I am a fan of science fiction involving time travel. My favorite TV show is Doctor Who. However, as a physicist time travel does not appear to be possible. It violates causality, and conservation of mass/energy. Paradoxes are a way to convince yourself of how paradoxical time travel would be.
The Grandfather Paradox is often quoted as a reason why time travel is complicated. It comes in many forms, but the one I have seen most often and was used in Futurama is as follows. You travel back in time to a time before your grandfather conceived any children and kill him. Then how can your parent be born? Futurama resolved this by having the main character, Fry, be his own grandfather after accidentally killing his grandfather.
There are many ways to think of this paradox, but paradoxes with even simpler formulation can be made. If you follow the above link there is a reference to a paradox created by physicists. You have a computer that can send signals back in time to itself. It sends a signal to shut itself off before the signal is sent. Think about that one for a while.
I have created a version of a time travel paradox that I find even simpler. It is similar to the pool table paradox developed by Joseph Polchinski but simpler. You have a time travel machine in the form of a box that has a hole in one end. You set the box to send anything that enters the hole back in time by ½ second with the sign of the velocity reversed. If you can send things back in time reversing the velocity is trivial.
Now place this box on a pool table with the hole facing you. Then take the cue ball and roll it into the hole. What is the result? The cue ball cannot enter the box because it would come out and stop itself. Are there two identical cue balls left on the table? Are there no cue balls left?
The fundamental problem with time travel is that mass/energy is not conserved. The appearance of an object earlier in time violates any kind of mass/energy conservation. Technically, General Relativity is the most general theory and it conserves the 4D divergence of the 4×4 energy-momentum tensor. This divergence should be zero. The sudden appearance of something out of nothing violates this more general version of mass/energy conservation.
Finally, General Relativity allows for worm holes that can connect different points in space-time. However, making such worm holes stable (aka traversible) requires large amounts of negative energy which does not appear to exist in nature. It is interesting to note that theoretical warp drives also require this negative energy. In quantum field theory particles with negative energy exist in the form of virtual particles, but they always occur in conjunction with a positive energy particle and only for a brief time. We know of no way to isolate the negative energy needed to keep a worm hole open.
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