论文标题
时空扩展的独特性和非唯一性结果
Uniqueness and non-uniqueness results for spacetime extensions
论文作者
论文摘要
给定一个函数$ f：如果扩展存在并且是连续的，那么当然，它是在$ a $的关闭上唯一确定的。对于lorentzian歧管的一般相对论，出现了一个类似的问题，而不是$ \ mathbb {r}^m $上的函数。然而，众所周知，即使洛伦兹歧管$（m，g）$的扩展是分析性的，在边界上通常可以选择各种选择。本文为全球双曲线洛伦兹流形的扩展建立了独特的条件，重点是低规律性：任何两个扩展是由不可扩展的因果弯道$γ$γ：[-1,0）\在两种范围内的限制，因为$γ$的限制，因为$γ$的限制，因为$γ$的限制，因为很长一定要限制，因此，这是一定的，因为很长一定的限制，这是一定的限制。 Lipschitz连续。我们还表明，这很敏锐：锚定扩展名通常是Hölder连续做的，通常不享受这种局部独特性结果。
Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined on some open subset $A \subseteq \mathbb{R}^m$, it is a classical problem of analysis to investigate whether the function can be extended to all of $\mathbb{R}^m$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on $\mathbb{R}^m$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions which are anchored by an inextendible causal curve $γ: [-1,0) \to M$ in the sense that $γ$ has limit points in both extensions, must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions which are only Hölder continuous do in general not enjoy this local uniqueness result.
