Use of G\"odel Universe to Construct A New Zollfrei Metric with $R^2 \times S^1$ Topology

Abstract

A new example of $(2+1)$-dimensional Zollfrei metric, with the topology $R^2 \times S^1 $, is presented. This metric is readily obtained from the celebrated $(3+1)$- dimensional rotating G\"odel universe $G_{3,1}$. This is because $G_{3,1}$ has the interesting property that, the light rays which are confined to move on the plane perpendicular to the rotation axis, return to their origin after a time period $T = \frac{2 \pi}{\omega}[\sqrt{2}-1]$ -where $\omega$ is the angular velocity of the universe. Hence by - the topological identification of pairs of points on the time coordinate, seperated by the time interval $T$. and droping the flat $x_3$ coordinate - which is directed along the rotation axis; one obtains the $(2+1)$-dimensional Zollfrei metric with the $R^2 \times S^1$ topology.