The Luminiferous Ether

Every wave the nineteenth century knew had something to wave. Sound is a jostling of air; ripples are a heaving of water. Pump the air from a bell jar and the bell inside rings in silence. So when Young and Fresnel proved, between 1801 and 1821, that light interferes and diffracts and polarises exactly as a wave must, physics inherited an obligation it could not put down: find the medium. It was christened the luminiferous — light-bearing — ether. The trouble is that light told the ether what to be, and the answer was monstrous. Light is a transverse wave: its oscillation is strictly perpendicular to its travel, as Fresnel established from the interference of polarised beams. Gases and liquids cannot carry transverse waves at all — they offer no resistance to shear. Only an elastic solid can, and a solid carries shear waves at $v = \sqrt{G/\rho}$ . Demand $v = c$ and you demand a rigidity $G = \rho c^{2}$ so ferocious that even a wisp of ether a million times thinner than air out-stiffens steel. Yet the planets have swept through this substance for billions of years without losing a second of orbital period. Rigid as steel; frictionless as vacuum. And it got stranger. Fresnel showed in 1818 that a moving transparent body must drag light along — but only partially, by the factor $1 - 1/n^{2}$ : neither the full drag a solid medium would give, nor the zero drag of a body slipping through untouched. Fizeau measured it in 1851, and the number was right. The ether was absurd, and it worked.

Key takeaways

The luminiferous ether was the medium supposed to carry light waves, demanded by the discovery that light interferes, diffracts and polarises exactly as a wave does.
Light is transverse, so the ether needed shear rigidity: an elastic solid, since fluids and gases carry no transverse waves at all.
A transverse wave in an elastic solid runs at $v = \sqrt{G/\rho}$ ; demanding $v = c$ forces $G = \rho c^{2}$ , beating steel's shear modulus even at a millionth of air's density.
Yet the same ether had to offer zero drag to the planets. Rigid as steel and perfectly frictionless is not a substance; it is a contradiction.
Fresnel's partial-drag coefficient $f = 1 - 1/n^{2}$ : a body of index $n$ moving at $w$ drags light along at only $fw$ — $0.435\,w$ for water, $0.556\,w$ for glass. Fizeau's 1851 experiment confirmed it within experimental error; Michelson and Morley's 1886 repeat measured it precisely.
Key idea: the ether was not merely undetected: no ordinary elastic solid could do everything it had to do. Transverse light demands rigidity $G = \rho c^{2}$ ; orbiting planets demand zero viscosity; moving matter demands a drag neither total nor absent but exactly $1 - 1/n^{2}$ . Every clause was forced by experiment, and escaping them all forced increasingly contrived frequency-dependent ethers.

The core insight

Light is a transverse wave, so its medium must resist shear. A transverse wave in an elastic solid travels at $v = \sqrt{G/\rho}$ , and setting $v = c$ forces the ether's shear modulus to $G = \rho c^{2}$ — stiffer, per unit density, than steel by nearly ten billion. That same medium must let the Earth plough through it at $29.8\ \mathrm{km\,s^{-1}}$ for four billion years without slowing. The double demand — infinite rigidity, zero friction — is the ether's fatal wound, and it was inflicted not by a null experiment but by the wave theory's own successes.

Where this lives today

The ether is gone, but the physics it was built from is not — every word of the elastic-solid argument is correct; it was merely applied to the wrong medium. Transverse waves really do require shear rigidity, and the Earth itself is the proof: earthquakes launch longitudinal P-waves and transverse S-waves, and the S-waves cross the mantle at $\sqrt{G/\rho} \approx 4.5\text{–}7\ \mathrm{km\,s^{-1}}$ and then stop dead. They cannot pass the outer core. That shadow, mapped by seismometers worldwide, is how we know the outer core is liquid: a liquid has $G = 0$ , and a medium with $G = 0$ carries no shear wave. The nineteenth century's criterion for the ether is the twenty-first century's tool for X-raying a planet. Fresnel's coefficient has had an even longer afterlife: the Fizeau effect is measured today in flowing liquids, spinning glass discs and optical fibres, and it is why a fibre-optic gyroscope must account for the motion of its own glass. As for the ether, the question it was invented to answer — light travels at $c$ relative to what? — did not vanish when the ether did. It sharpened, and it drives Ether Drift and the Absolute Rest Frame.

The physical picture

Start with the fact that killed the ether before any interferometer was built: light is transverse. Fresnel and Arago found that two beams polarised at right angles refuse to interfere — cross them at any phase you like and no fringes appear. That is impossible for a longitudinal wave, whose oscillation lies along the ray and so has no "sideways" to distinguish. It is exactly what you expect if the oscillation is perpendicular to the travel. Now ask what a transverse wave needs from its medium. Displace one layer sideways relative to the next. In air or water nothing pulls it back — a fluid has zero shear modulus by definition; it flows, and the disturbance dies. Only a solid, which resists being sheared, snaps the layer back and hands the disturbance on. That restoring force is the shear modulus $G$ , the inertia is the density $\rho$ , and the wave speed is $\sqrt{G/\rho}$ — the formula that gives a steel bar its $3.2\ \mathrm{km\,s^{-1}}$ shear-wave speed. The ether must do the same job at $c$ .

Figure

Two irreconcilable demands. Left — light is transverse, so the ether must resist shear: v = √(G/ρ), and v = c forces G = ρc². Right — the Earth must sweep through that same ether at v = 29.8 km/s (β ≈ 10⁻⁴) for four billion years with no measurable braking.

That is the whole indictment, and it was known long before anyone looked for an ether wind. But the ether's defenders had a sharper problem. If the ether fills space, what happens to it inside a piece of moving glass? Full drag — the glass carrying its ether like a bucket of water — is refuted by stellar aberration. Zero drag is refuted by Arago's 1810 prism experiment, which should have shown starlight refracting differently as the seasons changed the Earth's speed through the ether, and did not. Fresnel's answer, sent to Arago in 1818: a moving body drags the light inside it partially, by the fraction $f = 1 - 1/n^{2}$ of its own speed. That single idea kept the ether alive for seventy more years.

Figure

The Fresnel partial-drag coefficient f = 1 − 1/n² against refractive index n. Neither total drag (f = 1, the Stokes ether) nor none at all (f = 0, a body slipping through): light in a medium moving at w is carried along at exactly f·w. Water n = 1.33 → f = 0.435; glass n = 1.5 → f = 0.556 — the numbers Fizeau confirmed within experimental error in 1851, and Michelson and Morley measured precisely in 1886.

Look where the curve starts. At $n = 1$ — vacuum, or near enough, air, for which $f = 5.9\times10^{-4}$ — the drag is essentially nil, so the ether streams through the atmosphere and through a telescope tube unmoved. That is what makes the annual tilt of starlight possible at all. Fresnel wrote this curve down thirty-three years before anyone measured a point on it.

The core equations

Begin with the mechanics of any elastic solid. Push one layer sideways relative to its neighbour and the solid resists with a stress proportional to the shear strain; the constant of proportionality is the shear modulus $G$ . Working against the inertia $\rho$ , that restoring stress gives a wave travelling across itself — and the equation that sets the speed of an earthquake S-wave in rock sets the speed of light in the ether, if the ether is a solid.

v_{\perp} = \sqrt{\frac{G}{\rho}} \qquad (1)

where: $v_\perp$ — speed of a transverse (shear) wave; $G$ — shear modulus, in pascals; $\rho$ — density. Physical reading: stiffness on top, inertia underneath. A fluid or gas has $G = 0$ identically, so $v_\perp = 0$ : no transverse wave exists at all. Air carries sound only because sound is longitudinal. Now impose the one number light hands us: light is transverse and travels at $c$ , so the ether's transverse wave speed must be $c$ . Square equation (1) and rearrange — the ether's rigidity stops being a free parameter.

G_{\text{ether}} = \rho_{\text{ether}}\,c^{2} \qquad (2)

where: $G_{\text{ether}}$ — the shear modulus the ether must possess; $\rho_{\text{ether}}$ — its density; $c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}$ . Physical reading: what is pinned is the ratio $G/\rho = c^{2} = 8.988\times10^{16}\ \mathrm{m^{2}\,s^{-2}}$ . Steel has $G/\rho = 1.02\times10^{7}$ , so the ether must be $8.8\times10^{9}$ times stiffer per unit density than steel — and still let a planet through frictionlessly. The second pillar is Fresnel's. He supposed that matter condenses the ether: since $v \propto 1/\sqrt{\rho}$ at fixed elasticity, and light slows from $c$ to $c/n$ inside glass, the ether density there must be $\rho_n = n^{2}\rho_0$ . A moving body then carries only the excess it condensed, not the free ether streaming through — so the ether inside drifts at a definite fraction of the body's speed $w$ , and the light rides on that drift.

u = \frac{c}{n} \pm w\left(1 - \frac{1}{n^{2}}\right) \qquad (3)

where: $u$ — speed of light in the laboratory, inside a medium of index $n$ moving at speed $w$ ; the sign is $+$ with the flow and $-$ against it; $f \equiv 1 - 1/n^{2}$ — the Fresnel drag coefficient. Physical reading: the medium hands the light a fraction $f$ of its own velocity and no more — $0.435$ for water, $0.556$ for glass, $5.9\times10^{-4}$ for air, exactly $0$ for vacuum. Both extremes, full drag ( $f = 1$ ) and no drag ( $f = 0$ ), are excluded by experiment.

Where these results come from

Derivation AWhy the ether had to be an elastic solid

The goal is to show that the transversality of light is not a decorative detail but a structural constraint that fixes the ether's mechanical properties — and destroys it. We start from the observed fact that beams polarised at right angles cannot interfere, derive the transverse wave speed in an elastic medium from Newton's second law applied to a sheared slab, then set $v = c$ . The conclusion is not subtle: the ether must out-stiffen steel while resisting a planet no more than empty space does.

Step 1. Superpose two beams of the same frequency polarised at right angles. Experiment — Arago and Fresnel, from 1816 — says no fringes appear, at any relative phase. Why: a longitudinal wave oscillates along the ray and has no two independent "sideways" states. Only a transverse oscillation has orthogonal states that can fail to interfere.

Step 2. Take a slab of the medium, thickness $\mathrm{d}x$ , area $A$ , displaced sideways by $\psi(x,t)$ . The restoring force is the shear stress on its two faces, $\sigma = G\,\partial\psi/\partial x$ , giving a net $A\,G\,\psi_{xx}\,\mathrm{d}x$ ; Newton's second law on its mass $\rho A\,\mathrm{d}x$ collapses to $\psi_{tt} = (G/\rho)\,\psi_{xx}$ , a wave equation of speed $v_\perp = \sqrt{G/\rho}$ . Why: the slab is pulled back only if the medium objects to being sheared — if $G = 0$ it just slides, as a gas or liquid does. And $\psi_{tt} = v^{2}\psi_{xx}$ has travelling solutions $\psi(x \mp vt)$ , so the square root of the coefficient is the speed.

Step 3. Demand that this be $c$ : $G = \rho c^{2}$ . Let the ether be as dense as air, $\rho = 1.2\ \mathrm{kg\,m^{-3}}$ : then $G = 1.08\times10^{17}\ \mathrm{Pa}$ , some $1.3\times10^{6}$ times steel's. Thin it to a millionth of air: $G = 1.08\times10^{11}\ \mathrm{Pa}$ — still stiffer in shear than steel. Why: thinning lowers $G$ and $\rho$ together, so it is no escape. What is fixed is $G/\rho = c^{2}$ , exceeding steel's by $8.8\times10^{9}$ .

Step 4. Now put a planet in it: the Earth ploughs through this substance at $29.8\ \mathrm{km\,s^{-1}}$ , and the length of the year is stable to parts in $10^{10}$ . Why: rigidity and fluidity are the same property read two ways. A substance stiff enough to snap back against shear at $c$ is one a planet must fight to move through; a substance a planet does not fight cannot snap back at all. No ordinary elastic solid can meet both demands at once; the one escape — a frequency-dependent medium, rigid to fast shear and fluid to slow, the route Stokes and Kelvin took — grew ever more contrived, and Kelvin was still presenting ether–matter relative motion as an open problem (Cloud No. 1) in 1901. *As a check: the slab wave equation yields $v = \sqrt{G/\rho}$ ; steel gives $3.19\ \mathrm{km\,s^{-1}}$ ; $G = \rho c^{2}$ gives $1.08\times10^{17}\ \mathrm{Pa}$ at air density and $1.08\times10^{11}\ \mathrm{Pa}$ at a millionth of it; $c^{2}/(G/\rho)_{\text{steel}} = 8.8\times10^{9}$ .*

Go deeper · Derivation B · 3 minThe Fresnel partial-drag coefficient 1 − 1/n²

Derivation BThe Fresnel partial-drag coefficient 1 − 1/n²

This is the article's owned structural result. The question is sharp: when a transparent body moves through the ether, does it carry the ether inside it along, or does the ether stream through untouched? Both extremes are refuted — full drag would abolish stellar aberration, while zero drag should have made Arago's 1810 prism experiment show a seasonal change in the refraction of starlight, and it did not. Fresnel's resolution: a body drags only the ether it has condensed. That single idea yields $f = 1 - 1/n^{2}$ , a definite number for every material, which Fizeau confirmed within experimental error in 1851, and Michelson and Morley's 1886 repeat measured precisely.

Step 1. Adopt Fresnel's reading of the refractive index: the ether's elasticity is the same everywhere, and matter acts only by condensing the ether inside a body. Since $v = \sqrt{G/\rho}$ at fixed $G$ , and light slows from $c$ to $c/n$ , we need $\sqrt{\rho_n/\rho_0} = n$ , so $\rho_n = n^{2}\rho_0$ . Why: the cheapest mechanical picture of refraction — one medium, one elasticity, one knob. It also explains why $n \ge 1$ for ordinary matter: you can pack ether in, but there is none to take out.

Step 2. Of the total $n^{2}\rho_0$ inside the glass, the portion $\rho_0$ is the free ether of space, which streams through untouched — otherwise a moving telescope would drag the ether in its tube and aberration would not exist. What the body owns is the excess it condensed, $(n^{2}-1)\rho_0$ , and that travels with it at $w$ . Why: the body is not a bucket carrying all the water; it is a sponge in a stream, and only the water bound into the sponge goes along.

Step 3. Compute the mean velocity of the ether inside — the medium the light rides on. Weight by density: the excess $(n^{2}-1)\rho_0$ moves at $w$ , the free $\rho_0$ stays at rest, so $\bar{w} = \dfrac{(n^{2}-1)\rho_0\,w}{n^{2}\rho_0} = w\left(1 - \dfrac{1}{n^{2}}\right).$ Why: a wave rides on its medium's bulk motion, and bulk motion is momentum over mass — a density-weighted mean. The $1/n^{2}$ is exactly the fraction of the internal ether that does not belong to the glass.

Step 4. Add that drift to the wave's speed through its own medium: light travels at $c/n$ relative to the ether inside, which itself drifts at $\bar{w}$ , so $u = c/n \pm w(1 - 1/n^{2})$ . Define $f \equiv 1 - 1/n^{2}$ . Why: to first order in $w/c$ this is ordinary addition of a wave speed and a medium drift — except the "current" is only the fraction $f$ of the body's own speed.

Step 5. Read the limits and the numbers. As $n \to 1$ , $f \to 0$ : the free ether is not dragged, so starlight keeps arriving tilted and aberration survives. As $n \to \infty$ , $f \to 1$ : full drag. Between them, water gives $f = 1 - 1/1.7689 = 0.435$ and glass $f = 1 - 1/2.25 = 0.556$ . Why: the coefficient is a pure function of the index, with no free parameters — a prediction rather than a fudge, which is what let Fizeau put it on trial.

Step 6. Note where it reappears. Relativity re-derives this coefficient, exactly, as the leading term of the relativistic law for combining the speed of light in a medium with the speed of the medium — with no ether anywhere in the argument. Fresnel got the right answer from a picture wrong in every particular. Why: the coefficient is a kinematic fact about how wave and medium speeds combine, and kinematics does not care what mechanical story you tell yourself — the clue that a deeper principle hides underneath, the theme picked up in Einstein's Two Postulates. *As a check: the density-weighted mean reproduces $1 - 1/n^{2}$ symbolically; $f(1.33) = 0.4347$ , $f(1.5) = 0.5556$ , $f(2.42) = 0.8292$ , $f(1.000293) = 5.86\times10^{-4}$ ; the limits $f(1) = 0$ and $f \to 1$ as $n \to \infty$ both hold.*

Limiting cases

Case	Result	Meaning
Fluid or gas ( $G = 0$ )	$v_\perp = 0$	no transverse wave at all — why the ether could not be a gas, however tenuous, and why light forced it to be a solid
$n \to 1$ (vacuum, air)	$f \to 0$	the free ether is not dragged — the condition that lets stellar aberration exist, and gives Ether Drift and the Absolute Rest Frame a signal to hunt
$n \to \infty$	$f \to 1$	total drag, the Stokes limit. No real substance approaches it
Galilean limit ( $w/c \to 0$ )	drag term vanishes linearly	the drag is first order in $w/c$ : for Fizeau's $7\ \mathrm{m\,s^{-1}}$ water it is one part in $7\times10^{7}$ of $c/n$
Relativistic replacement	the same $f = 1 - 1/n^{2}$ to first order	the coefficient survives the death of the ether — relativity keeps Fresnel's number and throws away his mechanism

Common trapA moving medium does not carry light along at its own speed

The reflex — strong, because it is right for sound — is that if the medium moves at $w$ then everything riding in it moves at $w$ , so light in flowing water travels at $c/n + w$ . That is full drag, and it is wrong. It is equally wrong to say the water does nothing and the light travels at $c/n$ regardless; that is zero drag, and Fizeau's fringes refute it. The measured answer is $u = c/n + w(1 - 1/n^{2})$ : only $43.5\%$ of the water's speed reaches the light. Two further slips. The coefficient is $1 - 1/n^{2}$ , not $1/n^{2}$ — at $n = 1.33$ those are $0.435$ and $0.565$ . And the drag is first order in $w/c$ : small because $w \ll c$ , not from any cancellation — which is why a modest laboratory flow could reveal it, while the two-way anisotropy of the ether wind needed Michelson's interferometer.

Worked examples

1 · How much does flowing water drag light?

Water of index $n = 1.33$ flows along a tube at $w = 7.0\ \mathrm{m\,s^{-1}}$ — roughly Fizeau's 1851 flow. By how much is the speed of light in the water altered, and could a nineteenth-century interferometer see it?

Step 1. Compute the coefficient. $n^{2} = 1.7689$ , so $1/n^{2} = 0.5653$ and $f = 1 - 0.5653 = \mathbf{0.435}$ . Only $43.5\%$ of the water's velocity reaches the light — not the $100\%$ of full drag, nor the $0\%$ of none.

Step 2. Convert to a velocity: $f\,w = 0.435 \times 7.0 = \mathbf{3.04\ m\,s^{-1}}$ . Light in still water travels at $c/n = 2.254\times10^{8}\ \mathrm{m\,s^{-1}}$ , so downstream it runs at $2.254\times10^{8} + 3.04$ and upstream at $2.254\times10^{8} - 3.04$ — a fractional change of $\mathbf{1.35\times10^{-8}}$ , one part in seventy-five million. No timing measurement of the era could touch that, which is why the experiment had to be an interference experiment.

Step 3. Turn it into fringes. Send two beams in opposite directions through the same total water path $\ell = 3.0\ \mathrm{m}$ (Fizeau's two tubes). Their travel times differ by

\Delta t = \frac{\ell}{c/n - fw} - \frac{\ell}{c/n + fw} \approx \frac{2\ell f w n^{2}}{c^{2}} = 3.59\times10^{-16}\ \mathrm{s},

which for sodium light ( $\lambda = 589\ \mathrm{nm}$ ) is a shift of $\Delta N = c\,\Delta t/\lambda = \mathbf{0.18\ fringes}$ .

Step 4. Sense-check it. A fifth of a fringe is small but plainly visible; full drag would have given $2.3$ times as much, and zero drag none. Fizeau's own 1851 shift came in near $0.23$ — partial drag confirmed, but only to within ten or fifteen percent; the computed $0.18$ is what Michelson and Morley's 1886 repeat measured, $0.184 \pm 0.02$ .

2 · What would the ether have to be made of?

Impose the two things the ether must do — carry transverse waves at $c$ , and let the planets through unhindered. Compare with steel: $G = 8\times10^{10}\ \mathrm{Pa}$ , $\rho = 7850\ \mathrm{kg\,m^{-3}}$ .

Step 1. Calibrate against a real solid. Steel's shear-wave speed is $\sqrt{8\times10^{10}/7850} = \mathbf{3.19\ km\,s^{-1}}$ — real solids carry transverse waves at a few kilometres per second, the speed of an earthquake S-wave in rock.

Step 2. Demand $c$ instead: $G = \rho c^{2}$ , with $c^{2} = 8.988\times10^{16}\ \mathrm{m^{2}\,s^{-2}}$ . Grant the ether the density of air, $\rho = 1.2\ \mathrm{kg\,m^{-3}}$ : then $G = \mathbf{1.08\times10^{17}\ Pa}$ — $1.3\times10^{6}$ times steel's shear modulus. A million-fold stiffer than steel, in something as thin as air.

Step 3. Try to escape by thinning it. At a millionth of air's density, $\rho = 1.2\times10^{-6}\ \mathrm{kg\,m^{-3}}$ , we get $G = 1.08\times10^{11}\ \mathrm{Pa}$ — still $\mathbf{1.3\times}$ steel's shear modulus. The escape route is closed: thinning cuts $G$ and $\rho$ in lockstep.

Step 4. State the ratio that cannot be dodged: equation (2) pins $G/\rho = c^{2}$ . Steel's ratio is $1.02\times10^{7}\ \mathrm{m^{2}\,s^{-2}}$ ; the ether needs $8.988\times10^{16}$ , a factor of

\frac{c^{2}}{(G/\rho)_{\text{steel}}} = \frac{8.988\times10^{16}}{1.019\times10^{7}} = \mathbf{8.8\times10^{9}}.

Step 5. Now recall the other requirement: this near-infinitely rigid solid must be swept through by the Earth at $29.8\ \mathrm{km\,s^{-1}}$ with no detectable loss of orbital energy over the age of the Solar System. Drag on a body moving through an elastic solid is not optional; it is what elasticity means. The ether was asked to be the stiffest substance in the universe and no substance at all.

Physics of the portrait

The ether's most damaging critic and its most brilliant defender were the same man, and he lived to see neither verdict.

Origins · 2 minAugustin-Jean Fresnel and how this was discovered

Augustin-Jean Fresnel

1788–1827 · Paris and the Corps des Ponts et Chaussées

Fresnel was a road engineer. He trained at the École Polytechnique and the École des Ponts et Chaussées, spent his working life supervising highway construction in the French provinces, and did physics in the margins of the day — poorly paid, often ill, for years outside the Paris establishment. Between 1815 and 1821, in that spare time, he rebuilt optics: a quantitative wave theory of diffraction that won the Académie's 1819 prize (over a corpuscularian jury, one of whom, Poisson, derived from Fresnel's own equations the "absurd" bright spot at the centre of a circular shadow — whereupon Arago went and observed it); the experiments with Arago from 1816 showing that beams polarised at right angles do not interfere, hence that light is purely transverse; the Fresnel equations, still used in every optical coating; and, in an 1818 letter to Arago, the partial-drag coefficient $1 - 1/n^{2}$ that is this article's centrepiece. The Royal Society's Rumford Medal reached him in 1827 while he was dying of tuberculosis; he was thirty-nine. His transversality result made the ether a mechanical monster; his drag coefficient kept it alive for seventy more years; and relativity, when it came, kept the coefficient and buried the ether.

History

1690

— Christiaan Huygens publishes the Traité de la Lumière (presented 1678): light as a pulse spreading through a subtle ether.

1810

— François Arago finds the refraction of starlight by a prism unchanged by the season or the Earth's motion — a null result the ether must explain.

1818

— Fresnel, in a letter to Arago, proposes that a moving body drags the ether inside it only partially, by the coefficient 1 − 1/n².

1821

— Fresnel establishes that light is a purely transverse vibration, with no longitudinal component — forcing the ether to be an elastic solid.

1845

— George Gabriel Stokes proposes the rival hypothesis that the Earth fully drags the nearby ether with it.

1851

— Hippolyte Fizeau runs light through moving water and measures the drag coefficient: within experimental error it is 1 − 1/n², as Fresnel said — Michelson and Morley's 1886 repeat makes the confirmation precise.

1865

— James Clerk Maxwell publishes A Dynamical Theory of the Electromagnetic Field: light is an electromagnetic wave, and the ether becomes the seat of the field.

Going further

Texts. Whittaker, A History of the Theories of Aether and Electricity, Vol. I, Chs. 4–5 · Born and Wolf, Principles of Optics, Historical Introduction · Rindler, Introduction to Special Relativity, 2nd ed., Ch. 1 · Landau and Lifshitz, Theory of Elasticity, §22. $v_\perp = \sqrt{G/\rho}$ is the standard elastic shear-wave result. The drag law $u = c/n \pm w(1 - 1/n^{2})$ and its ether-condensation derivation $\rho_n = n^{2}\rho_0$ originate in Fresnel's 1818 letter to Arago, confirmed by Fizeau in 1851. Key results: the symbolic identity $\bar{w}/w = (n^{2}-1)/n^{2}$ ; steel shear speed $3.19\ \mathrm{km\,s^{-1}}$ ; stiffness ratio $c^{2}/(G/\rho)_{\text{steel}} = 8.8\times10^{9}$ ; $G = \rho c^{2}$ at air density gives $1.08\times10^{17}\ \mathrm{Pa}$ and at a millionth of it $1.08\times10^{11}\ \mathrm{Pa}$ ; drag speed in Fizeau's flow $3.04\ \mathrm{m\,s^{-1}}$ with a $0.18$ -fringe shift. Anchor values: $c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}$ , Earth orbital $29.8\ \mathrm{km\,s^{-1}}$ ; steel $G = 8\times10^{10}\ \mathrm{Pa}$ , $\rho = 7850\ \mathrm{kg\,m^{-3}}$ ; $n$ : water $1.33$ , glass $1.5$ , air $1.000293$ .

Builds on. Galilean Relativity · Inertial Frames of Reference · Absolute Space and Absolute Time. Leads to. Ether Drift and the Absolute Rest Frame · The Michelson–Morley Experiment · Einstein's Two Postulates.

If the ether was so obviously contradictory, why did anyone believe in it?

Because the alternative was worse. In 1820 a wave with no medium was not a bold hypothesis but a category error, since waves are what media do. And the ether was not idle: it explained interference, diffraction, polarisation and refraction quantitatively, and after Maxwell electromagnetism too. Physicists knew all about the rigidity paradox; Kelvin, Green, MacCullagh and Stokes spent decades building baroque ethers — rotationally elastic, vortex-filled, rigid to fast shear but fluid to slow motion. What killed the ether was not its absurdity, which physics tolerates from a theory that works, but its undetectability.

Why does the drag coefficient depend on n rather than on the density of the material?

In Fresnel's picture $n$ is a density: the ether inside a body is denser than the ether outside by exactly $n^{2}$ . The atoms of glass do not drag light directly; they drag the excess ether they have condensed. So the coefficient comes out as a ratio of ether densities, $(\rho_n - \rho_0)/\rho_n = 1 - 1/n^{2}$ , with nothing in it about mass, chemistry or atomic weight. Empirically that is right: two materials of the same index drag light identically, whatever they are made of. The mechanism is fiction; the functional dependence is exactly what nature does.

Does the coefficient mean the ether really is being partly dragged?

No — and this is the deep joke of the subject. The formula is right and the picture behind it is entirely wrong. There is no ether, nothing is condensed inside glass, nothing is dragged. Relativity later showed that $u = c/n + w(1 - 1/n^{2})$ is simply the leading term you get on combining two velocities — the speed of light in the medium and the speed of the medium — with the correct relativistic rule instead of the Galilean one. Fresnel's $1/n^{2}$ is really a cross-term in a velocity-addition formula.

Couldn't the ether just be a very stiff gas?

No, and this is the most important structural point here. "Stiff gas" is a contradiction: what makes a gas a gas is that it has zero shear modulus — one layer slides past another with arbitrarily little force, given time. A gas resists compression (its bulk modulus, which is why it carries sound, a longitudinal wave) but does not resist shear at all. Since light is transverse, its medium must resist shear — so it cannot be a gas or a liquid, however dense, and however stiff against compression, you make it. The rigidity paradox is structural, not a matter of picking better numbers.

Why doesn't the Earth's atmosphere drag light and ruin astronomy?

Because air has $n = 1.000293$ , so its drag coefficient is $f = 1 - 1/n^{2} = 5.9\times10^{-4}$ — six parts in ten thousand. With the Earth moving at $29.8\ \mathrm{km\,s^{-1}}$ , the atmosphere drags the light inside it by only $0.018\ \mathrm{km\,s^{-1}}$ , a negligible perturbation on the aberration angle. That near-zero value is not an accident but a requirement: if air dragged light appreciably, the observed annual tilt of starlight would not match $v/c$ . Fresnel's coefficient is built so that $n \to 1$ gives $f \to 0$ and the sky stays clean.

Did Maxwell's electromagnetic theory make the ether better or worse?

Both, which is why it was so destabilising. Maxwell's 1865 theory made the ether respectable: no longer a mechanical jelly invented to carry light, but the seat of the electric and magnetic fields, with a definite energy density and stress. Yet it also produced $c = 1/\sqrt{\mu_0\varepsilon_0}$ — a wave speed built from laboratory constants alone, with no observer's velocity in it. That begs a question: at $c$ relative to what? The natural answer, "relative to the ether", converts the ether from a mechanical convenience into a claim about an absolute rest frame — a claim you can go out and test.