Mass, Geometric Inequalities, and Partial Differential Equations in General Relativity

广义相对论中的质量、几何不等式和偏微分方程

• 批准号: 1708798
• 负责人: Marcus Khuri
• 金额: $ 17.4万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1708798&HistoricalAwards=false
• 关键词: Mass; Geometric; Inequalities; Partial; Differential

The goal of this project is to study several important related conjectures in general relativity. This geometric theory of gravity is fundamental to our understanding of the large-scale structure of the universe, and has many practical applications such as to the fine tuning of global positioning system (GPS) technology. The PI will seek to establish families of geometric inequalities relating mass, charge, angular momentum, and horizon area, which probe the grand weak cosmic censorship conjecture. This conjecture asserts that whenever singularities arise in spacetime (which is a generic phenomenon) they must always be shrouded inside a black hole event horizon; this is intimately tied to whether general relativity is a proper deterministic theory. Special black hole solutions with symmetry (referred to as stationary axisymmetric and electro-vacuum) play a large role in our understanding of the theory, and this project seeks to classify them in higher dimensions relevant to string theory. In particular, the PI aims to prove existence and uniqueness for such black holes with exotic (lens space) topologies in five spacetime dimensions. Furthermore, new criteria for gravitational collapse and black hole formation will be studied, namely those due to concentration of angular momentum and/or charge. The PI will also examine proposed definitions of quasi-local mass in order to determine whether they are mathematically and physically viable.Based on earlier work with Bray, the PI has recently completed a systematic approach to treating the full family of Penrose-type inequalities by reducing each to a canonical system of elliptic partial differential equations (PDEs). Thus, the entire range of these geometric inequalities is within reach. A natural by-product, associated with the study of a certain subclass of these inequalities, concerns a general procedure for constructing singular harmonic maps having 2-dimensional hyperbolic space target, which naturally arise from the stationary axisymmetric vacuum Einstein equations in four dimensions. Together with G. Weinstein, who initiated the study of such harmonic maps with prescribed singularities, we have begun development of the tools necessary to substantially generalize the 4-dimensional results to allow for exotic topologies in higher dimensions as well as a wide range of symmetric space targets.  Moreover, in joint work with M. Anderson, the PI has established what may be considered as the first step of Bartnik's minimal mass extension conjecture, and our methods indicate what should be a successful approach to the remaining parts. The trapped surface/hoop conjecture, dealing with the conditions under which black holes may form, is highly sought after but not well understood. However, the PI's work on Penrose-type inequalities suggests new black hole formation criteria as well as certain isoperimetric-type inequalities for relativistic bodies.

这个项目的目标是研究广义相对论中几个重要的相关理论。引力的几何理论是我们理解宇宙大尺度结构的基础，并有许多实际应用，如全球定位系统（GPS）技术的微调。PI将寻求建立与质量、电荷、角动量和视界面积相关的几何不等式家族，以探索大弱宇宙监督猜想。这个猜想断言，无论何时时空中出现奇点（这是一种普遍现象），它们必须总是被黑洞事件视界所掩盖;这与广义相对论是否是一个正确的决定论密切相关。具有对称性的特殊黑洞解（称为静态轴对称和电真空）在我们理解理论中起着重要作用，本项目试图将它们分类到与弦理论相关的更高维度。特别是，PI的目的是证明存在性和唯一性，这样的黑洞与异国情调（透镜空间）拓扑在5个时空维度。此外，还将研究引力坍缩和黑洞形成的新判据，即角动量和/或电荷集中的判据。PI还将检查拟局部质量的拟议定义，以确定它们在数学和物理上是否可行。基于与Bray的早期工作，PI最近完成了一个系统的方法来处理整个Penrose型不等式族，将每个不等式简化为椭圆型偏微分方程（PDE）的规范系统。因此，这些几何不等式的整个范围都是可以得到的。一个自然的副产品，与这些不等式的某个子类的研究，涉及到一个一般的程序，用于构建奇异调和映射的二维双曲空间的目标，这自然产生于固定的轴对称真空爱因斯坦方程在四维。与G. Weinstein，谁发起了这种谐波映射与规定的奇点的研究，我们已经开始开发必要的工具，以充分推广的4维结果，以允许在更高的维度以及广泛的对称空间目标的奇异拓扑结构。安德森，PI已经建立了什么可以被认为是第一步的巴特尼克的最小质量扩展猜想，我们的方法表明什么应该是一个成功的方法，其余部分。被困表面/环猜想，处理黑洞可能形成的条件，是非常受欢迎的，但没有得到很好的理解。然而，PI关于彭罗斯型不等式的工作提出了新的黑洞形成准则，以及相对论性天体的某些等周型不等式。

项目成果

The conformal flow of metrics and the general Penrose inequality

度量的等角流和一般彭罗斯不等式

• DOI: 10.1155/2018/7390148
• 发表时间: 2019
• 期刊: Advanced lectures in mathematics
• 作者: Han, Qing;Khuri, Marcus
• 通讯作者: Khuri, Marcus

The positive mass theorem with angular momentum and charge for manifolds with boundary

边界流形的角动量和电荷的正质量定理

• 发表时间: 2019
• 期刊: Journal of mathematical physics
• 影响因子: 1.3
• 作者: Edward Bryden, Marcus Khuri

Transformations of asymptotically AdS hyperbolic initial data and associated geometric inequalities

• DOI: 10.1007/s10714-017-2323-7
• 发表时间: 2017-11
• 期刊: General Relativity and Gravitation
• 影响因子: 2.8
• 作者: Ye Sle Cha;M. Khuri

Asymptotically locally Euclidean/Kaluza-Klein stationary vacuum black holes in 5 dimensions

5 维渐近局部欧几里得/卡鲁扎-克莱因静止真空黑洞

• 发表时间: 2018
• 期刊: Progress of theoretical and experimental physics
• 影响因子: 3.5
• 作者: Marcus Khuri, Gilbert Weinstein

5-dimensional space-periodic solutions of the static vacuum Einstein equations

静态真空爱因斯坦方程的 5 维空间周期解

• DOI: 10.1007/jhep12(2020)002
• 发表时间: 2020
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Khuri, Marcus;Weinstein, Gilbert;Yamada, Sumio
• 通讯作者: Yamada, Sumio