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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.

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    $\begingroup$ The references of the Wikipedia entry on 'physical information' are probably a good place to start. $\endgroup$ – tfb Nov 30 '18 at 18:09
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    $\begingroup$ FWIW, the Subluminal Java applet by Greg Egan shows how a wave packet with superluminal group velocity cannot be used to transmit information faster than c. $\endgroup$ – PM 2Ring Nov 30 '18 at 18:44
  • $\begingroup$ It's energy that can't move faster than time. See this answer: physics.stackexchange.com/questions/22084/… $\endgroup$ – safesphere Nov 30 '18 at 22:44
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    $\begingroup$ @PM2Ring I've run that applet and dropped the shutter at various times. What am I supposed to notice? $\endgroup$ – garyp Dec 1 '18 at 21:00
  • $\begingroup$ The original statement with "nothing" is pretty satisfactory as-is. The problem is artificially claiming a phase velocity is a "something". $\endgroup$ – R.. Dec 1 '18 at 21:10
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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!]]

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    $\begingroup$ The dot of light is not "a something," it is many, many different somethings. They only look similar. $\endgroup$ – David Conrad Nov 30 '18 at 22:32
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    $\begingroup$ Could you spell out a physics-criterion for what you will count as a "something"? And then see how it comes out on this: by the criterion, does that big wave off Malibu on which I surfed yesterday. The wave (and I) was rolling forward at about 20 knots; the the water constituting the wave, of course, was not. Will the wave by your criteria count as a "something" (governed, so to speak, by the "no-superluminal-things" law)--or will it, like the dot-of-light, count as "many, many different somethings" (not governed by that law)? $\endgroup$ – SteveW Nov 30 '18 at 23:46
  • $\begingroup$ Regarding your double bracketed comment: No, that is exactly the opposite of what I want. What I am looking for is exactly what I asked for in my question: A rigorous mathematical definition of information. I understand the conceptual arguments (of which I have seen many) to my satisfaction. What I really want is a mathematical tool to prove theorems about causality (of which I have seen none), and for that purpose I need a rigorous definition of information. $\endgroup$ – Yly Dec 1 '18 at 3:59
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    $\begingroup$ Both the dot of light and the big wave off Malibu are made up of particles, and when the wave or the dot have "moved" to another place, it isn't the particles that have gone there; there are different particles there. In the case of the wave, earlier particles pushed on the later ones to move the wave along, but in the case of the dot sweeping across the surface of the Moon there isn't even that connecting the photons, they are only connected in our imagination. $\endgroup$ – David Conrad Dec 1 '18 at 16:02
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    $\begingroup$ All true and perhaps relevant. Provided we can somehow use this to answer the two questions: (Q1) Is the malibu wave a "something"--in your sense, or not? (I take it your answer is 'Yes' though you haven't said so); and (Q2) What is the now the criterion for being a "something" (rather than "many, many things")--in your sense. $\endgroup$ – SteveW Dec 1 '18 at 17:45
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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.

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  • $\begingroup$ The second paragraph is, to me, a lot of words in an unfamiliar language. I don't see the connection to signals travelling faster or slower than $c$. $\endgroup$ – garyp Dec 1 '18 at 20:57
  • $\begingroup$ @garyp: I don't see the connection to signals travelling faster or slower than c. The fact that Alice and Bob's measurements are spacelike in relation to one another means that you can't get from A to B at the speed of light. $\endgroup$ – Ben Crowell Dec 2 '18 at 1:17
  • $\begingroup$ +1 This is the best answer so far, as the only one to touch on actual mathematical foundations. $\endgroup$ – Yly Dec 2 '18 at 1:28
  • $\begingroup$ I was referring to the second paragraph. $\endgroup$ – garyp Dec 2 '18 at 3:21
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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!

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The rigorous idea that "information doesn't go faster than $c$" is trying to express is simply that observable quantities at a given spacetime point only depend on data in the past light cone. Equivalently, events cannot influence the results of future measurements outside their future light cone. The reason we emphasize "information" is because this principle is easy to misapply.

For example, consider the usual case of the entangled EPR particle pair. Measuring the spin of either particle will always give the same statistics: a 50/50 chance of spin up or spin down, regardless of whether the other particle was measured. So observable quantities are not being influenced faster than light. The subtlety is that when one particle is measured, the wavefunction of the two-particle state instantaneously changes. That is, our description of the system looks completely different, even though nothing actually observable changes.

This is what leads people to emphasize that no "information" is transmitted. Historically, this scenario created a lot of confusion, and it's still asked on this site every week.


If you want more mathematical formalism: in the classical case, the configurations of fields are governed by partial differential equations. Causality is the requirement that the field value at a point can be computed by initial data lying only on its past light cone. In the quantum case, the state of a subsystem of a composite system is described by its reduced density matrix. Causality is the requirement that measurement of part of a subsystem cannot influence the reduced density matrix of its other part faster than light.

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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.

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What precisely does it mean for “information to not travel faster than the speed of light”?

There is a branch of computer theory handling information transfer. It has been very influential there as well as in other science branches. The seminal paper is "A mathematical theory of communication" by Claude E. Shannon. Here the concept of information is boiled down to the bit, the fundamental unit of information. https://ieeexplore.ieee.org/document/6773024?arnumber=6773024

Although this is not exactly the same definition as used in the Quantuum world, the root of the concept information clearly comes from Shannon.

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Several interesting answers, which however may result overall confusing. Information is the knowledge of the state of a system. Consider a system which can be in $N$ discrete states ($N=2$: on/off, open/closed, $N=4$: facing north/south/east/west, $N=100$: position in integer centimeters along a ruler of 1m, etc.) [expansion to continous states is possible]

Then the entropy of information, defined as $S=log_2(N)$ is the minimum amount of bits ("0"s and "1"s) you need to define that system.

For example, assume I sent you a letter with a question and a given number of tickboxes which you can tick to answer. Ticked box means 1, un-ticked means 0. Now, $S$, is the minimum number of tickboxes I need to include in my letter for you to be able to answer me. If I include more, I waste space. If I include less, you will not be able to give me an answer (of course, we had to prepare a code before - but after that first code is decided, we only communicate via letter).

Assume I ask you: is your phone on or off? You can answer 1 for on, 0 for off. So you need to send me at least 1 bit (0 or 1) to answer me. But if I ask you: are you north/south/east/west of my position? (neglecting NE/NW/SE/SW etc.), to code that information and answer me, you need at least 2 bits ($S=log_2(4)=2$) because only "1" and "0" alone are not sufficient (you can only tell me two answers out of four). So your codification could be (for example) "11" for north, "10" for south, "01" for east and "00" for west. Notice that in this case you could even code it as "1" north, "0" south, "11" east, "00" west and think that, at least in two cases, you could use only one bit, but since, when I send the letter, I can't know where you are I still need to include at least 2 tickboxes.

Summing up, to code a system in bits, you need at least $S$ bits. So, going along this way, you can arrive to defining information. More, here.

Now, it may seem weird, but information is a physical quantity: there is an associated cost in deleting information, see Landauer's principle.

Regarding your question, while there is no definite way to define information (but it is in principle possible), transmitting it always involves sending bits (be it checkboxes, voltages, eletctromagnetic waves...). None of this stuff can walk faster than light, and if you see something that goes faster than light (e.g. phase velocity) you can be assured no bit-based communication is feasible with it. There is now way to transmit the information about one system to another person faster than the speed of light (except spooky quantum mechanics stuff still debated).

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  • $\begingroup$ Good answer up until the last sentence; the whole issue under consideration is the fact that spooky quantum stuff can transmit something faster than light, but that that something can't be information (bits). To be more specific, you can use quantum entanglement to create shared random data between two people at a distance instantly, but you can't use it to communicate. $\endgroup$ – rlms Dec 2 '18 at 16:08
  • $\begingroup$ Thank you, i will edit asap. Of course it was just to mention entanglement and other possibilities but it does not change the core of the answer! $\endgroup$ – JalfredP Dec 3 '18 at 9:05
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What is physical information?

Despite many answers, nobody has yet addressed my actual question: What is a definition of information for the purposes of physics.

This is pretty straightforward. To bring this collection of items related to physical information to a point: physical information simply is what you are used to when reasoning about physical objects - their existence; simple attributes like mass, position, velocity; or more difficult ones like amount of kinetic energy, quantum state, etc.

Examples:

  • In the center of this cubic meter of otherwise empty space, there is a single hydrogen atom.
  • This metallic bar is exactly $1m$ long.
  • Yly is made up mostly of $H$, $O$, $N$ and $C$, charged slightly negatively, and travelling at $5\frac{km}{h}$.
  • The electron is "here" with a probability of $75\%$.
  • The gravitational field looks like "this" in that area of space

(Not all of these examples necessarily are true...).

Note that it is completely normal that our use of language highly compresses and fuzzifies information. For example, the information "I sit on a chair" is an incredibly compressed statement about all the atoms of my body and them being electrostatically repelled by the atoms of my chair, while being squished against the floor by gravity caused by some planet - and so on, and so forth. And you don't know anything at all about "I" except that there is a non-zero probability that I own a keyboard to type this sentence... It is physical information, nevertheless, at some level of detail (or abstraction).

Destruction of information

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?

A test like you mention could be as follows:

  • Make up a model of a volume of space with some matter/energy/fields in it rigorously, at the highest detail level the theory allows. You end up with some amount of data (numbers, structural symbols etc. - practical, everyday, non-mystical symbols that you can write on a piece of paper or store in a computer). For your question, this state obviously needs to contain a black hole somewhere, and you need to model that as detailed as your theory allows, as well. Note that what the symbols represent can by anything - this is what the theory you are testing will tell you (e.g., you do not need to break Heisenberg's Uncertainty Principle if your theory can work just fine with probabilities).
  • Let time run, and calculate a new state after the black hole has swallowed whatever else is there (you can pick whatever "end condition" you like).
  • Repeat this with another initial state (which needs to differ in any arbitrary respect from the first one - one bit is enough - but the black hole needs to be exactly the same as before as we want to test a property of it).
  • If the results of the two "black hole operations" are equal, then the black hole has destroyed information.

Speaking from the view of theoretical computer science, you might run - depending on the complexity of the theory - into hard problems regarding the halting problem. You need to show that the theory ends up with the same result for at least one pair of different initial states. You might have to try a huge amount of them before finding one, so it may not be easily possible to actually run the algorithm given above in human time scales. But theoretically (or if you get very lucky, or use an "oracle" to somehow find a pair for you), you can test it that way, and more importantly, you can define it that way, which was your question.

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In a simple and intuitive, though not quite scientific way you can think of information as "knowledge". If an event occurs at a certain distance from you, you cannot find out about it sooner than it takes light (in vacuum) to travel that distance.

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