According to special relativity, being able to send something (information or matter-energy) faster than the speed of light (in vacuum) is equivalent to the ability to send it back in time. In order to understand why this is the case, one must first understand what we mean by past and future in relativity and the meaning of timelike and spacelike separation of events. These topics are discussed in the light cone node. It may also be helpful to read about relative simultaneity. First we will describe the general picture of what's going on, then look at some numbers to make it concrete, and finally write down the algebra. At the end we discuss whether there are ways to get around this result, if it might be possible to send things faster than light without the implications of time travel.
The Basic Concepts
For simplicity, let's talk about sending a superluminal signal (meaning one that moves faster than light), but the same arguments apply to sending matter-energy. The reason that sending a superluminal signal enables you to send a signal back in time is that the events connected by a faster than light worldline have a spacelike separation. Suppose Alice is sitting in her lab on her space station, and she sends out a superluminal signal. Let's call the event of sending the signal event A. Alice sends the signal to Bob, who is out on a space ship, and he receives it at event B. Because the signal is superluminal, events A and B have a spacelike separation, Δx > cΔt. As a result, they have a different order in time in different frames of reference, as is discussed in the light cone node. That means that, although according to Alice event A clearly happens before event B (the signal is sent before it's received), there are some reference frames in which event B happens at an earlier time than event A, so the signal is received before it is sent according to such observers. Thus, we've already found that it looks like the signal is being sent to an earlier time, and we have to begin to question our ideas of causality.
You might well object that the signal is being sent to "elsewhere", and, though it does appear to arrive at an earlier time in some frames of reference, it does not really go into the "past", in our relativistic sense. Remember that for event A the past is defined as all the events that fall inside the light cone of A and happen at an earlier time. So, basically, what we're arguing is that the sort of "back in time" communication we've described so far isn't very interesting, since it doesn't allow Alice to communicate with her past self or any of the stuff that leads to cool sci-fi stories and troubling paradoxes. Now it makes sense to ask, "Could Alice use superluminal signals to communicate with her past self?" The answer turns out to be yes, using the postulates of relativity.
We said that Alice sends a signal to Bob, who is on a distant space ship. Let's call Alice's frame of reference S, and Bob's frame of reference S'. If Bob's space ship is traveling away from Alice at a significant portion of the speed of light, then in S' event B can happen before event A, so according to Bob, the signal is received before it's sent, just as we said could be the case. More importantly, by choosing Bob's speed fast enough, we can make B occur as long before A as we want. Since, according to the principle of relativity, the laws of physics are the same in all inertial frames of reference, Bob should be able to use the same technique Alice did to send a superluminal signal back to Alice, and he can make it have the same speed according to S' that Alice's signal had according to S.
Ok, so here's the trick, if we make sure Bob is going really fast, then he can receive the signal long before it's sent, so when Bob sends his superluminal signal back to Alice it will arrive back at Alice's space station (event C) before Alice ever sent her signal in the first place (event A). What's more, since events A and C both happen at Alice's position, and Alice is moving at less than the speed of light (by assumption), then A and C have a timelike separation. This means that if C happens before A in frame S', then C happens before A in every frame of reference. Put another way, C is inside the past light cone of A, which will be true in every frame of reference. So as long as she has a fast enough moving relay station, like Bob's space ship, Alice can send messages into her own past.
When I first started learning about relativity, this idea confused me quite a bit. I couldn't quite wrap my mind around the connection between "faster than light" and "back in time". I hope I've made the first step fairly clear, how communication with a spacelike separated event allows Alice to send signals to an earlier time according to some observers. It's crucial, though, that in order for Alice to send signals to her own past we must have this moving relay station to send the signal back. It is changing frames that allows us to send the signal back to Alice's past, something I did not understand at first. Next, we'll put some numbers in to give a more concrete sense of what's going on.
A More Concrete Example
Suppose that according to Alice, in frame S, Bob is on a space ship moving at a speed u = 3/5 c (that's 3/5 the speed of light in vacuum) away from her space station. Alice sends out her superluminal signal, moving at a speed v = 5 c, to Bob. That's event A, and we'll say that Alice considers that time zero, so xa = 0 and ta = 0. Also, we'll say that Bob arranges his coordinate system so that event A also happens and time and position zero in S', just to make the math simpler1. Let's say that Bob is a distance of 20 light minutes away2 in frame S when he receives the first superluminal signal. So in S, event B happens at xb = 20 lmin and that implies the signal took 4 minutes to get there, meaning that B happens at tb = 4 min.
Now, according to the Lorentz transformations, the relationship between the time t of an event in S and the time t' of the same event in S' is
t' = γ(t - ux/c2)
where γ is the Lorentz transformation gamma, which for u = 3/5 c is just equal to 5/4. The means that in S', event B occurs at tb' = -10 minutes, that's 10 minutes before event A. We can determine the position of event B either by using the fact that the spacetime interval has to be the same for event B in S and S' or just by using the Lorentz transformation for position, which is
x' = γ(x - ut)
Either leads to xb' = 22 lmin.
Having figured out the position and time of event B in S', we now just have to figure out when the return signal gets back to Alice. As soon as he receives the signal, Bob sends a superluminal signal back to Alice that has a speed v = 5 c in S', and it arrives back at Alice's space station at event C. Now, in S' Alice is moving in the negative x-direction with speed u (since in S, Bob was moving with speed u in the positive x-direction). Alice is at x' = 0 at t' =0, so we know that event C must take place at
xc' = -utc'
and the superluminal signal will take a time
Δt' = (xc' - xb')/v = tc' - tb'
In the end that gives tc' = -50/7 min, which means that in S' the second signal reaches Alice about 7 minutes before she sent the first signal. According to Alice's watch in S, the return signal from Bob would reach her more than 5 minutes before she sends her first signal, taking into account time dilation. So, if Bob just sends back a copy of Alice's original message, then she receives it 5 minutes before she sends it. Pretty nice for picking the winning lotto numbers.
Now that we've worked out some numbers, let's go back through the algebra briefly to state the general result. Just as before, event A marks when the first superluminal signal is sent, event B is when the first superluminal signal is received and the second is sent, and event C is when the second superluminal signal is received. Event A marks the origin in both S and S', so
xa = xa' = 0 and ta = ta' = 0
As we've already said, we know
tb' = γ(tb - uxb/c2)
from the Lorentz transformations, with u being the relative velocity between S and S'. If v is the speed of the superluminal signal, then
xb = vtb
so tb' = γtb(1 - (u/c)(v/c))
From that expression, we can see that if v > c then v/c is greater than 1, and we can find a value of u so that (1 - (u/c)(v/c)) is less than zero, and event B happens before event A in S'. Moreover, γ goes to infinity as u approaches c, so by choosing an appropriate value for u we can obtain any value of tb' we like.
The return superluminal signal travels from event B to event C with a speed v in in the negative x-direction, as measured in frame S'. That means we know
xc' - xb' = -v(tc' - tb')
But event C happens when the superluminal signal gets back to where it started in S, so xc = 0. That means
xc' = γ(xc - utc) = -utc'
Combining the last two equations and eliminating xc', we can then solve for tc' to obtain
tc' = (xb'/v + tb')/(1 + u/v)
Then using the equations above for xb' and tb' we can re-write that expression as
tc' = γtb[2 - (u/c)(1+v/c)]/(1 + u/v)
That may look a bit messy, but the main detail we really care about is the numerator [2 - (u/c)(1+v/c)]. If that's less than zero (the value of ta'), then event C happens before event A. If v > c, then we can always choose a u > 2c2/(c+v) such that the numerator is less than zero.
This is a problem that's fairly hard to solve using spacetime diagrams. I certainly won't try to present one here, but I wanted to mention it, because I got rather confused in the past trying to draw a spacetime diagram of the process we've discussed. Drawing a diagram of the worldlines of Alice and Bob, as well as the first superluminal signal in the coordinates of S is fairly easy; however, drawing the second signal in is a bitch. This is because the second superluminal signal has speed v in the frame S', so in order to draw it in the coordinates of S, you'd either have to draw a diagram of the second signal in S' and then figure out from that how to draw it in S or do the algebra as we have done. Doing the algebra showed us that event C happens before event A in all frames of reference, so the worldline connecting B and C in frame S will actually show the signal propagating backward in time. This is quite strange, but again it is what the principles of relativity predict when we think about each frame. It seems that relativistic addition of velocities does some strange things with faster than light signals.
Implications
What are the implications of the fact that faster than light communication implies the ability to send information into the past? Most physicists would say this is one of the primary pieces of theoretical evidence3 that it's not possible to send information (or matter-energy) faster than the speed of light. The reasons are, consequentially, the same as those against the possibility of time travel. Namely, if you can send information into the past, you can setup all sorts of paradoxical chains of events. For example, Fry stands in a suicide booth and uses his superluminal cell phone to text message a command for the suicide booth to kill him one minute in the past. However, if the message is received, then Fry is killed and the message is never sent. So is Fry killed or not? This seems to have no reasonable solution, or at least it seems that any solution would require altering all the applicable laws of physics to allow for some sort of self-consistent scenario. It would also require us to completely reformulate our understanding of causality. I went to a philosophy of science talk in which the speaker proposed that such a scenario can be resolved without changing the other laws of physics but merely requires a modification of our understanding of cause and effect. I was not convinced by her arguments, but the issue may need more exploration. I cannot say for certain that the existance of these causal loops would invalidate all our causal physical laws, but that certainly seems to be the case and is the opinion of most physicists; thus, there's good reason to think it's impossible based on what is currently known.
The "causal loops" we've mentioned here bring up similar problems to those found in closed timelike loops, which can occur in properly curved spacetime. Our signals are spacelike, not timelike, but since they give rise to causal loops, many of the issues should be the same. In spacetime with closed timelike curves, it has been found that quantum field theory can't be well defined, because quantum fluctuations would get larger and larger in a runaway effect (thanks to tdent for that bit of info). The same thing will likely be true if you try to formulate a quantum field theory that includes superluminal interactions like the ones we've discussed. So it looks like we'd also have to invalidate our modern understanding of quantum mechanics and particle physics to accept these superluminal interactions.
Many people are interested in the idea of superluminal communication and/or superluminal travel. Certainly those would seem to represent very useful technologies if they existed. The fact that relativity tells us that we then have to consider all the paradoxes of time travel is discouraging, and many people wonder if there might not be a way to "get around" the argument we've made above. One counter-argument might be that, since we've never dealt with things going faster than light, we shouldn't necessarily expect special relativity to apply to them properly, just as Newtonian mechanics only applies properly for objects going much less than the speed of light. Except, notice that we don't require any information about how the information gets from Alice to Bob in the above argument, we only need to know what the endpoints of the transmission were, so the result doesn't really depend on "superluminal physics", just on the nature of space and time in the ordinary sorts of situations special relativity applies to. Thus, this argument is completely independent of the mechanism of superluminal communication.
The only way to circumvent our argument is to claim that special relativity is incorrect somehow. One of the simplest ways around our conclusion is to say that, in fact, the laws of physics are not the same in all frames of reference. If we say, for example, that these superluminal signals always travel with a speed v as measured in a certain preferred frame of reference S, then Alice can no longer send a signal into her own past. Physicists would refer to this as Lorentz symmetry breaking. Alice can still send a signal to Bob, to an event B with spacelike separation from event A, so that in some frames of reference the signal will appear to arrive before it was sent; however, if we have Bob send a signal back, we are saying it will have speed v as measured in S (not as measured in S'), in which case we can just do the whole problem in S. The signal goes out, which takes some amount of time, and comes back, which takes more time, and the total time elapsed is a positive number, so the return signal always comes back to Alice after she sent the original signal.
In the preferred frame of reference things go more or less as you would expect from Newtonian physics. So things still look funny in some frames, but we've removed the paradoxes we found earlier. Still, we do retain the possibly unpleasant feature that for events interacting with superluminal signals we can no longer identify which is cause and which is effect, since there's no definite time ordering. It should also be pointed out that abandoning the principle of relativity, that physics is the same in all inertial frames, is not a small concession. It is a principle that actually predates Einstein and relativity and underlies even Newtonian mechanics. This is not to say it has to be true, but it is one of the more fundamental ideas in physics and has held up well so far.
Clearly, science is empirical, and we cannot rule things out completely. We can only say what is consistent with the evidence we have observed. Thus, we cannot completely rule out the possibility of superluminal signals, but we can say that they have never yet been observed, and that all our observations that form the basis of special relativity also suggest an equivalence to sending signals into the past. It further seems that allowing signals to be sent into the past would require abandoning our understanding of many, if not most, of our observational evidence, so it seems very unlikely that it is possible to send anything faster than the speed of light.
1 In this coordinate system Bob is not at x' = 0 as you might expect. There's nothing profound in this, you can easily measure positions with respect to a map where you're not at the center, can't you? It's just to make the math simpler. Note, even if event A has the same coordinates in S and S', length contraction and time dilation ensure that they will not agree on the coordinates of any other event.
2 A light minute is the distance light travels in a minute in vacuum. Similar to the idea of a light year. 20 light minutes is about 2.5 times the distance from the Earth to the sun.
3 As opposed to the experimental evidence, namely that we've never seen anything to suggest such a phenomenon exists.
This has been the culmination of a sort of multi-node project, so I'm very eager to get feedback. My main goal was to make this subject at least somewhat accessible to people who aren't comfortable with much more than basic algebra. Please let me know if I've succeeded or how I've failed. Either is helpful.
Sources: My own knowledge of special relativity.
Though I didn't really consult any sources to write this, if you want to look at some, here are some suggestions:
The book I originally learned from (not necessarily recommended)
Special Relativity, A. P. French
A well respected introduction to special relativity from the view of geometry (spacetime diagrams) at an introductory/intermediate college level
Spacetime Physics,Taylor and Wheeler
Finally, most general physics text books will have a chapter somewhere near the back about relativity, which generally will discuss at least the idea of relative simultaneity and related topics.