If our base space, B, is Minkowski spacetime and our fibers are circles S^1 are the following constructions ways to put together B and S^1 and have a total space, T, that is considered a fiber bundle. Remove from Minkowski spacetime, M, a timelike line, L. At the remaining points of Minkowski spacetime attach a S^1 fiber. If possible I would like to "twist" the fibers such that if we make one 360 degree orbit around line L we advance or go back one turn in the S^1 space? Is there such a total space T? Again let our base space be Minkowski spacetime, and now let our S^1 fibers be cut. At each point of Minkowski spacetime attach one end of a fiber, the other end is attached a small distance dx away from the first end. So if you make one complete orbit in any fiber S^1 you move in spacetime some small distance dx. Is that something that could be considered a fiber bundle? Could dx be a function of spacetime? Thanks!