvin's contractile aether, k + 4n/3 = 0). Or they built an aether that has no such mode at all (MacCullagh's rotational aether). Only Cauchy expected it might be found, perhaps as heat. Maxwell's medium has no longitudinal light mode either: its waves come from curls, not gradients (section 3.3). Two cautions. Whittaker's phrases "identical property of the medium" and "enormously greater" do not occur in Ch. V; they are his own 1902-03 wording. And his 1903 closing ties the property to the electromagnetic transmission of light, not to the elastic-solid mode. So this history is context for why a very fast longitudinal mode in the light medium was a respectable thought in 1902, not a source Whittaker names. - The speed of gravity before relativity (PDF pp.16-23, 32-33, 56, 110-111, 154). Newton conjectured an aether density gradient but gave no speed. In Cotes' day (1713) gravity was taken as instantaneous. Le Sage (1782) explained the inverse square by mutual shadowing of a rain of "ultramundane corpuscles". Aepinus (1759) thought gravity might be a small residual of nearly balanced electric forces. Euler (1760-61) used aether pressure. Laplace set the lower bound of 10⁸ c (section 3.5). Tisserand (1872) put Weber's velocity-dependent electrical law, with propagation speed c, into the Sun's gravity and got 14″ per century of advance of Mercury's perihelion, "only 3/8" of the 38″ excess known then. Lévy (1890) got 3/4 with Riemann's law, and "a combination" gave all of it (PDF p.111, bk 208). The 3/8 + 3/4 = 9/8, so that must be a weighted mixture, not a sum [calc]. Whittaker's 38″ is his own figure; the modern excess is about 43″ [memory, high]. Finally, Kelvin's vortex atoms were praised because they made gravity instantaneous (PDF p.154). Whittaker's "may be enormously greater" belongs to this tradition: a very large speed was the usual way to escape Laplace's aberration problem. - Planck and Michelson (Ch. XIII, PDF pp.200, 202). Planck in 1899 considered a compressible aether "subject to gravity", compressed round the Earth like an atmosphere. Michelson in 1897 found no dependence of the speed of light on height. This is background for any idea in which gravity changes the medium near the Earth. - Faraday 1846 and 1851: one medium, several jobs (Vol. 1, PDF pp.103-104, bk 192-193; text layer read for this primer, not deep-read from page images). In his 1846 "Thoughts on Ray-Vibrations" Faraday suggested that an atom may be "nothing else than a field of force—electric, magnetic and gravitational—surrounding a point-centre", and that "light and radiant heat might be transverse vibrations propagated along these lines of force", so as to "'dismiss the aether'". So in the first field picture of light the vibrations are transverse, and gravitational lines of force are named but given no waves. Then, discussing the aether as a possible "vehicle of magnetic force", Faraday wrote in 1851: "it is not at all unlikely that if there be an aether, it should have other uses than simply the conveyance of radiations". Whittaker: "This sentence may be regarded as the origin of the electromagnetic theory of light." It is the earliest statement in the folder of the idea behind Whittaker's 1903 "identical property of the medium": one medium doing more than one job.

4.9 What Whittaker himself wrote in 1953 (History Vol. 2, Chapter V "Gravitation"). This is the inventor's own later, published view on exactly the questions his 1902-03 note raised. All [read: Vol. 2 deep read in deep/, book pages given, PDF = bk + 11; page images where marked in the deep read] unless tagged otherwise. It adds no experiment and no derivation that would change a verdict (arbiter, QE revision 2), but it changes what we know about Whittaker.

(a) Speed of gravity. Laplace's superluminal assumptions "do not to twentieth-century minds seem very plausible" (bk 149); relativity requires gravity "propagated with the speed of light", not contradicted by observation (bk 151); both are quoted in section 3.5. On waves: "If the distribution of matter in space is changed, e.g. by the circular motion of a plate in its own plane, gravitational waves are generated, which are propagated outwards with the speed of light", and "an electron exposed to gravitational waves must radiate" (bk 183, page image). Notice the source: in 1953 a gravitational wave is radiated by moving matter. In 1903 the waves were a time-independent superposition whose source was left unexplained ("this investigation does not explain the cause of gravity").

(b) The single scalar potential given up. See section 3.6: Einstein's one-third-work argument and general covariance, as Whittaker reports them (bk 156-157).

(c) From "structure of the aether" to "curvature", and gravity as a medium for light. Whittaker quotes FitzGerald (1894): "Gravity is probably due to a change of structure of the aether, produced by the presence of matter." Then, in his own voice: "The 'aether' of FitzGerald was called by Einstein simply 'space' or 'space-time': and FitzGerald's somewhat vague term 'structure' became with Einstein the more precise 'curvature'" (bk 157, page image). He continues: in a static gravitational field, "feeble" electromagnetic effects (too weak to change the curvature themselves) "happen exactly in accordance with the ordinary Maxwellian theory of electromagnetic phenomena taking place in a medium whose specific inductive capacity and magnetic permeability are aelotropic and vary from point to point". In modern words: a static gravitational field acts on light like a transparent material whose permittivity ε (old name "specific inductive capacity") and permeability μ change from place to place and may differ by direction ("aelotropic" is the old word for anisotropic). For a point charge near a point mass, he gives

ε = μ = (r + 1)³ / [r² (r − 1)],

crediting E. T. Copson, Proc. R.S. A 118 (1928) p.184 (bk 158, page image). The book does not say what r is measured in. The deep reader reconstructed it [calc]: r is the distance from the mass in units of GM/(2c²), measured in the "isotropic" coordinates in which space looks the same in every direction at each point. Far from the mass ε = μ ≈ 1 + 2GM/(c²ρ), with ρ the distance in metres. Section 5 Part H works this through and gets the full 1.75″ bending of starlight at the Sun's edge (an arcsecond, ″, is 1/3600 of a degree). Whittaker adds that FitzGerald may have meant exactly this, a change of ε and μ, with matter pulled toward higher ε "by analogy with the fact that in a liquid whose dielectric constant varies from point to point, an electrified body moves from places of lower to places of higher dielectric constant", an idea "later developed by E. Wiechert, Ann. d. Phys. 63 (1920) p.301" (bk 158).

How this bears on the Q07 hypothesis, "the same medium that carries light": this is the clearest passage in the folder in which Whittaker himself gives gravity a precise meaning as a property of the light-carrying medium, inside general relativity. But the property is a static index profile (the medium is a little "slower" near a mass), not a sum of longitudinal waves, and it implies no speed above c. Whittaker does not connect it to his 1903 note. An earlier version of this primer attributed the variable-medium comparison only to Whittaker's 1928 lecture, which called the resemblance "not quite perfect" [web: research/2026-10-07-cambridge-synge.md]. That is incomplete: Vol. 2 credits the point-charge result to Copson (1928) and the dielectric reading of FitzGerald to Wiechert (1920), and says "exactly" for static fields and feeble effects; it does not contain the words "not quite perfect". The two statements are compatible, and both stand. Copson was Whittaker's student and colleague at Edinburgh [memory, medium]. The polarizable-vacuum model of Dicke and Puthoff (ε = Kε₀, μ = Kμ₀) has the same structure with K = n [memory, high, per the deep read]; it is in this primer's evidence table as claim 4E.

(d) Lorentz 1900: gravity from very penetrating electromagnetic waves. Three years before Whittaker's note, Lorentz tried to replace Le Sage's corpuscles (section 4.8) by "trains of electric waves of very high frequency" crossing space in all directions: "Why should there not exist radiations far more penetrating than even the X-rays?" He found that a net force between two charged particles "could exist only if in some way or other electromagnetic energy were continually disappearing", and concluded that his assumptions "could not provide a satisfactory explanation of gravitation" (bk 149-150, page images). This is the energy-budget form of the classic objection to push-type gravity. Lorentz then took up an older idea (Mossotti, Weber, Zöllner): unlike charges attract slightly more strongly than like charges repel, and a difference of order 10⁻³⁵ would account for gravity. Whittaker says only that "no strong physical reason for an assumption of this kind could be given" in 1900; he does not say Lorentz rejected it (bk 150-151). Lorentz's model is different from Whittaker's: it is a pressure (shadowing) mechanism that needs absorption, while Whittaker's is a superposition that needs none. It is the nearest kin to claim 4 in the 1953 book, and like Maxwell's energy problem (section 3.7) it shows that any medium theory of gravity has to balance its energy books.

(e) The aether after relativity. Of Cunningham's 1914 scheme, which gives the aether a velocity of magnitude c at every point: "The scheme is relativistically invariant: so that an objective aether is not necessarily foreign to the point of view of the principle of relativity" (bk 248). In the gravitation chapter he says "free aether" in his own voice for field-free space. And the root of his 1951 "so rich in physical properties": Planck's second theory (1911-12), "the first appearance in theoretical physics of the doctrine of zero-point energy", and "in 1916 W. Nernst suggested that the aether everywhere might be occupied by zero-point energy" (bk 103-104) [read: text layer and context, not deep-read]. So the aether Whittaker keeps in 1951-53 is space-time with medium-like properties (varying ε and μ, zero-point energy), not a mechanical substance with its own longitudinal waves (arbiter, claim 5).

(f) His framing of relativity, beside a historian's. Chapter II is titled "The Relativity Theory of Poincaré and Lorentz". Einstein "published a paper which set forth the relativity theory of Poincaré and Lorentz with some amplifications"; the constancy of light speed was "an assertion which at the time was widely accepted, but has been severely criticised by later writers" (bk 40) [read: text layer; the rest of Chapter II rests on digests]. In Chapter V he credits Bateman (1909-10) with the variable quadratic form that Einstein and Grossmann then developed (bk 154-156). The historian Jaume Navarro (2020) characterizes the relativity chapter as shaped by Whittaker's commitment to the aether and a gravity-electromagnetism link [web: research/2026-10-07-sagepub.md, abstract only]. Whittaker's framing is the author's own; Navarro's is one historian's reading of it. The reader weighs both.

(g) Caution about general relativity in 1952, and the alternatives he records. Whittaker prints the eclipse measurements of light bending from 1919 to 1947 (1.61″ to about 2.24″, against Einstein's 1.75″) and concludes "at the present time (1952) there is a discordance"; Mercury's perihelion has "alternative explanations"; the red shift "does not, properly speaking, constitute a test" (bk 180-181, page images). He reports with respect three theories in flat (uncurved) space: - Whitehead (1922): action at a distance propagated at c; Synge (1952) studied it; "no practicable observational test has hitherto been suggested for discriminating between the two theories" (bk 175). [memory, high, per the deep read] Will (1971) later showed that Whitehead's theory predicts an Earth-tide anisotropy that gravimeters rule out. - Rosen (1940) and followers: GR mapped onto flat space, which, in Whittaker's word, they "claimed" might "account for certain unexplained residuals in repetitions of the Michelson-Morley experiment", citing D. C. Miller (1925) (bk 175). - A. G. Walker (1951): a particle in ordinary Euclidean space with one Newtonian potential φ and a kinetic energy T = ½ e^(−2φ/c²) v² (v the speed) never exceeds c and reproduces both GR's perihelion advance and its light bending (bk 181-182, page images). The φ-dependent kinetic term amounts to giving space a refractive index, which is how a single scalar escapes the no-bending problem of section 3.6. These models came before gravitational-wave and binary-pulsar data and were not tested against them. Whittaker's moral: "It is unwise to accept a theory hastily on the ground of agreement between its predictions and the results of observation in a limited number of instances" (bk 182). On method he also writes that "the truth of the Newtonian or any other law of gravitation cannot be tested by means of controlled experiments in a laboratory" (bk 146), and he rejects Mach's principle because it would need "enormous quantities of matter ... which have not been detected" (bk 168). The 1952 deflection "discordance" was later resolved in GR's favour: radio and VLBI (very-long-baseline interferometry) measurements confirm the bending to about 10⁻⁴, and Cassini's γ is in section 3.6 [memory, high for the radio and VLBI figure].

(h) What Vol. 2 does not say. The gravitation chapter never mentions his 1902 MNRAS note, his 1903 Math. Ann. section 5, or Stoney. A full read of the chapter and a text search of the whole volume found no "Stoney", no 1903 or 1902 citation, no Endeavour article, and no undulatory, longitudinal or standing-wave gravity. His own works cited in the chapter are all on other subjects: small divisors in celestial mechanics (1917, bk 146), light paths in a gravitational field (1928, bk 165), a variational paper (1927, bk 170), the de Sitter world (1931, bk 184 and 186), and Gauss's theorem in GR (1935, bk 194). The deep reader weighed three passages as possible allusions (the Laplace sentence, Lorentz's penetrating waves, the FitzGerald-Copson medium) and judged that reading them as allusions would go beyond the text. The record is silence, not denial: he neither retracts nor repeats "may be enormously greater", and the 1903 mathematics is untouched.

4.10 Side by side again: Whittaker 1902-03 and Whittaker 1953.

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5. Worked example: Whittaker's waves, Stoney's flicker, the standing-wave question and the light medium

Everything here is reproduced by example.py (run command in its header; it runs in about 20 seconds). Parts A-D use Whittaker's dimensionless units: source strength 1, distances and times in arbitrary units with V the wave speed. Parts E-H use SI units.

Part A. Whittaker's sum, done with a cutoff

The sum over all wavenumbers is

r V(r, t) = (2/π) ∫₀^∞ sin(μ(Vt − r)) / μ dμ.

Put a = Vt − r (positive inside the front) and substitute y = μa: the integral is ∫₀^∞ sin y / y dy = π/2 if a > 0 and −π/2 if a < 0. So r V = sgn(a): +1/r inside the front, −1/r outside [calc]. Whittaker states only the first half ("so that (Vt − r) is positive").

No physical spectrum runs to infinite frequency, so cut it off at a largest wavenumber M (shortest wavelength 2π/M). Then ∫₀^M sin(μa)/μ dμ = Si(Ma) exactly, and

r V = (2/π) Si(M a).

With M = 10 and r = 1 [calc, example.py Part 2]:

So with any real cutoff the field is only approximately steady: it overshoots just behind the front and rings, settling as the front moves away. The largest overshoot is (2/π) Si(π) = 1.1790, 18 percent too high, at a = π/M: the Gibbs phenomenon [calc]. Raising M moves the overshoot closer to the front but does not shrink it.

The same formula at r → 0 gives the source's strength: (2/π) Si(MVt) is −0.9996 at t = −1 and +0.9996 at t = +1 for M = 1000 [calc]. The source is a step: its strength flips from −1 to +1 at t = 0 (a static −1/r plus a switched-on 2/r).

Part B. The force, and Stoney's flicker in numbers

The force is F = −∂V/∂r. One band contributes (Stoney's equation 4, written as a force)

dF = (2 dμ / πμ) [ sin(μa)/r² + μ cos(μa)/r ].

The first part falls as 1/r² and the second as 1/r; both oscillate in time at frequency μV. That is Stoney's "flicker". Summing with the hard cutoff, using ∫₀^M cos(μa) dμ = sin(Ma)/a:

r² F = (2/π) [ Si(Ma) + (r/a) sin(Ma) ].

The first term is the steady inverse-square force from Part A. The second is what is left of the 1/r flicker: it oscillates at the cutoff frequency, and its amplitude (2/π)(r/a) does not depend on M. It fades only as the front recedes (a grows). If instead the spectrum is rolled off smoothly, with weight e^(−μ/M), the integrals are arctan(Ma) and M/(1 + M²a²), and

r² F = (2/π) [ arctan(Ma) + r M/(1 + M²a²) ],

which has no oscillation and approaches 1 smoothly. For r = 1, M = 10 [calc, example.py Part 3; both closed forms checked by direct numerical summation of the bands]:

What this shows. Stoney was right that every constituent flickers and that only the exact 1/μ weighting cancels it. The size of the leftover depends on how the spectrum ends, which the model does not specify. In the limit of all frequencies both versions become exactly 1/r² inside the front plus the spike 2δ(Vt − r)/r at the front: the flicker has not vanished, it has been gathered into the moving front. This is the arbiter's claim 4A (the mathematics, Established) and 4B (whether the constituents are physically real, Never properly tested).

Part C. Why the constituent forces are longitudinal

For any plane-wave term V = F(n·r − ct), example.py Part 1 computes −∇V × n and gets the zero vector: the force lies along the direction of travel, for every profile F [calc]. The same is true of each spherical wave in Part A: its force −∂V/∂r is radial, along the direction of travel. So the longitudinal character comes from describing gravity by one scalar potential. It would be the same for sound, for a scalar field of any kind, or for the Newtonian potential cut into Fourier pieces. It is not separate evidence about a medium.

Part D. The Q07 hypothesis, spectrum form: the steady field as standing waves

Use sin(μ(Vt − r)) = sin(μVt) cos(μr) − cos(μVt) sin(μr) (checked symbolically in example.py Part 4). Each product on the right is a standing spherical wave: a fixed spatial pattern times an oscillation in time. So Whittaker's field splits exactly into two standing-wave families, V = S₁ − S₂, with

S₁ = (2/πr) ∫ sin(μVt) cos(μr) / μ dμ, S₂ = (2/πr) ∫ cos(μVt) sin(μr) / μ dμ.

Product-to-sum formulas and the sine integral give, with cutoff M, r S₁ = (1/π)[Si(M(Vt + r)) + Si(M(Vt − r))] and r S₂ = (1/π)[Si(M(r + Vt)) + Si(M(r − Vt))]. With M = 1000 and Vt = 10 (front at r = 10) [calc, example.py Part 4]:

Inside the front, the entire steady 1/r is carried by the family cos(μr) sin(μVt)/r, a set of standing waves of every frequency, each one oscillating, whose sum is constant. So the Q07 phrase can be given an exact mathematical meaning: the static Newtonian potential is a superposition of longitudinal standing spherical waves with weights 2/(πμ). This is the physicist's point that "standing versus travelling is partly a presentation matter". The cost is the same as before: the sum is exactly Newton's field, so it predicts nothing a static measurement could distinguish. In this form the idea is a representation, not yet a hypothesis that can be tested.

Part E. The Q07 hypothesis, single-mode form: one standing wave

Now take one real standing wave with wavenumber κ (kappa), satisfying the scalar wave equation away from the origin:

V = cos(κr) cos(ωt) / r, with ω = c_w κ, where c_w is the wave speed.

At an instant when cos(ωt) = 1, the force is F = −∂V/∂r. SymPy's series gives

r² F = 1 + (κr)²/2 − (κr)⁴/8 + ...

so the inverse-square law holds only to a relative accuracy of about (κr)²/2 over distance r [calc, example.py Part 5].

How slow must it be? Ask that the force law deviate by at most 1 part in 10⁸ at r = 1 AU (an illustrative tolerance chosen by the mathematician, not a measured limit from a source). Then: 1. (κr)²/2 = 10⁻⁸, so κr = √(2 × 10⁻⁸) = 1.414 × 10⁻⁴. 2. κ = 1.414 × 10⁻⁴ / 1.496 × 10¹¹ m = 9.45 × 10⁻¹⁶ per metre. 3. Wavelength 2π/κ = 6.65 × 10¹⁵ m = 0.70 light year. 4. If the wave moves at c: ω = cκ = 2.83 × 10⁻⁷ rad/s, period 2π/ω = 2.22 × 10⁷ s = 257 days.

(The mathematician's original illustration used κr = 10⁻⁴, which gives a deviation of 5 × 10⁻⁹, not 10⁻⁸, and a period of about 363 days; the arbiter recorded the correction. Both say the same thing: months to a year or longer.) A faster wave would shorten the period by the ratio c_w/c.

What it predicts. A pure standing wave makes g proportional to cos(ωt). The Sun's pull at 1 AU (5.93 × 10⁻³ m/s²) would pass through zero twice per period, about every 128 days in this example, and reverse in between. That is plainly not what holds the Earth in orbit, so a pure single standing wave cannot be all of gravity. A workable version must be a steady 1/r part plus a small standing addition ε cos(κr) cos(ωt)/r, which predicts an oscillation of g at one frequency, in step everywhere it is not at a node. That is a definite thing to look for with superconducting gravimeters, atom interferometers and pulsar timing; and if light shares the medium, optical clocks should be modulated at the same frequency. No such model has been stated with a frequency and amplitude and tested by name (arbiter, claim 4F). Part G adds a second, weaker constraint from the 19th-century pulsation model: the sign of the force.

Part F. "May be enormously greater": what GW170817 says

GW170817 came from about 40 megaparsecs (Mpc) away, 1.23 × 10²⁴ m, a light-travel time of about 1.3 × 10⁸ years [memory, high for the distance; calc for the conversion]. If gravitational waves moved at 10c they would have arrived 1.17 × 10⁸ years ahead of the light; even at 1.0001c, 1.3 × 10⁴ years ahead. They arrived about 1.7 s before the gamma rays. The crude bound c × 1.7 s / D is 4 × 10⁻¹⁶; the LVK analysis, allowing for uncertain emission delays, gives −3 × 10⁻¹⁵ to +7 × 10⁻¹⁶ [calc, example.py Part 6; web: https://arxiv.org/abs/1710.05834]. This tests the speed of gravitational waves. For Whittaker's static field the speed is untestable, as section 3.2 showed; a faster channel for static fields alone would need its own moving-source theory, which Whittaker did not write.

Part G. The pulsation form: Bjerknes' quarter-wavelength condition

Whittaker's History reports Bjerknes' result (section 4.8): bodies pulsating in phase attract by the inverse square, but if the medium is compressible, "attractions would change to repulsions ... at distances greater than a quarter wave-length" [read: History 1951, PDF p.149]. So attraction between two bodies a distance d apart needs

d < λ/4, where λ = v T,

with λ the wavelength of the compressions, v their speed in the medium and T the common pulsation period. Rearranged: T > 4d/v. Whittaker names no speed. Take v = c purely as an illustration [calc, example.py Part 7]:

Three lessons: 1. The sign bound is the weak one. A period of a few hours keeps the force attractive out to Neptune. But keeping the force an accurate inverse square is much harder. Part E's r²F = 1 + (κr)²/2 asks for a period of order a year (257 days for 1 part in 10⁸ at 1 AU). The deep reader carries the (κd)² scaling over to the retarded Bjerknes force as [memory, medium]; the exact law is in Hicks (1879-80), which was not read. 2. Particle-scale pulsations need an almost rigid medium. If matter pulsated at a particle frequency, such as the proton's Compton frequency m_p c²/h = 2.27 × 10²³ Hz (m_p the proton mass, h Planck's constant), then d < λ/4 at 1 AU would need v > 4 d f = 1.4 × 10³⁵ m/s, about 4.5 × 10²⁶ c [calc]. That is Bjerknes' "incompressible" (instantaneous) case in practice, which is presumably why Korn kept his fluid incompressible [inferred in the deep read]. A medium at speed c needs common periods of about 10³ to 10⁸ s, unlike any known atomic or nuclear frequency. 3. The conditions are explicit and checkable. Every gravitating body must pulsate in a common phase, with an amplitude proportional to its mass, inside the near-field zone (d < λ/4). And the medium must be of a kind that gives attraction for like phases: Leahy found the opposite in an elastic medium. A modern laboratory cousin exists: the "secondary Bjerknes force" between gas bubbles pulsating in a sound field, where in-phase bubbles attract and out-of-phase bubbles repel [memory, high]. It shows that "bodies pushed together by a longitudinal oscillation of a medium" is real physics. It is an analogy, not evidence about gravity.

Part H. "The same medium that carries light", in Whittaker's 1953 sense

Section 4.9(c) gave Whittaker's 1953 statement (crediting Copson, 1928) that for light a static gravitational field is a medium with ε = μ = (r + 1)³/[r²(r − 1)] around a point mass. Here is where the formula comes from and what it does [calc, example.py Part 8].

1. The equivalent index. In general relativity the field of a point mass M can be written in coordinates in which space looks the same in every direction at each point ("isotropic" coordinates, with ρ the distance from the mass). Put m = GM/c² (a length: 1477 m for the Sun). Light then moves at the speed c × A/B, where A = (1 − m/2ρ)/(1 + m/2ρ) and B = (1 + m/2ρ)². You do not need to derive A and B; take them as given [memory, high]. A material with refractive index n slows light to c/n, so the equivalent index is n = B/A = (1 + m/2ρ)³/(1 − m/2ρ). A medium with ε = μ = n has exactly this index (index = √(εμ) for relative ε and μ) and reflects nothing, because the ratio μ/ε, which sets reflection, stays 1.
2. Matching Copson's formula. Put r = 2ρ/m, the distance in units of GM/(2c²). SymPy gives n = (r + 1)³/[r²(r − 1)], exactly the printed law. So Whittaker's r is the distance in units of half of GM/c².
3. Far from the mass. Expanding for large r: n = 1 + 4/r + 7/r² + ..., that is n ≈ 1 + 2GM/(c²ρ).
4. At the Sun's surface (R = 6.957 × 10⁸ m [memory, high]): 2GM/(c²R) = 4.25 × 10⁻⁶. The "vacuum" next to the Sun acts like glass of index 1.00000425: light is slower by about four parts in a million.
5. Bending of starlight. In a medium whose index changes sideways, a ray bends toward the higher index (as in a mirage). Adding up the sideways gradient of ln n along a straight ray that just grazes the Sun gives 1.7512″ by numerical integration, and the weak-field formula 4GM/(c²b) (b the closest distance of the ray, here R) gives the same 1.7512″. That is Einstein's 1.75″, which Whittaker quotes (bk 180); Soldner's Newtonian value was 0.85″, and a scalar theory that leaves the light speed unchanged (Nordström's) gives zero (section 3.6).

What this shows. There is an exact, textbook sense in which "gravity is a property of the medium that carries light": for light, a static gravitational field is a variable-index medium, and the index profile gives the measured bending. That is Whittaker's 1953 reading and it is standard general relativity (the "optical medium" form of a static field [memory, high]). It is not Whittaker's 1903 picture: nothing here is a longitudinal wave, nothing moves faster than c, and the medium description holds for static fields and weak light, not for gravitational waves, which are transverse and tensor (section 3.4). So the Q07 "same medium" question has a precise answer for the static part of gravity inside GR, and its light-bending prediction is measured (Cassini, section 3.6). Because this medium is GR rewritten, it cannot tell a medium from geometry: the arbiter's claim 4E (generic shared medium) stays Never properly tested in any form that differs from GR. The "longitudinal standing wave" part is a separate question (Parts D, E and G).

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6. Evidence table

Related claims kept in the record, not verdicts on Whittaker [web: sites.md, research/]: - C. F. Brush (1911-1929) proposed a kinetic aether-wave theory of gravity admitting longitudinal waves. He reported that heat-generating silicates fall slower by about 10⁻⁵ over 122 cm, with a Bureau of Standards check whose result has not been located (one source, rexresearch). Lodge objected in 1911. Later composition tests (MICROSCOPE, about 10⁻¹⁵) cover the same effect far more sensitively, though they are not a replication of Brush's design. - Múnera (2018), as reported by Titleman (2023), criticised the 1903 generality claim as "unacceptable due to the assumption of periodicity" [web: research/2026-10-07-qeios.md; Múnera 2018 not read]. The arbiter's weighing: periodicity in the angle variables is harmless for the integral (only one period, 0 to 2π and 0 to π, is used), so the objection bites only against a global "every solution everywhere" reading, which the paper does not prove and no use of it needs (claim 1 verdict). - Múnera (2019) "agrees with Whittaker" on longitudinal solutions and argues for composition-dependent gravity. Darmos (2022) claims composition dependence from Cavendish data, and a published response exists. Both were seen only in abstracts or snippets. - T. J. J. See (1917) proposed a wave theory of gravitation. Titleman (2022-2026, arXiv:2205.08309 and Qeios) develops Whittaker's potentials into a theory of gravity and electromagnetism; the Qeios preprint has open reviews (mean 3.25). Meyl, Gouveia (2025) and others use "longitudinal standing wave" language. Labels such as "fringe" belong to the sources that applied them.

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7. What a learner should take away

What is solid. - Whittaker's 1903 theorems are real, lasting mathematics. The trick is a null vector, a direction whose "length squared" is zero: complex for Laplace's equation, real (space and time cancelling) for the wave equation. Through Bateman this idea leads to Penrose's twistor theory, and Thompson, Swearngin and Bouwmeester (2014) used the "Whittaker solution" to build exact gravitational waves, tensor ones with no medium [web: https://arxiv.org/abs/1402.3806]. - The 1904 two-potential result is a correct and economical way to write Maxwell-Lorentz fields, and it is still used. - The gravity note is correct as mathematics: a static 1/r field can be written as waves that cancel. McCrea (1957, a colleague and obituarist) read it as showing "how a static gravitational field can be regarded as resulting from effects that are being propagated through the field", anticipating "some features of general relativity" [web: research/2026-10-07-mactutor.md]. In the narrow sense that changes in gravity spread at a finite speed, that is right. - For light, a static gravitational field really does act as a medium of varying index: Whittaker's 1953 statement of Copson's result is exact general relativity and gives the measured 1.75″ bending (section 4.9(c), Part H). That is the solid core of "the same medium that carries light". It is static and it travels at no speed above c.

What Whittaker himself changed. By 1953, in his own History, Whittaker had adopted gravity at the speed of light, gravitational waves at c radiated by moving matter, and Einstein's ten potentials in place of one scalar (section 4.10). He never said so about his own 1902-03 note, and the mathematics of that note needed no retraction. A learner can take two lessons from this: a careful scientist's offhand physical suggestion is not a lifelong theory, and the record of what someone later printed is evidence about what he came to think, though not proof of what he privately believed.

What fails, and why the mistake is instructive. - A static field cannot tell you what it is made of. Any decomposition of it into waves (travelling, standing, longitudinal) predicts exactly the same static field. The physical content of such a model lies entirely in how it treats changing sources, and Whittaker never wrote that part. - "Longitudinal" comes from the choice of a scalar potential, not from nature. The gravitational waves actually observed are transverse tensor waves at speed c, and the binary-pulsar energy loss matches tensor radiation to 0.2 percent. The specific forms of the idea that can be tested (longitudinal radiation, speed far above c, a scalar vacuum medium) do not fit those measurements. - Stoney's 1903 objection was well aimed: the constituents flicker, and only one exact spectrum cancels them. Part B shows the leftover depends on how the spectrum ends and that the cancelled parts collect at the moving front. The objection lands against the constituents being physically real, not against the mathematics.

What is genuinely open. - A shared medium for light and gravity in a tensor form, one that reproduces GR's speed, polarizations and radiation, is not excluded; nobody has shown it differs from GR observably. Whittaker himself, in 1928, compared gravity's effect on light to a medium with variable permittivity and permeability, adding that the resemblance is "not quite perfect" [web: research/2026-10-07-cambridge-synge.md]; in 1953 he wrote that for static fields and weak light the match is exact, crediting Copson (1928) and Wiechert (1920), and he reported flat-space alternatives (Whitehead, Rosen, Walker) with respect (section 4.9) [read]. So in his own text the medium-versus-geometry question was left open. Part H shows that the static medium reading is GR itself and gives the measured 1.75″; what is open is whether any medium differs from GR anywhere, above all for gravitational waves. - A single-frequency standing component of gravity (Part E), or a common-phase pulsation of all matter (Part G), is a definite, searchable prediction: an oscillation of g at one frequency, the same for every body, correlated with optical clocks if light shares the medium. No named model of either kind has been tested. The nearest search on record is the Parkes Pulsar Timing Array's: 26 pulsars, 2004-2016, found no statistically significant periodic signal from an oscillating gravitational potential due to ultralight scalar dark matter, for boson masses up to 10⁻²³ eV (nanohertz frequencies) [web: https://arxiv.org/pdf/1810.03227]. That covers one model in one frequency band. It is not a search for a Whittaker-type or Bjerknes-type wave, and it says nothing about other frequencies or couplings. Statements that superconducting gravimeters or atom interferometers have found nothing are [memory, medium] and are not relied on here. - Where the Q07 hypothesis lives best (arbiter, QE revision). It has two homes, to be kept apart. (1) Whittaker 1902-03, the origin of "gravity as waves in the light medium", whose static result is 1/r inside the front only. (2) Bjerknes, Korn and Larmor, the origin of "gravity from in-phase longitudinal oscillation of a medium", recorded in Whittaker's own History. "Longitudinal standing wave" is closer to (2): an oscillating near-field with a common phase. The honest form of the question: what common-phase pulsation, with what amplitude, period and coupling, would give the inverse square to 1 part in 10⁸ at 1 AU (a period of order a year or longer if the medium's speed is c)? And where would it show up as a line in gravimeter, pulsar-timing or clock data? Open sub-questions: what Hicks (1879-80), Korn (1898) and Larmor (1895) actually derived for the retarded pulsation force, and how they justified the common phase. - Small scalar (breathing) polarizations of gravitational waves are still allowed by current detector networks. - Where the record goes dark. After Stoney's 1903 critique, no reply from Whittaker has been found, and no further paper on wave gravity is known; the 1943-44 Endeavour articles are unread. By 1927 his gravity papers work inside GR, yet in 1951 he kept the word "aether", and Navarro (2020), working from Whittaker's letters, says he was "convinced that gravitation and electromagnetism were ultimately connected" [web: research/2026-10-07-sagepub.md, abstract only; a historian's characterization, not Whittaker's words]. Both volumes of his History are now read on this point, and both are silent about his own note: Vol. 1 (1951) mentions his 1904 F and G result but not his 1902-03 undulatory gravity (section 4.8), and Vol. 2 (1953) does not mention the note, the 1903 paper or Stoney (section 4.9(h)). Vol. 2 also puts on record his own later view: a far-superluminal gravity rests on implausible assumptions, gravity and gravitational waves travel at c, and a single scalar potential was rightly given up. Three explanations fit the record, and the 1953 text moves the weight between them without deciding: - (a) He regarded the note as a representation, not a theory. Consistent: he never presents the 1903 result as physics in the History, which fits "does not explain the cause of gravity". - (b) He gave it up as relativity arrived. More strongly supported than before, for the speed and the scalar potential, by his own 1953 statements. - (c) He kept a private or later conviction. Not excluded, and for the medium part still alive: in 1953 he keeps the word "aether", identifies it with space-time, gives gravity a medium (ε, μ) reading, reports flat-space alternatives and calls the 1952 light-bending evidence discordant. That fits Navarro's characterization; it equally fits plain historical thoroughness. Net (arbiter, QE revision 2): weight toward (b) for the speed and the scalar potential; (c) survives for the medium. Silence in a history written by the man himself is an anomaly to explain, not a verdict, and it fits all three. The deciding records still not obtained are the 1944 Endeavour article "Old and new ideas of gravitation", the Fisher Library papers (MS COLL 00219, Box 11), the 1902-03 press articles, the 1927 Proc. R.S. A 116 paper on electric phenomena in gravitational fields, and Copson's 1928 paper. Nothing here suggests suppression: the idea was published in two major journals and answered in print within a year. The record is thin, not hostile. - Brush's Bureau of Standards result (1911-29) has not been located.

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8. Check yourself

1. Why does any static potential automatically satisfy the wave equation, and what does that imply about the speed Whittaker assigned to his gravity waves? What did Whittaker himself say about the speed of gravity in 1953?
2. Whittaker's sum (2/πr)∫₀^∞ sin(μ(Vt − r))/μ dμ equals 1/r inside the front. What is it outside, and what does that tell you about the source?
3. A scalar plane wave V = cos(k(x − ct)) is used as a "gravitational" potential. Which way does the force point, and why does this make "longitudinal" a weak piece of evidence for a medium? In what different sense did Whittaker in 1953 say a static gravitational field acts as a medium for light, and what does that medium predict for starlight grazing the Sun?
4. A single standing wave V = cos(κr)cos(ωt)/r is proposed for the Sun's gravity. Give two reasons, one from the force law and one from time dependence, why it cannot be the whole story, and say what a workable version would predict.
5. Stoney said the constituent forces "flicker". Using r²F = (2/π)[Si(Ma) + (r/a) sin(Ma)], explain where the flicker goes as M → ∞, and why a smooth roll-off changes the finite-M answer.

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9. Further reading

Mainstream. - D. J. Griffiths, Introduction to Electrodynamics (potentials, Laplace's equation, waves, retarded potentials). - A. P. French, Vibrations and Waves (standing and travelling waves, longitudinal and transverse). - M. L. Boas, Mathematical Methods in the Physical Sciences (Fourier series and integrals; the Gibbs phenomenon). - E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, the chapter on Laplace's equation ("Whittaker's integral"). - B. Schutz, Gravity from the Ground Up; C. M. Will, Was Einstein Right? (tests of GR, gravitational waves, without tensors). - S. Carlip, "Aberration and the speed of gravity", Phys. Lett. A (2000), https://arxiv.org/abs/gr-qc/9909087. - LVK, GW170817 and GRB 170817A, https://arxiv.org/abs/1710.05834; tests of GR with GWTC-3, https://arxiv.org/abs/2112.06861. - J. M. Weisberg and Y. Huang, PSR B1913+16 (2016), https://arxiv.org/abs/1606.02744. - Thompson, Swearngin and Bouwmeester, gravitational waves from the Whittaker solution (2014), https://arxiv.org/abs/1402.3806. - M. Ibison, the polarizable-vacuum cosmology and the binary pulsar (2003), https://arxiv.org/abs/astro-ph/0302273. - Parkes Pulsar Timing Array search for an oscillating gravitational potential from ultralight scalar dark matter (2018), https://arxiv.org/pdf/1810.03227. - On the secondary Bjerknes force between pulsating bubbles (the laboratory cousin of section 5 Part G): any graduate acoustics or bubble-dynamics text, e.g. T. G. Leighton, The Acoustic Bubble [memory, medium].

The 19th-century pulsation and density models (cited in Whittaker's History, 1951, PDF pp.149-152; none read in this review): - C. A. Bjerknes, Göttinger Nachrichten (1876) p.245; Repertorium d. Mathematik i (1877) p.268; Comptes Rendus 84 (1877) p.1375. V. Bjerknes, Fields of Force (1906) [memory, high for the title]. - W. M. Hicks, Proc. Camb. Phil. Soc. 3 (1879) p.276; 4 (1880) p.29 (the mathematics of the force between pulsating spheres). - A. H. Leahy, Trans. Camb. Phil. Soc. 14 (1884) p.45. - A. Korn, Eine Theorie der Gravitation und der elektrischen Erscheinungen (Berlin, 1898). - J. Larmor, Phil. Trans. 186 (1895) p.697. - B. Riemann, Schwere, Elektricität und Magnetismus, ed. K. Hattendorff (1875), p.330 (the 1861 lectures).

Whittaker's own works. - "On the general solution of Laplace's equation and the equation of wave motions, and on an undulatory explanation of gravity", MNRAS 62 (1902) 617-620, https://academic.oup.com/mnras/article/62/9/617/1015465. - "On the partial differential equations of mathematical physics", Math. Ann. 57 (1903) 333-355 (in the project folder). - "On an expression of the electromagnetic field due to electrons by means of two scalar potential functions", Proc. London Math. Soc. (2) 1 (1904) 367-372 (in the project folder). - "On electric phenomena in a gravitational field", Proc. R. Soc. A 116 (1927) 720-735, https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0160. - "The influence of gravitation on electromagnetic phenomena", J. London Math. Soc. 3 (1928) 137-144 (not read; known from Synge 1958). - "Old and new ideas of gravitation", Endeavour 3 (1944) 39-42 (not read). - A History of the Theories of Aether and Electricity, Vol. 1 (1951, in the project folder; first edition 1910, also in the folder) and Vol. 2, The Modern Theories 1900-1926 (1953, in the project folder; Chapter V "Gravitation" deep-read). Vol. 2 Chapter V (bk 144-196) is the place to read Whittaker's own later view of gravity's speed, scalar gravity and the medium (section 4.9).

Sources Whittaker cites in Vol. 2 on gravity and the medium (not read in this review; references as printed by Whittaker): - H. A. Lorentz, "Considerations on gravitation", Proc. Amsterdam Acad. 2 (1900) 559. - A. Einstein and M. Grossmann, Z. f. Math. u. Phys. 62 (1913) 225 (section 7: the one-third-work argument against a single scalar potential). - E. Wiechert, Ann. d. Phys. 63 (1920) 301 (FitzGerald's aether "structure" as a change of dielectric constant). - E. T. Copson, Proc. R. Soc. A 118 (1928) 184 (a point charge in the field of a point mass; the ε = μ law of section 4.9). - J. L. Synge, Proc. R. Soc. A 211 (1952) 303 (Whitehead's flat-space theory). - A. G. Walker, Nature 168 (1951) 961 (a flat-space scalar model reproducing the perihelion and light-bending results). - N. Rosen, Phys. Rev. 57 (1940) 147, 150, 154 (flat-space general relativity).

The critique and later readings. - G. J. Stoney, "Examination of Mr. Whittaker's 'Undulatory Explanation of Gravity' from the physical standpoint", MNRAS 63 (1903) 424-428, https://academic.oup.com/mnras/article/63/8/424/1008995. - W. H. McCrea, "Edmund Taylor Whittaker", J. London Math. Soc. 32 (1957) 234-256. - J. L. Synge, "Whittaker's contributions to the theory of relativity", Proc. Edinburgh Math. Soc. (1958). - J. Navarro, History of Science 59 (2020) 287-314 (Whittaker, the aether and History Vol. 2). - M. Titleman, "The duality of Whittaker potential theory", https://arxiv.org/abs/2205.08309 (the modern longitudinal/standing-wave reading; see also review/Bearden in this project). - H. A. Múnera (2018), the periodicity critique of Whittaker's "general solution", known here only as reported by Titleman, "Representations and Implications of Papers Written by E.T. Whittaker in 1903 and 1904", Qeios (2023), https://www.qeios.com/read/Y3D1HM. - Trovon de Carvalho and Rodrigues, "The Non Sequitur Mathematics and Physics of the 'New Electrodynamics' Proposed by the AIAS Group" (2001), a critique of the AIAS/Bearden use of Whittaker (not read; known from Titleman's citation, research/2026-10-07-qeios.md).

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Answers

1. If V does not change with time, ∂²V/∂t² = 0, so ∇²V = 0 gives ∇²V = k²∂²V/∂t² for every k. Static fields therefore fix no wave speed. Whittaker's "any constant" and "may be enormously greater" cost nothing for a static field. The speed matters only for changing sources, and for gravitational waves GW170817 measured it equal to c within about 10⁻¹⁵. In 1953 Whittaker wrote that Laplace's reasons for a far higher speed "do not to twentieth-century minds seem very plausible" and that relativity requires gravity, and gravitational waves, to travel at c (Vol. 2 bk 149, 151, 183). He did not mention his own 1903 "enormously greater".
2. Outside the front (r > Vt) the integral of sin(μa)/μ with a < 0 is −π/2, so the sum is −1/r. The full answer is sgn(Vt − r)/r. At r → 0 the source strength is sgn(t): a source that switched from −1 to +1 at t = 0, whose news spreads outward at speed V. The steady 1/r exists only inside the expanding sphere.
3. −∇V = −∂V/∂x x̂ = k sin(k(x − ct)) x̂, which points along x, the direction of travel. Every scalar-potential wave gives a force along its wave vector, whatever the medium or speed, so "longitudinal" follows from the choice of description, not from a measured property of space. Whittaker's 1953 sense (crediting Copson, 1928) is different: for light, a static field acts like a medium with ε = μ = n ≈ 1 + 2GM/(c²ρ), slightly "slower" near a mass. That is a static index profile, not a wave, and it bends a ray grazing the Sun by 4GM/(c²R) = 1.75″, the measured value (Part H).
4. Force law: r²F = 1 + (κr)²/2 + ..., so the inverse-square law fails unless κr is tiny. A 10⁻⁸ tolerance at 1 AU needs a wavelength of about 0.7 light year and, at speed c, a period of about 257 days. Time dependence: a pure standing wave makes g ∝ cos(ωt), so the Sun's pull would vanish twice per cycle and reverse in between. A workable version is a steady 1/r plus a small standing part, which predicts a one-frequency oscillation of g (and of optical clocks, if light shares the medium) that gravimeters or pulsar timing could find.
5. The 1/r term (2/π)(r/a) sin(Ma) oscillates ever faster as M grows. In the limit (2/π)(r/a) sin(Ma) → 2r δ(a), so all the flicker is squeezed into the spike at the front r = Vt and nothing flickers behind it. At finite M the hard edge leaves an oscillating remainder of amplitude (2/π)(r/a), independent of M. A smooth e^(−μ/M) roll-off replaces it with a non-oscillating term of about 2r/(πMa²). So the leftover depends on how the spectrum ends, which the model leaves unspecified. That is a quantitative version of Stoney's complaint.