On the Twin Paradox in a Universe with a Compact Dimension

We consider the twin paradox of special relativity in a universe with a compact spatial dimension. Such topology allows two twin observers to remain inertial yet meet periodically. The paradox is resolved by considering the relationship of each twin to a preferred inertial reference frame which exists in such a universe because global Lorentz invariance is broken. The twins can perform “global” experiments to determine their velocities with respect to the preferred reference frame (by sending light signals around the cylinder, for instance). Here we discuss the possibility of doing so with local experiments. Since one spatial dimension is compact, the electrostatic field of a point charge deviates from 1/r. We show that although the functional form of the force law is the same for all inertial observers, as required by local Lorentz invariance, the deviation from 1/r is observer-dependent. In particular, the preferred observer measures the largest field strength for fixed distance from the charge.