Chronos, I agree that those solutions are available theoretically, but right now they are practically unavailable. Building something with negative energy just seems impossible at this point. However, the two solutions I’m talking about both are available with proper amounts of material available to us in the universe (we see it) if arranged in the right fashion. I admit the one I’m more familiar with is the cosmic string model which may seem a bit out in left-field (as no cosmic strings have been seen), but may be conceivably constructed out of materials we do observe. The alternative time-travel to the past solution I’ve seen that involves using only materials available to our universe also seems to indicate that it’s not feasible to go to a point before the solution existed.
These two solutions are resultant from investigations into constraints of actually being able to BUILD the time machine in order to contradict Hawkings hypothesis that nature conspires to prevent time travel to the past. In effect, these two solutions allow for time travel to the past in such a way that the causality ambiguity is still up in the air. The other more commonly quoted solutions for time-travel to the past involving negative energy densities… it is unclear whether they are actually impossible or not. Regardless, the problem of causal ambiguity is independent of whether or not these constraints of having epochs of time-travel to the past available or not and having to use large masses are applied. These were simply mentioned as current constraints that avoid Hawking’s nature forbidding past-time travel.