# The speed of information

Does this expression have any rigorous meaning?

Intuitively ,I feel that information about an event or a system in my environment propagates to me at a certain "speed".

Does this correspond to reality or am I using loose language to describe something more complex?

If this phrase is accurate however ,can we say that since light is the fastest "propagator of information" then we can say that the "speed of information" is limited by its fastest carrier(ie anything which travels at the same speed as light -so not just light) ?

(I am not happy with my tag -it is the best I can find)

• Information in physics is encoded in the (measured) state of systems. The fastest speed at which the state of systems can change is the speed of light. Is that your question? – CuriousOne Sep 11 '15 at 18:14
• I've actually wondered about this before. I think CuriousOne nailed it, certainly my version of it. – Matt Sep 11 '15 at 18:23
• @CuriousOne not really my question. .But your answer is more interesting than the question.I was trying to separate out the transmission of information per se from the carrier of the information.But I was ,needless to say out of my depth. – geordief Sep 11 '15 at 19:05
• There is an information tag, if that helps. – user81619 Sep 11 '15 at 19:14
• Interesting point. There is information and information carriers. In signal processing Shannon ' s theorem, Nyquist frequency applies. Is light information or the carrier? And if the latter, what does Shannon say at the speed of light? That information can only be transmitted at half the speed of light? – docscience Sep 11 '15 at 19:31

You seem to be confusing "transmission rate" with "speed of propagation". In communications engineering terms, the latter is linked to the notion of network latency, i.e. how long your message "disappears" in the network before it shows up at the receiver.

But both "information transmission rate" and "speed of propagation" are very well defined and precise terms. In particular, "information" and "speed of propagation" have an important relationship in special relativity.

You can transmit at an arbitrarily high rate: say at $10^{15}$ bits per second (this is roughly the theoretical maximum through a single mode optical fiber) and the message latency to Alpha Centauri will be four years, but when the message arrives its data rate is still $10^{15}$ bits per second. The Alpha Centaurians will still see data coming in at $10^{15}$ bits per second.

Information transmission rate is limited only by the bandwidth and the signal to noise ratio of the channel, as described by the Shannon Noisy Channel Coding Theorem and the Shannon Hartley Theorems.

The notion of "information" is actually crucial to the notion of the relativistic speed limit $c$. In particular, the transfer of "information" is a necessary condition for a causal link between two events. The mathematical properties of the Lorentz transformation are such that if the time order of two events are timelike, i.e. a signal travelling at speed $c$ or less can reach one to the other in one inertial frame, then the order that those two events happen in is the same for all inertial observers: even though those observers may disagree on the time between the events, the sign of that interval is the same. Therefore, no causal link propagating at $c$ or less can have its direction in time reversed simply by a change of inertial frame.

If, however, information could travel at a speed greater than $c$, it could be a causal link between two events that have spacelike separataion. The order of such events does depend on frame: some observers would see the effect coming before the cause! Since we believe this is impossible, we therefore conclude that faster than light signalling must be impossible. This is the very reason why physicists conclude that the maximum signalling speed, or maximum information propagation speed, must be $c$.

This is a rigorous a definition as I could find:

In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free digital data (that is, information) that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named after Claude Shannon and Ralph Hartley.

Intuitively ,I feel that information about an event or a system in my environment propagates to me at a certain "speed".Does this correspond to reality or am I using loose language to describe something more complex?

This to me is a subjective or philosophical set of questions, I don't think anyone could give you a physics based answer. Reality is whatever you think it is.

Information theory, to me anyway, looks awfully complex, but the speed at which you receive a message seems simple enough to deal with, until you get into how your brain processes it, then it gets complex again.

If this phrase is accurate however ,can we say that since light is the fastest "propagator of information" then we can say that the "speed of information" is limited by its fastest carrier(ie anything which travels at the same speed as light -so not just light) ?

The speed of light in vacuum is the fastest speed we know of, and so far, we know of nothing else except light that travels at that speed.

• I thought all massless particles traveled at the same speed as light in a vacuum.Can they be persuaded to carry information? – geordief Sep 11 '15 at 21:10
• @geordief Right, all massless particles travel at the speed of light. However, the photon is the only massless particle that we know of. Neutrinos were thought to be massless until recently, but this was found to be wrong less than 20 years ago. – Mark H Sep 11 '15 at 22:00
• I feel embarrassed to cite articles that I don't understand (or can vouch for) personally but it says here that the "Weyl Particle " is massless .sciencealert.com/… – geordief Sep 11 '15 at 22:14
• @geordief Hi no need for embarrassment, I would be red in the face all day when I think of all the stuff I don't know. If you read to the end of the article, those Weyl fermions are not regular particles, like electrons or neutrons are, they can only exist inside a crystal, and that means we can only transmit them within a crystal, whereas we can fire an electron anywhere we want. – user81619 Sep 11 '15 at 22:31

I am starting to doubt the premise of my question. That premise ,I think is that we are somehow "entitled" to have knowledge of an event (and that the universe is connected by these communications ).

A universe without this possibility of knowledge of other things seemed "impossible" to me .

But I am now starting to consider that there are events that we do not know (even ,perhaps cannot know?) and the universe does not collapse as a consequence.)

Another of the premises of my question may be that there are alternative methods (and speeds -I mean slower speeds ,not faster) of the transmission of information.

I am also starting to doubt this as it is starting to look to me as if light (em radiation) has cornered the market(for a vacuum).