The Novicov Consistency Principal relies on the Kakutani fixed-point theorem, which relies on convexness, which implies connectedness, which implies an infinite number of realities.
I admit I’ve never seen a proof of the Novikov Consistency Principle. I know that’s how I’d prove something like that, and I know I could easily come up with a case of time travel with no possible stable time loop of I’m given a discrete space, a non-compact space, etc.
Sorry, I cannot find a link, but feel free to ask what does not make sense. As for the stable time loop, I’m not sure that it is always possible to find, given that you apparently want to fix both initial and final conditions of a hyperbolic PDE, unless I misunderstand what is involved in constructing such a loop.
But if you had done such arduous research as, say, reading the Wikipedia page for the NCP, you would see that a sum over histories using only non-paradoxical timelines apparently works. Not that I really understand that in more than a superficial way, but it sure as hell sounds like an answer to your point.
There are only guaranteed to be non-paradoxical timelines if you have an infinite number of realities, which is what I was saying from the beginning.
You could look for all the timelines that are within delta of being a paradox. I think the shadowing theorem guarantees that, for small enough delta, this is epsilon-close to a non-paradoxical history. I don’t think it tells you what delta is, and I don’t think it’s guaranteed that every non-paradoxical history will be shadowed. This would mean that you’re not randomly picking the choice of history. More importantly, it might be that none of the non-paradoxical histories are shadowed, and you’ll have no idea what to look at.
Do random samples until you find enough, then. It wouldn’t be perfect, but it should be close enough that you wouldn’t notice with enough computing power, right?
The Novicov Consistency Principal relies on the Kakutani fixed-point theorem, which relies on convexness, which implies connectedness, which implies an infinite number of realities.
It does not.
I admit I’ve never seen a proof of the Novikov Consistency Principle. I know that’s how I’d prove something like that, and I know I could easily come up with a case of time travel with no possible stable time loop of I’m given a discrete space, a non-compact space, etc.
What does it rely on?
Solely on the uniqueness of the metric in GR.
Do you have a link to a better explanation?
Also, can you explain how this could be used to find a stable time loop?
Sorry, I cannot find a link, but feel free to ask what does not make sense. As for the stable time loop, I’m not sure that it is always possible to find, given that you apparently want to fix both initial and final conditions of a hyperbolic PDE, unless I misunderstand what is involved in constructing such a loop.
Couldn’t it be approximated?
The theorem doesn’t tell you how to find a fixed point. It only tells you that one exists.
But if you had done such arduous research as, say, reading the Wikipedia page for the NCP, you would see that a sum over histories using only non-paradoxical timelines apparently works. Not that I really understand that in more than a superficial way, but it sure as hell sounds like an answer to your point.
There are only guaranteed to be non-paradoxical timelines if you have an infinite number of realities, which is what I was saying from the beginning.
You could look for all the timelines that are within delta of being a paradox. I think the shadowing theorem guarantees that, for small enough delta, this is epsilon-close to a non-paradoxical history. I don’t think it tells you what delta is, and I don’t think it’s guaranteed that every non-paradoxical history will be shadowed. This would mean that you’re not randomly picking the choice of history. More importantly, it might be that none of the non-paradoxical histories are shadowed, and you’ll have no idea what to look at.
Do random samples until you find enough, then. It wouldn’t be perfect, but it should be close enough that you wouldn’t notice with enough computing power, right?