I'm writing a short story that involves time travel but I have a question about my plot. Both my of characters live in the year 2030. A robot from the future comes and tells Rob that the world will be engulfed in a nuclear war and that 98% of the world's population will perish. The robot says that he can take only one person with him to the future and he wants it to be Rob because he plays a huge role in the future. Rob, however, gives up his spot to Sally, his love interest. In the future, Sally sees how much love Rob has for her and decides to send the robot back in time to rescue Rob. Am I breaking a time travel rule? basically, the robot is sent the first time by Sally to rescue Rob in the first place.
Make your own rules so that you don't break them. Time travel is full of paradoxes and there are various ways of dealing with them. Either there is one flow of time and "it's always happened like this", or your time travel can change the world (the past, the future) and you end up in a world with a different past than you remember. It's up to you.
As long as the second time the robot goes back, he goes to a time later than he did first time around, there won't be a paradox. Just to clarify: (9:01 AM) Rob and Sally in 2030 (9:09 AM) robot comes back, asks for Rob, gets Sally (9:10 AM) Rob is by himself, wondering what the future holds for himself and Sally (9:12 AM) robot comes back to get Rob The times are just for illustration. Days, weeks or months could pass between each event in 2030.
This is not a paradox. If Rob is to be rescued because he plays a huge role in the future, he must play that role and therefore cannot be rescued. Sally must be rescued so she can send the robot back in time. The robot makes only one journey. Everything works, there are no contradictions.
You can't resolve a fictional paradox in a realistic way because paradoxes only exist in the abstract. Since you're already in that world, just make the situation believable enough.
The idea of time travel itself breaks the paradox defending that if such travel were possible, us living in the present would have already met time travelers coming from the future. As far as your concern, there is also a paradox stating that nature compensates the changes operated during time travel so that the future outcome isn't changed. I think your plot is perfectly consistent.
Well there is a theory that time is self correcting. Be it by splintering off from the main timeline, in a deviation where those events will not interfere with the latter. Looking into the John Titor story, he changes things in his own time line by approx. 2%, from what actually happens as a result of him time traveling. Ergo, he can not travel to the past and expect to travel back to his original 2037 time line. Something has been altered and therefor his time line has changed, even if my a minute amount. What about the theory of the multiverse. In a nutshell where there is an infinite amount of universes and an infinite amount of time lines running simultaneously. Meaning that every outcome, situation, etc. can/has/will be played out in all the universes. Granted the effect would vary from minor, to 180degrees the opposite, but every outcome no matter how small will be factored in. So there is an alternate you that could exist that is almost exactly like you down to the smallest detail, and there is an alternate you that shares very little at all in common with the you in this verse. And in some universes you don't even exist. So paradoxes are merely corrected by shifting you into the proper verse based upon your actions as you traverse through time in either direction. Crazy huh?
If you want to use the many worlds interpretation of QM, that would allow you to split your timeline so that events that create the robot still happen, just in another timeline. The robot goes back in time and gets Sally. That creates a new timeline. The original still exists, but the new one contains only Rob and Sally is in his distant future. That Sally would have to send the robot back just after she left, and then return with Rob. This creates a third split allowing both Sally and Rob to exist in the future. It sadly does create one timeline in which Sally and Rob are separated. This can not be resolved in a splitting timeline.