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I'm so confused. I've read in this book (page 28) that group velocity of light can exceed $c$ in certain gases. However a lot of people online and in the forums say that phase velocity can exceed $c$.

Who's right? According to that same book, group velocity is not signal velocity and it's signal velocity which cannot exceed $c$ without violating causality. So is everyone correct? Do both phase and group velocity change when entering a new material, and can they both exceed $c$?

I would really appreciate any help clearing things up here.

Quote from the book:

In a spectral region of ‘normal dispersion’, $(\mathrm d n_R/\mathrm d\omega)_{\omega_L} > 0$, the group velocity is less than the phase velocity $c/n_R$. Because $v_g$ can exceed $c$ in a region of anomalous dispersion, $(\mathrm dn_R/\mathrm d\omega)_{\omega_L} < 0$ (section 1.2), and because $v_g$ was generally thought to be the velocity of energy propagation, $v_g > c$ was, in the past, thought to be in conflict with the special theory of relativity. This conflict was resolved in large part when Sommerfeld and Brillouin proved that the signal velocity cannot exceed c even in a region of anomalous dispersion.

Info in case the link breaks in the future.

Title: Fast Light, Slow Light and Left-Handed Light
Author: P W Milonni
Published by: Institute of Physics Publishing, 2005

Wavefronts and phase velocity faster than $c$

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    $\begingroup$ Could you point at the page where it's stated that group velocity can exceed $c$? $\endgroup$
    – Ruslan
    Apr 3, 2020 at 18:14
  • $\begingroup$ In the link you provide, $v_y$ is not the phase velocity. $\endgroup$ Apr 3, 2020 at 18:40
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    $\begingroup$ Minor comment to the post (v3): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic
    Apr 3, 2020 at 19:09
  • $\begingroup$ @Ruslan Voilà ; ) $\endgroup$
    – Alex P.
    Apr 3, 2020 at 19:16
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    $\begingroup$ There’s no contradiction at all: both group and phase velocity can exceed $c$. $\endgroup$
    – knzhou
    Apr 3, 2020 at 22:46

1 Answer 1

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Yes, phase velocity can and does exceed c. This is not a contradiction with SR.

So when phase is treated correctly, it doesn't affect any observable. That's why SR doesn't care about phase velocity.

Wave group and phase velocities

However, group velocity should not exceed c except in certain cases, but even then, this does not mean information traveling faster then light.

In all these cases, however, there is no possibility that signals could be carried faster than the speed of light in vacuum, since the high value of vg does not help to speed up the true motion of the sharp wavefront that would occur at the start of any real signal. Essentially the seemingly superluminal transmission is an artifact of the narrow band approximation used above to define group velocity and happens because of resonance phenomena in the intervening medium.

https://en.wikipedia.org/wiki/Group_velocity

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  • $\begingroup$ Thank you for your reply, would it be correct to say that group velocity is usually the same thing a signal velocity, however, in regions of anomalous detection the two are decoupled, thus signal velocity cannot exceed c while group velocity does? $\endgroup$
    – Alex P.
    Apr 4, 2020 at 7:50
  • $\begingroup$ @AlexP. you say "anomalous detection", can you please tell me what you mean? $\endgroup$ Apr 4, 2020 at 16:16
  • $\begingroup$ I'm sorry, I meant anomalous dispersion. Which I believe is a range of wavelengths unique to each material for which the index of refraction increases with the wavelength instead of decreasing. At least that's what I understood from the Wikipedia page for dispersion and from that book. $\endgroup$
    – Alex P.
    Apr 4, 2020 at 18:00

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