# Regarding the possibility of Closed Timelike Curves

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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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 '14 at 17:51