{"title":"Uniqueness and Non-Uniqueness Results for Spacetime Extensions","authors":"Jan Sbierski","doi":"10.1093/imrn/rnae194","DOIUrl":null,"url":null,"abstract":"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 that are anchored by an inextendible causal curve $\\gamma : [-1,0) \\to M$ in the sense that $\\gamma $ 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 that are only Hölder continuous do in general not enjoy this local uniqueness result.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"208 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Mathematics Research Notices","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imrn/rnae194","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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 that are anchored by an inextendible causal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma $ 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 that are only Hölder continuous do in general not enjoy this local uniqueness result.
期刊介绍:
International Mathematics Research Notices provides very fast publication of research articles of high current interest in all areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.