AI 中文总结
本文针对低正则性时空与洛伦兹长度空间中类光测地线缺乏标准参数化的问题，讨论通过类时测地线极限定义仿参数化的方法，指出其存在非唯一结果的缺陷，并探索利用类时测地线构造彭罗斯型奇点定理的可能性。
AI 中文摘要
在低正则性时空与洛伦兹长度空间中，非时序因果曲线承担类光（预）测地线的作用。由于缺乏测地方程，它们不具备标准参数化。本文讨论通过仿参数化类时测地线的极限来定义仿参数化的概念，但指出其存在重大缺陷：存在实例使得该近似过程给出非唯一结果，甚至极限类光测地线的完备性或不完备性都无法明确定义。该实例涉及在类光超曲面上不连续地拼接两个洛伦兹度量，以得到一个性质良好的洛伦兹长度空间，且该实例与参数化问题本身一样，明显具有洛伦兹特性。鉴于此，本文探索利用类时测地线构造彭罗斯型奇点定理的可能性。
英文摘要
In low-regularity spacetimes and Lorentzian length spaces, achronal causal curves play the role of null (pre-)geodesics. Because of the lack of a geodesic equation, they do not come with a canonical parametrization. In this context, we discuss a notion of affine parametrization via limits of affinely parametrized timelike geodesics. However, we point out a major drawback: an example where this approximation procedures gives a non-unique result, in a way that even completeness or incompleteness of the limit null geodesic is not well-defined. This example involves discontinuous gluing of two Lorentzian metrics across a null hypersurface to give a well-behaved Lorentzian length space and as such is, just like the parametrization problem itself, manifestly Lorentzian. In view of this, we explore possibilities for a Penrose-type singularity theorem using timelike geodesics.
Comments22 pages, 3 figures
