Closed Timelike Curve - Contractible Versus Noncontractible

Contractible Versus Noncontractible

There are two classes of CTCs. We have CTCs contractible to a point (if we no longer insist it has to be future-directed timelike everywhere), and we have CTCs which are not contractible. For the latter, we can always go to the universal covering space, and reestablish causality. For the former, such a procedure is not possible. No closed timelike curve is contractible to a point by a timelike homotopy among timelike curves, as that point would not be causally well behaved.