In this work we study the local behavior of geodesics in the neighborhood of a curvature singularity contained in stationary and axially symmetric space-times. Apart from these properties, the metrics we shall focus on will also be required to admit a quadratic first integral for their geodesics. In particular, we search for the conditions on the geometry of the space-time for which null and time-like geodesics can reach the singularity. These conditions are determined by the equations of motion of a freely-falling particle. We also analyze the possible existence of geodesics that do not become incomplete when encountering the singularity in their path. The results are stated as criteria that depend on the inverse metric tensor along with conserved quantities such as energy and angular momentum. As an example, the derived criteria are applied to the Plebański-Demiański class of space-times. Lastly, we propose a line element that describes a wormhole whose curvature singularities are, according to our results, inaccessible to causal geodesics.

中文翻译：

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静止和轴对称时空曲率奇点附近的测地线

在这项工作中，我们研究了静止和轴对称时空中包含的曲率奇点附近测地线的局部行为。除了这些属性之外，我们将关注的度量还需要承认其测地线的二次一次积分。特别是，我们搜索空时和类时测地线可以达到奇点的时空几何条件。这些条件由自由下落粒子的运动方程决定。我们还分析了在路径中遇到奇点时不会变得不完整的测地线的可能存在。结果被表述为取决于逆度量张量以及能量和角动量等守恒量的标准。举个例子，导出的标准适用于 Plebański-Demiański 时空类。最后，我们提出了一个描述虫洞的线元素，根据我们的结果，因果测地线无法访问其曲率奇点。