利用Gödel论域构造一个新的具有$R^2 \times S^1$拓扑的Zollfrei度量
Use of Gödel Universe to Construct A New Zollfrei Metric with $R^2 \times S^1$ Topology
摘要：
给出了一个新的$(2+1)$维Zollfrei度量的例子，其拓扑为$R^2 \times S^1 $.这个度规很容易从著名的$(3+1)$维旋转哥德尔宇宙$G_{3,1}$中得到。这是因为$G_{3,1}$有一个有趣的性质，即被限制在垂直于旋转轴的平面上运动的光线，在一个时间段$T = \frac{2 π}ω[\sqrt{2}-1]$后返回到它们的原点-其中$ω$是宇宙的角速度。因此，通过-拓扑标识的时间坐标上的点对，由时间间隔$T$分开。以及丢弃平坦的$x_3$坐标-其沿着旋转轴定向;人们利用$R^2 \times S^1$拓扑获得$(2+1)$维Zollfrei度量。
英文摘要：
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 Gödel 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 $T = \frac{2 π}ω[\sqrt{2}-1]$ -where $ω$ 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.
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