From the sheen of metal to the blue of the sky, the white of snow, the rainbow's 42 degrees — and a spacetime you can put on a bench
The last episode. Take the tools built so far — forward scattering (Ep. 1), the effective metric (Ep. 2), the plasma frequency (Ep. 3), Fresnel and the evanescent wave (Ep. 4), Kramers–Kronig (Ep. 5), the group index (Ep. 6) — and go outside. The sheen of metal, the blue sky, the red sunset, the blue of deep water, the white of snow, the rainbow's 42 degrees, a mirage in the desert. All of them read. And having read them, we return to Episode 2 — a distribution of refractive index is a metric. What we "see" is not objects but that map. Design the map and you can hide things (an invisibility cloak); build a horizon into it and you can put an analogue universe on a laboratory bench. With one line held to the end — an effective metric is not real spacetime. But it obeys the same equations. That these two are simultaneously true is the conclusion of this series.
| What you see | What it is | Episode |
|---|---|---|
| Metals shine | visible light is below \(\omega_p\), so \(n\) is imaginary and it dies within 17 nm | 3 |
| The sky is blue | scattering off density fluctuations in the air, \(\propto\omega^4\) | 1 |
| The sunset is red | the other face of the same thing — a long path strips out the blue | 1 |
| Deep water is blue | absorption, not scattering. Water absorbs red (OH-stretch overtones) | 5 |
| Clouds are white | droplets larger than the wavelength scatter every colour equally (Mie) | 1 |
| Snow, foam and frosted glass are white | just a lot of interfaces; the material itself is transparent | 4 |
| The rainbow sits at 42° | minimum deviation of the once-reflected ray in a sphere, split by dispersion | 2, 5 |
| Mirages | temperature gradient → index gradient → the ray arcs as a geodesic | 2 |
| Phone fingerprint sensors | valleys are air (total reflection), ridges touch (FTIR lets light through) | 4 |
This one is worth doing carefully. Episode 1 said: in a dense, uniform medium the sideways scattered waves interfere and cancel, leaving only the forward direction. That is refraction.
If the cancellation were perfect the sky would be pitch black. It is not. Because air is not perfectly uniform.
Thermal motion makes the air's density fluctuate everywhere. To the extent of those fluctuations the cancellation is incomplete, and the residue comes out sideways — that is the colour of the sky (Einstein 1910, Smoluchowski).
In other words — refraction and scattering are the cancelled part and the uncancelled part of one and the same thing: dipole re-radiation. They are not separate phenomena.
The residue goes as \(\propto\omega^4\) (Rayleigh). Blue (450 nm) scatters \((650/450)^4\approx\)4.4 times as much as red (650 nm). Hence a blue sky, and reddened light for whatever came through.
"The sky is blue because oxygen and nitrogen molecules scatter blue light" is half right and half off. Individual molecules do scatter — but the point is that if they were uniformly arranged, the scattering would cancel by interference. What matters is not the presence of molecules but the fluctuations.
The evidence: liquid water and clear glass are far denser than air, and far clearer. If more molecules meant more scattering, it would be the other way around. The denser and more ordered the medium, the better the cancellation — exactly Episode 1's claim.
Snow, foam, frosted glass, crushed ice, paper, milk, clouds, white paint. Every one of those materials is transparent. They look white because they contain an enormous number of index-mismatched interfaces — Episode 4's Fresnel reflection happening thousands of times in every direction.
In the figure below, you choose a distribution \(n(x,y)\) and light runs through it. What is computed is the ray equation \(\dfrac{d}{ds}\!\left(n\dfrac{d\vec r}{ds}\right)=\nabla n\) — the geodesic equation of Episode 2's effective metric.
Note the fourth button, labelled gravity lookalike. Lay down the map \(n=1+r_s/r\) and light is drawn toward the centre — and the deflection angle agrees with general relativity's prediction \(4GM/bc^2\).
Episode 2 established that a distribution of index is a metric. Two directions open from there.
If rays are geodesics, then design a metric whose geodesics never enter a chosen region. No light reaches the object inside, and from outside the light emerges on the far side as if nothing had been there — an invisibility cloak (Pendry, Leonhardt, 2006).
Implemented in metamaterials and demonstrated at microwave frequencies. But not across the whole visible band — and the reason is Episode 5. Producing the required index needs resonances, and resonances bring absorption. Kramers–Kronig is standing guard here too.
If the medium flows, waves can only move at \(c/n\) against the current. A surface where the flow exceeds \(c/n\) is a horizon for the wave (Episode 2).
Since Unruh pointed this out for sound in 1981, "analogue horizons" have been built in surface waves, Bose–Einstein condensates, optical fibres and superconducting circuits, and phenomena corresponding to Hawking radiation have been reported. And the real causal cone stays at 45° throughout.
This series began from a doubt: "Refraction slows light down, and yet we are told the speed of light is invariant. Isn't that odd? Looked at in a \(c\cdot t=\)const frame, isn't it equivalent to light slowing down?"
Seven episodes later, the answer is this.
| Question | Answer |
|---|---|
| Is light slowing down? | The wave is. The photon is not (Ep. 1: the phase shift of forward scattering) |
| Can that be written as a metric? | Yes. \(ds^2_{\rm opt}=-(c^2/n^2)dt^2+d\vec x^2\) (Ep. 2) |
| Then what of invariance of \(c\)? | Untouched. The effective cone is a second cone inside the real one (Eps. 2, 3) |
| And when \(n<1\) makes the phase exceed \(c\)? | Still the front is exactly \(c\) (Ep. 3) |
| And when it is stopped? | What stops is atomic coherence, not light. The front is \(c\) (Ep. 6) |
So: the intuition was right. But as "another metric," not as "another frame."
\(c\) is the speed of causality and it does not move. \(c/n\) is the speed for this wave and it changes with the material. Both are true; they live on different levels.
An effective metric is not real spacetime. An analogue black hole does not test gravity. An invisibility cloak does not bend space. A map of refractive index is a local arrangement that acts on one kind of wave only, the electromagnetic field.
And even so — it obeys the same equations. And when the equations are the same, what you learned on one side genuinely works on the other. That is not a consolation prize; it is the reason physics functions at all.
Established: metallic reflection and the plasma frequency; the \(\omega^4\) Rayleigh law and the Einstein–Smoluchowski account in which sideways scattering cancels in a uniform medium and density fluctuations supply the residue; that water's absorption minimum is in the blue (about 418 nm, \(\alpha\approx0.004\ \mathrm{m^{-1}}\)) while red is absorbed (\(\alpha\approx0.6\ \mathrm{m^{-1}}\) at 700 nm); Mie scattering and the whiteness of clouds; whitening by multiple interfaces and clearing by index matching; the primary rainbow at about 42° (secondary 51°) with colours split by dispersion; gradient-index mirages; FTIR fingerprint sensors; the ray equation \(d(n\,d\vec r/ds)/ds=\nabla n\); that \(n=1+r_s/r\) reproduces general relativity's weak-field light deflection \(4GM/bc^2\) (1.75 arcseconds at the solar limb); transformation optics and invisibility cloaks (Pendry, Leonhardt 2006); analogue gravity (Unruh 1981 onward). All standard results.
Caveats: (1) The "gravity lookalike" index \(n=1+r_s/r\) is the weak-field equivalent index and is not correct in strong fields (near a horizon). The exact optical index in isotropic coordinates is \((1+r_s/4r)^3/(1-r_s/4r)\), which diverges at \(r=r_s/4\). The figure is for teaching and does not correctly reproduce the photon sphere or capture orbits. (2) The ray tracing is a numerical integration with step-size-dependent error; the deflection comparison compares two numbers computed here. (3) The rainbow's 42° assumes perfectly spherical drops and monochromatic plane waves; the real distribution of colour and brightness is more complicated (supernumerary bows require diffraction). (4) The blue-sky account assumes a dry, clear atmosphere; abundant aerosol or water vapour whitens it (that is Mie scattering). (5) Invisibility cloaks have been demonstrated in limited bands such as microwaves; nothing works across the broad visible spectrum (and Episode 5 is the reason, as stated in the text). (6) Analogue gravity is a different system obeying the same equations, not a test of spacetime. (7) "What you see is a map of refractive index" is a strong simplification of vision — absorption, scattering, fluorescence and polarization all carry information too.
Six episodes of tools made most of the visible world readable. The sheen of metal is \(\omega<\omega_p\) (Ep. 3); the blue sky is the uncancelled residue of density fluctuations (Ep. 1) — refraction and scattering are the cancelled and uncancelled parts of the same dipole re-radiation. Deep water's blue is absorption, not scattering (Ep. 5); snow's white is nothing but interfaces, the material being transparent (Ep. 4 — wet it and it clears); mirages are geodesics of a gradient index (Ep. 2).
And a distribution of index is a metric. Design the map so that geodesics avoid a region and you have an invisibility cloak; build it so the flow exceeds \(c/n\) and you have an analogue horizon. As the figure shows, the map \(n=1+r_s/r\) alone reproduces general relativity's \(4GM/bc^2\).
The answer to the opening question — the intuition "measure with \(c\cdot t/n\) instead of \(c\cdot t\)" was right. But it was the introduction of another metric, not a coordinate change. \(c\) is the speed of causality and does not move; \(c/n\) is the speed for this wave and depends on the material. Both are true, on different levels. And the line held to the end — an effective metric is not real spacetime, but it obeys the same equations. That both of these hold at once is why physics works.
Print / PDF: ⌘+P (Ctrl+P on Windows). Press the four buttons to switch the index map and watch a GRIN lens, a mirage, an optical fibre and a gravitational lens all come out of one and the same equation. "Show answer" reveals the solutions.