Sci-fi physics — refractive index less than one

Problem

Media exist (X-rays in crystals, plasma for radio waves) where $n < 1$. What happens to the phase velocity? Does it violate relativity? Explain by introducing the group velocity.

Solution

Phase velocity. The refractive index is defined by $n = c/v_f$, where $v_f$ is the phase velocity (the speed at which the crests of a monochromatic wave advance). If $n < 1$:

$v_f = \frac{c}{n} > c$

The phase velocity exceeds the speed of light in vacuum.

Does it violate relativity? No. Phase velocity carries no information: a perfectly monochromatic wave is infinite in time and space, and carries no “signal”. A real signal is a wave packet (a modulation), which travels at the group velocity $v_g$.

Group velocity. It is $v_g$ that carries energy and information, and in these media it always turns out that $v_g < c$. Relativity only forbids information and energy from exceeding $c$, not the phase of an ideal wave.

$\ev{v_f = c/n > c,\ \text{but no signal travels at } v_f;\ \text{information travels at } v_g < c:\ \text{relativity is safe.}}$

Linked atoms

Topics: Ottica Concepts: Rifrazione e legge di Snell Skills: Analisi di casi limite e fantafisica