Einstein originally thought that special relativity was about light and how it always travelled at the same speed. Nowadays, we think that special relativity is about the idea that there is some universal speed limit on the transfer of information (and experiments tell us that photons, the quanta of light, move with the largest speed, $c$).
But what if tomorrow we happen to observe a particle $X$ that travels with a speed $v>c$? What changes would have to be made to special relativity?
Answer
If (and that's a big if) tomorrow we had a $70\sigma$ detection in a repeatable experiment of a particle that travelled faster than $c$, then one of several things would be true.
1) We would be forced to conclude that $c$ is not, in fact, the limiting speed of information transfer; everything based on this assumption would have to be scrapped (pretty much all of research-level physics); and we would have to start over in developing even the mathematics that allows us to start re-describing the universe.
2) We would be forced to conclude that $c$ is not the limiting speed of information transfer; we would assume that special relativity and everything based on it is the special-case effective theory of much broader physical laws and behaviours; and we would have to find a way of modifying relativity (and basically everything that relies on it) so that it can causally allow for this particle to exist and yet have everything else we see still basically operate under the idea that $c$ is the max speed.
3) We find a way to use this particle to communicate with the past and future, travel faster than light, and then we go home every night and laugh at Einstein.
4) We perform the experiment thousands of times in different laboratories, find the same result, then go back and discover that there was a fundamental flaw with the theory. Once the flaw is corrected, we see that we are not actually observing a superluminal particle.
5) We also discover flying pigs, perpetual motion, and that we really can believe it's not butter. Then I wake up from my nightmare.
My money is on (4) with (2) being a close second (although (5) has happened before).
Note: This answer assumes that the speed $c$ referenced is the assumed maximum speed; the speed of a massless particle in a vacuum. This is why I did not include a "we determine the photon is not massless and then have to change EM" option.
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