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.Scott

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Which wormhole was that? Which time machine?

These are things of Science Fiction.

These are things of Science Fiction.

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No, it doesn't. There is only one event of "ball coming out of the wormhole and hitting itself, making itself go into the wormhole". The "earlier" ball leaves this event and goes into the wormhole; the "later" ball leaves this event and goes off somewhere else.This sequence keeps going.

In short, the ball has only one worldline in spacetime: it just happens to have a (single) "loop" in it.

In principle, yes, if closed timelike curves can exist, then there are self-consistent solutions in which objects can hit themselves. The Novikov Self-Consistency Principle is worth looking at in this connection:Is this kind of stuff possible if we ever have a time machine?

https://en.wikipedia.org/wiki/Novikov_self-consistency_principle

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The wiki shows the billiard ball has a causal loop, why is this?No, it doesn't. There is only one event of "ball coming out of the wormhole and hitting itself, making itself go into the wormhole". The "earlier" ball leaves this event and goes into the wormhole; the "later" ball leaves this event and goes off somewhere else.

In short, the ball has only one worldline in spacetime: it just happens to have a (single) "loop" in it.

In principle, yes, if closed timelike curves can exist, then there are self-consistent solutions in which objects can hit themselves. The Novikov Self-Consistency Principle is worth looking at in this connection:

https://en.wikipedia.org/wiki/Novikov_self-consistency_principle

https://en.m.wikipedia.org/wiki/Causal_loop

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Because it went through a wormhole that functions as a time machine.The wiki shows the billiard ball has a causal loop, why is this?

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So its just one loop, then?Because it went through a wormhole that functions as a time machine.

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Yes. Go back and read my post #3.So its just one loop, then?

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Just curious, do you think time machines like wormholes could ever be actually built or used?Yes. Go back and read my post #3.

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Since they would require exotic matter, and since there is no known or foreseen way to be able to make exotic matter, I think it's very unlikely.do you think time machines like wormholes could ever be actually built or used?

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Thanks, just one last question, if causality isn't a fundamental law of physics, then how to we know for sure whether or not the universe is causal? Many say if a time machine could be built, causality is out the window, but how do we know for sure causality is true or not, right now?Since they would require exotic matter, and since there is no known or foreseen way to be able to make exotic matter, I think it's very unlikely.

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The term "causality" is vague. In order to answer your question, it would need to be made precise: how would we test to see whether "causality" is true?how do we know for sure causality is true or not

On at least one interpretation of "causality", the one used to classify spacetimes in General Relativity, it depends on the spacetime geometry; basically, "causality" holds in a particular geometry if that geometry has no closed timelike curves. On this interpretation, we test whether causality holds by testing whether there are closed timelike curves or not. All of our data so far shows no evidence of CTCs, so as far as we can tell, there are no CTCs in the portion of spacetime that we can see. But nobody has been able to prove that, starting with a spacetime region containing no CTCs, it is impossible for the geometry to the future of that region to contain CTCs (such a proof would mean that, for example, it is impossible to build a time machine in a universe that does not already contain one).

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Would the second law of thermodynamics work if time machines were found to exist (eg. CTCs) in our universe?The term "causality" is vague. In order to answer your question, it would need to be made precise: how would we test to see whether "causality" is true?

On at least one interpretation of "causality", the one used to classify spacetimes in General Relativity, it depends on the spacetime geometry; basically, "causality" holds in a particular geometry if that geometry has no closed timelike curves. On this interpretation, we test whether causality holds by testing whether there are closed timelike curves or not. All of our data so far shows no evidence of CTCs, so as far as we can tell, there are no CTCs in the portion of spacetime that we can see. But nobody has been able to prove that, starting with a spacetime region containing no CTCs, it is impossible for the geometry to the future of that region to contain CTCs (such a proof would mean that, for example, it is impossible to build a time machine in a universe that does not already contain one).

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Paul Colby

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PAllen

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Actually, I think it is known that a spacetime can evolve from no CTC to having them, in an idealized classical sense. Consider an idealized collapse to a Kerr BH. Not known, so far as I know, is whether the instability of Kerr interior to tiny deviations from perfect axial symmetry eliminates CTCs.The term "causality" is vague. In order to answer your question, it would need to be made precise: how would we test to see whether "causality" is true?

On at least one interpretation of "causality", the one used to classify spacetimes in General Relativity, it depends on the spacetime geometry; basically, "causality" holds in a particular geometry if that geometry has no closed timelike curves. On this interpretation, we test whether causality holds by testing whether there are closed timelike curves or not. All of our data so far shows no evidence of CTCs, so as far as we can tell, there are no CTCs in the portion of spacetime that we can see. But nobody has been able to prove that, starting with a spacetime region containing no CTCs, it is impossible for the geometry to the future of that region to contain CTCs (such a proof would mean that, for example, it is impossible to build a time machine in a universe that does not already contain one).

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The second law does not exclude the possibility of perfectly reversible processes; it just says they're very unlikely. Any process that occurred along a CTC would have to be perfectly reversible (since the process would have to return to the same state), so in a universe where CTCs were common, I think the second law would at least have to be looked at very differently.Would the second law of thermodynamics work if time machines were found to exist (eg. CTCs) in our universe?

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Is there a known solution for this case? The only known exact collapse solution I'm aware of is the Oppenheimer-Snyder collapse, which is spherically symmetric and non-rotating, so the endpoint is a Schwarzschild BH, not a Kerr BH.Consider an idealized collapse to a Kerr BH.

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martinbn

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Wouldn't that go against the strong cosmic censorship? The CTC are in the maximal analytic extension beyond the Cauchy horizon. But the horizon should be unstable. A small perturbation should lead to a curvature singularity.Actually, I think it is known that a spacetime can evolve from no CTC to having them, in an idealized classical sense. Consider an idealized collapse to a Kerr BH. Not known, so far as I know, is whether the instability of Kerr interior to tiny deviations from perfect axial symmetry eliminates CTCs.

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First, an analogy: If you have a spacetime obeying Einstein's field equations with periodic boundary conditions in some direction (that is, you have a coordinate system ##x, y, z, t##, and absolutely everything--scalar, vector and tensor fields--are unchanged by the translation ##x \rightarrow x + L##), then you can get another solution to the field equations by "gluing" each point ##(x,y,z,t)## to the corresponding point ##(x+L, y, z, t)##. That's a way to get a universe that is not simply connected.

I assume that the same is true of closed timelike curves. If there is a solution to the field equations that happens to be periodic in time---in some coordinate system, everything has the same value at ##(x,y,z,t)## and ##(x,y,z,t+T)##---then you can glue those points together to get a universe with closed timelike curves. Is that right? So it's trivial to come up with an example: Empty Minkowsky space.

But the examples in Wikipedia include something more complicated---Misner space, which instead of identifying points at different times and the same spatial coordinates, it identifies events that are related by a specific boost:

##(x,y,z,t) \rightarrow (\gamma (x - v t), y, z, \gamma(t - \frac{vx}{c^2}))##

where ##v## is chosen so that ##\frac{v}{c} = tanh(\pi)##.

I don't understand the significance of that particularly boost, or why boosts are important at all for an example of a CTC.

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PAllen

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No, because I didn't say anything about whether the CTC is behind a horizon or not. As to the instability, I mentioned it in my post.Wouldn't that go against the strong cosmic censorship? The CTC are in the maximal analytic extension beyond the Cauchy horizon. But the horizon should be unstable. A small perturbation should lead to a curvature singularity.

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PAllen

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I don't think there is any known exact solution, only numerical solutions, some of which, with just the right initial conditions, suggest naked singularities and CTCs. This was why Hawking conceded the original cosmic censorship hypothesis and restated it in terms of a set of initial conditions of greater than zero measure in the parameter space leading to violations. The reformulated conjecture is open.Is there a known solution for this case? The only known exact collapse solution I'm aware of is the Oppenheimer-Snyder collapse, which is spherically symmetric and non-rotating, so the endpoint is a Schwarzschild BH, not a Kerr BH.

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martinbn

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I meant that there are no CTC before the Cauchy horizon. A perturbation should lead to a solution with a spcelike singularity instead of the Cauchy horizon so there will be no CTC .No, because I didn't say anything about whether the CTC is behind a horizon or not. As to the instability, I mentioned it in my post.

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PAllen

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I don't see a disagreement, except I am mentioning the idealized case of no perturbation, so there would be CTC inside the Cauchy horizon.I meant that there are no CTC before the Cauchy horizon. A perturbation should lead to a solution with a spcelike singularity instead of the Cauchy horizon so there will be no CTC .

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martinbn

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Oh, I misread your post.

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YouIf there is a solution to the field equations that happens to be periodic in time---in some coordinate system, everything has the same value at (x,y,z,t)(x,y,z,t)(x,y,z,t) and (x,y,z,t+T)(x,y,z,t+T)(x,y,z,t+T)---then you can glue those points together to get a universe with closed timelike curves. Is that right? So it's trivial to come up with an example: Empty Minkowsky space.

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The loop is a problem because it violates causality, not because it goes on forever. It may seem perpetual, but really it's just non-linear. Unless you're the eight ball.So its just one loop, then?