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It is well-known that associated with the Kerr solution which represents a rotating black hole, there can be a region of space-time where there are loops in space time (non simply connected paths which are navigable in principle). If this is so, it breaks causality and permits time travel in these regions of space-time.

The question is whether this is really so. Can these types of region of space time even be created? It is difficult to picture how a region with linear time could join to one with cyclic time. Topologically, joining a line to a circle is clearly impossible without an anomalous point, but perhaps the other 3 dimensions of space-time make it possible. [It's not so easy to picture general 4-dimensional topology, even without adding a light cone structure. ]

If one imagines a path where time progresses normally but then crosses itself, but where space-time is otherwise perfectly normal locally, there is a problem with people (or fermions) navigating it. Suppose you plan to take this path and set off. When you reach the point where the path will cross itself, you are likely to meet yourself. And by meet yourself, this means be in exactly the same place you want to go. So suppose you try this, but see yourself coming back. Obviously the laws of physics say you have to change your path slightly to dodge around yourself, so you as a physical object are prevented from crossing the same point twice. [Physicists might say it was the fermions in you that refuse to occupy the same state]. But I see no problem with observing yourself (as I don't believe in causality as a dogma, just as something which is a fact about all the regions of space time we have empirical knowledge of). So my best guess is with regard to loops in space time, people may meet themselves from their future, but may simply interact with them physically as if they were a different physical person. Yes, they could tell you things about your future, and they would have been told those same things when they were in the same place as you were (because they are you). No, they can't kill you, because the probability that they would be where they are if they had done so is zero. And the same goes for your grandfather, if your family had settled in this region of space time a long time ago. To me it's a bit like saying what is the probability of a photon being vertically polarised if it has been measured to be horizontally polarised. Yes, being vertically polarised is a thing a photon can do, but not if you already know it isn't.

There is another superficial paradox that if you have a person or fermion in such a loop, it would expect to meet infinite numbers of copies of itself. But we know even the first copy can't take exactly the same path, so if you try to stay in this loop, you will have to make detours to avoid as many other copies of yourself as you come across. Eventually there will not be room for any more.

But now we have a problem. If the first time you arrive at the loop you see a vaste swarm of copies of yourself and decide it was a bad idea, and you should not enter the loop at all, then suddenly all those copies of yourself never entered the loop either, so why are they there? Perhaps the answer is that in the Kerr solution, you don't have a choice about entering this region - it is deep in a black hole after all. I really don't know. The breach of causality does complicate things, but perhaps no more so than solving differential equations on a non-simply connected manifold. But it is possible that such this factor might make it more difficult to extend the solution of a differential equation, which is related to what we do when inferring the future from the past.

What the truth is, and what would really happen, I am not sure. But what's for sure is that

The question is whether this is really so. Can these types of region of space time even be created? It is difficult to picture how a region with linear time could join to one with cyclic time. Topologically, joining a line to a circle is clearly impossible without an anomalous point, but perhaps the other 3 dimensions of space-time make it possible. [It's not so easy to picture general 4-dimensional topology, even without adding a light cone structure. ]

If one imagines a path where time progresses normally but then crosses itself, but where space-time is otherwise perfectly normal locally, there is a problem with people (or fermions) navigating it. Suppose you plan to take this path and set off. When you reach the point where the path will cross itself, you are likely to meet yourself. And by meet yourself, this means be in exactly the same place you want to go. So suppose you try this, but see yourself coming back. Obviously the laws of physics say you have to change your path slightly to dodge around yourself, so you as a physical object are prevented from crossing the same point twice. [Physicists might say it was the fermions in you that refuse to occupy the same state]. But I see no problem with observing yourself (as I don't believe in causality as a dogma, just as something which is a fact about all the regions of space time we have empirical knowledge of). So my best guess is with regard to loops in space time, people may meet themselves from their future, but may simply interact with them physically as if they were a different physical person. Yes, they could tell you things about your future, and they would have been told those same things when they were in the same place as you were (because they are you). No, they can't kill you, because the probability that they would be where they are if they had done so is zero. And the same goes for your grandfather, if your family had settled in this region of space time a long time ago. To me it's a bit like saying what is the probability of a photon being vertically polarised if it has been measured to be horizontally polarised. Yes, being vertically polarised is a thing a photon can do, but not if you already know it isn't.

There is another superficial paradox that if you have a person or fermion in such a loop, it would expect to meet infinite numbers of copies of itself. But we know even the first copy can't take exactly the same path, so if you try to stay in this loop, you will have to make detours to avoid as many other copies of yourself as you come across. Eventually there will not be room for any more.

But now we have a problem. If the first time you arrive at the loop you see a vaste swarm of copies of yourself and decide it was a bad idea, and you should not enter the loop at all, then suddenly all those copies of yourself never entered the loop either, so why are they there? Perhaps the answer is that in the Kerr solution, you don't have a choice about entering this region - it is deep in a black hole after all. I really don't know. The breach of causality does complicate things, but perhaps no more so than solving differential equations on a non-simply connected manifold. But it is possible that such this factor might make it more difficult to extend the solution of a differential equation, which is related to what we do when inferring the future from the past.

What the truth is, and what would really happen, I am not sure. But what's for sure is that

*something*happens if such regions exist. And if so it looks like causality does get broken in such a region. If so, we should not consider it a dogma and be very cautious about using it as a given to prove something else. For example, perhaps at very small scales, causality is broken even in ordinary space-time. The universe may be loopy and, if so, we have to accept it.
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