The momentum light carries into glass

In empty space the momentum of light is not in dispute. A pulse of energy $E$ carries momentum $E/c$, and light has a pressure measures it on a mirror. The dispute begins at the surface of a piece of glass. Inside, light travels at $c/n$, and there are two simple expressions for its momentum, both derived by careful people from Maxwell’s equations. Hermann Minkowski’s of 1908 gives $nE/c$ — more than in vacuum. Max Abraham’s of 1909 gives $E/nc$ — less. For glass of index 1.5 they differ by a factor of 2.25.

The momentum of something that is not moving names this as one of the places where field momentum becomes delicate: two expressions differing by the square of the refractive index, each experimentally confirmed, and a resolution that is about which part of the system is being counted. The argument ran for about a century. The resolution is now fairly clean, and the most instructive way to reach it is to take each momentum seriously and follow it to a measurement that only it can explain.

A block that must move forward #

In 1953 Nandor Balazs gave an argument for Abraham’s value that needs almost nothing. A block of glass of mass $M$ and length $L$ sits on a frictionless table. A pulse of energy $E$ passes straight through it, entering one face and leaving the other, with both faces coated so that nothing reflects. The pulse spends longer inside than it would have spent covering the same distance in vacuum, so when it emerges it is behind where it would have been by $(n-1)L$.

Now use a theorem that holds for any isolated system: its centre of energy moves at constant velocity. A pulse that falls behind has dragged the centre of energy backwards, and the only way to restore uniform motion is for the block to move forward while the pulse is inside it, by just enough to compensate — $\Delta x = (n-1)LE/Mc^2$. The block moves because the light passed through it, and a block that moves has momentum.

This is Einstein’s argument from the box of light that weighs something with the light passing through the box instead of bouncing inside it. There, the centre of energy forced a mass on the energy of light. Here it forces a momentum on the glass.

The drawing does the argument the other way round, which is the stronger direction. It does not assume the centre of energy moves uniformly and deduce the block’s motion. It assumes a momentum for the light inside, gives the block whatever momentum is left over from the incoming $E/c$, integrates the block’s motion, and then checks the centre of energy. With Abraham’s $E/nc$ the block is left $(1 - 1/n)E/c$ and moves forward; its shift and the pulse’s lag cancel at every instant, and the centre of energy runs along a straight line. With Minkowski’s $nE/c$ the light has taken more momentum than arrived, the block must recoil backwards to pay for it, and the centre of energy ends 1.25 units off the line — which no isolated system can do.

So if “the momentum of the light” means the momentum that travels with its energy, the momentum whose flow is tied to where the energy is, then it is Abraham’s. The block moves forward, and a forward-moving block settles the kinetic question. The physical size of the effect is a separate matter: a joule of light moves a kilogram of glass by $5.6\times10^{-19}$ metres, a three-thousandth of a proton’s diameter, and nobody has measured it.

Two numbers that disagree by the square of the index #

The two expressions are reciprocal in a medium without dispersion, and the shaded region between them is the whole of the disagreement: $(n - 1/n)E/c$, more than half the vacuum momentum in water, more than four-fifths of it in glass, twice it in diamond. Each account puts that amount somewhere, and the way to read the century-long argument is that it was never really about the light. It was about the medium.

When light enters a dielectric, its field polarises the material, and the polarised material is pushed by the field. There is a force on the atoms at the front of the pulse as it arrives and at the back as it leaves, and a force wherever the pulse’s intensity changes. The force read off a surface that touches nothing shows that the force on everything inside a closed surface can be read from the stress in the field on that surface. The two accounts differ in how they split that same total into a part called the field’s and a part called the matter’s, and every experiment measures some combination.

Why each side had a good reason #

The argument lasted because each expression came with a principle that seemed to require it.

Minkowski derived his in 1908 from the requirement that the equations of electrodynamics in a medium look the same in every inertial frame — the programme the drag that was only an addition follows, in which Fresnel’s partial dragging of light by moving water turns out to be nothing but the relativistic addition of the light’s speed in the water to the water’s speed. His energy–momentum tensor for the field in matter transforms correctly between frames. It is not symmetric, though: the momentum density it gives is not the energy flow divided by $c^2$.

Abraham insisted on symmetry. A symmetric tensor ties the density of momentum to the flow of energy, and it is what makes angular momentum conserved when the field is included — the requirement at the centre of the angular momentum that is in nothing at all, where a field at rest carries angular momentum and the books balance only if it is counted. Abraham’s momentum density is the Poynting vector over $c^2$, the energy flow of where the energy of a field actually is divided by the square of the speed of light, which is exactly what makes the centre of energy behave.

Both arguments are right about what they establish, and neither establishes what it was taken to. Minkowski’s tensor is not symmetric because it leaves out part of the momentum of the matter, which is harmless for questions about the light and fatal for questions about the centre of energy. Abraham’s is symmetric because it counts the light alone and relegates everything the field does to the matter to a separate force density — the Abraham force — which is small, hard to measure and present. Each is a complete account once its matter side is written down; the dispute was between two ways of drawing the line between light and glass, argued as if it were a dispute about the light.

A ledger in which one account overflows #

The time-series makes the difference between the two quantities unmistakable. The total momentum of the pulse and the slab is $E/c$ before the pulse arrives, while it is inside and after it leaves; it cannot be anything else, because nothing outside acts. Abraham’s account divides that total between the light, 0.667, and the glass, 0.333, and returns everything to the light when it leaves.

Minkowski’s momentum does not divide the total at all. While the pulse is inside it reaches 1.50 $E/c$, half as much again as all the momentum there is. It can only be consistent if the same accounting gives the slab $-0.50\,E/c$, a backward momentum, while the light is inside. Minkowski’s momentum is not the light’s share of the total. It is a different kind of quantity, and asking which of the two is “the momentum of light” was the confusion.

The distinction the modern resolution draws is between kinetic momentum, mass times the velocity of the centre of energy, which is what Balazs’s block measures and which Abraham’s expression gives, and canonical momentum — the quantity whose conservation follows from the uniformity of the medium, whose value for a photon is Planck’s constant times the wavevector, $\hbar k = n\hbar\omega/c$. Stephen Barnett put it in this form in 2010. In empty space the two coincide, which is why the question never arises there, and why a photon with a momentum could establish $\hbar\omega/c$ from electrons in a vacuum with no ambiguity at all. In a medium the canonical one is sometimes called pseudomomentum, and it is the one that decides what happens when something inside the medium absorbs or emits light.

A cone only one momentum allows #

That last claim can be tested with a particle rather than a block, and the cleanest test is one nobody designed.

A charged particle moving through water faster than light travels in water emits light at a fixed angle to its path — Cherenkov radiation, the blue glow in a reactor pool, which the cone the source leaves behind builds as a wake of wavefronts. The same angle follows from particle kinematics, by treating the emission as the particle giving up one photon and asking what energy and momentum conservation allow. With the recoil neglected, a photon of energy $\hbar\omega$ and momentum $p$ emitted at angle $\theta$ by a particle of speed $v$ must satisfy $\hbar\omega = vp\cos\theta$.

With Minkowski’s momentum, $p = n\hbar\omega/c$, the condition becomes $\cos\theta = 1/n\beta$: Cherenkov’s angle, with its threshold at $\beta = 1/n$ and its limit of 41.2 degrees in water, which is what detectors like the giant water tanks used to see neutrinos measure every day. With Abraham’s momentum, $p = \hbar\omega/nc$, the condition becomes $\cos\theta = n/\beta$, which exceeds one for every speed below light’s in vacuum. With Abraham’s momentum the particle could never emit at all. The cone exists, it has the angle it has, and that is a measurement of the canonical momentum.

Other measurements agree. In 1954 R. V. Jones and J. C. S. Richards hung a mirror in a series of liquids, shone light on it and found the force proportional to the index of the liquid — Minkowski’s $n$ — to about one per cent, a result Jones later improved to parts in ten thousand. In 2005 a group at MIT let a Bose–Einstein condensate absorb photons and measured the recoil of the atoms; it was $n\hbar k$, with the index of the condensate itself. Both are objects embedded in the medium exchanging momentum with the light, and both see the canonical value. The mirror is pushed by the light and also by the liquid the light is pushing; the atom recoils from the photon and the medium around it takes up the rest.

And the kinetic value has had its experiment too. In 1975 Walker, Lahoz and Walker suspended a disc of dielectric in crossed, slowly varying electric and magnetic fields and measured the small torque on it — the force that Abraham’s form of the field momentum predicts inside matter and Minkowski’s does not. It was there. The light that pulls rather than pushes shows the same bookkeeping at work in optical tweezers, where the force on a bead is computed by summing momentum across the bead’s surface and the answer does not depend on which name the momentum inside the bead is given, as long as the forces on the bead’s material are counted consistently.

In real glass, not even reciprocal #

Everything so far assumed an index that does not depend on colour. Real glass disperses, and dispersion separates two indices: the phase index $n$, which fixes where the crests are and so the wavevector, and the group index $n_g$, which fixes how fast a pulse’s energy moves — the distinction the packet that moves at another speed draws for any wave.

The Balazs argument depends on the delay, and the delay is set by the group index; so the kinetic momentum is $E/n_gc$. The canonical momentum is $\hbar k$ per photon and $k$ is set by the phase index; so it is $nE/c$. In BK7 at 400 nanometres, deep in the blue where dispersion is strongest, the two indices differ by three and a half per cent, and the product of the two momenta is 0.966 of $(E/c)^2$ instead of exactly one. In the near infrared, where the glass disperses less, the product returns towards one.

That is the clearest sign that the two momenta are not two guesses at one quantity. If they were competing values of the same thing, dispersion would not assign one to the phase index and the other to the group index. They are two different quantities that happen to be reciprocal in the special case of a medium with no dispersion.

Where the two-momentum picture stops #

The medium is lossless and linear. Absorption transfers momentum to the medium in a way neither expression includes, and a strong enough pulse changes the index it sees. Both matter in real experiments with intense light, and both are corrections to the argument rather than parts of it.

Surface effects are set aside. The coated faces of the block remove reflection, and with it the large momentum transfer of a partial reflection — $2E/c$ times the reflectance, which for uncoated glass is several per cent of the incident momentum and dwarfs the effect being discussed. Real experiments fight this term more than any other.

The medium’s response is taken as instantaneous and local. Electrostriction — the tendency of a dielectric to be drawn into regions of high field — produces flows and pressure changes that depend on the shape of the beam and the stiffness of the material, and it is why experiments with liquid surfaces, where the surface can bulge, have been read as supporting both momenta at different times.

One photon at a time is a picture, not a claim. The canonical momentum $\hbar k$ per photon is sharply defined; the kinetic momentum of a single photon in a medium involves the photon and the polarisation it drags with it together, and is best thought of as a property of the dressed excitation. The figures compute everything for classical pulses.

Where in the glass the momentum sits #

None of the figures shows where in the glass the medium’s momentum is. The slab’s 0.333 $E/c$ in the ledger is a total; it is carried by the atoms under the pulse, moving forward very slightly while the pulse passes and stopping when it leaves, and the Balazs figure draws that as a single displacement of the whole block rather than as the travelling compression it is. The Cherenkov figure computes an angle from conservation and cannot show the physical mechanism — the polarisation of the medium by the passing charge, radiating coherently — that the cone the source leaves behind draws. And the experimental results are named rather than plotted, because each one comes with corrections for surfaces, flows and heating that a single curve would misrepresent.

Still open: whether the kinetic momentum has been seen directly #

The Balazs shift has never been measured. It is the one experiment that would see the kinetic momentum of light in a medium directly, as a displacement of the medium caused by nothing but the light’s passage, and at $10^{-19}$ metres per joule per kilogram it is beyond any current instrument. Proposals exist that would enhance it — very light, very long waveguides; mechanical resonators whose tiny motion is read out by the light itself; pulses in media with enormous group indices, in which light is slowed by factors of millions — and experiments in optical fibres have reported displacements claimed as evidence for one momentum or the other and then disputed. A clean measurement of a medium moving forward by the amount Abraham’s momentum requires would close the last experimental gap in a question the theory already regards as settled.

The habit worth carrying away is to ask what a conserved quantity is conserved by. Kinetic momentum is conserved because the whole system cannot push on itself; canonical momentum is conserved because the medium looks the same everywhere. In empty space the two are the same number. Inside matter they separate, and every experiment measures the one that its apparatus exchanges with the light — a free block the first, an atom or a charge embedded in the medium the second.