As far as I can see, almost all the major textbooks on electrodynamics and nonrelativistic quantum mechanics make the following statement somewhere:(adsbygoogle = window.adsbygoogle || []).push({});

The group velocity is (at least in lossless media) the speed of information propagation and it is always smaller than c.

In the past I have done several computer simulations of classical wave equations (by means of 2nd order finite differences), just to get an intuitive feeling of wave propagation (and by the way it is nice to see that today's computers can do it in real time ). From what I have seen during these simulations I tend to dispute the above statement about group velocity. Let me describe it for a one dimensional problem.

Just picture a one dimensional wave carrier, i.e. a classical string, that may be described by the d'Alembert equation

[tex]\partial_\nu \partial^\nu \psi = 0[/tex]

with a constant phase velocity c embedded in the relativistic derivative. Imagine that you can pick at the string (initially at rest) instantaneously, i.e. impose a delta peak somewhere on the string. Then, as you probably have expected due to prior knowledge, you will see two sharp peaks going off in opposite directions with speed c.

Now switch to a string that is described by the (classical) Klein-Gordon equation (ie. in every point of the string there is an additional harmonic force that drives this point back to its equilibrium position)

[tex]\partial_\nu \partial^\nu \psi = -m^2\psi[/tex].

(Edit: In physical terms you could take this as a simplified version of a massive photon field)

In response to the picking of the string you get again two spikes going off in opposite directions, but there are trails behind them, which can easily be understood as a consequence of the dispersion relation

[tex]\omega^2 c^2 = m^2+k^2 [/tex].

If you calculate the group velocity from this dispersion relation, this results in an expression that is always smaller than c. So far so good.Butif you look at the string again you see the foremost spikes travel precisely with c ! From the point of view of the simulation this is no surprise since the only mechanism that promotes information from grid point to grid point lies in the finite difference implementation of the d'Alembertian. No way for the mass to enter that mechanism.

So if you tell your brother to move to alpha centauri in order to blow up the star with a nuclear bomb right when he receives a light signal from you here, then he will do his job exactly at the same moment, regardless whether there is dispersion in the medium between earth and alpha centauri, or not (provided that he uses an ideal detector for seeing your signal). Needless to say that I don't consider c-variation by gravitational effects here.

Since you may encode arbitrarily complex messages in digital form as light pulses (where you get some bandwith restrictions depending on the used frequency of course), I have to come to the conclusion that the speed of information is neither v<c nor v>c (if you're foolhardy), but it's always c for a wave equation containing the d'Alembertian.

I am also aware of where this contradiction to common belief comes from: The concept of group velocity is based on wave packets thecentroidof which moves with named group velocity. And if you consider the "wave front+trail" in my Klein-Gordon example as a wave packet, then you see that the trail causes the centroid of the packet to lag behind the wave front.

But then, how come that all the textbooks hold the seemingly erratic opinion that group velocity = speed of information ?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Group velocity and speed of information

Loading...

Similar Threads - Group velocity speed | Date |
---|---|

A Improper and non-orthochronous group | Feb 3, 2018 |

I Proof that Galilean & Lorentz Ts form a group | Dec 8, 2017 |

Group velocity and phase velocity of a matter wave | Jun 20, 2013 |

Phase vs group vs signal velocities | Aug 8, 2010 |

Little problem with group velocity | Jul 2, 2008 |

**Physics Forums - The Fusion of Science and Community**