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?