Ykhjuy

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.

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!]]

There is more than one mathematical formulation for this, but here are sketches of a couple of examples.

In a quantum-mechanical context, consider the following variation of the EPR paradox. A nucleus having zero angular momentum undergoes symmetric fission into two fragments, each with $ell=1$. By conservation of angular momentum their angular momenta are opposite. Let's say that except for this correlation, the two angular momentum vectors are randomly oriented. The fragments are observed by Alice and Bob, and these two observations are spacelike in relation to one another. Suppose that Alice measures $ell_x$, but Bob measures $ell_z$. It shouldn't matter who goes first, but let's say that Alice does. Can Alice send information to Bob by deciding whether or not to measure her particle's $ell_x$? If we calculate Bob's probabilities, they actually end up the same regardless of whether or not Alice has done her measurement before he does his. So essentially the mathematical statement is that stuff at A can't affect the density matrix at B.

In a classical context, a pretty standard way of talking about this is in terms of wave equations and global hyperbolicity. We want our spacetime to be globally hyperbolic, which basically means that wave equations have solutions to Cauchy problems that exist and are unique. An example of a failure of global hyperbolicity would be if you have a naked singularity. If there is global hyperbolicity, then you can find the solution to a wave equation at a certain point in space by knowing only the initial conditions on a Cauchy surface that is within that point's past light cone. This approach is developed in detail in Hawking and Ellis. They use a wave equation for a scalar field, just because it's mathematically simple.

The first example, using the density matrix, corresponds pretty closely to the information-theoretic idea of information. The second one focuses more on propagating signals.

Already two insightful answers here, but I'll chip in.

This is very much about the notion of cause and effect. That notion is itself not as straightforward as one might initially think, but I won't get into the metaphysics. The main point is that the word 'information' is an attempt to capture the idea that if a change $delta$ happens at event A, then if as a result of that change things go differently for X, such that the change in X can influence what happens at event B, and make things transpire differently there than they would have done if $delta$ had not happened, then X cannot travel faster than light.

If a theory has some gauge freedom then it can be non-trivial to figure out whether it is respecting this. Sorry!

How about:

Consider two points in spacetime X and Y. If Y is not in X's light cone, then intervening at a hypothetical distribution over events Q at X cannot affect a distribution over events R at Y: P(R) = P(R | do(Q=q)) for all q.