Suppose Carol has a clock. At X it reads 0, and at Y it reads 100 (100 units in the future). From Bob's perspective, Carol's clock would also read 0 at X and 100 at Y despite the fact that she was at Y first.In other words, as Carol travels 'forward' in Bob's time, her clock runs backwards. Or, from Carol's perspective, Bob's clock would be running backwards.This isn't actually a problem in relativity, but is the definition of time travel.

 No, it's not a time travel. Bob will see that Carol travels backward and Carol will see that Bob travel backward, like we see EM emission from FTL electron in reverse order. FTL electron is not moving back in time, nor Bob or Carol.
 The notion of "backwards" in time is a bit fuzzy once you get into relativity, so the three person example I gave may not clearly show it.However, in the four person example, Dave is able to hand Carol her diary of the trip that she is about to make. If that is not time travel, I do not know what is.
 It's very unlikely. If Dave has Carol diary, then Carol made the trip already, so Carol will notice empty fuel tanks and big holes in the shield. She will may have problems with memory, but she is not stupid.
 No she didn't. X is the point in spacetime where Carol begins her trip. From her perspective, Dave came from the future.The only reason that this cannot happen is that X and Y are so far away in space, and close in time, that it is impossible to travel between them.
 Sorry, I cannot follow you here. Maybe it's well known thought experiment, but I'm unable to find it.

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