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The short answer is that it is valid, but that misses a lot of subtleties. Using three-vectors for velocity in special relativity is fairly unnatural even for ordinary speeds, and it's worse for tachyons, since their three-velocity can be "infinite" and, in a certain sense, "time reversed" (see below), and those situations can't be ...

2

An important caveat: FTL implies time travel only if the FTL mechanism obeys the principle of relativity (that is if there is no "absolute" speed, only relative speed). All known physical laws obey this principle, and it dates back to Galileo, so this is a reasonable assumption. For simplicity let's consider an arbitrary FTL communication mechanism,...

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The refractive index (and hence the optical density) of a medium is a function of its permittivity and permeability which in turn are functions of temperature. This may be the reason of variation of refractive index with temperature

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There is no need to devise new experiments to disprove your formula- the formula is clearly already incompatible with the existing experimental evidence for special relativity. In any case, your understanding of the experimental requirements is wrong. You do not have to accelerate a human to a speed approaching c in order to test relativistic effects- you ...

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