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I've been looking a lot at Closed Timelike Curves, and how if a theory allows for these curves it doesn't respect causality. I understand that about the curves themselves (Grandfather Paradox), but can't seem to fathom how a theory would allow for such structures, since they seem to be "geometrically" impossible in a spacetime.

To my understanding, CTC are simply worldlines that loop back on themselves, and are therefore closed. The problem comes in when I actually try to picture a CLOSED worldline: If I start at a point in Minkowski Spacespacetime and draw any closed curve, I end up always having a portion of it be spacelike, and therefore the curve is never fully timelike. Meaning It's impossible to draw a CTC.

So my question is, how can a theory allow for such worldlines, since the fundamental principles behind the geometry of spacetime simply prohibit it?

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In Special Relativity CTCs can't exist (or at least I don't think so) but General Relativity has solutions that include CTCs. The best known is probably Gödel's solution for a rotating universe. The Alcubierre drive could also be used to construct CTCs, as could any FTL mechanism.

Also see the Tipler cylinder, and probably many other examples I can't remember. Needless to say, none of these examples of CTCs are remotely realistic.

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You're right, they can't exist in Minkowski. Quite interestingly though, they are possible in Anti-de Sitter space... This is immediately seen by embedding $AdS_D$ in a $D+1$ dimensional Minkowski space and drawing a picture. –  Danu Mar 3 at 17:51
    
Thanks! That helps a lot! :D –  Disousa Mar 3 at 18:20
    
@Danu What physicists usually mean by AdS is the universal cover of AdS. –  ungerade Mar 3 at 19:25
    
@ungerade I am far from an expert, so please elaborate if you'd like! Maybe you can explain how this eliminates the CCT's? –  Danu Mar 3 at 20:17
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@Danu Yes. So AdS in physics is usually a universal cover of the real AdS –  ungerade Mar 4 at 8:37

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