Black Holes, Geometric Inequalities, and Partial Differential Equations

黑洞、几何不等式和偏微分方程

• 批准号: 2104229
• 负责人: Marcus Khuri
• 金额: $ 34.43万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104229&HistoricalAwards=false
• 关键词: Black; Holes; Geometric; Inequalities; Partial

The goal of this project is to study several important related conjectures in general relativity. This geometric theory of gravity proposed by Einstein 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. Furthermore, new criteria for gravitational collapse and black hole formation will be studied, and the PI will also examine proposed definitions of quasi-local mass in order to determine whether they are mathematically and physically viable. Lastly, fundamental questions concerning the shape (topology) of the cosmos will be addressed, including whether we live in a finite or infinite universe.Recently the PI, in collaboration with Bray, Kazaras, and Stern has found a new and simple proof of the positive mass theorem, a result which has played a seminal role in mathematical relativity since its initial proof by Schoen, Yau, and Witten 40 years ago. This new approach, which has also been generalized to the spacetime and hyperbolic settings, yields an explicit lower bound for the mass in terms of quantities associated with (spacetime) harmonic functions, and suggests a strategy to establish the conjectured stability or almost rigidity for this theorem. Based on an initial collaboration with Bray, the PI has completed a systematic approach to treating the full family of Penrose-type inequalities by reducing each to a canonical system of elliptic PDE, thus placing the entire range of these geometric inequalities within reach. Together with Yamada and Weinstein, who initiated the study of harmonic maps with prescribed singularities associated with the axisymmetric 4D Einstein equations in the 1990s, the PI has developed the tools necessary to substantially generalize the 4D results to allow for exotic topologies in higher dimensions as well as a wide range of symmetric space targets. This work suggests that it is possible to obtain the full classification of stationary axisymmetric black holes within vacuum and supergravity. In joint work with Alaee and Yau, the PI has begun a study of the proposed Bekenstein bounds based on the Wang-Yau quasi-local mass; these inequalities concern the entropy/information contained within a relativistic body and have wide ranging implications from thermodynamics to computer science. This initial study has provided the foundations to address the full Bekenstein conjecture, and is closely related via the approach to the trapped surface/hoop conjecture dealing with the conditions under which black holes may form. In addition, joint work with Anderson has established what may be considered as the first step of Bartnik's minimal mass extension conjecture, and the PI's methods indicate what should be a successful strategy for the remaining parts.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目的目标是研究广义相对论中几个重要的相关猜想。爱因斯坦提出的几何引力理论是我们理解宇宙大尺度结构的基础，并且有许多实际应用，例如全球定位系统（GPS）技术的微调。PI将寻求建立有关质量、电荷、角动量和视界面积的几何不等式家族，以探索大弱宇宙审查猜想。这个猜想断言，无论何时奇点在时空中出现（这是一种普遍现象），它们一定总是被黑洞事件视界所笼罩；这与广义相对论是否是一个恰当的确定性理论密切相关。具有对称性的特殊黑洞解（称为静止轴对称和电真空）在我们对理论的理解中起着重要作用，本项目试图在与弦理论相关的更高维度上对它们进行分类。此外，将研究引力坍缩和黑洞形成的新标准，PI还将检查拟局部质量的拟议定义，以确定它们是否在数学和物理上可行。最后，将讨论有关宇宙形状（拓扑）的基本问题，包括我们是否生活在有限或无限的宇宙中。最近，PI与布雷、卡扎拉斯和斯特恩合作，发现了一个新的、简单的正质量定理的证明，这个结果自40年前由舍恩、丘和威滕首次证明以来，在数学相对论中发挥了开创性的作用。这种新方法也被推广到时空和双曲环境中，它产生了与（时空）调和函数相关的量的质量的显式下界，并提出了一种建立该定理的推测稳定性或几乎刚性的策略。基于与Bray的初步合作，PI已经完成了一种系统的方法来处理penrose型不等式的整个家族，通过将每个不等式简化为椭圆PDE的规范系统，从而将这些几何不等式的整个范围置于可及的范围内。与Yamada和Weinstein一起，他们在20世纪90年代发起了与轴对称四维爱因斯坦方程相关的具有规定奇点的谐波映射的研究，PI开发了必要的工具，可以大量推广四维结果，以允许更高维度的奇异拓扑以及广泛的对称空间目标。这项工作表明，在真空和超重力条件下获得静止轴对称黑洞的完整分类是可能的。在与Alaee和Yau的联合工作中，PI已经开始研究基于Wang-Yau准局部质量的贝肯斯坦边界；这些不等式涉及到相对论体中包含的熵/信息，并具有从热力学到计算机科学的广泛含义。这一初步研究为解决完整的贝肯斯坦猜想提供了基础，并通过处理黑洞可能形成的条件的捕获面/环猜想的方法密切相关。此外，与Anderson的联合工作已经建立了可以被认为是Bartnik最小质量扩展猜想的第一步，PI的方法表明对于其余部分应该是一个成功的策略。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Stability of the spacetime positive mass theorem in spherical symmetry

球对称时空正质量定理的稳定性

• DOI: 10.1007/s12220-020-00431-0
• 发表时间: 2021
• 期刊: Journal of geometric analysis
• 影响因子: 1.1
• 作者: Bryden, Edward;Khuri, Marcus;Sormani, Christina
• 通讯作者: Sormani, Christina

Existence and uniqueness of stationary solutions in $5$-dimensional minimal supergravity

$5$维最小超重力中平稳解的存在性和唯一性

• DOI: 10.4310/mrl.2022.v29.n5.a1
• 发表时间: 2022
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Alaee, Aghil;Khuri, Marcus;Kunduri, Hari
• 通讯作者: Kunduri, Hari

Black Lenses in Kaluza-Klein Matter

Kaluza-Klein Matter 中的黑色镜片

• DOI: 10.1103/physrevlett.131.041402
• 发表时间: 2023
• 期刊: Physical Review Letters
• 影响因子: 8.6
• 作者: Khuri, Marcus A.;Rainone, Jordan F.
• 通讯作者: Rainone, Jordan F.

Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS *

具有明显视界边界的渐近双曲爱因斯坦约束方程和 Schwarzschild-AdS 扰动的彭罗斯不等式 *

• DOI: 10.1088/1361-6382/acb24b
• 发表时间: 2023
• 期刊: Classical and Quantum Gravity
• 影响因子: 3.5
• 作者: Khuri, Marcus;Kopiński, Jarosław
• 通讯作者: Kopiński, Jarosław

Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds

调和函数和 3 维渐近平坦黎曼流形的质量

• DOI: 10.1007/s12220-022-00924-0
• 发表时间: 2022
• 期刊: Journal of geometric analysis
• 影响因子: 1.1
• 作者: Bray, Hubert;Kazaras, Demetre;Khuri, Marcus;Stern, Daniel
• 通讯作者: Stern, Daniel