A Consistent, Simple Model of Time Travel

The basic idea is that timelines do not “already” exist. They start at the creation of the universe, instantaneously (because there’s no outside time) grow randomly, based on quantum mechanics, and new branches can be considered to exist “after” the original. When time travel occurs, two things happen:
1. (this happens first) If travel is into the past, a new branch of the timeline is created.
2. The rest of the normal timeline is created without the traveler in it. If it’s in the future, the traveler arrives at the target time.

Here’s an example: Travel into the past, then future:

1. First, we start with a nice, clean timeline.

The green line represents the traveler in his original time. The blue blog is the time travel event.

2. The original timeline continues without our traveler. The past splits, with a new branch having a reddish line representing the traveler who just transported. The green line is still the original, younger traveler. The traveler decides to return home (red)

3. The traveler is unable to escape the new branch, which is subtly (or not-so-subtly) changed by his arrival. The timeline is “projected” ahead and randomly determined. He arrives, only to find that:

a. His arrival didn’t change much; maybe he wasn’t noticed. However, the “present” is still vastly different because it was determined differently than the original. This is not the Butterfly Effect, this is just a new outcome to a new set of quantum coin flips.

b. He changed something; maybe he shot the school bully. The present is changed in the way he intended, or there’s an unintended effect, but at any rate there are also unforeseen changes due to the new timeline.

In both scenarios, the green line continues; his “original” self, who never traveled, is in this universe. He may or may not be in a similar job or lifestyle to the first timeline. The purple line, the new traveler, has no traveler-shaped hole to fill. Meanwhile, he is missing from the first universe.

Let’s apply this to, say, Terminator:
Original line, starting just before the first movie takes place:
1. Cyberdyne, or someone else, creates Skynet without Terminator bits. Judgment Day ensues. Sarah Connor has John, but his father isn’t Whatshisface, who hasn’t traveled “yet”.
2. There aren’t any Terminatobits around, so the Terminator isn’t created. Sarah Connor has John, or doesn’t. Remember, determinism has been basically disproven. Everyone is happy.

1.1. The Terminator is sent back in time, and kills Sarah Connor. John is never born in the new branch. Whatshisface is never sent back in time in any branch. In a new time branch, there is no resistance and humanity is completely screwed. In the old branch, Skynet is short one Terminator and still has to deal with John Connor.
John sends back Whatshisface out of confusion over timelines and creates a third line, branching off the first at a slightly later date, in which W.H.F. arrives to find Sarah perfectly safe and dies of confusion. Meanwhile, there’s still a “humanity is screwed” branch.

1.2. The Terminator is sent back in time, and fails. John is maybe born. In the new branch, Skynet is created, or not. Because of quantum uncertainty, we don’t know what happens, but the starting conditions for the new timeline are very close to those for the original branch.
John sends back WHF, who arrives to find Sarah safe. We now have three branches, just like 1.1, but in the second there is John Connor, possibly, to slow down Skynet.

Here are images, assuming that John’s birth is inevitable given Sarah’s survival (which isn’t neccesarily true):

Some notes:

-the green lines in the parallel branches are NOT the original John. They are assumed analogues, born from the same parents under slightly different situations, with genes that are different because of quantum randomness.

-I didn’t change colors here, instead assuming you can sort it out.

-In every timeline, Skynet exists. In those with Connor at the end, Skynet may try to assassinate his mother. This could have the following result:

At the first point, a bunch of timelines split off in which the Terminator kills Sarah Connor and Skynet is unstoppable. These are marked in red at the end.

At the second, a bunch split off in which Skynet sends back the Terminator to the first point, and John sends back WHF to the second. This has no apparent result. These are circled in green.

This absurd result is my prediction of how the Terminator scenario would play out, if
a. John’s birth is an inevitable result of Sarah’s survival
b. Skynet and John will both send back people to change history if given a reason
c. The Terminator will succeed in killing Sarah
d. Some other predictions.