The twin paradox and space topology

Abstract

If space is compact, then a traveller twin can leave Earth, travel back home without changing direction and find her sedentary twin older than herself. We show that the asymmetry between their spacetime trajectories lies in a topological invariant of their spatial geodesics, namely the homotopy class. This illustrates how the spacetime symmetry invariance group, although valid locally, is broken down globally as soon as some points of space are identified. As a consequence, any non-trivial space topology defines preferred inertial frames along which the proper time is longer than along any other one.