Gödel showed that backwards time travel is physically possible and compatible with General relativity by providing a solution to Einstein's field equations that allow for closed causal chains.
His solution is to note that the duration of the time and the separation in time between destination and arrival are measured in different frames of reference (times). What he calls earth-time corresponds to what Lewis calls external time and what he calls proper-time corresponds to what Lewis calls personal time (i.e. clock/wristwatch time of the traveler).
I am my younger self even though we have different properties. How does this not conflict with Leibniz's law? One thing to say is that the properties that differ, e.g. being on a baseball team then and not now, are relativized to times; I was on a team at some time but not others. No contradiction there. But what if I now travel back in time to when I was on a baseball team? Two of me are there in 1988, one on a team the other not. Contradiction!
We forgot we have two times: earth- and proper-time. I am not on a team relative to proper-time-2016, but I was on a team relative to proper-time-1988. Of course there are still two distinct objects in earth-time 1988, my time traveling self and my younger self, so we are not identical in the strict sense by Leibniz's law. Why then are we trying to maintain that Leibniz's hold between me-now and me-then? Horwich is not clear on this.
Changing the past is logically impossible, according to Horwich; influencing it is possible. If I travel back to the past and say "Hello" to someone x, I've influenced the past, but I haven't chagned it. Back in the past there was already the event of a traveler existing then saying "Hello" to x.
Horwich also distinguishes between one's not having done something and one's not having been able to do otherwise. If I travel back and don't do anything to change the past (because I can't), that doesn't mean I couldn't have changed the past. He doesn't spell out what this could mean, but there are various ways of making sense of it. E.g. in an indeterministic setting like Lewis's, we can think of there being a diverging world where I do change the past (i.e. a counterpart of mine does all the stuff I do up to a point and then does something different thereafter).
Horwich reads Gödel as implicitly endorsing a principle Horwich calls (D): It is impossible for X to believe at a single time (a) that it is certain that a state of affairs S will obtain, and (b) that he is at liberty to bring about S, or not-S as he chooses, and has not yet decided what to do. He says that in cases of time travel, even if (D) were correct (though he doesn't think it is), the traveler might not trust his memories beliefs, or he might feel free to make decisions that involve influencing the past.
The paradoxes involving agents show at most that time travel wouldn't be possible if there were agents like us around to do the traveling. It wouldn't show that there couldn't be closed causal chains involving e.g. rocks. Since rocks don't have agency, an absurdity like Gödel's wouldn't arise for a time traveling rock. So if time travel is to be defeated on the most general grounds, that grounds cannot essentially involve agency.
His argument involves a rocket ship programmed to fire a probe into the past (lobe of the null cone at the position of the rocket) if a safety switch is off, where the switch is off iff the return of the probe is not detected by a sensing device. Is the probe fired? Suppose it is. Then the rocket detects the probe at the same time the probe is fired, so the probe isn't fired given the way the rocket is programmed. It follows that the probe can't be fired. But then if it's not fired, the rocket doesn't detect the probe, in which case the switch is off. By the way the rocket is programmed, the probe is fired. Contradiction.
Earman then holds that since it's impossible that we have both time travel and the existence of such a rocket, and since it's very possible in principle to actually build such a rocket, it follows that there (actually) is no time travel. Horwich claims that this sort of reasoning embodies a modal fallacy. From the fact that (i) some conjunction 'P and Q' (there is time travel and the rocket exists) is impossible, and (ii) Q (the rocket exists) is possible, it does not follow that P is false. This reveals yet another paradox of time travel: if there is time travel then even though it seems we can build such a rocket, it follows that we somehow can't. Every attempt of ours to build such a rocket would fail, despite initial impressions. Also, we do build such a rocket, we will have thereby shown that there is no time travel, despite the fact that it seems the mere existence of such a rocket is insufficient to show that time travel doesn't occur.
They all fail to appreciate that time travel puts constraints on the sort of timelike curves there are, in the sense of the sort of causal chains that may be located along such curves. The fact that some chains are possible along certain open curves does not imply that the same chains are possible along closed curves: "any causal chain must satisfy consistency conditions imposed by its surroundings" [p. 443].