Group velocity and speed of information

[OOO]

As far as I can see, almost all the major textbooks on electrodynamics and nonrelativistic quantum mechanics make the following statement somewhere:

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

$$\partial_\nu \partial^\nu \psi = 0$$

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)

$$\partial_\nu \partial^\nu \psi = -m^2\psi$$.

(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

$$\omega^2 c^2 = m^2+k^2$$.

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. But if 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 the centroid of 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 ?

 

[pervect]

I think the first step is to find a textbook that discusses group velocity, and to see exactly what they say about it. I took a look in Goldstein, but I didn't see any mention of group velocity. Most of my earlier textbooks are packed away, somewhere.

 

[OOO]

I think the first step is to find a textbook that discusses group velocity, and to see exactly what they say about it.

Okay, I have looked at several textbooks, but none claims that the group velocity is the speed of information. E.g. Jackson J.D., Electrodynamics describes the group velocity as the speed of the center of the wave packet.

I've recently read the claim (http://en.wikipedia.org/wiki/Group_velocity) that v_g=speed of information in wikipedia (yes, not a reliable source in any case), but I swear I've read that before somewhere.

But independently of any false accusations against textbooks, isn't it surprising that information always travels at c regardless of mass, whereas energy travels at v_g<c ? Or am I wrong in identifying the wave front as a carrier of information ?

 

[JesseM]

I haven't looked it over too carefully, but this page seems to give an argument for why information should travel at the group velocity in a dispersive medium:

http://www.mathpages.com/home/kmath210/kmath210.htm

The propagation of information or energy in a wave always occurs as a change in the wave. The most obvious example is changing the wave from being absent to being present, which propagates at the speed of the leading edge of a wave train. More generally, some modulation of the frequency and/or amplitude of a wave is required in order to convey information, and it is this modulation that represents the signal content. Hence the actual speed of content in the situation described above is Dw/Dk. This is the phase velocity of the amplitude wave, but since each amplitude wave contains a group of internal waves, this speed is usually called the group velocity.

Physical waves of a given type in a given medium generally exhibit a characteristic group velocity as well as a characteristic phase velocity. This is because within a given medium there is a fixed relationship between the wave number and the frequency of waves. For example, in a transparent optical medium the refractive index n is defined as the ratio $$c/v_p$$ where c is the speed of light in vacuum and $$v_p$$ is the phase velocity of light in that medium. Now, since $$v_p$$ = w/k, we have w = kc/n. Bearing in mind that the refractive index is typically a function of the frequency (resulting in the "dispersion" of colors seen in prisms, rainbows, etc), we can take the derivative of w as follows

$$\frac{dw}{dk} = \frac{c}{n} - \frac{ck}{n^2} \frac{dn}{dk}$$

Hence any modulation of an electromagnetic wave in this medium will propagate at the group velocity

$$v_g = v_p [1 - \frac{k}{n} \frac{dn}{dk} ]$$

 

[OOO]

I haven't looked it over too carefully, but this page seems to give an argument for why information should travel at the group velocity in a dispersive medium:

http://www.mathpages.com/home/kmath210/kmath210.htm

This is basically what Jackson says about group velocity. In addition, this document claims that the speed of energy propagation is the same as the speed of information propagation (as I have criticized at the beginning of this thread), like wikipedia does (too bad they don't cite their refs). By the way this document as well as Jackson mainly puts emphasis on dispersion due to variation of c whereas I have considered a mass term as the cause of dispersion.

What I find interesting is the fact that the author contradicts himself within one and the same paragraph:

The propagation of information or energy in a wave always occurs as a change in the wave. The most obvious example is changing the wave from being absent to being present, which propagates at the speed of the leading edge of a wave train.

and

More generally, some modulation of the frequency and/or amplitude of a wave is required in order to convey information, and it is this modulation that represents the signal content. Hence the actual speed of content in the situation described above is Dw/Dk.

ie. the group velocity.

In a dispersive medium it is impossible that the "leading edge of the wave train" (the wave front) moves at the same velocity as the "modulation" (the wave group/packet) because it is at the heart of (normal) dispersion that wave packets broaden with time. So the distance between some feature (e.g. the "center") of the wave packet and the wave front must increase with time and thus the feature's velocity (the group velocity) must be smaller than that of the wave front. But I'm repeating myself.

I guess googling for "front velocity" is what I should do.

 

[OOO]

I think I know now where the problem is, or rather where it isn't.

The wave front as I have described it is a traveling discontinuity and as such it contains arbitrarily high frequencies. The low frequencies get dispersed with time and form the trail behind the wave front. The remaining very high frequencies get dispersed as well but if we look at the dispersion relation

$$\omega^2c^2 = m^2+k^2 \approx k^2$$ for large frequencies,

we see that they travel approximately at c. Thus it is no contradiction that information travels with group velocity and at the speed of light. It's simply a matter of the frequencies used for transmission. In the case of a pulse they are arbitrarily large und thus the group velocity gets arbitrarily near the speed of light.

 

[Demystifier]

I think I know now where the problem is, or rather where it isn't.

The wave front as I have described it is a traveling discontinuity and as such it contains arbitrarily high frequencies. The low frequencies get dispersed with time and form the trail behind the wave front. The remaining very high frequencies get dispersed as well but if we look at the dispersion relation

$$\omega^2c^2 = m^2+k^2 \approx k^2$$ for large frequencies,

we see that they travel approximately at c. Thus it is no contradiction that information travels with group velocity and at the speed of light. It's simply a matter of the frequencies used for transmission. In the case of a pulse they are arbitrarily large und thus the group velocity gets arbitrarily near the speed of light.

Brilliant explanation!

 

[f95toli]

AFAIK (and according to my books on microwave engineering, e.g. Pozar) group velocity is a meaningful concept only for signals with a very limited bandwidth (or in media with low dispersion). In e.g the case of AM this means that vg is only well-defined as long as the modulation frequency is much smaller than the carrier frequency (which is also often assumed in the derivation of the expression of vg).

 

[pervect]

As far as general relativity goes, I've seen discussion of the issue of the speed of signal propagation in the context of whether or not GR "has a well posed initial value formulation".

The mathematical details are rather intimidating, but the end answer is that it does. (I'm not positive I understand all the "fine print" in the conclusion). This is discussed, for instance, in Wald, "General Relativity", chapter 10. One of the examples they work out is, interestingly enough, the Klein-Gordon field.

I'll quote a bit of the discussion about what it means for a theory to have a well posed initial value formulation.

If a theory can be formulated so that "appropriate initial data" may be specified, (possibly subject to constraints) such that the subsequent dynamical evolution of the system is uniquely determined, we say that the theory posses an initial value formulation. However, even if such a formulation exists, there remain further properties that a physically viable theory should satisfy.

I've included the above, which defines what an initial value formulation is, for background. The next part is about what makes an initial value formulation "well-posed", which is the part we are interested in. There are two properties needed. I've snipped a detailed discussion of the first property needed for well-posedness, which relates to sensitivity of the result to small changes in the initial values, i.e. if the theory is "too chaotic", it's not well-posed.

Second, changes in the initial data in a region, S, of the initial data surface should not produce any changes in the solution outside the causal future J'(S) of this region. If such changes occurred, we should be able to use them to propagate signals "faster than the speed of light." This would undermine the entire framework of relativity theory. If a theory posses an initial value formulation which satisfies both of the above properties, we say that this initial value formulation is well posed.

I've seen the phrase "speed of information transfer" used a lot, I may have even used it myself. I'm not quite sure about the origin of this phrase, offhand I don't recall reading any detailed textbook discussion of what "the speed of information transfer" is. I have seen information defined as the log of the number of states, but I don't think that approach to defining information will ever get us to consider the propagation of the "front" of the waveform as "the speed of information". Rather, I think any such approach should wind up showing that the information is being "dispersed" by a dispersive medium. I guess the point is that the dispersion never acts in such a way as to make the initial value problem ill-posed, i.e. one never see any causal effects of changes of initial data outside the light cone.