"Slower" and "farther" are the same statement — and changing distances means changing the metric
Episode 1 ended on this: "slowing down" and "lengthening the optical path" are the same thing said twice. Optics calls travelling a distance \(d\) through index \(n\) an optical path length \(nd\), treating it exactly like \(nd\) of vacuum. The distance grew, in other words. And when the way you measure distance changes — the metric has changed. This episode pushes that reading all the way. Inside a medium the light cone narrows, light rays become geodesics of that metric, and Snell's law is a consequence of geometry. The same duality as "is gravity a force or is it curvature?" from the sister series Relativity That Clicks appears verbatim. With one non-negotiable point — what narrowed is the optical cone, and the causal cone has not shifted by a millimetre. The effective metric is a second cone built inside the real light cone.
The phase accumulated over distance \(d\) in index \(n\) is \(k_0nd\) (\(k_0\) is the vacuum wavenumber). Over distance \(L\) of vacuum it is \(k_0L\). These agree when
$$L=nd\qquad\text{— the }\textbf{optical path length}$$So "1 cm through \(n=1.5\) glass" is, as far as phase goes, indistinguishable from "1.5 cm of vacuum."
Saying the speed became \(1/n\) and saying the distance grew by \(n\) are the same statement.
Draw a spacetime diagram. In vacuum light follows \(x=\pm ct\) — lines at 45°. Inside a medium it follows \(x=\pm ct/n\) — steeper lines. The cone narrows.
Write "narrow cone" as a formula and it is a metric:
$$ds^2_{\rm opt}=-\frac{c^2}{n^2}\,dt^2+dx^2+dy^2+dz^2$$The curves with \(ds^2_{\rm opt}=0\) are exactly those with \(|dx/dt|=c/n\). Light rays are null geodesics of this effective metric.
This is Gordon's optical metric, introduced in 1923, and it is now the foundation of "analogue gravity" and of transformation optics (metamaterial invisibility cloaks).
The intuition "if you look at it in a \(c\cdot t=\)const frame, isn't this equivalent to light slowing down?" acquires its exact form here — what is equivalent is not a frame but a metric, and the replacement is \(c\cdot t \to c\cdot t/n\).
Optics has had Fermat's principle since 1662: light takes the path that minimises (strictly, extremises) the optical path length \(\int n\,dl\).
Reread it in modern language and it is literally the definition of a geodesic. Measure spatial distance as \(dl_{\rm F}=n\,dl\), and \(\int n\,dl\) is length in that metric. The curve of least length is the geodesic.
When the index depends only on \(y\), Fermat's principle yields exactly one conserved quantity:
$$n(y)\sin\theta=\text{const}\qquad(\theta\ \text{measured from the layer normal})$$At a boundary where \(n\) jumps from \(n_1\) to \(n_2\) this reads \(n_1\sin\theta_1=n_2\sin\theta_2\) — Snell's law.
The "law of refraction" was nothing but a conserved quantity of the effective metric's geodesics.
On the left is a spacetime diagram. The dashed 45° lines are the true causal cone; the solid lines are the medium's light cone. Raise \(n\) and they fold inward.
On the right is a medium whose index varies with height (a mirage, or an optical fibre). The ray is traced numerically from Fermat's principle, and along it we draw the local light cone — the ray bends because the cone opens differently at different places. That is the whole mechanism.
The "\(c\cdot t=\)const view" now has an exact form. And there is one thing that must not be dropped.
| Causal cone | Medium's light cone | |
|---|---|---|
| Opening | \(x=\pm ct\) (45°) | \(x=\pm ct/n\) (narrower) |
| Can you change it? | No | Yes — pick a material |
| What it constrains | every causal relation | only how this wave propagates |
| Where the effective metric sits | — | inside the causal cone (for \(n>1\)) |
The effective metric does not rewrite real spacetime. It draws a second, thinner cone inside the real one. Therefore —
• Invariance of \(c\) (the causal \(c\)) is completely untouched.
• And yet, for this wave, the effective metric is every bit as operative as the real one.
• And it is precisely the structure "a narrow cone inside" that makes the next section's analogue gravity possible.
Here is the interesting part. Suppose the medium is flowing. Waves travel at \(c/n\) relative to the medium, which moves at speed \(v\). If \(v>c/n\), then waves can no longer make headway upstream.
The surface where the flow exceeds \(c/n\) is a boundary the wave cannot cross — exactly the horizon structure of the sister series Temperature That Clicks, Bonus 1.
Unruh pointed this out for sound in 1981 (the sonic black hole), and "analogue horizons" have since been made in flowing water, Bose–Einstein condensates, optical fibres and superconducting circuits. Phenomena corresponding to Hawking radiation have been reported.
The real causal cone stays untouched while a horizon is built in the inner cone alone — that is the whole idea of analogue gravity, and why it fits in a laboratory.
Analogue systems are different systems obeying the same equations, not tests of spacetime itself. "Hawking radiation was seen in a sonic black hole" does not mean "Hawking radiation was confirmed." The same distinction appears in the sister series Tunneling That Clicks, Bonus 1.
Established: optical path length \(nd\), and the fact that "slower" and "farther" are indistinguishable in phase; Gordon's optical metric for a static isotropic medium (1923), \(ds^2_{\rm opt}=-(c^2/n^2)dt^2+d\vec x^2\), and light rays as its null geodesics; the equivalence of Fermat's principle \(\delta\int n\,dl=0\) with the geodesic equation of the spatial metric \(n^2\delta_{ij}\); the conserved quantity \(n\sin\theta\) in a stratified medium and Snell's law; the fact that in a moving medium the surface where the flow exceeds \(c/n\) acts as a horizon for the wave (analogue gravity, Unruh 1981 onward); the basis of transformation optics. All standard results.
Caveats: (1) The effective metric is not the real spacetime metric. The causal cone (\(x=\pm ct\)) does not change at all; the effective metric is a cone "for the wave" built inside it. Invariance of \(c\) is untouched. (2) The Gordon metric takes this simple form for isotropic, non-dispersive, non-absorbing media. With dispersion, \(n\) depends on frequency, so there is no single metric — there is a different one per frequency (Episodes 3 and 5). With anisotropy (birefringence) a scalar \(n\) does not exist at all. (3) Fermat's principle is the geometrical-optics approximation (wavelength much smaller than structures) and fails where diffraction matters. (4) The ray tracing in the figure integrates \(y''=n n'/C^2\) (derived from Fermat's principle); it does not reproduce a real atmosphere or fibre profile. (5) Analogue gravity is a different system obeying the same equations, not a test in spacetime. (6) Strictly, "looking at it in a \(c\cdot t\) frame" is not a coordinate change but the introduction of a different metric. A coordinate change alone changes no physics.
In phase, "slower" and "farther" are indistinguishable (optical path length \(nd\)). Changing how distance is measured means changing the metric, and indeed one can write \(ds^2_{\rm opt}=-(c^2/n^2)dt^2+d\vec x^2\) (Gordon's optical metric, 1923). Light rays are its null geodesics.
Then Fermat's 1662 principle \(\delta\int n\,dl=0\) becomes literally the definition of a geodesic, and Snell's law becomes the geodesic's conserved quantity \(n\sin\theta=\)const. The "force or curvature" duality of gravity has an exact counterpart in refraction.
And the intuition "in a \(c\cdot t=\)const frame this is equivalent to light slowing down" now has its exact form — what is equivalent is not a frame but a metric, with \(c\cdot t\) replaced by \(c\cdot t/n\). But the causal cone has not moved. The effective metric is a cone inside the real one, meant for this wave alone — and that very structure is why analogue horizons can be built in a laboratory.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, raise n on the left and watch the light cone fold inward; on the right, change the gradient and launch angle and watch the ray bend. Check that the small cones drawn along the ray open differently at different heights. "Show answer" reveals the solutions.