We revisit the theory of timelike and null geodesics in the (extended) Kerr spacetime. This work is a sequel to a recent paper by Cieślik, Hackmann, and Mach, who applied the so-called Biermann–Weierstrass formula to integrate Kerr geodesic equations expressed in Boyer–Lindquist coordinates. We show that a formulation based on the Biermann–Weierstrass theorem can also be applied in horizon-penetrating Kerr coordinates, resulting in solutions that are smooth across Kerr horizons. Horizon-penetrating Kerr coordinates allow for an explicit continuation of timelike and null geodesics between appropriate regions of the maximal analytic extension of the Kerr spacetime. A part of this work is devoted to a graphic visualisation of such geodesics.

中文翻译：

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渗透水平的 Kerr 坐标中的 Kerr 测地线：用 Weierstrass 函数描述


我们重新审视了（扩展的）Kerr 时空中的类时测地线和零测地线理论。这项工作是 Cieślik、Hackmann 和 Mach 最近一篇论文的续集，他们应用所谓的 Biermann-Weierstrass 公式来整合以 Boyer-Lindquist 坐标表示的 Kerr 测地线方程。我们表明，基于 Biermann-Weierstrass 定理的公式也可以应用于渗透水平的 Kerr 坐标，从而得到在 Kerr 水平上平滑的解。水平穿透的 Kerr 坐标允许在 Kerr 时空的最大解析扩展的适当区域之间显式延续类似时间的测地线和零测地线。这项工作的一部分致力于此类测地线的图形可视化。