Timelike geodesics on a hyperplane orthogonal to the symmetry axis of the Gödel spacetime appear to be elliptic-like if standard coordinates naturally adapted to the cylindrical symmetry are used. The orbit can then be suitably described through an eccentricity-semi-latus rectum parametrization, familiar from the Newtonian dynamics of a two-body system. However, changing coordinates such planar geodesics all become explicitly circular, as exhibited by Kundt’s form of the Gödel metric. We derive here a one-to-one correspondence between the constants of the motion along these geodesics as well as between the parameter spaces of elliptic-like versus circular geodesics. We also show how to connect the two equivalent descriptions of particle motion by introducing a pair of complex coordinates in the 2-planes orthogonal to the symmetry axis, which brings the metric into a form which is invariant under Möbius transformations preserving the symmetries of the orbit, i.e., taking circles to circles.

中文翻译：

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哥德尔时空，平面测地学和莫比乌斯地图

如果使用自然适合圆柱对称性的标准坐标，则在正交于Gödel时空对称轴的超平面上的时空测地线看起来像椭圆形。然后可以通过偏心-半滞后直肠参数化来适当地描述轨道，这是两体系统的牛顿动力学所熟悉的。但是，像Kundt的Gödel度量形式所展示的那样，诸如平面测地线这样的变化的坐标都明确地变为圆形。在这里，我们得出沿这些测地线的运动常数之间以及椭圆形和圆形测地线之间的参数空间之间的一一对应关系。我们还展示了如何通过在正交于对称轴的2平面中引入一对复坐标来连接两个等效的粒子运动描述，