This is something that's been bothering me for a while. The way we usually first hear about causality is that "nothing travels faster than $c$". But then you learn that phase velocities can sometimes be faster than $c$, so we revise the previous statement to "information never travels faster than $c$". But maddeningly, I've never seen anyone actually define what "information" means in this context. Without a mathematical definition of information, it seems to me that the preceding statement is totally vacuous.
Can someone please provide a rigorous definition of information in this context, so that e.g. given some dynamical equations of a relativistic theory (e.g. of electrodynamics) I can verify mathematically that the equations indeed do not allow information to travel faster than light.
If this is impossible, or if nobody knows how to define information in this way, please describe the situation.
EDIT:
Despite many answers, nobody has yet addressed my actual question: What is a definition of information for the purposes of physics. I know about the arguments (given by people like Griffiths in his quantum mechanics book) about how certain things that appear to travel faster than light cannot be used to communicate in a way that violates causality. That is not what I'm asking! I am looking for a way to generalize the potpourri of such examples into a sharp theorem, and to that end I need a proper definition of information.
As another point for consideration, another situation in which "information" is implied to have meaningful physical interpretation is in the black hole information paradox. The rough statement of this paradox is "do black holes destroy information?". One way to interpret this question rigorously is "do black holes violate unitarity?". But what I want to know is the following: Is there a meaningful, mathematical definition of "information", which would in principle allow one to take a hypothetical theory of quantum gravity and determine rigorously whether or not black holes in that theory destroy information?
If there is no such definition of information, please provide an authoritative explanation of why not (and provide sources if possible), rather than trying to give more examples.
Best Answer
An example might help. While not giving a "strict definition," it might be a step toward constructing one. (I think I am remembering this from Hans Reichenbach's classic Philosophy of Space and Time.) Here goes:
From earth, you can sweep a laser beam across the surface of the moon such that the "dot of light" on the moon's surface moves -- continuously -- from point A on one side of the moon to point B on the other at a speed faster than the speed of light. The dot of light is a "something" -- so it's false to say that nothing can move faster than the speed of light.
But that moving dot of light cannot be used to convey information from some person (or some machine) at Point A to another at Point B. That is, there is nothing Person A can do with the dot of light while it is at A, to tell Person B by some pre-arranged code whether he (person A) is, say, a 0 or a 1 (drunk or sober; male or female, etc). The moving "dot of light", while a something, is not the sort of "thing" that can be marked by Person A to as to inform Person B of some fact.
Now of course, by pre-arrangement, Person A and Person B might use the dot of light to synchronize something: Person A might agree to make a toast to B when he sees the dot of light, so when Person B sees it, he has in a sense been informed that he has just been toasted. So a good definition of "information" will need to make clear why this doesn't count. [[Two other early answers prompt this addition. As I saw it,the questioner's perplexity seems to arise less from lack of a definition of "information" (or from need for some mathematical way of verifying the "nothing bearing information can travel faster than light" law) than from simple bafflement about what it means to hedge this limit-law by saying that the limit is not on how anything can travel, but only on how fast an information-bearing thing can travel"* (or "be sent"). "How," the questioner seems to be asking,"is this not just a dodge? What is added when we qualify the limit-claim by specifying that it is only a limit on information-bearing entities?" Insofar as this is the sticking-point (the questioner might want to clarify this!), then what's needed is simple conceptual clarification. And here one later answer (by Steane) here helps resolve the residual puzzle I left hanging. When we say that some moving entity E can carry information from A to B, E must be such the entity that it can be used not just to synchronize, but to notify a receiver at Point B of some arbitrary change being effected at Point A. In the synchronized-toast puzzle I left hanging, the person at A cannot bring about some arbitrary change at A (say, decide whether or not to hoist a toast to person B), and then by the moving light-dot, notify B of this. I think this solves the residual puzzle!]]