标题： 超越时空边界 显示英文标题

标题： Beyond the Space-Time Boundary

摘要： 在广义相对论中，如果存在障碍阻止 $M$ 中的不完全曲线继续延伸，则称时空 $M$ 是奇异的。 通常，这样的时空被扩展以形成 $\bar{M} = M \cup \partial M$，其中 $\partial M$ 是 $M$ 的奇异边界。 标准微分几何工具在 $M$ 上不允许“穿过边界”。 然而，所谓的合成微分几何（SDG），一种基于直觉逻辑的标准微分几何的范畴版本，拥有允许这样做的工具。 由于无穷小量的存在，能够深入“流形的胚芽”，这些从标准视角是不可见的。 我们提出一个简单的模型，展示“边界之外”发生了什么以及奇点最终达到时的情况。 该模型纯粹是数学的，并且在数学上是严格的，但它并不声称与物理宇宙有关。 显示英文摘要

摘要： In General Relativity a space-time $M$ is regarded singular if there is an obstacle that prevents an incomplete curve in $M$ to be continued. Usually, such a space-time is completed to form $\bar{M} = M \cup \partial M$ where $\partial M$ is a singular boundary of $M$. The standard geometric tools on $M$ do not allow "to cross the boundary". However, the so-called Synthetic Differential Geometry (SDG), a categorical version of standard differential geometry based on intuitionistic logic, has at its disposal tools permitting doing so. Owing to the existence of infinitesimals one is able to penetrate "germs of manifolds" that are not visible from the standard perspective. We present a simple model showing what happens "beyond the boundary" and when the singularity is finally attained. The model is purely mathematical and is mathematically rigorous but it does not pretend to refer to the physical universe.