Logic of Time Travel
By Tom Brown
Consistent with known science possibilities of non-contradictory time-travel are examined, speculating with ideas from history, logic and philosophy.
Of course irreconcilable difficulties and absurdities in philosophy and logic are not an issue inherent to travel to the future, it is traveling into the past that gives you those glaring paradoxes. Relativity theory in itself does not prohibit time-travel to the future as is well known.
Again my terminology is rather novel and some of it made up to make concepts easier to understand, so that the meaning of words and definitions are not meant as rigid.
The note speculates on a kind of time-travel (backwards!) that might in fact be plausible and consistent and without the familiar problems and obvious contradictions. As far as that goes I do believe that in essence these possibilities are indeed logically sound this being our main objective.
Some of the ideas are not original for example in popular science fiction I've heard it spoken but rather more vaguely and in comparisons, in popular words such as ripples, branching time-lines, jumps and loops, time streams and convergence. There is quite a variety and all this does in fact have good merit.
This is an attempt at a beginning of a new and stronger footing plausible as far logic goes.
The present ideas should very strongly appeal to the imagination. That is not to say it can actually be done it all depends on the setting and the true laws of time and nature, of cause and effect, which we hardly can know at all. The point is that as far as we know time-travel may indeed be possible.
The existence of a mathematical theory of time-travel is suggested and particularly back into the past although clearly any idea of making such a machine must at the moment be purely fantasy.
The strongly related mathematical study of dynamical systems reveals very similar behavior especially in a number of appropriate recurrence results, such as already the very simple but surprising theorem of Poincaré illustrates.
Discontinuities and leaps
Typically a classical paradox would be going into the past and assassinating yourself as a baby. What now? A bit more subtle, an adult from a baby prevents its own conception and birth. One can make up many paradoxes such as these.
Perhaps you can get around the problem if, when traveling back to a point in your past and history, after a discontinuity (jump) landing and starting again on a completely new time-line. It can never intercept the original curve from which has fallen off to this new curve because that would always give contradiction situations.
It means that now a new trajectory as such starts again at a given point elsewhere, but at the same point in time, continuing and sailing around and bypassing your state at present wherever, and again tends all the closer toward the same asymptote, as does all other trajectories.
Thus you are in effect bumped off-course to a completely new time-curve by a discontinuity and now have a new history but this new curve slowly just approaches again and once more the very same asymptote, attracting the graph nearer and still nearer again. Distinct time-lines could in effect all share the same distant future in phase space.
Limits and non-interception
At last then, beginning very far-off in the past you have that all time curves eventually become indistinguishable. It means that every single trajectory eventually converges to that same curve of states in phase space, whatever it may be.
The starting point (our “present”) must be bypassed but for the graphs to later “join” again in a common future. Such an operator may be defined on the entire space of cause and effect and be convergent no matter of initial position. Infinity is more than enough, time is very patient indeed.
You may never get to the same place but with time you can get as near as you wish. Of course the point you fell back on and the discontinuity to your new trajectory would never meet again. In a way it can be said our history and present are very different but we all share a destiny.
Of course any instances of interception and “classic” time-travel has to be completely prohibited also in this setting. It would still not be possible to go back and manipulate history with the idea to change the present. This would then logically spoken, not be possible, ever.
Quantum mechanics and experiments
Similar kinds of such discontinuities and jumps are quite feasible in space time and reality for example as predicted by quantum theory. In fact such phenomena do exist. Certainly many are already observed in experiments at atomic and nuclear scale and the existence of these kinds of events is today common knowledge.
A trajectory from a given point may thus return into the past and back again forward and into future time bypassing the starting point as if it never existed, eventually meeting up with the original trajectory again and in the very far extremely distant future.
Thus backward travel becomes forward, or past becomes future until eventually all trajectories get so close to each other as to be indiscernible.