"...some physicists have presented arguments to suggest that a theory of quantum gravity which merged the two theories would eliminate those solutions in general relativity which allow for backwards time travel (see the chronology protection conjecture), of which the Alcubierre drive is one" [emphasis added]
The thought experiment: imagine a billiard table with a wormhole which curves around and goes three seconds into the past. Roll a billiard ball into the wormhole such that it will emerge three seconds earlier and knock itself off the path, so it doesn't enter the wormhole.
Instead, the ball emerges along a slightly different path, striking itself merely a glancing blow that allows it to enter the wormhole.
And why did it emerge with an altered path? Because it was struck a glancing blow.
You do aim it so it hits dead on. Then it emerges at some different angle which knocks itself off that dead-on path that you so carefully aimed. Now it's traveling an altered path that makes it emerge at that different angle.
Why does it do that? I get why the principle demands that it does and I get why it's a stable solution once it's been made to happen, but how did it actually make the jump from "straight on" to "at an angle"? Presumably not every self-consistent causal loop materializes itself out of thin air just because it would work if it did.
Edit: possibly lame example, but in Terminator 2, we find out that terminators were a secret government project that went awry. Where did they get the technology? Well, a terminator from the future came back and died in Terminator 1 and the parts it left behind allowed engineers to get a head start on building one. So there's a similarly self-consistent causal loop. But it doesn't really answer the question of how there came to be terminators.
I can't say I fully understand it. One thing I've seen stressed is that there's only one timeline. There's never a time when the ball continues straight through into the wormhole, only to somehow emerge with an altered path. It only travels that path to the wormhole once, and when it does, it gets hit a glancing blow.
The wiki entry I linked is intriguing. It says in some cases there are an infinite number of consistent solutions. They found they could analyze those cases with quantum mechanics, doing a sum-over-histories that only includes the consistent solutions. That gave a probability for each particular outcome.
So maybe that's fundamentally how causation actually works in the universe, and in the absence of time loops it reduces to normal causality. Now I can't help speculate whether actual quantum mechanics involves time loops somehow. (And in fact, Cramer's transactional interpretation does involve waves travelling backwards in time.)
Another way to look at it, maybe, and take with a grain of salt because I came up with it myself: Imagine there isn't just one timeline. In the paradox the universe runs an infinite loop, the ball first going into the wormhole, then not, then it does, then it doesn't. But given quantum mechanics, the position and momentum of the ball is uncertain. Given a very large number of trials, it's eventually going to shift significantly. At some point, it hits on a consistent solution, and the looping finally stops.
I can only imagine this kind of thing happening in a reversible, constant entropy system and a system without intentional processes.
For example, imagine a programmable drone. It is programmed to enter the singularity if and only if nothing seems to emerge and to then avoid collision with anything once it emerges - and transmit a signal announcing it's emergence to boot. Maybe you could do setup a fancy scenario for not winding up with two (or no) drones at the "end". But things not longer seem very plausible.
And you could probably arrange for a thing to carry entropy backwards if you had a varying entropy a system. But that's left as an "exercise for the reader".
(wikipedia)