Faced with the impossibility of travelling faster than the speed of light, what's a science fiction writer to do? The easiest solution is to ignore the problem entirely. The world of science fiction is full of "warp drives," "hyperdrives," travel through "subspace," and their relatives -- the writer simply assumes that someone, sometime, will figure out a way to circumvent the cosmic speed limit.
Science fiction is full of such ideas: breezy violations of various laws of physics that just too darned inconvenient to obey. Probably the two most implausible ideas in all of science fiction are also the most popular: faster-than-light travel and time travel. These ideas have become such a part of the fabric of science fiction that they no longer even require justification when they appear in a story. The very first time travel novel, courtesy of H.G. Wells, includes a lengthy explanation of how the time machine works, but a modern time travel story just takes the time machine for granted. And a ship can simply jump into subspace or switch on its hyperdrive with the wave of a hand, and no further explanation is necessary.
But a few writers have taken the trouble to try to justify their violations of Einstein's cosmic speed limit. Perhaps the most elaborate of these is Catherine Asaro's exploration, in her Skolian Empire series, of the consequences of allowing speeds to be the sum of a real number and an imaginary number. (Imaginary numbers arise from taking the square root of a negative number). Asaro (who has a Ph.D. in physics) even wrote a scientific paper on the subject, which was published in the American Journal of Physics.
And on the science end, the most notable recent attempt to evade Einstein is probably the "Alcubierre Warp Drive," proposed by physicist Miguel Alcubierre in this paper. Alcubierre showed that it is possible in the theory of relativity to warp space around a spaceship, so the ship can move arbitrarily fast. The only problem is that this space warp requires a kind of exotic matter with negative energy, which might not exist. Oops. (I should not throw stones, since I have worked on similarly unusual ideas myself). So Alcubierre's warp drive is consigned to a gray area: it can't be conclusively ruled out by relativity, but there's also no experimental evidence that it's possible. Next time I'll talk about wormholes, which occupy this same twilight zone.