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In the case of the Roman ring, a traveler is able to go through a wormhole mouth and emerge at some point in the past without violating causality, so long as the time traveled back is less than the time it would take for a speed of light signal to arrive at your previous reference frame. So, for a slower than light trip that traveled to some point, and then returned to its original reference frame at a faster than light speed, but respecting the above condition, would that qualify as a causality violation?

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closed as unclear what you're asking by Ben Crowell, Jon Custer, John Rennie, ZeroTheHero, Aaron Stevens Sep 17 at 14:38

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Could you give us a reference for what you mean by a "Roman ring?" I assume this is the Roman of the Ford-Roman inequality? $\endgroup$ – Ben Crowell Sep 12 at 19:24
  • $\begingroup$ Yes, the name was in honor of Tom Roman, but it was a system proposed in 1993 by Matt Vissier. Link: arxiv.org/abs/gr-qc/9702043 $\endgroup$ – CuriousDroid Sep 12 at 19:28
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    $\begingroup$ When someone asks for clarification on a question, the thing to do is to edit the question, not provide the clarification in comments. $\endgroup$ – Ben Crowell Sep 12 at 21:41
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A causality violation is a very specific condition. Causality is said to be violated if, given a point $p$ of your spacetime, there is a causal curve (ie a massive or massless particle) which can start from $p$ and end up at $p$ (a bit of a stronger condition is strong causality, where the curve starting at $p$ will come arbitrarily close to $p$ again). An observer can very much indeed go "back in time" without violating causality. In general relativity, as you may know, the notion of an event being in the past or the future is a bit shaky since it's always possible to rewrite coordinates in a way where they may be switched.

And so it is actually much harder to violate causality than you'd think on the surface. A lot of faster-than-light spacetime metrics do not actually violate causality because of this. A wormhole where the other mouth ends up one year in the past doesn't violate causality if the two mouthes are separated by two light years. The Alcubierre drive in most cases doesn't violate causality (in its simplest incarnation it's a spherically symmetric spacetime and those are never acausal if they are topologically trivial), and the same is true for the Krasnikov tunnel. In a similar fashion, a single free tachyon isn't going to violate causality (if we extend the notion of causality to also involve spacelike movements), since such a system cannot go back to a previous point in spacetime.

But the trap here is that, once you have such regions of space, it is very easy to construct causality violations by using several such things. And just like a system of tachyons and tardyons can form a causal loop, the same is true of a system of such causal wormholes, warp bubbles and Krasnikov tunnels, which individually respect causality, but an observer travelling along several such regions will be able to come back to a previous point in spacetime.

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