Light travels at c/1.5 inside glass. Why does that not break the invariance of the speed of light?
A straw in a glass of water looks bent. The reason given is "light is slower in water." And indeed water's refractive index is 1.33, so light travels at \(c/1.33\); glass gives \(c/1.5\), diamond \(c/2.42\) — barely 40% of the vacuum value. Yet relativity insists the speed of light is invariant. There is no contradiction, because the thing that slows down is not the photon. Photons flying between atoms travel at exactly \(c\), always. What is slow is a different wave — light mixed with the material's polarization. Follow that mixing and you see exactly where the number \(n\) comes from. You will also see why the textbook story — "atoms absorb the photon and re-emit it a moment later, so it takes longer" — is wrong.
| Material | n (visible) | Speed of light |
|---|---|---|
| Vacuum | 1 | 3.00×10⁸ m/s |
| Air | 1.0003 | essentially the same |
| Water | 1.33 | 2.25×10⁸ m/s |
| Glass | 1.5 | 2.00×10⁸ m/s |
| Diamond | 2.42 | 1.24×10⁸ m/s |
| Silicon (infrared) | 3.5 | 0.86×10⁸ m/s |
These are measurements. Inside diamond, light manages only 41% of its vacuum speed. So what happens to "the speed of light is invariant"?
Relativity says that the maximum speed at which causal influence propagates is \(c\). Light happens to travel at \(c\) because the photon is massless, and a massless particle has no other option.
What travels through matter is not pure light. So it need not travel at \(c\) — nothing is broken. The real question is what is travelling instead.
Let us clear away the widely circulated bad story first.
"An atom absorbs the photon and is excited; a little later it re-emits. Repeat this, and on the whole the light is delayed."
Appealing, but it fails for at least four reasons.
| Problem | Why it fails |
|---|---|
| Direction is not preserved | An excited atom re-emits nearly isotropically. If this were happening, glass would not be transparent — it would be white like ground glass. In reality the beam goes straight through |
| The timing is off | Atomic excited-state lifetimes are nanoseconds. With \(10^7\) layers in 1 cm of glass, the product is milliseconds. The measured delay is picoseconds |
| Phase is destroyed | Spontaneous emission destroys coherence. Interference and image formation would both become impossible |
| There is nothing to absorb with | Glass is transparent in the visible precisely because it has no resonance there. With no level to absorb into, "absorbs" is a non-starter |
In short, the photons are not actually being absorbed. So what is happening?
The incoming field shakes the electrons in each atom. Far from resonance this is not absorption but a driven oscillation. An oscillating charge radiates — that is the scattered wave.
The waves scattered by an enormous number of atoms add up in phase only in the forward direction and cancel in every other direction (because the atoms are packed far more densely than a wavelength). That is why light goes straight.
What survives is original wave + forward-scattered wave. And here is the decisive fact: off resonance, a driven oscillator responds 90° out of phase with the drive. Adding a small vector at right angles barely changes the length — it only rotates the direction.
Amplitude unchanged, phase shifted. That is the refractive index.
Crossing a thin slab of thickness \(dz\) adds \(i\,k(n-1)\,dz\) to the field (the \(i\) is the "90°"). Stacking slabs gives
$$E(z)=E_0\,e^{ik(n-1)z}$$If \(n\) is real this is a pure rotation in the complex plane — the length never changes. The phase lags the vacuum wave by \(k(n-1)z\), and we rephrase that as "the speed became \(c/n\)."
If \(n\) had an imaginary part, the phasor would rotate and shrink — that is absorption, the subject of Episode 5.
On the left below is the complex plane. The thick horizontal arrow is the incident wave; we add the small scattered contribution of one atomic layer at a time. Because each contribution is at 90°, the tip traces a circle — the length stays fixed while the direction turns.
On the right is the resulting wave compared with the vacuum wave. Only the crests are displaced; the height is identical. That is all "slowing down" ever was.
Turn up the absorption slider and the phasor spirals inward — that is absorption, a preview of Episode 5.
What propagates inside matter has a name: a polariton — a single wave that is a mixture of the electromagnetic field and the material's polarization (the shaking electrons).
Because part of the travelling wave is made of matter. Matter has inertia, and that inertia drags the whole thing.
The photons themselves always fly between atoms at \(c\). What lags is the phase of the mixed wave. That is precisely what "light does not slow down; the wave does" means.
This picture pays off in Episode 6 — push the mixture all the way over to the material side, and light drops to 17 metres per second, and then "stops." At which point what has stopped is no longer light.
For \(d=1\) cm of glass with \(n=1.5\), the difference from vacuum is
$$\Delta t=\frac{(n-1)d}{c}=\frac{0.5\times0.01}{3.0\times10^8}=1.7\times10^{-11}\ \mathrm{s}=17\ \text{picoseconds}$$In crests, that is about 9000 wavelengths of phase shift at 550 nm. Accumulate this across a curved surface and you have a lens forming an image.
For contrast — the "absorb and re-emit" story predicts ns × \(10^7\) layers, i.e. milliseconds. That is eight orders of magnitude away from the measured 17 ps. On that alone the story fails.
Established: that the refractive index arises from coherent superposition of the incident and forward-scattered waves (the standard treatment, e.g. Feynman Lectures vol. I ch. 31); that an off-resonant driven oscillator responds 90° out of phase with the drive, so it shifts phase without changing amplitude; \(E(z)=E_0e^{ik(n-1)z}\); that the propagating excitation in matter is a polariton (a photon–polarization hybrid); the measured refractive indices quoted. All standard optics and condensed-matter physics.
Caveats: (1) The rejection of "absorb and re-emit" applies to coherent transmission. Near resonance, genuine resonant absorption and spontaneous emission do occur, and light is then scattered and attenuated (the medium goes opaque). Restrict the claim to the transparent region. (2) "Scattered waves add only in the forward direction" holds when the atomic spacing is much smaller than the wavelength. When the wavelength becomes comparable to the spacing, as for X-rays, constructive interference appears in other directions too (Bragg diffraction). (3) The figure draws each layer's contribution as an equal discrete vector; it is a schematic, not a calculation for a real continuous medium. (4) "Polariton" covers different things in different contexts (exciton polariton, phonon polariton, and so on); here it means "a propagating mode mixing the electromagnetic field with material polarization." (5) The refractive index depends on frequency (dispersion); the tabulated values are representative visible-light figures.
Light travels at \(c/1.5\) in glass, and relativity is untouched — what is invariant is the speed of causality \(c\), and what moves through matter was never pure light in the first place.
The correct picture is coherent forward scattering. The incident field shakes electrons, the shaking charges radiate, and those waves add constructively only forward. Off-resonance the response is 90° out of phase with the drive, so adding it rotates without lengthening — \(E(z)=E_0e^{ik(n-1)z}\), pure phase shift. That is the refractive index.
So "atoms absorb and re-emit" is wrong — direction and phase would not survive, the timing is off by eight orders of magnitude, and in the transparent region there is no level to absorb into. What actually travels is a polariton, a hybrid of light and polarization, and it is slow because part of it is made of matter. Light does not slow down; the wave does.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, add layers one at a time and watch the phasor tip trace a circle; raise the absorption slider and it spirals inward (a preview of Episode 5). "Show answer" reveals the solutions.