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Hawking's Chronology Protection Conjecture says that the laws of nature must always conspire to prevent a CTC from forming. Why can't we conclude that this is proven? An inconsistent CTC is s contradiction. It is always possible to set up such a contradictory situation if any kind of backward time travel is permitted. You can always create a grandfather paradox. A theory can not contain a self-contradiction. So isn't this proof by contradiction of Hawking's conjecture?

Qmechanic
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JeffK
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    without a solid theory of time travel in not sure how you can say "It is always possible to set up such a contradictory situation if any kind of backward time travel is permitted. You can always create a grandfather paradox. " – shai horowitz Mar 11 '22 at 00:50
  • @shaihorowitz - It doesn't sound like you need much of a theory. Suppose you have a system that progresses from state a to b to c time $2\Delta t$ in a normal timelike curve. Suppose you put the system in a closed timelike curve of length $2\Delta t$. It progresses from a to b to c. But it is back when the state is a. Cause and effect lead to an inconsistency. This is a Grandfather paradox. – mmesser314 Mar 11 '22 at 02:39
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    $log z$ is multi-valued in the complex plane. You can move locally at any point in the complex plane and your branch of the logarithm will have no discontinuities unless you do unnatural stuff like 'branch cuts' in order to have a Frankestein version of the logarithm that is single-valued. The hidden assumption in the 'backward time travel' leads to contradictions assume things that can manifest said contradictions have special properties like being "single valued" across multiple time loops. That's a reasonable assumption in classical physics, but not in quantum theory – lurscher Mar 11 '22 at 03:08
  • @lurscher - Doesn't the wave function have to be single valued? – mmesser314 Mar 11 '22 at 03:29
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    @mmesser314 But even if you assume single-valuedness. The grandfather paradox is build upon the idea of free will, that if you have a time machine, you can go back and kill your grandfather before you were born, even though you know historically this is not what happened.

    But when you look at the paradox strictly physically, why would, say, an electron traveling backwards in time necessarily mean inconsistent states a and c? Why are you so sure the physical processes would not "sort themselves out"?

     – Umaxo Mar 11 '22 at 09:27
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    Related: https://physics.stackexchange.com/q/407826/123208 Also see https://en.wikipedia.org/wiki/Novikov_self-consistency_principle – PM 2Ring Mar 11 '22 at 11:52
  • @mmesser314 if you measure one electron, then travel back right before the wavefunction collapse and measure again, do you expect to obtain the same value? if the wavefunction hasn't collapsed yet, if you repeat the time loop several times, then the electron ought to collapse to different eigenvalues the 2nd time – lurscher Mar 11 '22 at 13:47
  • When I stated that a Grandfather Paradox can always be made to occur, I was thinking of a simplified version without people, to avoid the free will question. A ball enters a transversible wormhole and exits its other mouth nearby but 5 minute before it entered. It hits a lever that triggers a gate to cover the first mouth. So its future self can't enter the wormhole. But if it doesn't, the gate wouldn't have been triggered. The gate is just simple Newtonian mechanics, so it must work. The wormhole is our hypothetical time travel device. How could nature avoid or resolve the paradox? – JeffK Mar 13 '22 at 08:25

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There have been some research papers, somewhat speculative I think it is fair to say, which investigate what kind of physical behaviour might involve a CTC while remaining self-consistent. One obvious idea is that a CTC is only possible if it happens in such a way as to influence physical evolution such that the state of any system on a CTC evolves as $a$ to $b$ to $a$ rather than $a$ to $b$ to $c$. I am guessing this would break the equivalence principle but that principle is already broken when you combine quantum theory with general relativity, in situations where entanglement is happening over distance scales that are non-negligible compared to the smallest local radius of curvature of spacetime.

Andrew Steane
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