In series such as the Hyperion Cantos and Ender's Game, there exist fictional faster-than-light communication systems allowing either instantaneous, or near-instantaneous transmission of information.
Are there any (or what are some scientifically plausible) methods of faster-than-light communication, and what are some important things to consider when applying one of these methods?
Our society does not yet have sufficiently advanced science to be able to answer this question.
However, in most science fiction, there are three commonly used approaches. Note that I am NOT claiming any of these would actually work.
Wormholes, or something like wormholes that directly connect two points in space
Subspace, or some other kind of one-for-one mapped parallel space through which signals can be transmitted at speeds that are effectively faster than the speed of light in normal space.
Quantum entagled particles, where two particles have a quantum level connection such that when one particle is manipulated, it affects the other particle. Communication is instantaneous, but things such as FTL travel can break the entanglement (meaning that devices must be shipped on slower than light transports). For an example of this use: Singularity Sky by Charles Stross.
One thing we know won't work is gravity waves. In the Honor Harrington universe, gravity waves are faster than light, but recent advances have proven that gravity waves only travel at light speed.
Those who trot out tachyons need to keep reading:
The bottom line is that you can't use tachyons to send information faster than the speed of light from one place to another. Doing so would require creating a message encoded some way in a localized tachyon field, and sending it off at superluminal speed toward the intended receiver. But as we have seen you can't have it both ways: localized tachyon disturbances are subluminal and superluminal disturbances are nonlocal.
This result is part of the original papers that coined the term, but everybody stops reading with "faster than light" and assumes that you could use them in the familiar matter with the familiar rules.
There is also a good description by Leonard Suskind in a Stanford lecture on youtube. He shows they are like inverted pendulums tied together with springs, and a disturbance still propagates in the sub-luminal manner.
See also this Answer where I went into that already.
Now for ansibles etc. What is "instantaneous"? There is no universal agreement on synchronization, and for space-like separation (as in FTL communication) there is no agreeement on the ordering of events.
So you can't just say it's instantaneous because there is no such thing. Instantaneous in one reference frame, not so much in others, and time travel reverse-causality in the rest. The same events will be all of the above to different observers.
An easy solution is to have your anibles work in a single ansible reference frame, as discussed in this answer. (Note that comminication is just travel for messages.)
Plausible?
When Carl Sagen had that question, he asked the physics community and Kip Thorne came up with wormholes.
The Giants novels used microscopic wormholes that could pass photons with short wavelengths. The time-travel bit was pretty much ignored until the series decided to exploit it wholesale.
The closest thing to FTL communication is communication via a wormhole. This wormhole need not be large enough for anything but a photon to be able to pass through, but if the mouths of the wormhole were separated, the photon would enter one mouth and exit the other in less time than it would take to travel the real-space distance between the mouths.
It would be necessary to generate a wormhole, and then to carry one mouth of the wormhole to the intended destination of the communication. This could take some time.
This answer presupposes that it is scientifically plausible to generate and maintain a wormhole.
Donald.McLean's answer is very good. Just as an addition, there are theories about the existence of tachyons, but most Physicists think they cannot exist. Tachyons are (in these theories) particles that move only faster than light, and can not reach the speed of light, the same way normal massive particles can't.
It is known that you can transform energy into matter and vice-versa (so it is possible to transform light into matter and vice-versa).
Hypotethically, there could be a similar device for tachyons, making a conversion between particles and energy. There is (at least) a pair of short tales about that (these two are in spanish) out there, first one about how the technology which discover them and the second one about possible usage:
In short, you can use a laser (pure light) to create tachyons and modulate them. These would travel faster than light itslef and can be readed at the other end by a similar means.
An answer to this question that hasn't come up yet relies on a certain interpretation of quantum mechanics, namely, Bohmian mechanics or pilot-wave theory. In a pilot-wave theory (PWT), you have a wavefunction like in ordinary QM, but you also have point particles with definite locations (or fields with definite values, depending on how the idea is extended to QFT).
Typically in a PWT, when you have an ensemble of systems with identical wavefunction, the actual positions of the particles are distributed according to Born's rule, a condition which is called quantum equilibrium. But PWT also allows for the possibility that we could have an ensemble of systems which are not in quantum equilibrium. In such cases, a number of things become possible:
(The last two items basically break all modern cryptography schemes, by the way.)
For FTL communication, this would interact with relativity by effectively defining a reference frame (or more generally, a foliation of spacetime) in which the communication was instantaneous. It would not allow any time-travel loops or paradoxes.
Another interesting feature of this kind of FTL communication is that, since it doesn't propagate via a signal, it can't be blocked, intercepted, or interfered with in any way (though if the initial entanglement is established by transmitting entangled photons for example, those might be intercepted).
The details can be found in this paper by Antony Valentini: https://arxiv.org/abs/quant-ph/0203049
YES (but really no)
Empty space isn't really empty -- there are quantum-mechanical "vacuum fluctuations" that the light interacts with, theoretically slowing it down. The Scharnhorst effect ( https://en.wikipedia.org/wiki/Scharnhorst_effect )says that light itself should travel faster if going through a special vacuum -- one with fewer vacuum fluctuations.
That hasn't been demonstrated or proven mathematically, but seems plausible enough.
However the only way we know of to reduce vacuum fluctuations is to have two plates very, very close together. And the effect would be tiny ("A photon travelling between two plates that are 1 micrometer apart would increase the photon's speed by only about one part in 10^36"), so small it's beyond today's technology to verify or refute the claim.
So... We might be able to send a signal very, very slightly faster than c, if we're sending the signal about an atom's diameter in distance.
Western Union is reportedly unconcerned.
Nothing can travel faster than light speed, however you could simply use wormholes to send signals between two points at "faster than light speed".
JDługosz and Donald have given the best answers so far. I'll just add a small, extra suggestion: Alcubierre Drive mail.
The Alcubierre Drive is an hypothetical means to "contract" space around a body, so that a ship could artificially shorten the distance between any two points it would travel through. For all effects, such travel would appear to be FTL for outside observers.
So you could write a letter, post it into the nearest alcubierre mail post and wait. It would get there faster than a beam of light would.
However, the equations involved seem to indicate that anything around the ship will be destroyed as it moves:
Brendan McMonigal, Geraint F. Lewis, and Philip O'Byrne have argued that when an Alcubierre-driven ship decelerates from superluminal speed, the particles that its bubble has gathered in transit would be released in energetic outbursts akin to a sonic boom shockwave; in the case of forward-facing particles, energetic enough to destroy anything at the destination directly in front of the ship.
So you should be really careful about whom you are sending your messages to, unless you're mailing people you really don't like.
YES there is.
It's just a hypothetical particle, but the tachyon, if existing, is faster than light.
We can't tell if it exist, but at least it's plausible and this handy for any not fantasy based sci-fi scenario.
Einstein was talking about a Time-Telephone which would allow to send information back in the past, but not the future. This hypothetical device is based on tachyons.
Note that FTL communication breaks the theory of relativity since FTL is equivalent to time travel (under some frames). Thus the best scientific knowledge is that tachyons, quantum entanglement and so on cannot carry information faster than the speed of light.
Since scientific theories such as the Theory of Relativity are not mathematically proven it is "plausible" that they will be disproven. However, given the many answers that assume that relativity is wrong, it is interesting to note that the existence of pseudo-telepathy is a known scientific fact and not limited to the speed of light.
The standard example is two space captains (Alice and Bob) who want to meet only if they both agree. They have the option of travelling to star system A or B. They say they want to meet, and they both receive a message saying to travel to system A. They arrive at system A, and meet.
Now they have received something information-like, because they were able to agree to meet, not limited by the speed of light. However, this doesn't break causality. They each have a 50% chance of receiving "A" regardless of what the the other captain responds, so there receiving "A" by itself gives no information about the other space captain. No information is exchanged until the captains meet, which happens slower than the speed of light.
Quantum mechanics doesn't allow the captain problem to be solved with 100% accuracy, but still much better than random chance.
Depending on the feel of your universe pseudo-telepathy might work better than actual communication. Imagine oracles spread across the galaxy with some from of pseudo-telepathy device wired into their brain. It gives a 100% scientific reason to have oracles that give cryptic statements that tell you what you need to know (but you don't know why until after), if that's your thing. There are limits to the what can be achieved by pseudo-telepathy, but the oracles may not even really know when they are just guessing. The main characters don't necessarily know the exact details, they consult the oracle and it works out (or not) as required by the plot.
Quantum entanglement
in theory no useful information is transmitted but if you could find a way to measure whether a particle is superimposed or not (something that technically does not violate the Heisenberg uncertainty principle) you could send individual bits faster than light (technically at infinite speed).
