What is the analogous notion of Gromov–Hausdorff convergence for sequences of spacetimes? Since a Lorentzian manifold is not inherently a metric space, one cannot simply use the traditional definition. One approach offered by Sormani and Vega (Class Quant Gravity, 33:085001, 2016) is to define a metric space structure on a spacetime by means of the null distance. Then one can define convergence of spacetimes using the usual definition of Gromov–Hausdorff convergence. In this paper we explore this approach by giving many examples of sequences of warped product spacetimes with the null distance converging in the Gromov–Hausdorff sense. In addition, we give an optimal convergence theorem which shows that under natural geometric hypotheses a sequence of warped product spacetimes converge to a specific limiting warped product spacetime. The examples given further serve to show that the hypotheses of this convergence theorem are optimal.

时空序列的格罗莫夫-豪斯多夫收敛的类似概念是什么？由于洛伦兹流形本质上不是度量空间，因此不能简单地使用传统的定义。Sormani 和 Vega (Class Quant Gravity, 33:085001, 2016) 提供的一种方法是通过零距离定义时空上的度量空间结构。然后我们可以使用格罗莫夫-豪斯多夫收敛的通常定义来定义时空收敛。在本文中，我们通过给出许多扭曲乘积时空序列的例子来探索这种方法，其中零距离在格罗莫夫-豪斯多夫意义上收敛。此外，我们给出了一个最优收敛定理，该定理表明，在自然几何假设下，扭曲积时空序列收敛到特定的极限扭曲积时空。给出的例子进一步表明该收敛定理的假设是最优的。