The question about the Starship Enterprise and time travel came up at the Halloween party on Saturday, asked by people who had a friendly interest in my answer. Considering the difficulty or trickiness of the question and my quick answer, I think that I should try again.
The first property of the Starship Enterprise is its facility for warp speed. But in the inherent theorems of spacetime, premised only on the existence of the metric*, there is a prohibition of unrestricted transition to and from warp speed, faster than light. The way allowed by logical consistency is to enter an event horizon, whether black hole, or cosmological - the edge of the universe.
But once this is done, the denizens beyond the event horizon no longer agree on the metric to be used anywhere, on either side of the event horizon. The Starship sees its new side of the event horizon in ways that outsiders left behind can not agree on. In fact the observers left behind see the Starship, stuck still in red-shifted time, exponentially dimming and exponentially approaching the event horizon.
Spacetime is logically coherent, complete and causal as a matter of theorem, at least until this question, or a few others, come up. So the hypothetical return of the Starship from warp drive causes a logical crisis, to put it mildly. This is that the crew of the Starship would report on events experienced in warp drive using a metric that is contrary to the one that the outsiders are unanimously committed to. There would be even an irreconcilable disagreement on the direction of time for many sequences of events.
There is more than one screenplay in the Star Trek saga where a close turn around the Sun at high warp speed results in time travel. Yes indeed, this turn yields still more logical difficulty. A ninety degree turn poses a disagreement between the two courses on many sequences of time.
The adjacency to the Sun is really of no aid to the plot. But physicists do speak of a time machine mechanism. An extreme black hole might rotate so fast, and with so much of a temporary preponderance of bulk at its outer rim, that the event horizons on the axis of the donut shaped thing are very close. When the Starship is caught on one of the event horizons, it is not completely impossible to imagine that it might quantum-tunnel through to the other and become unstuck**. The tales then told by the crew, and by outsiders, I can not imagine. This is because quantum tunneling breaks the logic of spacetime. I can imagine, though, that there is some mechanism invoked by the universe to disguise the disagreements.
Action in higher dimensions would make this last scenario more plausible. But it seems that some quantum effect prevents human access to higher dimensions.
* The metric is a way for different observers to negotiate to an agreement on the spacetime interval between two events. It is a geometric object (because observers can agree on it), and in another sense it is a generalization of the Pythagorean formula.
** Quantum tunneling is contrary to the theorems of spacetime. But the function of quantum mechanics is to fill in where the premise of spacetime fails, where the metric goes out of existence.
Michael J. Burns