A paradox in a theory appears to contradict the theory itself but might be resolved if looked at carefully. Physicists in the early days would take the occurrence of a paradox as a chance to find something new, as Neils Bohr famously said, "How wonderful that we have met with a paradox. Now we have some hope of making progress." These days physicist would go to any limit in trying to resolve these paradoxes in theories in a way that always shows that the theory is perfect and no need to be questioned. The paradoxes in special relativity are a classical example of this [1].
A well known paradox in special relativity is the twin paradox. I once investigated the twin paradox because I was not able to wrap my head around the reason for the asymmetric aging of the two twins. One of the main postulates of the theory of relativity is that frames of references are equivalent so if one of the twins leaves the planet and comes back they both should have aged by the same amount because they are moving relative to each other. The equations of relativity do not predict any asymmetry between two references moving relative to each other.
Now lets see how physicists resolve this paradox. One explanation is that since the twin who leaves the planet accelerates, especially when he turns back, the postulates of relativity do not hold and the frames are not equivalent. Furthermore, you have to take general relativity into account for the part of the journey where he turns back. There are two reasons I do not find this explanation satisfying. One is that if equivalence of frames in relativity breaks because of acceleration, no two frames of references should be equivalent since every moving frame starts from zero velocity and accelerates to attain a non-zero velocity. The second reason is that the two references were equivalent for at least half of the journey so there should be some implications of this when the twin returns. I have even seen a paper which says that it is actually the twin on the planet, and not the one travelling, who is going to be younger because he stays in an inertial frame of reference.
Another paradox is the train and tunnel paradox. Lets say that a train is moving towards a tunnel which is a bit smaller than the train. An observer is sitting on the tunnel and wants to trap the train. If the train is moving close to the speed of light it is length contracted for the stationary observer and much smaller than the tunnel. Therefore, as soon as the train disappears in the tunnel the observer will close the gates of the tunnel and trap the train. From the trains perspective, however, the tunnel is much smaller so it is impossible to trap the train in it. Even if the observer closes the door it might chop the train into pieces but its impossible to trap the train. Both of these cases have very different outcomes. In the first case the train gets trapped, when it slows down becomes longer and breaks within the tunnel because the gates are closed (assuming that the gates are sturdy enough). In the second case the train splits into pieces and never gets trapped. This paradox is typically resolved using the relativity of simultaneity. Again, I do not find these explanations completely satisfying because the fact that the outcomes of the two observers are completely different implies that this paradox cannot be completely resolved.
Look, dont get me wrong, I am not saying that relativity is a wrong theory, no way. There is a lot of experimental evidence that agrees with the predictions of relativity, such as the lifetime of the muons. The point that I am trying to make here is that we should sometimes allow for explanations of these paradoxes that indicate that there might be loop holes in the theory. Physicists never question the theory of relativity because it is considered too sacred by the physics community and anyone who questions these ideas is considered a crackpot. This attitude of physicists feels more like religion than science where they would not even open the door to question theories unless experimental evidence is found. If experimental evidence is found, they loose their minds and would do any thing to explain it (like what happened in the case of ultra-relativistic neutrinos from OPERA). We should have an atmosphere where people, especially young physicists, should be able to question these theories and not fear being ridiculed.
[1] Here is a video that describes paradoxes in relativity. https://www.youtube.com/watch?v=kGsbBw1I0Rg
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