Follow-up Q06: Maxwell's longitudinal (scalar) terms, in vector form, and why Heaviside dropped them

Date: 2026-10-07. Follow-up investigator (STEP F), answered inline by the orchestrator (Chrome was in use by the Eskridge-Amy Investigator, so the web-fetch tool was used). Web time: about 3 minutes (two Wikisource pages of the 1873 Treatise). Everything else is from the project's sources: the 1865 paper (page images pp.46-48, printed pp.498-501) and Hunt's Physics Today article on Heaviside.

The question

"Maxwell's theory allows longitudinal solutions. Give the modern vector equivalent of his quaternion formulas for the longitudinal (scalar) part, as they would read if Heaviside had kept them. Explain why these terms had meaning for Maxwell and why Heaviside discarded them."

1. Where the longitudinal terms are in Maxwell

1865 has no quaternions. The 1865 paper is in Cartesian components (primer, "No quaternions in 1865" [read: page images pp.32-34]). The longitudinal question is still there, in components: §§98-100.

1865, §98-99 [read: page images pp.47-48, printed pp.499-500]. Having shown that F, G, H (the vector potential A) carry transverse waves at V = √(k/4πμ), Maxwell goes "back upon the equations in (94), in which the quantities J and Ψ occur, to see whether any other kind of disturbance can be propagated through the medium depending on these quantities which disappeared from the final equations." His definitions and steps, in his notation:

• J = dF/dx + dG/dy + dH/dz (the divergence of A; he writes it out as the operator ∇² acting on a new function χ): ∇²χ = J (73); F' = F − dχ/dx, G' = G − dχ/dy, H' = H − dχ/dz (74); so dF'/dx + dG'/dy + dH'/dz = 0 (75). This is the split of A into a transverse part A' (zero divergence) and a longitudinal part ∇χ (zero curl).
• The transverse part obeys kV²F' = 4πμ d²F'/dt² (78): waves "in the plane of the wave".
• The longitudinal part "is subject to no condition except" dΨ/dt + d²χ/dt² = 0, so Ψ = −dχ/dt + φ(x, y, z) (77).
• Taking ∇² of this: ke = dJ/dt − k∇²φ(x, y, z) (79), with e the free electricity.
• Conclusion: "Since the medium is a perfect insulator, e, the free electricity, is immoveable, and therefore dJ/dt is a function of x, y, z, and the value of J is either constant or zero, or uniformly increasing or diminishing with the time; so that no disturbance depending on J can be propagated as a wave."

1865, §100 [read: printed pp.500-501]. "The equations of the electromagnetic field, deduced from purely experimental evidence, show that transversal vibrations only can be propagated. If we were to go beyond our experimental knowledge and to assign a definite density to a substance which we should call the electric fluid ... then we might have normal vibrations propagated with a velocity depending on this density. We have, however, no evidence as to the density of electricity ... both sciences are at a loss when called on to affirm or deny the existence of normal vibrations."

So Maxwell's own result is two-sided: in a perfect insulator his equations give no longitudinal wave; and he explicitly left the door open for "normal" (longitudinal) vibrations if electricity has inertia (a density).

1873 Treatise, quaternion forms [web: https://en.wikisource.org/wiki/A_Treatise_on_Electricity_and_Magnetism/Part_IV/Chapter_IX, Arts. 616-619]. The quaternions appear here, at Tait's urging [read: Hunt]. Maxwell writes: - 𝔅 = V.∇𝔄 (magnetic induction is the vector part of ∇ acting on the vector potential); - 𝔈 = V.𝔊𝔅 − 𝔄̇ − ∇Ψ (electromotive intensity; 𝔊 is the velocity of the conductor); - ∇²𝔄 = 4πμℭ together with S.∇𝔄 = 0. On χ and J (Art. 616): "The quantity χ disappears from the equations (A), and it is not related to any physical phenomenon. If we suppose it to be zero everywhere, J will also be zero everywhere" [web, same page; quoted via the fetch tool, not seen as a page image].

Treatise, Arts. 783-784 [web: https://en.wikisource.org/wiki/A_Treatise_on_Electricity_and_Magnetism/Part_IV/Chapter_XX]. The same result as 1865: in a non-conductor "∇²Ψ, which is proportional to the volume-density of free electricity, is independent of t. Hence J must be a linear function of t, or a constant, or zero, and we may therefore leave J and Ψ out of account in considering periodic disturbances." [web, via the fetch tool]

2. The quaternion terms in modern vector form

Hamilton's product of two vectors has a scalar and a vector part: for vectors a and b, ab = −a·b + a×b. So the quaternion ∇ acting on a vector field gives both "longitudinal" and "transverse" information at once [memory: high]:

The same split applied to the electric field (this is a reconstruction; Maxwell did not write it this way): ∇𝔈 = −∇·E + ∇×E. The scalar part is the charge density (Gauss's law, ∇·E = ρ/ε₀); the vector part is Faraday's law (∇×E = −∂B/∂t).

Maxwell's longitudinal equations, "if Heaviside had kept them" (modern SI, a medium with permittivity ε, permeability μ, no gauge condition imposed):

1. Split of A (1865 eqs. 73-75): A = A' + ∇χ, with ∇·A' = 0 and ∇²χ = J = ∇·A.
2. Transverse waves (eq. 78): ∇²A' = με ∂²A'/∂t².
3. Longitudinal condition (eq. 77): φ + ∂χ/∂t = φ₀(x, y, z), a function of position only.
4. So the longitudinal electric field is E_L = −∂(∇χ)/∂t − ∇φ = −∇(φ + ∂χ/∂t) = −∇φ₀(x, y, z): static.
5. Gauss's law then gives ε∇²φ₀ = −ρ: in an insulator the free charge cannot move, so ρ and E_L are fixed in time.

Point 4 is the heart of it. The only longitudinal electric field Maxwell's equations allow in an insulator is the static field of the charges that are there. The function χ (and with it J) can be anything; it changes A and φ together and leaves E and B unchanged. This is what Maxwell meant by χ "not related to any physical phenomenon", and it is what is now called gauge freedom (primer §2, Potentials).

What Heaviside's four equations keep. Heaviside kept the longitudinal physics: ∇·D = ρ and ∇·B = 0 are exactly the scalar (longitudinal) parts. What he removed is the potentials A and Ψ and with them the gauge part χ (J). No electric or magnetic field is lost by removing χ.

3. The extended version: keep the scalar as a field

If the gauge part is not thrown away but treated as a real field, the natural scalar is S = ∇·A + (1/c²) ∂φ/∂t (the combination that Lorenz set to zero in 1867 [memory: high], and that FitzGerald wrote in his notebook at the 1888 Bath meeting, "essentially a gauge condition" [read: Hunt]). If the potentials obey the plain wave equations with sources and no gauge condition (□φ = ρ/ε₀, □A = μ₀J, with □ = (1/c²)∂²/∂t² − ∇²), then in vacuum:

∇·E = ρ/ε₀ − ∂S/∂t ∇×B − (1/c²) ∂E/∂t = μ₀J + ∇S □S = μ₀(∂ρ/∂t + ∇·J) (∇×E = −∂B/∂t and ∇·B = 0 unchanged)

These three lines were checked symbolically with SymPy for arbitrary potentials (2026-10-07). This is the "extended electrodynamics" reviewed in the van Vlaenederen Koen project (its primer §3 and arbiter, claims 1-2), where the same S field and its developers (Ohmura, Hively, Arbab and others) are listed.

A longitudinal wave in this extended form (worked example). Take A = (a₀ cos(kx − ωt), 0, 0), φ = 0, ω = ck. Then: - B = ∇×A = 0 (no magnetic field); - E_x = −∂A_x/∂t = −ωa₀ sin(kx − ωt), pointing along the direction of travel (x): longitudinal; - S = ∂A_x/∂x = −ka₀ sin(kx − ωt), travelling at c; - ∇·E + ∂S/∂t = 0, so the wave is allowed with no charge present (checked with SymPy). In Heaviside's form this field is forbidden, because ∇·E ≠ 0 where there is no charge. In the extended form it is allowed. The last line of the system above says S is created only where charge is not conserved (∂ρ/∂t + ∇·J ≠ 0). Charge conservation is among the best-tested laws in physics [memory: high], so in the extended theory ordinary circuits produce no S; this is the main obstacle the van Vlaenederen review records (arbiter: "extended theory: Plausible, untested; S measurable: Never properly tested").

4. Why the terms mattered to Maxwell

• The potentials were physical to him. In 1856 Maxwell identified A with Faraday's "electro-tonic state" (arbiter, claim 2 [read: 1856 paper]), and his electromotive force (D) is built from A and Ψ [read: page image p.32]. The potentials were his primary quantities, so he had to ask what their remaining parts do.
• Longitudinal waves were the known weakness of elastic-solid theories of light. An elastic solid carries compression waves as well as shear waves, and optics saw only transverse light [memory: high; the 1865 paper's own words are that electromagnetism and optics "both affirm the propagation of transverse vibrations"]. Showing that his equations give transverse waves only, without an ad hoc assumption, was a point in his theory's favour; that is why §§98-100 exist.
• He kept the question open. §100 says normal vibrations would follow if electricity had "a definite density", and that neither science could "affirm or deny" them. This is not a rejection.

What later physics found for Maxwell's open case [memory: high, not web-checked this run]. Charge carriers with mass and density in a medium do carry longitudinal electric oscillations: Langmuir and Tonks's plasma oscillations (1928-29), at the plasma frequency ω_p = √(n e²/(ε₀ m)), and in a warm plasma the Bohm-Gross wave, ω² = ω_p² + 3k²v_th², whose speed depends on the electron density and temperature. That is close to Maxwell's "normal vibrations propagated with a velocity depending on this density". These are waves in a medium of charges, not in empty space.

5. Why Heaviside discarded them

• The potentials did not locate the energy. Heaviside built his equations backwards from his energy-flow formula S = E×H. Maxwell's A and Ψ "did not locate the energy correctly, however, and Heaviside regarded them as quite distant from the real workings of the field" [read: Hunt]. He rewrote everything in E and H.
• The gauge part had no physical content. Maxwell himself had said χ "is not related to any physical phenomenon" (Art. 616) and that J and Ψ can be "left out of account" for waves (Art. 783). Removing the potentials removes that arbitrariness, so Heaviside's equations have no gauge freedom and no redundant equations (primer §5).
• Quaternions were "antiphysical and unnatural". Heaviside: "I dropped out the quaternion altogether, and kept to pure scalars and vectors" from 1883 [read: Hunt]. One concrete reason: in quaternions the square of a vector is negative (a² = −|a|²), and the product mixes a scalar and a vector into one object, which he found unnatural for physical quantities [memory: high for the sign rule; the "antiphysical" quote is read].
• The "murder of Ψ". At the 1888 British Association meeting in Bath, FitzGerald, Lodge, Thomson and Rowland argued over "what place, if any, the potentials ought to hold"; the account says the discussions were "not altogether conclusive" [read: Hunt].
• Belrose's sentence on Helmholtz. Belrose says Heaviside criticised Helmholtz "because Helmholtz'e electromagnetic theory allowed for the existence of longitudinal rather than solely transverse wave propagation" [read: Belrose p.7]. It is unsourced; Helmholtz's theory had an adjustable constant that, for values other than Maxwell's, allowed longitudinal electric waves [memory: medium]. If correct, Heaviside saw allowing longitudinal waves as a defect, which fits optics' transverse-only evidence of the time.

What was lost, and what was not. No electric or magnetic field was lost: the longitudinal (divergence) equations are Heaviside's ∇·D = ρ and ∇·B = 0. What was lost is the potentials as primary quantities. That loss was partly reversed later: the Aharonov-Bohm effect (1959, observed 1960 and 1986) shows A has measurable effects through its loop integral (contrarian claim 2; arbiter). The scalar S of the extended theory goes beyond Maxwell: Maxwell gave it no physical role, and it needs non-conservation of charge as a source.

Competing readings

(a) "Heaviside truncated Maxwell's longitudinal physics": not supported by the sources. The 1865 paper has no quaternions; Maxwell himself showed no longitudinal wave in an insulator, and called χ physically empty; Heaviside kept the divergence laws. (b) "Maxwell left longitudinal waves open": supported. §100 explicitly leaves normal vibrations open if electricity has density, and plasmas later showed such oscillations in media. (c) "The scalar part is physical and was thrown away": this is the extended-electrodynamics position. It goes beyond Maxwell (he gave J no physical role) and is assessed in the van Vlaenederen Koen review, where it is untested experimentally.

Leads for review (follow-up Q06)

• Treatise Arts. 616-619 and 783-784 should be added to the Maxwell project's sources (page images); the quotes above came through the web-fetch tool and are not yet checked against page images.
• Helmholtz's 1870 constant and longitudinal waves; Heaviside's own criticism (source for Belrose's sentence).
• Langmuir-Tonks 1929 and Bohm-Gross 1949 as the experimental realisation of Maxwell's §100 "electric fluid with density".

Sites

The Maxwell project has no sites.md (no Investigator run yet), so these are recorded here only.