Mass in General Relativity

广义相对论中的质量

• 批准号: 1007156
• 负责人: Marcus Khuri
• 金额: $ 27.86万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007156&HistoricalAwards=false
• 关键词: Mass; General; Relativity

The primary topics of this research include three important conjectures concerning mass in General Relativity: the Static Extension Conjecture, the full Penrose Inequality, and the Hoop Conjecture. The first of these arises from a particularly promising definition of quasilocal mass due to Bartnik, which seeks to localize the total (or ADM) mass. Although this definition satisfies most desired properties, and thus has the potential to be very useful, its abstract and nonconstructive nature yield severe limitations on understanding and applicability. In order to rectify this problem the Static Extension Conjecture asserts that the quasilocal mass may be calculated as the ADM mass of a solution to the Static Vacuum Einstein equations which satisfies a certain geometric boundary condition. In recent joint work of the author and M. Anderson, existence for this boundary value problem has been established under a reasonable nondegeneracy assumption on the given boundary data; this development has placed the conjecture within reach. The Penrose Inequality relates the total mass of a spacetime to the area of its event horizons (boundary of black holes) via the inequality: total mass squared is greater than or equal to the total area of the event horizons divided by 16 pi. This may be viewed as a conjecture for an arbitrary spacelike slice of a spacetime, and in this setting an important special case (when the second fundamental form of the slice vanishes) has been confirmed independently by Huisken and Ilmanen (one black hole) and by Bray (finitely many black holes). In recent joint work with Bray, the author has succeeded in reducing this problem to solving a canonical system of partial differential equations, and has confirmed existence in special cases. It is a major goal of this project to complete this program by proving a general existence result, and thus establishing the Penrose Conjecture.Although there are numerous proposed methods for calculating the gravitational plus matter energy content of a bounded domain (the so called quasilocal mass) in General Relativity, all seem to suffer from one affliction or another. In the case of Bartnik's mass, the sole problem is the abstract nature of its definition. If the Static Extension Conjecture were verified, this lone difficulty should be resolved, thus opening the way for several applications. For example, it is generally believed that whenever enough mass is concentrated in a sufficiently small region, gravitational collapse must ensue and result in a black hole - such a statement is often referred to as the Hoop Conjecture. While there are many ways to accurately describe the size of a region, until a proper notion of quasilocal mass is established, a rigorous and complete version of this conjecture will remain elusive. Another promising application of a properly defined notion of quasilocal mass, is to the long time existence problem for the Einstein Equations. As these equations form a hyperbolic system (after fixing the gauge), it is tempting to search for a theory based on an energy method analogous to the classical theory of the scalar wave equation. A major obstacle to realizing such an approach, is the lack of an appropriate notion of energy for the gravitational field (a constituent of quasilocal mass). Moreover the boundary value problem associated with the Static Extension Conjecture, is in fact an elliptic boundary value problem for the Ricci operator. Hence the techniques developed to study this conjecture should prove useful when investigating well-posed boundary value problems in several other settings, such as for instance, the Ricci flow on manifolds with boundary. As for the Penrose Inequality, it was originally put forth by Penrose to study what is perhaps the most important open question in General Relativity today, namely the Cosmic Censorship Conjecture (whether spacetime singularities are always enclosed by black holes), which is related to General Relativity's veracity as a physical theory. Heuristically, the Penrose Inequality is essentially a necessary condition for cosmic censorship to hold. Thus if the Penrose Inequality were to be confirmed it would add significantly to the general belief in the validity of cosmic censorship. Lastly, our methods developed for the full Penrose Inequality are expected to provide a new powerful tool for reducing questions concerning general initial data sets to the case of time symmetry, and therefore will have numerous applications to a wide range of other problems in General Relativity.

本研究的主要内容包括广义相对论中关于质量的三个重要猜想：静态延拓猜想、完全彭罗斯不等式和环状猜想。第一个问题源于Bartnik对准局部质量的一个特别有希望的定义，该定义寻求将总质量(或ADM)局域化。虽然这一定义满足了最理想的性质，因此有可能非常有用，但其抽象和非建设性的性质在理解和适用性方面造成了严重的限制。为了纠正这个问题，静态延拓猜想认为准局部质量可以计算为满足一定几何边界条件的静态真空爱因斯坦方程解的ADM质量。在作者和M.Anderson最近的合作工作中，在给定的边界数据上，在合理的非退化假设下建立了这个边值问题的存在性，这一发展使得这个猜想变得容易实现。彭罗斯不等式将时空的总质量与其事件视界(黑洞边界)的面积联系在一起：总质量的平方大于或等于事件视界的总面积除以16pi。这可以被看作是对时空的任意类空片的猜想，在这种情况下，一个重要的特殊情况(当片的第二基本形式消失时)已经被Huisken和Ilmanen(一个黑洞)和Bray(有限多个黑洞)独立地证实。在最近与Bray的合作中，作者成功地将这一问题归结为求解一个规范的偏微分方程组，并在特殊情况下证明了该问题的存在性。这个项目的一个主要目标是通过证明一个普遍存在的结果来完成这个计划，从而建立彭罗斯猜想。尽管在广义相对论中有很多计算有界域(所谓的准局部质量)的引力和物质能量含量的方法，但似乎都受到了这样或那样的困扰。就巴特尼克的质量而言，唯一的问题是其定义的抽象性质。如果静态扩张猜想得到验证，这个单独的困难就应该得到解决，从而为几个应用程序打开了道路。例如，人们普遍认为，当足够多的质量集中在一个足够小的区域时，引力崩塌必然随之而来，并导致黑洞--这样的说法通常被称为环状猜想。虽然有许多方法可以准确地描述一个区域的大小，但在建立适当的准局部质量概念之前，这个猜想的严格而完整的版本仍然难以捉摸。适当定义的准局部质量概念的另一个有希望的应用是解决爱因斯坦方程的长期存在问题。由于这些方程形成了一个双曲系(在确定规范后)，人们很容易寻找一种基于能量方法的理论，类似于标量波动方程的经典理论。实现这种方法的一个主要障碍是缺乏适当的引力场(准局部质量的一个组成部分)的能量概念。此外，与静态扩张猜想相关的边值问题实际上是Ricci算子的椭圆型边值问题。因此，用来研究这一猜想的技术在研究其他几种情况下的适定边值问题时应该被证明是有用的，例如，带边界的流形上的Ricci流。至于彭罗斯不等式，它最初是由彭罗斯提出的，目的是研究当今广义相对论中可能是最重要的公开问题，即宇宙审查猜想(时空奇点是否总是被黑洞包围)，这与广义相对论作为物理理论的准确性有关。启发式地说，彭罗斯不平等本质上是宇宙审查有效的必要条件。因此，如果彭罗斯不平等得到证实，它将大大增强人们对宇宙审查制度有效性的普遍信念。最后，我们为完全彭罗斯不等式开发的方法有望提供一个新的强有力的工具，将关于一般初始数据集的问题归结为时间对称的情况，因此将在广义相对论的其他广泛问题中得到广泛的应用。