I would be very grateful if you could recommend any scientific articles on theoretical models of the time machine.
Except for Gödel's rotating universe I unfortunately do not know of any other work that could serve as a theoretical model for time-travel. However I have given thought to a model of space-time in which time is circular, a certain temporal state of the universe in our future will be the same as and identified with a certain state in our past (think of R^3 x I with the ends glued, a 4-dimensional torus) . Causality will thus be circular and we can be our own creators without any (local) violation of the laws of physics. Thus as we naturally travel into the future we are also on the journey to our own past.
Clarence Lewis Protin Thank you for your reply! I am familiar with Gödel's concept, and what you write is also interesting. I, on the other hand, have in my mind the model of time as a quotient category integrating different forms of time perception (I make the assumption that our models that we create are a reflection of how we perceive the world and hence the integration), I also thought about the applications of monoidal categories and forgetting functors, although for now it is more of a philosophical concept and not mathematically worked out.
Agnieszka Matylda Schlichtinger I just remembered another idea I once had about time and physics. If we consider a space-time manifold in general then time may be seen as defining a foliation on this manifold so that for any point we can associate a 3-manifold consisting of all (synchronous) points sharing the same 'now'. It is the universe 'now'. Each point of such a 3-manifold, if the manifold is smooth. will have a normal vector, orthogonal in the space-time manifold to its tangent space. All these tangent spaces of points in a 'now' 3-manifold satisfy the Frobenius integrability condition which is what guarantees that they are in fact tangent spaces of a submanifold. But what if such a globally constructed time were not possible ? To each point p in the space-time 4-manifold we now associate a time vector $t_p$. That is, we are considering a vector field on the 4-manifold. This vector then determines a 3-dimensional vector space which only locally represents the local space for 'now' for that point p. But given any two points p and p' in the 4-manifold in general it will not necessarily make any sense to speak of one being temporally before, synchronous or after each other, because the 3-dimensional vector spaces determined by the vector field need not be integrable (as in the Frobenius theorem), that is, there is no foliation or decomposition of space-time into synchronous spaces. The future evolution of a point p is determined by integrating the vector field of the time vectors. It could well lead into one's own past or close to the past or future of other beings. In short, in this model there is no global coherent concept of time, time is local only, determined by a 4-vector.
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