How do we write mathematically that "information" cannot go faster than light? And along a similar line of thought, how do we relate "information" with special relativity. Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity). Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?
Special Relativity – How Information Relates to Special Relativity
faster-than-lightphase-velocityquantum-informationspecial-relativity
Related Solutions
If you're sitting in the rocket then you appear to be stationary and it's the Earth that's moving. Therefore the people in the rocket will see time passing slowly on Earth. Of course we here on Earth see time passing slowly on the rocket.
The situation has to be symmetrical because it's a fundamental part of special relativity that all frames are equal and there are no special frames. If the people in the rocket saw time passing more quickly on Earth that would prove to them that they were moving and the Earth was stationary, and SR forbids this.
You need to be careful about tossing around concepts like time dilation as it's easy to fall into conceptual traps and end up with a paradox. The only reliable way to do things is to sit down with a piece of paper, choose the events you're interested in then do the Lorentz transforms to find out what happens.
In the case of relativity, "information" refers to a signal that enforces causality. That is, if event A causes event B, then some signal must travel from A to B. Otherwise, how would B "know" that A had occurred?
Some examples:
- Light (signal) from a candle (A) hits your eye (B), causing you to see it.
- Electricity (signal) flows from a connected switch (A) to a light bulb (B), turning it on.
- Your friend (A) throws a wad of paper (signal) that hits you (B) in the back of the head, causing you to turn around to see who's trying to get your attention.
In all of these cases, the effect (B) comes after the cause (A) because there must be some signal from A that interacts with B to cause B to happen. The technical term for this is "locality." Over the centuries of studying how the universe works, scientists have found that all causes are local to their effects; nothing happens at a distance without something (light, sound, matter, etc.) acting as a go-between. [1] If you want to interact with some distant object (a friend, a planet, an enemy target), you either have to go there yourself or send something in your place (a letter, a satellite, a missile).
Let's consider the case of a laser beam swept across the face of the Moon. Let's further imagine that there are two astronauts, Alice and Bob, on the surface of the Moon with a large distance between them. The laser spot sweeps across the Moon and falls upon both Alice and later Bob, with the spot moving at faster than the speed of light. So, the question is, does that spot constitute a causality signal from Alice to Bob? The answer is no, because nothing Alice does will affect how the spot moves or when it moves or even if it moves. The cause of the light is on Earth and is not local to Alice. Nothing Alice does will change the spot that Bob sees.
There is a way that Alice can use the spot. She can hold up a mirror and reflect the laser beam towards Bob. The reflected laser beam is a causality signal because its origin is local to Alice. Alice can choose whether or not to reflect the beam at Bob. But, notice that this signal travels at the speed of light. It will arrive after the laser spot sweeping across the surface.
[1] This is why Einstein and others objected to quantum entanglement weirdness. It looks like signaling at a distance, but it's really not. Various mathematical and experimental discoveries show that not even entanglement's "spooky action at a distance" can transmit information faster than light. Quantum teleportation has been demonstrated in the lab, but there must be a slower-than-light signal between the sender and receiver to make the system work. There's far too much detail to go into here.
Best Answer
Since you are looking for an equation (you say "mathematically"), I would undoubtedly choose this: $$\left[\hat O (x),\, \hat O' (y)\right]=0, \, \mbox{if}\; x-y \; \mbox{is spacelike}$$ where $\hat O$ and $\hat O'$ are the (linear self-adjoint) operators corresponding to two physical observables —in particular, both may be the same observable and therefore the same operator ($\hat O=\hat O'$)—, and $x, y$ are two points in space-time. This equation summarizes the fact that information cannot travel faster than light because it says that the results of two experiments separated by a space-like interval cannot be correlated. And this is what "information" means since one codes information with physical effects. Please, see this Definitions: 'locality' vs 'causality' if you are interested in the different usages of the terms "causality" and "locality", which are physically more relevant or why entanglement do not imply faster than light propagation.
The previous formula assumes that the physical laws obey the principles of quantum mechanics and special relativity, and are thus quantum field theories. This is the case for the electromagnetic, the weak and the strong interactions and also likely for the case of the gravitational interaction in the weak field limit and in the sense of an effective field theory; which are the fundamental interactions that we know.
Sometimes defining the speed of a wave is tricky. The signal or information velocity is often the group velocity (which is the velocity of a wave packet), even though in some media (see http://en.wikipedia.org/wiki/Signal_velocity) it is not. But the phase velocity (the rate at which the phase of the wave propagates) cannot carry information (see http://en.wikipedia.org/wiki/Phase_velocity) and may be faster than $c$.
You may take any inertial frame provided its speed be lower than $c$. Note that according to special relativity one needs an infinite amount of energy to cross the speed of light $c$ threshold.
Edit: SMeznaric points out —and I agree with him— that space-like separated measurements may give correlated results. What is not possible is to send information one has control over, such as the choice of measurement operators.