AI 中文总结
研究具有紧致切片的宇宙时空收敛概念，通过假设$h_t$扩展性质，确定时空性质，研究单调序列得零距离等收敛，证明满足条件时空是因果零的并联系相关距离，举例说明假设必要性。
AI 中文摘要
时空的度量理论使用度量几何工具研究洛伦兹流形。这通过零距离来实现，零距离是由时空上的时间函数构建的确定距离。这使得对时空的格罗莫夫 - 豪斯多夫型收敛的研究成为可能，这是萨科维奇和索尔马尼最近启动的一个项目。本文研究具有紧致切片的宇宙时空的这种收敛概念，即$(a,b)\times M$赋予洛伦兹度量$-dt^2 + h_t$，其中$h_t$是紧致流形$M$上的一族黎曼度量。假设$h_t$具有温和的扩展性质，我们首先确定这些时空是因果零可紧致化且未来可发展的。然后我们研究空间直径有一致上界的单调序列，得到零距离的一致收敛以及相关时间度量空间在未来发展的格罗莫夫 - 豪斯多夫意义下的收敛。最后，我们证明满足温和因果可达性条件的因果零可紧致化时空是因果零的，并将极限距离诱导的因果零距离与（可能非光滑的）极限度量张量诱导的零距离联系起来。还提供了例子来说明我们假设的必要性。
英文摘要
The metric theory of spacetimes studies Lorentzian manifolds using tools of metric geometry. This is achieved via the null distance, which is a definite distance constructed from a time function on a spacetime. This enables the study of Gromov-Hausdorff-type convergence of spacetimes, a program recently initiated by Sakovich and Sormani. In this paper we study such notions of convergence for cosmological spacetimes with compact slices, i.e., $(a,b)\times M$ endowed with a Lorentzian metric $-dt^2+h_t$, where $h_t$ is a family of Riemannian metrics on the compact manifold $M$. Assuming mild extension properties of $h_t$, we first establish that these spacetimes are causally-null compactifiable and future developed. We then study monotone sequences with a uniform upper bound on the spatial diameter, obtaining uniform convergence of the null distances, as well as convergence of the associated timed metric spaces in the future developed Gromov-Hausdorff sense. Finally, we prove that causally-null compactifiable spacetimes satisfying a mild causal accessibility condition are causally-null, and relate the causally-null distance induced by the limit distance with the null distance induced by the (possibly non-smooth) limit metric tensor. Examples are provided to motivate the necessity of our hypotheses.
Comments34 pages, 4 figures
