The drag that was only an addition

Light in water travels at $c/n$. Move the water and the light is carried along with it — but not fully. It picks up a fraction $1 - 1/n^2$ of the water’s speed, which for water is about 0.44, and that fraction depends on the refractive index.

Fresnel wrote the coefficient down in 1818, because nothing else fitted the aberration of starlight, and it was an embarrassment: an ether that is dragged partially, by an amount depending on the optical properties of whatever is moving through it, is not a substance in any ordinary sense. Fizeau measured it in 1851 and confirmed it. It stayed unexplained until 1907.

The experiment #

The apparatus is worth describing because its design is the reason a nineteenth-century measurement could settle a quantitative question.

A beam of light is split in two. Each half is sent through a pair of parallel tubes carrying flowing water — one half going with the flow in the first tube and against it in the second, the other half the reverse — and the two are recombined. A difference in the light’s speed between the two directions becomes a difference in transit time, and a difference in transit time becomes a shift of the interference fringes.

The numbers are modest. The tubes were about a metre and a half long, the water was pumped at up to seven metres per second, and the light was the yellow of a sodium flame. Everything about the apparatus was available to any well-equipped laboratory of the period; what was not available was the idea that the question could be settled by a fringe count.

Nothing has to be measured absolutely. The apparatus is symmetric, so almost everything that could go wrong affects both halves equally; what is read is a shift when the pump is turned on and off. The whole effect is a fifth of a fringe at seven metres per second, and it is visible because the comparison is with the same apparatus a moment earlier rather than with a calculation.

The three hypotheses under test predict zero, a fifth of a fringe, and half a fringe. That is not a matter of a careful error analysis; it is a matter of whether the fringes move at all and by roughly how much.

What Fresnel was trying to save #

The coefficient was not invented to explain Fizeau’s experiment, which came thirty-three years later. It was invented to explain why the aberration of starlight works.

Stellar aberration is the annual wobble in the apparent direction of every star, of about twenty arcseconds, caused by the Earth’s motion around the Sun. Bradley found it in 1728 and it is straightforward if light travels through a medium the Earth moves through freely. The trouble is that a telescope filled with water should then show a different aberration, since the light inside it travels more slowly — and Airy filled a telescope with water in 1871 and found no difference at all.

Fresnel’s coefficient is precisely what makes the two cancel. The slower light inside the water would give a larger aberration; the partial drag of the water gives an equal and opposite correction; and the telescope reads what an empty one reads. That was the requirement the coefficient was reverse-engineered to meet, decades before anybody measured it directly.

A number invented to make one experiment come out right, and then found in a different experiment, is worth taking seriously even inside a wrong framework. Fresnel’s ether was not there. The coefficient was.

Why the coefficient is what it is #

The relativistic answer takes two lines and needs nothing about water.

Light in the water’s rest frame travels at $c/n$. The water moves at $v$ in the laboratory. Speeds do not add, they compose, as speeds that refuse to add sets out:

$$u = \frac{c/n + v}{1 + v/(nc)}$$

Expand in $v/c$ and the leading terms are $c/n + v(1 - 1/n^2)$. Fresnel’s coefficient falls out.

The two limits in that figure are the ones that make the result feel inevitable once it is derived rather than postulated. At $n = 1$ the coefficient is exactly zero — moving a vacuum is not an operation, and there is nothing for the light to be dragged by. And it approaches one only as the index grows without bound, so the light is never carried along completely.

In the ether framework neither limit had an explanation. Why should an ether be dragged more by dense glass than by thin air, and by exactly this function of the index? In the relativistic framework there is no ether to drag, and the index appears because $1/n$ is one of the two speeds being composed.

What the experiment could and could not decide #

It is worth being precise about what Fizeau’s result established, because it is often credited with more.

Fizeau’s water moved at about a part in $10^8$ of the speed of light. The difference between Fresnel’s coefficient and the exact composition is second order in that, which is a part in $10^{16}$ of the effect — against a measurement good to perhaps ten per cent.

So the experiment did not, and could not, distinguish special relativity from Fresnel’s theory. What it did was measure a number that Fresnel’s theory had to be given and that relativity predicts, which is a different kind of support and a strong one. Einstein cited it, and the reason he cited it is not that it discriminated but that it was a standing anomaly his framework removed for free.

That is the more common shape of evidence than a decisive experiment. A framework that produces an existing unexplained number without adjustment has done something, even where no measurement can tell it apart from the theory it replaces.

The term nobody could have guessed #

There is a correction of a few per cent, and it is the part of the story that could only come from the transformation.

The argument is short. The coefficient involves the index at the frequency the medium sees, and the medium is moving, so that frequency is shifted. If the index varied with wavelength there would be nothing to say; it does, so there is an extra term proportional to $\mathrm{d}n/\mathrm{d}\lambda$.

Lorentz derived it, and Zeeman measured it four decades after Fizeau with a much better interferometer, in water and then in glass — moving the glass by mounting it on a rod driven back and forth, since a solid cannot be pumped. The term is a few per cent of the main one, which is well below what Fizeau could have seen and comfortably above what Zeeman could.

A correction that requires the frequency to be different in the two frames is a test of the relation between the frames, which is what the whole subject is about. The main term can be reached by several wrong routes; this one cannot.

Why water #

Air’s index is 1.0003, so its coefficient is six parts in ten thousand, and a wind fast enough to move the fringes is not a wind. That closes off the obvious experiment — the one on the medium everybody at the time actually cared about, since a gas was the nearest thing to an ether that could be made to move — and forces the measurement onto a liquid.

Water is the compromise. It has an index high enough to give a coefficient of nearly a half, and it can be pumped through a tube at several metres a second. Carbon disulphide is better still and was used in later repetitions; glass is better again and has to be moved bodily.

That constraint is not a footnote about apparatus. It is the reason the experiment took the form it did, and the reason its interpretation was contested for fifty years: the result was about water, and what everybody wanted to know about was the ether.

The modern version, at a hundred metres a second #

The experiment is still done, and its descendants are instruments rather than tests.

A ring interferometer with a moving medium in it measures exactly the Fizeau shift, and the same geometry with the medium at rest and the apparatus rotating is a Sagnac interferometer — the device inside every fibre-optic gyroscope. The two are not analogies of each other; they are the same phase difference, arrived at by moving the medium in one case and the frame in the other, and a fibre gyroscope has to have the Fizeau effect designed out of it precisely because the fibre is itself a medium.

The other descendant is in accelerator and plasma physics, where a medium moving at an appreciable fraction of the speed of light is ordinary. There the first-order coefficient is not enough and the full composition is used — the regime the third figure’s right-hand end is about, and one that Fizeau’s arithmetic would get wrong by a measurable amount.

An effect measured once as a curiosity and now designed around in a mass-produced instrument is a common enough fate for a nineteenth-century optical result. What is unusual here is that the instrument’s designers care about the second-order term, which nobody could have measured for a century after the first.

What it looks like from the other side #

The modern reading turns the experiment inside out.

There is no dragging. There is a medium, at rest in its own frame, in which light travels at $c/n$ by the ordinary optics of the medium decides the speed — a wave slowed by the response of the charges it drives. Someone moving relative to that medium describes the same light with different coordinates, and the speed they assign is the composition of the two.

The coefficient is therefore not a property of the interaction between light and matter at all. It is a property of how two frames’ coordinates relate, evaluated for a particular pair of speeds, and the index enters only as a number in the formula.

That is the same demotion that happens to the magnetic field in magnetism is electricity seen sideways: a quantity that looked like a separate physical agent turns out to be a component of something else, seen from a frame that mixes the components. In both cases what was strange about the original account was an effect whose size depended on a parameter that had no business being in it.

Why the phase velocity is the right thing to compose #

There is a subtlety in the derivation that is easy to pass over and is worth a paragraph, because it is where a careless version of the argument goes wrong.

The quantity composed above is $c/n$, the phase velocity of light in the medium. An interferometer measures a phase difference, so a phase velocity is the right quantity for it — but a phase velocity is not the speed of anything material, and composing two velocities relativistically is a statement about the motion of objects.

The composition law survives the transfer because it is really a statement about coordinates rather than about objects: it says how a ratio of a distance to a time transforms, and a phase velocity is such a ratio. What does not survive is the intuition attached to it. In a medium with strong dispersion the phase velocity can exceed $c$, as the packet that moves at another speed describes, and the composition law duly returns a result above $c$ for a moving medium — which is correct and carries no information.

So the drag coefficient is a statement about phases and not about energy transport. Deriving it by asking how fast a photon is carried by the water gives the right answer for the wrong reason, and gives the wrong answer wherever the two velocities differ.

Where the model stops #

The medium is treated as non-dispersive except in one figure and non-absorbing everywhere. A real medium with a resonance has a complex index, and the group velocity and the phase velocity part company badly near it — which matters, because what an interferometer measures is neither exactly, and separating them is the subject of the packet that moves at another speed.

The flow is assumed uniform. Water in a tube has a profile, faster in the middle than at the walls, so the light samples a range of speeds. Fizeau’s analysis used the mean, and later repetitions with better flow control found small corrections that are exactly this effect.

Only motion along the beam is treated. A medium moving across the light also does something — it deflects it — and that transverse drag is a separate effect with its own coefficient, measured much later.

And nothing here is at a speed where the exact form matters. The composition is used in full, and its departure from the first-order result is computed, but every number quoted is in the regime where the two agree to fifteen digits.

What it took to be believed #

The result was not accepted quickly, and the reason is worth recording because it is not the usual one.

Fizeau’s measurement was repeated by Michelson and Morley in 1886, seventeen years before their more famous experiment, with a much better apparatus and a longer path. They confirmed the coefficient to a few per cent and said so, and the two of them then spent the following years on the experiment that found no ether drift at all — which is the same question asked of a different medium, and which got the opposite answer.

That pair of results is what made the situation intolerable rather than merely puzzling. Light in moving water is partially dragged; light in a laboratory moving through the ether is not dragged at all. Every attempt to build one medium that does both required the drag to depend on the optical properties of whatever happened to be present, which is not a property a substance has.

The resolution was to notice that only one of the two experiments is about a medium. The water is a medium and behaves like one; the ether was never there, and the second experiment was measuring nothing. Both results then follow from one composition law, and neither needs a substance to be dragged.

What the pictures cannot show #

The fringe figure draws a shift as a number and a fringe pattern is a picture. What Fizeau read was the displacement of a set of dark bands against a scale, by eye, with the pump running and then stopped, and the uncertainty in the result is an uncertainty about a judgement of position rather than a statistical quantity that can be plotted.

The coefficient figure draws a smooth curve through indices from one to two and a half, and the media that occupy that range are not interchangeable. The figure invites reading it as a design space and it is a plot of a formula; whether a given index can be realised in something that flows is the constraint the last figure exists to supply.

Where the ladder goes next #

The velocity-addition ladder began with speeds that refuse to add and the composition law, went on to the turn that two pushes leave behind, where composing two boosts in different directions leaves a rotation, and to the motion that measures faster than light and the space that speeds live in, where the composition law turns out to be the geometry of a hyperbolic plane. This rung takes the same law back to the experiment that first needed it, forty years before it existed.

The rung after it is composition in a medium that is itself accelerating or turning, where the frame the medium is at rest in changes from place to place and the drag becomes a field rather than a number. The habit worth carrying is the one this rung is built on: when a theory needs a coefficient it cannot derive, the coefficient is usually the first term of something the theory does not have.