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Inspired by this question about learning from FTL, I wonder
Is it even possible for anything to happen, that, if observed by us using methods available to us, will unmistakably and absolutely certainly be the evidence that the object experiences FTL? If yes, what would it be?
I.e. some event (chain of events) that if observed by us will have no other explanation but FTL?
A spaceship arrives at Earth. The crew insists that they have FTL travel, but we don't believe them, so they propose an experiment. We (Earth) will build a "probe" that will broadcast some signal known to us but unknown to them. They will then take the probe and drop it off at some distance where we can a) receive the signal and b) determine how far distant the probe is. Then they drop the probe off and come back. When we receive the signal, we can calculate that the probe must have been moved faster than light.
(Variations on this are possible. My original idea was for them to "predict" some stellar event whose light hasn't reached us yet, but then maybe we'd only believe that they could cause such an event. Granted, if the event is something like a supernova, that would scare the <redacted> out of us...)
Of course, this requires that the folks with FTL are actively interacting with us to demonstrate their capability, but the question didn't exclude that...
As the ship decelerates to a stop, it will appear to be in multiple places at once. First the light from the stopped ship (and any time it was moving slower than light) will reach us. Then after that, the light from its FTL travel will catch up to us, so the ship will appear in two places at once. From there, the "ghost" of the ship would travel backwards as the farther light reaches us and the previous light fades.
In short, the effects would be very unmistakable, not to mention the blueshift we would see (though that could be caused by sub-light speeds too).
Edit: Granted, proving that this is FTL would be tricky. However, based on analysis of the event (let's say it was all recorded in good quality) we could say with probably 95% certainty that it was indeed FTL, especially if we then see the ship take off again.
The short answer is no there is nothing that unmistakably and with absolutely certainly would be evidence that a material object experiences FLT travel through space as we understand things today. This is because accelerating an object increases its mass and in the limit as light speed is approached the mass increases exponentially whilst the velocity tends towards c. So a material reference frame cannot accelerate across the light barrier.
There could be some very strange and unexplained events that puzzled people. Some might come up with theories that suggested these events were evidence of FTL travel. Such theories would be argued over and (hopefully) subject to scientific testing at which point either Einstein is proved right or the new theory holds up and our understanding of physics changes.
We can’t sensibly speculate on what happens if something beyond known physics happens and expect any kind of unmistakable certainty because we do not know how any such new physics would work. Extrapolating existing theories may not provide the correct answer and lots of assumptions would have to be made about the causality, and what we can expect under what conditions. Perhaps matter becomes invisible, perhaps it allows entry into hyperspace, perhaps matter is converted into anti matter who knows? Maybe our understanding of causality is flawed. The scope for new physics is very large.
It depends on the nature of the FTL. For a best-case scenario, assume the ship has a Star Trek style use-anywhere in any direction FTL drive. To prove their case, you tell them you them to fly to some distant probe, get in front the probe's camera, display lights in a random pattern you give them, and come back to Earth.
Say the probe was one light-hour from Earth. If the image shows up in less than two hours after the ship leaves orbit, you know some form of FTL is needed.
There seems to be a misconception that we do not have direct experimental evidence that FTL travel is possible. We do, and it is not in any dispute. What we don't know is how to reproduce the effect artificially and, most importantly, we have no idea how to mitigate the serious side-effects.
The universe is 14 billion years old or thereabouts, and the observable universe is 93 billion light-years across or so. Since the newly born universe is known to have been very small, this directly implies that there must exist a means of travelling at least 47 billion light years in 14 billion years, with an average apparent speed not less than 3.3 times faster than light.
Astronomers can see distant objects which are at present much farther away than the distance light would have been able to travel since the time they were formed. For example, there is a galaxy far, far away, known as GN-z11, which is 13.4 billion years old and yet it is 32 billion light-years away. It directly follows that in the distant past GN-z11 was only 13.4 billion light-years away and in the mean time its distance from us has increased by some 18.6 billion light years, for an average apparent speed 1.5 times faster than light.
The end result is that we know that FTL travel is possible; the example of the galaxy GN-z11 shows that we can even point to specific objects which must have travelled faster than light to be where they are.
What our physics says is not that FTL travel as such is impossible, but only that any means of FTL travel must use some method of doing something with the space itself between the origin and the destination. That is, what our physics says is that nothing can traverse more that 300 thousand kilometers per second; but it does not say that to go from A to B one must necessarily cross every single meter of space between A and B, or that every single meter of that space must have its usual length. If one could do something to the space between A and B so that every meter to be traversed by the vehicle is somehow reduced in size to 10 centimeters, then the vehicle could very well span the distance with an average apparent speed 10 times faster than light.
