Applications of the Convergence of Riemannian Manifolds to General Relativity

黎曼流形收敛性在广义相对论中的应用

• 批准号: 1309360
• 负责人: Christina Sormani
• 金额: $ 11.6万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309360&HistoricalAwards=false
• 关键词: Applications; Convergence; Riemannian; Manifolds; General

AbstractAward: DMS 1309360, Principal Investigator: Christina A. SormaniThe PI will apply Intrinsic Flat convergence between Riemannian manifolds to better understand how close space-like manifolds studied in Mathematical General Relativity approximate the standard well known models. The Intrinsic Flat distance, first introduced by the PI with Stefan Wenger using methods of Ambrosio-Kirchheim, is particularly well-suited to some questions arising in General Relativity because increasingly thin gravity wells disappear under this convergence. In joint work with Dan Lee, the PI has shown that spherically symmetric Riemannian manifolds with increasingly small ADM mass converge to Euclidean space in the pointed intrinsic flat sense, and here proposes to generalize this result. In addition, the PI proposes to develop two new notions of convergence: the first will allow mathematicians to study Lorentzian manifolds directly, and the second will prevent regions from disappearing due to orientation and cancellation. Both notions are specifically adapted to questions arising in General Relativity.Einstein's Theory of General Relativity describes how space is curved by gravity. Even within our own solar system, when computing the trajectories of spacecraft heading to Mars, engineers must take into account the curvature caused by the mass of the planets and the sun. Each planet forms a gravity well. If the mass of a planet is small, one would like to know in what sense the space around it is almost flat. In fact, the space around a planet of arbitrarily small mass could be very highly curved (and have a very deep but thin gravity well). In joint work with Dr. Stefan Wenger, the PI has developed a new means of measuring the closeness between curved spaces and, in joint work with Dr. Dan Lee, she has estimated how close the space around a single perfectly spherical planet is to Euclidean space. In this project, she will develop tools allowing one to better understand the space around groups of planets which are not perfect spheres: like the ones in our own solar system.

摘要奖：DMS 1309360，主要研究者：Christina A. SormaniThe PI将应用黎曼流形之间的内在平坦收敛，以更好地理解数学广义相对论中研究的类空流形如何接近标准的众所周知的模型。 本征平坦距离，首先由PI与Stefan Wenger使用Ambrosio-Kirchheim的方法引入，特别适合于广义相对论中出现的一些问题，因为越来越薄的重力威尔斯在这种收敛下消失了。 在与Dan Lee的联合工作中，PI证明了ADM质量越来越小的球对称黎曼流形在指向的内在平坦意义上收敛到欧几里得空间，并在这里提出推广这一结果。 此外，PI还提出了两个新的收敛概念：第一个将允许数学家直接研究洛伦兹流形，第二个将防止区域由于方向和取消而消失。 这两个概念都特别适用于广义相对论中出现的问题。爱因斯坦的广义相对论描述了空间如何被引力弯曲。 即使在我们自己的太阳系中，当计算前往火星的航天器的轨迹时，工程师也必须考虑行星和太阳质量造成的曲率。每颗行星都形成一个重力井。 如果一颗行星的质量很小，人们想知道在什么意义上它周围的空间几乎是平的。 事实上，一颗质量任意小的行星周围的空间可能是非常弯曲的（并且有一个非常深但很薄的重力井）。 在与Stefan Wenger博士的合作中，PI开发了一种测量弯曲空间之间接近程度的新方法，并且在与Dan Lee博士的合作中，她估计了一个完美球形行星周围的空间与欧几里得空间的接近程度。 在这个项目中，她将开发工具，使人们能够更好地了解行星群周围的空间，这些行星不是完美的球体：就像我们太阳系中的行星一样。