Applications of the Convergence of Riemannian Manifolds to General Relativity

[摘要]获得：DMS 1309360，首席研究员：Christina A. sormanipi将应用黎曼流形之间的内征平面收敛来更好地理解数学广义相对论中研究的类空间流形是如何近似于标准已知模型的。由PI和Stefan Wenger使用Ambrosio-Kirchheim的方法首次引入的固有平面距离特别适合于广义相对论中出现的一些问题，因为越来越薄的引力井在这种收敛下消失了。在与Dan Lee的合作中，PI证明了ADM质量越来越小的球对称黎曼流形在点内平意义上收敛于欧几里得空间，并在此提出推广这一结果。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，第二个将防止区域因方向和抵消而消失。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内，当计算飞往火星的航天器的轨迹时，工程师们也必须考虑到行星和太阳的质量造成的曲率。每颗行星都形成一个引力井。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坦的。事实上，一颗质量任意小的行星周围的空间可能是高度弯曲的（并且有一个很深但很薄的引力井）。在与斯蒂芬·温格博士的合作中，PI开发了一种测量弯曲空间之间紧密度的新方法，在与丹·李博士的合作中，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，让人们更好地了解行星群周围的空间，这些行星群不是完美的球体：就像我们太阳系中的行星群一样。

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.