Within the worlds of science and science fiction alike, there are a number of different theories about the concept of time travel. Each theory has it's own set of rules which govern time; the implications and the repercussions associated with time travel. A popular discussion with regards to time is whether time is linear or tree structured in nature. Both sides have their own set of paradoxes, which I will be discussing.
In the linear time model, all events could potentially affect the future; in some cases this stems from the butterfly effect, whereby even something marginal could have disastrous consequences. Converesely in the tree-structure time model, time branches off into "alternate timelines" as a result of time-traveller actions. I consider this to be overly simplistic as it is making the assumption that time bends only to the will of humans. I posit that in the tree-structure time model, time would branch off on an infinite set of "What If" scenarios. Hence there would be a timeline where the dinosaurs did not go extinct, a timeline where Hitler never rose to power...etc.
Touching upon the linear time model; if I were to time travel into the past, perform some action and return to the present, what frame of mind would I come back with? Would I return with the same frame of mind I left with, or would it be altered as a result of my actions? Let us take a simple example; my mission is to travel into the past and prevent a particular book 'X' from being written and I choose to do this by murdering the author. If I were to do this, would I return to present aware of this book, or would the awareness of this book be erased from my mindset? Let us suppose that I would return with knowledge of the book 'X'; my mind would be aware of two different timelines. The original timeline (before I went back in time) and the altered timeline (after I went back in time) would both be clear in mind; hence this state of acknowledging a before and an after would be in direct violation of the linear time model.
Let us suppose that instead, I would return without knowledge of the book 'X'; what would I think of, stepping outside of the time travel device? If I were in any way aware that I had succeeded at my mission, I would again be aware of two different timelines as I would be aware of a time-related mission which I had succeeded at, hence having the concept of before and after time travel. The implications of this are interesting; I would not be aware of 'X' but I would know that I had travelled backwards in time to prevent 'X'. Due to the fact that we're observing a linear time model, this would mean that I am aware of something which never existed! Additionally, if the book has never existed, what would justify my need to have time travelled in the first place? In concept since I travelled backwards in time for a reason; eliminating that reason should eliminate the time travel. Eliminating the time travel would prevent 'X' from being stopped...ad nauseam. It is a paradox.
Touching upon the tree-structure time model, if I were to go back in time to stop 'X' from being written, my actions would fork time, creating an alternate timeline. In order to simplify things, we should assume that there are two timelines as a result of my actions; the timeline in which the book is and the timeline in which it is not. If I were to travel back to the present, which timeline would I be returning to? Alternatively, would I return to both, creating a reality in which I succeeded at my mission and a reality where the mission never existed? If I were to return to my original timeline, it would appear by all accounts that I have failed at my mission, as the original timeline has operated on the assumption of my actions never having taken place (book 'X' was written and all the repercussions associated with this book took place, creating the original reason for me to time travel). The alternate timeline or reality which was created as a result of me preventing 'X' from being written would not be something I would experience as I would not have returned via that timeline. If however, I were to travel on the new time branch, then it is necessary for time to be infinite in both directions along this new branch, even though it has just been created. Alternatively, time would need to be infinite in a forwards direction and then join with the original timeline just before I time travelled to that time. If the time is not infinite in both directions (forwards and backwards in time), then time travel would not be possible along this new timeline, causing me to be trapped in the past, which would now be my present. Hence, in order for time travel to work in this scenario, every single timeline must have a past which stretches to the beginning of time and a future that stretches until the end of time.
What would happen if I were to travel along multiple time branches as a result of preventing the book 'X' from being written? What would happen if as a result of preventing the book 'X' from being written, I would have prevented time travel from being invented in the first place? The very non-existence of time travel would cancel out my ability to time travel in the first place. But not having time travelled would mean that I did not prevent time travel from being invented, which would mean that I could time travel and hence I did...ad nauseam. It is a paradox. What would happen if I were to murder a younger version of myself, or in some way prevent my birth from taking place? Would I cease to exist? Would the timeline fork into two timelines; one timeline in which I am and one timeline in which I am not? Would my non-existence prevent my time travel, hence ending up in another paradox?