Electromagnetism and the Failure of Galilean Invariance

Maxwell's Equations in Vacuum establishes the fact this article is going to break physics with. Take the two curl laws, let them chase each other, and out falls a wave equation whose speed is $c = 1/\sqrt{\mu_0\varepsilon_0}$ — assembled, as The Electromagnetic Wave Equation shows, from nothing but the permittivity and the permeability of empty space. Both numbers come off a bench: $\varepsilon_0$ from the force between charges at rest, $\mu_0$ from the force between steady currents. Weber and Kohlrausch measured their combination in 1856 and got $3.107\times10^8\ \mathrm{m/s}$ without knowing what they were looking at. Maxwell, back in London in the autumn of 1861, recognised the number. Now ask the question nobody had asked. Newton's laws keep their form under a Galilean boost — that is the whole content of Galilean Transformations, and the reason the Earth's motion does not throw your coffee. Run the wave equation through the same boost, $x' = x - vt$ , $t' = t$ , and it does not keep its form. A cross term appears — a mixed derivative with coefficient $2v/c^2$ , simply not there before — so the equation an observer in $S'$ is obliged to write is no longer a wave equation. Worse: interrogate it for its wave speeds and it hands back $c-v$ forward, $c+v$ backward. Galilean velocity addition, disgorged by the field equation itself, unasked.

Maxwell's equations can be exactly true in at most one frame. The nineteenth century named that frame — the luminiferous ether — and went looking for it. It is not there.

Key takeaways

Maxwell's equations fix a speed $c = 1/\sqrt{\mu_0\varepsilon_0} = 2.9979\times10^{8}\ \mathrm{m/s}$ , built from two static bench constants. Nothing in the inputs knows about the motion of a source, an observer, or a bench.
Boost the vacuum wave equation and the operator becomes $(1-\beta^2)\partial_{x'x'} + (2v/c^2)\,\partial_{x'\!t'} - (1/c^2)\partial_{t't'}$ , $\beta \equiv v/c$ . The cross term is the whole story: absent in $S$ , and odd in $v$ — so no rescaling of the constants, all even, can absorb it.
Feed that equation a travelling wave $\psi = F(x'-ut')$ and the coefficient condition factorises to $(u+v)^2 = c^2$ . Both roots: $u = c-v$ , $u = -c-v$ . Velocity addition is not assumed here; it falls out.
So galilean invariance fails for electromagnetism, and three exits remain: break Maxwell, accept one preferred frame (the ether), or break the transformation.
Key idea: Newton's mechanics survives a Galilean boost; Maxwell's electromagnetism does not. A theory carrying a fixed speed in its bones cannot be form-invariant under a map in which speeds simply add. One of the two had to go — and it was not electromagnetism.

The core insight

A Galilean boost adds a velocity to everything. Maxwell's equations contain a velocity — $c$ , built out of $\varepsilon_0$ and $\mu_0$ — with no dial on it to be added to. Boost them and the wave operator sprouts a cross term $\propto v$ ; ask how fast light goes and it answers $c-v$ one way, $c+v$ the other. The two theories are not merely awkward together: they are incompatible, and closing that gap is what special relativity is for.

Where this lives today

The cross term of equation (2) is a live experimental target, not a historical curiosity. Modern tests of Lorentz invariance add to the Standard Model, by hand, a catalogue of exactly the frame-dependent terms a Galilean-covariant electrodynamics would need, then hunt for their effects. The flagship instruments are direct descendants of Michelson's stone slab: perpendicular optical cavities of cryogenic sapphire, rotated on air bearings, their resonant frequencies compared. Any orientation-dependence of $c$ would show up as a beat note locked to the turntable. None has appeared: the bound now stands near $\Delta c/c \lesssim 10^{-18}$ . The other half of the story is quieter and larger: Maxwell's equations are form-invariant under the right boost, so accelerator physicists design in the beam's rest frame and build in the laboratory's, transforming the fields between the two as routine — and the magnetic field of a current-carrying wire turns out to be an electrostatic field seen from a moving frame (Magnetism as a Relativistic Effect). Every synchrotron light source in the world is that argument, running at full power.

The physical picture

Charge a capacitor and measure the force between its plates: you have $\varepsilon_0$ . Hang two parallel wires, run a steady current through each, weigh the force between them: you have $\mu_0$ . Neither experiment moves. Neither contains a velocity — not the source's, not the receiver's, not the bench's. Yet $1/\sqrt{\mu_0\varepsilon_0}$ has the units of a speed, and it comes out at three hundred million metres per second. That is already strange. It becomes intolerable the moment you ask relative to what? Sound goes at $343\ \mathrm{m/s}$ — relative to still air; run toward the whistle and you measure more. Every speed the nineteenth century knew was relative to something. Maxwell's $c$ is handed to you by no medium at all — by two static measurements — and it sits in the field equations as a bare number.

Figure

The assembly line. Two static bench experiments hand you ε₀ and μ₀; the curl laws chase each other into a wave equation; its speed is 1/√(μ₀ε₀) = 2.998 × 10⁸ m/s. No velocity enters the inputs — no source, no observer, no ether.

Newtonian mechanics is untroubled by boosts because its laws mention only accelerations, which a constant velocity leaves alone — so a theory of accelerations can never tell you your speed. Maxwell's equations speak of a speed, and under a map that adds $v$ to every velocity in sight, that speed has nowhere to hide.

Figure

PUNCHLINE — the light cone of the SAME wave equation in S (symmetric, ±c) and after a Galilean boost at v = 0.3c (skewed: 0.7c forward, 1.3c backward). Below: the anisotropy of c this predicts for Earth's orbit, log scale, against what is measured. Like for like, the gap is 5 × 10⁹.

The skewed cone is not an ether wind blowing light about — it is what the field equation itself says once boosted. The ladder below it separates two quantities routinely merged. The one-way split between $c+v$ and $c-v$ is first order, a thumping $2\beta = 2\times10^{-4}$ — but you cannot get at it without clocks already synchronised at both ends of the path, and synchronising them requires the one-way speed of light. What an interferometer can reach is the two-way anisotropy, $\beta^2/2 = 4.9\times10^{-9}$ ; modern rotating resonators have driven that bound below $10^{-18}$ , five billion times smaller than the theory demands.

The core equations

In a source-free vacuum Maxwell's equations collapse to one statement: every field component obeys the same wave equation, with one speed. The Electromagnetic Wave Equation derives it — curl Faraday's law, substitute Ampère–Maxwell. We take that as given, and boost it.

\frac{\partial^{2}\psi}{\partial x^{2}} - \frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} = 0, \qquad c \equiv \frac{1}{\sqrt{\mu_0\varepsilon_0}} \qquad (1)

where: $\psi$ — any Cartesian component of $\mathbf{E}$ or $\mathbf{B}$ ; $\varepsilon_0 = 8.8542\times10^{-12}\ \mathrm{F/m}$ — the permittivity of free space, from static charges; $\mu_0 = 1.2566\times10^{-6}\ \mathrm{H/m}$ — the permeability, from steady currents. Physical reading: $c$ is not an input here but an output of two bench measurements — the same number for every component, frequency, direction and source. Now boost. The Galilean map slides the spatial origin and leaves the clock alone: $x' = x - vt$ , $t' = t$ . Chain-rule it and the space derivative survives untouched, $\partial_x = \partial_{x'}$ — but the time derivative does not, because a point held fixed in $S$ drifts through $S'$ : $\partial_t = \partial_{t'} - v\,\partial_{x'}$ . The clause $t' = t$ forces that convective term, and it is the clause that will have to go. Substitute into (1), expand the square, collect. This is the equation an observer in $S'$ is forced to write, if Maxwell holds in $S$ and Galileo connects the frames.

\left(1-\beta^{2}\right)\frac{\partial^{2}\psi}{\partial x'^{2}} \;+\; \frac{2v}{c^{2}}\,\frac{\partial^{2}\psi}{\partial x'\,\partial t'} \;-\; \frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t'^{2}} \;=\; 0 \qquad (2)

where: $v$ — speed of $S'$ relative to $S$ along $x$ ; $\beta \equiv v/c$ ; the middle term is a mixed second derivative, absent from (1). Physical reading: two things changed and only one matters. Rescaling $\partial_{x'x'}$ from $1$ to $1-\beta^2$ is survivable — an overall constant can be absorbed. The cross term cannot be: it is odd in $v$ , so no adjustment of the even constants $\mu_0$ and $\varepsilon_0$ produces or cancels it, and it dies only at $v = 0$ . This is what "galilean invariance fails" means, written out. Now ask the boosted equation the question that matters — how fast does it propagate? — by feeding it a rigid profile of unchanging shape, travelling at speed $u$ in $S'$ .

\psi = F(x' - ut') \quad\Longrightarrow\quad \left(1-\beta^{2}\right) - \frac{2uv}{c^{2}} - \frac{u^{2}}{c^{2}} = 0 \quad\Longleftrightarrow\quad (u+v)^{2} = c^{2} \qquad (3)

where: $u$ — the wave speed in $S'$ ; $F$ — an arbitrary twice-differentiable profile. Physical reading: the condition factorises exactly, and its two roots are $u = c-v$ and $u = -c-v$ . The boosted theory propagates light at $c-v$ forward, $c+v$ backward. That is Galilean velocity addition — and it was never assumed. It came out of the field equation.

Where the incompatibility lives

Derivation ABoosting the wave equation, and the cross term that will not go away

The claim is sharper than "the equations look different in a moving frame", which is true of almost any equation in almost any coordinates. It is that the vacuum wave equation is not form-invariant: an observer in $S'$ applying Maxwell's equations to her own coordinates writes a genuinely different differential equation, and no redefinition of the constants reconciles the two. We then ask what wave speeds the boosted equation predicts — whereupon it hands back Galilean velocity addition, unprompted.

Step 1. Chain-rule, then square. Under $x' = x-vt,\ t' = t$ we get $\partial_x = \partial_{x'}$ and $\partial_t = \partial_{t'} - v\,\partial_{x'}$ , so $(\partial_{t'} - v\partial_{x'})^2 = \partial_{t't'} - 2v\,\partial_{x'\!t'} + v^2\partial_{x'x'}$ . Substitute into (1) and collect: equation (2). Why the time derivative is the culprit: holding $x$ fixed means drifting backwards through $S'$ at $v$ , so "the rate of change at a fixed point of $S$ " mixes in a spatial gradient — and the cross-multiplication throws out a mixed derivative that a wave equation simply does not have.

Step 2. Prove it cannot be repaired. Suppose $S'$ declares different vacuum constants, so her law reads $\partial_{x'x'}\psi - c'^{-2}\partial_{t't'}\psi = 0$ for some $c' = 1/\sqrt{\mu_0'\varepsilon_0'}$ . Matching (2) to that form, up to an overall factor, requires the cross-term coefficient to vanish: $2v/c^2 = 0$ , hence $v = 0$ . Why no cleverness helps: that coefficient is odd in $v$ , while everything available to be tuned — $\mu_0$ , $\varepsilon_0$ , any overall scale — is even. You cannot build an odd function out of even ones, so the obstruction is structural and bites at every $v \neq 0$ .

Step 3. Extract the wave speeds. Try $\psi = F(x'-ut')$ for arbitrary $F$ : then $\partial_{x'x'}\psi = F''$ , $\partial_{x'\!t'}\psi = -uF''$ , $\partial_{t't'}\psi = u^2F''$ , and (2) collapses to $F''\left[(1-\beta^2) - 2uv/c^2 - u^2/c^2\right] = 0$ , so the bracket must vanish. Why an arbitrary $F$ is the right probe: a wave equation propagates any shape rigidly, so a speed that works for every $F$ is the theory's own.

Step 4. Solve. Multiplying by $c^2$ gives $c^2 - v^2 - 2uv - u^2 = 0$ — precisely $c^2 - (u+v)^2 = 0$ . Hence $(u+v)^2 = c^2$ , with both roots $u = c-v$ and $u = -c-v$ . Why this is the punchline: the Galilean-transformed Maxwell equations propagate light at $c-v$ along the motion and $c+v$ against it. We did not put velocity addition in — the field equation gave it to us, and with it the verdict that Maxwell holds exactly in at most one frame. *Verified symbolically in verify_aq-sr-electromagnetism-galilean-invariance-failure.py: the three boosted coefficients; the parity argument; the factorisation $c^2-(u+v)^2$ ; both roots; and $u = +0.7c,\ -1.3c$ at $v = 0.3c$ .*

Go deeper · Derivation B · 2 minWhy nobody could simply fix Maxwell, and the trilemma that remained

Derivation BWhy nobody could simply fix Maxwell, and the trilemma that remained

The natural nineteenth-century reflex is to let the constants be frame-dependent — let $\varepsilon_0$ and $\mu_0$ vary with the observer's motion, so each frame gets its own $c$ and Galilean kinematics is saved. This card shows why that escape is barred, and barred by the experiments that define the constants rather than by theoretical prejudice. Three exits then remain, each with a price.

Step 1. Look at what defines the constants. $\varepsilon_0$ is fixed by the Coulomb force between charges at rest, $F = q_1q_2/(4\pi\varepsilon_0 r^2)$ ; $\mu_0$ by the force per unit length between two long, straight, steady currents, $F/L = \mu_0I_1I_2/(2\pi d)$ . Why this settles it: count the velocities in those two experiments. There are none. A number that no velocity went into cannot come out velocity-dependent.

Step 2. Assemble them: $\mu_0\varepsilon_0 = 1.1127\times10^{-17}\ \mathrm{s^2/m^2}$ , so $1/\sqrt{\mu_0\varepsilon_0} = 2.9979\times10^{8}\ \mathrm{m/s}$ . Maxwell's Equations in Vacuum derives that identification; we recompute the number, we do not re-derive it. Why it was electrifying in 1861: Weber and Kohlrausch had measured the combination in 1856 as $3.107\times10^8$ and had no idea what it was. Maxwell did.

Step 3. Return to Step 2 of the previous card. Whatever values $S'$ assigns her constants, the cross-term coefficient she needs is zero and the one she has is odd in $v$ . There is no slot in Maxwell's equations in which to hide the observer's motion. Why the attempts failed in the laboratory too: a Galilean-covariant electrodynamics of moving bodies requires matter to drag the ether along completely — and Fresnel's partial-drag coefficient, measured by Fizeau in 1851, forbids precisely that (The Luminiferous Ether).

Step 4. So the trilemma. (i) Break Maxwell — modify the field equations until they are Galilean-invariant; nobody produced a version that survived the data. (ii) Keep Maxwell, accept one preferred frame — the equations then hold exactly only in the ether's rest frame, and the Earth's $29.8\ \mathrm{km/s}$ orbit must betray itself as an ether wind: the programme of The Luminiferous Ether and Ether Drift and the Absolute Rest Frame, executed by The Michelson–Morley Experiment, which found nothing. (iii) Keep both, break the transformation — abandon $t' = t$ . Why exit (iii) sounds insane and is correct: $t' = t$ is not a law of physics but an assumption so old nobody had noticed it was one. Removing it is the work of Einstein's Two Postulates — and Maxwell's equations, left untouched, come through form-invariant. *Verified numerically in verify_aq-sr-electromagnetism-galilean-invariance-failure.py: $\mu_0\varepsilon_0 = 1.1127\times10^{-17}$ and $1/\sqrt{\mu_0\varepsilon_0} = 2.99792\times10^{8}\ \mathrm{m/s}$ (matching the exact SI $c$ to two parts in $10^{14}$ ), and the deviations of the 1849, 1856 and 1862 values from it.*

Limiting cases

Case	Result	Meaning
Galilean limit $c \to \infty$	$2v/c^2 \to 0$ , $(1-\beta^2)\to 1$	nothing to break: mechanics is Galilean-invariant precisely because it contains no $c$
Wave speeds from (3)	$u = c-v$ and $u = -c-v$	Galilean velocity addition, produced by the field equation rather than imposed on it
$v \to c$	forward root $u = c-v \to 0$	a frozen light wave — the Galilean prediction the young Einstein could not swallow, and a solution Maxwell's equations do not have

Common trapThe failure is Galileo's, not Maxwell's — and the cross term is not "a small correction at low speed"

Two misreadings. First: "so Maxwell's equations are only approximately right." No — they are exactly right, and exactly form-invariant, under the correct transformation. What equation (2) exposes is a defect in the Galilean map, specifically the clause $t' = t$ . Second, subtler: "the cross term is tiny at everyday speeds, so nothing is really wrong." Tiny is not zero, and here the structure matters, not the size: a term odd in $v$ , absorbable by no redefinition of the constants, proves non-invariance at every $v \neq 0$ . How big the residue is — whether anyone can catch it — is a separate, experimental question, and the one The Michelson–Morley Experiment was built to answer.

Worked examples

1 · The speed of light, from a capacitor and a current balance

Nothing here looks at the sky. Take $\varepsilon_0$ and $\mu_0$ from static bench measurements, see what speed the vacuum hands you, and set it beside the values the opticians were quoting in the 1860s.

Step 1. Multiply: $\mu_0\varepsilon_0 = (1.2566\times10^{-6})(8.8542\times10^{-12}) = 1.1127\times10^{-17}\ \mathrm{s^2/m^2}$ . The units give the game away — $\mathrm{(H/m)(F/m) = s^2/m^2}$ , the reciprocal of a squared speed. Take the reciprocal square root: $c = \mathbf{2.9979\times10^{8}\ m/s}$ , matching the modern exact value $299\,792\,458\ \mathrm{m/s}$ to two parts in $10^{14}$ . Two static experiments have silently produced a velocity.

Step 2. Compare with the measurements of the day. Fizeau's toothed wheel (1849) gave $3.15\times10^{8}$ , high by $\mathbf{+5.1\%}$ ; the electrical route of Weber and Kohlrausch (1856) gave $3.107\times10^{8}$ , high by $\mathbf{+3.6\%}$ ; Foucault's rotating mirror (1862) gave $2.98\times10^{8}$ , low by $\mathbf{-0.6\%}$ .

Step 3. Read the verdict. A spinning cog, a spinning mirror and a Leyden jar agree within a few per cent — and the field equations reproduce all three from constants that have nothing to do with light. That is the identification: light is an electromagnetic wave. It is also why the clash with galilean invariance could not be shrugged off: you cannot dismiss a number that three independent benches keep handing you.

2 · The first-order ether signal you can never quite catch

If Maxwell's equations hold only in the ether frame, then over a one-metre path in a laboratory drifting at the Earth's orbital speed, light going with the drift and light going against it must arrive at different times. How different — and why did nobody simply measure it?

Step 1. Get $\beta$ . With $v = 29.8\ \mathrm{km/s}$ and $c = 2.998\times10^{8}\ \mathrm{m/s}$ , $\beta = 9.94\times10^{-5}$ , so the one-way speeds $c-v$ and $c+v$ differ fractionally by $2\beta = 2.0\times10^{-4}$ . First order — and two parts in ten thousand is enormous by optical standards.

Step 2. Turn it into a time. Over a path $L$ the difference between the two one-way trips is $\Delta t = \dfrac{L}{c-v} - \dfrac{L}{c+v} = \dfrac{2Lv}{c^{2}-v^{2}}$ . At $L = 1\ \mathrm{m}$ that is $\mathbf{0.66\ ps}$ .

Step 3. Turn that into an optical path: $c\,\Delta t = \mathbf{199\ \mu m}$ , which in sodium light ( $\lambda = 589\ \mathrm{nm}$ ) is $\mathbf{338\ wavelengths}$ — over a single metre of bench. Hundreds of fringes: colossal, unmissable, sitting right there.

Step 4. Now try to collect it, and hit the wall. Timing a one-way trip needs a clock at each end of the metre agreeing to better than a picosecond — and setting two distant clocks means sending a signal between them and correcting for its travel time, using the one-way speed of light: the very thing you are trying to measure. The circle will not close, and the failure is structural, not technological (Ether Drift and the Absolute Rest Frame; The Conventionality of Simultaneity). Timing $0.66\ \mathrm{ps}$ is trivial today with a frequency comb; the synchronisation is still not a measurement. Which is why Michelson surrendered the first-order signal for a second-order one some $2\times10^{4}$ times smaller, and built an instrument that goes out and comes back.

Physics of the portrait

Maxwell did not set out to destroy Galilean relativity, and never knew he had. He died in 1879 — before Michelson's first interferometer, before Hertz's sparks, twenty-six years before Einstein.

Origins · 2 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Edinburgh · Aberdeen · London · Cambridge

The relevant fact about Maxwell, for this article, is how the speed of light got into his theory: not by measurement, not by assumption, but as an accident of bookkeeping he could not ignore. Working the vortex-ether model at Glenlair, his family estate in Galloway, in the summer of 1861, he got a formula for his medium's elastic constant — but no number: "I worked out the formulae in the country," he wrote, "before seeing Weber's number." Only back in London that autumn could he feed the electrical number in: $310\,740$ km/s against Fizeau's optical $314\,858$ km/s, agreeing to $1.3\%$ . He printed the identification in Part III of On Physical Lines of Force (Phil. Mag., January 1862). In the 1865 Dynamical Theory of the Electromagnetic Field he kept the equations and threw the vortices away: the scaffolding was ether, the building was not. What survived is a field theory with an absolute speed baked into it — a speed Maxwell still read as relative to the ether, because there was no other way to read a wave speed. He came close to seeing the trouble: his 1879 letter to D. P. Todd judged any terrestrial optical test of the Earth's motion through the ether second order in $v/c$ , and hopeless. Right about the order, wrong about the hopelessness — Michelson read that letter and built the instrument anyway.

History

1861

— Maxwell derives his medium's wave speed at Glenlair, then in London that autumn puts Weber and Kohlrausch's number in and finds it within 1.3 per cent of Fizeau's measured speed of light.

1862

— Part III of "On Physical Lines of Force" prints the identification: light is an electromagnetic wave.

1865

— "A Dynamical Theory of the Electromagnetic Field": the equations stand alone, the vortices are discarded, light is electromagnetic.

1879

— Maxwell writes to D. P. Todd that a terrestrial optical test of the Earth's motion is second order in v/c and looks hopeless; the letter is read to the Royal Society and printed in Nature in 1880.

1884

— Heaviside recasts the theory as the four vector equations used today.

1887

— Hertz generates and detects the predicted waves at radio frequency; Michelson and Morley hunt the ether wind the preferred-frame resolution demands, and find nothing.

1904

— Lorentz publishes the transformation under which Maxwell's equations are exactly form-invariant, still reading it as a dynamical effect of motion through a real ether.

1905

— Einstein leaves Maxwell untouched and discards absolute time instead.

Going further

Texts. Griffiths, Introduction to Electrodynamics, 4th ed., §9.2 and Ch. 12 · Rindler, Introduction to Special Relativity, 2nd ed., §§1–3 · Jackson, Classical Electrodynamics, §11.1.

Provenance. The boosted operator and its cross term are standard (Rindler §§1–3; Griffiths Ch. 12); Every number is recomputed in verify_aq-sr-electromagnetism-galilean-invariance-failure.py from the band anchors $\varepsilon_0 = 8.8541878128\times10^{-12}$ , $\mu_0 = 1.25663706212\times10^{-6}$ , $c = 2.998\times10^8\ \mathrm{m/s}$ , $v_\oplus = 29.8\ \mathrm{km/s}$ , $\lambda_{\mathrm{Na}} = 589\ \mathrm{nm}$ . Two corrections are logged in the verifier's footer: Heaviside's four-vector form is dated 1884, as Maxwell's Equations in Vacuum states, not 1889; and the punchline figure sets the two-way prediction against the two-way bound, not a one-way prediction against a two-way measurement.

Builds on. Maxwell's Equations in Vacuum · The Electromagnetic Wave Equation · Galilean Transformations · The Principle of Relativity. Leads to. The Luminiferous Ether · The Michelson–Morley Experiment · The Constancy of the Speed of Light · Einstein's Two Postulates.

If the cross term is tiny at everyday speeds, why is this a crisis rather than a small correction?

Because the argument is structural, not numerical. The cross term carries coefficient $2v/c^2$ , odd in $v$ ; every parameter you could adjust to cancel it is even in $v$ . So it cannot be removed at any non-zero $v$ — the vacuum wave equation is simply not form-invariant. Its size is a separate, experimental question: it decides how hard the preferred frame is to find, not whether physics has a problem. Physics had the problem from 1865, measurable or not.

Doesn't the ether solve everything? Maxwell's equations hold in the ether frame, and that is that.

It is coherent, and was the majority view for forty years — but it is expensive. You must accept that the principle of relativity, which mechanics obeys exactly, simply fails for electromagnetism, so that a purely optical experiment in a sealed cabin could reveal your absolute velocity. And it makes a hard prediction: the Earth ploughs through the ether at $29.8\ \mathrm{km/s}$ , so there must be a wind. The Michelson–Morley Experiment went looking and came back empty-handed. Exit (ii) is not refuted by logic. It is refuted by an interferometer.

Where does the one-way anisotropy 2β actually go? An interferometer sees β², not β.

Into the round trip, where it cancels. Going out you are slowed to $c-v$ ; coming back you are sped to $c+v$ . The two first-order errors have opposite signs and, on a there-and-back path, very nearly annihilate — leaving a second-order residue. So the $2\beta = 2\times10^{-4}$ effect is entirely real, and is exactly what equation (2) predicts, but no closed-path experiment can see it. What survives is $\beta^2/2 \approx 5\times10^{-9}$ — what an interferometer is for. Reaching first order needs synchronised clocks, which needs the one-way speed of light. Round and round it goes.

Was Maxwell aware that his equations broke Galilean invariance?

Not in these terms, and it would be anachronistic to expect it. He worked inside the ether picture, where a wave equation with a fixed speed is unremarkable — sound has one too, and its speed is relative to air. That $c$ was a speed relative to the ether was, for him, an explanation rather than a problem. What he did see, in the 1879 letter to D. P. Todd, was the practical consequence: a terrestrial optical test of our motion through that medium would be second order in $v/c$ , and looked hopeless. Suspecting the transformation law itself needed someone willing to give up $t' = t$ — and nobody was until 1905.

Could you not declare Maxwell's equations to be the laws of the ether frame, with (2) the law everywhere else?

You can, and it is self-consistent — but look at what you have conceded. The laws of physics now take a different form in every inertial frame, and that form encodes your absolute velocity. Physics acquires a preferred frame, discoverable by experiment, and the principle of relativity becomes a coincidence of mechanics. This was the Lorentz programme: by 1904 Lorentz had the transformation that makes the equations invariant, but read it as a dynamical distortion of rods and clocks caused by motion through a real ether. The mathematics of 1904 and 1905 largely coincides; the physics does not.

If Maxwell's equations are Lorentz-invariant, why is this article called a "failure"?

The failure is Galileo's, not Maxwell's — the title records a nineteenth-century verdict, not a twentieth-century one. From where we now stand, equation (2) is not evidence against electromagnetism at all: it proves the Galilean transformation is an approximation, good to $O(\beta^2)$ , and that the true relation between inertial frames must mix space and time. Maxwell's equations were relativistic decades before relativity, and never needed a single amendment. What needed amending was the assumption — unstated, two centuries old — that all observers share one clock.