Cosmic string was proposed several decades ago and there are claims stating it can span across the length of the universe and is less than an atom thick. How does a broken pieces of these cosmic string with immerse gravitational force in theory can allow an observer to travel back into the past? Wouldn't Einstein's general relativity forbids it? Is this cosmic string a mere fairytale to coax young children to go to bed early? If this method of travel backwards through time is true how will it resolve the grandfather paradox?
Firstly, note that there are multiple objects that go by the name "cosmic string." Usually, it indicates some one-dimensional topological defect in the universe, but the other properties of it might vary. The theory to which you seem to be referring is one originally due to John Richard Gott III, a professor of theoretical astrophysics at Princeton. He points out that under the conditions of two infinitely long, rapidly separating strings with a kind of anti-gravitational tension, there exist paths in spacetime called closed time-like curves (CTCs); these CTC solutions are fully compliant with Einstein's general relativity. The time-travel associated with a CTC is the fact that one could travel along it at subluminal speeds for nonzero, finite time and, upon returning to one's starting point in space, find that one had also returned to one's starting point in time. Unfortunaly, I don't know the details well enough to elaborate much beyond that, but you might find this book of interest: Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time, J. Richard Gott, Boston: Mariner Books, 2002, ISBN 0-618-25735-7.
Strangely enough, the mathematics probably allows for a set of laws that in each point of space-time locally follows special relativity and yet there exists a state of the universe with no solution to what future state it will evolve into while following those laws. For example, there exists a set of laws other than those of general relativity that the universe could have obeyed that allows you build a track through space with time contraction inside the track allowing objects to travel faster than light in it as observed from outside the track but does not allow you to build such a track in the first place faster than light. It also permits you to start building a second track right beside it that contracts time in a different frame of reference whose velocity differs enough that it would actually create a closed light like curve and unless the state that led to a light like curve satisfies an exact condition, there is no solution for the state of any points in space-time that are in the future light cone of any point in that closed light-like curve other than those at the very edge. One possible solution is that, points of space-time that are in the future light cone of any of those points without being at the edge don't exist so the creation of a closed light-like curve will nucleate the destruction of the universe at the speed of light before a closed timelike curve even gets a chance to evolve. https://www.quora.com/If-a-car-runs-at-the-speed-of-light-will-the-headlights-work/answer/Timothy-Bahry I'm not sure if general relativity also allows for a state of the universe with no solution to what future state it will evolve into. Maybe a closed light like curve has formed in a black hole and it did nucleate the destruction of space but the destruction of space never escaped the black hole because it would have to travel faster than light to escape the black hole.