Cloudscape

Since Special Relativity has become a paradigm of modern physics, any phenomenon of faster-than-light communication has either been ignored or dismissed by physicists. Over the years there have been many experiments in which true superluminal velocities have been observed. Unable to dispute their results, mainstream scientists often represented them in such way that their significance was masked, using cyclical arguments or irrelevant metaphors. Scientists seem to fear that if proven possible, FTL travel would mark the end of Special Relativity.
There is no reason, however, why we should give up Special Relativity if we find that superluminal velocities exist; but perhaps we should nuance it somewhat. Nothing in the universe may be absolute. Natural laws tend to have exceptions. Natural laws are causes of phenomena, but they themselves are phenomena, and so we might assume that they as well have causes. Those causes might be changeable, like anything else in nature appears to be.
We live in a universe that appears to be "fine-tuned" to the possibility of life. For example, should the fine-structure constant (a dimensionless constant equal to 7,29 · 10^-3) be just 4% different, then stellar fusion could not produce carbon, and so life would be impossible. (There is, of course, a possibility of life that need other or less atoms, but the question is what life could do with just two non-inert atoms, hydrogen and lithium!)
This suggests two things.
The first is that there are, or have been, many other universes with other parameters. After all, if there were only one universe (or one region in the universe with distinct physical constants), it would be too much of a coincidence that it is inhabitable. That we live in a universe which happens to be fine-tuned to life because there could be no other universe to live in in the first place.
The second is that physical laws, or at least physical constants (which include the speed of light) are changeable (at least in very extreme circumstances), since they must have formed at some point in time to become what they are now. For our universe, this supposedly happened in the early stages of its birth (although there may be other universes where the laws of nature are still changing constantly). Some of the circumstances of the primeval universe are actually being simulated in particle accelerators, albeit over extremely small spaces. Who knows? Perhaps, one day, we'll get so far as to bend the laws of nature in our own particle accelerators.
In fact, such phenomena have already been observed.
Natural nuclear fission reactors are subterranean deposits of uranium which may undergo spontaneous nuclear reactions; one of the oldest of these nuclear fission reactors lies in Gabon, which has been discontinuously active for about 2 billion years. By analyzing the nuclear decay in these deposits, researchers have discovered that the fine-structure constant, which determines nuclear reaction cycles, has slightly changed over that time.

Phenomena Superluminal Phenomena

1) Inflation Period:
Perhaps the most dramatic instance of superluminal velocity in the history of the universe was at its very beginning: in the "inflation period," a period which lasted for a tiny fraction of the first second after the Big Bang, all the matter in the entire universe exploded at a speed far higher than the speed of light. In 10–33 seconds, the universe increased 1026 times in size. The diameter it had achieved at the end of this period was no more than 10 centimeters, but all this had happened in such a short time that this was by far the most extreme explosion ever. The velocity it involved was 1032 meters per second, or 3 · 1023 times the speed of light.
Many physicists would, as usual, dismiss this phenomenon by pointing out that space and time at this point were distorted by gravity. However, this argument confuses two unrelated frames of reference; from our frame of reference, the speed was faster-than-light. If this would happen again before our eyes and we could somehow observe it, then, supposing that it wouldn't destroy the entire world, we would observe a speed that is undeniably superluminal.

2) Gain-assisted superluminality:
Researchers have achieved faster-than-light communication by sending light pulses through a supercooled gas of exotic cesium atoms. The light pulse travelled so fast that it had actually exited the gas chamber before it had finished entering.
They claim that this leaves Special Relativity intact; the light pulse did not actually travel faster than light, they state, because the light pulse on the other side of the gas chamber was actually a reconstruction of the entering pulse, so that is not actually the same pulse. This interpretation does a poor job hiding the fact that either how, information was nonetheless passed through to the other side of the gas chamber faster than light. This contradicts Special Relativity, which says that under no circumstances information could travel faster than light; it may be that Wang and his colleagues were afraid to openly contradict Special Relativity, for fear of criticism or disregard of their research.

3) EPR paradox:
Observing either of a pair of entangled particles will instantly affect the other particle, no matter how far it is — a fact which has been experimentally verified. No matter how one interprets this, the information that either particle has been observed travels to the other particle instantly, or at least (and perhaps more likely) at a speed faster than we have as yet been able to measure.
It is claimed that this does not allow faster-than-light communication, yet regardless it has been proposed to use this phenomenon in quantum cryptography: by using entangled particles to transmit information, eavesdropping would be instantly detected since it would affect the other particle of the pair. In other words, the information of an instance of eavesdropping would instantly travel to the other particle; this is certainly communication.
The event of the observation of either of the particles has an immediate effect on the other. Einstein mockingly called this "Spokhafte Fernwirkung," and in the formulation of the EPR paradox claimed that this meant that quantum mechanics is incomplete. However, since this "spooky action at a distance" has been confirmed as factual, it would appear that it is rather Special Relativity which is incomplete.

4) Speed of gravity:
Although Einstein dismissed "spooky action at a distance" as impossible, the mainstream interpretation of General Relativity today itself uses "spooky action at a distance" to avoid faster-than-light speed. Tom van Flandern has calculated, based on observation of planets and binary quasars, that the speed of gravity must propagate at no less than 20 billion times the speed of light to accord with their angular momentum.
To avoid this, mainstream physicists interpret gravity as the curvature of space-time rather than an actual force of nature, like electromagnetism. While representing gravity in this way may render the nature of gravity more obscure, however, it does nothing to change the fact that, be it through space-time curvature or through an actual force, gravity propagates at a certain speed. If gravity is represented as space-time curvature, the fact remains, obviously, that mass has an effect on space-time curvature; this effect cannot be random, and therefore requires a signal from the mass that causes it: this signal must be faster than light.
The effect of mass on gravity is such that it can affect the other side of the observable universe in just two seconds. In other words, information is passed at faster-than-light speed, the information of gravity; in the space-time curvature representation, this is the information space-time needs to know just in what way it should curve in accordance to the mass that causes it to do so.
The only way to deny that this is a faster-than-light effect is by detaching the effect from its cause, in which cause one has to give up the entire idea of gravity. In gravity, cause and effect are related at a speed that is faster-than-light, and only through sophistry can one deny this, unless observations are somehow wrong.

5) Virtual particles
Similarly to gravity, electromagnetism appears to propagate at faster-than-light speed through virtual photons. The nature of virtual particles is still unknown, but they are thought by mainstream physicists to be a manifestation of Heisenberg's uncertainty principle. Again, most physicists dismiss the faster-than-light nature of virtual particles because they are virtual; that is, they exist only for a very short time. Outside of their interaction, they do not exist. However, this is irrelevant, as their interaction itself is superluminal; therefore, their interaction might also be used to allow superluminal communication.

6) Opposite or closing speeds:
Special Relativity states that regardless of the frame of reference, nothing can go faster than light. Actually, one doesn't need to think far at all to see that this is plainly impossible: when two photons move on a straight line towards or away from each other, than from our frame of reference they do so twice as fast as the "speed of light." When you tell this to a mainstream physicist, he will say that from the frame of reference of either of the photons themselves, the photons will not move faster than light, thereby changing your question so as to best suit an answer which accords to Special Relativity; the frame of reference of the photons is not very relevant, however. Since all motion is relative and all things in the universe are in motion, it is easy to travel towards something at faster-than-light speed from the frame of reference of the Earth. That is to say, when you have reached your destination which was 200 light years far, it might be that less than 200 light years have passed in the universe.
This is probably the least relevant of all faster-than-light phenomena, since it does little to bring us closer to faster-than-light travel or communication, but it is a faster-than-light phenomenon nonetheless. One can hardly move a star towards a spaceship: while it may be possible to move a star by means of a stellar engine (a type-2 Dyson sphere), it would not be very worthwhile to use this simply to accelerate space traffic.
The least that this means is that we should reformulate the Special Relativity: "Nothing can travel faster towards or away from an object than light would travel towards or away from it." At least, that is how it tends to be.

7) Current cosmic inflation:
The observable universe is 93 billion light years or 28 billion parsecs across in diameter. Every second, the universe expands by 2 trillion kilometers in diameter every second, or 20 million times the speed of light. The matter at one end of the universe moves away from the matter at the other end with the same speed.
Mainstream physicists argue to this that it is not the matter in the universe which is expanding, but rather its space. But be it through the three dimensions we know or through some esoteric "fourth dimension," this expansion of space is in itself a kind of movement, albeit the movement of space. This is, again, nothing but another interpretation of the same thing; but an interpretation which is so abstruse that is hard to find arguments against it. Again, however, matter is moved faster than the speed of light; that is to say, the distance between them grows at a speed faster than light could cover it.
Physicists keep finding new ways of formulating "movement" to mask faster-than-light phenomena. All right, so let's call "movement" "the expansion of space between two objects." In that case, in accordance with this new formulation let me likewise reformulate my question! Is it possible to expand or shrink the space between two objects so that the objects move towards or away from each other faster than light would?
That the observable universe is able to increase the space between its ends faster than light does seem to give us hope that we might ourselves find ways to expand the space between masses faster than light could pass the same space.
Actually, there are plenty of physicists, including Feynman, Dirac, and, earlier mentioned, Tom van Flandern, who did not give credence to the theory that gravity is caused by the curvature of space-time, believing it to be a force of nature just like any other. Either how, it is quite clear that, in whatever way, the space between two objects can expand faster than the speed of light, and it is happening every second. How one interprets this changes little about the fact.

7) Quantum tunneling:
Perhaps the most significant FTL experiment aside from Wang's gain-assisted superluminality was conducted in Köln. Unlike Wang, Mintz was less timid about the results of his experiments, but like Wang's, Mintz' research has not gotten as much credit as it deserved.
Critics found it more difficult to find arguments against Mintz' research, mostly because the experiment was so simple that it was hard to make it seem complicated: the setup of the experiment consisted of an amplifier, a 20 centimeter long tube, and Mozart's 40th symphony in the form of microwaves.
This experiment used quantum tunneling, which is manifested in the earlier mentioned virtual particles, in this case virtual photons. This proves that, despite claims of the opposite, virtual photons can effectively be used as a means of faster-than-light communication.
Quantum tunnelling is a phenomenon in which a particle can spontaneously pass a finite potential barrier in the form of a virtual photon, which is then reconverted into a standard particle. Mintz wave transducer made use of the Hartman effect, the effect that, if a barrier is thick enough, the tunneling time (the time it takes for a particle to get past the barrier through quantum tunneling) becomes independent of the thickness of the barrier and inclines towards a constant value.
The tube, which was called a wave transducer. The wave transducer, which was about 11 centimeters wide, was far too small for microwaves, which start at a wavelength of 30 centimeters, so that normally, they would net be able to get through. Virtual photons, however, could get past the wave transducer through quantum tunneling. On the other side, the tunneled photons went through an amplifier, which then played Mozart; not at very high quality, but enough to be recognizable as Mozart's 40th symphony. In this way, the symphony had been transmitted at 4,7 times the speed of light.

It is often argued that while the group velocity (the speed of the whole of the wave, which may change in dimensions) may exceed the speed of light, the front velocity (the speed of the front of the wave) always remains the same. There are, indeed, phenomena in which some kinds of wave velocities (be it phase velocity, group velocity, energy velocity or signal velocity) are superluminal yet the front velocity remains unchanged, such as negative refractions and atomic coherence effects, but when the wave is observed to have arrived at its destination before light in vacuum would normally have done so, surely this argument is no longer satisfactory. If the wave as a whole has reached at superluminal speed, then obviously so has its front, in violation with Special Relativity.

There have been so many observed FTL phenomena, and many more will follow in future, that we can no longer ignore them. Sooner or later, modern physics will be forced to review its principles, at least insofar as to nuance them. Again, natural laws tend to have exceptions.