The phase velocity exceeds c — relativity survives, and the photon acquires mass
So far we have assumed \(n>1\), because water, glass and diamond all satisfy it. But there are materials with \(n<1\) — glass for X-rays, the ionosphere, metals. The cone of Episode 2 then becomes wider than the real one, and the phase velocity exceeds \(c\). Alarming, but relativity is entirely untouched: information is carried not by the crests of the phase but by the front of the signal, and the front travels at exactly \(c\) in any medium. This is precisely the argument of Bonus 4 in the sister series Tunneling That Clicks. And there is a second gift here. Put the plasma dispersion relation \(\omega^2=\omega_p^2+c^2k^2\) next to relativity's \(E^2=(mc^2)^2+(pc)^2\) — the same form. Inside a plasma, the photon has mass.
| System | Index | What happens |
|---|---|---|
| Glass for X-rays (10 keV) | \(1-10^{-5}\) | Total external reflection at grazing incidence. X-ray telescope and synchrotron mirrors are built on it |
| Ionosphere (radio, \(f_p\approx9\) MHz) | below 1 | AM broadcasts bounce and travel far; FM goes straight through |
| Metal (visible light) | imaginary | The wave cannot enter, so it reflects — this is why metals are mirrors |
With free electrons at number density \(N\), the response gives
$$n(\omega)=\sqrt{1-\frac{\omega_p^2}{\omega^2}},\qquad \omega_p=\sqrt{\frac{Ne^2}{\varepsilon_0 m_e}}$$\(\omega_p\) is the plasma frequency. For \(\omega>\omega_p\), \(n<1\) and real; for \(\omega<\omega_p\), \(n\) is purely imaginary.
Numbers: a metal (\(N\sim10^{29}\ \mathrm{m^{-3}}\)) gives \(\hbar\omega_p\approx12\) eV (ultraviolet); the ionosphere (\(N\sim10^{12}\ \mathrm{m^{-3}}\)) gives \(f_p\approx9\) MHz.
From the plasma dispersion \(\omega^2=\omega_p^2+c^2k^2\),
$$v_{\rm phase}=\frac{\omega}{k}=\frac{c}{n}>c,\qquad v_{\rm group}=\frac{d\omega}{dk}=c\,nThe phase velocity is the speed of "where the crest is," and crests are unlabelled — every crest is like every other, so you cannot send information with their positions. Information rides on changes in the waveform, which travel roughly at the group velocity and, strictly, at the front.
Even the group velocity can exceed \(c\). In the anomalous dispersion region right beside an absorption line, the group velocity can exceed \(c\) or go negative, and this has been measured (2000). Relativity still does not break.
The very leading edge of the waveform — the instant a signal first arrives where there was nothing — travels at exactly \(c\), whatever the medium.
The reason is simple: no material can respond at infinite frequency, so \(n\to1\) as \(\omega\to\infty\). The front propagates as though through vacuum.
Only the front can carry information. "The phase velocity exceeded \(c\)" and "the group velocity went negative" are both statements that the waveform was reshaped — not that anything moved faster.
This is the best part of the episode. Set the two dispersion relations side by side.
With \(E=\hbar\omega\) and \(p=\hbar k\), these are the same equation. The correspondence is
$$m_{\rm eff}=\frac{\hbar\omega_p}{c^2}$$Inside a plasma the photon carries an effective mass. In a metal \(\hbar\omega_p\approx12\) eV, about one forty-thousandth of the electron mass (511 keV).
And for \(\omega<\omega_p\) — that is, \(E
Below is the plasma dispersion relation. The slope of the line from the origin to the point is the phase velocity; the slope of the tangent is the group velocity. The dashed line is the light line \(\omega=ck\).
Push \(\omega\) below \(\omega_p\) and the curve disappears — there is no propagating wave. The wave then only seeps out of the boundary as \(e^{-z/\delta}\) (evanescent), and the net result is total reflection. That is why metals are mirrors, and it is the trailer for next episode's evanescent waves.
For \(\omega<\omega_p\), \(n\) is purely imaginary and the wave decays as \(e^{-z/\delta}\), with \(\delta=c/\sqrt{\omega_p^2-\omega^2}\) — for a metal in visible light, about 17 nm. Fewer than fifty atoms deep. It cannot get in, so it reflects.
Once \(\omega>\omega_p\), \(n\) becomes real again and the metal is transparent. Alkali metals really are transparent to ultraviolet, as Wood observed in 1933. "Metals block light" is a statement about visible light only.
For the same reason, the ionosphere reflects AM broadcasts (a few MHz) and transmits FM (100 MHz). Reflection is why AM carries far at night; transmission is why satellite communication works. The boundary is the plasma frequency, \(f_p\approx9\) MHz.
Established: that the refractive index of matter for X-rays falls slightly below 1, giving total external reflection (the basis of grazing-incidence mirrors); the plasma index \(n=\sqrt{1-\omega_p^2/\omega^2}\) and plasma frequency \(\omega_p=\sqrt{Ne^2/\varepsilon_0m_e}\); \(v_{\rm phase}v_{\rm group}=c^2\); that the waveform's front always propagates at \(c\) (Sommerfeld–Brillouin 1914); that the plasma dispersion relation has the same form as a massive particle's, readable as \(m_{\rm eff}=\hbar\omega_p/c^2\); that the metal skin depth is of order 10 nm in the visible; ultraviolet transparency of alkali metals (Wood 1933); the ionospheric cutoff. All standard physics.
Caveats: (1) In the anomalous-dispersion region the group velocity too can exceed \(c\) or go negative (as measured by Wang–Kuzmich–Dogariu in 2000, among others). Information still does not exceed \(c\). "Group velocity < c" holds in weakly absorbing regions. (2) The plasma formula is a simple free-electron-gas model neglecting collisions and magnetic fields. In real metals bound-electron contributions and scattering make \(n\) complex, which needs the framework of Episode 5. (3) "The photon has mass" is meant in the effective sense: the photon in vacuum remains massless, and the medium merely changes the dispersion relation. No gauge symmetry is broken (in a superconductor the story is different — that is spontaneous symmetry breaking). (4) The X-ray \(n<1\) is extremely close to 1 (of order \(1-10^{-5}\)) and total-external-reflection critical angles are milliradians. (5) The figure is an idealised collisionless unmagnetised plasma, not the actual response of the ionosphere or of a metal.
Glass for X-rays, the ionosphere and metals all have \(n<1\) (or imaginary). With \(n<1\), the phase velocity exceeds \(c\). Relativity is untouched: crests carry no labels, so no information rides on them, and the front, which does carry information, travels at exactly \(c\) in any medium (Sommerfeld–Brillouin). This is exactly the argument of Bonus 4 in the sister series Tunneling That Clicks. In a plasma there is also the pleasing relation \(v_{\rm phase}v_{\rm group}=c^2\).
And the gift of this episode — the plasma dispersion \(\omega^2=\omega_p^2+c^2k^2\) is the same equation as a massive particle's \(E^2=(mc^2)^2+(pc)^2\). So inside a plasma the photon carries an effective mass \(m_{\rm eff}=\hbar\omega_p/c^2\). This is the weak version of the Meissner effect (Episode 5 of the tunnelling series, where the photon gains mass in a superconductor): a metal being a mirror and a superconductor expelling magnetic field are the same family.
Below \(\omega_p\) no wave enters; it decays within a skin depth of about 17 nm and reflects — that is a metal mirror. Raise \(\omega\) and the metal turns transparent (ultraviolet transparency of alkali metals); the ionosphere reflects AM and transmits FM. The boundary is always \(\omega_p\).
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, push ω below ω_p and watch the propagating wave vanish and turn into reflection. The presets jump to metals, the ionosphere and X-rays. "Show answer" reveals the solutions.