  In that case, we now have to worry about a question we have not yet confronted: what happens if our point particle hits exactly at the point P on the diagram? Does it time-travel or not? Confronting this question requires us to face up to a feature of the rather cheap way we implemented time travel in our toy models by cut-and-paste. The way we rejiggered the space-time structure had a rather severe consequence: the resulting space-time is no longer a manifold: the topological structure at the point P is different from the topological structure elsewhere. Mathematical physicists simply don't deal with such structures: the usual procedure is to eliminate the offending point from the space-time and thus restore the manifold structure. In this case, that would leave a bare singularity at point P, an open edge to space-time into which anything could disappear and out of which, for all the physics tells us, anything could emerge.