Metric Differential Geometry and Mathematical Gravity

公制微分几何和数学引力

• 批准号: 0405906
• 负责人: Gregory Galloway
• 金额: --
• 依托单位: University of Miami
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405906&HistoricalAwards=false
• 关键词: Metric; Differential; Geometry; Mathematical; Gravity

AbstractAward: DMS-0405906Principal Investigator: Gregory J. GallowayRecent developments in gravitational theory, both theoretical andempirical, have lead to an increased interest in the study ofasymptotically anti-de Sitter and asymptotically de Sitterspacetimes. In this project, techniques of Lorentzian andRiemannian geometry will be used to attack certain problemspertaining to such spacetimes, as well as some problemsconcerning new developments in the theory of black holes. One ofthe specific aims of this project is to continue development ofan approach to establishing positivity of mass for asymptoticallyhyperbolic Riemannian manifolds (with spherical conformalinfinity) which does not require spin assumption. This approach,which makes use of the brane action introduced by Witten and Yauin their work on the AdS/CFT correspondence, may also be usefulin settling some positive mass conjectures for asymptoticallyanti-de Sitter spacetimes with toroidal topology at infinity. ThePI also plans to continue investigations into uniqueness andrigidity phenomena, and the development of singularities inasymptotically de Sitter spacetimes. The classical definition ofa black hole, though mathematically elegant, is problematic inpractice (especially for numerical studies) in that it requiresknowing the full future evolution of spacetime all the way toinfinity. As an alternative, Ashtekar and Krishnan haveintroduced a new, fully nonlinear and dynamical, approach to thestudy of black holes, based on the quasi-local notion of adynamical horizon. Another aim of this project is to investigatesome open problems concerning the occurence, i.e., existence,uniqueness and position of dynamical horizons.Modern theories of gravity are geometrical in nature. Thegravitational field and other fields, black holes and relatedobjects, may be described and analyzed using geometric methods.In more general terms, this project is concerned with the studyof certain features of gravity of current scientific interestfrom this geometric point of view, utilizing the tools ofRiemannian geometry, a mathematical theory of space, andLorentzian geometry, a mathematical theory of spacetime. Thesetheories provide a method for studying the relationship amongthree fundamental aspects of the spacetime universe: curvature(i.e., the bending of space or spacetime), topology (i.e., theglobal shape and complexity of space or spacetime) and causalstructure (i.e., the large scale behavior of light rays andlight cones).

摘要：Gregory J. galloway最近引力理论的发展，无论是理论的还是实证的，都引起了人们对渐近反德西特时空和渐近德西特时空研究的兴趣。在这个项目中，洛伦兹和黎曼几何的技术将被用来解决一些关于这样的时空的问题，以及一些关于黑洞理论的新发展的问题。本项目的具体目标之一是继续发展一种不需要自旋假设的方法来建立渐近双曲黎曼流形（球面共形无穷）的质量正性。这种方法利用了Witten和Yauin在AdS/CFT对应性研究中引入的膜作用，也可以用于解决具有无穷远环面拓扑的渐近反德西特时空的一些正质量猜想。pi还计划继续研究唯一性和刚性现象，以及非渐近德西特时空奇点的发展。黑洞的经典定义虽然在数学上很优雅，但在实践中（特别是在数值研究中）是有问题的，因为它需要知道时空未来的全部演变，一直到无穷大。作为一种替代方案，Ashtekar和Krishnan基于准局部动态视界的概念，引入了一种新的、完全非线性和动态的方法来研究黑洞。本课题的另一个目的是探讨动力学视界的存在性、唯一性和位置等开放性问题。现代引力理论本质上是几何的。引力场和其他场、黑洞和相关物体可以用几何方法来描述和分析。更一般地说，这个项目是从这个几何的角度，利用黎曼几何（空间的数学理论）和洛伦兹几何（时空的数学理论）的工具，研究当前科学感兴趣的重力的某些特征。这些理论为研究时空宇宙的三个基本方面之间的关系提供了一种方法：曲率(即。例如，空间或时空的弯曲)，拓扑（即空间或时空的整体形状和复杂性）和因果结构（即光线和光锥的大尺度行为）。