用 Morse 函数实现拓扑变化：Borde–Sorkin 猜想的进展
Topology change with Morse functions: progress on the Borde-Sorkin conjecture
AI总结：
本文研究由 Morse 函数构造的拓扑变化时空，证明 Borde–Sorkin 因果连续性猜想的一个特例，并基于启发式论证提出对原猜想的修正版本。
摘要：
拓扑变化被一些学者视为量子引力的必要特征，而另一些学者则认为它不可能发生。反对它的主要论据之一是，空间拓扑发生变化的时空具有糟糕的因果性质。Borde 和 Sorkin 提出了一种摆脱这一困境的方法：考虑由 Morse 函数构造的拓扑变化时空，在这种时空中度规允许在孤立点处退化。他们猜想，只要 Morse 点的指数不等于 1 和 n−1，这些 Morse 时空就是因果连续的（因此表现得相当良好）。在本文中，我们证明了该猜想的一个特例。我们还以启发式方式论证了原始猜想实际上是错误的，并给出了它的一个精炼版本。
英文摘要：
Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde and Sorkin proposed a way to avoid this dilemma by considering topology changing spacetimes constructed from Morse functions, where the metric is allowed to vanish at isolated points. They conjectured that these Morse spacetimes are causally continuous (hence quite well behaved), as long as the index of the Morse points is different from $1$ and $n-1$. In this paper, we prove a special case of this conjecture. We also argue, heuristically, that the original conjecture is actually false, and formulate a refined version of it.
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