Geometric flows and general relativity

几何流和广义相对论

• 批准号: 203614-2012
• 负责人: Woolgar, Eric
• 金额: $ 1.82万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569976
• 关键词: Geometric; flows; general; relativity

This project aims to unite the mathematics of geometric flows with the physics of gravitation. Geometric flows describe ways to deform manifolds. Manifolds are curved spaces, such as the familiar round sphere, but whereas the curvature of a round sphere is very regular and uniform, an arbitrary manifold can have bumps and valleys with quite general curvature, and can have any number of dimensions. In physics, curvature in the universe is the cause of gravitation, and so is central to general relativity as well as to string theory. An important mathematical development in recent years is the study of geometric flow equations. These are partial differential equations that specify ways to deform the curvature of a manifold, often making the manifold more uniform, though sometimes making it more curved. A singularlity forms if the curvature becomes infinitely large. The study of these equations has led to exciting developments in mathematics, such as the proof of the famous Poincaré conjecture and, more recently, the diffeomorphic ¼-pinched sphere theorem. But these flows also have a role to play in physics, and that aspect has been less well developed. One aim of this project is to develop the theory of these flows with a view to physics applications. One such application is the notion of mass in general relativity. There is a good definition of total mass of an isolated gravitating system, but there are several competing notions of the quasilocal mass of parts of a system, and each has drawbacks. One of the most useful notions of quasilocal mass was presented by Robert Bartnik, but it is one of the most difficult to calculate. The effort to overcome this difficulty led to certain conjectures which are important in their own right. This in turn leads to the consideration of geometric flows with boundaries. Very little is known about these flows, and this proposal is intended to change that. The methods I propose to develop will also have applications to numerical geometric flows and to the study of static and, perhaps, stationary systems in relativity, such as a star that may rotate but does not emit gravitational radiation, or a black hole in equilibrium with a "brane".

这个项目的目的是将几何流的数学与引力物理学结合起来。几何流描述了使流形变形的方法。流形是弯曲的空间，例如我们熟悉的圆形球体，但是圆形球体的曲率是非常规则和均匀的，而任意流形可以有非常普通的曲率的凸起和山谷，并且可以有任意数量的维度。在物理学中，宇宙中的曲率是万有引力的原因，对广义相对论和弦理论来说也是如此。 近年来一个重要的数学发展是几何流动方程的研究。这些偏微分方程指定了使流形的曲率变形的方法，通常会使流形更加均匀，尽管有时会使其更加弯曲。如果曲率变得无限大，就会形成奇点。对这些方程的研究导致了数学上令人兴奋的发展，例如著名的Poincaré猜想的证明，以及最近的微分同胚1/4-Pinch球面定理的证明。但这些流动也在物理学中发挥了作用，而这一方面一直没有得到很好的发展。这个项目的一个目标是发展这些流动的理论，以期应用于物理学。 广义相对论中的质量概念就是这样一种应用。孤立引力系统的总质量有一个很好的定义，但关于系统各部分的准局部质量有几个相互竞争的概念，而且每一个都有缺点。准局域质量是罗伯特·巴特尼克提出的最有用的概念之一，但它也是最难计算的概念之一。克服这一困难的努力导致了某些猜想，这些猜想本身就很重要。这又导致了对具有边界的几何流的考虑。人们对这些流动知之甚少，这项提议旨在改变这一点。 我提议开发的方法也将应用于数值几何流动和相对论中静态系统的研究，例如可以旋转但不发出引力辐射的恒星，或者与膜平衡的黑洞。