Cosmology Cafe

Recently, Hugh Everett’s thesis of an infinite number of parallel worlds exfoliating as it were from the present has emerged as a response to the so called ‘grandfather paradox’ inherent in the classical conception of time travel. (GP: going back to the past and disrupting the sequence of events leading to one’s birth leads to paradox). Leaving aside the tiresome example of killing grandfathers, consider what happens if we go ‘back’ and change something in the past in the infinite multiverse… Instead of changing the course of events in one’s his own world-line, our traveler launches, or more precisely most likely, merges with a new world-line narrative, which will lead to a new future; In the narrative which includes the traveler’s future from which he came, our traveler would simply have left the scene and not returned.
Now if these timelines are independent of each other, killing his grandfather would still have the effect of preventing his existence in the world-line in which he kills his grandfather, which is where he is. So really, if the world-lines are independent in this way, the paradox continues.
The question boils down to whether having launched or joined a new world-line through interference with an existing word-line in the past, is our traveler then subject to the new world-line he has joined, or to the one he left, or in some hybrid state of them both?
One might wonder if one were then for a second time to jump into the future, would one be limited to the future of the new world-line? Could he not return to his ‘own’ (previous) future?
Likewise traveling back in time, presumably one can only access the past of one’s own world-line. Or?
And traveling into the future, is there something that keeps our traveler in a particular narrative line or would he be tossing himself into an infinity of possible futures exfoliated in the intervening time between the jump-off point our traveler left from and his destination in time?
Note: no matter how many infinite world-lines one is no longer part of, there are always innumerable future world-lines all meeting at, or emerging from, the present moment.