# In superluminal phase velocities, what is it that is traveling faster than light?

I understand that information cannot be transmitted at a velocity greater than speed of light. I think of this in terms of the radio broadcast: the station sends out carrier frequencies $\omega_c$ but the actual content is carried in the modulated wave, the sidebands $\omega_c\pm\omega_m$. The modulation envelop has its group velocity and this is the speed at which information is being transmitted. We also know, for example, that x-rays in glass have a phase velocity which is greater than the speed of light in vacuum.

My question is, what exactly is it that is travelling faster than the speed of light?

EDIT: I know it couldn't be anything physical as its superluminal. My problem is, what is it that has absolutely no information content yet we associate a velocity with it.

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Shine a flashlight on a wall. Rotate the flashlight so the illuminated spot moves.

Q: How fast does the spot move?
A: It depends how far away the wall is.

Q: How fast can the spot possibly move?
A: There is no limit. Put the wall far enough away, and the spot can move with any speed.

Q: What is moving across the wall?
A: Nothing. The light that makes up the spot at one instant is unrelated to the light that makes up the spot an instant later.

This is how a wave can be apparently superluminal: we interpret a series of unrelated events as a continuous 'wave'. Group velocity can also be superluminal; even though the individual chunks of energy are going at roughly $c$, the region where they superpose constructively (the 'crest of the wave') goes faster than $c$.

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Great answer, or at least I would give the same one. ;-) +1 –  Luboš Motl Mar 15 '11 at 18:37
Thanks, Luboš! Offtopic, I don't suppose I could tempt you to take a crack at my most recent magnet question? –  Andrew Mar 16 '11 at 19:51
You don't say so explicitly (and I don't have access to the PRL paper), but I assume they either didn't measure the signal velocity or measured it and found that it was less than c. A similar paper: arxiv.org/abs/quant-ph/0407155 –  Ben Crowell May 31 '13 at 20:55

Let me make an analogy which may be helpful.

The next time you're at the beach, watch the waves coming in. You will notice that the point where the wave breaks will often move faster than the waves do. This happens when the waves come in close to perpendicular to the beach.

The effect is that if you know the wave is breaking at point A, you can predict that it will break at point B soon afterward. Note that the short vector gives the wave movement while the long arrow shows where on the beach the wave is breaking. But this is not information that is moving; the wave has the information built into its wave structure:

The same thing applies to de Broglie (or quantum matter) waves which are also superluminal. The short answer is that there isn't anything that is traveling faster than light. It's just an effect of little physical meaning.

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The analogy is good!(and nice graph too) but I don't agree on the "little meaning", I think it has the greatest meaning that anything else can have about this matter, information could be sent at higher than c by this method, if and only if, your graph where precise (if you could generate the plane waves near A and send them in the beach direction, then you have a faster than C travel in the orthogonal direction to B), BUT perhaps plane waves can't be really generated by a source, they didn't exist.. that is what I think, because if those plane waves exist, we got faster than c info travel –  HDE Mar 15 '11 at 12:47
I do not understand this analogy quantitatively. I am not a coastal person and the few times I've been on a beach I've spent admiring anthropic forms ;) Anyhow, my problem is, that the shallow water phase velocity and the shallow water group velocity of the ocean waves is the same. So I do not understand your analogy. Here is a link oceanworld.tamu.edu/resources/ocng_textbook/chapter16/… –  Approximist Mar 15 '11 at 18:21
It's certainly not intended to be any sort of literal analogy to water physics. I'm just noting that the point at which the waves breaks can move with speed faster than the waves coming up the shore. This is very obvious and has been observed by all at the beach. Hmmm. Maybe I should add in a photo of actual waves... –  Carl Brannen Mar 15 '11 at 22:50
I will look into youtube and try to get this. It sounds interesting. And thanks a lot for answering. +1 –  Approximist Mar 16 '11 at 6:48

Look at a particular local maximum on the electromagnetic field. A small moment of time $dt$ later, there will be a local maximum a small distance $dx$ away. Superluminal phase velocity means $dx/dt > c$. The "thing" moving faster than light is not a thing at all, but a point at which the electromagnetic field is a maximum (or has some other, constant phase, i.e. a minimum, inflection point, etc.)

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Hi! Thanks for answering. I was having some trouble convincing myself that this propagating maximum carries no information content. See edit. –  Approximist Mar 14 '11 at 22:24

Just one comment. The fact that information cannot travel faster than light means that it cannot travel faster than the speed of light in vacuum. In media were the speed of light is reduced it is still possible that other high energy particles, e.g. electrons, move faster than the speed of light in this medium and thus information speed is faster than the propagation of light there. When the speed of the particle exceeds the phase velocity Cherenkov radiation is produced. This is analogous to the sonic boom of a supersonic aircraft. This effect is used in high energy experiments to determine the velocity of particles. (Look up Cherenkov Radiation at Wikipedia)

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