Paradoxes

Everything that can be invented has been invented.  -Charles H. Duell (1899)

Time travel, as presented in movies and books, results in a variety of paradoxes.   The most famous paradoxes surround the tendency of people to want to change things in the past.  Forward time travel offers some interesting implications of its own.  Just as today is affected by the events of yesterday, so must the future be affected by the events of today. The future has not been decided yet. Therefore, it becomes possible that merely attempting to time travel to future years has a significant effect on the way that the future will eventually be.

Below is a brief explanation of some of the better known paradoxes including the grandfather parardox, the free-lunch paradoxthe twins paradox, and another question.

Backward time travel 

    The Grandfather paradox

   The grandfather paradox, perhaps the most well known of time travel paradoxes, becomes a possibility with backward time travel.  If I decided to go back in time and kill my grandfather before he had a chance to conceive my mother or my father, then I would no longer exist.  But if I no longer existed, then I could not have killed my grandfather so he would be able to conceive my mother or my father.  This would lead to my existence, again allowing me to kill my grandfather.  Thus, a time loop is born in which I am perpetually killing my grandfather and being born; it becomes a circular problem.  This also manifests itself in other ways.  For example, in Back to the Future II, when Biff steals the Sports Almanac in 2015 and gives it to himself in 1955, he begins a variation on the grandfather paradox.  Biff has Doc Brown committed before he could invent the time machine.  That would make it impossible for old Biff to go back in time to give himself the Sports Almanac.  It creates another loop. 
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    The Free-Lunch paradox

The free-lunch paradox also arises from backward time travel.  It suggests that if I were to travel back in time and present Beethoven with all of the piano sonatas that make him famous today, and he, in turn, copied the sonatas from what I showed him, then the sonatas exist only because I went back in time and Beethoven never composed them in the first place.  This creates another loop because the sonatas only exist due to my decision to travel back in time.  This can be seen in the novel The Anubis Gates by Tim Powers.  Brendan Doyle memorizes "The Twelve Hours of the Night" in 1984, and he writes it down in a coffee shop in 1815.  In doing so he becomes the poet, but that can only be possible if he travels back in time.  So the poem's existence rests upon Doyle's ability to travel back in time.
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These paradoxes can be resolved in several ways.  One theory is that in traveling back in time, one might actually travel to the past of a parallel universe; so, for example, in one universe one is born and decides to go back in time and kill one’s grandfather, but in another one arrives back in time and actually does kill him preventing one’s birth.  Another theory provides that nature would contrive a way to prevent the traveler from changing the past: “What has happened has happened.  It cannot be changed[;]…it cannot be repeated twice in different ways” (Nova).  Time travelers could not, therefore, rewrite history.

Forward Time Travel

    The Twins Paradox

 Forward time travel could further lead to something known as the twin paradox.  The twins paradox is based upon Einstein's Theory of Relativity.  If one member of a pair of identical twins were to orbit the earth at close to light speed for a few hours, when he returned to earth, he might find that his twin had aged several years. Taking advantage of this theory leads to one of the theories about how time travel could work.  This paradox does not prevent forward time travel from being a very plausible notion.
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Another question

Another important question is raised if time travel to the past is possible—why aren’t we meeting time travelers from the future?  Some obvious solutions present themselves.  Visitors might be here without our knowing that they are present.  Backward time travel might be possible, but human civilization might destroy itself before it can ever be invented.  Another idea is that it may be impossible to travel backward to a point before time travel was invented. 
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