In one of the two temporal anomalies that are called Grandfather Paradox, a time traveler impregnates his grandmother, becoming his father's father.  This is a subtype of the Predistination Paradox, but itself has three forms, represented by the secret of transparent aluminum in Star Trek IV:  The Voyage Home, the pocketwatch in Somewhere in Time, and the main character in Time Rider.  This does not happen under parallel or divergent dimension theories, because the traveler is not in his own universe.

In Star Trek, Mr. Scott trades a metallurgical formula for a sheet of plexiglass.  In a passing explanation, Scotty says, "How do we know he didn't invent the thing?"  However, this means that in this timeline no one discovered transparent aluminum; the knowledge was passed from future to past, and it simply was.  Dr Nichols is credited for it, but was given the formula.

Under fixed time, this is a logical possibility, that it was neither invented nor discovered but the knowledge passed from future to past.  The process of discovery is not necessary.  This is the uncaused cause for which this theory is criticized, a simple predestination paradox.

Under replacment theory, Dr. Nichols will be the one history credits, the original discovery being erased from time.  In the original history, John Doe discovered this, and Scott learned the process years later; he introduces Nichols to the theory, and Nichols beats Doe to the patent office.  This will impact things in the past, including the distribution of wealth, but as long as Scott is born, becomes an engineer, and travels to the past, he will have the knowledge to give to Nichols, and so the resolved final history of the N-jump has Scott getting the information from Nichols and returning it to him.

The watch is more difficult.  We first see elderly Elsie McKenna give it to young Richard Collier, who takes it to the past and leaves it with young McKenna, who later gives it to Collier when she is old.  Again we have an object with no origin in time.  Again fixed time theory claims this is acceptable, and replacement theory puts the origin of the watch in the lost original history.  But unlike the information passed from Scott to Nichols, the watch ages.  When McKenna hands it to Collier, she has owned it for sixty years; he then owns it for a decade before giving it back to her.  When she got it, then, it was at least seventy years old--but when he got it, it must have been 130, and thus 140 when he gave it to her, increasing seventy years with each pass.

Fixed time expositions ignore this aging factor; the watch does not age, because if it is there it must be there.  Replacement theory, though, recognizes the deterioration of the watch as fatal to time.  Eventually the watch will decay to dust, McKenna will have nothing to give Collier, and time must revert to the original history for a new version of the watch to be introduced, which McKenna gives to Collier, restarting the cycle.

The most complicated example appears in Time Rider.  Lyle Swan meets a woman in the past whom he impregnates, and she gives birth to his son who is his father.  Lyle is his own grandfather.  The deterioration of looping objects takes a distinct form in this case; it can best be seen in the replacement theory resolution.  There must have been an original Stranger who met and impregnated Grandmother, fathering Father.  Father's DNA is 50% Stranger and 50% Grandmother.  He marries Mother, siring to Lyle, 50% Mother, 50% Father--which is 25% Stranger, 25% Grandmother.  Lyle then disrupts the meeting between Grandmother and Stranger, fathering Father.  Father's DNA is now 62.5% Grandmother, 25% Mother, and 12.5% Stranger.  Lyle's becomes 62.5% Mother, 31.25% Grandmother, and only 6.25% Stranger.  But Stranger is the only contributer of a Y chromosome here, so as his percentage falls to 3.125% of Father and 1.5625% of Lyle, and then <1% of Father and <0.5% of Lyle, eventually one of them will become a woman, and the chain will break.  Stranger returns, and the cycle restarts, a cycling causality, a sawtooth snap with an infinity loop termination.

Fixed time glosses the genetic problem; if Lyle is his own ancestor, it happened.  Theorists who disallow predestination paradoxes avoid this problem, on the same basis as they do other paradoxes:  it did not happen so it will not happen.

Next time we will look at the other grandfather paradox.