The grandfather paradox has been used to argue that backwards time travel must be impossible. However, other resolutions have also been advanced.
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2 Relative timelines resolution
3 Restricted action resolution
Parallel universes resolution
There could be "parallel universes" (see Everett many-worlds interpretation) and when you travel back in time and kill your grandfather, you do so in a parallel universe in which you will never be conceived as a result. However, your existence is not erased from your original universe. Alfred Bester's short story "The Men Who Murdered Mohammed" uses this premise. It is also used in James P. Hogan's novel "Thrice Upon a Time".
Relative timelines resolution
It could be that the universe does not have an absolute timeline. Each particle has its own timeline. This is similar to the theory of relativity, except that it deals with a particle's history, rather than its velocity.
Physical forces affect physical particles. If your body's physical particles go back in time, you will be able to kill your grandfather (no physical forces will mystically stop you) and nothing will physically happen to you as a result because there are no physical forces that can "figure out" what happened.
Your younger self does not need to be born in order to fulfill a destiny of going back in time because there is no written-in-stone absolute timeline that needs to be followed.
If you were able to find and observe the younger versions of the particles that make you up, they too would follow physical laws and hence wouldn't form into a younger version of you (because one of your parents wouldn't be there to form you).
This theory is similar to the parallel universes theory, except that it happens within one universe.
Restricted action resolution
Another resolution holds that, if one was to travel back in time, the laws of nature would simply forbid them from doing anything that could later result in their time travel not occurring. This theory might lead to concerns about the existence of free will (in this model, free will may be an illusion). This theory also assumes that causality must be constant: i.e. that nothing can occur in the absence of cause, whereas some theories hold that an event may remain constant even if its initial cause was subsequently eliminated. It is also possible that the time travellers intended action might be completed, but never successfully enough to result in cancellation - see Novikov self-consistency principle.